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  <row Id="1" PostTypeId="1" AcceptedAnswerId="5" CreationDate="2014-03-13T20:32:04.033" Score="65" ViewCount="2811" Body="&lt;p&gt;I teach calculus to college students, and find it very difficult to get them to speak up in class when I ask questions, or when I'm trying to get a pulse for how much they understand.  I think students are afraid of being wrong, or asking questions that seem too easy to other students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I firmly believe that an interactive classroom is far more conducive to learning than one in which I simply download information onto a blackboard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some techniques I can use that help foster a safe environment for students to speak up and interact with me, other students, and the material?&lt;/p&gt;&#xA;" OwnerUserId="21" LastEditorUserId="21" LastEditDate="2014-03-13T21:09:42.963" LastActivityDate="2020-08-28T20:33:55.720" Title="Encouraging class participation" Tags="|classroom-management|interactive-teaching|class-participation|calculus|" AnswerCount="14" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2" PostTypeId="1" CreationDate="2014-03-13T20:36:20.300" Score="30" ViewCount="4027" Body="&lt;p&gt;I know from my experience I learnt proofs myself way before I learnt them in school and I felt it gave me a far better understanding of math.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is a good point to start learning proofs? what are the pros and cons? does the whole curriculum have to be geared towards proofs?&lt;/p&gt;&#xA;" OwnerUserId="13" LastActivityDate="2020-08-31T03:23:23.570" Title="Is there a good age/level to start learning mathematical proofs?" Tags="|proofs|" AnswerCount="8" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3" PostTypeId="2" ParentId="1" CreationDate="2014-03-13T20:43:18.423" Score="34" Body="&lt;p&gt;The think-pair-share technique is an oldie but a goodie:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Pose a question&lt;/li&gt;&#xA;&lt;li&gt;Give students 1 minute to quietly think of and write down their answer (if you have a computer/projector setup you can use an onscreen timer to enforce the &quot;1 minute&quot; frame)&lt;/li&gt;&#xA;&lt;li&gt;Give students 2 minutes to exchange / compare solutions with a neighbor&lt;/li&gt;&#xA;&lt;li&gt;Ask for volunteers to share results with the whole class&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Devoting a minimum of 3 minutes explicitly to think/pair communicates to your class that your question is &lt;em&gt;not rhetorical&lt;/em&gt; and that you &lt;em&gt;expect an answer&lt;/em&gt;.  The &quot;pair&quot; stage discussion provides them with an opportunity to pre-screen their answer so that they will (one hopes) feel safer about speaking up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the downside of this strategy is that it takes more time.  If what you are hoping for is a quick check-in with your students to punctuate your presentation without interrupting the flow, this may not serve that purpose.  But if your goal is to actually get students to stop, think, work, and talk to you (and each other) this may help.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-03-13T20:43:18.423" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="4" PostTypeId="2" ParentId="2" CreationDate="2014-03-13T20:44:41.337" Score="15" Body="&lt;p&gt;My next door neighbor's daughter is in fifth grade, and she is enrolled in an extracurricular course with advanced math curriculum.  I sometimes help her with her homework, which includes proofs.  For example, the other day she was asked to prove that the square root of 5 is irrational.  I went through the standard argument with her, and even though she has some understanding of what prime factorization is, the notion of proof by contradiction was a bit beyond her, and she quickly lost interest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tell this story because I think being too rigorous too soon can be damaging to children's enjoyment of mathematics, which may give them a sour taste and never give them the chance to enjoy math later on.  I know that in California high schools, geometry is where the notion of proof is first introduced (albeit, in a very awkward style), and students usually end up hating it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think we should always prioritize enjoyment/curiosity/intrigue/wonder over rigor when teaching children mathematics.  Rigor can come later, when they thirst for it.  If they don't care about the foundations of mathematics right away, and just enjoy looking at nice pictures and doing interesting activities, great, run with that.  Of course, there will be a few who at an early age desire more, but we shouldn't force students to get there until they are ready.&lt;/p&gt;&#xA;" OwnerUserId="21" LastActivityDate="2014-03-13T20:44:41.337" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5" PostTypeId="2" ParentId="1" CreationDate="2014-03-13T20:49:29.807" Score="54" Body="&lt;p&gt;This is a small tip based on the obvious idea that it needs to &lt;strong&gt;feel safe&lt;/strong&gt; to answer questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you need to take the derivative of x sin x, but you want students to speak up about it in the flow of lecture. Here are three ways to do it:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;Now I need the derivative of x sin x. What should I do first?&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;Now I need the derivative of x sin x. Which rule should I use?&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;Now I need the derivative of x sin x. The answer is not 1 * cos x. Why not?&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The first question is too open-ended and scary for students to answer. If you ask questions of this type, you will only possibly get answers from the top of the class, and no one else gains anything from the process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second one is a lot better even though it is almost the same question, because it explicitly tells them the format of the answer. You might get some brave B students who will name the correct rule to answer this one, because it is unlikely for them to answer in the completely wrong format.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The third one is my favorite form of question: instead of asking students to venture forth and dare them to avoid the trap, you ask a question, point at the trap, and say &quot;don't step on that trap.&quot; Everyone can gather around the trap and many will think &quot;What? I would have stepped into that trap.&quot; More students gain from it and more students are likely to venture forth an answer.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-13T20:49:29.807" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6" PostTypeId="1" CreationDate="2014-03-13T21:02:34.937" Score="88" ViewCount="10837" Body="&lt;p&gt;I find that making homework meaningful is becoming increasingly challenging. Let us suppose that I am planning for next semester's first-semester or second-semester calculus course at my university.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In all likelihood, we will be using one of the common calculus texts because the university expects consistency, both from semester to semester and from teacher to teacher. Solutions manuals exist for most of the common calculus texts, so assigning problems from the text has downsides. But this point is slightly moot, because WolframAlpha can do essentially all the problems anyway (albeit with poor exposition). There is also an endless supply of answers awaiting at Math.SE and similar fora.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, writing my own exercises (which I currently do) doesn't prevent or penalize the use of WolframAlpha or online math fora.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possible solution would be to make homework count much less. But I dislike this idea, as I find that students who do homework are much more competent on average than those who don't; homework should be consequential enough for students to take it seriously.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are possible, good ways of overcoming the availability of free and proficient homework assistance?&lt;/p&gt;&#xA;" OwnerUserId="20" LastActivityDate="2020-08-29T09:40:18.430" Title="How to assign homework when answers are freely available or attainable online?" Tags="|course-design|homework|" AnswerCount="13" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7" PostTypeId="1" CreationDate="2014-03-13T21:04:33.947" Score="25" ViewCount="2257" Body="&lt;p&gt;I am an undergraduate. Other undergraduates sometimes ask me to tutor them in an introductory real analysis course that covers the equivalent of the first half-dozen chapters of Rudin's &lt;em&gt;Principles of Mathematical Analysis&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books/about/Principles_of_Mathematical_Analysis.html?id=kwqzPAAACAAJ&amp;amp;hl=en&quot; rel=&quot;noreferrer&quot;&gt;baby Rudin&lt;/a&gt;). We meet once a week for about an hour to discuss concepts and their problem sets. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the beginning of the semester, I seem to spend about half the time on concepts (mainly repeating definitions and giving examples) and the other half giving hints on the homework problems. Later in the semester (and in the second semester continuation), I spend most of the time giving them considerable help with the homework. I don't have any trouble explaining concepts. However, I have a lot of difficultly with providing the appropriate amount of help with problem sets. The biggest problem I see with my students is a lack of problem solving ability. Part of it is a result of a poor grasp on the material and part of it is laziness. They don't seem to be able to think about a problem constructively for more than 5 or 10 minutes before giving up. Further, many of the problems are relatively easy once you see the essential idea or 'trick'. So either I show them the 'trick' or they just get the answer from another student. I usually end up giving too much assistance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have suggested they try easier exercises to build up their problem solving skills. Abbott's &lt;em&gt;Understanding Analysis&lt;/em&gt; is a good source of these. However, they are unwilling to do much extra work on their own. Of course, this is ultimately their problem, but as a busy undergraduate myself, I am sympathetic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another common concern is preparing for exams. A 1-hour exam is a much different format than an untimed problem set and much less forgiving. My standard advice is to first to memorize all of the definitions, be able to give examples and non-examples (where relevant), and memorize the statements of all of the important theorems. After they have done this, I suggest solving easy exercises from their textbook and trying to prove simple propositions in their book without looking at the book's proof (e.g. a finite intersection of open sets in $\mathbb R^n$ is open). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How should I address these issues? Are there any other techniques I could use to make my tutoring more effective? I am particularly interested in ideas that are specific to real analysis. &lt;/p&gt;&#xA;" OwnerUserId="9" LastEditorUserId="8571" LastEditDate="2017-12-09T19:03:57.550" LastActivityDate="2020-08-29T09:40:05.210" Title="How should one tutor a student in undergraduate real analysis?" Tags="|tutoring|mathematical-analysis|undergraduate-education|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8" PostTypeId="1" AcceptedAnswerId="21" CreationDate="2014-03-13T21:10:27.630" Score="20" ViewCount="464" Body="&lt;p&gt;Some place emphasis on higher order thinking skills at the cost of lower emphasis on drill/practice. Others disagree. Personally, I agree that higher order thinking skills are necessary for effective application of mathematics to problems, but I do not want to ignore the fact that effective higher order mathematical thinking requires firm understanding of fundamental mathematical principles. Therefore, I would like to combine both… but I'm a bit unsure about the “how to”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any well-vetted way to find a good balance between both?&lt;br&gt;&#xA;If there isn’t, can you provide some hints/tips that might help me?&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-06-14T17:21:39.240" LastActivityDate="2017-07-10T02:38:37.053" Title="How to balance higher order thinking and drill/practice" Tags="|undergraduate-education|course-design|tutoring|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9" PostTypeId="1" CreationDate="2014-03-13T21:21:58.587" Score="23" ViewCount="579" Body="&lt;p&gt;When I want to point out to my students that getting the right result is not enough, I like to show them the example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{16}{64} = \frac{1\hskip-.1cm- \hskip-.4cm{6}}{-\hskip-.2cm{6}\,4} = \frac14.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This example works well in some of my classes where the error is just the right level of &quot;stupid&quot;: The students feel absolutely safe from making this error, and they also do not feel that this is a &quot;kindergarten&quot; example too far removed from their level of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I would like to have some more examples like this for students with less and with more knowledge about mathematics.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are some good simple preferrably amusing examples that getting the right result is not enough?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="43" LastActivityDate="2014-03-13T21:26:04.477" Title="What are some good simple examples that getting the right result is not enough?" Tags="|proofs|examples|" AnswerCount="0" CommentCount="17" ClosedDate="2014-03-15T00:03:53.977" ContentLicense="CC BY-SA 3.0" />
  <row Id="10" PostTypeId="2" ParentId="2" CreationDate="2014-03-13T21:25:50.410" Score="6" Body="&lt;p&gt;One difficulty that I have seen is that many mathematics teachers present a course as if all their students were going to become mathematicians, and forget that many, even most, will prefer to become builders, restaurant cooks, nurses, elementary educators, musicians, or advertising copywriters instead. They are studying mathematics because they believe, or have been told, that it will someday be useful for something they really want to do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's also impossible to tell in advance, who will take an interest in mathematics for its own sake and enjoy rigor.  In either case, if the proofs are difficult, tedious, or incomprehensible,  it will indeed put them off learning any mathematics at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that the time and place to introduce proofs (and disproofs) is when students ask &quot;How do we know this for sure?&quot;,  In that case, it's not so much a question of whether to teach proofs, but how.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-13T21:25:50.410" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12" PostTypeId="1" AcceptedAnswerId="17791" CreationDate="2014-03-13T21:40:30.170" Score="37" ViewCount="2274" Body="&lt;p&gt;All introductory calculus books that I have seen spend most of their chapters on differential calculus talking about derivatives, with at most a short section defining differentials as &lt;span class=&quot;math-container&quot;&gt;$dy = f'(x) \, dx$&lt;/span&gt;.  However, differentials are useful for understanding a lot of things, like linear approximation, the chain rule, integration by substitution, and (when you get to multivariable calculus) the change-of-variables formula and the various manifestations of Stokes' theorem.  One doesn't have to agree with everything that &lt;a href=&quot;http://www.physics.orst.edu/bridge/papers/CMJdifferentials.pdf&quot; rel=&quot;noreferrer&quot;&gt;Dray and Manogue&lt;/a&gt; say to want to try introducing and emphasizing differentials early in differential calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any calculus textbook which does such a thing?&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="205" LastEditDate="2020-01-16T00:38:16.230" LastActivityDate="2023-02-21T21:25:59.547" Title="A calculus book that uses differentials?" Tags="|calculus|textbooks|infinitesimals|" AnswerCount="7" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13" PostTypeId="1" AcceptedAnswerId="165" CreationDate="2014-03-13T21:41:56.633" Score="27" ViewCount="1058" Body="&lt;p&gt;Many believe (I think rightly so) that the presentation of counterexamples should play an important role in the teaching upper level mathematics courses such as real analysis and topology. Counterexamples show why the hypotheses of various theorems are important. Further, counterexamples very often add to students' intuition and ability to quickly recognize false propositions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some examples of counterexamples that could be provided in a first year calculus course include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Removing or varying hypotheses in L'Hopital's Rule,&lt;/li&gt;&#xA;&lt;li&gt;Removing or varying hypotheses in Intermediate Value Theorem&lt;/li&gt;&#xA;&lt;li&gt;Converse of Differentiability implies continuity.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Should counterexamples play an important role in a first year calculus course taken largely by non-majors?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="52" LastActivityDate="2014-03-14T22:58:40.630" Title="Counterexamples in first year calculus" Tags="|calculus|mathematical-pedagogy|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="14" PostTypeId="2" ParentId="2" CreationDate="2014-03-13T21:42:22.263" Score="11" Body="&lt;p&gt;There is really no age where you can learn mathematics, but where you cannot learn mathematical proofs if you do not confuse rigor with technical notation and language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, the younger the child and the smaller the mathematical knowledge base, the less technical the results need to be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The types of results that I am thinking of:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sum of two even numbers is even. Why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can I find out the sex and the home planet of the aliens in this puzzle? If not, why not?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And no, you do not need to introduce set theory and information theory to give a &quot;rigorous&quot; proof. It is just a matter of using an &quot;axiom system&quot; that is adapted to the child's level of knowledge. Hardly any research paper that is not about mathematics foundation refers to foundations to give a rigorous proof. Teaching students a myth as introduction to proofs is not helpful. I have also noticed that first-semester courses on the use of quantifiers usually tell the students the lie that $\Rightarrow$ is used only in the logical sense ($0=1$ implies all beer is free) and then go on to mark down student papers because their implication arrows do not correspond to an actually reproducible step in the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the question actually is: &quot;Is there a good age/level to start forcing everyone to learn proofs?&quot; then the answer is probably &quot;never&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am biased here, because my first introduction to &quot;rigorous proofs&quot; included months of Aristotle's classification of syllogisms (see &lt;a href=&quot;http://en.wikipedia.org/wiki/Syllogism#Examples&quot;&gt;here&lt;/a&gt; ), which is about as useful as learning different names for problems of the type $a-b=?$ and $b+?=a$.&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2014-03-13T21:42:22.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="15" PostTypeId="2" ParentId="6" CreationDate="2014-03-13T21:55:13.837" Score="16" Body="&lt;p&gt;I agree that this is a problem in teaching calculus and other classes where solutions manuals and online forums are readily available and I don't know that there exists any one thing that you can do. No matter what, a student can always find &quot;help&quot; somewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First some remarks&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If a student gets hold of a solutions manual, then it is of course easy for the student to copy down the solution. But the student still faces the problem of having to reword the solution since you (most likely) also have access to the solutions manual. When you see a solution that is a copy of the one in the solution, the student has cheated and you can follow your school's policy on what to do. And so, having a solutions manual still does require some level of effort on the students part. (Granted, there are problems where it is hard to write down a solution that is different from the one in the solutions manual).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As you note, you can solve the problem of solutions manuals by writing your own problems. This, as you note, still doesn't eliminate the student seeking help elsewhere. But again, seeking help online or with a tutor requires some effort.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So by saying this, I guess that I am trying to point out that it &lt;strong&gt;might&lt;/strong&gt; not be a huge problem. It is my opinion that as long as the students actually learn the material, I don't care if they studied solutions in a manual or got help online. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, to answer your question, what do I believe that you can do?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;As you also note, making the homework part of the final grade smaller of course takes care of it, but I agree that this goes against the desire for students to spend time outside of the classroom.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This is radical, but I suggest implementing &lt;em&gt;oral exams&lt;/em&gt; or presentations and letting them be part of the final grade. This, of course poses some practical problems that might be addressed in another question. This could serve as a motivation for the students to understand the problems that they are solving better.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Instead of having oral &lt;em&gt;exams&lt;/em&gt;, you could also simply tell the students that they have to present a problem from the homework on occasion. Maybe you could make every 30 min. every Friday &quot;present a problem Friday&quot;. You could determine in advance who should do it, or you could pick someone at random in class. That way there is an incentive to understanding your homework. This, of course, can be a bit controversial, but in many countries this is actually practiced and so it might be worth considering.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="53" LastEditorUserId="53" LastEditDate="2014-03-13T22:02:41.463" LastActivityDate="2014-03-13T22:02:41.463" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="16" PostTypeId="1" CreationDate="2014-03-13T22:01:05.157" Score="19" ViewCount="647" Body="&lt;p&gt;In the USA most mathematics undergraduate classes seem to base the final grade  on the following items&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Homework / quizzes&lt;/li&gt;&#xA;&lt;li&gt;Midterms / exams.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I believe that there can be a lot of value in oral exams, so my question is: What is a concrete model for implementing oral exams in an undergraduate class? How could the format look like?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can imagine that this will look differently for different classes, so specifically I would be interested in beginning calculus.&lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-03-14T12:52:30.047" Title="Implementing oral exams" Tags="|calculus|oral-exams|testing|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="17" PostTypeId="2" ParentId="16" CreationDate="2014-03-13T22:08:00.477" Score="9" Body="&lt;p&gt;What would your purpose be for doing oral exams?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My purpose is to allow students to explain something they bombed on a test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I allow students to retake tests in Caclulus I. On the first test, I ask them to explain the definition of the derivative, comparing it to slope. On the retake, I ask the question a different way. I can't ask that question any more ways. So if they want a second retake on that, they must come to me and explain. I have only had one or two students do this so far.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-13T22:08:00.477" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="18" PostTypeId="1" AcceptedAnswerId="23" CreationDate="2014-03-13T22:09:57.763" Score="42" ViewCount="10900" Body="&lt;p&gt;Some teachers make memorizing formulas, definitions and others things obligatory, and forbid &quot;aids&quot; in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students are expected to take questions home, study the problems in any way they want and then submit solutions a few days later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Naturally, the memory-oriented problem sets are relatively easier (modulo time limit), encourage less understanding and more proficiency (in the sense that the student has to be efficient in his approach). As the mathematics is in big part thinking, I think that it is beneficial to students to let them focus on problem solving rather than recalling and calculating (i.e. designing a solution rather than modifying a known one). There is a huge difference between work in time-constrained environment (e.g. medical teams, lawyers during trials, etc.) where the cost of &quot;external knowledge&quot; is much higher and good memory is essential.&#xA;However, math is, in general (things like high-frequency trading are only a small part math-related professions), slow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, memory-oriented teaching is far from being a relic of the past. Why is this so? As this is a broad topic, I will make it more specific:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are the advantages of memory-oriented teaching?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are the disadvantages of allowing aids during tests/exams?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="114" LastEditDate="2014-03-14T18:04:26.913" LastActivityDate="2020-08-29T10:39:02.030" Title="Formula sheets and books during tests and exams" Tags="|testing|aids|exams|" AnswerCount="11" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="19" PostTypeId="2" ParentId="8" CreationDate="2014-03-13T22:13:39.063" Score="7" Body="&lt;p&gt;I've found that a big reason for drill and practice is speed and facility.&#xA;For instance, if you're doing arithmetic, you could get the same results from the addition tables by counting on your fingers and toes but it's slower. And if you're doing multiplication, you could get the same results by repeated addition, but it's slow.  Now that we don't have to add long columns of numbers by pencil and paper in order to do accounting and the like, there's less need for extensive drill in arithmetic than there was a century ago, but adding 3+7 mentally is still faster than punching it into the calculator. It takes practice with a technique to be able to solve common problems quickly rather than having to laboriously work through them each time they are encountered. That facility eases comprehension of higher level, abstract concepts. Another reason is that the ability to comprehend abstract concepts is often based on being familiar with a variety of concrete examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say that the best way to know how much drill/practice is needed seems to be to decide what you need the speed and facility for. If it's a common technique that will be used over and over again, like solving linear algebraic equations in one variable, then spend some time on drill. If it's as a foundation for more advanced, abstract concepts, then it would be useful to practice until the more specific example is no longer a hard problem. If it's a nice but not essential feature, like Egyptian or Mayan numerals, introduce the concept and move on.&lt;/p&gt;&#xA;" OwnerUserId="40" LastEditorUserId="40" LastEditDate="2014-03-14T18:53:00.290" LastActivityDate="2014-03-14T18:53:00.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="20" PostTypeId="2" ParentId="18" CreationDate="2014-03-13T22:14:48.817" Score="18" Body="&lt;p&gt;I allow notes on tests, because math is less about memory than about understanding, and I don't want students to focus on the memory part. I don't allow notes on quizzes, because they are on just one problem type, and I want students to be ready to think it through. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may find this blog post helpful: &lt;a href=&quot;http://exzuberant.blogspot.com/2012/07/monkey-and-mathematician-learn-calculus.html&quot;&gt;http://exzuberant.blogspot.com/2012/07/monkey-and-mathematician-learn-calculus.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Getting more into particulars:&#xA;One disadvantage of allowing notes for tests is that students will spend too much time hunting for the answer in their notes. To offset that, I allow only a 3x5 card.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One advantage of memory-oriented teaching is that it can help students retain. But if that retention is bought at the expense of deeper understanding, you've made a serious mistake. In pre-calc, I ask student to show the end behavior of a function with their arms. This is a memory technique. My hope is that it complements the understanding of &lt;em&gt;why&lt;/em&gt; odd-degree functions have &quot;one arm up, one arm down&quot;. &lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2014-03-14T01:54:14.577" LastActivityDate="2014-03-14T01:54:14.577" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="21" PostTypeId="2" ParentId="8" CreationDate="2014-03-13T22:25:59.800" Score="13" Body="&lt;p&gt;One of the benefits of practice is a deeper understanding. I don't care how fast you get the answer, but I hope you are not adding by columns when you add 101+99. And that's why thinking of practice as equivalent to drill may be problematic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maria Droujkova talks about how you need to love something to learn it well. If we want kids to learn the times tables, we shouldn't be doing drill (unless a particular kid actually likes it), instead we should make it interesting: have them build a times table with heights of legos or cuisenaire rods showing the heights, have them draw ten-stars, and connect every third number for the threes (for example), search for iconic representations (e.g. a dozen eggs is 6x2, a six-pack of soda is 3x2, see some great flash cards &lt;a href=&quot;http://www.emit.org/flashcards.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, though kids could make their own), and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think students in the U.S. are very used to math classes that focus at lower levels. One way to get a better feel for a good balance is to study how Japanese and Chinese math classes operate. For elementary math, read Liping Ma's book, Knowing and Teaching Elementary Mathematics, comparing U.S. and Chinese teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What course do you have in mind?&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2014-03-17T20:54:15.987" LastActivityDate="2014-03-17T20:54:15.987" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="22" PostTypeId="1" AcceptedAnswerId="88" CreationDate="2014-03-13T22:26:03.640" Score="28" ViewCount="2820" Body="&lt;p&gt;I always had problems teaching the implicit function theorem in advanced analysis courses. This result is motivated by later applications, but it would be great to provide easily accessible examples to motivate the whole thing.&lt;/p&gt;&#xA;&lt;p&gt;I usually use Example 1.1.1 from &lt;a href=&quot;http://books.google.de/books/about/The_Implicit_Function_Theorem.html?id=ya5yy5EPFD0C&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kranz, Parks: The implicit function theorem (Birkhäuser)&lt;/a&gt;, which is&#xA;&lt;span class=&quot;math-container&quot;&gt;$y^5+16y-32x^3+32x=0.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;What else is there?&lt;/p&gt;&#xA;&lt;p&gt;ADDED in edit: Though I accepted a very nice answer, I would be happy for more answers and for more slist of possible examples.&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorDisplayName="user155" LastEditDate="2021-12-20T22:46:03.537" LastActivityDate="2022-01-04T18:32:45.753" Title="What are some good examples to motivate the implicit function theorem?" Tags="|examples|mathematical-analysis|graduate-education|concept-motivation|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23" PostTypeId="2" ParentId="18" CreationDate="2014-03-13T22:32:16.893" Score="31" Body="&lt;p&gt;It is easy to explain the most immediate disadvantage of allowing &quot;aids&quot; during exams: many students misjudge the situation, thinking that having books and/or papers means they can study less. In particular, they often misjudge information access time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But many students benefit from some form or degree of open-access exams, because they can relax a little about memorization and data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, indeed, in real life, although there are obvious benefits to having things in one's head, it's not &lt;em&gt;possible&lt;/em&gt; to have &lt;em&gt;everything&lt;/em&gt; there, and it's simply not the way things work. So it is a bit exaggerated to conduct exams in such artificial environments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, although allowing &quot;aids&quot; reduces stress (and misleads some students), the necessarily-repetitive aspects of memory-oriented teaching/learning have their own benefits, that seem not achieved by less &quot;primordial&quot; teaching/learning. The point is not only memorization per se, but the repetitive aspect, which is often misguidedly downplayed in contexts where it is pretended that we all have nearly-perfect memories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In brief: &quot;drill&quot; is pretty inhuman(e), or can be, but has benefits not achievable otherwise. Simultaneously, having books and papers available during exams is much more like real life, and changes the emphasis of the exam from memory to function (ideally), but will mislead a substantial number of students who've grown up trying to &quot;game&quot; the system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But perhaps the real point is the exam-design itself, not the circumstances.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own choice has been to always do open-book-open-notes exams, but/and designed to strongly favor people who have the information in their heads. This design criterion can be taxing to fulfill if one allows essentially unlimited internet access, for example, and the course is a cliched one like Calculus I. One device which I have found useful, both because it promotes useful skills and because it complicates &quot;gaming the system&quot; is the requirement of coherent writing, full sentences, good grammar. Not just skeletal formulaic junk splashed on a page... working toward a formula to be circled at the end. &lt;em&gt;Explanations&lt;/em&gt; are not (yet?) so easily found by Googling. Of course I tell people in advance that they will need to be able to do this, and the homework would give feedback.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="60" LastEditDate="2014-03-14T18:02:29.543" LastActivityDate="2014-03-14T18:02:29.543" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="24" PostTypeId="2" ParentId="18" CreationDate="2014-03-13T22:34:33.407" Score="29" Body="&lt;p&gt;I disagree with one of the other answers when saying that &quot;math is not about memory&quot;. Doing math is not only about memory, but remembering your definitions and theorems can be crucial to doing problems. The argument that a mathematician can just look of these things on books disregards the fact that when doing the problem, you need to collect all the aspects of the solution in your mind (in some way). If you want to solve a problem in group theory, you need a good understanding of what a groups is. That is, you need to have memorized the definition and have internalized it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would say that this is the answer to the question: &quot;&lt;em&gt;What are the advantages of memory-oriented teaching?&lt;/em&gt;&quot;. The advantage (or an advantage) is that to &quot;see&quot; a solution to a problem one needs to be able to mentally play with the ingredients.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;What are the disadvantages of allowing aids during tests/exams?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The disadvantages are, then, that students might not have internalized concepts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition to this, I also believe that certain elementary principles and rules need to be memorized. You are not going to do well on a calculus exam if you don't remember basic algebra rules. If you want to succeed as a mathematician (assuming that par of the goal of undergraduate mathematics education is to prepare future mathematicians), you will have to memorize things. It would be very difficult to have a conversation about mathematics if you don't remember things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To sum up, my answer is: I don't think that open-book exams are bad. I think they are great. But, there is much value in memorizing/remembering. &lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-03-13T22:34:33.407" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="25" PostTypeId="2" ParentId="13" CreationDate="2014-03-13T22:38:38.063" Score="20" Body="&lt;p&gt;I tend to think that the &lt;em&gt;plusses&lt;/em&gt; and wonderful power of calculus to address natural, simple questions about geometry and mechanics should be communicated before students are taught to &lt;em&gt;worry&lt;/em&gt;. After all, not counting the classical Greek protocalculus, from Fermat and Descarte, and Newton and Leibniz, and Bernoullis and Euler, up to the 19th century, mathematicians thought calculus was wonderful because of what it could &lt;em&gt;do&lt;/em&gt;, not because or despite pathologies or worries or counter-examples might come up. (Cf. Bishop Berkeley.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As it is, many universities' use of basic calculus is as a filter, with no intellectual purpose whatsoever. This is already undignified enough, and subliminally understood by many students. Bringing up pathologies or dangers, historically inaccurate anyway, will most often just seem like adding insult to injury. More crazy nonsense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being &lt;em&gt;positive&lt;/em&gt; about what calculus can &lt;em&gt;do&lt;/em&gt; I think is much better than using it as testbed for &quot;rigor&quot;... which no one would care about, anyway, if calculus could not do amazing things.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-13T22:38:38.063" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="26" PostTypeId="2" ParentId="2" CreationDate="2014-03-13T22:40:17.970" Score="11" Body="&lt;p&gt;Very young children can prove that two odd numbers will always add up to an even number. They can explain with pictures. There are other good questions young children can address with proofs, the style of which will be more juvenile, but which can still be &quot;rigorous&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math circles are a great way to introduce young children to logically figuring out a mathematical idea. This post at Talking Stick Math Blog might help you imagine what it looks like: &lt;a href=&quot;http://talkingsticklearningcenter.org/logic-for-the-very-young/&quot;&gt;http://talkingsticklearningcenter.org/logic-for-the-very-young/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-13T22:40:17.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="27" PostTypeId="1" AcceptedAnswerId="40" CreationDate="2014-03-13T23:18:50.390" Score="15" ViewCount="7194" Body="&lt;p&gt;I know that there is a visual demonstration of &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2=c^2$&lt;/span&gt; using a smalĺ piece of paper, but there are also a lot of variations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which visual or drawing demonstration of the Pythagorean theorem can I show to 14-year-old students?&lt;/p&gt;&#xA;" OwnerUserId="73" LastEditorUserId="12255" LastEditDate="2019-07-02T11:26:32.873" LastActivityDate="2022-02-02T15:39:20.503" Title="Visual Pythagorean demonstration" Tags="|geometry|visual-proof|" AnswerCount="8" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="28" PostTypeId="1" AcceptedAnswerId="133" CreationDate="2014-03-13T23:26:43.747" Score="13" ViewCount="126" Body="&lt;p&gt;I am doing most of my research in a relatively new area of mathematics called finite subdivision rules. I have an undergraduate student who has begun doing research with me on some of the properties of simple subdivision rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He would like to get more background on this topic, but because it is such a new field, all of the references are at a research level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your experience, how would you deal with this situation?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-03-15T11:28:03.870" LastActivityDate="2014-03-15T11:28:03.870" Title="How to bring an undergraduate researcher up to speed on a brand new topic" Tags="|undergraduate-research|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="29" PostTypeId="1" AcceptedAnswerId="921" CreationDate="2014-03-13T23:29:05.023" Score="20" ViewCount="506" Body="&lt;p&gt;(I previously asked this at &lt;a href=&quot;https://mathematicsteachingcommunity.math.uga.edu/index.php/685/the-water-triangle-proportional-reasoning-task&quot; rel=&quot;noreferrer&quot;&gt;The Mathematics Teaching Community&lt;/a&gt;, but I'm hoping it would attract further answers here.)&lt;/p&gt;&#xA;&lt;p&gt;The Wikipedia page on &lt;a href=&quot;https://en.wikipedia.org/wiki/Proportional_reasoning&quot; rel=&quot;noreferrer&quot;&gt;proportional reasoning&lt;/a&gt; mentions a &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Proportional_reasoning#Water_Triangle&quot; rel=&quot;noreferrer&quot;&gt;water triangle&lt;/a&gt;&amp;quot; task.  It seems that the task was created in the 1970s, possibly by Robert Karplus.  However, I was not able to find any reference to it.  Does anyone know of any publication that mentions its source?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Consider a container of colored liquid inside a right triangle where the triangle can be tilted and the water levels on the left and right side can be measured on a built-in scale.  This is called a &amp;quot;water triangle&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/n6SCr.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;The water triangle is rotated until it shows a measurement of 4 units on the left side and 6 units on the right side. Suppose the triangle is tilted even more until the water level on the right side is at 8 units. Predict what the water level in units will be on the left side.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-28T09:43:05.027" Title="The &quot;water triangle&quot; proportional reasoning task" Tags="|proportional-reasoning|reference-request|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="30" PostTypeId="2" ParentId="1" CreationDate="2014-03-13T23:30:54.837" Score="13" Body="&lt;p&gt;Disclaimer: I teach at university, it is much different than teaching younger students (e.g. I wouldn't really know how to handle them well).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The approach that works best for me is to &lt;em&gt;force&lt;/em&gt; them to answer and to do it so frequently as to make it natural for them to ask questions, while, at the same time, letting them know it's alright to make mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To given an example, I often ask questions like &quot;what technique would you want to use&quot; which, sometimes, I make deliberately hard. Then, after a bad answer I respond &quot;unfortunately, this approach will not work in this setting, however this is a perfectly reasonable answer&quot; and then explain why the idea could work (i.e. intuitions, sometimes even explaining similar problems where it does work), and finally I would demonstrate why our current task is different and why that particular method fails.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are particular personalities such that making them ask even a silly question helps. I remember one session where I made the whole class ask clownish things which, then, greatly eased the tension (it transformed into a sort of joke-questions contest) and activated even the most stubborn part of the group (and we did lose a lot of time, but I wouldn't say it was wasted).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After a two-three classes it becomes a standard thing and I don't even need to pick them. If someone is behind, it is often enough to apply a very gentle pressure (like a slightly longer eye-contact) to activate him/her. During the semester (i.e. when I earned enough respect) I sometimes taunt/tease them, even to the point of encouraging disobedience (tricking them into independent thoughts), e.g. proposing some bad solutions (sometimes blatantly bad, e.g. for comical effect, sometimes subtly wrong, to guard them against bad intuitions) or saying &quot;no&quot; to perfectly valid inferences (e.g. to emphasize which part of the process was the key part &quot;oh, now I understand&quot;, and perhaps later to recall some previously solved problems &quot;it is the same we did last week&quot; and extract similar patterns, perhaps alike in some unobvious way; or to make the solution more formal and show them some traps which could be overlooked and later lead to incorrect reasoning).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As with any approach, there are disadvantages. There was one group which learned that saying &quot;I don't know&quot; is ok too fast and I was basically helpless - the contact was poor, and as things I prepare depend on class participation, they learned less and less, and that, in turn, made them even more remote. I was unable to break from the evil cycle until the very end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing is that these people start asking many (too many) questions in classes taught by others which makes them (the teachers) angry at you (e.g. it disrupts their flow).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, it works better with mature people, i.a. ones which are conscious that it is in their interest to learn, rather than just earn grades. One time such a situation turned itself into really dissonant tones: I scorned a &quot;let-me-show-off&quot; question of a good grade-earner and praised a bit naive, but honest question from person who usually doesn't know what's happening. There were some resentful looks and I had to give the class a much harder challenge so that it would be difficult even for the top part, so that the aforementioned person could reclaim his status, but in the process had to ask honest questions to complete the task.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps this is not the best way, but it works for me (among others, it matches my style of teaching).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-14T19:55:24.663" LastActivityDate="2014-03-14T19:55:24.663" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="32" PostTypeId="1" AcceptedAnswerId="828" CreationDate="2014-03-13T23:32:39.443" Score="14" ViewCount="365" Body="&lt;p&gt;I am currently teaching the workshop for a class on chaos and fractals in a computer lab. The class is predominantly first year, first semester university students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Worksheets have been developed for the students to use to encourage them to engage with the material in a way that isn't feasible on paper (e.g. bifurcation diagram). This requires them using provided Matlab scripts and modifying them for the different questions on the worksheet. In a class survey, all students said that they were familiar with the usage of Matlab.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The approach I am taking for teaching this class is to remind the students where the material can be downloaded from on the class website, before individually engaging with each student to ensure that they are understanding the topic at hand. This approach is significantly better for this class, as I have been allocated 2 hours with a 22 student class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first two classes, some students have really engaged and are using the worksheet questions to understand what is actually going on with the underlying theory, what the mathematical terms used in class actually look like on a graph, etc. Meanwhile, other students simply work through the worksheet as quickly as possible and don't seem to really be understanding, despite the instructions on the worksheet stating that the worksheet should be worked through slowly to help the understanding of the topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is the best technique for making the students who are working through the worksheet quickly change their approach to help their understanding?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The worksheet for this week focused on identifing the relationship between the time-series plot $(n,x_n)$ and &lt;a href=&quot;http://en.wikipedia.org/wiki/Cobweb_plot&quot;&gt;cobweb plot&lt;/a&gt; for the &lt;a href=&quot;http://en.wikipedia.org/wiki/Logistic_map&quot;&gt;Logistic map&lt;/a&gt;. The students who engaged with the worksheet were able to identify the progression of the sequence between the two graphs and the relationship between them. The students who did not engage completed through the worksheet very quickly be generating the necessary graphs for the relevant questions. When I asked them to explain the relationship, they were unable.&lt;/p&gt;&#xA;" OwnerUserId="71" LastActivityDate="2014-03-25T21:36:45.970" Title="Engaging students in computer lab" Tags="|classroom-management|class-participation|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="33" PostTypeId="1" AcceptedAnswerId="66" CreationDate="2014-03-13T23:37:01.927" Score="14" ViewCount="291" Body="&lt;p&gt;I remember looking once at what chikdren in Kindergarten were expected to know, and it was quite a bit. I have a young son, and would like to know:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is a Kindergartener expected to know about numbers and shapes and other mathematical concepts?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="143" LastEditDate="2014-07-17T19:38:41.910" LastActivityDate="2014-07-17T19:38:41.910" Title="According to Common Core standards, what math skills are beginning Kindergarteners supposed to have?" Tags="|primary-education|common-core|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="34" PostTypeId="5" CreationDate="2014-03-13T23:38:20.557" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-13T23:38:20.557" LastActivityDate="2014-03-13T23:38:20.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="35" PostTypeId="4" CreationDate="2014-03-13T23:38:20.557" Score="0" Body="For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc." OwnerUserId="78" LastEditorUserId="60" LastEditDate="2014-03-17T14:06:31.970" LastActivityDate="2014-03-17T14:06:31.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="36" PostTypeId="1" AcceptedAnswerId="107" CreationDate="2014-03-13T23:41:39.927" Score="24" ViewCount="5527" Body="&lt;p&gt;I need any plot software on Linux or Windows that my students should use it for plotting 3D functions. &#xA;I want introduce any software that be free and useful for bachelor students.&lt;/p&gt;&#xA;" OwnerUserId="79" LastEditorUserId="182" LastEditDate="2017-04-05T17:42:40.237" LastActivityDate="2020-08-31T01:44:19.167" Title="What is a free and simple 3D plot software for students?" Tags="|technology-in-education|software|" AnswerCount="13" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="37" PostTypeId="2" ParentId="36" CreationDate="2014-03-13T23:44:19.007" Score="3" Body="&lt;p&gt;I highly recommend wolframalpha.com&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is free, available on the internet without downloads, and is useful for many other classes. &lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-13T23:44:19.007" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="38" PostTypeId="2" ParentId="1" CreationDate="2014-03-13T23:47:31.670" Score="28" Body="&lt;p&gt;Safety: &#xA;When a wrong answer is given, if you can figure out what would have made it right, you help the student feel safer. Teacher: 2*3 is...? Student: 5 Teacher: oh, I bet you're thinking about 2+3. In calculus, teacher: integral of sinx? Student: cosx. Teacher: If I were asking the derivative, you'd be exactly right. What's the derivative of your answers? (Now they get to tell me about the missed negative sign.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I call on students equally with index cards. After years of doing it, I began to feel it was too ... cold? ... rigid? I stopped. But this winter I was re-reading something I've written about supporting women in math. Female students are less likely to raise their hands, and then they miss out. (Read &lt;em&gt;Failing at Fairness&lt;/em&gt;, mostly about K12, to learn more about the scope of the problems female students face.)  So I went back to my 3x5 cards. And my calc I class with 45 students is benefiting tremendously. It's at 8am, but is a very lively class. Shy students are willing to contribute when called on. (I used to say it was ok to pass. Too many students would pass. I did not do that this semester. It's working fine.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other ways to get interaction:&#xA;If you have issues with clickers, you can do a low-tech alternative. I ask for students to give me thumbs up-down-or sideways many times during each class, to get feedback on how well they feel they understand something. It's very helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the answer is a small whole number, I ask them to indicate with fingers raised. (In pre-calc: What's the degree of this function?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some things can be shown with the body (and the movement helps students internalize the ideas). Show me the long-term behavior of this function with your arms. (y=x^5 has left arm down, right arm up.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use small groups whenever the task can best be done that way, and often have them pair up to compare answers.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2014-03-14T14:07:41.383" LastActivityDate="2014-03-14T14:07:41.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="39" PostTypeId="1" AcceptedAnswerId="179" CreationDate="2014-03-13T23:54:58.280" Score="13" ViewCount="453" Body="&lt;p&gt;I love Euclid's elements, and would like to base a course around them. Before I can pitch it to my supervisors, I need to know where it would fit in the curriculum. While it begins from elementary principles, it requires a high level of logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would be an appropriate prerequisite(s) for this class?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="39" LastEditDate="2015-06-06T14:37:43.840" LastActivityDate="2015-06-07T04:54:18.833" Title="What prerequisites would college students need for a course based primarily on Euclid's elements?" Tags="|undergraduate-education|course-design|curriculum|geometry|prerequisites|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="40" PostTypeId="2" ParentId="27" CreationDate="2014-03-14T00:02:22.887" Score="10" Body="&lt;p&gt;A number of proofs (43!) can be found at &lt;a href=&quot;http://www.cut-the-knot.org/pythagoras/index.shtml&quot; rel=&quot;noreferrer&quot;&gt;cut-the-knot.org&lt;/a&gt;. Some of these are described below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know how hands on you want the students to get with the visual aspect. I remember doing these proofs around that age. I recall this progression and it made sense to me why Pythagoras's Theorem was true.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Geometrical demonstration&lt;/strong&gt;: Pythagoras's Theorem can be easily demonstrated. Construct squares on each side of the triangle, as shown below. Then, the large squares can be subdivided into smaller squares, which can be counted to be the same quantity. Also, I recall cutting the $c$ square to show that it overlaps exactly with the $a$ and $b$ squares.&#xA;This technique can show it is true for particular values of $a,b,c$, but cannot prove the statement is true in general. (Image source: &lt;a href=&quot;http://www.myastrologybook.com/PythagoreanTheorem16c.gif&quot; rel=&quot;noreferrer&quot;&gt;http://www.myastrologybook.com/PythagoreanTheorem16c.gif&lt;/a&gt;)&#xA;&lt;img src=&quot;https://i.stack.imgur.com/9m1gF.gif&quot; alt=&quot;Geometrical proof&quot;&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Pythagorean proof&lt;/strong&gt;: This proof uses squares of equal sizes to show the desired relation. There is some manipulation of the squares required in this proof. (Image source: &lt;a href=&quot;http://en.wikipedia.org/wiki/File:Pythagore.jpg&quot; rel=&quot;noreferrer&quot;&gt;http://en.wikipedia.org/wiki/File:Pythagore.jpg&lt;/a&gt;)&#xA;&lt;img src=&quot;https://i.stack.imgur.com/GwFKr.jpg&quot; alt=&quot;Pythatorean proof&quot;&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Algebraic proof&lt;/strong&gt;: This leads naturally from the second proof as a way of confirming what is being seen in the images. A detailed proof is given at &lt;a href=&quot;http://www.mathsisfun.com/geometry/pythagorean-theorem-proof.html&quot; rel=&quot;noreferrer&quot;&gt;mathsisfun.com&lt;/a&gt;. I will outline the basic steps below.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Each of the squares in &lt;em&gt;Proof 2&lt;/em&gt; has area $(a+b)^2$. In the left square, the total area is the area of the square plus the area of the four triangles, which is $c^2+4\times\frac12ab$. In the right square, the area is the sum of the two smaller squares plus the area of the for triangles, which is $a^2+b^2+4\times\frac12ab$. Equating both sides and simplifying gives the desired result.&lt;/p&gt;&#xA;" OwnerUserId="71" LastEditorUserId="71" LastEditDate="2014-03-14T05:17:50.650" LastActivityDate="2014-03-14T05:17:50.650" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="41" PostTypeId="1" AcceptedAnswerId="171" CreationDate="2014-03-14T00:10:40.183" Score="86" ViewCount="29872" Body="&lt;p&gt;My students frequently mix up my $t$'s with my $+$'s and my $y$'s with my $4$'s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is a good handwriting font for distinguishing these and other easily confused symbols?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2015-04-28T10:53:52.790" LastActivityDate="2022-12-11T08:18:32.903" Title="What is a good handwriting font for mathematics?" Tags="|board-use|" AnswerCount="19" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="42" PostTypeId="2" ParentId="27" CreationDate="2014-03-14T00:17:58.743" Score="5" Body="&lt;p&gt;Check out the book &amp;quot;Proofs Without Words&amp;quot; for a bunch of nice visual proofs of the Pythagorean Theorem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20160930200321/http://www.maa.org/publications/books/proofs-without-words&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proofs Without Words: Exercises in Visual Thinking | Mathematical Association of America&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorDisplayName="user7990" LastEditDate="2020-08-30T15:41:54.620" LastActivityDate="2020-08-30T15:41:54.620" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="43" PostTypeId="2" ParentId="27" CreationDate="2014-03-14T00:20:52.310" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://plus.google.com/+TomEigelsbach/posts/GqgNFEEY3t8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's a link&lt;/a&gt; to a Google+ posting about exactly this. In addition to additional links, it has a GIF using water to demonstrate the equalness of squares.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2Dgyx.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2Dgyx.gif&quot; alt=&quot;Animated GIF demonstrating Pythagorean theorem&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="87" LastEditorUserId="87" LastEditDate="2015-12-31T05:04:47.247" LastActivityDate="2015-12-31T05:04:47.247" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="44" PostTypeId="2" ParentId="2" CreationDate="2014-03-14T00:22:44.107" Score="20" Body="&lt;p&gt;Not to be annoying, but what is a proof?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's my best take: a proof is an explanation that could convince all conceivable skeptics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof, then, is absolutely crucial at all levels of math instruction, because &lt;i&gt;explanation&lt;/i&gt; is crucial to learning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I figure that you're imagining something different, a moment when you would start asking kids to offer the sorts of formal, rigorous proofs that would satisfy higher level students and working mathematicians. I think that this is a mistake. If we want students to be ready to offer and receive more rigorous proofs in later years, we prepare them by creating environments where offering convincing arguments that make sense to us and everyone we can imagine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We teachers will push them to be a bit more skeptical of themselves than they're comfortable being, and in doing so push their proofs to an increasingly higher level of rigor. What's crucial is remembering not to rush them. It makes sense that our kids' proofs wouldn't satisfy us. My students aren't me, and their skeptics are different than mine.&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2014-03-14T00:38:21.240" LastActivityDate="2014-03-14T00:38:21.240" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="45" PostTypeId="2" ParentId="12" CreationDate="2014-03-14T00:25:35.117" Score="24" Body="&lt;p&gt;This might be taking things too far, but Keisler's book (&lt;a href=&quot;http://www.math.wisc.edu/%7Ekeisler/calc.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;available free online&lt;/a&gt;) does everything using infinitesimals, which make differentials literally immediate. The rigorous underpinning for infinitesimals is nonstandard analysis, but this book doesn't dwell on that. It just teaches how to use them correctly.&lt;/p&gt;&#xA;&lt;p&gt;I'm guessing this isn't exactly what you were looking for, but it might be worth checking out because it's free.&lt;/p&gt;&#xA;" OwnerUserId="87" LastEditorUserId="1608" LastEditDate="2022-12-08T17:36:09.020" LastActivityDate="2022-12-08T17:36:09.020" CommentCount="20" ContentLicense="CC BY-SA 4.0" />
  <row Id="46" PostTypeId="2" ParentId="27" CreationDate="2014-03-14T00:26:52.913" Score="6" Body="&lt;p&gt;Here is one of my favorite visualizations of the Pythagorean Theorem:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/RSkds.gif&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is also a fun activity here.  Have the kids draw two adjacent squares of any size of their choosing, and use the technique shown above to have them cut the squares into 5 pieces that can be arranged to form one larger square.  The fun part about this is that the kids get do determine the discrepancy in the sizes of the initial squares, and no matter how large or small the discrepancy, this technique always works.&lt;/p&gt;&#xA;" OwnerUserId="21" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-14T00:26:52.913" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="47" PostTypeId="2" ParentId="6" CreationDate="2014-03-14T00:31:01.330" Score="8" Body="&lt;p&gt;What about just giving kids the answers and asking them to rediscover them? It would be harder to check, though...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(In light of some down-votes, allow me to clarify:)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wouldn't recommend this as an always thing to do on homework, but I've seen teachers give their students answers to homework when they want their kids to focus on the process, not the actual numerical solution. And this can be real problem solving. Instead of &quot;What's the answer?&quot; the question becomes &quot;How is this the solution?&quot; If the problem is difficult enough, it'll still be challenging.&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2014-03-16T15:56:45.240" LastActivityDate="2014-03-16T15:56:45.240" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="48" PostTypeId="2" ParentId="7" CreationDate="2014-03-14T00:35:53.047" Score="8" Body="&lt;p&gt;With lots of pictures/diagrams — giving an intuitive understanding — and then connecting these  diagrams to the logic which can be used in proofs. Once students can convert logical sentences to intuitive pictures, and back again they will have &amp;quot;got&amp;quot; (at least the basics of) analysis.&lt;/p&gt;&#xA;&lt;p&gt;This study guide is also very helpful for students new to analysis: &lt;a href=&quot;https://www.maths.ox.ac.uk/system/files/attachments/study_public_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;How do undergraduates do mathematics? A guide to studying mathematics at Oxford University&lt;/a&gt;. It shows how to read logical sentences in a useful way.&lt;/p&gt;&#xA;" OwnerUserId="41" LastEditorDisplayName="user7990" LastEditDate="2020-08-29T09:40:05.210" LastActivityDate="2020-08-29T09:40:05.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="49" PostTypeId="1" AcceptedAnswerId="52" CreationDate="2014-03-14T00:40:02.060" Score="24" ViewCount="630" Body="&lt;p&gt;In a college math course one is bound to find a fairly broad range of students in terms of their quickness in understanding the material.  This is due to many reasons, including differing mathematical backgrounds.  Inevitably, there are always a handful of very bright students who don't show up to lecture/discussion because they don't feel like they need to in order to pass the class (honestly, they're often right in this regard.  In fact, they most often score the highest on exams).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, just because they're quick to understand the necessary material to score well in the class doesn't mean that we don't have anything to offer them.  I don't think we should be content with just letting them skate by because they have the natural ability to do so.  This leads to my question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can we maintain and grow the interest of the brighter students in a math classroom, even if the bulk of our attention and effort is (rightfully so) spent with the rest of the class.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="21" LastActivityDate="2020-08-31T14:58:29.020" Title="Keeping quicker students engaged and interested throughout a course" Tags="|calculus|classroom-management|class-participation|interactive-teaching|undergraduate-education|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="50" PostTypeId="2" ParentId="13" CreationDate="2014-03-14T00:40:54.667" Score="12" Body="&lt;p&gt;This question should be &quot;community wiki&quot;, if there is such a thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think counterexamples need to be used to make the &quot;obvious&quot; statements of calculus meaningful. For example, the Intermediate Value Theorem says that $f(x)=x^2-2$ has a root between 0 and 2, but in fact this is only true if you believe that irrational numbers exist. This example motivates proving that $\sqrt{2}$ is irrational, and it also affords an opportunity to tell the students that their instinct (the IVT is obvious) is good, but that the IVT is saying something subtle about real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students also find Rolle's theorem obvious, but examples like $g(x)=|x|^{|x|}$ (with the removable discontinuity at $x=0$ removed) demonstrate that maximums don't always go along with $g'(x)=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another favorite type of counterexample are the polynomials that are always positive but don't achieve their minimum. I'll leave that one to the reader.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example that can be used early, in my opinion, so that it doesn't seem so difficulty later, is to remove the singularity of $h(x)=e^{-1/x^2}$. Then show that every derivative of $h$ is 0 at $x=0$. This requires the limit definition of derivative, and (later) is an example of a function that has a Taylor Series that converges for all $x$, but doesn't converge to the function it comes from.&lt;/p&gt;&#xA;" OwnerUserId="87" LastActivityDate="2014-03-14T00:40:54.667" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="51" PostTypeId="2" ParentId="16" CreationDate="2014-03-14T01:00:20.023" Score="9" Body="&lt;p&gt;The problem with oral exams is that they (almost surely) &lt;em&gt;cannot be given simultaneously to a large group&lt;/em&gt; (unlike written exams).  If the questions are the same and are not known in advance, then those taking the exams earlier are at a disadvantage, because those taking them later might get additional information about the questions from those who took it earlier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another disadvantage of oral exams over written exams is that they would &lt;em&gt;take a longer time to administer&lt;/em&gt;.  Thus the examiner will be forced to make the exam short (shorter than a corresponding written exam), that is, to use fewer questions and cover a smaller scope of topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to do tackle these two disadvantages is to have &lt;strong&gt;&lt;em&gt;a large set of questions announced beforehand, where the actual oral exam consists of a random sample of that set&lt;/em&gt;&lt;/strong&gt;.  Everyone has the same information about the questions, the exam would take a shorter time to administer, and the student needs to understand or to be skilled in a wide range of topics to have a good chance of passing the test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also suggest that your oral exam consist of &lt;strong&gt;&lt;em&gt;questions that test for understanding and for skill&lt;/em&gt;&lt;/strong&gt;.  For example, if the topic is, say, limits, then example tasks that assess understanding could be to give a precise definition of a limit, to give some examples of practical applications that use the concept of limits, or to determine whether a given statement about limits is true or not.  An example task that assesses skill would be to find the limit of a given specific function or sequence.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-03-14T01:00:20.023" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="52" PostTypeId="2" ParentId="49" CreationDate="2014-03-14T01:01:48.883" Score="23" Body="&lt;p&gt;Once upon a time I was one of those brighter students. Two things helped. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One was the opportunity to help other people. I got started tutoring mathematics when people would come to me for help with their homework, and I learned that the effort to explain a concept helped me understand it better.  A couple of times, I found that I didn't understand a concept well enough to explain it successfully.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another was for the teacher to suggest additional or &quot;extra&quot; topics that I could work on independently. for instance, my HS Freshman algebra teacher loaned me a book on Computer programming.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-14T01:01:48.883" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="53" PostTypeId="2" ParentId="6" CreationDate="2014-03-14T01:07:45.160" Score="105" Body="&lt;p&gt;My suggestion is that you stop thinking of homework as an assessment tool, and instead find the true meaning of homework, which is a method for students to gain proficiency with the ideas on their own. The key is to design your other assessment tools so as to motivate the students to see that doing homework is a good idea. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My practice, for example, is to assign a lot of homework, and encourage students to do it, but I don't collect it. Rather, I give an open-book/open-notes quiz every lecture (or every other lecture when it is too much), based on the homework---usually just a single question, similar to but never identical to the homework, aimed at testing their understanding of the main concepts or technique for that lesson. The idea is that students who have actually done the homework will find the quiz straightforward, and those who haven't will have a hard time. I make these quizzes count for about 50% of the course grade, so there is ample motivation for the students to be prepared. And the best way to become prepared, they quickly see, is to do the homework, even though I don't collect it. This arrangement also has the advantage that I can encourage the students to form study groups and work together on the homework; studying together, they rise to a high level. Also, I can freely encourage my students to post any questions they might have about their homework to math.stackexchange, or to ask other students and professors, and there is no &quot;cheating&quot; issue, since I don't collect the homework, and instead they sincerely want to learn whatever it is they are asking about.&lt;/p&gt;&#xA;" OwnerUserId="90" LastEditorUserId="90" LastEditDate="2016-11-02T16:04:42.627" LastActivityDate="2016-11-02T16:04:42.627" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="54" PostTypeId="2" ParentId="33" CreationDate="2014-03-14T01:14:01.953" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://www.corestandards.org/Math/Content/K/introduction/&quot; rel=&quot;nofollow&quot;&gt;http://www.corestandards.org/Math/Content/K/introduction/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not aware of &lt;em&gt;incoming&lt;/em&gt; requirements for Kindergarten but the link above details the learning goals for CC math at grade K.&lt;/p&gt;&#xA;" OwnerUserId="91" LastActivityDate="2014-03-14T01:14:01.953" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="55" PostTypeId="1" CreationDate="2014-03-14T01:16:04.940" Score="24" ViewCount="1129" Body="&lt;p&gt;&lt;a href=&quot;http://www.mrao.cam.ac.uk/~clifford/&quot;&gt;The Cambridge University GA Research Group’s website&lt;/a&gt; along with &lt;a href=&quot;http://geocalc.clas.asu.edu/&quot;&gt;the “Geometric Calculus R &amp;amp; D Home Page”&lt;/a&gt; should serve as a good introductions to geometric algebra, along with &lt;a href=&quot;http://en.wikipedia.org/wiki/Geometric_algebra&quot;&gt;the Wikipedia reference – “Geometric algebra”&lt;/a&gt;. A somewhat more advanced exploration in the context of differential geometry can be found in &lt;a href=&quot;http://www.amazon.ca/Approach-Differential-Geometry-Cliffords-Geometric/dp/0817682821&quot;&gt; the book titled “A New Approach to Differential Geometry using Clifford's Geometric Algebra”&lt;/a&gt;. A development with more of a 'universalist' flavour (universal as in universal algebra) is presented at the geocalc.clas.asu.edu page &lt;a href=&quot;http://geocalc.clas.asu.edu/html/UGC.html&quot;&gt;“Universal Geometric Calculus ”&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Should geometric algebra be presented to undergraduates in engineering, physics and mathematics early in their curriculum instead of, or alongside, linear algebra?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good answers should try to analyze:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;which subject is more fundamental, in an attempt to identify where they should fit in in a curriculum&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;what changes would be required in the general curriculum in order to meaningfully support an early introduction of geometric algebra, and whether such changes would be feasible given the current infrastructure&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;why such curriculum changes have not been made thus far&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;examples of personal or institutional experiences resulting from an attempt to make such a curriculum change (if any exist) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;whether a problematic or beneficial slippery slope exists -- could an (escalated, seemingly more &quot;extreme&quot;) argument be made for the introduction of &quot;applied abstract algebra&quot; early in high school as a consequence of serious consideration given to the main question? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user89" LastEditorUserId="376" LastEditDate="2014-10-24T16:18:57.690" LastActivityDate="2014-10-24T16:18:57.690" Title="Should geometric algebra be presented early on in undergraduate education?" Tags="|undergraduate-education|curriculum|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="56" PostTypeId="2" ParentId="36" CreationDate="2014-03-14T01:19:00.663" Score="16" Body="&lt;p&gt;For undergraduate level, &lt;a href=&quot;https://www.gnu.org/software/octave/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gnu Octave&lt;/a&gt; is probably what you want. It is open source, cross-platform, and syntax-compatible with MATLAB. It's very useful for 2D and 3D plotting and for numeric linear algebra, and it's a tool that will be beneficial for students to know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, &lt;a href=&quot;https://www.geogebra.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;GeoGebra&lt;/a&gt; is excellent plotting software, and has recently added 3D graphing support. It is not as powerful a tool as Octave, but is well-suited to simple plotting and rendering tasks.&lt;/p&gt;&#xA;" OwnerUserId="92" LastEditorUserId="92" LastEditDate="2018-07-09T18:37:31.750" LastActivityDate="2018-07-09T18:37:31.750" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="57" PostTypeId="1" CreationDate="2014-03-14T01:22:31.810" Score="21" ViewCount="238" Body="&lt;p&gt;So far I've tried building my presentation from elements, each of which is differently paced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A lot of tutorials are given just by rapidly writing fully worked solutions on the board, thereby leaving little room for actual discussion of concepts. I don't like that kind of tutorial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The last tutorials I taught were based on the fact that the maths school would release fully worked solutions electronically at the end of the week and encouraged their tutors to have a more &quot;lesson&quot; based presentation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ended up with a presentation based on putting skeleton solutions comprising a list of sub questions that just referred to the appropriate concepts from the lectures. Then I asked the class to come up with the answer to each sub question so that they could see how a solution was built up and could contribute to its construction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then of course, even though you are keeping the ones who want to move faster happy by giving them the opportunity to participate and thus not get bored, you still have to regulate who's answering and how often and how to encourage the weaker students to have a go.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any thoughts from others on how they handle this are welcome.&lt;/p&gt;&#xA;" OwnerUserId="57" LastEditorUserId="57" LastEditDate="2014-03-14T01:40:23.063" LastActivityDate="2021-04-30T21:02:41.883" Title="How do you handle a wide ability range when delivering a 50 min tutorial with lots of material to get through?" Tags="|mixed-ability|time-management|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="58" PostTypeId="2" ParentId="41" CreationDate="2014-03-14T01:25:11.923" Score="13" Body="&lt;p&gt;I think you need to fix any problems you have by modifying a letter or two of the way you write. For example, I used to write my y's in two separate straight strokes, but they would get confused with my x's if I wasn't careful about where the stroke stops. To correct this, I started writing my y's more like the way you write a g, with a curly bottom. I also stroke my z's with a cross to keep them separate from my 2's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To fix your t's, add a curl on the bottom.&lt;/p&gt;&#xA;" OwnerUserId="92" LastActivityDate="2014-03-14T01:25:11.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="59" PostTypeId="1" CreationDate="2014-03-14T01:26:18.333" Score="32" ViewCount="15926" Body="&lt;p&gt;I would like to know the advantages and disadvantages of open-book exams as compared to closed-book exams, particularly in standard undergraduate courses like calculus or linear algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My practice (about 20 years of teaching) has always been to give open-book open-notes exams, allowing the students to use their books and notes during the exam. Part of my reasoning for this is that I hope to place a greater emphasis on understanding the material, rather than on memorized details. I don't mind if a student has to look up some detail quickly during the exam. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But perhaps there are advantages of closed-book exams I am overlooking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What are the main advantages and disadvantages of open-book versus closed-book exams? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be particularly interested in the result of any math education studies that compare the two exam methods.&lt;/p&gt;&#xA;" OwnerUserId="90" LastEditorUserId="42" LastEditDate="2014-03-14T01:45:27.897" LastActivityDate="2019-08-31T16:05:05.970" Title="What are the comparative advantages of open-book versus closed-book exams?" Tags="|reference-request|exams|" AnswerCount="7" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="60" PostTypeId="2" ParentId="59" CreationDate="2014-03-14T02:01:01.883" Score="22" Body="&lt;p&gt;There is a middle ground: closed-book, with some notes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The disadvantage to open-book exams is that students will waste time looking for answers in the book. I know this from experience. As I personally have a very bad memory, I wanted to keep that aspect out of it. But I saw many students wasting time during exams, flipping through the book. (Have you not seen that?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My solution has been to let them use notes. Even a page of notes allowed them to write so much that I watched students searching all over their note page. So now, they get a 3x5 card. (Except calc II. They get a page still.) This seems to offer them enough notes to not worry about memorizing, and keep the searching down.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-14T02:01:01.883" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="61" PostTypeId="1" CreationDate="2014-03-14T02:14:36.510" Score="8" ViewCount="313" Body="&lt;p&gt;When teaching multidimensional differentials (I'm assuming the students grasped the one-dimensional case), there are many useful parallels relating to spatial imagination.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, when explaining that the gradient of a function is the direction of the steepest slope and properties of other related vectors, it is inspiring to let them imagine a mountain path which keeps its height. Then, the path is in the direction perpendicular to gradient, and if we look up perpendicular to path, then we would often see the peak.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example might be to explain Lagrange multipliers and related optimization methods using an orange and the thickness of its peel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this does not goes through to students, which have poor spatial imagination. They are unable to see themselves in the mountains looking at the slope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The questions are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Are there any good, non-spatial parallels on multidimensional differentials?&lt;/strong&gt; E.g. for single-dimensional case there is position/velocity/acceleration, but this is much harder with multiple dimensions.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Are there any methods to make spatial analogies easier to students?&lt;/strong&gt; E.g. making a model could be one solutions, the problem is, it is often unfeasible.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-07-25T06:51:35.517" Title="Multidimensional differentials for students with poor spatial imaging" Tags="|calculus|mathematical-analysis|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="62" PostTypeId="2" ParentId="41" CreationDate="2014-03-14T02:21:34.400" Score="11" Body="&lt;p&gt;As others have suggested, adding a serif can make a big difference. I write my y's like g's, and I also stroke my z's through the middle. Lower-case l's are written cursive-style, with a loop, and I make sure to make the 'tail' of my q's big enough to ensure there's no confusion with 9's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also never use certain letters as variables (if I can help it) - especially o's.&lt;/p&gt;&#xA;" OwnerUserId="96" LastActivityDate="2014-03-14T02:21:34.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="63" PostTypeId="2" ParentId="61" CreationDate="2014-03-14T02:23:59.377" Score="4" Body="&lt;p&gt;If you have a piece of paper, you can model a 2-dimensional surface! Tape a piece of graph / grid paper on to a desk with an object underneath (a cell phone, face-down, perhaps?), and you can talk about the slope at different points on the paper and how it changes as you move.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few questions you could ask:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Put your finger on the paper and follow a line to the left. Is your finger moving closer to the ceiling, or further away from it?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Trace a line around the [object you put under the paper]. If you follow that line, does your finger move up or down? Why or why not?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Put your finger on the paper at the edge of your object. If you move to the left, does your finger move up or down? What if you move forwards or backwards?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="96" LastEditorUserId="96" LastEditDate="2014-03-14T02:40:12.783" LastActivityDate="2014-03-14T02:40:12.783" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="64" PostTypeId="1" CreationDate="2014-03-14T02:31:01.770" Score="14" ViewCount="167" Body="&lt;p&gt;I've worked with a few online programs that present students with drills / exercises / etc. for a period of time before deciding that they've 'done enough' for the day (e.g. &lt;a href=&quot;http://www.reflexmath.com&quot;&gt;Reflex Math&lt;/a&gt;). How do these sites (or their implementors) decide what's 'enough'? Is there research that says that children will only be able to retain a specific number of new facts in a period of time?&lt;/p&gt;&#xA;" OwnerUserId="96" LastEditorUserId="143" LastEditDate="2014-03-19T19:15:38.430" LastActivityDate="2014-03-19T19:15:38.430" Title="How do online practicing websites decide how much practice / repetition is 'enough'?" Tags="|primary-education|technology-in-education|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="65" PostTypeId="1" CreationDate="2014-03-14T02:32:04.733" Score="13" ViewCount="222" Body="&lt;p&gt;Functions and mappings are usually introduced late in the curriculum, and functions of arity two or more are considered &quot;advanced&quot; (many don't even see them before college).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, the concept of a mapping (&quot;the dogie gets a bone, the kitten gets a fish, the bear gets an apple&quot;) is intuitive and many of my non-math friends told me when asked, that mapping from pairs or triplets isn't really a hard thing, just that they never thought about it from this perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, the questions: why it is not feasible to teach multiple-argument functions earlier, perhaps even before the addition is introduced in a more systematic way (the moment where one learns that it is one of many possible operators, and that multiplication is one too, and that there is some order to apply those operators, etc.)?&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-14T03:00:26.423" Title="Teaching functions/mappings early" Tags="|primary-education|functions|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="66" PostTypeId="2" ParentId="33" CreationDate="2014-03-14T02:35:58.540" Score="7" Body="&lt;p&gt;From &lt;a href=&quot;http://www.corestandards.org/Math/Content/K/introduction/&quot;&gt;http://www.corestandards.org/Math/Content/K/introduction/&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Counting and Cardinality&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Know number names and the count sequence.&lt;/li&gt;&#xA;&lt;li&gt;Count to tell the number of objects.&lt;/li&gt;&#xA;&lt;li&gt;Compare numbers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Operations and Algebraic Thinking&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Number and Operations in Base Ten&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Work with numbers 11-19 to gain foundations for place value.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Measurement and Data&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Describe and compare measurable attributes.&lt;/li&gt;&#xA;&lt;li&gt;Classify objects and count the number of objects in each category&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Geometry&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Identify and describe shapes.&lt;/li&gt;&#xA;&lt;li&gt;Analyze, compare, create, and compose shapes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mathematical Practices&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Make sense of problems and persevere in solving them.&lt;/li&gt;&#xA;&lt;li&gt;Reason abstractly and quantitatively.&lt;/li&gt;&#xA;&lt;li&gt;Construct viable arguments and critique the reasoning of others.&lt;/li&gt;&#xA;&lt;li&gt;Model with mathematics.&lt;/li&gt;&#xA;&lt;li&gt;Use appropriate tools strategically.&lt;/li&gt;&#xA;&lt;li&gt;Attend to precision.&lt;/li&gt;&#xA;&lt;li&gt;Look for and make use of structure.&lt;/li&gt;&#xA;&lt;li&gt;Look for and express regularity in repeated reasoning.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="96" LastActivityDate="2014-03-14T02:35:58.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="67" PostTypeId="2" ParentId="2" CreationDate="2014-03-14T02:50:01.973" Score="11" Body="&lt;p&gt;The concept of 'proof' in mathematics (education) is overloaded with importance/fear/magnificence/awe. What is 'proof' if not just an explanation to that or another claim by a sequence of logical steps? How is it different then 'proofs' in history? Few educators of history would suggest teaching historical facts without understanding and arguing about their cause, effect or importance. Yet, in mathematics education there seems to be more room for debate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is a good point to start learning proofs? Well, whenever the student is showing an interest would be fine. Showing an interest means that, at least on some level, they are ready to tackle it (whatever it may be). Then, of course, some care needs to be taken as to the level of maturity the student has in order to decide on the approach to take. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are pros and cons? Mathematics without proof is, at best, a cook-book recipe for long and boring computations. Not understanding what one does may lead one to mistake mathematics for magical rules to be followed just because somebody said so. It robs the students from the ability to understand what they are doing and how to check their answers make sense. Of course, introducing long and tedious proofs is also dangerous. Before a typical student can appreciate the intricacies of, say, proving the reals exist, one must first come to appreciate the merits of such a proof. Not all things should be proven, it's quite alright to take some things on faith, as long as we are not oversimplifying anything. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does the whole curriculum needs to be geared towards proofs? Well, it should be geared towards understanding, and that means that it should include explanation. Some people call it proof. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-14T02:50:01.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="69" PostTypeId="2" ParentId="65" CreationDate="2014-03-14T03:00:26.423" Score="4" Body="&lt;p&gt;One must draw a line somewhere. I mean, if binary functions were taught early on, would we ask for ternary functions too? what about functions of infinitely many variables? What about relations? The question is what is one trying to teach about functions. Which aspects of functions are the focus. Functions of a single variable are somewhat simpler than their binary or ternary friends, even though everybody is familiar with binary functions (like addition and multiplication) very early on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also quite simple to draw graphs for single variable functions than it is for other functions, a very useful visual aid. One can argue that once one is comfortable with single variable functions, the passage to multi-variables is quite trivial and one looses basically no generality by only introducing multi-variables later on. That while introducing multi-variables at the get-go serves little purpose, and may confuse the student (higher dimensions etc.). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, there seems to be no loss in taking the small jump to multi-variables only at a later stage, after the ideas of functions are firm. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-14T03:00:26.423" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="70" PostTypeId="1" AcceptedAnswerId="173" CreationDate="2014-03-14T03:03:13.900" Score="12" ViewCount="258" Body="&lt;p&gt;Both universities I have taught at have tightly-stuctured calculus courses that follow a pattern of 3 sections of Stewart Calculus every week, with some sections afforded two days.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find it difficult to fit 3 lectures (intended for a MWF course) into two 80-min sessions. What is an effective way to divide the material? I feel that some sectins just can't be cut in half easily.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="114" LastEditDate="2014-03-31T12:55:17.557" LastActivityDate="2014-03-31T12:55:17.557" Title="How do I adapt a MWF class into a TuTh class?" Tags="|course-design|scheduling|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="71" PostTypeId="2" ParentId="36" CreationDate="2014-03-14T03:03:21.237" Score="15" Body="&lt;p&gt;You might want to look into Geogebra.  The current beta version (Geogebra 5 Beta) has a 3D mode and allows you to create 3D plots of a function of the form f(x,y) and a graphical interface for rotating, zooming, etc.  It is also open-source, free, and cross-platform.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-03-14T03:03:21.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="72" PostTypeId="2" ParentId="27" CreationDate="2014-03-14T03:15:32.470" Score="21" Body="&lt;p&gt;The best explanation I know comes from &lt;a href=&quot;https://math.stackexchange.com/a/676075/26306&quot;&gt;this answer&lt;/a&gt; by Emanuele Paolini (the only thing I did was to redo the pictures, please go and upvote Emanuele's post). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that squares in the usual picture&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/WZViX.png&quot; alt=&quot;squares&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;doesn't have to be squares. The only thing is that we need to use the area (so that is scales with a square of the scale), it could have been pentagons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/L5o53.png&quot; alt=&quot;pentagons&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it could also be appropriate right triangles:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/06VSd.png&quot; alt=&quot;triangles&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is the similarity of the small triangles that makes it work (and explains why the triangle has to be right for the theorem to work).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-03-14T03:15:32.470" CommentCount="1" CommunityOwnedDate="2014-03-14T04:24:32.653" ContentLicense="CC BY-SA 3.0" />
  <row Id="73" PostTypeId="2" ParentId="36" CreationDate="2014-03-14T03:42:48.077" Score="9" Body="&lt;p&gt;A free tool for Linux and Windows is &lt;a href=&quot;http://www.gnuplot.info/&quot; rel=&quot;nofollow noreferrer&quot;&gt;gnuplot&lt;/a&gt;. This same graphical interface can be used from within Octave (as in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/36/what-is-the-free-and-simple-plot-software-for-students/56#56&quot;&gt;mkasberg's answer&lt;/a&gt;), which is also free. I haven't used it in a while, so I don't recall the difficulty of use. It does have a set of tutorials available for download.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.sagemath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sage&lt;/a&gt; is also available on Linux and Windows. It is a python based program. The &lt;a href=&quot;http://matplotlib.sourceforge.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;matplotlib&lt;/a&gt; library provides graphing functions. I have not used this program, so I am unsure how simple it is to use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are graphing functions of the form $z=f(x,y)$, then &lt;a href=&quot;http://www.google.com.au/search?q=online+3d+plot&quot; rel=&quot;nofollow noreferrer&quot;&gt;a number of online tools exist&lt;/a&gt;. One such tool is &lt;a href=&quot;http://www.math.uri.edu/~bkaskosz/flashmo/graph3d/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="71" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-03-14T03:42:48.077" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="74" PostTypeId="1" CreationDate="2014-03-14T03:43:20.523" Score="21" ViewCount="437" Body="&lt;p&gt;It happens frequently in math that problems have multiple possible solutions. This might become troublesome, e.g. when students use some other approach, hence, not learning the current topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could say that it is the task that is wrong, but in many cases examples which require the more complex approach are too hard to use for introduction. This leaves us the &quot;solve using X and without using Y,&quot; but it seems artificial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also problems with grading - one would like to check the understanding of the advanced techniques, but appropriate problem might be to hard for a short test.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there methods for encouraging the use of a particular approach?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit 2:&lt;/strong&gt; For example when teaching logical connectives, students learn the &quot;table approach&quot; first, and then get to know some other laws, like De Morgan's, distributivity, etc. Then, many times, instead of using a series of reductions, they create a big table (e.g. if there are multiple variables) and do not practice the transformations enough. And then it happens again, when they learn sets and use repeatedly element-chasing instead of algebraic methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example could be from computer-science. It happens that during abstract algebra courses students learn about matroids and perhaps that some graph-theory concepts are expressible using those structures. Then, in some algorithm-design class the intended proof might be to create an algorithm which constructs the solution (and in the process prove the solution exists), but students might use the properties of matroids, frequently just remembering the facts and not understanding them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit 1:&lt;/strong&gt; One possible example is when you have to find the extrema of the function &#xA;$$f(x,y)=x^2+y^2-2x-4y+9.$$&#xA;You can solve this by calculus techniques with derivatives, but also by writing&#xA;$$f(x,y)=(x-1)^2+(y-2)^2+4.$$ &lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="262" LastEditDate="2015-07-03T20:22:52.167" LastActivityDate="2017-12-16T21:01:53.723" Title="Multiple Solutions Methods vs. Encouraging a Particular Approach" Tags="|proofs|problem-design|testing|" AnswerCount="4" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="75" PostTypeId="1" CreationDate="2014-03-14T03:56:49.943" Score="19" ViewCount="521" Body="&lt;p&gt;I take it that lecture is rarely the most effective way for a student to learn. Lecture is a case where, I believe, research on learning firmly backs up the common experience that lecture rarely helps students who aren't able to help themselves. And lecture is far from our only pedagogical tool. There are many other ways for students to learn math that are frequently on display in elementary, middle and high school classrooms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regardless: when &lt;em&gt;is&lt;/em&gt; it appropriate to lecture?&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="143" LastEditDate="2014-08-02T23:35:40.493" LastActivityDate="2014-08-06T23:37:31.457" Title="When is it appropriate to lecture?" Tags="|general-pedagogy|classroom-interaction|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="76" PostTypeId="2" ParentId="49" CreationDate="2014-03-14T04:07:47.700" Score="11" Body="&lt;p&gt;Differentiation is one of the hardest parts of teaching, for sure. One thing that has helped me a lot is reframing my questions so that they are more &amp;quot;open.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;For instance, instead of asking my students to factor a quadratic, I'll ask them &amp;quot;Say that I've got &lt;span class=&quot;math-container&quot;&gt;$x^2 + bx + 9$&lt;/span&gt;. What can &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; be if this expression is factorable?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;What's great about more open questions is that kids will approach them differently, depending on where they're coming from. A weaker student will wonder what the easiest binomials to multiply and get 9 are. A stronger student might start getting clever and think of some sort of generalization. We can have a good discussion with the class with a lot of different perspectives after giving the kids a bit of time to work their thoughts out.&lt;/p&gt;&#xA;&lt;p&gt;You might check out this book from your library: &lt;a href=&quot;https://www.nctm.org/store/Products/More-Good-Questions--Great-Ways-to-Differentiate-Secondary-Mathematics-Instruction/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction&lt;/em&gt;&lt;/a&gt;  by Marian Small and Amy Lin.&lt;/p&gt;&#xA;&lt;p&gt;Another way to give your top students what they need is to experiment with non-lecture or whole-group discussion formats. You might consider giving your students a great deal of class time to work on problems. Your faster students will naturally find their way to problems that are challenging and worth their while.&lt;/p&gt;&#xA;&lt;p&gt;Check out the Park City Math Institute's problem sets from their educator's course: &lt;a href=&quot;http://projects.ias.edu/pcmi/hstp/resources/course2009.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;IAS/PCMI 2009 Number Theory Course&lt;/a&gt; — course notes by Bowen Kerins and Darryl Yong.&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorDisplayName="user7990" LastEditDate="2020-08-31T14:58:29.020" LastActivityDate="2020-08-31T14:58:29.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="77" PostTypeId="2" ParentId="41" CreationDate="2014-03-14T05:59:09.740" Score="21" Body="&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/bHR0C.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\times\ \ +\ \ 4\quad 2\quad \omega\quad 9\quad a\quad 1\quad i\\&#xA;x\quad t\quad y\quad z\quad w\quad g\quad q\quad l\quad j$&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-03-14T05:59:09.740" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="78" PostTypeId="1" CreationDate="2014-03-14T06:10:43.903" Score="9" ViewCount="303" Body="&lt;p&gt;I will be teaching a calculus class, specifically, integration of common functions (e.g., polynomials, logarithms, exponentials and the like).  It has been my experience that if an abstract concept can be at least related to a real life context, then it becomes easier for the students to comprehend and apply the particular concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are effective real-life practical activities that will enhance student comprehension and ability to apply principles involved with logarithmic and exponential integrals?&lt;/p&gt;&#xA;" OwnerDisplayName="user106" LastEditorUserId="77" LastEditorDisplayName="user89" LastEditDate="2020-01-01T05:29:58.403" LastActivityDate="2020-01-01T05:29:58.403" Title="What activities can enhance student comprehension of concepts involved in logarithmic and exponential integrals?" Tags="|calculus|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="79" PostTypeId="2" ParentId="57" CreationDate="2014-03-14T06:58:05.340" Score="9" Body="&lt;p&gt;This situation sounds very similar to many of the classes that I have taught.  Like you, I dislike the kind of tutorial that is just based on worked solutions - but, what I found was that the worked solution, could be a part of a dynamic tutorial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is what I have done - initially, I go through a worked example on the board (interactive whiteboard tends to work well), all the time encouraging questions about each of the steps. (I colour code the processes as well).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, I have the students in small mixed ability groups (i.e. I choose the groups based on abilities) go through separate questions, following the worked example as a guide.  Gradually, having the groups determine solutions without a worked example on the board - using the skills, they have acquired.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found that the smaller groups were less confronting as the quieter students started to become more vocal within the groups (and occasionally in front of the class) with more tutorials.  Also, I found that the students became more confident to tackle questions on their own for practice/homework.&lt;/p&gt;&#xA;" OwnerDisplayName="user106" LastActivityDate="2014-03-14T06:58:05.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="80" PostTypeId="2" ParentId="78" CreationDate="2014-03-14T07:04:37.840" Score="-1" Body="&lt;p&gt;Since the integral of a postive function is the area under the function, a possible real-life application is the following situation: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you want to paint the wall of your room in a fancy manner. Namely, you want to draw a squiggly line (e.g the graph of $\frac{1}{x}$ or $\ln{x}$ in, say $[1,5]$) and paint the area below the line in red. How much red paint are you going to need for this?&lt;/p&gt;&#xA;" OwnerUserId="108" LastEditorUserId="108" LastEditDate="2014-03-14T08:10:16.237" LastActivityDate="2014-03-14T08:10:16.237" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="81" PostTypeId="2" ParentId="1" CreationDate="2014-03-14T07:27:13.463" Score="15" Body="&lt;p&gt;When presenting some example, I like to let them cast votes on the correct answer (e.g. for choosing methods of integration, or for the question how many solutions a given linear system has, after reducing it to row echelon form). I offer them 3 possible answers (sometimes including &quot;Who doesn't want to vote on this?&quot;). This gives me a quick way to see how well the students are following, and it provides some start of a discussion if the vote wasn't unanimous, because then, you can ask &quot;Why do you think it's A and not B?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, what was kind of hard to learn, was to allow the students some time to think of the answer, and make a deliberate pause after the questions. Silcence in the classroom can be hard to endure, but I think creating a gap where students have time to think and answer is also important.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-03-14T07:27:13.463" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="82" PostTypeId="1" AcceptedAnswerId="119" CreationDate="2014-03-14T07:35:09.417" Score="23" ViewCount="1914" Body="&lt;p&gt;Many of my students don't have a mathematical background, and are not comfortable with concepts such as limits, random variables and distributions. Is there any intuitive way of explaining why the means of a random variable will form a normal distribution around the true mean? The goal here is not to produce a rigorous proof, as they are not mathematically oriented, but to demonstrate that the theorem works and teach them the &quot;logic&quot; behind it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I thought about using computer programming and simulations to show that it works, but I feel that sampling from a simulated data is still too abstract to be intuitive. &lt;/p&gt;&#xA;" OwnerUserId="111" LastEditorUserId="111" LastEditDate="2014-03-14T14:44:47.747" LastActivityDate="2017-06-11T10:35:31.073" Title="Any tips in explaining the central limit theorem in statistics?" Tags="|undergraduate-education|statistics|" AnswerCount="9" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="83" PostTypeId="2" ParentId="22" CreationDate="2014-03-14T08:04:20.227" Score="5" Body="&lt;p&gt;I can't offer you a good and interesting example, but a nice geometric interpretation. My math professor (Rainer Wüst from TU Berlin) had presented the theorem in way which was easier to digest: He started with a function $f$ which is continuously differentiable in a domain $\mathcal G$ in ${\mathbb R}^2$. He writes in his book &quot;an approach which is completely unnecessary, from a mathematical point of view.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can give a geometric interpretation to the assertion of the implicit function theorem, which I'll repeat here in his formulation: If $f(y_0,x_0)=0$ and with some conditions on the partial derivatives, there exists a neighborhood $V$ of $x_0$ and a function $h$ which is continuously differntiable in $V$ with $f(x_0)=y_0$ and $f(h(x),x)=0$ for all $x\in V$. (Note the deliberate order of $y$ and $x$, which I copied from his book.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The set $$H=\{(y,x)\in \mathcal G: f(y,x)=0\}$$ should consist of curves, at least under 'reasonable' assumptions on $f$. The conditions of the theorem ensure that $(y_0,x_0)\in H$ and that the tangent to the curve at this point is not parallel to the $y$-axis. The therorem states that we can represent this curve, at least in a neighborhood of $x_0$, as a function of $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One specific example for this interpretation is $f(y,x)=x^2+y^2 +1$ where $H$ is the unit circle.&lt;/p&gt;&#xA;" OwnerUserId="108" LastEditorUserId="376" LastEditDate="2014-06-14T20:38:10.733" LastActivityDate="2014-06-14T20:38:10.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="84" PostTypeId="2" ParentId="1" CreationDate="2014-03-14T08:32:50.617" Score="36" Body="&lt;p&gt;One important point to make is that you should ensure that interaction is part of the &lt;em&gt;culture&lt;/em&gt; of your lectures.  It isn't enough to pose a question now and again and expect them to suddenly leap in to action to answer it.  So you need to be asking questions consistently through the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next point I'd like to make is that it will take time for the students to figure out that that is what is expected and to participate (and even longer to enjoy participating).  This has two practical consequences.  The first is that you should expect - and know how to deal with - an initial desire on the part of (most) students to not participate.  The second is that you should think about participation on a continuous scale not as an on/off state, so in the early days ask questions that can be answered almost without thinking and make it easier for the students to answer than not answer.  Then once they know what to expect, you can get them to do the more thoughtful questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A danger that is easy to fall in to with interactive environments is to focus your teaching on those who are interacting.  This is natural, but can back-fire as all the others feel even less involved than they would normally.  If the majority will not start the course feeling like they should interact, the initial interactors are out-liers and you shouldn't be shaping the course by their needs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the university level, I would start by telling them explicitly what is expected of them, and why.  They are legally adults now and need to start taking responsibility for their own learning.  Then every time I asked them a question, I would remind them of why I'm asking them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On to practicalities.  I'm currently using the &quot;voting&quot; method of asking questions.  I'm teaching a course to non-mathematicians which is very much a techniques course.  So the &quot;I probably didn't understand the question&quot; barrier to answering is quite high.  Add in the fact that I'm not teaching in their native language, and the chance of someone answering a complicated question is quite small.  So I ask simple questions and I get them to vote on them.  Here's what is characteristic about my style (in reference to the points above):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The questions are &lt;em&gt;really easy&lt;/em&gt;.  Quite frequently, the answer is really obvious, not just mathematically but from the very fact that I'm asking the question.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I make it clear that guessing is absolutely fine.  I frequently use the example of doing the exam: a guessed answer gives you a chance at some points, no answer gives you no chance.  (I keep quiet about Oscar Wilde's quote about keeping one's mouth shut.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't take &quot;no&quot; for an answer.  I mean, &quot;no participation&quot;.  After not many have voted the first time around, I berate them and ask again.  Then I'll ask &quot;Who didn't vote?&quot;, and &quot;Who didn't say that they didn't vote (but didn't vote)?&quot;.  The idea being that they're going to have to put their arm up anyway so they might as well put it up at the start and get it over and done with.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't use technology.  I want a phone-free environment in my lectures, and getting out phones or clickers every time is too much of a hassle.  In addition, I'm often spontaneous about when I ask these questions and don't want to be taking time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I use the answer they give.  If it's right, I'll make a joke about &quot;We live in a democracy so let's go with the popular vote.&quot;.  If it's wrong (or if a sizeable minority give the wrong answer), I'll take the time to explain why.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Lastly, let me share a different technique that I used quite successfully with a class of more mathematically minded students (so who could be expected to answer more complicated questions, though they were still actually quite straightforward).  I would pick a starting point in the lecture hall and then simply go along the row every time I had a question (picking up where I left off next lecture).  If someone passed, that was fine and I'd ask the next person.  This once worked spectacularly well when someone had fallen asleep: I started the &quot;answer snake&quot; a few people before him in the row, and his neighbour realised what was going to happen so nudged him awake.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-03-14T08:32:50.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="85" PostTypeId="1" AcceptedAnswerId="168" CreationDate="2014-03-14T08:42:30.260" Score="11" ViewCount="250" Body="&lt;p&gt;In very advanced courses, typically there are only a few students attending and sometimes they are not willing to put so much efford into the course to participate or to pass an exam. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I made the experience that hardly anyone will do the exercises. Of course, the students will not learn that much if they do not exercise during the course and at some point it will be very hard to follow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there some strategies to get as much content and methods as possible to the students? What are the advantages of e.g. repeating proofs, applying theorems, etc.? Should the tutor enforfe the students to do it themselves or should the tutor repeat it for them?&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-23T11:45:09.150" LastActivityDate="2017-06-08T05:41:27.757" Title="How to design exercise groups in very advanced course?" Tags="|course-design|tutoring|graduate-education|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="86" PostTypeId="1" AcceptedAnswerId="108" CreationDate="2014-03-14T08:49:54.287" Score="23" ViewCount="1354" Body="&lt;p&gt;At a course at the university, the students have to do homeworks every week which will be graded and discussed in exercise groups. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it a good idea to put &quot;official&quot; solutions of the homework on the homepage (or give it to the students in some other way)? Is there some study that this has some advantage?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally think that most people will not go through their solution since they have some solution and maybe will never more look at the solution. It would be nice to have a reference for my statement.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T12:57:22.390" LastActivityDate="2022-11-05T06:47:30.537" Title="Is it good to have solutions of homework published?" Tags="|undergraduate-education|secondary-education|homework|tutoring|publishing|" AnswerCount="9" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="87" PostTypeId="2" ParentId="75" CreationDate="2014-03-14T08:54:41.573" Score="16" Body="&lt;p&gt;I prefer to think of lectures not as the &lt;em&gt;worst&lt;/em&gt; way to teach but rather:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A lecture is the worst way to teach, apart from all the other ways we've tried.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That said, to understand when it is appropriate to lecture we have to understand what lectures do.  At its heart, a lecture is a time-efficient way of communicating &quot;stuff&quot; from the lecturer to the students.  The problems with this are manifold and mainly focus around the fact that the communication channel is one-way.  But that can also highlight when it is appropriate.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;When there is no other good channel of information (for example, no standard book or websites).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When changing the system would take more time than is reasonable (at the university level - which I presume this question is aimed at - then teaching is only a part of your job).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When the students resist other means of instruction (sort of similar to 2; it may be possible to change their expectations but it's only going to be a reasonable task if your faculty as a whole take part).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When the students are &lt;em&gt;mature&lt;/em&gt; enough to take responsibility for their own learning.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This last one is probably the most important one.  If &lt;em&gt;I&lt;/em&gt; wanted to learn something then going to an all-singing, all-dancing, interactive event would be the last thing I would choose.  I want to know what I need to know and I want it presented efficiently.  The point of a lecture, in that circumstance, is to tell me &lt;em&gt;what&lt;/em&gt; is important and to give me the benefit of someone else's experience (for example, a book tends to lay out all the options in a fairly neutral way but in a lecture the lecturer can say &quot;&lt;em&gt;This&lt;/em&gt; technique is out-moded and basically wrong, &lt;em&gt;that&lt;/em&gt; is what you should use.&quot;.  I can do all the &quot;self-learning&quot; bit on my own and at my own pace.  It's a waste of my time and the lecturer's to be doing lots of &quot;interactive activities&quot; in the lecture room.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, if the students already know &lt;em&gt;how&lt;/em&gt; to learn and the goal is to tell them &lt;em&gt;what&lt;/em&gt; to learn, then a lecture is appropriate.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-03-14T08:54:41.573" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="88" PostTypeId="2" ParentId="22" CreationDate="2014-03-14T09:02:41.217" Score="25" Body="&lt;h1&gt;Isolines and isosurfaces&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Isolines&quot; rel=&quot;noreferrer&quot;&gt;Isolines&lt;/a&gt; and &lt;a href=&quot;http://en.wikipedia.org/wiki/Isosurface&quot; rel=&quot;noreferrer&quot;&gt;isosurfaces&lt;/a&gt; (i.e., lines and areas of equal whatever) correspond to the graphs of implicit functions and are relevant in many sciences, e.g., isopotentials (physics), isobars and isotherms (metereology).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the best-known example of this kind are topographical contour lines (lines of equal altitude, see image below):&#xA;On a sufficiently rough scale and ignoring some geological tourist attractions, geographical altitude as a function of geographical longitude and latitude meets the prerequisites of the implicit-function theorem. From the mathematical point of view, it’s a function $h$ from $\mathbb{R}^2$ to $\mathbb{R}$ (ignoring the curvature of the earth) and its implicit functions are the contour lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ajo8V.jpg&quot; alt=&quot;Example for contour lines&quot;&gt;&#xA;(Source: &lt;a href=&quot;http://commons.wikimedia.org/wiki/File:Cntr-map-1.jpg&quot; rel=&quot;noreferrer&quot;&gt;Wiki Commons&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another noteworthy feature of most real-life isolines or -surfaces is: If whatever is equal on them acts as some sort of potential, the corresponding forces act perpendicular to the isolines or -surfaces. E.g., if you put a ball on some geography, it will start rolling perpendicular to the contour lines.&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2014-03-14T09:02:41.217" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="89" PostTypeId="2" ParentId="86" CreationDate="2014-03-14T09:04:09.930" Score="8" Body="&lt;p&gt;I think it is important to make the solutions available for the homeworks and exams administered in any course, perhaps after you think that the students have had sufficient time to think about the questions. A student who has spent enough time thinking about the question will have somewhere to look for; and those who do not think about will not bother any way but this is not a reason against having some solutions written. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In principle, a good homework will entertain student's creativity. In this case, you have somewhere to make additional remarks about the material covered in the class and seen on the homework. Almost always, you can present introductions to other &quot;areas&quot; of Mathematics/Sciences where these ideas pop up. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To ensure that the homeworks are taken sufficiently seriously, a strategy that one could follow is to have them prove some results whose proof is a slight extension of something they have seen on the homework. This will ensure that students actually spend time thinking about the assigned homework and also develop some familiarity with techniques and ideas they would be expected to know. &lt;/p&gt;&#xA;" OwnerUserId="109" LastActivityDate="2014-03-14T09:04:09.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="90" PostTypeId="1" AcceptedAnswerId="111" CreationDate="2014-03-14T09:08:44.010" Score="19" ViewCount="1054" Body="&lt;p&gt;There are some very motivated students who are very interested in math (in general), where the interest takes over most of their time. The problem is that they don't put enough time in the lecture they are attending, which leads to bad grades or even them failing the course (which is not fair for them and often leads to discouragement). [E.g. there was a student in a basic course about numerics who was understanding the basic concepts and who was always asking me questions about the axiom of choice. He lacked a deeper understanding in the course and did not perform very well at the exam]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I encourage to such students to concentrate on their (exam-related) study without discouraging them?&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="11" LastEditDate="2014-04-11T15:03:57.247" LastActivityDate="2014-04-11T15:03:57.247" Title="How to deal with very motivated students having &quot;off-topic&quot; interests?" Tags="|undergraduate-education|mathematical-pedagogy|student-motivation|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="91" PostTypeId="2" ParentId="74" CreationDate="2014-03-14T09:15:39.037" Score="5" Body="&lt;p&gt;Such a situation usually occurs when it is necessary to introduce technique of solving the general problem while the previously known methods are intended for particular cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, I always try to explain the reasons that require the introduction of more complicated method. I describe the limitations that &quot;old&quot; methods have and try to give examples that require general approach. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, due to time limitation, general examples sometimes can not be used for initial introduction. In this case it is possible to consider simple example for introduction and give more complex example for home assignment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main problems arise during testing. In this case one has to to give simple examples which may allow for an alternative solution. However if prior to the test to announce its purpose, e.g. &quot;test knowledge of particular technique&quot;, it can partially solve such problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, if the student will know the situations in which the &quot;new&quot; method would be the only possible technique, it will be a more serious approach to the study of this method.&lt;/p&gt;&#xA;" OwnerUserId="105" LastActivityDate="2014-03-14T09:15:39.037" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="92" PostTypeId="2" ParentId="6" CreationDate="2014-03-14T09:36:00.427" Score="13" Body="&lt;p&gt;You cannot prevent them from finding solutions, but there are some tricks to keep the difficulty of the homework and the make it more diffucult for those students to google solutions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If students have to proof theorems, don't name them in the question. If they have to perform an algorithm (like &quot;Perform three steps of the Euler iteration&quot;), don't name the algorithm, just write down the definition.&lt;/li&gt;&#xA;&lt;li&gt;You can ask the students to proof something in a specific way. For those who do not &quot;google&quot; the answer, this is even easier for them. Often, this is also a good way to use results of the course and not a standard solution from a book not using results from the lecture. For example, I like the derivation of Neumann series via Banach's fixed point theorem. Instead of asking to show that the following sum is a inverse of (I-A), you can ask to show that this is invertable using Banach's fixed point theorem and to define a appropriate fixed point function.&lt;/li&gt;&#xA;&lt;li&gt;(This happend once to me by accident. I don't know if this is a good thing to do on purpose.) I asked to students to proof a not so known theorem and by grading it, everyone was having the same solution which was wrong (A part was missing, something was proven by contradiction, but the contraction was wrong). I saw that in one book a gave as reference there was this proof with all the mistakes in it. All the students copying from that got a wrong answer.&lt;/li&gt;&#xA;&lt;li&gt;You can try to reformulate the result. For example, only let the student proof a special case. Or add some additional case.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-14T09:36:00.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="93" PostTypeId="2" ParentId="61" CreationDate="2014-03-14T09:39:57.073" Score="6" Body="&lt;p&gt;In my experience, being too tightly wed to geometry was actually a hinderance to me when I was learning multivariable differential calculus.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my mind, the essential feature of the multivariable derivative is that it gives a linear map between tangent spaces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I strongly suggest giving your students enough linear algebra to at least understand what a linear map is, and how you can associate a matrix with a linear map, before starting to teach them the multivariable derivative.  To do otherwise would be like teaching the single variable derivative before understanding the slope of a line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are interested, Jim Fowler and I are writing an online multivariable calculus course which takes this approach &lt;a href=&quot;http://ximera.osu.edu/course/kisonecat/m2o2c2/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.  We attempt to get the student to construct all of the mathematics through engaging exercises, with hints.  We will also be covering the rudiments of multilinear algebra to have a convenient algebraic package for higher order derivatives.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-03-14T09:39:57.073" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="94" PostTypeId="2" ParentId="78" CreationDate="2014-03-14T09:47:54.593" Score="3" Body="&lt;p&gt;In my answer to this question &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/498339/demystify-integration-of-int-frac1x-dx/498790#498790&quot;&gt;https://math.stackexchange.com/questions/498339/demystify-integration-of-int-frac1x-dx/498790#498790&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I show how the fact that the $$ \ln(x) = \int_1^x \frac{1}{t}dt$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;can be understood as naturally in terms of trying to solve exponential equations.&lt;br&gt;&#xA;You could convert this into a bonus assignment where they have to compute the solution to $3^x=2$ as a $16th$ century mathematician might:  using only addition and multiplication.  Let them make a spreadsheet or program, but require that they only use addition or multiplication in their programs. Then take them down the same path as in my story.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-03-14T09:47:54.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="95" PostTypeId="1" AcceptedAnswerId="96" CreationDate="2014-03-14T09:52:14.003" Score="15" ViewCount="264" Body="&lt;p&gt;When I have to give a big lecture with a lot of exercise groups, I have to choose the tutors who are grading the student's homework and explaining the lecture to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How can I find out if someone is suitable?&lt;/strong&gt; What are good ideas to test if someone is a good tutor without embarrassing them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, when I know the tutor or know someone who knows him, it is clear for me. Also, if someone did perform very bad in the past overall, this is a clear case. But sometimes, there are students who were very good in the given lecture but perform very bad at the exercise group (One one hand by not being good teachers, but sometimes also they cannot even solve the exercises themselves). &lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-28T18:41:09.633" LastActivityDate="2014-03-28T18:41:09.633" Title="How to choose/test future tutors?" Tags="|undergraduate-education|tutoring|graduate-education|" AnswerCount="2" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="96" PostTypeId="2" ParentId="95" CreationDate="2014-03-14T10:04:07.123" Score="11" Body="&lt;p&gt;The university I worked as a tutor at, involved an interview with a brief teaching situation. This was done for all students who applied to a job as a tutor at the math department, regardless of the intended course (e.g. linear algebra for engineers, analysis for physiscists, numerics for mathematicians).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You had to prepare a short presentation (5 -10 Minutes) of an excercise or a theorem. The audience consisted of one professor, one assisstant professor and on student who evaluated the abilities of the applicant together.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-03-14T10:04:07.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="97" PostTypeId="2" ParentId="61" CreationDate="2014-03-14T11:25:00.637" Score="2" Body="&lt;p&gt;You can make 3D images that can be manipulated by mouse and put them on the web.  There are a bunch of examples at &lt;a href=&quot;http://neil-strickland.staff.shef.ac.uk/courses/MAS243/pics/&quot; rel=&quot;nofollow&quot;&gt;http://neil-strickland.staff.shef.ac.uk/courses/MAS243/pics/&lt;/a&gt;.  These are done with a Java applet called JavaView (&lt;a href=&quot;http://www.javaview.de/&quot; rel=&quot;nofollow&quot;&gt;http://www.javaview.de/&lt;/a&gt;) which is a little unfortunate as browsers are getting rather aggressive about applet security policies these days.  At some point I should convert to some kind of HTML5 version that does not require any plugins, but I am not teaching the relevant course at the moment so I have not done that.&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-14T11:25:00.637" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="98" PostTypeId="1" AcceptedAnswerId="1181" CreationDate="2014-03-14T11:52:29.723" Score="19" ViewCount="693" Body="&lt;p&gt;As the following math stack exchange question (and answers) show:&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/703212/is-dxdy-really-a-multiplication-of-dx-and-dy&quot;&gt;https://math.stackexchange.com/questions/703212/is-dxdy-really-a-multiplication-of-dx-and-dy&lt;/a&gt;&#xA;There are a lot of different ways to think about the integral. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Any ideas about how the concept of the integral should be taught, with special reference to teach an understanding of the relationship between the different conceptual ways to define &quot;integral&quot; (thus, we are not talking about a first course teaching integration here)?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also, good references for comparison of different concepts of the integral would be nice.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; As an answer to the question in the comment: The question linked to above is about the meaning of the symbol  &quot;&lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;&quot; in the integral. I have in mind various notions related to the integral such as differential forms, Riemann sums, measures,  ...  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are relationship between Integral defined via differential forms, Riemann sums, measures, as a linear operator (Daniel integral?), geometric integral, and so on.  Especially relationship between this various forms of integral, and comparisons between them.&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="122" LastEditDate="2019-11-10T12:25:41.970" LastActivityDate="2019-11-10T12:25:41.970" Title="Comparison of different concepts of integral" Tags="|calculus|mathematical-analysis|reference-request|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="99" PostTypeId="1" AcceptedAnswerId="112" CreationDate="2014-03-14T11:54:12.337" Score="13" ViewCount="345" Body="&lt;p&gt;Much mathematics is intimately tied to recursion, be it in definitions (like factorials and integer powers) and  proofs by induction. This is also very relevant in computer science and programming. However, it is my experience that students have a hard time to understand what is going on, and even more to see where and how to apply the techniques. My impression is that by now, through decades of daily use, I'm not able anymore to see where the student's dificulties are. When presented with a problem asking for a induction proof, most are able to write down a reasonable answer, but is seems they still feel unconfortable with it. Unless specifically directed to use it, they will very rarely explore it's applicability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any suggestions on what to emphasize, what symptoms alert of particular types of misunderstandings?&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-16T11:59:53.617" Title="Any suggestions on how to approach recursion and induction?" Tags="|mathematical-pedagogy|proofs|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="100" PostTypeId="2" ParentId="95" CreationDate="2014-03-14T12:01:34.400" Score="4" Body="&lt;p&gt;Here we don't have any structured selection process, it is up to each one how to select tutors. The students I've selected as tutors are mostly those I've seen asking questions (be it in class or out of it), and which I've seen interacting with other students successfully, often by explaining the material to them or in informal study groups. Decent grades in the class are a must, obviously. I also take a look at their grades in other courses, not just related ones. Students who do well overall are less likely to see themselves burdened by other obligations.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-14T12:01:34.400" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="101" PostTypeId="2" ParentId="86" CreationDate="2014-03-14T12:11:24.813" Score="11" Body="&lt;p&gt;Homework (and exam) solutions should be published, at the very least to enable students to request regrading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, many students never look at the solutions (I even have a stack of homework and exams from last term that they didn't even bother to pick up). But that some don't bother isn't reason enough to deprive those who are interested of being able to look at solutions and try to learn from them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, to be able to grade you need to have complete solutions, publishing them is not much more work.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-14T12:11:24.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="102" PostTypeId="2" ParentId="74" CreationDate="2014-03-14T12:23:56.040" Score="8" Body="&lt;p&gt;This only works in some cases:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask not only for the final result but also some intermediate results which are only produced by the new method. This way the new method becomes more feasible, since the students are required to use it anyway to produce the intemediate results. In your example on finding extrema, you could, e.g., ask your students to also calculate the gradient of the function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A drawback of this is, however, that you guide your students along the way of solving the task to some extent.&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2014-03-14T12:23:56.040" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="103" PostTypeId="2" ParentId="59" CreationDate="2014-03-14T12:34:08.657" Score="6" Body="&lt;p&gt;As a student, I learned to hate open book exams... too much time wasted looking around. A case could be made for take-home exams (can look up whatever you want, but not the same time pressure).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to construct exams made of several (5 to 7) short questions, each one testing a specific concept. If some technique requires several steps, make a question for each (perhaps even &quot;If the result of step 1 above is &lt;em&gt;&lt;/em&gt;, do step 2 ...&quot;). If specific, easy to confuse/forget formulas are involved, they are printed on the back of the exam as official notes.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-14T12:34:08.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="104" PostTypeId="2" ParentId="36" CreationDate="2014-03-14T12:45:38.173" Score="4" Body="&lt;p&gt;The computer algebra system &lt;a href=&quot;http://maxima.sourceforge.net&quot; rel=&quot;nofollow&quot;&gt;maxima&lt;/a&gt; is nice (it is at the heart of open source packages like &lt;a href=&quot;http://sagemath.org&quot; rel=&quot;nofollow&quot;&gt;SAGE&lt;/a&gt; or &lt;a href=&quot;http://gap-system.org&quot; rel=&quot;nofollow&quot;&gt;GAP&lt;/a&gt;), and includes plotting functions. A version running on Android (even phones) is available too.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-14T12:45:38.173" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="105" PostTypeId="2" ParentId="7" CreationDate="2014-03-14T12:49:47.107" Score="10" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;memorize all of the definitions, be able to give examples and&#xA;  non-examples (where relevant), and memorize the statements of all of&#xA;  the important theorems&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'd say you are on the right track for definitions, but I'd like to add a tactic for learning theorems. To &quot;learn&quot; a theorem, you should encourage the students to try to break the theorem, as a way to remember how it is constructed. For example, here is the intermediate value theorem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A continuous function on a closed interval passes through all intermediate values.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Instead of &quot;memorizing&quot; this, students should memorize what the conclusion means, perhaps with a picture -- they will need to know precisely what intermediate values are, otherwise the theorem cannot make sense. But then they should try to break the theorem:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Seek out an example of a discontinuous function on a closed interval that fails to pass through the intermediate values.&lt;/li&gt;&#xA;&lt;li&gt;Seek out an example of a continuous function on an open interval that does not pass through the intermediate values. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This kind of testing-the-limits of a theorem is ultimately what leads to deeper understanding (and as a side effect, better recall).&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-14T12:49:47.107" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="106" PostTypeId="2" ParentId="16" CreationDate="2014-03-14T12:52:30.047" Score="6" Body="&lt;p&gt;For each homework, a small sample of answers is selected &quot;at random,&quot; and the students are required to explain what they handed in. The oral exam grade is then the grade for the homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rationale is that it is impossible to give oral exams to a group of 60 students in a reasonable timescale, and doing it in a fair way. And for homework I'm not interested really if they did the work, but if they understand how it is done. By hanging the threat of an exam over their heads, I hope to get them to at least look at what they turn in.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-14T12:52:30.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="107" PostTypeId="2" ParentId="36" CreationDate="2014-03-14T12:55:05.767" Score="15" Body="&lt;p&gt;In terms of free software, a large portion of the available choices are based on a Gnuplot backend; I however would probably not recommend directly using Gnuplot. Instead, your choices are mostly between the various front ends. In terms of the front end, depending on what your students are familiar with and are willing to learn, there are many options. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://matplotlib.org/&quot;&gt;&lt;code&gt;matplotlib&lt;/code&gt;&lt;/a&gt; is extremely powerful and configurable, but requires some amount of Python programming from the students. &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://sympy.org/en/index.html&quot;&gt;&lt;code&gt;Sympy&lt;/code&gt;&lt;/a&gt; also is based on the basic Python syntax, but its interface and use may be more familiar and accessible to people who have already some familiarity with a computer algebra system. &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://andrejv.github.io/wxmaxima/&quot;&gt;&lt;code&gt;maxima&lt;/code&gt; and its GUI &lt;code&gt;wxMaxima&lt;/code&gt;&lt;/a&gt; forms a freely available computer algebra system with a nice interface. In addition to using the gnuplot backend, Maxima also supports a different &lt;code&gt;xMaxima&lt;/code&gt; backend for plotting. &lt;/li&gt;&#xA;&lt;li&gt;The list goes on, but the &lt;a href=&quot;http://www.gnuplot.info/links.html&quot;&gt;&lt;code&gt;gnuplot&lt;/code&gt; project page&lt;/a&gt; maintains a list of front-ends and programming interfaces using it. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For 3D plots of &lt;em&gt;algebraic&lt;/em&gt; relations (polynomials etc.), a slightly less well-known program is &lt;a href=&quot;http://surf.sourceforge.net/&quot;&gt;&lt;code&gt;surf&lt;/code&gt;&lt;/a&gt;, which was intended as a visualization aid for real algebraic geometry. Unfortunately development seems to have stopped several years ago, and recently I have had some difficulty building the source code on my Gentoo Linux machine. Your mileage with it may very well vary. &lt;/p&gt;&#xA;" OwnerUserId="125" LastActivityDate="2014-03-14T12:55:05.767" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="108" PostTypeId="2" ParentId="86" CreationDate="2014-03-14T13:19:02.310" Score="17" Body="&lt;p&gt;I think a great reason to post homework solutions is so that students have a way of reviewing their homework before exams (it would be nice if they were to review homeworks whenever they get their graded work back, but this seems to be a rare practice). While a well-graded problem set should have comments pointing out any mistakes the student may have made, they might not make it clear what a correct solution would look like. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Providing homework solutions also gives me more license to ask somewhat harder questions on exams if they are related to homework questions. It is true that some (perhaps significant) number of students will not actually look at the solutions and will struggle, but this seems to me to be a good way to distinguish between the students who are actively preparing for class/exams, and those who are not. &lt;/p&gt;&#xA;" OwnerUserId="126" LastActivityDate="2014-03-14T13:19:02.310" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="110" PostTypeId="2" ParentId="1" CreationDate="2014-03-14T13:34:08.833" Score="9" Body="&lt;p&gt;In teaching calculus, I like to have something like 5% of the course grade be based on &quot;Participation&quot;. On the first day of class, when going through the syllabus, I go over what I mean by 'participating' (it includes, for example, &lt;em&gt;asking&lt;/em&gt; questions as opposed to only answering any questions I happen to pose to the class, coming to office hours, etc.) and in the first few days of class, when the students are still somewhat uncomfortable about speaking up, I sometimes reference their Participation Grade. I've also found that the more I meet with students in office hour and they get more comfortable interacting with me, the more likely they are to speak up in class. Almost all reasonable participation also gets a fair amount of positive reinforcement from me ('That's exactly right!' for correct things, or 'That's a good idea, let's figure out how this would work out...' for not-so-correct things.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't tell my students, but almost everyone gets a near-perfect/perfect participation grade (this having been my secret plan all along), unless they never show up to class/some other relatively drastic thing .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that grades should not be the primary reason that students are interested in doing well in any given course, but since they are something that people care about, why not put it to use. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I imagine that this practice does not work very well with large classes where it isn't possible for everyone to participate in any real sense and it is presumably hard to keep track of the folks that do participate.)&lt;/p&gt;&#xA;" OwnerUserId="126" LastActivityDate="2014-03-14T13:34:08.833" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="111" PostTypeId="2" ParentId="90" CreationDate="2014-03-14T13:42:08.237" Score="19" Body="&lt;p&gt;Some possible approaches:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Challenge the student. A possible reason why the student is having &quot;off-topic&quot; interests is that the student doesn't find the &quot;on-topic&quot; things interesting (he &quot;thinks&quot; it is too easy) or that the student doesn't see the point of the topic (he lacks a vision of the big picture). So the thing to do would be to divert the student's interest into your course. Tell him about some of the big unresolved problems related (even tangentially) to the content your course. Give him a difficult nugget to think about. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know enough about numerics to give a suggestion of a topic. But were the course basic real analysis, an old chestnut is to ask the student about the existence of &quot;a function $f$ of two variables $x,y$, such that for each fixed $y$, $f(\cdot,y)$ is a continuous function of $x$, and vice versa, but such that $f$ is nowhere continuous as a function of two variables&quot;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mentor the student. Rather than having the student wander here and there somewhat randomly in his musings, wouldn't it be better to make his interest more focused and organised? Offer to mentor the student in a reading class on the subject he is exploring on his own (and maybe even get departmental approval for it to be graded). Set out precise expectations for the reading class and establish a regimented schedule (you meet with him every week, for week 1 he should have read so and so). The trick is to set up the reading class in a way that his &quot;off-topic&quot; interests are sufficiently sated, while leaving him enough &quot;down time&quot; for him to work out the assignments for the class in his official curriculum. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have a talk with the student. Life is not always fair, and we don't always get what we want. (Say what? I have to take time out of research to apply for a grant?) Explain to the student that to be successful in life, some &quot;unpleasantries&quot; necessarily intrude. Ask him to read the part of Littlewood's &lt;em&gt;Miscellany&lt;/em&gt; where he talks about his undergraduate exam preparation at Cambridge. Show him clear evidence that at the rate he is going, he will likely fail the class. Tell him that he needs to put some effort into the class. Your student will hopefully be an adult about it and change his attitude. And if he doesn't... well, it's a life lesson. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="125" LastActivityDate="2014-03-14T13:42:08.237" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="112" PostTypeId="2" ParentId="99" CreationDate="2014-03-14T13:47:46.563" Score="9" Body="&lt;p&gt;Suppose you give a kid an input/output table such as {(0,1), (1,3), (2,7), (3,13)}, and ask your students to find a rule for this table. My experience has been that some students will try to find a closed-form rule, while others will instead notice something like a recursive pattern in the outputs. What's hard for students about recursion, I find, isn't the recursive reasoning in definitions, but instead the way of representing that in function notation. I've found it productive, in the past, to have kids informally state recursive definitions, and then validate those sloppy, informal definitions before formalizing them in accurate language and notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I have a really great Precalculus textbook (from the CME Project) that builds a students understanding of proof by induction on top of their understanding of offering recursive definitions. It's a great textbook with a lot of fantastic ideas for teaching math, even if you're teaching more advanced material.&lt;/p&gt;&#xA;" OwnerUserId="84" LastActivityDate="2014-03-14T13:47:46.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="113" PostTypeId="2" ParentId="99" CreationDate="2014-03-14T13:55:36.870" Score="8" Body="&lt;p&gt;For a long time, I felt like induction was silly - because of the way most textbooks deal with it - giving inductive proofs for things that have more useful constructive proofs. Teach recursion through examples that need it. I would love to accumulate some good examples here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've seen better examples in programming courses (which I taught long, long ago) than in math courses. I once tried to figure the odds of winning the dice game of Craps. That is best done recursively, I believe.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-14T13:55:36.870" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="114" PostTypeId="5" CreationDate="2014-03-14T14:14:00.520" Score="0" Body="&lt;p&gt;Questions about the teaching of analysis (real- or complex). There are some specific didactic problems when teaching concepts, theorems, proofs, examples and counterexamples related to limits, continuity, differentiability, integrability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Questions with this tag should be on the teaching of higher level mathematical concepts than calculus.&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="39" LastEditDate="2014-03-30T15:15:40.727" LastActivityDate="2014-03-30T15:15:40.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="115" PostTypeId="4" CreationDate="2014-03-14T14:14:00.520" Score="0" Body="For questions applying to analysis courses: Real and complex analysis. Typically a higher and more proof-based level than calculus." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-17T16:35:24.800" LastActivityDate="2014-03-17T16:35:24.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="116" PostTypeId="2" ParentId="86" CreationDate="2014-03-14T14:23:55.310" Score="3" Body="&lt;p&gt;Usually I give the odd-numbered problems as homework, so the students can check their answers immediately. If I am giving problems that they must think longer about, I would want to eventually provide good answers.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-14T14:23:55.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="117" PostTypeId="2" ParentId="78" CreationDate="2014-03-14T14:39:46.397" Score="2" Body="&lt;p&gt;Although he teaches at high school level and it sounds like you're at university level, Shawn Cornally has great ideas for getting students to really experience the meaning of the calculus concepts they're working with. Here's one post on the natural logarithm: &lt;a href=&quot;http://shawncornally.com/wordpress/?p=611&quot; rel=&quot;nofollow&quot;&gt;http://shawncornally.com/wordpress/?p=611&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can search his blog on 'exponential' to find lots more.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-14T14:39:46.397" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="118" PostTypeId="1" AcceptedAnswerId="167" CreationDate="2014-03-14T14:58:06.683" Score="17" ViewCount="4012" Body="&lt;p&gt;Suppose that you meet a student who:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;has a firm grasp of algebra and trigonometry and is at least moderately intelligent&lt;/li&gt;&#xA;&lt;li&gt;has read a book such as &lt;em&gt;Love and Math&lt;/em&gt; by Edward Frenkel so has some appreciation of math as a discipline including its scope and goals&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;knows&lt;/em&gt; they want to be a mathematician&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What sequence of mathematical topics would you advise they study. Set aside reality for a moment, this question is not about what is pragmatically best for the student in terms of career. Rather, what course sequence should the unfettered, self-motivated, hard-working, unboundedly-curious student pursue to optimize their understanding of core undergraduate mathematics. Feel free to redefine what the &quot;core&quot; ought to be. Suppose they are just entering university so they have 4 years to study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for your thoughts in advance!&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="50" LastEditDate="2014-03-15T16:59:15.630" LastActivityDate="2014-03-15T18:44:20.367" Title="What is the ideal course sequence for an advanced student of mathematics?" Tags="|undergraduate-education|course-design|advising|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="119" PostTypeId="2" ParentId="82" CreationDate="2014-03-14T15:01:12.117" Score="13" Body="&lt;p&gt;This is a preliminary, not about a distribution of means: How about having every student measure your desk? Ask them to keep their measurements secret, so they don't influence each other (an interesting issue in itself). Then use their answers as a data set. If you have enough students, you should get a normal distribution, whose mean is the proper answer. (This is helpful for seeing why statistics is so important to science, and is much like my second, more relevant suggestion, therefore preparing them for it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now do something similar with means. What is a mean that you and your students can identify with, something that they could get somewhat random samples for? Probably the hours that people watch TV doesn't clump too much. If they each ask 15 people (awfully small sample size, but not constructive to ask for more), and give the mean of their own sample, you can show the distribution of these means. They should form a normal distribution if your class is big enough. If your population isn't unusual in its TV watching habits, the mean of this distribution should be the true mean.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-14T15:01:12.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="120" PostTypeId="2" ParentId="59" CreationDate="2014-03-14T15:09:35.057" Score="6" Body="&lt;p&gt;This basically comes down to the depth versus breadth debate. With an open book exam, you can assess more content, but not know exactly how well they know it. A closed book exam gives you a much better idea of exactly what they know, but it limits you to a smaller amount of material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Right now, it occurs to me that an open book exam might work better as a formative assessment.&lt;/p&gt;&#xA;" OwnerUserId="132" LastActivityDate="2014-03-14T15:09:35.057" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="121" PostTypeId="2" ParentId="118" CreationDate="2014-03-14T15:15:57.270" Score="5" Body="&lt;p&gt;A course that require proofs will be vital, as their high school career probably did not ask them to do enough serious proving. Other than that, there are many directions they can take. Rather than ask for an ideal for all such students, I'd ask for a list of good starting points. It would be great if every such student had a mentor who could help them figure out the ideal sequence for them personally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus is beautiful and part of every mathematician's vocabulary. There's no sense not taking it if you plan to be a mathematician. But it doesn't &lt;em&gt;have&lt;/em&gt; to come first. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A proofs course that gives previews of lots of other courses would be very cool. (I remember reading course descriptions, and being unhappy that I didn't even understand the descriptions of the courses, because those descriptions detailed theorems by name for instance.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I particularly loved non-Euclidean geometry (in grad school). A smart undergraduate student could take that course and might find it opens their mind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I were mentoring such a student, I'd feel them out about whether these courses would be interesting:  Number Theory, Non-Euclidean Geometry, Linear Algebra.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-14T15:15:57.270" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="122" PostTypeId="1" AcceptedAnswerId="162" CreationDate="2014-03-14T15:40:56.193" Score="20" ViewCount="9820" Body="&lt;p&gt;I was tutoring a student on early trigonometry. Solving for the hypotenuse of a right triangle, but with sine, not Pythagoras. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student went through getting the sine of a 45° triangle, and gave me the correct answer to 3 digits after decimal. (Note, the two legs of the triangle were both 7) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My issue – she had 0.7071 as sine of 45°, but lost the radical as she got the decimal solution. Her answer 7.071 was right, but I felt that $5\sqrt{2}$ was a missed opportunity. I accept and understand there‘s a time for calculators and 3 digits, but felt that early on, one should keep radicals, and both understand and be able to show the numbers for the sines of 0°, 30°, 45°, 60° and 90°.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question - given that the US is going toward a common core approach, and given that students are allowed calculators early on, is there value to the pedagogical process to insist on keeping the irrational numbers through the solution? I am suggesting there will be greater understanding by keeping it, not indefinitely, but for understanding until the next level of math is introduced. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="262" LastEditDate="2015-07-03T20:29:12.050" LastActivityDate="2022-12-25T15:05:48.487" Title="Answers in exact form (e.g. including radicals) vs. Decimal Approximations" Tags="|usa|trigonometry|" AnswerCount="6" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="123" PostTypeId="2" ParentId="118" CreationDate="2014-03-14T15:44:57.670" Score="8" Body="&lt;p&gt;I'd put a proof-based linear algebra course at the top of the list.  I've found that people sometimes overlook the central role of linear algebra.  Linear algebra provides not only important tools and results, but the basic language for huge swaths of both pure and applied mathematics, including geometry (Euclidean geometry obviously, but non-Euclidean differential geometry is aptly characterized as &quot;parametrized linear algebra&quot;), abstract algebra, analysis, numerical analysis, etc.&lt;/p&gt;&#xA;" OwnerUserId="129" LastEditorUserId="129" LastEditDate="2014-03-14T16:00:07.843" LastActivityDate="2014-03-14T16:00:07.843" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="124" PostTypeId="2" ParentId="18" CreationDate="2014-03-14T15:56:19.860" Score="13" Body="&lt;p&gt;I'd like to expand on points that Thomas made.  Learning math is like learning a language, and a certain amount of memorization (note: not necessarily drilling!) is a necessary component of that.  To use a language, you need to have immediate mental access to the basic vocabulary and grammar.  (In math, that means not only the definitions, but the fundamental results of a given field.)  This is not just an issue when solving exam problems, but for further learning.  A student in a linear algebra class isn't going to get much understanding of the proof of the spectral theorem, for example, if they're still struggling to remember what a linear map is.&lt;/p&gt;&#xA;" OwnerUserId="129" LastActivityDate="2014-03-14T15:56:19.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="125" PostTypeId="2" ParentId="118" CreationDate="2014-03-14T16:07:53.763" Score="5" Body="&lt;p&gt;In anything remotely computer-related a solid foundation in discrete mathematics (combinatorics, graph theory, ...) is a must.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Number theory shows up in the most unexpected places, more so in computer science and related fields.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-14T16:07:53.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="126" PostTypeId="2" ParentId="8" CreationDate="2014-03-14T16:19:23.073" Score="6" Body="&lt;p&gt;I enforce Sue's question as for which course is this. I'll assume meanwhile that it is a college one (from calculus to advanced analysis).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Examples and computations.&lt;/strong&gt; Say you are teaching a multidimensional real analysis. Don't be afraid to open a calculus book and take your time through the computations in class, except that this time you're showing how all the machinery is working. They have seen how each piece was constructed and now they deserve to see the whole structure at work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're dealing with measure theory, even if you start with the abstract spaces and move on to integration don't forget to walk them through computations in $L_2$, $l_2$, $C([a,b])$ and other examples. Show that integration means summation in the nonnegative integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The higher order thinking is already present at advanced courses, what makes it fail to click is this lack of &quot;technical&quot; skill that are being able to show/compute explicit examples and motivate the theory using those as models. It's not about speed: it's about concept comprehension.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-03-14T16:19:23.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="127" PostTypeId="2" ParentId="122" CreationDate="2014-03-14T16:20:43.533" Score="9" Body="&lt;p&gt;While exact solutions involving radicals are estetically pleasing to the mathematical mind, they are next to useless for practical use (Newton was thrilled to have found an easy way of approximating roots with the generalized binomial theorem!). Besides, few angles give such pleasing results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a sad fact that what is required most of the time is just a precise enough numerical value.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-14T16:20:43.533" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="128" PostTypeId="2" ParentId="18" CreationDate="2014-03-14T16:27:02.040" Score="14" Body="&lt;p&gt;Many of the disadvantages of allowing aids can be, in principle, resolved by &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;requiring that the only aids the students have are handwritten by themselves and &lt;/li&gt;&#xA;&lt;li&gt;setting a length limit. (I've seen somewhere between one index card [for non Americans: a piece of paper around 10 x 15 cm squared] and 4 pages of A4 [for Americans: 4 pages of letter paper].)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Ideally this will cause the student to be structured in their preparation (choose the important things only!) and the copying by hand of the relevant formulae should in theory reinforce learning (at least memorisation). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice, however, some common downsides to this include&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Students grow to rely on the crib sheets. For more creative problems you would end up with students who spend the entire exam trying vainly to apply random items from their crib sheet to the problem, instead of actually trying to solve the problem. &lt;/li&gt;&#xA;&lt;li&gt;There's transcription error somewhere: either when the student copied the formulae from his notes or textbook to the crib sheet, or perhaps his handwriting is bad enough that he misreads it during the exam.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="125" LastActivityDate="2014-03-14T16:27:02.040" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="129" PostTypeId="2" ParentId="28" CreationDate="2014-03-14T16:28:07.220" Score="3" Body="&lt;p&gt;No way around it, have your student dig in. Perhaps discuss an overview of the paper(s) first, set aside time (and patience!) to go over dificulties afterwards. Look for accessible textbooks/lecture notes on any required material that isn't standard fare, you'll need to have it at hand (or recommend it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can, get other students interested in the area. Studying new material as a group is always much more rewarding (and effective!). It has the side effect that overall it frees up some of the time mentioned above ;-)&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-14T16:28:07.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="130" PostTypeId="1" AcceptedAnswerId="145" CreationDate="2014-03-14T16:28:14.757" Score="37" ViewCount="2650" Body="&lt;p&gt;At my university, the students in math are mixed up (1/3-1/2 are bachelor/master students, the rest are future high school teachers). A problem arising very often is the discussion dramatically summarized by &quot;I don't need this in high school/I already know enough, don't bother me with this abstract stuff&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Are there good examples/showcases/questions/arguments/etc. illustrating how important math education at university is for their life as high school teachers?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Expect from general discussion like &quot;You need to know more than the school kids/What if they change the content of high school education/What will &lt;em&gt;you&lt;/em&gt; tell the school kids when the say that they wanted to study something completely unrelated to math?&quot; - If you want to, you can also add general issues, but I am more interetesed in concrete examples)&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="77" LastEditDate="2017-12-26T16:04:45.443" LastActivityDate="2021-07-15T11:58:55.680" Title="Examples why university education is important for future high school teachers" Tags="|undergraduate-education|student-motivation|teacher-training|" AnswerCount="13" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="131" PostTypeId="2" ParentId="86" CreationDate="2014-03-14T16:30:26.600" Score="7" Body="&lt;p&gt;Yes. The idea is not only to give them well thought-out answers and solutions to the problems presented but also (ideally) to show how these were constructed. Enlightening solutions make explicit the intricacies of the problem and how to deal with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For maximum effect, the problem set should have been designed possibly with some open-ended questions requiring concept explanations (can you explain why these steps are true?) I remember fondly of a test where one of the questions consisted of a long proof and you had to (i) complete several steps, (ii) explain difficult ones (such as motivating the construction but not doing it). It led to a greater appreciation of the material, understanding of the necessity of the hypotheses and comprehension in general.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-03-14T16:30:26.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="132" PostTypeId="2" ParentId="7" CreationDate="2014-03-14T16:42:19.307" Score="5" Body="&lt;p&gt;Students reaching for real analysis more often than not do &lt;strong&gt;not&lt;/strong&gt; have a firm grasp of calculus as they should. What ends up happening is that they don't understand how most of that was necessary as a foundation for calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I advocate the following: show constructions where things are working and then exhibit counterexamples as what you get when you push boundaries. They were discovered when people tried (to use a pun) to pass to the limit. Most counterexamples aren't really intuitive: took years of amazingly capable people to realize they were possible and other peers still weren't convinced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One more tactic: relate what they're doing back to their computations; it's not just about proving how it works but seeing it in action.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-03-14T16:42:19.307" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="133" PostTypeId="2" ParentId="28" CreationDate="2014-03-14T17:07:03.120" Score="6" Body="&lt;p&gt;If your student would like more background, then there is only one way to get that: Study a lot. There is a reason that it often takes years to get a research degree (Ph.D.). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, if you by &quot;bringing someone up to speed&quot; mean that you want the student to learn the background, then I believe this would be very hard unless your student is very bright and highly motivated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you by &quot;up to speed&quot; mean that you want the student to be able to help in carrying out the research, then one way to do that would be to do a lot of concrete computations. I mean, you forget about the background and the larger perspective and learn about a concept from doing concrete examples. Much research involves specific computations that one doesn't need much background to actually do. While doing examples you can slowly build up the background. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;the challenge for you is that you have to break up the research into parts where some of these parts can be attacked by someone who doesn't have much background. This, of course, isn't easy in some areas, but maybe you could do this - in particular because your area is somewhat new.&lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-03-14T17:07:03.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="134" PostTypeId="1" AcceptedAnswerId="156" CreationDate="2014-03-14T17:10:56.213" Score="31" ViewCount="912" Body="&lt;p&gt;I am looking for resources for designing undergraduate mathematics classes that are &lt;em&gt;not&lt;/em&gt; lecture-based. (Bonus points if the design is for an introduction to proof course). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;a href=&quot;http://chronicle.com/blognetwork/castingoutnines/&quot;&gt;Robert Talbert&lt;/a&gt; blogs about flipped classes (calculus, and intro-to-proof). Do you know of others?&lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="262" LastEditDate="2015-07-03T20:31:28.633" LastActivityDate="2022-09-21T22:34:19.747" Title="Alternatives to University Lectures: Non-lecture Mathematics Classes" Tags="|undergraduate-education|course-design|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="135" PostTypeId="1" AcceptedAnswerId="172" CreationDate="2014-03-14T17:11:25.803" Score="30" ViewCount="2344" Body="&lt;p&gt;When students take a course on real analysis, they have likely learned about Riemann integrals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is a good motivation why they have to learn a new way to integrate?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student don't want to hear to something like &quot;You will now learn the new concept and in next years you will the importance of it&quot;. It should be some understandable example for the student (It is also okay to give an example after some weeks of the real analysis class, but avoid answers which a student at that level will not understand after the whole lecture.)&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-28T18:41:42.093" LastActivityDate="2019-08-04T09:40:28.703" Title="Good motivation for the introduction of Lebesgue integral?" Tags="|examples|mathematical-analysis|" AnswerCount="8" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="136" PostTypeId="2" ParentId="134" CreationDate="2014-03-14T17:24:50.870" Score="7" Body="&lt;p&gt;I really think that the Park City Math Institute's courses are fantastic, both as resources and as models for what it could mean to have students spend their time in class solving interesting problems.&lt;/p&gt;&#xA;&lt;p&gt;They run 3-week courses in the summer, often on a topic relating to number theory. For example, check &lt;a href=&quot;https://web.archive.org/web/20171113152516/http://mathforum.org/pcmi/hstp/resources/course2009.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this set of problems out&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Or, more generally, &lt;a href=&quot;https://web.archive.org/web/20171112074317/http://mathforum.org/pcmi/hstp/resources.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="6053" LastEditDate="2022-09-21T21:31:58.050" LastActivityDate="2022-09-21T21:31:58.050" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="137" PostTypeId="2" ParentId="1" CreationDate="2014-03-14T17:27:12.450" Score="8" Body="&lt;p&gt;This will, as others have pointed out, depend on the level. If the students are old enough, I would suggest having students present at the blackboard/whiteboard often. Make it part of your routine that you you call on people to go to the board and answer questions. Doing this regularly will create an atmosphere where speaking in class is &quot;just what we do on a daily basis&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, just call on people to answer questions. This can be difficult in the beginning and some students will not like it. But if it becomes part of your routine, then students will get used to it and it can completely change the environment in the classroom (for the better). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, as others have suggested, make sure all this is done in a way that doesn't make the students afraid. The classroom, of course, needs to be a safe place. You don't want to humiliate or embarrass students in any way.  &lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-03-14T17:27:12.450" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="138" PostTypeId="2" ParentId="134" CreationDate="2014-03-14T17:29:56.440" Score="18" Body="&lt;p&gt;There are a few ways to do this.  I recently did it in Carnegie Mellon's introductory proofs course.  I'll try to outline here a few different solutions that various places have tried.&lt;/p&gt;&#xA;&lt;p&gt;Carnegie Mellon (15-151): We alternated introducing new topics (usually via lecture) with what I called workshops.  A workshop is during normal lecture time.  Students are given 3-5 questions that range from conceptual questions to difficult proofs.  We created the groups randomly, and we changed them every few workshops. We gave them whiteboards (you can buy big sheets of whiteboards at home depot for like $5 and have them cut them down for you) and markers, and asked them to work together on the proofs in order.  As the lesson progressed, the TAs and I would walk around the room answering questions and reviewing proofs.  Depending on how many resources you have this could be very difficult, but we decided to split the large lectures down into smaller ones so this was feasible.&lt;/p&gt;&#xA;&lt;p&gt;MIT: &lt;a href=&quot;https://stellar.mit.edu/S/course/6/sp15/6.042/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Albert Meyer in 6.042&lt;/a&gt; takes a different approach.  The classes are still smaller, and I believe there are still &amp;quot;lab assistants&amp;quot; running around the room helping out, but he does the flipped classroom every lecture.  He's done it in several different ways over the years.  I believe originally, he gave mini lectures for the first 20ish minutes, and then the students worked for the rest.  He might have moved to asking the students to watch the lectures at home--in a true flipped classroom format--now.  One other thing to note is that the book Albert wrote is absolutely fantastic and completely open.  It's a great resource for good problems, good explanations, and expository material for students to read.&lt;/p&gt;&#xA;&lt;p&gt;Stanford:  I can't remember the course number or name right now. I will do my best to follow up later by adding a link.  Stanford takes a different approach than either of the other two.  They have a standard lecture series, and sometimes, they offer workshops &lt;em&gt;in the evening&lt;/em&gt; (which I believe are optional).  This approach is particularly nice in the college setting, because it requires fewer resources and less contention for classroom space.  Both of the CMU and MIT courses used a &amp;quot;special room&amp;quot; which allows for collaboration.  At CMU, 15-151 is held in the &lt;a href=&quot;https://web.archive.org/web/20160717084937/http://www.cmu.edu:80/computing/clusters/facilities/ctc.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;CTC&lt;/a&gt;, and at MIT, 6.042 is held in the &lt;a href=&quot;https://icampus.mit.edu/projects/teal/&quot; rel=&quot;nofollow noreferrer&quot;&gt;TEAL rooms&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There are other various strategies to do this, but these are the ones I've seen most commonly (and successfully) used for introductory proofs courses.  Albert's resources are very useful, and I would be happy to share some of mine as well.&lt;/p&gt;&#xA;" OwnerUserId="93" LastEditorDisplayName="user7990" LastEditDate="2020-09-07T16:15:11.643" LastActivityDate="2020-09-07T16:15:11.643" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="139" PostTypeId="2" ParentId="134" CreationDate="2014-03-14T17:33:28.727" Score="7" Body="&lt;p&gt;Jim Fowler and I created a hybrid linear algebra/multivariable calculus/multilinear algebra course at&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://github.com/kisonecat/m2o2c2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://github.com/kisonecat/m2o2c2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The structure of the course was:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; as an inner product space.&lt;/li&gt;&#xA;&lt;li&gt;Linear maps &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n \to \mathbb{R}^m$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;The derivative of a function from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n \to \mathbb{R}^m$&lt;/span&gt; as a linear map of tangent spaces.&lt;/li&gt;&#xA;&lt;li&gt;The derivative of a function &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n \to \mathbb{R}$&lt;/span&gt; as a covector valued function (differential 1 form).  The relationship of this covector to the gradient.&lt;/li&gt;&#xA;&lt;li&gt;Matrix with respect to a basis other than the standard basis and Eigenstuff.&lt;/li&gt;&#xA;&lt;li&gt;Understanding bilinear functions &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$&lt;/span&gt; and the associated matrix.  Connection with adjoints when considering inner product structure.&lt;/li&gt;&#xA;&lt;li&gt;Understanding the second derivative of a function from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R^n} \to \mathbb{R}$&lt;/span&gt; as a symmetric bilinear form.  2nd degree Taylor theorem, with applications to max/min (eigenvalues give definiteness of the form).&lt;/li&gt;&#xA;&lt;li&gt;General multilinear forms and the full Taylor theorem.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This course was based on problem solving.  Students largely discovered the mathematics through fill in the blank computational problems and also free response items for proofs which they could have conversations with each other about, and compare to the &amp;quot;official solutions&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately even compiling the source files seems really difficult now. I would have to match up the state of different github repos (the content source code with the processing system source code). So I think this work is basically lost forever now. I might find time to sort it out someday.  In the mean time, if you are really interested, you can just read the tex files for each topic.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-04-29T15:42:42.670" LastActivityDate="2021-04-29T15:42:42.670" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="140" PostTypeId="2" ParentId="86" CreationDate="2014-03-14T17:33:59.850" Score="6" Body="&lt;p&gt;While writing and releasing solutions is in general a really good idea, because it helps the students see their errors, one major problem with doing this is that it increases cheating in the future.  One has to be careful to balance releasing the solutions with the ability to re-use good problems in future instances of the course.&lt;/p&gt;&#xA;" OwnerUserId="93" LastActivityDate="2014-03-14T17:33:59.850" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="141" PostTypeId="2" ParentId="135" CreationDate="2014-03-14T17:36:22.817" Score="10" Body="&lt;p&gt;When learning a new concept, it might help to point out what's wrong with the old one. In this particular case, a function which cannot be integrated in the Riemann sense on any interval, but where one can assign a value inuitively:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Dirichlet function is given by $$f(x)=\begin{cases}1, &amp;amp; \mbox{ if } x \in \mathbb{Q}\\0, &amp;amp; \mbox{ if } x \in \mathbb{R}\setminus \mathbb{Q}.\end{cases}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since $\mathbb{Q}$ is dense in $\mathbb {R}$, the Riemann sums do not converge. However (and this is a part where handwavy arguments need to be inserted), this function is only not equal to zero on a countable subset, so its integral should be equal to zero.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-03-14T17:36:22.817" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="142" PostTypeId="2" ParentId="135" CreationDate="2014-03-14T17:48:47.183" Score="18" Body="&lt;p&gt;First, we have the two &quot;obvious&quot; motivations.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Integrating more functions (e.g. the indicator function for the rationals)&lt;/li&gt;&#xA;&lt;li&gt;Making the space of integrable functions complete&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Second, Stein and Shakarchi give the following motivations in the introduction to their book &lt;em&gt;Real Analysis&lt;/em&gt; (volume 3 of the Princeton Lectures in Analysis series). &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Fourier Series (establishing the $\ell^2/L^2$ isometry)&lt;/li&gt;&#xA;&lt;li&gt;Interchanging limits and integrals (e.g. the limit of a sequence of continuous functions may not be Riemann integrable)&lt;/li&gt;&#xA;&lt;li&gt;Length of Curves (finding the length of curves that are only rectifiable, not continuously differentiable)&lt;/li&gt;&#xA;&lt;li&gt;Differentiation and Integration (extending the reciprocity to absolutely continuous functions)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can easily cook up (or look up) interesting examples to illustrate these four ideas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stein and Shakarchi also discuss &quot;the problem of measure&quot; (rigorously defining the area or volume of arbitrary sets is hard!), and why the existence of non-measurable sets means we have to deal with $\sigma$-algebras. You ask about integration, but since measure theory and integration are so intimately related, you may also wish to discuss these topics. &lt;/p&gt;&#xA;" OwnerUserId="9" LastEditorUserId="9" LastEditDate="2014-03-14T17:54:41.790" LastActivityDate="2014-03-14T17:54:41.790" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="143" PostTypeId="2" ParentId="130" CreationDate="2014-03-14T18:19:22.760" Score="11" Body="&lt;p&gt;When my high school offered the AP course in Calculus, the teacher chosen to teach the class was the one who had studied it most recently. She was rusty and it showed. She was continually making mistakes in her demonstrations and the students wound up frequently correcting her.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I tutored college algebra, I frequently encountered prospective elementary teachers who did not like or understand mathematics, and suspected that their attitudes and lack of understanding would be reflected in their teaching. The same would be of course be true in high-school teachers.&lt;/p&gt;&#xA;" OwnerUserId="40" LastEditorUserId="40" LastEditDate="2014-03-14T20:03:59.847" LastActivityDate="2014-03-14T20:03:59.847" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="144" PostTypeId="2" ParentId="78" CreationDate="2014-03-14T18:35:07.823" Score="2" Body="&lt;p&gt;Practical examples may be found in various fields of science. Exponentially decreasing functions can be found in mechanics (a body falling in air, where the resistance is proportional to the velocity), electromagnetism (the charge on  capacitor), chemical equilibrium (where the rates of reaction are proportional to the amount of reactants), biology (exponential growth and decay). More advanced logarithmic and exponential functions naturally appear when the expressions involved are not simple but depend on something else.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-14T18:35:07.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="145" PostTypeId="2" ParentId="130" CreationDate="2014-03-14T18:50:03.393" Score="40" Body="&lt;p&gt;I'm not completely sure how well this addresses the question, but here is my best response.&#xA;A few years ago I was teaching a Methods course for preservice secondary math teachers.  Over the course of the semester there was a fair amount of griping, not about &lt;i&gt;my&lt;/i&gt; course but about the other courses they were required to take, and in particular about their Abstract Algebra course, which they felt was completely useless for their future careers.&lt;/p&gt;&#xA;&lt;p&gt;At a certain point in the semester we were doing some collaborative lesson planning for a hypothetical set of classes about exponential functions.  Since our Methods class was structured in part around the &amp;quot;backwards design&amp;quot; approach of Wiggins &amp;amp; McTighe, we spent some time at the beginning discussing the &amp;quot;big ideas&amp;quot; of exponential functions.  One of my students proposed as a &amp;quot;big idea&amp;quot; that there was an analogy between exponential growth and linear growth:  in the functions &lt;span class=&quot;math-container&quot;&gt;$f(x)=mx+b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)=AB^x$&lt;/span&gt;, the coefficient &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is analogous to the constant term &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; (both give the y-intercept or &amp;quot;starting value&amp;quot; for the function) and the exponential base &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is analogous to the slope &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; (both capture the idea of a &amp;quot;constant growth rate&amp;quot;, one additively and one multiplicatively).&lt;/p&gt;&#xA;&lt;p&gt;After talking about this for a while I said something like:  &amp;quot;You know, there is a precise name for 'analogies' like this.  A mathematician would call it an isomorphism.  In fact what you seem to be saying is that the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers.  In fact the properties of logarithms and exponents — like &lt;span class=&quot;math-container&quot;&gt;$e^{a+b} = e^a  e^b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\log(ab)=\log(a) + \log(b)$&lt;/span&gt; — really don't say anything more than that the log and exp functions are homomorphisms.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;One of my students got one of those &amp;quot;aha&amp;quot; looks and commented that I had just taught her more about abstract algebra in 2 minutes than she learned in an entire semester of the other course.&lt;/p&gt;&#xA;&lt;p&gt;(An even more basic version of this:  Students already know that the rules for multiplying positive and negative numbers are, in a sense, &amp;quot;just like&amp;quot; the rules for adding even and odd numbers.  Abstract algebra provides a language for describing exactly what this &amp;quot;just like&amp;quot; sense means.)&lt;/p&gt;&#xA;&lt;p&gt;My point is that there &lt;em&gt;are&lt;/em&gt; ways in which higher-level undergraduate math courses (like abstract algebra) can be directly relevant for teaching secondary mathematics...  But also that those applications are usually not made explicit.  So students are probably not wrong in feeling that those courses are useless for their future careers.  It is up to us to connect the dots and show them how and why these things matter.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edited, November 2020:&lt;/strong&gt;. I hope it is not considered shilling to add this here, but in the six years since I originally wrote this post I have written and published a mathematics textbook that is explicitly organized around the goal I articulated in the last paragraph: &amp;quot;to connect the dots [between advanced level mathematics and the secondary curriculum] and show them [i.e. preservice and/or inservice secondary teachers] how and why these things matter&amp;quot;.  The book is called &lt;a href=&quot;https://www.routledge.com/Secondary-Mathematics-for-Mathematicians-and-Educators-A-View-from-Above/Weiss/p/book/9781138294677&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Secondary Mathematics for Mathematicians and Educators&lt;/em&gt;&lt;/a&gt;, and is published by Routledge.  (More information available at the link in the previous sentence.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="6053" LastEditDate="2021-07-15T11:58:55.680" LastActivityDate="2021-07-15T11:58:55.680" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="146" PostTypeId="2" ParentId="122" CreationDate="2014-03-14T19:01:05.133" Score="5" Body="&lt;p&gt;It seems that it would be beneficial to know both the radical expression and the numerical answer. The radical expression is better for theoretical understanding, the numerical one for computational purposes and application. &lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-14T19:01:05.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="148" PostTypeId="1" AcceptedAnswerId="154" CreationDate="2014-03-14T19:20:51.580" Score="41" ViewCount="15127" Body="&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt; I am fairly new at teaching, and in every subject I have taught, I have had difficulty estimating the length and difficulty of an exam. I need to write an exam for a university senior-level knot theory course. The exam must last one hour due to university constraints.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the most effective way to write an exam that must fit a given time constraint?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastEditorUserId="114" LastEditDate="2014-03-28T19:56:00.597" LastActivityDate="2014-03-30T02:02:27.767" Title="How can I estimate the length of an exam?" Tags="|exams|testing|test-design|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="149" PostTypeId="2" ParentId="90" CreationDate="2014-03-14T19:24:31.680" Score="3" Body="&lt;p&gt;For some such students, the problem is that they don't understand the &quot;why&quot; of the more elementary steps. It may help to point out an interesting application from middle or end of the course and explain that to get there, they need to master the intermediate stuff, and to get there, they need to go through the &quot;easy&quot; stuff first. Maybe a sports analogy can help. Even natural athletes don't win marathons or go to the Olympics unless they train, and the &quot;easy&quot; stuff is their training.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-14T19:24:31.680" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="150" PostTypeId="1" AcceptedAnswerId="152" CreationDate="2014-03-14T19:25:17.543" Score="27" ViewCount="2997" Body="&lt;p&gt;In almost all countries all over the world, mathematics is a main subject in school. Maybe &lt;strong&gt;the&lt;/strong&gt; subject bringing trouble to families with kids. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is clear that scientist, engineers, etc. need mathematics as a part of their higher education at colleges. This is maybe the biggest argument (Since after finishing school everybody should be able to study whatever he or she wants, they have to know some basic mathematics). But there are also many subjects unrelated to mathematics (law, language studies, etc.) and young people heading to such subjects (being bad in mathematics) are frustrated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the arguments I can give to a school kid, to his parents or general to anyone why mathematics is so important - apart from the above argument? &lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="143" LastEditDate="2014-03-19T17:39:25.793" LastActivityDate="2014-04-22T16:49:24.290" Title="What arguments can I give a high school student why mathematics is important?" Tags="|secondary-education|student-motivation|" AnswerCount="7" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="151" PostTypeId="2" ParentId="148" CreationDate="2014-03-14T19:31:54.393" Score="26" Body="&lt;p&gt;Here's a few useful strategies:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Once you've written the exam, time how long it takes you just to write down the answers.  That gives you a baseline on how much time someone (you!) who already knows all the questions would take to answer the exam.  This is particularly useful for exams that are writing-heavy (like proofs and &quot;explain this&quot; questions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) If you have TAs, ask them to take the exam.  Give them some fraction of the time you intend to give the students.  Depending on the topic/course, this could be 1/5, 1/3, 1/2.  Your TAs (arguably) should get near perfect scores on your exams (unless you want a &lt;strong&gt;very&lt;/strong&gt; difficult question).  Make sure you tell them to actually time themselves and actually write the answers down.  Also, make sure you actually review what they've written.  Once, one of my TAs finished in the time he was given, and I hadn't reviewed his answers.  It turned out (after the students took it) that he'd accidentally answered something incorrectly, and the exam was harder than I had intended.  Ideally, you'd want to use undergraduate TAs for this, if at all possible, because they're &quot;closer&quot; in experience to the students in the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Try to mark your idea of a &quot;difficulty level&quot; for each question you write on the exam, as you add them.  Generally, the best exams have a combination of &quot;easy&quot;, &quot;medium&quot;, and &quot;difficult&quot; questions.  Certainly, if you don't balance this for &lt;em&gt;your&lt;/em&gt; idea of what this means, the exam might end up skewed to one side.  Then, if you have your TAs take it, ask them to rate the questions too.  If their perspective is totally different from yours, that might indicate a problem.&lt;/p&gt;&#xA;" OwnerUserId="93" LastActivityDate="2014-03-14T19:31:54.393" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="152" PostTypeId="2" ParentId="150" CreationDate="2014-03-14T19:45:44.647" Score="20" Body="&lt;p&gt;I really like this answer given by John Green:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=x78PnPd-V-A&quot;&gt;https://www.youtube.com/watch?v=x78PnPd-V-A&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;School is not about you! School does not exist for your benefit or&#xA;  for the benefit of your parents. Schools exist for the benefit of me.&#xA;  The reason why I pay taxes for schools even though I don’t have a kid&#xA;  in school is that I am better off in a well-educated world. Public&#xA;  education isn’t a charity project; I pay for your schools because I&#xA;  want you to grow up and make my life better. I want you to make me&#xA;  beautiful books that will bring me pleasure and consolation; I want&#xA;  you to make me cooler cars for me to drive in, drugs so I can live a&#xA;  longer, healthier life. I’m paying for your education in the hopes&#xA;  that you invent a microwave pizza with actually crispy crust and that&#xA;  you’ll spread the availability of the internet so I can get more&#xA;  youtube views in Zambia. Your education isn’t just about you; your&#xA;  nation is making an investment in you because they believe that you&#xA;  are worth it.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To that I'd add, the world is a better place when people &lt;em&gt;understand&lt;/em&gt; each other, and understand the full scope of humanity. I want to live in a world not only of mathematicians and engineers, but one where the poets and the journalists and the waiters and the doctors and the nurses and the truck drivers value and understand mathematicians. (And writers. And historians. And construction workers. And etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kurt Vonnegut said that we need more writers who were trained as engineers, and that's exactly right because we need fewer things keeping us from empathizing with each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One more thing: if you're looking at human progress, it's easy to look at the extremes, the Newtons, the Eulers, the Shakespeares, the Twains, and yes, those people obviously helped progress humanity. But human progress doesn't primarily come from geniuses and the famous -- it couldn't, because there are so few geniuses and so many nots. Real, sustainable human progress comes from the broad swath of humanity that is just trying to learn and appreciate the ideas of others. When we learn we are committing a creative act as important and vital to humanity as Einstein's accomplishments. Without us, where would Einstein be?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we understand an idea or a perspective, we have the ability to pass it on to others. Sometimes that passing on happens formally -- you're trying to teach something -- but just as often it happens informally. You talk about an idea and someone happens to over hear it, they run off and think it themselves. You raise a kid and accidentally make them smarter than you were. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For better or worse, ideas spread. It's our job to fill our head with as many good ideas as possible so that we spread the right ones.&lt;/p&gt;&#xA;" OwnerUserId="84" LastActivityDate="2014-03-14T19:45:44.647" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="153" PostTypeId="2" ParentId="148" CreationDate="2014-03-14T19:46:10.363" Score="12" Body="&lt;p&gt;I think there is no general solution, but here are some ideas which could help estimating the time:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do you have access to old exams in the same subject or in something related? Look at them (inform yourself if the students were allowed to bring notes, books, etc.) and compare to what you want to do. Maybe you can also ask colleagues from a different univercity for their exams.&lt;/li&gt;&#xA;&lt;li&gt;If you give homework during the term, ask the students how long they needed to solve the exercises. If you want to put a similar question on the exam, take only a certain amount of the time they said (The concept is not new to them; they should be prepared for the exam; plus students normally overestimate the time they needed [Since this seems to be an advanced class, this time should be quite fair].). You may also have a feeling how good the students are (or, at least, if there are some very good ones). A good student should be able to finish the exam in time.&lt;/li&gt;&#xA;&lt;li&gt;Use multiple choice questions. If time is too short, many will take the option which makes sense to them in the first place. If there is enough time, they will maybe try to prove the statements in order to be sure. You can then adjust the time fact during the grading.&lt;/li&gt;&#xA;&lt;li&gt;Use questions in the following style: &quot;Please prove one of the following theorems&quot;/&quot;Perform a calculation with the following algorithm or write a C++ code for it&quot;. Since the student can choose, it is more possible that there is something he or she has actually learned. (Don't exaggerate on this. Students need more time to read and to decide alternative questions.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Ask someone you trust (with background in the field) to perform the exam.&lt;/strong&gt; Depending on the kind of questions, the ratio between a good student and a PhD student should be somewhere between 2 and 5 (the more &quot;calculation&quot; questions there are, the smaller the ratio; the more &quot;proofs everyone in the field should know by heart&quot; you ask, the bigger the ratio.). You may also stop the time how long your colleague needed for each question.&lt;/li&gt;&#xA;&lt;li&gt;If you underestimate the time the students need, think about a plan B: What will you do (in terms of grading?) to balance the time issue? Is it possible to do so?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-14T19:46:10.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="154" PostTypeId="2" ParentId="148" CreationDate="2014-03-14T19:54:15.917" Score="25" Body="&lt;p&gt;My background is in high school teaching, so my experience may not directly transfer, since the types of exams are different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I have found a very useful rule of thumb to be this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After writing the exam, I sit for it myself, i.e. I sit down to write down full answers in one sitting. Most students will take six to eight times as long as I did.&lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2014-03-14T19:54:15.917" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="155" PostTypeId="2" ParentId="130" CreationDate="2014-03-14T19:55:00.863" Score="13" Body="&lt;p&gt;To teach mathematics, you need to know, first, what mathematics is, no exceptions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem of school-math being a whole different thing having little in common with mathematics, is caused, among others, by people who teach school-math, but have a bad idea about what mathematics is. However, it's a hard chance that someone will have contact with mathematics in high school, in fact, if some student does, with high probability he/she will go to college (perhaps with major other than math, but still).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This might change in the future, but right know, the best option to learn mathematics is to go to the academia. Of course, just attending university courses is not enough, but itself. However, there are opportunities, many times more than somewhere else. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To this end, I would gladly require any math teacher to do some research (I'm not suggesting that teachers should teach research in high school, only that knowing what research looks like in math gives you an appropriate perspective). It might be very small and insignificant, but still. On the other hand, I understand this is highly unfeasible, there aren't enough people that would be able to do it, after all, nowadays you can do a PhD without having an independent thought $\ddot\frown$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what's left is hope that going to some university will be enough. It's not encouraging and I wouldn't ever use such an argument to convince some future teacher (the effect might be just the opposite). Nevertheless, it is a real concern in present times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To conclude: going to universities is the future teachers' best chance to get acquainted with mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-14T19:55:00.863" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="156" PostTypeId="2" ParentId="134" CreationDate="2014-03-14T20:08:05.647" Score="17" Body="&lt;p&gt;I think you will definitely want to check out the &lt;a href=&quot;http://www.jiblm.org/&quot;&gt;The Journal of Inquiry-Based Learning in Mathematics&lt;/a&gt;. It is dedicated to being a &quot;resource for designing undergraduate mathematics classes that are not lecture-based.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another place to start that is overwhelmingly huge and mostly aimed at the K-12 level, but definitely has a lot of good ideas that are applicable to this question, is the huge community of math teacher bloggers. &lt;a href=&quot;http://mathtwitterblogosphere.weebly.com/a-few-good-blogposts.html&quot;&gt;Here&lt;/a&gt; is one place to start that exploration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also second Michael Pershan's recommendation to examine the PCMI problem sets. They provide a magnificent model for what it looks like to develop content through problems, and it's a little different from much of the stuff you'll find at the Journal of Inquiry-Based Learning, which is coming out of the &lt;a href=&quot;http://legacyrlmoore.org/method.html&quot;&gt;Moore Method&lt;/a&gt; tradition.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="140" LastEditDate="2014-03-14T21:35:50.420" LastActivityDate="2014-03-14T21:35:50.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="157" PostTypeId="2" ParentId="90" CreationDate="2014-03-14T20:15:44.880" Score="7" Body="&lt;p&gt;I strongly agree with @WillieWong, just wanted to add one more solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes you don't have enough time to deal with such a student (e.g. mentor him). On the other hand, there are those who have more time, for example, you might be able to find a PhD student who would happily take care of this guy (esp. if their interests are aligned). Such arrangement is beneficial to each side, since:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The guy might find a friend that shares similar interests/passion.&lt;/li&gt;&#xA;&lt;li&gt;A slightly older friend might have a better influence than seasoned professor.&lt;/li&gt;&#xA;&lt;li&gt;The PhD student might interest the guy in his/her research project, perhaps gain a collaborator or someone to continue the work.&lt;/li&gt;&#xA;&lt;li&gt;The PhD student will also gain an experience as a mentor, it is low-impact now, but will be useful later and failure doesn't cost much.&lt;/li&gt;&#xA;&lt;li&gt;You have dealt with your problem.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-14T20:15:44.880" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="158" PostTypeId="2" ParentId="148" CreationDate="2014-03-14T20:23:14.740" Score="11" Body="&lt;p&gt;A useful rule of thumb I got when I started up here (which has turned out surprisingly accurate) is &quot;Solve the exam yourself, multiply by four.&quot; In any case, when possible I schedule twice that (some time is lost until everybody is seated, exams are handed out, etc.; and there is &lt;em&gt;always&lt;/em&gt; some straggler arriving half an hour late...). But we don't have regulations on exam lengths&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-14T20:23:14.740" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="159" PostTypeId="2" ParentId="150" CreationDate="2014-03-14T20:26:09.793" Score="8" Body="&lt;p&gt;Because math helps rewire your brain. Most people wont become scientists or engineers, but the part of the brain math helps exercise is the piece that works on problem solving. One can claim that it's a rare job that never needs to solve any kind of problem at all. The school years of exercising that part of the brain will help the adult in whatever they wind up doing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If math classes are structured so the students have group time for solving, this time isn't all about math, per se, but about the skills needed to function in any group doing any job. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Update - The article must have been a paper one I can't find, not one I tagged or bookmarked. In the interest of keeping an open mind, an article supporting a somewhat opposite view - &lt;a href=&quot;http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?pagewanted=all&amp;amp;_r=0&quot;&gt;Is Algebra Necessary&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-03-15T10:21:09.403" LastActivityDate="2014-03-15T10:21:09.403" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="160" PostTypeId="1" CreationDate="2014-03-14T21:21:27.360" Score="27" ViewCount="1482" Body="&lt;p&gt;There is a high school in the city I live in which has some high-level math courses in their curriculum. It's a special math class mentored by some university lecturers, and the children basically do the first year of university math courses (single-dimensional analysis, linear algebra, basics of set theory and some miscellaneous things).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students who come from this high school are generally very bright people, frequently participate in national math contests and achieve successes in various areas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be all great, but there is a problem: there is a subgroup that never really had to learn (i.e. put in a serious effort), being smart was enough. When they come to the university, the first year is a breeze (while all the others learn hard work), but the second gets tough (because there is many new things and they are unable to manage their time) and in the third year they drop out (which is a shame because they were very promising). Also, there are issues with ego (they were always good, and now what), neglect (skipping homework, because &quot;I can always do it&quot;) and attitude (like &quot;of course I'm the best&quot;, or at the beginning not paying attention because &quot;it's obvious&quot; and later because &quot;I don't know what is going on, why bother catching up, I can always do it later&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have heard about similar observations from people from different places, always with regard to math or computer science. What's wrong?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The university offers a wide range of courses of varying difficulty, from basic to insanely hard, so it's not a problem of &quot;not enough challenge&quot;. Also, the university doesn't have resources to mentor them, so it's not an option. It seems a kind of problem of maturity (they frequently have some childish behaviors), like doing all this stuff in high school (their schedules had to be packed) would leave too little time for growing up, but that's just a hypothesis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any related research? Are there any solutions or suggestions? Is it a problem of the university, or perhaps the high-school?&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="376" LastEditDate="2014-04-05T12:23:47.350" LastActivityDate="2014-04-05T18:13:25.097" Title="Students who know high-level math before going to college" Tags="|mathematical-pedagogy|reference-request|gifted-students|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="161" PostTypeId="2" ParentId="1" CreationDate="2014-03-14T21:31:44.623" Score="17" Body="&lt;p&gt;There are a lot of good answers already. That said, we have just scratched the surface. Shifting the environment of math classrooms from one in which students attempt (usually only semi-successfully) to passively absorb, to one in which students actually &lt;em&gt;think&lt;/em&gt;, is a profound project and I think our profession is only partly underway with it. I would like to add just a few specific techniques and a general comment, and the caveat that there is a ton of research and writing on this subject already out there, and a ton of research and writing that has not been done yet and needs to be.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Specific techniques:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Asking not just questions but asking for students to &lt;strong&gt;summarize&lt;/strong&gt; ideas. This is a technique I learned recently from some &lt;a href=&quot;https://www.heinemann.com/authors/2299.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;master&lt;/a&gt; &lt;a href=&quot;https://deborahloewenbergball.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;teachers&lt;/a&gt; and it has transformed my practice. I blogged about it &lt;a href=&quot;https://researchinpractice.wordpress.com/2013/09/08/kids-summarizing/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Think-pair-share&lt;/strong&gt; has been mentioned above by mweiss but I want to second it as an absolutely indispensable tool. Students engage much more readily when they have an opportunity to take some time to put their thoughts together and to attempt to articulate them in the comparative safety of a one-on-one interaction with a classmate rather than in the full glare of the room's complete attention.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Ask for questions:&lt;/strong&gt; Ask, &amp;quot;what could a person ask about this?&amp;quot; Or, &amp;quot;what might bother somebody about this?&amp;quot; I often call on a student wearing a befuddled expression and ask, &amp;quot;what's on your mind?&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://web.archive.org/web/20180507154122/http://uakron.edu:80/cpspe/agpa-k12outreach/best-teaching-practices/wait-time&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wait time&lt;/a&gt;:&lt;/strong&gt; Several people have mentioned the importance of tolerating silence after asking a question, to give students a chance to think. I'd add that, if you going to call on a raised hand, it is important to often not call on the first hand that goes up.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://linda103-dailyassignment.blogspot.com/2011/04/daily-assignment-56-dipsticking.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dipsticking&lt;/a&gt;&lt;/strong&gt; was mentioned by Sue VanHattum. I used to use thumbs-up, thumbs-sideways, thumbs-down as a quick gauge of the class' sense of its understanding, and I think it's great. More recently (since I started with students summarizing, see above), I'm more likely to say, &amp;quot;raise your hand if you think you understand, not necessarily if you buy it, but just if you understand, the last thing that was said.&amp;quot; If less than 3/4 of the hands go up, I ask somebody who raised their hand to summarize the idea for the people who didn't.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Explicitly encouraging a culture of risk-taking. &lt;strong&gt;Praise all risks.&lt;/strong&gt; When a student asks an honest question, thank them, no matter how obvious the answer seems to you or anyone else. If a student's wrong answer, confusion, question, or expression of doubt leads to the clarification of some point, &lt;em&gt;call everyone's attention to this fact.&lt;/em&gt; Train students to see the value when they and other people are willing to ask their questions and express their thinking, by pointing out this value when it happens.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;General comment:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Every single one of the techniques mentioned above slows down the coverage of material. In my experience, they all, always, slow it down &lt;em&gt;more than I think they will&lt;/em&gt;. This makes them hard to implement consistently because of the pressure to get certain topics &amp;quot;done&amp;quot; and/or my enthusiasm to reach a certain goal I was excited about.&lt;/p&gt;&#xA;&lt;p&gt;Be that as it may, when I &lt;em&gt;do&lt;/em&gt; implement them consistently, I am always left with the strong conviction that the extra time was worth it. Often, for example, when dipsticking, or asking for student summaries, I find out that a point I thought I had taught well actually missed everybody. The time spent in discovering this and doing something about it was obviously worth it, because if I had pushed ahead, I would have been pretending the students had learned something they really hadn't.&lt;/p&gt;&#xA;&lt;p&gt;I say this to encourage stick-to-it-iveness in broadening your repertoire of techniques to encourage participation, and to be ready to face down the costs to speed of coverage. It's worth it.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorDisplayName="user7990" LastEditDate="2020-08-28T18:51:52.260" LastActivityDate="2020-08-28T18:51:52.260" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="162" PostTypeId="2" ParentId="122" CreationDate="2014-03-14T22:04:38.230" Score="19" Body="&lt;p&gt;My strongly held opinion is that &lt;em&gt;some&lt;/em&gt; exact solutions are conceptually fundamental.  Knowing that $\sin 45^\circ = \frac{1}{\sqrt{2}}$ (rather than approximately 0.7071) may not be important for applications in engineering &amp;mdash; although it is certainly necessary in higher mathematics, and I wouldn't be surprised to see it in, say, theoretical physics &amp;mdash; but it is an indicator of conceptual understanding.  Even if you're only going to be an engineer and do numerical calculations, you need to understand the meanings of mathematical concepts in order to know &lt;em&gt;what&lt;/em&gt; numerical calculations to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue that if you don't know that $\sin 45^\circ = \frac{1}{\sqrt{2}}$, you don't really understand the meaning of the sine function and of the pythagorean theorem.  The dividing line is a bit fuzzy &amp;mdash; I'm not entirely sure on which side of it $\sin 30^\circ = \frac{1}{2}$ falls &amp;mdash; but when learning trigonometry, I think $\sin 45^\circ = \frac{1}{\sqrt{2}}$ is definitely on the hither side.  The position of the line also depends on the class; when teaching calculus I don't insist on $\sin 45^\circ = \frac{1}{\sqrt{2}}$, since it's not my job to be teaching them trigonometry any more, but I do still insist on some exact answers like $\sin \frac{\pi}{2} = 1$ and $\cos 0 = 1$ and $e^0=1$.&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="47" LastEditDate="2017-01-29T08:36:50.977" LastActivityDate="2017-01-29T08:36:50.977" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="164" PostTypeId="2" ParentId="122" CreationDate="2014-03-14T22:33:39.997" Score="9" Body="&lt;p&gt;Some people would consider $1/\sqrt{2}$ as problematic or even wrong (in the sense of incomplete), as it should be further simplified to $\sqrt{2}/2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason why I consider this as related is that it is first of all a matter of convention. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is possible to imagine a situation where the expected solution is the decimal approximation and the exact solution would be considered as 'wrong,' e.g., when the point would be practicing how to use a calculator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my mind there are certainly reasons to insist on students knowing or being able to derive the exact value of $\sin 45^{\circ}$, mainly as it requires an understanding of the trigonometric functions. Yet, if one wants to know if they know this, they should be asked explictly, or it should be explicitly clear that this is expected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me a main 'problem' in the situation seems to be to pose an exercise designed to use trigonometric functions where the angle is 45°. This seems a bit artificial, and it are often artificial exercises that lead to students' confusion.   &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="376" LastEditDate="2017-01-28T11:36:00.910" LastActivityDate="2017-01-28T11:36:00.910" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="165" PostTypeId="2" ParentId="13" CreationDate="2014-03-14T22:58:40.630" Score="18" Body="&lt;p&gt;I think it depends on your goals with your calculus class.  If your goal is primarily to prepare students to apply calculus in other subjects, then perhaps counterexamples are not so important.  And if your goal is to &quot;filter&quot;, then perhaps you want to hit them with difficult stuff for no particular reason, including counterexamples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there are other goals that you might have.  For instance, at my university, among the goals of calculus is to teach logical thinking and the meaning of things like definitions and theorems in mathematics.  For this purpose, counterexamples (within reason) are essential.  I do not discuss things that I consider &quot;pathologies&quot;, such as bounded non-integrable functions or nowhere differentiable functions, but simple counterexamples showing that the converse of a theorem may not hold are necessary.  This isn't the same as &quot;rigor&quot;, e.g. we don't use the $\epsilon\delta$ definition of limits &amp;mdash; rather we're trying to convey an idea of the logical structure of mathematics, which must be understood &lt;em&gt;prior&lt;/em&gt; to any attempt at rigor.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2014-03-14T22:58:40.630" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="166" PostTypeId="2" ParentId="74" CreationDate="2014-03-14T23:01:02.807" Score="10" Body="&lt;p&gt;In the example you mention of computing truth tables rather than use algebraic manipulations to answer questions about boolean expressions/sets, it's actually a wonderful situation where the students chooses the long and tedious way, not noticing a much more convenient method works much more nicely. In such cases, I let the student waste time getting the answer in long way, and then I show the short and sweet way. This usually has the effect of burning into the student's brain since they just worked unnecessarily hard to do something and here is a really simple way of achieving the same end-result. Same holds for the example of finding minima/maxima. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm of the opinion that if you want the students to use a particular method, then there needs to be a reason to use that method and not any other they prefer to use. If that is the case, then you should be able to convince them, by demonstration, that it's worth their while to learn that other method. We all prefer the comfort of the methods he had already mastered. But we all appreciate learning a new trick if it can help us. Show them the trick. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-14T23:01:02.807" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="167" PostTypeId="2" ParentId="118" CreationDate="2014-03-14T23:02:42.630" Score="9" Body="&lt;p&gt;Well, there's a fairly standard answer to this question, which is used by most universities in the United States.  A student who has completed algebra and trigonometry and wants to be a math major should learn the following subjects:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Advanced topics in precalculus mathematics, including logarithms, functions, basic mathematical modeling, advanced algebra techniques, vectors, matrices, conic sections, finite sequences and series, permutations and combinations, some basic probability and statistics, and complex numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Single-variable calculus, including limits, differentiation, integration, and infinite sequences and series.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At this point, most students would benefit enormously from learning the basics of physics and computer programming.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Multivariable and vector calculus, including partial derivatives, the derivative as a matrix, multiple integrals, curvilinear coordinates, vector fields, divergence and curl, Green's theorem, Stokes' Theorem, and the divergence theorem.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Linear algebra, including vectors, matrices, row reduction, vector spaces and subspaces, linear independence and span, orthogonality, linear transformations, and eigenvalues and eigenvectors.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(Optional) The basics of differential equations, possibly including an introduction to integral transforms, PDE's, and Fourier series.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At this point, most students should spend some time working on their formal proofs, and learning to read formal mathematics.  Usually, this is best accomplished by learning how to write simple proofs in a relatively concrete mathematical setting, such as number theory, combinatorics, or graph theory.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An introduction to abstract algebra, including groups, rings, and fields.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An introduction to real analysis, including a rigorous treatment of calculus, as well as topics such as the construction of the real numbers, topology of the real line, metric spaces and norms, pointwise and uniform convergence, and possibly an introduction to measure theory and Lebesgue integration.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At this point, most students should continue by learning something about several more topics within mathematics.  Some possible choices of topics include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Complex analysis (perhaps mandatory).&lt;/li&gt;&#xA;&lt;li&gt;Point-set topology (perhaps mandatory).&lt;/li&gt;&#xA;&lt;li&gt;Probability and statistics (perhaps mandatory).&lt;/li&gt;&#xA;&lt;li&gt;More linear algebra (perhaps mandatory), including quadratic forms, change of basis, canonical forms, matrix groups, dual spaces, etc.&lt;/li&gt;&#xA;&lt;li&gt;Logic and the foundations of mathematics.&lt;/li&gt;&#xA;&lt;li&gt;Advanced theoretical courses in physics and computer science.&lt;/li&gt;&#xA;&lt;li&gt;Advanced topics in algebra, such as algebraic geometry, commutative algebra, representation theory, or Galois theory.&lt;/li&gt;&#xA;&lt;li&gt;Advanced topics in geometry/topology, including differential geometry, geometric topology, modern geometry, discrete or computational geometry, basic algebraic topology, and dynamical systems.&lt;/li&gt;&#xA;&lt;li&gt;Advanced topics in analysis, such as the theory of PDE's, basic functional analysis, basic harmonic analysis, Lebesgue integration, and so forth.&lt;/li&gt;&#xA;&lt;li&gt;Advanced topics in discrete mathematics.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-03-15T15:24:23.807" LastActivityDate="2014-03-15T15:24:23.807" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="168" PostTypeId="2" ParentId="85" CreationDate="2014-03-14T23:15:51.387" Score="8" Body="&lt;p&gt;The exact situation you are in is not completely clear to me, I thus give a two-part answer assuming two different scenarios. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Scenario 1.&lt;/em&gt; The exercises are given by somebody else (the instructor of the course), and you run only the excercices-session. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The middle ground between 'enforce the students to do it themselves' and  'the tutor repeat it for them' seems like a good solution to me. The students in such a course already should have attained a certain level of mathematical maturity and are likley also somehow good students. So, if they do not prepare the exercises try to solve them on-the-fly &lt;em&gt;together&lt;/em&gt; instead of just presenting them to them. Ask a lot of small questions, or ask for ideas they had/have. Also, do not shy away from trying ideas they have of which you know they will not work. To see the failing of the idea can be very instructive and lead to finding a better one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Scenario 2.&lt;/em&gt; You have full control. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something that can work is to run it almost like a (students') seminar. Instead of excercises hand out to each student (or small groups) a couple of pages from a book (or paper) of closely related material and let them prepare and present this material in class. Some points in favor: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;They might find this more interesting as it adds something new and is perhaps different in style from other courses. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It might better teach them skills (understanding new material and presenting it) they need at that point of their developpment than the classical excercices-setup.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They do not have to work each week a bit, but at some point a lot. This could be a better fit with their time-management at that point.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The approach in scenario 1 is not really limited to advanced students but I think it is more feasible than, and at least it tends to work better with small groups. So far I never found an occassion to try the thing I describe in scenario 2 myself but know some people that do this (they sometimes use the time-slot for excercises-seesion at the beginning of the term for addition courses to have some weeks towards the end for the student presentations).&lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-03-14T23:15:51.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="169" PostTypeId="2" ParentId="150" CreationDate="2014-03-14T23:17:38.417" Score="9" Body="&lt;p&gt;What goes into the school curriculum is to a huge extent a political decision backed by ever-changing research 'evidence' pointing here today and there tomorrow. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What goes into the math curriculum is like-wise supported by 'evidence' about the superiority of this or that technique, that piece of mathematics or the other. It is to a huge extent random. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, there are (at least) two questions here. What does one gain from the discipline of mathematics, and what does one learn from the specific topics covered. I'll address only the latter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typical off-the-shelf answers seem to be: learning mathematics will enhance your problem solving abilities and your logical thinking. Who doesn't want to be a better problem solving and logical thinker? The problem is there seems to be little evidence to support these claims. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, looking at the evidence, the best reason to learn mathematics is that if you are good at it, it impresses people. And when people are impressed by you, they tend to give you better jobs with more money. Of course, some may argue that an even better reason to learn mathematics is if you like it. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-14T23:17:38.417" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="170" PostTypeId="2" ParentId="118" CreationDate="2014-03-14T23:29:01.230" Score="13" Body="&lt;p&gt;At least as a counter-point to the other good answers, I must confess that I have misgivings about the standard undergrad math curriculum in the U.S., primarily because I think it presents the subject as a plodding, exaggeratedly fastidious version of the most elementary parts of it... pointedly ignoring genuine motivations, historical motivations, chronology, and too many other aspects, while not allowing itself to mention any substantial contemporary developments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, most often, the &quot;logical order&quot; precept has taken over, and students are taught to worry endlessly over potential pathologies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At another extreme, there was an older tradition of &quot;engineering math&quot;, which went too far in the opposite direction, but did have the virtue of addressing interesting questions, both internal to mathematics and external. Oddly, with the advent of good &quot;math software&quot;, it appears that much of what was done by hand by engineers a few decades ago is essentially programmable...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, I &lt;em&gt;am&lt;/em&gt; proposing that &quot;rigorous proof&quot; is not the highest goal in mathematics, although, of course, knowing with greater certainty is desirable. Making implicit assumptions overt is good methodology. But being paranoid about technicalities is not good methodology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A riff I use to explain to the best undergrads I see is that undergrad classes are essentially worthless as preparation for grad school, and the first-year grad courses are scarcely better. That is, quasi-ironically, &quot;we&quot; train beginners into a behavior pattern that we often seemingly endorse for them, but which is not at all what we &lt;em&gt;do&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, the &lt;em&gt;possibility&lt;/em&gt; of &quot;logical perfection&quot; in mathematics, unlike most other venues, inordinately catches our collective fancies, and become a goal in its own right. (And, possibly, it is worthwhile, but it is not the whole.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, my caution would be to disbelieve the worldview implicit in undergrad math curricula, and to distrust the choices of focus of it. Better if one can survive in relatively gentle graduate courses, even if they have degenerated (as is often so) into laundry-lists of must-mention topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words: mathematics is far simpler than as portrayed in undergrad courses.&lt;br&gt;&#xA;Best to become acquainted with basics privately and efficiently, unless one truly needs the carrot-and-stick thing.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-14T23:29:01.230" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="171" PostTypeId="2" ParentId="41" CreationDate="2014-03-14T23:31:23.913" Score="82" Body="&lt;p&gt;I changed my handwriting font years ago for precisely this purpose, and I have continued to tweak my letterforms over the years, using the algorithm of changing the form of whichever letters seem to generate the most confusion.  Here is my current font:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/WRoR8.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few notes about these letters:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Capital letters aren't listed, but my experience is that they're all fairly straightforward.  Just make sure to use print letters instead of cursive.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's crucial to have versions of a, b, d, p, and q that can be written using a single stroke.  If you draw your circles and stems separately, they will constantly get disconnected, which vastly decreases legibility.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I often omit the bottom &quot;tail&quot; on the f when writing it in the middle of a word, but I always include it when it's part of an equation.  Similar statements hold for the top curve on the i.  I will also draw a 1 as a simple vertical line if I think it's clear from context.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The l (ell) is my newest letter, and I'm not really sure about it yet.  I tried using a cursive $\ell$ for a while, but it never looked good inside of words, and it still wasn't very legible as a variable.  At present, I am often omitting the top and bottom curves when the l is part of a word.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In general, I've had bad experiences with vertical loops.  I've tried loops on $\ell$'s, g's, j's, d's, and q's, and all of them seemed to make the letters less recognizable.  (This is probably because such loops hardly ever appear in computer or typewritten fonts.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The tail curve on the t is crucial to avoid confusion with a + sign.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The initial curves for the v and w are quite helpful for legibility.  I also think v's and w's are more legible with relatively sharp angles, as opposed to a curvy cursive approach.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The initial curve on the x is absolutely essential.  This will be one of your most used letters, and it really helps to get it right.  Curves on any of the other three stems don't seem to improve legibility, and make the letter annoying to write.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I also think this curvy version of a y is easier to read than a two-sticks version.  Among other advantages, it can be drawn in a single stroke, which avoids disconnection problems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The line across the z helps avoid confusion with the number 2, and the line across the 7 helps avoid confusion with a variety of symbols.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't seem to be able to draw a curvy-bottom 9 that looks good.  I wish I could.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This version of a 2 is much clearer than any version with a loop on the bottom.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I tried drawing a final loop on my o's for a while (like a cursive o), and it didn't work out.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; By request, here are my capital letters.  Unlike my lowercase letters, my capitals are really quite standard, and I don't have much to say about them.  I've also included by Greek letters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/FQ1Kd.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2015-01-15T20:37:03.377" LastActivityDate="2015-01-15T20:37:03.377" CommentCount="20" ContentLicense="CC BY-SA 3.0" />
  <row Id="172" PostTypeId="2" ParentId="135" CreationDate="2014-03-15T00:30:27.050" Score="30" Body="&lt;p&gt;What is striking about the Lebesgue integral is how relatively nicely limits play together with this integral, things like the &lt;a href=&quot;http://en.wikipedia.org/wiki/Dominated_convergence_theorem&quot;&gt;Dominated convergenece theorem&lt;/a&gt; are great. I think one can appreciate this result (especially when contrasted with the more clumsy Riemann Integral analogues) right away. One could mention this right at the start, before everything else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Put differently, that it is desirable to have $\int \lim f_n = \lim \int f_n $ be true (and make sense) seems essentially self-evident. The 'new' integral is there to make this true under weaker/nicer assumptions (on $f_n$). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By contrast to me the 'one can integrate more functions' reason is somehow an &lt;em&gt;anti&lt;/em&gt;-reason. When talking about the relevance of the Lebesgue Integeral I of course mention a standard example like the Dirchlet function but only to add that to be able to integrate this/such functions is &lt;em&gt;not&lt;/em&gt; the point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue is that the clear-cut nature of the reason 'you can integrate this function now' tends to overshadow further reasons one might give. Yet then on second thought this reason is not very compelling because to what end would anybody want to integrate that function specifically in the first place. Thus I consider it of some value to actively stress this point.&lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-03-15T00:30:27.050" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="173" PostTypeId="2" ParentId="70" CreationDate="2014-03-15T01:08:23.390" Score="8" Body="&lt;p&gt;I taught two sections of a course last semester, one MWF (50 mins each) and one TuTh (75 mins each). It was not great.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main problem is not even splitting material across days. It's an issue of timing and attention. It takes much more effort on the teacher's part, both pre- and during class, to effectively engage for 75 mins. I found myself planning different activities and group work, or sometimes letting them run longer in the 75 min section. I don't think even that was good enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Honestly, the best thing to do is make a list of the major concepts and activities from the course and plot your days around that list. With a TuTh course, you might have to cut two or three ideas over the course of a semester, but you can give a fuller treatment of the others. For instance, with your Stewart calculus course, identify a handful of sections that could feasibly be skipped, or at least covered only cursorily. Allow yourself more time for each of the ideas you have left and work in some extra in-class activities. &lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-15T01:08:23.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="174" PostTypeId="2" ParentId="86" CreationDate="2014-03-15T01:33:16.893" Score="6" Body="&lt;p&gt;I like sharing solutions for a few reasons, many already mentioned by others: they are good study aids before exams, students will catch your mistakes/assumptions,  students can see what is considered &quot;ideal&quot;, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However,  I've noticed (through pageview tracking on course management sites) that few students actually look at them! So, I think it behooves us to &lt;em&gt;do something&lt;/em&gt; with solutions. Pick one problem per assignment and discuss solutions in class. This can be the most difficult problem,  the one with the most valid approaches, the one that tripped up students the most, the one that provides a segue to the next topic, etc. Or, have a followup assignment where students must pick one problem and write about how their solution differs from yours. Or, offer extra credit for students who notice their solutions are significantly different and who write them up and send to you to be included in future versions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your mileage will vary depending on your course's difficulty and size and how much class time you're willing to devote to this. Personally, I've started using activities like this in a 15ish student intro to proofs course, including class discussions of actual student solutions (anonymously) that I type up and distribute. They have really enjoyed this so far.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-15T01:33:16.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="175" PostTypeId="1" CreationDate="2014-03-15T01:49:35.777" Score="19" ViewCount="394" Body="&lt;p&gt;This will likely depend on the class, of course. But I've asked calculus students in the past if (a) they regularly read the textbook and (b) whether this is helpful for them and (c) whether they like the book. The responses have been mixed for (a) and (b), and (c) is usually negative. Currently, I'm teaching an intro to proofs course for potential math majors, and I'm assigning daily (for a Tuesday/Thursday class) readings and giving a quiz each class about the reading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'm wondering is whether there is significant evidence that assigning specific readings will help students, whether or not they are assessed on this. And furthermore, are these potential effects at all helped/harmed by followup assessment about reading, and if so, what are the best ways to do this (if at all)?&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="5220" LastEditDate="2020-06-04T16:36:52.587" LastActivityDate="2020-06-04T16:36:52.587" Title="Is required reading of the text effective, and how can it be assessed?" Tags="|mathematical-pedagogy|homework|textbooks|assessment|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="176" PostTypeId="1" CreationDate="2014-03-15T02:08:46.123" Score="46" ViewCount="3122" Body="&lt;p&gt;Many brilliant mathematicians seem to make average or even poor classroom teachers. Is this an accurate assessment? Has there been any research to explain the phenomena?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the difference between knowing mathematics and knowing to teach mathematics?  Can you provide any examples (either abstract or concrete) of this phenomenon occurring?&lt;/p&gt;&#xA;" OwnerUserId="95" LastEditorUserId="93" LastEditDate="2014-03-15T16:59:28.833" LastActivityDate="2021-09-12T05:12:50.403" Title="Knowing mathematics does not translate to knowing to teach mathematics. Why?" Tags="|mathematical-pedagogy|" AnswerCount="9" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="177" PostTypeId="2" ParentId="176" CreationDate="2014-03-15T02:51:40.987" Score="21" Body="&lt;p&gt;There's two parts to this question.  So, I'll give them separately:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why?&lt;/strong&gt;  There is a lot of education and psychology research on expertise that points to the idea of an &quot;expert blind spot&quot;.  An easy way of thinking about it is that someone who has been studying mathematics for a very long time has likely forgotten the struggles she went through when she initially learned the material.  This can make it very difficult to understand the issues that students have and why they have them.  It often presents itself in the feeling &quot;This is trivial!  I have no idea why they didn't get this correct.&quot;  Here's a &lt;a href=&quot;http://pact.cs.cmu.edu/koedinger/pubs/2001_NathanEtAl_ICCS_EBS.pdf&quot;&gt;paper&lt;/a&gt; that explains this phenomenon in more detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's probably worth noting that a shallow understanding of a topic can account for students' abilities to get perfect scores but not be able to explain their reasoning to someone else.  This could be relevant to your question if you're interested in why the best students in your class might not be very good at teaching or communicating with the other ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Examples.&lt;/strong&gt; I have no idea what level or area of math you're looking for here.  So, I'll try to give some varied examples and/or ways in which this can happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Teaching Induction.&lt;/em&gt; Often, when mathematical induction is being introduced for the first time, the induction step is stated as something like &quot;Prove $P(n) \implies P(n+1)$&quot;.  Students often respond to this with something like &quot;but that's just assuming the conclusion!&quot;, because internally they aren't fixing $n$ to be a &lt;em&gt;particular&lt;/em&gt; natural number--to them, it's more intuitive to just assume it for &lt;em&gt;every&lt;/em&gt; natural number.  Understanding this distinction (and this common misunderstanding) might lead to an instructor highlighting what $n$ is, rather than just saying it and moving on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Assuming Prior Knowledge.&lt;/em&gt; Someone who uses group theory regularly might try to motivate or teach any number of theorems (Fermat's Little Theorem, Wilson's Theorem, any variety of counting problems) using either generalizations or proofs that rely on thinking abstractly about mathematical operations (e.g. Fermat's Little Theorem is a trivial consequence of Euler's Theorem).  Someone else might assume that the bijection between $\mathcal{P}(\mathbb{N})$ and functions from $\mathbb{N}$ to $\{0,1\}$ that uses binary is trivial to students who might be new to bijections or never have seen binary before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Ignoring &quot;Trivial&quot; Steps.&lt;/em&gt; Another very common issue is that instructors sometimes accidentally leave out steps or details (because they view them as &quot;obvious&quot;).  Someone who has been differentiating functions for a very long time might not explicitly call attention to uses of the chain rule, product rule, etc.  It's often an oversight rather than the intention.  One could imagine an extension of this where the students are supposed to take a partial derivative, as well.&lt;/p&gt;&#xA;" OwnerUserId="93" LastActivityDate="2014-03-15T02:51:40.987" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="178" PostTypeId="1" AcceptedAnswerId="245" CreationDate="2014-03-15T03:26:48.613" Score="20" ViewCount="8537" Body="&lt;p&gt;What are the best textbooks to introduce measure theory and Lebesgue integration to undergraduate math majors? Many students in such a class will go on to graduate school and be required to take a similar course in their first year. As such, is it better to focus on $\mathbb{R}^n$ or more abstract spaces?&lt;/p&gt;&#xA;" OwnerUserId="52" LastEditorUserId="376" LastEditDate="2022-03-24T16:35:23.187" LastActivityDate="2022-03-24T16:35:23.187" Title="Best textbooks to introduce measure theory and Lebesgue integration?" Tags="|undergraduate-education|reference-request|textbooks|mathematical-analysis|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="179" PostTypeId="2" ParentId="39" CreationDate="2014-03-15T03:54:21.207" Score="7" Body="&lt;p&gt;I really think this depends on what you intend to cover and what you want students to learn from it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you hope for your students to work through the major parts of the books and develop a true appreciation for it? Good luck; you'll need to presume they already have a level of mathematical maturity and enthusiasm that is difficult to quantify and pin to a certain set of prerequisites.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you hope to inspire a group of not-necessarily-math-oriented folks to develop an appreciation for mathematical rigor and proofs? Perhaps you hope to inspire visual learners to dig deeper into mathematics, to see that it's not all about numbers and calculating? Then, you might not presume much more than an interest in pictures and some modicum of logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are you working with high school students? College students? Math majors or not? All of these will affect your answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, I think there is not much mathematical &lt;em&gt;content&lt;/em&gt; with which you need to presume familiarity. I disagree with your &quot;high level&quot; assessment of requisite logic: working with hypotheses and conclusions, as well as some understanding of &lt;em&gt;reductio ad absurdum&lt;/em&gt; should suffice. Other than that, a working familiarity with plane geometry should be an advantage but not technically a prerequisite.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-15T03:54:21.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="180" PostTypeId="2" ParentId="176" CreationDate="2014-03-15T06:15:55.470" Score="24" Body="&lt;p&gt;A good math educator is good not only at math but also at psychology - reading the students' body language to figure out what their response is, good at facilitating groups, good at encouraging students to work hard, good at understanding where someone is stuck (and different students do get stuck in different places), and how to explain not one way but two or three.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A wonderful book that explains what more a math educator needs to know to explain math ideas (at elementary level) is Liping Ma's &lt;em&gt;Knowing and Teaching Elementary Mathematics&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-15T06:15:55.470" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="181" PostTypeId="2" ParentId="160" CreationDate="2014-03-15T06:33:17.553" Score="14" Body="&lt;p&gt;The question you are asking has little to do with the particular subject in which the student excels and everything to do with student motivation. The students have not developed the skills needed to study and persevere through difficult classes because everything to this point has come naturally for them. They haven't needed to set time aside for a study group or tutor because they've never needed it before and they likely don't even think about it as a problem until it's too late.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, I can't find any reputable studies on this in the myriad of articles from other sources. It seems that every newspaper columnist has written about student motivation at some time. But it's my firm opinion that some students are not mature enough to attend college after they graduate high school. I know that I was not ready for college at 18 years of age and dropped out after a semester. It was the best decision of my life. If I had continued on that path, I would never have ended up on the road that I am on now to become a teacher. Now, I spend hours per night on homework and studying, then read about interesting math topics and write about math in my free time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anecdotally, I have seen four students this year drop out of the Math Ed program because they can't handle the workload. And they are all intelligent students who simply would not apply themselves to study and do the homework. I think that a teacher can help motivate students through a number of different methods and the success of each depends entirely on the individual student. I've seen teachers bribe students (turn in this homework by X day and get extra points on the test). I've seen teachers strike the fear of God into their students (A quiz with 3 of the most difficult problems from the homework once a week on random days). I've seen teachers plead, joke, or be apathetic. Regardless of the method, the student is ultimately the one who makes the decision to try. Imparting the wisdom of experience to a college student is frustratingly fruitless. But every now and then, they get it.&lt;/p&gt;&#xA;" OwnerUserId="151" LastActivityDate="2014-03-15T06:33:17.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="182" PostTypeId="2" ParentId="178" CreationDate="2014-03-15T06:46:23.890" Score="13" Body="&lt;p&gt;First, I highly recommend David Bressoud's book &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521711835&quot;&gt;A Radical Approach to Lebesgue's Theory of Integration&lt;/a&gt;&lt;/em&gt; for this. It's highly motivated and historically grounded, written with undergraduates as the intended audience, and has a clear, engaging expository style and plenty of good exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, In my opinion the case of $\mathbb{R}^n$ makes sense as the focus for the situation you're describing; in fact, it might make sense to focus mostly on $\mathbb{R}$. The abstract measure spaces they'll encounter in grad school will be both more comprehensible and more engaging after they've had some experience with the most important-to-them special case. (I'm thinking back on how much more sense the definition of a &lt;a href=&quot;http://en.wikipedia.org/wiki/Functor&quot;&gt;functor&lt;/a&gt; made to me &lt;em&gt;after&lt;/em&gt; I had some experience with the &lt;a href=&quot;http://en.wikipedia.org/wiki/Fundamental_group#Functoriality&quot;&gt;fundamental group functor&lt;/a&gt;.) That said, it's a matter of your taste as a teacher. If you are really excited about some abstract measure spaces, then that can make sense too.&lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2014-03-15T06:46:23.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="183" PostTypeId="1" AcceptedAnswerId="185" CreationDate="2014-03-15T07:08:24.447" Score="24" ViewCount="1858" Body="&lt;p&gt;What is a good strategy when you realize (e.g. while grading the exam) that a question on an exam was incomplete/wrong?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More concretely:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If it is decided that additional points should be given:&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How many additional points should one give to the students? &lt;/li&gt;&#xA;&lt;li&gt;And to which students? &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Are there other good solutions?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The problem here is that you can never tell who really tried this question and maybe who tried but did not write down anything for it (but maybe saw the problem coming). Other students might argue that the tried most of the exam time an that particular question, so they could not do other questions.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="125" LastEditDate="2014-03-18T16:03:53.593" LastActivityDate="2014-03-18T16:03:53.593" Title="Non-answerable questions on exam: What to do?" Tags="|mathematical-pedagogy|exams|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="184" PostTypeId="1" AcceptedAnswerId="519" CreationDate="2014-03-15T07:20:50.880" Score="21" ViewCount="281" Body="&lt;p&gt;I am seeking for some pedagogical literature dealing with the following question: Imaging you have an average class in college/university: &lt;strong&gt;What is a good balance between Laissez-faire  and pressure?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;i.e., a balance between &quot;I'm offering you some stuff and help and you can do it or not. You just have to pass the exam&quot; and &quot;I'm forcing you to make at least XX% on the homework grades, see if any two homework of the whole course are copies, make the homework not that easy, etc. - If you fail on the XX% on homework you will not be able to take the exam.&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Background: It is clear if you have fully responsible students, there is no need for pressure. In my university the focus is more on pressure than on Laissez-faire and many students (a lot of them math ed students) argue &quot;In pedagogy we learned that we can only learn anything in the complete absense of pressure&quot;. But none of them was able to give me literature to that statement)&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="143" LastEditDate="2015-02-11T00:43:13.243" LastActivityDate="2015-02-11T00:43:13.243" Title="Pressure vs. Laissez-faire: Literature dealing with balance in university-classes" Tags="|undergraduate-education|graduate-education|general-pedagogy|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="185" PostTypeId="2" ParentId="183" CreationDate="2014-03-15T07:23:16.060" Score="17" Body="&lt;p&gt;I feel that this will be handled on a case by case basis, but there are some guidelines that can help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;If a question is difficult due to bad wording or poor setup:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Throw out the question and grade the exam as if it weren't part of your total.&lt;/li&gt;&#xA;&lt;li&gt;Give bonus points to students who got it right or partially so.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;If a question is wrong:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Give full credit for the question to every student.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;or&lt;/em&gt; Give a quiz sometime after the exam worth points on the exam. (a mini make-up exam)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you have students complaining that they did not have enough time on a question, reevaluate the time allotted for the exam and use your best judgement based on class performance to determine if the question should be treated as a poorly setup question. If not, use this as a teaching opportunity about time management during an exam! Many students will spend time staring at a problem they haven't prepared for properly and then &quot;run out of time&quot; on the other questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tend to err on the side of generosity in these situations, but each should definitely be considered separately. &lt;/p&gt;&#xA;" OwnerUserId="151" LastActivityDate="2014-03-15T07:23:16.060" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="186" PostTypeId="1" AcceptedAnswerId="199" CreationDate="2014-03-15T08:11:02.587" Score="30" ViewCount="1947" Body="&lt;p&gt;In the first lectures about calculus/analysis, you should introduce real numbers. Let's assume students know that rational numbers are. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are the advantages or disadvantages in the different &quot;definitions&quot; of the real numbers?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm particulary interested in pedagodic arguments: At that point of the lecture, you want to have a good balance in&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;not killing the motivation of the students/overcharging the students at that level,&lt;/li&gt;&#xA;&lt;li&gt;having a good definition to work with,&lt;/li&gt;&#xA;&lt;li&gt;a good agreement with the concept of real numbers the students have from school.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I've seen many books where the introduction of the reals are by far the most difficult chapter at that point (Compare handling Dedekind cuts vs. obtaining convergence of a series) or books where they not really introduce the real numbers, but also assume that students know it already more or less. There are also books where the authors have done a abstract contruction, but state that one should skip that part on first reading. This is okay for a book, but somehow this does not align with the deductive structure of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;edit: As it was asked in the comments. A typical course would be the first course where you want to define real numbers (i.e., a course where typically math majors are not sharing with engineers, etc.) like an &quot;Analysis I&quot; course in central Europe.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-28T18:42:56.340" LastActivityDate="2020-12-15T22:07:49.390" Title="Good definition for introducing real numbers?" Tags="|undergraduate-education|mathematical-analysis|real-numbers|" AnswerCount="7" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="187" PostTypeId="2" ParentId="70" CreationDate="2014-03-15T09:00:15.957" Score="4" Body="&lt;p&gt;One specific problem I have with classes which meet only twice a week is that in the last few years I've been giving more frequent, lower-stakes exams.  (There are reams of research suggesting that this is better for students' learning and retention in any subject.)  I don't mind using up a lot of MWF class days on exams, but it's harder for two-day-a-week classes (which in my case have been MW only), with only 2/3 as many class meetings.  So I've sometimes given 45 minute exams followed immediately by 30 minutes of lectures.&lt;/p&gt;&#xA;" OwnerUserId="129" LastActivityDate="2014-03-15T09:00:15.957" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="188" PostTypeId="2" ParentId="176" CreationDate="2014-03-15T10:17:16.087" Score="41" Body="&lt;p&gt;There is yet another issue here, it is related to &quot;expert blind spot&quot;, but not the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be a good teacher, one needs to understand the pupils and their issues with the topic. This becomes hard if the teacher &lt;em&gt;thinks in a qualitatively different way&lt;/em&gt;. He may know exactly what causes the problem and why, but cannot explain it well, and his messages don't reach the students, because his explanations align with his thought process, not the students'. Such a teacher might refer to intuitions that are alien to students and thus not convincing. On the other hand it's hard for him to assume student's thought-process, because he might see it as deeply flawed and missing some essential characteristic of considered structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example where this happens frequently is &lt;a href=&quot;http://en.wikipedia.org/wiki/Category_theory&quot;&gt;Category Theory&lt;/a&gt;. Learning it involves some &quot;brain  rewiring&quot; and after it happens that person speaks a different language even if it appears to be normal. The most useful introductory texts I saw were&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;written by people &lt;em&gt;during&lt;/em&gt; the process of learning the subject,&lt;/li&gt;&#xA;&lt;li&gt;stuffed with &lt;em&gt;examples&lt;/em&gt; that would let the reader build their own intuitions, independent of authors explanations.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of course, after &quot;it clicked&quot; my opinion on them was different. I didn't have to be an expert to see them as unnecessarily verbose, skewed or having too many trivial, pointless examples. It was even funny how little it took to change the perspective and how hard it was to begin that change.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="376" LastEditDate="2016-02-22T18:34:14.847" LastActivityDate="2016-02-22T18:34:14.847" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="189" PostTypeId="1" AcceptedAnswerId="201" CreationDate="2014-03-15T12:06:11.937" Score="14" ViewCount="734" Body="&lt;p&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt; This question will use as an example a computer-science university program, because it is much easier for me to describe and give some statistics, but it applies to math as well. The reason I posted it here is that talent (i.e. mathematical gift) plays a much greater role in thinking-heavy areas than in, for example, biology or history.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Context:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The university welcomes all kinds of students with varying degree of mathematical/programming ability. They are split into groups of 15 (approximately), however, to teach the beginners programming and to keep the experienced engaged, each such group might be &quot;imperative&quot; (beginners learn some standard programming language) or &quot;functional&quot; (those who could code before learn some functional language). There is a test of their programming skill and then students are assigned accordingly to the results (if student wishes and there are no obstacles, he may apply for a change; in the past there was no test and the students could pick the group type as they wished). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summarizing: there are several &quot;imperative&quot; groups which contain mostly beginners and a few &quot;functional&quot; groups which include students that could code before. It is worth noting that the majority of the gifted students would usually participate in programming contests in high school, while the average and below-average students almost always can't code (even if they know the syntax, the test is too hard for them). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works quite well: with such a split the beginners learn the basics, while students from &quot;functional&quot; groups are engaged because of the more advanced topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works quite well, until the exams. The groups motivate/demotivate themselves internally and in result mediocre students from the &quot;functional&quot; groups have learned a way more than the good from the &quot;imperative&quot; groups. &#xA;It wouldn't be a big problem, because the results between &quot;functional&quot; and &quot;imperative&quot; groups are incomparable - the tests are of incomparable difficulty, hence are graded separately.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, as the schedules are inter-dependent, the groups formed there are kept also for other courses, i.e. these which didn't plan for a split at all. Again, groups motivate themselves internally (and also teachers are more engaged with brighter students), but this time all the students are graded according to the same key. The gap in performance is disastrous: the average score for the first group might be easily over $80\%$, while the average for the rest can be as low as $25\%$. Such results have impact on their learning, motivation and many other things, but most importantly, a big part that would attend the &quot;imperative&quot; would fail their exams and sometimes even drop out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could think that splitting the students by abilities is just a bad solution. On the other hand, their knowledge and experience varies so much that it would be impossible to keep all the parties interested. For example, this happens for electable courses, where the groups mix together, and pose a serious challenge for the teachers. It is so inconvenient that professors happen to make their optional classes intentionally very hard or very easy (the funny part is that it even feels natural, because those courses are in majority theoretical and practical respectively).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure that it's not a healthy state of affairs, but this is a hard problem and might lack a general solution. Also, there is the whole topic of sliding scale between teaching all something and teaching some everything, but that's out of scope of this question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there a way to avoid the negative effects of the split, in particular effects of good students from &quot;imperative&quot; groups averaging down and poor students from &quot;functional&quot; groups averaging up?&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2017-12-16T21:13:22.480" Title="Splitting the students by abilities" Tags="|undergraduate-education|mixed-ability|gifted-students|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="190" PostTypeId="2" ParentId="186" CreationDate="2014-03-15T12:28:38.017" Score="14" Body="&lt;p&gt;I've approached this a couple different ways in the past, and I haven't decided what I think is best.  But I'd like to point out some arguments in favor of &lt;em&gt;not&lt;/em&gt; including any construction of the real numbers (like Dedekind cuts or limits of equivalence classes of Cauchy sequences), and simply stating the axioms which are needed for the rest of the course.  (This may or may not include stating a theorem that there is, up to an appropriate notion of isomorphism, a unique mathematical structure satisfying those axioms.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You note that the introduction of the real numbers is the most difficult part of many analysis textbooks.  Beyond that, I would say that it is in many ways orthogonal to the rest of the course.  The mathematical issues and techniques involved in &lt;em&gt;defining&lt;/em&gt; the real numbers are mostly rather different from the issues and techniques involved in &lt;em&gt;working with&lt;/em&gt; the real numbers.  Therefore time and effort spent on defining the real numbers doesn't actually contribute much to students' understanding of real analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You also raise the point that omitting a definition of the real numbers &quot;does not align with the deductive structure of mathematics.&quot;  But you have to start with &lt;em&gt;something&lt;/em&gt;.  Most constructions of the real numbers start with the rational numbers.  But what are those?  Rational numbers can be defined starting from the integers; this construction (and its generalization to integral domains) is frequently taught in algebra courses.  But what are the integers?  They can be constructed from the natural numbers, which can be constructed in pure set theory…  Most people wouldn't think of starting an analysis class with the Peano axioms, let alone axiomatic set theory.  (And how many people teaching analysis classes, or writing analysis texts, are even qualified to teach axiomatic set theory?)  So an analysis class starts somewhere higher up the ladder, with a lot of structure assumed to be familiar to the students.  This admittedly &quot;does not align with the deductive structure of mathematics,&quot; but that's a necessary fact of life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, the last time I taught analysis I covered Dedekind cuts, because it seemed to let me get to the real content of the course more quickly than Cauchy sequences.&lt;/p&gt;&#xA;" OwnerUserId="129" LastActivityDate="2014-03-15T12:28:38.017" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="192" PostTypeId="2" ParentId="64" CreationDate="2014-03-15T15:55:58.017" Score="8" Body="&lt;p&gt;Many of these sites use a model called the &lt;a href=&quot;http://en.wikipedia.org/wiki/Spaced_repetition&quot;&gt;Spaced Repetition Model&lt;/a&gt;.  In particular, I'm most familiar with how Duolingo used to do it a few years ago, as I worked there. Khan Academy seems &lt;a href=&quot;https://code.google.com/p/khanacademy/issues/detail?id=6&quot;&gt;to do this&lt;/a&gt; as well.  In principle, though, the idea is the same.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a high level explanation, these site group together types of questions (e.g. for language each word is a type of question, for basic addition &quot;one-digit addition&quot; might be a type of question. As it gives you questions, everything starts out equally likely.  As you answer questions correctly, it &quot;schedules&quot; them to be given further and further in the future (usually on an exponential model).  Eventually, it asks you these things so rarely that students don't feel like they're getting things they already know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other sites (though, it seems increasingly rarely) use a &quot;mastery model&quot;.  Khan Academy used to do this, but I believe they've all-but phased it out.  The idea here is if you get a type of question correct $n$ times in a row (where $n$ varies on the implementation), then it &lt;strong&gt;never&lt;/strong&gt; asks it to you again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another article explaining Khan Academy's approach is here: &lt;a href=&quot;http://david-hu.com/2011/11/02/how-khan-academy-is-using-machine-learning-to-assess-student-mastery.html&quot;&gt;http://david-hu.com/2011/11/02/how-khan-academy-is-using-machine-learning-to-assess-student-mastery.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="93" LastEditorUserId="93" LastEditDate="2014-03-15T16:56:39.643" LastActivityDate="2014-03-15T16:56:39.643" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="194" PostTypeId="1" CreationDate="2014-03-15T16:08:15.150" Score="29" ViewCount="946" Body="&lt;p&gt;In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier relationship with math. But in the present situation, how can we help the students who come to our classes, which they are required to take, with fear and loathing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How do we help students overcome their math anxieties?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2014-03-16T01:32:06.253" LastActivityDate="2020-09-07T20:48:40.757" Title="How can we help students who are very anxious about math?" Tags="|mathematical-pedagogy|anxiety|student-motivation|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="195" PostTypeId="1" CreationDate="2014-03-15T16:13:33.663" Score="2" ViewCount="80" Body="&lt;p&gt;Where can research and resources be found, about study skills and strategies for university math of immediate impact? To wit, how do I efficiently find categorical, pragmatic advice about learning/studying math, of instant effect? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I exemplify a few examples which I lighted upon with Google and Amazon.com:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.amazon.co.uk/Study-Mathematics-Degree-Lara-Alcock/dp/0199661324&quot; rel=&quot;nofollow&quot;&gt;Lara Alcock, &lt;em&gt;How to Study for a Mathematics Degree&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.amazon.co.uk/How-Think-Like-Mathematician-Undergraduate/dp/052171978X&quot; rel=&quot;nofollow&quot;&gt;Kevin Houston, &lt;em&gt;How to Think Like a Mathematician&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.amazon.co.uk/Bridging-University-Mathematics-Edward-Hurst/dp/1848002890&quot; rel=&quot;nofollow&quot;&gt;Martin Gould and Edward Hurst, &lt;em&gt;Bridging the Gap to University Mathematics&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I originally posed this as an &lt;a href=&quot;http://area51.stackexchange.com/proposals/64216/mathematics-educators/64305#64305&quot;&gt;Example Question&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="50" LastEditDate="2014-03-16T08:04:24.483" LastActivityDate="2014-03-16T08:04:24.483" Title="Where to find resources about study skills and strategies of immediate impact?" Tags="|aids|" AnswerCount="0" CommentCount="2" ClosedDate="2014-03-29T00:32:39.610" ContentLicense="CC BY-SA 3.0" />
  <row Id="196" PostTypeId="1" CreationDate="2014-03-15T16:15:01.933" Score="4" ViewCount="81" Body="&lt;p&gt;What are ideas and strategies on improving at discovering counterexamples?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I originally posed this as an &lt;a href=&quot;http://area51.stackexchange.com/proposals/64216/mathematics-educators/64407#64407&quot;&gt;Example Question&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2020-09-07T20:48:27.060" Title="What are ideas and strategies on improving at discovering counterexamples?" Tags="|proofs|aids|" AnswerCount="1" CommentCount="4" ClosedDate="2014-03-15T20:53:09.173" ContentLicense="CC BY-SA 3.0" />
  <row Id="197" PostTypeId="2" ParentId="186" CreationDate="2014-03-15T16:15:25.303" Score="8" Body="&lt;p&gt;&lt;em&gt;Note: The question is broad, I agree with a lot of what Mark Meckes said. Below my take on the Cauchy sequences approach (mainly in favor to complement the other answer).&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The approach of constructing the real numbers as equivalence classes of Cauchy sequences of rational numbers modulo sequences with limit zero seems certainly not like the most intuitive one or the one most in line with earlier concepts, but it does not that bad in this regard either, &lt;em&gt;and&lt;/em&gt; has some other advantages. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are the advantages (in my mind):&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;One uses objects and notions that one needs later on, such as: Cauchy sequences, sequences with limit zero, the notion of an equivalence relation in a somewhat non-trivial way, definining operations (sum and product) on the spaces of sequences, checking some operation stays well defined under that equivalence relation. By contrast, a Dedekind cut seems completely confined to this context; granted some of the things I mention also come up in that context but still I feel it is more useful to have it for sequences. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The approach generalizes. The same approach of constructing a completion works for any metric spaces. More generally, if one wants, one can use such a course to build up some basic notions of General Topology.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why it is not that bad regarding intuition:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the most intuitive sem-rigorous way to think of a real (non-rational) number is as a sequence of (rational) numbers that approach it in some sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps one has an algorithm to calculate ever better approxiamtions, this would fit this intuition. Perhaps one has &quot;the decimal expansion&quot;, this would fit this intuition. Perhaps one has a sequence of ever smaller intervalls containing the number, this would essentially also fit this intuition (considering end-points). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, then,  from a certain point on one should never be too far away from what one wants to approach, whence the Cauchy criterion. And, if a sequecne goes to zero, then it &quot;is&quot;  zero. And, two things are &quot;equal&quot; if the difference is zero.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;On the problem of what to assume.&lt;/strong&gt; Two possible arguments. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think there is a point to be made that to assume the rationals but to construct the reals is apt for an Analysis course since the real numbers are in some sense an object of the analytic world. By contrast, in the evoked context often, another, possibly the other, large course followed by students is some (Linear) Algebra course. There they will also use real numbers more often then not working in $\mathbb{R}^n$ and so on, but it is hardly ever really relevant, so I would say to make a 'big deal' in that context about what the real numbers are would feel besides the point. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One &lt;em&gt;can&lt;/em&gt; start earlier. Starting from (naive) set theory one can build up 'everything', introducing students to proofs by induction and some related things on the way. (For example Schwartz's 'Analyse I' is subtitled set theory and topologie.) If this is feasible depends on the context. In some classical curricula there would be essentially Anaysis 1,2,3,4 spanning the first two years, often thaught by one and the same instructor. In such a context one has quite a bit of flexibility at the start and to start with set thoery, Peano axioms and so on is viable. If the students should know some actual analysis after the first term, then of course less so. I would likely not do it myself even in the former conext, but I was taught it like this: I enjoyed it, many others did not. &#xA;And, this level of (naive) set theory feels then really like something of a natural baseline as it is this level on which many mathematicians actually build/operate. (Perhaps this is an over generalisation, and things are also changing in this regard but I think that this is/was a justification for such choices.)      &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are disadvantages.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It takes a lot of time while not being that central/useful. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is not that easy to follow and some students might find it boring or pointless. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Turning an argument in favor around. Things like this will anyway be seen later on (say, in a Toplogy course). [But this is not always the case.]  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;End note:&lt;/em&gt; I never had the possibility so far to teach Analysis &quot;like I want/decide&quot; but in one form or another participated in instructions using each of Dedekind cuts, Cauchy sequences, and an axiomatic approach. Depending on time and students I would go either for Cauchy sequences in detail (lot of time, solid students) or the axiomatic approach (possibly offering some reading material for those interested in 'filling the gap').  &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-03-15T16:15:25.303" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="198" PostTypeId="1" CreationDate="2014-03-15T16:22:25.180" Score="14" ViewCount="232" Body="&lt;p&gt;As Geoff Pointer commented:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] As a composer I've learnt a lot from studying famous composers why wouldn't that also apply to studying maths and mathematicians of note as well? [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Are there any (auto)biographies, studies, or references of mathematicians on &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;how they learn and study, &lt;/li&gt;&#xA;&lt;li&gt;apprehend new maths, &lt;/li&gt;&#xA;&lt;li&gt;or how they had done so while they were students?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the Area 51 page, someone recommended &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1402078595&quot; rel=&quot;noreferrer&quot;&gt;this book&lt;/a&gt;, what are some other examples?&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="42" LastEditDate="2014-03-18T17:19:04.787" LastActivityDate="2020-09-11T21:40:03.143" Title="Books/(auto)biographies/references on how mathematicians study/studied (as students)?" Tags="|reference-request|advising|secondary-education|undergraduate-education|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="199" PostTypeId="2" ParentId="186" CreationDate="2014-03-15T16:23:50.397" Score="27" Body="&lt;p&gt;As far as I know, there are four basic possibilities for defining the real numbers:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The real numbers can be defined &lt;a href=&quot;http://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Synthetic_approach&quot;&gt;using axioms&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The real numbers can be defined using decimal expansions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The real numbers can be defined using &lt;a href=&quot;http://en.wikipedia.org/wiki/Dedekind_cut&quot;&gt;Dedekind cuts&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The real numbers can be defined as equivalence classes of &lt;a href=&quot;http://en.wikipedia.org/wiki/Cauchy_sequence&quot;&gt;Cauchy sequences&lt;/a&gt; of rational numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Approach #1 is probably the easiest, but it is unsatisfying, because you don't actually prove that the real numbers exist.  That is, you don't demonstrate the existence of any number system that satisfies the given axioms.  I think #1 is probably the best if a course is short on time, or if the students aren't mathematically sophisticated enough to benefit from seeing one of the constructive definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Approach #2 is what the students are expecting, but unfortunately the details are quite cumbersome, and the construction is not very natural (since it depends on a choice of base).   It also doesn't give the students much new insight into the nature of the real numbers---it just fills in the details of insight they already have.  In my opinion, the only reason to use #2 is if you are independently interested in talking about decimal expansions, e.g. if most of the students in the class are future high-school teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Approach #3 is quite simple and elegant, and is certainly the easiest of the constructive approaches.  It's wonderfully intuitive, and really cements in the students minds the idea that the real numbers are obtained by &quot;filling in&quot; holes in the rational number line.  The disadvantage of this approach is that it's primarily order-theoretic: students are unlikely to see a construction like this again unless they study order theory, so it might be better to use an approach that's connected to other things the students will be learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Approach #4 is complicated, but has three main advantages (1) Students in a real analysis course should probably learn about Cauchy sequences anyway, so the extra time isn't exactly wasted (2) It starts the course off by identifying sequences as the important thing, a theme that is likely to continue for the rest of the course (3) It is a construction that can be generalized to completions of arbitrary metric spaces, which is something that students will need to understand when they learn point-set topology or advanced analysis.  The disadvantage, of course, is that this is certainly the most complicated of the four options, and some students will be completely mystified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My opinion is that the choice of approach should be based on the audience and the nature of the course.  No matter what, I think it's worth pointing out that the other approaches exist, and perhaps briefly discussing them in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here's what I think:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;For a one-semester course in real analysis, it's probably best to stick with an axiomatic approach.  There are more important topics to cover than constructions of the real numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For a two-semester course in real analysis for typical math majors, it probably makes sense to use either Dedekind cuts or Cauchy sequences.  For average students, I think Cauchy sequences are a bit too difficult, so I would start with Dedekind cuts.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For a two-semester course in real analysis for future mathematicians (i.e. student planning to go to Ph.D. programs), I would recommend covering &lt;em&gt;both&lt;/em&gt; Cauchy sequences and Dedekind cuts, as well as mentioning the other two approaches.  This audience will benefit a lot from the Cauchy sequences approach, but they should also know about other possible approaches, since they may be teaching real analysis themselves one day!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For an analysis course aimed at future high-school teachers, I would think it would be a good idea to discuss the decimal number system a bit at the beginning.  Depending on the length of the course, it might also be helpful to discuss Dedekind cuts and/or the axiomatic approach.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;One final note: if you do use the axiomatic approach, there is still some choice in which version of completeness you include as an axiom.  The tradition seems to be to use the least upper bound property, but I think there is good reason to use either the convergence of Cauchy sequences or the nested closed intervals property instead.  The advantages of using a Cauchy-sequence-based axiom should be obvious, while the advantages of the nested closed intervals property are that it doesn't require any preliminary definitions, it connects well to notions of compactness, and it is perhaps the easiest of the three to understand.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="143" LastEditDate="2014-03-15T21:40:55.673" LastActivityDate="2014-03-15T21:40:55.673" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="200" PostTypeId="1" CreationDate="2014-03-15T16:24:46.173" Score="7" ViewCount="159" Body="&lt;p&gt;Under what situations is it more productive for math students to stop attending lectures and learn the material alternatively? What are some remedies? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I originally posed this as an &lt;a href=&quot;http://area51.stackexchange.com/proposals/64216/mathematics-educators/64623#64623&quot;&gt;Example Question&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2014-03-15T18:55:07.467" Title="When is it more advantaged for students to stop attending lectures? What to do then?" Tags="|advising|" AnswerCount="2" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="201" PostTypeId="2" ParentId="189" CreationDate="2014-03-15T16:38:12.707" Score="8" Body="&lt;p&gt;As per the comments above, I'm going to tackle this explicitly for &lt;strong&gt;discrete math and set theory&lt;/strong&gt; being a shared course.  What follows is some strategies I've tried and how well they worked (or didn't work!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Group Work.&lt;/strong&gt; Asking the students to do group work (either in class or on the homework), and assigning groups can really help this issue.  Imagine that you ensure the groups are mixed between the less experienced and more experienced groups; so, you force the students to get to know others.  A couple things happen when you do this.  The first is that they realize even if they're different skill levels at programming, they share other interests, and you can often foster friendships between the groups which helps a lot.  The second is that if you explicitly reward the experienced students who guide and help the unexperienced ones, and reward the unexperienced students who ask questions, you really do close the gap somewhat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Varied Difficulty on the Homework.&lt;/strong&gt;  You can write homeworks where the students don't have to complete every question.  Instead, they can complete several more rudimentary questions or one difficult one.  This helps the weaker students get better at the basics, and it keeps the stronger ones from getting bored.  If you were really brave, you could try doing something like this on the exams, but I'd suspect you'd end up with most students doing the easy questions to maximize their score.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Extra Credit.&lt;/strong&gt; Extra credit is another way of keeping the interest of the stronger students without helping the weaker ones.  I have often give difficult problems (with no credit whatsoever) and the interested students would come to me to talk about them.  One way I implemented this that I think worked really well was that the &quot;participation&quot; part of your grade could be achieved by coming to office hours to talk about slightly more advanced topics.  I found a mix of very weak and very strong students were all interested.  Either way, you keep everyone's interest (and it gives you a chance to help the weaker students who normally wouldn't come to office hours!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Language Agnostic Teaching.&lt;/strong&gt;  I'm not sure if you do any programming (or use any programming analogies) in your course, but I've found using a sort of pseudocode that is similar (but not identical!) to what the weaker students are learning to be very useful.  They still need to integrate the concepts from the discrete math/set theory course(s), but it's at least somewhat familiar to them.  I avoid explicitly using a language that one group knows, so that from their perspective, it looks like an even playing field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, if it's clear to the students that there's the &quot;advanced&quot; group and the &quot;not-so-advanced&quot; group, it makes it really difficult; so, I do my best to not call &lt;em&gt;explicit&lt;/em&gt; attention to the discrepancy as much as humanly possible.&lt;/p&gt;&#xA;" OwnerUserId="93" LastActivityDate="2014-03-15T16:38:12.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="202" PostTypeId="1" AcceptedAnswerId="217" CreationDate="2014-03-15T16:38:46.050" Score="14" ViewCount="662" Body="&lt;p&gt;Many linear algebra books include two versions of row reduction for solving systems of linear equations:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Reduce to echelon form, and then use back substitution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Reduce to reduced echelon form (i.e. Gauss-Jordan elimination) and then read off the answer directly from the matrix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: what are the pedagogical reasons for including both of these approaches?   In particular, why not just use the Gauss-Jordan approach from the beginning?  Is there some advantage to covering back substitution?&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2018-03-08T08:41:44.763" Title="Why teach back substitution with row reduction?" Tags="|undergraduate-education|mathematical-pedagogy|linear-algebra|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="203" PostTypeId="2" ParentId="196" CreationDate="2014-03-15T16:42:45.777" Score="4" Body="&lt;p&gt;The question is really broad and a bit vague, but some remarks hoping they match the intent.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A key to discovering &lt;em&gt;counter&lt;/em&gt;-examples is to &lt;em&gt;understand&lt;/em&gt; the examples. Taken an example and try to modify to get more examples. See when/if they break. If they do not break think about &lt;em&gt;why&lt;/em&gt; they did not break. Then 'attack' the point that prevented breaking in the next round.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Dually a key to discovering counterexamples is to understand why it is hard/impossible to prove the can &lt;em&gt;not&lt;/em&gt; exist. Try to &lt;em&gt;prove&lt;/em&gt; that a counterexample &lt;em&gt;cannot&lt;/em&gt; exist. Think about why you are unable to prove the nonexistence.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(Depending on context.) Keep an open mind. Perhaps if you really cannot find a counterexample it is because there really is none. This happens. A famous case is &lt;a href=&quot;https://en.wikipedia.org/wiki/Carleson%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carleson's theorem&lt;/a&gt;, he first tried to find  counterexamples to what is now his theorem (then Luzin's conjecture) but eventually got convinced there is none and proved this. I quote a &lt;a href=&quot;https://www.ams.org/notices/200702/comm-carleson.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;part from an interview with him&lt;/a&gt;, on this process (replace notions you might not know by something that makes sense, just to feel the route):&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I was at that time working on Blaschke products, and I said maybe one could use those to produce a counterexample. Zygmund was very positive and said &amp;quot;of course, you should do&#xA;that.&amp;quot; I tried for some years and then I forgot about it before it again came back to me. Then, in the beginning of the 1960s, I suddenly realized that I knew exactly why there had to be a counterexample and how one should construct one. Somehow, the trigonometric system is the type of system where it is easiest to provide counterexamples. Then I could prove that my approach was impossible. I found out that this idea would never work; I mean that it couldn’t work. If there were a counterexample for the trigonometric system, it would be an exception to the rule. Then I decided that maybe no one had really tried to prove the converse. From then on it only took two years or so. But it is an interesting example of &amp;quot;to prove something hard, it is extremely important to be convinced of what is right and what is wrong.&amp;quot; You could never do it by alternating between the one and the other because the conviction somehow has to be there.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="143" LastEditorDisplayName="user7990" LastEditDate="2020-09-07T20:48:27.060" LastActivityDate="2020-09-07T20:48:27.060" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="204" PostTypeId="2" ParentId="198" CreationDate="2014-03-15T16:52:27.847" Score="7" Body="&lt;p&gt;Here are two from famous mathematicians who have tried to explain how they approach mathematics:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/4871878309&quot; rel=&quot;noreferrer&quot;&gt;George Pólya&lt;/a&gt;.  This book is actually targeted at introductory students.  It has great examples, and an explanation of a method to approach them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://global.oup.com/academic/product/solving-mathematical-problems-9780199205608;jsessionid=B75ABCB48D260D2FD91A7A831BFD6C0F?cc=us&amp;amp;lang=en&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;Terence Tao&lt;/a&gt; is one of the best problem solvers of our time.  He explains what I believe you're looking for in great detail.&lt;/p&gt;&#xA;" OwnerUserId="93" LastEditorUserId="171" LastEditDate="2014-03-17T03:14:42.657" LastActivityDate="2014-03-17T03:14:42.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="205" PostTypeId="2" ParentId="200" CreationDate="2014-03-15T17:15:04.863" Score="2" Body="&lt;p&gt;There are a variety of lecture styles that often result in students, for either valid or totally invalid reasons, stopping coming to lecture.  Here's a quick outline of a few:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Many lectures consist of worked examples from the book.  These can be helpful for students who are confused, but, often times, students convince themselves that they can just read the book.  For some of them, this works out, but for others, they are completely unable to recover.  A nice, easy way of dealing with this is to survey the class on how they feel about this type of lecture.  If they all say &quot;they don't find it useful&quot;, that doesn't mean you shouldn't do it---just that you might consider mixing it up more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Another type of lecture in a proofs-based class is to just write up solutions to proofs.  This can be really tedious for the students--and worse, it can make them feel like they're learning when they're actually just mechanically writing down what's on the board.  Since they believe the &quot;value&quot; of these lectures is the notes, students will often ask their friends to write notes for them and just copy them (or not!) later.  This can be even worse when instructors give out typeset notes to the students.  One way of mitigating this is to intersperse the proofs themselves with other content--like explanations of how and why things are the way they are.  I've also found that giving students notes that they can follow through the lecture and explicitly downplaying the importance of the content of the proof itself helps students realize that just copying someone else's notes isn't particularly helpful.&lt;/p&gt;&#xA;" OwnerUserId="93" LastActivityDate="2014-03-15T17:15:04.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="207" PostTypeId="2" ParentId="194" CreationDate="2014-03-15T18:10:28.067" Score="6" Body="&lt;p&gt;It seems that the first step is to diagnose the problem. What, specifically, causes the fear or anxiety? Getting a history of the student's mathematical experience is a reasonable start. Helpful questions include courses the student has taken, what the student liked best or disliked most, what parts were easiest or hardest, and so forth. This often identifies specific concerns that can be remedied, which usually works better than vague, general reassurances. An organized, systematic approach to assessment usually works better than an intuitive, haphazard one. &lt;/p&gt;&#xA;" OwnerUserId="40" LastEditorUserId="40" LastEditDate="2014-03-16T02:01:30.837" LastActivityDate="2014-03-16T02:01:30.837" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="209" PostTypeId="2" ParentId="175" CreationDate="2014-03-15T18:25:34.227" Score="6" Body="&lt;p&gt;Mathematics textbooks are notoriously difficult to read. I've become convinced that mathematics is at least in part a language, usually a foreign language, and mathematical notation is an ideographic writing system. For students unfamiliar with the language, most mathematical textbooks might as well be written in Chinese. It is usually not sufficient to read a textbook: one must translate it.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-15T18:25:34.227" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="210" PostTypeId="2" ParentId="200" CreationDate="2014-03-15T18:28:09.170" Score="5" Body="&lt;p&gt;Class time is a designated time for students to sit and focus on the material of the course. From experience advising college students, here are the most valid rationalizations I've heard for skipping all classes:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The professor just reads from the book / copies down things from the&#xA;book. &lt;/li&gt;&#xA;&lt;li&gt;The class moves [too slowly / too quickly] and as a result I don't get anything from the&#xA;class. &lt;/li&gt;&#xA;&lt;li&gt;The professor disrespects the students and I do not feel&#xA;comfortable in the class.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These are all complaints about &lt;em&gt;how class time is being used&lt;/em&gt; -- not objections about spending time itself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, most students who use one of these reasons &lt;strong&gt;stop attending class but then do not replace it with anything similar&lt;/strong&gt;. Students who stop attending class still must have some minimum amount of time spent sitting and focusing on the material of the course if they hope to succeed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can imagine replacing all lectures with a weekly session in a college's tutoring center -- a weekly study group that meets in reach of a tutor who is familiar with the material. This feels to me like the easiest way to replace lecture time in a constructive way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, I would say that a necessary but not sufficient criterion for an answer to this question is: &lt;strong&gt;the solution must involve multiple hours per week of focused struggle with the material&lt;/strong&gt;, in addition to any assigned homework.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-15T18:28:09.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="211" PostTypeId="2" ParentId="198" CreationDate="2014-03-15T18:39:24.590" Score="7" Body="&lt;p&gt;&lt;a href=&quot;http://www.math.hawaii.edu/~lee/how-to.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Does One Do Mathematical Research?&lt;/a&gt; (Or Maybe How Not To), by Lee Lady&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://math.slu.edu/~srivastava/Halmos.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics as a creative art&lt;/a&gt;, by Paul Halmos&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387960783&quot; rel=&quot;nofollow noreferrer&quot;&gt;I Want to Be a Mathematician: an Automathography&lt;/a&gt;, by Paul Halmos&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, &lt;a href=&quot;https://math.stackexchange.com/questions/301763/what-is-mathematical-research-like&quot;&gt;this MSE thread&lt;/a&gt; (&quot;what is mathematical research like?&quot;) might be of interest.&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-03-15T18:39:24.590" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="212" PostTypeId="2" ParentId="202" CreationDate="2014-03-15T18:40:37.670" Score="-1" Body="&lt;p&gt;If both approaches are known, then one can be used as a check on the other. If the answers disagree, there has been a mistake somewhere. &lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-15T18:40:37.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="213" PostTypeId="2" ParentId="118" CreationDate="2014-03-15T18:44:20.367" Score="4" Body="&lt;p&gt;So far one important thing missing from the answers here has been some kind of mathematical &quot;play.&quot; By play I mean tinkering and struggling with problems to advance mathematical maturity, rather than learning any particular concept or subject area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe at least one of these must be core to an undergraduate math student's plan:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A research group&lt;/strong&gt; course, where students split into groups and are tasked with open-ended projects like &quot;The repeating decimal digits of 1/7, 2/7, 3/7, ... are all the same digits but in different orders. What's the deal with that? What other examples of this phenomenon can you find? Why does this happen?&quot; They don't need to discover &lt;em&gt;new&lt;/em&gt; mathematics to experience discovery in mathematics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A problem seminar&lt;/strong&gt; in which small groups of students deal with traditional math-contest problems: problems that students are not expected to &quot;already know how&quot; to solve. Math students should not get in the habit of seeing a problem and always expecting to immediately know the first step to take! &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A project course&lt;/strong&gt; in mathematics: mine was a mathematical modeling in biology course, where a physical situation is given and the students attempt to build a mathematical model, then critique it and talk about the model's strengths and weaknesses.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;An outreach group&lt;/strong&gt; that runs a math club or math events for younger students. You can do a surprising amount of mathematics with young children -- Euler Characteristic, Nim, Modular Arithmetic, etc are all easaily understood and discoverable by ten-year-olds, and helping with this process of discovery can be excellent experience.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Mathematics problems in the &quot;real world&quot; are often open-ended, and the open-ended problems and questions are the most interesting and inspiring anyway. &lt;strong&gt;It was very important for my mathematical development to devote some time each semester to play with mathematics&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-15T18:44:20.367" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="215" PostTypeId="2" ParentId="175" CreationDate="2014-03-15T18:56:34.777" Score="9" Body="&lt;p&gt;I have tried using &quot;pre-class quizzes.&quot; In my opinion, it was not a success.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea of the preclass quiz is to assign reading for every class period, then set up one easy question from the section that must be completed online. The question must be answered less than 24 hours before the start of the class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal was to make sure students had to interact with the textbook before each class period, in the hopes that they would gain something from it while it was open. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, &lt;strong&gt;every policy is an incentive system&lt;/strong&gt;, so students did exactly what they were incentivized to do -- they opened the textbook the night before class, skimmed it until they found the answer, and then closed the textbook. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Attempts to make the questions more difficult in order to force the students to actually read instead of skim were ineffective -- the way this policy shift sounded to the student was &quot;you have to learn this material on your own before each class.&quot; This alienated the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I look forward to hearing answers that include success stories, but negative results can also be useful.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-15T18:56:34.777" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="216" PostTypeId="2" ParentId="198" CreationDate="2014-03-15T18:59:59.540" Score="5" Body="&lt;p&gt;G. H. Hardy's &lt;a href=&quot;http://en.wikipedia.org/wiki/A_Mathematician%27s_Apology&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;A Mathematician's Apology&lt;/em&gt;&lt;/a&gt; is a nice read. &lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="171" LastEditDate="2014-03-17T00:55:08.957" LastActivityDate="2014-03-17T00:55:08.957" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="217" PostTypeId="2" ParentId="202" CreationDate="2014-03-15T19:39:07.667" Score="15" Body="&lt;p&gt;One reason to know back substitution is that it is relevant when doing numerical mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A standard procedure to numerically solve linear systems $Ax =b$ especially if one wants to solve for more then one $b$ (which is very common) is to perform an &lt;a href=&quot;http://en.wikipedia.org/wiki/LU_decomposition&quot; rel=&quot;nofollow noreferrer&quot;&gt;$LU$-decomposition&lt;/a&gt; of $A$ (or something similar like a &lt;a href=&quot;http://en.wikipedia.org/wiki/Cholesky_decomposition&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cholesky decomposition&lt;/a&gt;), that is one decomposes $A$ as a product of a lower triangular matrix $L$ and an upper triangualar matrix $U$ (sometimes one needs permutation matrices in addition but let us ignore this detail). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, one solves the two triangular systems $Ly=b$ and $Ux=y$, &lt;em&gt;via back substitution,&lt;/em&gt; which is very 'cheap' relative to solving a general linear system ($O(n^2)$ vs $O(n^3)$). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One might now ask why not solve the triangular systems via Gauss--Jordan elimination, and I do not fully oversee momentarily if there is even a difference depending on how one sets things up precisely. Still the two things, reducing to triangular systems and solving the triangular systems, are &lt;em&gt;conceptually&lt;/em&gt; quite different, and that this difference is stressed when one uses back substitution could be considered as an advantage.   &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="376" LastEditDate="2018-03-08T08:41:44.763" LastActivityDate="2018-03-08T08:41:44.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="218" PostTypeId="2" ParentId="194" CreationDate="2014-03-15T19:51:17.347" Score="19" Body="&lt;p&gt;There are a few strategies that are supported by experimental research which I will share here, but they all have to do with &lt;strong&gt;stereotype threat&lt;/strong&gt;.  I am sure there are other types of anxiety related to math which would not be helped by these strategies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, the &lt;a href=&quot;http://en.wikipedia.org/wiki/Stereotype_threat&quot;&gt;wikipedia article&lt;/a&gt; on stereotype threat is fairly comprehensive.  It describes some studies that have been done, as well as some strategies to combat stereotype threat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;I am &quot;not a math person&quot;.&lt;/strong&gt;  Students who believe that there are &quot;math people&quot; and &quot;people who aren't good at math&quot; feel that there is no way to grow.  This is much more likely to effect a student who is undergoing stereotype threat.  Students who are told that intelligence is &quot;malleable&quot; &lt;a href=&quot;http://ac.els-cdn.com/S0193397303001126/1-s2.0-S0193397303001126-main.pdf?_tid=c1e71ef4-ac7a-11e3-b970-00000aab0f01&amp;amp;acdnat=1394913079_ec13a21ef4c82db26d76d9110bedd537&quot;&gt;perform much better&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Awareness of stereotype threat.&lt;/strong&gt;  Making students aware of stereotype threat (that is, explicitly telling them &quot;your race/gender/finances makes you more likely to do badly&quot;) can &lt;strong&gt;help&lt;/strong&gt; them on easy tests while simultaneously &lt;strong&gt;hurting&lt;/strong&gt; them on harder tests.  There are many studies that support this, but &lt;a href=&quot;http://psp.sagepub.com/content/29/6/782&quot;&gt;here is one&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Self-affirmation.&lt;/strong&gt; Self-affirmation (that is, affirming a value important to the individual--not necessarily related to math) can &lt;a href=&quot;http://ac.els-cdn.com/S0022103105000545/1-s2.0-S0022103105000545-main.pdf?_tid=3fb88986-ac7a-11e3-a6eb-00000aacb35d&amp;amp;acdnat=1394912860_8de0d13d7ac53fc0b18aae56225a6a9d&quot;&gt;significantly increase performance&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Role Models.&lt;/strong&gt; Role models who are in the minority that students respect can &lt;a href=&quot;http://gpi.sagepub.com/content/14/4/447&quot;&gt;significantly decrease stereotype threat&lt;/a&gt;.  This study also points out that students who did not believe the role model deserved (in this case) her success were not helped (but also not hurt).&lt;/p&gt;&#xA;" OwnerUserId="93" LastEditorUserId="93" LastEditDate="2014-03-15T19:56:41.323" LastActivityDate="2014-03-15T19:56:41.323" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="220" PostTypeId="1" CreationDate="2014-03-15T20:08:16.563" Score="43" ViewCount="11814" Body="&lt;p&gt;Many examples of induction are silly, in that there are more natural methods available. Could you please post examples of induction, where it is required, and which are simple enough as examples in a course on proofs (or which includes proofs, e.g. a first course on discrete mathematics)?&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="38" LastEditDate="2015-12-08T04:16:56.760" LastActivityDate="2023-03-18T07:49:19.947" Title="Good, simple examples of induction?" Tags="|mathematical-pedagogy|proofs|examples|induction|" AnswerCount="28" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="222" PostTypeId="2" ParentId="202" CreationDate="2014-03-15T20:39:32.777" Score="11" Body="&lt;p&gt;When I teach the $L U$-decomposition, mentioned in quid's answer, I use the following mnemonic:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Single $\vec{b}$?  Use GE.&lt;br&gt;&#xA;  Several to do?  Use LU.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The point being that if you need to solve $A \vec{x} = \vec{b}$ for several $\vec{b}$ then the $L U$-method is more efficient than setting up a full Gaussian Elimination for each individual $\vec{b}$.  In terms of the algorithm, the &lt;em&gt;forward&lt;/em&gt; bit is the same for any $\vec{b}$ but the &lt;em&gt;back&lt;/em&gt; part depends on $\vec{b}$.  Yes, you can do the back part with a generic $\vec{b}$ and then substitute in for your actual $\vec{b}$, but that doesn't actually save you many computation steps and may cost you some depending on the actual values you're computing with.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-03-15T20:39:32.777" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="223" PostTypeId="2" ParentId="178" CreationDate="2014-03-15T20:44:42.563" Score="8" Body="&lt;p&gt;I think Terry Tao's analysis, measure theory and real analysis books (numbers 4,10, and 11 on &lt;a href=&quot;http://terrytao.wordpress.com/books/&quot;&gt;this list&lt;/a&gt;) are excellent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With special emphasis on his Analysis 1-2 books, they can be used in an excellent way to teach advanced undergraduates.&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-03-15T20:44:42.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="224" PostTypeId="1" AcceptedAnswerId="2719" CreationDate="2014-03-15T21:04:00.767" Score="29" ViewCount="2183" Body="&lt;p&gt;I am thinking about teaching a university-level &quot;introduction to proofs&quot; class (mainly for math and CS majors) making use of a computer proof assistant like Coq.  I feel like there is a lot of potential benefit here, in that the students can get immediate feedback about what works and what doesn't when interacting with the proof assistant, thereby learning about the &quot;world of mathematics&quot; in a similar way to how a gamer learns about the world of a video game by interacting with it.  However, most existing proof assistants are not really targeted at this level, so I worry that the difficulty of learning to use them could outweigh the advantages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What experiences and research have there been with this sort of thing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I am &lt;em&gt;not&lt;/em&gt; asking about using a proof assistant in the teaching of a class about &lt;em&gt;formal logic&lt;/em&gt;.  I know that this has been done, and experiences and research on it is not totally irrelevant to what I'm asking, but the audience is very different from a class like I'm considering; students usually already have much more mathematical maturity, and an existing understanding of what a &quot;proof&quot; is, coming into a formal logic class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; For further clarification, while I'm certainly interested to hear opinions and suggestions, what I'm really asking about is, as I said, &lt;em&gt;experiences&lt;/em&gt; and &lt;em&gt;research&lt;/em&gt;.  In other words, have you or someone else &lt;em&gt;done&lt;/em&gt; this?&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="143" LastEditDate="2014-03-19T21:34:25.010" LastActivityDate="2020-09-14T12:21:53.203" Title="Teaching logic with a proof assistant" Tags="|proofs|logic|technology-in-education|" AnswerCount="7" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="225" PostTypeId="1" CreationDate="2014-03-15T21:06:41.963" Score="5" ViewCount="424" Body="&lt;p&gt;I have discovered that current mathematics education often teaches students to emphasize all polynomial equations should have complex solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But starting from non-polynomial equations (e.g. $\sin x=x$), even in fact there are infinitely many complex solutions, the current mathematics education often teaches students to ignore them, even for the occasions on just needing the analyses of the number of the solutions. These attitudes make me upset as I for example feel that they have forgotten most transcendental functions are also defined in $\mathbb{C}$ and they are not respect the world of complex numbers.&lt;/p&gt;&#xA;" OwnerUserId="152" LastEditorUserId="152" LastEditDate="2014-03-15T22:50:03.820" LastActivityDate="2014-03-16T15:25:28.937" Title="Why does current mathematics education often ignore the analyses of complex number solutions on (systems) of non-polynomial equations?" Tags="|course-design|textbooks|secondary-education|undergraduate-education|algebra|" AnswerCount="2" CommentCount="4" ClosedDate="2014-03-16T22:26:51.767" ContentLicense="CC BY-SA 3.0" />
  <row Id="226" PostTypeId="5" CreationDate="2014-03-15T21:16:03.637" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-15T21:16:03.637" LastActivityDate="2014-03-15T21:16:03.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="227" PostTypeId="4" CreationDate="2014-03-15T21:16:03.637" Score="0" Body="Questions about comparison of textbooks, the use of textbooks in mathematical education, requests about textbooks dealing with a specific topic in a specific way." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T12:37:51.647" LastActivityDate="2014-03-31T12:37:51.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="228" PostTypeId="1" AcceptedAnswerId="239" CreationDate="2014-03-15T21:24:42.513" Score="17" ViewCount="723" Body="&lt;p&gt;I've found one of the hardest topics to introduce to students early on is abstract algebra.  Even if they've already written proofs, it's hard for them to work directly from axioms. They seem to have problems with understanding why we view sets and operations abstractly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, it can often be difficult to motivate why these topics are useful.&#xA;One reasonable motivation that I've seen is RSA, but this generally compounds a tenuous grasp of number theory with a shaky understanding of group theory.  This boils down to giving students who lack a mathematical background a reason to care about the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What other strategies and motivations have people tried to (1) motivate how and why we think abstractly, and (2) why abstract algebra is useful in practice?&lt;/p&gt;&#xA;" OwnerUserId="93" LastEditorUserId="114" LastEditDate="2014-04-11T11:42:54.927" LastActivityDate="2020-09-09T14:49:16.007" Title="What are some good ways to motivate and introduce reasoning abstractly about abstract algebra?" Tags="|undergraduate-education|examples|abstract-algebra|concept-motivation|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="229" PostTypeId="5" CreationDate="2014-03-15T21:38:43.627" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-15T21:38:43.627" LastActivityDate="2014-03-15T21:38:43.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="230" PostTypeId="4" CreationDate="2014-03-15T21:38:43.627" Score="0" Body="For questions about teaching college students or college courses." OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-03-17T14:07:11.603" LastActivityDate="2014-03-17T14:07:11.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="231" PostTypeId="2" ParentId="228" CreationDate="2014-03-15T21:46:09.637" Score="6" Body="&lt;p&gt;In general, issues with abstraction and axioms can boil down to losing the ability to &lt;em&gt;visualize&lt;/em&gt; what's going on. I'd suggest appealing to visual/tactile intuition as much as possible, especially in the beginning of the course. Present motivating examples of groups and other fundamental concepts that students can (quite literally) play with.&lt;/p&gt;&#xA;&lt;p&gt;For instance, talk about the Rubik's Cube! Or symmetry groups of simple polygons, and fractals. Act out permutation groups in small (ahem) groups.&lt;/p&gt;&#xA;&lt;p&gt;Also did some googling, and this looks interesting: &lt;a href=&quot;https://faculty.salisbury.edu/%7Edespickler/pascgalois/publications/Focus/Focus.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;PascGalois Project&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorDisplayName="user7990" LastEditDate="2020-09-09T14:49:16.007" LastActivityDate="2020-09-09T14:49:16.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="232" PostTypeId="2" ParentId="178" CreationDate="2014-03-15T21:54:40.157" Score="8" Body="&lt;p&gt;I really like &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0120839717&quot;&gt;A Primer of Lebesgue Integration&lt;/a&gt;&lt;/em&gt; by H.S. Bear.  The book is short and very readable, and it introduces Lebesgue integration on the real line in a very understandable way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I think that it is much better to introduce measure theory and Lebesgue integration in the specific context of the real line and $\mathbb{R}^n$, perhaps moving on to general measure spaces after this is done.  As with many things in mathematics, the most general case is very hard to understand if you haven't yet seen a motivating example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also really like Bressoud's &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521711835&quot;&gt;A Radical Approach to Lebesgue's Theory of Integration&lt;/a&gt;&lt;/em&gt;, which includes a generous helping of motivation and historical context.  Personally, I prefer to use Bear's (shorter) book as the textbook, and to use Bressoud's book to help prepare my lectures.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-15T21:54:40.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="233" PostTypeId="2" ParentId="220" CreationDate="2014-03-15T21:57:08.160" Score="21" Body="&lt;ul&gt;&#xA;&lt;li&gt;Proving DeMorgan's Laws for $n$ sets. I like this example because it&#xA;requires the $n=2$ case in the induction step.&lt;/li&gt;&#xA;&lt;li&gt;It's common to have students prove that $\sum_{k=1}^n k^3 =&#xA;   \left(\sum_{k=1}^n k\right)^2$. A great follow up is to assume you&#xA;have a sequence $\langle a_k\rangle$ that satisfies $\sum_{k=1}^n&#xA;   a_k^3 = \left(\sum_{k=1}^n a_k\right)^2$ and prove, by induction on&#xA;$n$, that necessarily $a_k=k$ for all $k$.&lt;/li&gt;&#xA;&lt;li&gt;Proving the Well-Ordering Principle of $\mathbb{N}$ by induction on whether the set in question has $n$ as an element. I designed a scaffolded version of this as a homework exercise, and it also has the students think about why induction on the &lt;em&gt;size&lt;/em&gt; of the set in question does not prove the claim!&lt;/li&gt;&#xA;&lt;li&gt;Have not personally used this, but perhaps &lt;a href=&quot;http://www.proofwiki.org/wiki/Cauchy%27s_Mean_Theorem&quot;&gt;Cauchy's &quot;backward induction&quot; proof&lt;/a&gt; of the AGM inequality is of interest.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-15T21:57:08.160" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="235" PostTypeId="2" ParentId="228" CreationDate="2014-03-15T22:04:28.703" Score="3" Body="&lt;p&gt;I was introduced to abstract algebra through it's application to number theory, using Richman's &quot;Number Theory: An Introduction to Algebra&quot; (Brooks/Cole, 1971) as a textbook. Sadly, the book is long out of print (already was in 1984). Having &quot;concrete&quot; examples of rings, groups, ... was essential.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-15T22:04:28.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="236" PostTypeId="2" ParentId="225" CreationDate="2014-03-15T22:31:22.017" Score="5" Body="&lt;p&gt;I think the answer to your question is that techniques for solving transcendental equations for complex variables are fairly complicated (with the exception, perhaps, of &lt;a href=&quot;http://en.wikipedia.org/wiki/Newton%27s_method&quot; rel=&quot;noreferrer&quot;&gt;Newton's method&lt;/a&gt;), so they tend not to be taught in elementary courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, analogs of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Bisection_method&quot; rel=&quot;noreferrer&quot;&gt;bisection method&lt;/a&gt; for functions of a complex variable include the &lt;a href=&quot;http://en.wikipedia.org/wiki/Lehmer%E2%80%93Schur_algorithm&quot; rel=&quot;noreferrer&quot;&gt;Lehmer-Schur algorithm&lt;/a&gt; and Wilf's algorithm, which uses the &lt;a href=&quot;http://en.wikipedia.org/wiki/Argument_principle&quot; rel=&quot;noreferrer&quot;&gt;argument principle&lt;/a&gt; from complex analysis.  This &lt;em&gt;might&lt;/em&gt; be mentioned in a very good undergraduate complex analysis course, but otherwise it probably doesn't come up until graduate courses on complex analysis or numerical methods.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-15T22:31:22.017" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="237" PostTypeId="2" ParentId="220" CreationDate="2014-03-15T22:49:18.293" Score="13" Body="&lt;p&gt;Induction receives far too much attention as a proof technique. But, if you are to give it attention, it is probably worth-while going through the equivalence of the principle of induction and the well-ordering of the naturals. In fact, when I teach induction I proceed as follows: Suppose an inductive property does not hold for all naturals. Look at the set of counter-examples, and pick the smallest one. Arrive at a contradiction and conclude the property did hold for all naturals. I find that this approach is very natural (pun not intended). The equivalent formalism of prove case $k=1$, then prove $P(k)\implies P(k+1)$ is something I only introduce later. For the life of me I don't understand why this formalism is the standard one and not the &quot;look at all counter-example....&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is probably also a good idea to talk about the status of the principle of induction. It is not a theorem but rather a proof principle. It is ok not to accept it (though most mathematicians do). It is important to understand that what we accept by proof by induction is that a single recipe for proofs (i.e., for each $k$ we have a recipe for how to concoct a proof for $P(k)$) is accepted as a single proof for an infinitude of statement (i.e., all the $P(k)$ for all natural $k$). It is a good idea to talk about this in relation to the demand that proofs are finite. It is also a good idea to explain this in the context of the axiom of choice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is indeed quite hard to find good examples of proof by induction (which is part of the reason why I claimed that induction receives far too much attention). Other than the classical (more or less silly) exercises, some that actually do require induction include: the generalized associativity rule and other generalized algebraic rules but also proving properties of recursively defined sequences (e.g., monotonicity, boundedness etc.). &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-15T22:49:18.293" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="238" PostTypeId="1" CreationDate="2014-03-15T22:56:08.850" Score="13" ViewCount="1137" Body="&lt;p&gt;Starting from a young age, children can explore deep mathematical questions and enjoy thinking about basic math within the context of a story. There are some real treasures out there. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parents often think of math as addition and multiplication facts - dry stuff. But it is so much more, even at a basic level. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're thinking of books for teens or adults, post them as answers for &lt;a href=&quot;https://matheducators.stackexchange.com/questions/294/what-are-some-great-books-for-exploring-mathematics-not-kids-books-and-not-te&quot;&gt;this parallel question&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With your answer, please explain what makes this book appealing and what its math content is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are some great books for inspiring children to explore mathematics?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2020-09-11T21:40:17.467" Title="What are some great books for inspiring children to explore mathematics?" Tags="|reference-request|reference-books|recreational-learning|" AnswerCount="10" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="239" PostTypeId="2" ParentId="228" CreationDate="2014-03-15T22:58:57.557" Score="14" Body="&lt;p&gt;When I introduce groups I first go over a very (e.g., 15 examples) long list of particular groups. I go over each example and verify the group axioms without naming it such. Then I present the problem of studying all of these examples together, motivated by simply saving labour. Instead of proving things again and again for each case, we'd like a single uniform formalism in which to study all of them. From there the road to the definition of abstract group is quite short. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A bit further into the course, when we get to polynomials, I explain that polynomials are an excellent algebraic way to encode combinatorics and therefore it is important in lots of places (since counting is fundamental). I present the students with the question &quot;Can you design two (biased) six-sided dice such that the probability for each sum between $2$ and $12$ is equal&quot;. The negative answer can be obtained by considering the generating polynomials. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, if such theorems as the Burnside Lemma in group theory is covered, then lots of real-life applications can be given. I also like to very briefly mention some of the applications of group actions in chemistry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also find that many students try to visualize what an abstract group is. This is impossible of course, and for students who never took any abstract course, and are so used to just computing or drawing graphs, this is a big psychological hurdle. I strongly advise, several times, not to even try to visualize. Rather, they need to understand that they need to develop a feel for the axioms and the proofs, of course, through practice. But don't visualize!&lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-15T22:58:57.557" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="240" PostTypeId="2" ParentId="238" CreationDate="2014-03-15T23:22:27.500" Score="9" Body="&lt;p&gt;I have about twenty kids' books listed at &lt;a href=&quot;http://mathmamawrites.blogspot.com/p/math-books.html&quot;&gt;the books page on my blog, Math Mama Writes&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I had to narrow it down, my absolute favorite is probably &lt;em&gt;&lt;a href=&quot;http://www.betterworldbooks.com/the-cat-in-numberland-id-9780812627442.aspx&quot;&gt;The Cat in Numberland&lt;/a&gt;&lt;/em&gt;, by Ivar Ekeland, a five-chapter picture book dealing with the story of the Hotel Infinity (with Mr. and Mrs. Hilbert running the hotel, of course). The confused cat is charming, Ekeland's retelling of the Hotel Infinity story is delightful and thought-provoking, and the ideas can easily be extended. (If all the negative numbers showed up, how would the Hotel Infinity deal with them?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other favorites:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/068983389X&quot;&gt;How Hungry Are You?&lt;/a&gt;&lt;/em&gt; by Donna Jo Napoli, on equal sharing, among more and more animal friends who are going on a picnic together. The little icons for who is talking make me want to do this as a play.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0716714094&quot;&gt;Powers of Ten&lt;/a&gt;&lt;/em&gt;, by Philip and Phylis Morrison, with a photo and a descriptive page for each power of ten (for size/distance). The &lt;a href=&quot;https://www.youtube.com/watch?v=0fKBhvDjuy0&quot;&gt;youtube video&lt;/a&gt; may be more dramatic than the book, but it's a great book to keep out on the coffee table.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;http://www.moebiusnoodles.com/TheBook&quot;&gt;Moebius Noodles&lt;/a&gt;&lt;/em&gt;, by Yelena McManaman and Maria Droujkova, for exploring symmetry, functions, grids, and numbers, will help parents expand their notions of what math is. [Disclaimer: Maria Droujkova will be publishing my book.]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-15T23:22:27.500" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="241" PostTypeId="2" ParentId="238" CreationDate="2014-03-15T23:43:20.763" Score="12" Body="&lt;p&gt;I am a very big fan of &lt;a href=&quot;http://en.wikipedia.org/wiki/The_Number_Devil&quot;&gt;The Number Devil&lt;/a&gt; written by a famous German author (normally known for his books about history or politics). The subtitle in the original German version is &quot;Ein Kopfkissenbuch für alle, die Angst vor der Mathematik haben.&quot; (=A pillow book for all fearing mathematic.&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a (nicele illustrated) story about a boy having fear of mathematics in general and fear of his math teacher. He dreams every night of the so-called number devil who will teach the boy the beauty of math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The very special thing about the book is the fact that mathematical terms are not called how we would call it, but with special colorful words (e.g., the number devil explains the boy why multiplication is important and how it is derived from addition. Then, the number devil does an analogy of it to introduce factorial - But the book calls it &lt;em&gt;&quot;Wumm!&quot;&lt;/em&gt;. So the  boy is teached to perform e.g. &lt;em&gt;&quot;6 Wumm!&quot;&lt;/em&gt;. Taking roots are called taking rutabaga, etc.). One can discuss if this is good or bad, but at least  this doesn't sound like mathematics known to students in that age and is maybe motivating students to have interest in mathematics. (There is also a table at the end of the book translating the new terms to known terms for us. This table can also be helpful to students who might say: &quot;Okay, I understood the concepts of the book...Wait... This rutabaga thing was the complicated root thing our teacher explained a few weeks ago?&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More or less, every night (I think, there are about 10?) is one &quot;lecture&quot;. The mathematical content is about mathematics teached to students in the age of 10-14 maybe (depends on countries, etc.), but also has some hints to college related questions: Firstly, the book introduces basic operations like factorial, roots, etc and also numbers systems like natural numbers, rational numbers, etc. (and even a small adventure to complex numbers). Then the book deals a lot with pascal's triangle, Fibonacci  numbers and related content. At the end, there is small journey into more advances subjects like topology, the traveling salesman problem, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hightly recommend the reading of the books - even (or in particular!) for adults.&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-15T23:43:20.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="242" PostTypeId="2" ParentId="220" CreationDate="2014-03-16T00:55:03.070" Score="18" Body="&lt;p&gt;One example that I recently came across was to prove that the function&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x)=\begin{cases}&#xA;e^{-1/x},&amp;amp;x&amp;gt;0\\&#xA;0,&amp;amp;x\leq0&#xA;\end{cases}&#xA;$$&lt;/span&gt;&#xA;is smooth (i.e. &lt;span class=&quot;math-container&quot;&gt;$f\in\mathcal{C}^\infty$&lt;/span&gt;). &lt;a href=&quot;https://web.archive.org/web/20111230130520/http://andromeda.rutgers.edu/%7Eloftin/difffal03/bump.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; is a link to a proof online.&lt;/p&gt;&#xA;" OwnerUserId="71" LastEditorDisplayName="user7990" LastEditDate="2020-09-09T13:09:38.067" LastActivityDate="2020-09-09T13:09:38.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="243" PostTypeId="1" CreationDate="2014-03-16T01:23:58.817" Score="13" ViewCount="426" Body="&lt;p&gt;In particular, if you use Cantor's diagonalization argument, do you ignore the repeating decimal annoyance?  Or prove that it's not a problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there another clean way that gives students intuition on the uncountability of the reals?&lt;/p&gt;&#xA;" OwnerUserId="93" LastEditorUserId="114" LastEditDate="2014-03-30T09:46:17.270" LastActivityDate="2014-03-30T18:39:09.733" Title="What different ways do people use to show students that $\mathbb{R}$ is uncountable?" Tags="|proofs|real-numbers|" AnswerCount="7" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="244" PostTypeId="1" AcceptedAnswerId="373" CreationDate="2014-03-16T01:53:37.543" Score="21" ViewCount="1129" Body="&lt;p&gt;In the United States, secondary education students generally progress through pre-algebra courses, then algebra, Euclidean geometry, more algebra/trigonometry, then calculus or statistics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am particularly interested in the place that geometry holds in this sequence. When did this become the standard way to teach in the United States?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="77" LastEditDate="2020-01-01T05:34:53.713" LastActivityDate="2023-01-17T07:36:54.070" Title="When did the American school system's progression of math classes take its current form?" Tags="|secondary-education|usa|history|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="245" PostTypeId="2" ParentId="178" CreationDate="2014-03-16T02:14:07.047" Score="10" Body="&lt;p&gt;Frank Jones' Lebesgue Integration in Euclidean Space is possibly the best math book I have ever read. It goes over the Lebesgue integral in small steps over many sections before moving on to applications and properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is appropriate for advanced undergrads and the exercises are excellent (I went through the book over the summer after my class ended and did every exercise because they helped me learn so well).&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-16T02:14:07.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="246" PostTypeId="2" ParentId="82" CreationDate="2014-03-16T02:23:37.040" Score="7" Body="&lt;p&gt;One thing that I think can be made intuitive is that averaging several samples of a distribution gives a new distribution that is 'nicer'. You can explain that taking the mean averages things out, and removes any weird assymetries, 'lumps' or 'corners' in the distribution (you can illustrate this by plotting the distribution of the mean of a random variable with increasing sample size.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, the means of all random variables will converge to the distribution that is most smooth and most symmetrical. At this point, you claim that the normal distribution is the smoothest (which is one interpretation of the fact that it is fixed by fourier transforms).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This discussion ignores the fact that one must divide by a factor to keep the distributions normalized. One could just pick a distribution and show pictures of the mean distributions for a few values of $n$, and show how they look more like each other after rescaling by an appropriate factr.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-04-08T11:23:36.403" LastActivityDate="2014-04-08T11:23:36.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="247" PostTypeId="1" AcceptedAnswerId="256" CreationDate="2014-03-16T03:06:42.930" Score="23" ViewCount="2043" Body="&lt;p&gt;I will be a postdoc in the fall and will be teaching my very first classes aimed at graduate students. One will be an intro class, and the other a topics class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are of course many differences between undergraduate and graduate courses. I'm interested in knowing how to take these into account when planning for and teaching a graduate -level course. &lt;/p&gt;&#xA;" OwnerUserId="126" LastEditorUserId="114" LastEditDate="2014-03-28T18:44:23.450" LastActivityDate="2016-06-25T01:23:23.330" Title="What are the differences between graduate and undergraduate classes, relevant to course design and teaching?" Tags="|undergraduate-education|course-design|graduate-education|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="248" PostTypeId="1" AcceptedAnswerId="251" CreationDate="2014-03-16T03:25:58.850" Score="30" ViewCount="2095" Body="&lt;p&gt;One standard definition of a normal subgroup is &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A subgroup $N \subset G$ is normal iff the set of left cosets $\{gN\}$ and right cosets $\{Ng\}$ coincide.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There's a class of similar definitions (every left coset is also a right coset, $gNg^{-1}=N$, etc). This definition doesn't seem to provide intuition to students about why we &lt;em&gt;care&lt;/em&gt;, nor is it immediately obvious that this is necessary to form a quotient group. One soon learns, though, that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A subgroup is normal iff it is the kernel of some homomorphism. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This latter statement answers both of the previous complaints; it shows both why we care and the necessity of normality to take quotients. The first definition is more malleable for proving things, and the latter provides better intuition. Hopefully both are emphasized in an introductory course on group theory, but which should be taught &lt;strong&gt;first&lt;/strong&gt;?&lt;/p&gt;&#xA;" OwnerDisplayName="user37" LastEditorUserId="143" LastEditDate="2014-03-25T23:56:52.737" LastActivityDate="2021-07-07T12:36:21.383" Title="How should normal subgroups be introduced?" Tags="|undergraduate-education|abstract-algebra|intuition|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="249" PostTypeId="2" ParentId="248" CreationDate="2014-03-16T04:21:36.150" Score="18" Body="&lt;p&gt;The way I like to approach this is as follows. After discussing subgroups, the natural question as to forming the quotient $G/H$ arises. I then proceed to look at the cosets and prove that if $gH\cdot g'H=gg'H$ is a well-defined operation, then the cosets become a group, which we call the quotient group. This is a very easy proof with basically nothing to do. So, one clearly sees that the obstruction to forming the quotient group is the well-defindedness of the operation above. And voiala, this operation is well-defined if, and only if, $gH=Hg$ (or any of the other equivalent conditions). I find that this motivates the definition of normality clearly. I then go over the standard equivalent definition, emphasizing what each is good for (i.e., for checking normality of a given subgroup use that condition, for proving general things about normal subgroups use that etc.). &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-16T04:21:36.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="250" PostTypeId="2" ParentId="130" CreationDate="2014-03-16T04:33:51.310" Score="16" Body="&lt;p&gt;The question is this: &quot;why do you want to teach math ?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll try to stay positive here, I think anyone who has had experience with potential teachers is well aware that bad teachers are probably first bad students. Moreover, as one of my colleagues said, difference between doctors and us: they bury theirs we just graduate them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That aside, I would like for them to be motivated by genuine curiosity for math. If they don't have it already then comments ought to be made to shame them into caring. What are numbers? Why does algebra work like it does? Why do we need calculus? What if you did trigonometry with some shape other than a circle? I read in this book that you can cut a volume into two pieces with the same volume, what's the deal with that? What does it mean to be in 4-dimensions? Why are functions important, who thought of them first?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are just some examples, but, I think we can agree that a student will be much more likely to respect you as a teacher if you can at least start down a path which tells them how they might want to think to answer such questions. In other words, they ought to study abstraction and origins with some fervor. If your best answer is for them to google it, why bother teaching? As teachers of math we need to know math. It's just that simple.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-03-16T04:33:51.310" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="251" PostTypeId="2" ParentId="248" CreationDate="2014-03-16T05:30:28.463" Score="29" Body="&lt;p&gt;First of all, I should point out that &lt;em&gt;the&lt;/em&gt; standard definition of a normal subgroup is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A subgroup $N \subset G$ is normal iff $g n g^{-1} \in N$ for all $n\in N$ and $g\in G$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When I say &quot;the&quot; standard definition, I mean that this is how working group theorists think of normal subgroups, and this is one of two basic ways to prove that a subgroup is normal.  (Note that this is a bit easier to use than the statement $N = gNg^{-1}$, since we don't have to prove containment in both directions.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there is also the statement&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A subgroup is normal iff it is the kernel of some homomorphism.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is widely regarded as a theorem, as opposed to a definition, although it is certainly both the motivation for the definition of normal subgroups and the explanation for their importance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The statement&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A subgroup $N\subset G$ is normal iff $gN = Ng$ for all $g\in G$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;appears as a definition in some books (Gallian being the chief culprit I am aware of), but for the life of me I can't figure out why.  This is not a very good definition for proving things, nor for developing intuition about normal subgroups, nor for explaining their importance, nor is it very standard, and frankly it's not even one of the most important properties of normal subgroups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I teach this subject in abstract algebra, I &lt;em&gt;start&lt;/em&gt; by introducing quotient groups, which I describe as a smaller group obtained from a partitioning a larger group into subsets.  Next, I observe that you need to be careful about which subsets you choose if you want to get a well-defined group operation.  This motivates the definition of a &lt;a href=&quot;http://en.wikipedia.org/wiki/Congruence_relation&quot;&gt;congruence relation&lt;/a&gt;, and I show that the equivalence classes for a congruence relation form a group.  This mirrors the approach that is usually taken for defining modular arithmetic, and so immediately connects the two in the students' minds.  It also has the advantage that the definition seems fairly natural.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I then go on to observe that the congruence class of the identity must be a subgroup that is closed under conjugation, which prompts the definition of a normal subgroup.  I also point out to the students that congruence relations aren't used very often in group theory, since all you really need to know for making a quotient is the normal subgroup (although group theorists often use the notation $gN = hN$ when they want to say that $g$ and $h$ are congruent).  I do homomorphisms a bit later on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I were prohibited from using this approach, I think I would start by introducing homomorphisms, and then observe that the kernel of a homomorphism is a subgroup that is closed under conjugation, prompting the definition of a normal subgroup.  I would &lt;em&gt;not&lt;/em&gt; define a normal subgroup as the kernel of a homomorphism, since that's really more of a theorem than a definition, but I &lt;em&gt;would&lt;/em&gt; use homomorphisms to motivate the definition.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-16T05:30:28.463" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="252" PostTypeId="1" AcceptedAnswerId="253" CreationDate="2014-03-16T05:57:42.390" Score="14" ViewCount="250" Body="&lt;p&gt;I am teaching mathematics for elementary school (even playgroup) to university students. Most of my students think of mathematics might not be necessary for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have an hypothesis that learning mathematics build our characters to be stronger in daily life problems. Shortly speaking, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there an evidence that learning mathematics influences our&#xA;  characters?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I want to use this evidence to convince them. &lt;/p&gt;&#xA;" OwnerUserId="153" LastEditorUserId="114" LastEditDate="2014-03-16T08:24:19.313" LastActivityDate="2014-07-22T10:08:54.457" Title="Is there an evidence that learning mathematics influences our characters?" Tags="|teachers|psychology|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="253" PostTypeId="2" ParentId="252" CreationDate="2014-03-16T07:25:41.743" Score="13" Body="&lt;p&gt;Such an implication as you suggest seems highly far fetched. Firstly, there is a chicken and egg problem here. Suppose that research showed those who study mathematics end up far more likely to have strong characters and be expert at day-to-day problems. How would you tell, without advanced brain imagery techniques, that it is not the predisposition to strong characters and being good at solving problems that is the reason why they did well?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't even begin to imagine what would be required of a study that will actually properly test for the relation you wish to find.&lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-16T07:25:41.743" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="254" PostTypeId="2" ParentId="82" CreationDate="2014-03-16T08:10:16.813" Score="8" Body="&lt;p&gt;I personally find a &lt;a href=&quot;http://en.wikipedia.org/wiki/Bean_machine&quot;&gt;Galton board&lt;/a&gt; more compelling than a computer simulation.  There are lots of YouTube demonstrations; &lt;a href=&quot;https://www.youtube.com/watch?v=03tx4v0i7MA&quot;&gt;this&lt;/a&gt; is one of the better ones I found after a very brief search.&lt;/p&gt;&#xA;" OwnerUserId="129" LastActivityDate="2014-03-16T08:10:16.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="255" PostTypeId="2" ParentId="252" CreationDate="2014-03-16T10:38:16.837" Score="7" Body="&lt;p&gt;I agree with the premise, but as Ittay's answer suggests, a study sufficient to prove this would be difficult. The issue of correlation vs causation comes into play with the task of separating them to be difficult. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, it's fair to say that young children learn more easily. Things like language that are far more difficult to learn say, in high school, can be picked up by a young child. I think that observation has been accepted as true. For that reason, the focus on math learning at the early ages can help set them on a more successful path. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(to repeat, I agree with the premise. It's not 'far fetched' in my opinion, only difficult to prove)&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-03-16T10:38:16.837" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="256" PostTypeId="2" ParentId="247" CreationDate="2014-03-16T12:49:55.533" Score="17" Body="&lt;p&gt;Of course there are many differences between undergraduates and graduates, including:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mathematical independence.&lt;/strong&gt; In a graduate class, one may generally presume a much greater level of mathematical independence and self-motivation than in an undergraduate class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mathematical sophistication.&lt;/strong&gt; Graduate students generally are more knowledgeable than undergraduates and will benefit from more sophisticated explanations. Graduate students are more likely to appreciate it when the instructor follows sideline topics in class for a bit, or unifies topics with ideas from other areas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Capacity to be challenged.&lt;/strong&gt; Graduate students have a far larger capacity to be challenged mathematically. One can give mathematical puzzles, and not be surprised to find that students struggled with them for hours and hours over the weekend. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of these differences, I often organize my graduate classes in a much different manner. For example, in order to introduce graduate students to the math research experience, I usually require my graduate students to write a term paper (see &lt;a href=&quot;https://mathoverflow.net/a/133643/1946&quot;&gt;my account of this here&lt;/a&gt;), which I then collect into a &quot;Proceedings of Graduate Set Theory&quot; volume at the end of the semester and have a session devoted to student talks. (With undergraduates, in my experience, this doesn't work as well.) Also, in lectures I am willing to follow tangents in the topics that might come up&#xA;for a longer time and more thoroughly than I do in my undergraduate classes, since in the undergraduate classes I feel more compelled to stay closer to the established syllabus.&lt;/p&gt;&#xA;" OwnerUserId="90" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-03-16T12:49:55.533" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="257" PostTypeId="1" CreationDate="2014-03-16T13:02:49.390" Score="13" ViewCount="158" Body="&lt;p&gt;In my graduate courses, I often have my students write term papers on original mathematical topics. I explain the process in &lt;a href=&quot;https://mathoverflow.net/a/133643/1946&quot;&gt;this answer over at MathOverflow&lt;/a&gt;. It works fairly well for me. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I'd be very interested in hearing about other experiences or approaches to using term papers in mathematics. What has worked for you? What doesn't work so well? Do you have students pick their own topics? Do all students write on the same topic or from a list of topics? Do you meet with students individually to discuss their papers? Do you have them submit drafts and then revised versions? Do you insist on $\TeX$? Do you have students work in groups? Do you have students give talks on their research topics? Does it matter whether the course is introductory or advanced? What special issues arise for using term papers in an undergraduate course?&lt;/p&gt;&#xA;" OwnerUserId="90" LastEditorUserId="-1" LastEditorDisplayName="user173" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-03-21T21:20:29.047" Title="Ideas for term papers in a graduate or advanced undergraduate class" Tags="|term-papers|writing|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="258" PostTypeId="1" AcceptedAnswerId="261" CreationDate="2014-03-16T13:19:54.613" Score="29" ViewCount="1652" Body="&lt;p&gt;I teach Precalculus to high school kids, and I know a lot of you all teach Calculus.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are some issues that your students have in Calculus classes that&#xA;  you wish had been addressed in a Precalculus class?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;e.g. &quot;Man, my kids never know how to factor, and that really holds them back.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;e.g. &quot;Informal experience with finding speed graphs from distance graphs would be huge.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;e.g. &quot;Nothing! My calc students are perfectly well prepared. Keep it up, precalc teachers!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;e.g. &quot;Downvote. Not specific enough.&quot;&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="114" LastEditDate="2014-04-08T08:41:41.600" LastActivityDate="2014-07-31T23:25:05.863" Title="What holds your students back in Calculus?" Tags="|calculus|precalculus|students-mistakes|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="259" PostTypeId="1" AcceptedAnswerId="265" CreationDate="2014-03-16T13:53:47.667" Score="14" ViewCount="433" Body="&lt;p&gt;When I was in high school I explained a lot of things to my class mates. Later during the first time of my studies, I was hired as a private tutor. Although I was earning some money, I had very mixed feelings about that job. My impression is that only a few students really learned something and most students are only attending since it was their parent's will. Or they needed to have a good grade in the next exam (Which was - to some limit - always possible if there were enough meetings during the weeks before the exam) - But even in that cases, I had the feeling that they only repecipes from me and were drilled to an particular type of questions, ans as soon as a new topic with different type of questions came up they were completly lost. This was maybe also the reason why many students are taking private lessons over a period of severeal years (if not longer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is there some reference or some evidence if taking private lessons in mathematics really helps? If yes, in which way or to what limit?&lt;/strong&gt; &lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-18T07:16:30.037" LastActivityDate="2014-03-18T16:28:18.720" Title="Do private lessons really help?" Tags="|secondary-education|private-lessons|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="260" PostTypeId="2" ParentId="258" CreationDate="2014-03-16T14:18:47.283" Score="11" Body="&lt;p&gt;My impression (as a former private tutor/current university employee and from my mind as a former student) is that mainly equation solving is a big problem for poor (sometimes also for advanced) students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is somehow some task one has to perform to do subtask in calculus (e.g., a new thing is to calculate a minimum of a function. The students learn that his is done by taking the derivative (=new thing in calculus!) and solve $f'(x)=0$.) Since f can now be any function, the students must have in mind how to solve a various number of equations, but normally they learn at some point how to solve quadratic equations, (forget it maybe), then learn how to solve exponential equations, (forget it maybe), learn how to solve trigonometric equations, etc. -- At that point, everything you want to do is limited how good students can solve equations and if you are unlucky, you have to spent more time on teaching/reminding how to solve equations than on the actual topic. - If one is aware of this, the pre-calculus teacher as well as the calculus teacher can do something to solve this issue by giving some homework in repeating that things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very similar analysis goes with the issue of manipulating terms. &lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-16T14:18:47.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="261" PostTypeId="2" ParentId="258" CreationDate="2014-03-16T14:37:01.957" Score="23" Body="&lt;p&gt;They are not usually well-prepared, but factoring is not a big issue. I would like students to be able to:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Make meaning from graphs&lt;/strong&gt;. [I want to get to introduce the connection between speed graphs and distance graphs, though, so I'd be happier if you focused on other graphs.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;See what is algebraically sensible&lt;/strong&gt;, and what's not. e.g. you can't cancel from one term in a fraction that has more than one term on top or bottom ((x+h)/h does not = x), square root of (x^2 + y^2) does not equal x+y, ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Understand the basic area and volume &quot;formulas&quot;&lt;/strong&gt;, instead of trying to memorize them. Then maybe they'll have a chance at solving optimization problems. [I asked the post office problem on the last test. (The post office will not allow packages to exceed 108 inches in combined length (longest dimension) plus girth (measure around the shortest way). Find the package, with square ends, of maximum volume.) Although I explained repeatedly what length plus girth meant, they bombed it. To me, that means they can't deal with thinking about something even a tiny bit new.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This list is starting to take on a theme. &lt;strong&gt;It's not the topics you teach, so much as how well you help them learn to think mathematically.&lt;/strong&gt; Except trig. Is trig part of your pre-calculus course? They need to be able to understand the trig functions. My students suffer more from their trig weaknesses in calc II than in calc I. I've told my pre-calc students that the main theme of the course is understanding how equations and graphs go together.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-16T14:37:01.957" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="262" PostTypeId="2" ParentId="198" CreationDate="2014-03-16T15:01:03.060" Score="5" Body="&lt;p&gt;The Princeton Companion to Mathematics has a section &lt;a href=&quot;http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Advice to a Young Mathematician&lt;/a&gt;, which seems available for free, where each of Atiyah, Bollobás, Connes, McDuff, and Sarnak give advice and most of it is of the form that it fits what is asked for in the question.&lt;/p&gt;&#xA;&lt;p&gt;Terence Tao's blog contains various related information under &lt;a href=&quot;http://terrytao.wordpress.com/career-advice/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Career Advice&lt;/a&gt; and &lt;a href=&quot;http://terrytao.wordpress.com/advice-on-writing-papers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Writing&lt;/a&gt;; note that this is mostly different from the book 'Solving Mathematical Problems' mentioned in another answer, which he wrote more than two decades ago.&lt;/p&gt;&#xA;&lt;p&gt;Cédric Villani has written a book &lt;a href=&quot;https://web.archive.org/web/20130214135015/http://www.grasset.fr/automne_romanesque_2012/villani.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Théorème Vivant&lt;/a&gt; that reads somewhat like a diary and generally feels very authentic, even including reproductions of some of his email correspondence. It might be of less practical value regarding finding advice or behavior to adopt, yet as far as conveying what it can be like &amp;quot;to live as a research mathematician&amp;quot; goes this is the best I know. The original is in French &lt;a href=&quot;http://www.cedricvillani.org/activites-scientifiques/born-be-alive&quot; rel=&quot;nofollow noreferrer&quot;&gt;according to some rather recent information on his website&lt;/a&gt;  it is also available in Italian, German, and Serbian while  English, Romanian, Bulgarian, Japanese and Korean versions are planned.&lt;/p&gt;&#xA;&lt;p&gt;As a final note reading such things one should never forget that (as written by Connes in the text mentioned above) &amp;quot;each mathematician is a special case&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorDisplayName="user7990" LastEditDate="2020-09-11T21:40:03.143" LastActivityDate="2020-09-11T21:40:03.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="263" PostTypeId="2" ParentId="259" CreationDate="2014-03-16T15:06:58.163" Score="8" Body="&lt;p&gt;In my high-school days I tutored a boy in elementary school for a year or so. My feeling at the time was that I did't really teach him about the subject matter, but forced him to work by being there, and (indirectly) discipline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During college, I often helped classmates (and students in lower classes) in solving particularly hard proposed problems. I also tutored several classes formally. I learned a lot, and I hope my explanations and solutions of proposed problems helped the students understand the subject matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did it really matter? Hard to say. The ones asking me, or the ones going to my tutoring sessions, seemed to think so.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-16T15:06:58.163" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="264" PostTypeId="2" ParentId="247" CreationDate="2014-03-16T15:17:22.207" Score="9" Body="&lt;p&gt;The difference between undergraduate and graduate is a matter of degree, not a clear cut. The difference between first and third year students are as large (or larger) than undergraduates and graduates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anything, graduate classes are easier, the students take the courses because they are interested, not because they have to take it as part of the curriculum. You can tell people to study or find out stuff by themselves, you don't have to give them everything predigested. But they are harder, because you have to cover more material, and the material isn't readily available, is much less ordered and worked over, cleaned up and has not yet found its &quot;optimal&quot; exposition.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="1147" LastEditDate="2016-06-25T01:23:23.330" LastActivityDate="2016-06-25T01:23:23.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="265" PostTypeId="2" ParentId="259" CreationDate="2014-03-16T15:19:59.930" Score="18" Body="&lt;p&gt;I am that tutor. I work in a math center where my job is not classroom lessons, but one on one help. This is high school level, ages 13-18, grades 9-12. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most student come of their own free will, and as this is my first year, word is still spreading. The teachers around me are skilled, but in any class the distribution of students means the teacher has to teach to the middle. 40 minutes to get a certain amount of work done. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They walk into my room, and I can ask them to show me exactly what they missed, and explain it in a bit of a different way. I have a steady flow of emails and notes from their teachers telling me the student's latest grade, one went from D's in September to B's and finally an A on her last test. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is my success 100%? Maybe not, but my goal is to get either a &quot;now I understand this&quot; or the prized &quot;that's so cool&quot; when my method of explaining something sinks in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I taught classes, I would seek permission for a one semester experiment - every lecture done via an online video, as the &quot;homework&quot; and all practice homework actually done in class so we'd have the back and forth discussion. I'd agree to scheduling an extra block, so the class would be two groups of 10 kids in the room.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, the tutoring isn't just valuable, it's priceless.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-03-16T15:19:59.930" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="266" PostTypeId="2" ParentId="225" CreationDate="2014-03-16T15:25:28.937" Score="8" Body="&lt;p&gt;Rather than explaining why complex solutions to transcendental equations are &lt;em&gt;not&lt;/em&gt; discussed, I think it is worth explaining why the complex solutions to polynomial equations &lt;em&gt;are&lt;/em&gt; - that, I think, comes down to the Fundamental Theorem of Algebra. The heavy emphasis on complex roots of polynomials could help serve as a precursor for the result, so that when students finally are introduced to it, they're not quite as surprised.  &lt;/p&gt;&#xA;" OwnerUserId="145" LastActivityDate="2014-03-16T15:25:28.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="267" PostTypeId="1" AcceptedAnswerId="276" CreationDate="2014-03-16T15:37:28.997" Score="20" ViewCount="12261" Body="&lt;p&gt;In later courses on automata theory, many students just seem incapable of getting a proof that a language isn't regular right, be it using the &lt;a href=&quot;http://en.wikipedia.org/wiki/Pumping_lemma&quot;&gt;pumping lemma&lt;/a&gt; (see also the many questions on the matter on &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;http://math.stackexchange.com&lt;/a&gt;) or the (often easier) use of closure properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I am also teaching a first course in discrete matematics, were we go over proof techniques, I have tried (rather unsuccessfully, regrettably) to get the idea across. I would like to have more easy to grasp examples where the technique is natural (and needed, contrived examples just lead to &quot;the prof is off his rocker again, this is trivial to do by ...&quot;, and so are more a distraction than a help).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2022-12-24T15:44:08.757" Title="Good examples of proof by contradiction?" Tags="|mathematical-pedagogy|proofs|examples|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="268" PostTypeId="2" ParentId="82" CreationDate="2014-03-16T15:54:28.147" Score="5" Body="&lt;p&gt;Perhaps start with an uniform distribution between $-1$ and $1$, and show what the distribution of the sum (or average) of 2, 3, ... IID variables looks like? Not too hard to do analytically, easy to do by simulation (have the students write a small program generating the values, and use e.g. a spreadsheet or a plotting program to show histograms). To get a &quot;feel&quot; requires many values, and getting them experimentally is hard.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-16T15:54:28.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="269" PostTypeId="1" CreationDate="2014-03-16T16:09:05.743" Score="20" ViewCount="1383" Body="&lt;p&gt;One of the main difficulties for a student learning abstract algebra is understanding the motivations behind concepts like groups, normal subgroups, rings , ideals etc. Also, many have difficulty understanding why algebra is important and how it has helped pure mathematics or is applied in mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I would like to know what are some of the nice applications of algebra that can be taught in first algebra course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two examples that come to mind are unsolvabillity of equations of degree greater or equal to 5 by radicals and algebraic number theory. But it is difficult to include them in a first algebra course due to time constraints. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any other applications of algebra which can be introduced easily in a first algebra course?&lt;/p&gt;&#xA;" OwnerUserId="124" LastEditorUserId="114" LastEditDate="2014-04-11T12:25:52.043" LastActivityDate="2018-03-23T17:39:56.513" Title="What are some good mathematical applications to present in an abstract algebra course?" Tags="|undergraduate-education|abstract-algebra|concept-motivation|" AnswerCount="8" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="270" PostTypeId="1" CreationDate="2014-03-16T16:14:50.937" Score="8" ViewCount="422" Body="&lt;p&gt;I know this subject is inflamatory... but I also believe it is something that requires broader discussion (and hopefully some consensus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own position (from personal experience) is that today you rarely (if ever) do calculations (sums, multiplications, ...) by hand. Remembering the hours I spent (not all exactly happy ones) being drilled in multiplication and division, doing so-so in that, but going on to end up &lt;a href=&quot;http://matheducators.stackexchange.com&quot;&gt;here&lt;/a&gt;, I'm looking for better use of that time. In college, I see student's (and teacher's) time being wasted on &quot;what clever trick gets an integral for ...&quot; and similar questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is then, can (and should) we take out this &quot;drill&quot; aspect of teaching, and what should we do with the freed time?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Follow-up question is what this means in formation of future teachers?&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="143" LastEditDate="2014-03-20T13:03:00.067" LastActivityDate="2014-08-01T09:01:23.587" Title="On the use of calculators in elementary/high school? Computer algebra systems later on?" Tags="|course-design|technology-in-education|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="272" PostTypeId="2" ParentId="267" CreationDate="2014-03-16T16:22:30.337" Score="4" Body="&lt;h2&gt;Example 1&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prove that there is only one circle with $AB$ as its diameter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assumption:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume that there are 3 circles $C_1$, $C_2$, and $C_3$ passing through the points $A$ and $B$. $C_1$ and $C_2$ are concentric and $C_1$ and $C_3$ are not concentric. $C_1$ and $C_2$ have different radii and $C_3$ has any radius. Let $C_1$ be on the midpoint of $AB$ such that $AB$ is its diameter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Contradicting Arguments:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;As $C_1$ and $C_2$  have different radii, points $A$ and $B$ cannot be on the circle $C_2$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As $C_3$ is not on the middle of $AB$, $AB$ cannot be its diameter.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Conclusion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So there is only one circle $C_1$ with $AB$ as its diameter.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Example 2&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prove that $\sqrt{2}$ is an irrational number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assumption:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $\sqrt 2$ be a rational number. So it can be represented as $\sqrt{2}=\frac{m}{n}$ where $m$ and $n$ are natural numbers without common factors other than $1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Contradicting Arguments:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Squaring both sides, we get&#xA;\begin{align}&#xA;2 &amp;amp;=\frac{m^2}{n^2}\\&#xA;m^2 &amp;amp;= 2n^2&#xA;\end{align}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because $m^2$ is a multiple of $2$ then $m^2$ is an even number. Recall that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The square of an even number is even.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;it implies that $m$ is also even. Let $m=2k$ where $k$ is any natural number. Substituting   it for $m$, we get&#xA;\begin{align}&#xA;(2k)^2 &amp;amp;=2n^2\\&#xA;4k^2 &amp;amp;= 2 n^2\\&#xA;n^2 &amp;amp;= 2k^2&#xA;\end{align}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the same reasoning, $n$ is even. As both $m$ and $n$ are even numbers, 2 becomes their common factors so it contradicts the assumption that they have no common factors other than 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conclusion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\sqrt 2$ cannot be represented as a ratio of two natural numbers without common factor other than 1. It implies that $\sqrt 2$ is irrational.&lt;/p&gt;&#xA;" OwnerUserId="153" LastEditorUserId="153" LastEditDate="2014-03-16T16:39:16.427" LastActivityDate="2014-03-16T16:39:16.427" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="273" PostTypeId="2" ParentId="220" CreationDate="2014-03-16T16:33:21.130" Score="8" Body="&lt;p&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not simple, but it might be useful. The context in which it happend is algorithms, but it would be great if someone could test if this approach is useful during other courses, e.g. when introducing induction. Reason why I'm writing about this here is, that when I did the exercise described below, my students (not all, but a lot) had this &quot;being enlightened&quot; face expression, and as it was not about the algorithm itself, then it had to be about the other thing and in that case it was induction (i.e. it seemed that they grasped something important about induction).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Exercise:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The task was on dynamic programming and was given as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A company plans to organize a party. The hierarchy of employees is a rooted tree (not necessarily binary), and each employee can be characterized by a form of sociability, given by a positive number. Write a program that will pick the guests so that:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;no guest would meet their immediate superior at the party,&lt;/li&gt;&#xA;  &lt;li&gt;the sum of sociability coefficients is maximal,&lt;/li&gt;&#xA;  &lt;li&gt;the company founder (the root of the tree) has to be at the party.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;The program should output a single number, namely the sum of sociability coefficients of the guests.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The algorithm to calculate it is easy, let the symbol $\mathcal{T}$ denote the set of abstract people with their hierarchies&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align*}&#xA;\newcommand{soc}{\mathtt{sociability}}&#xA;\newcommand{sub}{\mathtt{subordinates}}&#xA;\newcommand{with}{\mathtt{with}}&#xA;\newcommand{without}{\mathtt{without}}&#xA;\soc &amp;amp;: \mathcal{T} \to \mathbb{R}^+\\&#xA;\sub &amp;amp;: \mathcal{T} \to 2^\mathcal{T} \\&#xA;\end{align*}&#xA;then in mathematical notation it would be&#xA;\begin{align*}&#xA;\with(t)&amp;amp;= \soc(t) + \sum_{s\ \in\ \sub(t)} \without(s), \\&#xA;\without(t) &amp;amp;= \sum_{s\ \in\ \sub(t)} \max\big\{ \with(s), \without(s) \big\}.&#xA;\end{align*}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where the result will be given by $\with$ as the founder has to be included. Now, the challenge is to prove that it is optimal (i.e. it returns the correct result). The proof is via induction on the structure of the tree, that is, we start from leaves and then go up to the root, with the assumption that each time when make the step in some node, all its subtrees are already done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The base step is easy, as for each leaf we have $\sub(l) = \varnothing$ so&#xA;\begin{align}\with(l) &amp;amp;= \soc(l) \\&#xA;\without(l) &amp;amp;= 0 \end{align} what is obviously true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, for the induction step, $\with(t)$ requires all the subtrees to skip their roots, and besides that they are all independent, so we can just take the sociability of $t$ and add all the optimal subsolutions (we can use them because of the induction hypothesis) and that is exactly what the formula is doing. As for $\without(t)$ the $t$ itself isn't a guest, so from each subtree we can pick the best option and sum it all together. This works, but only because we know that the subsolutions were calculated correctly from induction hypothesis, the shape of the tree ensures, that there weren't any circular arguments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes students benefit from an alternative proof: take an optimal solution and prove that one returned by the algorithm isn't worse. Again, prove it via induction on the structure of the tree, the hypothesis being that our algorithm returns an optimal subsolution for each node of the tree. Base case is easy, because there is no choice available in the leaves. For the inner nodes we show that our algorithm returns a pick of guests at least as good as the optimal solution (in $\with$ it follows from induction hypothesis and the fact that all the numbers are positive, for $\without$ it follows from properties of $\max$ function).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Some commentary:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each time I did this it was an introductory course in programming, and for many the first time where they would see a non-linear induction (in fact this &lt;em&gt;can&lt;/em&gt; be proved with induction on the height of the tree, but I did complicate it on purpose). Drawing a diagram with arrows, describing how the induction progresses and how it relies on the subtrees being already calculated helps a lot. After this exercise the topic would come back again several times and students would have much easier time proving that some algorithm (e.g. the Ackermann function)  terminates. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, it is hard to draw any reliable conclusions from this experience (it might be just a coincidence that it worked for me), so it would be great if someone could test it themselves. I'd also love to hear your opinions and suggestions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-16T22:38:54.600" LastActivityDate="2014-03-16T22:38:54.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="274" PostTypeId="2" ParentId="176" CreationDate="2014-03-16T16:34:22.457" Score="11" Body="&lt;p&gt;One of the points is that &lt;em&gt;teaching&lt;/em&gt; is an expertise on its own. Part of it is innate, much can be learned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Important questions, that during your traditional undergraduate/graduate curriculum (at least if not directed at pedagogy) aren't ever mentioned, and which many of us never really considered, include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How do I design an exam? What kinds of questions should I include? How do I make the result easy/unambiguous to grade? How do I make sure a dumb mistake early on doesn't invalidate later work (and perhaps requires extra work grading, following a completely different path)?&lt;/li&gt;&#xA;&lt;li&gt;What should be the homework's contents?&lt;/li&gt;&#xA;&lt;li&gt;What proposed exercises (for self study) should be given?&lt;/li&gt;&#xA;&lt;li&gt;Which ones are the best examples for classroom use?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="123" LastEditorUserId="64" LastEditDate="2019-10-05T12:27:35.093" LastActivityDate="2019-10-05T12:27:35.093" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="275" PostTypeId="2" ParentId="270" CreationDate="2014-03-16T16:43:43.427" Score="9" Body="&lt;p&gt;I think &lt;em&gt;some&lt;/em&gt; practice in manipulating symbols and number is necessary for many mathematical activities. To do everything &quot;by machine&quot; is first inconvenient and second there is also some insight to be gained from doing it by hand. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think by solving some systems of linear equations by hand one gains some insight that one would not gain just by typing it into some CAS and getting the solution back out. (Leaving aside the aspect that some need to build and improve such things, as this is a minority.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same goes first for basic calculations, then for more complex ones, then for manipulating expression containing indeterminantes and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above being said, I stress the &lt;em&gt;some&lt;/em&gt;. I agree it is nowadays pointless (for most anyone) to develop high proficiency at these tasks. And, &lt;em&gt;yes&lt;/em&gt;, I do not understand either why some &lt;em&gt;still&lt;/em&gt; let students do so many integrals. Surely, one should know how to do integration by parts and substitution, but a few examples should suffice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally speaking, I think some practice in such tasks is important, but only to the extent that one can do things by hand if one must. Forced &quot;drill&quot; to do such things fast and reliably is besides the point in my opinion (I stressed the forced as, personally, I enjoyed to do calculations a lot).         &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-03-16T16:43:43.427" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="276" PostTypeId="2" ParentId="267" CreationDate="2014-03-16T16:45:02.777" Score="24" Body="&lt;p&gt;Here are some good examples of proof by contradiction:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Euclid's proof of the infinitude of the primes.  (&lt;strong&gt;Edit:&lt;/strong&gt; There are some issues with this example, both historical and pedagogical.  See Mike F.'s answer and the ensuing discussion.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The famous proof that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is irrational.  (I don't particularly like this one---there are better ways of proving this.  See my comment above.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The sum of a rational number and an irrational number is irrational.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cantor's diagonal argument that &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; is uncountable is a proof by contradiction.  (&lt;strong&gt;Edit:&lt;/strong&gt; As Santiago Canez points out in the comments, this example and the next are perhaps better stated as direct proofs.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similarly, there's the proof that there is no bijection from a set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; to the power set of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Russell's proof that there exists no set of all sets.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The proof of Gödel's incompleteness theorem.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/2746/is-there-an-elementary-proof-that-sum-limits-k-1n-frac1k-is-never-an-int/2752#2752&quot;&gt;This Math Stack Exchange post&lt;/a&gt; has a nice simple proof that &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\sum_{k=1}^n \frac1k$&lt;/span&gt; is never an integer for &lt;span class=&quot;math-container&quot;&gt;$n\geq 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There are lots of basic statements about Diophantine equations that can be proven by contradiction.  For example, the statement &amp;quot;the equation &lt;span class=&quot;math-container&quot;&gt;$4x^2-y^2 = 1$&lt;/span&gt; has no integer solutions for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&amp;quot; has a simple contradiction proof. (Factor the left side.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/614209/infiniteness-of-non-twin-primes/614412#614412&quot;&gt;This Math Stack Exchange post&lt;/a&gt; has a simple proof that there exist infinitely many primes &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$p+2$&lt;/span&gt; is &lt;em&gt;not&lt;/em&gt; prime.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The proof of &lt;a href=&quot;http://en.wikipedia.org/wiki/Eisenstein%27s_criterion&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eisenstein's criterion&lt;/a&gt; for irreducible polynomials is a proof by contradiction.  There are many simpler statements along these lines (e.g. proving that specific polynomials have no integer roots) that can be proved by contradiction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/385679/a-matrix-satisfying-ab-ba-b/385717#385717&quot;&gt;This Math Stack Exchange post&lt;/a&gt; has a simple proof using the trace that if &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$n\times n$&lt;/span&gt; matrices and &lt;span class=&quot;math-container&quot;&gt;$AB-BA = B$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; cannot be invertible.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There's a simple proof by contradiction that there does not exist a continuous function &lt;span class=&quot;math-container&quot;&gt;$f\colon\mathbb{R}\to\mathbb{R}$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$f(f(x))=-x$&lt;/span&gt;.  (It must be a bijection, so it's either increasing or decreasing, so . . .)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You can prove by contradiction that there's no embedding of the complete graph &lt;span class=&quot;math-container&quot;&gt;$K_5$&lt;/span&gt; in the plane using Euler's formula.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The solution to the &lt;a href=&quot;http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Seven Bridges of Königsberg&lt;/a&gt; problem is essentially a proof by contradiction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.cut-the-knot.org/pigeonhole/25BoysAndGirls.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;Twenty five boys and twenty five girls sit around a table. Prove that it is always possible to find a person both of whose neighbors are girls.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2020-09-21T00:37:57.057" LastActivityDate="2020-09-21T00:37:57.057" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="277" PostTypeId="2" ParentId="269" CreationDate="2014-03-16T16:47:40.417" Score="5" Body="&lt;p&gt;I'm not sure if this is what you want, but&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;monoids are used in automata theory,&lt;/li&gt;&#xA;&lt;li&gt;semirings and matroids are used in graph theory,&lt;/li&gt;&#xA;&lt;li&gt;groups and finite fields are used in cryptography and coding theory.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The examples are like this, because abstract algebra is frequently tied to the discrete world. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-16T16:47:40.417" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="278" PostTypeId="2" ParentId="269" CreationDate="2014-03-16T16:48:20.280" Score="9" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Burnside%27s_lemma&quot;&gt;Burnside's lemma&lt;/a&gt; can be covered in one day (or less) in an abstract algebra course, and has lots of wonderful applications to counting problems.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-16T16:48:20.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="279" PostTypeId="2" ParentId="183" CreationDate="2014-03-16T16:49:08.427" Score="2" Body="&lt;p&gt;If you find out &lt;em&gt;during&lt;/em&gt; the exam, announce that the question won't be graded, and explain any grading adjustments. Ask for partial work to be handed in regardless, to be handled as @Skytso's answer states.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-16T16:49:08.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="280" PostTypeId="1" AcceptedAnswerId="282" CreationDate="2014-03-16T17:03:39.960" Score="17" ViewCount="14398" Body="&lt;p&gt;I often find students who dislike algebra. They prefer to work with numbers in solving problems. I believe there are many problems that are hard to solve without algebra. For example:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Finding the value of $x$ such that the volume of a box without lid reaches a maximum value.&#xA;&lt;img src=&quot;https://i.stack.imgur.com/vIxwP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A tank contains 40 gallons of a solution composed of 90% water and 10%&#xA;alcohol. A second solution containing half water and half alcohol is added to the&#xA;tank at the rate of 4 gallons per minute. At the same time, the tank is being drained&#xA;at the rate of 4 gallons per minute, as shown in below. Assuming that&#xA;the solution is stirred constantly, how much alcohol will be in the tank after 10&#xA;minutes?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/qxz11.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Etc, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Could you provide me with other simple &quot;interesting and challenging&quot; examples so I can introduce them to my students?&lt;/p&gt;&#xA;" OwnerUserId="153" LastEditorUserId="123" LastEditDate="2014-04-14T18:28:06.340" LastActivityDate="2020-01-22T01:05:16.590" Title="Looking for simple &quot;interesting&quot; math problems that cannot be easily solved without algebra" Tags="|mathematical-pedagogy|" AnswerCount="11" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="281" PostTypeId="2" ParentId="183" CreationDate="2014-03-16T17:12:23.860" Score="8" Body="&lt;p&gt;Anticipating that such scenarios are inevitable, a certain amount of warning/advice to students _in_advance_ is helpful. But don't go overboard, or you may generate paranoia about flawed questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Especially in upper-division or graduate classes, encouraging critical/skeptical thinking as a matter of course is a good thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trying to &quot;correct&quot; an exam part-way through may be the worst reaction, insofar as some students may have already wasted time. Thus, either throwing the question out or giving everyone full credit is not &quot;fair&quot;. After seeing some debacles of this sort on PhD qualifying exams, it finally dawned on me that declaring the exam to simply be whatever it is, possibly with some flaws, rather than trying to repair it &lt;em&gt;during&lt;/em&gt; the exam itself, is saner than any compensation scheme.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Some ways of dodging the issue of flawed exams strike me as adversarial-to-students, e.g., the &quot;prove or disprove&quot; gambit, wherein implied context or forgotten minor hypotheses are &quot;gotchas&quot; potentially unrelated to the mathematics itself. Or use/mis-use of English articles &quot;a, an, the&quot; and modifiers &quot;all, any, every&quot;...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, I refuse to discuss exams &lt;em&gt;during&lt;/em&gt; the exam, so would not overtly admit a flaw in any particular problem... because someone may have already wasted time on it, while others haven't. If an exam is so flawed that it grossly fails to serve its design purpose, probably throwing it out entirely and doing a new exam is the only reasonable course. Even then, some students will have been stressed by the thrown-out exam...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-16T17:12:23.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="282" PostTypeId="2" ParentId="280" CreationDate="2014-03-16T17:15:28.743" Score="29" Body="&lt;p&gt;The box problem may be hard for students who aren't into algebra. I like using the silly number puzzles when I teach beginning algebra. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Pick a number between 1 and 25. &lt;/li&gt;&#xA;&lt;li&gt;Add 9 to it. &lt;/li&gt;&#xA;&lt;li&gt;Multiply the result by 3. &lt;/li&gt;&#xA;&lt;li&gt;Subtract 6. &lt;/li&gt;&#xA;&lt;li&gt;Divide by 3. &lt;/li&gt;&#xA;&lt;li&gt;Subtract your original number. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then you get to go around and quietly say, &quot;You got 7, right?&quot; to each one. Once they get it that everyone has 7, it becomes interesting to figure out why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I got the idea to use this in algebra class from &lt;a href=&quot;http://www.amazon.com/s/ref=nb_sb_ss_i_1_13?url=search-alias=aps&amp;amp;field-keywords=harold%20jacobs%20algebra&amp;amp;sprefix=harold%20jacobs,aps,430&quot;&gt;Harold Jacobs' &lt;em&gt;Elementary Algebra&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2014-03-16T17:58:28.937" LastActivityDate="2014-03-16T17:58:28.937" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="283" PostTypeId="2" ParentId="280" CreationDate="2014-03-16T17:21:41.440" Score="9" Body="&lt;p&gt;I believe &lt;a href=&quot;http://mathworld.wolfram.com/DiophantussRiddle.html&quot;&gt;Diophantus' riddle&lt;/a&gt; is a good example.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;God gave him his boyhood one-sixth of his life,&#xA;One twelfth more as youth while whiskers grew rife;&#xA;And then yet one-seventh ere marriage begun;&#xA;In five years there came a bouncing new son.&#xA;Alas, the dear child of master and sage&#xA;After attaining half the measure of his father's life chill fate took him.&#xA;After consoling his fate by the science of numbers for four years,&#xA;He ended his life.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-16T17:21:41.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="284" PostTypeId="2" ParentId="270" CreationDate="2014-03-16T18:15:30.793" Score="3" Body="&lt;p&gt;I have some documents discussing various issues about the use of Maple in teaching here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://neil-strickland.staff.shef.ac.uk/talks/maple2/&quot; rel=&quot;nofollow&quot;&gt;http://neil-strickland.staff.shef.ac.uk/talks/maple2/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This is actually quite old, but I think that most of it is still valid.)&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-16T18:15:30.793" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="285" PostTypeId="2" ParentId="183" CreationDate="2014-03-16T18:17:42.113" Score="8" Body="&lt;p&gt;When I teach a first year undergraduate class I try to emphasise exam technique, and mostly this boils down to them having to &lt;em&gt;communicate&lt;/em&gt; that they know what they are talking about to the marker. Mostly this is to stop them skipping lines of working (so when they get the wrong answer I know what they have done), but it is relevant here. If they spot the error then I would expect them to write down an explanation about what the issue is, certainly if they are wanting a good grade. If they understand the examinable material then they should spot the error! Therefore, if an unanswerable question slipped into one of my exams then I would re-write my marking scheme accordingly and mark to this scheme. If a student didn't write anything then I would give them zero. However, I would mark generously (and would read all their scored out stuff to see if they had the right idea etc.).&lt;/p&gt;&#xA;" OwnerUserId="169" LastActivityDate="2014-03-16T18:17:42.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="286" PostTypeId="2" ParentId="175" CreationDate="2014-03-16T18:37:38.870" Score="4" Body="&lt;p&gt;This is really two questions...&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Is reading the textbook effective?&lt;/strong&gt; It depends a lot on the text. Besides, students have different ways of understanding, the explanation one sees as crystal clear can be total greek to the next one, and viceversa; or (even worse) bore them to tears. So perhaps it is more important to have a &lt;em&gt;selection&lt;/em&gt; of sources available. Perhaps have a &quot;lead&quot; text, with a variety of &quot;support material&quot; suggested. Here Wikipedia or the SE sites are wonderful. Suggest some starting searches.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;How to assess reading the text?&lt;/strong&gt; What do you really want to know? If they read the text, or if they understood the subject matter (in however shallow manner)? Ask them to &lt;em&gt;solve a problem&lt;/em&gt; based on the section read (if they just follow the steps outlined there, at least they did it once), or to &lt;em&gt;criticize&lt;/em&gt; the text (think of alternative ways of doing what they explain, explain why the text uses a particular technique, ...). Can be a mini-homework, to be handed in at class start.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Yes, those suggestions are bound to stir up a mutiny among your students ;-). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps grade on a scale 0-2 (0: did nothing but show up/copied the editorial from the newspaper; 1: said something sensible, but incomplete/wrong; 2: right solution/shows insight; 3: shows ingenuity/creativity beyond what is expected). Not too hard to grade, fast feedback.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-16T18:37:38.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="287" PostTypeId="2" ParentId="244" CreationDate="2014-03-16T18:45:59.923" Score="1" Body="&lt;p&gt;At least around here in Chile, school curricula are defined by law/decree of the Ministry for Education. So to answer the question would mean digging through the official documents. To find out &lt;em&gt;why&lt;/em&gt; it came to be that way gets lost in some murkiness of (much off-the-record) discussions among &quot;interested players&quot; (who might, mostly not, have a clue). All seasoned with a healthy dose of whatever the current popular &quot;feeling&quot; is. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-16T18:45:59.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="288" PostTypeId="5" CreationDate="2014-03-16T18:54:12.000" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-16T18:54:12.000" LastActivityDate="2014-03-16T18:54:12.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="289" PostTypeId="4" CreationDate="2014-03-16T18:54:12.000" Score="0" Body="Please do NOT use this tag; instead use general-pedagogy or mathematical-pedagogy." OwnerUserId="60" LastEditorUserId="143" LastEditDate="2014-06-21T20:20:02.117" LastActivityDate="2014-06-21T20:20:02.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="290" PostTypeId="5" CreationDate="2014-03-16T18:59:12.237" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-16T18:59:12.237" LastActivityDate="2014-03-16T18:59:12.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="291" PostTypeId="4" CreationDate="2014-03-16T18:59:12.237" Score="0" Body="For questions related to teaching elementary school students (generally speaking, those of ages about 5 to 11)." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-02T03:51:34.893" LastActivityDate="2014-04-02T03:51:34.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="292" PostTypeId="2" ParentId="57" CreationDate="2014-03-16T18:59:50.320" Score="4" Body="&lt;p&gt;Perhaps the real solution to the conundrum is to take out one of the restrictions. Limit the material (it is just &lt;em&gt;impossible&lt;/em&gt; to cover much material in any depth in 50 minutes), stretch time (perhaps by asking the students to prepare beforehand, or leave homework(ish) questions for later), or match abilities (ask the more quick students to help out others, perhaps by setting up work groups for each class).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-16T18:59:50.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="293" PostTypeId="2" ParentId="258" CreationDate="2014-03-16T19:01:01.783" Score="8" Body="&lt;p&gt;There are three main areas where students entering my calculus have trouble: keep in mind I teach a more formal calculus than most US schools these days. I do spend about 30% of the lecture time on proofs and I start with the axioms of the real numbers and some discussion of basic topology on the real line. I teach primarily from my notes and I'm not sure how much longer I'll be able to teach as I am given the invariable decline of the students with time. As it currently stands, I have been able to attract some excellent students with our calculus. They like the challenge.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;trigonometry:&lt;/strong&gt; many of them have been taught with calculators. Others not at all. Generally speaking there is a lot of ignorance about things they ought not need think hard about: $\sin(0)=0$, $\cos(0)=1$, graphs of sine, cosine, tangent. I just gave a physics (calculus-based, so these kids have already been through calculus) where no one could solve $0.8\sin \theta + \cos \theta = 1.2$. More generally, the problem is the bar is set too low in the previous courses. I suspect they are taught to solve standard problems rather than being taught how to think. Have they ever encountered a problem on an exam which was not in homework? Is that even on the table? Were they expected to take notes or just to fill in some template? Was homework done at home or in class? These are all factors.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;set notation, reading definitions, original thinking.&lt;/strong&gt; I've touched on this in 1. already, but it deserves it's own point. When I teach calculus I make a point of giving definitions. The student is expected to read it and use it. I see students totally shut-down when I ask a really simple question which requires nothing more than reading the set-builder notation and literally some number-line thinking. They go hunting for a problem like that, so they can follow the guide. This is a problem. The books we use teach them problem solving strategies as opposed to helping them think for themselves. I realize babies need milk, but at some point you gotta start to chew. Sooner would be better for them.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;False confidence.&lt;/strong&gt; Some students have taken a highshcool calculus before where it was mostly symbol-pushing. Well, I still make them push symbols around, but there is much more analysis involved. Can they actually write mathematics? As in , use sentences to explain what they are doing. Often, I spend a lot of ink on the first test just trying to communicate to them that they must connect the equations properly. To summarize, if getting the answer was sufficient in their previous course then it's an uphill battle when they get to my class and I tell them that the answer is the whole argument.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2014-03-17T05:05:49.293" LastActivityDate="2014-03-17T05:05:49.293" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="294" PostTypeId="1" CreationDate="2014-03-16T19:25:37.357" Score="38" ViewCount="2768" Body="&lt;p&gt;People often think of math as facts and procedure - dry stuff. But it is so much more, even at basic levels. What books about mathematics have you been inspired by? There are some real treasures out there - what are they?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your interest is in children's books, please check out &lt;a href=&quot;https://matheducators.stackexchange.com/questions/238/what-are-some-great-books-for-inspiring-children-to-explore-mathematics&quot;&gt;this question&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With your answer, please explain what makes this book appealing and what its math content is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are some great books for exploring mathematics?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2020-09-10T14:09:40.143" Title="What are some great books for exploring mathematics? (not kids' books and not textbooks)" Tags="|reference-request|reference-books|recreational-learning|" AnswerCount="26" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="295" PostTypeId="1" CreationDate="2014-03-16T19:50:00.850" Score="18" ViewCount="428" Body="&lt;p&gt;I notice that in current mathematics education they always have sections teaching about finding the missing member(s) of the sequences e.g. in this way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1,2,4,8,16$ , the next term is what?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Someone would argue that the next term is $32$ since they claim that the pattern of this sequences is about the non-negative integer powers of $2$ .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Someone would argue that the next term is $31$ since they claim that the pattern of this sequences is about the circle division by chords.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in fact this type of questions are completely wrong since e.g. by the principle of interpolation, since we can find infinitely many numbers of the patterns of the existing terms, the values of other terms will be very depending on the patterns assumed and therefore have infinitely many possibilities, i.e. you can fill any values of other terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the current mathematics education always assume this type of questions should have exact answers, that means they break the conscientiousness of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why the current mathematics education still hold this misleading education of the sequences?&lt;/p&gt;&#xA;" OwnerUserId="152" LastActivityDate="2014-03-17T06:06:46.290" Title="Are teaching about finding the missing member(s) of the sequences really appropriate?" Tags="|course-design|secondary-education|algebra|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="296" PostTypeId="2" ParentId="294" CreationDate="2014-03-16T19:50:38.837" Score="13" Body="&lt;p&gt;Two good books that I liked when I read them years ago are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Simon Singh: &lt;em&gt;The Code Book&lt;/em&gt;. This is a great introductory book to cryptography. The book is not very mathematical heavy, but cryptography is very related to number theory, so I think the book works well and can function well as an inspirational book.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simon Singh: &lt;em&gt;Fermat's Enigma&lt;/em&gt;. This is a book about Fermat's Last Theorem/ Simon Singh does a great job (IMO) at surveying the history behind FLT. He manages well to bring the mathematics down to an elementary level.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would suggest also a book on mathematical logic. There a lot of good books out there. I think that studying mathematical logic can inspire because it looks behind mathematics to the world of logic that mathematics formally is built upon. There are some great formal treatments out there. One concrete suggestion is &lt;em&gt;Gödel's Proof&lt;/em&gt; by Ernest Nagel and James Newman. This book discusses the famous (infamous?) theorems of Gödel and works very well as a soft and gentle introduction to logic. There is also a list of references for mathematical logic here: &lt;a href=&quot;https://math.stackexchange.com/questions/140681/where-to-begin-with-foundations-of-mathematics&quot;&gt;https://math.stackexchange.com/questions/140681/where-to-begin-with-foundations-of-mathematics&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It seems that many mathematicians got inspiration to study mathematics from studying problem books. There are, of course, a bunch of these out there. One is &lt;em&gt;The USSR Olympiad Problem Book&lt;/em&gt;. I believe that a good problem book can foster an the inquisitive mind that is needed in mathematics. One experiences the joy of problem solving.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="53" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-03-16T19:50:38.837" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="297" PostTypeId="2" ParentId="295" CreationDate="2014-03-16T20:18:29.163" Score="18" Body="&lt;p&gt;Theoretically, there's no way to determine the next term in the sequence&#xA;$$&#xA;1,\quad 2,\quad 4,\quad 8,\quad 16,\quad\ldots&#xA;$$&#xA;It literally could be anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, it is a vitally important skill to be able to look at this sequence and say &quot;it looks like the powers of 2&quot;.  This answer is &lt;em&gt;correct&lt;/em&gt; in the sense that any mathematician looking at this sequence would have that response, and a student who doesn't have that response when looking at this sequence has a serious gap in their knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is this an important skill?  Because it's very common in mathematics that you don't know the answer to a question, but you can make a lot of progress by &lt;em&gt;guessing&lt;/em&gt; the answer and then trying to prove it.  If there's a sequence that you're trying to understand, and the first few terms are $1,2,4,8,16$, then the obvious thing to do is to try to prove that the $n$th term is equal to $2^n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a special case of the critically important skill of &lt;strong&gt;&lt;em&gt;pattern recognition&lt;/em&gt;&lt;/strong&gt;, which is necessary for solving almost any difficult problem.  Indeed, in mathematics research, forming the appropriate conjectures is in many cases more difficult than proving these conjectures.  This is a &quot;soft&quot; skill, in the sense that there's no precise definition of &quot;the right conjecture&quot;, but it's still vitally important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, recognizing patterns in sequences is so important that there is an entire website, the &lt;a href=&quot;https://oeis.org&quot;&gt;Online Encyclopedia of Integer Sequences&lt;/a&gt;, that is designed to help with this.  The idea of the website is that you go there, type in the first few terms of a sequence you have found, and the encyclopedia tells you whether anyone else has ever studied that particular sequence.  This website is commonly used by mathematics researchers, and can be a vital tool for discovering patterns and making connections between different areas of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-16T20:18:29.163" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="298" PostTypeId="1" AcceptedAnswerId="336" CreationDate="2014-03-16T20:23:07.337" Score="19" ViewCount="1870" Body="&lt;p&gt;Does anyone have any experience giving students incorrectly &quot;solved&quot; math problems and asking them to identify this error? Being self-critical is one of the skills that I would like my students to take away from a remedial/intro college-level math class (you know, beyond numeracy, etc), but I am not sure how to support this kind of learning (besides using problems like &quot;correct the following attempted solution.&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So two questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Have you ever asked these kinds of problems? Did it help?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do you know of other strategies to support students learning how to check their own work?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="98" LastActivityDate="2014-08-29T22:48:45.263" Title="&quot;Correct the following mistake&quot;-style questions?" Tags="|problem-design|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="299" PostTypeId="2" ParentId="298" CreationDate="2014-03-16T20:32:57.893" Score="5" Body="&lt;p&gt;This will only address your first question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a bit of experience with asking students to find mistakes. I have (only) done this in the classroom and have not asked students to find mistakes in anything written.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I sometimes do is to tell the students that I will write down the solution and make three mistakes that they have to find. Most of the time the mistakes will be forgetting to close a parentheses or forgetting to write $\lim_{n\to \infty}$ when doing an improper integral. I might even make spelling mistakes. I haven't been able to evaluate how effective this all is, but I do believe that it helps. It targets the importance of correct notation. It also, in particular, works well as a way to get students to wake up during class and can be used as a tool to do just that.&lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-03-16T20:32:57.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="300" PostTypeId="2" ParentId="130" CreationDate="2014-03-16T21:01:56.437" Score="8" Body="&lt;p&gt;This is but a part of the bigger picture and certainly not the most important one: &lt;strong&gt;Speeding up the legwork.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously you have to be better than your students in what you teach to, e.g., design and correct tests in a reasonable time. There are two ways of getting better at such things: Getting more routine with the techniques you are teaching or having a deeper understanding and knowing more advanced techniques. The former has its limits (and so does the latter) and thus you need the latter, at least to some extent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One illustrative example (that is unfortunately far from the high-school level and also not fully realistic):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is the following calculation correct? $$152345+10340+532045+139525+10530+123525=968313$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can decide about this having learnt only how to add by simply calculating the correct result and checking it. But most likely you will quickly notice that the sum of some numbers ending on 5 and 0 can never end on 3. But that already requires more knowledge than just being able to add numbers mechanically.&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2014-03-16T21:01:56.437" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="301" PostTypeId="2" ParentId="298" CreationDate="2014-03-16T21:09:06.807" Score="5" Body="&lt;p&gt;I use this regularly and have the impression it helps a lot in understanding arguments or getting missing points. Typical examples are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In induction proofs not checking the case $n=1$&lt;/li&gt;&#xA;&lt;li&gt;In solving equations dividing by zero&lt;/li&gt;&#xA;&lt;li&gt;Multiplying inequalities with negative numbers&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What I try to do is to stress that this is a bad argument also by formulating a proof for a trivially false statement (like $1=0$). I do this because I always have a feeling that some students may not notice the mistake and try to memorize the proof. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, this might be a slight problems: with this method you know how to recognize a mistake when you know there is one, but not necessarily become more self-critic.&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-03-16T21:09:06.807" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="302" PostTypeId="2" ParentId="238" CreationDate="2014-03-16T22:00:21.947" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://www.logicomix.com/en/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Logicomix&lt;/em&gt;&lt;/a&gt; is a Greek graphical novel about the quest for Foundations of Mathematics in the 20th century, from &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_problems&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert's problems&lt;/a&gt; at the 1900 conference in Paris to &lt;a href=&quot;https://en.wikipedia.org/wiki/Godel%27s_incompleteness_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gödel's incompleteness theorem&lt;/a&gt;, using the &lt;a href=&quot;https://en.wikipedia.org/wiki/Barber_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;barber paradox&lt;/a&gt; as an illustration.&lt;/p&gt;&#xA;&lt;p&gt;Basically, it is narrated by Bertrand Russell and the whole book is fascinating. It covers a span of sixty years from Russell's own childhood desire to prove everything  to Gödel's result that shocked the mathematical world and all mathematics-related material is covered with a didactic purpose.&lt;/p&gt;&#xA;&lt;p&gt;It was written by Apostolos Doxiadis and the CS researcher Christos Papadimitriou and translated into &lt;a href=&quot;https://www.logicomix.com/en/index30af.html?option=com_content&amp;amp;view=article&amp;amp;id=92&amp;amp;Itemid=28#twoj_fragment1-6&quot; rel=&quot;nofollow noreferrer&quot;&gt;many languages&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="171" LastEditorDisplayName="user7990" LastEditDate="2020-09-11T21:40:17.467" LastActivityDate="2020-09-11T21:40:17.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="303" PostTypeId="2" ParentId="295" CreationDate="2014-03-16T22:06:00.180" Score="6" Body="&lt;p&gt;As @JimBelik aptly says, such problems are about pattern recognition. An important qualification of this is that the question might/should ask about the next number produced by the &lt;em&gt;simplest&lt;/em&gt; pattern(s). That is, the &quot;formally correct&quot; notion that &quot;of course there is no unique next number&quot; is in fact not useful in practice. After the observation that &quot;in principle&quot; there is no unique next number, one should still pursue the issue of recognizing relatively-simple patterns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such questions could/should be asked in a form that requires more than just the bare number, of course. &quot;Give a reason&quot;. &quot;Give a simple formula&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I disagree with the premise of the original question that there is something innately &quot;unconscientious&quot; about this. In the example given, the simplest pattern is powers-of-two, as noted by Jim Belik. The second-simplest is arguably &lt;em&gt;polynomial&lt;/em&gt; (successive differencing) interpolation, which gives 31. What's the problem?)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-16T22:06:00.180" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="304" PostTypeId="2" ParentId="220" CreationDate="2014-03-16T22:50:30.950" Score="25" Body="&lt;p&gt;Two more examples.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Proving that a $2^n \times 2^n$ chessboard with a single square missing can be covered using L-shaped (made out of three squares) pieces.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Proving that a convex $n$-gon can be divided into $n-2$ triangles.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="18" LastEditorUserId="18" LastEditDate="2014-05-20T23:11:03.257" LastActivityDate="2014-05-20T23:11:03.257" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="305" PostTypeId="2" ParentId="220" CreationDate="2014-03-16T23:08:13.790" Score="15" Body="&lt;p&gt;I indeed think &lt;span class=&quot;math-container&quot;&gt;$\sum_{k = 1}^n k = \frac{n(n + 1)}2$&lt;/span&gt; is not a good first example as writing the sum forwards and backwards is more clever and natural for a newcomer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But proving “the sum of the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; first odd numbers is &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt;”, while really close to the equality above, is a better example, because the induction part of the proof is really simple: &lt;span class=&quot;math-container&quot;&gt;$\sum_{k = 1}^{n + 1}(2k-1)= \left(\sum_{k = 1}^n(2k-1)\right) + 2n + 1 = n^2 + 2n + 1 = (n + 1)^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof by induction is more necessary and useful in a computer science scenario, where some objects are defined in an inductive way: “a complete binary tree of height &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$2^n - 1$&lt;/span&gt; nodes”: straightforward for &lt;span class=&quot;math-container&quot;&gt;$n = 1$&lt;/span&gt;; for &lt;span class=&quot;math-container&quot;&gt;$n \geqslant 2$&lt;/span&gt;, its left and right child are binary trees of height &lt;span class=&quot;math-container&quot;&gt;$n - 1$&lt;/span&gt;, so it has &lt;span class=&quot;math-container&quot;&gt;$2(2^{n - 1} - 1) + 1 = 2^n - 1$&lt;/span&gt; nodes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example where a special method of inductive reasoning called &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Infinite_descent&quot; rel=&quot;nofollow noreferrer&quot;&gt;infinite descent&lt;/a&gt;&lt;/em&gt; is useful: “show that &lt;span class=&quot;math-container&quot;&gt;$\sqrt2$&lt;/span&gt; is irrational”. Let us assume there exist two natural numbers &lt;span class=&quot;math-container&quot;&gt;$p, q$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\sqrt2 = p/q$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$2q^2 = p^2$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is even, thus there exists some natural number &lt;span class=&quot;math-container&quot;&gt;$p'$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$p = 2p'$&lt;/span&gt;, therefore &lt;span class=&quot;math-container&quot;&gt;$2q^2 = 4p'^2$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$q^2 = 2p'^2$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is even as well, &lt;em&gt;i.e.&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$q = 2q'$&lt;/span&gt; holds for some natural number &lt;span class=&quot;math-container&quot;&gt;$q'$&lt;/span&gt;. We then get &lt;span class=&quot;math-container&quot;&gt;$2q'^2 = p'^2$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$(p', q') = (p / 2, q / 2)$&lt;/span&gt; is a strictly lesser representation of &lt;span class=&quot;math-container&quot;&gt;$\sqrt2$&lt;/span&gt;. As there is no infinite decreasing sequence of natural numbers, this leads to a contradiction, so &lt;span class=&quot;math-container&quot;&gt;$\sqrt2$&lt;/span&gt; is irrational.&lt;/p&gt;&#xA;" OwnerUserId="171" LastEditorUserId="11" LastEditDate="2019-07-08T00:08:48.180" LastActivityDate="2019-07-08T00:08:48.180" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="306" PostTypeId="2" ParentId="295" CreationDate="2014-03-16T23:11:05.933" Score="11" Body="&lt;p&gt;After reading the other answers and comments, it seems that there is a solution that combines the best of both worlds. Give students problems where patterns arise, then ask them to guess the pattern &lt;em&gt;and prove it&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The level of question/proof depends on the grade. &lt;a href=&quot;https://math.stackexchange.com/questions/406099/my-sons-sum-of-some-is-beautiful-but-what-is-the-proof-or-explanation&quot;&gt;This post&lt;/a&gt; gives an example of a pattern that algebra students should be able to prove.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example for undergraduates: There are $n$ parking spots on the side of the road. You can fill them with Lincoln town cars (which take up 2 spaces) or Yugos (which take up one space). How many ways are there of completely filling up all $n$ spaces?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus example: Find $\int_1^\infty x^n e^x dx$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This type of example encourages pattern finding as a means to an end.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-03-16T23:11:05.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="307" PostTypeId="2" ParentId="257" CreationDate="2014-03-16T23:13:36.403" Score="12" Body="&lt;p&gt;One professor (who I greatly respect) did as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;He made the papers a part of the exam, the second part being the talk/presentation.&lt;/li&gt;&#xA;&lt;li&gt;The papers had to be in some particular conference format (there were style files at the conference page).&lt;/li&gt;&#xA;&lt;li&gt;There were two dates in which students could present, and it would be organized in a mini-conference form (20 minutes for the talk, 5 minutes for questions, 5 minutes for connecting laptop).&lt;/li&gt;&#xA;&lt;li&gt;There were 8-9 topics listed that students could pick from (there were about 20 people in the class) which were simplified cases of some research problems.&lt;/li&gt;&#xA;&lt;li&gt;You could do your own topic if you could persuade the professor (actually anything reasonable worked).&lt;/li&gt;&#xA;&lt;li&gt;&quot;I took the XYZ approach and failed&quot; were valid results.&lt;/li&gt;&#xA;&lt;li&gt;Students worked alone (but there were exceptions). They could discuss their project during office hours, but were not required to. &lt;/li&gt;&#xA;&lt;li&gt;There were no drafts, but if you submitted a draft, your paper would get &quot;peer reviewed&quot; by the professor.&lt;/li&gt;&#xA;&lt;li&gt;Some of the works were later submitted to student session of the said conference.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In my opinion it worked pretty well. One key thing was that each topic was assigned to more than one student and so it could be later compared, it made an interesting discussion and some funny discoveries. It is worth to point out that the professor would never use students' data against them (even if it was clear that one of them did something wrong). Finally, I think it is significant that it was a practical course, namely &lt;em&gt;natural language processing&lt;/em&gt;, where there is a big supply of easy research questions (and is quite entertaining because of the &quot;insights&quot; of machine learning algorithms).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-20T12:10:15.687" LastActivityDate="2014-03-20T12:10:15.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="308" PostTypeId="2" ParentId="238" CreationDate="2014-03-16T23:20:31.187" Score="2" Body="&lt;p&gt;I really enjoy &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0987823914&quot; rel=&quot;nofollow&quot;&gt;Introductory Calculus for Infants&lt;/a&gt;. It is an abc book for young readers. It has very little rigorous content and some weird areas (one joke requires you to pronounce $ln$ as 'lawn'), but it has a lot of the interesting symbols and graphs that got me into math in the first place. Every page has a function and a graph corresponding to a letter. The X page has $f=\pm|x|$, for instance.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-16T23:20:31.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="309" PostTypeId="2" ParentId="294" CreationDate="2014-03-16T23:36:27.270" Score="10" Body="&lt;p&gt;Martin Gardner's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0393020231&quot;&gt;The Colossal Book of Mathematics&lt;/a&gt;. It contains many of his best columns from Scientific American on recreational mathematics. My favorite chapter is the April Fool's day chapter, which includes a 'counter-example' to the Four-Coloring Theorem.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="60" LastEditDate="2014-03-17T00:25:38.823" LastActivityDate="2014-03-17T00:25:38.823" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="310" PostTypeId="2" ParentId="280" CreationDate="2014-03-16T23:41:49.727" Score="3" Body="&lt;p&gt;This type of problem is almost impossible without algebra:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If John paints a house in 3 hours and Jane paints a house in 2 hours, how long does it take them to paint 5 houses together?&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-16T23:41:49.727" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="311" PostTypeId="2" ParentId="269" CreationDate="2014-03-17T00:01:03.680" Score="8" Body="&lt;p&gt;I think a nice way to introduce groups is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Rubik%27s_Cube_group&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rubik's Cube non-commutative group&lt;/a&gt;. Also &lt;a href=&quot;https://en.wikipedia.org/wiki/Tessellation&quot; rel=&quot;nofollow noreferrer&quot;&gt;tessellation&lt;/a&gt; can be a way to inspire the need of a simple underlying structure to represent some complex sets. Also the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dihedral_group&quot; rel=&quot;nofollow noreferrer&quot;&gt;dihedral group&lt;/a&gt; has the advantage of being visual.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As dtldarek pointed out, you can praise finite fields for their applications in asymmetric cryptography (&lt;a href=&quot;https://en.wikipedia.org/wiki/RSA_(cryptosystem)&quot; rel=&quot;nofollow noreferrer&quot;&gt;RSA&lt;/a&gt;) or error-correcting codes like Reed-Solomon with their application to &lt;a href=&quot;https://en.wikipedia.org/wiki/Reed-Solomon_code&quot; rel=&quot;nofollow noreferrer&quot;&gt;CDs&lt;/a&gt;. Or more generally the existence and ease of computing the inverse of any element of a finite field using &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclidean_algorithm#Extended_Euclidean_algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;the extended Euclidean algorithm&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My favorite hidden application of algebra is the &lt;a href=&quot;https://stackoverflow.com/questions/6240113/what-are-the-mathematical-computational-principles-behind-this-game&quot;&gt;Spot It!&lt;/a&gt; game, which uses &lt;a href=&quot;https://en.wikipedia.org/wiki/Projective_plane#Finite_field_planes&quot; rel=&quot;nofollow noreferrer&quot;&gt;projective planes over finite fields&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="171" LastEditorUserId="-1" LastEditDate="2017-05-23T12:41:54.857" LastActivityDate="2014-03-17T00:01:03.680" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="312" PostTypeId="2" ParentId="130" CreationDate="2014-03-17T00:03:26.720" Score="25" Body="&lt;p&gt;Here is some text from the paper &quot;Teacher Characteristics and Student Achievement Gains: A Review&quot; (Wayne and Youngs, 2003):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Three analyses take advantage of the detailed teacher data in NELS:88&#xA;  on degrees. The analysis by Goldhaber and Brewer (1997a) illustrates&#xA;  the key finding most clearly. No differences were evident when the&#xA;  authors examined whether 10th-grade mathematics students scored better&#xA;  when their teachers had master’s degrees. However, introducing&#xA;  information about the subject of the teachers’ degrees produced&#xA;  significant results. Mathematics students whose teachers had master’s&#xA;  degrees in mathematics had higher achievement gains than those whose&#xA;  teachers had either no advanced degrees or advanced degrees in&#xA;  nonmathematics subjects. In addition, students whose teachers had&#xA;  bachelor’s degrees in mathematics learned more than students whose&#xA;  teachers had bachelor’s degrees in nonmathematics subjects. The&#xA;  contributions of these two indicators of subject preparation were&#xA;  independent of several other teacher characteristics; the student&#xA;  achievement model controlled for certification, mathematics&#xA;  certification, and years of high school teaching experience. The two&#xA;  other analyses of the NELS:88 data replicated this finding, again for&#xA;  mathematics, but with slight variations. Goldhaber and Brewer (2000)&#xA;  used the 12th- rather than 10th-grade students and again found that&#xA;  students learned more from teachers with mathematics majors and from&#xA;  teachers with master’s degrees in mathematics. Rowan, Chiang, and&#xA;  Miller (1997) used a single variable to indicate whether the teacher&#xA;  had an undergraduate or graduate degree in mathematics. In addition,&#xA;  as described earlier, they added a crude control for teachers’ test&#xA;  scores in mathematics knowledge.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-17T00:03:26.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="313" PostTypeId="2" ParentId="280" CreationDate="2014-03-17T00:20:35.620" Score="9" Body="&lt;p&gt;Here is a great riddle:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A bottle of wine costs €20. The wine costs €19 more than the empty bottle. How much does the empty bottle cost?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Everyone will answer €1. But 1 + (19 + 1) = 21. Putting it into an equation will lead to the €0.50 answer.&lt;/p&gt;&#xA;" OwnerUserId="171" LastActivityDate="2014-03-17T00:20:35.620" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="314" PostTypeId="2" ParentId="294" CreationDate="2014-03-17T00:37:10.797" Score="6" Body="&lt;p&gt;There is a awesome list on similar topics on &lt;a href=&quot;https://math.stackexchange.com/questions/275/best-maths-books-for-non-mathematicians&quot;&gt;Maths Stack Exchange&lt;/a&gt;, even more awesome list on &lt;a href=&quot;https://mathoverflow.net/questions/8609/favorite-popular-math-book&quot;&gt;Math Overflow&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I heard a lot of &lt;a href=&quot;https://mathoverflow.net/a/8673/38962&quot;&gt;good things&lt;/a&gt; about &lt;a href=&quot;https://en.wikipedia.org/wiki/Raymond_Smullyan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smullyan's bibliography&lt;/a&gt;, and a lot of &lt;a href=&quot;https://mathoverflow.net/questions/8609/favorite-popular-math-book#comment11696_8613&quot;&gt;bad things&lt;/a&gt; about &lt;a href=&quot;https://en.wikipedia.org/wiki/Godel_escher_bach&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Gödel, Escher, Bach&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="171" LastEditorDisplayName="user7990" LastEditDate="2020-09-10T14:09:40.143" LastActivityDate="2020-09-10T14:09:40.143" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="315" PostTypeId="2" ParentId="224" CreationDate="2014-03-17T01:18:11.657" Score="4" Body="&lt;p&gt;I also think there might be a steep learning curve for &lt;a href=&quot;http://coq.inria.fr/a-short-introduction-to-coq&quot; rel=&quot;nofollow&quot;&gt;Coq syntax&lt;/a&gt;. As you're referring also to CS majors, you can introduce loop invariants along with proofs by induction using the platform &lt;a href=&quot;http://why3.lri.fr/&quot; rel=&quot;nofollow&quot;&gt;Why3&lt;/a&gt;. It is intended for deductive program verification but very few lines of code (OCaml here) are needed to demonstrate its use:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://toccata.lri.fr/gallery/division.en.html&quot; rel=&quot;nofollow&quot;&gt;Euclidean division&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://toccata.lri.fr/gallery/power.en.html&quot; rel=&quot;nofollow&quot;&gt;Fast exponentiation&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://toccata.lri.fr/gallery/bresenham.en.html&quot; rel=&quot;nofollow&quot;&gt;Bresenham line-drawing algorithm&lt;/a&gt; introduced in the &lt;a href=&quot;http://www.msr-inria.fr/wp-content/uploads/2013/06/slides_jc_filliatre.pdf&quot; rel=&quot;nofollow&quot;&gt;slides&lt;/a&gt; (the slide full of proof code is ironic)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If bad invariants are used, Why3 rejects the proof. If you're looking for more mathematical examples, you can use this system to prove by induction that a function computing factorials is correct, for example.&lt;/p&gt;&#xA;" OwnerUserId="171" LastActivityDate="2014-03-17T01:18:11.657" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="316" PostTypeId="2" ParentId="243" CreationDate="2014-03-17T01:22:30.147" Score="8" Body="&lt;p&gt;I think that the set of reals isn't the one you would like to start with. The problem with it is that they have their quirks and for many it is counter-intuitive that any open interval is of uncountable size. On the other hand, there are many other objects with which the Cantor's diagonal argument works as well (sometimes it works even better). To name a few:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Infinite binary sequences, functions $\mathbb{N} \to 2$ or $\mathcal{P}(\mathbb{N})$. We define $g(k) = f_k(k) \oplus 1$.&lt;/li&gt;&#xA;&lt;li&gt;Unbounded functions $\mathbb{N} \to \mathbb{N}$. We define $g(k) = f_k(k)+k+1$.&lt;/li&gt;&#xA;&lt;li&gt;Strictly increasing functions $\mathbb{N} \to \mathbb{N}$. We define $g(k) = \max_{i = 0}^{k}f_i(k)+1$.&lt;/li&gt;&#xA;&lt;li&gt;Infinite subsets of $\mathbb{N}$. We sort them and proceed as with strictly increasing functions/sequences.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To make the appearance even stronger, make the case how any natural number contains a finite amount of information (e.g. you can write it down in finite, even if arbitrarily long time), while some of the above objects hold infinite amount of information (functions/sequences $\mathbb{N} \to \mathbb{N}$ are perfect for this). This could be done by show how you can take any finite amount of information from it, and it is still the same (in a sense):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;if you take some bits from natural number, it will get smaller, and eventually, after finite number of steps, &lt;em&gt;will&lt;/em&gt; decrease to zero,&lt;/li&gt;&#xA;&lt;li&gt;similar thing happens to sequences which have only finitely many non-zero entries,&lt;/li&gt;&#xA;&lt;li&gt;on the other hand, if you truncate some prefix from a sequence of infinite support, the rest still retains an infinite amount of information.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now you can move to reals, perhaps using continued fractions (where rationals are exactly the fractions of finite length), or some other methods and show that there are at least as many reals as the aforementioned objects. Then, you could make an analogy between truncating the prefix and taking an arbitrarily small open interval. Also, note that we take bits of information from the element itself, not the set. For example the set of rationals would still look similar after we would take some elements from it, but no rational number would keep all its information after truncating some of its bits. On the other hand, there are some real numbers which could be written down completely, but more importantly there are those which wouldn't fit on any piece of paper ever. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, the numbers which make $\mathbb{R}$ uncountable are inaccessible to us, because as humans we can write down only finite strings and it's not enough. This changes if we had an access to an alphabet of uncountable size, e.g. by assuming that the world and our behavior is continuous and specifying the number as the length between two dots drawn on a sheet of paper. However, this gets disturbingly close to philosophy, so I shall not continue further.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-17T01:22:30.147" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="317" PostTypeId="2" ParentId="178" CreationDate="2014-03-17T02:22:52.587" Score="8" Body="&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=0Y5fAAAACAAJ&amp;amp;dq=inauthor%3a%22Halsey%20L.%20Royden%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=ulsmU6SOAYer2gXzuIDQBQ&amp;amp;ved=0CDwQ6AEwAQ&quot; rel=&quot;noreferrer&quot;&gt;Royden's &lt;em&gt;Real Analysis&lt;/em&gt;&lt;/a&gt; is a great book on Lebesgue measure. It is very accessible. I haven't seen the 4th edition yet, but the 3rd edition starts out by giving a short overview of set theory and goes over the contruction of the real numbers. Later in the book there is discussion of abstract measures.&lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-03-17T02:22:52.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="318" PostTypeId="1" AcceptedAnswerId="909" CreationDate="2014-03-17T02:28:48.900" Score="7" ViewCount="345" Body="&lt;p&gt;I am looking for both a course hierarchy of mathematics education (for example, &lt;em&gt;Galois theory&lt;/em&gt; is part of &lt;em&gt;abstract algebra&lt;/em&gt;) and a representation of all competences involved in learning mathematics (possibly with prerequisites links, for example a student needs to know what is a limit in order to understand derivation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only ontology I found for the latter is &lt;a href=&quot;http://i2geo.net/comped/showProcess.html?uri=Competency&quot; rel=&quot;nofollow&quot;&gt;there&lt;/a&gt;. And I think a good start to build a course hierarchy would be to use &lt;a href=&quot;https://github.com/stared/tag-graph-map-of-stackexchange/wiki&quot; rel=&quot;nofollow&quot;&gt;this tag graph of Mathematics Stack Exchange&lt;/a&gt;. Can you provide any other valuable resources?&lt;/p&gt;&#xA;" OwnerUserId="171" LastEditorUserId="171" LastEditDate="2014-03-17T09:49:52.810" LastActivityDate="2014-03-28T19:39:12.060" Title="Is there a tag/competence classification for mathematics education?" Tags="|mathematical-pedagogy|course-design|meta|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="319" PostTypeId="1" AcceptedAnswerId="323" CreationDate="2014-03-17T02:29:08.287" Score="4" ViewCount="145" Body="&lt;p&gt;For some objects there are alternate definitions, which are &quot;morphism-oriented&quot;. To give some examples, there are two definitions of a prime number:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$p$ is prime if it is greater than $1$ and has no positive divisors other than $1$ and itself,&lt;/li&gt;&#xA;&lt;li&gt;$p$ is prime if for any $a$ and $b$ we have $p \mid ab \implies p \mid a \lor p \mid b$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also, there are two definitions of the Cartesian product:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the product $A \times B$ is the set of ordered pairs $\langle a,b\rangle$ such that $a \in A$ and $b \in B$,&lt;/li&gt;&#xA;&lt;li&gt;the product $A \times B$ is any set $P$ such that there exists functions $\pi_A : P \to A$ and $\pi_B : P \to B$ such that for any set $Q$ and functions $f_A : Q \to A$ and $f_B : Q \to B$ there exists a unique function $f : Q \to P$ such that $f_A = \pi_A \circ f$ and $f_B = \pi_B \circ f$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The second definitions are those which I call &quot;morphism-oriented&quot;. At first those might be harder to understand, but they capture some essential properties which aren't so apparent with the simpler definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main question is: &lt;strong&gt;what is your experience using them (if you did)?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some support questions might be as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What are their advantages or disadvantages?&lt;/li&gt;&#xA;&lt;li&gt;Are there any examples where such definitions shine or perform very badly?&lt;/li&gt;&#xA;&lt;li&gt;Are such definitions suitable for younger students (i.e. pre-college).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-17T03:03:06.947" Title="Morphism-oriented definitions" Tags="|undergraduate-education|course-design|definitions|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="320" PostTypeId="2" ParentId="98" CreationDate="2014-03-17T02:30:00.697" Score="6" Body="&lt;p&gt;For some references on the differential forms approach, the following two math.SE questions might be helpful (because of their references): &lt;a href=&quot;https://math.stackexchange.com/questions/714646/construction-of-the-hyperreal-numbers&quot;&gt;https://math.stackexchange.com/questions/714646/construction-of-the-hyperreal-numbers&lt;/a&gt;, &lt;a href=&quot;https://math.stackexchange.com/questions/315768/how-to-create-the-set-of-hyperreal-numbers-using-ultraproduct&quot;&gt;https://math.stackexchange.com/questions/315768/how-to-create-the-set-of-hyperreal-numbers-using-ultraproduct&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, if you want to understand the integral, I would suggest starting with measure theory. Understanding abstract measure theory will be like turning on a light that illuminates all of integration theory. In this setup, the meaning of the $dx$ becomes clear. So, in my opinion one should start with measure theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might look at this question: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/178/best-textbooks-to-introduce-measure-theory-and-lebesgue-integration&quot;&gt;Best textbooks to introduce measure theory and Lebesgue integration?&lt;/a&gt; for references.&lt;/p&gt;&#xA;" OwnerUserId="53" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-03-17T02:30:00.697" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="321" PostTypeId="2" ParentId="243" CreationDate="2014-03-17T02:43:20.947" Score="7" Body="&lt;p&gt;If you ignore repeating decimals, you aren't giving a good proof.  As an alternative, ignore decimals altogether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $q_n$ be an enumeration of real numbers, starting with $n=1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $x_0=0$.  For each $i$, set:&#xA;$$x_{i+1}=x_i \ \ \ \ \ \ \ \ \ \ \ \text{ if } q_{i+1} \ge x_i + 4^{-i}/2$$&#xA;$$x_{i+1}=x_i + 4^{-i} \ \text{ if } q_{i+1} &amp;lt; x_i + 4^{-i}/2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $x$ be the limit of the $x_i$.  Then $|x-x_{i+1}| \le 4^{-i}/3$ and $|q_{i+1}-x_{i+1}| \ge 4^{-i}/2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So $|q_{i+1}-x| \ge 4^{-i}/6$, and $x$ is distinct from all the $q$'s.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-03-30T10:23:30.470" LastActivityDate="2014-03-30T10:23:30.470" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="322" PostTypeId="2" ParentId="270" CreationDate="2014-03-17T02:55:49.887" Score="2" Body="&lt;p&gt;For your college example:  &lt;strong&gt;remove the techniques of integration from the calculus curriculum.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teach numerical integration instead, e.g.:  What is&#xA;$$\int_0^1 e^{-x^2} dx$$&#xA;and how do you calculate it to within 0.001?&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-17T02:55:49.887" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="323" PostTypeId="2" ParentId="319" CreationDate="2014-03-17T02:55:52.367" Score="6" Body="&lt;p&gt;Your second definition for the Cartesian product is the definition of a product of objects in a category (as I am sure that you are aware). So the question basically (as I am hearing it) becomes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is teaching category theory useful and when can you start introducing it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think category theory is extremely helpful for a mathematics student who is trying to make sense of definitions across various areas of mathematics. The language of category theory is very helpful because it (IMO) provides a nice general approach to various definitions/ In category theory we have general definitions of products of objects and this in the category of sets becomes the usual Cartesian product. You can then show how your definition one is a way to concretely define a product in that category. But this also extends well to, for example, the product of topological spaces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When can you start introducing this? Some mathematicians (my self included) wish that they had been introduced to category theory earlier than they were. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, it is hard to get a grasp on categories without looking at examples. And so the student should probably be familiar with basic concepts like: sets, groups, rings, vector spaces, topological spaces. I think it becomes hard to provide examples without a good background in various topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In conclusion I would probably stick with defining the Cartesian product the first way you did it. But this, again, depends on the level of your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think something similar can be said about the definition of a prime number. Here again the second definition is helpful to when trying to understand why prime numbers are so important. And it can be an eye opener when you see this when talking about integral domains. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I don't think that these &quot;morphism definitions&quot; would be good in a pre-college setting simply because they require much to get at examples. They are great at getting a better understand of &quot;what is really going on&quot;, but they are just not very accessible. But as an exercise in abstraction, I would think that you could get something good out of it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think another example of this is the question of whether or not you could introduce group theory in high school. I think you can, but again it isn't easy because it is hard to give good examples. But then again, I think that since high school students do have some familiarity with rational numbers, real numbers, and some even complex numbers, it might just work out.&lt;/p&gt;&#xA;" OwnerUserId="53" LastEditorUserId="53" LastEditDate="2014-03-17T03:03:06.947" LastActivityDate="2014-03-17T03:03:06.947" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="324" PostTypeId="2" ParentId="243" CreationDate="2014-03-17T03:00:27.843" Score="5" Body="&lt;p&gt;One way is to show that no countable set can represent what we think of as the number line. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1.See if your students agree that an open interval $(a,b)$ has 'size' $b-a$.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask them if the size of a union of sets is less than or equal to the sum of their individual sizes (better done by drawing an exampke of overlapping intervals than by just talking).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For any countable set, put an interval of size one around the first element, size 1/2 around the second, and so on. The sum of the individual interval sizes is 2, but the size of the real numbers is infinite.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If they have difficulty visualizng the remaining set, you can show a picture of the Cantor set or the Sierpinski carpet to show how removing a dense set of intervals or disks can still leave points behind.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-17T03:00:27.843" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="325" PostTypeId="2" ParentId="267" CreationDate="2014-03-17T03:06:34.220" Score="6" Body="&lt;p&gt;Many of Jim Belk's answers are good.  But let me state for the record, because it always comes up:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Euclid's proof of the infinitude of primes is &lt;em&gt;not&lt;/em&gt; a proof by contradiction.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Look at Euclid's text, e.g. the translation &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html&quot;&gt;here&lt;/a&gt;:  &quot;Prime numbers are more than any assigned multitude of prime numbers.&quot;  The proof is constructive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found that proving this theorem by contradiction confuses students.  I would encourage others to prove it constructively instead.  In any case:  do not attribute the proof by contradiction to Euclid.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-17T03:06:34.220" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="326" PostTypeId="2" ParentId="269" CreationDate="2014-03-17T03:09:51.557" Score="5" Body="&lt;p&gt;You ask for applications of abstract algebra. One obvious application is in physics (Just google &quot;group theory and physics). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I also wanted to point out that one doesn't necessarily need applications to motivate a topic. If you want to motivate abstract algebra, you could start by discussing the operations you can perform with numbers. You can show that under multiplication, only $1$ and $-1$ have inverses in the natural numbers. But in the rational numbers all non-zero numbers have multiplicative inverses. This all leads very naturally to the definition of groups, rings, and fields. You could look at the integers $\mathbb{Z}$ modulo a given number $n$ (and then you have &lt;a href=&quot;http://en.wikipedia.org/wiki/Modular_arithmetic&quot; rel=&quot;noreferrer&quot;&gt;clock arithmetic&lt;/a&gt;). A very natural question then becomes: for which $n$ do we get a field?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point is that motivation of many mathematical concepts can come from the fact that they arise so naturally. You show something that is already familiar and then you ask a slightly more abstract question and then boom you have all of abstract algebra (exaggerating a bit here). &lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-03-17T03:09:51.557" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="327" PostTypeId="2" ParentId="198" CreationDate="2014-03-17T03:26:20.653" Score="4" Body="&lt;p&gt;Not to blow my own horn, but a significant portion of &lt;a href=&quot;http://deepblue.lib.umich.edu/handle/2027.42/64672&quot; rel=&quot;noreferrer&quot;&gt;my doctoral dissertation&lt;/a&gt; was devoted to an analysis of memoirs and biographies of mathematicians at work.  Chapter 2 is probably the most relevant. &lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-03-17T03:26:20.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="328" PostTypeId="2" ParentId="122" CreationDate="2014-03-17T03:28:07.750" Score="2" Body="&lt;p&gt;Whether you want a student to know $\sin(\pi/4)=1/\sqrt{2}$ depends on &lt;strong&gt;the goal of teaching trigonometry&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A century ago -- before radar, before aerial maps, before the wonders of modern calculation -- &lt;strong&gt;trigonometry was useful&lt;/strong&gt; for surveying and navigation and more.  That may explain its historical position in the high school curriculum.  Nowadays, trigonometry does not deserve anywhere near a dedicated semester of high school math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;I would advocate teaching the sine function as the key example of a periodic function.&lt;/strong&gt;  If a high school student can answer the question &quot;where does a circle centered at the origin intersect the line $y=x$?&quot;, it doesn't matter to me if they answer numerically or algebraically.  But I'd rather that U.S. high schools stop with trigonometry around there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would rather that high schools stay away from the pretty formula in the question.  Time on that could be better spent on other topics, such as probability and statistics.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-03-17T03:37:28.943" LastActivityDate="2014-03-17T03:37:28.943" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="329" PostTypeId="2" ParentId="270" CreationDate="2014-03-17T03:34:41.230" Score="9" Body="&lt;p&gt;I agree that in the current paradigm, there is no need to emphasize the skills of computing things by hand for the sake of it. Having said that, I totally agree with the answer above which says that &quot;there's a valuable insight to be gained in doing things by hand&quot;. I think that precisely because it is laborious, the act of writing things out forces you to take notice of them, and in doing this a student takes in more. If I am allowed to be poetic, I would say that the act of writing or computing things forges the first relationship between a student and symbols. Of course this loses its meaning when computing by hand also becomes mechanical. It is the same reason I prefer board talks to slide shows. As a student, I was more awake in the class where the instructor wrote things on the board, talking to us as he did. As an instructor too I am able to engage my audience when I am not feeding them pre-packaged slides.  Hence I don't think that asking students to do things by hand is a waste of their time. But again, balance is the key.&lt;/p&gt;&#xA;" OwnerUserId="177" LastEditorUserId="177" LastEditDate="2014-03-21T00:34:37.100" LastActivityDate="2014-03-21T00:34:37.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="330" PostTypeId="1" AcceptedAnswerId="337" CreationDate="2014-03-17T04:02:55.697" Score="9" ViewCount="275" Body="&lt;p&gt;I'm still a undergrad math student, and my experience in education in math is very limited, however I've been lucky enough to meet teachers that encourage students who are interested in teaching, like myself, to work with their students as support. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have helped beginners in calculus, probability and algebra, and the common denominator, when it comes to confusions, is that the order in which the topics come sometimes doesn't make much sense to them, so they tend to forget what they've just &lt;em&gt;learned&lt;/em&gt; and/or not use them at all after reading them. Here I have in mind only the kids who do study and really try. I keep telling them that for now these things may not make much sense, but they should see them as tools that they're biulding and saving for later, so when the heavy stuff comes they don't have to prove those things any more and can just use them as they need them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I was thinking of something based of an experience I had. Last semester I took an algebra seminar, where we studied the history of equations, and the teachers' approach was $100\%$ chronological, and based on the original texts of the authors. In that class I learned a lot of new things, but also the other things that I already knew started to make sense, especially group theory stuff, like permutations, normal groups, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not a big fan of teaching/learning chronologically, i.e., in the same order things started to appear in history, because I feel it doesn't make much sense, however the fact of reading the original papers, and the original questions helped me a lot to understand the motivation behind these things that we study in the classroom. So my question is, how good would it be for beginers to read selected original texts to understand new concepts?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I attached some of the texts that we covered on the seminar; it's not all of them because I don't remember all, but here we go:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Descartes, &lt;/li&gt;&#xA;&lt;li&gt;De Moivre (&quot;De sectione Anguli&quot;, Philosophical Transactions), &lt;/li&gt;&#xA;&lt;li&gt;Colson (&quot;Aequationum Cubicarum &amp;amp; Biquadraticarum, tum Analytica, tum Geometrica &amp;amp; Mechanica, Resolutio Universalis&quot;, Philosophical Transactions), &lt;/li&gt;&#xA;&lt;li&gt;Gauss (Disquisitiones Arithmeticae), &lt;/li&gt;&#xA;&lt;li&gt;Euler, &lt;/li&gt;&#xA;&lt;li&gt;Lagrange (Réflexions sur la résolution algébrique des équations), &lt;/li&gt;&#xA;&lt;li&gt;Vandermonde (Mémoire sur la résolution des équations), &lt;/li&gt;&#xA;&lt;li&gt;Cauchy (Sur le nombre des valeurs qu'une fonction peut acquérir lorsqu'on y permute de toutes les manières possibles les quantités qu'elle renferme), &lt;/li&gt;&#xA;&lt;li&gt;Abel (Mémoire sur les équations algébriques: où on démontre l'impossiblité de la résolution de l'equation générale du cinquième dégré), &lt;/li&gt;&#xA;&lt;li&gt;Galois (Mémoire sur les conditions de résolubilité des équations par radicaux), &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="99" LastEditorUserId="4746" LastEditDate="2015-09-17T11:28:40.600" LastActivityDate="2015-09-18T02:17:01.603" Title="Using original texts while introducing new concepts in class" Tags="|mathematical-pedagogy|undergraduate-education|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="331" PostTypeId="2" ParentId="238" CreationDate="2014-03-17T04:08:13.500" Score="7" Body="&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1568815786&quot; rel=&quot;noreferrer&quot;&gt;You Can Count on Monsters&lt;/a&gt;, by Richard Evan Schwartz, creatively illustrates prime factorizations for the first 100 numbers. Some of my godchildren are very into it. Though I think most of the math content is going over their head at this point, they are into counting the dots, and they're into the idea that numbers have personalities.&lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2014-03-17T04:08:13.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="332" PostTypeId="2" ParentId="330" CreationDate="2014-03-17T04:23:01.457" Score="5" Body="&lt;p&gt;It sounds at first like a great idea. However, in many cases the vocabulary and phrasing, notation, and concepts have changed, sharpened, and been generalized or  simplified considerably since they were first introduced. The selection would have to be done very carefully. &lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-17T04:23:01.457" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="333" PostTypeId="1" AcceptedAnswerId="359" CreationDate="2014-03-17T04:24:09.967" Score="15" ViewCount="2273" Body="&lt;p&gt;Rounding to the nearest even digit is very practical in a lot of areas (e.g. statistics, accounting), but is never taught anywhere from elementary school to college. Even in R, the go-to statistics language, the &lt;code&gt;round&lt;/code&gt; function uses this method. There are a few other methods with their own appropriate situations; see &lt;a href=&quot;http://en.wikipedia.org/wiki/Rounding#Tie-breaking&quot;&gt;http://en.wikipedia.org/wiki/Rounding#Tie-breaking&lt;/a&gt; for an overview.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a reason this is not taught? What level/area of math would it be wise to introduce this concept for a minimum of exposure to the idea?&lt;/p&gt;&#xA;" OwnerUserId="180" LastEditorUserId="143" LastEditDate="2014-10-13T20:53:03.833" LastActivityDate="2014-10-13T23:25:17.863" Title="Why is rounding half away from zero the only method taught?" Tags="|curriculum|statistics|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="334" PostTypeId="2" ParentId="333" CreationDate="2014-03-17T04:51:37.867" Score="4" Body="&lt;p&gt;Mathematics is logical. I do not see how always rounding up when you are halfway between two numbers is a logical thing to do. If a number is exactly halfway between two numbers, then rounding to either would make sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are looking at data, and want no change in the distribution, it seems to me that you would want to alternate between rounding up and rounding down. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a math educator, I think it's very important that math always make sense to students. Each step they take should have reasons that make sense to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In response to &quot;[At] what level/area of math would it be wise to introduce this concept for a minimum of exposure to the idea?&quot; It's clear from the complex answer provided by Willie Wong that this sort of analysis cannot come before college (or advanced high school) courses. Until then, students could discuss the purposes of rounding, and work together on what would be a sensible way to do it.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2014-03-17T16:20:55.023" LastActivityDate="2014-03-17T16:20:55.023" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="336" PostTypeId="2" ParentId="298" CreationDate="2014-03-17T05:10:05.107" Score="14" Body="&lt;p&gt;Yes! I have used these a lot in an &quot;intro to proofs&quot; course. Typically, each weekly homework assignment has at least one problem of this variety, and I've written many like this for assignments and a text. Some thoughts about your posed questions and other ideas:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Does this help? I think yes in several ways, but it's hard to tell. I don't put questions like these on exams, so the &quot;becoming a better proof-reader&quot; skill is something I expect students to pick up from the course, but it is not a major component on which they are assessed. I think that working on these exercises makes them better readers of their own work, but it seems nearly impossible to find a way to quantify or track this progress. I'm also not sure whether students are aware of whether these exercises are helpful. Many might see them as silly (&quot;If we know it's wrong, why bother reading it?&quot;), so I like to follow them up with a &quot;Fix the proof, or give a counterexample to show why the claim is wrong&quot;.&lt;/li&gt;&#xA;&lt;li&gt;Other strategies? Not really, short of reminding and encouraging them &lt;em&gt;constantly&lt;/em&gt; to do so. One idea, though: I've recently started devoting a bit of class time to discussing printouts that have actual student-written proofs from recent assignments. (I attached no names, but some students opted to share when it was their proof being discussed.) I noticed a very positive response from students after this, but I think this is because they're very curious about what the others are doing, and I'm not necessarily sure this will make them better readers of their &lt;em&gt;own&lt;/em&gt; work. Still, I plan on repeating this kind of activity more and more often, and tweaking it to get a better understanding of what they get out of it.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Other thoughts:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Too many of these &quot;find the flaw&quot; problems are constructed by choosing a common error (dividing by zero, square-rooting an expression that is later evaluated to be negative, etc) and &lt;em&gt;burying&lt;/em&gt; that error under symbolism and/or verbiage. This is unfortunate, and most students will then approach these problems from the viewpoint of, &quot;Okay, I just need to know the right trick here&quot;. So, I've tried to come up with more interesting exercises that either (a) are based on actual student errors observed in the past, or (b) incorporate a flaw that's based on currently-learned material. For example, I have a &quot;fun&quot; problem that attempts to prove $\bigcup_{k\in\mathbb{N}} \mathcal{P}(\{1,2,\dots,k\}) = \mathcal{P}(\mathbb{N})$ (where $\mathcal{P}(S)$ is the power set of $S$). This one works out quite well because, usually, the students have trouble at first seeing why the claim is false; once I prod them into thinking about that (and disproving it with an example), they spot the error more easily.&lt;/li&gt;&#xA;&lt;li&gt;Students have a strong tendency to merely disprove the claim, and not identify any logical/mathematical error in a proof. Sometimes, this can be obvious, even to them, but other times they get so twisted in their explanation of what went wrong that they end up writing a whole paragraph just to say, &quot;This conclusion is wrong because ...&quot;  So if we go over these examples in class, I assume the role of the very stubborn yet erroneous proof-writer; I pretend that I am totally convinced my proof is right, and they have to work hard to show me &lt;em&gt;why&lt;/em&gt; I'm wrong, and not just &lt;em&gt;that&lt;/em&gt; I'm wrong. This role-playing activity really drives that point home, I've found.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-17T05:10:05.107" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="337" PostTypeId="2" ParentId="330" CreationDate="2014-03-17T05:16:03.690" Score="14" Body="&lt;p&gt;I doubt this would be a good idea. The only exception might be a very recently-developed field, so that they're written in modern style and notation. For most texts and papers much older than the mid 1900s, they're just too difficult to read and would probably be off-putting to students, blocking any potential learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than introducing a new concept via an original text, I think you could use the original text to supplementarily motivate the material. Show the students why topic X was first studied and what people were doing with it. Show them how our understanding has advanced over time. (Indeed, I like pointing out, for example, when we study cardinality of sets, that as recently as 150 years ago, these ideas were so controversial that top mathematicians debated them for years, and now we teach them to undergraduates! The students appreciate that.) Show them how modern notation has allowed us to talk about and use topic X more easily. And so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, I think original texts can really supplement a topic, but only in extreme rare cases could they feasibly be used to introduce a concept.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-17T05:16:03.690" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="339" PostTypeId="2" ParentId="294" CreationDate="2014-03-17T05:30:20.083" Score="12" Body="&lt;p&gt;William Dunham's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/014014739X&quot;&gt;&lt;em&gt;Journey Through Genius&lt;/em&gt;&lt;/a&gt; is, ultimately, about a bunch of facts, but it's written very well and can be inspiring to a budding math student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0393310728&quot;&gt;How to Lie with Statistics&lt;/a&gt;&lt;/em&gt; is just a classic and deserves to be mentioned, even thought it's not really &quot;math-heavy&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Kaplans' &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1608198693&quot;&gt;&lt;em&gt;The Art of the Infinite&lt;/em&gt;&lt;/a&gt; is genuinely playful and seeks to present math as the opposite of dry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I think an infrequently-tapped way to draw interest to math is through stories about its people. To that end:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1596914521&quot;&gt;Logicomix&lt;/a&gt;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0786884061&quot;&gt;The Man Who Loved Only Numbers&lt;/a&gt;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-17T05:30:20.083" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="340" PostTypeId="2" ParentId="330" CreationDate="2014-03-17T05:30:57.207" Score="6" Body="&lt;p&gt;Consider the difficulties that students have reading modern books, now multiply it by the factors that are introduced by the older phrasing, spelling, et al. Letting students see something of the development of a concept through historical flow is a great idea. Original text pages could be interesting as an image but I wouldn't want to include it as teaching material. &lt;/p&gt;&#xA;" OwnerUserId="183" LastActivityDate="2014-03-17T05:30:57.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="341" PostTypeId="2" ParentId="294" CreationDate="2014-03-17T05:35:22.810" Score="7" Body="&lt;p&gt;At &lt;a href=&quot;http://mathmamawrites.blogspot.com/p/math-books.html&quot;&gt;the books page on my blog, Math Mama Writes&lt;/a&gt;, you can find a list that includes about a dozen great books at adult level. (Scroll down past the kids' books to see them.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I had to pick my favorite, I think it would be &lt;em&gt;Math Girls&lt;/em&gt;, by Hiroshi Yuki (along with &lt;em&gt;Math Girls 2&lt;/em&gt;). The unnamed protagonist is a boy in high school who loves math. He helps Tetra with her math, and is challenged by the problems Miruka poses. In Math Girls 2, a few more girls join the gang. The math is challenging in these books, and the storyline makes it all the more fun.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few other favorites:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Chances Are: Adventures in Probability&lt;/em&gt;, by Michael and Ellen Kaplan. History, philosophy, science, and statistics all come together in this delightful exploration of probability.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Euclid in the Rainforest&lt;/em&gt;, by Joseph Mazur. Logic, infinity and probability are the topics. Adventures in Venezuela, Greece, and New York furnish the background. Mazur has wide-ranging interests, and skillfully brings the math to life.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Surreal Numbers&lt;/em&gt;, by Donald Knuth (requires well-developed math skills), requires lots of work, doing the math, and what fun work it can be! Alice and Bill are enjoying their extended vacation on an isolated tropical beach, but are getting a bit bored, when they discover a rock with two 'rules' on it. Conway has invented number through these two rules, and Alice and Bill (and the reader) are sucked in, trying to figure out how it all works. This is higher math.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;The Man Who Knew Infinity: A Life of the Genius Ramanujan&lt;/em&gt;, by Robert Kanige, would be unbelievable if it were fiction or even slightly fictionalized. Ramanujan was too focused on his own mathematical work to do well in school - he was kicked out college when he failed exams in his other subjects. It took him years of working as a clerk to support himself before he managed to catch the attention of a famous mathematician in England, G.H. Hardy, whose interest in him suddenly changed his life. A year later he would sail to England to begin with Hardy the work of making his mathematical results comprehensible to others. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-17T05:35:22.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="342" PostTypeId="2" ParentId="294" CreationDate="2014-03-17T05:38:55.767" Score="10" Body="&lt;p&gt;In terms of a book which inspires, I will offer &lt;strong&gt;Love and Math: The Heart of Hidden Reality&lt;/strong&gt; by Edward Frenkel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is an autobiography. It explains how a knowing mentor used physics to lure him into deeper math. I don't want to say too much and spoil it, but, he goes on to explain symmetries, groups, Lie groups, Loop groups, braid groups, Lie algebras, Kac-Moody algebras in non-technical sort of heurstics in as much as is possible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, the thing that is contagious about the book, is the raw passion for pure math with a hint of theoretical physics. Ultimately, the point of the book is to give you some sense of what Langlands conjecture is, how it has already been seen and/or verified and how it is still a vastly open question. The book is an invitation to join in the joy of discovery of math. Not as a drudgery, rather as an art.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beyond this, I also like Roger Penrose's &lt;strong&gt;Road to Reality&lt;/strong&gt;. Although, I will confess, I have not finished it yet. It does not read as fast as Frenkel's text. What math does that text contain? Better question: what math does it not contain?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added 12-28-2014&lt;/strong&gt; I just finished my first read of &lt;strong&gt;Fearless Symmetry&lt;/strong&gt; by Avner Ash and Robert Gross. A review by Dino Lorentzini is found &lt;a href=&quot;http://www.ams.org/notices/200701/rev-lorenzini.pdf&quot; rel=&quot;nofollow&quot;&gt;on the AMS website&lt;/a&gt;. I found it to be a bit tougher to read than Frenkel's book, but, I think that is largely due to the fact this book really endeavors to give you some psuedo-technical insight into the problem of investigating the absolute Galois group to solve Diophantine equations. The major theme is to understand the structure of reciprocity theorems. Explicit examples are given. Fermat's Last Theorem is also discussed in some detail and placed in part of a larger story about a whole family of reciprocity theorems some just conjectured at this time. The book is full of wonderful stories and nice quotes. I recommend it for both style and as a leisurely way to see pretty far into modern number theory (of which I am no expert!)&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2014-12-29T01:39:35.583" LastActivityDate="2014-12-29T01:39:35.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="343" PostTypeId="2" ParentId="298" CreationDate="2014-03-17T05:48:14.687" Score="3" Body="&lt;p&gt;I have done this when I have noticed a student blindly following a formula without checking the various limitations and assumptions involved. (Occasionally, I myself have been said student). &lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-17T05:48:14.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="344" PostTypeId="2" ParentId="298" CreationDate="2014-03-17T05:50:42.913" Score="3" Body="&lt;p&gt;This is a wonderful formative assessment (FA) tool that creates a greater amount of mathematical reasoning/communicating between students. The key to this technique of FA is creating a culture of self assessors and peer assessors while increasing their awareness of the factors of the concept you are covering. This will increase the amount feedback students can get in their own solutions. It also counters the culture of teaching mathematics as 'show 'n tell' teacher modelling &quot;the way&quot; to solve a problem which we are so want to do. &#xA;I create 'student' work for the technique or pull the actual student work from their own approaches. Having a common set of solutions for students to discuss in their groups allow for a richer debrief if needed.  &lt;/p&gt;&#xA;" OwnerUserId="183" LastActivityDate="2014-03-17T05:50:42.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="345" PostTypeId="2" ParentId="333" CreationDate="2014-03-17T06:05:30.953" Score="7" Body="&lt;p&gt;The reason is that rounding half up is by far the simplest rounding rule, because you only have to look at the next relevant digit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, everyone agrees that you should round $12.5134$ up to $13$, since $12.5134$ is closer to $13$ than to $12$.  The issue only arises when you are rounding &lt;em&gt;exactly&lt;/em&gt; $12.5$, with no further digits.  In this case, it's easier to just remember the rule &quot;always round up 5's&quot; than to worry about the distinction between these two cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, it would be possible to discuss further rounding rules at some point, as well as the advantages and disadvantages of each rule.  But the reality is that it just isn't important enough to be worth a spot in the curriculum.  The few professionals who have to deal with such things (accountants, some computer scientists, etc.) are aware of the distinctions, as are those of us who appreciate mathematical arcana, but there's just not a pressing need to inform the average math student.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-17T06:05:30.953" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="346" PostTypeId="2" ParentId="295" CreationDate="2014-03-17T06:06:46.290" Score="7" Body="&lt;p&gt;As the others have stated, the key issue is having students defend their pattern extensions with their reasoning or their assumptions. Now tie this to the question based on mistake analysis, and have students see if there are any errors in the alternate solutions. This could lead to rich mathematical discussions and an increased awareness of alternative solutions for discrete mathematics preparation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another issue, as I see it, is to tie patterns with term or figure numbers in tables; to physical objects; to graphs; to formulae; to words explaining their formulae. I would start with physical patterns like grid fragments with a shaded border of increasing size. The sequence generated would perhaps count the number of shaded squares in successive figures. The method they use to 'count' the squares would generate their formulae. There are multiple answers here but are all transformable into each other since they count the same number but in different ways. This takes the mathematics as scratches off the page and teaches them that abstract looking equations are representative of real things and have meaning.  &lt;/p&gt;&#xA;" OwnerUserId="183" LastActivityDate="2014-03-17T06:06:46.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="347" PostTypeId="2" ParentId="86" CreationDate="2014-03-17T06:31:00.723" Score="7" Body="&lt;p&gt;I was a bit surprised to not see anyone taking issue with the &quot;collected and graded&quot; aspect of the &quot;homeworks&quot;. Research done by the SFU mathematics department (Gray &amp;amp; Mulholland) showed that the concept of grading 'practice' led to the occurrence of copying solutions. Students thought these homeworks were for the professors and not for students to learn. Therefore they were being completed by Wolfram Alpha, a tutor, online sites, friends etc just to have it done and assigned marks. They actually were not learning from it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Creating a culture of learning where students are assigned opportunities to develop the concepts of the course, do some intentional practice around these concepts in order to have a complete understanding of these for summative assessments. Provided answers provide feedback to guide students in this development and to indicate readiness or review needs. SFU changed to a series of quizzes based on the assignments that were graded instead of the actual assignments quickly led students to understand that the purpose was to learn.  &lt;/p&gt;&#xA;" OwnerUserId="183" LastActivityDate="2014-03-17T06:31:00.723" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="348" PostTypeId="1" AcceptedAnswerId="350" CreationDate="2014-03-17T07:01:25.380" Score="17" ViewCount="480" Body="&lt;p&gt;Imagine that you are teaching a high school class in the last years of high school, an undergraduate class in university, or you are a tutor of a small group at university.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should one provide examples of concrete calculations in front of the class, i.e., what is a good balance of &quot;telling what to do and just stating the result of an example&quot; and &quot;really calculating everything in the given example including calculations with numbers&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have in mind the following concrete situations:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You want to calculate a minimum value of a function and have to take the derivative of it, solve an equation and evaluate the function for a given concrete value.&lt;/li&gt;&#xA;&lt;li&gt;In the above example, you want to find the minimal value by evaluating the function at the minimum and want to perform the number calulations.&lt;/li&gt;&#xA;&lt;li&gt;You want to use the inverse function theorem for a concrete function and have to calculate the determinant of the Jacobian.&lt;/li&gt;&#xA;&lt;li&gt;You have explained and derived the abstract algorithm for the LU-decomposition and want to perform this for a concrete example.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-28T12:54:15.280" LastActivityDate="2023-08-18T05:07:24.173" Title="Amount of concrete calculations on board?" Tags="|undergraduate-education|secondary-education|tutoring|calculations|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="349" PostTypeId="2" ParentId="224" CreationDate="2014-03-17T07:17:24.527" Score="10" Body="&lt;p&gt;This is really more of an extended comment than an answer, but I couldn't resist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, I don't know of any proof assistant that would be helpful in an &quot;introduction to proofs&quot; class.  I would guess that no such proof assistant exists.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I agree that such an assistant would have the potential to be very helpful.  My experience with teaching introductory proofs is that students use the following algorithm to write proofs:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Write an incorrect proof.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Turn it in, unaware of the problem&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Find out later that it was wrong.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If I plead with the students enough to attend my office hours, this algorithm can be improved to:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Write an incorrect proof.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Show it to me.  I point out that's its wrong.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Write a slightly better proof.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Show it to me.  I point out that it's better but still wrong.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Write a correct proof.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Show it to me.  I verify it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Turn it in.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This second algorithm is of course much better, since the students actually learn to write proofs!  (I cannot stress enough how important office hours are for an introductory proofs class.)  Unfortunately, it's hard to have enough office hours to help all of the students through this process.  What they need is an automated assistant that can check their proofs.  This should be possible to program --- if there's one thing a computer ought to be able to understand, it's a proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To illustrate, here is a little snippet of I/O depicting a student interacting with a machine using this hypothetical program.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;&amp;gt;&amp;gt;&amp;gt; import BasicNumberTheory&#xA;&amp;gt;&amp;gt;&amp;gt; begin theorem1&#xA;    &amp;gt;&amp;gt; suppose m is even&#xA;     Error: Variable 'm' not recognized!&#xA;    &amp;gt;&amp;gt; let m be an integer&#xA;    &amp;gt;&amp;gt; suppose m is even&#xA;    &amp;gt;&amp;gt; let n be an integer&#xA;    &amp;gt;&amp;gt; suppose n is even&#xA;    &amp;gt;&amp;gt; then m+n is even&#xA;    &amp;gt;&amp;gt; end theorem&#xA;&amp;gt;&amp;gt;&amp;gt; print theorem1&#xA; Let m be an integer.&#xA; Let n be an integer.&#xA; Suppose m is even.&#xA; Suppose n is even.&#xA; Then m+n is even.&#xA;&amp;gt;&amp;gt;&amp;gt; prove theorem1&#xA;     1: m is an integer&#xA;     2: n is an integer&#xA;     3: m is even&#xA;     4: n is even&#xA;    &amp;gt;&amp;gt; By (3), m = 2j&#xA;     Error: Variable 'j' is not recognized.&#xA;    &amp;gt;&amp;gt; By (3), there exists an integer j so that m = 2j&#xA;     5: There exists an integer j so that m = 2j&#xA;    &amp;gt;&amp;gt; Let j be such&#xA;     6: j is an integer&#xA;     7: m = 2j&#xA;    &amp;gt;&amp;gt; By (4), there exists an integer k so that n = 2k&#xA;     8: There exists an integer k so that m = 2k&#xA;    &amp;gt;&amp;gt; Let k be such&#xA;     9: k is an integer&#xA;     10: n = 2k&#xA;    &amp;gt;&amp;gt; By (7) and (10), m + n = 2j + 2k&#xA;     11: m + n = 2j + 2k&#xA;    &amp;gt;&amp;gt; So m+n is even&#xA;     Error: How does this follow?&#xA;    &amp;gt;&amp;gt; So m+n = 2(j+k)&#xA;     12: m+n = 2(j+k)&#xA;    &amp;gt;&amp;gt; So m+n is even.&#xA;     Error: Missing hypothesis -- is it true that j+k is an integer?&#xA;    &amp;gt;&amp;gt; By (6) and (9), j+k is an integer&#xA;     13: j+k is an integer&#xA;    &amp;gt;&amp;gt; So m+n is even&#xA; Proof accepted: theorem1&#xA;&amp;gt;&amp;gt;&amp;gt; save theorem1&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;I think the idea is that students would only be &lt;em&gt;required&lt;/em&gt; to use the software for the first month or so, but I would encourage them to use it throughout the semester to check their reasoning.  I'm not absolutely convinced that this would work well, but I'd love to try it!&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-03-17T07:25:14.250" LastActivityDate="2014-03-17T07:25:14.250" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="350" PostTypeId="2" ParentId="348" CreationDate="2014-03-17T07:21:20.040" Score="6" Body="&lt;p&gt;I find that too much detail distracts poor students, in particular when calculating numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Almost every student in a lecture wants to understand something in the lecture.  Some students may not follow complicated proofs or discussions, and may pay more attention to examples.  If you perform an example with concrete numbers, then a lot of students will try to calculate with you and reproduce the same number values. &lt;strong&gt;But&lt;/strong&gt; the speed of their calculations varies. If you as lecturer are too fast, the students will try to follow anyway and then probably miss the purpose of doing the example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a teacher, I try to minimize big calculations (when the topic is &lt;strong&gt;NOT&lt;/strong&gt; the calculation itself), especially when calculating concrete numbers. For example, a minimal example can show the concept, e.g., performing an LU decomposition only for a 3x3 matrix with a lot of zeros. Students can work on more sophisticated examples as homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a tutor, you can have a student perform a calculation on the board and then instruct that student. This has the huge advantage that the explanation and the concept come from one person and the calculation comes from a different one. The other students can then recognize from the voices if they have to pay attention to the concepts or if they want to reproduce the numbers.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorDisplayName="user173" LastEditDate="2014-08-25T03:55:10.357" LastActivityDate="2014-08-25T03:55:10.357" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="351" PostTypeId="1" AcceptedAnswerId="356" CreationDate="2014-03-17T07:34:21.623" Score="12" ViewCount="309" Body="&lt;p&gt;What are the arguments in publishing or not publishing old exams? If yes, should they contain also a solution? Would you publish them directly after that particular exam or publish your old one's just before a new exam?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is somehow related to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/86/is-it-good-to-have-solutions-of-homework-published&quot;&gt;Is it good to have solutions of homework published?&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-06-21T11:42:31.293" Title="It is good to have old exams (with solutions) published?" Tags="|exams|publishing|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="352" PostTypeId="1" AcceptedAnswerId="357" CreationDate="2014-03-17T07:48:17.980" Score="31" ViewCount="5079" Body="&lt;p&gt;I think, a lot of students are bothered by learning multiplication tables by heart, in particular when it comes to numbers greater than 10.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why should one learn (or not learn) these things by heart?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorDisplayName="user507" LastEditDate="2021-04-16T13:42:00.050" LastActivityDate="2022-12-10T13:29:22.927" Title="What are the arguments for and against learning the multiplication table by heart?" Tags="|primary-education|multiplication-table|" AnswerCount="11" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="353" PostTypeId="1" AcceptedAnswerId="412" CreationDate="2014-03-17T08:22:51.313" Score="45" ViewCount="2883" Body="&lt;p&gt;One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting &quot;false implies false&quot; and &quot;false implies true&quot; as true sentences. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are good point of view, methods and tips to help students grasp the concept of logical implication?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To focus the question, I would like to restrict to math majors, although the question is probably equally interesting for other kind of students.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2023-06-12T19:54:28.150" Title="How to teach logical implication?" Tags="|undergraduate-education|logic|" AnswerCount="18" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="354" PostTypeId="2" ParentId="352" CreationDate="2014-03-17T09:15:12.733" Score="6" Body="&lt;p&gt;I always thought that multiplication tables above 10 were used in English-speaking countries because of non-decimal units: in Italy nobody bothered with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This said, I think tables up to 10&amp;times;10 are useful in real life, even if they are no more used for long multiplication; so they should be learned by heart.&lt;/p&gt;&#xA;" OwnerUserId="26" LastActivityDate="2014-03-17T09:15:12.733" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="355" PostTypeId="2" ParentId="351" CreationDate="2014-03-17T09:28:25.757" Score="2" Body="&lt;p&gt;Exam questions usually are similar to homework questions, but they are not quite the same. Especially for first-year courses, I think it's a very good thing to give an idea how the exam might look like. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you provide solutions, you will have to decide if you just provide brief answers or full solutions. The former allows the students to check their results, but doesn't help them to learn how to solve this excercise. The latter needs more work, and can give the impression that this solution is the only legitimate way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, I recommend that there is some way for the students to talk and ask about these old exams and the solutions - be it in the form of a TA session with a tutor, a dedicated part of the lecture or in the form of office hours provided by the lecturer.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-03-17T09:28:25.757" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="356" PostTypeId="2" ParentId="351" CreationDate="2014-03-17T09:29:42.890" Score="10" Body="&lt;p&gt;At my university publishing old exams is considered a good practice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some advantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students know what to expect during real test.&lt;/li&gt;&#xA;&lt;li&gt;Students can test their knowledge and check which topics they need to relearn.&lt;/li&gt;&#xA;&lt;li&gt;There is an additional problem supply for practice.&lt;/li&gt;&#xA;&lt;li&gt;Sometimes professors lose their old tests (e.g. disk failure, etc.), and publishing them might serve as additional backup.&lt;/li&gt;&#xA;&lt;li&gt;You get access to exams constructed by your predecessors and your successors will get an easy access to yours.&lt;/li&gt;&#xA;&lt;li&gt;As pointed out in the comments by Andrew Stacey, it makes sure that everyone has access to tests and so &quot;levels the playing field&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some disadvantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Creating a new exam requires more work.&lt;/li&gt;&#xA;&lt;li&gt;It might be hard to assess the difficulty of a new problem.&lt;/li&gt;&#xA;&lt;li&gt;New exams are harder to grade objectively.&lt;/li&gt;&#xA;&lt;li&gt;Students know that some areas might be more probable (e.g. theorem XYZ won't be used, because it was given last year) than others.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="112" LastEditDate="2014-03-17T12:53:17.670" LastActivityDate="2014-03-17T12:53:17.670" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="357" PostTypeId="2" ParentId="352" CreationDate="2014-03-17T09:39:37.953" Score="16" Body="&lt;p&gt;There is some relevant research (and bibliography) in this paper:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Improving basic multiplication fact recall for primary school students&quot; &#xA;(Wong and Evans, 2007)&#xA;&lt;a href=&quot;http://link.springer.com/article/10.1007/BF03217451&quot;&gt;http://link.springer.com/article/10.1007/BF03217451&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me the most striking thing is how low the scores reported in their Figure 2 are (36.89/60 at best), even after interventions that they consider to be valuable.  I had no idea that teaching of multiplication tables was so unsuccessful.  The study was done with 10 year old pupils in inner city schools in Sydney; I do not know how far it would generalise.&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-17T09:39:37.953" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="358" PostTypeId="1" AcceptedAnswerId="365" CreationDate="2014-03-17T09:39:55.393" Score="15" ViewCount="1454" Body="&lt;p&gt;In Secondary education in Australia, the general outline for introducing techniques to solve the quadratic equation&#xA;$$&#xA;x^2+bx+c=0&#xA;$$&#xA;is first to introduce the idea to find two numbers $p$ and $q$ such that $b=p+q$ and $c=pq$. After this, the quadratic formula is introduced, which is applicable to all quadratic equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why are these topics this taught in this order?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems more natural to me to introduce the quadratic formula, which works in the general case first, and then introduce the other technique which only applicable to special cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Secondly, why is synthetic division not taught at all? This would allow students to extend their knowledge of solving quadratic equations to higher degree polynomials.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="71" LastActivityDate="2014-08-25T21:15:47.150" Title="Factoring quadratic polynomials" Tags="|secondary-education|solving-polynomials|" AnswerCount="6" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="359" PostTypeId="2" ParentId="333" CreationDate="2014-03-17T09:51:40.050" Score="19" Body="&lt;p&gt;To properly understand and appreciate the different rounding rules, one really needs to have some grounding in statistics and probability theory. It is a little hard to describe the full theory in detail (the least being that I don't have a entirely coherent formulation of the theory off the top of my head), but here are some examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Naively if we idealistically think of data as living in $\mathbb{R}$ and the rounded counterparts as living in $\mathbb{Z}$, then &lt;strong&gt;tie-breaking rules do not matter&lt;/strong&gt;, as for all reasonable probability measures, we would be thinking about events of probability zero. &lt;/li&gt;&#xA;&lt;li&gt;However, tie breaking rules &lt;strong&gt;do&lt;/strong&gt; matter, and this is due to that our data really lives in $\mathbb{Z}$ and we are rounding to $\lambda\mathbb{Z}$ for some positive integer $\lambda$. The event that the data falls exactly halfway between the rounding points is now of positive probability, if $\lambda$ is even. (If $\lambda$ is odd, there's no halfway point.) &lt;/li&gt;&#xA;&lt;li&gt;But there are other subtleties in rounding rules: for physical science and statistical analysis, we want the rounding rules to be stable under addition of small random noise, so that we don't introduce rounding artifacts. In statistics the &quot;round to the nearest even integer&quot; rule for tie-breaking is useful, under the assumptions that &#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;we intend to sum/average our data&lt;/li&gt;&#xA;&lt;li&gt;the ones-digit of the data is uniformly distributed, at least in terms of evens and odds, and that the distribution is &lt;em&gt;independent of&lt;/em&gt; the digits after the decimal point, which&lt;/li&gt;&#xA;&lt;li&gt;we assume again to be uniformly distributed. &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;There are times however our data is multiplicative. In this case rounding from the half-way point is no longer appropriate: it is more appropriate to round $0.3$ down and $0.4$ up. The exact dividing line is $\frac{1}{10} \sqrt{10}$ which conveniently happens to be an irrational number, so in finite precision computations the problem with the tie-breaking rules are side-stepped somewhat. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The main point being that the correct rounding rules depend at least on &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A priori assumptions on your data: where they come from, what noises are inherent, what are reasonable &lt;em&gt;a priori&lt;/em&gt; probability distributions of the digits...&lt;/li&gt;&#xA;&lt;li&gt;What you intend to do with those data: are they to be summed? are they to be multiplied? are they to be binned (&lt;em&gt;thank to &lt;a href=&quot;https://matheducators.stackexchange.com/users/60/sue-vanhattum&quot;&gt;Sue VanHattum&lt;/a&gt; for pointing out this case&lt;/em&gt;)? Some more complicated functions? &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And a proper discussion of the strengths and weaknesses of different rounding rules will necessarily involve quite some bit of statistics and mathematical modeling. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This is, however, not to say that there are not portions of the theory that can be taught to elementary schoolers. But one has to be very careful about the presentation: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You do not want it to devolve into a discussion of &quot;different conventions&quot;. As this just makes two different things that the poor kid will have to memorise, instead of just one. &lt;/li&gt;&#xA;&lt;li&gt;The issue of tie-breaking rules never occur when dealing with a single datum. Are you sure the kids can appreciate that rounding errors can accumulate when one works with many data points and that this is undesirable? &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Let me give one more try about the theory:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to think about rounding (or approximations in general) is that we have some space of objects $X$ which are sometimes hard to deal with, but within it we have a subspace of objects $Y$ which are easier to handle. We also have a family of maps $\mathcal{F} = \{F_1, F_2, \ldots\}$ which are maps from some finite Cartesian power of $X$ to $X$ itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;ideal&lt;/em&gt; rounding rule is a map $\mathcal{R}:X \to Y$ (which induces also $\mathcal{R}_k: X^k \to Y^k$ by acting on components) such that &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It is idempotent: $\mathcal{R}\circ \mathcal{R} = \mathcal{R}$. &lt;/li&gt;&#xA;&lt;li&gt;It &quot;commutes&quot; with the family $\mathcal{F}$: for every $i$, suppose that $F_i: X^{k_i} \to X$, then &#xA;$$ \mathcal{R} \circ F_i \circ \mathcal{R}_{k_i} = \mathcal{R} \circ F_i $$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This would mean that we can approximate computations (actions of the family $\mathcal{F}$) on $X$ by computations in $Y$ with no ill effects, up to some well-defined finite error. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In reality finding such a rounding function $\mathcal{R}$ can be very hard. Consider just the rounding of real numbers ($X = \mathbb{R}$) to integers ($Y = \mathbb{Z}$). No tie-breaker rule for the half-way rounding commutes with simple addition. So we often, in applications, relax the &quot;commutation&quot; rule by asking that it be satisfied for &quot;generic&quot; data, so that only very unlikely outliers will cause the commutation rule to break down. But to understand the notion of generic and to get a quantitative understanding of how badly the commutation rule can break requires some knowledge in probability and statistics. &lt;/p&gt;&#xA;" OwnerUserId="125" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-03-17T16:25:34.957" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="360" PostTypeId="2" ParentId="224" CreationDate="2014-03-17T09:56:52.823" Score="0" Body="&lt;p&gt;You might like to look at this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://aim.shef.ac.uk/aim_php/html/builder_question.html&quot; rel=&quot;nofollow&quot;&gt;http://aim.shef.ac.uk/aim_php/html/builder_question.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is experimental; I have not used it with real students yet.  The page gives you a selection of phrases and asks you to construct a definition of convergence by dragging and dropping.  (The dragging and dropping does not work as well as it should yet; it is a bit fiddly but is OK once you get used to it.) When you click the 'Mark' button the system will critique your answer, first making sure that it can be parsed in a meaningful way, then checking whether it is actually the correct definition.  Of course it would be possible to do proofs as well as definitions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have built a reasonably extensive Javascript framework for this kind of thing, but the logic specific to this question still takes 800 lines of code.  Quite a lot of that is just blocks of LaTeX giving feedback for various types of errors.  I wrote these 800 lines by hand but for a production system one would want to have some kind of GUI in which the teacher could specify the marking system in a more intuitive way.&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-17T09:56:52.823" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="361" PostTypeId="2" ParentId="352" CreationDate="2014-03-17T10:02:49.253" Score="12" Body="&lt;p&gt;As I tutor students and watch those with proficiency in up to 10x10, there's a clear difference in their ability to get through the problems in a timely manner. &#xA;It's a distraction to use a calculator for the simple math (sometimes) required in algebra problems, say, 6 X 7. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not suggesting a cause/correlation of intelligence. Teachers around me are split on the calculator issue, those who are in favor clearly don't share my concerns. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My claim is this: Take two sets of students, those who mastered the tables and those that didn't. But screen those that didn't to produce those who are otherwise proficient, fitting the teacher's claims that the calculator is a tool and it's the solving process that counts. As these students move forward, the calculator is a crutch and time waster. This results in the calculator student getting slowed enough to had a disadvantage over the non-user. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer - I am strictly speaking of calculator for the pieces of longer problems where I would naturally just do the math in my head. Single digit math, and simple math on larger numbers. When I see a student keying in &quot;7+5&quot; on a calculator, yet producing a B or A grade exam, I ask myself how that student will fare on the SATs (The college entrance exam in the US, where Math was, in my day, scored on a scale to 800 being high score). Or even on exams in college, where some amount of the manipulation is simple as part of the larger problem. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-03-17T10:02:49.253" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="362" PostTypeId="1" AcceptedAnswerId="423" CreationDate="2014-03-17T11:02:09.900" Score="15" ViewCount="4679" Body="&lt;p&gt;I find it useful to give physical examples of theorems, especially in vector calculus - for example $\nabla f$ being the direction of maximum ascent on a surface $f$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is a good example for &lt;a href=&quot;http://en.wikipedia.org/wiki/Stokes%27_theorem&quot;&gt;Stokes' Theorem&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-03-18T13:55:53.360" LastActivityDate="2020-11-08T15:11:55.403" Title="What is a good physical example of Stokes' Theorem?" Tags="|undergraduate-education|examples|vector-calculus|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="363" PostTypeId="2" ParentId="358" CreationDate="2014-03-17T11:19:21.997" Score="11" Body="&lt;p&gt;To the first question, I suspect the primary reason is that the mathematical community learned to solve quadratics proceeded in this order - that is, mathematicians realized there were specific, solvable cases before proceeding to the general solution. But I think there are two related arguments to be made for maintaining that order, one historical, one practical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The historical argument goes something like: We should teach solving quadratics in this order (special cases to the general case) because that order reflects the historical trajectory of the progress mathematicians made while working on this problem. But this argument is stronger than it looks on the surface - the reason the historical trajectory took the path that it did is because, in general, that's often what it means to do mathematics. When confronted with a new and unsolved problem, one of the ways that we make progress on the problem is to impose some assumptions on the problem (e.g. special cases) and solve the more constrained problem. Only after playing in this sandbox for a while does the general solution emerge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second, more practical, argument is related: by presenting the general solution up front, students are robbed of the need for it and it just becomes another formula to memorize because the teacher said so. On the other hand, by starting with the special cases, students will soon run into quadratics that do not factor. The limitations of the techniques used in the special cases can motivate the need for the general solution.&lt;/p&gt;&#xA;" OwnerUserId="193" LastActivityDate="2014-03-17T11:19:21.997" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="364" PostTypeId="1" AcceptedAnswerId="2305" CreationDate="2014-03-17T11:59:32.557" Score="14" ViewCount="336" Body="&lt;h3&gt;Question&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;I am looking for suggestions of good resources (textbooks or lecture notes preferably) for teaching Riemann integration in $\mathbb{R}^d$ with $d\geq 2$ and also for Riemann integration along (smooth) hypersurfaces of $\mathbb{R}^d$ (actually, all we need is Riemann integration along graphical hypersurfaces)). The goal is to have covered enough material for the students to be able to make sense of Stokes' Theorem. &lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Target Audience&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;The target audience are students who have just finished the second semester of a sequence of classes on analysis. In the first semester they should have learned integral and differential calculus in one variable. By the end of the second semester they should have covered ordinary differential equations and differential calculus in multiple variables (rigorously with $\delta$-$\epsilon$ and such). &lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Remarks&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;While I am fairly confident that I can write down by myself a consistent set of definitions and whatnots for Riemann integration in higher dimensions, I have not personally taken a class covering this topic (integration theory in higher dimensions is done using the Lebesgue version when I was an undergraduate), and so do not have any handy references to make sanity checks against when developing my own lecture notes. Talking to my colleagues, it seems that with the exception of the Germans my experience is rather common place. (And for reasons of personal incompetence, German lecture notes${}^\dagger$ are rather less useful to me.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;${}^\dagger$ Such as Heuser &lt;em&gt;Lehrbuch der Analysis&lt;/em&gt; Volume 2 (Teubner), which would probably be what I'll try to parse if I can't find an English language reference. &lt;/p&gt;&#xA;" OwnerUserId="125" LastEditorUserId="50" LastEditDate="2014-05-15T14:12:53.737" LastActivityDate="2014-05-15T14:12:53.737" Title="Resources for teaching Riemann integration in higher dimensions and on submanifolds, with view toward Stokes' theorem?" Tags="|reference-request|textbooks|mathematical-analysis|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="365" PostTypeId="2" ParentId="358" CreationDate="2014-03-17T13:09:20.030" Score="10" Body="&lt;p&gt;To help students understand how we move from specific cases to the general case, I created a visual depiction of the process of completing the square. I used shapes analogous to Algebra Tiles, which are a good manipulative for building understanding of factoring as well. I then took the process one step further and illustrated the derivation of the Quadratic Formula. I think the visual is powerful in building understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/0jE9x.png&quot; alt=&quot;Completing the Square&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/nkoUB.png&quot; alt=&quot;Deriving the Quadratic Formula&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="198" LastActivityDate="2014-03-17T13:09:20.030" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="366" PostTypeId="1" CreationDate="2014-03-17T13:30:10.970" Score="24" ViewCount="811" Body="&lt;p&gt;I continue to be troubled by the amount of symbolic manipulation in a typical Algebra 2 course. Once a student has completed Algebra 1 and Geometry, shouldn't there be another option for them if a math/science field is not in their future? What would that look like? What else could we teach if we weren't so focused on the path to Calculus?&lt;/p&gt;&#xA;" OwnerUserId="198" LastEditorUserId="143" LastEditDate="2014-03-19T17:31:21.707" LastActivityDate="2014-04-13T13:43:36.353" Title="Do all high school students need the same 3-year sequence of math courses?" Tags="|secondary-education|algebra|curriculum|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="367" PostTypeId="2" ParentId="294" CreationDate="2014-03-17T13:57:12.953" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://www.amazon.in/Love-Math-Heart-Hidden-Reality-ebook/dp/B00BKRW5E6&quot; rel=&quot;noreferrer&quot;&gt;Love and Math&lt;/a&gt; , which is semi-autobiographical book by Edward Frenkel. It is not exactly a math book, but it does try to explain the Langlands program and its connections with QFT in lay's man terms. It also tries to convey the beauty of and excitement of doing  mathematics&lt;/p&gt;&#xA;" OwnerUserId="124" LastActivityDate="2014-03-17T13:57:12.953" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="368" PostTypeId="2" ParentId="353" CreationDate="2014-03-17T13:58:18.480" Score="13" Body="&lt;p&gt;Historically, quantificational logic came before propositional logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So maybe start with &lt;strong&gt;quantified&lt;/strong&gt; sentences instead.  It is reasonable to say both &lt;em&gt;&quot;Every ten-foot-tall person plays basketball&quot;&lt;/em&gt; and &lt;em&gt;&quot;No ten-foot-tall person plays basketball&quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leads to accepting both &lt;em&gt;Ten-Foot-Tall $\rightarrow$ Plays-Basketball&lt;/em&gt; and &lt;em&gt;Ten-Foot-Tall $\rightarrow \neg$ Plays-Basketball&lt;/em&gt;. &lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-17T13:58:18.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="369" PostTypeId="1" AcceptedAnswerId="7231" CreationDate="2014-03-17T14:35:09.307" Score="19" ViewCount="3760" Body="&lt;p&gt;This question has been kicking around in the back of my head for a couple of years, but the impetus to post it now came from reading the related question at &lt;a href=&quot;https://matheducators.stackexchange.com/questions/244/when-did-the-american-school-systems-progression-of-math-classes-take-its-curre&quot;&gt;When did the American school system&amp;#39;s progression of math classes take its current form?&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been trying for some time now to locate digital copies of the School Mathematics Study Group (SMSG) textbooks, colloquially known as the &quot;New Math&quot;, and have accumulated quite a good-sized collection, available &lt;a href=&quot;https://drive.google.com/folderview?id=0B4wpC387vwzeTzVzdlM1dVBFTkk&amp;amp;usp=sharing&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.  (My understanding is that all of these texts are in the public domain, and may be freely distributed.)  I should acknowledge that I found most of these texts in an archive formerly hosted by &lt;a href=&quot;http://mathematics.buffalostate.edu/faculty/tom-giambrone&quot; rel=&quot;noreferrer&quot;&gt;Tom Giambrone at Buffalo State College&lt;/a&gt; that seems to no longer be available online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the curious things about this archive is that it contains more than one text with the words &quot;First Course in Algebra&quot; in the title (including Parts I and II of both a Student's Text and Teacher's Commentary, plus a &quot;Programmed First Course in Algebra&quot;) but there does not seem to be a follow-up &quot;Second Course in Algebra&quot; or &quot;Algebra 2&quot; or something similarly named.  The description of the &lt;a href=&quot;http://www.lib.utexas.edu/taro/utcah/00284/cah-00284.html&quot; rel=&quot;noreferrer&quot;&gt;SMSG archive at the UT-Austin&lt;/a&gt; also seems to lack anything that is obviously recognizable as &quot;Algebra 2&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure whether there &lt;i&gt;was&lt;/i&gt; a &quot;Second Course in Algebra&quot; text that I simply can't find any trace of, either because it is no longer extant or because my Google-Fu isn't strong enough, or whether such a book never existed in the first place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Was there an SMSG &quot;Algebra 2&quot; textbook or the equivalent, and if so, what was it called?&lt;/li&gt;&#xA;&lt;li&gt;If so, is it available anywhere, either in print or digital form?&lt;/li&gt;&#xA;&lt;li&gt;If not, what was the SMSG course sequence for students after Algebra and Geometry?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2021-03-05T07:30:18.463" Title="Was there an SMSG (New Math) &quot;Algebra 2&quot; text?" Tags="|curriculum|new-math|" AnswerCount="9" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="370" PostTypeId="2" ParentId="294" CreationDate="2014-03-17T14:47:15.800" Score="9" Body="&lt;p&gt;I greatly enjoyed &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486431819&quot;&gt;&lt;em&gt;Another Fine Math You've Got Me Into&lt;/em&gt;&lt;/a&gt; by Ian Stewart.  Entertaining and at a pace that any level of mathematician or non-mathematician would be comfortable with, but nevertheless discusses some very interesting and beautiful topics.&lt;/p&gt;&#xA;" OwnerUserId="200" LastActivityDate="2014-03-17T14:47:15.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="371" PostTypeId="2" ParentId="366" CreationDate="2014-03-17T14:51:36.900" Score="6" Body="&lt;p&gt;A very reasonable question. If the aims of high-school math were purely educational, and &lt;em&gt;if&lt;/em&gt; we really knew who might eventually be in STEM fields or not, there'd certainly be many other mathematical topics more interesting, intellectually substantive, aesthetically engaging, etc., than more-of-the-same symbol pushing in Algebra II.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in the current state of affairs at many U.S. colleges and universities, &quot;calculus&quot; (often of a cook-book genre) is used as a &quot;filter&quot; for nearly everyone, regardless of eventual major and interests. Thus, the game as it is currently played would often penalize those who'd deviated. This is unfortunate, yes.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-17T14:51:36.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="372" PostTypeId="2" ParentId="294" CreationDate="2014-03-17T14:53:06.907" Score="9" Body="&lt;p&gt;The books by Willian Dunham (&quot;Journey through genius&quot; (Penguin, 1991), &quot;Euler, master of us all&quot; (MAA, 1999), &quot;The calculus gallery&quot; (Princeton University Press, 2008) are the ones I've read) are outstanding. They show mathemathics in terms of the original work (more or less, using modern notation), and motivate the subject matter well. They do require some background and work by the reader.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Delicious is Nahin's &quot;An imaginary tale: The story of $\sqrt{-1}$&quot; (Princeton University Press, 1998).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Almost forgot Aigner and Ziegler's &quot;Proofs from THE BOOK&quot; (Springer, 4th edition 2012), a collection of beautiful proofs, accessible to the interested high-school/beginning college student.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2014-03-17T14:58:38.450" LastActivityDate="2014-03-17T14:58:38.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="373" PostTypeId="2" ParentId="244" CreationDate="2014-03-17T14:57:32.083" Score="10" Body="&lt;p&gt;I've read a few interesting articles over the past few months in the Notices of the AMS that offer a brief discussion of this. The most notable is a critique and comparison of American and Chinese mathematics curriculum including the beginnings and development of each. It's titled &amp;quot;A Critique of the Structure of U.S. Elementary School Mathematics&amp;quot; by Liping Ma.&#xA;&lt;a href=&quot;http://www.ams.org/notices/201310/&quot; rel=&quot;nofollow noreferrer&quot;&gt;www.ams.org/notices/201310/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The jist I believe is that back in the 60's America (led by the NSF) wanted to revitalize its mathematics curriculum to compete with the Soviet style. California specifically followed suit and developed the basis of the curriculum we see today with the strands structure.&lt;/p&gt;&#xA;" OwnerUserId="201" LastEditorDisplayName="user18187" LastEditDate="2023-01-16T19:47:22.630" LastActivityDate="2023-01-16T19:47:22.630" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="374" PostTypeId="2" ParentId="358" CreationDate="2014-03-17T15:08:55.093" Score="0" Body="&lt;p&gt;Two questions, really...&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If you start by considering $(x - a) (x - b)$ as a multiplication problem, later the search for factors as stated is natural when solving equations. But then you run into cases where the split isn't apparent. Enter the general formula. (That was the way I got it taught.)&lt;/li&gt;&#xA;&lt;li&gt;Beats me. In high school we learned long division of univariate polynomials. Used it little, mostly to divide out factors when solving higher degree equations.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-17T15:08:55.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="376" PostTypeId="1" CreationDate="2014-03-17T15:39:56.463" Score="19" ViewCount="13113" Body="&lt;p&gt;I will be teaching a course focusing on multivariable integration soon, for the millionth time. The most difficult topic in such a course is certainly Vector Calculus, by which I mean line and surface integrals of vector fields. It is essential to present good applications of these so that students are motivated, but all examples I've ever used are standard physical ones. My course will have many economics/finance majors, and I would love to have some examples I could present along these lines. However my knowledge in these fields are lacking, so I ask: what are some common applications of vector calculus to economics and/or finance, ones which will keep students in these fields motivated? If possible, these applications should be understandable by someone who has (or will have) only an undergraduate background in economics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could probably come up with some ad-hoc example where I model the flow of resources from one sector of an economy to another using a vector field and then come up with some interpretation of a line or surface integral involving this, but I would really like some examples which actually come up in practice. Bonus if they use Stokes' or Gauss' Theorem.&lt;/p&gt;&#xA;" OwnerUserId="18" LastActivityDate="2023-10-19T16:42:56.720" Title="Applications of Vector Calculus to Economics/Finance" Tags="|vector-calculus|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="377" PostTypeId="1" AcceptedAnswerId="386" CreationDate="2014-03-17T15:56:59.147" Score="31" ViewCount="7812" Body="&lt;p&gt;During the 60s, people in the US (and also in Europe), school curricula introduces &lt;a href=&quot;http://en.wikipedia.org/wiki/New_Math&quot;&gt;New Math&lt;/a&gt; where students began with set theory in the first grade before learning to perform addition or multiplication. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somehow, this has ended. &lt;strong&gt;Can you explain why?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Of course, in the linked article there is some criticism, but almost everything the government does is criticized and politics don't really care. Why did they care here?) &lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2024-02-14T08:40:31.080" Title="What was the problem with New Math? Why did it end?" Tags="|primary-education|history|new-math|" AnswerCount="8" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="378" PostTypeId="2" ParentId="376" CreationDate="2014-03-17T16:19:56.970" Score="6" Body="&lt;p&gt;The most important applications of multivariable integration to economics and finance are in &lt;strong&gt;statistics, especially expectations with multivariate probabilities&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many colleges have enough economics and finance majors to support a multivariable calculus class designed on this basis.  It would assume a course in probability and statistics as a prerequisite.  It would skip integrals of vector fields entirely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I can't help you for this year.  You could ask an appropriate professor of economics to write a letter requesting such a class from the math department, and that might pave the way for the future.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-17T16:19:56.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="379" PostTypeId="2" ParentId="269" CreationDate="2014-03-17T16:31:40.290" Score="2" Body="&lt;p&gt;Pinter's &quot;A Book of Abstract Algebra&quot; (Dover, 2nd edition 2010) covers beginning abstract algebra, carefully setting up illustrative examples.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-17T16:31:40.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="380" PostTypeId="1" CreationDate="2014-03-17T16:45:30.680" Score="33" ViewCount="2203" Body="&lt;p&gt;I teach calculus to students who have almost all taken calculus before.  (Primarily first-year college students who took calculus in high school but didn't perform well enough to skip the course.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For many topics, my students think they know the subject already, but all they actually know are a bunch of short-cuts or special cases.  The problem is that when I try to illustrate the new ideas with simple problems, they fall back on their short-cuts rather than using what we've covered.  Then when they get to the harder problems, they're completely lost, because the short-cuts don't work, but they haven't gotten any practice with the new material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can tell them &quot;you have to solve this problem using this method&quot;, but that feeds into the lesson that math is an arbitrary pile of methods which we choose based on the professor's whim, and it causes them to resent the new method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is what are good tactics to break through this: to either convince students that it's in their interests to rethink how they approach the simple cases, or even better, a way to make the parts they know useful rather than an obstacle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The original version of this question gave about a concrete example of a subject where this happens, but it attracted a lot of off-topic answers focusing on the example rather than the question, so I've removed it for clarity.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-07-24T15:52:53.610" LastActivityDate="2022-09-27T16:59:37.980" Title="Revisiting topics from previous courses" Tags="|mathematical-pedagogy|calculus|students-mistakes|" AnswerCount="14" CommentCount="19" ClosedDate="2022-09-25T21:42:12.570" ContentLicense="CC BY-SA 3.0" />
  <row Id="381" PostTypeId="2" ParentId="353" CreationDate="2014-03-17T16:48:31.990" Score="9" Body="&lt;p&gt;I have come to understand this to be analogous to an order relationship among the truth values. P -&gt; Q should be understood to mean &quot;Q is at least as true as P&quot;, or &quot;Q is not less true than P&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, any statement at all (Q = t or f) is at least as true as a known falsehood (P=f), and a known truth (Q=t) is at least as true as any other statement (P= t or f).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can be presented in connection with the most common instance of valid reasoning, modus ponens, and distinguishing two parts of sound reasoning: true premises, and valid reasoning. &#xA;What P -&gt; Q to be true means is that it is logically valid: Using it will not introduce a false conclusion. If we know or assume that P is true, and &quot;if P then Q&quot; is valid, then it is safe to conclude Q.  We don't often consider the case of when P is false. However, if know or assume that &quot;If P then Q&quot; is valid, but P is false, that combination of facts tells us nothing about Q, and it could equally well be true or false. &lt;/p&gt;&#xA;" OwnerUserId="40" LastEditorUserId="40" LastEditDate="2018-03-24T17:41:21.780" LastActivityDate="2018-03-24T17:41:21.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="382" PostTypeId="2" ParentId="353" CreationDate="2014-03-17T16:53:32.263" Score="10" Body="&lt;p&gt;When I've taught propositional logic I acknowledge that this is a formalism that doesn't perfectly match the English usage, and use it as an opportunity to point out &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The evaluation of $\rightarrow$ has to be purely a property of truth values, whereas &quot;implies&quot; in English involves the meaning of the statements, not just whether they're true or false.  This is an important property of the propositional logic, and this is the first good opportunity to emphasize it.  (My students usually agree that, if $\rightarrow$ has to be truth functional, the accepted interpretation is probably the best that can be done.)&lt;/li&gt;&#xA;&lt;li&gt;They're in good historic company, and there are variations of propositional logic (like modal logic and relevant logic) which try to address exactly this issue, but they have to give up having everything be a function on truth values.  This is a nice chance to indicate why there's more than one notion of formal logic.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-03-17T18:17:03.850" LastActivityDate="2014-03-17T18:17:03.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="383" PostTypeId="2" ParentId="351" CreationDate="2014-03-17T17:01:28.613" Score="14" Body="&lt;p&gt;Other people have given other good reasons to have old exams be public, but I want to emphasize the one Andrew Stacey points out in comments: old exams often &lt;em&gt;are&lt;/em&gt; public, and pretending they're not only confuses yourself and punishes students with less access.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If students get to take their exams home (as they can at most schools, though I know there are exceptions), some organizations &lt;em&gt;will&lt;/em&gt; accumulate folders of them to give to future members.  (I know of frats of sororities doing this, but other organizations can as well.)  Releasing them officially levels the playing field, and maybe more importantly, it makes sure that &lt;em&gt;you&lt;/em&gt; remember that students have access to those exams when writing the next one.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-03-17T17:01:28.613" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="384" PostTypeId="5" CreationDate="2014-03-17T17:10:07.803" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-17T17:10:07.803" LastActivityDate="2014-03-17T17:10:07.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="385" PostTypeId="4" CreationDate="2014-03-17T17:10:07.803" Score="0" Body="For questions about exercise groups; tutors, TAs, exercise group leaders and everything related to that." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-17T18:41:30.777" LastActivityDate="2014-03-17T18:41:30.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="386" PostTypeId="2" ParentId="377" CreationDate="2014-03-17T17:12:51.227" Score="29" Body="&lt;p&gt;In the first place, the impetus to &quot;reform&quot; math education was motivated by politics, not by any serious observed deficit. By coincidence, there &lt;em&gt;was&lt;/em&gt; a &quot;new&quot; style in &lt;em&gt;higher&lt;/em&gt; mathematics, reflecting the previous 50-60 years assimilation of set theory and rewriting of many things in terms of set theory. But until the &quot;sputnik scare&quot; no one had incentive to pretend to incorporate set theory or not. The people who &quot;decided&quot; to promote that &quot;New Math&quot; were mostly not mathematicians of any sort whatseover, but, rather, semi-politicized people who needed to be able to &quot;show that they were doing something&quot;. After all, set theory was not new, was not what had made any difference in WWII or subsequently, nor was it what made the Moscow school of mathematics what it was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many parents objected not on scientific grounds, although it was veiled as such, but because the &quot;new math&quot; was alien to them, &lt;em&gt;and&lt;/em&gt; kids who were learning whatever &quot;new math&quot; purported to be were &lt;em&gt;not&lt;/em&gt; learning &quot;traditional math&quot;. Even though &quot;traditional math&quot; included (I was there...) an enormous amount of repetitive drill, arguably to the point of senselessness, it was orthodox and familiar. Parents could no longer help their kids with their homework...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More substantive than parents' discomfort (though that might have been the dominant political determiner) essentially nothing had changed at the college and university level (or even high school), and kids that knew something about set theory couldn't do the basic algorithmic math for basic chemistry, physics, nor the standard high school math curriculum (whatever the flaws of the latter). Certainly &quot;the new math&quot; did not warm people up to the traditional trigonometry, calculus, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more insidious problem was that few of the elementary-ed teachers (nor middle-school, nor high-school) had prior preparation in such stuff, ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, of course, if there were room in the curriculum for it, and if kids were ready for it developmentally, &lt;em&gt;both&lt;/em&gt; some sort of &quot;meaning&quot; and &quot;algorithms&quot; could be taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there's neither room in the curriculum, nor (in my observation) are kids ready for more conceptual things at that point. Perhaps it is harder to &lt;em&gt;teach&lt;/em&gt; concepts than algorithms, also.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was in school then, and my parents both taught high-school math, so I heard a lot about this. It is important to note that, for better or for worse, &quot;school boards&quot; (whose qualifications are mostly political) decide textbooks. The &quot;new&quot; books my parents brought home were more interesting to me than doing yet-more elementary arithmetic, which I could already do, but it was also clear that most of the other fifth-graders wouldn't have been in the right state for such stuff, since they were still having trouble &quot;following instructions&quot; about very concrete activities. (I do not have any citations on &quot;development&quot;...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, so far as I recall, it's not that there was scientific-grounds objection to set theory itself, but that it displaced indispensable things, ... and was new and scary to non-mathematicians. Also, some of the semi-politicized proponents were blitheringly incompetent math-wise, which made them amusing targets for actual scientists... even though that was not the same thing as a criticism of set theory, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, with just a nudge from practical issues, the opportunity for political action and argument both created and killed off &quot;new math&quot;. (More recently, an attempt to modernize calculus and related material has re-generated &quot;math wars&quot;, again with similar extra-mathematical and extra-educational factors dominating the action, ...)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-17T17:12:51.227" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="387" PostTypeId="1" CreationDate="2014-03-17T17:17:18.567" Score="18" ViewCount="1293" Body="&lt;p&gt;At our university we now require one semester of Pre-calculus instead of one semester of Algebra and one semester of Trigonometry before you take Calculus I (for those who do not test into Cal I).  Since we have implemented this, it seems that students who take Pre-calculus, are more likely to fail or withdraw from Calculus I than those students who have tested into Calculus I.  This must mean that Pre-calculus is not preparing students for Calculus I.  What are some of the underlying causes of this disconnect?&lt;/p&gt;&#xA;" OwnerUserId="203" LastEditorUserId="3237" LastEditDate="2015-05-03T15:29:29.843" LastActivityDate="2015-05-04T14:26:59.763" Title="What deficiencies are present in Precalculus curricula that causes so many students to fail Calculus I?" Tags="|undergraduate-education|classroom-management|precalculus|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="388" PostTypeId="2" ParentId="387" CreationDate="2014-03-17T17:39:36.263" Score="25" Body="&lt;p&gt;A blanket problem I've observed over-and-over is that the deficits that scuttle calculus students are even more fundamental than what is discussed in (typical) pre-calculus courses. Specifically, kids, as well as adults returning to school, often cannot do middle-school (pre-?) algebra. Either they get stuck and confused, or immediately and frequently commit ghastly computational errors... leaving very tiny probability that anything will turn out right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, inability to correctly manipulate numerical fractions, or inexperience with it, leaves many of that demographic completely unable to correctly manipulate ratios of polynomials or any such thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Correct manipulation of exponents similarly...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A common further complication is that many in this population are convinced that they &quot;know algebra&quot; because they recall the &quot;A&quot; they got in 8th grade. Thus, they are disinclined to listen to &quot;review of algebra&quot; at the beginning of calculus. (I've found that invariably including far more &quot;algebra&quot; steps than minimal, all along the way, is fairly effective at (re)training people to do that low-level stuff. Leaving the low-level steps to the students is often a serious error...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, if &quot;pre-calculus&quot; is still predicated on the optimistic notion that the population of kids who don't test into calculus surely at least remember middle-school algebra, it won't address their fundamental deficits.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-17T17:39:36.263" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="389" PostTypeId="2" ParentId="362" CreationDate="2014-03-17T18:29:03.487" Score="16" Body="&lt;p&gt;I don't know if this is what you are looking for, but for Green's theorem, when you discuss how you can change the integral $\int 1 dA$ into the line integral $\int x dy - y dx$, you can discuss several real life applications.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You can approximate the area of a lake by walking around it and recording your GPS coordinates $(x_i,y_i)$, then adding up $\Sigma (x_i \Delta y_i-y_i \Delta x_i)$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similarly, you can record the mouse cursor position of a computer and use the same approximation as in 1) to write a program that finds the area (in pixels) of any loop that you draw on a computer (my roommate did this once within 10 minutes after I told him the idea).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Planimeter&quot;&gt;Planimeters&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then you can &lt;em&gt;ask&lt;/em&gt; your students if you can use this same technique to find the area of general 3-d surfaces using Stoke's theorem; it could be an interesting discussion.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-17T18:29:03.487" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="390" PostTypeId="2" ParentId="353" CreationDate="2014-03-17T18:57:21.450" Score="34" Body="&lt;p&gt;I find it helpful to introduce the negation of conditional claims simultaneously. For one, this better helps them to understand the &quot;false implies false&quot; case; but also, this helps them understand how to logically negate conditional claims (which is essential when they go on to learn proof techniques for conditional claims).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The classic &quot;If it is raining, then I definitely have an umbrella with me&quot; is my go-to. I say to the students: &quot;I assert that conditional claim. How could you possibly call me out to be a liar?&quot; They talk amongst themselves and realize that the &lt;em&gt;only&lt;/em&gt; way to call me a liar is if they observe me walking around sans umbrella in the rain; all other situations do not yield a falsehood, so they must be true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Admittedly, this might be passing the buck to accepting the Law of the Excluded Middle, but I've found students are far more comfortable with &quot;True or False?&quot; than they are with &quot;'false implies false' is true&quot; :-)  )&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-17T18:57:21.450" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="391" PostTypeId="2" ParentId="380" CreationDate="2014-03-17T19:00:52.703" Score="16" Body="&lt;p&gt;Sometimes it can help to take things to a ridiculous extreme, like finding the derivative of $\ln(\sin(\cos(e^{x^2})))$. Tell them that the step by step process is like peeling off onion layers:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Take the derivative of the outside the function (the first layer), and leave the inside &lt;em&gt;completely alone&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{1}{(\sin(\cos(e^{x^2})))}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;Now multiply that by the derivative of the second function (the second layer), and leave everything inside of &lt;strong&gt;that&lt;/strong&gt; function completely alone:&#xA;$\frac{1}{(\sin(\cos(e^{x^2})))}\cos(\cos(e^{x^2}))$&lt;/li&gt;&#xA;&lt;li&gt;Now multiply by the derivative of the third `layer', and leave everything inside of that alone. Have students try it first, separately, then reveal the answer and see what they got:&#xA;$\frac{1}{(\sin(\cos(e^{x^2})))}\cos(\cos(e^{x^2}))(-\sin(e^{x^2}))$&lt;/li&gt;&#xA;&lt;li&gt;Have them complete the last few steps then check their answers:&#xA;$\frac{1}{(\sin(\cos(e^{x^2})))}\cos(\cos(e^{x^2}))(-\sin(e^{x^2}))e^{x^2}2x$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Doing a long problem likes this helps them see how mechanical and straightforward it is. Once you do this, have them try:&#xA;1. Find the derivative of $\sin(x+e^x)$, then&#xA;2. Find the derivative of $\sin(xe^x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they get stuck, tell them to remember: Take the derivative of the first layer first, leaving everything inside alone, then multiply by the derivative of the inside, &lt;em&gt;using any rules necessary&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would leave powers of functions for last (like $\sin^2(x)$), because they look so different from other functions, being applied on the right. If you taught them about layers, you can explain that the power is the outermost layer. Otherwise, one option is just to teach it as a completely different rule:&#xA;$$\frac{d}{dx} [f(x)]^n=n[f(x)]^{n-1}f'(x)$$&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-17T19:00:52.703" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="392" PostTypeId="2" ParentId="380" CreationDate="2014-03-17T19:16:00.147" Score="0" Body="&lt;p&gt;Wolfram Alpha Pro will take them through any derivative step-by-step for \$5 per month.  See &lt;a href=&quot;http://www.wolframalpha.com/input/?i=D%5BLn%5BSin%5Bx%5E2%5D%5D,x%5D&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; for your example of $\ln \sin x^2$, with the light blue rectangle saying &quot;step-by-step solution&quot;.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-17T19:16:00.147" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="393" PostTypeId="2" ParentId="366" CreationDate="2014-03-17T19:32:37.680" Score="16" Body="&lt;p&gt;I'd be careful about trying to identify students for whom a math/science field is not in their future.  At the college I work at, about half of our math majors do not intend to major in math or science when they arrive, but discover during their first few semesters that math is actually much more interesting than they had assumed.  In the last five years, I've advised math majors who originally intended to major in photography, music, creative writing, anthropology, history, literature, psychology, political studies, and philosophy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When high-school students don't take Algebra 2, it almost entirely closes off any possibility that they might go to college and study math or science.  I agree that there are certainly some students who would be better served by a high-school math course that focused more on quantitative reasoning, but &quot;closing the door&quot; on students who might one day go into science is quite troubling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the very least, I would hope that the majority of college-bound students would take an Algebra 2 course.  I would even say that it makes sense for most college-bound students to take Precalculus.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-03-17T19:39:53.803" LastActivityDate="2014-03-17T19:39:53.803" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="394" PostTypeId="2" ParentId="387" CreationDate="2014-03-17T19:37:40.470" Score="11" Body="&lt;p&gt;What those students need is a course that will help them see math differently. They need to learn to think mathematically. An excellent pre-calculus course can begin to do that. But at my college (for example), pre-calc has way too many topics. If I taught the content I'm supposed to, we could never dig in properly. I skip some topics, so that I can work more with students on how to think through a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the calculus teacher can help by going over why we take each step we do. Any review should be done not at the beginning of the course, but when it is needed. For example, when multiplying by the conjugate to simplify the expression we get for derivative of a square root (using the definition of derivative), we may need to review the idea of the conjugate (and note its usefulness in a wide variety of contexts). &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-17T19:37:40.470" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="395" PostTypeId="2" ParentId="364" CreationDate="2014-03-17T19:44:14.867" Score="4" Body="&lt;p&gt;While it's notorious for being dense and having a lot of `slick' definitions and proofs, I really enjoyed Walter Rudin's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/007054235X&quot; rel=&quot;nofollow&quot;&gt;Principles of Mathematical Analysis&lt;/a&gt; for its version of Riemann integration in higher dimensions (found on p. 245) and its presentation of differential forms. In fact, I couldn't understand the material in my second-semester analysis course (we used a Dover thrift edition book), and so I had to learn it on my own from this book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I highly recommend it, but I would recommend giving many examples in class to complement the books concise presentation.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-17T19:44:14.867" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="396" PostTypeId="2" ParentId="358" CreationDate="2014-03-17T19:56:12.043" Score="0" Body="&lt;p&gt;I teach it in high school (age equivalents 15 - 17) since it ties in so nicely to the concept of prime factoring in number theory to gain an understanding of what is making up a number. The parallel concept is what is making up the polynomial. The factoring of a polynomial lets us know about reducing algebraic fractions, performing algebraic fraction operations and for solving many quadratic equations. It is formative as it lays a foundation for understanding the general case and as shown by @Jennifer Silverman through completing the square and how there are some that produce integral and rational roots while others have irrational roots.&lt;/p&gt;&#xA;" OwnerUserId="183" LastActivityDate="2014-03-17T19:56:12.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="397" PostTypeId="1" AcceptedAnswerId="406" CreationDate="2014-03-17T20:04:35.080" Score="36" ViewCount="1375" Body="&lt;p&gt;As a guiding example, imagine an undergraduate Calculus II course where students have to complete a guided &quot;research project&quot; and write a &quot;paper&quot; about their work. This can be a shockingly new experience for them, even if the creativity component is minimized by heavy-handed guidance through the steps. Most of them just aren't used to having to present their thoughts/work in a way beyond &quot;here's a bunch of equations in a row and here's the final answer in a box&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any effective ways we can help them with the writing and presentation component of such a project, or something like it? In general, are there ways (short of meeting with every student individually and reading through several successive drafts of their work) that we can foster good, cogent presentation of a student's own mathematical thoughts and efforts?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in both personal experiences (good and bad) and research about this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I don't intend this question to be about getting students to write in &quot;math speak&quot; as if for a journal article, nor about doing creative research. I just want them to learn how to share their ideas.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2021-11-21T06:44:49.730" Title="How can we help students learn to write about their mathematics?" Tags="|mathematical-pedagogy|undergraduate-education|writing|" AnswerCount="9" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="398" PostTypeId="2" ParentId="364" CreationDate="2014-03-17T20:13:07.943" Score="6" Body="&lt;p&gt;For multidimensional real analysis I recommend the two-volume bible by Duistermaat:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521551145&quot;&gt;Multidimensional Real Analysis, Volume 1 - Differentiation&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521829259&quot;&gt;Multidimensional Real Analysis, Volume 2 - Integration&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Features: many examples from interconnected areas, concrete computations, geometric motivation, prefers to use linear algebra in proofs. Exercises vary from very simple (to acquire familiarity with notation and language) to pretty hard (it walks you through the Peano curve construction).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With your input, these two books have everything to guide your students.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-03-17T20:13:07.943" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="399" PostTypeId="2" ParentId="397" CreationDate="2014-03-17T20:19:57.760" Score="17" Body="&lt;p&gt;For several years I taught a crypto course, and an error-correcting codes course, that both emphasized coherent writing. These were for math-major juniors, seniors, and grad students in engineering departments. The homework and exams were mostly about execution of algorithms rather than &quot;proofs&quot;, but/and &lt;em&gt;explanation&lt;/em&gt;, at least some sort of &lt;em&gt;narrative&lt;/em&gt; was required. Coping with students' incredulity (&quot;the symbols speak for themselves!&quot; ?!?!) was time-consuming and frustrating, but seemed worth the effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did distribute &quot;approved solutions&quot; as models for what kind of writing-out was desired.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, each term, there was a &quot;writing project&quot; required. We did not do revisions... since I usually had 100 students in a class... but/and by the end of the term most of the kids were catching on to writing in English rather than math-eze. In fact, it was gratifying that the issue shifted a bit from &quot;coherent writing&quot; to &quot;what is plagiarism, versus using sources...&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This was quite a lot more work than for &quot;standard&quot; math courses, and the other people now teaching these courses have reverted to more typical structure...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is indeed hard to communicate the idea that mathematics should be imbedded in natural language, to populations who've been conditioned by typical k-12 that the goal is usually a number-in-a-box, and even &quot;showing your work&quot; is a crazy luxury of sorts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point of having people write about their execution of algorithms was &lt;em&gt;intended&lt;/em&gt; originally to have people write about more down-to-earth activities (as opposed to the stylized &quot;proof&quot; notion). I discovered to my dismay that many or most had at least initial difficulty getting the verbal part and the computational part of their brains to cooperate... it seemed. That is, in some amazing cases, people were unable (or maybe merely balked...) to say in words what they were doing in some computation, and/or why they chose one thing over another, and so on. Surprisingly subverbal or non-verbal... An example was to explain how someone knew that $15$ factors into primes as $3\cdot 5$. One kid I recall could say &quot;it's obvious&quot;, and could &lt;em&gt;not&lt;/em&gt; say what they'd do if the number were large enough that a factorization was not &quot;obvious&quot; to them. (This in the days after discussion of trial division and so on... certainly not in a vacuum...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the whole, it seemed beneficial to push students to write coherently, but took much more effort to give useful feedback... and with the ambient curriculum collapsing back to non-verbal mathematics (or stylized math-English), it was somewhat a quixotic battle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At my university there is nowadays a &quot;writing-intensive&quot; requirement, but it is isolated in the math curriculum and viewed as a tiresome burden. And some of the vehicles for its fulfillment are (in my opinion) somewhat anti-coherent, emphasizing an extreme dialect of math-eze.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2014-03-18T01:04:20.317" LastActivityDate="2014-03-18T01:04:20.317" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="400" PostTypeId="2" ParentId="380" CreationDate="2014-03-17T20:48:33.147" Score="7" Body="&lt;p&gt;I suggest to have a look at your students' concept of a function. Typically, they see a function as a process but not as an object that can be part of a larger object. (They will identify terms as parts of a larger term; however, they don't realize them to be functions in a process sense.)&#xA;There is a nice piece of theory by a group led by Ed Dubinsky, they call it APOS theory, referring to Action, Process, Object and Scheme as levels of understanding:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dubinsky, E., &amp;amp; Mcdonald, M. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. The teaching and learning of mathematics at university level, 275–282.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe you could try to make a bit more explicit that the use of the chain rule requests two functions to be chained. One could try to simply identify a &quot;chain of functions&quot; in terms like $\sin(x+1)$ or even $x^4=(x^2)^2$. As a powerful application you may let them try to work on $x^x$. Unfortunateley, the transition from process to object state of understanding has no clear way that always works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many problems rising from this problem, e.g. they might struggle with seeing a differential operator (or any other operator) is linear, since they get confused because there are two functions at different level: the operator itself and its input are both functions.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="199" LastEditDate="2014-03-17T21:10:26.107" LastActivityDate="2014-03-17T21:10:26.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="401" PostTypeId="2" ParentId="377" CreationDate="2014-03-17T21:38:34.810" Score="13" Body="&lt;p&gt;I think very roughly speaking: &quot;new math&quot; (also known in europe as a fearful period of time, especially for parents) followed a mathematical construction of mathematical knowledge rather than a psychological one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematically, you would introduce an abstract concept like an equivalence relation first and then introduce concept like terms or fractions as applications. Psychologically, you would go the other way. Obviously, many people can handle fractions without even knowing the word equivalence relation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, if a way seems mathematically straight, that doesn't necessarily mean it is psychologically the easiest way or shortest way in time. You may start with set theory to learn the natural numbers; however, simply counting 1, 2, 3,... seems more promising.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-17T21:38:34.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="402" PostTypeId="2" ParentId="397" CreationDate="2014-03-17T22:07:17.867" Score="8" Body="&lt;p&gt;I'll share my experiences, but definitely want to hear from others, as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've assigned these &quot;research project&quot; papers in Calc II. The actual math is mostly guided, with some open-ended questions and extra credit available. I tell students to meet with me while they're working, to make sure they're on the right track, and they &lt;em&gt;always&lt;/em&gt; ask something like, &quot;How am I supposed to write about this?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pasted below are the guidelines I put in the project descriptions. I've found them to be somewhat helpful as general guidelines, but they still require the students to think about what &lt;em&gt;they&lt;/em&gt; themselves need to say and how to say it. But this is part of the goal of the project, anyway. Mostly, I just keep stressing this for them: &quot;Put yourself in the shoes of a classmate reading your paper. Would you learn from it?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found the results to be generally quite good, actually! I think this is partly because this experience is new for them, so they just have to think about it more. I also think this is because (most of them) they do come to me and ask about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some things that need improvement ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's tough to get the students to &lt;em&gt;qualitatively&lt;/em&gt; describe things. Often, they write about their work by &quot;translating&quot; the steps into wordy sentences that actually don't complement the accompanying equations. Similarly, they'll include a graph with little description or one plotted on an ill-fitted scale. I think this reflects a general trend that the students (particularly at into calculus college level) still view their work as entirely &lt;em&gt;quantitative&lt;/em&gt; in nature. I try to address this in class by describing my own heuristic/qualitative views whenever they're applicable, but it's tough to address this phenomenon head on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also think they do very little editing. Once they figure out the right steps and write a description of it, they're done with that part and move on. Without seeing their works in progress, I cannot verify this, and can only guess based on final outputs. Furthermore, I wonder if this just reflects a broader trend of poor writing skills gained from current education. (I actively combat this in my &quot;intro to proofs&quot; course, but that course is much more heavily devoted to developing writing skills, as a whole.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the following pages, I will provide a list of projects for you to&#xA;  choose from, for the month of March. These projects will guide you&#xA;  through the solution of an interesting and relevant theoretical&#xA;  problem or an application of the material we see in class (or a&#xA;  combination of the two). You will write your results and discoveries&#xA;  in the form of a &quot;research&quot; paper that describes the steps in solving&#xA;  your particular problem, some reasons as to why it is interesting and&#xA;  related to what we've done in class, and a description of what you&#xA;  learned and what you found useful.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;You will be graded not only on the &lt;em&gt;correctness&lt;/em&gt; of your work, but&#xA;  also the &lt;em&gt;clarity&lt;/em&gt; of your descriptions. The goal is that a fellow&#xA;  classmate can pick up your paper, read and follow along with your&#xA;  work, and feel like they learned something, even if they didn't work&#xA;  on that particular project.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I want you to actually &lt;em&gt;write some words&lt;/em&gt; for your research paper, as&#xA;  opposed to just showing a bunch of lines of equations with some&#xA;  descriptions beside them. (That would be a &lt;em&gt;bad&lt;/em&gt; paper.) You should&#xA;  exhibit all of the important steps you took in solving your problem,&#xA;  and you should provide some (a) description as to what they represent,&#xA;  as well as (b) some motivation for &lt;em&gt;why&lt;/em&gt; you did things the way you&#xA;  did. Remember: the goal is that a classmate could read your paper and&#xA;  discover what you did and learn something.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I do not want to put a specific &quot;page limit&quot;, either a minimum or&#xA;  maximum, on this project. I would say that, on average, a final&#xA;  write-up will be anywhere from 3-6 pages. This depends on how much&#xA;  math you need to show, and also on how detailed you make your&#xA;  descriptions. If you have a lot to say, go for it and write more. If&#xA;  you're concise with your math and skip over some easy details, maybe&#xA;  your paper ends up fairly short. This is all fine. Just keep in mind,&#xA;  though: longer is not necessarily better, and I will not use the&#xA;  length as a factor in your grade; however, I do find it very hard to&#xA;  believe that all of your details and descriptions could be squeezed&#xA;  into a page and a half. (Not impossible, but very very rare.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-17T22:07:17.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="403" PostTypeId="1" CreationDate="2014-03-17T22:11:51.943" Score="6" ViewCount="625" Body="&lt;p&gt;When seeing groups and such for the first time, the abstraction often seems pointless and unnecessary to students. (Most students at my school leave their introductory abstract algebra class thinking it was pointless and unnecessary, too.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some nice questions that can be used during the class to motivate the development of groups (or rings), that groups solve nicely? I'd rather Galois theory wasn't used as an example - it takes a bit too much work and wouldn't fit into the type of introductory course I'm thinking of.&lt;/p&gt;&#xA;" OwnerDisplayName="user37" LastEditorUserId="114" LastEditDate="2014-04-11T12:26:01.940" LastActivityDate="2014-04-11T12:26:01.940" Title="What are some good motivating questions for introductory abstract algebra?" Tags="|undergraduate-education|abstract-algebra|concept-motivation|" AnswerCount="1" CommentCount="6" ClosedDate="2014-03-19T00:21:42.270" ContentLicense="CC BY-SA 3.0" />
  <row Id="404" PostTypeId="2" ParentId="403" CreationDate="2014-03-17T22:59:55.360" Score="4" Body="&lt;p&gt;I have never tried this, but I think solving a rubik's cube would be a fun application. Try: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20Rubik%27s%20Cube.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20Rubik's%20Cube.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="52" LastEditDate="2014-03-18T02:48:46.633" LastActivityDate="2014-03-18T02:48:46.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="405" PostTypeId="2" ParentId="397" CreationDate="2014-03-17T23:07:04.880" Score="12" Body="&lt;p&gt;Good writing requires style and an intended audience.  When we ask our students to explain their mathematics in writing without giving them any guidance, typical results include a string of equations and explanations that display a minimum of effort and reflection.  In order to get better writing from our students, we should be more explicit about what we want.  For instance, we could have them write for a particular audience:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Write this as an explanation to an average student who will be taking this same class next semester.&lt;/li&gt;&#xA;&lt;li&gt;Write this as a newspaper article that assumes its readers have very little background knowledge.&lt;/li&gt;&#xA;&lt;li&gt;Write this in the style of an example in your course textbook.&lt;/li&gt;&#xA;&lt;li&gt;Write this in the style of a formal written proposal for grant funding for an important project.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Developing good writing skills also takes a lot of practice (revise, revise, revise!) distributed over a long period of time.  If you're really serious about improving the mathematical writing skills of students, you should start by requiring them to explain their answers to homework questions using complete sentences with correct grammar, spelling, and punctuation.  Of course, this requires papers to be hand graded and for the grader to enforce this policy (e.g., by having some of the homework points be for mathematical correctness and other points dedicated to exposition).  Students who have consistently gotten feedback about their writing on the homework usually have fewer difficulties with longer writing assignments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students need explicit guidance on what needs explaining and what does not.  Initially, many students will need to be told not to over-explain (e.g., tediously writing a sentence for each algebraic manipulation) and not to under-explain (e.g., assume their reader knows too much).  Giving students a long list of rules to follow might unintentionally produce bad results.  In general, getting students to make decisions about what to include and exclude boils down to getting them to think about what their reader needs to know in order for things to make sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students need to be told repeatedly that they are expected to explain &lt;strong&gt;why&lt;/strong&gt; they chose to do what they did.  The final result of a mathematical computation only has meaning when the problem is put into context, the assumptions are made explicit, and all of the  mathematical operations are stated clearly and justified fully. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Creating and grading writing assignments is a daunting and difficult task, as is managing students during the writing process.  Good writing assignments take a ton of time to create and grade, and I would encourage you to ask you colleagues in the humanities for tricks they use to make the process go smoothly.  To ease your feedback and grading burden, I would encourage you to set aside time in class for students to exchange their work in small groups and give each other feedback on their writing.  Another trick is to use examples of good and bad writing (e.g., on a projector screen) in class to start a classroom discussion about the strengths and weaknesses of student work (make sure to get student's permission to show their work beforehand).  Using student examples in class also provides the opportunity to openly discuss what feedback you would give and how you would grade such work.  This feedback should happen before the final draft is due.  You may also want to consider having students turn in both the rough draft (with comments from other students) and the final draft at the same time, and assigning more points to the rough draft to get them to take it seriously.  Think very carefully about how to manage your students (e.g., the timeline and incentives to meet deadlines) as this can save you time and anguish when you eventually grade their work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, take advantage of technology.  For instance, would it be appropriate for your students to write their papers on &lt;a href=&quot;https://stackedit.io&quot;&gt;https://stackedit.io&lt;/a&gt; (which uses MathJax and LaTeX syntax for typesetting mathematics) and share them with you via Google Drive?  They can also make a PDF copy and print it, if you prefer hard copy.&lt;/p&gt;&#xA;" OwnerUserId="204" LastActivityDate="2014-03-17T23:07:04.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="406" PostTypeId="2" ParentId="397" CreationDate="2014-03-18T00:16:08.487" Score="22" Body="&lt;p&gt;I would like to suggest that one can incorporate writing in a calculus course as a matter of routine mathematical activity, rather than using it only as a part of a big &quot;research project&quot;. And integrating mathematical writing into the everyday parts of a course might help to make the point that writing well is a fundamental part of mathematical activity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have long held the view that one of the most important skills that students might learn in an undergraduate math class, such as calculus, is the ability to explain a technical idea with clarity and precision. Most of our students will not actually be computing integrals or testing series for convergence in their future careers (though some will), but most of them will probably have to explain something technical to their colleagues, boss or clients, and a mathematics class is a great place to learn and practice this extremely important skill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to do so, I insist that students submit their mathematical work on homework, quizzes and exams in the form of a mathematical explanations, written essay-style with properly punctuated complete English sentences, and with embedded mathematical computations or diagrams to support whatever explanation is being made. The textbook, of course, is full of examples of the kind of writing I mean, with the worked-out examples and theorems, and this is of course the standard way that mathematicians communicate ideas. Weaker students are sometimes mystified by my instructions, and they are sometimes stuck in the habit of writing a series of mathematical expressions separated by equal signs (whether they are equal or not) and a box around the &quot;answer&quot; at the end. So I try to unstick them from that habit. Occasionally students misunderstand my instructions as a request to justify every picky little step, and so I have to correct this, telling them that the more valuable explanations concentrate on the big idea of the solution method, emphasizing especially what might be different in this solution than in another similar one. In time, they seem to get it. (But to be honest, my expectations student writing in calculus are not actually very high, although they are much higher in more advanced courses.)&lt;/p&gt;&#xA;" OwnerUserId="90" LastActivityDate="2014-03-18T00:16:08.487" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="407" PostTypeId="2" ParentId="238" CreationDate="2014-03-18T00:39:55.423" Score="4" Body="&lt;p&gt;The &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0805062998&quot; rel=&quot;nofollow&quot;&gt;Number Devil&lt;/a&gt; is a favorite of mine. It has so much interesting math in it and is written in a visually appealing way.&lt;/p&gt;&#xA;" OwnerUserId="94" LastActivityDate="2014-03-18T00:39:55.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="408" PostTypeId="1" CreationDate="2014-03-18T01:06:27.307" Score="11" ViewCount="557" Body="&lt;p&gt;When designing slides for a class I am teaching, I like to derive inspiration from slides others have given before me. However, it is difficult to find concrete examples of such slides outside of a typical Google search. Can you give me examples of slides you or your colleagues have given for entire classes?&lt;/p&gt;&#xA;" OwnerUserId="179" LastEditorUserId="179" LastEditDate="2014-03-24T16:44:54.937" LastActivityDate="2014-03-25T18:40:38.797" Title="Showcase of Powerpoint / Keynote / Beamer Presentations" Tags="|technology-in-education|presentation|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="409" PostTypeId="2" ParentId="377" CreationDate="2014-03-18T01:22:40.793" Score="17" Body="&lt;p&gt;You might want to read Kline (1973).  I haven't read the book, but according to &lt;a href=&quot;http://en.wikipedia.org/wiki/New_Math#Criticisms&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1973, Morris Kline published his critical book &lt;em&gt;Why Johnny Can't Add: the Failure of the New Math&lt;/em&gt;. It explains the desire to be relevant with mathematics representing something more modern than traditional topics. He says certain advocates of the new topics &amp;quot;ignored completely the fact that mathematics is a cumulative development and that it is practically impossible to learn the newer creations if one does not know the older ones&amp;quot; (p. 17). Furthermore, noting the trend to abstraction in New Math, Kline says &amp;quot;abstraction is not the first stage but the last stage in a mathematical development&amp;quot; (p. 98).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You might also want to read &lt;a href=&quot;http://www.csun.edu/%7Evcmth00m/AHistory.html&quot; rel=&quot;noreferrer&quot;&gt;Klein (2003)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A substantial number of mathematicians had already expressed serious reservations relatively early in the New Math period. In 1962, a letter entitled &lt;em&gt;On The Mathematics Curriculum Of The High School&lt;/em&gt;, signed by 64 prominent mathematicians, was published in the &lt;em&gt;American Mathematical Monthly&lt;/em&gt; and &lt;em&gt;The Mathematics Teacher&lt;/em&gt;. The letter criticized New Math and offered some general guidelines and principles for future curricula.&lt;/p&gt;&#xA;&lt;p&gt;By the early 1970s New Math was dead. The National Science Foundation discontinued funding programs of this type, and there was a call to go &amp;quot;back to the basics&amp;quot; in mathematics as well as in other subjects. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Klein, David (2003). &amp;quot;A Brief History of American K-12 Mathematics Education in the 20th Century.&amp;quot; &lt;em&gt;Mathematical Cognition&lt;/em&gt;. Information Age Publishing.&lt;/p&gt;&#xA;&lt;p&gt;Kline, Morris (1973). &lt;em&gt;Why Johnny Can't Add: The Failure of the New Math&lt;/em&gt;. New York: St. Martin's Press.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-18T01:22:40.793" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="410" PostTypeId="2" ParentId="36" CreationDate="2014-03-18T01:25:41.553" Score="4" Body="&lt;p&gt;On Windows, Microsoft publishes their own software called &lt;a href=&quot;http://www.microsoft.com/en-us/download/details.aspx?id=15702&quot; rel=&quot;nofollow&quot;&gt;Microsoft Mathematics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Microsoft Mathematics provides a graphing calculator that plots in 2D and 3D, step-by-step equation solving, and useful tools to help students with math and science studies. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can see it in action in &lt;a href=&quot;http://youtu.be/LVuaRxTNZR8&quot; rel=&quot;nofollow&quot;&gt;this video&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Microsoft also publishes an &lt;a href=&quot;http://www.microsoft.com/en-us/download/details.aspx?id=17786&quot; rel=&quot;nofollow&quot;&gt;add-in&lt;/a&gt; for Microsoft Word and Microsoft OneNote that allows you to create these plots directly in the Word or OneNote interface.&lt;/p&gt;&#xA;" OwnerUserId="179" LastActivityDate="2014-03-18T01:25:41.553" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="411" PostTypeId="2" ParentId="353" CreationDate="2014-03-18T05:49:58.990" Score="4" Body="&lt;p&gt;To supplement Brendan's idea:&#xA;I like to connect quantified statements to unquantified in the following way:&#xA;Assume a statement like &quot;If X is a dog, then X has a head&quot;. Now, once you have found this to be true, you might want the truth not to depend on X. Thus replacing X for anything like &quot;my car&quot; should still give you a true statement. However, &quot;If my car is a dog, then my car has a head&quot; should be true then.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A second remark:&#xA;At this point one should make clear, that the truth of statements is not given by nature, but something we define. So the question should not be which truth value is right, but which one makes more sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: third remark:&#xA;There are statements which feel &quot;more true&quot; than others, althoug they are of the form &quot;A=&gt;B&quot; with A, B being wrong. Compare &quot;If I am the pope, then I am a woman.&quot; and &quot;If I am the pope, then I live in Rome&quot;. For me (not the pope, male, not living in rome), all parts of the statements are wrong. However, the second feels true, whereas the first one feels wrong. So it is worth a discussion if and how one should calculate truth values of conditional statemens from the truth values of their parts.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="199" LastEditDate="2014-03-19T15:30:34.497" LastActivityDate="2014-03-19T15:30:34.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="412" PostTypeId="2" ParentId="353" CreationDate="2014-03-18T05:51:11.627" Score="32" Body="&lt;p&gt;Various psychological studies have been done which show that most people (including university students, who are the most common subjects of psychological tests!) are very poor at grappling with the last two entries of the truth table for &lt;span class=&quot;math-container&quot;&gt;$A \implies B$&lt;/span&gt; in an abstract context, but they are much better with it in a situation in which the consequences of falsifying the implication are understood.&lt;/p&gt;&#xA;&lt;p&gt;When I taught (twice) a &amp;quot;transitions&amp;quot; course for undergraduate math majors, I gave them this essay question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;T3.1) a) You are shown a selection of cards, each of which has a single letter printed on one side and a single number printed on the other side. Then four cards are placed on the table. On the up side of these cards you can see, respectively, D, K, 3 and 7. Here is a rule: &amp;quot;Every card that has a D on one side has a 3 on the other.&amp;quot; Your task is to select all those cards, but only those cards, which you would have to turn over in order to discover whether or not the rule has been violated.&lt;/p&gt;&#xA;&lt;p&gt;b) You have been hired to watch, via closed-circuit camera, the bouncer at a certain 18-and-over club. In order to be allowed to drink once inside the club, a patron must display valid 21-and-over ID to the bouncer, who then gives him/her a special bracelet. In theory the bouncer should check everyone's ID, but (assume for the purposes of this problem, at least!) it is not illegal for someone who is under 18 to enter the club, so you are not concerned about who the bouncer lets in or turns away, but only who gets a bracelet. You watch four people walk into the club, but because the bouncer is so large, sometimes he obscures the camera. Here is what you can see:&lt;/p&gt;&#xA;&lt;p&gt;The first person gets a bracelet.&lt;br /&gt;&#xA;The second person does not get a bracelet.&lt;br /&gt;&#xA;The third person displays ID indicating they are 21.&lt;br /&gt;&#xA;The fourth person does not display any ID.&lt;/p&gt;&#xA;&lt;p&gt;You realize that you need to go down to the club to check some IDs. Precisely whose ID's do you need to check to verify that the bouncer is obeying the law?&lt;/p&gt;&#xA;&lt;p&gt;c) Any comments?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Part a) is (up to psychological isomorphism) the infamous &lt;a href=&quot;http://en.wikipedia.org/wiki/Wason_selection_task&quot; rel=&quot;noreferrer&quot;&gt;Wason selection task&lt;/a&gt;.  Part b) is a real-world analogue designed to be closer to the students' experience.  It makes perfect sense that we do not need to recheck the IDs of anyone who did not get a bracelet: we're trying to enforce the implication &amp;quot;If you get served drinks, you must have ID.&amp;quot;  People can understand that no one is going to get in trouble for the people that they didn't serve drinks to.&lt;/p&gt;&#xA;&lt;p&gt;One can see relations here to one of the other (good) answers.  For one, yes, it's good to think in terms of when &lt;span class=&quot;math-container&quot;&gt;$A \implies B$&lt;/span&gt; is false: there's just one possibility and that's what we care about, so in every other case we make it true.  But yes, I do introduce implication via the truth table.  It can also be helpful to define it as &amp;quot;(not A) or B&amp;quot;: that somehow seems less arbitrary, and gives them good practice seeing that the negation is &amp;quot;A and (not B)&amp;quot;.  But then we have the burden of explaining why we call this &amp;quot;implies&amp;quot;....and I've found that if you emphasize that the one possibility you need to exclude is that A is true and B is false, then it is not in fact so terribly hard for students to swallow.  I follow up with the concept of &amp;quot;vacuously true&amp;quot;, namely the implication is true because the hypothesis is false.  This becomes a key proof technique later in the course: sometimes you need to begin to analyze an implication &lt;span class=&quot;math-container&quot;&gt;$\forall x \in A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P(x) \implies Q(x)$&lt;/span&gt; by first figuring out for which &lt;span class=&quot;math-container&quot;&gt;$x \in A$&lt;/span&gt; it is the case that &lt;span class=&quot;math-container&quot;&gt;$P(x)$&lt;/span&gt; is true.&lt;/p&gt;&#xA;" OwnerUserId="176" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-18T05:51:11.627" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="413" PostTypeId="2" ParentId="186" CreationDate="2014-03-18T06:07:54.090" Score="5" Body="&lt;p&gt;There are different ways to introduce the real numbers. Even if you choose an axiomatic introduction, you have to choose an appropriate formulation of the completeness theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since you are german, I suggest you to have a look the following article:&#xA;Zur Behandlung der reellen Zahlen im Oberstufenunterricht. In: H. Schröder (Hrsg.): Der Mathematikunterricht im Gymnasium. Schroedel, Hannover 1966, 215-227.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some 50 years ago, the same question was discussed concerning school. :-)&#xA;Kirsch preferes the axiomatic introduction; however, one should connect the new knowledge to the decimals which the students already know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since often, they think they already know the reals and thus do not pay attention, I would suggest to irritate them in the lecture by asking questions like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is $0.999... =1$?&lt;/li&gt;&#xA;&lt;li&gt;What is $0.999...+0.111...$?&lt;/li&gt;&#xA;&lt;li&gt;Are there infinitely small numbers?&lt;/li&gt;&#xA;&lt;li&gt;How is $\sqrt2^{\sqrt2}$ defined? How can it be calculated?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For an interesting introduction of the reals based on the integers (not the rationals!) you might have a look at &quot;eudoxos reals&quot;:&#xA;&lt;a href=&quot;http://arxiv.org/abs/math/0405454v1&quot; rel=&quot;noreferrer&quot;&gt;http://arxiv.org/abs/math/0405454v1&lt;/a&gt;&#xA;However, I would not recommend this for your teaching.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-18T06:07:54.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="414" PostTypeId="1" AcceptedAnswerId="536" CreationDate="2014-03-18T06:59:50.200" Score="7" ViewCount="214" Body="&lt;p&gt;I want give some extra homework about Lagrange multipliers and constrained optimization.  I am interested in examples that are complex (for example, those having more than two variables and more than one restriction).  Where I can find sufficient examples for students' practice? &lt;/p&gt;&#xA;" OwnerUserId="79" LastEditorUserId="77" LastEditDate="2020-01-01T05:39:29.290" LastActivityDate="2020-01-01T05:39:29.290" Title="Where can I find example of Lagrange multipliers and constrained optimization?" Tags="|homework|optimization|constrained-optimization|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="415" PostTypeId="1" AcceptedAnswerId="421" CreationDate="2014-03-18T07:02:16.630" Score="14" ViewCount="410" Body="&lt;p&gt;In university courses with compulsory homework, I quite often find students copying their work from others. Now, you can simply ignore this since they are responsible for their learning. If not, usually a very annoying game starts, where students try to slightly modify their soultions and teachers become detectives. I am not sure if this is helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is there any literature or best practice experience on the problem of how to deal with copying of mathematics students?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="376" LastEditDate="2016-01-22T06:15:18.727" LastActivityDate="2016-01-22T06:15:18.727" Title="How to deal with students copying?" Tags="|undergraduate-education|homework|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="416" PostTypeId="1" AcceptedAnswerId="456" CreationDate="2014-03-18T08:11:28.033" Score="29" ViewCount="1620" Body="&lt;p&gt;From high school to introduction courses in university, the expression $0^0$ is some (psychological) problems. High school students just apply it to their calculator and either the result is $1$ or undefined. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In calculus, you may define $0^0$ as a limit of $x_n^{y_n}$ for $x_n,y_n\to 0$, but the limit depends on the given sequences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you give arguments which explain/legitimate the &quot;definition&quot; of $0^0:=1$? The arguments should be accessible to the students at their point of knowledge, not contain anything like &quot;If you want to proof some theorem in the next two years, then you don't have to specify on that particular case that the base is $0$&quot;.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-28T18:47:46.197" LastActivityDate="2014-12-27T00:21:59.290" Title="What are argument one can give to students on the definition $0^0$?" Tags="|undergraduate-education|calculus|secondary-education|powers|" AnswerCount="10" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="417" PostTypeId="2" ParentId="416" CreationDate="2014-03-18T08:25:01.830" Score="10" Body="&lt;p&gt;Since there is much choice in Terms of limits as you said, a definition should be reasonable in terms of convenience. &#xA;If you define the exponential function via the series $e^x:=\sum_{n=0}^{\infty} \frac{x^n}{n!}$, then in $e^0$ you need $0^0=1$. The same is true for the binomial theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Also have a look at&#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero&quot;&gt;http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero&lt;/a&gt; resp.&#xA;&lt;a href=&quot;http://de.wikipedia.org/wiki/Potenz_%28Mathematik%29#Null_hoch_Null&quot;&gt;http://de.wikipedia.org/wiki/Potenz_%28Mathematik%29#Null_hoch_Null&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-18T08:25:01.830" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="418" PostTypeId="2" ParentId="416" CreationDate="2014-03-18T09:07:22.863" Score="30" Body="&lt;p&gt;The most intuitive reason I know for $0^0 = 1$ comes from interpretation in terms of functions, namely&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{There are } |B|^{|A|} \text{ functions } A \to B \text{ for any finite $A$ and $B$}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, there are no functions $\{\spadesuit\} \to \varnothing$, so $0^1 = 0$, but there exists exactly one function $\varnothing \to \varnothing$ (its set of input-output pairs is empty), so $0^0 = 1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-18T09:07:22.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="419" PostTypeId="2" ParentId="416" CreationDate="2014-03-18T09:47:24.780" Score="10" Body="&lt;p&gt;If it is feasible, you could start by introducing the big product operator ($\prod$) and use it to define (or just write) such things as the factorial, the exponentiation of natural numbers or binomial coefficients:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$n! := \prod\limits_{i=1}^n i;\quad a^n := \prod\limits_{i=1}^n a;\quad \binom{n}{k} := \prod\limits_{i=1}^k \frac{n+1-i}{i}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, it is convenient¹ to define the empty product as 1, since it allows us to get the following results from the above definitions without needing to define an exception²:&#xA;$$0! = \prod\limits_{i=1}^0 i = 1;\quad 2^0 = \prod\limits_{i=1}^0 2 = 1;\quad \binom{n}{0} = \prod\limits_{i=1}^0 \frac{n+1-i}{i}=1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular for the binomial coefficients, this makes some sense in “application”, as a set with $n$ elements has exactly one subset with 0 elements, namely $\emptyset$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From this, you also get:&#xA;$$0^0 = \prod\limits_{i=1}^0 0 = 1$$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;¹ And I would make clear that this is nothing more.&lt;br&gt;&#xA;² Also, it is somewhat consistent to have all the big operators return the neutral element of their respective operation, when empty:&#xA;$$\sum\limits_{i=1}^{0}a_i = 0;\quad \prod\limits_{i=1}^{0}a_i = 1;\quad \bigcup\limits_{i=1}^{0}a_i = \emptyset $$&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2014-03-18T10:07:13.457" LastActivityDate="2014-03-18T10:07:13.457" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="420" PostTypeId="2" ParentId="416" CreationDate="2014-03-18T10:05:20.623" Score="17" Body="&lt;p&gt;In the context of polynomials, $0^0 = 1$ is a convenient &lt;em&gt;convention&lt;/em&gt; to be able to write $p(x) = \sum_{0 \le k \le n} a_k x^k$ and have the formula valid for $x = 0$ (perhaps explain that here the $0$ is fixed, while $x$ is variable; so the &quot;indeterminate unless the exact path followed in $x^y$&quot; just doesn't come up).&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="42" LastEditDate="2014-03-18T16:21:53.657" LastActivityDate="2014-03-18T16:21:53.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="421" PostTypeId="2" ParentId="415" CreationDate="2014-03-18T10:35:57.963" Score="9" Body="&lt;p&gt;At my university the focus is moved to tests and exams. In larger courses homework has usually no or very little significance, while in smaller classes very few attempt copying (it is too easy to spot).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, programming homework is handled via plagiarism software (we have our own, its internals being kept secret; the output is reviewed manually), this is a routine procedure, as homework is usually tested automatically, and it is a standard step in various national-level contests that our university handles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, to answer your question, how to handle copying when it happens?&#xA;I know of three ways:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;By a formal process.&lt;/em&gt; The statute of our university says it's forbidden to do so (I suspect this is not unique), and such a student could be expelled. In practice, as far as I can remember, it had never happened. I heard that there had been such attempt, however, the student council had caused enough trouble that it all had come to nothing. This might have something to do with the fact, that the STEM-campus and the humanities don't like each other very much and the student council is in majority from the latter.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Fail the course&lt;/em&gt;. Some insist that such students shouldn't be allowed to attend the make-up session exam, but this would result in a bureaucratic mess. Instead, by plagiarism the student achieves an equivalent of $-\infty$ points and it all fits into professor's &quot;freedom of grading&quot;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;By some agreement.&lt;/em&gt; This might be not the most fair, but during their first semester, it might be the most appropriate. Students come from various neighborhoods, their high-school had different policies and are frequently immature. A good talking-to has more positive effects than punishment, which you could still apply later if he/she hasn't learned from the mistakes. Fortunately, at my university copying is rare enough that we can afford such a lenient treatment (and you don't even need to keep track, because people remember).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="376" LastEditDate="2016-01-22T06:11:19.953" LastActivityDate="2016-01-22T06:11:19.953" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="422" PostTypeId="2" ParentId="362" CreationDate="2014-03-18T10:58:07.380" Score="7" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Divergence_theorem&quot;&gt;&lt;strong&gt;divergence theorem or Gauß’s theorem&lt;/strong&gt;&lt;/a&gt; can be nicely linked to reality:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of &lt;strong&gt;hydrodynamic flows&lt;/strong&gt; you could start with the following statement:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The total signed water flow through the border of an area or surface of a volume (without sinks and sources) is zero – what goes in, must come out.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then you can add sinks and sources, i.e., points with a divergence that is not zero to the scope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A little bit more abstract are &lt;a href=&quot;http://en.wikipedia.org/wiki/Force_field_%28physics%29&quot;&gt;&lt;strong&gt;force fields&lt;/strong&gt;&lt;/a&gt;, such as the gravitational or electrical field:&lt;br&gt;&#xA;For those, sinks and sources correspond to masses or charges inside your surface and the vector field itself corresponds to the local field strength, which in turn is proportional to the force experienced by an object of a given mass or charge, respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, if your mass or charge distribution is radially symmetric, the force experienced by an object only depends on the total amount of masses or charges which are closer to the center of symmetry but not on their distribution. So just going by the gravitational field we cannot decide whether the earth corresponds to a solid sphere with a homogenuous mass distribution or a hollow one with all the mass agglomerating close to the surface. Also, if you are inside a hollow ball, you do not experience any gravitational force from it, even if you are close to the surface.&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2014-03-18T10:58:07.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="423" PostTypeId="2" ParentId="362" CreationDate="2014-03-18T12:05:08.810" Score="11" Body="&lt;p&gt;I always think of &lt;span class=&quot;math-container&quot;&gt;$\int_D \nabla \times v = \int_{\partial D} v$&lt;/span&gt; in terms of water flow. You have a bunch of water flowing around: It's velocity at a given position &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; is given by the vector field &lt;span class=&quot;math-container&quot;&gt;$v(x,y)$&lt;/span&gt;. Stick a tiny paddle wheel into the water at position &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; and let it start spinning: Its speed is &lt;span class=&quot;math-container&quot;&gt;$\nabla \times v$&lt;/span&gt;. (Probably times the length of the paddle wheel arms or something, since this answer doesn't have the right units.)&lt;/p&gt;&#xA;&lt;p&gt;Now take a great big inner tube and plop it down on the surface of the water. Let &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; be the area enclosed by the inner tube. It will rotate at &lt;span class=&quot;math-container&quot;&gt;$\int_{\partial D} v$&lt;/span&gt;. (Again, I am probably missing some physical factor that makes the units work out.) So the result is that the rotation of the inner tube is the total of the rotations from all the paddle wheels within it.&lt;/p&gt;&#xA;&lt;p&gt;I remember a good discusson of this near the beginning of Volume 2 of the &lt;i&gt;Feynmann Lectures&lt;/i&gt;, and in Courant &lt;i&gt;Differential and Integral Calculus&lt;/i&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Alternate answer (&lt;a href=&quot;https://www.smbc-comics.com/comic/2014-02-24&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pxYBf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pxYBf.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2020-11-08T15:11:55.403" LastActivityDate="2020-11-08T15:11:55.403" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="424" PostTypeId="1" CreationDate="2014-03-18T12:15:51.030" Score="10" ViewCount="407" Body="&lt;p&gt;Online courses, particularly massive open online courses (MOOC) are the current rage. I can see that having students going through the material presented at their own pace, whenever they have time, and to be able to &quot;rewind&quot; are valuable. But I think some form of one-on-one contact (even indirectly, through grading, commenting on the work, answering questions, etc) is required anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the question is: What are experiences in this area? How much (or little) extra work (contact with students, grading, ...) is required? What part of that (if any) can be left to teaching assistants (or similar)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps deeper: How does this compare to more traditional technologies, i.e. a good textbook (perhaps even one like Bogart's &lt;a href=&quot;http://www.math.dartmouth.edu/news-resources/electronic/kpbogart/ComboNoteswHints11-06-04.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;Combinatorics through Guided Discovery&quot;&lt;/a&gt;) and self study, seasoned with questions or just browsing &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;http://math.stackexchange.com&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="143" LastEditDate="2015-02-13T14:05:01.997" LastActivityDate="2019-06-18T04:41:36.747" Title="Experiences with online courses, specially MOOC?" Tags="|technology-in-education|general-pedagogy|interactive-teaching|online-instruction|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="425" PostTypeId="1" AcceptedAnswerId="432" CreationDate="2014-03-18T12:18:24.060" Score="10" ViewCount="379" Body="&lt;p&gt;The number $e$ has many interesting and important properties, many of which are related to calculus. How can I explain what $e$ is and why it is important to those who have not had calculus (or even, when introducing it in calculus for the first time before derivatives are defined).&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="114" LastEditDate="2014-04-11T12:26:23.827" LastActivityDate="2014-04-11T12:26:23.827" Title="How can you explain the importance of $e$ to those who have not taken calculus?" Tags="|concept-motivation|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="426" PostTypeId="1" AcceptedAnswerId="768" CreationDate="2014-03-18T12:23:55.080" Score="29" ViewCount="1290" Body="&lt;p&gt;&lt;strong&gt;Original post:&lt;/strong&gt; I have a math PhD. In the United States, I can teach at a 4-year university or a community college without any additional training. However, to teach mathematics in high-school I must receive additional state-based certification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question has two parts:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Why is certification required at the high-school level?&lt;/li&gt;&#xA;&lt;li&gt;Why is certification not required at the college level?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I like Willie Wong's improvement to this post: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there research backing up the empirically observed trend that teacher-training is necessary for teaching primary and secondary school, but not necessary for teaching at university level and above?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-03-18T13:28:21.097" LastActivityDate="2014-12-26T16:17:08.433" Title="Why does high school teaching in the USA require a teaching certificate while college/university teaching does not?" Tags="|secondary-education|usa|certification|" AnswerCount="7" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="427" PostTypeId="5" CreationDate="2014-03-18T12:24:41.547" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-18T12:24:41.547" LastActivityDate="2014-03-18T12:24:41.547" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="428" PostTypeId="4" CreationDate="2014-03-18T12:24:41.547" Score="0" Body="For questions about educating teachers." OwnerUserId="75" LastEditorUserId="75" LastEditDate="2014-03-18T13:06:31.670" LastActivityDate="2014-03-18T13:06:31.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="429" PostTypeId="5" CreationDate="2014-03-18T12:31:28.680" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-18T12:31:28.680" LastActivityDate="2014-03-18T12:31:28.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="430" PostTypeId="4" CreationDate="2014-03-18T12:31:28.680" Score="0" Body="For questions about examples for some mathematical subject – usually for purposes of motivation and illustration." OwnerUserId="75" LastEditorUserId="75" LastEditDate="2014-03-18T13:03:51.387" LastActivityDate="2014-03-18T13:03:51.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="431" PostTypeId="2" ParentId="425" CreationDate="2014-03-18T12:42:44.477" Score="12" Body="&lt;p&gt;This might not be what you are looking for, but if the students know about derivatives (which comes before derivatives) you could tell your students about compound interest. If an initial $P_0$ dollars invested at $100r\%$ (APR) compounded $n$ times a year then the value of the investment after $t$ years is&#xA;$$&#xA;P = P_0\left(1 + \frac{r}{n}\right)^{nt}.&#xA;$$&#xA;The funny thing is that if you compounded the interst more and more frequently, that is, if you let $n$ go to infinity, then you get&#xA;$$&#xA;P = P_0e^{rt}.&#xA;$$&#xA;This, of course uses that&#xA;$$&#xA;e = \lim_{n\to \infty} \left(1 + \frac{1}{n}\right)^{n}.&#xA;$$&#xA;So, this is really an example of how naturally $e$ shows up.&lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-03-18T12:42:44.477" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="432" PostTypeId="2" ParentId="425" CreationDate="2014-03-18T12:45:05.317" Score="12" Body="&lt;p&gt;I think the number $e$ is inherently analytical in the sense that every definition I know of it uses a limit somewhere.  So, I think from a purely mathematical standpoint, it's very difficult to explain the importance of $e$ without calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there are other standpoints.  The one I usually try to use with my college algebra students is to stress its utility - it comes up all over the place.  For example, it arises when modeling population growth, or radioactive decay, or the temperature in a cup of a coffee, or the money in a savings account.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To back this assertion up in my college algebra classes, I usually will show how the &quot;limit&quot; as $n$ goes to $\infty$ of the expression $P(1+\frac{r}{n})^{nt}$ &quot;turns into&quot; $Pe^{rt}$.  Of course, they don't really know what a limit is (and I don't use that terminology), but this is one of those cases where limits really are intuitive, and I think most of them follow it.  Then they get to see for themselves where the $e$ comes from.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, many of my college algebra students don't really care about where things come from, but for many of them, simply saying &quot;it comes up everywhere&quot; is as good enough.&lt;/p&gt;&#xA;" OwnerUserId="149" LastEditorUserId="149" LastEditDate="2014-03-18T14:13:02.290" LastActivityDate="2014-03-18T14:13:02.290" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="433" PostTypeId="1" CreationDate="2014-03-18T12:58:22.513" Score="27" ViewCount="780" Body="&lt;p&gt;I read a paper in my &quot;Research Issues in Mathematical Education&quot; class that I have applied to the Undergraduate Calculus I and Calculus II class that I teach.  I take five minutes to explain the history behind the material we learn each day.  So far, and similar to the results of the paper, I feel that students are more engaged when we get to the actual lesson.  Has anyone else tried this approach?  What were your results?  &lt;/p&gt;&#xA;" OwnerUserId="203" LastEditorUserId="76" LastEditDate="2014-03-18T15:26:16.020" LastActivityDate="2019-12-27T18:19:45.343" Title="Would taking 5 minutes to explain the history behind a mathematical idea help stimulate learning the idea?" Tags="|mathematical-pedagogy|calculus|undergraduate-education|history|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="434" PostTypeId="5" CreationDate="2014-03-18T13:05:06.643" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-18T13:05:06.643" LastActivityDate="2014-03-18T13:05:06.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="435" PostTypeId="4" CreationDate="2014-03-18T13:05:06.643" Score="0" Body="For questions concerning the history of mathematical education and the use of historical topics in teaching mathematics." OwnerUserId="114" LastEditorUserId="39" LastEditDate="2016-10-13T19:36:15.370" LastActivityDate="2016-10-13T19:36:15.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="436" PostTypeId="2" ParentId="425" CreationDate="2014-03-18T13:05:53.890" Score="3" Body="&lt;p&gt;&lt;strong&gt;What is it?&lt;/strong&gt;  I would start by defining it as $$1+\frac{1}{1}+\frac{1}{2}+\frac{1}{6}+\cdots$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why is it important? 1)&lt;/strong&gt; &#xA;In almost every subject with numbers, there are formulas with $e$'s:  the biologist's exponential growth, the economist's value of stock options, the physicist's Planck law of radiation, the statistician's bell curve, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why is it important? 2)&lt;/strong&gt; &#xA;The constant also appears in contexts that don't sound mathematical, like the &lt;a href=&quot;http://en.wikipedia.org/wiki/Secretary_problem&quot; rel=&quot;nofollow&quot;&gt;marriage problem&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you see candidates one at a time, and want to choose the best one immediately.  The optimal algorithm is to see the first $1/e$ of the candidates, and then pick the first one who is better than all of them.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-03-18T13:18:47.957" LastActivityDate="2014-03-18T13:18:47.957" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="437" PostTypeId="2" ParentId="433" CreationDate="2014-03-18T13:10:07.103" Score="15" Body="&lt;p&gt;My experience with giving history and chronology and people-involved has mostly (but not entirely) been disheartening: many students, from calculus to grad students, don't count that material (and the perspective it affords) as having much value. &lt;em&gt;Some&lt;/em&gt; do value it, but the more-typical response is more-or-less polite waiting for the &quot;real content&quot; to begin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of this is surely due to the eternal &quot;will it be on the final?&quot; criterion for the students' attention, which is not an irrational response in an environment in which they feel overwhelmed, I guess.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some others' reaction is connected to their affection for mathematics, perhaps extreme among academic subjects, as being allegedly disconnected from the (sordid?) everyday affairs of human beings... and the reminder that mathematics is a human activity is not welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some view history-of-mathematics as positively antithetical to contemporary mathematics, in a sort of trans-Whiggish attitude, namely, that things are so much improved by now that it is actually destructive to discuss how we arrived at this point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: but I forgot to say that I still &lt;em&gt;do&lt;/em&gt; give some background, despite unenthusiastic response. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-18T13:10:07.103" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="438" PostTypeId="2" ParentId="433" CreationDate="2014-03-18T13:13:55.810" Score="16" Body="&lt;p&gt;In Germany, there recently was a Project at two universities which tried to strengthen historical connections in analysis courses, especially for teacher education. For them, history can be an important source of motivation if used with real interest. (Bad use: &quot;Even Leibniz made these mistakes&quot; - no learning from history. Good use: &quot;The concept of continuity changed over time. Let's compare definitions and discuss their use.&quot;) &#xA;Unfortunatley, they didn't publish in English.&#xA;Anyway, for interested persons:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;G. Nickel: &lt;a href=&quot;http://link.springer.com/chapter/10.1007/978-3-658-00992-2_16&quot;&gt;Vom Nutzen und Nachteil der Mathematikgeschichte für das Lehramtsstudium&lt;/a&gt;. In: H. Allmendinger et al. (Hrsgg.): Mathematik verständlich unterrichten. Springer Spektrum, Wiesbaden 2013, 253-266. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Since vonbrand asked me, here a small summary of the paper, but not the project which included much more measures and cannot be summarised within a few lines:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally, the author criticises that mathematics is often taught in a very ahistorical way, whereas he sees history of mathematics as content in its own right.&#xA;Benefits:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Anecdotical use to comfort people, which has to be used pointed and not too often&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Genetic use, explaining how definitions, theorems and proofs were&#xA;worked out: getting a much deeper image of mathematics in the making,&#xA;connections, etc. Costs time, however. Also good to identify learning&#xA;obstacles!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Alienate topics: Can let you experience how hard things are (like the naturals in non-decimal systems: egyption, babylonian, roman numbers).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Exemplary use: Let students gain there own insight in research processes, which may be easier in historical issues.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;However, an anecdotical use can lead to very superficial discussions, rather on persons than on mathematics. That wouldn't be good. Caricatures like mathematicians as &quot;heores&quot; with an enormous mind or as morons making stupid mistakes do neither serve history nor mathematics. And you sometimes should reduce the historical complexity for sake of learning mathematics, not history.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="199" LastEditDate="2014-03-18T18:54:43.143" LastActivityDate="2014-03-18T18:54:43.143" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="439" PostTypeId="5" CreationDate="2014-03-18T13:14:43.640" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-18T13:14:43.640" LastActivityDate="2014-03-18T13:14:43.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="440" PostTypeId="4" CreationDate="2014-03-18T13:14:43.640" Score="0" Body="For questions about contents, order, background, alternatives in curricula." OwnerUserId="114" LastEditorUserId="143" LastEditDate="2014-03-19T17:56:37.837" LastActivityDate="2014-03-19T17:56:37.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="441" PostTypeId="5" CreationDate="2014-03-18T13:18:50.517" Score="0" Body="&lt;p&gt;&quot;New Math&quot; refers to a wave of mathematics curricular reforms launched in the late 1950s and throughout the 1960s, primarily in the United States and Europe.  New Math curricula were characterized by the introduction of modern vocabulary and theoretical foundations; for example, the language of &lt;em&gt;sets&lt;/em&gt;, &lt;em&gt;functions&lt;/em&gt;, etc. was woven into the curriculum at all level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the elementary level, New Math texts emphasized the difference between &lt;em&gt;numbers&lt;/em&gt; and &lt;em&gt;numerals&lt;/em&gt;, and introduced students to different number-base numeration systems.  At more advanced levels, many New Math textbooks emphasized rigorous proof, and some taught the basics of abstract algebra (e.g. &lt;em&gt;groups&lt;/em&gt;, &lt;em&gt;rings&lt;/em&gt;, and &lt;em&gt;fields&lt;/em&gt;).  Some of the innovations of the New Math (e.g. inequalities and simple matrix operations) remain as standard components of the curriculum today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a more detailed description see also &lt;a href=&quot;http://en.wikipedia.org/wiki/New_Math&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/New_Math&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="29" LastEditDate="2015-11-13T16:03:06.037" LastActivityDate="2015-11-13T16:03:06.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="442" PostTypeId="4" CreationDate="2014-03-18T13:18:50.517" Score="0" Body="For questions about the New Math school reforms introduced in the United States and Europe during the 1960s." OwnerUserId="114" LastEditorUserId="29" LastEditDate="2015-11-13T16:03:06.037" LastActivityDate="2015-11-13T16:03:06.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="443" PostTypeId="2" ParentId="424" CreationDate="2014-03-18T13:28:50.000" Score="10" Body="&lt;p&gt;I would suggest a distinction: A MOOC really should be massive, that means some 1000 participants or even more. In this case your problems will be about server capacity and technical things. The work like answering questions will then be done by the community (like in this forum). This, however, always requires some people to be online in your forum. Blended learning including some online-elements can even work with small groups, but then you need much more personal effort. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key point is students' motivation: In a large setting maybe 50% will drop-out but the rest will make it through forum activities. Getting answers immediately and helping people with your knowledge - this makes the difference to reading a book your library (which people can for a hundred years, but hardly anyone does).&#xA;In a small setting you want everyone to keep up with your course and you will need the whole group for interpersonal exchange of their learning experience. Thus, you will have to invest much more, e.g. in weekly meetings. You then might rather look for &quot;inverted classroom&quot; / &quot;flipped classroom&quot; concepts.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="1147" LastEditDate="2019-06-18T04:41:36.747" LastActivityDate="2019-06-18T04:41:36.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="444" PostTypeId="1" AcceptedAnswerId="448" CreationDate="2014-03-18T13:34:26.937" Score="21" ViewCount="959" Body="&lt;p&gt;Most mathematicians would agree that &#xA;$$e^{i\pi}+1=0$$ &#xA;is one of the most impressive &lt;a href=&quot;http://en.wikipedia.org/wiki/Euler%27s_formula&quot;&gt;formulas&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine your students have just learned about the definition of $e$ or $i$ (just assume it's $e$, normally $i$ comes later in curricula); $\pi$ should be known for sure at that point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it a good motivation for the new quantity (here $e$ or $i$) for the students if you tell them about the formula above (and maybe short explaining the unknown quantity) to emphasize the importance of the new variable ($e$ or $i$)? Or is such a thing distracting or even unhelpful since you overload the students with stuff they don't know at that point (and maybe don't care)? &lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-11T12:26:34.380" LastActivityDate="2015-05-31T22:53:53.173" Title="Is $e^{i\pi}+1=0$ a good motivation for introducing $e$ or $i$? Why (not)?" Tags="|undergraduate-education|secondary-education|concept-motivation|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="445" PostTypeId="2" ParentId="433" CreationDate="2014-03-18T13:42:57.183" Score="6" Body="&lt;p&gt;In a calculus class, I'd rather start with 5 minutes of an application than with 5 minutes of history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;History in 5-minute increments will probably:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;follow the textbook's ordering (destroying the chronology),&lt;/li&gt;&#xA;&lt;li&gt;use modern notation (usually destroying the geometrical context for anything before 1700),&lt;/li&gt;&#xA;&lt;li&gt;omit even basic context (e.g. what language did they use?  how did they get the     resources for this?).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;History in 20-minute increments can be more serious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many 5-minute applications have some nice historical names attached, and it might help the motivation to say &quot;Archimedes did something similar around 250 B.C.&quot;.  But I wouldn't call that history, and I'd keep the focus on the application.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-03-18T15:55:16.600" LastActivityDate="2014-03-18T15:55:16.600" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="446" PostTypeId="2" ParentId="198" CreationDate="2014-03-18T13:53:44.960" Score="3" Body="&lt;p&gt;For paedagogical use, the book &quot;&lt;a href=&quot;http://www.springer.com/education+&amp;amp;+language/mathematics+education/book/978-0-387-37141-2&quot; rel=&quot;nofollow&quot;&gt;Amongst mathematicians&lt;/a&gt;&quot; by Elena Nardi might serve your intentions.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-18T13:53:44.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="447" PostTypeId="2" ParentId="444" CreationDate="2014-03-18T14:04:28.927" Score="4" Body="&lt;p&gt;Usually not.  The constants $e$ and $\pi$ have great applications, which should be their motivation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe in an algebra class that introduces $i$, if you're discussing the geometry of the complex plane, then Euler's formula could be relevant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definitely in a differential equations class, where solutions often involve exponentiating complex numbers, Euler's formula is important.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-03-18T14:36:48.773" LastActivityDate="2014-03-18T14:36:48.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="448" PostTypeId="2" ParentId="444" CreationDate="2014-03-18T14:05:49.917" Score="28" Body="&lt;p&gt;I have never quite understood why it was impressive and/or beautiful, and it always frustrates me when people claim that it is. Therefore, I would say &quot;no, it is not good motivation&quot;, because beauty is subjective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if you explain to students that the formula is based on $e^{i\theta}=\cos\theta+i\sin\theta$ and this essentially allows you to combine circles with complex numbers and hence prove many wonderful things (&lt;a href=&quot;http://en.wikipedia.org/wiki/Residue_theory&quot;&gt;e.g.&lt;/a&gt;), then I agree that this is good motivation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point is, introducing the formula on its own is pointless. Rather, it is what the formula tells you, and the subsequent applications, which are important and, perhaps, motivate the study of $e$ and $i$. I mention this point because it is often omitted when people mention the formula: they simply claim that it is beautiful &quot;because it contains $e$, $\pi$ and $i$&quot;. However, (I believe that) the beauty lies in the application...&lt;/p&gt;&#xA;" OwnerUserId="169" LastEditorUserId="169" LastEditDate="2014-04-28T11:32:35.623" LastActivityDate="2014-04-28T11:32:35.623" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="449" PostTypeId="2" ParentId="426" CreationDate="2014-03-18T14:16:49.600" Score="3" Body="&lt;p&gt;Teachers' unions are the main political forces behind certification requirements in U.S. public high schools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If certification is as effective as graduate degrees in promoting teaching quality, it's &lt;a href=&quot;http://news.heartland.org/newspaper-article/2013/02/26/study-teacher-quality-not-linked-advanced-degrees-or-experience&quot; rel=&quot;nofollow&quot;&gt;not very effective&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I previously said &quot;certification is required in U.S. public high schools because the teachers' unions have promoted it,&quot; but I retract that after the historical criticism in the comments.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-12-26T16:17:08.433" LastActivityDate="2014-12-26T16:17:08.433" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="450" PostTypeId="5" CreationDate="2014-03-18T14:23:30.023" Score="0" Body="&lt;p&gt;&lt;em&gt;Proportional reasoning&lt;/em&gt; is reasoning in a system of two variables between which there exists a linear functional relationship.  This leads to conclusions about a situation or phenomenon that can be characterized by a constant ratio.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2014-03-18T14:44:50.880" LastActivityDate="2014-03-18T14:44:50.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="451" PostTypeId="4" CreationDate="2014-03-18T14:23:30.023" Score="0" Body="For questions about reasoning in a system of two variables between which there exists a linear functional relationship." OwnerUserId="77" LastEditorUserId="77" LastEditDate="2014-03-18T14:44:56.053" LastActivityDate="2014-03-18T14:44:56.053" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="452" PostTypeId="1" CreationDate="2014-03-18T14:44:27.997" Score="4" ViewCount="146" Body="&lt;p&gt;For a postgraduate class on category theory, how fast must one teach category theory? What is the pace one must take? Also, what must one assume that the students know?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks! &lt;/p&gt;&#xA;" OwnerDisplayName="user207" LastEditorUserId="114" LastEditDate="2014-03-28T18:45:57.890" LastActivityDate="2014-03-28T18:45:57.890" Title="How fast must one teach category theory?" Tags="|graduate-education|category-theory|" AnswerCount="0" CommentCount="4" ClosedDate="2014-03-22T15:57:18.297" ContentLicense="CC BY-SA 3.0" />
  <row Id="453" PostTypeId="5" CreationDate="2014-03-18T15:31:24.207" Score="0" Body="&lt;p&gt;This tag is for questions which relate to exams as viewed from the point of view of an instructor.  Questions related to any part of the examination process should be labeled with this tag, including (but not limited to) questions about&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the writing or construction of exams,&lt;/li&gt;&#xA;&lt;li&gt;the administration and proctoring of exams,&lt;/li&gt;&#xA;&lt;li&gt;the process of scoring and grading exams,&lt;/li&gt;&#xA;&lt;li&gt;the grading and scoring of exams, and&lt;/li&gt;&#xA;&lt;li&gt;the analysis of exam results.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This tag might be used in conjunction with other tags, such as &lt;a href=&quot;/questions/tagged/test-design&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;test-design&amp;#39;&quot; rel=&quot;tag&quot;&gt;test-design&lt;/a&gt; and &lt;a href=&quot;/questions/tagged/oral-exams&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;oral-exams&amp;#39;&quot; rel=&quot;tag&quot;&gt;oral-exams&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2019-10-13T22:02:54.300" LastActivityDate="2019-10-13T22:02:54.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="454" PostTypeId="4" CreationDate="2014-03-18T15:31:24.207" Score="0" Body="This tag is intended for questions about exams as viewed from the point of view of an instructor." OwnerUserId="114" LastEditorUserId="8571" LastEditDate="2019-10-13T22:03:02.680" LastActivityDate="2019-10-13T22:03:02.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="455" PostTypeId="2" ParentId="433" CreationDate="2014-03-18T15:33:05.840" Score="9" Body="&lt;p&gt;I would never &lt;em&gt;start&lt;/em&gt; with 5-minute historical introduction, main reasons being:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It's not the content of the subject.&lt;/li&gt;&#xA;&lt;li&gt;It's additional information that is not directly relevant and distracts the students.&lt;/li&gt;&#xA;&lt;li&gt;Many of historical math results are incomplete if not plain wrong (e.g. proof by example is not a proof, frequently the reasonings were informal, etc.).&lt;/li&gt;&#xA;&lt;li&gt;Wrong intuitions are contagious (e.g. I would never &lt;em&gt;start&lt;/em&gt; describing solar system with geocentric view).&lt;/li&gt;&#xA;&lt;li&gt;The evolution of the idea might be presented without historical context.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;On the other hand, I had presented a brief historical context on a few occasions, &lt;em&gt;during the class&lt;/em&gt;, that is, once the basics have been laid out (e.g. to explain inconsistencies or backward-compatibility). That technique is especially useful when talking about sequential improvements, like with lower/upper bounds, or time-complexity of algorithms. First, you present basic technique, or naive approach, then gradually introduce more complex ideas and move to cutting-edge research if time allows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give an example (what I'm describing wouldn't fit in a single lecture, but other examples were too short and weren't clear enough), in a graph theory course and topic being &lt;a href=&quot;http://en.wikipedia.org/wiki/Matching_%28graph_theory%29&quot;&gt;matchings&lt;/a&gt;, one would start with definitions, the bipartite case, then move to naive algorithms, then Edmonds (1965), Hopcrof-Karp (1973), mention Micali-Vazirani (1980,2013), and then move on to algebraic approaches finishing with Mucha-Sankowski (2004) and latest results of Mądry (2013) using linear-programming techniques.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen this technique during a lecture about paradigms of programming languages where the professor would describe programming languages and their features gradually building from machine code up to newest advances in compilers. It worked pretty well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet, I had to admit, I've never seen such an approach during mathematical course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, at my university it is compulsory for future teachers to attend &quot;history of mathematics&quot; course (fortunately the old professor who teaches it has a gift, the lectures are interesting and fun).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-18T15:33:05.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="456" PostTypeId="2" ParentId="416" CreationDate="2014-03-18T15:36:42.653" Score="32" Body="&lt;p&gt;I agree with vonbrand that it is important to stress that this is a &lt;i&gt;convention&lt;/i&gt; that is used &lt;i&gt;sometimes&lt;/i&gt; but not others.  But I would add the emphasis that all conventions are local.  There are places where it is helpful to adopt this convention but other settings in which it would be a disaster.  My preference would be to make sure students understand that the expression $0^0$ places two properties in direct conflict with one another.  First, there is the fact that $x^0=1$ for $x&amp;gt;0$.  Second, there is the fact that $0^x=0$ for $x&amp;gt;0$.  There is no way to define $0^0$ so that both of these are preserved and extended, so any definition you adopt is going to lead to some kind of undesirable discontinuity.  Understanding &lt;i&gt;that&lt;/i&gt; is, in my opinion, much more important than understanding why in some particular contexts one convention is more useful than the other.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="114" LastEditDate="2014-03-18T15:37:37.127" LastActivityDate="2014-03-18T15:37:37.127" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="457" PostTypeId="5" CreationDate="2014-03-18T15:42:03.453" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-18T15:42:03.453" LastActivityDate="2014-03-18T15:42:03.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="458" PostTypeId="4" CreationDate="2014-03-18T15:42:03.453" Score="0" Body="For questions about creating a new course or altering the format of an existing course. " OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-03-18T16:11:12.593" LastActivityDate="2014-03-18T16:11:12.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="459" PostTypeId="2" ParentId="444" CreationDate="2014-03-18T16:05:03.890" Score="8" Body="&lt;p&gt;Unless you have introduced or are about to introduce exponentiation of complex numbers, this is no more than interesting trivia. It doesn't mean anything and doesn't help them do anything. &lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-18T16:05:03.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="460" PostTypeId="1" CreationDate="2014-03-18T16:22:36.463" Score="13" ViewCount="210" Body="&lt;h3&gt;The Question&lt;/h3&gt;&#xA;&lt;p&gt;Quite often one is faced with the task of running a &lt;em&gt;large&lt;/em&gt; exercise group (think 40+ students in the room). What are some strategies to make the most of the (limited) time for the students?&lt;/p&gt;&#xA;&lt;h3&gt;The motivation&lt;/h3&gt;&#xA;&lt;p&gt;In my time as a student and now as an instructor, I've seen and been part of three main types of large exercise groups.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The instructor came prepared to talk about problem X, which may or may not be on the homework sheet, and may or may not be exemplary of the homework exercises. The instructor talks about it (and only it) in great detail.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The instructor stands in front of the class, polls the class for questions. Answers them on the board.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The instructor roams the classroom and answer questions as they come up. The students are either assigned to work in small groups or they coagulate in working groups by themselves.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;There are some pros and cons of each method:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Avoids dead air: something is going on throughout the session, and it feels like time is not wasted.&lt;/p&gt;&#xA;&lt;p&gt;The problem explained however may be something different from where the actual difficulties are.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The students are in control of what they want to hear.&lt;/p&gt;&#xA;&lt;p&gt;The order and the choice of questions discussed often is somewhat haphazard, so sometimes the discussion gets messy.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The students receive quite personal attention.&lt;/p&gt;&#xA;&lt;p&gt;Often the same problem will end up being explained multiple times to multiple groups.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Are there better alternatives?&lt;/p&gt;&#xA;" OwnerUserId="125" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-08-25T23:07:18.540" Title="How to effectively run large exercise groups?" Tags="|tutoring|homework|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="461" PostTypeId="2" ParentId="259" CreationDate="2014-03-18T16:28:18.720" Score="5" Body="&lt;p&gt;I began tutoring as a fellow student when I was asked for help with homework, continued when I began getting referrals to help student athletes, was employed by tutoring centers and the mathematics department, and finally attempted to set up a private tutoring service. It was my experience that tutoring could help considerably, especially when I was able to provide a remedy for a specific block to their mathematical understanding or progress. In several cases, the students gained much greater confidence in their ability to learn and do mathematics and went from struggling to excellent.  I was less able to help students who did not attempt their homework or assigned exercises, or who were severely deficient in their preparation for a course. &lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-18T16:28:18.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="462" PostTypeId="2" ParentId="460" CreationDate="2014-03-18T16:50:57.920" Score="2" Body="&lt;p&gt;Maybe you should try to add clicker on your list.&#xA;They work like in any tv-show. The tutor asks a question and the audience may individually choose out of four (or any number of) answers. Technically, this either needs so called clicker devices, which cost some money, or a webserver and students with mobile phones, both being standard, I think.&#xA;Concerning the latter, pingo is a system having some experiences documented in papers, e.g. &lt;a href=&quot;http://aisel.aisnet.org/icis2012/proceedings/ResearchInProgress/30/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. Unlike the paper, the &lt;a href=&quot;http://pingo.upb.de/&quot; rel=&quot;nofollow&quot;&gt;website&lt;/a&gt; is in german.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The use of the clickers can activate students during the session, especially when you give them some time after(!) they chose one answer for discussions in small groups. In this case, they often try to argue for their decision rather than waiting for the tutor's answer. &#xA;Clickers are sometimes used in the wider context of just-in-time-teachint (jitt).&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-18T16:50:57.920" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="463" PostTypeId="1" CreationDate="2014-03-18T17:09:18.977" Score="7" ViewCount="860" Body="&lt;p&gt;Once in a while an exam happens where nobody correctly solves (i.e. there might be partial solutions) that trivial problem of yours (and I'm not talking about intentionally hard and impossible tasks used, for example, in &quot;testing waters&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, it is my fault (i.e. when I wanted them to solve it and nobody did; it's another matter when I let students surprise me for bonus points). There are a lot of possible causes, e.g. an especially good day (like when everything seems simple), non-standard intuition, sometimes a missing dependency, like a formula from another course they should have known, but didn't (and I should have checked they do).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; However, after it happened, how one should grade such task? In particular, should the &quot;maximum&quot; be kept as intended, or maybe moved to progress point of the best solution?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example:&lt;/strong&gt; The task might be about throwing a fair coin some number of times, and bounding the number of obtained heads from below with given confidence. The basic solution could use the &lt;a href=&quot;http://en.wikipedia.org/wiki/Chebyshev%27s_inequality&quot; rel=&quot;noreferrer&quot;&gt;Chebyshev's inequality&lt;/a&gt;, but you can obtain a better bound with, for example, &lt;a href=&quot;http://en.wikipedia.org/wiki/Chernoff_bounds&quot; rel=&quot;noreferrer&quot;&gt;Chernoff bound&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-18T19:41:32.847" LastActivityDate="2014-03-18T22:07:54.923" Title="Grading problems that nobody solved" Tags="|exams|grading|" AnswerCount="2" CommentCount="1" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="464" PostTypeId="5" CreationDate="2014-03-18T17:10:46.113" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-18T17:10:46.113" LastActivityDate="2014-03-18T17:10:46.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="465" PostTypeId="4" CreationDate="2014-03-18T17:10:46.113" Score="0" Body="For questions about the study and teaching of abstract algebra, including topics such as groups, rings, fields, and vector spaces." OwnerUserId="40" LastEditorUserId="40" LastEditDate="2014-03-18T17:29:46.950" LastActivityDate="2014-03-18T17:29:46.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="466" PostTypeId="1" AcceptedAnswerId="476" CreationDate="2014-03-18T17:43:42.863" Score="20" ViewCount="903" Body="&lt;p&gt;In mathematics education, &quot;proof&quot; is widely used for many kinds of argumentation. For instance, one example could be called a proof, if it is paradigmatic in the following sense: The argument can immediately be transferred to the general case. (To be more concrete, assume you want to prove the sum of two even numbers is even again. Say we have $4+6$, then we see $2\cdot(2) + 2\cdot(3) = 2\cdot(2+3)$, not really depending on the numbers chosen.)&#xA;The other extreme is given by formal proofs, where every(!) step needs an argument. This is sometimes practiced in courses on logic and it takes half a page to see $2+1=1+2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the proofs in professional mathematics are far away from both poles; however, I think they will be closer to the formal pole. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question: &lt;strong&gt;Do you know any literature (book, paper, website, blog, ...) which tries to define what a proof is for the working mathematician or students at (under)graduate level?&lt;/strong&gt;&#xA;I would like to restrict the answers to the literature and not personal opinions of which I have heard a lot. :-)&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="39" LastEditDate="2015-08-30T08:52:34.557" LastActivityDate="2015-09-03T17:30:11.380" Title="What do mathematicians call a proof?" Tags="|undergraduate-education|proofs|undergraduate-research|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="467" PostTypeId="2" ParentId="463" CreationDate="2014-03-18T17:43:59.757" Score="7" Body="&lt;p&gt;I doubt there can be a general rule for this case. If it is &lt;em&gt;my&lt;/em&gt; mistake in asking something they couldn't possibly answer (or even if a few answered because by some chance they knew about the topic), I'd just take the question off the grade (probably keeping the points of those who did answer as bonus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they &lt;em&gt;should&lt;/em&gt; have been able to answer, I'd try to find out what happened, but would probably keep grades as-is.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-18T17:43:59.757" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="468" PostTypeId="1" AcceptedAnswerId="477" CreationDate="2014-03-18T17:58:09.240" Score="35" ViewCount="1564" Body="&lt;p&gt;Textbook writers are blessed with only solving problems with neat answers. Numerical coefficients are small integers, many terms cancel, polynomials split into simple factors, angles have trigonometric functions with known values. Pure bliss.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;real life&quot; is different (as any of us knows).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Giving such questions for homework runs the risk that the student knows she is wrong when some crooked formula/value shows up. On the other hand, checking messy results (unless perhaps directly numerical values) is harder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you think about posing questions which don't have neat derivations/results? I presume the answer could depend on the subject matter, the level of the students, and perhaps on exactly what the question should teach.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="114" LastEditDate="2014-03-18T21:33:13.817" LastActivityDate="2018-03-23T13:19:49.350" Title="Questions with &quot;round&quot; answers only?" Tags="|mathematical-pedagogy|calculations|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="469" PostTypeId="1" CreationDate="2014-03-18T18:02:28.037" Score="11" ViewCount="331" Body="&lt;p&gt;How do you approach a class taught by somebody else? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are multiple reasons why one could find him/herself in such a situation:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;an instructor is sick,&lt;/li&gt;&#xA;&lt;li&gt;there is a scheduled rotation of teachers,&lt;/li&gt;&#xA;&lt;li&gt;you are visiting some other university and want to exchange teaching experiences (never experienced on my own, but seems possible),&lt;/li&gt;&#xA;&lt;li&gt;you are a specialist on some topic and came just for this one lecture,&lt;/li&gt;&#xA;&lt;li&gt;and so on...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;With biology or history you might just continue where your predecessor ended. On the other hand, with math there are additional issues, to name a few:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You might be used to different notation.&lt;/li&gt;&#xA;&lt;li&gt;You might be used to different definitions (e.g. something which is a definition for you, would be a theorem for them).&lt;/li&gt;&#xA;&lt;li&gt;They might lack some intuition you think of as essential.&lt;/li&gt;&#xA;&lt;li&gt;You might rely on some specific order of topics (e.g. there is this nice explanations for $X$, but it needs $Y$ which has not been covered yet).&lt;/li&gt;&#xA;&lt;li&gt;Students might be taught using some analogy which you think is wrong.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are pros and cons to each of these, e.g. it's best to talk to students in a language they understand (i.e. use their notation), but also there might be additional benefit in letting them see a different perspective (i.e. review some older/already done topics).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So, how do you approach a class taught by somebody else?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="143" LastEditDate="2015-02-13T13:49:10.827" LastActivityDate="2015-02-13T13:49:10.827" Title="Teaching somebody else's class" Tags="|course-design|general-pedagogy|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="470" PostTypeId="2" ParentId="433" CreationDate="2014-03-18T18:05:17.240" Score="7" Body="&lt;p&gt;I use 30 seconds here or there to give brief, colourful, historical notes. It gives something of value to the students who are waiting for me to go on, and gives some extra time for the students struggling to keep up. I have never tried giving planned, longer historical explorations, only spontaneous quick tangents, as my interest sees fit.&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2014-03-18T18:05:17.240" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="471" PostTypeId="2" ParentId="460" CreationDate="2014-03-18T18:14:47.927" Score="3" Body="&lt;p&gt;What we used here with some success last term in a beginning discrete math course (the jury is still out, need more data) is to have 3 questions for the group for each session. One is solved by the teaching assistant (with help from the students); another one is given to solve by the students (perhaps in smaller groups), with active participation of the TA (answering questions or nudging into the right track, mostly); and a third (short) question has to be solved individually by each and turned in. For the last the grading is just 0 (not turned in/no show), 1 (turned something in), 2 (turned something in that looks sensible at first glance). This is easy/fast to assess, and the resulting grade is a 5% of the final grade, so it doesn't weigh too much. It has made students at least glance at the material, and not relay purely on the &quot;cram the last two days&quot; technique (fatal in any course covering much new subject matter). &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-18T18:14:47.927" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="472" PostTypeId="2" ParentId="466" CreationDate="2014-03-18T18:36:59.427" Score="9" Body="&lt;p&gt;I'd go first for the literature on mathematical writing, e.g. Knuth, Larrabee and Roberts' &lt;a href=&quot;http://www-cs-faculty.stanford.edu/~knuth/klr.html&quot;&gt;&quot;Mathematical writing&quot;&lt;/a&gt; (MAA, 1989). Mathematics writing is &lt;em&gt;not&lt;/em&gt; just &quot;proofs,&quot; in the end, you want to convince your reader that what your reasoning is correct (and not bore them to tears in the process), and (hopefully) convince them that it is relevant. It very much depends on the audience, for starting undergraduates the proof may need to be much more detailed and explained than the brief sketch when talking to a coleague, and that one probably won't ever be accepted by a journal either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For another view, refer to the journals devoted to completely formal proofs...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-18T18:36:59.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="473" PostTypeId="2" ParentId="248" CreationDate="2014-03-18T18:37:48.180" Score="6" Body="&lt;p&gt;I spend some time talking about this in my question here &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/41955/does-any-textbook-take-this-approach-to-the-isomorphism-theorems&quot;&gt;https://mathoverflow.net/questions/41955/does-any-textbook-take-this-approach-to-the-isomorphism-theorems&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I really think defining normal subgroups as kernels of homomorphisms is a great idea.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also find a lot of support for introducing normal subgroups as kernels here: &lt;a href=&quot;https://mathoverflow.net/a/7796/1106&quot;&gt;https://mathoverflow.net/a/7796/1106&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-03-18T19:08:32.743" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="474" PostTypeId="2" ParentId="469" CreationDate="2014-03-18T18:48:25.817" Score="7" Body="&lt;p&gt;If you are an invited speaker, do as asked: Talk on your own terms on your area of expertise. What came before or after doesn't matter at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some coleagues here (in some completely different subject matter) teach an upper-level undergraduate course by halves: A teaches the first half, B teaches the remainder. Even two courses simultaneously: at mid-term, they switch groups. Only works for independent material, obviously. Works very well for them, as it makes the most of their respective areas of expertise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are taking over a class, it is important to (try to) keep continuity. I.e., check notation beforehand (or ask the students), etc. If this is scheduled, it is easier as you can coordinate with the coleague; if it is an emergency it's harder. It also depends on the duration. I.e., covering a class or a week is quite different than taking over a class mid-term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Respect missing intuition/definitions/notation, perhaps ask the class. But many students, when asked, will tell you they never learned anything beyond $2 + 2 = 4$... use your judgement/ask whoever taught them before.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-18T18:48:25.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="475" PostTypeId="2" ParentId="468" CreationDate="2014-03-18T18:53:53.933" Score="15" Body="&lt;p&gt;I agree that such exercises have a built in risk of giving the students the wrong idea what is the generic situation and what is quite a special case. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, &quot;the risk that the student knows she is wrong when some crooked formula/value shows up&quot; you mention is more a feature in my opinion. It is not a bad thing to do heuristic checks that suggest one &lt;em&gt;might&lt;/em&gt; have made an error and to then double-check. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, for some things it is more or less unavoidable to present constructed exercises. A polynomial will not have a multiple root just so,  you need to take care it has one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, in linear algebra one has to choose the matrices somewhat carefully if one wants to illustrate certain things. And, it is a real pain to calculate even with moderately sized matrices if they involve radical expressions, and if you go for decimal approximations you 'lose' the clear notion of singularity of a matrix and would actually need to do something more advanced. So, somehow one is stuck with matrices whose eigenvalues are &quot;too nice&quot; (and this even leaving the issue of factoring the polynomial aside.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I mostly tend to give exercises where things work out nicely, yet I do talk with students about the fact that this is what happens and that this does not capture what would be a &quot;typical&quot; situation were one to choose things &quot;randomly.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, I think it is important to raise awareness of the issue of &quot;artificially nice&quot; problems with students, but as mentioned above consider it as unavoidable to use them in certain cases.  &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="376" LastEditDate="2018-03-23T13:19:49.350" LastActivityDate="2018-03-23T13:19:49.350" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="476" PostTypeId="2" ParentId="466" CreationDate="2014-03-18T19:24:48.350" Score="18" Body="&lt;p&gt;This collection of essays:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=MBK-83&quot;&gt;Reuben Hersh: Experiencing Mathematics: What do we do, when we do mathematics?, Amer. Math. Soc., 2014&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;contains, among other topics, also lots of excellent discussion of the problems you mention. I highly recommend it, some of the articles are even available online if you google for them.&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-03-18T19:24:48.350" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="477" PostTypeId="2" ParentId="468" CreationDate="2014-03-18T19:37:09.173" Score="49" Body="&lt;p&gt;When teaching linear algebra, I rely heavily on carefully constructed examples where everything can be done with rational numbers of small denominator.  (It is faintly amusing that the construction of such examples often requires mathematics that is far harder than the actual content of the course.)  However, I also show them the following slide:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/mgNnU.png&quot; alt=&quot;nasty eigenvalues&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also tell them explicitly and repeatedly about what is generic, emphasising that repeated eigenvalues are only likely to arise when forced by symmetry, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it is OK to stick with nice examples in this context, because the whole point is just to illustrate the conceptual theory.  If you actually need complex examples then you should obviously use a computer for the calculations anyway.&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-18T19:37:09.173" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="478" PostTypeId="2" ParentId="468" CreationDate="2014-03-18T20:05:50.180" Score="2" Body="&lt;p&gt;It may be useful to present at least one application that is solvable but does not have a simple answer. As an example, quadratic equations with rational coefficients that cannot be easily solved by factoring and require use of the quadratic formula arise in chemistry.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-18T20:05:50.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="479" PostTypeId="2" ParentId="426" CreationDate="2014-03-18T20:43:00.573" Score="-2" Body="&lt;p&gt;It related to universities policy. It assumed that all departments hire course experts. Then departments must classify and distinguish the pedagogic ability and scientific capability. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1- Interdependency of colleges in practice cause this matter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2- Also applied mathematics for each departments refuse any standard mathematics. Then departments hire teachers that are dominant about applications of mathematics there.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3- In assumption teachers in colleges are educated at high levels and have sufficient pedagogic and course knowledge.&lt;/p&gt;&#xA;" OwnerUserId="79" LastEditorUserId="79" LastEditDate="2014-03-18T21:51:59.420" LastActivityDate="2014-03-18T21:51:59.420" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="481" PostTypeId="2" ParentId="468" CreationDate="2014-03-18T21:06:09.743" Score="8" Body="&lt;p&gt;&lt;strong&gt;I would encourage to have a significant amount of non-&quot;round&quot; numbers in your homework and also in exams.&lt;/strong&gt; Some reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Except when you would teach calculations with numbers, normally the way you do things is important, not actual calculations. Even if the students feel that something is wrong with the numbers: As long as his/her way was correct, it is only a very small part of points not gained due to miscalculation.&lt;/li&gt;&#xA;&lt;li&gt;If your students know that the only possibility is that number results are &quot;nice&quot;, you add some sort of bias to the result: If the calculation is not too complex, students could try to do basic calculations with the given numbers in order to arrive at a &quot;nice&quot; result and can so perform &quot;reverse engineering&quot; in order to get the right idea only by guessing and trying.&lt;/li&gt;&#xA;&lt;li&gt;Even when you construct a &quot;nice&quot; example with &quot;round&quot; numbers, there could be a chance that there are two possible correct ways: One with your constructed nice numbers and one with not so nice ones. (For example, if you perform a &lt;a href=&quot;http://en.wikipedia.org/wiki/QR_decomposition#Using_Householder_reflections&quot; rel=&quot;nofollow noreferrer&quot;&gt;QR decomposition via Householder reflection&lt;/a&gt;, there a freedom in the sign of one real number (on the wikipedia-link this is called $\alpha$). Depending on the choice of the sign, you get different values. There is a recommendation how to choose the sign, but in most examples this does not play a role and you are really free in the choice of the sign.).&lt;/li&gt;&#xA;&lt;li&gt;You can train other skills in finding out if a calculation is completely wrong like perfoming rough calculation and estimating lower and upper bounds for some result. In real life this should be also very useful.&lt;/li&gt;&#xA;&lt;li&gt;If you allow a calculator in the exam and there is nothing where you really need it, this could be misleading to the students.&lt;/li&gt;&#xA;&lt;li&gt;As it was mentioned by the OP and in other answers, it is misleading that there will be only nice values as a result of a calculation since real world values are not nice.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;However, it is nice to have &quot;round&quot; results on many examples (in particular those you perform on the board) because you can save time and not distract students from the actual topic.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="376" LastEditDate="2018-03-23T10:02:19.110" LastActivityDate="2018-03-23T10:02:19.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="483" PostTypeId="2" ParentId="426" CreationDate="2014-03-18T21:31:29.480" Score="21" Body="&lt;p&gt;I'm sorry not having any empirical literature, but I try to explain the question by commong knowledge. (Feel free to downvote if this is not appropriate :))&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beside political reasons, I think there are two rules of thumb&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The more mature your students are, the less pedagogical knowledge you need.&lt;/li&gt;&#xA;&lt;li&gt;The more mature your students are, the more expertise on the subject you need&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;answering the question of the OP. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A little more detailed:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If you want to teach younger students, you have to know a lot about cognitive development (&quot;What task is suitable at which age&quot;?). Also, the students are not that motivated due to e.g. puberty. If the students go the college/university, they have to decide for some subject and don't have to attend subjects they dislike. It is easier to explain a student something when he/she attends your class by choice. You have to concentrate more on technial aspects regarding the subject and less on motivatory aspects.&lt;/li&gt;&#xA;&lt;li&gt;A very small child is normally getting answer to all his/her questions by the parents - no matter what their job is. Just to have an answer to questions, you need in general no special profession. The more knowledge someone gained, the more difficult questions he/she can ask. So you really need to know as much as possible. (I think, there are (not only a few) people in academics saying that it is a waste of time of having a teaching certification since you &quot;lost time&quot; which could better be spent in gaining knowledge about your subject).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(I'm beware that this is more a &quot;perfect world&quot; explaination where all college/university students are mature.)&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-18T21:31:29.480" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="484" PostTypeId="1" AcceptedAnswerId="498" CreationDate="2014-03-18T21:56:39.320" Score="35" ViewCount="1934" Body="&lt;p&gt;In a freshman calculus course for non-majors;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is it to the benefit of the students to include discussion of epsilons and deltas?&lt;/li&gt;&#xA;&lt;li&gt;To what extent, if any, should they be used? For example, just to prove simple limits, or to prove limit laws, or more?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some authors jettison the rigorous definition to an appendix and never mention it again, even when discussing limit laws. I know that in such a course many teachers never mention the rigorous definition.&lt;/p&gt;&#xA;" OwnerUserId="52" LastEditorDisplayName="user7990" LastEditDate="2019-07-28T01:50:39.160" LastActivityDate="2023-08-23T05:41:56.130" Title="Epsilons and deltas in a first calculus course" Tags="|calculus|undergraduate-education|" AnswerCount="11" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="485" PostTypeId="2" ParentId="463" CreationDate="2014-03-18T22:07:54.923" Score="2" Body="&lt;p&gt;There is no definite answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As long as the question was formulated in a non-confusing way and the students were provided with the right theorems to solve the questions, I don't see any urge to change grading or even remove the questions from grades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why? In an exam you want to find out if the students have learned a significantly enough. IF all fail in this, they haven't learned it and should not get better grades then (exept when the needed theorem was not covered).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an extreme situation, maybe almost all students would answer most questions only to a certain level and then stop which would maybe proof that fulfill the requirements of your course but did learn nothing in addition from your course. Would you let them pass the exam then? (I hope not).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, you can grade more merciful (especially when you have the feeling that something went wrong during the lecture or the kind of asking was maybe to sloppy). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also include in grading other factors seeking for adjustment to this decision how to grade that particular question: Was the exam maybe too long? How was the overall performance? etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Regarding your example&lt;/strong&gt;: I depends heavily on how you asked. If the questions was &quot;Derive the &lt;strong&gt;best&lt;/strong&gt; lower bound for ...&quot; and they used one method which was not the best, I would keep the grades how they are, but maybe give almost all points (but not all) if someone did so. If the word &lt;strong&gt;best&lt;/strong&gt; would be missing, a student would think, he/she has finished since there was a (not so bad) lower bound derived. It that case, you can think about giving away all points.&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-18T22:07:54.923" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="487" PostTypeId="1" CreationDate="2014-03-18T22:35:31.270" Score="19" ViewCount="693" Body="&lt;p&gt;When presenting the $3$-dimensional proof of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Desargues%27_theorem#Three-dimensional_proof&quot;&gt;Desargues' theorem&lt;/a&gt; an average student might have, speaking informally, a &quot;WTF moment&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is an extreme case, but a similar situation could happen in an ordinary setting: what is interesting for gifted students might be inconceivable for the rest of the class. In fact, it might be a very good student presenting his solution.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How to present a solution which contains an element which could not have been conceived by an average student? &lt;/li&gt;&#xA;&lt;li&gt;Is it possible to avoid negative afterthoughts?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Some more examples for clarification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Cantor's diagonal argument seems conceivable (thanks to hindsight bias): all he did was to construct a number which was different from all the others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The definition of continuous function also seems conceivable if the student understands what we are trying to achieve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A solution to a geometry problem might seem conceivable if the sequence of transformations was &quot;reasonable&quot; in each step, and no unknown element was added (if there was a line added, then it connected two, already existing points, perhaps also having some relation to some other element which is already there).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if a geometry problem requires adding several lines/points with no apparent reason behind them and shines only after all of them have been drawn, such solution is far to alien and no line of thought could convince the students that it could be, one day, in their reach.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-19T07:49:19.937" LastActivityDate="2014-09-22T14:29:15.920" Title="Presenting a solution with a stroke of genius" Tags="|mathematical-pedagogy|mixed-ability|gifted-students|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="488" PostTypeId="2" ParentId="426" CreationDate="2014-03-18T22:55:33.900" Score="5" Body="&lt;p&gt;I took the qualifying exams in my state to teach. All teachers need to pass the general literacy, along with the subject they intend to teach. For me, it was middle and high school math. (The former, about ages 10-13, the latter 13-18) My undergraduate degree was BSEE, electrical engineering, and even though I wasn't working as an engineer, I never lost my love of math. I had aced the math SAT (the exam most US students take to apply to college) with a score of 800. This is background. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was on line to get in to take the high school level exam, having passed the middle school a few weeks prior. A woman next to me is visibly nervous, and shares with me that she teaches English, but wants to teach middle school math. She said she is back for her third attempt. I can't really describe the emotion I felt. Safe to say, I hope she failed. Would you want your 8th grader to be taught by someone who needed 3 tries at passing an exam like this? More important, if you can't pass this the first time, do you really feel comfortable enough to teach the subject? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To circle back - teaching grade or high school needs the course work to have teaching skills presumably learned as part of the certification process. In my opinion, that's fine, but the level of understanding required to teach the subject should be higher. The high school level exam contained virtually no calculus, and in hindsight, barely differed from the middle school exam. &#xA;I'll take an individual with a masters in mathematics over a certified teacher who bared passed the exam any day. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-03-25T13:43:58.467" LastActivityDate="2014-03-25T13:43:58.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="489" PostTypeId="1" AcceptedAnswerId="490" CreationDate="2014-03-18T23:23:02.443" Score="5" ViewCount="60" Body="&lt;p&gt;You want to host a seminar where students will present either&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a chapter from a book, or&lt;/li&gt;&#xA;&lt;li&gt;a (part of a) scientific article.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is the best way&lt;/strong&gt; (=such that the students learn as much as possible from the seminar without increasing the efford of the adviser too much) &lt;strong&gt;to schedule the seminar&lt;/strong&gt;? Should there be one presentation per week or should there only a few meetings with a lot of presentations per day? Why? And does it depend on the type of seminar?&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="143" LastEditDate="2014-03-19T21:36:12.943" LastActivityDate="2014-03-19T21:36:12.943" Title="How to schedule a seminar (weekly based vs. one big meeting) to maximize learning?" Tags="|course-design|seminar|scheduling|graduate-education|" AnswerCount="2" CommentCount="1" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="490" PostTypeId="2" ParentId="489" CreationDate="2014-03-18T23:42:32.490" Score="2" Body="&lt;p&gt;Personally, I feel I take away a lot more from x isolated talk than from x talks at one conference day. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the seminar, especially if it is a sequence of talks from one book or article where one &quot;works through&quot; something together I would especially recommend several meetings. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the best case, students will try to actively digest the talks afterwards and/or prepare for the next one. &#xA;It is then also easier to address questions that might come up yet cannot be answered on the spot, by reporting them to the beinning of the next session. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, in the worst case, students do not care at all and forget everything from one meeting to the next. But, first, this is pessimistic a view and second one can work against this by briefly coming back to last weeks talk at the beginning of the session.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there are also considerations against weekly meetings: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes attendance can be a practical issue. Either since students simply do not show-up sometimes if it is weekly, or since it might be difficult for them to do so (in which case one might try to help them voa a different form of organizing it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the presentations are by and large independent there might be reasons for fewer longer session, perhaps even a full day, and to seomhow give it a mini-coferrence feeling. Yet, also here, there are practical considerations, such as: Is finding a room for a full day perhaps a problem? Can students be available for a full day?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In brief, in some cases like &quot;working through&quot; a book I would strongly favor weekly meetings; in the other cases, I'd say it depends mainly on local, practical circumstances.&lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-03-18T23:42:32.490" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="491" PostTypeId="2" ParentId="484" CreationDate="2014-03-18T23:56:04.793" Score="13" Body="&lt;p&gt;I teach at a community college, and I love teaching calculus. I hope my course is at least as good as university courses. I think it is a big benefit to &lt;em&gt;introduce&lt;/em&gt; them to the deep logic of the epsilon-delta definition of limits, but I do not prove anything with them, and don't think that's very valuable for beginners.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Timing&lt;/strong&gt;&lt;br&gt;&#xA;I think any attention to limits at the beginning of the course is not wise. Limits are strange new creatures for the students, and at the beginning of the course they have no idea why this weirdness is necessary. For most students it will be alienating to spend time there before they've learned about derivatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I start with a full unit on understanding derivatives as rate of change, slope, velocity, ... (Anton doesn't have nearly enough of what I want, so I use lots of material from  &lt;a href=&quot;http://faculty.gvsu.edu/boelkinm/Home/Open_Calculus.html&quot;&gt;Boelkins' free text&lt;/a&gt; for this part.) We use the regular limit notation, but I just say that it means infinitely close. (And I say that that's not really enough. But it's about what Newton did, and mathematicians took 150 years to do better, so it's good enough for a start.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, after the units on rules of derivatives (I call them properties) and applications, once the students see how valuable derivatives are, I introduce limits more formally. I do not ask any test questions regarding limits, but I do introduce the epsilon-delta definition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;br&gt;&#xA;I call it the sentence that took 150 years to write, and I write it almost like a poem, one short phrase on each line:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;lim (x-&gt;c) f(x) = L &lt;br&gt;&#xA;means&lt;br&gt;&#xA;for any &amp;#949; &gt; 0 &lt;br&gt;&#xA;there is a &amp;#948; &gt; 0 &lt;br&gt;&#xA;such that&lt;br&gt;&#xA;if 0 &amp;lt; abs(x-c) &amp;lt; &amp;#948; &lt;br&gt;&#xA;then abs(f(x)-L) &amp;lt; &amp;#949;.&lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I draw diagrams with colorful bands, much like what you see in any good textbook, and explain with lots of pointing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example&lt;/strong&gt; &lt;br&gt;&#xA;Then I give a colorful example to help the students understand the machinery. (I got it from a colleague many years ago. Thanks, Diane.) It uses a cookie crispness index on the y-axis, and the time your cookies are in the oven on the x-axis. Before I get into it, I ask the students what their favorite cookies are and how crisp they like them on a scale from 0 to 10, where 0 is doughy, 3 is still very soft and chewy, 6 is getting more crispy than chewy, and 9 is starting to get burned. Then I use one student's preference for my example. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I draw a line for the function that connects cooking time to crispness, and from that I estimate the time I need to bake. Then I play with them about how perfect the cookie will be (&amp;#949;). Now I get to draw my bands. If I want the cookie that close to perfect, how close to perfect must my timing be, when I pull that tray out of the oven?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No testing, no proofs, just a good example to chew on.  ;^)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-18T23:56:04.793" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="492" PostTypeId="2" ParentId="294" CreationDate="2014-03-19T00:12:11.993" Score="10" Body="&lt;p&gt;I strongly reccomend &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/052154677X&quot;&gt;The Cauchy-Schwarz Master Class&lt;/a&gt;&lt;/em&gt; by J. Michael Steele. It could be read by  advanced high school students who did well in calculus and have a strong interest in mathematics although it is probably better suited for first year undergraduate math majors. It reads like a novel that contains plenty of challenging exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another book that is usually brought up in such conversations is &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486284247&quot;&gt;&lt;em&gt;Concepts of Modern Mathematics&lt;/em&gt;&lt;/a&gt; by Ian Stewart. It is my understanding that it is geared towards a popular audience and attempts to give a brief overview of modern mathematics, but I have yet to read it so I can't give it any more of a review.&lt;/p&gt;&#xA;" OwnerUserId="52" LastEditorUserId="52" LastEditDate="2014-06-21T13:10:00.310" LastActivityDate="2014-06-21T13:10:00.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="493" PostTypeId="2" ParentId="484" CreationDate="2014-03-19T00:29:53.803" Score="5" Body="&lt;p&gt;I have taught calculus several times and I have always included the $\epsilon-\delta$ definition of limit. And I haven't really ever had any problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I have done is to first introduce limits of functions be using graphs. I explain what it intuitively means that the limit of a function exists. I make very clear that the initial presentation of the concept is just to get an intuition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that we do a lot of examples using this method. We talk about limits that are pretty obvious and some where rewriting the function makes the existence of the limit clear. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After this, I then state the precise definition. I should the several examples of how to prove that the limit of a function is a given number. I make sure to keep the format of the proofs the same. So I always start by writing &quot;proof&quot; and the &quot;Let $\epsilon &amp;gt; 0 $ be given&quot; and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After showing several examples of even quadratics, I go back and end with some linear functions. For the exams I just require them to know how to prove the limits of linear functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And all this in general works out pretty well. I will admit that the first example or two are very confusing for the students, but after realizing that we keep doing the same thing over and over, they come to appreciate the rigor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you could(should?) include examples where you have a specific value of $\epsilon$ given. The students then have to find a/the corresponding value of $\delta$ that will work. In my experience I have done this right after the definition or after you have done all the examples. I am probably leaning towards doing the specific value examples right after the definition as a help to understanding what we are doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, so from my experience it really isn't too hard to teach the $\epsilon-\delta$ definition. And the nice thing is that you can prove some of the rules that we use for limits, like $\lim_{x \to a} c = c$. You might even require your students to memorize some of these simpler proofs.&lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-03-19T00:29:53.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="494" PostTypeId="2" ParentId="484" CreationDate="2014-03-19T00:50:34.260" Score="16" Body="&lt;p&gt;Given that the specific (and not at all unique) &quot;rigorization&quot; of calculus took 200 years (from Fermat, Descartes c. 1650, up to 1850), as a reflection of human perception, it seems to me completely unreasonable to insist that novices &quot;appreciate&quot; the rigorization-throes of the mid-to-late 19th century. (Indeed, for all the citing of K. Weierstrass as exemplar of &quot;rigor&quot;, his self-citing absolutely does &lt;em&gt;not&lt;/em&gt; emphasize foundational things, but, rather, work on several complex variables, abelian functions, and other very-easily-arguably-good, but not &quot;elementary&quot;, topics!!!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I have suggested at other moment in similar venues, if we were to pretend that students were critical intellectuals (in the best sense), they'd surely not bend to mere authority or commands to comply, but would ask &quot;what advantage?&quot; (which I would universally encourage).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then it does become quite difficult to create any &lt;em&gt;imperative&lt;/em&gt; to do epsilon-delta... because all the &lt;em&gt;live&lt;/em&gt; issues, about modeling mechanics, or even the simplest calculus, do not seem to need confirmation!!!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, the two centuries of amazing &lt;em&gt;discoveries&lt;/em&gt; enabled by the very idea of calculus were the motive force. &quot;Rigorization&quot; was a luxury, and of interest only to specialists... even though, yes, Cantor was led to Set Theory by worrying about convergence of Fourier series. Cool, yes, and very interesting at some point, but not the critical factor in Calc I.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-19T00:50:34.260" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="495" PostTypeId="2" ParentId="484" CreationDate="2014-03-19T00:59:00.257" Score="1" Body="&lt;p&gt;Based on my own experience and the students I tutored in calculus, it took me several tries to &quot;get&quot; the epsilon-delta definition of a limit, including writing out the definition in formal logical symbolism and figuring out what it meant if the limit did not equal a particular number. The epsilon-delta technique is an inordinately cumbersome method for finding simple limits for simple functions, and the various limit theorems that can be proved with it are simpler and more useful.   I was baffled at finding an appropriate delta for non-linear functions, and found no further use for the technique until I took a course in analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a beginner in a non-rigorous course, I would find it appropriate to introduce limit theorems without formal proof at first, and save the formal definition as an  optional advanced topic for those who are doing well with the basics.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-19T00:59:00.257" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="497" PostTypeId="2" ParentId="466" CreationDate="2014-03-19T01:27:42.837" Score="24" Body="&lt;p&gt;From &lt;a href=&quot;http://arxiv.org/abs/math/9404236&quot; rel=&quot;noreferrer&quot;&gt;Bill Thurston&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When I started as a graduate student at Berkeley, I had trouble imagining how I could “prove” a new and interesting mathematical theorem. I didn’t really understand what a “proof” was. By going to seminars, reading papers, and talking to other graduate students, I gradually began to catch on.&lt;/p&gt;&#xA;&lt;p&gt;Within any field, there are certain theorems and certain techniques that are generally known and generally accepted. When you write a paper, you refer to these without proof. You look at other papers in the field, and you see what facts they quote without proof, and what they cite in their bibliography. You learn from other people some idea of the proofs. Then you’re free to quote the same theorem and cite the same citations. You don’t necessarily have to read the full papers or books that are in your bibliography. Many of the things that are generally known are things for which there may be no known written source. As long as people in the field are comfortable that the idea works, it doesn’t need to have a formal written source.&lt;/p&gt;&#xA;&lt;p&gt;At first I was highly suspicious of this process. I would doubt whether a certain idea was really established. But I found that I could ask people, and they could produce explanations and proofs, or else refer me to other people or to written sources that would give explanations and proofs. There were published theorems that were generally known to be false, or where the proofs were generally known to be incomplete. Mathematical knowledge and understanding were embedded in the minds and in the social fabric of the community of people thinking about a particular topic. This knowledge was supported by written documents, but the written documents were not really primary.&lt;/p&gt;&#xA;&lt;p&gt;I think this pattern varies quite a bit from field to field. I was interested in geometric areas of mathematics, where it is often pretty hard to have a document that reflects well the way people actually think. In more algebraic or symbolic fields, this is not necessarily so, and I have the impression that in some areas documents are much closer to carrying the life of the field. But in any field, there is a strong social standard of validity and truth. Andrew Wiles’s proof of Fermat’s Last Theorem is a good illustration of this, in a field which is very algebraic. The experts quickly came to believe that his proof was basically correct on the basis of high-level ideas, long before details could be checked. This proof will receive a great deal of scrutiny and checking compared to most mathematical proofs; but no matter how the process of verification plays out, it helps illustrate how mathematics evolves by rather organic psychological and social processes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So essentially, his view is that what a proof is is essentially based off of societal norms backed up by paper documents when necessary.&lt;/p&gt;&#xA;&lt;p&gt;In a classroom, I feel this translates to 'a proof is an argument which convinces the grader that you have either written all the details or are able to provide them.&lt;/p&gt;&#xA;&lt;p&gt;Formatting of the quote may be off.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-09-03T17:30:11.380" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="498" PostTypeId="2" ParentId="484" CreationDate="2014-03-19T01:29:44.520" Score="35" Body="&lt;p&gt;For some reason there is a wide spread view that the $\epsilon-\delta$ definition of a limit is an obscure thing, relevant only to mathematicians, and that the only reason to care about them is to make limits ``rigorous''.  This could not be further from the truth.  $\epsilon-\delta$ analysis is all about learning how to control error in the outputs of a function.  This is crucial for applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give an $\epsilon-N$ example, say that you are an engineer who needs to write a numerical integrator to compute the number of calories someone is expending when they use an exercise bike.  You want to make sure that the result is accurate to 10 calories per hour of use.  The question is:  how many subdivisions will I need to use?  This is incredibly relevant.  The $\epsilon-N$ analysis just takes it one step further:  If I ask you to be accurate within $\epsilon$ calories, how many rectangles will you need to use?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This applies to the standard $\epsilon-\delta$ style proofs as well. If $S(r)$ gives the strength of a beam as a function of the radius, many peoples lives might depend on knowing how precisely you must machine the radius to land within an acceptable range of strengths for the output.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For many of our students, the functions they meet in their day to day life will not be given by formula, but by numerical data.  At this point all the rules of differentiation and integration go out the window, and the most important thing becomes understanding how good an approximation you are getting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\epsilon-\delta$ analysis is hard, but it is not only there to satisfy mathematicians with an insane desire for &quot;rigor&quot;:  it is the very practical and important skill of learning to bound errors.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-03-19T01:29:44.520" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="499" PostTypeId="2" ParentId="484" CreationDate="2014-03-19T01:38:27.087" Score="8" Body="&lt;p&gt;My college first-year calculus class was by a professor who did teach the subject because he loved it. After a clear explanation of what limits are about (sorry, it was 40 years back, no details left by now), he explained the definition, and did a few simple ones, like $\lim_{x \to 2} x$ and $\lim_{x \to 2} x^2$. He made quite a show out of his &quot;secret technique to prove limits&quot; (the typical proof where you are given an $\epsilon$, conjure a complicated formula for $\delta$ out of thin air, substitute and get a clean ${} &amp;lt; \epsilon$ at the end). It was simply to work backwards &quot;on scratch paper, but make sure none of the spies from the parallel classes sees this!!&quot; and the &quot;for clean&quot; retrace of the steps. All seasoned with a few extra tricks, and finally discussion of $\lim_{x \to 0} \frac{\sin x}{x}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As other answers say, he soon dismissed this as &quot;reserved just for mathematicians, for you engineers we have simpler tools&quot; and introduced (and proved) the classical theorems on limits. He occasionally went back to the definition (sometimes just as a reminder, sometimes to prove a tricky limit where theorems weren't applicable).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the critical things was that he carefully made sure each had understood each step before going ahead. I suspect we didn't cover the complete official contents of the course (did't matter, he was dean of the Science Faculty at the time).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-19T01:38:27.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="500" PostTypeId="2" ParentId="487" CreationDate="2014-03-19T01:52:36.877" Score="6" Body="&lt;p&gt;A strange question. Almost every time you show something new to the students, it will be something that would never have occurred to them. Try to remember the first time you saw the derivation of the formula for the quadratic proved, or the $\epsilon - \delta$ definition of limit, or Cantor's diagonal argument. So this isn't the exception, it is the norm in the classroom. Part of what makes it rewarding, and also what can be so frustrating.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-19T01:52:36.877" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="501" PostTypeId="1" CreationDate="2014-03-19T02:14:02.700" Score="10" ViewCount="411" Body="&lt;p&gt;Assume that you have incoming college freshmen, so you might have access to their high school courses and grades, as well as their results on a placement test offered by your college. Also, assume that these are students who will need to progress along this sequence for their major (in chemistry or economics, e.g.) and will take the course in which they're placed during their first semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some effective methods to assess the abilities of incoming students and properly place them into these courses? Should you incorporate other data (e.g. high school courses and grades)? When designing a placement test, what are some considerations to be made to make it a good and effective test? What kinds of questions should be asked, and how should they be scored? What would you make you more apt to &quot;bump up&quot; a student beyond their placement, if they asked for it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested both in research about these questions (although I suspect it might be limited) as well as personal anecdotes/opinions/recommendations, &lt;em&gt;especially&lt;/em&gt; if you were involved in designing and implementing a placement test.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2020-01-23T09:35:17.290" Title="Placing incoming college freshmen: basic algebra, pre-calc, calc" Tags="|undergraduate-education|test-design|assessment|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="502" PostTypeId="2" ParentId="489" CreationDate="2014-03-19T02:15:25.047" Score="1" Body="&lt;p&gt;To relieve part of the burden, ask the students themselves to participate actively (with real responsibilities) in the organization. Lighten up/complement the seminar with some invited speakers. Depending on the breadth and depth, and the number of presentations, it miight need to be programmed beforehand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leave plenty of time between presentations, for discussion and relaxing before the next one. In conferences they are interested in cramming as many presentations as possible into the available time (and, cross your heart, you &lt;em&gt;don't&lt;/em&gt; go to conferences to see the presentations, but to talk informally with the authors).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-19T02:15:25.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="503" PostTypeId="2" ParentId="484" CreationDate="2014-03-19T03:00:59.353" Score="14" Body="&lt;p&gt;I've taught calculus with and without &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; techniques. In particular:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;at a large state school with a nearly endless supply of engineering students. Here math majors were a rare event... I would characterize the main goal of the course just to get the students up to speed computationally. For many students, they only really learn algebra in calculus. I say learn in the sense that they apply algebra to solve problems which cannot be fit into some standard form. In calculus, the algebra is wild, untamed, it goes where it wishes and you better be able to deal with it. This struggle ideally produces students with greater algebraic maturity and, naturally, a working understanding of the conceptual accomplishments of calculus. Proofs were rarely given and even when they were I would often give a &amp;quot;physicsy proof&amp;quot;. For example &lt;span class=&quot;math-container&quot;&gt;$\sin(h) /h \approx 1$&lt;/span&gt; is obvious if you just look at a tiny triangle incribed in the unit circle. I suppose my perspective when first teaching was in part the natural outgrowth of my origins story; I was raised by physicsists. So, the question for me in teaching was how can I prepare students to face anything which may be thrown their way in junior/senior level courses? My goal was to prepare them for those trials and tribulations.&lt;/li&gt;&#xA;&lt;li&gt;at a private liberal arts school. Here, proportionally, I have a nontrivial population of math majors. So, I cannot ignore the need for laying a foundation. I still care about the calculations, computations and applications. But, I make room for proofs. In 2011, I made a major revision to how I teach the course. That summer I wrote course notes which make an effort to prove as much as possible for the first course. Partly, I just want to demystify the idea of a proof for them. I want them to see that a proof is not some rigid construct as we sometimes encounter in highschool geometry. I also want them to see that we can give a nearly complete account for why things are true. Of course, I do not test too heavily on the proof aspect of the course. For example, in treating &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; I only require the linear and quadratic cases for the test. I prove several limit laws from the precise definition. Now, for the majority of students, this is probably annoying. But, honestly, they have the same feeling about limit laws. Every approach has it's advantages and disadvantages and I think what is best depends on your particular audience. That said, I found this approach has attracted students from other majors. There exist students who are looking for something deeper. All of this said, I'd trade the proofs in calculus for a required second course of linear algebra in a heart beat.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="128" LastEditorUserId="205" LastEditDate="2023-08-23T05:41:56.130" LastActivityDate="2023-08-23T05:41:56.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="504" PostTypeId="1" AcceptedAnswerId="507" CreationDate="2014-03-19T03:06:14.347" Score="27" ViewCount="2097" Body="&lt;p&gt;There are several groups of people that would benefit from learning LaTeX in college. Future teachers can use it to write exams, scientists and mathematicians can write papers, and everyone can write theses with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are three ways to introduce LaTeX:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A formal class devoted to LaTeX&lt;/li&gt;&#xA;&lt;li&gt;Portions of other classes focused on LaTeX&lt;/li&gt;&#xA;&lt;li&gt;Students picking it up on their own&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These are arranged in decreasing required time investment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What, from your actual experiences or empirical data, is the most effective way for students to learn LaTeX?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-03-19T19:51:23.177" LastActivityDate="2014-03-19T21:04:07.767" Title="How should LaTeX be taught to university students?" Tags="|undergraduate-education|course-design|curriculum|specific-software|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="505" PostTypeId="2" ParentId="469" CreationDate="2014-03-19T03:23:14.053" Score="5" Body="&lt;p&gt;By far the easiest method I have found for this is for the usual teacher to prepare lecture notes containing the entire lecture as they would teach it. Many US professors already have such notes or are in the process of building a collection of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the other professors notes, one knows, at least, what they would have taught. You can then work within that framework, and only change things if the other professor is comfortable with it and if you feel strongly about it.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-19T03:23:14.053" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="506" PostTypeId="1" AcceptedAnswerId="511" CreationDate="2014-03-19T04:02:59.883" Score="27" ViewCount="1163" Body="&lt;p&gt;I had this happen today, and several times before. I had had difficulty preparing the material, because I knew my students strengths and it didn't seem suited to them. However, it seemed the right material at the time I was preparing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I gave the lecture, I noticed the students seemed unusually bored. There were few comments, and as I lectured I doubted my choice of lesson plan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a senior-level course without a fixed syllabus. I could have abandoned ship and went on to other material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What should be done in this situation? Is it ever advisable to openly abandon a lesson plan when it isn't working?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="143" LastEditDate="2015-02-14T20:24:05.930" LastActivityDate="2015-02-14T20:24:05.930" Title="What do you do when you realize mid-lecture that your lesson plan is not working?" Tags="|general-pedagogy|teacher-preparation|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="507" PostTypeId="2" ParentId="504" CreationDate="2014-03-19T05:08:01.270" Score="24" Body="&lt;p&gt;In my experience, students in sophomore or junior level math courses usually have very little trouble picking up LaTeX on their own.  They typically require the following assistance:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Some guidance in downloading and installing it, e.g. links to user-friendly distributions for both the Mac and the PC.  I post links to my course webpage.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A sample LaTeX source file, 1-2 pages in length, using simple commands, and similar in format to a homework assignment for the course.  Again, I just make this downloadable from my course webpage.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A relatively short first homework assignment.  Most of the first &quot;homework&quot; is actually just getting LaTeX working!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Occasional help with LaTeX during office hours.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I have required LaTeX for all written homework when teaching introductory proofs, introductory abstract algebra, introductory point-set topology, and introductory real analysis, and in each case there was essentially no problem.  Usually one or two students per class had trouble getting it up and running before the first weekly homework assignment was due, but they would all have it working by the end of the second week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I really think that LaTeX helps students to write better proofs.  It makes them spend more time on the writing, and it also gives them the ability to their edit proofs a bit after they write a first draft.  Students often show me a LaTeX version of a proof during office my hours, and I suggest ways in which the writing could be improved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another huge advantage of requiring LaTeX is that students can simply e-mail you their homework assignments.  I have a tablet and a stylus that I use for grading, so I just write my comments directly on the PDF and then e-mail it back to them.  In addition to being faster, this has the nice advantage that you retain copies of the students' assignments, which can be helpful in a variety of situations.  Without a tablet, it would work just fine to simply print out the students' assignments and hand them back as paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, requiring LaTeX is somewhat more work for the students.  Fortunately, most of the extra work for the students happens during the first few weeks of class, which means that it isn't much of a burden for them.  I've never had students complain about the extra work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One more note: I don't think it would work very well to require LaTeX in a course such as graph theory where students commonly have to draw pictures on their homework assignments.  Even in topology, I had to drop the LaTeX requirement near the end of the course when we started talking about geometric topology, because the students' homeworks were full of pictures!&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-19T05:08:01.270" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="508" PostTypeId="1" AcceptedAnswerId="510" CreationDate="2014-03-19T05:24:52.590" Score="17" ViewCount="609" Body="&lt;p&gt;It is very tempting to assign extra credit problems that are (a) substantially more difficult than others on the problem set, (b) ask for the history of something we discussed in class, or (c) are related to something just barely outside of the scope of the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I have never done so because, in my experience, very few professors give extra credit.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Why do so few professors assign extra credit (or is this not the case)?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;What are the arguments against assigning extra credit problems?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="52" LastActivityDate="2014-03-20T10:06:48.233" Title="Is extra credit appropriate?" Tags="|undergraduate-education|homework|grading|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="510" PostTypeId="2" ParentId="508" CreationDate="2014-03-19T06:16:59.420" Score="10" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Why do so few professors assign extra credit?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In my experience, the attitude towards extra credit is consistent throughout the department. Nearly every professor in the education department at my university puts extra credit questions on the test, but only a few in the math department do. After chatting with other students and professors, this seems to be common in other universities as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://www.facultyfocus.com/seminars/extra-credit-an-undeserved-gift-or-a-second-chance-to-learn/&quot;&gt;this article on Faculty Focus&lt;/a&gt;, &lt;em&gt;of the 362 people who answered the question “Do you give extra credit?” 42 percent said sometimes, 31 percent said often and 27 percent said never.&lt;/em&gt; While this doesn't give a breakdown of what type of extra credit, it's interesting to see that 73% of respondents give extra credit at least some of the time.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;What are the arguments against assigning extra credit problems?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www.facultyfocus.com/articles/teaching-and-learning/does-extra-credit-have-a-place-in-the-college-classroom/&quot;&gt;an article on that seminar&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;According to Weimer, some instructors oppose extra credit because they believe that:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;It reinforces students’ beliefs that they don’t have to work hard because whatever they miss they can make up with extra credit.&lt;/li&gt;&#xA;  &lt;li&gt;Students who ask for extra credit tend to be those who aren’t working very hard — or those who hope they won’t have to work hard — because some easy extra credit options will be available to them.&lt;/li&gt;&#xA;  &lt;li&gt;Time spent on extra credit means less time spent on regular assignments.&lt;/li&gt;&#xA;  &lt;li&gt;Extra credit (especially if it’s easy) lowers academic standards.&lt;/li&gt;&#xA;  &lt;li&gt;It’s inherently unfair to students who work hard and get it done right the first time.&lt;/li&gt;&#xA;  &lt;li&gt;It means more work for already busy teachers.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;But I disagree with those issues!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like giving extra credit problems. I believe extra credit problems that are significantly more difficult and require more analytic thinking on the part of the student are fantastic as long as they're used properly. If you're planning on using extra credit problems, consider what you're trying to achieve by providing those problems. Personally, I want to use extra credit to separate the best students and provide those who have worked extremely hard to understand the concepts some leeway on simple mistakes.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Don't weigh them too heavily&lt;/strong&gt; - Extra credit should give a good student some extra points to offset &quot;stupid&quot; mistakes but not allow a student to pass without understanding the basics.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Do make them difficult&lt;/strong&gt; - You're not giving away free points. You want to reward students who understand the concepts you've taught them above and beyond what is required.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Do put them at the end and label them properly&lt;/strong&gt; - It would be terrible for a student to get stuck on the more difficult extra credit and not have time to finish the main problems in a test.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="151" LastActivityDate="2014-03-19T06:16:59.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="511" PostTypeId="2" ParentId="506" CreationDate="2014-03-19T06:57:56.380" Score="13" Body="&lt;p&gt;The first lesson I ever taught was on Standard Deviation to a high school Intro to Engineering class. It went fabulously and I stuck to the lesson plan covering every point on my slide show and getting through every activity only having to shorten the pair and share activity due to time constraints. The students understood everything, responded to discussion well, and the observing teacher gave me a glorious review.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I had to present the same lesson to my Education class. Within the first five minutes, I realized that there was only a single person in the entire class with the background needed to understand the mathematics I was presenting. This made the warm-up exercise and discussion extremely one-sided. In that moment, I decided to abandon my lesson plan and launched into a discussion about crappy furniture. Every college student can understand crappy furniture even if none of them understood what variance had to do with it. By the end of the lesson, most of the students had a reasonably good grasp on the concept of standard deviation and how it relates to Ikea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Any time you're speaking to an audience, you have to pay attention to their attitude and capture their attention.&lt;/strong&gt; In my example it's a bit different than most teaching situations because I had prepared for a class that had the background to understand my content but the idea can carry over to any classroom. If either they don't get it or you are having an off teaching day. It could be time to switch gears and doing that midway through a lesson is not a terrible idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Making a habit of going off-topic is ill-advised.&lt;/strong&gt; I've heard horror stories of professors who just go off on tangents for most of the lesson. This not only creates discontinuity in your teaching but is a waste of valuable instruction time. While this may be interesting every now and then, if it's your go-to teaching method, there is a problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;If at all possible, have a plan B.&lt;/strong&gt; When I prepared my lesson on Standard Deviation, I had weeks to work on the lesson plan and had a variety of topics to generate discussion in case the class decided to be sleepy and quiet that morning. So when my education class was not responsive, it wasn't hard to come up with something that they could relate to but still stay relatively on topic. When working on a lesson plan, considering the question &lt;em&gt;'What if this bombs?'&lt;/em&gt; is not a terrible one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't believe there's an easy method of knowing if a class is bored or lost or hungover but I think the best way to tell if you should jump ship on your lesson plan is to &lt;strong&gt;know your students&lt;/strong&gt;. Could you drop the lesson plan and just have a discussion on the topic? Would they be able to follow and participate in such a discussion? Is that something that would interest them? Would they get more out of a discussion than the lecture right now?&lt;/p&gt;&#xA;" OwnerUserId="151" LastActivityDate="2014-03-19T06:57:56.380" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="512" PostTypeId="2" ParentId="504" CreationDate="2014-03-19T08:09:07.667" Score="2" Body="&lt;p&gt;This semester I, for the first time, require all my students to submit their assignments typed and to use LyX for that. I find LyX to be much more easy to digest for students and it teaches both LyX and Latex at the same time. I devote labs to introduce LyX to the students. Most students have no problem working with LyX but some struggle considerably. I find that LyX, as compared to Latex, has a very minor learning curve. After the first few weeks most students are proficient enough and are able to deal with typing up their solutions just fine. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-19T08:09:07.667" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="513" PostTypeId="2" ParentId="353" CreationDate="2014-03-19T09:56:53.737" Score="8" Body="&lt;p&gt;My approach is like that of others but I like to use math instead of everyday language.  I get them to agree that we want this statement to be true: &quot;if &lt;em&gt;x&lt;/em&gt; is a perfect square then &lt;em&gt;x&lt;/em&gt; is not prime&quot; simply because &lt;em&gt;x=y*y&lt;/em&gt; is a factorization.  Then we use various &lt;em&gt;x&lt;/em&gt;'s to get the different lines of the truth table.&lt;/p&gt;&#xA;" OwnerUserId="213" LastEditorUserId="213" LastEditDate="2014-03-19T21:31:49.423" LastActivityDate="2014-03-19T21:31:49.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="514" PostTypeId="2" ParentId="353" CreationDate="2014-03-19T11:33:47.960" Score="5" Body="&lt;p&gt;One of the things that really helped me (not to learn it, but to appropriately apply it), back when I originally learned this myself was the equivalence between $p\implies q$ and $\neg p \vee q$. I constantly reminded myself of it when manipulating logic statements. Make sure the students are aware of this equivalence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What it did was give me a basis of comparison to something I already understood well. It gave me something to check in case I was worried I have made a mistake. It made negation of implication a lot easier to understand as well.&lt;/p&gt;&#xA;" OwnerUserId="208" LastActivityDate="2014-03-19T11:33:47.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="515" PostTypeId="5" CreationDate="2014-03-19T13:19:02.237" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-19T13:19:02.237" LastActivityDate="2014-03-19T13:19:02.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="516" PostTypeId="4" CreationDate="2014-03-19T13:19:02.237" Score="0" Body="For questions that (at least in part) are specific to the United States of America." OwnerUserId="143" LastEditorUserId="39" LastEditDate="2014-06-11T23:17:48.717" LastActivityDate="2014-06-11T23:17:48.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="517" PostTypeId="2" ParentId="504" CreationDate="2014-03-19T13:52:14.297" Score="4" Body="&lt;p&gt;In my experience, the only way to learn to use tools is to use them for real. A LaTeX course on its own will be forgotten by the time they need it. At LSU the library organized a series of short courses (2 or 3 sessions, longest was a week or so) on several aspects of using the library effectively. They were voluntary, and repeated irregularly during the year. It worked well, AFAIU. Somtehing like that for the wider audience, and perhaps dedicate the first week's session with the teaching assistant to get up to speed with LaTeX (or other tools in use in the course).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here we use &lt;a href=&quot;https://moodle.org&quot; rel=&quot;nofollow&quot;&gt;moodle&lt;/a&gt;, students are encouraged to turn in homework (in LaTeX) through it. I schedule a homework number 0 (worth half a regular homework) to get the acquainted with the system, get their account in order, and all the other administrative overhead. It must be turned in in LaTeX, for follwing homework LaTeX gives extra points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having the homeworks saved in the system has the extra benefit that they are available, and the exact instant time of turn in is recorded.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-19T13:52:14.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="518" PostTypeId="2" ParentId="508" CreationDate="2014-03-19T14:08:52.513" Score="1" Body="&lt;p&gt;I give extra credits routinely, mostly to reward not-asked-for work (point out an error in the class notes, suggest an interesting alternative solution for an exam or homework problem, ...) and sometimes for extra homework assigned. In my (by now extensive) experience, students who go the mile to get extra credit would get good or outstanding grades even without the extra points. A very few of those hanging in the balance get over the bump by this, but then they have shown enough insight to merit a passing grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In exams I asign a few extra points, but not a problem marked specially. That gives some leeway for mistakes and still getting a perfect score. My exams are typically a fairly simple, general question (&quot;do you even know what we are talking about?&quot;) and the remainder several more difficult problems covering the material about evenly. Sometimes questions have a &quot;bonus track&quot; part, if it makes sense to look somewhat further.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-19T14:08:52.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="519" PostTypeId="2" ParentId="184" CreationDate="2014-03-19T15:24:50.597" Score="8" Body="&lt;p&gt;Well, I think the answer is &quot;there is no literature on your question&quot;; however, this question is somehow unanswerable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me explain: For my PhD, I spent the last 3 years working with literature on how students learn mathematics best at university. The first (much weaker) argument is, that I couldn't find any literature that matches your question. The second (stronger) argument is, that educational psychologist and researcheres in math education would presumably go more into detail, what a &quot;good balance&quot; should be. One could sketch possible learning outcomes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;memorisation of facts, formulae, etc.&lt;/li&gt;&#xA;&lt;li&gt;deeper understanding of mathematical connections&lt;/li&gt;&#xA;&lt;li&gt;more appropriate use of learning strategies&lt;/li&gt;&#xA;&lt;li&gt;change in mathematical beliefs&lt;/li&gt;&#xA;&lt;li&gt;feelings of self-efficacy&lt;/li&gt;&#xA;&lt;li&gt;intrinsic motivation and interest&lt;/li&gt;&#xA;&lt;li&gt;...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What you offer is extrinsic motivation. This will surely help for the first item and surely not help for the sixth. &#xA;So from the research perspective, one would rather concentrate on one or two of these learning outcomes and then try to mesaure the effect of extrinsic motivation in a specified context (like math class at university).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning the &quot;good balance&quot; you asked for: this is a rather normative question. You have a trade-off between the different learning outcomes. Thus, the optimal solution depends on your valuation function. Answers to such a question are generally rare.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, your students are wrong if they say there is no learning as long as there is pressure. Have a look at learning in school. :-)&#xA;I think, however, this was not exactly your point...&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-19T15:24:50.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="520" PostTypeId="1" AcceptedAnswerId="521" CreationDate="2014-03-19T15:29:26.870" Score="26" ViewCount="3210" Body="&lt;p&gt;Courses about linear algebra make great demands on looking for eigenvalues and transforming matrices to diagonal matrices (or, at least, to Jordan normal form). This is somehow a technical, recipe-like part where many students don't see a deeper understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is &quot;the best&quot; showcase why one needs eigenvalues? Or: What is a good motivation to come up with where they will be needed (should be understandable for the student at that point)?&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-11T12:26:43.010" LastActivityDate="2021-05-21T15:35:54.557" Title="What is a good motivation/showcase for a student for the study of eigenvalues?" Tags="|examples|concept-motivation|linear-algebra|eigenvalues|" AnswerCount="13" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="521" PostTypeId="2" ParentId="520" CreationDate="2014-03-19T15:43:57.320" Score="20" Body="&lt;p&gt;Obtaining formulas for the $n$-th term in a linear recurrence, such as Fibonacci numbers, is one application that certainly does not overtly mention linear algebra in the set-up.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-19T15:43:57.320" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="522" PostTypeId="2" ParentId="520" CreationDate="2014-03-19T15:50:42.020" Score="14" Body="&lt;p&gt;Imagine a linear mapping &lt;span class=&quot;math-container&quot;&gt;$f: R^2 \to R^2, e_1 \mapsto (1.5, 0.5), e_2 \mapsto (0.5, 1.5)$&lt;/span&gt;. (As long as &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; contains the numbers &lt;span class=&quot;math-container&quot;&gt;$1.5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$0.5$&lt;/span&gt;, it could be any ring. The real numbers serve as the most convenient example, however.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can we &quot;see&quot;, what the mapping does? Can we &quot;see&quot; what &lt;span class=&quot;math-container&quot;&gt;$f^5$&lt;/span&gt; does? Given a basis of the two eigenvectors, &lt;span class=&quot;math-container&quot;&gt;$(1,-1), (1,1)$&lt;/span&gt; we find the first one has eigenvalue 1, the second one eigenvalue 2. &#xA;Take the time to sketch &lt;span class=&quot;math-container&quot;&gt;$e_1, e_2$&lt;/span&gt; and their images on the blackboard. Students will hardly see any structure. Now, take the basis of eigenvectors and sketch their images. Students will immediateley &quot;see&quot; the mapping (hopefully). They then can describe &lt;span class=&quot;math-container&quot;&gt;$f^5$&lt;/span&gt; without calculations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As one application you might take the fibonacci numbers. Their growth can be described using matrix multiplication (see &lt;a href=&quot;http://en.wikipedia.org/wiki/Fibonacci_number#Matrix_form&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt;) and determining the eigenvector means determining an explicit formula for the implicitly defined series.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="143" LastEditDate="2018-09-30T15:54:28.040" LastActivityDate="2018-09-30T15:54:28.040" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="523" PostTypeId="2" ParentId="520" CreationDate="2014-03-19T16:01:15.857" Score="7" Body="&lt;p&gt;Another cool application is to figuring out whether a given matrix is positive definite.  This is useful for the second derivative test, for example.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-03-19T16:01:15.857" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="524" PostTypeId="2" ParentId="520" CreationDate="2014-03-19T16:15:02.193" Score="8" Body="&lt;p&gt;One of the standard examples is from stability analysis of dynamical systems. In the linear approximation you ask whether the $0$ solution is stable for &#xA;$$ X_{j} = A X_{j-1} $$&#xA;(discrete time) or&#xA;$$ \frac{d}{dt} X = B X $$&#xA;(continuous time). &lt;/p&gt;&#xA;" OwnerUserId="125" LastActivityDate="2014-03-19T16:15:02.193" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="525" PostTypeId="2" ParentId="487" CreationDate="2014-03-19T16:16:06.580" Score="7" Body="&lt;p&gt;Yes, I think most arguments are &quot;routine&quot; in the sense of being very similar to others in the relevant context, while now-and-then there is something &quot;out of the blue&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The latter can be deconstructed a bit, by admitting that all the goofy attempts that &lt;em&gt;failed&lt;/em&gt; were not shown, but only the one that succeeded. Thus, possibly, if all the other attempts were depicted as well, the eventual success might seem not so disconnected from the &quot;routine&quot; line of thought, or, at least, might not seem so magical with a preamble of many failures. (And one should also point out that &quot;failures&quot; are usually informative, and in this sense mathematics is experimental, rather than &quot;deductive&quot;, despite our collective mythology.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-19T16:16:06.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="526" PostTypeId="2" ParentId="520" CreationDate="2014-03-19T16:30:18.870" Score="12" Body="&lt;p&gt;I have always found the standard motivations for eigenvalues to be a little artificial. The primary application for eigenvalues is ultimately diagonalization and there are several ways you &lt;em&gt;could&lt;/em&gt; try to motivate diagonalization:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Taking large powers of matrices seems to be a popular one. But its not immediately obvious what this is used for.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An extension of taking large powers using diagonalization is using it to find solutions to linear recurrences. But chances are you have never talked about linear recurrences before this point. Introducing a concept and then a problem that the concept solves is a little backwards. The problem should come before the solution.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Steven Gubkin's suggestion of using it for the definiteness of a matrix is also a good one. But again, definiteness is often not part of a first course in linear algebra and many students study linear algebra before they do multivariable calculus where they need to test the definiteness of the Hessian.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Using it to solve differential equations is yet another example, and yet again, many students may not have seen differential equations yet.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My point is, all of the standard attempts to introduce diagonalization are application based and such attempts are heavily dependent on audience. Worse yet, when improperly executed the examples appear contrived. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Diagonalization is ultimately a structure theorem. Anyone can appreciate the fact a diagonal matrix is much easier to study than a non-diagonal matrix. Diagonalization then offers a power tool to study any property which is invariant through similarity by reducing a matrix which we know little about to a matrix which is zero everywhere except the diagonal. That is motivation enough in my experience, and you could always fall back onto one of the standard motivating examples if your students are still not convinced.&lt;/p&gt;&#xA;" OwnerUserId="97" LastActivityDate="2014-03-19T16:30:18.870" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="527" PostTypeId="2" ParentId="487" CreationDate="2014-03-19T17:05:39.683" Score="12" Body="&lt;p&gt;I think it helps to make it abundantly clear whether or not you would expect an average student in your course to come up with such a &quot;stroke of genius&quot;. If you're presenting something that might induce such a &quot;WTF?!&quot; moment, be very up front about it. I even like to put myself on the students' level if it's a particularly ingenious stroke, and will say something like, &quot;I think this is beautiful, and even I would never ever come up with this, but that doesn't keep me from appreciating its ingenuity!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whenever possible, I also try to give some indication of &lt;em&gt;why&lt;/em&gt; the person might have thought to do what they did. Maybe I can motivate/explain a few of the steps but not the bigger picture, or the entire sequence. For example, I've shown classes some of Euler's proofs of the Basel Problem, and those are monstrosities of ingenuity, but they have followable ideas within them. For instance, the mathematical legality of factoring sine as an infinite product sails over their heads, but they understand roots and linear factors. Likewise, the changes of variables Euler introduces are insanely clever, but the students can appreciate taking the log of both sides, because that converts the infinite product into an infinite &lt;em&gt;sum&lt;/em&gt;, and we're trying to prove something about a series ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, ultimately, context and expectations are key. Do you expect the students to follow the idea but not come up with it? Say so. Do you expect them to follow the idea and get a glimmer of why one would think to do it? Say so. Do you expect them to be able to come up with this on their own? Say so. All along, marvel with them at the ingenuity of others.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-19T17:05:39.683" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="528" PostTypeId="2" ParentId="520" CreationDate="2014-03-19T17:12:24.310" Score="18" Body="&lt;p&gt;I like Markov chains and Google PageRank (which is essentially a special kind of Markov chain).  It doesn't take very long to explain and motivate Markov chains and to argue that the probability distribution at time $n$ is the $n$'th power of the transition matrix times the distribution at time $0$.  You can then start talking about how to calculate powers of matrices.  You can draw pictures of Markov chains, which is more pleasant than just writing down matrices.  You can make a compelling argument for the usefulness of very large matrices, which can seem gratuitous in other contexts.  You can draw Markov chains that have repeated eigenvalues for obvious reasons of symmetry, which is instructive.  You can show that $1$ is always an eigenvalue, using an argument that brings together various other ideas in an interesting way.  You can make heuristic arguments about long term behaviour, and justify them algebraically with reference to eigenvectors of eigenvalue $1$.  It is not hard to convince the students that they should care about PageRank.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My notes covering this are at &lt;a href=&quot;http://neil-strickland.staff.shef.ac.uk/courses/MAS201/&quot;&gt;http://neil-strickland.staff.shef.ac.uk/courses/MAS201/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-19T17:12:24.310" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="530" PostTypeId="1" CreationDate="2014-03-19T18:05:17.470" Score="24" ViewCount="6133" Body="&lt;p&gt;Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can both be most effectively taught in order to maximize student's understanding of the use of the circle constant?&lt;/p&gt;&#xA;" OwnerUserId="180" LastEditorUserId="2083" LastEditDate="2015-03-16T13:05:30.743" LastActivityDate="2016-03-14T00:35:31.213" Title="Pi or Tau? How should the circle constant be taught?" Tags="|secondary-education|geometry|trigonometry|" AnswerCount="10" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="531" PostTypeId="2" ParentId="530" CreationDate="2014-03-19T18:38:16.700" Score="38" Body="&lt;p&gt;(Disclaimer: I peronally really don't care if one uses $\tau$ or $\pi$, both are just numbers for me.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;But I would strongly recommend to use $\pi$.&lt;/strong&gt; Why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In every technical literature, in many popular literature, the people always use $\pi$ (even worse: $\tau$ is used for different things than $\tau=2\pi$ which would confuse when reading those literature). If $\pi$ is not taught, the students would not be able to read older literature. Plus, it is confusing when they wanted to talk about it with parents, frieds, etc. In terms of consistency, everyone should know $\pi$. However, if you want you can also talk about $\tau$, but due to my experience students will mostly react like &quot;And now? It's just the double value of $\pi$?!&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar thing is there in physics, where the direction of electric currents was defined in an other way than the electron flow. Due to old technical literature and technical reports, manuals, etc. the definition is still valid (and will be for the future), compare &lt;a href=&quot;https://physics.stackexchange.com/questions/17109/why-is-the-charge-naming-convention-wrong&quot;&gt;https://physics.stackexchange.com/questions/17109/why-is-the-charge-naming-convention-wrong&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:17.873" LastActivityDate="2014-03-19T18:38:16.700" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="532" PostTypeId="2" ParentId="530" CreationDate="2014-03-19T18:38:52.900" Score="23" Body="&lt;p&gt;One should teach $\pi$. One might discuss that there is a choice that is made that is somewhat arbitrary, and there are also reasons for a different choice but I do not see this as that relevant to make much ado about this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It should perhaps also be noticed that there are two conflicting proposals for $\tau$. Eagle (1958) proposed $\pi/2$ and Palais (2001) proposed $2 \pi$ (for detailed references see &lt;a href=&quot;http://en.wikipedia.org/wiki/Tau_%282%CF%80%29#tau&quot;&gt;the Wikipedia page on Tau&lt;/a&gt;) and chance are there are others (at least others that proposed the same if not still other values).  This further illustrates that the situation is not that clear, though &lt;a href=&quot;http://www.math.utah.edu/~palais/pi.html&quot;&gt;&quot;$\pi$ is wrong!&quot;&lt;/a&gt; certainly makes interesting points. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, it is not quite clear &lt;em&gt;when&lt;/em&gt; a choice differing from the mainstream one should happen in the teaching. To have somewhat good reasons to motivate a choice different than $\pi$ one needs more advanced mathematics than one typically has available when first needing a cricle constant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Purely 'geometrically' the standard choice is at least as good, in my opinion. On the one hand, one can make an argument that area is the more straight forward notion than length, leading rather to $\pi$;  second; if one goes with length to compare to the diameter does also no harm in that context and again has some merit, it might even be more natural.&lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-03-19T18:38:52.900" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="533" PostTypeId="2" ParentId="520" CreationDate="2014-03-19T18:48:19.413" Score="6" Body="&lt;p&gt;One clear motivation is the relationship to optimization:  The largest eigenvalue of a real symmetric matrix is the maximum value of the (normalized) quadratic form&#xA;$$&#xA;    \frac{x^{\mathsf T} A x}{x^{\mathsf T} x}&#xA;$$&#xA;This is used for instance in spectral graph theory and elsewhere.&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="3272" LastEditDate="2015-08-23T15:08:59.340" LastActivityDate="2015-08-23T15:08:59.340" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="534" PostTypeId="2" ParentId="380" CreationDate="2014-03-19T19:00:12.833" Score="4" Body="&lt;p&gt;With problems such as $\sin^3 x^2$ I tell them to explicitly introduce the helper function $u(x)=x^2$ and rewrite it as $(\sin u(x))^3$ or, if necessary, more helper functions. That seems to help!&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2014-03-19T19:00:12.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="535" PostTypeId="1" CreationDate="2014-03-19T19:11:14.510" Score="29" ViewCount="3589" Body="&lt;p&gt;(This question developed from &lt;a href=&quot;https://matheducators.stackexchange.com/questions/380/revisiting-the-chain-rule/392?noredirect=1#comment680_392&quot;&gt;a comment&lt;/a&gt; in the thread &lt;a href=&quot;https://matheducators.stackexchange.com/questions/380/revisiting-the-chain-rule&quot;&gt;&quot;Revisiting the chain rule&quot;&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students know that WolframAlpha and other software/computational resources exist and will make use of them as they see fit. Is there anything we can change about the way we teach calculus (in particular, for this thread) to make use of these tools? Ideally, we would like to simultaneously:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Prevent students from committing academic dishonesty (however it be defined in your context) and, hopefully, discover when it occurs&lt;/li&gt;&#xA;&lt;li&gt;Show students the immediate benefits of using such resources: save time in computation, visualize a graph, check our work&lt;/li&gt;&#xA;&lt;li&gt;Show students the long-range benefits of using such resources: explore similar problems (e.g. sensitivity to initial conditions for diff eqs), experimenting with problems to identify patterns, getting a &quot;solution&quot; to discover a method that would yield such an answer&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm interested in examples where you have made use of WolframAlpha (or others) in class or on an assignment, and whether the students seemed to gain anything from it. Did you address some of the points I listed above? Are there other potential positive/negative aspects that you noticed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also somewhat interested in how you address questions like, &quot;Why do I need to learn this when WolframAlpha can do it for us?&quot;, but I'd prefer if the thread didn't devolve into &lt;em&gt;only&lt;/em&gt; discussing this topic. Instead, have you addressed such a query by showing students the benefits of bringing mathematical knowledge and intuition to a problem so as to make it tractable for software?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume all of this is in the context of a college calculus course.&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-12-24T20:53:57.060" Title="Students use WolframAlpha. Can we change calculus instruction to exploit it while discouraging 'cheating'?" Tags="|undergraduate-education|calculus|technology-in-education|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="536" PostTypeId="2" ParentId="414" CreationDate="2014-03-19T19:22:04.943" Score="2" Body="&lt;p&gt;Some simple examples you can try:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Maximize the product of $n$ positive numbers, given their sum is one.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2)Maximize entropy of $p_1,\dots, p_n$ (all $p_i&amp;gt;0$, summing to one) that is maximizing  $-\sum p_i \log p_i$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3)  More conditions: Maximizing entropy, but with the extra condition that &#xA;    $ \sum a_i p_i = \mu$.  You can also try introducing more conditions (variance?). The $a_i$s are known constants.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) You can find a few more good problems here: &lt;a href=&quot;https://www.math.ucdavis.edu/~thomases/W11_16C1_lec_2_4_11.pdf&quot; rel=&quot;nofollow&quot;&gt;https://www.math.ucdavis.edu/~thomases/W11_16C1_lec_2_4_11.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5)  A few more good problems here:  &lt;a href=&quot;http://homepages.math.uic.edu/~dcabrera/practice_exams/topics/lagrangemultipliers.html&quot; rel=&quot;nofollow&quot;&gt;http://homepages.math.uic.edu/~dcabrera/practice_exams/topics/lagrangemultipliers.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2014-03-19T19:22:04.943" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="537" PostTypeId="5" CreationDate="2014-03-19T19:29:41.493" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-19T19:29:41.493" LastActivityDate="2014-03-19T19:29:41.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="538" PostTypeId="4" CreationDate="2014-03-19T19:29:41.493" Score="0" Body="Questions on software (including web-based one) for teaching and related to teaching and the relevant hardware. Video  and audio for the classroom. Usage of electronic tools in teaching.      " OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-03-19T19:32:11.630" LastActivityDate="2014-03-19T19:32:11.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="539" PostTypeId="2" ParentId="150" CreationDate="2014-03-19T20:14:27.127" Score="6" Body="&lt;p&gt;Mathematics is the language of number, pattern, and form. In our society that depends so heavily on technology and inventions, every educated person needs to be able to read and understand that language. Advanced studies in every field of knowledge, from history to politics to language to business depend heavily on mathematics. Not knowing mathematics limits what you can study and learn.&lt;/p&gt;&#xA;" OwnerUserId="40" LastEditorUserId="40" LastEditDate="2014-03-20T18:56:02.670" LastActivityDate="2014-03-20T18:56:02.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="540" PostTypeId="2" ParentId="501" CreationDate="2014-03-19T20:48:49.287" Score="4" Body="&lt;p&gt;What we found at our university was that most students who took the placement test were taking it to try to get into a higher mathematics course than their ACT score would allow.  Taking these kids out of the mix, the next largest group would be the adults who are returning to school after some time off.  What we have come to realize, is that you don't do the students any favors by putting them in a class for which they are not prepared.  We have found that the ACT test does a fairly good job predicting mathematical ability.  For those reasons, we rarely let a student into a higher mathematics course than what they actually tested into.  If we do, we require the student (or parent of the student) to sign a paper that states &quot;with your placement test /ACT test scores of &lt;strong&gt;&lt;em&gt;_&lt;/em&gt;__&lt;/strong&gt; we disagree with your decision to take &lt;strong&gt;&lt;em&gt;_&lt;/em&gt;__&lt;em&gt;_&lt;/em&gt;__&lt;em&gt;&lt;/strong&gt; and suggest that you reconsider your decision and take &lt;strong&gt;&lt;/em&gt;__&lt;em&gt;_&lt;/em&gt;__&lt;em&gt;_&lt;/em&gt;__&lt;/strong&gt; instead, which is more in line with your placement test scores. &lt;/p&gt;&#xA;" OwnerUserId="203" LastActivityDate="2014-03-19T20:48:49.287" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="541" PostTypeId="2" ParentId="504" CreationDate="2014-03-19T21:04:07.767" Score="3" Body="&lt;p&gt;If all you want your students to do is write mathematics using LaTeX syntax using only default packages, you could circumvent many of the logistical problems of a full LaTeX installation altogether by using MarkDown and MathJax.  For instance, the online text editor &lt;a href=&quot;https://stackedit.io&quot; rel=&quot;nofollow&quot;&gt;StackEdit&lt;/a&gt; combines the functionality of MarkDown (which has almost no learning curve) with MathJax (which renders math symbols via LaTeX syntax) to produce a way to type mathematics that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Requires no software installation and is free to use&lt;/li&gt;&#xA;&lt;li&gt;Has a very easy learning curve (apart from math syntax in LaTeX)&lt;/li&gt;&#xA;&lt;li&gt;Allows students to learn only the math syntax of LaTeX rather than all of LaTeX at once&lt;/li&gt;&#xA;&lt;li&gt;Avoids writing and compiling complete TeX files (which has a fairly steep learning curve)&lt;/li&gt;&#xA;&lt;li&gt;Can be used offline since it saves files in the web browser's cache&lt;/li&gt;&#xA;&lt;li&gt;Allows files to be downloaded as text (MarkDown .md files) or pdf files&lt;/li&gt;&#xA;&lt;li&gt;Can synchronize documents with Google Drive and DropBox (and can therefore be shared easily)&lt;/li&gt;&#xA;&lt;li&gt;Automatically saves documents every few minutes&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If your goal is to teach them how to install and use a full LaTeX system, then the lightweight solution proposed here is not what you're looking for.  Note: there are also other MarkDown + MathJax editors out there that are packaged as software to be installed on a computer hard drive rather than used in the cloud.&lt;/p&gt;&#xA;" OwnerUserId="204" LastActivityDate="2014-03-19T21:04:07.767" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="542" PostTypeId="2" ParentId="530" CreationDate="2014-03-19T21:30:33.623" Score="16" Body="&lt;p&gt;I cannot imagine a context where it would benefit a student to know about $\tau$ over and above knowing about $\pi$.  Every text book a student will ever encounter will exclusively talk about $\pi$. Every calculator a student will ever use will have the constant $\pi$ pre-programmed in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although there are perfectly lovely arguments to show that $\tau$ is a nicer constant, it is just $2\pi$, so it really doesn't matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At most I would introduce $\tau$ to higher ability students as an interesting aside, but I cannot see the value in actually teaching it as a useful constant.&lt;/p&gt;&#xA;" OwnerUserId="221" LastEditorUserId="169" LastEditDate="2014-03-20T10:52:10.677" LastActivityDate="2014-03-20T10:52:10.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="543" PostTypeId="2" ParentId="36" CreationDate="2014-03-19T21:32:37.913" Score="11" Body="&lt;p&gt;If you type a function into a Google search, you get a pretty nice graph.  In order to get interactive 3d graphics, you will need a WebGL enabled browser (with WebGL actually enabled, of course).  Try these for starters:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.google.com/#q=y%3Dsin%28x%29%2Fx&quot;&gt;y=sin(x)/x&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;z=-x^2+y^2&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="204" LastActivityDate="2014-03-19T21:32:37.913" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="544" PostTypeId="2" ParentId="530" CreationDate="2014-03-19T21:41:57.910" Score="32" Body="&lt;p&gt;I feel that it is perhaps a little irresponsible to teach $\tau$ instead of $\pi$. As a first introduction, it is the norm which should be taught: teaching a rare alternative to $\pi$ only serves to confuse students, especially when almost all available resources use $\pi$ instead of $\tau$. Imagine a student's confusion when they see $\tau$ in class and $\pi$ everywhere else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally don't even know why the $\tau$ vs $\pi$ topic is even debatable; any miniscule advantage (if there even is an advantage) in using $\tau$ over $\pi$ is heavily offset by the work needed to establish an entirely new cultural norm.&lt;/p&gt;&#xA;" OwnerUserId="97" LastActivityDate="2014-03-19T21:41:57.910" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="545" PostTypeId="2" ParentId="520" CreationDate="2014-03-19T23:30:23.983" Score="6" Body="&lt;p&gt;Suppose for simplicity that the matrix $A$ of some linear mapping $\mathbb{R}^n \to \mathbb{R}^n$ is symmetric (hence diagonalizable) and all the eigenvalues $\lambda_1,\ldots,\lambda_n$ are different. Then, there is an orthogonal base such that in that base&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{$A$ behaves like scaling with factors $\lambda_1,\ldots,\lambda_n$ respectively.}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, the eigenvalues tell us how the mapping looks like, is it a bit like a symmetry (e.g. when $\lambda_i &amp;lt; 0$), or perhaps like a projection (e.g. when $\lambda_i = 0$) or maybe a contraction (e.g. $\lambda &amp;lt; 1$), etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is kind of similar to the decomposition theorem for abelian groups, i.e. it lets you simplify the mapping into one which can be more easily handled, e.g. has no weird cross-dependencies and each part is by itself well understood. No wonder it is so important!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-20T01:50:24.853" LastActivityDate="2014-03-20T01:50:24.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="546" PostTypeId="2" ParentId="520" CreationDate="2014-03-20T01:12:10.137" Score="10" Body="&lt;p&gt;For a real showcase, I recommend a scenario where resonance frequencies play a role.&lt;/p&gt;&#xA;&lt;p&gt;Suspension bridges are real-world objects which are delicate enough that soldiers are usually not allowed to march over bridges (the German traffic law StVO states this in § 27 Abs. 6). The reason for this is that small, regular excitations with a certain frequency will cause the bridge to oscillate substantially. This can be seen in &lt;a href=&quot;http://www.youtube.com/watch?v=eAXVa__XWZ8&quot; rel=&quot;nofollow noreferrer&quot;&gt;this video&lt;/a&gt; of the Millennium bridge in London.&lt;/p&gt;&#xA;&lt;p&gt;In order to calculate these so-called resonance frequencies, one can try to describe such a bridge with the help of a differential equation; in the easiest case as a vibrating string. When you solve the corresponding differential equation, you will get a matrix whose eigenvalues are exactly the resonance frequencies.&lt;/p&gt;&#xA;&lt;p&gt;Depending on the knowledge on differential equations one may or may not start to work with these.&lt;/p&gt;&#xA;&lt;p&gt;Note that the &lt;a href=&quot;http://www.youtube.com/watch?v=CsKKDLKYsVU&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galloping Gertie&lt;/a&gt; did not collapse due to resonance, but because of &lt;a href=&quot;http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_%281940%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aeroelastic flutter&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you don't like bridges, playing with electrical circuits might also provide nice visible or audible results.&lt;/p&gt;&#xA;" OwnerUserId="108" LastEditorUserId="470" LastEditDate="2021-05-20T22:13:26.633" LastActivityDate="2021-05-20T22:13:26.633" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="547" PostTypeId="2" ParentId="508" CreationDate="2014-03-20T01:48:34.430" Score="3" Body="&lt;p&gt;&lt;strong&gt;Extra credit often leads to problems with curves.&lt;/strong&gt;  I've heard dialogues like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  &lt;em&gt;I got 88% overall, I think I deserve an A.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  &lt;em&gt;But lots of other students got 90's by answering extra credit.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here the grade without extra credit is being compared against a standard of grades with extra credit.  So the credit was not &quot;extra&quot; at all, but just a badly-labeled part of the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To avoid this, keep two sets of grade books, and don't look at the one with extra credit until you've set grade cutoffs from the one without extra credit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, more easily and more transparently, don't give extra credit at all.  Your suggestions a, b, and c sound good for short required written assignments.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-20T01:48:34.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="548" PostTypeId="2" ParentId="270" CreationDate="2014-03-20T02:14:33.983" Score="6" Body="&lt;p&gt;In addition to the other useful answers... there is a general &quot;problem&quot; that computers, given the long-known algorithms, ironically many not really human-executable, can do almost all of the algorithmic parts of k-12 and undergraduate and some once-upon-a-time-grad-level math. So it seems silly to struggle to teach kids to do things that machines can do better (the superiority of calculators at arithmetic was just the beginning...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;genuine&lt;/em&gt; sociological and pedagogical problem is that humans are not getting smarter or more capable to keep ahead of things that can be implemented algorithmically on machines. And, to add insult to injury, the same kids that had trouble getting the algebra right in by-hand-calculus will typically have similar troubles getting the input right to have software do it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, math software mostly helps the most able people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Returning to some aspect of the question at hand: yes, it is ostensibly silly to teach algorithms that are far-better executed by machines. (I don't know whether anyone's bothered to write Latin or classical-Greek or classical-Hebrew grammar-checkers...) Ideally, yes, it would be better to &lt;em&gt;use&lt;/em&gt; the machinery and allow kids to look at higher-level issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, sadly, those higher-level issues are intellectually more taxing than (pointless, yes) execution of algorithms. I am disappointed to report that my experience with trying such things generates such reports. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, the kids who'd make sign errors or not be able to do middle-school algebra will also be flummoxed by trivial computer-language syntax, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, it is better to be honest, to be genuine, to have math courses address real issues rather than traditional-fake. But we must brace ourselves for the disappointment that there'll still be a cadre of kids who can't do it... now for just-slightly-different reasons than the reasons for which they couldn't do hand-executed algorithms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, one should not insist on hand-execution of algorithms, but giving that up and doing better things will make it harder, overall, for the students. But that doesn't mean we shouldn't move in that direction. A tough situation, quite seriously, in my opinion.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-20T02:14:33.983" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="549" PostTypeId="1" AcceptedAnswerId="879" CreationDate="2014-03-20T03:13:24.307" Score="22" ViewCount="472" Body="&lt;p&gt;I read an &lt;a href=&quot;http://mathbabe.org/2014/01/20/the-coming-calculus-mooc-revolution-and-the-end-of-math-research/&quot;&gt;interesting article&lt;/a&gt; earlier this year about the future of mathematics. It claimed that math research is possible in the United States because of service courses (especially calculus). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus is by far the most common course taught at research universities, so their argument holds some water. It's hard to make this into one specific question, but here goes (feel free to suggest alternative formulations):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To what extent is the budget of a research department in the United States correlated to the number of students taking calculus each semester?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-26T20:11:37.270" Title="To what extent is math research in the United States subsidized by calculus teaching?" Tags="|calculus|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="550" PostTypeId="1" CreationDate="2014-03-20T03:40:33.260" Score="16" ViewCount="350" Body="&lt;p&gt;The most terrifying thing about teaching my own class is &lt;em&gt;'do I actually know enough to effectively teach students?'&lt;/em&gt; I know this is anxiety brought on by a lack of experience but finding the tools to deal with crisis moments before they occur is key to handling any situation well. To that end, I was inspired by a comment on &lt;a href=&quot;https://matheducators.stackexchange.com/questions/506/what-do-you-do-when-you-realize-mid-lecture-that-your-lesson-plan-is-not-working&quot;&gt;Brian Rushton's question&lt;/a&gt; about damage control during a lesson.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;What do you do when you realize two weeks into the course that your course plan is not working?&lt;/strong&gt; – &lt;a href=&quot;https://matheducators.stackexchange.com/users/90/joel-david-hamkins&quot;&gt;Joel David Hamkins&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I know that (in the US) most new teachers have an older teacher assigned to mentor them and all course and lesson plans will have to be approved, but I could see a situation in which I have a fantastic idea for how a course should progress and it simply falls apart. Some examples of issues could be:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Students do not have the necessary background knowledge&lt;/strong&gt; - This shouldn't be the case, but if a majority of the class doesn't understand basic algebra, it would make it extremely difficult to teach Pre-calculus.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Students are bored by the material&lt;/strong&gt; - I feel that it is the instructor's responsibility to engage the students but if the material itself is boring because it is not challenging, what could be done?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The schedule is completely inaccurate&lt;/strong&gt; - Again, I know that as a new teacher, any syllabus &lt;em&gt;should&lt;/em&gt; be approved before I'm allowed to teach it, but what if after two weeks, I am 3 or 4 days behind?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These are only a few examples of issues I could see coming up and not meant to be a comprehensive list of things that require damage control. I'm hoping for an overall concept that can then be applied to specific situations.&lt;/p&gt;&#xA;" OwnerUserId="151" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-24T18:03:17.150" Title="How do you fix a broken course plan?" Tags="|course-design|classroom-management|teacher-preparation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="551" PostTypeId="2" ParentId="520" CreationDate="2014-03-20T04:25:14.473" Score="4" Body="&lt;p&gt;Markov chains have already been mentioned, that is a nice direction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, zero is an eigenvalue iff $A$ is singular. I'm not sure I'm creative enough to take this as a start point for motivating the eigenvector concept. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way to discover them is simply by taking a point $x_o$ and a matrix $A$ and calculating $x_1=Ax_o$,... $x_k = Ax_{k-1}$. Since we're looking for a visualization, you want to use a $2 \times 2$ matrix. You'll find that the points are drawn into the eigen-axes or, if it happens to have complex e-value it spirals. This requires not too much curiousity and it immediately reveals real and complex e-values existence for real $2 \times 2$ matrices. See page 221-223 of &lt;a href=&quot;http://www.supermath.info/math321v2.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.supermath.info/math321v2.pdf&lt;/a&gt; for my not so pretty attempt at this discussion/discovery section.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm editing those notes in the past week or so when it hit me, it would also be good to emphasize the simple geometric example of the axis-vector of a rotation. Perhaps it is interesting that the axis may reside in the space $n=3$ or may not $n=2$. Again, the necessity of both real and complex eigenvalues is revealed by a topic which is a natural curiousity (rotations in $n$-dimensions ). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this said, the application of Jordan chains to unravel the matrix exponential and assemble the general solution to $x'=Ax$ is to my taste a powerful showcase. However, the intended audience needs some desire and/or familiarity with ODEs to appreciate it. While we're at it; I have found the connection between generalized eigenvectors and the double-root solution to be a satisfying justification for studying them in years past. I mean, I want to be able to solve all the ODEs with constant coefficients, not just the problems which happen to be diagonalizable. I wish more of us thought generalized e-vectors were worth teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have a physicsy audience it's easier. Eigenvectors are the observable states. Eigenvalues are the values of energy, momentum, spin etc... of the system which the states represent. That's fundamental physics. The question of simultaneous diagonalization becomes interesting as it is the question of being able to simultaneous know the values of a pair of observables for a given pair of physical quantities. For a non-example, momentum and position $[x,p] \neq 0$ hence we cannot find simultaneous eigenstates of $x$ and $p$. This leads to the famous &lt;a href=&quot;http://en.wikipedia.org/wiki/Uncertainty_principle&quot; rel=&quot;nofollow&quot;&gt;Heisenberg Uncertainty Principle.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-03-20T04:25:14.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="552" PostTypeId="1" AcceptedAnswerId="566" CreationDate="2014-03-20T04:31:32.053" Score="13" ViewCount="942" Body="&lt;p&gt;Some professors assign online homework in lower level courses, such as through services like &lt;a href=&quot;http://webwork.maa.org/&quot;&gt;WeBWorK&lt;/a&gt; or &lt;a href=&quot;http://www.webassign.net/&quot;&gt;WebAssign&lt;/a&gt;. Online homework has some clear advantages for instructors. However, in my experience, such systems receive largely negative feedback from students. Further, some instructors worry that online homework may not be as effective as paper homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is there any research that indicates online homework is better or worse for students?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="52" LastEditorUserId="143" LastEditDate="2014-03-20T08:08:57.337" LastActivityDate="2014-03-21T00:57:39.193" Title="online homework vs paper homework" Tags="|undergraduate-education|homework|technology-in-education|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="553" PostTypeId="1" AcceptedAnswerId="560" CreationDate="2014-03-20T09:07:06.577" Score="13" ViewCount="487" Body="&lt;p&gt;There are some intelligent jokes also explaining (in a very extreme way) how mathamticas works, for exameple&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A mathematician and a physicist are sitting in the teachers lounge. Suddenly the furniture catches fire. The physicist is able to handle to situation. Both shocked, the physicist makes some coffee for both of them. The next day, the mathematician has the desire for some coffee and sets the furniture on fire, thus reduced the problem the previously solved one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I often tell such a joke when it comes to a situation when either you reduce a problem to a big theorem or a previously solved one or if you want to motivate to do something directly without the use of a big machinery. Most of the students are normally stunned or look very confused, but some of them are very impressed since the know how the things work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is it a good thing to tell such jokes?&lt;/strong&gt; Or are you even doing bad since this is confusing and anti-motivating (&quot;Oh, those weird mathematicians. I don't want to get to &lt;strong&gt;that&lt;/strong&gt; point!&quot;)?&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="143" LastEditDate="2015-02-14T20:24:57.230" LastActivityDate="2015-02-16T14:33:46.090" Title="Use of mathematical humor suitable for motivation/explaination?" Tags="|student-motivation|general-pedagogy|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="554" PostTypeId="1" CreationDate="2014-03-20T09:31:13.487" Score="10" ViewCount="601" Body="&lt;p&gt;I'm considering increasing the online component of my mathematics courses (I teach at the university level).  I'd like to know what my options are to avoid reinventing the wheel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The features I would like are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A unified system&lt;/strong&gt;.  I could cobble together a system using a bit of this and a bit of that, but it would be better if it were all under the same roof to make it easier for students to navigate.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The ability to put actual material and quizzes in the system&lt;/strong&gt;.  The material might be in the form of a video or a webpage.  I'm using &quot;quiz&quot; quite generically with nothing beyond &quot;Questions to be answered&quot; in mind.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The ability to get feedback both for the students and the instructor&lt;/strong&gt;.  The latter is crucial in this endeavour.  I don't simply want to put stuff on the web and leave it there, I want to know how it is being used and more.  Ideally, there would be the opportunity for students to leave comments (for the instructor) as they go through the material and for the instructor to get summaries of the results of the quizzes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Easy to do maths&lt;/strong&gt;.  There are systems that do all the above, but maths support is somewhat lacking.  This is not just about being able to &lt;em&gt;write&lt;/em&gt; maths, in the quizzes then the students should be able to enter mathematical answers and any answer-checking needs to be mathematically robust.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I realise that this is something of a &quot;big-list&quot; question - though I suspect that the list is actually quite small - but it is also something that is hard to find out about, particularly with regard to the last criterion.  I'm particularly interested in systems that have been used by mathematicians and what they found most useful and most frustrating about the system.&lt;/p&gt;&#xA;" OwnerUserId="112" LastEditorUserId="1892" LastEditDate="2014-07-09T19:41:36.223" LastActivityDate="2014-07-09T19:41:36.223" Title="What course web software is usable by mathematicians?" Tags="|technology-in-education|software|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="555" PostTypeId="2" ParentId="508" CreationDate="2014-03-20T09:48:25.663" Score="3" Body="&lt;p&gt;I dislike &quot;extra credit&quot; for many reasons, many of which have been mentioned already. But mostly, I think that the argument of &lt;em&gt;Students like bonus problems...I think they [with the reward of extra credit] encourage doing more than was necessary while studying&lt;/em&gt; (which I believe to be roughly the standard argument, but this is (essentially) a quote from Tim Seguine in the comments) just isn't good enough. In the real world money is not the only motivation, and I would not like it if, for example, the police force was driven by money rather than a desire to make where I live a safe place. Similarly, if we replace &quot;money&quot; with &quot;final grade&quot; then I do not think it is healthy to motivate students solely by their final grade. I believe an alternative motivation can and should be used in place of &quot;extra credit&quot;. For example: Chocolate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More concretely, I tell my students that if they can find an error in my notes or in my problem sheets then they should come to my office and I will reward them with chocolate (in their answer, vonbrand says that they reward this with extra credit). This means that they can argue all they like about whether or not they deserve chocolate (they point out the errors, but they do it in lectures!), but ultimately the arguments are in good humour as they are arguing over...a piece of chocolate. Additional &quot;extra credit&quot; questions which are meant to motivate the students now become competitions where the winners get, say, a not-quite-the-cheapest box of chocolates. Now, because the rewards are chocolate everything is fun while the students remain motivated (well, enough of them remain motivated to keep me happy), so this all adds to the student experience. I believe that if I replaced &quot;chocolate&quot; with &quot;extra credit&quot; then there would be no improvement in the student's grasping of the subject and I would enjoy lecturing &lt;em&gt;much&lt;/em&gt; less.&lt;/p&gt;&#xA;" OwnerUserId="169" LastEditorUserId="169" LastEditDate="2014-03-20T10:06:48.233" LastActivityDate="2014-03-20T10:06:48.233" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="556" PostTypeId="2" ParentId="554" CreationDate="2014-03-20T11:15:26.750" Score="4" Body="&lt;p&gt;You are in luck!  I am part of a team creating &quot;Ximera&quot;, a system for converting latex documents into interactive web based learning courses.  We are still sort of at the early stages, but the homepage is here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://ximera.osu.edu/&quot; rel=&quot;nofollow&quot;&gt;http://ximera.osu.edu/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the link to a multivariable calculus course we are running in this system is here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://ximera.osu.edu/course/kisonecat/m2o2c2/course/&quot; rel=&quot;nofollow&quot;&gt;http://ximera.osu.edu/course/kisonecat/m2o2c2/course/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Features: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Questions with solutions which are functions or matrices of functions&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Multiple choice questions&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Free response questions, where they have access to a Latex editor.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Python coding exercises, which have machine checked answers (Ex: implement power iteration to approximate eigenvalue)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hints for questions, which can themselves contain more questions (for a more &quot;socratic&quot; learning experience)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Forum attached to each page&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ability to view source of each page, along with a link to github.  We have already had several students correct typos and such, in the first week!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Easy to embed youtube videos&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Currently the whole system is not really user friendly enough that I would recommend you start using it now.  Writing the content is fine, but we have not streamlined the process of taking the latex files and converting them into html pages.  I would imagine that the whole system should be streamlined enough for general use by the beginning of the next academic year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Multivariable calculus course will cover the following topics, just in case you are interested enough to follow its development:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Week 1: Linear transformation, matrices, dot product&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Week 2:  Derivative as a linear transformation, partial derivatives,gradient, chain rule&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Week 3: Abstract vector spaces, subspaces, eigenvectors&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Week 4:  Adjoints, bilinear maps, real spectral theorem&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Week 5: Second derivative as a bilinear map, second derivative test, lagrange multipliers&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Week 6: Multilinear functions, higher derivatives as multilinear functions, full taylors theorem&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2014-03-20T11:32:20.930" LastActivityDate="2014-03-20T11:32:20.930" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="557" PostTypeId="1" CreationDate="2014-03-20T11:59:04.397" Score="12" ViewCount="627" Body="&lt;p&gt;Before I begin this question properly I want to say that I feel responsible for my students, and I want them to do as well as they can do. I mention this because my question is about technology and distractions. I feel that many students are distracted by technology: they would sit in class texting and checking facebook if allowed. The issue that I (and presumably many others) face then is that students cannot reach their potential because they are distracting themselves. This frustrates me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What can a lecturer do to stop students being distracted by technology in their lectures?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are two obvious approaches: ban technology, or pretend it is not an issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Ban technology:&lt;/strong&gt; Pretend that we live in y2k. The main advantage is that the students cannot be distracted. Some disadvantages are that if students want to look at my slides at their own pace then they have to print them out (I have no scrolling boards so my lectures are frustratingly slide-heavy), which is far from ideal, and that some students who will not be engaged anyway will become disruptive (this can happen with or without taking attendance). Finally, &quot;banning&quot; technology doesn't do much to foster a happy or encouraging learning environment, as the students claim that they are grown ups and can decide for themselves while the ban needs to be enforced which generates fear (for some value of &quot;fear&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Ignore technology:&lt;/strong&gt; Pretend that noone is checking facebook. The advantage is that students can view the slides and take notes on their tablets. Technology is good and they are embracing it! Of course, many students will become distracted and I find this frustrating. Moreover, often students will become distracted by other people using technology, which is something I certainly don't want to happen (for example, during my undergrad I remember a very forward-thinking student bringing his laptop to a lecture. He sat in the front row playing Quake. The lecturer didn't notice, but everyone else was transfixed by this guys wonderful Quake-playing abilities. He was rather good...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Currently, I have a sort of middle ground where I allow technology but ask frequent questions and get students to work on exercises at regular intervals. This helps distract them from their technology and lets me patrol for facebook users. But I still think that this is far from ideal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I would love to hear any good ideas for how to stop students being distracted by technology in maths lectures.&lt;/p&gt;&#xA;" OwnerUserId="169" LastEditorUserId="2083" LastEditDate="2019-01-23T19:45:14.340" LastActivityDate="2019-01-23T23:04:34.337" Title="Technology and distractions" Tags="|undergraduate-education|technology-in-education|classroom-management|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="558" PostTypeId="2" ParentId="484" CreationDate="2014-03-20T12:11:31.447" Score="-1" Body="&lt;p&gt;With so many applications to teach, so many numbers to calculate, so many functions to graph, so many applications....  No, I wouldn't find it beneficial to discuss rigorous limits very much; they would be low-priority for me.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-03-20T12:24:22.760" LastActivityDate="2014-03-20T12:24:22.760" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="559" PostTypeId="2" ParentId="553" CreationDate="2014-03-20T12:19:16.890" Score="4" Body="&lt;p&gt;If you're good at telling jokes, you might as well tell some mathematical ones in class:  it will keep the students engaged.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're not good at telling jokes -- and the process of becoming a math teacher selects for quite a different set of skills -- then there's no point.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-20T12:19:16.890" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="560" PostTypeId="2" ParentId="553" CreationDate="2014-03-20T13:22:39.567" Score="7" Body="&lt;p&gt;I find jokes are useful for helping students to remember things. For example, the joke that Steve Gubkin tells in the comments will help the students remember how to check that &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is the inverse of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. I am sure if you pick your fact then there is a joke for it. My favorite remembering-joke is the following.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Lecturer: What is &lt;span class=&quot;math-container&quot;&gt;$\int\frac1{\text{cabin}}d\text{cabin}?$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Students: It is a log cabin. Ha ha.&lt;/p&gt;&#xA;&lt;p&gt;Lecturer: No. It is a holiday home. Log cabin plus c...&lt;/p&gt;&#xA;&lt;p&gt;Students: Groan.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And yet, they will want to tell someone this horrible joke which their their lecturer told them. In order to tell the joke they have to remember how to integrate &lt;span class=&quot;math-container&quot;&gt;$1/x$&lt;/span&gt;. As an added bonus, they might even remember to write &lt;span class=&quot;math-container&quot;&gt;$+c$&lt;/span&gt; in their exam...&lt;/p&gt;&#xA;" OwnerUserId="169" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-23T08:29:38.080" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="561" PostTypeId="1" AcceptedAnswerId="622" CreationDate="2014-03-20T18:19:26.283" Score="25" ViewCount="1042" Body="&lt;p&gt;When teaching Calculus, one thing that many teachers emphasize is that the variable of integration is a `dummy variable' that is unimportant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Around the same time, we introduce integrals with variables in the limits of integration. Students are told not to use the same variable as a limit and as a variable of integration, i.e.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\int_0^x x^2 dx$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is a concrete way to show students that this is a bad idea, without contradicting the fact that the variable of integration is a dummy variable?&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2019-09-08T07:12:15.300" Title="How can you explain to students that they should not use the same variable in an integrand and in the limits of integration simultaneously?" Tags="|calculus|examples|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="562" PostTypeId="1" AcceptedAnswerId="569" CreationDate="2014-03-20T18:24:41.720" Score="32" ViewCount="1854" Body="&lt;p&gt;There are many, many ways of writing the derivative of a function $y=f(x)$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{d}{dx}y, \frac{dy}{dx},\frac{d}{dx}f(x), \frac{df}{dx}, \dot y, D_x f,f',y',f'(x),f_x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students often feel uncomfortable switching back and forth between the various forms. And in fact, it is not necessary to use more than one form of notation in most beginning calculus courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, knowing more than one form of notation may be helpful in future classes.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Should an instructor encourage their students to use different forms of derivative in their writing?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastActivityDate="2020-02-12T00:18:01.373" Title="Should students be asked to use more than one notation for the derivative in an introductory calculus class?" Tags="|calculus|notation|" AnswerCount="10" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="563" PostTypeId="2" ParentId="554" CreationDate="2014-03-20T18:32:03.950" Score="6" Body="&lt;p&gt;Have you considered the online homework system &lt;a href=&quot;http://webwork.maa.org/wiki/Main_Page&quot;&gt;WeBWorK&lt;/a&gt;?  From the &lt;a href=&quot;http://webwork.maa.org/&quot;&gt;MAA's WeBWorK site:&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;WeBWorK is an open-source online homework system for math and sciences courses. WeBWorK is supported by the MAA and the NSF and comes with a National Problem Library (NPL) of over 20,000 homework problems. Problems in the NPL target most lower division undergraduate math courses and some advanced courses. Supported courses include college algebra, discrete mathematics, probability and statistics, single and multivariable calculus, differential equations, linear algebra and complex analysis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is how the features of WeBWorK measure up with your desired features: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A unified system:&lt;/strong&gt; WeBWorK can be &lt;a href=&quot;http://webwork.maa.org/wiki/Moodle_Integration&quot;&gt;integrated with Moodle&lt;/a&gt; and &lt;a href=&quot;http://webwork.maa.org/wiki/Blackboard&quot;&gt;Blackboard&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The ability to put actual material and quizzes in the system:&lt;/strong&gt; WeBWorK now has a &lt;a href=&quot;http://webwork.maa.org/wiki/National_Problem_Library&quot;&gt;problem library of over 25,000 questions&lt;/a&gt; (the MAA quote above is dated) and it is easy to create homework sets and &lt;a href=&quot;http://webwork.maa.org/wiki/Gateway_Tests_and_Quizzes&quot;&gt;timed quizzes&lt;/a&gt;.  You can also &lt;a href=&quot;http://webwork.maa.org/wiki/Category%3aAuthors&quot;&gt;write your own questions&lt;/a&gt; and easily &lt;a href=&quot;http://webwork.maa.org/wiki/HtmlLinks&quot;&gt;include links to videos, web pages, etc.&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The ability to get feedback both for the students and the instructor:&lt;/strong&gt;  The WeBWorK statistics, student progress, and gradebook pages offer students and instructors access to student progress in real time.  At the bottom of each homework problem, students can click an &quot;email instructor&quot; button to share their work and questions about the problem with their instructors.  It is also possible to write &lt;a href=&quot;http://webwork.maa.org/wiki/Essay_Questions&quot;&gt;essay questions&lt;/a&gt; that allow students to input mathematics using LaTeX syntax, but these questions require a human to grade them.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Easy to do maths:&lt;/strong&gt;  WeBWorK was originally built by mathematicians at the University of Rochester (Michael Gage and Arnold Pizer) for mathematics students and has expanded over the past ~15 years to meet other STEM education needs.  In my opinion, WeBWorK has the most reliable and robust answer checkers thanks to the hard work of Davide Cervone at Union College (who also created MathJax).  Since WeBWorK allows any answer that uses standard math syntax (and gives students feedback when their syntax is incorrect), students who have worked with calculators before find the transition to WeBWorK very easy.  The current WeBWorK user interface is good and easy enough to use, but it will be improved in the near future (hopefully available by Fall 2014) so that it becomes more intuitive and streamlined.  Also, for instructors who do not want to put together their own homework sets from WeBWorK's problem library, WeBWorK has begun to put together &lt;a href=&quot;http://webwork.maa.org/wiki/Model_Courses&quot;&gt;model courses&lt;/a&gt; that are essentially pre-packaged homework sets (which can of course be modified to suit the needs of a course).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As a WeBWorK developer and long time user, I have a strong positive bias for it.  I was forced to use another commercial online homework system for one semester, and it was clearly inferior to WeBWorK in terms of quality of questions available and reliability / robustness of answer checking.  Students of mine who have used WeBWorK and another online homework system have universally said that WeBWorK was better.&lt;/p&gt;&#xA;" OwnerUserId="204" LastActivityDate="2014-03-20T18:32:03.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="564" PostTypeId="2" ParentId="535" CreationDate="2014-03-20T18:48:07.430" Score="5" Body="&lt;p&gt;If use of  is a blessing or a curse depends entirely on what you want them to learn. If you want them to find indeterminate integrals by hand, using a slew of tricky substitutions, it is clearly a curse (sort of, having something against which to check results is valuable in itself). If you want them to know when to compute an integral, and do something with the result, the mere computation better be taken over by the computer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I contend we should see how to concentrate on the second aspect. Leave the &quot;find/apply clever tricks to compute ...&quot; to people interested in symbolic computation.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-20T18:48:07.430" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="565" PostTypeId="2" ParentId="562" CreationDate="2014-03-20T18:59:46.740" Score="15" Body="&lt;p&gt;Yes, students will encounter several forms of derivative notation, inside and outside of classes.  Therefore, a calculus class should give them fluency in reading these.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether they stick to one notation in their writing is not a big deal.  More importantly:  the lectures and questions in calculus classes should use more than one notation.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-03-25T18:24:41.857" LastActivityDate="2014-03-25T18:24:41.857" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="566" PostTypeId="2" ParentId="552" CreationDate="2014-03-20T19:04:49.557" Score="9" Body="&lt;p&gt;Here is one study from &lt;a href=&quot;http://www.math.rutgers.edu/~weibel/webwork.html&quot;&gt;Rutgers University&lt;/a&gt; that analyzes the effectiveness of WeBWorK, and there may be other useful studies or results available &lt;a href=&quot;http://webwork.maa.org/wiki/Assessment&quot;&gt;on this webpage&lt;/a&gt; and &lt;a href=&quot;http://webwork.maa.org/wiki/In_the_news&quot;&gt;this webpage&lt;/a&gt;.  If you are interested in learning more about the effectiveness of WeBWorK in particular, you should contact Vicki Roth at the University of Rochester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, the most effective homework systems, whether paper or online, get students to actually do their homework and persist until they have achieved mastery of a topic.  Anecdotally, after an initial adjustment period, my students who have used WeBWorK actually appreciate that it gives them instantaneous feedback (they love seeing the green bar that says and answer is correct) and allows them as many attempts as they need to achieve mastery (before the due date and on open-ended questions, of course).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are things that instructors can do to make online homework more effective for students.  For instance, instructors should encourage students to keep a homework notebook where they write down complete answers that will be useful for studying later.  Also,  instructors need to choose online homework questions carefully so that they are relevant, authentic, and on-topic.  Online homework questions of low quality that were assembled by someone else are often not aligned with course and do not help students learn (and produce students who are dissatisfied with online homework).&lt;/p&gt;&#xA;" OwnerUserId="204" LastActivityDate="2014-03-20T19:04:49.557" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="567" PostTypeId="2" ParentId="562" CreationDate="2014-03-20T19:18:29.320" Score="8" Body="&lt;p&gt;&lt;strong&gt;Short answer:&lt;/strong&gt; I don't think that is a good idea to encourage students to use different notation on their own. But I would encourage the lecturer and the TAs to do so and tell the students different notations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;More detailed answer:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For many students, the topic itsself is hard enough. I don't think you should distract them from the most important things for notation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But you can tell them, that there are different ways to write down things and the most important point is the notation should be consistent. For example, I've seen it several times that at the point of the definition, there were several methods mentioned to write down the new object (here: the derivative) and then in the lecture &lt;strong&gt;one&lt;/strong&gt; of these notations was used, but on the exercices another notation was used (this is - as I later learned - due to the fact that someone else was makeing exercises, but he hasn't talked about notation with the professor). Students will learn that different people use different notation and every notation has some advantages and disadvantages. Of course, it could be that students are confused at the first point (&quot;What does that strange symbol in the exercises mean?&quot; - You can explain it and you can write more than a symbol in the first exercise where it appears. Later on it should be clear). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;You should encourage the students to make up their minds what a good notation is to their understanding.&lt;/strong&gt; And the students should use that notation consequently as long as they are satisfied. (This changes from time to time. I for example was a big fan of the $\frac{\partial}{\partial x}f$ until I took a class about partial differential equations where I discovered that my old notation was a nightmare. Than I changed my whole notation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think, it is important that students should be flexible and open-minded to notation: It can be that in every lecture there is a different notation, in books and later in papers, people use different notation (in bad ones, the notation chances from chapter to chapter). But one should never overemphasize the discussion of notation since there are more important things to teach. &lt;strong&gt;And you should not jump arround with notation&lt;/strong&gt; (otherwise the notation gets the focus and students will ask you about the meaning of the symbols rather than about the content).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark:&lt;/strong&gt; Most of my arguments given is not specific to the notation of derivatives, but also in a general context. Maybe the most easy examples where some students have problems with notation is the example where an equation should not be solves for $x$, but for a different variable. Another example would be the use of another symbol than $f$ for a function.&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-20T19:18:29.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="568" PostTypeId="2" ParentId="557" CreationDate="2014-03-20T19:22:00.953" Score="9" Body="&lt;p&gt;I'm assuming this is a university level question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best way to approach this is on day one, where course standards are set. Trying to institute something in the middle of the semester can only start arguments (&quot;It's not in the syllabus I can't do this!&quot; or &quot;But we've been doing this all semester to take notes!&quot;), but if necessary, give them some forewarning to the new policy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find that, particularly in math courses, it is difficult to take notes on a laptop unless the instructor provides the slides (and if those should be used in a class is another question entirely). Additionally, many classes are now going online for their homework, so students will use class time to work on homework. There are a few questions here. Is my course easy to take lecture notes on a computer with? Do I want students working on homework in class? What do I do if students are using technology inappropriately?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answers depends on the the structure of the course. In all cases, I will make an announcement on the first day of class stating my policy (no laptops, limited laptops, etc) and what I will do if someone breaks the policy (ask to put it away, stop class, ask to leave, etc). You need to set clear boundaries on distractions to the students and have them know the consequences. They know that classes are for learning and that you are there to foster that. A reasonable student, when asked to stop playing Quake for example, will stop. If you have TA's, I would ask if they would sit in the back and inform you if someone is being distracting. Else I would occasionally move around the room during lecture and to pay attention to where students eyes are looking. It should be fairly obvious when someone isn't working.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, the level of technology is course dependent. As I am not too familiar with power point style lectures, I would leave that up to your discretion and decide if you post the lectures online before or not to have laptops. Personally in math, I find that the use of hand written lectures (either on a projector or chalk board) is the most used. In these, I find that it is near impossible for students to type up notes at a comparable rate to your lectures. In this case, I would put in the syllabus that laptops and other devises are not allowed in lecture and that I expect students to copy notes into a notebook. If students pulls out a laptop, I would just politely ask them to put it away and remind them of the policy. Now some days you can make exceptions (review days, etc), but always let them know at least a lecture in advance. With this, I believe that homework should be done outside of class and is no excuse for a laptop. If I do a problem in class, students should just copy it to the notebook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Technology in class can be beneficial, but I find it is more engaging than the instructor most of the time (particularly if they are reading off slides). To fight off student boredom, keep them engaged and interacting with you. Ask them questions, poll them for answers, and get them to work together will help push off their want of technology. I've seen an increased use of flipped classrooms and clickers to poll students and keep them engaged. You can't win against phones (just ask for them to put them away if seen), but if they are enjoying class, they are less likely to check them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you say you are more or less stuck to using slides, why not see about getting a tablet and use computer written notes? Your IT department might have some for loan for use in lectures. I find that using slides is the leading cause of distractions if you let them use laptops.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-03-20T19:22:00.953" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="569" PostTypeId="2" ParentId="562" CreationDate="2014-03-20T19:32:08.160" Score="34" Body="&lt;p&gt;The different notations are to a considerable extent outgrowth of slightly different ways of looking at &quot;the derivative.&quot; To me &lt;em&gt;these different ways&lt;/em&gt; is the important thing to discuss,  the notation is a by product. And, I think it is quite useful to spend some time on discussing these different ways and to mention the different notations in that context. (However I understand this is not always possible, see the end.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the notation $f'$ and $\dot{y}$ stresses the (physical) idea of having some function that describes something (some process) and one does something to it to analyze/understand its properties.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the notation $D_x f$ emphasizes the fact that one can think of the situation of having an operator (a function!) $D_x$ that assigns to each (under suitable conditions) function $f$ another function $D_xf$. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the notation $\frac{df}{dx}$ emphasizes the definition as a limit of the quotients of differences.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the notation $df(x)$, in particular when used like $f(x+h) = f(x) + df(x)h + r(h)$ stresses that the derivative at a some point is actually a linear function that just happens to be identifiable with its slope.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There would  also be different ways to split things up and to motivate things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one does this the different notations come more naturally and one then can and I think should vary one's own usage and exercises a bit. Still, I would &lt;em&gt;not insist&lt;/em&gt; or even much encourage students &lt;em&gt;using&lt;/em&gt; different notations themselves. There is some clear merit in having them &lt;em&gt;understand&lt;/em&gt; the different notations; for them being able to use different ones (fuently) this is much less clear to me. There is also a small risk that they would use them, but would use them not really 'idiomatically,' as the notations arein some sense not completely interchangable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example even an example in OP seems slightly of to me (though some place else different conventions might be in place): as far as I know, the dot-notation is mainly used by physicist and alike, and does typically not denote just some derivative, but writing $\dot{y}$ &quot;means&quot; derivative by &lt;em&gt;time&lt;/em&gt;, which &quot;would never&quot; be denoted $x$ but $t$. [For me, as a pure mathematician, the above is strange to write, but I really think that if $y=xt^2$, then to some it is completely obvious that $\dot{y}=2xt$ and not $\dot{y}=t^2$.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answer assumes of course that one has time to do this discussion, and implicitly also that the students or a considerable part there of will actually see the higher dimensional theory for example. If this is not the case, I think I would not bother much with different notations. Pick one (based on your own preference or possibly informed by the predominant convention in whatever other field the students might study, if there is one) and stick to it, and just mention once that there are different notations, so that students are not completely surprised when the open some other book. However, if some student prefers to use a different one, accept this, too (as long it is clear and coherent).  &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-03-20T22:23:37.703" LastActivityDate="2014-03-20T22:23:37.703" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="570" PostTypeId="1" AcceptedAnswerId="573" CreationDate="2014-03-20T20:15:10.747" Score="28" ViewCount="4027" Body="&lt;p&gt;There are many proofs where the whole idea can be expressed by a picture and often naturally translated into a correct formal proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often one has to argue with students that a picture is not a proof (but a good thing to get a proof started). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is the most impressive (but at most undergraduate level) example, where the picture-proof seems okay, but the formal proof goes wrong, to convince a student in paying attention to do a formal proof?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2017-05-09T09:17:11.220" Title="Impressive examples where a &quot;proof by picture&quot; goes wrong" Tags="|undergraduate-education|examples|proofs|" AnswerCount="11" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="571" PostTypeId="2" ParentId="570" CreationDate="2014-03-20T20:29:54.023" Score="10" Body="&lt;p&gt;Not exactly an answer to your question, but my favorite example where a picture proof can go wrong is the fake proof which shows that &lt;a href=&quot;http://en.wikipedia.org/wiki/Mathematical_fallacy#Fallacy_of_the_isosceles_triangle&quot; rel=&quot;nofollow noreferrer&quot;&gt;every triangle is isosceles&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="61" LastEditDate="2017-05-09T09:17:11.220" LastActivityDate="2017-05-09T09:17:11.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="572" PostTypeId="2" ParentId="570" CreationDate="2014-03-20T20:34:02.623" Score="12" Body="&lt;p&gt;&lt;a href=&quot;http://sbjoshi.wordpress.com/2013/03/27/paradox-about-high-dimensional-spheres/&quot;&gt;This one&lt;/a&gt; can be shown to any student that understands the distance formula, and has a willingness to think about $n$-dimensional space for $n&amp;gt;3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(It is, more specifically, an indication that &quot;inducting on pictures&quot; is invalid, and less so of a &quot;pay close attention to the picture&quot;. But I think it's striking enough to warrant inclusion here.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw four circles with $r=1$ on the plane at the points $(\pm 1,\pm 1)$. Draw the box which perfectly encloses these circles, i.e. a square with side length 2. Now, draw a circle that is tangent to all four of the circles. Notice that this circle is inside the box.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can repeat this for $n=3$. Draw spheres with $r=1$ at the eight points $(\pm 1,\pm 1,\pm 1)$. Draw the box that encloses these, i.e. a cube of side length 2. Now, draw a sphere that is tangent to all those spheres. Again, you can see it sits inside the box.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Repeat this in higher dimensions. When you get to $n\geq 10$, the &quot;inner&quot; sphere &lt;em&gt;leaks outside the box!&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-20T20:34:02.623" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="573" PostTypeId="2" ParentId="570" CreationDate="2014-03-20T20:43:45.577" Score="32" Body="&lt;p&gt;This one can be presented to students at any level, really, although the way to explain &quot;repeat to infinity&quot; will certainly change for your audience. It can be used to teach them that weird things happen with limits and we can't just pass things through to the other side. It's also a good way to jumpstart a discussion of definitions: what's a proper way to define &quot;convergence&quot; here? Does it really converge to the circle?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/znQDV.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-03-20T21:18:45.470" LastActivityDate="2014-03-20T21:18:45.470" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="574" PostTypeId="2" ParentId="570" CreationDate="2014-03-20T21:01:17.863" Score="6" Body="&lt;p&gt;Another famous example is cutting the chocolate bar to increase its size, like in &lt;a href=&quot;http://www.moillusions.com/chocolate-pieces-puzzle/&quot;&gt;this&lt;/a&gt; example. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since essentially the same answer has been given afterwords by others, let me add an other one in the same spirit which is a fake picture proof of the &lt;a href=&quot;http://www.cut-the-knot.org/pythagoras/FaultyPythPWW.shtml&quot;&gt;Pythagorean theorem&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="61" LastEditDate="2014-03-20T21:33:09.220" LastActivityDate="2014-03-20T21:33:09.220" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="575" PostTypeId="1" CreationDate="2014-03-20T21:11:57.347" Score="5" ViewCount="154" Body="&lt;p&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a theoretical computer-science question to see how the community will react, relevant meta discussion can be found &lt;a href=&quot;https://matheducators.meta.stackexchange.com/questions/57/how-does-computer-science-education-fit-into-the-community-if-at-all&quot;&gt;here&lt;/a&gt;. I picked the topic to be formal languages, as question from this area are frequently asked at math.se and answered without any trouble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Context:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are three main approaches to &lt;a href=&quot;http://en.wikipedia.org/wiki/Regular_language&quot; rel=&quot;nofollow noreferrer&quot;&gt;regular languages&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;operational, by &lt;a href=&quot;http://en.wikipedia.org/wiki/Finite_automaton&quot; rel=&quot;nofollow noreferrer&quot;&gt;finite automata&lt;/a&gt;,&lt;/li&gt;&#xA;&lt;li&gt;using &lt;a href=&quot;http://en.wikipedia.org/wiki/Regular_expression&quot; rel=&quot;nofollow noreferrer&quot;&gt;regular expressions&lt;/a&gt;,&lt;/li&gt;&#xA;&lt;li&gt;algebraic, via &lt;a href=&quot;http://en.wikipedia.org/wiki/Syntactic_monoid&quot; rel=&quot;nofollow noreferrer&quot;&gt;syntactic monoids&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Each has its strengths and weaknesses, for example&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;finite automata are easy to grasp, but then the proofs become tedious,&lt;/li&gt;&#xA;&lt;li&gt;regular expressions are practical, but hard to reason-about,&lt;/li&gt;&#xA;&lt;li&gt;algebraic approach is clean and elegant, but might be non-intuitive for students.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the best order in which to introduce the above concepts?&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-03-16T16:41:53.880" LastActivityDate="2015-01-01T20:15:06.517" Title="Three approaches to regular languages" Tags="|mathematical-pedagogy|computer-science|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="576" PostTypeId="2" ParentId="570" CreationDate="2014-03-20T21:12:29.527" Score="9" Body="&lt;p&gt;The missing square puzzle is a nice one. Maybe it doesn's fit your question perfectly, since no one would use this for a proof. But you might take it as a proof for the theorem, that the area of a triangle is not well-defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/583nP.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://en.wikipedia.org/wiki/Missing_square_puzzle&quot; rel=&quot;noreferrer&quot;&gt;wikipedia&lt;/a&gt;. However, this parallels András Bátkais answer.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-20T21:12:29.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="577" PostTypeId="2" ParentId="570" CreationDate="2014-03-20T21:12:58.853" Score="24" Body="&lt;p&gt;The &lt;a href=&quot;http://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml&quot; rel=&quot;noreferrer&quot;&gt;Curry Paradox&lt;/a&gt; is a classic. This animation resolves it:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/rGEoJ.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-20T21:12:58.853" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="578" PostTypeId="2" ParentId="570" CreationDate="2014-03-20T21:44:32.197" Score="6" Body="&lt;p&gt;I found a page with a few fallacious geometric proofs:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Imler/EMAT%206690%20Essay%20on%20Proofs/html%20files/Fallacious%20proof%204.html&quot;&gt;This one proves that all right angles are obtuse.&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Imler/EMAT%206690%20Essay%20on%20Proofs/html%20files/Fallacious%20Proof%202.html&quot;&gt;This one proves that all points inside a circle lie on the circle.&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;They both rely on inaccurate drawings of a geometric construction.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-20T21:44:32.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="579" PostTypeId="2" ParentId="570" CreationDate="2014-03-20T21:47:25.880" Score="7" Body="&lt;p&gt;Maybe, one could argue that $f(x)=x^2-2$ has a rational zero using the number line. Just imagine its graph in $\mathbb{Q}^2$. You cannot &quot;see&quot; whether the completeness axiom of $\mathbb{R}$ is given or not. &#xA;Historically, this problem was fixed by introducing the completeness axiom. &#xA;Maybe many problems of &quot;proof by picture going wrong&quot; were solved by choosing the right axioms.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-20T21:47:25.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="580" PostTypeId="2" ParentId="562" CreationDate="2014-03-21T00:31:57.107" Score="3" Body="&lt;p&gt;I'm for using different notations (particularly $\frac{\mathrm{d} f}{\mathrm{d} x}$ and $f'(x)$, as they are useful for emphasizing different aspects; and occasionally $\mathrm{D} f$ if the operator aspect is central). But don't overdo it, saying the same thing in five different ways &quot;just because we can&quot; will only confuse the reader. That they are expressions of different ways of seeing derivatives, from much before they were really understood, is incidental. And those views might even harm understanding, better leave them out until the concept is crystal clear and solid.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-21T00:31:57.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="581" PostTypeId="1" CreationDate="2014-03-21T00:35:58.703" Score="7" ViewCount="1450" Body="&lt;p&gt;Suppose you're the instructor of a large university course split into several sections, each of which has a TA. (Think of a giant lecture-style calculus course with several hundred students that meet in smaller &quot;recitations&quot; on other days.) Homework is graded by the TAs. Presumably, each has their own method/style/harshness of grading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is an effective method to incorporate homework grades that is fair to the students?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should this be addressed from a forward-looking perspective, by providing detailed grading rubrics to each TA? Should you follow up and oversee their grading somehow? Or, should this be addressed retroactively, by somehow analyzing the grades statistically and &quot;normalizing&quot; with some scheme? Or is there a feasible combination of these methods? Or something else entirely?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in what you have done in the past and how effective it was, especially if you have a good sense for how well your scheme worked (data analysis, student feedback, TA feedback, etc.). I am also interested if you have any statistical insight/research on this question. Mostly, I am interested in fairness to the students, as well as how to handle this problem &lt;em&gt;efficiently&lt;/em&gt; (that is, getting maximum fairness for a reasonable time commitment, not just minimizing time).&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T20:41:54.410" LastActivityDate="2014-04-01T20:41:54.410" Title="How to standardize grading across several sections of a course?" Tags="|undergraduate-education|course-design|grading|assessment|large-lecture|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="582" PostTypeId="2" ParentId="575" CreationDate="2014-03-21T00:42:24.383" Score="1" Body="&lt;p&gt;In applications, regular expressions are most directly used (as search patterns). To implement them, e.g. in compilers, you use automata. Automata are needed anyway to describe context free languages (with corresponding grammars, again needed to understand e.g. programming languages and their implementation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above from the Computer Science view. A &quot;pure&quot; matematician's needs/inclinations could well be different.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-21T00:42:24.383" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="583" PostTypeId="2" ParentId="581" CreationDate="2014-03-21T00:51:07.357" Score="4" Body="&lt;p&gt;As  student I was TA in a first-year course with a total of around 600 students each time, divided into classes with around 40 to 60 students each, and weekly quizzes in 20 student groups, a different question for each of the 3 or 4 days recitation was held. No homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The exams were divided among teachers and grading assistants by question (each one graded one question only, for part of the students), and had very detailed grading instructions. The quizzes were graded by us TAs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall this worked out pretty well, but required an enormous effort in setting up exams and quizzes, and in logistics.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-21T00:51:07.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="584" PostTypeId="1" CreationDate="2014-03-21T00:51:31.990" Score="29" ViewCount="1948" Body="&lt;p&gt;What are different ways we can get women to study mathematics? In my own experience, the higher the math class, the less women in the class. Most women tend to go on the math education track and do not have to take the more advanced classes. These more advanced classes can seem like hostile places for women, especially if they are the only female in the class, or only of a few female students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;As an educator, how can we encourage women to study math and become mathematicians?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="228" LastEditorUserId="228" LastEditDate="2014-03-21T06:16:45.573" LastActivityDate="2017-06-11T12:56:38.647" Title="How to encourage women to study mathematics?" Tags="|student-motivation|advising|gender|women|" AnswerCount="6" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="585" PostTypeId="2" ParentId="552" CreationDate="2014-03-21T00:57:39.193" Score="2" Body="&lt;p&gt;One of the reasons an online system was discontinued here for beginning math classes was that the questions had been badly selected (had little correlation with what was going on in class). Collecting a large enough (and relevant enough) corpus of questions, and organizing them, is &lt;em&gt;much&lt;/em&gt; work. Besides, some students hired older students to do their homework for them, and (while that is also seen for paper homework) that became too pervasive. Perhaps due to the above.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-21T00:57:39.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="586" PostTypeId="2" ParentId="584" CreationDate="2014-03-21T01:07:08.383" Score="10" Body="&lt;p&gt;Two fundamental issues, in my observation: pathetically, quite a few (male) people subconsciously decide that, while they are not athletes or whatever, their machismo can be proven in ... mathematics. By math contests and being aggressive in class. Vehicle for ego. As a consequence, there's the &quot;oop, it's not macho if chix can do it, too&quot; unfortunate-riff...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, the next point is that there is indeed a substantial tradition of exhorting kids to engage by invoking machismo.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aaaand, there's the traditional &quot;omuerta&quot; of &quot;not asking questions&quot; (because it reveals weakness, which a good Spartan would never do, blah-blah-blah), which seems statistically less &quot;reasonable&quot; to females than males. And, srsly, what is the point of &quot;classes&quot; if asking the natural questions is belittled? Crazy! From an intellectual, pedagogical, and social perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This ground presents serious difficulties for encouraging women (or any person not feeling at home in the nerdish pseudo-macho environment) to engage with mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An uncomfortable corollary is that the traditional quick-nerdy-combative kids have to be &lt;em&gt;discouraged&lt;/em&gt;, or redirected. This is not easy, given the ambient culture that reinforeces this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are further subsequent problems... too many to explain in short. Perhaps people who're interested in my take on this should email me.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-21T01:07:08.383" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="587" PostTypeId="1" CreationDate="2014-03-21T01:21:42.560" Score="15" ViewCount="400" Body="&lt;p&gt;I taught my first university course last Summer. It was integral calculus that usually has about 200-300+ students during the school year, but my Summer class had only about 35 students. I taught by lecturing during the whole class time. I am teaching again this Summer, but I really want to try something different. The main reason I want to avoid lecturing the whole time is because I thought about my own math education and how I hate sitting through lectures. I actually slept through most of the calculus lectures I attended as an undergrad.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that because the Summer courses are a lot smaller, I can try other methods of teaching besides only lecturing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a list of things I wanted to try this summer for my integral calculus course:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Lecturing for short periods and then giving the students a problem about the short lecture and letting them work on it for 5-10 minutes&#xA;2) Lecturing most of the time and leaving a period at the end to work on problems&#xA;3) Have the students present problems during lecture time&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering if anyone has other suggestions or different ways of making class more interactive that has been successful.&lt;/p&gt;&#xA;" OwnerUserId="228" LastEditorUserId="228" LastEditDate="2014-03-21T01:32:39.427" LastActivityDate="2014-03-21T16:03:57.347" Title="How to make calculus lecture time more interactive?" Tags="|calculus|interactive-teaching|class-participation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="588" PostTypeId="1" AcceptedAnswerId="591" CreationDate="2014-03-21T01:30:39.930" Score="18" ViewCount="2280" Body="&lt;p&gt;At my university, exams take place in a lecture hall, but it is usually pretty packed. Along with alternating versions of tests, it is pretty easy for a student to look to the next row or their neighbor for inspiration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do you confront a student with a wandering eye? I do not want to embarrass the student or distract others during the exam, but I also want them to know that it is not okay to do. In the past, I would usually tell the student to keep their eyes on their own exam. There have been cases where the students with wandering eyes are just spacing out. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have asked instructors at my university and they tend to just ignore it and not say anything. &lt;/p&gt;&#xA;" OwnerUserId="228" LastActivityDate="2014-04-08T06:46:10.957" Title="What to do when students are not keeping their eyes on their own test?" Tags="|classroom-management|exams|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="589" PostTypeId="2" ParentId="581" CreationDate="2014-03-21T01:36:17.060" Score="3" Body="&lt;p&gt;I'll share my experience, as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Context:&lt;/strong&gt; As a graduate student, I was the lead instructor for a 120-student lecture course with 4 recitation sections amongst 3 TAs. Lectures were MWF and recitations TuTh. I assigned weekly homework assignments collected and returned in recitation. This was our &quot;intro to proofs&quot; course, a 100-level course entitled &quot;Concepts of Math&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Method:&lt;/strong&gt; I wrote and distributed detailed grading rubrics for homework problems. I also assigned specific &lt;em&gt;problems&lt;/em&gt; to each TA, rather than have them grade everything from their own section. (Logistically, this was accomplished via a locked filing cabinet in a designated room that everyone had access to. Homework was turned in on Th, TAs would visit over the next few days to grade, and papers were returned on Tu.) My goal was to standardize grades across the sections throughout the semester, rather than at the end via statistical normalizing. My advisor had always used the end-of-term statistical method, but I worried this might foster jealousy of other sections with easy graders / contempt for TAs with high standards. I also helped out by grading one problem per assignment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used the same procedure for exams. I wrote detailed rubrics, including provisions for multiple methods of solution, predictions of common mistakes and how to handle them, and tips for providing good feedback beyond just a score. I graded a couple of questions on each exam, and read through a bunch of them to get a sense for how the students were doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For homework grading questions/complaints, I told the students to talk to their TA first, who would then direct them to the TA (or me) who graded the question. If that could not be resolved, they could come to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For exams, I gave the students exactly one week after return in which they could write up a document to explain their issue, their understanding of the relevant problem, and how their demonstration of knowledge should deserve more points. I would then read each of these and, when appropriate, raise the grade; either way, I wrote a response back to help them understand the grading as it was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Results:&lt;/strong&gt; I honestly think this was a great success, and am very proud of how the logistics of this course were handled, looking back. I never had complaints from students about unfairness, and I spoke with some of them often enough on a personal level that I like to think I could have determined whether there was an undercurrent amongst them that was just unspoken. When I did receive grading complaints, they were often of the form &quot;I don't understand why I lost points here, can you explain more?&quot; and I would do so on a case-by-case basis. Sometimes, their complaints were valid (and, indeed, there was one grader who was sloppy and unfair on early assignments, so I had to have a long meeting with them and show them how to better; thereafter, they improved greatly). Sometimes, they were just grade-grubbing, but in those cases, I was glad I had those formal rubrics written out, so I could point to the problem and say, &quot;This part was worth X points, and you didn't address issue Y. There's nothing I can do, in fairness to other students.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Thoughts:&lt;/strong&gt; As much of a success as this was, it required a &lt;em&gt;huge time commitment&lt;/em&gt; from me as the primary instructor. I spent a lot of time designing the rubrics, taking into account my experiences grading similar problems in the past as a TA myself. I spent a lot of time answering TA questions, in person and via email, about grading issues. I spent a lot of time talking to students about homework problems (and how they were graded). I spent a lot of time just outright grading. But that was my choice. Ultimately, I enjoyed it all and don't regret it, simply for the experience that I gained from it all. I was glad to have TAs who cared as much about grading fairness as I did and didn't see it as just a teaching job on the side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this is grossly infeasible. This was the &lt;em&gt;one course&lt;/em&gt; I was teaching at the time while finishing my dissertation and, even then, it was cutting into my life. Like I said, I enjoyed all of it, but it just doesn't scale upwards nicely at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the positive side, I now have a cache of homework/exam problems as well as their solutions and grading rubrics, and could happily share these with future instructors. So, perhaps this doesn't have to be a perpetual problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Recommendations:&lt;/strong&gt; Frankly, I would strive for this but make sacrifices where they are needed to save time/effort. In the future, if I have more than one course like this, I would first cut the actual grading of problems myself. Or, at the least, I would make sure one problem is completely trivial to grade, almost like a multiple-choice problem. By reading through those submissions, I could at least get some sense of how the students are doing. I would then cut time in the development of the rubrics. As it was, I spent much time considering varied approaches to problems and trying to nip grading considerations in the bud a priori. This isn't necessary, if you're willing to put trust in your TAs and potentially spend the time answering their email questions as they come up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, I feel strongly that the statistical method is flawed, and personally prefer a method that attempts to standardize grading across sections. Specifically, I actively promote the &lt;em&gt;each person grades a problem, not a section&lt;/em&gt; method. I believe this is the only way to be truly fair, and I don't think it requires &lt;em&gt;that&lt;/em&gt; much more effort on the part of the instructor/TAs to be implemented.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-21T01:36:17.060" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="590" PostTypeId="2" ParentId="562" CreationDate="2014-03-21T01:46:39.953" Score="12" Body="&lt;p&gt;I think it is important to introduce the &lt;span class=&quot;math-container&quot;&gt;$d/dx$&lt;/span&gt;, prime and, to a much lesser extent, dot notations for the following reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}$&lt;/span&gt; makes it quite clear which variable is being differentiated. Therefore, it is quite clear that &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}(a+x^2)=2x$&lt;/span&gt; whereas &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{da}(a+x^2) = 1$&lt;/span&gt;. In both of these calculations I assume that the complementary variable is held fixed in the differentiation. What do these look like in the prime notation? Well,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (a+x^2)' = 2x $$&lt;/span&gt;&#xA;is what my students would probably write. Perhaps, on occassion &lt;span class=&quot;math-container&quot;&gt;$(a+x^2)'=1$&lt;/span&gt; would be offered by the sarcastic outlier. To actually be clear, we have to include the parenthetical evaluation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (a+x^2)'(x) = 2x \qquad \&amp;amp; \qquad (a+x^2)'(a) = 1. $$&lt;/span&gt;&#xA;Even so, some might be tempted to write &lt;span class=&quot;math-container&quot;&gt;$(a+x^2)'(a) = 2a$&lt;/span&gt;. Which is it? The prime notation has danger in this regard.&lt;/li&gt;&#xA;&lt;li&gt;the prime notation is efficient. I think this may be one of the greatest aspects:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (f+g)' = f'+g' \qquad (cf)'=cf' \qquad (fg)'=f'g+fg' $$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \left(\frac{f}{g} \right)' = \frac{f'g-fg'}{g^2} \qquad (f \circ g)'=(f' \circ g)g' $$&lt;/span&gt;&#xA;It does require more writing to get these across in the &lt;span class=&quot;math-container&quot;&gt;$d/dx$&lt;/span&gt; notation. Although, I do prefer to include the argument for the chain rule &lt;span class=&quot;math-container&quot;&gt;$(f \circ g)'(x)=f'(g(x))g'(x)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;the &lt;span class=&quot;math-container&quot;&gt;$\dot{x}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\ddot{x}$&lt;/span&gt; are fun. I just like to say &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-double-dot. It takes me back to physics and so it has a certain nostalgic value for me personally. Pragmatically, these dot formulas are very pretty for expressing kinetic energy formulas for Lagrangian mechanics. When you use &lt;span class=&quot;math-container&quot;&gt;$\dot{x}$&lt;/span&gt; as a variable it is a lot nicer than &lt;span class=&quot;math-container&quot;&gt;$x'$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$dx/dt$&lt;/span&gt;. In variational calculus, the &lt;span class=&quot;math-container&quot;&gt;$\dot{x}$&lt;/span&gt; appears as an &quot;independent&quot; variable from &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="128" LastEditorUserId="408" LastEditDate="2018-09-23T17:45:20.627" LastActivityDate="2018-09-23T17:45:20.627" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="591" PostTypeId="2" ParentId="588" CreationDate="2014-03-21T02:12:16.727" Score="15" Body="&lt;p&gt;There are a few reasonable approaches, and they vary mostly in (a) intrusiveness towards the students and (b) effectiveness&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Walk intently in the direction of the student, cough, and stare at them for a minute.&lt;/strong&gt; From the perspective of the &quot;potentially cheating student&quot;, this is quite effective. There's no way they'll pull any funny business while you're actively looking at them and no one else. And they now know you're aware of them, so they might be less likely to try something later when you're across the room but have them on your mind. However, this can call attention to you and the student you're watching, depending on how loud you make your ahemming and how intent you make your staring. Although, if the eye-wandering is particularly glaring, you might &lt;em&gt;want&lt;/em&gt; to make your staring obvious and deterrent.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Speak to the room at large, reminding them that they should keep their eyes on their own paper and cheating will be punished&lt;/strong&gt;. The great thing about this is that you can do it whether you see one pair or twelve pairs of wandering eyes. You can say it not only while scanning your eyes across the room but also while the students &lt;em&gt;see you scanning&lt;/em&gt; your eyes across the room, so they don't feel individually called out for crimes (falsely or otherwise) and they also feel like you're being vigilant. The bad thing about this is that it can disrupt focused students. If I were intently working on a problem and were all of a sudden snapped out of my concentration to be reminded not to cheat (which I wasn't doing anyway), I might be miffed.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Call out a potential offender and either move them to a new seat or take away their test&lt;/strong&gt;. This is quite disruptive, of course, but can be effective if you are sure of your accusations. I have subtly done this, by asking students to spread out amongst an otherwise unoccupied row and then hovering near them for the rest of the hour. This is certainly disruptive to them, but I presume they were already suffering the cognitive dissonance of academic dishonesty and my ascertainment of that won't adversely affect them. Of course, if I'm wrong, this can totally backfire. So it behooves one to be &lt;em&gt;absolutely sure&lt;/em&gt; that cheating almost occurred (or did occur).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have tried variants of these methods quite effectively, depending on the situation. The following idea has not been tested by me, but I believe it could be effectively implemented.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Have a pre-determined method for handling &quot;cheating accusations&quot; in your syllabus, and make students sign it.&lt;/strong&gt; For instance, include a paragraph in your syllabus that says something like, &quot;If I see you doing something during an exam that constitutes academic dishonesty, I will jot it down and dock you 10 points on your score [or whatever penalty is appropriate, you feel]. You will learn of this when you receive your exam back; I will not call you out during the exam to avoid distracting you and others. If you wish to dispute the penalty, we can approach the appropriate channels [i.e. the 'Academic Integrity Panel' or the 'Student Conscience Board' or what have you].&quot; As I said, I have not tried this, so I wonder about its efficacy and, especially, its transparency to students (&quot;Is this a blanket statement for him to penalize me unwarranted?&quot; &quot;Can I dispute his claim without contrary evidence? Can he even make a claim without evidence of his own&quot; etc.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-21T02:12:16.727" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="592" PostTypeId="5" CreationDate="2014-03-21T02:57:50.803" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-21T02:57:50.803" LastActivityDate="2014-03-21T02:57:50.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="593" PostTypeId="4" CreationDate="2014-03-21T02:57:50.803" Score="0" Body="Grading questions ask how to properly, effectively, and fairly assign grades to assignments/exams/quizzes." OwnerUserId="80" LastEditorUserId="39" LastEditDate="2014-06-11T23:17:37.187" LastActivityDate="2014-06-11T23:17:37.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="594" PostTypeId="5" CreationDate="2014-03-21T02:59:59.740" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-21T02:59:59.740" LastActivityDate="2014-03-21T02:59:59.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="595" PostTypeId="4" CreationDate="2014-03-21T02:59:59.740" Score="0" Body="Assessment questions ask how to properly, effectively, and efficiently ascertain the abilities of students. This doesn't necessarily involve numerical grades, but rather is focused on an overall characterization of knowledge/ability." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-03-21T06:47:35.460" LastActivityDate="2014-03-21T06:47:35.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="596" PostTypeId="2" ParentId="588" CreationDate="2014-03-21T05:04:26.037" Score="5" Body="&lt;p&gt;The challenge you are weighing here is that you want to be able to move people to more secure seats without falsely accusing them of cheating. One possible plan is to announce the following at the start of the test:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;During this exam, I will move between two and four people to the front of the room to help prevent cheating on this exam. If I think I see a student looking at another student's test, I will move one of those two people to the front of the room. This means that &lt;strong&gt;when I move you, I am not accusing you of cheating.&lt;/strong&gt; Let me repeat this: When I move you to a new seat, please do not panic: it does not mean I think you are cheating. I am only trying to keep the exam running smoothly while preventing any potential problems, and I apologize in advance for the inconvenience.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can move people with even the slightest twinge of suspicion, and not really worry about it at all.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-21T05:04:26.037" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="597" PostTypeId="1" CreationDate="2014-03-21T05:29:09.787" Score="10" ViewCount="234" Body="&lt;p&gt;There are many cinematic and documentary movies about mathematics and mathematicians in every field of mathematics. Particularly in logic, set theory and philosophy of mathematics,  which are my research fields one can find plenty of such interesting movies easily. These movies explain complicated technical mathematical and philosophical ideas in a simple language. Also they can introduce the historical development of the subject and show the mathematical and non-mathematical aspects of the life of the professional mathematicians. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have an idea to use these movies for motivating undergraduate students for doing simple new researches in the scientific aspects of the movie during the semester and give a lecture about the results at the end but I am hesitated about the possible positive or negative aspects of such a teaching idea.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it a good idea to use movies for motivating undergraduate students? In the other words, what kind of benefits and harms such an idea can have? and, how should a teacher use these movies to have maximum benefits and less harms? Are these movies useful for graduate students too? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="114" LastEditDate="2014-03-28T18:45:25.037" LastActivityDate="2014-03-28T18:45:25.037" Title="Can math movies help us in teaching?" Tags="|undergraduate-education|course-design|student-motivation|undergraduate-research|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="598" PostTypeId="1" AcceptedAnswerId="611" CreationDate="2014-03-21T06:12:15.217" Score="13" ViewCount="255" Body="&lt;p&gt;Math clubs and groups tend to be small and prone to dying out. Once the organizers or members graduate, the club no longer exists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At my university, I have seen the creation and death of two different math groups. I myself started a math group for women, but I am worried that it will not last long after I graduate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What can be done to ensure the longevity a math group or club?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="228" LastEditorUserId="228" LastEditDate="2014-03-21T06:17:30.640" LastActivityDate="2014-03-24T03:03:43.557" Title="How to create and support a mathematics club or group that will last?" Tags="|undergraduate-education|student-motivation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="599" PostTypeId="1" CreationDate="2014-03-21T06:29:28.157" Score="14" ViewCount="2239" Body="&lt;p&gt;Some students do not do well under pressure or testing situations and cannot demonstrate their knowledge of a math subject with a written exam. On the other hand, I think that students that do not have a good enough grasp of the material are able to do well on a written exam also.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are testing alternatives that allow an instructor to evaluate a student's knowledge?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, maybe having students present problems on the board, as a sort of oral exam, but not calling it an exam so it does not induce anxiety, along with a written exam. Or no exam at all, where the grade is based entirely on homework scores.&lt;/p&gt;&#xA;" OwnerUserId="228" LastActivityDate="2014-04-09T15:46:26.017" Title="What are effective alternatives to a written math exam to evaluate knowledge?" Tags="|course-design|exams|testing|test-design|oral-exams|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="600" PostTypeId="2" ParentId="581" CreationDate="2014-03-21T07:50:55.900" Score="4" Body="&lt;p&gt;In all the exercise groups where I was a TA or a instructor for other TAs, we made a lot of efford how questions were graded. The problem is, however, even if you spent even more time, the problems arise when you see the homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students did some mistakes destroying the structure of the questions (when you said &quot;When someone arrives at that certain point, he/she will get 2 points, etc. - And they did some alternative way with a nice idea you did not think about, but not comparable to other solutions). Or they did something else you didn't plan for. In these cases, TAs have to make up their minds and everyone will problably grad in different ways. Or students did some subtle error which can be overlooked by inexperienced TAs. Lets face reality: Some TAs correct also every deep (They take responsiblity of their work) and some correct sketchy (due to time or competence). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you really want to maximize this, there are several possibilities:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You can divide the homework into questions and every TAs has to correct only one questions. If he/she e.g. overlooks some subtle error, then he/she it for everyone. The problem here is that students have to visit several different TAs if they have questions about the grading. If your TAs have in general a free weekend, you can also do the extreme way that one TAs is grading all the homework (this is possible in courses with at most 100-150, but not with the mentioned 700). &lt;/li&gt;&#xA;&lt;li&gt;You can ask the TAs to write emails to everyone reflecting each equivalent class of solution. If there is some good responsible TAs, he/she will get most of the errors and suggest the grading for such a solution and the rest can follow. The big problem is that you have to describe and summarize solution and read maybe 50-100 emails every week. I have seen this sometimes when only a few TAs have to grad exams for a lot of students. &lt;/li&gt;&#xA;&lt;li&gt;You can have extra grading sessions where all TAs meet and discuss directly if something appears. This is very effective, but you have to find a date where all TAs are free. Also, the problem is that maybe someone is faster than the others etc. If the groups are really small or the homework is not much, this is maybe the best solution.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;However, I think you can never attain the maximum by this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Our solution&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last time we had a bigger course (about 100-150 students, which is big in our department), there were not too few complains about grading although there was weekly very long meeting with all TAs and they were forced to come to me if anything is not clear, etc. We also saw this in statistics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Luckily, the date of the exercise groups was the same for all TAs and we &lt;strong&gt;decided to rotate the TAs between the different groups.&lt;/strong&gt; The message was: &quot;Yes, the world is not fair and in case of doubt, some persons will give a better grade and other persons will give a worse grade. In order to have to same condition for every student, we switch TAs every X weeks.&quot; (In smaller courses you can try that every student and every TAs have met; in bigger courses you maybe have to find a out a good mixture of severe and slack TAs with the contraint of their timetable.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;In a nutshell, I think the method worked really well.&lt;/strong&gt; Some problems occured:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For psychological reasons, you should communicate this at the beginning of the course. For some students, their feelings boiled over due to the &quot;surprise&quot;.&lt;/li&gt;&#xA;&lt;li&gt;The main focus of the groups should be to learn something, to ask questions, etc. -- If there is a regular change for the TAs, many students will not dare to ask questions. There should not be too many changes. I tried a intervall with 5 meetings (From a different viewpoint: If the students get some TAs they don't like, they can then count to 5 and get a new one).&lt;/li&gt;&#xA;&lt;li&gt;After the first change, there were many complaints: Most of them were either from good students having problems with a inexperienced TAs or poor students now getting worse grades. The TAs should be strongly encouraged to take over their responsibility (and you may combine the &quot;change-method&quot; with some of the methods above) in order to avoid such situations.&lt;/li&gt;&#xA;&lt;li&gt;Some students wanted to change the group in order to &quot;travel&quot; with the TAs. You should think about how to deal with such a situation before doing this &quot;change-method&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-21T07:50:55.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="601" PostTypeId="2" ParentId="588" CreationDate="2014-03-21T08:09:55.030" Score="5" Body="&lt;p&gt;When I was a TA, I have had similar experience like the OP. The problem is that it is very difficult to prove something from wandering eyes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most things when you arrive in such a situation was already said in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/588/what-to-do-when-students-are-not-keeping-their-eyes-on-their-own-test/591#591&quot;&gt;brendansullivan07's answer&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is so difficult to prove and disturbs the peace during exam that you should under all circumstances avoid the situation! The best solution to avoid such situation is to not let the situation happen: If you have only 1-2 free seats to the next person, it is very likely that you have wandering eyes (not necessary on purpose).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since I take responsiblity for exams, we take bigger lecture halls and sit the students in such a way that there are three rows and fours seats between them and the next student which makes it more or less impossible to look and someone else's solution. You should also walk around and keep an eye on them (some of them communicate through the lecture hall). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(In case someone complains about you taking big lecture halls, I say them, he should email his complain and I will forward it to university authorities or to the press that this particular person supports cheating. Since we had a big policital affair with cheating in Germany, no one has then ever said a word against this anymore.)&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-03-21T08:09:55.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="602" PostTypeId="1" AcceptedAnswerId="603" CreationDate="2014-03-21T08:12:40.847" Score="15" ViewCount="1103" Body="&lt;p&gt;This is sort of an opposite question to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/49/keeping-quicker-students-engaged-and-interested-throughout-a-course_&quot;&gt;keeping quicker students engaged?&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes I encounter students who are overenthusiastic about the course; they're constantly moving faster than the rest of the class and anticipating material (say, asking a question a minute before I prove a theorem that answers it) or answering ever question I ask the classroom. This is disruptive to the rest of the class, but I don't want to ruin their enjoyment of the class by asking them to not participate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's a good solution that keeps such students enthusiastic without allowing them to make learning harder for other students?&lt;/p&gt;&#xA;" OwnerDisplayName="user37" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-03-21T20:14:43.173" Title="How should I deal with overenthusastic students?" Tags="|classroom-management|class-participation|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="603" PostTypeId="2" ParentId="602" CreationDate="2014-03-21T08:38:51.300" Score="9" Body="&lt;p&gt;In any case, you should invite the student to a little chat at your office.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should probably tell him about your concern and ask him if there is something you can do to please him and to give him input for his motivation. In worst case, the student appreciate your honesty and gets a compliment how good he is and that it is a good way to be enthusiastic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a few things, you can offer him/her (maybe even before you ask him):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You can recommend a good book dealing with his/her interests. The book should be written in a self-contained way and dealing with your course topic, but much deeper and maybe covers more content. Tell him/her to come back to you or a PhD student of yours if there are questions or after he/she has read the first N chapters. Then talk about those chapters. You can also compara your lecture with the book together with him/her.&lt;/li&gt;&#xA;&lt;li&gt;Tell him something about the courses coming next and needing the content of your lecture. Recommend a book or a parallel lecture for it.&lt;/li&gt;&#xA;&lt;li&gt;Offer him/her to have a small session where you or your PhD students tell him about your/their research.&lt;/li&gt;&#xA;&lt;li&gt;Offer him/her to help him/her plan the courses he/she will attend the next semester.&lt;/li&gt;&#xA;&lt;li&gt;Appoint some extra credit task at a high difficulty level (which may also contain to read some source). Such questions may usually be open. Add some comment like: &quot;Solutions to this task have to discussed with the professor/the TAs instructor&quot; or similar. (I think about such a example. Image your course is about derivatives for functions $f:\mathbb{R}^n\to\mathbb{R}^n$. Then you could ask him to extend to concept for (infinity) dimensional Banach spaces (if this is unknown, add a resource where Banach spaces are introduced).)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-21T08:38:51.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="604" PostTypeId="1" AcceptedAnswerId="605" CreationDate="2014-03-21T09:38:43.560" Score="39" ViewCount="2545" Body="&lt;p&gt;Formally, if $f$ is a function, $f(x)$ is a value. So for instance, $f$ can be continuous, but not $f(x)$. &#xA;In teaching at school and university, notation is quite often mixed up, e.g. the function is called $f(x)$. This may be problematic but can also have benefits. &lt;strong&gt;What good reasons for the use of the different notations do you know?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question raised from the discussion of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/562/should-students-be-asked-to-use-more-than-one-notation-for-the-derivative-in-an&quot;&gt;this discussion on notation of derivatives&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2024-03-02T16:40:55.717" Title="Reasons for (not) distinguishing $f$ from $f(x)$" Tags="|calculus|mathematical-analysis|notation|" AnswerCount="11" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="605" PostTypeId="2" ParentId="604" CreationDate="2014-03-21T10:06:03.257" Score="22" Body="&lt;p&gt;To distinguish not too strictly between &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; allows us to operate more easily with functions built up from other functions.&lt;/p&gt;&#xA;&lt;p&gt;For example, one might want to say things like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let us consider the function &lt;span class=&quot;math-container&quot;&gt;$\sin (3 x^2)$&lt;/span&gt; on the interval &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This can be considered as sloppy in a formal sense, but I think it is still clear and in some contexts preferable over more correct alternatives, in my opinion.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="23536" LastEditDate="2024-03-01T14:49:13.293" LastActivityDate="2024-03-01T14:49:13.293" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="606" PostTypeId="2" ParentId="604" CreationDate="2014-03-21T10:08:51.453" Score="9" Body="&lt;p&gt;In most contexts, it makes sense to say that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x + \sin(x)$&lt;/span&gt; is a continuous function&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Consider the alternatives:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;the function taking &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x+\sin(x)$&lt;/span&gt; is a continuous function&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\text{id} + \sin$&lt;/span&gt; is a continuous function&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Take your pick.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-21T10:37:11.457" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="607" PostTypeId="2" ParentId="604" CreationDate="2014-03-21T10:28:42.183" Score="26" Body="&lt;p&gt;It was mentioned in other answers that having a more sloppy notation is better to not complicate the communication. This is okay for people who had really understood the concepts of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have the feeling that understanding what a function is one of the those things students don't really understand. Common mistakes (and in my opinion all of them are led by the fact of the sloppy notation) are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;A function is something you can write down&quot;. For example, most students have a lot of problems if you define a function in a more indirect way: For every argument, the function takes as a value the minimizer of some functional (dependent on the argument). In abstract algebra or differential geometry there are also very sophisticated examples of a very abstract usage of functions.&lt;/li&gt;&#xA;&lt;li&gt;By all the time taking $x$ as argument, some students are not able to think about another variable as argument of a function which is the case in many applications.&lt;/li&gt;&#xA;&lt;li&gt;The lack of not distinguishing between function and its evaluation leads often to a lack of the ability to use theorems, as it was also pointed in this question about the chain rule: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/380/revisiting-the-chain-rule&quot;&gt;Revisiting topics from previous courses&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If students fail to understand the deep meaning what a function is, there is no chance that they can understand basic concepts in e.g. functional analysis, where &quot;values&quot; are functions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-03-21T11:32:58.533" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="608" PostTypeId="2" ParentId="604" CreationDate="2014-03-21T10:29:26.973" Score="15" Body="&lt;p&gt;Though I also like to use the notation &quot;the function $x^3$&quot; when appropriate, sometimes this causes problems. For example when I teach inverse functions, it is difficult to grasp that the $x$ in $x^3$ is not the same $x$ as in $\sqrt[3]{x}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always have problems with first year students who are used to the notation that $x$ is in the domain and $y$ is in the range of the function.&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-03-21T10:29:26.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="609" PostTypeId="2" ParentId="584" CreationDate="2014-03-21T10:34:28.730" Score="15" Body="&lt;p&gt;I think there is no clear answer, although there has been some research on this topic.&#xA;I remember one study which focussed on gender differences of university math students:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mischau, A., Blättel-Mink, B., Daniels, J., &amp;amp; Lehmann, J. (2004). Doing gender in mathematics: indications for more gender equality in German universities? Bielefeld: IFF.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The researchers found that women have partly different interests, e.g.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Women do less emphasize the intellectual challenge and also important applications as reason for their interest.&lt;/li&gt;&#xA;&lt;li&gt;With regard to applications of mathematics, women are more interested in &quot;social&quot; applications like teaching in school, medicine and psychology. Men stronger focus on technical applications (physics, engineering, economics, computer science). Men also report slightly stronger interest in research.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Roughly speaking: the purer the maths, the less women are interested in it.&#xA;Presumably, even women who decided to study mathematics enter university with different interest profiles. Consequently, we should have a look on gender differences at school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Research showed differences even in primary school. We have more boys than girls taking part in math competitions, although there is no (or hardly any) overall difference in mathematical ability (see &lt;a href=&quot;http://www.tandfonline.com/doi/pdf/10.1080/00131720309335034&quot;&gt;Hanna&lt;/a&gt;, &lt;a href=&quot;http://dericbownds.net/uploaded_images/hyde.pdf&quot;&gt;Hyde et al.&lt;/a&gt;, &lt;a href=&quot;http://psycnet.apa.org/journals/bul/107/2/139/&quot;&gt;Ma&lt;/a&gt;, &lt;a href=&quot;http://psycnet.apa.org/journals/bul/136/6/1123/&quot;&gt;Lindberg et al.&lt;/a&gt;).&#xA;Differences can be found if we look at attributions (e.g. when they solved a task, girls might think &quot;the task was easy&quot; or &quot;I was lucky&quot; whereas boys would rather think &quot;I am generally good at maths&quot;.), see &lt;a href=&quot;http://www.pabst-publishers.de/psychology-science/1-2006/ps_1_2006_3-16.pdf&quot;&gt;Dickhäuser &amp;amp; Meyer&lt;/a&gt;. This might be due to behaviour of parents, teachers and peers (in short: the society). But that's a private speculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All in all, it looks like the phenomenon you described is just the tip of the iceberg - maybe the result of hundreds of situations in childhood. &#xA;I have no clear answer, but maybe I could illustrate it's not so easy to get one.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-21T10:34:28.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="610" PostTypeId="2" ParentId="587" CreationDate="2014-03-21T10:53:18.080" Score="7" Body="&lt;p&gt;Two suggestions:&#xA;The first is to try &lt;a href=&quot;http://www.kmel-journal.org/ojs/index.php/online-publication/article/viewArticle/200&quot;&gt;clickers&lt;/a&gt;. They work like the voting system in &quot;who wants to be a millionaire?&quot;. You can activate students and foster discussions on questions. There is no need for clicker hardware as long as your students have internet access via tablet, smartphone, etc. see &lt;a href=&quot;http://aisel.aisnet.org/icis2012/proceedings/ResearchInProgress/30/&quot;&gt;PINGO software&lt;/a&gt;.&#xA;The second suggestion is &lt;a href=&quot;http://educationnext.org/files/ednext_20121_BTucker.pdf&quot;&gt;flipped classroom&lt;/a&gt;. Students shall prepare at home. You might want to upload videos or hand-out a text, refer to books, whatever.&#xA;There is no need to come together just for listening to the teacher and reading what he is writing. Your lecture therefore becomes a place for discussions, since the students now are prepared.&#xA;There are a few mistakes not to be made, e.g. you should not stand at the blackboard trying to lead the discussion. Your students will then wait for you to do something. You rather stay back behind them and only act if real problems arise which the students can't answer. Succesfull experiences were gained naming one student to just moderate what the others say and one more student to simply write on the blackboard everything they say.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More literature on flipped classroom can be found &lt;a href=&quot;http://www.citeulike.org/user/cspannagel/tag/invertedclassroom&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-21T10:53:18.080" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="611" PostTypeId="2" ParentId="598" CreationDate="2014-03-21T11:12:16.463" Score="10" Body="&lt;p&gt;It is a very nice and very very hard question because it depends on many social parameters mainly unrelated to mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am founder of a logic group in our university. It is two years old now and we are extensively under development in both scientific and administrative aspects. We are somehow famous now and some other universities are following a same pattern.  Let me to explain more about some parameters which I guess could be useful for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1. I have a dream.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First note that you are building a new society in an old society so you should have something new to say. Thus you need an exact plan including all of your goals in short and long time and all possible ways which can help your group to reach them. Determining all of the scientific and non-scientific purposes, designing the structure of your organization and also defining possible opportunities which the group can provide for its members are the main duties of your &quot;president&quot; at the first step. These parameters are so important and you should pay enough attention to design them. Write them all exactly in a &quot;constitution&quot; and ask people to read it before joining your group. This document will be your first tool for defining a new abstract identity like a math group &quot;independent&quot; from its members. The best example is a football club. In fact any football club is just a production of its members including players, coaches, managers, fans, etc. but it has an &quot;independent&quot; identity from its members too. At first a football club is just a name and it is the founders who are its real identity but after a while when all of its founders are retired, the club itself is still operational and can give a new identity to its members and managers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2. A beautiful name, a beautiful mind!&lt;/strong&gt;   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to emphasize on the identity of your math group choose an adequate name, design a pretty logo and find an impressive motto. These simple parameters are very important because they are the first &quot;representatives&quot; of your group everywhere. You should choose them carefully and use these everywhere in your website, documents, colloquiums, mails, posters, etc. Use the advisement of professionals to choose the best. Being too informal in these cases is not recommended because maybe you will participate in official communications with the other mathematical groups in/out of your university and it is better to be formal in these communications. In comparison with managing a country your constitution is your &quot;law&quot;, your logo is your &quot;national flag&quot; and your motto is your &quot;national anthem&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3. Build internal and external networks.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By an external network I mean communicating with the other similar scientific/math groups. This helps your group to remain up to date. It also helps the others to be familiar with your activities and this encourages your members to be more active.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By an internal network I mean something like a forum/website just for your members and a very regular announcement system with a particular organizer (a public communication manager) which announces the last news of the group to all members. This helps the members to remain connected with the group even if they are graduated and far from the university for years.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4. Regular Events.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The scientific activities of the group should be very regular. Define different levels of activities in each year and separate the audiences for each part. There are two main levels of activities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Minor Events:&lt;/em&gt; As like as weekly seminars/colloquiums and study/research programs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Organize such events per week/month/semester. In this case the audiences are &quot;local&quot; including your members, department faculties, university students, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Major Events:&lt;/em&gt; As like as organizing national/international conferences.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Organize such events per year/two years.In this case, the audiences are &quot;global&quot; including people from other universities all around of the country/world. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;regularity&quot; is too important. People should count on your activities as a habit. Also these regular colloquiums will help your members to take the identity of the group more serious. This point of view remains alive even if a part of your members be graduated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5. Teachers as Members.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Invite your teachers/supervisors to be a member of your group. They can help you scientifically and supervise activities with a better view. Also you can offer a president position to one of them (in your case maybe one of the female faculty members of your department) which helps the group to keep its organization safe from graduating its student members because the faculty members are in the university for a longer time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;6. Remain Neutral to be Popular!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding to your comment on a math group &lt;em&gt;for women&lt;/em&gt;, I think your idea for &lt;em&gt;helping women to become researchers in mathematics&lt;/em&gt; is very interesting and surely reasonable but I am not sure that organizing a math group just for women is useful in this direction. I think there are better ways for reaching this goal. In principal I believe that if we define a x-group (here x=math) this community should be &lt;em&gt;neutral&lt;/em&gt; in any other aspect y different from x in order to be as popular as possible amongst people. I mean a math group should not be politically right or left, religiously up or down, racially white or black, sexually masculine or feminine, etc. Every body who likes math can be a member of a math group and limiting the membership to women seems inappropriate. Of course having a neutral math group in any non-mathematical aspect doesn't mean that each one of your members should be neutral in each non-mathematical aspect. It just says that you should be careful about the population balance of the group because people don't like to be one of the minorities in each community. Thus for example if you have a member with a right political point of view you should search for another member with a left political point of view to keep the political balance of the group safe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;7. Be Active, Be Exciting!&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your scientific activities are the best representatives of your math group but as I said you should have something exciting and new for your audiences. Think about any possible way for encouraging people in general and women in particular.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wish you a successful math group! Good Luck!&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-24T03:03:43.557" LastActivityDate="2014-03-24T03:03:43.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="612" PostTypeId="2" ParentId="599" CreationDate="2014-03-21T11:22:49.107" Score="7" Body="&lt;p&gt;In teacher education I had some positive experiences with term papers. Students had to write 10-20 pages where they discussed a topic related to the course and their future teaching. Topics were like &quot;Is there a usefull value that could be given to $0^0$?&quot; and &quot;On the different use and notation of sequences at school and university&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this works only, when you want your students to exemplarily go deeper into one question. If the whole course is to be covered, this won't help.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-21T11:22:49.107" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="613" PostTypeId="2" ParentId="599" CreationDate="2014-03-21T11:26:35.180" Score="10" Body="&lt;p&gt;Here is a small list of alternatives to an exam. Lets assume you are not constrainted by your university to some specific kind of progress. First lets collect some criteria we want:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Everyone understood the topic should pass the course.&lt;/li&gt;&#xA;&lt;li&gt;Good motivated should should pass with a good grade.&lt;/li&gt;&#xA;&lt;li&gt;Students who, e.g., only learned some definition by heart, should fail.&lt;/li&gt;&#xA;&lt;li&gt;It should be fair in &lt;/li&gt;&#xA;&lt;li&gt;At least as possible anxiety should be introduced (that's in fact the question here. Due to my experience, you would only move anxiety to a different point as long as there is a progress with a grade at the end. If you don't call it &lt;em&gt;exam&lt;/em&gt;, the students will know it is some kind of anyway.).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Oral exam&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pros:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If a question is formulated in a confusing way, there is a chance to ask.&lt;/li&gt;&#xA;&lt;li&gt;You can see if someone only learned things by heart or is really thinking about them.&lt;/li&gt;&#xA;&lt;li&gt;You can switch to a different topic when there is the impression that the students is really familiar with one topic.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Cons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Not possible for big courses.&lt;/li&gt;&#xA;&lt;li&gt;If someone fails, you have to say it into the student's face and you have (exept from maybe one witness) no proof of the bad performance of the student (in a written exam, you can let colleagues look at it). Let's face it: Poor students will normally pass the oral exam.&lt;/li&gt;&#xA;&lt;li&gt;The impression depends highly on your actual mood and the order of students. At the end you will maybe grade different than in the beginning.&lt;/li&gt;&#xA;&lt;li&gt;You do only grade good ideas. If someone is not able to write down anything, this is not even noticed.&lt;/li&gt;&#xA;&lt;li&gt;The questions either differ in difficulty or someone can be prepared for your questions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Grades from homework&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pros:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It does not depend on a one-day performance.&lt;/li&gt;&#xA;&lt;li&gt;You can be sure that every topic was somehow included in the grade.&lt;/li&gt;&#xA;&lt;li&gt;Everyone has the same questions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Cons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You cannot ensure that the students have done their homework by their own.&lt;/li&gt;&#xA;&lt;li&gt;The grade may depend on the TA, compare this question: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/581/how-to-standardize-grading-across-several-sections-of-a-course&quot;&gt;How to standardize grading across several sections of a course?&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Students will bargain for points during the whole course.&lt;/li&gt;&#xA;&lt;li&gt;Homework should be there for learning and making mistakes. It is okay (and I also recommend this) to have a threshold being achieved to ensure basic knowledge.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mixture of written exam, oral exam and homework&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pro: You get a good mixture of the pros above and the Cons above will be weaker since the are mixed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Con: A lot more work for you and your group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;No grades at all, but a written report&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, you write down some thoughts about the students relying on your personal view (from the lecture, your office hour), the view of the TA, as well as the homework. This report can be shown to other professors in order to apply for, e.g., a master's thesis. Your university will problably not like this since are should give grades in each course. But interestingly enough: Such a model is used in first grades of elementary school &lt;strong&gt;and&lt;/strong&gt; &#xA;when someone is applying for (tenure) academic position. But normally not in between :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Visiting the group for (half) a day&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a very very small course, it could be a possibility that the students have to visit to group for (half) a day and talk with everyone from the group. After every student was there, the group meeets and decides about grades. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have never heard of anyone doing this (it takes a lot of time and distracts everyone from working), but a similar method is used when several people are applying for grants and some comitee visits them for a day or more and decides weeks later (This is the procedure at least in Germany).&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-08T06:47:45.250" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="614" PostTypeId="2" ParentId="584" CreationDate="2014-03-21T12:40:43.217" Score="4" Body="&lt;p&gt;One way to interest women in mathematics is to &lt;strong&gt;teach applications, including statistics&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the following &lt;a href=&quot;http://www.ams.org/profession/data/annual-survey/2012Survey-SREC.pdf&quot; rel=&quot;nofollow&quot;&gt;survey&lt;/a&gt; of 2012 doctorates in the U.S. in the mathematical sciences:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{matrix}&#xA;\text{Men} &amp;amp;\text{Women} &amp;amp;\text{% Women} &amp;amp;\text{Type of Program}\\&#xA;653&amp;amp; 233 &amp;amp; 28\% &amp;amp;\text{public math programs}\\&#xA;204&amp;amp;  62 &amp;amp; 23\% &amp;amp;\text{private math programs}\\&#xA;105&amp;amp;  56 &amp;amp; 35\% &amp;amp;\text{applied mathematics}\\&#xA;195&amp;amp; 118 &amp;amp; 38\% &amp;amp;\text{statistics}\\&#xA; 87&amp;amp;  85 &amp;amp; 49\% &amp;amp;\text{biostatistics}\\&#xA;\end{matrix}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So 45% of the women are getting PhDs in applied fields, while only 31% of the men are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The choice of topics to teach is only one way of encouraging women to study mathematics, but it is an important way.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-21T12:40:43.217" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="615" PostTypeId="2" ParentId="604" CreationDate="2014-03-21T12:50:18.857" Score="22" Body="&lt;p&gt;It is interesting to see how computer algebra systems deal with this kind of thing.  In Maple, for example, you can do the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Define &lt;code&gt;f := x^3&lt;/code&gt;&lt;/li&gt;&#xA;&lt;li&gt;Enter &lt;code&gt;diff(f,x)&lt;/code&gt; to get the derivative&lt;/li&gt;&#xA;&lt;li&gt;Enter &lt;code&gt;subs(x=3,f)&lt;/code&gt; to evaluate.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Alternatively:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Define &lt;code&gt;g := (x) -&amp;gt; x^3&lt;/code&gt;&lt;/li&gt;&#xA;&lt;li&gt;Enter &lt;code&gt;D(g)&lt;/code&gt; to get the derivative&lt;/li&gt;&#xA;&lt;li&gt;Enter &lt;code&gt;g(3)&lt;/code&gt; to evaluate&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can convert between the two idioms by &lt;code&gt;f := g(x)&lt;/code&gt; and &lt;code&gt;g := unapply(f,x)&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When using Maple for research I find myself mixing the two idioms according to what is convenient for the immediate purpose.  This feels faintly unsatisfactory, but it works.  I used to teach a Maple-based undergraduate course, which also used both idioms.  The difference between them did cause some confusion, but it was not especially prominent compared to other things that the students found difficult.&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-21T12:50:18.857" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="617" PostTypeId="2" ParentId="598" CreationDate="2014-03-21T13:41:16.740" Score="6" Body="&lt;p&gt;I have seen many groups form and dissolve, not in the math field but I believe the observations apply across the board.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Make sure there is always a replacement generation. A club out of classmates ensures they all graduate together and the club is history.&lt;/li&gt;&#xA;&lt;li&gt;Define a clear goal and general lines by rough consensus. Be prepared for the group to evolve in completely unexpected directions.&lt;/li&gt;&#xA;&lt;li&gt;All members are (somewhat) equal. Give serious responsibilities even to junior members (at least soon). Just being allowed to be there, but having no say, discourages everyone.&lt;/li&gt;&#xA;&lt;li&gt;Be selective with new members, but not too much. Too much selectiveness gives a small group, which runs the danger of extinction by random events. Too little selectiveness, and it loses focus.&lt;/li&gt;&#xA;&lt;li&gt;Any group needs at least a modicum of resources to function (a place to meet, some secretarial support, etc.). Make sure that is available.&lt;/li&gt;&#xA;&lt;li&gt;A group of people wants to meet for non-technical reasons too. Organize periodical (say once a term) informal meetings, where shop talk is forbidden.&lt;/li&gt;&#xA;&lt;li&gt;Be &lt;em&gt;very&lt;/em&gt; careful when handling unruly members.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The whole stackexchange network is a prime example of how this is be done successfully (the setting is different, but much of what is done here online translates directly to meatspace).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-21T13:41:16.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="618" PostTypeId="2" ParentId="604" CreationDate="2014-03-21T13:45:52.157" Score="25" Body="&lt;p&gt;Consider the following observations:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;People shorten things with increasing frequency of usage.&lt;/em&gt; For example, the most frequent words are short. This is a way of making communication more effective, e.g. see &lt;a href=&quot;http://en.wikipedia.org/wiki/Huffman_coding&quot; rel=&quot;nofollow noreferrer&quot;&gt;Huffman coding&lt;/a&gt;. For example &lt;a href=&quot;http://en.wiktionary.org/wiki/Wiktionary:Frequency_lists/TV/2006/1-1000&quot; rel=&quot;nofollow noreferrer&quot;&gt;this list&lt;/a&gt; has its first 5-letter word at 39th place (first three are for comparison):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{array}{c|c|c|c}&#xA;\textbf{place} &amp;amp; \textbf{word} &amp;amp; \textbf{letters} &amp;amp; \textbf{count} \\\hline&#xA;1&amp;amp;\text{you}&amp;amp;3&amp;amp;1222421\\&#xA;2&amp;amp;\text{I}&amp;amp;1&amp;amp;1052546\\&#xA;3&amp;amp;\text{to}&amp;amp;2&amp;amp;823661\\&#xA;39 &amp;amp; \text{about}&amp;amp;5 &amp;amp; 142750 \\ &#xA;70 &amp;amp; \text{really} &amp;amp; 6 &amp;amp; 68311 \\ &#xA;93 &amp;amp; \text{something} &amp;amp; 9 &amp;amp; 54736 \\&#xA;95 &amp;amp; \text{because} &amp;amp; 7 &amp;amp; 53566\end{array}$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;If there is only one free variable in expression &lt;span class=&quot;math-container&quot;&gt;$\mathcal{E}$&lt;/span&gt;, saying &amp;quot;function &lt;span class=&quot;math-container&quot;&gt;$\mathcal{E}$&lt;/span&gt;&amp;quot; is unambiguous.&lt;/em&gt; The key word here is &amp;quot;function&amp;quot; which let us know that what follows should be treated as a function, and not just as an expression (with appropriate context it might be not necessary). Such formula might be unambiguous even with multiple free variables, if the convention designates one as such (e.g. &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; versus &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;), compare&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp;\text{function } &amp;amp;x^2&amp;amp;+\sin x &amp;amp;\text{ok}\\&#xA;&amp;amp;\text{function } &amp;amp;ax^2&amp;amp;+\sin bx&amp;amp;\text{ok}\\&#xA;&amp;amp;\text{function } &amp;amp;c^2&amp;amp;+\sin bd&amp;amp;\text{ambiguous} \\&#xA;&amp;amp;\text{function } &amp;amp;d\mapsto c^2&amp;amp;+\sin bd&amp;amp;\text{ok}&#xA;\end{align}&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Human thinking is fuzzy&lt;/em&gt;. Our brains do not process mathematical concepts formally (even if we are capable of it). There is a whole thread about &lt;a href=&quot;https://math.stackexchange.com/q/264610/26306&quot;&gt;notation abuse&lt;/a&gt; at math.se, in particular it includes a great quote by Gila Hanna:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}\text{The student of mathematics has to develop a tolerance}\\\text{for ambiguity. Pedantry can be the enemy of insight. }\end{align}&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I suspect that there are also multiple other reasons, but the above seem to be the most important. That being said, I think it is alright to tolerate such abuse in speech (which is ambiguous and ill-suited for math anyway) or if time constraint is severe. However, I would not accept such a style in homework assignments. Moreover, I urge the teachers to write correctly on the board during their classes, it's not a significant burden and it might help, e.g. to avoid confusion.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="2083" LastEditDate="2024-03-02T10:29:57.343" LastActivityDate="2024-03-02T10:29:57.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="619" PostTypeId="2" ParentId="587" CreationDate="2014-03-21T15:03:29.693" Score="2" Body="&lt;p&gt;You can assign problems for students to work on in class, in pairs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If students get stuck, you can provide hints.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some challenging possibilities, which can be modified to suit the class:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Numerical problems&lt;/strong&gt; with graphing calculators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a. Where is the minimum of $f(x)=x^2+e^{\cos(x)}$?  How many digits of accuracy can you get for $x$ and for $f(x)$ at that point?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b. What is the area between the axes and the curve $x^2+x^3+y^4+y^5=1$?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Smartphone research&lt;/strong&gt; with Google.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a. Why did the ancient Greeks study parabolas?  What does this have to do with derivatives?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b. What is the importance of the number &lt;code&gt;.67449&lt;/code&gt; in statistics?  What does this have to do with integrals? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then you can &lt;strong&gt;discuss&lt;/strong&gt; their different approaches and answers.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorUserId="35" LastEditDate="2014-03-21T16:03:57.347" LastActivityDate="2014-03-21T16:03:57.347" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="620" PostTypeId="1" AcceptedAnswerId="636" CreationDate="2014-03-21T15:07:02.087" Score="7" ViewCount="1165" Body="&lt;p&gt;Racism is not an easy subject to talk about so let me to explain my question by a more soft example. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question originally belongs to Michael Wazowski an undergraduate student of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Monsters_University&quot; rel=&quot;noreferrer&quot;&gt;Monsters University&lt;/a&gt; (MU). He want to become a Scarer. Scarers are heroes of monsters world. They belong to elite floor of the society and provide the energy of monsters world by going to humans world and scaring human children. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to reach his purpose Mike works hard and goes to the Scaring Department of Monsters University. In the university he continues his studies very well and becomes one of the most informed Scaring Theory students of the Scaring Department. But Scaring Department dean Prof. Hardscrabble who is a famous great scarer believes that &quot;&lt;em&gt;scariness is the true measure of a monster. So if you are not scary what kind of a monster are you?&lt;/em&gt;&quot; Based on this philosophy she wants to &quot;&lt;em&gt;make great students greater, not make mediocre students less mediocre.&quot;&lt;/em&gt; and this &lt;em&gt;mediocre&lt;/em&gt; students have no place in her Scaring Program. In her philosophy a part of being a &lt;em&gt;great&lt;/em&gt; student is the student's scary body and face. Unfortunately she believes that Mike has not a scary face and so he never can be a good scarer even if he work on scaring theory for years. Thus dean Hardscrabble simply removes Mike from her scaring program.       &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now back to mathematics. Unfortunately one can find some math professors in some math departments all around the world who believe that having an inborn special ability is absolutely necessary for becoming a good mathematician and one's voluntary efforts cannot help him/her too much in this direction and efforts just can make a &quot;potentially&quot; good math student better and cannot help students who haven't this mathematical &quot;potential&quot; at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't want to discuss on truth or falsity of this point of view. Also I don't want to categorize this philosophy as a kind of racism but the fact is that such a view harms some groups of math students including women, blacks, etc. because these math professors are not interested in &lt;em&gt;wasting&lt;/em&gt; their time on them and notice to their &lt;em&gt;efforts&lt;/em&gt;. They prefer to find their inborn great students and help them to become greater. Here I want to ask my main question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; How can we prevent harming the scientific career of these groups of math students? Precisely if you are a math student/faculty/dean in a mathematics department and one of your professors is a &quot;Hardscrabble&quot; what kind of solutions do you suggest for changing his/her philosophy or at least decreasing the impacts of his/her personal point of view on &quot;Wazowski&quot;s of your department?         &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/TB1Qq.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-04-03T19:24:40.160" LastActivityDate="2023-10-23T19:56:44.923" Title="How do I become a Scarer?" Tags="|undergraduate-education|teachers|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="621" PostTypeId="1" AcceptedAnswerId="645" CreationDate="2014-03-21T15:28:08.197" Score="11" ViewCount="672" Body="&lt;p&gt;When students ask me for the use of the formal and abstract theory, I often would like to give answers they wouldn't understand. For instance, one application of abstract vector spaces and the banach fixed-point theorem is solving differential equations by applying the theory to function spaces. That's too much for a beginner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A nice one is what is called the &quot;football theorem&quot; in german (Satz vom Fußball). Compare a football lying on the center spot before the game starts to the same football lying on the center spot before the 2nd half starts. There will be (at least) 2 point which are exactly in the same position, how ever much the ball was turned during 1st half. The proof is given by the fact that the composition of rotations still is a rotation having an eigenvector in three-dimensional space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is there more like this?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I think of first-year undergraduate students which are confronted with formal definition and proof for their first time and ask &quot;why?&quot;. &lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="75" LastEditDate="2014-03-23T10:43:22.627" LastActivityDate="2015-11-21T10:31:51.887" Title="Example for a theorem where the (more) formal proof is easier than other argumentation (e.g. imagination)" Tags="|undergraduate-education|examples|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="622" PostTypeId="2" ParentId="561" CreationDate="2014-03-21T15:50:36.203" Score="11" Body="&lt;p&gt;Elaborating my comment into a full answer. To me such an error is mostly a notational issue, where the students who instinctively know what they're doing and do get to the right answer, just write it poorly. This is an issue where students aren't use to the formal writing of math just yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually such a formulation would occur after they have seen the fundamental theorem of calculus in a Calc 1 course (as I have seen in my courses at least). If I noticed students doing this, I would provide the example &lt;span class=&quot;math-container&quot;&gt;$F(x) =\int_0^x x^2dx$&lt;/span&gt; and evaluate &lt;span class=&quot;math-container&quot;&gt;$F(1)$&lt;/span&gt; to show how their overuse of the variable x confuses things. Interpreting as a valuation of the area under a curve, this &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is a function on the width of the area under the curve, not the curve itself. Thus they should have differing variables, perhaps written as &lt;span class=&quot;math-container&quot;&gt;$F(x) = \int_0^x t^2 dt$&lt;/span&gt;, and compare this to the fundamental theorem of calculus's statement on this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think mathematics needs a &quot;local variable&quot; concept, the instructor just needs to make it more clear what each variable is of (i.e., x is of width of area (or limit of integration) and t is the integrand). Many students at this level are not use to several variables being used, so renaming them is not a first instinct (book might use x as integrand for example). I find that going to the geometric interpretation helps conceptually ground the multivariate issue here.&lt;/p&gt;&#xA;" OwnerUserId="201" LastEditorDisplayName="user12641" LastEditDate="2019-09-07T21:18:23.270" LastActivityDate="2019-09-07T21:18:23.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="623" PostTypeId="5" CreationDate="2014-03-21T16:08:10.207" Score="0" Body="&lt;p&gt;Motivation is a reason (or reasons) for acting or behaving in a particular way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Motivation describes the internal and external factors that stimulate desire and energy in people to be continually interested and committed to a job, role or subject, or to make an effort to attain a goal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Motivation results from the interaction of both conscious and unconscious factors such as &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the intensity of desire or need, &lt;/li&gt;&#xA;&lt;li&gt;the incentive or reward value of the goal, and &lt;/li&gt;&#xA;&lt;li&gt;the expectations of the individual and of his or her peers. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These factors are the reasons one has for behaving a certain way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example is a student that spends extra time studying for a test because he or she wants a better grade in the class.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-21T16:21:51.530" LastActivityDate="2014-03-21T16:21:51.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="624" PostTypeId="4" CreationDate="2014-03-21T16:08:10.207" Score="0" Body="Motivation is a reason (or reasons) for acting or behaving in a particular way." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-21T16:21:43.090" LastActivityDate="2014-03-21T16:21:43.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="625" PostTypeId="5" CreationDate="2014-03-21T16:15:02.810" Score="0" Body="&lt;p&gt;A proof is evidence or an argument establishing a fact, the non-existence a fact, the truth of a statement, and/or the untruth of a statement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast to scientific proof, which is mostly based on empirical evidence, mathematical-proof tends to be based on logical evidence.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-21T16:21:34.180" LastActivityDate="2014-03-21T16:21:34.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="626" PostTypeId="4" CreationDate="2014-03-21T16:15:02.810" Score="0" Body="For questions about mathematical proofs in an educational context." OwnerUserId="35" LastEditorUserId="39" LastEditDate="2015-01-11T03:37:56.933" LastActivityDate="2015-01-11T03:37:56.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="627" PostTypeId="5" CreationDate="2014-03-21T16:15:52.613" Score="0" Body="&lt;p&gt;A reference request is a request to be provided with (links to) documentation, official papers, and specs related to one or more specific algorithms or mathematical procedures, to provide a trusted base for what's being said or written.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-21T16:34:20.527" LastActivityDate="2014-03-21T16:34:20.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="628" PostTypeId="4" CreationDate="2014-03-21T16:15:52.613" Score="0" Body="A reference request is a request to be provided with (links to) documentation, official papers, and specs related to one or more specific algorithms or mathematical procedures, to provide a trusted base for what's being said or written." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-21T16:34:15.133" LastActivityDate="2014-03-21T16:34:15.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="629" PostTypeId="2" ParentId="561" CreationDate="2014-03-21T16:18:09.367" Score="5" Body="&lt;p&gt;I would say that the x in the variable of integration helps define the shape of the fence, while those at the limit are the fenceposts. Using the same variable name confuses them.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-21T16:18:09.367" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="630" PostTypeId="1" AcceptedAnswerId="906" CreationDate="2014-03-21T16:45:54.433" Score="26" ViewCount="1814" Body="&lt;p&gt;Almost by definition, innovation requires that things be done differently than established custom has it, and comes from the young more often than from the old.&#xA;In a field as old and established as mathematics, it can be difficult to find something genuinely new that hasn't been discovered, and it may require fairly advanced education to get to the forefront of an area were new mathematics is being created and studied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the emphasis on mathematics education is typically on getting the right answer, and often in doing it the right way, and many potentially creative students are likely to be stifled and discouraged long before getting a PhD and getting published.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, how can educators encourage creativity without sacrificing correctness?&lt;/p&gt;&#xA;" OwnerUserId="40" LastEditorUserId="143" LastEditDate="2014-05-24T22:14:52.913" LastActivityDate="2015-08-07T19:27:07.363" Title="How can mathematics educators encourage innovation and creativity?" Tags="|creativity|mathematical-pedagogy|" AnswerCount="9" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="631" PostTypeId="5" CreationDate="2014-03-21T16:47:15.493" Score="0" Body="&lt;p&gt;Interactive teaching is a pedagogical approach that incorporates things like social networking and computing into course design and delivery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interactive teaching incorporates a multitude of teaching styles and goals beneath a single roof and interactive classes are designed around a simple principle: &lt;em&gt;without practical application, students often fail to comprehend the depths of the study material&lt;/em&gt;. The according interactive training styles provide four basic forms of feedback:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Measurable student accomplishments&lt;/strong&gt;&lt;br&gt;&#xA;Teachers making use of interactive teaching styles are better equipped to access how well students master a given subject material.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Flexibility in teaching&lt;/strong&gt;&lt;br&gt;&#xA;Applying training methods that involve two-way communications enable the teacher to make quick adjustments in processes and approaches.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Practice makes perfect&lt;/strong&gt;&lt;br&gt;&#xA;Interactive instruction enhances the learning process.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Student motivation&lt;/strong&gt;&lt;br&gt;&#xA;Two-way teaching dispels student passivity.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-21T17:30:18.807" LastActivityDate="2014-03-21T17:30:18.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="632" PostTypeId="4" CreationDate="2014-03-21T16:47:15.493" Score="0" Body="Interactive teaching is a pedagogical approach that incorporates things like social networking and computing into course design and delivery." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-21T19:10:22.367" LastActivityDate="2014-03-21T19:10:22.367" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="633" PostTypeId="5" CreationDate="2014-03-21T16:53:49.753" Score="0" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Computer_science&quot; rel=&quot;nofollow&quot;&gt;Computer science&lt;/a&gt; is the study of the principles and use of computation. It spans topics that are theoretical in nature and very mathematical to more applied topics that veer towards engineering.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fields of computer science include logic, information theory, theory of computation, algorithms, programming language theory, artificial intelligence, computer architecture, graphics and vision, databases, security and cryptography, databases, software engineering, etc.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="123" LastEditDate="2014-05-25T02:21:40.130" LastActivityDate="2014-05-25T02:21:40.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="634" PostTypeId="4" CreationDate="2014-03-21T16:53:49.753" Score="0" Body="Computer science is the study of the principles and use of computation." OwnerUserId="35" LastEditorUserId="170" LastEditDate="2014-05-25T02:21:45.223" LastActivityDate="2014-05-25T02:21:45.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="635" PostTypeId="2" ParentId="630" CreationDate="2014-03-21T18:03:47.973" Score="4" Body="&lt;p&gt;My brief answer would include: interdisciplinary, integration with arts, open-ended problems. Of course, right answers are important as long as we can nurture the idea that not every question has one perfect answer. Instead of What is 2+3? I like asking children to find all dominoes that have total of 5 dots. I ask the same questions future teachers, broadening the domain ...&lt;/p&gt;&#xA;" OwnerUserId="95" LastActivityDate="2014-03-21T18:03:47.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="636" PostTypeId="2" ParentId="620" CreationDate="2014-03-21T18:28:53.773" Score="7" Body="&lt;p&gt;I do not think you can change the opinions of professors who hold these views.  All you can do is be a human:  interact with undergrads, and let them know that you do not think inherent ability is needed to do good mathematics, that it is mostly about hard work and perseverance, that mathematics needs all kinds of people (great expositors, people writing software that assists mathematics, highly visual thinkers, highly non-visual thinkers, etc...).   Tell them to read Terry Tao's article about genius in mathematics.  Just do all you can do to make people feel okay loving math!  The more people who do this, the more likely that the culture will change in the long run.  This is a battle for a cultural shift, and those take time.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-03-21T18:28:53.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="637" PostTypeId="2" ParentId="597" CreationDate="2014-03-21T19:24:36.630" Score="2" Body="&lt;p&gt;As &lt;a href=&quot;http://en.wikipedia.org/wiki/Sturgeon%27s_law&quot; rel=&quot;nofollow&quot;&gt;Sturgeon's revelation&lt;/a&gt; (&quot;90% of everything is crap&quot;) teaches us, it depends.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd say most of what you'll find off-the-shelf can motivate a general appreciation of , and perhaps get some key points across. To really teach requires a much more organized and larger effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My favorite example of this is Sesame Street. It manages to teach small children with some success. But the group behind its production is &lt;em&gt;massive&lt;/em&gt;, and the resources devoted equally so. And they have the advantage that it is quite unlikely that a breakthrough changes the order of the letters in the alphabet any time soon, while in our subject matter we expect such all the time (perhaps not in the subject itself, but at least in how to teach it effectively).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-21T19:24:36.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="638" PostTypeId="2" ParentId="602" CreationDate="2014-03-21T19:35:04.520" Score="7" Body="&lt;p&gt;Talk to them, make them see that you know they know the answers, but that you want to ensure that other (slower) students think it over by themselves before giving the answer. This has done the trick the times I had such a problem. Make a point of asking others first (sometimes/most of the time), then give them the opportunity to speak up. Perhaps ask just them about some tricky point now and then. Or some silly, &lt;em&gt;simple&lt;/em&gt; point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes they are just too eager to talk, and teaching them a bit of restraint helps them, not just for your class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Managed well, they can become an asset.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-21T19:35:04.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="639" PostTypeId="2" ParentId="588" CreationDate="2014-03-21T19:44:21.490" Score="4" Body="&lt;p&gt;As &lt;a href=&quot;https://matheducators.stackexchange.com/a/601&quot;&gt;@MarkusKlein says&lt;/a&gt;, get a bigger room (or rooms). If several rooms, allocate students to rooms &quot;randomly,&quot; e.g. by last name. Before starting the exam, mix them up. I.e., those sitting in the last row while the first is empty get invited forwards. Break up pairs of buddies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A colleage here makes exams with two (or even more) versions, to be handed out by columns (or rows). Sure, it is harder to make up two equivalent exams, but if I know that my neighbor has a different test than mine, looking around helps little.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is more problematic is the use of cellphones, during one test of mine some tried to download the lecture notes... wouldn't have done them too much good, the questions weren't in them anyway.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-04-08T06:46:10.957" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="640" PostTypeId="2" ParentId="575" CreationDate="2014-03-21T19:44:31.943" Score="3" Body="&lt;p&gt;In this class one of the most important things to work for is naturalness.  Coming in on the first day and writing first thing &quot;A FSA is a five-tuple ..&quot; seems to me to say to students that you are going to state and they are going to write, and there will be very little active learning.  So I prefer to start the course by developing a definition of Turing Machines with a combination of discussion of the history and prompting them about what could be meant by &quot;mechanical computation.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that I introduce studying Finite State Automata as focussing on the CPU of a TM.  To give it input it needs a read tape, but to make it not a TM it needs a read-only tape. That seems natural to students.  Then, with some experience (and some leading examples) students catch on to what languages can be accepted, and when I prompt they give the essence of regular expressions themselves.  I personally don't present an algebraic approach at all.&lt;/p&gt;&#xA;" OwnerUserId="213" LastActivityDate="2014-03-21T19:44:31.943" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="641" PostTypeId="2" ParentId="530" CreationDate="2014-03-21T19:46:12.940" Score="21" Body="&lt;p&gt;There is a simple solution to this debate:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://xkcd.com/1292/&quot; rel=&quot;noreferrer&quot; title=&quot;Conveniently approximated as e+2, Pau is commonly known as the Devil&amp;#39;s Ratio (because in the octal expansion, &amp;#39;666&amp;#39; appears four times in the first 200 digits while no other run of 3+ digits appears more than once.)&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hXQjd.png&quot; alt=&quot;1.5 pi = pau&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="-1" LastEditDate="2015-03-14T03:55:01.460" LastActivityDate="2015-03-14T03:55:01.460" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="642" PostTypeId="2" ParentId="602" CreationDate="2014-03-21T20:14:43.173" Score="5" Body="&lt;p&gt;I was prone to do this myself, in 6-8th grades. My teacher finally told me that I was going to be the one he went to last, when no one else had the right answer.&#xA;I realized that I didn't have to be the first to answer a question, I started paying attention to the other students (as well as the teacher) so I could figure out whether I indeed had the right answer, and that my first answer wasn't always right.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-21T20:14:43.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="643" PostTypeId="1" CreationDate="2014-03-21T20:39:57.503" Score="16" ViewCount="1142" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Peer_instruction&quot;&gt;Peer Instruction&lt;/a&gt; is a method developed by Eric Mazur in Harvard, designed with a student-centered approach in mind. In a nutshell, the core of the method is that when presented with a problem, students who readily overcome the obstacles necessary to understand the solution are in a much better place to explain and convince other students of what is wrong with their reasoning, dealing with specific obstructions in acquiring the new concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Could this peer-instruction method be successfully adapted for widespread use in a mathematics classroom? Could it be effectively employed as a guiding principle for a course?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On this site, we have dealt with questions regarding tutoring and class participation. This question is a combined form of those, in a sense. In real time, we seek to find out who is understanding the concepts, what misconceptions exist amongst the students, and how to get them to learn from each other. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good answers might share experiences you have had with this method, on any scale, although direct experience with using this for an entire course is preferable. Good answers also might include opinions/research about peer learning, in general, and suggestions for how to optimize student learning via these ideas.&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="80" LastEditDate="2014-03-22T20:19:05.043" LastActivityDate="2014-03-27T01:01:29.867" Title="Is Peer Instruction suited to mathematics classroom?" Tags="|mathematical-pedagogy|undergraduate-education|course-design|interactive-teaching|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="644" PostTypeId="2" ParentId="257" CreationDate="2014-03-21T20:42:09.920" Score="3" Body="&lt;p&gt;This was the term-paper assignment for the graduate class I took on differential geometry:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take a paper in differential geometry published in the past 10 years.  Explain what it draws on, what it means, and why it is important.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-21T20:42:09.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="645" PostTypeId="2" ParentId="621" CreationDate="2014-03-21T20:46:40.187" Score="8" Body="&lt;p&gt;To give a concrete example, consider the following task in geometry:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Let $A,B,C,D$ be four distinct intersection points of two parabolae with perpendicular axes of symmetry. Prove that $A,B,C,D$ are concyclic.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is an almost trivial formal proof using analytic geometry. Any other approach I know is close to nightmare. You can find more about it at math.se: &lt;a href=&quot;https://math.stackexchange.com/q/134366/26306&quot;&gt;intersection of two parabolae&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you dislike this particular example, there are combinatorial identities which are quite easy to prove formally (e.g. by induction or some manipulation), but rather hard via counting arguments. There were even some open problems (I don't know if those are open anymore) about identities for which there was no combinatorial interpretation known.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another counter-intuitive example is about vector spaces (see &lt;a href=&quot;https://mathoverflow.net/a/23501/48565&quot;&gt;here&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\dim(U +V + W) \neq \dim U + \dim V + \dim W - \dim (U \cap V) - \dim (U \cap W) - \dim (V \cap W) + \dim(U \cap V \cap W)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;despite quite obvious&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\dim (U + V) = \dim U + \dim V - \dim (U \cap V).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although it is possible to describe it intuitively, it's much better to just give a counterexample (which constitutes a formal proof).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-03-21T20:46:40.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="646" PostTypeId="2" ParentId="530" CreationDate="2014-03-21T21:39:13.043" Score="4" Body="&lt;p&gt;I only became aware of τ a few weeks ago. It's clever, and in theory, might simplify the teaching a bit. If one could wave a wand and rewrite every text book, and change every pi key on a calculator to tau, I'm in. Even though it would be tough for me to rewire my brain away from the pi related equations I've come to memorize. Not to mention the &lt;a href=&quot;http://en.wikipedia.org/wiki/Piphilology&quot; rel=&quot;nofollow&quot;&gt;mnemonic poem&lt;/a&gt; that helped me memorize the first 20 digits of pi.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Short of that, the question is one of transition. There will be a period of time one needs to learn both numbers as a way to bridge the gap. I am still getting over the trauma of losing an entire planet. I wonder when the last textbook calling Pluto a planet will be retired. How do you propose to replace every math text? &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-10T12:31:17.947" LastActivityDate="2015-03-10T12:31:17.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="647" PostTypeId="1" AcceptedAnswerId="648" CreationDate="2014-03-21T21:45:55.890" Score="24" ViewCount="2308" Body="&lt;p&gt;For example, I gave an exam earlier today with a problem that ended in the sentence&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Use the chain rule to find $(f\circ g)'(3)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;During the exam, one of the students asked me what the circle between the $f$ and $g$ means, and I answered that it represents the composition of functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later on, another student was working on a question that contained the phrase&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;. . . where $V$ represents the volume of water in the tank.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and also the phrase&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given that water initially drains from the tank at a rate of $2.0\;\mathrm{L}/\mathrm{min}$ . . .&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The student asked whether $2.0\;\mathrm{L}/\mathrm{min}$ is the value of $\dfrac{dV}{dt}$, and I said I couldn't help with that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would you have answered these students' questions?  How do you decide what to do in these situations?  I usually handle these on a case-by-case basis, but I'm curious if anyone has any good general rules for deciding what information to give.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-03-21T21:58:55.987" LastActivityDate="2014-04-14T15:40:22.220" Title="What are some good rules for handling student questions during exams?" Tags="|classroom-management|exams|" AnswerCount="6" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="648" PostTypeId="2" ParentId="647" CreationDate="2014-03-21T22:10:08.320" Score="15" Body="&lt;p&gt;In some cases (e.g., during an open-book exam), you can opt for no-questions policy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I think that case-by-case is the best option. Each time a student asks a question, what I ask myself is (with some stretch, you could call this a “general rule”),&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{Would the answer give the student unfair advantage?}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such an approach might mean that&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;some clarifications would be told to the whole group if spotted early;&lt;/li&gt;&#xA;&lt;li&gt;even sensible questions could be left unanswered if asked late.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To consider your examples, in my opinion explaining notation would not give the student an unfair advantage (i.e., the content of the tasks should be widely accessible). On the other hand, acknowledging the connection between $2.0 \mathrm{L/min}$ and $\frac{dV}{dt}$ would (this is already beyond the basic, literal understanding of the problem).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-28T18:52:30.203" LastActivityDate="2014-03-28T18:52:30.203" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="649" PostTypeId="2" ParentId="647" CreationDate="2014-03-21T22:31:14.463" Score="4" Body="&lt;p&gt;In a course I taught a while back I made it a rule: &quot;I ask &lt;em&gt;you&lt;/em&gt; 4 questions, each of you gets to ask &lt;em&gt;me&lt;/em&gt; at most 4 questions.&quot; Did wonders to cut down on dumb questions. And obviously, the only questions that get answered (but all count as asked) are questions about what is being asked. &quot;Can this be done by...&quot; gets no answer, counts as a question. This only works for relatively small groups (I had 15 to 20 then).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-21T22:31:14.463" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="650" PostTypeId="2" ParentId="647" CreationDate="2014-03-21T22:37:37.763" Score="15" Body="&lt;p&gt;Since &lt;a href=&quot;https://matheducators.stackexchange.com/a/648&quot;&gt;dtldarek's answer&lt;/a&gt; addresses well the issues of fairness to students, I'll mention another consideration. When writing exam questions, I try to make sure each question has a certain &lt;em&gt;intent&lt;/em&gt;, that it probes the student's knowledge of a certain concept/definition/technique (or some combination). When a student asks a question during the exam, I have these goals in mind, and I will not reveal any information about what is supposed to be probed by the relevant question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, consider your example with $(f\circ g)'(3)$. Are you probing the student's ability to recognize the applicability of, and subsequently use, the chain rule? If that is your goal of the question, then reminding them that $f\circ g$ means composition is totally fine. Or, have you been stressing the importance of notation in class? Is this question more about probing their ability to understand various ways of writing functions and compositions? If so, I would rather &lt;em&gt;not&lt;/em&gt; tell them what the circle means and instead say, &quot;I cannot answer that. That's the point of the question.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, ultimately, I'm advocating a second part to dtldarek's general rule: &lt;strong&gt;Would my answer adversely affect the intent of the exam question?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-07T20:56:04.533" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="651" PostTypeId="1" CreationDate="2014-03-21T22:51:42.043" Score="11" ViewCount="444" Body="&lt;p&gt;I have a motto, &quot;My class is not an Olympic swimming pool! It is a museum full of the masterwork paintings! Relax and enjoy all of this mathematical beauty which I am drawing on the blackboard!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By this sentence I mean that in my point of view &lt;em&gt;the best mathematicians are those who enjoy beauty of math more not those who enjoy to be called as a great mathematician.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't like the &lt;em&gt;competition&lt;/em&gt; atmosphere in my class because students effort to prove themselves as the best mathematicians of the class is contradictory with my teaching philosophy. Also it harms the team work morale of this small society. In this direction I try to avoid any kind of unnecessary &lt;em&gt;comparison&lt;/em&gt; between my students because it commences a competition amongst them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand I try to encourage them to team works for exploring mathematical wonderland and discovering more and more beauties of the teaching subject. The personal improvement of each student follows from this team works naturally and what is much more important is their ethical improvement during these group activities which learn them how to work together for the benefits of all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1.&lt;/strong&gt; How can I encourage my students to work together as a team? What are the practical ways for achieving this goal? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2.&lt;/strong&gt; Tests in any sense (including exams, quizzes, etc.) increase the comparison (and so the competition) atmosphere of the community of my students. How can I design a test free course with fair enough marks which are acceptable for both students and me at the end of semester? In the other words, is there a more &quot;soft&quot; testing method different from the classic exams with an acceptable accuracy which measures the team work morale of each student?     &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark.&lt;/strong&gt; I don't believe in numerical marks for measuring students at all. I think it is too simple-minded (and sometimes too harmful) to refer the abilities of a human just by a number. But unfortunately it seems this inappropriate method will be the common method of almost all math departments at least in the short future because there is no other known effective substitute for it.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please explain your ideas and suggestions even if they are strange, unnatural and completely new.&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastActivityDate="2018-10-04T22:46:28.937" Title="How can I encourage my students to team work?" Tags="|course-design|exams|testing|test-design|grading|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="652" PostTypeId="2" ParentId="651" CreationDate="2014-03-22T00:04:06.660" Score="7" Body="&lt;p&gt;Well, the method of &lt;a href=&quot;http://en.wikipedia.org/wiki/Randy_Pausch&quot;&gt;Randy Pausch&lt;/a&gt; (or perhaps of &lt;a href=&quot;http://en.wikipedia.org/wiki/Walt_Disney_Imagineering&quot;&gt;Disney Imagineering&lt;/a&gt;) seems to be a perfect answer to your questions. It is worth noting that his course was practical, hence the approach might not work for more theoretical courses.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There were multiple projects during the semester (for Randy Pausch it was 5).&lt;/li&gt;&#xA;&lt;li&gt;Tight deadline (i.e. two weeks to design, implement and test).&lt;/li&gt;&#xA;&lt;li&gt;Randomly chosen teams (4 people per team), change per project.&lt;/li&gt;&#xA;&lt;li&gt;The projects would be presented at the end.&lt;/li&gt;&#xA;&lt;li&gt;Students would give their peers marks on &quot;how easy it was to cooperate with them&quot;.&lt;/li&gt;&#xA;&lt;li&gt;There would be a teamwork-feedback after each project.&lt;/li&gt;&#xA;&lt;li&gt;The peer-evaluation would be transformed into a ranking during the final presentation (from what I understood, it was not a part of the grade).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are more details given in his famous &lt;a href=&quot;https://www.youtube.com/watch?v=ji5_MqicxSo&quot;&gt;last lecture&lt;/a&gt;, a lot of worthy content, inspirational and deeply moving; I recommend it very much!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;To anyone who had actually attend one of his courses, or knows more details from any other source:&lt;/em&gt; &#xA;&lt;br/&gt;I would greatly appreciate sharing your experiences/information.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-22T00:15:22.463" LastActivityDate="2014-03-22T00:15:22.463" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="653" PostTypeId="2" ParentId="553" CreationDate="2014-03-22T01:48:49.507" Score="12" Body="&lt;p&gt;I once asked Ted Slaman, after one of his (typically) excellent colloquium talks, what is the secret to giving such a great talk, and his advice to me was: you've got to think like a comedian. What he meant, of course, is not that one is supposed to tell jokes, but rather, that one should explain mathematical ideas with an appreciation for timing and the  arc of the story line. You've got to set the stage for your mathematical point, so that the audience (or students) attain the Aha! moment right as your explanation reaches its crescendo, just as a comedian sets up and then delivers the punchline. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three logicians walk into a bar, and the bartender asks, &quot;do all three of you want a drink?&quot; The first one says, &quot;I don't know...&quot;; the second one says, &quot;I don't know...&quot;; and then the third one says, &quot;Yes!&quot;.&lt;/p&gt;&#xA;" OwnerUserId="90" LastActivityDate="2014-03-22T01:48:49.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="654" PostTypeId="1" AcceptedAnswerId="731" CreationDate="2014-03-22T03:02:40.223" Score="30" ViewCount="5544" Body="&lt;p&gt;Mathematicians use (Powerpoint/Beamer) slides for their lectures. My question is about using slides for teaching math. There are several positive and negative arguments about teaching by slide show. e.g.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Positive:&lt;/strong&gt; One can show many mathematical processes by animations in Powerpoint. This will help students to find a better intuition about what is happening in the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Negative:&lt;/strong&gt; Students cannot follow slides as good as blackboard because the lecture speed in slide based lectures is higher than blackboard based teaching method. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What are possible advantages and disadvantages of teaching math by slides? How should one use slides for teaching math such that its disadvantages become minimum.&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-23T01:56:37.327" LastActivityDate="2020-05-04T10:26:31.630" Title="Teaching by Slides, Yes or No?" Tags="|course-design|technology-in-education|interactive-teaching|" AnswerCount="9" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="655" PostTypeId="5" CreationDate="2014-03-22T09:14:40.597" Score="0" Body="&lt;p&gt;An assignment given to a student to be completed outside the regular class period (aka &lt;em&gt;“at home”&lt;/em&gt;; which is why it is called “homework”).&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-22T10:55:34.260" LastActivityDate="2014-03-22T10:55:34.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="656" PostTypeId="4" CreationDate="2014-03-22T09:14:40.597" Score="0" Body="For questions concerning homework, such as good problems for specific topics, amount and difficulty of homework, grading homework, cheating" OwnerUserId="35" LastEditorUserId="114" LastEditDate="2014-04-05T14:40:12.417" LastActivityDate="2014-04-05T14:40:12.417" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="657" PostTypeId="5" CreationDate="2014-03-22T09:18:46.490" Score="0" Body="&lt;p&gt;Algebra is the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;x + 3 = 7&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;where &lt;code&gt;x&lt;/code&gt; is unknown… often called the “variable”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case the value of &lt;code&gt;x&lt;/code&gt; can be found by subtracting 3 from both sides of the equal sign:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;x + 3 = 7&#xA;x + 3 - 3 = 7 - 3&#xA;x + 0 = 4&#xA;&#xA;Answer: x = 4&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-22T15:19:02.920" LastActivityDate="2014-03-22T15:19:02.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="658" PostTypeId="4" CreationDate="2014-03-22T09:18:46.490" Score="0" Body="Algebra is the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-22T15:18:54.167" LastActivityDate="2014-03-22T15:18:54.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="659" PostTypeId="2" ParentId="654" CreationDate="2014-03-22T10:18:52.170" Score="10" Body="&lt;p&gt;One advantage of slides (I don't use PowerPoint, I use Beamer) is that it is a memory prompt that keeps me on track and in order.  However, like many people I use a combination: I might have a definition on the slide and do examples at the board.  That helps me keep to a pace they can follow and also helps me avoid being just a slide reader.&lt;/p&gt;&#xA;" OwnerUserId="213" LastActivityDate="2014-03-22T10:18:52.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="660" PostTypeId="2" ParentId="654" CreationDate="2014-03-22T14:53:18.650" Score="7" Body="&lt;p&gt;Creating slides is a lot of work. Using them exclusively makes handling random questions form the audience difficult (specially when they require a different derivation). And I like to adjust examples to data collected from the audience (or even have the audience conjure a problem, say build a graph or propose a regular expression), that becomes impossible with slides. Such interaction keeps them on their toes, and solving an arbitrary problem proposed by them with their help reinforces that whatever method is discussed works in general.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-22T14:53:18.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="661" PostTypeId="1" AcceptedAnswerId="664" CreationDate="2014-03-22T15:20:17.250" Score="9" ViewCount="1710" Body="&lt;p&gt;I've taught several upper-level courses recently, and have considered giving take-home exams to reduce the time pressure in the classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, their homework already consists of proving various exercises similar to what I would imagine to be on a takehome exam.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How is a takehome exam different from regular homework? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastEditorUserId="105" LastEditDate="2014-03-23T13:39:42.277" LastActivityDate="2014-03-23T13:39:42.277" Title="How does a take-home exam differ from a homework assignment?" Tags="|undergraduate-education|exams|homework|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="662" PostTypeId="5" CreationDate="2014-03-22T15:31:03.023" Score="0" Body="&lt;p&gt;Writing is a medium of communication that represents language through the inscription of signs and symbols.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics writing is different from ordinary writing because – in addition to the requirements of ordinary good writing – there are additional constraints and conventions in mathematics. For example: an additional constraint is that mathematics follows much more demanding rules of logic than ordinary discourse, and mathematical writing must follow and display this logic. Also, additional conventions are those for defining new concepts and those for organizing the material through theorems and examples.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-03-22T16:12:11.540" LastActivityDate="2014-03-22T16:12:11.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="663" PostTypeId="4" CreationDate="2014-03-22T15:31:03.023" Score="0" Body="For questions about student writing in mathematics, including considerations about assigning, grading, motivating, and improving students' written work." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T20:43:14.970" LastActivityDate="2014-04-01T20:43:14.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="664" PostTypeId="2" ParentId="661" CreationDate="2014-03-22T15:42:06.383" Score="12" Body="&lt;p&gt;A homework assignment is there to help the students learn.  A takehome exam is a test of the students' abilities and understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a practical standpoint, a few of the main differences are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I usually encourage students to work together in groups on solving homework problems.  By contrast, students must work individually when solving problems on a takehome exam.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When a student comes to office hours with questions about a homework problem, I will usually give them a hint every time they get stuck, working with the student until they are finally able to solve it.  By contrast, I will usually not give any significant hints for problems on a takehome exam, except for very basic things like &quot;you might find some of the theorems in chapter 3 helpful&quot;, or &quot;there was a homework problem kind of like this a few weeks ago&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similarly, I am usually willing to critique rough drafts of written proofs for homework problems during office hours.  I will not critique any writing for a takehome exam until I grade it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I usually make the problems on a takehome exam a little easier than homework problems, and the takehome exam counts more than a weekly homework.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Philosophically speaking, I tend to think of the students' homework grades as measuring the amount of effort they put into the course.  A student who works with other students, comes to all of my office hours, shows me rough drafts of their proofs and so forth really ought to be able to solve all of the homework problems and write them up well.  By contrast, a takehome exam really tests the student's understanding of the material, general problem-solving ability, and ability to write proofs clearly without help.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-03-22T15:48:34.903" LastActivityDate="2014-03-22T15:48:34.903" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="665" PostTypeId="2" ParentId="150" CreationDate="2014-03-22T15:49:43.137" Score="9" Body="&lt;p&gt;From &quot;&lt;a href=&quot;http://www.ams.org/notices/201005/rtx100500608p.pdf&quot;&gt;What is Mathematics For?&lt;/a&gt;&quot; by Underwood Dudley (&lt;em&gt;Notices of the AMS&lt;/em&gt;, vol. 57, no. 5, May 2010):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So that there is no confusion, let me say that by “mathematics” I mean algebra, trigonometry, calculus, linear algebra, and so on: all those subjects beyond arithmetic. There is no question about what arithmetic is for or why it is supported. Society cannot proceed without it. Addition, subtraction, multiplication, division, percentages: though not all citizens can deal fluently with all of them, we make the assumption that they can when necessary. Those who cannot are sometimes at a disadvantage.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Algebra, though, is another matter. Almost all citizens can and do get through life very well without it, after their schooling is over. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Jobs do not require algebra. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Even those who are not burdened with the error that algebra is necessary to hold many&#xA;  jobs support the teaching of algebra. Everyone supports the teaching of algebra. The public wants more mathematics taught, to more students. The requirements keep going up, never down. The reason for this, I am convinced, is that the public knows, or senses, that mathematics develops the power to reason. It shows, better than any other subject, how reason can lead to truth. Of course, other sciences exhibit the power of reason, but there’s all that overhead—ferrous and ferric, dynes and ergs—that has to be dealt with. In mathematics, there is nothing standing between the problem and the reasoning.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Economists reason as well, but sometimes two economists reason to two different conclusions. Philosophers reason, but never come to any conclusion. In mathematics problems can be solved, using reason, and the solutions can be checked and shown to be correct. Reasoning needs to be learned, and mathematics is the best way to learn&#xA;  it.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What mathematics education is for is not for jobs. It is to teach the race to reason. It does not, heaven knows, always succeed, but it is the best method that we have. It is not the only road to the goal, but there is none better. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-03-22T15:49:43.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="667" PostTypeId="1" CreationDate="2014-03-22T16:52:11.827" Score="9" ViewCount="179" Body="&lt;p&gt;Why is it so uncommon to find material that shows students how everything is connected?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A recurrent question from students is why are polynomial so important? If we shown them how these are related to real functions and to differential equations, they would be more interested in learning I think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Otherwise, processing polynomials is just a mechanical ability. The same applies to second degree equations. Many, really many students know how to solve them, but it's quite rare to find those who know what real world phenomena they describe. &lt;/p&gt;&#xA;" OwnerUserId="237" LastActivityDate="2014-03-23T01:57:40.063" Title="Motivating student by showing them where we are heading to" Tags="|student-motivation|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="668" PostTypeId="2" ParentId="667" CreationDate="2014-03-22T17:20:17.140" Score="7" Body="&lt;p&gt;Unfortunately, it is difficult to give such a general treatment as you describe without at least a little groundwork first.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the connections between polynomial and differential equations is interesting, but it would hard (or perhaps impossible) for a student to appreciate this without first knowing what a polynomial is, and what a differential equation is. By all means, when discussing linear differential equations it's a great idea to emphasise the connection between them and polynomials - but is it reasonable to do motivate the solutions to polynomials with differential equations, before the student even knows what differentiation is?&lt;/p&gt;&#xA;" OwnerUserId="145" LastActivityDate="2014-03-22T17:20:17.140" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="669" PostTypeId="2" ParentId="661" CreationDate="2014-03-22T17:24:54.183" Score="3" Body="&lt;p&gt;I'm not too excited about take-home exams, precisely for the reasons mentioned by OP. I prefer to have a clear-cut difference between homework (show that you can do the computations, write down clean and documented proofs, ...) and exams (show that you understand the subject matter, are able to explain it and note relationships, say what is applicable to a given situation, ...). If the matter turns out just too long, what I have done is to split one (or even various) problems up. I.e., do step one starting from this; assume that at the beginning of an iteration you have ... and show the state at the end of the iteration; if the state is this one, is the iteration done (and what is the result). You don't want to check if they can do all the calculations in detail, you are interested in assessing their understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, time pressure is a problem. I schedule exams for saturdays (rooms are empty,it is easy to get enough so the &quot;look a the neighbor's work&quot; isn't too much of a problem, and it is also possible to reserve rooms for two lecture blocks in a row (to have leeway for getting everyone seated, exams handed out, any last minute instructions, possible stragglers, ...). This reduces time pressure. If the estimate of the time turns out too short, there is no risk of getting thrown out by the next class.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-22T17:24:54.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="670" PostTypeId="2" ParentId="630" CreationDate="2014-03-22T17:29:25.793" Score="4" Body="&lt;p&gt;It helps getting students interested in some of the more &quot;recreational mathematics&quot; type sites and problems. Solving some not-completely-serious problem that interests them is as much creative and intellectually demanding (and often less demanding in specific background, a impassable roadblock for beginners) than &quot;serious&quot; work.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-22T17:29:25.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="671" PostTypeId="2" ParentId="654" CreationDate="2014-03-22T17:31:01.177" Score="15" Body="&lt;p&gt;I know a lot of people use slides to teach, but I cannot imagine doing so myself.  Here's why:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Slides are boring.  It's vital to good teaching to keep the students interested in what's happening, and one of the best ways to do this is to keep things spontaneous.  Slides kill spontaneity.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Slides are one-way.  They are entirely about the professor conveying information to the students.  But this isn't how a classroom should work at all -- there should be a give and take between the students and the professor, with the professor responding to students' questions, and deciding what to do next based on how well the audience seems to understand things so far.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Slides are inflexible.  What do you do during class if a student asks a question and you decide to talk about something unexpected for 20 minutes?  What if your lesson plan isn't working and you decide that you need to talk about some prerequisite topic for most of class?  Using slides forces you to stay on a certain track.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;That being said, there is a huge advantage to having good pictures or animations for certain topics, and in some classes I routinely have my laptop attached to the projector for this purpose.  But I see a huge difference between showing pictures and animations vs. having &quot;bullet-list&quot; style PowerPoint slides.  Among other things, pictures and animations are interesting, and bulleted lists of talking points are not.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-22T17:31:01.177" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="672" PostTypeId="2" ParentId="654" CreationDate="2014-03-22T22:26:26.453" Score="9" Body="&lt;p&gt;Unfortunately, nowadays there are more and more university lecture rooms where there is no proper blackboard, so we are facing this question even if we do not want to do anything with technology. &#xA;I have been using slides (powerpoint) for over ten years now in large lectures. They have certainly advantages and disadvantages. Some were mentioned earlier, so there will be some repetition here. &lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Pro:&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I try to combine traditional blackboard lecture with slides.&#xA;I do not use slides as static objects, but use them with a tablet device which allows me to use m handwriting. I usually comment, underline, write out details when necessary. I usually do not work out every detail in advance so that I can adapt my presentation to the needs of the audience.&#xA;Slides are also great to include graphics and animations, even small movies into your presentation. This is something phenomenal if you teach multi-variable calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can do the same by combining presentation with traditional blackboard (if there is one), you do not need a tablet.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Con:&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;It is much easier to give a boring lecture with powerpoint than with blackboard. Having the slides prepared, there is a great temptation not to prepare for the lecture. The missing mental tension makes the whole game tame. At least for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also a great temptation, as already mentioned, to be too quick. I usually build up my formulas in several steps to imitate the speed of handwriting. I have to admit that I do not always succeed.&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-03-22T22:26:26.453" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="673" PostTypeId="2" ParentId="550" CreationDate="2014-03-23T00:14:49.023" Score="8" Body="&lt;p&gt;Well, of course, if your plan is not working, then you have to adjust it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is easier to do, of course, if you are in the happy situation where you find that the students are able to handle much more difficult material. One can simply cover the planned material more quickly and supplement with more sophisticated asides or extensions of that material. This is relatively easy to do, and often quite enjoyable, since it is always a pleasure to teach students who are able and eager to learn more. One simply provides them with more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A much more difficult case, in contrast, occurs when one realizes that the students are weaker than had been expected and perhaps not able to master the material that had been intended for the course. In this case, the instructor may find him or herself at a loss. The matter may come to a head if the course is part of a long chain of courses, where the students must really know the stuff before the next course. In that case, one thing to do is go into emergency mode, concentrating the course on the most essential easiest parts, and try to give the students what they need to advance. But my considered opinion is that it is actually rarely the case that students must really master a set syllabus, even when the course is part of a long chain of courses. So instead, one can adjust the course to the students, providing them with suitable material that enriches their mathematical lives, even if this isn't what was originally planned. One must re-focus the course to be suitable for the students taking it. (This situation would probably be accompanied at the end by lower grades than usual for the course, in light of the reduced level of the material.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal style of teaching tends to have a lot of flexibility, where I follow new topics that come up in class, or respond to students ideas by exploring a related topic more fully. So I find it easy to make this kind of change in stride, both for advancing the course and for slowing it down, depending on what is called for. But I know that other instructors like to have a more rigid plan, with topics and assignments planned long in advance, and I can see that this would make it more difficult to accommodate changes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apart from those kinds of issues, another kind of a difficulty can arise when one realizes part way into the course that the chosen textbook is not suitable for some reason. Perhaps one realizes after the course has begun that the textbook is too difficult or focuses too much on the wrong topics, in a way that cannot be easily overcome. Fortunately, this has happened to me only once in my teaching career (about twenty years), but what a problem when it does happen! What I did was to essentially abandon the unsuitable textbook, and begin essentially to use another text. I provided print-outs to the students of the new material, as well as more thorough notes than usual. In the end, the course became refocused on the ideas I had wanted, in a way that was appropriate for the students. But of course, it would have been better if I had realized the unsuitability of the text much earlier.&lt;/p&gt;&#xA;" OwnerUserId="90" LastActivityDate="2014-03-23T00:14:49.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="674" PostTypeId="2" ParentId="667" CreationDate="2014-03-23T01:02:58.687" Score="0" Body="&lt;p&gt;Perhaps it is just that there are too many connections. What is intensely relevant to me (mostly in computer science) would have gotten a blank stare from my classmates in other fields. What even some run-of-the-mill themes like polynomials' relevance is to me has almost no relation to their use in e.g. chemical engineering.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-23T01:02:58.687" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="676" PostTypeId="2" ParentId="667" CreationDate="2014-03-23T01:57:40.063" Score="3" Body="&lt;p&gt;It is certainly important for motivation to know where you are heading in teaching, and why. But perhaps trying to reach too far is counterproductive. Sure, if I'm learning mathematics as such, I'd like to have an overview of the whole soon (but as comments/other answers say, that requires some non-trivial background). For non-specialists, the outlook will necessarily be more narrow (but might need to include a wider view of another field altogether).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-23T01:57:40.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="677" PostTypeId="1" CreationDate="2014-03-23T02:17:22.550" Score="17" ViewCount="334" Body="&lt;p&gt;&lt;strong&gt;How can you design an adequate course for a subject you never formally had classes?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is similar to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/469/teaching-somebody-elses-class&quot;&gt;teaching somebody else's class&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, suppose you are an algebraic geometer and you are asked to prepare an introductory algebraic geometry course but you are entirely self-taught in the area (perhaps you only had advanced classes).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are good methods for planning&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Learning outcomes;&lt;/li&gt;&#xA;&lt;li&gt;Homework sets;&lt;/li&gt;&#xA;&lt;li&gt;Bibliography used;&lt;/li&gt;&#xA;&lt;li&gt;What breadth/depth to strive for.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="47" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-03-24T03:14:35.627" Title="Teaching a class you never took" Tags="|course-design|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="678" PostTypeId="1" CreationDate="2014-03-23T02:22:55.593" Score="14" ViewCount="241" Body="&lt;p&gt;What is the best way to use online mathematics forums, such as mathematics.stackexchange.com or Khan Academy, in mathematics instruction? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I regularly encourage my undergraduate students to participate on mathematics.stackexchange.com (and my graduate students are all on mathoverflow.net). I tell them to post any questions they might have about their homework or other topics we've discussed in class, and I am confident that they can get high-quality mathematical guidance by doing so. Many of my students have learned a lot this way.  I give extra credit to any student who can accumulate sufficient reputation points, and I am significantly more impressed by students who do by answering questions rather than merely asking them.  Some of my students have gotten hooked and are now active participants. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What kind of experiences do other instructors have using math.stackexchange as part of their instructional experience? Has it been successful? How have you encouraged your students to take part? Do you give credit for earning reputation points? Do you worry about the students conspiring to vote each other up? Do you discuss explicitly in class the proper way to ask a mathematics question in such a forum? What are the main stumbling issues and success stories? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For my part, it has mostly been really great, with many students getting help with issues with which they were struggling. Some of my undergraduate students, however, are initially stung by a few downvotes or critical remarks on math.SE as they ask their first few questions, when they do not yet know the proper way to ask. Some are put off by this and are too timid to participate more fully, but I always encourage them to try to preservere. Others carry on and end up asking and answering a lot of math questions there, and I believe they really learn a lot that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have less experience with Khan Academy with my students, since the level is much lower, but I know from experience with my elementary school children that the Khan Academy is an excellent way to learn math. Please post an answer if you have experience using this or some other interactive online mathematics forum in your classes.&lt;/p&gt;&#xA;" OwnerUserId="90" LastActivityDate="2014-03-23T08:22:40.643" Title="How shall we use online math forums such as math.SE in mathematics instruction?" Tags="|online-instruction|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="679" PostTypeId="2" ParentId="535" CreationDate="2014-03-23T03:41:14.143" Score="6" Body="&lt;p&gt;I think it is better to highlight the limitations of WolframAlpha. For example using the free trial version can only get the basic calculations. Unless you open a pro account (which costs money), the &quot;Show Steps&quot; function is often not available for most calculations.  WolframAlpha easily refuses the calculations when it finds that it cannot express the results in terms of built in functions (e.g. WolframAlpha cannot calculate the series form solution of&#xA;$$\int x^x~\,\mathrm{d}x$$&#xA;thought it does in fact have a quite nice &lt;a href=&quot;https://math.stackexchange.com/questions/716324/series-expansion-of-an-integral/719408#719408&quot;&gt;series form solution&lt;/a&gt;. WolframAlpha cannot calculate&#xA;$$\int\dfrac{1}{ae^{bx^2}+c}\,\mathrm{d}x\qquad(a,b,c\neq0)$$&#xA;in terms of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Incomplete_polylogarithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;incomplete polylogarithm function&lt;/a&gt; because incomplete polylogarithm function is not in the built in function list. WolframAlpha cannot guarantee that the output results are really in the simplest form (e.g. when calculating&#xA;$$\int\sqrt{x^3+1}\,\mathrm{d}x,$$&#xA;WolframAlpha will give the quite awful elliptic functions result rather than gives the relatively simpler &lt;a href=&quot;http://en.wikipedia.org/wiki/Incomplete_beta_function#Incomplete_beta_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;incomplete beta function&lt;/a&gt; or &lt;a href=&quot;http://en.wikipedia.org/wiki/Hypergeometric_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;hypergeometric function&lt;/a&gt; results).  WolframAlpha cannot perform calculations with restrictions on constant parameter inputs (e.g. when calculating &#xA;$$\int\sin^nx\,\mathrm{d}x,$$&#xA;WolframAlpha will only treat $n$ as a complex number and does not have the wisdom to treat $n$ as e.g. an natural number).  Advise students that WolframAlpha is not really as omnipotent as they feel.  The sensible students will naturally not rely on WolframAlpha.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Always remember that the developments of the mathematical software by the mathematical software manufacturers often only care for the business, they don't care for the proper attitudes of studying mathematics.&lt;/p&gt;&#xA;" OwnerUserId="152" LastEditorUserId="8571" LastEditDate="2017-12-24T19:46:19.857" LastActivityDate="2017-12-24T19:46:19.857" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="680" PostTypeId="2" ParentId="678" CreationDate="2014-03-23T07:38:11.987" Score="7" Body="&lt;p&gt;This is an essential question. Thank you for asking it. Let me to mention some points on interactions of mathematics forums and usual academic mathematics instruction.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1. Love this point? Read more here!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A very successful way for introducing online math forums to people including students and teachers (which I usually use) is inciting their &lt;em&gt;sense of curiosity&lt;/em&gt; to know what is going on in these forums. To do this one can use several methods. The simplest way is explaining an interesting and &quot;unusual&quot; idea which can be found in MO/MSE &quot;uniquely&quot;. The uniqueness is important. They should be convinced that &lt;em&gt;there are something special in online math forums which cannot be found in the usual texts and courses.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2. Search for ideas not for solutions!&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Form the point of view of your students to consider online math forums as a &lt;em&gt;source of interesting ideas&lt;/em&gt; not &lt;em&gt;an easy tool for solving their problems&lt;/em&gt;. The first one leads them to become great adventurer researchers interested in new point of views and unknown realms of knowledge and the second one leads them to become lazy students who are searching for shortcuts continuously.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3. Need your teacher's help? Find him/her online when you want!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In a math department when some of its faculty members are continuously present in a particular online forum like MO it is a good idea to let students know that they can have a direct access to their teachers/advisers by participating in these forums. Surely it could be very interesting for both graduate and undergraduate students because each teacher has a limited meeting time per week. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4. Teaching MathOverflow as a part of Teaching Math.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It could be a good idea if those teachers who are active users of online math forums explain good asking/answering techniques in one of their lectures during their courses. One lecture per each course is completely enough and natural because one can say:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Dear students, in the last lecture I told you there are some useful online math forums which can help you in many aspects but like any other thing you should know how to use it! Let me to explain more... &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5. Don't be so shy! No one is perfect!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Online math forums like real societies are full of the different persons with different personalities. One should be aware of this point that any situation is possible in these communities and so he/she should know how to react these actions. Frankly speaking unfortunately in some aspects the culture of MO (and MSE) is too unfriendly. There are users who &lt;em&gt;like&lt;/em&gt; to call the others &lt;em&gt;silly&lt;/em&gt; (to prove that they are so clever!). Some others are too &lt;em&gt;dominative&lt;/em&gt;. Somebodies are &lt;em&gt;narcist&lt;/em&gt;. Some others are good but had a bad day! There are people who don't know when they should use an implicit sentence (instead of an explicit one) to refer a specific problem on somebody's act/post/comment and so on. These aggressive behaviors could be done against anybody even against the top users. (Unfortunately recently in one of the MO set theory posts I found a bad comment by one of the good users of MO against one of your helpful comments. I don't want to mention the case explicitly but this shows that an aggressive act could be happened even for the best users of online professional forums). What one of your new students should do in such a situation? Surely you can help them about it with your advisement. What I can say to them is:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Don't struggle with people at all. If you are not agree with their opinion simply ignore their critic remarks and do what you want. &lt;/li&gt;&#xA;  &lt;li&gt;Note to friendly admonitions and use them but don't take the critics &lt;em&gt;too&lt;/em&gt; serious. No one is perfect even the bests.&lt;/li&gt;&#xA;  &lt;li&gt;Some people are xenophobic against new users of the community and conservative about new ideas and methods. If you think you have something new to say don't be so shy to explain and defend it peacefully.   &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;6. Cheating Here! Cheating There! Cheating Everywhere!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If you are encouraging your students to participate in MO/MSE (which is very good) please don't make it too formal for them. Math forums should be interesting for students by their usefulness as a place which young students can find precious ideas, talk with the bests of the field and feel free to share their own ideas with professionals. These are enough to encourage them for contributing continuously in the community and considering the &lt;em&gt;quality of their participation&lt;/em&gt; as a part of their grades or a type of homeworks causes an expected phenomenon, cheating! Using an inaccurate formal numerical criterion like MO/MSE &lt;em&gt;reputation&lt;/em&gt; makes this approach more harmful. Your face to face comments about your students contribution on MO/MSE could be much more helpful. For example:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Alice! I like your last post in MSE very much, if you are interested in exploring the idea you can find these references useful. Please let me know about the result of your research. What is your idea about having a short talk about the subject next week? I am sure it could be interesting for your friends too. &lt;/li&gt;&#xA;  &lt;li&gt;Bob! There is a very nice idea in your recent MO post but they closed it because you didn't use some asking techniques which I explained in my &quot;How to ask a question?&quot; lecture last month. Let me to show you how to do this.     &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;7. A Science Fiction Future.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Network is changing the world. Let's consider 2050 A.D. when we are grandpas. It is possible to imagine online mega universities (at least in theoretical sciences like mathematics) with thousands of students and hundreds of teachers all around the world. You can wake up in your home in an island of pacific ocean, participate in set theory course of Prof. Chen at China in morning, attend the lecture of &lt;a href=&quot;http://en.wikipedia.org/wiki/Contact_%281997_film%29&quot;&gt;Dr. Arroway&lt;/a&gt; in US at afternoon and talk to Prof. X in &lt;a href=&quot;http://en.wikipedia.org/wiki/Vega&quot;&gt;Vega&lt;/a&gt; at the evening because based on Dr. Arroway's lecture, you are curious to know more about &quot;the notion of infinity on the infinitely distant planets&quot; (Hamming's famous paper, &lt;em&gt;&lt;a href=&quot;http://www.jstor.org/discover/10.2307/2589247?uid=2129&amp;amp;uid=2&amp;amp;uid=70&amp;amp;uid=4&amp;amp;sid=21103551842901&quot;&gt;Mathematics on a Distant Planet&lt;/a&gt;&lt;/em&gt; is related). The current online math forums are grandpas of such world wide universities. Nowadays we can buy and sell things online so it is completely reasonable to imagin a not too far day which we can learn and teach mathematics online.       &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-23T07:54:42.433" LastActivityDate="2014-03-23T07:54:42.433" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="681" PostTypeId="2" ParentId="678" CreationDate="2014-03-23T08:22:40.643" Score="7" Body="&lt;p&gt;I started strongly encouraging my students to participate in math.SE about a year ago. Excluding this year it was all with third year students taking group theory and/or complex analysis. This year I'm also teaching discrete maths to first year students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I notice is that those students who take the few hours to climb the learning curve for properly using math.SE profit enormously. It is clear that it is not primarily the strong or weak students who opt for using math.SE and they all struggle in the beginning until they figure it out. I try to help them out by giving encouraging answers and comments and improving their questions by editing them. I experimented for a short while with giving extra credit for reputation earners but I found it to be too much trouble for me so I stopped with that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last year I even held a workshop for undergrads on how to use math.SE but I don't have any records on the participants and how/if they used it later on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My impression is that most students, if not promised extra credit, are too intimidated to participate. Many are ashamed to ask questions in an open forum (and then I recommend to them to use a nickname). However, those students who do figure it out perform far better than their peers. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-23T08:22:40.643" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="682" PostTypeId="1" AcceptedAnswerId="687" CreationDate="2014-03-23T11:17:42.110" Score="14" ViewCount="1217" Body="&lt;p&gt;There are two young lovers in one of my crowded calculus classes. Sometimes when I am explaining an idea I see they are kissing each other and some of the other students are watching them and pay no attention to what I am explaining. I love lovers but I think their behavior is interrupting my teaching and there are better places for showing love! I really don't know what I should do in this case because the situation is too strange and unusual and the usual managing approaches are not useful.   &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I cannot say &quot;Quiet, please!&quot; because they don't produce any noise.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I cannot say &quot;Excuse me, please don't kiss each other!&quot; because it is too ridiculous!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maybe I can talk to them privately but I think it is a bit impolite because it is somehow related to their personal relations but I am thinking about this as a final solution.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another possible solution which I have in mind is finding an appropriate related joke for telling in these situations to back the attention of the students to teaching subject and implicitly tell the lovers that they are interrupting my teaching. Do you have any suggestion for such a joke?  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I &lt;em&gt;fear&lt;/em&gt; to say &quot;Please leave the classroom!&quot; because few years ago in a same situation one of my colleges did this and the girl cried and said &quot;you are a monster!&quot; The story spread all over the university in the light speed and she became a hero! After that almost all students call that poor college &quot;Prof. Monster&quot; and all of his attempts for removing the tragedy from the minds were unsuccessful. Now some of the students tell this legend for the younger ones and say that they are proud of being there when &quot;queen&quot; named him &quot;monster&quot;!    &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What is the appropriate approach for solving such a problem? Can you suggest any new solution? Were you in a similar situation? If yes, what did you do?&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-23T13:31:18.963" LastActivityDate="2014-08-25T15:25:34.307" Title="What should I do with kissing problem?" Tags="|classroom-management|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="683" PostTypeId="2" ParentId="677" CreationDate="2014-03-23T11:18:33.187" Score="8" Body="&lt;p&gt;As long as the course is somehow standard, which is the case in the example, one can ask colleagues what they teach in such a course and/or one can try to find similar courses on the internet; many individuals and institutions have a lot of information on their courses available on the internet. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some care is needed however to judge if the context is really similar, not to have found oranges while actually needing apples. This is especially true regarding prerequisites (and in particular for a course on algebraic geometry; it should make a difference if the students know some commutative algebra or not). In this vein, it also might only help that much having taken such a class oneself (except if the context was also very similar).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also not unlikely there is some specification of the content in the curriculum; though sometimes what is written in such documents is more a description of an 'ideal world' so it is not clear what is written there is really achievable. And, if there is not you are more flexible.      &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the course is not standard it typically will also not be a very rigid (and chances are one is an expert in the field). In this case one might only start with a very rough plan, and just make the thing up as one moves along (from one week to the next). There is a risk in this that one gets of track, but then as discussed in  &lt;a href=&quot;https://matheducators.stackexchange.com/questions/550/how-do-you-fix-a-broken-course-plan&quot;&gt;How do you fix a broken course plan?&lt;/a&gt; this risk also exists if one has a clear plan. &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-03-23T11:24:05.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="684" PostTypeId="2" ParentId="654" CreationDate="2014-03-23T11:37:18.673" Score="11" Body="&lt;p&gt;&lt;strong&gt;Slides are great...&lt;/strong&gt; if you know how to use them correctly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some advantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It works without blackboard (e.g. at a conference, or unprepared lecture room, sometimes even a pale wall will do if at some unusual venue).&lt;/li&gt;&#xA;&lt;li&gt;You can present pretty pictures (and animations) that would take ages to draw (or would be plain impossible).&lt;/li&gt;&#xA;&lt;li&gt;It helps to organize the class (e.g. students will know where are we (to enhance this aspect, some of my friends use applications like &lt;a href=&quot;https://prezi.com/&quot;&gt;prezi&lt;/a&gt;), how much material there is yet to cover, which parts they should write down, and so on).&lt;/li&gt;&#xA;&lt;li&gt;In some cases it is possible to print out copies and let the students make the notes there.&lt;/li&gt;&#xA;&lt;li&gt;Typed fonts are usually more readable than handwriting (and it's easier to use colors, esp. useful with source codes).&lt;/li&gt;&#xA;&lt;li&gt;You can do interactive-demonstrations, e.g. complex calculations with input from the students that would take too much time by hand, but it's possible with prepared programs (you can argue that it's not &lt;em&gt;slides&lt;/em&gt; anymore, but it fits into general laptop+projector use).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some disadvantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It doesn't work without electricity, projector, laptop, etc.&lt;/li&gt;&#xA;&lt;li&gt;It takes &lt;em&gt;a lot&lt;/em&gt; to prepare good slides.&lt;/li&gt;&#xA;&lt;li&gt;Slides tempt you into many bad habits (e.g. too fast, no proofs, one-way communication).&lt;/li&gt;&#xA;&lt;li&gt;Slides tempt the students into many bad habits (e.g. don't take notes, fall asleep, ask no questions).&lt;/li&gt;&#xA;&lt;li&gt;Slides without commentary often do not constitute a good help by themselves, e.g. when publishing online (so it would take even more work to make the slides and then apply appropriate commentary).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-23T11:37:18.673" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="685" PostTypeId="1" AcceptedAnswerId="696" CreationDate="2014-03-23T11:44:15.823" Score="5" ViewCount="172" Body="&lt;p&gt;Imagine a class which is split completey regarding previous knowledge which is in some way needed for the class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How can you deal with such a class? How can you - without giving too much workload to those without previous knowledge - bring those students to the level which is needed in order to understand new things - without bothering those who know this already?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Background: In my university, the future high school teachers don't have to take a class in real analysis and measure theory. But they have to take a class in probability theory together with the bachelor students who did take the real analysis class. Of course you don't need to know everything about measures for the probability theory class, but some background is more than helpful. How much content and methods of measure theory would you repeat?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is a course about partial differential equations where some students might have attended classes in real analysis and functional analysis; and some students haven't attended either of them. Here, you have usally very motivated students who want to take the course but they were not informed in the past.)&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-02T06:37:58.870" LastActivityDate="2014-04-02T06:37:58.870" Title="How can one deal with classes split up (in terms of previous knowledge)?" Tags="|undergraduate-education|mathematical-pedagogy|course-design|graduate-education|mixed-ability|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="686" PostTypeId="1" AcceptedAnswerId="698" CreationDate="2014-03-23T11:58:37.497" Score="19" ViewCount="1185" Body="&lt;p&gt;Many students complain about how old the things in mathematics are. When students finish their undergraduate studies, there are usually not able to state results and prove them which were found after the 1950s (or even older). Of course, the notation has improved and maybe there are some nicer proof for theorems proven far back. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Are there examples of theorems (including proofs) which are as new as possible (at least newer than 1950/1960) and accessible to undergraduate students?&lt;/strong&gt; Good students should be able to understand the meaning of the theorem and the tools which are needed for the proof should be available – the proof of Fermat’s Last Theorem would not fit into the examples here, although the statement is very clear.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="75" LastEditDate="2014-03-23T13:20:46.373" LastActivityDate="2014-04-17T11:17:44.657" Title="What is fairly new theorem one can teach (and prove) to an undergraduate student?" Tags="|undergraduate-education|student-motivation|examples|" AnswerCount="15" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="687" PostTypeId="2" ParentId="682" CreationDate="2014-03-23T12:21:06.480" Score="16" Body="&lt;p&gt;To approach this situation I would recommend you to imagine the same situation but that it is not kissing that is happening, but something else, say, two students were playing cards; so something that would also not cause any noise, but prevent the two from paying attention, and  might also  be disruptive to others in the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps, in this case you might not see a problem with the analogue of your second point, but also you do not really need to name the &lt;em&gt;precise&lt;/em&gt; activity that is disruptive. Perhaps you cannot see if they are really playing cards or rather trading stickers, yet in any case you might remind them (and everybody) that they should pay attention to the course. You might not even have to address the two directly.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same applies to your point three. Do not agonize about whether or not it is appropriate to talk to them about it due to the precise activity. You might just say something like: it seems like you two are not paying full attention during the course, it might be better you stop sitting next to each other during class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What precisely you should do is I think difficult to answer in abstract, as it should depend in several ways on the general cultural context. Yet, in any case, I think you should not focus too much on the kissing aspect of it. But instead consider it at what it is: two students engage in some activity that stops them (the two) from paying attention to the course and that in addition might be disruptive for others as well. Then think about how you would handle this situation. (If you also have a problem with th more general situation, you might ask another question.)&lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-03-23T12:21:06.480" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="688" PostTypeId="2" ParentId="686" CreationDate="2014-03-23T12:35:49.750" Score="15" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Art_gallery_theorem#Chv.C3.A1tal.27s_art_gallery_theorem&quot; rel=&quot;noreferrer&quot;&gt;Chvátal's art gallery theorem&lt;/a&gt;, a visibility problem in computational geometry, was proposed and proved in the 1970s.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\left\lfloor n/3 \right\rfloor$&lt;/span&gt; guards are always sufficient and sometimes necessary to guard a simple polygon with &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/eYFXG.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;http://en.wikipedia.org/wiki/Art_gallery_theorem#Fisk.27s_short_proof&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Fisk (1978) proves the art gallery theorem as follows.&lt;/p&gt;&#xA;&lt;p&gt;First, the polygon is triangulated (without adding extra vertices). The vertices of the polygon are then 3-colored in such a way that every triangle has all three colors. To find a 3-coloring, it is helpful to observe that the dual graph to the triangulation (the undirected graph having one vertex per triangle and one edge per pair of adjacent triangles) is a tree, for any cycle in the dual graph would form the boundary of a hole in the polygon, contrary to the assumption that it has no holes. Whenever there is more than one triangle, the dual graph (like any tree) must have a vertex with only one neighbor, corresponding to a triangle that is adjacent to other triangles along only one of its sides. The simpler polygon formed by removing this triangle has a 3-coloring by mathematical induction, and this coloring is easily extended to the one additional vertex of the removed triangle.&lt;/p&gt;&#xA;&lt;p&gt;Once a 3-coloring is found, the vertices with any one color form a valid guard set, because every triangle of the polygon is guarded by its vertex with that color. Since the three colors partition the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices of the polygon, the color with the fewest vertices forms a valid guard set with at most &lt;span class=&quot;math-container&quot;&gt;$\lfloor n/3\rfloor$&lt;/span&gt; guards.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-23T12:35:49.750" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="689" PostTypeId="1" CreationDate="2014-03-23T13:03:54.853" Score="21" ViewCount="1409" Body="&lt;p&gt;Cantor's discovery of the existence of more than one infinity was a revolutionary change in human knowledge. He defined the notion of counting by bijections and showed that one can use infinities as numbers for counting mathematical objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am planning a &quot;Set Theory for Children&quot; project to familiarize elementary school students with the basic notions of set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no problem with giving an intuitive definition of infinity.  Children in elementary school are already familiar with counting by natural numbers.  One can convince them that the set of all natural numbers is infinite because it &quot;never ends&quot; and one can find a number larger than any given number by adding 1 to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there are few concrete examples for infinities larger than $\aleph_0$.  Also introducing basic transfinite arithmetic in a naive intuitive way can easily lead to paradoxes like Hilbert's Hotel. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; How can I explain the existence of infinities of different sizes and transfinite arithmetic so elementary school students will understand it? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark.&lt;/strong&gt; I believe transfinite numbers and their arithmetic are as natural as finite numbers and finite arithmetic.  The fact that they seem strange even for professional mathematicians has a root in our elementary school education and our incorrect/incomplete basic intuition about the notions of &quot;number&quot;, &quot;enumerating&quot; and &quot;arithmetic&quot;.  So there is a natural way to teach these natural notions to students even in elementary school.  We just need to discover such a teaching method.&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="143" LastEditorDisplayName="user173" LastEditDate="2017-04-25T09:50:40.970" LastActivityDate="2023-10-11T13:19:30.820" Title="How can I familiarize elementary school students with infinities larger than $\aleph_0$?" Tags="|course-design|primary-education|set-theory|infinity|" AnswerCount="13" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="690" PostTypeId="2" ParentId="686" CreationDate="2014-03-23T13:19:13.037" Score="9" Body="&lt;p&gt;There are various results of this form, you can find some &lt;a href=&quot;http://maths.mq.edu.au/~alf/45.pdf&quot;&gt;in a related MO question&lt;/a&gt;. But, I agree this is a problem that students often do not learn such results, or might not realize how recent some things are. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example is &lt;a href=&quot;http://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_theorem&quot;&gt;Apéry's theorem (1979)&lt;/a&gt; establishing the irrationality of $\zeta(3)$, that is, the result that $\sum_{n=1}^{\infty} \frac{1}{n^3}$ is an irrational number.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What makes this result in my opinion especially well-suited for the intended purpose is that: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It fits together with things students should know anyway, namely $\sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$, and possibly the evaluation of  $\zeta(2k)$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is easy to motivated given the above that one seeks to understand the situation for odd values two. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The result touches on number theory &lt;em&gt;and&lt;/em&gt; analysis.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There was continued activity around this problem since then and open questins remain, such as, already: is $\zeta(5)$ irrational? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For an overview see the link above and for an actual exposition of the proof written at that time see &lt;a href=&quot;http://maths.mq.edu.au/~alf/45.pdf&quot;&gt;&quot;A proof that Euler missed&quot;&lt;/a&gt; by the van der Poorten; and meanwhile even more streamlined arguments exist.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-03-23T13:28:05.147" LastActivityDate="2014-03-23T13:28:05.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="691" PostTypeId="2" ParentId="689" CreationDate="2014-03-23T13:19:27.863" Score="33" Body="&lt;p&gt;I think that actually trying to get students at this age to contemplate infinities in a rigorous way is probably ill advised.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think that exploring counting from both an &quot;ordinal&quot; and a &quot;cardinal&quot; point of view is probably a good idea.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example for a 5 year old: Something you could do is have 20 stuffed animals,  only 18 of them wearing hats.  You can ask:  are there more hats or stuffed animals?  Get them to explain their reasoning:  How could you know there are more stuffed animals than hats without counting them?  These kinds of questions introduce them to the ideas of injection, surjection, etc,  at a level appropriate to the age.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2014-03-23T21:31:09.527" LastActivityDate="2014-03-23T21:31:09.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="692" PostTypeId="2" ParentId="689" CreationDate="2014-03-23T14:06:08.697" Score="2" Body="&lt;p&gt;One can give them some intuition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw a line segment of length 'one'. Ask them how long it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then draw a single point. Ask them how much room it takes up (the answer is zero).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw two points; how much room do they take up? (Zero)&#xA;Keep drawing poiints one by one. Say, 'No matter how far I go, even if I go forever, to normal infinity, they take up no room.'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;'So there must be an even bigger infinity of points in the line segment'.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-23T14:06:08.697" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="693" PostTypeId="2" ParentId="686" CreationDate="2014-03-23T14:21:29.957" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there examples of theorems (including proofs) which are as new as&#xA;  possible (at least newer than 1950/1960) and accessible to&#xA;  undergraduate students?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I understand that you're asking for historical proofs, but this idea that math is something that was done in the past might not be best ameliorated with examples from the recent past.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The easiest way to end up with new proofs and theorems is to help your students ask and answer their own mathematical questions. My favorite bit of writing on this sort of pedagogy is from this post:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://christopherdanielson.wordpress.com/2012/10/12/the-hierarchy-of-hexagons/&quot;&gt;http://christopherdanielson.wordpress.com/2012/10/12/the-hierarchy-of-hexagons/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't really see how this sort of questioning environment is compatible with a lecture-based course, though, so maybe this isn't helpful for your needs.&lt;/p&gt;&#xA;" OwnerUserId="84" LastActivityDate="2014-03-23T14:21:29.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="694" PostTypeId="2" ParentId="689" CreationDate="2014-03-23T14:36:42.457" Score="13" Body="&lt;p&gt;You can easily make them draw $\aleph$s; however the rest is much more demanding. There is a nice analysis from a researcher group taking a constructivist perspective. They distinguish &lt;em&gt;potential infinity&lt;/em&gt; from &lt;em&gt;actual infinity&lt;/em&gt;. The first one is covered by the idea of &quot;counting on forever&quot; the second one needs infinitely many things to exist in your thinking at the same time. Taking the step from potential to actual infinity seems to be even more demanding than taking the step from natural numbers as ordinals to natural numbers as cardinals (both times, the view has to change from process to object). The latter step takes children (roughly) two years. Consequently, the literature they cite does not find people up to the age of 15, who manage the transition to understanding actual infinity. This might be due to teaching as you indicated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that understanding different infinities needs at least understanding $\aleph_0$ as actual infinity. So your goals seem very ambitious. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Literature&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dubinsky, E., Weller, K., Mcdonald, M. A., &amp;amp; Brown, A. (2005). Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos-Based Analysis: Part 1. Educational Studies in Mathematics, 58(3), 335–359. &lt;a href=&quot;http://dx.doi.org/10.1007/s10649-005-2531-z&quot; rel=&quot;nofollow&quot;&gt;doi:10.1007/s10649-005-2531-z&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dubinsky, E., Weller, K., McDonald, M. A., &amp;amp; Brown, A. (2005). Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos Analysis: Part 2. Educational Studies in Mathematics, 60(2), 253–266. &lt;a href=&quot;http://dx.doi.org/10.1007/s10649-005-0473-0&quot; rel=&quot;nofollow&quot;&gt;doi:10.1007/s10649-005-0473-0&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="276" LastEditDate="2014-06-27T14:14:21.610" LastActivityDate="2014-06-27T14:14:21.610" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="695" PostTypeId="2" ParentId="643" CreationDate="2014-03-23T14:37:36.013" Score="8" Body="&lt;p&gt;Well, yeah, why not?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mazur's innovation is to import and adapt insights on learning and teaching from k-12 and apply them to the large college lecture. This is a hugely difficult task, because what we know works best for students is made difficult by a lot of the structures of college courses. For instance, we know that when students have an opportunity to attempt to solve a problem, they'll better appreciate an explanation of the solution. But where do you find time for that in a class that meets just twice a week? How do you support students when they get stuck on a problem when you have 40 people enrolled in your class? And won't this go &lt;em&gt;slowly&lt;/em&gt;? How will we finish all the material on the syllabus?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would have a lot of trouble applying what I do with 4th, 9th, and 11th graders to a university math course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Mazur has figured out a way to make some of these things work in a way that's palatable to the university environment. So, why not? His techniques aren't based on idiosyncratic principles. Here are some general principles of learning that his approach is consistent with:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Teachers need to know what their students actually think.&lt;/li&gt;&#xA;&lt;li&gt;People better understand an explanation if they feel a need for that explanation.&lt;/li&gt;&#xA;&lt;li&gt;Conversation is a great context for thinking about a problem.&lt;/li&gt;&#xA;&lt;li&gt;Informal experience with an idea is a great preparation for formal experience with an idea.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(For more about general principles of teaching and learning, see the free NRC book &quot;How People Learn&quot;. &lt;a href=&quot;http://www.nap.edu/openbook.php?isbn=0309070368&quot;&gt;http://www.nap.edu/openbook.php?isbn=0309070368&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anything that's consistent with general principles of learning/teaching is going to be good for your kids. So, go for it! Let us know how it goes.&lt;/p&gt;&#xA;" OwnerUserId="84" LastActivityDate="2014-03-23T14:37:36.013" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="696" PostTypeId="2" ParentId="685" CreationDate="2014-03-23T14:56:17.530" Score="4" Body="&lt;p&gt;If the course is required for graduation, and if the curriculum does not adequately prepare the students, then I strongly feel you should teach to the less prepared students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am teaching a capstone course based on knot theory. It is the last course before graduation for math and mathed majors, and it is the only section. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mathed majors have not had many (or perhaps any) formal proof courses and quickly fell behind in the homework. I couldn't just fail half of the class, so I moved towards computational problems, which both groups excelled at.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It wasn't a difference of innate intelligence; they both did great at computations. They just hadn't been adequately prepared.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the course is not essential, I believe you should set a certain level of requirements and make it very clear at the outset that you expect everyone to have met those requirements. An easy way to do this is to have a pre-test with a required minimum score.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-23T14:56:17.530" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="697" PostTypeId="2" ParentId="686" CreationDate="2014-03-23T15:36:28.757" Score="6" Body="&lt;p&gt;The entire area of fractals was born after computers became easily available. Once the idea of fractals came up, it was easy to prove a lot of theorems about them that were only new because noone thought to pose them before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The definition of the Mandelbrot set and its properties in particular are a good option for undergraduates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See this paper for some sample theorems:&#xA;homepages.warwick.ac.uk/~masdbl/dimension-total.pdf&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-03-23T16:50:18.170" LastActivityDate="2014-03-23T16:50:18.170" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="698" PostTypeId="2" ParentId="686" CreationDate="2014-03-23T16:06:26.403" Score="17" Body="&lt;p&gt;There aren't many new theorems in old subjects that are accessible to undergraduates.  However, as others have mentioned, there are many relatively new subjects within mathematics that are accessible.  These include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Graph Theory, which really dates back to &lt;a href=&quot;http://en.wikipedia.org/wiki/D%C3%A9nes_K%C5%91nig&quot;&gt;Dénes Kőnig's&lt;/a&gt; 1936 textbook.  Many of the fundamental results were proven in the 1930's, but many others were proven several decades later.  For example, the &lt;a href=&quot;http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem&quot;&gt;Max Flow Min Cut Theorem&lt;/a&gt; was first proven in 1956, and the definition and basic properties of &lt;a href=&quot;http://en.wikipedia.org/wiki/Treewidth&quot;&gt;treewidth&lt;/a&gt; were not introduced until 1976.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Dynamical Systems picked up steam slowly, and many of the fundamental results were proven in the second half of the 20th century.  For example, the theorem that &lt;a href=&quot;http://mathworld.wolfram.com/PeriodThreeTheorem.html&quot;&gt;period three implies chaos&lt;/a&gt; for a one-dimensional system was proven in 1975, though it was a special case of &lt;a href=&quot;http://en.wikipedia.org/wiki/Sharkovskii%27s_theorem&quot;&gt;Sharkovskii's Theorem&lt;/a&gt;, which was proven in 1964 but not well known until later.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similarly, most of the fundamental results of Complex Dynamics and Fractal Geometry are quite new.  The first pictures of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Mandelbrot_set&quot;&gt;Mandelbrot set&lt;/a&gt; did not appear until 1978, and many important results are newer still.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Game Theory is a relatively new subject.  &lt;a href=&quot;http://en.wikipedia.org/wiki/Nash_equilibrium&quot;&gt;Nash equilibria&lt;/a&gt; were first defined in general by Nash in 1951, so most of the known results are newer than this.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Knot Theory has been developing quickly in the last half of the 20th century.  For example, the tabulation of knots with up to 11 crossing was not completed until 1974, and the &lt;a href=&quot;http://en.wikipedia.org/wiki/Jones_polynomial&quot;&gt;Jones polynomial&lt;/a&gt; was not defined until 1984.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-23T16:06:26.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="699" PostTypeId="1" AcceptedAnswerId="702" CreationDate="2014-03-23T16:23:27.400" Score="10" ViewCount="347" Body="&lt;p&gt;I like using quotations of great mathematicians as a source of inspiration for young students. I think even a short sentence could have a great influence on forming their research interests and point of views in future. Just as an example look at the following sentences of Hugh Woodin and his student Joel Hamkins. Can you see any sign of inspiration?   &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Quoted from BBC documentary movie &quot;&lt;em&gt;To Infinity and Beyond&lt;/em&gt;&quot;:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;&lt;strong&gt;Prof. Doron Zeilberger (Rutgers University):&lt;/strong&gt; Infinity may or may not exist. God may or may not exist. But in mathematics there should not be any place for neither infinity nor God!&lt;/li&gt;&#xA;  &lt;li&gt;&lt;strong&gt;Prof. Hugh Woodin (Harvard University):&lt;/strong&gt; I am so sorry for those who deny existence of infinity. It is a beautiful subject. I can say the &lt;em&gt;stars&lt;/em&gt; don't exist and stay inside and always look down but I don't see the &lt;em&gt;beauty of the stars&lt;/em&gt;!&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;Quoted from Hamkins' paper &quot;&lt;em&gt;The Set Theoretic Multiverse&lt;/em&gt;&quot;:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;&lt;strong&gt;Prof. Joel David Hamkins (City University of New York):&lt;/strong&gt;&#xA;  Set theory appears to have discovered an entire cosmos of set-theoretic universes, revealing a category-theoretic nature for the subject, in which the universes are connected by the forcing relation or by large cardinal embeddings in complex commutative diagrams, like &lt;em&gt;constellations filling a dark night sky.&lt;/em&gt;&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; Is there a book/website which includes a collection of interesting quotations of mathematicians? It would be very interesting if it is classified by mathematical fields. e.g. Sentences about set theory, logic, algebra, analysis, probablity, etc.   &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-23T16:36:34.940" LastActivityDate="2014-06-25T11:48:47.950" Title="Quotations of Great Mathematicians as a Source of Inspiration for Young Students" Tags="|course-design|student-motivation|reference-request|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="700" PostTypeId="2" ParentId="686" CreationDate="2014-03-23T17:01:52.777" Score="4" Body="&lt;p&gt;Some famous theorems in mathematical statistics was proven arounf 1950.&#xA;Some examples, the Rao-Blackwell theorem (see:  &lt;a href=&quot;https://en.wikipedia.org/wiki/Rao-Blackwell_Theorem&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Rao-Blackwell_Theorem&lt;/a&gt;)  is interesting and accessible. Also Basus theorem should be, and that is even later:&lt;a href=&quot;https://en.wikipedia.org/wiki/Basu_theorem&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Basu_theorem&lt;/a&gt;&#xA;That is from 1955.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, a lot of results from distribution theory are new (distribution here in the probability sense)&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2014-03-23T17:01:52.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="701" PostTypeId="1" AcceptedAnswerId="703" CreationDate="2014-03-23T18:20:20.967" Score="18" ViewCount="274" Body="&lt;p&gt;Creativity is the core of research mathematics. However, most introductory math consists of learning fixed rules to perform basic, essential mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, for many elementary school students, math is perceived as a rote and routine subject. This doesn't turn everyone off; a lot of people love math into their teens. But creativity is not seen as part of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can creativity be introduced at the elementary level?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My teachers did this exceptionally well; one of my favorite memories involves the classic 'four fours' puzzle: using the symbol 4 four times and unlimited math operations, create all numbers from 1 to 100. I remember working on this with my grandmother.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What general principles can be used to promote creativity in elementary school mathematical students?&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-24T22:02:55.117" Title="How can creativity be incorporated into elementary school mathematics?" Tags="|primary-education|creativity|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="702" PostTypeId="2" ParentId="699" CreationDate="2014-03-23T19:13:34.557" Score="9" Body="&lt;p&gt;Peter Cameron has a &lt;a href=&quot;http://cameroncounts.wordpress.com/quotes/&quot; rel=&quot;nofollow&quot;&gt;nice collection of quotes related to mathematics&lt;/a&gt; (&lt;a href=&quot;http://www.maths.qmul.ac.uk/~pjc/comb/quotes.html&quot; rel=&quot;nofollow&quot;&gt;old version&lt;/a&gt;) sorted by subject (as requested) or rather keywords, like Infinity, Axioms, Numbers but also some general sections. Not all of them are from mathematicians, but many are. He also links to some outher resources. &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-06-25T11:48:47.950" LastActivityDate="2014-06-25T11:48:47.950" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="703" PostTypeId="2" ParentId="701" CreationDate="2014-03-23T19:29:03.663" Score="7" Body="&lt;p&gt;I have a distinct memory of my second grade teacher handing out a &lt;a href=&quot;http://en.wikipedia.org/wiki/Logic_puzzle#Logic_grid_puzzles&quot;&gt;logic grid puzzle&lt;/a&gt; to the class one day near the end of the year.  I remember thinking, &quot;This is fun, but what does it have to do with math?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, logical puzzle-solving &lt;em&gt;is&lt;/em&gt; an important kind of math, and should be explicitly included in the curriculum.  Students of all ages should be taught about logic grid puzzles, tangrams, mazes, magic squares, sodoku, minesweeper, Rubik's cubes, towers of Hanoi and so forth in math class, as well as games such as nim, dots and boxes, sprouts, etc.  Puzzle-solving is a basic mode of creative thought, and it needs to be explicitly included in the curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition to such explicit logic puzzles, teachers at the elementary school level should endeavor to include small mathematical puzzles that the students can potentially solve.  Especially if they work in groups of two or three, fourth or fifth grade students ought to be able to solve a simple puzzle like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find two numbers whose sum is 22 and whose product is 96.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We teach explicit methods for these things when they get to algebra class, but in elementary school students ought to be clever enough to figure out a puzzle like this without having any explicit method.  Other puzzles like this include&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How many different ways are there to make 36 cents in change? (I'm assuming U.S. currency.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A solid, four-inch cube of wood is coated with blue paint on all six sides.&#xA;  Then the cube is cut into smaller one-inch cubes. &#xA;  These new one-inch cubes will have either three blue sides, two blue sides, one blue side, or no blue sides. How many of each will there be?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A general rule for such puzzles is that the children should &lt;em&gt;not&lt;/em&gt; be solving them using some method or algorithm that they were explicitly taught.  Math is not a rote subject, and puzzles like these are much more representative of real mathematics than multiplying three-digit numbers or reducing fractions to lowest terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, in addition to puzzles, there are many possible geometry-based toys that children can use creatively, including &lt;a href=&quot;http://www.polydron.co.uk/&quot;&gt;polydrons&lt;/a&gt;, &lt;a href=&quot;http://zometool.com/&quot;&gt;zomes&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Tangram&quot;&gt;tangrams&lt;/a&gt;, and so forth.  These could lead to all sorts of interesting classroom activities -- e.g. a contest to see which group can construct the most interesting closed shape (polyhedron) using polydrons.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-23T19:29:03.663" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="704" PostTypeId="2" ParentId="689" CreationDate="2014-03-23T21:41:54.400" Score="3" Body="&lt;p&gt;If the idea of &quot;larger than infinity&quot; caused a riot among grown up mathematicians, &lt;em&gt;accustomed to rarefied abstract thinking for years if not decades&lt;/em&gt;, this will just make their little heads explode.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my automata theory classes I have to get my students (by then mangled though school and two years of mathematics at college level) to grasp Cantor's diagonal argument, and I feel I'm unsuccessful most of the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Yes, it is an extremely beautiful result. &lt;strong&gt;But not everyone is prepared to appreciate it as such.&lt;/strong&gt; Tough luck, move on.)&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-23T21:41:54.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="705" PostTypeId="2" ParentId="686" CreationDate="2014-03-23T21:42:44.723" Score="5" Body="&lt;p&gt;Here is a nice (basically linear algebra!) theorem from the representation theory of quivers proved by Gabriel: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; The number of indecomposable representations of a quiver is independent of the orientation of the edges of the underlying digraph. A connected graph gives rise to a quiver of finite type if and only if it is a simply laced Dynkin diagram. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/BGG-CoxeterF-Usp.pdf&quot;&gt;Here&lt;/a&gt; is the proof due to Bernstein, Gelfand and Ponomarev which is usually discussed. Another very accessible exposition of this result may be found in &lt;a href=&quot;http://math.mit.edu/~etingof/replect.pdf&quot;&gt;these&lt;/a&gt; online notes (See §1.8 and Chapter 5). &lt;/p&gt;&#xA;" OwnerUserId="109" LastActivityDate="2014-03-23T21:42:44.723" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="706" PostTypeId="1" AcceptedAnswerId="792" CreationDate="2014-03-23T21:45:37.570" Score="35" ViewCount="3865" Body="&lt;p&gt;I always read about the &lt;a href=&quot;http://en.wikipedia.org/wiki/Moore_method&quot;&gt;Moore method&lt;/a&gt; with great enthusiasm. Somehow I always felt that it should be how we do it in an ideal world, but it is impossible to use because of time and other constrains. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most important point is that users of the method claim that the students may have less factual knowledge of the material, but their understanding and the deepness of their knowledge pays for everything: they can easily master any missing part afterwords when needed. This seems to me impossible to do with a fixed syllabus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The wikipedia page, however, has an impressive list of places where this method is still in use. Do you use the Moore method? Are there (preferably online) resources helping to learn how to do it as an instructor? How do you overcome the problem that your students have to master all the material cited in the syllabus at the end of the semester?&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="61" LastEditDate="2014-04-16T21:22:13.420" LastActivityDate="2014-12-21T21:03:57.970" Title="What happened to the Moore method?" Tags="|undergraduate-education|course-design|classroom-management|graduate-education|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="707" PostTypeId="2" ParentId="685" CreationDate="2014-03-23T21:55:29.137" Score="1" Body="&lt;p&gt;If the knowledge really is required for the class (and can't be supplemented with a short &quot;fill in&quot; for those who lack it), and there is no way to redefine the evaluation as @BrianRushton says, the course is just badly defined (or has the wrong requisites). Even in the mentioned cases I believe you should think of defining the course anew.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that for someone familiar with the subject matter and related areas it might come very natural to suppose some knowledge as given, like real analysis and measure theory, and talk in terms of Stieltjes integrals simplifies probability enormously. But not everybody has this knowledge, and for the vast mayority of those who want to use this all those niceties are just superfluous baggage. Just using hand-crafted derivations (even if longer and more restrictive than needed) might well be enough for non-specialists. Besides, once you  as much as mention non-Riemann integrals, you'll have lost them for good anyway.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-23T21:55:29.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="708" PostTypeId="2" ParentId="677" CreationDate="2014-03-23T22:03:28.530" Score="-2" Body="&lt;p&gt;Hopefully you are given enough time beforehand... apply what is said here (and at math.SE) about self-learning. Check out a few potential textbooks, grab a selection of lecture notes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analyze the courses that come before, talk to who teaches them (anything can be said in course decriptions); also what comes after (if any). See what the students are going to need on the subject for their future professional life.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-23T22:03:28.530" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="709" PostTypeId="2" ParentId="682" CreationDate="2014-03-23T22:16:04.817" Score="6" Body="&lt;p&gt;If somebody disrupts the class (be it playing on the computer, chatting with their neighbor, whatever) I just ask them to please take their disruptive activity elsewhere if they aren't interested in the class, out of respect for their classmates who are interested. Solves the problem nicely.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-23T22:16:04.817" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="710" PostTypeId="1" AcceptedAnswerId="804" CreationDate="2014-03-24T01:28:39.933" Score="15" ViewCount="4441" Body="&lt;p&gt;My dissertation involved, among other things, the East Asian way of teaching and learning mathematics. (See, for example, Leung (2001).)  I was particularly interested in the &lt;a href=&quot;http://en.wikipedia.org/wiki/Kumon_method&quot;&gt;Kumon method&lt;/a&gt;.  Although I found a nice paper in a refereed journal that describes it (Ukai, 1994), and some informal studies comparing it with other ways of teaching mathematics, I was unable to find any experimental studies (those involving random sampling or assignment).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What papers in refereed journals have presented experimental studies comparing the Kumon method (or something similar to it) with other ways of teaching and learning mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leung, F. K. S. (2001). &lt;a href=&quot;http://www.cimm.ucr.ac.cr/ciaem/articulos/evaluacion/internacionales/In%20search%20of%20an%20east%20asian%20identity%20of%20mathematics%20education.%2aLeung,%20Frederick.%2aLeung,%20F.%20In%20search%20of%20an%20east%20asian...2002.pdf&quot;&gt;In search of an East Asian identity in mathematics education&lt;/a&gt;.&#xA;&lt;em&gt;Educational Studies in Mathematics&lt;/em&gt;, &lt;em&gt;47&lt;/em&gt;, 35-51.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ukai, N. (1994). &lt;a href=&quot;http://www.jstor.org/discover/10.2307/132785?uid=374191101&amp;amp;uid=3738824&amp;amp;uid=2&amp;amp;uid=3&amp;amp;uid=67&amp;amp;uid=374191071&amp;amp;uid=62&amp;amp;sid=21103836494793&quot;&gt;The Kumon approach to teaching and learning&lt;/a&gt;. &lt;em&gt;Journal of Japanese&#xA;Studies&lt;/em&gt;, &lt;em&gt;20&lt;/em&gt;, 87-113.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Here is additional information as requested in a comment.  (I'm quoting from my dissertation.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Students in East Asian countries outperform their counterparts in other countries in large-scale international studies of mathematics achievement (Leung, 2006).&#xA;  Leung (2006) asserted that this is because their teachers have good conceptual understanding and procedural skills, their instruction is teacher-dominated, and their&#xA;  culture strongly emphasizes the importance of education, which creates high expectations for them and their teachers. It also seems that East Asian teachers and students&#xA;  acquire their mathematics competence from instruction that focuses on procedures&#xA;  and repeated practice with variation (Leung, 2006).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;How does this kind of education result in the high mathematics achievement observed? Leung (2006, p. 43) proposed this explanation:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The process of learning often starts with gaining competence in the procedure, and then through repeated practice, students gain understanding. Much of the mathematics in the school curriculum may need to be practiced without thorough understanding first. With a set of practicing exercises that vary systematically, repeated practice may become an&#xA;  important &quot;route to understanding&quot; [...].&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The view that repeated practice of procedures leads to conceptual understanding is especially apparent in the program known as Kumon Math.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Kumon Math, published by Kumon North America, Inc., is a supplemental mathematics curriculum for students in preschool through secondary school. The curriculum is not structured by age or grade level, but by a student's own pace, and is composed of hundreds of short assignments that progress through increasingly difficult mathematics skills. Students complete one assignment every weekday and attend a Kumon center for two sessions per week. Every assignment is timed and graded.&#xA;  Students master skills through repetition and cannot progress to a new skill before completing an assignment within a set amount of time and with close to 100% accuracy. New skills and exercises build on previously mastered concepts. (WWC, 2009)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Ukai (1994) gives a detailed description of the Kumon method and the philosophy behind it. In Kumon Math, &quot;[c]oncepts are not explicitly taught. Rather, through repetition, learners experience insight. For example, the means of solving such problems as $48\div 6$ or $132\div 12$ will be patently clear, without verbal or written explanation, to a child who has completed several hundred multiplication problems and to whom $6\times 8$ and $11\times 12$ are operations that have been 'overlearned' &quot; (pp. 92-93).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2014-03-24T02:57:20.573" LastActivityDate="2022-11-27T12:49:39.950" Title="What experimental studies have been done on the Kumon method of teaching and learning mathematics?" Tags="|reference-request|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="711" PostTypeId="2" ParentId="682" CreationDate="2014-03-24T02:34:34.577" Score="5" Body="&lt;p&gt;First, make a public announcement, preferably not while they are engaged in the activity so as not to draw attention to it: something like  &quot;I am not sure whether the people who become involved in certain extracurricular activities (e.g. poker games, courtship rituals, cell phone conversations) during class time would prefer privacy without interruption, public critique, or audience participation, but for any of these, there are better times and places than right here, right now. We are all supposedly here to study mathematics and most of the class wants exactly that. I do not wish to single out anyone for public embarrassment, but if they must continue, I must in turn insist that they find such a time and place where they will not distract the other students.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then back this up with a private conversation with the offender(s): &quot;The principle applies generally but I was particularly addressing you because you were doing thus and such which is disruptive.&quot; If they make excuses about other students, &quot;I can watch so and so. But you were the one who stuck out&quot;.  If the uninvolved ask for more details, address their own behavior (commendable or otherwise) and decline comment on other students. Those who were involved as either participants or observers already know or suspect. Those who weren't, don't need to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then present the offender(s) with a choice of alternatives: To concentrate on mathematics while they are in your class (which is what you would prefer) or have something less pleasant, (call them out during class, have them removed from class, or whatever disciplinary measures your institution has in place.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then follow up on whatever they have chosen. if they stop the disruptive behavior, problem solved. If they continue the behavior after agreeing to drop it or become defiant, let the consequences follow. Dropping the hammer before you have given warning, or being unduly severe makes you look unfair (with reason): Consequences that are more entertaining than serious and threats that you fail to back up make you look weak (also with reason).&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-24T02:34:34.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="712" PostTypeId="1" AcceptedAnswerId="2267" CreationDate="2014-03-24T02:43:15.387" Score="61" ViewCount="5968" Body="&lt;p&gt;I am a young math educator. I've TAed four semesters of calculus for various instructors. Some instructors have required me to grade selected problems in homework sets. Another required me simply to note if it was turned in. Yet another did not require homework handed in at all. In all cases, the students were tested on weekly quizzes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My impression was that the students learned just as well without having graded homework. The teaching burden was reasonably reduced without grading the homework and I always found the quizzes I wrote easier to grade. But I say this without knowing much. I would guess that more detailed comments are better than less but time is finite. So here is my question: Is it worth grading calculus homework?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on some clarifying questions in the comments, here's what I mean by &quot;grading&quot; and &quot;worth it&quot;. Grading is collecting homework and specifically marking any number of problems for correctness or otherwise in greater detail. Not grading would be collecting homework but only marking for completion (for the whole assignment) or not collecting at all. Based on one of the answers I guess there is a middle ground of marking papers but not assigning &quot;correctness&quot; or a grade. By worth it, I am considering the efficiency of grading. I presume more effort or more skilled grading will produce better results. But if you spend too much time the gains will be marginally worse. To measure effectiveness, I guess exam scores, from the perspective of a TA, and actual understanding from a broader perspective are what we should consider. I'd allow alternative interpretations of these terms.&lt;/p&gt;&#xA;" OwnerUserId="244" LastEditorUserId="2083" LastEditDate="2019-01-10T16:17:37.087" LastActivityDate="2019-01-10T16:17:37.087" Title="Is it worth grading calculus homework?" Tags="|undergraduate-education|calculus|homework|grading|assessment|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="713" PostTypeId="2" ParentId="677" CreationDate="2014-03-24T03:14:35.627" Score="11" Body="&lt;p&gt;Most of the courses I teach I didn't take as an undergraduate, including:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Calculus I and II&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Abstract  Algebra&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Dynamical Systems&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Differential Geometry&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Introduction to Proofs&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Point-Set Topology&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In all cases, I use the following algorithm to figure out what to teach:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Choose a textbook.&lt;/strong&gt;  You can usually figure out what the choices are by asking around or searching for the course title on Google and Amazon.com.  (The sales rankings on Amazon.com can be particularly helpful for figuring out which books are popular.) After you've assembled a list of standard choices, order some desk copies so that you can take a look at them, and read reviews to figure out how they differ.  You can choose a book based on (1) what topics are covered (2) whether reviews are generally positive or negative and (3) your perception of the difficulty of the text relative to the level of the students.  Make sure not to choose a graduate text for an undergraduate course unless you're &lt;em&gt;sure&lt;/em&gt; it's a good idea.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Find some syllabi.&lt;/strong&gt; You can usually find some syllabi online for courses that other professors have taught using the textbook.  Just Google the name (or author) of the textbook together with the word &quot;syllabus&quot; or &quot;course description&quot; or &quot;homework&quot;.  You can usually find several week-by-week descriptions of what various other courses covered, and you can use these to help plan your own course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, especially the first time you teach a course and especially if you didn't take the course as a student, you may want to err on the side of following the &quot;standard&quot; curriculum.  It's fine if you have your own vision for how material should be covered, but it may be a good idea to try the &quot;standard&quot; approach once before you try to invent your own approach.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Choose some course goals.&lt;/strong&gt; This part of course construction is often overlooked.  It is &lt;em&gt;vital&lt;/em&gt; to figure out your pedagogical goals for a course, and how those goals are related to the material that you plan to cover.  For example, my goals for the students in abstract algebra are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Teach the basic material of abstract algebra that they will need for future math courses.&lt;/li&gt;&#xA;&lt;li&gt;Reinforce the proof-writing skills that the students learned in Introduction to Proofs.&lt;/li&gt;&#xA;&lt;li&gt;Introduce them to the idea that you must sometimes &lt;em&gt;play&lt;/em&gt; with a math problem (especially specific examples) before you are ready to write a proof.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My goals in dynamical systems are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Teach the students how to use &lt;em&gt;Mathematica&lt;/em&gt; to model complicated mathematical objects.&lt;/li&gt;&#xA;&lt;li&gt;Help the students learn to make conjectures and draw conclusions based on experimental evidence.&lt;/li&gt;&#xA;&lt;li&gt;Reinforce the students' understanding of calculus and linear algebra &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note that none of the goals for dynamical systems are related to the actual content.  I don't really &lt;em&gt;care&lt;/em&gt; if the students learn certain specific topics dynamical systems -- the purpose of the course is for the students to improve their overall mathematical sophistication in the ways I have described.  (Learning something about dynamical systems is a happy side effect.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way to choose goals is a combination of thinking about (1) how the course you're teaching fits into the overall curriculum (2) what course goals will mesh well with the material you plan to cover, and (3) what kind of students you will have, and what they need to work on to become better at math.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Make a tentative plan.&lt;/strong&gt; Make a tentative schedule of which sections of the book to cover on which days.  You can base this plan off of syllabi you find, as well as your own sense of what you plan to cover.  Make a note of some topics that are optional and could conceivably be skipped if you start running behind.  You should also make the other usual course decisions, e.g. what kinds of exams there will be, whether there will be weekly quizzes, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Play it by ear.&lt;/strong&gt; The first time I teach a course, I don't give the students the course schedule beforehand, and I don't announce homework assignments until about a week before they are due.  When you're teaching a course for the first time, you &lt;em&gt;have&lt;/em&gt; to allow for the possibility that your original plan may have been overly ambitious, or that you didn't realize which aspects of the material the students would need to spend more time on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's also very important the first time you teach a course to have a lot of interaction with the students, to make sure that you know how difficult they find the course so far, which aspects of the course they are struggling with, and so forth.  Don't be afraid to explicitly ask students how the course is going, and whether they have any suggestions.  You need to make sure to find out about any problems &lt;em&gt;before&lt;/em&gt; they become serious issues.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I hope all of this helps!&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-24T03:14:35.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="714" PostTypeId="1" CreationDate="2014-03-24T04:23:09.337" Score="20" ViewCount="636" Body="&lt;p&gt;I do not mean for this thread to be a discussion of whether or not students should show work on an exam, whether they should be docked points for just &quot;seeing&quot; an answer and writing it down, whether using a calculator is fine, etc. etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather, I want to know if there are good, detailed, and clear ways to get across &lt;em&gt;your&lt;/em&gt; requirements/opinions/policies about &quot;showing work&quot; so that your students will...&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Understand what is expected of their writeups on both homeworks and exams, in terms of how they will be graded&lt;/li&gt;&#xA;&lt;li&gt;Follow along with &lt;em&gt;why&lt;/em&gt; you expect them to show what you ask of them (at least enough to buy into it for the context of the course)&lt;/li&gt;&#xA;&lt;li&gt;Feel like they are being accurately and fairly assessed on their knowledge and implementation of techniques (or whatever is relevant for the course)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For responses to these queries, I'm looking for specific details about things you have tried, and an idea of how successful they were. This might include a standard &quot;disclaimer&quot; or &quot;prompt&quot; you put on every exam. Or, this might include a common, repeated discussion you have with your classes that reminds them about your policies.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-03-25T13:30:01.677" Title="Making standards for &quot;showing work&quot; explicitly clear to students" Tags="|course-design|student-motivation|technology-in-education|grading|assessment|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="715" PostTypeId="1" AcceptedAnswerId="718" CreationDate="2014-03-24T05:35:26.930" Score="17" ViewCount="2330" Body="&lt;p&gt;I showed my calculus students two circular cones with the same base and height, but one of them &quot;right&quot; and the other slanted, and asked which had a greater volume.  They all answered correctly that both have the same volume, but a lot of them had just memorized the formula for the volume of a cone, so the question mostly failed at causing them to think.  What would be a good example of two solids that have the same volume by &lt;a href=&quot;http://en.wikipedia.org/wiki/Cavalieri%27s_principle&quot;&gt;Cavalieri's principle&lt;/a&gt;, but where this fact is not otherwise obvious or deducible from frequently memorized formulae?&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2023-03-08T18:35:23.253" Title="Surprising examples of Cavalieri's principle" Tags="|calculus|examples|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="716" PostTypeId="2" ParentId="714" CreationDate="2014-03-24T05:49:47.560" Score="16" Body="&lt;p&gt;One technique which is fairly obvious, but (at least for some of us) surprisingly difficult to implement consistently, is to just model for them in class what you expect them to write on their own.  When I solve a problem in class, I try to show the same work and write the same explanations that I expect them to show.  I also try to talk about it as I do it, explaining from time to time why we need to write such-and-such in order to make it clear to the reader what's happening.  If in the process of solving a problem in class we do some &quot;scratch work&quot;, I try to make it clear that the scratch work is separate from the actual written-up solution, perhaps by using a different colored marker.  And I try to also write these &quot;meta&quot; comments down on the board as well, rather than just saying them verbally.  All of these take time and effort, and I'm nowhere near perfect yet, but I've found that the more consistently I engage in this behavior, the fewer complaints I get about grading on clarity and showing work.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2014-03-24T05:49:47.560" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="717" PostTypeId="2" ParentId="706" CreationDate="2014-03-24T05:58:03.667" Score="22" Body="&lt;p&gt;The Moore method is used at the University of Chicago in some sections of &quot;Honors Calculus&quot;, which is really an introductory real analysis course for top incoming freshmen.  I assisted with it a couple of times and taught it on my own once.  It absolutely depends on having a well-constructed sequence of notes to use; we started with a truly excellent set of questions that develop the topology of the continuum from three or four simple axioms and I was amazed at how successful it was.  Designing such a set of notes does require intimate familiarity with the material; teaching from them only a little less so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for covering material, the class in question is a whole year long (a three-quarter sequence), and although we spent a &lt;em&gt;lot&lt;/em&gt; of time in the fall quarter on topology and learning to write proofs while the other sections of Honors Calculus were zipping on through limits and derivatives, we pretty well caught up to them in winter and spring.  Perhaps we didn't go quite as deeply into the winter and spring topics as we might have, but the students were so solid in the foundations that I didn't worry about their ability to pick up anything we missed.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2014-03-24T05:58:03.667" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="718" PostTypeId="2" ParentId="715" CreationDate="2014-03-24T06:20:25.433" Score="20" Body="&lt;p&gt;One interesting example along these lines involves the following three solids:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Half of a sphere (the upper half) with a radius of $r$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A circular cone (vertex at the bottom) with base of radius $r$ and height $r$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A circular cylinder with base of radius $r$ and height $r$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The statement that's true is that the areas of the cross sections of shape #1 plus the areas of the cross sections of shape #2 are equal to the areas of the cross sections of shape #3.  It follows that the volumes add, i.e.&#xA;$$&#xA;\frac{2}{3}\pi r^3 \,+\, \frac{1}{3}\pi r^3 \;=\; \pi r^3.&#xA;$$&#xA;This shows that you can derive the formula for the volume of a sphere from the formula for the volume of a cone, and vice-versa.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you like, you make this into an example of the kind you requested by having one of the solids be the complement of either #1 or #2 in #3. That is, have the first solid be a half-sphere and the second be the complement of a cone in a cylinder, or vice-versa.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Another interesting example is the following pair of solids:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The paraboloid obtained by rotating the region $x^2 \leq y \leq 1$ in the $xy$-plane around the $y$-axis, with circular cross sections.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A triangular prism on a $1$-$1$-$\sqrt{2}$ triangle, with depth $\pi$, using cross sections parallel to one of the $1\times \pi$ rectangular faces.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This example lets you compute the volume of a paraboloid without using calculus (assuming you don't count Cavalieri's principle as calculus).&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-03-24T17:14:57.167" LastActivityDate="2014-03-24T17:14:57.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="719" PostTypeId="1" AcceptedAnswerId="800" CreationDate="2014-03-24T06:45:35.037" Score="4" ViewCount="76" Body="&lt;p&gt;In some cases the historical development of a mathematical subject/tool is not straightforward. Mathematicians define a particular notion and work in an accepted direction. After a while they come across some problems/complexities and refine/redefine their methods/tools/notions in a completely new way and so on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Texts are not based on historical development of the mathematical subjects necessarily because the recent refined approaches are more regular and well-designed but these approaches don't reflect the &lt;em&gt;original motivations&lt;/em&gt; and &lt;em&gt;essential problems&lt;/em&gt; of the field properly. These roots are very useful for introducing subjects to students in a &quot;natural&quot; way.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; Which one should I choose when there is a contradiction between historical and official development processes of a particular subject? What are the advantages and disadvantages of each method? Is it possible to have an effective combination of these two methods?&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-24T10:27:59.283" LastActivityDate="2014-03-25T13:43:01.217" Title="Historical Development vs Official Development" Tags="|course-design|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="720" PostTypeId="1" AcceptedAnswerId="725" CreationDate="2014-03-24T06:53:41.400" Score="10" ViewCount="232" Body="&lt;p&gt;There is a lot of mathematical literature in some non-English languages (French, German, Spanish, etc.) that students from these countries don't need to read English literature (at least) for their undergraduate studies. However, in more advanced classes or when a student wants to get closer to research, there is mostly only English literature, i.e., at some point he/she has to start reading mathematical literature written in English.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;When should a student be encouraged to read mathematical literature written in English? Why?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;(I'm beware that this is not a big problem in countries where there is rarely any native literature.)&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="8571" LastEditDate="2020-08-31T15:21:14.577" LastActivityDate="2020-08-31T15:21:14.577" Title="When to encourage students to read mathematical literature written in English (in non-English speaking countries)?" Tags="|undergraduate-education|graduate-education|cultural-differences|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="721" PostTypeId="1" AcceptedAnswerId="722" CreationDate="2014-03-24T06:59:53.697" Score="12" ViewCount="437" Body="&lt;p&gt;In German universities, the first-year students typically start their analysis courses with introducing the real numbers. Most commonly, the incompleteness of $\mathbb{Q}$ is discussed using the example of $\sqrt{2}$. Then the real numbers are mostly introduced in an axiomatic way or sometimes constructed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunateley, when it comes to the written exams at the end of the term, this topic is usually not covered. It seems to be not easy at all, since we don't just want our students to learn the axioms off by heart. We want them to be aware of the fact, that the reals aren't the rationals and that completeness is not natural at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there a good way to ask for the knowledge on real numbers in a written exam?&#xA;Good answers would give examples for questions or describe experiences with this topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: The knowledge I want to test includes at least: Knowing that the real numbers have to be introduced and do not exist per se (as unique concept extending the rationals, see nonstandard analysis); knowing the property of completeness; understanding that the every real number has a decimal representation which yet is not always unique; knowing that &quot;infinitly small numbers&quot; don't exist as real numbers; being aware of the fact, that analysis has no use if it is restricted to $\mathbb{Q}$. &lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="199" LastEditDate="2014-03-24T14:13:17.550" LastActivityDate="2019-01-15T15:47:57.597" Title="How to test knowledge on the real numbers in a written exam?" Tags="|undergraduate-education|exams|mathematical-analysis|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="722" PostTypeId="2" ParentId="721" CreationDate="2014-03-24T07:25:54.923" Score="5" Body="&lt;p&gt;I agree with you that this is in general not really questioned. But I also think that is is very difficult in contrast to other topics in such a course. In my opinion this should be more a part of an oral exam than a written exam. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your questions, I've seen the following examples in exams:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Multiple choice: True or false? &quot;Every bounded subset of $\mathbb{R}$ has an infimum.&quot;&lt;/li&gt;&#xA;&lt;li&gt;You can ask for consequences of the construction/definition of the real numbers, e.g., questions involving the density of $\mathbb{Q}$ in $\mathbb{R}$. For example, asking if a function like&#xA;$$f(x) = \begin{cases} x, &amp;amp; x\in\mathbb{Q}, \\ -x, &amp;amp; x\in\mathbb{R}\setminus \mathbb{Q} \end{cases}$$ &#xA;is continuous.&lt;/li&gt;&#xA;&lt;li&gt;Depending on the method you used to introduce the real numbers, you can hide some part of the construction into a questions. For example, if you use equivalance of Cauchy sequences to construct the real numbers, you can define a similar equivalence relation (or just exactly this) and let the students show that it is an equivalence relation. &lt;/li&gt;&#xA;&lt;li&gt;You can ask to show that the real numbers are uncountable (Okay, this was maybe also learned by heart, but since this is not a very short proof, you can ask them only of a part of it).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-24T07:25:54.923" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="723" PostTypeId="2" ParentId="719" CreationDate="2014-03-24T09:33:20.960" Score="4" Body="&lt;p&gt;The &quot;official&quot; development (with modern notation, notions, ...) is for a reason. It simplifies understanding, makes relationships that were noticed much later aparent, generalizes results that were very narrow, and in general eases the path for the casual wanderer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, it does violence on my sense of historical accuracy. It is often linked with scientific &quot;biographies&quot; that are wide off the mark, misrepresenting the real struggle and triumphs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in final summary, I'd stick with the &quot;official story,&quot; just reminding students that this is not the real history, but is heavily edited.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-24T09:33:20.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="724" PostTypeId="2" ParentId="712" CreationDate="2014-03-24T09:33:58.087" Score="6" Body="&lt;p&gt;This is an &quot;in my experience&quot; answer. However, I should say that my experience is slightly odd in that I teach students from a &quot;wide variety&quot; or backgrounds (I have fewer than 10 students from any individual country in my foundation first year courses).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, &lt;em&gt;in my experience&lt;/em&gt;, grading is pointless but marking is useful (I would argue even &quot;necessary&quot;). This is because it is your first opportunity to give the students feedback on their work and how they present it, and it is also (probably) your only opportunity before their exam...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I once had a student write three pages of mostly correct maths for an exam where some people used three answer booklets. They were a bright student, and they just &lt;em&gt;passed&lt;/em&gt; the exam, but as answer books are anonymous you wonder if the anonymous person was cheating and they lost marks right, left and center for not showing their working. If I had given this student feedback then this would have been less of an issue. So, lesson learned, I will always set fresh-faced students work which I can mark and return to them.&lt;/p&gt;&#xA;" OwnerUserId="169" LastActivityDate="2014-03-24T09:33:58.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="725" PostTypeId="2" ParentId="720" CreationDate="2014-03-24T09:41:12.383" Score="5" Body="&lt;p&gt;As soon as possible. They &lt;em&gt;will&lt;/em&gt; need to read English sooner rather than later (not just for math!). Most of the time the saying that last month's results are circulated privately (or via blogs), what was done the last term is in the conference, papers are published 2 or 3 years later, the papers are summarized in a textbook some 5 years after that, and translations of successful texts at least 3 years later. In any fast-moving field relying on the later is just suicide. The whole filtering down process can introduce errors, having access to the original sources can be vital. Besides, the amount and depth of English resources is just so much larger it isn't funny. For one, there are no math.SE or MO in, say Spanish, the coverage in Wikipedia is much better.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-24T09:41:12.383" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="726" PostTypeId="2" ParentId="714" CreationDate="2014-03-24T10:05:07.427" Score="4" Body="&lt;p&gt;Here's how I address this issue on my course syllabi:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Throughout this class, you need to explain your answers even when the problem doesn't explicitly ask for a proof; this typically means writing &lt;strong&gt;in complete English sentences&lt;/strong&gt;. When deciding how much detail to include, here's the standard to keep in mind: your solution to a problem should be complete and clear enough that one of your classmates, who has paid attention in class but hasn't thought about that specific problem yet, could read your solution and understand exactly how it works. If you only try to convince the grader that you understand the solution, then you almost certainly won't write enough.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="129" LastActivityDate="2014-03-24T10:05:07.427" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="727" PostTypeId="2" ParentId="714" CreationDate="2014-03-24T10:50:42.183" Score="3" Body="&lt;p&gt;It does help if &lt;em&gt;complete&lt;/em&gt; solution to previous homework/exams is published as a guide. Also add a paragraph or two to homework handouts on what is expected as a solution (perhaps the first few ones only).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-24T10:50:42.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="728" PostTypeId="2" ParentId="712" CreationDate="2014-03-24T10:57:30.750" Score="5" Body="&lt;p&gt;I take homework to be &quot;show you can &lt;em&gt;do&lt;/em&gt; stuff with what has been taught&quot; while exams are &quot;show you &lt;em&gt;understand&lt;/em&gt; what it means, and how it hangs together.&quot; In that sense, both should be graded as they asess different aspects of the student's mastery of the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that marking should be rather detailed (and furthermore complete solutions published soon after the deadline), and returned as soon as possible as feedback (that many don't pick up their work later is frustrating, &quot;got the grade, forgot the subject&quot; is all too common...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grading only part of the homework cuts down on the work, but requires several &quot;equally hard&quot; problems, and cooking &lt;em&gt;them&lt;/em&gt; up is very complicated.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-24T10:57:30.750" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="729" PostTypeId="1" CreationDate="2014-03-24T11:28:01.950" Score="2" ViewCount="66" Body="&lt;p&gt;Teaching is a social activity and very sensitive to any change in social parameters. Even the simplest acts of a teacher could cause some positive/negative reactions of students and conversely. Each one of us has his own &lt;em&gt;personal experiences&lt;/em&gt; in possible good and bad teaching styles and we are sharing them here in the forum. It is very helpful for us. Also the &lt;em&gt;scientific analysis&lt;/em&gt; of the academic teaching done by professional sociologists and psychologists could be very useful too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What are good references for researches about the best teaching styles done by psychologists/sociologists?&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastActivityDate="2014-03-24T15:55:00.413" Title="Psychological and Sociological Researches on Teaching" Tags="|reference-request|" AnswerCount="2" CommentCount="1" ClosedDate="2014-03-25T01:33:43.823" ContentLicense="CC BY-SA 3.0" />
  <row Id="730" PostTypeId="2" ParentId="729" CreationDate="2014-03-24T11:41:09.533" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://teachpsych.org/ebooks/asle2014/index.php&quot; rel=&quot;nofollow&quot;&gt;Here&lt;/a&gt; is a link to the new (free) e-book &lt;em&gt;Applying Science of Learning in Education: Infusing Psychological Science Into the Curriculum&lt;/em&gt;, edited by Victor A. Benassi, Catherine E. Overson, and Christopher M. Hakala, which I haven't read yet but intend to.&lt;/p&gt;&#xA;" OwnerUserId="129" LastEditorUserId="129" LastEditDate="2014-03-24T14:17:06.340" LastActivityDate="2014-03-24T14:17:06.340" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="731" PostTypeId="2" ParentId="654" CreationDate="2014-03-24T11:55:42.320" Score="43" Body="&lt;p&gt;First, a little background.  I switched to slides a few years ago when teaching a 3rd year course at my university.  Because of how teaching works at my university, this course is one of the first where as a lecturer I can assume that the students taking it are interested in mathematics as a subject in its own right.  That is, they aren't students from another program taking it because it is required of them.  This doesn't mean that they are all mathematically sophisticated -- in fact, they tend not to be -- but rather that they are expected to learn about how to do mathematics from the inside.  So there are proofs, and proof techniques, and the like.  It's also a fairly diverse course with topics from metric spaces to linear algebra to Hilbert spaces.  (Finding a common theme through that lot was one of my difficulties in the early days!)  Lastly, for background, I teach in English but the majority of my students have Norwegian as their primary language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I taught the course using chalk the first time that I taught it, and then switched to slides the second time.  I then taught it three times using slides before it passed on to another lecturer.  The last time that I taught it, I was awarded the faculty prize for &quot;Teacher of the Year&quot;, primarily for that course.  I won't say that it was the slides that won it, I mention that to show that it was part of a general strategy &lt;em&gt;that worked&lt;/em&gt;.  On that, I can also mention that enrolment in this course also increased over the four years that I taught it, to almost double its initial numbers.  Also worth knowing are that I experimented with slides &lt;em&gt;before&lt;/em&gt; this course with a half-and-half method (on a different course) and it was the feedback from that which convinced me to try the full method, and that after teaching that course then I also taught a lower level course with slides and for that course then I have moved back to using chalk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's what I was thinking when I switched, plus some commentary on whether it worked as I expected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My primary consideration in switching was to consider the question &quot;Where is a student's attention during the lecture?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I completely agree with the theory that students learn by &lt;em&gt;doing&lt;/em&gt;, and I love the idea of an inquiry-based course and of interactive lectures with communication in both directions.  However, I'm not convinced that the &lt;em&gt;lecture&lt;/em&gt; has to be where the students &lt;em&gt;learn&lt;/em&gt;.  In a course like this one, I saw the lecture primarily as the &lt;em&gt;demonstration&lt;/em&gt; where the students would see what they should do and then they would go to their own labs and try it out.  So the lecture is primarily to communicate mathematical thinking.  Yes, a lot of knowledge as well, but never bare facts (definitions or theorems or even proofs) - every definition comes with a story[1] as to why it is that definition and not another, every proof is ladled with detail as to why each step is the natural one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this, then, the desired focus of the students is on me.  I am trying to demonstrate what is going through a mathematician's head when (s)he contemplates a mathematical problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what happens with a chalk talk?  I say something, the students obediently listen, then I write something which the students obediently copy down.  I then wait for them to finish writing, which some do quickly and some do slowly, before saying the next thing which half of them miss because they're still copying down the last thing I wrote.  In addition, some of them are having difficulty copying stuff down because I wrote it in English and their mental error-checking codes are set up for Norwegian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using slides gives me the following advantages:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;No time is wasted through &lt;em&gt;writing&lt;/em&gt;.  What I write appears on the screen so I don't waste time.  I &lt;strong&gt;always&lt;/strong&gt; make my slides available in advance (as both &quot;presentation&quot; and &quot;handout&quot; versions) so the students do not need to copy things down during the lecture: they can print them out and bring them along and make additional notes if they want but don't have to.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I know &lt;em&gt;exactly&lt;/em&gt; what the students are looking at.  Yes, when new stuff appears on the slide then the students look at the slide and read it, but I control when that appears and I know what it will be.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The students know what we will cover.  They can see the slides beforehand and read them (if they want).  I also provide references to the text book(s) or other resource pages for background material.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The students know that they will have the notes that I meant them to have.  So they don't need to worry about missing a word in a definition or misreading a symbol.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I'm not quite sure where to put this next point so I'll put it here for lack of anywhere else.  I only switched to slides when I knew that the technology could support it.  I did &lt;strong&gt;not&lt;/strong&gt; want to remove the flexibility of chalk so I needed to know that I could go &quot;off piste&quot; if necessary.  To that end, I started out using a graphics tablet with (a slightly hacked version of) xournal.  Once I had an iPad, I switched to that.  My handwriting leaves a little to be desired, but nevertheless is workable.  In addition, I supported the in-lecture material with a wiki and a forum to provide out-of-lecture support.  This is how I got around the perceived inflexibility of slide lectures.  The annotated slides were published after the lecture, so if I wrote something on the slide then the students didn't have to copy it down as they knew they'd get access to it later that day (this is one reason why I heartily dislike using slides &lt;em&gt;and&lt;/em&gt; chalk, the other reason is to do with the lighting in the lecture hall).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once I'd embarked on using slides, I encountered other advantages:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It forced me to prepare in advance and prepare well.  Since my plan was to get the lecture notes up in time for the students to print off before the lecture, I had to have the lecture prepared the day before, not the hour before.  In addition, as the notes were what the students would see and not just hints for what I wanted to write, I had to check them carefully.  So I couldn't assume &quot;Oh, I know how to do this&quot; and not bother to prepare each example carefully.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It forced me to keep to time.  I integrated my slide preparation with my lesson plan so I had planned how long to spend on each topic.  If I changed the timings in my lecture then I had to have a very good reason for doing so.  Note that I &lt;em&gt;could&lt;/em&gt; change my timings, but wouldn't do so on a whim.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It forced me to think very carefully about &lt;em&gt;what&lt;/em&gt; the students could see.  It quickly became clear that the best slide was a blank slide, and that it stops being optimal when you start adding content.  So the trick is to figure out just what you want the students to be able to see at any given point and put just that on the slide.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It forced me to think from the students' perspective.  Since slides put me in full control of the lecture hall, I had to think a lot more about how the students would react and what they would need to know and when.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It gave me more flexibility in how to present material.  Of course, once can always go too far (and some may argue that I did), but I think that it is much easier to be creative with slides than with chalk.  An example that springs to mind is stepping through the proof of why the space of continuous functions is complete.  I would have the same &quot;slide&quot; with small changes each time and a commentary on what was happening in each step.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In addition, once the computer (or iPad) is in the room &lt;em&gt;and switched on&lt;/em&gt;, it becomes easy to use further functionality, such as animations or videos.  It's all very well to talk about the SVD of a matrix, but there's a nice animation of why it works for 2x2 which makes it much clearer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The main disadvantage that I encountered with using slides was the time taken to prepare the material.  And even though I could re-use them the next time, I still wanted to tweak them and so it didn't cut down as much as I'd hoped.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that said, as I mentioned above then I've stopped using them in the lower level course that I currently teach.  I'd say that my main reason for doing so was that I felt that the students didn't know how to cope with maths being taught via slides.  I don't think that they knew what to do with themselves in the lecture hall: they came to get a set of lecture notes and since that was being provided for them without them needing to do anything else, they didn't do anything else.  So ultimately, whether they work or don't will depend on the students.  The goal is to get the students learning and if they can't or won't learn from slides, there's no point in using them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find slides from my lectures from links on &lt;a href=&quot;http://www.math.ntnu.no/~stacey/Teaching/&quot;&gt;my teaching page&lt;/a&gt;.  The course that I'm talking about above is TMA4145; TMA4190 was my half-and-half experiment; TMA4115 is the lower level course.  Some of the notes are located on the mathsnotes wiki, linked from the above teaching page.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] I don't mean anecdote here.  I mean more in the flavour of an explanation.  &lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-03-24T11:55:42.320" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="735" PostTypeId="2" ParentId="715" CreationDate="2014-03-24T14:02:20.123" Score="7" Body="&lt;p&gt;&lt;strong&gt;Spiral paper blocks and regular paper blocks&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.de/search?q=%22spiral+paper+block%22&amp;amp;tbm=isch&quot;&gt;This image search&lt;/a&gt; should provide some example pictures of spiral paper blocks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would not know of a technique of calculating the volume of the shape of a spiral paper block which does not use Cavalieri’s principle or something higher. However, with the principle it becomes directly clear that the volume is the same as that of a regular paper block (with the same paper size and height). As a bonus, the individual sheets correspond to the individual slices, not to forget that the principle itself is nicely illustrated as both blocks contain the same amount of paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Didactically, you might consider introducing the shape abstractly first, without presenting or mentioning the real-life thing – but whether that’s a good idea depends on several factors.&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2014-03-24T14:02:20.123" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="736" PostTypeId="1" CreationDate="2014-03-24T14:18:26.093" Score="4" ViewCount="131" Body="&lt;p&gt;I am trying something new this semester in my Calculus classes.  I currently have over 400 students, and there is no way I can find the time to grade and give constructive criticism on all the homework they turn in each week.  So this semester, I decided to use mylabsplus for the homework, strictly to make sure they are keeping up with the material, and then assign chapter-wide mini projects that are due on the test day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In these mini projects, I ask students to solve a &quot;realworld-like&quot; problem  or a &quot;Reasearch-like&quot; problem using the tools from the entire chapter.  For example, on the chapter covering Jacobians, I ask them to &quot;find a method to derive the formula for an n-dimensional hypersphere&quot;. or I may ask them to use Calculus to find both the straight line difference between 5 points on the globe versus the arc length distance between these points.  In this manner I am able to give constructive criticism on each assignment, and the students seem to actually enjoy the projects much more than rote homework.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far, test scores also seem to be somewhat better than in classes where I did not do the mini-projects, but I don't know if this is due to a better understanding of the material versus smarter students.          &lt;/p&gt;&#xA;" OwnerUserId="203" LastEditorUserId="78" LastEditDate="2014-03-24T15:32:24.773" LastActivityDate="2014-03-25T13:59:06.680" Title="Would chapter mini-projects be more beneficial than weekly homework for large classes n&gt;100" Tags="|mathematical-pedagogy|undergraduate-education|calculus|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="737" PostTypeId="1" AcceptedAnswerId="742" CreationDate="2014-03-24T14:29:18.843" Score="21" ViewCount="2438" Body="&lt;p&gt;I see a lot of students struggling when they have to show that something is well-defined. I have the feeling that this is often not understood. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When defining a sequence $x_n= g(x_{n-1})$ (for some function $g$) and asking to show that $\lbrace x_n\rbrace$ is well-defined (since there is a root or a fraction somewhere in the definition), students often show that the sequence converges to something (under the assumption that it is well-defined) and do not see that they have to ensure that there is something bad.&lt;/li&gt;&#xA;&lt;li&gt;When you want to define a function, say $g:A\to B$ (for some sets $A$ and $B$) with $g(x):=\ldots$, they forget to show that $g(x)$ holds the properties to be an element of $B$. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have the feeling that some are not even aware of what &lt;em&gt;well-defined&lt;/em&gt; means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why do students have this problem?&lt;/li&gt;&#xA;&lt;li&gt;How can one (or the one who introduces functions) do to gain a better understanding of the concept of well-defined functions/sequences/...?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastEditorUserId="1147" LastEditDate="2016-12-15T07:05:10.463" LastActivityDate="2019-01-22T00:06:28.163" Title="Why do students have problems with showing that something is well-defined? How can this be improved?" Tags="|undergraduate-education|students-mistakes|definitions|" AnswerCount="9" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="738" PostTypeId="2" ParentId="737" CreationDate="2014-03-24T15:17:44.133" Score="14" Body="&lt;p&gt;I've never tried this in a classroom, but I suspect a lot of the trouble with functions is that students haven't been taught the vocabulary to deal with things that are weaker than functions. For example, they are trying to define a function $f: A \to B$. What they have written down probably defines SOMETHING: Perhaps a subset $R$ of $A \times B'$ for some $B' \supset B$. What do they then need to check: That for each $a$ in $A$, there is precisely one $b \in B$ so that $(a,b) \in R$. But if they can't use set theoretic language well enough to figure out what they have constructed, it will be hard for them to see what more they need to do. &lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2014-03-24T15:17:44.133" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="739" PostTypeId="2" ParentId="729" CreationDate="2014-03-24T15:55:00.413" Score="3" Body="&lt;p&gt;The National Research Council has two (free!) books that are worth checking out.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.nap.edu/openbook.php?isbn=0309070368&quot; rel=&quot;nofollow&quot;&gt;&quot;How People Learn&quot;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.nap.edu/catalog.php?record_id=11101&quot; rel=&quot;nofollow&quot;&gt;&quot;How Students Learn: Mathematics in the Classroom&quot;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I'd also recommend going beyond what psychologists and sociologists have found about teaching math. Psychologists especially are going to end up with somewhat reductive pictures, and a human activity as complex as learning or teaching also benefits from a holistic approach. There's some excellent research done, for instance, on how students learn relatively elementary concepts such as fractions and multiplication. I am a huge fan of the &quot;Cognitively Guided Instruction&quot; series, personally.&lt;/p&gt;&#xA;" OwnerUserId="84" LastActivityDate="2014-03-24T15:55:00.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="740" PostTypeId="2" ParentId="721" CreationDate="2014-03-24T16:03:19.123" Score="7" Body="&lt;p&gt;Some possible exam questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Prove that if $x&amp;lt;1/n$ and $x&amp;gt;-1/n$ for all natural numbers $n$, then $x=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Prove or disprove:  If $x$ and $y$ are reals with $x&amp;lt;y$, there is a rational $q$ with $x&amp;lt;q&amp;lt;y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Construct an injective map $f:\mathbb{Q}\times \mathbb{Q} \times \mathbb{Q \rightarrow R}$, and prove that it is injective.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-24T16:03:19.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="741" PostTypeId="2" ParentId="737" CreationDate="2014-03-24T16:14:44.433" Score="7" Body="&lt;p&gt;On the question &lt;em&gt;why?&lt;/em&gt;, I think the answer is clear enough:  &lt;strong&gt;mathematicians often speak and write as if all of their formulas are well-defined, and students emulate them&lt;/strong&gt;.  Usually we write the additive rule of derivatives as:&#xA;$$f'+g'=(f+g)'$$&#xA;when the rigorous version would be:&#xA;$$f'\negthinspace\negthinspace\downarrow \thinspace \&amp;amp; \thinspace \ g'\negthinspace\negthinspace\downarrow \ \rightarrow \ \ f'+g' = (f+g)'.$$&#xA;Solomon Feferman has &lt;a href=&quot;http://www.academia.edu/160323/Definedness&quot;&gt;shown&lt;/a&gt; how to rigorize the introduction of partial functions in this way.  If the notation for &quot;is well-defined&quot; looks unfamiliar, that may prove my point.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-03-24T16:33:29.293" LastActivityDate="2014-03-24T16:33:29.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="742" PostTypeId="2" ParentId="737" CreationDate="2014-03-24T16:20:40.650" Score="28" Body="&lt;p&gt;Maybe, your students have a belief problem. They will rarely (maybe never) have encountered problems where something was not well-defined. &#xA;If you have never been in trouble since everything you were shown was well-defined, then you don't even understand the problem! (Even harder: after proving that something is well-defined, the world looks right like it was before, since they don't see the gain. Thus, they might try to do something that has more use for them.)&#xA;You might try to give them explicitly not well-defined functions. Say, the sum of a fraction $\frac{a}{b}$ is $a+b$. Then let them check whether different representations of a rational number give the same value. Let them find a representation equal to 1/1 which has the sum 100.&#xA;You could then discuss what typically can go wrong (problems with different representations, formulae that aren't defined for all values, values that do not lie within the range, ...?).&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="199" LastEditDate="2014-03-26T14:08:15.517" LastActivityDate="2014-03-26T14:08:15.517" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="743" PostTypeId="2" ParentId="408" CreationDate="2014-03-24T16:48:09.463" Score="1" Body="&lt;p&gt;Below is a list of slides assembled by the community, organized by subject area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Algebra and Combinatorics&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Analysis, Geometry, and Topology&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mathematical Biology&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mathematical Physics&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Numerical Analysis and Scientific Computing&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Optimization and Control&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Ordinary Differential Equations and Dynamical Systems&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Partial Differential Equations&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(Pre)Calculus&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Probability, Stochastic Processes, and Financial Mathematics&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Symbolic Computation&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="179" LastActivityDate="2014-03-24T16:48:09.463" CommentCount="1" CommunityOwnedDate="2014-03-24T16:48:09.463" ContentLicense="CC BY-SA 3.0" />
  <row Id="744" PostTypeId="2" ParentId="701" CreationDate="2014-03-24T17:32:27.213" Score="3" Body="&lt;p&gt;&lt;strong&gt;Resources&lt;/strong&gt;&lt;br&gt;&#xA;You might like the book, &lt;a href=&quot;http://www.moebiusnoodles.com/TheBook&quot; rel=&quot;nofollow&quot;&gt;Moebius Noodles&lt;/a&gt;, to use with young kids. I'd also recommend checking out math circles. &lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=MCL-5&quot; rel=&quot;nofollow&quot;&gt;This book&lt;/a&gt; is a delightful start. There's lots more. And you might share &lt;em&gt;The Number Devil&lt;/em&gt; with the young people you work with. The book I'm working on, &lt;em&gt;Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers&lt;/em&gt;, will also be helpful once it's out. (Actually, you're welcome to request a copy of the manuscript now. Just email me.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One creative problem, that I'll be doing as a math circle with college students (starting next week), is the question of how the cards for Spot It! were created. I blogged about it &lt;a href=&quot;http://mathmamawrites.blogspot.com/2012/03/oakland-math-circle-does-spot-it.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;http://mathmamawrites.blogspot.com/2012/06/spot-it-circle-today.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, and &lt;a href=&quot;http://mathmamawrites.blogspot.com/2012/01/math-adventures-thinking-about-spot-it.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. It doesn't require algebra, but is related to a field of higher math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Principles&lt;/strong&gt; &lt;br&gt;&#xA;Your second question is interesting: What general principles can be used to promote creativity in elementary school mathematical students?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps this passage from my book illustrates what I see as the first principle, that &lt;strong&gt;math is more than arithmetic, and even young kids can see its other sides&lt;/strong&gt;: &lt;br&gt;&#xA;What is math? &#xA;Most people think it’s adding, subtracting, multiplying, and dividing; knowing your times tables; knowing how to divide fractions; knowing how to follow the rules to find the answer. These bits are a tiny corner of the world of math. Math is seeing patterns, solving puzzles, using logic, finding ways to connect disparate ideas, and so much more. People who do math play with infinity, shapes, map coloring, tiling, and probability; they analyze how things change over time, or how one particular change will affect a whole system. Math is about concepts, connections, patterns. It can be a game, a language, an art form. Everything is connected, often in surprising and beautiful ways. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second principle: If we want to promote creativity, we need to let kids know that math is playful. (I think those two principles probably sum it up for me.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-24T17:32:27.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="745" PostTypeId="2" ParentId="584" CreationDate="2014-03-24T17:51:01.743" Score="12" Body="&lt;p&gt;First, we need to be aware of the ways that women are discouraged, and when those are not related to what mathematics is, we need to change the culture. Does math have to be approached as a competitive sport? No. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related to this concern, please note that I am the only obvious woman in the top 40 users here. There are plenty of women blogging about math and math education. What about this culture alienates women? I think people do not try very hard to be diplomatic when they disagree with a poster: &quot;This question does not belong.&quot;, &quot;This is not an answer.&quot; I have a particular goal, and see myself as a central actor in the math education online discussion world (as a blogger at least), so I have persevered. But if I were less motivated, I might have decided the aggravation wasn't worth it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My story may help others understand subtler issues which are not related to obvious sexism. I got my masters in a program that was easy, and enjoyed it immensely. I decided to go for a phd, and attended UCSD. It was very hard (as was U of Michigan, where I did my undergrad), and I didn't like it. I knew going in that that might happen, and quit very quickly. My interest was in teaching, so for me it wasn't a bad thing. But I am considered one of the casualties, since I didn't 'go on'. And I love math. But I also want to help others in the work I do. My interest in math education satisfies my passion for math, along with my interest in making a difference in the world. Is there a way to change the way we structure the study of higher math so that someone like me would want to continue? Maybe. I'm not sure yet what it would look like.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-24T17:51:01.743" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="746" PostTypeId="1" CreationDate="2014-03-24T19:03:33.967" Score="24" ViewCount="569" Body="&lt;p&gt;Back when I was an undergrad calculus TA, I participated in a general TA training class. We were taught to be mindful of different &quot;learning types,&quot; such as visual learners, audio learners, and kinesthetic learners.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It wasn't until recently that I learned &lt;a href=&quot;http://www.changemag.org/Archives/Back%20Issues/September-October%202010/the-myth-of-learning-full.html&quot;&gt;there is no evidence that these learning styles exist&lt;/a&gt;, (a link to an &lt;a href=&quot;https://www.psychologicalscience.org/index.php/news/releases/learning-styles-debunked-there-is-no-evidence-supporting-auditory-and-visual-learning-psychologists-say.html&quot;&gt;actual study&lt;/a&gt; for completeness). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I began to think about the different ways my calculus students now approach and accept learning. Some people listen to lectures, go to recitations, and are fine. Others read the book and could manage just fine without any listening to me at all. Still others prefer Khan Academy or this style of presentation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I wonder:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there any evidence that mathematical &quot;learning styles&quot; exist, either characterized by what I mentioned above or otherwise?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20" LastActivityDate="2020-01-01T05:41:22.387" Title="Any support for mathematical &quot;learning types?&quot;" Tags="|reference-request|presentation|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="747" PostTypeId="1" AcceptedAnswerId="766" CreationDate="2014-03-24T19:29:35.803" Score="5" ViewCount="114" Body="&lt;p&gt;A large portion of 4-year university mathematics jobs are advertised on mathjobs.org.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a similar, centralized location for secondary education mathematics jobs? If not, what are the standard techniques to use in such a job search?&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-05-06T16:47:03.930" Title="How are secondary-education mathematics jobs advertised in the United States?" Tags="|secondary-education|usa|employment|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="748" PostTypeId="2" ParentId="746" CreationDate="2014-03-24T19:43:35.167" Score="8" Body="&lt;p&gt;I wouldn't count it as evidence, but there is a great &lt;a href=&quot;http://books.google.com/books?id=B60AAAAAMAAJ&amp;amp;pg=PA2&amp;amp;lpg=PA2&amp;amp;dq=felix%20klein%20mathematical%20types%20logical%20geometrical%20lectures%20evanston&amp;amp;source=bl&amp;amp;ots=u2uBm3RYJC&amp;amp;sig=WeAfJ0Xz8wLcgpvPb_Aufhp0z1s&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=h4kwU4WtE-vC0AHhloHoBA&amp;amp;ved=0CGIQ6AEwBw#v=onepage&amp;amp;q=felix%20klein%20mathematical%20types%20logical%20geometrical%20lectures%20evanston&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;reference&lt;/a&gt; on this showing Felix Klein's opinions from 1893:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/bRJbL.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-24T19:43:35.167" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="749" PostTypeId="2" ParentId="584" CreationDate="2014-03-24T20:07:51.573" Score="19" Body="&lt;p&gt;(I am a woman in my last year of a PhD in low-dimensional topology, I'll be a postdoc in the fall, and I have research aspirations. I'm mostly trying to articulate how I got to this stage in my life. In a real sense, my answer shouldn't matter very much since the many things that keep women away from mathematics clearly didn't work on me too well, or on the several other women that are currently in mathematics; but perhaps it is still valuable to see what worked for me.) &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As a woman and an Indian in mathematics, the following is one of my 'favorite' phenomena: &lt;a href=&quot;http://en.wikipedia.org/wiki/Stereotype_threat#cite_note-23&quot;&gt;stereotype threat&lt;/a&gt;. Here is a more detailed &lt;a href=&quot;http://en.wikipedia.org/wiki/Stereotype_threat#cite_note-23&quot;&gt;non-Wikipedia source&lt;/a&gt;. Briefly, there have been studies involving Asian-American girls, where the group who were reminded that they were of Asian descent performed better than the control group on a given math test, and the group that was reminded that they were girls performed worse. It's unclear to me how we should respond to this phenomenon as educators. One possibility is to straight up mention it to students, assuming that awareness might dampen its effects. Another might be to be entirely indifferent to the genders of our students (and play up other similarities. A woman in your class might be the only woman there, but she presumably belongs to shared groups with some of the other students.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I grew up in upper middle class urban India, in a part of the country where it simply never occurred to me that women and men could have (markedly) differing mathematical abilities - gender discrimination abounds in India, but at least in our socio-economic class and region, it wasn't pronounced in terms of intellectual ability; in particular, my elder brother suffered from math anxiety - I expect that my desire to compete with him had at least something to do with my interest in the subject.  In any case, intellectual pursuits were strongly encouraged for everyone. We revered the academically successful students (class ranks were a matter of avid discussion) and the top positions seemed equally distributed (in my high school, there was an award for highest aggregate score in math and physics in our 12th grade (national) exam, and I recall the genders being equally represented. Nonetheless, there were various such pursuits where I was often the only female participant (inter-school quiz competitions, etc.). The OP mentions such environments as being perceived as hostile to girls and women, but I did not particularly feel this way -- while my fellow participants were not the same gender, we had bunches of other similarities, for example, interest in participating in the relevant activities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, as a child I didn't know that one could do mathematics for a living. I like to plan ahead for things (and I believe this is correlated with gender), and my youthful planning for my life didn't include mathematics since I just didn't even consider it an option. My love for mathematics was kindled at a very early age by my grandfather, whose games with my toddler-self consisted of doing additions and subtractions using little circles. Coming to the US as an undergraduate I was excited to learn about the concept of 'double majors' and my decision to pick up a second major in mathematics was motivated in part due to my interest in it, but also due to being a typical over-achiever. My decision to switch to mathematics for graduate study was motivated by the following: math was more intellectually challenging than my primary major (biochemistry), mathematical results contain way more certainty than anything one would be doing in biochemistry, and I just liked my mathematics professors more. As an undergraduate I did attend the &lt;a href=&quot;http://www.math.unl.edu/~ncuwm/16thAnnual/&quot;&gt;Nebraska Conference for Undergraduate Women in Mathematics&lt;/a&gt; which was a valuable experience just to know how many women there are in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Outside of my own personal experience, I really like the idea of &lt;a href=&quot;https://sites.google.com/site/awmmath/programs/kovalevsky-days&quot;&gt;Sonia Kovalesky Days&lt;/a&gt;. I was recently involved in an iteration as a panelist, and it was really exciting to see so many young women participating and seemingly having a great time with mathematics. One of the young women asked the panelists to give them names of notable women mathematicians, and I think even knowing about women mathematicians (and women in other STEM fields) would be neat for them. On which note, I'm also a fan of the &lt;a href=&quot;https://sites.google.com/site/awmmath/programs/essay-contest&quot;&gt;AWM essay contest&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;tl:dr&lt;/strong&gt; Here are the factors which I personally credit for my being in mathematics:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Being introduced to mathematics as a fun game as a young child&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Not&lt;/strong&gt; being exposed to the notion that 'women cannot do mathematics' or that 'men are just better at mathematics than women'&lt;/li&gt;&#xA;&lt;li&gt;Ability to find other points of similarity with my fellow mathfans, as opposed to gender&lt;/li&gt;&#xA;&lt;li&gt;I am personally not too upset by competitiveness.&lt;/li&gt;&#xA;&lt;li&gt;Mathematics is generally difficult, so I consider it a challenge&lt;/li&gt;&#xA;&lt;li&gt;Mathematical results are long-lasting. If I've proved something correctly, it lasts forever, unlike most things in the other sciences.&lt;/li&gt;&#xA;&lt;li&gt;I like pretty much all the mathematicians I've met (y'all are really nice to young people)&lt;/li&gt;&#xA;&lt;li&gt;On most days, I'm not averse to standing out in a crowd, so I'm not too upset about looking quite different from the &quot;average&quot; mathematician&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I attribute any success I might have to having excellent advisors, teachers, and mentors throughout my life; they were almost all male, which is unsurprising, since most mathematicians are. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some possible steps one could take as an educator to interest more women in mathematics:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Do not buy in to the notion that 'women simply cannot do math' (there is some amount of research showing that teachers call on boys more often than girls, etc. Presumably being aware of such findings might help us be more equitable)&lt;/li&gt;&#xA;&lt;li&gt;Point out, early on, that mathematics is something one can do for a living, and in fact it has tons of applications in all sorts of places.&lt;/li&gt;&#xA;&lt;li&gt;Point out examples of notable female mathematicians&lt;/li&gt;&#xA;&lt;li&gt;Encourage participation in/or organize events like the &lt;a href=&quot;https://sites.google.com/site/awmmath/programs/kovalevsky-days&quot;&gt;SK Days&lt;/a&gt;, the &lt;a href=&quot;http://www.math.unl.edu/~ncuwm/16thAnnual/&quot;&gt;NCUWM&lt;/a&gt;, or the &lt;a href=&quot;https://sites.google.com/site/awmmath/programs/essay-contest&quot;&gt;AWM Essay contest&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="126" LastActivityDate="2014-03-24T20:07:51.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="750" PostTypeId="1" CreationDate="2014-03-24T20:37:31.260" Score="19" ViewCount="1793" Body="&lt;p&gt;What are some mathematical problems that are feasible for preschool children to stimulate their intellectual development?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are multiple natural laws that are not apparent to them, for example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the conservation of number/quantity,&lt;/li&gt;&#xA;&lt;li&gt;sets are not disjoint classes (i.e. might overlap),&lt;/li&gt;&#xA;&lt;li&gt;fallacy of circular arguments.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I posted the examples as answers, so they aren't privileged. Also, &#xA;it would be great if we could have &lt;strong&gt;one problem per answer&lt;/strong&gt;, so the votes would represent our opinions more precisely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the ideas for the examples I posted come from a great book &lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=mcl-5&quot;&gt;&quot;Math from Three to Seven&quot;&lt;/a&gt; by &lt;a href=&quot;http://www.labri.fr/perso/zvonkin/&quot;&gt;A. Zvonkin&lt;/a&gt;, which I strongly recommend to anyone interested in this topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; First, I would not ask a child a bare question. Some posts suggests games and games are great. In fact, all of examples I posted are easily transformable into games (the symmetry-related actually is a game already). In this light, the question could be understood:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If I had to ask a question or set a mathematical problem (in whatever form, game or not, that would be appropriate for the child), which questions and problems would be most suitable for a preschooler?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm looking for &lt;em&gt;concrete examples&lt;/em&gt;. Posts like &quot;use another approach&quot;, or &quot;use questions that arise because of natural situations&quot;, or &quot;just let it play some Lego&quot; are &lt;em&gt;not&lt;/em&gt; valid answers. In other words, I &lt;em&gt;do&lt;/em&gt; acknowledge Lego as a great educational toy/tool, but &quot;Lego&quot; by itself does not answer the question. If you insist on games, how and &lt;em&gt;which&lt;/em&gt; mathematical ideas would you incorporate into your play?&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-04-04T12:02:01.703" LastActivityDate="2018-03-27T13:06:56.987" Title="Mathematical problems for preschoolers" Tags="|mathematical-pedagogy|preschool-education|" AnswerCount="12" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="751" PostTypeId="2" ParentId="750" CreationDate="2014-03-24T20:37:31.260" Score="6" Body="&lt;p&gt;&lt;strong&gt;Odd one out&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a bird, a plane, a square with a hole and a cloud, which one does not fit?&#xA;&lt;img src=&quot;https://i.stack.imgur.com/4LkoD.png&quot; alt=&quot;bird,plane,square,cloud&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of this problem is to highlight that &lt;em&gt;explanations&lt;/em&gt; are as important as answers. &#xA;In&amp;nbsp;fact, any object can be singled out:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;the bird&lt;/em&gt;, because it's the only animal;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;the plane&lt;/em&gt;, because it's the only mechanical object;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;the square&lt;/em&gt;, because it's abstract, or because it does not fly;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;the cloud&lt;/em&gt;, because it's gas/liquid (immaterial, but not abstract).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-24T20:45:31.607" LastActivityDate="2014-03-24T20:45:31.607" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="752" PostTypeId="2" ParentId="750" CreationDate="2014-03-24T20:38:43.313" Score="3" Body="&lt;p&gt;&lt;strong&gt;Counting&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the pictures below, are there more cats or dogs?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/h1FYG.png&quot; alt=&quot;6cats&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/wJlhg.png&quot; alt=&quot;7dogs&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of this problem is to teach that moving or rearranging objects does not change the count. If the child does not see this, we remove some cats (spreading them far enough so they take &quot;more space&quot; than dogs) until arrive at an absurd situation (two cats versus bunch of dogs).&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="63" LastEditDate="2014-03-25T00:46:28.973" LastActivityDate="2014-03-25T00:46:28.973" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="753" PostTypeId="2" ParentId="750" CreationDate="2014-03-24T20:39:45.190" Score="2" Body="&lt;p&gt;&lt;strong&gt;Sets and subsets&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is there more of in the whole world: cats and dogs, or animals?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/pqkrP.png&quot; alt=&quot;cats_and_dogs&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/KgGGg.png&quot; alt=&quot;animals&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of this problem to teach the children that &quot;the whole is bigger than its part&quot; and to stimulate them to grow out of the disjoint-classes thinking.&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-24T20:39:45.190" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="754" PostTypeId="2" ParentId="750" CreationDate="2014-03-24T20:40:52.820" Score="3" Body="&lt;p&gt;&lt;strong&gt;Which tower is taller?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which tower is taller (measuring from the base of the tower)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/fsbgK.png&quot; alt=&quot;2towers&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of this problem is to teach that height (and perhaps other measurements) are translation-invariant.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-25T09:09:01.827" LastActivityDate="2014-03-25T09:09:01.827" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="755" PostTypeId="2" ParentId="750" CreationDate="2014-03-24T20:41:58.517" Score="6" Body="&lt;p&gt;&lt;strong&gt;Symmetry&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose the blue line is a mirror, how the figure would look like in it (i.e. put the blocks at the right side to recreate the left side)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/3mtGK.png&quot; alt=&quot;symmetry&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of this problem is to train pattern recognition. There are multiple follow-up challenges (point-symmetry, scaling, etc.).&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-24T20:41:58.517" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="756" PostTypeId="2" ParentId="686" CreationDate="2014-03-24T21:57:28.690" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Mason%E2%80%93Stothers_theorem&quot; rel=&quot;nofollow&quot;&gt;Mason-Stothers Theorem&lt;/a&gt;, the polynomial analogue of ABC conjecture for integers.&lt;/p&gt;&#xA;" OwnerUserId="109" LastEditorUserId="114" LastEditDate="2014-03-25T11:56:18.523" LastActivityDate="2014-03-25T11:56:18.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="757" PostTypeId="2" ParentId="701" CreationDate="2014-03-24T22:02:55.117" Score="3" Body="&lt;p&gt;Have them measure their world in creative ways:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How much&lt;/strong&gt; water does your bathtub hold at home?  If the bath is nearly full, and you get in the water up to your neck, how much does the water level go up?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a map, draw the route that you take to get to school.  Draw the route that a bird would take flying directly.  &lt;strong&gt;How long&lt;/strong&gt; is the bird's route?  How long is your route?  Where are the routes furthest apart?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw a floor plan of your room at school, and a room at your house.  Where are there circles, rectangles, squares, triangles or other shapes?  &lt;strong&gt;What shapes&lt;/strong&gt; would you like to have?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's lots of opportunity for creative mathematical thought along these lines.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-24T22:02:55.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="758" PostTypeId="2" ParentId="643" CreationDate="2014-03-24T22:21:34.047" Score="7" Body="&lt;p&gt;I am using a form of &quot;peer instruction&quot; in my university calculus classes for the first time this year.  I basically use &quot;clicker questions&quot; such as those available from &lt;a href=&quot;http://mathquest.carroll.edu/resources.html&quot;&gt;carroll college&lt;/a&gt;.  I don't use them every class, but maybe 2-3 times a week.  Usually I ask a question, give them a few minutes to respond, show the bar chart of votes, and give them a few more minutes to discuss with their neighbors and vote again.  The rest of the time I teach in a more traditional &quot;interactive lecture&quot;.  I do think some variety in teaching method is a good thing in its own right, since different students learn best in different ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not using physical &quot;clickers&quot; but instead the free online software &lt;a href=&quot;http://pingo.upb.de/&quot;&gt;pingo&lt;/a&gt; that the students can connect to through their laptops, tablets, or smartphones.  This has the advantage of not costing them any money, and additionally has a bit more functionality (e.g. it accepts numerical answers in addition to multiple-choice ones).  Of course, it has the disadvantage of encouraging them to have their phones out during class, and also it doesn't (yet) give me the information of &lt;em&gt;who&lt;/em&gt; voted, so nonparticipation is a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, I am very happy with the experiment and I'm going to continue doing it in the future, at least in lower-division courses.  I feel that with well-chosen questions, I get a very good feel for what the students are stuck on conceptually, and it forces them to address such issues and talk them out with their peers and with me.  It takes some skill and experience to be able to choose and write good questions, of course, and I'm getting better at it over time.  I do have an easier time finding and writing questions for some topics than others.  I would say that finding/writing good questions, and dealing with the nonparticipation problem, are the main issues that I'm struggling to overcome.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2014-03-24T22:21:34.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="759" PostTypeId="2" ParentId="408" CreationDate="2014-03-24T23:07:34.050" Score="1" Body="&lt;p&gt;Material for all my courses is at &lt;a href=&quot;http://neil-strickland.staff.shef.ac.uk/courses/&quot; rel=&quot;nofollow&quot;&gt;http://neil-strickland.staff.shef.ac.uk/courses/&lt;/a&gt;.  For the following courses, there are complete sets of lecture slides:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Methods for differential equations&lt;/strong&gt;: I will be teaching this in Nanjing starting next week, so there are various bits of Chinese in the slides. There are a large number of diagrams, often with some kind of animation.  Separate from the slides, there are also web pages with interactive animations of phase portraits.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Mathematics IV (Electrical)&lt;/strong&gt;: This is basically vector calculus for electrical engineers.  Again, there are many diagrams, often with some kind of animation.  Separate from the slides, there are also web pages with three-dimensional diagrams that can be rotated by mouse.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Linear mathematics for applications&lt;/strong&gt;: In this case, only the handout version of the slides is currently on the web.  Various bits of these handouts have been blanked out and replaced with light green patches.  The students were encouraged to work out for themselves what should go in those patches, or failing that, fill them in during the lecture.  Some of my colleagues have been using that system for a while, but last semester was the first time I tried it.  I am not sure how successful it was.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-24T23:07:34.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="760" PostTypeId="2" ParentId="750" CreationDate="2014-03-24T23:23:08.593" Score="10" Body="&lt;p&gt;For preschool children I suggest &lt;strong&gt;games&lt;/strong&gt; instead of &lt;strong&gt;questions&lt;/strong&gt;. Children come across some mathematical questions when they are playing games. They will learn how to solve these problems &lt;strong&gt;in practice&lt;/strong&gt; (instead of abstract situations) and without any need to continuous supervision of an adult trainer (which is necessary for the case of asking mathematical questions). As a consequence they will &lt;strong&gt;learn by playing&lt;/strong&gt; again and again. After a while you can see that they are &lt;strong&gt;inventing&lt;/strong&gt; new games based on their new point of view. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lego could be one of the best examples of a game with mathematical aspects. It will introduce the geometric notions of &lt;em&gt;area&lt;/em&gt; and &lt;em&gt;volume&lt;/em&gt; to children in an easy and effective way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ahydg.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastActivityDate="2014-03-24T23:23:08.593" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="761" PostTypeId="2" ParentId="686" CreationDate="2014-03-25T00:49:16.437" Score="6" Body="&lt;p&gt;I can think of at least these two good sources:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Number theory over function fields:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many undergraduates learn about rings, fields, the ring of integers of a function, and related topics in a second semester of algebra and/or a first semester of non-elementary number theory. Once you have these topics, you know all you need to learn about the zeta function over the field $\mathbb{F}_q(x)$. This would likely even solidify often-mistaken concepts about function fields and perhaps inspire some towards number theory or algebraic geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Resultwise, it's easy to prove the Riemann Hypothesis for this zeta function (which is nice and &quot;cool&quot;), and from that, the prime number theorem with optimal error term for function fields. Much of the material is approachable to undergrads, and it's all very recent - the best resource for inspiration would be Rosen's &lt;a href=&quot;http://www.springer.com/mathematics/numbers/book/978-0-387-95335-9&quot;&gt;book&lt;/a&gt;, although this book would probably be a bit scary for the undergrads themselves. (That's the best part of course design, right? And why they want experts to digest material?).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Arithmetic/Complex Dynamics&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This field of arithmetic dynamics is less than 50 years old. The best resource would be Silverman's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387699031&quot;&gt;book&lt;/a&gt;, albeit with the same caveat as above. Most of the early proofs are extremely elementary, though recent, and have many connections to other areas of mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, a lot of the material from Beardon's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387951512&quot;&gt;Iteration of Rational Functions&lt;/a&gt; is morally recent. It ties in nicely to a study of fractals, which is entirely new. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20" LastActivityDate="2014-03-25T00:49:16.437" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="764" PostTypeId="2" ParentId="18" CreationDate="2014-03-25T01:31:26.947" Score="9" Body="&lt;p&gt;One disadvantage of allowing &quot;memory aids&quot; is that there will then be little or no credit given for the appropriate formula. This makes it more difficult for a student with only a modest level of understanding to get a C, or even to pass.&lt;/p&gt;&#xA;" OwnerUserId="256" LastActivityDate="2014-03-25T01:31:26.947" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="765" PostTypeId="1" AcceptedAnswerId="781" CreationDate="2014-03-25T01:36:52.807" Score="6" ViewCount="606" Body="&lt;p&gt;In a &quot;foundations of math&quot; undergraduate course when I was talking about the axioms of set theory I said: &quot;Well! The most trivial example of a &lt;em&gt;set&lt;/em&gt; is a set which has &lt;em&gt;no elements&lt;/em&gt; we call it the &lt;em&gt;empty set&lt;/em&gt; and this axiom simply says that it exists in the world of mathematical objects.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suddenly the students began to discuss on impossibility of the existence of &lt;em&gt;a set with no elements&lt;/em&gt; and when I tried to convince them by showing the empty intersection of two disjoint sets they simply said that &quot;OK! But this just shows that two disjoint sets have no intersection!&quot; (In the other words they were trying to say that the intersection of two disjoint sets is not a set necessarily. Equivalently the world of mathematical objects is not closed under the intersection operator. In fact this is a reasonable statement even when we add empty set to the world of mathematical objects because $\cap \emptyset=V$ which is not a set.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We had a similar discussion when I explained Russell's paradox on non-existence of set of all sets and I felt that it is completely unnatural theorem for students because they believe that the set of all sets &lt;em&gt;trivially exists&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1.&lt;/strong&gt; What are effective ways for convincing undergraduate students that the current foundation of math is really &quot;natural&quot;? (If it is really natural at all!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact one can build mathematics in both upward and downward directions by assuming empty set and set of all sets as the basic building block of the world respectively (it suffices to change the interpretation of $\in$ relation). Also there are axiomatic foundations of mathematics which imply the existence of the set of all sets (e.g. Quine's New Foundation $NF$) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark 1.&lt;/strong&gt; Note that the upward or downward &lt;em&gt;direction of building mathematical world&lt;/em&gt; changes our point of view about our actual world too. For example counting apples in a &lt;strong&gt;set of all sets - based&lt;/strong&gt; point of view could be something like this: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All apples,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All apples minus one apple, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All apples minus two apples, (in fact &quot;All apples minus one apple minus one apple&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All apples minus three apples, (in fact &quot;All apples minus one apple minus one apple minus one apple&quot;)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\vdots$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In our current &lt;strong&gt;empty set-based&lt;/strong&gt; point of view we enumerate apples of the actual world in this way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No apples, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;No apples plus one apple, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;No apples plus two apples, (in fact &quot;No apples plus one apple plus one apple&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No apples plus three apples, (in fact &quot;No apples plus one apple plus one apple plus one apple&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\vdots$   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that we don't need to define the notions &quot;one&quot;, &quot;two&quot;, &quot;three&quot;, ... or even &quot;plus&quot; or &quot;minus&quot; to make the above statements meaningful. We are just using the notions of &quot;preceding&quot; and &quot;succeeding&quot; (in the sense of successor operator in Peano Arithmetic). Using this change in our thinking foundation we can look to the world in an inverse (or maybe correct) way! Living in such an &lt;em&gt;inverse universe&lt;/em&gt; could unfold something interesting for us.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2.&lt;/strong&gt; Is there any research to show that which one of the notions of set of all sets and empty set are more natural intuitively? By research I mean a poll amongst the society of those who are not aware of the formal foundations of math including little children, usual people, students who don't work on math and math students who didn't see any axiomatic approach to foundations of math before. Something like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Which one is more natural?&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;(1) A set which all sets are its elements.&lt;/li&gt;&#xA;  &lt;li&gt;(2) A set which has no elements.&lt;/li&gt;&#xA;  &lt;li&gt;(3) Both.&lt;/li&gt;&#xA;  &lt;li&gt;(4) None.  &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My personal limited research shows that the &lt;em&gt;popularity&lt;/em&gt; of the empty set amongst people is too low and amazingly the &lt;em&gt;strange&lt;/em&gt; answers (3) and (4) are more frequent amongst little children! (It is somehow strange to live in a mathematical world which is endless in both up and down directions also it is a bit hard for me to imagine the world with an end in both up and down directions!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark 2.&lt;/strong&gt; As a set theorist I am one of the fans of set of all sets (and its philosophical consequences). Also I am interested in non-well founded set theory. The question is because I want to define an undergraduate course on &quot;Non-well founded foundations of math&quot; and I am  interested in being familiar with students &quot;natural&quot; background on the subject to decide on the best approach. Fortunately our department allows me to teach &quot;Non-well founded foundations of math&quot; course even before &quot;foundations of math&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 3.&lt;/strong&gt; If the researches show that the &lt;em&gt;natural thinking system&lt;/em&gt; of undergraduate students is non-well founded, is it a good idea to begin teaching the axiomatic foundations of math by a course in non-well founded set theory? (e.g. Teaching $NF$ instead of $ZF$) What about teaching some non-well founded set theory beside usual &quot;foundation of math&quot; course? (e.g. Teaching some basic facts about Quine's Atoms $x=\{x\}$ and $AFA$, $BAFA$, $SAFA$ anti-foundation axioms)&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-25T11:58:10.660" LastActivityDate="2014-03-25T12:11:06.600" Title="Which one is more natural, empty set or set of all sets?" Tags="|course-design|undergraduate-education|" AnswerCount="3" CommentCount="8" ClosedDate="2014-03-25T22:20:57.217" ContentLicense="CC BY-SA 3.0" />
  <row Id="766" PostTypeId="2" ParentId="747" CreationDate="2014-03-25T01:38:34.510" Score="6" Body="&lt;p&gt;There are some good websites for specific states (e.g., edjoing.org for California and expanding to a few more states next school year. You can just find them through googling) but I never came across to a national secondary education mathematics jobs website (probably the closest website is indeed.com, which is more like a meta-search engine for job postings) . I think it is because of moving between states is not that easy once you have a teaching credential from one state  (it usually requires you to retake some classes and reapply through the state department of education) and such a website will not be very functional (especially for the hiring school district).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the unique landscape of US secondary education (i.e., independent school districts that can do whatever they want), I would expect every school district to advertise jobs differently and to have different recruitment procedures. In my experience, my current school district (a medium California central valley USD) puts all qualified applicants into a pool from where principals can start interviewing. My previous school district (a large Southern California USD) had the same procedures but, if you wanted a job, your best option was to contact directly a school with an opening and talk with the principal before going thorough HR. I also applied to a small school district which was accepting only paper application and the opening were just posted in the job bulletin board at the district office.&lt;/p&gt;&#xA;" OwnerUserId="255" LastActivityDate="2014-03-25T01:38:34.510" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="767" PostTypeId="2" ParentId="689" CreationDate="2014-03-25T02:04:53.313" Score="2" Body="&lt;p&gt;To get playing with one-to-one correspondence, &lt;em&gt;The Cat in Numberland&lt;/em&gt;, by Ivar Ekeland, is marvelous. Of course that only helps with ℵ0, but it's a step in the right direction. I wonder if there's a way to tell a story about ℵ1...&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-25T02:04:53.313" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="768" PostTypeId="2" ParentId="426" CreationDate="2014-03-25T02:31:17.277" Score="17" Body="&lt;p&gt;That is because teaching math and knowing math are different skills. Unfortunately, having a PhD in mathematics will not guarantee that a candidate will also be able to manage a class of 35 students, be able to design pedagogically and developmentally adequate lessons that progress in units, how to assess students, deal with the requirements for special education (i.e., IDEA) or English-language learners (e.g., California requires a special certification for ELD), and in general survive your first years of teaching without giving up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These skills are usually taught during a teacher preparation program which is accredited by the state and follows teacher preparation standards (e.g., TPES for California). After competing a teacher preparation program, teacher candidates are recommended to the credentialing commission for their state (For this reason, it is not possible to automatically transfer credentials between states. Even if I am a teacher in California, I cannot just walk in an Oregon school and get a job). For example, in California you usually have to take 2/3 semesters worth of post-baccalaureate coursework to be able to qualify for a teaching credential. Each candidate has also to pass 4 qualifying exams (CalTPAs) if you want a multiple subject credential), a basic skills exam (CBEST), and 3/4 subject specific exams (subject CSET) before being recommended for a credential.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are new programs that put you directly in a class (transition to teaching or TfA), but you will have to complete all the coursework if you want to have a credential (you can work temporarily with an emergency credential, but that will preclude you from tenure and to its perks). So, you can be in a classroom without a teaching credential but you will not able to keep your job. On top of that, a public school is not allowed to hire a teacher that is not cleared by the state teaching commission (especially after No Child Left Behind).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Private schools (not public charter schools) are exempt from these requirements and can hire who they deem more fit for the job. They act more like universities in that sense where evaluations by all stakeholders are more important that a state certification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the history of teaching credentials, basic skills exams date to the early 1800s and the introduction to teaching credentials to the 1930s. The credentials were modeled after the medical board licences or bar licences and were intended to strengthen the requirements to entering the teaching profession. Federal and state governments are, however, still working out some issues with the credentialing system. For example, NCLB (2002) introduced the requirement for a content area teacher to be highly qualified in the content area that he or she is teaching. This has been translated in requiring a math teacher to have a math teaching credential. Before that, a teacher with a teaching credential could teach whatever he or she was comfortable teaching (including an art teacher moving to math because her job was being cut) and to have a teaching credential in something (California use to issue general teaching credentials up until 1970's, so it is not uncommon to have old teachers with a general credential). Teacher unions do not have anything to do with credentialing teachers or with the introduction of requirements to become a teacher (they are usually more interested in protecting teachers with tenure). A good reading on the history of teaching in the US is 'Tinkering towards Utopia' by Tyack and Cuban.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that having a teaching credential does not guarantee that you are a good teacher or know the subject you are teaching, but the same is true for doctors or lawyer. Nonetheless, if you want to practice medicine or law you need to have a licence. The same is true for teaching in the United States.&lt;/p&gt;&#xA;" OwnerUserId="255" LastActivityDate="2014-03-25T02:31:17.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="769" PostTypeId="2" ParentId="746" CreationDate="2014-03-25T02:59:29.747" Score="6" Body="&lt;p&gt;I do not know of any article or study that &quot;proves&quot; that multiple intelligences exist for math (or for what is worth, any other subject). Such a study will be difficult to design and its results will just show some causation between teaching and learning methods, rather than something physiological within the brain. On top of that, research in math education is focused on the elementary grades and most of the articles that cover middle, high school, or college are very rare.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gardner (1983) has been the biggest proponent of multiple intelligences and his work seems reasonable. The basic idea is that people prefer to learn and do different things based on their dominant intelligence. For that reason, I am good at math (logical-mathematical intelligence), while my sister is good at making friends (intrapersonal intelligence). &lt;a href=&quot;https://web.archive.org/web/20170731232349/http://education.jhu.edu/PD/newhorizons/strategies/topics/mi/&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Hopkins University&lt;/a&gt; has a good page with references on this subject if you wish to research this in more depth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a math teacher, it is your goal to leverage each student's preferred intelligence to have him or her learn math. What I see every day is students that don't have a strong logical intelligence give up solving a math problem but they could explain you word-by-word what they think about a problem or students that are able to solve flawlessly a numerical problem grind to a halt when they need to sketch a net of a solid. This might show some correlation between their strongest intelligence and task performance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give you another example, let's consider polynomial multiplication. To multiply two binomial, you can use either the distributive property, a rectangle (or area) model, or even stack them and multiply &quot;the long way&quot;. The math is the same but its representation on paper is different. I have students preferring one of these three methods. I do not know why, but it could be related to their dominant intelligence (distributive property for logical-mathematical and rectangle model for visual-spatial intelligence).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the university level, I would expect students to pretty much be sorted out of math if they do not have a strong logical-mathematical intelligence. After all, you will not go very far in algebra if you are not able to use math symbols or follow algorithms. You can, however, see the different learning styles/intelligences in action in an elementary class where students use different methods to perform basic operations. One could argue that the selection of math discipline could be an example of different intelligences in math (analysis vs. topology for example).&lt;/p&gt;&#xA;" OwnerUserId="255" LastEditorUserId="-1" LastEditDate="2020-01-01T05:41:22.387" LastActivityDate="2020-01-01T05:41:22.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="770" PostTypeId="2" ParentId="706" CreationDate="2014-03-25T03:40:31.013" Score="16" Body="&lt;p&gt;I took a number theory class at University of Cincinnati that was taught using a modified Moore method.  The class size was pretty small, and credit-wise it was an &quot;upper level mathematics elective&quot; (i.e. not a core class), so our professor had some room to be experimental.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like with the Moore method, the students presented all course material. The instructor sat in back, asked questions every so often if the student was not clear enough, helped us if we got impossibly stuck, and participated in the discussion when he had something insightful to say.  We rotated teaching between three groups of two or three people.  Our presentations were allowed to last multiple class sessions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The class differed from the classic Moore method by loosening the requirement that students prove all theorems themselves, to address the time/constraints issue.  We did have a book, but it was a computer science oriented text that left large holes in the proofs (or omitted them completely).  We had to complete proofs at least well enough to field questions from each other and the professor, which often forced us to create our own lemmas.  If anyone couldn't prove something at the board, the whole class had to finish it for homework.  We were also encouraged to come up with our own examples, and add in whatever cool applications, algorithms, or observations we could find.  Among other things, I ended up proving Chebyshev's theorem (the one leading up to the prime number theorem) after a whole bunch of lemmas, at a depth which I never would have otherwise encountered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It turned out that most students learned the material from their own sections very deeply, but did not understand others' presentations as well as their own.  I guess that's inevitable.  Another complication was that some students were better at communicating their ideas than others.  I bet that this method would work particularly well with a group of motivated students that had some previous practice working together.  In any case, I would say I learned more in this class than other traditional classes at around the same level, so I support this adaptation of the method.  It could use more development, but at the very least, I don't think it turned out worse than standard instruction.&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2014-12-21T21:03:57.970" LastActivityDate="2014-12-21T21:03:57.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="771" PostTypeId="1" AcceptedAnswerId="1238" CreationDate="2014-03-25T03:57:05.800" Score="14" ViewCount="787" Body="&lt;p&gt;I've been teaching remedial mathematics for the better part of a decade, and I've noticed a big trend in my classes lately. Many of my students are able to grasp the more complex ideas but still struggle with the basic facts (e.g., adding/subtracting/multiplying/dividing negative numbers; properties of 0 and 1; basic multiplication and long division).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My Question: &lt;strong&gt;Have other CC teachers noticed a similar trend? For those of you who have--and maybe have more experience dealing with this problem--what have you done when confronted with this issue?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps an example will help to clarify my question some more and to narrow down the broader questions. In some of my current remedial math courses, I took time out of the regular curriculum to focus specifically on the times table, properties of integer operations, and properties of the identities. Here's an excerpt from one of the worksheets I gave them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complete the following sentences by filling in the blanks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A negative number multiplied or divided by another negative number is...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A positive number multiplied or divided by a negative number is... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A number divided by one is ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A number divided by zero is ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A number divided by itself is ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compute the answers to each of the following.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1\div 1=$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$-1 \div (-1)=$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$-1\div 1=$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1\div (-1)=$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1\times 1=$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$-1\times (-1)=$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="267" LastEditDate="2014-04-01T15:54:32.490" LastActivityDate="2016-05-15T05:44:43.843" Title="Community College Remedial Mathematics difficulties for students" Tags="|undergraduate-education|usa|primary-education|community-colleges|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="772" PostTypeId="2" ParentId="686" CreationDate="2014-03-25T04:15:51.093" Score="7" Body="&lt;p&gt;I think that Bhargava's generalized factorial functions fit the bill perfectly. This construction, and the theorems he proves using them, are recent, topical, and completely undergraduate-accessible. Bhargava himself gives a good survey &lt;a href=&quot;http://www.math.upenn.edu/~ted/620F09/Notes/Bhargava/2695734.pdf&quot;&gt;HERE&lt;/a&gt;, and there is a nice blog-post explanation of generalized factorials &lt;a href=&quot;http://notaboutapples.wordpress.com/2010/02/08/bhargavas-factorials/&quot;&gt;HERE&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analogues of various number-theoretical results involving factorials, such as the fact that the gcd of elements in the image of a degree $k$ primitive polynomial with integer coefficients is divisible by $k!$, remain true when the integers are replaced by a subset $S$ of the integers and the factorial function is replaced by Bharagava's generalized factorial function $k!_S$. One of my students has been looking at these theorems, including the proofs, and I think that the level has proven to be just right for her to have understood everything and also to have learnt some useful mathematics.&lt;/p&gt;&#xA;" OwnerUserId="185" LastActivityDate="2014-03-25T04:15:51.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="773" PostTypeId="2" ParentId="584" CreationDate="2014-03-25T04:36:04.563" Score="6" Body="&lt;p&gt;I see the original question as actually &lt;em&gt;two&lt;/em&gt; questions asked here:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How do we engage women into studying mathematics?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How do we get women to become mathematicians?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Studying mathematics for the goal of becoming mathematicians (pure, applied, etc.) is a subset to the first question. For the first question, I have read about some of Jo Boaler's research in gender and mathematics education:&#xA;&lt;a href=&quot;http://gender.stanford.edu/news/2012/sugar-and-spice-and%E2%80%A6-math-under-achievement&quot;&gt;http://gender.stanford.edu/news/2012/sugar-and-spice-and%E2%80%A6-math-under-achievement&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Essentially, it comes down to &lt;em&gt;how&lt;/em&gt; classes are being taught. I have the impression that mathematics education is traditionally set up in a competitive environment, which has the tendency to turn off the collaborative learning aspects that academia is supposed to foster. If everything in the learning environment is geared to working individually and getting the technically correct answers without having the space and time to develop deeper understanding and allowing a safe environment to compare and discuss answers, then it tends to turn off the girls (and possible future mathematicians). Content has less of a bearing to motivation than the delivery and environment of mathematics education.&lt;/p&gt;&#xA;" OwnerUserId="261" LastActivityDate="2014-03-25T04:36:04.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="774" PostTypeId="2" ParentId="294" CreationDate="2014-03-25T04:48:26.090" Score="2" Body="&lt;p&gt;Some of the best areas for mathematics, and I believe to get an interest in mathematics too, is the field of probability and statistics. Statistics is learnt fairly intuitively early on when people learn to count and do frequency tables.&lt;a href=&quot;http://bayesianthink.blogspot.com/2012/12/the-best-books-to-learn-probability.html&quot; rel=&quot;nofollow&quot;&gt;Here&lt;/a&gt; is a list of books that are good to learn probability and statistics.&lt;/p&gt;&#xA;" OwnerUserId="258" LastActivityDate="2014-03-25T04:48:26.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="775" PostTypeId="2" ParentId="689" CreationDate="2014-03-25T04:55:06.103" Score="9" Body="&lt;p&gt;Consider a Sumerian person, living around 2500BC, who owns a flock. She hires shepherds to take the flock to pasture. Not being a scribe, she does not know Sumerian numbers- instead, for each sheep that passes her, she places a stone in an urn. When the shepherd returns, for each sheep that comes back, she removes a stone from the urn. If the last sheep passes and the urn still has stones in it, the shepherd has been doing something funny. If the stones run out, but there are still sheep, it's her lucky day. If stones and sheep run out at the same time, all is in order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Sumerian person in question can compare urns in the same way- remove one stone from each, at the same time. A finite urn can be emptied by removing stones one at a time. An infinite urn cannot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider now two infinite urns. If we can match each stone in urn A to a stone in urn B, then urn B is at least as large as urn A. But if, after matching up each stone in urn A to a stone in urn B, there are still unmatched stones in urn B left over, then urn B is larger than urn A. You can visualize elements in urn B as stones and elements in urn B as Greek-style points with no volume to add conceptual intuition.&lt;/p&gt;&#xA;" OwnerUserId="185" LastActivityDate="2014-03-25T04:55:06.103" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="776" PostTypeId="2" ParentId="714" CreationDate="2014-03-25T05:04:10.703" Score="2" Body="&lt;p&gt;I haven't practiced this myself as an instructor, but as a student I benefitted from my professors occasionally singling out someone's homework as an exemplar, and distributing a photocopy of it.&lt;/p&gt;&#xA;" OwnerUserId="87" LastActivityDate="2014-03-25T05:04:10.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="777" PostTypeId="2" ParentId="765" CreationDate="2014-03-25T05:13:51.990" Score="6" Body="&lt;p&gt;Pedagogical answers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, it is very natural to think of a wallet that contains no money; with this analogy my students have no problem with the empty set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of a wallet that contains not only all money but all wallets (including itself), this is an easy sell to dissuade students from walking down this road.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Mathematical answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a lot of scary things out there, so taking all of them as a default seems backwards to me.  For example, I'd rather start with simple functions (polynomials) and add more complicated ones, than start with all functions (including &lt;a href=&quot;http://en.wikipedia.org/wiki/Conway_base_13_function&quot;&gt;1&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Weierstrass_function&quot;&gt;2&lt;/a&gt;, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Cantor_function&quot;&gt;3&lt;/a&gt;) and remove things.&lt;/p&gt;&#xA;" OwnerUserId="270" LastActivityDate="2014-03-25T05:13:51.990" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="778" PostTypeId="2" ParentId="771" CreationDate="2014-03-25T05:22:33.180" Score="5" Body="&lt;p&gt;Generalizing from your personal experience with a small, local group of students to trends in &quot;mathematics education in the US&quot; is a tremendous leap.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The evidence you have for this &quot;trend&quot; could just as well be evidence that over the last 10 years you have become better at identifying students' weaknesses in arithmetic. &lt;/p&gt;&#xA;" OwnerUserId="270" LastActivityDate="2014-03-25T05:22:33.180" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="779" PostTypeId="1" CreationDate="2014-03-25T06:25:46.070" Score="8" ViewCount="513" Body="&lt;p&gt;A good way for motivating young students in undergraduate level is telling them that &lt;strong&gt;you can do great works!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What are good examples of research level mathematical discoveries done by undergraduate students? &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-25T16:46:09.083" LastActivityDate="2018-06-02T17:59:54.780" Title="Examples of Research Level Math Discoveries Done by Undergraduate Students" Tags="|undergraduate-education|student-motivation|reference-request|undergraduate-research|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="780" PostTypeId="1" CreationDate="2014-03-25T07:27:27.677" Score="10" ViewCount="1101" Body="&lt;p&gt;As a crazy idea in a coffee break after a boring lecture I suggested my students and colleges that we can have some fun in our department by participating in a &quot;Mathematical Olympic Games&quot;. They supported the idea extensively and now we are in choosing/designing games phase. Beside usual games with mathematical aspects like Sudoku, Rubik Cube, Origami, etc. I suggested that it could be very useful if we invent some games inspired by usual material of undergraduate courses like Algebra, Analysis, Number Theory, Graph Theory, etc. because participating in  such games helps students to become more familiar with actual applications of what their teachers are trying to teach them. Also if teachers are amongst participants/referees they can help students by their advisement too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What are good examples of mathematical games inspired by the usual subjects of undergraduate courses? Please specify that for improving abilities in which undergraduate courses your suggested game is good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark.&lt;/strong&gt; All other suggestions for interesting games are welcome.   &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="1685" LastEditDate="2014-07-14T04:49:58.967" LastActivityDate="2014-09-20T03:44:27.670" Title="Examples of Good Mathematical Games for a Mathematical Olympic" Tags="|undergraduate-education|student-motivation|math-puzzle|games|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="781" PostTypeId="2" ParentId="765" CreationDate="2014-03-25T07:45:59.673" Score="11" Body="&lt;p&gt;Regarding the empty set: if your students object to its existence, and you suspect they might, set them up for a surprise. Introduce &lt;em&gt;first&lt;/em&gt; the separation axiom, then ask whether there are any sets. Then ask if there is the empty set. When they say &quot;no&quot;, prove that it exists by separation and the claim that there is a set. You can now pompously declare that they are illogical (hint: wear pointy ears that day). Warn them that one has to follow the logic, no matter how much it goes against their beliefs. They may question initial assumptions, but not their consequences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the set of all sets, again, this is an opportunity to show that things are not that obvious. Let them have the set of all sets, and derive from it that you are the pope. After all, you are a saint already.&lt;/p&gt;&#xA;" OwnerUserId="165" LastEditorUserId="165" LastEditDate="2014-03-25T08:07:13.217" LastActivityDate="2014-03-25T08:07:13.217" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="782" PostTypeId="2" ParentId="737" CreationDate="2014-03-25T08:24:50.017" Score="11" Body="&lt;p&gt;I think that you hit the nail on the head, when you said &lt;em&gt;some are not even aware of what well-defined means&lt;/em&gt;. As Anschewski suggests the problem may be that students have not encountered enough non-well-defined operations to fully appreciate the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This Spring I was teaching freshman algebra, and while explaining equivalence relations (prior to getting started with congruences) I tried the following example. I had explained that being a namesake gives an equivalence relation among human beings. But if you try and define a function called &lt;em&gt;weight&lt;/em&gt; from the set of equivalence classes Humans/Namesakes, you run into the difficulty of namesakes often having different weights.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately the student response was a bit mixed. Some of them commented in their teacher evaluation forms that the example with namesakes was great, but I failed to follow up on it and give something similar when explaining all the other concepts they encountered. Well, if I think of something, I will use it next time. &lt;/p&gt;&#xA;" OwnerUserId="282" LastActivityDate="2014-03-25T08:24:50.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="784" PostTypeId="2" ParentId="780" CreationDate="2014-03-25T09:23:19.070" Score="5" Body="&lt;p&gt;Some possibilities coming from algebra are&lt;br&gt;&#xA;&lt;a href=&quot;https://mathoverflow.net/questions/93276/a-game-on-noetherian-rings&quot;&gt;https://mathoverflow.net/questions/93276/a-game-on-noetherian-rings&lt;/a&gt;&lt;br&gt;&#xA;and&lt;br&gt;&#xA;&lt;a href=&quot;http://arxiv.org/abs/1205.2884&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://arxiv.org/abs/1205.2884&lt;/a&gt;&lt;br&gt;&#xA;though it depends on how much algebra the students will have been introduced to at that point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another possibility is the game Hex (&lt;a href=&quot;http://en.wikipedia.org/wiki/Hex_(board_game)&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Hex_(board_game)&lt;/a&gt;) since here the impossibility of a draw is related to the Brouwer fixed point theorem.&lt;/p&gt;&#xA;" OwnerUserId="286" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-03-25T09:23:19.070" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="785" PostTypeId="2" ParentId="41" CreationDate="2014-03-25T09:46:35.610" Score="6" Body="&lt;p&gt;We discussed this, or more specifically the subject of handwritten $x$, a while ago &lt;a href=&quot;http://aperiodical.com/2012/08/lets-talk-about-x/&quot; rel=&quot;noreferrer&quot;&gt;at The Aperiodical&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I still don’t understand why the two-curves $x$ never caught on in the US. Newton wrote $x$ the way modern Brits do, so it predates American independence. Curious!&lt;/p&gt;&#xA;" OwnerUserId="288" LastEditorUserId="75" LastEditDate="2014-03-25T11:45:34.590" LastActivityDate="2014-03-25T11:45:34.590" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="786" PostTypeId="2" ParentId="746" CreationDate="2014-03-25T09:49:52.697" Score="-2" Body="&lt;p&gt;I think about this often, not just for math but for school in general.  Maybe we should group kids not by ability, but on preference of teaching style.  Some love the lecture and need quiet time to process what's being presented.  Some are hands-on learners, some need to move, some need to mix it up.  We've all done one activity that worlds great with some kids but not with others and I think there's a reason for that.  Teachers have different styles, as well.  Maybe we should match styles with style.&lt;/p&gt;&#xA;" OwnerUserId="289" LastActivityDate="2014-03-25T09:49:52.697" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="787" PostTypeId="2" ParentId="765" CreationDate="2014-03-25T10:34:42.927" Score="3" Body="&lt;p&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt; This answer uses &lt;em&gt;lists&lt;/em&gt;, not &lt;em&gt;sets&lt;/em&gt;, quite different objects, but I guess there are enough similarities to make it meaningful. As the notion of sets of all sets has its own problems, I will use the set of natural numbers instead, i.e. I change the question to: &lt;em&gt;what is more intuitive, the empty set, or the infinite set of all the natural nubmers?&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For lists, it is: &lt;em&gt;what is more intuitive, the empty list, or the infinite list of all the natural numbers?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt; It seems that inductive approach is more intuitive.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The process of construction (in your language building from empty set) is frequently called &lt;a href=&quot;http://en.wikipedia.org/wiki/Structural_induction&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;induction&lt;/em&gt;&lt;/a&gt;, and the deconstruction (starting from &quot;full&quot; sets and going down) is called &lt;a href=&quot;http://en.wikipedia.org/wiki/Coinduction&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;coinduction&lt;/em&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I taught both approaches during an introduction to programming course, for example, consider &lt;a href=&quot;http://en.wikipedia.org/wiki/List_%28computing%29&quot; rel=&quot;nofollow&quot;&gt;lists&lt;/a&gt; and their duals &lt;a href=&quot;http://en.wikipedia.org/wiki/Stream_%28type_theory%29&quot; rel=&quot;nofollow&quot;&gt;streams&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\mathrm{List}\ \alpha$, that is, the type of lists of elements of type $\alpha$, describes an inductive data structure defined by two constructors&#xA;\begin{align}&#xA;\mathrm{Nil} &amp;amp;\;: \mathrm{List}\ \alpha\\&#xA;\mathrm{Cons}&amp;amp;\;: \alpha \to \mathrm{List}\ \alpha \to \mathrm{List}\ \alpha&#xA;\end{align}For example $l \equiv [1,2,3]$ would be $l = \mathrm{Cons}(1,\mathrm{Cons}(2,\mathrm{Cons}(3,\mathrm{Nil})))$.&lt;/li&gt;&#xA;&lt;li&gt;A stream type $\mathrm{Stream}\ \alpha$ describes a coinductive data structure defined by two destructors:&#xA;\begin{align}&#xA;\mathrm{head} &amp;amp;\;: \mathrm{Stream}\ \alpha \to \alpha\\&#xA;\mathrm{tail}&amp;amp;\;: \mathrm{Stream}\ \alpha \to \mathrm{Stream}\ \alpha&#xA;\end{align}For example the stream $s \equiv \langle 0,1,2,3,\ldots \rangle$ of natural numbers would be defined by $(\mathrm{head} \circ\mathrm{tail}^n)(s) = n$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Each time the effect was that lists were easy to understand, while streams posed some problems. It was simple to introduce lists and how to manipulate them, while streams caused confusion and were considered &quot;an advanced topic&quot;. Any student were able to code a program to produce list of the $n$ first natural numbers (this is inefficient, but I wanted to make the codes similar)&#xA;\begin{align}&#xA;&amp;amp;\verb`let natl n = `\\&#xA;&amp;amp;\quad\verb`if n == 0 then []`\\&#xA;&amp;amp;\quad\verb`else 0 : (map (+1) (natl (n-1)))`&#xA;\end{align}&#xA;but definitions like the following would take time to digest $$\verb`let nats = 0 : map (+1) nats`$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To conclude, it seems that inductive approach is more intuitive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-25T12:11:06.600" LastActivityDate="2014-03-25T12:11:06.600" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="788" PostTypeId="2" ParentId="689" CreationDate="2014-03-25T10:45:34.040" Score="3" Body="&lt;p&gt;Power sets, maybe? I was just pondering this because of course preschoolers don't understand decimals.&lt;/p&gt;&#xA;" OwnerUserId="295" LastActivityDate="2014-03-25T10:45:34.040" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="789" PostTypeId="2" ParentId="737" CreationDate="2014-03-25T11:03:41.827" Score="8" Body="&lt;p&gt;In my opinion this problem arises due to the fact that students are never told that, when introducing a new symbol which isn’t a variable (here I’m not using &lt;em&gt;variable&lt;/em&gt; in the formal system sense of the word) and which does not depend on previously defined symbols, then one must prove that there exists exactly one object with the given property.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example take the square root of a non-negative real number. Showing it is well defined requires one to first prove $\forall x\in \left[0, +\infty\right[ \; \exists !y\in \left[0, +\infty\right[ \;\left(y^2=x\right)$, otherwise why whould $\sqrt x$ have any meaning at all?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An additional problem happens with equivalence relations. In this context, let us consider $\equiv _p$ over the integers. Given $m,n\in \mathbb Z$, the function $+_{\equiv _p}$ maps $\left([m]_{\equiv p}, [n]_{\equiv p}\right)$ to $[m+n]_{\equiv p}$. Even after proving the analogous property to the example above, this has another problem, namely that the function isn't defined with respect to variables, but rather with respect to the representation of variables. Ideally one would write $f\colon A\to B, a\mapsto f(a)$, but in this context we’re actually writing $f\colon A\to B, g(a)\mapsto f(a)$. Why it can be done is a trivial matter, but it should be observed.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I have the feeling that some are not even aware of what well-defined means.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I agree with Jyrki on this, but I don't think it is just ‘some’. Knowing what well-defined means is being aware of all of the above. I graduated only recently and I’d bet an organ that less than 6% of my former fellow undergraduate students are aware of all the subtleties. &lt;/p&gt;&#xA;" OwnerUserId="287" LastEditorUserId="287" LastEditDate="2016-12-14T22:14:08.377" LastActivityDate="2016-12-14T22:14:08.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="790" PostTypeId="2" ParentId="294" CreationDate="2014-03-25T11:10:37.747" Score="9" Body="&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0201038129&quot;&gt;Surreal Numbers&lt;/a&gt;  by Donald Knuth is a story about two people discovering the &lt;a href=&quot;http://en.wikipedia.org/wiki/Surreal_numbers&quot;&gt;surreal numbers&lt;/a&gt; and proving theorems about them.&lt;/p&gt;&#xA;" OwnerUserId="298" LastActivityDate="2014-03-25T11:10:37.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="791" PostTypeId="2" ParentId="689" CreationDate="2014-03-25T12:15:45.070" Score="4" Body="&lt;p&gt;I would expect children can understand the idea that there are as many even numbers as there are natural numbers, as long as it's presented in a lively style, asking them questions and  drawing pictures.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How many numbers are there? &lt;em&gt;Infinity.&lt;/em&gt; (Write the first few naturals)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Can you tell me what an even number is? What are the first few? (Write&#xA;  them down) How many even numbers are there? &lt;em&gt;Infinity!&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Are there more numbers than there are even numbers? &lt;em&gt;Yes, hmm, they're&#xA;  both infinity&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;But look, if I write the even ones under the ordinary numbers like&#xA;  this...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The trouble with &lt;em&gt;higher&lt;/em&gt; infinities is that there are no uncountable sets that children are familiar with. Even middle school students may be uncomfortable with the idea that there are infinite decimals. The simplest example would be the set of all infinite sequences of ones and zeroes, but I suspect that would be too abstract for small children (and possibly some adults) to grasp without some difficulty. You might replace the ones and zeroes with something more colorful, ideally something that could be wrapped up in a story, but the concept of &quot;an infinite set &lt;em&gt;of infinite sets&lt;/em&gt;&quot; might never stop being difficult. The diagonalisation argument is accessible for high school students and probably middle school students, but I can't imagine an 8 year old being able to maintain all the structures in their head through to the conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if we're resigned to the idea that &lt;em&gt;proving&lt;/em&gt; the existence of uncountable sets is too hard, can we at least communicate what their existence &lt;em&gt;means&lt;/em&gt;? I think that to do this, you should first establish the idea that there are a lot of countably infinite sets. I'd expect kids to be surprised that there are as many even numbers as natural numbers (surprise them further by repeating the argument with multiples of a billion million trillion, rather than multiples of two). What about an infinite checkerboard (with a top-left-hand corner)? First number the first row with the natural numbers, to make it look like there are way more squares than natural numbers. Then ask if they can think of a way to number the squares so that they all get a number, and if they don't come up with anything, show them the correct answer. The crucial part is to set it up so that they're &lt;em&gt;surprised&lt;/em&gt; that an infinite checkerboard is countable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There might be a few other simple countable sets, and once that's done, you can ask them if they think &lt;em&gt;all&lt;/em&gt; infinities are the same size. Then you can drop the existence of uncountables on them. I would finish with something like &quot;these bigger infinities were first discovered by a man named Georg Cantor about a hundred and fifty years ago&quot;. I feel it's important that they understand that the fact that certain sets are uncountable is something that a person can &lt;em&gt;prove&lt;/em&gt;, and simply giving the name of the first human to do so might be enough to communicate that, on an emotional level at least.&lt;/p&gt;&#xA;" OwnerUserId="115" LastActivityDate="2014-03-25T12:15:45.070" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="792" PostTypeId="2" ParentId="706" CreationDate="2014-03-25T13:04:45.513" Score="26" Body="&lt;p&gt;The Moore Method is alive and well, and so are a great many variants. These days the community is more likely to use the term Inquiry-Based Learning (IBL), because the Moore Method can be seen as a restrictive set of practices, and people use the same underlying ideas in a lot of different class structures. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to learn more and meet people who are using IBL in college classrooms, I suggest the following resources:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;a href=&quot;http://www.inquirybasedlearning.org/&quot;&gt;The Academy of Inquiry Based Learning&lt;/a&gt; has information about workshops and obtaining mentors&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;a href=&quot;http://www.jiblm.org/&quot;&gt;The Journal of Inquiry Based Learning in Mathematics&lt;/a&gt; has sets of course notes that have been class-tested and peer-reviewed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Each summer there is a conference called &lt;a href=&quot;http://legacyrlmoore.org/events.html&quot;&gt;the Legacy of RL Moore Meeting&lt;/a&gt;. This summer we meet in Denver on June 19-21, 2014. (Full disclosure: I am program co-chair for the upcoming meeting.) This meeting is sponsored by the Educational Advancement Foundation and the Mathematical Association of America.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Many of the larger meetings of the MAA (MathFest &amp;amp; JMM) have had sessions devoted to IBL teaching in the last few years. This is the case for at least the next year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to the meat of your question: I use IBL in all of my courses in some form. This includes proof-based courses for majors (where what I do looks a lot like a Moore method classroom), down to classes for liberal arts students who are required to take one quantitative reasoning course (where my approach is similar in spirit, but very different in practice).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Right now, I am learning to do IBL in a Linear Algebra course. It is definitely more challenging to figure out what to do in a course where the list of content goals is long and the focus is not on writing complete proofs. The key insight I have so far I got from a mentor (Ed Parker): find the appropriate level of rigor. Basically, you have to answer the question: “What counts as a convincing argument for students at a pre-rigorous stage of development?” For this term, I am experimenting with the idea of building examples or non-examples as a replacement for formal proof writing.&lt;/p&gt;&#xA;" OwnerUserId="309" LastEditorUserId="309" LastEditDate="2014-03-25T14:57:37.313" LastActivityDate="2014-03-25T14:57:37.313" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="793" PostTypeId="2" ParentId="6" CreationDate="2014-03-25T13:24:52.077" Score="7" Body="&lt;p&gt;You might be interested in the article &lt;em&gt;&lt;a href=&quot;https://www.jstor.org/stable/10.5951/mathteacher.107.7.0528&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra Homework: A Sandwich!&lt;/a&gt;&lt;/em&gt; by D. Bruce Jackson in the March 2014 issue of Mathematics Teacher. Despite the title, it's not really specific to algebra and it provides one way to address exactly the problem you're talking about. Basically, the idea is to make a homework problem a 3-step process. First, do the problem, then check the answer, and if it's wrong do it again. The students effectively self-grade, from the perspective of getting the correct answers.&lt;/p&gt;&#xA;&lt;p&gt;I don't remember what the article suggests in terms of how to translate that into a homework grade. Personally, I would want to do some spot-checking to make sure the student is really checking all of their answers (and the penalty for an uncorrected wrong answer would be harsh) and checking that the work that leads to a correct answer is correct.&lt;/p&gt;&#xA;" OwnerUserId="311" LastEditorDisplayName="user7990" LastEditDate="2020-08-29T09:40:18.430" LastActivityDate="2020-08-29T09:40:18.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="794" PostTypeId="2" ParentId="714" CreationDate="2014-03-25T13:30:01.677" Score="11" Body="&lt;p&gt;The math classroom standard &quot;show your work&quot; is really just a version of &quot;communicate your reasoning&quot; or &quot;explain yourself&quot;, required in &lt;em&gt;any&lt;/em&gt; profession.  We sometimes do a disservice by implying that math classes have a special show-your-work requirement that is somehow not used in other disciplines.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do model this communication in my own writing in class.  But also, a standard I apply to homework and (somewhat) to exams is: &quot;&lt;em&gt;Your work should be understandable by others&lt;/em&gt;.&quot;  If Bob doesn't understand how to do problem 5, could I show Alice's work to Bob, have him read it, and then have Bob say, &quot;Ah, I get it now&quot;?  If so, then Alice's communication is good.  I apply this standard more rigorously in upper level college classes.  With lower level (first-year) college classes, I tend to emphasize that, along with being readable by a peer, student work must &lt;em&gt;foremost&lt;/em&gt; be readable by the student herself!  Many errors in lower level math would be corrected if the student carefully wrote out the work and then casually skimmed it before handing it in.  Students should &quot;show their work&quot; for their own benefit!!  I stress this repeatedly, pointing out examples of this in exams and homework, as I grade.&lt;/p&gt;&#xA;" OwnerUserId="313" LastActivityDate="2014-03-25T13:30:01.677" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="795" PostTypeId="2" ParentId="654" CreationDate="2014-03-25T13:30:07.830" Score="4" Body="&lt;p&gt;I use slides as an accompaniment to a lecture, like a music video to a pop song. This is something I can only do with slides- there is no way to do this using a blackboard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only a small percentage of what is on the slide is mathematical content- the rest is pictures (downloaded from google, drawn myself, or scanned in). The goal is to give a compelling visual background to words I'm speaking, and to provide visual intuition to content. I'll always provide pictures of relevant personages, graphs of relevant functions, pictures of places where things were discovered... examinable statements will be in red with a box around them, completely tangential assides will be in blue or green. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't only use slides of course. When I want to make a computation or to write a proof, I write directly onto a piece of paper on a document camera, which forces me to slow down and keeps the students on board. Important formulae and things I want people to remember are on handouts, and the assumption is that students have read the relevant parts of the textbook before coming to class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I taught Complex Analysis using this approach, and I found it to be quite successful. Its drawback is that it's very time consuming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in conclusion, I would recommend slides for lots of pictures, animations, and visual accompaniments, and to colour-code and visually indicate relative importance of statements and their role in the &quot;big picture&quot; of the course. But for computations and for proofs, I &lt;b&gt;never&lt;/b&gt; use slides- I instead write directly only a piece of paper placed on a document camera.&lt;/p&gt;&#xA;" OwnerUserId="185" LastActivityDate="2014-03-25T13:30:07.830" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="796" PostTypeId="2" ParentId="561" CreationDate="2014-03-25T13:35:53.747" Score="4" Body="&lt;p&gt;I think the &lt;span class=&quot;math-container&quot;&gt;$F(x) =\int \limits_0^x x^2\mathrm dx, F(1)=?$&lt;/span&gt; trick just shifts the problem . Once you do this you give the students the right to ask, &quot;well, set &lt;span class=&quot;math-container&quot;&gt;$F(t)=\int \limits_0^xt^2\mathrm dt$&lt;/span&gt;, what is &lt;span class=&quot;math-container&quot;&gt;$F(1)=?$&lt;/span&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that integration is an operator that takes &lt;strong&gt;functions&lt;/strong&gt; as inputs. Given an integrable function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; no one would have any problem with &lt;span class=&quot;math-container&quot;&gt;$\int \limits _0^xf$&lt;/span&gt;. When one writes &lt;span class=&quot;math-container&quot;&gt;$\int \limits_0^x t^2\mathrm dt$&lt;/span&gt; you're in a sense saying that &lt;span class=&quot;math-container&quot;&gt;$t^2$&lt;/span&gt; is a function, or that it is a function of &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; or something like that. But &lt;span class=&quot;math-container&quot;&gt;$t^2$&lt;/span&gt; isn't a function, &lt;span class=&quot;math-container&quot;&gt;$t\mapsto t^2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\{(t,t^2)\colon t\in\text{somewhere}\}$&lt;/span&gt; are. And you can happily abbreviate any of the notations above with a letter (&lt;span class=&quot;math-container&quot;&gt;$f,g,h, \varphi, \ldots$&lt;/span&gt;). It is also important to note that &lt;span class=&quot;math-container&quot;&gt;$x\mapsto x^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t\mapsto t^2$&lt;/span&gt; are one and the same function. The concept 'function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;' isn't defined (of course you can define it, but that brings a whole set of unnecessary problems into the mix).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question asks&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can you explain to students that they should not use the same variable in an integrand and in the limits of integration simultaneously?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;while the above implies what I mean as answer: explain what is actually being done and there's no reason to think they would have any problems with the notation.&lt;/p&gt;&#xA;" OwnerUserId="287" LastEditorUserId="2083" LastEditDate="2019-09-08T07:12:15.300" LastActivityDate="2019-09-08T07:12:15.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="798" PostTypeId="1" CreationDate="2014-03-25T13:38:19.287" Score="19" ViewCount="1280" Body="&lt;p&gt;In my home province Discovery Learning is getting a &lt;a href=&quot;http://www.theglobeandmail.com/news/national/parents-push-alberta-back-to-math-basics/article17654746/&quot;&gt;substantial amount of pushback&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been trying to follow the discussions, but have been struggling because I can't seem to get a clear answer as to what exactly is meant by Discovery Learning. The &lt;a href=&quot;http://en.wikipedia.org/wiki/Discovery_learning&quot;&gt;wiki&lt;/a&gt; page ends with me more confused than I came in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My understanding is that the teaching is based on attempting to have children understand &lt;em&gt;why&lt;/em&gt; mathematical properties hold through 'discovery'. Yet parents seem to be &lt;strong&gt;very upset&lt;/strong&gt;, and it seems to be more than just their children not knowing their multiplication-tables. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what it is discovery learning, and why so controversial?&lt;/p&gt;&#xA;" OwnerUserId="314" LastActivityDate="2019-07-22T17:34:58.250" Title="What is Discovery-Learning, and why is it so controversial?" Tags="|primary-education|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="799" PostTypeId="2" ParentId="779" CreationDate="2014-03-25T13:40:00.737" Score="10" Body="&lt;p&gt;You can point them to &lt;a href=&quot;http://msp.org/involve/2014/7-2/index.xhtml&quot;&gt;Involve&lt;/a&gt;, a journal &lt;em&gt;all&lt;/em&gt; of whose articles involve substantial contributions by undergraduates. [Disclaimer: I am on its editorial board]&lt;/p&gt;&#xA;" OwnerUserId="270" LastActivityDate="2014-03-25T13:40:00.737" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="800" PostTypeId="2" ParentId="719" CreationDate="2014-03-25T13:43:01.217" Score="3" Body="&lt;p&gt;There is &lt;em&gt;always&lt;/em&gt; a significant difference between the historical development of a subject and the logical, mathematical reasoning taught in the classroom.  The historical development involved lots of twists and turns, dead-ends and loops, occurring possibly over centuries.  It is rarely practical to teach a topic (to first-time learners) in a historically valid fashion.  Take advantage of the hindsight and use limits in calculus, for example.  (If you are driving to the west coast for the first time, take the interstate!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, the precise sterile logic of the topic hides the excitement of mathematical exploration.  So if you get a chance, digress for a &lt;em&gt;short&lt;/em&gt; time to describe the history (Newton's fluxions and fluents!) and show that it was not straightforward.  (Pull off the interstate and drive a few miles on route 66 before going back onto the interstate.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have little historical vignettes I stick into my lectures to try and add a human side to the topic.  But the topic itself should be taught as logically as possible, so that the logic is reasonable and understandable.  (Cantor's first proof on the cardinality of the reals is probably NOT what you want to teach to newbies learning cardinality!  But you can say something about the impact Cantor's work had on both Cantor and others in the late nineteenth century.)&lt;/p&gt;&#xA;" OwnerUserId="313" LastActivityDate="2014-03-25T13:43:01.217" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="801" PostTypeId="2" ParentId="798" CreationDate="2014-03-25T13:47:36.970" Score="11" Body="&lt;p&gt;Discovery learning has each child build his or her own individual understanding of a concept.  The teacher is there to guide and reinforce this process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This stands in contrast to the traditional model for teaching, in which the teacher knows a secret, reveals it to the students, and the students recite it back verbatim.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Discovery learning has been demonstrated more effective not only at conveying understanding and motivating students, but building meta-understanding (i.e. knowing what you know).  However, letting students build their own understandings comes with risk -- they way they understand things might not be the same way that their parents understood them.  This becomes a problem when parents see how the children are doing arithmetic and complain that the algorithms that they, the parents, were taught to memorize do not have the same emphasis (or are even missing). And indeed if taken to excess discovery learning risks rediscovering the wheel; some amount of secret-sharing is necessary.&lt;/p&gt;&#xA;" OwnerUserId="270" LastActivityDate="2014-03-25T13:47:36.970" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="802" PostTypeId="2" ParentId="682" CreationDate="2014-03-25T13:54:35.910" Score="13" Body="&lt;p&gt;I learn my students' names.  (All 100 of them.)  But I learn the names of &quot;problem&quot; students first!  If you are reading a newspaper in my class (in ways visible to all) or constantly talking to a friend, or making out with your girlfriend, I make sure to get your name by the end of class. (I may ask you your name during class. Or, as a last resort, I may simply ask, as you leave, &quot;I'm sorry, I don't know your name&quot; and I make sure that I have it from then on.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that, I say things like, &quot;Carl, are you with me here?&quot; or &quot;Leticia, are you following me?&quot;  Students react quite differently if you KNOW who they are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Final step -- if this does not change behavior, I talk to students outside of class and simply explain that in class they are to focus on the material, engage in the lecture and worksheets, etc., and that their behavior in class needs to change.  (But if I know the students' names, I usually don't have to get to this step.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In larger classes, I use a seating chart to help me learn names rapidly. Then I practice the names as I return materials.  I've twice taught classes of 150 or more and I learned all the names by the third week.  In large classes, students appreciate me calling them by name when they raise their hand and it clearly improves both morale and student behavior.&lt;/p&gt;&#xA;" OwnerUserId="313" LastActivityDate="2014-03-25T13:54:35.910" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="803" PostTypeId="2" ParentId="736" CreationDate="2014-03-25T13:59:06.680" Score="2" Body="&lt;p&gt;You have conducted an experiment, with your past sections of this course as control.  However, you have broken the cardinal rule of the scientific method and made two substantial changes:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Homework has moved from instructor-graded to mylabsplus-graded.&lt;/li&gt;&#xA;&lt;li&gt;Projects have been added.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;With just this data point, it is impossible to tell which of these two changes, if either, causes any effect that you see.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my personal experience, the biggest challenge facing students in large lower-division courses is motivation.  The greatest ROI with such students is from keeping them focused on the course.  Frequent homework/quizzes is one such way, engaging classtime is another such way, etc.  It's possible that your projects have made some students very excited.  It's also possible that the change in homework grading is helping students keep pace.&lt;/p&gt;&#xA;" OwnerUserId="270" LastActivityDate="2014-03-25T13:59:06.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="804" PostTypeId="2" ParentId="710" CreationDate="2014-03-25T14:05:27.410" Score="8" Body="&lt;p&gt;Outside of what is found in refereed journals (which one can generally find by just searching Google Scholar) a good source - if you have access - is ProQuest's &amp;quot;Digital Dissertations.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Searching there, I found a few theses on Kumon: Here are citations and abstracts for two of them; the bold text was added by me.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1.$&lt;/span&gt; Thijsse, L. J. (2003). The effects of a structured teaching method on mathematics anxiety and achievement of grade eight learners.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Abstract:&lt;/strong&gt;  The hypothesis that a structured, sequenced, approach to mathematics learning, based on the application of learnt facts, decreases mathematics anxiety and increases mathematics achievement is tested. A literature study and an empirical investigation were conducted with respect to the relationships between maths anxiety, maths achievement and teaching methods. A qualitative research design which focussed on the cross-case analysis of different case studies was used.&lt;/p&gt;&#xA;&lt;p&gt;The qualitative case study involves multiple methods such as interviews, observations and a pretest, posttest design. &lt;strong&gt;It analyses and compares the effects of the Kumon method, used as the intervention programme, on maths anxiety and maths achievement of an experimental group and a control group.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The results of this research indicate that learners on the intervention programme who showed a decrease in anxiety, showed an increase in achievement. This has implications for the teaching methods used in South Africa.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2.$&lt;/span&gt; Medina, S. L. (1989). A study of the effects of the kumon method upon the mathematical development of a group of inner-city junior high school students.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Abstract:&lt;/strong&gt; In this research, 103 Hispanic junior high school students were instructed using the Kumon Mathematical Method. Instruction took place 1 hour per day, 5 days a week in a public school setting. Instruction extended over an eight month period. Student subjects were pretested and posttested on the Mathematics batteries of the California Achievement Test (CAT). The amount of progress in mathematics was determined by computing gain scores for each section of the examination. Data included in this report includes: (1) CAT math concepts scores; (2) CAT computation and applications scores; (3) CAT total math scores; (4) school attendance; (5) Kumon Examination scores; (6) Kumon time scores; (7) number of packets completed; (8) Kumon level advancement; (9) class period; (10) academic track; and (11) grade level. Correlational relationships between variables are also discussed. Reported were significant gains in math computation, math concepts, and math applications scores at the seventh grade level; grade 8 students maintained their percentile rankings for the duration of the study; subjects significantly increased their speed on the Kumon exam; and CAT math gain scores were greater for the seventh graders than for the eighth graders. CAT gain scores were correlated with post Kumon exam scores. CAT gain scores were not correlated with Kumon exam time scores, number of Kumon packets completed, or Kumon advancement.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And here is one non-dissertation (disclaimer: the paper is &lt;a href=&quot;http://www.kumon.ca/miscellaneous/resourcesfiles/kumon_resources_roeper%20review%20article.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;linked&lt;/a&gt; from the Kumon site).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3.$&lt;/span&gt; McKenna, M. A., Hollingsworth, P. L., &amp;amp; Barnes, L. L. (2005). Developing latent mathematics abilities in economically disadvantaged students. &lt;em&gt;Roeper Review, 27&lt;/em&gt;(4), 222-227.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This Javits grant evaluation study sought to develop latent abilities in economically disadvantaged students by providing opportunities for mathematics development and acceleration. &lt;strong&gt;This study examined the effects of Kumon instruction, a supplementary, highly sequential, individualized method of developing mathematics skills. Whole classes of Title I elementary school students from grades two through five were divided into two groups, those with Kumon instruction and those without.&lt;/strong&gt; All students continued with traditional textbook mathematics. Pre- and posttests were administered to all participants to assess progress, compare standardized test results, and examine levels of acceleration. Results showed that Kumon group students improved their mathematics skill levels more than non-Kumon group students, and they scored significantly higher than non-Kumon group students two years after their Kumon instruction ended.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-26T00:03:08.403" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="805" PostTypeId="1" CreationDate="2014-03-25T14:09:17.093" Score="87" ViewCount="16092" Body="&lt;p&gt;Is it better to design an exam with fewer questions and relaxed timing or with more questions and a resulting time-pressure? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One the one hand, it seems that students who really know the stuff will be able to complete an exam quickly, and a time-pressure exam detects this, allowing me to sort the students by this measure. On the other hand, perhaps it would be better to have the exam test only for knowledge and not speed. But with fewer or easier questions this would naturally mean that fewer mathematical ideas are tested on the exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice, my choice boils down to this: should I drop questions from my exam, even though this means fewer topics on the exam, in order that more students will not experience time-pressure during the exam?&lt;/p&gt;&#xA;" OwnerUserId="90" LastEditorUserId="376" LastEditDate="2015-12-25T13:17:36.927" LastActivityDate="2021-04-28T12:15:18.767" Title="Should I design my exams to have time-pressure or not?" Tags="|exams|test-design|" AnswerCount="15" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="806" PostTypeId="2" ParentId="369" CreationDate="2014-03-25T14:33:57.950" Score="8" Body="&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt; I had believed not, but see Paul Stanley's &lt;a href=&quot;https://matheducators.stackexchange.com/a/7231/262&quot;&gt;&lt;strong&gt;response&lt;/strong&gt;&lt;/a&gt; above.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;A good source for information about twentieth-century textbooks on algebra and geometry is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Donoghue, Eileen F. &quot;Algebra and geometry textbooks in twentieth-century America.&quot; A history of school mathematics 1 (2003): 329-398.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the section entitled &quot;SMSG's &lt;em&gt;First Course in Algebra&lt;/em&gt;&quot; (beginning on p. 366) Donoghue writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Among the members of the [ninth-grade SMSG writing] group were Eugene Northrop of the University of Chicago, Mary Dolciani of Hunter College in New York City, and Henry Pollak, a research mathematician at Bell Laboratories. The group decided that algebra ought to form the core content of ninth-grade mathematics. In their view, all citizens could benefit from studying algebra; for the college-bound student, it was essential.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The key names above are Northrop, Dolciani, and Pollak. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Northrop went on to write a textbook, &lt;em&gt;Fundamental Mathematics&lt;/em&gt;, which dedicated about one third of the corresponding class time to Algebra; you can read more about it in my earlier MO answer &lt;a href=&quot;https://mathoverflow.net/questions/51285/where-and-when-did-transition-to-abstraction-courses-start/151910#151910&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pollak has now been a (&quot;visiting&quot;) professor at Teachers College Columbia University, where Donoghue earned her doctorate, for the past 25 years. He has not written any textbooks, but he has personally remarked to me about the great skill possessed by Dolciani.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dolciani &lt;em&gt;did&lt;/em&gt; go on to write another textbook on Algebra. It is the subject of the subsequent section in Donoghue's article, entitled: &quot;Dolciani's &lt;em&gt;Modern Algebra: Structure and Method, Book One&lt;/em&gt;.&quot; She later wrote other books (which you can search out online) so that you may wish to look into work by &lt;a href=&quot;http://en.wikipedia.org/wiki/Mary_P._Dolciani&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Mary Dolciani&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(See also &lt;a href=&quot;http://dolcianihalloran.squarespace.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;this&lt;/strong&gt;&lt;/a&gt; wonderful link provided by &lt;strong&gt;James S. Cook&lt;/strong&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-04-13T21:05:26.980" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="807" PostTypeId="1" CreationDate="2014-03-25T14:47:36.207" Score="19" ViewCount="1002" Body="&lt;p&gt;When you encounter a very bright student in a first-year (college/university) class (and who is therefore bored), what do you do?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leaving them to their own devices can be problematic. It can lead to a loss of motivation, or, at the other end of the spectrum, enthusiastic crankery (I currently have a student who is trying to define the &lt;em&gt;number&lt;/em&gt; infinity). Therefore, some nurturing is required. However, I am busy (as I suppose most people here are) and so I cannot lend such students very much of my time and I so cannot personally &quot;lead&quot; their thoughts and ideas as much as I might wish. For example, I suggested that my &quot;infinity-defining&quot; student might want to look up ordinal numbers, but I simply don't have the time to spend the hours that it would require to sift through his workings or to find the most appropriate concept of &quot;infinity&quot; for what he is trying to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Telling them to attend second year lectures or giving them an advanced book to read is again problematic. It just defers their boredom for another year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A final approach, and is the approach I have been favoring, is to give the students a good -but mostly irrelevant- book. That is, a book which has a small intersection with the courses they will take in the future, but one which they might still find useful. However, such books are hard to find...(my current one out on loan is Yaglom's &lt;em&gt;Geometric transformations&lt;/em&gt;, in the hope that when (okay, if...) they encounter Fuchsian groups they will understand them a bit better...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be interested to hear your thoughts and ideas on how to help nurture such students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;strong&gt;Note:&lt;/strong&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/questions/160/students-who-know-high-level-math-before-going-to-college&quot;&gt;This question&lt;/a&gt; is similar, but disjoint.)&lt;/p&gt;&#xA;" OwnerUserId="169" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-05-08T00:21:40.423" Title="How to nurture a good student?" Tags="|undergraduate-education|tutoring|undergraduate-research|gifted-students|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="809" PostTypeId="2" ParentId="805" CreationDate="2014-03-25T14:59:23.873" Score="83" Body="&lt;p&gt;The time-pressure not only sorts students by knowledge and speed, but also by who is susceptible to math anxiety. Mathematics Educator &lt;a href=&quot;http://en.wikipedia.org/wiki/Jo_Boaler&quot;&gt;Jo Boaler&lt;/a&gt; comments in a &lt;a href=&quot;http://www.edweek.org/ew/articles/2012/07/03/36boaler.h31.html&quot;&gt;popular media piece&lt;/a&gt; (which includes a few links to more formal research) that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;research ... has shown that timed tests are the direct cause of the early onset of math anxiety.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Her comments are mostly about K-12 students, particularly on the younger end, as she also remarks:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is critical that we take a moment to review the emerging evidence on the impact of timed testing and the ways in which it transforms children’s brains &lt;a href=&quot;http://pss.sagepub.com/content/23/5/492&quot;&gt;(1)&lt;/a&gt; &lt;a href=&quot;http://www.mccc.edu/~jenningh/Courses/documents/math_anxiety.pdf&quot;&gt;(2)&lt;/a&gt;, leading to an inevitable path of math anxiety and low math achievement.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Policies in education rarely draw from research knowledge. But I would argue that this particular policy—of giving young children timed math tests—is one of the clearest ways schools damage children, and we now have evidence of the extent of the damage. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I do not think it is a leap to suggest that timed tests could have a similar impact on undergraduate (or even graduate...) students. More precisely, timed tests may not only interfere with students' affect during the test-taking process (note &lt;em&gt;affect&lt;/em&gt; is crucial to mathematical success in the classroom; cf. e.g., &lt;em&gt;beliefs and belief systems&lt;/em&gt; in Schoenfeld's &#xA;(1985) &quot;Mathematical Problem Solving&quot;) but afterwards as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a closing remark, I leave you with a quotation from G. Perelman's graduate advisor, Y. Burago:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;[Perelman] was not fast. Speed means nothing. Math doesn't depend on speed. It is about deep.&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So: My vote goes to &lt;strong&gt;fewer questions and relaxed timing&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-03-25T14:59:23.873" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="810" PostTypeId="2" ParentId="805" CreationDate="2014-03-25T15:27:11.837" Score="27" Body="&lt;p&gt;A common approach in elementary school math is the use of &amp;quot;mad minutes&amp;quot; wherein a child attempts to madly complete as many arithmetic problems as they can. This is an extreme example of a timed assessment.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;http://www.edweek.org/ew/articles/2011/05/18/31math_ep.h30.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this description of some research&lt;/a&gt;, the authors suggest that when students have math anxiety, some of the mental resources they have available are used to deal with the anxiety, rather than work on the mathematics. Hence, if we want all students to be able to utilize their full mental resources for an assessment (which seems like a reasonable goal) we should do what we can to reduce anxiety, and &lt;a href=&quot;https://www.semanticscholar.org/paper/Research-Suggests-that-Timed-Tests-Cause-Math-Johnson-Pearson/584c637689348ab642fb8df48d21a4d7f47ccc49&quot; rel=&quot;nofollow noreferrer&quot;&gt;timed tests are one practice that likely results in anxiety&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.edudemic.com/how-one-teaching-practice-ruined-math-for-students/&quot; rel=&quot;nofollow noreferrer&quot;&gt;In this article&lt;/a&gt;, the author talks about how her belief that she could not do mathematics started with the practice of mad minutes in elementary school. At 6 years old, she was convinced she could not do mathematics.&lt;/p&gt;&#xA;&lt;p&gt;This next point comes from my personal experience. The way we assess students, and our emphasis on these assessments, tells students what we value in mathematics class. I have assessed students a variety of different ways, and found that students define &amp;quot;what it means to do mathematics&amp;quot; based on what is valued in class. Hence, if we want students to learn that one view of mathematics is that it involved careful, patient, creative problem solving, we need to demonstrate that with the tasks we ask them to do.&lt;/p&gt;&#xA;" OwnerUserId="254" LastEditorUserId="-1" LastEditDate="2021-04-28T12:15:18.767" LastActivityDate="2021-04-28T12:15:18.767" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="811" PostTypeId="2" ParentId="779" CreationDate="2014-03-25T16:00:28.003" Score="5" Body="&lt;p&gt;An example from 2002 is the polynomial-time &lt;a href=&quot;http://en.wikipedia.org/wiki/AKS_primality_test&quot; rel=&quot;noreferrer&quot;&gt;algorithm&lt;/a&gt; to decide primality.  Neeraj Kayal and Nitin Saxena were two of the three authors, and had just gotten their undergraduate degrees that year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example from 1956 is the &lt;a href=&quot;http://books.google.com/books?id=y8n76ztNVlMC&amp;amp;pg=PA132&amp;amp;lpg=PA132&amp;amp;dq=intermediate%20degrees%20recursive%20friedberg%20muchnik%20senior%20thesis&amp;amp;source=bl&amp;amp;ots=EwFZchDIv-&amp;amp;sig=p9O39uBZ3_qkAQfmutvWZK1fz18&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=2aYxU4_XC46gkQfT54GgDQ&amp;amp;ved=0CCkQ6AEwAA#v=onepage&amp;amp;q=intermediate%20degrees%20recursive%20friedberg%20muchnik%20senior%20thesis&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;construction&lt;/a&gt; of a set of intermediate degree, by Richard Friedberg in his senior thesis at Harvard, and also by a not-much-older Albert Mucnik in Russia.  This is a set which is neither recursive nor strong enough to allow the halting problem to be computed recursively from it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These were both influential results in logic.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-25T16:00:28.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="812" PostTypeId="2" ParentId="294" CreationDate="2014-03-25T16:01:09.183" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/William_Poundstone&quot; rel=&quot;nofollow&quot;&gt;William Poundstone's&lt;/a&gt; book &lt;a href=&quot;http://store.doverpublications.com/048649098x.html&quot; rel=&quot;nofollow&quot;&gt;The Recursive Universe&lt;/a&gt;. It's an engaging read for any intelligent reader. It's about Conway's Game of Life amongst other things.&lt;/p&gt;&#xA;" OwnerUserId="291" LastEditorUserId="143" LastEditDate="2014-03-26T11:14:54.733" LastActivityDate="2014-03-26T11:14:54.733" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="813" PostTypeId="7" CreationDate="2014-03-25T16:35:08.783" Score="0" Body="&lt;p&gt;&lt;b&gt;Mathematics Educators Stack Exchange&lt;/b&gt; is a question and answer site for those involved in the field of teaching mathematics. It's built and run &lt;i&gt;by you&lt;/i&gt; as part of the &lt;a href=&quot;https://stackexchange.com&quot;&gt;Stack Exchange&lt;/a&gt; network of Q&amp;amp;A sites. With your help, we're working together to build a library of detailed answers to every question about teaching mathematics.&lt;/p&gt;&#xA;" OwnerUserId="-1" LastEditorUserId="16457" LastEditDate="2021-07-13T11:25:47.477" LastActivityDate="2021-07-13T11:25:47.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="814" PostTypeId="2" ParentId="805" CreationDate="2014-03-25T16:36:01.960" Score="13" Body="&lt;p&gt;Disclaimer: I'm not an educator, so my basis for this answer is quite limited anecdotal evidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you apply time-pressure to mathematics, you switch in some sense from testing what the student can do, to testing what the student can do &lt;em&gt;easily&lt;/em&gt;. I would think that most of the time you want to test the former. The difference between &quot;not knowing&quot; and &quot;knowing&quot; is usually far more important than the difference between &quot;knowing&quot; and &quot;really knowing&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd speculate that the latter depends to a large extent on something that most of the time is quite uninteresting: inherent aptitude. I'm not saying you don't want to know who in your classes is a mathematical prodigy, but I'm pretty sure you have an inkling of that anyway without needing to destruction-test the ones who aren't. Mathematics is a subject in which it's really quite important to stretch the best students with additional material[*], I think not so much with the same material done quicker.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wasn't challenged much by mathematics testing before I went to university, and I almost always finished exams early. I don't think it's unusual to see quite a wide range of speed in mathematics (and other subjects). So to me time-pressured exams would have been more interesting and would have showed off what I could do, since some other people would have got much worse marks and I would only have got slightly worse marks due to a somewhat increased rate of errors. But so what? How does that help the class? It's not the only and probably not even the best way to identify the most able students, and anyway one should not design routine examinations for the benefit of the best students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One summer when I was a child (can't remember how old) I did do some time-pressured tests of multiplication tables. IIRC it worked like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;my mother wrote the numbers 2-12 across the top and side of a grid, in different random orders.&lt;/li&gt;&#xA;&lt;li&gt;I filled in the products as quickly as possible.&lt;/li&gt;&#xA;&lt;li&gt;I tried to beat my time the next day.&lt;/li&gt;&#xA;&lt;li&gt;I can't remember for sure the rules on errors, I think that any error was a disqualification.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;However the purpose was training by repetition (so that I would know my multiplication tables, not just know them in order). Fairly quickly it became a test of hand-writing under time pressure, to avoiding writing any 0 that looked like a 6. The purpose was &lt;em&gt;not&lt;/em&gt; assessment. OK, it's an extreme example of testing &lt;em&gt;purely&lt;/em&gt; by time-pressure since the task itself was trivial (to me at that age) absent time pressure. But I think it's clear that it would not be useful as a classroom exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[*] &quot;what subject isn't?&quot; chime in all the non-mathematicians. Well, I doubt there's any subject where the best can't benefit from advanced studies, but mathematics seems to have a particular quality that any given material will be frustratingly challenging for one third of the class and boringly easy for another third. At least in English the whole class can study the same novel at different depths :-)&lt;/p&gt;&#xA;" OwnerUserId="321" LastEditorUserId="321" LastEditDate="2014-03-25T18:03:20.023" LastActivityDate="2014-03-25T18:03:20.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="815" PostTypeId="2" ParentId="530" CreationDate="2014-03-25T16:40:38.050" Score="15" Body="&lt;p&gt;I feel as though a lot of this misses the point. Teaching the unit circle and trig functions in terms of &quot;one turn&quot; makes a lot more intuitive sense to young students than do arbitrary variables tau or pi. It just so happens that Tau is equivalent to one turn. Once students grasp the concept of the unit circle, then the pi conventions can be explored.&lt;/p&gt;&#xA;" OwnerUserId="307" LastActivityDate="2014-03-25T16:40:38.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="816" PostTypeId="2" ParentId="2" CreationDate="2014-03-25T17:14:34.967" Score="11" Body="&lt;p&gt;With regard to Math Education literature on proofs: a person to look to is &lt;a href=&quot;https://education.utexas.edu/faculty/eric_knuth&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eric Knuth&lt;/a&gt; (&lt;a href=&quot;https://scholar.google.com/scholar?q=proof%20EJ%20knuth&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=0,33&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Scholar&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;However, it may be more fruitful to shift from talking about writing (formal) mathematical proofs to discussions around incorporating &lt;em&gt;sense making&lt;/em&gt; into the mathematics classroom.&lt;/p&gt;&#xA;&lt;p&gt;Sue VanHattum comments in another response here that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Very young children can prove that two odd numbers will always add up to an even number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For an example of this, consider the excerpt below, which mentions &lt;a href=&quot;http://en.wikipedia.org/wiki/Deborah_Loewenberg_Ball&quot; rel=&quot;nofollow noreferrer&quot;&gt;Deborah Ball&lt;/a&gt;, and comes from &lt;a href=&quot;https://journals.library.columbia.edu/index.php/jmetc/article/view/620&quot; rel=&quot;nofollow noreferrer&quot;&gt;an article&lt;/a&gt; by Alan Schoenfeld (citation: Schoenfeld, A. Mathematical Modeling, Sense Making, and the Common Core State Standards. &lt;em&gt;The Journal of Mathematics Education at Teachers College, 4&lt;/em&gt;(2).)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In contrast, here’s a sense making example that comes from Deborah Ball’s third grade class. Deborah had a bunch of kids who were playing with numbers—adding, subtracting, doing various things. They noticed that every time they added two odd numbers, they got an even number. One of the kids said, &amp;quot;Is that always going to happen?&amp;quot; She then stopped and said, &amp;quot;But the odd numbers go on forever. We can never test them all, so we can never know.&amp;quot; That’s pretty damned good for a third grader.&lt;/p&gt;&#xA;&lt;p&gt;Another kid says, &amp;quot;Well, I was thinking about seven plus nine, and actually I knew that the sum was going to be even before I added them up. If you look at seven, it’s a bunch of pairs with one left over, and if you look at the nine, it’s a bunch of pairs with one left over (Figure 1). So when you put them together, the pairs are going to stay the same—but the two leftovers become a pair, so everything together is in pairs, so it’s going to be even&amp;quot; (Figure 2). The girl stopped at that point and she said, &amp;quot;But wait! It doesn’t have to be seven and nine. No matter what that first odd number is, it's going to be a bunch of pairs with one left over. The second odd number is also going to be a bunch of pairs with one left over. And when you put them together, you have all the pairs you started with plus the pair you made, so it’s going to be even.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Now that’s a third grader. I guarantee you that every mathematician I know would say that the student produced a completely rigorous mathematical proof. That’s what I call mathematical sense making, and that’s what I’d like to see in our classrooms. The real challenge we face is to support sense making in our classrooms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The topic of &lt;em&gt;sense making&lt;/em&gt; comes up a fair amount in Math Education; it also features prominently in the Common Core State Standards for Mathematics (&lt;a href=&quot;http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF&lt;/a&gt;) as a Standard for Mathematical Practice:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h1&gt;Mathematics | Standards for Mathematical Practice&lt;/h1&gt;&#xA;&lt;p&gt;The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important &amp;quot;processes and proficiencies&amp;quot; with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;1. Make sense of problems and persevere in solving them.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, &amp;quot;Does this make sense?&amp;quot; They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course, sense making is something that can (should!) be taught from the earliest years.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorDisplayName="user7990" LastEditDate="2020-08-28T18:50:54.740" LastActivityDate="2020-08-28T18:50:54.740" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="817" PostTypeId="2" ParentId="408" CreationDate="2014-03-25T17:35:56.917" Score="3" Body="&lt;p&gt;I am happy to share my Beamer slides, available through each course webpage at &lt;a href=&quot;http://qcpages.qc.edu/~chanusa/courses/index.html&quot; rel=&quot;nofollow&quot;&gt;my homepage&lt;/a&gt;. The courses that have slides are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Combinatorics&lt;/strong&gt;: A course in enumeration, including simple counting, combinatorial proofs, bijective combinatorics, and generating functions.  The page also has information about the structure of the course project involving individual research and poster presentation. (&lt;a href=&quot;http://people.qc.cuny.edu/faculty/christopher.hanusa/courses/636fa13/Documents/636fa13ch21.pdf&quot; rel=&quot;nofollow&quot;&gt;Example set of notes&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Graph Theory&lt;/strong&gt;: Basic graph theory, graph statistics and properties, coloring, planarity, graph algorithms.  There is a course project where students research and write about mathematicians. (&lt;a href=&quot;http://qcpages.qc.edu/~chanusa/courses/634sp14/notes/634sp14ch11.pdf&quot; rel=&quot;nofollow&quot;&gt;Example set of notes&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Mathematical Modeling&lt;/strong&gt;: The focus is on the modeling process from start to finish.  Students learn to use &lt;em&gt;Mathematica&lt;/em&gt; and apply it to function fitting, probabilistic models, queuing theory, and linear optimization.  The course project is a group research project where students take some real world situation and try to understand it mathematically. (&lt;a href=&quot;http://qcpages.qc.edu/~chanusa/courses/245sp14/notes/245sp14ch13.pdf&quot; rel=&quot;nofollow&quot;&gt;Example set of notes&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Multivariable Calculus&lt;/strong&gt;: Parametric and Polar curves, including area and arc length.  Vectors, vector functions, functions of multiple variables, gradients, multiple integrals.  These notes are the most newly created (as of 2014), so they have a more advanced use of color and graphics.  In &lt;a href=&quot;http://qcpages.qc.edu/~chanusa/courses/201sp14/notes/201sp14ch115.pdf&quot; rel=&quot;nofollow&quot;&gt;this example&lt;/a&gt; I use color to explain the chain rule and include graphics about the directional derivative.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Slides are broken down by lecture and are intended to encourage active learning by leaving out certain details and examples that will be worked out on the board.  Students download the notes before class and bring them to class on paper or on their electronic devices.&lt;/p&gt;&#xA;" OwnerUserId="324" LastEditorUserId="75" LastEditDate="2014-03-25T18:40:38.797" LastActivityDate="2014-03-25T18:40:38.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="818" PostTypeId="2" ParentId="807" CreationDate="2014-03-25T18:00:25.843" Score="5" Body="&lt;p&gt;The undergraduate-research tag suggests you know the answer already. Get this student doing something engaging and interesting. It doesn't have to be &lt;em&gt;you&lt;/em&gt; that mentors this person. Can you help match him or her with someone who can spare a little time? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the US we have a set of Research Experience for Undergraduates Programs. That might be a way to go. If you are not in the US, perhaps you have something similar?&lt;/p&gt;&#xA;" OwnerUserId="309" LastActivityDate="2014-03-25T18:00:25.843" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="819" PostTypeId="1" AcceptedAnswerId="1778" CreationDate="2014-03-25T18:14:11.787" Score="12" ViewCount="344" Body="&lt;p&gt;(This question was &lt;a href=&quot;https://math.stackexchange.com/questions/90349/probability-textbooks&quot;&gt;posted more than two years ago&lt;/a&gt; on math.stackexchange.com and, although there were some worthwhile answers, none actually answered the question as phrased.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone compiled a moderately comprehensive list on the web or elsewhere of textbooks on probability&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;For students who have not been introduced to the subject before&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;That introduce both discrete and continuous probability distributions and their cumulative distribution functions, and include things like&#xA;the Poisson limit theorem, the central limit theorem (say the former&#xA;with proof and the latter not necessarily), and &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;That perhaps cover the simplest stochastic processes like the Poisson process or&#xA;infinite sequences of Bernoulli trials?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Experience using these texts and a short description of how it went would also be welcome.&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2017-04-20T08:52:32.550" Title="Probability textbooks repository" Tags="|textbooks|probability|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="820" PostTypeId="2" ParentId="805" CreationDate="2014-03-25T18:16:09.793" Score="2" Body="&lt;p&gt;What about taking a page from the &lt;a href=&quot;http://en.wikipedia.org/wiki/SAT&quot; rel=&quot;nofollow&quot;&gt;SAT&lt;/a&gt;? The exam is designed such that it's not necessarily expected that the student completes every problem within the time allotted (although there are certainly students who do). Instead of starting at 100% and losing points for every incomplete or incorrect answer, the student starts at 0%, gains points for every correct answer, and loses a &lt;em&gt;fraction&lt;/em&gt; of that for every incorrect answer¹ – incomplete answers are ignored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This style is probably superior for gauging where a student is at over a wide range of topics, possibly to help direct further teaching energy, as opposed to testing full comprehension of a single topic.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;¹ According to &lt;a href=&quot;http://en.wikipedia.org/wiki/SAT_Reasoning_Test&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;, the SAT will stop deducting points for incorrect answers in 2016.&lt;/p&gt;&#xA;" OwnerUserId="328" LastEditorUserId="75" LastEditDate="2014-03-25T18:38:42.187" LastActivityDate="2014-03-25T18:38:42.187" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="821" PostTypeId="2" ParentId="426" CreationDate="2014-03-25T18:32:00.960" Score="4" Body="&lt;p&gt;This seems to me to be two questions.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there research backing up the empirically observed trend that&#xA;  teacher-training is necessary for teaching primary and secondary&#xA;  school, but not necessary for teaching at university level and above?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Not as far as I know. I do know there is research that strongly suggests that different teaching methods at the university level lead to different learning outcomes from the students. For a summary of some of this research, see &lt;a href=&quot;http://www.youtube.com/watch?v=WwslBPj8GgI%E2%80%8E&quot; rel=&quot;nofollow&quot;&gt;Eric Mazur (warning: long) talk about how his teaching changes&lt;/a&gt;. &lt;a href=&quot;http://www.publicaffairs.ubc.ca/2011/05/12/interactive-teaching-methods-double-learning-engagement-in-large-undergraduate-physics-class/&quot; rel=&quot;nofollow&quot;&gt;There is also research suggesting&lt;/a&gt; that &quot;interactive teaching methods&quot; &quot;double&quot; learning gains for students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both of these are relevant because they suggest that there is a body of knowledge that describes good teaching at the university level, and so it stands to reason that this body of knowledge is something that can be taught to people, and one way to do that would be through an teaching credential. I have some friends who are working on this exact issue at a few different universities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the commenter who says that good teaching is not necessary at higher grade levels, I completely disagree. For example, Carnegie Mellon University had a 5% pass rate for their remedial math courses. With an increased emphasis on math mindsets, they have increased this pass rate by 10 times in the first year alone (unpublished research, look for it in a year or so). This says that how and what one teaches matters, even at the higher levels!&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why do universities and colleges not require teaching credentials?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For this, I would look at how universities and colleges get their money to operate. They receive part of their money from tuition (ie. which is dependent at least partially on the outcomes of their teaching), but it is my understanding (although I do not have time now to search for the relevant research), that universities receive much of the money they use to operate from operating grants &amp;amp; endowments (often of limited accountability on any measure) and from funded research. Only one of their funding resources has anything to do with the outcomes of their teaching, and so there is less incentive to focus on improving their teaching, which would be the obvious purpose of introducing a teaching certificate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for k-12 schools, basically the only point of the money given to them (aside from the fact it gives millions of students something to do while their parents work) is that they are expected to teach students. Hence, the k-12 system has a much stronger incentive to require capacity for teaching, which again, the assumption is that a teaching certificate improves.&lt;/p&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-03-25T18:32:00.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="822" PostTypeId="2" ParentId="805" CreationDate="2014-03-25T18:36:47.500" Score="21" Body="&lt;p&gt;While I agree in large parts with the other answers posted already, I would like to say a little something in defence of time pressure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience teaching at the university level, many students fall into the trap of feeling like they understand a concept that they really ought to think about more. I fall into this trap myself sometimes: everything seems nice and clear and straightforward while someone is guiding me through it, and I only realize the gaps in my knowledge later, when I'm forced to do it on my own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While this phenomenon is not in itself an argument for time-pressured tests, I do believe that such tests can be used quite effectively (and efficiently) to help nudge students into realizing when they've fallen into such a trap. I've found that sprinkling a few shorter tests or quizzes here and there with relatively easy questions but a tight time constraint can help students see for themselves what they &lt;em&gt;really&lt;/em&gt; understand and what they just &lt;em&gt;think&lt;/em&gt; they understand.&lt;/p&gt;&#xA;" OwnerUserId="327" LastActivityDate="2014-03-25T18:36:47.500" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="824" PostTypeId="2" ParentId="771" CreationDate="2014-03-25T20:58:37.600" Score="3" Body="&lt;p&gt;There is probably not a single answer to this, but there are fads and fashions in the practice of elementary education. Some of the participants here have seen some of them come and go, and the value of practice and drill at basic skills has been questioned by many teachers as well as students. Without slighting the competence or dedication of teachers, there are some philosophies that are better grounded in reality than others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A systematic failure in a particular educational philosophy may be difficult to detect immediately. It is more likely be detected at a later stage when the student needs the skills that once were taught at an earlier stage. Tutors and teachers in remedial education are more likely to encounter the failures than teachers of advanced mathematics. &lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-03-25T20:58:37.600" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="825" PostTypeId="1" CreationDate="2014-03-25T21:07:19.680" Score="10" ViewCount="292" Body="&lt;p&gt;What resources are available for any grade level K- 12 that are aligned with the Common Core Mathematics Standards and Mathematical Practices that have sets of problems or problem banks that can be used by teachers for instruction or homework?&lt;/p&gt;&#xA;" OwnerUserId="336" LastEditorUserId="50" LastEditDate="2014-03-25T21:19:06.780" LastActivityDate="2014-04-20T23:48:35.780" Title="Problems sets for instruction" Tags="|secondary-education|homework|primary-education|common-core|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="826" PostTypeId="2" ParentId="807" CreationDate="2014-03-25T21:20:56.363" Score="6" Body="&lt;p&gt;The book that I found as a young student that propelled my mathematical career forward was &lt;a href=&quot;http://en.wikipedia.org/wiki/On_Numbers_and_Games&quot;&gt;On Numbers and Games&lt;/a&gt; by Conway, and then later &lt;a href=&quot;http://en.wikipedia.org/wiki/Winning_Ways_for_your_Mathematical_Plays&quot;&gt;Winning Ways for your Mathematical Plays&lt;/a&gt; by Conway, Berlekamp, Guy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact that games can be solved by mathematics, and that mathematics can be built out of games, is the perfect kind of ridiculous abstract concept for an accelerated student to grapple with. If you want the same topic but in a more entertaining way, &lt;a href=&quot;http://www-cs-faculty.stanford.edu/~uno/sn.html&quot;&gt;Surreal Numbers&lt;/a&gt; by Knuth is excellent, and has exercises!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd actually argue that my undergraduate research projects were mostly useless, as far as &quot;nurturing&quot; goes, compared to the extreme mathematical entertainment I got out of those books.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-25T21:20:56.363" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="827" PostTypeId="2" ParentId="32" CreationDate="2014-03-25T21:27:07.473" Score="1" Body="&lt;p&gt;I think that the first thing to do is to give only a fraction of the maximum grade (let's say 60-70%) for students who are just completing the worksheet without being able to explain what that mean from the theoretical point of view. Of course I am assuming that a large part of the final grade will come from evaluating their lab work. &lt;/p&gt;&#xA;" OwnerUserId="330" LastActivityDate="2014-03-25T21:27:07.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="828" PostTypeId="2" ParentId="32" CreationDate="2014-03-25T21:36:45.970" Score="6" Body="&lt;p&gt;My mother tells an excellent story from when I was learning to read. She finds that I am reading very quickly considering my age. After asking me some questions, she discovers that I thought &quot;reading&quot; meant &quot;looking at and understanding each word on the page.&quot; I was able to do that very quickly!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your students sound like they are doing the same thing -- completing each question, but not thinking about what is going on around those questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, maybe we can learn from the English teachers here and use their solution -- comprehension questions. At the end of the worksheet, instead of asking practice questions that are retooled versions of the questions they just did, ask them questions that are more about the &quot;sentences&quot; and &quot;paragraphs&quot; of the worksheet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, you should explicitly label them &lt;strong&gt;Comprehension Questions&lt;/strong&gt; -- after a couple runs through the worksheets where the students found they could not answer the Comprehension Questions, they might notice what that implies about their work!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-25T21:36:45.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="829" PostTypeId="1" AcceptedAnswerId="837" CreationDate="2014-03-25T21:52:26.660" Score="10" ViewCount="166" Body="&lt;p&gt;When an excellent student is working on an honors or independent study project, the advisor's job is to not only serve as the content expert, but also to &lt;strong&gt;motivate the student&lt;/strong&gt; to complete the project.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My first research experience as an undergraduate was very disappointing, in hindsight. I arrived at each weekly meeting completely unprepared after having spent almost no time on the project. Since it wasn't graded nor did it have deadlines, it never seemed important enough to work on at any particular time, and so it never got worked on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I've mentored students working on independent projects in the past, the result has often been various flavors of the same outcome. The student arrives unprepared and the project does not move forward, even if specific goals were set out at the previous meeting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are good strategies (besides the obvious &quot;set a goal for each weekly meeting&quot;) for organizing an independent study / research project for a busy student?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://matheducators.stackexchange.com/questions/807/how-to-nurture-a-good-student&quot;&gt;How to nurture a good student?&lt;/a&gt;. I'm speaking about undergraduate students, but I can imagine answers applying to advanced high school students (or perhaps graduate students)!&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-03-26T22:25:09.183" Title="How does an advisor effectively motivate progress on an independent project?" Tags="|student-motivation|undergraduate-research|advising|gifted-students|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="830" PostTypeId="2" ParentId="825" CreationDate="2014-03-25T22:29:03.577" Score="11" Body="&lt;p&gt;Can we include rich math tasks in our answers? If so, look at:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://visualpatterns.org&quot;&gt;http://visualpatterns.org&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://graphingstories.com&quot;&gt;http://graphingstories.com&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://map.mathshell.org&quot;&gt;http://map.mathshell.org&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.illustrativemathematics.org/&quot;&gt;http://www.illustrativemathematics.org/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://threeacts.mrmeyer.com&quot;&gt;http://threeacts.mrmeyer.com&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.estimation180.com/&quot;&gt;http://www.estimation180.com/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.dailydesmos.com/&quot;&gt;http://www.dailydesmos.com/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://betterassessments.wordpress.com/&quot;&gt;http://betterassessments.wordpress.com/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://maththinking.org/&quot;&gt;http://maththinking.org/&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.collaborativemathematics.org/&quot;&gt;http://www.collaborativemathematics.org/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-03-25T22:29:03.577" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="831" PostTypeId="2" ParentId="807" CreationDate="2014-03-25T22:29:44.347" Score="7" Body="&lt;p&gt;In defense of enthusiastic crankery, I feel it's very valuable to try to define &quot;infinity&quot; as a number. You can ask the student critical questions, like how to deal with the &quot;$\infty=\infty+1\Rightarrow 0=1$&quot; dilemma. Let him resolve the apparent contradictions on his own; he'll either conclude that infinity isn't a number, or make up some other algebraic rule preventing you from making that equation. You could indirectly motivate, for example, Rudin's number system $\mathbb{R}\cup \{±\infty\}$, where $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$, and help him develop the necessary rules of algebra to make it work.  It's extremely instructive to a student to invent his own mathematical language, and especially with gifted students it's important to allow creative exploration. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You say you don't have time to sift through his stuff: I think that's good. Make him figure out the parts of his theory that do and do not work. It's more valuable for him to do so than you, and if he's already motivated enough to pursue this independently, he'll only go after it harder once you poke a few holes in his theory. If his ideas really are crankish, it should only take a minute or so to find an inconsistency, something that I'm sure could be managed during office hours.&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2014-03-25T22:34:47.550" LastActivityDate="2014-03-25T22:34:47.550" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="832" PostTypeId="2" ParentId="825" CreationDate="2014-03-25T23:28:14.457" Score="4" Body="&lt;p&gt;The CME Project! It's a full four-year high school curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://cmeproject.edc.org/&quot; rel=&quot;nofollow&quot;&gt;http://cmeproject.edc.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="84" LastActivityDate="2014-03-25T23:28:14.457" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="833" PostTypeId="1" AcceptedAnswerId="846" CreationDate="2014-03-26T00:26:24.297" Score="17" ViewCount="3551" Body="&lt;p&gt;Now, the title of the question definitely needs some background!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In some courses, everyone can get an A and it's no problem (especially later grad courses).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, many times it is good to have a wide spread of grades. Most often, this is because they will use the material in future courses, and if they are not prepared, they should know it. In my current university, the engineering department requested us to make the grades lower because the engineers couldn't do calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My problem is that I always overestimate the difficulty of my quizzes and exams and end up giving too good of grades at the beginning, and then face the task of either trying to lower their grades by being much tougher or not fulfilling departmental expectations.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What should I do when I realize mid-semester that students who haven't learned the material are still getting A's in a core class?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Like Jim said in the comments, this can be better put as,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can I know how hard to make exams and quizzes?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-03-26T11:43:59.257" LastActivityDate="2022-09-08T08:58:58.617" Title="What to do when your students are all getting A's" Tags="|grading|" AnswerCount="8" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="834" PostTypeId="2" ParentId="833" CreationDate="2014-03-26T01:56:09.790" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What should I do when I realize mid-semester that students who haven't&#xA;  learned the material are still getting A's in a core class?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the crux of the matter. I'd have no issue if a class got nothing but As. It would seem an outlier, but if the class were prescreened, i.e. a group of top performers who were all able to master the material, it's possible for them all to deserve an A. But your quote above implies otherwise. That those not quite understanding the material are still getting As. The questions on the exams need to be chosen in a way that to get them right, one must actually understand the work required. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-03-26T01:56:09.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="835" PostTypeId="2" ParentId="228" CreationDate="2014-03-26T01:59:32.450" Score="5" Body="&lt;p&gt;I learned algebra from Hungerford's undergraduate &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1111569622&quot; rel=&quot;noreferrer&quot;&gt;text&lt;/a&gt;, with &lt;strong&gt;rings before groups&lt;/strong&gt;.  This is salient for your question in a few ways.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;The examples are more concrete and more familiar.&lt;/strong&gt;  Students will&#xA;have seen $\mathbb{Z, Q, R, C}$, and polynomials and matrices.  That&#xA;plus arithmetic modulo $n$ gives plenty of examples, all easier to&#xA;explain than the symmetric or dihedral or alternating group on $n$&#xA;elements.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;The variety of examples shows the advantages of a general theory&lt;/strong&gt; that deals with&#xA;all of them.  This motivates &quot;how and why we think abstractly&quot;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;The applications come sooner for number theory and cryptography.&lt;/strong&gt;&#xA;This shows &quot;why abstract algebra is useful in practice.&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-26T01:59:32.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="836" PostTypeId="2" ParentId="798" CreationDate="2014-03-26T02:14:44.950" Score="17" Body="&lt;p&gt;I have been keeping up with the Canadian mathematics education battles from the last few years, especially in how the media discusses it. In fact, my last graduate assignment had everything to do with this issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that the &quot;Math Wars&quot; in Canada and in the United States are closely related. The debate has to do with a &quot;traditional&quot; vs, a &quot;reform&quot; mathematics education, and stems from an even longer history of mathematics education which Schoenfeld describes very well in his paper, Math Wars: &lt;a href=&quot;http://www.math.cornell.edu/~henderson/courses/EdMath-F04/MathWars.pdf&quot;&gt;http://www.math.cornell.edu/~henderson/courses/EdMath-F04/MathWars.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As presented by the media, discovery learning seems to be a method of student-centered teaching and learning, coupled together with a corresponding style of assessment and evaluation. The prevailing media image is that given a problem, a student can &quot;discover&quot; a mathematical concept on their own, with very minimal guidance from the teacher, and be able to extend their abstractions to solve other problems. The motivation behind this method is to have students take initiative in their own learning and gain experiential knowledge. Dr. Jawaharlal explains guided discovery learning in this blog post: &lt;a href=&quot;http://www.huffingtonpost.com/dr-mariappan-jawaharlal/teaching-discovery-learning_b_856463.html&quot;&gt;http://www.huffingtonpost.com/dr-mariappan-jawaharlal/teaching-discovery-learning_b_856463.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The pushback from parents is from a misguided notion of what this looks like. Parents see &quot;new math&quot; homework and don't understand what they are looking at because it is unlike their own educational experience. The idea that their children are not getting direct instruction and might end up &quot;discovering&quot; a wrong understanding of concepts leads to more confusion and backlash.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the missing element in this misunderstanding of discovery learning is the teacher. This is where classroom design has to be done very carefully - the teacher is supposed to challenge understanding and correct reasoning. Parents' notions that traditional teacher-centered learning would prevent this from happening is also misguided - it is easy to just assume that dispensing correct information from the source would allow students to get it right and move on. Follow-up in both situations is essential.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since most parents don't get to see or understand current educational research, their only sources come from what they hear in the media. As you might imagine, media is not very good at delivering a balanced message about this because they often rely on the loudest voices to inform them, and anti-reform advocacy groups will be the one to take up the pitchforks and torches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: I neglected to give the journal reference to Schoenfeld's paper:&#xA;Schoenfeld, A. H. (2004). The Math Wars. Educational Policy, 18(1), 253-286.&lt;/p&gt;&#xA;" OwnerUserId="261" LastEditorUserId="261" LastEditDate="2014-03-27T16:06:44.643" LastActivityDate="2014-03-27T16:06:44.643" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="837" PostTypeId="2" ParentId="829" CreationDate="2014-03-26T02:47:52.943" Score="10" Body="&lt;p&gt;I've mentored roughly a dozen year-long undergraduate senior research projects, and I've always used a mix of the following techiques to keep students motivated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Set clear goals, both short and long term.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students often flounder when they don't understand quite what they should be doing.  Research is hard to figure out, and students often don't know how to &quot;work&quot; on a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whenever I meet with a student, I make sure near the end of the meeting to make a clear list of things the student ought to do by next week.  Usually these have the form &quot;work out a few more examples of this&quot;, &quot;see if you can figure the same thing for all of these&quot;, &quot;try applying this to this&quot;, and &quot;see if you can write up this proof&quot;.  You should try to set at least three goals for each week, with the hope that the student will accomplish at least two of them.  When a student doesn't make progress on a goal for a week, you can spend part of the following meeting discussing the specifics of what the student needs to do to work on the goal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also helps to keep the student aware of long-term goals for the project.  Sentences like &quot;hopefully you can finish up this proof by December, so that you will have time next semester to prove two or three more theorems like this&quot; can work wonders.  Students should constantly be aware of what they need to get done if they want to have a good project.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Ask the student to keep a research journal.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student should have journal with all of their scratchwork that keeps track of what they thought about on each day of the project.  Near the beginning, you should tell the student that they should try to work on their project every day, and they should make sure to write down whatever they think about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You don't &quot;check&quot; the journal, but usually the student should have it open during your research meetings.  If you find during a meeting that you don't have much to talk about, you can ask something like &quot;Well, do you have your journal?  What have you been thinking about?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you use this approach, students will have trouble hiding it when they don't get work done, so they will tend to honestly admit that they didn't have much time this week.  When this happens, you should tell them that it's fine---it's okay if they're busy once in a while, as long as they get some good work done for the next week.  If they go two weeks in a row without doing much work, you should remind them of the goals for the project, and say that you're worried that they may have a hard time accomplishing these goals if they can't manage to get more work done.  Try not to criticize the student directly: this will sour your working relationship, which is essential to getting the project done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Try to be available between meetings.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I often try to encourage my research students to stop by my office hours for a few minutes between weekly meetings.  For example, this semester I have a student that I meet with on Wednesdays, but I usually ask her to stop by my office hours for a few minutes on Fridays and Mondays as well.  We usually only get to talk for 5 or 10 minutes during my office hours, but in many cases I'm able to help her with something she's struggling with enough to give her a few more days of productivity.  I also find out on Friday whether she's done anything since Wednesday, and I find out on Monday whether she's done anything over the weekend.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your office hours aren't convenient for this, you could try to get your student to stop by at the end of one of your classes each week, or some other convenient meeting time.  But it really does help to have short, 5-minute meetings between the weekly hour-long meetings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Be excited about the project.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Excitement is infectious, and it helps a lot to be excited about the student's project during the meetings.  You should express enthusiasm whenever the student accomplishes anything, and you should talk excitedly about potential results that the student might be able to get.  (By the way, I'm not suggesting that you fake excitement here.  You really should be excited about what the student is doing.  If you're not, maybe you should find something more interesting for them to work on.)&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-26T02:47:52.943" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="838" PostTypeId="2" ParentId="805" CreationDate="2014-03-26T03:40:17.613" Score="10" Body="&lt;p&gt;Long ago, I read some fascinating research that I have not been able to find online. The title (or part of it) was in French, though the article was in English. It started with something like y'a'til. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It discussed the disparate impact of standardized testing on women and men. A large calculus course at a university with relatively high cutoffs of SAT scores was looked at. Among women and men who had the same SAT scores, the women were earning A's and the men were earning C's. This was apparently because of the timed nature of the SAT.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking now, I find &lt;a href=&quot;http://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=6&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CFkQFjAF&amp;amp;url=http://www.stanford.edu/~niederle/NV.JEP.pdf&amp;amp;ei=x0oyU5XmMY_1oATxz4CIAg&amp;amp;usg=AFQjCNEUI1XreCAsyOXK2Esqaz1RY7B9qQ&amp;amp;sig2=dvTDoGBPaK_jb-bAQmcvKA&quot;&gt;this research article&lt;/a&gt; suggesting that timed tests have a disproportionately negative impact on women.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And why do you want to make the test harder? I allow my students to retake tests (using a new version each time), so they can show what they know. If they know what they're supposed to, I want their grade to reflect it.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-03-26T03:40:17.613" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="839" PostTypeId="1" AcceptedAnswerId="849" CreationDate="2014-03-26T05:24:59.000" Score="14" ViewCount="2213" Body="&lt;p&gt;I like students' nice questions too much (sometimes much more than their nice answers). It shows that they are thinking on what I am teaching and I am successful to make them curious about the subject. Also it allows me to explain on the subject deeper. Thus I usually encourage my students to ask as much as they can and I appreciate their questions by an extra grade for the best questions at the end. But the fact is that all questions are not really good for example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I explained the answer before.&lt;/li&gt;&#xA;&lt;li&gt;It is not related to the current subject.&lt;/li&gt;&#xA;&lt;li&gt;They didn't think on the question by their own.&lt;/li&gt;&#xA;&lt;li&gt;They didn't understand the definitions correctly.&lt;/li&gt;&#xA;&lt;li&gt;And almost all other reasons which makes a question silly. &lt;/li&gt;&#xA;&lt;li&gt;MO and MSE criterions about an inappropriate question could be useful. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I want to find a good way for decreasing the silly questions without decreasing the number of questions or the students' motivation for asking many questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What is a wise reaction to a silly question?&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="143" LastEditDate="2014-10-17T17:56:15.500" LastActivityDate="2014-10-21T21:15:23.990" Title="What is a wise reaction to a silly question?" Tags="|classroom-management|general-pedagogy|class-participation|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="841" PostTypeId="2" ParentId="839" CreationDate="2014-03-26T06:44:51.820" Score="7" Body="&lt;p&gt;For off-topic questions or just questions that we can't devote the time to answer right away, I have a devoted 'question wall'. The students have access to post-its and use them to post any question to the wall -- silly, content-focused, it doesn't matter to me. The result is if a student interrupts with a silly question I can quickly reply &quot;Put it on a post-it&quot; and we're back on track with the lesson.&lt;/p&gt;&#xA;" OwnerUserId="104" LastActivityDate="2014-03-26T06:44:51.820" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="842" PostTypeId="2" ParentId="643" CreationDate="2014-03-26T07:42:57.507" Score="5" Body="&lt;p&gt;There is an emergent literature around peer tutoring used for mathematical instruction in urban settings.&lt;/p&gt;&#xA;&lt;p&gt;A nice article to begin with is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Walker, E. N. (2007). The structure and culture of developing a mathematics tutoring collaborative in an urban high school. &lt;em&gt;The High School Journal, 91&lt;/em&gt;(1), 57-67. &lt;a href=&quot;http://muse.jhu.edu/journals/high_school_journal/v091/91.1walker.html&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Abstract:&lt;/strong&gt; This conceptual article describes a model of a school-based, student-led initiative that uses peer tutoring to address underachievement in mathematics. The model is three pronged: a) it suggests a site-based approach to building on existing student excellence in mathematics to drive improved student mathematics achievement; b) it seeks to address the lack of teacher knowledge about urban students and their mathematics understanding; and c) it aims to deepen existing mathematics knowledge, confidence and interest among high school students. In the article, I share analyses of the interactions among tutors, tutees, advisors, and teacher; the mathematical discourse within those interactions; and the hierarchical and collaborative relationships that emerged over time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The piece above fits into a larger discussion of building &lt;em&gt;mathematics communities&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For an article in this vein, see:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Walker, E. N. (2006). Urban high school students' academic communities and their effects on mathematics success. American Educational Research Journal, 43(1), 43-73. &lt;a href=&quot;http://aer.sagepub.com/content/43/1/43.full.pdf&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Finally, you can find an entire book by &lt;a href=&quot;http://www.tc.columbia.edu/faculty/walker/&quot; rel=&quot;noreferrer&quot;&gt;Erica Walker&lt;/a&gt; entitled &lt;a href=&quot;http://store.tcpress.com/0807753289.shtml&quot; rel=&quot;noreferrer&quot;&gt;Building Mathematics Learning Communities&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As you can see from the description there, the book calls for mathematics teachers and students to collaborate in building these communities, and identifies three fundamental principles:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Urban students want to be a part of academically challenging environments.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teachers and administrators can inadvertently create obstacles that thwart the mathematics potential of students.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Educators can build on existing student networks to create collaborative and non-hierarchical communities that support mathematics achievement.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I believe that details surrounding the third principle could be of help in answering your question here.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-26T07:42:57.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="843" PostTypeId="2" ParentId="750" CreationDate="2014-03-26T08:04:11.410" Score="8" Body="&lt;p&gt;A good math curriculum to check out for Pre-K to K is &lt;em&gt;Big Math for Little Kids&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find some information about it in &lt;a href=&quot;http://www.scholastic.com/teachers/article/early-childhood-today-interviews-dr-herb-ginsburg-math-education-young-children&quot; rel=&quot;noreferrer&quot;&gt;this interview&lt;/a&gt; with co-developer &lt;a href=&quot;http://www.tc.columbia.edu/academics/?facid=hpg4&quot; rel=&quot;noreferrer&quot;&gt;Herb Ginsburg&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ginsburg is also working on software that will include problems for the range you have specified. In particular, he and others (including doctoral students at Teachers College Columbia University) are working on a set of computer activities called &lt;a href=&quot;http://www.tc.columbia.edu/mathemantics/index.asp?Id=Home&amp;amp;Info=About%20MathemAntics&quot; rel=&quot;noreferrer&quot;&gt;MathemAntics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of his students have also gone on to create their own apps. Two worth mentioning are &lt;a href=&quot;http://www.teachley.com/&quot; rel=&quot;noreferrer&quot;&gt;Teachley&lt;/a&gt; and &lt;a href=&quot;http://tiggly.com/&quot; rel=&quot;noreferrer&quot;&gt;Tiggly&lt;/a&gt;. Although there are many apps and instances of software put out for the teaching of Pre-K students, it is very rare to find cases in which the development of these programs is based on actual research about early childhood mathematics. In this respect, the links above are exceptions to the norm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, if you are a bigger fan of book-learning, I must recommend the classic &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0906212391&quot; rel=&quot;noreferrer&quot;&gt;Mirror Puzzle Book&lt;/a&gt; by Marion Walter. (This is the same Walter who co-wrote, with Stephen Brown, &quot;The Art of Problem Posing.&quot;) &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In response to the OP's &lt;strong&gt;edit&lt;/strong&gt;, here is a concrete example. The source of this is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ginsburg, H. P. (2009). Early mathematics education and how to do it. &lt;em&gt;Handbook of child development and early education: Research to practice&lt;/em&gt;, 423-428.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/jkQAN.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The description continues:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/0Iems.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Final note:&lt;/strong&gt; It is important to realize that just allowing children to engage in free-play is not the best way to promote mathematical thinking. There should be some amount of structure and scaffolding by the teachers, which requires a real knack for observation and interaction. With regard to the latter, those who teach mathematics to Pre-K students may wish to familiarize themselves with the clinical interview technique. This approach is in keeping with the general thrust of Piaget's work (another good source for problems to give to students of a young age). For more on the clinical interview, see:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ginsburg, H. (1997). &lt;em&gt;Entering the child's mind: The clinical interview in psychological research and practice&lt;/em&gt;. Cambridge University Press.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-04-04T18:06:16.283" LastActivityDate="2014-04-04T18:06:16.283" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="844" PostTypeId="2" ParentId="833" CreationDate="2014-03-26T09:13:41.493" Score="7" Body="&lt;p&gt;If all students gets good grades, this is a bad thing for the real good students (who will, at least for their grades, not be recognized). It is also bad for poor students since the get the feeling that the understood everything and don't have to repeat the material of the course again (&quot;Hey, nothing is better than an A. Why should I repeat that?&quot;). This seems to be true, especially if someone from the next courses is complaining.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, in order to answer&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can I know how hard to make exams and quizzes?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would suggest to put a (very) hard question on the test; it should be something where you combine two topics of the course and where you can say: &quot;If someone is able to this, he/she has a deep understanding and my respect.&quot; - In this case, they deserve the good grade. &lt;em&gt;Even if&lt;/em&gt; no student was able to answer the hard question, you can tell the students that there are able to perform some basic things, but should repeat the course in order to have a deeper understanding which will be needed for the next courses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I good rule of thumb (which I often use) is maybe the following: Construct questions in such a way that&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If the students are able to perform standard questions which they have seen several times during the course and where the way you ask is completely clear (e.g., &quot;take the derivative of the following (not sophisticated) function&quot;). If there are a many questions where students have to calculate something, define the limit where a student passes the exam as that points minus a few points (&quot;It's okay to have small mistakes in calculations&quot;).&lt;/li&gt;&#xA;&lt;li&gt;Put in &lt;strong&gt;one&lt;/strong&gt; very hard question. If someone fails to solve the question completely, he/she should not be able to get an A (But, if there is at least a small idea and everything was solved perfectly, an A should be possible).&lt;/li&gt;&#xA;&lt;li&gt;The rest of the exam should contain questions with either a known way of questioning (but more sophisticated example) or questions with an unknown way of questioning which can be easily performed in a straight-forward way.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;An exam more or less constructed in this will lead to a bigger scattering of grades (leading to have deserved good grades and to let students know how &quot;good&quot; they are compared to others which then &lt;em&gt;maybe&lt;/em&gt; leads to repeating the course).&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-26T09:13:41.493" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="846" PostTypeId="2" ParentId="833" CreationDate="2014-03-26T09:30:24.760" Score="18" Body="&lt;p&gt;The part of this question that raises red flags for me is the line:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;However, many timesit [sic] is good to have a wide spread of grades.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Why?  This seems backward reasoning to me wherein you know the distribution of grades that you want to give and are figuring out how to design the test to fit that distribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your assessment should be &lt;strong&gt;criterion based&lt;/strong&gt;.  Look at your learning objectives for the course and design the assessment accordingly.  For example, &lt;a href=&quot;http://mathsnotes.math.ntnu.no/mathsnotes/show/scheme+of+work+2011&quot;&gt;here&lt;/a&gt; is my scheme of work for one course.  Note in the introduction how the skills are knit to the grades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In designing the exam, the question that I have always before me is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;What competence can I use this question as evidence for?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also imagine the following scenario.  Suppose that a year later, another lecturer storms in to my office and demands to know why I gave a particular student a particular grade.  What will I answer?  That they scored 52 points?  That they hit a particular number of standard deviations away from the mean?  Or that in their exam then they demonstrated that they met the criteria for the given grade?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when I have complete freedom to set the exam and the assessment criteria, then I design my exam by these principles.  Each indecomposable unit on the test will be used as evidence for their grade, but each such unit comes with a &lt;em&gt;maximum&lt;/em&gt; grade that it can be used for.  So when assessing for an E, I ask &quot;Has this student shown sufficient evidence that they deserve an E?&quot; meaning that I count the number of answers rated &quot;E&quot; or above.  If this is sufficient, they have secured an E.  Next, I look at the &quot;C&quot; grade.  Any answers rated &lt;em&gt;only&lt;/em&gt; E are now dropped.  Note that some answers &lt;em&gt;could only get an E&lt;/em&gt; so these are automatically dropped.  I do the same at the &quot;A&quot; grade.  For &quot;B&quot; and &quot;D&quot;, I take these to mean &quot;almost the grade above&quot;.  So a &quot;B&quot; is a &quot;low A&quot;, not a &quot;high C&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, &lt;a href=&quot;http://mathsnotes.math.ntnu.no/mathsnotes/files/TMA4145h2011eng.pdf&quot;&gt;here's an exam&lt;/a&gt; designed on these principles.  I don't say that it is perfect.  Note the first question &lt;em&gt;only&lt;/em&gt; asks about definitions and statements.  So the answers to this question can only count for an E, but I give them lots of opportunities to show that they have achieved that level.  The other questions basically allow for Cs and As (one part of each question can be given an A, I'll leave it as an exercise to work out which part).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, if you are worrying about the distribution of grades then you are worrying about the wrong thing.  If you are worrying that the grades are not knit to the desired competencies of the course, then you are worrying about the right thing and the solution is obvious: test for the desired competencies, not for anything else.  And &lt;strong&gt;never&lt;/strong&gt; grade by just adding up points.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-03-26T09:30:24.760" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="847" PostTypeId="2" ParentId="833" CreationDate="2014-03-26T12:25:34.703" Score="4" Body="&lt;p&gt;When I design a test, I will typically designate questions (using the homework in the book as a guide) as basic skills, moderate, moderate-difficult, and separates A's from B's.   If I am giving a 10 question test I will typically give 1 basic skill question, 4 moderate questions, 4 moderate difficult questions, and 1 separate A's from B's question.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I see I made a test too difficult, then I will offer to let the students make corrections to earn back 1/4 of the points they lost.  There is no partial credit on corrections, it's either right or wrong. If I find that I made a test too easy (non-normal distribution) then the next test I will either remove the basic skill question and replace it with a moderate difficult.  &lt;/p&gt;&#xA;" OwnerUserId="203" LastActivityDate="2014-03-26T12:25:34.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="848" PostTypeId="2" ParentId="294" CreationDate="2014-03-26T12:39:54.520" Score="5" Body="&lt;p&gt;an old one, but still a good one&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Courant &amp;amp; Robbins, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0195105192&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;What is Mathematics?&lt;/em&gt;&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It explains what mathematics is all about, for someone who may not know, but is willing to spend the time to learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In getting the link, I was delighted to discover that the latest edition is &quot;Revised by Ian Stewart&quot;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2014-03-26T12:39:54.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="849" PostTypeId="2" ParentId="839" CreationDate="2014-03-26T12:41:38.483" Score="33" Body="&lt;p&gt;&lt;em&gt;Every&lt;/em&gt; student question should be treated like a gift. It gives insight into student's thinking. Even a disrepectful question is an opportunity for you to &lt;em&gt;teach&lt;/em&gt;, except that in such a case you don't teach math. There is no such thing as a &quot;only math&quot; teacher. Every teacher teaches life just by standing in front of the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say it is actually &lt;em&gt;wrong&lt;/em&gt; to qualify questions as &quot;silly&quot; and &quot;not silly&quot;. To paraphrase, silliness is in the eye of the beholder. Surely the student doesn't think the question was silly, does he? Surely, you wouldn't answer with &quot;this is a silly question&quot;, would you?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you feel that a question is silly, you should analyze it further, as you do in your question. And the response then depends on how the question is categorized. In your cases:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;I explained the answer before:&lt;/em&gt; if you have time, explain it again, otherwise just be nice and apologize for not having time. Only us teachers are crazy enough to think that people learn by hearing, seeing or trying just once. Are piano lessons repetitive? Is martial art training repetitive? How about learning a new langauge, are you told each word just once?&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;It is not related to the current subject:&lt;/em&gt; the thing to worry about is &lt;em&gt;why&lt;/em&gt; the student thinks it is related to the current subject. And even if it isn't related to the current subject, so what? Only us teachers are crazy enough to think that people learn by doing one thing for a long time. That's boring as hell. So just answer the question, or be nice if you don't have time.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;They didn't think on the question by their own.&lt;/em&gt; That's completely irrelevant, the thing to ask is &lt;em&gt;why&lt;/em&gt; are they asking anyway? If they want to know, then who cares if they didn't think of the question. If they don't want to know, well maybe they're just sucking up to you, but that would be a different ME question.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;They didn't understand the definitions correctly.&lt;/em&gt; That is &lt;em&gt;not&lt;/em&gt; a silly question! You have just discovered that you failed to explain the definitions correctly. Go back and fix the damage &lt;em&gt;you&lt;/em&gt; created.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;And almost all other reasons which makes a question silly.&lt;/em&gt; There are no silly questions as far as students are concerned.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;MO and MSE criterions about an inappropriate question could be useful.&lt;/em&gt; A question is only inappropriate if it is insulting, generally inappropriate, or intentionally designed to cause trouble. If a student asked me whether I believed in God, I would never say &quot;that's inappropriate&quot;. I would use the opportunity as best as I could. For example, I would say: &quot;Ah, but that's not something you need to believe in. Leibniz already proved God existed. Here is how ...&quot; And we'd have an excellent lesson in logic that the students would actually enjoy and pay attention to.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are no silly questions.&lt;/p&gt;&#xA;" OwnerUserId="165" LastEditorUserId="165" LastEditDate="2014-03-26T16:21:19.673" LastActivityDate="2014-03-26T16:21:19.673" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="852" PostTypeId="1" AcceptedAnswerId="854" CreationDate="2014-03-26T13:05:18.600" Score="21" ViewCount="8098" Body="&lt;p&gt;This is a typical problem in my undergraduate Calculus I class.  Many of these freshmen come in having made an A in their High-School Calculus class, and believe they know everything they need to about Calculus.   They love to try to challenge my explanations or they always have a &quot;better way&quot; and they feel the need to share it with the rest of the class interrupting my lecture. Most of the time I can talk to the student after class and explain to them that the way I teach will set them up for success in Calculus II and III, but I have had one student in particular who refuses to admit that he can learn anything from me.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked him if he would like to me to assign him a more challenging homework set instead of the current class set, but then he accused me of trying to fail him (he currently has an A in the class).  I also suggested that he switch to another class, but mine is the only one that fits in his schedule.  What techniques can I use help this student understand there is always something that you can learn even if you know it all?      &lt;/p&gt;&#xA;" OwnerUserId="203" LastEditorUserId="143" LastEditDate="2014-09-21T10:24:31.543" LastActivityDate="2014-09-21T10:24:31.543" Title="What is a method you use to handle the students who thinks they &quot;know it all&quot;?" Tags="|undergraduate-education|calculus|classroom-management|general-pedagogy|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="853" PostTypeId="2" ParentId="750" CreationDate="2014-03-26T13:07:05.853" Score="10" Body="&lt;p&gt;To supplement the &quot;games not problems&quot; answer above (I love that answer): Certain games provide interesting opportunities for discovery...even if you don't know anything about the rules!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example is &lt;a href=&quot;http://www.tantrix.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tantrix&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/EXgu9.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The tiles feel good in one's hands, and most preschoolers will spontaneously start tiling if given some of these. If the tiling continues, automatically paths and loops form. It is possible that the child will even try to form monochromatic paths and loops. My three-year-old was alternately tiling with these the other day, and pretending that they were cookies. As the child gets older, there are other great activities to do with these tiles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although it is controversial to say, certain video games are good practice for mathematical rule-following for preschoolers. A simple example is &lt;a href=&quot;http://www.mathsisfun.com/games/labyrinth-puzzle.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this maze game&lt;/a&gt;. The other games on the mathisfun.com site are quite good for engaging little ones. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a rule, I try to be on the lookout for things that encourage &quot;combinatorial play&quot;. These are things that have multiplicity and can be combined and recombined and promise discovery by doing so. Classic lego blocks are the example everybody knows, but found items with combinatorial richness are also good.&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2014-03-26T13:07:05.853" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="854" PostTypeId="2" ParentId="852" CreationDate="2014-03-26T14:02:21.947" Score="21" Body="&lt;p&gt;You should avoid a &quot;I am smarter than you are&quot; war with the student, the teacher must be above that. (You're already avoiding it, I am saying so for the benefit of the Internet.) In Japan you could just smile mysteriously, but I suspect we're talking about the Western world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first and most important thing to resolve is this: &lt;em&gt;Is your student right?&lt;/em&gt; If his or her suggestions are actually improvements over what you do, or if they can lead to improvements, then you should suspend your ego and use them. And give the student proper credit, and figure out how not to inflate his ego beyond the size of the solar system. Teaching is tough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming your student is actually suggesting irrelevant modifications of teaching material, the nastiest thing you can do to your student is to relativize his little world:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;There may not be the best way to explain or teach any given topic in math.&lt;/em&gt; What is best depends on previous experience, background knowledge, personal taste etc. You can tell the student that &lt;em&gt;he&lt;/em&gt; is correct to think that for &lt;em&gt;him&lt;/em&gt; another way is best, although he should not presume that everyone else has exactly the same background as he does.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;There is nothing wrong with knowing how to do a thing in two different ways.&lt;/em&gt; The student already knows one way to attack the topic. You're offering another for him to compare. He should therefore listen, and in the end think not just in terms of &quot;absolutely better than&quot; but also in terms of &quot;what are the pros and cons of each way&quot;. (See first point.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Regarding interruption of your classroom, you can use your authority to tell the student that such interruptions should be replaced by off-line conversations.&lt;/p&gt;&#xA;" OwnerUserId="165" LastActivityDate="2014-03-26T14:02:21.947" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="855" PostTypeId="2" ParentId="833" CreationDate="2014-03-26T14:21:39.380" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;In my current university, the engineering department requested us to make the grades lower because the engineers couldn't do calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now, I'm no expert in maths, making exams or anything, but why don't you just ask the engineering department what the students need to be able to do there exactly? Maybe even take some questions from there? &#xA;If engineering is one possible followup class and they fail there miserably despite good grades, engineering should be the standard you should hold your students against. &lt;/p&gt;&#xA;" OwnerUserId="361" LastActivityDate="2014-03-26T14:21:39.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="856" PostTypeId="1" AcceptedAnswerId="860" CreationDate="2014-03-26T14:40:20.810" Score="15" ViewCount="2323" Body="&lt;p&gt;It often happens that students answer (partly) a questions in an exam (lets assume this part is okay), but then add something completely off-topic or something very wrong in their answer (see examples below).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How would you deal with such an issue and why?&lt;/strong&gt; Should one ignore the wrong things and only grade everything which is okay? Or should I bring to account that the additional part of the answer shows that he/she hasn't understood much?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a few examples I have in mind:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Assume the question is split into two parts. The answer for the first part was okay, but then on the second part the student discloses big deficits in their knowledge (e.g., he/she uses the fact that $\mathbb{R}$ is compact; or he/she calculates $(A+B)^{-1}=A^{-1}+B^{-1}$, etc., i.e., wrong arguments where you see that not much was understood).&lt;/li&gt;&#xA;&lt;li&gt;Assume someone answered a question right, but then add something is is which has nothing to do with the rest of the question (once when I was a TA, it happens that a question was just an exercise, the student copied the solution which was given in the exercise, but he did not finish when he arrived at the end and he also copied some part of the next exercise into his exam solution).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-08T08:42:05.807" LastActivityDate="2014-04-08T08:42:05.807" Title="How to deal with answers containing completely off-topic/random/very wrong arguments?" Tags="|undergraduate-education|secondary-education|exams|grading|students-mistakes|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="858" PostTypeId="1" AcceptedAnswerId="932" CreationDate="2014-03-26T14:58:56.720" Score="28" ViewCount="743" Body="&lt;p&gt;Educators are constantly evaluated by at least the following methods:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Anonymous evaluations collected from students by the school&lt;/li&gt;&#xA;&lt;li&gt;Anonymous evaluations posted online by students&lt;/li&gt;&#xA;&lt;li&gt;Grades or pass-rates of the students in the class&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Experienced educators have all learned to &lt;strong&gt;not read too much into any of these&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, if you would like to evaluate your own performance, what is an effective method? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think some good answers may be specific to mathematics, otherwise I would post the question elsewhere.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-01-26T19:32:37.190" Title="What is a healthy and effective way for a math educator to evaluate his or her performance?" Tags="|teacher-evaluations|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="859" PostTypeId="2" ParentId="746" CreationDate="2014-03-26T15:32:23.693" Score="3" Body="&lt;p&gt;I've discussed this several times with other education researchers and it does not seem easy. A learning style (or maybe thinking style) should scarcely depend on the knowledge someone has or specifities of a task or topic. It should rather be a personal preference. As such, it is very hard to describe. If you choose a geomtrical way to learn - is it due to your abilities in geometry, due to the fact that you are learning Pythagoras' theorem or due to your style? To surely identify such styles, you would have to choose tasks with different possible solutions, people with comparable knowledge in the different topics and then try to see if there is a pattern different from random. To my knowledge, that doesn't exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, there is a math educator whose research is mainly on thinking styles. Her name is Rita Borromeo-Ferri and unfortunateley she mostly writes in German. Fortunately, I found one paper in her literature list that is in English:&#xA;&lt;a href=&quot;http://link.springer.com/article/10.1007%2Fs13138-010-0009-8&quot; rel=&quot;nofollow&quot;&gt;Borromeo Ferri, Rita (2010). On the influence of mathematical thinking styles on learners' modelling behaviour. In: Journal für Mathematikdidaktik, 31 (1), S. 99-118&lt;/a&gt;. &#xA;Maybe this will help you.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="376" LastEditDate="2015-12-18T11:03:57.337" LastActivityDate="2015-12-18T11:03:57.337" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="860" PostTypeId="2" ParentId="856" CreationDate="2014-03-26T15:40:18.550" Score="12" Body="&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/vOJd6.png&quot; alt=&quot;Grading a random argument.&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is my interpretation of your question: The student in my above grading example achieved the correct answer, and therefore feels cheated at losing points for the extra (incorrect) simplification. Should we be sympathetic to the student's complaint and give full credit, excusing the extra work done?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My Best Argument Against Taking Off a Point&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you take off a point, the student learns his or her lesson, and you find a later quiz with the following answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/n9Fj3.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you are in trouble. If you take off points for not completing the question, the student feels cheated again, and believes that you are just being arbitrary. They were playing it safe since they got in trouble simplifying before, and the question didn't say to simplify! This kind of thing can cause a student to &quot;check out&quot; of the class entirely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My Best Argument for Giving Half Credit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do not take off a point, that student loves you and takes your class again next semester. They turn in a quiz with the following answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Z6LBX.png&quot; alt=&quot;Lesson not learned.&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You would love to curse that student's college algebra instructor for not emphasizing taking care with arithmetic, but that college algebra instructor is &lt;strong&gt;you&lt;/strong&gt;. So you curse yourself.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;In conclusion,&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've felt both of these bad outcomes. The second bad outcome feels much worse. I'd stick with &lt;strong&gt;always taking off points for incorrect things&lt;/strong&gt;. The student's future instructors will thank you -- and the student's future instructor might be you!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-26T15:40:18.550" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="861" PostTypeId="2" ParentId="856" CreationDate="2014-03-26T15:43:26.247" Score="8" Body="&lt;p&gt;I would absolutely take points off in those situations.  If I ask &quot;what's 2+2?&quot; and one student responds &quot;4&quot; and the other responds &quot;4, because $\pi$ is rational&quot;, the first answer is definitely better than the second, so it should be scored higher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a more idealistic level, my job as a grader is to determine the student's understanding of the material as precisely as I can.  If the student has revealed that he has a big gap in his knowledge, or that he's just memorizing someone else's words without the slightest understanding of what's being said, then his score should be lowered. One could argue that the student who answered &quot;4&quot; might have also thought that this was implied by the rationality of $\pi$, yet simply didn't write it, but even the fact that he had the good sense not to write this nonsense means he's a little suspicious of it too, and that's worth something.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, in the second case you listed, where a student bleeds memorized verbatim solutions together, I would be inclined to take off more than the first, since there's a hint of dishonesty there (or at least bad faith).  I don't think demanding an honest effort towards a genuine understanding the material is unreasonable.  Conversely, not enforcing this with grading tells the student it's okay to mindlessly memorize everything, even if the teacher doesn't like it.&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2014-03-29T21:02:17.937" LastActivityDate="2014-03-29T21:02:17.937" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="862" PostTypeId="2" ParentId="6" CreationDate="2014-03-26T15:47:12.230" Score="6" Body="&lt;p&gt;In my last couple years of teaching I did not grade or count as any credit towards a grade homework completed as an assignment.  I did collect it and mark it as COMPLETED ON TIME (COT).  If a student had completed on time all homework assignments they were given the option of retaking the test after an additional review assignment was completed.  At the end of the grading period if all homework is COT they were given a perk to be negotiated such as dropping a quiz or test grade or bending the grading scale slightly.  The availability of homework help should not be a detriment to doing the homework.  If the student does the homework they should do better on tests and assessments of any kind.  If they cheat themselves during the homework phase and do poorly on assessments the question becomes-&quot;why didn't you know you were having trouble when you did the homework?&quot;&lt;/p&gt;&#xA;" OwnerUserId="336" LastActivityDate="2014-03-26T15:47:12.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="863" PostTypeId="2" ParentId="856" CreationDate="2014-03-26T15:47:48.930" Score="4" Body="&lt;p&gt;I usually give credits for everything that is right, ignoring what ist wrong, &lt;strong&gt;iff&lt;/strong&gt; things are not contradictory (I wouldn't give credits for &quot;answer is right or wrong&quot;).&#xA;If things are contradictory (e.g. two solutions given), I mark the first one only. This is told to my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not happy with that as I sometimes would rather like to downgrade people writing useless stuff. However in Germany (where you also work) some courts decided that in multiple-choice, additional information may not lead to less points (see &lt;a href=&quot;http://www.justiz.nrw.de/nrwe/ovgs/ovg_nrw/j2008/14_A_2154_08urteil20081216.html&quot; rel=&quot;nofollow&quot;&gt;here, line 72&lt;/a&gt;).&#xA;In some sense, students  have a right for guessing, as they practiced in school. You can't prevent it anyway.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="199" LastEditDate="2014-03-28T09:27:44.140" LastActivityDate="2014-03-28T09:27:44.140" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="864" PostTypeId="2" ParentId="833" CreationDate="2014-03-26T16:03:06.240" Score="2" Body="&lt;p&gt;My exams usually are 4 to 6 questions (it depends on the amount of material covered, duration is designed for give or take one and a half hour, we schedule two hours to account for stragglers, people who are just slower, ... and the inevitable miscalculation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often the exam is a simple question (worth some 25 points) to check basic understanding of definitions and &lt;em&gt;trivial&lt;/em&gt; application of the concepts. Often 3 to 5 short questions, graded mostly yes/no each. The rest of the questions range from moderately easy to moderately difficult, with a difficult question thrown in to &quot;differentiate A from B&quot; as was said here. To give leeway for minor mistakes, there are usually 120 points total (grades are 0 to 100 here).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grades for each question (except the &quot;did you even &lt;em&gt;glance&lt;/em&gt; at the material?&quot; one) are typically distributed with a peak on &quot;very little&quot; and another one at &quot;good&quot; spreading to &quot;perfect.&quot; If it turns out like that, I'm satisfied. I'd like it more with just one peak at &quot;good,&quot; but so is life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously much depends on selection of questions. Particularly critical is to make sure one doesn't repeat questions from previous years, students &lt;em&gt;will&lt;/em&gt; have them (and solutions), whatever the policy of publishing them might be. It might be possible to alter a question enough to make it &quot;not the same&quot; but still &quot;covers the same stuff, and isn't much harder/easier,&quot; but  that has to be designed carefully. You can ask variants (or extensions) on work done in homework, gives an incentive to do it carefully. Works best if announced beforehand.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-26T16:03:06.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="865" PostTypeId="1" CreationDate="2014-03-26T16:25:06.423" Score="7" ViewCount="317" Body="&lt;p&gt;I'm interested to read writings about mathematics education written by famous mathematicians. By famous mathematicians, I mean roughly anybody with a result or object named after them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure what the format of answers should be, but perhaps one link per answer would be good, so that votes and discussions can be on individual articles, and perhaps each answer could include a short excerpt from the article to avoid having just a long list of links.&lt;/p&gt;&#xA;" OwnerUserId="369" LastEditorUserId="369" LastEditDate="2014-04-12T12:38:26.913" LastActivityDate="2014-04-12T12:38:26.913" Title="Writings about mathematics education by famous mathematicians" Tags="|mathematical-pedagogy|" AnswerCount="3" CommentCount="12" ClosedDate="2014-03-26T17:38:13.487" ContentLicense="CC BY-SA 3.0" />
  <row Id="866" PostTypeId="2" ParentId="865" CreationDate="2014-03-26T16:30:02.833" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://pauli.uni-muenster.de/%7Emunsteg/arnold.html&quot; rel=&quot;noreferrer&quot;&gt;On teaching mathematics&lt;/a&gt; by V. I. Arnold&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun).&lt;/p&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;p&gt;The theorem of classification of surfaces is a top-class mathematical achievement, comparable with the discovery of America or X-rays. This is a genuine discovery of mathematical natural science and it is even difficult to say whether the fact itself is more attributable to physics or to mathematics. In its significance for both the applications and the development of correct Weltanschauung it by far surpasses such &amp;quot;achievements&amp;quot; of mathematics as the proof of Fermat's last theorem or the proof of the fact that any sufficiently large whole number can be represented as a sum of three prime numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="369" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-26T16:30:02.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="867" PostTypeId="2" ParentId="858" CreationDate="2014-03-26T16:31:56.147" Score="18" Body="&lt;p&gt;Though faculty are often skeptical of student evaluations, research suggests that student evaluations &lt;em&gt;are&lt;/em&gt; an effective tool for evaluating teaching.  Here is a link to a meta-study that surveys the results of approximately 1500 other studies on student evaluations:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://eric.ed.gov/?id=ED402338&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cashin, &amp;quot;Student Ratings on Teaching: The Research Revisited&amp;quot;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here are some of the major conclusions of the paper:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Correlation to Exam Scores&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In courses with multiple sections and a common exam, there is between a 0.3 and 0.5 correlation between an instructor's evaluation scores and the students' scores on the common exams.  The paper notes that it is rare to ever find correlations above 0.5 in the social sciences when studying complex phenomena.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Correlation to Instructor Self-Ratings&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;When instructors are asked to rate their own teaching effectiveness, the answers they give have approximately a 0.29 correlation with student evaluations.  In addition, when instructors are asked to rate how their effectiveness varied over different courses they have taught, &lt;em&gt;&lt;strong&gt;the instructor's answers have roughly a 0.45 correlation with student variation in evaluations&lt;/strong&gt;&lt;/em&gt; across those courses.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Correlation to Ratings of Administrators and Colleagues&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;When instructors are rated by either their colleagues or by administrators, the scores they receive have between a 0.45 and 0.7 correlation with student evaluation scores.  Of course, colleagues and administrators might be basing their score on student evaluation scores that they have seen.&lt;/p&gt;&#xA;&lt;p&gt;However, ratings by trained classroom observers (who have not seen student evaluation scores) also have 0.5 correlation with student evaluation scores.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Further Discussion&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Based on this research, I think that it &lt;em&gt;is&lt;/em&gt; reasonable to take student evaluation scores into account when analyzing the effectiveness of your own teaching.  From a particular instructor's point of view, the 0.45 correlation highlighted above has the most relevance: it means that you can tell &lt;em&gt;which courses&lt;/em&gt; you did a good job in based on student evaluations.&lt;/p&gt;&#xA;&lt;p&gt;Of course, there are some other effects that need to be taken into account.  Classes full of highly motivated students tend to give higher ratings than classes of reluctant students.  Students in advanced courses also tend to give higher ratings than students in introductory courses.  If your abstract algebra students consistently rate you higher than your calculus students, that doesn't necessarily mean that you're doing a better job in abstract algebra.&lt;/p&gt;&#xA;&lt;p&gt;Finally, it should be pointed out in any discussion of evaluating teaching that there is no universally accepted definition of good teaching.  You may perceive advantages or disadvantages in your teaching style that the students do not see, and there's no objective way to determine whether you are right.  Ultimately, it's part of every teacher's job to form their own opinions on what constitutes good teaching.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit: Who graded the exams?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;dtldarek asks below whether the correlation between student evaluations and student exam scores might be due to instructors grading their own class's exams.&lt;/p&gt;&#xA;&lt;p&gt;Cashin's study cites two other meta-studies for the results on exam scores: one by &lt;a href=&quot;http://www.jstor.org/stable/1170209&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cohen&lt;/a&gt; and one by &lt;a href=&quot;http://www.jstor.org/stable/40195923&quot; rel=&quot;nofollow noreferrer&quot;&gt;Feldman&lt;/a&gt;.  I haven't looked through the Feldman study, but Cohen's meta-study analyzes several methodological features, including who grades the exams.  Of the 68 studies that Cohen analyzes, 10 had each teacher grade their own class's exam, 28 had some external evaluator grade all of the exams, and the remaining 30 had some other system (e.g. each instructor grades one question from all of the exams).  From Cohen:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The only other methodological variable that correlated significantly with effect size was evaluation bias. The correlation between ratings and achievement was large when an external grader was used or when each instructor evaluated one part of the test for all students. For 10 studies in which achievement tests were evaluated by students' own instructors, the effect size was much smaller.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That is, the correlation between good evaluations and good exam scores is stronger when the exams are &lt;em&gt;not&lt;/em&gt; graded by the teacher themselves.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2023-01-26T19:32:37.190" LastActivityDate="2023-01-26T19:32:37.190" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="868" PostTypeId="2" ParentId="865" CreationDate="2014-03-26T16:33:26.580" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://arxiv.org/pdf/math/0503081.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Education&lt;/a&gt; by Bill Thurston&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The typical response of American adults, on meeting a mathematician, is one of dismay. They apologetically recall the last mathematics course they took, which is usually the one where they lost their grip on the subject matter.&lt;/p&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;p&gt;I remember as a child, in fifth grade, coming to the amazing (to me) realization that the answer to 134 divided by 29 is 134/29 (and so forth). What a tremendous labor-saving device! To me, ‘134 divided by 29’ meant a certain tedious chore, while 134/29 was an object with no implicit work. I went excitedly to my father to explain my major discovery. He told me that of course this is so, &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; divided by &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are just synonyms. To him it was just a small variation in notation. [...] Mathematics is full of this kind of thing, on all levels. It never stops.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="369" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-26T17:15:49.870" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="869" PostTypeId="2" ParentId="833" CreationDate="2014-03-26T16:36:27.537" Score="16" Body="&lt;p&gt;You wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My problem is that I ... end up giving too good of grades at the beginning, and then face the task of either trying to lower their grades by being much tougher or not fulfilling departmental expectations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My advice is that, as a practical matter, things turn out much better when one is tougher in the beginning of a course, easing off if necessary only later on, rather than starting out easy and then hoping to get tough. Students are grateful when a hard course gets easier, but feel cheated when an easy course becomes difficult. Isn't this basic human nature? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should set high standards initially---you may be surprised that many students meet them. And if they don't, then it is easy to relax a bit later when the situation is more clear. But it is extremely difficult to raise standards after the students realize that you probably don't really mean it; they will call your bluff. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I often use a quiz-every-lecture format for assessment, and when I do, I make sure to announce it as quiz-every-lecture and start with a quiz at the second lecture, and really have one every lecture for the first few weeks, before I start relaxing to every-other-lecture or so (which was all along my real intention), once the culture has been established and the students are motivated to stay on top of their homework in preparation. If I were relaxed at first, then I think the students wouldn't be motivated to do the work.&lt;/p&gt;&#xA;" OwnerUserId="90" LastEditorUserId="90" LastEditDate="2014-03-26T18:01:56.527" LastActivityDate="2014-03-26T18:01:56.527" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="870" PostTypeId="2" ParentId="856" CreationDate="2014-03-26T16:50:07.837" Score="7" Body="&lt;p&gt;When I first started teaching and I was grading homework or exam solutions, I would often write something like, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I don't understand your argument here.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;at a place where the student's argument was incoherent and made no sense (and they would get a correspondingly low mark). I think I probably wrote that phrase out of a desire to be polite or to protect the student's feelings. I had felt that it would be too harsh to tell a student that their argument was totally wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I no longer use that phrase at all. I stopped using it when a student came into my office and tried to &quot;explain&quot; his totally incorrect argument to me. Of course, he hadn't gotten my subtle drift, namely, that his argument MADE NO SENSE! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So now, when a student's argument makes no sense, I just write:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Your argument makes no sense.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And perhaps explain why it makes no sense. Although this kind of feedback is blunter, nevertheless the student learns more from this honest assessment of the situation. They need the accurate feedback that their argument was totally off-base.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So don't mince words, and tell your students that their argument is totally wrong, when indeed it is.&lt;/p&gt;&#xA;" OwnerUserId="90" LastActivityDate="2014-03-26T16:50:07.837" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="871" PostTypeId="2" ParentId="865" CreationDate="2014-03-26T16:58:22.013" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://www.ams.org/notices/199705/comm-mumford.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus Reform – For the Millions&lt;/a&gt; by David Mumford&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Watching my children move through the mathematical curriculum of elementary and high schools has been very instructive for me. For instance, I believe there is no universal best way to teach mathematics which applies to all the basic skills. It is very tempting to adopt some pedagogical theory or intellectual standpoint and convince yourself that this is the yellow brick road leading to understanding. I am not convinced that the experts who study pedagogy in mathematics have a deeper insight into what works than most concerned parents.&lt;/p&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;p&gt;A very interesting point relating to the use of formal definitions has been raised by Saunders Mac Lane and other critics of the Gleason-Hallett text. They object to the definition of a continuous function—“the closer &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; gets to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, the closer &lt;span class=&quot;math-container&quot;&gt;$f (x)$&lt;/span&gt; gets to &lt;span class=&quot;math-container&quot;&gt;$f(a)$&lt;/span&gt;” in this book—raising the example of &lt;span class=&quot;math-container&quot;&gt;$x \sin{1/x}$&lt;/span&gt;. [...] For the sentence above which purports to define continuity, I suggest you ask a mathematically naive friend whether they find anything odd about the sentences: “A clock pendulum is slowing down from friction. As it does so, it gets closer and closer to the vertical position.” Or ask whether the assertion that “runner &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is getting closer and closer to a new world’s record in the 100-meter” implies that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; never has a bad day? I certainly agree that a footnote clarifying the Gleason-Hallett definition to say that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; need not &lt;em&gt;go straight&lt;/em&gt; to &lt;span class=&quot;math-container&quot;&gt;$f (a)$&lt;/span&gt;, but may wobble on the way, is appropriate. But whether the Gleason-Hallett definition is correct as it stands is not a well-posed question: virtually the only sentences in English with an unambiguous interpretation (not requiring the use of common sense by the reader) are those written in mathematical jargon.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="369" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-26T16:58:22.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="873" PostTypeId="2" ParentId="643" CreationDate="2014-03-26T17:35:33.340" Score="7" Body="&lt;p&gt;Below are a few of my personal experiences (as a student) with this, followed by some (heavily qualified) possible lessons that can be extracted from them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my high school math teachers would do the day's lesson at the board, then give us the remainder of the period to start on the night's homework.  People would get going on the homework right away because they knew they could get help immediately if they were getting stuck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A line of students would form at the teacher's desk as people got into the assignment and needed help.  After a while, a second line would also form at my desk.  The teacher never even acknowledged that this was happening; neither encouraged it nor interfered, nor even monitored as far as I was able to tell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a student, I was often able to understand what another student was confused about when that student asked a question, while the teacher (maybe because of the age/generation difference, or because the teacher was so far temporally from learning it herself that she had forgotten what was confusing about it) misunderstood the student's real question and answered something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some teachers would let me break in and answer the actual question; in other cases, I would have to wait, then pretend to be confused about the same thing but ask it in such a way that the teacher would understand and address the actual source of the confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One other instance I remember is in 8th grade (generally, the school year during which you turn 14, for those unfamiliar with the American system) we had a section of the class that was accelerated.  One day we had a substitute who didn't know the material and I taught the lesson to the non-accelerated part of the class since we had seen it a few days previous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[NOTE: large-ish edit follows, I just realized that some experience from grad school was perhaps more relevant than anything I'd said so far]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, let me relate something from graduate school.  I was pursuing a Master's degree in mathematics, and by this time had become reflective enough about education that I started actively observing myself in my interaction with the learning environment.  One thing that I observed was that I was never able to keep up with lecture.  Another was that mathematics is taught completely backwards from how it is naturally learned (for example:  when you are working on a new concept, the first thing that is taught are axioms and definitions; when you are discovering mathematics, you never even think about defining things until you've started to see some significant relationships that you feel need a name, and axioms probably come last).  But to return to the point--it hit me very hard one night at a study group that the only natural way for me to learn was to have someone else start asking questions about something that I kind of have some understanding of, at which point I would start to explain it as far as I knew, and &lt;strong&gt;then&lt;/strong&gt; my brain would kick into high gear as it attempted to refine the explanation--at which point I would start actually figuring out the material for real.  In theory, this would work if I could convince myself to attempt to explain something to, say, an inanimate object.  In practice, the only time I do this is when I see someone struggling with the concept and I start trying to help them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A proof is in some sense an explanation, and that did work, too.  But I had to drag myself to the task of writing a proof, whereas if I could see another human being needing help with it, I would work on it until they dragged me away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was one of many people who hit graduate school and suddenly realize that they don't know how to study, because until then nothing had really challenged them.  It was a huge breakthrough for me to see that there was a way that I could grapple with the material that was a natural and pleasant experience instead of &quot;why did I think I wanted to pursue a degree in this at all?&quot;-level drudgery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there is an overarching generalization to be made about this series of anecdotes with a self-selected mathophile (and obviously there may not be), it might be that &lt;strong&gt;peer instruction &lt;em&gt;can&lt;/em&gt; happen naturally, if you don't do anything to interfere with it&lt;/strong&gt;.  Another takeaway might be my school-kid observation that &lt;strong&gt;the teachers often didn't understand what the kids were asking, while I (another kid) did&lt;/strong&gt;.  This experience was repeated many, many times for me, and not just in math, and I saw other students doing similar things (understanding the asker's question better than the teacher did, I mean).  And, the graduate school stuff I mentioned suggest another possibility--&lt;strong&gt;there may be kids that &lt;em&gt;only&lt;/em&gt; learn when they are explaining&lt;/strong&gt;.  In other words, it's possible that not only is peer instruction appropriate, it's the only optimal way for certain student/subject combinations.&lt;/p&gt;&#xA;" OwnerUserId="371" LastEditorUserId="371" LastEditDate="2014-03-26T21:51:59.503" LastActivityDate="2014-03-26T21:51:59.503" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="874" PostTypeId="2" ParentId="858" CreationDate="2014-03-26T17:51:57.873" Score="9" Body="&lt;p&gt;This is just a minor addition, but I talk to my students about it.  In class I sometimes stop to take little show of hands surveys, e.g. &quot;do you guys like it better if I start with an example and move on to the general formula, or vice versa?&quot; After each exam and quiz I ask students personally how they felt about it, whether it was too hard or too easy, when they come up and turn it in. I don't mind criticism at all, and I encourage them to be honest with me about how things are going. And I have more lengthly discussions during office hours with those who show up.  Of course this isn't a one hundred percent reliable way to quantify performance, but it helps me make little qualitative improvements to my teaching, with the side effect of making the students feel more empowered.&lt;/p&gt;&#xA;" OwnerUserId="263" LastActivityDate="2014-03-26T17:51:57.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="875" PostTypeId="1" AcceptedAnswerId="878" CreationDate="2014-03-26T18:13:34.643" Score="41" ViewCount="2261" Body="&lt;p&gt;I've asked basically the same question previously on on math.SE, then &#xA;cogsci.SE without much response, surely here is the place to ask this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As anecdotal evidence is plentiful, but unfortunately for whatever conclusion you wish to draw, I'm hoping for answers that are backed by actual education or cognitive psychology research.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There seems to be a prevalent view among mathematicians and math educators &#xA;that forcing students to struggle is an effective way to improve retention and understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since I first asked the question, several points have become clearer:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There's sufficient (for me) evidence that required math courses are sometimes used to &quot;weed out&quot; weaker students as a form of &quot;population control&quot;. I'm ignoring those.&lt;/li&gt;&#xA;&lt;li&gt;There are researchers who teach (historically, surely) who consider teaching to be a distraction from research and so have an incentive to embrace any teaching philosophy that minimizes their efforts, such as &quot;let them struggle with it and prevail&quot;. Putting aside ethical and/or utilitarian arguments on the&#xA;wisdom of such an approach, here too the nature of the course is not motivated by a belief that difficulty increases the benefit for students. I'm ignoring these too.&lt;/li&gt;&#xA;&lt;li&gt;There are various &quot;non-traditional&quot; approaches such as Problem-based learning, peer-learning, constructivist approaches, the moore method (and variants), lecture-skepticism (my term) and others which claim, and can often demonstrate quantitatively, improved performance by students (yet often, interestingly, decreased confidence). They share a de-emphasis on the traditional lecture in favor of students taking a more active role in the learning process (So in a sense, working harder). They often simply require a larger investment of time by the students which might itself account for the&#xA;difference.&lt;/li&gt;&#xA;&lt;li&gt;Research suggests that humans have a natural tendency to &quot;skim&quot; when a&#xA;suitable opportunity presents itself, so that increasing the difficulty involved in processing the material (even extremely artificially, as in using&#xA;a less readable font) forces more engagement from the reader/student&#xA;and improves retention (see papers). Similarly, when students have a strong preexisting intuition on the subject, such as when studying newtonian physics (which we all have much experience with), students appear to resist the learning of new concepts that replace their existing intuitions unless actively confronted with their shortcomings (see papers). This also can be interpreted as evidence that learning is improved when students are forced to break out of a tendency to use &quot;cruise-control&quot;. &lt;/li&gt;&#xA;&lt;li&gt;The theory of &quot;cognitive load&quot; has been pointed out as relevant on cogsci.SE, but from what I've skimmed (naturally) so far, the idea is that under increased&#xA;cognitive load learning breaks down. I've not seen yet results demonstrating&#xA;that increasing cognitive load improves learning other then, as just mentioned,&#xA;by forcing engagement.&lt;/li&gt;&#xA;&lt;li&gt;Vygotsky's theory on the &quot;Zone of proximal development&quot; is intriguing in it's claim that learning is most effective when students tackle problems which they &quot;cannot solve unaided, but can solve with some guidance&quot;. The discussion is centered on children's education, it's unclear to me how widely applicable it is to a collegiate math setting. &lt;/li&gt;&#xA;&lt;li&gt;The related idea of a teacher as providing &quot;scaffolding&quot; also matches up with research on human memory: we remember new facts better when we connect them to existing memories and concepts. It seems plausible to claim that a student who derives a result on his own has built up a more substantial structure around the discovered fact and is therefore more likely to remember it.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;That's an outline of 3 different senses of &quot;difficulty&quot; in learning:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Challenge (&quot;He sets impossible standards, so you work really hard or crash. or both.&quot;)&lt;/li&gt;&#xA;&lt;li&gt;Engagement (&quot;I really have to pay attention to make sense of it, &#xA;it's exhausting!&quot;)&lt;/li&gt;&#xA;&lt;li&gt;Time spent (&quot;It took me 3 days to do that assignment, &#xA;it was really difficult!&quot;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Engagement seems specifically to be an issue of growing importance in&#xA;recent decades, vis the debate on the effects of TV/video games and the&#xA;increased diagnosis rate for attention/hyperactivity disorders.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm looking for answers that shed more light on the issue, point out &#xA;research results and generally make the issue clearer. Though I am&#xA;not an educator (math or otherwise), a good guideline are answers indicating research that would help someone with teaching duties, in math, at the college level, form an informed policy on the difficulty level of a course intended&#xA;(one would hope) to help students achieve their best. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question is admittedly quite too broad, but I care about &#xA;the answer and so I'm asking anyway.  I've done my best to make it constructive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original question on Math.SE was sparked by a bunch of quotes&#xA;I've accumulated through readings on math. I've included them here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first is by Lebesgue:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When I was a rather disrespectful student at the Ecole Normale we used&#xA;  to say that 'If Professor Jordan has four quantities which play exactly the&#xA;  same role in an argument he writes them as $u$, $A''$, $\lambda$ and $e_{3}'$ Our criticism&#xA;  went a little too far but, nonetheless, we felt clearly how little Professor&#xA;  Jordan cared for the commonplace pedagogical precautions which we&#xA;  could not do without, spoiled as we were by our secondary schools.&#xA;  &amp;lt;...&gt;&#xA;  Professor Jordan's only object is to make us understand the facts of&#xA;  mathematics and their interrelations. If he can do this by simplifying the&#xA;  standard proofs, he does so; &amp;lt;...&gt; But he never goes out of his way to reduce the reader's&#xA;  trouble or compensate for the reader's lack of attention.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &quot;Moore Method&quot; (and variants thereof) is a well known approach to math education.  Here's an excerpt from P.R. Halmos' autobiography &quot;I want to be a mathematician&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Can the mathematician of today be of any use to the budding mathematician of tomorrow? Yes.&#xA;  We can point a student in the right direction, put challenging problems&#xA;  before him, and thus make it possible for him to &quot;remember&quot; the &#xA;  solutions. Once the solutions start being produced, we can comment on&#xA;  them, we can connect them with others, and we can encourage their&#xA;  generalizations. Almost the worst we can do is to give polished lectures&#xA;  crammed full of the latest news from fat and expensive scholarly journals&#xA;  and books—that is, I am convinced, a waste of time.&#xA;  You recognize, I am sure, that I am once more advocating something&#xA;  like the Moore method. Challenge is the best teaching tool there is, for&#xA;  arithmetic as well as for functional analysis, for high-school algebra as&#xA;  well as for graduate-school topology.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Lastly, here's a quote from the preface to &quot;Mathematics Made Difficult&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;there is no doubt that an absolute ignoramus (not a mere qualified ignoramus, like the &#xA;  author) may become slightly confused on reading this book. Is this bad? On the contrary,&#xA;   it is highly desirable. &amp;lt;..misleading redaction...&gt; it is hoped that this book may help to &#xA;  confuse some uninitiated reader and so put him on the road to enlightenment, limping along &#xA;  to mathematical &lt;i&gt;satori&lt;/i&gt;. If confusion is the first principle here, beside it and &#xA;  ancillary to it is a second: pain. For too long, educators have followed blindly the &#xA;  pleasure principle. This over-simplified approach is rejected here. Pleasure, we take it, &#xA;  if for the initiated; for the ignoramus, if not precisely pain, then at least a kind of  generalized &lt;i&gt;Schmerz&lt;/i&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;h2&gt;Previous questions elsewhere&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;math.stackexchange.com/questions/702044&lt;/li&gt;&#xA;&lt;li&gt;cogsci.stackexchange.com/questions/5921&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h2&gt;Related Papers&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Designing Effective Multimedia for Physics Education - www.physics.usyd.edu.au/super/theses/PhD(Muller).pdf by &lt;em&gt;Derek Muller&lt;/em&gt;&#xA;TED talk: www.youtube.com/watch?v=RQaW2bFieo8&lt;/li&gt;&#xA;&lt;li&gt;Fortune favors the bold - web.princeton.edu/sites/opplab/papers/Diemand-Yauman_Oppenheimer_2010.pdf by &lt;em&gt;Diemand-Yauman &amp;amp; Oppenheimer&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user370" LastEditorDisplayName="user370" LastEditDate="2014-03-26T19:28:53.253" LastActivityDate="2014-04-10T17:09:38.947" Title="What does math education research know about difficulty vs. effectiveness?" Tags="|mathematical-pedagogy|psychology|education-research|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="877" PostTypeId="2" ParentId="805" CreationDate="2014-03-26T18:37:39.493" Score="6" Body="&lt;p&gt;For a number of reasons, I would say time pressure is not generally good for math exams.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It forces to students to be more concerned with getting done quickly than with checking their work. In any real-world job that uses math, being correct is much more important than being blazingly fast. I'm much more concerned with teaching an engineer to check his work thoroughly than teaching him to do calculations very, very quickly, for instance. The same goes for accountants and, really, almost other profession that makes significant use of mathematics skills. Being really, really fast at solving math problems is simply not a particularly valuable skill, especially today when we have computers that can do it much, much faster than you no matter how fast you are. We don't hire people to sit there and perform computations very quickly. We use computers for that. He hire people to use mathematics to create new solutions to practical or theoretical problems, a skill which doesn't strongly depend on one's ability to quickly perform computations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It measures things completely unrelated to mathematics, including how quickly the student can read the questions (which can be limited by eyesight and, at least judging by my anecdotal experience, is not strongly correlated to students' mathematical abilities.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As others have mentioned, it further disadvantages students who have test or math anxiety, which is more likely to discourage them than help them in any way.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So, in short, no, you shouldn't design your exams to have time pressure.&lt;/p&gt;&#xA;" OwnerUserId="375" LastActivityDate="2014-03-26T18:37:39.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="878" PostTypeId="2" ParentId="875" CreationDate="2014-03-26T18:39:24.320" Score="24" Body="&lt;p&gt;Perhaps the key-word needed here is not just &lt;em&gt;struggle&lt;/em&gt; but &lt;em&gt;productive struggle&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hiebert and Grouws (2007) discuss two key features of mathematical teaching/instruction for &quot;promoting conceptual understanding&quot; (p. 383). Their paper can be found here:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hiebert, J., &amp;amp; Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. Second handbook of research on mathematics teaching and learning, 1, 371-404. &lt;a href=&quot;https://carnegiefoundation.org/sites/default/files/Hiebert_Grouws.pdf&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To the point, their second feature is called &lt;em&gt;Students Struggle with Important Mathematics&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The authors trace this line of thought back at least to &lt;a href=&quot;http://en.wikipedia.org/wiki/John_Dewey&quot; rel=&quot;noreferrer&quot;&gt;John Dewey&lt;/a&gt; (p. 388):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/AHO2L.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article continues with references to work by cognitive theorists such as &lt;a href=&quot;http://en.wikipedia.org/wiki/Lev_Vygotsky&quot; rel=&quot;noreferrer&quot;&gt;Lev Vygotsky&lt;/a&gt; (whose &lt;em&gt;Zone of Proximal Development&lt;/em&gt; is mentioned in the OP), the mathematician &lt;a href=&quot;http://en.wikipedia.org/wiki/George_polya&quot; rel=&quot;noreferrer&quot;&gt;George Pólya&lt;/a&gt; (whose &quot;How to solve it&quot; effectively laid the groundwork for the study of mathematical problem solving), and the developmental psychologist &lt;a href=&quot;http://en.wikipedia.org/wiki/Jean_Piaget&quot; rel=&quot;noreferrer&quot;&gt;Jean Piaget&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The authors continue with a discussion of some material on work done outside of mathematics before returning to studies specific to mathematics and mathematics education. You remark that your &quot;question is in natural peril of being too broad,&quot; but hopefully this source (and the references contained therein) can get you started on tackling it. &lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-03-29T20:29:04.797" LastActivityDate="2014-03-29T20:29:04.797" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="879" PostTypeId="2" ParentId="549" CreationDate="2014-03-26T20:11:37.270" Score="13" Body="&lt;p&gt;I'm not sure what a complete answer would look like to this question, but since I know how to find the data for the University of Illinois at Urbana-Champaign, I'll answer for that school. Hopefully others will be able to answer in other ways or from other schools. The data I used is publicly available at  &lt;a href=&quot;http://dmi.illinois.edu/cp/&quot; rel=&quot;noreferrer&quot;&gt;http://dmi.illinois.edu/cp/&lt;/a&gt;, bless their hearts. Disclaimer: I worked there teaching calculus and advising undergraduates for three excellent years before moving on to a full-time teaching role at another school.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary of What Follows&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use publicly-available employment data to estimate the amount of math research that was done at University of Illinois from 2004-2013. Then I use publicly-available course data to estimate the amount of calculus taught on the same time period. If you don't care about the details of how my estimates happened, you can skip to the results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The results section suggests that there has been a massive increase in calculus taught with no corresponding increase in math research done. The conclusion, which is essentially from my personal experience, is that an increased calculus population is much more likely to drive up class sizes and academic professional employment rather than lead to the hiring of many new math researchers.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;An Estimate of the Amount of Math Research Done at UIUC, 2004-2014&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll start with this chart of Full-Time Equivalent Employees of UIUC's Math Department, 2004-2014:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;                        '13-14  '12-13  '11-12  '10-11  '09-10  '08-09  '07-08  '06-07  '05-06  '04-05&#xA;----------------------------------------------------------------------------------------------------&#xA;Tenure Track            65.25   62.25   62.17   64.67   62.78   65.73   70.74   67.87   66.67   65.67&#xA;Visiting Faculty        9.00    9.75    8.50    4.00    3.00    2.00    7.00    6.50    5.00    9.25&#xA;Other Instructors       21.12   20.20   12.06   16.42   18.18   19.98   14.53   20.00   17.97   22.52&#xA;Academic Professionals  17.10   15.60   14.35   10.85   10.35   10.35   9.85    9.55    7.80    6.05&#xA;Teaching Assistants     69.56   66.88   64.60   64.80   69.69   66.55   63.79   65.30   61.52   60.07&#xA;Research Assistants     5.75    8.25    8.37    6.96    8.36    7.00    9.50    4.50    2.83    6.35&#xA;Civil Service Staff     10.75   9.75    10.19   9.75    9.75    9.75    9.75    9.75    9.75    9.25&#xA;----------------------------------------------------------------------------------------------------&#xA;**Total**               199.53  193.68  180.23  177.94  182.11  181.86  185.16  183.47  171.91  179.16&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The units here are &quot;full-time equivalents.&quot; So a part-time employee working 28 hours per week would count as 0.7 FTE's toward their row. What you should notice is that there is a &lt;strong&gt;nearly-constant amount of math research&lt;/strong&gt; being done, as best we can tell from this data: besides one outlier year involving a retirement policy, there have always been between 74 and 82 full-time-equivalents of math researchers employed by the department. I added the tenure-track faculty, visiting faculty, and research assistants to find an estimate of &quot;math research done&quot;:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;                        '13-14  '12-13  '11-12  '10-11  '09-10  '08-09  '07-08  '06-07  '05-06  '04-05&#xA;----------------------------------------------------------------------------------------------------&#xA;Tenure+Visit+Research   80      80.25   79.04   75.63   74.14   74.73   87.24   78.87   74.5    81.27&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;An estimate of the amount of calculus taken at UIUC, 2004-2013&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can start with the number of Instructional Units taught by the math department. An instructional Unit is awarded when a student completes one credit hour of a math course.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;                '13-14  '12-13  '11-12  '10-11  '09-10  '08-09  '07-08  '06-07  '05-06  '04-05&#xA;100 level               8119    9104    8751    9556    9940    10004   10901   12477   11695&#xA;200 level               46530   44647   41731   39170   38897   32870   29437   28964   25211&#xA;300 level               1486    1213    1192    1204    1234    3438    6363    6360    7024&#xA;400 level               13195   12165   10730   10648   9848    9360    9481    9043    9860&#xA;500+ level              5086    5138    5089    5055    4906    4806    5193    5591    5740&#xA;----------------------------------------------------------------------------------------------------&#xA;Total                   74416   72267   67493   65633   64825   60478   61375   62435   59530&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;We are lucky in that the vast majority of the 200-level classes are calculus classes (exceptions are other &quot;service courses&quot; like Discrete Math, beginning Linear Algebra, and Differential Equations, which presumably fit under the same question anyway), and all of the calculus classes are 200-level, so the publicly available data is pretty good for our purposes: the row of 200-level data is almost exactly the &quot;amount of calculus taught,&quot; in units of credit hours.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Results&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the above methods, I arrive at the following estimates (no '13-14 data for courses yet, so drop it):&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;                                    '12-13  '11-12  '10-11  '09-10  '08-09  '07-08  '06-07  '05-06  '04-05&#xA;----------------------------------------------------------------------------------------------------&#xA;Math Research (in FTEs)             80.25   79.04   75.63   74.14   74.73   87.24   78.87   74.5    81.27&#xA;Calculus Taught (in credit hours)   46530   44647   41731   39170   38897   32870   29437   28964   25211&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Sorry about all the data being in reverse order; that's how it is on their website. At least my chart is in the correct order:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/P2Dvp.png&quot; alt=&quot;UIUC Estimates of Math Research vs Calculus Taught, 2004-2013&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not a complete answer to the question of course, but at least for one research institution for the last ten years, you can see that there has been a massive increase (nearly doubling) in the amount of calculus taught over the past ten years, with no corresponding increase in the amount of employment for math researchers. This suggests that at least at this school, a possible future crash in the amount of calculus taught would also not have a corresponding disastrous impact on the amount of math research completed at the school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, calculus increases (at this school) were mostly handled by increased class sizes and by increases in the &quot;Academic Professional&quot; categories -- employees who are wonderful usually-full-time employees who are teaching calculus part-time and doing other things (advising, recruiting, careers, directing initiative) but &lt;strong&gt;not&lt;/strong&gt; doing math research.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I welcome and wish for criticism on my methods and conclusions!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-26T20:11:37.270" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="880" PostTypeId="2" ParentId="856" CreationDate="2014-03-26T20:14:23.383" Score="6" Body="&lt;p&gt;I think an important consideration in deciding how to grade something is to ask yourself whether the answer even &lt;em&gt;makes sense&lt;/em&gt;. I mean this in the more formal use of the word &lt;em&gt;sense making&lt;/em&gt; in Mathematics Education; a few relevant remarks can be found in my earlier MESE response &lt;a href=&quot;https://matheducators.stackexchange.com/a/816/262&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as implications for your question: The decision as to how one handles a very wrong answer depends on numerous factors. At least one of them is the setting (i.e., the class and the student). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way I would respond to a primary school student who makes an error shortly after learning about the multiplication of fractions is very different from how I would mark the work of an undergraduate engineering major who made an analogous error in a course on mathematical modeling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To illustrate this point better, here is an excerpt from COMAP's &lt;a href=&quot;http://www.iitgn.ac.in/mcm/cd/Mathematical%20Modelling%20Handbook/pdf/Modeling_Handbook_Full.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Modeling Handbook&lt;/a&gt;, taken out of &lt;a href=&quot;http://www.maa.org/about-maa/governance/maa-presidents/henry-otto-pollak-1975-1976-maa-president&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henry Pollak&lt;/a&gt;'s introduction &quot;What Is Mathematical Modeling?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/I7sxb.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own preference is for Side 2: when one can reasonably claim that a solution &quot;shows no judgment at all,&quot; it suggests that something has gone quite awry with regard to the student's learning. (Though, if such mistakes are widespread among members of a class, then perhaps the true error lies in the teacher's instruction...)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-03-26T20:14:23.383" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="881" PostTypeId="2" ParentId="348" CreationDate="2014-03-26T20:20:15.960" Score="4" Body="&lt;p&gt;You can also plug in your computer and show the automated approach, e.g. in Mathematica:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathtt{f[x\_] := x^4 + x^3}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathtt{Plot[f[x],\ \{x, -1, 1\}]}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathtt{df = D[f[x],\ x]}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathtt{Solve[df == 0,\ x]}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathtt{f[-3/4]}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathtt{Minimize[f,x]}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathtt{NMinimize[f,x]}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathtt{jac = D[Sin[x\ y],\ \{\{x, y\}, 2\}\ ]}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathtt{Det[jac]}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This allows you to show several examples, including ones too hard to do by hand.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-26T20:20:15.960" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="882" PostTypeId="2" ParentId="771" CreationDate="2014-03-26T21:36:48.743" Score="-5" Body="&lt;p&gt;Just what does dividing &lt;strong&gt;by&lt;/strong&gt; a negative number mean in real life?.....  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people are just better at things with real life meaning but are being forced to get a degree rather than train for a trade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So are you seeing a trend in more people going to collage that would have done “on the job” training in past decades?&lt;/p&gt;&#xA;" OwnerUserId="381" LastActivityDate="2014-03-26T21:36:48.743" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="883" PostTypeId="2" ParentId="294" CreationDate="2014-03-26T21:38:57.793" Score="4" Body="&lt;p&gt;I've recently read T.W. Körner's &lt;a href=&quot;https://www.amazon.co.uk/Pleasures-Counting-T-W-K%C3%B6rner/dp/0521568234&quot; rel=&quot;nofollow&quot;&gt;&quot;The Pleasures of Counting&quot;&lt;/a&gt; and I consider it a true gem, unlike anything I've read before. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Largely, it is a book about the historical uses of applied mathematics &#xA;which manages to be accessible to a clever high-school student yet&#xA;rewarding to a math/science major with some courses under her belt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among topics covered are fighting cholera epidemics in Europe using &#xA;statistics, and the birth of operations research in WW2 (the organization of convoys, how not to train pilots, why you should repair your &#xA;planes less often). He explores predator-prey models through &#xA;epidemiology as well as Darwinism. Dimensional analysis is used &#xA;to derive power laws from mathematical biology and physics (humans &#xA;and fleas can jump roughly the same height, how on earth can that be &#xA;can that be??). He discusses Richardson and the genesis of weather modeling. &#xA;Euclid and Galileo are in there too, as is as the obligatory &#xA;section on the Enigma machine. It's an amazing book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The math content of the stories is relished rather then elided. There &#xA;are occasional many exercises fueled by the history (Requiring anything &#xA;common sense to trig, probability, calculus, ODEs, group theory). &#xA;Occasionally, there's a choose-your-own-adventure moment right before &#xA;a crucial point in the historical narrative, when you're given a &#xA;chance to reflect on a problem before learning how it was actually &#xA;solved and lives saved. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's filled with example after example of real life crisis where&#xA;the use of simple mathematical tools coupled with ingenuity&#xA;lead to insights that made a huge, undeniable difference. Examples&#xA;showing that actually &lt;em&gt;measuring&lt;/em&gt; things can &lt;em&gt;prove&lt;/em&gt; common-wisdom to&#xA;be wrong. The book manages to do this without resorting to synthetic, &#xA;trivialized problems such as would be found in your average &#xA;calculus book these days. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found it to be a truly inspirational book; it got me curious and excited&#xA;about math, and made me want to learn more. It was also a thoroughly &#xA;good read. I'm convinced it has the potential to be a life-changing read to &#xA;a certain kind of person at the right age.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing to point out is that the exercises are positively &#xA;riddled with errors, so you better arm yourself with a copy of the &#xA;(lengthy) errata and maintain vigilance. Other then that, I cannot &#xA;recommend it enough.&lt;/p&gt;&#xA;" OwnerDisplayName="user370" LastActivityDate="2014-03-26T21:38:57.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="884" PostTypeId="2" ParentId="468" CreationDate="2014-03-26T21:44:51.730" Score="5" Body="&lt;p&gt;Like others I more often than not craft the problems to have nice solutions. Just to keep the students on their toes I occasionally insert something not so nice. As we are largely discussing eigenvalue problems, let me propose the following trick I picked up from a senior colleague. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use a problem, where the precise eigenvalues are not needed, only their signs. For example you can ask whether a given 3-variable quadratic form is positive definite or not. You probably should warn students that they may need to estimate the zeros of the characteristic polynomial as opposed to find them. Also the students need to recall how to do that - always nice to use stuff from courses with very different topics. Your task as the problem designer is then to make sure that the zeros have distinct integral parts, so that it won't be too arduous a task to locate intervals with integral endpoints each containing a single eigenvalue.&lt;/p&gt;&#xA;" OwnerUserId="282" LastActivityDate="2014-03-26T21:44:51.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="885" PostTypeId="2" ParentId="805" CreationDate="2014-03-26T21:46:28.813" Score="8" Body="&lt;p&gt;Disclaimer: Most answers say in a nutshell &quot;No, you should not have time pressure due to the following (comprehensible) reason(s)&quot;. I agree with most of the arguments there, but want to add some arguments that time-pressure is not such a bad things, at least on university level.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Assuming a fixed time constraint for the exam, you have three posibilities: Either you risk to have time pressure, or you only ask about a minor part of the course, or you avoid any deeper questions and ask questions about (almost) every part of the course but without going in to deep. &#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The second possibility is bad for those who lack in that particular part of the course and at the same time maybe some students are lucky since they only learned that part. In my opinion, such an exam is unfair (exept maybe, there is one particular main topic which is obvious, or you have announced that particular topic as being relevant for the exam). Moreover, if someone has to build up on your course and needs some other topics, there is a possibility that students might have no clue of that topics.&lt;/li&gt;&#xA;&lt;li&gt;The third possibility is maybe a way between, but if you do not have &lt;em&gt;any&lt;/em&gt; questions with deeper understanding, this is also bad for the following courses as well as you cannot distinguish between good and excellent students. This argument was also elaborated in some answers in the question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/833/what-to-do-when-your-students-are-all-getting-as&quot;&gt;What to do when your students are all getting A&amp;#39;s&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If you want to have a broad knowledge and some sort of deep understanding, you might run out of time.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Most of the students will be working in industry and economy after their study, where the hiring is mostly based on their final grad in their study. Since it is not only important that someone is able to things right, but also to do things quickly, it is not unfair to also test how quick a student is in solving problems. Also, if two students are able to solve the same level of problems, but one of them is significant quicker, he/she should get a better grade.&lt;/li&gt;&#xA;&lt;li&gt;Even for mathematicans in academia, (a certain amount of) speed is to some way important. Of course, if you are a genius and can work on his own and solve big problems, you don't need speed. But most of the students will at some point write a thesis and have to talk to their advisor. At sessions with the advisor, you may do a brainstorming or discuss possible solutions for a problem where a student can in general not say: &quot;Let me think about it for a few hours and then I will come back&quot;, but a student who is capable of telling quickly if a solution might lead into a good direction has a lot of advantages here. The same holds for discussions with colleagues at conferences. The ability for a quick scan if ideas make sense or not cannot be trained within a few days or weeks and exams where time is also important can help to train this.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-03-26T21:46:28.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="886" PostTypeId="2" ParentId="852" CreationDate="2014-03-26T22:06:35.113" Score="6" Body="&lt;p&gt;Telling them &quot;this is the way to II and III and ...&quot; is doomed to failure, they &lt;em&gt;know&lt;/em&gt; their way is right (it got them straight As before, and they are so smart that they made it to college!). Ask them to discuss offline/during office hours, go over their technique and yours. Let them compare. Where relevant, point out what/how yours is more general (i.e., presumably applicable for next courses). Sometimes the difference is just in untenable basic assumptions (i.e., all functions are &lt;em&gt;obviously&lt;/em&gt; continuous/differentiable/...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the time this is due to an ego the size of the Earth's orbit, coupled with a sense of inadequacy they won't acknowledge to themselves (being intimidated by their new environment, where they aren't by far the smartest kid of the block, as they are accustomed from school back home).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Handling those people is difficult.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-26T22:06:35.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="888" PostTypeId="2" ParentId="829" CreationDate="2014-03-26T22:25:09.183" Score="2" Body="&lt;p&gt;One critical point is to remind them that they should stay in contact with the subject matter reasonably constantly, if only to go over the writeup to do minor corrections. Lost contact means a lot of time getting back on track. This means don't ever let a week go by without doing some real work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start writing up what you are doing from the first day. Perhaps keep different files for short comments/summaries of relevant papers/books/webpages, and different aspects of the own work. Keep scratch work and failed attempts around, obviously preliminarly cleaned up. Organizing the results, selecting what is relevant in retrospective, and such is easy (if somewhat boring) work to be done nearing the end. Until then, all this is (more or less) scratch paper. Trying to remember how you came to some conclusion, or how you proved something, or the steps in a computation a couple of months back makes you sweat blood.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If stuck advancing the work, you have something else to do (write up the morning's work, read through and summarize the latest issue of a journal, ...); if bored of writing, go back to work.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-26T22:25:09.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="889" PostTypeId="2" ParentId="839" CreationDate="2014-03-27T00:12:12.577" Score="11" Body="&lt;p&gt;There's already very nice answer of Andrej Bauer, but I would like to view the question from a slightly different perspective. Perhaps one should not call questions silly, but &lt;em&gt;there are questions which we wouldn't want to answer&lt;/em&gt;, the main reason usually being that it would not be the best response. To name a few concrete examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;As a teacher we have a responsibility to &lt;em&gt;all&lt;/em&gt; of our students, not only the one who has asked a question and gained our attention.&lt;/li&gt;&#xA;&lt;li&gt;Answering a question might induce bad habits (e.g. I don't need to think on my own).&lt;/li&gt;&#xA;&lt;li&gt;With huge volume of questions we are resource-bound to leave some unanswered (e.g. deferred after class, mail or office hours).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To give an extreme example, one semester, during a laboratory-class, I experimented with no-easy-answers policy. It was a curious experience, because during exercise-sessions I would encourage questions, even &lt;em&gt;silly&lt;/em&gt; ones (yes, students had asked questions which were silly in their own opinion or so it seemed). Instead, I told them &lt;em&gt;where and how to look for answers&lt;/em&gt; and them made them search themselves. When searches were unsuccessful, I would suggest better search terms, better sources, or how to check it via some simple test, and so on. Only after a significant effort was put in, I would answer the question myself (the &quot;significance&quot; threshold was low at first and went up as they got better and better).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The results were marvelous. We were able to do a lot of stuff, and the students made a tremendous progress (it helped that it was a really good group and we had a good contact). It happened that students did a job so good, that I had to explain some questions to the rest of the group (and why some intuitive answers didn't work) before actually answering them. Finally, the students' opinions were positive, often along the lines of &quot;the class was tough, but I learned a lot, I would recommend it&quot;, some even praising that approach (although &quot;in hindsight&quot;, that is, they hinted that at first it was not pleasant).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Naturally, it did work due to specific reasons, e.g.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;easy internet access (each student had a computer),&lt;/li&gt;&#xA;&lt;li&gt;majority questions were easy to find (for example in the programming language manual),&lt;/li&gt;&#xA;&lt;li&gt;I was able to motivate the students enough, so that time delays would not break the flow of the class,&lt;/li&gt;&#xA;&lt;li&gt;with only a few questions, I was able to focus more on the weaker part of the group, and to keep them &quot;online&quot; even during difficult exercises,&lt;/li&gt;&#xA;&lt;li&gt;it was a &lt;em&gt;really bright&lt;/em&gt; group.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Unfortunately it was just a single class, so it might have been just a coincidence that it worked and some other approach could have worked even better. Nevertheless, I &lt;em&gt;will&lt;/em&gt; try it a second time with right opportunity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what do I do with a question that I don't think I should answer?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;I answer it nevertheless.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Nevertheless, &lt;em&gt;I answer it.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Did I mention that &lt;em&gt;I answer it?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;I answer it after the class.&lt;/em&gt; This doesn't need an explanation.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;I let the class answer it.&lt;/em&gt; For example, if I fell that the particular student should put it more effort, it works wonders against some personalities.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;I answer it partially and let the student himself finish it.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Rarely, but it happens, &lt;em&gt;I let the student know that the question in its current form is inadequate/inappropriate.&lt;/em&gt; There are multiple variants, it might need a reformulation or it may be unsalvageable. Non-verbal cues are often enough.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-27T00:26:58.833" LastActivityDate="2014-03-27T00:26:58.833" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="890" PostTypeId="1" AcceptedAnswerId="893" CreationDate="2014-03-27T00:14:51.013" Score="26" ViewCount="6600" Body="&lt;p&gt;I've taught calculus in college for five years, and it's always interesting to see students coming in who already had calculus in high school. Many of them do very well, and don't even seem like they needed the course. Others struggle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I took Calculus in high school, and I don't remember doing Newton's method or his law of heating and cooling. In fact, even though my high school class was a 'double class' (counting for two credits), I don't remember covering as much material as we did in college; but I didn't feel behind in my next classes.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the United States, what are the main differences between the way high school Calculus and college Calculus are taught?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm interested in amount of time spent in class, average homework loads, topics covered, and any other differences you feel are important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I'm specifically talking about AP Calculus BC in highschool compared to a one year sequence of Calculus at a four-year university. Both as currently taught.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="376" LastEditDate="2016-10-01T17:45:50.157" LastActivityDate="2017-06-30T07:22:11.233" Title="How is teaching calculus in high school different from teaching calculus in college?" Tags="|undergraduate-education|calculus|secondary-education|usa|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="891" PostTypeId="2" ParentId="643" CreationDate="2014-03-27T01:01:29.867" Score="6" Body="&lt;p&gt;When I was an undergraduate at Georgia Tech, I was hired to be an organizer for &quot;Peer Lead Undergraduate Study&quot; (PLUS) for some calculus courses (in fact, I was one of the first two, piloting the program). The &lt;a href=&quot;http://www.success.gatech.edu/plus-sessions&quot;&gt;program&lt;/a&gt; was so successful that it has been expanded and implemented for all the calculus courses at Georgia Tech, and a few other courses as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The PLUS sessions were optional, and viewed as a recitation of sorts. In this way, the standard student of a Georgia Tech Calc I class might attend 3 hours of lecture, 2-3 hours of recitation, and 0-1 hour of PLUS per week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The PLUS sessions were student oriented and student run, with the coordinator being there almost as an icebreaker and inspirateur rather than teacher (in fact, we were explicitly told not to teach). In general, students who attended the PLUS sessions far outscored those who didn't (though this is almost certainly biased as those who attended are at least superficially more motivated). Overall, it allowed professor-led-lectures to be focused on presentation of material, TA-led-recitations to be more around harder questions and second viewings of material, and the PLUS sessions to be where students develop their problem-solving techniques and skills. From my experience, I'd say it was a moderate success, and could lead to something nice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar to this, I've heard of &lt;a href=&quot;http://www.pltl.org/&quot;&gt;Peer led team learning&lt;/a&gt;, which is a different model that has also been implemented in university math classes. The link is to the center of PLTL, which has additional resources (and which has been studied - which the PLUS model hasn't, as far as I can tell).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something that strikes me about both is the sheer time effort. The only reason it was a successful program at Georgia Tech is because it was acceptable for students to have 7 hours of class time per week for a (four credit) math course. This would not be acceptable at my current university (where they spend 3.5 hours per week in a math course). On the other hand, in my experience I find that the typical student finishing a second semester calculus course at Georgia Tech is far more proficient in both technique and knowledge of theory than the typical student finishing a second semester calculus course at my current university.&lt;/p&gt;&#xA;" OwnerUserId="20" LastActivityDate="2014-03-27T01:01:29.867" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="892" PostTypeId="2" ParentId="41" CreationDate="2014-03-27T02:19:08.007" Score="16" Body="&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ENLBf.png&quot; alt=&quot;alphabet&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/hexWa.png&quot; alt=&quot;alphabet with stroke order&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are what my alphabet and numbers have ended up looking like. For the sake of clarity, I've included a stroke order chart, where the first stroke is red, the second green, the third blue, and the fourth cyan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try to make distinctions between every character, but I agree with those above who have noted that &quot;o&quot; is, in general, a terrible name for a variable. I tend to only use the &quot;O&quot; with a flourish where I have to make a distinction between &quot;o&quot; and &quot;O&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use the double-story &quot;g&quot; to distinguish from &quot;y&quot;, and &quot;S&quot; with a serif to distinguish from &quot;5&quot;. I use serifs to distinguish between uppercase and lowercase (&quot;c&quot; and &quot;C&quot;, &quot;s&quot; and &quot;S&quot;, &quot;k&quot; and &quot;K&quot;, &quot;v&quot; and &quot;V&quot;, &quot;w&quot; and &quot;W&quot;, &quot;p&quot; and &quot;P&quot;), but where I don't have to make that distinction, &quot;c&quot; and &quot;s&quot; get serifs to distinguish from &quot;(&quot; and &quot;5&quot;, respectively.&lt;/p&gt;&#xA;" OwnerUserId="388" LastEditorUserId="388" LastEditDate="2014-03-27T02:51:43.387" LastActivityDate="2014-03-27T02:51:43.387" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="893" PostTypeId="2" ParentId="890" CreationDate="2014-03-27T04:18:59.140" Score="12" Body="&lt;p&gt;I took AP calculus BC in high school, and since TA's and taught calculus at two universities since (I'm a grad student, but I've lectured as well as TA'd). These experiences have amalgamated together into this answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had 170 days of calculus instruction in high school, each one hour long. We used a standard college text of Larson and Edwards (whatever edition it was at the time). There were weekly homeworks, a test roughly each month, and a final each semester (and an AP test afterwards, I suppose). We did not prove things, and we did not need to use the $\epsilon- \delta$ formulation of a limit. In practice, most functions were continuous and differentiable. In fact, most functions were elementary and smooth. We did learn Newton's law of cooling, some numerical integration bits like the the trapezoid rule and Simpson's rule, Taylor series with remainder, u-substitution, integration by parts, trig substitution (to mention topics of various difficulty that I feel might be omitted in some high school classes). We did not talk about many differential equations (we learned initial value problems, and Euler's method, but I simply cannot remember if we did much else).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very important component that was missing from my high school education was understanding &lt;em&gt;why&lt;/em&gt; anything was true. We might heuristic our way through things and prove a (small) subset of the things we came across, but in general there were few proofs. Math was a gift, set down whole before us like a swaddled babe from a stork.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With respect to the homework, I never did any calculus homework at home - many of us were able to finish it all during school hours. In this regard, we couldn't have done too much homework timewise. But the homework was graded completely and returned with notes, and we did an incredibly large amount number-of-problems-wise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, university calculus. At my current university, calculus meets for almost 4 hours per week: roughly 2.5 hours for lecture and roughly 1.5 hour for recitation. There are roughly 15 weeks weeks, so that there are approximately 60 hours of instruction per semester. My high school calculus BC course is the equivalent of two semesters of calculus at my university (roughly corresponding to differentiation and then integration), so we should really be thinking of 170 hours vs. 120 hours. (At my undergraduate university, calculus courses met for 6 hours per week and BC calculus was roughly the equivalent of the first semester and the first fourth of the second semester of calculus - the other three-fourths were actually linear algebra).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We teach Newton's law of cooling, but we do almost no numerical integration. In fact, we barely cover Riemann sums at all. The lectures will talk about Riemann sums insofar as to define integration, and they aren't really brought up again. (At my undergrad, we similarly skimmed over Riemann sums, though there was more variability: some professors care and others don't). Thus we don't talk about any numerical integration techniques. We don't talk about &quot;differentials,&quot; by which I mean approximating $f(x+\delta)$ by using some sort of $f(x) + f'(x)\delta$ style argument, whereas in high school we did, including some with multiple steps (like calculating $\arctan{1.5}$ using three steps of length $.5$ from $0$, or something). We similarly skipped Euler's method for approximating solutions to differential equations. This is a recurring theme: we skimp out on numericals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I partially attribute this to calculator usage. In high school, we used calculators as part of the AP test uses calculators. At my university, we use no calculators ever - so no numerics and usually the problems themselves are easier. I don't know whether this is a good thing or a bad thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At my university, we teach $\epsilon-\delta$ but do not test it (a conceptual hurdle that the powers that be have decided isn't worth the struggle, but which &lt;em&gt;is not&lt;/em&gt; skipped at my undergrad). We teach Taylor remainder estimation but do not test it (another numerical problem skipped). This was a large and challenging part of high school calculus entirely missed. (This &lt;em&gt;was not&lt;/em&gt; skipped at my undergrad).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the flipside, we do teach first and (some) second order differential equations during the second semester - students learn integrating factor and separable differential equations in particular. Sometimes, depending on the professor/speed of the class, we teach power series solutions to differential equations. The homeworks that we assign were very long (at my undergrad) and are moderately long at my current university. But the graders at both were hired undergrads, for whatever that is worth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And every lecturer I know proves everything from the ground up, even though the students are not held directly accountable to the proof material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I find high school calculus to be a much better course, with more contact hours, required attendance, more and more completely graded homeworks, and most topics (except $\epsilon-\delta$ and some differential equations bits that are covered in some university classes). In my university classes, we cover fewer subjects, but we cover them deeper (i.e. with proof), but there are fewer (though perhaps harder) homeworks that are graded by undergrads. And we omit most numerics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suspect a student who got a 5 on the AP calc BC test to get an A or B at the end of the second semester of calculus at my university. But before one concludes that I think high school better, I find that most of my students (including those who took calculus in high school) have almost no conceptual understanding of calculus coming in. Some do when they pass the course.&lt;/p&gt;&#xA;" OwnerUserId="20" LastActivityDate="2014-03-27T04:18:59.140" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="894" PostTypeId="1" CreationDate="2014-03-27T04:59:48.213" Score="18" ViewCount="1569" Body="&lt;p&gt;Each year my university tries to decide whether or not it will have calculator and CAS based introductory math courses (the calculus sequence, linear algebra, and ODE) or not. Other than some hearsay or personal preference (which looks like &quot;You need to do the arithmetic to get a real &lt;em&gt;feel&lt;/em&gt; for how things work&quot; vs. &quot;Wolfram|Alpha and MatLab can do all their &lt;em&gt;real&lt;/em&gt; work for them too - might as well get them used to using it now and not duplicate the effort&quot;), there is no basis for the decision.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering if there were any studies indicating a trend (or lack of a trend) between calculator usage in middle and high school and success in college math courses (or perhaps another measure of post-secondary school mathematics).&lt;/p&gt;&#xA;" OwnerUserId="20" LastEditorUserId="201" LastEditDate="2015-02-03T20:05:41.397" LastActivityDate="2017-12-07T16:27:42.753" Title="A study comparing effects of calculator usage on later math skills?" Tags="|reference-request|technology-in-education|calculators|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="895" PostTypeId="1" AcceptedAnswerId="902" CreationDate="2014-03-27T06:28:13.403" Score="25" ViewCount="1161" Body="&lt;p&gt;I’d like to start off with saying I am not a teacher so I don’t know how much of this is already trying to be addressed in math education throughout the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In talking to and tutoring fellow students, I have seen that most students think of math as solely applying formulas to given situations by rules given to you. Very few students see the beauty and creativity that goes into math. One consequence this seems to have is that students who find that they are bad at this rote application of formulas say that they are “bad at math”. Many of these people are creative and logical though, so they could possibly work with what in my opinion is the much more interesting math. Probably my favorite article on this is &lt;a href=&quot;http://worrydream.com/refs/Lockhart-MathematiciansLament.pdf&quot;&gt;&lt;em&gt;A Mathematician’s Lament&lt;/em&gt; by Paul Lockhart&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is what is being done about this in education and what can I do to help? I’m just a tutor, but I do try to show the students I work with the more interesting and fun parts of whatever subject they are doing. The more interesting proofs and properties and logic that goes with them.&lt;/p&gt;&#xA;" OwnerUserId="374" LastEditorUserId="75" LastEditDate="2014-03-27T10:55:57.790" LastActivityDate="2014-03-27T17:28:04.133" Title="Student Poisoned Experience with Math" Tags="|course-design|secondary-education|anxiety|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="896" PostTypeId="2" ParentId="41" CreationDate="2014-03-27T08:23:43.483" Score="16" Body="&lt;p&gt;I would like to add a point that I've not seen explicitly mentioned in the other answers.  That is that &lt;strong&gt;context&lt;/strong&gt; can often be your aid in clarifying your lettering.  As a trivial example, no one is going to misread &lt;code&gt;they&lt;/code&gt; as &lt;code&gt;+hey&lt;/code&gt;.  So the &quot;special alphabet&quot; is primarily for &lt;em&gt;mathematics&lt;/em&gt; in your lectures.  (That said, you should cultivate a clear hand writing for normal text but it doesn't need to be as extreme as for the mathematical part.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, if you never write $\times$ then the students will soon realise that &lt;code&gt;x&lt;/code&gt; means $x$ and not $\times$.  And I doubt that someone will mistake $x_k$ or $\vec{x}$ for $\times_k$ or $\vec{\times}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, &lt;em&gt;scale&lt;/em&gt; is often hard to see on a board so &lt;code&gt;X&lt;/code&gt; and &lt;code&gt;x&lt;/code&gt; are not easy to distinguish.  It can be awkward to avoid using these together (consider the common desire to have $x \in X$) and distinguishing them visually can be difficult, but then context can help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some practical tips:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Go in to a large lecture hall, write some random equations on the board, and then go to the back and try to read what you wrote.  Then modify your handwriting until you think it is clear.  This won't be perfect (since you know what you wrote), but it'll at least be a lot better than not doing it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Say what you write as you write it, and make sure that before you write anything on the board then the students are at the same point as you so that they are paying attention to what you say as you write it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cultivate writing &quot;the wrong way&quot;.  It's a bit trickier for those not blessed with sinister tendencies, but if you can learn to write so that you do not stand in front of what you just wrote then it will be much easier for the students to see what you write as you write and say it.  This doesn't mean writing with your left hand, it is possible to do this with the right hand but the body position takes a bit of getting used to.  This has the added bonus of making you more turned towards the students as you write.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Having just given a lecture, I'd add one more practical point: ensure that the board is clean before you write on it.  Stray squiggles left over from before you erased it can be confusing.&lt;/p&gt;&#xA;" OwnerUserId="112" LastEditorUserId="112" LastEditDate="2014-03-27T20:15:36.137" LastActivityDate="2014-03-27T20:15:36.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="897" PostTypeId="1" CreationDate="2014-03-27T08:38:35.760" Score="31" ViewCount="560" Body="&lt;p&gt;This site is - as far as I'm aware - for what I would term &lt;em&gt;working&lt;/em&gt; maths educators.  That is, on the whole the users of this site are not researchers in mathematics education.  Rather, we are the people who get on with the work of providing mathematics education every day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, there are some people out there who do actual research on mathematics education.  There are more who study education in general.  I have a hope that some of the results of this would be useful to people like me, but I find it hard to get hold of stuff that I can directly use.  In particular, what I want is ideas on how to improve my teaching that:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Are based on actual research.  That is, its effectiveness has been proved.  (Of course, this might be that it has been proved to be effective in teaching Australian Skiers how to do Differential Topology, but that's still better than &quot;It sounds like a good idea so let's give it a try.&quot;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are practical.  I've heard my fair share of general theories of learning and education and general theories are not all that useful to me because it takes time to translate them in to things that I can actually do in my lecture hall.  Moreover, the opportunities for experiment are somewhat limited because if I get it wrong, the students tend not to come back for a second attempt.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are mathematical.  Here the problem is that many techniques are subject-specific and what works well in a history class or philosophy class may not be so good in maths (&quot;Get in to groups of three and discuss the Gauss Elimination Algorithm.&quot;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What I'm looking for is a way in to the maths education literature.  I don't really want to start reading random articles because I don't have the necessary background to evaluate them.  Maybe eventually I'll get to that level, but for the moment I need some guidance.  However, although short of time and expertise, I do have a reasonable level of intelligence and motivation so I think that if someone could provide me with a road map then I'd do a reasonable job of finding my way from that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I don't want a list of all maths education departments or journals.  I want a recommended initial reading list.  But also that should be knit to practice since I don't want to become a maths education researcher myself, just that I want to make use of the fact that they exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, where do I start reading?&lt;/p&gt;&#xA;" OwnerUserId="112" LastEditorUserId="112" LastEditDate="2014-03-27T20:05:37.610" LastActivityDate="2019-04-10T12:43:02.937" Title="What Math(s) Ed literature is accessible to the working math(s) educator?" Tags="|education-research|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="898" PostTypeId="2" ParentId="807" CreationDate="2014-03-27T08:38:39.757" Score="13" Body="&lt;p&gt;I don't see my thoughts expressed by anyone here so perhaps I can chip in! I'm currently a graduate student but I was once that kind of bright bored student you are talking about. I had my fair share of boredom, lost motivation, exciting learning, disappointment and enthusiastic crankery. I can tell you that taking more interesting courses and finding good books to read will happen naturally - you don't have to help too much with that. Of course you can keep it at that but there's so much more you can do. Perhaps it won't take more than a couple of hours a week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students don't need more courses or books what do they need? They need a mentor and a friend. They need someone like themselves but who has achieved something in life (If you are a professor that's probably enough success right there) despite difficulties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They need a mentor to help them become a better student. They may be bright but how is their work ethic? How is their stamina and long-term motivation? Does the student know how to keep working on a problem day after day AFTER they've lost the motivation? Does the student know how to openly complain about their difficulties? Those are very necessary skills that they are not likely to develop naturally. An hour a week can make a huge difference for the student over the course of their undergraduate years and it probably beats answering another few e-mails. Not to mention that you get to help shape a promising student into something more than they would become on their own during the natural course of their studies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On of the most important things a mentor can do is to take initiative. Schedule to meet each other but not in your office. The office, the desk and the chairs are barriers to communication. With these things you are an authority figure and they'll have a hard time relating to you well. Meet elsewhere and eat lunch together. Talk about stuff - ideally it's not going to be just about math the first few times. The more real and honest you are about things, the better it'll be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bright students need to hear your stories of how you had difficult times or how you tried to prove insert-big-open-problem-here at some point before you even had the tools to work with. They need to hear about procrastination and what they can do about it. Ideally, you'll share some memories of failures past and what you did to cope with them. Hopefully the student will be inclined to share their personal story as well. You might even have trouble shutting them up. If the student complains to you about their difficulties you are on the right track. You'll be someone they can talk to and confide in when the difficult times come. This is especially true if they end up doing some sort of undergrad research and even more true if they go to graduate school later on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they have a good time with you at your meetings they will want to come back for more and they'll want to learn from you and perhaps become more like you. From there on it won't be difficult to inspire them and teach them how to become better mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately my experiences with professors have rarely been this good and I only know about the things I'm sharing with you because of my own struggles as a student and because I've had amazing mentors outside of academia.&lt;/p&gt;&#xA;" OwnerUserId="390" LastActivityDate="2014-03-27T08:38:39.757" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="899" PostTypeId="1" AcceptedAnswerId="911" CreationDate="2014-03-27T09:29:08.497" Score="15" ViewCount="3272" Body="&lt;p&gt;In our university both math and engineering students attend in the same calculus classes. There are arguments in our department about the possible influences of this approach on students. It seems based on the students' needs in their future career they need different types of calculus. For example in a calculus class for the math students, the teacher should emphasize on the abstract aspects, proofs, generalizations and the interactions with the other fields of mathematics and in a calculus class for engineering students, the applications and calculation are more important features of the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; Is there any difference in teaching calculus for the math and engineering students? Should math and engineering students attend in separated calculus classes?      &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-27T10:48:49.763" LastActivityDate="2017-06-11T07:41:44.327" Title="Is there any difference between teaching calculus for math and engineering students?" Tags="|mathematical-pedagogy|calculus|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="900" PostTypeId="2" ParentId="899" CreationDate="2014-03-27T09:44:46.603" Score="13" Body="&lt;p&gt;Tentatively, yes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the universities I've taught, there are substantial differences between those courses: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For engineering students, the courses in calculus (and linear algebra) provide some supplemental, rather abstract information which helps to understand the techniques taught in other classes, such as mechanics and electrodynamics. This is why I think that a good calculus course for engineering students has a curriculum which is coordinated with the engineering classes, uses a lot of examples from those subjects and is light on hard proofs (uses motivations and rough explanations instead).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus for mathematicians is a main subject on its own; it should be thorough and you can afford not to be in tune with some other classes which cover related subjects. Proofs should be precise, but examples can still help to understand the theorems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's worth noting that students of physics deserve their own calculus curriculum as well due the use of more advanced tools in modern theoretical physics compared to the engineering classes.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-03-27T09:44:46.603" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="901" PostTypeId="2" ParentId="805" CreationDate="2014-03-27T11:08:48.323" Score="17" Body="&lt;p&gt;I would like to bring up something for consideration that's really come from the discussion in the comments on &lt;a href=&quot;https://matheducators.stackexchange.com/a/809/112&quot;&gt;Benjamin Dickman's answer&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a problem with the dichotomy presented in the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;my choice boils down to this: should I drop questions from my exam, even though this means fewer topics on the exam, in order that more students will not experience time-pressure during the exam?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In Norway we seem to have done our best to do away with time pressure.  Our exams are 4 hours long and are not (in the lower level courses) designed to take anywhere near that length of time.  Nevertheless, it is rare that students leave before the time is up and it is still common to see &quot;I ran out of time&quot; written on exam scripts.  So simply making the exam shorter does not remove the time pressure.  It also has some perhaps undesirable side effects.  Since it is now feasible for a reasonable student to complete all the tasks, a good student is expected to do so and to do so well.  This makes the exam even more sensitive to &quot;computation error&quot; than before since a good student cannot compensate for making a few such errors by completing more questions.  Sometimes this is appropriate, but not always.  So shortening the exam not only does not completely alleviate time pressure but also increases the accuracy pressure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An alternative is that which is (or was - my experience is now approaching 20 years ago) practised by Oxford University.  The exams were designed so that it was &lt;strong&gt;impossible&lt;/strong&gt; to complete all the questions.  Even just writing out an answer to a question without having to think took about half an hour, so in a 3hr exam the most even the best students could do was about six questions.  There were nine on each exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To emphasise the point: in my year, the best student's best papers consisted of 6 good questions (given that the best of these consisted of Representation Theory and $C^*$-algebras, it's a bit confusing why he's gone in to Differential Topology, but there's no accounting for taste).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This had the effect of replacing &lt;em&gt;time&lt;/em&gt; pressure by &lt;em&gt;choice&lt;/em&gt; pressure.  You now had a choice as to which questions to answer, and thus could, to a limited extent, compensate for lack of ability in one area by excelling in another (of course, the questions were designed so that you couldn't get away with being completely ignorant of the core).  This was exaggerated by the fact that the final grade was cumulative across &lt;em&gt;all&lt;/em&gt; exams, not just one; but I recognise that that is not possible in the modular system practised by most universities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The emphasis on choosing questions to do well on was further strengthened by the exotic mark scheme that Oxford used (sadly, I hear no longer) which put far greater emphasis on completed questions than on partial questions.  So one complete question was worth &lt;strong&gt;four&lt;/strong&gt; half completed questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, the purpose of this type of exam is to give the good students sufficient space to show that they are good, whilst giving the excellent students room to show off, and giving the less able students every opportunity to show that they have learnt &lt;em&gt;something&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a system where you don't have complete freedom to revise the overall grading system, this could still be implemented in a course simply by revising the grading scheme for that exam.  The idea is to make sure that it is not out of 100 and to &lt;strong&gt;not scale the points&lt;/strong&gt;.  So an A is not 90% but is some number of points (actually, I wouldn't go for a point scheme at all, but that's a different issue), and a B is not 80% but is some slightly smaller number of points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the Oxford system, in my year, then one of the criteria for a first class degree was (roughly) to get at least 20 good questions and 20 fairly good questions.  That's spread over 13 papers, so that's just under 2 good and 2 okay questions per paper.  That's from a total of 9 questions per paper, so there's plenty of room there for a candidate to choose their questions wisely, and plenty of time for a good candidate to achieve that level.&lt;/p&gt;&#xA;" OwnerUserId="112" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-03-27T11:08:48.323" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="902" PostTypeId="2" ParentId="895" CreationDate="2014-03-27T11:38:02.463" Score="13" Body="&lt;p&gt;If you want to get a sense for what great math education looks like in 2014, you're blessed with an abundance of options.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pick up the &lt;a href=&quot;http://investigations.terc.edu/&quot;&gt;TERC Investigations&lt;/a&gt; curriculum, or the &lt;a href=&quot;http://cmeproject.edc.org/cme-project/precalculus&quot;&gt;CME Project&lt;/a&gt;, or grab a copy of some of the &lt;a href=&quot;http://www.nctm.org/publications/toc.aspx?jrnl=mtms&quot;&gt;NCTM journals&lt;/a&gt;. Or go check out some of the amazing work that educators are doing and &lt;a href=&quot;http://mathtwitterblogosphere.weebly.com/&quot;&gt;sharing for free online&lt;/a&gt;. Shoot, go check out &lt;a href=&quot;http://christopherdanielson.wordpress.com/&quot;&gt;Christopher Danielson's blog&lt;/a&gt;, or &lt;a href=&quot;http://blog.mrmeyer.com/&quot;&gt;Dan Meyer's blog&lt;/a&gt;, or &lt;a href=&quot;http://function-of-time.blogspot.com/&quot;&gt;Kate Nowak's blog&lt;/a&gt;, or &lt;a href=&quot;http://samjshah.com/&quot;&gt;Sam Shah's blog&lt;/a&gt;, or &lt;a href=&quot;http://nicoraplaca.com&quot;&gt;Nicora Placa's blog&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, you write that you try to show kids the &quot;interesting&quot; parts of math. I know what you mean -- things like knots or groups or puzzles and games, much in the spirit of Lockhart and those influenced by him. (I'm thinking of &lt;a href=&quot;http://christopherdanielson.wordpress.com/2014/02/13/edsurgemath/&quot;&gt;this article&lt;/a&gt;, in particular.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's remarkable about all the people that I linked to above is that they are innovating in education while teaching the same topics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How you can help #1:&lt;/strong&gt; Don't advocate for changing the curriculum. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This lets the entire system off the hook. The problem is &lt;em&gt;not&lt;/em&gt; that school math is a different subject than real math, or at least it's not because the subjects that kids learn aren't genuinely mathematical. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;School math is genuinely mathematical, or at least it should be. (The Common Core Standards &lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot;&gt;say as much&lt;/a&gt;, by the way.) Every curriculum or person I linked to above excels at giving kids room to think mathematically about school topics. That means that rather than asking kids to memorize times tables, they ask kids to use strategies that make sense. Those educators use all their creativity and brain-power to figure out ways of presenting kids with opportunities to be mathematically creative with traditional topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How you can help #2&lt;/strong&gt; Use amazing resources for your students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;No educator is an island, and it's important that we don't all think through our teaching alone. I'm not saying that you are --  it sounds like you're a really thoughtful teacher! Still, it must be particularly difficult for you, as a tutor, to plan an activity or a lesson for your students. I would still recommend preparing yourself for your sessions as well as a teacher would prepare for class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your student is struggling with linear equations? Go do some reading about the struggles kids have with algebra. (May I recommend &lt;em&gt;Fostering Algebraic Thinking&lt;/em&gt;?) If your student is struggling with rational functions? Go check out a high-quality textbook that's full of interesting problems and activities on the topic. (May I recommend &lt;em&gt;CME Precalculus&lt;/em&gt;?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How you can help #3&lt;/strong&gt; Show your students the multiplicity of mathematical talent. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It sounds like you're doing this already, but if school math often overemphasizes getting the right answer of remembering a procedure, then your students don't need you to do that. Instead, emphasize other mathematical talents. Value questions as much as answers. Value thinking as much as answers. Value ways of seeing, and not just ways of manipulating. Value explanations as much as answers. I'm sure others can add to this list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for being such a thoughtful and supportive tutor! Your students are very lucky to have you.&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2014-03-27T16:10:47.377" LastActivityDate="2014-03-27T16:10:47.377" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="903" PostTypeId="1" AcceptedAnswerId="961" CreationDate="2014-03-27T12:13:59.770" Score="3" ViewCount="220" Body="&lt;p&gt;When one wants to let students calculate (Riemann) integrals in calculus, what is a good term to call this task?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to focus on what the main task is, you may call it &quot;Calculate the definite integral&quot; or &quot;Integrate a function&quot;, but as long as the function is continuous, one might use terms like calculating the &quot;primitive function&quot; or &quot;antiderivative&quot; (In Germany the only science-related term is &quot;Stammfunktion&quot; (=primitive function), but some people came up with the non-scientific term artificial term &quot;Aufleitung&quot; (maybe, a good &quot;translation&quot; would be &quot;ascrivative&quot;) meant to be the opposite of a derivative (&quot;Ableitung&quot; in German)). However this covers only a part of real world integrals, since the concept of integration is much more general than only being the opposite of taking derivatives. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are (dis)advantages of using terms like &quot;antiderivative&quot; instead of &quot;integrating&quot; or &quot;primitive function&quot;?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-03-28T14:35:40.057" Title="Better use &quot;Integrate a function&quot;/&quot;Calculate the definite integral&quot; or use terms like &quot;primitive function&quot;, &quot;antiderivative&quot;, &quot;Aufleitung&quot; (German)?" Tags="|calculus|notation|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="904" PostTypeId="2" ParentId="897" CreationDate="2014-03-27T12:18:37.517" Score="20" Body="&lt;p&gt;For those with a background in mathematics, perhaps the following source would be of use:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Curtis C. McKnight (Ed.). (2000). &lt;em&gt;Mathematics education research: A guide for the research mathematician&lt;/em&gt;. American Mathematical Soc.. &lt;a href=&quot;http://books.google.com/books?id=K2buDjkL5a0C&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A review of the book can be found on the MAA website &lt;a href=&quot;http://www.maa.org/publications/maa-reviews/mathematics-education-research-a-guide-for-the-research-mathematician&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From this latter link:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In a nutshell, this book is a guide for reading and evaluating mathematics education research, especially undergraduate mathematics education research. Its purpose seems to be to encourage research mathematicians, who do much of the teaching of undergraduate mathematics, to use education research to actually inform their teaching. In order to use this research, though, mathematicians need to know how to find and evaluate the research already done, and this is the focus of this book.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is plenty more to the review, and the book itself is quite readable. (Unfortunately, only part is available through the Google Books link provided above; please edit in a full link if you find one!)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I might add that the MAA has a page on &lt;strong&gt;&lt;a href=&quot;http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/teaching-and-learning&quot;&gt;Teaching and Learning&lt;/a&gt;&lt;/strong&gt;. You could also &lt;strong&gt;&lt;a href=&quot;http://www.maa.org/search/node/education&quot;&gt;search&lt;/a&gt;&lt;/strong&gt; the site for &lt;em&gt;education&lt;/em&gt; to find relevant links. For example, the second result is the &quot;Searching for Common Ground&quot; reference provided in a separate answer here from M Jansen.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-04-23T15:34:29.287" LastActivityDate="2014-04-23T15:34:29.287" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="905" PostTypeId="2" ParentId="269" CreationDate="2014-03-27T12:26:44.963" Score="3" Body="&lt;p&gt;The book &quot;&lt;a href=&quot;http://jhupbooks.press.jhu.edu/ecom/MasterServlet/GetItemDetailsHandler?iN=9780801890123&amp;amp;qty=1&amp;amp;source=2&amp;amp;viewMode=3&amp;amp;loggedIN=false&amp;amp;JavaScript=y&quot; rel=&quot;nofollow&quot;&gt;Adventures in Group Theory&lt;/a&gt;&quot; by David Joyner develops abstract algebra and uses it to &quot;study certain toys and games from a mathematical modelling perspective&quot;. The main example is the Rubik's Cube, but there are plenty of others. For example, Chapter 4 is entitled &quot;A procession of permutation puzzles&quot; and looks at:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Sam Loyd's 14-15 puzzle. This is a 4x4 sliding grid with tiles numbered 1 to 15, and the bottom right square is empty. The 14 and the 15 have switched places, and the &quot;challenge&quot; is to solve it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The hockeypuck puzzle. Cut a hockeypuck into 12 pieces as you would a cake (but using a saw as opposed to a knife...), and number them. You can flip the pie along a cut, and you can rotate it (by multiples of $\pi/6$). Solve it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Rainbow Masterball&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Pyraminx&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The $2^3$ and $3^3$ Rubik's cubes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Skewb&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Megaminx&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am unsure how it is as a set text or as an introduction to group theory. However, as a source of ideas and examples it is very good! Also, the text is full of SAGE examples, which you may or may not consider to be a good thing. However, if you want to teach someone SAGE or similar then using these kinds of puzzles and games is perhaps a good way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, using puzzles allows you to (tentatively) introduce group actions. Which is a very important way of looking at groups (but then I am a biased geometric group theorist).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; David Joyner is one of the authors of the book &quot;&lt;a href=&quot;http://jhupbooks.press.jhu.edu/ecom/MasterServlet/GetItemDetailsHandler?iN=9780801878220&amp;amp;qty=1&amp;amp;viewMode=3&amp;amp;loggedIN=false&amp;amp;JavaScript=y&quot; rel=&quot;nofollow&quot;&gt;Applied Abstract Algebra&lt;/a&gt;&quot;. This looks interesting and very relevant, but I know nothing about it other than what Google can provide.&lt;/p&gt;&#xA;" OwnerUserId="169" LastEditorUserId="169" LastEditDate="2014-03-27T14:00:22.790" LastActivityDate="2014-03-27T14:00:22.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="906" PostTypeId="2" ParentId="630" CreationDate="2014-03-27T13:41:46.653" Score="13" Body="&lt;p&gt;&lt;strong&gt;Edit (5/24/14):&lt;/strong&gt; For the reader interested in a somewhat longer answer, I am including the literature review (and all references) from my thesis on conceptions of creativity with regard to problems posed from the multiplication table. A copy of the excerpt can be found &lt;a href=&quot;https://www.dropbox.com/s/8v9kzhslyczch54/Chapter%20II%20Dickman.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. My original, shorter answer remains un-edited below.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The story of &lt;em&gt;mathematical creativity&lt;/em&gt; is long and probably deserves an entire book dedicated to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One (perhaps subtle) difficulty is that it's tough to define what is meant by &lt;em&gt;creativity&lt;/em&gt;. For example, in&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Treffinger, D. J., Young, G. C., Selby, E. C., &amp;amp; Shepardson, C. (2002). Assessing Creativity: A Guide for Educators. National Research Center on the Gifted and Talented.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;the authors write: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Treffinger (1996) reviewed and presented more than 100 different definitions from the&#xA;  literature. Aleinikov, Kackmeister, and Koenig (2000) offered 101 contemporary&#xA;  definitions from children and adults (p. 5).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Furthermore, there is a debate about whether or not creativity is domain-specific (i.e., whether one can be &lt;em&gt;generally creative&lt;/em&gt; or whether one should instead use words like &lt;em&gt;mathematically creative&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than attempt to review the creativity literature in its entirety, let me hit a few high points:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1.$ Creativity research did not take off until about 1950, when the president of the APA, J.P. Guilford, gave his inaugural address (aptly) titled &lt;em&gt;Creativity&lt;/em&gt;, in which he called for the concept to be investigated - primarily in terms of various traits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$2.$ Preceding Guilford's address, related work came from the literature on &lt;em&gt;thought&lt;/em&gt;. Wallas' (1926) book &lt;em&gt;The Art of Thought&lt;/em&gt; is the first to put forth the five-stage model of problem solving: preparation, incubation, intimation, illumination, and verification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$3a.$ There are numerous descriptions of mathematical creativity that accord with the incubation model described by Wallas. The most famous example is Poincare stepping onto an omnibus and having a revelation about Fuschian functions; this is contained in his work &lt;em&gt;The Foundations of Science&lt;/em&gt; in a chapter (again, aptly) titled &lt;em&gt;Mathematical Creation&lt;/em&gt;. This book, and that chapter in particular, encouraged another mathematician to write his own book on mathematical creativity: Jacques Hadamard's &lt;em&gt;The psychology of invention in the mathematical field&lt;/em&gt;. (This is all suggested reading!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$3b.$ For other examples, see Cohen's write-up of his development of forcing (related to my MO answer &lt;a href=&quot;https://mathoverflow.net/questions/159935/sources-of-theorem-drafts-by-the-original-author/160508#160508&quot;&gt;here&lt;/a&gt;) or Gauss' comment about post-incubation illumination, cited in Klamkin's (1994) &lt;em&gt;Mathematical Creativity in Problem Solving and Problem Proposing II&lt;/em&gt; as recorded by K. Knopp:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...all the brooding, all the searching was for nothing; finally, a few days ago I succeeded. But not by long searches but by the sheer grace of God, I may say, like lightning strikes, the riddle was solved; I myself would not be able to find the connection between what I knew previously, with what I used for my last attempt and with what finally succeeded.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The lesson of $3a$ and $3b$ is that even great mathematicians are (often) metacognitively unaware as to what produced their final creative insight. What's clear, though, is that before the incubation period, one needs to work consciously on a problem (Wallas' first stage). To quote Poincare:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is another remark to be made about the conditions of... unconscious work: it is possible, and of a certainty it is only fruitful, if it is on the one hand preceded and on the other hand followed by a period of conscious work... Perhaps we ought to seek the explanation in the preliminary period of conscious work which always precedes all fruitful unconscious labor.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In a related vein, George Polya writes (semi-humorously):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Past ages regarded a sudden good idea as an inspiration, a gift from the gods.  You must deserve such a gift by work, or at least a fervent wish.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(For a very outdated and, in my opinion, poorly written article that also emphasizes work, see Adler's (1984) &lt;em&gt;Mathematics and Creativity&lt;/em&gt; in &lt;em&gt;Mathematics, People, Problems, Results v2&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$4a.$ Your question asks: &quot;How can educators encourage creativity without sacrificing correctness?&quot; I think the key is to conceive of creativity as produced by &lt;em&gt;conscious work&lt;/em&gt;. One account in the creativity literature to this end is provided by Howard Gruber; see his book &lt;em&gt;Darwin on Man&lt;/em&gt;, or compare the Gauss quotation above (with the frequently used metaphor of lightning) to the excerpt &lt;a href=&quot;http://books.google.com/books?id=ZYo5AAAAIAAJ&amp;amp;lpg=PA243&amp;amp;ots=OcakkuMrdd&amp;amp;dq=mount%20olympus%20gruber&amp;amp;lr&amp;amp;pg=PA243#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Much of Gruber's work helps to demystify creativity; to this end, helping students see that even great theoretical advances are a product of hard work and not some magical power can encourage innovation and creativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$4b.$ For a concrete suggestion as to how educators can encourage creativity, I would suggest incorporating problem posing (not just problem solving) into the classroom. This relates to creativity and the literature on &lt;em&gt;problem finding&lt;/em&gt;; you can find a more extended discussion on the relation between the two in the section &lt;em&gt;Problem posing as a feature of creative activity or exceptional mathematical ability&lt;/em&gt; in:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Silver, E. A. (1994). On Mathematical Problem Posing. &lt;em&gt;For the learning of mathematics, 14&lt;/em&gt;(1), 19-28.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Sources relating the above work to creativity can be found &lt;a href=&quot;http://scholar.google.com/scholar?q=creativity&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=5,33&amp;amp;sciodt=0,33&amp;amp;cites=11208674570557059518&amp;amp;scipsc=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, e.g., &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. &lt;em&gt;ZDM, 29&lt;/em&gt;(3), 75-80.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Finally, as far as problem posing, the classic text is Brown and Walter's &quot;The Art of Problem Posing,&quot; which provides a very accessible account of how to begin with a given mathematical scenario and use it to pose increasingly complex problems, as well as how this can be incorporated into an actual course.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-05-25T01:09:44.723" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="907" PostTypeId="1" CreationDate="2014-03-27T13:59:51.750" Score="20" ViewCount="1229" Body="&lt;p&gt;There is a great quote of Yitz Herstein:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A number of such problems can be found in Herstein's classic book &quot;Topics in Algebra&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given appropriate background, certain mathematical problems lead us to discover things we otherwise might not, often with elegance and surprise. These problems make us feel more clever than we actually are by our discovery of important and deep ideas. They have the delight and surprise of a competition problem, but reveal an important idea for a theory...and the discovery of the idea is accompanied by a lovely sense of ownership since the &quot;would-be solver&quot; has found it. (I'm thinking about a particular problem from Herstein's book that basically forces the idea of a quotient group on the investigator. I don't reveal which problem this is since I don't want to ruin it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyhow, the purpose of this question is to collect examples of such problems.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are some good problems that require the discovery of central and difficult concepts of a theory?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Since such problems usually require background, please include the requisite background with your answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please note: I'm not so sure it is a good idea to post excellent problems along with the concepts they lead to...if a problem does what I want it to, one should be able to discover the idea that is needed by working on the problem. &lt;em&gt;A good answer to this question might omit explicit mention of what idea the problem leads to.&lt;/em&gt; &lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2014-03-27T15:16:16.480" LastActivityDate="2016-12-22T18:32:10.923" Title="Good problems that uncover difficult points in a theory" Tags="|undergraduate-education|undergraduate-research|problem-design|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="908" PostTypeId="2" ParentId="907" CreationDate="2014-03-27T14:42:30.610" Score="14" Body="&lt;p&gt;&lt;strong&gt;Edit (Dec 2016):&lt;/strong&gt; Encouraged by a few comments on SE, and a few direct emails about this post, I wrote up the ideas below for a journal of math education. The citation, and linked pre-print, are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dickman, B. (2017). Enriching Divisibility: Multiple Proofs and Generalizations. &lt;em&gt;Mathematics Teacher, 110&lt;/em&gt;(6), 416-423. &lt;a href=&quot;https://www.researchgate.net/publication/303920376_Enriching_Divisibility_Multiple_Proofs_and_Generalizations&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Pre-Print&lt;/strong&gt;&lt;/a&gt; (no pay-wall).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(Adapted from a lecture described &lt;a href=&quot;https://mathoverflow.net/a/152300/22971&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Requisite background:&lt;/strong&gt; Secondary school mathematics for the problem, but moving forward is a nice way to introduce concepts from Number Theory and Abstract Algebra.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem:&lt;/strong&gt; Observe that &lt;span class=&quot;math-container&quot;&gt;$(n-1)n = n^2 - n$&lt;/span&gt; is always even. How would you generalize this result?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In particular, consider the following equivalent formulations:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Statement A:&lt;/strong&gt; The product of any &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; consecutive natural numbers is divisible by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Statement B:&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$n \in \mathbb{N}$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$n^2 - n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For me, generalization is a cornerstone of mathematical thinking and an important learning outcome for mathematics students. It is discussed in Polya's (1945) &lt;em&gt;How to solve it&lt;/em&gt;, and there are some nice related comments in Wertheimer's (1953) &lt;em&gt;Productive Thinking&lt;/em&gt;, but I shan't delve deeper into this point here.&lt;/p&gt;&#xA;&lt;p&gt;For the question at hand: I think the most natural generalizations come from replacing &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Generalization A:&lt;/strong&gt; The product of any &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; consecutive natural numbers is divisible by &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Generalization B:&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$n \in \mathbb{N}$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$n^k - n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Side-note: I once distributed forms to fifty prospective mathematics teachers in a course on Problem Solving - not taught by me - where twenty-five received Statement A and the other twenty-five received Statement B, and all were asked to state a generalization without needing to provide a proof. The result was that almost no student/prospective teacher was able to complete the task properly, suggesting there is some deep misunderstanding insofar as what constitutes a &lt;em&gt;generalization&lt;/em&gt; in mathematics. I won't include sample responses here, since I find them very disheartening. However, you might consider replicating such a task with your own students to see how they fare.)&lt;/p&gt;&#xA;&lt;p&gt;The two generalizations above now &amp;quot;require the discovery of central and difficult concepts of a theory.&amp;quot; More precisely, proving the first statement requires the very essence of modular arithmetic, which students might understand without having formally studied Number Theory, whereas the latter generalization is false as stated. (For example, take &lt;span class=&quot;math-container&quot;&gt;$k = 4$&lt;/span&gt;.) The natural follow-up is to classify &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; for which Generalization B holds, which is the content of Fermat's Little Theorem, proved most easily (in my opinion) as a corollary of Lagrange's Theorem in a first course on Group Theory.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-12-22T18:32:10.923" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="909" PostTypeId="2" ParentId="318" CreationDate="2014-03-27T15:28:32.573" Score="4" Body="&lt;p&gt;Since this question is still in the unanswered category, I'll repost my comment as an answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar to the tag graph of MathSE, there is the Mathematics Subject Classification of Math Reviews and Zentralblatt. The search/browse mask of the &lt;a href=&quot;http://zbmath.org/classification/&quot; rel=&quot;nofollow&quot;&gt;Zentralblatt&lt;/a&gt; looks nicer than the one of &lt;a href=&quot;http://www.ams.org/msc/msc2010.html&quot; rel=&quot;nofollow&quot;&gt;Math Reviews&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunatly, this is just a classification tree by topic, and not a competency tree.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-03-27T15:28:32.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="910" PostTypeId="2" ParentId="907" CreationDate="2014-03-27T15:47:39.593" Score="8" Body="&lt;p&gt;I think there are several good problems that can be explored (e.g., using the Moore method) by beginning with a word or term and trying to axiomatize it. I happen to think that assembling several of these words/terms and axiomatizing them would make for a nice textbook, but I digress...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to the question: Some examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What should &quot;bigger than&quot; mean?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Similarly: What should &quot;at least as big as&quot; mean?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, what should &quot;equivalent&quot; mean?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this answer, I will explore briefly what happens when you ask what &quot;closeness&quot; should mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Required Background:&lt;/strong&gt; Some experience with proof-writing; naive set theory.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Problem:&lt;/strong&gt; (to be explored with the Moore method or otherwise scaffolded) Let $X$ be a non-empty set and let $\mathbf{K}: P(X) \rightarrow P(X)$ be a function whose input and output are subsets of $X$. Given a subset $A \subset X$, we wish to define our function so that $\mathbf{K}(A)$ captures the set of elements &quot;close&quot; to $A$. What sort of properties should $\mathbf{K}$ have?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Next, one might proceed by considering $A, B \subset X$ and the various ways in which they can be related.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What could (or should) we say about $\mathbf{K}(A \cup B), \mathbf{K}(A \cap B), \mathbf{K}(A - B)$ or $\mathbf{K}(A)$ and $\mathbf{K}(B)$ when $A \subset B$? What about $\mathbf{K}(\emptyset)$ or $\mathbf{K}(X)$? What if we repeatedly apply $\mathbf{K}$ to some subset? Etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These questions can also be thought of (or translated into) sentences about closeness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the statement &quot;nothing is close to nothing&quot; corresponds to $\mathbf{K}(\emptyset) = \emptyset$. (1)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, it seems reasonable to assert something like &quot;everything &lt;em&gt;in&lt;/em&gt; a subset is also &lt;em&gt;close&lt;/em&gt; to that subset.&quot; This can be written as: For all $A \subset X$, we have $A \subset \mathbf{K}(A)$. (2)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition to (1) and (2) above, one can similarly motivate two more statements. Namely,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathbf{K}(\mathbf{K}(A)) \subset \mathbf{K}(A)$ (3) and $\mathbf{K}(A \cup B) = \mathbf{K}(A) \cup \mathbf{K}(B)$ (4). (Describe (3) and (4) in words!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the four items as above, we now have the &lt;a href=&quot;http://en.wikipedia.org/wiki/Kuratowski_closure_axioms&quot;&gt;Kuratowski closure axioms&lt;/a&gt;. In particular, our exploration of &quot;closeness&quot; has led us to an axiomatization equivalent to the standard one in Topology!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We might even make up some terminology to go along with our definitions. For example, suppose for some $A \subset X$ we find that $A = \mathbf{K}(A)$. In such a case, we could call $A$ &lt;em&gt;closed&lt;/em&gt; (with respect to $\mathbf{K}$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, this notion of being closed is equivalent to the standard topological one, as well.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-03-27T15:47:39.593" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="911" PostTypeId="2" ParentId="899" CreationDate="2014-03-27T15:54:12.213" Score="11" Body="&lt;p&gt;In my understanding, multiple representations of a concept benefit conceptual understanding. Hopefully the differences among departments can be negotiated to benefit students. Maybe a question to discuss is how we differentiate instruction for diverse learners in a single classroom. After all, even in a pure mathematics classroom all learners are at different places with their prior knowledge and different potential for learning.&#xA;So philosophically and in my opinion having students from engineering and mathematics (and physics, if you like) in the same classroom can benefit both groups. It does require competent teacher to be able to handle such class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do recognize significance of different contexts and different opinions, so, in reality some negotiations between departments have to happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A paper that might be of interest:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dunn, J. W., &amp;amp; Barbanel, J. (2000). One model for an integrated math/physics course focusing on electricity and magnetism and related calculus topics. American Journal of Physics, 68(8), 749-757.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some references (although mostly focused on k-12) in support of this kind of thinking:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Ellis, E., Gable, R. A., Gregg, M., &amp;amp; Rock, M. L. (2008). REACH: A framework for differentiating classroom instruction. Preventing School Failure, 52(2), 31-47.&lt;/li&gt;&#xA;&lt;li&gt;Nunley, K. (2004). Layered curriculum (2nd ed.). Amherst, NH: Brains.org&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="95" LastEditorUserId="95" LastEditDate="2014-03-27T16:10:20.583" LastActivityDate="2014-03-27T16:10:20.583" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="912" PostTypeId="2" ParentId="899" CreationDate="2014-03-27T16:03:18.943" Score="0" Body="&lt;p&gt;Sure.  For example, engineers don't mess with fields apart from the real numbers.  They only mess with quite particular Hilbert spaces.  They don't mess with Galois fields apart from GF(2) in communication and coding theory.  They don't work quaternions or generally non-commutative groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They'll not even learn the tools for computational analysis (like generating functions and everything around them).  If they do use generating functions, their variable is called $z^{-1}$.  If they use Fourier transforms, their domain variable is called $2\pi f$.  When they work with integral transforms, convergence is assumed.  They learn the formulas for main tension values, main inertial axes, and transformations of quadratic forms separately.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Number theory is not covered, tensors are mostly left to the physicists to play with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For three-phase electric systems, they have transforms used for diagonalizing circular impendance matrices without recognizing those transforms as a 3-point DFT.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, a lot of stuff is learnt multiple times under separate names, with separate scaling and naming constants and sign conventions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proofs are not really all that important.  It's more important for the engineers to be able to apply mathematical tools rather than to create them: a carpenter only needs rather limited knowledge of plant biology.&lt;/p&gt;&#xA;" OwnerUserId="405" LastActivityDate="2014-03-27T16:03:18.943" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="913" PostTypeId="2" ParentId="897" CreationDate="2014-03-27T16:16:23.853" Score="12" Body="&lt;p&gt;I very much welcome connections of research and teaching, although results from research still need or be interpreted in your specific situation. Let me give an example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might happen that your students can state the correct definition of a subspace of a vectorspace, but if you ask them if a specific set is a subspace, they &quot;try guessing&quot;. There is no research that says &quot;you should introduce the definition of a vector space in a specific way&quot;. However, there is research on how students work with definitions. Math educators mostly agree that for a given object or property, people usually have a picture in mind (maybe many pictures). We call this their &lt;em&gt;concept image&lt;/em&gt;. The students also might have a &lt;em&gt;concept definition&lt;/em&gt; in mind, hopefully the definition you gave to them. This distinction might help you see if your students try to compare the given subset with other subspaces they know, so they would work with their &lt;em&gt;concept image&lt;/em&gt; only. The question of how to introduce definitions thus becomes the question of how to help your students to work with definitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this example somehow matches your interest, I would suggest you to read the book &lt;a href=&quot;https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/8846/1/Alcock_Simpson_book.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ideas from Mathematics Education, (free download)&lt;/a&gt; by Alcock and Simpson. It was written explicitily for mathematicians. Both authors worked are researchers in mathematics education and they collaborate with teachers and researchers in mathematics. Concerning &lt;em&gt;concept image&lt;/em&gt; and &lt;em&gt;concept definitions&lt;/em&gt; they have recommendations for the examples you choose, what to emphasize when you work with definitions, etc.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="376" LastEditDate="2019-04-10T12:43:02.937" LastActivityDate="2019-04-10T12:43:02.937" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="914" PostTypeId="1" AcceptedAnswerId="920" CreationDate="2014-03-27T16:35:54.667" Score="8" ViewCount="365" Body="&lt;p&gt;I try to be not only a good teacher but also a good friend for my students. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I tell them that they can call me by my first name even in the formal colloquiums.&lt;/li&gt;&#xA;&lt;li&gt;I give them a personal phone number to call me about their math questions anytime.&lt;/li&gt;&#xA;&lt;li&gt;I try to spend some time each week/month for riding bicycles or watching a match with them. &lt;/li&gt;&#xA;&lt;li&gt;I ask about their possible personal problems and offer my help when I think there is a meaningful descending change on their grades.&lt;/li&gt;&#xA;&lt;li&gt;And so on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Once one of my colleges told me:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You pay too much attention to your students. They are just a part of your job as a university teacher. Your informal relations with students could be harmful for you and makes our job too hard because they expect all of us to treat them as like as you but we cannot respond this expectation properly and so they will think that we don't care them.      &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; How formal should I be as a university teacher? Can my informal behavior with my students have some disadvantages for me, my colleges, my students and my department? &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-27T18:15:59.983" LastActivityDate="2014-03-27T18:39:04.987" Title="How formal should I be as a university teacher?" Tags="|mathematical-pedagogy|" AnswerCount="1" CommentCount="19" ClosedDate="2014-03-27T20:09:22.360" ContentLicense="CC BY-SA 3.0" />
  <row Id="915" PostTypeId="2" ParentId="41" CreationDate="2014-03-27T16:49:29.647" Score="6" Body="&lt;p&gt;Taking my cue from printed math (and particularly TeX), I use different fonts for text mode and math mode.  The italic font (math mode) is cursive, while the roman font (text mode) is printed.  This helps distinguish ‘a’ from ‘$a$’, for example.  (And assuming that you and I are both using the default fonts on this site, then my handwriting looks pretty much like those two letters do on the screen.)  When I started this, I had a somewhat too exaggerated cursive style, so the fonts look less different now than they used to, but I maintain a difference in all letters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, after several false starts, I've settled on a little loop at the top of ‘0’ to distinguish it from ‘O’.  (Much as TeX uses merely slanted letters for math-mode capitals, I don't distinguish text-mode and math-mode capitals at all, saving the fancy cursive stuff for calligraphic font, which I use only in extreme circumstances.)&lt;/p&gt;&#xA;" OwnerUserId="408" LastActivityDate="2014-03-27T16:49:29.647" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="916" PostTypeId="2" ParentId="686" CreationDate="2014-03-27T16:55:12.883" Score="6" Body="&lt;p&gt;a neat theorem that highlights many emerging areas of research such as &lt;a href=&quot;http://en.wikipedia.org/wiki/Undecidable_problem&quot;&gt;(un)decidability&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Cellular_automaton&quot;&gt;cellular automata&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Turing_complete&quot;&gt;Turing completeness&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Emergence&quot;&gt;emergence&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Self-organizing_system&quot;&gt;self-organization&lt;/a&gt;, computer science etc: the proof that &lt;a href=&quot;http://en.wikipedia.org/wiki/The_Game_of_Life&quot;&gt;Life&lt;/a&gt; is Turing complete, originally by &lt;a href=&quot;http://en.wikipedia.org/wiki/John_Horton_Conway&quot;&gt;Conway&lt;/a&gt;. Conways original version seems possibly unpublished however there is a version by Rendell that is quite intuitive and there are even youtube videos. the original proof seems to date to the ~1960s.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rendell-attic.org/gol/tm.htm&quot;&gt;This is a Turing Machine implemented in Conway's Game of Life&lt;/a&gt; Rendell&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.ics.uci.edu/~welling/teaching/271fall09/Turing-Machine-Life.pdf&quot;&gt;A Turing Machine In Conway's Game Life&lt;/a&gt; pdf paper&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=My8AsV7bA94&quot;&gt;Game of life Turing machine youtube video&lt;/a&gt; &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="409" LastEditorUserId="409" LastEditDate="2014-03-27T17:00:58.317" LastActivityDate="2014-03-27T17:00:58.317" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="917" PostTypeId="2" ParentId="269" CreationDate="2014-03-27T17:27:43.647" Score="5" Body="&lt;p&gt;As dtldarek said telecommunications applications offer quite a few such examples. To make this more explicit I mention the following two (that I have tried in the past):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange&quot; rel=&quot;noreferrer&quot;&gt;Diffie-Hellman key exchange&lt;/a&gt;. After you have let the students prove that the group $\Bbb{Z}_p^*$ is cyclic (either always or for some specific suitable prime such as $p=107$ or $p=257$), then you can make them wonder, why the problem of solving $m$ from the congruence&#xA;$$ma\equiv b\pmod{p-1}$$ for a given $a$, coprime to $p-1$ and $b$ is easy, but the &quot;isomorphic&quot; problem of solving $m$ from the congruence&#xA;$$&#xA;g^m\equiv b\pmod p&#xA;$$&#xA;is difficult for a larger $p$ ($g$ a generator of $\Bbb{Z}_p^*$).&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://spacetimecodes.blogspot.fi/p/alamouti-code.html&quot; rel=&quot;noreferrer&quot;&gt;Alamouti coding&lt;/a&gt;. This is a method of adding diversity to radio transmission. Transmission from a single antenna will occasionally fade (a moving receiver will occasionally be in a node when a signal combines destructively with a reflected copy of itself). This technique uses two separate antennas transmitting the signal intelligently such that the signals of the two antennas always superpose constructively. It is based on a clever but relatively easy to describe application of the algebra of the division ring of Hamiltonian quaternions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My students commented on these in their teacher evaluation forms. The response was not unanimously positive (that will never happen), but some did say that such examples convinced them about the usefulness of abstract algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have mixed feelings about investing a lot of time on a course to examples like this. They do distract the students from the main material a little bit, they are not everybody's cup of tea, and the students may panic into thinking that they need to understand the example fully. The way I did it was to craft a few homework problems to serve as pons asinorum. Then in the homework problem sessions I took the stage (after a student had presented their solution), told them not to take notes, and spent may be 15-20 minutes outlining the application. I have enough control over my problem sessions so that this won't cause the session to run into overtime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also use permutation games as examples. 10-15 years ago a significant fraction of my students had played &lt;a href=&quot;http://www.mailcom.com/NokiaRotation.html&quot; rel=&quot;noreferrer&quot;&gt;Rotation&lt;/a&gt; on their cellphones. Its group theory is very simple. I am retrying it this Spring. I don't expect it to be a hit as the game is probably quite passé now.&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2014-03-27T17:34:31.323" LastActivityDate="2014-03-27T17:34:31.323" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="918" PostTypeId="2" ParentId="895" CreationDate="2014-03-27T17:28:04.133" Score="6" Body="&lt;p&gt;For new mathematics education research, I keep suggesting Jo Boaler's research, but only because I read &lt;strong&gt;What's Math Got to Do with It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject&lt;/strong&gt; and took the free online course that she gave last year on &quot;How to Learn Mathematics&quot; (a MOOC through StanfordX - it's open again soon but it is not free this time around).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boaler advocates the shift of mathematics education from the traditional, uninspired rote and memorization that many teachers and parents are used to seeing in the classroom (since they lived through those methods themselves), to the more active learning methods that challenges students to think of mathematics as a way to think about problems. Her book and course have more topics about this, ranging from shifting students' mindsets (neuroplasticity), creating collaborative classrooms, changing how we speak about achievement and assessments (making mistakes is actually learning), and organizing Number Talks to acknowledge how everyone has different methods in seeing and using mathematics. Boaler especially emphasized Carol Dweck's mindset research in &lt;em&gt;first&lt;/em&gt; letting students know that they can have a growth mindset; consequently this awareness can encourage better learning habits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, I encourage you to take all these topics into account for when you teach mathematics because sometimes it takes the right kind of encouragement to really change how people see and feel about mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As a side note, I get the &quot;I am bad at math&quot; comments from everyone, but once I let people in on this mindset idea they actually tell me that they start thinking about things differently!)&lt;/p&gt;&#xA;" OwnerUserId="261" LastActivityDate="2014-03-27T17:28:04.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="919" PostTypeId="2" ParentId="99" CreationDate="2014-03-27T17:35:45.430" Score="1" Body="&lt;p&gt;Let's remember that there are even advanced mathematicians still trying to work out inductive proofs. its a complex topic at the heart of math/CS also. Suggest studying the &lt;a href=&quot;http://en.wikipedia.org/wiki/Fibonacci_number&quot; rel=&quot;nofollow&quot;&gt;Fibonacci sequence&lt;/a&gt;. Suggest studying the different formulas for it, constrasting the recursive one versus Binet's formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suggest studying it from a CS point of view. Show computer experiments that compute recursive formulas/functions and then show &quot;debug output&quot; of them running. Show how an inductive proof is like a recursive program running. another great area for recursion/induction analysis is &lt;a href=&quot;http://en.wikipedia.org/wiki/Sorting_algorithm&quot; rel=&quot;nofollow&quot;&gt;sorting&lt;/a&gt; e.g. &lt;a href=&quot;http://en.wikipedia.org/wiki/Quicksort&quot; rel=&quot;nofollow&quot;&gt;quicksort&lt;/a&gt; and other recursive sort algorithms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many ideas/exercises in the following book, one of the classics that focuses on the tight coupling between induction &amp;amp; recursion in CS/math: &lt;a href=&quot;http://mitpress.mit.edu/sicp/&quot; rel=&quot;nofollow&quot;&gt;Structure &amp;amp; Intrepretation of computer programs&lt;/a&gt; Abelson/Sussman MIT which uses Scheme, a variant of Lisp (and built out of Lambda calculus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might help to study induction across different math fields, e.g. show an inductive proof in calculus, number theory, computer science one right after the other and show the basic &quot;architecture&quot; of the concept.&lt;/p&gt;&#xA;" OwnerUserId="409" LastEditorUserId="123" LastEditDate="2014-06-16T11:59:53.617" LastActivityDate="2014-06-16T11:59:53.617" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="920" PostTypeId="2" ParentId="914" CreationDate="2014-03-27T18:39:04.987" Score="7" Body="&lt;p&gt;This heavily depends on the culture that you are in. In the USA it is common for students to address their teacher by last name. If the teacher has a Ph.D., then you would call them Dr. Lastname. If the person has a military rank, then the rank overrules the academic degree. Even in the USA some schools have a tradition of being more informal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are from Europe, then there might be a different culture. If you are is South Korea, then there is a different culture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And there different cultures in different departments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it is impossible to answer how formal a university teacher should be with his/her students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think a good piece of advice (that I once got) is to ask your colleges about what they do. Go talk to your chairperson about what the standard is and the don't deviate from that. If one teacher is very formal with her students and you are very informal that might (I say might) create some conflict in a department. Students might think of the other teacher as being more strict and ask questions like &quot;Why isn't she more like ...&quot;. You don't want that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, here are some things that are worth considering when you are trying to figure out how formal/informal you want to be:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Does being more informal with student cause a problem if you have to report the student for academic misconduct in the future?&lt;/li&gt;&#xA;&lt;li&gt;Does being more informal put you at risk for sexual harassment? (You really want to take this one seriously!&lt;/li&gt;&#xA;&lt;li&gt;Does being more informal promote learning?&lt;/li&gt;&#xA;&lt;li&gt;Does being more informal put me at odds with the other faculty?&lt;/li&gt;&#xA;&lt;li&gt;Am I so close in age to my students that I might prefer to create a distance by being very formal?&lt;/li&gt;&#xA;&lt;li&gt;Is my school an exclusive private school where formality is assumed?&lt;/li&gt;&#xA;&lt;li&gt;Are my students underprivileged and might they actually benefit from be showing an interest in their personal lived?&lt;/li&gt;&#xA;&lt;li&gt;Will my students take advantage of a personal relationship?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(You can probably add more to this list yourself). Again: This is all culture dependent. The answers are going to look different. &lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-03-27T18:39:04.987" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="921" PostTypeId="2" ParentId="29" CreationDate="2014-03-27T20:32:44.610" Score="24" Body="&lt;p&gt;The instrument pictured was created by Barry Kurtz. He writes by email:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I completed my PhD under Bob Karplus at UC Berkeley.  I was his last PhD student.  My dissertation dealt with teaching for proportional reasoning.  I invented the idea of a &quot;water triangle&quot; to teach inverse proportions.  These were all made by the workshop at the Lawrence Hall of Science; they were not a commercial item.  I doubt any exist today; I certainly don't have any.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The thesis can be found through DigitalDissertations; the citation is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Kurtz, Barry Lloyd. &lt;em&gt;A Study of Teaching for Proportional Reasoning&lt;/em&gt;. University of California, Berkeley, ProQuest, UMI Dissertations Publishing, 1976.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An image of the water triangle appears on page 34:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/KNBWh.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an additional picture of an actual water triangle:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Y3zK2.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-03-28T09:43:05.027" LastActivityDate="2014-03-28T09:43:05.027" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="922" PostTypeId="2" ParentId="805" CreationDate="2014-03-27T22:27:53.210" Score="3" Body="&lt;p&gt;It depends on the type of math being tested and the students who are being tested. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, ALL exams are timed. However, some exams are designed to be easily finished in the time allowed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept of a timed exam is that it would be impossible or extremely challenging to complete the exam in the time allowed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of this should be to &lt;em&gt;measure a persons skill or ability&lt;/em&gt;, not to measure their &lt;em&gt;knowledge&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since most elementary schools measure knowledge, not skills or ability, then a timed exam would not be appropriate for them. Also, if the test is only a knowledge test, then again a timed exam is not appropriate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A timed exam is appropriate when you are measuring a person's mastery of skills and their ability to apply knowledge to complex problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT&#xA;After rereading your question, I think the true question is at the end - the true question is &quot;Should I drop topics to allow students time to finish.&quot; In fact, whether you drop topics or not, the students will not &quot;finish&quot; the exam, either due to lack of time or lack of questions!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the best solution is to create two exams if one exam cannot adequately cover all the topics, or arrange for a special double-long testing session.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, the distribution of your topics should be reconsidered if a certain period's teaching cannot be tested in one hour. If it cannot be tested in the allotted time, it probably cannot be learned in the alloted time either. If that is the case, then the student is doomed regardless of the testing format.&lt;/p&gt;&#xA;" OwnerUserId="421" LastEditorUserId="421" LastEditDate="2014-03-27T23:29:18.580" LastActivityDate="2014-03-27T23:29:18.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="923" PostTypeId="2" ParentId="130" CreationDate="2014-03-27T22:55:58.373" Score="16" Body="&lt;p&gt;I have faced exactly the same issue. I often teach masters level math courses to students getting a masters in education with a focus in mathematics; so these are future high-school math teachers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way I look at it, my real task is to provide them with mathematical enrichment and sophistication, to show them a larger context and provide a deeper perspective from which they might look upon the more elementary material with a greater degree of mathematical insight. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But indeed, they will not directly be teaching this more sophisticated material in their high-school classes. Nevertheless, I do believe that the enrichment I offer is indeed enriching, and develops important parts of their mathematical thinking, such as a greater appreciation for mathematical proof and a greater facility with abstract thinking. And this I believe is in turn beneficial for their own students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Faced with the occasional objection that they don't need that stuff (and it is usually only the weaker students who make such objections), I have found that I can almost always convince them simply by asking: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To teach 2nd grade reading well, does it suffice for the teacher to have only 3rd grade reading ability? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Everyone agrees, of course, that a 2nd grade (age 7) reading teacher should read at far better than 3rd grade (age 8) level. And similarly, for the same obvious reasons, a math teacher at any level should know far more than just the material they are teaching their students.&lt;/p&gt;&#xA;" OwnerUserId="90" LastActivityDate="2014-03-27T22:55:58.373" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="924" PostTypeId="2" ParentId="805" CreationDate="2014-03-27T23:12:56.327" Score="10" Body="&lt;p&gt;To start off, I'm a mere highschool student. There are a lot of things wrong with how education is done nowadays, but I'll keep focus on this issue at hand. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Time is quite a big deal. For the last three years I've been competing in higher level mathematics with students around the state of Florida, and I've come to realize that time isn't a very good test of who knows math and who doesn't. Each test is comprised of 30 questions, and an hour, making each question about 2 minutes long. I can say with confidence that on a given day, I can complete every single question on the test, just, maybe not in the allotted hour. On the last few tests, I've come to review my answers after the exam with my friends, as we're allowed to take our test with us after the exam, and more often than not do I notice that I made a simple mistake, like adding wrong. (Mind you this is calculus, so evaluating a definit integral of a higher degree polynomial is a lot of tedious arithmetic). Often do I sit in the testing room on one or two of the harder questions, and after 5 or 10 minutes, figure them out. It feels good, I'll say. There is no extra reward for knowing how to do any of the harder questions, and all of the grading is done by machine. What sets me apart from the students who win, is the fact that as much as I practice, I will still write 2*4cos(72) as my answer, and select the answer choice with 4cos(72). This rush for time throws out the smart students, and only rewards students for being able to be accurate, not clever, or intuitive. Quite a shame if I may say. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I spent my last summer studying mathematics at the local university, since I am quite passionate about it, and my highschool really isn't. All of the professors I had curved their tests. So nearly every test, when there was a problem I couldn't do, or was too lazy to do (I know, I know.) I just wouldn't do it. And the curve would boost me up to an A+ every single time. The system was broken, and it was proving that without minimal effort, you could succeed. &#xA;And an aside, the 95% of the class hated mathematics, and had an irrational fear of working with numbers. I suppose the stress brought on from schools did this. Very unfortunate.&lt;/p&gt;&#xA;" OwnerUserId="423" LastActivityDate="2014-03-27T23:12:56.327" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="925" PostTypeId="1" AcceptedAnswerId="927" CreationDate="2014-03-27T23:41:08.723" Score="33" ViewCount="8324" Body="&lt;p&gt;Sequences and series are related concepts but differ extremely from one another. I feel that students in integral calculus frequently mix them up. Part of the problem is that:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Sequences are usually taught only briefly before moving onto series.&lt;/li&gt;&#xA;&lt;li&gt;The definition of a series involves two related sequences (terms and partial sums).&lt;/li&gt;&#xA;&lt;li&gt;Both have operations that take in a sequence and output a number (the limit or the sum)&lt;/li&gt;&#xA;&lt;li&gt;Both have convergence tests for convergence (monotone convergence and squeeze theorem vs. root test, ratio test, etc.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What methods can I use to teach students to distinguish between sequences and series? Specifically, methods that adress the above concerns.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="114" LastEditDate="2014-04-08T08:42:12.537" LastActivityDate="2019-11-06T19:51:38.980" Title="How can I teach my students the difference between a sequence and a series?" Tags="|calculus|students-mistakes|series|sequences|" AnswerCount="11" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="926" PostTypeId="1" AcceptedAnswerId="946" CreationDate="2014-03-27T23:42:37.513" Score="76" ViewCount="11843" Body="&lt;p&gt;$$ \color{red}{(a+b)^2 = a^2+b^2}$$&#xA;$$ \color{red}{\sqrt{x^4+y^4} = x^2+y^2} $$&#xA;$$ \color{red}{e^{t^2+C} = e^{t^2}+e^C}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've observed this phenomenon -- wherein, implicitly, students say, &quot;Everything is linear! Just pass the operation through!&quot; -- in courses at all levels. High school students. Undergraduates in calculus. It's all over the place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested in two things about this phenomenon:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Where does it come from?&lt;/strong&gt; Is there a human tendency to view things linearly? More specifically, is there a reason that this occurs quite frequently with students of mathematics? Do we unknowingly &quot;teach&quot; this behavior, only to have to &quot;unteach&quot; it later?&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;What can we do about it?&lt;/strong&gt; Is there any evidence that this can be &quot;unlearned&quot; at a later age, and how can this be done effectively? Do we need to just accept that it is part of human learning to view the world linearly, and instead focus on ways of addressing it as it comes up? How can this be done?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This phenomenon was addressed in a recent math.se thread: &lt;a href=&quot;https://math.stackexchange.com/questions/630339/pedagogy-how-to-cure-students-of-the-law-of-universal-linearity&quot;&gt;Pedagogy: How to cure students of the “law of universal linearity”?&lt;/a&gt; It is a community wiki thread with many answers, but the main question there was essentially, &quot;Do you have a good explanation for when a student commits an error like this that will make them not do it again?&quot; Many answers are suggestions about showing the student why their reasoning is flawed. A couple start to address the &quot;Where does this come from?&quot; question, but not to the depth that I would like to hear about. Some answers amount to, &quot;There's no good answer&quot;, interestingly. And none of them (as of the posting of this current thread) cite any research about this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I'm hoping that this thread, on matheducators.se, can broaden the types of answers seen there. I really want to understand why and how this behavior develops (is it innate, is it a consequence of the way we teach, is it both?) and what we can do to address it effectively (both in the short term, like that other thread, but also in the long term). I'm sincerely hopeful that there is research out there about this phenomenon and that it can be shared here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addendum&lt;/strong&gt;: Please do not duplicate answers/suggestions from that math.se thread. I am looking for significant evidence of effectiveness of techniques, and do not want this thread to become a big list of suggestions.&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2021-12-24T13:45:32.397" Title="Whence the &quot;everything is linear&quot; phenomenon, and what can we do about it?" Tags="|mathematical-pedagogy|algebra|students-mistakes|education-research|" AnswerCount="11" CommentCount="21" ContentLicense="CC BY-SA 3.0" />
  <row Id="927" PostTypeId="2" ParentId="925" CreationDate="2014-03-27T23:54:05.777" Score="41" Body="&lt;p&gt;I have no evidence to back this up, nor do I know how I could obtain such a thing. But I strongly believe this is a vocabulary issue, and I would like to see the term &quot;series&quot; phased out of usage in this context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To many minds, &quot;sequence&quot; and &quot;series&quot; convey the same thing: a list of items. In modern parlance, we speak of a television series (a sequence of episodes) or a comic book series (a sequence of issues) or the World Series (a finite sequence of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; games where &lt;span class=&quot;math-container&quot;&gt;$4\leq n\leq 7$&lt;/span&gt;). &lt;a href=&quot;http://dictionary.reference.com/browse/series&quot; rel=&quot;noreferrer&quot;&gt;Dictionary definitions&lt;/a&gt; for &quot;series&quot; cite everything &lt;em&gt;but&lt;/em&gt; the mathematical definition, and even use all sorts of other mathematical terms like &quot;group&quot; or &quot;number&quot; or &quot;set&quot;. No wonder students are confused! &lt;em&gt;Only in math does &quot;series&quot; mean &quot;sum of terms&quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am going to stop using this term, and instead refer to them as &quot;infinite sums&quot;. It's exactly what they are, and it immediately triggers the fact that the terms are &lt;em&gt;added&lt;/em&gt;, as opposed to being &lt;em&gt;listed&lt;/em&gt;. I don't know whether this will solve the problem of students misunderstanding the differences between an infinite sum and its sequence of partial sums, but I know that this cannot possibly hurt their understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Addendum: 11/6/2019:&lt;/em&gt; As the comments below describe, it surely is important for students to recognize that a term (like &quot;series&quot;) may have a specific meaning in the context of mathematics that differs from its meaning in more colloquial contexts. So, I do not actually exclusively use the term &quot;infinite sums&quot; and avoid saying &quot;infinite series&quot;. Instead, when this topic first arises in a calculus course, I show this very post to my students!&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2019-11-06T19:51:38.980" LastActivityDate="2019-11-06T19:51:38.980" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="928" PostTypeId="2" ParentId="926" CreationDate="2014-03-28T00:06:31.750" Score="22" Body="&lt;p&gt;You can encourage your students to check their algebra with &lt;strong&gt;random numerical examples&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.g.:  Try setting $a=x=t=2$ and $b=y=C=3$.  Then the equations above give&#xA;$$25=13$$&#xA;$$9.85=13$$&#xA;$$1096.6=74.7$$&#xA;which are vividly wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one pair of numbers comes out wrong, your algebra was definitely wrong.  If a couple pairs of numbers come out right, your algebra was probably right.  It's a great heuristic.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-28T00:06:31.750" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="929" PostTypeId="2" ParentId="926" CreationDate="2014-03-28T00:08:29.187" Score="49" Body="&lt;p&gt;I don't view these common mistakes as 'universal linearity' assumptions. The mistake that $(a+b)^2=a^2+b^2$ is just a visually appealing statement. It is mistaken to be correct because it looks nice. Our brains tend to like things that look nice. Similarly, $\sqrt{a+b}=\sqrt a+\sqrt b$ is visually appealing and it resembles the correct formula $\sqrt {ab}=\sqrt a\cdot \sqrt b$. This is a fallacy of the kind &quot;I don't really understand why all these algebraic rules for square root are true, I never really took the time to see the proof and I never really stopped for a second to think about the equalities I was given. Instead I just vaguely tried to remember them, totally devoid of content. Thus, I kinda remember that square root behaves like that, kinda, I hope, and so I'll just compute that way. Moreover, I have so little ability to compute even the simplest things in my head, and I don't feel like pulling out my trusty pocket calculator, that I am completely incapable of detecting the falsehood of this silly claim.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are plenty of such fallacies, linearly looking or not. The cause, I believe, is simply a fundamental lack of understanding of the formulas coupled with a deep lack in ability to compute very simple things mentally. The way to deal with that is to include true/false questions of this type where the students need to prove or provide a counterexample, and disallow use of calculators. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-28T00:08:29.187" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="930" PostTypeId="2" ParentId="926" CreationDate="2014-03-28T00:12:32.197" Score="23" Body="&lt;p&gt;This became to big to be a comment.&lt;/p&gt;&#xA;&lt;p&gt;Layman's opinion.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Where does it come from?&lt;/em&gt;&lt;br /&gt;&#xA;It comes from the fact that universal linearity is useful to move forward in calculations even if it's wrong. Psychologically this is very attractive. The other option is being stuck. Moving forward has the added incentive that it can be right, that maybe the student can get some points. Another reason why this might happen is that typographically linearity is aesthetically pleasing, so it should be right...&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;What can we do about it?&lt;/em&gt;&lt;br /&gt;&#xA;A solution is to increase critical thinking overall. This can be done by not slacking off on detail, rigor and logic. Students should be taught to justify every step by going back to the definitions or previously proved properties. But this is school of thought isn't used outside of proof-based courses and this is what happens.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h1&gt;Edit&lt;/h1&gt;&#xA;&lt;p&gt;I'm now reading &lt;a href=&quot;https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thinking, Fast and Slow&lt;/a&gt; by the überpsychologist &lt;a href=&quot;https://en.wikipedia.org/wiki/Daniel_Kahneman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Danny Kahneman&lt;/a&gt;. In the book Kahneman provides a conceptual framework that attempts to describe how people's brains function and he does this by distinguishing 'between two modes of thought: &amp;quot;System 1&amp;quot; is fast, instinctive and emotional; &amp;quot;System 2&amp;quot; is slower, more deliberative, and more logical'. System 1 is what allows us to map the physical world, to react quickly to stimuli (as weather conditions, people's expressions and body language, driving, accidents about to happen, etc). System 2, on the other hand, is what is engaged when we actually think, when we solve a problem, when we perform a difficult task. I believe that my answer above is very much related to these ideas and can be summarized as follows.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Where does it come from?&lt;/em&gt;&lt;br /&gt;&#xA;It comes from System 1 being in the driver's seat.&lt;br /&gt;&#xA;&lt;em&gt;What can we do about it?&lt;/em&gt;&lt;br /&gt;&#xA;Train System 2 to takeover.&lt;/p&gt;&#xA;&lt;p&gt;Below is a section of the book which I think is very relevant and exemplificative of the issue at hand.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h2&gt;The Lazy System 2&lt;/h2&gt;&#xA;&lt;p&gt;One of the main functions of System 2 is to monitor and control&#xA;thoughts and actions “suggested” by System 1, allowing some to be&#xA;expressed directly in behavior and suppressing or modifying others.&#xA;For an example, here is a simple puzzle. Do not try to solve it but&#xA;listen to your intuition:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A bat and ball cost $1.10.&#xA;The bat costs one dollar more than the ball.&#xA;How much does the ball cost?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A number came to your mind. The number, of course, is 10: 10¢. The&#xA;distinctive mark of this easy puzzle is that it evokes an answer that&#xA;is intuitive, appealing, and wrong. Do the math, and you will see. If&#xA;the ball costs 10¢, then the total cost will be \$1.20 (10¢ for the&#xA;ball and \$1.10 for the bat), not \$1.10. The correct answer is 5¢. It&#xA;is safe to assume that the intuitive answer also came to the mind of&#xA;those who ended up with the correct number — &lt;strong&gt;they somehow managed to&lt;/strong&gt;&#xA;&lt;strong&gt;resist the intuition&lt;/strong&gt;.&lt;br /&gt;&#xA;Shane Frederick and I worked together on a&#xA;theory of judgment based on two systems, and he used the bat-and-ball&#xA;puzzle to study a central question: How closely does System 2 monitor&#xA;the suggestions of System 1? His reasoning was that we know a&#xA;significant fact about anyone who says that the ball costs 10¢: that&#xA;person did not actively check whether the answer was correct, and her&#xA;System 2 endorsed an intuitive answer that it could have rejected with&#xA;a small investment of effort. Furthermore, we also know that the&#xA;people who give the intuitive answer have missed an obvious social&#xA;cue; they should have wondered why anyone would include in a&#xA;questionnaire a puzzle with such an obvious answer. A failure to check&#xA;is remarkable because the cost of checking is so low: a few seconds of&#xA;mental work (the problem is moderately difficult), with slightly&#xA;tensed muscles and dilated pupils, could avoid an embarrassing&#xA;mistake. &lt;strong&gt;People who say 10¢ appear to be ardent followers of the law&lt;/strong&gt;&#xA;&lt;strong&gt;of least effort&lt;/strong&gt;. People who avoid that answer appear to have more&#xA;active minds.&lt;br /&gt;&#xA;Many thousands of university students have answered the&#xA;bat-and-ball puzzle, and the results are shocking. More than 50% of&#xA;students at Harvard, MIT, and Princeton ton gave the&#xA;intuitive—incorrect—answer. At less selective universities, the rate&#xA;of demonstrable failure to check was in excess of 80%. The&#xA;bat-and-ball problem is our first encounter with an observation that&#xA;will be a recurrent theme of this book: &lt;strong&gt;many people are overconfident,&lt;/strong&gt;&#xA;&lt;strong&gt;prone to place too much faith in their intuitions. They apparently&lt;/strong&gt;&#xA;&lt;strong&gt;find cognitive effort at least mildly unpleasant and avoid it as much&lt;/strong&gt;&#xA;&lt;strong&gt;as possible&lt;/strong&gt;.&lt;br /&gt;&#xA;Now I will show you a logical argument—two premises and a&#xA;conclusion. Try to determine, as quickly as you can, if the argument&#xA;is logically valid. Does the conclusion follow from the premises?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;All roses are flowers.&#xA;Some flowers fade quickly.&#xA;Therefore some roses fade quickly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A large majority of college students endorse this&#xA;syllogism as valid. In fact the argument is flawed, because it is&#xA;possible that there are no roses among the flowers that fade quickly.&#xA;Just as in the bat-and-ball problem, a plausible answer comes to mind&#xA;immediately. Overriding it requires hard work—the insistent idea that&#xA;“it’s true, it’s true!” makes it difficult to check the logic, and&#xA;most people do not take the trouble to think through the problem.&lt;br /&gt;&#xA;This experiment has discouraging implications for reasoning in everyday&#xA;life. It suggests that when people believe a conclusion is true, they&#xA;are also very likely to believe arguments that appear to support it,&#xA;even when these arguments are unsound. If System 1 is involved, the&#xA;conclusion comes first and the arguments follow.&lt;br /&gt;&#xA;Next, consider the following question and answer it quickly before reading on:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How many murders occur in the state of Michigan in one year?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The question,&#xA;which was also devised by Shane Frederick, is again a challenge to&#xA;System 2. The “trick” is whether the respondent will remember that&#xA;Detroit, a high-crime city, is in Michigan. College students in the&#xA;United States know this fact and will correctly identify Detroit as&#xA;the largest city in Michigan. But knowledge of a fact is not&#xA;all-or-none. Facts that we know do not always come to mind when we&#xA;need them. People who remember that Detroit is in Michigan give higher&#xA;estimates of the murder rate in the state than people who do not, but&#xA;a majority of Frederick’s respondents did not think of the city when&#xA;questioned about the state. Indeed, the average guess by people who&#xA;were asked about Michigan is lower than the guesses of a similar group&#xA;who were asked about the murder rate in Detroit.&lt;br /&gt;&#xA;Blame for a failure to think of Detroit can be laid on both System 1 and System 2.&#xA;Whether the city comes to mind when the state is mentioned depends in part on&#xA;the automatic function of memory. People differ in this respect. The&#xA;representation of the state of Michigan is very detailed in some&#xA;people’s minds: residents of the state are more likely to retrieve&#xA;many facts about it than people who live elsewhere; geography buffs&#xA;will retrieve more than others who specialize in baseball statistics;&#xA;more intelligent individuals are more likely than others to have rich&#xA;representations of most things. Intelligence is not only the ability&#xA;to reason; it is also the ability to find relevant material in memory&#xA;and to deploy attention when needed. Memory function is an attribute&#xA;of System 1. However, everyone has the option of slowing down to&#xA;conduct an active search of memory for all possibly relevant&#xA;facts—just as they could slow down to check the intuitive answer in&#xA;the bat-and-ball problem. The extent of deliberate checking and search&#xA;is a characteristic of System 2, which varies among individuals.&lt;br /&gt;&#xA;The bat-and-ball problem, the flowers syllogism, and the Michigan/Detroit&#xA;problem have something in common. &lt;strong&gt;Failing these minitests appears to&lt;/strong&gt;&#xA;&lt;strong&gt;be, at least to some extent, a matter of insufficient motivation, not&lt;/strong&gt;&#xA;&lt;strong&gt;trying hard enough&lt;/strong&gt;. Anyone who can be admitted to a good university is&#xA;certainly able to reason through the first two questions and to&#xA;reflect about Michigan long enough to remember the major city in that&#xA;state and its crime problem. These students can solve much more&#xA;difficult problems when they are not tempted to accept a superficially&#xA;plausible answer that comes readily to mind. &lt;strong&gt;The ease with which they&lt;/strong&gt;&#xA;&lt;strong&gt;are satisfied enough to stop thinking is rather troubling. “Lazy” is a&lt;/strong&gt;&#xA;&lt;strong&gt;harsh judgment about the self-monitoring of these young people and&lt;/strong&gt;&#xA;&lt;strong&gt;their System 2, but it does not seem to be unfair&lt;/strong&gt;. Those who avoid the&#xA;sin of intellectual sloth could be called “engaged.” They are more&#xA;alert, more intellectually active, less willing to be satisfied with&#xA;superficially attractive answers, more skeptical about their&#xA;intuitions. The psychologist Keith Stanovich would call them more&#xA;rational.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="287" LastEditorUserId="-1" LastEditDate="2020-11-28T22:02:33.230" LastActivityDate="2020-11-28T22:02:33.230" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="931" PostTypeId="2" ParentId="252" CreationDate="2014-03-28T00:26:44.837" Score="5" Body="&lt;p&gt;This is answer is self evaluation of what I think happened to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know about building characters to be 'stronger'. (What does 'stronger' or 'better' even mean here? What is the partial order?). But I do think it influences character.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently graduated from my BSc in mathematics and I can guarantee that I'm much more honest now that I graduated than what I was before I got into mathematics. I think it was because I studied mathematics. I sort of got obsessed with the truth, something that its in the core of mathematics, and telling a lie, even if it is the polite or politically correct thing to do, makes me cringe inside. I have become brutally honest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How exactly is this a consequence of studying mathematics, I can't really explain, but as I said before, the 'search for truth' is something that pervades through all of mathematics and not in other sciences. 'Facts' change all the time in other sciences, but not in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added much later:&lt;/strong&gt; In this &lt;a href=&quot;https://www.youtube.com/watch?v=Ng1W2KUHI2s&quot; rel=&quot;nofollow&quot;&gt;documentary&lt;/a&gt; about Perelman (see how the english subtitles came to be &lt;a href=&quot;http://meta.math.stackexchange.com/questions/12883/the-documentary-about-grigori-perelman-needs-english-subtitles-finished&quot;&gt;here&lt;/a&gt;), at &lt;a href=&quot;https://www.youtube.com/watch?v=Ng1W2KUHI2s#t=1907&quot; rel=&quot;nofollow&quot;&gt;31:47&lt;/a&gt; it can be read&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Perelman's &quot;teachers insisted that mathematics is not only the Queen of Sciences, but also the most moral science.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The next few moments after the given time mark are also relevant.&lt;/p&gt;&#xA;" OwnerUserId="287" LastEditorUserId="287" LastEditDate="2014-07-22T10:08:54.457" LastActivityDate="2014-07-22T10:08:54.457" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="932" PostTypeId="2" ParentId="858" CreationDate="2014-03-28T00:48:14.610" Score="11" Body="&lt;p&gt;One place to look is in the following source:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;National Research Council. &lt;em&gt;Evaluating and Improving Undergraduate Teaching in Science, Technology, Engineering, and Mathematics&lt;/em&gt;. Washington, DC: The National Academies Press, 2003.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is free online access, and Chapter 5 (&lt;a href=&quot;http://www.nap.edu/openbook.php?record_id=10024&amp;amp;page=71&quot; rel=&quot;nofollow noreferrer&quot;&gt;Evaluation Methodologies&lt;/a&gt;), includes a section specifically on &lt;a href=&quot;http://www.nap.edu/openbook.php?record_id=10024&amp;amp;page=86&quot; rel=&quot;nofollow noreferrer&quot;&gt;Self-Evaluation&lt;/a&gt; for STEM instructors at the undergraduate level.&lt;/p&gt;&#xA;&lt;p&gt;Chapter 6 (&lt;a href=&quot;http://www.nap.edu/openbook.php?record_id=10024&amp;amp;page=100&quot; rel=&quot;nofollow noreferrer&quot;&gt;Evaluation of Individual Faculty: Criteria and Benchmarks&lt;/a&gt;) may be worth a read, as well.&lt;/p&gt;&#xA;&lt;p&gt;For more information on the book, see the &lt;a href=&quot;http://www.nap.edu/openbook.php?record_id=10024&amp;amp;page=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Executive Summary&lt;/a&gt; and &lt;a href=&quot;http://www.nap.edu/openbook.php?record_id=10024&amp;amp;page=196&quot; rel=&quot;nofollow noreferrer&quot;&gt;Background on Committee Members&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="2083" LastEditDate="2021-06-11T09:19:47.883" LastActivityDate="2021-06-11T09:19:47.883" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="933" PostTypeId="2" ParentId="907" CreationDate="2014-03-28T01:53:00.340" Score="6" Body="&lt;p&gt;Here are a couple problems to shatter misconceptions about homomorphisms, while introducing the student to constructive thinking in group theory.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Are the following statements true?  Prove them or provide a counterexample.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$K_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$K_2$&lt;/span&gt; are isomorphic subgroups of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$G/K_1$&lt;/span&gt; is isomorphic to &lt;span class=&quot;math-container&quot;&gt;$G/K_2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$\varphi:G\rightarrow H$&lt;/span&gt; is a group homomorphism, a subgroup &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; exists such that &lt;span class=&quot;math-container&quot;&gt;$G/\operatorname{ker}(\varphi)\cong A$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To play fair, it's best to have introduced direct products and cyclic groups for the first one, and the quaternion group &lt;span class=&quot;math-container&quot;&gt;$Q_8$&lt;/span&gt; for the second one.&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-28T02:41:45.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="934" PostTypeId="2" ParentId="925" CreationDate="2014-03-28T02:01:06.450" Score="8" Body="&lt;p&gt;It is unfortunate the term &lt;strong&gt;convergent&lt;/strong&gt; has radically different meaning as it applies to sequences and series. A sequence $a_n \rightarrow L$ as $n \rightarrow \infty$ essentially means we can capture the tail of $a_n$ within as small an epsilon band as we desire. Whereas the series formed from $a_n$ converges iff the sequence of partial sums converges. Of course, we who are educated understand the convergence is in both cases the convergence of sequences. But, the fine print is often missed by the starting student who doesn't understand that $a_n$ converging to $1$ does not mean the series $a_n$ converges to $1$. Of course, I am being deliberately careless in my use of the term series in this post because this is how the abuse is accomplished in the minds of calculus students. If we used a wholly different term for convergence of series this would help separate the concepts. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-03-28T02:01:06.450" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="935" PostTypeId="2" ParentId="925" CreationDate="2014-03-28T02:26:13.677" Score="16" Body="&lt;p&gt;Mathematics is littered with terms that are commonly used in the real world that can mean very different things. &lt;em&gt;Sequence&lt;/em&gt; and &lt;em&gt;series&lt;/em&gt; are just two. &lt;em&gt;Group&lt;/em&gt;, &lt;em&gt;function&lt;/em&gt;, &lt;em&gt;kernel&lt;/em&gt;, &lt;em&gt;field&lt;/em&gt;, and &lt;em&gt;ring&lt;/em&gt; are just some examples of terms that have very exact meanings in various mathematical areas, but are used colloquially in English very differently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can emphasize this idea, and the fact that sequence and series are two very exact definitions, I think you may find some more success.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-03-28T02:26:13.677" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="936" PostTypeId="1" AcceptedAnswerId="939" CreationDate="2014-03-28T02:32:20.720" Score="12" ViewCount="504" Body="&lt;p&gt;In teaching Calculus III geometry plays a very important role. It is crucial that students get a good sense of how to visualize curves, surfaces, coordinate axis, frames to curves, vector fields and so forth. I think my pictures do some justice to the problem, but, sometimes, I just can't get the image across. Naturally software helps, however, I don't teach in a computer lab and short of attaching points to a task it is hard to get most students to take an assignment seriously. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I'd like to do more of is to give little projects in class which are little craft times where we get out some scissors, maybe some tape, a sheet of paper, perhaps some markers and make something to &lt;strong&gt;visualize&lt;/strong&gt; a concept or object in Calculus III. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I usually get them to take a strip of paper and twist it and tape it back. Before they tape it they draw a line on one side, then after the twisted strip is constructed the line is on the only side of the strip... of course, this is a model Möbius strip.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Can you think of any other &lt;em&gt;crafty&lt;/em&gt; ways to bring the geometry of Calculus III alive?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I teach standard American third semester of calculus, 4 credit hours, covers from vectors and three dimensional coordinate geometry through the basic vector calculus including Stokes' Theorem, however, I'm generally interested in any geometric craft to help ingrain principles of analytic geometry.)&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="123" LastEditDate="2014-04-10T08:41:21.910" LastActivityDate="2014-04-10T08:41:21.910" Title="Hands-on demonstration ideas for multivariate calculus" Tags="|analytic-geometry|vector-calculus|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="937" PostTypeId="1" AcceptedAnswerId="938" CreationDate="2014-03-28T02:38:10.863" Score="7" ViewCount="198" Body="&lt;p&gt;I'm currently beginning research that involves an aspect of Professional Development (PD) for Mathematics teachers, but I don't have much experience or knowledge of PD. What are some of the seminal works on PD for Mathematics teachers that I can include in my Literature Review?&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2015-01-28T20:53:45.093" Title="Seminal Professional Development writings" Tags="|teachers|teacher-training|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="938" PostTypeId="2" ParentId="937" CreationDate="2014-03-28T03:53:15.227" Score="7" Body="&lt;p&gt;In terms of researchers who I am personally familiar with: I would recommend work by &lt;a href=&quot;http://steinhardt.nyu.edu/faculty_bios/view/Orit_Zaslavsky&quot; rel=&quot;nofollow&quot;&gt;Orit Zaslavsky&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From her NYU-Steinhardt bio: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Orit Zaslavsky's research focuses on mathematics teacher education. She has been among the first scholars to study the development of teacher-educators, from both theoretical and practical perspectives. Within this broad research domain, she has investigated and conceptualized the nature of productive mathematics-related tasks and examples.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A search in google scholar &lt;a href=&quot;http://scholar.google.com/scholar?q=professional%20development%20mathematics%20education%20orit%20zaslavsky&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=0,33&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; has as its top two citations:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Zaslavsky, O., &amp;amp; Leikin, R. (2004). Professional development of mathematics teacher educators: Growth through practice. &lt;em&gt;Journal of Mathematics Teacher Education, 7&lt;/em&gt;(1), 5-32.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Zaslavsky, O., Chapman, O., &amp;amp; Leikin, R. (2003). Professional development of mathematics educators: Trends and tasks. In &lt;em&gt;Second international handbook of mathematics education&lt;/em&gt; (pp. 877-917). Springer Netherlands.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For mathematics teachers in general (as opposed to mathematics teacher-educators) see if you can track down the &lt;a href=&quot;http://eric.ed.gov/?id=ED374989&quot; rel=&quot;nofollow&quot;&gt;1994 NCTM yearbook&lt;/a&gt;, which had as its focus (and title) &quot;Professional Development for Teachers of Mathematics.&quot; It's already 20 years old, but should provide a good starting point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Incidentally, &lt;a href=&quot;http://scholar.google.com/scholar?q=professional%20development&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=5,33&amp;amp;sciodt=0,33&amp;amp;cites=2999199949224876472&amp;amp;scipsc=1&quot; rel=&quot;nofollow&quot;&gt;checking&lt;/a&gt; google scholar for articles on PD that cite the '94 yearbook yields the latter of Zaslavsky's articles cited above as the first result.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-03-28T17:48:18.300" LastActivityDate="2014-03-28T17:48:18.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="939" PostTypeId="2" ParentId="936" CreationDate="2014-03-28T04:03:30.280" Score="10" Body="&lt;p&gt;For crafty approaches:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;use &lt;strong&gt;oranges and balloons&lt;/strong&gt; for drawing spherical coordinates&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;use a &lt;strong&gt;lamp and lampshade&lt;/strong&gt; to make conic sections on the wall&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a &lt;strong&gt;temperature map&lt;/strong&gt; cut from a daily newspaper is a contour graph&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;They are nice complements to what you can do by hand, or with a computer and a projector. &lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-28T04:03:30.280" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="940" PostTypeId="2" ParentId="925" CreationDate="2014-03-28T06:11:23.493" Score="24" Body="&lt;p&gt;I second &lt;a href=&quot;https://matheducators.stackexchange.com/a/935/114&quot;&gt;Andrew Sanfratello's answer&lt;/a&gt;&amp;mdash;there's just no getting around the fact that technical terms have technical meanings which differ from their everyday meanings.  And when I teach series, I start each class by writing &quot;SERIES MEANS SUM&quot; at the top of the board every day.  (At one previous institution, I often taught in a room with several large boards, so I could easily keep that visible, in huge letters, for the whole lecture.)  A silly gimmick, but it helped.&lt;/p&gt;&#xA;" OwnerUserId="129" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-03-28T07:50:54.320" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="941" PostTypeId="2" ParentId="936" CreationDate="2014-03-28T07:21:14.000" Score="8" Body="&lt;p&gt;Here's a crafty but perhaps crazy way to convey some ideas to a class. Have all the students gather on the football field (or another field) in a grid on a mildly windy day. Each student carries a little stick with a strand of paper to measure the direction of the wind. Perhaps with some physics they can also estimate the speed of the wind. Have them make several measurements, compile the data and make a vector field of the wind pattern on the field at that point in time. Of course, for best effect, the students will be synchronized in their measurements to have more impact. You can use this exercises as a segue into many topics ranging from the difficulties of measurement in general, the imprecision of the real-world compared to the idealized calculus situations, and of course calculus ideas like vector fields. They'll now know from experience what a vector field is - they were part of one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possibly they can each bring a thermometer and measure the temperature in a hallway or something where there are multiple heat sources and sinks. This way they can create a heat-map. I'm sure there are many more examples one can think of.&lt;/p&gt;&#xA;" OwnerUserId="390" LastActivityDate="2014-03-28T07:21:14.000" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="942" PostTypeId="1" AcceptedAnswerId="960" CreationDate="2014-03-28T07:22:02.547" Score="20" ViewCount="1392" Body="&lt;p&gt;In mathematics notation plays an important role in clarifying the subject. A bad notation could be confusing. Recently I use a logic textbook which has a very nice approach and content but an inappropriate old notation which I never use personally. I decided to teach its content using my own notation but I see the students are a bit confused about this double notation method. They cannot translate between lecture and textbook notations quickly and this is somehow annoying for them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What should I do about this notation conflict? Does passing time help them and solve the problem or I should change my notation or my textbook? Is this problem serious at all? What happens if we continue in this situation? &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastActivityDate="2016-10-25T13:00:45.970" Title="Notation Conflict between Teachers and Textbooks" Tags="|mathematical-pedagogy|notation|logic|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="943" PostTypeId="2" ParentId="899" CreationDate="2014-03-28T07:50:24.670" Score="8" Body="&lt;p&gt;There is a danger that lies in tailoring a class too much towards a major and it is that it won't broaden the students' worldviews. What follows a calculus course is years of specialization. Why start early when the students will see all this stuff anyway in their degree? Take an engineer for example. They'll be doing engineering and speaking with their engineering peers and professors for several years. A more abstract calculus course can be a great opportunity to build strong foundations for those engineering students and this can only help them later on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This doesn't mean there shouldn't be an tailoring towards particular fields. More so, I believe that calculus courses should be taught the same way regardless of specializations. Teach everyone rigorously (as possible for such a level) and introduce examples from various fields. Most of the biology, business or engineering students I've tutored suffer from poor understanding of the fundamental calculus concepts and not of how they apply to their field. Once we cover those basics they usually have no problem with the applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the ideal homework would contain enough abstract examples to teach the concepts, followed by problems from various fields with an emphasis on the students' specialization.&lt;/p&gt;&#xA;" OwnerUserId="390" LastActivityDate="2014-03-28T07:50:24.670" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="944" PostTypeId="2" ParentId="942" CreationDate="2014-03-28T07:58:55.913" Score="2" Body="&lt;p&gt;I think the damage has been done because the notations have already been introduced and you can't really do away with one or the other at this point. I think you can make a lot of progress by motivating what the different notations mean and why one notation has become outdated. If the students understand the underlying principles well, it should help with translating quickly. There was a question about notation for differentiation &lt;a href=&quot;https://matheducators.stackexchange.com/questions/562/should-students-be-asked-to-use-more-than-one-notation-for-the-derivative-in-an&quot; title=&quot;herequot;&quot;&gt;here&lt;/a&gt; and one can surely use the lessons learned there in your situation. For example, the $\dot{y}$ notation is very similar to the $y'$ notation but conveys a slightly different meaning. Ideally one would not teach it right away and introduce it once the core idea is understood well-enough. However, if both have been introduced from the start it would help to differentiate (sorry) between the two so as to introduce a different point of view that will go along with said notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately I don't know the particulars of your situation with the logic textbook so perhaps I'm way off.&lt;/p&gt;&#xA;" OwnerUserId="390" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-03-28T07:58:55.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="945" PostTypeId="2" ParentId="942" CreationDate="2014-03-28T08:12:17.350" Score="12" Body="&lt;p&gt;In any case, you should warn them about the different notation in the textbook. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question is related to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/562/should-students-be-asked-to-use-more-than-one-notation-for-the-derivative-in-an&quot;&gt;Should students be asked to use more than one notation for the derivative in an introductory calculus class?&lt;/a&gt; - In my &lt;a href=&quot;https://matheducators.stackexchange.com/a/567/114&quot;&gt;answer&lt;/a&gt;, I wrote &quot;I think, it is important that students should be flexible and open-minded to notation&quot; and went into detail there (It will not be the first time that they will be confused with notation, once they read a textbook on their own or they have a different teacher, a similar problem arises). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in your situation, the problem is that the textbook is older and the notation there is out of date. But, also this will not be the last time that students will have a textbook or an article with outdated notation!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think, in general it depends on the &quot;type&quot; students you have. If this a class of engineers who will only have a few courses in mathematics, what was said above might not be true - and in this case, you should probably change your textbook; but if you have a class of math majors who will eventually do their master or even more in mathematics (which I think you have since you mentioned logic), at some point you &lt;strong&gt;have to&lt;/strong&gt; talk with about notation. Most textbook have some pages where they explain the notation needed to read the book and the rest of the notation is explained when intoducing. Explain your students how to use that and explain them the (dis)advantages of your notation and the textbook's notation. At least, they will learn to be more flexible (which is good for them in future as mentioned above) and they will also learn that mathematics is a historical progress and notation is part of this progress as well. If you have the feeling that your message does not arrive, you can give them examples from fields and notations they already know, e.g., &lt;a href=&quot;http://en.wikipedia.org/wiki/Nicolas_Bourbaki#Influence_on_mathematics_in_general&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bourbaki&lt;/a&gt; introduced some important notation like $\varnothing$.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-03-28T08:12:17.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="946" PostTypeId="2" ParentId="926" CreationDate="2014-03-28T09:04:01.197" Score="48" Body="&lt;p&gt;The problem you describe is well-known in mathematics education research. I cite the paper of &lt;a href=&quot;http://link.springer.com/article/10.1023/A:1021205413749&quot;&gt;De Bock, D., Van Dooren, W., Janssens, D., &amp;amp; Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50(3), 311–334.&lt;/a&gt; and give some of the references mentioned:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As the concept of linearity itself, the misuse of linearity has many&#xA;  faces: it has been found at different age levels and in a variety of&#xA;  mathematical domains (see, e.g., De Bock et al., 1999). In elementary&#xA;  arithmetic, the phenomenon of improper proportional reasoning is often&#xA;  related to a ‘lack of sense-making’ in the mathematics classroom&#xA;  (Gagatsis, 1998; Greer, 1993; Nesher, 1996; Verschaffel et al., 1994,&#xA;  2000; Wyndhamn and Säljö, 1997). When confronted with so-called&#xA;  ‘pseudoproportionality problems’ (such as, e.g. “It takes 15 minutes&#xA;  to dry 1 shirt outside on a clothesline. How long will it take to dry&#xA;  3 shirts outside?”), many students give answers based on direct &#xA;  proportionality (i.e., tripling the drying time because the&#xA;  number of shirts is tripled). [...] &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In secondary education, ‘linearity errors’&#xA;  are often reported in the fields of algebra and (pre)calculus.&#xA;  Students tend to overgeneralise what has been experienced as ‘true’&#xA;  for linear functions to non-linear functions (e.g. “the square root of&#xA;  a sum is the sum of the square roots” or “the logarithm of a multiple&#xA;  is the multiple of the logarithm”). This type of systematic errors has&#xA;  been discussed and illustrated by Berté (1987, 1993), Gagatsis and&#xA;  Kyriakides (2000) and Matz (1982). According to Matz (1992), these&#xA;  linearity errors result from students’ overgeneralisation of the&#xA;  distributive law. The immense number of occasions wherein students add&#xA;  and use the distributive law in arithmetic and early algebra is very&#xA;  likely to reinforce students’ acceptance of linearity.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Berté, A. (Réd.): 1987, Enseignement des mathématiques utilisant la&#xA;  ‘réalité’, Tome 1, IREM, Bordeaux.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Berté, A.: 1993, Mathématique dynamique, Nathan, Paris. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;De Bock, D., Verschaffel, L. and Janssens, D.: 1999, ‘Some reflections on the illusion of linearity’, Proceedings of the 3rd European Summer University on History and Epistemology in Mathematical Education, Vol. 1, Leuven/Louvain-la-Neuve, Belgium, pp. 153–167. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Gagatsis, A. and Kyriakides, L.: 2000, ‘Teachers’ attitudes towards their pupils’ mathematical errors’, Educational Research and Evaluation 6(1), 24–58. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Matz, M.: 1982, ‘Towards a process model for high school algebra errors’, in D. Sleeman and J.S. Brown (eds.), Intelligent Tutoring Systems, Academic Press, London, pp. 25–50.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="199" LastEditorUserId="199" LastEditDate="2014-03-28T13:58:48.247" LastActivityDate="2014-03-28T13:58:48.247" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="949" PostTypeId="2" ParentId="925" CreationDate="2014-03-28T09:23:00.167" Score="11" Body="&lt;p&gt;I suggest that this is more than a problem in wording. In mathematics, many objects are introduced as a process (e.g. functions as &quot;making a y out of a given x&quot;, sets as &quot;taking things together&quot;) but later need to be used as an object (e.g. derivation of functions, sets of sets where the &quot;inner&quot; set must be perceived as an object (topology, measures)).&#xA;This also seems the case here. Students maybe can see a sequence as a process, but not as an object. So the sequence of partial sums is too complex for their thinking as it includes infinitely many (finite) sequences. Quite often, &quot;objects of objects&quot; don't work with undergraduates: Operators in functional analysis (functions of functions), topologies (sets of sets), conditional statements in logic (statements of statements), ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you think this might fit your situation, have a look at &lt;a href=&quot;http://books.google.de/books?hl=de&amp;amp;lr=&amp;amp;id=knYjqCFRhlkC&amp;amp;oi=fnd&amp;amp;pg=PA77&amp;amp;dq=D.%20Mathews,%20M.%20A.%20McDonald%20and%20K.%20Strobel.%20Understanding%20sequences%3a%20A%20tale%20of%20two%20objects&amp;amp;ots=SB3n1ckqwn&amp;amp;sig=jmbI7JRMGYhYnI8fR7TFozZxcgI#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;this paper&lt;/a&gt; where APOS-theory which focusses on process-object duality is applied to series and sequences.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-28T09:23:00.167" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="950" PostTypeId="1" AcceptedAnswerId="954" CreationDate="2014-03-28T09:59:05.057" Score="10" ViewCount="2144" Body="&lt;p&gt;I see a lot of students who repeat the whole question (i.e., before starting with anything which has some content, they write down all assumptions, claims, etc.) in their answers in homework or exams. At least in exams, I have the feeling that they are losing time with this. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is there an explanation why students do so?&lt;/li&gt;&#xA;&lt;li&gt;Are there any (psychological) advantages why this is done?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-28T10:07:20.783" LastActivityDate="2014-03-28T17:10:14.430" Title="Advantages on repeating question in student's answer" Tags="|exams|homework|psychology|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="951" PostTypeId="1" CreationDate="2014-03-28T10:05:28.357" Score="15" ViewCount="3206" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Confirmation_bias&quot;&gt;Confirmation bias&lt;/a&gt; is a quality of human mental processes which makes us tend to think in terms of positive examples and tests that would confirm our working hypothesis, rather than negative examples or tests that would falsify the conjecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2-4-6 task:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The term was coined by P. C. Wason after his &lt;a href=&quot;http://dx.doi.org/10.1080%2F14640746808400161&quot;&gt;2-4-6 task experiment&lt;/a&gt;, where he asked the participants to find a rule which governed whether a triples of numbers would fit or not. First, they were told that $(2,4,6)$ fits. Then the subjects would ask about their own triples of numbers and the experimenter would answer if that triple fits or not. Only a small percent of people found the true rule, which was &quot;any increasing sequence&quot;, instead generating more specific rules like &quot;the middle is the average&quot;. The problem was that participants offered triples that would align with their hypothesis, rarely trying falsifying it. In fact some test subjects didn't get any &quot;negative results&quot;, that is, the rule could have been &quot;any triple of numbers&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Confirmation bias in math education:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Confirmation bias happens in math education in two flavors: student's and teacher's. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student's version is when the pupils try to solve some problem, but aren't critic about their own opinions. Perhaps this is most important to &lt;em&gt;science&lt;/em&gt; education, but in math pattern recognition plays a huge role and fallacious reasoning hurts progress.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I'm more interested in the teacher's flavor, which is of different kind. It regards how the teacher thinks of his/her students, for example,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;if the teacher suspects that the student has some problem, he will try to help before checking if it really is that problem;&lt;/li&gt;&#xA;&lt;li&gt;if the teacher thinks that the child does not understand some concept, he/she will try to prove that it indeed is the case;&lt;/li&gt;&#xA;&lt;li&gt;the teacher my want to stress why his/her solution is better than some other solution presented by the student (perhaps not even accepting it).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is the teacher's confirmation bias a serious problem?&lt;/li&gt;&#xA;&lt;li&gt;How could we make the teachers aware of it (not as frequently, but even colleagues from university fall for this)?&lt;/li&gt;&#xA;&lt;li&gt;How could I make some particular teacher aware of it (as a parent, or as a student)?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="42" LastActivityDate="2020-05-06T10:05:55.987" Title="Confirmation bias in math education" Tags="|secondary-education|primary-education|teachers|" AnswerCount="4" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="952" PostTypeId="2" ParentId="950" CreationDate="2014-03-28T10:12:13.640" Score="10" Body="&lt;p&gt;Personally I encourage students to write answers that make sense as a self-contained piece of writing, because I think that that is a valuable skill.  Certainly it is required when writing about mathematics in any context other than homework or exams.  This generally means that they need to reproduce some or all of the content of the question, possibly phrased in a different way.  Simply copying the question verbatim is not the best way, however.&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-28T10:12:13.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="954" PostTypeId="2" ParentId="950" CreationDate="2014-03-28T11:31:00.613" Score="10" Body="&lt;p&gt;In order to cultivate a greater appreciation for precision in one's mathematical statements, I ask my students not to copy the problem question, but rather to transform it from a question or problem statement into a theorem statement, which they then prove or solve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if the quiz question is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is the ring $\mathbb{Z}_2\oplus \mathbb{Z}_3$ an integral domain?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then the student is expected to write something like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; The ring $\mathbb{Z}_2\oplus \mathbb{Z}_3$ is not an integral domain.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Proof. This ring is not an integral domain, because it has zero divisors, namely, ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In a lower-level course, I ask my students to transform the question into a positive assertion of what they will do. &lt;/p&gt;&#xA;" OwnerUserId="90" LastEditorUserId="90" LastEditDate="2014-03-28T17:07:54.597" LastActivityDate="2014-03-28T17:07:54.597" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="955" PostTypeId="2" ParentId="936" CreationDate="2014-03-28T12:02:06.603" Score="1" Body="&lt;p&gt;I like to use ZomeTools to create 3D visualizations of coordinate axes and the interaction between lines and planes.  Henry Segerman has produced some amazing looking &lt;a href=&quot;http://www.shapeways.com/model/781790/calculus-surfaces-small.html&quot; rel=&quot;nofollow&quot;&gt;quadric surfaces&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="324" LastActivityDate="2014-03-28T12:02:06.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="957" PostTypeId="2" ParentId="950" CreationDate="2014-03-28T13:50:27.057" Score="3" Body="&lt;p&gt;I can't really comment on the psychological reasons, but I can say from personal experience that some students have practical reasons for doing so: by copying the problem from the textbook or question sheet onto work paper, this allows directing focus entirely to the paper the problem is being solved on rather than having to switch between the two to verify parts of the question. However, in my case that usually also involved removing all the &quot;extra&quot; stuff in the question if present and reducing it to just the parts necessary to solve the problem, or at least summarizing the question in a more notational form, and I usually don't do that when the questions are already provided on the answer paper and no scratch paper is provided, so I suppose this might not be all that applicable to your question given that I'm probably not the sort of student you're more interested in for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thinking back, I believe I developed that habit due to having had teachers in the past who required questions to be copied to homework/test papers (probably for the benefit of the grader[s], at least in part), wanting to keep the amount copied to a minimum, and later realizing that it did actually provide some benefit for me (and for graders) even in the minimal form; thus, I continued to do so even when it was not explicitly required.&lt;/p&gt;&#xA;" OwnerUserId="447" LastActivityDate="2014-03-28T13:50:27.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="958" PostTypeId="2" ParentId="951" CreationDate="2014-03-28T14:12:41.477" Score="2" Body="&lt;p&gt;&lt;strong&gt;Is the teacher's confirmation bias a serious problem?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's one of a list of assessment errors. It's specially noteworthy, because it hits the tension field &lt;em&gt;instruction vs. construction&lt;/em&gt;. As such, it is a problem, but not much more serious than other errors in this tension field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How could we make the teachers aware of it?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can't help teachers not accepting or devalueing alternative valid solutions. If they genuinely think them wrong, they are at the wrong place. If they argue not to confuse other students, they are principally right. In that case one needs to start on a more basic level of &lt;em&gt;instruction vs. construction&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can help a teacher projecting non-understanding onto other students by letting thus students take tests from other teachers/educators. They probably won't succeed fully, but they will show some competence that the biased teacher didn't test for or recognize. You can then show him these results or even show him how thus student managed some other problems with his competence shown.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can help a teacher overeagerly helping students by giving him time for the students. If a teacher is under time pressure (like &quot;Students must be able to do this calculation/solve that class of problems after $n$ weeks.“) then he won't risk a student trying out his own ways and thus &lt;em&gt;possibly&lt;/em&gt; constructing an alternative solution but also &lt;em&gt;possibly&lt;/em&gt; failing at all. The call for Construction need not only be, how teachers should make students learn, but also how teachers' success in letting students construct is evaluated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How could you make some particular teacher aware?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tell him of the student's success with his alternate solution or his method of learning.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-03-28T14:12:41.477" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="959" PostTypeId="1" AcceptedAnswerId="1019" CreationDate="2014-03-28T14:21:43.803" Score="6" ViewCount="199" Body="&lt;p&gt;In a traditional exam, there is a strong focus on facts and techniques. For instance, in a course on linear algebra, students are asked to diagonalize matrices and they have to check whether a given set forms a vector space, etc. Sometimes I think that a well-trained monkey should also do this well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently I read a report on a different kind of task in an exam which had the following style: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Assume $f$ and $g$ to be real-valued functions satisfying $f(1)=2$, $f(2)=3,$ $f(3)=6$ and $g(2)=5$, $g(5)=3$, $g(6)=3$. Is this information sufficient to calculate $f\circ g$ of the value $5$? If the answer is &quot;yes&quot;, then calculate the value, if not then argue why.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found this task very interesting since it seems very easy for students who are familiar with the composition of functions and it seems nearly impossible for students who are not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question: Where can I find more non-standard tasks for (written) exams that try to focus on deeper understanding?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to concentrate on undergraduate level.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="11" LastEditDate="2014-04-11T19:04:39.673" LastActivityDate="2014-04-11T19:04:39.673" Title="Where can I inform me about experiences with exam tasks heading for deeper understanding?" Tags="|undergraduate-education|exams|examples|problem-design|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="960" PostTypeId="2" ParentId="942" CreationDate="2014-03-28T14:24:52.790" Score="9" Body="&lt;p&gt;When I teach introductory logic, I usually show my class a list of alternative notations.  Even if the textbook and I agree on notation, students who look at literature beyond the textbook are likely to see other notations, and it's good for them to know that this can happen before it actually happens and confuses them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An especially bad source of confusion is that the same symbol can be used by different authors with entirely different meanings.  For example, one can find the symbol $\sim$ used for equivalence and also for negation.  One can find $|$ used for disjunction and for Sheffer's stroke.  &lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-03-28T14:24:52.790" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="961" PostTypeId="2" ParentId="903" CreationDate="2014-03-28T14:35:40.057" Score="4" Body="&lt;p&gt;&lt;strong&gt;Foreword&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students in secondary education will not leave the space of Riemann integrals (and symbolic Integrals). The existence of different integration concepts in higher maths is totally irrelevant here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Knowledge&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The operations &lt;em&gt;calculate the derivative&lt;/em&gt; and &lt;em&gt;calculate the integral&lt;/em&gt; (whether Riemann indefinite or symbolic) are inverse operations with some restrictions (one-way is translation invariant and thus not injective, discontinuities). These are to be made clear to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disregarding historic context, these restrictions are the only disadvantages of inverse name pairs for the two operations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you regard the historic context that created non-inverse names or not, is up to you. If you do regard the historic context, it's highly disadvantegous not to use the actual, history-driven names.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Communication competence&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students need to be capable to read texts and understand people which use the actual names.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If students use inverse name pairs, one of them should be the actual name, so that the people they talk to can understand them. People not accustomed to this problem will probably be confused at first but able to understand later-on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The disadvantage of using inverse name pairs is thus only serious, if the students lack passive understanding of the actual names.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-03-28T14:35:40.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="962" PostTypeId="2" ParentId="959" CreationDate="2014-03-28T14:39:42.293" Score="3" Body="&lt;p&gt;You can try problems of math olympiads and reduce them in difficulty or take them as inspiration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.imo-official.org/problems.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Problem archive of IMO&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, 43rd IMO (2002) Problem 5.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/HPc3b.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="439" LastEditDate="2014-03-28T16:52:28.453" LastActivityDate="2014-03-28T16:52:28.453" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="963" PostTypeId="2" ParentId="950" CreationDate="2014-03-28T14:53:14.157" Score="2" Body="&lt;p&gt;I'm going to hazard a guess - For 2 column geometry proofs, the repetition of every 'given' is required. When working with students on their Geo, it struck me that many times, the 'givens' are 2/3 of the solution, just a couple steps to finish the proof. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For algebra, the question is usually brief enough, it makes sense to re-write it on the answer sheet. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These two examples set the stage for a continuation of the process of re-write.    &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-03-28T14:53:14.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="964" PostTypeId="2" ParentId="852" CreationDate="2014-03-28T15:22:41.007" Score="20" Body="&lt;p&gt;Does this student really already know everything? I ask because of an experience I had many years ago, teaching a &quot;semi-honors&quot; second-semester calculus class in the first (i.e., fall) semester.  So the students were there because they had already done some calculus in high school and had done well on a placement test.  The first topic of the semester was integration, so I talked about the general notion of integral, applications to specific sorts of problems, and techniques of integration.  As far as techniques of integration were concerned, these students could integrate messy functions at least as well as I can; that and analogous computational skills were what got them through their high school calculus courses and the placement exam.  But the first exam I gave them included questions that required knowledge of what an integral actually is. Many of them had not learned that in high school and had ignored my attempts to explain it (attempts that they viewed as unnecessary prose surrounding the &quot;real math&quot; of computing integrals).  The grades on that first exam were quite a shock to these students, and most of them soon learned that the non-computational prose is also math.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your students are at all like this, I recommend showing them (preferably in a milder context than an exam) that they still have some things to learn from you.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-03-28T15:22:41.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="965" PostTypeId="1" CreationDate="2014-03-28T15:32:04.353" Score="6" ViewCount="169" Body="&lt;p&gt;Many of the notions, methods and theorems of the mathematical logic and its different sub-fields like set theory, model theory, etc. are closely related to some philosophical background. I believe understanding these philosophical aspects of mathematical arguments could be very useful for logic students to understand the subjects better. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I am teaching logic, I usually talk about the philosophical backgrounds of each concept/theorem briefly. But there is a problem here. Introducing philosophical subjects causes questions and discussions amongst students which is fine in principal but sometimes these discussions become hot and too long and change the direction of the arguments from mathematical part to philosophical part of the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; How can I explain philosophical aspects of the logical theorems without distracting attention of the students from the main teaching subject? In the other words how can I manage the discussion frame and time of the philosophical parts of my logic courses? Precisely I am searching for some discussion managing techniques which are useful for managing a philosophical discussion in a mathematical logic course.&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-28T15:56:40.583" LastActivityDate="2014-03-28T16:08:53.057" Title="Philosophical Subjects in Logic Courses" Tags="|mathematical-pedagogy|class-participation|logic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="966" PostTypeId="2" ParentId="965" CreationDate="2014-03-28T16:08:53.057" Score="5" Body="&lt;p&gt;Why not have the students talk about philosophical questions in small groups? That way you could allow them to grapple with the philosophical issues, but add more structure to your time together so that things don't spiral out of control.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's what I'm imagining:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;You ask kids: &quot;What do we mean by a tautology?&quot;&lt;/li&gt;&#xA;&lt;li&gt;You ask them to talk with the people they're sitting near and discuss this. You might even have a google doc open where their group can place their answer. You give them 5 minutes (10 minutes? whatever) to do this.&lt;/li&gt;&#xA;&lt;li&gt;You bring everyone together, and give some time to go over the answers that various groups gave.&lt;/li&gt;&#xA;&lt;li&gt;Then you say, &quot;OK, cool. This was a problem that really bugged Wittgenstein. Here was his answer.&quot; And then you show them truth tables or whatever and you keep on running class as you would normally.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(I'm obviously pretty rusty on my history of logic and philosophy.)&lt;/p&gt;&#xA;" OwnerUserId="84" LastActivityDate="2014-03-28T16:08:53.057" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="967" PostTypeId="2" ParentId="630" CreationDate="2014-03-28T16:12:48.840" Score="6" Body="&lt;p&gt;Ask students to solve problems, pose their own questions, make their own definitions, etc... This is about creating opportunity for creative work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it is also crucial that you then take the student's ideas seriously! Give the ideas respect. Encourage other students to do the same. Don't replace student thinking or explanation with your own, instead ask questions to help the student refine their ideas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students need a safe environment (intellectual and psychological) so that they can feel free to take risks. Then you can give them opportunity and freedom. But unless they feel respected and comfortable, you won't get anywhere.&lt;/p&gt;&#xA;" OwnerUserId="309" LastActivityDate="2014-03-28T16:12:48.840" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="968" PostTypeId="2" ParentId="950" CreationDate="2014-03-28T17:10:14.430" Score="5" Body="&lt;p&gt;I recommend this to most people I have helped or tutored over the past years for a few reasons.&lt;/p&gt;&#xA;&lt;h3&gt;Problems are complicated when you don't understand them.&lt;/h3&gt;&#xA;&lt;p&gt;Basic problems are not as easily digestible to students, especially those who are less confident.&lt;/p&gt;&#xA;&lt;p&gt;I have found people consistently make mistakes based on the problem definitions &lt;em&gt;even if they are explicitly given&lt;/em&gt; and as a result I strongly recommend people to write them out. Even if it's repeating information.&lt;/p&gt;&#xA;&lt;p&gt;This is true even for simple problems. As problems get more complicated this is even more important.&lt;/p&gt;&#xA;&lt;h3&gt;People don't read well.&lt;/h3&gt;&#xA;&lt;p&gt;I eventually tired of people making mistakes resulting from simply misreading or misinterpreting assumptions or even given information.&lt;/p&gt;&#xA;&lt;p&gt;Forcing them to write this down at least somewhat helps, though, of course there's always opportunity for stupid mistakes.&lt;/p&gt;&#xA;&lt;h3&gt;It can be part of problem solving.&lt;/h3&gt;&#xA;&lt;p&gt;I can't count the number of times in the past I've had no clue how to approach a problem myself, but followed my normal approach of writing the problem down and as I was basically rewriting it, ended up with significant insight into how to solve the problem.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This is also the &amp;quot;if I write down lots of stuff it'll look like I tried&amp;quot; perspective, though, this is more &amp;quot;student led&amp;quot; than something I ever recommend..&lt;/p&gt;&#xA;" OwnerUserId="456" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-28T17:10:14.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="969" PostTypeId="2" ParentId="630" CreationDate="2014-03-28T17:11:52.160" Score="7" Body="&lt;p&gt;This is tangential, but I'll submit it as an answer because I cannot comment yet.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the emphasis on mathematics education is typically on getting the right answer, and often in doing it the right way&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It shouldn't be controversial to want the right answer (to an unambiguous question). The 'right way', however, presents serious problems. The 'right way' mentality teaches students that math class is an exercise in memorizing and executing a flow chart. This discourages students from actively thinking about what they're doing, and in turn greatly reduces their chances at becoming creative mathematical thinkers.  In order to encourage creative thinking about mathematics, we really just need to encourage &lt;strong&gt;thinking&lt;/strong&gt; about mathematics rather than doing &lt;strong&gt;recall&lt;/strong&gt; of the flow chart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've had success in junior and senior high school aged students by challenging their flow-chart operations on &lt;em&gt;extremely simple questions&lt;/em&gt; (what this means will vary from student to student). For example, I've seen many students approach equations such as $\frac{x}{2} = 5$ by drawing the $1$ into $\frac{x}{2} = \frac{5}{1}$ so that they can execute 'cross multiplication'. The same people might successfully find a common denominator and add $\frac{5}{1} + \frac{3}{2}$ but fail to do so with $5 + \frac{4}{3}$. These sorts of problems are excellent litmus tests for people executing flow-charts rather than thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To challenge the flow chart, one fruitful place to start is the concept that &lt;strong&gt;doing something to both sides of an equation yields a new equation which is exactly as true as the previous one&lt;/strong&gt;. When a student is on board with this (as in, actually has an intuitive understanding of its truth), they're not far from putting cross-multiplication behind them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This must be done with &lt;strong&gt;prodding questions&lt;/strong&gt; rather than instruction, which is time consuming and, to be honest, a little mentally taxing (trying to find just the right question for student X in moment Y is tough). It's worth it.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2014-03-28T17:11:52.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="970" PostTypeId="2" ParentId="942" CreationDate="2014-03-28T18:21:12.250" Score="5" Body="&lt;p&gt;It is okay to deviate from a notation in a course textbook as long as &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The notation you introduce is still a standard notation.&lt;/li&gt;&#xA;&lt;li&gt;You clearly demonstrate and emphasize notational differences.&lt;/li&gt;&#xA;&lt;li&gt;You explain where/why (and when, if one has perhaps fallen out of style) each notation is used.&lt;/li&gt;&#xA;&lt;li&gt;You &lt;em&gt;remind&lt;/em&gt; students every now and then about your notation vs. the book's notation.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For example, in a class field theory class learning about Tate cohomology, notation can be all aflutter. Neukirch, Lang, Milne, and Serre all use different notations for the cohomology (including $H^0$, $H_T^0, \hat{H}^0$ - and that's not even mentioning the names for the various groups and modules).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember this because the professor who taught me class field theory was careful to point out that the notation he uses differs from Neukirch (our text) because Neukirch's notation was out of date, and he told us about the other notations and who used which. Interestingly, I don't remember which one Neukirch uses nor which one is actually used now, which I think is an indication of good, notation-independent teaching (and a bad memory on my part).&lt;/p&gt;&#xA;" OwnerUserId="20" LastActivityDate="2014-03-28T18:21:12.250" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="972" PostTypeId="2" ParentId="647" CreationDate="2014-03-28T19:13:14.327" Score="15" Body="&lt;p&gt;If you are worried about fairness, I'd offer the answer to any students' questions to the whole class, whether announced to the class verbally, or written down on the board.&lt;/p&gt;&#xA;" OwnerUserId="307" LastActivityDate="2014-03-28T19:13:14.327" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="973" PostTypeId="2" ParentId="942" CreationDate="2014-03-28T19:42:03.247" Score="2" Body="&lt;p&gt;Perhaps you should write up (the most important) results of the text in your notation as a complement to the text. Perhaps &quot;translate&quot; salient exercises too, and do a &quot;dictionary.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as other answers say, it very much depends on your students. First years, I'd &lt;em&gt;never&lt;/em&gt; consider doing something like this (and I take care to use the same notation used in other courses, as far as possible). Advanced/graduate students, specially ones for whom the subject is a specialty course, &lt;em&gt;must&lt;/em&gt; be able to read alternative notations, so there it is much less critical.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-28T19:42:03.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="974" PostTypeId="2" ParentId="926" CreationDate="2014-03-28T19:59:08.500" Score="18" Body="&lt;p&gt;As a student myself, I'd say that, while I'm not representative of all students, some of it is the intimidating or dismissive way some lecturers or teachers might treat questions relevant to simple things like these, or not teach off the exam specification. This leads to a subconscious or even conscious bias against asking these sort of questions which we're unsure of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if I asked a question such as why you can separate roots with multiplication signs, I might be told to just &quot;accept it and learn it&quot; or if I asked if you could do something incorrect, such as one of the examples you gave, I'm likely to receive a response like &quot;Of course not! Don't be silly!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think these two factors especially lead people to just try to memorise as much of the method, but not logic or reasoning, behind the mathematics, and as a result end up getting it wrong, improvising, or guessing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I think it's a lot due to teachers not concerning themselves with a student's future, only getting the student past the current exam hurdle, which results in generations of students who are &lt;em&gt;assumed&lt;/em&gt; to know something relatively simple, but since the teachers deemed it unnecessary to explain since they don't need the explanation for the exam at hand, never learnt it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Glancing back at this, it occurs to me I should mention that I would probably blame government policy more than teachers, who no doubt are under lots of unhelpful pressure and stress which may make it much more difficult to teach &quot;well&quot; - this is just what I have observed from my teachers from an objective standpoint.&lt;/p&gt;&#xA;" OwnerUserId="469" LastEditorUserId="469" LastEditDate="2017-07-17T21:58:05.303" LastActivityDate="2017-07-17T21:58:05.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="975" PostTypeId="2" ParentId="942" CreationDate="2014-03-28T20:23:15.033" Score="1" Body="&lt;p&gt;As a graduating college student, I have gone through countless variations of logic notation since middle school. The only thing that ever bothered me was being forced to use a specific notation. Certain teachers would mark answers incorrect for being in a different, but otherwise completely valid, format. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tell them various notations exist, let them use what they want, and everything will be fine.&lt;/p&gt;&#xA;" OwnerUserId="472" LastActivityDate="2014-03-28T20:23:15.033" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="976" PostTypeId="2" ParentId="926" CreationDate="2014-03-28T20:24:13.413" Score="6" Body="&lt;p&gt;&lt;strong&gt;&lt;em&gt;What can we do about it?&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make them &lt;a href=&quot;http://mathcurmudgeon.blogspot.com/2014/01/do-this-and-bunny-dies.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;feel bad&lt;/a&gt; about it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But seriously...I don't think it's necessarily a linearity thing. I agree that those &quot;simplifications&quot; look nice, resembling the correct shortcuts they've been shown.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've tried some things to overcome this: having students numerically verify all simplifications below a certain level (pros - they must learn to use a calculator effectively and they know when they've made a mistake; cons - many of them do not get a handle on using their calculator, this doesn't show them where they went wrong with the algebra anyway), having students focus on identifying errors by having them create multiple-choice tests (pros - they have to name the common errors and put them in context, they get practice answering the tests; cons - the skills are single-step simplifications outside of a larger problem, they may spend too much time thinking about potential errors).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;...which results in generations of students who are assumed to know something relatively simple, but since the teachers deemed it unnecessary to explain since they don't need the explanation for the exam at hand, never learnt it.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that not enough review material is built into our courses, except for the first day or two. I feel like I have to really hustle to keep my students practicing &quot;old&quot; material so they don't take a vague memory and improvise or guess their way through it when they need it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My answer: Tell your students that you expect they will correctly use exponent rules (or whatever), diagnose where they are at (even if it is an advanced class where they &quot;should&quot; know how to do it), then give them resources to build them up to where they need to be. &lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2017-06-27T21:32:37.330" LastActivityDate="2017-06-27T21:32:37.330" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="978" PostTypeId="1" CreationDate="2014-03-28T21:02:05.560" Score="41" ViewCount="9640" Body="&lt;p&gt;I'm not a teacher, I am a student. But in math, I am one of the best ones in my class so sometimes other people will ask me to explain stuff to them. And usually it works quite well: If I understood the topic, I am able to explain it quite well according to most of the class.&lt;/p&gt;&#xA;&lt;p&gt;But on my way home, via train, one of my fellow students always comes to me and asks me to teach her math. She is not a good student, she is not trying to be and she is, I believe, not able to because she fails again and again at the very same things. This, of course, is not meant offensive, but just observational. But after all, she &lt;em&gt;wants&lt;/em&gt; to be. We are all learning to get our &amp;quot;Abitur&amp;quot; (something like a &amp;quot;university entry exam&amp;quot;). She really wants to study something, so her goal is to make it.&lt;/p&gt;&#xA;&lt;p&gt;Now to specificially the math problems. She fails again and again at the very basic things. Once, we had the expression&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{63}{3}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and she shortened it to&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{61}{1}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I tried to explain to her why that does not work. I told her, at first, that she cannot just take one part of the number and make it away, but she didn't remember why. So we started more basic.&lt;/p&gt;&#xA;&lt;p&gt;I wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{63}{3} = \frac{3\cdot 3 \cdot 7}{3}, $$&lt;/span&gt;&#xA;you can make one of the three's away and replace it with 1, so you have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{3*7}{1} \qquad\text{which is}\qquad \frac{21}{1}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;She seemed to understand it as with another number, she was able to explain it to me again.&lt;/p&gt;&#xA;&lt;p&gt;A few lines later, she did the very same mistake. We exercised it over and over again, but she didn't get it. Though she could reproduce (in her own words) why she cannot do that, she could never apply it.&lt;/p&gt;&#xA;&lt;p&gt;And this is just one (probably bad) example. There are plenty of those.&lt;/p&gt;&#xA;&lt;p&gt;Another problem is her self-esteem. Whenever I tell her she is wrong here and there, she tries to argue with me as if that would change anything. And she cannot take an &amp;quot;I don't know&amp;quot; as an answer from herself. Once, when she had &lt;code&gt;2.5E10&lt;/code&gt; on her calculator and the teacher asked her what that means, the shrugged and told something about &amp;quot;emaginary numbers&amp;quot;, of which she had no understand at all, just so the does not need to admit her not knowing what that meant. And if she does not know the answer at all, she is just guessing. In about 1% of the time she is correct and then she brags for the rest of the time how bored she is and that she is so much faster than anyone else.&lt;/p&gt;&#xA;&lt;p&gt;I don't want to make her seem bad or something, I am just clueless and have no idea what to do anymore. I would really like to help her (though it is not my job), because she is a nice person, but as the idea that what she thinks might be wrong never enters her brain, I see barely anyway to help her.&lt;/p&gt;&#xA;&lt;p&gt;Is there any way to bring the idea that she might sometimes be wrong and has to learn quite a lot of stuff into her brain without insulting her?&lt;/p&gt;&#xA;" OwnerUserId="473" LastEditorUserId="8571" LastEditDate="2021-06-18T14:31:47.787" LastActivityDate="2021-06-18T16:45:14.297" Title="How to teach math to someone who is neither [really] willing nor able to understand it?" Tags="|mathematical-pedagogy|" AnswerCount="8" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="980" PostTypeId="2" ParentId="6" CreationDate="2014-03-28T21:27:22.153" Score="8" Body="&lt;p&gt;This problem was already solved at my university when I was an undergrad in the mid 80s. The assignments were fixed, in that they were the same for all sections, and unchanged along the years. They represented a training platform/self-assessment benchmark to know if you were ready for the tests. The marks came from the tests. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This had the additional advantage that it helped the professors to be more uniform in their teaching, because they were all preparing the students to tackle the same practice questions. &lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2014-03-28T21:27:22.153" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="981" PostTypeId="2" ParentId="978" CreationDate="2014-03-28T21:33:07.230" Score="7" Body="&lt;p&gt;The problem with this lady seems to be, second, a lack of talent for math (for which she is not to blame), and first, an unrealistic view towards her abilities, coupled with no self-criticism. There is very little you can do about the latter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Professional mathematicians and good professors have (or at least should have)  no issue saying &quot;I don't know&quot;. I say &quot;I don't know&quot; every day, and so do most of my colleagues. That a student wouldn't want to do it, is both pretentious and incredible naive. &lt;/p&gt;&#xA;" OwnerUserId="475" LastEditorUserId="475" LastEditDate="2014-03-28T22:05:02.243" LastActivityDate="2014-03-28T22:05:02.243" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="983" PostTypeId="1" CreationDate="2014-03-29T00:06:02.480" Score="10" ViewCount="661" Body="&lt;p&gt;How much detail should you include when providing example solutions for students that involve algebra? I realize that this will of course depend on the level that the students are at. So I guess my question is, how do you know when to show something like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align*}&#xA;4x - 3 &amp;amp;= 2x + 5\\&#xA;+3 &amp;amp; \hspace{2em} +3\\&#xA;4x &amp;amp;= 2x + 8\\&#xA;-2x &amp;amp; \hspace{1em} -2x\\&#xA;2x &amp;amp;= 8\\&#xA;x &amp;amp;= 4\\&#xA;\end{align*}&#xA;versus showing a solution like this:&#xA;\begin{align*}&#xA;4x-3 &amp;amp;= 2x+5\\&#xA;2x &amp;amp;= 8\\&#xA;x &amp;amp;= 4&#xA;\end{align*}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't want to lose my slower students but also don't want to bore my students with too much unnecessary detail. I should add that this question is targeted at Algebra I level specifically, but a similar idea would apply across most levels of math.&lt;/p&gt;&#xA;" OwnerUserId="92" LastEditorUserId="114" LastEditDate="2014-03-29T07:44:12.567" LastActivityDate="2014-03-29T11:22:40.150" Title="How much detail should you show in algebra steps while teaching?" Tags="|calculations|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="984" PostTypeId="2" ParentId="983" CreationDate="2014-03-29T00:10:59.613" Score="5" Body="&lt;p&gt;Answer 1: err on the side of too much detail. In my case at least, the students always want more detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer 2: I think the only way to know for sure is to ask your class what they are comfortable with. Students are often hesitant about volunteering that information, so I created an anonymous email account for them, so that they could provide feedback about this sort of thing (among others). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(p.s. in years of teaching, I have not yet received a single nasty email, either.)&lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="98" LastEditDate="2014-03-29T01:13:10.827" LastActivityDate="2014-03-29T01:13:10.827" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="985" PostTypeId="2" ParentId="978" CreationDate="2014-03-29T00:30:16.040" Score="9" Body="&lt;p&gt;Consider your assumptions with the student: &#xA;Are you assuming she does not have discalculia?  You might want to read up on this, it can afflict about 5% of the population.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have you tried using graphical or manipulative approaches?  For reducing fractions, you could use buttons or coins initially to model division, then bring the student the abstract 63/3 type written fractions.  She may not understand what 63/3 means, but if you put 15 coins down and split them three ways, she might get the idea that the fraction bar means divide.&lt;/p&gt;&#xA;" OwnerUserId="479" LastActivityDate="2014-03-29T00:30:16.040" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="986" PostTypeId="2" ParentId="983" CreationDate="2014-03-29T00:38:42.403" Score="3" Body="&lt;p&gt;You just have to learn to &quot;read&quot; your audience. Getting them to speak up is usually hard, and most of the time too late. Do one step by little step, and start omitting steps until you see they aren't following anymore. Or start the other way, adding steps if you see lost faces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask &lt;em&gt;them&lt;/em&gt; to do (a part of) an example. The level of detail shown is a clue.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-29T00:38:42.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="987" PostTypeId="1" CreationDate="2014-03-29T01:07:09.180" Score="8" ViewCount="288" Body="&lt;p&gt;This question is inspired by &lt;a href=&quot;https://matheducators.stackexchange.com/questions/148/how-can-i-estimate-the-length-of-an-exam&quot;&gt;How can I estimate the length of an exam&lt;/a&gt;. It is not always possible to accurately predict how long an exam will take. As such,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What should one do if an exams looks to be taking substantially longer than intended?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="52" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-03-29T11:38:01.440" Title="What to do if an exam takes longer than intended?" Tags="|undergraduate-education|exams|testing|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="988" PostTypeId="2" ParentId="983" CreationDate="2014-03-29T02:36:33.747" Score="2" Body="&lt;p&gt;By default, show the level of detail that you want to see from the average student in the class.  Be willing to show more detail if students ask (at least, up to a point), but don't ever show any less detail than you'd want to see from them.&lt;/p&gt;&#xA;" OwnerUserId="482" LastActivityDate="2014-03-29T02:36:33.747" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="989" PostTypeId="2" ParentId="987" CreationDate="2014-03-29T02:39:34.997" Score="3" Body="&lt;p&gt;Several options:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Keep the time restriction as-is and scale the scores to compensate.&lt;/li&gt;&#xA;&lt;li&gt;Extend the time restriction, keeping to a single contiguous session (if possible).&lt;/li&gt;&#xA;&lt;li&gt;Give additional time in a separate session.&lt;/li&gt;&#xA;&lt;li&gt;Give additional time in a separate session, but grade the work done in the original session and the extension session separately (have students use a different color for continuing, have students work on a photocopy of the original, etc.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="482" LastActivityDate="2014-03-29T02:39:34.997" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="990" PostTypeId="2" ParentId="983" CreationDate="2014-03-29T02:51:03.807" Score="12" Body="&lt;p&gt;The most effective manner is to ask the students to provide the steps as you carry them out at the board and use the resulting feedback to determine how much detail to provide going forward. That is, as you are writing at the board, you do not just carry on with the next step yourself, but rather, you say, &quot;and then...?&quot; until the students say &quot;add three to both sides&quot;, when you do that and say, &quot;and then...?&quot; and a student shouts out &quot;subtract $2x$ from both sides!&quot; and so on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that you should make no steps yourself that are not directly provided by the students, and sometimes you have to have the patience to wait for a suggestion, or to prompt suggestions by making things easier. This serves several purposes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Students become more actively engaged in the class, and learn the material more effectively. You can call on different students to explain their ideas, and everyone becomes involved. This is also a way of teaching at several levels at once, since the weaker students might understand the specific step, while advanced students appreciate the more general remarks you might explain surrounding it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You get vital feedback about what the students find easy or difficult. The importance of this information cannot be overstated. If all the students are shouting out &quot;add three to both sides!&quot;, then you know immediately that they know that part well and you needn't spend a lot of time on it; but if nobody can say &quot;subtract $2x$&quot; or the equivalent, then you know what needs more attention. And so when students have difficulty providing the step, you give more detail, and when it is easy for them, you may proceed more quickly.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This method works with instruction at virtually any level, whether it is calculus or linear algebra or advanced graduate-level topics. In my experience, many instructors can improve their teaching simply by adopting this one instructional technique, which gives them the important feedback on student understanding that they did not previously even realize they lacked.&lt;/p&gt;&#xA;" OwnerUserId="90" LastEditorUserId="90" LastEditDate="2014-03-29T03:04:12.000" LastActivityDate="2014-03-29T03:04:12.000" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="992" PostTypeId="1" CreationDate="2014-03-29T03:48:56.070" Score="18" ViewCount="640" Body="&lt;p&gt;I don't teach school math but as a part of my voluntary activities in some NGOs, sometimes I am in this special situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was in a school for special children who cannot see I came across a question about a possible problem in reading advanced math texts for those who want to follow research level mathematics in future. Both of writing and reading advanced math texts in the &lt;a href=&quot;http://en.wikipedia.org/wiki/Braille&quot;&gt;Braille&lt;/a&gt; writing system are hard (if it is possible at all). It takes too much time in both writing and reading phases. Fortunately nowadays there are text scanners which can read books and transform the written words to voice. But the main problem of such devices is that they cannot read &quot;unusual&quot; characters like math symbols. This is an essential problem because the meaning of a professional math book/paper is closely related to its math symbols. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; Is there any math text reader device to help math students who cannot see? If no, is it possible to design such a useful device? Do you know anybody (e.g. a software/hardware engineer) who can help us in this direction?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark.&lt;/strong&gt; For the second part of the above question note that most of the professional math texts are written in LaTeX language which is very near to our natural language in some sense. For example when we want to write $\emptyset$ symbol we simply write \emptyset between two $ symbols. Maybe a device which can detect the LaTeX code of each math symbol by scanning, could provide the first step of designing a complete math text reader device. Such an instrument could be a revolutionary change in special education and a great help for those who want to see the beauty of mathematics.         &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="143" LastEditDate="2014-03-31T19:16:15.933" LastActivityDate="2015-03-11T23:05:30.490" Title="Is there any math text reader device to help children who cannot see?" Tags="|mathematical-pedagogy|technology-in-education|special-needs-education|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="993" PostTypeId="1" CreationDate="2014-03-29T04:08:35.007" Score="13" ViewCount="526" Body="&lt;p&gt;This question is asked from a self-teacher standpoint(I am currently trying to learn more about non-standard analysis on my own), but I'd think it could be applicable to educators also.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are good textbooks for non-standard analysis that goes in depth on how to construct the hyperreals and explores the properties of them? I have been working with Keisler's great &lt;a href=&quot;http://www.math.wisc.edu/~keisler/calc.html&quot;&gt;calculus book&lt;/a&gt; based on infinitesimals with associated &lt;a href=&quot;http://www.math.wisc.edu/~keisler/foundations.html&quot;&gt;monograph&lt;/a&gt;. This doesn't seem to go over some of what I think are the more interesting topics like microcontinuity and other properties that the hyperreals have that the reals do not or are extensions with some changes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone taught a course in this and can recommend best ways to learn it and good books to go with it?&lt;/p&gt;&#xA;" OwnerUserId="374" LastEditorUserId="1385" LastEditDate="2017-04-21T14:57:18.377" LastActivityDate="2017-04-21T14:57:18.377" Title="Teaching Infinitesmals and Non-Standard Analysis" Tags="|textbooks|curriculum|mathematical-analysis|infinitesimals|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="994" PostTypeId="2" ParentId="992" CreationDate="2014-03-29T04:23:14.067" Score="9" Body="&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; There is a related meta.MO post &lt;a href=&quot;https://meta.mathoverflow.net/questions/1595/about-blind-mathematicians?cb=1&quot;&gt;here&lt;/a&gt;. (This MESE thread is linked to in a comment there.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Check out &lt;a href=&quot;http://www.csi.cuny.edu/nsf/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt; and, if applicable, you might consider contacting some of the professors involved.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;A Graphical Calculus Course For Blind Students&lt;/strong&gt;: For blind students seeking education and a career in&#xA;  science, engineering and mathematics, the calculus has&#xA;  presented a formidable barrier. This is not due to the&#xA;  intrinsic difficulty of the subject, which is obstacle enough&#xA;  for most students. The additional hurdle for blind students&#xA;  is the substantial graphical component of the typical calculus&#xA;  course. There are two major aspects of that graphical component:&#xA;  primarily, there is the representation of geometrical objects,&#xA;  especially the graphs of functions: then, there is the&#xA;  presentation of mathematical formulas as a graphical display.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Under a grant from the National Science Foundation, the&#xA;  Computer Science Department(CSD) of the College of Staten&#xA;  Island and the Computer Center for Visually Impaired People&#xA;  (CCVIP) of Baruch College are developing text materials and&#xA;  providing an environment to offer blind and visually impaired&#xA;  students technologically assisted access to the graphical&#xA;  content of calculus. The goal of the project is to equal or&#xA;  exceed the quality of courses for students with unimpaired&#xA;  vision. We are installing facilities for reading mathematical&#xA;  text and graphics directly without the help of sighted readers.&#xA;  It is only quite recently that any such technology has become&#xA;  practical and affordable for institutions. We can expect that&#xA;  it will become so for individuals before long. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Somewhat relatedly, there is apparently a way to create LaTeX documents using one's voice. &lt;a href=&quot;https://www.youtube.com/watch?v=xwQlY2XNCLU&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-03-17T10:13:05.280" LastActivityDate="2014-04-09T10:41:12.550" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="995" PostTypeId="1" AcceptedAnswerId="1003" CreationDate="2014-03-29T04:39:49.410" Score="28" ViewCount="2416" Body="&lt;p&gt;I am trying to teach my calculus students to apply integration by &lt;em&gt;thinking&lt;/em&gt; about what they are integrating rather than just applying formulas.  Calculus books are full of formulas like &quot;to find the volume of a solid of revolution obtained by rotating region A around axis B, write down integral C&quot;.  I don't want my students to just memorize these formulas, but to understand where they come from and to be able to apply integration in contexts where these formulas don't necessarily apply.  What are some good examples of applications of integration that don't fit into the textbook formulas?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a simple example: find the area of a circle by dividing it into concentric rings and integrating the area $2\pi x\, dx$ of each ring from 0 to $r$.  Of course you can do the same finding the volume of a sphere.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2022-06-07T20:14:08.257" Title="Unusual applications of integration" Tags="|calculus|examples|" AnswerCount="10" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="996" PostTypeId="2" ParentId="983" CreationDate="2014-03-29T06:16:52.730" Score="4" Body="&lt;p&gt;The amount you write in the calculation of things should be inversely proportional to how many times the students have seen these calculations. The first example or two should have everything. Later examples should omit the easiest steps. I think a rule of thumb should be to ask yourself &quot;Can the students show the missing steps in example N by looking at examples N-1 and below?&quot;. There are a couple of things I'd like to mention though.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Are the examples shown in different lectures? Will the students still remember the previous lectures (perhaps, but not fully)? If there was homework assigned in between did the students do it (probably not)? In this case the decay of steps shown should probably be on the lower side. You can repeat an example from the last lecture with the same amount of detail so as to get students up to speed and make them more comfortable to go to the next step.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are the steps you want to omit touching on knowledge learned in the previous term/year (This is just like 1. but on a longer time scale)? If so, perhaps it is safe to assume that students don't remember perfectly and it's better to err on the side of repeating yourself.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I know you are targeting Algebra I specifically but I'd like to highlight that there are a lot of first year university students who are not comfortable with something like your second example (albeit the examples faced may involve a more complicated expression with fractions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, when I tutor students the usual complaints are that teachers either don't show enough properly worked examples that they can then mimic and learn the material from or the examples are too easy and the exercises they need to work on are too complex comparatively.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;don't show enough properly worked examples that they can then mimic and learn the material from&lt;/li&gt;&#xA;&lt;li&gt;or the examples shown are too easy and the exercises they need to work on are too complex comparatively &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="390" LastActivityDate="2014-03-29T06:16:52.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="997" PostTypeId="1" AcceptedAnswerId="1030" CreationDate="2014-03-29T07:15:53.773" Score="7" ViewCount="404" Body="&lt;p&gt;This question is inspired by a set of questions at the end of &lt;a href=&quot;http://en.wikipedia.org/wiki/Timothy_Gowers&quot; rel=&quot;nofollow noreferrer&quot;&gt;Timothy Gowers&lt;/a&gt;' book:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0192853619&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics: A Very Short Introduction, Oxford University Press, 2002.&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The possible relation between mathematics and music is an old open question. In the ancient Greece some philosophers like Pythagoras and Plato strongly believed that there is a mysterious relation between these two completely different subjects. Their arguments were mainly based on the abstract nature of mathematics amongst different fields of science and abstract nature of music amongst different fields of art. Also they argued on the possible similarity between &quot;mathematical beauty&quot; and &quot;beauty of music&quot; based on the philosophical similarity between notions of &quot;pattern&quot; and &quot;harmony&quot; in math and music respectively. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the recent years the new approach to this old question is mainly focused on the possible positive impacts of listening and training music on improving mathematical abilities of children and adults. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1.&lt;/strong&gt; Are there any scientific math education research which proves a direct relevance between listening/training music and a meaningful improvement in children/adults' mathematical abilities?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2.&lt;/strong&gt; If the answer of the question 1 is positive and there is such a relation, do different genres of music (e.g. classic, without vocal, pop, rock, etc.) have different impacts? What the statistics shows? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 3.&lt;/strong&gt; Does listening/training music regularly have a notable impact on notable mathematical and scientific discoveries? Precisely, is there any statistical research which determines the frequency of musical interests of great mathematicians and scientists? e.g. Both Cantor and Einstein were good violinists.               &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/gwwzo.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-04-03T19:12:11.850" LastActivityDate="2014-04-03T19:12:11.850" Title="Can music improve math abilities?" Tags="|mathematical-pedagogy|psychology|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="998" PostTypeId="2" ParentId="894" CreationDate="2014-03-29T10:44:09.767" Score="15" Body="&lt;p&gt;&lt;strong&gt;Brief Remarks:&lt;/strong&gt; It is difficult to find longitudinal studies on calculator use as specified by the OP. One of the reasons for this is that tracking students from, e.g., high school till college is quite complicated. Another reason is that studies on technology use are often fodder for theses, which are completed in too short a timeframe to provide such an account (i.e., one over several years). A third reason is that technology changes pretty quickly, so that, e.g., computer use five years ago and computer use today look very different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The National Center for Education Statistics has some data on their page &lt;a href=&quot;http://nces.ed.gov/pubs92/web/92060.asp&quot;&gt;Calculators and Computers&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some information is available there from the National Assessment of Educational Progress (NAEP)  administered in 1990; for example, the site says:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Teachers and administrators reported that computer access was difficult for about half of the students at grades 4 and 8. Teachers and students agreed that school use of computers was greater at grade 4 than at grade 8, but usage in general was quite limited. A positive overall relationship was seen to exist between the availability of computers and average mathematics proficiency at grade 4; this relationship tended to hold across ability groupings at that grade level. At grade 8, however, there appeared to be no relationship between the availability of computers and mathematics performance.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Calculator use in Calculus:&lt;/strong&gt; With regard to Calculus and calculators, one paper of interest is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Penglase, M., &amp;amp; Arnold, S. (1996). The Graphics Calculator in Mathematics Education: A Critical Review of Recent Research. &lt;em&gt;Mathematics education research journal, 8&lt;/em&gt;(1), 58-90. &lt;a href=&quot;http://eric.ed.gov/?id=EJ528433&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The text provided at the link above says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Reviews recent research into the effectiveness of the graphing calculator as a tool for instruction and learning within precalculus and calculus, specifically in the study of functions, graphing, and modeling. Contends that much research fails to provide clear guidance or informed debate regarding the role of graphing calculators in mathematics teaching and learning. Contains 103 references. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Further Remarks:&lt;/strong&gt; Though longitudinal data are not easy to find, there are certainly a &lt;em&gt;lot&lt;/em&gt; of studies on calculator use (consider that the one source above has over a hundred references). Below I include six theses that I found on the first page of ProQuest while searching for &lt;em&gt;calculator calculus mathematics education&lt;/em&gt;. Even with just six studies and only excerpts from their abstracts, you can see that the length grows very quickly. Overall, I think the lesson is that there is not substantial evidence for the use of calculators in teaching Calculus; perhaps this may change moving ahead, but these dissertations (from a range of years between 1993 and 2011) do not provide conclusive data for calculator adoption.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;&#xA;Searching Google Scholar (as suggested in the comments) is one way to find a lot of sources. Just to give a slight feel for some of the literature out there, here are some abstract &lt;em&gt;excerpts&lt;/em&gt; from recent dissertations about calculator use as it relates to Calculus instruction. &lt;strong&gt;Bold text&lt;/strong&gt; was added by me.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Everett, K. M. (2011). Does the use of a graphing calculator tutorial affect the attitudes, achievement, and calculator ability of non-mathematics majors in a calculus course? (Order No. 3477163, The University of Southern Mississippi). ProQuest Dissertations and Theses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Abstract:&lt;/strong&gt; This study shows the effects of a graphing calculator tutorial for non-mathematics major students taking calculus as a required course. Qualitative methods were used in order to provide information about the types of students taking a calculus course for nonmathematic students. The tutorial was used to help the student better understand the use of a graphing calculator while learning calculus. The study tested changes in attitudes, achievement, and ability with respect to the use of a graphing calculator through the use of a post-survey, a pre/post-quiz, and overall grade comparison. The tutorial was not heavily used by the students. &lt;strong&gt;This finding suggests that students are not dependent on the use of a graphing calculator to better understand calculus and its applications.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hunter, J. S. (2011). The effects of graphing calculator use on high-school students' reasoning in integral calculus. (Order No. 3463473, University of New Orleans). ProQuest Dissertations and Theses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Abstract:&lt;/strong&gt; This mixed-method study investigated the impact of graphing calculator use on high school calculus students. reasoning skills through calculus problems when applying to concepts of the definite integral and its applications. The study provides an investigation of the effects on reasoning when graphing calculators are used, since it is proposed that, through reasoning, conceptual understanding can be achieved. Three research questions were used to guide the study: (1) Does the use of the graphing calculator improve high school calculus students' reasoning ability in calculus problems applying the definite integral? (2) In what specific areas of reasoning does use of the graphing calculator seem to be most and least effective? and (3) To what extent can students who have used the graphing calculator demonstrate ability to solve problems using pencil and paper methods? The study included a quantitative, quasi-experimental component and a qualitative component. Results of the quantitative and qualitative analysis indicate that (1) graphing calculators had a positive impact upon students' reasoning skills (2) graphing calculators were most effective in the areas of initiating a strategy and monitoring progress (3) students' reasoning skills were most improved when graphing calculators were used together with the analytic approach during both instruction and testing and &lt;strong&gt;(4) students who used the graphing calculator performed equally as well in all elements of reasoning as those who used pencil and paper to solve problems.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Nasari, G. Y. (2008). The effect of graphing calculator embedded materials on college students' conceptual understanding and achievement in a calculus I course. (Order No. 3296875, Wayne State University). ProQuest Dissertations and Theses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Abstract:&lt;/strong&gt; The two types of treatment used were: (a) Using TI-83 graphing calculator embedded materials to teach four units of Calculus I course to the experimental group. (b) Using traditional methods that did not use TI-83 graphing calculator embedded materials to teach four units of Calculus I to the control group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Data analyses revealed the following: (a) there was a statistically significant difference in &lt;strong&gt;favor of the experimental group&lt;/strong&gt; found on the skills oriented problems in the limit and their properties unit; (b) there was statistically significant difference in &lt;strong&gt;favor of the experimental group&lt;/strong&gt; found on the conceptually oriented problems in the limit and their properties unit; (c) there was &lt;strong&gt;no statistically significant difference&lt;/strong&gt; between the experimental and control groups found on the skills oriented problems in the differentiation unit; (d) there was a statistically significant difference in &lt;strong&gt;favor of the experimental group&lt;/strong&gt; on the conceptually oriented problems in the differentiation unit; (e) there was a statistically significant difference in &lt;strong&gt;favor of the experimental group&lt;/strong&gt; on the skills oriented problems in the application of differentiation unit; (f) there was a statistically significant difference in &lt;strong&gt;favor of the experimental group&lt;/strong&gt; on the conceptually oriented problems in the application of differentiation unit; (g) there was &lt;strong&gt;no statistically significantly difference&lt;/strong&gt; between the experimental and control groups on the skills oriented problems in the integration unit; (h) there was a statistically significant difference in &lt;strong&gt;favor of the experimental group&lt;/strong&gt; on the conceptually oriented problems in the integration unit. &lt;strong&gt;The final analysis showed that the use of graphing calculator embedded materials improved students' grades in the Calculus I course.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ocak, M. A. (2006). College calculus students' experiences with, attitudes towards, and uses of the graphing calculator: Interactions from cognitive flexibility perspective. (Order No. 3221094, State University of New York at Albany). ProQuest Dissertations and Theses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Abstract:&lt;/strong&gt; This study found that &lt;strong&gt;mere availability of the graphing calculator in the problem solving process does not affect or change students' solving problem strategies&lt;/strong&gt;. Rather, the results suggest that using the graphing calculator and mathematical understanding must work together for the solution of the problem. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Castillo, T. F. (1997). Visualization, attitude, and performance in multivariable calculus: Relationship between use and nonuse of graphing calculator. (Order No. 9824883, The University of Texas at Austin). ProQuest Dissertations and Theses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Abstract:&lt;/strong&gt; Results indicated that the treatment and nontreatment groups had no statistically significant difference $(\alpha = .05)$ in their mathematical processing preference. The four sub-scales used from the Fennema-Sherman Mathematics Attitudes Scales also did not register a statistically significant difference between groups. Assigned grade for overall performance between the treatment and nontreatment groups was statistically significant at $\alpha = .05$ in favor of the treatment group [TI-85 group]. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Emese, G. L. (1993). The effects of guided discovery style teaching and graphing calculator use in differential calculus. (Order No. 9316149, The Ohio State University). ProQuest Dissertations and Theses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Abstract:&lt;/strong&gt; The research design was a three group experimental study in a university freshmen differential calculus course with the following group specifications. Group 1: Use of graphing calculators and discovery approach, Group 2: Use of graphing calculators without discovery, Group 3: Traditional instruction. In the discovery section, part of the new material was covered using worksheets, where a sequence of questions/problems led to the new concept, relationship or technique. Students worked in groups, pairs or individually. They could get help from the hint-sheet, solution-sheets, their classmates and/or from the instructor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;No statistically significant differences&lt;/strong&gt; were found on the computational, conceptual, and transfer skills parts of the pretest, &lt;strong&gt;nor&lt;/strong&gt; on the following background variables: placement level, the year in which students took the placement test, their precalculus grade and the year in which they took precalculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analyses of covariance were used for student achievement comparisons. The scores on the corresponding subtest of the pretest served as covariates. Students' time spent on the course and the extent to which students worked with their classmates outside of class were compared. &lt;strong&gt;Statistically significant differences were not found between the groups on any of these variables. No instructional method proved superior to the others on comparison.&lt;/strong&gt; &lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-03-29T22:36:09.747" LastActivityDate="2014-03-29T22:36:09.747" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="999" PostTypeId="2" ParentId="983" CreationDate="2014-03-29T11:22:40.150" Score="3" Body="&lt;p&gt;You can test, how detailed the students' understanding is, by using alternative exercises. Examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Give them a rather long solution without the transformations or with some steps missing. Ask them to provide the missing transformations or steps.&lt;/li&gt;&#xA;&lt;li&gt;Give them a solution with mistakes in several places. Ask them to find and correct the solution. You should also give one solution with unnecessary or wrong leading transformations, which are nonetheless carried out correctly.&lt;/li&gt;&#xA;&lt;li&gt;Let them talk each other trough a rather long solution and listen to what transformations and steps they name or leave out and which ones the student talked to asks for.&lt;/li&gt;&#xA;&lt;li&gt;One students presents a solution uninterrupted and the others give feedback after a small break.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-03-29T11:22:40.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1000" PostTypeId="2" ParentId="995" CreationDate="2014-03-29T11:30:32.813" Score="5" Body="&lt;p&gt;Let them guess the mass of a mountain by showing them some pictures of it. They then can then model the mountain in different ways:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;cone&lt;/li&gt;&#xA;&lt;li&gt;rotational body&lt;/li&gt;&#xA;&lt;li&gt;irregular pyramid&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Additionally they can model the tip of the mountain in different ways&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;flat&lt;/li&gt;&#xA;&lt;li&gt;flat with inverse semi-ball for the caldera&lt;/li&gt;&#xA;&lt;li&gt;flat with inverse cone for the caldera&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can also give them elevation data and let them model the mountain as a stack of layers.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-03-29T11:30:32.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1001" PostTypeId="2" ParentId="987" CreationDate="2014-03-29T11:38:01.440" Score="6" Body="&lt;p&gt;Depending on the moment of realization, there are different options.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Early in the exam:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;kill some tasks&lt;/li&gt;&#xA;&lt;li&gt;declare some tasks to be chosen between&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Near the end of an exam before anybody has left&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;give extra time in the same session (beware not to kill breaks between lessons)&lt;/li&gt;&#xA;&lt;li&gt;let students choose one task to be killed for them (count those points as bonus points for students who did all the tasks)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;After somebody has left or after the exam&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;rescale the scores&lt;/li&gt;&#xA;&lt;li&gt;let students redo some of the tasks/similar tasks without telling them before&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-03-29T11:38:01.440" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1002" PostTypeId="1" AcceptedAnswerId="1013" CreationDate="2014-03-29T12:05:14.570" Score="18" ViewCount="21973" Body="&lt;p&gt;When introducing logic in a first semester university course, the examples I use are often quite artificial. One example: One of three kids (Annie, Bob, Chris) has broken a window. Annies says &quot;it was Bob&quot;, Bob says &quot;it was Chris&quot;, Chris says &quot;exactly one of them is lying&quot;. You know that all three kids are lying. Who has broken the window?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, such examples (how ever complicated you design them) are far away from real-world problems. I remember I once heared that in some expensive medical tests, you mix up material from different persons to see whether one of them is positive (don't know if this is true). So I became curious if there might be more realistic applications of logic (e.g. by mixing different samples).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Are there real-world applications of propositional calculus? (Maybe very different from what I suggested?) Good answers should refer to problems which already exist (for instance in applied sciences) and are more easily handeled with the use of propositional calculus. I do not mean problems which were just generated as examples for the use of logic.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="199" LastEditDate="2014-03-29T16:05:39.960" LastActivityDate="2018-03-01T01:33:12.090" Title="Real-World Applications of Logic" Tags="|undergraduate-education|examples|logic|" AnswerCount="12" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1003" PostTypeId="2" ParentId="995" CreationDate="2014-03-29T12:27:08.630" Score="18" Body="&lt;p&gt;An example that illustrates a slightly different integral is the following. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/oKbY3.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An observer at point $O$ is exposed to traffic noise from a long straight highway (the line $AP$). The task is to show that the total noise level at $O$ is inversely proportional to the distance $r=|OA|$ from the highway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use the following approximations/simplifications. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The traffic density is constant along the road (a kind of an average).&lt;/li&gt;&#xA;&lt;li&gt;The level of &quot;atomic noise intensity&quot; at $O$ follows the inverse square law (this is probably a good approximation you can easily justify).&lt;/li&gt;&#xA;&lt;li&gt;The total noise intensity is the sum of the infinitesimal/atomic noise intensities (i.e. the quantity measured in Watts per square meter is additive. You only switch to the logarithmic decibel scale afterwards if ever, as things are not additive on the decibel scale).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So if you place $x$-axis along the road, and $P$ is at $x$, then the noise generated along a segment of the road of length $dx$ around $P$ has (at a unit distance from $P$) intensity $K\,dx$ for some constant $K$ proportional to traffic density (= number of passing vehicles per hour). When we take the distance $|OP|=\sqrt{r^2+x^2}$ into account we get&#xA;$$&#xA;dI=K\,\frac{dx}{r^2+x^2}&#xA;$$&#xA;for the contribution of this atomic noise to the total observed noise level at point $O$.&#xA;The total noise intensity is thus&#xA;$$&#xA;I=\int dI=\int_{-\infty}^{\infty}K\,\frac{dx}{r^2+x^2}.&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you get to illustrate/discuss:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;improper integrals&lt;/li&gt;&#xA;&lt;li&gt;indefinite integral of a simple rational function&lt;/li&gt;&#xA;&lt;li&gt;discuss what goes wrong when $r\to0+$ (the approximations fail, and you cannot be at distance zero from a continuous source of the noise)&lt;/li&gt;&#xA;&lt;li&gt;point out that a similar (but also different) integral emerges when you calculate the field generated by an electric current along a long straight wire&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have tried this both as an in class and as a homework example. As a HW problem it did create some discussion as the physics majors protested at first (assuming that I was adding decibels together). I guess the non-physics majors may have been bored. You know your customers (and if somebody can add an example from chemistry I will be very grateful).&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2017-06-25T07:10:10.173" LastActivityDate="2017-06-25T07:10:10.173" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1004" PostTypeId="1" AcceptedAnswerId="1011" CreationDate="2014-03-29T12:57:23.007" Score="7" ViewCount="295" Body="&lt;p&gt;I'm fairly new to teaching and am currently teaching high school students about vector geometry. I am currently making a test for them for next week and it is mostly about the scalar product and the vector product. For this, I went through my old exams and exercises (which are a lot) and chose the ones I thought were important to solve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now there is one exercise that I think would be a good exercise for them, however it has really really ugly numbers as a result whereas all other exercises in the exam have very nice numbers. It reads as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The point $M = (6,1,6)$ is midpoint of the regular octahedron $ABCDEF$. Further, one of the edges is part of the line $g$ consisting of the points $P = (0,-6,6)$ and $Q = (9,0,3)$. Find the volume of the octahedron without calculating any futher points. Then, find the coordinates of the six vertices $A,B, C, D, E, F$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What software would be suited to create such exercises? Basically, what I need is an octahedron with a &quot;nice&quot; volume, its vertices at nice points in $\mathbb{R}^3$ and one of its edges running through nice points. By nice, I mean points with coordinates in $\mathbb{Z}$ or maybe very simple fractions.&lt;/p&gt;&#xA;" OwnerUserId="493" LastEditorUserId="1892" LastEditDate="2014-07-09T19:41:30.540" LastActivityDate="2014-07-09T19:41:30.540" Title="How are exercises in vector geometry created?" Tags="|exams|software|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1005" PostTypeId="2" ParentId="1002" CreationDate="2014-03-29T13:07:43.653" Score="7" Body="&lt;p&gt;Nearly all examples for perceived equivalences which are only implications. The classic (at least in Germany) is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If it rains, the road gets wet.&lt;/p&gt;&#xA;&lt;p&gt;If the road ist wet, has it rained?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You'll stumble across these things everyday if you look out for them.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-29T13:07:43.653" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1006" PostTypeId="2" ParentId="1004" CreationDate="2014-03-29T13:10:27.780" Score="2" Body="&lt;p&gt;&lt;strong&gt;Trial-and-Error solution&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Model this situation with any Dynamic Geometry Software, e.g. &lt;a href=&quot;http://www.geogebra.org&quot; rel=&quot;nofollow&quot;&gt;GeoGebra&lt;/a&gt;, using parameters. Change the parameters, e.g. using slow animation, watching for &quot;nice&quot; solutions.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-03-29T13:10:27.780" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1007" PostTypeId="2" ParentId="959" CreationDate="2014-03-29T13:40:04.413" Score="6" Body="&lt;p&gt;I disagree with the &quot;diagonalize a matrix&quot; type &lt;em&gt;exam&lt;/em&gt; questions. Such questions call for accurate, longish computation, not understanding. That I leave for homework. In an exam I'd ask &lt;em&gt;why&lt;/em&gt; you'd diagonalize a matrix (e.g. explain how to do something using this, step by step, not do it). Set questions up so that computation mistakes don't invalidate the work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comming up with questions testing understanding, that the student can rapidly grasp, and can solve in a few minutes if they know the subject, is &lt;em&gt;hard&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-29T13:40:04.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1008" PostTypeId="2" ParentId="993" CreationDate="2014-03-29T14:01:34.910" Score="8" Body="&lt;p&gt;Alain Robert's book on non-standard analysis (following Edward Nelson's IST approach) was what finally convinced me that non-standard analysis could be packaged in an effective form. It is a small book, and the narrative is very unpretentious and informal, yet touches several further topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Nonstandard analysis in practice&quot;, edited by Diener and Diener is very good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Nonstandard methods in stochastic analysis and mathematical physics&quot;, edited by Albevario, Fenstad, et al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A. Robinson's original book is in fact quite good, too.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-29T14:01:34.910" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1010" PostTypeId="2" ParentId="926" CreationDate="2014-03-29T15:03:56.603" Score="13" Body="&lt;p&gt;In addition to other good answers and comments... I think it should be noted that &quot;linear mathematics&quot; at higher levels is the part of mathematics that we (collectively) understand relatively well, while &quot;non-linear mathematics&quot; is often intractable... except to the extent we can usefully approximate it linearly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is &lt;em&gt;both&lt;/em&gt; symbol patterns and the mathematical assertion(s) given by the simplest symbol patterns that are both appealing and very handy ... if correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Paraphrasing, and as in some of the other answers, if the choice is between &quot;being stuck&quot; and &quot;making progress&quot;, often an admittedly dubious assumption of linearity, or some other mildly outrageous optimistic assumption, is necessary to avoid getting stuck. That is, methodologically, linearity assumptions (and other such) are entirely reasonable... at least as a transitional device.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, after all, &quot;differentiability&quot; of a function is in many regards the assertion that the function can be locally approximated by a linear function. The Newton-Raphson method shows how &lt;em&gt;iteration&lt;/em&gt; of a nearly-linear device achieves excellent effects.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-29T15:03:56.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1011" PostTypeId="2" ParentId="1004" CreationDate="2014-03-29T15:29:52.877" Score="7" Body="&lt;p&gt;I'm not sure how software would be helpful here.  To make a problem of this form, you need to:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Choose any center point $M$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Choose two orthogonal vectors $\textbf{v}$ and $\textbf{w}$ of the same length.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Compute $M + \textbf{v}$ and $M+\textbf{w}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Find two other points on the line through $M+\textbf{v}$ and $M+\textbf{w}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The volume will end up being $\dfrac{4}{3}\|\textbf{v}\|^3$.  If you want to avoid radicals during the intermediate calculations, the only important consideration is that $\textbf{v}$ and $\textbf{w}$ have integer lengths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some orthogonal triples of integer-length vectors of equal lengths that you can use to pick $\textbf{v}$ and $\textbf{w}$.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;$(3,4,0)$, $(4,-3,0)$, and $(0,0,5)$.  More generally, $(a,b,0)$, $(-b,a,0)$, and $(0,0,c)$ for any Pythagorean triple $(a,b,c)$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$(1,2,2)$, $(2,1,-2)$, and $(-2,2,-1)$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$(2,3,6)$, $(6,2,-3)$, and $(3,-6,2)$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$(4,4,7)$, $(1,-8,4)$, and $(8,-1,-4)$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you look for them, these triples can be found in many vector geometry problems.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-03-29T19:22:09.173" LastActivityDate="2014-03-29T19:22:09.173" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1012" PostTypeId="2" ParentId="1002" CreationDate="2014-03-29T16:28:22.840" Score="12" Body="&lt;pre&gt;&lt;code&gt; if (real world applications are those which make money){&#xA;   then if (programming makes money){&#xA;     print(&quot;Programming is a real world application of propositional logic&quot;)&#xA;     }&#xA;   }   &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;You definitely have to know how to evaluate truth values of various statements to accomplish even very basic programming tasks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am adding extra text to prevent the constant spam attacks on this answer.  I think this should do it.  I believe I am now over 400 characters.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2014-08-04T02:00:34.220" LastActivityDate="2014-08-04T02:00:34.220" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1013" PostTypeId="2" ParentId="1002" CreationDate="2014-03-29T16:39:22.950" Score="12" Body="&lt;p&gt;There's a couple nice directions you could go here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To discuss questions that are most similar to your example but more &quot;authentic&quot;, you could look at any variety of constraint satisfaction problems.  For example, &lt;strong&gt;Sudoku&lt;/strong&gt; can be formulated as a bunch of constraints (each grid has exactly one of each number, no number appears multiple times, etc.)  All of these can be formulated in &lt;strong&gt;propositional calculus&lt;/strong&gt;.  Another nice example is &lt;strong&gt;scheduling classrooms&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For something totally different, you could point out how ambiguous english can be and motivate logic as a precise system.  My favorite way to do this is to choose some particularly hairy paragraph from a very old philosopher (e.g. some part of Pascal's wager), ask one of the students to read the whole thing &lt;em&gt;quickly&lt;/em&gt;, and then ask the students if they understood anything.  This isn't directly what you're asking for, I think, but it definitely helps motivate why mathematicians might care about logic to students who have read confusing prose (which is most of them).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, using the SMT solver Z3, this is code to solve Sudoku by defining logical formulae:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;# define the puzzle as a 2D array here&#xA;from z3 import *&#xA;solution = Function('solution', IntSort(), IntSort(), IntSort(), BoolSort())&#xA;&#xA;constraints = []&#xA;&#xA;for i in range(len(puzzle)):&#xA;    for j in range(len(puzzle)):&#xA;        if puzzle[i][j] != 0:&#xA;            # The result must match what we already know.&#xA;            constraints.append(solution(i, j, puzzle[i][j]))&#xA;&#xA;        # Every cell must have some value&#xA;        constraints.append(Or([solution(i, j, n) for n in range(1, 10)]))&#xA;&#xA;        #...but only one value&#xA;        constraints.append(And([&#xA;          Implies(&#xA;            solution(i, j, n),&#xA;            And([Not(solution(i, j, np)) for np in range(1, 10) if np != n])&#xA;          )&#xA;        for n in range(1, 10)]))&#xA;&#xA;&#xA;# Every row must have some value&#xA;constraints.append(And([&#xA;  And([&#xA;    Or([solution(i, j, n) for j in range(9)])&#xA;  for i in range(9)])&#xA;for n in range(1, 10)]))&#xA;&#xA;# Every column must have some value&#xA;constraints.append(And([&#xA;  And([&#xA;    Or([solution(i, j, n) for i in range(9)])&#xA;  for j in range(9)])&#xA;for n in range(1, 10)]))&#xA;&#xA;# Every box must have some value&#xA;constraints.append(And([&#xA;    And([&#xA;      Or([solution(i + boxi*3, j + boxj*3, n) for i in range(3) for j in range(3)])&#xA;    for boxi in range(0, 2) for boxj in range(0, 2)])&#xA;for n in range(1, 10)]))&#xA;&#xA;s = Solver()&#xA;s.add(constraints)&#xA;&#xA;if s.check() == sat:&#xA;    m = s.model()&#xA;    r = [[ [n for n in range(1, 10) if is_true(m.evaluate(solution(i, j, n)))][0]  for j in range(9)] for i in range(9)]&#xA;    print_matrix(r)&#xA;else:&#xA;    print &quot;failed to solve&quot;&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="93" LastEditorUserId="93" LastEditDate="2014-03-31T17:00:56.580" LastActivityDate="2014-03-31T17:00:56.580" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1014" PostTypeId="2" ParentId="925" CreationDate="2014-03-29T16:43:26.087" Score="3" Body="&lt;p&gt;The reason that similarities arise in sequences and series is that in discussing the convergence of series and other properties, you're actually working with a &lt;em&gt;partial sum&lt;/em&gt; of a series, like &quot;the first n terms&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These partial sums, however, are not themselves a series: they in fact form a &lt;em&gt;sequence&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, the partial sums of 1 + 2 + 3 + 4 + ... form the sequence 1, 3, 6, 10, 15.  If such a sequence converges on a value, then the series converges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The partial sums, and the sequence which they form, are not objects which are not interchangeable with the original series from whose fragments they are derived.&lt;/p&gt;&#xA;" OwnerUserId="500" LastActivityDate="2014-03-29T16:43:26.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1015" PostTypeId="2" ParentId="995" CreationDate="2014-03-29T17:03:52.870" Score="10" Body="&lt;p&gt;Physics provides with a lot of examples, whose main disadvantage is that they might be considered too much physics by some, i.e., require too much thinking or knowledge about something other than mathematics. (Also engineers and physicists might rather consider them the &lt;strong&gt;usual&lt;/strong&gt; applications of integration.) I kept the examples simple, and as a result they might still be solved without understanding by guessing what to integrate, but they can easily made more difficult to avoid this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lengthening of an elastic rope under its own weight&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q: An elastic rope of length $l$ has a weight per length of $ρ$ (e.g., $ρ=1\,\text{kg}/\text{m}$) – both if unstretched. If a total of mass $m$ is pulling on a piece of rope, it extends to $f(m)$ of its unburdened length (typically $f(m)=1+Em$). To what length does the rope extend under its own weight?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A: A infinitesimally short piece of rope of length $\mathrm{d}x$ located at point $x$ (measured from the bottom of the rope in the unstretched case) is burdened with a weight of $ρx$ and thus its burdened length is $f(ρx)\mathrm{d}x$. The burdened length of the entire rope thus is the integral $\int_0^l f(ρx) \mathrm{d}x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A difficulty lies in considering the rope unstretched for most purposes (but this can be given as a hint). To “spice up” this problem, one could attach weights to the ropes (at the end or other positions) or have the rope vary in thickness (i.e., make $ρ$ as well as $f$ depend on $l$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Uniformly accelerated object&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q: An apple starts falling from a tree at $t=0$. Its speed $v(t)$ at time $t$ is $gt$ (with $g$ being the &lt;a href=&quot;http://en.wikipedia.org/wiki/Gravity_of_Earth&quot; rel=&quot;noreferrer&quot;&gt;local gravity of Earth&lt;/a&gt;). How far has it fallen after time $T$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A: The distance covered by the apple during an infinitesimally short time $\mathrm{d}t$ is $v(t)\mathrm{d}t$ and thus the total distance covered up to time $T$ is $\int_0^T v(t)\mathrm{d}t = \int_0^T at\mathrm{d}t = \tfrac{1}{2}aT^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To “spice up” the problem, one could consider more complicated acceleration functions (e.g., considering air resistance), initial speeds and so on.&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2014-03-30T18:34:12.460" LastActivityDate="2014-03-30T18:34:12.460" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1016" PostTypeId="2" ParentId="6" CreationDate="2014-03-29T17:45:20.707" Score="11" Body="&lt;p&gt;Some ideas I've developed over the years:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Give the answer (for example: sin(x)/2) and have the student find two or three questions (ex.&quot;what is the function whose derivative is&quot;) with this answer, of course before studying integral calculus.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Audrey thinks that the integral of a product of functions is the product of the integrals of the functions. Write her a letter giving two counterexamples (one simple, one more complicated) to show that this is false.&quot; If the subject is the definite integral, one of the counterexamples should also have a graphic explanation.&lt;/li&gt;&#xA;&lt;li&gt;The same idea can be used for derivatives, trigonometric and algebraic identities.&lt;/li&gt;&#xA;&lt;li&gt;When teaching functions of several variables: &quot;Find 5 examples of functions of 2 variables from different sciences. You can look anywhere you want, but you've got to identify your variables.&quot; Usually, one or two A= 2*Pi*r would appear as an answer. When giving back the assignment, it was a good time to discuss the difference between variables and constants.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="503" LastEditorUserId="11" LastEditDate="2014-03-29T20:01:11.317" LastActivityDate="2014-03-29T20:01:11.317" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1017" PostTypeId="2" ParentId="1002" CreationDate="2014-03-29T19:24:27.793" Score="3" Body="&lt;p&gt;Forensic Science is probably one of the answer. There you have evidences which are logically evaluated to come to a conclusion. Sometimes logic is the only option &lt;em&gt;to fill in the gap&lt;/em&gt; to continue/reach the final conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://studiegids.uva.nl/web/uva/sgs/nl/c/11457.html&quot; rel=&quot;nofollow&quot;&gt;Formal Course&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user505" LastEditorDisplayName="user505" LastEditDate="2014-03-30T14:12:49.613" LastActivityDate="2014-03-30T14:12:49.613" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1018" PostTypeId="2" ParentId="1002" CreationDate="2014-03-29T20:13:56.997" Score="10" Body="&lt;p&gt;I think a good real-world example for an undergraduate course on logic is to discuss the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/CbaXP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The source of this is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;McGee, V. (1985). A counterexample to modus ponens. &lt;em&gt;The Journal of Philosophy&lt;/em&gt;, 462-471.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In a somewhat related vein, I think it is wise to discuss the difficulty that can be run into when trying to rephrase language from our regular lives into &quot;logical&quot; notation. &lt;a href=&quot;http://en.wikipedia.org/wiki/Susanna_S._Epp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Susanna S. Epp&lt;/a&gt; gives the following example, in which different interpretations of &quot;unless&quot; are provided by three different mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/osTPP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-03-29T20:13:56.997" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1019" PostTypeId="2" ParentId="959" CreationDate="2014-03-29T20:16:18.990" Score="8" Body="&lt;p&gt;My method for creating a concept-testing question is as follows. Start with the most open-ended question possible on the topic, then slowly refine it until it has a &quot;correct answer.&quot; I've outlined an example for a conceptual derivative question below.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Start with the generic question: &quot;&lt;strong&gt;Tell me about the derivative,&lt;/strong&gt;&quot; and recognize that the generic question is not very good because it still has many terrible yet technically-correct answers. For example, a student might answer &quot;Tell me about the derivative&quot; with the correct but unsatisfying&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The derivative of a function is another function, like the derivative&#xA;  of x^2 is 2x and the derivative of 4 is 0.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Refine the question to eliminate the possibility of empty answers. In my example, you are displeased because the student didn't mention what the derivative measures! So you might end up with the question &quot;&lt;strong&gt;The derivative of a function is another function. Discuss what the new function represents.&lt;/strong&gt;&quot; Recognize potential bad answers that are still technically correct. In this case, we have the correct but unsatisfying answer&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The new function tells you how to get the derivative at each point. The derivative of x^2 is 2x and so if you want the derivative of the function at x=3, it would be 2*3 = 6.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If you catch yourself just trying to get them to say a particular word, your question is actually a short-answer question in disguise. You were trying to write a conceptual question. In this case we catch ourselves hinting at the word &quot;slope.&quot; Forget that. Put it in the question and ask why that is in there. Our new question would become &quot;&lt;strong&gt;The derivative of a function is a new function that gives the slope of the original function. Explain why, by using the definition of the derivative.&lt;/strong&gt;&quot; This question is getting a lot better but you aren't being clear enough about what you want to hear. So, for example, you might get the following answer from a really good student:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If you look at a line, like y=3x+2, then the derivative is [calculation of the derivative using the definition] which comes out to 3. This is the slope, so the derivative measures the slope.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I am now realizing that what I really wanted was for the student to pick apart the definition of the derivative and tell me why it is what it is -- not show me that the derivative works. So I settle on &quot;&lt;strong&gt;The definition of the derivative of f(x) at x = c is lim h-&gt;0 [f(c+h) - f(c)] / h. (a) Draw a sketch of f(x) = x^2. Label c, h, f(c), f(c+h), and f(c+h) - f(c) on the picture. (b) In one sentence and using part (a), point out how the derivative formula is really a slope formula.&lt;/strong&gt;&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I used essentially the last version of that question on my exam 1 this semester in freshman calculus. I was mostly pleased with the results.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-29T20:16:18.990" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1020" PostTypeId="2" ParentId="997" CreationDate="2014-03-29T20:54:22.900" Score="4" Body="&lt;p&gt;A small contribuition; This might help discussion going. I do not have a comprehensive direct answer, however, at the moment I am working on something else that allowed me to see some of the potentially relevant information:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“…Klinedinst (1991), who also found links between retention in an instrumental music programme and scholastic ability, reading achievement, and mathematics achievement.”&#xA;[from: Evans, P., McPherson, G. E., &amp;amp; Davidson, J. W. (2013). The role of psychological needs in ceasing music and music learning activities. Psychology of Music, 41(5), 600-619. doi: 10.1177/0305735612441736, p.1]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Klinedinst, R. E. (1991). Predicting performance achievement and retention of fifth-grade instrumental students. Journal of Research in Music Education, 39, 225–238.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also know that it has been lately research in neuroscience with positive answers to Q1, just don't have that reference handy. I will try to find it.&lt;/p&gt;&#xA;" OwnerUserId="95" LastActivityDate="2014-03-29T20:54:22.900" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1021" PostTypeId="2" ParentId="997" CreationDate="2014-03-29T21:03:57.753" Score="0" Body="&lt;p&gt;In fact there are only some categories of music (mainly in music engineering or acoustical engineering) really relate to mathematics. For example we will expect the musical instrument manufacturers, the loudspeakers manufacturers or the acoustical engineers should have capable mathematics backgrounds but will not expect the singers should have capable mathematics backgrounds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the strong bondings between mathematics and music formed only when the developments of e.g. music engineering and acoustical engineering, the ancient greeks' feelings of the relationship between mathematics and music are in fact too farfetched and you should ignore them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if you really want to improve mathematical abilities through music, you can for example join some music engineering or acoustical engineering works, but for example listening music are certainly not helpful.&lt;/p&gt;&#xA;" OwnerUserId="152" LastEditorUserId="152" LastEditDate="2014-03-29T21:45:13.913" LastActivityDate="2014-03-29T21:45:13.913" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1022" PostTypeId="2" ParentId="995" CreationDate="2014-03-29T22:28:02.447" Score="3" Body="&lt;p&gt;Newton's law of cooling: The rate at which the cup of tea cools is proportional to it's temperature difference to the environment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compute the areas/volumes of curved figures/solids. I.e., the area under a parabola, the area of a circle, the volume of a cone, a cylinder, a sphere. Can even introduce/prove &lt;a href=&quot;http://en.wikipedia.org/wiki/Cavalieri%27s_principle&quot; rel=&quot;nofollow&quot;&gt;Cavalieri's principle&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-29T22:28:02.447" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1024" PostTypeId="2" ParentId="1002" CreationDate="2014-03-29T22:29:36.427" Score="7" Body="&lt;p&gt;Note that a processor (like in your computer, or cellphone, or the one in every nontrivial electronic device that you have) is just a huge proposition written in propositional calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If that's not a real world application of logic, I don't know what is.&lt;/p&gt;&#xA;" OwnerDisplayName="user508" LastActivityDate="2014-03-29T22:29:36.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1025" PostTypeId="2" ParentId="243" CreationDate="2014-03-29T22:51:28.677" Score="5" Body="&lt;p&gt;There might be better or worse choices depending on how you actually build up $\mathbb{R}$. Are you using Dedekind cuts? Are you using (Cauchy) convergent sequences of rational numbers modded out by the appropriate equivalence relation? Are you using an axiomatization with the completeness axiom?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any event, one way to prove the reals are uncountable that I find somewhat fun is via the game described &lt;a href=&quot;http://www.cut-the-knot.org/proofs/Uncountability.shtml&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. You will have to check through the proof to see what concepts need to be discussed in your class (e.g., intervals, convergence of increasing sequences that are bounded above, perhaps some game theory around &quot;winning strategies&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find the original article on this game in an &lt;a href=&quot;http://arxiv.org/pdf/math/0606253.pdf&quot; rel=&quot;noreferrer&quot;&gt;arXiv article&lt;/a&gt;, which includes generalizations beyond the uncountability of $[0,1]$ (hence $\mathbb{R}$). It also concludes the uncountability proof by remarking:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This argument is in many ways much simpler than Cantor’s original proof!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(You can decide whether or not this is the case.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-03-29T22:51:28.677" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1026" PostTypeId="1" AcceptedAnswerId="1080" CreationDate="2014-03-29T23:03:27.160" Score="17" ViewCount="3969" Body="&lt;p&gt;To me, first-semester calculus is a &quot;big ideas&quot; course, whereas second-semester calculus is about a certain bag of tricks. How much memorization is it appropriate to require in the first semester?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one gives open-notes and/or open-book exams, then a clear message is sent to the students that it's not all about memorization, and I think that's a good thing. I think instructors often use an overemphasis on memorization in order to keep students happy, since students universally accept a requirement to memorize, but may get very angry about a requirement to think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for example I often encounter students who don't remember, one semester later, that the derivative of $e^x$ is $e^x$. This complete lack of technical facility would seem to make it impossible for them to &lt;em&gt;use&lt;/em&gt; their calculus for anything. It could be like turning out piano students who can't locate F-sharp on the keyboard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there some basic body of knowledge that a calc student should be expected to memorize? If so, what does it consist of, and what are effective methods for enforcing this expectation? I have at least one colleague who has a short test on calculus facts that requires memorization and that is completely separate from the other tests in the course. Is this a good idea?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To put this another way, suppose that a student who got a C from me in first-semester calc shows up in your second-semester course. Is there a particular fact X such that, if the student didn't know X, you'd curse me for not having high enough standards for a C? In the case where X is $d(e^x)/dx=e^x$, I would think that an A student would understand that this is essentially the whole &lt;em&gt;idea&lt;/em&gt; behind the exponential function, and therefore wouldn't have to memorize it -- but this might not be a reasonable expectation for a C student.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-06T19:19:17.973" Title="How much memorization should be required in a first-semester calculus course?" Tags="|calculus|testing|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1027" PostTypeId="2" ParentId="1026" CreationDate="2014-03-29T23:39:15.670" Score="6" Body="&lt;p&gt;As little as possible. We do have technology (books, Wikipedia, CAS, ...) which are more than able to replace the &quot;learn by rote&quot; that was critical for a medieval scholar, who could call himself lucky to have access to a dozen books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides, I've heard the phrase &quot;but that is the subject of a course we already passed!&quot; too many times already when asking for something simple like the derivative of a simple rational function or a trivial integral in my second year discrete mathematics class...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-29T23:39:15.670" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1028" PostTypeId="1" AcceptedAnswerId="13989" CreationDate="2014-03-29T23:57:36.000" Score="26" ViewCount="2443" Body="&lt;p&gt;I've heard the phrase &quot;we passed that course already!&quot; too many times when asking for e.g. the derivative of a simple rational function or a simple integral, getting blank stares, and digging deeper. This in a second-year discrete mathematics course, after they took at least an introductory calculus course. The gist is that students consider a course passed as subject matter to be forgotten as soon as possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two, probably related questions: How can I make them aware of the mathematical prerequisites, in a concrete way? Adding a line in the course description saying you shouldn't take course XYZ-203 if you haven't passed ABC-101 clearly doesn't work. I don't have the time to struggle with such trivial stuff in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The wider question is how to string courses (particularly courses from different areas, here common Math and Discrete Math as a specialty course for Computer Science) together? How can we convince our students that what they are being taught is a &lt;em&gt;series&lt;/em&gt; building on previous courses, not just a set of scattered, unconnected subjects?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Yes, I know this is very broad. If somebody has specific suggestions for a Discrete Math course, covering proof techniques (again, should be covered in the first math course, but isn't really), a glimpse of generating functions (thus at least the mechanics of derivation/integration and power series), and a bit of number theory (using a bit of groups/rings) to give some understanding on modern cryptgraphy, I'd be thrilled.]&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="114" LastEditDate="2014-03-30T09:45:47.033" LastActivityDate="2018-04-21T15:39:31.443" Title="&quot;We already passed that course!&quot; How to overcome this?" Tags="|undergraduate-education|course-design|secondary-education|" AnswerCount="8" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="1029" PostTypeId="2" ParentId="1026" CreationDate="2014-03-30T00:08:31.423" Score="10" Body="&lt;p&gt;My experience, even with grad students, strongly indicates that students do not realize that &quot;passing the course&quot; (or even &quot;getting an A&quot;) is irrelevant, insofar as we want to demand something like perfect recollection of all details from all prior math courses. Apart from gaming-the-system issues, kids just can't fathom this, as it is quite contrary to other threads in their education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If those remarks seem not to be responding to the question... this incomprehension on the part of the students has the consequence that they &lt;em&gt;certainly&lt;/em&gt; will not &lt;em&gt;review&lt;/em&gt;, and are not receptive to in-class review, of courses in which they received a passing grade. It is mildly amazing to me, but I have witnessed it many times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems to be an argument in favor of substantial memorization in lower-level courses, because otherwise the most rudimentary facts will have slipped out of students' heads. I think that to some extent this is correct, insofar as memorization is a dubious approximation to understanding, but &lt;em&gt;is&lt;/em&gt; an approximation, and does imbed things in peoples' heads. In contrast, an incomprehending disengagement visibly (in my experience) does &lt;em&gt;not&lt;/em&gt; stick things in peoples' memories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A common device to reward having-things-in-their-heads is to emphasize speed. Indeed, having things in one's head speeds things up. However, &quot;speed&quot; is &lt;em&gt;also&lt;/em&gt; not what we should want to declare a &quot;mathematical virtue&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most conscionable answer I've come up with is in-class repetition of critical riffs. The idea is to &lt;em&gt;do&lt;/em&gt; the repetition &lt;em&gt;for&lt;/em&gt; them, without declaring that they should &quot;do drills&quot;. This is especially relevant at the graduate level, I think! Yes, somewhat less material is covered, but maybe a much greater fraction is &lt;em&gt;remembered&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, it is obviously not intellectually sensible, and not humane, to demand lots of memorization, even though it &lt;em&gt;would&lt;/em&gt; benefit students in the long run. Rather, the repetition that will ingrain things in their memories might be created in-class, ... without announcing that that's what happening? :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I've even had students complain that they &quot;see what I'm doing&quot;, and are somewhat annoyed that I have caused them to remember things that they didn't really want to remember after the exam was over... A strange objection.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-30T00:08:31.423" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1030" PostTypeId="2" ParentId="997" CreationDate="2014-03-30T00:40:11.867" Score="5" Body="&lt;p&gt;Responding to &lt;strong&gt;Q1&lt;/strong&gt;: At the grade-school level,&#xA;there is considerable evidence that music (and also dance)&#xA;can be used to teach fractions. E.g.,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Rhythm and Music Help Math Students.&quot; &lt;em&gt;Scientific American&lt;/em&gt;. 2012. &lt;a href=&quot;http://www.scientificamerican.com/podcast/episode/rhythm-and-music-help-math-students-12-03-27/&quot;&gt;Article link&lt;/a&gt;:&#xA;  &quot;Grade school kids who learned about fractions through a rhythm-and-music-based curriculum outperformed their peers in traditional math classes.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Another article: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Fractions Curriculum Strikes Right Note In California.&quot; &lt;em&gt;NPR&lt;/em&gt;. 2012. &lt;a href=&quot;http://www.npr.org/2012/04/07/150126658/fractions-curriculum-strikes-right-note-in-california&quot;&gt;Article link&lt;/a&gt;.&#xA;  &quot;A recent study found that students who went through the program tested better on fractions. &quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-03-30T00:40:11.867" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1031" PostTypeId="2" ParentId="148" CreationDate="2014-03-30T01:35:11.637" Score="6" Body="&lt;p&gt;Once I've written an exam, I take it to determine my rubric or how I'm going to grade each question. If a question has $n$ parts/steps to its solution, then that question will take about $n$ minutes. This is a rough estimate and some steps follow quicker than others, but it seems to average out and has worked well for me for the last handful of years. This works best in non-proof courses, but can also be used for very formulaic proofs. One good/bad aspect of this approach is that it gives students a hint as to how difficult/long the solution should be.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-03-30T01:35:11.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1032" PostTypeId="1" AcceptedAnswerId="1036" CreationDate="2014-03-30T01:57:10.113" Score="12" ViewCount="13128" Body="&lt;p&gt;Several times, I have come close to losing a student's Calculus exam. These exams are usually common exams, with everyone grading 100's of stacks. I've had exams go in wrong piles or get squished in the bottom of backpacks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If an exam were lost in such a class (with hundreds of students taking a common Calculus midterm), what could I do that would be fair to this student and to others? I'd like to have a plan before it happens.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-03-30T14:18:26.483" LastActivityDate="2018-02-17T01:17:36.777" Title="What should you do if you lose your student's exam?" Tags="|calculus|exams|large-lecture|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1033" PostTypeId="2" ParentId="148" CreationDate="2014-03-30T02:02:27.767" Score="5" Body="&lt;p&gt;Srsly, the expansion factor should not be 4 or 10 but something like 40. If you are a quick, fluent, experienced person, and if we're talking about mathematics at any level beyond cook-book/filter calculus, and if we're talking about situations with kids who're still &quot;not dry behind the ears&quot; mathematically, a huger expansion factor is appropriate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, as in other answers, one should be wary of self-deceit about the mere time to write out an appropriate response.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, more insidiously, further, even for kids who do approximately know &quot;the right answer&quot;, there will be a huge amount of dithering. And, there will be a huge amount of no-op self-doubt, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, as the level of the mathematics goes up, the expansion factor goes up, hugely. It is admittedly subtle in undergrad math, depending on circumstances, prior experience, etc. But, e.g., for PhD work, I allow an expansion factor of about 1,000. To explain why this is not a joke, and why this gives perspective on undergrad exams, I seriously comment that it is my (very considered) opinion that in starter-thesis situations, &quot;the advisor&quot; should &quot;see&quot; how to do any thesis problem suggested. &lt;em&gt;And&lt;/em&gt; for the student this &lt;em&gt;should&lt;/em&gt; involve much confusion, much reading, etc. So, if we ask about expansion factors, ... which, !!!, include self-education, etc., _of_course_ it'll take a novice 1,000 times longer than an expert. Hopefully one's advisor is sufficiently expert so that they &lt;em&gt;can&lt;/em&gt; do things that much faster... as dis-heartening as this might be to a naive beginner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we dial-back this deconstruction to undergrad stuff, the lesson still has weight...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The dumbed-down summary's rhetorical version is &quot;whatever factor you think is good, you should multiply by 5&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(But/and think how to avoid &quot;time constraints&quot;... which test things that are irrelevant...)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-30T02:02:27.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1034" PostTypeId="2" ParentId="1032" CreationDate="2014-03-30T02:04:10.000" Score="1" Body="&lt;p&gt;Very unfortunate scenario, indeed, but I think the squeaky-clean thing is to give them a perfect score. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, if you screwed up, ... well, ... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: in light of other remarks... this presumes you do have a system to certify that the exam was handed in, so that students are not gaming you. Even in a large class, it's worth doing this. Having a list of names to check off can succeed even with a large class and in a large room, since you can take the viewpoint that there was no way for any exam to get lost _through_your_malfeasance_. Thus, in that situation, any missing exam is concluded to have not been handed in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, subsequently, if someone's exam &quot;is missing&quot; (that was certified present earlier), &lt;em&gt;you&lt;/em&gt; must have lost it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if the student was not stellar, it would be presumptious (or worse) to exclude the possibility that they did well on the exam... Giving them the mean or any other quasi-uniform replacement score seems to me completely insufficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact that &lt;em&gt;other&lt;/em&gt; students might resent ... your not having lost &lt;em&gt;their&lt;/em&gt; paper? ... is a very secondary issue. I would think that, instead, they should understand that you feel responsible to the student whose paper you lost!&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2014-03-30T12:50:24.593" LastActivityDate="2014-03-30T12:50:24.593" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1035" PostTypeId="2" ParentId="247" CreationDate="2014-03-30T03:09:22.147" Score="11" Body="&lt;p&gt;One difference that is often overlooked is the availability (or non-availability) of mathematical vocabulary.  When teaching an advanced course, I can talk essentially as I would talk to a colleague (or to myself).  In a lower-level course, I have to watch my language more carefully to avoid using terms that are second nature to me (and to other mathematicians) but unknown to the students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some years ago, I was assigned to visit the first-year calculus class being taught by a new junior faculty member, in order to assess the quality of his teaching and, if necessary, to make suggestions for improvement.  He did a great job of teaching except for one thing: He used the phrase &quot;necessary and sufficient condition&quot; several times (and pronounced it as if it were a single word).  My one suggestion for improvement was to point out that a typical calculus student would, at best, understand that phrase only with some reflection (and at worst would not understand it at all) --- so it is necessary (but not sufficient) to bear in mind the vocabulary that the students are familiar with.  &lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-03-30T03:09:22.147" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1036" PostTypeId="2" ParentId="1032" CreationDate="2014-03-30T03:19:47.780" Score="14" Body="&lt;p&gt;Get a sheet signed by each student who took the exam, use that to check nothing is amiss. I collect exams (and keep them during the whole grading process) into large, sturdy envelopes. So no sheet can wander away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I separate exams by question, each question to be graded separately (some by TAs). Students are required to turn in &lt;em&gt;all&lt;/em&gt; questions, with a sheet identifing the question and the student, even if otherwise blank. No chasing &quot;where is question 5 from ...&quot; (at least much, much less), and no suspect &quot;I forgot to turn in question 3, and found it today&quot; type requests.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Massive classes (hundreds of students, many graders) require a well-thought-out, orderly, and rigurously followed logistics process. Not only in handling the exams, but in making sure grading is uniform (detailed grade assignment, covering not only the typical solution but also alternatives is a must), and also reliable recording of the resulting grades.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-30T03:19:47.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1037" PostTypeId="1" AcceptedAnswerId="1041" CreationDate="2014-03-30T07:35:10.473" Score="20" ViewCount="2678" Body="&lt;p&gt;Students' messy exam and homework sheets (which are messy because of a bad handwriting or an irregular logical argument in proofs) cause many serious problems. e.g.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Reading their arguments takes too much time.&lt;/li&gt;&#xA;  &lt;li&gt;Messy sheets could be confusing. Understanding the truth or falsity of a particular mathematical argument could be so hard if it is not written properly.&lt;/li&gt;&#xA;  &lt;li&gt;It is a bad habit if they continue their bad math writing in future.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I assigned some extra grades for good writings but it seems insufficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; How can I overcome the messy sheets problem?   &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="143" LastEditDate="2015-02-25T23:13:32.707" LastActivityDate="2016-10-28T23:02:26.687" Title="Students' Messy Sheets: The Big Problem of Exams and Homeworks" Tags="|exams|homework|general-pedagogy|writing|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1038" PostTypeId="2" ParentId="243" CreationDate="2014-03-30T07:51:02.830" Score="5" Body="&lt;p&gt;A lot of attempts at avoiding the &quot;non-unique decimal representation&quot; problem from the start have the 'problem' of being less visual. How the normal construction is supposed to work is immediately &lt;em&gt;apparent&lt;/em&gt; once you see the attempt at a list being made--or at least much more apparent than dodging around explicitly writing any numbers out. So I generally &lt;em&gt;don't&lt;/em&gt; avoid it, at least not at the beginning, and just correct for it afterward. So this essentially goes&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Convince class that reals in $[0,1]$ are almost the infinite bit strings, but the repeated representations of some numbers are an issue.&lt;/li&gt;&#xA;&lt;li&gt;Based on the 'almost', start by proving the uncountability of the infinite bitstrings through Cantor's argument normally. You're now most of the way there.&lt;/li&gt;&#xA;&lt;li&gt;Show that the 'repeated representation' cases are merely countable, and use basic results about unions of countable and uncountable sets to show that the reals are uncountable as well.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So you show that an easy argument (if you consider the normal diagonal argument easy) &lt;em&gt;almost&lt;/em&gt; gets you what you want, and correct for the 'almost' later. I still haven't settled on a &quot;best&quot; way of doing 3 though. &lt;a href=&quot;https://math.stackexchange.com/questions/271118/characterization-of-non-unique-decimal-expansions&quot;&gt;Alex Becker gives a nice one here&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like this for a couple reasons. Firstly because it advocates the proof &quot;technique&quot; of pressing onward with close but not exact arguments and seeing how one can try to continue. Beginning proof students often subconsciously think that a proof must be clear in their head from the very start, and it's useful to see that playing around with similarities can be properly productive. The second is because it shifts the messy part of the proof to resolving a technical detail, and leaves the main thrust of the proof as a clear separate step. So it divides the process clearly into &quot;this is the main idea&quot;, and &quot;these are the technical issues you need to resolve&quot;, rather than potentially losing the main point behind such technical issues.&lt;/p&gt;&#xA;" OwnerUserId="522" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-03-30T07:51:02.830" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1039" PostTypeId="2" ParentId="1028" CreationDate="2014-03-30T08:08:29.577" Score="6" Body="&lt;p&gt;I think one there is a bunch of problems. First, university curriculum tends to treat every topic exactly one time and then assumes people know how to deal with it. Second, as long as students may pass the second course although they forgot everything from the first (or maybe never attended it), other students will try the same. Third, they've passed it. Why are we giving students good grades who cannot do the basic stuff three months later? (I don't want to blame you for this, I believe this problem to be general.) I think there are even more problems involved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what to do? School's answer to the first problem is a curriculum which often treats roughly the same thing (slightly extended; spiral approach). I am not sure if that is good at university level. The second and third problem are connected since they are both on passing exams. The simple answer would be &quot;be more demanding in your exams&quot;. However, this might go wrong if too many students would drop-out then.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I don't have a clear answer but I have an idea in mind for some months now. What would happen if we can manage that students have to pass every exam twice. The better grade counts for their certificate. Most of them should try to improve their grade when they have their second exam so they would repeat all the stuff which is somehow spiral. You might then be able to raise standards a bit. In addition, they might believe that learning for understanding is more appropriate than learning everything off by heart, since they should not forget everything after exam 1.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-03-30T08:08:29.577" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1040" PostTypeId="1" AcceptedAnswerId="1058" CreationDate="2014-03-30T10:23:15.327" Score="36" ViewCount="3714" Body="&lt;p&gt;I tutor high school algebra and I’ve noticed that a lot of my students don’t seem to understand what they’re doing when they “convert” between different ways of writing numbers involving perfect squares.  For instance, I have seen students simplify $\sqrt9$ to $\sqrt3$ time after time, even after I’ve just corrected them on $\sqrt4\neq\sqrt2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, if I show them the difference of squares formula $(a^2-b^2)=(a-b)(a+b)$ and then write $(x^2-9)=?$ I will inevitably get $(x^2-9)=(x-9)(x+9)$ (sometimes with the preliminary step of $(x^2-9)=(x^2-9^2)=(x-9)(x+9)$, which is truly baffling). Showing similar examples with other numbers doesn’t seem to help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody have any tricks on how to teach this subject and snap students out of the idea that $\sqrt3 = \sqrt9 = 9 = 9^2=\ldots$?&lt;/p&gt;&#xA;" OwnerUserId="525" LastEditorUserId="75" LastEditDate="2014-03-30T10:39:52.233" LastActivityDate="2021-03-07T19:04:44.273" Title="How to cure students from the idea that root and squaring are identity operators?" Tags="|tutoring|algebra|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1041" PostTypeId="2" ParentId="1037" CreationDate="2014-03-30T11:22:28.520" Score="17" Body="&lt;p&gt;First, make sure they know that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The purpose of exams is to &lt;em&gt;test&lt;/em&gt; students' knowledge and understanding.&lt;/li&gt;&#xA;&lt;li&gt;The burden of proof is on &lt;em&gt;their&lt;/em&gt; side, that is, blank/unreadable sheets work against them.&lt;/li&gt;&#xA;&lt;li&gt;The teachers might &lt;em&gt;choose&lt;/em&gt; to decipher some of the messy work, but this choice is intrinsically unreliable, erratic and may produce unfair results. &lt;/li&gt;&#xA;&lt;li&gt;The teachers grade papers according to their &lt;em&gt;whimsy&lt;/em&gt; and the fairness relies on trust in the faculty. The fact that some professor is your teacher is a display of institution's trust in that person. There has to be some amount of goodwill involved, and it &lt;em&gt;is&lt;/em&gt; there (i.e. in the teachers).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are several methods I have seen in practice, some better, some worse, decide what fits your style by yourself.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Use &quot;clarity points&quot;.&lt;/em&gt; For two correct proofs, the one which is easier to understand is simply better. Let the students know there are such points. Personally I had used $1\text{pt}$ on $[0\text{pt},10\text{pt}]$ scale, students didn't liked it, but it did make the solutions better. It also had some funny side effects, like more students preferring to return an empty sheet (100% clarity) than a bit of something (often 0% clarity), if they were uncertain about it.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Let the students explain&lt;/em&gt; (e.g. assign $0$ points and make them come to your office). This is time consuming and tricky, because students might want to update their solution. However, according to my experience, it is frequently the &lt;em&gt;most fair way&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Time-limit your grading process.&lt;/em&gt; If you can't understand it, it is unclear; inference with huge mental leaps is invalid.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Assign enough time,&lt;/em&gt; so they can rewrite their solutions (unclear solution is worth 0 points).&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Request a &quot;clear solution rewrite&quot;&lt;/em&gt; with no strokes, linearly ordered, etc. This is related to previous bullet, some students still won't do it, but many will, and it is justifiable to have much higher &quot;unreadable&quot; threshold (for example: Mr. Student, please look at this &amp;lt;huge stack of nice, clean solutions&amp;gt;, are you sure that your work is as easily readable as these?).&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Request solutions in LaTeX&lt;/em&gt; (e.g. for homework or take-home exams). Many students rebel against it, but in my experience (from both student and teacher perspectives) it is one of the best solutions available.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Use answer sheets.&lt;/em&gt; Unsuitable for wide range of question types.&lt;/li&gt;&#xA;&lt;li&gt;(for computer-science exams) &lt;em&gt;Require code indentation.&lt;/em&gt; The rules are clear enough that it's fairly objective to decide if the code is correctly indented or not. Helps &lt;em&gt;a lot&lt;/em&gt;. Some require underlining keywords, but that's kind of extravagant.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-30T11:27:59.510" LastActivityDate="2014-03-30T11:27:59.510" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1042" PostTypeId="1" CreationDate="2014-03-30T11:46:15.460" Score="25" ViewCount="19540" Body="&lt;p&gt;I am thinking of giving my high school students some pure maths projects to do. It is a lot easier to think of some interesting stats projects but not in pure maths. The students' maths background are weak, they might not really understand the relationship between integration and area finding. I am planning of giving a short project that could be done in 1-2 weeks time (for 1-2 pages long) that emphasises on conceptual understanding but if possible exciting and upon completing the projects students should feel more confident tackling more problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my experience when I studied maths back then in high school, I preferred to understand the concepts first before trying to do any problems. But nowadays, I think it is a common phenomenon that students tend to jump straight into doing problems without bothering the purposes, motives, and relationships from one concept to the other. In other words, students are not interested in connecting the dots, they just want to do and pass the tests. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The topics that we are discussing now are: differential and integral calculus of trigonometric and logarithmic functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some ideas that I have in mind:&lt;br&gt;&#xA;1. Summarise different interpretations of derivatives (for example: geometric interpretation, algebraic interpretation, physical interpretation, etc)&lt;br&gt;&#xA;2. Relate the idea of (definite) integration (continuous) and summation (discrete) and give some properties or equations which are very similar in both cases.&lt;br&gt;&#xA;3. Summarise common trig identities and prove them. For example, $sin^2\theta+cos^2\theta=1$ can be proved using Pythagoras.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would really appreciate if anyone could share some other ideas or examples which might be helpful in enhancing students' understanding of concepts, I have a very limited knowledge on the current research, maybe there are some exciting research topics which do not require tons of advanced maths. Many thanks!&lt;/p&gt;&#xA;" OwnerUserId="404" LastEditorUserId="404" LastEditDate="2014-03-30T12:25:15.013" LastActivityDate="2019-06-11T21:47:22.620" Title="Ideas for high school pure maths projects" Tags="|secondary-education|examples|" AnswerCount="13" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="1043" PostTypeId="1" AcceptedAnswerId="1063" CreationDate="2014-03-30T12:25:53.067" Score="13" ViewCount="884" Body="&lt;p&gt;Sometimes I come across some exam answers which describe a proof sketch or a counterexample very well but are not written formally. Such proofs show that a particular student understands the general picture of the question completely. Furthermore in the case of counterexamples sometimes these descriptive proofs are really interesting and one can find essentially new examples to refute a particular conjecture (or a possible theorem without one of its essential assumptions) using such descriptions.&lt;/p&gt;&#xA;&lt;p&gt;In the usual process of the course, I encourage my students to understand the general picture of a question/phenomenon and describe it sketchy on their own. I appreciate this approach too much. But I don't know what I should do when I receive such creative descriptive proofs of my best students in an exam, especially when some of the other students write the usual formal proof correctly.&lt;/p&gt;&#xA;&lt;p&gt;I am worried about &amp;quot;forcing&amp;quot; my students to write their true ideas in exams too formally because I think it could affect their motivation for being creative. On the other hand, formal proofs are necessary parts of formal mathematics and they should adopt themselves with this culture.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1.&lt;/strong&gt; What should I do with inexact true descriptive proofs in exams? Should I give them a full mark (which I think such proofs deserve it) or considering a full mark could be harmful to encourage them to be more formal?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2.&lt;/strong&gt; How can I encourage my students to write their ideas as &lt;em&gt;formal&lt;/em&gt; as possible but think about them &lt;em&gt;descriptively&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="14466" LastEditorDisplayName="user230" LastEditDate="2020-08-27T15:06:35.747" LastActivityDate="2020-08-27T15:06:35.747" Title="Descriptive Thinking vs. Formal Writing" Tags="|mathematical-pedagogy|exams|proofs|creativity|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1044" PostTypeId="1" CreationDate="2014-03-30T13:23:32.480" Score="17" ViewCount="2048" Body="&lt;p&gt;Eisenstein’s criterion:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If you have a polynomial with integer coefficients (and a non-zero constant term) and there’s some prime number $p$ such that&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;$p$ goes into every coefficient but not the first (or last) and&lt;/li&gt;&#xA;  &lt;li&gt;$p^2$ doesn't go into the last (or first, respectively)&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;then your polynomial doesn’t factor into two polynomials with integer coefficients.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(But the converse isn’t true. That is, if the criterion fails, the polynomial might nonetheless not factor.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is true of polynomials of any degree, but is most useful for quadratics, for two reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Those are the polynomials that are most frequently factored over the integers in high school.&lt;/li&gt;&#xA;&lt;li&gt;It’s easy to see &lt;em&gt;why&lt;/em&gt; the criterion holds true for quadratics.&lt;sup&gt;[1]&lt;/sup&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Its use is as a quick check for non-factoring, so as not to waste time trying to factor. Thus, for example, a student can tell at a glance that $x^2+9x+24$ doesn't factor (because $3|9$ and $3^2 \nmid 24$), without checking which of the many factors of $24$ might sum to $9$. And can tell that $x^2-10x+60$ doesn't factor without checking factors of $60$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this criterion doesn’t seem to be taught in high school at all, which makes me wonder whether doing so is pedagogically unsound. So my question is: Should one teach this in a high-school algebra class? For quadratics at least? Why or why not?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;[1]&amp;nbsp;&lt;/sup&gt;It’s especially easy to see for $x^2+bx+c$. There, that $p$ goes into the $b$ means that the $r,s$ in $(x+r)(x+s)$ must both or neither be divisible by $p$; but then $c=rs$ must be divisible by $p^2$ or not by $p$ at all.&lt;/p&gt;&#xA;" OwnerUserId="450" LastEditorUserId="78" LastEditDate="2014-04-04T00:57:29.527" LastActivityDate="2021-02-25T09:25:04.683" Title="Should Eisenstein’s criterion be taught in high-school?" Tags="|secondary-education|algebra|solving-polynomials|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1045" PostTypeId="2" ParentId="1032" CreationDate="2014-03-30T13:24:04.343" Score="7" Body="&lt;p&gt;Offer them a choice of retaking the exam or averaging their course grade without it. And apologize, of course.&lt;/p&gt;&#xA;" OwnerUserId="12" LastActivityDate="2014-03-30T13:24:04.343" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1046" PostTypeId="2" ParentId="1002" CreationDate="2014-03-30T13:49:47.267" Score="4" Body="&lt;p&gt;Besides digital hardware, which others have mentioned, propositional logic has applications in software.  For better or worse, this is usually three-valued logic; however for the purposes of a logic course, this fact can be glossed over.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit: Example added.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a real example. It comes from the merge phase of the merge sort algorithm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the command&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;if q = k or (p not= j and B[p] &amp;lt; B[q])&#xA;then (A[r] := B[p]; p:=p+1)&#xA;else (A[r] := B[q]; q:=q+1)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Suppose that  you know that &lt;code&gt;not(q = k and p = j)&lt;/code&gt;, show that the &quot;else&quot; part is executed exactly if&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;p = j or (q not= k and B[q] _&amp;lt; B[p])&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The only thing you need to know about programming is that the else part will be executed exactly if the guard &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;q = k or (p not= j and B[p] &amp;lt; B[q])&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;is false. So the logical problem consists of showing that, given &lt;code&gt;not(q = k and p = j)&lt;/code&gt;, the following equivalence holds&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;not( q = k or (p not= j and B[p] &amp;lt; B[q]) ) = ( p = j or (q not= k and B[q] _&amp;lt; B[p]) )&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;In terms of propositional logic, the problem is: Given &lt;code&gt;not(P and Q)&lt;/code&gt; show&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;not( Q or (not P and R)) = (P or (not Q and not R))&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="527" LastEditorUserId="527" LastEditDate="2014-04-01T11:08:12.313" LastActivityDate="2014-04-01T11:08:12.313" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1047" PostTypeId="2" ParentId="1044" CreationDate="2014-03-30T13:51:21.793" Score="11" Body="&lt;p&gt;One drawback I see is that it is only marginally better than the standard algorithm of factoring $c$ and seeing which factors add up to $b$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To apply Eisenstein's, you must find a factor of b or c and then test it on the other factor. So factoring occurs in either situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While Eisenstein's is a little bit faster, it doesn't lead to a solution in the case it does factor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see Eisenstein's as more useful in the cubic case, and I think it would be useful to introduce for the quadratic if only to generalize to higher degrees.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-03-30T13:51:21.793" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1048" PostTypeId="2" ParentId="1044" CreationDate="2014-03-30T14:08:51.870" Score="1" Body="&lt;p&gt;I think it is perfectly fine to introduce as early as possible. It is a useful criterion, insofar as it can be executed quickly without holding much information in one's mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the main reason it is not part of the traditional high school curriculum is just inertia, both in the textbooks and in the education of high school math teachers.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-30T14:08:51.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1049" PostTypeId="2" ParentId="1028" CreationDate="2014-03-30T14:25:19.080" Score="9" Body="&lt;p&gt;One thing I tend to do is to mention fairly often in class when students will need current material in future courses of various types and also how we are relying on previous material in the current course. This doesn't completely ameloriate the problem, but I hope it raises awareness of the often vertical structure of mathematics and the fact that the courses are taught in a certain order for a reason.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2014-03-30T14:25:19.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1050" PostTypeId="2" ParentId="1037" CreationDate="2014-03-30T14:26:14.437" Score="10" Body="&lt;p&gt;The way I choose to combat this is to make my -&gt; &lt;a href=&quot;http://qc.edu/~chanusa/courses/245/16/guidelines.html&quot; rel=&quot;nofollow&quot;&gt;Homework Guidelines&lt;/a&gt; &amp;lt;- very clear from the start of the semester.  I expect the students to follow the guidelines and have the disclaimer that &quot;If your turned-in homework takes too much effort to read, it will not be graded!&quot;  Since I have instituted my guidelines, the homework assignments have been pristine.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I make sure that my students are accountable for the guidelines by asking them to read through that webpage (and the rest of my syllabus) and answer questions on a &quot;Syllabus Quiz&quot; for their first homework assignment.  This makes sure we are all in sync to start off the semester.  &lt;/p&gt;&#xA;" OwnerUserId="324" LastEditorUserId="324" LastEditDate="2016-10-26T17:55:18.303" LastActivityDate="2016-10-26T17:55:18.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1051" PostTypeId="2" ParentId="1044" CreationDate="2014-03-30T14:34:49.117" Score="7" Body="&lt;p&gt;No, one should not teach Eisenstein's criterion in a high school class, because that puts far too much focus on factoring.  In my opinion, a high-school algebra class ought to give students mathematical skills that are useful in non-mathematical contexts, and this would detract from that goal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an alternative, one can ask questions like:  How does the graph of $x^2-10x+24$ compare with the graphs of $x^2-10x+23$, $x^2-10x+25$, and $x^2-10x+26$?  What are the numerical values of the roots, and how do they differ for the different polynomials? &lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-30T14:34:49.117" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1052" PostTypeId="1" AcceptedAnswerId="1065" CreationDate="2014-03-30T14:47:32.283" Score="7" ViewCount="2839" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Occam%27s_razor&quot; rel=&quot;nofollow noreferrer&quot;&gt;Occam's Razor&lt;/a&gt; (also written as Ockham's razor from &lt;a href=&quot;http://en.wikipedia.org/wiki/William_of_Ockham&quot; rel=&quot;nofollow noreferrer&quot;&gt;William of Ockham&lt;/a&gt; (c. 1287 – 1347), and in Latin lex parsimoniae) is a principle of parsimony, economy, or succinctness used in problem-solving. It states that among competing hypotheses, the one with the fewest assumptions should be selected. Other, more complicated solutions may ultimately prove correct, but - in the absence of certainty - the fewer assumptions that are made, the better.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Quoted from Wikipedia&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In mathematics there are theorems which have several proofs, some short and some long. Based on the limitation of time I cannot explain more than one proof for each multi-proof theorem in my courses but there is a problem here.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; Which proof should I choose?&lt;/p&gt;&#xA;&lt;p&gt;Based on the Occam's razer I should choose the shorter one because it is simpler and less confusing but it seems some of the longer proofs for a particular theorem have more intuitive content. They explain the nature of the phenomenon better. Also they show the complexity and importance of the theorem clearly. Maybe students can gain a deeper intuition using some of these longer proofs. In other words my question is about possible (dis)advantages of using Occam's Razor in mathematics. Precisely, is there any exception for Occam's Razor principal in mathematics? Is there a longer proof of a particular theorem which has a significant educational advantage to all other shorter proofs?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example.&lt;/strong&gt; In my &amp;quot;Introduction to Large Cardinal Theory&amp;quot; graduate course, there are several approaches to Hanf-Tarski theorem which says &lt;em&gt;The least inaccessible cardinal is not measurable&lt;/em&gt;. The current standard proof in textbooks (e.g. &lt;em&gt;Drake, An Introduction to Large Cardinal Theory&lt;/em&gt;) is based on a very short simplified proof but Tarski and Hanf's original sophisticated argument (using introducing the notion of a weakly compact cardinal and analyzing the expression power of infinitary language &lt;span class=&quot;math-container&quot;&gt;$\mathcal{L}_{\kappa,\kappa}$&lt;/span&gt; for weakly compact cardinal &lt;span class=&quot;math-container&quot;&gt;$\kappa$&lt;/span&gt;) is really useful for understanding the complexity of this theorem which was one of the most important open problems of set theory for many years.&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="-1" LastEditorDisplayName="user230" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-02T00:02:16.620" Title="Ockham's Razor &amp; Mathematical Proofs" Tags="|mathematical-pedagogy|course-design|proofs|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1053" PostTypeId="2" ParentId="925" CreationDate="2014-03-30T14:51:17.623" Score="11" Body="&lt;p&gt;Historically it seems that &quot;sequence&quot; is the interloper.  &quot;Series&quot; was used to denote both concepts going as far back as Wallis, usually with the qualifier &quot;infinite&quot;; sometimes &quot;progression&quot; is used to denote what we call a sequence.  Our current use of &quot;sequence&quot; in the theory of series did not become common until later, perhaps around 1900, as did the prominence of the term &quot;partial sum.&quot;  In modern terms, one might say that a sequence or a series is simply a function $f \colon {\bf N} \rightarrow {\bf R}$ -- the same in both cases.  I think the significance of these observations is that a sequence and a series are not (readily) distinguished by what they &lt;em&gt;are.&lt;/em&gt;  They are distinguished by how they are &lt;em&gt;used.&lt;/em&gt;  We speak of the &quot;sum&quot; of series and the limit of the &quot;terms.&quot;  The words &quot;sequence&quot; and &quot;partial sum&quot; began to be used, I suppose, to help clarify the intended use.  You do not &lt;em&gt;sum&lt;/em&gt; a &lt;em&gt;sequence&lt;/em&gt;, and you do not find the &lt;em&gt;limit&lt;/em&gt; of a series; but you do find the &lt;em&gt;limit&lt;/em&gt; of the &lt;em&gt;sequence&lt;/em&gt; of &lt;em&gt;partial sums&lt;/em&gt; of a series.  To me, the notion of a sequence and a series are intrinsically difficult to keep straight, the capital sigma being the main difference (or the plus dot-dot-dot).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Difficult is not the same as impossible.  Certainly &quot;the sum of a series&quot; and &quot;the limit of a sequence&quot; are easily distinguished.  I warn my students ahead of time that synonyms are going to be used for different things, and that they will have to make an effort to learn the difference.  I probably repeat it a few times, and I repeat it anytime someone makes a mistake in class.  There did not seem to be a big problem with the terms.  Unpacking the definition of the sum into the limit of the partial sum causes the most difficulty (leaving aside the other complexities of the topic).  I do a few things that probably help the students learn the meanings: quizzing and testing on the definitions and testing them on the proofs of the n-th term test and integral test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure how much &quot;convergence&quot; adds to the potential for confusion.  It certainly adds some, but perhaps only a little.  The terms are used analogously for improper integrals, so the connection should make it easier to learn their meanings.  One approach, which did not catch on, is to start first-year calculus with sequences (see Courant, rev. Courant and John, &lt;em&gt;Intro. to Calc. and Anal.,&lt;/em&gt; or Hardy, &lt;em&gt;A Course of Pure Math.,&lt;/em&gt; for examples).  Then sequences are a long way from series in the development of the course.  I suppose getting to derivatives quickly, for the sake of science and engineering students, is the reason for the current popular sequence of topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do find that students' performances on the first test on sequences and series to be the lowest on average.  There are many reasons.  One is that for the students from AP Calc AB courses, it's the first test for which they have to learn new concepts.  Another is that most of the problems have a different nature -- show whether this series has an answer rather than find the answer.  Finally there is the range of tests of different natures: some are for divergence, some are for convergence, and some are for both.  Nomenclature is definitely a factor but not the major factor.  This probably depends on how the subject is taught, though.  Perhaps I have minimized the impact of sequence/series confusion at the expense of something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think we owe it to students to teach them to be conversant with mathematical lingo.  I tutored a middle school student, whose teacher did not use the word &quot;cancel.&quot;  The student in all seriousness suggested that the next step was to &quot;s----&quot; it. It sounded like a mix &quot;smush&quot; or &quot;scrunch&quot; -- I don't recall clearly now.  It was a made-up word.  That seems extreme to me.  In any case, language evolves so perhaps the lingo will change.  Perhaps it's worth pointing out that the root of the meaning of &quot;series&quot; comes from a word meaning to join up.  So &quot;series&quot; means the terms &quot;joined up&quot; (by addition) to form a whole.  Sequence, of course, comes from a word meaning to follow and implies that the terms are viewed as separate things.  One might use the etymology to help students think about the meaning of the mathematical terms.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2014-03-30T18:31:34.797" LastActivityDate="2014-03-30T18:31:34.797" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1054" PostTypeId="2" ParentId="1040" CreationDate="2014-03-30T14:51:28.110" Score="21" Body="&lt;p&gt;First: I do not think this is really an issue with a lack of understanding of the square root function. When someone writes $\sqrt9 = \sqrt3$ it means they are not thinking about what the equals sign even means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have had a small amount of success with the following method, which relies on the students &quot;believing in their calculator&quot; as a source of truth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose a student has the following work, with the square root error that you outlined:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{-2 + \sqrt{20}}{2} = \frac{-2 + \sqrt4 \sqrt 5}{2} = \frac{-2 + 4 \sqrt 5}{2} = -1 + 2\sqrt5$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They look up the answer and find that the correct answer is $-1 + \sqrt5$, and ask for assistance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can tell them to &lt;strong&gt;check each step with their calculator&lt;/strong&gt;. They know they were supposed to get $-1 + \sqrt5$ as the answer, and their calculator happily translates that correct answer into 1.236.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;When they type the first fraction into their calculator, it also comes out 1.236. This means they were correct at that step.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When they plug the second fraction into their calculator, it also comes out 1.236. This means they were correct at that step too.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But when they plug the third fraction into their calculator, it comes out 3.472. This shows that the simplification error was made between the second and third steps.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;What is a student supposed to take away from this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd argue that when using this method, you are teaching the student to begin to grasp that each mathematical expression they write is actually a value, and that as they simplify, they should be doing things that do not change that value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Expert-level mastery of this concept would allow them to look at the 4 and the $\sqrt4$ next to each other and realize &quot;wait... $4$ and $\sqrt4$ are &lt;strong&gt;different numbers&lt;/strong&gt; -- but first students need the insight that $4$ and $\sqrt4$ are &lt;strong&gt;numbers that could be different&lt;/strong&gt; at all!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-30T14:51:28.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1055" PostTypeId="2" ParentId="1052" CreationDate="2014-03-30T15:12:46.470" Score="12" Body="&lt;p&gt;The issue is not length itself, but intelligibility, I think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On one hand, over time, proofs are polished and &quot;perfected&quot;, in part exactly to make them as short and efficient as possible. Obviously this is good in some ways, but it does sometimes have the bad side effect of making the details become unmotivated. Indeed, since the details of the perfected proof became clear only in hindsight, that is, only upon study of the not-perfected arguments, a student reading the perfected argument without experience with the imperfect one may reasonably be baffled by motivations and choices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, &lt;em&gt;sometimes&lt;/em&gt; the improvements in the argument are very great, and represent conceptual improvements in their own right, and part of our task is to improve our intuition to match.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, in general, I find it desirable to &lt;em&gt;try&lt;/em&gt; a &quot;natural&quot; line of argument, clumsy or inefficient though it may be, rather than wait for or present some seemingly-magical, ineffable argument whose motivations are obscure except after having tried a more natural approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, methodologically, admitting to students that it's reasonable to &quot;try the obvious thing&quot;, even if revision is required or advantageous, is helpful and honest.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-30T15:12:46.470" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1056" PostTypeId="1" AcceptedAnswerId="1469" CreationDate="2014-03-30T15:24:35.950" Score="9" ViewCount="1757" Body="&lt;p&gt;If it is known from context that variables $x$ and $y$ represent integers, an open Boolean formula such as $x \ge y \Rightarrow x+1 &amp;gt; y$ evaluates to true regardless of the value assigned to variables $x$ and $y$, at least in the standard interpretation of arithmetic.  What should (or could or would) one call such a formula?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been using the term &quot;valid&quot;, but am not very happy with this.  Logicians (e.g. Boolos and Jeffery&quot;) use &quot;valid&quot; for closed formulas that are true regardless of the interpretation. I could just call it &quot;true&quot;, but this seems to fail to acknowledge that there are free variables. It might be called a &quot;theorem&quot;, but I'd rather stay away from issues of provability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just for context: This is not a course in logic, but a course that uses logic as a tool for proving things about software.  So I'd like to be consistent with standard logical terminology without getting into the possibility of multiple interpretations.&lt;/p&gt;&#xA;" OwnerUserId="527" LastEditorUserId="3272" LastEditDate="2014-10-24T15:38:29.163" LastActivityDate="2021-10-07T19:32:06.197" Title="Standard word for a formula that is always true" Tags="|logic|terminology|" AnswerCount="3" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="1057" PostTypeId="2" ParentId="1052" CreationDate="2014-03-30T15:37:21.283" Score="1" Body="&lt;p&gt;I don't particularly know the Hanf-Tarski theorem or its proofs, but based on what you say:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teach the shorter proof.  A complicated proof shows that it used to be a complicated theorem, but apparently it isn't today.  Whatever your goals in teaching set theory, complexity should not be a goal in itself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Pythagoarean theorem has so many short proofs that it can make sense to teach more than one.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-03-30T15:37:21.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1058" PostTypeId="2" ParentId="1040" CreationDate="2014-03-30T15:57:03.633" Score="63" Body="&lt;p&gt;I suspect that there are several different and interacting things going on here.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It's likely that these students don't understand the &amp;quot;equals&amp;quot; sign in the same way that you do.  An extensive body of research over several decades has documented that for many students the symbol &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; does not indicate a &lt;em&gt;relationship&lt;/em&gt; (that the thing on the left is identical in value to the thing on the right) but rather an &lt;em&gt;operation&lt;/em&gt;:  They interpret &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; as a signal to do something, produce a result.  (I don't have access to a bibliography right now but Carolyn Kieran's 1981 article on elementary schoolchildren's conceptions of the equals sign was the first in this field; it has been extensively followed up on by researchers at the secondary and college level who have found that  the operator conception can significantly impede students' success in algebra.)  So your students may not be understanding the expression &lt;span class=&quot;math-container&quot;&gt;$(x^2-9)=(x-9)(x+9)$&lt;/span&gt; as meaning that the two quantities are equal to each other; rather they see the LHS as a question, and the RHS as an answer.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Second, it's almost certain that your students' understanding of what they are doing is limited to a set of rules about what you are supposed to write down where, and is disconnected from any understanding of what the symbols mean.  A phenomenon related to the one you describe:  Students are asked to simplify &lt;span class=&quot;math-container&quot;&gt;$\sqrt{12}$&lt;/span&gt;. They write &lt;span class=&quot;math-container&quot;&gt;$\sqrt{12} = \sqrt{4 \times 3} = 3 \sqrt{4}$&lt;/span&gt;.  When asked what they mean, they say &amp;quot;Wait, which number is supposed to go on the outside and which one is on the inside?&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A third issue (closely related to both of the above) may have to do with decoding the written symbols into verbal form and vice versa.  I have often heard students read &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt; aloud as &amp;quot;sin times x&amp;quot;; it may be that your students are doing something similar, using the notation &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3}$&lt;/span&gt; to say &amp;quot;&lt;em&gt;The square root is 3&lt;/em&gt;&amp;quot;.  (&amp;quot;Is&amp;quot;, not &amp;quot;Of&amp;quot;).  In more detail, they may be reading the initial expression &lt;span class=&quot;math-container&quot;&gt;$\sqrt{9}$&lt;/span&gt; as an instruction (&amp;quot;Find the square root of 9&amp;quot;), know (remember) that the answer is 3, saying to themselves &amp;quot;the square root is 3&amp;quot;, and write that down as &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The connection among these is (in part) that the square root symbol designates completely different things on the different sides of the equals sign:  On the left, it means a question, and on the right, it labels the answer.  In neither case is it connected with what square roots or squaring actually mean.&lt;/p&gt;&#xA;&lt;p&gt;One strategy that can help with all three of these issues is to emphasize that these operations ought to be reversible.  If the student thinks that &lt;span class=&quot;math-container&quot;&gt;$(x^2-9)=(x-9)(x+9)$&lt;/span&gt;, ask them to multiply the RHS out and see if they get the LHS.  (They are less likely, I think, to repeat the error in the other direction, because they will be thinking &lt;span class=&quot;math-container&quot;&gt;$9 \times 9$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$9^2$&lt;/span&gt;.)  If a student thinks that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{12} = 3\sqrt{4}$&lt;/span&gt;, ask them what they &lt;em&gt;should&lt;/em&gt; get if they multiply the RHS times itself, and then ask them what they &lt;em&gt;do&lt;/em&gt; get if they multiply the RHS times itself.&lt;/p&gt;&#xA;&lt;p&gt;(Aside:  I once asked a group of 8 high school teachers how they would respond to a student who made the mistake of writing &lt;span class=&quot;math-container&quot;&gt;$\sqrt{12} = 3\sqrt{4}$&lt;/span&gt;.  Almost all of them said that they would go over the correct process again.  One creative thinker suggested putting both the LHS and the RHS into a calculator and showing that they are not equal.  When I proposed multiplying the RHS times itself and getting not 12 but 36, several of them expressed surprise, not at the method but at the fact that it had never occurred to them before.)&lt;/p&gt;&#xA;&lt;p&gt;Another strategy that you may find helpful (over the long term) is to try to get into the habit of giving them problems that are like those they are accustomed to, but with the LHS and RHS swapped (i.e., of the form ____ = &lt;span class=&quot;math-container&quot;&gt;$\sqrt 9$&lt;/span&gt;).  This may help break the associations &amp;quot;left side = question&amp;quot; and &amp;quot;right side = answer&amp;quot;; at the very least, it may help to bring it to the surface, which can help confirm the diagnosis.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="6053" LastEditDate="2021-03-07T19:04:44.273" LastActivityDate="2021-03-07T19:04:44.273" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="1059" PostTypeId="2" ParentId="243" CreationDate="2014-03-30T16:15:50.137" Score="3" Body="&lt;p&gt;I may be misunderstanding the question, but I assume that by the &quot;repeating decimal annoyance&quot; you mean that the diagonalization process may end up producing a bit string that eventually becomes an infinite sequence of 1s, which is equal to a terminating bit string, and therefore the &quot;new&quot; real number that the diagonalization process is &lt;em&gt;supposed&lt;/em&gt; to produce might actually replicate a real number that &lt;em&gt;already appears&lt;/em&gt; in your enumeration but in a different form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If that is what you mean, it seems to me that the problem is not intrinsic to the diagonalization argument itself, but rather is an artifact of the (unnecessary) decision to represent real numbers as binary sequences.  If instead you represent real numbers in the customary way as &lt;em&gt;decimal&lt;/em&gt; sequences, the diagonalization construction protocol can be easily designed so avoid repeating sequences of 9s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explicitly:  Let $ \{ r_i \}$ be any enumeration of real numbers in the interval $(0,1)$.  Let each $r_i$ have the decimal expansion $0. a_{i1} a_{i2} ....$ where each $a_{ij}$ is a digit from the set {0, 1, ... , 9}.  We construct a real number $s= 0. b_1 b_2 b_3 ...$ as follows:  We let $b_i = 5$ unless $a_{ii}=5$, in which case we let $b_i=1$.  Then $s$ has a decimal expansion consisting only of 1s and 5s, but $s$ cannot be the same as any of the decimal expansions in the set $ \{ r_i \}$, because it differs from each $r_i$ in the $i$th decimal place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;repeating decimal annoyance&quot; is avoided altogether because in base 10 repeating decimals only lead to different decimal representations of the same real number if the repeating digit is 9, and we have avoided 9s completely in this process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course there's nothing special about 1 and 5; any two digits will work as long as we avoid 9.  And there's nothing special about base 10 either; the argument will work fine in any base 3 or higher.  The only base in which this &lt;em&gt;doesn't&lt;/em&gt; work is base 2, because we are limited to only two digits, and there is therefore no way to pre-empt the possibility of ending up with a string of repeating 1s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the reason people prefer to use the binary representation for this proof is because it provides a tight connection (via the use of characteristic functions for subsets) between this result (that the reals are uncountable) and the more general result that the power set of any set $S$ is always larger than $S$ itself.  But depending on your audience and the purpose of the class, that more general result may be both more general than you really need, and too abstract for people to follow.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-03-30T16:15:50.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1060" PostTypeId="2" ParentId="1028" CreationDate="2014-03-30T16:48:26.600" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How can I make them aware of the mathematical prerequisites, in a concrete way?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One way which has been proven effective time and time again would be to do a pretest, from that you can glean their experience in that level of mathematics as well as provide the student the knowledge of whether he/she SHOULD take that course. Require them to test somewhere around the halfway point of summer, long enough for them to forget everything they didn't learn, but with enough time to drop-out of the class or get tutoring if they need to.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The wider question is how to string courses ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Perhaps the easiest way to get your students to really grasp the connections is to simply have them write, read, debug, modify, and just experiment with code, higher languages, assembly, as well as other interactive projects, anything that can get them closer to real-world applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:&#xA;You could have an 'analyze linux day' where you decode exactly what those original 17 lines meant and how they function with respect to the conceptual foundations they've been learning. Have them write an essay that outlines the OS's limitations, algorithmic complexity, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm assuming you probably only have a small block of time with each group of students, and I would encourage you, and any educator to create a blog/website/forum for the class (if they don't have one already). This will help ease the amount of work you need to do tremendously! It's also a good idea to record, or allow students to record, and post videos of your lectures onto said website for reference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are still in contact with any of your old students, I would definitely talk to them and see what suggestions they might have, especially if they got a job in the CS field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My final 2-penny thought: The easiest (IMO) and most effective method for learning is the 'keep them entertained' approach of most grade-schools. It's not going to get any better for classical educators with the amount of information vying for student's attention these days.&lt;/p&gt;&#xA;" OwnerUserId="529" LastEditorUserId="529" LastEditDate="2014-03-30T18:00:04.990" LastActivityDate="2014-03-30T18:00:04.990" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1061" PostTypeId="2" ParentId="1043" CreationDate="2014-03-30T17:10:38.180" Score="8" Body="&lt;p&gt;This depends to a large extent on the background of your students, which varies a lot from country to country.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the United States, many students who take calculus have not seen any formal proofs before, while others had some limited exposure to two-column proofs in geometry class.  In this context, it is nearly impossible to require students to consistently write formal mathematical proofs during calculus.  At this stage in their mathematical development, students are only starting to become aware that proofs are required in mathematics, and you should be happy when students write arguments that are informal, but convincing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the reason for this is that calculus is a course that's primarily about calculus, not about formal proofs.  Typically calculus textbooks are relatively low on formal proofs, and do not convey the sense that the entire subject takes place within a formal framework.  Indeed, most calculus textbooks include a large number of relatively informal arguments designed to help the students' intuition for the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other countries, it is much more common for calculus students to have experience with formal proofs in their previous math classes, in which case it may be reasonable to expect student proofs to be both convincing and formal.  The same may be true in the United States for students in an honors calculus class -- even if such students don't have much experience in proofs, it is possible for very talented students to learn how to write formal proofs within the context of calculus, especially if this pedagogical goal is supported by both the professor and the textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for most students in the United States, writing formal proofs is a skill that will need to wait for later math classes.  Many colleges have a proof-writing class for sophomores for precisely this purpose, while others emphasize proof-writing in linear algebra and subsequent classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, my recommendation would be to accept as correct any argument that is coherent and convincing.  Think of it as more like an essay question -- it's fine to deduct for poor writing, but any mathematically correct argument that's written in an understandable way should be given full credit.  Students need to get used to making arguments like these before formality can be added into the mix.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-03-30T17:10:38.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1062" PostTypeId="2" ParentId="1043" CreationDate="2014-03-30T17:27:33.877" Score="5" Body="&lt;p&gt;It depends on how you have dealt with these proofs during the course. You have written, that you appreciate the approach of describing proofs in sketchy way. In that case, you should accept any coherent proof in exams, regardless of its formality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If formality was part of your course, you should give part of the points to coherent, informal proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Catching up Jim Belks evaluation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Germany, formality is mostly dealt with by the Algebra profs along Algebra itself whereas Calculus profs have to deal with the conceptual difficulties of inequalities and infinitesimal math outside of pure limits.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-03-30T17:27:33.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1063" PostTypeId="2" ParentId="1043" CreationDate="2014-03-30T17:32:41.987" Score="14" Body="&lt;p&gt;I have come to view mathematical proofs as legal briefs,&#xA;intended to convince a certain audience that the theorem is true.&#xA;The degree of formality very much depends on the target audience.&#xA;To echo Jim, the argument should be &quot;coherent and convincing&quot; to that audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One phrase that encapsulates this view is: &lt;em&gt;mathematical proofs are social constructs&lt;/em&gt;.&#xA;I first encountered this viewpoint in a paper published by three theoretical&#xA;computer scientists that caused quite a stir at the time (1979):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;De Millo, Richard A., Richard J. Lipton, and Alan J. Perlis. &quot;Social processes and proofs of theorems and programs.&quot; &lt;em&gt;Communications of the ACM&lt;/em&gt; 22.5 (1979): 271-280.&#xA;  (&lt;a href=&quot;https://www.ics.uci.edu/~pattis/misc/socialproofs.pdf&quot;&gt;PDF download link&lt;/a&gt;).&#xA;  &quot;We believe that, in the end, it is a social process that determines whether&#xA;  mathematicians feel confident about a theorem.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Discussions of this topic continue to this day. Here's a 2012 blog entry addressing&#xA;the issue: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Cathy O'Neil. &lt;a href=&quot;http://mathbabe.org/2012/08/06/what-is-a-proof/&quot;&gt;What is a Proof?&lt;/a&gt;:&#xA;  A proof is &quot;a convincing argument of why you think something is true....It is the responsibility of the person proving something to convince his or her audience that it's true.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-03-30T18:53:47.830" LastActivityDate="2014-03-30T18:53:47.830" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1064" PostTypeId="2" ParentId="1052" CreationDate="2014-03-30T17:40:12.873" Score="3" Body="&lt;p&gt;Firstly, Occam's razor is completely unsuitable to Mathematics and Natural sciences. Propositions and Theorems have different qualities:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;few assumptions&lt;/li&gt;&#xA;&lt;li&gt;many results&lt;/li&gt;&#xA;&lt;li&gt;intelligibility&lt;/li&gt;&#xA;&lt;li&gt;practical applications&lt;/li&gt;&#xA;&lt;li&gt;scientific usefulness&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Occam's razor reduces all that to only one quality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, proofs are no hypotheses and they are not based on assumptions. They are based on technique and knowledge, and this is something, that you should regard:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do the students know and understand all the techniques used in the proof? (Techniques, which you delibaretly want to introduce by the proof, don't count.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And lastly, although shortness may be a quality of proofs it is no &lt;em&gt;didactic&lt;/em&gt; one. Choose the proof, that illustrates better, what you want the students to learn.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-03-30T17:40:12.873" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1065" PostTypeId="2" ParentId="1052" CreationDate="2014-03-30T17:44:53.653" Score="7" Body="&lt;p&gt;This is a fairly complicated question, it has several aspects, each contributing to the final answer. Consider&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Is the intuition from the longer proof better?&lt;/em&gt; It might explain some phenomena of why the question was hard, but it could also limit other, perhaps even better explanations.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Is it possible to supplement the shorter proof with appropriate intuition?&lt;/em&gt; Perhaps you could elaborate on each part so the student can learn more, but keep the final formal derivation short.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Which proof is easier to understand?&lt;/em&gt; Sometimes the shorter proof is nicer, cleaner and have a more intuitive feel, but the longer one could be still more accessible, e.g. it doesn't need any complex machinery, just the most basic and mundane tools are enough.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Which proof would let the students learn more?&lt;/em&gt; It might happen, that student's are yet unable to appreciate that truly brilliant idea in the shorter proof, while the boring, longer proof has a number of mid-results that would still be useful to students at this point.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Do you have enough time?&lt;/em&gt; It is possible that the students could benefit more from something else that wouldn't fit if the longer proof was chosen.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;That's not all, but let's finish here. It's easy to guess, there's no general solution, but perhaps this list might clarify some issues. If you have serious doubts which one to choose, you could also try to present both proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-03-30T17:44:53.653" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1066" PostTypeId="2" ParentId="1002" CreationDate="2014-03-30T17:51:07.350" Score="0" Body="&lt;p&gt;Another application is negation of universal and existential quantification. Take for example an advertisement:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;All our second-hand cars come with working air-con.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You need only to find one these cars without working air-con to disprove the sentence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And another application is contraposition. For example, teachers often say to the student's parents&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If your student works harder, he'll improve.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which is easily accepted. When he won't improve and the parents come back to the teacher, he will answer:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He didn't improve, that means he didn't work harder.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-03-30T17:51:07.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1067" PostTypeId="2" ParentId="1052" CreationDate="2014-03-30T17:59:01.940" Score="5" Body="&lt;p&gt;I'd choose the proof which helps to understand the theorem the most. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the long one is too technical, the message might get lost. If the short one is too mysterious, the time saved with the more elegant version might need to be spent to digest the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know one book on calculus &amp;amp; linear algebra for physicists (in German, &lt;a href=&quot;https://portal.dnb.de/opac.htm?method=showFullRecord&amp;amp;currentResultId=partOf%3D965090620%26any&amp;amp;currentPosition=0&quot;&gt;R. Wüst: Mathematik für Physiker und Mathematiker, Bd. 2&lt;/a&gt;) where the implicit function theorem is first stated and proven in a two-dimensional version, and then dealt with in the general case. As R. Wüst mentions, an approach which is completely unnecessary from a mathematical point of view. However, I must add, it is really helpful from a pedagogical point of view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In lectures, elegance is still a factor, but one should not sacrifice legibility in favor of elegance.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-03-30T17:59:01.940" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1068" PostTypeId="1" AcceptedAnswerId="1073" CreationDate="2014-03-30T18:05:29.617" Score="26" ViewCount="1209" Body="&lt;p&gt;I have the oportunity to talk to a highschool class about mathematics, the topic I got to present are the integers modulo $n$, ie, $\Bbb Z/n\Bbb Z$ , however I don't want to be very heavy and formal, because I don't want to lose them. For the talk I also have limited time, since this is just  talk, so my hands are a little tied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At first I wanted to present the idea of equivalence class, so the fact that an infinity of numbers &lt;em&gt;fit&lt;/em&gt;  in a few &lt;em&gt;boxes&lt;/em&gt; make sense, however I'm having a little bit of problem with that, the example I wanted to use was the hours of the day, or else, the hours of all time (sounds weird doesn't it?) which, for purposes of the discussion, we are assuming are countable infinite (and also, without them knowing we're also assuming the Axiom of choice). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long story short, I find myself having a lot of issues, because whenever I encounter a very abstract mathematical concept (modular arithmetic, equivalence classes, infinites, etc.) I feel that I'm not actually going to help their intuition nor their interest (I aspire to get a: &quot;I never saw it like that!&quot;) What recommendations and/or reference can you suggest about this topic?&lt;/p&gt;&#xA;" OwnerUserId="99" LastEditorUserId="99" LastEditDate="2014-03-30T18:40:22.233" LastActivityDate="2015-10-19T10:03:47.823" Title="How to present $\Bbb Z/n\Bbb Z$ to highschool level audience" Tags="|mathematical-pedagogy|secondary-education|intuition|" AnswerCount="6" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1069" PostTypeId="2" ParentId="243" CreationDate="2014-03-30T18:39:09.733" Score="2" Body="&lt;p&gt;I use Cantor's diagonal argument. To avoid non-uniqueness issues (which I certainly discuss) I first restrict to $[0,1]$, then agree to always write the expansion with infinitely many 9's rather than 0's. Then when constructing the not listed element I just make sure to use digits other than '0' and '9'. I then wait for the better students to ask &quot;why did you use '4' and '7'?&quot; and I explain that it is arbitrary as long as we don't choose '0' or '9'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find this to be an extremely visual argument and most students cope well with it. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-30T18:39:09.733" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1070" PostTypeId="2" ParentId="1068" CreationDate="2014-03-30T18:48:37.533" Score="14" Body="&lt;p&gt;Modular arithmetic is beautiful and simple but tends to be frighteningly clotted with detail and notation at first. Whenever I present it I start off by telling the audience they already know modular arithmetic. I proceed to ask them: if it is now 1 o'clock (and then I write a '1' on the board), what time will it be in 2 hours (and I write a &quot;+ 2&quot; next to the '1'). Everybody answers '3', so I complete the equation &quot;1+2=3&quot;. Then I say, if it is now 11 o'clock (and I write '11'), what time will it be in 2 hours? this gives rise to the equality &quot;11+2=1&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I formalize this in the language of arithmetic modulo $12$, and then I give the general definition, compute some things, prove some things, show that $x=1 \mod n$ implies $x^n=1 \mod n$, and derive some divisibility criteria (i.e., a number is divisible by $3$ iff the sum of its digit is). &lt;/p&gt;&#xA;" OwnerUserId="59" LastEditorUserId="-1" LastEditDate="2014-03-30T21:05:02.823" LastActivityDate="2014-03-30T21:05:02.823" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1071" PostTypeId="2" ParentId="135" CreationDate="2014-03-30T19:36:11.003" Score="8" Body="&lt;p&gt;Why not motivate it be being a bit more general, maybe introducing Lebesgue-Stieltjes integration?  That way it is easy to see it is very useful in, for instance, probability theory, by giving a unified theory for continuous and discrete probability ( and even finding some new phenomena, continuous distribution not being absolutely continuous, like the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cantor_distribution&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Cantor distribution&quot;&lt;/a&gt;.)   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then one could even continue asking if those new, strange, singular distributions maybe have some use? and discover &lt;a href=&quot;https://mathoverflow.net/questions/163325/singular-distributions-applications-and-instances&quot;&gt;https://mathoverflow.net/questions/163325/singular-distributions-applications-and-instances&lt;/a&gt;  or &lt;a href=&quot;https://en.wikipedia.org/wiki/Attractor#Strange_attractor&quot; rel=&quot;nofollow noreferrer&quot;&gt;Strange attractors&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="122" LastEditDate="2019-08-04T09:40:28.703" LastActivityDate="2019-08-04T09:40:28.703" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="1072" PostTypeId="2" ParentId="1043" CreationDate="2014-03-30T19:49:38.763" Score="6" Body="&lt;p&gt;If I had to simplify this problem to one question, then what I would ask myself is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;strong&gt;To what extent the student convinced me that he&lt;br/&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;knows and understands the required material?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a class that required formal proofs (e.g. it was stated in the task) writing just descriptions wouldn't convince me at all. On the other hand, if we were studying something else, and the description was accurate enough (e.g. there were no ambiguous expressions, no guessing and the mental structure was clear), then the student would receive a perfect score (for this part).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, if there is a proof with a clear intention, then this is all well; even if the student would make some typos, I would fix it for him. However, for a &lt;em&gt;correct&lt;/em&gt; formal derivation that contains a lot of &quot;noise&quot; (unnecessary formulas, inadequate diagram, etc.), I would ask the student to explain and only after that I would acknowledge his solution (or not, if he does not understand his own work). Of course, there might not be enough time to do this in large courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your second question comes, in my opinion, from an entirely different area. Many mathematicians consider something done once they know the proof, there are even jokes and anecdotes about it. To make students &lt;em&gt;want&lt;/em&gt; to write a formal proof, you need to make it worth it. The downside is that requiring a formal proof decreases the incentive to think first in intuitive terms. Moreover, formal proofs need much more work than descriptions and there might be not enough time during an exam (or you might have to give easier problems, but this is undesirable). Perhaps you could make it into two assignments graded separately, and the formal derivation could be done at home.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-03-30T20:44:31.440" LastActivityDate="2014-03-30T20:44:31.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1073" PostTypeId="2" ParentId="1068" CreationDate="2014-03-30T20:04:30.313" Score="23" Body="&lt;p&gt;I did this once with a high school group.  My approach was to use the metaphor of a scientist doing a controlled experiment:  You study a complicated system by introducing a single change, while trying to keep everything else constant, and watch how that change affects the system overall.  In this case, the system is arithmetic, and the &quot;one change&quot; we are going to make is that we are going to replace the number &lt;em&gt;line&lt;/em&gt; with a number &lt;em&gt;circle&lt;/em&gt;:  When we count, we are going to &quot;loop back around&quot; after a certain pre-chosen value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like Ittay (despite our names we are not related, by the way!) I pointed out that this approach is actually familiar, and used the &quot;arithmetic on a clock&quot; model to demonstrate what happens if after 12 you go back to 1.  I also introduced the idea of a &quot;7-clock&quot;, &quot;5-clock&quot;, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We practiced writing out some basic arithmetic on a 7-clock:  for example, 4 * 5 = 6 on a 7-clock, and also 5 * 6 = 2, etc.  I gave some puzzles, like:  On what kind of clock is the equation 5 * 8 = 2 true?  We discuss the fact that each number on a clock can be given many different names; for example, on a 12-clock we can think of 5 as 5, or 17, or 29, but also as -7, -19, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we started asking some more interesting questions, like &quot;Which elements are invertible on a 12-clock?&quot; and &quot;Which elements are invertible on a 7-clock?&quot;  This starts to bring out the way in which prime numbers are different than composite numbers.  Another interesting one is &quot;How many square roots does 4 have on a 12-clock and on a 7-clock?&quot;  This latter one is an example of how changes in the ground rules for the system can have surprising ripple effects.  In particular:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students learn in high school that a quadratic equation has at most two real solutions, and on a 7-clock the equation $x^2=4$ has only two solutions (which can be recognized as ± 2, just like in &quot;ordinary&quot; algebra), but on a 12-clock the equation $x^2=4$ has &lt;em&gt;four&lt;/em&gt; solutions.  So that means that the Fundamental Theorem of Algebra -- a Theorem that high school students may learn &lt;em&gt;about&lt;/em&gt; but do not learn the proof of -- must, at some deep level in its proof, rely on the difference between a 12-clock and a 7-clock.&lt;/li&gt;&#xA;&lt;li&gt;Another difference that they can recognize is the fact that the &quot;Zero Product Property&quot; (which is the conceptual hinge on which turns the whole high school theory of solving quadratic equations) isn't true on a 12-clock.&lt;/li&gt;&#xA;&lt;li&gt;On the other hand the equation $x^2 = -1$, which does &lt;em&gt;not&lt;/em&gt; have any solutions in &quot;ordinary&quot; (i.e. real-valued) algebra, does have solutions on a 5-clock.  So on a 5-clock $\sqrt{-1}$ is not imaginary!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Things that I &lt;em&gt;avoided&lt;/em&gt;:  I did not say a thing about equivalence classes or well-definedness of operations; nor did I try to provide a formal definition or proof for anything I said.  If you are going to have the opportunity to come back and develop the ideas more systematically over a longer time frame those would be appropriate, but for a &quot;done in one&quot; lecture to a general audience I would stick with clear, specific, and surprising (if possible) examples.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2015-10-18T03:47:53.950" LastActivityDate="2015-10-18T03:47:53.950" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1074" PostTypeId="2" ParentId="995" CreationDate="2014-03-30T20:11:02.920" Score="3" Body="&lt;p&gt;This example is actually from a textbook I am using.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are in a submarine and want to know how long you have gone in some time&#xA;(there are no km marks to read under water!).  You have a speedometer, so you can just write down the speedometer writings, and integrate them numerically with respect to time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(You can do that in a car too, but ther there is no point in doing it).&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="122" LastEditDate="2017-06-25T06:59:45.870" LastActivityDate="2017-06-25T06:59:45.870" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1075" PostTypeId="2" ParentId="1044" CreationDate="2014-03-30T21:02:25.007" Score="9" Body="&lt;p&gt;I maintain that what you want to focus on is sense making &lt;a href=&quot;https://matheducators.stackexchange.com/a/816/262&quot;&gt;(MESE 1)&lt;/a&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/a/880/262&quot;&gt;(MESE 2)&lt;/a&gt;. If you can present Eisenstein's criterion as a way of tackling such problems that helps to promote students' critical thinking about algebra, then I would say go for it. (I'd also be interested to look at your curricular materials!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To illustrate my more general feeling, I look at your example: Is $x^2 + 9x + 24$ reducible?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even with Eisenstein's criterion, one must still find a prime $p$; of course, here the choice $p=3$ is pretty immediate, so you can, indeed, employ this method of solution to show that the expression is irreducible. More generally, though, there can be a question of which prime to use...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my vantage point, I'd rather a student remark: $b^2 - 4ac = 9^2 - 4(1)(24) = 81 - 96 &amp;lt; 0$, so there are no real solutions. (In particular, the polynomial is irreducible [over e.g. $\mathbb{Z[x]}$].)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, I wouldn't mind seeing a solution like: $x^2 + 9x + 24 = (x+\frac{9}{2})^2 + $ something positive; so there are no real solutions (hence no integer solutions) and the polynomial cannot be factored in $\mathbb{Z}[x]$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(To be clear: I would not use the $\mathbb{Z}[x]$ notation in a high school class.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a third method, a student might note that a hypothetical factored expression would have to look like $(x-a)(x-b)$ for $a, b &amp;gt; 0$; you need $ab = 24$ and $a + b = 9$, but the latter constraint means the maximum of $ab$ occurs when $a = b = 9/2$; observe $(9/2)^2 = 81/4 &amp;lt; 96/4 = 24$. Therefore, no such $a, b$ exist; so the expression cannot be factored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The three methods suggested here are similar on a deep level, but how related would your high school students find them? What other methods could be used? There is already a lot that students can explore without adding on an additional criterion (at least one that I did not see until a course on Group Theory).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, I would especially like it in any of the suggested alternatives if the student could give an explanation of why these methods are viable, i.e., at a level that makes them understandable to other students in the course who would not think of such approaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt; Incorporating Eisenstein's criterion into a high school course on algebra would be nonstandard but not (for any reasons I see) obviously worth objecting to; the question remains whether or not you can make this criterion meaningful, and do so in such a way that students who understand the approach can give their own explanations - ones that are understandable to others in the class.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-04T03:11:08.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1076" PostTypeId="2" ParentId="1044" CreationDate="2014-03-30T21:38:32.800" Score="5" Body="&lt;p&gt;I see little sense in introducing Eisenstein's criterion (a pretty big cannon) only to have a slight advantage when trying to factorize quadratics. It's really not justified. The strength of Eisenstein's criterion is in proving large polynomials irreducible. This certainly can be taught at the high school level since it's not that complicated and quite intuitive. But then the students need to be asked for the irreducibility of such polynomials as $x^{100}+2$ or $x^{100}-1$. It is then a very nice example of how to solve questions where a naive direct approach would lead to disaster. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-03-30T21:38:32.800" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1077" PostTypeId="2" ParentId="1042" CreationDate="2014-03-30T21:57:55.340" Score="14" Body="&lt;p&gt;&lt;a href=&quot;https://www.york.cuny.edu/~malk/&quot;&gt;Joseph Malkevitch&lt;/a&gt;, based out of CUNY York College but also a visiting professor at Columbia University Teachers College, has a fair bit on his website about (high school) student research. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on the student's mathematical level, one could point to Malkevitch's pages on &lt;a href=&quot;http://www.york.cuny.edu/~wu/jm/&quot;&gt;Mathematics Research Projects&lt;/a&gt; or a couple other pages directed at undergraduate students &lt;a href=&quot;https://www.york.cuny.edu/~malk/problems.html&quot;&gt;(1)&lt;/a&gt; &lt;a href=&quot;https://www.york.cuny.edu/~malk/researchproblems3.html&quot;&gt;(2)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other helpful links and pieces of writing can be found on his main site (click his linked name above).&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-03-30T21:57:55.340" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1078" PostTypeId="2" ParentId="135" CreationDate="2014-03-30T22:29:52.740" Score="9" Body="&lt;p&gt;Reiterating points made in the other good answers: first, we want to have a principle that is, approximately, $\lim_n \int f_n = \int \lim_n f_n$. An immediate issue is clarification of in what senses (plural!?!) we might take limits of &lt;em&gt;functions&lt;/em&gt;. Pointwise? Uniformly pointwise? These seem only-slightly different, but, as we know by now, are wildly different in their effect. Uniform pointwise is too strong a constraint... we often simply don't have it. While uniform pointwise limits of nice (e.g., continuous) functions are again nice, that's not so strongly true for merely-pointwise limits. (Of course there is considerable landscape in-between.) A related issue is, then, that not-necessarily-uniform pointwise limits of continuous functions need not be continuous... to say the least. So, for the principle of interchanging limit and integral even to have &lt;em&gt;sense&lt;/em&gt;, we have to have some &lt;em&gt;compatible&lt;/em&gt; (not merely definitional) notion of integral for more general functions... which we want substantially to make sense of these interchange-of-limits theorems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One side-effect conclusion is that we want descriptions of various classes of functions with metric or other topologies making them &lt;em&gt;complete&lt;/em&gt;, in the first place, and then have &quot;integrals&quot; that are continuous linear functionals in that topology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A meta-point that I find significant is that we really did have some issues needing resolution by extending notions of &quot;integral&quot;. Not just trying to extend things for laughs or out of boredom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.g., although I was once charmed by the line of patter about integrating the characteristic function of the rationals, we surely can agree that this was not a burning issue needing resolution. It could be made into a more relevant form if we observe that that function &lt;em&gt;is&lt;/em&gt; a pointwise limit of &lt;em&gt;continuous&lt;/em&gt; functions!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(And, by the way, &quot;the Riemann integral&quot; slightly mis-represents the attitudes of people pre-1850, who knew quite well what they &lt;em&gt;wanted&lt;/em&gt;, how to differentiate and integrate. It seems that in their context, the fundamental theorem of calculus, that the derivative of $\int_{x_o}^x f(t)\,dt$ is $f(x)$ for reasonable functions, was the decisive &lt;em&gt;additional&lt;/em&gt; feature of integrals... beyond computing areas and such. In particular, they didn't have a &quot;definition&quot; of integral...)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-30T22:29:52.740" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1079" PostTypeId="2" ParentId="1044" CreationDate="2014-03-30T23:10:02.317" Score="2" Body="&lt;p&gt;Firstly for quadratic equations, the quadratic formula will tell all. Is &lt;span class=&quot;math-container&quot;&gt;$b^2-4ac$&lt;/span&gt; a perfect square? I think that is easier than checking for common primes.  Secondly you didn't state the criteria right: &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; may not divide the coefficient of the highest term; however, &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; but not &lt;span class=&quot;math-container&quot;&gt;$p^2$&lt;/span&gt; may divide the constant term. The point of Eisenstein's criteria is to give a criteria for factorization in cases when finding the roots of the polynomial isn't reasonable (think degree &amp;gt; 4). I wouldn't think about Eisenstein's criteria unless faced with a polynomial that I couldn't factor. That's my two cents.&lt;/p&gt;&#xA;" OwnerUserId="541" LastEditorUserId="376" LastEditDate="2021-02-25T09:25:04.683" LastActivityDate="2021-02-25T09:25:04.683" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="1080" PostTypeId="2" ParentId="1026" CreationDate="2014-03-31T01:41:50.880" Score="17" Body="&lt;p&gt;All the answers so far are very general, so here's a very specific, practical answer.&lt;/p&gt;&#xA;&lt;p&gt;Students entering a second-semester calculus course need to be very good at taking derivatives.  This includes memorizing the following parts of the differentiation algorithm:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The power rule (including the meanings of fractional and negative exponents)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The product rule&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The chain rule (specifically, they should be good at using it)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The derivatives of &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Of course, they must also understand the structure of complicated formulas well enough that they can figure out which rule to apply.&lt;/p&gt;&#xA;&lt;p&gt;The reason students have to know this is that most of Calculus 2 is integration, which means doing differentiation backwards.  You literally have to know the details of differentiation off the top of your head if you want to do it backwards.  It is vital for checking the results of integration, and it's also used during both substitution and integration by parts.  It is also used during subsequent topics, including differential equations and power series.&lt;/p&gt;&#xA;&lt;p&gt;A few more notes:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It's usually ok if students don't memorize the quotient rule.  This rule mostly not relevant for integration, which means that it rarely comes up in Calculus 2.  (Also, I happen to think that students are better off using the &lt;span class=&quot;math-container&quot;&gt;$A/B = AB^{-1}$&lt;/span&gt; method than the quotient rule.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Depending on what techniques of integration are covered in Calculus 2, the students may need to know the derivatives of the other trig functions (&lt;span class=&quot;math-container&quot;&gt;$\tan x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cot x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sec x$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\csc x$&lt;/span&gt;) as well as the inverse trig functions.  While students who have these memorized coming into Calculus 2 may be slightly better off, it's usually not necessary, and they can always pick them up along the way.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, of these rules, the most important is the derivative of &lt;span class=&quot;math-container&quot;&gt;$\tan^{-1}x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="50" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-03-31T01:41:50.880" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1081" PostTypeId="2" ParentId="1043" CreationDate="2014-03-31T03:39:20.973" Score="2" Body="&lt;p&gt;In an &lt;em&gt;exam&lt;/em&gt; (unless specifically about formal proofs) I'd want a reasonably coherent, even if rather informal, argument. In an exam I want to check general understanding (there is the time limit, and you want to cover a range of subject matter). &lt;em&gt;Homework&lt;/em&gt; is another kettle of fish, there I expect orderly, reasonably formal arguments. No time pressure, can check other sources (citing/acknowleding them!), can redo the work as needed.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-03-31T03:39:20.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1082" PostTypeId="1" CreationDate="2014-03-31T04:02:45.923" Score="14" ViewCount="329" Body="&lt;p&gt;I am teaching Calculus I this semester and it's not going well. There are roughly 30 students registered for the course but for any given lecture, only 1-4 students show up. Attendance has been consistently declining throughout this semester. I'm not sure why this is the case, but it is disconcerting. What is worse, is that despite plenty of notice, only six students showed up to the second exam this week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What should I do to encourage class attendance and what should be done regarding the students who missed the exam?&lt;/p&gt;&#xA;" OwnerUserId="545" LastEditorUserId="545" LastEditDate="2014-03-31T04:54:04.610" LastActivityDate="2014-05-05T06:47:57.183" Title="What to do to get a class back on track?" Tags="|undergraduate-education|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1083" PostTypeId="2" ParentId="1052" CreationDate="2014-03-31T06:24:47.947" Score="0" Body="&lt;p&gt;My reading of Occam's Razor is that it is a heuristic principle about (essentially) nature or the world: it is sometimes quoted as &quot;Do not multiply hypotheses&quot;; in other words if a simple assumption will explain something then there is no need for more complicated assumptions.  I don't think it really applies to mathematical proofs.  For proofs, certainly one driving force is to cut down the number of assumptions to a minimum so as to find the widest domain of applicability of a theorem, but this may result in a more complicated proof (for example, constructive proofs that avoid unrestricted application of the excluded middle).  As other posters have said, different proofs may serve different purposes, and having more than one proof may bring a range of insights about the theorem under consideration.&lt;/p&gt;&#xA;" OwnerUserId="547" LastActivityDate="2014-03-31T06:24:47.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1084" PostTypeId="1" AcceptedAnswerId="1135" CreationDate="2014-03-31T06:30:14.300" Score="5" ViewCount="1046" Body="&lt;p&gt;It seems there are few well-known professional mathematicians in Africa. It is mainly because of the poor quality of elementary/undergraduate mathematics education in African countries. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1.&lt;/strong&gt; Is there any math education research which shows the past, current and future status of mathematics and math education in African countries?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2.&lt;/strong&gt; Is there any charity NGO in EU or US specially for help on improving the quality of math education in African countries at different levels?  &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="114" LastEditDate="2014-03-31T07:23:02.593" LastActivityDate="2014-03-31T18:23:17.990" Title="Mathematics Education in Africa" Tags="|mathematical-pedagogy|reference-request|africa|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1085" PostTypeId="2" ParentId="1084" CreationDate="2014-03-31T06:56:10.773" Score="4" Body="&lt;p&gt;There is a scheme &quot;Mentoring African Research in Mathematics&quot; run by the London Mathematical Society (which is effectively the UK national maths society) and the  African Mathematics Millennium Science Initiative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some details are here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.lms.ac.uk/news-entry/02092013-1601/mentoring-african-research-mathematics-marm&quot; rel=&quot;nofollow&quot;&gt;http://www.lms.ac.uk/news-entry/02092013-1601/mentoring-african-research-mathematics-marm&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-03-31T06:56:10.773" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1086" PostTypeId="1" CreationDate="2014-03-31T09:11:37.750" Score="6" ViewCount="140" Body="&lt;p&gt;&lt;strong&gt;What and how much scaffolding is necessary for a motivated undergraduate student who doesn't have the ideal prerequesites for work?&lt;/strong&gt; This could be for a number of reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;He didn't realize the purpose of certain classes until later.&lt;/li&gt;&#xA;&lt;li&gt;He had troubles grasping concepts on his own and didn't have as much help as he should.&lt;/li&gt;&#xA;&lt;li&gt;Really bad classes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Whatever they are, he's willing to catch up. How to deal with this situation?&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-03-31T09:11:37.750" Title="How to nurture an unprepared student" Tags="|mathematical-pedagogy|undergraduate-research|" AnswerCount="0" CommentCount="6" FavoriteCount="0" ClosedDate="2014-04-03T16:59:55.527" ContentLicense="CC BY-SA 3.0" />
  <row Id="1087" PostTypeId="5" CreationDate="2014-03-31T09:12:09.913" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T09:12:09.913" LastActivityDate="2014-03-31T09:12:09.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1088" PostTypeId="4" CreationDate="2014-03-31T09:12:09.913" Score="0" Body="For questions about mathematical education related to countries in Africa." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T12:37:43.797" LastActivityDate="2014-03-31T12:37:43.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1089" PostTypeId="5" CreationDate="2014-03-31T09:12:48.867" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T09:12:48.867" LastActivityDate="2014-03-31T09:12:48.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1090" PostTypeId="4" CreationDate="2014-03-31T09:12:48.867" Score="0" Body="For questions related to studies for graduate or more-advanced students and courses." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T13:23:15.167" LastActivityDate="2014-03-31T13:23:15.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1091" PostTypeId="5" CreationDate="2014-03-31T09:16:11.813" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T09:16:11.813" LastActivityDate="2014-03-31T09:16:11.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1092" PostTypeId="4" CreationDate="2014-03-31T09:16:11.813" Score="0" Body="For questions about good use of notation, comparison of specific notation, motivation of notation." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T12:37:58.120" LastActivityDate="2014-03-31T12:37:58.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1093" PostTypeId="5" CreationDate="2014-03-31T09:20:15.257" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T09:20:15.257" LastActivityDate="2014-03-31T09:20:15.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1094" PostTypeId="4" CreationDate="2014-03-31T09:20:15.257" Score="0" Body="For questions about giving advice to students in general, as well as advising a student's thesis." OwnerUserId="114" LastEditorUserId="80" LastEditDate="2014-04-01T20:43:59.310" LastActivityDate="2014-04-01T20:43:59.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1095" PostTypeId="5" CreationDate="2014-03-31T09:22:15.447" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T09:22:15.447" LastActivityDate="2014-03-31T09:22:15.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1096" PostTypeId="4" CreationDate="2014-03-31T09:22:15.447" Score="0" Body="Questions related to mathematical education in the topics of optimization in general, optimal control, discrete optimization." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T12:37:35.280" LastActivityDate="2014-03-31T12:37:35.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1097" PostTypeId="2" ParentId="1042" CreationDate="2014-03-31T09:27:55.717" Score="2" Body="&lt;p&gt;At my daughters high school, last year every student must do a project of student research. They can choose the subject themselves (my daughter choose biology). Some math projects which have been done:&#xA;-Fractals, programming fractals.&#xA;-determinants and the eight-queen-problem&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2014-03-31T09:27:55.717" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1098" PostTypeId="2" ParentId="1084" CreationDate="2014-03-31T09:48:21.683" Score="7" Body="&lt;p&gt;The &lt;a href=&quot;http://www.nexteinstein.org/&quot;&gt;African Institute for Mathematical Sciences&lt;/a&gt; (AIMS) aims to create an &quot;African Einstein&quot;. AIMS is a network of what are essentially maths universities in Africa, and is a joint venture between, I believe, the University of Stellenbosch, University of Cambridge, University of Cape Town, University of Oxford, University of Paris-Sud, and University of the Western Cape. I quote from their web page,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;AIMS is a pan-African center for post-graduate training and research, providing advanced, broadly applicable mathematical skills to talented students recruited from all over Africa.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There used to be money floating around the UK for people to give short courses at the AIMS centers. Also, I know that Heriot Watt University has ties with AIMS. For example, one of their former PhD students is currently doing a postdoc at AIMS Ghana, while I believe there are two current PhD students there who are products of AIMS.&lt;/p&gt;&#xA;" OwnerUserId="169" LastActivityDate="2014-03-31T09:48:21.683" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1099" PostTypeId="2" ParentId="1042" CreationDate="2014-03-31T11:15:16.220" Score="9" Body="&lt;p&gt;There are some very nice questions in &lt;a href=&quot;http://en.wikipedia.org/wiki/Art_gallery_theorem&quot; rel=&quot;noreferrer&quot;&gt;art gallery theorems&lt;/a&gt; that can be explored&#xA;via paper-and-pencil experimentation.&#xA;Below, placing guards at the three blue vertices visually guard&#xA;the entire interior of this polygonal &quot;art gallery&quot;:&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Wu79P.png&quot; alt=&quot;3-coloring&quot;&gt;&lt;sub&gt;(Wikipedia image)&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-03-31T11:15:16.220" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1100" PostTypeId="1" AcceptedAnswerId="1158" CreationDate="2014-03-31T11:20:23.897" Score="12" ViewCount="746" Body="&lt;p&gt;Logic and its sub-fields are closely related to philosophy. There is an undeniable mutual interaction between one's philosophical point of view and his/her approach in teaching mathematical logic. In fact you can teach a particular course material in essentially different ways based on your philosophical interests. Some logic teachers teach logic with implicit/explicit &lt;em&gt;indoctrination&lt;/em&gt; of their personal philosophical point of views to their students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example once I watched a course video of one of my intuitionist colleagues in his undergraduate &quot;Foundations of Math&quot; class. In that lecture he described the Axiom of Choice as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Axiom of Choice &lt;em&gt;is not intuitively reasonable&lt;/em&gt; because it suggests assuming the existence of an invisible entity. This axiom doesn't say anything about the way which one can produce a choice function. It just says that such a function exists somewhere in the ideal mathematical universe.     &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some years later some of the students of that class participated in my &quot;Advanced Set Theory&quot; graduate course as a part of their voluntary graduate courses. It was clear that they don't like the uncountable cardinals at all. Also everywhere which I proved the essential use of Axiom of Choice in a usual mathematical theorem using forcing construction, they said something like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...Thus this theorem &lt;em&gt;is not intuitively reasonable&lt;/em&gt; and we should remove it from our mathematics.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;None of these students were working on logic. They just passed a few elementary courses in their undergraduate curriculum. I don't know if this is just a coincidence or it is an impact of participating in a logic course of an intuitionist teacher. Based on such facts there are arguments among logicians of our department about possible impacts of this &lt;em&gt;ideological teaching&lt;/em&gt; on students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Those who are agree with ideological teaching in logic courses say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;It is an undeniable right of each teacher to introduce the teaching material by their own philosophical point of view.&lt;/li&gt;&#xA;  &lt;li&gt;The impacts of a teacher's personal philosophical point of view on the students is minor. They can see different arguments in different references which are accessible everywhere. &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Those who are not agree with ideological teaching in logic courses say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;It is nice to introduce philosophical aspects of mathematical theorems in logic courses but the teachers should be neutral in explaining these ideas because a logic teacher's duty is teaching logic to students not indoctrinating his/her personal point of views.&lt;/li&gt;&#xA;  &lt;li&gt;The impacts of a teacher's personal philosophical point of view on the students could be essential because the teacher is students' main reference and so he/she has a great influence on forming their research interests and scientific future. Thus by an ideological teaching, a teacher restricts the freedom of his/her students to choose their own philosophical point of views.    &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What are possible (dis)advantages of the ideological teaching in logic courses? Could it be harmful/useful for students? Does it have a notable impact on students' research interests in future? Is there any math education long term research to determine the similarity of students and teachers' point of views in logic and philosophy of mathematics?      &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-31T11:25:55.690" LastActivityDate="2014-04-01T00:12:02.123" Title="Ideological Teaching in Logic Courses" Tags="|mathematical-pedagogy|course-design|reference-request|logic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1101" PostTypeId="1" AcceptedAnswerId="1105" CreationDate="2014-03-31T11:31:16.627" Score="22" ViewCount="1983" Body="&lt;p&gt;There are some good students who understand a lot and are very critical. Such students tend to think that they will only understand abstract algebra if they have followed a course about logic; or they think they can only understand an analysis course if they have attended a course about abstract topology, etc. - They tend to be only satisfied if they follow the structures of the books of Bourbaki where nothing is used unless if was proven before, a very strict deductive way of doing mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a good way of thinking and it is good to not believe something unless one has a proof for it. However, at most universities, the curriculum is not build in such a way. Mostly, you learn about the topology of $\mathbb{R}$ before attending a course about abstract topology where this is the main example everyone can deal with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is the advice I can give to students with that attitude?&lt;/strong&gt;  &lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T12:48:46.440" LastActivityDate="2018-03-10T18:38:38.307" Title="How to advise students who want to do a &quot;Bourbaki&quot;-style study?" Tags="|undergraduate-education|curriculum|gifted-students|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1102" PostTypeId="5" CreationDate="2014-03-31T11:44:52.093" Score="0" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Secondary_education&quot; rel=&quot;nofollow&quot;&gt;Secondary Education&lt;/a&gt; is non well-defined concept and differs a lot between (and even &lt;em&gt;within&lt;/em&gt; different countries). Mathematical education in secondary education contains the introduction of variables, abstract concepts in calculations, the definition of functions, pre-calculus, basic algebra (e.g., solving equations), basic geometry (calculation areas and volumes of geometric objects).&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T14:32:12.143" LastActivityDate="2014-03-31T14:32:12.143" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1103" PostTypeId="4" CreationDate="2014-03-31T11:44:52.093" Score="0" Body="For questions about teaching mathematics in secondary education (in most countries approx. ages 10-18). " OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T14:31:57.010" LastActivityDate="2014-03-31T14:31:57.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1104" PostTypeId="2" ParentId="1084" CreationDate="2014-03-31T12:15:30.910" Score="3" Body="&lt;p&gt;The book &lt;em&gt;Mathematics and Science Education in Developing Countries&lt;/em&gt; (edited by M. Nagao, J. Rogan, and M. Magno and published by The University of the Philippines Press in 2007) has a few chapters on mathematics education in Africa.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chapter 7 is entitled &quot;Improving the Teaching of Science and Mathematics in Basic Schools in Ghana: Challenges and Lessons from Technical Cooperation Assistance&quot; (by Kwame Akyeampong and Norihiro Kuroda).  From p. 211:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Before reviewing JICA's STM approach and achievements, we shall present briefly some evidence about Ghanaian basic school pupils' performance in science and mathematics to highlight the extent of the problem.  Next, we shall present an overview of the curriculum for basic education of mathematics and science teachers in Ghana.  Our intention is to highlight the weaknesses in traditional pre-service (PRESET) teacher education programming and the need for strong linkages with in-service education and training (INSET) to change teachers' classroom practices.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Chapter 10 is entitled &quot;Localization Process of Mathematics Education in Post-independence Kenya&quot; (by Takuya Baba).  From pp. 276-277:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This chapter describes the beginning stage of an international project co-funded by Kenyan and Japanese governments---Strengthening Mathematics and Science at Secondary Education (SMASSE)---with a focus on problems encountered when attempting to introduce localization.  In Kenya, by the year 1998, there were a few scattered initiatives to train secondary teachers to upgrade the quality of teaching.  A major concern addressed by the project was that after twelve years of primary and secondary education, students did not understand the basic mathematics necessary to function effectively in society, and that the performance of mathematics in the national examination was low compared to other subjects.  The project's aim was to strengthen mathematics and science at the secondary level through the systematic in-service training (INSET) of teachers.  Although both mathematics and science are targeted in the project, in this chapter the analysis is limited to mathematics education.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(The book has other chapters that involve South Africa but these are about science education.)&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-03-31T12:15:30.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1105" PostTypeId="2" ParentId="1101" CreationDate="2014-03-31T12:31:41.440" Score="19" Body="&lt;p&gt;I've run into a few students like this. I usually try to convey a few messages.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It is great that you are so interested in foundations and there is absolutely a place in math for people with this perspective. Followed by a recommendation of books suited to their interests: At a variety of levels, I might include Spivak's &lt;i&gt;Calculus&lt;/i&gt;, an intro set theory book, various Bourbaki volumes or the &lt;a href=&quot;http://stacks.math.columbia.edu/&quot;&gt;Stacks Project&lt;/a&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Most mathematicians work at a higher level, and even mathematicians who work in foundations need to be aware of the work above them. You should spend some of your time reading work which takes a broader, less thorough view.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematical writing contains statements of theorems. You should be able to trust the theorems. It is perfectly all right to see a theorem in a reputable book and not read the proof, in order to spend time learning about how that result is used.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="51" LastActivityDate="2014-03-31T12:31:41.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1106" PostTypeId="5" CreationDate="2014-03-31T12:50:49.410" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T12:50:49.410" LastActivityDate="2014-03-31T12:50:49.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1107" PostTypeId="4" CreationDate="2014-03-31T12:50:49.410" Score="0" Body="For questions about psychological background of mathematical education." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T14:32:15.233" LastActivityDate="2014-03-31T14:32:15.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1108" PostTypeId="5" CreationDate="2014-03-31T12:51:26.747" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T12:51:26.747" LastActivityDate="2014-03-31T12:51:26.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1109" PostTypeId="4" CreationDate="2014-03-31T12:51:26.747" Score="0" Body="For questions how to design and create questions, exercises and examples." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T14:32:03.390" LastActivityDate="2014-03-31T14:32:03.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1110" PostTypeId="5" CreationDate="2014-03-31T12:56:22.337" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T12:56:22.337" LastActivityDate="2014-03-31T12:56:22.337" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1111" PostTypeId="4" CreationDate="2014-03-31T12:56:22.337" Score="0" Body="For questions related to eigenvalues and eigenvectors." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T14:32:18.587" LastActivityDate="2014-03-31T14:32:18.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1112" PostTypeId="1" AcceptedAnswerId="1133" CreationDate="2014-03-31T13:20:16.633" Score="9" ViewCount="420" Body="&lt;p&gt;From education theory, there are different concepts how to grade an exam: Either you can say that someone gets only by correct answers or you can say: No matter how good someone is, the best 10% get an A, the best 30% get a B, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the arguments in using one or the other concept? Would it be a good thing to use both concepts at once?&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2021-10-15T14:38:30.563" Title="Argument for and against giving two different grades at the end of a course" Tags="|undergraduate-education|grading|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1113" PostTypeId="1" AcceptedAnswerId="1120" CreationDate="2014-03-31T13:25:36.463" Score="29" ViewCount="5025" Body="&lt;p&gt;Several religious universities, such as Brigham Young University, ask all instructors in every area to try to make their courses spiritually uplifting. This is something included in student ratings.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When asked to by a department, how can one make a calculus or linear-algebra class spiritually uplifting? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems harder to do for math than for history or even economics or physics (I feel that physics, taught right, inspires students with awe for the universe).&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2019-01-16T14:44:01.007" Title="When asked to by a religious university, how can an instructor make a mathematics course spiritually uplifting?" Tags="|religious-institution|" AnswerCount="11" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="1114" PostTypeId="1" AcceptedAnswerId="1139" CreationDate="2014-03-31T13:36:08.377" Score="23" ViewCount="1214" Body="&lt;p&gt;It lectured a stochastics course for pre-service teachers last semester and had a teXed manuscript for myself which grew as the course was running. After some debate with the students, I made a version accessible online. In university courses, my students are free to attend the lessons or not. I think some thought then they would not have to come at all, some stopped copying from the blackboard, some struggled with my typos. That's the bad side of the coin. On the other side, students could check the lecture when they had not been there (some of them had another course overlapping), they now could think when I wrote and copying from the blackboard may generate new typos.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there any founded discussion (mabye research) on the effects of handing-out lecture notes?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know some lecturers do hand-out notes, some not, some hand-out notes with gaps to be filled in the lecturer. However, their reasons mainly lie in personal experience but not scientific debate.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-05-11T18:59:52.413" Title="Hand out lecture notes or not?" Tags="|undergraduate-education|course-design|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1115" PostTypeId="2" ParentId="1113" CreationDate="2014-03-31T13:40:27.387" Score="15" Body="&lt;p&gt;One can explore the spiritual motivations of a variety of mathematicians.  Leonhard Euler was the son of a Protestant pastor and was presumably to go into the ministry when Johann Bernoulli intervened.  Euler went on to write prolifically about math, science and Christianity (see &lt;a href=&quot;http://en.wikipedia.org/wiki/Letters_to_a_German_Princess&quot;&gt;Letters to a German Princess&lt;/a&gt; as an example.) Euler was an outspoken Christian. Newton was a deist and that religious view motivated some of his pursuit to understand the universe.  If one wants nonChristian examples, there is Ramanujan (apparently a devout Hindu) and Omar Khayyam (Muslim)…. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many mathematicians were motivated by a view of mathematics as transcendent, touching on the language of God, giving insight into a divine Creation.  I think it is fairly natural to touch on this motivation, especially in a religious setting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is customary in our rational age to downplay the religious motivations of people like Euler, Newton, Cantor, Ramanujan or Khayyam.  Some translations of &quot;Letters to a German Princess&quot; apparently leave out Euler's Christian views; Hardy downplayed Ramanujan's Hindu beliefs and tried to view Ramanujan as agnostic, but I think the truth was otherwise.&lt;/p&gt;&#xA;" OwnerUserId="313" LastActivityDate="2014-03-31T13:40:27.387" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1116" PostTypeId="2" ParentId="1113" CreationDate="2014-03-31T14:01:31.740" Score="5" Body="&lt;p&gt;I've never made spiritual uplift a goal of my teaching, but if asked, I would do it using history.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Discuss Newton's General &lt;a href=&quot;http://isaacnewton.ca/newtons-general-scholium/&quot; rel=&quot;noreferrer&quot;&gt;Scholium&lt;/a&gt; to the Principia, his greatest application of calculus:  &quot;This most beautiful System of the Sun, Planets, and Comets, could only proceed from the counsel and dominion of an intelligent and powerful being.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Discuss Berkeley's &lt;a href=&quot;http://en.wikipedia.org/wiki/The_Analyst&quot; rel=&quot;noreferrer&quot;&gt;Analyst&lt;/a&gt;, with its criticism of the unrigorous calculus and frequent comparisons to religion.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Discuss &lt;a href=&quot;http://en.wikipedia.org/wiki/Magic_square#History&quot; rel=&quot;noreferrer&quot;&gt;magic squares&lt;/a&gt; in a linear algebra class, perhaps reformulating the conditions in terms of matrices.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This has the advantage of connecting spirituality to particular topics in the math classes.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-03-31T14:20:02.600" LastActivityDate="2014-03-31T14:20:02.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1117" PostTypeId="2" ParentId="1028" CreationDate="2014-03-31T14:03:33.483" Score="6" Body="&lt;p&gt;My syllabus describes the prerequisites and usually elaborates on the importance of the prerequisites.  And then in classes like precalculus, calc 1 and calc 2, I stubbornly give a quiz the first day over some of the simpler prerequisite material.  Nothing enforces my prerequisite statements like a &lt;em&gt;grade&lt;/em&gt; over earlier material!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I usually send out an email a day or two before the first class &quot;reminding&quot; students about the prerequisite material and alerting them to this first quiz.  But regardless, I publicly insist that I gave a &quot;nice&quot; quiz over &quot;easy&quot; stuff and I move on.  (Privately, in my office, I help a few students &quot;remember&quot; the prerequisite material.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Getting on a soapbox -- I have colleagues who spend 2-3 weeks &quot;reviewing&quot; the prerequisite material.  In my opinion that merely reinforces the student belief that we don't really expect people to remember the previous course.  I don't do that -- I start fast in calc 2 with new material but then I occasionally stop for a moment (5-10 minutes of lecture) to &quot;recall&quot; some material &quot;we all know.&quot;  I find that these regular short reviews are helpful to the students but they don't give the impression that old material can be forgotten.&lt;/p&gt;&#xA;" OwnerUserId="313" LastActivityDate="2014-03-31T14:03:33.483" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1118" PostTypeId="2" ParentId="1113" CreationDate="2014-03-31T14:18:08.317" Score="10" Body="&lt;p&gt;For calculus, talk a lot about infinity. This is a topic which is tremendously theologically interesting- from Zeno's paradox (potential infinity) to Rabbi Hasdai Crescas (actual infinity) to Bolzano (a priest by profession, who proved in IVT and saw the necessity for completeness of the real numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact that there is more than one infinity is something which divinity students seem to love and get very excited by- and it ties right in with completeness of the reals.&#xA;So, in a word, if you make calculus rigourous and give a proper discussion of how to construct the real numbers, completeness of the reals, the philosophy behind epsilon-delta (a line as a collection of overlapping neighbourhoods rather than as a collection of points), the definition of a limit, and suchlike, stressing the philosophical insight behind this worldview, I think that the students will find it quite spiritually uplifting!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For linear algebra, I think talking about Hermann Grassmann and his ideas might be a direction to think about, but I haven't thought about it seriously. &lt;/p&gt;&#xA;" OwnerUserId="185" LastActivityDate="2014-03-31T14:18:08.317" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1119" PostTypeId="2" ParentId="1113" CreationDate="2014-03-31T14:19:33.323" Score="10" Body="&lt;p&gt;Mentioning other mathematician's motivations (as Ken W. Smith did in his answer) might be indeed a good starting point. I'd like to highlight one particular case where beliefs and mathematics come together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are the first lines of the introduction of M. Aigner, G. Ziegler: &quot;Proofs from THE BOOK&quot;, Springer, 1998 (3rd Ed. - the current one is the 4th, though).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Paus Erdős liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdős also said that you need not believe in God, but, as a mathematician, you should believe in The Book.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think that this anecdote helps a lot put the concept of beauty and elegance in mathematics into a larger context which touches spiritual topics: For religious students, you have one possible answer to the question &quot;Why is the proof done in this way?&quot; For non-religious students, this might offer an oppurtunity to understand spiritual beliefs better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a largely unrelated note: Ziegler and Aigner don't claim to have written THE BOOK - &lt;a href=&quot;http://www.dailymotion.com/video/x1uuvj_tenacious-d-tribute_music&quot;&gt;it's just a tribute&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-03-31T14:19:33.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1120" PostTypeId="2" ParentId="1113" CreationDate="2014-03-31T14:27:58.910" Score="16" Body="&lt;p&gt;For the Christian perspective, a general approach which is useful (these comments apply to any math course which is not merely &quot;plug-and-chug&quot;):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;mathematics is part of the general revelation of God&lt;/strong&gt;: in particular, it does provide all cultures a sense of awe. Granted, this probably assumes a platonic viewpoint. A simple way to implement this in the course is to pray each class at the start to appreciate the beauty of the mathematics which is to be covered that day. This attitude of thanksgiving for math could go a long way in the minds of students who are rebellious towards math, but open minded towards the work of God in their life.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;training your mind&lt;/strong&gt;: mathematics requires us to think in a way which challenges our assumptions. It helps us develop logical skills which allow us to avoid pitfalls which might otherwise trap a less practiced mind. Ideas of homomorphism verses isomorphism etc... give precise language to capture relationships and structures which in ordinary language take a lot more analogies to say a lot less. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;historical motivations&lt;/strong&gt;: many giants of math past were practicing Christians. However, it is not a required trait! Certainly, any culture which searches for truth can find God's truth. Or, if you refuse a platonic view, since we are created in the image of God we have the power to create and this is reflected in the practice of mathematicians across all cultures. Mostly, the point to make here is that Christianity is not specifically at odds with mathematics (here I mean Christianity in the abstract, certainly certain eras of human history there have been specific events where math was held back by various religious institutions)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-03-31T14:27:58.910" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1121" PostTypeId="1" AcceptedAnswerId="1581" CreationDate="2014-03-31T15:18:35.337" Score="14" ViewCount="587" Body="&lt;p&gt;I am teaching, among other things, a college algebra course this semester. In this course we do a lot of conceptual things but we also do some techniques to prepare students for calculus. One of the techniques is dealing with equations that involve squaring or rooting both sides.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Solve $(x - 3)^2 = 16$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When showing them how to solve this, we have the method: &lt;strong&gt;when you square root both sides, you get a plus or minus sign&lt;/strong&gt;. This is the thing we need them to know. I can't have them going into calculus finding only one solution to an equation like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conceptually, the reason this method is needed is that $\sqrt{x^2}$ is not x. Instead $\sqrt{x^2} = |x|$. However, this concept is significantly more difficult than the method, and you don't necessarily need the concept to complete the problems.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The advice I have received so far from other educators on this topic has been to &lt;strong&gt;sometimes skip the concept&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;My question: Is there any good framework for deciding when it is a good idea to &quot;skip the concept&quot; and &quot;just teach the method,&quot; knowing that students will appreciate the concept only much later in life, or possibly never?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't even tag this &quot;undergraduate-education&quot; because the question works for elementary schools too -- do you really show children &lt;em&gt;why&lt;/em&gt; long division works? And if you do, do you intend them to retain that concept? Why is this skipping-of-the-concept sometimes good and sometimes bad?&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="262" LastEditDate="2014-04-12T19:47:22.670" LastActivityDate="2014-04-22T17:45:12.577" Title="When is it a good idea to avoid talking about why something works?" Tags="|concept-motivation|curriculum|education-research|" AnswerCount="5" CommentCount="29" ContentLicense="CC BY-SA 3.0" />
  <row Id="1122" PostTypeId="2" ParentId="1113" CreationDate="2014-03-31T15:42:31.167" Score="4" Body="&lt;p&gt;In addition to the above, you might mention that mathematics is the science of material-independent phenomena.  On the simple end, three apples, cars, or orang-utans are all &quot;three&quot;, but more fundamentally, the same equations describe for example, turbulence, in water, air, or solar plasma.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More: Mathematics demonstrates emergent properties, where simple rules can give rise to incredibly complex structures.  Patterns such as the spirals and Fibonacci arrangements of a flower derive purely from the flowers' motions and patterns of growth -- the flower doesn't arrange them, but its buds grow according to simple rules, which produce the patterns.  More purely mathematical &quot;emergent phenomena&quot; include fractals such as the Mandelbrot set.&lt;/p&gt;&#xA;" OwnerUserId="560" LastActivityDate="2014-03-31T15:42:31.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1124" PostTypeId="2" ParentId="1082" CreationDate="2014-03-31T16:35:10.120" Score="11" Body="&lt;p&gt;As Benjamin says in the comments, your class is totally off-track, if not a disaster. For whatever reason, most of your students have found themselves unable or unwilling to learn from you, and have either dropped the class, transferred to another section or just given up. This is a very strong signal that there was something fundamentally wrong with the way the course was being run. My main recommendation, therefore, is that you use this situation to reflect seriously upon what went wrong, to consider how you might run a better class in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a number of ways that an instructor might drive an entire class away: an instructor could have totally unrealistic expectations; an instructor could be completely unapproachable; an instructor might be unfair or abusive; perhaps an instructor's remarks are unintelligible. I think the most common explanation for a situation where students aren't learning and the instructor doesn't realize it is that the instructor is teaching on too advanced a level, and doesn't get feedback about how little is being understood.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably none of those reasons applies to you. But since clearly something is wrong, you should try to think about what the problem was, so that you can fix the problem going forward.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as fixing the problem for this semester, perhaps some of your students have already dropped the class, and in this case you won't be getting them back. For the others, you should try to contact them---do you have a class email list to send an email inviting everyone back? (If not, then in the future make sure to get such a list.) It may be that some of them haven't realized that the exam was scheduled then, so you can remind the class of the exams. (In the future make sure the exam schedule is more clear.) With the few students you have left, make sure to teach in such a way that you get feedback during the class about whether they are understanding the material or not. Ask the students questions during the lecture (I do so dozens of times) concerning what you are lecturing on, since when they are unable to provide the answers, you can adjust the topic and pace. &lt;/p&gt;&#xA;" OwnerUserId="90" LastActivityDate="2014-03-31T16:35:10.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1125" PostTypeId="5" CreationDate="2014-03-31T17:55:23.580" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T17:55:23.580" LastActivityDate="2014-03-31T17:55:23.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1126" PostTypeId="4" CreationDate="2014-03-31T17:55:23.580" Score="0" Body="For questions about the mathematical education in the first years in school (ages approx. 5-10)." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T18:13:51.780" LastActivityDate="2014-03-31T18:13:51.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1127" PostTypeId="5" CreationDate="2014-03-31T17:57:12.707" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T17:57:12.707" LastActivityDate="2014-03-31T17:57:12.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1128" PostTypeId="4" CreationDate="2014-03-31T17:57:12.707" Score="0" Body="For questions related to the issue of concepts of definitions." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-01T01:36:16.277" LastActivityDate="2014-04-01T01:36:16.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1129" PostTypeId="5" CreationDate="2014-03-31T17:57:57.670" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T17:57:57.670" LastActivityDate="2014-03-31T17:57:57.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1130" PostTypeId="4" CreationDate="2014-03-31T17:57:57.670" Score="0" Body="For questions related to women in mathematics." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T18:14:05.513" LastActivityDate="2014-03-31T18:14:05.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1131" PostTypeId="5" CreationDate="2014-03-31T17:58:37.647" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T17:58:37.647" LastActivityDate="2014-03-31T17:58:37.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1132" PostTypeId="4" CreationDate="2014-03-31T17:58:37.647" Score="0" Body="For questions about seminars where students have to present talks." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-03-31T22:42:52.877" LastActivityDate="2014-03-31T22:42:52.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1133" PostTypeId="2" ParentId="1112" CreationDate="2014-03-31T18:14:29.363" Score="8" Body="&lt;p&gt;I recall an undergraduate thesis presentation on grade inflation at a particular liberal arts college in the United States (completed for the Department of Economics).&lt;/p&gt;&#xA;&lt;p&gt;The thesis can be found &lt;a href=&quot;https://www.amherst.edu/media/view/18883/original/Gottlieb.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;; it is quite readable, and includes the following relevant citations:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Johnson, V. E. (1997). &lt;a href=&quot;https://projecteuclid.org/journals/statistical-science/volume-12/issue-4/An-alternative-to-traditional-GPA-for-evaluating-student-performance/10.1214/ss/1030037959.full&quot; rel=&quot;nofollow noreferrer&quot;&gt;An alternative to traditional GPA for evaluating student performance&lt;/a&gt;. &lt;em&gt;Statistical Science, 12&lt;/em&gt;(4), 251-278. (DOI: 10.1214/ss/1030037959)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sabot, R., &amp;amp; Wakeman-Linn, J. (1991). &lt;a href=&quot;http://www.jstor.org/stable/1942708&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grade inflation and course choice&lt;/a&gt;. &lt;em&gt;The Journal of Economic Perspectives&lt;/em&gt;, 159-170.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A popular media piece in Slate entitled &lt;a href=&quot;http://www.slate.com/articles/life/do_the_math/2002/10/dont_worry_about_grade_inflation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Stop worrying about grade inflation&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This last piece remarks:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Indeed, a &lt;a href=&quot;http://nces.ed.gov/pubs2002/2002168.PDF&quot; rel=&quot;nofollow noreferrer&quot;&gt;2000 Department of Education study&lt;/a&gt; found that just 14.5 percent of undergraduates nationwide had a GPA above 3.75. And Henry Rosovsky and Matthew Hartley, in their well-reasoned &lt;a href=&quot;http://www.amacad.org/publications/monographs/Evaluation_and_the_Academy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;monograph&lt;/a&gt;, found &amp;quot;no large body of writings in which, for example, employers or graduate schools complain about lack of information because of inflated grades.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hopefully the six references here help give at least a partial answer. To close, I quote from the senior thesis above (&lt;span class=&quot;math-container&quot;&gt;$\S$&lt;/span&gt;1.3) in which the Johnson reference is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; above and the Ellenberg reference is &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;What are the consequences of grade inflation?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Absolute grade inflation is problematic because it narrows the range of available grades, which makes grades less precise and comparisons between students less fine-grained and more error-prone (though the importance of this&#xA;has been disputed; see [Ellenberg2002]). Relative grade inflation is even&#xA;more problematic, of course. [Johnson1997] summarizes the problems fairly&#xA;well: students will tend to sort into the classes and / or departments that&#xA;are graded more easily, professors face pressure to inflate their own classes’&#xA;grades to compete for students and receive high marks in evaluations by&#xA;students, and this pressure tends to flatten grades, which diminishes the rewards that can be given to the most excellent students [Johnson1997, 266].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="5220" LastEditDate="2021-10-15T14:38:30.563" LastActivityDate="2021-10-15T14:38:30.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1134" PostTypeId="1" CreationDate="2014-03-31T18:19:11.313" Score="13" ViewCount="377" Body="&lt;p&gt;Sometimes a teacher prefers to use several textbooks in his/her courses because he/she thinks the arguments of each book is better in a part of course material or there is no comprehensive textbook in the subject.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What are possible advantages/disadvantages of using several textbooks in a course? Based on the different approaches and preliminaries of different textbooks, could it be confusing for students? How should a teacher choose his/her set of textbooks to avoid this confusion?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is intended for both graduate and undergraduate levels. But the problem is more complicated in undergraduate case; sometimes it is a real problem for me.&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="143" LastEditDate="2018-04-02T10:02:46.257" LastActivityDate="2018-04-02T16:22:13.073" Title="Using Several Textbooks in a Course" Tags="|undergraduate-education|mathematical-pedagogy|textbooks|course-design|graduate-education|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1135" PostTypeId="2" ParentId="1084" CreationDate="2014-03-31T18:23:17.990" Score="7" Body="&lt;p&gt;This is my personal view of the problem. I think sending NGO's out to Africa is a very very bad idea. It will perpetuate Africa being the undeveloped mess it is and continual subservience to the western world (unless that's the plan of course). If people really want to help, the best first step is to let Africans try their own ideas first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, my background: I was a professional mathematician from Africa doing research in Algebra, now working is computational finance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my country, the problem is not so much with the education system. The biggest problem, like all problems in Africa, are the powers-that-be. Unfortunately aid to Africa only exacerbates the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The high school education syllabus was first rate, covering a far wider ranger of mathematics than the US syllabus. We had math clubs, magazines etc.. Every year the undergraduate system produced bright math students with potential. I believe my university produced 10 to 15 math honors students, the vast majority who went for higher degrees overseas. Based on my undergrad education I was able to ace US qualifying exams, that's how good the system was. One of the biggest advantages we had was that European and American professors used to visit our university during their summer breaks. I got to meet many great mathematicians that way. Stupidly, the university has now adopted the US calendar just to look &quot;developed&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The biggest problem is this: The people in power want to exploit the everybody, so after doing an undergrad in math the only work available was teaching in rural areas for next to nothing, unless you could get a way out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal belief is that EU and US aid is a very bad thing. It perpetuates the beggar mentality of Africa (unless that's the plan of course). All those scholarships given by the western world are distributed to the politicians' kids. The truly deserving are left with nothing. I saw this personally because I was one of those who tried to get a scholarship and failed for bizarre reasons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had really wanted to go back when I finished my PhD. The system there made it not worthwhile. I would become and overworked underpaid professor. so I chose to remain in the US.&lt;/p&gt;&#xA;" OwnerUserId="555" LastActivityDate="2014-03-31T18:23:17.990" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1137" PostTypeId="1" AcceptedAnswerId="1144" CreationDate="2014-03-31T18:41:02.833" Score="13" ViewCount="1341" Body="&lt;p&gt;In a freshman course in mathematics for biology students, I have had the issue that basic algebra (e.g. simplifying $\frac{a/b}{c/d}$) is far from being mastered. On the one hand, it seems ill-suited to teach more advanced matters to student failing on the basics. On the other hand, if we take too much time to cover the material allegedly covered in high school, we can hardly hope to cover what is needed to these student, and we enforce the impression that not mastering high-school mathematics is ok.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How much effort should we spend in class for studying material that is&#xA;  supposed to be mastered but in practice is not?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-01T05:57:56.067" Title="How much time should we spend on material that should be already known by students?" Tags="|undergraduate-education|" AnswerCount="6" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1138" PostTypeId="2" ParentId="581" CreationDate="2014-03-31T18:55:36.343" Score="3" Body="&lt;p&gt;Presumably at least the exams are graded in a standardized way such as &quot;each TA grades a problem.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means you can &lt;strong&gt;use exam grades to calibrate the students' homework grades&lt;/strong&gt;. Split the students up by TA, and find the mean and standard deviation of the exam grades, and the mean and standard deviation of the homework grades for just that TA's students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Check for significant differences in rank (i.e. TA X has the top homework grades but the second-worst exam grades), and if there is a massive difference, add homework points to certain sections. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This has clear downsides, but it has a place: it is only slightly more work than &quot;standardize all sections so that the homework average is 80%,&quot; but it is substantially more fair. Of course many of the other answers here are significantly more fair, but they are also significantly more work!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-03-31T18:55:36.343" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1139" PostTypeId="2" ParentId="1114" CreationDate="2014-03-31T18:57:42.087" Score="14" Body="&lt;p&gt;&lt;a href=&quot;http://files.eric.ed.gov/fulltext/ED544239.pdf&quot; rel=&quot;nofollow&quot;&gt;Writing&lt;/a&gt; is an essential part for most of us when learning mathematics. I think it is essential to get the students to learn how to take notes and, in an ideal world, after each class to write out in every detail the content of that lecture. We should do everything to encourage students to take notes and write more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1348/026151007X185329/abstract;jsessionid=D9A1C3389FD557FCBF2D831C6A9A79A6.f01t02?deniedAccessCustomisedMessage=&amp;amp;userIsAuthenticated=false&quot; rel=&quot;nofollow&quot;&gt;Some&lt;/a&gt; even think that handwriting is essential and we should encourage it in contrast to taking notes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of this, if possible, I try not to distribute my notes or my slides. This may also be a serious problem for those who cannot attend the lecture. I feel bad about this. I try, however, to make a balance between taking notes and discussing, developing the material together. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I worked out some parts of the material and I feel it is better than the textbook, I usually put that part when we finished the lecture on my homepage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;ADDED:&lt;/strong&gt; I just &lt;a href=&quot;http://pss.sagepub.com/content/early/2014/04/22/0956797614524581.abstract&quot; rel=&quot;nofollow&quot;&gt;came across&lt;/a&gt; this article which also discusses the advantages of handwritten note-taking (though in a different context).&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="61" LastEditDate="2014-05-11T18:59:52.413" LastActivityDate="2014-05-11T18:59:52.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1140" PostTypeId="2" ParentId="1137" CreationDate="2014-03-31T19:03:13.903" Score="6" Body="&lt;p&gt;I think if you notice this problem, you should briefly revise it, go on with the material, stress that it is essential that they know the basic stuff, and hand out extra exercises on the basic material which will not be graded but which you are ready to discuss in your office hour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This way you offer a chance to learn the basics but stress that your job is to teach something more advanced. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience, however, is mixed. They acknowledge my goodwill to help and even recognize that they have serious problems, but it is very very rare that someone comes to my office. They are not mathematics students, they have more important things to do...&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-03-31T19:03:13.903" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1141" PostTypeId="5" CreationDate="2014-03-31T19:13:53.043" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-03-31T19:13:53.043" LastActivityDate="2014-03-31T19:13:53.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1142" PostTypeId="4" CreationDate="2014-03-31T19:13:53.043" Score="0" Body="For issues related to teaching courses with a large audience, say, 100+ students. " OwnerUserId="143" LastEditorUserId="39" LastEditDate="2015-08-31T00:47:10.437" LastActivityDate="2015-08-31T00:47:10.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1143" PostTypeId="2" ParentId="1137" CreationDate="2014-03-31T19:36:24.357" Score="4" Body="&lt;p&gt;Some strategies could be useful for improving students' basic information without wasting time in the main process of the course.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Use a TA and ask him/her to work on students' basic information during some sessions.&lt;/li&gt;&#xA;&lt;li&gt;Surely there are some students in your class who know the basics better than others. Divide the students to some groups such that each group includes an informed student as a head. Ask heads to help their friends on basic subjects in some sessions. Of course you should motivate these heads by some extra grades or something like this.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-03-31T20:26:19.030" LastActivityDate="2014-03-31T20:26:19.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1144" PostTypeId="2" ParentId="1137" CreationDate="2014-03-31T20:02:00.990" Score="13" Body="&lt;p&gt;This is a situation where using an online homework system is particularly useful; usually there will be an early section of review material from which you can assign problems. I would advise issuing such an assignment as early as possible in the term (preferably on the first day), making sure the students realize it is for a grade and that as it covers prerequisite material for the course, no in class time (or very little) will be spent on these problems. Note that these homework systems will also give students examples and detailed instruction on how to do the assigned problems, so the students should be able to (re)learn the necessary material on their own. &lt;/p&gt;&#xA;" OwnerUserId="568" LastActivityDate="2014-03-31T20:02:00.990" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1145" PostTypeId="2" ParentId="1121" CreationDate="2014-03-31T20:30:35.413" Score="6" Body="&lt;p&gt;Here are my responses for the examples you and Jim Belk gave:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For $(x-3)^2=16$, I would &lt;strong&gt;explain&lt;/strong&gt; by graphing $y=(x-3)^2$ and discussing it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For long division, I would &lt;strong&gt;not explain&lt;/strong&gt;, but use that time for calculator-based examples.  (If Justice Scalia has been on the Supreme Court for 10,000 days, how many years is that?  If a pest advances by 100 ft every day, how many miles will it advance in a year?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For determinants, I would &lt;strong&gt;promise an explanation much later&lt;/strong&gt;:  &quot;if you want to know why $\det(AB)=\det(A)\det(B)$, you can look up exterior algebras, or go to math grad school and get a good explanation there.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can probably reason other cases by analogy with these examples.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorUserId="114" LastEditDate="2014-03-31T20:39:25.427" LastActivityDate="2014-03-31T20:39:25.427" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1146" PostTypeId="2" ParentId="353" CreationDate="2014-03-31T20:42:31.587" Score="6" Body="&lt;p&gt;Let me add the tricks I used already to the interesting ones proposed in the other answers. They are partly, but not completely redundant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, implication is often best understood in quantified propositions; &lt;a href=&quot;https://matheducators.stackexchange.com/a/368&quot;&gt;this has already been pointed out by Matt F.&lt;/a&gt;, but I am not too found of allegedly real-life assertions, which usually sound rather artificial. I rather use statements like: &quot;every integer $n$ that is divisible by 6 is also divisible by 3&quot; or &quot;for any integer $n$, if $n^2$ is even then so is $n$&quot;. One of the best of this kind is &quot;for any integer $n$, if $n$ is a multiple of 4 then $n^2$ also is a multiple of 4&quot;: you have all three cases where implication is true by taking $n=2,3,4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second angle is to give an example where one wants to state and use an implication without knowing the truth value of its terms. Here ironically, an allegedly real-life assertion does the job quite beautifully: assume you are investigating a murder that took place in London Saturday and that Colonel Mustard is a suspect. You can say confidently that &quot;if Colonel Mustard is guilty, then she was in London Saturday&quot; even if you do not know whether she is guilty nor whether she was in London Saturday. What is especially good in this example is that it is a good way to introduce contrapositives: the statement &quot;if Colonel Mustard was not in London Saturday then she is not guilty&quot; shows convincingly that the contrapositive has the same truth value than the original implication. I learned this example from Viviane Durand-Guerrier, a researcher in educational studies in mathematics in Montpellier. &lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-03-29T21:14:09.923" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1147" PostTypeId="1" AcceptedAnswerId="1151" CreationDate="2014-03-31T20:55:36.297" Score="11" ViewCount="265" Body="&lt;p&gt;I think, the most standard way is to introduce measures in real analysis is to &lt;a href=&quot;http://en.wikipedia.org/wiki/Measure_%28mathematics%29#Definition&quot;&gt;define them via the usual properties like $\sigma$-additivity&lt;/a&gt;, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if the students are familiar with functional analysis at that point, one can introduce (Borel) measures via the &lt;a href=&quot;http://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem&quot;&gt;Riesz representation theorem&lt;/a&gt; as the dual space of the set of continuous functions, $\mathcal{C}(\Omega)$ (i.e., the theorem becomes a definition and one has to prove the standard properties of a measure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is there a textbook dealing measure theory like this?&lt;/li&gt;&#xA;&lt;li&gt;What are - beside having Riesz representation theorem - the (dis)advantages in doing so?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-04-02T19:49:42.470" Title="Introducing (Borel) measures via Riesz representation theorem" Tags="|graduate-education|mathematical-analysis|functional-analysis|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1148" PostTypeId="2" ParentId="1101" CreationDate="2014-03-31T21:07:19.867" Score="11" Body="&lt;p&gt;I was a bit like that in my first three years if study, then I read the first volume of Schwartz' Analysis, where he introduces the ZFC axioms. This is the point when I understood that &lt;em&gt;one simply cannot make sense out of set theoretical axioms before having manipulated higher-level mathematics&lt;/em&gt; (here &quot;higher-level&quot; is to be understood in the sense of higher-level programming). The analogy with programming is quite good I think: who would first learn machine language, before learning higher-level languages? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Advanced mathematics consist in both building more complicated structure, and digging the foundations of what has already been built. Students in mathematics should realize that what they feel is well founded in their background knowledge in fact rests on a lot of maths they did not yet learned. Even the construction of the natural numbers is not as trivial as they might think, and in fact they have been used to reason from (high-level) axioms they have accepted. Accepting a Theorem because in some given course its proof is not central as its usage is not that different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the way one founds mathematics is not that important to higher-level maths. When we do mathematics, we all think in a strongly typed way while set theory is almost not typed at all (with the exception of classes vs sets). As a side note, I am very intrigued by the homotopy type theory as a strongly typed mathematical foundation, but have not have had enough time to dive in it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As another side note, in the old debate as to which integration theory use in the first undergraduate years, my favorite answer is &lt;em&gt;none at all&lt;/em&gt;: state the axioms of an integration theory we need for the course, and wait until Lebesgue integral is introduced. This is a healthy way to remind that we do rely on unproved thing almost all the time; better be warned about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point, I did not really answered the question. Why not let them read Bourbaki? Either they will end up hating it pretty fast and accept more easily a few result and hand-waving, or they will love it and will learn a lot from it.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2014-04-01T09:16:34.013" LastActivityDate="2014-04-01T09:16:34.013" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="1149" PostTypeId="1" AcceptedAnswerId="1178" CreationDate="2014-03-31T21:20:27.243" Score="23" ViewCount="2446" Body="&lt;p&gt;Many colleges offer an an elementary linear algebra class for sophomore math, science, and economics majors.  Such a class typically covers a chapter on determinants, including the following aspects:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Cofactor expansion using any row or column.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How the determinant changes when you perform row operations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The fact that a determinant can be viewed as the volume of a parallelepiped.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The fact that $\det(AB) = \det(A) \det(B)$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cramer's rule.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The formula for the inverse of a matrix using the classical adjoint.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Usually this material needs to be covered relatively quickly, e.g. in a week or so of class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there any way of covering determinants quickly that fully explains all of this material?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In my experience, it's usually possible to give explanations for &lt;em&gt;some&lt;/em&gt; of this material, but there's no one way of looking at determinants that provides a simple explanation for all of it.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2024-01-11T12:02:30.640" Title="Is there a good way to explain determinants in an elementary linear algebra class?" Tags="|mathematical-pedagogy|undergraduate-education|linear-algebra|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1150" PostTypeId="2" ParentId="1149" CreationDate="2014-03-31T21:43:30.940" Score="5" Body="&lt;p&gt;1) the definition of the determinant as a volume function easily gives rise to &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) the formula for the determinant using permutation, which easily gives rise to &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) cofactor expansion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 and/or 3 give rise to&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) how the determinant changes with elementary row operations, which gives rise to&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) a proof that the determinant is multiplicative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6) both Cramer's rule and the classical adjoint formula are easily derived directly by computing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have two lecture hours a week then covering all of this may be a little tight, but doable. Certainly if you delegate some of this to an exercise (e.g., Cramer's rule and the adjoint formula, or the proof of the multiplicative property (with some guidance)). &lt;/p&gt;&#xA;" OwnerUserId="59" LastEditorUserId="59" LastEditDate="2014-04-01T07:44:51.583" LastActivityDate="2014-04-01T07:44:51.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1151" PostTypeId="2" ParentId="1147" CreationDate="2014-03-31T21:54:06.250" Score="6" Body="&lt;p&gt;Look for &quot;Radon integrals&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure about any advantages, but the book by &lt;a href=&quot;http://link.springer.com/book/10.1007/978-1-4612-1007-8&quot;&gt;Gert Pedersen: Analysis now&lt;/a&gt; contains a chapter (Chapter 6) on integration theory. This is I think in the style you want: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This chapter has two functions: Throughout the book it has served as&#xA;  an Appendix, to which the reader was referred for definitions,&#xA;  arguments, and results about measures and integrals. It will now serve&#xA;  as a functional analyst's dream of the ideal short course in measure&#xA;  theory. Thus, we shall develop the theory of Radon integrals on a&#xA;  locally compact Hausdorff space, assuming full  knowledge of topology&#xA;  and topological vector spaces.  This theory takes as point of&#xA;  departure an integral (a positive linear functional) on  the minimal&#xA;  class of topologically relevant functions on  $X$, namely, the class&#xA;  $C_c(X)$  of continuous functions with compact supports.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The author himself is however sure that you should not start measure theory this way:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In all honesty the author will admit that the reader should have had&#xA;  an ordinary course (however dull) in measure and integration theory in&#xA;  order to appreciate fully the high-tech approach here. He should also&#xA;  be aware that the theory is richer than the spartan exposition might&#xA;  lead to believe. A study of one or more of the classical areas of&#xA;  application, harmonic analysis, prob­ ability, potential theory, and&#xA;  ergodic theory, is advisable, in order to under­ stand the&#xA;  significance of integration theory as a cornerstone in that dread&#xA;  Temple of Our Worth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-03-31T21:54:06.250" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1152" PostTypeId="2" ParentId="1137" CreationDate="2014-03-31T22:40:19.687" Score="6" Body="&lt;p&gt;I've previously written online algebra tutorials for 2nd year bio students, with online quizzes at the end. It seemed to help. The advantage is you can base the learning materials on the course content to some extent. Also you can let students know early that they may have a problem before its too late to pass or withdraw. Set it as an early mandatory thing to complete as an assessment of prerequisities.&lt;/p&gt;&#xA;" OwnerUserId="570" LastActivityDate="2014-03-31T22:40:19.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1153" PostTypeId="2" ParentId="1134" CreationDate="2014-03-31T23:05:53.107" Score="11" Body="&lt;p&gt;Based on extensive (if anecdotal) experience, undergrads really cannot cope with more than a single reference, which must be traversed in order, possibly omitting some sections. Having any other source is somehow beyond imagination. Subject=course=textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grad students can do somewhat better, but are not happy about having to operate at a higher level. Perhaps understandably. Obviously habits from undergrad have inertia. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most bizarre negative side effect of not &quot;just following a standard textbook&quot; is the occasional suspicion from the students that the instructor doesn't know what they're doing (no matter that they might be a respected expert among their peers, etc). I recall being taken off guard when an undergrad couldn't believe that, in fact, I &lt;em&gt;was&lt;/em&gt; the author of the text being used for the course. The idea of author as authority looms very, very large in peoples' minds. (Some etymological connection there, too.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, at lower levels, using several sources (in a math class as opposed to literature, etc) seems to so violate expectations that the instructor's credibility is lost in the minds of a substantial fraction of the student population. Somewhat less so with grad students, but still non-trivial non-sequitur problems. E.g., especially before I had the advantage of a gray beard and bald head, divergence from the students' preferred reference was viewed as lack of preparation, or negligence, on my part, even if/when I explained to them that we can do better than the usual sources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the graduate level, especially beyond first-year things, I've tried to change the perspective... by writing my own notes (yes, a bit of work), with historical references, explaining chronology, and so on, the idea being to get students thinking in terms of the &lt;em&gt;subject&lt;/em&gt; rather than secondary or tertiary sources. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unclear to me how successful this is, but I feel professionally obligated to give a push in the right direction. &lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-31T23:05:53.107" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1154" PostTypeId="2" ParentId="1149" CreationDate="2014-03-31T23:25:04.613" Score="5" Body="&lt;p&gt;I'd say it's just plain tough to get through all this quickly. For an elementary class I typically dump out most proofs of why various properties hold and focus more on how the ideas hang together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I advocate starting with $1 \times 1$, $2 \times 2$, and $3 \times 3$ (just expanding along the top row for the last one). Then show off the &lt;a href=&quot;http://www.coolmath.com/algebra/14-determinants-cramers-rule/02-determinants-3x3-method-1-02.htm&quot;&gt;$3 \times 3$ determinant trick&lt;/a&gt; (down diagonals minus up diagonals). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then discuss how $2 \times 2$ determinants give signed areas of parallelograms (the sign encoding orientation: clockwise vs. counter-clockwise). $3 \times 3$ giving signed volumes of parallelepipeds (the sign encoding orientation: right-hand vs. left-hand rules). Then mention $n \times n$ give $\pm$ $n$-dimensional volume of the corresponding parallelotope. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, general row/col expansions (picking out examples with lots of zeros to make it worth while). Briefly mention how one can use permutations to define the determinant. &lt;em&gt;Maybe&lt;/em&gt; show off how this works for the $2 \times 2$ case. In a class where proofs are appropriate, maybe indicate how decomposing the symmetric group allows one to get these various expansions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, without doing precise proofs, I mention multi-linearity and skew-symmetry/alternating properties. Then show how elementary operations effect the determinant (flowing from the properties). Once this is done I like to sketch out how one can compute determinants by doing a forward pass of Gauss-Jordan elimination (do type I &amp;amp; III operations - no scales - until upper-triangular, multiply diagonal, adjust the sign according to the number of swaps). Mention how much faster this is when we have a sizable matrix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Elementary operations yield an easy proof that the determinant is multiplicative. Make sure to mention why $\det(cA) = c^n\det(A)$ not $c\det(A)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I then like to quickly sketch Cramer's rule and how the classical adjoint inverse formula works. Emphasizing how the only division performed when finding an inverse is division by the determinant. Mention how an integer matrix with determinant $X \not=0$ has an inverse involving fractions with denominator $1/X$. [Nice professors give matrices whose determinants are $\pm 1,\pm 2$. :)] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mostly as I go through this I skip any detailed proof that involves matrix components. If I feel such a proof is warranted, I'll just do the proof for $2\times 2$ or $3 \times 3$ matrices. I think it best serves the students to emphasize the connections between various properties and not the nuts-and-bolts nasty symbol shuffling proofs to get those properties - at least for an &lt;em&gt;elementary&lt;/em&gt; class. &lt;/p&gt;&#xA;" OwnerUserId="453" LastActivityDate="2014-03-31T23:25:04.613" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1155" PostTypeId="2" ParentId="1137" CreationDate="2014-03-31T23:32:42.717" Score="9" Body="&lt;p&gt;In addition to other useful answers... this sort of problem will recur forever, since students (even with good attitudes) chronically misjudge the importance of background courses, and chronically misjudge the degree of facility and competence really desirable for later work. E.g., sometimes students see a passing grade (C+?) as showing they're ready to move forward... while, in reality, that's terrible from the viewpoint of competence and reliability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there'll be no room in your course to rant about the failings of the ambient cultural attitudes, blah-blah-blah. Nor will many students be willing to agree with the premise that they in fact do not know things they thought they know... with &quot;know&quot; as certified by a grade in a course in the past.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, for me both in undergrad courses at all levels, and in grad courses, I &quot;recall&quot; things ... sometimes in gruesome detail ... pretending to be apologetic for boring people who know it all too well... to at least passively drill everyone in the absolutely critical riffs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, I am implicitly doubting that formal reviews are as effective as one would want, exactly because of the inaccurate self-perceptions of the students. It is quicker, and not toooo burdensome to &quot;in-line&quot; the review/recapitulation of things that... yes, we'd like to be able to assume the kids know, but they just don't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The meta-comment is that this is a huge issue, and we can't expect any sort of &quot;solution&quot;, but only a strategy to fight back on an on-going basis...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-03-31T23:32:42.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1156" PostTypeId="2" ParentId="1147" CreationDate="2014-03-31T23:38:30.023" Score="4" Body="&lt;p&gt;Students should learn &lt;em&gt;many&lt;/em&gt; versions of the development of &quot;integral&quot;.  Simultaneously, we should ask &quot;why bother?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A Lebesgue &quot;formal&quot; theory of integration, or the &lt;em&gt;possibility&lt;/em&gt; of such a development (even on spaces without a topology) is interesting.  However, mostly we don't care about such situations; the Riesz representation theorem explains most common scenarios well enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We want and need a notion of integral that behaves well with respect to limit processes, such as $\lim_n \int f_n = \int \lim_n f_n$, under suitable hypotheses. This will allow us to integrate limits of functions, and so on...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This is yet another bit of propaganda against definitions that do not admit ulterior goals.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorDisplayName="user173" LastEditDate="2014-04-01T03:27:47.247" LastActivityDate="2014-04-01T03:27:47.247" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1157" PostTypeId="2" ParentId="1114" CreationDate="2014-03-31T23:45:50.670" Score="15" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/1139&quot;&gt;@AndrasBatkai's answer&lt;/a&gt; surprised me a little, and I don't disagree with it. But/and contemplating the plausible factoids of his remarks makes me a little unhappy, perhaps only &quot;unhappy with reality&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, certainly, after he pointed this out, I can agree that making PDFs (or whatever format) available ahead of time, or promised, does lead a significant fraction of the student population to disengage. Not to take advantage of the &lt;em&gt;further&lt;/em&gt; possibilities, such as making notes on printed-out notes, or doing the tablet version of this, but, ... just disengaging, or not going to class at all, and not really looking the notes, either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this reminds me of a different wording for the question... in effect, to what extent are we (educators, mentors) responsible for preventing self-sabotage by students?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real problem arises when a practice that hugely benefits 75% of the population has a bad resonance with the other 25%. Then what?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, making my notes available is mostly beneficial, but seems to fatally confuse a small but non-trivial fraction of the population. Dang. But is this the real &quot;final exam&quot; for them? Or is it just a misunderstanding? At this point, I still don't really know, although I do know that predisposition to fall into this trap is not a survival trait.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-04-01T06:21:51.967" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="1158" PostTypeId="2" ParentId="1100" CreationDate="2014-04-01T00:12:02.123" Score="6" Body="&lt;p&gt;What are possible (dis)advantages of the ideological teaching in logic courses? First I'd say that ideological teaching is not restricted to logic. Granted, it is more likely to encounter ideology in a logic class, but not only there. I've had a lecturer in differential equations who told us that all of mathematics is basically concerned with solving differential equations (this exaggeration did not serve to make the course more interesting than it was (and it was, perhaps not to all students, but it was)). I've heart algebra enthusiasts bad-mouth the concreteness of analysis and analysts bad-mouth the abstractness of algebra. I guess it is only natural, but should be avoided. I think it is obvious that some of the (dis)advantages of ideological teaching is that it may alienate some students and draw others to the subject matter. The thing is you don't know what will happen to which student. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could it be harmful/useful for students? Yes, it can be both harmful and useful. If your ideology closely fits that of the mainstream and you thus push a good student to pursue working on a problem many people appreciate, then you (probably) increase their potential for landing a job. On the other hand, if you discourage a good student from following a seemingly tangential area of mathematics that ends up being really important then it may do harm. As said above, it is very difficult to know in advance. But, in my opinion, ideology should be presented as such. I may like or dislike the axiom of choice, but what I teach should reflect (largely) the mainstream. I may provide my own opinion of course, but present it as such. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does it have a notable impact on students' research interests in future? It certainly might (as the cases you describe would indicate). In my experience, some students are highly influenced by peers and experts while others are less so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any math education long term research to determine the similarity of students and teachers' point of views in logic and philosophy of mathematics? I have no idea. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-04-01T00:12:02.123" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1159" PostTypeId="2" ParentId="1114" CreationDate="2014-04-01T01:30:04.480" Score="6" Body="&lt;p&gt;I liked one class in which we were asked to read a chapter of the text, and do a set of exercises as preparation for class. Classes were questions about what had been read. I learned a lot more than if I had copied some stuff from a blackboard. Now, that is certainly &lt;em&gt;much&lt;/em&gt; harder on the teacher (they have to field all and any questions), and I certainly worked much more for that class than for &quot;regular&quot; ones. But most dropped out very soon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And even though I make notes available, most students come to class to copy stuff, or just don't show up. They seem to believe that &quot;cram from the notes two days before the exam&quot; is enough. It isn't.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-01T01:30:04.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1160" PostTypeId="2" ParentId="1114" CreationDate="2014-04-01T01:37:38.930" Score="17" Body="&lt;p&gt;I hand out notes in most of my courses. These notes parallel my lectures, sometimes precisely, sometimes just roughly. Some of them are typed, some of them are pdf-scans of my handwritten work. I usually post a day-by-day lecture schedule detailing where I'll be in the notes as best as I can forecast. I suppose, in principle, there is a possibility that students think:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;I can skip class because:&lt;/strong&gt; he posted notes and I already have the inside scoop beyond the text.. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;I can just write nothing in class because:&lt;/strong&gt; I already have these notes and printing them out makes the material stick in my mind just as much as actively writing them...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I suppose both of these are possible, but, I actively discourage both practices at the outset by:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;including more than is required in my lecture notes:&lt;/strong&gt; if they come to class they get a better sense of which parts of the notes are beyond the required core of material.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;adding examples not in my posted notes:&lt;/strong&gt; this works best in the lower level classes where it is safe to make up examples, obviously this requires additional work in higher courses. In any event, taking notes in my course adds to my posted notes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Both items above go to the point that I don't just read my notes. Rather, my notes are an indication of the totality of my thoughts on a topic, sometimes, there is just too much for lecture. My notes go into second or third order caveats whereas the lecture is more towards emphasizing the first order material. As a somewhat extreme example, I prove the n-th order variational parameters formulae via some tensor arithmetic. That is something I would not want to lecture in differential equations as it has little to do with my expectations for them in the course. On the other hand, if I can peak the interest of just one of them to the joys of tensor calculus then it's a win in my book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Warning:&lt;/strong&gt; they will complain. Some students are simply annoyed if you do anything but teach them how to do the calculations you will put on the test. I frankly don't care. I refuse to cater to this student as they endanger the whole concept of university. On the other hand, the students who are genuine in their curiosity typically appreciate the effort of the notes. It is for those students I post them. On the whole I still have students attend and take notes so in my opinion the hypothetical that it shuts down both activities is not observable in my situation.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-04-01T01:37:38.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1161" PostTypeId="1" AcceptedAnswerId="1163" CreationDate="2014-04-01T02:15:19.313" Score="18" ViewCount="905" Body="&lt;p&gt;Having seen many people go through abstract algebra courses, I've noticed that one factor for success is the ability to pull out examples to test ideas on (e.g. if two subgroups have the same index, are they the same? No, the Klein four-group is a quick counterexample). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, my example may seem trivial, but that's the point; many students who struggle with abstract algebra don't see such things as trivial, and try to prove or disprove using symbolic manipulations instead of quickly going through a list of examples.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are the core examples of groups that students should think of when testing ideas?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Please indicate the groups and why you think they are important.&lt;em&gt;emphasized text&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="114" LastEditDate="2014-04-01T06:41:18.583" LastActivityDate="2014-04-03T09:30:28.973" Title="What examples of groups should students in abstract algebra learn to test ideas on?" Tags="|examples|abstract-algebra|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1162" PostTypeId="2" ParentId="1113" CreationDate="2014-04-01T02:30:20.173" Score="-4" Body="&lt;p&gt;I like this one,for promoting a sense of awe and wonder..and inquiry: there is a 'gap' in the graph of the function - obviously at $n=1$,  of $y = \big(\frac{n}{n-1}\big)^n$... since it is a gap, it must be able to be 'bridged' you would think, but since it also is a gap of zero dimensions it can't be: it both can and can't 'exist' simultaneously.. it's 'there'.. but when we try to measure it in physical terms.. it's not!&lt;/p&gt;&#xA;" OwnerUserId="573" LastEditorUserId="11" LastEditDate="2014-04-01T03:01:34.193" LastActivityDate="2014-04-01T03:01:34.193" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1163" PostTypeId="2" ParentId="1161" CreationDate="2014-04-01T02:31:43.677" Score="20" Body="&lt;p&gt;$\mathbb Z_n$ since it is very easy to compute in and you have one of order $n$ for every $n\ge 1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$S_3$, since it's the smallest non-abelian group. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$S_4$, since $S_3$ is sometimes too small. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A_4$, Since it is the smallest group that does not contain a subgroup of every possible order dividing its order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathbb Z_2 \times \mathbb Z_2$ since it's the smallest non-cyclic abelian group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathbb Z$ since it's infinite. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathbb R$ since it's infinite but not cyclic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathbb Q$ since it's infinite and countable, but still not cyclic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$S_n$ due to their immense importance and their complexity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A_n$ since they have index two in $S_n$ (for $n&amp;gt;1$). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$D_n$ for their geometric content. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure I'm missing some more crucial groups...&lt;/p&gt;&#xA;" OwnerUserId="59" LastEditorUserId="59" LastEditDate="2014-04-01T21:02:13.783" LastActivityDate="2014-04-01T21:02:13.783" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1164" PostTypeId="1" CreationDate="2014-04-01T02:31:43.533" Score="14" ViewCount="1353" Body="&lt;p&gt;(This question was proposed during the &lt;a href=&quot;http://area51.stackexchange.com/proposals/64216/what-are-some-investigative-projects-i-can-assign-to-calculus-students-from-bio-c/64356#64356&quot;&gt;area51 phase&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's common for chemistry/biology/physics majors to be required to take certain calculus courses. At my school, chem/bio students must take up through Calc II (integration, separable diff eqs, infinite sequences/series) and physics students up through Calc III (vector calc). We don't have the resources necessarily to develop a separate &quot;Calc for science students&quot; course. Yet, this semester, I find myself with a section of 25 students entirely from other science majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to appeal to the interests of these students and, hopefully, demonstrate that calculus can be both useful and interesting for them. Alas, many textbook applications are somewhat contrived, or don't represent the way the students will have to handle data/problem-solving in their actual scientific work (i.e. the numbers are too &quot;round&quot;, the modeling function is already given with no motivation as to its applicability/accuracy, etc.), while other application problems are just too difficult to assign, given the range of abilities in the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Do you have any in-class activities or take-home assignments that demonstrate a calculus concept/technique in the context of a scientific problem and have effectively appealed to the interests of science majors in your course?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am specifically curious about activities/assignments you have &lt;em&gt;used before&lt;/em&gt;, and would like a description of how you think it was effective (and what could be improved, if need be), and to whom it appealed. I don't want this thread to be full of &quot;Try something with topic X, that might work.&quot;&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2018-10-24T15:29:27.230" Title="What are some activities/projects I can assign to calculus students from bio/chem/physics majors to specifically motivate their interest?" Tags="|undergraduate-education|calculus|student-motivation|class-participation|physical-sciences|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1165" PostTypeId="5" CreationDate="2014-04-01T02:55:46.017" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-01T02:55:46.017" LastActivityDate="2014-04-01T02:55:46.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1166" PostTypeId="4" CreationDate="2014-04-01T02:55:46.017" Score="0" Body="For questions about teaching students at the undergraduate (university) level." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T06:12:00.493" LastActivityDate="2014-04-01T06:12:00.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1167" PostTypeId="5" CreationDate="2014-04-01T02:56:34.977" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-01T02:56:34.977" LastActivityDate="2014-04-01T02:56:34.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1168" PostTypeId="4" CreationDate="2014-04-01T02:56:34.977" Score="0" Body="For questions about overseeing, directing, motivating, and effectively controlling student behavior while in the classroom." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T06:11:45.167" LastActivityDate="2014-04-01T06:11:45.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1169" PostTypeId="5" CreationDate="2014-04-01T02:57:22.360" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-01T02:57:22.360" LastActivityDate="2014-04-01T02:57:22.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1170" PostTypeId="4" CreationDate="2014-04-01T02:57:22.360" Score="0" Body="For questions about involving undergraduate (university) students in creative mathematical research." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T06:11:51.320" LastActivityDate="2014-04-01T06:11:51.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1171" PostTypeId="5" CreationDate="2014-04-01T02:58:42.847" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-01T02:58:42.847" LastActivityDate="2014-04-01T02:58:42.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1172" PostTypeId="4" CreationDate="2014-04-01T02:58:42.847" Score="0" Body="For questions about effectively training current and new educators." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T06:11:55.583" LastActivityDate="2014-04-01T06:11:55.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1173" PostTypeId="2" ParentId="1164" CreationDate="2014-04-01T02:58:55.700" Score="16" Body="&lt;p&gt;I took an excellent set of &lt;a href=&quot;https://faculty.bard.edu/belk/ScienceProblems.pdf&quot; rel=&quot;noreferrer&quot;&gt;related rates problems from Jim Belk's webpage&lt;/a&gt; and turned it into a nice worksheet. I spun the problems this way, to motivate them:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have now covered related rates. This means you now have an extra power that works in all your science classes. Anytime anyone gives you an equation -- a fact -- that is true about the world, &lt;strong&gt;you have secret knowledge&lt;/strong&gt; that gives you an extra bonus fact about the world! This is because you can take their equation and take the derivative of both sides with respect to time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$PV = nRT$&lt;/span&gt;? Why not differentiate both sides? Your chemistry teacher might not even know that &lt;span class=&quot;math-container&quot;&gt;$P\frac{dV}{dt} + V\frac{dP}{dt} = nR\frac{dT}{dt}$&lt;/span&gt;! And if the gas is actually gradually being created by a reaction, you can declare n to be a variable instead of a constant, and take the derivative that way to get a different formula.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Kinetic Energy is &lt;span class=&quot;math-container&quot;&gt;$E = \frac{1}{2}mv^2$&lt;/span&gt;? Why not differentiate both sides? &lt;span class=&quot;math-container&quot;&gt;$\frac{dE}{dt} = mv\frac{dv}{dt}$&lt;/span&gt;. What's that all about? I don't even know. It's Power = something, because I know the units of &lt;span class=&quot;math-container&quot;&gt;$\frac{dE}{dt}$&lt;/span&gt;. (It doesn't matter that I do know what it means because I looked it up; the effect is greater when I say that I don't know -- it gives the students more of a feeling of power, which is the whole point.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$pH = -\log_{10} [H^+]$&lt;/span&gt;? Why not differentiate both sides? &lt;span class=&quot;math-container&quot;&gt;$\frac{d(pH)}{dt} = \frac{1}{(\ln 10) [H^+]}\frac{d[H^+]}{dt}$&lt;/span&gt;. It... apparently tells you how fast the pH changes depending on how fast the concentration is changing. That could actually be useful! And it shows that a reaction happening at a constant rate doesn't cause the pH to change at a constant rate.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastEditorUserId="77" LastEditDate="2018-10-23T09:29:26.577" LastActivityDate="2018-10-23T09:29:26.577" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="1174" PostTypeId="2" ParentId="353" CreationDate="2014-04-01T04:53:03.573" Score="5" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Avoid real world analogies. They confuse students because real world means natural language. Instead remind them of implication and its truth table whenever you write a mathematical proposition on the board.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Remind students to work with the definitions. Over and over again.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Introduce the notion of a vacuous truth (or vacuous implication) as soon as possible, I prefer the first class of the semester (even if it's before anything related to propositional calculus), and point it out explicitly whenever it comes up.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="578" LastEditorUserId="578" LastEditDate="2014-04-02T08:31:15.377" LastActivityDate="2014-04-02T08:31:15.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1175" PostTypeId="2" ParentId="1134" CreationDate="2014-04-01T05:27:03.980" Score="7" Body="&lt;p&gt;It sounds like you're talking about requiring the students to use one book AND another book AND ..., rather than giving them a list of books and saying they can use this book OR this book OR ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One problem is cost. For many lower-division courses the cost of a &lt;em&gt;single&lt;/em&gt; textbook is already completely exploitative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In (physics) grad school, I remember really, really hating a couple of courses where the instructor did this. The reason was that the instructor basically wanted to give lectures on the blackboard, which were supposed to present all the material -- but this left us with no organized source of information, just a bunch of hand-written lecture notes without a table of context, index, etc. The multiple books were assigned as an afterthought, and they didn't cover the material appropriately. I felt completely lost in these courses.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-04-01T05:27:03.980" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1176" PostTypeId="2" ParentId="1113" CreationDate="2014-04-01T05:46:12.920" Score="2" Body="&lt;p&gt;Math, when examined/shown properly, is a beautiful thing, worthy of being called a work of art or natural wonder depending on your stance on things. So it definitely can be spiritually uplifting if taught well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to how to teach it like that (i.e. effectively, engagingly and enchantingly), well, if I knew that I'd be shouting it from the rooftops, not here. But speaking as someone who unfortunately was never gifted or particularly proficient at it but can sometimes see the beauty of math, I do have some advice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that the times in my education when I felt most disconnected, disinterested and even disgruntled with math were the times when I felt that I was being told to learn things without being helped or allowed to understand their beauty or utility, especially when I felt I wasn't being given the &lt;em&gt;time&lt;/em&gt; to do so. &#xA;I hated the times when I felt that I was being made to learn without regards for my curiosity instead of through it or because of it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the times that I felt the most uplifted and filled with wonder were when I was taught something that brought down barriers, that allowed me to solve problems I'd have written off as unsolvable before. For example, learning long division after spending so long dealing with remainders. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also uplifting are times when the amazing &lt;em&gt;structure&lt;/em&gt; and &lt;em&gt;endless order&lt;/em&gt; of math revealed itself to me. Playing with  fractions and finding the different relationships between the numbers chosen and the fraction's value as a decimal--for example, seeing the way that the decimal forms of a series like (2/1, 3/2, 4/3, 5/6, etc. 'walk' their way downwards and trying to find the rules that governed such things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Word problems can be engaging, but they should be something that makes the student care &lt;em&gt;more&lt;/em&gt; about the answer, not just something that gives a flimsy and petty pretext for why the math needs to be done. The latter can actually make doing the problem seem less worthwhile. Less &quot;A train leaves Detroit...&quot; and more &quot;Can this man get to the airport to stop the woman he loves leaving forever?!&quot; Less &quot;How much fence do you need for this garden&quot; and more &quot;What's the largest enclosure this tiger can have at this zoo without being able to escape with a running jump?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the times I loved were when I was being shown not just the what, when and where of numbers, but the why and the how. There's no need to include God in the &quot;who&quot; position, just try to let the students see the beautiful patterns in the numbers.&lt;/p&gt;&#xA;" OwnerUserId="579" LastActivityDate="2014-04-01T05:46:12.920" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1177" PostTypeId="2" ParentId="1137" CreationDate="2014-04-01T05:57:56.067" Score="1" Body="&lt;p&gt;You should give your students an assessment test to see what they have not mastered.&#xA;Then you may wish to give them some basic material for practice on their own time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once a professor asked me &quot;Didn't you take a course in differential equations&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I said &quot;Of course, twice&quot;.&lt;/p&gt;&#xA;" OwnerUserId="581" LastActivityDate="2014-04-01T05:57:56.067" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1178" PostTypeId="2" ParentId="1149" CreationDate="2014-04-01T07:38:32.790" Score="21" Body="&lt;p&gt;I subscribe to the &lt;a href=&quot;http://www.axler.net/DwD.html&quot;&gt;down with determinants&lt;/a&gt; school of thought.  In an elementary linear algebra course, particularly for non-mathematicians, then I think that determinants are as useful as your appendix[1].  Nevertheless, they are on the syllabus so I have to teach them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's what I do.  I tell the students the truth: that determinants used to be a key part of doing computations with matrices but now there are better methods, however that they will still need to compute some determinants on an exam.  Then I tell them how to compute them using Gaussian Elimination, and the tricks for 2x2 and 3x3 (emphasising that they are tricks, and that there is no such trick for 4x4 or higher).  If time, I might mention the relationship to volume but I wouldn't regard it as important particularly as up to this point we won't have talked about geometry at all.  I certainly wouldn't bother with Cramner's rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my current course, I have four hours set aside for determinants.  The above takes a little over one.  I use the rest of the time for more useful things like going back over topics that the students have had issues with earlier in the semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a couple of useful nuggets to reinforce the &lt;em&gt;down with determinants&lt;/em&gt; line.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Myth: &quot;Determinants are useful to figure out if a matrix is invertible or not.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, in &lt;em&gt;theory&lt;/em&gt;, maybe.  But in practice, no.  How do we compute the determinant?  Generally, by using Gaussian Elimination but remembering row swaps and row scales.  But Gaussian Elimination tells us &lt;em&gt;directly&lt;/em&gt; whether or not the matrix is invertible.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Myth: &quot;Determinants are useful for finding eigenvalues.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, this is true in theory, but not in practice.  It's fine with 2x2, and with 3x3 if you give them a running start (such as an obvious root).  But any higher and you're in dubious territory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, a few years ago I looked up how one of the major mathematical pieces of software found roots of polynomials.  What I found surprised me, and put the final nail in determinant's coffin.  To find the roots of an arbitrary polynomial, this program figures out a matrix with that as its characteristic polynomial.  Then does an iterative method to find the eigenvalues of that matrix, and returns them as the roots of the polynomial.  I leave it as an exercise to the reader to see how this shows the idiocy of teaching determinants as a method for finding eigenvalues.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;[1] I want to make it clear that this is about determinants in such a course.  As a differential topologist, I'm fully aware of the usefulness of determinants in other arenas.  But the time to introduce them is in those other arenas, not in an elementary linear algebra course.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-04-01T07:38:32.790" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="1179" PostTypeId="2" ParentId="1114" CreationDate="2014-04-01T11:05:46.183" Score="9" Body="&lt;p&gt;There is a middle ground here, I think. I was often in courses which provided &quot;abridged&quot; notes, and it worked pretty well. The idea is to leave off details and make it clear to the students that not everything that is required to know for the test is in the notes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the professors I have had, stated theorems and definitions in the handout notes, but didn't prove them or show examples. Since we were required to prove some of theorems on the final exam, it meant we either took notes or proved them ourselves. And since we didn't know in advance which ones we would be expected to prove, we effectively had to learn them all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A slight modification, if your lecture is not proof heavy, might be to give a summary version of the notes, with questions in place of certain facts which they should know. (as an example: Why doesn't the Central Limit Theorem apply here?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the key though is to make it clear to the students that they will be tested on the content of the lecture(anything and everything you say), which will cover things that aren't addressed in the notes. You can drive this point home by scheduling quizzes or giving homework assignments that ask about things not on the notes.&lt;/p&gt;&#xA;" OwnerUserId="208" LastActivityDate="2014-04-01T11:05:46.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1180" PostTypeId="2" ParentId="1149" CreationDate="2014-04-01T14:44:52.170" Score="5" Body="&lt;p&gt;&lt;strong&gt;Warning&lt;/strong&gt;:  What follows are the words of someone who has not taught a linear algebra course.  Take everything with the appropriate level of salt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A great theme to try and convey throughout a linear algebra course is that sometimes it is fruitful to try to understand a mathematical object by characterizing properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first example students would probably see in a first linear algebra course is the formula for matrix multiplication.  If you give the definition as the formula, it is incomprehensible.  Why would anyone ever perform this horrible operation?  On the other hand, if you define matrix multiplication as the matrix which results from composing the linear maps they represent, it is totally motivated.  This definition seems less &quot;computational&quot;, but it is actually just as good for computation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The determinant provides another great example of something which is best understood through properties which characterize it:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with the motivation that for vectors in $\mathbb{R}^n$, we want to define the signed&#xA;volume of the parallelepiped spanned by an &lt;strong&gt;ordered&lt;/strong&gt; list of vectors $(v_1,v_2,...,v_n)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We should have &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$D(e_1,e_2,...,e_n) = 1$&lt;/li&gt;&#xA;&lt;li&gt;Switching two inputs should negate the output (The determinant is &lt;strong&gt;alternating&lt;/strong&gt;)&lt;/li&gt;&#xA;&lt;li&gt;The determinant is linear in each vector input separately (This requires some&#xA;carefully drawn pictures)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Now have them break up into groups and try to compute some determinants using these properties.  Since the only value they know is on the ordered list of basis&#xA;vectors, they must proceed by using multilinearity reduce to basis vectors, and then the alternating property to put them into the correct order.  For example, you might have them compute&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$&#xA;\begin{align*}&#xA;D\left(\begin{bmatrix}2\\3\end{bmatrix},\begin{bmatrix}4\\5\end{bmatrix}\right)&#xA;&amp;amp;=D\left(2\begin{bmatrix}1\\0\end{bmatrix}+3\begin{bmatrix}0\\1\end{bmatrix},\begin{bmatrix}4\\5\end{bmatrix}\right)\\&#xA;&amp;amp;=2D\left(\begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}4\\5\end{bmatrix}\right)+3D\left(\begin{bmatrix}0\\1\end{bmatrix},\begin{bmatrix}4\\5\end{bmatrix}\right)\\&#xA;&amp;amp;=(2)(4)D\left(\begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}1\\0\end{bmatrix}\right)+(2)(5)D\left(\begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}0\\1\end{bmatrix}\right)+(3)(4)D\left(\begin{bmatrix}0\\1\end{bmatrix}, \begin{bmatrix}1\\0\end{bmatrix}\right)+ (3)(5)D\left(\begin{bmatrix}0\\1\end{bmatrix}, \begin{bmatrix}0\\1\end{bmatrix}\right)\\&#xA;&amp;amp;=2(5) -3(4)\\&#xA;&amp;amp;=-2&#xA;\end{align*}&#xA;$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I really think that this kind of computation, where each step is dictated by the properties of the determinant, is essential.  For some reason, I have not really seen this kind of computation in any linear algebra book.  It is usually only shown in the general case, as part of the proof that the determinant is characterized by these properties, and I think students often &quot;turn off&quot; for proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is as far as I wanted to go in my comment in the other thread that the &quot;computation of the determinant is clear from its characterizing properties&quot;.  I do think covering all of the things you mention in a week is probably a bit much.  So far, the above presentation has covered the volume aspect, and how to compute them.  It also follows cleanly from these properties that if one vector is in the span of the others, then the determinant is zero.  This is also clear geometrically.  This shows that if the corresponding matrix is not injective, the determinant must be zero.  It also shows why column operations do not effect the determinant. Cramer's rule is also a very short computation from these properties.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I must admit, the other topics you mention (row operations, cofactor expansion, and classical adjoint) all seem somewhat tricky from this perspective. &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-04-01T14:44:52.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1181" PostTypeId="2" ParentId="98" CreationDate="2014-04-01T14:48:46.643" Score="11" Body="&lt;p&gt;At least as a counter-point to other reasonable viewpoints: I am not such an enthusiast about measure theory as &quot;foundation&quot; for integrals, because it seems to me that measure theory over-emphasizes just one smallish part of what we really want. Namely, starting with (ambiguously) nice functions $f_n$, perhaps continuous and compactly supported on a Euclidean space, so that &lt;em&gt;any&lt;/em&gt; notion of integral makes sense for the individuals, _one_thing_ we want is that $\lim_n \int f_n=\int\lim_n f_n$. Part of the issue is in what sense the limit of the functions themselves is taken, and there is not a unique answer to this. But it is clear from examples that pointwise limits of continuous functions may be harder to sensibly integrate in naive terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another fundamental issue is the distinction between integrating differential forms and integrating functions, although again I claim this is more of a technical distinction than fundamental. Namely, while from a &quot;measure-theory&quot; viewpoint integrating $f$ on an interval $[a,b]$ does not even &lt;em&gt;mention&lt;/em&gt; direction, in usual calculus, and if we imagine we are integrating the differential form $f\,dx$, direction &lt;em&gt;does&lt;/em&gt; matter. (That is, in elementary calculus the integral $\int_a^b$ is the negative of $\int_b^a$.) This issue is still more notational than substantive even when doing path integrals and surface integrals. It seems to me that the issue is serious only when we are doing calculus on manifolds, where there is no canonical coordinate system, so that a differential form has no canonical local expression, and the only thing that is well-defined is an integral of an exterior $k$-form over a $k$-cycle or $k$-current, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, in all cases, the external properties and desiderata are very similar, so that all these things are more the same than different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even when (as in basic Lebesgue theory) we discover that &lt;em&gt;some&lt;/em&gt; restrictions are necessary to be able to pass a linear operator inside an integral, it seems to me that the development of mathematics has been to make sense of otherwise-dubious outcomes, whenever possible. For example, $\int_{\mathbb R} {e^{ix\xi}\;dx\over 1+x^2}$ cannot be differentiated with respect to $\xi$ very much without producing a not-classically-sensible integral. But as Fourier transform of a tempered distribution it's still completely fine.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-01T14:48:46.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1182" PostTypeId="2" ParentId="1101" CreationDate="2014-04-01T14:55:11.467" Score="14" Body="&lt;p&gt;I would discuss with these students the rather &lt;a href=&quot;https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;healthy point of view of Terry Tao&lt;/a&gt; on the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One can roughly divide mathematical education into three stages: The&#xA;  &quot;pre-rigorous&quot; stage, in which mathematics is taught in an informal,&#xA;  intuitive manner, based on examples, fuzzy notions, and hand-waving.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The &quot;rigorous&quot; stage, in which one is now taught that in order to do&#xA;  maths &quot;properly&quot;, one needs to work and think in a much more precise&#xA;  and formal manner.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The &quot;post-rigorous&quot; stage, in which one has grown comfortable with all&#xA;  the rigorous foundations of one's chosen field, and is now ready to&#xA;  revisit and refine one's pre-rigorous intuition on the subject, but&#xA;  this time with the intuition solidly buttressed by rigorous theory.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;All too often, one ends up discarding one's initial intuition and is&#xA;  only able to process mathematics at a formal level, thus getting&#xA;  stalled at the second stage of one's mathematical education.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The point of rigour is not to destroy all intuition; instead, it&#xA;  should be used to destroy bad intuition while clarifying and elevating&#xA;  good intuition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2018-03-10T18:38:38.307" LastActivityDate="2018-03-10T18:38:38.307" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1183" PostTypeId="1" CreationDate="2014-04-01T15:23:14.237" Score="9" ViewCount="190" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find that the best way for my mathematical modeling class to understand the modeling process is by having the experience of modeling some real world situation of their group's choosing.  I have a number of strategies so that the project topics are interesting and of the right scope to be doable in the amount of time allotted.  One of two main deliverables that I require is a 16&amp;ndash;20 page paper about their model, focusing heavily on the evaluation of the model.  (See here for &lt;a href=&quot;http://qcpages.qc.edu/~chanusa/courses/245sp14/project.html&quot;&gt;Project Description&lt;/a&gt; and &lt;a href=&quot;http://qcpages.qc.edu/~chanusa/courses/245sp14/homework/ProjectExpectations.pdf&quot;&gt;Project Expectations&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One day of the semester, I ask the students to submit a final draft to peer review.  The groups bring in three copies of their paper and the students spend the 75-minute class period reading and commenting on their peers' papers.  I give the students directed questions to address about the papers. (See here for &lt;a href=&quot;http://qcpages.qc.edu/~chanusa/courses/245sp14/homework/PeerReview.pdf&quot;&gt;Peer Review Sheet&lt;/a&gt;.)  The goal of this peer review process is to give the students a chance to improve upon their drafts before submitting them to me for grading a few weeks later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel that I have set up the Peer Review process for success but every year there are issues with the process.  For example, some groups get peer comments that their paper needs only minimal revisions but when I read the paper, it is clear that the main focus of the paper is not the model or the paper does a bad job with model evaluation.  This leads to frustration and disappointment with the grades that I assign, especially with all the effort that I know that the students put into the project.  (I do allow for a second revision to improve their grade afterward.)  I feel that the peer review process is not doing its intended job to improve the quality of the paper and giving a false sense of security.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have any suggestions for ensuring success in the Peer Review process?  &lt;/p&gt;&#xA;" OwnerUserId="324" LastActivityDate="2014-04-10T08:16:38.647" Title="Peer Review of Mathematical Essays" Tags="|undergraduate-research|interactive-teaching|writing|term-papers|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1184" PostTypeId="2" ParentId="750" CreationDate="2014-04-01T15:54:02.717" Score="2" Body="&lt;p&gt;One should start very easy!!! A long time ago, I where on a beach with my (then) four year old daughter. I asked &quot;How many grains of sand do you think it is on the beach?&quot;.  She thought a little and then said &quot;seventeen?&quot; Then we started to count, and it was soon obvios that guess was far off!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I guess my answer is:  just take advantage of natural situations!&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2014-04-01T15:54:02.717" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1185" PostTypeId="2" ParentId="1183" CreationDate="2014-04-01T16:30:40.410" Score="6" Body="&lt;p&gt;Here's my perspective as a project manager.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Weighting the different sections&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would help to break down the 16-20 pages into sections with page guidelines, e.g.:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;2 p. Introduction&lt;/li&gt;&#xA;&lt;li&gt;3 p. Assumptions&lt;/li&gt;&#xA;&lt;li&gt;3 p. Results&lt;/li&gt;&#xA;&lt;li&gt;3 p. Discussion: Strengths&lt;/li&gt;&#xA;&lt;li&gt;3 p. Discussion: Weaknesses and Expected Errors&lt;/li&gt;&#xA;&lt;li&gt;3 p. Discussion: Future Research&lt;/li&gt;&#xA;&lt;li&gt;1 p. Conclusion&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This keeps the units of guidance under 5 pages long.  (Also, you have lengths in a mixture of pages, paragraphs and words, so it would help to standardize.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Revised Questions on Content&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For each 3-page section of the paper, provide at least one question.  This will help ensure that both the papers and the feedback have the right emphasis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Content #3:  It may help to focus on clarity over length.  &quot;Did the paper explain the model’s assumptions clearly?  What other assumptions are important?&quot;  And then ask the same about the errors, in a different question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Content #4:  Identifying gross overstatements asks for too much judgment.  &quot;What are the boldest claims in the paper?  Are they adequately supported?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Other questions to remove&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overview #2, what to expand:  You may not need this if you have questions on each individual part identified above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overview #3, redundancy and fluff:  I wouldn't worry about this so long as the first draft is of the right length and the final draft is complete.  Students may explain one thing twice because they don't realize that two things are the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Structure #3, style:  This requires a vocabulary that you probably wouldn't develop in a math class.  The authors can decide for themselves which criticisms can be met by changing their style.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps -- let us know how it goes next time!&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-01T16:30:40.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1187" PostTypeId="2" ParentId="1113" CreationDate="2014-04-01T19:00:42.287" Score="-5" Body="&lt;p&gt;Well,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are the perfect numbers like 6 and 28 etc. 6 was actualy called &quot;Number of God&quot;. Because its a perfect number where the sum of there dividers always ends up to be the number himself. 1 + 2 + 3 = 6 and God created the world in 6 days !!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;28 is also a perfect number (1+2+4+7+14=28) and was always be refered to as the Moon-Cicle which could only come from god.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Humans are known for there wild interpretations :-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Funny though, I was able to remember this because of the wild interpretation !&lt;/p&gt;&#xA;" OwnerUserId="592" LastActivityDate="2014-04-01T19:00:42.287" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1188" PostTypeId="2" ParentId="224" CreationDate="2014-04-01T19:43:11.280" Score="3" Body="&lt;p&gt;Maybe this is relevant for your purpose: &lt;a href=&quot;https://www.cs.cmu.edu/%7Ealdrich/sasylf/&quot; rel=&quot;nofollow noreferrer&quot;&gt;SASyLF&lt;/a&gt; is an educational proof assistant for language theory:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;SASyLF has a simple design philosophy: language and logic syntax, semantics, and meta-theory should be written as closely as possible to the way it is done on paper.  SASyLF can express proofs typical of an introductory graduate type theory course.  SASyLF proofs are generally very explicit, but its built-in support for variable binding provides substitution properties for free and avoids awkward variable encodings.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;On the SASyLF webpage you can find an &lt;a href=&quot;https://www.cs.cmu.edu/%7Ealdrich/sasylf/fdpe08.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; describing among other things a case study from in-class use from around 2008. Currently &lt;a href=&quot;http://www.cs.uwm.edu/faculty/boyland/&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Boyland&lt;/a&gt; uses it in his Type Systems course based on Pierce's &lt;em&gt;Types and programming languages&lt;/em&gt;. A big part of the exercises consists in writing proofs in SASyLF. I've been attending the course and found SASyLF quite easy to learn, although the documentation could be expanded.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit June 2016:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some other places where logic is being taught interactively with the aid of computers:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://proofweb.cs.ru.nl/login.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;ProofWeb&lt;/a&gt;. Based coq and other proof checkers and suited among other things for use with the book &lt;a href=&quot;https://www.cambridge.org/gb/academic/subjects/computer-science/programming-languages-and-applied-logic/logic-computer-science-modelling-and-reasoning-about-systems-2nd-edition?format=PB&amp;amp;isbn=9780521543101&quot; rel=&quot;nofollow noreferrer&quot;&gt;Logic in Computer Science&lt;/a&gt; by Huth &amp;amp; Ryan.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;a href=&quot;https://www.gradegrinder.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Openproof project&lt;/a&gt; at Stanford's Center for the Study of Language and Information. Has several courseware packages.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.cis.upenn.edu/%7Ebcpierce/sf/current/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Software Foundations&lt;/a&gt; uses coq. Here's a &lt;a href=&quot;https://vimeo.com/6615365&quot; rel=&quot;nofollow noreferrer&quot;&gt;talk&lt;/a&gt; by Benjamin Pierce about his experience with this.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://web.archive.org/web/20180905122011/http://www.sfu.ca:80/llep/Ara/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Project Ara&lt;/a&gt; uses the book &lt;a href=&quot;https://broadviewpress.com/product/proof-and-consequence/#tab-description&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proof and Consequence&lt;/a&gt; by Jennings &amp;amp; Friedrich and a software called Simon.&lt;/li&gt;&#xA;&lt;li&gt;A introductory &lt;a href=&quot;https://avigad.github.io/logic_and_proof/&quot; rel=&quot;nofollow noreferrer&quot;&gt;course&lt;/a&gt; on Logic an Proof by Jeremy Avigad, Robert Lewis &amp;amp; Floris van Doorn using Lean.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I have no personal experience with any of these but I'm adding them for completeness. Some were mentioned in &lt;a href=&quot;https://mathoverflow.net/questions/44620/undergraduate-logic-textbook/&quot;&gt;this&lt;/a&gt; mathoverflow question.&lt;/p&gt;&#xA;&lt;p&gt;Also there is a conference called &lt;a href=&quot;http://ttl2015.irisa.fr/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tools for Teaching Logic&lt;/a&gt; whose &lt;a href=&quot;http://ttl2015.irisa.fr/TTL2015_proceedings.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;proceedings&lt;/a&gt; might contain more answers to the original question.&lt;/p&gt;&#xA;" OwnerUserId="590" LastEditorDisplayName="user7990" LastEditDate="2020-09-10T14:09:34.770" LastActivityDate="2020-09-10T14:09:34.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1190" PostTypeId="5" CreationDate="2014-04-01T20:33:14.037" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-01T20:33:14.037" LastActivityDate="2014-04-01T20:33:14.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1191" PostTypeId="4" CreationDate="2014-04-01T20:33:14.037" Score="0" Body="For questions about giving students written tests, typically in a classroom setting." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T20:43:01.563" LastActivityDate="2014-04-01T20:43:01.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1192" PostTypeId="5" CreationDate="2014-04-01T20:36:11.197" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-01T20:36:11.197" LastActivityDate="2014-04-01T20:36:11.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1193" PostTypeId="4" CreationDate="2014-04-01T20:36:11.197" Score="0" Body="For questions about designing effective tests and writing exam problems." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T20:42:52.010" LastActivityDate="2014-04-01T20:42:52.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1194" PostTypeId="5" CreationDate="2014-04-01T20:38:53.447" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-01T20:38:53.447" LastActivityDate="2014-04-01T20:38:53.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1195" PostTypeId="4" CreationDate="2014-04-01T20:38:53.447" Score="0" Body="For questions about effectively designing, giving, and assessing oral exams, where a student meets one-on-one with their instructor and demonstrates his/her abilities." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T20:42:38.350" LastActivityDate="2014-04-01T20:42:38.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1196" PostTypeId="5" CreationDate="2014-04-01T20:39:58.323" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-01T20:39:58.323" LastActivityDate="2014-04-01T20:39:58.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1197" PostTypeId="4" CreationDate="2014-04-01T20:39:58.323" Score="0" Body="For questions about assessing the abilities of mathematics instructors, including considerations about self-review, peer-review, and student evaluations." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T20:44:28.607" LastActivityDate="2014-04-01T20:44:28.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1198" PostTypeId="5" CreationDate="2014-04-01T20:41:05.047" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-01T20:41:05.047" LastActivityDate="2014-04-01T20:41:05.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1199" PostTypeId="4" CreationDate="2014-04-01T20:41:05.047" Score="0" Body="For questions about teaching calculus with functions of more than one variable (also perhaps known as&quot;Vector calculus&quot; or &quot;Calculus 3&quot;)." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-01T20:44:12.410" LastActivityDate="2014-04-01T20:44:12.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1200" PostTypeId="1" CreationDate="2014-04-01T21:59:42.767" Score="55" ViewCount="5937" Body="&lt;p&gt;I never gave this a second thought until a friend who works in education brought it up the other day.  Should we say that a fraction like $\frac{1}{2}$ &quot;is&quot; a number, or &quot;represents&quot; a number?  In particular, should we say that $\frac{1}{2}$ and $\frac{2}{4}$ &quot;are&quot; the same number, or &quot;represent&quot; the same number?  (Or something else entirely?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My immediate reaction on hearing the question was that &quot;obviously&quot; they &lt;em&gt;are&lt;/em&gt; the same number.  I would explain the difference between $\frac{1}{2}$ and $\frac{2}{4}$ as being analogous to the fact that one person can have two names.  If I go by &quot;Mike&quot; to my friends and &quot;Dr. Shulman&quot; to my students, and one of my friends is talking about me to one of my students, they might end up having to explain that &quot;Mike and Dr. Shulman are the same person.&quot;  They wouldn't say &quot;Mike and Dr. Shulman represent the same person.&quot;  They might say &quot;'Mike' and 'Dr. Shulman' are names for the same person&quot;, which conveys about the same meaning as the latter, but in that case the two names would be quoted, because we're talking about them as bits of syntax rather than about the objects they denote.  But when a name is not quoted, we're talking about its denotation rather than the name itself, e.g. when we say &quot;$\frac{1}{2}$ is between 0 and 1&quot; we don't mean that the symbol &quot;$\frac{1}{2}$&quot; is written in between the symbols &quot;0&quot; and &quot;1&quot; on a sheet of paper, we mean that the &lt;em&gt;number&lt;/em&gt; denoted by &quot;$\frac{1}{2}$&quot; lies in between the numbers denoted by &quot;0&quot; and &quot;1&quot; with respect to the ordering of rational numbers.  So we can say &quot;$\frac{1}{2}$ and $\frac{2}{4}$ are the same number&quot; but &quot;'$\frac{1}{2}$' and '$\frac{2}{4}$' represent the same number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this is just my intuitive reaction based on experience as a mathematician, and I've never actually tried to teach someone about fractions.  I certainly wouldn't want to try to explain sense and reference to a third grader!  So I'm curious about the consensus (or lack thereof) of people who do teach fractions and study people's understanding of them.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2018-04-19T20:30:59.790" Title="Should we say that fractions &quot;are&quot; or &quot;represent&quot; numbers?" Tags="|primary-education|fractions|" AnswerCount="14" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="1201" PostTypeId="2" ParentId="1113" CreationDate="2014-04-01T22:40:03.523" Score="1" Body="&lt;p&gt;It is a fundamental question. Let me to add some brief additional points to other useful answers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1. How can I add spiritual aspects to my mathematics?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Spirituality of your mathematics comes from your &lt;em&gt;meta-mathematics&lt;/em&gt;. By meta-mathematics I mean all parts of human knowledge which discuss &lt;em&gt;about&lt;/em&gt; mathematics including logic, philosophy of math, etc. It is a very important point. Your meta-mathematics determines the &lt;em&gt;interpretation&lt;/em&gt; of the mathematical theorems which you teach. In fact, one can talk about all beautiful theorems of mathematics and all religious mathematicians of the history without any spiritual impact on his students when he doesn't use a spiritual meta-mathematics. Simply because beauty of a subject/object cannot lead one to any spiritual experience necessarily. Can looking to a beautiful flower convince somebody that there are some spiritual beings? Not necessarily. The same situation happens when one tries to increase the spiritual aspects of his teaching just by showing the beauty of mathematical theorems to his student. Also in the case of talking about faithful mathematicians there are many counterexamples for each religious person. For example in logic if you mention Cantor and Godel as religious logicians another one can mention Russell and Turing as counterexamples. All of these persons are great logicians with brilliant deep works. Thus the technical theorems of logic/mathematics &lt;em&gt;cannot&lt;/em&gt; lead one to believe in God or any other immaterial existence necessarily. You need &lt;em&gt;endow&lt;/em&gt; your mathematics with an appropriate &lt;em&gt;philosophy&lt;/em&gt; to conclude such conclusions. If you choose a suitable spiritual philosophy of math as the standard philosophy of your courses then all of your lectures, theorems, proofs, discussions and activities seem really spiritual and leads your students to a spiritual experience. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2. What is the best meta-mathematics for a spiritual teaching?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly, &lt;a href=&quot;http://en.wikipedia.org/wiki/Platonism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Platonism&lt;/a&gt; is a good option because:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;As a possible consequence of Plato's travels to the Eastern countries including Jerusalem (and maybe Persia) and meeting with Rabbis and Zoroastrian clergies his philosophical point of view is designed based on a &lt;em&gt;pyramid&lt;/em&gt;. Plato's philosophical system implicitly leads one to a &lt;em&gt;unique&lt;/em&gt; point on the top of this pyramid (&lt;em&gt;Plato's God&lt;/em&gt;).&lt;/li&gt;&#xA;  &lt;li&gt;Platonism suggests a clear pattern for &lt;em&gt;uplifting&lt;/em&gt;. In Plato's point of view, the role of mathematics in one's spiritual travel from earth to the ideal world of ideas (&lt;em&gt;Plato's heaven&lt;/em&gt;) is very important. Mathematics is a &lt;em&gt;bridge&lt;/em&gt; in the route and maybe the last one of them in the modern age.&lt;/li&gt;&#xA;  &lt;li&gt;Based on Platonic point of view, mathematical objects and theorems are &lt;em&gt;discoveries&lt;/em&gt; not &lt;em&gt;inventions&lt;/em&gt;. For those who believe in Platonism, each new theorem is a fact about the &lt;em&gt;invisible&lt;/em&gt; world of mathematical objects.&lt;/li&gt;&#xA;  &lt;li&gt;In Platonism, doing mathematics is a kind of &lt;em&gt;praying&lt;/em&gt; (or meditation) which helps one to understand the &lt;em&gt;absolute truth&lt;/em&gt; of the world. &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark 1.&lt;/strong&gt; The ideas of Pythagoras about the interactions of mathematics and world are related. Also the philosophy and theosophy of &lt;a href=&quot;http://en.wikipedia.org/wiki/Plotinus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Plotinus&lt;/a&gt; and other philosophers of &lt;a href=&quot;http://en.wikipedia.org/wiki/Neoplatonism&quot; rel=&quot;nofollow noreferrer&quot;&gt;neoplatonism&lt;/a&gt; could be related and very useful.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3. How should I use the spiritual meta-mathematics in my teaching?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In one word, &lt;em&gt;implicitly&lt;/em&gt;. Generally, there is no need to talk about Socrates, Plato, Pythagoras, Plutinus, Cantor, Godel or any other philosopher or mathematician in your teaching. (In special cases, for example when you are teaching special courses like logic or set theory, it is relevant and useful to refer to life and belief of some persons directly but it is irrelevant in usual math courses like calculus and linear algebra). What you need is just a Platonic approach and explaining his ideas about mathematics and mathematical objects in your implicit comments. Just as an example when you are talking about a particular mathematical object/theorem you can use a sentence in the following form:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1963, Paul Cohen &lt;em&gt;discovered&lt;/em&gt; the Forcing method and a new model of ZFC and finally &lt;em&gt;found&lt;/em&gt; the solution of one of the most important questions of 20th century. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Compare with this one:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1963, Paul Cohen &lt;em&gt;constructed&lt;/em&gt; a new model of ZFC by &lt;em&gt;his&lt;/em&gt; Forcing method and &lt;em&gt;solved&lt;/em&gt; one of the most important questions of 20th century. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Asking questions&lt;/em&gt; instead of expressing a sentence explicitly could be another useful method to lead students to build their own spiritual world. For example you can say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What phenomenon does Cantor's theorem on existence of more than one infinity, describe? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Compare with this one:       &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Cantor's theorem on existence of more than one infinity says there are some infinite and immaterial worlds beyond our finite materialistic universe.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can find more useful techniques to make your teaching as &lt;em&gt;mysterious&lt;/em&gt; as possible. Let students be free to have their ideas and imaginations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark 2.&lt;/strong&gt; Note that spirituality is an internal experience of each person. You cannot and should not &lt;em&gt;force&lt;/em&gt; them to believe on existence of immaterial objects including God, Gabriel, $\pi$, $\aleph_1$, etc. What you can do is just &lt;em&gt;motivating&lt;/em&gt; them to &lt;em&gt;explore&lt;/em&gt; inside their minds and hearts to find these beings and unfold the invisible objects of the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark 3.&lt;/strong&gt; Note that what you want to do is an &lt;em&gt;ideological teaching&lt;/em&gt; in the sense of what I described in my post &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/1100/ideological-teaching-in-logic-courses&quot;&gt;Ideological Teaching in Logic Courses&lt;/a&gt;&quot;. This is not easy. You should be aware of possible negative and inverse impacts of this approach. Detect the feedback of your audiences continuously.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4. Do some fields of mathematics have more potential to be interpreted spiritually?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, of course. The nature of some parts of mathematics is very adequate for spiritual interpretations. You can use this potential in your teaching by leading your students to these realms. For example in set theory you don't need to do too much effort to bring spirituality in your teaching. Almost all textbooks are full of the comments with explicit or implicit references to religious and Platonic beliefs. Here are some of the quotations from usual textbooks of set theory.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;..., Kunen's result (Kunen inconsistency theorem) can best be viewed as an ultimate limitation imposed by the Axiom of Choice on the extent of reflection possible in the universe. ‎$‎ZFC‎$ ‎rallies ‎at ‎last ‎to ‎force a‎ ‎veritable ‎&lt;strong&gt;Gotterdammerung&lt;/strong&gt;‎ ‎for ‎large ‎cardinals!‎ [&lt;em&gt;Akihiro Kanamori, The Higher Infinite, Page 324&lt;/em&gt;]‎&lt;/li&gt;&#xA;  &lt;li&gt;...,‎Cantor came to view the finite and the transfinite as all of a piece, similarly comprehendable within mathematics, and delimited by what he termed the ‎&lt;strong&gt;absolute&lt;/strong&gt;‎ which he associated mathematically with the class of all ordinals and metaphysically with ‎&lt;strong&gt;God&lt;/strong&gt;‎. [&lt;em&gt;Akihiro Kanamori, The Higher Infinite, Page XII&lt;/em&gt;]‎&lt;/li&gt;&#xA;  &lt;li&gt;There are more things in‎ &lt;strong&gt;heaven&lt;/strong&gt; and Earth Horatio, than are dreamt of in your philosophy - Hamlet, I.v. 166-7.‎ [&lt;em&gt;Akihiro Kanamori, Menachem Magidor, The Evolution of&#xA;  Large Cardinal Axioms in Set Theory, Page 1&lt;/em&gt;]‎&lt;/li&gt;&#xA;  &lt;li&gt;..., Someone once said to me, doing set theory with a universal set is like believing in ‎&lt;strong&gt;God&lt;/strong&gt;.‎ [&lt;em&gt;Thomas Forster, Set theory with a Universal Set: Exploring an Untyped&#xA;  Universe, Page 11&lt;/em&gt;]&lt;/li&gt;&#xA;  &lt;li&gt;..., To use a medical metaphor set membership presents as an allegory of predication but it is also an allegory in more obscure ways, for example of the dominance relations of the numerous conceptual hierarchies that people have dreamed up from time to time, be they of language levels or &lt;strong&gt;levels of existence&lt;/strong&gt;, (&lt;strong&gt;humans&lt;/strong&gt;, ..., &lt;strong&gt;angels&lt;/strong&gt;,...,&lt;strong&gt;God&lt;/strong&gt;). [&lt;em&gt;Thomas Forster, Set theory with a Universal Set: Exploring an Untyped&#xA;  Universe, Page 11&lt;/em&gt;]&lt;/li&gt;&#xA;  &lt;li&gt;..., People living in ‎$‎‎M$ cannot construct a ‎$‎‎G$ which is ‎$‎‎‎\mathbb{P}‎$ - generic over ‎$‎‎M$. They may believe on ‎&lt;strong&gt;faith&lt;/strong&gt; that ‎&lt;strong&gt;there exists a being&lt;/strong&gt;‎ to whom their universe, ‎$‎M‎$‎, is countable. Such a being will have a generic ‎$‎‎G$ and ‎an $‎‎f_{G}= ‎\bigcup ‎G‎$. Then people in ‎$‎M‎$ ‎don't know what ‎$‎G‎$ ‎and ‎$‎f_{G}‎$ ‎are but they have names for them, ‎‎$\Gamma‎$‎‎ ‎and ‎$‎\Phi‎‎‎$. They may also read the preceding few paragraphs and thus figure out certain properties of ‎$‎‎G$ and ‎$‎‎f_{G}$;...More generally, they can construct a ‎&lt;em&gt;forcing language&lt;/em&gt;, where a sentence ‎$‎‎\psi$ of the forcing language uses the names in ‎$‎‎M^{‎\mathbb{P}‎}$ to assert something about ‎$‎‎M[G]$‎. [&lt;em&gt;Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, Page 193&lt;/em&gt;]&lt;/li&gt;&#xA;  &lt;li&gt;&lt;strong&gt;Remark 4.&lt;/strong&gt; ‎Note ‎that ‎what ‎Kunen ‎describes ‎in the ‎above ‎paragraph ‎as a‎ ‎&lt;em&gt;forcing language&lt;/em&gt; ‎for ‎the &lt;em&gt;people living in&lt;/em&gt; ‎$‎‎M$‎ ‎is very similar to what we call &lt;em&gt;religion&lt;/em&gt; in &lt;em&gt;our world&lt;/em&gt;. Precisely a religion consists from sentences about &lt;em&gt;names&lt;/em&gt; of some beings (such as &lt;em&gt;God&lt;/em&gt;, &lt;em&gt;Gabriel&lt;/em&gt;, &lt;em&gt;Satan&lt;/em&gt;, ...) which some people of &lt;em&gt;our world&lt;/em&gt; believe on &lt;em&gt;faith&lt;/em&gt; that they exist but they don't know precisely what these beings are. However people can describe some of properties and relations of these creatures using their &lt;em&gt;names&lt;/em&gt; such as: &quot;&lt;em&gt;Satan is the enemy of God&lt;/em&gt;&quot;. So religion and science are both a collection of sentences in the language of people living in our world with a difference that &lt;em&gt;using names of generic objects which don't belong to our world&lt;/em&gt; are allowed in religion but in science every sentence must use &lt;em&gt;names of real objects which are members of our world&lt;/em&gt; like any materialistic creature.&lt;/li&gt;&#xA;  &lt;li&gt;..., Philosophical positions on the foundations of mathematics have more marked impact on set theory than anywhere else in mathematics; and the reader should be well aware of the prejudices of the author he is reading. I have written this book from an   uncompromisingly realist or ‎&lt;strong&gt;Platonist&lt;/strong&gt; position; that is, I have taken the viewpoint that in some sense sets do exist, as objects to be studied, and that set theory is just as much about fixed objects as is number theory. ..., ‎It seems very difficult to me to give any reason‎ for the study of large cardinals without taking a viewpoint of this sort. &quot; [&lt;em&gt;Frank Drake, An Introduction to Large Cardinals, Page VIII&lt;/em&gt;]‎ &lt;/li&gt;&#xA;  &lt;li&gt;..., The ‎&lt;strong&gt;Platonists&lt;/strong&gt;‎ of ‎$‎2100‎$‎ may know whether ‎$‎CH‎$‎ is true, but neither ‎$‎CH‎$‎ nor ‎$‎\neg CH‎$‎ will ever be a formal theorem of ‎$‎ZFC‎$‎. Likewise, many &lt;strong&gt;Platonists&lt;/strong&gt; of today‎ &lt;strong&gt;believe&lt;/strong&gt; that inaccessible cardinals &lt;strong&gt;exist&lt;/strong&gt;, even though the statement that they exist is not a formal theorem of ‎$‎ZFC‎$‎. ‎[&lt;em&gt;Keneth Kunen, Set Theory: An Introduction to Independence Proofs, Page 11&lt;/em&gt;]‎&lt;/li&gt;&#xA;  &lt;li&gt;..., So our results are actually relative consistency results - i.e., predicated upon the assumption that ‎$‎ZF^{-}‎$‎ is consistent. These relative consistency results will be accomplished by completely finitistic means, whereas the consistency of $‎ZF^{-}‎$ will remain either an open question or an article of ‎&lt;strong&gt;faith&lt;/strong&gt;‎, depending upon one's philosophy. [&lt;em&gt;Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, Page 111&lt;/em&gt;] &lt;/li&gt;&#xA;  &lt;li&gt;&lt;strong&gt;Remark 5.&lt;/strong&gt;‎ You can find more information about the religious backgrounds of set theory in these books:&lt;/li&gt;&#xA;  &lt;li&gt;&lt;strong&gt;(1)&lt;/strong&gt; &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0743422996&quot; rel=&quot;nofollow noreferrer&quot;&gt;Amir Aczel, &lt;em&gt;The Mystery of the Aleph: Mathematics, the &lt;strong&gt;Kabbalah&lt;/strong&gt;, and the Human Mind.&lt;/em&gt;&lt;/a&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;strong&gt;(2)&lt;/strong&gt; &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0674032934&quot; rel=&quot;nofollow noreferrer&quot;&gt;Loren Graham, Jean-Michel Kantor, &lt;em&gt;Naming Infinity: A True Story of &lt;strong&gt;Religious Mysticism&lt;/strong&gt; and Mathematical Creativity.&lt;/em&gt;&lt;/a&gt;    &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5. How can I prove existence of spiritual beings to my students?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You &lt;em&gt;don't&lt;/em&gt; need to prove any thing! &lt;em&gt;The faith begins when the proof fails!&lt;/em&gt; In fact the scientific and mathematical proofs as &lt;em&gt;finite&lt;/em&gt; sequences of valid sentences based on a reasonable set of axioms, are too &lt;em&gt;short&lt;/em&gt; to reach the realm of transcendental objects. The same situation happens when we want to decide on truth or falsity of Continuum Hypothesis ($CH$) or existence of large cardinals. They are independent from our set of axioms ($ZFC$). Set theorists believe on &lt;em&gt;faith&lt;/em&gt; in one of $CH$ or $\neg CH$ based on their personal intuition about the Cantor's heaven. Of course this &lt;em&gt;faith&lt;/em&gt; has an important influence on their mathematical approaches and mathematical life too. Let me to explain more using the following pseudo-logical terminology:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;(Soundness) One should believe on the truth of every provable subject.&lt;/li&gt;&#xA;  &lt;li&gt;(Smallness) Scientific and mathematical proofs are too few and too short.&lt;/li&gt;&#xA;  &lt;li&gt;(Largeness) Transcendental subjects and objects are too many and too far from our intuitive set of mathematical and physical axioms.  &lt;/li&gt;&#xA;  &lt;li&gt;(Incompleteness) Most Transcendental subjects and objects (e.g. Large cardinals, Angels, God, etc.) lie beyond borders of scientific and mathematical proofs.&lt;/li&gt;&#xA;  &lt;li&gt;(Faithfulness) The only way to decide on existence of the transcendental objects and subjects is by intuition and faith. &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark 6.&lt;/strong&gt; Note that independent mathematical subjects like Continuum Hypothesis and existence of large cardinals are &lt;em&gt;not&lt;/em&gt; meaningless, useless or ignorable. They are just &lt;em&gt;undecidable&lt;/em&gt;. The same situation happens about the other transcendental subjects like angels and God.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, in order to lead your students to spiritual subjects you just need to &lt;em&gt;motivate&lt;/em&gt; them to think about these matters and try to talk about your (and other mathematicians) &lt;em&gt;faiths&lt;/em&gt;.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;6. Can I use formal language of mathematics to increase the spirituality of my teaching?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes. Beside their applications, mathematical notations and names could carry implicit meanings. Let me to explain by an example. In set theory we use several names and symbols which have religious meanings too. For example: Hebrew letters, cardinals, absolute, hierarchy, Cantor, etc. Also one can enlarge this list by some funny creative tricks. e.g. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You can pronounce the large cardinal symbol $0^{\dagger}$ as &lt;em&gt;zero-cross&lt;/em&gt; not zero-dagger because cross is more peaceful than dagger!&lt;/li&gt;&#xA;&lt;li&gt;You can find a relation between ordinal and &lt;em&gt;ordinance&lt;/em&gt;!&lt;/li&gt;&#xA;&lt;li&gt;You can rename &lt;a href=&quot;http://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;ordinal trichotomy theorem&lt;/a&gt; as &lt;em&gt;trinity&lt;/em&gt; theorem!  &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can use local notations and names of each field of mathematics effectively. This depends on your creativity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;7. Is there any special spiritual aspect in Linear Algebra?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes. Use the notion of &lt;em&gt;dimension&lt;/em&gt;. The spiritual beings live in the spiritual worlds. These worlds are &lt;em&gt;higher&lt;/em&gt; than our materialistic universe. What is the meaning of a &lt;em&gt;higher&lt;/em&gt; world and a being of a &lt;em&gt;higher type of existence&lt;/em&gt;? In some sense one can interpret it as a world with &lt;em&gt;higher dimensions&lt;/em&gt;. The linear algebra course is the first place which students learn about the notions of space, sub-space and dimension formally. Try to relate these notions to religious concepts. Also you can compare the power of a three dimensional being (like us) in the world of two dimensional creatures (e.g. those who live in a sheet!) with power of a four dimensional being in our three dimensional world. For example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;A human can put his finger somewhere on a sheet. The two dimensional inhabitants of the sheet think some new creature appeared in their world (the two dimensional intersection of finger and sheet). They come to see it! Then the human can withdraw his finger and put it somewhere else on the sheet far from the first place. They see that the strange creature disappeared suddenly and then appeared somewhere else. They may think this creature is traveling in their world like a ghost!  &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-01T22:40:03.523" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1202" PostTypeId="2" ParentId="175" CreationDate="2014-04-01T23:27:11.503" Score="4" Body="&lt;p&gt;With regards to your question, I can only offer some personal reflection. I think that a carrot/whip for reading some given text is a bad idea, unless your objective &lt;em&gt;really only&lt;/em&gt; is to make them read the text. Those who feel they benefit from reading already do it. Those who don't feel they benefit will either not read, or not get much out of it, and thus get the whip every time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, for the real reason I'm posting this: Your question raises some counter-questions. I don't mean to sound like I'm picking a fight, and perhaps there are cultural differences here, but some parts of your question seem strange to me.&#xA;(I'm not familiar with the school system in which you're teaching, and I don't know at what level you teach, so bear with me.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Why do you keep using a book that most students do not like? They'll be more likely to read it if they like it...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Is class time spent mostly on you giving lectures? If so, from the students' point of view, I'd expect you to cover at the very least everything needed to pass the course. I might turn to the book to clarify, or to help me remember what you said, or to prime me before your lecture, but I would not expect the book to be the only source of anything essential.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) How old are these kids? If they're somewhat grown up (say 18 yrs old or so), assignments and quizes every time you meet seems like a lot of interference in their learning. People learn in different ways, and the sooner they get to figure out how they learn effectively, the better. Offering a lecture plan, so that those who want to can read ahead and prepare themselves is helpful. Trying to force students to learn in &lt;em&gt;one&lt;/em&gt; particular way is probably not very effective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yeah.. I do apologize for not answering your question properly, and once again, I don't mean to sound like I'm picking a fight, so try not to take it that way. I'm just surprised by the picture I get when I read your question, but perhaps I'm interpreting things incorrectly due to some cultural bias or something.&lt;/p&gt;&#xA;" OwnerUserId="599" LastActivityDate="2014-04-01T23:27:11.503" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1203" PostTypeId="2" ParentId="1200" CreationDate="2014-04-01T23:28:55.210" Score="42" Body="&lt;p&gt;Questions of this nature were addressed in some part by &lt;a href=&quot;http://en.wikipedia.org/wiki/New_Math&quot; rel=&quot;noreferrer&quot;&gt;New Math&lt;/a&gt; in the late 50s and 60s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They were also used as a point of criticism by its opponents. Most notable is &lt;a href=&quot;http://en.wikipedia.org/wiki/Morris_Kline&quot; rel=&quot;noreferrer&quot;&gt;Morris Kline&lt;/a&gt;'s book &lt;em&gt;Why Johnny Can't Add&lt;/em&gt; in which he writes relevantly in Chapter 1 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Evidently the class is not doing too well and so the teacher tries a simpler question. &quot;Is 7 a number?&quot; The students, taken aback by the simplicity of the question, hardly deem it necessary to answer; but the sheer habit of obedience causes them to reply affirmatively. The teacher is aghast. &quot;If I asked you who you are, what would you say?&quot;&#xA;  The students are now wary of replying, but one more courageous youngster does do so: &quot;I am Robert Smith.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The teacher looks incredulous and says chidingly, &quot;You mean that you are the name Robert Smith? Of course not. You are a person and your name is Robert Smith. Now let us get back to my original question: Is 7 a number? Of course not! It is the name of a number. 5 + 2, 6 + 1, and 8 - 1 are names for the same number. The symbol 7 is a numeral for the number.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As you can see, your concern extends beyond rational numbers (and is at least a half century old).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today teachers are unlikely to draw such a fine distinction between whether e.g., 1/2 &lt;em&gt;is&lt;/em&gt; a number or &lt;em&gt;represents&lt;/em&gt; a number or is &lt;em&gt;the name of&lt;/em&gt; a number. Instead, they are likely to focus more on whether the students are speaking about the mathematics in a way that makes sense. (This relates to sense making; cf. the Common Core State Standards for Mathematics or some of my previous posts, e.g., &lt;a href=&quot;https://matheducators.stackexchange.com/a/1075/262&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the majority of early mathematics teachers will tell you 1/2 is a number: it's a fraction. 2/4 is also a number: it's a fraction. Furthermore, 1/2 and 2/4 are the same number even though they are written differently. Nothing like &lt;em&gt;equivalence relations&lt;/em&gt; would be formally covered in these classes (I'd write &quot;needless to say,&quot; but people try some wacky stuff) though the notion of &lt;em&gt;equivalent fractions&lt;/em&gt; may well be presented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Implicitly, it helps to talk about a fraction in its simplified form (which is effectively asking for a representative from the infinite equivalence class to which it belongs). This ought to be covered early on.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;With regard to your title question:&lt;/strong&gt; I would say, e.g., 1/2 &lt;em&gt;is&lt;/em&gt; a number. (Talking specifically about the context you have put forward, i.e., teacher discourse in elementary school education.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a teacher were to write 1/3 and 8/12 on the board and say, &quot;Okay, add up these numbers!&quot; then I would be concerned by students who get stuck upon (or object to) hearing 1/3 and 8/12 referred to as such. Much better, I think, would be to note that the latter is equivalent to 2/3 and say, &quot;1.&quot;&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-11-08T05:36:37.627" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="1204" PostTypeId="1" AcceptedAnswerId="1228" CreationDate="2014-04-01T23:35:31.897" Score="15" ViewCount="1050" Body="&lt;p&gt;I have not yet taught Linear Algebra, but I teach Computer Graphics regularly, &#xA;which uses linear algebra at many junctures,&#xA;and uses concepts such as the cross product.&#xA;I have often been disappointed to learn that even students who took Linear Algebra&#xA;and did well, have little (or no) familiarity with the cross product in $\mathbb{R}^3$.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;. Does the cross product properly belong to linear algebra, or is it a&#xA;  geometric diversion from the main thrust of linear algebra?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My sense is that mathematics departments prefer to keep linear algebra abstract,&#xA;to shy away from geometry in favor of formal rigor, often the first such rigorous&#xA;course for students. Whereas in my experience, I never fully &quot;grokked&quot; linear&#xA;algebra until I saw orthogonal transformations (etc.) geometrically. But that is my personal bias.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it can be instructive to explain (at some level) that the cross product&#xA;can only exist in its familiar form in in $\mathbb{R}^3$ and in $\mathbb{R}^7$:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Massey, W. S. &quot;Cross products of vectors in higher dimensional Euclidean spaces.&quot; &lt;em&gt;American Mathematical Monthly&lt;/em&gt; (1983): 697-701. (&lt;a href=&quot;http://www.jstor.org/discover/10.2307/2323537?uid=3739256&amp;amp;uid=2&amp;amp;uid=4&amp;amp;sid=21103780384137&quot; rel=&quot;nofollow noreferrer&quot;&gt;JSTOR link&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(My question is prompted by &lt;a href=&quot;https://matheducators.stackexchange.com/q/1149/511&quot;&gt;the recent question on determinants&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-06-14T22:26:58.693" Title="Should the cross-product in $\mathbb{R}^3$ be discussed in Linear Algebra?" Tags="|undergraduate-education|linear-algebra|" AnswerCount="5" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="1205" PostTypeId="2" ParentId="1204" CreationDate="2014-04-01T23:54:10.593" Score="8" Body="&lt;p&gt;One of the main reasons why the cross product is seldomly (to my experience) taught in linear algebra is not only the fact that this particular vector product only works well in $\mathbb{R}^3$ (a specific vector space which isn't the main focus of linear algebra), but that the arsenal of linear algebra provides already a tool to find vectors which are orthogonal to a given set of vectors: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Gram-Schmidt algorithm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you take $n-1$ vectors $v_1, v_2, \dots, v_{n-1} \in \mathbb{R}^n$ and want to know a vector which is perpendicular to each of these, i.e. the normal vector on the hyperplane spanned by them, then you find some vector $v_n$ such that it completes the set to a basis. Then you can start the Gram-Schmidt procedure: Subtract orthogonal projections, normalize, repeat. You end up with an orthonormal basis $w_1, \dots, w_n$ of $\mathbb{R}^n$ such that $$\text{span}\{v_1, \dots, v_i\}=\text{span}\{w_1, \dots, w_i\}, i=1, \dots, n.$$ In particular, $w_n$ is orthogonal to all $w_1,\dots,w_{n-1}$, and, thus to $v_1,\dots,v_{n-1}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a computational point of view, this might not be very effective, but from a mathematical point of view, there is a tool which does what the cross product does, and it's constructive and I think has an easier geometric interpretation than the cross product.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a final note, there are a &lt;a href=&quot;http://en.wikipedia.org/wiki/Cross_product#Generalizations&quot;&gt;number of generalizations&lt;/a&gt; from the cross product in different areas of abstract algebra (I am no algebraist, so I cannot tell how popular or important these are).&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-04-01T23:54:10.593" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1206" PostTypeId="2" ParentId="184" CreationDate="2014-04-01T23:59:47.227" Score="4" Body="&lt;p&gt;I'm answering this from the perspective of the student, and I don't have any citations, only my own experiences...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the courses I've taken have been laissez-faire. The basic setup has been 7 weeks of lectures that I don't have to attend if I don't feel like it, recommended reading (sections so and so in the book) and recommended exercises to solve (nothing to hand in), and then in the 8th week, an exam to determine my grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many courses have offered the option of doing things during those 7 weeks to gain bonus points for the exam. Maybe hand in some exercises, maybe do some sort of project using the skills taught in the course. This is generally good; those who don't have the time or motivation can ignore it with no ill effects, and the others can do the extra work, retain more knowledge and thus score better on the exam, in addition to the bonus points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few courses have had compulsory activities during the 7 weeks. This has generally been bad. A lot of the students are not used to having deadlines &quot;mid term&quot;, and end up planning poorly, and end up with a lower grade than they'd get if everything was tested in the final exam. Those who plan well end up doing more work for the same knowledge. Some courses really need this though, and for those courses, it's worth the trouble. Examples might be when the course is fairly &quot;hands on&quot;, not very theoretical, and you really need to have the students &quot;do the dirty work&quot; to know what it's like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One course had compulsory assignments to be handed in every 2 weeks, and did &lt;em&gt;not&lt;/em&gt; require the final exam to get a passing grade. This particular course was very much &quot;hands on&quot;; the theoretical content was mostly repetition of earlier courses. The exam was only required for higher grades, and those grades were only obtainable for students who had done well on the assignments. This set up is well suited to this type of course, but I wouldn't ever want it if the focus is the theory.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As you can see, most of my experience is with laissez-faire, and perhaps that means I'm biased, but I strongly prefer it to pressure-driven teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason is simple: People are different, and applying pressure will favor those who like &lt;em&gt;the particular form of pressure&lt;/em&gt; that you apply. For everyone else, the pressure is distracting at best, painful at worst. If you let the students do as they wish, they will have to figure out how they learn best, gaining both important insights about themselves and others, and also the important life skill of taking the reins and taking responsibility.&lt;/p&gt;&#xA;" OwnerUserId="599" LastActivityDate="2014-04-01T23:59:47.227" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1208" PostTypeId="1" CreationDate="2014-04-02T00:14:23.170" Score="7" ViewCount="3348" Body="&lt;p&gt;Talking about physical phenomena related to a particular field of mathematics can be interesting for students and might further motivate their study of the subject. For instance, there are interactions between physics and linear algebra, and discussion of these interactions could simultaneously motivate physics students to study the underlying mathematics and inspire mathematics students to appreciate the physical applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are good examples of physical phenomena related to linear algebra that are comprehensible by undergraduate students? Could these be feasibly incorporated into a linear algebra course?&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-04-02T15:36:18.390" LastActivityDate="2014-04-04T09:57:53.327" Title="Physics in Linear Algebra" Tags="|undergraduate-education|mathematical-pedagogy|linear-algebra|physical-sciences|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="1209" PostTypeId="2" ParentId="1208" CreationDate="2014-04-02T01:48:21.237" Score="2" Body="&lt;p&gt;In classical mechanics, the motion of a rigid body can be described completely by the motion of the center of mass and by the rotations around the center of mass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As opposed to linear velocity and momentum, the angular velocity does not always point in the same direction as the angular momentum. As mentioned on &lt;a href=&quot;http://kwon3d.com/theory/moi/prin.html&quot; rel=&quot;nofollow&quot;&gt;this site&lt;/a&gt;, this can lead to problems as it creates wobbling in rotating machinery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The principal axes theorem says that each rigid object has three perpendicular axes about which you can rotate the object and have the angular momentum match up with the angular velocity. These are the best axes to rotate an object by.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proof is nothing but the fact that a symmetric matrix can be diagonalized by an orthogonal change of basis.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-04-02T01:48:21.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1210" PostTypeId="2" ParentId="1200" CreationDate="2014-04-02T01:49:11.677" Score="11" Body="&lt;p&gt;I would say that $\frac12$ is a number and that &quot;$\frac12$&quot; represents (or names) that number.  I should therefore not say things like &quot;the denominator of $\frac12$ is 2&quot;, because the denominator depends not only on the number but on its name.  I confess, however, to being rather careless about such things most of the time.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-04-02T01:49:11.677" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="1211" PostTypeId="2" ParentId="992" CreationDate="2014-04-02T02:24:23.260" Score="6" Body="&lt;p&gt;If math is displayed in a web page using &lt;a href=&quot;http://www.mathjax.org/resources/articles-and-presentations/accessible-pages-with-mathjax/&quot;&gt;MathJax&lt;/a&gt;, which uses LaTeX syntax for math, then it is possible to configure MathPlayer to read the math text out loud.&lt;/p&gt;&#xA;" OwnerUserId="204" LastActivityDate="2014-04-02T02:24:23.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1212" PostTypeId="2" ParentId="1200" CreationDate="2014-04-02T02:56:27.197" Score="8" Body="&lt;p&gt;This makes me think of the intensional/extensional distinction. Here's a sentence you probably see all the time: &quot;Consider the function $f(n) = n^{2}$.&quot; Of course, on the natural numbers, this function yields the same set of ordered pairs as the rule $n \mapsto \sum_{k=1}^{n} 2k-1$ that says &quot;given $n$, return the sum of the first $n$ odd numbers&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are these the &quot;same function&quot;? Sometimes the right answer is yes, and sometimes it's no---it depends on the context. I don't think such subtleties should be entirely glossed over in a math class, but I do think it's important to cultivate in one's students an &lt;em&gt;ability to suffer abuses of terminology in stride&lt;/em&gt;, and moreover to default to the intended level of abstraction in standard contexts. This is the same ability an author relies upon any time they write (or fail to write) &quot;by an abuse of notation&quot;.&lt;/p&gt;&#xA;" OwnerUserId="327" LastActivityDate="2014-04-02T02:56:27.197" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1213" PostTypeId="5" CreationDate="2014-04-02T03:28:43.317" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-02T03:28:43.317" LastActivityDate="2014-04-02T03:28:43.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1214" PostTypeId="4" CreationDate="2014-04-02T03:28:43.317" Score="0" Body="For questions about participation from students in the classroom setting, either motivating them to do so or working with issues related to their participation." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-02T03:51:27.833" LastActivityDate="2014-04-02T03:51:27.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1215" PostTypeId="2" ParentId="1200" CreationDate="2014-04-02T03:31:38.243" Score="23" Body="&lt;p&gt;The other answers are good, and/but some of those points are reinforced by a diagnostic that has ever-more been important to me: is intention being communicated? And a different, more linguistic point mentioned in other answers: toleration of ambiguity, allowing context (where intent is adequately clear) to disambiguate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, yes, it is perverse to entrap/prank little kids by exaggerating the distinction between a thing and its name, a referent and the reference. Sure, it &lt;em&gt;can&lt;/em&gt; matter, but it's not usually the main issue. Such a distinction &lt;em&gt;can&lt;/em&gt; be presented as a funny thing, but not to prank kids. It's not constructive, although it has interest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics does lend itself to contriving formally-correct but pointless, non-constructive entrapment situations, &quot;gotchas&quot;. Pranking people, whether young or old, by perpetrating such stunts is close to immoral, and certainly mildly anti-social. Certainly misrepresents what I consider to be the (positive) true nature of mathematics (which, yes, I admit is not much reflected in k-12 or undergrad math, but...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, operationally, what do we &lt;em&gt;do&lt;/em&gt; with fractions &quot;in real life&quot;? Not what we might do formally, formal-linguistically, formal-mathematically, formal-logically. Don't create fake issues, fake distinctions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, then, there is the sad side-effect of belief in the essential formality of mathematics that we are supposedly &quot;not allowed&quot; to use context-dependent or ambiguous language. Of course we are &quot;allowed&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... but while small kids probably do have experience with context-disambiguation of usage of ordinary language, they will &lt;em&gt;not&lt;/em&gt; have been taught (give the ambient tradition) that they should exercise common sense for mathematical issues. Pity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps just as &quot;naive set theory&quot; is much more useful than &quot;formal set theory&quot; to the rest of mathematics, and similarly &quot;naive category theory&quot; (my own term) rather than &quot;formal...&quot;, perhaps &quot;naive mathematics&quot; is more useful than &quot;formal mathematics&quot;. For me this is a guide to how to explain things... Honesty, rather than &quot;formal correctness&quot;. &lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-02T03:31:38.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1216" PostTypeId="5" CreationDate="2014-04-02T04:14:41.117" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-02T04:14:41.117" LastActivityDate="2014-04-02T04:14:41.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1217" PostTypeId="4" CreationDate="2014-04-02T04:14:41.117" Score="0" Body="For questions about teaching and motivating students who are engaged in studying the physical sciences, namely biology, chemistry, and physics." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-02T06:36:35.080" LastActivityDate="2014-04-02T06:36:35.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1218" PostTypeId="2" ParentId="1204" CreationDate="2014-04-02T04:26:50.980" Score="2" Body="&lt;p&gt;Our eyes and visual cortex are fabulous tools for modeling scenarios, processing information, and communication. It's a disservice to students to ditch the visual approach because it's a &quot;geometric diversion&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 space and 3 space are familiar and can be visualized. Presenting coordinates as vertices of polygons or polyhedra can give a visually oriented student a handle on what the numbers can represent. Showing that matrices can scale, rotate, flip or skew these objects can engage the interest of a student that might otherwise see linear algebra as abstract concepts with little application.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your question, if vector representation of 3-space is used, it's a good idea to talk about cross products. Not only is the cross product a &lt;a href=&quot;http://hopsblog-hop.blogspot.com/2013/07/study-of-eschers-print-gravity.html&quot; rel=&quot;nofollow&quot;&gt;fun and useful tool for making stuff&lt;/a&gt;, but it also gives an opportunity to talk about determinants.&lt;/p&gt;&#xA;" OwnerUserId="603" LastEditorUserId="603" LastEditDate="2014-04-02T16:21:00.320" LastActivityDate="2014-04-02T16:21:00.320" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1219" PostTypeId="5" CreationDate="2014-04-02T04:40:39.437" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-02T04:40:39.437" LastActivityDate="2014-04-02T04:40:39.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1220" PostTypeId="4" CreationDate="2014-04-02T04:40:39.437" Score="0" Body="For questions about &quot;gifted&quot; students, i.e. students who are talented, motivated, and predisposed to studying mathematics." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-02T08:35:54.987" LastActivityDate="2014-04-02T08:35:54.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1221" PostTypeId="5" CreationDate="2014-04-02T04:42:32.733" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-02T04:42:32.733" LastActivityDate="2014-04-02T04:42:32.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1222" PostTypeId="4" CreationDate="2014-04-02T04:42:32.733" Score="0" Body="For questions about addressing students with varying abilities, e.g. a course with both math majors and non-majors, or a course where some students succeeded in a prerequisite while others didn't." OwnerUserId="80" LastEditorUserId="39" LastEditDate="2014-12-13T21:19:50.550" LastActivityDate="2014-12-13T21:19:50.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1223" PostTypeId="5" CreationDate="2014-04-02T04:43:34.023" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-02T04:43:34.023" LastActivityDate="2014-04-02T04:43:34.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1224" PostTypeId="4" CreationDate="2014-04-02T04:43:34.023" Score="0" Body="For questions about identifying, promoting, and fostering intellectual creativity in students of mathematics." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-02T08:35:20.863" LastActivityDate="2014-04-02T08:35:20.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1225" PostTypeId="5" CreationDate="2014-04-02T04:44:46.363" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-02T04:44:46.363" LastActivityDate="2014-04-02T04:44:46.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1226" PostTypeId="4" CreationDate="2014-04-02T04:44:46.363" Score="0" Body="For questions about effectively motivating and teaching the concepts of trigonometry, including the unit circle, the sin/cos/tan functions, and other related ideas." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-02T08:35:44.967" LastActivityDate="2014-04-02T08:35:44.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1227" PostTypeId="2" ParentId="1200" CreationDate="2014-04-02T08:19:44.697" Score="8" Body="&lt;p&gt;This is strictly context dependent. Consider the following sentence:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Is banana a fruit?&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Well, is it? The answer, of course, depends on what exactly you're talking about. It can be interpreted in two ways:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;1. Is the physical object known as banana a fruit?&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;In this case, yes, it obviously is. But consider this interpretation:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;2. Is the string of characters &quot;banana&quot; a fruit?&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;In this case, again, &lt;em&gt;obviously&lt;/em&gt;, answer is no. The string &quot;banana&quot; merely represents a fruit called banana.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in conclusion, I'd say that both &quot;X is a number&quot;, and &quot;X represents a number&quot;, are corect, in their suitable contexts. Which one you should use is depended on what you are trying to comunicate.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2014-04-02T08:19:44.697" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1228" PostTypeId="2" ParentId="1204" CreationDate="2014-04-02T08:48:52.413" Score="18" Body="&lt;p&gt;The problem is that &lt;em&gt;there are multiple main thrusts&lt;/em&gt; of linear algebra. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;For many students (especially pure math majors), linear algebra is their first introduction to &lt;em&gt;abstract algebra&lt;/em&gt;. When a course is designed with this in mind, rigor and formalism takes higher precedence as we want as much of the material to be reusable in the context of modules over rings and also multilinear algebra&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;On the other hand, even in the world of pure mathematics, linear algebra is fundamental in the development of differential geometry, functional analysis (&quot;infinite dimensional linear algebra&quot;), and differential equations (including dynamical systems, stability analysis, and PDEs). With these applications in mind, it would only make sense to introduce to the students appropriate &lt;em&gt;geometric intuition&lt;/em&gt; behind linear transformations. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Outside of pure mathematics, linear algebra is often viewed as synonymous with &quot;stuff having to do with matrices&quot;. For these applications the crucial points of understanding are how to manipulate matrices to facilitate analysis and computation (think PCA and SVD, computationally extracting information from the matrix &lt;em&gt;without&lt;/em&gt; inverting it). &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you are somehow able to teach a class addressing all of the above, more the power to you. But the trends that I have been familiar with is to have linear algebra classes designed with different applications in mind: at the very least often times math majors end up taking a different version of the class compare to the engineers and economists. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Going back to your question about cross-products: using a strict definition of linear algebra as the study of linear transformations of vector spaces, then cross products do not belong to linear algebra. It requires necessarily some &lt;em&gt;multilinear&lt;/em&gt; input, be it a quadratic form (an inner product, so we can talk about orthogonality), or a volume form (to convert from two-vectors to one-vector by Hodge dual). On the other hand, these topics may already be covered in a linear algebra class (which usually do not restrict itself to the strict definition anyway), and in this case cross product may be mentioned as an application. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;But&lt;/strong&gt; if the cross product is mentioned, care must be given to the fact that it behaves &quot;weirdly&quot; under coordinate transformations. (The result of a cross product is a pseudovector.) One of the themes of linear algebra (at least for the first two &quot;thrusts&quot; that I describe) is the possibility of thinking about things in a manner that is coordinate independent (which should certainly be part of any &quot;geometric intuition&quot;). The formula used to compute the cross product, as it is usually taught, is certainly &lt;em&gt;not&lt;/em&gt; coordinate independent. (I have seen students trying to compute the cross product in a spherical coordinate system.) There are a lot of subtle issues involved that may bring the discussion in conflict with some of the earlier themes of the class, and care must be given to resolve them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I taught linear algebra, I did not talk about the cross product in any great detail (it was not on the centrally-set syllabus; a student brought it up when we talked about computing determinants for 3-by-3 matrices). Looking at the above, it would be hard to fit it into an already tight schedule: to do justice to the subject it will have to be at least one full lecture just on its properties. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect somehow that this is the heart of your problem: many of your students must have seen the cross product before. But the amount of geometry related to the cross product and its properties for a computer graphics course is likely a bit more involved than the cursory discussion the cross product is afforded in any other introductory level mathematics class. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As a side remark: the first time I saw cross products was not in a math class at all! This was in a high school &lt;em&gt;physics&lt;/em&gt; class where it naturally came up in the discussion of torque and angular momentum. (Yes, &lt;em&gt;American&lt;/em&gt; high school.) But I wouldn't claim to have a good understanding of the cross product until having studied multilinear algebra and differential geometry.&lt;/p&gt;&#xA;" OwnerUserId="125" LastActivityDate="2014-04-02T08:48:52.413" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1229" PostTypeId="2" ParentId="1200" CreationDate="2014-04-02T10:02:42.313" Score="6" Body="&lt;p&gt;In elementary education, &quot;is&quot; would be the right term since students hardly know what &quot;represent&quot; means. However in teacher education when discussing the rationals, it's different. Once you understood equivalence relations, you should notice that they appear many times. Teachers should be aware of the fact that operations might not be well-defined in these contexts. This can prevent mistakes and even help students to see that some operations (like cancelling digits in a fraction) cannot be correct.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-04-02T10:02:42.313" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1230" PostTypeId="2" ParentId="1200" CreationDate="2014-04-02T10:19:53.980" Score="1" Body="&lt;p&gt;I'd personally say the hint is in the name &quot;fraction.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A fraction of a cake is still cake. So, a fraction of a number is still a number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But also, seriously, a fraction is essentially a sum, a half is (1/2), id est... a number. But this produces the argument, does a sum represent the resultant number, or is it in itself THE resultant number?&lt;/p&gt;&#xA;" OwnerUserId="606" LastActivityDate="2014-04-02T10:19:53.980" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1231" PostTypeId="2" ParentId="995" CreationDate="2014-04-02T11:45:59.900" Score="12" Body="&lt;p&gt;Chemical reaction rates and how they relate to changes in temperature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use this example when teaching chemical engineering students about integrals.&#xA;The Boltzmann distribution describes the velocity distribution of molecules in a gas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f(v) = \sqrt{\frac{2}{\pi} \left( \frac{m}{k_B \cdot T} \right)^3}&#xA;		\cdot v^2 \cdot \exp\left( \frac{-m v^2}{2 k_B T} \right)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The integral $\int_a^b f(v) dv$ is then the fraction of the gas molecules with a velocity in the range  $[a, b]$.&#xA;For a certain reaction to occur there must be a minimal amount of kinetic energy present (the activation energy, $E_a$), from that the lowest possible velocity of a gas molecule with this kinetic energy can be determined by&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$v_\textrm{min} = \sqrt{\frac{2 E_a}{m}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fraction of gas molecules that can participate in the reaction is then given by &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_{v_\textrm{min}}^{\infty} f(v) \, dv$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can be used to investigate how changes in temperature alters the fraction of gas molecules that have sufficient kinetic energy to overcome the activation energy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One application is to test the rule of thumb, which states that an increase in temperature of 10 kelvin will roughly double the reaction rate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below is Mathematica code for solving this&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;assump = Assumptions -&amp;gt; {k &amp;gt; 0, T &amp;gt; 0, &#xA;   m &amp;gt; 0}; MaxwellBoltzmannSpeedDistribution =&#xA; Sqrt[2/Pi (m/(k T))^3] v^2 Exp[-m v^2/(2 k T)];&#xA;FractionOfParticlesWithEnergyAbove = &#xA;  Integrate[&#xA;   MaxwellBoltzmannSpeedDistribution, {v, minVelocity, Infinity}, &#xA;   assump];&#xA;TemperatureDependency = &#xA; FractionOfParticlesWithEnergyAbove //. {minVelocity -&amp;gt; &#xA;    Sqrt[2 10^-19/m], k -&amp;gt; 1.3806488*10^(-23), m -&amp;gt; 1.6726 10^-27}&#xA;res = Table[{T, TemperatureDependency} /. {T -&amp;gt; Tval}, {Tval, 293, &#xA;    343, 10}];&#xA;res // TableForm&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The above code gives the following output&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;293   1.05132*10^-10&#xA;303   2.33909*10^-10&#xA;313   4.94258*10^-10&#xA;323   9.96656*10^-10&#xA;333   1.926*10^-9&#xA;343   3.58024*10^-9&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="607" LastActivityDate="2014-04-02T11:45:59.900" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1232" PostTypeId="2" ParentId="1002" CreationDate="2014-04-02T13:52:56.650" Score="1" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Fuzzy_logic&quot; rel=&quot;nofollow&quot;&gt;Fuzzy logic&lt;/a&gt; &quot;is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. Compared to traditional binary sets (where variables may take on true or false values) fuzzy logic variables may have a truth value that ranges in degree between 0 and 1.&quot;  It &quot;has been applied to many fields, from control theory to artificial intelligence.&quot;  You may have heard of appliances such as washing machines that use fuzzy logic.  An example of a fuzzy logic statement is &quot;If the temperature is hot then speed up the fan.&quot;  (Note that &quot;hot&quot; and &quot;speed up&quot; take on a range of values.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Temporal_logic&quot; rel=&quot;nofollow&quot;&gt;Temporal logic&lt;/a&gt; &quot;is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time.&quot;  I can't give a reference right now, but I've seen it used to describe the operation of elevators.  An example of a temporal logic statement is &quot;The elevator will eventually stop.&quot;  (Note that the truth of a statement can vary with time.)&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-04-02T13:52:56.650" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1233" PostTypeId="2" ParentId="992" CreationDate="2014-04-02T14:33:05.997" Score="9" Body="&lt;p&gt;The correct approach to making accessible mathematics is to publish it in &lt;a href=&quot;http://www.w3.org/Math/&quot;&gt;MathML&lt;/a&gt;.  This is the web standard for mathematics and is what all web browsers are &lt;strong&gt;meant&lt;/strong&gt; to be working towards to support.  Sadly, &lt;a href=&quot;http://www.cnet.com/news/google-subtracts-mathml-from-chrome-and-anger-multiplies/&quot;&gt;not all&lt;/a&gt; &lt;a href=&quot;http://news.dessci.com/2013/10/microsoft-cripples-display-math-ie10-11.html&quot;&gt;are&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are lots of ways of creating MathML content, some work directly from LaTeX input (I've written one of these) and some are a little more creative.  MathJaX is a bit of a hybrid in that the conversion only happens in the browser (so the content transferred over the internet need not be MathML).  This sounds fine, except that not everyone wants to read their documents in a web browser and MathML is used in more technologies, such as ePub 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as reading goes, once it is in MathML then a screen reader &lt;strong&gt;ought&lt;/strong&gt; to be able to read it.  This will depend on the screen reader and the web browser/other software being used.  Again, the ideal and the reality are not yet in alignment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many people who know a lot about this, and far more than I do.  A lot of them subscribe to the &lt;a href=&quot;http://www.nfbnet.org/mailman/listinfo/blindmath_nfbnet.org&quot;&gt;blindmath mailing list&lt;/a&gt; (US based, but not in terms of discussion).  Other links worth investigating are &lt;a href=&quot;http://access2science.com/index.html&quot;&gt;Access 2 Science&lt;/a&gt; and &lt;a href=&quot;http://access2science.com/index.html&quot;&gt;Design Science&lt;/a&gt; (the latter is a commercial company who do a lot of work in this line).&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-04-02T14:33:05.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1234" PostTypeId="1" CreationDate="2014-04-02T14:35:47.573" Score="12" ViewCount="1038" Body="&lt;p&gt;ALEKS (&lt;a href=&quot;http://www.aleks.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.aleks.com/&lt;/a&gt;) is a good way of learning procedural math, because it is very systematic and forces you to master the dependencies of a kind of problem before working on that kind of problem itself. Also the interface is very clean and user-friendly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The method of ALEKS seems especially well-suited to introductory logic (truth tables, symbolizing English sentences, truth trees, proofs in SL or PL, etc.), but ALEKS doesn't have a logic program. Has anyone else made something like this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note: I asked this question on math.SE and nobody has answered it for a long time so I'm asking it here. Here is the link to the post on math.SE: &lt;a href=&quot;https://math.stackexchange.com/questions/722363/is-there-a-program-like-aleks-for-mathematical-logic&quot;&gt;https://math.stackexchange.com/questions/722363/is-there-a-program-like-aleks-for-mathematical-logic&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="612" LastEditorUserId="-1" LastEditorDisplayName="user230" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2017-01-25T10:29:04.447" Title="Is there a program like ALEKS for mathematical logic?" Tags="|reference-request|technology-in-education|logic|curriculum|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1235" PostTypeId="2" ParentId="175" CreationDate="2014-04-02T15:29:33.813" Score="7" Body="&lt;p&gt;Yes, required reading of the text is effective. I know because my students say so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's that, you say? Your students not only do the reading, but also say that they like doing it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes. And here's how I make them do it: 10% of their grade is ``examples.'' They are required to bring to class, 10 times through the semester, a handwritten copy of the examples in the part of the book that we are covering that day. (you can obviously change these numbers). I skim them to ensure they're actually what I ask for, then return them, so it's not a lot of extra work for me. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hand-copying the examples is essentially impossible without at least trying to learn, and for most students it's the first time they've EVER tried reading a math book, which they don't realize means doing out the examples. So I just demand the latter, and it turns out that in this process, they (normally) do the former.&lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-04-02T15:29:33.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1236" PostTypeId="1" AcceptedAnswerId="1394" CreationDate="2014-04-02T16:32:56.743" Score="18" ViewCount="3479" Body="&lt;p&gt;This question could be very broad, and so I'd like to make it more specific.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the United States, mathematics educators generally work in secondary education or college education (elementary school specialists exist but are rare).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondary educators tend to have scheduled work for most of every day, as well as grading homework during preparation time and at home.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;College educators can range from research professors teaching two or three classes a year (each of which meets for 3 hours a week) to adjuncts teaching up to 24 hours a week (as one colleague of mine does).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To those who have been employed in both secondary and college education, is there a level of college teaching that feels similar to high-school teaching (i.e. is teaching high school similar to teaching 4 college classes a semester)? If college education is too dissimilar to compare to high school/middle school, what are the key differences that set them apart?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Please feel free to edit this question to make it more understandable.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="2083" LastEditDate="2021-04-15T19:14:17.877" LastActivityDate="2021-04-15T19:14:17.877" Title="How does the workload of a high-school mathematics teacher compare to a university-level instructor?" Tags="|undergraduate-education|secondary-education|usa|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1238" PostTypeId="2" ParentId="771" CreationDate="2014-04-02T17:03:38.200" Score="6" Body="&lt;p&gt;The question is broad, so my answer will be as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students who have been in school for a sufficient length of time have probably learned highly effective strategies for accomplishing their scholastic goals -- whatever those may be. You (it sounds to me) have identified that the learning strategies many of your students have adopted (and likely adopted for rational reasons) are ineffective for them when it comes to learning what you want them to learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Put pithily, this might also be said &quot;what you are trying to teach is simply different from what they are trying to learn.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This cannot be simply corrected by informing them that their learning strategies are broken. Their learning strategies were adopted due to their success in the past, and most students will not re-evaluate these strategies until they witness their failure. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, generalities aside, how do you correct this? I use a bit of dishonesty. When I've identified a type of material that I want my students to learn, but that their learning strategies prevent them from learning, I take the following four step procedure in teaching that material:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I teach the material, being explicit about the types of questions I expect them to be able to answer.&lt;/li&gt;&#xA;&lt;li&gt;I give them a sample exam/quiz question with the correct answer to it as a &quot;study guide.&quot;&lt;/li&gt;&#xA;&lt;li&gt;I ask them in an exam/quiz a question that is almost exactly (or exactly) the same as the one on the study guide. Many students will fail at this point, despite having the study guide. This is (I believe) not mostly due to laziness, as we are quick to assume, but due to the students not recognizing that their learning strategies were ineffective. This failure represents their being confronted by that fact.&lt;/li&gt;&#xA;&lt;li&gt;Finally, the dishonest bit, I forgive the initial test (which I knew all along I would do, but they did not), and allow a second attempt at a nearly-identical test. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In my experience this procedure accomplishes two things: (a) it ends with a larger percentage of the students knowing the material and (b) they learn a new way to learn. Both of these are valuable.&lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-04-02T17:03:38.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1239" PostTypeId="2" ParentId="1149" CreationDate="2014-04-02T17:43:47.577" Score="9" Body="&lt;p&gt;I spend 4 (50 minute) days on determinants and one homework assignment. This is a course mostly made up of engineers and some economists/social scientists, taught out of &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0321796942&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bretscher's book&lt;/a&gt;.  Here is what I said the most recent time I did it:&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Day One&lt;/b&gt; Derive how &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; determinants come up in computing areas of parallelograms and triangles, in inverting &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; matrices and in checking whether &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; matrices are invertible. Assert without proof that the &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; determinant formula has similar properties. State the multilinear properties of determinant and work through them from as many perspectives as I can: How can we prove them from the formulas?  Why is area multilinear (up to sign)? The area discussion again with a topologically different configuration of vectors, so they can see the signs popping up. If &lt;span class=&quot;math-container&quot;&gt;$(u,v,w)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(u,v,w')$&lt;/span&gt; are dependent, why should &lt;span class=&quot;math-container&quot;&gt;$(u,v,a w+ a' w')$&lt;/span&gt; be dependent for scalars &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a'$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Day Two&lt;/b&gt; Assert that there is a multlinear formula, like determinant, for &lt;span class=&quot;math-container&quot;&gt;$n \times n$&lt;/span&gt; matrices. Show how to compute it using row reduction just from multilinearity, and how to take short cuts for upper triangular or block upper triangular matrices.&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Day Three&lt;/b&gt; Give the formula in terms of a sum of &lt;span class=&quot;math-container&quot;&gt;$n!$&lt;/span&gt; permutations but emphasize that we almost never want to use it for &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;3$&lt;/span&gt;. Show that it is multlinear and alternating; don't worry if they don't really follow this. Emphasized that I am only presenting this so that they know determinant exists; they shouldn't think in terms of this formula.&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Day Four&lt;/b&gt; Determinants as volume. To some extent, this repeats day 1, but by now they are used to the formulas involved. Talk about why you might want to divide a region into a nonsquare grid and sum to numerically compute an integral, so they need to know volumes of the grid pieces. (Multivariable calc isn't formally a prereq, so I can't do anything deep with multivariate integrals, but a lot of them have it.)  Describe the geometry of the formula &lt;span class=&quot;math-container&quot;&gt;$\det (AB) = (\det A)(\det B)$&lt;/span&gt;. (I also prepared a &lt;a href=&quot;http://www.math.lsa.umich.edu/%7Espeyer/417/DetMult.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;handout&lt;/a&gt; with a rigorous proof for those who were interested.)&lt;/p&gt;&#xA;&lt;p&gt;Of your list, I never do Cramer's rule or adjugates. They are useful for algebraists (like me!) but their applied use is made obsolete by computers. I spend a lot of time on using multilinearity and on geometric thinking. Cofactor expansion isn't presented as a separate subject, but when I talk about multilinearity, I do examples like:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\det \begin{pmatrix} 1 &amp;amp; 2 &amp;amp; 3 \\ 4 &amp;amp; 5 &amp;amp; 6 \\ 7 &amp;amp; 8 &amp;amp; 9 \end{pmatrix} = &#xA; \det \begin{pmatrix} 1 &amp;amp; 0 &amp;amp; 0 \\ 4 &amp;amp; 5 &amp;amp; 6 \\ 7 &amp;amp; 8 &amp;amp; 9 \end{pmatrix} &#xA;+\det \begin{pmatrix} 0 &amp;amp; 2 &amp;amp; 0 \\ 4 &amp;amp; 5 &amp;amp; 6 \\ 7 &amp;amp; 8 &amp;amp; 9 \end{pmatrix} &#xA;+ \det \begin{pmatrix} 0 &amp;amp; 0 &amp;amp; 3 \\ 4 &amp;amp; 5 &amp;amp; 6 \\ 7 &amp;amp; 8 &amp;amp; 9 \end{pmatrix} =$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\det \begin{pmatrix} 5 &amp;amp; 6 \\ 8 &amp;amp; 9 \end{pmatrix} - 2 &#xA;\det \begin{pmatrix} 4 &amp;amp; 6 \\ 7 &amp;amp; 9 \end{pmatrix} +3\det \begin{pmatrix} 4 &amp;amp; 5 \\ 7 &amp;amp; 8 \end{pmatrix} . $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;Update: Thanks to COVID, I now have lecture videos to go with these! I've rearranged the order since I wrote this answer.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/rTI-SAeN-b4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Video 1&lt;/a&gt; (&lt;a href=&quot;https://dept.math.lsa.umich.edu/%7Espeyer/LinearAlgebraVideos/Lecture7a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;slides&lt;/a&gt;): The properties of the determinant. More or less day 2 above.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/nLgvYTsWKsI&quot; rel=&quot;nofollow noreferrer&quot;&gt;Video 2&lt;/a&gt; (&lt;a href=&quot;https://dept.math.lsa.umich.edu/%7Espeyer/LinearAlgebraVideos/Lecture7b.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;slides&lt;/a&gt;): Geometry of determinants. More or less days 1 and 4.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/l3zGv5c-lxE&quot; rel=&quot;nofollow noreferrer&quot;&gt;Video 3&lt;/a&gt; (&lt;a href=&quot;https://dept.math.lsa.umich.edu/%7Espeyer/LinearAlgebraVideos/Lecture7c.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;slides&lt;/a&gt;): The formula for the determinant. More or less day 3.&lt;/p&gt;&#xA;&lt;p&gt;This class had prerecorded lecture videos (like these), plus  interactive Zoom sessions with group work and discussion. If I recall correctly, a number of the motivational pictures in video 2 were drawn informally on zoom on the first day.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2024-01-11T12:02:30.640" LastActivityDate="2024-01-11T12:02:30.640" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="1240" PostTypeId="1" AcceptedAnswerId="1282" CreationDate="2014-04-02T19:03:43.037" Score="13" ViewCount="295" Body="&lt;p&gt;In a typical course (proof-based, for the context of this question) I have about 50 exercises that I might want to use when I teach the course again. Each exercise might be used in the same way as before, or perhaps in a modified form (for difficulty or dependence on course material/pacing). Sometimes, I might even have exercises that are based on old ones, so I must not forget to include the prerequisite exercises earlier if I want to assign the later ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is a good method to keep track of old exercises?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, a good method should allow one to:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;easily store exercises to be recycled later (even perhaps in a different course),&lt;/li&gt;&#xA;&lt;li&gt;search for exercises related to a certain topic,&lt;/li&gt;&#xA;&lt;li&gt;search for exercise that make use of a certain technique or idea,&lt;/li&gt;&#xA;&lt;li&gt;keep track of any dependencies,&lt;/li&gt;&#xA;&lt;li&gt;make note of difficulty levels&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and perhaps more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have any methods that have proven effective at addressing these goals, or others? Is there software that would aid such a task?&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="80" LastEditDate="2014-04-02T19:18:07.470" LastActivityDate="2014-04-07T10:32:43.287" Title="How to keep track of your old exercises in a heavy proof based course?" Tags="|exams|homework|technology-in-education|organization|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1241" PostTypeId="2" ParentId="1240" CreationDate="2014-04-02T19:45:35.263" Score="6" Body="&lt;p&gt;I have a colleague who TeXed up a whole bunch of questions as \newcommands in one file. This way he can make his homeworks/quizzes/exams by including this source file, and quickly calling the questions. With appropriate naming of the questions, it becomes easy to look back and see what you've used and haven't used already.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-04-02T19:45:35.263" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1242" PostTypeId="2" ParentId="1147" CreationDate="2014-04-02T19:49:42.470" Score="0" Body="&lt;p&gt;The RRT approach provides a significant advantage when you are trying to define Lebesgue measure. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The standard method is to construct Lebesgue measure using exterior measure. This is a little messy. In particular, it involves proving the Caratheodory extension theorem. You then must show that integrating a nice function against this measure gives the same result as the Riemann integral. The proof of this equivalence is messy and unintuitive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, suppose we already have the RRT in hand. Consider the positive function $L(f)$ on $C_c(\mathbb R^n)$ given by $f\rightarrow \int f$, where the integral is the Riemann integral. By the RRT, this gives you a measure, and it is clear by construction that integration against this measure gives the same result as the Riemann integral for all suitably nice functions. Simple! &lt;/p&gt;&#xA;" OwnerUserId="9" LastActivityDate="2014-04-02T19:49:42.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1243" PostTypeId="2" ParentId="22" CreationDate="2014-04-02T19:55:43.363" Score="8" Body="&lt;p&gt;The most striking easy application of the implicit function theorem is in my opinion the regularity of roots of polynomials in term of their coefficients. The only issue is that you have to deal with multiplicity to state the full version, but understanding what happens around a simple root is already interesting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, you can interpret the implicit function theorem as saying that &quot;a smooth family of equations have solutions that vary smoothly&quot;, and it is easy to build &lt;em&gt;ad hoc&lt;/em&gt; examples.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-02T19:55:43.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1244" PostTypeId="2" ParentId="1161" CreationDate="2014-04-02T20:04:59.763" Score="6" Body="&lt;p&gt;Student should know some groups coming from geometry, which are often overlooked (sure, $A_4$, $S_n$, $A_5$, $D_n$ and others previously cited do arise in geometry...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion the most important examples are the affine groups $\mathrm{Aff}(\mathbb{R}^n)$ (in fact, already $\mathrm{Aff}(\mathbb{R})$ is great): it helps remember how semidirect products work (which subgroup acts on which one? Linear maps act on translations; which one is normal? Affine maps have a linear part so the set of translations is the kernel of a map, hence is normal). I consider I really understood skew product when I understood  $\mathrm{Aff}(\mathbb{R})$.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2014-04-03T09:30:28.973" LastActivityDate="2014-04-03T09:30:28.973" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1245" PostTypeId="1" AcceptedAnswerId="1248" CreationDate="2014-04-02T20:07:24.210" Score="33" ViewCount="5261" Body="&lt;p&gt;I've noticed that series are one of the most difficult portions of calculus for new students to learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the level of abstraction has to do with this. Limits, derivatives, and integrals, as well as continuous functions, all have helpful visual meanings that give students good intuition (derivatives are slopes, integrals are area, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This intuition helps them remember certain facts. Why is the derivative of a constant zero? Because the slope of a horizontal line is zero. Why is $\int_a^a f(x)\, dx=0$? Because a line segment has zero area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There seems to be no similar intuition in my calculus courses. If I ask, &quot;Does $\Sigma_{n=0}^\infty 1$ converge or diverge?&quot;, many students have to use some test and often get it wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even more, to the students, there seems to be no motivation behind series. Derivatives are velocities, and integrals give area and volume, but series don't seem to many students to have similar uses. This compounds the intuition problem.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What visual intuiton or real-life motivation will help students understand series the same way that slopes or velocities tell students understand derivatives? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastEditorUserId="114" LastEditDate="2014-04-11T12:26:51.287" LastActivityDate="2023-08-29T16:40:50.590" Title="For calculus students, what should be the intuition or motivation behind series?" Tags="|calculus|concept-motivation|intuition|series|" AnswerCount="12" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1246" PostTypeId="2" ParentId="258" CreationDate="2014-04-02T20:24:20.297" Score="13" Body="&lt;p&gt;Here's one I haven't seen mentioned so far: students coming into calculus are very uncomfortable with the idea that there might be more than one way to do something, and especially that they might have to make decisions, especially decisions that depend on context rather than just a rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two concrete examples of this phenomenon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Simplifying.  I see students routinely oversimplifying in the middle of problems---half-way through, they'll do a bunch of work to put an expression into some canonical form, even if it makes the next step in the problem much harder.  They're happy to learn &quot;don't simplify at all ever&quot;, or even &quot;simplify in these problems but not these&quot;, but it's a real struggle to get them to ask, &quot;Can I simplify here, and if so, will it help or hurt what I do next?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Lines.  My students believe that the only way to write the equation of a line is in slope-intercept form.  Present a line in another form and they're not even sure it is a line; ask them for a line in point-slope form and they'll typically force it into slope-intercept anyway.  (For specific reasons about material, this is a particular obstacle: working with the linear approximation to a function lends itself very naturally to dealing with the point-slope form.)  And they're very uncomfortable with the idea that there are several ways to write lines, and that one should &lt;em&gt;choose&lt;/em&gt; which way based on the application you have in mind.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-04-02T20:24:20.297" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1247" PostTypeId="2" ParentId="1245" CreationDate="2014-04-02T20:27:16.330" Score="9" Body="&lt;p&gt;Unfortunately, my example is not a full answer to your question, but I think, it helps students to be at least beware of what can happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can explain series as a inifite summation of areas (at least as long as everything is non-negative). And convergence of a series means: Is there a big rectangle such that all the given areas fit in? Intuitively, most students would say in general: That is not possible since there is always coming more to the sum. But a simple example of the geometric series can be given in a picture:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/tY1r8.png&quot; alt=&quot;geometric series&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also take some more sophisticated picture like the &lt;a href=&quot;http://en.wikipedia.org/wiki/Koch_snowflake&quot; rel=&quot;noreferrer&quot;&gt;Koch snowflake&lt;/a&gt; and ask about the area. (This example illustrates the concept of partial sum: At some step n of the snowflake, the given area is the partial sum while in the series you add up the area of &lt;em&gt;all&lt;/em&gt; triangles of a given size.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/3Zgyv.gif&quot; alt=&quot;Koch snowflake&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-02T20:40:05.877" LastActivityDate="2014-04-02T20:40:05.877" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1248" PostTypeId="2" ParentId="1245" CreationDate="2014-04-02T20:28:30.773" Score="23" Body="&lt;p&gt;In my experience, one of the problems with series is that usually you have two sequences if you investigate the series $\sum(a_n)$: the sequence $(a_n)$, and the sequence of partial sums $S_n=a_1+\ldots + a_n$. I noticed that trying to stress this distinction helps a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To the intuition, I like R. Péter: &lt;a href=&quot;http://books.google.hu/books?id=6V3wNs4uv_4C&amp;amp;lpg=PA110&amp;amp;ots=BjUWaHdPbb&amp;amp;dq=infinite%20series%20chocolate%20example&amp;amp;hl=de&amp;amp;pg=PP1#v=onepage&amp;amp;q=infinite%20series%20chocolate%20example&amp;amp;f=false&quot;&gt;Playing with Infinity&lt;/a&gt;, the chocolate bar example on page 105.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A well-known mathematician (while still a child) interpreted the&#xA;  meaning of the sum of an infinite geometric series in the following&#xA;  way.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;There was a type of chocolate which was popularized by putting a&#xA;  coupon in the wrapping, and anyone who could produce 10 such coupons&#xA;  would get another bar of chocolate in exchange. If we have such a bar&#xA;  of chocolate, what is it really worth?&#xA;       Of course it is worth more than just one bar of chocolate since there is a coupon in it, and for each coupon you can get $1/10$ of a&#xA;  bar of chocolate. But with this  $1/10$ of a bar will go one tenth of&#xA;  a coupon, and if for one coupon we get $1/10$ of a bar of chocolate,&#xA;  for $1/10$ of a coupon we get $1/100$ of a bar of chocolate. To this&#xA;  $1/100$ of a bar of chocolate belongs $1/100$ of a coupon, and for&#xA;  this we again get $1/10$ as much chocolate, i.e.  $1/1000$ of a bar of&#xA;  chocolate, and so on... This process can go infinitely, so that my one&#xA;  bar of chocolate together with its coupon is in fact worth &#xA;  $$1+ 1/10 +1/100 + 1/1000+...$$ bars of chocolate.&#xA;       On the other hand, this is exactly 10/9 of a bar of chocolate. 1 is the value of the actual chocolate and the coupon  that goes with it&#xA;  is 1/9  of a bar of chocolate. Indeed 9 coupons are worth one bar of&#xA;  chocolate because you could ask for such a bar and tell that you will&#xA;  pay after you eat it. So you eat it, take out the accompanying coupon,&#xA;  and give 10 coupons back which is exactly what is needed for one&#xA;  chocolate bar.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-04-02T20:28:30.773" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1249" PostTypeId="2" ParentId="1234" CreationDate="2014-04-02T20:28:40.823" Score="5" Body="&lt;p&gt;The most relevant piece of software of which I'm aware is called &lt;a href=&quot;http://www.cs.amherst.edu/~djv/pd/pd.html&quot;&gt;Proof Designer&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Proof Designer is a Java applet that writes outlines of proofs in elementary set theory, under the guidance of the user. It is designed to help students learn to write proofs. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It was created by &lt;a href=&quot;http://www.cs.amherst.edu/~djv/&quot;&gt;Dan Velleman&lt;/a&gt;, who (as you can see on his site) was the &lt;a href=&quot;http://www.maa.org/publications/periodicals/american-mathematical-monthly&quot;&gt;AMM&lt;/a&gt; editor 2007-2011.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first page linked to above includes a list of 64 suggested problems. (&lt;a href=&quot;http://www.cs.amherst.edu/~djv/pd/help/Problems.html&quot;&gt;Direct link&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(He has, at least in the past, used it as a part of his course on Mathematical Logic at Amherst College.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-04-02T20:28:40.823" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1250" PostTypeId="2" ParentId="1245" CreationDate="2014-04-02T21:26:21.603" Score="9" Body="&lt;p&gt;Another point that oughtn't be neglected was that, historically, numerical computation/approximation used Taylor-Maclaurin expansions (not to mention Newton-Raphson when convenient) to approximate root-taking. Newton was apparently very happy with his discovery of the binomial expansion for general exponents, although surely not only for numerical purposes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For historical verity, though, one perhaps should mention Poincare's discovery that some of the venerable series used in astronomical calculations were really only &lt;em&gt;asymptotic&lt;/em&gt; expansions, so gave good-up-to-a-point approximations, but were actually divergent. This phenomenon is already visible (once one knows to look) with irregular singular points of second-order O.D.E.'s&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-02T21:26:21.603" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1251" PostTypeId="2" ParentId="1245" CreationDate="2014-04-02T22:10:57.857" Score="15" Body="&lt;p&gt;(I have not done this exact presentation, so I cannot vouch for its efficacy. But I have used the main idea before, and it seems to help some students, and is at least a bit of fun. Also, this is meant to address the &lt;em&gt;intuition&lt;/em&gt; aspect of the question, not the motivation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Demonstrate by walking! Convergence means there's a spot that will be approached to any desired level of proximity, given infinite time. Divergence means a journey for which we cannot identify one specific destination.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Write the series $1+\frac12+\frac14+\frac18+\frac{1}{16}+\cdots$ on the board. Start at one wall of the classroom and take one giant step forward. Announce this to be your standard measurement of one step. Get a volunteer student (let's call him Joe) to go stand at the spot that is 1 &quot;step unit&quot; away from where you are now. Announce that Joe is your destination.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, take a step forward that is half as big as your first one. Ask the students how many &quot;step units&quot; you have traveled so far. Get another volunteer (Sam) to write this on the board. Take a $\frac14$ step forward. Ask them how many &quot;step units&quot; you have gone so far. Sam writes this down. Keep doing this, having Sam record this sequence of partial sums (you can even call it that now). After several iterations, they'll see that you're standing right in front of Joe. Ask them what will happen if you keep going.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Announce: &quot;This series &lt;em&gt;converges&lt;/em&gt; to 2. If I were to keep taking paces like this forever, I would never be able to step beyond Joe. I will never actually be standing &lt;em&gt;exactly&lt;/em&gt; where Joe is, but if you want me to get within a distance $d&amp;gt;0$ of Joe, I can definitely get there, no matter how small you make $d$. But the smaller $d$ you choose, the longer you'll have to wait. Mathematically, the &lt;em&gt;sequence of partial sums approaches 2&lt;/em&gt;.&quot; (At this point, you can even show them the pattern that the partial sums obey: $s_n=2-\frac{1}{2^n}$. Ask them for $\lim_{n\to\infty} s_n$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Go back and write this series on the board: $1+1+1+1+\cdots$. (I chose this based on your example in the question.) Start walking by taking paces of size &quot;1 step unit&quot;, having Sam record your sequence of partial sums. Walk straight out the door. Ask the students loudly from the hallway, &quot;What is my destination now?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Come back and announce: &quot;This series &lt;em&gt;diverges&lt;/em&gt;. The sequence of partial sums [point to $s_n=n$] grows without bound. This is an infinite journey that will not approach a specific place. I cannot tell Joe where to go and wait for me because, no matter where he stands, I will &lt;em&gt;eventually&lt;/em&gt; walk straight past him.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this will be a striking presentation and will at least demonstrate to students the different natures of con/divergence. You could even do this with other series (and later on, with any series that comes up, you can remind them of this discussion by just starting to take paces across the room).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take, for example, $1-1+1-1+1-1+\cdots$ and show that you just jump back and forth by 1 step and never settle on a specific place for Joe to wait for you. Then, take $1-\frac12+\frac14-\frac18+\cdots$ and tell Joe to wait at the spot corresponding to $\frac23$. Show that you'll step back and forth around him, getting closer and closer all the while. (Refer back to this when you do the &quot;alternating series test&quot;.) You could even bring in Sam for this part and show what the sequences of partial sums are doing.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As for the motivation behind series, I like to mention the idea of Taylor series right away, and state that they are both very powerful and very common, and can be used to approximate functions accurately and efficiently. I don't say anything yet about how to find them, but I try to give the students an idea of where/how/why they're useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I point out that the approach to many physics problems (at least in the courses they'll take) is &quot;Write the Taylor series and chop it off at $n=2$ (or even $n=1$).&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I ask them what they think their calculator is doing when they type in $\sin(0.35)$, say. How does it know what that number is? It's not like it has a big list of all the values of the sine function. There's infinitely-many! How does it spit out an answer?&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-02T22:16:44.490" LastActivityDate="2014-04-02T22:16:44.490" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1252" PostTypeId="1" CreationDate="2014-04-02T22:24:48.330" Score="25" ViewCount="2731" Body="&lt;p&gt;How to convince a middle school student that $0.50=0.5=0.500=\cdots$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used the fact that $0.50=\frac{5}{10}+\frac{0}{100}=\frac{5}{10}=0.5$ but that far from intuitive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I tried to explain that $50$ apples to $100$ apples is the same as $5$ apples to $10$ apples (the half): $0.50=\frac{50}{100}=\frac{1}{2}=\frac{5}{10}=0.5$. It worked in this simple example but I'm seeking an intuitive way to use for all numbers (e.g. to explain why $520.100500=520.1005$ and not $52.15$ or $520.15$).&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="114" LastEditDate="2014-04-08T08:44:21.100" LastActivityDate="2021-09-11T04:59:46.363" Title="Redundant zeros" Tags="|students-mistakes|real-numbers|" AnswerCount="11" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1253" PostTypeId="2" ParentId="1252" CreationDate="2014-04-02T22:46:20.823" Score="14" Body="&lt;p&gt;Check out Berkeley mathematician H.H. Wu's &lt;a href=&quot;http://math.berkeley.edu/~wu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;homepage&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, see his textbook drafts for &lt;a href=&quot;http://math.berkeley.edu/~wu/Pre-Algebra.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pre-Algebra&lt;/a&gt; (pdf) and &lt;a href=&quot;http://math.berkeley.edu/~wu/Algebrasummary.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to School Algebra&lt;/a&gt; (pdf).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, see p. 20 and the discussion of decimals as &quot;a class of fractions&quot; in the former text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the book is intended for teachers of 6th-8th graders and &lt;em&gt;not&lt;/em&gt; for students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The exposition can, at times, be more rigorous than what you might be used to or expect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the least, I think this can help clarify some of the instructor's thinking around topics like decimals; after that, it will be up to you how best to adapt the presentation so as to make it useful for your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Specific example:&lt;/strong&gt; See p. 28 in the Pre-Algebra text; I will paste a screen-shot below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Hb7DJ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-04-10T20:33:58.193" LastActivityDate="2014-04-10T20:33:58.193" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1254" PostTypeId="2" ParentId="1200" CreationDate="2014-04-03T00:08:16.000" Score="5" Body="&lt;p&gt;When I was in elementary school, we distinguished between &lt;em&gt;numbers&lt;/em&gt; and &lt;em&gt;numerals&lt;/em&gt;.  A number is a number, and a numeral is the symbol we use to represent it.  So we can represent the number five with the numeral '5' in the Arabic system, or the numeral 'V' in the Roman system, or in any other number of ways.  This took about thirty seconds to explain and I don't think anyone in my class found it confusing.  &lt;/p&gt;&#xA;" OwnerUserId="623" LastActivityDate="2014-04-03T00:08:16.000" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1255" PostTypeId="2" ParentId="220" CreationDate="2014-04-03T01:18:23.047" Score="21" Body="&lt;p&gt;&lt;strong&gt;Historical remark.&lt;/strong&gt; Though it has already been mentioned in an answer, I can't resist posting a bit more about the following wonderful example of a proof by induction.&lt;/p&gt;&#xA;&lt;p&gt;I quote directly from the original printing of &lt;em&gt;Polyominoes&lt;/em&gt; (1965) by Solomon Golomb:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;T R O M I N O E S&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is impossible to cover an &lt;span class=&quot;math-container&quot;&gt;$8 \times 8$&lt;/span&gt; board entirely with &lt;em&gt;trominoes&lt;/em&gt;, polyominoes of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; squares, because &lt;span class=&quot;math-container&quot;&gt;$64$&lt;/span&gt; is not divisible by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. Instead, it shall be asked: Can the &lt;span class=&quot;math-container&quot;&gt;$8 \times 8$&lt;/span&gt; board be covered with &lt;span class=&quot;math-container&quot;&gt;$21$&lt;/span&gt; trominoes and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; &lt;em&gt;monomino&lt;/em&gt; (a single square)? (p.  20).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this particular problem, Golomb is talking about trominoes that look like extended dominoes. He demonstrates that this is only possible when the monomino is placed in one of four different squares, and then begins his discussion of L-shaped trominoes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When another type of tromino is considered, the result is surprisingly different: No matter where on the checkerboard a monomino is placed, the remaining squares always can be covered with &lt;span class=&quot;math-container&quot;&gt;$21$&lt;/span&gt; right trominoes [i.e., L-shaped trominoes] (p. 21).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He then proceeds to prove this for the &lt;span class=&quot;math-container&quot;&gt;$8 \times 8$&lt;/span&gt; board, but also for the &lt;span class=&quot;math-container&quot;&gt;$2^n \times 2^n$&lt;/span&gt; board, remarking:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The proof just given proceeds by &lt;em&gt;mathematical induction&lt;/em&gt;... (p. 22).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As a spoiler, here is a proof:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/JfhAV.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;The pasted image above comes from a nice article based on earlier work by Norton Starr:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Soifer, A. (2010). Norton Starr’s 3-Dimensional Tromino Tiling. &lt;em&gt;Geometric Etudes in Combinatorial Mathematics&lt;/em&gt;, 221-225. Springer New York.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You can find further information about this problem on Starr's site &lt;a href=&quot;http://www3.amherst.edu/%7Enstarr/trom/intro.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. In particular, you can find a list of related references as well as a gamefied version of the problem that can be played online (or purchased as an actual puzzle - a nice gift!).&lt;/p&gt;&#xA;&lt;p&gt;Though the latter tromino problem is solved using induction, the former one is also quite nice. The first to formulate that problem was, apparently, prolific problem poser &lt;a href=&quot;http://en.wikipedia.org/wiki/Murray_S._Klamkin&quot; rel=&quot;noreferrer&quot;&gt;Murray S. Klamkin&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...back in &lt;span class=&quot;math-container&quot;&gt;$1948$&lt;/span&gt; ... I had begun to submit a rather large number of problem proposals to the American Mathematical Monthly. ...Since probably at that that time, in retrospect, most of my proposals were not particularly good, they just got filed away. Unfortunately for me, this included one favorite one. This was to determine whether or not one could remove one square from a chess board and cover the deleted board with &lt;span class=&quot;math-container&quot;&gt;$21$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$3 \times 1$&lt;/span&gt; trominoes. This was no loss to problem solvers, since several years later, Solomon Golomb's book on polyominoes came out which included this problem plus many more goodies (p. 73).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This remark comes from (&lt;strong&gt;N.B.&lt;/strong&gt; Good luck tracking down part one of this article...):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Klamkin, M. S. (1994). Mathematical Creativity in Problem Solving and Problem Proposing II. &lt;em&gt;In Eves' circles&lt;/em&gt;, (34), 73-95.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Sorry if the tangent here seems unrelated; problem histories can be surprisingly tough to figure out, so I thought it might be nice to include a bit of the back-story online.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-03T01:18:23.047" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1256" PostTypeId="2" ParentId="380" CreationDate="2014-04-03T02:39:35.237" Score="16" Body="&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Since another response mentions using APOS theory to analyze what happens in learning about the chain rule, I thought I would direct you to a source you might find of interest:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Clark, J. M., Cordero, F., Cottrill, J., Czarnocha, B., DeVries, D. J., St John, D., &amp;amp; Vidakovic, D. (1997). Constructing a schema: The case of the chain rule? &lt;em&gt;The Journal of Mathematical Behavior, 16&lt;/em&gt;(4), 345-364. &lt;strong&gt;&lt;a href=&quot;http://homepages.ohiodominican.edu/~cottrilj/chnrl.pdf&quot; rel=&quot;noreferrer&quot;&gt;Free PDF link&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some of the sources contained therein or subsequent &lt;strong&gt;&lt;a href=&quot;http://scholar.google.com/scholar?q=%22chain%20rule%22&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=40000005&amp;amp;sciodt=0%2C22&amp;amp;cites=9419220315590516769&amp;amp;scipsc=1&quot; rel=&quot;noreferrer&quot;&gt;articles&lt;/a&gt;&lt;/strong&gt; that cite this paper might aid you in your revisiting of the chain rule. Here is an image of the abstract; below the break is my original answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/DTUxb.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;You close by parenthetically remarking:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Of course, general answers are helpful, as are ones specifically concerned with the chain rule. The broader question here, if you want to give a more general answer, is not &quot;how should the chain rule be taught&quot;, it's &quot;how should we deal with students who have previously covered topics in unhelpful ways&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I will attempt to provide a more general answer, which could be applied to the chain rule in particular:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;George Polya's (1945) &lt;em&gt;How to solve it&lt;/em&gt; contains a list of strategies (&quot;heuristics&quot;) for mathematical problem solving. One of these strategies is to &quot;think of a related problem.&quot; In the case you are describing, it seems as if students are not realizing that the problems that they are faced with are &lt;em&gt;chain rule problems&lt;/em&gt;; more precisely, they are not able to see the relation between something like the latter examples you give and earlier problems solved with the chain rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the reasons Polya's book was so successful is that its content resonated with mathematicians, and students who do particularly well in mathematics classes. More generally, though, one might ask whether mathematicians (including experienced mathematics teachers - not just those who do research) and students would interpret &lt;em&gt;related problem&lt;/em&gt; in the same (or at least a similar) way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was a topic of research around 1980, and the result was that mathematicians were inclined to relate problems by looking at deeper structural features, whereas students were more inclined to &quot;relate&quot; problems according to superficial features. (I will provide a few &lt;em&gt;related&lt;/em&gt; references below!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A way to intervene (generally or with the chain rule) could be as follows: Give your students a collection of problems and ask them to sort the collection by the approach they would use to solve the problems. Examine the sortings, and see if those of the students accord with your own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they do not (e.g., problems solved with the chain rule are &lt;em&gt;not&lt;/em&gt; grouped together) then discuss the problems and their methods of solution, and explain why you sorted the problems as you did. Follow-up by having them classify another collection (or even the initial collection...) and see whether their approaches to sorting are closer to what you would expect/hope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[If you should carry out such experiments, I would be interested to hear the results!]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Schoenfeld, A. H., &amp;amp; Herrmann, D. J. (1982). Problem perception and knowledge structure in expert and novice mathematical problem solvers. &lt;em&gt;Journal of Experimental Psychology: Learning, Memory, and Cognition, 8&lt;/em&gt;(5), 484. &lt;a href=&quot;http://psycnet.apa.org/journals/xlm/8/5/484/&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;2.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Silver, E. A. (1979). Student perceptions of relatedness among mathematical verbal problems. &lt;em&gt;Journal for research in mathematics education&lt;/em&gt;, 195-210. &lt;a href=&quot;http://www.jstor.org/stable/748807&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;3.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Silver, E. A., &amp;amp; Smith, J. P. (1980). Think of a related problem. &lt;em&gt;Problem Solving in School Mathematics, NCTM, Reston, Virginia&lt;/em&gt;, 146-156.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-08-18T20:47:33.063" LastActivityDate="2014-08-18T20:47:33.063" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1257" PostTypeId="2" ParentId="1252" CreationDate="2014-04-03T02:54:55.770" Score="21" Body="&lt;p&gt;When it comes to convincing younger students of something, I find that analogy can be quite useful, even if you have to squint a bit to make it technically rigourous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I imagine a younger student reasoning something like this: &quot;Hey, I know 5 is different from 50, and 50 is different from 500. When you add zeros at the end of number, it changes the number. So why shouldn't this be true after the decimal point?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a potential antidote to this line of reasoning: &quot;It's true that 50 is different from 500, but what about 050, or 0050? What are those numbers? We usually don't write zeros before the leftmost number, right? Because it doesn't change anything. Really what's happening is that we just assume it's all zeros to the left. Similarly, after the decimal point, we don't write zeros after the rightmost number because it doesn't change anything---we just assume that it's all zeros to the right.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me say right away that I concede this explanation is somewhat less than satisfying from a technical perspective. It's not incorrect, but it doesn't exactly shed light on every aspect of decimal notation that's relevant here. Nonetheless, I think it's a reasonable first step at an explanation, something that might be revisited and fleshed out in future lessons, or in later years. My thinking is, if you hit them with too much detail while they're skeptical, no matter how good your intentions, you stand a good chance of (effectively) just making an argument from authority. They might not really understand your explanation, but decide to just accept it's true for mysterious reasons. I tend to prefer to offer them a simple analogy that they can use to overcome their initial skepticism, opening the way for a more detailed explanation later on.&lt;/p&gt;&#xA;" OwnerUserId="327" LastActivityDate="2014-04-03T02:54:55.770" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1258" PostTypeId="2" ParentId="220" CreationDate="2014-04-03T03:36:39.267" Score="35" Body="&lt;p&gt;The problem with induction proofs is that too often the problem is given by &quot;Prove that...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After a few examples and explanations of induction, if the students know elementary calculus, the following sequence might prove interesting:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Find the first ten derivatives of $x\cdot e^x$.&lt;/li&gt;&#xA;  &lt;li&gt;What seems to be the formula for the $n$th derivative of $x\cdot e^x$?&lt;/li&gt;&#xA;  &lt;li&gt;Prove that your formula is right by induction.&lt;/li&gt;&#xA;  &lt;li&gt;Find and prove a formula for the $n$th derivative of $x^2\cdot e^x$. When looking for the formula, organize your answers in a way that will help you; you may want to drop the $e^x$ and look at the coefficients of $x^2$ together and do the same for $x$ and the constant term.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'd have the students work this out in class and in teams of two and their work was corrected and counted as a homework. Of course, the use of a CAS helped getting rid of the messy part of finding derivatives. Not all students found this interesting. But those who did started to learn what it was to do (elementary) research.&lt;/p&gt;&#xA;" OwnerUserId="503" LastEditorUserId="5664" LastEditDate="2016-03-06T20:00:02.300" LastActivityDate="2016-03-06T20:00:02.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1259" PostTypeId="2" ParentId="1200" CreationDate="2014-04-03T04:23:42.797" Score="5" Body="&lt;p&gt;I would like to say that 1/2 and 2/4 and 3/6 and $\pi /2\pi$ are different &lt;strong&gt;fractions&lt;/strong&gt; however they are all the same number. There are many different ways to express the number, but they are all the same number. This is what we mean by number. Implicit within the concept of number is the assumption of equivalence of fraction. Therefore, when I write $1/2 = 2/4$ I mean to say they are the &lt;strong&gt;same&lt;/strong&gt; number. Or, when I write $0.999\dots =1$ again, I mean to say they are the same number. Are they different expressions? Sure. Do they indicate differing mathematical structures. Absolutely. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I simply object to making students doubt themselves on questions like &quot;is $1/2$ really $2/4$ ?&quot; There is a time and a place to examine such questions, in my view, that time and place is when you have the tools to properly do such. That is in the context of a course where you are building numbers from the base up. Perhaps a second course in algebra, but certainly not in elementary school. That is a time to build up confidence about how to do things with numbers, not to breed uncertainty about questions which they cannot hope to properly dissect.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is always something foundational to learn. However, I don't think that everything I know now is worthless just because there are gaps in my foundations. I hope that I am aware of the gaps, especially when I task myself on a problem where the gaps have bearing. But, more to the point, it's quite important to be able to do math at different levels of abstraction. Often, it is more important to understand the structure of an object than to be able to give an explicit construction thereof. For example, what is $a+ib$ where $a,b \in \mathbb{R}$ and $i^2=-1$? Is it an extension field of $\mathbb{R}$? Is it the set of $2 \times 2$ matrices $\left[ \begin{array}{cc} a &amp;amp; -b \\ b &amp;amp; a\end{array} \right]$? Is it a pair $(a,b)$ with the multiplication defined in the manner of Gauss? I would offer, for most purposes, the answer that $a+ib$ is the quantity that when multiplied with $c+id$ gives $(a+ib)(c+id) = ac-bd+i(bc+ad)$. Of course, this can be derived from any of the &lt;strong&gt;models&lt;/strong&gt; of complex numbers I mention. Moreover, apart from dispelling a certain air of mystery as to &lt;em&gt;what&lt;/em&gt; a complex number &lt;em&gt;is&lt;/em&gt; I don't think it a good use of time in an elementary complex course to belabor the construction. Much better to use them and get to applications and theorems which show the utility and beauty. It is the same with real numbers. Just use them, if you want to discuss their construction, save it for a time when you have tools.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-04-03T04:23:42.797" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1260" PostTypeId="1" CreationDate="2014-04-03T09:38:15.877" Score="6" ViewCount="1102" Body="&lt;p&gt;Puzzles are interesting form of exercises. They help students to learn the teaching material in a funny way. Particularly in logic, puzzles could be very useful to show the complexity of the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are good examples of logic puzzles for undergraduate logic courses, focused on propositional logic and set theory?&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="75" LastEditDate="2014-04-04T17:47:30.347" LastActivityDate="2014-05-06T09:32:44.520" Title="Puzzles for Logic Courses featuring propositional logic and set theory?" Tags="|mathematical-pedagogy|course-design|homework|logic|math-puzzle|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1261" PostTypeId="2" ParentId="1252" CreationDate="2014-04-03T10:11:37.543" Score="5" Body="&lt;p&gt;If the student struggles with this task, what do &lt;em&gt;you&lt;/em&gt; think &lt;em&gt;he&lt;/em&gt; (or she) thinks of a decimal number? What &lt;em&gt;is&lt;/em&gt; .37 for him? Let him try to explain the meaning of .37 in terms of examples or calculations where this number is used. A number &lt;em&gt;must&lt;/em&gt; have a deeper meaning than just being a sequence of digits in order to understand how we use them. &#xA;One of your examples ($520.100500=52.15$) would indicate even problems with the naturals ($520=52$?)! The student should also be able so say that $520 = 5\cdot100+2\cdot 10+0\cdot1$. Based on this description of a natural number, fractions like $1/10$ in decimals are not too far away.&#xA;You may convince your student and it may work for some technical tasks. However, for understanding, there is no way but fractions. Thus, I recommend Benjamins answer.&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="199" LastEditDate="2014-04-03T10:36:24.003" LastActivityDate="2014-04-03T10:36:24.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1262" PostTypeId="2" ParentId="1204" CreationDate="2014-04-03T10:37:22.447" Score="2" Body="&lt;p&gt;Where I am working now (EPFL, Switzerland), first year students have a dedicated two semester geometry course in parallel with the usual calculus, analysis, and linear algebra sequence. &#xA;I have not taught these courses but they seem to do a lot of plane and space geometry using the various tools at hand (inner product, cross products, determinants, angles...). They also cover some elementary spherical geometry and geometry of curves and surfaces.&lt;/p&gt;&#xA;" OwnerUserId="400" LastActivityDate="2014-04-03T10:37:22.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1263" PostTypeId="1" AcceptedAnswerId="1289" CreationDate="2014-04-03T11:39:51.200" Score="26" ViewCount="3560" Body="&lt;p&gt;In advanced graduate courses, there are often no textbooks on the material being covered, and sometimes no good introductory material at all. Even when there are textbooks, many professors prefer to write their own notes on the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a fairly new instructor, the idea of creating a course from scratch seems intimidating. Fortunately, I don't have to do it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question has two parts:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are the most important considerations when writing your own lecture notes for a course?&#xA;  What is the motivation for writing your own notes in a course where textbooks are available?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If this is too broad, let me know and I can degeneralize it.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-04-06T22:41:17.640" Title="How do you go about writing your own lecture notes for a new course?" Tags="|course-design|textbooks|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1264" PostTypeId="2" ParentId="1252" CreationDate="2014-04-03T12:09:51.180" Score="6" Body="&lt;p&gt;Let me play devil's advocate.&#xA;They are not identical. In situations where complex calculations result in a number such as 6.3856, and the science teacher is asking for 4 significant digits after decimal, one wouldn't round at all. If the answer happened to be 5.0000, the student should keep the zeros to indicate the level of accuracy, that no rounding was involved. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My middle schooler learned about significant digits in a science class, and I could see a potential conflict as she applied this reasoning when it wasn't really necessary. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-04-03T12:48:51.873" LastActivityDate="2014-04-03T12:48:51.873" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1265" PostTypeId="2" ParentId="1245" CreationDate="2014-04-03T12:54:20.813" Score="13" Body="&lt;p&gt;It seems to me that the reason for learning series in calculus is to analyze functions through power series representations.  There are other reasons series are important in mathematics, but in the standard calculus course, it seems to me that series are introduced for the sake of explaining power series.  The key is then, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Can one motivate power series before even covering series?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think so.  To some degree, you have to &lt;em&gt;sell&lt;/em&gt; it, and then deliver later in the course.  I tell the students that series have been important in understanding difficult functions, such as algebraic and transcendental functions.  I sometimes make rather broad claims that are only roughly correct.  For instance, when you press a button on a calculator, how do the engineers know that the digits are correct?  (I guess I often motivate something by asking questions that I don't answer; sometimes that I can't answer.)  I also tell them they will be able to see how a function is related to all its derivatives and see how L'Hôpital's rule, montonicity, and concavity work.  &lt;a href=&quot;https://matheducators.stackexchange.com/users/63/paul-garrett&quot;&gt;@paul garrett&lt;/a&gt; has mentioned Newton's binomial series.  One can also do some calculations that suggest the development of series. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For instance, given a small number $x$ with $|x|$ much smaller than $1$, approximate $\sqrt{1+x}$ with a linear polyomial $1+px$.  (Pick $x = 1/10$ or $1/100$ for concreteness, and plug in to the results and compare with a calculator value as you go along.)  So we would like to pick $p$ such that $(1+px)^2\approx 1+x$.  If we match coefficients in order from lower to higher degree (i.e., from greater to smaller terms in order of magnitude), we get $p = 1/2$, with the left over term $p x^2$ being the error of the approximation; but $px^2$ will be much smaller than the linear term $2 p x = x$.  Next improve the approximation by adding a quadratic term:&#xA;$$ \left(1 + {1\over2}\,x+q\,x^2 \right)^2 \approx 1+x$$&#xA;Expanding and matching coefficients yields $q = -1/8$.  And so on.  One can see that this can be continued indefinitely.  Is there an easy way to do it?  Does it always get closer?  [Since one probably does not care whether students can carry out these calculations, they could be written out on notes for the students to follow or done on a computer.]  Alternatively, one might do the algebra all at once:  Expanding&#xA;$$ \left(1 + p\,x + q\,x^2 + r\,x^3 + s\,x^4 \right)^2 \approx 1+x$$&#xA;yields&#xA;$$p={{1}\over{2}},\ q=-{{p^2}\over{2}},\ r=-p q,\ s={{1}\over{2}} \left(-2 p r-q^2\right)\,.$$&#xA;The solution can be completed through substitution.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Another example is that the repeated integration from $0$ to $x&amp;gt;0$ of $\sin x \le 1$ or $e^{-x} \le 1$ yields inequalities that show that $\sin x$, $\cos x$, and $e^{-x}$ lie between successive Taylor polynomials. (I would omit the word Taylor in class.)  Since $n!$ grows so fast, you can approximate sine and cosine rather well over the first quadrant.  Again it leads to questions about the indefinite continuation.  This example can be done in Calculus I, if you want to give the students advertisement for Calculus II.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;One question that motivates series, not dealing with power series, is the following:  If you could&#xA;&lt;a href=&quot;http://mathworld.wolfram.com/BookStackingProblem.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;stack as many books&lt;/a&gt; as you wish, how far out over the edge of the book on the bottom could the book at the top extend without the books falling over?&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-04-12T21:00:40.317" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1266" PostTypeId="1" AcceptedAnswerId="1278" CreationDate="2014-04-03T13:39:28.417" Score="23" ViewCount="1439" Body="&lt;p&gt;I've noticed that many of my calculus students (all college students) will write, e.g., $1/3x$ to mean $(1/3)x$. This is an inherently ambiguous notation which I'd like them to avoid. Is simply pointing this out in class effective? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Here's a handwritten version of the ambiguous notation: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/pAIyG.png&quot; alt=&quot;Example offense.&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="344" LastEditorUserId="114" LastEditDate="2014-04-08T08:45:32.930" LastActivityDate="2019-04-03T20:36:30.600" Title="Students using ambiguous notation" Tags="|students-mistakes|notation|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1267" PostTypeId="2" ParentId="1252" CreationDate="2014-04-03T13:53:14.923" Score="3" Body="&lt;p&gt;It seems to me there is a conceptual mistake hidden in this misunderstanding: many students (and, sadly, probably many educators) misidentify numbers with their decimal expansions. Number are &lt;em&gt;represented by&lt;/em&gt;, not &lt;em&gt;identical to&lt;/em&gt; their decimal expansions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you convince someone of this? No. It's rather like the plus/quus-type problems in the Kripkenstein literature. On the other hand, finding a student making this type of misidentification is the chance for a great learning opportunity; I think the previous responses have identified many ways to bring out the consequences of this mistake: integers become ambiguous; applications cease to make sense, etc. In this way, you are confronting the student with the cost of maintaining their point of view. Hopefully they find it too expensive.&lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-04-03T13:53:14.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1268" PostTypeId="1" CreationDate="2014-04-03T14:00:50.327" Score="13" ViewCount="570" Body="&lt;p&gt;There has long been debate about whether a first year undergraduate course in discrete mathematics would be better for students than the traditional calculus sequence. The purpose of this question is not to further that debate, but to inquire whether any text has tried to teach calculus by emphasizing probability as a primary motivating example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that such a text would introduce integrals before derivatives, which I know many authors have tried.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am asking this question because it seems &lt;a href=&quot;https://mathoverflow.net/questions/162125/what-areas-of-pure-mathematics-research-are-best-for-a-post-phd-transition-to-in&quot;&gt;industry has use for people proficient with probability&lt;/a&gt;, and pure mathematics makes great use of these ideas as well (think of probabilistic methods in number theory, for example). It seems that a good course of the above type would be cosmopolitan enough to attract students with tastes in either pure or applied mathematics to major in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: In the absence of a perfect text for this, it may be worth a collaborative effort by mathematicians to write such a text. Is there a way to &quot;crowdsource&quot; writing a text like this as a wiki?&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-10-23T02:14:41.383" Title="First year undergraduate text that teaches calculus using probability as a primary motivating example?" Tags="|undergraduate-education|calculus|textbooks|probability|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1269" PostTypeId="2" ParentId="1263" CreationDate="2014-04-03T14:26:24.473" Score="11" Body="&lt;p&gt;A motivation for writing course notes, when textbooks/monographs already exist, is to &lt;em&gt;omit&lt;/em&gt; some things, first. This might seem strange, until one observes that many standard texts which go through several editions often succumb to the natural pressure, amplified by publishers, to become ever-more encyclopedic. The latter has its uses as reference, but perhaps not so much as a guide to the actual threads of a live course. Further, _making_choices_ of what to pay attention to in the course, as opposed to what to ignore (in the course), imparts considerable information to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As positive side-effect of omitting some iconic-but-not-immediately-essential things, sometimes there is room to include newer or novel things that otherwise wouldn't find a spot, or happen not to be included in the standard texts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, one's own notes can more accurately support and/or complement one's own lectures and plot-line for a course, so one could argue that picking a standard text and just indicating a subset of it for students could suffice. Even if that is the case, people will see all the other stuff and wonder why you're ignoring it. Intentional, or accidental? Some distractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And self-generated notes can be made freely available, and available in electronic formats, and... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to &quot;important considerations&quot;, ... certainly facility with an adequate typesetting software system comes first, or it's hopeless. For a first pass through, don't keep everything in one big file, both because it takes longer to do the inevitable frequent re-typesettings, and because it's hard to find things. Use your computer's file system, etc., to organize things. Keep backups. Some form of &quot;version control&quot; can be useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;About content: especially if you're trying to lighten the burden of encyclopedic texts, don't fall into the (sometimes appealing) trap of &quot;completeness&quot;, or of giving all the iconic examples, or perhaps even working out all details of all proofs of all small things. Instead of contriving artificial exercises, if you've given representative prototypes of relevant arguments, subsequent similar arguments can be left to the reader. Of course, one should be honest with the reader, rather than merely lazy, and be honest with oneself about why something's omitted. (Not to mention competent to confirm that it really is just &quot;more of the same&quot; and routine.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A different sort of point: write at a level and in a style to address the students you actually have, rather than ones you wished you had, or some idealized students. E.g., don't omit arguments or explanations that you know perfectly well most of your students couldn't fill in, don't couch things in terms you know are more sophisticated than your audience, ... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Coming to some questions of taste: the pressures for traditionally published texts to be fairly formal are not entirely helpful to students or mathematics. That is, there is stylistic pressure to be &lt;em&gt;formal&lt;/em&gt;, and to follow a certain (quite recent, modern!) style of definition-lemma-proof-lemma-...-theorem-proof... exercises. Candidly, one often discovers that the exercises are cryptic exhibition of the formative examples that motivated the whole business, but the natural causal sequence has been &lt;em&gt;inverted&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, yes, a certain amount of that inversion can be &lt;em&gt;helpful&lt;/em&gt;, clarifying, but the standard full-blown version is not. (To name-drop) Gelfand was a consistent advocate of giving the key examples so that &quot;definitions and theorems and proofs&quot; become almost afterthoughts. E.g., definitions should be &quot;justified&quot; by examples...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Especially in fairly sophisticated mathematics, terminology is heavy, usage is inconsistent or slightly abused, etc., and trying to &quot;repair&quot; the situation by cranking up the formality is attractive, but often doomed or at least incredibly expensive. Instead, giving the watershed examples, rather than trying to make &quot;rules&quot; or rigidify language, seems to me to more directly address the genuine issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And be prepared to defend (on scientific and pedagogical grounds) any deviation from the iconic sources, because it seems that people are quite happy to believe in the authority of iconic authors' writings. Many people in the math business do seem to believe that definitions given in iconic texts are &quot;sacred&quot;, too, and are beyond discussion, so don't be surprised when people are inflexible in this regard! Have explanations ready... Yes, some of your energy will be spent defending your choices to students, and this &lt;em&gt;can&lt;/em&gt; be made mostly productive, if you are prepared.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-03T14:26:24.473" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1270" PostTypeId="2" ParentId="1200" CreationDate="2014-04-03T15:59:07.593" Score="-1" Body="&lt;p&gt;&lt;strong&gt;I think it's helpful to separate representation from semantics.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe if we rephrase the question this can help accomplish that goal.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is 1/2 a number (an abstraction of quantity) or &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is 1/2 a representation (a process or symbol) of a number.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What &quot;is&quot; a number, as others have explained it is an abstract concept of quantity. Those abstractions can be assigned symbols for us to understand and discuss, for example quantity 2, can written in symbolic form as &quot;2&quot;, &quot;II&quot;, or &quot;..&quot;, etc... Different symbols, which indicate the same abstract quantity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;There are two meanings of representation that we have to distill.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definition of representation (as you mean it):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The symbol that identifies something (&quot;2&quot;, &quot;0.5&quot;, &quot;pi&quot;).&lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;process&lt;/em&gt; that identifies something.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Ask yourself is &quot;1/2&quot; actually a symbol like &quot;2&quot;? Do you have a &quot;1/2&quot; key on your keyboard? No.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, &quot;1/2&quot; actually represents a &lt;em&gt;process&lt;/em&gt; to determine a number. Think about it, &lt;strong&gt;&quot;1/2&quot; is actually a combination of 3 distinct &lt;em&gt;symbols&lt;/em&gt; that when combined have inherent meaning as to a process. That process is mathematical division.&lt;/strong&gt; It is literally &quot;1 divided by 2.&quot; From this thought process I would conclude that &quot;1/2&quot; is &lt;em&gt;NOT&lt;/em&gt; the number, but a representation (&lt;em&gt;definition #2&lt;/em&gt;) of it. This is distinctly different than looking at &quot;0.5&quot;, this &lt;em&gt;is&lt;/em&gt; the symbol that identifies that value (there is no &lt;em&gt;procedure&lt;/em&gt; to this representation). Now some numbers (irrational ones, for example) cannot be shown symbolically in decimal form without losing accuracy, and in that case the fractional &lt;em&gt;representation&lt;/em&gt; may be the most accurate one, but it still fundamentally represents a process, not the number itself. So I think this question hinges on your quest for semantics, which is interesting to think about. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Disclaimer: I'm not a mathematician, so take my answer worth a grain of salt. :)&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="648" LastActivityDate="2014-04-03T15:59:07.593" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1271" PostTypeId="2" ParentId="258" CreationDate="2014-04-03T17:24:23.043" Score="11" Body="&lt;p&gt;When I teach calculus, there are two preparedness issues that frequently surface:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Understanding functions: I don't mind if my students can't write down the definition of a function (though ideally they could), but if I say something like &quot;let $f$ and $g$ be functions,&quot; they shouldn't be horrifically confused. Similarly, if I ask very basic questions about functions (e.g. if $f$ is a function from the reals to the reals, and $x$ is a number, what type of thing is $f(x)$?), I expect they will have at least a nose for the answer.&lt;/li&gt;&#xA;&lt;li&gt;An openness to being confused. Many students in my calculus classes expect to never struggle, never not know precisely how every step of a problem will be done in advance, etc. I try to emphasize to them that non-confusing things aren't worth spending time learning, but most of them have so internalized the inference chain &quot;confusion leads to getting things wrong leads to getting bad grades leads to living under a bridge'' that they run from confusion at all costs, even when that means never engaging with confusing material that I've promised will be on exams.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-04-03T17:24:23.043" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1272" PostTypeId="2" ParentId="1200" CreationDate="2014-04-03T18:42:20.830" Score="3" Body="&lt;p&gt;This veers towards &lt;a href=&quot;http://en.wikipedia.org/wiki/Haddocks%27_Eyes&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Haddocks%27_Eyes&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that distinctions have to serve differences. I very much doubt that you will help an elementary-level student by calling '1/2' a process or not a number, or getting involved in whether '1/2' and '2/4' are two numbers with the same value or one number with two representations. Learning math is all about harnessing naive intuition when possible, and correcting it when it hurts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My view is that you should not need to be concerned with the distinction between numbers and their representations until you get involved in multiple bases or the reals. When discussing the reals, it's quite important to recognize that .999999... &lt;em&gt;is the same&lt;/em&gt; as 1.0, which very strongly leads in the direction of distinguishing representation from the thing represented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Until you get there, you are perilously close to teaching Platonism to Penguins.&lt;/p&gt;&#xA;" OwnerUserId="562" LastEditorUserId="114" LastEditDate="2014-04-03T19:03:30.593" LastActivityDate="2014-04-03T19:03:30.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1273" PostTypeId="2" ParentId="1263" CreationDate="2014-04-03T18:44:22.870" Score="6" Body="&lt;p&gt;One important thing is to think ahead about the content. Clarify the goals, both in terms of target theorems and of methods. Try to have a nice goal for the end of the course, and to clarify what you want your students to master, to what level of details, and for each point what you want to show and what you want them to do by themselves. Writing a detailed syllabus should makes you do all that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About how to proceed without being too intimidated: I usually like to write the title of chapters, then write the title of sections, then write the title of subsections, and only then starting to go into the real writing. Of course, the titles may change during the writing, but then at any time you're not writing a whole book, only a subsection of it. Much less intimidating. Writing all titles first helps you knowing where you are headed at any time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About the motivation to write your own lecture notes when textbook are available is to &lt;em&gt;learn what you will teach&lt;/em&gt;. Even if you know it well, writing it makes you understand it much better because you are forced to go into the details very thoroughly. But this motivation is especially important when you choose to teach an advanced course on a topic you do not master yet, which is probably the most efficient way to learn. &lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-03T18:44:22.870" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1274" PostTypeId="2" ParentId="1252" CreationDate="2014-04-03T18:44:59.797" Score="13" Body="&lt;p&gt;Turn the problem around.  Say to the student:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If &quot;5.00&quot; is not the same number as &quot;5&quot;, then one of them is larger than the other.  Which, and by how much?  You're the one who claims there is a difference.  Then tell me, literally &quot;what is the difference&quot;?  Does a movie ticket priced at &quot;five dollars and no cents&quot; cost more or less than one priced at &quot;five dollars&quot;, or are those the same price expressed two different ways?&lt;/p&gt;&#xA;" OwnerUserId="650" LastActivityDate="2014-04-03T18:44:59.797" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1275" PostTypeId="2" ParentId="1026" CreationDate="2014-04-03T18:49:17.243" Score="3" Body="&lt;p&gt;I don't quite know if this answers the question, but I ask no more memorization of my students than I find myself capable of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps due to (ahem) &lt;em&gt;chemical indiscretions&lt;/em&gt; in late adolescence/early adulthood, I find myself not to be the greatest memorizer. Things like the sum-of-angles formula for $\sin$ and $\cos$ I find myself hopelessly unable to retain for more than about twenty minutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know, however, that this is idiosyncratic. Some people are capable of remarkable feats of memorization. So I often end up teaching two tracks: I tell people in my calculus courses that, if they are only interested in gaining &lt;em&gt;knowledge&lt;/em&gt; (as opposed to understanding) and are &lt;em&gt;capable memorizers&lt;/em&gt;, then knowing e.g. this or that list of facts will likely suffice for passing (almost certainly without an A) my course. On the other hand, for those who want &lt;em&gt;understanding&lt;/em&gt; or are (like me) &lt;em&gt;incompetent memorizers&lt;/em&gt;, here is a bag of tricks for rebuilding all the information in this course from very little. Yes, you still have to remember the very little, but after using the bag of tricks for some time, you develop &quot;muscle memory&quot; that can get you from this to any given thing you might need.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of this is the use of De Moivre's formula to figure out the aforementioned sum-of-angles formula. It's the only hope for people like me.&lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-04-03T18:49:17.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1276" PostTypeId="2" ParentId="220" CreationDate="2014-04-03T19:12:36.973" Score="9" Body="&lt;p&gt;I suggest the following counter-example. (&lt;strong&gt;Added in edit:&lt;/strong&gt; I agree with kcrisman comment that this may be misleading to beginners, and should probably be kept for people that master induction relatively well and start being lazy verifying the base case).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; In a (finite) color pencils box, all pencils must have the same color.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Proof.&lt;/em&gt; We proceed by induction on the number of pencils. If the box has one pencil, then obviously all its pencils have the same color. Now,&#xA;  assume that in all boxes of $n$ pencils, the pencils are all of the&#xA;  same color and let $X$ be a box with $n+1$ pencils. Order the pencils&#xA;  arbitrarily and consider the $n$ first pencils: by the induction&#xA;  hypothesis, they all have the same color, call it $c$. Consider now the&#xA;  $n$ last pencils: by the induction hypothesis, they all have the same&#xA;  color, call it $c'$. Since the $n-1$ central pencils have color $c$&#xA;  and $c'$, we have $c=c'$ so that all $n+1$ pencils have the same&#xA;  color.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The goal here is to stress that if one fails to properly identify and test the base case, then things can go dead wrong.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2016-03-08T16:56:38.163" LastActivityDate="2016-03-08T16:56:38.163" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1278" PostTypeId="2" ParentId="1266" CreationDate="2014-04-03T20:06:25.797" Score="32" Body="&lt;p&gt;A colleague of mine includes on all his tests a line that reads (something like) &quot;you will be graded on what you actually write down, not what I think you may have meant by what you wrote.&quot; He spends time carefully discussing what this means (examples like the one you gave are among them) before and after testing times. Normally after the second round (of four), students have developed the ability to reflect on how their own writing will look to others, and can avoid this type of mistake.&lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-04-03T20:06:25.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1279" PostTypeId="2" ParentId="1200" CreationDate="2014-04-03T21:16:40.107" Score="0" Body="&lt;p&gt;They are not the same.  For example, $$1\over1+x$$ is clearly a fraction, but it is not in general a number but rather a variable expression.  We usually represent rational numbers as a fraction of integers, and if we want to have a unique representation, in lowest terms and with a positive denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that does not mean that rational numbers are the same as fractions.&lt;/p&gt;&#xA;" OwnerUserId="405" LastActivityDate="2014-04-03T21:16:40.107" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1280" PostTypeId="2" ParentId="1208" CreationDate="2014-04-03T21:42:54.580" Score="8" Body="&lt;p&gt;Here is a list of how I actually used linear algebra as a physics major. The general techniques are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Change of basis (including calculating reciprocal vectors, which in turn require calculating determinants)&lt;/li&gt;&#xA;&lt;li&gt;Projections onto subspaces&lt;/li&gt;&#xA;&lt;li&gt;Finding the eigenvalues and eigenvectors (including calculating determinants)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some examples&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;calculating the intersections of lines and planes&lt;/li&gt;&#xA;&lt;li&gt;Finding the axis of rotation (eigenvector)&lt;/li&gt;&#xA;&lt;li&gt;Calculating the rotation matrix (Euler angles)&lt;/li&gt;&#xA;&lt;li&gt;calculating the volume of a crystal cell (determinant)&lt;/li&gt;&#xA;&lt;li&gt;calculating a cross product as a determinant in R3 (this is a neat trick as the cross product is not always used to find the normal to a plane, but if you present it as the area of an oriented 2-blade then it generalizes Rn)&lt;/li&gt;&#xA;&lt;li&gt;calculating the coefficients of the metric tensor in a different&#xA;coordinate system. After calculating the partial derivatives, the&#xA;metric tensor can be transformed by a similarity transformation.&lt;/li&gt;&#xA;&lt;li&gt;calculating the reciprocal basis vectors. Students may be familiar with orthonormal systems, but lots of questions arise when dealing with curvilinear systems. Writing the gradient operator in terms of the reciprocal basis vectors clears up everything. Reciprocal vectors are also used in solid state (crystals have natural axes which may not be orthogonal and natural length scales which may be different along each axis).&lt;/li&gt;&#xA;&lt;li&gt;Projecting vectors onto orthonormal bases (easy with geometric vectors, but not recognized when generalized to functions)&lt;/li&gt;&#xA;&lt;li&gt;Projecting vectors onto orthogonal bases. Students are familiar with spherical/cylindrical coordinates using orthonormal basis vectors. But calculating gradient, divergence, curl, and laplacian, are all magic. It is a good way to introduce reciprocal vectors and less familiar orthogonal coordinate systems. Make sure to generalizing from geometric vectors to non-normalized eigenfunctions.&lt;/li&gt;&#xA;&lt;li&gt;Projecting vectors onto non-orthogonal bases. This requires calculating reciprocal vectors. This is useful in solid-state, as well as special and general relativity. It also encourages students to make up coordinate systems, like ellipsoidal coordinates z=a r cos q, x=b r sin q cos f, y=c r sin q sin f.&lt;/li&gt;&#xA;&lt;li&gt;calculating the principal moments and principal axes of the moment of inertia.&lt;/li&gt;&#xA;&lt;li&gt;calculating the (eigen)modes and modal frequencies of vibrating systems. This deals with Fourier series and orthogonal functions.&lt;/li&gt;&#xA;&lt;li&gt;expanding a wavefunction in terms of stationary states (eigenfunctions). Calculating the probability of a measuring a particular eigenvalue. Various problems can help make the connection between Fourier series, legendre polynomials and spherical harmonics as just various orthogonal bases.&lt;/li&gt;&#xA;&lt;li&gt;Multipole expansions in various fields (acoustics, e&amp;amp;m). Yet another example of projecting onto bases.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="652" LastActivityDate="2014-04-03T21:42:54.580" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1281" PostTypeId="2" ParentId="1208" CreationDate="2014-04-03T22:00:27.620" Score="7" Body="&lt;p&gt;I have two good examples, they are somewhat similar, but one is solved more analytically, and the other more computationally:&lt;/p&gt;&#xA;&lt;h1&gt;Masses on a spring:&lt;/h1&gt;&#xA;&lt;p&gt;From classical mechanics:  a mass on a spring with one end fixed can be described by the classical equation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \mathbf{F} = m\ddot{\mathbf{x}} = -k\mathbf{x} $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If we have two masses on a spring, (in 1d)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ m_1 \ddot{x}_1 = -k(|x_1 -x_2| - \ell_{\text{eq}})$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ m_2 \ddot{x}_2 = -k(|x_1 -x_2| - \ell_{\text{eq}})$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;When you get to more springs, this is more easily posed as a linear algebra problem:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \mathbf{M} \ddot{\mathbf{x}} = \mathbf{K}\mathbf{x}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where now, &lt;span class=&quot;math-container&quot;&gt;$\mathbf{M}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{K}$&lt;/span&gt; are matrices describing the masses of the different objects and the spring constants between them respectively, and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{x}$&lt;/span&gt; describes the positions of the various objects (with the equilibrium distance &lt;span class=&quot;math-container&quot;&gt;$\ell$&lt;/span&gt; subsumed into it).&lt;/p&gt;&#xA;&lt;p&gt;If you look at this as a simple eigenvalue equation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \ddot{\mathbf{x}} = \mathbf{M}^{-1}\mathbf{K} \mathbf{x} $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The eigenvectors (remember that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is really time dependent), describe the normal modes of the system, and the eigenvalues are the frequencies of these modes.  Demonstrating this with a system of 3 or so masses isn't THAT hard, and is very cool.&lt;/p&gt;&#xA;&lt;h1&gt;Schrödinger Equation steady state:&lt;/h1&gt;&#xA;&lt;p&gt;This equation is so fundamental to physics that it can really drive home how ridiculously important linear algebra is to Physicists.  Unfortunately, describing what the wave function represents can be difficult.  The best way is to describe the square of the wave function as a probability distribution.  Thankfully as well, the steady state problem is real, and you don't have to deal with complex numbers. We can write down the Schrödinger equation as follows:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ {\cal H}\Psi(t) = \left(-\frac{\hbar^2}{2m} \boldsymbol{\nabla} + V(\mathbf{x}) \right) \Psi(t) = -i \hbar \partial_t \Psi(t)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;From here, we can choose a basis by separating out the time dependence:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \left(-\frac{\hbar^2}{2m} \boldsymbol{\nabla} + V(\mathbf{x}) \right) \Psi_e = e \Psi_e $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And then we can write:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ {\cal H}\Psi_e = e \Psi_e $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;${\cal H}$&lt;/span&gt; is the finite difference expression of &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; on a grid.  For example, for the 1D problem with a simple harmonic oscillator:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ {\cal H}\Psi = -\frac{\hbar^2}{2m h^2} \left(\Psi(x_{i-1}) - 2\Psi(x_{i} + \Psi(x_{i+1}) \right) + \frac{k}{2} x_i^2 \Psi(x_{i}) $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; is an index on the evenly spaced discretized grid.&lt;/p&gt;&#xA;&lt;p&gt;Then you can find the eigenvalues and eigenvectors (unfortunately, for anything on a reasonable grid, this requires too many points to do by hand).  These eigenvalues describe the steady state energies of the system!  And the eigenvectors describe the steady state wave functions!  Interestingly enough, this particular problem describes vibrational levels in molecules remarkably well (especially for the lowest energy states).&lt;/p&gt;&#xA;" OwnerUserId="621" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-03T22:36:45.020" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1282" PostTypeId="2" ParentId="1240" CreationDate="2014-04-03T22:52:45.587" Score="5" Body="&lt;p&gt;I went through my old exams (and homeworks), and extracted the questions (and, where available) the answers to them. As I write in LaTeX, it was mostly copy&amp;amp;paste (some old texts used Latin-1, now UTF-8, or ancient LaTeXisms, those I fixed; also updated notation, and expanded on the answers). The collection I publish as &quot;suggested exercises&quot; (students &lt;em&gt;have&lt;/em&gt; all old exams and homeworks, don't kid yourself). I also added some interesting questions and either my own answers or the answers from MSE. Each new exam gets added to the collection, if merited.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is an invaluable resource at the moment of making up exams and homework, but as the originals are published, each question has to be extensively tweaked for reuse. So it is &lt;em&gt;far&lt;/em&gt; from &quot;reusable questions,&quot; sadly.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-03T22:52:45.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1283" PostTypeId="2" ParentId="1208" CreationDate="2014-04-03T23:21:54.093" Score="3" Body="&lt;p&gt;This should rivet students' attention!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the MathOverflow question,&#xA;&quot;&lt;a href=&quot;https://mathoverflow.net/q/81960/6094&quot;&gt;The “Dzhanibekov effect”--an exercise in mechanics or fiction? Explain mathematically a video from a space station&lt;/a&gt;,&quot;&#xA;and especially the beautifully informative response by Terry Tao &lt;a href=&quot;https://mathoverflow.net/a/82020/6094&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also known as the &lt;a href=&quot;http://en.wikipedia.org/wiki/Tennis_racket_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tennis racket theorem (Wikipedia)&lt;/a&gt;.&#xA;(The remarkable) video &lt;a href=&quot;https://www.youtube.com/watch?feature=player_embedded&amp;amp;v=L2o9eBl_Gzw&quot; rel=&quot;nofollow noreferrer&quot;&gt;URL here&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/HtQOY.png&quot; alt=&quot;Snapshot&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-04-04T09:57:53.327" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1284" PostTypeId="1" AcceptedAnswerId="1291" CreationDate="2014-04-04T00:26:15.550" Score="7" ViewCount="804" Body="&lt;p&gt;This is a simple question, which I haven't found the answer to online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Due to the increasing number of Spanish-speaking students in American universities, I've been interested in teaching Spanish-language courses in mathematics, such as calculus or linear algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What US universities, if any, hold mathematics courses in Spanish?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="123" LastEditDate="2014-04-04T13:14:45.120" LastActivityDate="2014-04-04T13:14:45.120" Title="What universities in the United States have Spanish-language mathematics classes?" Tags="|undergraduate-education|curriculum|usa|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1285" PostTypeId="2" ParentId="1252" CreationDate="2014-04-04T00:47:55.787" Score="4" Body="&lt;p&gt;Perhaps are the student more used to real world rounded values where 0.5 and 0.500 &lt;strong&gt;are&lt;/strong&gt; different. The former just mean some number between 0.45 included and 0.55 excluded while the latter is more precise, between 0.4995 and 0.5005.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should then tell them this being a Math course, you are not talking about the rounded approximations used for example in Science and Engineering but about mathematical unambiguous representations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next and tougher exercise would be to convince them 0.5 and 0.4999999999999999999... represent strictly identical numbers ... ;-)&lt;/p&gt;&#xA;" OwnerUserId="655" LastEditorUserId="655" LastEditDate="2014-04-08T14:36:38.557" LastActivityDate="2014-04-08T14:36:38.557" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1286" PostTypeId="2" ParentId="1266" CreationDate="2014-04-04T01:26:37.407" Score="7" Body="&lt;p&gt;I would take off one percent if I thought the wrong meaning was unlikely, and more if I thought the wrong meaning was possibly what they meant. I would write: &quot;This looks like 1/(3x). You may have meant 1/3 * x, but it looks like your x is in the denominator.&quot;&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-04-04T01:26:37.407" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1287" PostTypeId="2" ParentId="1252" CreationDate="2014-04-04T02:35:38.583" Score="3" Body="&lt;p&gt;I developed my personal understanding of zeros similar to how Adam describes it only a bit different. I understand zeros in the following way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The number 1.1 can be represented as:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;01.1&#xA;001.1&#xA;0001.1&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;They are the same number therefore the leading zeros can be discarded. Similarly, it can also be represented as:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;1.10&#xA;1.100&#xA;1.1000&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;They are the same number therefore the trailing zeros can be discarded.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The important part is the dot. Zeros close to the dot mean something since they tell you the position of the digit and therefore its magnitude. Zeros far away are redundant and can therefore be discarded.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not entirely sure if others find the above notion intuitive but that's how I understood it when I was a kid.&lt;/p&gt;&#xA;" OwnerUserId="656" LastActivityDate="2014-04-04T02:35:38.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1288" PostTypeId="2" ParentId="1245" CreationDate="2014-04-04T02:36:55.580" Score="2" Body="&lt;p&gt;Perhaps you can use infinite series to expose your students to the notion that math can be beautiful with no practical applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you add up an infinite number of terms, you would think you'll just get infinity. But this is not the case - as the simple example of $\sum 1/2^n$ shows, some infinite sums converge. This raises what I think is an inherently interesting question: which sums converge, and which ones diverge? To convince them that this question really is interesting, show them the harmonic series. Thought a series would converge as long as its terms tend to zero? Guess again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Show them the fact that when subtraction gets involved, you can't always rearrange the terms in the series anymore. Treat the whole subject as a trip to the zoo: get them interested in all the odd phenomona that can happen when you add an infinite number of terms, and in how unpredictable the convergence or divergence of a particular series can be. The sum of the reciprocals of the integers diverges. But the sum of the reciprocals of &lt;em&gt;square&lt;/em&gt; integers converges. What about the sum of the reciprocals of an arbitrary sequence of integers? Of the primes, for instance?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this type of approach may put an apparant gap between series and the rest of the material in the course, which might be presented as being much more practical.&lt;/p&gt;&#xA;" OwnerUserId="115" LastActivityDate="2014-04-04T02:36:55.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1289" PostTypeId="2" ParentId="1263" CreationDate="2014-04-04T07:21:25.270" Score="19" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What are the most important considerations when writing your own lecture notes for a course?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would say that the single most important consideration is to make life easy for yourself.  It is easy to fill your time with writing lecture notes but that isn't always the best use of that time.  Nevertheless, it can be useful for the students so here are my tips.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Make sure you'll be teaching the course several times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's likely that the notes won't come together until after the end of the course, so you should be thinking a bit that you're writing to help the next lot of students.  To ensure that this isn't unnecessary work, make sure that the next lot of students are your students as well.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Write as you go along.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can never plan everything from the start, particularly at the detail needed for notes.  Writing as you go along means that you are writing it knowing what the current lot of students need, so your efforts will be concentrated on where they are most needed.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Write as little as possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's tempting to write as much as possible, but as others have already said this is a temptation to be resisted.  Write what the students need right now.  If you get a moment to add more later, fine, but don't let that be your top priority.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Add value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there are textbooks around, but they aren't really usable as a book for this course, think about how to make your notes be the bridge between the course and the book rather than replacing the book.  Even a simple glossary can be of immense value, or a &quot;look up list&quot; of where students can find particular topics (useful if the book is rather large compared to what you're teaching).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use a wiki.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a practical tip.  Write your notes on a wiki (preferably a mathematically-enabled one).  The advantages of this are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Psychologically, it's easier to write a short, incomplete wiki page than a, say, PDF document.  So writing on a wiki means that you get a little bit up quickly and then gradually improve it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hyperlinking your notes is an immense added value.  Just imagine a theorem in which all the key terms are hyperlinked to their definitions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Student's can fix typos themselves instead of continually barraging you with emails.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The notes no longer need to be linear.  You can add optional extras, such as a page where you go into greater detail for those interested.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the motivation for writing your own notes in a course where textbooks are available?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I wrote notes for a course which was a &quot;cut and shut&quot; course.  When I took it over, it used three textbooks (one only a chapter) and only one of those was actually any good.  Although I found better replacements, I still felt unhappy about asking students to buy three textbooks for one course (as partial mitigation, I found free online textbooks for some parts).  Also, things didn't go smoothly between the books.  So I wrote supplemental notes to help bridge the gap between the lectures and the books, and in some places to replace the books altogether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find my notes online on &lt;a href=&quot;http://mathsnotes.math.ntnu.no/mathsnotes/show/TMA4145+Home+Page&quot;&gt;my course wiki&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-04-04T07:21:25.270" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1290" PostTypeId="2" ParentId="1266" CreationDate="2014-04-04T07:25:48.793" Score="6" Body="&lt;p&gt;At Warwick University then there were explicitly given 5 points per homework (out of 25, I think) for &quot;Clarity and Style&quot;.  By really emphasising this in the first year, I found that with my students (I was a graduate student at the time, so working as a sort of TA) then this sort of problem soon went away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I don't have an answer to the &lt;em&gt;specific&lt;/em&gt; question (&quot;Is simply pointing this out in class effective?&quot;), but I can say that putting it in the rubric for homeworks and backing that up in tutorial time is effective.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-04-04T07:25:48.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1291" PostTypeId="2" ParentId="1284" CreationDate="2014-04-04T08:36:04.957" Score="9" Body="&lt;p&gt;Excerpt from a 2011 Insider Higher Education &lt;a href=&quot;http://www.insidehighered.com/news/2011/05/24/liberal_arts_college_in_kentucky_experiments_with_spanish_language_immersion_program&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; (bold text added by me):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In an effort that some are calling &lt;strong&gt;the first program of its kind&lt;/strong&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Georgetown_College&quot; rel=&quot;noreferrer&quot;&gt;Georgetown College&lt;/a&gt;, a Christian liberal arts college in Georgetown, Ky., is hoping the same high interest in foreign languages in K-12 schools can be translated to the postsecondary level. This fall, the 1,300-undergraduate college will welcome its first cohort of 15 students to its &lt;a href=&quot;http://www.georgetowncollege.edu/international/ie/iegc-courses/&quot; rel=&quot;noreferrer&quot;&gt;Inmersión en Español program&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;These students&lt;/strong&gt; still will be able to pursue a major in any of the college's departments, but they &lt;strong&gt;will take about one-third of their core requirements in Spanish instead of English&lt;/strong&gt;. Specifically, five of the courses that are part of Georgetown's Foundations and Core Program will be offered in Spanish. &lt;strong&gt;They will include&lt;/strong&gt;, at first, required courses in critical thinking, literature, philosophy, political science and &lt;strong&gt;mathematics&lt;/strong&gt;. The courses will be taught, in large part, by professors who are versed in both Spanish and the core discipline. For example, the college recently hired a Ph.D. in Spanish who has a master's degree in philosophy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The article continues:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...&lt;strong&gt;the initiative at Georgetown is thought to be unique&lt;/strong&gt;, said Emily Spinelli, executive director of the American Association of Teachers of Spanish and Portuguese.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;“Although there might be other colleges or universities that offer required general education courses in Spanish, I am not personally aware of them,”&lt;/strong&gt; she wrote in an e-mail.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I altered the link to the program above to direct you specifically to the course offerings.&lt;/p&gt;&#xA;&lt;p&gt;Philosophy is already listed, and under &amp;quot;Upcoming Courses&amp;quot; you will also see:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Other courses planned include: Political Science, Biology, and &lt;strong&gt;Math&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The comments from the Inside Higher Education article also mention a former school, Elbert Covell College (ECC), at the &lt;a href=&quot;http://en.wikipedia.org/wiki/University_of_the_Pacific_%28United_States%29&quot; rel=&quot;noreferrer&quot;&gt;University of the Pacific&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Though ECC has since closed, it once had the sort of offering you describe.&lt;/p&gt;&#xA;&lt;p&gt;A school description can be found &lt;a href=&quot;http://www.pacific.edu/Academics/Schools-and-Colleges/School-of-International-Studies/About/School-History/Cluster-Colleges.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, where the relevant section begins:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Elbert Covell College&lt;/strong&gt;, a liberal arts and sciences college also, was unique because all courses were taught in Spanish.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Finally, an October of 1963 &lt;a href=&quot;http://www.boveecruz.com/raymond/timemag.html&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; in Time Magazine remarks:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This month [College President] Burns opened another college that appears to be &lt;strong&gt;unique in the U.S., one teaching everything in Spanish&lt;/strong&gt;. The goal of Elbert Covell College is &amp;quot;education for life in the Americas in the 20th century.&amp;quot; It will stress &lt;strong&gt;math&lt;/strong&gt;, science, business and school teaching. Equally important, it will throw together 250 dissimilar students, two-thirds of them from Latin America, the rest Americans fluent in Spanish. Already on hand are 60 students from the U.S. and 14 Latin American countries. &lt;strong&gt;Faculty is still a problem. Covell has spent months trying to find a Spanish-speaking physicist, for example. &amp;quot;The very difficulty we've had shows how much this program is needed,&amp;quot; says Director Arthur J. Cullen.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-04T08:36:04.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1292" PostTypeId="2" ParentId="1268" CreationDate="2014-04-04T08:46:12.053" Score="4" Body="&lt;p&gt;I think this one may suit your needs, at least partially: Richard W. Hamming, &lt;em&gt;Methods of Mathematics Applied to Calculus, Probability, and Statistics&lt;/em&gt;, &lt;a href=&quot;http://store.doverpublications.com/0486439453.html&quot; rel=&quot;nofollow&quot;&gt;http://store.doverpublications.com/0486439453.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="660" LastEditorUserId="498" LastEditDate="2014-04-04T10:35:42.950" LastActivityDate="2014-04-04T10:35:42.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1294" PostTypeId="1" AcceptedAnswerId="1299" CreationDate="2014-04-04T12:58:21.113" Score="13" ViewCount="4795" Body="&lt;p&gt;I am teaching those topics (arithmetic/geometric series) just now, and want some &#xA;not so standard (fun) examples, which can be used essentially at high school/beginning calculus level.  I'm considering some, and will post them as answers, but first I'll wait to see what others can come up with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With standard examples of geometric series, for instance, I think about applications to interest/loan/annuity calculations, and so on.&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="123" LastEditDate="2014-04-04T13:13:52.827" LastActivityDate="2014-06-12T02:21:47.873" Title="What are some fun/nonstandard examples of arithmetic/geometric series?" Tags="|reference-request|examples|series|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1295" PostTypeId="2" ParentId="1263" CreationDate="2014-04-04T13:30:36.007" Score="6" Body="&lt;p&gt;Mine started out as a student (who recognizes having a horrible handwriting) got into the habit of transcribing my scribblings on the blackboard to LaTeX. He shared them with me via &lt;a href=&quot;http://git-scm.com&quot;&gt;git&lt;/a&gt;, I fixed up his copying mistakes (and my much more frequent glitches on the blackboard), completed derivations, added examples, ... and published the result for the class. It sort of grew from there, adding new material as the course's contents changed from term to term, filling out with ancilliary matter not directly used but which was needed to check coherence, and ocassional interesting sidetracks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for starting this was that the syllabus was a coherent collection of themes, but no book covering them decently could be found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has turned out a very long project, running (on and off) for a few years now, with no end in sight...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I got much help from my students, which get extra credits for pointing out mistakes, streamlining proofs (or directly making them understandable), or suggestions like new examples. Often I gave out extra homework to read and criticize the sections covered in class (or sometimes sidetracks), that helped them learning the material and got me high-quality feedback.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-04T13:30:36.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1296" PostTypeId="1" AcceptedAnswerId="1298" CreationDate="2014-04-04T14:55:35.470" Score="7" ViewCount="335" Body="&lt;p&gt;I'm planning to ask my students (first-year college calculus) to turn in some problems in a &quot;final draft&quot; form: rewritten from scratch, in linear order, with exactly the necessary steps included.  I suspect other people have tried similar assignments, and I'm wondering what lessons other people have learned, and what pitfalls I should be watching out for.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-04-04T15:37:08.690" Title="Asking for Final Drafts of Homework" Tags="|undergraduate-education|homework|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1297" PostTypeId="2" ParentId="1294" CreationDate="2014-04-04T15:09:33.017" Score="8" Body="&lt;p&gt;As an engineering student, I dealt with binary all the time. A good calculator would convert from decimal to binary, hex, octal.&#xA;When converting 1/3 to binary, I noted that it's .01010101... Which if you are used to reading binary past the decimal point is simply 1/4+1/16+1/64....&#xA;This seemed interesting to me. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-04-04T15:09:33.017" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1298" PostTypeId="2" ParentId="1296" CreationDate="2014-04-04T15:37:08.690" Score="5" Body="&lt;p&gt;I do this occasionally with some of my classes, and while I haven't done it with Calculus students, I can inform you of some common errors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The term &lt;strong&gt;final draft&lt;/strong&gt; doesn't seem to ring with all students when it pertains to mathematics homework. Determine what you would consider a final draft for this assignment. I have received a wide variety of submissions that I would not classify as final drafts. When grading these assignments, I find it is best to not give much leniency and to let the students know clearly where their final drafts need to be changed. If I am harsh at first, it makes it clear to my students exactly what I am looking for, which is care and precision when completing assignments.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-04-04T15:37:08.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1299" PostTypeId="2" ParentId="1294" CreationDate="2014-04-04T17:45:46.637" Score="9" Body="&lt;p&gt;The classic examples of the first $n$ natural numbers, first $n$ even numbers, and first $n$ odd numbers are all nice introductions. Here are some pictures I created to help explain them using blocks:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Natural numbers:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/OI5dK.jpg&quot; alt=&quot;Naturals&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Even numbers:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/bZDQX.jpg&quot; alt=&quot;Evens&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Odd numbers:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/d3jna.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Similar to the binary example given earlier, here is one using ternary (base 3):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$0.020202\ldots = \sum_{n = 1}^{\infty} \frac{2}{3^{2n}} = \sum_{n = 1}^{\infty} \frac{2}{9^n} = \frac{1}{4}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though the question concerns the beginning Calculus level, those later in their mathematical learning may note that the line above indicates $1/4$ is in the Cantor set.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Since you asked for nonstandard:&lt;/strong&gt; Rarely have I seen these problems asked in reverse. For example, Given a natural number $k$, what's an algorithm that could be used to determine if there is some $n$ such that the sum of $1, 2, \ldots, n$ gives $k$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Does this approach work?&lt;/strong&gt; Suppose you are given $666$. The sum of $1, 2, \ldots, n$ is $n(n+1)/2$, so we now ask whether there is an $n$ for which $2\cdot 666 = n(n+1)$. Taking the square root of the left side, we find $\sqrt{2\cdot 666} = 36.49657\ldots$, which suggests checking $n = 36$. (And it works!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Similarly&lt;/strong&gt;, suppose you are given $902$. Reasoning as above, we find $\sqrt{2\cdot 902} = 42.47352\ldots$, which suggests checking $n = 42$. (But no: $42(43)/2 = 903$. A narrow miss!)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;One more:&lt;/strong&gt; If you will allow for modifications similar to the $41 + \sum_{k=0}^{n}2k$ example, then consider:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$23 + \sum_{k=0}^{n}6k$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observe that this can be re-written as $3n^2 + 3n + 23$, which is prime for $0 \leq n \leq 21$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A nice follow-up problem is to prove that there cannot be a real-valued polynomial producing primes when evaluated at every natural number.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-04-06T09:40:26.823" LastActivityDate="2014-04-06T09:40:26.823" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1300" PostTypeId="1" AcceptedAnswerId="1310" CreationDate="2014-04-04T18:04:39.427" Score="15" ViewCount="795" Body="&lt;p&gt;I'm wondering about the use of word problems on exams which are &quot;cute&quot;: they have a slightly funny story, or some sort of pop culture reference, or tie into a running theme of some kind.  (As an example of the last category, I recall hearing about a professor who was, or at least professed to be, obsessed with some animal, perhaps a porcupine, and routinely gave word problems in which porcupines were the main actors.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some advantages are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Many students seem to enjoy them (I know I did when I was a student).&lt;/li&gt;&#xA;&lt;li&gt;They can provide a small break from the monotony and stress of an exam.&lt;/li&gt;&#xA;&lt;li&gt;They can make a problem more memorable and get students to read it more carefully.  (For instance, the one time I did this in a class, students asked about the &quot;Indiana Jones problem&quot; from the rest of the quarter.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some concerns are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They may unfairly disadvantage students who aren't native speakers&lt;/li&gt;&#xA;&lt;li&gt;They may distract students (especially weaker students) from the essential information (for instance, a sufficiently nervous student might not realize an unfamiliar pop culture reference is purely fluff)&lt;/li&gt;&#xA;&lt;li&gt;Students who dislike the class may be further annoyed by the apparent frivolity&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some questions (closely related enough, I hope, to count as a single question):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is there any actual research on the effect of such questions on student performance?&lt;/li&gt;&#xA;&lt;li&gt;Am I missing important benefits/problems?&lt;/li&gt;&#xA;&lt;li&gt;Does anyone have useful anecdotal evidence in favor of or against such questions?&lt;/li&gt;&#xA;&lt;li&gt;Does anyone have tips for using such questions in a way which avoids those concerns?  (In particular, I don't think kinds of problems I'm talking about are equally problematic.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-04-06T14:37:54.130" Title="Cute Word Problems" Tags="|exams|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1301" PostTypeId="2" ParentId="1263" CreationDate="2014-04-04T21:07:56.767" Score="6" Body="&lt;p&gt;I have built into my course requirements the expectation that students will, on a rotating basis, neatly rewrite/retype their notes and upload them to a class Wiki or discussion forum, and that other students will add comments and/or questions about those notes.  This serves multiple purposes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It forces graduate students to learn how to produce neatly formatted mathematical formulas;&lt;/li&gt;&#xA;&lt;li&gt;It ensures that if a student is absent for a legitimate purpose then he or she is able to find out what was missed;&lt;/li&gt;&#xA;&lt;li&gt;It gives me a way to keep track of improvised departures from my plans;&lt;/li&gt;&#xA;&lt;li&gt;It gives me an opportunity to add after-the-fact corrections, clarifications, or better examples;&lt;/li&gt;&#xA;&lt;li&gt;It gives me some idea of what is being correctly understood or misunderstood by my students;&lt;/li&gt;&#xA;&lt;li&gt;and (what is relevant for this question) &lt;strong&gt;it yields a reasonably complete rough draft of a set of lecture notes at the end of the semester&lt;/strong&gt;, which I can then reuse, edit, revise, or put to another purpose.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-04-04T21:07:56.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1302" PostTypeId="1" AcceptedAnswerId="1414" CreationDate="2014-04-04T22:11:10.697" Score="26" ViewCount="2355" Body="&lt;p&gt;For my background, I'm a software engineer currently studying for his master's degree in information security.  But when that's all done, I plan on going back to mathematics to keep me busy.  But with wife and kids, taking more college classes is purely out of the question.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a solid foundation in number theory/combinatorics/graph theory, and basic algebra and calculus.  Is there a solid lesson plan to take someone from number theory/comp sci to real analysis without a college course?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The winning answer will include:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Textbook suggestion with justification.  (perfect answer would be one written for a similar background)&lt;/li&gt;&#xA;&lt;li&gt;Three common pain points that empirically tend to require instructor intervention (perfect answer would be roadblocks comp sci students tend to have)&lt;/li&gt;&#xA;&lt;li&gt;list of your own length of theorems that make the transition from number-theory simpler.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This does seem fairly broad, but when I was reading on the meta site, it seems that education affords a certain expected level of subjectivity.  However I don't know at all where else I can go to ask questions to a large group of educators.  I have tried to narrow down the focus, but I'll understand if the question gets canned.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;===========MOTIVATION============&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J.W. asked a really good question in the comments, enough I'll quote it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do you mind if I ask what your motivation is for studying real&#xA;  analysis? Would you like to study it for its own sake or do you see it&#xA;  as a stepping stone to another topic (or both)? Where you're trying to&#xA;  get is arguably as important as where you've come from.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My motivation is simple:  I want to know &lt;em&gt;how&lt;/em&gt; calculus &lt;em&gt;works.&lt;/em&gt;  I tend to be a rather annoying student.  For example, when taking college algebra, it wasn't enough to see the term &lt;code&gt;x^2&lt;/code&gt; in an exercise and just do operations on it.  I needed to know &lt;em&gt;why&lt;/em&gt; &lt;code&gt;x^2&lt;/code&gt; was &lt;code&gt;x^2&lt;/code&gt;.  Both College Algebra and Calculus more or less introduced me to all this stuff that &quot;works by fiat.&quot;  Well, Discrete Math/Combinatorics taught me &lt;em&gt;how&lt;/em&gt; algebra works. (Not all the way of course, but it felt far less like magic!)  Calculus still appears to be magic to me.  It isn't enough for me to know that smart people proved it works... so analysis it is!  After I climb this mountain I know there will be other peaks but for now I'll limit myself to just understanding how calculus works.  I will likely study Abstract Algebra after this so I can deeply understand hyperelliptic curves.  But its also possible that the study of analysis will provoke me to study a deeper topic within analysis too... I'm keeping my options open.  &lt;/p&gt;&#xA;" OwnerUserId="676" LastEditorUserId="676" LastEditDate="2014-04-05T17:10:09.790" LastActivityDate="2020-08-19T15:07:36.157" Title="Lesson plan to self-teach real analysis to student with comp-sci background" Tags="|mathematical-pedagogy|mathematical-analysis|self-learning|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="1303" PostTypeId="2" ParentId="1294" CreationDate="2014-04-04T22:40:26.050" Score="5" Body="&lt;p&gt;I like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$41+\sum_{k=0}^{0}2k$,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$41+\sum_{k=0}^{1}2k$,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$41+\sum_{k=0}^{2}2k$,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;… through&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$41+\sum_{k=0}^{40}2k$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each sums to a prime except for the 41st sum which is $41^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Attached is a pic of this sequence of primes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/QGKlA.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="603" LastActivityDate="2014-04-04T22:40:26.050" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1304" PostTypeId="1" CreationDate="2014-04-04T22:45:47.903" Score="12" ViewCount="190" Body="&lt;p&gt;I don't use a textbook for my Algebra I class and am always spending hours looking for good math tasks for this course. Does anyone know of any printed materials that I can access where I can get ahold of good math &quot;application&quot; problems. I already use teacherspayteachers.com but even they do not have many good &quot;word problems&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="677" LastActivityDate="2014-04-16T20:13:01.383" Title="Looking for good Algebra I tasks" Tags="|algebra|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1305" PostTypeId="2" ParentId="1245" CreationDate="2014-04-04T23:06:03.553" Score="7" Body="&lt;p&gt;Decimal versions of fractions give very simple examples. For example 1/3 = .333333333…..&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And what is that decimal but $\sum_{k=1}^{\infty}3/(10^k)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or 1/11 is .09090909... Which is $\sum_{k=1}^{\infty}9/(100^k)$&lt;/p&gt;&#xA;" OwnerUserId="603" LastActivityDate="2014-04-04T23:06:03.553" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1306" PostTypeId="2" ParentId="750" CreationDate="2014-04-05T05:18:54.600" Score="3" Body="&lt;p&gt;A toy my kids have enjoyed lately (most popular with my 5 year old) is an interesting set of plastic turtles which interlock in numerous ways. These turtles illustrate many different symmetry patterns. &#xA;&lt;img src=&quot;https://i.stack.imgur.com/OHvXw.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They are a lot of fun. Beyond that, there are tons of pattern recognition games that can encourage children to begin thinking about abstraction. When I was little, my brother and I used to look for triangles, we found them everywhere. The manipulatives to illustrate 1, 10, 100 and 1000 are also a big hit. Although, I mostly see them being used to build towers, I'm not sure how much math is there just yet. Honestly, the most important thing is just to nurture a love of learning in general. Math will follow when the time is ripe.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-04-05T05:18:54.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1307" PostTypeId="2" ParentId="1300" CreationDate="2014-04-05T10:56:53.653" Score="6" Body="&lt;p&gt;You are &lt;strong&gt;missing&lt;/strong&gt; two important &lt;strong&gt;benefits&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;These problems visualize, that the theory has &lt;em&gt;many&lt;/em&gt; and &lt;em&gt;diverse&lt;/em&gt; applications and it's up to us humans to find even more applications. Anything can be modelled mathematically.&lt;/li&gt;&#xA;&lt;li&gt;If this kind of question becomes a routine, you can use it to make your students use some routine (way of solving, heuristic, think outside of the box, …)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One &lt;strong&gt;anecdote&lt;/strong&gt; here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During a math course for 11th and 12th graders at the German school in Egypt, I work for, I repeatedly used the same person in my word problems: Mahmoud. Whenever the students read a problem involving Mahmoud, they knew it's gonna be about practical applications and they should think practically as well as purely mathematically. Of course, these problems were harder than purely formal tasks. They even told the students of the other German schools, who did the same exams: &quot;If there is a task involving Mahmoud, forget about it, you won't solve it.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I regard this as evidence in favour but have the ambiguity of it in mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My regard of your &lt;strong&gt;concerns&lt;/strong&gt;: They are invalid.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Non-native speakers have the same problem with &quot;cute word problems&quot; as with &quot;normal word problems&quot;. If you connect the &quot;cute word problems&quot; with some routine, you can even help non-native speakers. Besides, it's part of the challenge of math.&lt;/li&gt;&#xA;&lt;li&gt;Distraction from essential information: This is a reason &lt;em&gt;for&lt;/em&gt; this kind of problems. It's part of the challenge of math to extract mathematical information from non-mathematical. This is the very essence of modelling.&lt;/li&gt;&#xA;&lt;li&gt;Students, who dislike the class: Forget about them. They won't like the class because of &quot;normal word problems&quot; instead of &quot;cute word problems&quot;. You cannot help these students, except offering different approaches to math. This is only a concern against only using cute word problems.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Concerning &lt;strong&gt;research&lt;/strong&gt; on &lt;strong&gt;students' performances&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would be rather difficult, as &quot;cute word problems&quot; are a wanted difficulty in itself. Solving a &quot;cute word problem&quot; may be a greater performance than solving a &quot;normal word problem&quot;.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-05T10:56:53.653" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1308" PostTypeId="2" ParentId="1294" CreationDate="2014-04-05T11:15:29.827" Score="4" Body="&lt;p&gt;Good answers!! What I had in mind was this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fractals, as the Koch snowflake:   &lt;a href=&quot;https://en.wikipedia.org/wiki/Koch_snowflake&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Koch_snowflake&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The area is a convergent geometric series, while the circumference is infinite, &#xA;represented by a divergent geometric series. The geometric series arises in such problems because of the self-similarity of the figure.                                &lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="122" LastEditDate="2014-05-30T11:32:04.943" LastActivityDate="2014-05-30T11:32:04.943" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1309" PostTypeId="2" ParentId="750" CreationDate="2014-04-05T11:22:58.517" Score="9" Body="&lt;p&gt;&lt;strong&gt;Addition/subtraction by measurement&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure whether this is at all what you had in mind, but here comes anyway. When I was something like 2-3 years old my Dad taught me to add and subtract single digit numbers by sawing (and sanding for safety) pieces of wood of dimensions about half an inch x $2$ inches x $2n$ inches, where $n$ is the integer to be presented. He also drew the digit in the middle in red ink. The idea was then to add, say $2+3$, by juxtaposing the sticks representing $2$ and $3$ and then compare the total length to that of longer sticks and find the one that matches. He also defined $a-b$ as a solution of the equation $b+x=a$ that can similarly be solved by finding the stick that fits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A generation later I repeated the exercise. It worked to my satisfaction and to my son's amusement. When my son started school (in Finland first graders are aged 7) his teacher asked parents for ideas for an &quot;evening at school&quot;. So I did this number again. The kids picked it up immediately, and while they were excited for the allotted 5 minutes or so, it was apparent that it was too easy for most of them (some kids remained silent, so I cannot give any stats). The teacher was observing me with a group of kids, and he suggested afterwards that an average 7-year-old's cognitive ability has, indeed, developed enough to deal with the quantity concept more abstractly, and this was thus easy. He did think that this model is beneficial for pre-schoolers. If somebody knows enough about the cognitive development of children, I would appreciate a confirmation/reference/link.&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2018-03-27T13:06:56.987" LastActivityDate="2018-03-27T13:06:56.987" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1310" PostTypeId="2" ParentId="1300" CreationDate="2014-04-05T11:30:29.223" Score="8" Body="&lt;p&gt;A related example that comes to mind at the (strong) secondary level is &lt;a href=&quot;http://en.wikipedia.org/wiki/Yellow_Pig%27s_Day#Yellow_Pigs_Day&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yellow Pigs Day&lt;/a&gt; at Hampshire College; this includes &lt;a href=&quot;http://en.wikipedia.org/wiki/David_Kelly_%28mathematician%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Kelly&lt;/a&gt;'s focus on the number 17.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the (undergraduate) tertiary level, I am reminded of &lt;a href=&quot;http://en.wikipedia.org/wiki/Frank_Morgan_%28mathematician%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frank Morgan&lt;/a&gt;'s focus on soap bubbles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Earlier in school mathematics, teachers often use sports questions (e.g., ones related to baseball) to teach various courses in mathematics (e.g., Statistics). I don't think &quot;baseball&quot; lends itself to &quot;cute word problems&quot; the same way that, e.g., yellow pigs, soap bubbles, and porcupines do, but there is a similar issue insofar as some students may simply be disinterested in the recurring theme that you choose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regard to research, I am not aware of anyone exploring the notion of a &quot;cute&quot; problem. (Probably you would need to be more precise with regard to what you mean...) I will say, though, that there is some relation to &lt;a href=&quot;https://matheducators.stackexchange.com/a/1256/262&quot;&gt;my answer&lt;/a&gt; to your earlier question (on the chain rule): namely, novice problem solvers (here, students - much of the time) differ from expert problem solvers (here, math teachers - much of the time) in how they think of related problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More precisely, there is a tendency among the novices to associate problems based on superficial features, such as grouping together &lt;em&gt;animal problems&lt;/em&gt; as being mathematically similar instead of, say, &lt;em&gt;chain rule problems&lt;/em&gt;. (See my linked answer for relevant references.) From this perspective, it may be that disentangling the &lt;em&gt;cuteness&lt;/em&gt; from the &lt;em&gt;mathematics&lt;/em&gt; is not entirely straightforward. This worry is captured somewhat by your second concern; indeed, it is a reasonable consideration.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-05T11:30:29.223" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1311" PostTypeId="1" AcceptedAnswerId="1314" CreationDate="2014-04-05T14:19:43.370" Score="11" ViewCount="433" Body="&lt;p&gt;In the question &lt;a href=&quot;https://matheducators.stackexchange.com/q/1/114&quot;&gt;Encouraging class participation&lt;/a&gt; there are some answers how to encourage class participation in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes, there are some smaller lectures (in particular in graduate courses) or you have smaller exercise groups where you know most of the students, sometimes even by name or from other courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are the (dis)advantages of asking questions directly to specific students?&lt;/strong&gt; E.g., you can ask: &quot;[Student's name], you should remember that from last year's lecture where we done something similar in the proof of Theorem X.&quot; or &quot;[Student's name], I've seen that you solved a similar problem very well in your homework. Can you explain to the other students how to start?&quot;.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-17T20:06:42.403" Title="Should one ask questions directly to specific students in a lecture or exercise group?" Tags="|undergraduate-education|classroom-management|graduate-education|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1312" PostTypeId="5" CreationDate="2014-04-05T14:29:08.497" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-05T14:29:08.497" LastActivityDate="2014-04-05T14:29:08.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1313" PostTypeId="4" CreationDate="2014-04-05T14:29:08.497" Score="0" Body="For questions on teaching and introducing real numbers and properties of real numbers." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-05T14:39:55.217" LastActivityDate="2014-04-05T14:39:55.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1314" PostTypeId="2" ParentId="1311" CreationDate="2014-04-05T14:44:34.980" Score="8" Body="&lt;p&gt;&lt;strong&gt;Advantages&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Full attention of said student&lt;/li&gt;&#xA;&lt;li&gt;Possibly shared attention of other students due to empathy&lt;/li&gt;&#xA;&lt;li&gt;Personal connection&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Disadvantages&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Exposure of said student (might still be an inconvenience in undergraduate courses)&lt;/li&gt;&#xA;&lt;li&gt;Possibly loss of attention of other students due to non-concern&lt;/li&gt;&#xA;&lt;li&gt;Overvalueing of said students answer by other students (his status is lifted to something between student and teacher)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Workaround&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, ask all questions to all students. Then, ask said student seemingly randomly. Depending on the situation, reason your choice of student to answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;BTW&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't worry to much. It's not high school. The students are adults and can decide for themselves, if they want to compare their own answer/solution to that of said student.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-05T14:44:34.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1315" PostTypeId="2" ParentId="1311" CreationDate="2014-04-05T14:47:32.340" Score="8" Body="&lt;p&gt;Some ideas:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Literature (sorry, don't have a reference handy) says that when you ask one student, other students stop thinking. So, if you are going to do that, pose the question to the whole group, give a bit of wait time (6-10 seconds)so everyone is thinking about the problem.&lt;/li&gt;&#xA;&lt;li&gt;Spending some time at the beginning of the course to establish communication and trust rules/culture in your classroom is a good investment :)&lt;/li&gt;&#xA;&lt;li&gt;I would also soften the phrase &quot;you should remember&quot;; maybe &quot;you might remember...?&quot; or &quot;we have done ...&quot;...&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Here is an article related to this topic: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bond, N. (2007). Questioning Strategies that Minimize Classroom Management Problems. Kappa Delta Pi Record, 44(1), 18-21. &lt;a href=&quot;http://www.tandfonline.com/doi/pdf/10.1080/00228958.2007.10516486&quot;&gt;doi: 10.1080/00228958.2007.10516486&lt;/a&gt;; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article has a nice list of supporting references about questioning strategies in the classroom that are also worth exploring. &lt;/p&gt;&#xA;" OwnerUserId="95" LastEditorUserId="446" LastEditDate="2014-05-14T14:47:29.620" LastActivityDate="2014-05-14T14:47:29.620" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1316" PostTypeId="1" AcceptedAnswerId="1321" CreationDate="2014-04-05T14:47:35.263" Score="31" ViewCount="3066" Body="&lt;p&gt;I'm working in the field of applied mathematics (optimization and numerics) and I meet a lot of students saying that they are allergic to applied mathematics or that they hate it or some quotes like &quot;I'm here to learn how to think and become a thinker and that's why I only want to do pure mathematics&quot;, &quot;Can't the engineers do all that?&quot;. I think, the students split in two groups: One group contains poor students who somehow worked out to pass basic courses, but are not able to apply the basics and want to blame something for that. The other group consists normally of very good students who like the concept of &lt;em&gt;Definition -&gt; Theorem -&gt; Proof&lt;/em&gt; and miss that concept when it comes to modelling, looking at specific algorithm, comparing ideas etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, due to the curriculum, they &lt;strong&gt;have&lt;/strong&gt; to attend lectures about applied math, so I could say that and be fine. But somehow I feel that one can argue at least with the second group of students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How should one react to such quotes? Should one even react or could it make the situation worse? If yes, what are good arguments?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="376" LastEditDate="2014-09-24T16:59:01.087" LastActivityDate="2014-09-24T17:47:15.683" Title="How to react to students saying that they are allergic to applied mathematics?" Tags="|undergraduate-education|student-motivation|general-pedagogy|applied-mathematics|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1317" PostTypeId="5" CreationDate="2014-04-05T14:52:27.950" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-05T14:52:27.950" LastActivityDate="2014-04-05T14:52:27.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1318" PostTypeId="4" CreationDate="2014-04-05T14:52:27.950" Score="0" Body="For questions concerning the motivation of students and helping them to motivate for their study in general." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-05T14:59:02.313" LastActivityDate="2014-04-05T14:59:02.313" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1319" PostTypeId="2" ParentId="1236" CreationDate="2014-04-05T14:59:09.220" Score="7" Body="&lt;p&gt;I can only answer for Germany.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;High-school maths teacher&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;25-27 lessons (each 45 min) per week. Depending on teaching style, additionally grading homework or grading tests (once a month) and exams (2 to 6 times per year). Supervision of projects etc. Overhead: class management, parents management, talks to students, coordinating with colleagues, school issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In sum, you very easily come to 45-50 hours per week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Professor&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6 to 8 lessons (each 45 min) per week. Coordinating your graduate assistants. Coordinating 2-4 exams per year. Consultation hours: 2 per week. Overhead: Course management.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In sum, you'll come to 10-12 hours per week. But be aware, that in Germany, a Professor is equally lecturer and researcher.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="439" LastEditDate="2014-04-05T18:14:16.880" LastActivityDate="2014-04-05T18:14:16.880" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1320" PostTypeId="2" ParentId="1316" CreationDate="2014-04-05T15:06:59.087" Score="-3" Body="&lt;p&gt;If you want to motivate your students, you have to address that issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe, you can motivate them via modelling. Show them, that what they thought is pure math, is already modelling.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Set theory is modelling the human psyche.&lt;/li&gt;&#xA;&lt;li&gt;Peano is modelling linear, stepwise algorithms.&lt;/li&gt;&#xA;&lt;li&gt;The axioms and basic theorems about real numbers are modelling human reception of the surrounding.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Maybe you can catch them with &lt;a href=&quot;http://en.wikipedia.org/wiki/Gabriel%27s_Horn&quot; rel=&quot;nofollow&quot;&gt;Gabriel's horn&lt;/a&gt; or &lt;a href=&quot;http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox&quot; rel=&quot;nofollow&quot;&gt;Banach-Tarski-paradox&lt;/a&gt; and the gap between math and physics and the need to close that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can continue with &quot;truly applied math&quot; by motivating with the need to model something.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-05T15:06:59.087" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1321" PostTypeId="2" ParentId="1316" CreationDate="2014-04-05T16:03:46.920" Score="19" Body="&lt;p&gt;In the first place, it is &quot;only&quot; common course-naming conventions (and the AMS and NSF subject classifications) are to blame for the impression that there is some meaningful schism between something called &quot;pure&quot; and something called &quot;applied&quot; mathematics. Second, as noted, as much as anything people rationalize their own limitations or failings by blaming something external. And identifying an &quot;other&quot; and vilifying it has a long tradition in human history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could explain to the students that the definition-lemma-proof-theorem-proof picture is usually created with much &lt;em&gt;hindsight&lt;/em&gt;, after many bouts of experiment, heuristics, revision. So our modern definition often came &lt;em&gt;last&lt;/em&gt; in real life, while the desired/needed proto-theorem was presented to us by exterior circumstances. That is, &quot;axiomatic mathematics&quot; is most often done &lt;em&gt;after&lt;/em&gt; a long period of experiment and revision.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sure, some parts of school-math present only the end-product, but it would promote a methodological error to allow people to take a &quot;Whiggish&quot; view of these developments. That is, it's not that pre-axiomatic ideas were foolish or became irrelevant or obsolete. Rather, they constitute the phenomena-of-interest that were later organized by axiomatization. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Terry Tao &lt;a href=&quot;https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/&quot;&gt;https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/&lt;/a&gt; and others, such as &lt;a href=&quot;http://m-phi.blogspot.com/2013/03/terry-tao-on-rigor-in-mathematics.html&quot;&gt;http://m-phi.blogspot.com/2013/03/terry-tao-on-rigor-in-mathematics.html&lt;/a&gt; have reasonably labelled developmental stages of mathematical thinking as pre-rigorous, rigorous, and &lt;em&gt;post&lt;/em&gt; -rigorous, in application to individuals. This has observable substance in application to individuals, but application to the larger enterprise is less clear.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And we can point out to our students that few of the historically well-known mathematicians made a distinction between &quot;pure&quot; and &quot;applied&quot;, nor did they artificially confine themselves either in action or self-labelling. Most of the really interesting developments in mathematics arguably help us understand both very literal objects as well as idealized objects, and the whole range between.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And there is also the caution that some of the mid-to-late 20th century sense of &quot;applied math&quot; really meant numerical solution of PDEs, whether truly directly &quot;applied&quot; to create marketable products and local jobs or not. It may be that the WWII and study of non-linear waves to make big bombs leant special gravitas to numerical solution of PDE. But by this year we find that nearly all parts of mathematics have bluntly crass utility in business settings. Certainly it is not the case that these applications necessarily follow anyone's whimsical set-ups of axiom systems, but mathematics itself never really did random things, either.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-05T16:03:46.920" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1322" PostTypeId="2" ParentId="1236" CreationDate="2014-04-05T17:25:55.957" Score="14" Body="&lt;p&gt;This answer is really more of an extended comment, so please do not award me the bounty.&lt;/p&gt;&#xA;&lt;p&gt;I think this question may be difficult to answer objectively, because the answer varies so much depending on (a) where you are teaching, (b) what specific courses you are teaching, and (c) what level of quality you aspire to.  Of these, I think (c) may be particularly relevant.&lt;/p&gt;&#xA;&lt;p&gt;For example, in 1999-2000, MIT professor &lt;a href=&quot;https://en.wikipedia.org/wiki/Walter_Lewin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Walter Lewin&lt;/a&gt; taught a  two-semester introductory physics course.  &lt;a href=&quot;https://ocw.aprende.org/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/video-lectures/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Videos of this course&lt;/a&gt; were placed online, &lt;a href=&quot;https://www.youtube.com/watch?v=NeXIV-wMVUk&quot; rel=&quot;nofollow noreferrer&quot;&gt;some of which&lt;/a&gt; have garnered millions of views on YouTube.&lt;/p&gt;&#xA;&lt;p&gt;His goal for the course was to teach &lt;em&gt;the&lt;/em&gt; introductory physics course.  According to &lt;a href=&quot;http://www.nytimes.com/2007/12/19/education/19physics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this New York Times article&lt;/a&gt;, he prepared almost 25 hours for each lecture, which means 50 hours of preparation for two lectures a week.  In addition to preparing marvelous experimental demonstrations, he also prepared nearly word-for-word exactly what he would say during each lecture, and then memorized this &amp;quot;script&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;What this means is that he taught only one four-credit course each semester, and he worked more than full time on this.  How does this compare to the workload of the typical high-school teacher?  Or of a typical professor?&lt;/p&gt;&#xA;&lt;p&gt;My impression is that most good teachers put &lt;em&gt;more&lt;/em&gt; work into their classes than is absolutely required, in an effort to help the students learn.  For example, sometimes when I teach I write up extensive lecture notes for the students, even though they already have the lecture as well as the textbook.  These notes seem to help the students quite a bit, but writing up such notes isn't exactly required of me.&lt;/p&gt;&#xA;&lt;p&gt;I know this is true at the high school level as well.  My mother was a high-school Spanish teacher for many years, and she would regularly put quite a lot of effort into preparing interesting and fun activities for the students that would introduce them to aspects of the Spanish and Mexican cultures.  I don't think this was required of her, but she did it because she was fulfilling &lt;em&gt;her&lt;/em&gt; interpretation of what it means to be a good teacher.&lt;/p&gt;&#xA;&lt;p&gt;So my conclusion is that &amp;quot;workload&amp;quot; isn't exactly a relevant concept for teachers.  Most teachers choose for themselves how much work to do, based on their own personal standards for what constitutes good teaching.  The more classes you have to teach, the less time you will have for each, and so the less well you will do your job.  But I'm not sure there's any objective way to compare the workloads of different kinds of teachers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; All that being said, I do agree with the spirit of the question.  Like Brian Rushton, I would love to hear from someone who has taught at both the high school and college levels.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="11" LastEditDate="2021-04-15T15:43:52.023" LastActivityDate="2021-04-15T15:43:52.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1323" PostTypeId="2" ParentId="160" CreationDate="2014-04-05T18:13:25.097" Score="13" Body="&lt;p&gt;Apart from quite general issues about smart kids' study skills or lack thereof, my observations over many years gives me the impression that mathematics and computer science offer special hazards/advantages. I'll address mathematics, since most of my experience lies there, but I suspect similar observations apply to computer science (which I've also paid some attention to since late 1960s).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, beyond the obvious connection to immediate tactile reality, mathematics allows &quot;precocity&quot; and forms of &quot;problem-solving&quot; that do not necessarily depend heavily on wider experience, in contrast to, for example, history or philosophy. And timed low-level problem-solving contests, in which speed plays a key role? Nothing like that in much else. Even music performance, which can be very much &quot;of the moment&quot; and allowing precocity, is not just about playing as many notes as possible. The &quot;problem&quot; is that subsequent development of mathematics in school or in real life &lt;em&gt;does&lt;/em&gt; depend more and more on wider experience (even if only somewhat interior to mathematics), and considerably less on raw quickness or cleverness. The sense of pace changes enormously. Kids may reasonably see this as a bait-and-switch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The typical school curriculum (what I've seen in the U.S.) and ambient cultural attitudes (manifest in the curriculum) reinforce a peculiarly shallow picture of what mathematics is, and what it means to excell at it. Contests are understandable, but more substantive things aren't. Use of low-level math (&quot;(pre) calculus&quot;?) to &quot;filter&quot; for other life paths, is rather strange, but accounts for the size of the math departments in U.S. public universities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, kids can be declared &quot;successes&quot; in mathematics already at a strangely young age, based on criteria that mislead them, thus selecting for traits that are not ... in fact ... best longer-term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there'd not really be anything good accomplished by eliminating contests, for example. The fact that this aspect is easily and popularly misinterpreted seems to me inescapable. There should be a strong counterpoint to this attitude, but our school system itself traditionally rewards similar traits, so...?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a further, complementary problem, in my observation, that (in the U.S., anyway) &quot;the system&quot; often selects &lt;em&gt;against&lt;/em&gt; mathematical traits that would in fact be beneficial long-term, in cases where the quick-clever-contest-math attributes are not strongly present. It's so easy to over-simplify, and identify quickness with talent, that even in math grad school lack of facility in timed exams is misconstrued as lack of mathematical talent. I confess that it took me a long time to recover from this misunderstanding, myself, and I have no good recommendation about over-coming either side of this false identification, except to keep it in mind and remind others.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-05T18:13:25.097" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1324" PostTypeId="2" ParentId="18" CreationDate="2014-04-05T19:27:34.090" Score="12" Body="&lt;p&gt;I’m a high school maths teacher and I use a way to test where crib sheets are allowed; yet pupils are rewarded for not using it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the start of the test, the pupils use black ink and are not allowed to use the test aids (such as crib sheet and calculator). When they want to, they indicate to me that they want to use their aids, at which point they must now continue their exam in blue ink. They can only do this swap once, the black pen is removed from their desk. Solutions written in black carry more weight than solutions written just in blue (typically indicated on the exam sheet).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many extra advantages. The pupils like to know that at some point they can check their work. They can even correct their work if they find they have made a mistake (which gains credit). Because of this, students learn to present their work better, since they themselves have to come back to their work to fill in the gaps at a later time in the exam. Students who are algebraically strong get their due credit. Weaker students can let the calculator do the algebra, so they can still solve the questions but knowingly obtain fewer points. If a student knows a formula, they can get credit for it. Finally, it is easier for me to write the tests as I don’t need to write two parts: one without aids and one with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admit, it takes one test to get the students to appreciate how the system works and how they can benefit from it. The feedback I have had is mostly positive.&lt;/p&gt;&#xA;" OwnerUserId="685" LastEditorUserId="75" LastEditDate="2014-04-05T20:05:04.480" LastActivityDate="2014-04-05T20:05:04.480" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1325" PostTypeId="1" AcceptedAnswerId="1339" CreationDate="2014-04-05T20:04:31.287" Score="76" ViewCount="19373" Body="&lt;p&gt;Statistics are used everywhere; politicians, companies, etc. argue with the help of statistics. Since calculations are needed for the interpretation of statistics, such things should be taught in mathematics in school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are the most impressive common misleading interpretations of statistics that students should be aware of?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-07T08:53:53.620" LastActivityDate="2023-11-10T08:06:20.547" Title="Impressive common misleading interpretations in statistics to make students aware of" Tags="|undergraduate-education|secondary-education|statistics|mathematics-in-daily-life|" AnswerCount="20" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1326" PostTypeId="2" ParentId="1325" CreationDate="2014-04-05T20:04:31.287" Score="6" Body="&lt;p&gt;I think, a very commong example are newspapers writing something like &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The economic performance decreases: Last year's economic growth was 3%, now it is only 2.2%.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My interpretation: Although probably not knowing what a derivative is, they mix up decreasing of the first derivative (=economic growth) with the decreasing of the function (=economic performance).&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-08T11:34:29.327" LastActivityDate="2014-04-08T11:34:29.327" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1327" PostTypeId="2" ParentId="1325" CreationDate="2014-04-05T20:25:14.153" Score="25" Body="&lt;p&gt;Percentages are a source of many, many, many common mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One that is very common is believing that percentages can be added. An example: one of our presidents increased its salary by 172%; the next president decreased the presidential salary by 30%. It was commented that compared to the salary before the raise, it was still a 142% increase.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another one is to confuse 300% of a price and a raise by 300%. One of our former ministers made this mistake on twitter (while in office), sticking by it when corrected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another mistake, which is not really about percentage, is to invert roles in ratios when considering the correlation between two characters. E.g.: &quot;30% of convicted criminals have purple hairs&quot; is sometimes translated into &quot; 30% of purple haired people are convicted criminals&quot;.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-05T20:25:14.153" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1328" PostTypeId="2" ParentId="1325" CreationDate="2014-04-05T20:33:00.383" Score="10" Body="&lt;p&gt;Unfortunately, it is in German, but the book &lt;a href=&quot;http://www.amazon.de/dp/3446177507&quot; rel=&quot;noreferrer&quot;&gt;Angewandte Statistik: Eine Einführung für Wirtschaftswissenschaftler und Informatiker&lt;/a&gt; by Kröpfl, Peschek and Schneider contains many typical mistakes that you can make. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My favorite example is that you can show a strong geographic correlation in Germany between the number of stork nests and the number of newborn children.&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="394" LastEditDate="2014-04-06T07:29:44.683" LastActivityDate="2014-04-06T07:29:44.683" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1329" PostTypeId="2" ParentId="1316" CreationDate="2014-04-05T20:38:43.803" Score="10" Body="&lt;p&gt;I do not have a true answer, but only a personal anecdote that backs up the point of teaching applied maths to mathematicians-to-be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I work on fundamental mathematics, mostly differential geometry. My best work to date (in collaboration with Greg Kuperberg) has a differential geometric statement, but uses heavily linear programming which, as a fundamental mathematician, I have never been taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I managed to learn what I needed for the project (and I had already met with LP in previous research), but I now consider a bit sad not to have been more educated in what is often called applied mathematics, because it is really mathematics (and often very beautiful mathematics at that).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should stress that the use of LP in our work is related to the so-called &quot;Delsarte method&quot;. Delsarte was French, one of the founders of Bourbaki, but it is Greg that learned me about him. He was really amazed to see how little known he and is work is in France, among pure mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-05T20:38:43.803" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1330" PostTypeId="1" AcceptedAnswerId="1332" CreationDate="2014-04-05T20:54:00.717" Score="32" ViewCount="2502" Body="&lt;p&gt;This is something I have always had issues with. Generally, three approaches are used:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;The geometric path:&lt;/strong&gt; this follows the standard way how you would introduce these functions in school. The problem is that it is almost impossible to do this in a rigorous way and you have to introduce difficult concepts (like arc-length) what you would normally do later on.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;The power-series road:&lt;/strong&gt; the advantage is that this is quick, elegant, and you can prove all the properties in a quick and not too painful way. The problem is that for many students this is ad-hoc, the definition is unmotivated, and there is no connection with the geometric meaning.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;The functional equation road:&lt;/strong&gt; to define them through functional identities and continuity, and then prove properties. This is halfway motivated because at least addition formulae can be assumed from school, but, as I had it this way as I was a student, I have some reservations because I remember that this approach left some odd feeling.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If your students had a calculus class, then clearly the second approach can be used and this is what I do with those students who have this class after a calculus class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But generally, in my university, it is common that first year students start their studies with a rigorous analysis course. Which approach do you consider as the best in this case, where I cannot assume much familiarity with calculus knowledge and also do not want to wait till power series appear?&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="123" LastEditDate="2020-02-09T23:33:04.707" LastActivityDate="2020-06-10T21:08:34.513" Title="The best way to introduce trigonometric functions in a rigorous analysis course" Tags="|undergraduate-education|mathematical-analysis|trigonometry|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="1331" PostTypeId="2" ParentId="380" CreationDate="2014-04-05T20:58:47.227" Score="4" Body="&lt;p&gt;In France this problem is made especially acute by the fact that the concept of composed functions are not part of the high-school curriculum, so that student learn a bunch of separate rules to differentiate special cases of composed functions (one for $u^2$, one for $u(ax)$, one for $\ln(u)$, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I try to do is to use student's natural lazyness by telling them I will teach them a rule that contains all of these rules, so that they will be authorized to forget them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that could help them grasp it is to list a lot of functions they learned to differentiate using the chain rule, and ask them to find the common pattern between the formulas instead of giving it yourself. This may take a little time, but that would be well-used time.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="1274" LastEditDate="2014-05-02T17:57:13.400" LastActivityDate="2014-05-02T17:57:13.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1332" PostTypeId="2" ParentId="1330" CreationDate="2014-04-05T21:29:31.333" Score="39" Body="&lt;p&gt;I suppose my preference would be&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;strong&gt;4. The differential equations road.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, define $x(t) = \cos t$ and $y(t) =  \sin t$ to be the solutions to the following initial value problem:&#xA;$$&#xA;\frac{dx}{dt} = -y,\qquad \frac{dy}{dt} = x,\qquad x(0) = 1,\qquad y(0)=0.&#xA;$$&#xA;This assumes that you've proven the existence and uniqueness theorem for solutions to ODE's, which I would certainly want to cover in any two-semester analysis course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This treatment has the following advantages:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It is 100% rigorous.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is related to geometry.  The differential equations essentially say that $(x,y) = (\cos t,\sin t)$ is a unit-speed parametrization of the unit circle.  Indeed, the idea of a unit-speed parametrization of the unit circle is essentially a rigorous definition of &quot;angle&quot;, so this is really very close to the geometric path.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It leads directly to the power series for $\cos t$ and $\sin t$.  Specifically, it's very easy to show that the power series for sine and cosine are solutions to the given initial-value problem, so they're equal to $\cos t$ and $\sin t$ by the uniqueness theorem for ODE's.  Actually, this is the most coherent explanation I know for why the sine and cosine are analytic functions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A very similar treatment is available for $e^x$, so you can use the same approach for all of your transcendental functions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It provides a nice application of the existence and uniqueness theorem, which otherwise seems strangely disconnected from the rest of analysis.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-04-05T21:29:31.333" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1333" PostTypeId="2" ParentId="1294" CreationDate="2014-04-05T22:10:20.233" Score="10" Body="&lt;p&gt;Here's one you've probably seen. But I'll post it since I like it a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\sum_{k=0}^{\infty}(1/\phi^2)^k=\phi$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $\phi$ is the golden mean $1+(\sqrt{5}-1)/2$, approximately 1.618&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/RVcjY.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="603" LastActivityDate="2014-04-05T22:10:20.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1334" PostTypeId="2" ParentId="1325" CreationDate="2014-04-05T23:16:42.523" Score="45" Body="&lt;p&gt;Sally Clark (&lt;a href=&quot;http://en.wikipedia.org/wiki/Sally_Clark&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Sally_Clark&lt;/a&gt;) was convicted in the UK of murdering both her infant sons, when in fact it is much more likely that they died of natural causes.  The case against her was largely based on invalid statistical reasoning.  The Royal Statistical Society made a &lt;a href=&quot;https://web.archive.org/web/20110824151124/http://www.rss.org.uk/uploadedfiles/documentlibrary/744.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;statement&lt;/a&gt; about at at the time, which begins as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the recent highly-publicised case of R v. Sally Clark, a medical&#xA;expert witness drew on published studies to obtain a figure for the&#xA;frequency of sudden infant death syndrome (SIDS, or &amp;quot;cot death&amp;quot;) in&#xA;families having some of the characteristics of the defendant's family.&#xA;He went on to square this figure to obtain a value of 1 in 73 million&#xA;for the frequency of two cases of SIDS in such a family. This approach&#xA;is, in general, statistically invalid. It would only be valid if SIDS&#xA;cases arose independently within families, an assumption that would&#xA;need to be justified empirically. Not only was no such empirical&#xA;justification provided in the case, but there are very strong a priori&#xA;reasons for  supposing that the assumption will be false. There may&#xA;well be unknown genetic or environmental factors that predispose&#xA;families to SIDS, so that a second case within the family becomes much&#xA;more likely.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After more than three years in prison Sally Clark was released following a second appeal, but she died of alcohol poisoning a few years later.  This is a very sad but instructive story.&lt;/p&gt;&#xA;" OwnerUserId="76" LastEditorUserId="76" LastEditDate="2023-11-10T08:06:20.547" LastActivityDate="2023-11-10T08:06:20.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1335" PostTypeId="2" ParentId="1330" CreationDate="2014-04-05T23:33:53.580" Score="13" Body="&lt;p&gt;This aspect is discussed in great detail in §7.2 of &lt;a href=&quot;http://books.google.co.in/books?id=F8zgg3I_xsQC&amp;amp;lpg=PR2&amp;amp;ots=sl17563eoY&amp;amp;dq=Course%20in%20Calculus%20and%20Mathematical%20Analysis%20Ghorpade&amp;amp;pg=PR2#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Course in Calculus and Mathematical Analysis&lt;/a&gt; by Ghorpade and Limaye. The discussion must be accessible to students undergoing their first rigorous course in analysis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discussion is well-motivated: the authors point out in §7.1 that one can integrate all functions of the form $x^r$ for all rational $r \neq 1$ in terms of functions of the same kind. So to integrate $x \mapsto 1/x$, they introduce the logarithm function and study its properties. Now, of course, one has to verify that this is a genuinely new function: they show that $x \mapsto \ln(x), x &amp;gt; 0$ is not a rational function (cf. Lemma 7.26, loc. cit.): in fact, it is not even an algebraic function (cf. Proposition 7.27, loc. cit.). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the next natural question is integrating inverses of quadratic polynomials. If the polynomial factors into (necessarily) linear factors, one might use partial fractions. However, if the polynomial does not factor over $\mathbf{R}$, then, we are stuck again! One may show using the method of partial fraction that it suffices to learn to integrate the function $x \mapsto 1/(1+x^2)$. They then introduce the arctangent function as the integral of $1/(1+x^2)$ and go on to prove its familiar properties. The other trigonometric and inverse trigonometric functions are defined from just this function (by algebraic formulas or by using the monotonicity to establish the existence of an inverse, as applicable). Once again, the authors prove that the trigonometric functions are transcendental (cf. Proposition 7.29, loc. cit.). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, let me reproduce the following quote from the linked book:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The approach outlined above gives not only a precise definition of the logarithmic, exponential and the trigonometric functions, but also a genuine motivation for introducing the same. In most texts on calculus, the trigonometric functions are ‘defined’ by drawing triangles and mentioning that angles are ‘measured’ in radians. The main problem with this approach is succinctly described by Hardy [...], who writes: “The whole difficulty lies in the question, what is the x which occurs in cos x and sin x.” Hardy also describes different methods to develop an analytic theory (cf. [...]) and the approach we have chosen is one of them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Finally, I would like to point out that this method is very similar to how we introduce the number systems to students (by showing them equations &quot;in those systems&quot; you cannot solve by staying there!). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The authors also point out: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Failure to be able to integrate a function has often led to interesting developments in mathematics. For example, a rich and fascinating theory of the so-called elliptic integrals and elliptic functions arises in this way.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I hope this helps. &lt;/p&gt;&#xA;" OwnerUserId="109" LastEditorUserId="109" LastEditDate="2018-03-10T07:06:17.037" LastActivityDate="2018-03-10T07:06:17.037" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1336" PostTypeId="2" ParentId="1294" CreationDate="2014-04-05T23:41:58.393" Score="2" Body="&lt;p&gt;This might be a bit ambitious but you could describe how the geometric series enters the &lt;a href=&quot;http://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function&quot; rel=&quot;nofollow&quot;&gt;Euler Product Formula&lt;/a&gt; for the Euler-Riemann zeta function (you could just introduce this as a real-variable function and handle the products formally). &lt;/p&gt;&#xA;" OwnerUserId="109" LastActivityDate="2014-04-05T23:41:58.393" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1337" PostTypeId="2" ParentId="1325" CreationDate="2014-04-06T01:03:15.473" Score="22" Body="&lt;p&gt;Just two (now &lt;strong&gt;three&lt;/strong&gt;, see below), to whet the appetite. Stating the mistakes:  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Correlation implies Causation&quot;: &lt;strong&gt;it doesn't&lt;/strong&gt;. The finding of statistical correlation between two variables may strengthen a pre-existing theoretical/logical argument of existing causal links. But it may also reflect the existence of an underlying third variable that affects both and thus creates the correlation. When correlation is unexpectedly found, it indicates the possible existence of hitherto unknown causal links and should initiate a deeper (and not necessarily statistical) investigation -but it does not imply causation from the outset.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;If we take a larger and larger sample from a population, its distribution will tend to become normal (Gaussian) no matter what it is initially&quot;: &lt;strong&gt;it won't&lt;/strong&gt;. The Central Limit Theorem, the misreading of which is the cause of this mistake, refers to the distribution of &lt;em&gt;standardized sums of random variables&lt;/em&gt; as their number grows, not to the distribution of a &lt;em&gt;collection&lt;/em&gt; of random variables.  Alternative statement of the mistake: &quot;Everything has a bell-shaped distribution&quot; -let alone the fact that the normal distribution does not always look so bona fide bell-shaped.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The research paper &lt;em&gt;Students’ misconceptions of statistical inference: A review of the&#xA;empirical evidence from research on statistics education&lt;/em&gt; from &lt;em&gt;Ana Elisa Castro Sotos&lt;/em&gt; et al., reviewing research papers on the matter can be downloaded from&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;ftp://ftp.soc.uoc.gr/Psycho/Zampetakis/%D3%F4%E1%F4%E9%F3%F4%E9%EA%DE%20%C9/Useful%20Papers/statistical%20terms.pdf&quot;&gt;ftp://ftp.soc.uoc.gr/Psycho/Zampetakis/%D3%F4%E1%F4%E9%F3%F4%E9%EA%DE%20%C9/Useful%20Papers/statistical%20terms.pdf&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;ADDENDUM April 8 2014&lt;/strong&gt;&lt;br&gt;&#xA;I am adding a third one, which is really dangerous, since it relates in a more general sense to reasoning and inference, not necessarily statistical inference.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3.&quot;My sample is representative of the population&quot;: &lt;strong&gt;it isn't&lt;/strong&gt;. Ok, it may be, but you need to try hard to achieve that (or to be lucky), so don't take it as a given. It may look an &quot;ordinary&quot; day to you, with nothing special, but this does not mean that it is a &lt;em&gt;representative&lt;/em&gt; day. So counting during just this one day, the numbers of red and of blue cars passing outside your window, won't give you a reliable estimate of the average number of red and blue cars, or of the average proportion of red and blue cars, or of the probability that the cars will be red or they will be blue per day... This is sometimes called, sarcastically, &quot;&lt;a href=&quot;http://www.businessdictionary.com/definition/law-of-small-numbers.html&quot;&gt;the law of small numbers&lt;/a&gt;&quot; (but the Poisson distribution is sometimes also called that), and it points out the pitfalls of doing any kind of inference based on too little information, persuading yourself that this information is nevertheless &quot;representative&quot; of the whole picture, and so it suffices to reach valid conclusions. People do it all the time, even when statistics do not appear to be involved. Fundamentally, it has to do with the difficulty we have of understanding and accepting the phenomenon of &lt;em&gt;random variability&lt;/em&gt;: it does not need a reason to occur, it just occurs (at least given our current state of knowledge).&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2014-04-08T13:41:58.447" LastActivityDate="2014-04-08T13:41:58.447" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1338" PostTypeId="2" ParentId="1325" CreationDate="2014-04-06T01:32:43.113" Score="9" Body="&lt;p&gt;If a coin is biased to land heads with probability $p$ and $(a,b)$ is a $95\%$ confidence interval for $p$ then $p$ is in $(a,b)$ with probability $95\%$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added in edit -&lt;/strong&gt; While it is often argued that the difference is only philosophical, this distinction is of huge practical importance because the latter phrase is usually interpreted as if $p$ were the random variable, while actually $(a,b)$ is the only random variable if we are in a non-Bayesian setting. In a Bayesian setting, then $p$ is a random variable but the probability that it belongs to the confidence interval depends on the prior. To makes things clearer, let us adapt the first item of Noah's answer to this case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine the coin is taken randomly uniformly from a jar known to contain coins for which $p=5\%$ and coins for which $p=95\%$ (and assume the coins auto-destruct after being flipped, otherwise we can flip the same coin a number of times; for a more realistic framework, see Noah's answer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, if we know the two possible values (but ignore the distribution of the two kind of coins in the jar), a natural confidence interval $I$ (which I recall is far from being unique) is to take $I=\{0.95\}$ if we observe heads and $I=\{0.05\}$ if we observe tails. We can here replace these singletons by small intervals around the same values if we so decide.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This $I$ is random, as it should: it depend on the random outcome of the experiment. Let us show that this indeed gives a $95\%$ confidence interval. Consider all possible situations and their probabilities, denoting by $a$ the proportion of $p=0.95$ coins:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;we drew a $p=0.95$ coin and got heads (odds: $0.95\, a$),&lt;/li&gt;&#xA;&lt;li&gt;we drew a $p=0.05$ coin and got heads (odds: $0.05\,(1-a)$),&lt;/li&gt;&#xA;&lt;li&gt;we drew a $p=0.95$ coin and got tails (odds: $0.05\, a$),&lt;/li&gt;&#xA;&lt;li&gt;we drew a $p=0.05$ coin and got tails (odds: $0.95\,(1-a)$).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Precisely in the first and last case will $I$ contain $p$, and this sums up to a probability of $95\%$: our design for $I$ indeed ensured that in $95\%$ of the cases, the experiment would lead us to choose a $I$ that contains $p$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us go further: if the outcome is heads, we take $I=\{0.95\}$ and the &lt;em&gt;a posteriori&lt;/em&gt; probability that $I$ contains the actual value of $p$ is&#xA;$$ \frac{0.95\, a}{0.95\, a+0.05\,(1-a)}=\frac{0.95\, a}{0.90\, a+0.05}$$&#xA;&lt;strong&gt;which can be anywhere between $0$ and $1$&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, assume that $a=1/1000$. If heads turns up, the (conditional) probability that $I$ contains $p$ is less than $2\%$: the overwhelming prior makes that a &quot;heads&quot; outcome is much more likely to result from a little of bad luck with a $p=0.05$ coin than from a very rare $p=0.95$ coin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This example might seem artificial, but when a scientist assumes $p$ is in her or his computed confidence interval with $95\%$ probability, as if $p$ where random, it may leads to false interpretations notably in presence of bias in his or her experiments. &lt;strong&gt;I thus prefer to say &quot;$I$ lands around $p$ with $95\%$ probability&quot; to make more explicit the randomness of the confidence interval.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="640" LastEditorUserId="187" LastEditDate="2017-01-03T14:33:46.240" LastActivityDate="2017-01-03T14:33:46.240" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1339" PostTypeId="2" ParentId="1325" CreationDate="2014-04-06T01:43:04.710" Score="60" Body="&lt;p&gt;Here are two well known examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If someone tests positive for a rare disease (say its prevalence is 1 out of 100,000) with a test that has a 1% false positive rate, it is tempting to say that we are 99% sure they have that disease. This isn't true if you go through the numbers; they probably don't have that disease and are a false positive. (Bayes)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you look at a list of cancer rates by county, you see that counties with the lowest rates of cancer tend to have a much lower population than average. Students will speculate all sorts of reasons for this - &quot;healthier country living&quot; etc. But you can also look at the counties with the highest cancer rates. You find they too are the least populated. If you show that to students first they will have all sorts of reasons why that makes sense. But what is really going on is that &lt;a href=&quot;https://scholar.google.com/scholar?cluster=16415358650142164964&quot; rel=&quot;noreferrer&quot;&gt;the standard error is larger for smaller samples&lt;/a&gt;. Standardized test results from schools &lt;a href=&quot;https://scholar.google.com/scholar?cluster=9456853755545208886&quot; rel=&quot;noreferrer&quot;&gt;show the same effect&lt;/a&gt;. I &lt;a href=&quot;http://marginalrevolution.com/marginalrevolution/2010/09/the-small-schools-myth.html&quot; rel=&quot;noreferrer&quot;&gt;heard&lt;/a&gt; the Gates Foundation invested millions in small high performing schools before realizing that this effect was in play.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Here is a great article with a much better explanation of the first error and a lot of other examples of statistical confusion:&#xA;&lt;a href=&quot;http://web.mit.edu/5.95/readings/gigerenzer.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://web.mit.edu/5.95/readings/gigerenzer.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt;&#xA;I recently discovered that p-values are a lot more subtle than I had thought. The Wikipedia page for p-values lists 6 common misconceptions, most of which I had. It references &lt;a href=&quot;http://xa.yimg.com/kq/groups/18751725/636586767/name/twelve+P+value+misconceptions.pdf&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt; on twelve common misconceptions about p-values. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Psychologists Tversky and Kahneman have studied various misconceptions in statistics. &lt;a href=&quot;http://isites.harvard.edu/fs/docs/icb.topic470237.files/articles%20spring%202008/Judgement%20under%20uncertainty%20readings/belief%20in%20the%20law%20of%20small%20numbers.pdf&quot; rel=&quot;noreferrer&quot;&gt;Here is one of their better known papers.&lt;/a&gt; They found even trained statisticians often ignore base rates when calculating probabilities and engage in a version of the gamblers fallacy, expecting small samples to have the same standard error as large ones.&lt;/p&gt;&#xA;" OwnerUserId="690" LastEditorUserId="3499" LastEditDate="2017-01-05T07:24:56.827" LastActivityDate="2017-01-05T07:24:56.827" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1340" PostTypeId="2" ParentId="1325" CreationDate="2014-04-06T02:04:36.540" Score="36" Body="&lt;p&gt;Simpson's paradox: see &lt;a href=&quot;http://en.wikipedia.org/wiki/Simpson%27s_paradox&quot; rel=&quot;noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Simpson%27s_paradox&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarize the Berkeley Admissions example: in 1973, 43% of men applying to graduate school at Berkeley were admitted, but only 35% of women. But, broken down across the six departments, women either did better than men, or the difference was not significant. The paradoxical result appeared because women were more likely than men to apply to the most competitive departments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/1327&quot;&gt;Benoît Kloeckner's answer&lt;/a&gt; mentions some other problems that arise from averaging percentages.&lt;/p&gt;&#xA;" OwnerUserId="640" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-06T06:17:52.753" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1341" PostTypeId="2" ParentId="1263" CreationDate="2014-04-06T05:26:11.090" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What are the most important considerations when writing your own lecture notes for a course?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My aim is to pick a good path through the syllabus.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It must be logically sound and well-motivated. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Important examples and algorithms should go near the start. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ideally there will be a running example that can illustrate several different ideas.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It will fit well enough with previous years' problem sheets that most questions can be asked again. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Most students are going to get lost at some point: I try to postpone this point as far as possible, even if it means some compression of harder material at the end of the course.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I then write outline notes that have complete definitions and statements of theorems, but only a few examples, and usually no proofs. Then in lectures I can spend time on short questions, examples, and proofs. This helps keep the lectures fresh and interesting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an example see my Spring 2013 &lt;a href=&quot;http://www.ma.rhul.ac.uk/~uvah099/other.html#LectureNotes&quot; rel=&quot;nofollow&quot;&gt;Notes on Error Correcting Codes&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the motivation for writing your own notes in a course where textbooks are available?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;b&gt; It helps me think about the material&lt;/b&gt;. Writing outline notes is time-consuming, but it makes certain that I have thought hard about all of the material. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;b&gt; It saves time, in the long run&lt;/b&gt;. Having the big picture fixed reduces the amount of preparation needed for each lecture, sometimes to almost nothing. When preparing I am free to concentrate on small scale problems in particular examples / proofs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;b&gt; It makes sure the syllabus is covered in an efficient way&lt;/b&gt;. Textbooks usually cover more material than is needed, and there is the problem of finding good books at the right level. If multiple textbooks are required they will probably have competing conventions, which have to be reconciled. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There is also the less respectable reason that, in the UK system at least, students  expect to be able to rely solely on their lecture notes and problem sheets issued by the lecturer, and are unprepared to learn from textbooks. Of course this is a bad thing, but it is a long and hard fight to change.&lt;/p&gt;&#xA;" OwnerUserId="640" LastEditorUserId="640" LastEditDate="2014-04-06T22:41:17.640" LastActivityDate="2014-04-06T22:41:17.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1342" PostTypeId="2" ParentId="1266" CreationDate="2014-04-06T05:30:06.067" Score="11" Body="&lt;p&gt;Many people simply do not realize that written math has notational conventions, and despite the fact that the algorithms and proofs &lt;em&gt;are&lt;/em&gt; logical and complete, the notation is just a language and has holes and ambiguities like anything else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As plenty of troll questions on Facebook show, many people will defend their interpretation of &quot;what is the value of 5 + 1/2(6)&quot; to the death, acting like PEMDAS is a complete mathematical rule handed down by the gods unto man and not just an incomplete (doesn't include juxtaposition!), arbitrary convention that makes it easier to figure out what some other person wrote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best way I've ever seen this solved is just to write a bunch of really easy but ambiguous expressions on the board. Tell the class to solve them on a sheet of paper, have them hand it forward, and then take a tally of how many completely different but still correct answers you get for each problem. Then use this as a learning experience about how math requires clear communication just like everything else, and that despite it being a logical field, not everything is an invariant theorem so you really &lt;em&gt;don't&lt;/em&gt; know what they mean when they write confusing and ambiguous expressions.&lt;/p&gt;&#xA;" OwnerUserId="521" LastActivityDate="2014-04-06T05:30:06.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1343" PostTypeId="2" ParentId="1325" CreationDate="2014-04-06T05:45:36.253" Score="4" Body="&lt;p&gt;Related to the answer of Mark, it is also a widely believed misconception (even among MDs) that if an AIDS test has 99% sensitivity, then someone testing positive is with 99% probability ill. Acually, the numbers can be extremely low...&#xA;This is true for all illnesses which are erlatively rare in our society, see &lt;a href=&quot;http://uhavax.hartford.edu/bugl/treat.htm#notions&quot;&gt;http://uhavax.hartford.edu/bugl/treat.htm#notions&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-04-06T05:45:36.253" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1344" PostTypeId="2" ParentId="1316" CreationDate="2014-04-06T08:56:51.023" Score="14" Body="&lt;p&gt;When in doubt, I often decide simply to quote others! A nice choice, in this case, would be someone who started as a pure mathematician, then worked in applied mathematics, and ultimately moved into mathematics education. Luckily, precisely such a person exists in &lt;a href=&quot;http://www.maa.org/about-maa/governance/maa-presidents/henry-otto-pollak-1975-1976-maa-president&quot; rel=&quot;noreferrer&quot;&gt;Henry Pollak&lt;/a&gt;: Ph.D under Lars Ahlfors, then Director of the Mathematics and Statistics Research Center at Bell Laboratories, and, along the way, wrote a fair bit about mathematics education (and has been a long-time &quot;visiting&quot; professor at Teachers College Columbia University).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a wonderful interview with Pollak that touches upon some issues related to applied mathematics. Before pasting a few relevant excerpts, one citation of the interview is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Pollak, H., Albers, D. J., &amp;amp; Thibodeaux, M. J. (1984). A conversation with Henry Pollak. &lt;em&gt;The College Mathematics Journal, 15&lt;/em&gt;(3), 194-217. &lt;a href=&quot;http://www.jstor.org/stable/2686329&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Three excerpts that I hope can help shape your responses to such students:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/PNktJ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$2.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/dSU7d.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$3.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/3Bwpo.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-04-06T08:56:51.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1345" PostTypeId="1" AcceptedAnswerId="1402" CreationDate="2014-04-06T09:08:12.247" Score="13" ViewCount="980" Body="&lt;p&gt;Oral exams take place at two different points in during their study: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;As final examination to end their study (i.e., when the students have finished all necessary courses, they have an oral exam about 2-3 courses in a field, e.g., an oral exam about PDE and numerics of PDE).&lt;/li&gt;&#xA;&lt;li&gt;As an alternative of exams for specific courses (i.e., there is are timeslots on 1-2 given days after the course has ended, and the students come there for a 20-30 minutes oral exam).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you have maybe about 10 students in your course, you are not in charge to set a lot of rules what and how to ask at such an oral exam, but for more people it is hard to be fair since you either ask exactly the same questions (where the students coming at last have advantages) or the oral exams differ too much in the topics of the course or in difficulty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In oral exams which end the study is it even more difficult since you examine the same content over years and decades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are good guidelines to have fair oral exams in both cases?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-06T09:33:08.390" LastActivityDate="2014-04-07T16:13:28.693" Title="How to design a fair oral exam for &gt;10 students?" Tags="|undergraduate-education|graduate-education|oral-exams|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1346" PostTypeId="2" ParentId="1260" CreationDate="2014-04-06T10:58:49.883" Score="4" Body="&lt;p&gt;For propositional logic (especially truth tables and implication) I suggest Raymond's Smullyan's wonderful book, &lt;i&gt;What is the name of this book?&lt;/i&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From memory: one example has a prosecutor in a court say to the defendant: 'If you committed the crime, then you must have had an accomplice'. The defendant hotly denies this. But $A \implies B$ is false only if $A$ is true, and $B$ is false, so his denial is also an admission of guilt.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;b&gt; Pirates (disjunctive/conjunctive normal form)&lt;/b&gt;. Another puzzle I like has&#xA;five pirates, who want to padlock their treasure chest so that it can be opened if and only if (at least)&#xA;four of them are present. How many padlocks are needed? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Writing $p_i$ for the proposition 'Pirate $i$ is present', &#xA;the condition is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$  (p_1 \wedge p_2 \wedge p_3 \wedge p_4) \vee (p_1 \wedge p_2 \wedge p_3 \wedge p_5) \vee \cdots \vee (p_2 \wedge p_3 \wedge p_4 \wedge p_5). $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This holds if and only if no two pirates who are absent, so a logically equivalent condition is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ (p_1 \vee p_2) \wedge (p_1 \vee p_3) \wedge \cdots \wedge (p_4 \vee p_5) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in conjunctive normal form. The corresponding (minimal) solution puts 10 padlocks on the chest, labelled by&#xA;the $2$-subsets of $\{1,2,\ldots, 5\}$, and gives the keys to the padlock labelled $\{r,s\}$ to&#xA;pirates $r$ and $s$.&lt;/p&gt;&#xA;" OwnerUserId="640" LastEditorUserId="640" LastEditDate="2014-05-06T09:32:44.520" LastActivityDate="2014-05-06T09:32:44.520" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1347" PostTypeId="1" CreationDate="2014-04-06T11:33:01.477" Score="9" ViewCount="370" Body="&lt;p&gt;In our department, something is growing up into the darkness. A dark power which seduces my colleagues and their students one by one and leads them into the shadow. Those whose  hearts are corrupted by this culture try to prove their research field is the best and deepest field of mathematics, &lt;em&gt;The Lord of the Fields!&lt;/em&gt; All around the department, these students and teachers try to disparage the research fields of their colleagues in their discussions and lectures implicitly and explicitly. Analysts against Algebraists, Algebraists against Topologists, pure mathematicians against applied mathematicians and so on. This phenomenon really harms the scientific development of our institute and restricts the possible joint works between researchers in different fields. Also, it prevents students from seeing mathematics as a whole and using the tools of a particular field in another one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fortunately there are some researchers in our department who resist this culture. Recently I asked them to participate in an upcoming council to decide on the possible policies which could help us to overcome this situation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What can we do to overcome the lord of the fields dark culture? How can a teacher teach his/her students to appreciate all fields of mathematics regardless of their beloved research area? How can he/she convince his/her colleagues to stop poisoning their students by this culture?   &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-04-06T17:44:06.193" LastActivityDate="2014-04-06T17:44:06.193" Title="The Lord of the Fields" Tags="|mathematical-pedagogy|course-design|classroom-management|" AnswerCount="1" CommentCount="16" ClosedDate="2014-04-07T02:06:26.870" ContentLicense="CC BY-SA 3.0" />
  <row Id="1348" PostTypeId="1" CreationDate="2014-04-06T12:14:24.640" Score="19" ViewCount="3433" Body="&lt;p&gt;I'm a first year student at a university college, and I'm currently teaching 7th grade for a few weeks as a part of my education. I'm struggling a bit with finding models and strategies to develop an understanding of fractions within the class, specifically when multiplying a fraction with a fraction, and dividing fractions by a fraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Up until now I've been using a lot of squares to try to visualize the the parts, dividing them horizontally and vertically, to show how we get to the denominator and which numerator we end up getting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is this, do you guys know and strategies or visual models that are good at developing a profound understanding of fractions that I can use in class? Either through use of the blackboard, or assignments and activities? Have you been taught fractions through a method that you in retrospect found to have been particularly effective?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Thank you all so much for your feedback. I've been able to use a lot of your insight in practice, and although I have a long way to go yet as a teacher, this has aided immensely in my development. &lt;/p&gt;&#xA;" OwnerUserId="695" LastEditorUserId="695" LastEditDate="2014-04-09T16:00:21.357" LastActivityDate="2019-10-18T13:26:03.030" Title="Models and strategies for teaching fractions in 7th grade mathematics" Tags="|secondary-education|fractions|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1349" PostTypeId="2" ParentId="1325" CreationDate="2014-04-06T12:20:29.780" Score="6" Body="&lt;p&gt;Sampling issues occur frequently in opinion polls.  Errors were the result of bias. Nonresponse bias has been mentioned lately as well as other factors concerning the sampling of &lt;em&gt;potential&lt;/em&gt; voters predicting the behavior of actual voters.  See &lt;a href=&quot;http://www.pewresearch.org/2009/06/25/perils-of-polling-in-election-09/&quot; rel=&quot;nofollow&quot;&gt;Perils of Polling in Election ’08&lt;/a&gt;, the Pew Research Center.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2014-06-13T14:10:22.603" LastActivityDate="2014-06-13T14:10:22.603" CommentCount="0" CommunityOwnedDate="2014-06-13T14:10:22.603" ContentLicense="CC BY-SA 3.0" />
  <row Id="1350" PostTypeId="2" ParentId="1304" CreationDate="2014-04-06T13:26:52.553" Score="5" Body="&lt;p&gt;Check out:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.visualpatterns.org/&quot;&gt;visualpatterns.org&lt;/a&gt; - a great way to get them creating algebraic&#xA;expressions to describe how something is growing&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.estimation180.com/&quot;&gt;estimation180.com&lt;/a&gt; - though estimation isn't a part of algebra,&#xA;this will strengthen their number sense&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, there are lots of wonderful blogs by high school teachers. One way to use these to search for specific content is through the &quot;virtual filing cabinets&quot; a number of bloggers have created. &lt;a href=&quot;http://samjshah.com/worksheets-projects/&quot;&gt;Sam Shah's&lt;/a&gt; refers to others, so it's a good place to start.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-04-06T13:26:52.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1351" PostTypeId="1" CreationDate="2014-04-06T13:52:41.777" Score="30" ViewCount="7632" Body="&lt;p&gt;Once when I was talking about &lt;a href=&quot;http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox&quot;&gt;Banach-Traski&lt;/a&gt; theorem (paradox) I said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;OK! This is Banach-Tarski's theorem which is against our intuition but provable from our intuitive axioms! It says you can decompose a small diamond to finite parts and then attach them together and build a diamond as large as a mountain! &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Suddenly one of the students happily said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Amazing! Let's do it!   &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And I said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ah! Unfortunately you never can do it! Go, think and try on it! I offer a full mark to the best research.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;At the end of the course Students' reasons were very confusing and strange! Some of their arguments were based on physics, some others related to philosophy, some inspired by religious beliefs, etc. I have my own reasons about impossibility of paradoxical decompositions in the physical world but I don't know if they are the best possible reasons or not. Here I want to ask about any possible argument which I can use to convince my curious students about impossibility of paradoxical decompositions in actual world.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What do you say to students who want to apply Banach-Tarski theorem in practice?!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark.&lt;/strong&gt; Note that if we use an argument which &lt;em&gt;destroys&lt;/em&gt; the bridge between mathematical and physical worlds in the case of Banach-Tarski theorem, we should be able to answer a question in the following form:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We know many parts of mathematics which use Axiom of Choice in their main theorems (e.g. Analysis, Linear Algebra, Differential Equations, etc.) have amazingly correct applications in the actual world. Why Banach-Tarski theorem is an exception? Is it really an exception?!  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-05-06T18:33:47.227" LastActivityDate="2017-11-16T16:03:47.500" Title="What do you say to students who want to apply Banach-Tarski theorem in practice?" Tags="|mathematical-pedagogy|classroom-management|logic|set-theory|" AnswerCount="13" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1352" PostTypeId="2" ParentId="1302" CreationDate="2014-04-06T14:05:41.733" Score="5" Body="&lt;p&gt;I think the main advantage of your background is that you have had exposure to proofs and concepts such as sets, functions and relations. Being familiar with this &quot;Chapter Zero&quot; material will help you get started with analysis, but keep in mind J. M. Cargal's &lt;a href=&quot;http://www.cargalmathbooks.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;advice&lt;/a&gt; to start with an elementary text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cargal's website lists numerous real analysis texts, although I would add &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0387952977&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Real Mathematical Analysis&lt;/a&gt; by Charles Pugh, possibly as a second text. See also MSE posts such as &lt;a href=&quot;https://math.stackexchange.com/questions/50444/teaching-introductory-real-analysis&quot;&gt;this one&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/questions/188255/good-first-course-in-real-analysis-book-for-self-study&quot;&gt;this one&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest trying at least a couple of texts and seeing what works best for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Initial difficulties in learning analysis include seeing the need to prove &quot;obvious&quot; results from calculus. A good supply of counterexamples can help, e.g. &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486428753&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Counterexamples in Analysis&lt;/a&gt; (or &lt;a href=&quot;http://www.maa.org/publications/ebooks/counterexamples-in-calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Counterexamples in Calculus&lt;/a&gt;, as Dave Renfro kindly suggested), although too many can be distracting. Related to this is learning to pay careful attention to definitions. You may find &lt;a href=&quot;https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/8846/1/Alcock_Simpson_book.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ideas from Mathematics Education&lt;/a&gt; by Alcock &amp;amp; Simpson worth looking into, or Alcock's wonderful little book, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0198723539&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to Think About Analysis&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Working with inequalities and finding suitable bounds for various quantities can also be a stumbling block. This is characteristic of analysis.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2019-04-10T08:09:39.977" LastActivityDate="2019-04-10T08:09:39.977" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="1353" PostTypeId="2" ParentId="1351" CreationDate="2014-04-06T14:18:32.963" Score="11" Body="&lt;p&gt;I think, the most important point is to emphasize that the proof relies on the axiom of choice. If this connection is made, you &quot;only&quot; have to convince the students that the use of the axiom of choice is not constructive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would ask them the following question: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Take an arbitrary subset $A\subset \mathbb{Q}$. Please give me a concrete constructive way to choose one element of $A$. Take into account that $A$ has in general no special element, i.e., no natural numbers, no minimal or maximal or &quot;middle&quot; element or similar. If you can give me a rule how to choose an element no matter how $A$ looks like, we can go on to $\mathbb{R}$, do the same or similar and then you could use Banach-Tarski.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The students should try things out, but all examples they give normally rely on the existence of a special element of $A$ and you can lead them to a contradition of their rule to find a choice function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, you should not be too direct. Otherwise the students will never want to use axiom of choice again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: As @mweiss pointed out in the comments, the question has an answer on $\mathbb{Q}$ (but not on $\mathbb{R}$). &lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-06T18:39:42.143" LastActivityDate="2014-04-06T18:39:42.143" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1354" PostTypeId="2" ParentId="1351" CreationDate="2014-04-06T14:24:39.627" Score="9" Body="&lt;p&gt;&lt;strong&gt;Failing argument&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Where one body is, there cannot be another. (Physical axiom)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you wanted to decompose and recompose the brilliant, you'd have to move some parts of it &lt;em&gt;through&lt;/em&gt; some others, which is physically not possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know the proof, but I'm told, that there are decompositions and recompositions which avoid intersecting word lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Valid argument in classical physics&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At least one of the decomposed sets is not Lebesgue-measurable. But the atoms of physical bodys are Lebesque-measurable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Non-classical physics&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess, it's not a paradox at temperature 0, but who knows. &lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-06T14:24:39.627" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1355" PostTypeId="2" ParentId="1325" CreationDate="2014-04-06T14:31:11.973" Score="17" Body="&lt;p&gt;Sometimes extreme sample bias. Here is an example (numbers made up, but realistic): In some country with a population of 100 million people, every year 100 people are bitten by poisonous snakes and 50 of these die. Every year 50 people are given treatment against snake bites, and 10 of these die (40 die without getting treatment). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your chances of dying from snake bite if you are not given treatment is 40 in 100 million or 1 in 2.5 million. Your chance of dying from snake bit if you are given treatment is one in five. Clearly you should strongly refuse snake bite treatment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here the error is quite obvious, but there are medical situations where something similar happens. For some medical condition, there are two medications. One is more effective but may increase your blood pressure. The other is slightly less effective but won't increase your blood pressure. If your doctor sees that you have high blood pressure, you will be given the second medication to avoid a risk of the blood pressure getting too high. Now if you examine the statistics, people given the second medication will statistically end up with higher blood pressure, and totally wrong conclusions can be drawn. &lt;/p&gt;&#xA;" OwnerUserId="697" LastActivityDate="2014-04-06T14:31:11.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1356" PostTypeId="2" ParentId="1300" CreationDate="2014-04-06T14:34:44.363" Score="3" Body="&lt;p&gt;Please consider whether your examples typically favor one gender. For example, questions about sports (&lt;a href=&quot;https://matheducators.stackexchange.com/a/1310/114&quot;&gt;@BenjaminDickman mentioned baseball&lt;/a&gt;) will turn off some students, more females than males. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree with &lt;a href=&quot;https://matheducators.stackexchange.com/a/1307&quot;&gt;@Toscho, that making up memorable situations helps students learn to visualize better&lt;/a&gt;. During a unit on exponential functions (and logarithms), I use &lt;a href=&quot;http://mathmamawrites.blogspot.com/2010/04/murder-mystery-project-for-logarithms.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;a murder mystery that my students must solve&lt;/a&gt;. I think they learn a lot from it. Ironically, I've also seen evidence that their learning hasn't transferred well - they often do better on murder problems on the exam than the easier population growth problems. &lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-06T14:37:54.130" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1357" PostTypeId="2" ParentId="1351" CreationDate="2014-04-06T14:44:45.003" Score="6" Body="&lt;p&gt;It depends on the context of the course where Banach-Tarski is being presented. My answer relies on knowing a bit of measure theory. Let's call our intial set which we're going to split into some pieces $A$, and the reassembled version of the pieces $B$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $A$ is a bounded and measurable set in $\mathbb{R}^3$ and $A= \bigcup_{k=1}^n A_k$ is a pairwise disjoint partition of $A$ consisting of measurable sets, then the Lebesgue measure of $A$, $\mu(A)$ is equal to $\sum_{k=1}^n A_n$. This is the $\sigma$-additivity of the Lebesgue measure. Together with the translation invariance of $A$, this shows $\mu(A)=\mu(B)$, i.e.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;the paradoxical example of recombining pieces of one sphere into two spheres cannot work if we restrict ourselves to measureable sets.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point, you can ask a (mean) counter-question: Construct a non-measurable set. This is quite hard; some courses in measure theory you show that there are actually non-measurable sets. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This way, Banach-Tarski shows how nasty and unintuitive non-measurable sets are. There's nothing wrong with the theorem, it's just that it shows how nice measurable sets are.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-04-06T14:44:45.003" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1358" PostTypeId="5" CreationDate="2014-04-06T14:48:34.387" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-06T14:48:34.387" LastActivityDate="2014-04-06T14:48:34.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1359" PostTypeId="4" CreationDate="2014-04-06T14:48:34.387" Score="0" Body="For questions related to linear mappings, matrices, basis, determinants, eigenvalues and other topics commonly done in a linear algebra course." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-06T15:18:02.350" LastActivityDate="2014-04-06T15:18:02.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1360" PostTypeId="2" ParentId="1351" CreationDate="2014-04-06T16:10:27.433" Score="4" Body="&lt;p&gt;We don't know if the axiom of choice is true or false. It might just be that in a black hole, you can actually appeal to the axiom of choice; on the other hand there are papers about counterexamples to quantum mechanics from certain failures of the axiom. And who knows, maybe in the future...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the point that the axiom of choice is false in the physical universe is as valid as the point that the Banach-Tarski is impossible to do in the physical universe. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is true is that we have no means to utilize the Axiom in the physical world. And since everything we are capable of doing has a volume, it doesn't allow us to make the infinitely precise cuts (over boundaries we cannot even define) that is required to make the Banach-Tarski theorem work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end, my belief is that you shouldn't convince them it's impossible or false in the real world, but rather that mathematics talks about idealized balls in an idealized space; and not a pea or the sun in our physical space. Then ask them how they intend to &lt;em&gt;construct&lt;/em&gt; something which is non-constructive, like non-measurable sets. Even if possible through some unknown physical phenomenon, it's unknown and we can't do it. For now.&lt;/p&gt;&#xA;" OwnerDisplayName="user508" LastActivityDate="2014-04-06T16:10:27.433" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1362" PostTypeId="2" ParentId="1351" CreationDate="2014-04-06T16:32:16.523" Score="14" Body="&lt;p&gt;&lt;strong&gt;Matter is no bounded set of points with a non-empty interior&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sets in the Banach–Tarski theorem have to fulfill some requirements:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;They need to be bounded.&lt;/li&gt;&#xA;&lt;li&gt;They have to have a non-empty interior.&lt;/li&gt;&#xA;&lt;li&gt;They need to be, well, sets of points. In particular their elements are identical, i.e., do not contain any other information than position.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;No matter how you interprete matter as a set (which is questionable on its own), a piece of real matter spectacularly fails to fulfill requirement 3, as it certainly isn’t homogeneous on the atomic level (which cannot be ignored for the inevitable non-measurable sets). Also, if matter is composed of anything particle-like, it is discrete set and thus requirement 2 is broken. Finally, quantum-mechanical probability densities do not have a bounded support and thus requirement 1 is broken.&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2014-04-06T16:32:16.523" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1363" PostTypeId="2" ParentId="1347" CreationDate="2014-04-06T16:39:07.743" Score="1" Body="&lt;p&gt;As an individual, I think one place to start is by learning about other fields than your own and teaching your students about the interconnectedness and interapplicability of mathematics. Perhaps, gradually, your influence on your colleagues will also grow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example of a crossover field, with applications within and beyond mathematics, is algebraic topology. See, for instance, the work of &lt;a href=&quot;http://www.math.upenn.edu/~ghrist/&quot; rel=&quot;nofollow&quot;&gt;Robert Ghrist&lt;/a&gt; and numerous others.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2014-04-06T16:39:07.743" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1364" PostTypeId="2" ParentId="1348" CreationDate="2014-04-06T16:51:49.853" Score="7" Body="&lt;p&gt;This depends on what meaning you give to fractions. One approach is to give fractions the meaning of shares or portions &lt;strong&gt;of&lt;/strong&gt; some unit e.g. a rectangle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Multiplication&lt;/strong&gt; is mostly equal to the word &lt;strong&gt;of&lt;/strong&gt;.&#xA;Examples: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{5}{8}=\frac{5}{8}\cdot 1$ are five eights &lt;strong&gt;of&lt;/strong&gt; the unit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{2}{3}\cdot\frac{3}{4}$ means two thirds &lt;strong&gt;of&lt;/strong&gt; $\frac{3}{4}$. Show a rectangle, divide it into 4 equal parts and mark three of them red. Now, take two thirds of the red area.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Variant 1: Divide the red area into three equal parts. Hey, it's already divided into three equal parts. Mark two of them blue. The blue area is $\frac{2}{3}\cdot\frac{3}{4}$ &lt;strong&gt;of&lt;/strong&gt; the rectangle. If you ignore the red part, you can see, that it's also $\frac{2}{4}=\frac{1}{2}$ of the rectangle.&lt;/li&gt;&#xA;&lt;li&gt;Variant 2: Divide each of the quarters into three equal parts. How many small parts is the rectangle composed of? It's $3\cdot 4=12$, so we're talking twelths here. In each of the red quarters, mark two of the three parts blue. How many are those? It's $3\cdot 2=6$. So, what's the blue share of the full rectangle? It's $\frac{6}{12}=\frac{1}{2}$. That's the same as variant 1, so our thinking was right.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you write the numbers in different colors, you can show the students by this generalizable example, that $\frac{a}{b}\cdot\frac{c}{d}=\frac{a\cdot c}{b\cdot d}$ and that they can cancel out the 3s as they are not needed in the end of each variant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this should be &lt;a href=&quot;http://en.wikipedia.org/wiki/Constructivism_%28psychological_school%29&quot; rel=&quot;noreferrer&quot;&gt;constructed&lt;/a&gt; as much as possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Division&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast to multiplication, we work formally here, revisiting the technique of inverse operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: $\frac{5}{7}:\frac{2}{3}=\frac{a}{b}$ and we want to know, what $a$ and $b$ are. The inverse operation to division is multiplication. $\frac{a}{b}$ is the fraction, for which $\frac{2}{3}\cdot\frac{a}{b}=\frac{5}{7}$. We know now, that then $2\cdot a$ must be $5$ and $3\cdot b$ must be 7, but that's not possible for integers $a$ and $b$. But expanding the fraction is our friend here. We just expand $\frac{5}{7}$ by $2$ and by $3$, then it's no problem anymore to find integers $a$ and $b$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{2}{3}\cdot\frac{a}{b}=\frac{5}{7}=\frac{5\cdot 2}{7\cdot 2}=\frac{5\cdot 2\cdot 3}{7\cdot 2\cdot 3}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So now we have $2\cdot a =5\cdot 2\cdot 3$, which means $a=5\cdot 3$, and $3\cdot b=7\cdot 2\cdot 3$, which means $b=7\cdot 2$. So:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{5}{7}:\frac{2}{3}=\frac{5\cdot 3}{7\cdot 2}=\frac{5}{7}\cdot\frac{3}{2}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compare the leftmost and the rightmost side and you can see:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One divides by a fraction by multiplying with the reciproke of this fraction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Let them do another example this way on their own. Let them countercheck their results by multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Important notice&lt;/strong&gt; Use lots of colors. Every number or associated variable gets the same color.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="439" LastEditDate="2014-04-06T17:08:10.270" LastActivityDate="2014-04-06T17:08:10.270" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1366" PostTypeId="2" ParentId="1348" CreationDate="2014-04-06T18:21:02.610" Score="1" Body="&lt;p&gt;You might want to check some of virtual manipulatives sites:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://nlvm.usu.edu/&quot; rel=&quot;nofollow&quot;&gt;National Library of Virtual Manipulatives&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.glencoe.com/sites/common_assets/mathematics/ebook_assets/vmf/VMF-Interface.html&quot; rel=&quot;nofollow&quot;&gt;Virtual Manipulatives - Glencoe&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.mathplayground.com/math_manipulatives.html&quot; rel=&quot;nofollow&quot;&gt;Math Playground&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="95" LastActivityDate="2014-04-06T18:21:02.610" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1367" PostTypeId="2" ParentId="1345" CreationDate="2014-04-06T18:30:57.140" Score="6" Body="&lt;p&gt;Usually, the course material is more than you could talk about in 20 - 30 minutes. This means that you have a large array of questions at hand, and you usually won't be able to ask every student the same set of questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the oral exams I took (iirc, it was linear algebra) had an interesting mechanism which removed some initial-quesion bias: Before the actual exam, you were presented two boxes of index cards. One for definitions, one for theorems, but the cards only contained the names. You (randomly) picked two of these and were to prepare these on a sheet of paper within 10 mins. After that time span, the actual exam started. These written-down versions were somewhat a conversation starter, and showed either some need to delve further, or skip to a different topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally speaking, it should be somewhat clear to the students which questions are to be expected, and which theorems are the important ones. If you don't want the students to be talking behind your back about the questions, just be up-front! Important theorems deserve special attention, and remarks like &quot;When I was TA, many oral exams involved this theorem.&quot;, or &quot;This kind of example could be asked for in an exam.&quot; are a sign of a good lecture, imho.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A final note: If you do oral exams regularly, students will start to write down what was asked and share this with other students. At one university, this went so far that for one professor, there wasn't only a binder with the mind protocols of the students, but actually a TeX'd &quot;walkthrough&quot; as well.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-04-06T18:30:57.140" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1368" PostTypeId="2" ParentId="1345" CreationDate="2014-04-06T18:59:30.820" Score="4" Body="&lt;p&gt;(Migrating my comments since they appear to serve as an answer.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why are the &quot;students coming at last&quot; in an advantageous position? Can they hear all the questions being asked of the students before them? If so: Does it suffice to administer half the examination student by student, and then administer the latter half in the reverse order? (E.g., if your students are A, B, C, then the order of testing is A-B-C-C-B-A.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Your response is that this would be helpful in case one (a particular course) but not case two (where examinations are similar over a period of years or even decades). You also note that students mention questions and sometimes answers to one another during the examination period. To this I respond:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;First, let students know that sharing questions/answers during the examination period is a violation of the school's (or your) academic policy, and will lead to failure if discovered. Second, I do not believe there is a shortage of problems (e.g., in PDEs), but if you can distill the essence of a subject down to a small enough collection of ideas such that memorizing a set-list of heuristics will allow a student to pass, then it seems to me like this would be a good thing. I would not consider it a cause for concern.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-04-06T18:59:30.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1369" PostTypeId="2" ParentId="1325" CreationDate="2014-04-06T20:10:00.563" Score="63" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Anscombe%27s_quartet&quot;&gt;Anscombe's quartet&lt;/a&gt; is pretty good:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/WqvOy.png&quot; alt=&quot;from Wikipedia&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All four of these sets have almost identical mean and variance for both x and y coordinates, correlation, and best-fit linear regression. But they're obviously very different!&lt;/p&gt;&#xA;" OwnerUserId="710" LastEditorUserId="114" LastEditDate="2014-04-06T20:13:06.927" LastActivityDate="2014-04-06T20:13:06.927" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1370" PostTypeId="2" ParentId="686" CreationDate="2014-04-06T20:33:18.967" Score="3" Body="&lt;p&gt;In metric fixed point theory, you have Darbo (and Sadovski) theorems.  They are generalizations of Schauder fixed point theorem, but instead of assuming that the set (or the mapping) is &lt;em&gt;compact&lt;/em&gt;, one assumes that it maps sets into &quot;more compact&quot; sets (using some kind of a &lt;em&gt;measure of noncompactness&lt;/em&gt;).  (I don't have any reference with me at home now, but one classical source is Granas' and Dugundji's monograph on fixed point theory.)  For instance, in Darbo's theorem, you assume that you have a continuous mapping $F$ on a non-empty bounded closed convex subset $C$ of a Banach space and a constant $k&amp;lt;1$ such that $\alpha(F(B))\le k\alpha(B)$ for each $B\subset C$, where $\alpha$ is the Kuratowski measure of noncompactness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The notion of measure of noncompactness goes back to 1930s (Kuratowski was the one who introduced it, without relation to fixed point theory), Darbo's and Sadovski's theorems are (AFAIK) from 1950s (maybe a bit later).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a related note, there are quite a few theorems (some fixed-point ones and some others) about &lt;em&gt;hyperconvex metric spaces&lt;/em&gt; (a notion introduced in the 1930s by Aronszajn, but first appearing in a published paper in 1956) whose proofs employ simple tools from metric topology, but are highly nontrivial.  A beautiful example is Baillon's theorems:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An intersection of a chain of bounded hyperconvex spaces is nonempty and hyperconvex.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any nonexpansive (i.e., having Lipschitz constant $1$) mapping of a bounded hyperconvex space into itself has a fixed point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both can be proves using quite elementary methods (but rely heavily on the axiom of choice), but the proofs are far from trivial.  Both were proved in a paper published in late 1980s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(For more info on hyperconvex spaces, see e.g. &lt;a href=&quot;http://en.wikipedia.org/wiki/Injective_metric_space&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Injective_metric_space&lt;/a&gt; and references therein, in particular the original paper of Aronszajn ana Panitchpakdi and the article by Espinola and Khamsi).&lt;/p&gt;&#xA;" OwnerUserId="704" LastActivityDate="2014-04-06T20:33:18.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1371" PostTypeId="2" ParentId="686" CreationDate="2014-04-06T20:58:47.507" Score="4" Body="&lt;p&gt;And I've just recalled one more thing: &lt;a href=&quot;http://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral&quot;&gt;Henstock–Kurzweil integral&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;New enough?  Introduced in 1957.  Check.&lt;/li&gt;&#xA;&lt;li&gt;Easy to understand?  The definition is a slight variation on the Riemann integral, and the theory is only slightly more difficult (the difficulties being mostly technical), so (at least theoretically) it may be taught to first-year students.  Check.&lt;/li&gt;&#xA;&lt;li&gt;Bonus points: interesting?  Let's see: it's conceptually much simpler than the Lebesgue integral, though more powerful (it integrates all Lebesgue-integrable functions and then some – among others, &lt;em&gt;all&lt;/em&gt; derivatives are H–K-integrable, and the Newton-Leibniz formula holds without exceptions); identical to Lebesgue integral when the latter is defined, hence can be used to define Lebesgue measure in an elementary way; and more.  For me: check.&lt;/li&gt;&#xA;&lt;li&gt;Bonus points II: are there any open problems in the theory?  Yes.  For instance, the dual space of the space of H–K-integrable functions (with Alexiewicz norm) is known only in the case of functions on a &lt;em&gt;bounded&lt;/em&gt; interval.   The unbounded case is AFAIK &lt;em&gt;terra incognita&lt;/em&gt;.  Check.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="704" LastActivityDate="2014-04-06T20:58:47.507" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1372" PostTypeId="2" ParentId="1325" CreationDate="2014-04-06T21:01:40.843" Score="8" Body="&lt;p&gt;Problems with interpreting $P$-values are discussed by Regina Nuzzo, &quot;Statistical Errors,&quot; &lt;em&gt;Nature&lt;/em&gt; &lt;strong&gt;504&lt;/strong&gt; (13 February 2014), 150-152. &lt;a href=&quot;http://www.nature.com/news/scientific-method-statistical-errors-1.14700&quot;&gt;[link]&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a nutshell: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&quot;Most scientists&quot; [sic] interpret a $P$-value as the probability of being wrong in concluding the null hypothesis is false. &lt;/li&gt;&#xA;&lt;li&gt;The $P$-value is often valued more highly than the size of the effect.  &lt;/li&gt;&#xA;&lt;li&gt;&quot;$P$-hacking&quot;: &quot;Trying multiple things until you get the desired result.&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="498" LastActivityDate="2014-04-06T21:01:40.843" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1373" PostTypeId="1" AcceptedAnswerId="1405" CreationDate="2014-04-06T21:14:41.043" Score="15" ViewCount="1339" Body="&lt;p&gt;&quot;Intuition&quot; is the best friend and worse enemy of mathematicians! Sometimes using intuitive arguments could be very helpful to understand the nature of a phenomenon. Many of the deepest true conjectures were just an intuitive argument at the beginning. (e.g. Fermat's Conjecture)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand in some realms of mathematics it seems &lt;em&gt;nothing is as it seems!&lt;/em&gt; The theorems are against our natural intuition and a naive approach could lead one to bad mistakes. (e.g. Naive set theory and some parts of geometry)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also in some relams of mathematics there is an uncertain situation. For example some consequences of the Axiom of Choice like &lt;em&gt;existence of a basis for each vector space&lt;/em&gt;, are completely intuitive and some other consequences of this axiom like &lt;em&gt;existence of a well-ordering on every set&lt;/em&gt;, are completely strange. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am hesitated what I should say to my students about the general role of their naive intuition in their mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1.&lt;/strong&gt; Which one of the below options are better to say to a student? What are possible advantages/disadvantages of each option? &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Your naive mathematical intuition is your best leader in the mathematical research. Follow it often but be careful for some &lt;em&gt;rare&lt;/em&gt; anti-intuition &lt;em&gt;exceptions&lt;/em&gt;.&lt;/li&gt;&#xA;  &lt;li&gt;In mathematics &lt;em&gt;nothing is as it seems!&lt;/em&gt; Don't trust on your naive mathematical intuition at all. Just follow the logical reasoning not intuitive guessing. &lt;em&gt;Almost all&lt;/em&gt; important mathematical theorems are against natural intuition. This is the point which makes them &lt;em&gt;special&lt;/em&gt; and &lt;em&gt;important&lt;/em&gt;.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2.&lt;/strong&gt; If the real situation is a combination of both of the above options, is there any criterion to determine what situation is true in a particular field of mathematics? In other words, can we say that a particular mathematical field (e.g. Linear Algebra) is intuitive and another one (e.g. Set Theory) is anti-intuition? If yes, how can we determine it?     &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-04-06T21:24:28.813" LastActivityDate="2014-04-07T22:09:14.027" Title="When should I say &quot;nothing is as it seems&quot;?" Tags="|undergraduate-education|mathematical-pedagogy|intuition|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1374" PostTypeId="2" ParentId="1373" CreationDate="2014-04-06T21:35:06.403" Score="4" Body="&lt;p&gt;For the first question, without a doubt neither. You should explain to them that mathematical intuition is developed through working with definitions. When they first encounter something they shouldn't guess, and they should always work with the definitions in front of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When they sit to revise the material, or solve their homework assignments, open the notes from class and look at the definitions. When sitting in the exam, write the relevant definitions (for what you are given, and what you need to show) in the draft pages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, my experience is that the more abstract the field is, the more it is important to work with the definitions. What is true for a first course in logic and set theory, might not be true for the first course in linear algebra. But in both cases &lt;strong&gt;it is important to work with the definitions&lt;/strong&gt;, and by doing so develop intuition.&lt;/p&gt;&#xA;" OwnerDisplayName="user508" LastActivityDate="2014-04-06T21:35:06.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1375" PostTypeId="2" ParentId="1236" CreationDate="2014-04-06T22:21:34.350" Score="5" Body="&lt;p&gt;I am both a high school and college instructor in Los Angeles. Here's  my two cents:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;High School Math Teacher&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In high school, it's all about classroom management. I recognize all students who range in math skills, ability and disposition towards the subject and offer accommodations for all. For example, I had 2 high students in my previous algebra 1 classes whom demonstrate very low skills and math comprehension. I provided accomodations for these students: extended office hours, extended time on assessments, typed or printed notes, visual aids, graphic organizers, and partner seating arrangement with students who can assist them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On my whiteboard, the standards, objectives, agenda with the use of formative (five finger check for understanding, thumbs up/thumbs down understanding/class discussion) and summative assessments (ticket out, quiz, test etc.) is written on the board for students to see and how they will be engaged therefore minimizing level of distraction for students which could really help with behavior problems. My agenga is usually a warm up, warm up debrief, homework review, collaborative activity, and ticket-out assessment. At the end of the week, if I feel (from ticket outs) that students are ready for a quiz, I will give a quiz.  For grading, I usually only grade ticket outs. Their classwork/homework is collected but I only grade certain problems - sometimes (because there's 150 students) I have them exchange papers and grade it for me. Also, a lot of collaborative assignments are given therefore minimizing my grading. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A high school teacher must also be aware of pre-requisite knowledge and students misconceptions with respect to the standards discussed. A high school teacher must also include in his lesson plan how he will address the needs of subgroups and individual students such as English Language Learners, students with Learning Disabilities, and Gifted students.  An example of all the aforementioned is here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/xv1Gz.png&quot; alt=&quot;enter image description here&quot;&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;College Instructor&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is my &quot;easy&quot; job. Similar to high school. No behavior issues therefore classroom management is more laid back. No parent conferences. No calls home etc. More teacher centered than that of high school (which is both teacher centered, student centered, one on one instruction, and direct instruction). We have Audial, Visual, and Kinesthetic learners here too so we must still use a combination of teacher and student centered approaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I establish rapport with students first day of class (student survey), communicate my expectation/procedures, and I motivate students to learn. But since time is an issue and you can't really give 'classwork', I have to really keep my instruction mostly doing explicit instruction 80%. Here's my lesson plan for college instruction:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Warmup Lecture (Performing an instruction strategy &quot;I do, we do, you do&quot;)&#xA;2)Lecture&#xA;3) Guided Practice&#xA;4) Lecture&#xA;5) Guided Practice&#xA;6) Ticket out (to inform me of whether students were learning what I put on the board - this is important because the following lecture I will outline any student misconceptions on my warmup)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All homework assignments are on their syllabus. They turn in all homework last day of lecture for that week. A test is given for every 2 chapters with the ability to make them up if they didn't pass. I have a philosophy that Math is not about &lt;em&gt;when&lt;/em&gt; you get it, as &lt;em&gt;long&lt;/em&gt; as you get it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="712" LastEditorUserId="712" LastEditDate="2014-04-06T23:15:46.220" LastActivityDate="2014-04-06T23:15:46.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1376" PostTypeId="2" ParentId="1373" CreationDate="2014-04-06T22:27:22.500" Score="7" Body="&lt;p&gt;I would be inclined to choose my words more delicately. Namely, it's not that intuition-per-say is misleading, but that &lt;em&gt;untrained&lt;/em&gt; or &lt;em&gt;inexperienced&lt;/em&gt; intuition is... untrained and inexperienced. &lt;em&gt;Of&lt;/em&gt; _course_ many reasonable, normal things (in this complicated world) will be surprises to a naif. That doesn't mean they're truly counter-intuitive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would advocate &lt;em&gt;always&lt;/em&gt; taking as preliminary position that one &lt;em&gt;must&lt;/em&gt; trust one's intuition, all the while recognizing the possibility that one is mistaken/naive. To tell people to &lt;em&gt;distrust&lt;/em&gt; their in tuition I think is a very bad thing, especially insofar as it tells them that they should, in effect, accept &lt;em&gt;anything&lt;/em&gt; from a sufficient authority, without much questioning. When this becomes suspension of critical faculties, it is obviously methodologically disastrous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also disagree with the examples of naive set theory and so on. &lt;em&gt;Mostly&lt;/em&gt; naive set theory is perfectly fine in practice, in applications to mathematics. The fact that it can be over-revved, pushed beyond safe limits, should not be construed as counter-intuitive. Yes, it is interesting to see how to axiomatize (=set rules) to avoid seeming paradox... but I note that it is very difficult to arrange rules sufficiently encompassing so as to prevent all stupid choices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the real point is that intuition is not static. It is &lt;em&gt;trained&lt;/em&gt;, not innate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I think that it's not exactly &quot;definitions&quot; (or axioms, nearly the same thing) that train intuition at a most fundamental level, but &lt;em&gt;examples&lt;/em&gt;, especially the examples that show both the good side, and the bad side, of issues eventually iconified into definitions or axioms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the historical fact that almost universally examples have preceded &quot;definitions&quot;, and that definitions tend to have more explanatory power than do definitions (which always leave me immediately asking myself what situation the dang definition was trying to avoid or get into... why not just tell me directly, instead of being coy?), I strongly feel that definitions are in effect a &lt;em&gt;summary&lt;/em&gt; of important examples, and make scarce sense and should not be given prior to examples... and, by that point (as I am fond of saying) the formal definitions, the main theorems (which were what we wanted all along, if only we admitted it up front) and their proofs (illustrated more simply in examples) should be almost entirely clear.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-06T22:27:22.500" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1377" PostTypeId="2" ParentId="1373" CreationDate="2014-04-06T22:27:29.860" Score="16" Body="&lt;p&gt;I think it is important to develop mathematical intuition in one's work, but not necessarily of the naive sort.&lt;/p&gt;&#xA;&lt;p&gt;Of course, one's intuition may turn out, ultimately, to be incorrect. An important outcome in this case is not to assume that you have &lt;em&gt;bad&lt;/em&gt; intuition, but instead to think about how you can make the end result sync up better with what could be reasonably guessed.&lt;/p&gt;&#xA;&lt;p&gt;Borrowing terms from (Piagetian) developmental psychology, the (unpleasant) experience of acquiring new information that does not accord with one's existing cognitive schemas is known as disequilibrium; the process of updating one's schematic understanding so as to fit in this new information more naturally is known as equilibration, and this is when &lt;em&gt;learning&lt;/em&gt; occurs.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;Allow me to give a &lt;strong&gt;concrete example:&lt;/strong&gt; Ask secondary school students to use their intuition to guess the percentage of entries in a &lt;span class=&quot;math-container&quot;&gt;$10 \times 10$&lt;/span&gt; multiplication table that are odd. In all likelihood, they will guess &lt;span class=&quot;math-container&quot;&gt;$50\%$&lt;/span&gt;; this is probably based on some intuition around numbers alternating between odd and even.&lt;/p&gt;&#xA;&lt;p&gt;However, the actual answer is &lt;span class=&quot;math-container&quot;&gt;$25\%$&lt;/span&gt;. Why? A proof is that a product of two numbers is odd iff both numbers are odd; in this case, we have five choices for odd numbers (&lt;span class=&quot;math-container&quot;&gt;$1, 3, 5, 7, 9$&lt;/span&gt;) and so a total of &lt;span class=&quot;math-container&quot;&gt;$25$&lt;/span&gt; out of the &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; entries will be odd. With regard to intuition, one might note that taking products can introduce new factors, and a number is only odd if you never get a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;; this is tough. Alternatively, one might note that among the four cases even&lt;em&gt;even, even&lt;/em&gt;odd, odd&lt;em&gt;even, and odd&lt;/em&gt;odd, only &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; of the &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; gives an odd number; so seeing &lt;span class=&quot;math-container&quot;&gt;$1/4 = 25\%$&lt;/span&gt; feels right &lt;em&gt;intuitively&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;At a higher level, suppose you are a teacher thinking about other nonstandard problems to pose with the multiplication table. How many distinct entries are there in an &lt;span class=&quot;math-container&quot;&gt;$n \times n$&lt;/span&gt; table? This question looks pretty reasonable, and one can quickly guess how to start it: check small values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. But now step back and use your (Number Theoretical) intuition: the multiplication table is a way of encoding composite numbers and prime numbers (e.g., a number &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is prime iff it appears exactly twice in an &amp;quot;extended&amp;quot; multiplication table). By updating one's intuition to conceive of the table as being related to primality, it comes as less of a surprise that this question is far too difficult for a secondary school student. (In fact, what we have here is the Erdos multiplication table problem; it is quite hard: &lt;a href=&quot;https://mathoverflow.net/a/108939/22971&quot;&gt;MathOverflow link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;As for distinguishing between different areas of mathematics by using intuition as a lens, I do not think this is a fruitful approach. It may be that some people are primed to have certain ways of thinking, or &lt;em&gt;modalities of thought&lt;/em&gt;, cf. Facet 4 in the following reference:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gruber, H. E., &amp;amp; Wallace, D. B. (1999). The case study method and evolving systems approach for understanding unique creative people at work. &lt;em&gt;Handbook of creativity, 93&lt;/em&gt;, 115.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But sometimes one's way of thinking differing from the general trend in an area of mathematics can lead to a breakthrough. For a concrete example, consider that Paul Cohen managed to prove the independence of the Continuum Hypothesis (from ZFC) with a line of thought (related to &amp;quot;decision procedures&amp;quot;) that directly contradicted Godel's work. (&lt;a href=&quot;https://mathoverflow.net/a/160508/22971&quot;&gt;MathOverflow link&lt;/a&gt;.) In the end, it was not Cohen's set theoretical intuition that allowed him to tackle Hilbert's first problem, but rather intuition developed elsewhere. That said, other Mathematical Logicians and Set Theorists, once they equilibrated in response to Cohen's work, were quickly able to push forward the notions introduce by Cohen. For more on this history, see:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kanamori, A. (2008). Cohen and set theory. &lt;em&gt;The Bulletin of Symbolic Logic&lt;/em&gt;, 351-378.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this case, we have advances in set theory due both to those with mathematical intuition from elsewhere, as well as due to those with mathematical intuition specifically from set theory and mathematical logic. Though I only offer this one example here, I am quite sure that others abound throughout the many areas of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-07T09:51:37.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1378" PostTypeId="2" ParentId="1351" CreationDate="2014-04-07T00:49:14.897" Score="38" Body="&lt;p&gt;Whether one can or can not double the brilliant in real life has nothing to do with the Banach-Tarski paradox. Various mathematical objects are models of various aspects of our universe. $\mathbb R^3$ is typically taken as a model of three-dimensional space. But, it is only a model. Some things that are true in the model are false in space and some that are false in the model are true in space (just like any good model, it is not a precise replica of the thing being modeled, it distorts some things, neglects others, all so that it simplifies the original thing so that it is amenable to be studied by formal means). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Banach-Tarski paradox is an illustration of (one of) the limitations of $\mathbb R^3$ as a model of the familiar (yet bizarre) ambient space we live in. There are plenty other such incompatibilities. For instance, perfect circles exist in $\mathbb R^3$, you may want to invite your student to construct one in real life. Physics seems to prohibit that just as much it prohibits doubling lumps of gold just by rearranging them piece-wise. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To conclude, models are models. If a model predicts something about whatever it models, then that prediction should be understood within the limitations and distortions made during the modeling process. An incompatibility between the model and reality may entail lots of things. Perhaps a need to abandon the model or a need to adjust it. But sometimes it should just be put away, stored on the shelf of those odd little things that the simplifications assumptions we made bring forth. To your students (or mine, as I often give this example in a first year calculus course), I would say: go ahead and decide how to categorize the Banach-Tarski paradox: is it a serious flaw in the model, a minor disagreement with reality, or an oddity.  &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-04-07T00:49:14.897" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1379" PostTypeId="2" ParentId="1316" CreationDate="2014-04-07T02:16:31.940" Score="8" Body="&lt;p&gt;This quote from Jerry Marsden's 1980 &lt;em&gt;AMS Bulletin&lt;/em&gt; review of Dieudonné's &lt;em&gt;Treatise on Analysis&lt;/em&gt; is relevant:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;History screams at pure mathematicians not to ignore applications&lt;/em&gt;; the origins of such important topics as calculus, Fourier series, operator theory and dynamical systems were all closely related to applications. Some of the greatest mathematicians, Gauss, Hilbert, Poincaré, and von Neumann, were all rooted in applications.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="717" LastActivityDate="2014-04-07T02:16:31.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1380" PostTypeId="1" AcceptedAnswerId="1382" CreationDate="2014-04-07T02:22:42.263" Score="21" ViewCount="12238" Body="&lt;p&gt;As lifelong students of mathematics, we find problem-solving to be absolutely essential to enhance our understanding of the subject. Teaching others what we know serves to reinforce our existing knowledge and disseminate information to learners.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;However, how does one go about creating &quot;good&quot; problems?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &quot;good&quot;, I mean thought-provoking, inspiring problems with solutions that are extensible to other domains. Also, this builds up to the level of olympiad problems, for which problem writers seem to have a remarkable degree of ingenuity and creativity in devising novel problems.&lt;/p&gt;&#xA;" OwnerDisplayName="user718" LastEditorUserId="143" LastEditDate="2014-04-11T17:49:48.837" LastActivityDate="2014-04-11T17:49:48.837" Title="How does one create &quot;good&quot; math problems?" Tags="|problem-design|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1381" PostTypeId="2" ParentId="1380" CreationDate="2014-04-07T03:12:54.863" Score="4" Body="&lt;p&gt;Two suggestions: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Attend workshops and conferences and seek out problem solving sessions or presenters who are sharing their &quot;favorite problems.&quot; When the problems and solutions are discussed unique methods and approaches appear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Build a library and make time to read. Collect books, pdfs and sources. A textbook not suitable for the students can be a great source of problems. (Use Amazon and eBay to get used versions that are much cheaper.) Modify the textbook version as needed. Creativity in creation of problems comes from flipping through sources.      &lt;/p&gt;&#xA;" OwnerUserId="627" LastEditorUserId="-1" LastEditDate="2014-04-07T10:41:19.827" LastActivityDate="2014-04-07T10:41:19.827" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1382" PostTypeId="2" ParentId="1380" CreationDate="2014-04-07T04:28:40.910" Score="15" Body="&lt;p&gt;Since your question is very broad, here is a somewhat broad answer: Read about problem posing.&lt;/p&gt;&#xA;&lt;p&gt;Three key pieces are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Silver, E. A. (1994). On Mathematical Problem Posing. &lt;em&gt;For the learning of mathematics, 14&lt;/em&gt;(1), 19-28.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and the book&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Brown, S. I., &amp;amp; Walter, M. I. (2005). &lt;em&gt;The art of problem posing&lt;/em&gt;. Psychology Press.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The latter is a re-print of a book that first came out in 1983. You can also find a related book edited by Brown and Walter; a citation for the most recent version is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Brown, S. I., &amp;amp; Walter, M. I. (Eds.). (2014). &lt;em&gt;Problem posing: Reflections and applications&lt;/em&gt;. Psychology Press.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Start with these three documents, their references, and (searching on google scholar) other papers and articles that cited them.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;To sketch out very roughly Brown and Walter's suggestion: Start with a mathematical scenario, list assumptions, vary constraints (in their terms: &amp;quot;What-if-not-ing&amp;quot;), and then ask questions. You can even &amp;quot;cycle&amp;quot; through this process repeatedly in order to produce problems of increasing complexity.&lt;/p&gt;&#xA;&lt;p&gt;Of course, problem posing brings with it the danger of not knowing the answer to what you are asking.&lt;/p&gt;&#xA;&lt;p&gt;For example, your starting scenario might use the Pythagorean Theorem:&lt;/p&gt;&#xA;&lt;p&gt;Find all integer solutions for &lt;span class=&quot;math-container&quot;&gt;$x^2 + y^2 = z^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This particular example is explored in Brown and Walter's book, but it seems to me a reasonable assumption to list is that the exponent everywhere is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, and to ask for integer solutions when the exponent is &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;.... or, if one feels particularly daring, to generalize and ask for exponent &lt;span class=&quot;math-container&quot;&gt;$k \geq 3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;At a glance, this might seem like a reasonable question; but, if you are familiar with Fermat's Last Theorem, then you will realize that this is not an appropriate problem for most students.&lt;/p&gt;&#xA;&lt;p&gt;You can find some of my brief remarks about problem posing and creativity in part &lt;span class=&quot;math-container&quot;&gt;$4b$&lt;/span&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/a/906/262&quot;&gt;here&lt;/a&gt;, and a couple of other examples relating problem posing and intuition in the &lt;strong&gt;concrete example&lt;/strong&gt; section &lt;a href=&quot;https://matheducators.stackexchange.com/a/1377/262&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;&lt;strong&gt;A final remark:&lt;/strong&gt; You start off by mentioning the &amp;quot;essential&amp;quot; role of problem &lt;em&gt;solving&lt;/em&gt; in enhancing our understanding of mathematics. It may be worth noting that problem &lt;em&gt;posing&lt;/em&gt; plays an important role in solving; consider Polya's list of heuristics and how many of them are questions: What's a related problem? What's a simpler problem? How can I generalize this problem? Etc. (Historically, both Silver, in the first piece cited above, and Kilpatrick, on &lt;em&gt;problem formulation&lt;/em&gt;, trace this observation, i.e., that problem posing is an integral part of problem solving, at least back to a 1945 paper by Karl Duncker.)&lt;/p&gt;&#xA;&lt;p&gt;As Cantor (1867) wrote in his doctoral thesis:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“In re mathematica ars proponendi pluris facienda est quam solvendi”&lt;/p&gt;&#xA;&lt;p&gt;(“In mathematics the art of asking questions is more valuable than solving problems”).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-08T00:56:39.243" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1383" PostTypeId="1" AcceptedAnswerId="1396" CreationDate="2014-04-07T04:41:20.560" Score="23" ViewCount="1079" Body="&lt;p&gt;Lie groups are among the most important examples of groups in mathematics and physics, but they are rarely discussed in introductory undergraduate abstract algebra courses, which tend to focus on finite groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Partially, this is because you can't even &lt;em&gt;define&lt;/em&gt; a Lie group without knowing what a differentiable manifold is, which requires some amount of differential geometry or topology.  In addition, most of the main results about Lie groups involve the notion of a Lie algebra, which would require a significant detour to develop.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is, can anything coherent be said about Lie groups in a first abstract algebra course?  For example, if I were willing to spend a week or so discussing topics related to Lie groups, what could I cover that would be meaningful? Does anyone have any experience including material along these lines?&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2015-05-12T19:25:28.020" Title="What can be said about Lie groups in a first abstract algebra course?" Tags="|undergraduate-education|curriculum|abstract-algebra|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1384" PostTypeId="2" ParentId="1351" CreationDate="2014-04-07T04:44:50.787" Score="19" Body="&lt;p&gt;This explanation from &lt;a href=&quot;http://www.kuro5hin.org/story/2003/5/23/134430/275&quot; rel=&quot;noreferrer&quot;&gt;http://www.kuro5hin.org/story/2003/5/23/134430/275&lt;/a&gt; might be useful:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One important difference between S and a real, physical sphere is that S is &lt;em&gt;infinitely&lt;/em&gt; divisible. Mathematically speaking, S contains an infinite number of points. This is not true of a physical sphere, as there are a finite number of atoms in any given physical sphere; so a physical sphere is &lt;em&gt;not&lt;/em&gt; infinitely divisible.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In fact, if we assume that spheres are &lt;em&gt;not&lt;/em&gt; infinitely divisible, then the Banach-Tarski paradox doesn't apply, because each of the &quot;pieces&quot; in the paradox is so infinitely complex that they are not &quot;measurable&quot; (in human language, they do not have a well-defined volume; it is impossible to measure their volume). Immeasurable pieces can only exist if the sphere can be cut into infinitely-detailed pieces; this obviously isn't true for real spheres, since you cannot cut atoms into arbitrary shapes, especially not into infinitely complex shapes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In other words; the mathematical &quot;objects&quot; are infinitely dense, but physical objects are not. Any physical analogies would involve division into two objects with the same volume but half the density of the original.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, Banach-Tarski doesn't really apply to physical objects, because the nature of physical objects puts them outside the domain of that theorem (&lt;a href=&quot;https://matheducators.stackexchange.com/a/1362/680&quot;&gt;Wrzlprmft's answer&lt;/a&gt; also describes this). You may have inadvertently opened the door to difficult questions from students by giving the brilliant =&gt; mountain analogy to begin with without prior explanation of its shortcomings. The closest result of that analogy would be a brilliant as large as a mountain but with the same mass as the original object, which isn't quite accurate to the theorem. To explain why it doesn't apply, you can touch on the differences between a mathematical &quot;object&quot; and a physical one wrt. infinite divisibility. To pre-emptively address the issues in the future, you can precede your physical analogy with a disclaimer that it is only a very rough analogy for the reasons given above.&lt;/p&gt;&#xA;" OwnerUserId="680" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-07T19:38:00.340" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="1385" PostTypeId="2" ParentId="1325" CreationDate="2014-04-07T08:52:28.620" Score="4" Body="&lt;p&gt;The book &lt;a href=&quot;http://plus.maths.org/content/tiger-isnt&quot; rel=&quot;nofollow&quot;&gt;The tiger that isn't&lt;/a&gt; is what you're looking for.&lt;/p&gt;&#xA;" OwnerUserId="729" LastActivityDate="2014-04-07T08:52:28.620" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1386" PostTypeId="2" ParentId="1316" CreationDate="2014-04-07T08:57:03.153" Score="9" Body="&lt;p&gt;Question: What's different about applied mathematics that you don't enjoy?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice the difference between that and asking why they don't like applied mathematics. One demands an accurate description of applied mathematics, the other can be answered in terms of one's aspirations or perception of value, and that distinction can be explained if it's unclear or not acknowledged.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Motivation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Deep theorems often come from intuitions. Intuitions have to come from somewhere. Sometimes you get them from working elementary examples' features into your brain, but this can be near-sighted. Instead, they often come from an appeal to a natural connection between two subjects, and the overdetermining of properties induced by the multiple sets of semantics - every group is a permutation group, for example, is a formalization of the intuition about groups being transformers as well as algebras. The Sylow theorems and several properties of interest are capturing periodicity and self-similarity decompositions of group structure arising from viewing them as or expecting them to be the collections of symmetries of an object. Applied mathematics functions always in a context, and cross-pollination of semantics is an inherent part of such naturalistic environments. As such, it's an important source of intuition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Projective geometry, for instance, arose from applied mathematics and the intuition it brought, and its importance in the structure of abstract objects (algebraic geometry, projective representations, manifolds, homology, hyperbolic geometry, continued fractions, the modular group) is worth the thinking it takes to justify without any trace of solipsism.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applied mathematics can lead to these new fields because it poses problems that wouldn't be considered problems or be thought up by someone working only off their existing knowledge. It can also pose problems that are unmanageably difficult, which in mathematics suggests there are underlying paradigms left to be uncovered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a modern example, crack analysis can now be done using fractal measures and other abstract ways of reasoning, except when it comes to intersections of propagating cracks, these theories tend to break down abruptly and the finite-element method must be used to proceed with the description of the cracks approximately.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's also one of the best ways to motivate a special case, informing the specific calculations that will allow one to become adept at a field. Most (classes of) differential equations are posed as tasks for their representation of a physically observed behavior, and in that way arbitrary calculations obtain a life of their own through application. So, for a pure mathematician - and by that I mean one which isn't strictly concerned with real-world tasks, applied contexts are rich interfaces to mathematics. Knowing any field in such detail helps gauge your depth of knowledge in other fields, and an application will evade giving a sense of security until that point is reached and the motivating problems have been solved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All mathematics is descriptive of something, so how is non-applied mathematics to be intrinsically distinct from applied mathematics? It might be they disagree with having a non-mathematical goal, or have the impression that someone primarily does applied mathematics when mathematical goals aren't inherently valuable to them. It might be because educators often treat students as if mathematics is opaque to them whenever it isn't shrouded in a real-world scenario. These are cultural reactions. Perhaps their problem is actually with the experience they expect of applied mathematics, or the type of output they expect to be giving. The diagnostic should come before the prescription. Much of what people are trying to avoid by professing these alignments can be a problem in pure mathematics, in a generalized form, and if you have the wrong reasons to avoid something you narrow your tools for gaining experience and proficiency in a field in which you're mostly left to your own devices.&lt;/p&gt;&#xA;" OwnerUserId="727" LastActivityDate="2014-04-07T08:57:03.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1387" PostTypeId="2" ParentId="6" CreationDate="2014-04-07T10:16:25.113" Score="15" Body="&lt;p&gt;Here's another idea: Give a completely different type of homework assignment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, this year, collect all of your homework as usual. Look through the homework for examples of solutions that are clearly done by students, and which have mistakes in them. Choose representative samples of the student homework which have evidence of common misconceptions in them. Typeset them in LaTeX to make them anonymous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The homework assignment then becomes; for these three example problems, find the mistake in reasoning in this example homework, and explain what you think the reasoning is that led to this mistake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has the benefit of directly giving students counter-evidence to the likely mistakes and misconceptions they have, and being an assignment that cannot be Googled (remember you have LOTS of examples of homework assignments, you can easily change them up for the following year).&lt;/p&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-04-07T10:16:25.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1388" PostTypeId="2" ParentId="1304" CreationDate="2014-04-07T10:25:45.813" Score="2" Body="&lt;p&gt;I don't know if this is acceptable question-answering etiquette, but &lt;a href=&quot;https://matheducators.stackexchange.com/questions/825/problems-sets-for-instruction/830#830&quot;&gt;this list of rich math tasks&lt;/a&gt; from another question is an excellent starting place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Aside: Is it better to link to the other list, or to copy and paste it into this question?)&lt;/p&gt;&#xA;" OwnerUserId="254" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-16T20:13:01.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1389" PostTypeId="2" ParentId="1240" CreationDate="2014-04-07T10:32:43.287" Score="2" Body="&lt;p&gt;What about using a spreadsheet, with a column for links to the tasks themselves? Each column of the spreadsheet contains information about each problem you want to store (like a quick description, title, related problems, prerequisite problems, etc...), and you can just use the filter function on a column to limit the list to the problems you want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could even put the LaTeX commands for the specific problem into a column of the spreadsheet for future copying and pasting.&lt;/p&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-04-07T10:32:43.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1390" PostTypeId="2" ParentId="506" CreationDate="2014-04-07T11:01:40.153" Score="4" Body="&lt;p&gt;My suggestions (which assume that you are a lecturer in a large university class):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Design lesson plans with hinge questions that check for important understandings. In a large lecture hall, you can use individual student response devices so that every student votes on their answer to the question, and you get to see the distribution of responses. This allows you to make your decision about how to proceed with the lesson plan based on data, rather than just your intuition about what they understand. Sometimes people are bored and do not ask questions because there is nothing new in what is being presented. For more information about this, see Dylan Wiliam's book &quot;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/193400930X&quot; rel=&quot;nofollow&quot;&gt;Embedded Formative Assessment&lt;/a&gt;.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This is related to the strategy from #1. When the results from students come in from your voting system, they will often be quite mixed. One immediate strategy to use here is to ask students to discuss their solutions with each other before moving on to explain why A (or whatever) is the answer. See Eric Mazur's &quot;&lt;a href=&quot;http://mazur.harvard.edu/research/detailspage.php?rowid=8&quot; rel=&quot;nofollow&quot;&gt;Peer Instruction&lt;/a&gt;.&quot; The benefit here is that, in general, students will move to better understanding if they have an opportunity to get feedback on their thinking, and discourse with someone else is an excellent way to get feedback.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Design your class so you regularly get feedback from your students on what they understand, and use this feedback to help design your lessons. This will help reduce the probability that the lesson you design doesn't meet their needs, because you will have much more knowledge about their needs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is really difficult to deviate from something you have planned that is not working, and then on-the-fly come up with a better plan. It is better to have prepared two or three possible plans (one of which is well-prepared and the other two of which might be more loosely sketched out) planned ahead of time, so that you are not stuck going in a completely different direction with no idea what it will look like.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-04-07T11:01:40.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1391" PostTypeId="2" ParentId="6" CreationDate="2014-04-07T12:11:07.283" Score="48" Body="&lt;p&gt;The underlying issue is something that is hinted at in JDH's answer, but not explicitly stated.  So I'd like to state it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key is to decide what your homework is for.  The big words are &lt;a href=&quot;http://en.wikipedia.org/wiki/Formative_assessment&quot;&gt;&lt;strong&gt;Formative&lt;/strong&gt;&lt;/a&gt; and &lt;a href=&quot;http://en.wikipedia.org/wiki/Summative_assessment&quot;&gt;&lt;strong&gt;Summative&lt;/strong&gt;&lt;/a&gt;.  In short, &lt;em&gt;formative&lt;/em&gt; assessment is designed to help you learn, &lt;em&gt;summative&lt;/em&gt; assessment is designed to figure out what you've learnt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The distinction is not clear cut, and I'm no expert so I won't pretend to be.  In truth, all assessment is a mixture since unless we know what we have learnt, how can we figure out how to learn it better?  And if we don't use the assessment of what we've learnt to find out more, what was the point in measuring it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, we can broadly assign assessment to one of the two classes by its primary use.  I suspect that the majority opinion would be that homeworks are for &lt;em&gt;formative&lt;/em&gt; assessment.  We only grade them to encourage students to do them, reasoning that if we don't give the students credit for doing them then they won't choose to do so of their own free will (but note quid's comment on the question!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, this sends out the wrong message.  By counting them as part of the final grade, the students view the homeworks as &lt;em&gt;summative&lt;/em&gt;: a measure of what they have learnt, and therefore view them as something to achieve rather than as something to reflect upon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also unfortunately, the default assumption on students' part seems to be that all assessment is summative.  This is because the majority are not aware of these nuances, and are more aware of the occasions where they have undergone summative assessment rather than formative (the latter being, by its nature, less stressful and more subtle, and so less noticeable).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore it is not enough to simply say &quot;Do the homework for your own good&quot;.  You will need to set up a culture of reflection.  Here are some broad guidelines.  Again, I'm not an expert and have not implemented (all of) these myself.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If you must include a homework part in the overall assessment, make the bar as low as possible.  Essentially, if they try the homework they should get the credit.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use feedback from the homeworks in your teaching.  Use the homeworks to find out what people found difficult and then go over that again in class time.  Make it so that the students see the link between the information that they give you through the homeworks and the help that they get afterwards.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In my course, the rule is that they need 8 out of 12 homeworks &quot;approved&quot; to be able to take the final exam.  Other than that, the homeworks don't count.  Nevertheless, because we don't really do (2), a lot of students simply do the first 8 and then stop.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-04-07T12:11:07.283" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1392" PostTypeId="2" ParentId="1380" CreationDate="2014-04-07T12:20:05.293" Score="7" Body="&lt;p&gt;For me there are perhaps three main types of problems which I assign:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Routine skill building&lt;/strong&gt;: either are modeled on a computation which I have shown similar problems solved, or, are a proof problem which is just a natural consequence of definition with little extra technique required. For a proof course, many problems are little more than an invitation to care about what the notation actually means.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Breadth discovery&lt;/strong&gt;: in every course there are certain topics for which we do not have enough time in lecture. It is a very rewarding experience for students to be guided through a short module of problems where they discover the essential features of a topic which is not covered in depth by lecture and other materials.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Challenge&lt;/strong&gt;: here there are no rails, no box, no expectation anyone in the course solves it. Sometimes these are used to show the limitations of a current family of techniques to solve problems, sometimes these involve some fuzzy intuition which guides a creative leap.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I suspect most of the problems I write and/or assign fit into either 1 or 2, but students often accuse me of 3. Honestly, one of the reasons I try to surf the MSE a fair amount is to assess what is covered in my courses at other universities. Also, the international flavor of the MSE helps me get some cross-section of what is happening at schools all over the world. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-04-07T12:20:05.293" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1393" PostTypeId="1" AcceptedAnswerId="1398" CreationDate="2014-04-07T12:42:17.437" Score="6" ViewCount="202" Body="&lt;p&gt;Sometimes, if you want students to manipulate some sort of special matrices (like, asking if they are invertible or ask for a LU decomposition, etc.), you have to possibility to state all properties of that special matrix in a rigorous way. Moreover, since it is a matrix, you can additionally show a picture of that matrix with a lot of $\ldots$ in it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem, however, is that for some students the picture is helpful; for other students, it is distracting or misleading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;An example question:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $A$ be a tridiagonal matrix (i.e., $a_{ij}=0$ for $j&amp;gt;i+1$ and for $i&amp;gt;j+1$) with an non zero element $a_{n1}\neq 0$, i.e., the matrix looks like&#xA;  $$A = \begin{pmatrix} * &amp;amp; * &amp;amp; 0 &amp;amp; \ldots &amp;amp; \ldots &amp;amp; 0  \\ * &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp;  &amp;amp; \vdots \\ 0 &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \vdots \\ \vdots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; 0 \\ 0 &amp;amp; &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; * \\ * &amp;amp; 0 &amp;amp; \ldots &amp;amp; 0 &amp;amp; * &amp;amp; * \end{pmatrix}$$&#xA;  where $*$ means that on this location, there is an arbitrary real number which can be different from zero. Let $LU=A$ be a LU-decomposition of $A$. Perform an LU decomposition of $A$ in order to find the locations  in $L$ and $U$ which are zero for sure.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Several students manipulated the matrix and arrived at some point to calculate things like&#xA;$$\frac{*}{*}=1, \quad *-*=0$$&#xA;etc. which is not true since the interpret $*$ as a fixed number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How can one avoid such misinterpretations of the picture? What went wrong when students perform calculations like above? Should one avoid pictures like the above? Should one write even more text to make sure the meaning of $*$?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-07T15:33:26.163" LastActivityDate="2014-04-07T17:44:24.620" Title="Should one visualize properties of a matrix or/and state its properties?" Tags="|exams|linear-algebra|test-design|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1394" PostTypeId="2" ParentId="1236" CreationDate="2014-04-07T12:55:38.773" Score="19" Body="&lt;p&gt;&lt;em&gt;How does the workload of a high-school mathematics teacher compare to a university-level instructor?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give you some context, I taught grades 7 - 9 for three years and I have taught in one way or another at the university level for 15 years (first as a grad student and now as a professor). My current job is Associate Professor of Mathematics Education in an education department, but I do teach mathematics courses (mathematics content for elementary teachers) as a part of my job.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The workload itself is very different in general. As a junior high mathematics teacher, I taught 5 sections a year (two sections of one course and three sections of another course), and each course had about 30 students in it. I also had other roles, such as teacher duties (lunch duty, detention duty) and coaching (or advising student groups) responsibilities. I had one preparation period each day for which I had to complete all grading, planning, copying, etc. And of course that meant I took a lot of work home. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a college professor at a research institution, I teach two or fewer classes each semester (depending on whether a research grant has bought me out of teaching a course), with no more than 35 students in each class, probably less. But I have more administrative duties now as a professor, and the research expectation is large enough to take up a non-trivial portion of my time. Plus, there is a great deal of one-on-one teaching while working with graduate students -- advising them on dissertation work, etc. So the general nature of the work is tough to compare. Both roles require a lot of time beyond a standard 40 hour work week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have never felt more physically exhausted than when I was a junior high teacher. I have never felt more mentally drained than in my current job. Both kinds of draining also have unique feelings of invigoration associated with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;To those who have been employed in both secondary and college education, is there a level of college teaching that feels similar to high-school teaching (i.e. is teaching high school similar to teaching 4 college classes a semester)? If college education is too dissimilar to compare to high school/middle school, what are the key differences that set them apart?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ah, so the real question is to compare the work of teaching, then?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Preparing to teach feels similar now, although I think I am better at it just because more time has passed -- I have learned more about how to target my teaching to connect to my students' thinking, how to create lessons that support conceptual understanding in balance with procedural fluency. But I'd like to think I would have gotten better at this and learned more regardless of my role. So, planning to teach might be similar if we compare class to class and assume equal time was available.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The act of delivering a lesson -- assume the same number of students in each class (which is a big assumption in some cases) -- and one major difference is probably obvious: classroom management. K-12 education in the USA is compulsory up until age 16, right? And college students have more of a choice of whether or not to be there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But also teachers still must motivate and engage learners at every level, because all students can choose to check out mentally (either choosing not to physically show up in college or not mentally be there in secondary school).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also agree with the person above who said no parent calls or contact. I would do this as a middle school teacher. I am literally not allowed to discuss student progress with a parent at the university level, because the student is a legal adult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not agree that college teaching has to be more teacher-centered. I think that college teaching can build upon students' thinking, be interactive, and be tailored to the individual needs of students -- particularly if the class is smaller in size -- but college instructors are not required to do this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I collaborate more with colleagues on planning and debriefing from instruction and also I conduct research on my undergraduate students' learning now as a college professor. But I would have liked to collaborate like this and to conduct action research as a teacher. I didn't have time as a newer teacher, and none of us in my department made this kind of work a priority then.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do probably spend more time planning and grading as a professor than I did as a teacher, but perhaps because I have more time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Class-to-class, I probably spend more time on teaching two classes now than I did when teaching two classes as a junior high teacher. But I had more students and more classes when I taught junior high.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a college professor, we are not evaluated by our students' test scores. Our raises are not affected by student performance (although perhaps the administration could take students' performance into account a bit more?). Instead, I am evaluated by student evaluations and my reflections on my own teaching. I ramp this up by conducting publishable research related to my students' learning, however, so I am able to present data to administrators that takes students' learning into account, as least for some of my courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is how a typical lesson of mine went as a junior high teacher: Students came in and work on a warm up problem or two. We discussed that problem. We went over the previous night's homework. I presented some material, they practiced some problems with some guidance from me, we discussed the problems -- including some students coming to the board to share their work and thinking, then the students had time to work independently on the problems to get started on their homework during class. Sometimes (once a week or every two weeks), we had a more exploratory problem solving lesson involving working together in groups using manipulatives to understand concepts and develop mathematical understandings on more challenging tasks that required working together in groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is how a typical lesson of mine (for my mathematics courses for prospective teachers) goes as a college professor: Students come in and sit in their groups and start sharing their homework solutions with each other. We spend time at the beginning of the class going over homework -- mostly with the undergrads coming up to the document camera explaining their thinking and talking with each other about the mathematics. I am more of a facilitator during the going-over-homework process than I used to be. Then, I give them a task that is within their zones of proximal development (they have enough prior knowledge to get started, but they are learning new mathematics by working on the task) to solve with their groups. After they work for a while, we have a class discussion that involves students going to the chalkboard and the document camera to present their work and thinking and the class discusses. Again, I am more of a facilitator and my role is to highlight and clarify and bring out the most important ideas and reinforce them. Then, students receive homework to practice what they learn. I assign fewer homework problems now, maybe about 7, when I assigned about 30 per night to junior high students, but each of the 7 tasks requires a great deal of diagramming, explaining, etc., because the goals of the course are to develop conceptual understanding among the prospective teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The nature of my teaching changed because I changed how I think about teaching over time, and I had a heavier balance on procedural fluency than conceptual understanding or problem solving as a junior high mathematics teacher (this would be different if I taught junior high today!), and I have a heavier balance on conceptual understanding now as a college professor of mathematics for future teachers -- because this is what my students need. They already have strong procedural fluency. What they need is to understand the meanings behind the procedures in order to be more effective teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the work of teaching not only is shaped by our level of teaching (junior high, high school, college), but also it is shaped by our goals for our students' learning. That's such an important point that I'd like to end with that. (Thanks for reading all of this, if you did!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edited to add: I would like to &lt;a href=&quot;https://www.dropbox.com/s/on3w54k4p85fe0r/JansenMiddleton_MotivationMatters_Ch9.pdf&quot;&gt;share a book chapter&lt;/a&gt; in which I reflected upon the ways in which I think about teaching mathematics differently now than I used to think about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Middleton, J. A., &amp;amp; Jansen, A. (2011). Motivation Matters, and Interest Counts: Fostering Engagement in Mathematics, Reston, VA: National Council of Teachers of Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="739" LastEditorUserId="739" LastEditDate="2014-04-10T13:30:55.620" LastActivityDate="2014-04-10T13:30:55.620" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1395" PostTypeId="2" ParentId="1393" CreationDate="2014-04-07T13:24:43.837" Score="3" Body="&lt;p&gt;My approach:  Give them a matrix with random entries, and ask for the zeroes in the decomposition of that particular matrix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should be able to generate a suitably generic matrix after a few tries.  If you use Mathematica, you can test this as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\mathtt{&#xA;B = Table[Random[Integer, \{1, 9\}], \{6\}, \{6\}]&#xA;}$$&#xA;$$\mathtt{&#xA;C = \{\{1, 1, 0, 0, 0, 0\}, \{1, 1, 0, 0, 0, 0\}, \{0, 0, 0, 0, 0, 0\},\\ \{0, &#xA;   0, 0, 0, 0, 0\}, \{0, 0, 0, 0, 1, 1\}, \{1, 0, 0, 0, 1, 1\}\}&#xA;}$$&#xA;$$\mathtt{A = B\ C}$$&#xA;$$\mathtt{&#xA;LUDecomposition[A]&#xA;}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Find an $A$ whose decomposition has zeroes only in the right places, and then just give them that $A$, e.g.&#xA;$$&#xA;2\ 5\ 0\ 0\ 0\ 0 \\&#xA;6\ 3\ 0\ 0\ 0\ 0 \\&#xA;0\ 0\ 0\ 0\ 0\ 0 \\&#xA;0\ 0\ 0\ 0\ 0\ 0 \\&#xA;0\ 0\ 0\ 0\ 1\ 3 \\&#xA;6\ 0\ 0\ 0\ 2\ 5&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 1:  The definition of $A$ here used the Mathematica code for element-wise product, where $\mathtt{B.C}$ would have given the matrix product instead. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 2:  I think the example shows that I understood the pattern incorrectly, but I hope the approach is clear anyway!&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-07T13:24:43.837" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1396" PostTypeId="2" ParentId="1383" CreationDate="2014-04-07T13:36:03.017" Score="23" Body="&lt;p&gt;It is possible to usefully mention &quot;Lie groups (and Lie algebras)&quot; in an introductory course, if one does not give formal definitions, but, rather, examples. It is not necessary (or advisable) to &quot;define&quot; smooth manifolds, which seems to have considerable baggage-of-abstraction of its own. Just give important examples, noting that they do seem to have a lot in common: circle group, $SU(2)$, $SO(3,\mathbb R)$, $SL(2,\mathbb R)$, letting them act on spheres, and maybe by the &quot;linear fractional transformation&quot; action on the upper half-plane?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Lie algebra and exponentiation are easy to do in the case of linear Lie groups, and illustrate many interesting points... and the Lie bracket presents itself as an artifact.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience indicates that such a discussion will stimulate some students, and, as usual, upset those students who are already barely keeping up. All the worse that the feasible style cannot be as formal (or else the start-up cost becomes prohibitive), and, ironically, getting away from the usual exaggeratedly formal style can be psychologically disturbing to some students, if only because it's a change from the usual.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But introducing such ideas is a positive thing, and serves very well any students who will be continuing seriously in mathematics or any science that uses serious mathematics. For that matter, those who might have been bored and disenchanted because the all-too-common sterilized-formal abstract algebra was too easy may be provoked a bit, in a good way, by the interaction of more than one idea at a time.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-07T13:36:03.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1397" PostTypeId="1" AcceptedAnswerId="1400" CreationDate="2014-04-07T13:58:15.967" Score="19" ViewCount="2758" Body="&lt;p&gt;Sometimes when assigning homework, it is possible that the homework can be found in some books or lecture notes, sometimes even referenced by yourself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As long as the proofs and arguments there are good and/or students bring up their own thoughts about the question, there is nothing bad in grading it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem comes up if there is a bigger part missing in reference book or if there is even something wrong there and students copy that without adding the missing parts (One time, a whole course with more than 80 students did exactly copy a solution which was solved in some lecture note with a contradiction proof, but the contradiction did only cover a small part of the statement. Only very few students got that.). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is now: &lt;strong&gt;How should one grade such solutions? If you do not give full credit, how can you explain that the students without making them angry (Since you gave that particular book as reference)?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is related to &lt;a href=&quot;https://matheducators.stackexchange.com/q/6/114&quot;&gt;How to assign homework when answers are freely available or attainable online?&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2017-06-16T09:25:02.147" Title="&quot;But I copied that from a book-reference you gave! How can that be wrong?&quot;, or how can you grade homework which is solved poorly in some book?" Tags="|homework|grading|general-pedagogy|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1398" PostTypeId="2" ParentId="1393" CreationDate="2014-04-07T14:22:20.497" Score="5" Body="&lt;p&gt;Well, here's some possibly relevant experience.  I gave the following question on a recent (in-class) exam in a sopohmore linear algebra course.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose that&#xA;  $$&#xA;\newcommand{\?}{\scriptsize\text{?}\,}&#xA;A \;=\; \left[\begin{array}{@{\;\;}r@{\;\;}r@{\;\;}r@{\;\;}}&#xA;2 &amp;amp; \? &amp;amp; \phantom{+}x \\[6pt]&#xA;\? &amp;amp; 4 &amp;amp; \? \\[6pt]&#xA;\? &amp;amp; -1 &amp;amp; \?&#xA;\end{array}\right]&#xA;\qquad\text{and}\qquad&#xA;A^{-1} \;=\;&#xA;\left[\begin{array}{@{\;\;}r@{\;\;}r@{\;\;}r@{\;\;}}&#xA;7 &amp;amp; \? &amp;amp; 3  \\[6pt]&#xA;\? &amp;amp; \phantom{+}2 &amp;amp; \phantom{+}0 \\[6pt]&#xA;1 &amp;amp; \? &amp;amp; 1&#xA;\end{array}\right],&#xA;$$&#xA;  where each ? indicates an unknown entry.  Use this information to find the value of $x$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Between the use of question marks and the explanatory text, I didn't seem to have any trouble with students misinterpreting the question.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-04-07T14:27:28.270" LastActivityDate="2014-04-07T14:27:28.270" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1399" PostTypeId="2" ParentId="1397" CreationDate="2014-04-07T14:55:18.077" Score="13" Body="&lt;p&gt;Make clear that the &quot;make sure the proof is correct&quot; is part of the work to be done in the homework. If it is my proof, or yours, or from &amp;lt;famous textbook&amp;gt; that is wrong, the answer is wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Yes, need to emphasize that even the above cited authorities get it wrong sometimes).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-07T14:55:18.077" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1400" PostTypeId="2" ParentId="1397" CreationDate="2014-04-07T15:19:48.543" Score="37" Body="&lt;p&gt;Students are used to other people being the source of truth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even in an algebra class, they will do something (incorrect, at least in the common context) like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$(x + 3)^2 = x^2 + 9$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and then &lt;em&gt;ask me if it is correct&lt;/em&gt;, or if it actually goes a different way. The implication is that I know the truth and they cannot know it without me. My goal then becomes the following: &lt;strong&gt;show the students that they can independently identify objective truths.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I think the answer here is the same as it is there: Regardless of what is printed in some book or another, the correctness of a proof is essentially an objective fact -- one that they must learn to measure themselves. This is a difficult point! The students may rebel and use the book as evidence that the truth is in fact &lt;em&gt;subjective&lt;/em&gt;, determined by a semi-arbitrary grader. This is the underlying misunderstanding to dispel. &lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="143" LastEditDate="2017-06-16T09:25:02.147" LastActivityDate="2017-06-16T09:25:02.147" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1401" PostTypeId="2" ParentId="686" CreationDate="2014-04-07T16:00:26.980" Score="2" Body="&lt;p&gt;The whole field of NP completeness in computer science dates from around 1970. Some results are easy to understand (but the proofs tend to be gargantuan until some basic results are in).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-07T16:00:26.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1402" PostTypeId="2" ParentId="1345" CreationDate="2014-04-07T16:13:28.693" Score="6" Body="&lt;p&gt;Cook up say 40 approximately equally hard problems covering the subject area, and have the students pick say 3 at random, could even go for without replacement (but that requires more problems). Might also go for say 15 problems in each area, and have them pick one of each. Or have a shading of problem complexities (2 easy ones, 1 medium hard). Make your imagination run wild.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beauty is that your collection of problems will grow over time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, problems will get known (that's why you should go on adding new ones constantly, and perhaps retire ones &quot;too many times asked&quot; or some other criteria). But if someone is able to learn enough to solve a wide spattering of 40 problems or so, I'd contend they merit a passing grade anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make grading fair/uniform, set up a grading schema (might as well publish it): Solve the problem correctly, 6 points; use correct notation/notions, 1 points; answer 3 questions from the committee, 1 point each. Or something similar. Perhaps each of the points on a scale of 0 (nothing), 1 (partial), 2 (complete); I'd even add 3 (more than was asked for). Agree on a reasonably precise meaning of each grade among the grading committee(s) to ensure fairness (each crop and along time).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-07T16:13:28.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1403" PostTypeId="1" CreationDate="2014-04-07T17:15:04.660" Score="37" ViewCount="2309" Body="&lt;p&gt;Some university-level courses have no specific prerequisites, yet are mathematically involved to the extent that someone with little to no experience in math will probably find themselves in over their heads. Such courses are often cross-listed and appeal to students in disciplines outside of math like philosophy, computer science, economics, psychology, etc. For example, certain courses in logic or decision theory often fit this bill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One standard way of &quot;warning&quot; the students about the mathematical nature of the course is by saying something like, &quot;The only prerequisite is mathematical maturity.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, this seems like an apt characterization: they're going to be asked to prove things on the homework, to absorb formal definitions fairly rapidly, etc. But just because &lt;em&gt;I&lt;/em&gt; have an intuitive grasp on the meaning of &quot;mathematical maturity&quot; doesn't mean &lt;em&gt;they&lt;/em&gt; do. I can't help but feel that this warning falls short with some and goes too far with others. I know from experience that some students &lt;em&gt;underestimate&lt;/em&gt; this warning, and remain in the course despite, e.g., having no idea what it means to rigourously prove something. Unsurprisingly, these students tend to struggle greatly. On the other hand, I like to view such courses as good alternatives to more traditional courses like, say, calculus, for helping to strengthen and deepen general mathematical facility. It would be a shame if some students who would otherwise find the course quite useful &lt;em&gt;overestimate&lt;/em&gt; the meaning of &quot;mathematical maturity&quot; and shy away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How can I adequately convey to students the meaning of &quot;mathematical maturity&quot;?&lt;/strong&gt; In some sense, it's one of those, &quot;If you don't know it, you haven't got it&quot; type things, but I'd like to have something more tangible than this, perhaps a method of assessment or a representative class of examples I could offer to students who are unsure.&lt;/p&gt;&#xA;" OwnerUserId="327" LastActivityDate="2014-06-15T20:29:40.740" Title="How to convey the meaning of &quot;mathematical maturity&quot;?" Tags="|undergraduate-education|examples|assessment|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1404" PostTypeId="1" AcceptedAnswerId="1412" CreationDate="2014-04-07T17:32:37.420" Score="11" ViewCount="405" Body="&lt;p&gt;In a course I am teaching this semester, I am required to have my students write a large (6-12 page or so) paper worth 40% of their grade. Many other colleges have courses with similar requirements, e.g. a capstone major course involving writing, or a history of mathematics course, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where papers are required in a course, how can they be fairly graded? My concerns are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How much emphasis (if any) to place on correct grammar/spelling/formatting&lt;/li&gt;&#xA;&lt;li&gt;How much emphasis to place on good exposition vs mathematical rigor (which are, of course, not contradictory goals)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My concern is that I am not trained as an English teacher, and my students are not taking an English course. However, the paper should be readable. What has worked in your experience?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="261" LastEditDate="2014-04-09T14:29:10.973" LastActivityDate="2021-04-05T02:36:26.923" Title="In math courses with writing requirements, how should large papers be graded fairly?" Tags="|undergraduate-education|grading|assessment|writing|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1405" PostTypeId="2" ParentId="1373" CreationDate="2014-04-07T17:32:54.177" Score="7" Body="&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Intuition is one of the things that distinguishes humans from machines. We should use it, especially to guess what we would like to prove. However, then, we ought to prove our claims.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Introduction:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to start with dividing mathematical problems into two groups:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;straightforward problems&lt;/em&gt;, where mechanical work is enough,&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;complex problems&lt;/em&gt;, where there has to be something more in the proof.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Intuition comes with a cost of introducing errors, and while dealing with problems from the first category we should avoid it unless it outperforms the cost of manipulating overcomplicated expressions. However, while solving a problem from the second group we &lt;em&gt;have to use intuition&lt;/em&gt;, there is no other way (well, we could ask someone or do a brute-force search, but in most cases this is infeasible).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Quite long digression about automated theorem proving:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a whole research area called &lt;a href=&quot;http://en.wikipedia.org/wiki/Automated_theorem_proving&quot; rel=&quot;nofollow noreferrer&quot;&gt;automated theorem proving&lt;/a&gt; which tries to make computers prove theorems. It is worth noting that there are results, like &lt;a href=&quot;http://en.wikipedia.org/wiki/Robbins_conjecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robbins conjecture&lt;/a&gt; which eluded humans, but were proved by machines. However, there is a vast number of results that are inaccessible to machines (existence of such problems is implied, for example, by undecidability of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Halting_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;halting problem&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to emphasize one aspect which makes theorem proving hard. It is not exactly right, because right now the use of many various heuristics allows computers to solve a multitude of practical problems. Nevertheless, the core issue did not disappear and shows itself in more complicated cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The aforementioned issue is that, if the proof behaves in some non-monotone way, the computers are unable to guess the appropriate middle formula that would connect the premises and the thesis. For example, the proofs are usually of the form &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\textit{premises} \to \textit{something} \to \textit{thesis}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proof would be straightforward if one could deduce &lt;em&gt;something&lt;/em&gt; going forward from &lt;em&gt;premises&lt;/em&gt; or going backward from &lt;em&gt;thesis&lt;/em&gt; (perhaps doing both). However, it might happen that the inference looks like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\textit{premises} \to \ldots \to p \land (p \to q) \to q \to \ldots \to \textit{thesis}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observe that there is a some form of non-monotonicity between $p$ and $(p \to q)$, namely, the weaker $p$ is, the easier it is to prove it from the premises, but the harder it is to prove $(p \to q)$; the stronger $p$ is, the easier is for $(p \to q)$ but the more difficult for $p$. Independent of whether we go forwards from &lt;em&gt;premises&lt;/em&gt; or backwards from &lt;em&gt;thesis&lt;/em&gt;, we might have to check an impossible number of paths before we decide if the theorem is true (produce a proof) or not (produce a counterexample).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if we had to prove two theorems, first $\textit{premises} \to p \land (p \to q)$ and the second $p \land (p \to q) \to \textit{thesis}$, then it would be much easier (provided there's no similar situation inside the subparts).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why it matters:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a professor who said, that it is the &lt;a href=&quot;http://en.wikipedia.org/wiki/Modus_ponens&quot; rel=&quot;nofollow noreferrer&quot;&gt;modus ponens / rule of detachment&lt;/a&gt; where the genius in the proof hides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The hardness is to know the middle lemma that makes the proof work. In other words, the difficulty lies in &lt;em&gt;what we try to prove&lt;/em&gt;, all the rest is straightforward (of course, that does not mean easy). A great example of this phenomenon are theorems which we have to strengthen to make them easier prove (see &lt;a href=&quot;https://math.stackexchange.com/q/448967/26306&quot;&gt;here&lt;/a&gt; for some concrete examples).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it is the &lt;em&gt;intuition&lt;/em&gt; that tells us what to prove. It is a necessary component of any research that deals with more that just a straightforward tasks. It &lt;em&gt;is&lt;/em&gt; the best leader we have – we don't have anything else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it is great to use intuition and to encourage the students to use it. &#xA;Every time I can, I encourage them to train their intuition. &#xA;Nevertheless, there is an important part: we have to actually prove our results. Intuition might be great, but it will not substitute for a proof. There is nothing wrong with having bad intuition if we learn from our mistakes. Once we will start forming a proof, we gain more and more understanding that will correct our intuition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some suggested that they alternate between trying to prove the hypothesis and finding a counterexample. From my perspective both actions deepen our understanding and indirectly correct our intuitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Intuition should be trained and that can't be done without using it. Perhaps our solution will be inferior or suboptimal, but thanks to that, next time our approach we will be better. On the other hand, if we were to keep using the same techniques again and again, we won't advance as much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your second question, I don't believe there is any general way to determine how much intuition do you need in any particular domain. I would say &lt;em&gt;experience&lt;/em&gt; is the best you can do, but even then the new generations might have a whole different view, i.e. new tools and techniques might allow for new points of view that might turn everything upside-down. In fact, if you understand some result deep enough, it &lt;em&gt;is&lt;/em&gt; intuitive (despite how non-intuitive it might seem to others).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-04-07T22:09:14.027" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1406" PostTypeId="2" ParentId="1393" CreationDate="2014-04-07T17:44:24.620" Score="4" Body="&lt;p&gt;The pitfall of this notation is: In mathematics, if something has the same (variable) name, it is the same thing. Because of that, this notation needs to be read in a relaxed way, which probably only comes when you have some kind of rough understanding of the topic already.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is why I'd say it depends on the context if such a notation is a good choice. I think it's a nice choice to illustrate an example or something similar in a lecture or tutorium. In this context it can be made clear that this is an illustration, and if there is something unclear, you can talk about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an exam or exercise sheet, there is no easy way to communicate that this matrix is just a relaxed illustration and not an actual written-down version of the matrix. I think for situations like this, it's better to be on the pedantic side and write &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$A = \begin{pmatrix} a_{11} &amp;amp; a_{12} &amp;amp; 0 &amp;amp; \ldots &amp;amp; \ldots &amp;amp; 0  \\ a_{21} &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp;  &amp;amp; \vdots \\ 0 &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \vdots \\ \vdots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; 0 \\ 0 &amp;amp; &amp;amp; \ddots &amp;amp; \ddots &amp;amp; \ddots &amp;amp; a_{n-1 \ n-1} \\ a_{n\ 1} &amp;amp; 0 &amp;amp; \ldots &amp;amp; 0 &amp;amp; a_{n \ n-1} &amp;amp; a_{n n} \end{pmatrix}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The extra effort in typing is not too much, but the benefit is a huge one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That having said, it surprises me as well that your students regarded $*$ as an actual name for a variable.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-04-07T17:44:24.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1407" PostTypeId="2" ParentId="1403" CreationDate="2014-04-07T18:19:02.380" Score="2" Body="&lt;p&gt;Some reasonable phrases are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For an economics class: &lt;em&gt;Prerequisite: some familiarity with the fundamental theorem of calculus&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;For mathematical logic: &lt;em&gt;Required background: understanding one proof of the Pythagorean theorem&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;For decision theory: &lt;em&gt;This course is appropriate for students who know why the digits test works for divisibility by 3&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Most students who have any inclination to rigorous analysis will have seen these theorems before they look at this new class.  Then they can judge for themselves.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-07T18:19:02.380" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1408" PostTypeId="2" ParentId="1397" CreationDate="2014-04-07T19:32:32.040" Score="12" Body="&lt;p&gt;By recommending some book you implicitly acknowledge all of its contents (unless specified otherwise). In fact it would be similar if the student used a solution &lt;em&gt;you&lt;/em&gt; presented, which happens to be wrong, but you only discovered the flaw when grading the exam. What would you do?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, I would not penalize the student for copying the solution from the book, actually I would deal with it as if the student happen to write exactly the same solution by himself/herself. The reason is that the modern math is built upon previous results and reinventing the wheel is not a good strategy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, there are three solutions I would consider:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Give full credit.&lt;/em&gt; The solution might be wrong, but you don't expect the students to recognize the difficulty (perhaps even the teacher missed it at first). One can give bonus points to those who did.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Give partial credit.&lt;/em&gt; The fact that some respectable author got it wrong is &lt;em&gt;some&lt;/em&gt; indication that it might be a &quot;reasonable mistake&quot;, that is, the approach could be mostly correct but for some subtlety that is non-intuitive and hard to spot. Partial credit is always tricky, though.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Grade it like everything else (perhaps no credit).&lt;/em&gt; Wrong solution is not a valid solution independent of where it comes from. If the students think the solution is good enough to copy it, they are wrong and should be graded accordingly.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My preferred one is the third: in math the meaning of &lt;em&gt;authority&lt;/em&gt; is slightly different than in other areas and students should understand that nobody is exempt from following only valid inferences. On the other hand, it would be weird to enforce this rule unless you told the students about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Partial credit is the most common solution from my experience. Many professors grade not only what was written down, but also what kind of understanding the work implies. The solution might be incorrect, but the intuition behind it might be still valid. However, it is inherently hard to grade understanding and what is obvious from one perspective might be difficult from another point of view. The fact that the book's author got it wrong suggests that there is some other intuition that works just as good as yours (e.g. the rest of the book is great), but handles this particular case badly. Therefore, it could still seem like an obvious mistake to me, but I might opt to give some more points nevertheless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rarely, but it happens, you may want to give full credit. I don't like it, but I see why someone might want to use this solution. One case is when you don't want to bother students with subtleties, for example &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;in some engineer classes, where theory is secondary at best, you could accept solutions that miss the fact that some matrix could be singular in a problem where a random distortion of input data fixes the problem;&lt;/li&gt;&#xA;&lt;li&gt;in a discrete math course, some series-related theorems work only with proper convergence arguments, however, appropriate techniques/formalizations might be too hard or not that important (e.g. the topic wasn't covered in depth); a correct, formal solution would be to guess the result using the incorrect derivation, and then proceed by induction (or whatever), but some deem such a level of scrutiny unnecessary.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Another case is when the problem set wasn't tested enough. There was an exam once in my student times where the lecturers did a mistake about what works and what does not. In result, one task came out much, much harder than it was meant to be. Most students missed the subtlety (i.e. used the same solution that the lecturers thought it worked), others did notice and presented a counterexample. There was an announcement and both types were accepted with a perfect score.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-04-07T19:47:10.413" LastActivityDate="2014-04-07T19:47:10.413" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1409" PostTypeId="2" ParentId="1351" CreationDate="2014-04-07T19:36:40.940" Score="6" Body="&lt;p&gt;&lt;strong&gt;1.&lt;/strong&gt; One main point that has been made is that Banach-Tarski Paradox needs to use non-mesurable sets; I want to insist on this by comparing with another similar looking statements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem (Wallace-Bolyai-Gerwien).&lt;/strong&gt; Given any two polygons of equal area, it is possible to cut one of them into finitely many pieces and rearrange the pieces into the second polygon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here the pieces can even be taken to be polygons. This theorem can be applied: you can produce examples in your Fablab. Explaining how different this is from Banach-Tarski is half the job of teaching Banach-Tarski, and I do not think that talking about the unknown possible future discovery in physics is more than barely relevant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2.&lt;/strong&gt; Another major point is that the pieces in the Banach-Tarski decomposition are not mathematically definable in any explicit way (we know that Banach-Tarski fails in some models of ZF: AC is really needed, not an artifact of the proof). Asking for the physical feasibility is at least one step beyond the true issue. And one should not let student think that the axiom of choice is just as any theorem; and in fact the Banach-Tarski paradox can be a good motivation to exclude it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you say: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We know many parts of mathematics which use Axiom of Choice in their&#xA;  main theorems (e.g. Analysis, Linear Algebra, Differential Equations,&#xA;  etc.) have amazingly correct applications in the actual world. Why&#xA;  Banach-Tarski theorem is an exception? Is it really an exception?!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;you are missing an important point: the parts of analysis, linear algebra, differential equations that really use the axiom of choice have certainly no physical relevance or practical application either; I defy anyone to give one convincing one. These topics can all be derived without choice (but then not all linear space have a basis, of course).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-07T19:36:40.940" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1410" PostTypeId="2" ParentId="1383" CreationDate="2014-04-07T19:36:47.560" Score="9" Body="&lt;p&gt;I think in a first abstract algebra course the goal is simply to make students aware that such things exist, give a couple of examples, and let them know that there is much, much more that can be learned in future classes.  With that in mind:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with $SO(2)$.  Note that all elements of $SO(2)$ can be represented in terms of a continuously-varying parameter $\theta$, but not uniquely, and use that fact to interpret $SO(2)$ as the points on a circle.  Focus on two key ideas:  (i) The group contains infinitely many elements, but is unlike the &quot;discrete&quot; examples (like $\mathbb{Z}$) that they already know about because of its intrinsic &quot;continuity&quot; (I would not formally define or prove anything about the continuity but simply rely on the intuition that circles are smooth curves, and leave it at that.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second example:  $U(1)$.  Give basic definition, interpret it geometrically as a circle, and then presto, $U(1) \cong SO(2)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third and fourth examples:  $SO(3)$ and $SU(2)$.  The only goal should be to show that both groups can be parametrized by four continuous variables, and that those parameters satisfy the equation of a 3-sphere.  Any more than that is overdoing it.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-04-07T19:36:47.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1411" PostTypeId="2" ParentId="1403" CreationDate="2014-04-07T19:51:08.163" Score="1" Body="&lt;p&gt;I think you answered it youself when you state the need of &quot;proving something rigurously&quot; and &quot;absorb definitions fairly rapidly.&quot; You might even give examples of the kinds of proof (formality) required, or (as &lt;a href=&quot;https://matheducators.stackexchange.com/a/1407&quot;&gt;@MattF's answer&lt;/a&gt; says) give some examples of results they should understand the &lt;em&gt;proof&lt;/em&gt; of, and some definitions they should understand the &lt;em&gt;why&lt;/em&gt; behind them.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-08T06:33:14.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1412" PostTypeId="2" ParentId="1404" CreationDate="2014-04-07T20:16:33.540" Score="11" Body="&lt;p&gt;Set up a grading schema, awarding points for grammar, spelling, correct use of the technical terms. Central should be points for correct, logical overall structure, and enough explanations to lead the reader from one step to the next.&lt;/p&gt;&#xA;&lt;p&gt;You can publish the grading schema for their guidance, and perhaps write up an example of how  their work should look like. A few, short, examples of what &lt;em&gt;not&lt;/em&gt; to do (a rambling stretch of algebra with no explanation whatsoever, another one with totally redundant comments like &amp;quot;now we add 1 to &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; to give &lt;span class=&quot;math-container&quot;&gt;$n + 1$&lt;/span&gt;.&amp;quot; I'm sure in a term or two you'll have hair-raising examples enough to distill into more &amp;quot;don't do this&amp;quot; texts.&lt;/p&gt;&#xA;&lt;p&gt;There are lots of texts on how to write mathematics around, I like Knuth, Larrabee, Roberts' &lt;a href=&quot;http://www-cs-faculty.stanford.edu/%7Euno/klr.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Mathematical Writing&amp;quot;&lt;/a&gt; a lot, or Chang's &lt;a href=&quot;http://cheng.staff.shef.ac.uk/proofguide/proofguide.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;How to write proofs: a short guide&amp;quot;&lt;/a&gt; should be required reading.&lt;/p&gt;&#xA;&lt;p&gt;If this is their first collision with this, make up a &amp;quot;test drive homework&amp;quot; (for, say 5% of the grade) asking them to write up some of the proofs in class (or assigned &amp;quot;randomly&amp;quot; from a pool of not-so-brilliantly-written texts), to be graded like the 40% requirement, so you can fling zero grades around at leisure when it goes &amp;quot;for real.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2021-04-05T02:36:26.923" LastActivityDate="2021-04-05T02:36:26.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1413" PostTypeId="1" AcceptedAnswerId="1415" CreationDate="2014-04-07T20:48:15.413" Score="7" ViewCount="943" Body="&lt;p&gt;I am teaching calculus for undergraduate students in humanities departments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Will Powerpoint presentations be better than using marker and whiteboard?  Will students learn more efficiently? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am using computer-based presentations only to describe 3D functions and constrained optimizations. &lt;/p&gt;&#xA;" OwnerUserId="79" LastEditorUserId="1892" LastEditDate="2014-07-09T17:04:43.023" LastActivityDate="2014-07-09T17:04:43.023" Title="Teaching by powerpoint presentation" Tags="|technology-in-education|software|presentation|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1414" PostTypeId="2" ParentId="1302" CreationDate="2014-04-07T20:52:13.443" Score="17" Body="&lt;h1&gt;Some suggestions&lt;/h1&gt;&#xA;&lt;p&gt;Michael Spivak's &lt;strong&gt;Calculus&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is really an honors level calculus text, but it might be useful to have around. It's pretty expensive, however.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Stephen Abbott's &lt;strong&gt;Understanding Analysis&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Back in &lt;a href=&quot;https://groups.google.com/g/sci.math/c/6jMusV56ELQ/m/ko7igljijZIJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;January 2003, in sci.math&lt;/a&gt;, I wrote: &lt;em&gt;I think it's the best written introductory analysis book that's appeared in the past couple of decades&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;David M. Bressoud's &lt;strong&gt;A Radical Approach to Real Analysis&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Takes a more historical approach than any other standard text currently in use. Heavy focus on pointwise and uniform convergence of sequences of functions (power series, Fourier series).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Victor Bryant's &lt;strong&gt;Yet Another Introduction to Analysis&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;All exercises have solutions. Full of pictures, diagrams, and other pedagogical aids. Excellent as a direct sequel to the standard calculus sequence and the book is carefully written with this in mind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Commentary&lt;/h1&gt;&#xA;&lt;p&gt;I've been thinking about your specific concerns and I cannot think of anything inherently unique about your situation or your background that would suggest a specific approach. My feeling is that typical differences among students in general would be first order effects, while issues facing someone coming from a computer science background would be second order effects that get drowned out (at the population level) by the first order effects. In fact, your background in number theory and combinatorics probably make you better prepared for real analysis than many (most?) students are after the standard calculus, ODE, and linear algebra courses taken in the first two years of college.&lt;/p&gt;&#xA;&lt;p&gt;What I suggest is that you visit a university library, bringing a pen and paper to write on, and spend two or three hours browsing the shelves. Real analysis texts will be mostly located in the vicinity of QA 300 to QA 330 Library of Congress classification. Read the prefaces of the texts on the shelves, and perhaps even photocopy the prefaces along with the tables of contents. Jot down titles of books that look interesting, either for primary study or for secondary reading. Make a note of how frequently the book was checked out or how worn out from reader use the book seems. Then do some online searching to see what others say about the various books. Keep in mind that a book no one talks about could still be very good. Most people are only aware of the books they used and perhaps a few other books, and most people commenting online probably have little knowledge of anything that hasn't been widely used in the last 30 years. For example, I think John F. Randolph's &lt;strong&gt;Basic Real And Abstract Analysis&lt;/strong&gt; is one of the best beginning graduate level real analysis texts I know of, but because it appeared in 1968 and because it is an awkward fit for most mathematics programs (too advanced for most undergraduate courses, too leisurely for most graduate courses; in this respect it is a bit like Neal L. Carothers' text &lt;strong&gt;Real Analysis&lt;/strong&gt;), there is virtually no discussion of Randolph's book anywhere online.&lt;/p&gt;&#xA;&lt;p&gt;A good time to visit such a library is a week or two after final exams, because a lot of books that had been checked out will be returned and the library staff will have had time to reshelve the books.&lt;/p&gt;&#xA;&lt;p&gt;Early this morning I went through a lot of papers I have related to the teaching of real analysis and made a list of those that could be of use to you if you're especially interested in discussions about students' difficulties in real analysis and people's thoughts on how to minimize these difficulties. I am not including papers that would be mainly of interest to those whose primary focus is on mathematics education rather than on mathematics, because despite this being a mathematics education group, your interest is in mathematics. [The difference for me is whether the paper appeared to be primarily about mathematical ideas or to be primarily about psychological and sociological aspects of learning mathematical ideas.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;ADDITIONAL BOOKS THAT MAY BE WORTH LOOKING AT&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Robert Lee Brabenec, &lt;a href=&quot;http://www.maa.org/publications/maa-reviews/resources-for-the-study-of-real-analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Resources for the Study of Real Analysis&lt;/strong&gt;&lt;/a&gt;, Classroom Resource Materials, Mathematical Association of America, 2004, xii + 231 pages.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Andrew Michael Bruckner, Judith Brostoff Bruckner, and Brian Sheriff Thomson, &lt;a href=&quot;http://classicalrealanalysis.info/com/FREE-PDF-DOWNLOADS.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Elementary Real Analysis&lt;/strong&gt;&lt;/a&gt;, Prentice-Hall, 2001, xvi + 677 + 58 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The final 58 pages consist of appendixes: (A) &lt;em&gt;Background&lt;/em&gt; (B) &lt;em&gt;Hints for Selected Exercises&lt;/em&gt; (C) &lt;em&gt;Subject Index&lt;/em&gt;. Like Bressoud's book above, this book does not suffer from the problem pointed out by Bressoud in his review of another book below. Also, it's now freely available on the internet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Kenneth Ownsworth May (editor), &lt;strong&gt;Lectures on Calculus&lt;/strong&gt;, Holden-Day, 1967, vii + 180 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Contains some very interesting selections, written by 9 different people. I encourage looking at this if/when you visit a university library to browse real analysis books. Some (but not all) of the chapters will probably be too advanced for a beginning real analysis student.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Joanne Erdman Snow and Kirk Edward Weller, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0883857340&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Exploratory Examples for Real Analysis&lt;/strong&gt;&lt;/a&gt;, Classroom Resource Materials, Mathematical Association of America, 2003, xv + 141 pages.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[5]&lt;/strong&gt; Robert Michael Young, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0883853175&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Excursions in Calculus. An Interplay of the Continuous and the Discrete&lt;/strong&gt;&lt;/a&gt;, Dolciani Mathematical Expositions #13, Mathematical Association of America, 1992, xiv + 417 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This book is an extremely well-cited encyclopedic exposition with 463 references, one of which is Young's book itself. In addition, on pp. 397--408 a reference citation is given for each of the book's 417 problems (or the problem is stated as currently unsolved).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;MISCELLANEOUS ITEMS RELATED TO THE TEACHING OF REAL ANALYSIS&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[6]&lt;/strong&gt; &lt;a href=&quot;http://web.archive.org/web/20010708093131/http://www.bham.ac.uk/ctimath/talum/austin/analysis/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching Mathematical Analysis to Students&lt;/a&gt;, Teaching and Learning Undergraduate Mathematics (TALUM) archive.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Articles written by members of the TALUM content working group: Johnston Anderson, Keith Austin, Janet Jagger, Frank Jellet, David Tall.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[7]&lt;/strong&gt; David Marius Bressoud, &lt;em&gt;Review of Undergraduate Analysis: Second Edition by Serge A. Lang&lt;/em&gt;, &lt;strong&gt;Mathematical Intelligencer&lt;/strong&gt; 20 #1 (Winter 1998), 76-77.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;From Bressoud's review&lt;/strong&gt; &amp;quot;Throughout this book, all of the motivations and the intuitions and the connections have been wrung out of the mathematics.&amp;quot; The 2nd half of Bressoud's review discusses his experience with teaching a second semester undergraduate course in real analysis that made heavy (but not exclusive) use of Thomas Hawkins's book &lt;strong&gt;Lebesgue's Theory of Integration. Its Origins and Development&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[8]&lt;/strong&gt; Thomas Arthur Alan Broadbent and others, &lt;a href=&quot;http://www.jstor.org/stable/3608026&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The first encounter with a limit&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 19 #233 (May 1935), 109-123.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is a written report of an extended discussion on introducing limits to students that occurred at the January 1935 Annual Meeting of the [British] Mathematical Association. Most of the participants were teachers at various university preparatory schools, the exceptions being Broadbent (at University of Reading) and brief appearances by Godfrey Harold Hardy and George Neville Watson. See also the &lt;em&gt;letters to the editor&lt;/em&gt; in the same issue of this journal by F. C. Boon (on pp. 131-132) and N. M. Gibbins (on pp. 132-134), as well as Dockeray (1935) below.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[9]&lt;/strong&gt; N. R. C. Dockeray, &lt;a href=&quot;http://www.jstor.org/stable/3606321&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The teaching of mathematical analysis in schools&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 19 #236 (December 1935), 321-336.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is a written version of a paper given on 23 February 1935 at a meeting of the London Branch of the [British] Mathematical Association. Immediately following this article, on pp. 336-340, is a written report of the discussion at the meeting that followed the talk. Related to this is Broadbent (1935) above.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[10]&lt;/strong&gt; Marianne Freundlich, &lt;a href=&quot;http://www.ams.org/notices/199805/comm-smith.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Eventually&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Notices of the American Mathematical Society&lt;/strong&gt; 45 #5 (May 1998), 599-600.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Personal reflections about the author's struggles to understand epsilon-delta definitions and proofs in analysis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[11]&lt;/strong&gt; Gurcharan Singh Gill, &lt;a href=&quot;http://www.jstor.org/stable/2315581&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A survey of texts for a one year course in real analysis at the advanced undergraduate or beginning graduate level&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;American Mathematical Monthly&lt;/strong&gt; 75 #9 (November 1968), 1033-1035.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[12]&lt;/strong&gt; E. J. Moulton, &lt;a href=&quot;http://www.jstor.org/stable/2972040&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The content of a second course in calculus&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;American Mathematical Monthly&lt;/strong&gt; 25 #10 (December 1918), 429-434.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dated, but contains useful observations about the teaching of certain topics in real analysis. Note the follow-up response: David Clinton Gillespie, &lt;em&gt;Advanced calculus or differential equations&lt;/em&gt;, &lt;strong&gt;American Mathematical Monthly&lt;/strong&gt; 26 #5 (May 1919), 189-190.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[13]&lt;/strong&gt; Edgar G. Phillips, &lt;a href=&quot;http://www.jstor.org/stable/3607075&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;On the teaching of analysis&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 14 #214 (December 1929), 571-573.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[14]&lt;/strong&gt; Kenneth Allen Ross, &lt;a href=&quot;http://www.ams.org/notices/201309/rnoti-p1179.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Genesis of elementary analysis courses&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Notices of the American Mathematical Society&lt;/strong&gt; 60 #9 (October 2013), 1179-1180.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[15]&lt;/strong&gt; Henry Scheffé, &lt;a href=&quot;http://www.jstor.org/stable/2304114&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;At what level of rigor should advanced calculus for undergraduates be taught?&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;American Mathematical Monthly&lt;/strong&gt; 47 #9 (November 1940), 635-640.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[16]&lt;/strong&gt; David Bernard Scott, &lt;em&gt;On teaching analysis&lt;/em&gt;, &lt;strong&gt;Mathematics Teaching&lt;/strong&gt; 42 (Spring 1968), 34-41.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[17]&lt;/strong&gt; Rolph Ludwig Edward Schwarzenberger and David Orme Tall, &lt;em&gt;Conflicts in the learning of real numbers and limits&lt;/em&gt;, &lt;strong&gt;Mathematics Teaching&lt;/strong&gt; 82 (March 1978), 44-49.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[18]&lt;/strong&gt; David Bernard Scott, &lt;em&gt;On teaching analysis&lt;/em&gt;, &lt;strong&gt;Mathematics Teaching&lt;/strong&gt; 42 (Spring 1968), 34-41.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2020-08-19T15:07:36.157" LastActivityDate="2020-08-19T15:07:36.157" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="1415" PostTypeId="2" ParentId="1413" CreationDate="2014-04-07T21:12:10.467" Score="7" Body="&lt;p&gt;There are different kinds of 'learning types'. Some people learn best by listening and talking, some people learn best by understanding the material with the help of pictures, and some learn best by writing things down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 'classical method' of presenting formulas, proofs and theorems by writing them down gives the students the time to write it down themselves, and to digest the content. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, using computer-based illustrations will help students who are not familiar working with formulas. For this particular course, well-designed plots can help to understand the idea behind certain definitions and theorems. For instance, you can show that, in order to show convergence of a sequence, it doesn't suffice to check the first few elements. Consider three series like $$a_n = \frac{1000}{n} - 1, \quad b_n= \sin\left(\frac{n}{900}\right), c_n= \frac{n}{1000},$$ where you can see that their behavior shouldn't be judged on the first 10 or 100 elements. And with a plot, you could just show the pictures (eg for 10, 100 and 1000 elements) and discuss them, and maybe show the actual definitions of the series after that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that for a maths for humanities course you can't be illustrative and accessible enough, so using computer graphics is a good idea. You should however be wary that using slides instead of writing stuff down may speed up your lectures towards non-understandability.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-04-07T21:12:10.467" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1416" PostTypeId="2" ParentId="1325" CreationDate="2014-04-07T23:40:12.330" Score="26" Body="&lt;p&gt;Multiple hypothesis testing is a common one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's say you run a study where you try to link some genetic marker to cancer rates. You look at perhaps 80 different genes and see if any of them have a correlation with occurrence of cancer. Lo and behold, one does! With p-value = 0.03! You conclude that there is a strong correlation (and seek to prove causation since you know the difference).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Seems pretty reasonable, but here's an analogous example: Let's say you want to find out if any of 80 people have the ability to predict the future. You ask each of them to predict 6 coin flips. And one predicts all 6 correctly! The p-value of this occurrence is 0.03 again! The stats seem to reject the null hypothesis that &quot;John Doe cannot predict the future.&quot; But obviously this individual just got lucky. Your null hypothesis should be &quot;someone can predict the future&quot; but your p-values don't give any data about this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a number of ways to adjust p-values because of this. The simplest (but not very powerful) of these is &lt;a href=&quot;http://en.wikipedia.org/wiki/Bonferroni_correction&quot;&gt;Bonferroni correction&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is perhaps one of the most common violations of statistics in published academic papers.&lt;/p&gt;&#xA;" OwnerUserId="753" LastActivityDate="2014-04-07T23:40:12.330" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1417" PostTypeId="1" AcceptedAnswerId="1421" CreationDate="2014-04-08T00:35:47.813" Score="6" ViewCount="212" Body="&lt;p&gt;I was a grad student at Brigham Young University, which has a very high marriage rate, and a large number of grad students (if not most) were parents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of different parents brought their newborn child to class for a semester (an undergraduate class once and a graduate class the other time).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people felt uncomfortable or distracted by the child. I thought the parents did a great job, and soothed the child whenever it was fussy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After a few semesters that it happened, teachers started putting 'No babies allowed in the classroom' on the syllabus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of these parents had only one semester left, and had no access to childcare for such a young infant.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How should this have been handled to everyone's satisfaction? If babies were not allowed, how and when should this have been communicated to the parents?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-04-08T02:00:53.693" Title="How to handle babies in the classroom" Tags="|classroom-management|" AnswerCount="1" CommentCount="6" ClosedDate="2014-04-10T19:16:09.113" ContentLicense="CC BY-SA 3.0" />
  <row Id="1420" PostTypeId="2" ParentId="1403" CreationDate="2014-04-08T01:48:54.440" Score="6" Body="&lt;p&gt;This question is tough to answer since I do not know the course you have in mind; I am imagining something that involves proof-writing, possibly naive set theory, and, more generally, the ability to write convincing arguments and criticize unconvincing ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regard to:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a representative class of examples I could offer to students who are unsure.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I might use the course listing to link to a site with a few sample problems from early on in the course, a bit more about the relevant background, and your contact information in case they want to discuss the choice with you one on one. If you want example content for a proof-based course, I might use:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove in two different ways: The sum of the first $n$ odd numbers is $n^2$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove in two different ways: The number of diagonals in a regular $n$-gon ($n &amp;gt; 2$) is $\frac{n(n-1)}{2}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove in two different ways: $N$ a finite set with $n&amp;gt;0$ elements; then its power set has $2^n$ elements.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you can understand all of the language above, then this is probably an appropriate course for you. If you can carry out most of the proofs above (at least one per question) then you will be fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, consider the following examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If $a$ and $b$ are even, then $a+b$ is even. [Proof: Since $a$ is even, we can write $a = 2n$ for some integer $n$; similarly, we can write $b = 2m$ for some integer $m$. Adding the two together, we find that  $a+b = 2n+2m = 2(n+m)$. Since $n+m$ is an integer, the sum is two times an integer, i.e., the sum is even. Therefore, $a+b$ is even as desired. Q.E.D.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Follow-up A: Prove that if $a$ and $b$ are odd, then $a+b$ is even. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Follow-up B: Prove that if $a$ is even and $b$ is odd, then $a+b$ is odd.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Follow-up C: Prove that if $a$ is odd and $b$ is even, then $a+b$ is odd. (Can you use Follow-up B?)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose $a$ and $b$ are rational. Prove that $a + b$ is rational. [Insert teacher proof.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Follow-up X: Suppose $a$ and $b$ are irrational. Must $a+b$ be irrational?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Follow-up Y: Suppose $a$ is rational and $b$ is irrational. Prove that $a + b$ is irrational.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Follow-up Z: Suppose $a$ is irrational and $b$ is rational. Prove that $a + b$ is irrational.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-04-08T01:48:54.440" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1421" PostTypeId="2" ParentId="1417" CreationDate="2014-04-08T02:00:53.693" Score="8" Body="&lt;p&gt;This strikes me as fundamentally a matter of opinion, but for what it is worth, here is mine:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A baby in a classroom is not the same as a baby on an airplane.  If a baby on an airplane cries, there is no escape and nothing one can do.  If a baby in a classroom cries, the parents can quietly slip out through a door into the hallway and try to soothe the baby there.  I think as a practical matter most parents would prefer not to have to bring a baby with them to class, so I would give them the benefit of the doubt and assume that they do not have better alternatives.  If so, it seems to me that the appropriate and humane thing to do would be as supportive and understanding as possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than try to ban babies from the classroom or write an explicit policy into the syllabus, it would probably be reasonable to have a private conversation with the students at the beginning of the semester.  Let them know that they are welcome to bring their children with them so long as it does not become a distraction to other students, and perhaps suggest that they sit on an aisle seat, not far from an exit, so that they can slip out discretely if and when it is necessary.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-04-08T02:00:53.693" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1422" PostTypeId="2" ParentId="506" CreationDate="2014-04-08T03:03:19.023" Score="3" Body="&lt;p&gt;Absolutely! Abandon your lesson plan. If its not working, it's not working.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've had this happen numerous times. I literally stop teaching and go back to the beginning of the lesson and start over using formative assessment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know my students understood some of it, I just need to figure out which part. Then, once I figure out where the misconception is, I'm able to ask questions to help students make connections and move on or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just the other day, I had to completely stop, think of another way to explain it, and start over. No one understood the first way, but 85% (formative assessment) understood the second way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well worth stopping, rethinking, and starting over.&lt;/p&gt;&#xA;" OwnerUserId="760" LastActivityDate="2014-04-08T03:03:19.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1423" PostTypeId="2" ParentId="1413" CreationDate="2014-04-08T03:37:12.573" Score="5" Body="&lt;p&gt;I think the main thing to remember, regardless of which method you choose, is that &lt;em&gt;you&lt;/em&gt; are the most important part of the presentation. You could spend hours/days/weeks/months preparing the best slides ever, but if the delivery is no good, then it will all be for nothing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Slides can certainly be an important accessory, but they should only ever be there to provide support for &lt;em&gt;you&lt;/em&gt;, and for what you are saying; it should never be the other way round.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ultimate fear in this context (as the presenter) is to cause &lt;a href=&quot;http://en.wikipedia.org/wiki/Microsoft_PowerPoint#.22Death_by_PowerPoint.22&quot; rel=&quot;noreferrer&quot;&gt;death by powerpoint&lt;/a&gt;, so it is incredibly important to 'mix things up'; slides are ok for some content now and again, but you have to keep the audience engaged. You could, for example, aim to transition between slides, board work, and (if room space, time, institutional ethos allow) group work. &lt;/p&gt;&#xA;" OwnerUserId="761" LastActivityDate="2014-04-08T03:37:12.573" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1424" PostTypeId="2" ParentId="1348" CreationDate="2014-04-08T04:06:06.127" Score="9" Body="&lt;p&gt;&quot;Fraction Bars&quot; may be helpful to you. It's a virtual space for exploring bar-shaped fraction models. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.kaputcenter.umassd.edu/products/software/fractionbars/&quot;&gt;Fraction Bars Page&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On that page you'll find a link to launch the software as well as a software guide which includes some appropriate activities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A version of this software has been used in classes to prepare 7th grade teachers in Georgia to (among other things) teach fraction and ratio concepts using a heavily representation and inquiry-based approach. You can find some of the &lt;a href=&quot;http://intermath.coe.uga.edu/topics/nmcncept/fractns/rec.htm&quot;&gt;recommended investigations here on the InterMath website&lt;/a&gt;. Many of the activities are standalone. You may find the software helpful, though, for some of the investigations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The software is free to use, with no strings and also no user support. I don't use the software in my own research (I'm writing my dissertation on a different topic), but I would still be interested to hear whether you found it of use (I am one of the software developers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cheers and wellwishes!&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-04-08T04:06:06.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1425" PostTypeId="2" ParentId="1380" CreationDate="2014-04-08T04:29:04.773" Score="2" Body="&lt;p&gt;You didn't specify a specific level, but I think your question has merit in any case. I will take it at the K-8 level. First I want to address your specific requirement:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By &quot;good&quot;, I mean thought-provoking, inspiring problems with solutions that are extensible to other domains. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I will interpret &quot;inspiring&quot; to mean that students will have a motivation to engage in the mathematics of the problem. For &quot;thought-provoking&quot; I will assume that you mean that the problems have a high likelihood to require students to engage in productive mathematical reasoning. These are essential characteristics of good investigations in a curriculum. That is to say, a good curriculum should contain activities and investigations that satisfy these.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once asked the a well-known high-quality curriculum developer how she knew her curriculum problems fit the requirements of &quot;&lt;a href=&quot;http://dme.colorado.edu/fius/rme_tour.pdf&quot; rel=&quot;nofollow&quot;&gt;realistic mathematics education&lt;/a&gt;&quot; (which was the approach that inspired her curriculum. She replied that they had to try each activity with real students many times in the research and development process. While the first drafts may have been based on theory, in reality the finished curriculum was heavily tested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, find and collect problems developed by good curriculum designers. If necessary, build your own library of such problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One final note: you suggested you wanted problems whose solutions were extensible to other domains. I suggest you be careful with this sort of assumption in looking for problems. What they come to understand in the process of problem posing and solving may help them form connections between contexts. However, you may find it difficult to support the notion of &quot;domain transferrable solutions&quot; in good math education literature. Focus more on what sort of mathematical reasoning the students will have the opportunity and resources to engage in. &lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-04-08T04:29:04.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1426" PostTypeId="2" ParentId="1252" CreationDate="2014-04-08T05:59:52.770" Score="5" Body="&lt;p&gt;This would appear to be an essential problem with understanding the place value system. If the student hasn't already understood place value, showing and explaining is unlikely to be effective. First off, you may be unaware of what the student's conception of the value of these numbers is. An activity might help you get to the heart of that, if not help the student see through the misconception.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A number line representation, if the student already understands it, can be used to scaffold this place value understanding. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But first, it might help to have a representation that shows the symmetry of naming of places around the one's place. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/O0Iru.gif&quot; alt=&quot;diagram of place names&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask the student to consider, what does a zero mean in any of those places? The answer may be helpful to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, draw a number line like so:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/6pVhv.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have the student place numbers on this line to illustrate relative positions. Make sure that 5.0 and 5.00 are among the numbers, but try also numbers with values in the tenths and hundredths place. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What does it mean if numbers (5.0, 5, and 5.00) are in the same spot on this number line?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, a student could place these numbers at juuuuuust before 5.1, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then it would appear you need to have the student focus more heavily on each place of place value in the questions he answers. He may not see 105.3 as One hundred(s) AND 5 ones AND three tenths. Tell him that you want him to talk a while using this long name of this type of number. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is 5.0? What is 5.00?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What does it mean to have no tenths? No hundredths? What is five (plus/and) no tenths?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cheers.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-04-08T05:59:52.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1427" PostTypeId="1" AcceptedAnswerId="1777" CreationDate="2014-04-08T07:39:02.987" Score="10" ViewCount="293" Body="&lt;p&gt;What's the appropriate &lt;a href=&quot;http://en.wikipedia.org/wiki/Educational_stages&quot; rel=&quot;nofollow noreferrer&quot;&gt;education level&lt;/a&gt; for the following concise but non-trivial geometry problem?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Points $A$, $B$, $C$ are &lt;a href=&quot;http://en.wikipedia.org/wiki/Collinear&quot; rel=&quot;nofollow noreferrer&quot;&gt;collinear&lt;/a&gt;; $\|AB\|=\|BD\|=\|CD\|=1$; $\|AC\|=\|AD\|$.&lt;br&gt;&#xA;  What is the set of possible $\|AC\|$ ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;To check one's answer, hover mouse over the following very partial spoiler.&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; The mean of the elements of the set of solutions is $\approx 1.08$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Link to &lt;a href=&quot;https://i.stack.imgur.com/d0fbK.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;near-complete spoiler illustration&lt;/a&gt;; ctrl-click for new window or tab.&lt;/p&gt;&#xA;" OwnerUserId="763" LastEditorUserId="262" LastEditDate="2014-04-17T23:45:10.413" LastActivityDate="2014-04-17T23:45:10.413" Title="Appropriate education level for this geometry problem" Tags="|secondary-education|algebra|geometry|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1428" PostTypeId="1" AcceptedAnswerId="1431" CreationDate="2014-04-08T08:20:56.333" Score="8" ViewCount="836" Body="&lt;p&gt;I think, it is a very common problem that some students have huge problems with definitions when there appears a quantification. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Of course the sequence is bounded, because every part of the sequence is a real number and this real number is bounded by itself.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Look, here is a finite covering for this set, hence it is compact.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;A lot more examples related to $\delta$-$\epsilon$ criterion, etc.&lt;/li&gt;&#xA;&lt;li&gt;Also, many students don't quantify new introduced variables which leads then to bigger problems during their proofs.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What goes wrong if students interchange (or leave open) the quantification in the definitions? How can this be fixed or at least improved?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-04-08T09:17:39.307" Title="What goes wrong when students interchange &quot;there exists&quot; and &quot;for all&quot; randomly? How to fix this?" Tags="|undergraduate-education|logic|students-mistakes|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1429" PostTypeId="5" CreationDate="2014-04-08T08:23:04.153" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-08T08:23:04.153" LastActivityDate="2014-04-08T08:23:04.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1430" PostTypeId="4" CreationDate="2014-04-08T08:23:04.153" Score="0" Body="For questions investigating students mistakes, clarifying their origin and asking for advice to fix or improve the mistakes." OwnerUserId="114" LastEditorUserId="39" LastEditDate="2014-09-22T20:43:48.317" LastActivityDate="2014-09-22T20:43:48.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1431" PostTypeId="2" ParentId="1428" CreationDate="2014-04-08T09:17:39.307" Score="11" Body="&lt;p&gt;One way to increase awareness of this issue is to explore what happens if the quantors are changed from exists to for all, and vice versa.&lt;/p&gt;&#xA;&lt;p&gt;On math.SE, there was a &lt;a href=&quot;https://math.stackexchange.com/questions/683590/a-game-with-delta-epsilon-and-uniform-continuity&quot;&gt;question which dealt precisely with this&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The following exercise is supposedly from Makarov's &lt;em&gt;Selected Problems in Real Analysis&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Describe the set of functions &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb R \rightarrow \mathbb R$&lt;/span&gt; having the following properties (&lt;span class=&quot;math-container&quot;&gt;$\epsilon, \delta,x_1,x_2 \in \mathbb R$&lt;/span&gt;) :&lt;/p&gt;&#xA;&lt;p&gt;a) &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon ,\qquad\qquad \exists \delta&amp;gt;0 ,\qquad |x_1-x_2| &amp;lt; \delta \Rightarrow |f(x_1)-f(x_2)|&amp;lt;\epsilon$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;b) &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt;0 ,\qquad \exists \delta ,\qquad \qquad |x_1-x_2| &amp;lt; \delta \Rightarrow |f(x_1)-f(x_2)|&amp;lt;\epsilon$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;c) &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt;0 ,\qquad \exists \delta&amp;gt;0 ,\qquad (x_1-x_2) &amp;lt; \delta \Rightarrow |f(x_1)-f(x_2)|&amp;lt;\epsilon$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;d) &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt;0 ,\qquad \forall \delta&amp;gt;0 ,\qquad |x_1-x_2| &amp;lt; \delta \Rightarrow |f(x_1)-f(x_2)|&amp;lt;\epsilon$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;e) &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt;0 ,\qquad \exists \delta&amp;gt;0 ,\qquad |x_1-x_2| &amp;lt; \delta \Rightarrow |f(x_1)-f(x_2)|&amp;gt;\epsilon$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;f) &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt;0 ,\qquad \exists \delta&amp;gt;0 ,\qquad |x_1-x_2| &amp;lt; \epsilon \Rightarrow |f(x_1)-f(x_2)|&amp;lt;\delta$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;g) &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt;0 ,\qquad \exists \delta&amp;gt;0 ,\qquad |f(x_1)-f(x_2)| &amp;gt; \epsilon \Rightarrow |x_1-x_2|&amp;gt; \delta$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;h)  &lt;span class=&quot;math-container&quot;&gt;$\exists \epsilon &amp;gt;0 ,\qquad \forall \delta&amp;gt;0 ,\qquad |x_1-x_2| &amp;lt; \delta \Rightarrow |f(x_1)-f(x_2)|&amp;lt;\epsilon$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;i)  &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt;0 ,\qquad \exists \delta&amp;gt;0 ,\qquad x_1-x_2 &amp;lt; \delta \Rightarrow f(x_1)-f(x_2)&amp;lt;\epsilon$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="108" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-08T09:17:39.307" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1432" PostTypeId="5" CreationDate="2014-04-08T11:00:28.997" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-08T11:00:28.997" LastActivityDate="2014-04-08T11:00:28.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1433" PostTypeId="4" CreationDate="2014-04-08T11:00:28.997" Score="0" Body="For questions related to the use of mathematics or mathematical arguments in daily life." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-08T12:35:29.723" LastActivityDate="2014-04-08T12:35:29.723" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1434" PostTypeId="5" CreationDate="2014-04-08T11:26:34.733" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-08T11:26:34.733" LastActivityDate="2014-04-08T11:26:34.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1435" PostTypeId="4" CreationDate="2014-04-08T11:26:34.733" Score="0" Body="For questions related to organization of a course and organization of own material." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-08T12:36:16.783" LastActivityDate="2014-04-08T12:36:16.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1436" PostTypeId="5" CreationDate="2014-04-08T11:27:47.030" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-08T11:27:47.030" LastActivityDate="2014-04-08T11:27:47.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1437" PostTypeId="4" CreationDate="2014-04-08T11:27:47.030" Score="0" Body="For questions about functional analysis in infinite-dimensional spaces including concepts of dual spaces, Hahn-Banach, etc." OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-08T12:36:23.893" LastActivityDate="2014-04-08T12:36:23.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1438" PostTypeId="2" ParentId="380" CreationDate="2014-04-08T13:37:26.117" Score="2" Body="&lt;p&gt;I always thought compacting notation helps in such cases (it helps me not to make mistakes in complicated cases). To consider the &quot;ridiculous example&quot; of @Brian Rushton (to whose answer I believe mine is just an extension):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y_1=\ln(\sin(\cos(e^{x^2})))$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would make the following &lt;em&gt;definitions&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y_2 \equiv \sin(\cos(e^{x^2})),\qquad y_3 \equiv \cos(e^{x^2}),\qquad y_4\equiv e^{x^2},\qquad y_5 \equiv x^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So now I would be looking at&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y_1=\ln(y_2),\qquad y_2 = \sin(y_3),\qquad y_3=\cos(y_4),\qquad y_4= e^{y_5}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and so I could write the &quot;direct&quot; derivatives&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac {\partial y_1}{\partial y_2} = \frac 1{y_2}\qquad \frac {\partial y_2}{\partial y_3} = \cos(y_3),\qquad \frac {\partial y_3}{\partial y_4} = -\sin(y_4),\qquad \frac {\partial y_4}{\partial y_5} = e^{y_5},\qquad \frac {\partial y_5}{\partial x} =2x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Visually this helps connect with the abstract chain rule&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac {\partial y_1}{\partial x} = \frac {\partial y_1}{\partial y_2}\cdot \frac {\partial y_2}{\partial y_3}\cdot\frac {\partial y_3}{\partial y_4}\cdot\frac {\partial y_4}{\partial y_5}\cdot\frac {\partial y_5}{\partial x}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and then we insert the simple derivatives we calculated&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac {\partial y_1}{\partial x} = \frac 1{y_2}\cdot \cos(y_3)\cdot [-\sin(y_4)]\cdot e^{y_5}\cdot 2x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and then substitute for the $y$'s from the definitions we have made in the beginning.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2014-04-08T13:37:26.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1439" PostTypeId="1" CreationDate="2014-04-08T14:21:36.627" Score="5" ViewCount="397" Body="&lt;p&gt;I need help finding a way to teach the standard of reproducing a scale drawing at a different scale for my 7th grade math class.&lt;/p&gt;&#xA;" OwnerUserId="780" LastEditorUserId="114" LastEditDate="2014-04-08T14:33:05.063" LastActivityDate="2014-04-10T02:45:13.100" Title="What did you use to teach scale drawings?" Tags="|secondary-education|technology-in-education|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1440" PostTypeId="1" AcceptedAnswerId="1473" CreationDate="2014-04-08T15:18:08.593" Score="12" ViewCount="1093" Body="&lt;p&gt;I'm getting down to details here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;When a quiz or exam is the last thing in the class period, students can leave when they are done. If not, there is necessarily a time when some students are done with the quiz but others are not.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have tried multiple different approaches to dealing with this and am dissatisfied with all of them so far:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Have students turn in the quiz at the front so they can do other things.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is ineffective because the &quot;last person to turn in the quiz&quot; has to do so by walking up to the front of the room after everyone else, which is demoralizing. Also the &quot;first person to turn in the quiz&quot; is equally demoralizing to the class. This &quot;first person&quot; also hopes to impress the instructor by turning in the quiz first instead of checking their answers.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Have students keep the quiz and all pass it up at the end.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is ineffective because students have an assessment in front of them, so they are less willing to take a break or read a book even if they are explicitly told it is okay -- they are much more likely to put their head on the desk to rest or stare forward with hostility, poisoning the atmosphere of the class.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Put a very hard extra credit problem on the quiz that is worth one point.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is ineffective because it creates unwanted time pressure for B students and takes the focus away from the important concepts being tested.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Put an unrelated riddle on the board for bored students to solve, to keep their brains turned on.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is ineffective because I don't have enough interesting riddles to use, and I'm not sure this is a good use of my prep time. I basically have one interesting riddle that I use on a particularly sleepy quiz day.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;How do you give a start-of-class quiz without totally losing the class?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-04-09T21:36:47.060" Title="How to keep students engaged during a quiz that is not at the end of class?" Tags="|classroom-management|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1441" PostTypeId="2" ParentId="380" CreationDate="2014-04-08T15:54:02.190" Score="10" Body="&lt;p&gt;I no longer teach, but I used to delay the general chain rule for a while by making use of special cases. For example, after introducing the rule &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}x^{n}=nx^{n-1}$&lt;/span&gt; (the &lt;em&gt;power rule&lt;/em&gt;) and giving some examples with polynomials (and what I used to call &amp;quot;polynomial-like functions&amp;quot;, which includes things like &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{\sqrt{x}} – 2x\sqrt{x} + 5x^{1/3} + x^{3}),$&lt;/span&gt; I would introduce the &lt;em&gt;general power rule&lt;/em&gt;, which is &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}u^{n}=nu^{n-1}\frac{du}{dx}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; is some function of &lt;span class=&quot;math-container&quot;&gt;$x.$&lt;/span&gt; This was usually abbreviated as &lt;span class=&quot;math-container&quot;&gt;$(u^n)'=nu^{n-1}u',$&lt;/span&gt; and I would indicate how the power rule is the special case when &lt;span class=&quot;math-container&quot;&gt;$u(x)=x.$&lt;/span&gt;  Each of the functions &lt;span class=&quot;math-container&quot;&gt;$e^{x},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\ln{x},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\sin{x},$&lt;/span&gt; etc. come with their own general version --- the &lt;em&gt;general&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$e^{x}$&lt;/span&gt; &lt;em&gt;rule&lt;/em&gt;, the &lt;em&gt;general logarithm rule&lt;/em&gt;, the &lt;em&gt;general sine rule&lt;/em&gt;, etc.&lt;/p&gt;&#xA;&lt;p&gt;On tests (especially on gateway type tests) and on 5-minute quizzes I would include what I called &lt;em&gt;abstract types&lt;/em&gt;, such as find the derivative with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$\sin{f(x)}$&lt;/span&gt; or find the derivative with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$\;x^{2} \ln{(f(x) + x^{3})}.$&lt;/span&gt; Of course, these types were done in class and given in homework, the latter being supplementary problems assigned (this code for that dirty word, &lt;em&gt;worksheets&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;It was only after students were (for the most part) fairly comfortable with these rules, both the specific versions and the general versions, and they were comfortable with evaluating derivatives of &amp;quot;abstract types&amp;quot;, that I would bring up the even more abstract idea of &lt;strong&gt;having a general function on the outside&lt;/strong&gt;. Thus, the derivative with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; (no surprise), and the corresponding general version is &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}f(u) = f'(u)u'.$&lt;/span&gt; But before giving this, I would make a column on the board to show the pattern of all our general rules. The column would have 5 or 6 functions like &lt;span class=&quot;math-container&quot;&gt;$e^{u},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\ln{u},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\sin{u},$&lt;/span&gt; etc. lined up so that in one vertical line we'd see all the &lt;span class=&quot;math-container&quot;&gt;$f'(u)\,$&lt;/span&gt;'s appearing and just to the right of these we'd see in another vertical line all the &lt;span class=&quot;math-container&quot;&gt;$u'\,$&lt;/span&gt;'s appearing.&lt;/p&gt;&#xA;&lt;p&gt;At some point I would give one or more of the standard &amp;quot;analogy justifications&amp;quot; (e.g. bicycle gears), as well as various alternate notations such as the &amp;quot;little circle&amp;quot; composition notation and the Leibniz version where fraction cancellation seems to occur. However, all this came well after when I felt most students had fully internalized the formal technique. In this way the focus could be on the forest, without the tree leaves getting in the way.&lt;/p&gt;&#xA;&lt;p&gt;What follows is a slightly edited version of my &lt;a href=&quot;https://web.archive.org/web/20170106213248/http://mathforum.org/kb/message.jspa?messageID=5353194&quot; rel=&quot;nofollow noreferrer&quot;&gt;4 November 2006&lt;/a&gt; post the &lt;em&gt;Chain Rule&lt;/em&gt; thread in the Math Forum discussion group &lt;em&gt;AP Calculus&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here's what I used to do regarding the &lt;em&gt;mechanics&lt;/em&gt; of carrying out the chain rule.&lt;/p&gt;&#xA;&lt;p&gt;If the function was, for example, &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sin{(3x^{2} + 2)},$&lt;/span&gt; then I'd write &lt;span class=&quot;math-container&quot;&gt;$f'(x) = \cos{(3x^2 + 2)} \cdot 6x,$&lt;/span&gt; with a semi-circular arrow drawn over the top. The arrow's tail began just above (and roughly centered on) the &lt;span class=&quot;math-container&quot;&gt;$3x^2 + 2$&lt;/span&gt; expression inside the cosine. The arrow curved in a concave down fashion (like the graph of &lt;span class=&quot;math-container&quot;&gt;$y=-x^2)$&lt;/span&gt; to where the &lt;span class=&quot;math-container&quot;&gt;$6x$&lt;/span&gt; was. I drew the arrow's head so that it pointed right at the &lt;span class=&quot;math-container&quot;&gt;$6x.$&lt;/span&gt; When I was actually doing this, I drew the arrow after writing &lt;span class=&quot;math-container&quot;&gt;$\cos{(3x^2 + 2)}$&lt;/span&gt; and before writing the dot and the &lt;span class=&quot;math-container&quot;&gt;$6x.$&lt;/span&gt; I drew the arrow to &amp;quot;remind us&amp;quot; that there was more to do --- there's the derivative of the inside that still needed to be taken care of.&lt;/p&gt;&#xA;&lt;p&gt;Every &amp;quot;inside&amp;quot; got its own arrow.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d}{dx}\sin{(3x^2 + e^{2x})} \;\; = \;\; \cos{(3x^2 + e^{2x})} \ldots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;got &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; arrows -- one arrow was for the sine (tail started at the &lt;span class=&quot;math-container&quot;&gt;$3x^2 + e^{2x}$&lt;/span&gt; expression inside the cosine) and another arrow was for the exponential (tail started at the &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; that was inside the exponential which showed up &lt;em&gt;after&lt;/em&gt; I had written down the term that the first arrow pointed to).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d}{dx}\exp\,\{\,\sin\,[\cos\,(x^2)\,]\,\} \;\; = \;\; \exp\,\{\,\sin\,[\cos\,(x^2)\,]\,\} \ldots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;got 3 arrows -- one for the &lt;span class=&quot;math-container&quot;&gt;$\exp,$&lt;/span&gt; one for the &lt;span class=&quot;math-container&quot;&gt;$\sin,$&lt;/span&gt; and one for the &lt;span class=&quot;math-container&quot;&gt;$\cos.$&lt;/span&gt; In this case, the arrows humped horizontally across like the graph of &lt;span class=&quot;math-container&quot;&gt;$y = |\sin{x}|$&lt;/span&gt; does on the closed interval &lt;span class=&quot;math-container&quot;&gt;$[0, \, 3\pi].$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Almost all of my students wrote their answers this way on tests, which was fine with me. I didn't require them to rewrite the derivative leaving out the arrows or require them to erase the arrows.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, when I first introduced the product rule (or when a student later had difficulty), I found it helpful to insert a &lt;em&gt;parentheses template&lt;/em&gt; before going further. Thus, when showing the use of the product rule for&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x) \;\; = \;\; (3x^2 - 2x)^{3} \cdot \cos{3x},$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'd first write&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f'(x) \;\; = \;\; (\;\;\;\;\;)'*(\;\;\;\;\;) \;\; + \;\; (\;\;\;\;\;)*(\;\;\;\;\;)'.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The idea was to get the students to see the big picture before we got into the details. Some teachers do this using rectangular boxes.&lt;/p&gt;&#xA;&lt;p&gt;Then I'd write the appropriate factor inside each pair of parentheses, not rewriting or doing anything else -- just copy the factor &lt;strong&gt;exactly&lt;/strong&gt; as it was written.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ (\,(3x^2 - 2x)^3\,)' * (\,\cos{3x}\,) \;\; + \;\; (\,(3x^2 - 2x)^3\,) * (\,\cos{3x}\,)'$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(\,\text{one derivative goes here}\,) * (\,\cos{3x}\,) \;\; + \;\; (\,(3x^2 - 2x)^{3}\,) * (\, \text{other derivative goes here} \,)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then I told them: &amp;quot;Now we're &lt;span class=&quot;math-container&quot;&gt;$3/4$&lt;/span&gt; done!&amp;quot; Most of the time we really weren't &lt;span class=&quot;math-container&quot;&gt;$3/4$&lt;/span&gt; done, but this often generated laughs (especially when it was clear there was quite a bit involved in taking care of the two indicated derivatives), and I think it helped to put the more math-phobic students at ease. The &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$3/4$&lt;/span&gt;&amp;quot; comes from having half of the problem taken care of with the setting up and filling in of the product rule tinplate, and having half of the parentheses parts already done (i.e. &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; of the &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; parentheses parts were freebies). If necessary, I'd go to another part of the board in order to work out each of the indicated derivatives and put the results where I wrote &amp;quot;one derivative goes here&amp;quot; and &amp;quot;other derivative goes here&amp;quot; in the last displayed expression above. I usually only did this (calculate the derivatives of the factors on another part of the board) the first day we covered the product rule, or when someone was having a lot of trouble with it later on, or when the function itself seemed to be scary looking to the students.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="6053" LastEditDate="2022-09-24T19:51:22.713" LastActivityDate="2022-09-24T19:51:22.713" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="1442" PostTypeId="2" ParentId="380" CreationDate="2014-04-08T16:36:43.460" Score="7" Body="&lt;p&gt;When helping friends with this who don't understand the chain rule, I actually show them chain rule from the very beginning and make them apply every derivative rule to things that don't necessarily need it such as $f(x) = x$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f(x) = x^1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f'(x) = (1)x^{1-1}\cdot(1)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= x^0 \cdot (1)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= 1 \cdot 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We then move on to $f(x) = x + 1$, and then to an $f(x) = 3(x+1)$, and then to a $f(x) = 3 (x^2 + 1)$, slowly getting more complex. They'll hate you for it at first, but then will start to see that you should show every step every time, no matter what.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By showing them that it works for even the easiest problems, I've found that something seems to click in their heads letting them know that they should use the chain rule for everything. I think the hardest part about chain rule is that people don't understand why they use it sometimes and not others, so showing them that they should use it all the time makes it become habit as problems become more complex. So if you start out saying &quot;This is the chain rule. Use it on every derivative problem, even if you don't think you need to,&quot; it should become habitual rather than a daily struggle.&lt;/p&gt;&#xA;" OwnerUserId="782" LastEditorUserId="143" LastEditDate="2014-04-09T18:40:14.487" LastActivityDate="2014-04-09T18:40:14.487" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1443" PostTypeId="2" ParentId="1440" CreationDate="2014-04-08T18:28:43.980" Score="2" Body="&lt;p&gt;I have a suggestion which is related to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/805/should-i-design-my-exams-to-have-time-pressure-or-not/822#822&quot;&gt;my answer to this question&lt;/a&gt; regarding time pressure. I typically design 5-10 minute quizzes in such a way so that there's a bit of a time crunch. Of course, this might not be ideal in some cases, depending on how you're using the quizzes. But personally, I tend to put relatively easy questions on the quizzes so that, combined with the time crunch, they function as &quot;do you have this down cold?&quot; wake-up calls (and are worth a correspondingly small fraction of their final mark).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This might help with your problem since it will reduce the number of people who finish early as well as the average extra time they have to twiddle their thumbs. Judging by a visual survey, I'd say typically only 1-3 students finish with more than a minute to spare. I've never actually experienced students &quot;staring forward with hostility&quot;—perhaps because they don't have to wait too long?&lt;/p&gt;&#xA;" OwnerUserId="327" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-08T18:28:43.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1444" PostTypeId="2" ParentId="1325" CreationDate="2014-04-08T20:12:05.750" Score="11" Body="&lt;p&gt;If you succumb to the temptation of ejecting, say, a 5-sigma outlier from a n=10 sample taken from what you believe to be a normally distributed source, then you are discarding 50% of the sample's &lt;a href=&quot;http://en.wikipedia.org/wiki/Information_content&quot;&gt;information content&lt;/a&gt;. Not so harmless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: I'll give it a go:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Low-probability events carry more information (a.k.a. surprisal) than high-probability events. E.g. &quot;The building is on fire.&quot; carries more information than &quot;The building is not on fire.&quot; This &quot;information&quot; is quantified by information theory, and appreciated by inductive inference. Solomonoff inductive inference is rooted in Bayesian statistics and is the optimal procedure for drawing strong-as-possible conclusions from available data, which is what frequentist statistical inference (as I understand its aims, vaguely) also tries to do, albeit less efficiently. Since information is a commodity valued by (Solomonoff) inductive inference, and since frequentist statistical inference seems to be a largely parallel pursuit of the same aims under a different theoretical framework, we would expect information to be a valuable commodity to statistical inference in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get back to the point: Outliers are unlikely events (according to anyone tempted to dismiss them as fluke events, anyway) and therefore carry the most information within the sample (to that person) and therefore should be seen by that person as the &lt;em&gt;most&lt;/em&gt; valuable members of a sample. The desire to &quot;clean the data&quot; and get rid of them is diametrically misguided (unless an explanation has been provided for them, in which case they would no longer carry much information anyway since under that explanation they are no longer such low-probability events).&lt;/p&gt;&#xA;" OwnerUserId="783" LastEditorUserId="783" LastEditDate="2014-04-08T23:39:57.063" LastActivityDate="2014-04-08T23:39:57.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1445" PostTypeId="2" ParentId="1440" CreationDate="2014-04-08T22:00:10.077" Score="5" Body="&lt;p&gt;I have found that &lt;em&gt;group&lt;/em&gt; quizzes work remarkably well in all of the (Community College-see below) levels of Mathematics that I have taught. I typically give them at the end of class, but I imagine it would work nearly as well at the beginning. The rules are always the same: no notes, no books, but you can work together, walk around the room, and work at the boards.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I stress to the students that one of the outcomes of this quiz is collaboration, and that I don't want them to leave anyone on their own; I try to encourage folks to touch base with people, and make sure that everyone understands what's going on as much as possible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has, generally speaking, helped to form a stronger community in the classroom- when there are points for a grade at stake, people generally seem to work harder. Furthermore, when I set non-graded activities, the students are more likely to work together because of the quiz experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I initially worried that, because the students could work together, they would all get perfect scores. This has never been the case--I still receive a wide range of scores and the quality is still varied; the main benefit is that it seems to ease some of the tension and anxiety that students feel when faced with a quiz.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;Community College level mathematics covers a fairly wide range of Mathematics, from developmental (pre-GCSE/high school) to first &amp;amp; second year University level. Community College students are typically adults, and vary widely in demographics.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="761" LastActivityDate="2014-04-08T22:00:10.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1446" PostTypeId="2" ParentId="599" CreationDate="2014-04-08T23:09:58.390" Score="4" Body="&lt;p&gt;I like to structure all exams with two equally-weighted components:  a traditional in-class portion and a take-home portion.  The in-class portion generally consists of routine exercises, not too many, and are meant to demonstrate basic proficiency; the take-home portion will typically have five or six multi-part, more challenging questions, from which students are asked to choose a subset (typically $n-2$).  I usually give a week on the take-home problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the big risk is that students will cheat on the take-home portion.  I try to forestall that up front by being very explicit about what is and is not okay:  looking up information and ideas in published sources (whether in print or online) is okay, asking someone for help is not (whether in person or online).  I have them sign an honor statement affirming that they have understood and complied with those expectations, and I make a big speech about how I am willing to take the risk because I believe they are deserving of my trust, etc.   So far it has worked out okay.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-04-08T23:09:58.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1447" PostTypeId="1" AcceptedAnswerId="1454" CreationDate="2014-04-09T00:47:13.243" Score="27" ViewCount="1787" Body="&lt;p&gt;Equivalence classs are very useful in mathematics, but many of the applications require further background, like quotient spaces in topology or quotient groups in algebra. One good example is residue classes (I.e. $\mathbb{Z}/n\mathbb{Z}$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is another good example of an equivalence class that I could use to motivate the definition?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="114" LastEditDate="2014-04-11T12:27:00.037" LastActivityDate="2021-07-15T14:37:10.317" Title="How to motivate equivalence classes" Tags="|concept-motivation|set-theory|" AnswerCount="16" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1448" PostTypeId="1" AcceptedAnswerId="1492" CreationDate="2014-04-09T00:51:37.837" Score="11" ViewCount="373" Body="&lt;p&gt;What topics are appropriate for an Introduction to Proofs course which is:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Aimed at Freshman who have taken integral calculus and nothing else&lt;/li&gt;&#xA;&lt;li&gt;Is designed to introduce them to formal reasoning and writing proofs&lt;/li&gt;&#xA;&lt;li&gt;Includes a unit/chapter on proof techniques (such as contrapositive, cases, etc.)?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-04-09T01:00:53.103" LastActivityDate="2016-01-08T00:34:04.940" Title="What topic can I use in an Introduction to Proofs course that would introduce students to a wide variety of proof methods?" Tags="|undergraduate-education|course-design|proofs|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1449" PostTypeId="2" ParentId="1447" CreationDate="2014-04-09T01:02:35.440" Score="15" Body="&lt;p&gt;I would think a good first example is the rational numbers. (Note the &quot;quotient&quot; terminology here, too.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, the rationals can be written as the set of integer pairs $(a,b)$ with $b\neq0$, modulo the following equivalence relation: $(a,b) \equiv (c,d)$ iff $ad = bc$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the rule described here is commonly referred to as &quot;cross-multiplication.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are plenty of other examples you can give to assess whether or not students have understood the definition. (Would &quot;is a brother of&quot; be an equivalence relation? Why or why not? Would &quot;has the same first name as&quot; be an equivalence relation? Why or why not? Is &lt;em&gt;equality&lt;/em&gt; e.g. of integers an equivalence? Why or why not?) For each &quot;yes,&quot; you could follow by discussing partitions w.r.t. an equivalence relation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The last crucial early example from mathematics that comes to mind is the definition of the real numbers as equivalence classes formed by looking at (Cauchy) convergent sequences of rational numbers.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-04-09T01:02:35.440" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1450" PostTypeId="2" ParentId="1447" CreationDate="2014-04-09T03:52:46.220" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://www.youtube.com/watch?v=sqEyWLGvvdw&quot; rel=&quot;nofollow&quot;&gt;These&lt;/a&gt; lectures in real analysis motivate equivalence classes by constructing the rational numbers as a set of equivalence classes, which is a good motivation as it requires little more than some basic set theory and high-school level mathematics. I have also seen equivalence classes introduced more formally in &lt;a href=&quot;http://www.youtube.com/watch?v=WDZsXW7nDns&amp;amp;list=PLA58AC5CABC1321A3&quot; rel=&quot;nofollow&quot;&gt;Harvard's abstract algebra class&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="382" LastActivityDate="2014-04-09T03:52:46.220" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1451" PostTypeId="2" ParentId="1447" CreationDate="2014-04-09T03:58:13.463" Score="3" Body="&lt;p&gt;What about a non-mathematical motivation, namely, the idea of &lt;em&gt;classification&lt;/em&gt;?  A simple example I use with my students: define an equivalence class on the set of all animals (yes, this &lt;em&gt;is&lt;/em&gt; going to be imprecise and a bit fuzzy, but we're talking about intuition here, not formal maths!) defined by $x\equiv y$ iff $x$ and $y$ have the same number of legs.  This way, we get a &lt;em&gt;classification&lt;/em&gt; of animals with respect to the number of legs: the equivalence classes are the sets of animals with no legs, with one leg etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this may be &lt;em&gt;not&lt;/em&gt; the way we &lt;em&gt;usually&lt;/em&gt; use equivalence classes in mathematics (but it &lt;em&gt;does&lt;/em&gt; happen, of course), but it shows that the idea of an equivalence relation and classes is a natural one for the human mind.&lt;/p&gt;&#xA;" OwnerUserId="704" LastActivityDate="2014-04-09T03:58:13.463" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1452" PostTypeId="2" ParentId="1447" CreationDate="2014-04-09T04:03:50.910" Score="5" Body="&lt;p&gt;(I'm giving this as another answer, since it is completely unrelated to my other one.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very simple example of a natural equivalence relation is (for two straight lines on a plane) to be &lt;em&gt;parallel&lt;/em&gt;.  The equivalence classes may be thought of as &lt;em&gt;directions&lt;/em&gt; on the plane.  This may have an added benefit of showing beginners that in mathematics, &lt;em&gt;properties&lt;/em&gt; (i.e., &quot;having such and such direction&quot;) are basically &lt;em&gt;sets&lt;/em&gt; (well, they may be &lt;em&gt;classes&lt;/em&gt;, but there's no need to talk about formality of ZFC and paradoxes here).&lt;/p&gt;&#xA;" OwnerUserId="704" LastActivityDate="2014-04-09T04:03:50.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1453" PostTypeId="2" ParentId="1447" CreationDate="2014-04-09T04:49:47.000" Score="10" Body="&lt;p&gt;I like this question very much. But I think the best approach is via a plethora of examples meant to demonstrate the variety of uses of equivalence classes. I doubt there is a singular example that can open every student's mind to the concept of equivalence classes. That said, here are some examples I have used effectively:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1)&lt;/strong&gt; I have taught a &quot;transition to proofs&quot; course for a few years, and have included the following sequence of exercises. During the first few weeks or so, after working with sets and their notation, I assign this exercise:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;What's $\mathbb{Z}$ Point Of This Problem?&lt;/strong&gt;: In this problem, we are going to ''prove'' the existence of the negative integers! I say ''prove'' because we won't really understand what we've done until later but, trust me, it's what we're doing.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Because of this goal, you cannot &lt;em&gt;assume&lt;/em&gt; any integers strictly less than 0 exist, so your algebraic steps, especially in part (d), should not involve any terms that might be negative.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;That is, if you consider an equation like $x+y=x+z$, we &lt;em&gt;can&lt;/em&gt; deduce that $y=z$, by subtracting $x$ from both sides, since $x-x=0$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;However, if we consider an equation like $x+y=z+w$, we &lt;em&gt;cannot&lt;/em&gt; deduce that $x-z=w-y$. Perhaps $y&amp;gt;w$, so $w-y$ does not exist in our context... On to the problem!&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Let $P=\mathbb{N}\times\mathbb{N}$. Define the set $R$ by&#xA;  $$ R = \{((a,b),(c,d))\in P\times P\mid a+d=b+c\} $$&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Find three different pairs $(c,d)$ such that $((1,4),(c,d))\in R$.&lt;/li&gt;&#xA;  &lt;li&gt;Let $(a,b)\in P$. Prove that $((a,b),(a,b))\in R$.&lt;/li&gt;&#xA;  &lt;li&gt;Let $((a,b),(c,d))\in R$. Prove that $((c,d),(a,b))\in R$, as well.&lt;/li&gt;&#xA;  &lt;li&gt;Assume $((a,b),(c,d))\in R$ and $((c,d),(e,f))\in R$. Prove that $((a,b),(e,f))\in R$, as well.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I pose this mainly as a &quot;can you understand new notation and write a proof about it&quot; problem, and say as much to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But a few weeks later on, when we're talking about equivalence relations, I bring up this exercise again. I even write a passage in our book about this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Remember that crazy exercise from Chapter 3 that had you prove something about a set of pairs of pairs of natural numbers, and we claimed that was proving something about the existence of the integers? What was that all about? Look back at the exercise now, Exercise [reference]. You'll see that the last three parts of the problem have you prove that the set $R$ we defined is an &lt;em&gt;equivalence relation&lt;/em&gt; on the set $P=\mathbb{N}\times\mathbb{N}$. Look at that! You proved $R$ is reflexive, symmetric, and transitive.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What that exercise showed is that (essentially, we are glossing over some details here) any negative integer is represented as the &lt;em&gt;equivalence class&lt;/em&gt; of pairs of integers whose difference is that negative integer. That is,&#xA;  $$ -1\;\; \text{''}=\text{''}\;\; [(1,2)]_R = \{(1,2),(2,3),(3,4),\dots\} $$&#xA;  and, for another example,&#xA;  $$ -3\;\;\text{''}=\text{''}\;\; [(1,4)]_R=\{(1,4),(2,5),(3,6),\dots\}$$&#xA;  This is only an intuitive explanation and not rigorous, mathematically speaking, but that's the idea!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For the students who might already be inclined to think abstractly and want to pursue higher math, this is a great teaser, and has led to many discussions in office hours about set theory, logic, and so on. For other students, it's a reminder that exercises from the past weren't done in a vacuum; they have a meaning, and can teach us new things. And for every student, it's at least a reminder that math is interconnected in ways we might not expect, a priori.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2)&lt;/strong&gt; An in-class example I like to discuss involves comparing an equivalence relation to a similar (non-equivalence) relation that is meant to &quot;encode the same information&quot;. I include the following example in the text, and follow up on it with an in-class discussion, which also serves to point out the distinction between &quot;a relation from $A$ to $B$&quot; and a &quot;relation on $A$&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $S$ be the set of students in our class. Define a relation $R_1$ between $S$ and $\mathbb{N}$ by saying $(s,n)\in R_1$ if person $s\in S$ is $n$ years old.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Now, define a relation $R_2$ on $S$ itself by saying $(s,t)\in R_2$ if persons $s$ and $t$ are the same age (in years).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;How do the relations $R_1$ and $R_2$ compare? Do they somehow ''encode'' the same information about the elements of the set $S$? Why or why not?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3)&lt;/strong&gt; You mention $\mathbb{Z}/n\mathbb{Z}$, naturally. I think it behooves us to show &lt;em&gt;why&lt;/em&gt; this is useful, not just &lt;em&gt;that&lt;/em&gt; it's an equivalence relation. In the past, I have demonstrated exemplary uses via the Chinese Remainder Theorem, Fermat's Little Theorem, etc. But I've found that the most striking (and convincing) uses are ''smaller'', computationally, and serve to show how previously tedious arguments can be cleaned up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, a standard induction problem asks a student to show $\forall n\in\mathbb{N}$ that $6\mid n^3+5n$. A standard induction argument requires some algebraic manoeuvering, and ends up being entirely unenlightening for the beginning student (for whom this is meant to be practice relating to the &lt;em&gt;inductive&lt;/em&gt; nature of such relationships). Instead, I go back and use mod 6 and say, ''There are 6 cases. Either $n$ is congruent to 0,1,2,3,4,5 modulo 6. In each case, we see...'' And there they have it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Likewise, consider proving that any perfect square is either a multiple of 4 or one more than a multiple of 4. This can be good practice working with formal definitions (''multiple of 4 means there exists $k\in\mathbb{Z}$ such that...'') but it's far more ''fun'' for them to just see that $0^2\equiv 2^2\equiv 0$ and $1^2\equiv 3^2\equiv 1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, divisibility tricks are fun, too. I find that college students are well aware of the ''casting out 3s/9s'' trick, but are wholly unaware of how it works. Setting up a mod 10 congruence to prove it really shows some &quot;aha&quot;s and smiles.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In summary, I don't think it's necessary to show a typical undergraduate the sophisticated concept of ''modding out'' a set by an equivalence relation. In a classroom setting, it usually suffices to whet their appetite by showing students the &lt;em&gt;utility&lt;/em&gt; of equivalence relations in various settings. (In the examples above, this means (1) formally defining $\mathbb{Z}$ from $\mathbb{N}$, (ii) comparing an equivalence relation and a ''regular'' relation, and (iii) using equivalence classes to clean up a formerly ''messy'' argument.) If particular students are intrigued by this, then you might foster further discussions, either in class or in office hours (depending on the popularity/prevalence).&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-04-09T04:49:47.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1454" PostTypeId="2" ParentId="1447" CreationDate="2014-04-09T08:03:59.650" Score="31" Body="&lt;p&gt;I think that the following story is quite illuminating for introduction. Naturally, one can/should adapt/change it to better fit the audience, I just wanted to sketch the general idea.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose you came here by bus. Which bus was that? You say it was the &lt;em&gt;42&lt;/em&gt;, which is the line that goes from the main station to the university. However, was it really the &lt;em&gt;42&lt;/em&gt;? Is there only a single bus carrying this number?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;No, there are multiple buses marked &lt;em&gt;42&lt;/em&gt;, but you don't care which one you rode, because they all take the same route. You &lt;em&gt;abstracted&lt;/em&gt; the differences between particular cars, because they were &lt;em&gt;equivalent&lt;/em&gt; for your purposes. You have just used the concept of &lt;em&gt;equivalence classes&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After that, I try to explain the intuition using graphs (informally), like &lt;a href=&quot;https://math.stackexchange.com/a/740084/26306&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-04-09T15:27:07.577" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1455" PostTypeId="1" AcceptedAnswerId="1460" CreationDate="2014-04-09T08:15:17.670" Score="35" ViewCount="20051" Body="&lt;p&gt;I don't know whether you guys have made a similar experience but it just happened to me: I made a very stupid mistake in front of the class. I can't really tell you how it happened and I feel too ashamed to even tell you what it was (and it doesn't really matter) but anyway&lt;/p&gt;&#xA;&#xA;&lt;p&gt;my questions are:&lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How should one best handle the situation in the following cases: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;you realize it yourself more or less immediately&lt;/li&gt;&#xA;&lt;li&gt;you realize it only much later because something (e.g. a proof) is not working properly &lt;/li&gt;&#xA;&lt;li&gt;you don't realize it at all but a student tells you in front of the class (this happenend to me)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So my questions really aim at best practices of experienced educators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Seeing all the great answers and the upvotes I am really relieved that it is not only me - Thank you all for these great tips! :-)&lt;/p&gt;&#xA;" OwnerUserId="292" LastEditorUserId="292" LastEditDate="2014-04-11T09:54:54.777" LastActivityDate="2019-04-22T15:10:46.520" Title="How to handle the situation when you made a stupid mistake in front of the class?" Tags="|classroom-management|" AnswerCount="14" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="1456" PostTypeId="2" ParentId="1455" CreationDate="2014-04-09T08:27:46.097" Score="9" Body="&lt;p&gt;It happened to me and I think to most of us. The most important thing is that this should not happen too often, then you can handle it. If this happen too often, then you lose credibility.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you react the correct way, there is no problem if this happens once. I think it is important to acknowledge your mistake and make a correction as soon as possible. Students are no fools, and we would like to teach them to think critically. The whole scientific world relies on acknowledging our mistakes and searching for the truth. We are no gods, and if this happens only once then this can be even beneficial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you recognize the mistake immediately, then it is the easiest: you just correct yourself. If a student recognizes your mistake shortly after you made it, thank her and correct the mistake. Write out the correct part in detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you recognize an earlier mistake, then the same applies, but, if it is important, you should even hand out notes on the corrected statement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the most important thing is, as I said, that you should not let this happen too often.&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="143" LastEditDate="2015-11-19T08:58:05.500" LastActivityDate="2015-11-19T08:58:05.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1457" PostTypeId="2" ParentId="1455" CreationDate="2014-04-09T09:02:15.900" Score="7" Body="&lt;p&gt;Making a mistake is OK! In fact, I think your students would react positively if you explain that you made a mistake, and challenge them to find it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been praised for the way I handle my own mistakes in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're especially afraid of making mistakes, try implementing mistakes on a more regular basis, on purpose, and still challenge the class to find it. That way, when you make a mistake for real, and a student notices it, you can just say &quot;good job&quot; after having them elaborate on why it's wrong. It'll give you time to find out if it's a true mistake, or if the student is mistaken.&lt;/p&gt;&#xA;" OwnerUserId="796" LastEditorUserId="804" LastEditDate="2014-04-09T14:28:26.513" LastActivityDate="2014-04-09T14:28:26.513" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1458" PostTypeId="1" CreationDate="2014-04-09T09:11:18.513" Score="6" ViewCount="276" Body="&lt;p&gt;I've lectured integration from the very basics, having covered the following;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What are integrals, and what do they represent?&lt;/li&gt;&#xA;&lt;li&gt;Indefinite integrals as the opposite of derivatives.&lt;/li&gt;&#xA;&lt;li&gt;Using the power rule for derivatives to provide the power rule for integration.&lt;/li&gt;&#xA;&lt;li&gt;Integral as infinite sum.&lt;/li&gt;&#xA;&lt;li&gt;Area under curve, and area between two curves.&lt;/li&gt;&#xA;&lt;li&gt;Volume of solid of revolution.&lt;/li&gt;&#xA;&lt;li&gt;Integration by substitution.&lt;/li&gt;&#xA;&lt;li&gt;Integration by parts.&lt;/li&gt;&#xA;&lt;li&gt;Integration by partial fraction expansion.&lt;/li&gt;&#xA;&lt;li&gt;Improper integrals&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;All this is a total of 41 short (5-10 min) lectures (videos), accompanied by example problems etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm wondering what I should do next, such that it will be natural step forward. Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="796" LastActivityDate="2014-04-26T11:28:23.413" Title="Teaching integration (online). Logical progression in subject?" Tags="|calculus|tutoring|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1459" PostTypeId="2" ParentId="1448" CreationDate="2014-04-09T09:36:36.123" Score="5" Body="&lt;p&gt;I find that basic inequality proofs are very well recieved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, given that $x^2 \geq 0$ for $x \in \mathbb R$, prove that $a^2 + b^2 \geq 2ab$ for all $a,b \in \mathbb R$. Bonus points for finding all sets of $a,b$ that satisfy the equality. This is very basic, and requires only some fundamental algebra to sort out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And from there, there's a wide variety of equalities to prove, that are both important to know, and interesting to discover.&lt;/p&gt;&#xA;" OwnerUserId="796" LastEditorUserId="796" LastEditDate="2014-04-09T21:15:54.200" LastActivityDate="2014-04-09T21:15:54.200" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1460" PostTypeId="2" ParentId="1455" CreationDate="2014-04-09T10:13:38.547" Score="31" Body="&lt;p&gt;&lt;strong&gt;Edit (Nov 2015):&lt;/strong&gt; I feel it would be disingenuous not to mention that my views on this matter have evolved since posting my original answer, which remains un-edited, below. I suppose the provided answer could be contextualized specifically in the setting of a mathematics course for undergraduate mathematics majors that is primarily lecture-based. If you are a stick-to-the-script sort of lecturer, and are trying to get through material at a fast pace, then the suggestion may still be useful. If your class incorporates student discussion and you are willing to draw from scripts but also &lt;em&gt;improvise&lt;/em&gt; in your teaching, then I think the approach of, after getting lost in a mistake, deciding to state a result holds by &lt;em&gt;fiat&lt;/em&gt; and moving on - then providing written notes for the next class meeting - may be unnecessarily rigid, and, ultimately, sub-optimal. From a &amp;quot;meta&amp;quot;-perspective, I would consider my earlier answer &lt;em&gt;mistaken&lt;/em&gt;, and, personally, I find the remarks of JPBurke &lt;a href=&quot;https://matheducators.stackexchange.com/a/1461/&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; to be much more enlightening.&lt;/p&gt;&#xA;&lt;p&gt;Since my answer was &lt;em&gt;accepted&lt;/em&gt;, I cannot (per StackExchange rules) delete it. Since it was up-voted a fair amount, I think editing over its content significantly would be misleading. Hence this disclaimer.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;&lt;strong&gt;Anecdote:&lt;/strong&gt; I know a very experienced and well-liked professor (emeritus, now) of mathematics who would send a small square of chocolate via snail mail to any student who pointed out a mistake. The professor brought in typed up versions of what he was going to cover each class meeting, and the mistake could be anything said aloud, written on the board, or typed up in his notes. (Related: Donald Knuth's &lt;a href=&quot;http://en.wikipedia.org/wiki/Knuth_reward_check&quot; rel=&quot;noreferrer&quot;&gt;policy&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Responses:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How should one best handle the situation in the following cases:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1.$&lt;/span&gt; you realize it yourself more or less immediately&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Either correct it immediately or pause and tell the class that you made a mistake. Ask everyone to spend a minute looking for it. (Of course, this will only make sense in certain scenarios; for example, it probably would not make sense in writing down the givens before proving a theorem, since students don't know what result you are aiming for.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2.$&lt;/span&gt; you realize it only much later because something (e.g. a proof) is not working properly&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If it's straightforward to go through and correct it (e.g., a minus sign is missing in several places) then either follow the advice of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; above or draw a hard line on the board, remark what the error is, and write what's correct. If you aren't even sure what the correct conclusion should be by then, then admit the mistake, note that everyone makes them (it would be nice for students to get 100s on all tests, but unlikely) and abandon ship to focus on the next result.&lt;/p&gt;&#xA;&lt;p&gt;In any event, type up a clean version of the result and its proof to hand out (and possibly discuss) during the next class period.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3.$&lt;/span&gt; you don't realize it at all but a student tells you in front of the class (this happened to me)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thank the student, and ask him or her (if possible) to explain why it is an error to the rest of the class. Ask if another student can suggest a way to fix it. Then decide whether going back over what has been written would be a good use of class time, or whether you should ask that they accept the conclusion on faith for the time being. (You might even put such a decision to a majority vote.)&lt;/p&gt;&#xA;&lt;p&gt;Once again, in any case you should type up a clean version of the related mathematics to distribute (and possibly discuss) during the next class period.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-11-18T23:21:58.540" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1461" PostTypeId="2" ParentId="1455" CreationDate="2014-04-09T11:34:58.810" Score="44" Body="&lt;p&gt;In my view, &lt;em&gt;mathematics education&lt;/em&gt; is not about the authority of the teacher, it's about developing the students as their own mathematical authorities. Ultimately, the authority in mathematics is that any one of us can use what we have learned, even via different methods, to verify what other people have done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematics classrooms, we do not learn only mathematical rules and structure. We learn to think mathematically. And this involves learning something about our identity relative to &quot;mathematics.&quot; Valuable things for your students to learn:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Every one of them can wield mathematical authority; that's the power of learning mathematics.&lt;/li&gt;&#xA;&lt;li&gt;In mathematics (and in science, and in life) we have some responsibility to trustworthiness, which means an attitude of eagerness to be challenged. &lt;/li&gt;&#xA;&lt;li&gt;Justification is not an inconvenient response to a challenge, it is a vital mathematical practice&lt;/li&gt;&#xA;&lt;li&gt;We should avoid anything that teaches the student that making a mistake is something to be ashamed of; when students learn that, they begin to attribute their own mistakes to &quot;not being good at math.&quot; They start to believe that someone &lt;em&gt;good&lt;/em&gt; at math would not have made that mistake. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A teacher who is relying on cultivating the air of an infallible mathematical authority will have a hard time conveying important aspects of mathematics to the students. But worse, that concern seems to put one's own personal emotional well-being ahead of the students' learning. If this façade helps you avoid challenges from the students, is it for their good, or is it ego-driven? It's a question educators should at least consider. Who benefits?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If students learn from you that even teachers make mistakes, but yet they persevere in correcting them, they may see themselves as possibly reaching your level, with some work. Isn't that, ultimately, our goal as mathematics educators? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I thought I would offer one peek into the implications of the development identity in mathematics classrooms, since I am interested broadly in STEM education and how people come to see themselves as entering STEM fields. Jo Boaler has done research on the implications of different models of teaching in math classrooms, and one perspective she has used to compare traditional and more heavily discussion-based classrooms is to look at the developing identities of the students. One brief excerpt from &lt;a href=&quot;http://www.sfu.ca/~sbc7/joboaler.pdf&quot;&gt;one of Prof. Boaler's papers&lt;/a&gt; illustrates some of what students say when they begin to think mathematics is not for them:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The disaffected students we interviewed were being turned away from&#xA;  mathematics because of pedagogical practices that are unrelated to the&#xA;  nature of mathematics (Burton, 1999a, b). Most of the students who&#xA;  told us about their rejection of mathematics in the 4 didactic&#xA;  classrooms – 9 girls and 5 boys, all successful mathematics students –&#xA;  had decided to leave the discipline because they wanted to pursue&#xA;  subjects that offered opportunities for expression, interpretation and&#xA;  human agency. In contrast, those students who remained motivated and&#xA;  interested in the traditional classes were those who seemed happy to&#xA;  ‘receive’ knowledge and to be relinquished of the requirement to think&#xA;  deeply:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;J: I always like subjects where there is a definite right or wrong&#xA;    answer.  That’s why I’m not a very inclined or good English student.&#xA;    Because I don’t  really think about how or why something is the way it&#xA;    is. I just like math  because it is or it isn’t. (Jerry, Lemon school)&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In other words, there was a sort of filtering process going on in heavily didactic classrooms, teaching students to select themselves out of mathematics if they were interested in having control of their own choices (human agency), self-expression, and interpretation. The students who stayed interested were ones who felt relieved at the lack of complexity of &quot;one right answer.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By contrast, the students in a more discussion-oriented calculus classroom did not see mathematics learning as conflicting with their developing identities as expressive, interpretive people with personal agency. Part of the difference between these two types of classroom is in how authority is seen. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's more than just how mistakes are handled, of course. I offered this example to show that there is much more going on in math classrooms, and part of it is very important to how students determine what they are going to put their effort into.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something to consider!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Works Cited&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boaler, J. (2002). &lt;a href=&quot;http://www.sfu.ca/~sbc7/joboaler.pdf&quot;&gt;The development of disciplinary relationships: knowledge, practice and identity in mathematics classrooms.&lt;/a&gt; For the Learning of Mathematics, 22(1), 42–47.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-04-09T12:11:28.893" LastActivityDate="2014-04-09T12:11:28.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1462" PostTypeId="2" ParentId="1455" CreationDate="2014-04-09T13:03:37.327" Score="14" Body="&lt;p&gt;Making a mistake in maths is a positively good thing to do.  There are lots of students these days that seem to think maths is something they can never be good at and that if you are good at maths it means you are some sort of weirdo with a brain that functions completely differently to theirs.  However this isn't true and it can give the students confidence in their own ability to see that you make mistakes as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also teach programming and quite often I use lecture time to demonstrate how to go about writing and debugging programs by writing a non-trivial bit of code live during the lecture.  Of course I make mistakes, and frequently the student spots them before I do.  One of the benefits of this is that I can show them how to go about writing programs in a way that limits your opportunity for mistakes and that makes fixing them as easy as possible.  The same is true for maths, the fact that you make mistakes is a good way of teaching the importance of checking your maths as yo go, and if there is time to see if you can get the same result by a different method.  If we never made mistakes, this sort of thing wouldn't be &quot;good practice&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So for the three cases:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;you realize it yourself more or less immediately&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ask the students if they have spotted it - good opportunity for involving the students in the class&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;2&quot;&gt;&#xA;  &lt;li&gt;you realize it only much later because something (e.g. a proof) is not working properly&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Use it as an opportunity for demonstrating how to go about finding out what the problem is, and how to check as you go along.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;3&quot;&gt;&#xA;  &lt;li&gt;you don't realize it at all but a student tells you in front of the class (this happened to me)&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Commend the student, point out that we all make mistakes when doing maths and that this is to be expected and nothing to be embarrassed about.  If maths was easy there wouldn't be any fun in it, and the fact maths is hard means you are entitled to enjoy your successes.&lt;/p&gt;&#xA;" OwnerUserId="802" LastEditorUserId="6053" LastEditDate="2019-04-16T12:37:46.950" LastActivityDate="2019-04-16T12:37:46.950" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="1463" PostTypeId="2" ParentId="1404" CreationDate="2014-04-09T13:07:46.090" Score="4" Body="&lt;p&gt;I do not have precise, practical advice for how you should grade, but my comment will hopefully be of some general value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What, specifically, do you hope your students will develop over the course of your class? My advice would be to keep this closely in mind and make &lt;em&gt;that&lt;/em&gt; the focus of your grading. Ask: to what extent do the papers demonstrate the essential aspect of our class?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It will, of course, be helpful to the students to know what you're focusing your attention on when you are grading. So, let them know what you will be paying attention to, and give them a rubric if you can. That way they can put their effort into showcasing how they have coherently learned the subject matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The overlap of your content and ELA, of course, is in comprehensibility and coherence. It is possible for the paper to fail to communicate; if you are unable to decipher points of the paper, then they have not merely failed in an ELA sense. So, technically, grading them on their ability to make a coherent argument and express it clearly still falls well within your class. They can only succeed if they were able to express their ideas effectively, even if they break some ELA rules along the way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this doesn't prevent you from marking up any ELA problems you happen to notice as you are grading. While it may not be part of their grade consideration, and it need not be something you spend additional time on, you can probably provide some useful ELA feedback to them. You may note that there are grammar problems, that spelling is an issue, that there is a problem with the coherence from paragraph to paragraph, etc. These comments can be useful to students even if a grade is not dependent on them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cheers!&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-04-09T13:07:46.090" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1464" PostTypeId="2" ParentId="1325" CreationDate="2014-04-09T13:57:13.420" Score="54" Body="&lt;p&gt;A book I remember has the title &amp;quot;the egg-laying dog&amp;quot;. The titular dog enters a room where we placed 10 sausages and 10 eggs. After a while the dog leaves the room, and we observe, that the percentage of eggs relative to the sausages increased, so we conclude that the dog must have produced eggs.&lt;/p&gt;&#xA;&lt;p&gt;It's easy to spot the mistake in the above example, because the image of a dog laying eggs is absurd. However, consider the following case: a few decades ago a new medicine against heart diseases was developed. It worked well. However, 10 years later someone observed, that the rate of people dying of cancer was much higher among those who have been treated with the new medicine, in fact, the rate of cancer increased by a significant margin. Mass hysteria ensues: the new medicine causes cancer! Bans are being issued, companies are sued, etc. until a better look at the statistics showed that the situation was exactly the same as in the case of the egg-laying dog: people are not immortal, and sooner or later they tend to die of &lt;em&gt;something&lt;/em&gt;. As fewer people died of heart diseases, they died of other causes years or decades later, and cancer, being a leading cause of death especially among older people, was one of them. The new medicine was not causing cancer at all, it just decreased the rate of another disease.&lt;/p&gt;&#xA;&lt;p&gt;Other interesting, commonly occurring examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Regression toward the mean: people, especially bosses tend to think that scolding people when they perform badly improves their effectiveness and complimenting them when they perform well decreases it. It's easy to see the problem. Take a 6-sided die and start throwing it. Every time you throw a 1, scold the die why did it give such a bad result. Observe, that after your scolding, in over 80% of the cases, the result of the next throw was better. Was it because of the scolding?&lt;/li&gt;&#xA;&lt;li&gt;Improper scaling in graphs. A common election tactic, you put two bar graphs next to each other, one very low for your opponent and one very high for yourself. What people tend to miss, is that the values don't start from zero. In fact you created 537 new jobs, while your opponent only 519. Not a big difference, but if you start the graph from 500, it seems quite large.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;While not strictly statistics-related, it's worth to mention the &amp;quot;fallacy fallacy&amp;quot;. If someone uses a fallacy to prove or defend a statement, this fact alone is not a proof that the statement is wrong.&lt;/p&gt;&#xA;" OwnerUserId="803" LastEditorUserId="803" LastEditDate="2020-07-22T10:04:05.567" LastActivityDate="2020-07-22T10:04:05.567" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="1465" PostTypeId="2" ParentId="1447" CreationDate="2014-04-09T14:35:01.457" Score="3" Body="&lt;p&gt;To give another inner-mathematical example from another subdiscipline, which I always liked:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f≡g :\Leftrightarrow \int \left | f(x)-g(x) \right | \,\mathrm{d}x = 0,$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where the integral is defined such that it ignores what happens to the integrand on sets with measure zero, e.g., the Lebesgue integral or the Henstock–Kurzweil integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The corresponding equivalence classes are, e.g., essential for defining convergence almost everywhere and convergence by measure, which need to be defined for these equivalence classes (and not for individual functions) if they shall have a unique limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A drawback of this example is that it may not be easily accessible to your students, depending on their prerequisites.&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2014-04-09T14:35:01.457" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1466" PostTypeId="2" ParentId="1447" CreationDate="2014-04-09T15:25:24.883" Score="5" Body="&lt;p&gt;As an alternative to non-mathematical or very complex examples:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to go back to the very basics: terms. &#xA;Is $1+2$ the &lt;em&gt;same&lt;/em&gt; as $2+1$? As an expression, it's not. The first one begins with a $1$, but not the second one. For a child, it has to be learnt that these terms are the same, although they look different. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematically, it's annoying to &lt;em&gt;define&lt;/em&gt; equivalence here. However, psychologically the big advantage is that your students &lt;em&gt;know&lt;/em&gt; which terms (of natural numbers, say) are equivalent. So, this example helps to get the feeling that &quot;being equivalent&quot; means &quot;being the same although not looking so&quot;.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-04-09T15:25:24.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1467" PostTypeId="2" ParentId="599" CreationDate="2014-04-09T15:46:26.017" Score="1" Body="&lt;p&gt;Random list of ideas:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;board races - this is not great for grades but is a good evaluation of where the students stand against each other to monitor progress on the topics&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;in class worksheets...worth more than a homework but less than a test or quiz&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;making study guides - have students create their own study guides in class with example problems that they create and step-by-step solutions, vocabulary, symbols, etc. turn this in for a grade, much like a term paper would be, but more fun. Possibly allow them to work in small groups?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;make your own test - set simple parameters such as you must cover a problem from these five topics we learned this unit and have students create their own tests and answer keys. Let them know it will be graded by correctness as well as complexity.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="782" LastActivityDate="2014-04-09T15:46:26.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1468" PostTypeId="1" AcceptedAnswerId="1475" CreationDate="2014-04-09T15:46:54.903" Score="17" ViewCount="1214" Body="&lt;p&gt;I am currently going through the &lt;em&gt;Topics in Algebra&lt;/em&gt; by I.N. Herstein. The problems are pretty good, but there are no answers. The same is the case with &lt;em&gt;Mathematical Analysis&lt;/em&gt; by Rudin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why is this?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though some would argue that it robs the question of its charm, sometimes it is necessary to check out the answer just to get confirmation that what you have done is correct. Moreover it becomes necessary in an environment where you study on your own: Even if I have solved an exercise, there is always a doubt regarding the correctness of the process or answer.&lt;/p&gt;&#xA;" OwnerUserId="808" LastEditorUserId="75" LastEditDate="2014-04-09T17:14:24.077" LastActivityDate="2017-06-11T12:58:49.813" Title="Why do most of the books in mathematics not include answers?" Tags="|undergraduate-education|self-learning|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1469" PostTypeId="2" ParentId="1056" CreationDate="2014-04-09T16:08:00.763" Score="7" Body="&lt;p&gt;I asked&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What should (or could or would) one call such a formula?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Several good suggestions came up in the comments&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;a law&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;a fact&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;a tautology&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;true&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;universally true&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I'd like to thank everyone who commented. Given that these are all good suggestions, I'm willing to consider the question answered.&lt;/p&gt;&#xA;&lt;p&gt;Many people up voted &amp;quot;tautology&amp;quot;. In fact, it is a word I've used in the past.  I'd like to warn that there is at least one competing definition for &amp;quot;tautology&amp;quot; applied to a first order formula. This is that a formula is a tautology if and only if it is an instance of a propositional tautology. For example &lt;span class=&quot;math-container&quot;&gt;$x=0 \vee \neg(x=0)$&lt;/span&gt; is a tautology, as it is an instance of &lt;span class=&quot;math-container&quot;&gt;$P\vee \neg P$&lt;/span&gt;, but &lt;span class=&quot;math-container&quot;&gt;$2+2=4$&lt;/span&gt; is not a tautology. As I said in the OP, I wanted to be consistent with the standard logical terminology, so that ruled out &amp;quot;tautology&amp;quot; for me.&lt;/p&gt;&#xA;&lt;p&gt;I ended up with &amp;quot;universally true&amp;quot;.  For context, this is a computing course and the kinds of formulas we typically deal with describe states of the computer. E.g. &lt;span class=&quot;math-container&quot;&gt;$x=y$&lt;/span&gt; describes all states in which the location called &amp;quot;x&amp;quot; contains the same value as the location called &amp;quot;y&amp;quot;. Usually the right question to ask about such a formula is not whether it is &amp;quot;true&amp;quot; or not, but which states is it true of.  This is the reason I preferred not to use simply &amp;quot;true&amp;quot;. &amp;quot;Law&amp;quot; and &amp;quot;fact&amp;quot; are to me slightly more suggestive than &amp;quot;true&amp;quot;, but I didn't think that they went far enough in emphasizing the universality.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I'll note that Dijkstra and Scholten in their book &lt;em&gt;Predicate Calculus and Program Semantics&lt;/em&gt; use the square brackets as an operator which they call &amp;quot;everywhere&amp;quot;. This operator is applied to what they call &amp;quot;Boolean structures&amp;quot;, which are essentially the same as what most of us would call Boolean functions. For them a condition &lt;span class=&quot;math-container&quot;&gt;$x\ge y\Rightarrow x+1 &amp;gt; y$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are integer variables, is a &amp;quot;Boolean structure&amp;quot; which represents a set of states, while &lt;span class=&quot;math-container&quot;&gt;$[x\ge y\Rightarrow x+1 &amp;gt; y] = \mathit{true}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="527" LastEditorUserId="527" LastEditDate="2021-10-07T19:32:06.197" LastActivityDate="2021-10-07T19:32:06.197" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="1470" PostTypeId="2" ParentId="1455" CreationDate="2014-04-09T17:47:39.250" Score="19" Body="&lt;p&gt;Well, at least in cases 1 and 3, I usually say (in a serious tone): &quot;Well, I did it on purpose, to see if anybody is actually paying attention to what I'm saying&quot;.  It usually takes the students about 3 times (sometimes over a couple of weeks) until they realize it's a joke.  Then, I usually explain to them that it's normal, and in the process of learning one is actually &lt;em&gt;expected&lt;/em&gt; to make mistakes - this is just natural.  It is probably a good idea to proceed then to explaining that the difference between a student and a teacher is not that the latter does &lt;em&gt;not&lt;/em&gt; make mistakes, but that the teacher (as a more mature human being, at least in theory;)): (i) makes them less often, (ii) can more easily see whether something is correct or not, and (iii) does not build his/her self-esteem on whether s/he makes mistakes.  (That said, I don't try to claim that I always live up to the ideal described here; in fact, I'm still far from it...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I find that making a bit fun of it is a good way to cope with the natural stress of such a situation.  A few other running gags I employ are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Well, you see, I'm an idiot, but don't tell that to my dean, 'cause he's going to fire me.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;I'm lucky I already passed this course.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I think that (besides being funny, though your mileage may vary on that) it's quite formative for students to see that their teacher: (i) also is a human being and makes mistakes, (ii) does not have a problem with admitting it, and (iii) does not treat himself &lt;em&gt;too&lt;/em&gt; seriously.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, it's not &lt;em&gt;that&lt;/em&gt; easy, and requires some effort to develop such an attitude (and even then, it does not work all the time, obviously).  But my experience tells me that especially the trait described in (iii) is very useful here (and also in general).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, some time ago I did an &lt;em&gt;extremely&lt;/em&gt; stupid mistake of making a bad drawing (it was an elementary geometry problem) and discarding a &lt;em&gt;correct&lt;/em&gt; answer because it didn't fit to my bad drawing.  Even though I was &lt;em&gt;very&lt;/em&gt; embarrassed (so it must have been a really stupid mistake), since it was an introductory course for freshmen, I decided that it's too good an opportunity to waste it, and I did a mini-lecture on how bad drawings can hinder our reasoning.  So I think that both a student's mistake and a teacher's one may make a good opportunity to teach something.&lt;/p&gt;&#xA;" OwnerUserId="704" LastActivityDate="2014-04-09T17:47:39.250" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1471" PostTypeId="2" ParentId="1302" CreationDate="2014-04-09T20:38:45.147" Score="3" Body="&lt;p&gt;I'm a bit surprised nobody mentioned Rudin's &lt;em&gt;Principles of Mathematical Analysis&lt;/em&gt;.  It's a difficult book (mostly because it's very concise for its level), but I think that, since you have experience with mathematical proofs &lt;em&gt;per se&lt;/em&gt;, you &lt;em&gt;might&lt;/em&gt; be satisfied with it.  (Though I would strongly suggest borrowing a copy from a library, studying a chapter or two and only then making the decision to buy it or not.)  In fact, its conciseness might even be an advantage - there won't be many distractions along the way to the &quot;how does calculus work&quot; part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I have some experience of self-teaching (not a lot of maths though, but a foreign language), and I have to say that what worked for me really well was to study the same topics from 2-3 books simultaneously.  This would almost certainly be tougher idea with analysis, since different authors may use different (but equivalent) definitions of e.g. Riemann integral (in this case, one man's definition is another man's theorem) - but again, this &lt;em&gt;might&lt;/em&gt; as well give you more insight.&lt;/p&gt;&#xA;" OwnerUserId="704" LastActivityDate="2014-04-09T20:38:45.147" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1472" PostTypeId="1" CreationDate="2014-04-09T21:21:38.710" Score="14" ViewCount="638" Body="&lt;p&gt;From aesthetical point of view, using two symbols for the same concept during the same course is obviously a &lt;em&gt;very&lt;/em&gt; bad idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, especially when I teach freshmen, I often &lt;em&gt;deliberately&lt;/em&gt; mix different notations.  The rationale is that the ability to cope with such a situation is an important part of &lt;em&gt;mathematical maturity&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is that a good practice?  Are there any other ways to teach students that they should be flexible wrt. notations, e.g. when studying from different sources?  (A similar case can be made for different words, like &quot;increasing&quot; vs. &quot;weakly increasing&quot; vs. &quot;nondecreasing&quot;, all three of them meaning the same thing in different books.)&lt;/p&gt;&#xA;" OwnerUserId="704" LastActivityDate="2019-07-16T06:15:37.000" Title="Is using different notations in one course a good idea?" Tags="|notation|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1473" PostTypeId="2" ParentId="1440" CreationDate="2014-04-09T21:36:47.060" Score="4" Body="&lt;p&gt;In a comment you asked for some examples of extra credit questions I've given.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 1:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Roxelle Porter-Knight is driving an early 23rd century gravity-pulsed car with $700$-factor inertial dampers along a one mile course laid out in the desert salt flats and manages to average only $28$ $\frac{\text{mi}}{\text{hr}}$ for the first half mile. How fast must she average over the next half mile so that her average speed for the entire one mile trip is $55$ $\frac{\text{mi}}{\text{hr}}?$ Give your answer to the nearest $0.1$ $\frac{\text{mi}}{\text{hr}}.$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I asked variations of this one in classes such as college algebra, precalculus, calculus 1, and occasionally others. I often asked a different version again in the same class, letting students know in advance that I would. I did this because I thought the underlying principle was important enough (average of velocities doesn't have to equal average velocity), especially in calculus 1, and it was an easy way to get most students to master something somewhat relevant to the material without putting in a lot of class time. Usually in college algebra and precalculus classes I would work it on the board for them the first time, and in calculus classes I left it up to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 2:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\lim_{n \rightarrow \infty}\frac{(2n^3 – n^2 + 6)^4(3n-7)^3}{n(1+7n)(2+3n^2)^2(3+4n^3)^3}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I often put something like this on a calculus 2 quiz on a day when I wanted to review limits of rational functions (e.g. for ratio test with series convergence), with the review being at a more mathematically mature level than was probably the case when they covered this in precalculus or calculus 1. By &quot;more mathematically mature level&quot;, I mean knowing the idea of using dominate terms and being able to quickly find the dominate terms. I'd work the extra credit problem to demonstrate the idea:&#xA;$$\frac{(2n^3 – n^2 + 6)^4(3n-7)^3}{n(1+7n)(2+3n^2)^2(3+4n^3)^3} \;\; = \;\; \frac{(2^4n^{12} + \cdots)(3^3n^3 + \cdots)}{n(7n + \cdots)(3^2n^4 + \cdots)(4^3n^9 + \cdots)}$$&#xA;$$= \;\; \frac{2^43^3n^{15} + \cdots}{3^24^3\cdot7n^{15} + \cdots} \;\; \longrightarrow \;\; \frac{2^43^3}{3^24^3\cdot7} \;\; = \;\; \frac{2^43^3}{2^63^2\cdot7} \;\; = \;\; \frac{3}{2^2\cdot7} \;\; = \;\; \frac{3}{28}$$&#xA;This specific example came from a quiz given to a very strong high school class, by the way. Note that it is easy vary the difficulty of something like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 3:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Evaluate the following, where $a$ is a constant greater than $1.$ Your answer will be an expression involving $a.$ Show appropriate work! (A suitable graph along with some high school geometry area calculations is acceptable.)&#xA;  $$\int_{0}^{a}|ax-a|\,dx$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 4:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Expanding as much as possible using the chain rule, find the 2nd derivative of $f \circ f \circ f.$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 5:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\int (6x – 1)^2 \sqrt{2x-1}\,dx$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is a good type to use near the beginning of covering integration by $u$-substitution. It's a standard type, but students will remember the trick better if they struggle with it a bit on a quiz (or hour test). The idea, of course, is that the square root function is not an additive function, so you want to let $u = 2x-1$ (and not $u = 6x-1).$ Then you have to &quot;force&quot; the substitution by solving for $x$ in terms of $u$ in order to express $(6x – 1)^2$ in terms of $u.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 6:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find the unique solution to the following ODE that contains the point $\left( 0,\,1\right).$ Express your answer as $y$ solved in terms of $x,$ making use of the error function. The &lt;em&gt;error function&lt;/em&gt; is defined as $\text{erf}\,(x) =\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt.$ [Footnote: Your text discusses the error function on pp. 84-85 and in Section 165 (pp. 511-518).]&#xA;  $$y' - 2xy\;=\;1$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I put this on a differential equations quiz on a day that I wanted to talk about using unfamiliar functions and indicated integrations to express solutions. I had planned to work this problem as one of my lecture examples, but then decided it would be worth the time to put it as an extra credit problem on that day's quiz. I'm certain the extra time I gave them for the quiz (I probably give them 20 minutes for what was otherwise a 10-minute quiz) was more than made up by their retention of the idea and by their attention to my working it out after the quiz.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-04-09T21:36:47.060" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1474" PostTypeId="2" ParentId="1472" CreationDate="2014-04-09T21:50:36.603" Score="-1" Body="&lt;p&gt;Don't do it. It just helps confusing them. In case several alternative notations are common (apear in textbooks, ...) &lt;em&gt;do&lt;/em&gt; point them out, but stick to one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Graduate classes (&quot;more (mathemathical) maturity&quot;) it might be fine, but I'd prefer to stick to one (there must be a &quot;best&quot; one for you, or pick &quot;arbitrarily&quot;).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-09T21:50:36.603" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1475" PostTypeId="2" ParentId="1468" CreationDate="2014-04-09T22:17:13.407" Score="26" Body="&lt;p&gt;For a little historical perspective, which at least explains why older books on non-elementary mathematics rarely have &quot;solutions&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine first that there's no internet, that long-distance phone calls are prohibitively expensive, and that photocopying is not only expensive but mostly inaccessible to students. Imagine also that people use the same homework and exam questions over-and-over. The only step needed to prevent gaming the system is to not put solutions in the book (or in the library, either). Done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In combination with that immobility of information, there was a sort of macho culture of not telling anyone who couldn't &lt;em&gt;already&lt;/em&gt; do a problem how to do it. This is perverse from an educational or scientific viewpoint, of course, but that doesn't mean that humans won't do it. That is, homeworks and exams would be graded, and points subtracted, and things marked &quot;nonsense&quot;, &lt;em&gt;without&lt;/em&gt; any &quot;approved solution&quot; being given, much less &quot;published&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The common student complaint that they can't tell whether their own &quot;solution/proof/computation&quot; is correct... raises a different point, namely, that students should be very strongly encouraged to develop exactly the sensibilities to know at least roughly whether they're doing the right thing. (The culture of math as rules imposed by an ineffable external authority doesn't help.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ignoring that misguided complaint, &lt;em&gt;my&lt;/em&gt; complaint is that there are many important, useful, traditional questions/issues with essentially no expert-written treatments... because they're always assigned as exercises... but/and crappy very-inexpert solutions or pseudo-solutions or non-solutions &lt;em&gt;do&lt;/em&gt; circulate, thus misleading, certainly not &lt;em&gt;helping&lt;/em&gt;, myriad yet-more-junior students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, by now, given the mobility of information, I think good, that is, &quot;expert&quot;, solutions &lt;em&gt;should&lt;/em&gt; be available. Yes, this does entail that &quot;assessment&quot; methods have to change. So be it. Also, people can read and write.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2014-04-09T22:31:47.967" LastActivityDate="2014-04-09T22:31:47.967" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1476" PostTypeId="2" ParentId="1472" CreationDate="2014-04-09T22:26:50.840" Score="7" Body="&lt;p&gt;I think that there is enough notation chaos between courses/books/papers that you don't need to make special effort in this direction. Surely, it would be beneficial to mention that there are different notations representing the same thing, so the students won't be surprised when reading a book by yet another author, but I suggest consistence, i.e. stick to one way during a single course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, there are times when you &lt;em&gt;want&lt;/em&gt; to use different notation for the same concept, the most important reason being a different perspective or emphasis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, during a programming course I used &lt;em&gt;four&lt;/em&gt; different notations for the same concept, in this case function.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The standard notation, mostly for dealing with mathematics, named and simple functions; the advantage is the familiarity, the disadvantage can be seen easily in the second example;&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\mathrm{id}(x) &amp;amp;= x \\&#xA;\mathrm{apply}(f)(x) &amp;amp;= f(x)&#xA;\end{align}&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;An alternate notation for high-order or unnamed functions; one has to get used to this notation, but afterwards it just &lt;em&gt;feels good&lt;/em&gt;;&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\mathrm{id} &amp;amp;= x \mapsto x \\&#xA;\mathrm{apply}(f) &amp;amp;= x \mapsto f(x) \\&#xA;\mathrm{apply} &amp;amp;= f \mapsto \big(x \mapsto f\ x\big)&#xA;\end{align}&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;-notation for terms of lambda-calculus (which are also functions); used to stress lambdiness, also for historic reasons; it's awkward;&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\mathrm{id} &amp;amp;= \lambda x.\ x\\&#xA;\mathrm{apply} &amp;amp;= \lambda f.\ \lambda x. f\ x&#xA;\end{align}&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Programming notation, when dealing with code (that would also include functions); to emphasize that students could type it into an interpreter and it would run; uses the syntax of some programming language, here &lt;a href=&quot;http://www.haskell.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Haskell&lt;/a&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\mathtt{id\ x}&amp;amp;\mathtt{= x} \\&#xA;\mathtt{apply\ f\ x}&amp;amp;\mathtt{= f\ x}&#xA;\end{align}&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Another example could be &lt;em&gt;semantics of programming languages&lt;/em&gt;, where there are four different notations that describe essentially the same thing (i.e. the semantic of some collection of expressions), e.g. the meaning of &lt;code&gt;if-then-else&lt;/code&gt; could be written down as&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;in operational semantics with small steps&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\Gamma \mid b \leadsto \mathtt{true}}{\Gamma \mid \verb`if`\ b\ \verb`then`\ e_1\ \verb`else`\ e_2, s \leadsto e_1, s}, \frac{\Gamma \mid b \leadsto \mathtt{false}}{\Gamma \mid \verb`if`\ b\ \verb`then`\ e_1\ \verb`else`\ e_2, s \leadsto e_2, s}$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;in operational semantics with large steps&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\Gamma \mid b \leadsto \mathtt{true}, \Gamma \mid e_1,s \leadsto e_3, s'}{\Gamma \mid \verb`if`\ b\ \verb`then`\ e_1\ \verb`else`\ e_2, s \leadsto e_3, s'}, \frac{\Gamma \mid b \leadsto \mathtt{false}, \Gamma \mid e_2,s \leadsto e_3, s'}{\Gamma \mid \verb`if`\ b\ \verb`then`\ e_1\ \verb`else`\ e_2, s \leadsto e_3, s'}$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;in denotational semantics (there are actually two versions here, with or without fix-point operator, but it doesn't matter for &lt;code&gt;if-then-else&lt;/code&gt;)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\newcommand{llb}{\big[\hspace{-2.5pt}\big[}&#xA;\newcommand{rrb}{\big]\hspace{-2.5pt}\big]}&#xA;\mathcal{E}\llb\verb`if`\ b\ \verb`then`\ e_1\ \verb`else`\ e_2\rrb\ \varrho\ s = \begin{cases}&#xA;\mathcal{E}\llb e_1\rrb\ \varrho\ s &amp;amp;\text{ if } \mathcal{B}\llb b \rrb\ \varrho\ s = \mathtt{true}\\&#xA;\mathcal{E}\llb e_2\rrb\ \varrho\ s &amp;amp;\text{ if } \mathcal{B}\llb b \rrb\ \varrho\ s = \mathtt{false}&#xA;\end{cases}&#xA;$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note, that in each of those cases there are very specific uses for introduced notations. Each time there is a class that draws parallels between such concepts, how they are similar and why we treat them differently. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concluding, there are uses for different notations during the same course or same class, but those are &lt;em&gt;rare&lt;/em&gt;. Introduce a different notation only when you have a concrete reason.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="2083" LastEditDate="2019-07-16T06:15:37.000" LastActivityDate="2019-07-16T06:15:37.000" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="1477" PostTypeId="2" ParentId="1472" CreationDate="2014-04-09T22:41:00.890" Score="15" Body="&lt;p&gt;There is a certain benefit to &quot;confusing&quot; students; I alluded to the ideas of &lt;em&gt;disequilibrium&lt;/em&gt; and the resulting &lt;em&gt;equilibration&lt;/em&gt; in an earlier MESE &lt;a href=&quot;https://matheducators.stackexchange.com/a/1377/262&quot;&gt;post&lt;/a&gt;. More comments about Piaget can be found on &lt;a href=&quot;http://www.simplypsychology.org/piaget.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this site&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the context about which you are asking: I think that if you want to introduce multiple notations, then you should, at least, abide by two principles:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Make it very clear which notation &lt;em&gt;you&lt;/em&gt; will use and be consistent.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Help students to understand why other notation styles exist.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;With regard to the second principle, this could be as simple as the $\div$ sign, but you might also consider an example such as how to denote the cyclic group of order three, e.g.,  $C_3, \mathbb{Z}_3,$ or $\mathbb{Z}/3\mathbb{Z}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each of these has its own advantages and disadvantages. For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$C_3$ is easy to remember because the $C$ can stand for cyclic, but its use is not so widespread.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathbb{Z}_3$ is succinct and shows the similarity with $\mathbb{Z}$, but it can also be read as the $3$-adic integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathbb{Z}/3\mathbb{Z}$ emphasizes the quotient group aspect, but is often introduced before quotient groups. (Plus, the notation is a bit more cumbersome than either of the previous two.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar remarks could be made about increasing and nondecreasing, or countable and enumerative.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Your question reminds me of a paradigm-shift in theories about language learning. In the past, many thought that teaching a child two different languages would confuse him or her, and, therefore, be disadvantageous. Today, I believe a more common interpretation is that while this can be confusing, it is the &lt;em&gt;resolution&lt;/em&gt; of this confusion that can be cognitively beneficial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regard to the mathematics classroom: I don't think that varying the use of notation is the best way to induce the sort of productive struggle that leads to learning. Instead, I think discussing &lt;em&gt;why&lt;/em&gt; different mathematicians use different notations (as described above) and helping students practice the &quot;bilingualism&quot; involved in moving between, e.g., set theoretical notation and paragraph style proofs are both better ways to raise students' mathematical maturity.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-09T22:41:00.890" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1478" PostTypeId="2" ParentId="1455" CreationDate="2014-04-09T23:54:13.780" Score="5" Body="&lt;p&gt;There is absolutely nothing wrong with making a mistake in front of the class.  I even claim that it may be helpful for the students to see you make mistakes.  Think about it, we work from examples that we have planned in advance.  The students don't see us struggle with problems, we make it look easy on the board.  Then they go home to work on homework and don't understand why they struggle with it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also a psychological aspect to this as well.  Students tend to judge themselves (and their mathematical ability) on whether or not they can get the right answer.  Furthermore, they also need something to compare their mathematical ability against, and usually they will compare themselves against the instructor.  I tell my students the first day of class, that I am human, I don't know everything there is to know about mathematics, (or whatever course I am currently teaching) and I &lt;strong&gt;will&lt;/strong&gt; make mistakes on the board.  I also tell them that I will sometimes make intentional mistakes to see if they can catch them. The main thing is not to make a big deal of it.  I made a mistake, now the learning can start.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The jist is that students need to see that we make mistakes also, they need to know that we struggle with our level mathematical work just like they struggle with their homework. &lt;/p&gt;&#xA;" OwnerUserId="203" LastActivityDate="2014-04-09T23:54:13.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1479" PostTypeId="1" AcceptedAnswerId="1484" CreationDate="2014-04-10T00:16:18.910" Score="30" ViewCount="4413" Body="&lt;p&gt;It seems to me that an open-source model could work quite well for textbooks, with issues being raised by the users of the book and different forks of the project being created for different preferences, such as early transcendentals in calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quick search brings up the &lt;a href=&quot;http://aimath.org/textbooks/&quot;&gt;Open Textbook Initiative&lt;/a&gt;, but I have no experience with any of these and in fact had not heard of them before my quick research.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone here have any experience with open-source books or with an attempt to adopt an open-source book? What are some surprising upsides and downsides to this approach? I'm specifically interested in calculus, statistics, or college algebra, but I imagine the challenges would be shared regardless of the specific book used.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-08-24T08:06:22.890" Title="Open-Source Math Textbooks" Tags="|calculus|textbooks|algebra|statistics|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1480" PostTypeId="2" ParentId="1472" CreationDate="2014-04-10T00:26:04.563" Score="6" Body="&lt;p&gt;I always try to use several notations. If you just mention once that another notation for something exists, they will forget it. Then in later courses, or when reading, when they come across the other notation, they will be confused; they might vaguely recall that they should know what it means, and have a tentative idea of how to interpret it, but it will always be surrounded with a fog of uncertainty and doubt, which is harder to dispel after the fact than to prevent in the first place. Instead, you should strive to use various notations freely enough&#xA;so that they're all instinctive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I fell prey to this back when I was taking undergraduate classes, and I wish my professors had made a point to use more notations, so that they would be natural to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I'm teaching, this is primarily about using both Newton and Leibniz notation for differentiation. There are good reasons why we still have both: each is more convenient for certain cases, and the students should know both.&lt;/p&gt;&#xA;" OwnerUserId="748" LastActivityDate="2014-04-10T00:26:04.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1481" PostTypeId="2" ParentId="1479" CreationDate="2014-04-10T01:34:59.603" Score="16" Body="&lt;p&gt;This semester I adopted &lt;a href=&quot;http://joshua.smcvt.edu/linearalgebra/&quot;&gt;Linear Algebra&lt;/a&gt; by Jim Hefferon for the junior-level linear algebra course. I didn't have any particular difficulty adopting it, the bookstore was able to arrange printing through some print shop and it was fairly easy given the free availability of the text. I'm not sure how many students actually purchased a copy. Once nice thing about this particular text is there is an solution manual for nearly all the exercises which is also freely available as a pdf. It has all the problems ordinary textbooks have: it doesn't fit my idiosyncrasies and it doesn't have all the content I want. On the other hand, I have little room to complain because:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;it is free&lt;/li&gt;&#xA;&lt;li&gt;the raw $\LaTeX$ source is available (but, I am unable to compile it, however, those with more adept computer skills have managed)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I actually follow my course notes fairly closely and at 350 pages they're more or less a book without exercises.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-04-10T01:34:59.603" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1482" PostTypeId="2" ParentId="1439" CreationDate="2014-04-10T02:45:13.100" Score="3" Body="&lt;p&gt;An important concept in scaling figures at this level is the connection to proportionality (which is obvious to many advanced math users, but is not obvious to many students and even teachers). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's unclear what sort of activity you're thinking about. I have produced Geometer's Sketchpad sketches that are used with Sketch Explorer on the iPad, and these sketches were used as dynamic activities in interview items as part of a study of 6th and 7th grade math teachers' knowledge of concepts of proportion, scale, fraction, and ratio. I have found that, while constant of proportionality is obviously an important concept in understanding the relationship between similar but differently-scaled drawings, this is not used as commonly as one would expect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, it might be useful to your students to manually construct scaled drawings (construct drawing B from drawing A), determining a constant of proportionality in the relationship between the corresponding linear measures. Even 7th grade students may be surprised that there is a constant relationship in all the corresponding linear measures (including widths, heights, perimeters, circumferences, etc). Once this relationship is discovered / understood, any scaled drawing can be reproduced. Angles need not be measured; any point in the scaled drawing can be located by finding its horizontal and vertical distances from the nearest sides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cheers!&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-04-10T02:45:13.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1483" PostTypeId="2" ParentId="557" CreationDate="2014-04-10T02:52:49.557" Score="4" Body="&lt;p&gt;At least worth considering: devote class time more to discussion, argumentation, and work that requires heavy interaction. Provide the lectures in some video form that can be watched outside of class. Then, lecture-watching becomes the responsibility of the student, and they are expected to be actively engaged in some kinds of mathematical work involving mathematical reasoning in class rather than listening to a lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not trying to criticize the practice of lecturing at all, just pointing out that the classroom's &quot;activity structures&quot; are a contributing factor to all sorts of possible distraction sources. I'm sure that before Facebook, we have all seen people doing homework for another class during a lecture. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may not entirely be a new problem, or as related to technology as we might assume. &lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-04-10T02:52:49.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1484" PostTypeId="2" ParentId="1479" CreationDate="2014-04-10T03:00:12.297" Score="13" Body="&lt;p&gt;While I haven't used an open-source text for a semester-long university class, I have used an open-source calculus text for a summer program with high school students, and I'm currently modifying a number-theory text for a different summer program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summary, with extended discussion below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Upsides&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I can modify the text to suit my needs.&lt;/li&gt;&#xA;&lt;li&gt;I can modify the text in response to student reactions. &lt;/li&gt;&#xA;&lt;li&gt;My students don't need to pay excessive amounts of money.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Downsides&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It is less tenable, at least from my position, to work with open-source texts with large courses that have multiple professors, for many reasons.&lt;/li&gt;&#xA;&lt;li&gt;The general open-source text available is supbar, in my opinion, and needs to be modified. &lt;/li&gt;&#xA;&lt;li&gt;Actually modifying/writing a textbook is very time-consuming and hard.&lt;/li&gt;&#xA;&lt;li&gt;(If you care) - it is not currently feasible to use one of the online-homework systems without using the accompanying book.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The great thing about both texts is that I can modify the source to fit the course, which I much prefer to modifying my course to fit the text/jumping around the text. In fact, this is extremely useful. It also allows me to modify and change wording, examples, and problems based on direct feedback from my students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I must admit that I am also very happy to not force students to buy these texts, given that the information itself is readily available online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the option, I would choose to use an open-source text for essentially any lower-level class (precalc, calc, linear algebra, statistics, etc.). I don't know about upper-level classes, for which there often exist great books that are extremely well-written and which don't change from year to year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, I don't usually have that option, at least not from my current position. And many professors I know don't have that option either. The university expects consistency, and lower-level classes are those often taught by multiple people at a time. For all of us to use a modified version of an open-source text requires all of us to agree on modifications/the text. At the very least, university policy and bureaucracy make this much harder. (I suppose it would just take the person in charge, if there is a person in charge, to make a solid decision and to commit - but this has not been my experience).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, in my experience, I find that most open-source textbooks &lt;em&gt;need&lt;/em&gt; some revision for one reason or another. This limits them to the ones whose source is actually available, rather than those that are just free and which get lumped in. And the problem here is that authors get to make up their own notations/TeX conventions, which can make it very unpleasant to actually modify their text. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For that matter, writing a text at all is time-consuming and hard. It's hard to write clearly, it's hard to make pretty pictures (&lt;strong&gt;really hard&lt;/strong&gt;). It really is much easier to just use some generic calculus text (the big ones are all the same, and all such classes are isomorphic) than to find, modify, and use an open-source one. Or rather, it's hard to &lt;em&gt;first&lt;/em&gt; go out, find, modify, and use an open-source one. Having done it once, it's now very easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps soon a good variety of good open-source texts will become available, which will make the transition cost lower and make it more likely that a university might accept it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I talked to the teaching head of my department (so we actually do have a guy in charge), and he's of the opinion that the future isn't in open-source, but in online homework (like webassign, or other forms). We don't actually use that at my university, but it was used at my undergrad, and I know that many other schools use online homework. &lt;/p&gt;&#xA;" OwnerUserId="20" LastActivityDate="2014-04-10T03:00:12.297" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1485" PostTypeId="2" ParentId="1447" CreationDate="2014-04-10T05:45:31.023" Score="2" Body="&lt;p&gt;Consider giving the student something of a meta-motivation.  (In addition to what has been proposed here about using examples from outside of mathematics, like the bus one.  That is really what will turn the light on in their heads.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explain that in mathematics, there are many times when we want to talk about things being &quot;equal-ish&quot;.  They aren't actually &lt;em&gt;the same thing&lt;/em&gt;, but, functionally, for our particular purposes at that time, they &quot;do the same thing&quot; or &quot;look the same&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, since this is math, we have to wave our hands very precisely, so here's how we're going to define it (give definition).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now show how &quot;Bus is numbered 42&quot; fits by going through all the parts of the definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three examples from mathematics is a great motivation to you, but likely not meaningful to your students.  The equivalence relation is a new concept; have them learn the new concept by seeing how it applies to something they are familiar with so that they are only dealing with one new thing and one familiar thing.&lt;/p&gt;&#xA;" OwnerUserId="371" LastActivityDate="2014-04-10T05:45:31.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1487" PostTypeId="2" ParentId="1447" CreationDate="2014-04-10T05:51:16.590" Score="3" Body="&lt;p&gt;The example I always use is the equivalence relation, on a set of humans, &quot;born in the same year&quot;.&lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2014-04-10T05:51:16.590" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1488" PostTypeId="2" ParentId="1351" CreationDate="2014-04-10T07:19:38.573" Score="4" Body="&lt;p&gt;When we talked about Banach-Tarsky in an introductory topology course back in the day, the professor explained it informally something like, &quot;You can take a sphere and cut it into two pieces, and then reassemble those pieces to make two spheres of the same size!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student, looking a bit confused, asked, &quot;How do you cut it?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He thought for a moment and replied, &quot;Very, very carefully!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I thought this was both hilarious and a pretty good explanation at a layman level.  (Others have explained the technical aspects of it very well above.)  He included a few hand gestures that made it pretty clear to everyone that we weren't going to be able to make this cut with a pair of kindergarten safety scissors and a ping-pong ball.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a layman level, that gets to a very basic reason why you can't do this in real life: You just won't be able to make the cut.  For starters, the required cut is infinitely long.  It's also infinitely curvy.  Both of those are actually impossible in the real world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During that same class period another student asked, &quot;What do you mean when you say it's a paradox?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;A paradox,&quot; replied our professor, &quot;A paradox is . . . a surprise.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About as succinct an explanation as you're likely to find . . . &lt;/p&gt;&#xA;" OwnerUserId="819" LastActivityDate="2014-04-10T07:19:38.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1489" PostTypeId="2" ParentId="557" CreationDate="2014-04-10T07:40:36.437" Score="3" Body="&lt;p&gt;Given that in a university setting students are in class voluntarily, I handle people doing non-class-related activities in the same way: ask them to put it away or do it elsewhere, particularly if it disrupts the class in some way. It is normally obvious who is using the computer or tablet for taking notes or checking the class notes.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-10T07:40:36.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1490" PostTypeId="2" ParentId="1448" CreationDate="2014-04-10T07:58:59.427" Score="2" Body="&lt;p&gt;Each problem is different, I think it is best to illustrate proof techniques with a variety of problems. The problems themselves should be simple, easy to understand, perhaps even well known to the audience. I emphasize writing proofs in a schematic way: announce what strategy will be used, and write down the steps, explaining each one. I like Hammack's &lt;a href=&quot;http://www.people.vcu.edu/~rhammack/BookOfProof&quot; rel=&quot;nofollow&quot;&gt;The book of proof&lt;/a&gt; as a guide.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A full course on this is probably nonsense, I'm a firm believer in that you learn stuff only if you use it in real settings, by applying it. Give a short overview at the beginning of the class, use the schemas introduced consistently, and make frequent references.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-10T07:58:59.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1491" PostTypeId="2" ParentId="1183" CreationDate="2014-04-10T08:16:38.647" Score="2" Body="&lt;p&gt;The &quot;peer review&quot; term is really misleading, what it is really about (at least for the first papers you submit) is having someone experienced read and comment on the text. Basic understandability and other gross characteristics can be assessed by class mates, finer points only by a TA or teacher. You'd need to combine both methods. Having their text read by classmates (and seeing their mistakes firsthand) should help both parts, probably as a required step to be certified somehow as really done, but not part of the grade.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-10T08:16:38.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1492" PostTypeId="2" ParentId="1448" CreationDate="2014-04-10T09:25:00.570" Score="5" Body="&lt;p&gt;I would say that one of the main things to make sure is that the material covered satisfies two things: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The material itself (i.e. those statements they end up proving as part of practicing proofs) will be useful for them later.&lt;/li&gt;&#xA;&lt;li&gt;There will be plenty of statements that are not quite obvious, but which are easy to prove by applying the correct technique together with the definitions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The reason for the first is that this makes it much easier to justify spending a course learning this stuff, and therefore the students will hopefully not think it is a waste of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for the second is that many students will get desillutioned if all they are able to prove are statements that are completely obvious, and at the same time, proving something which is &quot;obvious&quot; can in fact be incredibly hard unless you are already very experienced (the students will often end up very confused about which facts they are allowed to use and which ones not).&lt;br&gt;&#xA;That the statements should not be too hard to prove when using the right technique and using the definitions seems obvious, as the students will otherwise be stuck not being able to think of the correct &quot;trick&quot;, and will end up not getting the proper feeling of what it takes to prove something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above then of course begs the question: &quot;Which topics satisfy this&quot;, to which my best reply will be from personal experience, having taken a course with these topics myself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that course, the curriculum was:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Naive logic, so everyone knew the precise meaning of the various connectives and what it took to prove a statement involving them.&lt;/li&gt;&#xA;&lt;li&gt;Naive set theory to get all the notation in that context settled.&lt;/li&gt;&#xA;&lt;li&gt;Relations, especially orderings, equivalence relations and functions.&lt;/li&gt;&#xA;&lt;li&gt;Induction and recursion.&lt;/li&gt;&#xA;&lt;li&gt;Construction of the reals using Cauchy sequences.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The first two topics are very important to get everyone on the same page when it comes to the notation used, and introducing the various proof techniques fits in well while introducing this notation (since it is related to the rules for how one can manipulate the connectives).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Induction is of course vitally important for all mathematicians to know, and getting introduced to it more formally can hopefully make the students more aware of precisely why and how it works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Relations, especially the special types covered, are of course something everyone will end up using a lot, and so getting a formal introduction is nice, especially as this is a topic that allows for a huge amount of not quite obvious but not really hard exercises (such as equivalent definitions of being injective or surjective, equivalence relations corresponding to partitions and so on). This topic also allows for a lot of examples, many of which the students will have seen without the formality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The final topic, construction of the reals, is a little bit different, in that many of the proofs involved will be a lot harder.&lt;br&gt;&#xA;But it does introduce some quite nice things to the students, since the construction of something as the set of equivalence classes of something else is done a lot in mathematics (in some sense one might call it the way mathematicians &quot;patch&quot; sets to obtain missing objects, such as in this case obtaining the limits of sequences that should converge but do not, by declaring the sequence itself to represent its limit).&lt;br&gt;&#xA;At the same time, this topic will have the students do several epsilon-delta proofs, which is something most students at this level will still need to practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The course in question used the book Chapter Zero by Carol Schumacher, which I heartily recommend.&lt;/p&gt;&#xA;" OwnerUserId="286" LastActivityDate="2014-04-10T09:25:00.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1493" PostTypeId="2" ParentId="1448" CreationDate="2014-04-10T09:37:23.097" Score="2" Body="&lt;p&gt;Some combinatorics: For example the identity $\binom{n}{r} = \binom{n-1&#xA;}{r} + \binom{n-1}{r-1}$ can be proven using the formula involving factorials. It can also be proved using combinatorial interpretation. Similarly Vandermonde's identity can be presented in two ways.&lt;/p&gt;&#xA;" OwnerUserId="548" LastEditorUserId="123" LastEditDate="2016-01-08T00:34:04.940" LastActivityDate="2016-01-08T00:34:04.940" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1494" PostTypeId="2" ParentId="1447" CreationDate="2014-04-10T09:51:27.830" Score="5" Body="&lt;p&gt;You might attempt to motivate them using partitions. Present them with the attitude that the three properties (reflexivity, symmetry, and transitivity) are simply a convenient way to check that something is indeed a partition. Partitions are easy to visualize, so even if a student doesn't know any concrete applications yet, it should be plausible to them that such a thing could be useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every other answer just gives examples of equivalence relations. This answer is more about the general &lt;em&gt;point of view&lt;/em&gt; from which you look at them.&lt;/p&gt;&#xA;" OwnerUserId="115" LastActivityDate="2014-04-10T09:51:27.830" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1495" PostTypeId="1" AcceptedAnswerId="1499" CreationDate="2014-04-10T12:48:18.100" Score="14" ViewCount="509" Body="&lt;p&gt;In most universities that I am aware of, graduate courses in Mathematics are taught almost exclusively by research mathematicians. My question has two parts:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is the justification for this practice?&lt;/p&gt;&#xA;&lt;p&gt;Has research been done showing any correlation between research ability and students' learning in graduate courses?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-10T19:02:58.040" Title="Is there any research on the correlation between a professor's research ability and their ability to teach graduate level courses?" Tags="|graduate-education|education-research|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1496" PostTypeId="2" ParentId="1455" CreationDate="2014-04-10T12:55:01.163" Score="5" Body="&lt;p&gt;Case 1: If it is a minor mistake (e.g. wrong sign) point it out, correct it and go on. If the mistake is a grave one, stop and explain what went wrong (maybe even ask students to spot the mistake themselves)&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Case 2: Depends on how much later. If it is still in the current class, correct it immediately (see case 1). If it was in the latest class mention and correct the error at the beginning of the next class. If the mistake was made far earlier try to hand out corrected papers and/or take some time at the next class to address it, especially if it isn't something as simple as a wrong sign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Case 3: Congratulate the student who found it, proceed like case 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reason: Everybody can and does make mistakes from time to time, but it is handled best if you can spot and correct the mistake instantly before the students have time to internalize (memorize) it. Better yet, explain how to avoid that kind of mistake - especially if your students can't find the mistake on their own.&#xA;&lt;hr/&gt;&#xA;I personally can relate to the third case, but as the student who spotted the mistake. My teachers handled it quite well, correct the error and proceed with the lesson as planned. They didn't take it personal, they even encouraged others to find their mistakes. As a student this was kind of a relief: one was sure to get a &quot;right&quot; result and one could see that the teacher isn't perfect either, which gave some students a morale boost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally I think it's a good idea to explain how to spot and avoid most common mistakes, since this is the problem most students have when studying on their own. Most teachers don't plan to do this at all because normally their prepared classes work out nicely so there aren't many possibilities for mistakes to happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S.: English isn't my native language, feel free to correct me if something isn't clear/easy to understand ;)&lt;/p&gt;&#xA;" OwnerUserId="826" LastActivityDate="2014-04-10T12:55:01.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1497" PostTypeId="1" CreationDate="2014-04-10T13:12:09.213" Score="10" ViewCount="319" Body="&lt;p&gt;I once read a book on infinity that was written by an American female maths writer. Her writing was very easy to read and she was a great explainer of concepts. A very distinct aspect of her books was that there were only a few words on each line to assist with reading quickly. (For these reasons, I think the author and her works would be of interest to folks reading this site. Once this question has been accurately answered, I encourage you to check out her books.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone tell me her name? &lt;/p&gt;&#xA;" OwnerUserId="827" LastEditorUserId="1385" LastEditDate="2017-04-26T13:34:04.943" LastActivityDate="2017-04-26T13:34:04.943" Title="Looking for a specific female maths writer, who has written on infinity" Tags="|reference-books|infinity|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1498" PostTypeId="1" AcceptedAnswerId="1501" CreationDate="2014-04-10T13:13:11.070" Score="14" ViewCount="1972" Body="&lt;p&gt;I think the title says it all, but in case it does not:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is being a math teacher gratifying?&lt;/li&gt;&#xA;&lt;li&gt;If yes, what is gratifying in being a math teacher? (If not, why...)&lt;/li&gt;&#xA;&lt;li&gt;Does the feeling lasts, or perhaps fades or blossoms with age?&lt;/li&gt;&#xA;&lt;li&gt;What events that happened to you as a teacher were most gratifying?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The above questions are just suggestions, what I'm looking for is opinions or testimonies that could inspire. For example, one could use it during training new instructors, or explaining to a new grad student some benefits of teaching. The only requirement is &lt;em&gt;self-containment&lt;/em&gt;, that is, the posts should include any context that is necessary to understand why it is/was gratifying for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel this is a borderline off-topic and should be made at least CW. On the other hand, there's no better place to gather such stories but here. I guess it will be of substantial help to anyone who needs to hear why being a teacher is great and worth its sweat (e.g. prospective new teachers or current teachers feeling down at the moment).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the question is too broad for you, you could always try sharing the moment that was most gratifying in your career as a mathematics teacher.&#xA;Relevant meta post is &lt;a href=&quot;https://matheducators.meta.stackexchange.com/questions/310/on-the-question-what-is-gratifying-in-being-a-mathematics-teacher&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the scope, I'm mainly interested in university-level education. However, illuminating examples from high school or even elementary school are also welcome (especially if they regard gifted students during some math circles, etc.).&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-03-16T16:41:53.880" LastActivityDate="2023-02-02T15:33:06.370" Title="What is gratifying in being a mathematics teacher?" Tags="|undergraduate-education|student-motivation|graduate-education|teachers|" AnswerCount="8" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1499" PostTypeId="2" ParentId="1495" CreationDate="2014-04-10T13:44:23.783" Score="12" Body="&lt;p&gt;Researchers have &lt;a href=&quot;http://www.math.ksu.edu/~bennett/onlinehw/qcenter/ballmkt.pdf&quot;&gt;philosophized about&lt;/a&gt; and &lt;a href=&quot;http://sitemaker.umich.edu/lmt/files/hillrowanball.pdf&quot;&gt;demonstrated that&lt;/a&gt; there is a specific kind of knowledge that teachers need that is likely different from just knowing the content itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, there is data that demonstrates that knowledge of content alone is not what good teachers need. Rather, they need knowledge of things like how students make sense of the content, how to design tasks that align with knowledge of how students think and move their thinking forward, how to create representations that help students see mathematical relationships, that sort of thing. This knowledge aligns with what &lt;a href=&quot;https://archive.org/details/childandcurricul00deweuoft&quot;&gt;John Dewey&lt;/a&gt; called &quot;psychologizing the curriculum,&quot; to connect the logic of the discipline with how the learner thinks and experiences the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that the common assumption is that subject matter knowledge is enough to be an effective teacher -- experts in the discipline could explain what they know and then students will learn it. A faulty aspect of this assumption is the idea that teaching is more than transmitting knowledge. Teaching involves translating knowledge in ways that make sense to the student and giving students an experience to make sense of the knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I don't know of research that correlates between knowledge of the discipline and ability to teach. I just knew of research that shows that it's more than knowledge of the discipline that matters, but instead an additional, more specialized knowledge for teaching that matters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I recognize that the research I am sharing is about teaching at the K-12 level, but I think the general principle -- that knowledge needed to teach well is more than content knowledge located in the discipline -- carries over and transfers into the university level.)&lt;/p&gt;&#xA;" OwnerUserId="739" LastActivityDate="2014-04-10T13:44:23.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1500" PostTypeId="2" ParentId="1497" CreationDate="2014-04-10T13:50:32.280" Score="10" Body="&lt;p&gt;Lillian R. Lieber&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Author of &lt;a href=&quot;http://books.google.com/books?id=AhI1Uea_Pb8C&amp;amp;lpg=PP1&amp;amp;dq=inauthor%3a%22Lillian%20R.%20Lieber%22&amp;amp;pg=PA18#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;Infinity: Beyond the Beyond the Beyond&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-04-10T13:50:32.280" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1501" PostTypeId="2" ParentId="1498" CreationDate="2014-04-10T14:30:34.587" Score="17" Body="&lt;p&gt;In searching for information about hours of work per week by teachers at the secondary vs. tertiary level (for an earlier question) I came across a nice report by Scholastic and the Bill and Melinda Gates Foundation. A summary-type page can be found &lt;a href=&quot;http://www.scholastic.com/primarysources/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;; the full report (pdf) can be found &lt;a href=&quot;http://www.scholastic.com/primarysources/PrimarySources3rdEditionWithAppendix.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(From the same study) A nice summary of &amp;quot;teachers on teaching&amp;quot; can be found &lt;a href=&quot;http://www.scholastic.com/primarysources/teachers-on-teaching.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;; it begins:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/vpZKy.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/kl3ur.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Much more can be found in the full report.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;For now, I exclude in-depth discussion of my own personal experiences, though I can briefly answer your four questions: I find teaching to be very gratifying; for me, the gratification comes primarily from the first six bars in the histogram above; I'm at an early stage in my career, but thus far the feeling has only grown; the most gratifying experiences, of course, have come from seeing those who I helped (by &amp;quot;teaching&amp;quot; or, as I prefer, &amp;quot;working alongside&amp;quot;) demonstrate meaningful learning has occurred.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Closing comment:&lt;/strong&gt; With regard to one of the above-described sources of gratification (&amp;quot;to help students reach their full potential&amp;quot;) there is a nice piece called &amp;quot;An Exhortation to Learning&amp;quot; 《劝学》 by the Confucian philosopher &lt;a href=&quot;http://en.wikipedia.org/wiki/Xunzi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Xunzi&lt;/a&gt; (荀子). In this piece (ca. mid-200s BCE) he writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My translation: The dye of the indigo plant is even bluer than the plant itself&lt;/p&gt;&#xA;&lt;p&gt;Original wording: 青，取之于蓝，而青于蓝&lt;/p&gt;&#xA;&lt;p&gt;Modern phrasing: 青出于蓝而胜于蓝&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is written in regard to teaching, where the goal, stated metaphorically, is to produce students whose abilities exceed those of the teachers (just as the plant produces a dye bluer than itself). More than two millennia later, I believe this is still a valuable goal to keep in mind; and, for me, it is the most gratifying outcome of being a teacher.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorDisplayName="user18187" LastEditDate="2023-01-30T13:38:03.910" LastActivityDate="2023-01-30T13:38:03.910" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="1502" PostTypeId="1" CreationDate="2014-04-10T16:12:38.493" Score="10" ViewCount="459" Body="&lt;p&gt;Here is a common process I witness in office hours:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Student struggles with a difficult concept.&lt;/li&gt;&#xA;&lt;li&gt;Student gains understanding of the concept.&lt;/li&gt;&#xA;&lt;li&gt;Student, now bewildered since the concept is mastered, declares &quot;Oh! That's it??.&quot; &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;They not only learned the concept, but they have declared the concept to be &lt;strong&gt;easy&lt;/strong&gt;!!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student then maintains the following incorrect perception: &lt;strong&gt;&quot;I only understand easy math.&quot;&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I hear this all the time, at every level. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Some percentage of students in Calculus II think that Calc I was easy, which is why they got an A, but now Calc II is hard, so they are not succeeding.&lt;/li&gt;&#xA;&lt;li&gt;Some percentage of students in algebra think that arithmetic was easy, which is why they got an A, but now algebra is hard. &lt;/li&gt;&#xA;&lt;li&gt;To be honest, even I personally have the perception that everything before Hartshorne's Algebraic Geometry is relatively easier, which is why I did well in it!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;How many of these people are correct? Is there any research or profound wisdom on this issue out there? I'd prefer research but I'm not sure how it would even be conducted.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;All math is approximately the same difficulty, from arithmetic to algebraic geometry. People slip up because of a bad semester somewhere along the way and that is why anyone ever gets stuck in math, or&lt;/li&gt;&#xA;&lt;li&gt;There are &quot;hard spots&quot; in math where you can get stuck; the hard spots become more and more dense as you progress further in math, or&lt;/li&gt;&#xA;&lt;li&gt;Something else??&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2014-04-11T14:58:41.330" LastActivityDate="2014-04-11T14:58:41.330" Title="Students who have succeeded before but are not succeeding now" Tags="|advising|student-motivation|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1503" PostTypeId="2" ParentId="897" CreationDate="2014-04-10T16:59:15.910" Score="8" Body="&lt;p&gt;I am a mathematics educator who conducts research on teaching and learning, and I also teach mathematics courses that are mathematics content courses for future teachers. I endorse the references given here! Great ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another option: &lt;a href=&quot;https://www.dropbox.com/s/kv20232ca7xtjeo/Mathematics%20and%20Mathematics%20Education%20ebook.pdf&quot;&gt;Mathematics and Mathematics Education: Searching for a Common Ground&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="739" LastActivityDate="2014-04-10T16:59:15.910" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1504" PostTypeId="2" ParentId="875" CreationDate="2014-04-10T17:09:38.947" Score="8" Body="&lt;p&gt;Here is an &lt;a href=&quot;http://www.npr.org/blogs/health/2012/11/12/164793058/struggle-for-smarts-how-eastern-and-western-cultures-tackle-learning&quot;&gt;NPR article&lt;/a&gt; that discusses how teachers' efforts to engage learners in productive struggle (or not) may be culturally situated. (Of note, Benjamin cites Jim Hiebert above, who has written The Teaching Gap with Jim Stigler, interviewed in this article.) Jim and Jim have conducted research on how &lt;a href=&quot;http://timssvideo.com/sites/default/files/Math%20Teaching%20in%20the%20US%20Today%20and%20Tomorrow.pdf&quot;&gt;instructional approaches differ culturally between nations&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="739" LastActivityDate="2014-04-10T17:09:38.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1505" PostTypeId="2" ParentId="194" CreationDate="2014-04-10T17:19:58.207" Score="11" Body="&lt;p&gt;I agree with the idea that different people might be anxious about mathematics for different reasons...&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.npr.org/blogs/health/2012/11/12/164793058/struggle-for-smarts-how-eastern-and-western-cultures-tackle-learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Culturally&lt;/a&gt;, in the United States, we tend to look at capabilities in mathematics as determined by &amp;quot;ability&amp;quot; and from a &amp;quot;fixed&amp;quot; ability&amp;quot; perspective –  some people can do math and others cannot. Other nations attribute capabilities in mathematics to effort – everyone can improve at doing mathematics if they put in time and effort. It is likely that anyone with a &amp;quot;fixed ability&amp;quot; mindset will encounter anxiety when they struggle to understand something (oh no! I am struggling to make sense of this! I guess I am not as smart as I thought I was / I guess I'm not good at math!). [Carol Dweck's book – &lt;a href=&quot;https://www.mindsetworks.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mindset&lt;/a&gt; – gets at this issue!]&lt;/p&gt;&#xA;&lt;p&gt;So, one thing we can do as teachers, is promote a growth mindset – that effort combined with opportunity to learn (with support) leads to capabilities. A small thing: When people say, &amp;quot;I don't understand!&amp;quot; Instead, it's: &amp;quot;I don't understand YET.&amp;quot; More things – look up promoting a growth mindset in math. For instance: &lt;a href=&quot;https://blogs.edweek.org/teachers/classroom_qa_with_larry_ferlazzo/2012/10/response_classroom_strategies_to_foster_a_growth_mindset.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. And Dweck has research that indicates that intervening to promote a growth mindset makes a positive different.&lt;/p&gt;&#xA;" OwnerUserId="739" LastEditorDisplayName="user7990" LastEditDate="2020-09-07T20:48:40.757" LastActivityDate="2020-09-07T20:48:40.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1506" PostTypeId="1" AcceptedAnswerId="1594" CreationDate="2014-04-10T17:20:34.493" Score="14" ViewCount="810" Body="&lt;p&gt;Consider the following way to teach division:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Division works this way: any product equation &lt;span class=&quot;math-container&quot;&gt;$xy = z$&lt;/span&gt; can be rewritten as a quotient equation &lt;span class=&quot;math-container&quot;&gt;$x = \frac{z}{y}$&lt;/span&gt;. Just move the numbers in that way.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is true, but is a horrible way to think about division! Instead we show students that they can &lt;em&gt;divide both sides of an equation by y&lt;/em&gt;, and then that &lt;em&gt;multiplication is the inverse operation of division&lt;/em&gt; so the two resulting y's on the left cancel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the following way to teach logarithms (which is &lt;strong&gt;very&lt;/strong&gt; common):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Logarithms work this way: any exponential equation &lt;span class=&quot;math-container&quot;&gt;$x^y = z$&lt;/span&gt; can be rewritten as a logarithmic equation &lt;span class=&quot;math-container&quot;&gt;$\log_x z = y$&lt;/span&gt;. Just move the numbers in that way.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is true, but isn't it equally bad? Shouldn't we be demanding that students take the &lt;span class=&quot;math-container&quot;&gt;$\log_x$&lt;/span&gt; of both sides?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Many algebra textbooks contain questions as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Convert the following logarithmic equation into an exponential equation: &lt;span class=&quot;math-container&quot;&gt;$\log_5 x = 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They imply that the way to deal with logarithms is to &lt;em&gt;move symbols around&lt;/em&gt;. Is there some [historical? pedagogical?) reason this approach is so widely accepted even though the same approach seems ridiculous when talking about multiplication and division?&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="64" LastEditDate="2019-03-24T18:12:48.200" LastActivityDate="2023-02-02T10:30:10.690" Title="The &quot;rearranging&quot; approach to teaching logarithms" Tags="|mathematical-pedagogy|algebra|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="1507" PostTypeId="2" ParentId="1447" CreationDate="2014-04-10T18:04:55.180" Score="2" Body="&lt;p&gt;I am not sure if this is a valid answer to this question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I would probably say that equivalence relation is a fancy name for a partition of sets and equivalence class is a fancy name for a cell of this partition. (A partition of a set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is an expression for &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; as a disjoint union of non-empty subsets of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Indeed, one has the following &amp;quot;&lt;em&gt;theorem&lt;/em&gt;&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; There is a one-to-one correspondence between equivalence relations on a set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and partitions of the set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Proof.&lt;/em&gt; Note first that if &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; is an equivalence relation on &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, then, &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt;-classes (non-empty by definition) partition &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;. Conversely, if &lt;span class=&quot;math-container&quot;&gt;$X = \sqcup_{\lambda \in \Lambda} X_\lambda$&lt;/span&gt; is a partition of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, define a relation &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$x \sim y$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$x, y \in X_\lambda$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;. Note then that &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; is an equivalence relation and &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt;-classes are precisely the &lt;span class=&quot;math-container&quot;&gt;$X_\lambda$&lt;/span&gt;'s.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then, this becomes a matter of language as it should.&lt;/p&gt;&#xA;" OwnerUserId="109" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-10T18:04:55.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1508" PostTypeId="1" AcceptedAnswerId="1528" CreationDate="2014-04-10T18:25:50.053" Score="12" ViewCount="918" Body="&lt;p&gt;I often see students confused/mystified by definitions (and proofs) that allow/consider &quot;useless&quot; cases. A case in point is the definition of a DFA (deterministic finite automaton), which allows states that aren't reachable at all from the starting state; also the proof that any NFA (nondeterministic finite automaton) can be &quot;simulated&quot; by a DFA whose states are subsets of the states of the NFA, again, the vast majority of those states are useless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When defining DFAs I point out that such superfluous states are OK by the definition, but my students stumble over this anyway. They ask why the definition allows such nonsense. The obvious answer is that not forbidding them makes for a simpler definition, much easier to work with; but it grates their (engineering outlook) sense of parsimony the wrong way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure other areas have similar phenomena. What do you do about this?&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-23T00:22:59.153" Title="Definitions/proofs that allow &quot;useless&quot; cases?" Tags="|mathematical-pedagogy|proofs|" AnswerCount="2" CommentCount="1" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1509" PostTypeId="2" ParentId="1508" CreationDate="2014-04-10T18:45:16.030" Score="8" Body="&lt;p&gt;I think the most useful analogy is standard arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can express the number &quot;4&quot; in many ways. The most obvious is &lt;code&gt;4&lt;/code&gt;, but we also get &lt;code&gt;2+2&lt;/code&gt;, &lt;code&gt;8/2&lt;/code&gt;, &lt;code&gt;9/3 + 1&lt;/code&gt;, &lt;code&gt;3.99999...&lt;/code&gt; and that's not even scratching the surface of the (literally!) infinite ways I can express &lt;code&gt;4&lt;/code&gt;. I challenge the students to explain why arithmetic allows such useless ways to express &lt;code&gt;4&lt;/code&gt;. Why does basic arithmetic allow such nonsense? Why have so many equivalent ways to express such a simple concept as 4? Is it an oversight? Wouldn't it be easier if the only way to express &lt;code&gt;4&lt;/code&gt; was to just write that one symbol?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think overall the best parallel to &quot;empty states&quot; in DFAs is multiplication by 1 and addition of 0. &lt;code&gt;4*1*1*1*1*1... + 0 + 0 + 0 + 0...&lt;/code&gt; is probably one of the silliest statements imaginable, and yet it's allowed and a side effect of very useful properties of math. With DFAs it's the same, it can be a side effect of combining DFAs or converting NFAs that states become unreachable and useless. Culling them can be seen as the same thing they did in Algebra I when they simplified an equation for the first time and their teacher made them change &lt;code&gt;4x/2 + 3 - 4&lt;/code&gt; into &lt;code&gt;2x - 1&lt;/code&gt;. The fact that you &lt;em&gt;can&lt;/em&gt;, if you want, define completely superfluous unreachable states for giggles is merely a side effect, just like adding &lt;code&gt;0&lt;/code&gt; to a number over and over. And just like adding 0 and multiplying by 1, sometimes  it can have surprising benefits (see: multiplying by &lt;code&gt;a/a&lt;/code&gt; and &lt;code&gt;b/b&lt;/code&gt; so you can do the operation &lt;code&gt;1/a+1/b&lt;/code&gt;).&lt;/p&gt;&#xA;" OwnerUserId="521" LastActivityDate="2014-04-10T18:45:16.030" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1510" PostTypeId="2" ParentId="1404" CreationDate="2014-04-10T18:45:47.023" Score="10" Body="&lt;p&gt;For a &quot;writing in the disciplines&quot; course, which it sounds like this is, think about what it means to be a writer in this discipline.  For a writing in mathematics course, I'd expect writing of a quality that could be published in a mathematical journal.  This means that, yes, I would expect proper spelling/grammar/punctuation/formatting.  But primarily I'd expect clear exposition of the mathematics being presented, and mathematical rigor.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does your school not have guidelines for grading in the &quot;writing in the disciplines&quot; classes?  Since this is probably a graduation requirement it would make sense for the school to specify what its goals are for students upon completion of your course.  Use those goals to guide your grading.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In all cases, provide a clear rubric to the students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I assume that in addition to grading students on their mathematical writing, you are expected to &lt;em&gt;teach&lt;/em&gt; the students to be good mathematical writers (and not just assign a paper and expect them to figure it out).  The pointers that others have posted here will be helpful.  In the course of teaching students to be good writers of mathematics, you will naturally touch on the points that end up in your rubric.  So students should have a good idea of how to achieve the goals you set for them in the rubric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-04-10T18:45:47.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1511" PostTypeId="2" ParentId="1502" CreationDate="2014-04-10T18:47:51.887" Score="5" Body="&lt;p&gt;Perhaps because all math is simple once you understand it? ;-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a bit of reflection, you'll see that each area of human endeavour advances until the going gets though for the geniuses in the area, who struggle at the forefront. They are followed by bright people who struggle to understand them, and sometimes try to make the genius' findings understandable to the unwashed masses. Thus in the end for us integration by parts is routine, but it was a very complicated procedure to Leibnitz (IIRC, Dunham in one of his books tells the story). The going gets harder as you approach the bleeding edge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On there being specific tough spots, I've heard it said that abstract algebra &quot;separates men from boys,&quot; and in my own experience (personal and as a teacher) many people have a &lt;em&gt;very&lt;/em&gt; hard time wrapping their brain around abstract concepts with no direct connection to everyday experience (thus struggle with group theory, for example, or in my case in Computer Science with automata and languages). Once the hurdle is overcome, it is smooth sailing. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-10T18:47:51.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1512" PostTypeId="2" ParentId="1498" CreationDate="2014-04-10T18:58:26.130" Score="4" Body="&lt;p&gt;In part, we stick to it because we like it. We would be doing something else otherwise... and pay is way better elsewhere. Small wonder that the survey turns out as it does.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal reasons to teach (not math, in general) at the university is that it keeps you in touch with very bright, motivated people (no, I don't believe the old chant of &quot;students are even dumber this year around&quot;). Your work is ever changing, it doesn't get boring. To a large extent, you define your own tasks, a freedom seldom seen elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And it is certainly a big satisfaction when you see your ex-students succeed in life later on.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-10T18:58:26.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1513" PostTypeId="2" ParentId="1495" CreationDate="2014-04-10T19:02:58.040" Score="2" Body="&lt;p&gt;Graduate school &lt;em&gt;is&lt;/em&gt; about research, what the students are supposed to learn is how to do it on their own (and in uncharted areas, to booth). The only people who have shown they know about research, and know where uncharted areas are, are active researchers. Most presumably suck baby elephants through straws as teachers, but it's all you have got...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-10T19:02:58.040" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1514" PostTypeId="1" CreationDate="2014-04-10T19:17:28.310" Score="18" ViewCount="1023" Body="&lt;p&gt;No one would argue against the idea/ observation that proofs are very important in mathematics. Some people are trying to make their notations on a blackboard during a lecture as consistent as possible with their course notes or textbooks; including full reproduction of rigorous and detailed proofs for all theorems, corollaries and so on. On the other hand, some believe that the theory with detailed proofs should be in textbooks, while it is crucially important to explain main ideas via examples. So, my question is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;How do you think, what is more important at a lecture: 1) to give a rigorous proof or 2) only formulate a statement and give an intuitive understanding via illustrative examples?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.:&lt;/strong&gt; I understand that it would be great to do both. But in real life we have natural time limitations. And feel free to edit this post because of my bad English :).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; this is for undergraduate and graduate level.&lt;/p&gt;&#xA;" OwnerUserId="506" LastEditorUserId="409" LastEditDate="2015-03-31T03:19:21.907" LastActivityDate="2015-03-31T03:19:21.907" Title="Rigorous proofs vs. illustrative examples" Tags="|mathematical-pedagogy|course-design|examples|proofs|" AnswerCount="5" CommentCount="8" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1515" PostTypeId="2" ParentId="1514" CreationDate="2014-04-10T19:26:52.030" Score="10" Body="&lt;p&gt;What your class should aim at is &lt;em&gt;understanding&lt;/em&gt;. There are some topics (theory of automata, for example) where most proofs of intuitively obvious results are long-winded multiple inductions. In such cases, stick to intuiton on the blackboard, remind students they should at least glance over the proofs, and hopefully follow one of them in detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the result has the happy coincidence of an understandable, short proof, go for it on the blackboard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also think of what you are trying to accomplish: A course for non-mathematicians should emphasize intuition and applicability, a course for specialists should dig into proofs. The &quot;understanding&quot; alluded to above can happen at very different levels.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-10T19:26:52.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1516" PostTypeId="2" ParentId="1502" CreationDate="2014-04-10T19:50:53.617" Score="9" Body="&lt;p&gt;It seems to me that you're wondering about two different issues:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;are some topics or areas in mathematics more or less difficult to understand?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the nature of developing an understanding.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I can't speak to the first point, but the second point... there is something about arriving at an understanding such that you cannot ever imagine what it was like to NOT understand. The insight is so great, the sense of knowing is strong, and so you almost immediately forget that you didn't know, feel like it was weird that you never knew it. It's not because the topic was inherently easy that you understood -- but that once you understand something, it seems &quot;easy!&quot; It's an attribution error to assume that if you understand something, then it must be easy.&lt;/p&gt;&#xA;" OwnerUserId="739" LastEditorUserId="739" LastEditDate="2014-04-10T19:56:20.713" LastActivityDate="2014-04-10T19:56:20.713" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1517" PostTypeId="2" ParentId="1498" CreationDate="2014-04-10T20:02:04.590" Score="9" Body="&lt;p&gt;What I find gratifying about teaching, math specifically, is learning from my students. I cannot count how many times I have had students come to me and ask me to check their work because they were not sure they were doing it correctly, and they end up teaching me a method of solving the problems that I have never thought of before. I love showing students who do not believe in themselves, that they can, in fact, &quot;do&quot; math. One of the worst things about beginning a new school year is having students with little to no confidence in their math abilities and then helping them realize that all they have to do is try.&lt;/p&gt;&#xA;" OwnerUserId="839" LastEditorUserId="64" LastEditDate="2016-04-19T00:10:39.707" LastActivityDate="2016-04-19T00:10:39.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1518" PostTypeId="2" ParentId="1514" CreationDate="2014-04-10T20:06:05.550" Score="15" Body="&lt;p&gt;I like vonbrand's emphasis on &quot;think of what you are trying to accomplish.&quot; If the goal is to have the students writing proofs, doing a proof in class is not actually about the theorem, it's about you demonstrating the process of writing a proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often the goal is conceptual understanding, which means that an illustrative example, that hopefully has access so that the students can play with it, is often far superior to a proof. The exception, to me, is a good constructive proof that gets at the mathematical structure. An existence proof by construction, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As the OP points out, time in class is precious! Don't spend it on coverage, spend it on engaged activity.&lt;/p&gt;&#xA;" OwnerUserId="12" LastActivityDate="2014-04-10T20:06:05.550" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1519" PostTypeId="1" CreationDate="2014-04-10T20:22:42.750" Score="6" ViewCount="230" Body="&lt;p&gt;This question is similar to &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/805/should-i-design-my-exams-to-have-time-pressure-or-not&quot;&gt;Should I design my exams to have time-pressure or not?&lt;/a&gt;&quot;, but is, I think, sufficiently different to deserve its own space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Naively, there are two benefits to placing (any, or at least strict) time limits on tests: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It saves &lt;em&gt;me&lt;/em&gt; time giving tests (not a noble end, but not so ignoble it deserves to be completely discarded either), and&lt;/li&gt;&#xA;&lt;li&gt;Students who really &lt;em&gt;really&lt;/em&gt; &quot;get&quot; the material will have not struggle with time limits, so I can single those students out by placing (severe-enough) time restrictions on the exam.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I've found ways to make item 1 less important, but I've come to suspect item 2 is just plain false in many cases. I am curious if there is any research supporting this. So we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; is there any research relating to performance on time-restricted mathematics tests and (somehow suitably defined) a student's grasp of the material they are being tested over. (Both positive and negative results on the efficacy of time-restriction are welcome.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;However, on a slightly different note, I'm also dubious whether it's all that important to separate those who really &lt;em&gt;really&lt;/em&gt; &quot;get&quot; the material from those who are &quot;merely competent enough.&quot; So we also have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; is it all that important to separate the &quot;ultra-high-performing&quot; students from the &quot;pretty-darn-good&quot; students, and if not, is there any other reason at all (aside from item 1 above) for giving time-restricted mathematics exams?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;((Please feel free to edit the question, I feel its clarity can be improved.))&lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-04-11T00:02:15.053" Title="Timed Exams, More Generally" Tags="|exams|assessment|time-management|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1520" PostTypeId="1" CreationDate="2014-04-10T20:25:38.450" Score="14" ViewCount="3227" Body="&lt;p&gt;Apologies if this question has some obvious answer. Why are degrees still taught as a measure for angles, to be replaced later by radians (probably confusing many people), rather than just starting with radians in the first place?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;e.g. one could define the circumference of a circle to be $\tau$ or $2\pi$ and then immediately say &quot;let's define the size of a central angle as $\tau$ times the proportion of the circumference made by the arc that it is subtended by&quot;...&lt;/p&gt;&#xA;" OwnerUserId="369" LastEditorUserId="369" LastEditDate="2015-09-29T20:03:52.853" LastActivityDate="2015-09-30T06:28:04.023" Title="Why are degree angle-measurements taught?" Tags="|trigonometry|" AnswerCount="8" CommentCount="19" ContentLicense="CC BY-SA 3.0" />
  <row Id="1521" PostTypeId="2" ParentId="1520" CreationDate="2014-04-10T20:36:58.920" Score="5" Body="&lt;p&gt;Degree measure is used in practical applications such as civil engineering, surveying, and astronomy.  Other trigonometric conventions used in these applied branches are different from those in mathematics: for instance, angles are conventionally measured clockwise from North, rather than counterclockwise from the positive X axis. &lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-04-10T20:36:58.920" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1522" PostTypeId="2" ParentId="1514" CreationDate="2014-04-10T20:49:01.327" Score="9" Body="&lt;p&gt;Although, you're aiming at undergraduate, I'd like to point out some aspects of secondary:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your task as a teacher/prof/lecturer is not to write things at the blackboard, that the students can read in a book. A book is far cheaper than you are. Your task is to make the students learn/understand (a.k.a. teach):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;foundation of knowledge of your subject&lt;/li&gt;&#xA;&lt;li&gt;basic methodology of your subject (in math usually: finding new results, prooving new results, apply new results + quality of definitions)&lt;/li&gt;&#xA;&lt;li&gt;aims, applications, self-concept of your subject&lt;/li&gt;&#xA;&lt;li&gt;where and how to further their knowledge if they are interested in&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So into what aim does a proof fall? Depends on the proof:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If the proof itself is a classical example showing the fundamental technique, then do it. If it takes long, then do one of this kind and let the students find a similar one for another result. (Then you're teaching 2.)&lt;/li&gt;&#xA;&lt;li&gt;If the proof itself is boring/technical/extremely long, but the result is important, then leave the proof out but illustrate the result in its importance/applications/whatever. (Then you're teaching 1. or 3.)&lt;/li&gt;&#xA;&lt;li&gt;If neither proof nor result itself are interesting, then simply name the result and a source for it and the proof. The students should read that source, try to understand it and question it. Check that the next course. (Then you're teaching 4.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-10T20:49:01.327" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1523" PostTypeId="2" ParentId="1520" CreationDate="2014-04-10T21:01:20.690" Score="9" Body="&lt;p&gt;This is a good example for the &lt;a href=&quot;http://en.wikipedia.org/wiki/Spiral_approach&quot;&gt;Spiral approach&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Angle and degrees are introducted in late primary or early secondary. At that stage, students only know integers and proportions are too abstract.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later on in mid-secondary, they learn about arc length and wedge area to be proportional to the angle. Depending on how well the students understand proportions and rescaling, one could introduce radians here as a proportional rescaling of degrees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And finally in late-secondary, they learn about trigonometric functions, circle movements and oscillations, where radians has advantages over degrees. At this stage, they should learn to use both, because both units are just tools for other results. Here, you may redefine an angular measure.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-10T21:01:20.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1524" PostTypeId="2" ParentId="1479" CreationDate="2014-04-10T21:10:25.353" Score="8" Body="&lt;p&gt;My colleagues are very active in this area. They have an IBL calculus and communication in mathematics (proof writing) texts. Another colleague is writing an open-source intermediate algebra text this semester on her sabbatical. These are classroom tested here as well as in other schools, and support more active learning than many commercial textbooks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some resources to investigate:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Matt Boelkin's blog: &lt;a href=&quot;http://opencalculus.wordpress.com/&quot;&gt;http://opencalculus.wordpress.com/&lt;/a&gt; with links to the calculus&#xA;text &lt;/li&gt;&#xA;&lt;li&gt;Ted Sundstrom's Mathematical Reasoning text:&#xA;&lt;a href=&quot;https://sites.google.com/site/mathematicalreasoning3ed/&quot;&gt;https://sites.google.com/site/mathematicalreasoning3ed/&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;George Cain's&#xA;list of online, free university math texts:&#xA;&lt;a href=&quot;http://people.math.gatech.edu/~cain/textbooks/onlinebooks.html&quot;&gt;http://people.math.gatech.edu/~cain/textbooks/onlinebooks.html&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;Albert Schueller's &lt;a href=&quot;http://www.openmathbook.org/&quot;&gt;http://www.openmathbook.org/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12" LastActivityDate="2014-04-10T21:10:25.353" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1525" PostTypeId="2" ParentId="1252" CreationDate="2014-04-10T21:34:35.237" Score="4" Body="&lt;p&gt;Use this opportunity for revisiting place-value-systems and other systems. The 0 in the decimal system was originally just a gap, an empty space. The eastern arabic numerals still only have the zero as a centered dot.&lt;/p&gt;&#xA;&lt;p&gt;Another example are roman numerals. What's the difference between&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;XVI&#xA;and&#xA;X VI?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What's the difference between&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;123 and 1_23?&lt;/p&gt;&#xA;&lt;p&gt;123 and 1023?&lt;/p&gt;&#xA;&lt;p&gt;123 and 0123?&lt;/p&gt;&#xA;&lt;p&gt;123 and 1230?&lt;/p&gt;&#xA;&lt;p&gt;12.3 and 12.03?&lt;/p&gt;&#xA;&lt;p&gt;12.3 and 12.30?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Answer a 0 has no value in itself, but pushes other digits outward onto different values and thus changes the value of a numbers. But if the 0 is at the outward end of a number, it can't push other digits outward. So it doesn't change the value.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-10T21:34:35.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1526" PostTypeId="2" ParentId="1520" CreationDate="2014-04-10T22:06:52.407" Score="13" Body="&lt;p&gt;As others said, degrees are taught, since they are still used. So, the question becomes why are they still used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To purely work with fractions would not be very convenient for various somewhat everyday things, since many people are more used to/better at operating with integers. So to really use $\tau$ and fractions thereof seems incovenient, and one somehow will want to rescale. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, one could say, this is actually how degrees arise, one rescales by a factor of $360$.&#xA;Now the question becomes, why $360$ and not something else, such as perhaps $100$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first answer is that some &lt;em&gt;do&lt;/em&gt; rescale by something else, there is another unit called &lt;a href=&quot;http://en.wikipedia.org/wiki/Gradian&quot;&gt;gradian&lt;/a&gt; where a right angle has $100$ gradian, which came up along with the metric system. There is (are in fact) also &lt;a href=&quot;http://en.wikipedia.org/wiki/Angular_mil&quot;&gt;mil&lt;/a&gt; and a couple other things, see the Wikipedia page on &lt;a href=&quot;http://en.wikipedia.org/wiki/Angle&quot;&gt;angle&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second answer is that $360$ is &lt;em&gt;quite reasonable&lt;/em&gt;, more so than things more based on $10$, if one is interested in geometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To construct a hexagon is an extremely natural thing, so obviously a sixth of a turn is important and should be 'nice.' And a right angle being important too, this should also be 'nice.' So aready the scale should be divisible by $12$. Now, throw in the mix that also the constrcution of a regular pentagon is known since ancient times, and you are at multiples of $60$.     &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-04-10T22:06:52.407" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1527" PostTypeId="2" ParentId="1520" CreationDate="2014-04-10T22:09:42.543" Score="8" Body="&lt;p&gt;First, &lt;strong&gt;it’s a historically grown and widely used standard&lt;/strong&gt; and as such it is as difficult to change as our weird time system, the decimal number system, QWERTY-based keyboards and English orthography. In particular, there are far too many people who only know the degree system and with whom children could not communicate, had they only been taught radians. Also note the failed attempt of introducing &lt;a href=&quot;http://en.wikipedia.org/wiki/Gradian&quot;&gt;gradians&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, If one tried to introduce such a system, it would arguably be a good idea to introduce $π$ like any other unit and tell the students much later that it actually is a number. But even then, &lt;strong&gt;it’s not &lt;em&gt;that&lt;/em&gt; advantageous&lt;/strong&gt;, as the mostly used angles would be $π/4$, $π/3$ and so on, which is not that practical for most people. Of course one could introduce a new unit corresponding to $π/12$ or such, but that would be not be so different from degrees anymore.&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2014-04-10T22:20:58.633" LastActivityDate="2014-04-10T22:20:58.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1528" PostTypeId="2" ParentId="1508" CreationDate="2014-04-10T22:36:00.987" Score="17" Body="&lt;p&gt;Think of a maze, like the ones you can find in kids' books that you trace through with a pencil. This may sound like a silly question, but what's the &lt;em&gt;definition&lt;/em&gt; of a maze? If you ask your students this question, I suspect you'll get some puzzled reactions. I mean, you know a maze when you see one, right? But that very intuition &lt;em&gt;supports&lt;/em&gt; the inclusion of &quot;useless cases&quot; in the definition: if you define a maze in such a way as to exclude those setups where not every corridor is reachable from the entrance, then you definitely &lt;em&gt;don't&lt;/em&gt; know a maze when you see one—indeed, to verify maze-hood in any given case you'd have to manually check that no corridors are unreachable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you make a good case for not excluding mazes with unreachable sections from the definition, it's not such a large leap to not excluding unreachable states from DFAs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But now suppose you run a company that designs mazes for theme parks. In other words, these mazes are actually going to be built, and so it'd be pretty silly for some sections to be unreachable. What this demonstrates is: (1) when you're thinking about the abstract definition of a concept, including &quot;useless&quot; cases can make good sense, but (2) when you're thinking about &lt;em&gt;implementation&lt;/em&gt;, &quot;useless&quot; cases are often absurd. You might have more luck selling this kind of analogy to a crowd with an &quot;engineering outlook&quot;.&lt;/p&gt;&#xA;" OwnerUserId="327" LastActivityDate="2014-04-10T22:36:00.987" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1529" PostTypeId="2" ParentId="1498" CreationDate="2014-04-10T23:16:34.887" Score="7" Body="&lt;p&gt;For me, the strongest immediate gratification comes when a student suddenly understands something: &quot;Oh, now I get it!&quot;  The strongest delayed gratification is when students and former students do well over the years.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-04-10T23:16:34.887" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1530" PostTypeId="2" ParentId="1519" CreationDate="2014-04-11T00:02:15.053" Score="4" Body="&lt;p&gt;I think that it is important, in this situation, to ask yourself the question &quot;why am I even giving students tests at all?&quot;, and the answer to this question will shed light onto the question of timing your exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously we're giving students tests to ensure that they learn the material, but the reason we are ensuring that they learn the material is an economics problem.  High Schools want their students to be accepted to colleges, so they need a grading system, and Colleges want their students to get good jobs.  If there was not a system for telling whether a student learned the material, then employers would not care that a student had a degree or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As educators, we naturally want students to learn the material because 1. We think that the material is important in and of itself and 2. We assume that students will love the material for what it is.  Usually 2 is simply false, especially in high school and undergraduate mathematics, and in any case, neither of these is really a good reason to give exams at all.  We would give interesting homework problems for the students to ponder in order to learn deeply, instead of exams which don't really teach a student anything, but are simply a measurement tool.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if we're giving exams (whether we like it or not) because we are measuring students so that employers can fill positions more efficiently, and you're the sort of person who thinks that this is important, then timed exams ensure that not only does a student know the material, but can utilize the material in a meaningful way in a reasonable amount of time, which is crucial for the 'real world'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're the sort of person who doesn't care about being a sorter for the industrial world, then there's no reason to give exams at all.  At least, to me, it seems that there are better instructional tools for students than exams.  Even if one does view exams as an effective instructional tool, there's no good reason to time them, outside of a reasonable constraint to ensure that the undisciplined student actually completes the exam.&lt;/p&gt;&#xA;" OwnerUserId="842" LastActivityDate="2014-04-11T00:02:15.053" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1531" PostTypeId="2" ParentId="1514" CreationDate="2014-04-11T00:54:31.387" Score="0" Body="&lt;p&gt;It's not an &quot;either or&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think there should be two branches of mathematics taught; theoretical and computational.&#xA;They should have their own chain of prerequisites.&lt;/p&gt;&#xA;" OwnerUserId="843" LastActivityDate="2014-04-11T00:54:31.387" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1532" PostTypeId="2" ParentId="1514" CreationDate="2014-04-11T03:36:58.873" Score="3" Body="&lt;p&gt;My experience in Linear Algebra: while trying to explain how the co-ordinates/matrix  change for a different basis I started with  a 3d-rotation matrix for a basis consisting of a vector in its axis and two vectors orthogonal to it; then carried out the appropriate computation for similarity change to express it for standard basis. Filling out a board with $a_{ij}, b_{kl}$ etc would have been messy. I generally do  more illustrative computations than a formula with lot of notations. &lt;/p&gt;&#xA;" OwnerUserId="548" LastActivityDate="2014-04-11T03:36:58.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1533" PostTypeId="1" AcceptedAnswerId="1534" CreationDate="2014-04-11T03:56:44.673" Score="5" ViewCount="1311" Body="&lt;p&gt;I need to compile a relatively short &quot;Crash Course&quot; in high school mathematics packet, whose aim is to be a quick learning guide for academic competition (think &lt;a href=&quot;http://en.wikipedia.org/wiki/United_States_Academic_Decathlon&quot; rel=&quot;noreferrer&quot;&gt;Aca-dec&lt;/a&gt;) for students with mixed mathematical backgrounds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know (to some degree) what topics I want to cover in the crash course, but I am not sure what format would be the most useful for learning large amounts in short times (these are meant for prep in about two weeks)*. I was thinking of doing a &quot;Problem-Solution-Explanation&quot; (Henceforth PSE), in order to let the reader become the most accustomed to the Problem-Solution format of the competition. However, I know in standard pedagogical writing this is bad form, and intuition helps solidify concepts more than PSE, and so I am not sure. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;*I know from a pedagogical standpoint this is nowhere near the most useful methodology, and I would love to write a whole textbook intuitively explaining high school math, but in this case it is what is necessary. &lt;/p&gt;&#xA;" OwnerUserId="382" LastActivityDate="2014-04-11T10:12:36.440" Title="How to compile a crash course on Mathematics?" Tags="|secondary-education|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1534" PostTypeId="2" ParentId="1533" CreationDate="2014-04-11T04:43:29.810" Score="4" Body="&lt;p&gt;I'm not sure if this is helpful, but the Art of Problem Solving Textbooks are written in a style similar to the one you seem to be proposing, and they are excellent.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.artofproblemsolving.com/index.php?mode=books&quot; rel=&quot;nofollow&quot;&gt;http://www.artofproblemsolving.com/index.php?mode=books&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Problems are chosen to guide the students to discovering the mathematical principles that the section aims to teach.  A set of problems is given, then each problem is solved with a discussion of the important features of the solution.  Often a Key Concept is summarized at the end of the solution.  There is no other explanation.  These are followed by exercises, and more difficult review and challenge problems at the end of the chapter.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These texts are written with gifted math students in mind, and many of the problems are taken from math competitions.  The texts are indeed frequently used by students studying for math competitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course these are complete textbooks and not a &quot;crash course&quot; but my experience with these textbooks is that this problem/solution approach to introducing topics is very effective for the types of students who are typically interested in math competitions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest using both simple problems to motivate and illustrate the topic, and then provide some challenging problems that use or extend the topic, so that the students can see how the various topics might get applied creatively in a competition problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you write it, I'd love to see what you come up with!&lt;/p&gt;&#xA;" OwnerUserId="817" LastEditorUserId="817" LastEditDate="2014-04-11T04:53:18.813" LastActivityDate="2014-04-11T04:53:18.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1535" PostTypeId="2" ParentId="366" CreationDate="2014-04-11T04:50:39.283" Score="7" Body="&lt;p&gt;I've always thought we should teach students more statistics.  Including &quot;uses and abuses&quot; thereof.  Serious statistics does require calculus, of course, but I'd throw a non-calculus-based statistics course into the standard HS curriculum, if I were in charge of the world.  Statistics is important for all citizens, to be able to understand a lot of what is reported in the news.  I would consider it more useful than Algebra II for most students.  (However... I also agree with the arguments that we don't want to assume which students don't have higher math in their future.)&lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-04-11T04:50:39.283" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1536" PostTypeId="2" ParentId="1520" CreationDate="2014-04-11T07:46:53.347" Score="7" Body="&lt;p&gt;I used to be firmly in the &quot;Grr, what use are degrees?&quot; camp until I actually started to &lt;em&gt;use&lt;/em&gt; angles and then I switched to a &quot;Why would anyone &lt;em&gt;ever&lt;/em&gt; use radians?&quot; attitude.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in short, my current attitude is that if someone is likely to want to use actual angles, teach them degrees.  If their interest is theoretical, then they might be able to cope with radians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My use-case that switched my attitude was that of graphical programming.  I frequently want to draw something on the screen rotated by some angle.  I'm far more likely to want to rotate it by some fraction of a circle than anything else, so I want to work with proportions of a full circle.  At this point, I could use degrees, gradians, or radians equally well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key, though, is in the accuracy of specification and storage of that angle.  I want to know that if I rotate by a sixth of a circle and do it six times then I will rotate by a full circle.  Now, for a sixth then the discrepancy is unlikely to be that great, but in fact I'll do this for much smaller angles.  However I specify the angle, the resulting matrix is not going to be exact due to precision issues.  So continually composing the matrix will eventually diverge from what it should be.  Therefore my best strategy is to keep track of the angle and compute the matrix from the angle each time.  But this won't work if my angle is not precise.  So I want to be able to store the angle as precisely as I can.  And if I store &amp;pi; or &amp;tau; or some fraction thereof, it will only ever be an approximation.  But if I store 10 or 20 or 30 then I can store it exactly and do exact calculations on it, thereby minimising the error in the final transformation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then why degrees instead of gradians or just proportions?  The reason to discard just proportions is that most computer programs do not handle fractions exactly &quot;out of the box&quot; whereas they do handle integers exactly.  So for the range of values we're actually interested in, it is more efficient to work with multiples of a fixed small angle than work with fractions of the whole.  As for gradians, well, the Babylonians were right: 360 is a brilliant number of pieces to divide something into because there are so many factors that we can handle.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-04-11T07:46:53.347" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1537" PostTypeId="2" ParentId="1520" CreationDate="2014-04-11T08:51:53.677" Score="2" Body="&lt;p&gt;Measuring in radians is useful when doing math, but it is terribly awful in the real life. It's not a chance that we still use a non-decimal measurement for degrees, since it is quite easy to divide in various numbers of parts.&lt;/p&gt;&#xA;" OwnerUserId="26" LastActivityDate="2014-04-11T08:51:53.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1538" PostTypeId="1" AcceptedAnswerId="1686" CreationDate="2014-04-11T08:54:09.983" Score="25" ViewCount="1715" Body="&lt;p&gt;Let me motivate my question with the following situation. While teaching the concept of continuity, I usually start with motivating the concept. Then, when we see that there is an important and intuitive concept here, we start formalizing it. After arriving at the standard definition, it is usual that some counterintuitive facts surface. For example, with the definition I use for continuity, the following facts cause problems:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A function defined at an isolated point is continuous at that point (trivial &quot;empty&quot; consequence of the $\varepsilon$-$\delta$ formulation)&lt;/li&gt;&#xA;&lt;li&gt;The function $f(x)=\frac{1}{x}$ is a continuous function (it is continuous at every point in its domain). (This last one is also a problem because &lt;a href=&quot;http://amzn.com/0321637429&quot; rel=&quot;noreferrer&quot;&gt;some calculus books&lt;/a&gt; make ambiguous statements about its continuity.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Students usually find these statements strange, and I try to convince them that the definition is so good that we should accept these facts. Making a new definition which excludes these facts does no t make sense. I am, however, not too satisfied with this explanation. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Can I do better? &lt;/li&gt;&#xA;&lt;li&gt;How do you handle this situation?&lt;/li&gt;&#xA;&lt;li&gt;What do you do in a similar analogous situation?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This question is (in a distance) related to &lt;a href=&quot;https://matheducators.stackexchange.com/q/1508/61&quot;&gt;this other&lt;/a&gt; question on empty cases in definitions.&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2017-09-01T04:43:47.080" Title="Counterintuitive consequences of standard definitions" Tags="|examples|mathematical-analysis|concept-motivation|" AnswerCount="8" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="1539" PostTypeId="2" ParentId="1538" CreationDate="2014-04-11T10:04:14.537" Score="3" Body="&lt;p&gt;In the end, definitions are for &lt;em&gt;use&lt;/em&gt;. If in some outlandish situations, which aren't &quot;normal use&quot; of the definition, something strange happens, so be it. It's not worth &quot;fixing&quot; (i.e., complicating) the definition for such cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The continuity of $\frac{1}{x}$ is fine, that the function goes to infinity at $x = 0$ is of no consecuence as long as you stay away from that point. A function defined just at one point is of no real interest when using continuity, so it is also fine.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-11T10:04:14.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1540" PostTypeId="2" ParentId="1533" CreationDate="2014-04-11T10:12:36.440" Score="3" Body="&lt;p&gt;This should be a comment, but it came out too long.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I think that doing several years of math in two weeks is a hopeless endeavor, mainly because students need time to digest the content. Some successful approaches to math workshops that I am aware are as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;review&lt;/em&gt;: the students already learned the material and now we are only refining their understanding;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;idea-overstuff&lt;/em&gt;: the teachers cram as much different ideas into students' minds and then the pupils digest it for the next six months;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;limited-content&lt;/em&gt;: the workshops cover one aspect in detail, like contest math geometry, counting arguments, etc.;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;training-camp&lt;/em&gt;: there is no general idea, the sole purpose is to gather students in a place free of distractions, so they can prepare for the incoming competition;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;social event&lt;/em&gt;: math is secondary, we want the students to have good time and find peers which share similar interests (for example, it's not easy when one lives in a small town).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It seems that none of the above fits your aim. Could you explain the context a bit more? In particular, what is your purpose? What are the prerequisites, what is the desired outcome?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-04-11T10:12:36.440" CommentCount="2" CommunityOwnedDate="2014-04-11T10:12:36.440" ContentLicense="CC BY-SA 3.0" />
  <row Id="1541" PostTypeId="2" ParentId="1538" CreationDate="2014-04-11T11:26:15.683" Score="6" Body="&lt;p&gt;I don't have idea to do better, but some to explain such consequences of definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some thoughts:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Definition have, of course, a historical development and come from a lot of examples and attempts to prove theorems and properties of examples. If students are not happy with the definitions, you can give them note how mathematics works and that definitions are coming at last in order to summarize technical properties needed for some theorems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematicians should have intuition to work. However, intuition should come from working a lot with definitions and experience; &quot;real world ideas&quot; can sometimes be very confusing. If everything comes from real world intuition, why do you need people doing mathematics?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An important point is: What are definitions good for? Of course, you can introduce some definition and find out what functions hold that property and which don't. But if you define something, you want to work with it! The task is not to find out which functions are continuous, but to find out: For which function can I use the 100 theorems which holds if the function is continuous! (A similar example: When I was in school, I always wondered if zero is a natural number of not and I believed that mathematicians are doing research about that topic; but later I realized that this is insignificant: There are theorems stating: Something holds for every natural number and sometimes, I have to exclude zero and sometimes zero has to be included.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sometimes, there are (like in your first example) &quot;empty&quot; consequences from a definition. Good point is that it does not contradict the definition. At first, to say that these function is continuous is like an answer to a question no one would like ask. At that point, you can explain (like I said before): You are interested to use some often used arguments like: &lt;em&gt;If $f(x)\neq 0$ holds for some $x$, then this holds in a sufficiently small neighborhood of $x$.&lt;/em&gt; This is true if the function is continuous. If you have a function which is continuous only as an &quot;empty&quot; consequence, even the question is useless, since &lt;em&gt;there is not neighborhood of $x$, where the function is defined&lt;/em&gt;. At that point, students might suggest a modification of the definition of continuity to only define it for domains having not singular points. You can then say that is is possible to include that, but then the definition is longer and for all excluded cases, the consequences are &lt;strong&gt;not wrong, but irrelevant&lt;/strong&gt;. And you can give an outlook that there a generalizations (in topology) where continuity can in general not be related to some real world intuition and you don't want to get rid of the additional empty examples (e.g., $f(K)$ is compact if $K$ is compact and $f$ is continuous). [I'm not sure how much one wants to go into detail there.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;According to your second example, I must say that I was very annoyed as a student for such &quot;counterexamples&quot;, since most students would say: Okay, let's look at the same function on a different domain and everything works fine. Plus, you normally have the definition of continuous in one point and continuous as a function on a given domain.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-04-11T11:26:15.683" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1542" PostTypeId="5" CreationDate="2014-04-11T11:45:55.250" Score="0" Body="&lt;p&gt;For questions how to motivate a mathematical concept (i.e., the motivation and examples of definitions, theorems, etc.) or general concepts of mathematics. Please use the &lt;a href=&quot;/questions/tagged/student-motivation&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;student-motivation&amp;#39;&quot; rel=&quot;tag&quot;&gt;student-motivation&lt;/a&gt; tag for questions about how to motivate students in general.&lt;/p&gt;&#xA;" OwnerUserId="6053" LastEditorUserId="6053" LastEditDate="2019-12-31T00:53:01.617" LastActivityDate="2019-12-31T00:53:01.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1543" PostTypeId="4" CreationDate="2014-04-11T11:45:55.250" Score="0" Body="For questions how to motivate a mathematical concept (i.e., the motivation and examples of definitions, theorems, etc.) or general concepts of mathematics. Please use the [student-motivation] tag for questions about how to motivate students in general." OwnerUserId="114" LastEditorUserId="6053" LastEditDate="2019-12-31T00:52:55.023" LastActivityDate="2019-12-31T00:52:55.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1544" PostTypeId="2" ParentId="1538" CreationDate="2014-04-11T11:46:48.080" Score="1" Body="&lt;p&gt;The case of continous functions is a little bit different than the case of natural numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basis is the very local definition of continuity at a point, which shouldn't raise any discussions. Then there are two ways to derive global definitions which serve different purposes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;define continuity for all points of a given interval (or other set). This serves the purpose to inspect that very interval (or set). Why? Don't know, ask the physicist who wanted to know.&lt;/li&gt;&#xA;&lt;li&gt;define continuity for all points of the functions domain. This serves the purpose to understand the function as a whole better.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You could also define continuity for all points in the convex hull of the function's domain. You can ask the students, what purpose this definition serves. Answer: It determines, if you can draw the graph in one stroke (possibly taking infinitely long or being infinitely fast).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What definition you take, depends on the purpose. It's irrelevant, if some of its examples are exotic. It's relevant, if some of its examples should be counterexamples.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-11T11:46:48.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1545" PostTypeId="1" AcceptedAnswerId="13519" CreationDate="2014-04-11T11:59:28.947" Score="11" ViewCount="10456" Body="&lt;p&gt;A standard example of motivating constrained optimization are examples where the setup is described in a lot of lines, e.g., when you own a company and the company is making some products out of resources and are mixed in a certain ratio, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Are there more easy (i.e., to explain and to understand) examples from daily life which motivate constrained optimization?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The examples should be nonlinear if possible, they don't have to be solvable, but it should be clear how to translate daily life language into the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A great answer should also explain the need of constrained optimization tools (i.e., the constrains should not be solvable explicitly) and maybe also demonstrate that the gradient is not zero &lt;em&gt;without calculating everything, but from the (first) view of the example&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: This question is related to &lt;a href=&quot;https://matheducators.stackexchange.com/q/1550/114&quot;&gt;Optimization problems that today&amp;#39;s students might actually encounter?&lt;/a&gt;, where more advanced problems should be discussed.&lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2020-09-30T20:33:26.960" Title="What are easy examples from daily life of constrained optimization?" Tags="|examples|concept-motivation|constrained-optimization|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1546" PostTypeId="2" ParentId="1545" CreationDate="2014-04-11T12:03:52.973" Score="5" Body="&lt;p&gt;This example is maybe the most easy, but in my opinion it does not highlight the necessity to use methods of constrained optimization since the constrained equation is explicitly invertible.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A gardener has 20 meters of fence-material and wants to fence a rectangle shaped area with maximal area.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The problem reads then as: &lt;em&gt;Maximize $f(a,b)=a\cdot b$ subject to $2a+2b=20$.&lt;/em&gt; &lt;/p&gt;&#xA;" OwnerUserId="114" LastEditorUserId="114" LastEditDate="2014-04-11T12:47:42.127" LastActivityDate="2014-04-11T12:47:42.127" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1547" PostTypeId="2" ParentId="1468" CreationDate="2014-04-11T14:22:06.523" Score="8" Body="&lt;p&gt;In my experience whenever answers are provided a huge majority of students will rush for the answers before even giving it a try. Unfortunately the best you learn from an exercise, especially a non routine one, sits in the gloomy time during which you're trying to figure out what to do and have no clues... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think giving immediately all solutions to students is enforcing a bad habit. It is a fact that when a problem is difficult it is human to face some form of lazyness. Not providing solutions is a way to stimulate people to overcome this laziness. You do not learn much from reading a solution if you did not REALLY try to solve the problem yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it ios for this reason that at times solution are provided in a separate form. Personally (i.e. undergrad level) whenever I put some exercises on line from my students I postpone providing solutions of a certain amount of time...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;@paul garrett Nowadays with internet and all the likes gaming the sistem is impossibile due to the mobility of informations... my students are much less proficient than I am in finding solved exercises online. &lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2014-04-11T14:22:06.523" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1548" PostTypeId="2" ParentId="1520" CreationDate="2014-04-11T14:31:16.370" Score="11" Body="&lt;ul&gt;&#xA;&lt;li&gt;360 has 24 divisors, more than any smaller number (this is called a &lt;a href=&quot;http://en.wikipedia.org/wiki/Highly_composite_number&quot;&gt;highly composite number&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;li&gt;None of its prime factors is larger than 5 (&quot;5-smooth&quot;).&lt;/li&gt;&#xA;&lt;li&gt;It can be divided by every natural number between 1 and 10, with the exception of 7. No smaller number can.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Plus a couple more characteristics that make it very much suited for &lt;strong&gt;subdivision&lt;/strong&gt;. Thus, many applications of &quot;degrees&quot; allow the use of &lt;strong&gt;integers&lt;/strong&gt; instead of fractions, which most people are far more comfortable with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember all the people working with angles all day that &lt;em&gt;do not&lt;/em&gt; have an academic degree to do fancy maths: Carpenters, masons, seamstresses... they prefer a natural number over multiples or fractions of some funny constant any time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, that's also the reason why we subdivide time by 24 and 60 instead of the more &quot;natural&quot; 10, or &quot;fractions of day&quot;: Ease of subdivision. Thankfully, our forbearers were a lazy lot. ;-)&lt;/p&gt;&#xA;" OwnerUserId="849" LastEditorUserId="849" LastEditDate="2014-04-11T15:11:50.650" LastActivityDate="2014-04-11T15:11:50.650" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1550" PostTypeId="1" AcceptedAnswerId="1592" CreationDate="2014-04-11T14:42:13.387" Score="58" ViewCount="71759" Body="&lt;p&gt;Our students are not fencing in farm fields, cutting wires and folding them, or designing windows, so they are often uninspired by the optimization problems we give them. They seem like something that &quot;someone, somewhere&quot; might use, but the examples feel distant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are good examples of constrained optimization problems (perhaps not simple!) that today's students might actually encounter in their lives?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your goal is to find problems that are more easily accessible, see also the sister question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1545/what-are-easy-examples-from-daily-life-of-contrainted-optimization&quot;&gt;What are easy examples from daily life of constrained optimization?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2018-05-17T07:13:37.667" Title="Optimization problems that today's students might actually encounter?" Tags="|calculus|examples|concept-motivation|constrained-optimization|" AnswerCount="24" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="1551" PostTypeId="1" CreationDate="2014-04-11T15:56:03.243" Score="6" ViewCount="554" Body="&lt;p&gt;Nothing more quickly disspitates the myth that most people aren't interested in math than hitting them with a good puzzle and watching the instinctive human urge to solve it get to work. To be fair, this doesn't apply to literally &lt;em&gt;all&lt;/em&gt; people, but there's certainly a lot more people that enjoy riddles and puzzle games than there are mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've always thought that if I were to teach students at the pre-college level (who we can therefore assume are there because they have to be, not because they already like math), I would use puzzles to get them interested in mathematics, emphasizing that the fascination they (hopefully) feel towards these puzzles is exactly how mathematicians feel about their work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;But does this actually work?&lt;/strong&gt; Or, if I gave a class of pre-algebra 13 year olds a puzzle like &quot;Given the sum and difference of two numbers, can you work out the numbers themselves?&quot;, would I just get a roomful of groans and no attempts at a solution?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My definition of a &quot;puzzle&quot; is roughly as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A puzzle is not a routine application of the class material, it requires insight and creative thinking.&lt;/li&gt;&#xA;&lt;li&gt;A puzzle can be solved using techniques learned in class or about to be learned in class, although the exact connection might not be obvious.&lt;/li&gt;&#xA;&lt;li&gt;It may be too hard for the entire class to solve it, the goal is more to get them thinking about the problem (to get them interested in the class, and motivate the techniques you're teaching them).&lt;/li&gt;&#xA;&lt;li&gt;The statement of a puzzle is elegant and designed to be inherently interesting (rather than &quot;here is a very specific geometrical scenario, calculate this very specific paramater relating to it&quot;).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="115" LastEditorUserId="61" LastEditDate="2014-04-16T20:43:27.000" LastActivityDate="2014-04-20T04:08:28.770" Title="Are puzzles a good way to interest pre-college students?" Tags="|mathematical-pedagogy|secondary-education|math-puzzle|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1552" PostTypeId="2" ParentId="1551" CreationDate="2014-04-11T16:11:19.190" Score="1" Body="&lt;p&gt;Well, your sum/difference question is awful, but then the 'puzzle' definition is spot on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's a first layer where real life math makes sense. Ex - as an interim step, a student had to calculate half of 1500. I told him he could do this in his head, no calculator. He protested, and I said &quot;what's half of $15?&quot; He replied, '7.50' instantly. (I can't make this up)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The trick to get to the next level is to have the puzzle be really interesting, that they'll want solve for the sake of the challenge, and will push them a bit to learn something new, or apply what they know, but in a new way.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-04-11T16:11:19.190" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1553" PostTypeId="2" ParentId="1551" CreationDate="2014-04-11T16:44:56.883" Score="9" Body="&lt;p&gt;(This is more an extended two-part comment than explicit answer.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Part one.&lt;/strong&gt; Finding &quot;puzzles&quot; (often with a mathematical flavor) is not too difficult. Martin Gardner is probably the main go-to, but &quot;The Art and Craft of Problem Solving&quot; (Zeitz) or the classic &quot;Polyominoes&quot; (Golomb) might fit the bill, too. Some other books I like are &quot;The Tokyo Puzzles&quot; (Fujimura) and &quot;The Simple Book of Not-So-Simple Puzzles&quot; (Grabarchuks). There is yet another book, which has a title meant somewhat as a &quot;jape,&quot; called &quot;More Games for the Super-intelligent&quot; (Fixx). Here is an excerpt:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The mere fact that you are reading this book [on puzzles] by no means grants you license to urge its contents on those who are not similarly committed. The only safe procedure is to acknowledge the fact that in your liking for puzzles you are different from other people. Revel privately, if you want to, in that difference, but do not - under any circumstances - undertake even the gentlest of proselytizing. That way lies nothing but disaster. Remember: you are an intellectual at play, not a missionary... I think, we must choose our puzzles companions with care. We must not assume, until we have seen irrefutable evidence of it, the presence of an enthusiasm that may not exist at all. Rather, those of us who share an interest in puzzles should stick quietly together, welcoming outsiders whenever they chance to turn up, but never taking it for granted that they will (vi-vii).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Whether or not I agree entirely with Fixx, I can at least say that there are plenty of students for whom puzzles are not enjoyable, so one must be cautious in expecting any sort of interest-transfer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Part two.&lt;/strong&gt; Looking at your definition of &lt;em&gt;puzzle&lt;/em&gt;, it essentially accords with the idea of a &lt;em&gt;problem&lt;/em&gt; as contrasted with an &lt;em&gt;exercise&lt;/em&gt; (in the Mathematics Education literature). To this effect, I think books on problem solving or old examinations (such as the Putnam - but maybe stick to A1, A2, B1, B2 for pre-college...) could be useful sources for what you are looking for. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In distinguishing between a &lt;em&gt;problem&lt;/em&gt; and &lt;em&gt;exercise&lt;/em&gt;, one way to define the former is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a question for which the method of solution is unknown at the outset&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This ought to satisfy your first three conditions, and asking that the problem be &lt;em&gt;good&lt;/em&gt; (elegant, interesting, etc.) should be enough for the fourth and final condition.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-04-11T16:44:56.883" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1554" PostTypeId="2" ParentId="1550" CreationDate="2014-04-11T16:45:09.967" Score="10" Body="&lt;p&gt;An answer depends on what you mean with &lt;strong&gt;their lives&lt;/strong&gt;. I think, it is more likely that students will face such problems in their working (not academic) life. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The Netflix Prize example&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My example is the &lt;a href=&quot;http://en.wikipedia.org/wiki/Netflix_prize&quot;&gt;Netflix Prize&lt;/a&gt;, which was awarded by Netflix in the mid 2000s. At that time, Netflix was a DVD rental service and users did rate the movies they have seen. Netflix had some algorithm which gave recommendations which movies a particular user could like and awarded 1 mio $ to the team which improved the algorithm by 10%. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rating was as follows: For all users $u\in U$ and all movies $m\in M$, there is a (possible) rating $r(u,m)\in\lbrace 0,1,2,3,4,5\rbrace$. There is a subset $K\subset U\times M$, for which the rating function is known. Netflix did a partition of $K$ into $K_{\text{known-to-everyone}}$ and $K_{\text{known-to-netflix}}$. The problem was then:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find an approximating rating function $r_{\text{appr}}$ such that $\Vert r_{\text{appr}}(u,m)-r(u,m)\Vert_{K_{\text{known-to-netflix}}}$ subject to $r_{\text{appr}}(u,m)=r(u,m)$ on $K_{\text{known-to-everyone}}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That means, everyone got the known data and should make some model to make good predictions (depending on a specific given norm) on unknown data only known to Netflix (who by that data found out which algorithm was good).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The documentation of the best algorithm is published; some other teams published their approach as well and used methods from constrained optimization. I was attending a talk where someone presented his solution which uses a low rank approximations and optimization methods on manifolds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why could that be important to students in their future lives?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The prize was awarded and the million is gone. But I can image that such &quot;data fitting&quot; task are common in insurance economy where they have a lot of data and want to find models in order to derive unknown data, which will be important for the insurance company to model their prices.&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-04-11T16:45:09.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1555" PostTypeId="2" ParentId="1550" CreationDate="2014-04-11T16:46:13.663" Score="2" Body="&lt;p&gt;&lt;strong&gt;Profit maximation!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's easy to understand, not so hard to calculate and even testable. Let them sell lemonade or whatever at the next school festival and see, if they can find the optimal price for the lemonade.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-11T16:46:13.663" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1556" PostTypeId="2" ParentId="1550" CreationDate="2014-04-11T16:58:10.357" Score="13" Body="&lt;p&gt;This is from a &lt;a href=&quot;https://math.stackexchange.com/questions/410331/using-math-for-interior-decorating-with-lamps&quot;&gt;post&lt;/a&gt; of mine on Math Stackexchange.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When I was in college, I owned three lamps and had a dark apartment. I kept trying to position them in different areas of the room, but it was still dark. Then I decided to model the problem with math: given three light sources of equal strength in a rectangular solid of a room, where can you place them to maximize the average value of the amount of light reaching a point in the solid?&lt;/p&gt;&#xA;&lt;p&gt;Then I realized that the asymptotes at the light bulbs themselves would make the average diverge.&lt;/p&gt;&#xA;&lt;p&gt;Recently, I thought of a reformulation:&lt;/p&gt;&#xA;&lt;p&gt;Given three light sources in a rectangular solid of a room (assume height of ten feet, width of 15 and length if 25),all at height 6 ft., where should they be placed to maximize the average value (Edit: Minimum value) of the intensity of the light on the plane at height 3.5 ft.? (Since the couches are that height).&lt;/p&gt;&#xA;&lt;p&gt;Alternatively, if someone can give a more accurate reformulation of this problem (perhaps involving reflections of the wall) and solve it, that would be much cooler.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-11T16:58:10.357" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1558" PostTypeId="2" ParentId="1550" CreationDate="2014-04-11T17:06:43.703" Score="8" Body="&lt;p&gt;&lt;strong&gt;Edit (4/29/14):&lt;/strong&gt; A popular media piece in the New York Times on &lt;a href=&quot;http://www.nytimes.com/2014/04/29/science/to-divide-the-rent-start-with-a-triangle.html?_r=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;rent division&lt;/a&gt; gives a great example of how &lt;a href=&quot;http://en.wikipedia.org/wiki/Sperner%27s_lemma&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sperner's Lemma&lt;/a&gt; can be used for fair division problems (as related to cake cutting, the history of which can be found in a pasted excerpt at the end of my response here). The AMM article drawn upon is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Su, F. E. (1999). Rental Harmony: Sperner's Lemma in Fair Division. &lt;em&gt;American mathematical monthly, 106&lt;/em&gt;(10), 930-942. &lt;a href=&quot;http://www.math.hmc.edu/%7Esu/papers.dir/rent.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt; (no paywall).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;A March 2014 PLOS &lt;a href=&quot;http://www.plos.org/wp-content/uploads/2014/04/plcb-10-03-forger.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; describes a mathematical model using optimal control theory and a system of ODEs to build an app that combats jet-lag. A popular media piece can be found &lt;a href=&quot;http://www.latimes.com/science/sciencenow/la-sci-sn-jet-lag-entrainment-app-20140410,0,1771735.story&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, with the headline: &lt;strong&gt;A new mathematical model can cut jet-lagged time in half&lt;/strong&gt;. Even if the details elude students, this is certainly a problem that today's students might run into; moreover, they can now take part in &lt;em&gt;testing&lt;/em&gt; the app to determine if it works.&lt;/p&gt;&#xA;&lt;p&gt;Excerpt from the latter link:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[Biological mathematician] Forger admits that the schedules his model spits out may sometimes seem counterintuitive, but intuition, he says, is not the point.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;We're trying to move the science beyond your grandmother's advice of 'wake up late' or 'avoid carbohydrates' to something that can be rigorously tested,&amp;quot; he said. &amp;quot;All I know is these schedules are optimal according to the mathematics.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Forger spent 10 years building his model based on data collected from sleep studies done at Harvard and the University of Michigan. He had no idea what type of schedules the model would come up with when he first started, so he was pleasantly surprised that according to the math, the best way to beat jet lag is to adjust the time of your dawn and dusk each day.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See more in the LA Times write-up or in the methods section (p. 10) of the actual paper.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;In case the cake-division discussion below does not suffice, there is the question of how one physically carries out the division for a different food: matzah.&lt;/p&gt;&#xA;&lt;p&gt;See the short &lt;em&gt;Wired&lt;/em&gt; article &lt;a href=&quot;http://www.wired.com/2014/04/scale-free-mathematics-in-matzah/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, and perhaps the question of how to break matzah optimally (for two people: in half) can provide a real-life optimization problem.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;I would probably start by looking to literature on mathematical modeling. An example source is the COMAP book &amp;quot;For All Practical Purposes&amp;quot; which includes sections on optimal entry (in elections), optimal production policies, and optimal schedules. There are plenty of other problems that don't have the word &amp;quot;optimal&amp;quot; in their name (e.g., bin packing). The book provides a wealth of examples; another nice one is fair division, which can be applied to scenarios ranging from company mergers to inheritance to organ transplant policies. Some of these are probably a bit removed from students' daily lives, but others are run into frequently. For instance, you could read over the literature on cake-division.&lt;/p&gt;&#xA;&lt;p&gt;For three cake eaters, there is the Steinhaus Proportional Procedure; for four or more, there is the Banach-Knaster proportional procedure. The book contains a sixty-year history of cake cutting. I have pasted an image of this below (written up as p. 421 in the eighth edition); the word &amp;quot;California&amp;quot; is, for some reason, omitted between the two columns:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/TRJSe.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have included only one actual excerpt here, but I recommend this book - and other materials on mathematical modeling, e.g., COMAP's Mathematical Modeling Handbook - as sources for optimization problems encountered (either literally, such as when you want to divide up a cake, or mathematically, when you see another problem to which you can apply these models/methods).&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-29T13:45:13.440" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1559" PostTypeId="2" ParentId="1550" CreationDate="2014-04-11T17:11:04.160" Score="11" Body="&lt;p&gt;As I mentioned in my other answer, it highly depends what is meant by &lt;strong&gt;their lives&lt;/strong&gt;. Here is a every day problem which I, at least, often encounter: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If you do the dishes and finish, there is a lot of foam in your sink. Arrange the position and strength of the water-tap in order to let the foam disappear as fast as possible.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The problem sounds easy, but is indeed an optimal control problem with partial differential equations (Navier-Stokes) and boundary control, which is not very easy to solve :)&lt;/p&gt;&#xA;" OwnerUserId="114" LastActivityDate="2014-04-11T17:11:04.160" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1560" PostTypeId="2" ParentId="1455" CreationDate="2014-04-11T17:14:04.510" Score="5" Body="&lt;p&gt;You just tell the truth and say something like, &quot;Oh, I made a mistake&quot;, and correct it by saying something like &quot;it should be ...&quot;. You are the instructor and you are required to deliver the truth. If students don't do something correctly, you correct them. I you don't do something correctly, you correct yourself. If the students correct you, you accept it. No matter how ridiculous the mistake, my students are completely accepting it. Each time, the event passes as if nothing happened. It's not a big deal. The whole point is to deliver the correct information. The truth is more important than any ego.&lt;/p&gt;&#xA;" OwnerDisplayName="user854" LastEditorUserId="6053" LastEditDate="2019-04-16T12:38:00.563" LastActivityDate="2019-04-16T12:38:00.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1561" PostTypeId="2" ParentId="1550" CreationDate="2014-04-11T18:55:17.737" Score="58" Body="&lt;p&gt;Here's the example I had which inspired me to post the question in the first place:&lt;/p&gt;&#xA;&lt;p&gt;The game &lt;a href=&quot;http://en.wikipedia.org/wiki/League_of_Legends&quot; rel=&quot;noreferrer&quot;&gt;League of Legends&lt;/a&gt; was the most-played PC game, in number of hours played, in North America and Europe in 2012. There is a good chance that League of Legends is a part of many of your students' daily life, especially if you are teaching engineering calculus. It doesn't have any sexual or deviant themes -- it is a game about teams of &amp;quot;champions&amp;quot; fighting in an arena -- so I've found it to be reasonable to bring it up in class. The reason to go this route is that &lt;strong&gt;many of your students might actually be encountering this optimization problem every day&lt;/strong&gt;. Here's the problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In League of Legends, a player's Effective Health when defending against physical damage is given by &lt;span class=&quot;math-container&quot;&gt;$E = \frac{H(100+A)}{100}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; is health and &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is armor.&lt;/p&gt;&#xA;&lt;p&gt;(1) Health costs 2.5 gold per unit, and Armor costs 18 gold per unit. You have 3600 gold, and you need to optimize the effectiveness E of your health and armor to survive as long as possible against the enemy team's attacks. How much of each should you buy?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You can actually go further if you want to make it more complicated (and more accurate to what actually happens in the game!)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(2) Ten minutes into the game, you have 1080 health and 10 armor. You have only 720 gold to spend, and again Health costs 2.5 gold per unit while Armor costs 18 gold per unit. Again the goal is to maximize the effectiveness E. Notice that you don't want to maximize the effectiveness of what you purchase -- you want to maximize the effectiveness E of your &lt;em&gt;resulting&lt;/em&gt; health and armor. How much of each should you buy?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And the last one is much more challenging, but it is something that even professional players of League of Legends (these exist!) can get incorrect:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(3) Thirty minutes into the game, you have 2000 health and 50 armor. You have 1800 gold to spend, and again Health costs approximately 2.5 gold per unit while Armor costs approximately 18 gold per unit. Again the goal is to maximize the effectiveness E of your &lt;em&gt;resulting&lt;/em&gt; health and armor. How much of each should you buy?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;My favorite part of this problem is that the game is actually more complicated than this -- in reality you need to defend against both physical and magic damage by buying both armor and magic resistance, which adds another variable. So the &amp;quot;punchline&amp;quot; of the exercise is that &lt;strong&gt;you can't actually play the most popular PC game properly until you understand Lagrange multipliers. I'll see you in Multivariable Calculus&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Instructors answers (not student solutions!):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1) You do not spend equal money on A and H: &lt;span class=&quot;math-container&quot;&gt;$E = 3H - \frac{1}{720}H^2$&lt;/span&gt; so the maximum is at &lt;span class=&quot;math-container&quot;&gt;$H = 1080$&lt;/span&gt;, plug back in for &lt;span class=&quot;math-container&quot;&gt;$A = 50$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(2) One way to do this is to realize from number 1 that we know an optimal configuration is &lt;span class=&quot;math-container&quot;&gt;$H = 1080$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A = 50$&lt;/span&gt;, so right now we have way too much health and not enough armor. The answer to this is that we should spend all the money on 40 armor, to get exactly back to the optimized answer to #1.&lt;/p&gt;&#xA;&lt;p&gt;(3) If H and A are the amount they plan to buy, the effectiveness is &lt;span class=&quot;math-container&quot;&gt;$E = \frac{(H + 2000)(100 + (50 + A))}{100}$&lt;/span&gt; since they started with 2000 and 50 respectively.  The critical point appears at H = -100, so the maximum actually occurs at one of the endpoints, not at the critical point at all. Again the player should spend all the money on armor.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-11T18:55:17.737" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="1562" PostTypeId="2" ParentId="1121" CreationDate="2014-04-11T20:19:49.347" Score="3" Body="&lt;p&gt;The criteria for deciding when to teach the &quot;why&quot; in addition to the &quot;how&quot; has to be how complicated the &quot;why&quot; is relative to what the students can reasonably handle. What the students can reasonably handle is dependent both on their age and their educational background. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For elementary age kids, the &quot;why&quot; is almost never called for. They would not appreciate it, almost certainly wouldn't understand it, and have no need of it. I remember the John Milton quote, &quot;...forcing the empty wits of children to compose Themes, verses, and Orations,&#xA;which are the acts of ripest judgement and the final work of a head filled&#xA;by long reading, and observing, with elegant maxims, and copious invention.&#xA;These are not matters to be wrung from poor striplings, like blood out of the&#xA;nose, or the plucking of untimely fruit.&quot; - from &quot;Of Education&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The analogy in mathematics is that the &quot;why&quot; should not be forced on very young children. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For middle school on up, the &quot;why&quot; is much more important. The kids are going to ask about the why, and it is definitely time to begin asnwering those questions. Even then, in middle school, &lt;em&gt;following&lt;/em&gt; the logic of the mathematics is more important than begin able to &lt;em&gt;come up with&lt;/em&gt; the logic themselves. The imagination, that most important faculty of the mathematician, is somewhat dormant in the middle school years. Without it, you simply cannot write proofs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, in the high school years, you have kids who have hopefully learned how to follow logic, but now have recovered somewhat of their imaginations, and can see how to get from A to B. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These ideas, by the way, do not originate with me, except insofar as I have applied them to teaching mathematics. The ideas themselves are from the Dorothy Sayers speech &quot;The Lost Tools of Learning&quot;, which you can find online. &lt;/p&gt;&#xA;" OwnerUserId="196" LastActivityDate="2014-04-11T20:19:49.347" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1563" PostTypeId="2" ParentId="1121" CreationDate="2014-04-11T23:07:55.743" Score="8" Body="&lt;p&gt;I think the precise wording of the question accidentally prejudices the potential answers. Namely, at all levels of fanciness-of-math, and at all levels of development of kids'/peoples' thinking, the thread of &quot;how can we &lt;em&gt;do&lt;/em&gt; these things&quot; and the thread of &quot;why did that work to accomplish our goals?&quot; and the thread of &quot;should I be curious about both discovery of, and substance of, discovery of amazing algorithms or principles ??!??&quot; should not be put in apparent conflict.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At every stage of development, few people are able to &quot;ask truly insightful questions&quot;, and, perhaps, to understand answers to those questions which they haven't managed to ask.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, I think mathematics teachers should encourage questioning, while not making it &quot;mandatory&quot;. Sketch explanations and history without grilling kids on it. Make it clear that mathematics is a human endeavor, can be legitimately questioned, but/and has many good answers. Many kids will not care much, but the meta-point that asking questions is legit is important to make by itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For that matter, with a talented expositor, the &quot;why&quot; can very often be made far more interesting than &quot;drill in execution of algorithm&quot;. In my own direct experience, going back to childhood, the execution was not difficult, and was completely boring apart from the &quot;points scored&quot; with authorities for perfect papers, for being quick in &quot;blackboard races&quot;, and so on. And, yes, it is sometimes (not always) useful to have a quick mind and good memory... but those are not things that are directly taught, and by now it seems to me perverse to &quot;test&quot; or &quot;teach&quot; in ways that too heavily reward inalterable natural features.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I suspect many kids are ignoring what the teacher's saying &lt;em&gt;anyway&lt;/em&gt;, statements of principle that apparently legitimize skepticism or disinterest might be good PR for &quot;the Establishment&quot;, in any case, as opposed to continued emphasis on obedience to authority.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point that various algorithms actually succeed, not merely because of some authority, but because of &lt;em&gt;fact&lt;/em&gt;, should be the goal. And that we might like to have an acquaintance with these remarkably reliable truths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I'd not want to pre-emptively refuse to explain the &quot;why&quot;... at least acknowledge that there is an answer, and a &lt;em&gt;good&lt;/em&gt; answer, to the &quot;why&quot;, for any interested parties. And the answers are &quot;right&quot; not by authority of the teacher, but from reality. In fact, this is an amazing point in comparison and contrast to much else that people see these days (as well as days long gone).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: opportunities for further explanations for interested parties, and admission in general that there are explanations for those interested. Emphasis that it's not &quot;rules from authority&quot;, but &quot;rules as &lt;em&gt;description&lt;/em&gt; of &lt;em&gt;reality&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-11T23:07:55.743" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1564" PostTypeId="2" ParentId="1121" CreationDate="2014-04-12T03:32:47.337" Score="2" Body="&lt;p&gt;I see the solution of $(x-3)^2=1$ as $x-3 = \pm 1$ hence $x = 3 \pm 1$ as the proper use of a &lt;strong&gt;theorem&lt;/strong&gt;. In particular:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Theorem:&lt;/strong&gt; If $a,b \in \mathbb{R}$ and $a^2=b$ then $a = \pm \sqrt{b}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proof:&lt;/strong&gt; Suppose $a,b \in \mathbb{R}$ and $a^2=b$. Observe $a^2 \geq 0$ hence $b \geq 0$ thus $b=(\sqrt{b})^2$. Note then $a^2-\sqrt{b}^2=0$ hence $(a-\sqrt{b})(a+\sqrt{b})=0$. But, the product of two real numbers is zero iff one of the factors is zero. Thus, $a-\sqrt{b}=0$ or $a+\sqrt{b}=0$. Consequently, $a = \pm \sqrt{b}$. $\Box$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On occasion you may encounter an instructor who says $x^2=4$ implies $x = \pm 2$ is not good work. Instead, you must show $x^2-4=(x-2)(x+2)=0$ thus $x = \pm 2$. Personally, I reject this philosophy because I think as we mature we ought to learn from our previous work and as we converse with individuals of similar training there is no need to rehash things which are already settled.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you first learn factoring you ought to be forced to prove this theorem a few times. However, sometime, not too long after that, we should also learn that those details are so obvious that there is no common expectation those details are shown. In the same way, we teach difference quotients to begin our discussion of derivatives. But, only a silly person insists to use difference quotients and limits on problems which allow algebraic techniques from the family of &lt;strong&gt;theorems&lt;/strong&gt; we develop early in calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want students who leave my courses to know for certain the best way to calculate standard problems. If a substandard method is needed to motivate the definition then I want to make certain the demarcation between the motivating substructure and the optimal (theorem-based) calculation is crystal clear. I see the same for algebra: there are tricks (theorems) which help us to see past multiple steps which are closely tied to axioms of real numbers. If you can jump reliably from one point to another then it hardly makes sense to tip-toe just because you can. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, it is just as important to &lt;strong&gt;know&lt;/strong&gt; theorems as to be able to prove them. This continues to be true at higher levels of mathematics. Unfortunately, this reality is sometimes clouded by our tendency to over-emphasize the definition-proof aspect of the subject over the intuition-theorem-application of higher courses.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-04-12T03:32:47.337" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1565" PostTypeId="2" ParentId="1550" CreationDate="2014-04-12T05:51:32.373" Score="8" Body="&lt;p&gt;A lot of examples exist in industry.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Manufacturing plants use optimization to figure out how to best run their machinery, buy raw materials, ship finished goods, etc.&lt;/li&gt;&#xA;&lt;li&gt;Airlines and other passenger transportation services use optimization to determine their schedules.&lt;/li&gt;&#xA;&lt;li&gt;The cargo transportation industry (trucking, trains, etc.) uses optimization to determine how to transport goods as quickly and inexpensively as possible.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I had an entire series of classes in my undergraduate program centered around an example of optimization in transportation. (I was part of a program that combined computer science and business, so the administrators had a lot of leeway designing classes that went together.) We had to write a program for a fictitious train company that determined how to transport orders in the cheapest way possible using the simplex algorithm. We had to write our own implementation of the simplex solver in a programming class and learned about the algorithm in detail in a math-heavy business course.&lt;/p&gt;&#xA;" OwnerUserId="864" LastActivityDate="2014-04-12T05:51:32.373" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1566" PostTypeId="2" ParentId="1550" CreationDate="2014-04-12T11:22:51.933" Score="7" Body="&lt;p&gt;Here's a problem that I found interesting: I listen to audiobooks. When you import them from CD, you get lots of tiny chapters (often 2 or 3 minutes) which is not very practical. Also, audiobooks can be very long - some are 30 hours - which is also not very practical. I prefer audiobooks to be split into individual volumes of not more than 8 hours, and chapters around 10 minutes. Assume that you can combine consecutive chapters into one chapter (chapters are indivisible), and that you can combine consecutive chapters into volumes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problem 1: Define an optimisation problem. Find a function that, if minimised, produces audiobooks in the way that I (or you) prefer them. I'd want volumes as long as possible, but never more than 8 hours (unless a chapter is longer than 8 hours and it becomes unavoidable), and volumes of approximately equal length. I'd want chapter length around ten minutes, and preferably equal lengths. Defining a function to minimise is not trivial. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problem 2: Find an optimal solution according to the function to be minimised. Decide how important execution speed is (hint: Actually creating an audiobook according to the solution takes a long time). Obviously the way you combine chapters affects the possible lengths of volumes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problem 3: Does the optimal solution actually match my/your preferences? If not, is it because finding a good solution was hard, or because the optimisation problem wasn't chosen well? If needed, go back to problem 1. &lt;/p&gt;&#xA;" OwnerUserId="697" LastActivityDate="2014-04-12T11:22:51.933" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1567" PostTypeId="2" ParentId="1550" CreationDate="2014-04-12T13:08:45.313" Score="5" Body="&lt;p&gt;Finance has lots of good examples.  Here is one version of the portfolio optimization problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose two assets have mean excess returns $\mu, \mu'$, standard deviations $\sigma, \sigma'$ and a correlation $\rho$.  You have cash to distribute among these two assets in proportions $a, a'$ where $a+a'=1$.  What values of $a,a'$ maximize the Sharpe ratio of your investment:&#xA;$$\frac{a\mu + a'\mu'}{\sqrt{a^2 \sigma^2 + a'^2\sigma'^2 + 2\rho\, aa'\sigma\sigma'}}?$$&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-12T13:08:45.313" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1568" PostTypeId="2" ParentId="130" CreationDate="2014-04-12T13:52:29.617" Score="15" Body="&lt;p&gt;Of course, as already mentioned, it is important for a teacher to know more than her or his students. But one thing where this principle is especially important is exercise (and test) designing. Let me give a few simple examples.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How do teachers manage to chose the second degree polynoms that have not-too-complicated roots? They start from the factorization, which needs a slightly deeper understanding of polynoms than root finding methods.&lt;/li&gt;&#xA;&lt;li&gt;How can a teacher improvise a $2\times 2$ affine system with a unique solution (or with no or multiple solutions) at the board, whenever needed (e.g. some students finished all exercises already)? They use determinant. Variant: finding $3$ affine equations on $2$ variables that have solutions by using row manipulations, which needs some understanding of independence in linear algebra.&lt;/li&gt;&#xA;&lt;li&gt;We know how to compute sequences defined by rank-two induction by reducing the corresponding matrix. This can be used to design a problem (e.g. on Fibonacci sequence) where the relevant change of variable is given to the students.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There are obviously a whole lot of such examples, where deeper understanding of somehow more abstract mathematics than taught is needed for teachers simply to design good exercises.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-12T13:52:29.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1569" PostTypeId="2" ParentId="130" CreationDate="2014-04-12T14:29:04.050" Score="16" Body="&lt;p&gt;How can you hope to clearly explain something you yourself do not understand well?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found the following line of reasoning is quite convincing for many skeptics:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How hard did precalculus seem when you took it? After taking a semester of calculus, how did your perception change? How about after taking 3 semesters of calculus?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics is cumulative. The more you learn, the more you reinforce the basics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the time you finish the standard sequence of several semesters of calculus, differential equations and linear algebra, you start to find that algebra and the basic precalculus material is equally difficult with basic arithmetic. When first confronted with derivatives, maybe they seemed weird/difficult/tricky, but after a couple of semesters of material building on the basics, taking a derivative is like adding 2 and 2 or breathing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is, the more you know, the more you will consider basic/easy/trivial. When teaching a subject, you want to be able to focus on how you are presenting the material - how to best communicate this to you students. You &lt;em&gt;don't&lt;/em&gt;  want to be fighting with understanding it yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Along the same lines, the more you know, the better perspective you have as to why different topics are important. The more math you learn, the better you will be armed when confronted with a skeptical student of your own who asks, &quot;Why do I need to learn this?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Admittedly modern algebra as taught at most universities is a hard sell for you typical unmotivated future highschool teacher (why you would enter such a profession half-hearted, I do not know, but it seems common). If you focus on ring theory, you have a chance at selling the connections to factorization and the like. Groups are trickier because, honestly, that specific material may very well be completely unapplicable. However, students who spend a semester really engaging their modern algebra (or any other proof oriented) class should leave with at the very least bullet proof logic skills. This of course is of immeasurable value when teaching mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, that's how I've approached this topic with many students (and been successful as far as I can tell).&lt;/p&gt;&#xA;" OwnerUserId="453" LastActivityDate="2014-04-12T14:29:04.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1570" PostTypeId="2" ParentId="1538" CreationDate="2014-04-12T16:44:18.147" Score="8" Body="&lt;p&gt;Here's an analagous situation: &lt;a href=&quot;http://en.wikipedia.org/wiki/Clopen_set&quot;&gt;http://en.wikipedia.org/wiki/Clopen_set&lt;/a&gt;. Intuitively, open and closed seem like mutually exclusive concepts, but their definitions allow for 'clopen' sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, I'd argue that it's both useful and more elegant to have definitions which are &lt;em&gt;inclusive&lt;/em&gt;, rather than &lt;em&gt;exclusive&lt;/em&gt;. In other words, if we tried to define continuity in such a way that we removed all the seemingly &quot;weird&quot; (i.e. unusual, vacuous) cases, then we'd end up with a convoluted definition. And that's not to mention the difficulty of systematically identifying and eliminating the &quot;weird&quot; cases.&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2014-04-12T16:44:18.147" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1571" PostTypeId="2" ParentId="130" CreationDate="2014-04-12T16:53:15.310" Score="9" Body="&lt;p&gt;&lt;strong&gt;Examples from Abstract Algebra&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The reason why the product of two negative numbers is positive boils down to a proof from abstract algebra.  $(-3)(4)$ should be the opposite of $(3)(4)$ because you add the two to get zero.  Then $(-3)(-4)$ should in turn be the opposite of $(-3)(4)$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why negative numbers exist, and why subtraction is just adding a negative.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why fractions exist.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why zero behaves oddly with multiplication and division.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why $-1$ is not a prime number.  Why $1$ is not a prime number.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why $\infty$ is often not considered a number.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What we mean in general when we talk about a &quot;number&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\cdots$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;With all of the above questions, I think only a trained mathematician would know how to answer the question honestly if they needed to explain it to a child.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure there are many examples from analysis too, but abstract algebra in particular is practically the direct study of everything taught in basic elementary school math: integers, addition and multiplication, divisibility, fractions, and so on.&lt;/p&gt;&#xA;" OwnerUserId="200" LastEditorUserId="200" LastEditDate="2016-05-15T18:40:12.480" LastActivityDate="2016-05-15T18:40:12.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1572" PostTypeId="1" CreationDate="2014-04-12T17:13:57.610" Score="29" ViewCount="7358" Body="&lt;p&gt;I read &lt;em&gt;Innumeracy&lt;/em&gt; by John Allen Paulos and would like to share more up-to-date and relevant examples of innumeracy to motivate my grade 8, 9 &amp;amp; 10 students. Are there any websites, blogs, books, etc. with lots of examples of innumeracy in the form of pictures, reporting, news articles, etc.?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are just 2 examples of what I'm thinking about, I just want to find LOTS more:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://johnquiggin.com/2011/05/08/two-billion-examples-of-innumeracy/&quot; rel=&quot;noreferrer&quot;&gt;http://johnquiggin.com/2011/05/08/two-billion-examples-of-innumeracy/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/p0HK8.jpg&quot; rel=&quot;noreferrer&quot;&gt;https://i.stack.imgur.com/p0HK8.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;UPDATE:&lt;/strong&gt; Some MESE members recently started a new site devotedd to innumeracy: &lt;a href=&quot;http://innumeracy.net/welcome/&quot; rel=&quot;noreferrer&quot;&gt;http://innumeracy.net/welcome/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="867" LastEditDate="2016-06-18T09:21:03.773" LastActivityDate="2022-09-27T21:55:11.060" Title="Examples of Innumeracy" Tags="|examples|student-motivation|" AnswerCount="17" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="1573" PostTypeId="2" ParentId="1572" CreationDate="2014-04-12T17:28:33.540" Score="26" Body="&lt;p&gt;&lt;strong&gt;Re-Re-Edit (May 2019):&lt;/strong&gt; Found in a selection of tweets &lt;a href=&quot;https://twitter.com/lizardbill/status/1127005323636686848&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; but pasted as images to preserve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Credit for the first one goes to &lt;a href=&quot;https://twitter.com/lizardbill&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;@lizardbill&lt;/strong&gt;&lt;/a&gt; and to the rest to &lt;a href=&quot;https://twitter.com/GeneticJen&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;@GeneticJen&lt;/strong&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LgqaP.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LgqaP.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FXT8y.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FXT8y.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CtCmQ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CtCmQ.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qv4Ly.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qv4Ly.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Wcmof.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Wcmof.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Re-Edit (Jan 2016):&lt;/strong&gt; Perhaps this does not quite qualify, but I was rather surprised to spot the following question (#6 in the image below) in a recent airplane &lt;em&gt;Mensa&lt;/em&gt; quiz:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RaNBg.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RaNBg.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Side-note: #2 part 1 has two answers, as does part 4.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Take a moment to solve #6 before reading on.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are the &quot;answers&quot; from that same page (American Way, Jan 2016):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Gbyat.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Gbyat.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The description is not &lt;em&gt;wrong&lt;/em&gt;, but it &lt;em&gt;is&lt;/em&gt; a bit different from the phrasing I had in mind (adding 18).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit (Jan. 2015):&lt;/strong&gt; After seeing &lt;a href=&quot;https://matheducators.stackexchange.com/q/7200/262&quot;&gt;&lt;strong&gt;MESE 7200&lt;/strong&gt;&lt;/a&gt;, a quick google yielded the following from a blog called &lt;strong&gt;&lt;a href=&quot;https://threesixty360.wordpress.com/2011/08/24/granola-fail/&quot; rel=&quot;noreferrer&quot;&gt;360&lt;/a&gt;&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/tzRjP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Looking at your examples, I recall a story that had been trending online for quite some time: It alleged that Samsung paid Apple about \$1 billion (USD) in nickels (\$0.05 coins) as carried by &quot;30 trucks.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After seeing this shared on facebook far too many times, I posted the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/kpbK2.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems the story goes back at least to August of 2012, and was also debunked by &lt;a href=&quot;http://www.snopes.com/politics/satire/samsung.asp&quot; rel=&quot;noreferrer&quot;&gt;Snopes&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The Snopes estimate is 2,755  eighteen wheelers, but is based off of the judgement being \$1.05 billion, as opposed to the \$1 billion claim in the image above. Scaling my estimation up by 5% gives 2,625 eighteen wheelers. So, the numbers are pretty close. In any event: There is some &quot;innumeracy&quot; here.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2019-05-12T23:13:33.267" LastActivityDate="2019-05-12T23:13:33.267" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="1575" PostTypeId="2" ParentId="1572" CreationDate="2014-04-12T17:55:20.333" Score="9" Body="&lt;p&gt;A YouTube video of an &lt;a href=&quot;http://m.youtube.com/watch?v=PgukT8yCCtc&quot; rel=&quot;nofollow&quot;&gt;Illustrious Senator&lt;/a&gt; Talking about the cost of health care, 500 trillion dollars. This is more than all the world's wealth, and nearly 8 times all the wealth in the US. I guess he meant Billion, but in Washington, no one is listening anyway. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-05-30T00:54:44.887" LastActivityDate="2015-05-30T00:54:44.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1576" PostTypeId="2" ParentId="444" CreationDate="2014-04-12T18:04:26.590" Score="5" Body="&lt;p&gt;Let me take on a different meaning of &quot;introducing&quot; and argue why I think Euler's identity is a &lt;em&gt;great&lt;/em&gt; way to introduce the concept:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach primarily remedial students in grade 8, 9, and 10, and I put the following classroom poster:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ \sqrt-1 \space 2^3 \space \sum \pi...$ and it was delicious!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.spreadshirt.com/1-2-i-8-sum-pi-i-ate-some-pie-shirt-C3376A10250297&quot; rel=&quot;nofollow&quot;&gt;http://www.spreadshirt.com/1-2-i-8-sum-pi-i-ate-some-pie-shirt-C3376A10250297&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;None of my students need to know about $i$ or $\Sigma$ (yet) -- and a few of them are disappointingly clueless about $2^3$ and $\pi$ -- but nearly &lt;em&gt;all&lt;/em&gt; of them were interested in the poster and wanted to know what it meant, and some students even did a bit of research outside of class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I have not &lt;em&gt;introduced&lt;/em&gt; these concepts in a mathematically meaningful way. My students don't know what half of the symbols mean. But by casually mentioning to them that $i$ is not on but &lt;em&gt;above&lt;/em&gt; the number line and $\Sigma$ is like + but not really, I hope I can pique their curiosity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the same way, I think that using Euler's identity to dive into a long lecture might undermine students' enjoyment, but perhaps leaving the identity for students to think about as they progress through learning about $e$, $\pi$, and $i$ &lt;em&gt;would&lt;/em&gt; be motivating or at least interesting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, it must be noted that Euler's identity really is much better than the lame math pun I put up in my classroom because it's true! That fact might be lost on students, but it shouldn't!&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="867" LastEditDate="2015-05-31T22:53:53.173" LastActivityDate="2015-05-31T22:53:53.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1577" PostTypeId="2" ParentId="1572" CreationDate="2014-04-12T18:42:15.807" Score="23" Body="&lt;p&gt;There's the &lt;a href=&quot;http://verizonmath.blogspot.com/2007/08/original-recording-of-verizon-customer.html&quot;&gt;Verizon &quot;0.002 cents versus 0.002 dollars&quot; mishap&lt;/a&gt;, wherein an unhappy customer calls to complain that he was billed 0.002 $/kB after being told the rate is 0.002 cents/kB. The confusion is perhaps deeper than expected.&lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="98" LastEditDate="2014-04-12T18:53:33.677" LastActivityDate="2014-04-12T18:53:33.677" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1578" PostTypeId="2" ParentId="1572" CreationDate="2014-04-12T19:19:01.253" Score="10" Body="&lt;p&gt;A lot of the xkcd &quot;what if&quot; posts, for example &lt;a href=&quot;http://what-if.xkcd.com/85/&quot;&gt;this one&lt;/a&gt; about hitting golf balls off a spaceship in order to reach escape velocity, seem surprising to me in part, I suspect, because of my own innumeracy. (It turns out, in this case, you might well need a bag of golf balls about 100 billion miles in diameter...)&lt;/p&gt;&#xA;" OwnerUserId="327" LastActivityDate="2014-04-12T19:19:01.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1579" PostTypeId="2" ParentId="1572" CreationDate="2014-04-12T19:23:25.253" Score="11" Body="&lt;p&gt;Common accounts in popular press and TV and on-line about &quot;the rate of increase of X is slowing&quot;, with varying interpretations, all too often mistaking this for X itself decreasing, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As in &quot;unemployment&quot; or &quot;inflation&quot; or &quot;debt&quot; or ...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-12T19:23:25.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1580" PostTypeId="2" ParentId="1572" CreationDate="2014-04-12T19:27:54.397" Score="19" Body="&lt;p&gt;Let me offer a different type of response, a student's answer to a problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question offered the height of a building, the equation for distance of a falling object, and asked to calculate the time till a rock dropped off the building would fall to the ground. The student used his calculator and the answer was 900 seconds. I asked if that was right, and tried to get him to apply common sense. 900 sec = 15 minutes. Do you think you can see your friend drop the rock, go to Starbuck's, get a coffee, and step back out before it hits the ground? Of course not. His answer was off by a factor of 100. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd read &lt;em&gt;Innumeracy&lt;/em&gt; a long time ago, but recall that this was one of the author's lessons, the ability to estimate orders of magnitude as being correct or way off. Part of my goal is to ask students if the answer makes sense, in cases where it's not just numbers but real life situations.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-08-06T02:45:09.210" LastActivityDate="2014-08-06T02:45:09.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1581" PostTypeId="2" ParentId="1121" CreationDate="2014-04-12T19:39:39.210" Score="5" Body="&lt;p&gt;&lt;strong&gt;Framework for Concept Introduction:&lt;/strong&gt; A general framework can be adapted from the &lt;a href=&quot;http://en.wikipedia.org/wiki/Zone_of_proximal_development&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zone of Proximal Development&lt;/a&gt; or ZPD, which is an instructional concept (due initially to Lev Vygotsky) that continues to evolve, especially in its treatment within the world of education and relation to &lt;a href=&quot;http://en.wikipedia.org/wiki/Instructional_scaffolding&quot; rel=&quot;nofollow noreferrer&quot;&gt;scaffolding&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The Wikipedia definition provided (quoting Vygotsky) is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or in collaboration with more capable peers&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the two constraints of (1) whether or not the concept falls within a student's ZPD, and (2) how much time can be spent on working with the student, are probably enough to give a very basic framework for introducing a concept. In particular, if either of these constraints is &lt;em&gt;not&lt;/em&gt; satisfied, then the concept should be avoided for the time being.&lt;/p&gt;&#xA;&lt;p&gt;This approach is admittedly simplistic, since there are lots of things that &lt;em&gt;could&lt;/em&gt; be covered in a course, i.e., that are potentially within students' ken given enough time, but at the expense of other material. To develop such a framework further, then, one must at least have some idea of the course goals.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Frogs, Birds, and Chunking:&lt;/strong&gt; The &amp;quot;frogs and birds&amp;quot; metaphor refers to a notion introduced by Freeman Dyson in a &lt;a href=&quot;http://www.ams.org/notices/200902/rtx090200212p.pdf?q=and&quot; rel=&quot;nofollow noreferrer&quot;&gt;talk&lt;/a&gt; where he begins:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Some mathematicians are birds, others&#xA;are frogs. Birds fly high in the air and&#xA;survey broad vistas of mathematics out&#xA;to the far horizon. They delight in concepts that unify our thinking and bring&#xA;together diverse problems from different parts of&#xA;the landscape. Frogs live in the mud below and see&#xA;only the flowers that grow nearby. They delight&#xA;in the details of particular objects, and they solve&#xA;problems one at a time. I happen to be a frog, but&#xA;many of my best friends are birds. The main theme&#xA;of my talk tonight is this. Mathematics needs both&#xA;birds and frogs. Mathematics is rich and beautiful&#xA;because birds give it broad visions and frogs give it&#xA;intricate details. Mathematics is both great art and&#xA;important science, because it combines generality&#xA;of concepts with depth of structures. It is stupid&#xA;to claim that birds are better than frogs because&#xA;they see farther, or that frogs are better than birds&#xA;because they see deeper. The world of mathematics is both broad and deep, and we need birds and&#xA;frogs working together to explore it (p. 212).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Dyson is discussing mathematicians, but you may wish to consider whether or not the discussion can be applied to students of mathematics, as well. Something like: Can students in mathematics classes be meaningfully divided into tadpoles and hatchlings?&lt;/p&gt;&#xA;&lt;p&gt;My sense is that the answer, generally, is &lt;em&gt;no&lt;/em&gt;: There might exist certain preferences, but early on students are not to be classified in this way. Instead, you must elect to provide a mix of &amp;quot;concepts that unify our thinking&amp;quot; and &amp;quot;details of particular objects.&amp;quot; To share my own perception (and perhaps out myself as somewhat ranine), I think one must alternate between minor details, as objects are introduced, and broader comments as students' arsenal of examples grows.&lt;/p&gt;&#xA;&lt;p&gt;With regard to &amp;quot;chunking,&amp;quot; I refer to studies on chessboard perceptions; an early one, comparing the ability to reconstruct positions among players at different levels, can be found &lt;a href=&quot;http://matt.colorado.edu/teaching/highcog/fall8/cs73.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. In this study and others that follow, it seems that experts are able to &amp;quot;chunk&amp;quot; concepts in a different way than novices. This can be helpful in situations that rely on one's memory capacity, but is also helpful more generally when clumping together broad ideas that can then be deconstructed.&lt;/p&gt;&#xA;&lt;p&gt;To clarify: I bring up these three items (frogs, birds, chunking) to suggest that students be taught about new objects and some of the related details (frogs), and that more general conceptual explanations (birds) be introduced to unify the objects in a way that allows for better chunking. This allows for mastery to be attained in a domain (here, mathematics).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Loose Ends:&lt;/strong&gt; Your &lt;em&gt;question&lt;/em&gt; is actually four different ones.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there any good framework for deciding when it is a good idea to &amp;quot;skip the concept&amp;quot; and &amp;quot;just teach the method,&amp;quot; knowing that students will appreciate the concept only much later in life, or possibly never?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do you really show children why long division works?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And if you do, do you intend them to retain that concept?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why is this skipping-of-the-concept sometimes good and sometimes bad?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I hope that I have provided helpful thoughts about 1, though I make no remarks here about &lt;em&gt;utility&lt;/em&gt;; this is a needed consideration, but difficult to pin down when you do not know students' future paths. The answers to 2 and 3 are &lt;em&gt;yes&lt;/em&gt;, assuming that the students have sufficient procedural fluency for this concept to be within their ZPD.&lt;/p&gt;&#xA;&lt;p&gt;As for 4, skipping a concept can be bad because (as indicated above) it robs students of the opportunity to gain the bird's-eye-view that leads to chunking and, eventually, mastery/expertise. Skipping a concept can be good because, in most cases, an individual item covered can be generalized far beyond what could reasonably be expected in any single course. (Consider an example like &lt;a href=&quot;http://en.wikipedia.org/wiki/Bloch%27s_Principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bloch's principle&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;As a final note, I think a good example of a concept that should be initially skipped is &lt;em&gt;why&lt;/em&gt; you can write any rational number in lowest terms. It is true that, for a given rational number, you will be able to do this. However, to assert this for all rational numbers, one will need approximately the strength of the Principle of Mathematical Induction (PMI) or the Well-Ordering Principle (WOP). When teaching students (whether in elementary school or a community college developmental mathematics course) about rational numbers and discussing the process of simplification, I would &lt;em&gt;not&lt;/em&gt; start talking about PMI or WOP.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-22T17:45:12.577" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1582" PostTypeId="2" ParentId="444" CreationDate="2014-04-12T20:38:53.283" Score="6" Body="&lt;p&gt;I don't think the equation itself is a good motivation. To be seen as beautiful the students first need to have an understanding of $e$ and an understanding and its Taylor series (which is motivated by lots of applications). Then they need some appreciation of the algebra of the complex numbers (which again can be motivated in many ways). Then, once they know what $i$ and $e$ are, and they would generally agree the two constants are unrelated, you can do the magic of plugging in a complex number into the Taylor series for $e^x$ and get the lovely formula. Plug in $i$ and you get the (now beautiful) equation. If you just drop the equation on them, then they will see no beauty in it. It's beauty lies in the relationship it makes between two seemingly unrelated constants. The students need to first think the constants are unrelated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An analogy would be: is it a good idea to present a Weierstrass nowhere differentiable continuous functions as motivations for differential calculus. Well, if the students don't know anything about continuity of differentiability, then they won't find it surprising that Weierstrass functions exist. To be surprised by their existence the students need to first develop the wrong feeling for these concepts, only so that their understanding can be honed by the surprises. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-04-12T20:38:53.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1583" PostTypeId="2" ParentId="1551" CreationDate="2014-04-12T20:58:06.357" Score="3" Body="&lt;p&gt;As a grade 8, 9, and 10 maths teacher I can testify that good puzzles often do motivate my students. Here are a few observations I have about the puzzles I've given in class:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;strong&gt;A good puzzle must be framed very well&lt;/strong&gt;. A puzzle about the sum and product of two integers is (as stated) too boring for all but my most highly motivated students. However, an engaging puzzle has a real world context, a game, an argument, a manipulative, open ended questions (but still structured), etc. I've found that there is relatively little correlation between how interesting I find a puzzle and how interesting my students find it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;A puzzle must attract and keep my weaker students' attention&lt;/strong&gt;. It's relatively easy to gain the attention of my top students (if I just give them a Martin Gardner book they can often occupy themselves with puzzles) but much harder with my weaker students. For them, point (1) is critical, but it's also important for them that I give the puzzle good structure so they don't give up. By that I mean that I try to present puzzles with lots of encouragement, defining terms clearly, providing a simple example and counterexample if possible, and sometimes -- but never before at least a little frustration has set in -- giving one or two good hints.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &lt;strong&gt;The serious math discussion is best when it comes with a surprise ending.&lt;/strong&gt; For example, we discussed the U.S. men's college basketball tournament and at the end I demonstrated that the probability of picking 63 games in a row correctly at random is 1 in $2^{63}$. If I begin the exercise by talking about scientific notation, I will most likely get groans, but by discussing basketball, gambling, competition, etc. first some of my students were at least a little impressed when they found out that it would take everyone on the earth counting one number per second for over over 8 months to reach $2^{63}$.&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2014-04-12T20:58:06.357" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1584" PostTypeId="2" ParentId="1550" CreationDate="2014-04-12T21:00:31.440" Score="10" Body="&lt;p&gt;The original classic real-life example is to lay out a shop floor with multiple workstations and workers, so as to minimize travel distance of the work piece given a required sequence of operations. This can be generalized to consider a &lt;em&gt;standard work load&lt;/em&gt; distributed amongst more than one type of workpiece, with different sequences of operations, and different times spent at each workstation.&lt;/p&gt;&#xA;&lt;p&gt;Some links:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.bain.com/about/client-results/revitalizing-profits-performance-by-transforming-manufacturers-shop-floor.aspx&quot; rel=&quot;noreferrer&quot;&gt;Revitalizing profits and performance by transforming a manufacturer's shop floor&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.slideshare.net/deepbaazigar/optimization-of-operations-in-the-job-shop&quot; rel=&quot;noreferrer&quot;&gt;Optimization of Operations in the Job Shop&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.thinkinglean.com/img/files/PAPER_9.pdf&quot; rel=&quot;noreferrer&quot;&gt;Improvization (sic) of Productivity Through Layout Optimization in Pump Industry.&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Any number of additional links could probably  be found by Googling &amp;quot;Lean Kai-Zen&amp;quot; and &amp;quot;optimizing shop floor layout&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;:&lt;br /&gt;&#xA;A second classic example is the optimization of paper-flow through an accounting department. Two other examples were these techniques were used to great effect in WWII are noted &lt;a href=&quot;http://en.wikipedia.org/wiki/Operations_research#Second_World_War&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; (&lt;strong&gt;my emphasis&lt;/strong&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;While performing an analysis of the methods used by RAF Coastal Command to hunt and destroy submarines, one of the analysts asked what colour the aircraft were. As most of them were from Bomber Command they were painted black for night-time operations. At the suggestion of CC-ORS a test was run to see if that was the best colour to camouflage the aircraft for daytime operations in the grey North Atlantic skies. Tests showed that aircraft painted white were on average not spotted until they were 20% closer than those painted black. &lt;strong&gt;This change indicated that 30% more submarines would be attacked and sunk for the same number of sightings.[19]&lt;/strong&gt; As a result of these findings Coastal Command changed their aircraft to using white undersurfaces.&lt;/p&gt;&#xA;&lt;p&gt;A Warwick in the revised RAF Coastal Command green/dark grey/white finish&#xA;Other work by the CC-ORS indicated that on average if the trigger depth of aerial-delivered depth charges (DCs) were changed from 100 feet to 25 feet, the kill ratios would go up. The reason was that if a U-boat saw an aircraft only shortly before it arrived over the target then at 100 feet the charges would do no damage (because the U-boat wouldn't have had time to descend as far as 100 feet), and if it saw the aircraft a long way from the target it had time to alter course under water so the chances of it being within the 20-foot kill zone of the charges was small. It was more efficient to attack those submarines close to the surface when the targets' locations were better known than to attempt their destruction at greater depths when their positions could only be guessed. &lt;strong&gt;Before the change of settings from 100 feet to 25 feet, 1% of submerged U-boats were sunk and 14% damaged. After the change, 7% were sunk and 11% damaged.&lt;/strong&gt; (If submarines were caught on the surface, even if attacked shortly after submerging, the numbers rose to 11% sunk and 15% damaged). Blackett observed &amp;quot;there can be few cases where such a great operational gain had been obtained by such a small and simple change of tactics&amp;quot;.[20]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="873" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-13T21:18:23.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1585" PostTypeId="2" ParentId="366" CreationDate="2014-04-12T21:46:22.773" Score="4" Body="&lt;p&gt;NO! Absolutely not!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answer became resoundingly clear to me when I moved from learning high school and undergraduate math in the U.S. to teaching high school maths in South Africa. The differences in the way maths is taught in the two countries is astonishing, yet the outcomes (at least for good students going to good schools) is more or less comparable. Let me highlight two of the most important differences between the two curricula:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The South African curriculum is standardized nationally. There is overall significantly less room in South Africa for textbook writers, schools, or teachers to decide what content should or shouldn't be taught. Though the U.S. is slowly becoming more standardized with the rise of Common Core standards, I suspect that teachers and curriculum organizers in the U.S. still have much more latitude to teach what and how they want than in South Africa&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most interesting part of South Africa's mathematics education system, however, is their alternate math curriculum called &quot;Mathematical Literacy.&quot; &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Maths lit is required through grade 12 for all students not taking the higher grade maths. &lt;/li&gt;&#xA;&lt;li&gt;Its purpose is to ground students in &quot;life related applications&quot; and &quot;enable learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems.&quot; (From p. 19 in the following document: &lt;a href=&quot;http://www.education.gov.za/LinkClick.aspx?fileticket=25TyZ4RTmKc%3D&amp;amp;tabid=246&amp;amp;mid=594&quot; rel=&quot;nofollow&quot;&gt;http://www.education.gov.za/LinkClick.aspx?fileticket=25TyZ4RTmKc%3D&amp;amp;tabid=246&amp;amp;mid=594&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Maths lit has been criticized widely in South Africa for not being rigorous enough. Indeed, South Africa scores among the very worst in its maths scores, an unfortunate and lasting legacy of apartheid. Nevertheless, I believe that while maths lit might not be done correctly in South Africa, I'm wide open to the idea that mathematics can be taught and learned in many different ways.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="867" LastActivityDate="2014-04-12T21:46:22.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1586" PostTypeId="2" ParentId="647" CreationDate="2014-04-12T22:43:38.263" Score="3" Body="&lt;p&gt;Don't neglect the students in your class with unasked questions! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one student in your class asks you a question, then it's safe to assume that at least one or two also have the question somewhere in mind. If this is the case, then I would argue that it is unfair to answer a question for only the students who are willing to ask. If we really want to avoid giving &quot;unfair advantages&quot; then surely it is unfair to bias against those students without the courage or volition to ask a question even when their question isn't really what the the test is &quot;about.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The school I teach in currently has a &quot;no questions allowed&quot; policy during exams, including when there is a flagrant or misleading mistake in the question paper. At first I thought this was harsh, but I've come to think that this really is the most &lt;em&gt;fair&lt;/em&gt; way of handling things for everyone. Additionally, this really motivates teachers to moderate their exams and carefully check for wording and clarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my class tests, if a student asks a question I will respond in &lt;strong&gt;only&lt;/strong&gt; two ways:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) I provide general encourage along the lines of &quot;You can do it!&quot; This (I hope) is a way of calming the nerves of those with test anxiety or a memory block.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) If I find that I have made a really bad mistake in writing a class test, then I'll write the correct question on the board and explain the mistake to the entire class.&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2014-04-12T22:43:38.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1587" PostTypeId="1" AcceptedAnswerId="1588" CreationDate="2014-04-12T22:46:11.317" Score="11" ViewCount="5846" Body="&lt;p&gt;Someone I know recently took an online intermediate algebra course to prepare for college algebra.&lt;/p&gt;&#xA;&lt;p&gt;Thus course had 70+ sections, each with 10-30 poblems, beginning with set-builder notation and going on to associative/distributive/commutative laws, etc. and in general proceeding axiomatically.&lt;/p&gt;&#xA;&lt;p&gt;Most college math courses proceed axiomatically, to maintain rigor.&lt;/p&gt;&#xA;&lt;p&gt;But in the same high school courses this is intended to replace, the emphasis is on 'learn this useful formula and apply it 30 times until you get it'.&lt;/p&gt;&#xA;&lt;p&gt;In my experience, the best students, who would actually appreciate the rigor, never take the rigorous college courses, but instead do great in high school and move directly into college-only courses such as linear algebra and vector calculus. While those who take intermediate or college algebra in college tend to be put off by the overly rigorous approach.&lt;/p&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is there a significant difference in the rigor of these courses as taught in college and high school (i.e., is it not just my imagination)?&lt;/p&gt;&#xA;&lt;p&gt;If there is a difference, is there any published research supporting such a difference, or statements by those in charge of curriculum at an institution giving an explanation?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-12-21T05:51:15.380" Title="Why are university algebra courses often harder than the corresponding high school courses?" Tags="|undergraduate-education|secondary-education|algebra|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1588" PostTypeId="2" ParentId="1587" CreationDate="2014-04-12T23:23:37.583" Score="14" Body="&lt;p&gt;Although someone else might agree that &quot;difference in the rigor&quot; is the crucial aspect, my own observation is that this is not the issue, although one can argue that there &lt;em&gt;is&lt;/em&gt; some sort of &quot;difference of rigor&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather, the best students in today's high schools are treated very gently grade-wise, which they have learned is the only truly relevant currency as a measure of success. Thus, they subliminally infer a level of effort, a &quot;good intention&quot;, that suffices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Upon arrival at a substantial university in the U.S., there are several upsets. First, although the students think this is a good thing, the structure to their personal lives created by living at their parents' house is gone, and this makes everything else more volatile. Second, instead of being high-school seniors perhaps at the top of their graduating class, they've become the lowest-status creatures on the whole campus... which has a population 20 times greater than their high school, etc. On top of all of &lt;em&gt;that&lt;/em&gt;, suddenly their Pre-Calc instructors don't seem to have their own self-esteem wrapped up in the success of their students... but almost the opposite, in many cases. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In many cases, the use of low-level math courses is not to educate anyone, but to &lt;em&gt;filter&lt;/em&gt;, so that otherwise-alarming failure rates are not immediately considered a bad thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By this point, seriously, issues of &quot;rigor&quot; are not the critical ones... but general ambience, adversarial relationship of student and instructor, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I do not have literal stats, but have been a concerned observer for 30+ years...)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-12T23:23:37.583" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1589" PostTypeId="1" AcceptedAnswerId="1593" CreationDate="2014-04-12T23:35:48.630" Score="34" ViewCount="4622" Body="&lt;p&gt;I am aware of three proofs of the fundamental theorem of algebra, using:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Liouville's theorem&lt;/li&gt;&#xA;&lt;li&gt;The fundamental group of the punctured plane, or&lt;/li&gt;&#xA;&lt;li&gt;Multiplicativity of field extensions together with the intermediate value theorem&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If I had a small group of undergraduates who had taken Calc III, is there any of these proofs or another proof that I could show them over a period of 3 1-hour sessions or less?&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2023-11-18T22:26:57.240" Title="Is there any proof of the fundamental theorem of algebra that can be introduced to undergraduates who have just completed Calc III?" Tags="|undergraduate-education|prerequisites|" AnswerCount="11" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1590" PostTypeId="2" ParentId="1589" CreationDate="2014-04-12T23:52:52.670" Score="16" Body="&lt;p&gt;You can find a very elementary and direct proof &lt;a href=&quot;http://www.ime.usp.br/~oliveira/FTAAUTHOR.pdf&quot;&gt;here&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, while I think that any of the more sophisticated proofs is very cool the direct proof is valuable since it really shows the fundamental theorem of algebra is not a theorem of algebra but rather of the complex numbers, and that it really is fundamental in the sense that it is very elementary (and important). It shows the proof is really not that complicated and amounts to a relatively simple analysis exercise. Using other methods (in my humble opinion) is using a sledgehammer where it really is not needed. &lt;/p&gt;&#xA;" OwnerUserId="59" LastActivityDate="2014-04-12T23:52:52.670" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1591" PostTypeId="2" ParentId="1589" CreationDate="2014-04-13T01:34:31.783" Score="13" Body="&lt;p&gt;Your students might find it useful to see this &quot;visual approach&quot; to proving the FTA:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Velleman, D. J. (2007). The Fundamental Theorem of Algebra: A Visual Approach. &lt;a href=&quot;http://www.cs.amherst.edu/~djv/FTAp.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For a more rigorous approach by the same author, see:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Velleman, D. J. (1997). Another proof of the fundamental theorem of algebra. &lt;em&gt;Mathematics Magazine&lt;/em&gt;, 216-217. &lt;a href=&quot;http://www.jstor.org/stable/2691268&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(The latter proof uses &quot;the fact that entire functions can always be represented by power series.&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, an entire list of proofs can be found &lt;a href=&quot;http://mathfaculty.fullerton.edu/mathews/c2003/FunTheoremAlgebraBib/Links/FunTheoremAlgebraBib_lnk_2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. I have not read closely through these, so I can neither make a more specific recommendation nor vouch for all of them; in particular:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark.&lt;/strong&gt; In the list above, #60 &quot;A topological proof of the fundamental theorem of algebra&quot; (Arnold, 1949) is known to have errors. This is why he published a correction paper a couple years later (#58); the main idea, though, of using the Brouwer Fixed Point Theorem to prove the FTA &lt;em&gt;has&lt;/em&gt; been carried out (though perhaps this is a result your post-Calc III students have not yet seen). In case this is useful, see:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fort, M. K. (1952). Some properties of continuous functions. &lt;em&gt;American Mathematical Monthly&lt;/em&gt;, 372-375. &lt;a href=&quot;http://www.jstor.org/stable/2306806&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(The last two sources I became aware of while writing up an answer &lt;a href=&quot;https://mathoverflow.net/a/132080/22971&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Unsurprisingly, there is a &lt;a href=&quot;https://mathoverflow.net/q/10535&quot;&gt;list&lt;/a&gt; of ways to prove the FTA on MathOverflow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is likely to include most any suggestion that could be posted here (on MESE) but the reverse inclusion is sure not to hold: The majority of these proofs are not written for students who just completed Calc III. On the other hand, three lectures with the FTA as a sole goal could cover a fair bit of material...&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-04-13T04:51:42.417" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1592" PostTypeId="2" ParentId="1550" CreationDate="2014-04-13T02:39:07.090" Score="40" Body="&lt;h2&gt;Bad Optimization Problems&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I thought that Jack M made an interesting comment about this question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There aren't any. There may be situations where it's possible to apply optimization to solve a problem you've encountered, but in none of these cases is it honestly worth the effort of solving the problem analytically. I optimize path lengths every day when I walk across the grass on my way to classes, but I'm not going to get out a notebook and calculate an optimal route just to save myself twelve seconds of walking every morning. Mathematics beyond basic arithmetic is simply not useful in ordinary life. But I'm not sure if that's exactly what you mean. &amp;ndash; JackM&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To some extent, I agree with this comment.  With few exceptions, mathematics beyond basic arithmetic is simply not useful in everyday life.  Students know this, and you'll have trouble convincing them otherwise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of this, I've always found &quot;everyday&quot;-style calculus problems a little artificial.  Consider the following problem from Stewart's &lt;em&gt;Calculus: Concepts and Contexts&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building.  What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The proper response to this question is: who cares?  Is there any reason to calculate this length precisely?  Why would anyone ever use calculus to compute this?  If you have an actual building and an actual ladder, you could just try it and see if the ladder fits.  If you don't have a specific ladder in mind (e.g. you are buying a ladder), the thing to do would be to &lt;em&gt;draw&lt;/em&gt; the situation on paper and then use a ruler to estimate the minimum length.  Of course, it's neat that you can use calculus to solve this problem precisely, but this is more of a curiosity than a legitimate application.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chris specifically mentions the farmer fence problem, the wire-cutting problem, and the Norman window problem as not relevant to the students' lives.  I agree&amp;mdash;none of these problems are relevant.  But it's not because the students aren't farmers, or wire-cutters, or architects.  Even in a class full of future farmers, the fence problem would still be bad, because &lt;em&gt;farmers don't use calculus to plan their fences&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Good Optimization Problems&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;What calculus &lt;em&gt;is&lt;/em&gt; useful for is science, economics, engineering, industrial operations, finance, and so forth.  That is, it's useful for &lt;em&gt;all the things that make our society run&lt;/em&gt;.  Most students who take calculus at a university are planning to go into one of these fields, so calculus will be relevant in their lives&amp;mdash;specifically in their future studies and in their professions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's something that's closer to a real-life optimization problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When a critically damped RLC circuit is connected to a voltage source, the current $I$ in the circuit varies with time according to the equation&#xA;  $$&#xA;I \;=\; \biggl(\frac{V}{L}\biggr)te^{-Rt/(2L)}&#xA;$$&#xA;  where $V$ is the applied voltage, $L$ is the inductance, and $R$ is the resistance (all of which are constant).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Suppose an RLC circuit with a resistance of 30.0&amp;nbsp;volt/amp and an inductance of 0.400&amp;nbsp;volt&amp;middot;sec/amp is attached to a 12.0-volt voltage source.  Find the maximum current that will occur in the circuit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is at least close to something that a physics or engineering student might actually come across in their future studies.  It's real in a way that the farmer fence problem isn't, and even students who don't plan to study physics can sense that this is a legitimate application.  (By the way, if you have good students, you might even ask them to come up with a formula for the maximum current, without giving them specific numbers for $V$, $L$ and $R$.  This has the advantage that it can't simply be solved using a graphing calculator.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this isn't actually a &lt;em&gt;constrained&lt;/em&gt; optimization problem&amp;mdash;it's just an optimization problem.  I'm not actually aware of any place in science that simple constrained optimization problems arise, although there are examples from economics (maximizing utility), finance (optimal portfolios), and industrial design (e.g. shape of a can type problems).  When I cover constrained optimization in calculus, I usually stick to industrial-type problems (best cans, best shipping crates/boxes, best pipeline across a river, etc.), but that's probably just because I don't know enough about economics or finance to make up problems that involve them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I should mention that I've never found the optimization portion of Calculus I particularly compelling.  It's good to introduce the idea of optimization, but setting the derivative equal to zero isn't actually a very useful optimization technique by itself.  It only really works for simple formulas&amp;mdash;for anything complicated it just replaces one essentially numerical problem (finding the maximum of a function) with another (finding roots of a function).  I agree that it should be covered, but it's far from the most important application of calculus.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-04-13T02:50:05.523" LastActivityDate="2014-04-13T02:50:05.523" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1593" PostTypeId="2" ParentId="1589" CreationDate="2014-04-13T02:42:49.320" Score="27" Body="&lt;p&gt;I think that the proof based on the fundamental group of the punctured plane can be re-wrigged, just by omitting references to homotopies and the fundamental group, to be convincing to students who don't even necessarily know calculus, as long as they have sufficient comfort with the complex plane to be able to think about how a polynomial maps loops about the origin. From an aggressively technical point of view this introduces some handwaving, but IMHO the underlying topological theorems are sufficiently intuitively reasonable that this more or less counts as a proof anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More specifically:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Do you think the students could (be made to) see that for $t$ very small, $f(te^{i\theta})$ (for $\theta\in[0,2\pi]$) traces a very small loop of some kind about the constant term of $f$, that therefore gets nowhere near the origin?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Do you think they could see that for $t$ large, $f(te^{i\theta})$ traces some kind of incredibly huge $n$-fold loop that encloses the origin?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the answers are both yes, then I bet they will also see that if $t$ varies continuously from very small to very large, the image of $f(te^{i\theta})$ is going to vary continuously and therefore have to pass through the origin.&lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2014-04-13T02:42:49.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1594" PostTypeId="2" ParentId="1506" CreationDate="2014-04-13T04:00:50.177" Score="20" Body="&lt;p&gt;I think the approach to division (and logarithms) that you complain about is actually the right one.  It seems to me that the statement&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$&#xA;x = \frac{z}{y}\qquad\text{is the same as}\qquad xy=z.&#xA;$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is essentially the definition of division.  For example, if you were to ask a third grader the question&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it true that 12 divided by 4 is 3?  If so, why?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;you would hope to get the answer&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Yes, since 3 times 4 is 12.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This would be much better than the answer&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Yes, since we can divide both sides of the equation $3\times 4 = 12$ by 4 to get the equation $3 = 12 \div 4$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the same way, the proper answer to the question&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it true that $\log_{10}(100) = 2$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Yes, since $10^2 = 100$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A student who has a good understanding of logarithms should be able to do this in one step in their minds, without writing down any equations or thinking about applying operations to both sides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, you complain that this approach involves &quot;moving symbols around&quot;, but I counter that the symbols in mathematics have &lt;em&gt;meaning&lt;/em&gt;.  In this case, the way that we're moving the symbols around directly reflects how we think about logarithms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I should point out that this really just has to do with two different definitions of inverse functions.  Here they are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Definition 1.&lt;/strong&gt; Two functions $f\colon X\to Y$ and $g\colon Y\to X$ are inverses if&#xA;$$&#xA;f(x) = y\qquad\Leftrightarrow\qquad g(y) = x&#xA;$$&#xA;for all $x\in X$ and $y\in Y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Definition 2.&lt;/strong&gt; Two functions $f\colon X\to Y$ and $g\colon Y\to X$ are inverses if&#xA;$$&#xA;g(f(x)) = x\qquad\text{and}\qquad f(g(y)) = y&#xA;$$&#xA;for all $x\in X$ and $y\in Y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that you prefer using definition 2, but I think definition 1 is simpler.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-04-13T04:00:50.177" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1595" PostTypeId="2" ParentId="444" CreationDate="2014-04-13T08:47:41.483" Score="5" Body="&lt;p&gt;I find it brilliant, wonderful, beautiful, but I should think it's a big turn-off for most students because it's so abstract; it's impossible to explain what it actually means.&lt;/p&gt;&#xA;" OwnerUserId="877" LastActivityDate="2014-04-13T08:47:41.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1596" PostTypeId="1" AcceptedAnswerId="1619" CreationDate="2014-04-13T09:37:34.477" Score="9" ViewCount="432" Body="&lt;p&gt;I teach grade 8, 9, and 10 maths students who are blissfully ignorant of great mathematical tools available to them such as WolframAlpha or graphing calculators (which are not used in our school). At present I am aware and grateful that I don't have to worry about any issues related to cheating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do feel that these students should be introduced to WolframAlpha, since it is a great (not to mention interesting) tool they should use even if they don't study maths in university. Besides, they're likely to find out about it before grade 12 anyway. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;But here's my question: &lt;em&gt;When&lt;/em&gt; and &lt;em&gt;how&lt;/em&gt; should I introduce my students to WolframAlpha?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd be especially grateful if someone could suggest a particular topic or question for which WolframAlpha could provide my students appropriate insight.&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="123" LastEditDate="2014-04-14T23:23:11.253" LastActivityDate="2014-04-25T19:15:15.740" Title="How and when should I introduce my students to WolframAlpha?" Tags="|technology-in-education|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1597" PostTypeId="2" ParentId="1042" CreationDate="2014-04-13T10:13:45.410" Score="8" Body="&lt;p&gt;Here are a few projects done at my high school:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Given a quadratic in the form $y=ax^2+bx+c$, determine what changing $a$, $b$ and $c$ does to the turning point (vertex) of the parabola and prove your answer. Most students are able to figure out $a$ and $c$ relatively quickly, and they are good practice for $b$, which turns out to be much harder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Prove that 7 congruent circles form 12 points of tangency as shown in the diagram below. Challenge students to prove this at least two ways using separate branches of maths. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/9wtjX.gif&quot; alt=&quot;&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Taxicab_geometry&quot; rel=&quot;noreferrer&quot;&gt;Taxicab Geometry&lt;/a&gt;&lt;/strong&gt;. Resources and questions abound online. &lt;strong&gt;&lt;a href=&quot;http://wwwp.cord.edu/faculty/andersod/TaxicabWorksheets.pdf&quot; rel=&quot;noreferrer&quot;&gt;This&lt;/a&gt;&lt;/strong&gt; is one example of a good guiding worksheet. I haven't done Taxicab Geometry with my students, but I think it's an interesting way to challenge students to rethink all of their basic geometry definitions and axioms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Have students revisit every maths theorem and formula used in high school. Have them summarise the theorem in their own words and give two examples of how it is used. (This project was given to the &quot;bottom set&quot; of students, with the feeling that they needed more practice, whereas the top students were more prepared for an extension.)&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="867" LastEditDate="2014-06-04T06:16:11.933" LastActivityDate="2014-06-04T06:16:11.933" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1598" PostTypeId="1" AcceptedAnswerId="1614" CreationDate="2014-04-13T10:28:55.037" Score="35" ViewCount="5123" Body="&lt;p&gt;I have heard the story (may be an urban legend?) of a top professor who occasionally wanted to teach freshman analysis. He believed in the method of letting students see how a mathematician's mind works, so he came to the lectures largely unprepared, and was proving theorems as they were met in the textbook. Consequently he would not always follow the straightest path to the destination. Occasionally he ran afoul full speed, and had to backtrack a bit or even start over.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does this method help a large enough group of students to make it worth our while?  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My views in what follows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pros&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students do get to see how a trained mathematician approaches the task of proving a theorem.&lt;/li&gt;&#xA;&lt;li&gt;They learn that doing math is not a single-track exercise.&lt;/li&gt;&#xA;&lt;li&gt;Students see the real way of doing math as opposed to fully digested polished presentation in the book.&lt;/li&gt;&#xA;&lt;li&gt;This slows down the progress a bit, which may help some.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cons&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Taking notes (if a student chooses to do so meticulously) becomes a nightmare, when the teacher is hopping from one step to another&lt;/li&gt;&#xA;&lt;li&gt;This slows down the progress a bit making it harder to cover all the material.&lt;/li&gt;&#xA;&lt;li&gt;Doesn't invite student participation (unless you are willing to invest even more time on it).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What I have tried/done&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Accidentally doing this (krhm).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;After a motivational example I go straight to the proof of the main result of the day's lecture. I keep adding the necessary assumptions to the statement of the theorem as we encounter them. Also I get these lightbulb moments &quot;Ahh, this is why the colleague who wrote these lecture notes had that lemma a few pages back. Comes in handy here.&quot; Naturally I had skipped that lemma, but go through it now - distracting some, but also giving a more realistic impression of how math gets done.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary of the discussion&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The views expressed by participants in this discussion converge to AFAICT:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Letting students see dead-ends of trains of thought has, indeed, a lot of pedagogical merit, and could be turned into a tool of inviting student participation.&lt;/li&gt;&#xA;&lt;li&gt;But these wrong turns should be crafted as carefully as the polished examples. They should not be random by-products of un(der)preparation. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Good teachers get their results in the old-fashioned way of earning them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;All: Thanks for sharing your ideas and suggestions in how to best implement this in practice.&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="143" LastEditDate="2014-08-03T11:15:44.363" LastActivityDate="2018-04-08T00:09:02.443" Title="Lecturers &quot;(intentional) mistakes&quot; as a teaching tool" Tags="|undergraduate-education|mathematical-pedagogy|general-pedagogy|" AnswerCount="13" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1599" PostTypeId="2" ParentId="1550" CreationDate="2014-04-13T11:49:48.997" Score="7" Body="&lt;p&gt;Here's an optimisation problem that really should have solutions readily available, but hasn't: Consider a SatNav that tells you the best way to get from A to B. Now assume the distance is quite long, and you won't be able to get from A to B without going to a petrol station. So what I want: Find the shortest way from A to B visiting &lt;em&gt;any&lt;/em&gt; petrol station within X miles of my current location. They all are capable of showing you nearby petrol stations, but that's not what I want if I'm 200 miles away from B and have fuel for 150 miles in the tank. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way: Finding the shortest path from A to B is actually an interesting problem in itself. SatNavs (at least the cheap ones that you will tend to use) are not capable of actually finding what I would consider good solutions, which take into account time, fuel consumption, wear and tear on the car, driver stress. (For example a short journey with many turns where I have to give way to other drivers means I stop and accelerate a lot and have more stress). &lt;/p&gt;&#xA;" OwnerUserId="697" LastActivityDate="2014-04-13T11:49:48.997" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1600" PostTypeId="2" ParentId="1596" CreationDate="2014-04-13T13:23:06.497" Score="4" Body="&lt;p&gt;You should introduce Wolfram Alpha as soon as you like. Some students will already know about it, and it might be an unfair advantage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, students will always find ways to cheat if getting the correct answer is all that is needed. See Joel's answer here: &lt;a href=&quot;https://matheducators.stackexchange.com/a/53/78&quot;&gt;https://matheducators.stackexchange.com/a/53/78&lt;/a&gt; . This doesn't really apply directly to your age group; but one idea to take from it is that you could give quizzes with no calculators allowed, or worksheets without calculators, and emphasize how important it is to know how to do all the steps by yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One fun exploration is to have them graph different polynomials to figure out how many 'bumps' you can have for a polynomial with a specific degree.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-13T13:23:06.497" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1601" PostTypeId="2" ParentId="1598" CreationDate="2014-04-13T13:27:25.370" Score="8" Body="&lt;p&gt;It has happened to me too. I believe it leaves the impression that you didn't prepare properly (guilty as charged) and doesn't really help understanding. What does seems to work is to have the class make up a problem (or when they propose an interesting sidetrack) and solve that with the required backtracking, asking the class for input. Just making sure they know what is exploratory (I write &quot;scratch&quot; and use another color for this, perhaps on a separate area of the board) while/before writing up the definitive version. If it is interesting/important enough, or no satisfactory solution is reached, I'll write ìt up formally (scratch work included) to publish for them.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-13T13:27:25.370" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1602" PostTypeId="2" ParentId="1596" CreationDate="2014-04-13T13:36:34.573" Score="3" Body="&lt;p&gt;There are several CAS around, including my personal favorite &lt;a href=&quot;http://maxima.sourceforge.net&quot; rel=&quot;nofollow&quot;&gt;maxima&lt;/a&gt;, which incidentally also is available on smartphones...&#xA;Use the computer to do the routine jobs, save the brain for understanding and creating. Sure, &lt;em&gt;some&lt;/em&gt; familiarity with the routine work is needed, but I'd concentrate on the &quot;higher level&quot; tasks.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-13T13:36:34.573" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1603" PostTypeId="2" ParentId="366" CreationDate="2014-04-13T13:43:36.353" Score="1" Body="&lt;p&gt;I believe that the math needs of folks are different (just ask different people when they last computed a derivative or integral...). The hard problem is to find out what each one &lt;em&gt;really&lt;/em&gt; needs, and offer personalized curricula... Mostly I believe the resources just aren't there to do this.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-13T13:43:36.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1604" PostTypeId="2" ParentId="1598" CreationDate="2014-04-13T14:13:27.803" Score="22" Body="&lt;p&gt;Here is something that you will hear from students who attend clear, perfectly-prepared lectures:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;When I'm in class watching you do it, it all makes perfect sense and seems really easy. But then I go home and work on it and I don't know how to do &lt;em&gt;anything&lt;/em&gt;.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is a disastrous outcome in introductory courses because it makes your students think that you are smart and they are dumb.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in my opinion, even in a non-proofs class, a lecturer must do all of these:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Make mistakes, cross them out in red, and fix them&lt;/li&gt;&#xA;&lt;li&gt;Stop, acting puzzled, at a place where a student might get stuck&lt;/li&gt;&#xA;&lt;li&gt;Show what it looks like when a technique is being misapplied&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Basically the goal is to give the students' brains something to process during the lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd also say that some of this actually does invite &lt;strong&gt;more&lt;/strong&gt; class participation:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If every single time you solve $x^2 = C$ you make the &quot;mistake&quot; of only finding one solution, it becomes a kind of game to the students to notice the error first. &lt;/li&gt;&#xA;&lt;li&gt;When the correct answer to &quot;what should I do next&quot; is sometimes &quot;give up on this approach??,&quot; it takes a lot of pressure off of students who are used to the airwaves only being filled with perfectly correct reasoning.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-04-13T14:13:27.803" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1605" PostTypeId="2" ParentId="651" CreationDate="2014-04-13T14:18:04.637" Score="2" Body="&lt;p&gt;Getting people to really work together is hard. Encourage them to form study groups, so they start working together on small projects (solve individual problems), learn how to organize themselves (where, when to meet, what to prepare). Make them organize a short seminar on some topic (contact invited speakers, but have them handle the rest). Encourage independent groups/clubs, even in totally unrelated matters. Ask for help organizing welcome for new students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As in everything, you learn not by seeing how it is done, but by doing (and making your own mistakes on the way, preferably where they don't matter much).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-13T14:18:04.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1606" PostTypeId="2" ParentId="280" CreationDate="2014-04-13T14:33:33.847" Score="10" Body="&lt;p&gt;I recently came across the riddle that $\frac{3}{16} - \frac{3}{19} =\frac{3}{16} \cdot \frac{3}{19}$, and thus the question what values of the variables give the remarkable coincidence&#xA;$$&#xA;\frac{a}{b} - \frac{a}{d} =\frac{a}{b} \cdot \frac{a}{d}&#xA;$$&#xA;The point is that algebra gives a simple explanation to a mystifying phenomenon.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-13T14:33:33.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1607" PostTypeId="2" ParentId="1042" CreationDate="2014-04-13T14:51:27.143" Score="1" Body="&lt;p&gt;Ask  them to show that the following procedure constructs all rhombuses (rhombi?) of a fixed side  length: take two circles of the same radius. Let them be placed to intersect at two distinct points. Now the two centres and the intersection points together form vertices of a rhombus. Now 'pull the circles apart' slightly or push them closer and obtain more of them. &lt;/p&gt;&#xA;" OwnerUserId="548" LastEditorUserId="548" LastEditDate="2014-04-13T15:04:40.297" LastActivityDate="2014-04-13T15:04:40.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1608" PostTypeId="1" AcceptedAnswerId="1623" CreationDate="2014-04-13T15:31:13.743" Score="8" ViewCount="506" Body="&lt;p&gt;I attended a really &lt;em&gt;great&lt;/em&gt; university with excellent math professors, engaged and intelligent classmates, and great community. However, my education classes were &lt;em&gt;horrendous&lt;/em&gt; to say the least. While I took hard and enjoyable math classes, my education classes were taught by boring, preachy, and uninterested profs who gave trivial homework and rote-learning tests all about the importance of learning styles and assessment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since setting out on my own teaching journey, I'm slowly recovering from my education class malaise by reading Neil Postman, learning about KIPP schools, and visiting this site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What inspires you to more deeply &lt;em&gt;think&lt;/em&gt; and &lt;em&gt;learn&lt;/em&gt; about education and pedagogy? What books, authors, ideas, articles, etc. have guided you to this end?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="262" LastEditDate="2014-04-13T19:20:24.130" LastActivityDate="2014-04-16T00:16:52.690" Title="Inspiration for Learning Math Pedagogy" Tags="|mathematical-pedagogy|teacher-training|teacher-motivation|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1609" PostTypeId="1" AcceptedAnswerId="1616" CreationDate="2014-04-13T16:12:45.083" Score="37" ViewCount="5005" Body="&lt;p&gt;Neil Postman wrote an interesting (and freely available) article called &quot;&lt;a href=&quot;http://neilpostman.org/articles/Postman-TheEducationistAsPainkiller.pdf&quot;&gt;The Educationist as Painkiller&lt;/a&gt;.&quot; I highly recommend you read the article for your own enjoyment and as a background to this question, especially pp. 4-6. Here is one of his main ideas, from p. 4 (emphasis added):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This, then, is the strategy I propose for educationists—that we&#xA;  &lt;em&gt;abandon our  vague, seemingly arrogant, and ultimately futile attempts&#xA;  to make children intelligent,  and concentrate our attention on&#xA;  helping them avoid being stupid&lt;/em&gt;... The&#xA;  physician knows about sickness and can offer  specific advice about&#xA;  how to avoid it. Don't smoke, don’t consume too much salt or &#xA;  saturated fat, take two aspirins, take penicillin every four hours,&#xA;  and so forth. I am  proposing that the study and practice of education&#xA;  adopt this paradigm precisely. The  educationist should become an&#xA;  expert in stupidity and be able to prescribe specific  procedures for&#xA;  avoiding it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Throughout the rest of the article, Postman addresses a few historical writings about stupidity and summarizes what people have said about stupidity in history.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I grant that, unlike the study of sickness and injustice, the study of&#xA;  stupidity  has rarely been pursued in a systematic way. But this does&#xA;  not mean that the subject  has no history...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here's my question: &lt;strong&gt;What are the main &lt;em&gt;types&lt;/em&gt; of mistakes do we encounter in our classrooms?&lt;/strong&gt; Of course we can all think of many, many mistakes that our students have made, but perhaps  it will be helpful to classify, categorize these mistakes for a &quot;Lexicon of Math Mistakes&quot; which our students should avoid.&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="77" LastEditDate="2016-08-21T02:34:27.570" LastActivityDate="2016-08-21T02:34:27.570" Title="A Lexicon of Math Mistakes" Tags="|mathematical-pedagogy|curriculum|students-mistakes|" AnswerCount="7" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1611" PostTypeId="2" ParentId="1608" CreationDate="2014-04-13T16:55:34.680" Score="6" Body="&lt;p&gt;While my answer is not a book, author, idea, or article, it is a source of ideas: &lt;strong&gt;conferences of people who are interested in such things&lt;/strong&gt; are very good for inspiring you to learn. The American Mathematical Association of Two-Year Colleges (&lt;a href=&quot;http://www.amatyc.org&quot;&gt;AMATYC&lt;/a&gt;) has &lt;a href=&quot;http://www.amatyc.org/?page=Affiliates&quot;&gt;affiliates in each region&lt;/a&gt; and many of them host annual conferences. My first experience at the Illinois conference was excellent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't speak to specific conferences outside of the community college scene, but I assume such things exist for math education at other levels.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-04-13T16:55:34.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1612" PostTypeId="2" ParentId="1572" CreationDate="2014-04-13T17:09:24.870" Score="17" Body="&lt;p&gt;True story: I ordered new carpet flooring for a room in my house. The length of the room was 13 foot 11 inches. The employee took his calculator and typed &quot;13.11 x 30.48 =&quot; to convert into centimetres. I didn't actually manage to convince him of his mistake, but had to ask for a more experienced colleague. Would have been a nasty surprise if I hadn't noticed and they had delivered a piece of carpet 10 inch too short. &lt;/p&gt;&#xA;" OwnerUserId="697" LastActivityDate="2014-04-13T17:09:24.870" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1613" PostTypeId="2" ParentId="1609" CreationDate="2014-04-13T17:12:26.427" Score="19" Body="&lt;p&gt;As a first guess, I'd give the following categories:&lt;/p&gt;&#xA;&lt;h2&gt;Overgeneralization&lt;/h2&gt;&#xA;&lt;p&gt;of known theorems or techniques. Application of these theorems or techniques in unsuited situations.&lt;/p&gt;&#xA;&lt;h2&gt;Overconfidence in intuition&lt;/h2&gt;&#xA;&lt;p&gt;Making intuition into false theorems. Applying intuition where it contradicts known theorems.&lt;/p&gt;&#xA;&lt;h2&gt;Nonapplication&lt;/h2&gt;&#xA;&lt;p&gt;of known theorems or techniques. (Mistakes in problem solving, that leads to not solving the problem.)&lt;/p&gt;&#xA;&lt;h2&gt;Noncorrection&lt;/h2&gt;&#xA;&lt;p&gt;of committed mistakes, although obivously wrong or by testing.&lt;/p&gt;&#xA;&lt;h2&gt;Nonrigorosity&lt;/h2&gt;&#xA;&lt;p&gt;in proofs or arguments. Leaving gaps, unresolved issues or forgetting cases.&lt;/p&gt;&#xA;&lt;h2&gt;Nonconformity&lt;/h2&gt;&#xA;&lt;p&gt;to mathematical standards, thereby writing something false down, although maybe something correct was thought.&lt;/p&gt;&#xA;&lt;h2&gt;Algorithmic mistakes&lt;/h2&gt;&#xA;&lt;p&gt;Mistakes in the ordering or execution of steps of an algorithm.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-13T17:12:26.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1614" PostTypeId="2" ParentId="1598" CreationDate="2014-04-13T17:14:24.603" Score="45" Body="&lt;p&gt;Yes, showing your students that you too can make mistakes, and that real mathematics is not a linear process, are both very good ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;However, this &lt;em&gt;does not&lt;/em&gt; mean that you should come to the lecture unprepared!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I seem a bit vehement about this, it's because I've met all too many math lecturers who seem to feel a need to &quot;prove their manlyhood&quot; (or at least their awesome math skills) by refusing to prepare their proofs ahead of the lecture and instead proving everything on the spot.  Trust me, spending 30 minutes out of a two-hour lecture watching the lecturer look for a sign error they made 15 minutes earlier gets old &lt;em&gt;real&lt;/em&gt; fast.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that any lecture time you spend backtracking and correcting errors is time you &lt;em&gt;could've&lt;/em&gt; spent teaching the students something new.  I've never heard of a lecturer complaining about having &lt;em&gt;too much&lt;/em&gt; time to cover everything they'd like, so good lecturers usually need to carefully budget their time, and any time spent finding and fixing mistakes in front of the class takes time out of this budget that has to be made up somehow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, while &lt;em&gt;showing&lt;/em&gt; your students a carefully planned example of how a mathematician would really approach a new problem can be very educational, that's &lt;em&gt;not&lt;/em&gt; the same as just having the students twiddle their thumbs while you stare at the blackboard in confusion, which is what's more likely to happen if you &lt;em&gt;actually&lt;/em&gt; get stuck in the middle of a proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, here's a few ideas you could try, if you think your lectures are too polished and would benefit from more mistakes and spontaneity:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Fake it:&lt;/strong&gt; Prepare your lectures as carefully as you can, but make occasional deliberate mistakes when writing on the blackboard.  (You could even write something like &quot;make mistake here&quot; in your notes.)  Encourage your students to try and spot these mistakes (with small rewards, if necessary).  If they &lt;em&gt;don't&lt;/em&gt; spot them in a few minutes, pretend to notice them yourself and fix them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this also helps encourage your students to spot any &lt;em&gt;genuine&lt;/em&gt; mistakes you might make.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Make it an event:&lt;/strong&gt; Pick a specific problem that you're fairly sure you can solve on the spot, and allocate some extra time for it.  Make sure to tell your students that you haven't prepared notes for that specific proof in advance &amp;mdash; this also has the useful effect of helping your students realize that you &lt;em&gt;do&lt;/em&gt; normally prepare your proofs in advance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make sure to keep describing what you're thinking while solving the problem out loud, and encourage your students to offer suggestions if you get stuck.  Optionally, prepare a &quot;cheat sheet&quot; in case you really do get stuck, or pick the problem so that you can leave completing the proof as a bonus exercise.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Act it out:&lt;/strong&gt; Same as above, but actually have a plan for your &quot;spontaneous&quot; proof.  That is, when preparing the lecture, prove the theorem in advance, but write down not just the final &quot;tidy&quot; solution, but also a log of any mistakes and dead ends you encounter.  Then re-enact the full proof, dead ends and all, in front of the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if you don't feel like actually re-enacting your solving process in every detail, you can still include some deliberate dead ends in your lecture plan.  For example, once you've shown how to prove the stability of a stable node using phase-plane analysis, try doing it for a stable focus instead.  Or pick a pumpable non-regular language and try to prove its nonregularity using the pumping lemma, before giving up and using Myhill&amp;ndash;Nerode instead.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also, even if you don't feel like using any such tricks, one thing you can do to make your usual &quot;tidy&quot; proofs seem a bit less linear is to sprinkle them with details about possible dead ends (&quot;The obvious way to try and prove this would be to use lemma 1, but that doesn't work here, because...&quot;) or historical notes (&quot;The original proof of this theorem was 50 pages long; the short proof I'll show you was only found 38 years later by...&quot;).&lt;/p&gt;&#xA;" OwnerUserId="526" LastActivityDate="2014-04-13T17:14:24.603" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1615" PostTypeId="2" ParentId="1609" CreationDate="2014-04-13T17:56:43.890" Score="11" Body="&lt;p&gt;Perhaps one of the largest mistakes is &lt;em&gt;failing to check your work&lt;/em&gt;. This has at least 3 manifestations:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Failing to check arithmetic by estimation. If you perform a large multiplication or addition, you should check your answer against a guesstimate for number of digits, etc.&lt;/li&gt;&#xA;&lt;li&gt;Failing to check a solution to an equation by plugging into the original. Sometimes this is hard to do, but on an exam it is easy to plug in to check your work.&lt;/li&gt;&#xA;&lt;li&gt;Failing to test a proof on an example. Many proposed proofs can be seen to be false when to apply them to something simple. For instance, is there a unique subgroup of any given order dividing a group's order? The Klein 4-group answers that trivially. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-04-13T17:56:43.890" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1616" PostTypeId="2" ParentId="1609" CreationDate="2014-04-13T19:12:43.053" Score="23" Body="&lt;p&gt;&lt;strong&gt;Edit 9/5/14:&lt;/strong&gt; It has recently come to my attention that another helpful paper is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Confrey, J. (1990). A review of the research on student conceptions in mathematics, science, and programming. &lt;em&gt;Review of research in education&lt;/em&gt;, 3-56. &lt;a href=&quot;http://www.jstor.org/stable/1167350&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Though there are sure to be more technology-related errors today, you can find a late 70s article on this subject from the JRME that discusses the topic at hand:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Radatz, H. (1979). Error analysis in mathematics education. &lt;em&gt;Journal for Research in Mathematics Education&lt;/em&gt;, 163-172. &lt;a href=&quot;http://www.jstor.org/stable/748804&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Here is a quick list of some error types identified in the paper above:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Errors Due to Difficulties in Obtaining Spatial Information&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Errors Due to Deficient Mastery of Prerequisite Skills, Facts,&#xA;and Concepts&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Errors Due to Incorrect Associations or Rigidity of Thinking&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Errors Due to the Application of Irrelevant Rules or Strategies&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Note that for #3 the author cites a German paper of Pippig (1975) that further classifies these errors as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A. Errors of perseveration&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B. Errors of association&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C. Errors of interference&lt;/p&gt;&#xA;&#xA;&lt;p&gt;D. Errors of assimilation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E. Errors of negative transfer from previous tasks&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All topics are given further discussion at the main source.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Radatz paper is mainly about errors among children, which is why it mentions authors such as Ginsburg (&lt;a href=&quot;https://matheducators.stackexchange.com/a/843/262&quot;&gt;MESE 1&lt;/a&gt;); however, you might also check the papers on errors that cited it - found &lt;a href=&quot;http://scholar.google.com/scholar?q=errors&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=5,33&amp;amp;sciodt=0,33&amp;amp;cites=6659278460213463960&amp;amp;scipsc=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the first paper to show up there is a JRME article co-authored by Zaslavsky (&lt;a href=&quot;https://matheducators.stackexchange.com/a/938/262&quot;&gt;MESE 2&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Movshovitz-Hadar, N., Zaslavsky, O., &amp;amp; Inbar, S. (1987). An empirical classification model for errors in high school mathematics. &lt;em&gt;Journal for Research in Mathematics Education,&lt;/em&gt; 3-14. &lt;a href=&quot;http://www.jstor.org/stable/749532&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The authors of this last paper note:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The only basic assumption was that most of the errors high school students commit in mathematics are not accidental and are derived by a quasi-logical process that somehow makes sense to the student. This assumption is an extension to high school mathematics of&#xA;  Ginsburg's (1977, p. 129) principles underlying errors in arithmetic.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Citation&lt;/strong&gt; Ginsburg, H. (1977). &lt;em&gt;Children's arithmetic: The learning process&lt;/em&gt;. New York: Van Nostrand.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The high school categories of errors are:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Misused data&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Misinterpreted language&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Logically invalid inference&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Distorted theorem or definition&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Unverified solution&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Technical error&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Each category is characterized/discussed and has examples provided in the paper.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-09-05T19:20:07.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1617" PostTypeId="2" ParentId="1609" CreationDate="2014-04-13T19:13:19.657" Score="16" Body="&lt;p&gt;Through the repetitive drilling of pointless, boring, exercises many people develop a disconnect between the mathematical formalism and semantics. Many people have lost the ability to reason logically about simple mathematical entities since they don't understand them as entities. This is, in my view, a horrible situation and the biggest mistake, as in a sense it is; the misconception of that mathematics is all about computation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, when one of my students mis-read a &quot;find a counterexample or prove the following....&quot; as &quot;find a counterexample and prove the following....&quot; and proceeded to correctly give three counterexamples, showing the claim is false, and then immediately set down to give a formal proof for the validity of the claim, simply going through the motions (assume that, let this, notice that apply this...). Complete disconnect of reason. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I propose: a-reasonalism. &lt;/p&gt;&#xA;" OwnerUserId="59" LastEditorUserId="59" LastEditDate="2014-04-13T19:19:15.803" LastActivityDate="2014-04-13T19:19:15.803" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1618" PostTypeId="2" ParentId="1608" CreationDate="2014-04-13T20:44:31.197" Score="6" Body="&lt;p&gt;In my particular case, I was lucky to have some very inspiring teachers, and to be teaching assistant in large, well-organized courses. Yes, other teachers were awful, but I learned how to teach from the ones I admire (and I hope I'm doing them justice in my work). I learned a lot from colleagues, even (or perhaps particularly) from ones in other fields. Reading a lot of different textbooks has shown different approaches, and thus to be able to select different techniques when one doesn't seem to work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I joined here in the hope to add a bit to the toolbox (and also to help out others). Up to here it has been worthwhile, I've learned a bit and got useful suggestions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the most important step on the road of becomming a better educator is to just recognize that having a fancy degree doesn't mean squat. Make sure you understand where the student's thinking process goes awry, and how to straighten it up. See where the learning process gets off track, and see how to keep it running (in my courses, with mostly completely unfamiliar subject matter, the standard &quot;cram the last two days&quot; technique is a recipe for disaster, so in the weekly exercise session with the TAs we propose three problems, one to be solved by the TA with the class, one to be solved by the class with help from the TA, and a third one to be turned in for an aggregate 5% of the final grade. This problem is just graded as &quot;looks like it makes sense,&quot; no detailed checking. All problems and solutions are published the same week. It made wonders to the results last term, but I'll run it a few more times.).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-13T20:44:31.197" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1619" PostTypeId="2" ParentId="1596" CreationDate="2014-04-13T21:05:39.957" Score="15" Body="&lt;p&gt;Never.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that in 2014 we can teach students to use opensource projects: &lt;a href=&quot;http://sagemath.org&quot;&gt;Sage&lt;/a&gt; does a very good job: &lt;a href=&quot;http://www.sagenb.org/&quot;&gt;Sage Notebook&lt;/a&gt; is couple times better than Wolfram Alpha since you can use it with no subscription, save your worksheets and it's based on much better programming language than Alpha is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will probably very likely find some other free (really free, not like Wolfram's stuff that doesn't allow you to copy-paste a number) and good alternatives, but most of them are topic-specific and already included in Sage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we generalize the question to: &lt;strong&gt;How and when should I introduce my students to online mathematics softwares?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then my opinion is: As soon as you feel they are able to learn to use it. They should have some basics in programming, since learning both programming and mathematics tools is quite difficult. I agree that for WolframAlpha you don't need to do any programming, but you don't get much interesting stuff out of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How to introduce it?&lt;/strong&gt; Some ideas:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Make a lecture on it, and then give a (very simple) homework that is solvable without the software, but is easier to solve with the software. Examples: roots of a polynomial of higher degree that can be guessed, but it takes some effort; computing integrals; performing RSA method; solving linear equations; working with complex numbers; compute the 25th Fibonacci number ... This shows that it can &lt;em&gt;ease our job&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use it yourself to verify results of some exercises you do with the students, use it to plot functions you introduce, generate nice fractals if speak about them, etc. That all shows that it can &lt;em&gt;significantly help us do things that are (close to) impossible otherwise&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="818" LastEditorUserId="818" LastEditDate="2014-04-13T21:46:10.910" LastActivityDate="2014-04-13T21:46:10.910" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1620" PostTypeId="2" ParentId="1598" CreationDate="2014-04-13T21:11:01.393" Score="9" Body="&lt;p&gt;Here's an anecdote from being an undergraduate student working in a tutorial center.  Students could come in and ask us questions about a fairly long list of standard introductory courses (we each wore badges indicating which courses we were (hopefully) competent to help with).  I was terrified to do this at first, because I figured I'd be asked to help with lots of problems I wouldn't remember how to do (having actually been in that class in 2-3 years prior, and not everything sticks or gets built on).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we tutors ended up doing most of the time was doing a lot of what you suggest -- winging it obviously, but in the process showing how to go about figuring out a tough math problem.  So we'd ask them what they'd learned about in class that week (often a good clue as to what theorems might be being exercised) and basically had them get out their textbooks and we'd look through the section they were in to find something relevant to the problem.  And yes, sometimes we'd get a student started on an approach that seemed like it would work but ended up not working out. And sometimes we had to call over another tutor to help us figure things out.  At first I felt really bad about having to do that instead of just &quot;knowing&quot; what to tell them, but eventually I realized that we were modeling the type of thinking and problem solving they'd need to learn to do to be able to solve the problems on their own. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to your question:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the interest of time, I wouldn't suggest lecturers come to class unprepared.  However, I would suggest that maybe your preparation consist of &quot;winging it&quot; in advance, and keeping track of your approach, your wild goose chases, etc. and presenting a somewhat cleaned up summary of &quot;how I solved this problem&quot; in lecture.   You can share some of your wrong turns, maybe in the context of, &quot;at first I thought I could try X -- can anyone see why that wouldn't work?&quot; (and if necessary give a quick summary of why that didn't work).  That mitigates the issue of you wasting too much time on unproductive asides, while still helping them see what kinds of things you thought about when trying to solve the problem (or prove the theorem, or whatever you're demonstrating).  And it saves the problem for students about notes being unintelligible because you can announce in advance that you're going to go off on an unproductive tangent, when you choose to do so for illustrative purposes.&lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-04-13T21:11:01.393" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1621" PostTypeId="2" ParentId="1550" CreationDate="2014-04-13T22:01:49.263" Score="3" Body="&lt;p&gt;Creating a timetable, all students should understand the issues.   However mathematical solutions don’t work as well as computer sci solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Choosing a set of options, so as to get a degree while having the lease dead time, the dead time being when you are waiting for the next activity to start, but don’t have long enough to do someone useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the US when degrees can take a undefined number of years, deciding how much part time work to do, so as to minimise the debt level at age 30.     More part time work, increase the time to get the degree, but decreases the debt earn per year.    Completing the degree sooner, allows the repayment of the debts to start sooner.&lt;/p&gt;&#xA;" OwnerUserId="381" LastEditorUserId="-1" LastEditDate="2014-04-13T23:12:55.357" LastActivityDate="2014-04-13T23:12:55.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1622" PostTypeId="2" ParentId="1598" CreationDate="2014-04-13T22:07:40.050" Score="16" Body="&lt;p&gt;What a good question this is.  Others have already given good answers.  I think Ilmari Karonen's answer is closest to my own heart.  For now let me just respond to one aspect of the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I have heard the story (may be an urban legend?) of a top professor who occasionally wanted to teach freshman analysis.  He believed in the method of letting students see how a mathematician's mind works, so he came to the lectures largely unprepared, and was proving theorems as they were met in the textbook. Consequently he would not always follow the straightest path to the destination. Occasionally he ran afoul full speed, and had to backtrack a bit or even start over.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I find this &quot;top professor&quot;'s teaching practice obnoxious.  That's a matter of opinion, but the proposed justification contains (perhaps intentionally!!) several points which I find fallacious.  Namely:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\bullet$ I believe in letting my students see how a mathematician's mind works too.  But whenever a mathematician is involved in teaching a course, one sees, in one way or another, how a mathematician's mind works.  The idea that under-preparation gives a truer view to the mathematician's mind is ridiculous.  Much of what a mathematician does is to think things out in advance and take advantage of what she has figured out.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\bullet$ &quot;...and were proving the theorems largely as they were met in the textbook&quot;  Already a problem: in closely following a textbook students are -- at best -- seeing how the textbook author's mind works, not the instructor's.  In a course which consists primarily of theorems and proofs (I wonder in what land &quot;freshman analysis&quot; is such a course; it is not the United States, certainly; we rarely teach analysis in any form to freshmen), strictly adhering to the logical sequence of the textbook turns the lecture into a dramatic reading of the textbook.  This is so much less than what an instructor should be doing in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\bullet$ Running afoul at full speed and having to backtrack or start over entirely are reasonable classroom events.  To see them happen at least once in a given course is probably a good thing: I have heard students react surprisingly (to me) positively when this sort of thing happens in my own courses.  But as others have pointed out, when it happens once or twice it is heartening.  When it causes an entire lecture to become derailed and then repeated the next time without a fatal sign mistake, it is clearly not good use of class time, and it will be frustrating to students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other fallacy here is that one needs to take such a completely derelict &quot;let it all hang out&quot; lecturing attitude in order for such mistakes to occur in the lecture.  I have been teaching for many years, and I still make mistakes in my calculus lectures, let alone in graduate courses in which I am presenting the material for the first time.  A certain positive rate of errors seems all but inevitable: if I asked you to prepare complete lecture notes, got a team of experts to check them over, gave them back to you and repeated until we were satisfied that they are seamlessly correct and complete, and then allowed you to bring the notes into the lecture and give a lecture which consisted purely of writing the notes on the board, then you will still make errors a positive percentage of the time.  (In fact, as one learns, there is an interior maximum for this with respect to preparation: if you prepare too much then you stop thinking about what you're doing, and that makes errors more likely to occur.)  The idea that one needs to make sure to be unprepared so as to get a desirable density of errors is really risible to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me close by saying that the pedagogical practice of &lt;strong&gt;intentional mistakes&lt;/strong&gt; is a good one, and I have used it myself: if I want to introduce a subtle point in an advanced course, I will often introduce it by doing an example or a computation in which this subtle point is ignored and a contradiction is reached.  I think this is an excellent example of a &quot;mathematician's mind at work&quot;.  The difference is that I have nevertheless prepared this in advance and the mistake is intentional.  (At least, the intended mistake is intentional! I remember that earlier this semester I gave an argument in which I warned in advance that I was going to make a mistake in the proof.  After the lecture was over, someone came to me and pointed out that I seemed to assume what I wanted to prove.  I smacked myself in the head and wondered how I could have done something so terrible.  She responded: &quot;Well, you said there would be a mistake.&quot;)  &lt;/p&gt;&#xA;" OwnerUserId="176" LastEditorUserId="176" LastEditDate="2014-04-14T22:07:19.357" LastActivityDate="2014-04-14T22:07:19.357" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1623" PostTypeId="2" ParentId="1608" CreationDate="2014-04-13T22:12:19.933" Score="6" Body="&lt;p&gt;For math pedagogy, I recommend exploring some of the blogs out there, in particular Dan Meyer's &lt;a href=&quot;http://blog.mrmeyer.com/&quot;&gt;blog&lt;/a&gt;. He also has a great &lt;a href=&quot;https://www.youtube.com/watch?v=BlvKWEvKSi8&quot;&gt;TEDx talk&lt;/a&gt;. The whole online community of math educators, sometimes called the mathtwitterblogosphere, is worth exploring. What's out there ranges from virtual filing cabinets of creative lessons, to folks venting about common frustrations, to more philosophical and academic reflections. I recommend using an RSS reader like feedly to maintain some subscriptions and when you find bloggers who are especially thoughtful, look at their &quot;blog rolls&quot; - lists of blogs they follow - to find other good sources. There's the potential to have a conversation through comments, through blogging yourself, or through Twitter.&lt;/p&gt;&#xA;" OwnerUserId="690" LastActivityDate="2014-04-13T22:12:19.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1624" PostTypeId="2" ParentId="1598" CreationDate="2014-04-13T22:22:02.893" Score="4" Body="&lt;p&gt;I love this approach, and I use it consistently. Particularly during problem solving.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keep in mind, I teach using online videos. The feedback I get is 99.9% positive (the negative comments being largely weird, unserious comments).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Particularly, I get good feedback from students saying that they are motivated by realizing that teachers aren't, in fact, robots whose sole purpose is to repeat the text book. They are human beings, capable of making mistakes. And seeing your teacher make mistakes reminds you of this. It also shows how to bounce back from errors, and &lt;strong&gt;critiquing your own work when the answer seems blatantly wrong.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="796" LastActivityDate="2014-04-13T22:22:02.893" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1625" PostTypeId="1" AcceptedAnswerId="1631" CreationDate="2014-04-13T22:32:57.760" Score="11" ViewCount="788" Body="&lt;p&gt;I'd like to propose a class on fractals to my department in the next few years. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One issue is that there seems to be no consensus on what a fractal is (see the wikipedia talk page on fractals, for instance). The following are all things that I would consider fractals or fractal-like:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Subsets of $\mathbb{R}^n$ whose Hausdorff dimension does not agree&#xA;with the topological dimension.&lt;/li&gt;&#xA;&lt;li&gt;Space-filling curves&lt;/li&gt;&#xA;&lt;li&gt;Infinite self-similar tilings (like the Penrose tiling and other tilings from subdivision rules)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Perhaps the first class of fractals would be the most important, while the other two would be short appendices at the end. Within the first class, one could cover&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Julia sets&lt;/li&gt;&#xA;&lt;li&gt;The Mandelbrot set&lt;/li&gt;&#xA;&lt;li&gt;The Cantor set, Sierpinski carpet, and Menger sponge&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;And so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue is, these topics are more or less all over the place, and require complex analysis and topology. I'd prefer to keep this to a senior-level course, so that means cutting some things out.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What subset of the above topics (or additional concepts about fractals) can be reasonably covered in a one-semester course aimed at senior undergraduates who have taken real analysis and abstract algebra? Also, should complex analysis be a prerequisite?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastActivityDate="2020-09-25T15:25:38.277" Title="What topics could be covered in a course on fractals?" Tags="|course-design|curriculum|fractals|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1626" PostTypeId="2" ParentId="1589" CreationDate="2014-04-13T23:06:40.797" Score="4" Body="&lt;p&gt;Following &lt;a href=&quot;https://matheducators.stackexchange.com/a/1591/888&quot;&gt;this answer&lt;/a&gt;, students can experiment with &lt;a href=&quot;http://www.cs.amherst.edu/~djv/FTAp.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Velleman, D. J. (2007). The Fundamental Theorem of Algebra: A Visual Approach&lt;/a&gt; using &lt;a href=&quot;http://puzlet.com/m/b00dd&quot; rel=&quot;nofollow noreferrer&quot;&gt;interactive versions of the paper's plots&lt;/a&gt;. (Disclosure: I am a dev for the site.)&lt;/p&gt;&#xA;" OwnerUserId="888" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-04-14T01:03:19.823" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1627" PostTypeId="2" ParentId="1608" CreationDate="2014-04-14T00:20:06.180" Score="3" Body="&lt;p&gt;A perhaps unusual recommendation: Lakoff and Núñez's book &lt;em&gt;Where Mathematics Comes From&lt;/em&gt;, despite suffering from being mildly repetitive, has made me think &lt;em&gt;very&lt;/em&gt; differently about mathematics education, as long as this is defined as the activity of instilling mathematical &lt;em&gt;understanding&lt;/em&gt; in students (versus &quot;mere&quot; mathematical knowledge). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lakoff and Núñez, more clearly than any other authors I am aware of, explore exactly what mathematical understanding is. It may change &lt;em&gt;everything&lt;/em&gt; about what you think your task as a mathematical educator is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(P.S. sorry for the excess of &lt;em&gt;italics&lt;/em&gt;. I'm a glutton for emphasis.)&lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-04-14T00:20:06.180" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1628" PostTypeId="2" ParentId="1609" CreationDate="2014-04-14T04:05:32.787" Score="10" Body="&lt;p&gt;This is sort of an extended comment, but I find I need space to give what I think is a possibly useful answer that addresses underlying assumptions rather than answering the question as asked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll give you the summary first: Confronting misconceptions may be productive in some cases, and indeed the focus on student misconceptions in math and science was central to much research in mathematics education in the 70's and 80's. However, there are reasons that this approach is problematic for instruction. I have a research reference, and an anecdote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Smith, diSessa, and Roschelle (1993) discussed some of what had been accomplished with understanding student learning using the model of misconceptions, and then gave some critique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the section &quot;Is Confrontation an Appropriate Model of Classroom &#xA;Learning?&quot; p. 126, they wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There are both strengths and weaknesses in this conceptualization of&#xA;  classroom instruction. We need energetic classroom discussions in&#xA;  which students take positions, make sense of and explain problematic&#xA;  phenomena, and articulate justifications for their ideas. Activities&#xA;  that produce states of cognitive conflict are certainly desirable and&#xA;  conducive to conceptual change. However, as judged by constructivist&#xA;  standards, confrontation suffers from important deficits as either a&#xA;  phase of conceptual change or a model of instruction. As cognitive&#xA;  competition, it cannot explain why expert ideas win out over&#xA;  misconceptions. The rational replacement of one conception with&#xA;  another requires criteria for judgment. As knowledge, those criteria&#xA;  must be constructed by the learner, and neither confrontation nor&#xA;  replacement explains the origins of such principles for choosing&#xA;  concepts, crucial data, or theories. In fact, change in how one&#xA;  decides in favor of one conception over another is a complex part of&#xA;  conceptual development (Kuhn &amp;amp; Phelps, 1982; Schauble, 1990), and&#xA;  confrontation is an implausible mechanism for changing principles that&#xA;  decide, for example, the relevance of data to theory. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Confrontation is also problematic as an instructional model. In&#xA;  contrast to more evenhanded approaches to classroom discussions in&#xA;  which students are encouraged to evaluate their conceptions relative&#xA;  to the complexity of the phenomena or problem, confrontation&#xA;  essentially denies the validity of students' ideas. It communicates to&#xA;  students that their specific conceptions and their general efforts to&#xA;  understand are fundamentally flawed. The metaphor of confrontation is&#xA;  also inconsistent with the pedagogical sensitivity and care required&#xA;  to negotiate new understandings in the classroom (Yaekel, Cobb, &amp;amp;&#xA;  Wood, 1991). Finally, some misconceptions are powerful enough to&#xA;  influence what students actually perceive, thereby decreasing the&#xA;  chances that planned confrontation and competition will be successful&#xA;  (Resnick, 1983).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My anecdote (much abbreviated): A few years ago I was assisting a teacher in a 4th grade class by helping two students who were having difficulty comparing fractions. They thought 3/4 was greater than 4/5. A number of ways of approaching this occurred to me, and to keep the story short, I will just say that I had them construct a physical model that you and I would likely agree showed that 4/5 is greater than 3/4. However, as time ran out on us, they looked at their work and said, &quot;yes, see? 3/4 is more than 4/5.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure you could find things to critique in the details of my approach. But two things struck me later as I was driving home. The first was that sometimes seeing depends on what you are sure you know. But the second is even more relevant: I still did not know much about &lt;em&gt;why&lt;/em&gt; those students thought 3/4 was greater than 4/5. Spending all my time trying to help them reach a productive state of confrontation, I didn't learn about what they knew. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When students know something, it represents knowledge that somehow made some sense to them in some other context. But it's not working appropriately in the desired context. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is research showing that a focus on student conceptions, rather than mistakes, is not only productive for student learning, it also can be a helpful model in preparing teachers for thinking about student thinking (Fennema et al., 1996).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, these conceptions students have are tied to specific content knowledge. Therefore, a taxonomy of mistakes (de-contextualized from specific content) may not necessarily be useful in the action of classroom instruction. Certainly useful is knowing with some accuracy the types of ways students come into the class making sense of the content in a way that is limiting them (yet it makes some sense to them). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cited:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.uta.edu/faculty/tjorgens/pastcourses/WNO/jrme.pdf&quot; rel=&quot;nofollow&quot;&gt;Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., &amp;amp; Empson, S. B. (1996). A Longitudinal Study of Learning to Use Children’s Thinking in Mathematics Instruction. Journal for Research in Mathematics Education, 27(4), 403–434.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://people.ucsc.edu/~gwells/Files/Courses_Folder/ED%20261%20Papers/Misconceptions%20reconceived.pdf&quot; rel=&quot;nofollow&quot;&gt;Smith, J. P., diSessa, A., &amp;amp; Roschelle, J. (1993). Misconceptions Reconceived: A  Constructivist Analysis of Knowledge  in Transition. The Journal of the Learning Sciences, 3(2), 115–163.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="5563" LastEditDate="2016-08-20T15:14:48.713" LastActivityDate="2016-08-20T15:14:48.713" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1629" PostTypeId="2" ParentId="1545" CreationDate="2014-04-14T08:30:19.400" Score="8" Body="&lt;p&gt;Determine the minimum distance from a parametric equation $(x(t), y(t))$ to a given point $(x_0, y_0)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eg. at which point should a person leave a road (described by the parametric equation), such that the walking distance to the point $(x_0, y_0)$ from the road is minimized?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is then:&#xA;Minimize $\sqrt{(x-x_0)^2 + (y - y_0)^2}$ subject to $x = x(t)$ and $y = y(t)$.&lt;/p&gt;&#xA;" OwnerUserId="607" LastActivityDate="2014-04-14T08:30:19.400" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1630" PostTypeId="2" ParentId="1598" CreationDate="2014-04-14T08:53:06.943" Score="3" Body="&lt;p&gt;I only teach smaller groups, but have found that structuring my exercises so that I deliberately put a trap or two in there for the students to discover and learn from is extremely beneficial.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The trick is to make sure the trap is small and (relatively) ineffective and provide the clues they need to find their own way out without being too obvious.  One way of providing those clues is to solve a similar category of problem for them and emphasise certain key processes I follow to get back on track.  It might be that I notice someone has a problem and I provide the answer they need while talking to another student nearby.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need your students to discover the solutions themselves, but it took the maths community many years to make the discoveries that you want to teach them in a few weeks/months, so you also need them to remain resourceful and provide the clues close enough so they can join them together. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So making mistakes as a lecturer is useful where your students mentally copy your recovery strategies.  (teaching them how you identify/recognise that you have gone off track, and what the heck you do to get back on a more productive path, how did you know it was just a messed up sign, or a totally different approach was required)&lt;/p&gt;&#xA;" OwnerUserId="910" LastEditorUserId="910" LastEditDate="2014-04-15T00:34:32.740" LastActivityDate="2014-04-15T00:34:32.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1631" PostTypeId="2" ParentId="1625" CreationDate="2014-04-14T09:10:10.127" Score="7" Body="&lt;p&gt;Some years ago, I attended a course where Julia and Mandelbrot sets were part of. It was called &quot;Chaotic Dynamical Systems&quot;, and was held once a week for 90 minutes for a total of 14 lectures. It was made for master students in math, although advanced bachelor students could also have participated. This doesn't completely fit the topics of your course, but it might be a solid base. Without understanding of complex numbers, this course wouldn't have worked, although complex analysis wasn't strictly required. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The contents of the course were:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Introduction of discrete dynamical systems; the definition mentions topological spaces, but the script says &quot;Usually, we have more than that&quot;. Orbit, invariant sets, conjugate systems, examples are usually on $\mathbb{R},\mathbb{R}^+$ and intervals. Main toy example is the logistic system/family.&lt;/li&gt;&#xA;&lt;li&gt;Fixed points, periodic points, stability of orbits, attractive points, repellent points, characterization of these points for functions which are continouosly differentiable on $\mathbb{R}$ or $\mathbb{C}$, Bifurcation for a family of dynamical systems, Sharkovsky's theorem (no proof), theorem of Li-Yorke (with proof)&lt;/li&gt;&#xA;&lt;li&gt;Definition of a chaotic system (Devaney, see references), symbolic dynamics &lt;/li&gt;&#xA;&lt;li&gt;Complex dynamic systems: Newton's method, Julia sets, Fatou set, Mandelbrot set. proofs partly omitted, partly shortened to a sketch&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;References for the course:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0805316019&quot; rel=&quot;nofollow&quot;&gt;R. L. Devaney: An introduction to chaotic dynamical systems&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0070252033&quot; rel=&quot;nofollow&quot;&gt;D. Gulick: Encounters with chaos&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The desired topics were only covered in the last few weeks of the course, but I think that if you go lighter on the chaotic part, you might be able to give more emphasis to the sets you are interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that a course on discrete dynamical systems is a good context for these kind of sets - unfortunately I can't quite tell how to fit in Cantor/Sierpinski/Merger - usually I'd include these in a course on measure theory.&lt;/p&gt;&#xA;" OwnerUserId="108" LastEditorUserId="108" LastEditDate="2014-04-15T05:39:12.757" LastActivityDate="2014-04-15T05:39:12.757" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1632" PostTypeId="2" ParentId="1589" CreationDate="2014-04-14T09:12:26.947" Score="8" Body="&lt;p&gt;You can find a proof that goes back to Gauss, which is based only on multivariable calculus (double integrals and partial derivatives) at &lt;a href=&quot;http://www.math.uconn.edu/~kconrad/blurbs/fundthmalg/fundthmalgcalculus.pdf&quot;&gt;http://www.math.uconn.edu/~kconrad/blurbs/fundthmalg/fundthmalgcalculus.pdf&lt;/a&gt;. I think the proof will come across as rather mysterious even if it can be followed line by line. The last paragraph of the file gives an indication of what proof it is related to in complex analysis. &lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2014-04-14T09:12:26.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1633" PostTypeId="2" ParentId="651" CreationDate="2014-04-14T12:41:33.433" Score="4" Body="&lt;p&gt;Elizabeth Cohen and Rachel Lotan of Stanford University have written about an approach called &lt;a href=&quot;http://cgi.stanford.edu/group/pci/cgi-bin/site.cgi&quot; rel=&quot;nofollow&quot;&gt;Complex Instruction&lt;/a&gt; that supports students &lt;a href=&quot;http://www.uvm.edu/complexinstruction/about%5fci.html&quot; rel=&quot;nofollow&quot;&gt;working together and learning from each other&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the principles involve:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Group worth tasks: The tasks must be challenging enough so that you need to work with others, cannot be solved easily on your own&lt;/li&gt;&#xA;&lt;li&gt;Assigning different roles and responsibilities to different students (again, promotes interdependence)&lt;/li&gt;&#xA;&lt;li&gt;Assessing both the process of working together and the final product -- group and individual accountability as well&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This approach also involves addressing issues of status -- who is the &quot;smart&quot; one -- as when the tasks are challenging, different people can contribute something (a new way to see the question, a unique representation that allows for a connection) so that different people are smart in different ways, and we all need each others' forms of smartness.&lt;/p&gt;&#xA;" OwnerUserId="739" LastActivityDate="2014-04-14T12:41:33.433" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1634" PostTypeId="2" ParentId="1598" CreationDate="2014-04-14T12:45:13.673" Score="4" Body="&lt;p&gt;Everyone makes mistakes. I think the value in showing mistakes and talking through them is important not only to illustrate that you can be an intelligent, smart mathematician and still make mistakes... rather, the value in discussing mistakes is the learning opportunity involved related to developing an understanding of the content. Discussing why the mistake doesn't work and doesn't make sense is a useful counter example to compliment that examples that do work and do make sense. &lt;/p&gt;&#xA;" OwnerUserId="739" LastActivityDate="2014-04-14T12:45:13.673" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1635" PostTypeId="1" AcceptedAnswerId="1681" CreationDate="2014-04-14T13:03:05.020" Score="23" ViewCount="4818" Body="&lt;p&gt;I am a bit afraid to ask this, but the question has bothered me for some time now. I have a student in my analysis class having a medical certificate of &lt;a href=&quot;http://en.wikipedia.org/wiki/Dyscalculia&quot; rel=&quot;nofollow noreferrer&quot;&gt;dyscalculia&lt;/a&gt;. This entitles her to write tests and exams in a special environment and using extra time. This is ok and I do not have questions regarding this case. My question is more general.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What studies are there to show connections between advanced mathematical ability (needed for example to be a maths major) and dyscalculia?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Would it be wise to advise students with this illness not to choose mathematics as a major?&lt;/p&gt;&#xA;&lt;p&gt;I know this is a sensitive subject and I do not want to open a discussion about it. Please, provide me links to medical/pedagogical literature supporting the practice (or opposing) we have at our university.&lt;/p&gt;&#xA;&lt;p&gt;In many subjects if you have a medical certificate you cannot study that particular subject. Most notable in our university is sports. Clearly, mathematics and dyscalculia are not comparable to this, but what is the evidence supporting this feeling?&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="2083" LastEditDate="2021-06-09T16:56:31.203" LastActivityDate="2021-06-09T16:56:31.203" Title="Dyscalculia and studying mathematics (as major)" Tags="|undergraduate-education|dyscalculia|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="1636" PostTypeId="2" ParentId="1589" CreationDate="2014-04-14T14:36:28.927" Score="7" Body="&lt;p&gt;You can short-cut the Liouville proof as follows.  Assuming that $f$ is a monic polynomial of degree $d&amp;gt;0$ with no roots, define &#xA;$$ h(r) = \int_{\theta=0}^{2\pi} f(r\,e^{i\theta})^{-1}\,d\theta. $$&#xA;If you are willing to differentiate under the integral sign, then it is not hard to show that $h'(r)=0$, so $h$ is constant.  When $r$ is large enough the $z^d$ term in $f(r\,e^{i\theta})$ will dominate and we see that $|h(r)|$ is approximately $2\pi r^{-d}$ at most.  It follows that $h(r)\to 0$ as $r\to\infty$, but $h$ is constant, so $h(0)=0$, which is impossible as $h(0)=2\pi/f(0)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Various things need to be justified to turn this into a complete proof, but I think that they are all quite plausible and intuitive.&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-04-14T14:36:28.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1637" PostTypeId="2" ParentId="1572" CreationDate="2014-04-14T14:41:40.263" Score="8" Body="&lt;p&gt;One example that annoys me is when science stories in newspapers (especially stories about high energy physics or astronomy) insist on writing out large numbers, such writing &lt;span class=&quot;math-container&quot;&gt;$1,000,000,000,000,000,000,000,000,$&lt;/span&gt; or writing things like &lt;em&gt;trillion trillion&lt;/em&gt;, when neither is very useful to a reader with a high school education and neither would make any sense to anyone else. Why not write &lt;span class=&quot;math-container&quot;&gt;$10^{24}?$&lt;/span&gt; Scientific notation is taught (in the U.S.) to students who have not yet begun the study of school algebra, and it is used in high school science classes.&lt;/p&gt;&#xA;&lt;p&gt;Below are two other examples that I've previously posted about in the past.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 1:&lt;/strong&gt; &lt;a href=&quot;https://edition.cnn.com/2005/US/10/14/Franklin/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Atlanta mayor: In resettling evacuees, FEMA no help&lt;/a&gt;, CNN news article, 14 October 2005. [I previously posted a different version in &lt;a href=&quot;https://web.archive.org/web/20161206232246/http://mathforum.org/kb/message.jspa?messageID=4021230&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 15 October 2005 sci.math post&lt;/a&gt;.]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;CNN anchor Miles O'Brien on Friday spoke about the challenges facing one city with Atlanta Mayor Shirley Franklin. Atlanta took in 42,000 families fleeing the disaster.&lt;/p&gt;&#xA;&lt;p&gt;O'BRIEN: All right, let's talk about this, 42,000 families. You're a big city. It's a prosperous city, but that still puts a burden on the city, doesn't it?&lt;/p&gt;&#xA;&lt;p&gt;FRANKLIN: Well, it certainly does, but I don't think it's a burden that FEMA [Federal Emergency Management Agency] can't help us to address.&lt;/p&gt;&#xA;&lt;p&gt;The Congress and the president have allocated 62 billion [dollars]. Our estimates are that a family needs assistance for about six months in order to stabilize themselves and that would cost about 11,000 [dollars] per family. The city of Atlanta can't absorb that cost, but we can certainly work with FEMA, if they were willing, to help families get resettled in the city and the metropolitan area.&lt;/p&gt;&#xA;&lt;p&gt;O'BRIEN: So 11,000 times 42,000. I can't do that kind of math on the fly here. But how much of that money have you seen?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; This is trivially estimated via &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; followed by &lt;span class=&quot;math-container&quot;&gt;$4+4=8$&lt;/span&gt; zeros, or &lt;span class=&quot;math-container&quot;&gt;$400$&lt;/span&gt; million. By missing this, and hence also the fact that &lt;span class=&quot;math-container&quot;&gt;$60$&lt;/span&gt; billion divided by &lt;span class=&quot;math-container&quot;&gt;$400$&lt;/span&gt; million is &lt;span class=&quot;math-container&quot;&gt;$(1.5)(100)=150,$&lt;/span&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Miles_O%27Brien_(journalist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Miles O'Brien&lt;/a&gt; (a well known broadcast news journalist who, incidentally, specializes in science, technology, and aerospace reporting) missed a good opportunity to make a point I suspect he would have liked to make.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 2:&lt;/strong&gt; Formerly posted in &lt;a href=&quot;https://web.archive.org/web/20161206232220/http://mathforum.org/kb/message.jspa?messageID=6598178&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 8 February 2009 math-teach post&lt;/a&gt; at Math Forum:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I wonder if the author of the article below has any awareness of just how mathematically illiterate one of the comments below makes professional news writers sound. The author writes &amp;quot;CNN checked McConnell's numbers with noted Temple University math professor and author John Allen Paulos&amp;quot; for something that any college-bound middle school student should be able to do, even without a calculator. Although some of Paulos' comments are nice, especially his speculation &amp;quot;People tend to lump [million, billion, trillion] together, perhaps because they rhyme&amp;quot;, going to him in order to check McConnell's numbers is like asking a university linguist for the correct spelling of the word &amp;quot;especially&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I mention this because I've seen many examples of this over the years, especially newspaper writers consulting mathematicians for something that is nothing more than an easy high school level probability or combinatorics problem (easier than many of the problems in standard precalculus and college algebra texts). I don't know whether the reporters really don't know how to work these problems (like checking McConnell's numbers below) or whether they are just using the occasion to get some possibly interesting remarks from someone well known and don't realize how stupid their rationale sounds to a large percentage of their readership.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://edition.cnn.com/2009/LIVING/02/04/trillion.dollars/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Numb and number: Is trillion the new billion?&amp;quot;&lt;/a&gt; by Christine Romans CNN's American Morning&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;To put a trillion dollars in context, if you spend a million dollars every day since Jesus was born, you still wouldn't have spent a trillion,&amp;quot; McConnell said.&lt;/p&gt;&#xA;&lt;p&gt;CNN checked McConnell's numbers with noted Temple University math professor and author John Allen Paulos.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;A million dollars a day for 2,000 years is only three-quarters of a trillion dollars. It's a big number no matter how you slice it,&amp;quot; Paulos said. Here's another way to look at it.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;A million seconds is about 11.5 days. A billion seconds is about 32 years, and a trillion seconds is 32,000 years,&amp;quot; Paulos said. &amp;quot;People tend to lump them together, perhaps because they rhyme, but if you think of it in terms of a jail sentence, do you want to go to jail for 11.5 days or 32 years or maybe 32,000 years? So, they're vastly different, and people generally don't really have a real visceral grasp of the differences among them.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="745" LastEditorUserId="6053" LastEditDate="2022-09-27T21:55:11.060" LastActivityDate="2022-09-27T21:55:11.060" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="1638" PostTypeId="2" ParentId="647" CreationDate="2014-04-14T15:40:22.220" Score="3" Body="&lt;p&gt;For questions like &quot;what does the circle between $f$ and $g$ mean&quot; in a calculus course, I would tell the student and then write on the board (for everyone else) something like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In problem 5, $(f \circ  g)(x)$ stands for $f(g(x))$ (composition of functions).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I wouldn't do this in a precalculus course, however, where this notation is one of the things being tested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for questions that I don't think are appropriate to answer, I usually make an assessment of how much I think knowing the answer is worth for the problem at hand. For example, for a 10-point applied related rates problem in which your second question came up, I might tell the student that I'll provide the answer for a 3-point penalty, which they can then choose to accept or reject. I explain this sort of bargaining in advance, when reviewing the syllabus the first day of class and/or when reviewing for the test before the test is given. Personally, I'd much rather do this (and mark on their test, in ink and in my handwriting, &quot;$-3$ for hint&quot;) than to later deal with the much more time consuming task of deciding how to grade a problem that I think they could have worked if they knew such-and-such, but it's difficult to tell for sure because what they don't know and what they do know is mixed together in their solution.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-04-14T15:40:22.220" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1639" PostTypeId="1" AcceptedAnswerId="1641" CreationDate="2014-04-14T15:50:47.430" Score="54" ViewCount="11557" Body="&lt;p&gt;I'm currently TAing a Linear Algebra class where a significant portion of the class is struggling, oftentimes getting marked down on homeworks or tests because they misunderstand some concept (rather than some trivial mistake in calculation). The students have the following out-of-class options for help available to them:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Office hours (both mine and those of the professor teaching the course)&lt;/li&gt;&#xA;&lt;li&gt;Free tutoring available both day and night-time through the university&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Very few students take advantage of these options (I'd say about 20% of an 80-person class has used any of these options so far), which leads me to a more general question:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How do I encourage struggling students to go to office hours or available tutoring?&lt;/li&gt;&#xA;&lt;li&gt;What are some possible reasons they &lt;strong&gt;aren't&lt;/strong&gt; coming to office hours or tutoring hours?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user37" LastActivityDate="2015-10-02T21:06:59.663" Title="How do I motivate my students to go to office hours?" Tags="|undergraduate-education|student-motivation|tutoring|struggling-students|" AnswerCount="13" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1640" PostTypeId="2" ParentId="1639" CreationDate="2014-04-14T16:12:33.023" Score="16" Body="&lt;p&gt;As a TA, attend one of the class sessions and personally advertise your office hours. If the students see you in person, they may be more willing to come to you with questions. Also making yourself available outside of normal/regular office hours, giving them your email, and generally making yourself more approachable are all things you can do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like for a lot of students, especially students who found high school math easy, feel ashamed to be struggling once they are in college. Asking for help for the first time in something you usually excel at can be very difficult. &lt;/p&gt;&#xA;" OwnerUserId="307" LastActivityDate="2014-04-14T16:12:33.023" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1641" PostTypeId="2" ParentId="1639" CreationDate="2014-04-14T17:26:25.500" Score="38" Body="&lt;p&gt;When I'm the instructor for the course, I remind them of the time and location of my office hours by writing it on the board at the beginning of &lt;em&gt;every lecture&lt;/em&gt; (if you're not the instructor, you can ask the instructor to do this). It sounds like overkill, but in my experience part of the reason students don't attend office hours is because it's just not a salient option to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also find that attendance is much higher before an assignment or test. If there's a weekly assignment due on Friday, say, choosing to hold your office hours on Wednesday or Thursday will probably result in higher attendance than holding them on Monday.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps most importantly, some students don't attend office hours because they're embarrassed. This is the main reason why &lt;em&gt;I&lt;/em&gt; typically didn't attend office hours as an undergraduate (a fact I regret to this day). Although in office hours the students don't have to contend with feeling stupid in front of too many of their peers, they are, on the other hand, less able to &quot;melt into the crowd&quot;. A lot of students feel that to attend office hours they have to have &quot;good&quot; questions prepared, so if they're too confused or have fallen too far behind, they won't go.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to counteract this? I try to be as open and explicit about it as possible. I say things to the class like, &quot;If you're so confused you don't even know what question to ask, then office hours are for you. Chances are you're not alone,&quot; or, &quot;You don't have to come to office hours with specific question; you can just come and tell me you're confused about a particular section.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, depending on your grading scheme, the following can be a very effective inducement to attend office hours: I will work through and explicitly solve homework questions in office hours. More precisely, I work through specific questions that I'm asked to on the board, with involvement from the students, regardless of whether they're on the homework or not. Thus, students who attend office hours will almost certainly do better on the homework. I don't mind doing this because I view the assigned homework as an inducement to practice more than a way of assessing skill, and if students put in the time during office hours, that ought to count as practice.&lt;/p&gt;&#xA;" OwnerUserId="327" LastActivityDate="2014-04-14T17:26:25.500" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1642" PostTypeId="2" ParentId="1609" CreationDate="2014-04-14T19:50:23.293" Score="9" Body="&lt;p&gt;A few mistakes I observe from time to time while teaching.  Note that these are as true of professional mathematics as mathematics education, at least occasionally.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Formalism; or, the belief that a rigorous style is necessarily rigorous mathematics, or even correct, or even intuitively valid.&lt;br&gt;&#xA;&lt;em&gt;Of course, style cannot save a bad argument.  Furthermore, it might not rescue it from implicit omissions, such as, for example, &quot;A triangle is the union of three lines&quot;, despite using a precise concept such as &quot;union&quot;.  The idea might not even be reasonable, like, say, &quot;Let $x$ be the smallest positive real number&quot;.&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Platonism; or, the belief that an appeal to the intuitive essence of an idea is sufficient replacement for rigor, or even correct, or even universally understood.&lt;br&gt;&#xA;&lt;em&gt;Famously, intuition in calculus cannot always be made rigorous, such as the original definitions of Newton and Leibniz in terms of infinitesimals (they are not to be blamed much, as rigor was different then).  At least these could be rescued, unlike &quot;proofs&quot; of the parallel postulate that rejected the consequences of what turned out to be non-Euclidean geometry as being contrary to the concept of a line.  Similarly to the first, the definition of continuity &quot;$f(x) - f(a)$ is small&quot; can mean almost anything you want.&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Luddism; or, the rejection of mathematical &quot;technology&quot;, i.e. refined methods.&lt;br&gt;&#xA;&lt;em&gt;Students often insist that what they think they know is more reliable than what they are taught.  Even researchers may feel that an elementary technique they can get away with using is preferable to one that is specifically appropriate, but theoretically more difficult.&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use of English; or, stringing a complex concept together with words alone, rather than encapsulating its ideas precisely and referring to them with jargon.&lt;br&gt;&#xA;&lt;em&gt;This can be as basic as the antique style &quot;The sum of twice something with an additional quantity of three has the total amount of five, and therefore that thing is itself unity&quot;.  In the now-rare Euclidean geometry proof class, you see phrases like &quot;corresponding parts of congruent triangles are congruent&quot; used as a mantra.&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Misuse of symbols; or, stringing a complex deduction together without any words at all.&lt;br&gt;&#xA;&lt;em&gt;Parse this: &quot;$f(x) = x + \sin(x) = f'(x) = 1 + \cos(x) = 0 = \cos(x) = -1 = x = n \pi \text{ ($n$ odd)}$&quot;.  There are three related computations in there, plus the use of one theorem (on critical points) and a classification of trig function values, which should all be separated.  I have heard this described as &quot;knowing only 'equals' as a mathematical verb&quot;.  Actually, I hope that this is never seen in professional math.&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nearsightedness; or, not following your nose; or, getting stuck because the next step looks like it is necessarily hard.&lt;br&gt;&#xA;&lt;em&gt;Any problem where, say, one suddenly has to solve a system of several linear equations.  Tedious, yes, but there is a straightforward procedure and it must be done, or if not, finding the alternative is possibly even more work.  Numerous exams have been submitted where the solution ends at just such a spot.&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Farsightedness; or, not simplifying the question because it looks too hard as a whole.&lt;br&gt;&#xA;&lt;em&gt;I knew of a professor who would answer most questions with &quot;Use the definition!&quot;.  If it weren't for this fallacy, no one would ever have trouble with set-theory exercises like &quot;$f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)$&quot;.  Tim Gowers and Mohan Ganesalingam have written a program designed to solve this kind of problem, in part, by not making this mistake.&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Not generalizing; or, doing exercises without getting any better at doing non-exercises.&lt;br&gt;&#xA;&lt;em&gt;Every semester I have students who ask me after exams how they could study better even though they do all the exercises, sometimes even more than I ask.  My current advice is that they should be mindful of the implicit lessons to be derived from the experience, but I don't know if this is even helpful.&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As the article in the question says, the variety of &quot;stupidity&quot; is infinite.  However, I think that people who avoid these eight mistakes are typically among the best students in any class, while those who make even three or so have serious problems.  If only I could teach people how not to do them, rather than have them come into my class already knowing...&lt;/p&gt;&#xA;" OwnerUserId="513" LastActivityDate="2014-04-14T19:50:23.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1643" PostTypeId="2" ParentId="1625" CreationDate="2014-04-14T20:11:23.610" Score="4" Body="&lt;p&gt;You certainly can have even a one-year course on fracals without complex analysis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, even homotheticaly self-similar fractals are rich, with easy-to-state open questions; e.g. the precise behavior of the Favard length (average measure of $1$-dimensional projections) of the four-corner Cantor set in the plane (product of two middle-half Cantor sets on the line) is still unknown.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also investigate the notion of rectifiability, or study snowflaked metrics, i.e. metrics as $d(x,y)^\alpha$ where $d$ is the Euclidean metric on $\mathbb{R}^n$ and $\alpha\in(0,1)$. It is known that such metric spaces can be bilipschitz embedded into $\mathbb{R}^N$ for some $N=N(n,\alpha)$ (a theorem of Assouad) but apart when $n=1$, we do not know how big $N$ must be (of course, it must be at least $n/\alpha$, the Hausdorff dimension of $d^\alpha$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More basically, already exploring the notion of Hausdorff dimension can be interesting (e.g. constructing two sets of dimension $0$ whose product has dimension $1$). Mattila has a great book to learn these things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mentioned open problems because they can be motivating to students, but of course there are uncountably many proved results to teach in these directions. I mentioned more than can be covered in a one-semester course, but all this is highly flexible.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-14T20:11:23.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1644" PostTypeId="2" ParentId="1625" CreationDate="2014-04-14T20:58:38.923" Score="6" Body="&lt;p&gt;Background: In my senior year of undergrad, I was a TA for our school's &amp;quot;Fractal Geometry&amp;quot; course, having worked with the professor before on a research project in fractals. We followed some of the materials provided (publicly!) in &lt;a href=&quot;http://classes.yale.edu/fractals/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yale's Fractal Geometry course&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There was a particular focus on &lt;em&gt;Iterated Function Systems&lt;/em&gt;, which are great ways to generate fractals of &amp;quot;type 1&amp;quot;, per OP's list. (Indeed, the Yale course was created by Michael Frame, who was a collaborator on our research into IFS.) There are some great visuals and applets provided to explore the main ideas and appeal to some more visual/hands-on learners. I highly recommend at least looking through the course's contents, whether or not you end up using all of it.&lt;/p&gt;&#xA;&lt;p&gt;If you are looking to address a student audience of senior math majors, you might want to home in on some of the following sections. Note: This list is certainly not exhaustive; this is just what I found scanning through now. But these will at least require the students to use some of their math knowledge and maturity.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://classes.yale.edu/fractals/FracAndDim/DimAlg/DimAlg.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra of Dimensions&lt;/a&gt;: Does knowing the dimension of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; tell us anything about the dimensions of &lt;span class=&quot;math-container&quot;&gt;$A\cap B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A\cup B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A\times B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; projected onto &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://classes.yale.edu/fractals/MandelSet/MandelCombinatorics/MandelCombinatorics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Combinatorics of the Mandelbrot set&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://classes.yale.edu/fractals/MandelSet/ComplexNewton/ComplexNewton.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's Method for complex numbers&lt;/a&gt;, followed by a &lt;a href=&quot;http://classes.yale.edu/fractals/MandelSet/MandelUniversality/MandelUniversality.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;use of such a method on the Mandelbrot set&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Various topics in the chapter on &lt;a href=&quot;http://classes.yale.edu/fractals/Chaos/welcome.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chaos&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In addition, there some appendix resources:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://classes.yale.edu/fractals/LP/LP.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lesson plans and a list of required mathematical topics, sorted by fractal topic&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://classes.yale.edu/fractals/Labs/Labs.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Lab&amp;quot; activities&lt;/a&gt; for exploring certain topics (could serve as in-class activities or take-home projects)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Regarding your question about whether to require complex analysis: You don't have to. This will depend on how much material on the Mandelbrot/Julia sets you want to cover, and to what &lt;em&gt;depth&lt;/em&gt; you do so. I would recommend tending towards the side of not requiring much familiarity with complex analysis, and only a working knowledge of the algebra of complex numbers; hopefully, this will make more students want to take this course!&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="376" LastEditDate="2020-09-25T15:25:38.277" LastActivityDate="2020-09-25T15:25:38.277" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="1645" PostTypeId="2" ParentId="1635" CreationDate="2014-04-14T21:06:23.707" Score="6" Body="&lt;p&gt;I teach middle and high school math students. It is my understanding that dyscalculia is the math equivalent to dyslexia with reading. While I was in school for my teaching certificate, I was directed to this website. &#xA;&lt;a href=&quot;http://www.ncld.org/types-learning-disabilities/dyscalculia/what-is-dyscalculia?utm_source=dyscalculia-interest-email&amp;amp;utm_medium=email&amp;amp;utm_content=dearreaderimage&amp;amp;utm_campaign=dyscalculia-interest-email&quot;&gt;http://www.ncld.org/types-learning-disabilities/dyscalculia/what-is-dyscalculia?utm_source=dyscalculia-interest-email&amp;amp;utm_medium=email&amp;amp;utm_content=dearreaderimage&amp;amp;utm_campaign=dyscalculia-interest-email&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The website gave a very good overview of what students diagnosed with dyscalculia may be experiencing. I have also found that students diagnosed with is may have the ability to overcome it just as students diagnosed with dyslexia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another website that I found that goes into more detail regarding college age students is:&#xA;&lt;a href=&quot;http://www.dyscalculia.org/college-dyscalculia&quot;&gt;http://www.dyscalculia.org/college-dyscalculia&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Through my education courses, it was unadvised to advise students on whether or not they should pursue certain goals in life, such as dropping out of school, taking certain classes, etc. I would assume that the same would apply to college level students as well. I was told that advise for students about their choices should be left to the adviser, teaching the students should be left to the teacher. If a teacher starts giving advise to the students, it may contradict something the adviser has already talked to the student about, or worse, it could interfere with a personal goal/dream that the student has, or a family based decision. You, as the teacher, also are unaware of whether or not the student is trying to achieve something, and your advise is what pushes them to do something bad, like give up on their goals, drop out of school, or even try to harm themselves or others.  &lt;/p&gt;&#xA;" OwnerUserId="839" LastActivityDate="2014-04-14T21:06:23.707" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1646" PostTypeId="2" ParentId="280" CreationDate="2014-04-14T21:07:55.813" Score="10" Body="&lt;p&gt;Here's a nice way to square a number ending in a &quot;5&quot;; let's take 75 as an example.  You cut off the &quot;5&quot;; what remains is 7.  You then multiply it by its successor: 7×8=56.  Then you write &quot;25&quot; after the &quot;56&quot; and get the result: 5625.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Show it to high-schoolers, and they'll shout: &quot;It's magic!&quot; (tested!).  Now show (and explain) them this:&#xA;$$(10a+5)^2=100a^2+100a+25=100a(a+1)+25$$&#xA;and they (hopefully) get the &quot;aha! so algebra isn't useless after all...&quot; moment.&lt;/p&gt;&#xA;" OwnerUserId="704" LastActivityDate="2014-04-14T21:07:55.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1647" PostTypeId="2" ParentId="280" CreationDate="2014-04-14T21:13:17.793" Score="3" Body="&lt;p&gt;And here's my another, but unrelated answer.  I sometimes did (more or less) the following: I showed high-schoolers the quadratic formula (i.e., the formula for the discriminant and solutions for a quadratic equation) and told them something along the lines: &quot;Here's a formula that gives a way to solve this equation.  Now try to explain the &lt;em&gt;way&lt;/em&gt; to solve it &lt;em&gt;without algebra&lt;/em&gt;.&quot;  Then, I open up the Wikipedia page with the formula for the solutions for the quartic and say &quot;And good luck with this one!&quot;.  Maybe it's not interesting (for them), but I find it convincing that algebra can actually &lt;em&gt;solve&lt;/em&gt; problems, not &lt;em&gt;create&lt;/em&gt; them (in this case, the problem of concisely and precisely writing down the way of doing some computation).&lt;/p&gt;&#xA;" OwnerUserId="704" LastEditorUserId="60" LastEditDate="2020-01-22T01:05:16.590" LastActivityDate="2020-01-22T01:05:16.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1648" PostTypeId="2" ParentId="1639" CreationDate="2014-04-14T21:16:36.867" Score="17" Body="&lt;p&gt;When I teach courses, I usually see the majority of my students in office hours at least once a week.  Here are some strategies for promoting office hours:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Get a room.  My office isn't nearly large enough to accommodate 10 students, and students who have to wait outside don't feel welcome.  By reserving a classroom or other large room, I make sure there's enough space for everybody.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Go at least two hours at a time.  Students often have trouble making significant progress on difficult homework assignments, so you need to schedule at least a two hour block if you expect to be helpful.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have lots of office hours.  The more office hours you have, the more likely it is that students will be able to make it to one of them.  I try to have at least 5 office hours a week, with an average of 7 or 8 when I'm teaching three classes.  In some semesters, my office hours have been so full that I've gone up to 10.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Schedule your office hours for times that the students will find convenient.  At my college, this means either the evenings or on Fridays.  In addition, it's very important to schedule office hours for right when the students want to be working, i.e. the day before homework is due, or the day before a quiz.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Make sure that the students perceive office hours as helpful.  Every student who attends office hours must feel like they got some noticeable help with the homework assignment.  In many cases, this means that you have to &lt;strong&gt;&lt;em&gt;give significant help with homework problems&lt;/em&gt;&lt;/strong&gt;.  Of course, talking with you about the course material is one of the best things a student can do, but students may not perceive the value in this, so the only way to make sure that they keep coming back is to give them hints, check over their answers, and do other tangible things to help them complete the homework assignments.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Announce your office hours every day in class.  I usually write the times of my office hours on the board at the beginning of every class, and I leave them up for at least 10-20 minutes.  I also mention my office hours verbally at the beginning of every class, as well as at the end of class if I remember.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Emphasize that successful students go to office hours.  Especially near the beginning of the semester, I usually emphasize that it's hard to be successful in the class without going to office hours, that you should plan to come to office hours at least once a week, etc.  I also like to say that the homework is made particularly difficult on purpose, because it really helps to get experience solving difficult problems, and that I'm expecting that students will get stuck and then need to come to office hours.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-04-14T21:16:36.867" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1649" PostTypeId="1" CreationDate="2014-04-14T23:48:14.257" Score="22" ViewCount="3363" Body="&lt;p&gt;There are quite a lot of tricks/shortcuts enabling doing calculations efficiently in your head.  (One of them that came to me today is the &quot;squaring a number ending with a 5&quot; trick I wrote about in &lt;a href=&quot;https://matheducators.stackexchange.com/a/1646/704&quot;&gt;this answer&lt;/a&gt;).  In a comment, user &lt;a href=&quot;https://matheducators.stackexchange.com/users/12/john-golden&quot;&gt;John Golden&lt;/a&gt; generously shared a few other tricks like that.  These are quite useful, especially in high school, e.g. as a way of showing students stuff &lt;em&gt;they&lt;/em&gt; find interesting, and also as a way of promoting &lt;em&gt;numeracy&lt;/em&gt; (in a sense of being able to compute or at least estimate things without a calculator or a similar device).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I though that asking for more such tricks - and hence gathering a few of them here - might be useful for MESE users.  (I hope the question isn't considered &quot;too broad&quot;, even though it does not have a definite answer - on TeX.SE such questions are quite common and can be both interesting and useful.  It might be a good idea, though, to convert it to CW.)&lt;/p&gt;&#xA;" OwnerUserId="704" LastEditorUserId="5987" LastEditDate="2017-08-22T01:36:31.487" LastActivityDate="2017-09-04T22:11:12.700" Title="Tricks for computing things in your head" Tags="|math-puzzle|mental-math|" AnswerCount="16" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1650" PostTypeId="2" ParentId="1649" CreationDate="2014-04-15T00:06:41.810" Score="4" Body="&lt;p&gt;I agree that the &quot;this is the way this &amp;lt;cool trick&amp;gt; works&quot; can be very motivating, but I see it much more important to learn to do (and use!) what Jon Bentley in his &lt;a href=&quot;http://www.cs.bell-labs.com/cm/cs/pearls&quot; rel=&quot;nofollow&quot;&gt;&quot;Programming Pearls&quot;&lt;/a&gt; calls &quot;back of the envelope computations&quot; (or even just getting rough estimates, so you aren't embarrased when the finance minister says that &quot;well, 10% of 10% is ... something like... uh, 0.1%&quot;).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-15T00:06:41.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1651" PostTypeId="2" ParentId="1639" CreationDate="2014-04-15T00:10:59.250" Score="9" Body="&lt;p&gt;Have you considered using a more modern solution?  They might be busy, embarassed, or don't want to walk to campus again.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few ideas:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Instead of having them come to you, what about having electronic&#xA;office hours?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use screen share on Skype, along with a pen input device, to go through problems.  They can tell you which problems they missed in chat.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask the students, via email, which problems or concepts they're having trouble with.  Then post a Youtube video of you solving the problems or explaining a concept in detail.    If you have limited time, you can just do the most requested issues.  You can also password protect the Youtube video, to prevent the public from seeing it.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Go through the tests or homework and see if there are some common problems that people are getting wrong, then do a video on just those.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use Google Docs to post problem work-throughs and detailed explanations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Send a weekly email update, or pass out a handout in class, about what you've posted online.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="911" LastActivityDate="2014-04-15T00:10:59.250" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1652" PostTypeId="2" ParentId="1649" CreationDate="2014-04-15T00:55:25.083" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://books.google.com/books/about/Mathemagics.html?id=DyUPAAAACAAJ&quot; rel=&quot;nofollow&quot;&gt;This book of Art Benjamin&lt;/a&gt; is fantastic. There is a new version of it, but I cannot remember the title.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should mention that although the book contains many specific &quot;tricks for calculation&quot; as requested by the OP, I think the value of the book is to give a taste for mental constructions for calculation that feel an awful lot like what mathematicians do when working. Benjamin gives insight into how to think about mental calculation and how to use one's working memory (incorporating auditory reinforcements) to carry out computations. The book gives the reader an appreciation for the aesthetic of mental calculation that I haven't found in other books on the topic. &lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2014-04-15T14:47:11.920" LastActivityDate="2014-04-15T14:47:11.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1653" PostTypeId="2" ParentId="1649" CreationDate="2014-04-15T02:31:17.113" Score="6" Body="&lt;p&gt;To tell if a number is divisible by&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;2, ask whether the last digit even.&lt;/li&gt;&#xA;&lt;li&gt;3, if there is one digit, ask whether it is 0, 3, 6, or 9, otherwise add the digits and ask whether the sum is divisible by 3.&lt;/li&gt;&#xA;&lt;li&gt;4, if the second last digit is even, ask whether the last digit is 0, 4, or 8, otherwise ask whether it the last digit is 2, 6.&lt;/li&gt;&#xA;&lt;li&gt;5, ask whether the last digit is 0 or 5.&lt;/li&gt;&#xA;&lt;li&gt;6, ask whether it is divisible by both 2 and 3.&lt;/li&gt;&#xA;&lt;li&gt;7, chop off the last digit and subtract twice that digit from the remaining number, then ask whether that number is divisible by 7. E.g. 203 --&gt; 20-2*3 --&gt; 14. (You can also add 5 times the last digit. E.g. 98 --&gt; 9 + 5*8 --&gt; 49.)&lt;/li&gt;&#xA;&lt;li&gt;8, if the third last digit is even, ask whether the last two digits are divisible by 8,  otherwise ask whether the last two digits are divisible by 4, but not 8.&lt;/li&gt;&#xA;&lt;li&gt;9, add the digits and ask whether the sum is divisible by 9.&lt;/li&gt;&#xA;&lt;li&gt;11, if the number is less than 100, ask whether the two digits are the same, otherwise add the digits in groups of two (e.g. abcdef -&gt; ab + cd + ef) and ask whether the result is divisible by 11.&lt;/li&gt;&#xA;&lt;li&gt;and so on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="527" LastEditorUserId="-1" LastEditDate="2014-07-22T13:30:06.620" LastActivityDate="2014-07-22T13:30:06.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1654" PostTypeId="2" ParentId="1639" CreationDate="2014-04-15T06:57:10.810" Score="12" Body="&lt;p&gt;When I was a calculus TA last semester, I was able to recruit tons of students to my office hours. Here are some tips:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Hold &lt;strong&gt;test reviews&lt;/strong&gt; just before midterms/finals. These were my best attended hours! Also, some students who came to these started coming to other office hours later in the semester.&lt;/li&gt;&#xA;&lt;li&gt;At your test reviews, &lt;strong&gt;provide materials for the students to work on together&lt;/strong&gt; (i.e. practice exams, extra problems.) That way, students won't feel responsible to come with their own questions. It makes it easier for them to come. Announce ahead of time that you are providing materials.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Evenings&lt;/strong&gt; may be popular, especially if for more than one hour. My Wednesday 7-9 pm hours were very crowded.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Food&lt;/strong&gt;. Sometimes I would advertise ahead of time that we would all pitch in and order pizza during the office hour. This was also popular. Dinner and a math problem ;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you can get lots of students in your office hour at once, they will start helping each other. Very efficient!&lt;/p&gt;&#xA;" OwnerUserId="914" LastActivityDate="2014-04-15T06:57:10.810" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1655" PostTypeId="1" AcceptedAnswerId="1663" CreationDate="2014-04-15T07:20:31.553" Score="49" ViewCount="15603" Body="&lt;p&gt;I had one very vocal student in my Calculus recitation last year. Sometimes she would point out if I made a mistake in the lecture. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, sometimes she would &lt;em&gt;insist&lt;/em&gt; that I had made a mistake, even if I concluded it was NOT a mistake after careful consideration. She would continue to contradict me in front of the class, which was disruptive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I am a graduate student, so I probably seem less authoritative.) I want her to keep thinking critically and carefully during lecture. But disrupting the lecture excessively is a problem.  How would you handle this situation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one example. I stated that the cotangent function is decreasing where it's defined, and gave some explanation based on the sine and cosine functions. She insisted cotangent was increasing. The encounter was rather short and I moved on quickly, but she stopped coming to recitation soon after. I think she was offended.&lt;/p&gt;&#xA;" OwnerUserId="914" LastEditorUserId="2083" LastEditDate="2018-12-05T07:32:16.073" LastActivityDate="2020-10-20T14:46:51.207" Title="How to handle the situation where a student insists I am wrong during the class?" Tags="|classroom-management|students-mistakes|gifted-students|" AnswerCount="10" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="1656" PostTypeId="2" ParentId="280" CreationDate="2014-04-15T07:24:19.213" Score="9" Body="&lt;p&gt;The most striking example of this to me is a reasonably complicated first-degree equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's that, you say? There are no complicated first-degree equations? They're all just $ax+b=0\implies x=-b/a$? That's because you know algebra - it's not at all obvious they can be put into that standard form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3\cdot(2\cdot\_\_+5)-2\cdot(\_\_+5)=3\cdot\_\_+14$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What number can go in the blanks to make this true?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I find this so striking is because, if you don't know any algebra at all, the above looks intractably difficult, but &lt;em&gt;with&lt;/em&gt; algebra, it's so easy you can do it in your head in under a minute, with a bit of practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to think that a simple &quot;fill in the blank&quot; puzzle is simple and interesting enough to motivate all but the most math-phobic students, as long as you start simple.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\_\_+5=12$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$7\cdot\_\_=42$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Easy, especially once you work out you can get the answer without guessing by doing division and subtraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3\cdot\_\_+9=90$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A little trickier, but it doesn't take formal algebra training to see that you can get this with a subtraction followed by a division. Note that the problem is now getting too hard to make guessing as viable. Then, once you get to something like what I posted above, the situation is hopeless if you don't know about distributivity and balancing equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember when I was about 10, I knew roughly what &quot;solve for $x$&quot; meant, and I wanted to come up with something to stump my dad. I need something simple enough that I myself could solve it. If I remember correctly, my best effort after about 15 minutes of brainwracking was something like $x+1=2x$, if it wasn't even simpler. He solved it in his head in about half a second, and I was stunned.&lt;/p&gt;&#xA;" OwnerUserId="115" LastEditorUserId="115" LastEditDate="2014-04-15T13:58:03.760" LastActivityDate="2014-04-15T13:58:03.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1657" PostTypeId="2" ParentId="1649" CreationDate="2014-04-15T07:36:12.037" Score="10" Body="&lt;p&gt;Some tricks I've seen:&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Tricks with notable products&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;$(a + b)^2 = a^2 + 2ab + b^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This formula can be used to compute squares. Say that we want to compute $46^2$. We use $46^2 = (40+6)^2 = 40^2+2\cdot40\cdot6 +6^2 = 1600 + 480 + 36 = 2116$. You can also use this method for negative $b$:&#xA;$ 197^2 = (200 - 3)^2 = 200^2 - 2\cdot200\cdot3 + 3^2 = 40000 - 1200 + 9 = 38809 $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The last subtraction can be kind of tricky: remember to do it right to left, and take out the common multiples of 10:&#xA;$ 40000 - 1200 = 100(400-12) = 100(398-10) = 100(388) = 38800 $&#xA;The hardest thing here is to keep track of the amount of zeroes, this takes some practice!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also note that if we're computing $(a+b)^2$ and a is a multiple of $10^k$ and $b$ is a single digit-number, we already know the last $k$ digits of the answer: they are $b^2$, then the rest (going to the right) are zeroes. We can use this even if a is only a multiple of 10: the last digit of $(10 * a + b)^2$ (where $a$ and $b$ consist of a single digit) is $b$. So we can write (or maybe only make a mental note that we have the final digit) that down and worry about the more significant digits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also useful for things like $46\cdot47 = 46^2 + 46 = 2116 + 46 = 2162$. When both numbers are even or both numbers are uneven, you might want to use:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(a+b)(a-b) = a^2 - b^2$&#xA;Say, for example, we want to compute $23 \cdot 27$. We can write this as $(25 - 2)(25 + 2) = 25^2 - 2^2 = (20 + 5)^2 = 20^2 + 2\cdot20\cdot5 + 5^2 - 4 = 400 + 200 + 25 - 4 = 621$.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Divisibility checks&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Already covered by Theodore Norvell. The basic idea is that if you represent numbers in a base $b$, you can easily tell if numbers are divisible by $b - 1$, $b + 1$ or prime factors of $b$, by some modular arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Vedic math&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;A guy in my class gave a presentation on &lt;a href=&quot;http://en.wikipedia.org/wiki/Vedic_Mathematics_%28book%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vedic math&lt;/a&gt;. I don't really remember everything and there probably are a more cool things in the book, but I remember with algorithm for multiplication that you can use to multiplicate numbers in your head.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/orCcx.jpg&quot; alt=&quot;gelosia or lattice multiplication&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This picture shows a method called &lt;a href=&quot;http://en.wikipedia.org/wiki/Lattice_multiplication&quot; rel=&quot;nofollow noreferrer&quot;&gt;lattice or gelosia multiplication&lt;/a&gt; and is just a way of writing our good old-fashioned multiplication algorithm (the one we use on paper) in a nice way. Please notice that the picture and the Vedic algorithm are not tied: I added the picture because I think it helps you appreciate and understand the pattern that is used in the algorithm. The gelosia notation shows this in a much nicer way than the traditional notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The algorithm the guy explained is essentially the same algorithm as we would use on paper. However, it structures the arithmetic in such a way that we never have remember too many numbers at the same time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's illustrate the method by multiplying $456$ with $128$, as in the picture. We work from left to right: we first compute the least significant digits and work our way up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We start by multiplying the least significant digits:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$6 \cdot 8 = 48$: the least significant digit is $8$, remember the $4(0)$ for the next round (of course, I don't mean zero times four here but four, or forty, whatevery you prefer: be consistent though, if you include the zero here to make forty, you got do it everywhere).&#xA;$ 8 \cdot 5(0) = 40(0) $ &lt;br&gt;&#xA;$ 2(0) \cdot 6 = 12(0) $ &lt;br&gt;&#xA;$ 4(0) + 40(0) + 12(0) = 56(0) $: our next digit (to the left of the $8$) is $6$: remember the $5(00)$ &lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ 8 \cdot 4(00) = 32(00) $ &lt;br&gt;&#xA;$ 2(0) \cdot 5(0) = 10(00) $ &lt;br&gt;&#xA;$ 1(00) \cdot 6 = 6(00) $ &lt;br&gt;&#xA;$ 5(00) + 32(00) + 10(00) + 6(00) = 53(00) $: our next digit is a $3$, remember the $5(000)$ &lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pfff... starting with 2-digit numbers is a better idea, but I wanted to this longer one to make the structure of the algorithm clear. You can do this much faster if you have practiced, since you don't have to write it all down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ 2(0) \cdot 4(00) = 8(000) $ &lt;br&gt;&#xA;$ 1(00) \cdot 5(0) = 5(000)$ &lt;br&gt;&#xA;$ 5(000) + 8(000) + 5(000) = 18(000)$: next digit is an $8$, remember the $1(0000)$&lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ 1(00) \cdot 4(00) = 4(0000) $ &lt;br&gt;&#xA;$ 1(0000) + 4(0000) = 5(0000) $: the most significant digit is a $5$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we have $58368$.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Quadratic equations&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;There are multiple ways to solve a quadratic equation in your head. The easiest are quadratic with integer coefficients. If we have $x^2 + ax + c = 0$, try to find $r_{1, 2}$ such that $r_1 + r_2 = -a$ and $r_1r_2 = c$. It is also possible to solve for non-integer solutions this way, but it is usually too hard to actually come up with solutions this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way is just to try divisors of the constant term. By the rational root theorem (google it, I can't link anymore &lt;em&gt;sigh&lt;/em&gt;) all solutions to $x^n + ... + c = 0$ need to be divisors of $c$. If $c$ is a fraction $\frac{p}{q}$, the solutions need to be of the form $\frac{a}{b}$ where $a$ divides $p$ and $b$ divides $q$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this all fails, we can still put the abc-formula in a much easier form:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ ux^2 + vx + w = 0 $ &lt;br&gt;&#xA;$ x^2 + \frac{v}{u}x + \frac{w}{u} = 0 $ &lt;br&gt;&#xA;$ x^2 + \frac{v}{u}x + \frac{w}{u} = 0 $ &lt;br&gt;&#xA;$ x^2 - ax - b = 0 $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ x^2 = ax + b $ &lt;strong&gt;(This is the form that I found easiest to use!)&lt;/strong&gt; &lt;br&gt;&#xA;$ (x - \frac{a}{2})^2 = (\frac{a}{2})^2 + b $ &lt;br&gt;&#xA;$ x = \frac{a\pm\sqrt{a^2 + 4b}}{2} = \frac{a}{2} \pm \sqrt{(\frac{a}{2})^2 + b} $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure there are also a lot of techniques for estimating products and the like, but I'm not really familiar with them.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Tricks that aren't really usable but still pretty cool&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://faculty.randolphcollege.edu/tmichalik/feynman.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;this excerpt&lt;/a&gt; from Feynman's &quot;Surely you're joking, Mr. Feynman!&quot; about how he managed to amaze some of his colleagues, and also &lt;a href=&quot;http://www.youtube.com/watch?v=y8acoaakvPM&quot; rel=&quot;nofollow noreferrer&quot;&gt;this video from Numberphile&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="915" LastEditorUserId="915" LastEditDate="2015-10-24T23:30:38.433" LastActivityDate="2015-10-24T23:30:38.433" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1658" PostTypeId="2" ParentId="520" CreationDate="2014-04-15T07:59:28.077" Score="4" Body="&lt;p&gt;Another way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can introduce students to the recurrence relation:&#xA;$$ a_{n + 1} = a_n + 2b_n $$&#xA;$$ b_{n + 1} = a_n + b_n $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And show that this recurrent relation can be represented as a matrix multiplication:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ v_{n+1} = Av_n $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $ A = \binom{1 \ \ 2}{1 \ \ 1}$ and $ v_n = \binom{a_n}{b_n} $. Then show (just demonstrate, not prove) that $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = \sqrt{2}$. Then you can do a visual representation that shows that the matrix $A$ is a skew (and reflection, I guess) matrix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using eigenvalue and eigenvectors, you can show convergence to $\sqrt{2}$, and (if you put some effort into it) make similar recurrent relations for other squareroots (and argue about their convergence: we want the eigenvector that it needs to converge to to have big eigenvalue and the other eigenvector a small eigenvalue). Also, if there is a basis of eigenvectors (in this case, there is), you can give an explicit formula for each point in the recurrent relation which is kinda cool.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can motivate eigenvalues as one way to analyze this kind of recurrent relations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not the easiest way to handle recurrent relations, but it gives students a geometrical example, and shows an application at the same time. I think it is also good to present this idea that matrix multiplication can be used to describe recurrent relations.&lt;/p&gt;&#xA;" OwnerUserId="915" LastActivityDate="2014-04-15T07:59:28.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1659" PostTypeId="2" ParentId="1042" CreationDate="2014-04-15T08:19:09.363" Score="5" Body="&lt;p&gt;Solving quadratic and cubic equations. At least for cubics, this really does require that your students have seen complex numbers.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Solving the quadratic by completing the square is simple, clever, and you may not have ever shown them the proof in class.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For the ambitious, Cardano's method for solving the cubic is quite straight forward, based on the lemma that for any two numbers $S$ and $P$, there are two (possibly complex) numbers whose sum is $S$ and whose product is $P$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Staying within the realm of algebra, some basic things about complex numbers could make for a decent two-page project. Students could prove the basic geometrical facts about complex numbers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Addition corresponds to vector addition on an Argand diagram.&lt;/li&gt;&#xA;&lt;li&gt;Multiplication correponds to multiplying the lengths and adding the angles.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Proving those facts could be page 1. Page 2 could be using them to prove another geometrical fact about complex numbers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There are exactly $n$ $n$-th roots of $1$, and they form the vertices of an $n$-gon centered at $0$. From this you can deduce that there are exactly $n$ $n$-th roots of &lt;em&gt;any&lt;/em&gt; number.&lt;/li&gt;&#xA;&lt;li&gt;You can use complex numbers to prove certain trig identities.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="115" LastActivityDate="2014-04-15T08:19:09.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1661" PostTypeId="2" ParentId="1649" CreationDate="2014-04-15T11:18:57.973" Score="5" Body="&lt;p&gt;The squares of the first couple of numbers using only the digit one show a nice pattern and are thus easy to remember. They are collected in the list below.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align}&#xA;11\cdot11&amp;amp;=121&#xA;\\ 111\cdot111&amp;amp;=12321&#xA;\\ 1111\cdot1111&amp;amp;=1234321&#xA;\\ 11111\cdot11111&amp;amp;=123454321&#xA;\\ 111111\cdot111111&amp;amp;=12345654321&#xA;\\ 1111111\cdot1111111&amp;amp;=1234567654321&#xA;\\ 11111111\cdot11111111&amp;amp;=123456787654321&#xA;\\ 111111111\cdot111111111&amp;amp;=12345678987654321&#xA;\end{align}$$&lt;/p&gt;&#xA;" OwnerUserId="584" LastEditorUserId="5664" LastEditDate="2015-10-18T16:23:53.593" LastActivityDate="2015-10-18T16:23:53.593" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1662" PostTypeId="2" ParentId="1655" CreationDate="2014-04-15T11:31:09.233" Score="5" Body="&lt;p&gt;It is not good enough for you to &quot;think carefully&quot; and conclude and state that you did not make a mistake. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to prove to yourself and the class that you are correct. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you cannot do this on the spot show a bit of humility and tell the class that you think you are correct and you will confirm via email ASAP after the lecture.&lt;/p&gt;&#xA;" OwnerUserId="584" LastActivityDate="2014-04-15T11:31:09.233" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1663" PostTypeId="2" ParentId="1655" CreationDate="2014-04-15T11:56:45.917" Score="50" Body="&lt;p&gt;It depends on how much time I can afford to spend on the problem.  I check, as you did, whether I see a mistake.  If not, I try to explain the math in a different way: a different conceptual approach, which is good for everyone anyway; perhaps replace the variables with numerical values, if appropriate; or replace a general function $f$ by a specific function.  If there is time and the student seems fixed in her idea, I might ask her for an example.  Often the student is assuming something extra or false, and her example might reveal it.  Also, I can check the example and confirm again that there is no mistake.  Finally, if there isn't time to work it out, I ask the student to see me after class or in my office hours.  (&quot;Maybe we can work this out after class.  Are you free?&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, it's very helpful to remain calm, receptive to the criticism, and seem happy to look into it.  Controlling the tension in my voice and the look on my face took me a few years to learn.  I now tend to seem excited by investigating mistakes.  In part, I know I'm doing at least one of two good things: clarifying the lesson or dispelling someone's misconception.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2014-04-15T11:56:45.917" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1664" PostTypeId="2" ParentId="1655" CreationDate="2014-04-15T12:25:23.983" Score="21" Body="&lt;p&gt;I have had this problem before with students who always think they're right.  If a student continues to insist you made a mistake, when you know that you haven't, then tell the student to hold the thought and ask them to discuss it with you after class.  Once the class is over, write the original problem on the board and ask the student to solve it.  If their answer is correct, congratulate them on their thinking.  If their answer is incorrect, show them why they are incorrect.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, make sure you plan your recitation and do the homework.  I had a student that kept contradicting me one day in class, even after I showed him where his thinking was flawed.  I told him to wait after class.  It turned out that he had used the Chegg website to get his answer, and when we went to the website to look at it together, the answer was incorrect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Put the onus back on them to prove to you they have a correct answer. This is very common in College Calculus, especially from students who took AP Calculus in High School.     &lt;/p&gt;&#xA;" OwnerUserId="203" LastEditorUserId="526" LastEditDate="2014-04-15T20:29:23.307" LastActivityDate="2014-04-15T20:29:23.307" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1665" PostTypeId="2" ParentId="1655" CreationDate="2014-04-15T12:27:05.727" Score="14" Body="&lt;p&gt;Taking some extra care to verify the details of a confusing bit of material (preferably with class involvement, intuition-building examples, etc.) is usually an excellent use of class time. Offering to talk about the issue further after class or during office hours is also a reasonable tactic when time is short or when dealing with a particularly insistent student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I think there are also times when it is appropriate to take a firmer hand. A balance must be struck between nurturing critical thinking and being fair to the rest of the class—the loudest student should not get to dictate the course of the lecture. It's one thing if a good portion of the class is confused and one student is simply voicing their collective concerns. But it also happens that one student decides to treat the class as if it's their own personal tutoring session. In this case, it's appropriate to tell them to stop: &quot;I understand you disagree, and like I said I'll be happy to assuage your concerns after class, but right now I need to finish this chapter, so please hold your comments.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an ideal world, perhaps, classes have no official &quot;end&quot; time, the students are there to learn, and they simply stay until they are satisfied with their own understanding. But unfortunately, that's not the way most university systems work, and the types of classes graduate students are likely to be TAing tend to come with &lt;em&gt;very&lt;/em&gt; tight time constraints. Something has to give.&lt;/p&gt;&#xA;" OwnerUserId="327" LastActivityDate="2014-04-15T12:27:05.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1666" PostTypeId="2" ParentId="1649" CreationDate="2014-04-15T13:21:40.343" Score="1" Body="&lt;p&gt;This might be of interest to you - &lt;a href=&quot;http://en.wikipedia.org/wiki/Vedic_Mathematics_(book)&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Vedic_Mathematics_(book)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/File:Vedicmathematics.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vvpnS.jpg&quot; alt=&quot;Vedic Mathematics&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Vedic Mathematics&lt;/em&gt;&lt;/strong&gt; is a book written by the high-ranking Hindu cleric Bharati Krishna Tirthaji and first published in 1965. It contains a list of mental calculation techniques claimed to be based on the Vedas. The mental calculation system mentioned in the book is also known by the same name or as &quot;Vedic Maths&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source: &lt;a href=&quot;http://en.wikipedia.org/wiki/Vedic_Mathematics_%28book%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Vedic Mathematics&lt;/em&gt; (book)&lt;/a&gt; on Wikipedia&lt;/p&gt;&#xA;" OwnerUserId="920" LastEditorUserId="160" LastEditDate="2014-05-29T21:30:00.780" LastActivityDate="2014-05-29T21:30:00.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1667" PostTypeId="2" ParentId="1538" CreationDate="2014-04-15T13:34:43.653" Score="6" Body="&lt;p&gt;You can do better using &lt;em&gt;uniform continuity&lt;/em&gt;:  $f$ is uniformly continuous on $[a,b]$ if $$\forall\epsilon\ \ \exists\delta\ \ \forall x,y\in[a,b], \ \ |x-y|&amp;lt;\delta \rightarrow |f(x)-f(y)|&amp;lt;\epsilon$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has several advantages:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;strong&gt;It makes more sense in applications.&lt;/strong&gt;  E.g.:  If astronomers want to determine the volume of a star to within $\epsilon$, how accurately do they need to determine the radius?  It does not help to give a $\delta$ which depends on the exact and unknown $r$.  It makes more sense to give a $\delta$ which works for any $r$ within the preliminary estimate of $[a,b]$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;It allows a more honest proof&lt;/strong&gt; that continuous functions are integrable.  The proof from pointwise continuity appeals to uniform continuity anyway, either leaving that implication unproved or providing a proof too advanced for calculus students to appreciate.  Starting from uniform continuity instead provides all the data for an integration algorithm, and the proof follows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &lt;strong&gt;It avoids your counterintuitive statement&lt;/strong&gt; by leading to the sensible statement &quot;for every $a&amp;gt;0$, the function $1/x$ is uniformly continuous on the intervals $(-\infty,a]$ and $[a,+\infty)$&quot;.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This (especially #2) is Errett Bishop's insight.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-15T13:34:43.653" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1668" PostTypeId="2" ParentId="1639" CreationDate="2014-04-15T14:20:34.637" Score="3" Body="&lt;p&gt;Talk with your professor and suggest that extra credit be given to students who come for lab hours. If your professor agrees, then, he can advertise that in his class and I'd be great if you also ahow up in class and advertice your office hours.&lt;/p&gt;&#xA;" OwnerUserId="923" LastActivityDate="2014-04-15T14:20:34.637" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1669" PostTypeId="2" ParentId="280" CreationDate="2014-04-15T15:25:06.227" Score="7" Body="&lt;p&gt;You can find a lot of such problems (freely available on the internet) among the &lt;em&gt;Arithmetic Problems&lt;/em&gt; in the early issues of the &lt;strong&gt;American Mathematical Monthly&lt;/strong&gt; and in 1800s algebra textbooks. Below are three such examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Solution to Arithmetic Problem #116&lt;/em&gt;, &lt;a href=&quot;http://books.google.com/books?id=iIDxAAAAMAAJ&quot;&gt;&lt;strong&gt;American Mathematical Monthly&lt;/strong&gt; 6&lt;/a&gt; #10 (October, 1899), 238-239. [The problem also appears on p. 120 of &lt;a href=&quot;http://books.google.com/books?id=g9dHAAAAIAAJ&quot;&gt;&lt;strong&gt;Advanced Algebra&lt;/strong&gt;&lt;/a&gt; by Joseph Victor Collins (1918).]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Problem Statement:&lt;/strong&gt; Two candles are of the same length. The one is consumed uniformly in $4$ hours, and the other in $5$ hours. If the candles are lighted at the same time, when will one be three times as long as the other?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Joseph Ray, &lt;a href=&quot;http://books.google.com/books?id=bOVEAAAAIAAJ&quot;&gt;&lt;strong&gt;Elements of Algebra&lt;/strong&gt;&lt;/a&gt;, 1865.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Problem 24 on page 117&lt;/strong&gt; A tank is supplied with water from three pumps. The first and second will fill it in $30$ hours, the first and third in $40$ hours, and the second and third in $50$ hours. In what time can each fill it separately?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Horatio Nelson Robinson, &lt;a href=&quot;http://books.google.com/books?id=5tY2AAAAMAAJ&quot;&gt;&lt;strong&gt;Elementary Treatise on Algebra&lt;/strong&gt;&lt;/a&gt;, 1846.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Problem 24 on p. 64 (variables changed to numerical values by me):&lt;/strong&gt; A person engaged to work $24$ days on these conditions: For each day he worked he was to receive $25$ cents, for each day he was idle he was to forfeit $15$ cents. At the end of $24$ days he received $320$ cents. How many days was he idle?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-04-15T15:25:06.227" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1670" PostTypeId="2" ParentId="1649" CreationDate="2014-04-15T17:02:39.600" Score="2" Body="&lt;h1&gt;Quick &quot;multiplication&quot;&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Ask your students to call out digits until you have the entire width of the board filled. Then write $\times\,5$ on the next line. Proceed to right the full answer out from left to right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the usual algorithm for multiplying by hand where the answer is produced right to left, it looks like you've already computed the product when you start writing the answer. But if you use division, then you start writing the answer left to right, so the trick is to take advantage of the fact that multiplying by five is the same as multiplying by 10 and then dividing by two.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;(Another) Feynman Method&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;In his memoris, Richard Feynman actually means a few tricks he saw performed during his time at Los Alamos. It's interesting because the tricks are of a similar form to &quot;multiplication&quot; by five or other tricks involving binomial expansion: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;some easy transformation into parts that are easier to work with (multiplying by 10, decomposition into some sum of easy numbers)&lt;/li&gt;&#xA;&lt;li&gt;new easier operations (sometimes the operation changes completely, e.g. the example above, sometimes it doesn't, e.g. certain methods of squaring numbers near a certain other number) &lt;/li&gt;&#xA;&lt;li&gt;transformation to the original form (possibly an identity transformation, as in my example)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A discussion of such tricks would actually be a really nice way to lead into ideas like homo- and isomorphy. Logarithms are a classic example of isomorphisms, but are rather difficult to compute by hand. If you have a handy table or mechanical device, i.e, slide rule, then an expensive transformation becomes an easy one, and the computational step also becomes an easy one -- addition instead of multiplication. It's easy enough to make a simple slide rule in the classroom, depending on the age of your audience, and it really brings together a lot of important concepts.  &lt;/p&gt;&#xA;" OwnerUserId="926" LastActivityDate="2014-04-15T17:02:39.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1671" PostTypeId="1" AcceptedAnswerId="1682" CreationDate="2014-04-15T17:56:55.460" Score="11" ViewCount="1152" Body="&lt;p&gt;I've noticed that my students (primarily first-year students taking their first college math course) have a lot of trouble with inequalities.  Not the math, the purpose: they seem to have a preconception that math is supposed to be about finding things exactly, and every time we talk about a bound, I get multiple people asking why anyone would ever care about such a thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been sort of waving my hands and saying &quot;Well, if you build a bridge, you care about knowing the smallest weight which might cause it to break...&quot; but I think it might help to be more concrete.  I'm wondering if anyone can suggest good techniques or examples which illustrate the importance of bounds.  (Given that the students aren't mathematicians, uses outside of math which make sense even in a world with computers that produce calculations accurate to 30 decimal places are particularly useful.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-06-23T20:11:54.377" Title="Introducing bounds" Tags="|undergraduate-education|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1672" PostTypeId="2" ParentId="1671" CreationDate="2014-04-15T18:16:30.407" Score="10" Body="&lt;p&gt;I had the exact same problem as the students (I disliked analysis because the bounds seemed so arbitrary). I overcame this by seeing very precise results coming from imprecise bounds (one example is the Squeeze theorem). However, it might be easier to show them they use bounds all the time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Examples In Real life&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Imagine you are going to McDonalds. You see a man with a briefcase full of hundred dollar bills. He pulls one out and pays for his sandwich with it. He says, &quot;I didn't know which meal I would get, so I wanted to make sure I have enough money&quot;. This is a stupid example, right? But if your students start talking about &lt;em&gt;why&lt;/em&gt; it's stupid, that could help.&lt;/li&gt;&#xA;&lt;li&gt;If you have a washer full of laundry, it seems that you can always fit in one more sock. Can you fit in infinitely many socks? Give me a number of socks that can't fit into a washer. Can you give me the &lt;em&gt;smallest&lt;/em&gt; number of socks that can't fit into a washer?&lt;/li&gt;&#xA;&lt;li&gt;For what temperatures does ice stay frozen (at normal pressures, etc)? Can you express this without using inequalities or bounds?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-04-15T18:16:30.407" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1673" PostTypeId="2" ParentId="1671" CreationDate="2014-04-15T18:39:00.437" Score="1" Body="&lt;p&gt;An example they might already know from secondary education are definite integrals. The integral is bounded in both directions by partial sums. For every precision $h$, the partial sums aren't exact. But for infinite precision, the limits are exact (for Riemann-integrable functions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students don't know this definition of definite integrals but are fit in modelling, you can introduce it as modelling the area by rectangle/trapezoids.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-15T18:39:00.437" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1674" PostTypeId="2" ParentId="1639" CreationDate="2014-04-15T19:16:33.007" Score="7" Body="&lt;p&gt;I'd like to add a suggestion on location.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I currently hold my office hours in one of the student cafés.  When I had students from a particular program and those students had a common room, then I held them in a corner of that room.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea there is to go where the students are.  It can take a bit of courage to go to a lecturer's office, particularly if it is a bit out of the way (my office is on the 12th floor).  But a café is neutral and with lots of exits, and places where one can sit and listen to others' questions.  Also, it's possible to be sitting nearby working on the homework and only coming over for help when needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Plus it gets me out of my office, and in the vicinity of decent coffee.)&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-04-15T19:16:33.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1675" PostTypeId="2" ParentId="1649" CreationDate="2014-04-15T19:17:26.017" Score="9" Body="&lt;p&gt;I think $(a+b)(a-b) = a^2 - b^2$ is the most underutilized trick. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A bit of practice and it's easy to see how the square is $b^2$ bigger than the rectangle, and conversely, the rectangle $b^2$ smaller than the square. So 45 squared is (40*50)+25 or 2025. And any multiplying where you can latch on to an easy head math where the true numbers are +/- the difference to the easy result, makes for some fast calculations. And 39*41 is just 1600-1=1599.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a side note, I've observed that schools have become complacent with calculator use, and this effort (the head math trick) isn't promoted. In my time tutoring student at my High School job, I walk a fine line, encouraging them not to use the calculator for what they can do in their head, but not applying too much pressure. On the other hand, I'll sometimes blurt out a result, showing them that head math is far faster than the time it takes on a calculator. Those bits of time add up when taking our state's standardized tests or the SATs (US based college entrance exam), and in my opinion, that time is going to impact the score. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet, there's not going to be any conclusive way to prove such an assertion, as those with head math skills are the kids who are likely to perform better anyway, so no way to produce a study that will satisfy the objective observer. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-04-15T19:17:26.017" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1676" PostTypeId="2" ParentId="1655" CreationDate="2014-04-15T19:58:18.560" Score="7" Body="&lt;p&gt;Humbly accept that you could be wrong, and be open to feedback. Turn it into a teaching moment when possible by inviting the student to come up and show their solution. If they are wrong, don't tell them why, but instead ask they class if they see any problem with the approach. It will not only get the class involved, but it will humble the student from trying to show off their knowledge. If you are the one that is wrong, simply admit it, explain the error in your thinking, thank the student and move on. This way it moves away from an &quot;I'm right and you are wrong&quot; argument, but a way for other students to recognize where mistakes are possible and to learn from it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student is consistently disruptive, speak to them personally and ask them not to be disruptive, but to please bring their concerns to you privately. Obviously, you want the class to know when/if you are making mistakes, and you should humbly tell the class when the student does reveal an error in your logic so they can learn from it. But you don't want the disruptions to become a distraction from the materials you need to get through.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Look at it this way: would you rather focus be on proving the student wrong and defending your reputation, or would you rather focus on helping the class to learn, regardless of who is making mistakes? We all make mistakes. Help your students to learn from them.&lt;/p&gt;&#xA;" OwnerUserId="934" LastEditorUserId="6053" LastEditDate="2019-04-08T08:37:42.333" LastActivityDate="2019-04-08T08:37:42.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1677" PostTypeId="2" ParentId="1671" CreationDate="2014-04-15T20:37:06.100" Score="2" Body="&lt;p&gt;You might consider appealing to &lt;em&gt;computationally intractable problems&lt;/em&gt; that can be wrangled by mathematical bounding. This could double as an argument against those students who might say, &quot;Why bother learning math? Can't I just make my computer solve things for me?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that students of a variety of ages can understand an argument that $R(3,3)=6$. Depending on whether you're talking to high-schoolers or undergrad math majors, you might phrase the setup of the problem differently, or go into different depths in the argument. But they can all follow along.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Follow this up with asking about $R(4,4)$. If they can't find the exact number &quot;by hand&quot;, could they program a computer to do it? Work with them to realize how many possible 2-colorings of $K_n$ there are, and estimate how long it would take to check one graph for a monochromatic $K_4$. (You can be hand-wavey here, depending on the audience.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Continue for $R(5,5)$ and $R(6,6)$. Try to explain how the computational time required &lt;em&gt;explodes&lt;/em&gt;. Explain how we have no hope of simply letting a computer run to figure out an answer. By proving better and better bounds, we can narrow the &lt;em&gt;search space&lt;/em&gt;, but without doing that, we're essentially knowledge-less.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, hit 'em with this quote (idea attributed to Erdős, words by Joel Spencer):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of $R(5,5)$ or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for $R(6,6)$. In that case, he believes, we should attempt to destroy the aliens.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-04-15T20:37:06.100" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1678" PostTypeId="2" ParentId="1671" CreationDate="2014-04-15T20:42:21.097" Score="3" Body="&lt;p&gt;Another interesting example appears when evaluating infinite series. I have always personally found it fascinating that, for example, with an alternating series, we can use the terms of the series to accurately estimate the final sum; more importantly, we can declare how accurate our estimation is &lt;em&gt;without even knowing the final sum&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many calculus textbooks have this idea expressed as an inequality: &quot;For $\sum (-1)^n a_n$, where $a_n\geq 0$, the remainder of the $n$-th partial sum obeys $|R_n|\leq a_n$.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I think far too few stress this idea: we can say something like, &quot;Adding up the first 100 terms yields 10 decimal places of accuracy, even though we don't know what number we're approximating so accurately!&quot;&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-04-15T20:42:21.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1679" PostTypeId="2" ParentId="1671" CreationDate="2014-04-15T21:12:56.790" Score="1" Body="&lt;p&gt;You mentioned two examples, and I'm enthusiastic about one of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For using the tangent line to approximate a function, I sympathize with the students.  &quot;Use the derivative of $y=\sqrt{x}$ at $x=25$ to approximate $\sqrt{26}$ by hand&quot; is not impressive.  So I wouldn't focus much on this topic in a calculus class.  If students continue mathematically, they'll see natural situations without closed-form expressions, and that will be a better time to see the power of linearization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for using sums to approximate integrals, I would give them hard problems.  What is $\int_0^1 (e^{1-x^2}-1)\ dx$ or $\int_1^e\negthinspace \sqrt{1-\ln y}\ dy$ or $\int_0^1 x \sin(1/x)\ dx$, or the area between the axes and $x^2+x^3+y^4+y^5=1$?  Bounds are easier to appreciate for problems that can only be solved numerically.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-15T21:12:56.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1680" PostTypeId="1" CreationDate="2014-04-15T21:47:10.227" Score="8" ViewCount="677" Body="&lt;p&gt;For which undergraduate and graduate mathematics courses is &lt;em&gt;multivariable analysis*&lt;/em&gt; an essential prerequisite?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;$\text{*}$ That is, a rigorous follow up to a first real analysis course at the level of baby Rudin.&lt;/p&gt;&#xA;" OwnerUserId="941" LastEditorUserId="376" LastEditDate="2014-06-15T09:10:59.067" LastActivityDate="2014-06-15T09:10:59.067" Title="What courses require multivariable analysis?" Tags="|mathematical-analysis|prerequisites|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1681" PostTypeId="2" ParentId="1635" CreationDate="2014-04-15T21:47:44.970" Score="10" Body="&lt;p&gt;I am not an expert in dyscalculia, and, in fact, my area of study in mathematics education has not predisposed me to focus on the idea of mathematical disorders. Because of this, I think I have a point of view that may at least offer a useful way to view these questions. [Note: there are a lot of conjectures here, but I think that is not unusual for questions of this type. I will try to be explicit about it.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Item 1&lt;/strong&gt; Assumption: everyone can learn mathematics. As far as I know, there is no dispute about this. This does not contradict the possibility that some people may have deficits related to certain parts of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Item 2:&lt;/strong&gt; dyscalculia is specifically a deficit in learning arithmetic (Butterworth, Varma, &amp;amp; Laurillard, 2011).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Item 3:&lt;/strong&gt; there is mathematical reasoning apart from arithmetic reasoning, such as quantitative reasoning described by Smith and Thompson (2008). Such reasoning can be used in the development of algebraic reasoning, apart from arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Answer pt. 1:&lt;/em&gt;&lt;/strong&gt; I do not know of specific studies you ask for, but with the items above, we can conjecture that it is possible for someone with deficits in arithmetical reasoning to develop mathematical thinking that relies heavily on quantity rather than arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Answer pt. 2:&lt;/em&gt;&lt;/strong&gt; I would suggest it is not wise to dissuade students against a mathematics if such deficits are real, causing students to rely on other skills. The reason being: a motivated student may well find new ways to think about mathematics, relying on different skillsets. That person may well discover ways of thinking about mathematics that contribute not only to mathematics, but to mathematics teaching and learning. We can't know for certain, but diversity in thinking has a practical value, apart from the idea of social justice. At least, there is an argument there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Works Cited:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.lonklab.ac.uk/cms/files/jce/author/science-2011-butterworth-1049-53.pdf&quot; rel=&quot;noreferrer&quot;&gt;Butterworth, B., Varma, S., &amp;amp; Laurillard, D. (2011). Dyscalculia: from brain to education. Science, 332(6033), 1049–1053.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.pat-thompson.net/PDFversions/2006SmithThompsonEarlyAlg.pdf&quot; rel=&quot;noreferrer&quot;&gt;Smith III, J. P., &amp;amp; Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. W. Carraher, &amp;amp; M. L. Blanton (Eds.), Algebra in the early grades (pp. 95–131). New York, NY: Taylor &amp;amp; Francis Group.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-04-15T21:47:44.970" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1682" PostTypeId="2" ParentId="1671" CreationDate="2014-04-15T21:56:49.067" Score="7" Body="&lt;p&gt;Here's something I've done in 2nd semester calculus classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the blackboard draw a big schematic graph of $y = \frac{1}{x}$ from $x=1$ to $x=n,$ with $1,\,2,\,\ldots,\,n-1,\,n$ labeled on the $x$-axis. Looking at the right and left hand Riemann sums with $\Delta x = 1$ gives&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n-1} + \frac{1}{n} \;\; &amp;lt; \;\; \int_{1}^{n} \frac{1}{x}\,dx \;\; &amp;lt; \;\; 1 +\frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n-1}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Letting $S(n)\; = \; 1 +\frac{1}{2} + \cdots  + \frac{1}{n}$ be the sum of the first $n$ terms of the harmonic series, we get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$S(n) - 1 \; &amp;lt; \; \ln{n} \; &amp;lt; \; S(n) - \frac{1}{n}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The left inequality gives $S(n) &amp;lt; \ln{n} + 1.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The right inequality gives $\ln{n} + \frac{1}{n} &amp;lt; S(n).$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, we get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\ln{n} + \frac{1}{n} \; &amp;lt; \; S(n) \; &amp;lt; \; \ln{n} + 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, we see that $0&amp;lt; S(n) - \ln{n} &amp;lt; 1.$ This allows us to obtain extremely accurate estimates (relatively speaking) to the sum of the first $10^{100}$ terms of the harmonic series or the first $10^{1000}$ terms of the harmonic series. It also provides a jumping off point to talking about whether this difference has a limit when $n \rightarrow \infty,$ and what that limit might be.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-04-15T21:56:49.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1683" PostTypeId="2" ParentId="1649" CreationDate="2014-04-15T23:41:26.547" Score="17" Body="&lt;p&gt;I never divide or multiply by 5. I'll double and divide by 2 or divide by 10 and double. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a similar manner, I was in China last summer and found a quicker way to calculate the temperature in Fahrenheit when given the Celsius temperature: (1) Double the temperature in Celsius, (2) subtract off 10% (something that is easy to calculate), (3) add 32.&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="267" LastEditDate="2014-04-15T23:47:30.433" LastActivityDate="2014-04-15T23:47:30.433" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1684" PostTypeId="2" ParentId="1598" CreationDate="2014-04-15T23:53:07.663" Score="10" Body="&lt;p&gt;Getting students engaged in the act of catching and repairing mistakes in proofs and other solutions to problems can have a great deal of educational value. Lara Alcock has written a lot about &quot;proof validation&quot; and its role in undergraduate math education, &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0732312305000143&quot;&gt;for example in this paper&lt;/a&gt;, and there's quite a few others who also have done research on this. The tl;dr version is that activities that ask students to decide &lt;em&gt;whether&lt;/em&gt; a proof is valid, and then find and repair the error if there is one, significantly improve students' abilities to then go and construct correct proofs on their own. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However: There is a huge difference between setting up intentional, well-designed proof validation activities one the one hand, and doing what this apocryphal professor did on the other hand, which is to come into class and start a proof cold without any idea where s/he is going. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For one thing, in the urban legend case the professor is still the center of attention, not the student; the professor surely can contribute to helping students understand the perspective of an expert problem-solver. But it's probably best to wait until &lt;em&gt;after&lt;/em&gt; the students have had a chance to do it on their own, so that students will have a basis for paying attention to how the professor does it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For another thing, it just seems like carelessness to go into a class unprepared. Basically you should never do that, no matter how good of a mathematician you are, out of professionalism. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, it can backfire. I've seen situations where professors try this sort of thing and then get hopelessly lost, and they lose the students' trust. And once you've lost that, it is very hard to gain it back, and learning is hard when you don't trust the prof. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So -- rather than take a freewheeling approach, design good proof activities that target specific misconceptions or decision-making processes you'd like to bring out in class, then hand those to students and let them evaluate the proofs firsthand. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-04-15T23:53:07.663" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1685" PostTypeId="2" ParentId="1608" CreationDate="2014-04-16T00:16:52.690" Score="3" Body="&lt;p&gt;The best way is to surround yourself with people who are creative, smart, and actively thinking about and using effective pedagogy in the classroom. Twitter can be great for this. I'm fortunate to be in a department full of such people, such as &lt;a href=&quot;http://deltascape.blogspot.com/&quot; rel=&quot;nofollow&quot;&gt;David Coffey&lt;/a&gt;, &lt;a href=&quot;http://mathhombre.blogspot.com/&quot; rel=&quot;nofollow&quot;&gt;John Golden&lt;/a&gt;, and others. I can just walk down the hall and embroil myself in a fascinating conversation about pedagogy any time I want. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to writing, I find Lara Alcock's research to be fantastic and fascinating and inspiring. Also the work of my former boss and mentor Linda Nilson, who basically taught me how to teach and whose book &lt;em&gt;Teaching at its Best&lt;/em&gt; was my very first introduction to the scholarship of teaching and learning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think most of all, though, my inspiration comes from working with my students, who are always pushing me out of my comfort zone and inspiring me to get better at my craft and improve on my ideas. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-04-16T00:16:52.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1686" PostTypeId="2" ParentId="1538" CreationDate="2014-04-16T01:17:14.413" Score="6" Body="&lt;p&gt;It is unfortunate that the students are taught that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a continuous function is one for which you can draw the graph with a single stroke of the pencil.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course, this is not consistent with the general definition of continuous as it intersects first semester calculus. We say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a function is continuous if it is continuous on its domain. Moreover, this means $f(x) \rightarrow f(a)$ as $x \rightarrow a^{\pm}$. We require only left or right limits as appropriate to boundary points of the domain. If the point is isolated then the function is continuous at such a point by definition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As the OP points out $f(x)=1/x$ is continuous, but &lt;strong&gt;students are taught this is discontinuous in some algebra or precalculus courses&lt;/strong&gt;. Of course, you do need to lift your pencil to graph $y=1/x$. So, how can we reconcile this? Frankly, we can't. The previous course was in error. That said, I try to smooth things out by emphasizing that when we restrict the focus to &lt;strong&gt;connected domains&lt;/strong&gt; aka &lt;strong&gt;intervals&lt;/strong&gt; then their intuition is restored. It is true that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the graph of a continuous function with a connected domain is drawn without lifting your pencil.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The old way (precalculus) of thinking is that by default a function has domain $\mathbb{R}$ then as we discover defects in the formula $f(x)$ points are removed due to various types of discontinuities; removable, jump, vertical asymptotes are the usual suspects. This is intuitive, but, this is not how we carefully work with functions $f:A \rightarrow B$. The points where $f$ is not defined are not taken away, there not even there at the outset. I face this issue with both eyes open and don't shy away from it in calculus because I want students to pay more mind to the idea that a function is not just a formula. The domain and codomain must be specified to be careful. Our theorems hold for sets of numbers, the type of set matters. Connected is key.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-04-16T01:17:14.413" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1687" PostTypeId="2" ParentId="1538" CreationDate="2014-04-16T02:02:56.707" Score="3" Body="&lt;p&gt;I'll attempt to answer your question in reverse order. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would I do in a similar situation? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The choices for what we now consider standard definitions were refined countless times before the a single definition was accepted. It is easy to forget that and it is partly our job as educators to explain this from time to time to the students (least they start thinking mathematics is a static discipline). So, such situations were students show a distrust in a particular definition that we know is a really good one is an excellent opportunity to engage the students with the dynamic nature of mathematics. I usually encourage the students then to scrutinize the definition, making their objections explicit. Depending on how much time I wish to spend on such things I then decide whether or not to look into their objections more carefully or not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do I handle this situation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The two particular difficulties you mention I would treat as follows. Difficulties on whether or not $f(x)=\frac{1}{x}$ is continuous are the result of being careless with specifying the domain. This is thus an excellent opportunity to explain that the question &quot;is $f(x)=\frac{1}{x}$ is continuous&quot; is meaningless until one specifies the domain. If there are some books that fail to do this properly, then it's a good opportunity to warn the students of lousy books. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second situation, that of vacuous situations, can indeed be quite confusing but I would not spend too much time on it since such things rarely show up in nature. Moreover, once students are a bit more proficient with mathematical arguments they will easy dispense with such trivialities themselves, so the problem solves itself. If students are really troubled by this I would tell them to spend 15 minutes thinking about it and resolve it themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can I do better?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, if one can no longer improve, then one is dead, so yes, you can also do better :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The particular notion of continuity is actually a very tricky one, at least historically. Our intuition for what continuity means is quite bad. It is common to say that a function is continuous if its graph can be drawn without lifting the pen from the paper. This is wrong of course, as it more accurately describes functions that are infinitely differentiable at all but finitely many points. It is actually quite difficult to geometrically understand continuous functions. This is something the students should be able to relate to - after all it is difficult for them to understand this concept. Also, historically, mathematicians struggled long and hard with this concepts with many disagreements between Lagrange, Cauchy, Bolazano, and Weierstrass. Some of these historical accounts can also be conveyed to the students. It can also be mentioned that the standard Cauchy definition is not the only way to formalize the notion of continuity, just to tell the students that more than one road exists.&lt;/p&gt;&#xA;" OwnerUserId="59" LastEditorUserId="57" LastEditDate="2017-09-01T04:43:47.080" LastActivityDate="2017-09-01T04:43:47.080" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1688" PostTypeId="2" ParentId="1680" CreationDate="2014-04-16T02:32:09.410" Score="13" Body="&lt;p&gt;By multivariate analysis, I'm assuming that you're talking about a course that covers the following topics in a rigorous way:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Differentiation of functions between vector spaces&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The inverse and implicit function theorems, and related topics (e.g. rank of the derivative, immersions and submersions).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Critical points and regular points of multivariable functions&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Multiple integrals and some version of Fubini's Theorem&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Differential forms and Stokes' Theorem&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Possibly an introduction to differentiable manifolds&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My opinion is that such a course isn't an &lt;em&gt;essential&lt;/em&gt; prerequisite for any later courses, for the simple reason that many (or perhaps most) colleges do not offer a course on multivariate analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, I think such a course would be very helpful for students who are planning to go into geometry, topology, or related subjects.  A typical graduate course on differentiable manifolds (out of, say, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1441999817&quot;&gt;Lee's &lt;em&gt;Introduction to Smooth Manifolds&lt;/em&gt;&lt;/a&gt;) covers some of this material very quickly near the beginning, and covers differential forms quickly in the middle and in the more general setting of manifolds.  As with many graduate courses, such a course is theoretically self-contained, but it helps a lot to have seen some of the material before.  Other graduate courses that use this material might include courses on Riemannian Geometry, Differential Topology, Algebraic Topology (out of, say, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387906134&quot;&gt;Bott and Tu&lt;/a&gt;), Lie groups, or even courses on Riemann Surfaces or Algebraic Geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An undergraduate differential geometry course might also assume some of this material, but probably at most on the level of multivariable calculus.  For example, it might assume that students have seen the inverse function theorem, but it wouldn't assume that students have seen the proof.  This is probably true of almost any undergraduate course.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-04-16T02:32:09.410" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1689" PostTypeId="1" AcceptedAnswerId="1732" CreationDate="2014-04-16T02:34:09.960" Score="21" ViewCount="885" Body="&lt;p&gt;I was tutoring a student today and we were doing basic factoring of quadratics and expanding terms like &lt;span class=&quot;math-container&quot;&gt;$(x+2)(x+5)$&lt;/span&gt;. Now he ended up being able to do this by the end of our 2 and a half hour session, but it was clear that he didn't understand why he should be learning this and didn't see much of a point to it. Now I could try and tell him that factoring is useful later on for simplifying things, partial fractions for integration, etc, but I don't think this is useful as these students don't know about integration and its uses. Is there any application which a high-schooler could understand that would help motivate learning this material (or other applications which just require high school algebra)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only thing I could think of was shooting a projectile off a cliff horizontally and trying to figure out how far it goes. &lt;/p&gt;&#xA;" OwnerUserId="530" LastEditorUserId="8571" LastEditDate="2018-10-30T15:16:53.243" LastActivityDate="2018-10-30T15:16:53.243" Title="How to help motivate math when tutoring low level algebra (High school)" Tags="|secondary-education|algebra|student-motivation|tutoring|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1690" PostTypeId="1" CreationDate="2014-04-16T05:21:35.620" Score="13" ViewCount="2893" Body="&lt;p&gt;I like to ask true-false questions on exams, because I feel that they can be an efficient way to assess students' understanding of concepts and ability to apply them to somewhat unfamiliar situations.  In general, I'm very happy with true-false questions, but there is one annoyance that I have never figured out how to deal with correctly.  Namely, should students be asked to justify or explain their answers to true-false questions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some reasons to ask for explanations:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It may inspire them to think harder.&lt;/li&gt;&#xA;&lt;li&gt;It makes it possible to give partial credit.&lt;/li&gt;&#xA;&lt;li&gt;It reduces the incentive to guess randomly.&lt;/li&gt;&#xA;&lt;li&gt;It's easy to give the right answer for a wrong reason, and if I don't ask for an explanation I may never notice.  (I find this reason the most compelling.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Some reasons not to ask for explanations:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Writing an explanation takes a lot more time than just writing &quot;true&quot; or &quot;false&quot;, partially negating the efficiency of true-false questions versus &quot;free-response&quot; questions.  In particular, a student who has no clue may spend a long time writing rubbish in hopes of partial credit.&lt;/li&gt;&#xA;&lt;li&gt;For some otherwise nice true-false questions, it's not clear what sort of &quot;explanation&quot; could be given.  Sometimes I find that one of &quot;true&quot; and &quot;false&quot; (often the correct one) has a much easier explanation than the other.  It's also hard to communicate to the students what level of detail and precision is expected, which makes it hard to grade fairly.&lt;/li&gt;&#xA;&lt;li&gt;Often students will give an explanation that's &lt;em&gt;basically&lt;/em&gt; correct, but also contains inaccuracies or false statements.  If I let it slide, then they won't learn to be careful with language; but it's hard to take off something while still giving credit for basic correctness when I usually assign only a couple of points per true-false question.  As with #1, this pushes the true-false question towards more of a free-response question.&lt;/li&gt;&#xA;&lt;li&gt;Not a really &lt;em&gt;good&lt;/em&gt; reason, but for completeness, I'll include it: it takes longer to grade.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Is there a good solution?  Do you ask your students to give explanations for true-false questions?  How do you deal with the issues raised above?&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2014-09-02T23:18:15.287" Title="Should true-false questions ask for an explanation?" Tags="|test-design|" AnswerCount="8" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1691" PostTypeId="1" CreationDate="2014-04-16T06:45:37.450" Score="17" ViewCount="1161" Body="&lt;p&gt;I'm teaching an Intro to PDEs course (I'm an analyst, but PDEs are a bit outside my bailiwick) and I'm covering the basic examples: Heat, Wave, and Laplace.  How should I move from these examples to motivate the definitions of hyperbolic, elliptic and parabolic?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see the connection between the PDEs and the equations of conic sections, but why is that important?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am under the impression that one of the big differences between the wave equation and the heat equation is that solutions to the wave equation are periodic in time (somehow conserving 'energy' or 'information' perhaps?) and the solutions to the heat equation diffuse and tend to a steady state (somehow losing energy or information).  Is something like that generally true about hyperbolic and parabolic PDEs?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there other important motivations/explanations I should be aware of?&lt;/p&gt;&#xA;" OwnerUserId="223" LastEditorUserId="61" LastEditDate="2014-04-16T20:27:14.653" LastActivityDate="2018-09-25T08:37:45.493" Title="What is the motivation for characterizing second order linear PDEs as hyperbolic, elliptic, or parabolic?" Tags="|mathematical-analysis|concept-motivation|differential-equations|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1692" PostTypeId="2" ParentId="1690" CreationDate="2014-04-16T08:11:26.270" Score="5" Body="&lt;p&gt;First of all, it depends on the situation. I use true-false questions mainly because of your point 4, as a complement to a standard exam. With 150-200 mathematics major students, it is common practice at our university to complement &quot;real&quot; exam questions with tests, and there true-false questions are great for the reasons you mention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In out practice, there is a part of the exam consisting only of true-false questions, and those who reach a required point (like 75%) can take part at the second part of the exam, which may be oral or written depending on the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this setup it is important that your true-false questions are straightforward, do not need deep thinking but the understanding of concepts, definitions or theorems. Then you also do not really need deep explanations. I think of statements like (from an calculus course designed for maths majors):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If a function is continuous on $(a,b)$, then it is differentiable.&lt;/li&gt;&#xA;&lt;li&gt;If a function is differentiable on $(a,b)$, then it is continuous.&lt;/li&gt;&#xA;&lt;li&gt;$\int f^2(x)dx = \frac{f^3}{3}+C.$&lt;/li&gt;&#xA;&lt;li&gt;If $f(x)=\sqrt{5-x}$  with $x\leq 5$ and $g(x)=5-x^2$, then $(f \circ g)(x)=x.$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="61" LastEditorUserId="61" LastEditDate="2014-04-16T10:31:40.300" LastActivityDate="2014-04-16T10:31:40.300" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1693" PostTypeId="2" ParentId="1690" CreationDate="2014-04-16T08:11:43.167" Score="7" Body="&lt;p&gt;Your question is in fact three questions.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there a good solution?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do you ask your students to give explanations for true-false questions?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How do you deal with the issues raised above?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My answer for 1 is that I would &lt;strong&gt;not&lt;/strong&gt; include pure true or false questions.&lt;/p&gt;&#xA;&lt;p&gt;My answer for 2 is &lt;strong&gt;yes&lt;/strong&gt;, I would ask for explanations; however, with regard to pure T/F questions, I think some are better than others, and will include some of my own commentary below.&lt;/p&gt;&#xA;&lt;p&gt;My answer for 3 is twofold. &lt;strong&gt;First&lt;/strong&gt;, I might ask questions that don't require justification (see below) but not where the choices are T/F. &lt;strong&gt;Second&lt;/strong&gt;, I think a better general approach is to hand out a review sheet with many statements that are either true or false, and to include an &amp;quot;answer key&amp;quot; with the truth value for every (or every other) statement.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;Here are a couple of examples of true or false questions and how I would either re-phrase them into a better T/F question (and why I think it is better) or how I would re-phrase it into a question that similarly does not require a full proof or counterexample.&lt;/p&gt;&#xA;&lt;p&gt;To start with, here is a question that I think is a bad example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;True or false:&lt;/strong&gt; Every cyclic group with non-composite order is isomorphic to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}/p\mathbb{Z}$&lt;/span&gt; for some prime &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If instead you made this a &amp;quot;prove or disprove&amp;quot; question, then - it would still be bad - but at least students who tried to prove the answer is &lt;strong&gt;true&lt;/strong&gt; could earn themselves a fair bit of partial credit. The issue here is that the true or false answer is more of a subtle trick, namely, noting that a trivial group of order 1 is cylic, but that 1 is not prime; so the correct answer is &lt;strong&gt;false&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is, I think, a pretty good example of a true or false question:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;True or false:&lt;/strong&gt; Every nontrivial cyclic group of prime order is isomorphic to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}/p\mathbb{Z}$&lt;/span&gt; for some prime &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Again, this could make for a nice iff proof; however, my sense is that this is a fact that students should know cold, and, while the proof can be made pretty succinct, writing out all the details during a timed examination seems to me like more trouble than it is worth.&lt;/p&gt;&#xA;&lt;p&gt;Here is another bad example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;True or false:&lt;/strong&gt; For every finite group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, if every subgroup is normal, then &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is abelian.&lt;/p&gt;&#xA;&lt;p&gt;I think this is bad because it &lt;em&gt;feels&lt;/em&gt; like it is probably false, but actually dreaming up the counterexample is not so straightforward unless it has been given some thought beforehand.&lt;/p&gt;&#xA;&lt;p&gt;A better question might be:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A.&lt;/strong&gt; Classify (without justification) all groups up to isomorphism for orders 1 through 7.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;B.&lt;/strong&gt; Give an example (without justification) of an 8 element nonabelian group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; with every subgroup normal.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;C.&lt;/strong&gt; Identify two nontrivial and non-isomorphic proper subgroups of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;; prove they are not isomorphic, and then prove that each is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In this case, parts &lt;strong&gt;A&lt;/strong&gt; and &lt;strong&gt;B&lt;/strong&gt; are both quick and not too tough to guess, though writing out the details (e.g., of why there are only two groups of order 6) is a bit of a hassle. Part &lt;strong&gt;C&lt;/strong&gt; also allows students to get deeper into the problem in a way that discourages guessing for &lt;strong&gt;B&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Lastly, if you are concerned about students guessing, you could always start the test with, say, 5 true or false questions: make them +2 if correct, -3 if incorrect, and 0 if left blank. (This is a general suggestion: Adjust the actual numbers as you see fit.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-18T08:52:23.100" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1694" PostTypeId="2" ParentId="530" CreationDate="2014-04-16T08:23:29.083" Score="19" Body="&lt;h1&gt;$\tau$ should be taught in schools&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;There's plenty of material arguing why $\tau$ is a much more intuitive and easier to teach concept (some of my favorites: &lt;a href=&quot;http://www.maa.org/sites/default/files/pdf/Mathhorizons/apr12_aftermath.pdf&quot;&gt;1&lt;/a&gt;,&lt;a href=&quot;https://www.youtube.com/watch?v=jG7vhMMXagQ&quot;&gt;2&lt;/a&gt;,&lt;a href=&quot;http://tauday.com/tau-manifesto&quot;&gt;3&lt;/a&gt;) and I don't want to rehash their arguments, but if you think that $\tau$ is just to make equations look nicer, please check out those resources.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question at hand is whether math educators should teach $\tau$.  &lt;strong&gt;I will assume that $\tau$ is a more intuitive and didactically nicer concept than $\pi$ for the sake of argument - if you disagree, you can stop reading here.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Many arguments against $\tau$ are appeals to tradition&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;There seems to be a lot of: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\pi$ is what we've always used, it is what we currently use, so we should continue using it.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In fact, many of the arguments are generic enough to be able to apply to any proposed change.  For example, imagine when the first few countries were switching to the metric system, I'm sure exactly the same arguments were used.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is part of EuYu's answer, but with $\tau$ and $\pi$ replaced with &quot;the metric system of units&quot; and &quot;the British system&quot;, respectively:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I feel that it is perhaps a little irresponsible to teach the metric system of units instead of British units. As a first introduction, it is the norm which should be taught: teaching a rare alternative to the British units only serves to confuse students, especially when almost all available resources use British units instead of metric units. Imagine a student's confusion when they see meters in class and feet everywhere else.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;h1&gt;Thinking long-term&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Some of the arguments for the status-quo have seemed a bit short-sighted. Many changes have a switching cost, but pay perpetual benefits after the switch is over, as in switching to metric units.  I think we should be asking ourselves: &quot;Do we really want to be using a didactally and intuively inferior circle constant for the next 500 years?&quot; instead of &quot;Do we want to have to rewrite our textbooks?&quot;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Yes, textbooks would need to be rewritten, but textbooks get updated all the time with new, modern notation.  This is a good thing.&lt;/li&gt;&#xA;&lt;li&gt;Yes, &quot;the students would not be able to read older literature&quot; as Markus Klein points out in his answer.  True, but when I read a very old journal paper, I already have a hard time.  Again, it's because notation changes, usually for the better.  I'm glad that people are no longer using the same exact math/physics notation that was used 100 years ago.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h1&gt;Educators can make this change happen&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Perhaps more people would agree to switch to $\tau$ if everyone were to do it at once - say, starting in 2015, all professors, all publishers, all researchers, etc. would switch to $\tau$ in one fell swoop.  I'm down with that, but that's not going to happen if I've understood human nature correctly.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has to start somewhere and educators are the gatekeepers of knowledge - educators teach the youngsters and write the textbooks.  If the change is going to happen, it will happen thanks to them.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;My Personal experience of teaching with $\tau$&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;I'm the TA for a Math Methods for Physics course at an American univeristy, and I use $\tau$ in my discussions.  I wasn't sure how the students would react, but being nimble-minded, young students, they caught on pretty quickly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also felt it wasn't a huge burden on any students who were opposed to $\tau$ - I reminded them that they could always just write down &quot;$2\pi$&quot; in their notes every time they see me write &quot;$\tau$&quot;.  And I reminded them that they were allowed to ask questions using $\pi$ if they preferred (although most of them just asked questions using $\tau$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why don't you give $\tau$ a try in your class?  See if your students like it.  You may find yourself pleasantly surprised with how easily your students become fluent in $\tau$.&lt;/p&gt;&#xA;" OwnerUserId="933" LastEditorUserId="933" LastEditDate="2014-04-16T08:45:34.853" LastActivityDate="2014-04-16T08:45:34.853" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1695" PostTypeId="2" ParentId="1639" CreationDate="2014-04-16T08:50:28.153" Score="7" Body="&lt;p&gt;I have TAed several content courses for &lt;strong&gt;graduate students in mathematics education&lt;/strong&gt;. In this respect, my interactions with students may differ from the scenarios faced by others who have served as Teaching Assistants, e.g., for undergraduate mathematics students/majors taking Calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The courses I have TAed: Abstract Algebra, Problem Posing, Problem Solving, Proof Writing, Real Analysis, Set Theory, Topology; I've also TAed a separate Proof Writing course for undergraduate mathematics education majors, and an Intro to Calculus course for undergraduate [mostly] math majors.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the techniques I have used include:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Begin the semester by distributing an information sheet about office hours; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Set up a discussion board (through Moodle) so that students can ask questions and discuss them with one another;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Type up notes corresponding to what is covered in office hours - many students in education are also full-time teachers, so it can be especially tough for them to make the hours;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Encourage students to email me with questions - as long as they include their own thinking along with whatever they are asking - and tell them that if they organize study sessions somewhere on campus, then they should feel free to let me know (but that I cannot guarantee I will be available to drop by).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;With regard to 1, I will paste below a template of the letter that goes out to every student on the first and/or second class meeting. With regard to 2, I usually give students a chance to answer each other's questions first; if a question has stagnated for more than a day or two, then I will address it. With regard to 3, I will link to two examples from an Abstract Algebra course I TAed: generally, I go through the assigned problems one by one, but often I will get them to modify the questions (e.g., what if we replace &quot;cyclic&quot; with &quot;abelian&quot;?) or to provide alternative proofs; sometimes I will turns comments into theorems. Finally, with regard to 4, I have noticed that asking a student to include his or her reasoning along with the questions can vastly decrease the volume of emails, and give both the student &lt;em&gt;and&lt;/em&gt; you an insight into where the difficulty lies.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Example CA (course assistant) notes: &lt;a href=&quot;https://www.dropbox.com/s/6a1dyctk07ug34q/10.03%20Topics%20in%20Algebra%20CA.pdf&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Intro to Group Theory&lt;/strong&gt;&lt;/a&gt; and &lt;a href=&quot;https://www.dropbox.com/s/eixw3ihc986bmtk/12.05%20Topics%20in%20Algebra%20CA.pdf&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Intro to Ring Theory&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Template.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To the esteemed students of [course name] -- &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will be available as a TA for your class starting [date/time] at [location].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some things I am happy to do:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-address lingering confusion from material covered in class&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-discuss problems from the book: old, new, complex, basic&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-review past exams, help prepare for future exams, create sample exam questions&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-brainstorm about incorporating topics from [course name] into a class you teach or might teach&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-play Scrabble, if you bring a board and no one else is around&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one thing I will not do:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-discuss anything you haven’t already given some thought to&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My email address is [ ].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are planning on coming in, it would be best to email me first with what you want to talk about. If you haven’t emailed me but get a sudden urge to drop by, that’s okay too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Official Hours/Location: [ ]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If no one has appeared within 15 minutes or emailed me to say they’ll come in later, then I will leave.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If other commitments preclude you from coming in at this time, then please email me to reschedule.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2015-10-02T21:06:59.663" LastActivityDate="2015-10-02T21:06:59.663" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1696" PostTypeId="1" AcceptedAnswerId="1697" CreationDate="2014-04-16T13:02:47.940" Score="13" ViewCount="1027" Body="&lt;p&gt;I took functional analysis from Conway's book. I thought it was just as abstract and dry as homological algebra, if not more. I knew of no applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I learned that standing waves on a drum were related to functional analysis, and then that quantum mechanics (as taught in Dirac's classic book) is nothing but functional analysis, with the structure of the atom and the nature of particles all tied up with functional analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd love to see a textbook for functional analysis that proceeds through the theory while giving numerous real-life applications. As of now, Dirac's book on quantum mechanics is the best example I can think of, giving a lot of background theory.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there an application-heavy functional analysis textbook?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-04-17T20:20:21.340" Title="An application-heavy functional analysis textbook?" Tags="|textbooks|functional-analysis|applications|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1697" PostTypeId="2" ParentId="1696" CreationDate="2014-04-16T13:25:41.240" Score="9" Body="&lt;p&gt;There is &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0125850506&quot;&gt;Methods of Modern Mathematical Physics by Reed and Simon&lt;/a&gt;, which is a 4-volume book which teaches functional analysis, with a focus on operators in Hilbert spaces. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Its main aim is to provide a sound mathematical background for the methods used in quantum mechanics, but it serves well as a textbook on functional analyis (for example, it covers some measure theory, the dual of banach spaces with the Hahn-Banach theorem, locally convex spaces). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The theorems are usually rather brief and might require expansion on the lecture. For instance, I remember a lecture (from volume 4) where we did a proof for 2x90 minutes - the book's version of the proof was only half a page, but iirc, this is a rather extreme example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To sum up, Reed/Simon is application-centered, but not strictly application-heavy, and it's 'only' application is quantum mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another book worth looking into might be &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0821846604&quot;&gt;Mathematical Methods in Quantum Mechanics by Gerald Teschl&lt;/a&gt; - it's influenced by Reed/Simon as well as the German books of Weidmann, but a bit more recent. &lt;a href=&quot;http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/index.html&quot;&gt;On his webpage&lt;/a&gt; G. Teschl provides a link to the pdf version of the book, if you want to have a peek at it. Like Reed/Simon, it is a proper functional analysis textbook, aimed at applications in theoretical physics.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-04-16T13:25:41.240" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1698" PostTypeId="2" ParentId="1538" CreationDate="2014-04-16T14:14:51.753" Score="2" Body="&lt;p&gt;When I first learned about $p$-adic metric which leads  to a situation where every triangle is isosceles, two open balls  can intersect only if one of them is contained in the other, I was shocked. I was told metric spaces are not easy it can have such examples. Later I realised that the definition of a distance function the  three requirements for a function to be called a metric  are not restrictive enough to exclude such counter-intuitive $p$-adic metrics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now it is  a matter of convention whether Archimedean condition is to be imposed for all metrics. &#xA;If one can develop  a theory without that extra assumption and even if the counter-intuitive $p$-adic  spaces other wise satisfy many other properties we can live with that.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same argument can be given for the requirements on a topology on  a space: should Hausdorff separation condition  be included in the definition? If the theory is rich enough to provide lot of examples where many properties can be proved then there is no need to bring in that condition. Let a few counter-intuitive cases co-exist peacefully.&lt;/p&gt;&#xA;" OwnerUserId="548" LastActivityDate="2014-04-16T14:14:51.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1699" PostTypeId="2" ParentId="1696" CreationDate="2014-04-16T14:20:51.513" Score="6" Body="&lt;p&gt;Peter Lax: &lt;a href=&quot;http://amzn.com/0471556041&quot;&gt;Functional Analysis&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is sometimes difficult to use because of its cryptic style, but it is a great source of applications and a great source of historical references on applications which motivated functional analytic concepts.&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-04-16T14:20:51.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1700" PostTypeId="1" CreationDate="2014-04-16T14:21:51.290" Score="20" ViewCount="7371" Body="&lt;p&gt;There are a lot of ways of handling late homework submissions, of which I've only tried a few. The general policy I've settled on is something like the following.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Homework must be submitted at the beginning of class on the day it's due.&lt;/li&gt;&#xA;&lt;li&gt;Late homework is not accepted.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This is pretty harsh, I think. But with this policy in place, I make unofficial exceptions on a case-by-case basis (it's easier to start with an overly harsh policy and make exceptions in the students' favour than the other way around), for example, if a student happens to be late to class on a day that homework is due.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want students to be motivated to hand in their work on time. &lt;em&gt;Highly&lt;/em&gt; motivated. That's why I make the official policy so strict. But I feel bad about actually adhering to such draconian measures (they should get zero for missing the bus?), which is why I make exceptions. But overall, I'm dissatisfied with the ad hoc nature of this approach. &lt;strong&gt;How can I motivate students to submit homework by the deadline without being &quot;overly harsh&quot;?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="327" LastActivityDate="2016-01-27T23:19:17.240" Title="What's a good policy for accepting late homework?" Tags="|student-motivation|homework|" AnswerCount="8" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1701" PostTypeId="2" ParentId="1690" CreationDate="2014-04-16T14:27:45.223" Score="4" Body="&lt;p&gt;Instead of &quot;true/false&quot; it might be better to have mutiple choice questions. Around here the physics people are fond of them, but ask for explanation of the result. AFAIU, only questions with the correct choice are graded. Not my choice at all, but somebody you might consider.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I prefer to give  one question (out of 5 or 6) that has a few &lt;em&gt;very short&lt;/em&gt; subquestions, like asking for a definition or a very simple, one line application of an important result. This way, if they even know what the course is all about, they navigate away from the dreaded 0 grade... In any case, I try to get it across that the correct answer with a wrong derivation (or a derivation that can't be deciphered) is wrong. The aim in my field (Computer Science, programming particularly) is to get &lt;em&gt;understood&lt;/em&gt; (by yourself fixing the program a month later, by whoever gets to win the lottery of having to fix it when you are long gone). Mathematics (really all science and engineering) should strive for this in the very first place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About level of detail requested, I make that clear in the question. Students have access to older exams and solutions, that should give them some guidance. As a general rule, I ask for at most one handwritten page (most of the time less), and they know that. &quot;Last will and testament as an answer gets discounted points for each extra sheet&quot; (not really).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One last comment: When there are longer &quot;true of false, and comment on your answer&quot; questions, I tell them explicitly that often there is no &quot;right&quot; answer, that they get full credit &lt;em&gt;for the discussion&lt;/em&gt;, not the &quot;correct choice.&quot;&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-16T14:27:45.223" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1702" PostTypeId="2" ParentId="1696" CreationDate="2014-04-16T14:34:24.223" Score="9" Body="&lt;p&gt;At the introductory level, Erwin Kreysiz's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471504599&quot;&gt;&lt;strong&gt;Introductory Functional Analysis with Applications&lt;/strong&gt;&lt;/a&gt; is excellent. See the reviews at amazon.com. It's probably a bit elementary for you at this point (still, it could be very suitable for others reading this thread for recommendations), but I recommend at least looking through a copy at the library. I've actually had a course using Kreysiz's book, for what it's worth. And a course using Conway's book, by Conway himself, for that matter.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2014-04-16T14:43:47.563" LastActivityDate="2014-04-16T14:43:47.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1703" PostTypeId="2" ParentId="1700" CreationDate="2014-04-16T14:37:57.650" Score="5" Body="&lt;p&gt;My policy is that homework has to be turned in by the due time, for each day late 20 points (of 100) are discounted, after the third day no more submissions are allowed and the solution is posted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As OP says, there is a bit of leeway for exceptional cases, i.e., if somebody was sick or otherwise unavailable they might be allowed to turn it in late with no penalty, or get a replacement homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, a (somewhat) harsh policy, with fixed dates for homework turn in and exams, dates that are only changed in case of extreme circumstances, is best, it isn't really hard and lightens work by not having it pile up at the end of the term.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-16T14:37:57.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1704" PostTypeId="2" ParentId="1700" CreationDate="2014-04-16T15:21:08.600" Score="3" Body="&lt;p&gt;The comments that follow pertain mainly to classes at or below the level of the beginning calculus sequence, elementary linear algebra, and a first course in differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To minimize the effect of situations where someone couldn't make it to class for some reason or had legitimate questions about one  or more of the problems, over the years I gradually began to take up the homework in class when it was due, but still accept homework delivered to my mailbox or slid under my office door. I established a &lt;em&gt;absolute due time&lt;/em&gt;, which typically was that it should be there before I get to my office and check mail the next morning. I warned them that I sometimes got to work very early (5:00 a.m. or earlier), and sometimes not until after 7:00 a.m., so to not take any chances, they should essentially regard the absolute deadline as the night before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, as much as possible (these being departmental or course supervisor constraints), I tried to avoid actually taking up and grading homework as much as possible. Instead, I found it much simpler to have frequent short (5 to 15 minutes) quizzes, averaging at least one quiz per week for the semester, and sometimes averaging nearly two quizzes, these averages being for classes that meet 2 or 3 times a week. &lt;em&gt;Sometimes&lt;/em&gt; the quizzes were what others call &quot;homework quizzes&quot; (students get to use their homework on the quiz), but I would instead just make it an open notes quiz. I did this because I did not want to go around to every desk and police whether they were really using their homework (and not class notes), especially when a lot of students did their homework in the same spiral bound notebook they took class notes in. And yes, I realize that some students used photocopies of other students' notes and homework, but I didn't police that either. The only thing I did try to police in these situations was to make sure that they didn't pass notes to each other. You have to watch carefully for this, because once a note is passed and received, you have no way of telling (without careful scrutiny of all the papers on a student's desk) whether a certain student may have gotten worked quiz solutions from a nearby student.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-04-16T15:21:08.600" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1705" PostTypeId="2" ParentId="1700" CreationDate="2014-04-16T16:07:09.570" Score="1" Body="&lt;p&gt;I'm with VonBrand, mostly.  My way of dealing with it is to say that if you turn in your homework on time you can resubmit before the end of the semester to raise your grade for half credit.  So if I give out a ten problem set, and a student turns in five problems that are 100% correct that's still 50%, F.  Three weeks later the student can come in with the remaining five problems.  Assuming they're 100% correct, minus 50% late penalty = 75% or C, but still better than an F.  Of course if the student turns in nothing at the due date then the student gets zero. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imparts a sense of understanding the importance of meeting deadlines while allowing for some flexibility.  I know that some students save up for the week of Spring Break to catch up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also flexible on before-the-fact excuses.  Come to me two days before the homework is due and say Fluffy had an accident and needs to go to the vet, fine, I'll grant an appropriate extension.  Come to me on the day the assignment is due or after the assignment is due, sorry, no extension.&lt;/p&gt;&#xA;" OwnerUserId="953" LastActivityDate="2014-04-16T16:07:09.570" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1706" PostTypeId="2" ParentId="1690" CreationDate="2014-04-16T16:21:35.690" Score="7" Body="&lt;p&gt;I rarely asked true/false questions, but when I did I usually made them worth $2$ or $3$ points each (not $1$ point each), and incorrect answers with a somewhat appropriate and at least a somewhat correct explanation could earn one or two points back. In fact, if it was clear from their explanation that they accidentally circled the wrong &quot;T/F&quot; letter (rare, but it did happen), they would get full credit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;(ADDED NEXT DAY)&lt;/em&gt; In reply to your comment about how I would handle the issues you raised, these issues did not really show up for me in the lower level classes where I sometimes asked T/F questions, such as trigonometry (e.g. T/F: $\pi$ does NOT belong to the domain of $\cot{x}).$ In graduate level courses I often asked (on in-class tests) a type of question that is somewhat similar to T/F: &quot;For each of the following statements, give a brief justification or give a counterexample, according as whether the statement is true or false.&quot; Maybe you can use this type instead. To take care of your concerns of students not knowing what is appropriate to write, you can try doing what I did -- give lots of examples of appropriate justifications in advance (write up answers to short quiz problems, solve sample test problems of this type when appropriate during lecture, etc.).&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2014-04-17T16:22:17.430" LastActivityDate="2014-04-17T16:22:17.430" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1707" PostTypeId="2" ParentId="1700" CreationDate="2014-04-16T16:36:07.113" Score="2" Body="&lt;p&gt;Since I'm not exactly timely in grading homework, my policy is: You have until I grade the stack to hand it in. If you do so, I just put it into the middle of the stack and no big deal. But if I'm done grading the stack (or took the stack home before you slid yours under my door), you're out of luck. This means that there's sort of a probabilistic decay in the chances that a late homework will be accepted. Students realize the risk of waiting too long, but appreciate the ability to sometimes hand things in just a tad late.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer: If I'm publishing solutions to be studied from for an exam, then the deadline is firm on being in before solutions are posted.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-04-16T16:36:07.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1708" PostTypeId="5" CreationDate="2014-04-16T17:06:23.440" Score="0" Body="&lt;p&gt;Questions on fractions, numbers of the form $p/q$ where $p$ and $q$ are integers, and $q$ is not zero.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:24:53.370" LastActivityDate="2014-04-16T17:24:53.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1709" PostTypeId="4" CreationDate="2014-04-16T17:06:23.440" Score="0" Body="For questions about the teaching of fractions and their interpretations. Fractions are numbers of the form $p/q$, where $p$ and $q \neq 0$ are integers." OwnerUserId="35" LastEditorUserId="39" LastEditDate="2015-08-23T15:08:36.030" LastActivityDate="2015-08-23T15:08:36.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1710" PostTypeId="5" CreationDate="2014-04-16T17:07:16.477" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-16T17:07:16.477" LastActivityDate="2014-04-16T17:07:16.477" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1711" PostTypeId="4" CreationDate="2014-04-16T17:07:16.477" Score="0" Body="Questions on fractals, irregular-looking mathematical objects that display the property of self-similarity." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:42:24.307" LastActivityDate="2014-04-16T17:42:24.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1712" PostTypeId="5" CreationDate="2014-04-16T17:08:01.230" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-16T17:08:01.230" LastActivityDate="2014-04-16T17:08:01.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1713" PostTypeId="4" CreationDate="2014-04-16T17:08:01.230" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-16T17:08:01.230" LastActivityDate="2014-04-16T17:08:01.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1714" PostTypeId="5" CreationDate="2014-04-16T17:09:21.840" Score="0" Body="&lt;p&gt;For questions about the teaching of &lt;a href=&quot;http://en.wikipedia.org/wiki/Probability&quot; rel=&quot;nofollow&quot;&gt;probability&lt;/a&gt;, dealing with students misconceptions in probability, and explaining probability theory paradoxes.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="39" LastEditDate="2015-01-11T19:21:35.087" LastActivityDate="2015-01-11T19:21:35.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1715" PostTypeId="4" CreationDate="2014-04-16T17:09:21.840" Score="0" Body="For questions about the teaching of probability, dealing with students misconceptions in probability, and explaining probability theory paradoxes." OwnerUserId="35" LastEditorUserId="39" LastEditDate="2015-01-11T19:21:26.630" LastActivityDate="2015-01-11T19:21:26.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1716" PostTypeId="5" CreationDate="2014-04-16T17:11:20.993" Score="0" Body="&lt;p&gt;For questions about set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies, etc.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:25:10.973" LastActivityDate="2014-04-16T17:25:10.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1717" PostTypeId="4" CreationDate="2014-04-16T17:11:20.993" Score="0" Body="For questions about set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies, etc." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:43:01.853" LastActivityDate="2014-04-16T17:43:01.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1718" PostTypeId="5" CreationDate="2014-04-16T17:12:36.643" Score="0" Body="&lt;p&gt;For questions about differential and integral calculus with more than one independent variable.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:42:22.447" LastActivityDate="2014-04-16T17:42:22.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1719" PostTypeId="4" CreationDate="2014-04-16T17:12:36.643" Score="0" Body="For questions about differential and integral calculus with more than one independent variable." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:42:56.630" LastActivityDate="2014-04-16T17:42:56.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1720" PostTypeId="5" CreationDate="2014-04-16T17:14:45.213" Score="0" Body="&lt;p&gt;For questions about exponentiation; a mathematical operation which produces a &lt;em&gt;power&lt;/em&gt; $a^n$ from a &lt;em&gt;base&lt;/em&gt; $a$ and an &lt;em&gt;exponent&lt;/em&gt; $n$. The objects involved are usually numbers, but the procedure can be generalized to matrices, elements in algebraic structures, sets, etc.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:42:11.883" LastActivityDate="2014-04-16T17:42:11.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1721" PostTypeId="4" CreationDate="2014-04-16T17:14:45.213" Score="0" Body="For questions about exponentiation; a mathematical operation which produces a *power* $a^n$ from a *base* $a$ and an *exponent* $n$. The objects involved are usually numbers, but the procedure can be generalized to matrices, elements in algebraic structures, sets, etc." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:43:20.307" LastActivityDate="2014-04-16T17:43:20.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1722" PostTypeId="5" CreationDate="2014-04-16T17:18:27.713" Score="0" Body="&lt;p&gt;For questions about polynomials. Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials… like adding, multiplying, polynomial long division, factoring and solving for roots.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="123" LastEditDate="2014-04-16T18:19:40.163" LastActivityDate="2014-04-16T18:19:40.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1723" PostTypeId="4" CreationDate="2014-04-16T17:18:27.713" Score="0" Body="For questions about solving polynomials. Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials… like adding, multiplying, polynomial long division, factoring and solving for roots." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:42:32.960" LastActivityDate="2014-04-16T17:42:32.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1724" PostTypeId="5" CreationDate="2014-04-16T17:20:40.717" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-16T17:20:40.717" LastActivityDate="2014-04-16T17:20:40.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1725" PostTypeId="4" CreationDate="2014-04-16T17:20:40.717" Score="0" Body="For questions about a feeling of worry, nervousness, or unease about something with an uncertain outcome." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:49:15.523" LastActivityDate="2014-04-16T17:49:15.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1726" PostTypeId="5" CreationDate="2014-04-16T17:21:47.380" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-16T17:21:47.380" LastActivityDate="2014-04-16T17:21:47.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1727" PostTypeId="4" CreationDate="2014-04-16T17:21:47.380" Score="0" Body="For questions about the practice or science of collecting and analysing numerical data in large quantities, for instance for the purpose of inferring proportions in a whole from those in a representative sample." OwnerUserId="35" LastEditorUserId="39" LastEditDate="2016-10-03T18:08:31.873" LastActivityDate="2016-10-03T18:08:31.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1728" PostTypeId="5" CreationDate="2014-04-16T17:24:01.993" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-16T17:24:01.993" LastActivityDate="2014-04-16T17:24:01.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1729" PostTypeId="4" CreationDate="2014-04-16T17:24:01.993" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-04-16T17:24:01.993" LastActivityDate="2014-04-16T17:24:01.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1730" PostTypeId="5" CreationDate="2014-04-16T17:25:53.920" Score="0" Body="&lt;p&gt;For questions related to geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:49:22.177" LastActivityDate="2014-04-16T17:49:22.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1731" PostTypeId="4" CreationDate="2014-04-16T17:25:53.920" Score="0" Body="For questions related to geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-16T17:42:13.057" LastActivityDate="2014-04-16T17:42:13.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1732" PostTypeId="2" ParentId="1689" CreationDate="2014-04-16T17:29:11.707" Score="15" Body="&lt;p&gt;I think at this level, and for a tutor (not a teacher), the main carrot (really more of a stick) is getting good grades, getting into a good college, etc. Most people &lt;strong&gt;really will never use this later&lt;/strong&gt;, and for most others it's mostly a stepping stone to more advanced topics.&lt;/p&gt;&#xA;&lt;p&gt;As for how to present it to someone, try the handshake method (every term in one factor &amp;quot;shakes hands&amp;quot; with every term in the other factor) for how to carry it out, and try the rectangle area method for a justification at their level. By &amp;quot;rectangle area method&amp;quot; I mean, when expanding &lt;span class=&quot;math-container&quot;&gt;$(x+2)(x+5),$&lt;/span&gt; draw a rectangle with one side of length &lt;span class=&quot;math-container&quot;&gt;$x+2$&lt;/span&gt; (the side consists of abutting intervals on the same line, one labeled as length &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and the other labeled as length &lt;span class=&quot;math-container&quot;&gt;$2),$&lt;/span&gt; and an adjacent side of length &lt;span class=&quot;math-container&quot;&gt;$x+5$&lt;/span&gt; (the side consists of abutting intervals on the same line, one labeled as length &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and the other labeled as length &lt;span class=&quot;math-container&quot;&gt;$5).$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For basic trinomial factoring, look up something called the &lt;em&gt;AC Method&lt;/em&gt;. I like this method for weaker students because it makes things a little more systematic for weaker students, and it reinforces the idea of &amp;quot;factoring by grouping&amp;quot;, which students are often weak with.&lt;/p&gt;&#xA;&lt;p&gt;However, if you really want to give applications, maybe psudo-applications like the following will be enough:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 1:&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$(997)(1003)$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$(1000-3)(1000 + 3) = (1000)^{2} - 3^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 2:&lt;/strong&gt; A &lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt; percent increase followed by a &lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt; percent decrease is a net decrease since &lt;span class=&quot;math-container&quot;&gt;$\left(1 + \frac{17}{100}\right)\left(1 - \frac{17}{100}\right)$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$1 - \left(\frac{17}{100}\right)^{2} &amp;lt; 1.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 3:&lt;/strong&gt; To find the prime factorization of &lt;span class=&quot;math-container&quot;&gt;$12^{11} + 12^{12} + 12^{13},$&lt;/span&gt; factor as &lt;span class=&quot;math-container&quot;&gt;$12^{11}\left(1 + 12 + 12^{2}\right),$&lt;/span&gt; which equals &lt;span class=&quot;math-container&quot;&gt;$4^{11}3^{11}(157) = 2^{22} \cdot 3^{11}\cdot 157.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 4:&lt;/strong&gt; Determine the exact value of &lt;span class=&quot;math-container&quot;&gt;$a^2 - b^2$&lt;/span&gt; if&#xA;&lt;span class=&quot;math-container&quot;&gt;$$a = 7.0241132301442003123012230341430201$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$b = 6.0241132301442003123012230341430201$$&lt;/span&gt;&#xA;Calculators are allowed for #4.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="745" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-16T17:29:11.707" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1733" PostTypeId="2" ParentId="530" CreationDate="2014-04-16T17:41:13.553" Score="2" Body="&lt;p&gt;I feel like $\tau$ is very useful, but of course everybody uses $\pi$ in 'the real world'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wouldn't hurt to mention $\tau$ one of two times though, and show to students that if they are having trouble with visualizing the length of circle segments or find $\tau$ convenient for other reasons, they can always use $\tau$ 'internally', as long as they formulate their final answers in $\pi$ (this is a trivial conversion and should not cause confusion to any student).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or you could even allow answers formulated in $\tau$, but I don't know if it's a good idea to let students get used to $\tau$. Maybe I'm a little too sceptical: it may be very easy for students to use both $\tau$ and $\pi$ without getting confused, but I wouldn't risk it.&lt;/p&gt;&#xA;" OwnerUserId="915" LastActivityDate="2014-04-16T17:41:13.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1734" PostTypeId="1" AcceptedAnswerId="1745" CreationDate="2014-04-16T18:04:15.643" Score="47" ViewCount="31153" Body="&lt;p&gt;Many universities are changing up the way that they teach math service courses. 1-3 semesters of calculus and maybe a course in linear algebra are often included in majors (such as computer science) that will not use the majority of the material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, these courses are often justified by saying that they promote creative and mathematical thinking, and I agree that they are good precisely because of that. However, many fields of mathematics outside of calculus have the same effect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to hear from a professional computer programmer's viewpoint what topics could be covered in a yearlong `Mathematics for Computer Programmers' course such that:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;This course would be the only math course (besides remedial courses) that computer science students take.&lt;/li&gt;&#xA;&lt;li&gt;The course needs to introduce them to the same level of mathematical rigor as a typical 2-semester calculus sequence.&lt;/li&gt;&#xA;&lt;li&gt;The course covers the majority of mathematics that computer programmers tend to run into.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Additional requirements can be added if these are too vague.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="263" LastEditDate="2014-04-17T06:16:55.860" LastActivityDate="2014-04-18T16:48:18.637" Title="What should be included in a freshman 'Mathematics for computer programmers' course?" Tags="|undergraduate-education|course-design|curriculum|computer-science|" AnswerCount="12" CommentCount="21" ContentLicense="CC BY-SA 3.0" />
  <row Id="1735" PostTypeId="2" ParentId="1304" CreationDate="2014-04-16T18:23:00.840" Score="1" Body="&lt;p&gt;Tons of good stuff &lt;a href=&quot;http://algebra.mrmeyer.com/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. (&lt;a href=&quot;http://blog.mrmeyer.com/&quot; rel=&quot;nofollow&quot;&gt;Dan Meyer&lt;/a&gt;'s algebra I curriculum materials.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I heartily second Sue VanHattum's recommendation to check out Sam Shah's catalogue of math teacher bloggers' virtual filing cabinets.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="140" LastEditDate="2014-04-16T18:40:58.910" LastActivityDate="2014-04-16T18:40:58.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1736" PostTypeId="2" ParentId="825" CreationDate="2014-04-16T18:25:45.453" Score="2" Body="&lt;p&gt;David Wees already mentioned in but I'd like to highlight &lt;a href=&quot;http://www.illustrativemathematics.org/&quot; rel=&quot;nofollow&quot;&gt;Illustrative Mathematics&lt;/a&gt; - it's designed precisely for this purpose. (It's the brainchild of Bill McCallum, one of the lead authors of the Common Core math standards.)&lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2014-04-16T18:25:45.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1737" PostTypeId="2" ParentId="1700" CreationDate="2014-04-16T18:35:12.323" Score="7" Body="&lt;p&gt;My policy is that late work is not accepted, but I also give students multiple ways to turn in work, most of which revolve around electronic submissions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The vast majority of work that I assign students is done electronically -- computer programs, Geogebra constructions, LaTeX-ed up proofs, etc. and so all they have to do is attach the file to an email and send it. Having electronic submissions cuts down on physical stuff happening, like missing the bus, that would prevent timely submission. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last semester I experimented with having students use Dropbox folders to hold all their work and then sharing the folder with me. As they worked, they kept their works-in-progress in the Dropbox folder. Once they were done, they just added the word FINAL to the file name so I would know what to grade. This way they never &quot;turned in&quot; homework -- all the versions of the work were in the Dropbox folder at all times, and they didn't have to &quot;remember&quot; to turn something in. I've been doing the same thing with SageMath Cloud this semester with my discrete structures students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you use paper submissions, then tell students that if they are prevented from being physically present to submit their work on paper, they can (and should) scan it with their phones and send it to you. Not every student has a smart phone, but most do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;TL;DR -- &quot;Late homework gets a 0&quot; is a perfectly fine policy to have as long as you give students multiple avenues of submitting work. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-04-16T18:35:12.323" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1738" PostTypeId="2" ParentId="1734" CreationDate="2014-04-16T18:57:27.680" Score="5" Body="&lt;p&gt;Disclaimer: My answer is based on me being a computer science major, with an obscene interest in mathematics, to the point that I teach it, both on my own, and as a TA.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During regular programming, one thing I always come across, is the need to do some basic &quot;regression&quot;. In quotation marks because really, it all comes down to, &quot;I have n points, and need to find a [linear/exponential/whatever] function that crosses these n points&quot;. This is particularly something I find useful during development of software with graphical interfaces. Especially games, but really, anything with moving objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of course, if you're doing anything in 3D, Linear Algebra is soooo useful! Learning how to view matrices as linear transformations (functions) with the stuff you normally learn about in algebra and calc. Like the fact that matrices can be injective, surjective, and have domains and ranges. And learning about eigenvalues and -vectors, and how they can be visualized.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, of course, discrete math is super important. Learning how to apply combinatorics and create and improve algorithms with decreased complexity has been a godsend.&lt;/p&gt;&#xA;" OwnerUserId="796" LastActivityDate="2014-04-16T18:57:27.680" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1739" PostTypeId="2" ParentId="1734" CreationDate="2014-04-16T19:34:33.357" Score="7" Body="&lt;p&gt;&quot;Computer programmers&quot; is a bit broad, and I think the answer to this question depends on what exactly your university teaches in its own CS or Software Engineering department but doesn't technically label &quot;math&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing to keep in mind is that Computer Science (and by extension anything in programming that's not engineering best practices, code organization, and such) &lt;em&gt;is&lt;/em&gt; math, in that the theory of computation is in large part a formalization of &lt;em&gt;how&lt;/em&gt; we do math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd say that by far the most important thing that any programmer can be expected to know are under the discrete math banner. Graph theory (esp. trees and graph search), set theory, and formal logic (mostly propositional or first order) are backbones of programming. Going deeper, automata theory is important as well, but not as integral. &lt;strong&gt;However&lt;/strong&gt;, it's likely that these topics are covered in the CS department already. If it's not, they're excellent candidates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing that's important is asympototic analysis, mostly in the form of Big O, Big Omega, and Big Theta. Again, this is usually introduced in a required CS course, but not necessarily. This is important enough to know that it tends to come up on interviews for software engineering positions. The ability to find the asymptotic behavior of recursive functions is also important, and less taught than the ability to do it for iterative processes (which are often trivially converted into polynomials). Obviously you only need to focus on more trivial recursive functions, proving a function's asymptotic behavior can be legitimately difficult and is very expansive, but single-variable recursion that can be solved with the &quot;find the pattern&quot; variety of problem solving is pretty good to teach. Finding the behavior of stuff like $M(1)=1;$ $M(n) = M(\lfloor\frac{1}{2}n\rfloor) + M(\lceil\frac{1}{2}n\rceil) + 2n + c;$ $n \in \mathbb{Z}^{1+};c\in \mathbb{Z}^{0+}$ can be immensely useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you start getting into the nitty gritty: statistics, regression, calculus, probability, and such are all very important, but only to certain subfields or people planning on going deeper into &quot;CS&quot; territory than &quot;programming the front page of a comedy website&quot;. Being able to work with big data tends to require a lot of probability calculus (as does any machine learning). These skills are very in demand right now, but I'm not sure if you can cover enough to give the foundation in a single survey course. I mean, you're just plain not going to get to meaningfully explain gradients to anyone if you're going for reasonable breadth and replacing basic calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Likewise linear algebra, which is important in computer graphics and computational geometry. You can find a good survey of the kind of math used in graphics by looking at any good &lt;a href=&quot;http://www.opengl-tutorial.org/&quot; rel=&quot;nofollow&quot;&gt;Open GL tutorial, here's one for reference&lt;/a&gt;. It's mostly basic 2D and 3D matrix math and a tiny bit of quaternions, but even things like understanding what a basis is, whether a matrix is singular, or how to do an orthographic projection is very useful. Again, though, this may be covered in the university's computer graphics course. Granted if you put this in the course it could significantly lower the load on teaching this math in graphics and allow time for more advanced topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Combinatorics is also useful, if only because being able to count the number of unique password combinations of $n$ characters is useful. Cryptography is also traditionally a mathematical field, and is useful for all sorts of reasons, but may be a &lt;em&gt;taaad&lt;/em&gt; advanced for a low level survey course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, as vonbrand pointed out, whichever combination of these you choose, formal instruction in proofs is very important. It can be done for any or all of these, and the merits of proving correctness is immense, even if they never &quot;formally&quot; do it again. Proofs also tend to introduce the ability to clearly express mathematical thoughts in words, even if those words are still surrounded by notation. That's a skill that isn't always valued highly enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, there's a lot of useful math. The most useful is by far discrete, but the other math can be extremely useful to know depending on what the department already covers, what it wishes it didn't have to cover, and what sort of electives and specializations they offer.&lt;/p&gt;&#xA;" OwnerUserId="521" LastEditorUserId="521" LastEditDate="2014-04-17T22:46:55.087" LastActivityDate="2014-04-17T22:46:55.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1740" PostTypeId="2" ParentId="1696" CreationDate="2014-04-16T20:34:58.593" Score="5" Body="&lt;p&gt;If you know German, then the book by&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://amzn.com/3835100262&quot; rel=&quot;noreferrer&quot;&gt;Harro Heuser&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is the one you are looking for. It contains many many applications and mixes the theory with relevant applied material. An English translation of an earlier edition is &lt;a href=&quot;http://amzn.com/0471100692&quot; rel=&quot;noreferrer&quot;&gt;available&lt;/a&gt; from Wiley.&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-04-16T20:34:58.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1741" PostTypeId="2" ParentId="1680" CreationDate="2014-04-16T20:56:01.423" Score="9" Body="&lt;p&gt;Motivated by Jim Belk's answer, let me mention that a rigorous advanced PDE course builds heavily on the material you mention. In particular, &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;generalizations of integration by parts (Gauss/Green/Stokes..) are&#xA;used for the weak formulation of PDEs.&lt;/li&gt;&#xA;&lt;li&gt;Technical theorems about Sobolev spaces and  traces use elementary facts of basic differential geometry (partition of unity, local coordinates, etc.)&lt;/li&gt;&#xA;&lt;li&gt;Variants of the implicit function theorem are used in many local existence theorems (like the Nash-Moser iteration technique)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-04-16T20:56:01.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1742" PostTypeId="2" ParentId="1734" CreationDate="2014-04-16T21:03:35.367" Score="7" Body="&lt;p&gt;I think the general trend is that computer science initially was a subfield of mathematics, and has grown more and more apart from mathematics, to the point that there are universities that offer computer science programs &lt;em&gt;without any math courses at all&lt;/em&gt;, my home university being one of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note, I think this development is very bad, as programming an sich doesn't seem to be much of a science to me (programming websites and apps is very cool but I just wouldn't classify it as a science). I would argue that the university should not offer such programs. I think that computer science is more of a subfield of math than the other way around, which is why I might give you an overload of topics here ;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The topics that you include should obviously depend on the rest of the curriculum. Of course, it's impossible to cover any of this topics comprehensively, but you should identify the elements that are important in the other courses that are offered and make the students comfortable with the ideas and math (with 'math', I mean the kind of deductions and/or calculations that are commonly done in this field).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This list probably differs from what professional programmers would say. A financial programmer would probably focus on probability, a game programmer on linear algebra (and maybe calculus), a cryptographer on number theory and information theory, and I can go on for a while.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to order the list &lt;em&gt;extremely loosely&lt;/em&gt; in order of $ \frac{\text{usefulness}}{\text{difficulty}} $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;General skills&lt;/strong&gt;&lt;br&gt;&#xA;These are very important and not hard to learn. One of many possible uses is in algorithm analysis, where students are expected to do calculations with fractions, powers and logarithms. To my surprise, many of the computer science students had trouble with this (but of course, not everyone has the same background in math, so maybe you should do an extra section for the people that need extra practice with the basic math skills, just to not bother the students that have a sound knowledge with stuff that is way too easy for them). You can also fit things like proving something with induction in here and maybe more advanced skills such as finding a solution for a recurrent equation (just because they don't fit in anywhere else).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Asymptotics&lt;/strong&gt;&lt;br&gt;&#xA;Big O and it's friends are used in most courses about algorithmics, and limits provide a sound way to reason and provide rigor in calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Graph theory&lt;/strong&gt;&lt;br&gt;&#xA;Graph theory is a very hot topic in computer science (and it will probably stay to be that way for a few decades). There is a large amount of algorithms important for computer science (this becomes very interesting if you combine it with probability: this way you can do stuff like facebooks friend suggestions), but in one course I would just focus on a few elementary algorithms (maybe something with node coloring or min/max cut, tree searches and a little bit on parallel algorithms).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Linear algebra&lt;/strong&gt;&lt;br&gt;&#xA;Extensively used in graphics and just a basic skill that comes in handy when doing anything geometric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Combinatorics&lt;/strong&gt;&lt;br&gt;&#xA;Combinatorics pop up in a lot of places in algorithmics. For one course I would give some example of combinatoric proofs, and teach some basic stuff about binomial coefficients. Generating functions are very useful IMO, but probably way out of scope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Probabililty&lt;/strong&gt;&lt;br&gt;&#xA;Probabilty is another topic that pops up all over the place. There is a lot you can cover as well. You can combine discrete probability with combinatorics and continuous probability with calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Calculus&lt;/strong&gt;&lt;br&gt;&#xA;Students should be able to have a grasp of differential and integral calculus. In my country, this is mostly high school stuff, but high schools are of varying quality and there is some new stuff as well (partial integration, integration by substitution).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Number theory&lt;/strong&gt;&lt;br&gt;&#xA;Some basic number theory is used for encryption and sometimes comes in handy while analyzing algorithms. Basic topics are greatest common divisors, smallest common multiples, modular arithmetic...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Differential equations&lt;/strong&gt;&lt;br&gt;&#xA;Needed for simulation of real-world phenomenons. Programmers using this would be programmers working with graphics or simulations, and they would typically use finite-difference methods, as opposed to a mathematical approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Information theory&lt;/strong&gt;&lt;br&gt;&#xA;Used for encryption, compression and reliable transmission over noisy channels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This maybe a little bit of a rant and it probably displays my view that for every mathematical field, you have a subfield in algorithmics (which in its turn is a part of computer science). I think that a student should have a good understanding of each of these topics to have a firm mathematical basis. In your situation it's probably a good idea to look at the courses and how many students take them, and maybe ask other professors on their opinion (for example, if there is a graphics course that has a low percentage of students that succeed, it may have to do with the amount of linear algebra, if there are no courses on cryptography or compression you can skip the number theory and information theory, etc...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After this long list you can probably see why I think it's a bad idea to have just one (or even none) mathematics course in a computer science curriculum...&lt;/p&gt;&#xA;" OwnerUserId="915" LastEditorUserId="915" LastEditDate="2014-04-17T01:53:50.817" LastActivityDate="2014-04-17T01:53:50.817" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1743" PostTypeId="2" ParentId="1655" CreationDate="2014-04-16T21:26:37.030" Score="17" Body="&lt;p&gt;Your student reminds me of me in my first algebra class, in 8th grade. I insisted that my answer was a 'better' solution to a homework problem. The teacher was an ex-Marine, who took the time to step outside the room with me and say something to the effect of &quot;I understand you disagree with me about this problem, but I'm responsible for teaching this class, and I can't have you undermining my authority in the classroom, nor can we hold up everybody else's learning.  I'm happy to talk about it with you, outside of class.&quot;  It was the most respectful treatment I'd ever received from a teacher, and it meant a lot to me that he was willing to speak frankly to me this way.  A math classroom isn't &lt;em&gt;just&lt;/em&gt; about being right or wrong, it's a social activity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Projecting wildly from my own life, the student might be insecure, have been rewarded excessively for being 'smart' or 'right', and be a bit behind the curve in social skills. Frankly pointing out the non-technical aspects of the situation can be a big help to a student like that ;-)&lt;/p&gt;&#xA;" OwnerUserId="963" LastActivityDate="2014-04-16T21:26:37.030" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1744" PostTypeId="2" ParentId="1734" CreationDate="2014-04-16T22:22:02.263" Score="17" Body="&lt;p&gt;After reading the previous answers, I'd add logic and proofs. To reason about a program's (in)correctness &lt;em&gt;is&lt;/em&gt; proofs, even if you don't go to the length of proving correctness formally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students struggle with the idea of recursion, recursively defined functions/data structures, and the related proofs by induction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More on the soft side, being able to express oneself clearly in English (or whatever the language used locally might be) seems to be correlated with being a good programmer. Besides, presenting an argument to non-specialists is often part of the job. So I'd insist on clearly written and reasoned homework (and not too sloppy exams).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And for the love of anything holy to you, &lt;em&gt;do not overdo it&lt;/em&gt;. Yes, it would be extremely nice if they knew combinatorics, and probability, and linear algebra, and a bit of multivariable calculus, and enough number theory to understand RSA, a smattering of algebra for elliptic curve groups, and complexity theory, and... [sadly, our calculus curriculum came from collecting &quot;wouldn't it be nice&quot; comments from all engineering departments, with seemingly no criticall winnowing] concentrate on a &lt;em&gt;few&lt;/em&gt; topics that are truly basic, mesh well with what they already know, and go smoothly over into courses taken in paralell/just afterwards. Offer a selction of optional courses for those who want to go into non-standard areas.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-16T22:22:02.263" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1745" PostTypeId="2" ParentId="1734" CreationDate="2014-04-17T01:34:58.243" Score="50" Body="&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt; 2-semester course is not enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt; I write this as a computer-scientist that uses math a lot in his work (I'm a research assistant at a university).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Introduction:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are three (overlapping) aspects of math in computer science:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Math that is actually useful.&lt;/li&gt;&#xA;&lt;li&gt;Math that you can run into, and is generally good to know.&lt;/li&gt;&#xA;&lt;li&gt;Math that lets you build more awesome math.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;First is essential, because the students need to be able to do stuff. Second is important, because you cannot teach students everything, and at the same time trying to get into a new field all by yourself is quite hard (i.e. it's good to know the barest basics of everything). The third are these which aren't directly useful, but present meta-concepts that happen all the time (a bit like design patterns in programming); you can live without them, but intuition you gain there makes life much easier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The list was sorted by (subjectively defined) importance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Math that is actually useful.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Logic.&lt;/em&gt; How to handle quantifiers and read formulas, tautologies, satisfiability; less important: free variables, substitutions and unifiers.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Asymptotics.&lt;/em&gt; This is a must. The big-oh notation, etc., term manipulation, basic limits, sequences and series, both $\to \infty$ and $\to 0$.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Graph theory.&lt;/em&gt; This is a must. Graphs are so prevalent and ubiquitous nowadays that it would be a major setback to not know the basics of graph theory (e.g. searching algorithms, connectedness and flows).&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Linear algebra.&lt;/em&gt; This is a must. What is a linear space, matrices, determinants, orthogonalization, linear programming, perhaps even eigenvalues.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Probability.&lt;/em&gt; Strongly desirable. Discrete spaces, Bayes theorem, expected values, Markov's, Chebyshev and Chernoff inequalities, the law of large numbers.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Data processing.&lt;/em&gt; Strongly desirable. It's important to know how to process a gathered set of data without destroying some of its vital properties. This includes a bit of statistics, some filters, means, clustering, basic approximation, modelling and regression.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Math that you can run into, and is generally good to know.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Numerical analysis.&lt;/em&gt; Floating-point arithmetic and error-analysis are a must.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Universal algebra.&lt;/em&gt; It's essential to make the students not afraid of abstract structures. No advanced stuff, no need for all the groups/rings/skew fields definitions. The barest essentials so that students won't run scared when they see algebraic data types in functional programming, monoids in automata theory or group theory in cryptography.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Statistical and probabilistic modelling.&lt;/em&gt; This is strongly desirable. Markov chains and Monte-Carlo methods.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Differential equations.&lt;/em&gt; Differential equations happen to explain a lot about world's interactions and are used frequently in multitude of practical applications. This is strongly desirable, because it's just hard to learn them with no help.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Information theory and cryptography.&lt;/em&gt; This is a must for modern programmers. Symmetric and public-key cryptography, handshakes, entropy, notion of entropy extractor, Kolmogorov complexity, impossibility of perfect compression, lower bound on comparation-based sorting complexity.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Signal processing.&lt;/em&gt; Fourier transform is strongly desirable. Kalman filters would be good, but there's not enough time. Just anything that would let them reduce the noise in a practical setting.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Math that lets you build more awesome math.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Term rewriting.&lt;/em&gt; Notion of term rewriting is very powerful and underappreciated. A lot of processes can be explained in terms of term rewriting and it's good to know the relevant intuitions. It's useful for showing termination property of algorithms. &lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Automata theory.&lt;/em&gt; The notion of finite automaton is a must.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Complexity theory.&lt;/em&gt; Basic complexity classes and inter-dependencies (including randomized algorithms classes). Maybe other models of computations like $\lambda$-calculus or circuits.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Number theory.&lt;/em&gt; The theory being beautiful and enriching is enough for me here.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Game theory.&lt;/em&gt; You can find game theory in unexpected places, it's good to have this perspective. Nash equilibrium, prisoner dilemma, second-price auction.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's apparent that there's just too many things to cover during 2-semester course.&#xA;If I had to pick, then I would choose the whole first group and nothing else (cryptography can be self-learned). However, I have no idea how to make there 2 semesters seem coherent. On the other hand, I think that it is possible to do in 4-semester course (e.g. one year, but double the load),&#xA;and I would recommend it even at the cost of practical cs curriculum. With a stretch you could group it as follows: logic and asymptotics; graph theory, abstract algebra, automata, cryptography and information theory; numerical analysis, linear algebra, differential equations; probability, data processing, statistical and probabilistic modelling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-04-17T19:30:09.280" LastActivityDate="2014-04-17T19:30:09.280" CommentCount="21" ContentLicense="CC BY-SA 3.0" />
  <row Id="1746" PostTypeId="2" ParentId="1598" CreationDate="2014-04-17T02:15:11.150" Score="9" Body="&lt;p&gt;With regard to the education literature (note that this is not specific to mathematics education) Keith Sawyer writes about the paradoxical balance of &lt;em&gt;improvising&lt;/em&gt; in the classroom and having &lt;em&gt;structured scripts&lt;/em&gt; in mind with regard to what is to be covered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He also writes about &lt;em&gt;teaching as performance&lt;/em&gt;, &lt;em&gt;teacher expertise&lt;/em&gt;, and &lt;em&gt;creativity in teaching&lt;/em&gt;, and certain paradoxes that emerge: the &lt;em&gt;teacher paradox&lt;/em&gt;, the &lt;em&gt;learning paradox&lt;/em&gt;, and the &lt;em&gt;curriculum paradox&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roughly speaking, the idea is that expert teachers are able to bring with them enough &quot;scripts&quot; so that even when there are mistakes or instances in which the topics covered in a class meeting veer off course, the teacher is sufficiently prepared to improvise in a way that leads to positive learning outcomes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sawyer writes a short introduction covering these topics for his book, with citation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Sawyer, R. K. (2011). What makes good teachers great? The artful balance of structure and improvisation. &lt;em&gt;Structure and improvisation in creative teaching&lt;/em&gt;, 1-24.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have uploaded a link to the chapter &lt;a href=&quot;https://www.dropbox.com/s/ddsiy9668otabez/Sawyer%20Good%20Teachers%20Great.pdf&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The intended take-away is that &lt;strong&gt;improvisation need not imply a lack of preparation&lt;/strong&gt;; this is consistent with answers already provided in this thread, which I enjoyed and thought would be well-complemented by pointing interested readers to an additional reference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As for &lt;strong&gt;improvisation without preparation&lt;/strong&gt;: This is lazy at best, and probably pedagogically damaging.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-04-17T02:15:11.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1747" PostTypeId="2" ParentId="1639" CreationDate="2014-04-17T02:23:07.680" Score="7" Body="&lt;p&gt;If they don't come to the office, move the office to where they are. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a TA, I used to go to the department's library during the time I knew my students will be there, working on my next assignment, and engage in some light reading (say, the latest issue of my favorite math journal). Pretty soon, someone will be over with &quot;Sorry to interrupt, do you have a minute to answer this question?&quot;.&lt;/p&gt;&#xA;" OwnerUserId="100" LastActivityDate="2014-04-17T02:23:07.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1748" PostTypeId="2" ParentId="1427" CreationDate="2014-04-17T02:39:32.037" Score="4" Body="&lt;p&gt;It depends on how difficult you want the problem to be.  This seems like it would be a good &quot;math contest&quot; type problem for high-school students.  For example, this could easily be on the &lt;a href=&quot;http://en.wikipedia.org/wiki/American_Mathematics_Competitions&quot; rel=&quot;nofollow&quot;&gt;AMC 10 or AMC 12&lt;/a&gt;.  It might also work well as a longer-term or &quot;project&quot; type homework assignment in a geometry or trigonometry class.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-04-17T02:39:32.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1749" PostTypeId="1" CreationDate="2014-04-17T03:19:21.970" Score="15" ViewCount="3577" Body="&lt;p&gt;First, why bother: I teach and play math with my son in the mornings. It's a lot about letting him enjoy learning, so the curriculum (my intentionally vague vision of how we'll proceed) is flexible. If he finds something interesting and wants to spend time thinking about it, we usually do. He thinks it'd be cool to memorize the times tables up to 100 - and it seems he'd be committed to doing it if we started. But the merits of doing it aside, are there any methods for doing this other than rote memorization?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Hal&lt;/p&gt;&#xA;" OwnerUserId="961" LastEditorUserId="262" LastEditDate="2014-04-19T15:19:47.170" LastActivityDate="2014-05-10T00:29:36.883" Title="Teaching 100 x 100 times tables" Tags="|primary-education|private-lessons|multiplication-table|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="1750" PostTypeId="2" ParentId="1749" CreationDate="2014-04-17T05:01:42.417" Score="19" Body="&lt;p&gt;I realize that your question is about the &lt;span class=&quot;math-container&quot;&gt;$100 \times 100$&lt;/span&gt; table: But since you ask about approaching the multiplication table in ways other than by rote learning alone, I thought I would leave you with a list of problems I generated based on the &lt;span class=&quot;math-container&quot;&gt;$10 \times 10$&lt;/span&gt; table.&lt;/p&gt;&#xA;&lt;p&gt;[&lt;strong&gt;Edit 5/9/14:&lt;/strong&gt; You can find some of the problems below in an informal paper of mine; the citation is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dickman, B. (2014). Problem Posing with the Multiplication Table. &lt;em&gt;Journal of Mathematics Education at Teachers College, 5&lt;/em&gt;(1).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You can also access a free copy online &lt;a href=&quot;http://journals.tc-library.org/index.php/matheducation/article/view/986&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.]&lt;/p&gt;&#xA;&lt;p&gt;The inspiration for these problems can be found in a paper by Trivett (1980) in which he writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The recommendation here is that the multiplication table should be viewed, apparently for the first time by most people, as a dynamic synergetic combination of patterns, a veritable repository of mathematical relationships waiting as it were to gush forth from kindergarten through the secondary grades (p. 21).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Citation&lt;/strong&gt; Trivett, J. (1980). The Multiplication Table: To Be Memorized or Mastered? &lt;em&gt;For the Learning of Mathematics&lt;/em&gt;, 21-25. &lt;a href=&quot;http://www.jstor.org/stable/40247697&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(For A, B, and G, see the &lt;strong&gt;concrete example&lt;/strong&gt; section in my earlier post &lt;a href=&quot;https://matheducators.stackexchange.com/a/1377/262&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;A. Estimate how many distinct numbers are in the 100 boxes of a 10x10 multiplication table.&lt;/p&gt;&#xA;&lt;p&gt;B. How many of the entries in the 10x10 multiplication table are odd?&lt;/p&gt;&#xA;&lt;p&gt;C. Starting at the 1 in a multiplication table, you can take steps to adjacent boxes: right or left, up or down, but not diagonally. What is the biggest number you can arrive at in 10 steps?&lt;/p&gt;&#xA;&lt;p&gt;D. Consider the 2x2 sub-grid cutouts from a multiplication table. For which x can you fill in the remaining three numbers with complete surety?&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/c6Uf8.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;E. Find the only 49 in the multiplication table. Add to it the six numbers above in its column, and the six numbers to the left in its row. What is the total?&lt;/p&gt;&#xA;&lt;p&gt;F. Add up the numbers from all 100 boxes in a 10x10 multiplication table. What is the total?&lt;/p&gt;&#xA;&lt;p&gt;G. Using a multiplication table, explain what it means for a number to be “prime.”&lt;/p&gt;&#xA;&lt;p&gt;H. Look at the numbers on the south-east diagonal starting from either 2 entry in the 10x10 times table: 2, 6, 12, 20, 30, 42, 56, 72, 90. What pattern or patterns do you see?&lt;/p&gt;&#xA;&lt;p&gt;I. Starting at the 3 in row one of the multiplication table, jump like a knight (from Chess) to move one square down and two to the right. Doing this repeatedly, you get the following numbers: 3, 10, 21, 36. What pattern or patterns do you see?&lt;/p&gt;&#xA;&lt;p&gt;J. What kinds of symmetry can you find in the 10x10 multiplication table?&lt;/p&gt;&#xA;&lt;p&gt;K. How would you extend the 10x10 multiplication table to cover the negative numbers?&lt;/p&gt;&#xA;&lt;p&gt;L. Which numbers are missing from the 10x10 multiplication table and why?&lt;/p&gt;&#xA;&lt;p&gt;M. How many one syllable number names are found in a 10x10 multiplication table?&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-05-10T00:29:36.883" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1751" PostTypeId="2" ParentId="604" CreationDate="2014-04-17T05:48:47.460" Score="12" Body="&lt;p&gt;Perhaps it is worth mentioning here there is another notation which at first glance seems non-rigorous. For example, $f = x^2+\sin(y)+z$. Here $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ is a function built from the &lt;strong&gt;coordinate functions&lt;/strong&gt; $x,y,z$ each of which is itself a function defined in the obvious manner:&#xA;$$ x(a,b,c)=a, \qquad y(a,b,c)=b \qquad z(a,b,c) = c. $$&#xA;At first I resisted this notation, but as the semester wears on I grow to enjoy its simplicity. I encountered in in Barrett O'neill's Elementary Differential Geometry. There is always a tension for me in teaching manifold theoretic topics, it's hard to strike the right balance of notation; too much kills intuition, too little makes calculations seem impossible.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2014-04-17T17:02:17.273" LastActivityDate="2014-04-17T17:02:17.273" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1752" PostTypeId="2" ParentId="1042" CreationDate="2014-04-17T05:53:06.887" Score="12" Body="&lt;p&gt;Just a list of some cool things:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Recurrent relations&lt;/strong&gt;&lt;br&gt;&#xA;There are a lot of different ways to find solutions for recurrent relations. &lt;a href=&quot;http://mathworld.wolfram.com/LogisticMap.html&quot;&gt;Logistic maps&lt;/a&gt; are also very neat way to illustrate what's going on.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Bernoulli numbers&lt;/strong&gt;&#xA;This is what I gave a short presentation on. It may be a bit boring for highschoolers sinec there is not really a cool way to explain or illustrate what you're doing, but maybe for the students who are more interested in pure math.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Linear algebra&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Fractals&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Complex numbers&lt;/strong&gt;&lt;br&gt;&#xA;Some very basic complex analysis can be suited for high-schoolers (of course, formal proofs for all the theorems that are commonly used is way too much, but maybe some arguments or some informal proofs would be feasible).&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Infinite series&lt;/strong&gt;&lt;br&gt;&#xA;There was an excellent pdf on this, which presented the difference between the different types on convergence for infinite series, together with some convergence tests, all in a very intuitive way very suited for highschoolers, but unfortunately I can't find it now.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Cardano's formula for solving (some)&#xA;cubic equations&lt;/strong&gt;&lt;br&gt;this can be done geometrically like the quadratic formula, but probably requires guidance.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Fourier theory&lt;/strong&gt;&lt;br&gt;&#xA;An illustrated approach could be really cool, together wth an explanation of some of the basic concepts of Fourier analysis.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Calculus&lt;/strong&gt;&lt;br&gt;some semi-rigorous proofs using limits and limits of summations can probably made by high school students and can be very interesting.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Trigonometric identities&lt;/strong&gt;&lt;br&gt;like cos and sin addition and multiplication formulas&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Platonian solids&lt;/strong&gt;&lt;br&gt;&#xA;There is a very nice proof that there are only 5 platonian solids, and there are some nice properties for 3-dimensional solids as well (F + V = E + 2), and some basic linear algebra for calculating areas, volumes and angles would be appropriate.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Ramsey theory&lt;/strong&gt;&lt;br&gt;This can be explained quite nicely by using the analogy of people in a group who are either friends or not. Also, the proofs have a nice graphical representation. Graph theory in general has some really cool topics for high school students, so there are definitely more topics here.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Programming an AI for a simple game&lt;/strong&gt;&lt;br&gt;It's always fun to play a game against a computer, and it's impressive when you can program a computer in such a way that it performs better in a game then most humans can.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Maps and projections&lt;/strong&gt;&lt;br&gt;&#xA;Things like the Mercator projection and stuff. I don't know the specifics, but I know a guy who did this as his high school project.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Non-euclidian geometry&lt;/strong&gt;&#xA;I don't know too much about this, so this may be too hard, but this is a really cool demonstration of how axiomatic systems work and a fun mind-experiment that, suprisingly, can be embedded in a picture.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Game theory&lt;/strong&gt;&lt;br&gt;&#xA;Game theory provides a lot of interesting question. For example, I was just playing that 2048 game, and I found myself repeating: right, down, right, down, ... until i got stuck, then I did left or up, and started over again. Provided you have the probabilities of the new blocks appearing (else you could estimate them or just create your own variant in which the probabilities are known) you can, for example, try to analyze the average score. Then you could also try to find a pattern that ends the game as quickly as possible, or one that ends the game with the lowest average score (and are these patterns the same?). The nice thing is that even when a mathematical analysis is not feasible, you can just let your computer play for a day and come up with a very good estimate.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I've taken some subjects that I found interesting myself, and some others from a course where everybody had to give a presentation on a mathematical subject. Some of these subjects may be too hard or maybe too specialized to write a 100-page paper about, but I'll let you be the judge of that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've mainly focussed on things that have either a nice geometric representation (because a picture usually makes things more interesting and easier to understand) or some nice things that are not to hard to understand but still are considered interesting and/or nontrivial. I think all but the first three have elements that allow for a nice graphical explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, the students probably require a considerable amount of guidance. Most of the material they'll find is not suited for them, so the teacher either needs to have some material that is appropriate, or help out in another way that allows the students to do the research relatively independent.&lt;/p&gt;&#xA;" OwnerUserId="915" LastEditorUserId="915" LastEditDate="2014-04-25T20:53:24.543" LastActivityDate="2014-04-25T20:53:24.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1753" PostTypeId="2" ParentId="1734" CreationDate="2014-04-17T06:29:28.817" Score="8" Body="&lt;p&gt;I'm not a professional programmer, but I worked in scientific programming for a couple years.  Here are my picks.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Basic calculus&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that calculus is overemphasized in many programs, but we can distill the useful parts. (And to study most higher level math, we need to.) Teach what limits, derivatives, and integrals are.  The most important thing would be to emphasize their conceptual meaning and how they can be used.  The student should come out with enough understanding to use calculus in real scenarios given access to computational algebra software like Mathematica.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It should be sufficient to teach how to do these operations on polynomials and trig functions (since they are easy and common), some basic properties (such as linearity), and the Fundamental Theorem of Calculus.  Leave out techniques like partial fractions and trigonometric substitution - these are not conceptually important, they can use Mathematica for that.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Numerical Analysis&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Emphasize iterative methods and order of convergence&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Include an in depth discussion of Monte Carlo integration and its generalization to other Monte Carlo algorithms.  Teach them to solve problems by simulating data.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cover asymptotics, Newton's method, truncation error (including Taylor's theorem here if it hasn't been covered already), analysis of algorithms&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In a dream world, I'd also include discrete fourier transform on here, but this may not be feasible with only a pragmatic understanding of calculus.  Perhaps best to leave for a later course for those who wish to learn it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Logic (and other &quot;proofs class&quot; topics)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Logic itself is directly applicable to computer science.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Elementary set theory should be very intuitive to anyone who is used to working with arrays and lists all the time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Direct, contrapositive, and contradiction proof methods will be needed for further topics.  There should be a &lt;em&gt;heavy&lt;/em&gt; emphasis on induction. (Recursion is basically induction.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Combinatorics&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Lexographic, colexicographic, and quasi-lexographic ordering, both in general and for monomials&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Of course, some graph theory.  Include depth/breadth first searches, Dijkstra's algorithm, Kruskal's algorithm.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Numerical Linear Algebra&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The basics of linear algebra would, of course, need to be introduced before getting into the numerical part.  Solving systems of linear equations is pervasive, and solving them quickly is definitely important.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For an example of applications, return to a graph theory for a moment to show them a linear algebraic graph drawing algorithm&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Linear programming / convex analysis if there's time&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Vector calculus&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would be the last topic. This should be presented as a natural synthesis of everything learned so far. Anything in graphics programming requires a good understanding of vectors, as do many scientific programming applications.  Learning how to take normals, how to compute projections, how to navigate vector fields, etc. all has direct practical value.  Learning how to think about vectors in general does too.  Due to time constraints, I wouldn't try to get any more in depth than that.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;That's it.  Those six topics should be enough for a very productive year long, five days a week class.  I would caution against adding any more, as I think it's better to learn a few topics in depth than fly through a ton with only a shallow understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should also note that each of these subjects would be great for both short-term and ongoing programming projects.  I would make these worth quite a lot of the grade, and save the rest for weekly quizzes and some written homework.&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2014-04-18T16:19:04.857" LastActivityDate="2014-04-18T16:19:04.857" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="1754" PostTypeId="2" ParentId="1734" CreationDate="2014-04-17T07:19:54.120" Score="4" Body="&lt;p&gt;Something which I don't see mentioned here so far is mathematics in non-base 10. there are legitimate situations where bit shifting, binary algebra and hexadecimal are quite a benefit to coding. For example, if the device you're developing for has limited resources (embedded systems, legacy programs), using a bit shift can sometimes save precious memoryspace and processing cycles.&lt;/p&gt;&#xA;" OwnerUserId="775" LastActivityDate="2014-04-17T07:19:54.120" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1756" PostTypeId="1" AcceptedAnswerId="1776" CreationDate="2014-04-17T09:26:46.000" Score="17" ViewCount="517" Body="&lt;p&gt;I'm not too sure if this is the right place for my question, but I couldn't think of anywhere more suited.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am a first year university Math student, I am currently revising for six end of year examinations. My problem is that I am struggling to make the step up in time for exams. I would like to know how you suggest it is best to revise efficiently for Math exams. I have spent a lot of time making notes, but when it comes to tackling papers I don't seem to be able to get very far without them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I appreciate your time to answer my question. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="973" LastEditorUserId="75" LastEditDate="2014-04-17T13:08:15.290" LastActivityDate="2014-04-24T12:20:07.443" Title="How do you revise?" Tags="|undergraduate-education|student-motivation|self-learning|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1757" PostTypeId="2" ParentId="1639" CreationDate="2014-04-17T11:14:26.380" Score="9" Body="&lt;p&gt;I hope this answer isn't too short, but nobody else has yet mentioned one of the most obvious reasons why students won't seek extra help.  They may be afraid that if they turn up and ask basic questions, then you, the instructor, will realise just how bad they are, and this will affect their final degree result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could try to counter this belief indirectly, by making it very clear how marks are awarded.&lt;/p&gt;&#xA;" OwnerUserId="894" LastActivityDate="2014-04-17T11:14:26.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1758" PostTypeId="2" ParentId="686" CreationDate="2014-04-17T11:17:44.657" Score="3" Body="&lt;p&gt;Theorem about geometric numerical integration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays introductory ODE courses (naturally) usually take a dynamical systems point of view and asymptotic/geometric properties of the solutions play an ever increasing role.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that an introductory numerical analysis course where time discretization of ODEs are treated should respect this and there are many easily accessible results in this direction that can be presented already on an undergraduate level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The monograph by &lt;a href=&quot;http://amzn.com/364205157X&quot; rel=&quot;nofollow&quot;&gt;Hairer, Lubich and Wanner&lt;/a&gt; is hardly accessible for undergraduates but a good instructor should be able to extract results which can be presented. Most of these results are from the last four decades.&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-04-17T11:17:44.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1759" PostTypeId="2" ParentId="1427" CreationDate="2014-04-17T11:56:13.780" Score="2" Body="&lt;p&gt;I agree with Jim Belk that this would be a good &quot;math contest&quot; type problem for high-school students.  For example, this could easily be on the AMC 10 or AMC 12.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would not assign the problem in an ordinary class, where the difficulty in checking the answer would seem to promote more frustration than payoff.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-17T11:56:13.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1760" PostTypeId="1" CreationDate="2014-04-17T11:58:45.553" Score="20" ViewCount="1694" Body="&lt;p&gt;Increasingly, I see computational topology being applied to problems involving sensor networks, robotics, data analysis, signal processing and various other areas. The topics I mention are interesting to many of my engineering students, but they lack the background in abstract mathematics, especially group theory, to be able to comprehend material such as homology. I try to get them to the borders by way of some graph theory and computational geometry in the context of robust sensor networks, even touching on the Vietoris-Rips complex, but then the wall of abstraction looms up. I have also taught a short data analysis course focusing on cluster analysis, with a touch of anomaly detection, but so far I have stopped short of introducing topological data analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Does anyone have any suggestions or experience in dealing with teaching material of this nature to advanced undergraduates who are not specializing in mathematics? What are good ways of making the abstract mathematics involved a little more concrete and manageable?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Related questions from Mathematics Stack Exchange:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/30557/introduction-to-computational-topology&quot;&gt;Introduction to Computational&#xA;Topology&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/505670/references-for-topology-with-applications-in-engineering-computer-science-robo&quot;&gt;References for Topology with applications in Engineering, Computer&#xA;Science,&#xA;Robotics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;An incomplete list of online resources on computational topology:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://calculus123.com/wiki/Topology_Illustrated&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peter Saveliev's page on Topology Illustrated book&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.math.upenn.edu/~ghrist/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Ghrist's home page&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.cse.ohio-state.edu/~tamaldey/course/CTDA/CTDA.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tamal Dey's Computational Topology and Data Analysis course&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.drmichaelrobinson.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michael Robinson's home page&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.sas.upenn.edu/~vnanda/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vidit Nanda's home page&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=2PSqWBIrn90&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matthew Wright's introduction to persistent homology&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(Removed &lt;a href=&quot;http://comptop.stanford.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://comptop.stanford.edu/&lt;/a&gt; as this site seems to have been taken down. Clicking on the link now redirects you to &lt;a href=&quot;http://appliedtopology.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://appliedtopology.org/&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2020-06-09T13:10:51.417" LastActivityDate="2021-04-20T15:00:54.490" Title="Computational topology for engineers" Tags="|undergraduate-education|course-design|applied-mathematics|engineering-mathematics|topology|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="1761" PostTypeId="2" ParentId="1734" CreationDate="2014-04-17T13:48:46.100" Score="5" Body="&lt;p&gt;One of the challenges here is that there is a big difference between software engineering and computer science.  The latter has very rigorous mathematical underpinnings, whereas the former can often get away with very little math, depending on what area of industry you wind up working in.  I'm assuming this is a program that is geared toward engineering.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I've personally found the most useful in my work are linear algebra and signal analysis (Fourier transforms, convolution with filter kernels, etc.)  Again, though, this is largely due to industry in which I work (industrial automation), where I deal with geometric transformations and interpretation of signals.  If I was in process engineering, I might solve more differential equations.  If I was in data mining or optimization, I might need to know more about linear programming and simplex methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I will say is that I learned a lot more from my focused courses than I did from the 'survey' courses.  My major included a required survey engineering course that covered a wide range of topics, but it covered each topic so briefly (no more than a week on a given area) that there was barely time to even get a taste of what the subject offered.  I can understand the intention of the course (to help undergraduates discover what focus they might want to pursue), but I didn't learn anything significant from it that stuck with me.  As such, it might be better to just teach Calculus, Linear Algebra or some other classic subject in full introductory depth, perhaps giving the student the option of which one to take.  The student will retain more from the increased depth of the course and will likely develop better mathematical skills as a result of having to tackle more challenging problems than could be raised in a survey course.&lt;/p&gt;&#xA;" OwnerUserId="979" LastActivityDate="2014-04-17T13:48:46.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1762" PostTypeId="2" ParentId="1734" CreationDate="2014-04-17T14:33:42.520" Score="7" Body="&lt;p&gt;dtldarek's answer is excellent. The only thing I would add to it is a good foundation in proofs. While I don't run into proofs on the job, I've seen a tremendous number of proofs in my CS classes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Experience with rigorous proofs also helps with logic and algorithms -- have you covered all the bases? Are there any places where this falls apart? Have you made any assumptions you shouldn't have? All good things to have in mind when writing programs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[I have a BS in Math, did a career change to programming about 5 years ago, and am working on my MS in CS while holding a full time job as a software developer.]&lt;/p&gt;&#xA;" OwnerUserId="908" LastActivityDate="2014-04-17T14:33:42.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1763" PostTypeId="1" CreationDate="2014-04-17T14:51:34.250" Score="21" ViewCount="6807" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Pigeonhole_principle&quot; rel=&quot;nofollow&quot;&gt;Pigeonhole Principle&lt;/a&gt; (or &lt;a href=&quot;http://mathworld.wolfram.com/DirichletsBoxPrinciple.html&quot; rel=&quot;nofollow&quot;&gt;Dirichlet's box&lt;/a&gt; principle) is a method introduced usually quite early in the mathematical curriculum. The examples where it is usually introduced are (in my humble experience) usually rather boring and not too deep.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is well-known, however, that there are great and deep applications of it in research mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What applications of the pigeonhole principle would you consider in an &quot;Introduction to proofs&quot; course for university students? They should be non-trivial but accessible for undergraduate students, and an interchange between different mathematical fields is always welcome.&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="-1" LastEditDate="2014-09-15T09:58:05.167" LastActivityDate="2014-09-15T09:58:05.167" Title="Examples and applications of the pigeonhole principle" Tags="|proofs|problem-design|" AnswerCount="6" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1764" PostTypeId="2" ParentId="1763" CreationDate="2014-04-17T15:51:25.237" Score="12" Body="&lt;p&gt;Dirichlet's theorem that an irrational number can be approximated to within $1/q^2$ for a sequence of rationals $p/q$ exemplifies this principle.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-17T15:51:25.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1765" PostTypeId="2" ParentId="1763" CreationDate="2014-04-17T16:03:34.373" Score="5" Body="&lt;p&gt;Show that among any $n + 1$ numbers out of $1, \ldots, 2n$ there are two that are relatively prime, and there is one that divides the other.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-17T16:03:34.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1766" PostTypeId="2" ParentId="1756" CreationDate="2014-04-17T16:17:40.023" Score="7" Body="&lt;p&gt;The only advise I can give is to work on the subjects constantly. Don't have any &quot;zero weeks,&quot; weeks where you didn't do something on some subject, getting (back) on track gets &lt;em&gt;hard&lt;/em&gt; very rapidly. You have my permission to take a day off to do something totally different once a week (it does wonders for not ending up all stressed out).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Collect earlier homework and exams (the 'web teems with them), give them a shot. Look for lecture notes, they usually contain interesting examples and suggested problems. You learn by &lt;em&gt;doing&lt;/em&gt;, not by reading/looking. Check for interesting questions around &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;MSE&lt;/a&gt;, ask if you get stuck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For other than this (very general) advice (might as well tell this your friend who's mayor is English...), you'll have to give a lot more detail.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-17T16:17:40.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1767" PostTypeId="2" ParentId="1639" CreationDate="2014-04-17T16:56:37.230" Score="6" Body="&lt;p&gt;&lt;strong&gt;TL;DR:&lt;/strong&gt; Make the rest of the course so interesting that the office hours will become appealing, too; or incidentally make yourself a very good viral advertisement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not being a teacher I can only give account on what motivated me in my student years to go and see my prof/TA during office hours. Now I wish I had used the oportunity far more foten, but hey, what looks like a mistake now I percieved as perfectly all right then.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Namely there were two noteworthy occasions when I enjoyed office hours.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;This story is kind of long but I need you to walk through it to see the big picture. In my introductory course for calculus, I was very lucky to have &lt;em&gt;very&lt;/em&gt; talented and enthusiastic professor. He himself designed a very clever e-learning system on which even the tests were based, so chances were when you spend enough time and effort with it, you'd have a very good chance reaching a top score in the test. When tests were written, there were math books available at a makeshift desk on a grass patch in the middle of the campus, and when you needed a help from a book, you simply walked off the test room (where around 200 students were writing their tests), had a go with the book and then resumed with your test. On the desk, there was a big cardboard sign with F1 written on it (the course was meant for aspiring software engineers, you see). The lectures were superbly conducted and fun to be at, too. Yet the turnup for office hours (if I may call it so) was never below 150. Why? The biggest lecture room was booked so that everyone could come. There was the professor and his TAs, too, so that councensual advice and reasoning could be presented 'live'. The office hours took place at 20:00 to 22:00 on the night before every test. The e-learning system allowed for sumbiting problems directly to a database that would be looked through at the office hours. Once, when a very &lt;em&gt;important&lt;/em&gt; football (or soccer, if you like) match collided with the schedule, the professor brought a projector (what an asset, back then, with the lamp costing two month's worth of students' accomodation!) ant let the students watch the &lt;em&gt;important&lt;/em&gt; moments (and watched with them). To that he noted domething like &lt;em&gt;You have books, seminars, our e-learning thing, online forum* with all the teachers, well, you have pretty much everything. My duty as a teacher is to motivate you, to get you to learning at any cost.&lt;/em&gt;&lt;br&gt;&#xA;*This was in 2005, when online fora and e-learning for maths was not as common as it is today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, you see, the courses were not only very successful, enlighting and encouraging (some students even switched to maths from computers!), but also they were fun and inviting and adventurous (we all know here how &lt;em&gt;adventurous&lt;/em&gt; maths can be, but then we were still virtually untouched). And, what I find very important - everybody from other universities envied us when we told stories about our calculus course, so we eagerly went in for more! And got a good calculus basis in the process.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Once I failed an exam miserably when I had been very confident about my knowledge. I asked after the exam if I can come to the office hours. To this the answer was &lt;em&gt;But off course, I had been waiting the whole semester for you to come! If you have time, we can look at it right now!&lt;/em&gt; I had time so I went for it. And as he reviewed my test, two things became apparent: I did really vastly overestimate my knowledge and understanding, and I wasn't a complete failure either; and at the end I got a grade acceptable for me. Word about that spread quickly - no one after me got his grade updated during the office hours, but overall the grades of those who went to consult were improved.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="981" LastActivityDate="2014-04-17T16:56:37.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1768" PostTypeId="2" ParentId="1763" CreationDate="2014-04-17T17:15:02.953" Score="12" Body="&lt;p&gt;&lt;strong&gt;Application 1:&lt;/strong&gt; Every rational number has a repeating decimal expansion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Application 2:&lt;/strong&gt; Each infinite decimal expansion has the property that there exists a $10^{100}$-length sequence of digits that is repeated infinitely often in the expansion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Application 3:&lt;/strong&gt; If $x$ is irrational, then at least two digits appear infinitely often in the decimal expansion of $x.$&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2014-04-17T19:47:25.567" LastActivityDate="2014-04-17T19:47:25.567" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="1769" PostTypeId="2" ParentId="1763" CreationDate="2014-04-17T18:17:49.523" Score="19" Body="&lt;ul&gt;&#xA;&lt;li&gt;If $\gcd(a,b)=1$, there exists a multiplicative inverse for $a$ modulo $b$. (Otherwise, look at the $b-1$ multiples of $a$, namely $a,2a,3a,\dots,(b-1)a$. They must fall into congruence classes that aren't 0 or 1, but there are only $b-2$ of those.)&lt;/li&gt;&#xA;&lt;li&gt;$R(3,3)\leq 6$, and other Ramsey-style arguments&lt;/li&gt;&#xA;&lt;li&gt;Give any domino tiling of a $6\times 6$ checkerboard, there exists a way to split the board along some row or column that does not disturb the tiling.&lt;/li&gt;&#xA;&lt;li&gt;The Erdos-Szekeres result: Every sequence of length $ab+1$ contains a monotone increasing subsequence of length $a$ or a monotone decreasing subsequence of length $b$.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Lossless data compression algorithms cannot guarantee compression for all input data sets. In other words, for any lossless data compression algorithm, there will be an input data set that does not get smaller when processed by the algorithm, and for any lossless data compression algorithm that makes at least one file smaller, there will be at least one file that it makes larger.&quot; (source and proof: &lt;a href=&quot;http://en.wikipedia.org/wiki/Lossless_data_compression#Limitations&quot;&gt;Wikipedia&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If this is in an intro to proofs course, I also recommend stating the following version and having the students prove it. It's a good example of a contradiction argument: &quot;Given any $n$ real numbers, at least one of them is as large as their average.&quot;&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-04-17T18:17:49.523" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1770" PostTypeId="2" ParentId="1689" CreationDate="2014-04-17T19:03:26.827" Score="7" Body="&lt;p&gt;How many of your students are interested in space exploration?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quadratic equations became a lot more interesting to me when I learned paths about the sun are ellipses, parabolas or hyperbolas with the sun as a focus. Paths of moons or satellites about a planet are also conic sections with the planet center at a focus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, the visual presentation of quadratics as conic sections also made them more tangible and less remote and abstract.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if the student becomes interested in quadratics, he might also become interested in factoring them.&lt;/p&gt;&#xA;" OwnerUserId="603" LastActivityDate="2014-04-17T19:03:26.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1771" PostTypeId="2" ParentId="1734" CreationDate="2014-04-17T20:16:41.740" Score="2" Body="&lt;p&gt;Actually there is &lt;b&gt; NO REASON &lt;/b&gt; to include calculus level mathematics for a computer science degree. I repeat.. &lt;b&gt;NO REASON.&lt;/b&gt;&#xA; WHY is that, you might ask?? &#xA;Because very few coders use heavy duty math (or even simple math) in their daily work. I've been coding for 16 years at both large and small corporations, and even when I worked at banks and mortgage companies, I hardly EVER needed anything more complex than multiplication. &#xA; Put simply, coding is mostly logic and problem solving. &#xA; As for those coders claiming that learning complex forms of math, teaches logic, I still have to disagree. &#xA; If you want to learn coding logic, do some actual coding, or play a game of chess, or build a website in HTML and add some javascript functionality to it. There are far more interesting and practical ways to learn coding than bending your brain around rarely used formulas and procedures. So..other than earning colleges lots of money and denying tons of people, I am strongly against any kind of math for programmers. It just slows the real learning down.&lt;/p&gt;&#xA;" OwnerUserId="990" LastActivityDate="2014-04-17T20:16:41.740" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1772" PostTypeId="2" ParentId="1696" CreationDate="2014-04-17T20:20:21.340" Score="5" Body="&lt;p&gt;Brezis: &lt;a href=&quot;http://amzn.com/0387709134&quot;&gt;Functional Analysis, Sobolev Spaces and Partial Differential Equations&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book is a great introduction into functional analysis but it is built up to show how FA results can be used to solve partial differential equations. So if you count PDEs as applications, then this is a great book.&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-04-17T20:20:21.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1773" PostTypeId="2" ParentId="1028" CreationDate="2014-04-17T20:27:15.047" Score="7" Body="&lt;p&gt;I teach middle and high school math students. I have encountered the issue of students saying they have already covered the material on several occasions. As a result, I would give the students a pretest at the beginning of the academic year. This way I would have an idea of what the students know coming into my classroom, and it would also give the students a glance at what they would be learning. This by itself allowed the students to see that they would be 1) using previously learned information, and 2) learning more than what they already know. Another plus to giving a pretest is that if all the students are familiar with a topic, say solving one step variable equations, they you do not have to reteach the material, but rather give a review of the topic, which will allow you to focus more on the new material that hey students may have difficulties with. Because you have given a pretest, you can use that as your data for why you are not &quot;teaching&quot; the section. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A way you can make the point concrete that something is a prerequisite, is by giving them a problem that they should be able to solve by the end of the course, you could tell them that they have already learned some of the methods needed to solve the problem, and the course they are in will teach them the rest of the methods necessary to accomplish the task. You can also use this as a way of stringing other courses that the students have taken together, as a way of emphasizing that Math, as a whole, builds upon itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are talking more specifically about how you can show math is related to other courses, like engineering or computer science, you could give some examples from an engineering textbook, and emphasize the point that it is from another field of study, or state that the methods they are learning are also necessary for other fields of study while being specific as to what fields, courses, or particular practices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To comment on your last note about some techniques for Discrete Math: I am currently teaching a student who is in Discrete Math, but we do not have a Discrete Math textbook. So as a result, I am using the Calculus, Algebra 2, and Applied Calculus textbooks for material. You could do the same or something similar as a way of showing your students that the courses are related, if you want. I find that the calculus book is a very good source for example and explanations of derivation/integration, power series, and number theory.   &lt;/p&gt;&#xA;" OwnerUserId="839" LastActivityDate="2014-04-17T20:27:15.047" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1774" PostTypeId="2" ParentId="1734" CreationDate="2014-04-17T20:43:33.233" Score="5" Body="&lt;p&gt;None of the other answers have mentioned it thus far, so I'll add that &lt;strong&gt;lambda calculus&lt;/strong&gt; would be a useful field to include.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, several programming languages have introduced lambda functions to their syntax in the past several years. Java 8 was just released last month, introducing the concept to Java; C# added lambda functions in .Net 3.5(?) in 2007; C++ gained the functionality with C++11 in 2011.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Working as a TA as an undergraduate for a freshman-level course, this feature was definitely the &lt;em&gt;most&lt;/em&gt; confusing piece of syntax the students encountered.&lt;/p&gt;&#xA;" OwnerUserId="328" LastActivityDate="2014-04-17T20:43:33.233" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1775" PostTypeId="2" ParentId="1690" CreationDate="2014-04-17T20:56:01.237" Score="1" Body="&lt;p&gt;You have some really good pros and cons for explanations on a true/false test. If that is the method that you would like to keep using, you could always try having students explain why the find a statement/problem true or false. For example, have the directions read something like: Mark the statement either true or false, if you find the statement to be false, explain your reasoning. Or you could have them explain themselves if they find it true. because I do not know whether or not you give tests that consist of just true/false questions, I would like to suggest that a combination of test structures, i.e. multiple choice, free response, true/false, etc. This would allow your students to have other avenues to express their understanding of the material than just &quot;do you think this is correct or not?&quot; But it is your classroom and you know how much time you have to dedicate to test grading, and what test structure you prefer.&lt;/p&gt;&#xA;" OwnerUserId="839" LastActivityDate="2014-04-17T20:56:01.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1776" PostTypeId="2" ParentId="1756" CreationDate="2014-04-17T21:00:53.913" Score="10" Body="&lt;p&gt;My first suggestion is not to prepare for your exams alone. Prepare for your exams with some of your colleagues, but make sure each of you reflects on your own progress toward readiness independently; the role of you and your colleagues is to provide feedback for each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My next suggestion is to identify areas of concern and focus your efforts on these areas. Which topics are the most challenging for you, and which topics do you actually have evidence you understand (do not rely on &quot;oh I remember how to do this, try practice problems and verify you can produce the same answers as your source of the problems&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should also regularly quiz yourself on the topics. There is evidence to suggest that our brains memory emphasizes information which is regularly retrieved, which means that just taking a quiz helps you remember that information, since you essentially force your brain to recall it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, you should ideally attempt non-standard problems in the domain of mathematics you are attempting. A recent randomized-control study suggests that if you follow a sequence of steps someone else provides in order to solve a problem, you are less successful at remember your attempts for later than if you just struggle with problems (presumably collaboration is helpful here) even if you do not get the answer to the problem. A non-standard problem without a known solution that lies within the domain of mathematics you are studying should help simulate this effect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Find examples of work you did unsuccessfully in the past (or ideally, find a collection of such examples with you and your colleagues) and take the time to reflect on what worked in what you did, and what did not work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Construct a 1 page summary of everything you learned in that content area, even if you do not get to use it during your exam. This summary should be clear and easy to read (don't use 2 point font!). Doing this will help you focus on the essential elements you need to understand for this unit, and creating it will help you recall those elements later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make sure in the days leading up to your exam that you relax. If you have taken the other suggestion from this post and spread our your preparation over the weeks in advance of your assessment, you will not need the last day to cram. You should take a break, get some rest, and be well hydrated (Hopefully this is obvious, but don't drink or do drugs during this day, both will negatively affect your performance) before your exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-04-17T21:00:53.913" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1777" PostTypeId="2" ParentId="1427" CreationDate="2014-04-17T21:09:06.157" Score="6" Body="&lt;p&gt;I disagree with the other two answers. I would assign this problem in an introductory high school class, but I would not use it as an assessment item. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's why:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Most high school students will understand what this problem is asking.&lt;/li&gt;&#xA;&lt;li&gt;There is some evidence that suggests that struggling with a problem, when one is very clear about the expectations of the problem, improves ones ability to solve problems and importantly, to remember the information required to solve the problem later.&lt;/li&gt;&#xA;&lt;li&gt;Too often lower to mid-level students are given problems which do not require them to think. If we don't ask these students to think, we cannot expect them to improve their ability to do so.&lt;/li&gt;&#xA;&lt;li&gt;This problem is what I would call an open middle problem. It is relatively easily stated, and has a single possible answer, but in the middle students can approach it in a variety of different ways. This will help prompt students, particularly if they work in groups, to share different strategies for approaching the problem with each other, and to get feedback on those strategies. Even if students don't get the right answer, they will learn from this experience.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-04-17T21:09:06.157" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1778" PostTypeId="2" ParentId="819" CreationDate="2014-04-17T21:11:52.857" Score="5" Body="&lt;p&gt;Perhaps: &lt;a href=&quot;http://collegeopentextbooks.org/opentextbookcontent/open-textbooks-by-subject/statisticsandprobability&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://collegeopentextbooks.org/opentextbookcontent/open-textbooks-by-subject/statisticsandprobability&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or as JW suggested:&#xA;&lt;a href=&quot;http://www.cargalmathbooks.com/#Probability&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.cargalmathbooks.com/#Probability&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beyond these list, I have not yet found one.&lt;/p&gt;&#xA;" OwnerUserId="839" LastEditorUserId="122" LastEditDate="2017-04-20T08:52:32.550" LastActivityDate="2017-04-20T08:52:32.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1779" PostTypeId="2" ParentId="1028" CreationDate="2014-04-18T00:08:21.070" Score="11" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I've heard the phrase &quot;we passed that course already!&quot; too many times when asking for e.g. the derivative of a simple rational function or a simple integral, getting blank stares, and digging deeper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the only appropriate response to a statement like that is &lt;strong&gt;&quot;Excellent, then you should have no trouble answering this!&quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then &amp;mdash; assuming, of course, that the student in question &lt;em&gt;does&lt;/em&gt; in fact have trouble answering the question &amp;mdash; you can take some time to explain to your students that, yes, they &lt;em&gt;are&lt;/em&gt; going to need the skills they learned in earlier courses if they want to keep up with this one, and that, yes, this was explained in the course description, and finally that, if they've forgotten those skills, now would be a &lt;em&gt;very&lt;/em&gt; good time to pick up their old textbooks and refresh their memory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This &lt;em&gt;is&lt;/em&gt; how you convince your students that the courses they're taking really do build upon each other.  Nothing teaches a lesson quite as well as experience, and the realization that all the stuff they crammed and forgot last semester is actually something they'll need to re-learn now will hopefully teach them to pay at least a little better attention to future courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully, this is something they'll encounter fairly early, preferably during their first year at college, or at least at the beginning of the second year.  It won't be easy for many of them &amp;mdash; let's face it, this habit of study-and-forget can become very strongly ingrained during secondary school, where much of the material taught &lt;em&gt;is&lt;/em&gt; something the students can forget after the exam without any &lt;em&gt;immediate and concrete&lt;/em&gt; consequences &amp;mdash; and some of them will complain, but they'll also gradually learn better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, you'll probably have to incorporate &lt;em&gt;some&lt;/em&gt; remedial teaching and refreshing of earlier material into your course, if you want to ensure that most of your students will be able to follow along.  One option can be to prepare and hand out an extra set of optional &quot;warm-up&quot; exercises that briefly cover the things you're expecting the students to already be able to do.  (If your course covers a broad set of topics, you may want to do this several times during the course, as new prerequisites become relevant.)  It could also be a good idea to encourage the students to help each other, perhaps even by offering extra points to students who volunteer to help others catch up.&lt;/p&gt;&#xA;" OwnerUserId="526" LastActivityDate="2014-04-18T00:08:21.070" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1780" PostTypeId="2" ParentId="1763" CreationDate="2014-04-18T01:01:26.953" Score="9" Body="&lt;p&gt;Here is one application in introductory abstract algebra:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A finite integral domain is a field.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="263" LastActivityDate="2014-04-18T01:01:26.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1781" PostTypeId="1" CreationDate="2014-04-18T01:08:58.567" Score="17" ViewCount="566" Body="&lt;p&gt;Anecdotally, I've heard it said that in (Australian) grades 11 and 12 calculus needs to be taught in a procedural way involving merely recipes for doing calculus, rather than teaching for understanding common in lower grades. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In primary school there is a lot of emphasis on relational understanding, perhaps at the expense of rote learning (eg times tables) or procedures (eg long division). But on reaching senior school this is not expected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there too much content in the curriculum?&#xA;Is there an emphasis on exam preparation, including revision at the expense of understanding?&#xA;Is the content not amenable to relational understanding (cf complex numbers just are)? &#xA;Perhaps relational understanding only comes after a mastery of procedural techniques (of differentiation and integration). &#xA;Does it need just to be taken on faith?&#xA;What role does proof play? &#xA;What role do algorithms play (Differentiation if easy but integration is hard)?&#xA;Is relational understanding even necessary? - historically calculus was used successfully without proper underpinnings.&#xA;Is instrumental understanding necessary  without introducing further abstraction or esoteric machinery (logic and set theory)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is anyone aware of arguments or research supporting any of these claims?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A related question is regarding teacher (and student) attitudes to these things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Of course, I believe in the importance of imparting a relational understanding, but (for an essay I was writing) I was looking for some evidence of teachers attitudes regarding the necessity of teaching calculus in an instrumental way.) &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I'm sorry if any of you misinterpreted my inquiry as dissent. I was merely looking for evidence of what teachers actually thought. I was looking for a representative sample of what teachers thought not just the opinion of a few.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also I was looking for evidence to support the proposal that calculus needs to be taught in a particular way. I wasn't making that proposal!&lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="143" LastEditDate="2014-04-30T07:56:01.790" LastActivityDate="2014-04-30T07:56:01.790" Title="Research supporting &quot;recipe-style&quot; calculus in senior high school?" Tags="|calculus|secondary-education|reference-request|education-research|teacher-motivation|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="1782" PostTypeId="2" ParentId="1690" CreationDate="2014-04-18T01:54:36.420" Score="1" Body="&lt;p&gt;I would definitely say that there needs to be some part of the question asking justification. The sole purpose of the test is to see if they have sufficiently learnt the material and giving the ability to just plug and chug or even guess isn't doing mathematics. Also if the student is like me they may have made programs on their calculators to do the work for them which ultimately in the long prohibits their ability to come to the conclusion on a step by step basis. Just by the way I answer this as a college freshman, not a teacher. Thought a student's perspective may help.&lt;/p&gt;&#xA;" OwnerUserId="937" LastActivityDate="2014-04-18T01:54:36.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1785" PostTypeId="2" ParentId="1760" CreationDate="2014-04-18T03:03:22.497" Score="17" Body="&lt;p&gt;I'm just finishing up a graduate course in computational topology which could be adapted very effectively for this purpose. We're focusing on topological data analysis and computational homology.  All the topology in the course has been self-contained, meaning that essentially no previous experience in topology was required.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book we're using is  &lt;em&gt;Computational Topology: An Introduction&lt;/em&gt;, by H. Edelsbrunner and J. Harer. It's &lt;a href=&quot;http://www.ee.oulu.fi/research/imag/courses/Vaccarino/Edels_Book.pdf&quot;&gt;available online&lt;/a&gt;, or you buy it &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0821849255&quot;&gt;in print&lt;/a&gt; (I did this and found it worthwhile). The author works in computer science, and it is written in a style that engineers would appreciate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The professor began the class by explaining our basic problem: if we have a bunch of data points from some topological space, how do we figure out what space the data came from?  He continued by showing us various motivating examples to explain why this question is well-posed, e.g. a ton of points obviously taken from a circle.  He moved on with illustrations and animations to explain (at least visually, in the 2D case) how the Vietoris-Rips complex works, then worked up to bar codes / persistence diagrams.  We were exposed to essentially everything we were going to do over the course of the semester, so we had the &quot;big picture&quot; in our heads right away.  This was very important, especially because I had no background in algebraic topology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We spent the next couple weeks learning the basics of abstract simplicial complexes and their embeddings in $\mathbb{R}^n$.  Everything here was intuitive. Most examples had dimension $2$ (i.e. visually, graphs with some triangles filled in). In this context, homology groups were easily motivated by Betti numbers, which conceptually bridge the gap to group theory.  &quot;What we want is to know the number of components, the number of missing 'circles', the number of missing 'spheres', etc.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get the Betti numbers, we needed to learn how homology groups worked.  Working in this discrete context everything is very constructible, so we could actually make the $C_p$, $Z_p$, and $B_p$ groups (cf p.96 of Edelsbrunner's book), along with the $\delta$ maps, for small examples.  I went home and did this for a whole bunch of examples until I understood how ranks of the matrices related to the &quot;missing pieces&quot; interpretation.  Shortly after this, the professor gave us homework where we had to compute the $\bmod{2}$ and $\bmod{3}$ Betti numbers of the Torus, Klein Bottle, and projective plane from given triangulations (which of course was expected to be done with computational algebra software, not by hand.)  This really cemented the whole point of the course for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After some more theory about persistence, we started using &lt;a href=&quot;https://code.google.com/p/javaplex/&quot;&gt;javaplex&lt;/a&gt; (in conjunction with MATLAB, for most of us) to actually get our hands dirty with some data.  A few weeks ago, we were given a few data sets, told which manifolds they came from, and told find the Betti numbers using a barcode analysis.   For our final assignment, we were given several data sets, but no other information, and tasked with figuring out what topological space they came from.  (One of the data sets was an embedding of a certain well known $3$-manifold in $\mathbb{R}^9$.)  So, we got to take the whole thing to fruition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyhow, it probably wouldn't take much to adapt this course towards engineering students.  A few less proofs, a few more computation-based homeworks and lectures, and earlier introduction to javaplex (or another topology package) would suffice.  The most critical aspect of adapting the course would be to keep the big picture in everyone's mind at all times, so that the definitions and influx of information doesn't become overwhelming.  Including real world examples should keep the abstraction terror at bay.&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2014-04-21T00:19:37.743" LastActivityDate="2014-04-21T00:19:37.743" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1786" PostTypeId="2" ParentId="1763" CreationDate="2014-04-18T05:03:25.573" Score="11" Body="&lt;p&gt;&lt;strong&gt;Problem&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Each point in the plane is colored one of $n$ colors. Prove that there&#xA;  exists a rectangle whose four vertices are the same color.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Both the problem and the solution are very simple, yet those unfamilar with the Pigeonhole Principle would likely be at a complete loss to solve it.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Solution&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider a grid of points with $n+1$ rows and $n^{n(n+1)/2}+1$ columns.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Since there are $n$ colors, by the Pigeonhole Principle, for each column there must be a pair of the $n+1$ grid points with the same color.&lt;/li&gt;&#xA;&lt;li&gt;Each column has $\frac{n(n+1)}2$ possible positions for this same-colored pair. Since there are $n$ colors, each column has one of $n^{n(n+1)/2}$ possible same-colored pairs. By the Pigeonhole principle, two of the $n^{n(n+1)/2}+1$ columns  have the same pair, forming a rectangle.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This problem works even better when explained visually.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The USA Mathematical Talent Search asked this &lt;a href=&quot;http://www.usamts.org/Solutions/Solution4_1_18.pdf&quot; rel=&quot;nofollow&quot;&gt;problem&lt;/a&gt; for the case $n=3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want, you can extend this to a hypercube in $k$ dimensions.&lt;/p&gt;&#xA;" OwnerUserId="1005" LastEditorUserId="1005" LastEditDate="2014-04-20T03:31:43.427" LastActivityDate="2014-04-20T03:31:43.427" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1787" PostTypeId="2" ParentId="41" CreationDate="2014-04-18T05:49:25.363" Score="4" Body="&lt;p&gt;How about using cursive handwriting as shown below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/9I1qU.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="392" LastActivityDate="2014-04-18T05:49:25.363" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1788" PostTypeId="2" ParentId="1690" CreationDate="2014-04-18T06:43:35.727" Score="3" Body="&lt;p&gt;I recall one test I had as an undergraduate for a Linear Algebra course. It was, bar none, the most educational test I'd ever taken. It was a set of 50 questions, of which we only had to answer 30. Every single one was a conjecture, which we had to answer T/F and either formally prove or provide a counterexample for. Naturally, some of these questions were far easier to answer than others. You got 1 point for guessing T/F correctly, and 2 points for a correct explanation (half credit for kind of getting there). He'd grade the first consecutive 30 you made an attempt at, to prevent too much guessing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why was do I adore this test so much?&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It provides flexibility, and allows you to play to your strengths. Since you only have to answer a subset of questions, you don't have to worry about knowing everything. Still, you have to answer a majority of questions so it doesn't allow you to have too many weaknesses either.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It requires you to &lt;em&gt;identify&lt;/em&gt; which questions are even feasible for you to answer. I'm not saying put &quot;T/F P=NP&quot; on your test to gull weaker students into trying to prove things they can't. I'm saying that even being able to recognize trivial counterexamples and do those before they start on the more elaborate proofs (or trickier counterexamples) is a great skill and measures (IMO) a lot about their grasp of the material. I'm not usually a fan of time pressure on tests, but it really works here, and is made up for by the fact that they don't have to answer every question. Heck, this fact alone makes it a good candidate for an open book test.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It encourages simple, concise, elegant proofs. Of course, we all know that theorems don't spring up out of the aether, fully formed, with no rough drafts. Those simple proofs in books were worked and polished to death. However, the time pressure forces students to really consider &quot;what is the easiest way to show this is true/false?&quot; Even if it doesn't get you the best proof possible, it gets them thinking in that direction.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Obviously I'm not saying that this is the one true test. Different tests have different goals. What I do think is that tests like these not only test mastery, but also forces the student to think in new ways and enhances their education in addition to just testing them. It uses time pressure in a constructive way. It uses it to get them to think critically, rather than just staring at them angrily for rederiving a 30/60/90 triangle from an equilateral instead of memorizing it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't speak highly enough of how much I think tests like this do tests correctly from both a teaching and a learning perspective. Though it may be a bit intense for students conditioned to only expect &quot;standard&quot; math exams. Though one downside is that it does disproportionately penalize students particularly susceptible to the sunk cost fallacy that keep wasting their time on one question they just can't answer because DAMMIT I SPENT THIS MUCH TIME ALREADY I NEED TO FIGURE THIS OUT.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Disclaimer: it did take him like a month and a half to grade this doozy)&lt;/p&gt;&#xA;" OwnerUserId="521" LastEditorUserId="521" LastEditDate="2014-04-18T06:56:50.567" LastActivityDate="2014-04-18T06:56:50.567" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1789" PostTypeId="2" ParentId="41" CreationDate="2014-04-18T09:35:44.540" Score="10" Body="&lt;p&gt;In response to Chris Cunningham's request, I think this sort of 5 is unlikely to be mistaken for an s:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Bd9K8.png&quot; alt=&quot;5&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key is not just the right angle at the top left, but the sharp angle directly below it.  The danger here is to make sure you don't come so close to closing up the bottom that it looks like a 6.&lt;/p&gt;&#xA;" OwnerUserId="129" LastActivityDate="2014-04-18T09:35:44.540" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1790" PostTypeId="1" AcceptedAnswerId="1791" CreationDate="2014-04-18T12:06:28.453" Score="11" ViewCount="268" Body="&lt;p&gt;When I teach constant coefficient linear differential equations, the usual guess of an exponential can be motivated because it is &quot;approximately&quot; a fixed point for the differentiation operator. The aesthetic here being that mathematicians look at fixed points to exploit symmetry to find solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one considers a 2x2 first order matrix linear differential equation of the form $$X'=AX$$ when $A$ has a repeated eigenvalue with only a one-dimensional eigenspace, one can still discover what needs doing using the approximate fixed point idea while appealing to an analogy with the one-dimensional procedure of &quot;reduction of order&quot;. This is done &lt;a href=&quot;https://www.math.psu.edu/srikrish/math251/notes/repeatedeigs.pdf&quot;&gt;here&lt;/a&gt;, for example, in the way most of us probably teach it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always feel like I'm making a stretch when teaching this particular material, and was wondering if anyone had a more global explanation for the appearance of generalized eigenvectors here? I always felt that there should be, but can't think of a more natural way to explain this than the process found in the link above.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="262" LastEditDate="2014-04-18T13:14:47.523" LastActivityDate="2014-04-18T13:14:47.523" Title="A more natural motivation for the appearance of generalized eigenvectors in linear system with repeated eigenvalue" Tags="|undergraduate-education|concept-motivation|linear-algebra|differential-equations|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1791" PostTypeId="2" ParentId="1790" CreationDate="2014-04-18T13:06:05.903" Score="8" Body="&lt;p&gt;So, there is some point where they just propose multiplying the given solution by $t$ and then work out the generalized e-vector of order two condition. Pretty in its way, but I prefer the perspective of the matrix exponential.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Define $e^{tA}=I+tA+(t^2/2)A^2+ \cdots$. It is easy to show that the matrix exponential is a solution matrix and it is an invertible matrix. Thus, the columns of the matrix exponential form a fundamental solution set; contained in $e^{tA}$ are all the solutions of $x'=Ax$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question then, how to actually calculate the columns without explicit computation of the series. One technique is suggested by the following formula:&#xA;$$ e^{tA} = e^{\lambda t}(I+t(A-\lambda I)+(t^2/2)(A-\lambda I)^2+(t^3/3!)(A-\lambda I)^3+ \cdots ) $$&#xA;This identity holds for any complex scalar $\lambda$. However, it is most useful when we consider eigenvectors and then generalized e-vectors. It produces the usual cases given in the document linked by the OP as well as an infinite family of possibilities for higher order problems. Of course, the Jordan cannonical form completes the thought of this discussion. I can say more, but this is standard stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this said, I'm not sure this is the motivation you seek. Another idea, $y''=0$ has solution $y = c_1+c_2t$ by calculus I. Then do reduction of order and study the structure of the resulting solutions. You'll find our beloved eigenvector solutions as well as generalized e-vectors of order 2. This might motivate. You might look at:&#xA;&lt;a href=&quot;https://math.stackexchange.com/q/206967/36530&quot;&gt;https://math.stackexchange.com/q/206967/36530&lt;/a&gt; as the inquiry and discussion there is very much related to your question.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-04-18T13:06:05.903" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1792" PostTypeId="2" ParentId="1734" CreationDate="2014-04-18T14:31:38.647" Score="8" Body="&lt;ul&gt;&#xA;&lt;li&gt;I have been a professional programmer since I graduated in 2000.&lt;/li&gt;&#xA;&lt;li&gt;My degree is an MMath. I never studied CS.&lt;/li&gt;&#xA;&lt;li&gt;I will accept your premise of a single course, not give you a bucket list of everything potentially useful that adds up to at least a 1-year full-time syllabus.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have used &lt;em&gt;very close to none&lt;/em&gt; of the material from my degree in my professional work:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The notion of mathematical rigour and what it means to prove something is close to essential. But whatever you study, if you aren't doing this then it really isn't university-level mathematics, so it doesn't help us choose a syllabus.&lt;/li&gt;&#xA;&lt;li&gt;A few specific concepts such as complexity are very important &lt;em&gt;for communicating with other programmers&lt;/em&gt;. Complexity is also somewhat useful for noticing early that your code won't scale, but actually real computer programmers test that. You do not, strictly speaking, need to know what Omega(n^2) means in order to realise that looking at every pair of items will get slow eventually.&lt;/li&gt;&#xA;&lt;li&gt;At one point I was working in and around an entropy collector. I found it useful that I had taken a course on information theory, I actually knew what this &quot;entropy&quot; thing was that most of the programmers had no idea how to estimate rigorously. But that was just a fluke of course choice. I could just as well have read a book if I hadn't taken the course and if it had been all that important to my employer.&lt;/li&gt;&#xA;&lt;li&gt;One of my courses defined and proved the correctness of an algorithm for computation (the simplex method). If this kind of analysis would not be covered anyway by the CS syllabus, then it should be in the mathematics syllabus, and of course analysing that particular algorithm requires some abstract algebra. I've never used the simplex method professionally, but I have tried to work out whether my algorithm is correct.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The way I think and analyse problems, which was partly shaped by my degree, is fundamental to my work. Mathematics is certainly a respectable way to train for programming, but it's far from the only way, and I don't think there's any one thing on my university syllabus that I'd reject a CV just because it was missing. So I'd worry that this single course does half a job &lt;em&gt;not&lt;/em&gt; because there isn't time to cover the necessary material, but because there isn't time to teach someone to think like a mathematician. If you crack the latter then the former is just picking up the right book when you need it. Or of learning geometry in the computer graphics course, mechanics in the physical modelling course, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I don't know whether your CS course does what I thought of as CS back in the day, or whether it's primarily concerned with producing employable software/hardware engineers ready to go. The mathematics requirements of each are rather different, but it might be that in the CS case, a lot of stuff that I consider to be mathematics will in fact be taught in the course, it just won't be called mathematics. So I don't know quite how to interpret &quot;this is the only mathematics course they'll take&quot;. If they really do no other mathematics then to me it's not CS: call it software engineering. But I don't want to argue over what you call your courses, I'm just trying to understand what's called for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, this may be highly controversial, but my instinct is that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If the CS syllabus includes theoretical computer science, then the mathematics syllabus need only provide the necessary grounding for that. Consult the TCS syllabus to figure out what that is. There should also be content beyond that, but it doesn't actually matter what it is. Either elementary calculus or algebra provide plenty of opportunity for rigorous proof. Mechanics or probability/statistics might be more practically useful in life, but I think introductory courses don't really have time to avoid hand-waving important results so you risk sacrificing rigour. As long as the students have been taught what &quot;doing mathematics&quot; is, the specific content will appear in the courses it relates to, whether that's TCS or games programming.&lt;/li&gt;&#xA;&lt;li&gt;If the CS syllabus is all programming with very little TCS, then the &quot;mathematics&quot; syllabus should be TCS in disguise because that's the only way to present any theoretical material at all. Ditch the algebra (except that they'll need to know what vectors and matrices are) and calculus, do what you can from: computability (including classes of formal language and the machines that decide them), complexity, combinatorics, graph theory, information theory (entropy and codes), maybe some cryptology (which will bring in a bit of number theory). One course, two semesters, I guess you can introduce at least 4 fields in enough depth to matter? Teaching 'em mathematical physics so they can write games or the software for X-ray interferometers is someone else's problem.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm also guessing a bit what's considered &quot;remedial&quot;. I'd expect an 18-year-old who wants to do a CS degree to already have mathematics education, and I would expect that education to include basic: geometry, trigonometry, dynamics, probability, calculus (know the fundamental theorem of calculus, a list of indefinite integrals, and the product rule. Maybe know l'Hôpital's rule. Probably not able to prove any of them but might be able to prove the derivatives of polynomials from first principles), proof by induction, maybe some abstract algebra, maybe some non-trivial statistics (&quot;compute the mean&quot; is trivial, by non-trivial I mean, &quot;what's a two-tail test?&quot; on upwards). But my expectations are based on a different education system, so might be out of line for your students. If any of that isn't remedial, it might be worth throwing it in just so that your degree provides a broad mathematical grounding in addition to the specialized CS. Unfortunately this turns back into that bucket list. I don't believe you can expect to take students who have never heard the word &quot;integrate&quot; except in the context of Brown v. Board of Education, and give them broad mathematics in the time available. So you might have to give up the aspirations of the mathematics department and cater to the CS department directly. I guess you're more or less at the mercy of how much time they give you?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any or all applied mathematics &lt;em&gt;could&lt;/em&gt; be useful to a programmer:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;you won't get far writing a physics engine if you don't know mechanics&lt;/li&gt;&#xA;&lt;li&gt;you can't be a quant without sophisticated statistical modelling and some economics / game theory&lt;/li&gt;&#xA;&lt;li&gt;without numerical analysis all your floating-point code blows up due to the &quot;obvious&quot; algorithm being numerically unstable&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But you actually can't provide all of applied mathematics in the compulsory courses of a single undergraduate degree anyway, so why try. Ultimately it's domain knowledge for some programmers, it's not essential to the actual programming. You could just as well say that programmers &lt;em&gt;might&lt;/em&gt; need to know tax law ;-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth, IIRC the &quot;computation&quot; (there was no &quot;computer science&quot;) undergraduates at my university did something like half the first-year mathematics syllabus and then had to take a &quot;foundations of mathematics&quot; course including set theory and formal logic. How we laughed. But the intention of the course was to prepare them for academic computer science if anything: I don't think there was required to be any great advantage in preparing to be a programmer compared with the mathematics course. Or physics for that matter.&lt;/p&gt;&#xA;" OwnerUserId="321" LastEditorUserId="321" LastEditDate="2014-04-18T16:48:18.637" LastActivityDate="2014-04-18T16:48:18.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1793" PostTypeId="2" ParentId="1749" CreationDate="2014-04-18T20:11:45.007" Score="6" Body="&lt;p&gt;If he wants to memorize them, then obviously there is no other way than to memorize them. (I would let him estimate the number of entries in the bigger table compared to the usual $10\times10$-table first, though.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If he just wants to be able to quickly multiply two-digit numbers in his head, then I would start out differently:&#xA;Memorize the squares, powers of 2, 3, 5, 7, then move on to tricks like $a(a+2)=(a+1)^2-1$, memorize products of primes $19\times 23$, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A reasonable compromise is to memorize the table up to $25\times 25$ and exploit this to get the rest with arguments like $23\times 88 = 2300-23\times 12$.&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2014-04-18T20:11:45.007" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1794" PostTypeId="1" CreationDate="2014-04-19T00:11:44.177" Score="25" ViewCount="1714" Body="&lt;p&gt;I have always loved the beauty of mathematics and physics. However I'm severely dyslexic and find it hard to keep numbers in my head, any more than 4 numbers at a time and they melt together and lose their meaning. (You know that feeling when you wake up after a lucid dream but as you think about it, it just evaporates? Well that's what it's like.) However, I'm pretty good at understanding concepts like Cantor Sets, mathematical logic, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have degrees in the philosophy of science and have studied epistemology, so I'm comfortable with abstract concepts BUT, I want to do some mathematics, I want to try algebra and trigonometry, to play with equations but I just don't know how to start.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any books designed for the likes of me? Can someone who finds it hard to add two two-digit numbers together in his head, and even takes a second to decipher a 4-digit number on paper be taught to do advanced math? Is it like teaching a 1-legged man to run or a pig to fly?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested in hearing the opinions of people who love math, who can do math and maybe understand what a lot of my friends don't - namely why I at least want to try.  Thanks.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Thanks for your comments,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me rephrase my original question if it helps.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Being dyslexic. I have a very poor working memory (like a computer with very little ram if you like) the processing power required to recognise the shape of numbers written down as characters (i.e. 1, 5, 9, 3, 2, 8, etc) and then understand what they represent requires all my processing power. That means if I want to calculate two numbers in my head, 7+5 for example it takes a second but I can do it. However, if I want to add 14+63 my brain will spin and crash. Ive not been able to move past what most 7 year olds can do in math class.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;That said, in common with many dyslexics I’m very good at understanding complex and abstract theories and ideas (spatial reasoning). I have degrees (BA and MA form Durham) in subjects that can broadly be described as philosophy. I have been fascinated with subjects like mathematics and physics since I was a child and I have always read popular books on the subject. I can understand, for example, ‘set theory’ .. at a conceptual level, mathematic proof, .. at a conceptual level. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What you have written is encouraging, but math - even at a basic level is a foreign land to me. I would love to do some sort of higher math (no, I don’t really know what that means). but I would love to try to understand equations. But can I learn to do algebra or trigonometry and beyond when I can’t recite the 6x table or add 34 and 58 in my head? If I can, and I think you are saying that I might just be able to, where do I start? &lt;/p&gt;&#xA;" OwnerUserId="1016" LastEditorUserId="262" LastEditDate="2015-07-10T19:33:36.827" LastActivityDate="2020-09-21T06:18:18.673" Title="The impact of dyslexia on learning mathematics, and available resources" Tags="|reference-request|recreational-learning|special-needs-education|dyscalculia|" AnswerCount="3" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="1795" PostTypeId="2" ParentId="1794" CreationDate="2014-04-19T00:40:40.187" Score="6" Body="&lt;p&gt;I look forward to the responses to this interesting question.&#xA;Instead of answering directly,&#xA;permit me to make an analogy to art.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Frank_Stella&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frank Stella&lt;/a&gt; is a renowned artist whose influence and compositional talent cannot be questioned.&#xA;Nevertheless, I happen to know from reliable second-hand reports, that&#xA;he cannot &quot;draw&quot; in the sense that, say, Leonardo, or even a contemporary&#xA;cartoonist, can &quot;draw.&quot; This is roughly equivalent to limited mental arithmetic&#xA;abilities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just as the essence of art is the compositional inventiveness with shapes and colors&#xA;that resonate with viewers&amp;mdash;not drawing talent, similarly&#xA;the essence of mathematics is patterns of interrelated, fertile concepts&amp;mdash;not calculation abilities. &#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/A0JzN.jpg&quot; alt=&quot;FrankStellaWikipedia&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-04-19T01:42:25.600" LastActivityDate="2014-04-19T01:42:25.600" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1796" PostTypeId="2" ParentId="1042" CreationDate="2014-04-19T06:22:45.523" Score="2" Body="&lt;p&gt;If you understand some French, you can go to the site of &lt;a href=&quot;http://mathenjeans.free.fr/amej/accueil.htm&quot; rel=&quot;nofollow&quot;&gt;Math en Jeans&lt;/a&gt; and look at the subjects other people have worked at with high school students.&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2014-04-19T06:22:45.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1797" PostTypeId="2" ParentId="1690" CreationDate="2014-04-19T08:53:51.303" Score="2" Body="&lt;p&gt;I use such questions frequently, but in a limited fashion. Typically my exam contains five or six question, and one of those consists of, say, four true/false questions. The way I do it is that I make four claims, and students are to either give a quick proof or give a counterexample. I design them in a way that either it is easy to construct a counterexample using the body of examples covered in class, or there is a two line argument reducing the claim to a result in the lecture notes. I think explanations are a must.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I also include trick questions. Then a good explanation for a wrong choice earns partial credit. For example, if a student gives an argument that covers the generic case, but fails in a special case that actually leads to a counterexample, then I will be relatively generous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example in first semester calculus I might ask something like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fact or fiction?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A) If the sequence $(a_n)$ converges, then so does the sequence $(a_n/n)$.&lt;/li&gt;&#xA;&lt;li&gt;B) If the sequence $(a_n/n)$ converges, then so does the sequence $(a_n)$.&lt;/li&gt;&#xA;&lt;li&gt;C) If the sequences $(a_n)$ and $(a_nb_n)$ both converge, then so does the sequence $(b_n)$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Here a student is expected to A) for example appeal to the result about the product of two converging sequences being convergent itself, B) come up with a counterexample, C) come up with a counterexample, but will get partial credit for realizing that $(b_n)$ should converge as a quotient of two converging sequences, but forgets about the possibility of an attempt to divide by zero (that actually is crucial in finding a counterexample).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In second semester calculus I might ask something like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fact or fiction?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A) If a function is differentiable on the interval $[0,1]$, then it is also Riemann integrable over that interval.&lt;/li&gt;&#xA;&lt;li&gt;B) If the sequence $(a_n)$ converges, then the power series $\sum_n a_nx^n$ converges for all $x$ with $|x|&amp;lt;1$.&lt;/li&gt;&#xA;&lt;li&gt;C) If the sequence $(a_n)$ converges, then the power series $\sum_n a_nx^n$ converges for all $x$ with $|x|\le1$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A student should be able to prove A by recalling from first semester that a differentiable function is continuous and combine that with a theorem from second semester telling that a continuous function is integrable over a closed interval. Parts B and C simply test whether they absorbed the basic facts from the chapter of series, power series, and the body of examples covered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sad to say I often include claims testing whether students understand basic &quot;if..then&quot; clauses at all. The way I grade these comes as a surprise to some, so to soften the blow, I include a sample question like this in the HW set prior to midterms, and explain the philosophy.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In case you were wondering: I teach a section for non-math majors. Some of them major in theoretical physics or some such discipline, and I need to cater to their need to be at least somewhat prepared for more advanced courses. Some of the students shouldn't be in that course, but are there because they did well in high school math, and think they can waltz through this. Yet they wouldn't recognize a basic logical fallacy if it hit them on the nose. Schizophrenic, yes. But somehow we cope.&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2014-04-19T09:04:04.157" LastActivityDate="2014-04-19T09:04:04.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1798" PostTypeId="2" ParentId="1756" CreationDate="2014-04-19T10:12:59.757" Score="7" Body="&lt;p&gt;My strategy for exam preparation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Write increasingly short outlines of the course, then of the content/proof and check that you can get from each version to the longer version.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us say that you have a proof of the existence of an Eulerian circuit in a graph with appropriate conditions, then the roughest sketch of the proof might read at most something like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Start anywhere and go on as long as possible.&lt;/li&gt;&#xA;&lt;li&gt;Never stuck until you return (even degrees).&lt;/li&gt;&#xA;&lt;li&gt;Repeat with unsatisfied vertices and glue them to the first circuit (connected!).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now, I check that &quot;Proof of the existence of an Eulerian circuit&quot; triggers this list of keywords. Then I check that I can convert each step into formal proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the proof is more complicated, you can have one more intermediate stage of unfolding the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, this strategy corresponds to the way memory works. Good chess players are much better than normal people when asked to remember a chess position, but they are almost equally bad at memorising a &lt;em&gt;random&lt;/em&gt; placement of chess figures on a chess board.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why? Because they remember something like &quot;In the left half, this is basically the game of Karpov vs. Kasparov in 1985 except for one pawn, in the right half, it is the classical position after ...&quot;, so they reduce the position to something like five items where each item again stands for several items.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It will be very, very hard to remember a proof if you cannot represent it in your head with 3 to 7 informal steps. I would go so far to guess that the problem is that you think that &quot;following a proof&quot; (in the sense of seeing that each step is correct) is enough. However, if you cannot partition the proof into steps, you have not understood it in any meaningful sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could argue that the professor should do this structuring and I do usually provide it. However, you will remember it even better if you manage to find the structure yourself.&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2014-04-19T10:12:59.757" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1799" PostTypeId="2" ParentId="1550" CreationDate="2014-04-19T11:07:32.000" Score="9" Body="&lt;p&gt;Can't believe no one's mentioned the coke-can problem!&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You are the designer of drinks cans for Coke, how can you make them the most money?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Pretty quickly it comes out that you want a cylinder of volume 330ml with minimal surface area (use the least metal possible to enclose your beverage).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a classic problem and one kids can very easily relate to on all number of levels, especially if you hype it a little bit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The interesting punchline is that Coke cans don't look at all like your design (which is predictably as close to a cube as a cylinder can get), for reasons that transpire to be aesthetic. But paint cans totally do though... This is because the metal must be significantly thicker in this case.&lt;/p&gt;&#xA;" OwnerUserId="1023" LastActivityDate="2014-04-19T11:07:32.000" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1800" PostTypeId="1" CreationDate="2014-04-19T16:59:14.120" Score="5" ViewCount="378" Body="&lt;p&gt;In courses that rely primarily on &quot;lecture notes&quot; over a text book, I've primarily seen two different ways to go:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Write slides that intentionally serve as notes (either by adding more detail after the fact or writing the presentation in this way intentionally)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Write a document with a standard &quot;book-like&quot; feel to it that contains the ideas from lecture, generally in a section/paragraph format.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we assume (unrealistically) that there were infinite time to generate slides, lecture notes, etc, I am wondering what makes one format better over another.  Here are some (definitely incomplete) initial ideas:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pro Slides:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Animations and pictures are easier to write for this format&lt;/li&gt;&#xA;&lt;li&gt;Slides provide a &quot;pre-chunked&quot; format of the material (in the sense that the student is looking at a small piece at a given time, rather than just reading through pages)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Pro &quot;book-like&quot;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students are, generally more familiar with this format&lt;/li&gt;&#xA;&lt;li&gt;It can often be more thorough&lt;/li&gt;&#xA;&lt;li&gt;The notes are more easily editable if the lecture itself changes due to student questions and interests.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So, finally, the question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any research backing one of these formats over the other?  Or lacking that, has anyone tried both formats and noticed students are more engaged and likely to use one over the other?&lt;/p&gt;&#xA;" OwnerUserId="93" LastActivityDate="2014-04-20T16:06:04.867" Title="Lecture Notes: Slides with extra detail or longer PDF?" Tags="|education-research|presentation|lecture-notes|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1801" PostTypeId="2" ParentId="1794" CreationDate="2014-04-19T17:59:41.010" Score="7" Body="&lt;p&gt;I'm not exactly sure what's being asked here, since you say you want to to learn how &quot;to do advanced math,&quot; but also that you &quot;want to try algebra and trigonometry, to play with equations.&quot; I can't tell exactly what you mean by &lt;em&gt;do advanced math&lt;/em&gt;, but you say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I'm pretty good at understanding concepts like Cantor Sets, mathematical logic, etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So:&lt;/strong&gt; It seems to me like you have &lt;em&gt;already&lt;/em&gt; &quot;done [some] advanced math.&quot; Furthermore, you write:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I have degrees in the philosophy of science and have studied epistemology, so I'm comfortable with abstract concepts&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So:&lt;/strong&gt; Sure, if those degrees are legitimate, and your study of [some area of] epistemology was more than superficial, then I expect you could handle abstract mathematics as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;That said:&lt;/strong&gt; I get the sense that you are concerned about a combination of dyslexia and dyscalculia interfering with your learning. To this extent, there are some comments by Poincare and Hadamard about how neither of them was a particular capable &quot;calculator&quot; (in the sense of doing mental computations). The &lt;a href=&quot;http://en.wikipedia.org/wiki/Jacques_Hadamard&quot;&gt;latter&lt;/a&gt; goes on to remark in his book &quot;The Psychology of Invention in the Mathematical Field&quot; (1945) that &lt;em&gt;very few&lt;/em&gt; of the mathematicians who he knows think in words or even algebraic symbols. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More precisely, Hadamard writes that nearly all of the mathematicians who he knows &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;avoid not only the use of mental words but also, just as I do, the mental use of algebraic or any other precise signs; also as in my case, they use vague images&quot; (p. 84). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Finally, Hadamard discusses physical intuition and mental pictures of a kinetic sort as being relevant to mathematicians' ways of thinking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Thus:&lt;/strong&gt; If you can think abstractly but are tangled up by words and symbols, then you could still be in good mathematical company.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;For completeness:&lt;/strong&gt; Hadamard does write about a few notable exceptions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He contrasts the use of image-based thinking with the following three mathematicians:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;George D. Birkhoff, &quot;who is accustomed to visualize algebraic symbols and to work with them mentally,&quot; Norbert Wiener, who &quot;happens to think either with or without words,&quot; and Jesse Douglas, whose &quot;research thought is in connection with words, but only with their rhythm, a kind of Morse language where only the numbers of syllables of some words appear&quot; (p. 84). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Lastly, he mentions the rare exception of George Polya, whose thoughts rely in some large part on words and letters. Hadamard quotes Polya describing his own thinking: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;I believe… that the decisive idea which brings the solution of a problem is rather often connected with a well-turned word or sentence… The right word, the subtly appropriate word, helps us to recall the mathematical idea…&quot; (p. 84). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Hadamard himself continues: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Moreover, [Polya] finds that a proper notation – that is, a properly chosen letter to denote a mathematical quantity – can give him similar help; and some kind of puns, whether of good or poor quality, may be useful for that purpose&quot; (p. 85).  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(In case it is unclear: I highly recommend this book!)&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-04-19T17:59:41.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1802" PostTypeId="2" ParentId="1800" CreationDate="2014-04-19T18:17:39.867" Score="4" Body="&lt;p&gt;I can only offer my personal opinion on the matter, having witnessed both formats as a student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that the material you use should be designed for the specific use you want them for. Slides are great for talks, presentations and to supplement lectures. They are usually designed in a way that you can see all the important things at a glance. Unwieldy calculations can be chipped into gradually uncovering formulas. Good slides complement the things you talk about and should work best when accompanied by a talker. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Books and lecture notes, on the other hand are self-contained and are designed for people who read them on their own. There is few ambiguity, and unwieldy calculations are put into in-line formulas, blocks of equations and are interlaced with comments. They are sometimes printed out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because their use is so different, I suggest to pick the mode of presentation which fits best for its intended use: Slides style for talks/lectures, book style for lecture notes.&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-04-19T18:17:39.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1803" PostTypeId="2" ParentId="1800" CreationDate="2014-04-19T18:18:38.087" Score="2" Body="&lt;p&gt;With your assumption of &quot;infinite effort available&quot; the &quot;easier to edit&quot; is of no consecuence...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, slides should just point out the important aspects, &quot;making them into lecture notes&quot; isn't the right use of the medium. They should &lt;em&gt;supplement&lt;/em&gt; the detailed notes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you distribute your lecture notes as e.g. PDF (not printed on paper), you can certainly include links (internal to the document, to external URLs), animations, and even interactive applets (I know that it can be done e.g. in the beamer LaTeX package to create slides, which generates PDF as output). So this really isn't a limitation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or (as has been said here IIRC) you could just go for a website, complete with cross links, animations, interactive applications, the works. See for example &lt;a href=&quot;http://www.cut-the-knot.org&quot; rel=&quot;nofollow&quot;&gt;Cut the knot&lt;/a&gt;, like it's &lt;a href=&quot;http://www.cut-the-knot.org/ctk/Water.shtml&quot; rel=&quot;nofollow&quot;&gt;water problem&lt;/a&gt; discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given infinite time to do it, this won't be a problem ;-)&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-19T18:18:38.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1805" PostTypeId="1" AcceptedAnswerId="1814" CreationDate="2014-04-19T20:20:39.317" Score="15" ViewCount="1169" Body="&lt;p&gt;I realize this is a very specific question, but it is something I have noticed over the last few semesters teaching sequences and series in Calculus 2 to undergraduates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of the Limit Comparison Test is to take an intuition, such as&#xA;$$\text{The series }\sum_{n=1}^\infty \frac{5\sqrt{n}-2}{n+1}\text{ ''behaves like'' }\sum_{n=1}^\infty \frac{1}{\sqrt{n}}\text{ , which we know is divergent}$$&#xA;and make it more rigorous. I take great care in explaining how the limit&#xA;$$\lim_{n\to\infty} \frac{(5\sqrt{n}-2)/(n+1)}{1/\sqrt{n}} = \lim_{n\to\infty} \frac{5n-2\sqrt{n}}{n+1} = \lim_{n\to\infty}\left(\frac{5n}{n+1} - \frac{2\sqrt{n}}{n+1}\right) = 5-0 = 5$$&#xA;yielding a &lt;em&gt;finite, nonzero&lt;/em&gt; result shows that the &lt;em&gt;terms&lt;/em&gt; of the two series approach 0 at rates that are approaching &lt;em&gt;constant multiples&lt;/em&gt; of each other, thereby showing that their convergent/divergent behavior is identical. We go through several examples, each time saying, &quot;Well, it looks like this series is 'kinda like' $\sum 1/n^2$, so let's use Limit Comparison to test it!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is how the test is presented in every calculus textbook I have seen, and how I have taught it in the past. I try to give more of an explanation as to how the test works, but I definitely point out how it &lt;em&gt;needs&lt;/em&gt; to be used to test our intuitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perennially, though, I notice at least a few (and sometimes shockingly many) students &quot;inventing&quot; their own form of the Limit Comparison test and using it on homework and exams. It goes something like this:&#xA;$$ \lim_{n\to\infty} \frac{5\sqrt{n}-2}{n+1} = \text{(by l'Hôpital)} \lim_{n\to\infty} \frac{5/(2\sqrt{n})}{1} = \lim_{n\to\infty} \frac52 \cdot\frac{1}{\sqrt{n}} \text{ and we know $\sum \frac{1}{\sqrt{n}}$ diverges}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quite often, this method is not even presented so nicely, showing something more like this:&#xA;$$\frac{5\sqrt{n}-2}{n+1} \rightarrow \frac{5/(2\sqrt{n})}{1}\rightarrow \frac52 \cdot\frac{1}{\sqrt{n}}  \rightarrow \text{diverges}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a few questions about this phenomenon:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Have you also noticed this before? How did you react?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Why does this occur? Are there texts/courses/teachers out there who teach this as a valid method? If not, are students tapping into some common ideas that make them come up with this (seemingly) independently every semester?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;(Whether or not this is currently being taught) are there good reasons to do so?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;If this is not a good approach to use, how can we show this to students?&lt;/strong&gt; It does seem to be &quot;effective&quot; somehow, so how might we dissuade them? Should we show them outright and say, &quot;Don't do this&quot;? Or do we handle it only with those who re-invent this particular wheel?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="80" LastEditorUserId="123" LastEditDate="2014-04-22T20:09:08.263" LastActivityDate="2014-04-22T20:09:08.263" Title="Students using l'Hôpital's Rule on the terms of a series, instead of the Limit Comparison Test" Tags="|undergraduate-education|calculus|students-mistakes|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1806" PostTypeId="2" ParentId="1550" CreationDate="2014-04-19T20:21:49.920" Score="3" Body="&lt;p&gt;A real-life optimization problem: It takes about 20 clicks and 2 minutes to shoot a wolf. 200 dead wolves will get you enough experience points to get to the next level. It takes about 50 clicks and 10 minutes to steal gold from a palace. 50 burglaries will get you enough experience points to get to the next level. However, you would get carpal tunnel syndrome after 3000 clicks which would force you to stop playing for a day. What is the shortest possible time you can get to the next level?&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2014-04-19T20:21:49.920" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1807" PostTypeId="2" ParentId="604" CreationDate="2014-04-19T20:46:07.757" Score="10" Body="&lt;p&gt;Several arguable points where made in previous answers, mostly that the abusive notation $x+\sin x$ for $x\mapsto x+\sin x$ is in many circumstances lighter, while the more formal notation avoids some confusions that are common among students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the most important point is to &lt;strong&gt;adapt the notations and the rigor to the kind of students we deal with&lt;/strong&gt;. In front of first and second year students, I try to always distinguish functions from values (or from expressions with free variables), because the very notion of function is shaky in their minds. I ask them to write things rigorously too, until they master them. With higher level math majors, I start to accept more sloppiness, as seems convenient, and sometimes reminding students the abuse made to notations. In a graduate course in differential geometry, abuses would probably be everywhere (but I did not teach such a course yet).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-19T20:46:07.757" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1808" PostTypeId="2" ParentId="978" CreationDate="2014-04-20T03:25:45.220" Score="3" Body="&lt;p&gt;I think you need to present this lady with the information in it's simplest form, for 63/3 forget about fractions, we are trying to break 63 things into 3 groups, act this out with 63 things, or at least draw it on paper so you can teach her the meaning of these symbols.  In this case I have 63 Easter eggs and want to divide them between  you, me and your friend (3), ensure you also write the equation down as you need to also establish in her mind that they mean the same thing.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the general problem you want to take 2 steps back and explain the concept in non-mathematical ways. What are we doing and why.  The more relevant the example is to her the better she will understand and remember it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember also no-one like to admit they are wrong or stupid,  many people have a false idea of their skills and abilities at many things, (Not you or me obviously, but those other people :)  This is often sustained by ignoring any evidence they lack skill, and emphasising anything that sustains their belief.  Generally the smarter/more able someone is the more likely they are to admit their short falls.  This speeds up their learning as being aware of a problem or short falls allows you to focus on it.  This lady appears to realise she needs help understanding, and has recognised you do understand and can help her, the more empathy you have with her, the easier it will be for you to help her.&lt;/p&gt;&#xA;" OwnerUserId="910" LastActivityDate="2014-04-20T03:25:45.220" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1809" PostTypeId="2" ParentId="1551" CreationDate="2014-04-20T04:08:28.770" Score="3" Body="&lt;p&gt;When I was in high school we had a brilliant teacher who gave us an assignment every second week and a test on the alternant weeks.  In the assignment he has 10 or so boring questions about whatever we were learning, then he had his alpha, beta and gamma questions.  The Greeks were worth bonus marks, in fact if you answered all the alphas you usually made enough bonus marks to not need to do the boring questions, and a solid attempt at a gamma or 2 would also be sufficient to pass.  I am not sure how he sold it to us, but I always started with the Greeks, at least reading them and thinking about how I would start solving the problem, I never remember doing any of the boring questions and always got at least a pass mark.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reflecting on the problems years later I realised the Greeks were also structured so they generally either required a deeper understanding of whatever we were learning, or provided a historical or practical usage of the underlying concepts, or he introduced whatever we were going to learn next, or merged something we had just learnt with something we had previously learnt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly in the class of 25, 20 came in the top 100 in the state in our final exam.  I am not sure if this was because of the fun we had, or the fact our teacher loved maths, and taught us how much fun it was and how easy it was. When I got to university I discovered that I had learnt most of what they were trying to teach 2 years before and never realised this was meant to be hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also don't know how my teacher dealt with the slower students, except perhaps they didn't start with the Gammas like I insisted on doing.&lt;/p&gt;&#xA;" OwnerUserId="910" LastActivityDate="2014-04-20T04:08:28.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1812" PostTypeId="1" CreationDate="2014-04-20T13:35:56.963" Score="8" ViewCount="250" Body="&lt;p&gt;We'll reach vector calculus very soon and the following problem presents itself: &lt;strong&gt;how can I help students distinguish curves, surfaces and volumes as separated entities?&lt;/strong&gt; I've seen they hold the implicit belief that any surface bounds volumes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Examples might help. One common problem in vector calculus is to use theorems to simplify or substitute obstruse computations for simpler ones, such as you want the flux of a horrible vector field over a hemisphere. To use Gauss's theorem you close that surface (such as taking $z=0$ assuming you have $x^2+y^2+z^2=r^2$ for $z \geq 0$), use it, and then compute the flux through the plane normally. However I usually see students apply Gauss's theorem blindly and think the result is the desired answer. &lt;strong&gt;They don't see that the hemisphere does not bound a volume and that there are infinite ways to do so.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="47" LastEditDate="2014-04-23T02:35:35.623" LastActivityDate="2014-04-23T02:35:35.623" Title="Surfaces and volumes for vector calculus" Tags="|undergraduate-education|mathematical-pedagogy|vector-calculus|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="1813" PostTypeId="2" ParentId="1800" CreationDate="2014-04-20T16:06:04.867" Score="2" Body="&lt;p&gt;First: I'm not aware of any research on this sort of thing. Neither is specifically designed to create opportunities for students to think mathematically, so I doubt there is a significant math ed basis (or interest) for looking into this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My guess is that a lot of the use of one or the other technology will come down to preference and convenience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as that goes, my opinion on the choice of technologies for presentation assuming infinite time would be something like a &lt;a href=&quot;http://prezi.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Prezi presentation&lt;/a&gt;. Assuming we are talking about something relatively agnostic to the subject area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It solves a number of the drawbacks of the slide technology. For instance, whereas slides are essentially a one-dimensional sequentially-arranged (though randomly accessible) structure, Prezi allows for any 2 dimensional arrangement. Though 2 dimensionality doesn't confine your structure to 2 dimensions (i.e. you're not stuck to a grid structure; you could use a tree or other structure). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have used a Prezi to create a main, sequential thread for a lecture (picture it as horizontally arranged) with side-trips off each slide which can go off in another direction temporarily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/dZNYw.jpg&quot; alt=&quot;schematic of Prezi structure&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's one possibility, but obviously there are others based on whatever type of structure you can think of that an be laid out in 2 dimensions and makes sense for the organization and interrelatedness of your subject matter. Another &quot;dimension&quot; is size. You can compress blocks of explanatory text within slides so that they are visible, but not intrusive. Clicking on them will zoom into them. For example, definitions that are useful to have at hand, or elaborations that you would not go into detail with in class would be available for the students to use when they are reviewing the lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can define a main &quot;path&quot; through the lecture, but alternate paths are possible just by clicking around. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not the best example for what you're asking about, but &lt;a href=&quot;http://www.prezi.com/eanmkdg71pto/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here is a Prezi I made&lt;/a&gt; fairly quickly (from templates) to aid in an informal presentation I was collaborating on for a student-group discussion about one of Alan Schoenfeld's research studies. It shows some of the zooming in, 2D placement of &quot;slides&quot;, inclusion of pictures. YouTube videos can also be incorporated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a pretty flexible tool for presentation, and what I find is the main limitation is how long it can take to produce a good Prezi... but you offered unlimited time, so. :)&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-04-20T16:06:04.867" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1814" PostTypeId="2" ParentId="1805" CreationDate="2014-04-20T19:30:06.327" Score="14" Body="&lt;p&gt;I have also observed students doing this, and then I gave a 'proof' of it, and told a friend I was going to publish it. Then I did a literature search and discovered that it is false. Even though the limit before and after L'Hopitals must be the same, the rate of convergence can change. See &lt;a href=&quot;https://math.stackexchange.com/questions/77024/lhopitals-rule-and-series-convergence&quot;&gt;this question&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/questions/39979/when-does-lhospitals-rule-work-for-series&quot;&gt;this question&lt;/a&gt; on Math Stackexchange, where they give examples and counterexamples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that this approach comes from a fundamental misunderstanding of the difference between sequences and series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that you should emphasize that L'Hopitals rule has to go from start to finish, i.e. it can't be used to change one function into another unless you plan on taking the limit and getting a number at the end. Once you get that number, the intermediate steps cannot be used for an other purpose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would use the linked examples (especially the first) to show them it doesn't work; although some students may have difficulty with the counterexample, in which case you can use a CAS to 'prove' that the counterexample doesn't work.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-04-20T19:30:06.327" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1815" PostTypeId="2" ParentId="1756" CreationDate="2014-04-20T21:17:33.437" Score="2" Body="&lt;p&gt;The most helpful aid to my undergraduate exam preparation was &lt;strong&gt;writing 3 x 5 notecards&lt;/strong&gt;. I usually wrote the name of a theorem/term/concept on the front and on the back gave the definition/description and/or an example. I think notecards were helpful to me for the following reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The act of writing the cards helped me to remember my notes and look over them carefully without wasting time.&lt;/li&gt;&#xA;&lt;li&gt;Notecards help condense concepts into a short space so that I was not overwhelmed by too many pages of notes.&lt;/li&gt;&#xA;&lt;li&gt;Using the notecards before the test helped me see which areas I needed to spend the most time reviewing and which I already understood.&lt;/li&gt;&#xA;&lt;li&gt;Notecards are also a useful reference once a class is over in case you want to go back for a definition, key concept, or another test.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="867" LastActivityDate="2014-04-20T21:17:33.437" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1816" PostTypeId="1" CreationDate="2014-04-20T21:25:25.577" Score="10" ViewCount="5894" Body="&lt;p&gt;&lt;strong&gt;Where can I go to find open source high school level textbooks?&lt;/strong&gt; I'm especially interested in geometry and algebra, though probability and trig are also useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested in rearranging and adapting these to my school's integrated math curriculum to fit the South African national curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question is basically the same as &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1479/open-source-math-textbooks&quot;&gt;Open-Source Math Textbooks&lt;/a&gt;, except I'm looking specifically for high school - not undergraduate - textbooks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example of what I'm looking for is &lt;a href=&quot;http://en.wikibooks.org/wiki/Subject:K-12_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikibooks&lt;/a&gt; but unfortunately most of these books are unfinished.&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2020-03-05T20:31:28.620" Title="Open Source High School Textbooks" Tags="|secondary-education|textbooks|" AnswerCount="8" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1817" PostTypeId="1" CreationDate="2014-04-20T21:37:24.810" Score="102" ViewCount="20401" Body="&lt;p&gt;I enjoy talking about Pythagoras when I teach the Pythagorean theorem. I sometimes mention Descartes when introducing Cartesian coordinates. And Leibniz and Newton are mentioned in many calculus classes. But all of these famous mathematicians accessible to high school students are male.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What &lt;em&gt;female&lt;/em&gt; mathematician can I introduce to my high school students? And what mathematical concept did she work with?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="867" LastEditDate="2014-04-20T21:43:13.320" LastActivityDate="2020-05-08T14:18:48.613" Title="What female mathematician can I introduce to my High School students?" Tags="|secondary-education|examples|history|women|" AnswerCount="35" CommentCount="26" ContentLicense="CC BY-SA 3.0" />
  <row Id="1818" PostTypeId="2" ParentId="1812" CreationDate="2014-04-20T21:55:34.170" Score="7" Body="&lt;p&gt;I think the situation is clearer in $\mathbb{R}^2$ than in $\mathbb{R}^3$.  Draw an open curve and ask students how much area it enclosed.  Presumably they will recognize that the question has no real answer -- there is no way to tell what is &quot;inside&quot; and &quot;outside&quot; an open curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Perhaps one way to dramatize this is using a Paint-type program...  They typically have a &quot;fill&quot; tool that lets you fill an enclosed region with a solid color.  If you draw a curve that is not closed and try to &quot;fill&quot; it the paint pours out and fills the entire drawing area.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to why they have this problem, I suspect it is because the axes themselves, together with horizontal and vertical gridlines, create visual &quot;noise&quot; that is hard to distinguish from the actual curve or surface.  For example, when the graph of $y = \sqrt{1-x^2} + 2$ is drawn it looks like an open upper semicircle, but when the graph of $y = \sqrt{1-x^2}$ is drawn looks like a &lt;em&gt;closed&lt;/em&gt; semicircle because the x-axis appears to be part of the picture.  I imagine something analogous happens in 3-dimensions; I note that in the example you gave, students had difficulty recognizing that the plane $z=0$ was not already part of the figure, and I suspect this may have had something to do with how the surface was sketched or plotted (did the illustration include some kind of shading to show where the xy-plane was?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some thing that might help is to use more examples where the &quot;opening&quot; to the surface does not easily align with an axis or other visual element that could masquerade as part of the surface.  Of course, this technique could make the computations more complicated, as we typically prefer examples that have precisely that kind of arrangement because they are easier to work with!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;  It occurs to me that another reason students have trouble with the notion that open surfaces do not enclose a definite volume is because that idea runs utterly counter to everyday, normal experience.  Consider measuring cups, cereal bowls, 2-gallon paint pails.  Our very notion of &quot;volume&quot; is, in the world of real life, based on filling &lt;em&gt;open containers&lt;/em&gt; like graduated cylinders or beakers.  Even in calculus classes we give them questions where they are supposed to find the dimensions of an open-top box or cylinder containing a maximal volume under certain constraints.  So of &lt;em&gt;course&lt;/em&gt; they think that a surface that is open encloses a volume.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What makes this notion so stubborn?  Perhaps it stems in part from the fact that when we do consider an open surface it typically has a boundary curve that lies in a plane, and therefore there is a &quot;natural&quot; way to cap it.  That, after all, is how real-life containers work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the best way to deal with issues like this -- where there is a technical usage that differs significantly from the way a word or concept is used in everyday experience -- is to confront it head-on:  before discussing integration over surfaces or volumes, it is essential to have a discussion about the difference between open and closed surfaces, and to be explicit about the precise way the phrase &quot;enclosed volume&quot; or &quot;volume of the interior&quot; is used in this context.  And in every example you demonstrate you should begin by asking:  Is this surface open or closed?  If open, how can we close it off to create a well-defined interior?  etc.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2014-04-23T02:05:33.380" LastActivityDate="2014-04-23T02:05:33.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1819" PostTypeId="1" AcceptedAnswerId="1827" CreationDate="2014-04-20T21:56:21.110" Score="15" ViewCount="1072" Body="&lt;p&gt;Does anyone have any experience using either &lt;a href=&quot;https://en.wikipedia.org/wiki/GAP_%28computer_algebra_system%29&quot;&gt;GAP&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Magma_%28computer_algebra_system%29&quot;&gt;Magma&lt;/a&gt; in a first abstract algebra course?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The context here is a course for sophomore/junior math majors that focuses primarily on groups.  I'm curious:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;What are the advantages (and disadvantages) of introducing either GAP or Magma in such a course?  For example, does it somehow make groups more concrete for students?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What sorts of homework problems involving GAP/Magma would be reasonable to assign?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How much instruction do the students need in GAP/Magma before they can solve meaningful problems?  Is it something they can figure out on their own, or is it necessary to spend a class day in a computer lab?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Does anyone have any resources along these lines that they would be willing to share?  References to books or articles would also be appreciated.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-12-21T21:08:45.640" Title="Can GAP/Magma be helpful for a first abstract algebra course?" Tags="|undergraduate-education|technology-in-education|abstract-algebra|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1820" PostTypeId="1" AcceptedAnswerId="1825" CreationDate="2014-04-20T22:40:50.120" Score="24" ViewCount="11870" Body="&lt;p&gt;I've always been puzzled by the unequal treatments of square roots and logarithms in school mathematics.  In the United States, most students know what a square root is before they enter high school (9th grade, age 14 or so), and can answer questions such as&#xA;$$&#xA;\sqrt{25} \,=\, \text{?}\qquad\text{or}\qquad \sqrt{64} \,=\, \text{?}&#xA;$$&#xA;Most students also learn about the algebraic properties of square roots in a first algebra course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, it seems to be very rare for 8th grade students to be able to answer questions like&#xA;$$&#xA;\log(100) \,=\, \text{?}\qquad\text{or}\qquad \log(10\text{,}000) \,=\, \text{?}&#xA;$$&#xA;Indeed, students don't usually learn anything about logarithms until Algebra 2 or even Precalculus.  One result of this is that calculus students always seem very comfortable with square roots, but have a very shaky knowledge of logarithms, even though the two concepts have about the same difficulty level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is, why the unequal treatment?  Is the idea that square roots are somehow more common or more useful than logarithms?  Or is there something that's harder about logarithms that I'm missing?&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2014-04-20T23:00:01.053" LastActivityDate="2020-01-01T16:01:46.003" Title="Why aren't logarithms introduced earlier?" Tags="|secondary-education|curriculum|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="1821" PostTypeId="2" ParentId="1817" CreationDate="2014-04-20T22:51:03.113" Score="91" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Emmy_Noether&quot;&gt;Emmy Noether&lt;/a&gt; comes first to mind, as one of the most influential mathematicians in abstract algebra, specifically in the development of Noetherian rings (along with many properties of ideals).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One aspect of her work that high school students might like is from another area, analysis. &lt;a href=&quot;https://en.wikipedia.org/wiki/Noether%27s_theorem&quot;&gt;Noether's theorem&lt;/a&gt; says that every symmetry of the laws of nature (or the universe) gives rise to a Conservation Law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, energy is conserved because of time symmetry (meaning that the laws of nature don't change after time).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Momentum is conserved because of translation symmetry (meaning that the laws of nature are the same at every point).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Angular momentum is conserved because of rotation symmetry (meaning that the laws of nature are the same in every direction).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Recently I've learned that, in quantum mechanics, Noether's theorem has quite a few more implications. For instance, the fact that the phase factor is redundant (i.e. the direction of the complex number that you square to get a probability) gives rise to conservation of charge, and quite a few other things get conserved in QFT.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-08-21T01:19:03.473" LastActivityDate="2014-08-21T01:19:03.473" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1822" PostTypeId="2" ParentId="1816" CreationDate="2014-04-20T23:24:42.170" Score="2" Body="&lt;p&gt;If you have such a specific curriculum in mind, with presumably many interested parties, your best bet would be to try to band together a group of like-minded people and start writing away. Just look around for what makes open source projects successful, consider the language to use (I'd suggest LaTeX with the AMS packages, and a few ground rules) and where to host the project (some of the public git hosting sites, or perhaps &lt;a href=&quot;http://sourceforge.net&quot; rel=&quot;nofollow&quot;&gt;http://sourceforge.net&lt;/a&gt;). Open source works by &lt;em&gt;you&lt;/em&gt; scratching your own itch.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-20T23:24:42.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1823" PostTypeId="2" ParentId="825" CreationDate="2014-04-20T23:48:35.780" Score="1" Body="&lt;p&gt;The Khan Academy has been developing a &lt;a href=&quot;https://www.khanacademy.org/commoncore/map&quot; rel=&quot;nofollow&quot;&gt;common core map&lt;/a&gt;. They have been working with Smarter Balanced and Illustrative Mathematics on the project.&lt;/p&gt;&#xA;" OwnerUserId="1103" LastActivityDate="2014-04-20T23:48:35.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1824" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T00:06:54.823" Score="14" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Alicia_Boole_Stott&quot;&gt;Alicia Boole Stott&lt;/a&gt;, the daughter of George Boole (Boolean Algebra), had a deep understanding of 4D geometry.  She got married and lived the life that entailed back then (1890s and on).  &lt;a href=&quot;http://en.wikipedia.org/wiki/Harold_Scott_MacDonald_Coxeter&quot;&gt;Coxeter&lt;/a&gt; gives her husband some credit for connecting her to &lt;a href=&quot;http://en.wikipedia.org/wiki/Pieter_Schoute&quot;&gt;Pieter Schoute&lt;/a&gt;.  They worked together and published some papers on 4D polytopes.  Coxeter's book, &lt;a href=&quot;http://books.google.com/books?id=2ee7AQAAQBAJ&amp;amp;printsec=frontcover&amp;amp;dq=Regular+Polytopes&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=lV5UU8LRDsWU8QGZz4CIBg&amp;amp;ved=0CCsQ6AEwAA#v=onepage&amp;amp;q=Regular%20Polytopes&amp;amp;f=false&quot;&gt;Regular Polytopes&lt;/a&gt; has a brief biography.  Coxeter worked with her later in her life.  George Boole's household was rather interesting according to Coxeter.  Alicia was introduced to 4D geometry through &lt;a href=&quot;http://en.wikipedia.org/wiki/Charles_Howard_Hinton&quot;&gt;C.H. Hinton&lt;/a&gt;, who married her sister and wrote &lt;a href=&quot;http://books.google.com/books?id=_ZG3MA1wvjIC&amp;amp;printsec=frontcover&amp;amp;dq=Charles+Hinton+geometry&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=6F9UU4KsJerT8gGjlIDwDQ&amp;amp;ved=0CDUQ6AEwAg#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;a book&lt;/a&gt; on the fourth dimension.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2014-04-21T00:06:54.823" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1825" PostTypeId="2" ParentId="1820" CreationDate="2014-04-21T00:19:44.530" Score="16" Body="&lt;p&gt;The &lt;a href=&quot;http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf&quot;&gt;appendix A in the Common Core State Standards for Mathematics&lt;/a&gt; seems to agree that traditional curricula put logarithms somewhere after Algebra I. CCSSM isn't a curriculum, and it doesn't suggest teaching logarithms later, it just offers &quot;pathways&quot; that are supposed to help districts using traditional curriculum find a way to meet the standards. I mention it just to show that, yes, this seems to agree that a traditional time to introduce the subject is later than Algebra I. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Massachussetts Curriculum Frameworks put logarithms into &quot;Algebra I&quot; under a category like &quot;Functions.&quot; Although this is also considered to be &quot;Common Core Aligned.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One short answer is that &quot;it's traditional.&quot; The order you describe is seen as early as 1898 in &lt;a href=&quot;https://archive.org/details/newschoolalgebra00wentrich&quot;&gt;Bull Wentworth's &lt;em&gt;New School Algebra&lt;/em&gt;&lt;/a&gt;. (Donoghue, 2003)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Curriculum design is (should be) about structuring the subject matter in a way that students can form a coherent understanding. At the least, it is an arrangement that the designers find coherent. This means that it's not necessarily true that the options are: square roots are more common, square roots are more useful, or square roots are easier. Nor is it necessarily true that there is one correct way. So, one way may be produced, become traditional, and then be perpetuated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reform curricula have come up against this time and again. Sometimes, the way we do things in math education are traditional and people don't &lt;em&gt;want to&lt;/em&gt; question them, even in the face of evidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I think you have a good question, perhaps a more forward-looking question might be whether we can find reasons to teach logarithms earlier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Squares and square roots are connected to a particular 2D representation (squares) that can be used as a model for multiplication. It's popular for assisting students beyond the tenacious idea of multiplication as repeated addition. A representational leg up like that may be (just speculating here) one convenient reason that squares and square roots can be placed early in a curriculum &quot;pathway.&quot; I'm really only guessing here; IANACD (I Am Not A Curriculum Designer). But I doubt that's a hard and fast rule that &lt;em&gt;cannot&lt;/em&gt; be broken. Some people have successfully introduced algebraic thinking to early grades students (Smith III &amp;amp; Thompson, 2008). Although getting students to reason algebraically is a little bit different (in intention) from introducing a particular concept or operation at a different time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cited:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Donoghue, E. F. (2003). Algebra and geometry textbooks in twentieth-century America. In G. M. A. Stanic &amp;amp; J. Kilpatrick (Eds.), A history of school mathematics (Vol. 1, pp. 329–398). Reston, VA: National Council of Teachers of Mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Smith III, J. P., &amp;amp; Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In &lt;a href=&quot;http://www.kaputcenter.umassd.edu/products/publications/algebra_early_grades/&quot;&gt;J. Kaput, D. W. Carraher, &amp;amp; M. L. Blanton (Eds.), Algebra in the early grades&lt;/a&gt; (pp. 95–131). New York, NY: Taylor &amp;amp; Francis Group.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-04-21T11:10:55.740" LastActivityDate="2014-04-21T11:10:55.740" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1826" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T00:35:59.627" Score="53" Body="&lt;p&gt;Perhaps not strictly a mathematician in the traditional sense, but I think &lt;a href=&quot;http://en.wikipedia.org/wiki/Ada_Lovelace&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Ada Lovelace&lt;/strong&gt;&lt;/a&gt; might be a great woman to start with in today's digital world. She even has an important programming language named after her: &lt;a href=&quot;https://stackoverflow.com/tags/ada/info&quot;&gt;&lt;strong&gt;Ada&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Augusta Ada King, Countess of Lovelace (10 December 1815 – 27 November&#xA;  1852), born Augusta Ada Byron and now commonly known as Ada Lovelace,&#xA;  &lt;strong&gt;was an English mathematician&lt;/strong&gt; and writer chiefly known for her work on&#xA;  Charles Babbage's early mechanical general-purpose computer, the&#xA;  Analytical Engine. Her notes on the engine include what is recognised&#xA;  as the first algorithm intended to be carried out by a machine.&#xA;  &lt;strong&gt;Because of this, she is often described as the world's first computer programmer&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or, if you choose to go with the ancients, I'd suggest &lt;a href=&quot;http://en.wikipedia.org/wiki/Hypatia&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Hypatia&lt;/strong&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hypatia (/haɪˈpeɪʃə/ hy-PAY-shə; Ancient Greek: Ὑπατία; Hypatía) (born&#xA;  c. AD 350 – 370; died 415) was a Greek Alexandrine Neoplatonist&#xA;  philosopher in Egypt &lt;strong&gt;who was one of the earliest mothers of&#xA;  mathematics. As head of the Platonist school at Alexandria,&lt;/strong&gt; she also&#xA;  taught philosophy and astronomy.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;As a Neoplatonist philosopher, she belonged to the mathematic&#xA;  tradition of the Academy of Athens, as represented by Eudoxus of&#xA;  Cnidus; she was of the intellectual school of the 3rd century thinker&#xA;  Plotinus, which encouraged logic and mathematical study in place of&#xA;  empirical enquiry and strongly encouraged law in place of nature.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Recent scholarship has determined that certain important commentaries were indeed written by Hypatia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://news.bbc.co.uk/dna/place-lancashire/plain/A481006&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Hypatia also wrote&#xA;  commentaries&lt;/strong&gt;&lt;/a&gt;&#xA;  on the Arithmetica of Diophantus, the Conics of Apollonious and edited&#xA;  part of her father's Commentary on the Almagest by Ptolemy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Hypatia&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Hypatia also met with a rather unfortunate end&lt;/strong&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;According to the only contemporary source, Hypatia was murdered by a&#xA;  Christian mob after being accused of exacerbating a conflict between&#xA;  two prominent figures in Alexandria: the governor Orestes and the&#xA;  Bishop of Alexandria. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;More details @ &lt;a href=&quot;http://en.wikipedia.org/wiki/Hypatia#Death&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Events leading to her murder&lt;/strong&gt;&lt;/a&gt;. She was at times something of a popular romantic icon as well. See: &lt;a href=&quot;http://en.wikipedia.org/wiki/Hypatia#19th_century&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;In the 19th century, interest in the &quot;literary legend of Hypatia&quot; began to rise&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the dramatic aspects of Hypatia's life could be deemed distractions from her accomplishments in mathematics, when dealing with HS students. On the other hand, since her story is interesting and provocative, it might well serve to foster students' interest in her accomplishments.&lt;/p&gt;&#xA;" OwnerUserId="1105" LastEditorUserId="-1" LastEditDate="2017-05-23T12:41:54.857" LastActivityDate="2014-04-22T18:39:28.400" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1827" PostTypeId="2" ParentId="1819" CreationDate="2014-04-21T00:52:30.213" Score="13" Body="&lt;p&gt;I TA'd a first course in abstract algebra during my senior year of undergrad.  The professor wanted a computational flavor to the course, so we introduced Magma right off the bat.  We wanted to allow the students to experiment with permutations without needing to actually do large computations themselves.  As part of my contribution to the course, I was put in charge of the Magma side of things, both making the assignments and teaching them how to use the software.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After a few in class lectures on Magma syntax and a couple ungraded, basic-syntax mini-assignments, I gave the students &lt;a href=&quot;http://plaza.ufl.edu/gruberan/GroupTheory_S13/CA1.pdf&quot; rel=&quot;nofollow&quot;&gt;this&lt;/a&gt;, which they finished in about two weeks.  I had to help them a lot by email.  Everyone finished the second problem once they figured out the list syntax.  In retrospect I think the first problem was a bit too difficult. I had expected the students to have at least a little programming experience, and it turned out most of them didn't, so the Monte Carlo type solution was not intuitive. They just weren't comfortable enough with conditionals and control structures to use them to tackle unfamiliar concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I gave two more small projects like this, but I don't have copies online.  The second one involved verifying the Chinese remainder theorem, calculating isomorphism classes of abelian groups of some given orders, and taking quotients of abelian groups.  The third one was about alternating groups and conjugacy classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It turned out that the students found Magma to be more trouble than it was worth.  They didn't learn as much from the assignments as I had expected them to.  The goal had been to allow students to see more complex group structure in action without actually having to go through the trouble of calculating everything on their own, but it seems that this difficulty was simply replaced by the difficulty of learning the new language. I had expected they would enter an &quot;exploratory phase&quot; after learning the basic tools where they learned a whole lot of group theory (because that's what happened to me after I learned Magma), but it didn't happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my experience suggests that even several days in the computer lab was not sufficient to introduce Magma.  I suspect it may only be practical to use Magma for a first course in group theory when you &lt;em&gt;know&lt;/em&gt; that students have at least some previous programming experience.  If they do, I still think it could be a great way to learn. It also might be worth considering use in a first year graduate level course in abstract algebra, where the students are committed enough to put in the time to learn programming concepts if they don't know how to yet.  (Also, at least for me, the most educational part of learning Magma was messing around with the SmallGroups package, which is probably out of the scope of a first year course in group theory.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S., to answer your final question, the only resource I have ever personally used to learn Magma has been the &lt;a href=&quot;http://magma.maths.usyd.edu.au/magma/handbook/&quot; rel=&quot;nofollow&quot;&gt;Magma handbook&lt;/a&gt;, which has been so tremendously helpful that I have never felt the need to seek an alternative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: I remembered that I had made some Magma notes for my students, a &lt;a href=&quot;http://plaza.ufl.edu/gruberan/GroupTheory_S13/Magma_Notes_1.txt&quot; rel=&quot;nofollow&quot;&gt;quick reference&lt;/a&gt; and some &lt;a href=&quot;http://plaza.ufl.edu/gruberan/GroupTheory_S13/Magma_Lecture_2.txt&quot; rel=&quot;nofollow&quot;&gt;short programs&lt;/a&gt; we went over in class.&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2014-12-21T21:08:45.640" LastActivityDate="2014-12-21T21:08:45.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1828" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T01:01:34.970" Score="45" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Sophie_Germain&quot;&gt;Sophie Germain&lt;/a&gt; and her work on Fermat's Last Theorem.&lt;/p&gt;&#xA;" OwnerUserId="18" LastEditorUserId="1144" LastEditDate="2014-04-21T19:54:48.117" LastActivityDate="2014-04-21T19:54:48.117" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1829" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T01:24:58.290" Score="23" Body="&lt;p&gt;Classically speaking, &lt;a href=&quot;http://en.wikipedia.org/wiki/Maria_Gaetana_Agnesi&quot;&gt;Maria Agnesi&lt;/a&gt; is the best classical mathematician to study.  She published calculus texts that expanded and reflected upon the works of Leonhard Euler.&lt;/p&gt;&#xA;" OwnerUserId="1100" LastActivityDate="2014-04-21T01:24:58.290" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1830" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T01:27:28.160" Score="28" Body="&lt;p&gt;Sonya Kovalevsky, correspondent of Weierstrass, for example.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-21T01:27:28.160" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1831" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T01:56:43.797" Score="52" Body="&lt;p&gt;Florence Nightingale, elected to the Royal Statistical Society and (honorarily) to the American Statistical Association, for her work on the importance of statistical data and statistical graphics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Her statistics and her graphics persuaded the British government to improve sanitation in military hospitals, saving many soldiers' lives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She was a statistician rather than a mathematician, but since statistics is part of high school math classes, she seems worth talking about.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-21T01:56:43.797" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1832" PostTypeId="2" ParentId="1816" CreationDate="2014-04-21T03:17:38.133" Score="7" Body="&lt;p&gt;Off the top of my head, these are the only two examples that I can think of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.mathematicsvisionproject.org&quot;&gt;Mathematics Vision Project&lt;/a&gt;. It is an integrated curriculum aligned with the new common core standards.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.engageny.org/mathematics&quot;&gt;EngageNY&lt;/a&gt; has also modules for high school.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Both are released under Creative Common &lt;a href=&quot;http://creativecommons.org/licenses/by-nc-sa/3.0/&quot;&gt;by-nc-sa&lt;/a&gt; license.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a side note, (at least in the US) most curricula need to be approved by the state boards of education before they are adopted by school districts, so there aren't many open source projects started because teachers wouldn't be able to use them in their classes.  For example, in California there is a law that require schools to provide students with the most recent instructional materials (Williams vs State of California). That is oftentimes translated in students having access to a semi-modern textbook at school and at home. Using materials that a teacher found on the internet could be interpreted as being out of compliance with the court order, no matter the quality of the materials.&lt;/p&gt;&#xA;" OwnerUserId="255" LastActivityDate="2014-04-21T03:17:38.133" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1833" PostTypeId="2" ParentId="1816" CreationDate="2014-04-21T03:40:59.347" Score="6" Body="&lt;p&gt;&lt;a href=&quot;http://www.ck12.org/teacher/&quot; rel=&quot;noreferrer&quot;&gt;CK-12&lt;/a&gt; is a great source of material. Math for every high school topic, with options to edit material to suit your needs.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-04-21T03:40:59.347" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1834" PostTypeId="1" AcceptedAnswerId="1840" CreationDate="2014-04-21T03:51:46.337" Score="29" ViewCount="5311" Body="&lt;p&gt;I find that soon I'll be working with high school students that are struggling with math. In particular, we'll be talking a lot about algebra and some basic trigonometry. The latter I have experience with (via working with students in calculus and &quot;pre-calculus&quot;), but I have legitimately no idea how one would teach algebra. If I see $3x+5=14$, it's obvious to me what to do, and unlike, say, calculus, I can't really even see how someone would get confused on that (even though I know they do!) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a bit broad, but how do you teach introductory algebra? Do you have any references for new teachers?&lt;/p&gt;&#xA;" OwnerDisplayName="user37" LastActivityDate="2019-09-18T23:48:38.973" Title="How do I teach algebra?" Tags="|secondary-education|algebra|" AnswerCount="15" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1835" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T04:27:58.310" Score="65" Body="&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Julia_Robinson&quot;&gt;Julia Robinson&lt;/a&gt;!&lt;/strong&gt; I recommend her for a high school audience for a few reasons:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Mathematical reasons:&lt;/em&gt; She is best known for her work towards the solution of &lt;a href=&quot;http://en.wikipedia.org/wiki/Hilbert%27s_Tenth_Problem&quot;&gt;Hilbert's 10th Problem&lt;/a&gt;, regarding an algorithm for solving Diophantine Equations. High school students can absolutely recognize and solve particular Diophantine Equations. Furthermore, and more relevant to Robinson's work, I believe high school students can appreciate (upon seeing examples) the subtle dependence of such equations on their coefficients.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, ask a high school algebra student to experiment and find all the solutions to, say, $6x+15y=33$. Then, ask them to solve $6x+15y=34$. What changes? Give them some random coefficients for $ax+by=c$. What would they do? Can they generalize to more variables?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, work with them on a quadratic equation, something like $x^2+1=y^2$. Can they find any solutions? What about $3x^2-5y^2=2$? Do they see how hard it is to get a general method?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, you can start to talk about what Hilbert's 10th Problem says. This can facilitate several interesting discussions for a high school audience&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Solving equations in integers (Why have mathematicians been interested in solving them since pretty much antiquity? Where do they appear in real life?)&lt;/li&gt;&#xA;&lt;li&gt;Algorithms and decision procedures (What is an algorithm? What makes one better than another? Do we care about how &lt;em&gt;efficiently&lt;/em&gt; they can run, or how &lt;em&gt;easy&lt;/em&gt; they are to describe/implement?)&lt;/li&gt;&#xA;&lt;li&gt;Logic and philosophy of mathematics (What does it mean to &lt;em&gt;prove&lt;/em&gt; something? What does it mean to &lt;em&gt;disprove&lt;/em&gt; something? What does it mean to prove that there is &lt;em&gt;no procedure&lt;/em&gt; that could solve every Diophantine equation?)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(I understand this doesn't &lt;em&gt;really&lt;/em&gt; address the full depth of Hilbert's 10th, but if you're looking to pique the interest of a high school audience, I believe this should suffice.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Historical reasons:&lt;/em&gt; Julia Robinson is one of the more notable and renowned mathematicians (no need for &quot;female&quot;) of the 20th century, especially in America. Despite this, she had difficulty securing an academic position. But despite &lt;em&gt;that&lt;/em&gt;, she went on to obtain such a position and became the first female president of the AMS. Along the way, she devoted much of her time to other interests, including political campaigns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surely, she can spark many interesting discussions for a high school audience!&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="1162" LastEditDate="2014-04-22T06:59:23.320" LastActivityDate="2014-04-22T06:59:23.320" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1836" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T05:24:34.493" Score="29" Body="&lt;p&gt;&lt;strong&gt;Edit (Jan 2018)&lt;/strong&gt; I recommend checking Annie Perkins' page:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://arbitrarilyclose.com/2016/08/21/the-mathematicians-project-mathematicians-are-not-just-white-dudes/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Mathematicians Project: Mathematicians Are Not Just White Dudes&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you scroll down, then you will find a section entitled &lt;strong&gt;Women (alphabetical by last name)&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are some great sources/names there, and - as a bonus - the project keeps evolving!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ $$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Marjorie Rice has recently passed away; for more on her life, see the Quantum story &lt;a href=&quot;https://www.quantamagazine.org/marjorie-rices-secret-pentagons-20170711/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I think &lt;a href=&quot;http://en.wikipedia.org/wiki/Marjorie_Rice&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marjorie Rice&lt;/a&gt; is a great example of an amateur mathematician who managed to make nontrivial discoveries with regard to tessellations. Interestingly, her notation was deciphered by a female professor of mathematics, &lt;a href=&quot;http://en.wikipedia.org/wiki/Doris_Schattschneider&quot; rel=&quot;nofollow noreferrer&quot;&gt;Doris Schattschneider&lt;/a&gt; (also &lt;a href=&quot;http://www.agnesscott.edu/lriddle/women/schatt.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;), who helped to lead the development of Geometer's Sketchpad. (Perhaps you use this piece of software in your school?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another great choice is &lt;a href=&quot;http://en.wikipedia.org/wiki/Mary_P._Dolciani&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mary Dolciani&lt;/a&gt; (also &lt;a href=&quot;http://www.nctm.org/resources/content.aspx?id=2912&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) who was a Ph.D mathematician especially known for her teaching abilities, and a prolific writer of textbooks and curricula (e.g., through the School Mathematics Study Group, SMSG, which developed New Math). I included a bit about her in an earlier MESE &lt;a href=&quot;https://matheducators.stackexchange.com/questions/369/was-there-an-smsg-new-math-algebra-2-text/806#806&quot;&gt;answer&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2018-01-26T23:41:23.153" LastActivityDate="2018-01-26T23:41:23.153" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1837" PostTypeId="2" ParentId="1834" CreationDate="2014-04-21T05:41:07.190" Score="8" Body="&lt;p&gt;I think you will need to be very cognizant of student conceptions of how to solve algebraic problems. It may be useful to not try and immediately show them how to solve problems, but rather to ask them how they would go about solving the problems. This will enable you to learn about their mathematical thinking and possible misconceptions they may have. In the example you gave, a student my try to divide both sides by 3 but then simplify it to x + 5 = 14/3. Students solving linear equations often forget to apply the operation to both entire sides and are very focused on eliminating a particular coefficient or term.&lt;/p&gt;&#xA;" OwnerUserId="1111" LastActivityDate="2014-04-21T05:41:07.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1838" PostTypeId="2" ParentId="1820" CreationDate="2014-04-21T05:56:58.660" Score="13" Body="&lt;p&gt;&lt;strong&gt;Historical comments.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Early on, the study of logarithms and logarithmic tables was incorporated into trigonometry. For more on this background from the perspective of the history of trigonometry education, search through the following dissertation for the word &lt;em&gt;logarithm&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Van Sickle, J. (2011). A History of Trigonometry Education in the United States: 1776-1900 (Doctoral dissertation, Columbia University). &lt;a href=&quot;http://academiccommons.columbia.edu/catalog/ac:132221&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example:&lt;/strong&gt; You can find a book from 1914 entitled &quot;Trigonometry with the Theory and Use of Logarithms.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an excerpt:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/iFxDt.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the least, the author seems to realize that finding the hypotenuse using the Pythagorean Theorem is more sensible; but it is strange that this would even need saying. JPBurke's informative answer mentions Eileen Donoghue, which made me wonder about the history of logarithms with regard to the New York Regents. (Donoghue is an excellent source for historical information about curricula, textbooks, and the Regents in particular; I mentioned the same 2003 article by her in an earlier MESE &lt;a href=&quot;https://matheducators.stackexchange.com/a/806/262&quot;&gt;post&lt;/a&gt;.) It seems that four years after the above-mentioned book was published, a 1918 article in The Mathematics Teacher argues that logarithms should be learned in intermediate algebra, rather than &quot;consigning&quot; it to trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The citation for this article is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Decker, F. F. (1918). The New York State Regents Syllabus in Intermediate Algebra. &lt;em&gt;The Mathematics Teacher, 11&lt;/em&gt;(1), 1-8. &lt;a href=&quot;http://www.jstor.org/stable/27950161&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jstor&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A free copy of the article can be found &lt;a href=&quot;http://archive.org/stream/jstor-27950161/27950161_djvu.txt&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, from which I quote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The introduction to logarithms is placed in intermediate algebra. Consigning this topic to trigonometry has several disadvantages. Many students who carry their mathematical study through the course in trigonometry seem to get the idea that the usefulness of logarithms is confined to trigonometry. Those who do not reach trigonometry lose what is at once an important tool and an elegant system. In the separation of the treatment of exponents and logarithms the theory seems to contain a discontinuity. The laws for commensurable exponents are carefully proved and then the theory of logarithms is made to appear to rest on that of exponents. But most of the logarithms used are incommensurable! The plan here proposed is to introduce incommensurable exponents with reference to the continuity of an exponential curve and to pass immediately to logarithms. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The author also cites a separate document arguing that logarithms be studied along with exponents:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[A report] of the Committee on the Teaching of Mathematics to Students of Engineering, calls for the teaching of logarithms in logical connection with the subject of exponents rather than with trigonometry.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This report is foot-noted as: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Syllabus of Mathematics,&quot; Society for the Promotion of Engineering Education, 1912.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;but an APA citation is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;American Society for Engineering Education. Committee on the Teaching of Mathematics to Students of Engineering. (1912). &lt;em&gt;Syllabus of mathematics: a symposium.&lt;/em&gt; New Era Printing Co..&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The report can be found on google books; here is the relevant excerpt:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/OQsZK.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Evidently, the study of logarithms within trigonometry is widespread enough by 1912 to warrant comments from a group of engineering educators. For now, I will not pursue the connections between these two subjects any further, but I would not be surprised if the disentangling of logs from trig resulted in the former being placed later in the curriculum than you might consider ideal from a more modern perspective.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-29T15:36:36.847" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1839" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T05:58:37.497" Score="18" Body="&lt;p&gt;&lt;a href=&quot;http://it.wikipedia.org/wiki/Maria_Gramegna&quot; rel=&quot;noreferrer&quot;&gt;Maria Gramegna&lt;/a&gt;, the brilliant student of Giuseppe Peano.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you use matrices to solve systems of differential equations, you rely in many ways to &lt;a href=&quot;http://books.google.hu/books?id=U3k8yfchaPYC&amp;amp;lpg=PA505&amp;amp;ots=8YEC34fNK8&amp;amp;dq=maria%20gramegna%20peano&amp;amp;hl=de&amp;amp;pg=PA504#v=onepage&amp;amp;q=maria%20gramegna%20peano&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;her ideas&lt;/a&gt;. She defined the exponential function of a matrix through its power series and used it as we do it now. Though this is not strictly speaking high school mathematics, you can mention her story to every undergraduate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She also generalized this to infinite systems and integrodifferential equations, and many ideas of her 1910 thesis belong to the foundation of functional analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that, she became a school teacher and died in an earthquake in 1915.&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="61" LastEditDate="2014-04-22T20:10:46.057" LastActivityDate="2014-04-22T20:10:46.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1840" PostTypeId="2" ParentId="1834" CreationDate="2014-04-21T07:07:48.167" Score="46" Body="&lt;p&gt;As a personal tutor, I’ve been teaching algebra to kids from ages 8 to 16 for many years.  Mostly I find myself in the position of picking up the pieces when the kids are failing and fearing more failure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The root of the problem, in my experience, is the way algebra is taught as something alien, and in particular, different from arithmetic, which it really isn’t (at least in the early years).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So first off, &lt;strong&gt;constant emphasis on the fact that “$x$ is just a number you don’t know yet”&lt;/strong&gt;.  So it behaves like a number, and you can do all the stuff to it, that you can do to numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, the nature of equality $2 + 3 = 4 + 1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And from there, the fact that when you do the same thing to both sides, you still end up with two things that are equal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Always&lt;/strong&gt; “do the same thing to both sides” (since this is clearly based on the nature of equality), &lt;strong&gt;never&lt;/strong&gt; “move this from one side to the other and change the sign” (which is a magic rule that makes no sense until you have a deeper understanding).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you get them happy with the idea that doing the same thing to both sides is the way to go, you can give them suggestions for which things to do in which order, but stress that provided they rigorously write down the consequence of the thing they decide to do to both sides, they won't go wrong (although some ways are harder – look out for these as a pointer that choosing another way will be easier).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The manipulation of each line is easy, once you’ve got them to decide what they’re going to do at each stage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, in the example, $3x + 5 = 14$:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First, decide what to do to both sides (subtract 5)&lt;/li&gt;&#xA;&lt;li&gt;Write down first what you have ($3x + 5$), then do what you’ve decided.  So you get $3x + 5 - 5$, and on the RHS, $14 - 5$.&lt;/li&gt;&#xA;&lt;li&gt;Then collect terms and simplify to get $3x = 9$.&lt;/li&gt;&#xA;&lt;li&gt;Then repeat for division by 3.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Emphasise that once you’ve decided what to do at each stage, there’s very little thinking, since you’re just writing – starting with what you had on the previous line, and adding on the chosen operation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Figuring out what to do (add or multiply, subtract or divide) needs to come after they are truly grounded in the principle that doing the same thing to both sides is the key.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They will also need help with things like why $3x/3 = x$.  Again, use numbers to illustrate, and stress that $x$ is just a number, so it behaves the same way as a number.&lt;/p&gt;&#xA;" OwnerUserId="980" LastEditorUserId="75" LastEditDate="2014-04-21T08:21:05.037" LastActivityDate="2014-04-21T08:21:05.037" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="1841" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T08:09:43.517" Score="5" Body="&lt;h2&gt;Shakuntala Devi a.k.a Human Calculator&lt;/h2&gt;&#xA;&lt;p&gt;Her talents eventually earned her a place in the 1982 edition of The Guinness Book of World Records.As a writer, Devi wrote a number of books, including novels and non-fiction texts about mathematics, puzzles, and astrology.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Shakuntala_Devi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link to wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.csmonitor.com/Innovation/2013/1104/Shakuntala-Devi-and-other-human-calculators&quot; rel=&quot;nofollow noreferrer&quot;&gt;Other Human calculators&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1117" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-21T08:09:43.517" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1843" PostTypeId="1" CreationDate="2014-04-21T08:21:51.973" Score="8" ViewCount="158" Body="&lt;p&gt;I came across &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.136.228&amp;amp;rep=rep1&amp;amp;type=pdf&quot;&gt;Small-group composition and peer effects&lt;/a&gt; (Wilkinson &amp;amp; Fung, 2002) which got me thinking about the pros and cons of grouping students based on ability levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When is it a good idea to keep strong students with strong students and weaker students with weaker students in the math classroom? And when is it beneficial to have a mixture of strengths within a collaborative group? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested to know what other literature exists on this subject, but personal perspectives are welcome as well.&lt;/p&gt;&#xA;" OwnerUserId="104" LastActivityDate="2014-04-21T09:07:54.577" Title="What are some best practices for creating student groups?" Tags="|education-research|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1844" PostTypeId="2" ParentId="1843" CreationDate="2014-04-21T09:07:54.577" Score="5" Body="&lt;p&gt;You seem to be asking about within-class grouping by ability, but there is a fair amount of literature about what happens when the entire class is grouped by ability, e.g., when mathematics classes for a certain year are divided into a standard version and an advanced version.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These &quot;versions&quot; are also called &lt;em&gt;tracks&lt;/em&gt;, and the process of separating classes in this manner is known as &lt;em&gt;tracking&lt;/em&gt;; analogously, the practice of removing the tracks and putting everyone into a single course is known as &lt;em&gt;detracking&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the effects that are witnessed with regard to tracking/detracking are the same sort you would see if you grouped by ability within-class, though perhaps the impact would be smaller in the latter case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for tracking/detracking literature: My first thoughts are an article by Loveless and another by Oakes.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Citation&lt;/strong&gt; Loveless, T. (1999). Will Tracking Reform Promote Social Equity? &lt;em&gt;Educational Leadership, 56&lt;/em&gt;(7), 28-32.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've uploaded a copy &lt;a href=&quot;https://www.dropbox.com/s/2w4xsypl5xfqf81/Loveless%20Will%20Tracking%20Reform%20Promote%20Social%20Equity.pdf&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Citation&lt;/strong&gt; Oakes, J. (1995). Two cities' tracking and within-school segregation. &lt;em&gt;The Teachers College Record, 96&lt;/em&gt;(4), 681-690.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've uploaded a copy &lt;a href=&quot;https://www.dropbox.com/s/2w4xsypl5xfqf81/Loveless%20Will%20Tracking%20Reform%20Promote%20Social%20Equity.pdf&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Those two sources should get you started; look at their citations and the papers that cited them (especially the latter: see &lt;a href=&quot;http://scholar.google.com/scholar?q=mathematics&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=5,33&amp;amp;sciodt=0,33&amp;amp;cites=5549883790432591850&amp;amp;scipsc=1&quot;&gt;here&lt;/a&gt;) for further results. (The former paper is more casual, but gives a nice overall idea.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lastly:&lt;/strong&gt; My general impression from the literature is that when you mix together students of different levels, the result is that the bottom and middle tier students do better, and the top tier students do slightly worse. The negative portion of this trade-off can probably be mitigated by paying some extra attention to the more advanced students and ensuring that they are sufficiently challenged (possibly by introducing supplementary materials). One of the consequences of top tier students not feeling challenged enough (addressed in the Loveless piece) is that their parents may wish to send them elsewhere; this can result in &lt;em&gt;bright flight&lt;/em&gt;. As Loveless writes: &quot;It's difficult to imagine how any school can benefit by alienating and driving out its best students&quot; (p. 30).&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-04-21T09:07:54.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1846" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T09:53:57.580" Score="13" Body="&lt;p&gt;I think for young girls &lt;a href=&quot;http://en.wikipedia.org/wiki/Ruth_Lawrence&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Ruth Lawrence&lt;/strong&gt;&lt;/a&gt; is a great role model since she got her phd at age of &lt;strong&gt;17&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;At the age of 9, Ruth Lawrence gained an O-level in mathematics,    setting a new age record.&lt;/li&gt;&#xA;  &lt;li&gt;Also at the age of 9 she achieved a Grade A at A-level Pure    Mathematics.&lt;/li&gt;&#xA;  &lt;li&gt;In 1981 Ruth Lawrence passed the Oxford University interview entrance    examination in mathematics, coming first out of all 530&#xA;  candidates    sitting the examination, and joining St Hugh's College&#xA;  in 1983 at the    age of just twelve.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Another example would be &lt;a href=&quot;https://en.wikipedia.org/wiki/Shelly_Harvey&quot; rel=&quot;nofollow noreferrer&quot;&gt;Shelly Harvey&lt;/a&gt; from Rice University.&lt;/p&gt;&#xA;" OwnerUserId="1123" LastEditorUserId="1123" LastEditDate="2017-06-15T06:40:12.023" LastActivityDate="2017-06-15T06:40:12.023" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1847" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T13:03:52.597" Score="11" Body="&lt;p&gt;More famous for computer science than maths, but a strong mathematician none the less and creator of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Liskov_substitution_principle&quot;&gt;Liskov substitution principle&lt;/a&gt; (the L in &lt;a href=&quot;http://en.wikipedia.org/wiki/SOLID_%28object-oriented_design%29&quot;&gt;SOLID&lt;/a&gt;), &lt;a href=&quot;https://en.wikipedia.org/wiki/Barbara_Liskov&quot;&gt;Barbara Liskov&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1129" LastEditorUserId="1129" LastEditDate="2014-04-21T20:42:15.710" LastActivityDate="2014-04-21T20:42:15.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1848" PostTypeId="1" CreationDate="2014-04-21T13:44:16.650" Score="7" ViewCount="867" Body="&lt;p&gt;I am an Undergraduate student ready to enter into Graduate School. I will be studying Mathematics. In my Undergraduate studies, I came across various Problem books and followed some of them. The results have been outstanding. I wanted to know if there are any more problem books which I missed or should follow apart from the ones listed below:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Problems in Mathematical Analysis by Kaczor &amp;amp; Nowak, Vol 1,2,3&lt;/li&gt;&#xA;&lt;li&gt;Berkeley Problems in Mathematics&lt;/li&gt;&#xA;&lt;li&gt;Challenging Problems in Linear Algebra by Fuzhen Zhang&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="808" LastEditorUserId="77" LastEditDate="2017-12-13T13:46:05.437" LastActivityDate="2017-12-13T13:46:05.437" Title="Problem books at undergraduate level" Tags="|undergraduate-education|textbooks|reference-books|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1849" PostTypeId="2" ParentId="1834" CreationDate="2014-04-21T13:45:21.253" Score="15" Body="&lt;p&gt;I saw this question and laughed, &quot;That is way too broad!&quot;, but I've been in your position. I was a classroom teacher for 10yrs in the public school system and was often tasked with teaching something that I hadn't had training in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you are looking for initially is a &quot;Scope and Sequence&quot; - a guide showing the steps in teaching a subject. Your 'expert blindness' makes it hard to make one on your own, but it is also redundant - experts have already done this. You can put together a S&amp;amp;S by look at roughly 2 sources:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;State or privately developed curriculums - some states offer there curriculum online in the form of &quot;Standards&quot;. You can look at what is required at each grade level and get an idea of what you need to teach. You'll need to assess your student against current grade level requirements and then work backwards until you get to the point they understand. That reveals their 'deficiency'. Then you remediate. So, the curriculum will tell you at 8th grade they need to know 'this' and at ninth, 'this'. You teach what they sequentially through the curriculum.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Books and guides - academic textbook are often set up in a proper sequence that will show you a framework of what needs to be learned first. You can obtain these often at libraries, but you may need to dig. Ideally you can find the books the students have used in their classes. Homeschooling resources are also readily available and can be found to meet a lot of different special needs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This is a tough nut to crack. It really highlights the fact that 'Teaching' is far more than knowledge of a subject! Teaching is its own skill. Good luck! &lt;/p&gt;&#xA;" OwnerUserId="1130" LastActivityDate="2014-04-21T13:45:21.253" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1850" PostTypeId="2" ParentId="1848" CreationDate="2014-04-21T13:53:13.540" Score="11" Body="&lt;p&gt;There are certainly many more. The following list is not meant to be exhaustive but meant just to give you a selection of the many more books that you have missed:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Halmos: Linear Algebra Problem Book, A Hilbert Space Problem Book.&lt;/li&gt;&#xA;&lt;li&gt;George Polya and Gabor Szegö: Problems and Theorems in Analysis - I, II.&lt;/li&gt;&#xA;&lt;li&gt;Ram Murty: Problems in Analytic Number Theory.&lt;/li&gt;&#xA;&lt;li&gt;Jody Esmonde, Ram Murty: Problems in Algebraic Number Theory.&lt;/li&gt;&#xA;&lt;li&gt;Gelbaum: Problems in Analysis, Problems in Real and Complex Analysis. &lt;/li&gt;&#xA;&lt;li&gt;D. Aliprantis and O. Burkinshaw, Problems in Real Analysis.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="109" LastEditorUserId="109" LastEditDate="2014-04-22T19:24:28.500" LastActivityDate="2014-04-22T19:24:28.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1851" PostTypeId="2" ParentId="1834" CreationDate="2014-04-21T14:39:16.443" Score="26" Body="&lt;p&gt;For some students, the difficulty with solving $3x+5=14$ is even more basic than figuring out what operations to do in what order in order to reach the goal.  Before getting to that, they need to know what the goal is.  &quot;Everybody knows&quot; that, when solving an equation with one variable $x$, the goal is to end up with a statement of the form $x=$ some specific number. Unfortunately, this &quot;everybody&quot; doesn't really include everybody; some students have never had the goal made clear.  Moreover, in some cases, once they understand the goal, they're remarkably good at finding strategies for working toward it.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-04-21T14:39:16.443" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1852" PostTypeId="2" ParentId="1834" CreationDate="2014-04-21T15:03:21.117" Score="7" Body="&lt;p&gt;Emphasize &lt;strong&gt;word problems&lt;/strong&gt;.  If I have \$14 to spend on 3 toys and a hat, and the hat will cost me \$5, how much can I spend on each toy?&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-21T15:03:21.117" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1853" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T15:25:46.147" Score="15" Body="&lt;p&gt;&lt;a href=&quot;http://www.agnesscott.edu/lriddle/women/young.htm&quot;&gt;Grace Chisholm Young&lt;/a&gt; seems overlooked so far (13 answer so far) and in my opinion she is worth considering. She worked mostly in real analysis and what is sometimes called classical point set theory (among other things, she's the &quot;Young&quot; in the Denjoy-Young-Saks theorem and she wrote a well known survey paper on nowhere differentiable continuous functions in 1916), and she played a large role in her husband's 200+ papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides math, she knew 6 languages, completed all the requirements for a medical degree except for residency, had 6 children in a period of 9 years, and wrote a book on reproduction for children. Despite all her activities, she was also very devoted to her children (teaching each to play a musical instrument), who went on to become: both a son and a daughter were fairly well known mathematicians (the son was also a chess grandmaster), a daughter became a medical doctor and the first female member of the Royal College of Surgeons, a son earned a Ph.D. in chemistry at University of Oxford and later pursued public finance and diplomacy, a daughter completed an undergraduate degree in math and was an Associate Professor of French at Bryn Mawr from 1927 to 1935, a son earned an undergraduate engineering degree and was killed as a pilot in World War 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend looking through Ivor Grattan-Guinness' 1972 paper &lt;em&gt;A mathematical union: William Henry and Grace Chisholm Young&lt;/em&gt;, which contains many interesting details about her life.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-04-21T15:25:46.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1854" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T15:51:24.127" Score="10" Body="&lt;p&gt;The story of &lt;a href=&quot;http://en.wikipedia.org/wiki/Sarah_Flannery&quot;&gt;Sarah Flannery&lt;/a&gt; may interest high school students, as she worked on non-trivial mathematics related to codes as a sixteen year old.  &lt;/p&gt;&#xA;" OwnerUserId="1133" LastActivityDate="2014-04-21T15:51:24.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1855" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T16:12:30.210" Score="5" Body="&lt;p&gt;Whitman, Betsey S. Women of Mathematics: A Biobibliographic Sourcebook, Louise Grinstein and Paul Campbell, Editors, Greenwood Press, 1987.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bailey, Martha J. American Women in Science: A Biographical Dictionary, ABC-CLIO, 1994.&lt;/p&gt;&#xA;" OwnerUserId="1134" LastActivityDate="2014-04-21T16:12:30.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1856" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T17:37:49.803" Score="11" Body="&lt;p&gt;I have several daughters and I like talking to them about my female friends in the computer graphics industry. In particular, an acquaintance of mine, Kelly Ward, who did the physics (and math) for the hair in Disney's Tangled.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few links:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.gpb.org/blogs/passion-for-learning/2012/06/06/disneys-tangled-an-exercise-in-physics-and-computer-animation&quot;&gt;http://www.gpb.org/blogs/passion-for-learning/2012/06/06/disneys-tangled-an-exercise-in-physics-and-computer-animation&lt;/a&gt;&#xA;&lt;a href=&quot;http://unc.edu/meet-a-tar-heel/kelly-ward/&quot;&gt;http://unc.edu/meet-a-tar-heel/kelly-ward/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1138" LastActivityDate="2014-04-21T17:37:49.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1857" PostTypeId="1" CreationDate="2014-04-21T19:18:42.190" Score="32" ViewCount="4267" Body="&lt;p&gt;I've never seen any really good expositions of elementary mathematics (middle school or earlier). A good college-level textbook, written for people with an interest in mathematics, reads like a novel or an impassioned essay. The most famous example I can think off the top of my head of is Spivak's Calculus - the author has a vision for the subject, and wants to communicate their passion to the student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, the rule for multiplying fractions together takes some quite clever and pretty arguments to prove, but in math class and in classroom textbooks, this is just presented as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Fraction multiplication rule:&lt;/strong&gt; Multiply the numerators and the denominators together separately.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Looking at the difference in presentation, you'd almost think elementary arithmetic and geometry aren't part of mathematics. Where are the proofs? Is there even a single book in existence that proves the above fact at all?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any books about elementary math written like actual math books? Note that &lt;strong&gt;I am &lt;em&gt;not&lt;/em&gt; looking for things like &quot;The Number Devil&quot;&lt;/strong&gt;, which (to be fair I haven't read it) you probably wouldn't mistake for an actual textbook. I'm looking for systematic expositions of basic math, covering (and proving!) all the theorems, rules of computation, and definitions covered in elementary to high school (not necessarily all in one book - that's a lot of material).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't necessarily want it to be readable by 8 year olds, that might be asking a lot, but it would be nice to have something you could recommend to a bright 12 year old to give them a more solid foundation and inspire more of an interest in math. Something well written, that wants to make the subject interesting and beautiful, is preferred to a dry going-through-the-motions proof compendium, but of course that's basically like saying &quot;make sure the book is actually good&quot;.&lt;/p&gt;&#xA;" OwnerUserId="115" LastEditorUserId="262" LastEditDate="2014-05-05T17:04:09.433" LastActivityDate="2020-08-05T20:46:07.183" Title="Books about elementary mathematics written like a good undergraduate textbook" Tags="|reference-request|textbooks|primary-education|" AnswerCount="4" CommentCount="23" ContentLicense="CC BY-SA 3.0" />
  <row Id="1858" PostTypeId="2" ParentId="1857" CreationDate="2014-04-21T20:20:44.537" Score="20" Body="&lt;p&gt;I think you'll find some of what you want on Berkley mathematician H.H. Wu's &lt;a href=&quot;http://math.berkeley.edu/~wu/&quot; rel=&quot;noreferrer&quot;&gt;homepage&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More precisely, see: &lt;a href=&quot;http://math.berkeley.edu/~wu/Pre-Algebra.pdf&quot; rel=&quot;noreferrer&quot;&gt;Pre-Algebra&lt;/a&gt; (pdf) and &lt;a href=&quot;http://math.berkeley.edu/~wu/Algebrasummary.pdf&quot; rel=&quot;noreferrer&quot;&gt;Introduction to School Algebra&lt;/a&gt; (pdf).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; I mentioned the same homepage (and the two pdf textbooks) in an earlier MESE post &lt;a href=&quot;https://matheducators.stackexchange.com/a/1253/262&quot;&gt;here&lt;/a&gt;; I would have just re-posted this as a comment, but I believe it is the actual answer to your question (!). I also ought not mark your question as a duplicate of the one I answered earlier, since they are quite different.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-21T20:20:44.537" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1859" PostTypeId="2" ParentId="353" CreationDate="2014-04-21T20:49:26.587" Score="1" Body="&lt;p&gt;I am really impressed by the answers given by others here; I will definitely keep them in mind when teaching freshmen next semester.  But I also have my 2 cents to add, since I haven't seen anything like that in them.  I realize that this is only some kind of vague intuition, and it would probably confuse a lot of students, but it &lt;em&gt;might&lt;/em&gt; as well help some.  (It &lt;em&gt;did&lt;/em&gt; help &lt;em&gt;me&lt;/em&gt; at some point, at least.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So let us assign a &quot;$0$&quot; to false statements and a &quot;$1$&quot; to true ones (this is a common convention at least here in Poland).  Now &quot;$p\implies q$&quot; is true iff $p\le q$ (note that in the first formula we treat $p$ and $q$ as &lt;em&gt;propositional variables&lt;/em&gt;, and in the second as &lt;em&gt;numeric variables&lt;/em&gt;, which is clearly an abuse of notation!).  This way one can view a material implication as a way of saying that &quot;one sentence ($p$) &lt;em&gt;implying&lt;/em&gt; another one ($q$) means that the latter one must be &lt;em&gt;at least as true&lt;/em&gt; (whatever that means!) as the former one&quot;.  In other words, when going from the antecedent to the consequent, we cannot &quot;lose knowledge&quot;, only gain it.  (Now this is really stretching things from philosophical point of view, and logicians would probably torture and kill me for that; but then again – this is only a (vague) intuition I'm talking about).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way I present this to students (if I do!) is more or less this (with a wink): &quot;So just like $p\land q$ is &lt;em&gt;somehow similar&lt;/em&gt; to (but different from!) $p\cdot q$ – so that even we call $p$ and $q$ the &quot;factors&quot; of the conjunction [at least this is what we do in Polish; we also sometimes call a conjunction a &quot;logical product&quot;, and we do similar things with &lt;em&gt;alternative&lt;/em&gt; (i.e., use words &quot;logical sum&quot; and &quot;summands&quot;)] – in a similar vein, $p\implies q$ is &lt;em&gt;somehow similar&lt;/em&gt; to $p\le q$.  But you know, better forget about it, since &quot;truth&quot; is not &quot;one&quot; and &quot;falsehood&quot; is not &quot;zero&quot; anyway, so what I've just told you is more or less a lie anyway.&quot;&lt;/p&gt;&#xA;" OwnerUserId="704" LastActivityDate="2014-04-21T20:49:26.587" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1860" PostTypeId="1" AcceptedAnswerId="1965" CreationDate="2014-04-21T21:51:11.363" Score="28" ViewCount="8877" Body="&lt;p&gt;I'm teaching an &quot;intermediate algebra&quot; college course ($\approx$ junior high school or beginning high school algebra) and we have a bunch of problems on rationalizing denominators. How do I motivate this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About the best I can think of is that it's a collection of tricks that are handy later, but I'd like to know something I can say or do that demonstrates some kind of intellectual need to my students &lt;em&gt;now&lt;/em&gt;. &lt;/p&gt;&#xA;" OwnerUserId="1148" LastActivityDate="2023-11-26T19:36:54.767" Title="How to justify teaching students to rationalize denominators?" Tags="|secondary-education|student-motivation|concept-motivation|algebra|" AnswerCount="10" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="1861" PostTypeId="2" ParentId="1834" CreationDate="2014-04-21T22:02:54.967" Score="7" Body="&lt;p&gt;Teach the students that they can (and should) check their own work.  A student who knows that their check-by-substitution worked will be a lot more confident that they learned that day's lesson than a student who is waiting until the next day to find out they got some answers wrong.  Also, it is great practice for professions (like accounting and programming) that need to &quot;tie out&quot; or &quot;unit test&quot; their work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is how I was taught to check my work.  In the examples, most of the &quot;·&quot; signs are optional:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1)  Write out my answer, such as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;x = 3&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If it is the answer to a story problem, include a note about what the answer means, such as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;x = 3 \$/toy.  Each toy can cost an average of 3 dollars.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;2)  Circle the answer in a fluffy cloud.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3)  Write &quot;CBS:&quot; below the answer.  (CBS stands for check-by-substitution.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4)  Substitute in the answer into the original problem.  Put a question mark over the equals sign.  For example,&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;3 toys · 3 &lt;span class=&quot;math-container&quot;&gt;$/toy  + 1 hat · 5 $&lt;/span&gt;/hat   ≟ 14 $&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;5)  Do the math on both sides of the equals sign.  Keep the question mark over the equals sign until it is obvious that the equation is true.  Put each version of the equation on a following line, and try to line up the equals signs.  For example,&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;3 · 3 &lt;span class=&quot;math-container&quot;&gt;$ · toy/toy + 1 · 5 $&lt;/span&gt; · hat/hat ≟ 14 &lt;span class=&quot;math-container&quot;&gt;$&#xA;3 · 3 $&lt;/span&gt;           + 1 · 5 &lt;span class=&quot;math-container&quot;&gt;$           ≟ 14 $&lt;/span&gt;&#xA;    9 &lt;span class=&quot;math-container&quot;&gt;$           +     5 $&lt;/span&gt;           ≟ 14 &lt;span class=&quot;math-container&quot;&gt;$&#xA;                       14 $&lt;/span&gt;           = 14 $&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;6)  When/if it becomes obvious that the equation is true, put a check over the equals sign, and congratulate yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7)  If it becomes obvious that the equation is not true, either &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7243/teaching-students-to-find-and-correct-their-own-errors#7244&quot;&gt;try to find&lt;/a&gt; the &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7243/teaching-students-to-find-and-correct-their-own-errors#7254&quot;&gt;mistake&lt;/a&gt;, or start over, or try a different problem.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2019-09-16T22:17:35.160" LastActivityDate="2019-09-16T22:17:35.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1862" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T22:44:21.253" Score="-3" Body="&lt;p&gt;Since you are referring to High School students, I suggest a different approach. The women mentioned so far probably appeal to girls who are already interested in Math, but not to the masses. If you want to change the perception is that Math is &quot;nerdy&quot; and &quot;uncool&quot;, here are some women (not Math necessarily, but technology) who are alive, cool, possibly even &quot;hot&quot; (A role model for teenage girls?)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Limor &quot;Ladyada&quot; Fried&#xA;She founded the Electronics company in 2005 and appeared on a hangout with president Obama in 2013&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nixie Pixel (not sure about her real name)&#xA;She makes Linux tutorials on Youtube&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Veronica Belmont&#xA;A technology host for Revision3.com&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Proof that a woman can be smart and &quot;cute&quot; at the same time!&lt;/p&gt;&#xA;" OwnerUserId="1150" LastActivityDate="2014-04-21T22:44:21.253" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1863" PostTypeId="2" ParentId="1834" CreationDate="2014-04-21T23:14:12.440" Score="13" Body="&lt;p&gt;First, I want to comment on something that ChrisA seemed to have glossed over in his &lt;a href=&quot;https://matheducators.stackexchange.com/a/1840/267&quot;&gt;detailed description.&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For instance, in the example, $3x+5=14$:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;First, decide what to do to both sides (subtract 5)&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In my experience as a teacher and tutor, I have noticed that &lt;em&gt;this&lt;/em&gt; is not easy for novice Algebraists. However, I have found that there is a way that you can help to make this &quot;decision.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are all familiar with the order of operations and many of us use the mnemonic PEMDAS to determine which operations to perform in which order when evaluating an expression. This is applicable for evaluating expressions and complicated/fabricated arithmetic problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can be used in Algebra as well, though it is not something that I have seen often. The decision on what to do is the reverse of the order of operations. The two operations which are acting upon $x$ in the given example are multiplication and addition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As an aside, multiplication is represented with the $3$ immediately in front of it; multiplication has many forms and these forms may be something that you may want to discuss with your students as well if they have trouble recognizing each of the forms. Addition is represented with the $+5$, though you may wish to have a deeper discussion when dealing with a problem with subtraction, as it can be thought of as either subtraction or addition of a negative number.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the order of operations and PEMDAS, multiplication comes before addition, and if this were a problem with only numerals, then that is the order that you would have to do things. However in Algebra when we solve for a variable, we are attempting to unravel the operations being performed on the variable so we can read the variable alone and use the property of equality to determine its equivalent value. This unraveling is done by observing what is happening to the variable, and performing the inverse operation to be left with only an identity (in the case of multiplication and division, that identity is $1$ while in addition and subtraction it is $0$; this is another topic you may wish to go into more detail about). Because identities provide equivalent values, we often do not write these. It is this reason that in Algebra we reversing the steps of the order of operations and PEMDAS, and because these operations are reversed, so too is the order of the operations. This is why we need to do the opposite of addition (subtraction) first, and the opposite of multiplication (division) second. I have found that making this thought process explicit has helped some of my students more easily determine this &quot;decision.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do not be afraid to use the technical terminology either (e.g., inverse, identity), as I have found that this actually helps to clarify things.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;---------&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Second, there is a lot of value in rewriting the equations in two different ways. I have seen students who prefer each style, so you may want to try both:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Method 1:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation} 3x+5=14 \end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation} \qquad \color{red}-\color{red}5 = \color{red}-\color{red}5 \end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation} \quad 3x=9 \end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Method 2:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation} 3x+5=14 \end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation} 3x+5\color{red}-\color{red}5 =14\color{red}-\color{red}5 \end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation} \quad 3x =9 \end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using color here with this second method is particularly helpful, and is something that I use whether in a room with a white board or a blackboard.&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-12-23T08:23:06.140" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1864" PostTypeId="2" ParentId="1817" CreationDate="2014-04-21T23:56:22.713" Score="25" Body="&lt;p&gt;If your main interest is to provide a role model that students can identify with, you might want to look at &lt;a href=&quot;http://www.danicamckellar.com/&quot; rel=&quot;noreferrer&quot;&gt;Danica McKellar.&lt;/a&gt;  According to &lt;a href=&quot;http://en.wikipedia.org/wiki/Danica_McKellar&quot; rel=&quot;noreferrer&quot;&gt;her Wikipedia entry&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;McKellar studied mathematics at UCLA, graduating &lt;em&gt;summa cum laude&lt;/em&gt; in 1998.&lt;/p&gt;&#xA;&lt;p&gt;As an undergraduate, she coauthored a scientific paper with Professor Lincoln Chayes and fellow student Brandy Winn entitled &amp;quot;Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}^2$&lt;/span&gt;.&amp;quot; Their results are termed the 'Chayes–McKellar–Winn theorem'.&lt;/p&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;p&gt;McKellar has authored several mathematics-related books primarily targeting adolescent readers [those in middle-school and high-school] interested in succeeding at the study of mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-21T23:56:22.713" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1865" PostTypeId="1" CreationDate="2014-04-22T00:03:28.600" Score="11" ViewCount="661" Body="&lt;p&gt;Multivariable limits are harder than their one-variable counterparts, and textbooks examples usually focus on limits that don't exist when approaching from different straight lines. This gives the false impression that testing straight lines is &quot;the method&quot; to use when trying to discover if a given limit exists. Counterexamples as $$\lim_{(x,y) \to (0,0)} \frac{x^2y}{x^4+y^2}$$ don't convince students. The question is: &lt;strong&gt;what are effective strategies to convince them that there is no single way to attack these problems?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-04-23T14:49:09.387" Title="Multivariable limits" Tags="|undergraduate-education|mathematical-pedagogy|vector-calculus|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1866" PostTypeId="2" ParentId="1860" CreationDate="2014-04-22T00:25:42.750" Score="14" Body="&lt;p&gt;Making my comment an answer: rationalizing fractions was useful when computers weren't so readily available. Division using integers is way more feasible than dividing by rationals. Even today, if you want to have a reasonable estimate by hand you don't compute $1/\sqrt{2}$ but you do quickly $\sqrt{2}/2$.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-04-22T00:25:42.750" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1867" PostTypeId="2" ParentId="1819" CreationDate="2014-04-22T00:57:29.907" Score="8" Body="&lt;p&gt;I wouldn't normally post this as an answer but I don't have enough rep to leave a comment. This will mainly address your last bullet point since I can only speak from the perspective of a student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my first abstract algebra course we used the open-source textbook by Judson (see link below since I can only post two links due to rep). It had a large collection of Sage exercises for the end of each chapter and these were due (roughly) once per week. Much of what Sage implements is done as a wrapper to GAP. The exercises, along with significant commentary, can be found at&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://abstract.pugetsound.edu/sage-aata.html&quot;&gt;http://abstract.pugetsound.edu/sage-aata.html&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everyone in the course had seen at least a one semester introductory programming course. A significant portion of the class had taken linear algebra from a professor who also integrates Sage everywhere. If you're interested you can &lt;a href=&quot;http://linear.pugetsound.edu/sage-fcla.html&quot;&gt;see the linear textbook&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the exercises could prove incredibly frustrating at times, in hindsight I do feel they were worth the time investment. I don't feel that any of the group theory ideas were made more concrete through the Sage exercises, but the general comfort level I was forced to develop with Sage has lead me to return to it whenever I want to play around with an idea for a counterexample, or experiment with a theorem I don't fully understand on first reading, etc.&lt;/p&gt;&#xA;" OwnerUserId="1154" LastActivityDate="2014-04-22T00:57:29.907" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1868" PostTypeId="2" ParentId="1860" CreationDate="2014-04-22T01:06:07.427" Score="13" Body="&lt;p&gt;One idea: Calculate 1/(sqrt(3.0000001)-sqrt(3)) and compare with the answer of calculating the rationalized form (sqrt(3.0000001)+sqrt(3))/0.0000001.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Adjust the number of 0's to the precision of the calculator or software used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Maple with Digits:=10, the first expression gives: 3.571428571*10^7 while the second gives 3.464101644*10^7.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would be a good occasion to show that calculators or computers don't always give the &quot;right&quot; answer. A discussion could also be held as to what it means to give a &quot;right&quot; answer.&lt;/p&gt;&#xA;" OwnerUserId="503" LastActivityDate="2014-04-22T01:06:07.427" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1869" PostTypeId="1" CreationDate="2014-04-22T01:59:30.977" Score="21" ViewCount="1057" Body="&lt;p&gt;In short, my question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What percentage of American adults know what a prime number is?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Since this question is very specific, and my interests are a bit broader, I'd also be happy with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a good source for statistics on adult mathematical literacy?  I have in mind things like knowledge of basic &lt;em&gt;abstract&lt;/em&gt; mathematical concepts, like divisibility, primality, etc.  I'm less interested in practical knowledge (interpreting graphs, statistics, that sort of thing).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Let me give some motivation for this question.  When writing &quot;popularly accessible&quot; mathematics, I'd like to know something about my audience's background; it's also useful to have some idea of the average person's background when discussing math with a layperson.  But I've had a lot of trouble finding statistics about people's knowledge of basic mathematical concepts, and people I've asked have given wildly different estimates of the answer to the question above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I'm looking for some reliable information, e.g. in the form of a poll.  Of course I would also appreciate statistics about other countries, or sources of data of this type.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good answer would be numerical data about this question or a similar one.&lt;/p&gt;&#xA;" OwnerUserId="1156" LastEditorUserId="1156" LastEditDate="2014-04-25T18:34:43.850" LastActivityDate="2014-04-25T19:00:34.417" Title="Adult Mathematical Literacy" Tags="|mathematical-pedagogy|statistics|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1870" PostTypeId="2" ParentId="1860" CreationDate="2014-04-22T02:03:55.320" Score="6" Body="&lt;p&gt;In the olden days before calculators and computers were commonplace, mathematicians had thick books of precomputed values, like $\sqrt{2}$, which they could reference to compute their answers to an accuracy of several decimal places.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without having to actually purchase one of these books and try it, you can still mention that using a book to approximate the value of $\sqrt{2}/2$ was considerably easier than finding $1/\sqrt{2}$, although a calculator today would do both just as easily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why is it relevant today to rationalize denominators?  Well, math contests (especially American ones such as the American Regions Mathematics League) want answers given in simplest form, with rational denominators.  Not knowing how to do this would lose easy marks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you meant rationalizing denominators in expressions like $3/(\sqrt{2}+\sqrt{3})$ by multiplying top and bottom by $(\sqrt{2}-\sqrt{3})$, then it's because this technique can be used later on to greatly simplify expressions or derive proofs for higher math.&lt;/p&gt;&#xA;" OwnerUserId="1157" LastEditorUserId="1157" LastEditDate="2014-04-22T02:13:54.970" LastActivityDate="2014-04-22T02:13:54.970" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1871" PostTypeId="2" ParentId="1860" CreationDate="2014-04-22T02:16:30.463" Score="11" Body="&lt;p&gt;(Hi, Dan!) Although there are various not-unreasonable motivations to &quot;rationalize denominators&quot;, I think mostly it is mostly just another not-very-well-motivated rule to test students' &lt;em&gt;compliance&lt;/em&gt;. It is true that polynomials in radicals, with rational coefficients, are simpler than rational expressions in radicals, unless something simplifies surprisingly, it really makes little difference. Perhaps an example of one of the few times this might &lt;em&gt;really&lt;/em&gt; matter is testing exact equality, where the &quot;reduced&quot; polynomial expression is canonical and unique, while rational expressions are not unique. Nevertheless, when looking at polynomials in several algebraic numbers, it quickly becomes non-trivial to decide on nice minimal/canonical polynomial expressions, also. This already happens with $n$th roots of unity with highly composite $n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For that matter, what about denominators involving several square roots? Or, worse, several roots of higher degrees? These are tractable, but the usual curriculum stops surprisingly soon... giving more credence to the suspicion that it's just a thing to browbeat students about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, a &lt;em&gt;larger&lt;/em&gt; version of this reduction/simplification is useful, but the version almost universally taught stops far too soon, and thus (at least passively) has the effect merely of a test of obedience to rules. Not my favorite thing to test.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-22T02:16:30.463" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1872" PostTypeId="2" ParentId="1781" CreationDate="2014-04-22T04:24:46.680" Score="12" Body="&lt;p&gt;Why does it matter what the research shows?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At best, a study might have found that students who were blindly fed recipes did better in the short term on examinations for calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then, what is the ultimate goal of education? For students to pass tests? I really hope not. The goal of any teacher, &lt;em&gt;especially a teacher in mathematics&lt;/em&gt;, should first and foremost be to make the students understand &lt;em&gt;why&lt;/em&gt; things are true, how they work, how to prove them... I personally feel that mathematics teachers have a moral duty to at least &lt;em&gt;attempt&lt;/em&gt; to let their students really understand the material - otherwise the students are just being programmed like computers (they do it correctly, but they have no idea what they're doing).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;calculus needs to be taught in a procedural way&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is simply not true. I know for a fact that it does not &lt;em&gt;need&lt;/em&gt; to. I know this because I was once a grade 12 students who loved maths, and I know such students exist today. I know that for them, not only should it &lt;em&gt;not&lt;/em&gt; be taught that way, it &lt;strong&gt;must not&lt;/strong&gt; be taught that way!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students want to understand. Deep down, all of us have this urge. Giving up and just asking how to get the answer is what a student does when they are frustrated and do not think they can overcome a conceptual difficulty. Teaching &quot;procedurally&quot; only encourages this kind of failure. If students can be taught that they have the inner potential to understand &lt;em&gt;anything&lt;/em&gt; - and I believe all who like maths do - then they will take infinitely more away from a class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember that mathematics did not come out of a text book. It was created by real people, who were trying to solve real problems, who often had to try hundreds of solutions before finally coming up with something that worked. They knew that it worked because they &lt;em&gt;proved&lt;/em&gt; it. Mathematics is not a collection of magical formulas that &quot;just work,&quot; and it should never be portrayed as such.&lt;/p&gt;&#xA;" OwnerUserId="1161" LastActivityDate="2014-04-22T04:24:46.680" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1873" PostTypeId="2" ParentId="1860" CreationDate="2014-04-22T05:25:38.087" Score="1" Body="&lt;p&gt;Having an irrational number in the denominator presents problems in computation, whether you are doing it by hand or with a calculator. If an irrational number is non-terminating, using a rounded or terminated approximation always introduces a computational error.  It's harder to estimate or control that error when it appears in the denominator of a fraction than in the numerator, and especially if the denominator is anywhere close to zero.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2014-04-22T05:25:38.087" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1874" PostTypeId="2" ParentId="1869" CreationDate="2014-04-22T07:04:29.197" Score="13" Body="&lt;p&gt;I don't see any studies of this sort on prime numbers, though I'm sure you could conduct an informal one and get a good estimate relatively quickly. Instead, I tackle your final note:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A good answer would be numerical data about this question or a similar one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How about: &lt;strong&gt;Is zero even?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Citing a &lt;a href=&quot;http://www.bbc.com/news/magazine-20559052&quot; rel=&quot;noreferrer&quot;&gt;popular media piece&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;According to Dr James Grime of the Millennium Maths Project at Cambridge University, reaction time experiments in the 1990s revealed people are 10% slower at deciding whether zero is odd or even than other numbers.&lt;/p&gt;&#xA;&lt;p&gt;Children find it particularly difficult to recognise if zero is odd or even. &amp;quot;A survey of primary school children in the 1990s showed that about 50% thought zero is even, about 20% thought it was odd and the remaining 30% thought it was neither, both, or that they don't know,&amp;quot; explains Dr Grime.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And it gets worse:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It's not just the public who have struggled to recognise zero as an even number. During the smog in 1977 in Paris, car use was restricted so that people with licence plates ending in odd or even numbers drove on alternate days.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The police did not know whether to stop the zero-numbered licence plates and so they just let them pass because they didn't know whether it was odd or even,&amp;quot; says Dr Grime.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;As for primes:&lt;/strong&gt; Are you wondering about the percentage of people who can &lt;em&gt;precisely&lt;/em&gt; define when a number is prime? If so, I think asking them if &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is prime is probably enough to reduce this number down to something miniscule.&lt;/p&gt;&#xA;&lt;p&gt;(Consider &lt;strong&gt;&lt;a href=&quot;http://www.newscientist.com/article/dn25452-game-of-primes-ends-as-mathematics-gets-too-difficult.html&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;&lt;/strong&gt; popular media piece on the polymath project post-Zhang, in which the author writes: &lt;em&gt;Over the past year, mathematicians have been battling it out in a game involving primes, numbers that are only divisible by themselves and 1.&lt;/em&gt; Even allowing for &amp;quot;numbers&amp;quot; to refer to natural numbers, this definition would still include 1 as a prime.)&lt;/p&gt;&#xA;&lt;p&gt;In any event, I think it is nice when popular media pieces on mathematics start with some basic definitions. I don't even think it's a stretch to include a definition of prime numbers in a graduate textbook on number theory.&lt;/p&gt;&#xA;&lt;p&gt;In the most recent issue of the AMM, there is an article entitled &lt;em&gt;The primes that Euclid forgot&lt;/em&gt; (&lt;a href=&quot;http://www.math.uga.edu/%7Epollack/mullin.pdf&quot; rel=&quot;noreferrer&quot;&gt;link&lt;/a&gt;). Though 'prime' is not explicitly defined, 'quadratic residue' is: see the start of section two. Of course, this latter term is sure to be quite unfamiliar to the general populace; but I expect it is known by a very large proportion of the &lt;em&gt;American Mathematical Monthly&lt;/em&gt;'s readership. If the authors there feel including a reminder-definition is worthwhile, then I suggest you dedicate a line to defining the term 'prime' in any relevant popular media piece.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-25T19:00:34.417" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1876" PostTypeId="2" ParentId="1817" CreationDate="2014-04-22T07:31:18.633" Score="9" Body="&lt;p&gt;Émilie du Châtelet ,(17 December 1706  – 10 September 1749) was a French mathematician, physicist, and author during the Age of Enlightenment. Her crowning achievement is considered to be her translation and commentary on Isaac Newton's work Principia Mathematica. The translation, published posthumously in 1759, is still considered the standard French translation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Voltaire, one of her lovers, declared in a letter to his friend King Frederick II of Prussia that du Châtelet was &quot;a great man whose only fault was being a woman&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(From Wikipedia)&lt;/p&gt;&#xA;" OwnerUserId="1165" LastEditorUserId="1165" LastEditDate="2014-04-22T07:46:37.997" LastActivityDate="2014-04-22T07:46:37.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1879" PostTypeId="1" CreationDate="2014-04-22T12:11:18.450" Score="5" ViewCount="1305" Body="&lt;p&gt;It is that time of the semester again (after the final drop date) and my colleagues and I are once again having the discussion on whether or not the two remaining tests should be made so they are less difficult in order to help the remaining students raise their grades.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is department policy not to curve grades and the current average of the remaining students is 61%; furthermore, the grade distribution is skewed left and not normal.  I argue that as long as we are not compromising the goals of the class, and that the students can demonstrate that they understand the Calculus, then we should make the individual test problems a little less difficult and let them raise their grades (my colleagues disagree). Is this a valid solution or would it be better to weight the tests less, and the homework and quizzes more?     &lt;/p&gt;&#xA;" OwnerUserId="203" LastActivityDate="2014-04-24T19:52:23.577" Title="Should tests given after the drop date be made less difficult in order to help the remaining students raise their grades?" Tags="|undergraduate-education|calculus|testing|test-design|" AnswerCount="4" CommentCount="3" ClosedDate="2014-04-29T11:00:18.390" ContentLicense="CC BY-SA 3.0" />
  <row Id="1880" PostTypeId="1" CreationDate="2014-04-22T12:14:29.237" Score="4" ViewCount="1651" Body="&lt;p&gt;I've been thinking about how you would go about teaching fractions, and there seems to be a problem in that every basic fact needs to be proven/explained &lt;em&gt;twice&lt;/em&gt;, using two different layers of definition. Let me illustrate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rule for multipliying fractions is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Where $a, b, c, d$ are integers, $\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To prove this takes an argument that relies entirely on the fact that $a, b, c$ and $d$ are integers, but then you define what it means to divide fractions and realize that a more general fact is true...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Where $a, b, c, d$ are &lt;em&gt;rational&lt;/em&gt;, $\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The same thing applies for every basic fact about fractions: cancelling, adding, dividing, equivalence of $\frac{a}{b}$ and $a\frac{1}{b}$... There are two problems, here.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The formal side. It's a bit annoying to have to prove fact (1) only to have to prove its generalization. The only way I can think of is to explicitly write out $a, b, c$ and $d$ in numerator/denominator form and then apply rule (1), so you need the more specific case to prove the general case. &lt;strong&gt;What is the most efficient way to prove all of these facts?&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The conceptual side. If this isn't explained properly, it's going seem like some sort of fantastic coincidence! The multiplication rule (1) is hardly obvious, and its proof takes a bit of cleverness. And then when this strange fact turns out to be true in a more general context, and the proof is just a formal manipulation of symbols, it's going to seem very mysterious. Actually, it seems mysterious even to me. &lt;strong&gt;How can these generalizations be presented in a way that doesn't make them seem like magical coincidences?&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="115" LastActivityDate="2014-04-23T03:08:26.253" Title="Teaching fractions: the generalization problem" Tags="|proofs|fractions|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1881" PostTypeId="2" ParentId="1865" CreationDate="2014-04-22T12:15:50.820" Score="2" Body="&lt;p&gt;I'd go by the definition (an $\delta$ neighborhood of $(0, 0)$ in your case) a few times. Perhaps cook up some limit that is easy with straight lines in a different coordinate system (but don't tell them the weird curves you choose are straight there until the end!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is always useful to give a homework comparing several approaches, e.g. prove the limit using the definition, use straight lines in all directions (essentially polar coordinates), go by a spiral, ... If for each way you can find problems that are easy to solve and hard in the others, you'll probably get what you want. &lt;em&gt;Do&lt;/em&gt; mention them here ;-)&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-22T12:15:50.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1882" PostTypeId="2" ParentId="150" CreationDate="2014-04-22T12:30:15.570" Score="6" Body="&lt;p&gt;Mathematics is an enormously powerful cultural achievement that (generally speaking) expands our ability to see the world and to act in the world. Mathematics education is our time-consuming effort to welcome you, the individual, into this culture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not a passive effort, and it doesn't happen all at once. You have to &lt;em&gt;engage&lt;/em&gt; with the world of mathematics, and you will do so &lt;em&gt;gradually&lt;/em&gt; as you gain the ability to mathematize in the world (the ability to use math in how you see and think about your world). &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Sidebar:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Take, for example, knowledge of ratios and proportionality. Assuming&#xA;  you already know some arithmetic, you can learn to read a word problem&#xA;  so that you place three numbers this way:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/TPOK4.png&quot; alt=&quot;2/3 = ?/12&quot;&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;And you can use arithmetic to find the missing value. But people who&#xA;  have an understanding of ratios and proportion don't see this problem the&#xA;  same way. Those people see a relationship between two quantities&#xA;  becoming a property or quantity in itself (in mathematics, we call&#xA;  this &lt;a href=&quot;http://en.wikipedia.org/wiki/Intrinsic_and_extrinsic_properties&quot; rel=&quot;noreferrer&quot;&gt;an extrinsic property or quantity&lt;/a&gt;). In appropriate&#xA;  situations, this becomes a powerful thing to know. It's related to&#xA;  rate, and it can be used to think about and compare proportional&#xA;  situations in many contexts.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It's the difference between being able to follow instructions and just&#xA;  mix the ingredients of lemonade together so that you get the recipe&#xA;  right vs. thinking mathematically about how the ingredients come&#xA;  together to create the &lt;em&gt;flavor&lt;/em&gt; of the lemonade.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When your view of the world changes it's as if you can see things that were invisible to you before. And once you can see things, you can make decisions based on them. Imagine being in big room in the dark, and the room is full of big obstacles. You want to move around, but you have to stumble around, feel your way around obstacles slowly, be careful not to trip. You can make progress moving around, but it seems pointless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, turn the lights on. Not only can you see where the obstacles are, but you can see the opposite wall of the room, and doors. You can choose to walk directly toward them -- that's a goal you weren't even aware of before. Seeing the objects in the room didn't just allow you to move around more quickly; you can now move with purpose. There's a point to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the most important thing: as the knowledge becomes your own, &lt;em&gt;you&lt;/em&gt; are the one who answers the question &lt;strong&gt;what is the point of learning this mathematics?&lt;/strong&gt; Math educators can only help turn the lights on. The rest is really up to you and what you want to do.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-04-22T12:30:15.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1883" PostTypeId="1" CreationDate="2014-04-22T12:57:25.120" Score="25" ViewCount="3838" Body="&lt;p&gt;Complex numbers are, or at least were to me, generally introduced like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There's no number whose square is negative. That's a shame! Well, whatever - we'll make one up! Set $i^2=-1$ and declare that all the usual rules of the real numbers apply.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is a simple approach, and close to how the complex numbers were stumbled upon historically. But I don't think it gets to the heart of &lt;em&gt;what complex numbers are&lt;/em&gt;, and less subjectively, I think it leads to confusion.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;For one thing, if we're being rigorous, it's nonsense. What precisely does &quot;all the usual rules apply&quot; mean? What is your proof that a structure obeying the above rules even exists?&lt;/li&gt;&#xA;&lt;li&gt;For me at least, it led to a feeling of unease, basically because I was intuitively aware of point (1).&lt;/li&gt;&#xA;&lt;li&gt;It makes the geometrical facts about complex numbers seem very mysterious.&lt;/li&gt;&#xA;&lt;li&gt;It leads to &lt;a href=&quot;https://math.stackexchange.com/questions/125186/why-not-to-extend-the-set-of-natural-numbers-to-make-it-closed-under-division-by&quot;&gt;this&lt;/a&gt; type of confusion. In the words of 19th century mathematician John T. Graves: &lt;em&gt;&quot;I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. (...) If with your alchemy you can make three pounds of gold, why should you stop there?&quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My personal approach would be something like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If we take the plane with a polar coordinate system, and define the following fairly natural geometric operations on points, we find that they obey the same algebraic laws as multiplication and addition. We express this observation by calling these points complex &lt;em&gt;numbers&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would start by writing down $a+bi$ as $(a, b)$ before quickly defining $i=(0, 1)$, identifying the horizontal axis with $\mathbb R$ and switching to the conventional notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this a poor way to teach complex numbers to a student who is encountering them for the first time (such as a high school student)? Is it more difficult for students to understand than the conventional approach? I can see how it might be difficult to explain why the horizontal axis &quot;is&quot; $\mathbb R$. And what about students who go on to do math or science in college? Is there any reason why they might struggle?&lt;/p&gt;&#xA;" OwnerUserId="115" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2020-10-16T21:38:14.690" Title="Are there disadvantages to teaching complex numbers as purely geometrical objects?" Tags="|secondary-education|geometry|definitions|complex-numbers|" AnswerCount="8" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="1884" PostTypeId="2" ParentId="1883" CreationDate="2014-04-22T13:43:09.707" Score="7" Body="&lt;p&gt;To answer the literal question of the title: yes, there are some disadvantages in portrayal of the complex numbers as &quot;purely geometric&quot; objects&quot;, although perhaps these disadvantages are more philosophical than mathematical. E.g., it is unclear whether or not we &quot;have license&quot; to declare that &quot;the plane&quot; is &quot;numbers&quot;... although we might have reasons to want to do so. That is, further, we might be unhappy that the arithmetic of complex numbers depends on two-dimensional geometry and geometric intuition for its sense and apparent self-consistency.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, the appearance of complex numbers was taken more seriously in the solution of cubics with three real roots, but for which the formula irremediably involved complex numbers. (Taking square roots of negative numbers was a thing that could simply be prohibited or declared nonsensical.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hamilton's search for a way to give numerical sense to three-space led to the quaternions as non-commutative (division algebra) structure on four-space, and, in particular, showed the impossibility of putting such a structure on three-space. Graves found the octonions, and so on. All this work quite blatantly had the motivation of &quot;identifying&quot; geometry and numbers. Grassmann's work was similarly motivated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, yes, many people were uneasy. Kronecker's construction of $\mathbb C$ as a quotient $\mathbb R[x]/\langle x^2+1\rangle$ is not as elementary as one might wish, and not geometric, but &lt;em&gt;is&lt;/em&gt; arguably a more bullet-proof argument for existence without appeal to geometric intuition. No, it doesn't explain why we cared in the first place, but it may give a skeptic more confidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as what beginning students should be told... probably not the Kronecker version, since this does not convey the intuitions that motivated things in the first place.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-22T13:43:09.707" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1885" PostTypeId="2" ParentId="1879" CreationDate="2014-04-22T13:53:07.680" Score="14" Body="&lt;p&gt;The situation you describe is pretty dire. At &lt;a href=&quot;http://en.wikipedia.org/wiki/Academic_grading_in_the_United_States#Numerical_and_letter_grades&quot; rel=&quot;nofollow&quot;&gt;most US insitutions&lt;/a&gt;, an uncurved 61 is a D or F. If your mean is a D or F, and the median is below that (I think that's what you mean by leftward skew?) then most of the class is likely to fail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the first thing you should ask yourself is whether the tests were fairly evaluating the goals you set for your students. If they were too hard for other reasons (not enough time given, numerical problems too difficult, poor directions), then you should fix these problems in future tests, which will have the effect of making your tests less difficult. You will probably also need to adjust your grading scheme to deal with the damaged tests, probably by weighting them less. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you decide the tests were fair, then your students are not meeting the goals you set for them. &lt;b&gt;I would encourage you to lower your goals.&lt;/b&gt; There might be a few cases were it is legitimate for an instructor to fail half a class. (A flight instructor shouldn't certify anyone who can't fly.) But these cases are really rare. Normally, I think that you should adjust the goals of the class to a level that most students can meet, so they are getting some benefit from the class. Lowering your goals for the class will have the effect of making exams more easy, but it will have the more important effect of making it relevant for your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also think you should speak to your chair about whether this policy against curving is absolute. Solving the problems with your test and curricular goals, as described above, are the big tasks, but a curve could also be a helpful bandaid, and your administration doesn't want half your class to fail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;b&gt;ADDED 4/24/2014&lt;/b&gt; I just wanted to note that I wrote this answer before the OP clarified that this was a course with a common curriculum over several years and a consistent testing history, whose students had recently become unable (for whatever reason) to meet the previous standards. That's an incredibly messy situation, and I don't claim to have advice for it without actually knowing what larger forces are causing this. The only advice I would give strongly in that setting is that the OP should notify his higher ups of what is going on.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2014-04-24T19:52:23.577" LastActivityDate="2014-04-24T19:52:23.577" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1886" PostTypeId="2" ParentId="1879" CreationDate="2014-04-22T14:00:11.420" Score="6" Body="&lt;p&gt;I would like to complement Speyer's answer with this: &lt;strong&gt;encourage their feedback about the course goals, tests and related activities.&lt;/strong&gt; This is a critical time to take in account what they have to say as well. As inarticulated their opinions may be, they may carry (part of) a solution to what's going on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lowering your goals and/or reevaluating how your tests are measuring them will have greater effect if you manage to see from their perspective as well.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-04-22T14:00:11.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1887" PostTypeId="2" ParentId="1879" CreationDate="2014-04-22T14:04:28.060" Score="10" Body="&lt;p&gt;Fairness suggests that the grade a student gets shouldn't depend on the semester they happen to take a course in.  So the first question I'd ask is what sorts of grades have students typically gotten in this course.  If they haven't typically been averaging 61% at this point, then you need to know what's changed: are these actually a weaker crop of students, or were the tests harder than before, or did something else cause this difference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most likely explanation (though not the only) is that the tests were a bit harder this year.  (This is the most likely explanation because if you're teaching a large number of students, the law of large numbers says they should be pretty similar to last year's---and 100 or so is usually large enough.  On the other hand it's very easy to drop the average on an exam by 10 points by making one or two questions just a little harder in ways that cause students to waste time on them.)  In that case you should absolutely reconsider the remaining tests, and probably make them easier to compensate.  (Actually, you should curve the earlier tests, but apparently that's not allowed.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-04-22T14:04:28.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1888" PostTypeId="2" ParentId="604" CreationDate="2014-04-22T14:05:19.630" Score="8" Body="&lt;p&gt;To my mind, this abuse of notation is more similar to how I think when I'm doing physics. Imagine a bottle of gas. It has a temperature $T$, a volume $V$, a pressure $P$, a number of moles $N$ and an entropy $S$. They are related by various relations such as the &lt;a href=&quot;http://en.wikipedia.org/wiki/Ideal_gas_law&quot;&gt;ideal gas law&lt;/a&gt;. Which one is a function of which? Well, any one can be a function of any other, under varying circumstances. Insisting that I am going to write everything as a function of temperature, or everything as a function of volume, or whatever, only introduces confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also tend to think in terms of many related quantities, no one a function of any other, when doing algebraic geometry (my research area) and differential geometry, but those aren't relevant to students in an ordinary calculus course.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2014-04-22T15:22:55.427" LastActivityDate="2014-04-22T15:22:55.427" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1889" PostTypeId="2" ParentId="1857" CreationDate="2014-04-22T14:15:11.163" Score="5" Body="&lt;p&gt;Below are three books for what I think you want. At one time these books were all very well known, at least in the U.S., probably because virtually every public library (even small town libraries) used to have copies of these books.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0442005288&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Mathematics, Its Magic and Mastery&lt;/strong&gt;&lt;/a&gt; by Aaron Bakst (here is the &lt;a href=&quot;https://web.archive.org/web/20130216020037/https://www.livingmath.net/Reviews/ReviewsHSAdultMathLit/MathematicsItsMagicMastery/tabid/1079/language/en-US/Default.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Table of Contents&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0395065666&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Realm of Numbers&lt;/strong&gt;&lt;/a&gt; by Isaac Asimov&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0449243982&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Realm of Algebra&lt;/strong&gt;&lt;/a&gt; by Isaac Asimov&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2020-08-05T20:46:07.183" LastActivityDate="2020-08-05T20:46:07.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1890" PostTypeId="1" AcceptedAnswerId="1904" CreationDate="2014-04-22T14:42:55.760" Score="8" ViewCount="344" Body="&lt;p&gt;When leaving homework to students, should we state that there is an excercise with a special level of difficulty?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen this done in a lot of physics books, and some books on mathematics have their section of &quot;advanced excercises&quot;. Is it convenient al all to use this in classroom?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make the question more specific, I'm talking about from the point of view of the student, do they benefit from it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example: I think that it may make students to feel challenged and more satisfied if they solve it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Papers on the subject are welcomed.&lt;/p&gt;&#xA;" OwnerUserId="1151" LastActivityDate="2014-04-22T22:22:21.777" Title="Is it convenient to classify problem by level of difficulty?" Tags="|homework|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1891" PostTypeId="2" ParentId="1865" CreationDate="2014-04-22T15:49:53.753" Score="11" Body="&lt;p&gt;A problem I used to assign on take-home tests back when I was teaching gifted high school students (late 1990s) might be of interest.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Show that &lt;span class=&quot;math-container&quot;&gt;$f(x,y) =y^{x}$&lt;/span&gt; has the same limit along any 1st quadrant polynomial approach towards &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; (i.e. along any path having the form &lt;span class=&quot;math-container&quot;&gt;$y=ax^{n},$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a positive integer), but the limit varies if approach paths of the form &lt;span class=&quot;math-container&quot;&gt;$y=\exp \left(-\frac{a}{x}\right)$&lt;/span&gt; are used, where &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;0.$&lt;/span&gt;  (Why do I need &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;0?)$&lt;/span&gt; As part of your solution include carefully drawn graphs, for &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;x&amp;lt;0.5,$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$y=x^{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=x^{3},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$y=x^{4},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$y=\exp \left(-\frac{0.5}{x}\right),$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$y=\exp \left(-\frac{1}{x}\right),$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=\exp \left(-\frac{2}{x}\right)$&lt;/span&gt; together on the same coordinate axis. You may sacrifice some accuracy in order to give a more informative and schematic presentation of the behavior of these graphs as &lt;span class=&quot;math-container&quot;&gt;$x \rightarrow 0^{+}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Extra Credit) Verify analytically that if &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is any positive integer, then &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{x\rightarrow 0^{+}}\left[ \frac{\exp \left(-ax^{-1}\right)}{x^{n}}\right] =0.$&lt;/span&gt; You will find that a direct application of L'Hopital's rule doesn't help. However, this limit can be evaluated by using the changing variables method given (somewhere) in my handwritten handout titled &amp;quot;Some Techniques for Evaluating Limits&amp;quot;. But feel free to use another (mathematically justified) method if you find one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; The problem itself can be solved easily by noting the following.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$y^x \;\; = \;\; \left(ax^{n}\right)^x \;\; = \;\; a^{x} \cdot x^{nx} \;\; = \;\; a^{x} \cdot \left(x^x\right)^n \;\; \rightarrow \;\; a^0 \cdot 1^n \;\; = \;\; 1 $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ y^x \;\; = \;\; \left(e^{-\frac{a}{x}}\right)^x \;\; = \;\; e^{-a} \;\; \rightarrow \;\; e^{-a}  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The point of the extra credit is to show analytically what the graphs should show, namely that the exponential approaches slip underneath every one of the graphs &lt;span class=&quot;math-container&quot;&gt;$y = ax^n$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x \rightarrow 0^{+}.$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-22T15:49:53.753" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1892" PostTypeId="2" ParentId="1890" CreationDate="2014-04-22T16:14:28.090" Score="1" Body="&lt;p&gt;No, I don't think it's convenient, and worse, I think it may be a waste of time.&#xA;I think there are better things to do with your time than classify questions like this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difficulty level can be counterproductive, setting up a student to give up on a &lt;em&gt;difficult&lt;/em&gt; problem, or to feel stupid for not solving one that's deemed &lt;em&gt;easy&lt;/em&gt;. &#xA;Last, a given problem might not strike every student the same, e.g. some happen to be better at geometry, so for them a &lt;em&gt;hard&lt;/em&gt; geometry problem might be far easier than for student who is otherwise at the same overall level. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd be curious to understand the benefits of such classification. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-04-22T20:37:35.123" LastActivityDate="2014-04-22T20:37:35.123" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1893" PostTypeId="1" CreationDate="2014-04-22T16:29:58.017" Score="9" ViewCount="458" Body="&lt;p&gt;&lt;strong&gt;Updated&lt;/strong&gt;: The first version of this question was implicitly assuming that we know in advance roughly what the distribution of grades in a class will be (for instance, roughly the same as last year).  This assumption wasn't actually essential to the question, and was bothering some people, so I've updated it to remove that assumption.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Inspired by &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1879/should-tests-given-after-the-drop-date-be-made-less-difficult-in-order-to-help-t&quot;&gt;this question&lt;/a&gt;, how difficult should an exam be?  For instance, assume you've decided about what skills should denote a B- student, and that you're free to norm exam scores to course scores as needed.  By adjusting the questions you put on an exam, you can adjust what score a student with B- skills will typically get.  Where should you aim for?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some answers I can think of, which some plusses and minuses:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Aim for the score traditionally correlated with that grade.  For instance, in the US there's a custom that a B- corresponds to a grade in the 80-82 range, so you could try to write an exam where a score of 80-82 will typically reflect the skills of a B- student.  (Actually, if it's a class where non-exam items will tend to increase grades, aim a bit lower.)  The advantage is that, if the exam, there's &quot;no curve&quot;, and students tend to feel like they've done well.  A disadvantage is that if you accidentally make the exam too easy (say, a question has a short cut and doesn't reveal as much as you intended), your options are limited.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Aim a bit lower.  Say, aim for an exam where a score of around 70 reflects a B-.  More importantly, this gives a bit more separation at the top of the class (you have room for a question or two to distinguish the top students from the very good).  On the other hand, students seem to feel like their grades are arbitrary, and they may feel more competitive (and less willing to work and study together) if they feel like their friends doing well will cut into their curve.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Aim much lower---say, put a B- around 50 and curve heavily.  This makes the cut-offs between final grades much wider (the gap between a B- and a B might be something like 7 points instead of 3), and therefore a single small mistake has a smaller impact on a score.  Students get quite discouraged by these, in my experience.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My question is whether there's any actual research suggesting that some of these are clearly good/bad for student learning, and whether there are ethical or practical issues I'm overlooking.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-04-22T22:21:54.843" Title="How should exam scores correlate to final grades?" Tags="|undergraduate-education|exams|grading|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1894" PostTypeId="2" ParentId="220" CreationDate="2014-04-22T16:47:59.750" Score="0" Body="&lt;p&gt;Every natural number is odd or even.&lt;/p&gt;&#xA;" OwnerUserId="1178" LastActivityDate="2014-04-22T16:47:59.750" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1895" PostTypeId="2" ParentId="150" CreationDate="2014-04-22T16:49:24.290" Score="7" Body="&lt;p&gt;I personally feel that this question, as posed, is a bit of an oxymoron. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But there are also many subjects unrelated to mathematics (law, language studies, etc.) and young people heading to such subjects (being bad in mathematics) are frustrated.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is this not understandable? I, as a student, was heading from the start to mathematics. This meant that I was frustrated by nearly everything else: speeches on making apple pie, tests on government, 5-paragraph essays with lots of transitions, ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no possible justification to give a student who knows they will never use math in their real life. And in fact, most people do not. Many of the most influential people in the world - politicians, lawyers, business people, artists - will never touch a single equation once they leave school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact is that &lt;strong&gt;mathematics is &lt;em&gt;not&lt;/em&gt; so important!&lt;/strong&gt; It is important to those who love it and use it, to people like me. But the same goes for everything else - History is vastly important to &lt;em&gt;those who use it&lt;/em&gt;, but not to people like me who would rather not read another history book for the remainder of our existence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The worst possible choice would be to give them terrible reasons as to why they will need mathematics someday. Students are intelligent enough to know that is not true. You should just tell them that working hard and pushing themselves to complete tasks which may not be to their liking, are good life skills to have. But you should also encourage them to pursue what &lt;em&gt;they&lt;/em&gt; love, no matter what anyone tells them.&lt;/p&gt;&#xA;" OwnerUserId="1161" LastActivityDate="2014-04-22T16:49:24.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1896" PostTypeId="1" CreationDate="2014-04-22T16:50:49.773" Score="32" ViewCount="4710" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;This is a very basic fact from...&lt;/p&gt;&#xA;&lt;p&gt;It then clearly follows that...&lt;/p&gt;&#xA;&lt;p&gt;Obviously, we have...&lt;/p&gt;&#xA;&lt;p&gt;The proof is trivial...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I could add plenty of other phrases to this list that mathematicians are prone to use when trying to communicate that a particular concept is so simple that they refuse to discuss the details.  I can't think of a time when words like these helped me.  Usually they just make me realize how lost I am, and how much more the professor/author knows than I do.  Even if I agree that the fact is basic, or the proof is trivial, it didn't help me to hear that it was easy or trivial, because I already thought that.&lt;/p&gt;&#xA;&lt;p&gt;To get to my question, consider the following amendments:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is a fact from...&lt;/p&gt;&#xA;&lt;p&gt;It then follows that...&lt;/p&gt;&#xA;&lt;p&gt;We have...&lt;/p&gt;&#xA;&lt;p&gt;The proof is left as an exercise.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is there ever a context in which it is more helpful for students to hear the first set of sentences as compared to this second set?  Are words like &amp;quot;easy,&amp;quot; or &amp;quot;trivially,&amp;quot; ever constructive?  In what setting?  As math educators, should we work to weed them out of our written/spoken vocabulary?&lt;/p&gt;&#xA;" OwnerUserId="21" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-10-30T14:26:51.367" Title="Are the words &quot;easy,&quot; &quot;basic,&quot; &quot;clearly,&quot; &quot;obviously,&quot; etc., ever helpful?" Tags="|mathematical-pedagogy|terminology|" AnswerCount="11" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1897" PostTypeId="2" ParentId="1896" CreationDate="2014-04-22T17:00:02.067" Score="15" Body="&lt;p&gt;I can think of a few instances where it &lt;em&gt;might&lt;/em&gt; be useful:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To situate the current piece of the concept among others coming up&lt;/li&gt;&#xA;&lt;li&gt;To call-back to something earlier in the course that really &lt;em&gt;should&lt;/em&gt; be easy to them at this point&lt;/li&gt;&#xA;&lt;li&gt;To intentionally make fun of something you know they thought was stupid (e.g. high school? if you're teaching a proofs course)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;And, examples of each of these:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To start off this proof, we prove this easy (or easier) theorem...  (again, the goal is to tell them that something &lt;em&gt;harder&lt;/em&gt; is coming soon)&lt;/li&gt;&#xA;&lt;li&gt;Obviously, we have 2n+1 is odd, by the definition of odd.  (again, later in the course--not if you've just introduced quantifiers)&lt;/li&gt;&#xA;&lt;li&gt;And now, to finish off this proof, we do a trivial computation of this obnoxious integral.  You all learned this in high school, right?  (here, the goal would be for people to roll their eyes)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I do agree with you on the general idea though.  We use these terms far too often when we're explaining things, and they are &lt;em&gt;almost always&lt;/em&gt; not a good idea.&lt;/p&gt;&#xA;" OwnerUserId="93" LastEditorUserId="143" LastEditDate="2014-04-23T18:03:11.217" LastActivityDate="2014-04-23T18:03:11.217" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1898" PostTypeId="2" ParentId="1112" CreationDate="2014-04-22T18:00:28.930" Score="9" Body="&lt;p&gt;Your first option---&quot;someone gets [a grade] only by correct answers&quot;---hides some problems.  How does one convert a fraction of correct answers into a grade, and correct answers on what problems?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unless you have some sort of externally given bank of problems with a built in curve, both of these choices are essentially arbitrary.  (There happens to be a customary answer to the first question---90% is an A, 80% a B, and so on---but there's nothing particularly canonical about it, and it doesn't address the second question, which questions should a student get 90% of right in order to get an A.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In some courses you may have some sort of external guide to what adequate performance is; for instance, you might be norming against an exam provided by some external source, or you might have a clear view of what should count as A work.  A crucial property of this scenario is that your exam questions wouldn't change if you knew student scores; if all the students get D's this year, you'll conclude that they hadn't learned enough, give them D's, and ask comparable questions next year (possibly while reconsidering how you were teaching).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If that's not the case---if your response to a year where all the students got D's would be to adjust the questions---then implicitly you are setting the grades against an expected class performance.  If you look at scores in the 60s and think &quot;This year everyone gets a D, but next year I'll make it easier so that doesn't happen again&quot;, you're effectively applying a curve, you're just doing so inconsistently (and in a way which is unfair to some of the students).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any time the class deviates from the expected performance, you have to ask &quot;Is this deviation a reflection in the abilities of the students, or the quality of the exam?&quot;  The latter is actually much more variable than the former, so we often attribute most of the variation to the exam, and adjust class grades accordingly.  (Depending on the system, there may be intermediate options; I generally curve exams to an expected median, but some years there are indicators that it's a stronger or weaker group---for instance, I may have particular problems I can do a direct comparison with other year's performances---in which case I adjust the curve a little.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Phrased differently, curving a class (for instance, by saying in advance what percentage will get each grade), is often characterized as being about some kind of competition amongst students.  This isn't necessarily the case.  Curving a class is often using &lt;em&gt;an assumed distribution of student performance to norm the exam&lt;/em&gt;. &lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-04-22T18:00:28.930" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1899" PostTypeId="1" AcceptedAnswerId="2119" CreationDate="2014-04-22T18:07:20.177" Score="22" ViewCount="747" Body="&lt;p&gt;I'm curious if anyone knows of any books, studies, or other resources on the &lt;em&gt;philosophy&lt;/em&gt; of creating and grading mathematics exams.  After working as a graduate TA for 4 years and dealing with a wide variety of issues creating and grading precalculus and calculus exams, I feel like I've developed a lot of personal opinions about how things &quot;should&quot; be done.  I've even had quite a few &quot;discussions&quot; (borderline arguments) about exam fairness and so forth, so now I'm wondering if there is any well-repsected consensus opinions out there. I'm being deliberately non-specific, so perhaps this question might work best as a community wiki?&lt;/p&gt;&#xA;" OwnerUserId="227" LastEditorUserId="262" LastEditDate="2016-06-18T02:38:56.363" LastActivityDate="2016-06-21T12:01:17.917" Title="Exam philosophy" Tags="|undergraduate-education|exams|grading|assessment|testing|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1900" PostTypeId="2" ParentId="1880" CreationDate="2014-04-22T18:08:01.583" Score="5" Body="&lt;p&gt;I think it would be best to focus on explanations for &lt;strong&gt;specific examples&lt;/strong&gt; rather than generalities, which can come later. For example, let's consider multiplying $\frac{4}{7}$ and $\frac{3}{5}.$ (However, I suggest first doing simpler examples, such as $\frac{1}{2} \times \frac{1}{3}$ and $\frac{1}{3} \times \frac{2}{3}.)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can view $\frac{4}{7}$ of something as dividing the &quot;something&quot; into $7$ equal pieces and then keeping $4$ of those pieces. We want to do this for $\frac{4}{7}$ of $\frac{3}{5}.$ Viewing $\frac{3}{5}$ as $\frac{1}{5} + \frac{1}{5} + \frac{1}{5},$ we want to divide $\frac{1}{5} + \frac{1}{5} + \frac{1}{5}$ into $7$ equal pieces and then keep $4$ of those pieces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do we divide $\frac{1}{5} + \frac{1}{5} + \frac{1}{5}$ into $7$ equal pieces? One way is to simply divide each the three $\frac{1}{5}$'s into $7$ equal pieces, and keep one of these pieces for each of the three $\frac{1}{5}$'s. O-K, so what do we get if we divide $\frac{1}{5}$ into $7$ equal pieces? [Insert appropriate discussion about why this makes each piece be $\frac{1}{35}.]$ Therefore, dividing $\frac{1}{5} + \frac{1}{5} + \frac{1}{5}$ into $7$ equal pieces gives us three pieces each of size $\frac{1}{35},$ or $\frac{1}{35} + \frac{1}{35} + \frac{1}{35}.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remembering that we were to divide $\frac{1}{5} + \frac{1}{5} + \frac{1}{5}$ into $7$ equal pieces &lt;strong&gt;and then keep 4 of those pieces&lt;/strong&gt;, we are led to:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\left(\frac{1}{35} + \frac{1}{35} + \frac{1}{35}\right) \; + \; \left(\frac{1}{35} + \frac{1}{35} + \frac{1}{35}\right) \; + \;\left(\frac{1}{35} + \frac{1}{35} + \frac{1}{35}\right) \; + \;\left(\frac{1}{35} + \frac{1}{35} + \frac{1}{35}\right) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this is a total of $12$ many $\frac{1}{35}$'s, or $\frac{12}{35}.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point we can think about what changes are needed to obtain a similar argument for other situations, such as $\frac{4}{17} \times \frac{3}{7}$ or $\frac{7}{11} \times \frac{6}{12}.$&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-04-22T18:08:01.583" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1901" PostTypeId="2" ParentId="1890" CreationDate="2014-04-22T18:28:28.203" Score="0" Body="&lt;p&gt;Either add a challenging problem for &quot;bonus points,&quot; or just asign points according to difficulty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Average students will just do enough to get a passing grade anyway.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-22T18:28:28.203" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1902" PostTypeId="2" ParentId="1893" CreationDate="2014-04-22T18:35:25.483" Score="5" Body="&lt;p&gt;In my experience, many students will just coast along the second half if they get a good grade in the midterm. So I make the midterm a bit harder, and it counts less than the final in the overall grade. This makes sense otherwise too, as the final covers the whole course, and the midterm just the first half or so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There should be no &quot;desired grade&quot; for an exam, I consider it my duty to certify that whoever passes my course has shown a sufficient mastery of the subject matter, as defined by the syllabus. If everybody passes with A or B, great! If few pass, there is something wrong (too ambitious syllabus, exams were too hard, bad teacher, ...).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-22T18:35:25.483" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1903" PostTypeId="2" ParentId="1817" CreationDate="2014-04-22T18:37:53.813" Score="2" Body="&lt;p&gt;35 years ago Florence Nightingale was taught at my school to be the female mathematician to look up to because of her many disciplines.  However, I've replaced that image with Lady Ada Lovelace who is considered the first computer programmer. &lt;/p&gt;&#xA;" OwnerUserId="1180" LastActivityDate="2014-04-22T18:37:53.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1904" PostTypeId="2" ParentId="1890" CreationDate="2014-04-22T18:46:46.903" Score="5" Body="&lt;p&gt;As a counterpoint to Joe's answer, I would say that classifying problems can be very helpful.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without this kind of gradation, the student does not know your expectations:  if they struggle on some problems but not others, they may assume that the things they are struggling with are &quot;challenge problems&quot;, rather than something which is supposed to be considered basic for the course.  Then, when this comes up on a test, or in later material, they are lost.  Also the reverse problem can occur:  a student may think that their lack of success with a challenging problem indicates they are not doing well, which could be discouraging. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you clearly state which problems are basic and which are challenging, then they know that to do well in the course they must at least understand all of the &quot;basic&quot; problems.  Also, the students who are doing well will know not to feel too bad when the challenging problems are, well, challenging.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarize, if you do not classify the problems, the students will make their own classification, and they will probably classify them incorrectly.  This leads to you and your students having a different idea of what success in the course means.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-04-22T18:46:46.903" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1905" PostTypeId="2" ParentId="1896" CreationDate="2014-04-22T19:44:41.187" Score="6" Body="&lt;p&gt;They aren't any four-letter words in the set, so they are allowed in polite company. But use them sparingly, and when you really mean it. When you say something is &quot;simple,&quot; you should make sure it really is simple for most of your audience. Be careful, what is trivial to you (presumably &lt;em&gt;the&lt;/em&gt; world expert on the topic you are writing about) can very well be a profound mistery to the average reader.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If in doubt, err on the side of explaining (a bit) too much.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-22T19:44:41.187" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1906" PostTypeId="2" ParentId="1865" CreationDate="2014-04-22T19:50:57.783" Score="5" Body="&lt;p&gt;I think the whole problem is connected to the problem of quantifiers, like &quot;for all&quot; and &quot;there is&quot;. See also &lt;a href=&quot;https://matheducators.stackexchange.com/q/1428/61&quot;&gt;this question&lt;/a&gt; which I feel is relevant here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The important point is to stress that the definition of the limit ($\varepsilon$-$\delta$ or sequences, does not really matter) means that you have to get the same limit when approaching the point independently of the way how you approach it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To prove that a limit &lt;strong&gt;does not&lt;/strong&gt; exist, it is sufficient to show that there are two possible ways to get close and get different limits, and the easiest possible way is to do this is by checking straight lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But to show existence, you have to prove it for every possible approach. This makes these exercises far more difficult if the limit exists, because to prove the converse it is sufficient to see one example, but to have the limit, you have to have it all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples like the one you mention and also in other answers are great to stress that if you do not find a counterexample it does not mean there is no. &lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-04-22T19:50:57.783" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1907" PostTypeId="2" ParentId="1896" CreationDate="2014-04-22T20:01:08.580" Score="15" Body="&lt;p&gt;The main point here is that these words/expressions should not be used as substitute for an argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They obviously have some negative effects:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You evaluate your students by them and not in the positive sense. If you say &quot;obviously&quot;, then it has a message that &quot;it better be obvious&quot;. And if it is for someone not clear, then it is an evaluation that this person is not up to the course. This can turn into humiliation of your audience (or reader, if it is in notes) quickly.&lt;/li&gt;&#xA;&lt;li&gt;If you turn out to be wrong, the you loose your face quickly. This is a consequence of my previous point: you humiliated your audience, and it turned out that you did it wrongly. You are not a person to be taken seriously.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You should, however, use these words in context, explanation. For example, to stress that though a calculation looks complicated, it is actually a simple idea what is behind. &lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2014-04-22T20:01:08.580" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1908" PostTypeId="2" ParentId="1879" CreationDate="2014-04-22T20:03:23.010" Score="3" Body="&lt;p&gt;I consider it my duty to certify that whoever passes, has a decent grasp of the subject. Within that constraint, you can reevaluate what you are doing. Are exams too hard? Do students waste much time doing computations? (In my first-year chemistry class, before calculators and such, it was a hard rule that numerical results were worth 5%. Everything was set up so you could leave the numbers and operations written down and do them at the end if you had time.) You could consider giving extra homework (read and critizice a section of the lecture notes, or a recent paper in the area, ...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm against changing the rules (i.e., adding an extra exam covering earlier material or such), but in extreme cases it might be warranted (I had a repeat exam once, when the results of the original were beyond horrible).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd seek guidance from more senior colleagues locally.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-22T20:03:23.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1909" PostTypeId="2" ParentId="1817" CreationDate="2014-04-22T20:43:38.810" Score="29" Body="&lt;p&gt;&lt;a href=&quot;http://www.youtube.com/user/Vihart&quot;&gt;Vi Hart&lt;/a&gt;, the self-termed &lt;em&gt;Mathemusician&lt;/em&gt;. I especially enjoy her &lt;a href=&quot;http://www.youtube.com/playlist?list=PLC20F52B96F3E8206&quot;&gt;Doodling in Math Class&lt;/a&gt; YouTube series.&lt;/p&gt;&#xA;" OwnerUserId="1003" LastActivityDate="2014-04-22T20:43:38.810" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1910" PostTypeId="2" ParentId="220" CreationDate="2014-04-22T22:09:25.410" Score="16" Body="&lt;p&gt;I like having students show that the Euler characteristic ($\chi=V-E+F$, where $V$, $E$, and $F$ are the number of vertices, edges, and faces, including the exterior face, of the graph) of a planar graph is always 2. Planar graphs are easy to define for those who have not seen them before, and the proof is a (relatively) straightforward induction on the number of edges in a graph.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, after giving the definition of a planar graph, there is ample opportunity for students to explore and come up with a conjecture on their own (assuming you have the time for this). This seems a much more natural way to show how induction is used by mathematicians, rather than just stating &quot;Prove that...&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another benefit is giving a concrete example of a topological invariant, likely the first seen by your students. Finally, it's not too difficult to motivate why non-math majors should care about graphs. There are dozens and dozens of applications (e.g. neural networks). It is likely that whatever the major of the student, there is some problems in that field for which graphs are useful tools.&lt;/p&gt;&#xA;" OwnerUserId="651" LastActivityDate="2014-04-22T22:09:25.410" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1911" PostTypeId="2" ParentId="1890" CreationDate="2014-04-22T22:22:21.777" Score="2" Body="&lt;p&gt;What I was doing in my (undergraduate statistics) class was to release the rubrics to the students, explaining what they are expected to know at the &quot;C&quot; proficiency level (usually, mechanics), and what they are expected to know at the &quot;A&quot; proficiency level (understanding where a given technique applies). With that, I would &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;give a list of problems from the text book to solve, and ask them to rate their difficulty level;&lt;/li&gt;&#xA;&lt;li&gt;since most of the textbook problems where C level (mechanics), anyway, on some homeworks I would give the list of C problems from the textbook, and give my own A problems.&lt;/li&gt;&#xA;&lt;li&gt;I would also ask them to compose their own problem, and rate its level of difficulty. In the beginning of the next class, I would have them trade problems with the neighbor for an (ungraded) D-I-Y quiz. &quot;A&quot;-level students were sometimes able to come up with A-difficulty problems that I used on the tests :).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If we are talking about derivative of trig functions, say, then&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Find $\frac{d}{dx} \sin^3 5x$ &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;is obviously a C-level problem: mechanics, boring, not that much to do other than applying the rules. But something like&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;True fact: $\frac{d}{dx} \sin x = a \sin x + b \cos x$ for some $a$ and $b$. (1) Using the standard trig identities between sine and cosine, and the chain rule, find $\frac{d}{dx} \cos x$ in terms of  $\sin x$, $\cos x$, $a$ and $b$; (2) using the chain rule for $\sin 2x$ and $\cos 2x$, as well as derivatives of products and squares, find $a$ and $b$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;-- well, that's a noble, if not upscale, &quot;A&quot; problem (which incidentally also reminds/tests solving a system of two linear equations with two unknowns, with coefficients being symbols rather than numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you cannot solve &quot;A&quot; problems on the homeworks, it is fairly unlikely you'd do one on the test (which also followed the difficulty levels outlined in rubrics, and problems were explicitly listed as &quot;C&quot;, &quot;B&quot; or &quot;A&quot; difficulty).&lt;/p&gt;&#xA;" OwnerUserId="742" LastActivityDate="2014-04-22T22:22:21.777" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1912" PostTypeId="2" ParentId="1042" CreationDate="2014-04-22T23:53:45.660" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Continued_fraction&quot; rel=&quot;nofollow&quot;&gt;Continued fractions&lt;/a&gt; can produce a vast range of good projects, which you can give at any age from (willing) middle school to (reluctant) high school students. One simple way to motivate the concept is to ask students to discover approximations to pi that were in wide use &lt;a href=&quot;http://en.wikipedia.org/wiki/Approximations_of_%CF%80&quot; rel=&quot;nofollow&quot;&gt;in ancient world&lt;/a&gt;. Sure, everybody knows $\pi \approx 22/7$, but can you do better than that? How do you proceed from $22/7$ to the next approximation? How do you convert continued fractions to &quot;regular&quot; ones, and vice versa?  What if you allow not only the fractions like $1/a_n$, but also $2/a_n$, $3/a_n$, etc? E.g., if your first approximation is $\pi \approx 3.14 = 157/50 = 3 + 7/50$, how can you improve it using the continued fractions approach? Has anybody discovered this approximation? Can you produce a reasonably looking approximation that is not listed on Wikipedia?&lt;/p&gt;&#xA;" OwnerUserId="742" LastActivityDate="2014-04-22T23:53:45.660" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1913" PostTypeId="2" ParentId="220" CreationDate="2014-04-23T00:32:07.650" Score="6" Body="&lt;p&gt;Computational geometry is a good source for basic induction proofs where non-inductive methods are either impossible or hard to conceive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also graph theory. Someone already mentioned Euler's Formula which is great. It's also simple to prove that every connected simple graph can be 6-colored using induction. (You can assume in such a graph that there exists a vertex whose degree is less than or equal to 5 -- or prove it separately.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a CS-oriented audience, proofs of correctness for recursive algorithms are a great way to motivate induction -- especially because induction and recursion go hand in hand. Here's a nice video proof from Udacity on the correctness of the Russian peasant's algorithm for integer multiplication: &lt;a href=&quot;https://www.youtube.com/watch?v=GGBEiIuheTY&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=GGBEiIuheTY&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-04-23T00:32:07.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1914" PostTypeId="1" CreationDate="2014-04-23T00:39:26.907" Score="11" ViewCount="433" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/1528/42&quot;&gt;This answer&lt;/a&gt; describes an analogy between finite state machines and mazes. This allows for some playful exercises, like &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Draw a representation of a word accepted by the following automaton...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which sums up to &quot;solve this maze&quot; in more &quot;respectable&quot; terms. Many kids like solving mazes and such a special content might be desirable on some occasions (e.g. special exam on April 1st).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know any other examples of this kind? In other words, are there any other puzzles that are likable which we can formulate using serious theories?&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-06T00:38:52.210" Title="Playful problems in mathematics" Tags="|undergraduate-education|math-puzzle|" AnswerCount="3" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="1915" PostTypeId="1" CreationDate="2014-04-23T01:43:00.497" Score="5" ViewCount="1522" Body="&lt;p&gt;I don't know how to teach algebraic expressions to the students who have heard the term for the first time. $(a+b)^2$ results in $a^2+2ab+b^2$, we know that, but they ask me why it is so. I tried to make them understand by expanding the expression to $(a+b)*(a+b)$ and multiplying, but  still some of them find it hard to understand. So how do I make them understand that? Are there any other ways?&lt;/p&gt;&#xA;" OwnerUserId="1172" LastEditorUserId="1172" LastEditDate="2014-04-23T02:10:48.520" LastActivityDate="2014-04-23T17:12:08.947" Title="How to teach algebraic expressions?" Tags="|algebra|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1916" PostTypeId="2" ParentId="484" CreationDate="2014-04-23T02:30:29.993" Score="2" Body="&lt;p&gt;Yes, teach the entire proof.  Here's why ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Limits are the underpinning of the entire field of the Calculus of a single variable.  They bind the inverse operations of integration and differentiation together through the limit of the difference quotient, and the limit of the Riemann sum, together with the Mean Value theorem for Integrals.  Without limits, the Fundamental Theorem[s] cannot be proven.  The concept of a limit is used for nearly every convergence and divergence test, L'Hopitals Rule, definitions of asymptotes, &lt;em&gt;the proof that continuous compounding equates to the law of exponential change&lt;/em&gt;, &lt;em&gt;required for a rigorous proof of the chain rule&lt;/em&gt;, required for proof of all the derivative formulas, required for proof of all of the integration techniques, etc., etc.  &lt;strong&gt;&lt;em&gt;It literally binds the entire field together&lt;/em&gt;&lt;/strong&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an AP Calculus AB and BC teacher in high school, I initially treated the instruction of limits much like it was treated when I was student, namely, rushed over, given only a cursory look, and slandered as 'kid stuff' or 'simplistic,' concepts that were only necessary for the 'big stuff' up ahead.  Not surprisingly every time I found myself teaching the later material, I began to realize, as an educator, that every contention I attempted to make about the subtlety of the calculus relied on limits as the underpinning of those arguments.  Don't get me wrong ... I had always proved nearly every algorithm in the course, and stressed the importance of epsilon delta proofs, etc., but had skipped over that of limits.  I am now firmly of the opinion that we do the students a &lt;strong&gt;&lt;em&gt;huge disservice&lt;/em&gt;&lt;/strong&gt; when we do this ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the beginners understanding of all the concepts in the list I provided above ... and then consider telling the beginner the truth.  That you refuse to prove that any of it is true ... Well, that's essentially what you are doing as an educator when you don't demonstrate the epsilon delta proof of limits.  Why?  Because that proof underpins the entire canon of the calculus.  As a student, it left a bad taste in my mouth that I never fully appreciated until I started teaching the course ... To be sure, there are some things that are too difficult to teach first year calculus students, e.g., a rigorous understanding of the ramifications of the implicit function theorem and how that can be extended to the concept and basis of implicit differentiation.  That's certainly a topic I forego ...  BUT ... limits are the bellybutton of calculus ... teach and demonstrate the full proof, master it to teach it swiftly and with grace, and hopefully students will understand the different phenotypes of limits and have a more robust appreciation of the calculus ... That's what I do now with my BC students and I am starting it with the AB students next year, for what it's worth ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regards, Jonathan Haack&lt;/p&gt;&#xA;" OwnerUserId="1100" LastActivityDate="2014-04-23T02:30:29.993" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1917" PostTypeId="2" ParentId="1915" CreationDate="2014-04-23T02:41:23.597" Score="10" Body="&lt;p&gt;Difficulty with an advanced concept is often due to not grasping a prerequisite concept.  Here are some prerequisite concepts that students should have grasped before trying to grasp expanding $(a+b)^2$...you could ask questions about these concepts to see if you need to review any of them before moving on:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What are variables  (ie, letters can be used to represent numbers that vary, like the dimensions of a room which vary as the architect draws and adjusts it)&lt;/li&gt;&#xA;&lt;li&gt;What is an expression  (ie, $(a+b)^2$ says if we knew the actual numbers, we would add them and square the result)&lt;/li&gt;&#xA;&lt;li&gt;How to evaluate an expression when we know the actual variable values&lt;/li&gt;&#xA;&lt;li&gt;Can we do anything useful with expressions without knowing the actual variable values? (yes! eg, factor/expand/simplify to get an equivalent expression.)&lt;/li&gt;&#xA;&lt;li&gt;Bonus for best comprehension: As a cross-check whether two expressions are equivalent, evaluate them for arbitrary variable values (is this foolproof? how can we make it more robust?)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1186" LastActivityDate="2014-04-23T02:41:23.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1918" PostTypeId="2" ParentId="1915" CreationDate="2014-04-23T02:44:34.473" Score="12" Body="&lt;p&gt;Draw a square with sides of length $a+b$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then divide the square into $4$ parts: the $a\times a$ square, the $b\times b$ square, and the two $a\times b$ rectangles. Explain that to find the area of the square - that is, $(a+b)(a+b)=(a+b)^2$ - you have to add all the parts up: $$(a+b)^2=a^2+b^2+ab+ba=a^2+2ab+b^2$$ Then talk about the distributive property, and perhaps the meaning of the term &quot;square&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1161" LastActivityDate="2014-04-23T02:44:34.473" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1919" PostTypeId="2" ParentId="1896" CreationDate="2014-04-23T02:52:14.050" Score="3" Body="&lt;p&gt;I would argue that they are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When one reads a word such as &quot;clearly&quot;, it is a sign that an argument has been omitted, and that said argument should be relatively easy to find. I think it engages the reader and makes them &quot;participate&quot; with the material more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wouldn't worry about making people feel bad when they don't get something - everyone eventually has to learn to get over that feeling (and we all experience it, time and time again). Students who are assertive will ask you if they didn't understand something you said or wrote.&lt;/p&gt;&#xA;" OwnerUserId="1161" LastActivityDate="2014-04-23T02:52:14.050" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1920" PostTypeId="2" ParentId="1880" CreationDate="2014-04-23T03:08:26.253" Score="2" Body="&lt;p&gt;Perhaps an even deeper problem is that, in school, students will frequently make use of the same identity when $a, b, c, d \in \mathbb{R}$. H.H. Wu discusses this in his Pre-Algebra text (&lt;a href=&quot;https://matheducators.stackexchange.com/a/1253/262&quot;&gt;MESE 1&lt;/a&gt;, &lt;a href=&quot;https://matheducators.stackexchange.com/a/1858/262&quot;&gt;MESE 2&lt;/a&gt;) as follows:&#xA;&lt;hr&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/I6vfp.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The main difference between the situation Wu describes and what you describe is that students could actually prove the identity you've put forward. The reason for what is called &lt;strong&gt;FASM&lt;/strong&gt; above is that students have not yet seen limits (in particular, have not yet seen real numbers as limits of convergent rational sequences modulo the appropriate equivalence relation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a teacher, you now choose how to proceed with your situation. One choice is to let students work out concrete examples, as suggested by DL Renfro; this is probably one of the best ways to start demystifying the generalization you have mentioned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another choice would be to write a proof. If you have students who you think are ready to grasp such a concept (i.e., for whom such a task falls in their ZPD), then you could scaffold it as appropriate, but (as JP Burke points out in a comment) you have now moved into the realm of proving. Moreover, even with a general proof of the fact you've written out, it may not be so clear how directly it can be applied to an example such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{\frac{2}{3} - \frac{1}{2}}{\frac{6}{5} - \frac{1}{4}} \cdot \frac{7 - \frac{8}{11}}{\frac{12}{5} - 2}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, no numerator or denominator is (written as) a fraction. Does this law still hold? If so, why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question and other subtleties (what happens when one or more of $a,b,c,d$ is negative?) suggest you at least ensure students have sufficient procedural fluency with regard to working out examples before delving more deeply into generalizations.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-04-23T03:08:26.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1921" PostTypeId="2" ParentId="1883" CreationDate="2014-04-23T03:09:37.227" Score="3" Body="&lt;p&gt;I think there are several significant disadvantages to such a presentation of complex numbers.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There's no number whose square is negative. That's a shame! Well, whatever - we'll make one up! &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;OK, don't do it like that though. What do you mean by &quot;there's no number whose square is negative&quot;? I could just as well say, &quot;there's no number whose square is $2$ - well, I'll just make one up and call it $\sqrt 2$!&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quote &quot;God created the integers, all the rest is the work of man&quot; is more or less true. All numbers, even starting with rationals, are (basically) fabrications of the imagination. They are &lt;em&gt;helpful concepts&lt;/em&gt; that do not necessarily make any sense to our intuitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in this sense, the statement &quot;there's no number whose square is negative&quot; is neither true nor false. It depends entirely on whether we have &lt;em&gt;created&lt;/em&gt; such a number, and whether that creation has proved consistent and useful for doing mathematics.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Might I say next that they are called &quot;complex/imaginary &lt;strong&gt;numbers&lt;/strong&gt;&quot; for a reason. They are basically numbers, not geometrical objects. I don't think students are going to take very well to the idea that geometric objects can be the solutions to quadratic equations. They are used to &lt;em&gt;numbers&lt;/em&gt; being the solutions to equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the Argand Plane is, more than anything, a &lt;em&gt;representation&lt;/em&gt; of the complex numbers - not the complex numbers themselves. The real number line, for that matter, is also a &lt;em&gt;representation&lt;/em&gt; of the real numbers, but those numbers are not literally the points on that line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have introduced $i$, simply explain that complex numbers are traditionally written in the form $$a+bi$$ This expression has two real parameters, and therefore we can think of the set of complex numbers as a set of pairs of real numbers - in other words, the complex plane. Then point out that doing various operations to these &quot;vectors&quot; in the plane results in interesting things - for example, multiplying yields a new vector with angle the sum of the inputs' angles and length the product of the inputs' lengths. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;To summarize, in my opinion: you should certainly not exclude the &quot;normal&quot; presentation of $i$ when introducing it to your students, but you should also incorporate the geometric aspect of complex numbers to give the students another way of thinking about them - and if you feel that you can more effectively explain them from a geometric standpoint, then by all means spend more time on that! Teach what you're best at, but I wouldn't leave them entirely unprepared for the idea that complex numbers are actually &quot;numbers&quot; in some peoples' minds.&lt;/p&gt;&#xA;" OwnerUserId="1161" LastActivityDate="2014-04-23T03:09:37.227" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1922" PostTypeId="1" CreationDate="2014-04-22T19:21:44.483" Score="14" ViewCount="1964" Body="&lt;p&gt;I am a parent of a HS freshman who is taking precalc/trig now. His teacher is pretty much useless; my son has to come to me or my wife (let's just say that each of us has about 100+ college and graduate credits in math, and Ph.D.s in math intensive fields) to ask for explanations, and after almost a year of trig, his active formulae include mostly $\sin^2 x + \cos^2 x=1$ (which he knew anyway as Pythagorean theorem), and I am not sure he can do $\sin(x+y)$ without referring to his cheat sheet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am seriously considering going to the high school math department and asking to pull my son out to homeschool him for AP Calc BC using the materials that I trust (which would probably have to include the proofs of Lebesgue-Borel lemma and the extreme value Weierstrass theorem; I learned them when I was 15 or 16, so I don't see a big deal here). My big question is: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;how do I approach high school math department without completely pissing them off?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My son still has to be there for three more years. His precalc teacher actually runs the school math team (although I think they missed organizing AMC 10 this year, partly because the school was closed due to snow on the primary competition day). Finally, I want somebody to write a ref letter to attest to my son's math skills when he graduates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The specific complaint my son has about the teacher is that he does not explain things enough, and he does not explain things well. My understanding of a typical way the teacher interacts with the students is that he gives a bunch of definitions, and then gives assignments, many of which are defined as vaguely as &quot;Well, here is a full set of 60 problems. Work about half of them; there will be 12 from this set on the test&quot;. All of the assignments are on the publisher's website. Technology is a blessing, of course: if a student cannot factor $2x^2-5x-3$, the system will hold student's hand, show calculations step by step, and then generate another $x^2 + 3x +2$ to let students show that they've learnt something. However, in my view, the technology should untie teacher's hands so that they are offer &lt;strong&gt;more&lt;/strong&gt; explanations and examples worked through in class. The technology can be used to test the basic knowledge and afford a C grade; however, the grade of A should be earned by doing proofs and derivations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The worst thing, though, is that I finally found out what the dreaded rote memorization is. I heard everybody talk about it as a problem in American math education; I now know what this phrase actually means. I don't think it hit my son as hard in any of the earlier classes as it did in this precalc. The most recent example from his precalc class is the partial fraction decomposition:&#xA;$$&#xA;\frac{x^2 - 3x + 5}{x^3 - 2x^2 + x - 2} = &#xA;\frac{\mbox{stuff}_1}{\mbox{linear binomial}} + \frac{\mbox{stuff}_2}{\mbox{quadratic polynomial}} + \mbox{may be more stuff like that}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The teacher then threw in the concepts of distinct linear factors, repeated linear factors, distinct quadratic factors, repeated quadratic factors -- my son had the terms in the notes, but not the definitions (OK, let's give the benefit of the doubt and assume that the teacher actually gave these definitions in class, and my son failed to copy them down properly; in the end, we had to Google them to make sure we got them right.) The teacher then said that this material is important because they will use this later in integration. Fine, I agree that it is useful when integrating rational functions; but why on earth throw this useless information at this stage, when integration is like a year ahead, and nobody in the class has any clue what the teachers just said? Why asking for these terms on the test? Knowing that $\frac{x+3}{(x^2+5)^2}$ is called &lt;em&gt;repeated quadratic factor&lt;/em&gt; is not helping a student to do the calculation proficiently. I would introduce this decomposition when I would need it when we touch the integrals of $1/(x+a)$ and $1/(x^2+a^2)$. Without context, this information and computational method is going to be lost by the time the (next) class arrives at these integrals, so the (next) teacher would have to work through this, anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A minor complaint that my son had was that the teacher did not break down the class efficiently into small groups when small group activities were offered. In my son's view, a better division would be to stratify the class by knowledge/ability, so that more advanced students would work on more complex problems. (In that partial fraction decomposition, what should stuff1 and stuff2 consist of? What degrees other irreducible polynomials could have? Can you create, and solve, a meaningful example with repeated irreducible factors of a sixth degree?). I hear my son here, although in my own teaching experience (a few years of undergraduate teaching in a couple of US universities), I would sometimes make sure that I have at least one strong student in each small group to really explain to other students what's going on. May be my son's teacher does the same; may be he does not really care how the knowledge is being transferred.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can form two sorts of implications for the calculus class that is coming up next fall, and none of them is promising.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The math department / district does not have a quality control system in place. My son said that he wanted to give his negative feedback the way college students do in their student evaluations (and, I believe, in evaluations he gave in some of his other educational institutions). If this is true, then there is no guarantee that the next teacher will be any better.&lt;/li&gt;&#xA;&lt;li&gt;The math department / district is simply fine with math being taught that way. If this is true, I don't want them to touch my son's math education, as I still hope he'd grow to love math.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Either way, I see my son better off taking his calc class off the grid -- with me using proper Russian books, or through AoPS, JHU CTY or Stanford EPGY or Coursera or whatever is going to both challenge him and provide better explanations than his school could.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My big plan for my son is that he takes AP Calc BC next year, and then takes stuff like Linear Algebra and Abstract Math online.&lt;/p&gt;&#xA;" OwnerUserId="742" OwnerDisplayName="StasK" LastEditorUserId="47" LastEditDate="2014-04-23T04:21:10.520" LastActivityDate="2017-06-11T13:24:50.607" Title="Interacting with high school teachers (US)" Tags="|secondary-education|student-motivation|" AnswerCount="7" CommentCount="28" ContentLicense="CC BY-SA 3.0" />
  <row Id="1924" PostTypeId="2" ParentId="1922" CreationDate="2014-04-23T00:51:14.180" Score="18" Body="&lt;p&gt;Ah, yes. My sympathies... maybe the question should be moved to another site, but, pending that... and/but the U.S. environment will not change significantly any time soon, for many reasons. I have my own acquaintance through my kids with better-quality [sic] U.S. education, at the supposed high end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that context: first, also I am acquainted (through colleagues and general awareness) of other countries' education systems and, especially, mathematics. The &quot;former Soviet&quot; system, as well as many &quot;Soviet bloc&quot; countries, but also to a lesser degree Western European countries, compelled students to take mathematics seriously. The point is ... &quot;or be left behind&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not only does the current U.S. system honorably emphasize &quot;breadth&quot;, but it does also not judge so harshly. For some decades, and, even now, in the U.S. people have many further chances after earlier failures. There is no nation-wide stigma/reward. This is a good thing in many ways, but it is bad in that it allows a certain fraction of people to not try hard, to think that there will always be one more chance, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that context... :) ... there is simply no one to turn to for high-school tutoring at the level of the old Soviet and eastern-bloc mathematical establishment. At the same time, ... did it ever really matter? To the extent that &quot;mathematical prowess&quot; was a low-overhead proof of machismo or value-to-the-state, or to corporate entities, aggressive orthodox low-level mathematics (which includes Olympiads, btw) was only of interest as a demonstration of aggression and quickness, not of wisdom about structure of the universe, to say the least.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, as in my own childhood, hoping that a kid has access to books (or, ... the internet?), there is no lack of &lt;em&gt;opportunities&lt;/em&gt; to learn many things, including much mathematics. The question of learning to feel an &lt;em&gt;obligation&lt;/em&gt; to learn such things is different. This sense of obligation is not currently available in the U.S. Not clear to me that it should be... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do explain to my own daughter that the &quot;standards&quot; promulgated by k-12 schools, and even good universities, and even graduate schools, are ... sub-minimal. The current culture makes understanding such things an abrupt transition. No easy way, I fear. But, in her early 20s, my daughter does seem ever-more able to understand my point to her (for almost 20 years!) that she (and anyone) should &lt;em&gt;embrace&lt;/em&gt; their studies, not thinking first about what's on the exam, and how to minimize effort (as game-theoretically rational as that surely is!), and view &quot;school&quot; as an annoying side-effect or incidental. I tell the same to my grad students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to paraphrase, it's not about the school curriculum at all... which should be only some formality like a driver's license test... but what one (=the kid) &lt;em&gt;wants&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(And, of course, kids have a hard time understanding such a message, given the ambient culture.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best wishes! :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oop, and the operative advice about how to approach the H.S. teachers to get them on a better track... is... don't. They are not equipped to understand the mathematics, and will at best be alienated by your raising the issue. Srsly. I've been there... offering my advice to my local high school, and been rebuffed on the grounds that, while I have a Ph.D. in math from a fancy school, I do not have a license for teaching H.S. math in Minnesota. That kind of thing. Better not to make yourself crazy, and not irritate the locals.&lt;/p&gt;&#xA;" OwnerUserId="63" OwnerDisplayName="paul garrett" LastActivityDate="2014-04-23T00:51:14.180" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1925" PostTypeId="2" ParentId="1883" CreationDate="2014-04-23T03:16:09.743" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If you're concerned what we really mean by saying the &quot;usual rules&quot; apply, you really are concerned with the &lt;a href=&quot;http://en.wikipedia.org/wiki/Real_number_system#Axiomatic_approach&quot; rel=&quot;nofollow noreferrer&quot;&gt;axioms of the real number system&lt;/a&gt;, and how they compare to the &lt;a href=&quot;http://en.wikipedia.org/wiki/Complex_number#Field_structure&quot; rel=&quot;nofollow noreferrer&quot;&gt;axioms of the complex number system&lt;/a&gt;.  The answer is a bit complex but can be understood thoroughly by knowing the axiomatic definitions of fields.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Your &quot;geometric&quot; approach sounds good to me, but you omitted the part about complex numbers which makes them a field: the ability to multiply* any two complex numbers and meaningfully attain a new one.  I grasped complex number multiplication by &lt;a href=&quot;http://www.google.com/search?q=complex+multiplication&amp;amp;tbm=isch&quot; rel=&quot;nofollow noreferrer&quot;&gt;looking at graphic examples&lt;/a&gt; where you can see that it amounts to multiplying the magnitudes while adding the angles.  And you can show that this new angular addition concept of multiplication still gives the correct answers for multiplying real numbers such as (1)(1)=1, (1)(-1)=-1, (-1)(-1)=1.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;*EDIT: Or divide, except by 0. Thanks to Incnis Mrsi's comment!&lt;/p&gt;&#xA;" OwnerUserId="1186" LastEditorUserId="1186" LastEditDate="2017-05-30T22:06:02.237" LastActivityDate="2017-05-30T22:06:02.237" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1926" PostTypeId="2" ParentId="1922" CreationDate="2014-04-23T03:34:27.037" Score="15" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;how do I approach high school math department without completely pissing them off?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If this is really the question, then I suggest you don't approach them at all...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What exactly are you expecting? Chances are that the high school teacher who does the Calc class will be ok/not great. You might luck out and get an awesome teacher, but don't bet on it. If your son really wants to start learning math on his own, then he should just learn math on his own. There are a plethora of online resources you could use, and many state universities run enrichment programs for advanced math students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you simply explain to your high school that your son would like to enroll in such a program, I'm sure they would be more than happy to accommodate him. My experience has been that your typical high school is not very capable of handling &quot;off the charts&quot; students, but they get quite excited when they can help to arrange for such students to pursue their studies elsewhere. I think your son's teachers want him to succeed, even if they may not be able to mediate that progress themselves.&lt;/p&gt;&#xA;" OwnerUserId="1161" LastActivityDate="2014-04-23T03:34:27.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1927" PostTypeId="2" ParentId="1834" CreationDate="2014-04-23T03:46:39.247" Score="5" Body="&lt;p&gt;If your curriculum allows you the flexibility to do this, I prefer to start with what are &lt;strong&gt;variables&lt;/strong&gt; (a letter representing a number that varies), then what are &lt;strong&gt;expressions&lt;/strong&gt; (a plan what you'll do once you know the variable's value), then how can we &lt;strong&gt;evaluate&lt;/strong&gt; the expression for a particular variable value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stick with various expressions for at least a few days before turning the page to &lt;strong&gt;equations&lt;/strong&gt;.  With all this practice evaluating expressions, the guessing-game nature of equations will be clear: you guess the value of $x$,  &lt;em&gt;evaluate&lt;/em&gt; the LH expression, &lt;em&gt;evaluate&lt;/em&gt; the RH expression, and see if they're equal, meaning  the $x$ you guessed is a valid solution to the original equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once the problems get too hard to solve by guessing, finally follow ChrisA's answer to teach a methodical way to solve equations, always &lt;strong&gt;preserving equality&lt;/strong&gt; by doing the same thing to both sides.&lt;/p&gt;&#xA;" OwnerUserId="1186" LastActivityDate="2014-04-23T03:46:39.247" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1928" PostTypeId="2" ParentId="1883" CreationDate="2014-04-23T05:43:44.927" Score="19" Body="&lt;p&gt;I think there are serious pedagogical problems with such an approach.  Here is a good general rule for explaining any kind of math:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Skipping over the motivation doesn't make something easier to understand.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As math educators, our instinct is always to simplify the math that we are presenting as much as possible.  We always want students to understand the core idea first, and then build from that idea towards an understanding of the details.  When presenting a new topic, we always try to strip away anything that seems extraneous or distracting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But motivation isn't extraneous, and it isn't distracting.  Motivation is necessary for human beings to understand mathematics.  It's not part of the logical structure of mathematics, but it's crucial for developing &lt;em&gt;intuition&lt;/em&gt; about the subject.  If you don't know what something is &lt;em&gt;for&lt;/em&gt;, or &lt;em&gt;why&lt;/em&gt; it was invented, then you don't really have any hope of understanding it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The narrative that negative numbers don't have square roots, so we should try making some up, is &lt;em&gt;the&lt;/em&gt; motivation for complex numbers.  It's where they came from.  It's the only way for them to make sense.  Absent this motivation, complex numbers will come across as completely arbitrary, and therefore incomprehensible.  Students will feel like there's something they're missing, and they will be right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You raise some objections to presenting this narrative, on the grounds that it leads to &quot;confusion&quot;. I disagree with the spirit of these objections, for the following reason:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A student who asks questions is not necessarily &quot;confused&quot;.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often, students need to understand a subject pretty well to be able to ask intelligent questions about it.  When a student asks how something they just learned can be generalized, or how to tell whether something really works, it means that they already understand it well enough to think about it critically.  The goal of education is not to leave students dumbfounded.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me respond to your individual objections:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For one thing, if we're being rigorous, it's nonsense. What precisely does &quot;all the usual rules apply&quot; mean? What is your proof that a structure obeying the above rules even exists?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;These are excellent questions, and I think any good explanation of complex numbers should address these directly, at least to some extent.  The definition of complex numbers highlights the fact that we need to codify the rules of algebra (leading, ultimately, to abstract algebra), and raises the question of whether it suffices to have axioms for a mathematical object, or whether we really need to construct it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, this discussion is a bit too sophisticated for students at this level, but it should be possible to say something about it.  A student who realizes that such issues arise in mathematics understands things much better than a student who thinks mathematics is always cut-and-dried.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It makes the geometrical facts about complex numbers seem very mysterious.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They are very mysterious.  It's one of the great mysteries of mathematics that complex numbers work so well, that they are so intimately related to geometry, and that they arise in so many places.  How did we start with square roots of negative numbers and arrive at Euler's formula?  Why is complex analysis so much simpler and more beautiful than real analysis?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If something truly is mysterious, it's ok for students to perceive it that way.  Point out explicitly how amazing and unexpected the geometry is, given that we didn't start with something geometric at all.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It leads to &lt;a href=&quot;https://math.stackexchange.com/questions/125186/why-not-to-extend-the-set-of-natural-numbers-to-make-it-closed-under-division-by&quot;&gt;this&lt;/a&gt; type of confusion. In the words of 19th century mathematician John T. Graves: &quot;&lt;em&gt;I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. (...) If with your alchemy you can make three pounds of gold, why should you stop there?&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Again, this isn't confusion.  The student who asked the question on Stack Exchange has grasped the fundamental technique of complex numbers, i.e. &lt;em&gt;making up new number systems to do what we want&lt;/em&gt;, and wants to know how far it goes.  Intellectually, they are doing as well as Graves, and they are on the conceptual path that eventually leads to abstract algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student who &lt;em&gt;doesn't&lt;/em&gt; ask a question like this after learning about complex numbers hasn't really grasped them.  If you manage to teach complex numbers in a way that doesn't prompt students to ask this question, that means that you've taught less, not more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, you won't have time to fully answer questions like this, but that's ok.  Students at the high-school level sometimes get the feeling that mathematics is some kind of closed book, and this is a great opportunity to point out how much mathematics there really is in the world to explore.  Graves' question is one of the most important questions in mathematics, the motivation for huge swaths of modern algebra, and in some sense it still hasn't been fully answered.  It's not a question to avoid --- it's the kind of question you should go out of your way to mention!&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-04-23T05:52:27.637" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="1929" PostTypeId="2" ParentId="1883" CreationDate="2014-04-23T07:28:46.503" Score="5" Body="&lt;p&gt;I think that including the geometry from the start is very important, as it is the geometry that explains why the complex numbers are &lt;em&gt;useful&lt;/em&gt; and therefore why we bother teaching about them.  After all, if it were all about the algebra then we'd continue with the quaternions, octonians, sontarans, and so forth.  But we don't, because complex numbers find themselves used far more than those others and it is their interaction with geometry that is the reason for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm going to sketch out an approach that starts with the geometry and leads to the algebra.  The key is to realise complex numbers as an orientation-preserving conformal linear transformations of the plane.  Why should we be interested in such?  Many reasons, but the basic idea is that these preserve the notion of &quot;Turn Left&quot; in that &quot;turning left&quot; commutes with such transformations[1].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, any such transformation consists of a rotation and a uniform scaling.  The rotation is encoded by its angle and the scaling by its scale factor.  So we need two numbers: $r$ and $\theta$.  Composition of transformations results in $(r,\theta) (s,\phi) \mapsto (r s, \theta + \phi)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is all very well, but it is more complicated than it need to be.  Although the composition rule is nice, the application rule is not.  That is, if we apply $(r,\theta)$ to $\begin{bmatrix}u \\ v \end{bmatrix}$ then the result is $\begin{bmatrix} r \cos(\theta) u - r \sin(\theta) v \\ r \sin(\theta) u + r \cos(\theta) v\end{bmatrix}$.  So in terms of application, it is the numbers $x = r\cos(\theta)$ and $y = r\sin(\theta)$ that are important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that notation, the application rule becomes $\begin{bmatrix} u \\ v \end{bmatrix} \mapsto \begin{bmatrix} x u - y v \\ y u + x v \end{bmatrix}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is also the composition rule, and that's because the vector $\begin{bmatrix} u \\ v \end{bmatrix}$ is the result of rotating the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ by a certain angle and scaling it by a certain length.  So each vector in $\mathbb{R}^2$ corresponds to applying a rotation and scaling to $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$.  Thus we can identify $\mathbb{C}$onformal orientation-preserving linear transformations with $\mathbb{R}^2$ as the action is full and faithful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Associativity of multiplication, aka composition, is obvious.  Commutativity follows from the geometry in that rotations obviously commute with each other.  Distributivity of multiplication over addition is also obvious since addition is translation and it's geometrically obvious how to change a &quot;translate then rotate&quot; (aka $z(w + c)$) in to &quot;rotate then translate&quot; (aka $z w + z c$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the square root of minus one, well if you rotate by a quarter turn and rotate by a quarter turn again then you end up pointing backwards.  Simple.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] I know that this shifts the burden of motivation from complex numbers to that of &quot;knowing how to turn left&quot;, but that's hopefully easier to motivate directly from physics and engineering.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-04-23T07:28:46.503" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="1930" PostTypeId="2" ParentId="1915" CreationDate="2014-04-23T09:56:40.490" Score="7" Body="&lt;p&gt;This particular expression makes sense as a &lt;strong&gt;numerical pattern&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$1003^2 = \ \ 1006009$&lt;/li&gt;&#xA;&lt;li&gt;$2003^2 = \ \ 4012009$&lt;/li&gt;&#xA;&lt;li&gt;$3003^2 = \ \ 9018009$&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$4003^2 = 16024009$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$1004^2 = \ \ 1008016$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;$\ldots$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can look at these numbers and ask students to figure out the pattern.  They may find it cool that the pattern gives a short cut for calculating $2014^2$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they find it cool, or if you've given them drills on this, you can show them $(a+b)^2=a^2+2ab+b^2$ as a good way of writing down the pattern.  When $a$ is a multiple of 1000, it gives an especially good shortcut because you can see all the terms $a^2$, $2ab$, $b^2$ in the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can check what the pattern does with $103^2$, $203^2$, etc.  The formula $a^2+2ab+b^2$ gives a good shortcut here too.  And even for $a=11$ and $b=3$, you can check that $(a+b)^2=a^2+2ab+b^2$ is not a helpful shortcut, but still accurate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This interpretation may appear contrived, but you can do something similar for any polynomial identity with positive coefficients.  The underlying technique of guessing formulas from special cases of inputs, and testing them on other inputs, is surprisingly useful.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-23T09:56:40.490" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1931" PostTypeId="2" ParentId="484" CreationDate="2014-04-23T11:08:29.610" Score="5" Body="&lt;p&gt;I'll probably be downvoted or flamed for this, but after teaching calculus for 20 years both with and without $\epsilon-\delta$ definitions of limits, I've decided that there is a time and place for such treatment of limits, but it is &lt;strong&gt;not&lt;/strong&gt; in an introductory course for non-majors. The payoff is simply not worth the investment at this stage. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I've experienced is: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students at this level are still developing their mathematical maturity and are not (generally speaking) at a level where $\epsilon-\delta$ stuff truly makes sense. Hence the majority of students simply parrot the instructor without true understanding. &lt;/li&gt;&#xA;&lt;li&gt;Students lock in single-mindedly on these problems as if the whole of their understanding of calculus depended on it, which at this level it does not, and then miss out on understanding concepts that &lt;em&gt;are&lt;/em&gt; within their grasp -- like the simple notion that a derivative indicates a rate of change. &lt;/li&gt;&#xA;&lt;li&gt;The overall instructional goals for an introductory course in calculus for nonmajors do not typically encompass understanding of $\epsilon-\delta$ proofs. Put simply, this is not what students in the course are supposed to be getting out of the course, and our client disciplines are telling us this as well. Check out the MAA CRAFTY report for what our client disciplines &lt;em&gt;are&lt;/em&gt; asking for and you will not see &quot;a deeper understanding of mathematical proof&quot; among any of the things that are listed.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can make a calculus course without $\epsilon-\delta$ quite rigorous and demanding, and make it so students come away with a strong understanding of both concepts and computation that they can then take back to their home disciplines and use effectively. This is the main goal of such a course. Adding in $\epsilon-\delta$ in this situation becomes more like one of those memory-hogging apps that runs in the background on your phone or computer -- it takes more away from the course than it adds. Instead, save that stuff for the &lt;em&gt;second&lt;/em&gt; time through calculus (= advanced calculus/real analysis). &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-04-23T11:08:29.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1932" PostTypeId="2" ParentId="1834" CreationDate="2014-04-23T11:24:12.097" Score="10" Body="&lt;p&gt;A practical introduction is always a good idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The box method.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here we have three sealed boxes. Each box contains the same number of counters. I will label each box with an x.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I give Anna the three boxes and five counters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I give Bob fourteen counters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I will tell you that if Anna was allowed to open the boxes she would have the same number of counters as Bob.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without opening the boxes how can we work out how many counters there are in the box?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See if the class can come up with a way of solving this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT Further to this idea I have a &lt;a href=&quot;http://countonmaths.blogspot.co.uk/&quot; rel=&quot;noreferrer&quot;&gt;blog&lt;/a&gt; that is developing it further &lt;/p&gt;&#xA;" OwnerUserId="1191" LastEditorUserId="1191" LastEditDate="2014-12-17T16:36:06.873" LastActivityDate="2014-12-17T16:36:06.873" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1933" PostTypeId="2" ParentId="1922" CreationDate="2014-04-23T11:49:24.527" Score="27" Body="&lt;p&gt;Talk to math teachers as you would talk to any other professional with whom you share a common interest (the education of your son). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, I have addressed this to respond to you specifically in some areas, and I have had to make some assumptions in some places. I did this, in part, to try to make this answer more generally helpful (because the title is so general). So, please don't take some of my later assumptions as me presuming your thought processes.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Be clear about what you want&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A lot of the response you've gotten so far is sympathetic, agreeing with your concerns about school math, or focused on suggestions for what to do with your son's education. That's because the question itself has a bit of a &quot;blowing off steam&quot; quality to it rather than a &quot;focused on a specific outcome&quot; quality. In a way, that's good. Better that you do that here, because your son's math teacher does not want to be a sounding board for you blowing off steam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In parts, your question seems to want working groups chosen differently, though you don't entirely agree with your son. You also seem to want the teacher not give certain information at a certain time. You also talk about pulling your son out of school. And wanting a letter written to attest to his math knowledge. Is it all of the above?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Want something realistic&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The question is vague on what you expect the teacher to do as a result of your interaction, so I can only guess at what it is you hope will change. The list of things I was able to discern from your question had items that were in conflict with each other, so right off the top I would say to make sure what you want is something the teacher is able to provide.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I tell my kids, always ask the person who can actually help you. Likewise, ask for things a person can actually provide.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Frame your question in a efficient way&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Here I am assuming your goal is to address what the teacher is teaching.&lt;/em&gt; Whatever class your son is in, there are learning outcomes defined for that class. This is part of what is guiding the teacher. Some of that may be dictated by standards, but an advanced class may go well beyond state standards. Whatever the situation is, you won't know until you talk to the principal or perhaps even the teacher to get some idea of what the end-goal is for the class. If the class goals are different from what you would prefer, it may not entirely be in the teacher's hands. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But my point here is that a better understanding of what you want will help you then seek information to help you frame your question (i.e. in the context of these learning goals for the class, you may be concerned that what your son is currently doing will not help him reach those goals.) Frame the question in a way that helps the teacher respond to you constructively.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Don't talk about your PhD&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Credentials aren't a prerequisite for talking to your son's math teacher. In reality, what does mentioning that actually establish? In some contexts we might mention our credentials to establish a basis for our expertise. But your concern here is as a parent. Your &lt;em&gt;experience&lt;/em&gt; and &lt;em&gt;knowledge&lt;/em&gt; is certainly helpful, but it's helpful to &lt;em&gt;you&lt;/em&gt; and not to the teacher. If the fear is that you will not be taken seriously, your authority is in your status as a parent, not a mathematician. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point is, examine how you position yourself so that it is focused on smooth communication, a sense of common purpose, and how the teacher can help with the goals you are interested in. If your intention is &lt;em&gt;not&lt;/em&gt; to intimidate the teacher, then you want to make certain you are not doing so inadvertently. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Assume professional expertise / plan for a conversation&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Teachers are constantly challenged as professionals. Their sense of profession is under siege from both parents and policy makers. Be aware of this, because if you say things that indicate you are essentially there to add to the chorus, you'll just sound like the chorus. You want to be someone who has a concern you both can address together constructively. You don't want to be part of a chorus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an accomplished math learner, you may underestimate the role of pedagogical content knowledge (Shulman, 1986) in teaching. The teacher &lt;em&gt;may&lt;/em&gt; know quite a bit about teaching math that you do not. Hopefully, the teacher also has some understanding of your son's math learning that you do not have. If you're not ready to accept this, then you are going in ready to dictate rather than to have a conversation. Communication is not one way. As you titled your question, you are seeking to &lt;em&gt;interact&lt;/em&gt; with the teacher. You want to tell the teacher something, but the teacher may also have something to tell you. Approach the teacher with your goals, but recognize that the teacher may have suggestions you have not yet considered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;And finally&lt;/strong&gt;, I hope your son has a great high school experience and a smooth transition toward who it is he wants to be. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best of luck to you.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Cited:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shulman, L. S. (1986). &lt;a href=&quot;http://ccnt5.wcu.edu/WebFiles/PDFs/Shulman.pdf&quot;&gt;Those who understand: Knowledge growth in teaching&lt;/a&gt;. &lt;em&gt;Educational Researcher&lt;/em&gt;, &lt;em&gt;15&lt;/em&gt;(2), 4.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-04-23T16:44:35.133" LastActivityDate="2014-04-23T16:44:35.133" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1934" PostTypeId="1" AcceptedAnswerId="1974" CreationDate="2014-04-23T13:00:03.877" Score="14" ViewCount="1057" Body="&lt;p&gt;I know from my own education that the information understood from traditional lectures is usually much less than what the teacher is trying to convey. This is especially true in &quot;hard&quot; lectures such as mathematics and physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore me and a group of students created a tool for the teacher to visualize how many of their students understand what is being said at every moment of the lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we're starting to wonder if this is a problem for other students/lecturers/schools as well. What is the efficiency of a lecture? How effective is the transfer of information? How large proportion of the students do actually understand what the teacher is talking about? Is there any research which has tried to estimate this?&lt;/p&gt;&#xA;" OwnerUserId="1194" LastEditorUserId="93" LastEditDate="2014-04-24T15:59:18.847" LastActivityDate="2014-04-25T03:38:01.847" Title="How much of a lecture do the students understand on average" Tags="|large-lecture|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1936" PostTypeId="2" ParentId="1896" CreationDate="2014-04-23T13:44:53.523" Score="6" Body="&lt;p&gt;Yes, they are useful, but they can be over-used, or used when not true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you state &quot;it is a basic fact from...&quot;, and the reader does not see &lt;em&gt;why&lt;/em&gt; it follows, then they know that they're not following your argument as you intended it to be followed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you merely state &quot;it is a fact from...&quot;, and the reader does not see &lt;em&gt;why&lt;/em&gt; it follows, then they might think that it's a deep result and that you intend either that they should accept it on trust, or that you're presenting a lemma to be proved later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compare:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;It is a very basic fact from distributivity/associativity/commutativity that $(1-x)(1+x) = 1 - x^2$&quot; -- the reader should be able to &lt;em&gt;immediately&lt;/em&gt; visualize the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;It is a very basic fact from arithmetic that $(1-x)\sum\limits_{i=0}^{n-1}x^i = 1 - x^n$&quot; -- the reader can immediately see how a proof for a fixed $n$ might be carried out, and you're telling them that the calculations do indeed work out to save them the bother of writing it. The detail around the implied &quot;for all $n$&quot; is being ignored, which is a little shady. Really you want a proof that retains the summations and therefore involves distributivity over the summation symbol rather than just distributivity over binary addition, and that's still within the reach of the reader.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;It is a fact from the Axiom of Choice that every set has a well-order&quot; -- the reader might well know or be able to construct a proof, but they're not expected to instantly produce it. It is not a &quot;very basic&quot; consequence except perhaps to an audience of skilled set theoreticians who are indeed all expected to trot out that proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;It is a fact from the Axiom Choice that a sphere can be decomposed into finitely many pieces, which themselves can be rotated and translated to produce two spheres of equal size to the first. This is the so-called 'Banach-Tarski paradox'&quot; -- Information presented for interest and not proved. The student probably thinks &quot;okaay, I can't even imagine how to prove that&quot;, but the proof might come in a sufficiently specialized undergraduate course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a similar difference between &quot;the proof is trivial&quot; and &quot;the proof is left as an exercise&quot;. If someone attempts the exercise and finds themself part way into a proof that isn't trivial, then if you've said it's trivial then &lt;em&gt;they know there's a better proof they've missed&lt;/em&gt;. If you haven't they don't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Clearly follows&quot; and &quot;obviously&quot; are similar, although I think they're frequently mis-used for things that objectively are neither clear nor obvious to part of the audience. It's just that the speaker thinks they ought to be clear or obvious and therefore doesn't care to spend time and screen real-estate on them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's another such phrase, &quot;it follows immediately&quot;, which asserts that no new gizmos need to be introduced to the proof. To stretch the use of the term, it may turn out that &quot;immediate&quot; actually means a few lines of multiplying out brackets, but that's OK.&lt;/p&gt;&#xA;" OwnerUserId="321" LastEditorUserId="321" LastEditDate="2014-04-23T14:06:40.587" LastActivityDate="2014-04-23T14:06:40.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1937" PostTypeId="2" ParentId="1922" CreationDate="2014-04-23T14:19:56.140" Score="8" Body="&lt;p&gt;Sorry, you don't get to decide how the teacher breaks down the class into groups, or at what point they deal with partial fractions. Nor does your son.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your son is capable of much more, and you and your wife have advanced degrees in mathematics, the two of you should be teaching him math. At least what $\sin(x+y)$ is, and potentially the Extreme Value Theorem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point your son's only problem is that he has to spend some time in a class where he already understands everything (not too much if you make sure that you add depth and rigor to the topics his teacher is teaching, rather than work ahead too far) and he has to spend a small amount of time doing very boring homework. He should be able to cope with this, and certainly won't be the first person to have done so.&lt;/p&gt;&#xA;" OwnerUserId="659" LastActivityDate="2014-04-23T14:19:56.140" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1938" PostTypeId="2" ParentId="1865" CreationDate="2014-04-23T14:49:09.387" Score="3" Body="&lt;p&gt;This idea is not (yet) supported by experience, but one thought that occurs to me is to emphasize the &quot;infinitesimal&quot; definition of limit: $\lim_{(x,y)\to(0,0)} f(x,y) = L$ if $f(x,y)$ is infinitely close to $L$ whenever $x$ and $y$ are both infinitely close to $0$ (but not both equal to it), i.e. whenever $(x,y)$ is in a punctured two-dimensional &quot;infinitesimal region&quot; around $(0,0)$.  The point is then that the ratio of two infinitesimals can itself be infinitesimal or infinite.  E.g. for $f(x,y) = \frac{x^2y}{x^4+y^2}$, if $\epsilon$ is infinitesimal, then so is $\epsilon^2$, but we have $f(\epsilon,\epsilon^2) = \frac{1}{2}$ which is not infinitesimal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could avoid mentioning infinitesimals by instead talking about &quot;small&quot; or &quot;close&quot; numbers.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2014-04-23T14:49:09.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1939" PostTypeId="2" ParentId="1922" CreationDate="2014-04-23T14:54:37.293" Score="13" Body="&lt;p&gt;Here is an analogy:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I mailed some ice cream last week through UPS, but they took almost a week to deliver the package and the ice cream melted. This is ridiculous, because I was only mailing it to my next-door neighbor.&lt;/p&gt;&#xA;&lt;p&gt;Can you believe that they picked up the package, took it to their local delivery center, processed it for an &lt;em&gt;entire day&lt;/em&gt;, and then claimed that they could not deliver the package because my neighbor &lt;em&gt;was not home&lt;/em&gt;? They didn't deliver it until my neighbor answered his doorbell to accept the package.&lt;/p&gt;&#xA;&lt;p&gt;I could have just taken the package over to my neighbor and it would have been no problem. UPS is terrible. How can I talk to them without completely pissing them off?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you are in a unique situation to deliver mathematics to your son, then you should not angrily bust into the school offices. That is not their fault.&lt;/p&gt;&#xA;&lt;p&gt;They have a system in place that works, &lt;em&gt;on average, pretty well&lt;/em&gt; considering that it teaches everyone in an entire country. The system that works &lt;em&gt;on average, pretty well&lt;/em&gt; is almost by design not very good at dealing with certain side cases.&lt;/p&gt;&#xA;&lt;p&gt;Teaching mathematics to the son of a mathematician and delivering ice cream to your next-door neighbor are side cases that should not be handled using the general method.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-23T14:54:37.293" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1940" PostTypeId="1" AcceptedAnswerId="1941" CreationDate="2014-04-23T15:10:58.387" Score="13" ViewCount="6236" Body="&lt;p&gt;As a school project we've developed a web based tool (similar to a clicker) which helps the teacher understand how large proportion of their students actually understand what she/he says. Now we've been approached by our school's business incubator who thinks we should commercialize the idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What makes our solution unique right now is our focus on extreme simplicity both for teachers and students, but there are many similar ideas on the market. That makes me wonder why they haven't succeeded in making clickers omnipresent. Only a small percentage of lecturers use these tools despite claims from their manufacturers that they improve learning dramatically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why doesn't every teacher use a clicker or similar tool? Is it just tradition or are there practical issues to using them that discourage most teachers?&lt;/p&gt;&#xA;" OwnerUserId="1194" LastEditorUserId="1194" LastEditDate="2014-04-23T18:47:41.533" LastActivityDate="2014-04-28T15:26:24.300" Title="Why don't all teachers use clickers?" Tags="|interactive-teaching|large-lecture|" AnswerCount="10" CommentCount="27" ContentLicense="CC BY-SA 3.0" />
  <row Id="1941" PostTypeId="2" ParentId="1940" CreationDate="2014-04-23T15:16:56.567" Score="31" Body="&lt;p&gt;Every time I add a new technological object to my classroom, I increase the chances that my class does not happen that day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine if you could only make copies for your students in class in front of your students, but the copies came out very quickly. This would be extremely stressful, because when you brought a quiz to class, there would be a small chance that the copier would not work that day, and so your class would be compromised.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clickers and all other technological advances in the classroom are thus (for me) automatically suspect since they make the class inherently less reliable.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-04-23T15:16:56.567" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1942" PostTypeId="2" ParentId="1834" CreationDate="2014-04-23T15:31:44.500" Score="4" Body="&lt;p&gt;I would recommend looking at Dan Chazan's excellent book, &lt;em&gt;Beyond Formulas in Mathematics and Teaching: Dynamics of the High School Algebra Classroom&lt;/em&gt;, which grapples with many of the issues you raise.  In particular Chazan narrates the challenges of working with struggling students like the one you anticipate working with, and he spends a &lt;em&gt;lot&lt;/em&gt; of time unpacking fundamental issues like &quot;What does an equation mean?&quot;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-04-23T15:31:44.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1943" PostTypeId="2" ParentId="1896" CreationDate="2014-04-23T15:36:39.897" Score="1" Body="&lt;p&gt;I think the most important thing to keep in mind is your audience. &lt;em&gt;Obviously&lt;/em&gt; if you were teaching a calculus class you wouldn't have to remind them on how to add fractions. Well, in this case the correct word might be &quot;hopefully&quot;. Hopefully you don't have to remind them, but I've had calculus students that couldn't add 2 fractions to save their lives. So one should really think about one's audience first. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the use of such phrases builds &lt;strong&gt;confidence&lt;/strong&gt; if used appropriately. I remember when I was a student and my professors would use &quot;trivial&quot; or &quot;obvious&quot; ...sometimes I would agree and other times I would disagree. When I agreed, I felt that all my studying paid off. The problem is when the student disagrees. In that case, it's up to both the student and the professor to bridge the gap. This is why it's important for both students and teachers to ask questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is just one facet of the question that I feel wasn't mentioned.&lt;/p&gt;&#xA;" OwnerUserId="533" LastActivityDate="2014-04-23T15:36:39.897" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1944" PostTypeId="2" ParentId="1896" CreationDate="2014-04-23T15:39:51.727" Score="10" Body="&lt;p&gt;In addition to what @andras-batkai said, seeing words like 'obviously', 'clearly', and so on, in assignments or texts raises red flags and scepticism with me.  (Recall that, for thousands of years, it was obvious that the earth was flat.  Also, &lt;a href=&quot;http://www.smbc-comics.com/index.php?db=comics&amp;amp;id=2204#comic&quot;&gt;&lt;em&gt;obviously&lt;/em&gt; a set contains all of its elements.&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As both a former student of formal logic and an occasional tutor, this is a pet peeve of mine.  I feel these words should not be in educational or instructional texts (manuals, course notes, troubleshooting):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;They don't improve comprehension.  If something is obvious, by its nature, it doesn't need to be pointed out.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They put the burden of mathematical rigour on the reader, rather than the author.  This makes it more likely for hidden assumptions to leak into proofs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They sweep complexity under the rug.  If it really is obvious, wouldn't the explanation fit into a footnote?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They discourage discussion.  If the assumption does turn out to be wrong, students will be more likely to assume they misunderstand than they are to ask about it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I did notice that this practice seems to be going out of style.  At least in institutions I've worked with, lecturers are advised to take teaching classes, and it looks like this topic is given some attention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. The author experimented with the word 'obviously' in providing IT support to lecturers.  The experiment was not well received.&lt;/p&gt;&#xA;" OwnerUserId="1193" LastActivityDate="2014-04-23T15:39:51.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1945" PostTypeId="2" ParentId="1883" CreationDate="2014-04-23T15:56:21.540" Score="17" Body="&lt;p&gt;It's funny you should ask this now, because I just taught a Math Ed graduate class on this topic the other day, so a lot of these thoughts are fresh in my mind.&lt;/p&gt;&#xA;&lt;p&gt;The pedagogical sequence we conventionally follow with respect to complex numbers -- first introducing them in a &amp;quot;just pretend&amp;quot; way, and only later (for some students) providing a geometric interpretation/construction of them -- largely replicates the historical development of the subject.  From the time complex numbers were first introduced in the 16th century, until Hamilton's reconstruction of the theory in the 19th century, &amp;quot;imaginaries&amp;quot; were generally regarded as a useful fiction, instrumental for the purpose of extracting &lt;em&gt;real&lt;/em&gt; solutions to problems with &lt;em&gt;real&lt;/em&gt; quantities in them, but intrinsically meaningless themselves.  It was not until Hamilton introduced what we would now describe as a &amp;quot;vector&amp;quot; framework, showing that the field of complex numbers can be constructed out of real ingredients, that they began to escape their tainted reputation.&lt;/p&gt;&#xA;&lt;p&gt;One drawback to the approach you suggest is that although you describe the multiplication operation as &amp;quot;fairly natural&amp;quot; -- which I agree it is when points/vectors are represented using polar coordinates -- there is a far more natural multiplication operation when points/vectors are represented in rectangular coordinates, namely &lt;span class=&quot;math-container&quot;&gt;$(a,b) \cdot (c,d) = (ac,bd)$&lt;/span&gt;, in precise analogy with how vector addition is defined (and I do not think you can plausibly argue that vector addition is best represented in polar coordinates).  In contrast the &amp;quot;correct&amp;quot; definition &lt;span class=&quot;math-container&quot;&gt;$(a,b) \cdot (c,d) = (ac-bd,ad+bd)$&lt;/span&gt; is harder to remember and seemingly unmotivated, except in hindsight.  So this approach does away with the &amp;quot;let's pretend&amp;quot; quality of the complexes but at the cost of basing the theory on a seemingly arbitrary multiplication law.&lt;/p&gt;&#xA;&lt;p&gt;Indeed, explicitly acknowledging this fact -- that there are at least two different ways one can define multiplication of vectors, one natural and one seemingly arbitrary -- can lead to some interesting investigations.  How &lt;em&gt;many&lt;/em&gt; different ways can multiplication on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; be defined?  What properties do those multiplication laws have?  How many of them are fields?  I still remember how exciting it was when I figured out that the &lt;em&gt;only&lt;/em&gt; way to define multiplication on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; that yields a field is, up to a change of basis, Hamilton's definition.  It was like a missing piece of a jigsaw puzzle clicked into place.  For me, &lt;em&gt;that&lt;/em&gt; is the &amp;quot;motivation&amp;quot; for the complex numbers.&lt;/p&gt;&#xA;&lt;p&gt;But insight like that can only come to people who already have familiarity with working with these numbers in a less formal way first.  While I certainly do not think as educators we must be slaves to the historical development of a subject, I think there is probably some truth to the idea that Hamilton's insights would not have been possible if they had not been preceded by three centuries of people working formally with numbers whose true &amp;quot;meaning&amp;quot; even the most brilliant mathematicians did not really understand.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edited, October 2020:&lt;/strong&gt; I am resurrecting and updating this six-year old answer to respond to a request in the comments.  Way back in May 2017, user &amp;quot;Andrew&amp;quot; asked:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Can you link to a proof that &amp;quot;the only way to define multiplication on &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^2$&lt;/span&gt; that yields a field is, up to a change of basis, Hamilton's definition.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am happy to say that such a proof -- in fact three different proofs! -- have now been published in Chapter 6 of the textbook &lt;a href=&quot;https://www.routledge.com/Secondary-Mathematics-for-Mathematicians-and-Educators-A-View-from-Above/Weiss/p/book/9781138294677&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Secondary Mathematics for Mathematicians and Educators: A View from Above&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;More specifically, in that chapter you will find the following theorem:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt;.  There are infinitely many different ways to define a multiplication law on &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^2$&lt;/span&gt;. However, up to a change of basis, each of these possible multiplicative structures is isomorphic to one of the following three possibilities:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt;, the field of complex numbers;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathbb R \oplus \mathbb R$&lt;/span&gt;, the direct sum of two copies of the real number line;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathbb R [\epsilon]$&lt;/span&gt;, the ring of dual numbers.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Of these three possibilities, only the first is a field.&lt;/p&gt;&#xA;&lt;p&gt;This result is proved three different ways:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;First, it is proved using nothing more complicated than very basic linear algebra (specifically, the ideas of a &lt;em&gt;basis&lt;/em&gt;, and &lt;em&gt;coordinates relative to a basis&lt;/em&gt;, and how to use determinants to tell whether a given system of 2 equations in two unknowns has a unique solution.&lt;/li&gt;&#xA;&lt;li&gt;Next, it is proved again using the high school technique of &amp;quot;completing the square&amp;quot;, and some clever change-of-variables tricks.&lt;/li&gt;&#xA;&lt;li&gt;Third, it is proved using basic ideas of quotient rings, and the Chinese Remainder Theorem.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;All of the above can be found on pp. 296-315 of the print edition of the book.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2020-10-16T21:38:14.690" LastActivityDate="2020-10-16T21:38:14.690" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="1946" PostTypeId="1" AcceptedAnswerId="1971" CreationDate="2014-04-23T15:59:46.550" Score="7" ViewCount="246" Body="&lt;p&gt;I've never had any trouble bringing things in to my geometry, trig, or calculus courses. I think it's very important to do so that students have something that sticks out to them years later. Unfortunately, my problem is that I can't seem to think of anything for algebra. Sure I can talk to my classes about functions that model real world situations, but I can't actually bring in a rocket that's modeled by a certain function, or a car traveling at a certain rate, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I'm looking for ideas or things that you bring in to show your algebra classes. Thanks. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm motivated by these videos and this professor. &lt;a href=&quot;https://www.youtube.com/watch?v=97oTDANuZco&quot;&gt;https://www.youtube.com/watch?v=97oTDANuZco&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="533" LastActivityDate="2014-04-24T20:21:31.067" Title="Interactive Algebra Lessons and Experiments" Tags="|algebra|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1947" PostTypeId="2" ParentId="1915" CreationDate="2014-04-23T17:12:08.947" Score="4" Body="&lt;p&gt;Here is one possible approach. (I also like the geometric rectangle method, but user1161 has already mentioned this.) The main idea is to forget about issues such as does the argument work for rational numbers in general, does the argument work for irrational numbers, etc. and simply focus on showing students how the rules arise in a natural way from what they know. Besides being more concrete for the students, this approach also gives them the tools to reconstruct a correct method of expanding, although I suspect very few will do this. (My experience is that those who would find it neat to know that one can discover the rules in this way are sufficiently &quot;mathy&quot; that they know the rules anyway.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Notation:&lt;/strong&gt; &lt;em&gt;In what follows, ellipses (i.e. $\cdots$) denote the continuation of a finite sum, not the continuation of an infinite sum.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Begin by drilling the the distributive property with specific positive integer examples, such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(2)(2+3) \;\; = \;\; (2+3)+(2+3) \;\; = \;\; (2+2) + (3+3)$$&#xA;$$(3)(2+3) \;\; = \;\; (2+3)+(2+3)+(2+3) \;\; = \;\; (2+2+2) + (3+3+3) $$&#xA;$$(4)(2+3) \; = \; (2+3)+(2+3)+(2+3)+(2+3) \; = \; (2+2+2+2) + (3+3+3+3) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's throw letters in and do it again:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(2)(a+b) \;\; = \;\; (a+b)+(a+b) \;\; = \;\; (a+a) + (b+b)$$&#xA;$$(3)(a+b) \;\; = \;\; (a+b)+(a+b)+(a+b) \;\; = \;\; (a+a+a) + (b+b+b) $$&#xA;$$(4)(a+b) \; = \; (a+b)+(a+b)+(a+b)+(a+b) \; = \; (a+a+a+a) + (b+b+b+b)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general (throw another letter in), we see that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(n)(a+b) \; = \; (a+a+\cdots [n \; \text{many} \; a\text{'s}]) \; + \; (b+b+\cdots [n \; \text{many} \; b\text{'s}]) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus (explain that $n$ many $a$'s added will be $na),$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(n)(a+b) \;\; = \;\; na + nb$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, we can now see how the following arises:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(n)(a+b+c+\cdots) \; = \; (a+a+\cdots [n \; \text{many} \; a\text{'s}]) \; + \; (b+b+\cdots [n \; \text{many} \; b\text{'s}]) \;+\; (c+c+\cdots [n \; \text{many} \; c\text{'s}]) \; + \; \cdots $$&#xA;Therefore,&#xA;$$(n)(a+b+c+\cdots) \;\; = \;\; na + nb + nc + \cdots$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By replacing $n$ with $A+B+C+\cdots,$ we can now make the jump to expanding multinomials:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(A+B+C+\cdots)(a+b+c+\cdots) \;\; = \;\; (A+B+C+\cdots)(a) \; + \; (A+B+C+\cdots)(b) \; + \; (A+B+C+\cdots)(c) \; + \; \cdots$$&#xA;$$ = \;\; (a)(A+B+C+\cdots) \; + \; (b)(A+B+C+\cdots) \; + \; (c)(A+B+C+\cdots) \; + \; \cdots$$&#xA;$$ = \;\; (aA+aB+aC+\cdots) \; + \; (bA+bB+bC+\cdots) \; + \; (cA+cB+cC+\cdots) \; + \; \cdots$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point tell them how this possibly bewildering maze of symbols can be summarized by noticing each term in the first original factor gets paired (or shakes hands with) each pair in the second original factor. The rectangle approach in user1161's answer can also be used to get this. Let one side of the rectangle be $A+B+C+\cdots$ and a perpendicular side be $a+b+c+\cdots.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how do we expand $(a+b)(a+b)$? Like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(a+b)(a+b) \; = \; aa + ab + ba + bb \;=\; a^2 + 2ab + b^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To continue this to more complicated expansions, well beyond what one would do in a classroom but what one might do for that rare student who is especially interested in math, see my answer to &lt;a href=&quot;https://math.stackexchange.com/questions/358604/how-to-expand-a-0a-1xa-2x2-a-nxn2&quot;&gt;How to expand $(a_0+a_1x+a_2x^2+...a_nx^n)^2$?&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-04-23T17:12:08.947" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1948" PostTypeId="2" ParentId="1940" CreationDate="2014-04-23T18:30:43.053" Score="16" Body="&lt;p&gt;&lt;em&gt;Only a small percentage of lecturers use these tools despite claims from their manufacturers that they improve learning dramatically.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe people who actually use these tools have a different opinion on their effectiveness to that of people trying to sell these tools?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a part of the network support team in a college, I don't deal directly with classroom support for things like this all that much but I talk to staff who do support it, and lecturers who are actually in the classrooms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Chris says, it's one more thing to go wrong - not just in the local sense of &quot;batteries won't work&quot; but for a web-based tool like you have (?) it's at the mercy of someone like me having a bad morning and blocking half the internet by accident (it can happen). It's something else that might get lost, stolen, or whatever. It's one more thing to set up. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These might sound like small issues but there is a wildly varying level of comfort and confidence in using IT technology in lecturers and for those who are not comfortable with technology, no matter how simple you make these things, they're still a trial to endure rather than a tool to help them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And that's not even considering that some lecturers will be quite happy using alternative methods. Despite being an IT person myself and quite comfortable with using technology, if you're ever in a lecture room listening to me talk about something expect me to write on all the whiteboards many times over before expecting to see anything like a clicker appear.&lt;/p&gt;&#xA;" OwnerUserId="1202" LastActivityDate="2014-04-23T18:30:43.053" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="1949" PostTypeId="2" ParentId="1940" CreationDate="2014-04-23T18:48:03.757" Score="25" Body="&lt;p&gt;Here's what I've found, based on my own clicker use for the last 5-6 years and by doing clicker workshops: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Cost&lt;/strong&gt;. Clickers (and their software BYOD variants) cost money and many profs don't want to ask students to buy Yet Another Item for class when textbooks are so expensive already. Also, many IT departments don't want the hassle of managing fleets of clickers. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Shelf life&lt;/strong&gt;. To make matters worse, there's often no standardization among clicker devices on the same campus, so one class will require &lt;a href=&quot;http://www.turningtechnologies.com/response-solutions/responsecard-rf&quot;&gt;Turning Point RF&lt;/a&gt; clickers and another will require a &lt;a href=&quot;http://tophat.com&quot;&gt;Top Hat&lt;/a&gt; license while another will require an iClicker. In this case students end up laying down serious amounts of money for essentially a unitasking device -- or rather, several of them, which will likely never see the light of day after the course is over. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Not having a sense of how best to use the things&lt;/strong&gt;. There are some amazing things you can do pedagogically with clickers, but many faculty don't see these or fail to think outside the lecture box. Or, the faculty know what can be done but it sounds time-consuming to convert lectures over to clicker-driven classes, so they bail. (Which is too bad since this is not entirely true.) &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For me, having just spend a semester using a web-based BYOD type of clicker implementation, I'm not so sure I'm going to stick with it. The web-based tool I used had some neat features that physical clickers can't do, but I lost track of how many times someone's wifi connection would drop out in the middle of a lesson or something like that. Physical clicker devices &lt;em&gt;just work&lt;/em&gt; IMO. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-04-23T18:48:03.757" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1950" PostTypeId="2" ParentId="1817" CreationDate="2014-04-23T18:59:27.557" Score="4" Body="&lt;p&gt;I always make sure to mention Hypatia in my classes and at least show them parts of this video (The Story of Maths - Marcus Du Sautoy BBC). I've linked to the part where she's mentioned. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://youtu.be/rDBdT1Dl_QY?t=55m6s&quot; rel=&quot;nofollow&quot;&gt;http://youtu.be/rDBdT1Dl_QY?t=55m6s&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="533" LastActivityDate="2014-04-23T18:59:27.557" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1951" PostTypeId="1" CreationDate="2014-04-23T19:02:36.637" Score="10" ViewCount="533" Body="&lt;p&gt;In &lt;a href=&quot;https://matheducators.stackexchange.com/a/1949/704&quot;&gt;this answer&lt;/a&gt;, user &lt;a href=&quot;https://matheducators.stackexchange.com/users/942/robert-talbert&quot;&gt;Robert Talbert&lt;/a&gt; stated that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There are some amazing things you can do pedagogically with clickers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'd like to see some examples.  (Not that I'm eager to try that myself, but I'm just curious.)  As I stated in one of the comments to the linked question,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I can't see much point in using such devices, at least in the context of teaching.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;– but I'm probably mistaken.  Please enlighten me, and at the same time give MESE users some neat ideas! :)&lt;/p&gt;&#xA;" OwnerUserId="704" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-04-23T22:38:23.310" Title="What is a reasonable use of clickers in the context of teaching?" Tags="|interactive-teaching|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1952" PostTypeId="1" CreationDate="2014-04-23T19:31:07.670" Score="12" ViewCount="1210" Body="&lt;p&gt;In a classical course on Euclidean, compass-and-ruler geometry, &lt;em&gt;Thales' theorem&lt;/em&gt; has always had a prominent place.  However, as the &lt;a href=&quot;http://en.wikipedia.org/wiki/Intercept_theorem&quot;&gt;Wikipedia article&lt;/a&gt; says,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is equivalent to the theorem about ratios in similar triangles,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which seems to be conceptually easier for most students (I don't have any hard data for that, but this is both my and my colleagues' experience).  So my question is: should we teach Thales' theorem?  If yes, should we expect the students to actually use this particular result (as opposed to ratios in similar triangles)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I am strongly opposed against purely pragmatic approach to teaching – I think that we &lt;em&gt;should&lt;/em&gt; teach people many things that are not directly &lt;em&gt;useful&lt;/em&gt;, but are an important part of our culture (Latin language being a classical (pun intended;-)) example) – so an argument like &quot;it is not &lt;em&gt;needed to solve problems&lt;/em&gt;, so it should not be taught&quot; is not at all convincing.&lt;/p&gt;&#xA;" OwnerUserId="704" LastEditorUserId="262" LastEditDate="2015-09-03T05:02:38.427" LastActivityDate="2020-03-08T17:12:55.557" Title="Reasons to teach Thales' theorem" Tags="|concept-motivation|geometry|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1953" PostTypeId="2" ParentId="1940" CreationDate="2014-04-23T19:50:15.880" Score="12" Body="&lt;p&gt;I've used clickers and helped other faculty with clickers for over 10 years. I'm at a very large, very research-oriented public university. Here are the reasons I've seen for why research faculty don't use clickers when they teach:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They team-teach, which means all instructors have to agree to use&#xA;them&lt;/li&gt;&#xA;&lt;li&gt;They teach only 5 weeks a year, and have to re-learn the technology each time. They get very frustrated with changes in software and hardware &lt;/li&gt;&#xA;&lt;li&gt;They have a low tolerance for student confusion with technology. Without an expert TA to do the grading and solve student problems, clickers are too much work.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Our teaching faculty (senate-level lecturers) all teach much more often and all use some sort of audience response device. This group is generally willing to try anything and can make anything work. The major difficulty with BYOD web-clickers is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The lecture hall wifi can't handle 440 students submitting at once&lt;/li&gt;&#xA;&lt;li&gt;Not every single student wants to bring a device to class. This is less of a problem as smartphones become nearly ubiquitous, but still. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1207" LastActivityDate="2014-04-23T19:50:15.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1954" PostTypeId="2" ParentId="1951" CreationDate="2014-04-23T19:52:45.577" Score="6" Body="&lt;p&gt;Several times in my high school Computer Science classes, my teacher used devices like this to administer quizzes &lt;em&gt;in parallel&lt;/em&gt; with a lecture. That is, as he covered material, he would put up quiz questions on a projector, and we would answer with the clickers. The questions in general tested comprehension of what the teacher was saying, rather than parroting the words that had just come out of his mouth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The specific devices we had were basically IR remotes, however, making their usage difficult. With one receiver and a couple dozen remotes, we had to fight over each others' signals to get our responses in. The fact that the remotes needed to be pointed fairly accurately towards the receiver was also annoying. These problems would likely be solved by different physical technology being put to use, though.&lt;/p&gt;&#xA;" OwnerUserId="328" LastActivityDate="2014-04-23T19:52:45.577" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1955" PostTypeId="2" ParentId="1952" CreationDate="2014-04-23T19:57:16.973" Score="6" Body="&lt;p&gt;Sadly, the time you (and your students) have available for this is finite. So one has to weigh usefulness carefully. Also, easier to understand is often paramount in teaching. So I'd go for similar triangles (wider usefulness, easier to understand).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-23T19:57:16.973" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1956" PostTypeId="2" ParentId="1951" CreationDate="2014-04-23T20:07:40.620" Score="5" Body="&lt;p&gt;Warning. Biologist is answering.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our instructors are often in very large, very sleepy lecture halls. Clickers provide a stimulus for student discussion and trigger learning through testing effects.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Common uses:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The instructor is about to begin a new subject. She opens a clicker&#xA;question that contains a common misconception as one of the answer&#xA;options. Students answer incorrectly. The instructor can now address&#xA;the misconception and correct it. Students are given additional, new&#xA;problems to answer, either individually or in groups, and then click&#xA;in.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The instructor asks students to rank a list of procedures, processes, steps in order. Can be done individually, and then in groups. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If the instructor has assigned pre-class reading, students can take a 5-minute clicker quiz when they walk in to reinforce the assignment and remind them that the instructor will not be covering the basics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Newer, web-based audience response systems can require students to draw a curve on a graph or type in a short answer. Some (Learning Catalytics) can determine which students answered correctly or incorrectly, and tell Marcos to &quot;turn to Jane on your left and discuss your answer,&quot; maximizing student teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Attached is an example of the research on the subject in biology:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Smith, M. K., Wood, W. B., Krauter, K., &amp;amp; Knight, J. K. (2011). &lt;a href=&quot;http://www.lifescied.org/content/10/1/55.full&quot;&gt;Combining peer discussion with instructor explanation increases student learning from in-class concept questions.&lt;/a&gt; CBE-Life Sciences Education, 10(1), 55-63.&lt;/p&gt;&#xA;" OwnerUserId="1207" LastActivityDate="2014-04-23T20:07:40.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1957" PostTypeId="2" ParentId="1817" CreationDate="2014-04-23T20:26:13.450" Score="1" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Mary_Kenneth_Keller&quot; rel=&quot;nofollow&quot;&gt;Mary Kenneth Keller&lt;/a&gt;, also more of a computer scientist than a mathematician, but still.&lt;/p&gt;&#xA;" OwnerUserId="704" LastActivityDate="2014-04-23T20:26:13.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1959" PostTypeId="2" ParentId="1951" CreationDate="2014-04-23T22:38:23.310" Score="12" Body="&lt;p&gt;First of all I would highly recommend &lt;a href=&quot;http://amzn.com/B002UHTTYO&quot;&gt;Derek Bruff's definitive book on this subject&lt;/a&gt;. There are more good ideas in that book than any one faculty member can expect to implement. If there's anything I do with clickers in the classroom that works, it's probably appropriated from Derek in some way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, here are three ways I use clickers on a regular basis: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Warmup/entry quizzes&lt;/em&gt;. Students are given 2-3 questions at the beginning of class to activate knowledge that they will need from the day, usually some combination of review from the previous class or some major point from the pre-class reading and exercises that they were supposed to finish. Since I usually teach with a flipped classroom format, I grade these (well, the clicker software grades them and I just download the data) for a light preparation grade. &lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Peer instruction&lt;/em&gt;. This is a teaching style that Eric Mazur of Harvard University Physics pioneered. The class is organized around a series of small lectures that set up conceptual questions that are targeted at common or important misconceptions. Students use the clickers to vote on the questions, then if there's widespread disagreement on the answer (which there usually is if the question is good) the students are put into pairs and tasked to defend their choice to the other person for two minutes. Then students vote again on the same question after they've had a chance to discuss. Quite often, the second round of voting converges either on near-unanimity on the right answer or else it's 50/50. There's an enormous literature on the effectiveness of peer instruction if you want to Google it and learn more. &lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Formative assessment during a lecture&lt;/em&gt;. If your instructional tastes are more aimed at traditional lecture, you can insert &quot;concept check&quot; questions at significant moments in the lecture to check for student understanding. That way you have real-time data -- collected without having the personal or social biases inherent in hand-raising and other methods of getting student feedback -- that show whether your lecture is hitting home or not. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Again, though, Derek's book is awesome and encyclopedic, so check that out. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-04-23T22:38:23.310" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1960" PostTypeId="2" ParentId="1940" CreationDate="2014-04-23T23:38:12.057" Score="13" Body="&lt;p&gt;As in other answers and comments: while in gigantic &quot;classes&quot; one might hope to either get meaningful responses, or do quick polling/quizzing via in-real-time responses, ... techno-glitches make it impossible to depend on these, in the first place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The potential plus that feedback can be anonymous, allowing perhaps more sincere response than otherwise, would be a wonderful result to realize. However, in those huge-class scenarios, and given our contemporary academic culture, there is scant reason to believe that clicker responses are ... that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The faux-benefits of anonymous response are misleading, I think. Instead, I've always implemented systems that allow me to learn everyone's name even in classes of up-to-150. Even if &quot;kids&quot; seem to desire a sort of anonymity, it seems to me that they &lt;em&gt;benefit&lt;/em&gt; from belief that the instructor is aware of them as an individual, not as a stat on a clicker.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reading facial expressions and body language gives me quite a lot of information. Techno-vulnerable yes-no stuff is not only limited and high-overhead, but too narrow a pipe for what I want to know about the people in the room.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I'm not at all a technophobe, btw, having written a web browser in Perl, etc. But not a technophiliac, either. Trying to be a realist.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-23T23:38:12.057" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1961" PostTypeId="2" ParentId="1952" CreationDate="2014-04-24T00:58:32.687" Score="1" Body="&lt;p&gt;&lt;strong&gt;NB&lt;/strong&gt;. This reply mistakenly focused on a different &lt;a href=&quot;http://en.wikipedia.org/wiki/Thales&amp;#39;_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thales' Theorem&lt;/a&gt;. So it is not relevant to the posed question.&#xA;&lt;hr /&gt;&#xA;I would just like to point out that Thales' Theorem plays a key role in &lt;a href=&quot;http://en.wikipedia.org/wiki/Delaunay_triangulation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Delaunay triangulations&lt;/a&gt;, which are a central topic in computational geometry and meshing algorithms.&#xA;Were Thales' Theorem connected to these cutting-edge topics,&#xA;it might be seen as more relevant and important.&#xA;How to reach high-school teachers with this connection&#xA;is a significant challenge...&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Y2OWV.png&quot; alt=&quot;VD&amp;amp;DT&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;sub&gt;(Red dual graph is the Delaunay triangulation.&lt;/sub&gt;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;sub&gt;Image from &lt;a href=&quot;http://cs.smith.edu/~orourke/DCG/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discrete and Computational Geometry&lt;/a&gt;, which invokes Thales' Theorem four times.)&lt;/sub&gt;&#xA;&lt;hr /&gt;&#xA;Let me try to address Jim Belk's (excellent) question.&#xA;What is important is the relationship&#xA;depicted below: &#xA;$A &amp;gt; B &amp;gt; C$.&#xA;That it can be proved by similar triangles does not diminish the importance&#xA;of the nonobvious conclusion.&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/yGS8P.png&quot; alt=&quot;Thales&quot;&gt;&#xA;&lt;br /&gt;&#xA;To know that Thales' Theorem plays a role in what students&#xA;see in &lt;em&gt;The Lord of the Rings&lt;/em&gt; or &lt;em&gt;The Game of Thrones&lt;/em&gt;&#xA;could make it more relevant.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-04-24T10:38:37.833" LastActivityDate="2014-04-24T10:38:37.833" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1962" PostTypeId="2" ParentId="1922" CreationDate="2014-04-24T02:53:44.783" Score="13" Body="&lt;p&gt;The teacher may be doing better than you can imagine, teaching a large (likely) group of students, very few of whom have any interest in math. I think it is silly that you blame the teacher. (Sure, the teacher might be bad, but you give me no evidence to think so.) Yes, the American system has problems. (I am very impressed with the math skills of the people I meet from Russia.) But that is not the fault of one teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I heard very little in your question about what your son desires. Have you been trying to hold him to Russian standards? Does he enjoy this? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I love math. But no one pushed me. (And I do not like working at the pace required by elite universities. Because of that I don't have a PhD, just a masters' degree. I teach at a community college, where we have a number of great teachers. My guess is that you would not judge us as great, because we don't do things the way you think is best.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might be trying to feed his hunger for interesting math. That's great. You could homeschool him. You can sign him up for classes with Art of Problem Solving. You might be able to get him excused from his high school math classes, while he stays in school for the rest. Find creative ways to solve the problem, without blaming the teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do agree that it's silly to teach partial fraction decomposition in precalc, but I can't imagine a teacher who would choose that. Someone higher up made that terrible decision. The teacher would be taking a risk to leave it out. And, as you said, it would be pretty hard to do right by the topic at that level. If you get past your bad attitude about the teacher, there's one topic the two of you would probably agree on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you &lt;em&gt;might&lt;/em&gt; have to offer is not your knowledge of math, but your knowledge of a different system of teaching. If you can be patient, and think about building connections, you might be able to help with the math team at some point. You might be able to make a positive difference for a number of kids at the school - &lt;em&gt;if&lt;/em&gt; you learn to respect the hard job every one of the teachers there is required to do.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-04-24T02:53:44.783" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1963" PostTypeId="2" ParentId="1940" CreationDate="2014-04-24T05:00:58.907" Score="6" Body="&lt;p&gt;I originally wrote that there is independent data showing no difference in outcomes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was based on a presentation at Stevens Institute of Technology where they tested clickers and found that students were more enthusiastic in class, and that attendence rose.&#xA;At the time, they said that scores did not rise.  Scores after all, are the objective goal.  If students don't understand any better, then the immediate goals are more ego-gratification for the professor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today, 10 years later, the professor says they use clickers, and find that they still achieve the immediate improvements.  He cannot tell me about grades because they don't do any studies on it.  It is hard to believe that grades cannot improve based on better attendance, but without objective measurement, I still believe their initial impressions which were based on comparison of year on year scores.  Another way of looking at this is that as the studies by the ASU modeling curriculum found, students of physics don't believe what they are writing.  When asked what happens in a &quot;real&quot; situation they give Aristotelian answers.  In other words, the lecturer can improve lecture mechanics with clickers, but unless they change the way they teach, the fundamental scores won't change because the students don't really believe the physics.  Anyway, sorry, but that's peculiar to physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach programming, and I find that multiple choice questions are of limited utility. I can sometimes ask the right question.  But the best test of computer programming ability (aside from asking for an entire program) is to see whether people know what to type in a particular place without prompting.  This is short answer, and complicated short answer since multiple answers can sometimes be acceptable.  I will attach some sample questions I ask below.&lt;img src=&quot;https://i.stack.imgur.com/PZm5V.png&quot; alt=&quot;Example of CLOZE questions&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, using clickers takes time, and I have moodle and can tell them to review things at home and answer my questions there.  That way, I get them to study out of class, review what we did, and it reinforces it without taking up classtime.  I find that works better for me, but in any case I am teaching graduate classes, without access to uniform clickers.&lt;/p&gt;&#xA;" OwnerUserId="1032" LastEditorUserId="1032" LastEditDate="2014-04-28T15:26:24.300" LastActivityDate="2014-04-28T15:26:24.300" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1964" PostTypeId="2" ParentId="1940" CreationDate="2014-04-24T05:23:01.923" Score="5" Body="&lt;p&gt;I don't use clickers nor anything like that because I teach on the assumption that each person seated in front of me is an adult, in charge of their own learning. Of course that is not true in many cases these days, but to me the fact that university students want to behave like children is still no reason to treat them as such. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of the above, and since the students have already lots of means to communicate with me (personally before, during, and after class; office hours; email) plus other resources (the course web page, their notes, the textbook, tutoring, online searches), I find no use for clickers in my class. &lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2014-04-24T05:23:01.923" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1965" PostTypeId="2" ParentId="1860" CreationDate="2014-04-24T05:23:26.460" Score="23" Body="&lt;p&gt;My thinking is that it is just so damn useful for students to be aware of these tricks. The examples/exercise should allow them to develop a sense of when and how it is helpful to simplify an expression in this way, &lt;strong&gt;BUT&lt;/strong&gt; also when it is &lt;strong&gt;NOT&lt;/strong&gt; necessary.&lt;/p&gt;&#xA;&lt;p&gt;Leading up to it by looking at fractions. Should the students write a rational number in the form &lt;span class=&quot;math-container&quot;&gt;$3\frac17$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\frac{22}7$&lt;/span&gt;? The answer IMHO depends on what you want to do with this number. If it is the final answer to a problem, then in this case either one is probably fine, but in some cases it will be helpful to have the integer part and fractional parts separated, because then you immediately get a good idea of the magnitude, and can answer questions like &amp;quot;Is the answer larger than &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;?&amp;quot; BUT. If this is an intermediate step, then converting &lt;span class=&quot;math-container&quot;&gt;$\frac{22}7$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$3\frac17$&lt;/span&gt; is usually not necessary, and may be a colossal waste of time, if for example the next step in the problem requires you to divide this intermediate result by &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We surely want to teach the students&#xA;the process of converting &lt;span class=&quot;math-container&quot;&gt;$3\frac17$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\frac{22}7$&lt;/span&gt; and back, not as an end, but as a tool for saying something useful about the answer. We also develop their sense of how fractions and quotients work so that they have a sporting chance to learn to manipulate symbolic fractions later in their studies. But that's a bonus, and difficult to use as motivation at the point when they are learning to manipulate rational numbers.&lt;/p&gt;&#xA;&lt;p&gt;On with the radicals. IMO the salient questions are &amp;quot;what are we expected to do with this result?&amp;quot; and &amp;quot;can we expect a significant simplification?&amp;quot; Consider the following assignments.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Plot the graph of the function&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x)=\frac{x}{\sqrt2-1}-\frac{x}{\sqrt2+1}&#xA;$$&lt;/span&gt;&#xA;by calculating the value of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; at the eleven points &lt;span class=&quot;math-container&quot;&gt;$x=0, 0.1, 0.2,\ldots, 1.0$&lt;/span&gt;. The usefulness of first rationalizing the denominators becomes clear, if the student has developed a sense that there may be significant simplifications lurking in there. Calculating those fractions with a calculator requires several key strokes, properly placed parens et cetera, and is thus error-prone.&lt;/li&gt;&#xA;&lt;li&gt;The same with a more symbolic expression like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;g(x)=\frac1{\sqrt x+1}-\frac1{\sqrt x-1}.&#xA;$$&lt;/span&gt;&#xA;This is an analogue of the grade school drill of doing a lot of manipulations with fractions, so that it is easier to later learn to manipulate rational functions, when the calculator won't do it for ya.&lt;/li&gt;&#xA;&lt;li&gt;OTOH, when doing a lot of trigonometry with the usual fractional multiple of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, you can &amp;quot;almost&amp;quot; treat &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt2$&lt;/span&gt; as having a rational denominator. If we are to next square the intermediate result, I doubt anyone would first convert &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\sqrt2/2$&lt;/span&gt;. I usually won't bother to rationalize that &lt;span class=&quot;math-container&quot;&gt;$\sqrt2$&lt;/span&gt; in the denominator myself, so I obviously should not expect my students to do so either.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;TL;DR; The projected way of using the intermediate result, or the end result, gives hints as to whether an effort in simplification is warranted. The students should learn these rationalizing tricks in order to develop a sense of whether it is worth their while to do it in a given situation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.gocomics.com/peanuts/1979/02/20&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.gocomics.com/peanuts/1979/02/20&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2023-02-09T10:15:06.310" LastActivityDate="2023-02-09T10:15:06.310" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="1966" PostTypeId="2" ParentId="1940" CreationDate="2014-04-24T11:09:19.940" Score="8" Body="&lt;p&gt;People have mentioned lots of good thoughts but I'll add a bit.  Even staying away from ancient history and sticking to recent years, in addition to clickers (I don't know the vendor they want me to use here), people have been telling me that I should record my classes on video (we have Tegrity), digitize all my whiteboard work (we have SmartBoard), flip my classroom (I think they want me to use Kahn Academy), or fold in MOOC's (there seems to be interest in EdX) or Open Educational Resources (the one at Carnegie Mellon is often mentioned), and use a learning management systems (we have recently switched to Canvas, so things built in the prior one may well be lost).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No one wants to be &lt;em&gt;that guy&lt;/em&gt;.  But aren't you skeptical?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are two specific things that are in my cloud of help-make-me-skeptical thoughts. (1) These things are all a ratcheting-up, that all add degrees of franticness to the class and therefore threaten a focus on the work at hand.  (2) Going back to ancient history, the best guide to success for typical students has been to come to class and to do the homework.  Perhaps new stuff is a big win, and faculty that are interested should definitely experiment (I've tried some things on the above list; for example last semester I flipped a Business Calc class), but I often take a pass myself. &lt;/p&gt;&#xA;" OwnerUserId="213" LastActivityDate="2014-04-24T11:09:19.940" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1967" PostTypeId="2" ParentId="1756" CreationDate="2014-04-24T12:20:07.443" Score="4" Body="&lt;p&gt;If I would give only one advice to focus on, it would be to find someone to discuss the problems with. Preferably someone on your own level of understanding like a classmate. The reason is simple:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your brain has to form a mental model on how everything works. When you listen to a lecturer, or read a text, your brain can basically put itself on standby if it wants to. You don't ever have to prove that you actually understand the stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But when discussing, you have to start with your mental model and then try to express it in words, making it impossible to not know your stuff. And the amazing thing is that you can start with the small bits and pieces you do know and merge them with what the other person knows to create a new mental model quite easily.&lt;/p&gt;&#xA;" OwnerUserId="1194" LastActivityDate="2014-04-24T12:20:07.443" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1968" PostTypeId="2" ParentId="1869" CreationDate="2014-04-24T14:05:19.537" Score="12" Body="&lt;p&gt;Opening section of &lt;a href=&quot;http://www.iom.edu/~/media/Files/Activity%20Files/PublicHealth/HealthLiteracy/Commissioned-Papers/Numeracy-and-the-Affordable-Care-Act-Opportunities-and-Challenges.pdf&quot;&gt;this document&lt;/a&gt; refers to a number of studies of numeracy of US population. One of the major international efforts in providing standardized, comparable-across-countries data on adult literacy and numeracy skills is OECD's &lt;a href=&quot;http://www.oecd.org/site/piaac/&quot;&gt;Programme for the International Assessment of Adult Competencies (PIIAC)&lt;/a&gt;. The US country report is &lt;a href=&quot;http://nces.ed.gov/pubs2014/2014008.pdf&quot;&gt;here&lt;/a&gt;; I know some of the authors through the professional circles, and can ask informally for details. The &lt;a href=&quot;http://www.oecd.org/site/piaac/International%20Codebook_PIAAC%20Public-use%20File%20%28PUF%29%20Variables%20and%20Values.xlsx&quot;&gt;codebook&lt;/a&gt; (meta-data about questions) does not seem to have any questions about primes, but I would still need to see the specific instrument. You can even &lt;a href=&quot;http://www.oecd.org/site/piaac/educationandskillsonlineassessment.htm&quot;&gt;take the test yourself&lt;/a&gt; if you like, although the website only support Firefox (and declined to operate in my Chrome... ah well). Media outlets that &lt;a href=&quot;http://www.usatoday.com/story/news/nation/2013/10/08/literacy-international-workers-education-math-americans/2935909/&quot;&gt;reported on the study&lt;/a&gt; also &lt;a href=&quot;http://www.usatoday.com/story/news/nation/2013/10/08/education-workers-literacy-sample-tests-math/2936323/&quot;&gt;reposted&lt;/a&gt; the typical questions. The single number that the media picked up is 9% of US adults ages 16 to 65 being proficient at the highest level. That will probably be the best answer to your question, lacking the specific study on knowledge of prime numbers.&lt;/p&gt;&#xA;" OwnerUserId="742" LastActivityDate="2014-04-24T14:05:19.537" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1970" PostTypeId="1" AcceptedAnswerId="1973" CreationDate="2014-04-24T19:07:54.570" Score="15" ViewCount="864" Body="&lt;p&gt;This question might sound vague, but I'm really just looking for particular examples that worked for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my experience, it seems like a large portion of &quot;weak&quot; students remain weak despite their best efforts. I understand that part of this is due to things that have happened earlier on in their lives. Perhaps they didn't learn a fundamental subject well enough and were passed anyway. Perhaps they have poor study skills that haven't been corrected:  copy notes mindlessly, look at solutions before attempting the problem, don't memorize, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides just reminding them and showing them by example, what can be done?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;In particular, my question is this: What have you done in your classroom that has helped instill good study habits or good habits in general. Or alternatively, what have you done to break bad habits?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My go-to weapon is office hours. I can talk to individual students about their study habits, ask them to work problems in front of me, etc. but it seems very inefficient to have to deal with students on a case-by-case basis...and hard enough to get them to show up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this a lost cause or can you teach an old dog new tricks?&lt;/p&gt;&#xA;" OwnerUserId="533" LastEditorUserId="262" LastEditDate="2014-04-25T04:08:53.273" LastActivityDate="2014-04-26T16:05:15.880" Title="What can be done to instill good habits in students?" Tags="|undergraduate-education|secondary-education|student-motivation|classroom-management|struggling-students|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1971" PostTypeId="2" ParentId="1946" CreationDate="2014-04-24T20:21:31.067" Score="7" Body="&lt;p&gt;&lt;strong&gt;I made &quot;logarithm sticks&quot; for my college algebra class.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I bought some dowels at home depot and cut them to the following lengths:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\log_{10} 10$ feet&lt;/li&gt;&#xA;&lt;li&gt;$\log_{10} 12$ feet&lt;/li&gt;&#xA;&lt;li&gt;$\log_{10} 15$ feet&lt;/li&gt;&#xA;&lt;li&gt;$\log_{10} 20$ feet&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;etc. Then I labeled them as such with stickers, so they look like small flags.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Then when you put the stick that is $\log_{10} 20$ next to the one that is $\log_{10} 10$, they add up to the same length as the $\log_{10} 200$ stick. This visually demonstrates the logarithm multiplication rule. More specifically it tries to convince them that $\log_{10} 15$ is &lt;strong&gt;a number&lt;/strong&gt; which is actually the part that is hard for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The numbers I actually chose were 10, 12, 15, 20, 30, 50, and then all the products you can generate from those numbers. Make sure you make more of the smaller ones, and make lots of 10s, 100s, and 1000s because those make sense first: they are 1, 2, and 3 feet long. I recommend this list:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;10, 12, 15, 20, 30, 50, 100, 120, 150, 180, 200, 240, 300, 360, 450, 500, 600, 750, 1000&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I haven't done this yet, but you could make a different-sized set by just using a different base -- then you could maybe visually motivate the change-of-base formula?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-04-24T20:21:31.067" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1972" PostTypeId="2" ParentId="1970" CreationDate="2014-04-24T20:41:55.167" Score="1" Body="&lt;p&gt;The causes are probably a hodgepodge of all of the above, and a few more. First step is clearly to find out the most important factor in each case... and that takes one-on-one time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is special tutoring (even in general math) available? If so, make sure the students go there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suggest/encourage/enforce studing habits:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Take a look at the notes after each class&lt;/li&gt;&#xA;&lt;li&gt;Don't try to cram three days before the exam, it just won't work. It is my impression that some variation of this is at the root of most bad performance.&lt;/li&gt;&#xA;&lt;li&gt;Make up a regular schedule for studying, and stick to it. If no homework or other work, go through your notes, solve proposed exercises. Count yourself lucky, I didn't have the Internet to trawl for material...&lt;/li&gt;&#xA;&lt;li&gt;Leave a day off from studying each week (does wonders against ending all stressed out)&lt;/li&gt;&#xA;&lt;li&gt;Don't study for the exam the day before. You may study other subjects, but not for tomorrow's exam (the last day you won't learn anything new, you'll only make yourself nervous).&lt;/li&gt;&#xA;&lt;li&gt;Make sure you are well-rested for the exam (i.e., no allnighters the day before!)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;With my TAs here we came up with the idea of proposing three problems for each weekly tutoring session: one solved by the TA (asking for input from the class), one to be solved by the class in open discussion (with hints/help from the TA), and one to be solved individually and handed in. Solutions to all are posted after the session. The problems handed in are checked just for rough relevance to the subject, not really graded, and their yes/no make up a 5% of the final grade. Seems to have made wonders in the one occasion we have used it up to here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Give short homework, often. Keep them involved with the subject. If a week goes by without looking at the subject, reconnecting becomes very hard (even hopeless).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-24T20:41:55.167" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1973" PostTypeId="2" ParentId="1970" CreationDate="2014-04-24T20:44:12.967" Score="9" Body="&lt;p&gt;One thing I have done is to have taken the courses I have helped with (Teaching Assistant, Course Assistant, Research Assistant, Instructor) very seriously. A summary of my approach to TAing and office hours (your &quot;go-to weapon&quot;) can be found in an earlier MESE post &lt;a href=&quot;https://matheducators.stackexchange.com/a/1695/262&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of the approaches mentioned there, one that I would like to re-emphasize here is the importance of asking that students who would like assistance: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Give their questions due consideration before asking them, and &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Indicate how they are thinking about their questions as a part of the asking process.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Some of the so-called &lt;em&gt;bad habits&lt;/em&gt; that you cite can be addressed to a reasonable extent by ensuring that students are not viewing you, their teacher or even their peer, as a crutch, but rather as someone to help them in their struggle (ideally a &lt;em&gt;productive struggle&lt;/em&gt;, cf. &lt;a href=&quot;https://matheducators.stackexchange.com/a/878/262&quot;&gt;MESE 2&lt;/a&gt;) to learn the material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an &lt;strong&gt;actual email excerpt&lt;/strong&gt; from a graduate student in mathematics education (previously a kindergarten teacher!) who decided to take a course in Topology for which I was the TA. (For the sake of clarity, I added in the TeX below.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Are basis elements in $\mathbb{R}_d \times \mathbb{R}$ of the form $(a \times b, a \times d)$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is why I'm guessing basis elements in $\mathbb{R}_d \times \mathbb{R}$ are $(a \times b, a \times d)$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;•$\mathbb{R}_d$ is the discrete topology. The discrete topology is the collection of all subsets of $X$, which means all subsets are open, which means $\{p\}$ is open for all $p \in \mathbb{R}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;•For $\mathbb{R}$ we have the standard topology, which means open intervals $(a,b)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;•So the basis of $\mathbb{R}_d \times \mathbb{R}$ is $\{p\} \times (a,b)$ or, written another way, $(a \times b, a \times d)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this correct? Is this completely wrong? I'm doing #9 p. 92 [in Munkres' &lt;em&gt;Topology 2e&lt;/em&gt;] and if the above is correct, then the dictionary ordered topology on $\mathbb{R} \times \mathbb{R}$ and product topology on $\mathbb{R}_d \times \mathbb{R}$ have the same basis elements. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I include the above excerpt not for the specifics, but rather because this student initially overwhelmed me with questions. Eventually I asked her to include her thoughts - even if wrong - whenever emailing me. In the above example, note that she includes a &lt;em&gt;guess&lt;/em&gt; and her &lt;em&gt;reasoning&lt;/em&gt;. In this case, they are pretty correct; moreover, from around this time, the volume of emails dwindled substantially as the student was able to gain a fuller sense of agency.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The question of whether Topology should be taught in Mathematics Education programs is one I leave to another thread...)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-24T20:44:12.967" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1974" PostTypeId="2" ParentId="1934" CreationDate="2014-04-24T21:14:27.230" Score="7" Body="&lt;p&gt;Question: What is the efficiency of a lecture?&#xA;Answer: Assuming that the learning comes from studying the notes generated by the lecture, students can recall about 80% of a list of facts for a day or so, and then this declines -- often down to low levels like 40% -- after a few weeks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, difficult concepts are not learned as well to begin with, and concepts that are repeatedly tested are retained longer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some classic studies of this issue are comparing retention of straight lecture vs alternative, more active teaching. Some examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Pausing during the lecture can increase retention.&lt;/li&gt;&#xA;&lt;li&gt;Memory of studied material increases with testing&lt;/li&gt;&#xA;&lt;li&gt;General overview on how to be more effective than lecturing in engineering&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Your tool is a good idea, and rather hot in edtech right now. Apps like &quot;&lt;a href=&quot;http://letsgeddit.com/&quot;&gt;Geddit&lt;/a&gt;&quot; are designed for K-12 classrooms. And instructors who want to assess understanding can use clickers and ASK how students are doing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Citations &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Ruhl, K. L., Hughes, C. A., &amp;amp; Schloss, P. J. (1987). Using the pause procedure to enhance lecture recall. Teacher Education and Special Education: The Journal of the Teacher Education Division of the Council for Exceptional Children, 10(1), 14-18.&lt;/li&gt;&#xA;&lt;li&gt;Roediger, H. L., &amp;amp; Karpicke, J. D. (2006). Test-enhanced learning taking memory tests improves long-term retention. Psychological science, 17(3), 249-255.&lt;/li&gt;&#xA;&lt;li&gt;Prince, M. (2004). Does active learning work? A review of the research. Journal of engineering education, 93(3), 223-231.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1207" LastActivityDate="2014-04-24T21:14:27.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1975" PostTypeId="2" ParentId="1896" CreationDate="2014-04-24T21:26:04.650" Score="5" Body="&lt;p&gt;There's two separate issues here.  The first is whether there's important work which the word clear/trivial/obvious is doing when used in mathematical explanations.  As explained by Jack M. and Steve Jessop and several of the MO posts, the answer to that question is yes: it indicates to the reader/listener that there's a very short argument that's being skipped over rather than something deep.  The second issue is whether &quot;clear/trivial/obvious&quot; are good &lt;em&gt;choices of words&lt;/em&gt; to use for the concept.  I think in the context of teaching the answer is no.  (In a research context they're fine, because all readers will know that they're technical terms of art that don't actually mean what clear/trivial/obvious mean in everyday speech.)  However, I don't know what would be a better choice of word which would be less likely to be misunderstood.  Has anyone found a word choice that works better with students?&lt;/p&gt;&#xA;" OwnerUserId="345" LastEditorUserId="345" LastEditDate="2014-04-24T22:20:15.777" LastActivityDate="2014-04-24T22:20:15.777" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1976" PostTypeId="1" CreationDate="2014-04-24T21:31:51.980" Score="14" ViewCount="1011" Body="&lt;p&gt;Overheard in the Math Office while another Professor was helping a student with Statistics: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Always use a calculator when doing decimal arithmetic because you'll eventually make a mistake if you do it all by hand.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My thought:&#xA;&lt;strong&gt;You could just as easily make an input mistake on your calculator.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quick Google Scholar search turned up a book of interest by W. Stephen Wilson and Daniel Q. Naiman entitled &quot;K-12 Calculator Usage and College Grades.&quot; The abstract reads:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We find that students in the big mathematics service courses at the Johns Hopkins University who were encouraged to use calculators in K-12 have somewhat lower grades than those who weren't.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Other partially relevant studies have come up, with many of them stating how calculators can greatly enhance student understanding. I can't help but think about how an over reliance on calculators might affect student performance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know of any relevant research out there that might help to answer this question: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How does a &lt;em&gt;reliance&lt;/em&gt; on calculators for basic arithmetic tasks affect student performance in a mathematics course with deeper mathematical ideas (e.g., Statistics)?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="262" LastEditDate="2014-04-30T16:04:55.290" LastActivityDate="2014-04-30T16:04:55.290" Title="How does a reliance on calculators affect student performance?" Tags="|reference-request|students-mistakes|calculators|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="1977" PostTypeId="5" CreationDate="2014-04-24T22:46:39.293" Score="0" Body="&lt;p&gt;Questions about severe difficulty in making arithmetical calculations, as a result of brain disorder.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-25T00:08:57.983" LastActivityDate="2014-04-25T00:08:57.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1978" PostTypeId="4" CreationDate="2014-04-24T22:46:39.293" Score="0" Body="Questions about severe difficulty in making arithmetical calculations, as a result of brain disorder." OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-04-25T00:08:49.893" LastActivityDate="2014-04-25T00:08:49.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1979" PostTypeId="2" ParentId="1976" CreationDate="2014-04-25T00:53:12.143" Score="5" Body="&lt;p&gt;Yes very much so from what I have observed. I tutor math at my college and I had one person today who literally could not do the math by hand. The objective was to solve a system of equations by using Gaussian elimation. I tried walking the student through each step and justifying each one along the way on paper. The student was beyond lost as to how I got to the answer by hand. I knew tried showing them how to manipulate matrices with their Ti83. I discovered that they could do the problem with the Ti83 because they had memorized what buttons to press to get the answer but it was more or less like magic to them because they couldn't begin to understand what they trying to do anyway. They just knew to press buttons and fill in the blank. The student was obviously taught to pass not userstand in any way. On the other hand I also had a student who didn't understand how to use the calculator at all. This student was supposed to be comparing very similar graphs and picking out which one is which. I taught to them how to graph and zoom in and out etc. so that they could simply look at the function and postulate various things like intercepts and what direction it's going in. I taught them how to utilize it as a tool not a problem solver. Everything I showed them could be done by hand but doing that on a test with many questions it wouldn't be practical. So I do believe calculators have their place. They should be used as tools to help understand not as a magical device to tell you what to write. &lt;/p&gt;&#xA;" OwnerUserId="937" LastEditorUserId="937" LastEditDate="2014-04-28T02:14:30.547" LastActivityDate="2014-04-28T02:14:30.547" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1980" PostTypeId="2" ParentId="1817" CreationDate="2014-04-25T03:01:46.623" Score="10" Body="&lt;p&gt;While in high school &lt;a href=&quot;http://en.wikipedia.org/wiki/Britney_Gallivan&quot;&gt;Britney Gallivan&lt;/a&gt; folded a piece of (very special) paper twelve times, when most people thought it couldn't be folded more than 7 or 8 times, and &lt;a href=&quot;http://pomonahistorical.org/12times.htm&quot;&gt;wrote a paper about it&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like the amateur mathematician, Marjorie Rice, mentioned in another answer, she shows that people who are deeply engaged in a problem can make advances, and that mathematics has room for new discoveries.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-04-25T03:01:46.623" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1981" PostTypeId="2" ParentId="1817" CreationDate="2014-04-25T03:19:24.223" Score="20" Body="&lt;p&gt;Adm. &lt;a href=&quot;http://en.wikipedia.org/wiki/Grace_Hopper&quot;&gt;Grace Hopper&lt;/a&gt; earned a Ph.D. in mathematics at Yale (1934), helped program the Mark I (1944), developed the first compiler (1952) and some early computer languages, and worked on the development of the UNIVAC I.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2014-04-25T03:19:24.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1982" PostTypeId="2" ParentId="1934" CreationDate="2014-04-25T03:38:01.847" Score="4" Body="&lt;p&gt;As a recently &quot;retired&quot; student, this is my perspective:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You seem to be tacitly implying that the efficiency is somewhere around &quot;very poor&quot;. If so, I am certainly not surprised.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found very early on in my mathematics education that if I went to a mathematics lecture in which the material was entirely or almost entirely new, I would comprehend very little of what was said. I would perhaps be able to follow the first few (largely preliminary) definitions and arguments given by the professor, but after that I would get progressively more lost. By the end of such a lecture, I was usually looking at something vaguely reminiscent of Greek, and not at all appealing :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what did I do? I took up the habit of learning things &lt;strong&gt;entirely before&lt;/strong&gt; the lecture started. My goal was always to come into the lecture hall having a solid grasp of all the topics covered in that day's section. I found that this helped &lt;strong&gt;enormously&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think it's fair to say that &quot;lectures are worthless&quot;. But, at least for me, &lt;em&gt;the primary benefit from lectures was seeing a proficient expert work examples that I already had a rough idea of how to solve&lt;/em&gt;. I don't think anyone but an intellectual daredevil could simply walk into a graduate-level mathematics class and learn the material cold, with no background whatsoever. Very few people absorb concepts with such rapidity, and even fewer retain them after such brief exposure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I definitely believe that lectures have the potential to bestow massive intellectual benefits on their students, but those students have to engage at an appropriate level. Passive listening, at least in mathematics, seems rarely to work (if at all).&lt;/p&gt;&#xA;" OwnerUserId="1161" LastActivityDate="2014-04-25T03:38:01.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1983" PostTypeId="1" AcceptedAnswerId="2002" CreationDate="2014-04-25T12:26:26.150" Score="6" ViewCount="1003" Body="&lt;p&gt;Recently I have been perplexed by a non-mathematical problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my Indian student-friend is preparing for the next year's IMO exam but he says that he doesn't like simply doing hard and challenging problems. To him it is kind of boring to &lt;em&gt;only&lt;/em&gt; solve problems. For example he likes to read the Logical Foundations of Geometry rather than solving geometry problems. He has recently appeared for a number of M. C. Q. exams but did not qualify in any of them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He couldn't answer even the half of the questions because he didn't want to answer any question which he couldn't prove. I don't think in this way he can crack the IMO exam because in IMO the skill of problem solving is acquired by solving harder and harder problems and its the thing that is required most. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only thing that he has to his credit is that he has given an elementary proof of &lt;strong&gt;Sylvester-Gallai Theorem&lt;/strong&gt; (a modification of my wrong proof that I asked for verification in Mathematics Stack Exchange), some theorems on Elementary Number Theory (among which one deals with a non-trial method of finding the order of an integer modulo some integer)  which are perhaps new (I don't know precisely because I am not an expert in this field). He wants to be judged by his works, not by passing any exam (even the IMO!). He even tried to send his works to some journal and to some mathematicians also but either it was rejected or he got no reply from the corresponding mathematicians. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basing upon the incidents, I think that the best way to motivate him would be to have recommendation for his work from some renowned mathematician of his interested fields which are only Number Theory, Logic and Euclidean Geometry but I don't think that it will be logical for any mathematician to waste time on his work without knowing that it is something significant. I can tell this to him but probably it will do much harm to him rather than doing good. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any other way to tell him the truth or is there some other way to motivate him to do mathematics rather than trying to seek its logical consistency?&lt;/p&gt;&#xA;" OwnerUserId="1241" LastEditorUserId="1241" LastEditDate="2014-04-26T03:14:39.230" LastActivityDate="2014-04-26T16:24:29.680" Title="Problem of a talented student who doesn't like to solve too much problems" Tags="|student-motivation|undergraduate-research|gifted-students|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1984" PostTypeId="2" ParentId="1983" CreationDate="2014-04-25T13:37:16.987" Score="6" Body="&lt;p&gt;If &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He wants to be judged by his works&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;, perhaps point out that being able to replicate other people's proofs isn't intrinsically any more original than just using the results of the proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most true craftsmen start off by using existing tools before they design their own, and they're probably more competent for all that as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So there should be no shame in becoming proficient in the existing toolset.&lt;/p&gt;&#xA;" OwnerUserId="980" LastActivityDate="2014-04-25T13:37:16.987" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1985" PostTypeId="2" ParentId="1983" CreationDate="2014-04-25T14:04:58.697" Score="6" Body="&lt;p&gt;Perhaps he should hang out in &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;MSE&lt;/a&gt; a bit, and ask for confirmation/refutation/&quot;this is well known, check ...&quot; on results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a fact that the standard &quot;tell people of some results/have them do exercises/test them&quot; cycle isn't for everyone. But it is the only accepted way to select the few whith the abilities to go farther. Yes, there have been exceptions, but &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, remind them the oft-quoted &lt;a href=&quot;http://en.wikipedia.org/wiki/Standing_on_the_shoulders_of_giants&quot;&gt;standing on shoulders of giants&lt;/a&gt;; if you want to check everything out strictly by yourself, you won't get very far. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-25T14:04:58.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1987" PostTypeId="2" ParentId="1940" CreationDate="2014-04-25T18:09:36.350" Score="6" Body="&lt;p&gt;A colleague of mine recently tried out a very cheap low-tech alternative: university students were provided with red and green pieces of cardboard, to be folded by the students themselves. They were asked to place these on their desk, one hidden inside the other, and could thus very visually indicate their answer to the basic question “do you understand”. The results were quite good, so I think I might adapt that approach when next I'm facing a larger audience.&lt;/p&gt;&#xA;" OwnerUserId="1248" LastActivityDate="2014-04-25T18:09:36.350" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1988" PostTypeId="2" ParentId="1596" CreationDate="2014-04-25T18:15:38.280" Score="6" Body="&lt;p&gt;[Long winded response!] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I do enjoy Sage, I have to agree with others that it is not quite user-friendly enough for high schoolers to tolerate just yet. I would also echo those that discount W|A due to it not being free. Maxima being on smartphones interests me, but I offer another alternative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The free dynamic geometry software, &lt;a href=&quot;http://www.geogebra.org&quot; rel=&quot;noreferrer&quot;&gt;Geogebra&lt;/a&gt;, is perfect for high schoolers. It has a strong (and intuitive) CAS to be able to do anything they'll want to do and combines the power of geometry with algebra, as should be happening in the classes. They're working on phone/tablet versions (just released). The commands for things are simple, and the software helps guide you for input if you're stuck. Please check it out! [Don't use the browser version.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question of whether to use these tools or not... I'm kind of surprised by many answers here. To not introduce mathematics students in this century to technology and have them actively using it is just shocking and a great disservice. We should be embracing these tools to provide more legitimate mathematics to our students. Let's take an example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A cookbook problem that exists in a class with no calculators/software could be something (for example) like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solve $x^2-5x+6 = 0 $. And maybe a few of these are necessary to get a feel. However, what is ventured by spending days doing these things by hand with ever more ridiculous problems like $2415x^2+433x-20=0$? That is not a real problem. It comes from nowhere, and no one does these kinds of things without software -- neither mathematician nor applied scientist.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A real problem might result in the cubic $−0.0080556t^3 + 0.11881t^2 − 0.30671t + 3.36$ (from &lt;a href=&quot;http://msemac.redwoods.edu/~kyokoyama/math106/Math380pdf/sect5-3.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, example 1) where we are taking data and fitting a curve. Now I might ask (because this came from data about gasoline prices over time) when the prices were increasing. I can ask that of someone in Algebra 2! Even in Calculus if this is asked, a calculator/software is going to be REQUIRED to find derivatives and arrive at actual numbers. Before Calculus, they can use technology to firstly -- find that cubic(!) with the given data and then discuss what it means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Point of honesty: Of course I was taught the cookbook stuff! I thought it was awesome! Let's simplify rational expressions and verify trig identities all day long, in my opinion. But I'm a mathy, nerdy teacher. That just isn't doing (real!) mathematics nor applications justice.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And to finish my rant, I would argue that the technology can more quickly allow teachers to introduce students to conjecturing about very hard things more readily than if they don't have such access. Please see the journal &quot;Mathematics Teacher&quot; for more cogent and detailed arguments in this vain. &lt;/p&gt;&#xA;" OwnerUserId="1249" LastEditorUserId="1249" LastEditDate="2014-04-25T19:15:15.740" LastActivityDate="2014-04-25T19:15:15.740" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1989" PostTypeId="2" ParentId="1817" CreationDate="2014-04-25T18:22:30.923" Score="39" Body="&lt;p&gt;&lt;strong&gt;Any Living One&lt;/strong&gt; who is friendly enough to come talk with them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Seriously, learning about &quot;people in books&quot; can sometimes be inspiring. But actual live role models are best. Write a local college, university, or business to find a woman who self-identifies as a mathematician. Invite her to your school to spend some time with your students. You want a real person who happens to be a mathematician, a woman, and reasonably happy about that arrangement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel awkward about leaving out women who might be working as mathematics instructors in your school, because I think they count as part of the community, too. But students are used to the whole &quot;female high school teacher&quot; archetype, and you seem like you want to adjust their view of normal jobs for women mathematicians, so you are better off finding someone new.&lt;/p&gt;&#xA;" OwnerUserId="309" LastActivityDate="2014-04-25T18:22:30.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1990" PostTypeId="2" ParentId="1860" CreationDate="2014-04-25T18:58:13.107" Score="15" Body="&lt;p&gt;What follows is an edited and expanded version of comments, and a list of examples, that I posted &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=665512&quot;&gt;12 June 2001&lt;/a&gt; (and later in &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=5926953&quot;&gt;26 September 2007&lt;/a&gt;, in a more abbreviated form) in the Math Forum discussion group &lt;em&gt;AP-calculus&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe rationalizing the denominator was originally positioned so early in the curriculum --- algebra 1 and geometry for division by $\sqrt{n},$ and algebra 2 for division by something like $m + \sqrt{n}$ --- was partly for reasons having to do with &lt;em&gt;numerical calculation&lt;/em&gt;, and partly for reasons having to do with &lt;em&gt;algebraic combination and simplification of exact numerical values&lt;/em&gt;. Incidentally, if you look at textbooks written 50 to 150 years ago, you don't really see much of an expectation that radicals were numerically approximated (this view being based on worked examples in the text and answers to exercises), except for trigonometry texts. However, the numerical aspect becomes much more important in applications that occurred &lt;em&gt;outside&lt;/em&gt; of mathematics (mainly in science courses), so I suspect what happened is that the training in appropriately rewriting radical expressions so that square root tables and such could be easily used was left to the math courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally think there has been too much emphasis on rationalizing the denominator in the past 40 years (perhaps in the past 20 years the emphasis has been more appropriate), especially in classes below the precalculus level, but I also think it's easy to forget just how often the technique of rationalization shows up in math, even if we restrict ourselves to the lower undergraduate level. As for me, when departmental and/or course supervisor constraints allowed me to do so, I &lt;strong&gt;DID NOT REQUIRE&lt;/strong&gt; answers to be in denominator-rationalized form in high school or college algebra classes, or in precalculus classes. However, I felt it was an important skill for anyone getting at least as far as calculus. Thus, in calculus courses, I tried to make up for this inattention to rationalization (both by me and by other teachers) by working the topic in at a number of places. I did this mainly by working examples in class and by assigning problems (with an appropriate hint) like #1-6 below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;MISCELLANEOUS LIST OF EXAMPLES FOR RATIONALIZING&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1.&lt;/strong&gt; These limits can be evaluated without taking derivatives if you first apply a binomial rationalization step:&#xA;$$\lim_{x \rightarrow 1}\frac{x^2 - 1}{\sqrt{2x+2\,} \; - \; 2} \;\;\; \text{and} \;\;\; \lim_{x \rightarrow 0}\frac{1 - \cos x}{x} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2.&lt;/strong&gt; To rewrite&#xA;$$ \ln \left( \frac{x + \sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}} \right) \;\;\; \text{as} \;\;\; 2\ln\left (x + \sqrt{x^2 - 1}\right),$$&#xA;it helps if you first rationalize the numerator. &lt;strong&gt;Note:&lt;/strong&gt; Putting $x = \sec \theta$ gives an identity that is sometimes useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3.&lt;/strong&gt; To differentiate $x^{\frac{1}{2}},$ $x^{\frac{1}{3}},$ etc. using the limit definition of the derivative, you'll want to rationalize numerators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4.&lt;/strong&gt; The derivative of&#xA;$$ \frac{\sqrt{a-x} \; + \; \sqrt{a+x}}{\sqrt{a-x} \; - \; \sqrt{a+x}} $$&#xA;is much easier to put into the more useful form&#xA;$$ \frac{a^2 \; + \; a\sqrt{a^2 - x^2}}{x^2\sqrt{a^2 - x^2}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;if you rationalize the denominator BEFORE differentiating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5.&lt;/strong&gt; Let $a \neq 0,$ $b,$ and $c$ be real number constants. To verify that&#xA;$$ \lim_{n \rightarrow \infty} \left( \sqrt{an^2 + bn} \; - \; \sqrt{an^2 + cn} \right) \;\;\; = \;\;\; \frac{1}{2\sqrt{a}}(b-c),$$&#xA;it helps to rationalize the numerator first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;6.&lt;/strong&gt; The linearization of&#xA;$$\frac{1+x}{1-x} \;\; \text{at} \;\; x=0$$&#xA;is easy if you begin by multiplying both the numerator and denominator by $1+x.$ After doing this, you get to ignore the $x^2$ terms that appear additively with constants or with multiples of $x.$ The result will be $1 + 2x.$ More generally, rationalization ideas can be used to obtain the quotient rule for derivatives by multiplying/dividing by an appropriate conjugate and ignoring all but first order terms, and similar methods can be used to approximate $f(x+h,\,y+k)$ for rational functions $f(x,y)$ when $h$ and $k$ are close to $0.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the same way, one can show by rationalization methods that for $\delta$ and $\epsilon$ near $0,$ we have&#xA;$$\frac{1}{1 + \delta} \; \approx \; 1 - \delta \;\;\; \text{and} \;\;\; \frac{1}{1-\delta} \; \approx \; 1 + \delta \;\;\; \text{and} \;\;\; \frac{1+\epsilon}{1+\delta} \; \approx \; 1 + \epsilon - \delta $$&#xA;These and other approximations are discussed in Philip L. Alger's 1957 text &lt;strong&gt;Mathematics for Science and Engineering&lt;/strong&gt; (see pp. 145-155 of Chapter 6: &lt;em&gt;Numerical Calculations&lt;/em&gt;) and in William Charles Brenke's 1917 text &lt;strong&gt;Advanced Algebra&lt;/strong&gt; (see Chapter IX, Section 146: &lt;em&gt;Useful Approximations&lt;/em&gt;, pp. 126-127). These approximations are often more important for giving approximations that are valid over a range of variable values than for giving individual and isolated numerical approximations. This is especially useful when an exact algebraic form is difficult to work with, such as in a differential equation (recall the pendulum equation). For instance,&#xA;$$\tanh M \;\; = \;\; \frac{e^{M} \; - \; e^{-M}}{e^{M} \; + \; e^{-M}} \;\; = \;\; \frac{1 \; - \; e^{-2M}}{1 \; + \; e^{-2M}} \;\; \approx \;\; 1 - 2e^{-2M}$$&#xA;is an approximation that is correct to $16$ decimal places when $M = 10.$ This particular approximation for $\tanh M$ is obtained in the same manner I've just shown, and then used to find the lowest eigenvalue in the high barrier limit for a quantum mechanical particle confined to a double potential well, in Charles S. Johnson and Lee G. Pedersen's 1974 &lt;strong&gt;Problems and Solutions in Quantum Chemistry and Physics&lt;/strong&gt; (see Problem 4.8(b) on pp. 105-106).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example can be found in Jerry B. Marion's 1970 text &lt;strong&gt;Classical Dynamics of Particles and Systems&lt;/strong&gt; (see p. 270). Marion uses the approximation&#xA;$$\theta \;\; = \;\; \frac{2\pi}{1 - \frac{\delta}{\alpha}} \;\; \approx \;\; 2\pi\left(1 + \frac{\delta}{\alpha}\right) \;\; = \;\; 2\pi + \frac{2\pi\delta}{\alpha}$$&#xA;near the end of a derivation of the precession of Mercury's orbit as predicted by Einstein's Theory of Relativity. The term $\frac{2\pi\delta}{\alpha}$ represents the approximate precession per orbit, which in Mercury's case works out to approximately $43$ seconds (angle measure) per century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;7.&lt;/strong&gt; To express the quotient of two complex numbers in rectangular form, when each of the complex numbers is given in rectangular form, you'll want to use a &quot;rationalization of the denominator&quot; technique. Related to this is finding the real and imaginary parts of a rational function of a complex variable (e.g. verifying the Cauchy-Riemann equations, finding a harmonic conjugate of a rational function, investigating certain orthogonal families of curves, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;8.&lt;/strong&gt; For numerical purposes (e.g. reducing round-off errors during a computer computation), the quadratic formula&#xA;$$ x \;\; = \;\; \frac{-b \; \pm \; \sqrt{b^2 - 4ac}}{2a}$$&#xA;is in some cases more usefully expressed as&#xA;$$ x \;\; = \;\; \frac{2c}{-b \; \pm \; \sqrt{b^2 - 4ac}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;9.&lt;/strong&gt; To show that ${\mathbb Q}[\sqrt{2}]$ (i.e. real numbers of the form $r + s\sqrt{2}$ where $r$ and $s$ are rational numbers) is a field, a  &quot;rationalization of the denominator&quot; technique is useful when verifying the multiplicative inverse part of the definition of a field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;10.&lt;/strong&gt; Rationalizing techniques are useful to obtain non-radical forms for the general equation of a hyperbola and an ellipse directly from their geometric definitions. Related to this is the general idea of rationalizing an algebraic equation (say, for an algebraic curve or an algebraic surface -- see Cayley's 1868 paper &lt;em&gt;On Polyzomal Curves, otherwise the Curves&lt;/em&gt; $\sqrt{U} + \sqrt{V} +$ &amp;amp;c. $=0,$ which begins on p. 470 &lt;a href=&quot;http://books.google.com/books?id=-5dQAAAAYAAJ&quot;&gt;here&lt;/a&gt;, for some eye-opening stuff) and of solving radical equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;11.&lt;/strong&gt; It is easy to find a simple expression for the following sum if each denominator is rationalized:&#xA;$$\frac{1}{\sqrt{1} + \sqrt{2}} \; + \; \frac{1}{\sqrt{2} + \sqrt{3}} \; + \;\frac{1}{\sqrt{3} + \sqrt{4}} \; + \; \cdots \; + \; \frac{n}{\sqrt{n} + \sqrt{n+1}}$$&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-04-25T18:58:13.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1991" PostTypeId="2" ParentId="1383" CreationDate="2014-04-25T19:11:03.753" Score="11" Body="&lt;p&gt;I suggest having a look at the following book by John Stillwell: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/144192681X&quot;&gt;Naive Lie Theory&lt;/a&gt;. A fast-paced week or (if you're lucky enough... or believe it's worth it as I would) two weeks could be created by pulling the big ideas from chapters 1, 2, and 4. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue that this shouldn't be the first time the students are seeing groups like SO, SU, etc. Those should be introduced as soon as possible as examples of different groups in the beginning of the course (see, for example, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1133599702&quot;&gt;Gallian&lt;/a&gt;, chapter 2). Then as you near the end of your tour of group theory, you could introduce the ideas from Stillwell which would bring in concepts from Calculus (tangent space!) that might really interest those future mathematicians of yours. How often do you get to see analysis and algebra interacting at these stages?! Handwaving is your friend here, in my opinion. This is a sampling of the future awesomeness. &lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2014-04-25T19:11:03.753" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1992" PostTypeId="1" AcceptedAnswerId="1994" CreationDate="2014-04-25T19:12:56.930" Score="12" ViewCount="437" Body="&lt;p&gt;Next semester I plan to experiment with &quot;flipping&quot; a classroom, by assigning required readings from the textbook (perhaps with supplementary notes) so that lecture doesn't have to &quot;cover all the material&quot;, freeing up some class time for discussions, questions, group work, etc.  In planning the course, I have run into a meta-obstacle: I'm not sure how big the &quot;chunks&quot; of assigned reading should be and how and how frequently they should be assigned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose (for the sake of argument) that I want to cover one chapter of the book per week (they're short chapters).  Should I ask students to read the whole chapter for that week by Monday (say)?  Or should I break it up and assign section 1 for Monday, section 2 for Wednesday, and section 3 for Friday (say)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, when should I give out the assignments?  Should I give out a list of assigned readings at the beginning of the semester (recognizing that it may need to be updated as the class progresses)?  Should I tell them the next week's readings each Friday?  Or the next class's readings each class meeting?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I expect that probably the only general answer that can be given is &quot;it depends&quot;, but lacking any personal experience with this sort of thing, I'd like to hear about the factors that ought to influence my choice.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2014-04-25T20:21:03.813" Title="How to divide up readings in a flipped classroom?" Tags="|course-design|textbooks|interactive-teaching|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1993" PostTypeId="2" ParentId="1992" CreationDate="2014-04-25T20:18:31.983" Score="9" Body="&lt;p&gt;This depends a fair amount on the class and the textbook.  I'm going to answer for the only sort of class I'm familiar with, an early college level calculus course.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;First, as brendansullivan07 points out in the comments, reading a math textbook is a skill in its own right, and one that your students likely don't have yet.  You should indeed plan on teaching them how to read, and you should also have low expectations, especially early in the course, for what they'll actually get out of reading it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Relatedly, I'd recommend breaking it up day by day, as part of a general practice of giving a lot of guidance about what reading entails.  In other words, don't say &quot;Read Chapter 1 by Friday&quot;, say &quot;Read pages 182-186 for Monday.  When you're finished, you should know what the definition of a Blah is and be able to give two examples, and you should understand how to solve problems similar to Examples 1 and 2.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I'd recommend trying to announce reading in multiple ways---giving a list of readings at the beginning of the semester (updating if needed), and also announcing them either daily or weekly.  Many students do still have their schedule planned at a week by week level, so some of them will want to do their reading for the week in advance, or do both the Wednesday and Friday reading on Tuesday so they can participate in some activity Thursday evening or something like that.  So you want them to be able to get ahead if they want to.  On the other hand, if you only give them a list at the start of the semester, a bunch of them will lose it and be too shy to ask for a new copy (and forget to look at the website), so they won't know what the assignments are.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-04-25T20:18:31.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1994" PostTypeId="2" ParentId="1992" CreationDate="2014-04-25T20:21:03.813" Score="12" Body="&lt;p&gt;I've found it more natural, and more helpful for the students, to arrange the out-of-class work around the learning objectives you have for your instructional units rather than the number of pages in a book. There's not always a natural linear relationship between page numbers and concepts -- sometimes a concept that takes a couple of pages takes twice as long as simpler concepts that take up more space. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://chronicle.com/blognetwork/castingoutnines/2014/03/05/creating-learning-objectives-flipped-classroom-style/&quot;&gt;Here is a blog post I recently wrote&lt;/a&gt; about learning objectives and the design of pre-class activities. The tl;dr version is&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For each particular class meeting, decide what learning objectives you would like to see students address in their work. &lt;/li&gt;&#xA;&lt;li&gt;Order that list from lowest cognitive complexity to highest cognitive complexity using something like Bloom's Taxonomy. &lt;/li&gt;&#xA;&lt;li&gt;Decide where the cutoff should go that demarcates what students should be responsible for before class and what they should expect to see in class. Not all learning objectives for a given day are good material for pre-class work. &lt;/li&gt;&#xA;&lt;li&gt;Organize the reading or video watching or combination of these around the &quot;basic&quot; objectives that students are responsible for prior to class. Organize your class work around the more advanced objectives. &lt;/li&gt;&#xA;&lt;li&gt;Design good in-class work about the learning objectives and assess before, during, and at the end of class on how students are doing with those. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There's a link in my blog post to the finished product of a pre-class activity on the Chain Rule for my flipped calc 1 class, if you want to see the finished product. We use the book &lt;a href=&quot;http://faculty.gvsu.edu/boelkinm/Home/Download.html&quot;&gt;Active Calculus&lt;/a&gt; by my colleague Matt Boelkins (free to download) and a whole bunch of videos that a colleague and I made for calculus, &lt;a href=&quot;http://youtube.com/GVSUCalculus&quot;&gt;found on YouTube here&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-04-25T20:21:03.813" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1995" PostTypeId="2" ParentId="1458" CreationDate="2014-04-26T06:40:17.387" Score="2" Body="&lt;p&gt;As Michael E2 mentions, a next step could be numerical integration, introducing methods such as the rectangle rule, trapezoidal (trapezium) rule and Simpson's rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I have made this answer Community Wiki.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2014-04-26T06:40:17.387" CommentCount="0" CommunityOwnedDate="2014-04-26T06:40:17.387" ContentLicense="CC BY-SA 3.0" />
  <row Id="1996" PostTypeId="2" ParentId="1458" CreationDate="2014-04-26T11:28:23.413" Score="4" Body="&lt;p&gt;I would &lt;strong&gt;move integral as infinite sum&lt;/strong&gt; to the second topic, with crude numerical calculations, or by drawing a curve on graph paper and counting how many squares are covered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would &lt;strong&gt;add&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;techniques for numerical integration**, e.g. trapezoidal rule and Simpson's rule.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;integrating by computer, for both symbolic and numerical results.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;applications in physics:  distance from velocity, impulse from force, moments.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;applications in biology or economics, e.g. population growth as the integral of population changes, or GDP growth as the integral of changes in GDP.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;applications in statistics, e.g. normal probabilities as the area under the normal distribution.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I would &lt;strong&gt;remove&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;integration by partial fractions, which I find unimportant and distracting for a course at the level of &quot;What are integrals?&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;integration by parts, for the same reason&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;volumes of revolution, if pressed for time, because it fits better in the multivariable class&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;improper intergals, if pressed for time, because students are unlikely to appreciate the subtleties&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;That keeps the total lectures at roughly the same length, with a different emphasis.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-26T11:28:23.413" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1997" PostTypeId="2" ParentId="1970" CreationDate="2014-04-26T12:47:15.977" Score="6" Body="&lt;p&gt;Give frequent quizzes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I say frequent, I mean every day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quizzes should be very brief: one basic problem from the lesson of the previous day. The grading will not be too onerous if the quiz has just one basic problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Give the quiz at the start of class, after briefly answering questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make the point value of the quizzes very small. Drop several of the lowest quiz scores at the end of the term. No matter how small the point value of the quizzes, students will take the quizzes seriously.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to instill good habits in students, you have to first decide exactly what good habits you want your students to have. Surely most of us agree that the best habit is for students to keep up with their work, and the worst habit is to fall behind and give up. Frequent quizzes help to instill the good habit of keeping up with their work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assigning homework is not enough by itself. If you assign homework but don't collect it, the good students will do the homework, and the other students will not. If you assign homework and do collect it and grade it, the good students will do the homework, and the other students will cheat off the good students. So in either case, the good students do the homework and the other students don't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By giving frequent quizzes, more students begin to realize the value of doing their homework and keeping up with their work. And it is not easy for students to cheat on quizzes, if you are a good proctor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No technique works perfectly to instill good habits in all students, but giving frequent quizzes is very valuable.&lt;/p&gt;&#xA;" OwnerUserId="306" LastActivityDate="2014-04-26T12:47:15.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1998" PostTypeId="1" AcceptedAnswerId="1999" CreationDate="2014-04-26T13:56:58.207" Score="11" ViewCount="303" Body="&lt;p&gt;I recently asked a question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1940/why-dont-all-teachers-use-clickers&quot;&gt;on the use of clickers&lt;/a&gt; and one comment I got was:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I have a very low-tech clicker technology I use in all my classes. When a student doesn't understand something I say, they raise their hand and ask a question.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But in my experience, this is not always true. In large classes I suspect many questions remain unasked and many people lose track of the subject due to a fear of &quot;looking stupid&quot; in front of a large crowd. In math classes I've taken, I've noticed that small classes (10-20 students) are more similar to a dialogue between students and teacher and large classes (100-200 students) are usually more of a monologue(or a dialogue between teacher and a very small subset of the class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We even thought the problem was so large that we made a web app to improve the teachers insight on how much students understood.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this phenomena known by other students or is this a cultural thing of Sweden where I come from? And if it's a known phenomena, what is the critical class size?&lt;/p&gt;&#xA;" OwnerUserId="1194" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-27T15:47:54.267" Title="Is there a critical class size for student participation (students perspective)?" Tags="|technology-in-education|class-participation|large-lecture|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1999" PostTypeId="2" ParentId="1998" CreationDate="2014-04-26T15:14:51.980" Score="7" Body="&lt;p&gt;In my experience, everyone participates in a class of 10.  By the time you get to 20, there's maybe a couple of students in the back who aren't really engaged, and class of 30 has a noticeable contingent of disengaged students.  I haven't taught any classes in the 40 - 70 range, but a class of 80 really feels like a large lecture, with only a few students in the front able to participate.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-04-26T15:14:51.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2000" PostTypeId="2" ParentId="1998" CreationDate="2014-04-26T15:50:36.387" Score="4" Body="&lt;p&gt;Honestly, I have not seen a major difference in class participation in courses with as few as 10 students than in courses with as many as 100. To be clear, I'm referring to the tendency of students to ask questions when confused---certainly in larger courses there are more students not paying attention, but I don't think being in a smaller class would necessarily change their behavior much. I think it's fair to say that the percentage of students asking questions in my courses has not depended much on class size, I think the reason being that I try to make all of my students feel very comfortable in class, in order to eliminate the fear of &quot;looking stupid&quot;.&lt;/p&gt;&#xA;" OwnerUserId="18" LastActivityDate="2014-04-26T15:50:36.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2002" PostTypeId="2" ParentId="1983" CreationDate="2014-04-26T16:24:29.680" Score="3" Body="&lt;p&gt;There's an important distinction to make.  It's not clear to me if your friend is failing to really learn at all (reading about abstract topics, feeling satisfied and not doing any problems at all, but not really grasping the subject---I know I'm prone to this mistake), or if your friend is in fact learning in his preferred way, but is struggling with obtaining the recognition needed to proceed to the next academic step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since you only raise the latter topic, I'm only going to try to address that.  It sounds like your friend is in a difficult position because in order to advance academically, he needs to demonstrate a different set of skills than the ones he's actually good at (even though his actual skills may justify advancement).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The IMO is prestigious, but it also focuses on a relatively narrow kind of mathematical problem---the sorts of problems that require a great deal of cleverness but not a lot of deep theory.  So while doing well on the IMO is a good indicator of mathematical ability, not doing well doesn't mean much.  From what you say of your friend, it's not his sort of math, so it's unsurprising he's not motivated by it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, your friend's desire to only solve problems when he can prove the answer sounds fairly ordinary to me, and many professional mathematicians share that attitude.  (I'd much rather read about the logical foundations of geometry than solve geometry problems!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I imagine focusing on the suggestion that he's doing math wrong is counterproductive and causing him to dig in his heels.  It might be better to acknowledge that these questions aren't interesting or mathematically important to him, and that the source of motivation has to be doing something boring to get into ISI.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I should add that the attitudes he's showing can be taken too far---some amount of problem solving is necessary for understanding.  The details you gave don't reflect that, but perhaps he's doing that as well.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-04-26T16:24:29.680" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2003" PostTypeId="2" ParentId="1940" CreationDate="2014-04-26T17:15:27.720" Score="5" Body="&lt;p&gt;I really liked using clickers myself. Ultimately (and this was about 8 years ago) I only had two classes that used them for attendance and a daily quiz. Physics. The problem with them was that they were somewhat expensive, didn't always work, and if you forgot yours you were SOL for that lecture's participation points. The one thing that stood out in my mind was that between the two physics classes we had to have two different clickers. They had changed the type of clicker used between the two classes even though they were parts 1 and 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I imagine if clickers were used more often they would require at lease some uniformity between departments and from the school's purchasing. I personally really did think they were a great idea. Perfect for instant feedback from students.&lt;/p&gt;&#xA;" OwnerUserId="1262" LastActivityDate="2014-04-26T17:15:27.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2004" PostTypeId="2" ParentId="1976" CreationDate="2014-04-27T13:23:38.770" Score="9" Body="&lt;p&gt;This answer/comment might be glib, but my motto is: &lt;em&gt;If it's not in your head then you can't think with it.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another image (I paraphrase) that is helpful in this regard is due to Alain Connes: If I fly from A to B then I see much less of the landscape than if I walked from A to B.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These things said, we should regard routine calculator use as analogous to skipping steps in an argument. If a person can easily fill in the steps when pressed, then the use of the calculator is probably fine. The trouble is, if a student has the ability to fill in the steps then it is often easier to do so (I am thinking of the many times I've seen students whip out their calculator to add zero or multiply by 1).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Intelligent use of technology is a different story, of course. Computers can be used to develop mathematical intuition. This is the best use of technology…to gain material with which to think. If technology is used to eliminate reflection then it is not good.&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2014-04-27T13:23:38.770" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2005" PostTypeId="1" AcceptedAnswerId="2008" CreationDate="2014-04-27T14:16:15.407" Score="9" ViewCount="479" Body="&lt;p&gt;I am looking for studies which compare students who did not receive mathematical education beyond basic mathematics and those that learned maths upto introductory calculus, with the assumption that both groups received similar education in other subjects such as social sciences and natural sciences uptil average high school standards. Has it been found that there is a quantifiable difference in understanding, analytical ability etc between the two groups? In other words, what evidence is there that learning maths beyond the basics has benefited them at the stage of just having completed high school?   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand a basic science curriculum in this case to include a little mathematics, which both groups should know, and for this purpose a notion of solving linear equations (and hence elementary algebra) besides arithmetic is included in basic mathematics. However, there is no trigonometry or geometry in a basic mathematics course, and in general a person learning basic mathematics knows no more then is the essential to understand basic science.&lt;/p&gt;&#xA;" OwnerDisplayName="user101" LastActivityDate="2014-04-27T15:32:26.570" Title="Studies which compare students with and without math training" Tags="|secondary-education|education-research|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2006" PostTypeId="1" CreationDate="2014-04-27T14:55:42.773" Score="11" ViewCount="255" Body="&lt;p&gt;Most discussions about teaching often assume that the learning outcome is the important variable (compare evaluations, discussions about clickers, discussions about syllabi, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I find that there are teaching methods that I would not employ for either long-term or even ethical reasons.&#xA;For an extreme example, let me point to the novel &quot;The Wave&quot; where the effective discipline would probably improve learning outcomes for many types of content but has certainly unethical implications and would hinder goals that are less easy to measure.&#xA;For another example, on the recent thread on clickers, someone commented that they would not want to use clickers to remove peer-pressure because at some point, students have to learn to deal with peer-pressure, anyway.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do you know any books or other resources that offer an analysis or a terminology for this problem?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am only aware of people trying to make the more ephemeral goals (like &quot;critical thinking&quot;) also measurable which might be quite hard for &quot;maturity&quot; and similar things.&#xA;However, in a discussion it might already be helpful to be able to point to some studies on this issue that teaching focussed on measurable short-term goals does not provide the best long-term outcomes.&lt;/p&gt;&#xA;" OwnerUserId="43" LastEditorUserId="143" LastEditDate="2014-06-21T20:11:07.647" LastActivityDate="2014-06-30T19:06:39.587" Title="How to balance short-term learning outcomes with long-term goals and ethical considerations" Tags="|reference-request|general-pedagogy|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="2007" PostTypeId="2" ParentId="1860" CreationDate="2014-04-27T15:04:49.900" Score="5" Body="&lt;p&gt;&lt;strong&gt;Why assume that your students &lt;em&gt;should&lt;/em&gt; learn to rationalize the denominator?&lt;/strong&gt; It's true that there are instances when rationalizing a denominator is helpful, but my guess is that in your college algebra context you're probably trying to have your students write $\sqrt{2}/2$ instead of $1/\sqrt{2}$ for their final answers. As others have pointed out, rationalizing the denominator was important back before calculators, but nowadays both answers can be estimated as decimals just as quickly using a calculator. In other words, $\sqrt{2}/2$ isn't really a &quot;simplified&quot; form of $1/\sqrt{2}$ like it used to be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My suggestion&lt;/strong&gt;: Let your students answer their questions with or without rationalizing the denominator. It may cost you a little extra effort when you grade answers, but also free up more of your class time to teach more important concepts.&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2014-04-27T15:04:49.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2008" PostTypeId="2" ParentId="2005" CreationDate="2014-04-27T15:08:33.217" Score="10" Body="&lt;p&gt;Course-taking beyond the level of advanced algebra (what is normally called &quot;Algebra 2&quot; in the United States) has been strongly linked to future academic and economic attainment; for example, a U.S. Department of Education study (Adelmann, 1999) found that students who complete mathematics coursework beyond the level of a course in advanced algebra are more than twice as likely to pursue and complete a postsecondary degree.  Similarly, Horn and Nunez (2000) report that students whose parents never attended college are more than twice as likely to enroll in four-year colleges if they take courses beyond advanced algebra; and Carnevale &amp;amp; Desrochers (2003) found that “Algebra II is the threshold mathematics course taken by people who eventually get good jobs in the top half of the earnings distribution”, despite the fact that relatively few actually make use of that mathematical knowledge on the job.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I realize that these studies focus on economic outcomes, rather than &quot;understanding, analytic ability etc&quot; as you asked, but I think this is the closest you are likely to get in terms of measurable benefit to students.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However -- and this is a huge caveat -- these results show a &lt;em&gt;correlation&lt;/em&gt;, not &lt;em&gt;causation&lt;/em&gt;, and the question you ask is probably impossible to answer.  That is because course-taking beyond the basic level is a &lt;em&gt;consequence&lt;/em&gt; of many factors, like intelligence, socioeconomic status, etc., and so it is unsurprising that the students who take those advanced courses go on to have better outcomes -- they were already having better outcomes before they took the courses, which is why they took them in the first place!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To actually demonstrate that learning higher-level math has a beneficial outcome you would need to do some kind of controlled experiment in which kids were randomly assigned to one of two educational tracks, and see what happens.  Research like that would be (obviously) unethical and hence cannot be done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, there is a kind of naturally-occurring version of this experiment unfolding in the US right now.  Several states have enacted new, tougher high school graduation requirements in which &lt;em&gt;all&lt;/em&gt; students must take and pass Algebra 2 in order to graduate.  In most states, there is no such requirement.  So in principle one could study the outcomes in those states and try to see if those new policies lead to discernible effects in their graduates, as contrasted not only with other states but also with the status quo ante.  (Algebra 2 is perhaps not as high-level as you have in mind, but I don't think there would be much support for requiring all students to take Calculus in order to graduate high school.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edited to add:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Afterthought:  The hypothetical study I described in my last paragraph would probably not show much if any effect, for the following reason:  Faced with a new obligation to get all students into, and successfully through, a course that large numbers of them would not have taken in previous years could reasonably be expected to have the unintended consequence of the course being somewhat watered-down.  There's no evidence yet that this has happened, but it's not implausible to expect the system to accommodate the new requirements in ways that the designers of those requirements were not hoping for.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2014-04-27T15:32:26.570" LastActivityDate="2014-04-27T15:32:26.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2009" PostTypeId="1" CreationDate="2014-04-27T15:29:45.320" Score="8" ViewCount="834" Body="&lt;p&gt;I am a junior math major in NY and will apply to PhD programs next year. I am terrified about getting rejected as I have not yet put any serious thought into a backup plan. I've tutored high school mathematics for the past two years and very much enjoyed this so I wouldn't mind becoming a (private or public) high school math teacher in the case of univesral rejection from graduate schools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I only recently began considering this option. I haven't taken any education courses and do not know a single thing about the certification process for high school teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is it too late to prepare for this possibility? If not, what should I do to prepare?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="1269" LastEditorUserId="2083" LastEditDate="2019-09-23T09:12:50.000" LastActivityDate="2019-09-23T09:12:50.000" Title="Devising backup plan to become HS teacher as a late undergrad" Tags="|teacher-preparation|usa|certification|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2010" PostTypeId="1" AcceptedAnswerId="2020" CreationDate="2014-04-27T15:45:32.023" Score="14" ViewCount="763" Body="&lt;p&gt;This summer, I am teaching 8-week calculus courses that meet a total of 16 times in &lt;strong&gt;210 minute class periods&lt;/strong&gt;. This sounds insane to me since &lt;a href=&quot;http://www.imdb.com/title/tt0167260/&quot;&gt;Return of the King&lt;/a&gt; is only 201 minutes long, but it is the plan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that when the class period is this long, it will be essentially impossible to &quot;cover&quot; 1/16 of the course material in a class period, since so much of the material builds on previous material. When the fourth(!) hour of class begins, the material from the first hour will be very difficult for students to access.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can a calculus course be adapted to this format? I would like answers from people who actually have experience teaching calculus in this kind of extreme fast-paced format.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-04-27T22:06:33.333" Title="How is calculus possible in a small number of very long class periods?" Tags="|course-design|calculus|homework|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2011" PostTypeId="2" ParentId="1998" CreationDate="2014-04-27T15:47:54.267" Score="3" Body="&lt;p&gt;Small classes are higher courses, which people take by choice, not forced by a sequence of required ones. The difference in maturity and drive makes a large difference in the attitude towards the course and thus their active participation. Having taught both large and small, upper and lower level, part of the speciality and complementary classes, I have seen this difference personally many times. Unless you find some study that separates out such effects, I'd be very wary of any conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And there is the at first sight surprising fact that the &quot;personality&quot; of a class or a whole generation of students is defined mostly by a few driving personalities. Win them over to be active and ask in class, the class is active; if they aren't interested or hostile, you won't go anywhere.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-27T15:47:54.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2012" PostTypeId="2" ParentId="2009" CreationDate="2014-04-27T15:56:35.833" Score="12" Body="&lt;p&gt;Undergraduate students should be aware of their academic advising offices. Essentially every school employs full-time academic advisers to answer questions precisely like this one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you go to a larger school, the math department (or at least the college of education) at your school will have a professional adviser just for that content area. You need to talk to this person immediately. Designing backup plans is the adviser's favorite pastime (I know, because I was one for three years). Here are some tips for interacting with your math adviser:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do not send an email and assume that it will be worked out that way. As soon as you send an email about a topic, it becomes impolite to appear in their office or call them about the same thing. Telling an adviser &quot;I emailed you a couple days ago, but it turns out I would like an appointment&quot; sounds like an accusation that they did not answer their email fast enough. This will not be received well. &lt;strong&gt;Get an appointment to talk to them from the start.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;If something they say to you does not make sense, ask a question. &lt;strong&gt;Do not pretend to know what is going on.&lt;/strong&gt; When they ask &quot;You know about applying for graduation, right?&quot; you need to stop and think about it before answering &quot;yes,&quot; regardless of whether you &quot;already should know&quot; the answer.&lt;/li&gt;&#xA;&lt;li&gt;If you suspect that the adviser does not actually know anything about your question, &lt;strong&gt;make it easy for them to refer you to someone else&lt;/strong&gt;. You can actually open the conversation with &quot;I'm not sure you are the right person for this, but I am hoping you will know who is the best person for me to ask.&quot;&lt;/li&gt;&#xA;&lt;li&gt;If the adviser begins to scold you for your lack of planning, &lt;strong&gt;respond as an adult&lt;/strong&gt;. Advisers are used to dealing with children. A constructive response t osomething you perceive as scolding is a very calm and collected &quot;I agree that additional prior planning would have been good, but unfortunately I am currently in this position. I'm not expecting everything to work out perfectly, but I hope you can help me make the right decisions from here on out given my current situation.&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-04-27T15:56:35.833" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2013" PostTypeId="1" CreationDate="2014-04-27T17:08:34.977" Score="14" ViewCount="3716" Body="&lt;p&gt;&lt;strong&gt;What is the most efficient way to create neat and consistent geometric drawings which I can insert into tests, worksheets, etc.?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are my primary concerns:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) A simple drawing should be quick to &lt;em&gt;draw, label, and insert&lt;/em&gt; into a document.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) A drawing should be &lt;em&gt;well formatted&lt;/em&gt; so that the lines are the right thickness, and the picture is cropped correctly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) A drawing format should be &lt;em&gt;consistent&lt;/em&gt; so that all drawings I put into a document look like they were drawn the in same way (like in any decent textbook) and not like they come from 17 different places.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At present I use GeoGebra. I like it because it is free and very easy to use, but I doubt it is very efficient, and I especially dislike the way the pictures tend to blow up in size when I insert them into a document. Perhaps someone who uses GeoGebra can provide insight into how to best format GeoGebra so that the drawings fit well into Word documents.&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="143" LastEditDate="2014-04-27T17:43:24.457" LastActivityDate="2022-07-16T11:49:43.110" Title="How do I Create Geometric Drawings Efficiently?" Tags="|technology-in-education|presentation|software|lecture-notes|" AnswerCount="7" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2014" PostTypeId="2" ParentId="2013" CreationDate="2014-04-27T17:32:06.343" Score="5" Body="&lt;p&gt;I use &lt;em&gt;Adobe Illustrator&lt;/em&gt;, but my school has a license so I don't have to purchase it myself. It is extremely versatile, but there is a learning curve. One can paste drawings and images created by other software into Illustrator, and then control many aspects in Illustrator before exporting into PDF or JPEG or whatever format is best for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one example, rather plain, but I hope effective:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/TBpxh.png&quot; alt=&quot;Fig.5.15 from DCG&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-04-27T17:32:06.343" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2015" PostTypeId="1" AcceptedAnswerId="2016" CreationDate="2014-04-27T18:02:48.830" Score="8" ViewCount="1281" Body="&lt;p&gt;In an introductory trignometry course, there are many options for introducing trigonometric functions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;As ratios of sides of right triangles&lt;/li&gt;&#xA;&lt;li&gt;As coordinates (or ratios of coordinates) of intersections of the unit circle with rays from the origin&lt;/li&gt;&#xA;&lt;li&gt;As graphs that are periodic (and wavelike in the sin/cos case)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I was taught #1 first in high school, and then graphs. I saw #2 in college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel that #1 is the traditional secondary-education method of introducing trigonometric functions, which has the benefit of coming a year or two after Euclidean geometry in the United States.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel that #2 is traditional in more rigorous textbooks used in University level courses, where radians are used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Method #3 has the advantage that many students benefit from graphing calculators and visualize a function based on its graph.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Which of these three methods (or another unmentioned method like power series) would be best to introduce a college freshman with no math past algebra to trigonemetric functions with the goal of eventually covering the other two methods?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastActivityDate="2016-05-03T13:36:58.110" Title="Unit circle vs. ratios of right triangles vs. wave functions for introducing trig functions" Tags="|course-design|trigonometry|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2016" PostTypeId="2" ParentId="2015" CreationDate="2014-04-27T18:22:13.167" Score="7" Body="&lt;p&gt;&lt;strong&gt;Physics approach&lt;/strong&gt; (recommended, if the students have physics background or are simultaneously educated in the relevant physics):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Introduce the &lt;em&gt;operations&lt;/em&gt; as the coordinates of a point ($\sin$ and $\cos$) resp. slope of the lines ($\tan$ and $\cot$) moving in a uniform motion along a circle of radius 1. Best example: Earth moves (nearly) uniformly along a (near) circle of radius 1 (AU). Give them other examples as well (car going in a curve, clock hand going around the clock).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Reason:&lt;/strong&gt; Circular motions usually have more real life connection to the students than ratios of sides of triangles.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Afterwards, introduce the &lt;em&gt;functions&lt;/em&gt; via harmonic mechanical oszillations. That way, you keep to the context of motions.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-27T18:22:13.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2017" PostTypeId="2" ParentId="2013" CreationDate="2014-04-27T19:22:21.960" Score="7" Body="&lt;p&gt;I'm fond of &lt;a href=&quot;http://asymptote.sourceforge.net&quot;&gt;asymptote&lt;/a&gt;, but it is not geared towards geometric diagrams. For me it is important that it integrates well with LaTeX (uses the same fonts, even complete math formulas). I use it more for general drawing. Take a look at the gallery, there are several geometric diagrams.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-27T19:22:21.960" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2018" PostTypeId="2" ParentId="2010" CreationDate="2014-04-27T19:35:40.973" Score="8" Body="&lt;p&gt;Never been in such a bind teaching (as a student I had a full term compressed into some six weeks once, but I don't remember much of that), but I'd look for ways to split the subject &quot;sideways&quot;, I.e. make it into two or three concurrent courses on related subjects. So you can switch subjects and don't wear out your students (hopefully with a short break). Alternate between covering the material and solving problems using what was covered in groups or open discussion. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-27T19:35:40.973" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2019" PostTypeId="2" ParentId="1330" CreationDate="2014-04-27T20:59:59.373" Score="6" Body="&lt;p&gt;My answer goes a bit sideways: I would say that it is not &lt;em&gt;necessary&lt;/em&gt; to rigorously define the trigonometric functions in a rigorous analysis course that uses them. And in fact, this applies much more broadly (e.g. one can do a course on integration without using any particular integration theory).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, we never, ever define everything we use in courses. There always are black boxes, as we do not go back to founding all mathematics from axioms before many years of study (if ever). And this ok: of course we learn how to add and multiply numbers before we rigorously construct them. We should do rigorous proofs before having founded everything from axioms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An important point is to make as clear as possible what you do not prove, and what you ask your student to accept (until they know more and can clear it up). It is in principle possible to use any approach to trigonometric function: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Consider what you need from them and, more importantly, what kind of proofs you feel are most useful to show in your course, and choose the&#xA;  definition for trigonometric function accordingly.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If the chosen definition feels unnatural to you, or unrelated to previous definitions student have seen, then call it a &quot;starting property&quot; instead and ask your student to accept that sin, cos, etc. has this property and that it gives a characterization of the given function (of course, try to give them a feeling why this property is not unexpected given their current knowledge).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it would be a loss to get constrained into proofs or considerations that are not central to the course by a definition.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-27T20:59:59.373" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2020" PostTypeId="2" ParentId="2010" CreationDate="2014-04-27T22:06:33.333" Score="7" Body="&lt;p&gt;Concurring with @vonbrand, try to split the material &quot;sideways&quot;, so that the subsequent hours do not depend on supposed (instantaneous) mastery of the first hour, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've taught summertime classes that met two hours a day, five days a week, instead of what would have been a single hour on MWF, for a full term. When I was young and foolish, I thought that in the upper-division courses of this sort it would be &lt;em&gt;better&lt;/em&gt;, since there'd not be such gaps between episodes. But, in fact, exactly as the question anticipates, it was horrible. Not only is there the literal incapacity to adequately assimilate the first hour's material as prerequisite for the second... but there is another terrible problem, namely, few students are acquainted with such a pace and all it entails.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, many students, perhaps a majority, can't help but use their usual &quot;pace&quot; even in doubly or triply speeded-up courses. This includes procrastination at the very beginning, typically, ... which, while unwise in ordinary circumstances, creates an impossible situation in speeded-up instances. Even the students' &quot;catch-up&quot; chops are woefully inadequate for a speeded-up situation, since they are usually not-so-good at catch-up even at normal speeds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect all-the-worse if you don't actually meet with the students every day, because many will misinterpret the &quot;off-days&quot; as days to not think about your course, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, I'd suggest nearly-oppressive amounts of homework to get their attention on the off-days, or they won't take you (the course) seriously. And, again, in your three hours, as you anticipate, it would be foolish to hope for immediate assimilation. As @vonbrand suggests, making things not too-intensely depend on each other is highly desirable. (Thus, the traditional &quot;strict logical ordering&quot; is infeasible.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very unfortunate and impractical pedagogical situation, for sure.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-04-27T22:06:33.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2021" PostTypeId="1" AcceptedAnswerId="2028" CreationDate="2014-04-27T23:14:51.073" Score="22" ViewCount="8672" Body="&lt;p&gt;The Fibonacci numbers are one of the first sequences given as examples of sequences in many calculus textbooks as they have a definition that does not obviously have a closed form and they have many real-world applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that last part has been very controversial in recent years, with people maki g extravagant claims for and against its appearance in nature.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What claimed properties of the Fibonacci sequence (mathematical or in real life) are both factual and interesting to first year calculus students?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastEditorUserId="738" LastEditDate="2014-04-29T12:25:53.953" LastActivityDate="2017-03-27T04:32:05.847" Title="What interesting properties of the Fibonacci sequence can I share when introducing sequences?" Tags="|calculus|sequences|" AnswerCount="11" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2022" PostTypeId="2" ParentId="2009" CreationDate="2014-04-27T23:58:24.067" Score="6" Body="&lt;p&gt;1) I wouldn't ask &lt;strong&gt;&lt;em&gt;will I get in to graduate school&lt;/em&gt;&lt;/strong&gt;, but instead &lt;strong&gt;&lt;em&gt;will a graduate school admit me which is worth my time and money to attend?&lt;/em&gt;&lt;/strong&gt;  In some cases the answer will be yes and in some no.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To figure this out, you can ask graduate schools that admit you for statistics:  how many of their entering students graduate, how long recent graduates spent at the school, what recent graduates have done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These statistics are especially important because plenty of people encounter the graduate-school version of your situation:  &lt;strong&gt;&lt;em&gt;I am terrified about getting rejected from academic jobs as I have not yet put any serious thought into a backup plan&lt;/em&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;Private schools&lt;/strong&gt; will be an easier option than public schools for long-term employment without teaching credentials.  However, the public schools have usually offered better &lt;a href=&quot;http://nces.ed.gov/fastfacts/display.asp?id=55&quot;&gt;salaries&lt;/a&gt; and benefits.  Also, Teach For America and similar programs do not require teaching credentials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) You have many &lt;strong&gt;options outside of teaching&lt;/strong&gt;!  You have spent the past twenty years of life in educational institutions, but that may not be a reason to spend the next twenty doing the same.  Look at finance (accounting, banking, insurance, trading), data science (in marketing or politics or journalism), security (military or cryptographic), statistics, consulting.... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-04-28T00:15:07.790" LastActivityDate="2014-04-28T00:15:07.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2024" PostTypeId="2" ParentId="2009" CreationDate="2014-04-28T00:25:44.243" Score="3" Body="&lt;p&gt;On the website for schools in NY state (if you're hoping to stay there), there are many options suggested for alternate routes to certification that you may be able to pursue. However, these do have deadlines so you should certainly look at these sooner rather than later. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The website is here: &lt;a href=&quot;http://schools.nyc.gov/teachnyc/certification/alternatives.htm&quot; rel=&quot;nofollow&quot;&gt;http://schools.nyc.gov/teachnyc/certification/alternatives.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have to make an honest statement too that programs such as Teach for America are... less than spectacular for a variety of reasons, not the least of which is the disturbingly high rate of those having gone through leaving teaching rapidly. There are more math-focused programs as you can see. &lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2014-04-28T00:25:44.243" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2025" PostTypeId="2" ParentId="2021" CreationDate="2014-04-28T00:30:03.707" Score="3" Body="&lt;p&gt;&lt;em&gt;Proofs That Count&lt;/em&gt; by Benjamin and Quinn has some interesting statements about Fibonacci numbers. They establish a correspondence between Fibonacci numbers and the number of ways to tile an $ n \times 1$ strip with tiles that are either $ 1 \times 1$ or $ 2 \times 1$, so every claim they present has a visual/combinatorial interpretation. A lot of the statements relate Fibonacci numbers to binomial coefficients. Here is one statement from the book: four consecutive Fibonacci numbers generate a Pythagorean triple:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ (f_{\text{n-1}}f_{\text{n+2}})^{2} + (2f_{\text{n}}f_{\text{n+1}})^{2} = (f_{\text{2n+2}})^{2}  $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real appeal of the book for me isn't any particular statement but the correspondence the authors establish. I personally find it interesting, but I think PVAL's comment has a lot of merit. &lt;/p&gt;&#xA;" OwnerUserId="690" LastActivityDate="2014-04-28T00:30:03.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2026" PostTypeId="2" ParentId="2021" CreationDate="2014-04-28T00:37:06.333" Score="1" Body="&lt;p&gt;How about the fact that the n+1th number in the sequence divided by the nth number approaches phi? I always thought that was cool. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-04-28T00:37:06.333" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2027" PostTypeId="2" ParentId="926" CreationDate="2014-04-28T00:40:22.820" Score="9" Body="&lt;p&gt;Just to steer in a different direction. I think it's the way we learn mathematics (and I personally don't think this is a good thing). From a very young age, we learn to assume that everything is linear. In elementary school, we get problems like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If John paints one house in five hours and Mary paints one house in three, how long does it take them to paint one house together?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some kids will say: &quot;Well, I don't know. Maybe they have to spend some time to divide the work between them, and maybe they spend a lot of time bringing their stuff in. You can't really divide this work unless they both bring half their stuff, but then they would have to share their equipment and they may have to wait for eachother...&quot;. But no, according to the teacher and the textbook they are wrong, and the answer should be $\frac{15}{8}$ hours. This reinforces the idea that you should just make the assumption of linearity when the problem is complicated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case probably the only way to do an actual computation is to assume linearity. I think the despair of the teachers to show them that mathematics is useful has driven them to this kind of examples. But, it doesn't stop in primary school, linearity is almost always assumed. I think I even saw questions in mathematics and physics courses where you had to assume linearity to solve the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I think the easiest examples that I can think of have to with pressure and the speed of water streaming out.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now linearity is not a bad thing (even if the thing we are trying to compute isn't exactly linearly we can still use it as an approximation), but the hidden assumption of it is a bad thing. Students will get confused and assume things are linear when they don't. The sentence 'a manager is someone who expects that two women can deliver a baby in 5 months' comes to mind now (but actually, you can substitute most students for the manager, and all non-linear things for 'delivering the baby').&lt;/p&gt;&#xA;" OwnerUserId="915" LastEditorUserId="915" LastEditDate="2014-05-07T21:37:13.590" LastActivityDate="2014-05-07T21:37:13.590" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2028" PostTypeId="2" ParentId="2021" CreationDate="2014-04-28T00:56:47.587" Score="28" Body="&lt;ul&gt;&#xA;&lt;li&gt;Since $\varphi$ is rather close to the conversion rate between miles and kilometers, one can use the Fibonacci numbers to convert: if $f_n$ is the distance in miles, then $f_{n+1}$ is (roughly) the distance in kilometers. You can use this to facilitate a discussion about, first of all, the convergence of these ratios $\frac{f_{n+1}}{f_n}\to\varphi$, and also about how the accuracy of the conversion gets better with larger $n$. What is the &lt;em&gt;rate&lt;/em&gt; of convergence of the ratios? I think students will find these discussions interesting.&lt;/li&gt;&#xA;&lt;li&gt;Summation properties of the Fibonacci sequence can serve as quasi-interesting facts for the students, but they can also serve you as a way to gently introduce them to induction without really spending an entire unit on it. For instance, work with the students to first observe and then prove that $\sum_{k=0}^n f_k = f_{n+2}-1$. Discuss your proof and why it's necessary, why we can't just observe the pattern and say, &quot;Oh yeah, it keeps going, trust me ...&quot;&lt;/li&gt;&#xA;&lt;li&gt;If you have some students with computer science interests, you can talk about &lt;a href=&quot;http://en.wikipedia.org/wiki/Zeckendorf%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zeckendorf normal form of integers&lt;/a&gt;, wherein you encode an integer by expressing it as a sum of Fibonacci numbers. Amazingly, every integer has a &lt;em&gt;unique&lt;/em&gt; such representation where no two &lt;em&gt;consecutive&lt;/em&gt; Fibonacci numbers are used! Without the consecutivity, existence is trivial to prove but uniqueness is not guaranteed. This topic can facilitate a discussion of existence and uniqueness in mathematics, in addition to considerations of computational efficiency.&lt;/li&gt;&#xA;&lt;li&gt;This combinatorial idea is also mentioned by &lt;a href=&quot;https://matheducators.stackexchange.com/a/2025/80&quot;&gt;Noah's answer&lt;/a&gt;, but I'll modify it slightly and expand: $f_n$ is the number of ways to tile an $n\times 2$ chessboard with dominoes. You either take a tiling of an $(n-1)\times 2$ board and append a vertical domino, or take a tiling of an $(n-2)\times 2$ board and append two horizontal dominoes. This kind of simple counting argument can be followed by calculus students, and they might see how these kinds of sequences come from natural questions about &quot;How many ways are there to do $X$ with property $Y$?&quot; You could even expand on this and point out the appearance of Fibonacci numbers along the diagonals of Pascal's Triangle:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/2Kjz5.gif&quot; alt=&quot;Fibonacci numbers are sums of shallow diagonals of Pascal&amp;#39;s triangle&quot;&gt;&lt;/li&gt;&#xA;&lt;li&gt;Without getting too deeply into the general ideas, you can present the &lt;em&gt;generating function&lt;/em&gt; for the Fibonacci numbers. You could hold off on this until you talk about power series later on in the course, or later in the calculus sequence.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-04-28T00:56:47.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2029" PostTypeId="2" ParentId="2021" CreationDate="2014-04-28T03:15:53.270" Score="2" Body="&lt;p&gt;Unless it shows up naturally (or it's generating function makes a good example, ...) better leave it out. There are plenty of other topics, more easily related to the main subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You see from the other answers that there are lots of cool combinatoric interpretations, but the properly &quot;analytical&quot; are too deep for a first course. Sadly. They even have their own journal, The Fibonacci Quarterly.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-28T03:15:53.270" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2030" PostTypeId="2" ParentId="2021" CreationDate="2014-04-28T04:44:18.613" Score="16" Body="&lt;p&gt;I think the Fibonacci numbers &lt;em&gt;should&lt;/em&gt; be mentioned in a first-year calculus class, though they should not be discussed at length.  I wouldn't try to stress the real-world applications&amp;mdash;these range from convoluted to dubious, and aren't really the reason that Fibonacci numbers are famous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some good reasons to mention them:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;They are a famous sequence, and being aware of them is part of general mathematical literacy.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They provide an excellent example of a recursively defined sequence, another topic that deserves to be mentioned.  (My other favorite recursively defined sequence is the seqeunce defined by $a_1 = 1$ and $a_n = \cos(a_{n-1})$, which &lt;em&gt;converges&lt;/em&gt;.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They are related to many of the ideas that come up during the sequences &amp;amp; series unit.  Indeed, there are several places where Fibonacci numbers provide an interesting example of the material under discussion.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Here are a few examples supporting the last point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; Does the series $\displaystyle\sum_{n=1}^\infty \frac{1}{f_n}$ converge?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Answer&lt;/em&gt;: It turns out that $f_n$ goes to infinity geometrically as $n\to\infty$.  In particular,&#xA;$$&#xA;\lim_{n\to\infty} \frac{\varphi^n}{f_n} = \sqrt{5}&#xA;$$&#xA;where $\varphi$ is the golden ratio.  Thus the series converges by the limit comparison test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, you can use the ratio test since&#xA;$$&#xA;\lim_{n\to\infty} \frac{f_{n+1}}{f_n} = \varphi&#xA;$$&#xA;or the root test, since&#xA;$$&#xA;\lim_{n\to\infty} \sqrt[n]{f_n} = \varphi.&#xA;$$&#xA;The advantage of bringing this up is to get the students to think about geometric growth a little abstractly.  That is, the Fibonacci numbers provide a nice example of a sequence that grows geometrically, but for which this isn't obvious.  I wouldn't attempt to prove any of these statements, I would just state them as empirical facts, and then make the connection to convergence of series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What's the radius of convergence of the series $\displaystyle\sum_{n=1}^\infty f_n x^n$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Answer.&lt;/em&gt; It's $1/\varphi$.  There are plenty of other interesting radius of convergence problems that involve an $f_n$ somewhere in the formula for the coefficient.  Again, the purpose of mentioning this would be to get the students to think about the relationship between the coefficient of $x^n$ and the radius of convergence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What's the sum of the series $\displaystyle\sum_{n=1}^\infty f_nx^n$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Answer.&lt;/em&gt; There's a neat trick for this.  Let&#xA;$$&#xA;f(x) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + \cdots&#xA;$$&#xA;Then&#xA;$$&#xA;x f(x) = x^2 + x^3 + 2x^4 + 3x^5 + 5x^6 + 8x^7 + \cdots&#xA;$$&#xA;and&#xA;$$&#xA;x^2 f(x) = x^3 + x^4 + 2x^5 + 3x^6 + 5x^7 + 8x^8 + \cdots&#xA;$$&#xA;By inspection&#xA;$$&#xA;f(x) \;=\; x + x f(x) + x^2 f(x)&#xA;$$&#xA;and solving for $f(x)$ gives&#xA;$$&#xA;f(x) \;=\; \frac{x}{1-x-x^2}.&#xA;$$&#xA;I like this trick, because it's a nice variation on the usual trick for finding the sum of a geometric series.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-04-28T04:44:18.613" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2031" PostTypeId="1" AcceptedAnswerId="2035" CreationDate="2014-04-28T09:26:45.563" Score="19" ViewCount="3370" Body="&lt;p&gt;While talking about different types of equations, a student in my class asked: &lt;strong&gt;Why we use the word quadratic to refer to second-degree equations?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is some context:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Linear ($x^1$) equations make sense, because the graph of a linear equation is a straight line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Exponential ($a^x$) equations make sense, because $x$ is in the exponent, and my students will soon learn to recognize graphs of exponential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Cubic ($x^3$) equations make sense, because $x$ is being cubed, and my students will soon learn to recognize graphs of cubic equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... so why do we use the word &quot;quadratic&quot; to refer to second degree equations? My students already know that the graph of a quadratic equation can be called a &quot;parabola,&quot; and the word QUADratic sounds like it has something to do with the number four, rather than squaring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Follow up question:&lt;/strong&gt; Would it be better to call $x^2$ equations &quot;parabolic&quot; referring to the shape they make when graphed? Or why not call them &quot;squared&quot; equations in the same way we use for higher-degree equations?&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2014-05-03T14:02:31.137" Title="Why use the word Quadratic?" Tags="|mathematical-pedagogy|student-question|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2032" PostTypeId="2" ParentId="2021" CreationDate="2014-04-28T09:50:20.150" Score="6" Body="&lt;p&gt;Our math teacher came with the fun fact that the sum of the first 10 terms equal to 11 times the 7th term no matter what starting numbers you choose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:&#xA;1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 11 * 13.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason it works with any starting numbers is because you can rewrite it as:&#xA;(a) + (b) + (a + b) + (a + 2b) + (2a + 3b) + (3a + 5b) + (5a + 8b) + (8a + 13b) + (13a + 21b) + (21a + 34b) = 55a + 88b = 11 * (5a + 8b)&lt;/p&gt;&#xA;" OwnerUserId="1281" LastActivityDate="2014-04-28T09:50:20.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2033" PostTypeId="2" ParentId="2031" CreationDate="2014-04-28T09:55:31.213" Score="27" Body="&lt;p&gt;It may be as simple as the fact that the word quadratic derives from the Latin &lt;em&gt;quadratus&lt;/em&gt; which means square. A &lt;em&gt;square&lt;/em&gt; has 4 sides, but the word &lt;em&gt;squared&lt;/em&gt; is used to mean &quot;to the power of 2&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer the follow on - No. Changing anything means re-writing the complete body of text to use the new designation, and un-learning the old definitions for all existing students. See discussion of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/530/pi-or-tau-how-should-the-circle-constant-be-taught&quot;&gt;tau vs pi&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-29T17:30:37.420" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2035" PostTypeId="2" ParentId="2031" CreationDate="2014-04-28T10:47:11.047" Score="21" Body="&lt;p&gt;The situation for quadratic equations is in one sense exactly analogous to the one for cubic equations, except that for reasons of the development of the language some discrepancy between square, squaring and related and quadratic arose.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not well placed to discuss this in English but see on the English.SE site for the etymology of &quot;quadratic&quot;: &lt;a href=&quot;https://english.stackexchange.com/questions/151217/why-does-quadratic-describe-second-power-while-quad-usually-describes-four&quot;&gt;Why does “quadratic” describe second power while “quad” usually describes “four”?&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, note that in other languages this split did &lt;em&gt;not&lt;/em&gt; happen. In German a square is a &quot;Quadrat,&quot; and to square is &quot;quadrieren,&quot; and quadratic is &quot;quadratisch.&quot; In fact from the link above it seems to quadrate &lt;em&gt;was&lt;/em&gt; used in English too (long ago).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus the naming linear, quadratic, cubic follows the logic line, square, cube (as said by Joe Taxpayer) only that the development of language took one of the terms slightly apart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would rather advise against using the terminology parabolic equation, because this in fact breaks the logic of the naming.&#xA;To add the &quot;s&quot; and to call them something like squared equations, as you propose, would seem relatively better to me, though also not good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to avoid any naming issues I suggest to call them first degree, second degree, third degree equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To sum this up:  Either explain the origin of the terminology to make the (hidden) logic clear or use a simple systematic naming. But avoid introducing the term parabolic in this context. &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="-1" LastEditDate="2017-04-13T12:38:45.993" LastActivityDate="2014-04-29T13:54:58.493" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2036" PostTypeId="2" ParentId="2031" CreationDate="2014-04-28T12:19:39.333" Score="13" Body="&lt;p&gt;When something has a standard name, use the standard name, especially when it's something so fundamental. Calling quadratic equations anything other than quadratic equations just means you end up with a class of students who become confused when they discuss quadratic equations with anyone who wasn't taught by you (i.e., almost everybody else on the planet).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As mentioned in the other answers, explaining to your students why the term &quot;quadratic&quot; is used, should help them to remember and understand it.&lt;/p&gt;&#xA;" OwnerUserId="1285" LastEditorUserId="1285" LastEditDate="2014-04-28T13:31:12.220" LastActivityDate="2014-04-28T13:31:12.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2037" PostTypeId="2" ParentId="2021" CreationDate="2014-04-28T12:39:20.083" Score="2" Body="&lt;p&gt;Possibly &lt;a href=&quot;http://primes.utm.edu/glossary/xpage/LamesTheorem.html&quot; rel=&quot;nofollow&quot;&gt;Lamé's theorem&lt;/a&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some extra characters because the system won't let me input only the prior paragraph. &lt;/p&gt;&#xA;" OwnerUserId="213" LastActivityDate="2014-04-28T12:39:20.083" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2038" PostTypeId="2" ParentId="2021" CreationDate="2014-04-28T14:06:55.850" Score="4" Body="&lt;p&gt;Some number theoretic properties: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For any number $N$ there exists a Fibonacci number divisible by $N$, which also means that for any $N$ there exists a Fibonacci number ending in $N$ zeroes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$F_n$ divides $F_m$ iff $n$ divides $m$ except for $F_2 =1$ which divides more stuff, this requires that you start with $F_0 =0$ and $F_1 = 1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Following a similar vein we get that $\gcd(F_n,F_m)= F_{\gcd(m,n)}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a corollary $F_n$ can be a prime number only when $n$ is prime. One might ask: Are there exist infinitely many prime Fibonacci numbers? Heuristics suggest that the answer should be yes but it is an open question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A slight variation of that with a different outcome is that $34$ is the only Fibonacci number that is twice a prime number.&lt;/p&gt;&#xA;" OwnerUserId="1286" LastActivityDate="2014-04-28T14:06:55.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2039" PostTypeId="1" CreationDate="2014-04-28T14:19:24.153" Score="7" ViewCount="340" Body="&lt;p&gt;Dynamic geometry software (DGS) has been around for decades; Geometer's Sketchpad (not the first, but probably the most well-known commercial product) has been around since 1986.  &lt;a href=&quot;http://en.wikipedia.org/wiki/Dynamic_geometry_software#GeoGebra&quot;&gt;Wikipedia lists more than thirty examples of the genre&lt;/a&gt;.  I know that in the math ed literature there have been tons of studies (thirty years' worth, going back to the original research on the Geometric Supposer) on how DGS &lt;em&gt;can&lt;/em&gt; be used (or misused) in teaching, on how students conceive of and interact with interactive diagrams, and on the factors (material and attitudinal) that constrain or enable teachers' adoption of this technology.  But I have not found a straightforward answer to the following question:  &lt;strong&gt;Just how many secondary geometry teachers actually use DGS, in some way, in their classrooms?&lt;/strong&gt;  Are we talking 20%?  50%?  Less than 3%?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own anecdotal experience seems to suggest that software of this type has only had limited penetration into the marketplace; that is to say, my impression is that most Geometry classes in most schools in the United States do not make any use of this software at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course &quot;use DGS&quot; can mean different things to different people.  I imagine there are several different categories of DGS use:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;DGS software is not used at all&lt;/li&gt;&#xA;&lt;li&gt;DGS software is used occasionally, but only by the teacher, either for demonstration purposes or for creating static diagrams to include on printed documents like tests and worksheets&lt;/li&gt;&#xA;&lt;li&gt;DGS software is used occasionally by the whole class, but such use is limited to special projects or activities and not tightly-integrated into the rest of the curriculum&lt;/li&gt;&#xA;&lt;li&gt;DGS software is used consistently throughout the year and is tightly-integrated into the rest of the curriculum.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Ideally I'd like data that not only gives a single number for market penetration, but also gives some idea about how those adopters of the technology actually use it, either along the lines of the list above or using some other categorization.  But frankly I'd settle for a simple number, even if it masks most of the important subtleties.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-05-05T20:00:29.957" Title="Adoption / penetration rates for dynamic geometry software in secondary school" Tags="|secondary-education|technology-in-education|geometry|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2040" PostTypeId="2" ParentId="2009" CreationDate="2014-04-28T15:37:12.883" Score="2" Body="&lt;p&gt;First, unless your credentials are &lt;em&gt;very poor&lt;/em&gt;, getting accepted shouldn't be the issue. Instead, it's getting financial support and making sure that you apply to at least one or two nearly &quot;sure bets&quot; for you. There are a lot of Masters only schools that you should have no problem getting money for, and many (most?) of these cater mainly for those intending to teach than to research or to work in industry. There are also Ph.D. granting schools of this type, which you can find by googling: &quot;Ph.D. in college teaching&quot; {AND} &quot;math&quot;. For your backup plan, I &lt;strong&gt;very strongly suggest&lt;/strong&gt; you go this route and then try high school teaching, rather than having your backup plan be teaching high school right away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This way, with at least a Masters in hand, you will be a much better candidate for schools such as the better private schools and those belonging to the &lt;a href=&quot;http://ncsss.org/membership/institutional-members&quot; rel=&quot;nofollow noreferrer&quot;&gt;National Consortium of Secondary STEM Schools (NCSSS)&lt;/a&gt;. Community college teaching can also be very rewarding (but don't plan on doing much research), especially if you can find a good fit for you. However, you will probably never come across really talented mathematics students while teaching community college, so if this is an important consideration for you, then you will want to focus on the kinds of high schools where such students can be found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it is of any interest, I've done (part of) the alternative certification route at a very low performing school (school was taken over by the state board of education) and I have also taught several years at an NCSSS school.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2019-09-22T20:15:53.227" LastActivityDate="2019-09-22T20:15:53.227" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="2041" PostTypeId="1" AcceptedAnswerId="2064" CreationDate="2014-04-28T20:29:41.533" Score="5" ViewCount="812" Body="&lt;p&gt;I'd like to process data like the following, from my most recent seven-question exam.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;       Item     1     2     3     4     5     6     7&#xA;PtsPossible     9    13    11    12    12    11    12&#xA;     Median     8     9     8     7    10    10    12&#xA;       Mean  7.10  9.10  7.25  6.65  9.27  9.39 10.57   &#xA;      Stdev  2.22  2.05  3.10  2.86  2.46  1.94  1.89          n = 51&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Essentially I'd like to know what this data table can tell me about my exam design.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read a very small amount about item analysis in general, but I lose patience with the authors immediately when I see things like: &quot;The question difficulty is defined as the percentage of students who selected the correct response.&quot; This definition makes sense if the question is asking for factual recall, but mathematics questions with partial credit are not like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What useful item analysis can be done post-exam, besides seeing which questions were the &quot;hardest?&quot;&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2014-04-28T20:38:29.663" LastActivityDate="2014-04-29T21:02:56.560" Title="Item Analysis in Mathematics Exams" Tags="|exams|test-design|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2042" PostTypeId="2" ParentId="2013" CreationDate="2014-04-28T20:48:30.897" Score="4" Body="&lt;p&gt;I personally use &lt;a href=&quot;http://ipe7.sourceforge.net/&quot; rel=&quot;nofollow&quot;&gt;IPE&lt;/a&gt; to draw diagrams for insertion into TeX documents. Some benefits about it that I like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It is free.&lt;/li&gt;&#xA;&lt;li&gt;It generates .pdf outputs, easily inserted into TeX files or usable/viewable on their own.&lt;/li&gt;&#xA;&lt;li&gt;It renders LaTeX text so you can label parts of diagrams easily.&lt;/li&gt;&#xA;&lt;li&gt;The &quot;grid snap&quot; feature is great; quoting from the site linked above: &quot;It is easy to align objects with respect to each other (for instance, to place a point on the intersection of two lines, or to draw a circle through three given points) using various snapping modes.&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It also addresses your three points:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I find it quite quick and easy to use, and it is especially easy to &lt;em&gt;jump right in&lt;/em&gt; without the need to learn a bunch of stuff beforehand.&lt;/li&gt;&#xA;&lt;li&gt;You can choose thickness of lines, and the output will automatically crop around only the area you drew in.&lt;/li&gt;&#xA;&lt;li&gt;It's easy to make things consistent across different diagrams. You can copy/paste parts into new files, use the same features every time, etc.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-04-28T20:57:09.637" LastActivityDate="2014-04-28T20:57:09.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2043" PostTypeId="2" ParentId="2031" CreationDate="2014-04-28T20:52:32.210" Score="4" Body="&lt;p&gt;I don't have any scholarly justification for this suggestion, but it seems very plausible that &lt;em&gt;quadratic&lt;/em&gt; equations are so called because they split the rectangle up into &lt;em&gt;quadrants&lt;/em&gt; as so&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/hOB5l.png&quot; alt=&quot;Quadratic&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously not all quadratics can be expressed in the form $(x+a)(x+b),$ but the ones that were considered solvable when the word quadratic was first being used (i.e. before the invention/acceptance of imaginary numbers) all could.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just to add to an already interesting debate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's an idea. Cubes are bounded by $6$ faces, so let's call $x^{3}$ a hexic. No, wait. That should be $x^{6},$ surely. In which case $x^{3}$ should be called... triadic? Or even better, $x^{4}$ could be called a tesseractic! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm only half joking here. The point is that trying to create too profound a link between the language used to describe things and the things themselves can result in linguistic or logical hiccups. Squares have a lot to do with the number four, it being the number of vertices, edges, lines of symmetry and number of rotational symmetries, but this breaks down for cubes, and so the language instead settled on extending the verb 'cubed' into the adjective 'cubic'. Poor language, it tries so hard to be non-arbitrary, but it was seduced by the 'fourness' of squares when, really, two is the more important number.&lt;/p&gt;&#xA;" OwnerUserId="901" LastEditorUserId="901" LastEditDate="2014-05-03T14:02:31.137" LastActivityDate="2014-05-03T14:02:31.137" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2044" PostTypeId="2" ParentId="2013" CreationDate="2014-04-28T21:06:18.603" Score="12" Body="&lt;p&gt;Over the years I have tried multiple tools including commercial and free, gui-based, script-based and LaTeX-based, vector, raster and 3D renderers. For the last three years I've mostly used&#xA;&lt;strong&gt;Inkscape&lt;/strong&gt; &#xA;(&lt;a href=&quot;http://inkscape.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;official website&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Inkscape&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia entry&lt;/a&gt;), which I strongly recommend.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Advantages:&lt;/strong&gt; easy to use, enough effects, free, versatile.&lt;br&gt;&#xA;&lt;strong&gt;Disadvantages:&lt;/strong&gt; not as streamlined as commercial products, a bit rough font handling.&lt;br&gt;&#xA;&lt;strong&gt;Stability:&lt;/strong&gt; it crashed a few times (~10) during 3 years of use, each drawing was successfully recovered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a great general vector program. Regarding geometry, it doesn't have features like GeoGebra (automatic constructions, linking points, etc.). On the other hand, the first picture below took about 5-10 minutes of work, which is not much. The middle one was easy, ~3 minutes, while the last was 30-40, because I couldn't get the curves right (these are not actual geodesics, i.e. were drawn by hand, not computer generated). For more complicated pictures, I sometimes use scripts to generate part of the drawing, and then finish by hand in Inkscape (the SVG being text-based helps a lot).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/2MvPn.png&quot; alt=&quot;geometry&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/yQBXd.png&quot; alt=&quot;automaton&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/sXEPs.png&quot; alt=&quot;cone&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should you wish to see more examples, take a look at these math.SE posts:&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/455953/26306&quot;&gt;gradient triangles&lt;/a&gt;,&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/272250/26306&quot;&gt;angles&lt;/a&gt;,&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/467010/26306&quot;&gt;grid&lt;/a&gt;,&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/707380/26306&quot;&gt;hexagonal grid&lt;/a&gt;,&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/446133/26306&quot;&gt;areas&lt;/a&gt;,&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/492387/26306&quot;&gt;simple graphs&lt;/a&gt;,&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/457863/26306&quot;&gt;more complex graphs&lt;/a&gt;,&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/350507/26306&quot;&gt;more complicated geometry&lt;/a&gt;,&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/439223/26306&quot;&gt;function&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding your points:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To insert into LaTeX, export to PDF (use &quot;File &gt; Save a copy&quot;), to insert into MS Word use EMF, for open office/libre office use SVG; if something does not work, you can always use a raster format like PNG (use &quot;File &gt; Export bitmap&quot;, use 300dpi for standard print, 1200dpi for quality print, 70-150dpi for computer screen).&lt;/li&gt;&#xA;&lt;li&gt;The biggest problem here is the picture size, and it is up to you to make it reasonable (whatever tool you use). Even with constant thickness of lines for each drawing, it might look bad if the pictures were created too big or too small. A way to get around this is to set the paper size &lt;em&gt;before&lt;/em&gt; drawing. To crop the final result I use &quot;File &gt; Properties &gt; Fit&quot; with a small margin for each side.&lt;/li&gt;&#xA;&lt;li&gt;For the diagrams to be consistent, &lt;em&gt;you&lt;/em&gt; have to be consistent. This is easy if don't change the settings too much. On the other hand, no tool will help you if you choose to use a different technique and colors each time.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-04-30T09:30:31.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2045" PostTypeId="2" ParentId="2031" CreationDate="2014-04-28T21:14:05.090" Score="9" Body="&lt;p&gt;&lt;strong&gt;Quoting &lt;a href=&quot;http://en.wikipedia.org/wiki/Quadratic_function#Origin_of_word&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The adjective quadratic comes from the Latin word quadrātum (&quot;square&quot;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summarizing succinctly:&lt;/strong&gt; Consider objects of side-length $a \in \mathbb{R}^{+}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a^1$ gives the size (&quot;length&quot;) of a &lt;em&gt;line segment&lt;/em&gt;, hence &lt;em&gt;linear&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a^2$ gives the size (&quot;area&quot;) of a &lt;em&gt;square&lt;/em&gt;, hence &lt;em&gt;squared&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a^3$ gives the size (&quot;volume&quot;) of a &lt;em&gt;cube&lt;/em&gt;, hence &lt;em&gt;cubed&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-04-29T03:15:58.917" LastActivityDate="2014-04-29T03:15:58.917" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2046" PostTypeId="2" ParentId="2013" CreationDate="2014-04-28T21:18:14.753" Score="5" Body="&lt;p&gt;Regarding this part of the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Perhaps someone who uses GeoGebra can provide insight into how to best format GeoGebra so that the drawings fit well into Word documents.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are two distinct issues you might be having, and I am not sure which one is your main problem.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When you export the Geogebra diagram to a picture (either via the clipboard or to a saved file) the output includes more of the canvas than you want, requiring you to crop the resulting image inside Word.&lt;/li&gt;&#xA;&lt;li&gt;When you paste the exported Geogebra diagram into the Word file it scales to fill the entire width of the page, requiring you to scale it back down to the size you want it to be.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Both of these problems are fairly easy to solve.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When you export a Geogebra diagram to a picture file or to the clipboard, the exported image contains everything that is currently visible within the Geogebra window.  When I work in Geogebra, I typically have my window filling the screen, but my actual construction is usually confined to a small portion of the window -- that way I have room to work in the margins as I construct and tweak things.  So when it is time to export, I have to remind myself to manually shrink the window down so that &lt;em&gt;only the portion of the diagram I want&lt;/em&gt; is visible.&lt;/li&gt;&#xA;&lt;li&gt;On the other hand if you are referring to the scale of the output, note that there is more than one way to export a Geogebra file to a graphic.  The simplest way is to use &quot;Graphics view to Clipboard&quot; (under the Edit menu), but this gives you no control over the size or scale of the output.  On the other hand if you use &quot;File &gt; Export... &gt; Graphics view as picture&quot; there is an explicit setting for the size and scale of the result.  So if you want the exported image to be only 6 cm wide, you can set it to be so.  Then when you paste the image into Word it ought to be just the size you want.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-04-28T21:18:14.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2047" PostTypeId="2" ParentId="444" CreationDate="2014-04-28T22:22:20.277" Score="2" Body="&lt;p&gt;the key is &quot;..have just learned about the definition of e or i&quot;. I think it would be total overload to introduce this equation at this point. I would save it for the end of the school year, making clear that this is not going to be on an exam. I like user1729's point about the trigonometric identity, especially when there is a path parameter like t and you can make things go around in a circle..&#xA;Also, I think it is important to talk about the power series expansion in this context. I find that part most beautiful-seeing the connection between the e-function and sine and cosine, seeing how two infinite series are combined into one.&lt;/p&gt;&#xA;" OwnerUserId="1150" LastActivityDate="2014-04-28T22:22:20.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2048" PostTypeId="2" ParentId="2013" CreationDate="2014-04-28T23:34:42.173" Score="4" Body="&lt;p&gt;Google Docs has a very convenient &quot;Create drawing&quot; option that has mist of the core functionality of Illustrator or Inkscape and requires no software download. I've used it for several papers and Wikipedia articles. As a total amateur, I find it easy to make lame drawings like this in 10 minutes: &lt;img src=&quot;https://i.stack.imgur.com/PtSiF.png&quot; alt=&quot;&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or this in 20 minutes: &#xA;&lt;img src=&quot;https://i.stack.imgur.com/V3Zm2.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-04-28T23:34:42.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2049" PostTypeId="2" ParentId="176" CreationDate="2014-04-29T00:02:25.023" Score="16" Body="&lt;p&gt;Yours is an important question in mathematics education, and especially in mathematics teacher preparation. Subject area content knowledge (as mentioned in the question) is of some importance, as well as generalized pedagogical knowledge (reflecting the general knowledge about what it means to be a good teacher.&lt;/p&gt;&#xA;&lt;p&gt;However, Shulman (1986) identified other forms of knowledge important to teaching. For example, Pedagogical Content Knowledge:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A second  kind  of  content&#xA;knowledge  is  pedagogical knowledge,  which  goes  beyond  knowledge&#xA;of  subject  matter per  se  to  the  dimension  of  subject  matter&#xA;knowledge  for  teaching.  I  still  speak  of  content  knowledge&#xA;here,  but  of  the  particular  form  of content  knowledge  that&#xA;embodies  the  aspects  of  content  most germane  to  its&#xA;teachability. (p. 9)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Knowledge of representation is an important form of PCK. For example, a mathematician may not need to know the sorts of visual models that help a 4th grader grasp aspects of ratio and proportion as a part of their learning fractions (as opposed to teaching them to manipulate fractions).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Within  the  category  of  pedagogical  content  knowledge  I&#xA;include,  for  the  most  regularly  taught  topics  in  one's  subject&#xA;area,  the  most  useful  forms  of  representation  of  those  ideas,&#xA;the  most powerful  analogies, illustrations,  examples,&#xA;explanations,  and  demonstrations—in  a  word,  the  ways  of&#xA;representing  and  formulating  the  subject  that  make  it&#xA;comprehensible  to  others. (p. 9)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is not &lt;em&gt;just&lt;/em&gt; knowledge of students. There is legitimate mathematical content knowledge that is more important for teaching than it is for doing mathematics. For instance, few mathematicians have a reason to distinguish between quotative and partitive division. Yet they are distinct mathematical conceptions of division that also happen to make it easier to understand how young students see division.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.learner.org/series/learning-math-number-and-operations/meanings-and-models-for-operations/meanings-and-relationships-of-the-operations-40-minutes-addition/&quot; rel=&quot;nofollow noreferrer&quot;&gt;This page goes into some detail on the difference between these conceptions, including a video example of work with a student.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Shulman also discussed &amp;quot;Curricular Knowledge&amp;quot; which specifically refers to instructional materials (and would be of little interest to experts in the subject area).&lt;/p&gt;&#xA;&lt;p&gt;Other researchers focusing on teacher knowledge have broken the categories down further. Hill, Ball, and Schilling (2008) proposed an overall domain map known now as &amp;quot;MKT&amp;quot; (Mathematical Knowledge for Teaching). There is some evidence linking forms of knowledge described in this model to classroom practice (Shechtman, Roschelle, Haertel, &amp;amp; Knudsen, 2010). It breaks down into many, very specific, categories that researchers continue to investigate (Common Content Knowledge, Specialized Content Knowledge, Knowledge at the Mathematical Horizon, Knowledge of Content and Students, Knowledge of Content and Teaching, and Knowledge of Curriculum).&lt;/p&gt;&#xA;&lt;p&gt;Even more recently, Schoenfeld has asked questions about how teachers make decisions (2010) and how we should observe the quality of a particular mathematics classroom based on what we see going on there (2013). The latter has resulted in a theoretical model of classroom quality involving multiple dimensions that are related to what he sees as going into educational decision-making (teacher knowledge, teacher orientations, routines, etc.). Not surprisingly, the classroom itself is quite a bit more complex than what goes into teacher decision making.&lt;/p&gt;&#xA;&lt;p&gt;In any case mathematical content knowledge is distinct from what makes a good math teacher. Others in their responses here have covered the idea that expertise sometimes makes it harder to actually bridge a communication gap in understanding (referred to by @adamblan as a blind spot). The differences go well beyond that, and even beyond knowledge of students and teaching. Some of the references here should help you delve into how researchers investigate (and have theorized about) the differences alluded to in your question.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Cited:&lt;/p&gt;&#xA;&lt;p&gt;Hill, H. C., Ball, D. L., &amp;amp; Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.&lt;/p&gt;&#xA;&lt;p&gt;Schoenfeld, A. H. (2010). How We Think: A Theory of Goal-Oriented Decision Making and Its Educational Applications. Studies in Mathematical Thinking and Learning Series. ERIC.&lt;/p&gt;&#xA;&lt;p&gt;Schoenfeld, A. H. (2013). Classroom observations in theory and practice. ZDM, 45(4), 607–621.&lt;/p&gt;&#xA;&lt;p&gt;Shechtman, N., Roschelle, J., Haertel, G., &amp;amp; Knudsen, J. (2010). Investigating links from teacher knowledge, to classroom practice, to student learning in the instructional system of the middle-school mathematics classroom. Cognition and Instruction, 28(3), 317–359.&lt;/p&gt;&#xA;&lt;p&gt;Shulman, L. S. (1986). &lt;a href=&quot;https://www.jstor.org/discover/10.2307/1175860?uid=3739696&amp;amp;uid=2&amp;amp;uid=4&amp;amp;uid=3739256&amp;amp;sid=21103937174967&quot; rel=&quot;nofollow noreferrer&quot;&gt;Those who understand: Knowledge growth in teaching.&lt;/a&gt; Educational Researcher, 15(2), 4.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorDisplayName="user7990" LastEditDate="2020-09-05T21:42:41.287" LastActivityDate="2020-09-05T21:42:41.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2050" PostTypeId="2" ParentId="2041" CreationDate="2014-04-29T00:50:55.643" Score="5" Body="&lt;p&gt;I've got no suggestions for item analysis on your presented data; stop reading if the rest is irritating because it doesn't answer your question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read a fair bit on test design, and the vast majority of it is based upon analysis of multiple-choice questions... which you clearly do not have. And, sadly, I don't think the total points you've shown will help give you the formative assessment you want and deserve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've done some analysis of short-answer biology questions, and the errors come down to things like a) not understanding the biology, b) understanding the biology but being unable to work through the logic needed to determine the answer, or c) understanding the biology and likely able to do the work but not understanding the question. Obviously, &quot;c&quot; requires immediate rectification (even when grading that exam), and the other two provide feedback for me to improve the next iteration of the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A classic element of test analysis compares the errors of top students to the errors of bottom students. How hard would it be for you (or a TA) to parse out where the points were lost for each question, particularly for the top 10 and bottom 10 students? That might be a shortcut to find the difference between simple unpreparedness and poor question-writing.&lt;/p&gt;&#xA;" OwnerUserId="1207" LastActivityDate="2014-04-29T00:50:55.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2051" PostTypeId="2" ParentId="2041" CreationDate="2014-04-29T07:23:17.473" Score="3" Body="&lt;p&gt;The comment of Mon Kee Poo raises an important point: The actual distributions of the points per item might offer some more insight than just the median and the standard deviation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if there is a 'hit or miss' question, the points will not be normally distributed, but rather bimodal, i.e. one portion of students will have almost 0 points and the other portion of students will have almost full points. Such situation can occur, for example, if a specific isolated technique is asked for, such as finding an integrating factor to solve a differential equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having this in mind, the mean and the sd. might not be the best way to discribe your distributions. In particular, you don't see if the results are skewed to the right or skewed to the left (although you can calculate the &lt;a href=&quot;https://en.wikipedia.org/wiki/Skewness#Relationship_of_mean_and_median&quot; rel=&quot;nofollow&quot;&gt;nonparametric skew&lt;/a&gt;, which gives an idea of the skewness, but is imprecise).&lt;/p&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-04-29T07:23:17.473" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2052" PostTypeId="2" ParentId="2021" CreationDate="2014-04-29T10:09:23.483" Score="11" Body="&lt;h1&gt;Mathematics&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;(People have already mentioned many of the great properties of this sequence at the time of answering. These are a couple of good ones that were missed.)&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Combinatorics&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The Fibonacci sequence is the number of strings of characters of length $n$, containing only &lt;code&gt;a&lt;/code&gt; and &lt;code&gt;b&lt;/code&gt;, where two &lt;code&gt;a&lt;/code&gt;'s cannot be consecutive. For $n=4$ there are $8$ strings for example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;abab abba abbb baba babb bbab bbba bbbb&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Relation&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Another nice factoid is that $F_{n-1} F_{n+1} = F_n^2 + (-1)^n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This relation, and lots of other nice ones, including the limiting behavior, follow from the closed form &#xA;$$ F_n = \phi_1^n + \phi_2^n $$&#xA;where $\phi_1$ and $\phi_2$ are the two solutions to $x^2 - x - 1 = 0$ (and the positive one of these is the usual 'golden ratio' $\phi$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solving the recursion to find this closed form is neat and not too complicated, although it does require fluency with algebraic manipulation, and some tricky concepts. If you can do this, you can see why all Fibonacci sequences, regardless of the first two numbers, have the same limiting ratio between two adjacent terms, and how all the closed forms are related.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't want to find the closed form, you can probably still obtain the relation, just using the recursive definition.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Phi&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The irrational number which can be least well approximated by fractions of a given size of denominator, is $\phi$. The best approximations at any size are given by $\frac{F_n}{F_{n-1}}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For most irrational numbers, if you try to find fractions which are close to that number, but whose denominator is not too big, you occasionally stumble across fractions which are much close than the 'average'. Eg $\pi$ can be written approximately as $\frac{3}{1}$, $\frac{22}{7}$, and then $\frac{355}{113}$. The last one is much, much better than you would hope for - the error is one over 5 million whereas you should statistically only expect an error of one in a few tens of thousands, for a denominator of that size.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With $\phi$ you never 'get lucky' - fractional approximations, even the best possible fractional approximations, always fall roughly the expected distance away from $\phi$. They also alternate between too big and too small.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To fully understand the math behind this, you have to learn about &lt;em&gt;continued fractions&lt;/em&gt;, which are extremely interesting, but probably off-topic for your class. The math is not very difficult, but it's not the kind of thing you can explain in a 10 minute parenthesis.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Pseudo-mathematics&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;A lot (perhaps most) of what is claimed about the Fibonacci sequence and the (very closely related) Golden Ratio is bogus. A nice paper is here: &lt;a href=&quot;http://community.dur.ac.uk/bob.johnson/fibonacci/miscons.pdf&quot;&gt;http://community.dur.ac.uk/bob.johnson/fibonacci/miscons.pdf&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some pointers:&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Natural world&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The Fibonacci sequence has a pretend real-world justification, in terms of rabbits reproducing. No-one sensible has ever claimed that these number were observed in real rabbit populations.&lt;/li&gt;&#xA;&lt;li&gt;The Fibonacci sequence does appear in some plant physiology, notably numbers of branches, petals and seeds for certain plants. There are some good reasons for this: some plants branch in simple ways which are analogous to the 'rabbit family trees'. &lt;/li&gt;&#xA;&lt;li&gt;Some of these patterns also involve the 'Fibonacci spiral', eg sunflower heads. This has a slightly more complex mathematical justification, which is not at all controversial.&lt;/li&gt;&#xA;&lt;li&gt;Lots of people have found Fibonacci numbers, the golden ratio and Fibonacci spirals in many, many natural objects. Most of these claims are either coincidence or deliberate (perhaps unconscious) mismeasurement (unless the universe works very, very differently to the way most scientists think it does).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h2&gt;Examining 'fringe' claims&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Some of these people are just on the wrong track, or doing something of interest to them which isn't quite mathematics or science, some of them are probably insane. It's often hard to tell papers written by these people apart from serious writing about the links between mathematics and say, zoology.&lt;/li&gt;&#xA;&lt;li&gt;Some of them might be right, and have spotted observed something about the mathematics of say, a zebra's stripes, that wasn't previously known to professionals. What is very unlikely is that all of these claims are true.&lt;/li&gt;&#xA;&lt;li&gt;The Golden ratio is probably the worst offender. If you make enough different measurements of parts of an object, you are pretty certain to find close approximations to any ratio you care to choose.&lt;/li&gt;&#xA;&lt;li&gt;Finding small values from the Fibonacci sequence (such as 5 or 8) is another common trick.&lt;/li&gt;&#xA;&lt;li&gt;Some writing which purports to find several occurrences of $\phi$ or the Fibonacci sequence in a given natural object, actually finds one such occurrence, and then transforms it in various simple mathematical ways and claims that these are independent observations.&lt;/li&gt;&#xA;&lt;li&gt;The ratio of the side to the diagonal of a pentagon is $\phi$. Natural objects with fivefold symmetry (some fruit, starfish) will probably contain the golden ratio without this meaning much, except that they have fivefold symmetry.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h2&gt;Art and culture&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Some historical and contemporary architects, artists and designers have intentionally used $\phi$ or the Fibonacci sequence. The most famous example is the Parthenon in Athens.&lt;/li&gt;&#xA;&lt;li&gt;Some of these people believed that these ratios and numbers were specially beautiful, some that they had some kind of mystic power, and some were playing around with interesting math.&lt;/li&gt;&#xA;&lt;li&gt;Some of the intentional use by artists might have degenerated into the belief that the numbers are present in physical objects. For example, artists might have chosen to use $\phi$ for the ratio of navel height to height when drawing figures, and this might have led people later to believe that this exact ratio actually occurs in all humans.&lt;/li&gt;&#xA;&lt;li&gt;As with natural objects, it has also been claimed that $\phi$ and Fibonacci appear in almost all buildings, paintings, sculptures and music in various disguised or non-disguised forms. Much of the 'evidence' of this works the same way as with natural objects - measure many different parts of a painting, and work out all the ratios until you find the ones you want.&lt;/li&gt;&#xA;&lt;li&gt;There is no evidence that our brain is wired to select these patterns unconsciously or to find them especially beautiful.&lt;/li&gt;&#xA;&lt;li&gt;Nor is there a scientific consensus that $\phi$ and Fibonacci are ubiquitous in all human endeavour at all times in history because of divine inspiration, secret knowledge handed down by aliens, or an extremely wide-reaching conspiracy. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you discuss this with your students you will probably hear a lot of the bogus claims above which have passed in the popular consciousness, partly as a result of 'The Da Vinci Code'. I think examining these is great math. Depending on how focussed on your syllabus you have to be, you might be able to discuss some or all of the following with your class:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How accurate does the ratio between a person's height and the height of their belly button have to be, before we can be sure that it is $1.618$ and not $1.638$ or $1.5922$? Is the related claim that the height of the belly button and the vertical distance between the top of the head and the belly button are in the same ratio further evidence, or is it redundant? How many people should we measure? How many daisies should we count petals of to decide if only Fibonacci numbers appear, and how close to the exact numbers do we need to be? What signs within a math paper and from its context should we examine to decide if it can be trusted? Is the distinction between mainstream math and pseudo-mathematics valid or is pseudo-mathematics just mathematics by people outside the mainstream?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might or might not be interested in talking to your class about this. I had some success when students in a Calc I class asked me about appearances of $\pi$ in Egyptian pyramids. However even if you don't want to go into it, you should say just enough when talking about this topic that your students understand that not every claim they might read is reliable.&lt;/p&gt;&#xA;" OwnerUserId="659" LastEditorUserId="659" LastEditDate="2014-04-29T14:42:30.550" LastActivityDate="2014-04-29T14:42:30.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2053" PostTypeId="2" ParentId="2021" CreationDate="2014-04-29T14:23:01.337" Score="7" Body="&lt;p&gt;If I were to cover the Fibonacci Sequence in introducing sequences to Calculus students, I would probably avoid some of the more obscure properties. I would focus on what early Calculus students should know: for example, how to prove that a sequence converges, and if it does, then how to find what it converges to. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note also the places, not necessarily specified below, in which one needs to remind students about the basic properties of sequences and limits, and when things can be &quot;split up&quot; only when convergence has already been demonstrated.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, one can show the ratio $\frac{f_{n+1}}{f_n}$ is monotonically increasing and bounded above by $2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is enough to conclude that the ratio converges to some limit as $n$ tends to infinity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Call this limit $\phi$. Then consider two ways of writing $\frac{f_{n+2}}{f_n}$ as $n \rightarrow \infty$. The first way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\lim_{n \rightarrow \infty} \frac{f_{n+2}}{f_n} = \lim_{n \rightarrow \infty} \frac{f_{n+1} + f_{n}}{f_n} = \lim_{n \rightarrow \infty} \frac{f_{n+1}}{f_n} + \frac{f_{n}}{f_n} = \lim_{n \rightarrow \infty} \frac{f_{n+1}}{f_n} + 1 = \phi + 1.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\lim_{n \rightarrow \infty} \frac{f_{n+2}}{f_n} = \lim_{n \rightarrow \infty} \frac{f_{n+2}}{f_{n+1}} \cdot \frac{f_{n+1}}{f_n} = \lim_{n \rightarrow \infty} \frac{f_{n+2}}{f_{n+1}} \cdot \lim_{n \rightarrow \infty} \frac{f_{n+1}}{f_n} = \phi \cdot \phi = \phi^2.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since these expressions are equal, we find that $\phi^2 = \phi + 1$, whence we can solve for $\phi$ using the quadratic equation. This will give two possibilities: one positive, and one negative. Noting that $\phi &amp;gt; 0$, we find the limit to which our ratio converges.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;On the other hand&lt;/strong&gt;, if you are looking for a somewhat nonstandard problem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Call a binary string &quot;tripletless&quot; if it never contains three consecutive $0$s or $1$s.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How many tripletless binary strings are there of length $n$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I worked this problem out some time ago, and found the answer is $2f_{n+1}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Length $1$: $\{0, 1\}$. Total: $2$, i.e., $2f_{2} = 2\cdot1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Length $2$: $\{00, 01, 10, 11\}.$ Total: $4$, i.e., $2f_{3} = 2\cdot2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Length $3$: $\{001, 010, 011, 100, 101, 110\}.$ Total: $6$, i.e., $2f_{4} = 2\cdot3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Length $4$: $\{0010, 0011, 0100, 0101, 0110, 1001, 1010, 1011, 1100, 1101\}.$ Total: $10$, i.e., $2f_{5} = 2\cdot5$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(If more details on the proof would be helpful then I would be happy to provide them, but I think it is a nice problem to work out. For pedagogically appropriate materials, though, I think the example above is a much better choice.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit (4 April 2016):&lt;/strong&gt; An irresistible mathematical side-note. The &quot;somewhat nonstandard problem&quot; here essentially asks for a count of&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;base $2$ strings of length $n$ without $3$ consecutive occurrences of the same digit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A natural generalization would be to ask for a count of&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;base $a$ strings of length $b$ without $c$ consecutive occurrences of the same digit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I asked this very question way back in &lt;strong&gt;Apr 2014&lt;/strong&gt; on MSE (&lt;a href=&quot;https://math.stackexchange.com/q/775863&quot;&gt;&lt;strong&gt;775863&lt;/strong&gt;&lt;/a&gt;) and it now has a nice &lt;a href=&quot;https://math.stackexchange.com/a/1726069&quot;&gt;&lt;strong&gt;answer&lt;/strong&gt;&lt;/a&gt; from &lt;strong&gt;Markus Scheuer&lt;/strong&gt; that uses generating functions and the &quot;Goulden-Jackson Cluster Method&quot; (&lt;a href=&quot;http://arxiv.org/abs/math/9806036&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;arXiv&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit (27 March 2017):&lt;/strong&gt; I recently explored with a class of mine the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$1+\frac{1}{1}, 1 + \frac{1}{1 + \frac{1}{1}}, 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1}}}, \ldots $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where each subsequent term is $1$ plus $1$ over the previous term. These yield&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{2}{1},  \frac{3}{2}, \frac{5}{3}, \ldots$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and, more generally, one can show that you get ratios of consecutive Fibonacci numbers. So the next one would be $8/5$, then $13/8$, etc. When you really do this out by hand, e.g., observing that $1 + 1/(8/5)$ is $1 + 5/8 = (8+5)/8 = 13/8$, you not only get to grasp &lt;em&gt;why&lt;/em&gt; the Fibonacci numbers are emerging, but by taking the &quot;limit&quot; of the sequence, you have that $x = 1 + 1/x$, which allows you to solve for $x$ and pick the positive root of this disguised quadratic to figure out precisely what the ratio of consecutive Fibonacci terms converges to.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2017-03-27T04:32:05.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2055" PostTypeId="2" ParentId="1914" CreationDate="2014-04-29T15:59:41.480" Score="5" Body="&lt;p&gt;There are many examples of games that children play (in the same way that kids might enjoy mazes) which have deep mathematics underlying them. An oft-cited example of this is the game of &lt;a href=&quot;http://en.wikipedia.org/wiki/Hex_%28board_game%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Hex&lt;/strong&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/t9iqf.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the board above, the red and blue players take turns putting down hexagons of their own color. Whoever first traces a connected path between his or her two sides (always opposite one another) wins. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example: blue has won the game above because there is a connected blue path going from one blue side to the other.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Can the game of Hex ever end in a tie?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The answer is &lt;strong&gt;no&lt;/strong&gt;, but if you have never seen this game before, then you might wish to pause and see if you can prove &lt;em&gt;why&lt;/em&gt; it cannot end in a tie.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an image of a game board without any tiles placed; more on the &lt;strong&gt;question&lt;/strong&gt; can be found below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/WfMZM.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have just tried for the first time to prove that the game of Hex cannot end in a draw, then you have probably realized that it is not so straightforward. A couple of independent proofs of this fact were found, one of which is due to John Nash (of &lt;em&gt;A Beautiful Mind&lt;/em&gt; fame). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting fact, familiar perhaps only to those who know of this game, is that one can use Hex to prove the &lt;a href=&quot;http://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brouwer fixed-point theorem&lt;/a&gt;! The citation for this proof can be found here:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gale, D. (1979). The game of Hex and the Brouwer fixed-point theorem. American Mathematical Monthly, 818-827. &lt;a href=&quot;http://www.jstor.org/stable/2320146&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-04-29T15:59:41.480" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2056" PostTypeId="2" ParentId="2021" CreationDate="2014-04-29T16:02:19.883" Score="5" Body="&lt;p&gt;The &lt;a href=&quot;https://www.youtube.com/watch?v=SjSHVDfXHQ4&quot; rel=&quot;noreferrer&quot;&gt;relation with squares&lt;/a&gt; is pretty nice:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\sum_{i=1}^n {F_i}^2 = F_{n} F_{n+1}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The video link shows how suitable this concept might be for consumption by a general audience.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2014-04-29T18:05:53.700" LastActivityDate="2014-04-29T18:05:53.700" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2058" PostTypeId="2" ParentId="1914" CreationDate="2014-04-29T17:55:27.027" Score="2" Body="&lt;p&gt;The game of &lt;a href=&quot;http://en.wikipedia.org/wiki/Nim&quot; rel=&quot;nofollow&quot;&gt;Nim&lt;/a&gt; is often used as introduction to binary numbers and $\mathbf{Z}\ /\ 2\mathbf{Z}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It uses the curious addition on $\mathbf{N}$ defined by $a+b=c$ iff $a_i+b_i=c_i \pmod 2$ for all $i$, where $a_i$ is the $i^\text{th}$ digit in the base 2 expansion of $a$.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorUserId="42" LastEditDate="2014-04-29T20:10:48.553" LastActivityDate="2014-04-29T20:10:48.553" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2059" PostTypeId="2" ParentId="1068" CreationDate="2014-04-29T18:50:52.357" Score="6" Body="&lt;p&gt;The motto I use for teaching is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Subsets are what you get when you throw elements away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quotients are what you get when you glue elements together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of $\mathbb{Z} / n\mathbb{Z}$, we glue together any pair of numbers if the number of steps to get from one to the other is a multiple of $n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like this interpretation better because you don't have to be as abstract, considering whole sets as your elements. It is, for newcomers to math, very difficult to imagine &quot;the set of all multiples of $n$&quot; as a concrete, tangible number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So instead, we just work with the representatives of the set as if they were the coset themselves. This is conceptually easier, now, because you never stop working with numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The cost of this, of course, is you have to take a second to explain that, when working with elements that are glued together, everything has to treat them as equal. For instance, if you want to look at all the &quot;even&quot; numbers in $\mathbb{Z} / 3\mathbb{Z}$, you run into an unexpected surprise: 2 is glued to 5, and if 2 is even, that must mean that 5 is also an even number! Why? Because 2 = 5 once you glue them together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So some ideas stop working when you glue. But other ideas keep working just fine. We know that 2 = 5 in $\mathbb{Z} / 3\mathbb{Z}$, so what happens if we add 1? Well, 2 + 1 = 3, and 5 + 1 = 6. But since 2 = 5, it better be the case that 3 = 6! And sure enough, the difference of 6 and 3 is a multiple of 3. So addition &quot;respects&quot; gluing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, multiplication does too. An example of this would be that 2 * 2 = 4 and 2 * 5 = 10, and sure enough, 4 and 10 differ by a multiple of three.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This style of presenting quotients is much less common, but I have personally found it more appealing.&lt;/p&gt;&#xA;" OwnerUserId="1018" LastActivityDate="2014-04-29T18:50:52.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2060" PostTypeId="1" AcceptedAnswerId="2065" CreationDate="2014-04-29T18:51:01.210" Score="46" ViewCount="32307" Body="&lt;p&gt;It's common for majors in biology to take calculus courses, and many calculus textbooks (and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1164/what-are-some-activities-projects-i-can-assign-to-calculus-students-from-bio-che&quot;&gt;calculus professors&lt;/a&gt;) try to cater to these students by including applications to biology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, &lt;strong&gt;in what specific ways is a calculus course actually helpful for biology majors?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, are there any courses typically taken by biology majors that involve ideas from calculus?   If so, what ideas come up?  Do any biology courses actually require students to take derivatives, compute integrals, or solve differential equations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also curious in what ways a two-semester calculus course could be made more helpful for biology majors.  For example, would it help to cover basic multivariable functions and partial derivatives? Applications of calculus to probability and statistics? Systems of differential equations? Fourier series?&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-12-06T20:08:27.390" Title="How is calculus helpful for biology majors?" Tags="|undergraduate-education|course-design|calculus|biology|" AnswerCount="16" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2061" PostTypeId="2" ParentId="2060" CreationDate="2014-04-29T19:30:07.853" Score="5" Body="&lt;p&gt;Differential equations are used to model e.g. predator/prey interactions in ecology, spread of diseases in epidemiology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much of (molecular) biology is chemical reaction kinetics, again calculus/differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[The above just as somebody with an interest in biology in general, no formal relation to the subject.]&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-29T19:30:07.853" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2062" PostTypeId="2" ParentId="2060" CreationDate="2014-04-29T19:46:19.847" Score="16" Body="&lt;p&gt;Most bio majors won't need calculus in their bio classes.  They will take chemistry classes in which understanding rates of change is useful, so:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;partial derivatives will help them.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;More importantly, many bio majors will work in quantitative areas in the life sciences, where data science is key&lt;/strong&gt;.  Think of developing drugs from chemical compounds, or clinical tests of drugs, or genomics.  A calculus class with this in mind will definitely include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The normal curve -- since the expression&#xA;$$\frac{1}{\sigma\sqrt{2\pi}}\Large e^{\Large-(x-\mu)^2/2\sigma^2}$$&#xA;and its integrals, which are ubiquitous in statistical thinking, will not become natural to them in any other way.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Transforming data with log and exp, e.g. reading log-log plots.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Different ways to visualize functions, e.g. contour graphs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-04-29T19:46:19.847" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2063" PostTypeId="2" ParentId="2060" CreationDate="2014-04-29T20:19:28.617" Score="15" Body="&lt;p&gt;I am not a biologist, and this question asks for the contribution of a biologist, nevertheless I might be contribute about the practice in our university in Budapest. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have a special two-semester calculus-type mathematics course for biologists developed together with biology departments. The curriculum is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;First semester:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;complex numbers, matrices, eigenvalues, Leslie model&lt;/li&gt;&#xA;&lt;li&gt;elements of one-and higher dimensional calculus (very quickly, mostly through examples)&lt;/li&gt;&#xA;&lt;li&gt;discrete dynamical systems&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Second semester:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;differential equations (mostly geometric theory with phase diagrams on computer), Lotka-Volterra model&lt;/li&gt;&#xA;&lt;li&gt;elements of probability theory&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This looks extremely quick for a mathematician but we have to solve somehow the problem that some parts of biology need deep mathematical results but there is no time to develop the theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later and in the master/PhD program they can choose specialized courses held by biologists about game theory in ecology and population models (based on Lotka-Volterra type models), disease transition or tumor growth models use heavy ODE theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt; Here are some links to Hungarian course materials (at least the literature is in English).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://ramet.elte.hu/~ramet/oktatas/populbiol.html&quot; rel=&quot;noreferrer&quot;&gt;population biology&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://ramet.elte.hu/~ramet/oktatas/evoljatek.html&quot; rel=&quot;noreferrer&quot;&gt;evolutionary game theory&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="696" LastEditDate="2014-05-01T15:37:25.347" LastActivityDate="2014-05-01T15:37:25.347" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2064" PostTypeId="2" ParentId="2041" CreationDate="2014-04-29T20:55:21.190" Score="6" Body="&lt;p&gt;To answer your direct question, not much. But, assuming you have available the analytical grades and not just the statistics:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the specific example, questions are distinguished  by their &lt;em&gt;maximum value&lt;/em&gt; &quot;PtsPossible&quot;. So they are different objects, because they have a possibly different range -the &quot;maximum points awarded&quot; are not necessarily the same (even though the variability is not large). Now, you may have a detailed analysis and discussion to offer about the &quot;difficulty&quot; of each answer (importance, length of requirements to answer, nature of the question etc etc), but &quot;&lt;a href=&quot;http://en.wikipedia.org/wiki/Frankly,_my_dear,_I_don%27t_give_a_damn&quot; rel=&quot;nofollow&quot;&gt;frankly my dear I don't give a damn&quot;&lt;/a&gt; :) -no matter what your considerations were, for my distant statistical eyes, you synthesized them and compacted them all in one single value - the &quot;PtsPossible&quot; per question.&lt;br&gt;&#xA;Although this ranking is important, for the moment, &quot;standardize&quot; your data to make the grades comparable:&#xA;Express the grading of each answer by each student as a percentage of the maximum points that the specific question gives. You have transformed your sample into &quot;relative achievement scores&quot; (or whatever you want to call it). Then recalculate the descriptive statistics of your sample. Now all your data range in 0-1 (or 0-100%), and now they are comparable.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now use the fact that each question has a different maximum value: order the questions ranking them by their &quot;PtsPossible&quot;, and see whether the descriptive statistics follow the same pattern  (e.g. the more &quot;valuable&quot; the question the higher the average relative achievement), or the inverse pattern, or no pattern at all. Although at this level of aggregation you have only 7 data points (the 7 &quot;PtsPossible&quot; and the 7 values of each descriptive statistic), you can calculate their correlation coefficient. Even better,  graph the 7-point series of PtsPossible pair-wise with each series of a descriptive statistic in a X-Y scatter-plot, to see visually whether there is a clear linear or non-linear association. If there is an association, how can you interpret it? E.g., if more valuable questions exhibit &lt;em&gt;lower&lt;/em&gt; variability in the grades (e.g. lower st.dev.), it is an indication that you have designed the test so that the students had a relatively easy day... Is that so? etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Create histograms with bars of length 0.05 or 0.1 for each question and look at them -are they unimodal, bimodal? Unimodal shapes will tell you that students are relatively more homogeneous regarding their knowledge/performance, compared to a question whose answer grades have a bimodal histogram. Does this accords with, or contradicts, any related opinion you had prior to the test?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Get &lt;em&gt;centered&lt;/em&gt; visual insights: subtract from all data points the relevant sample mean (per question), obtaining a zero-mean data sample. Create here too histograms, and graph together, not the bars, but all the lines that each connects the middle point of each bar (i.e a simple empirical &quot;probability density function&quot; for each question), for this data sample. You will immediately see whether they more or less &quot;look the same&quot; or whether some have a visibly different shape, &lt;em&gt;without&lt;/em&gt; being confused by possible different mean values. This will tell you immediately whether the distribution of the questions differ in terms of skewness, i.e. you will obtain visually the information that the comparison between mean and median usually gives.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you will be doing all these, your brain will start to combine the new information with the presumably good knowledge you have about the intricacies of each question, as well as about your students -and conclusions will emerge, usually richer than &quot;which question was the hardest&quot;. And richer conclusions require richer analysis.&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2014-04-29T21:02:56.560" LastActivityDate="2014-04-29T21:02:56.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2065" PostTypeId="2" ParentId="2060" CreationDate="2014-04-29T21:27:45.563" Score="42" Body="&lt;p&gt;I'm an old-school biologist (animal physiology) who works with mostly cell biologists. I sent out an email to a bunch of grad students and postdocs I work with. Here is the data so far:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Senior undergrad, pharmacology major: absolutely no calculus used in biology courses. She actually laughed when I asked her. &lt;/li&gt;&#xA;&lt;li&gt;Grad student: Undergrad &lt;strong&gt;biophysics&lt;/strong&gt; course used &lt;strong&gt;modeling with differential equations&lt;/strong&gt;. Graduate class in &lt;strong&gt;systems cellular biology&lt;/strong&gt; used modeling with differential equations.&lt;/li&gt;&#xA;&lt;li&gt;Grad student: Undergrad physical chemistry used calculus, no biology&lt;/li&gt;&#xA;&lt;li&gt;Grad student: none, other than watching some derivatives and integrals in an engineering-level physics. Suggests perhaps a course on bioinformatics might use calculus.&lt;/li&gt;&#xA;&lt;li&gt;Grad student: none. Suggests systems biology might have some.&lt;/li&gt;&#xA;&lt;li&gt;Grad student: none. Some algebra for bacterial growth curves.&lt;/li&gt;&#xA;&lt;li&gt;Postdoc: no actual calculus used, but calculus helpful for understanding diffusion of molecules in space&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I will add to the list (open-source data!) as emails come in, but it seems safe to say that calculus is rarely used by biology students outside of calculus class. &lt;/p&gt;&#xA;" OwnerUserId="1207" LastEditorUserId="1207" LastEditDate="2014-04-30T17:27:09.720" LastActivityDate="2014-04-30T17:27:09.720" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2066" PostTypeId="2" ParentId="1068" CreationDate="2014-04-29T22:31:47.643" Score="4" Body="&lt;p&gt;I think there are lots of ways of doing this. One good and interactive activity could be to divide the class into four 'teams' (or more) and pick one member from each team and line them up at the front. The remaining folks in each team take it in turn to say a number larger than 12 (or some larger number) and the lined up folks have to count up to that number always going from left to right. The team scores points if they get the &lt;em&gt;last&lt;/em&gt; number called out to land on their player. Eventually it should crystallise that it is only the &lt;em&gt;remainder&lt;/em&gt; that counts and that numbers $4n + r$ and $4m+s$ are essentially the same in this context if $r=s.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This could be extended by saying that now, instead of always starting from the leftmost player in the line-up, you carry on counting from the last player who spoke a number in each round. This should get them thinking about modular addition. I'm sure you could imagine loads of variations/extensions of this game. There are probably a few 'hacks' in this game where students can bypass thinking about modularity, but so long as there is a good discussion which explicitly brings these strategies to the fore afterwards I think it would be quite fruitful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some form of imagery such as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/sh8B4.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;could be good to bring the point home visually because the 'groups of four' just go together to make more groups of four. Depends on their level but this could quite easily lead to some algebra. [I know ~ is more standard than $\widetilde{=}$ in mathematics to show equivalence of elements in classes but I think having the equals sign there expresses the idea of 'equal in a sense' - let's not argue about it just change it if you don't like it!]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This being done, it could be great to lead them to discover a proof that the sum of an odd number and an even number is always odd using the language of modular arithmetic (some could be amazed when they realise that odds and evens, which they have been using forever, are just a special case of a higher concept - 'even' in the above example would mean divisible by four, and there are three 'types' of 'odd': remainder 1, remainder 2 and remainder 3).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just a few musings but I hope they might give you something. I'd be interested to know how you get on!&lt;/p&gt;&#xA;" OwnerUserId="901" LastActivityDate="2014-04-29T22:31:47.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2067" PostTypeId="2" ParentId="2060" CreationDate="2014-04-30T00:02:10.377" Score="8" Body="&lt;p&gt;An all-inclusive neurobiology class, which is normally appropriate for upper-division undergraduates, will present the physiology of excitable membranes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modeling at this level can be as simple as the Nernst equation for the equilibrium potential of a particular ionic species:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/6dGf4.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Nernst_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Nernst_equation&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By taking into account ion permeability, the Goldman–Hodgkin–Katz equation can be used to illustrate the reversal potential for a given membrane:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/N4feS.png&quot; alt=&quot;enter image description here&quot;&gt;&#xA;&lt;sup&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Goldman_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Goldman_equation&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Neither of these models uses calculus explicitly, but more advanced students (especially those who are interested in computational modeling) can be introduced to the Hodgkin-Huxley model:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/1Ou5V.png&quot; alt=&quot;enter image description here&quot;&gt;&#xA;&lt;sup&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As mentioned in some of the other answers, a thorough knowledge of statistics is incredibly useful to students pursuing undergraduate research or those with plans to continue their education, but the aforementioned example is an opportunity for students to directly employ differential equations-based models in the undergraduate biology curriculum.  &lt;/p&gt;&#xA;" OwnerUserId="16" LastActivityDate="2014-04-30T00:02:10.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2068" PostTypeId="1" CreationDate="2014-04-30T02:36:12.587" Score="6" ViewCount="184" Body="&lt;p&gt;How should I teach students the trigonometric ratios who are indeed studying trigonometry for the first time in their life? The course has said that there are six ratios based on sides in a triangle and that each ratios is given special names and that they need to memorize $\sin(\text{ref. angle})=p/h$ etc. Now They ask me what sine, cosine and tangents are? Now, what do I say them in order to make them understand what sine, cosine and tangents are?&lt;/p&gt;&#xA;" OwnerUserId="1172" LastEditorUserId="1172" LastEditDate="2014-04-30T11:49:02.947" LastActivityDate="2014-04-30T11:49:02.947" Title="Trigonometric ratios" Tags="|trigonometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2069" PostTypeId="1" CreationDate="2014-04-30T03:44:07.433" Score="26" ViewCount="2794" Body="&lt;p&gt;I have a background in math but no contact with secondary education or kids. I hear all sorts of stories ... horror stories mostly ... about the &lt;a href=&quot;//en.wikipedia.org/wiki/Common_Core_State_Standards_Initiative&quot;&gt;Common Core&lt;/a&gt; math curriculum in the USA. Then I hear that it will lead to more mathematical thinking so it's actually better. Or maybe that it's better for the smart kids but it's elitist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can I get some opinions from people who have actually seen or dealt with the Common Core material? I'm really wondering if this is the end of the world or the beginning of an improvement in the way we teach math.&lt;/p&gt;&#xA;" OwnerUserId="1314" LastEditorUserId="262" LastEditDate="2014-05-04T22:36:40.437" LastActivityDate="2014-07-17T18:32:48.330" Title="Common Core, threat or menace? Or maybe ok after all?" Tags="|mathematical-pedagogy|curriculum|usa|common-core|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2070" PostTypeId="2" ParentId="1896" CreationDate="2014-04-30T04:56:19.067" Score="2" Body="&lt;p&gt;I would say that &quot;trivial&quot; is okay, the others are probably less so. While &quot;trivial&quot; has connotations of &quot;exceptionally easy&quot;, in math speak we often use it to roughly mean &quot;an exceptionally simple statement, requiring little complexity to prove or define&quot;. I'd probably shy away from &quot;the proof is trivial&quot; a little more, but even so, if it's a tiny 3 line lemma, even if it's not easy and involves a huge leap of logic, it's still &quot;trivial&quot; by my metric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I view &quot;trivial&quot; like &quot;simple&quot;, and &quot;simple doesn't mean easy&quot; is practically a catch phrase of mine at this point. Things can be simple in that they only rely on two or three axiomatic facts, but actually thinking of which two or three axioms to use isn't always easy. It may pay off to remind students of this fact. &quot;It's not exceptionally complex, but can be hard to understand&quot; goes a long way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basic is the next I'd be okay with, but only in limited cases. I'd pretty much solely use &quot;basic&quot; to refer to things like so-called &quot;basic facts&quot; (2+2=4; you can't divide by zero, etc), or more generally literal axioms of the system you're working in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Obviously&quot; and &quot;clearly&quot; I'd avoid at all costs, unless &lt;strong&gt;maybe&lt;/strong&gt; you're saying &quot;2+2=4&quot;. Though I've been known to use &quot;obviously&quot; in humor, especially with contradictions (&quot;And, well, obviously 0=4 so...&quot;). It just alienates people who didn't get the intuition. Unfortunately, these two words are almost the proof-writer's version of &quot;um...&quot;. People don't notice they're saying them, they're filler words. I always try to proof read them out of anything I write, it actually tends to make things more readable in addition to not alienating the students who don't understand a concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps one tiny caveat is that sometimes look more complex than they are, and it can be useful to reassure students that this gnarly thing isn't really that scary. Newton-Raphson comes to mind as something that made my brain overflow when I first saw the formal definition of it in a textbook we were using. Being assured that, yes, it really is just this simple iterative process was helpful in making it less scary. See also: functions with 25 Greek letters where 24 of them are stupid constants.&lt;/p&gt;&#xA;" OwnerUserId="521" LastActivityDate="2014-04-30T04:56:19.067" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2071" PostTypeId="2" ParentId="2069" CreationDate="2014-04-30T06:00:35.763" Score="38" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I'm really wondering if this is the end of the world or the beginning of an improvement in the way we teach math.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'll answer this first, and then talk a little bit about the rest of the question. Please excuse my bluntness. I will try to give you the opinion you ask for by weighing what I think may be most important relative to quality of math education. Black and white is not my usual style.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you say &quot;Common Core&quot; you must, primarily, be referring to the &lt;a href=&quot;http://www.corestandards.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Common Core State Standards&lt;/a&gt; (CCSS). And in this context, the &lt;a href=&quot;http://www.corestandards.org/Math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Common Core State Standards for Mathematics&lt;/a&gt; (CCSS-M)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;No&lt;/strong&gt;, it is not the end of the world. &lt;strong&gt;Yes&lt;/strong&gt;, it is (or should be) the beginning of an &lt;em&gt;improvement in the practice of math education&lt;/em&gt; here in the US.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;WHY?&lt;/strong&gt; The shortest answer as to why this is definitely an improvement lies in the CSS-MP, the &lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot; rel=&quot;nofollow noreferrer&quot;&gt;practices portion of the Common Core State Standards&lt;/a&gt;. In short, the CCSS-M include a list of eight practices which, as a math person, you would likely recognize as good mathematical habits of mind related to the sense-making aspect of mathematics. Summarized, they are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Problem solving and perseverence&lt;/li&gt;&#xA;&lt;li&gt;Quantitative reasoning&lt;/li&gt;&#xA;&lt;li&gt;Mathematical argumentation (reasoning and proof)&lt;/li&gt;&#xA;&lt;li&gt;Mathematical modeling with data&lt;/li&gt;&#xA;&lt;li&gt;Strategic use of tools (representations, technology, etc.)&lt;/li&gt;&#xA;&lt;li&gt;Attention to appropriate precision in calculation, communication, claims, definitions&lt;/li&gt;&#xA;&lt;li&gt;Seek and make use of structure&lt;/li&gt;&#xA;&lt;li&gt;Generalize from repeated reasoning&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(Some may not like my summaries, so it's best just to look at &lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the descriptions themselves&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is the case that not all states had these practices (or the notion of an explicit focus on mathematical practices) in mind in their existing frameworks and standards. So, this standardizes the recognition that there should be an explicit focus on these aspects of mathematical thinking in mathematics education &lt;em&gt;rather than&lt;/em&gt; standards simply existing as a list of what content comes in what order and hoping that the students develop good habits of mind along the way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://vimeo.com/21317961&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here you can listen to Prof. Alan Schoenfeld briefly make a similar argument.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may not be the argument you hear most often in defense of the CCSS. There is the general defense of standardization, which is &lt;a href=&quot;https://vimeo.com/51933492&quot; rel=&quot;nofollow noreferrer&quot;&gt;described in this short video&lt;/a&gt;. Essentially, that &lt;a href=&quot;http://ies.ed.gov/&quot; rel=&quot;nofollow noreferrer&quot;&gt;IES&lt;/a&gt; noticed great variability in the quality of education from state to state. &lt;a href=&quot;http://nces.ed.gov/nationsreportcard/about/&quot; rel=&quot;nofollow noreferrer&quot;&gt;NAEP testing&lt;/a&gt; revealed that the states had varying expectations, the results of which were only uncovered in NAEP testing done across the nation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is an important issue in itself, but in the potential CCSS-M has for improving math instruction, I see the mathematical practices as their best contribution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;NOW&lt;/strong&gt; - These are &lt;em&gt;standards&lt;/em&gt;, not &lt;em&gt;curriculum&lt;/em&gt;. For political reasons, it was important to not dictate curriculum or specific pedagogy. Therefore, how a state adopts these standards, what curriculum they choose, and whether they put any effort toward supporting teachers in doing what is necessary to develop students' mathematical habits of mind is a local decision. And it is a decision with fiscal and other resource implications. There are some serious political implications here that I will not divert into. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suffice it to say, &lt;strong&gt;YES&lt;/strong&gt; to that part of your question, though states and districts will have to take the practice standards to heart, lead teachers to &lt;a href=&quot;http://map.mathshell.org/materials/stds.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;appropriate formative assessment lessons&lt;/a&gt; that both develop student mathematical reasoning and help teachers determine their students' progress in that development. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;But why do I &quot;hear all sorts of stories ... horror stories mostly ... about the Common Core math curriculum?&quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, the reason you're hearing &quot;Common Core math curriculum&quot; because there is curriculum that claims to be &quot;aligned with the common core.&quot; The actual problems you may see with this curriculum may have nothing to do with the standards. You may even be seeing problems arising from a misinterpretation of what the standards call for. Or a pedagogy that does little to encourage the practices, but it claimed to be aligned simply because it took a traditional approach and mapped it to the content standards.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, you could also be seeing an actual attempt to help students reason in some way aligned with one or more of the practices, but at an appropriate grade level. For instance, before students can do formal proving, they may be developing an understanding of the role of justification in mathematical reasoning, and engage in informal proving and argumentation. Students may well be asked to do things that their parents simply do not recognize as mathematics because they were never asked to, for example, reason quantitatively (Smith III &amp;amp; Thompson, 2008).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why have I maybe heard &quot;that it's better for the smart kids but it's elitist?&quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm actually not sure. I haven't actually heard this. It is possible that some are worried that higher standards, in some cases will cause test scores to temporarily drop. That may well be the case. But a period of transition may not be able to avoid this, and the point is really the learning, not the scores themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What other attacks are there on the CCSS?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Some see it&lt;/em&gt; as a &lt;strong&gt;states rights&lt;/strong&gt; issue. But you asked about education, and I think &lt;em&gt;that&lt;/em&gt; is the appropriate focus, unless it is a discussion among lawyers, which can happen elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Some want more &lt;strong&gt;local control&lt;/em&gt;&lt;/strong&gt; of the content standards. Today, a wise math ed researcher I know pointed out that the NCTM Principles and Standards for School Mathematics takes a wider grade-band approach, which would have given states, districts, and teachers more flexibility in their local use of the standards. My &lt;em&gt;personal&lt;/em&gt; opinion is that a desire to hold individual teachers accountable within a school year (i.e. the ability to use standardized assessments to apply a Value Added Model to teacher evaluation) is embedded in these standards and some of the standardized assessments that are being developed. This is something I wish they'd change about the standards (and something I think they &lt;em&gt;could&lt;/em&gt; change). Combine grades into bands and hand some flexibility to the states. But I'm not in charge of them. And I see the presence of the mathematical practices to be more significant in possibly improving instruction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Some are disturbed by all the &lt;strong&gt;standardized testing&lt;/em&gt;&lt;/strong&gt; that is going on. This is reflected in &lt;a href=&quot;http://truthinamericaneducation.com/common-core-assessments/what-states-have-pulled-out-of-their-common-core-assessment-consortium/&quot; rel=&quot;nofollow noreferrer&quot;&gt;some states that have withdrawn from the common core-aligned assessments&lt;/a&gt;. I will note that this is not &lt;em&gt;necessarily&lt;/em&gt; related to the question about improving math instruction, however, it definitely could be depending on how districts change their schools to prepare for tests. It is understandable that a burden of test preparation is a serious concern for teachers, and it is also understandable that it has so often been conflated with the standards themselves (we call them standardized tests, after all, and they are definitely related). However, getting rid of the CCSS-M wouldn't stop standardized testing. In that sense, attacking Common Core State Standards because of issues over testing is barking up the wrong tree. This is a bad thing because it won't lead to desired results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think there are no publications on this yet, but there are people &lt;a href=&quot;http://www.smarterbalanced.org/smarter-balanced-assessments/&quot; rel=&quot;nofollow noreferrer&quot;&gt;working on providing tests that score along multiple dimensions&lt;/a&gt; and help teachers gain an understanding of how to help their students rather than just one score that will be used to evaluate teacher performance. And there are people working to provide materials that help teachers &quot;teach to the test&quot; in a way that develops mathematical reasoning rather than just developing a higher score on the test. That sort of thing is possibly coming in the near future. But it depends on the type of tests states adopt. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Anything else?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes. There are people who fundamentally disagree with certain research-based approaches to math education, preferring a more traditional didactic approach. &lt;a href=&quot;https://matheducators.stackexchange.com/a/1461/759&quot;&gt;See my previous answer here&lt;/a&gt; for just one thread of research that shows how there are multiple dimensions to mathematics education, and implications in traditional approaches that we did not realize were interfering with both math learning and with how students decided to go on to further study in math and related fields. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are political wars that buffet math education, and math wars (Schoenfeld, 2004) that have produced political movements to quash certain kinds of curriculum and instruction reforms. Some of these people tour the country to fan the fires of their complaints, knowing that parents showing up at school boards has been effective for them in the past in opposing the kind of instruction reform they disagree with. While I respect differences of opinion, I cannot always say I respect the political approach to winning this sort of debate, or &lt;a href=&quot;http://www.stanford.edu/~joboaler/#sthash.cwuj2cDB&quot; rel=&quot;nofollow noreferrer&quot;&gt;other, even less noble, approaches&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long story short, some of this stuff will also show up on social media because some people are specifically attacking Common Core State Standards as the new manifestation of reform they have long opposed. Ironically, many of the reformers they oppose would not have written the standards this way. They're very much a compromise. I wanted to include this information mainly to explain that there is another motivation out there for some of these attacks/complaints.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Finally&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not gone into legitimate, educational complaints about the CCSS-M (other than the grade band issue) mainly because you asked for an opinion and I have provided the arguments to explain my opinion. There are ways to improve the standards that I think are worth exploring some time later. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Cited:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schoenfeld, A. H. (2004). &lt;a href=&quot;http://www.math.cornell.edu/~henderson/courses/EdMath-F04/MathWars.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Math Wars&lt;/a&gt;. Educational Policy, 18(1), 253–286. doi:10.1177/0895904803260042&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Smith III, J. P., &amp;amp; Thompson, P. W. (2008). &lt;a href=&quot;http://www.pat-thompson.net/PDFversions/2006SmithThompsonEarlyAlg.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantitative reasoning and the development of algebraic reasoning&lt;/a&gt;. In J. Kaput, D. W. Carraher, &amp;amp; M. L. Blanton (Eds.), Algebra in the early grades (pp. 95–131). New York, NY: Taylor &amp;amp; Francis Group.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-04-30T06:00:35.763" CommentCount="26" ContentLicense="CC BY-SA 3.0" />
  <row Id="2072" PostTypeId="2" ParentId="2068" CreationDate="2014-04-30T07:03:35.263" Score="1" Body="&lt;p&gt;As anything, show them what they are useful for. Perhaps the typical problem of computing the height of a building from distance and angle, and work from there. An understandable sort of application is surveying, setting up a base line, measure angles and compute the triangles.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-30T07:03:35.263" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2073" PostTypeId="2" ParentId="2060" CreationDate="2014-04-30T11:04:30.940" Score="5" Body="&lt;ol&gt;&#xA;&lt;li&gt;Mathematics courses encourage analytical thinking in a way that may be helpful for biology majors.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is some argument that calculus ought to be more widely known within the biology community.  For example, see the following infamous paper, which has gained over 200 citations according to Google scholar:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://fliptomato.wordpress.com/2007/03/19/medical-researcher-discovers-integration-gets-75-citations/&quot;&gt;M.M. Tai, A Mathematical Model for the Determination of Total Area Under Glucose Tolerance and Other Metabolic Curves. &lt;em&gt;Diabetes Care&lt;/em&gt;, Vol 17, Issue 2, 152&amp;ndash;154.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;mathematical model&quot; discussed in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Trapezoid_rule&quot;&gt;trapezoid rule&lt;/a&gt;, which is often covered in second-semester calculus courses.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1320" LastEditorUserId="50" LastEditDate="2014-04-30T14:42:50.343" LastActivityDate="2014-04-30T14:42:50.343" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2074" PostTypeId="1" CreationDate="2014-04-30T11:27:34.983" Score="32" ViewCount="7176" Body="&lt;p&gt;Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonn&amp;eacute;, &quot;A bas Euclide! Mort aux triangles!&quot; (see King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, p. 157), often associated in the popular mind with Bourbaki's general stance on rigorous, formalized mathematics (eschewing pictorial representations, etc.). See Dieudonn&amp;eacute;'s address at the &lt;a href=&quot;http://www.academia.edu/1645989/New_Thinking_in_School_Mathematics_by_J._Dieudonne_1959&quot; rel=&quot;noreferrer&quot;&gt;Royaumont seminar&lt;/a&gt; for his own articulated stance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In brief, the suggestion was to replace Euclidean Geometry (EG) in the secondary school curriculum with more modern mathematical areas, as for example Set Theory, Abstract Algebra and (soft) Analysis. These ideas were influential, and Euclidean Geometry was gradually demoted in French secondary school education. Not totally abolished though: it is still a part of the syllabus, but without the difficult and interesting proofs and the axiomatic foundation. Analogous demotion/abolition of EG took place in most European countries during the 70s and 80s, especially in the Western European ones. (An exception is Russia!) And together with EG there was a gradual disappearance of mathematical proofs from the high school syllabus, in most European countries; the trouble being (as I understand it) that most of the proofs and notions of modern mathematical areas which replaced EG either required maturity or were not sufficiently interesting to students, and gradually most of such proofs were abandoned. About ten years later, there were general calls that geometry return, as the introduction of the alternative mathematical areas did not produce the desired results. Thus EG came back, but not in its original form. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach in a University (not a high school), and we keep introducing new introductory courses, for math majors, as our new students do not know what a proof is. [Cf. the rise of university courses in the US that come under the heading &quot;Introduction to Mathematical Proofs&quot; and the like.] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in hearing arguments both FOR and AGAINST the return of EG to high school curricula. Some related questions: is it necessary for high-school students to be exposed to proofs?  If so, is there is a more efficient mathematical subject, for high school students, in order to learn what is a theorem, an axiom and a proof? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;b&gt;Full disclosure&lt;/b&gt;: currently I am leading a campaign for the return of EG to the syllabus of the high schools of my country (Cyprus). However, I am genuinely interested in hearing arguments both pro and con. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note.&lt;/strong&gt; This question was also asked in &lt;a href=&quot;https://mathoverflow.net/questions/152352/is-euclid-dead&quot;&gt;mathoverflow&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1321" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2022-06-03T12:33:03.860" Title="Is Euclid dead? or Should Euclidean geometry be taught to high school students?" Tags="|secondary-education|proofs|geometry|" AnswerCount="13" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2075" PostTypeId="2" ParentId="2074" CreationDate="2014-04-30T12:53:48.700" Score="14" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I am interested in hearing arguments both FOR and AGAINST the return of EG to high school curricula. Some related questions: is it necessary for high-school students to be exposed to proofs?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am going to address part of this question, specifically the changing role of the notion of proof as it appears in standards (influenced by research). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My brief response borrows heavily from the document I am referencing (Yackel &amp;amp; Hanna, 2003); by responding I intend to connect your question to something underlying the inclusion of mathematical practices (and/or mathematical processes) in recent standards documents here in the USA. This inclusion is based on mathematics education theory. I do not address arguments for including Euclidean geometry and geometric proofs as curriculum content. As proof and proving are not my particular area, I am hoping that what I can provide here points you towards research and arguments that help you address the part of your question that is tied to &quot;proofs.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many math ed researchers tightly associate learning and reasoning (see a quote from Thompson in Yackel &amp;amp; Hanna, 2003, p. 227). Von Glasersfeld (the prominent theorist) &quot;posited that knowledge is built up by the cognizing individual (2003, p. 227).&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The importance placed on mathematical reasoning in the learning process has been embraced by some recent research-based approaches to math education, Being able to communicate &lt;em&gt;how&lt;/em&gt; you know something now becomes an inherent part of mathematics instruction (and classroom activity for both teachers and students). Therefore, at all levels, argumentation and justification become a practice that must be suffused throughout the curriculum rather than a specific content target for a particular grade band. This is not only true for the NCTM Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000), but also for the recent &lt;a href=&quot;http://www.corestandards.org/&quot;&gt;Common Core State Standards&lt;/a&gt; with their &lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot;&gt;section on mathematical practices&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will find some research-based justifications for this approach to standards-based education in the Yackel and Hanna (2003) chapter. A more recent book (Stylianou, Blanton, &amp;amp; Knuth, 2009) also provides insights into how research informs the inclusion of proof throughout the grades, and as early as kindergarten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this acts as a decent starting point to sources that argue strongly for proof as a mathematical practice, which would provide a foundation for secondary-level proof and proving that might be more recognizable in traditional curricula. Yackel and Hanna (2003) note the difficulty some students face in secondary school when they have not had an underlying mathematical education that includes justification and argumentation (p. 234).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, we can see a research-supported change in the way  proof and proving is approached in K-12. We see the idea that argumentation and justification are practices students need throughout their mathematics education rather than being introduced to &quot;proofs&quot; as an activity late in their public school education. Additionally, that these practices support students in learning &quot;proof&quot; later on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The chapters in Stylianou, Blanton and Knuth (2009) make the argument for the broadening of the notion of proof, and are a much more detailed look at how and why researchers are seeing the need for this perspective change. This doesn't argue against Euclidean proof, but I think it does argue against that being the first introduction to the idea of proving, and of the practice of justification as an inherent part of knowing in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;ADDENDUM:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I checked, and there is proof used to &quot;refute or support conjectures&quot; in the &lt;a href=&quot;http://www.nctm.org/standards/content.aspx?id=3956#analyze&quot;&gt;9-12 Geometry Standard of the NCTM Principles and Standards for School Mathematics.&lt;/a&gt; (Link is behind a paywall)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Cited&lt;/p&gt;&#xA;&#xA;&lt;p&gt;National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stylianou, D. A., Blanton, M. L., &amp;amp; Knuth, E. J. (2009). Teaching and learning proof across the grades: A K-16 perspective. Routledge. Retrieved from &lt;a href=&quot;http://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=8RCRAgAAQBAJ&amp;amp;oi=fnd&amp;amp;pg=PP1&amp;amp;dq=proof+blanton&amp;amp;ots=c5wYvMHgOT&amp;amp;sig=-5-wM5dmwLG_ZWCJeLfIpyEUDpA&quot;&gt;http://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=8RCRAgAAQBAJ&amp;amp;oi=fnd&amp;amp;pg=PP1&amp;amp;dq=proof+blanton&amp;amp;ots=c5wYvMHgOT&amp;amp;sig=-5-wM5dmwLG_ZWCJeLfIpyEUDpA&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yackel, E., &amp;amp; Hanna, G. (2003). &lt;a href=&quot;http://393methods1.wikispaces.com/file/view/Yackel+&amp;amp;+Hanna.pdf&quot;&gt;Reasoning and proof&lt;/a&gt;. In J. Kilpatrick, W. G. Martin, &amp;amp; D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 227–236). Reston, VA: National Council of Teachers of Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-04-30T22:15:17.717" LastActivityDate="2014-04-30T22:15:17.717" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2077" PostTypeId="2" ParentId="2069" CreationDate="2014-04-30T13:45:15.913" Score="6" Body="&lt;p&gt;Working in a district that has been &quot;ahead of the curve&quot; in preparing teachers about the CC, I feel confident in what the Core has to offer. On the other hand, the parents are still unsure about it. The negative feedback is the typical stuff you see on FB, etc., coupled with the statement, &quot;My kid already knows how to &lt;strong&gt;______&lt;/strong&gt; (in math).&quot; The reality is that they have memorized a process, but don't understand the process or concept. Ok, so they are efficent at something -- they aren't quite sure what though. The sad examples of someone's attempt to show what CC is on social media is a joke. Plus, the dad's angry letter to his son's teacher about math homework shows how ignorant parents are about this topic.&lt;/p&gt;&#xA;" OwnerUserId="1323" LastActivityDate="2014-04-30T13:45:15.913" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2078" PostTypeId="2" ParentId="2069" CreationDate="2014-04-30T14:04:30.330" Score="10" Body="&lt;p&gt;&lt;strong&gt;Edit (7/17/14):&lt;/strong&gt; You can find a very informative flow chart about CCSSM &lt;strong&gt;&lt;a href=&quot;http://jeffbaumes.github.io/standards/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt; to show dependencies among standards. At the time of posting, the page covers the standards included for K-8.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;If by &amp;quot;material&amp;quot; you mean CCSSM lesson plans, then it is important to draw the distinction between &lt;em&gt;standards&lt;/em&gt; and &lt;em&gt;curricula&lt;/em&gt;. JPBurke makes this distinction in his answer, but I re-iterate here that there is a dearth of standards-aligned material (which will be important when students - and probably teachers, schools, and so forth - are assessed based on CCSSM).&lt;/p&gt;&#xA;&lt;p&gt;For example, a major difference between CCSSM and previous standards and curricula is its inclusion of &lt;strong&gt;mathematical modeling&lt;/strong&gt; (as both a Standard for Mathematical Practice at all ages, and its own strand at the secondary school level). I have yet to see real materials for teaching modeling; moreover, I think many teachers are not only unsure as to how they should teach modeling, but - in many cases - are not even sure what &amp;quot;mathematical modeling&amp;quot; &lt;em&gt;means&lt;/em&gt;. The short bit in the standards is not very illuminating, even with its diagrammatic representation of the process. Others have written about the subject (e.g., Blum, Ferri, Pollak) and there is a recent thesis by Gould on teachers' (mis)conceptions about modeling:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gould, H. (2013). Teachers’ Conceptions of Mathematical Modeling (Doctoral dissertation, Columbia University). &lt;strong&gt;&lt;a href=&quot;http://academiccommons.columbia.edu/item/ac:161497&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As for personal experiences with the Common Core State Standards for Mathematics: A former classmate of Gould's, &amp;quot;Master [mathematics] teacher&amp;quot; Heidi Reich, recently had her letter about CCSS impressions published in the New York Times. It is entitled: &lt;em&gt;A Teacher's View of Common Core&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;http://publiceditor.blogs.nytimes.com/2014/04/22/introducing-the-reader-spotlight-a-teachers-view-of-common-core/&quot; rel=&quot;noreferrer&quot;&gt;full version&lt;/a&gt; is available on the site (and was linked to by &lt;a href=&quot;http://dianeravitch.net/2014/04/26/the-new-york-times-publishes-heidi-reich-a-voice-for-teachers/&quot; rel=&quot;noreferrer&quot;&gt;this blog&lt;/a&gt;); here is the main part:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I am a teacher and have been for 15 years, which means I am right in the middle of my career. I have been recognized for my teaching by Math for America (I have been a “Master Teacher” for eight or so years now), am locally respected (sorry, no data to support that) and have loved my job for all of these years. Now I find that the nutty wacky whims of the Department of Education under Bloomberg and Klein have been dwarfed by NYS and the federal government’s desire to implement truly difficult standards in a matter of months. We (teachers) are required to write curriculum based on almost NO information, tailor said curriculum to testing about which there are NO data, and still teach our five classes of 34 students a day without skipping a beat.&lt;/p&gt;&#xA;&lt;p&gt;I imagine you are thinking, why do you need to tailor curriculum to tests, especially if the tests don’t even exist yet? Sure, it has something to do with our jobs being on the line if our students don’t surpass some standard or other (sorry, but to us it all seems just so very arbitrary), but more to the point, no reform means anything until you see what assessment is going to be. We are accustomed to writing our curricula by determining what it is we want our students to be able to do and then designing activities and lessons to convey those expectations and to train students to accomplish goals. It would be duplicitous for the powers that be to withhold those expectations from us if they were even close to having established them, but we are all too aware that, unfortunately, Pearson and others are scrambling madly to write tests (for billions and billions of dollars) that they have no time to field test, which has already resulted in chaos and utter confusion in lower grades in NY State. My colleagues and I have NO problem holding students to high standards as long as those standards are clearly conveyed to us and as long as we have time to develop appropriate curricula and activities. (We would have used the summer to do this if the standards had been available before September — not happily, but we would have done so.) The current situation is diametrically opposed to that. And I must reiterate my disappointment that The Times, the only paper of record as far as I am concerned, totally missed the point: that parents and students and educators are ALL up in arms about the Common Core, not just extremist politicians on both sides, because to us, the Common Core standards are not even standards. They are vague ideas being developed (for huge personal profit) by billionaires and testing companies, imposed upon teachers, students and parents with complete disregard for education, learning and progress.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-07-17T18:32:48.330" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2079" PostTypeId="1" AcceptedAnswerId="2094" CreationDate="2014-04-30T14:23:42.373" Score="12" ViewCount="7842" Body="&lt;p&gt;U.S. News and World Report publishes school &lt;a href=&quot;http://grad-schools.usnews.rankingsandreviews.com/best-graduate-schools&quot;&gt;rankings&lt;/a&gt; for many different disciplines, including mathematics. &lt;strong&gt;Is there any ranking for mathematics education graduate programs?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Weaker question: Is there a guide to selecting a math education graduate program, including a reasonably comprehensive list of programs?&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="262" LastEditDate="2016-03-09T21:52:03.753" LastActivityDate="2016-03-10T14:45:21.217" Title="Mathematics Education Graduate Program List and Rankings" Tags="|reference-request|graduate-education|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2080" PostTypeId="2" ParentId="1896" CreationDate="2014-04-30T15:04:20.693" Score="2" Body="&lt;p&gt;These words can be quite helpful, because it is important to know how difficult a skipped pproof is. The problem is that these words have contradictory meaning, they depend on context and are often used out of laziness.&lt;/p&gt;&#xA;&lt;p&gt;Since it can be quite frustrating to get stuck on an abused &amp;quot;trivial&amp;quot;, the trust of students and readers in this kind of word can be low, so it is almost always better to be more precise, either by giving more details on the difficulty:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The fully-written proof of this lemma would take half a page of intermediate difficulty among the proofs written in this book.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The reader can verify the base case of the induction (same difficulty as the one-star exercices in this book).&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or be more honest on the sense of trivial:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The proof of this result is elementary, uninspiring and long, so we skip it.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The proof of this result is so well-known that I expect my readers to know it.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;I am so scared about boring anyone that I would rather risk losing half the audience.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="43" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-04-30T15:04:20.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2081" PostTypeId="1" CreationDate="2014-04-30T15:26:18.280" Score="13" ViewCount="330" Body="&lt;p&gt;In an undergraduate Abstract Algebra class, it appears that there are two standard approaches: &quot;groups first&quot; or &quot;rings first&quot;. Most abstract algebra texts with a &quot;groups first&quot; approach start with some combination of groups in $\mathbb{C}$, groups from modular arithmetic, groups from permutations (including dihedral groups), matrix groups, and then generalize to abstract groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question 1: Why not free groups with relations from the beginning?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rationale: Working with a free group is almost as general as working with an abstract group, except that you get to explicitly state, for instance, that nothing is commutative, or what the order of an element is, etc. Like $S_n$, all groups can be constructed out of free groups; possibly in a way that's more intuitive than the $S_n$ construction. The popular focus on permutations is great when thinking about groups as acting on something, but it seems that free groups capture the more basic idea that groups are sets that have additional structure and that certain strings of elements are (or are not) equal to other strings of elements. Granted, this is very algebraic/symbolic. But compared to the standard geometric/permutation approach, it seems understanding the awesome generality of free groups would better aid in understanding the properties of a general group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question 2: So, does anyone know of an undergraduate abstract algebra text that starts with free groups, relations, presentations, etc?&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-05-03T22:36:43.990" Title="Free Groups First Approach" Tags="|abstract-algebra|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2082" PostTypeId="2" ParentId="1068" CreationDate="2014-04-30T16:22:54.623" Score="5" Body="&lt;p&gt;This is more of an &quot;add-on&quot; to existing answer than a full answer, but I think it needs more visibility than a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest beginning with something very concrete, like the idea of odd and even integers. You can observe that the sum of two even integers is even, the sum of an odd integer and an even integer is odd, etc. Then ask how one can be &lt;strong&gt;absolutely sure&lt;/strong&gt; that these formal identities always hold. For the sum of two even integers, you can observe that $2$ can be factored out of each number, and hence out of the sum (by the distributive law), but what about an odd plus an even?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, to summarize the rules for adding even and odd integers, you can make a $2$ by $2$ Cayley table, using &quot;O&quot; for odd and &quot;E&quot; for even. Then you can say that since mathematicians like numbers, they replace &quot;O&quot; with $0$ and &quot;E&quot; with $1.$ Maybe mention $0$ and $1$ are the remainders upon division by $2$ to motivate somewhat what the rule $1 +1 = 0$ is saying. (Perhaps insert a joke at this point about how mathematicians can't even add $1$ and $1$ correctly.) The nice thing about having these two tables is that, later on, if you have to talk about what isomorphic means, you can come back to this example and say it's basically renaming the objects like we did with &quot;even&quot; and &quot;odd&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get past the even and odd example, you can then ask if there is any reasonable way someone can think of to classify integers into three different categories, such as &quot;psudo-even&quot;, &quot;psudo-odd&quot;, and &quot;neither&quot;. Then talk about how this addition table works, then replace the objects with $0,$ $1,$ and $2.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After all this, &lt;em&gt;only then start talking about&lt;/em&gt; clocks and &quot;mod 12&quot; addition and the like.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-04-30T16:22:54.623" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2083" PostTypeId="1" AcceptedAnswerId="2086" CreationDate="2014-04-30T16:26:12.823" Score="6" ViewCount="723" Body="&lt;p&gt;I am looking for suggestions about ways to introduce the &lt;em&gt;preimage&lt;/em&gt; of a set under a function. My experience is that many students find it a confusing concept. The definition I use is as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose that $f:X\to Y$ is a function, and $A\subseteq Y$, then the &lt;em&gt;preimage of&lt;/em&gt; $A$ &lt;em&gt;under&lt;/em&gt; $f$ is $$f^{-1}[A]=\big\{x\in X:f(x)\in A\big\}\;.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that when teaching engineering students, I use the idea of a measuring device being a mapping from a physical state space $X$ to an observation space $Y$. You can then ask the question: Given a particular measurement/reading/observation, what physical states could have generated it? This approach is inspired by Steven M. LaValle's &lt;a href=&quot;http://msl.cs.uiuc.edu/~lavalle/papers/Lav11.pdf&quot;&gt;Sensing and Filtering: A Fresh Perspective Based on Preimages and Information Spaces&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2014-04-30T20:21:05.200" Title="Preimage of a set under a function" Tags="|undergraduate-education|functions|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2084" PostTypeId="2" ParentId="2079" CreationDate="2014-04-30T16:53:07.587" Score="1" Body="&lt;p&gt;As the comment by @JPBurke says, your criteria will be different than some blanket ones by a periodical for mass consumption.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To look for a program to join, look for a potential advisor or group. Look for people doing work in areas that interest you, check out their webpages, see if they belong to a group working together. This should narrow it down nicely. With a handful of options, check their graduate schools (admission requirements, and other miscellanea like housing and such). See if you can contact (past or present) students to get first-hand impressions.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-30T16:53:07.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2085" PostTypeId="2" ParentId="2081" CreationDate="2014-04-30T17:03:18.833" Score="3" Body="&lt;p&gt;I agree that free groups are &quot;cleaner,&quot; but what you are looking for here is to get (beginning) students to grasp the subject. This means showing concrete examples, and that the whole theory build up around groups/rings isn't just idle speculation but has down to earth applications. I learned much of algebra from Richman's &quot;Number Theory: An Introduction to Algebra&quot; (Brooks/Cole 1971; sadly long out of print, we got a photocopy of the book as text for the class way back in '85 or so). To &lt;em&gt;see&lt;/em&gt; how applying the ring or group ideas simplifies otherwise mysterious, unmotivated proofs is an eye opener (e.g. Euler's theorem is just a one-line corollary to Lagrange's theorem that the order of a subgroup divides the order of the group).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just my 2¢ as a non-specialist, mind you.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-04-30T17:03:18.833" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2086" PostTypeId="2" ParentId="2083" CreationDate="2014-04-30T17:05:55.297" Score="5" Body="&lt;p&gt;I have thought about this a lot and, specifically, have wondered about the effects of terminology and notation. I'm not sure if there is a &lt;em&gt;best&lt;/em&gt; answer, but here are some of my thoughts:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When I introduce image/preimage in an intro to proofs course, I avoid using the $f^{-1}$ notation, stressing for the students that we don't want to presume $f$ has an inverse. I understand this notation is fairly standard later on and elsewhere, but before a student really understands inverse functions, I think it's best to remove the potential opportunity to conflate these two completely different concepts that happen to share notation.&lt;/li&gt;&#xA;&lt;li&gt;A professor of mine used &quot;push-forward&quot; and &quot;pull-back&quot; to refer to &quot;image&quot; and &quot;preimage&quot; and, the more I think about it, the more I really like those terms and want to use them (almost) exclusively myself. They really convey the &lt;em&gt;operational&lt;/em&gt; aspect of the definitions: there's a function that does something, and we care about where it &lt;em&gt;pushes&lt;/em&gt; domain elements (for the image) or where it &lt;em&gt;pulls back&lt;/em&gt; to domain elements (for the preimage). Perhaps using these two terms in tandem with the standard terms will really tie the ideas together more coherently for students.&lt;/li&gt;&#xA;&lt;li&gt;I like your example about measurements and observations, but I would recommend using even &lt;em&gt;simpler&lt;/em&gt; examples, both of the arbitrary/abstract and applied varieties. Specifically, use some examples to show &quot;surprising&quot; behavior of the preimage operator. For instance, you can construct quickly (with students' help, even) a toy example to show that &quot;the preimage of an image of a set is not necessarily equal to the starting set&quot;. Draw some schematic diagrams with dots and arrows. Then, draw a graph of a function $f:\mathbb{R}\to\mathbb{R}$ and ask about preimages of certain intervals on the $y$-axis: Where are all the $x$-values whose output &quot;lands in&quot; this set? (You could use this to introduce the idea from topology that a function is continuous when &quot;the pull-back of an open set is open&quot;.)&lt;/li&gt;&#xA;&lt;li&gt;Generally, as many examples as possible is, I think, the only way to really get used to this. Make sure some homework problems address these concepts. Make sure students have to &quot;compute&quot; preimages of certain sets, given a function, but also make sure they have to &lt;em&gt;construct their own examples&lt;/em&gt;. Make them come up with a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f^{-1}[\mathbb{N}]=\varnothing$, say. Depending on the specific interests of your students (computer science, engineering, math) you can tailor some more examples/exercises.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-04-30T17:05:55.297" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2087" PostTypeId="2" ParentId="2060" CreationDate="2014-04-30T18:32:16.870" Score="-2" Body="&lt;p&gt;At the end of the day, all science is 'applied mathematics'...without the math supporting your observations, you greatly limit yourself in your chosen field.  Can you get through life in a science career without math? Sure...if all you care about is qualitative observations.  With post-trig math knowledge though (e.g.- Calculus, Differential Equations, Linear Algebra, etc)...you're provided a deeper, quantitative understanding of your chosen field.&lt;/p&gt;&#xA;" OwnerUserId="1327" LastActivityDate="2014-04-30T18:32:16.870" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2088" PostTypeId="1" CreationDate="2014-04-30T20:06:06.190" Score="11" ViewCount="314" Body="&lt;p&gt;I am teaching an exercice section for a group of students who have (rather difficult and long) exams every two weeks. The students are focussed on the exams and barely willing to take the time to understand something because they are so nervous about what they have to write on the next exam which is always this week or next week.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can I encourage them to still strive for understanding which will be very important for the years to come when the short-term incentives seem to them to be the exact opposite?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They want me to write exam-perfect answers on the blackboard instead of trying to explain to them HOW I find the answers. Also, they believe in the ONE perfect answer and look at me in a reproaching way when I tell them that at some point they have several options.&lt;/p&gt;&#xA;" OwnerUserId="43" LastEditorUserId="143" LastEditDate="2014-06-15T11:02:41.230" LastActivityDate="2014-06-15T11:02:41.230" Title="Encourage students to strive for understanding despite looming exams" Tags="|student-motivation|general-pedagogy|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2089" PostTypeId="2" ParentId="2074" CreationDate="2014-04-30T20:09:06.703" Score="26" Body="&lt;p&gt;I'd like to tackle the question from another point of view than JPBurkes answer: If you accept, that mathematical argumentation (whatever level) is an essential part of mathematics courses in K-12, than Euclidean Geometry is a great way to implement this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Visuality&lt;/strong&gt; Euclidean Geometry deals with objects that can be easily visualized. It can be properly served on all three represantative layers: enactive, iconic, symbolic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Scaling of Argumentation level&lt;/strong&gt; Theorems in Euclidean Geometry can be proven or argumented for on different argumentation levels: intuitively formal-rigorous, abstractedly formal-rigorous (Euclid's way), with generalizable examples, using intuitive knowledge (symmetry, movement invariance, …).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Loss of calculations&lt;/strong&gt; Many proofs in Euclidean Geometry include no calculations at all, others only as small substeps. Consequently, students can learn that way, that maths is not just calculations. This sounds trivial, but it is often a problem: They hesitate in Euclidean Geometry, because they cannot simply calculate a result. The result isn't even a number or a value, but the theorem itself.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-30T20:09:06.703" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2090" PostTypeId="2" ParentId="2083" CreationDate="2014-04-30T20:21:05.200" Score="1" Body="&lt;p&gt;Mapping between reality and observation demands a high level of &lt;em&gt;scientific&lt;/em&gt; abstraction, that &lt;em&gt;math&lt;/em&gt; students in such a course might not have. It's possibly easier to use physical (or other scientific) laws in the form of equations between values. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example $U$-$I$-relationship in NTCs: Should this be a function $U(I)$ or a function $I(U)$? Physics can't give an answer, so Math should deal with both. But the relation is not injective, so how to deal about it?&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-04-30T20:21:05.200" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2091" PostTypeId="2" ParentId="2060" CreationDate="2014-04-30T21:12:29.687" Score="21" Body="&lt;p&gt;I happen to have revised our calculus syllabus for first year biology majors about one year ago (in a French university, for that matter). I benefited a lot from my wife's experience as a math-friendly biologist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main point of the course is to get students able to deal with &lt;em&gt;quantitative models.&lt;/em&gt; For example, my wife studied the movement of cells under various circumstances.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A common model postulates that the average distance $d$ between two&#xA;  positions of a cell at times $t_0$ and $t_0+T$ is given by &#xA;  $$d = \alpha T^\beta$$ where $\alpha&amp;gt;0$ is a speed parameter and&#xA;  $\beta\in[\frac12,1]$ is a parameter that measures how the movement&#xA;  fits between a Brownian motion ($\beta=\frac12$)&#xA;  and a purely ballistic motion ($\beta=1$). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This simple model is a great example to show how calculus can be relevant to biology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My first point might be specific to recent French students: first-year students are often not even proficient enough with basic algebraic manipulations to be able to do anything relevant with such a model. For example, even asking to compute how $d$ changes when $T$ is multiplied by a constant needs to now how to &lt;strong&gt;deal with exponents&lt;/strong&gt;. In fact, we even had serious issues with the mere use of percentages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the main point of our new calculus course is to be able to &lt;strong&gt;estimate uncertainties&lt;/strong&gt;: in particular, given that $T=T_0\pm \delta T$, $\alpha=\alpha_0\pm\delta\alpha$ and $\beta=\beta_0\pm\delta\beta$, we ask them to estimate $d$ up to order one (i.e. using &lt;strong&gt;first-order Taylor series&lt;/strong&gt;). This already involves &lt;strong&gt;derivatives of multivariable functions&lt;/strong&gt;, and is an important computation when you want to draw conclusions from experiments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another important point of the course is the &lt;strong&gt;use of logarithms and exponentials&lt;/strong&gt;, in particular to interpret log or log-log graphs. For example, in the above model, it takes a (very) little habit to see that taking logs is a good thing to do: $\log d = \beta\log T+\log \alpha$ so that plotting your data in log-log chart should give you a line (if the models accurately represent your experiments).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This then interacts with &lt;strong&gt;statistics&lt;/strong&gt;: one can find the &lt;strong&gt;linear regression&lt;/strong&gt; in log-log charts to find estimates for $\alpha$ and $\beta$. But then one really gets an estimate of $\beta$ and... $\log\alpha$, so one should have a sense of how badly this uncertainty propagates to $\alpha$ (&lt;strong&gt;one variable first-order Taylor series&lt;/strong&gt;: easy peasy).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other main goal of the course is to get them able to deal with some (ordinary) differential equations. The motivating example I chose was offered to me by the chemist of our syllabus meeting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A common model for the kinetics of a chemical reaction&#xA;$$A + B \to C$$&#xA;is the second-order model: one assumes that the speed of the reaction is proportional to the product of the concentrations of the species A and B. This leads to a not-so-easy differential equation of the form&#xA;$$ y'(t) = (a-y(t))(b-y(t)).$$&#xA;This is a &lt;strong&gt;first-order ODE with separable variables&lt;/strong&gt;. One can solve it explicitly (a luxury!) by dividing by the second member, &lt;strong&gt;integrate&lt;/strong&gt; in $t$, do a &lt;strong&gt;change of variable&lt;/strong&gt; $u=y(t)$ in the left-hand-side, &lt;strong&gt;resolve into partial fractions&lt;/strong&gt; the rational fraction that comes out, and remember how log is an &lt;strong&gt;antiderivative&lt;/strong&gt; of the inverse function (and how to adjust for the various constants the appeared in the process). Then, you need some &lt;strong&gt;algebraic manipulations&lt;/strong&gt; to transform the resulting equation into the form $y(t) = \dots$. Unfortunately and of course, we are far from being able to properly cover all this material, but we try to get the student able to follow this road later on, with their chemistry teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, I would love to be able to do more quantitative analysis of differential equations, but it is difficult to teach since it quickly goes beyond a few recipes. For example, I would like them to become able to tell in a glimpse the &lt;strong&gt;variations&lt;/strong&gt; of solutions to&#xA;$$y'(t)=a\cdot y(t)-b \sqrt{y(t)}$$&#xA;(a model of population growth for colonies of small living entities organized in circles, where death occur mostly on the edge - note how basic &lt;strong&gt;geometry&lt;/strong&gt; makes an appearance here to explain the model) in terms of the initial value. Or to be able to realize that solutions to&#xA;$$y'(t)=\sqrt{y(t)}$$&#xA;must be sub-exponential (and what that even means...). For this kind of goals, one must first aim to basic proficiency in calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To sum up, &lt;strong&gt;dealing with any quantitative model needs a fair bit of calculus&lt;/strong&gt;, in order to have a sense of what the model says, to use it with actual data, to analyze experimental data, to interpret it, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To finish with a controversial point, it seems to me that, at least in my environment, biologists tend to underestimate the usefulness of calculus (and statistics, and more generally mathematics) and that improving the basic understanding of mathematics among biologists-to-be can only be beneficial.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-04-30T21:12:29.687" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2093" PostTypeId="1" AcceptedAnswerId="2096" CreationDate="2014-04-30T23:41:21.463" Score="17" ViewCount="1032" Body="&lt;p&gt;I remember when I took geometry in high school, like most students it's where I was formally introduced to proofs. However, the way we went about them was strange, it really felt like symbol manipulation. We had a big list of axioms and postulates, and proofs were fairly mechanical. They were done by a table, one side was the result, the other was what you invoked to get it&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Result                |     Method&#xA;________________________________________________&#xA;1 Lycanthropy         |     Given&#xA;2 A cute bunny        |     Given&#xA;3 A werewolf rabbit   |     Flibbergoth's theorem of lycanthropic mammals on (1) and (2)&#xA;________________________________________________&#xA;∴ Wolf isomorphism    | The harewolf-lupine equivalence axiom on (3)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;You can see an actual, non-silly simple example &lt;a href=&quot;http://www.sparknotes.com/math/geometry3/geometricproofs/section1.rhtml&quot;&gt;here&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I question whether or not this is a good way to introduce proof writing. Certainly the proofs that come out are valid (if done correctly). There's nothing inherently wrong, but even now they feel &lt;em&gt;foreign&lt;/em&gt; to me. I don't feel like I really got proofs until I took a discrete math course in college, where proofs were more free, we could choose our own approach, and we tended to write in paragraphs. Certainly we still invoked axioms and theorems and brought up givens, but it was much less structured and, for me, was easier to understand and perform.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Full disclosure: we did introduce more freeform, paragraph-like proofs by contradiction near the end of the course, but it was introduced as a novelty and never used on any exams or homework).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I am not the world. I was wondering if there was evidence to support this method for introducing proofs. Is there any evidence that this improves mathematical reasoning, or is easier for students to learn and execute than more free-form proofs? Certainly it's a lot more similar to the mechanical symbol manipulation most students have been doing up to that point, and I can see an argument that while argumentation and mathematical creativity is still involved, providing the structure to make it more mechanical should make it easier to perform given their likely education up to that point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, I can't shake that I had very little idea what was going on until I got to less structured looking proofs. What does the evidence say? Is introducing proofs in a more structured format helpful? Am I just atypical? I'll grant it's entirely possible I wouldn't have even understood freeform proofs if I hadn't done the structured approach first, but I still have trouble doing the same proofs in table format that I could do easily in a paragraph with sentences interspersed with notation, so I don't know. I've always personally had a more argumentative/wordy approach to doing math (post-arithmetic at least).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit: As &lt;strong&gt;mweiss&lt;/strong&gt; indicated, &quot;effectiveness&quot; isn't very well defined. I was trying to get at concepts such as &quot;does the 2-column method help foster an ability to prove mathematical statements, or does it hold back or confuse students more than the more freeform style of proving things?&quot; Overall, I think &lt;strong&gt;mweiss&lt;/strong&gt; attempt to answer the question in terms of whether it provides reasonable structure to guide students or else acts as a straightjacket that stifles their ability to freely think about problems is what I was getting at.&lt;/p&gt;&#xA;" OwnerUserId="521" LastEditorUserId="50" LastEditDate="2014-05-01T02:19:19.023" LastActivityDate="2014-05-01T02:19:19.023" Title="Is there any evidence about the effectiveness of &quot;table proofs&quot; in pre-college mathematics education?" Tags="|secondary-education|reference-request|proofs|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2094" PostTypeId="2" ParentId="2079" CreationDate="2014-05-01T00:24:10.197" Score="8" Body="&lt;p&gt;&lt;strong&gt;Update (2016 Mar 9).&lt;/strong&gt; The OP asks (emphasis added): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a guide to selecting a &lt;strong&gt;math education graduate program&lt;/strong&gt;, including &lt;strong&gt;a reasonably comprehensive list of programs&lt;/strong&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is a &lt;a href=&quot;https://docs.google.com/spreadsheets/d/15tWjjJO66qpUv0dCJuHI-jZ8Qhu24E5eJSBBa8NNovk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;List of Links to U.S. Doctoral Programs with a focus on Mathematics Education&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The linked google spreadsheet was generously compiled by &lt;strong&gt;Kenneth Bradfield&lt;/strong&gt; with the assistance of a few others (click through for attribution) and contains the following headers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Institution&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;College/School/Department/Program&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Degree Type&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Title&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Program Link&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;At the time of this update, there are 110 different institutions listed.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The most recent approximation of rankings (of which I am aware) is available &lt;a href=&quot;http://www.ams.org/notices/200710/tx071001283p.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that it only concerns the United States, and that the article is from 2007. Its citation is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Reys, R., Glasgow, R., Teuscher, D., &amp;amp; Nevels, N. (2007). Doctoral programs in mathematics education in the United States: 2007 status report. &lt;em&gt;Notices of the American&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The relevant chart:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ThwaF.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt; A fair amount has changed in the intervening years, and there are surely more changes in store for the future. Moreover, the value of a program will depend in some major way on the professors with whom you work, and the subject of your research. It may also be worth noting that the demand for those with doctorates in Mathematics Education is quite high (Inside Higher Education 2013 &lt;a href=&quot;http://www.insidehighered.com/news/2013/03/13/survey-shows-demand-still-outstrips-supply-doctorates-math-education&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;piece&lt;/strong&gt;&lt;/a&gt; and related AMS notices 2013 &lt;a href=&quot;http://www.ams.org/notices/201304/rnoti-p470.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;pdf&lt;/strong&gt;&lt;/a&gt;) and federal funding seems to be increasingly available for work in education (at the graduate level: an example is the extension of the &lt;a href=&quot;http://www.nsfgrfp.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;NSFGRFP&lt;/strong&gt;&lt;/a&gt; to STEM Education). Lastly, it seems clear that the Common Core State Standards (2010) and the retirement of older faculty members (who are prominent at some of these institutions) are both likely to have influenced these rankings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;FWIW:&lt;/strong&gt; Since the National Science Foundation began granting fellowships for graduate research in Mathematics Education in 2011, students at the following institutions have received them:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2015: Morehouse College; University of California-Los Angeles; North Carolina Agricultural &amp;amp; Technical State University; University of Texas at Austin; University of California-Berkeley&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2014: Michigan State University; University of Minnesota-Twin Cities&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2013: $\emptyset$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2012: University of California-Berkeley&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2011: University of California-Irvine; University of Illinois at Chicago; Northwestern University; Teachers College, Columbia University&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Listed above are award offers strictly in &lt;em&gt;STEM Education and Learning Research - Mathematics Education&lt;/em&gt;. For more information, see &lt;a href=&quot;https://www.fastlane.nsf.gov/grfp/Login.do&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;NSF FastLane :: GRFP&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2016-03-09T21:45:48.973" LastActivityDate="2016-03-09T21:45:48.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2095" PostTypeId="2" ParentId="2088" CreationDate="2014-05-01T02:06:55.207" Score="9" Body="&lt;p&gt;Part of the problem is simply that you're trying to combat &lt;em&gt;rationality&lt;/em&gt;. Regardless of what level you're teaching, grades are important. They control scholarships and school admissions (be they undergrads or high school students). Doing poorly on an exam can be a death sentence to a grade, and that can mean a lot of bad things for a good student. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most students really, truly understand that it's better to understand the material. They really do, however, the grade comes this or next week. And that grade will have an influence (however minor on its own) on their academic career. If they need to understand something, they can reread the textbook over the summer. I always try for understanding, but there have certainly been times where I had to say &quot;okay, either I can try to deeply understand this and possibly fail or I can spend the next two days memorizing these special cases so I pass the final&quot;. It's unfortunate, but rational and necessary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The solution really depends how much control you have over the course. Can you reduce some of the exams to weekly quizzes with less grade impact? That can be a good way to test basic competency and familiarity with the material without the grade pressure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you absolutely have to abide by the exam schedule (or it's external to your class e.g. a GRE prep or you're a tutor for someone else's course), you could make your tests require a deep understanding; structure questions as proofs and conceptual short answer questions. I do warn you, though, that this may backfire horribly for any number of reasons. Not the least of which is that it's hard to regularly and consistently make a large number of high quality conceptual questions that are really fair to beginners at a topic (often we become blind to the difficulty of conceptual questions about concepts we understand).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, I think this is a &lt;em&gt;very&lt;/em&gt; common question in all areas of education. &quot;How can I get students to try and understand instead of trying to min/max their chances of getting an A?&quot; And I don't think there's a perfect answer unless you're in a position to change the system. (Which isn't to say I'm for abolishing all grades, necessarily, but the entire institution is so wrapped up around various metrics that you can't really blame students for recognizing the reality of their situation).&lt;/p&gt;&#xA;" OwnerUserId="521" LastEditorUserId="521" LastEditDate="2014-05-01T02:27:28.063" LastActivityDate="2014-05-01T02:27:28.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2096" PostTypeId="2" ParentId="2093" CreationDate="2014-05-01T02:08:45.177" Score="21" Body="&lt;p&gt;The two-column proof form has been the dominant mode of presentation for proofs in secondary geometry in the United States for most of the past century.  You ask about its effectiveness; unfortunately, I think that question is ill-posed, because the goal state isn't clearly defined (effective &lt;em&gt;at what&lt;/em&gt;?) and so there's no way to measure whatever it is you want to know about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, there has been a &lt;em&gt;ton&lt;/em&gt; of scholarship on the two-column form.  Rather than attempt to address &quot;effectiveness&quot;, instead these studies address the qualitative question, &quot;How is the two-column form used?&quot;  For example, here is a paper I wrote with Patricio Herbst and Chia-ling Chen:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Weiss, M., Herbst, P., and Chen, C.  (2009)  &lt;a href=&quot;http://link.springer.com/article/10.1007/s10649-008-9144-2&quot;&gt;Teachers' perspectives on 'Authentic Mathematics' and the two-column proof form.&lt;/a&gt;  &lt;em&gt;Educational Studies in Mathematics&lt;/em&gt;, 70(3), 275-293.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a few excerpts from that paper, which include citations to other relevant studies for this question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This paper explores geometry teachers’ perspectives on how the two-column proof form can be both a resource and a constraint in engaging students in proving. The two-column form has been used in the USA to teach students how to do proofs since early in the twentieth century (Herbst 2002a). But the two-column form has been criticized for perpetrating a formalistic image of proving, and calls for reform in the 1980s (NCTM 1989) had recommended that teachers decrease attention to two-column proving. In this article we examine a counterproposal that emerges from listening to teachers: namely, that the two- column form can actually be productive when engaging students in proving, by acting not only a constraint, but also a resource enabling complex and non-linear reasoning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and from later:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;American geometry students often learn to write proofs by way of a numbered sequence of “Statements” and “Reasons” in parallel columns (Sekiguchi 1991). This form of writing proofs is quite different from how mathematicians ordinarily write their proofs (Lamport 1995; Schoenfeld 1988; Wu 1996). The two-column form of “statements” and “reasons” has been the standard for the writing￼of proofs since early in the twentieth century. Herbst (2002a, b; Herbst et al. 2008) has argued that this form, as it is deployed in the high school geometry course, is not merely a special way of recording a finished proof; in common classroom practice the two-column form is both a symbol of, and a tool for, a customary way of engaging students in proving. In most classrooms, a proof in two-column form is written out in a chronological sequence that correlates with the numbered assertions: line 1 is written first, line 2 is written second, and so on. Moreover, the two columns are written in alternating sequences: first the statement of line 1, then the reason for line 1; then the statement for line 2, then the reason for line 2; and so on. This turn-taking structure facilitates the teacher in engaging many students in the construction of a single proof. Thus it is not the case that a proof argument is developed first, and only later &lt;em&gt;recorded&lt;/em&gt; in the two-column form; rather, the two-column form becomes a structure that scaffolds the &lt;em&gt;crafting of the argument itself&lt;/em&gt;. More than a final summary of the reasoning, it often amounts to a chronological narrative of the development of the argument. For this reason, dispositions important for the working mathematician, such as working backwards, mapping a strategy, or structuring a proof, are poorly accommodated in the way geometry classes tend to use the two-column format (see Anderson 1982; Greeno 1980; Herbst 2002b; Leron 1983; Thurston 1994).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As you may be able to gather from those excerpts, most people who look at the teaching of proof are pretty critical of the two-column form, arguing that it puts too much emphasis on form rather than substance, and that it is an inauthentic representation of what mathematicians actually do when they prove.  Given this, one might wonder why the two-column form has stuck around as long as it has.  Patricio Herbst traced the evolution of this form and its gradual establishment as a norm over time, and argued that the 2CF made it possible for teachers to hold students accountable for learning to write proofs -- in effect, it solved a problem of teaching.  His paper is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Herbst, P. G. (2002). &lt;a href=&quot;http://link.springer.com/article/10.1023/A:1020264906740&quot;&gt;Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century&lt;/a&gt;. &lt;em&gt;Educational Studies in Mathematics&lt;/em&gt;, 49(3), 283-312.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main argument we made in our 2009 paper was that, yes, some teachers can and do use the 2CF in a way that inhibits flexible reasoning in the classroom; but others use the 2CF in a way that &lt;em&gt;enables&lt;/em&gt; flexible reasoning.  The form can function as either a straightjacket or as a safety net, depending on how it is deployed.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-05-01T02:08:45.177" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2097" PostTypeId="2" ParentId="2060" CreationDate="2014-05-01T02:44:54.953" Score="7" Body="&lt;p&gt;One division of Biology that can be quite mathematical is Ecology and Evolutionary Biology. There are definitely courses that require calculus and differential equations quite similar to what you would teach an engineer for example. From what I understand this can come as quite a surprise to the biology students who go into Ecology because they like the outdoors and the plants/animals. But if you want to understand something like how it is possible that different animals can occupy what seems like the same evolutionary niche, then mathematical models really are the best way to do it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the University of Arizona &lt;a href=&quot;https://uaccess.schedule.arizona.edu/psp/uazsaprd2/UA_CATALOG/HRMS/h/?tab=DEFAULT&quot;&gt;course catalog&lt;/a&gt; (that link will require some clicking around, sorry):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ECOL  447 - Introduction to Theoretical Ecology&#xA;Population growth and density dependence; predation; competition and apparent competition; coexistence mechanisms: niches, spatial and temporal variation; food web concepts and properties; applications. Emphasis on understanding through models and examples.&#xA;Prerequisite: Calculus I&lt;/p&gt;&#xA;" OwnerUserId="1336" LastActivityDate="2014-05-01T02:44:54.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2098" PostTypeId="1" AcceptedAnswerId="2103" CreationDate="2014-05-01T15:28:50.633" Score="31" ViewCount="6722" Body="&lt;p&gt;Many students take calculus or algebra courses in high school, then later take college courses of the same name. There are various reasons for this, but in most cases the students in a college calculus class are not already experts in calculus. Students in a college algebra class are not already experts in algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in these students' minds, they &lt;em&gt;&quot;already did all this in high school.&quot;&lt;/em&gt; This disengages them from lectures, activities, and homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is the most productive way to fix this misconception?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I've tried to directly address the misconception on the first day. This simply alienates students and puts them on the defensive. From then on, they are always trying to prove you wrong, which is completely counterproductive.&lt;/li&gt;&#xA;&lt;li&gt;I've tried letting poor grades speak for themselves, but often students have trouble connecting the poor grades to a lack of understanding. All errors look like &quot;stupid mistakes&quot; to someone who thinks they already know the material, and again directly addressing this causes a seemingly-counterproductive adversarial relationship.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-05-05T00:58:25.977" Title="What to do with students who think they &quot;already know it,&quot; but actually don't?" Tags="|undergraduate-education|student-motivation|advising|" AnswerCount="9" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2099" PostTypeId="1" CreationDate="2014-05-01T16:03:15.183" Score="18" ViewCount="1183" Body="&lt;p&gt;In a 3 minute &lt;a href=&quot;http://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education&quot;&gt;talk&lt;/a&gt; on ted.com, mathematician Arthur Benjamin made the argument that it makes sense to give emphasis on statistics instead of on calculus in school, after students have been given a basic background on algebra and arithmetic. Statistics is something most people can use in their day to day life, if only they knew how to. In contrast, people who don't study maths after school hardly find any use of calculus in their daily lives. Obviously such people don't do calculus related jobs, but most of the people are in that category. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;How can statistics be used in day to day life by common people if it&#xA;is emphasized more in the school level? What will some concrete&#xA;examples of this?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How should a two year post high school curriculum focusing on&lt;br&gt;&#xA;statistics instead of on calculus be designed to accomplish the&lt;br&gt;&#xA;above?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user101" LastEditorDisplayName="user173" LastEditDate="2015-01-04T07:51:47.890" LastActivityDate="2015-01-04T15:38:33.117" Title="Emphasizing Statistics instead of Calculus" Tags="|calculus|curriculum|statistics|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2100" PostTypeId="2" ParentId="2099" CreationDate="2014-05-01T16:49:35.250" Score="4" Body="&lt;p&gt;A question arose earlier about &quot;how is calculus applicable in students' daily lives?&quot;, and the fact is... it's not. We use basic arithmetic in our daily lives, and that's generally as far as it goes. Occupations aside, of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Statistical analysis, however, is pretty instantly applicable all over the place. You can use it to analyze how you spend your money, and use it to develop budgeting skills, and overall get a good economic sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can use statistics to measure your workouts and your diet, and find out what gives your body the effect you desire. It could be weight increase or decrease, generating muscle mass. If you consistently log your behaviour and measure your results, you can analyze this and become an expert in managing your physique. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I started doing this, and while I won't go on too much of a tangent, I'm very pleased with the results on my body. I used very basic statistics like noticing periods of change in my weight, checking my log, figuring out my &lt;strong&gt;average&lt;/strong&gt; caloric intake over that period (1-2 months), and jotting down the result. Started getting more advanced and did the same, give or take, over another period, and compared the &lt;strong&gt;standard deviations&lt;/strong&gt;. End tangent...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point is that basic statistics is immediately useful everywhere. Whether you're trying to increase your productivity, figure out how many hours of sleep render you rested and alert the next day, or even figure out what kind of diet makes your finger nails grow faster; statistics is the way to go. Simple logging and very basic, almost menial calculations, can give you sooo much information.&lt;/p&gt;&#xA;" OwnerUserId="796" LastActivityDate="2014-05-01T16:49:35.250" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2101" PostTypeId="2" ParentId="2098" CreationDate="2014-05-01T16:57:23.877" Score="7" Body="&lt;p&gt;Some ideas that may help&lt;/p&gt;&#xA;&#xA;&lt;p&gt;~&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Help them understand the point of any difference in approach&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the emphasis turns to proof or derivation from more fundamental principles, you can make the analogy of &quot;you know how to drive a car, let's learn how the engine works&quot;. They can argue that being a better driver makes you a better mechanic and visa-versa but they can't argue that knowing one means you know the other. This analogy exemplifies the link and the difference. Calling it all &quot;motoring&quot; isn't helpful in the first instance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;~&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Name topics or even whole courses to more clearly indicate new content&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A shift in direction often causes tension in the students, where they felt on solid ground with applying memorised rules in predictable contexts, but are very uncomfortable questioning why or delving into how, especially if this seems needlessly complicated to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case it's important for them to understand they're studying something other than what they were before (the engine rather than the driving) Flagging this up with language and explaining what's happening reduces the upset that students feel when the thing they're good at morphs into a different beast and also the overconfidence that comes from the familiarity of some of the answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd advocate calling topics like differentiating by taking limits things like &quot;differentiation under the hood&quot;,  &quot;the truth behind differentiation&quot; or similar phrases emphasising slightly more esoteric content. The more you can make clear what more you're expecting than their existing knowledge, the better you have communicated the content.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;~&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Acknowledge prior learning where appropriate&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taking the same calculus with limits example, you can satisfy the students' desire to have their existing knowledge recognised by saying &quot;but of course you know what the final simplified form of the derivative will be from the rules you memorised at high school, so you'll be able to quickly and easily check your final answer with high school math&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where the content is recap, making this clear can also lend more credibility to the claims of newness elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;~&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Examples from my experience lecturing in the UK&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taking some examples from UK undergraduate courses with which I'm more familiar, and where the topic is rather radically different to what they learned at high school, you could call highlight this by calling Algebra 1 &quot;Abstract Algebra 1&quot; or even just &quot;Sets, Groups, Rings and Fields 1&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I used to work at one of the UK's top mathematics departments where we ran a similarly titled course in the first term for undergraduates with the specific aim of helping them to make the transition to university mathematics. There was some resistance at first, including to the name, but it was an effective course.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the &lt;strong&gt;end&lt;/strong&gt; of the course you can then admit that at university we collectively call Group Theory, Ring Theory, Field Theory etc &quot;Algebra&quot;, and say that it's strange that we use the same name, but you can see there's a sort of link there somewhere. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Calculus&quot; could be (as appropriate) &quot;Multidimensional Calculus&quot; or &quot;Applied Real Analysis&quot;, as specific as you can be, again flagging with some words that there's something very new about this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;~&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;(edited version of answer to correct out some assumptions I'd made based on UK rather than US undergraduate courses)&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="1340" LastEditorUserId="1340" LastEditDate="2014-05-02T21:48:13.500" LastActivityDate="2014-05-02T21:48:13.500" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="2102" PostTypeId="2" ParentId="2098" CreationDate="2014-05-01T17:47:31.773" Score="19" Body="&lt;p&gt;I was unofficially advising a student the other day who told me &quot;I am struggling, but I know more than my friends in the class.&quot; This was the excuse she used for not attending meetings of a popular study group. How could she increase her grasp of the subject if her friends couldn't help her?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;An&lt;/em&gt; answer to this question (if not &lt;em&gt;the&lt;/em&gt; answer, as I believe there will be multiple possibilities) and to yours is to find some way to:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Increase interaction&lt;/li&gt;&#xA;&lt;li&gt;Increase opportunities for skills to be presented&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I told this student that she should be involved with the study group even if she would be the one helping all the other students. You are no doubt aware of the benefits to learning that lie in the act of instructing, but I was more interested in the possibility that her self-assessment would be challenged if she must use her knowledge to interact in a group on-task.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key is that students like this probably need opportunities to:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;See what their peers know, for comparison&lt;/li&gt;&#xA;&lt;li&gt;Discuss what they know, to explore their knowledge and strengthen what they do know&lt;/li&gt;&#xA;&lt;li&gt;Encounter the edges of what they know&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I can't say whether these students need a hard confrontation with the edges of their knowledge, or a slow dawning in a comfortable context in which they can encounter skill limitations, see a need for effort in that direction, start to address where to focus their efforts, and finally improve their skills and understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the key here is, as you said, the belief is &quot;in their mind.&quot; But human activity is out here in the world where we can observe it. We have to get math out of their minds, in a way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To address this practically:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I told the student that &quot;going over what you know in the presence of others will help your friends, but most of all it will help you strengthen what you know&quot; this was the truth, but I hoped it could do more for her if her self-assessment is not accurate. On the basis of that, she is going to meet with the study group. So, one course of action is to convince a student that a tutorial role in a social context helps them. This is the truth, and it is not incompatible with their beliefs about themselves, but could lead to a change in those beliefs.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-05-01T17:47:31.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2103" PostTypeId="2" ParentId="2098" CreationDate="2014-05-01T18:22:10.480" Score="12" Body="&lt;p&gt;Let me suggest two things. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First: I think a lot of the misjudgments about how much students know stem from having an incomplete or wrong idea about what the learning objectives for the course are. For example, calculus students think that the sole objective of calculus sometimes is learning how to differentiate polynomials by hand. That's part of it, but there are many other things you want students to be able to do, both computational and conceptual. If you lay these things out for them in the form of unambiguous, action verb-oriented learning objectives and then make those objectives clear to students, maybe they will see that what they know (or think that they know) is just a small subset of what needs to be known. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: Here's my learning objective list for the section on the Chain Rule (section 2.5 in Matt Boelkins' &lt;a href=&quot;http://faculty.gvsu.edu/boelkinm/Home/Download.html&quot;&gt;Active Calculus&lt;/a&gt; book): &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;(&lt;em&gt;Review&lt;/em&gt;) Apply all the derivative rules from Sections 2.1--2.4 with fluency. &lt;/li&gt;&#xA;&lt;li&gt;(&lt;em&gt;Review&lt;/em&gt;) Identify composite functions (that is, functions of the form (y = f(g(x)))) and identify the “inner” and “outer” functions. &lt;/li&gt;&#xA;&lt;li&gt;State the Chain Rule and explain how it works in English. &lt;/li&gt;&#xA;&lt;li&gt;Identify situations where the Chain Rule should be used when taking a derivative. &lt;/li&gt;&#xA;&lt;li&gt;Use the Chain Rule to differentiate a simple composite function, for example a composition of a polynomial and a power function (e.g.,  $f(x) = (x^2 + x + 1)^{1/2}$).&lt;/li&gt;&#xA;&lt;li&gt;Use the Chain Rule to differentiate a composite function involving two (more complex)  functions. &lt;/li&gt;&#xA;&lt;li&gt;Use the Chain Rule to differentiate a composite function involving three or more functions. &lt;/li&gt;&#xA;&lt;li&gt;Use the Chain Rule in combination with other rules from earlier in this chapter. &lt;/li&gt;&#xA;&lt;li&gt;Use the Chain Rule to differentiate a composite function in which at least one of the functions in the composite is given as a graph or a table of values. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I run calculus in a flipped classroom model and as such, students get these learning objectives when they are assigned the pre-class exercises to do prior to the class meeting on the chain rule -- so the students know what the learning objectives are. Students from a garden-variety AP Calculus course in high school probably have seen 2/3 of these objectives and are fluent with maybe 1/2 of them. What they need to notice is that there are some objectives that are not simple, like the last couple. Just knowing what needs to be known is a prerequisite for noticing that you don't really know it all yet. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second: Use lots of formative assessment during class to give nearly-live updates to students on what they &lt;em&gt;actually&lt;/em&gt; know -- short individual exercises, group quizzes, clicker questions, etc. that are aimed at common tasks and common misconceptions. Students say that they know more than they do because for many of them knowledge is a &lt;em&gt;feeling&lt;/em&gt; rather than an objective fact. Give them data about what they &lt;em&gt;actually&lt;/em&gt; know and their feelings should fall into line. And it will give you more info about how to teach better to meet their needs  &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-05-01T18:22:10.480" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2104" PostTypeId="2" ParentId="2099" CreationDate="2014-05-01T18:27:03.990" Score="2" Body="&lt;p&gt;Here's today's &lt;a href=&quot;http://foresteruniversity.net/webinar-simple-linear-regression.html&quot; rel=&quot;nofollow&quot;&gt;ad&lt;/a&gt; for statistics, aimed at people working with wells and sewage. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Introduction to Simple Linear Regression (SLR) &amp;amp; R&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Whether you're determining the trends of pollutant levels in&#xA;  contaminated water bodies to evaluate your watershed or looking for&#xA;  contaminant concentration reductions to assess efficiency of your&#xA;  groundwater remediation system, without statistics...your data&#xA;  analysis is dead in the water.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;So regression – your software can compute it...BUT do you understand&#xA;  it?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Join Joseph Ofungwu to get the skinny on Simple Linear Regression&#xA;  (SLR)....&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The audience of common people who might be interested in statistics is broad.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2015-01-04T07:42:17.220" LastActivityDate="2015-01-04T07:42:17.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2105" PostTypeId="1" CreationDate="2014-05-01T20:57:28.863" Score="19" ViewCount="858" Body="&lt;p&gt;From time to time, I have come across course ideas emphasizing the discrete over the continuous, such as Peter Saveliev's &lt;a href=&quot;https://calculus123.com/wiki/Fantasy_math&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fantasy Math curriculum&lt;/a&gt; (update: see also his &lt;a href=&quot;https://calculus123.com/wiki/Discrete_calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;material&lt;/a&gt; on discrete calculus) and Kelemen et al's article &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.110.2883&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Has Our Curriculum Become Math-Phobic?&lt;/a&gt;. For instance, Saveliev refers to discrete functions and data coming in digital format (no formula, just function), leading to a discrete calculus based on discrete differential forms. Kelemen et al discuss early discrete mathematics for computer science students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions&lt;/strong&gt;: What are the pros and cons of promoting the discrete and, to some extent, de-emphasizing or postponing the continuous in the first year or two of undergraduate education? What are your experiences, if any, of teaching in such an environment?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://mathoverflow.net/questions/27076/why-does-undergraduate-discrete-math-require-calculus&quot;&gt;Why does undergraduate discrete math require calculus?&lt;/a&gt; on Math Overflow Stack Exchange back in 2010.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Speculatively, it might be possible, albeit unconventional, to teach an early undergraduate course using graph theory as a springboard to introduce a range of concepts, such as linear algebra and network analysis (e.g. Kirchhoff's circuit laws),  algorithms, mathematical induction, simplicial complexes, elementary combinatorial topology, relevant aspects of discrete/computational geometry, and possibly some elements of discrete calculus. The topics could be kept as low-dimensional, visual and concrete as feasible. Graph theory is fairly rich in applications these days, so the course could potentially appeal to those with both pure and applied interests.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2019-11-21T15:05:51.123" LastActivityDate="2019-11-21T15:05:51.123" Title="Emphasizing the discrete in early undergraduate education?" Tags="|undergraduate-education|calculus|course-design|discrete-math|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2106" PostTypeId="2" ParentId="2098" CreationDate="2014-05-01T21:50:30.473" Score="27" Body="&lt;p&gt;Allow me to contribute from the student perspective. I've taken College Algebra classes three times at three different levels and schools due to unusual circumstances. To make things even more interesting, I started out as an English Lit major, and later became a computer programmer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, you have already realized that confronting the students in an aggressive manner on the first day sets you against them, and challenges them to &quot;prove&quot; that you're wrong. If you walk over to the English department you may overhear a saying popular in creative writing classes. Don't tell me, show me. One of the most common mistakes in amateur writing is to tell the reader how a person feels, rather than showing the emotion through action and description.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would submit that telling a math student that the high school class taken does not give them a complete understanding of algebra is a &quot;tell me&quot; rather than &quot;show me&quot; approach. The difference between knowledge and wisdom lies in the ability to apply knowledge to relevant situations at hand. Thus, the person who truly understands algebra is the one who can apply it successfully. In the understandably academic environment of college, I believe that sometimes gets lost. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;During my first college algebra class, I decided to look for a tutor and found a junior who was pursuing mechanical engineering. He impressed me in the first tutoring session by drawing out pictures of various engineering principles and how the equations and expressions in my textbook were useful for describing these actions. It was the first practical application anyone had ever shown me, and it completely rocked my perspective of mathematics. This guy understood how to apply the concepts, not just how to memorize or regurgitate them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider providing scenarios that can be solved by a correct understanding of the math rather than only providing equations to solve in familiar formats. Many students who memorized formulas will find that they struggle to apply their rote memorization to real world situations. Likewise, you may also discover a select few who genuinely do grasp the meaning and value of the principles they learned in high school classes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, I would propose that the value of the high school class they took is that it provided enough context and common language to begin having the discussions you will initiate at the college level. If the only conversation occurring is how well the student has memorized a particular fact or method, then both student and teacher have missed out on valuable interaction. In my opinion, your greatest opportunity is to affect the student's worldview and perspective of math, which is also where the greatest gap exists between teacher and student. I am proof that perspective can still be changed at the college level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are passionate about your craft, and you know how to apply it, then show them why it matters and what can be done with it. That alone should be enough for anyone who is humble to recognize their own limitations. Much like the difference between sharpshooting and hunting, algebra is a mathematical tool that in and of itself has little value until it is applied in some way for a useful purpose, usually to produce information that can be acted upon. It is the surprise of seeing a master do something previously unrecognized as possible or relevant that makes a novice realize and admit how little they really do know about a subject. &lt;/p&gt;&#xA;" OwnerUserId="1344" LastActivityDate="2014-05-01T21:50:30.473" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2107" PostTypeId="1" AcceptedAnswerId="2112" CreationDate="2014-05-02T00:11:05.777" Score="6" ViewCount="290" Body="&lt;p&gt;Anybody know where I could get some cuboid (but not cube) dice for use with teaching probability?  Or advice on how to make some without a 3d printer?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: In response to comments seeking clarification, I am indeed in search of the type of dice referred to in this question &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/581900/probability-of-rolling-a-cuboid-dice&quot;&gt;https://math.stackexchange.com/questions/581900/probability-of-rolling-a-cuboid-dice&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I first read about them in &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eichler, A., &amp;amp; Vogel, M. (2014). Three Approaches for Modelling Situations with Randomness. In Probabilistic Thinking (pp. 75-99). Springer Netherlands.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specific use case: I am leading a workshop for teachers on the 7th grade probability related common core standards, and such dice seem like a very nice way to illustrate the transition from the classical approach to probability to the frequentist approach, while also hitting standard &lt;a href=&quot;http://www.corestandards.org/Math/Content/7/SP/C/7/b/&quot; rel=&quot;nofollow noreferrer&quot;&gt;7.SP.C.7.b&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1346" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-05-02T17:30:11.897" Title="Where to get cuboid dice?" Tags="|probability|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2108" PostTypeId="2" ParentId="2098" CreationDate="2014-05-02T01:25:23.420" Score="4" Body="&lt;p&gt;You mentioned that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I've tried to directly address the misconception on the first day&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You should try to start with a complex and practical example, not just in order to see the reaction of your students but also to create gossip like &quot;that professor was doing some weird stuff so I was wondering if I was in the wrong place&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the next one&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I've tried letting poor grades speak for themselves&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can do your lecture more dynamic try to interact with them with simple and tricky questions, but this doesn't have the same effects if they aren't willing to talk so maybe &lt;em&gt;you should try something that is very useful in RPG games&lt;/em&gt;, which consists in give a small reward after some effort involved, this could mean &quot;extra points&quot; that are going to be managed for you so they can assure their good marks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Try to keep them interested on what you're teaching and showing them&lt;/strong&gt;. One can think that giving and update from time to time is strange on a lecture, but in this case I can bet that they will be impressed with the crazy things that are going on developing with basis in some basic theory, of course that this comes along with advanced topics, but this is just a thinking.&lt;/p&gt;&#xA;" OwnerUserId="1347" LastActivityDate="2014-05-02T01:25:23.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2109" PostTypeId="1" CreationDate="2014-05-02T03:25:13.270" Score="14" ViewCount="289" Body="&lt;p&gt;Shaffer and Kaput (1998) wrote that technology has the power to transform the mathematical thinking that is possible in classrooms, giving students new ways of experiencing mathematics as a sense-making activity:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We suggest here  that  the  ability  to  externalize  the &#xA;  manipulation  of  formal  systems changes  the  very  nature  of &#xA;  cognitive  activity.  These  changes  will  have  important&#xA;  consequences for mathematics education in coming decades. In&#xA;  particular, we argue that mathematics education in a virtual culture&#xA;  should strive to give students generative fluency to learn varieties&#xA;  of representational systems, provide opportunities to create and&#xA;  modify representational forms, develop skill in making and exploring&#xA;  virtual environments, and emphasize mathematics as a fundamental way&#xA;  of making sense of the  world,  reserving most exact computation and&#xA;  formal proof for those who will need those specialized skills.&#xA;  (p. 97)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My question is twofold:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How have you used technology in your classroom in a way that created opportunities for students to reason mathematically?*&lt;/li&gt;&#xA;&lt;li&gt;What result, if any, did you note? (For example, did students express insights related to mathematical relationships represented within the technology? Or did you have a memorable interaction related to mathematical reasoning?)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;[* Kaput (1992) describes four classes of mathematical activities in school mathematics: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Syntactically constrained transformations within a particular notation system, with or without reference to any external meanings,&lt;/li&gt;&#xA;  &lt;li&gt;Translations between notation systems, including the coordination of actions across notation systems,&lt;/li&gt;&#xA;  &lt;li&gt;Construction and testing of mathematical models, which amount to translation between aspects of situations and sets of notations,&lt;/li&gt;&#xA;  &lt;li&gt;The consolidation or crystallization of relationships and/or processes into conceptual objects or &quot;cognitive entities&quot; that can&#xA;  then be used in relationships or processes at a higher level of&#xA;  organization.&#xA;  (pp. 524-525)&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Technology could play a role in any of these types of activities. I provide them in case they help people to consider what role technology might play with respect to the mathematics in their classrooms.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Clarification:&lt;/strong&gt; I'm &lt;em&gt;primarily&lt;/em&gt; looking for people to share their personal experiences and impressions. Research you may have been involved with is, of course, also of interest.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Cited&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kaput, J. J. (1992). Technology and mathematics education. In D. Gurows (Ed.), Handbook of research on mathematics teaching and learning (pp. 515–556). New York, NY: Macmillan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shaffer, D. W., &amp;amp; Kaput, J. J. (1998). Mathematics and virtual culture: An evolutionary perspective on technology and mathematics education. Educational Studies in Mathematics, 37(2), 97–119.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-05-02T06:17:48.560" LastActivityDate="2014-05-05T16:47:03.347" Title="How have you used technology to create an opportunity for mathematical reasoning in your classroom?" Tags="|technology-in-education|classroom-interaction|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2110" PostTypeId="2" ParentId="2105" CreationDate="2014-05-02T07:30:12.827" Score="10" Body="&lt;p&gt;Analyzing data (probably by pushing a button in a statistics programme) instead of learning mathematics is not the same thing as &quot;emphasizing the discrete&quot; as in discrete mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as the discrete vs. continuous goes:&#xA;I do think that it is a good idea to teach more discrete mathematics early in the curriculum because it is kind of strange if students in the middle of their studies are comfortable with integrals yet wary of sums, understand functions, but are uncomfortable with recursive definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it is not true that discrete is generally easier to understand than continuous. Human beings have a good intuition for continuous processes from a continuous concept of space and from movement in space. Almost everyone accepts the intermediate value theorem as eminently reasonable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, from me a clear &quot;no&quot; to eliminating continuous content from the beginning of the curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as &quot;mathematics&quot; vs. &quot;data manipulation&quot; goes:&#xA;I think that it can be harmful to manipulate data without understanding, because it encourages the mindset of &quot;let us push the button for the linear approximation, if this does not work well, push other buttons&quot; among people who are actually studying important data.&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2014-05-02T07:30:12.827" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2111" PostTypeId="2" ParentId="2109" CreationDate="2014-05-02T10:55:58.567" Score="4" Body="&lt;p&gt;I am most happy about technology when it comes to dynamic processes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is just nothing like seeing something actually move, whether you rotate a transparency or whether you move a point in Geogebra/Cinderella/etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another big subject class where visualization is extremely enlightening is asymptotics. I have let my students plot the prime counting function (already implemented) and then &quot;zoom out&quot; and see how the graph gets smoother and smoother. (I let them do this prior to telling them anything about the result in the lecture.)&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2014-05-02T10:55:58.567" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2112" PostTypeId="2" ParentId="2107" CreationDate="2014-05-02T11:56:07.523" Score="5" Body="&lt;p&gt;I can think of a couple of possibilities, though I know of no ready-made source of cuboid dice.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.ericharshbarger.org/dice/&quot; rel=&quot;nofollow&quot;&gt;Eric Harshbarger makes dice for fun and profit&lt;/a&gt;. I, myself, have purchased a number of his math-related dice, including the fun &lt;a href=&quot;http://www.ericharshbarger.org/dice/#gofirst_4d12&quot; rel=&quot;nofollow&quot;&gt;&quot;Go First&quot; dice&lt;/a&gt;, four separate 12-sided dice which are guaranteed to give you a turn order for four people with one roll. Eric has a &lt;a href=&quot;http://www.ericharshbarger.org/dice/custom_dice.html&quot; rel=&quot;nofollow&quot;&gt;special page for custom dice&lt;/a&gt;, since he gets requests. I think your cuboid dice fall outside of his usual orders, but it may be worth contacting him.&lt;/li&gt;&#xA;&lt;li&gt;You could craft your own dice out of &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B001C4RTP2&quot; rel=&quot;nofollow&quot;&gt;Super Sculpey (Firm)&lt;/a&gt;. I and my daughters use this stuff all the time to make small, semi-durable objects. The firm version is a little tougher to work with (like heavy clay), but it holds its shape very well while curing. You could make a jig of some sort with the shape you want, cut off the approximate amount of Sculpey, force it into the jig, unmold, then bake. Possibly paint and lacquer as a finishing step. This is what I would do if I needed cuboid dice tomorrow (I keep this stuff on hand for crafting and object-making).&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.prettygamer.com/2010/09/arts-crafts-with-sculpey-how-to-make.html&quot; rel=&quot;nofollow&quot;&gt;Here's a &quot;How To&quot;&lt;/a&gt; for making stylized dice out of the stuff.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Making the dice: In thinking about the &quot;jig&quot; I'm not sure there is an easy way to do that. Perhaps a better method would be to use flat surfaces to form the dice. Start by cutting off the Sculpey from the block at the approximate size and shape you want. Weigh the thing with a decent scale that gives you fractions of grams so that you can more reliably make a second one of the approximate same size. Weight is going to be more helpful than anything else here. Then, use flat surfaces to gently square off the sides (it should already be the approximate cuboid shape when you cut it). Be gentle, so as not to deform it. Once you have the desired number of dice, bake. &lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-05-02T12:01:35.273" LastActivityDate="2014-05-02T12:01:35.273" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2114" PostTypeId="2" ParentId="2060" CreationDate="2014-05-02T13:48:20.440" Score="4" Body="&lt;p&gt;When you want to discuss the rate something happens, you will find that differential equations of calculus are helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some examples in biology:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;population growths: dx/dt = Rx, describes unlimited/exponential growth of a population that could be rabbits, cells, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;kinetics of a chemical reaction: &#xA;reversible [A][B] &amp;lt;-&gt; [AB].  d[AB]/dt = k1*[A][B]-k2[AB]&#xA;formation rate of d[AB]/dt slows as you use up [A] and [B]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1353" LastActivityDate="2014-05-02T13:48:20.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2115" PostTypeId="2" ParentId="2088" CreationDate="2014-05-02T15:35:49.667" Score="3" Body="&lt;p&gt;My answer is similar to Jsor's, so if it seems redundant, say so in the comments and I'll remove it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have weekly quizzes, each with minimal grade impact. In addition, their grade on a quiz is always replaceable by their grade on the exam nearest that quiz. Finally, the problems on my exams are only minorly different from the problems on my quizzes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Altogether, these policies encourage students to learn from mistakes they make on the quizzes and to be willing to make such mistakes, since they can recover those points by doing well on the exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This brings me to the philosophical point: in order to encourage students to seek understanding rather than knowledge, you must allow (or perhaps even encourage) mistakes. When we reflect, most of us realize that actually making mistakes and seeing why they were mistakes is among the most productive ways to gain understanding. If we actively discourage this in our classrooms, we are thereby actively discouraging one very fruitful way of gaining understanding.&lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-05-02T15:35:49.667" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2116" PostTypeId="2" ParentId="2081" CreationDate="2014-05-02T15:45:02.263" Score="8" Body="&lt;p&gt;I think the answer to Question 1 is that just as numbers are fundamentally about quantities and topology is fundamentally about space, groups are fundamentally about symmetry. We often overlook the &quot;what it's about&quot; part of a branch of mathematics when we look for clean ways to exposit the material, and we can then lose sight of the fact that if we don't know what the math is about then we don't &lt;em&gt;understand&lt;/em&gt; it at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, why don't we teach free groups first? For exactly the reason you think we should: &quot;[it] capture[s] the more basic idea that groups are sets that have additional structure.&quot; Unfortunately, from this point of view most theorems in group theory seem like they are silly. For example, why all the emphasis on homomorphisms rather than merely functions between groups? If they're really just sets, functions seem fine. From there, why all the emphasis on normal subgroups? Why not arbitrary subsets? Even if you had the idea to look at normal subgroups, why worry about factor groups? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end of the day, while &lt;em&gt;groups&lt;/em&gt; may not care how you present them, &lt;em&gt;group theory&lt;/em&gt; has teleology built into it. From the point of view of the developed theory of groups, a group is, fundamentally, something that codifies symmetry. If our job as instructors is to introduce our students to mathematical culture, then, we can't ignore the teleology of the theory when we present it.&lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-05-02T15:45:02.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2117" PostTypeId="2" ParentId="2039" CreationDate="2014-05-02T15:59:28.957" Score="3" Body="&lt;p&gt;I've found very little good information about how many classrooms use DGS, though Sketchpad does &lt;a href=&quot;http://www.dynamicgeometry.com/General_Resources/The_Sketchpad_Story.html&quot; rel=&quot;nofollow&quot;&gt;boast&lt;/a&gt; on its website: &quot;Each new school year, tens of millions of students around the globe encounter mathematics through Sketchpad's Dynamic Geometry.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't seem to find a percent, so let me instead argue indirectly why I think the number of classrooms using DGS is &lt;em&gt;lower&lt;/em&gt; than most people would expect. I'll begin my answer internationally, then discuss the U.S.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In many or most schools around the world, an important reason DGS is not adopted by schools is a lack of resources including but not limited to availability of computers and internet access.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Starting in South Africa (where I teach), there are serious problems with internet connectivity and student access to computers. Our school has a computer lab and (unreliable) internet access, but we have not adopted DGS despite the fact that we are spending far more on computers and internet than nearly other schools in South Africa could afford.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.infodev.org/infodev-files/resource/InfodevDocuments_429.pdf&quot; rel=&quot;nofollow&quot;&gt;This data&lt;/a&gt; from 2005 indicates that only about 20% of South African government schools have &quot;computers for teaching and learning.&quot; From there it's reasonable to infer that &lt;em&gt;very&lt;/em&gt; few schools in South Africa (and the rest of Africa) &lt;strong&gt;could&lt;/strong&gt; implement DGS level 3 or 4, even if the schools had reliable internet and administration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.uis.unesco.org/Communication/Pages/ict-education-report-asia.aspx&quot; rel=&quot;nofollow&quot;&gt;Here is data&lt;/a&gt; from 2012 showing student-to-computer ratios for Asia. The accompanying &lt;a href=&quot;http://www.uis.unesco.org/Communication/Documents/ICT-asia-en.pdf&quot; rel=&quot;nofollow&quot;&gt;report&lt;/a&gt; indicates that many countries (though India is a notable exception) have roughly enough computers dedicated to &quot;pedagogical purposes&quot; to make DGS feasible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar reports from UNESCO are available for &lt;a href=&quot;http://www.uis.unesco.org/Communication/Pages/ict-regional-survey-lac-2012.aspx&quot; rel=&quot;nofollow&quot;&gt;Latin America and the Caribbean&lt;/a&gt; and &lt;a href=&quot;http://www.uis.unesco.org/Communication/Pages/ict-survey-arab-states.aspx&quot; rel=&quot;nofollow&quot;&gt;5 Arab states&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the U.S., &lt;a href=&quot;http://www.census.gov/prod/2013pubs/p20-569.pdf&quot; rel=&quot;nofollow&quot;&gt;census data&lt;/a&gt; indicate that private internet use is relatively high, though it is still somewhat surprising to note that over 30% of Americans in 2011 used little to no internet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as U.S. schools are concerned, computer access shouldn't be too much of an issue, but there is some concern that &lt;a href=&quot;http://www.theatlantic.com/education/archive/2013/11/more-than-half-of-us-public-schools-dont-have-adequate-wireless-access/281410/&quot; rel=&quot;nofollow&quot;&gt;internet access is limited&lt;/a&gt; for &lt;a href=&quot;http://mashable.com/2013/08/18/digital-divide/&quot; rel=&quot;nofollow&quot;&gt;some schools&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;However&lt;/strong&gt;, full-scale DGS use requires &lt;strong&gt;HUGE&lt;/strong&gt; amounts of resources compared to nearly all other computer and internet related activities, such that mere access to a computer and internet is &lt;strong&gt;not&lt;/strong&gt; enough! For example, while online learning might initially seem more difficult than DGS, it is far more readily available nowadays to anyone with a computer and internet connection (e.g. Khan Academy, Coursera, etc.) than classroom DGS. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast to online learning, here are the &lt;em&gt;minimum&lt;/em&gt; resources a school needs in order to effectively reach DGS level 3 or 4:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;strong&gt;School computers and internet connection&lt;/strong&gt;. Note that this is the &lt;em&gt;only&lt;/em&gt; thing I've discussed above, since international data on this topic is more readily available.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;Supportive administration&lt;/strong&gt;. A dedicated teacher might make one class reach DCG level 3 or 4, but without both administrative and technical support, this will have little effect on the whole school. Note that this entails an additional cost (especially in terms of time and management) on top of #1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &lt;strong&gt;Teachers.&lt;/strong&gt; Teachers must understand and teach using DCG, then have patience to deal with students and ensure that they use DGS in a meaningful way. For most schools, DCG would require additional teacher training, entailing yet more costs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, I would argue that while DGS adoption will almost certainly increase with time, there are presently too many costs and hassles - even for affluent schools - and not enough incentives for schools to implement DGS above level 2.&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="867" LastEditDate="2014-05-02T16:06:29.363" LastActivityDate="2014-05-02T16:06:29.363" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2118" PostTypeId="2" ParentId="2107" CreationDate="2014-05-02T17:30:11.897" Score="1" Body="&lt;p&gt;How about not using dice, but a top or roulette? Mark sectors with colors/numbers; by adjusting the sizes you get whatever probability distribution you want.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-02T17:30:11.897" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2119" PostTypeId="2" ParentId="1899" CreationDate="2014-05-02T18:59:17.853" Score="13" Body="&lt;p&gt;Since you remark that your question is &quot;deliberately non-specific,&quot; here is a (necessarily) incomplete response: First are two links to documents about assessment that might be of interest, and then two grading schemes that I have encountered in mathematics courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Documents:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as the philosophy of creating examinations, early work on this was done by former &lt;a href=&quot;http://en.wikipedia.org/wiki/American_Educational_Research_Association&quot; rel=&quot;noreferrer&quot;&gt;AERA&lt;/a&gt; president &lt;a href=&quot;http://en.wikipedia.org/wiki/Michael_Scriven&quot; rel=&quot;noreferrer&quot;&gt;Michael Scriven&lt;/a&gt;. In particular, he was the first to distinguish between &lt;em&gt;formative assessment&lt;/em&gt; and &lt;em&gt;summative assessment&lt;/em&gt;; this was done in &quot;The Methodology of Evaluation&quot; (1966) which can be found in full (as a pdf) &lt;a href=&quot;http://www.comp.dit.ie/dgordon/Courses/ILT/ILT0005/TheMethodologyOfEvaluation.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. The citation is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Scriven, M. (1966). &lt;em&gt;The methodology of evaluation&lt;/em&gt; (Vol. 110). Boulder, CO: Social Science Education Consortium.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For a more recent discussion of formative vs. summative assessment, see, for example, the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Taras, M. (2005). Assessment–summative and formative–some theoretical reflections. &lt;em&gt;British Journal of Educational Studies, 53&lt;/em&gt;(4), 466-478. &lt;a href=&quot;http://www.jstor.org/stable/3699279&quot; rel=&quot;noreferrer&quot;&gt;Jstor link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is a relevant excerpt from the latter link:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Tsiqv.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Insofar as creating (mathematics) examinations is concerned, it will be important for you first to decide whether you intend the assessments strictly as summative instruments, or whether you wish them to serve a formative purpose as well. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Grading schemes:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are two that I have seen; I am sure other MESE users could provide many others.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Indicate to students roughly what will be covered; give three examinations; multiply the better two scores by 3, then take an average of the tests (i.e., if scores are $a \geq b \geq c$, then use $(3a+3b+c)/7$ as the student's average). Subsequently, use the course final as at most half of the student's grade; in particular, it can be worth more than half of the grade if the score is &lt;em&gt;higher&lt;/em&gt; than the average going into the test.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Set the total number of points for a course at $N$ and allow for some fixed number of points that can be earned during the semester (through examinations, quizzes, etc). Make the final examination worth $N-($student's earned points$)$. The consequence here is that everyone enters the final with the &lt;em&gt;possibility&lt;/em&gt; of getting full marks for the course, but also wanting to have accumulated points during the term.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-05-03T01:33:38.877" LastActivityDate="2014-05-03T01:33:38.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2120" PostTypeId="1" AcceptedAnswerId="2420" CreationDate="2014-05-02T21:49:42.877" Score="8" ViewCount="354" Body="&lt;p&gt;Since the beginning of time, candy has been used to illustrate the principles of distributions in early lessons on statistics.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.csis.ysu.edu/~chang/student_proj/S3743_s02/s3743s02Measel.pdf&quot;&gt;A recent informal study&lt;/a&gt; seems to indicate that the flavors of Skittles are not uniformly distributed.  While it would be a great exercise for more advanced students to ascertain the extent to which a given bag is distributed evenly, it does not serve as hefty a purpose in the very early stages of learning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have any educators or statisticians systematically studied different varieties and brands of candy expressly for their pedagogical value in primary- and secondary-level statistics?  &lt;/p&gt;&#xA;" OwnerUserId="16" LastActivityDate="2014-05-25T06:23:41.370" Title="Who put the skew in my Skittles? Have there been studies as to which candy has a fairly uniform distribution of flavors?" Tags="|secondary-education|primary-education|statistics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2121" PostTypeId="1" CreationDate="2014-05-03T00:38:09.950" Score="13" ViewCount="1086" Body="&lt;p&gt;My girlfriend is studying to become a math teacher, and asked me what I have ever used symmetries for. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a web developer, and do some designing from time to time, so I answered that symmetry have given me a general understanding of aesthetics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, that answer didn't quite work for any of us, so now I'm asking you:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the reason we are taught symmetry in school?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the practical usages of symmetries in our everyday lives?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Preferably I'm looking for documented answers, but any answer will do at the moment. &lt;/p&gt;&#xA;" OwnerUserId="1357" LastEditorDisplayName="user507" LastEditDate="2021-05-07T18:42:00.407" LastActivityDate="2021-05-08T02:19:22.733" Title="Symmetry - practical uses" Tags="|curriculum|primary-education|geometry|applications|" AnswerCount="13" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2122" PostTypeId="2" ParentId="2121" CreationDate="2014-05-03T02:38:41.387" Score="5" Body="&lt;p&gt;Schiralli (2007) points out that our sense of the aesthetic is part of our ability to engage in formal scientific and mathematical endeavors: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;The identification of pattern is, therefore, a fundamentally&#xA;  aesthetic apprehension that in systematic inquiry soon moves beyond&#xA;  the immediately perceptable towards the more formal conceptual&#xA;  connections with which scientific and mathematical theory is&#xA;  ultimately concerned.&quot; (p. 106)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For a practical example, Frank Wiczek notes in the introduction to Weyl's (2009) Philosophy of Mathematics and Natural Science:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Time-reversal symmetry asserts the equivalence of past and future, in&#xA;  the &lt;em&gt;microscopic laws of physics&lt;/em&gt;. Of course, both the concrete history&#xA;  of the universe and (at a mundane, but more specific and practical&#xA;  level) the laws of thermodynamics distinguish past and future. Yet the&#xA;  laws of Newtonian mechanics and Maxwellian electrodynamics--and&#xA;  indeed, all the basic laws of the microcosmos known in Weyl's day--do&#xA;  not. (p. xi)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wiczek explains that symmetry has played a role in theory, and that violations of theoretical symmetry are important in that they have lead to questions that powered major advances in physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But maybe you wanted a more geometric example. Patterns in math are so often useful, and symmetry is no different. Lets say my pool is some strange shape. I want to calculate the volume of water in the pool so I know how much chlorine to use, but I don't have a formula for the shape in question. But then I notice that the pool is symmetrical in some way (choose whatever type of symmetry you prefer). The symmetry involves a shape I &lt;em&gt;do&lt;/em&gt; have the formula for. So I calculate the volume of that shape, then iterate accordingly (or just multiply). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps you meant something different, but noticing symmetry in a system always gives you a way to reduce the complexity in how you represent the system. In the reverse, from a software or computer graphics standpoint, symmetry gives you a way to generate forms programatically by transformation of other forms.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Cited&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schiralli, M. (2007). The meaning of pattern. In N. Sinclair, D. Pimm, &amp;amp; W. Higginson (Eds.), Mathematics and the Aesthetic (pp. 105–125). Springer. Retrieved from &lt;a href=&quot;http://link.springer.com/chapter/10.1007/978-0-387-38145-9_6&quot; rel=&quot;noreferrer&quot;&gt;http://link.springer.com/chapter/10.1007/978-0-387-38145-9_6&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Weyl, H. (2009). Philosophy of Mathematics and Natural Science. Princeton University Press.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-05-03T02:38:41.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2123" PostTypeId="1" AcceptedAnswerId="2143" CreationDate="2014-05-03T03:18:40.613" Score="2" ViewCount="2093" Body="&lt;p&gt;Why are math textbooks so often boring?  It requires some mental discipline to do math; cartoons can help make the principles easier to digest. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Mathematical Association of America has many FUN artistic Advanced Math books. Maybe more Math textbooks could be written with an emphasis on interesting summaries first $BEFORE$ filling in the precise details? I think learning a lot of details first that you have to memorize tend to obscure a general appreciation of the  whole 'layout' of the subject. Look at the success of the 'For Dummies' series like Algebra For Dummies.. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in summery, why are math books uninspiring? Could Math Books be written with an emphasis on fun informal summaries of various concepts before the precise details are 'filled in'?&lt;/p&gt;&#xA;" OwnerUserId="1359" LastEditorUserId="60" LastEditDate="2015-07-24T04:39:43.410" LastActivityDate="2015-07-24T04:39:43.410" Title="Why are most math textbooks so uninspiring; unless you already like Math?" Tags="|textbooks|mathematical-pedagogy|" AnswerCount="3" CommentCount="15" ClosedDate="2014-05-05T02:14:44.970" ContentLicense="CC BY-SA 3.0" />
  <row Id="2124" PostTypeId="2" ParentId="2121" CreationDate="2014-05-03T03:43:16.317" Score="11" Body="&lt;p&gt;Symmetries give profound insight into structure.  This is demonstrated by &lt;a href=&quot;https://en.wikipedia.org/wiki/Noether%27s_theorem&quot;&gt;Noether's theorem&lt;/a&gt;, proved just under 100 years ago by Emmy Noether (probably the greatest female mathematician in history).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Noether's theorem states that every symmetry of a physical system is there because something is being conserved.  Examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Translational symmetry -&gt; conservation of momentum&lt;/li&gt;&#xA;&lt;li&gt;Rotational symmetry -&gt; conservation of angular momentum&lt;/li&gt;&#xA;&lt;li&gt;Time symmetry -&gt; conservation of energy&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="270" LastActivityDate="2014-05-03T03:43:16.317" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2125" PostTypeId="2" ParentId="2098" CreationDate="2014-05-03T06:40:08.320" Score="3" Body="&lt;p&gt;The quickest way to reframe this misconception is to early and repeatedly acknowledge/accept that your students already know a lot and together you are going to use that understanding to take them to the next level.  In fact you reassure them that it is good they have a solid base on which to build.  (Just be sure not to scare any students with a more shaky understanding)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You then need to confront them with reality, not by head butting them but by reviewing, you quickly review what you think they know, constantly testing they do actually understand the basic principals, and reassuring them that it is good they already know most of this stuff, until you hit the point where they become unsure.  You then change your language so that this is what they are going to learn and you give them a summary of what you are going to teach them.  Depending on the size of your class and the variability in them, you may acknowledge that some students already know this stuff, others are going to have to work really hard and catch up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to do this both at the start of a semester in a more general way and as we begin each new topic, it is almost a joke between me and my 'regular students'.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is always possible that some of your students actually do know that you are teaching them, you then have to find something to do with them, let them move on, use them to teach the others, or expand the curriculum so they do learn, I guess this depends on their and yours reason for doing the class. &lt;/p&gt;&#xA;" OwnerUserId="910" LastActivityDate="2014-05-03T06:40:08.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2126" PostTypeId="2" ParentId="2060" CreationDate="2014-05-03T07:42:39.927" Score="6" Body="&lt;p&gt;Some years ago I taught a one-semester course on mathematics for pharmacy students.  (They also got a semester of statistics in another course.)  I looked at some of the second and third-year prescribed books for the pharmacy degree and they had quite a lot of calculus in them.  Physical pharmacy: rates of diffusion of various things.  Interpreting the elimination of a drug given orally from the body by looking at measurements in the blood at different times: the drug goes first into the stomach and then into the bloodstream, so you end up with two coupled DEs (or even three, if some organ or tissue is acting as a reservoir).  Chemistry: in pharmacy you are generally dealing with weak acids and weak alkalis, so the situation is considerably more complicated than in the usual beginning chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly things like semi-log plots occurred quite a lot - not exactly calculus, but often taught with it.  And we did teach the trapezoidal rule!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There wasn't any other maths/stats as such except the two one-semester courses in the Pharmacy program.   They did a lot of chemistry and biology, and specialised courses on Pharmacy topics.  This course was in Australia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a bit surprised at the Pharmacology major mentioned above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I would say that anyone who is good at both maths and biology has got some fantastic opportunities.&lt;/p&gt;&#xA;" OwnerUserId="1361" LastActivityDate="2014-05-03T07:42:39.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2127" PostTypeId="2" ParentId="2121" CreationDate="2014-05-03T07:43:10.857" Score="9" Body="&lt;p&gt;People tend to use the presence of symmetry in a phenomenon to simplify a model of it. For example, in physics, systems with certain symmetries are the easiest to model (e.g., an infinite plane, an infinitely long cylinder, a sphere, etc.).  But we have to be careful here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Ian_Stewart_%28mathematician%29&quot;&gt;Ian Stewart&lt;/a&gt; has a very nice article on symmetry-breaking.  (The article is from his column in &lt;em&gt;Pour La Science&lt;/em&gt; and was reprinted as &quot;Curie's Mistake&quot; Chapter 10 of &lt;em&gt;Another Fine Math You've Got Me Into...&lt;/em&gt; (New York: W. H. Freeman and Company, 1992).)  Aside from presenting examples of symmetry found in nature, he also presents examples in nature where symmetry breaks (like why certain animals have stripes and why others have spots).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stewart discusses the question &quot;How does the symmetry of a system affect its behavior?&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A famous answer was given by the great physicist Pierre Curie [...].  In 1894 Pierre Curie gave two logically equivalent statements of a general principle from the folklore of mathematical physics.&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;If certain causes produce certain effects, then the symmetries of the causes reappear in the effects produced.&lt;/li&gt;&#xA;  &lt;li&gt;If certain effects reveal a certain asymmetry, this asymmetry will be reflected in the causes which give rise to them.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Stewart then goes to see if Curie was correct.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For example, in the Kensington Science Museum in London there is an engineering model of a passenger jet, used in a wind tunnel to study the flow of air around the aircraft.  Since the aircraft is bilaterally symmetric, the engineers built only half of the model---tacitly assuming that the air flow had to be bilaterally symmetric as well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It turns out that Curie was not correct.  There are many instances in nature where symmetry breaks, instances where a symmetric state becomes unstable.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2014-05-03T07:51:02.317" LastActivityDate="2014-05-03T07:51:02.317" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2128" PostTypeId="2" ParentId="2121" CreationDate="2014-05-03T09:23:03.020" Score="4" Body="&lt;p&gt;Symmetry in polynomials gives ways to find zeroes. I.e., &#xA;$$&#xA;x^n + a x^{n -1} + b x^{n -2} + \ldots + b  x^2 + a x + 1&#xA;$$&#xA;can be reduced in degree by $x = y + 1/y$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One often looks for symmetries in problems to simplify solving them, or to check solutions. I can't give a concrete example right now, but it is where I start.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-03T09:23:03.020" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2129" PostTypeId="2" ParentId="2098" CreationDate="2014-05-03T13:01:21.470" Score="2" Body="&lt;p&gt;Personally,  I address the issue in the following way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume for simplicity that the textbook has an initial chapter with preliminary material. I tell them &quot;How do you know if you are ready for the class?  You should be able to take exercises from that chapter, and you should be able to get the correct answer, on your own, without asking for help,  without looking at the text, and in a reasonable amount of time&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use this same idea for them to assess whether they are ready for a test.&lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2014-05-03T13:01:21.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2130" PostTypeId="2" ParentId="2081" CreationDate="2014-05-03T20:15:20.073" Score="10" Body="&lt;p&gt;Describing groups by generators and relations (i.e., as quotients of free groups) is not a good way to describe them most of the time. There are some exceptions, Coxeter groups, braid groups, but mostly groups do not arise in that fashion. Further, &quot;the word problem&quot; is undecideable, so it's not the case that there are algorithms (much less efficient ones) to tell whether different presentations give the same group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;applications&lt;/em&gt; of group theory to the rest of mathematics are mostly &lt;em&gt;not&lt;/em&gt; about the combinatorics of strings, which is what generators-and-relations defaults to. Invariably, there are structural features of groups-that-we-care-about, which are often not &quot;merely combinatorial&quot;, but refer to often-occurring supplemental features.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-05-03T20:15:20.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2131" PostTypeId="2" ParentId="2081" CreationDate="2014-05-03T21:40:55.520" Score="4" Body="&lt;p&gt;Lots of groups, important for lots of applications, are &lt;strong&gt;ugly&lt;/strong&gt; when presented as generators and relations.  Will you teach the key example of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the free abelian group on $\{a_n: n\in\mathbf{N}\}$ modulo the relations $n a_n = a_{n-1}$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If not this, how do you present $\mathbf{Q}$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do you get students comfortable with an uncountable set of generators for $\mathbf{R}$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What generators and relations would you choose for $M_n(\mathbf{C})$?&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-05-03T22:06:34.043" LastActivityDate="2014-05-03T22:06:34.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2132" PostTypeId="2" ParentId="2081" CreationDate="2014-05-03T22:12:20.527" Score="2" Body="&lt;p&gt;Question 1: &#xA;I wouldn't want to present everything in free groups. However, as an early example it could be useful as it takes you outside the world of things which are necessarily finite. Also, if your goal is to expose students to abstract examples, to take them outside their comfort zone, then it's probably a nice idea. So it depends on your audience. What kind of student are you challenging? Can they accept a challenge or do they need to stay closer to the world which they already have experience? The word problem, the problem of deciding if two presentations are equivalent is a bit unsettling to students who want cut and dried answers of how to solve problems. In short, it seems to me a few free groups in your lexicon of examples adds a certain depth to the course which is nice. Especially for those students who go on to work with free groups in later courses. That said, using only free groups introduces needless technical complications that would obfuscate the true geometric origin of many examples.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-03T22:12:20.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2133" PostTypeId="2" ParentId="2081" CreationDate="2014-05-03T22:36:43.990" Score="5" Body="&lt;p&gt;If you ask, &quot;Why not begin with free groups?&quot; You might first consider what a free group is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although you can define them in term of equivalence classes of words, their true definition is in terms of their universal property. I have a hard enough time getting grad students to digest this. Good luck with undergrads first learning what a group is. Related to this, have you considered how difficult it is to prove anything about a group defined by generators and relations? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dummit and Foote's Abstract Algebra introduces generators and relations informally early on, but even here we don't really ever prove things about these groups rigorously. Issues like how we know $A\not=B$ cannot be addressed without appealing to the universal property.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Plus if you want clean generality, why not just do permutations (i.e. 1800's style)?&lt;/p&gt;&#xA;" OwnerUserId="453" LastActivityDate="2014-05-03T22:36:43.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2134" PostTypeId="2" ParentId="2039" CreationDate="2014-05-03T23:16:46.447" Score="2" Body="&lt;p&gt;I do not believe a study has been done to determine the percentage of secondary teachers actually using DGS in their classrooms, with levels that indicate how the software is actually used. I did some poking around and the result is that I believe there is market data that does not answer your question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found &lt;a href=&quot;http://i2geo.net/files/i2g_status_quo_report_jan2008.pdf&quot; rel=&quot;nofollow&quot;&gt;this publication which surveys DGS use in other countries&lt;/a&gt;. Not very detailed, but I thought I would share it with you. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth. I asked people who had been involved with research in DGS, and nobody I asked indicated they were aware of such a study being done. Good luck.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-05-03T23:16:46.447" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2135" PostTypeId="2" ParentId="2098" CreationDate="2014-05-04T05:04:14.800" Score="1" Body="&lt;p&gt;Hmm, from a student's perspective there are a few things I want to point out.  I don't mean to be attacking here, but you say that the students believe their mistakes are trivial, aka, &quot;stupid mistakes.&quot;  I am curious how you have ascertained that they were not stupid mistakes?  This is something that can often be difficult to determine if a student does not show all or almost all of their work (and thought process) outside of speaking individually with every student (which is often impossible in larger intro courses).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing is that you may have students that have genuine understanding but poor memories.  In this case what is going on is that they understand, but because they understand they are perhaps not devoting enough practice to have the formulas and proofs ready in their heads when they need them.  This might sound like a lazy pupil, but there are many demands on a student's time from all directions and the first thing to take a back seat is going to be the one they feel most confident in.  Some have suggested that they should be able to solve problems without the book, that does not necessarily test understanding.  I'm someone that has quite a poor memory, in fact I failed memorization heavy subjects like Biochem. the first time around because of it, but my understanding is strong and I have done well enough on homeworks to carry me through and pass despite poor exams scores due to my bad memory in many math courses.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion the best things you can do is to have as much in class practice as possible with immediate feedback as another poster suggested.  Also, really try to schedule your office hours for directly after classes so that students that didn't get problems that were just done in class can get further explanations and help while the problems is still fresh in their mind.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not really seeing how they can think they understand but not understand that they don't unless you're giving them problems that are more ambiguous than you realize (given your bias as someone that already knows the intended solution).  My most fair professors have been those that would evaluate your answer based on your stated assumptions as long as the assumptions are not counter to the problem statement (aka in a somewhat ambiguous case you can pick one and go with it without being penalized).  To be a little more clear, a lot of profs I have met think they're being very clear when they are in fact being quite ambiguous and it would only be clear if you considered the context of recent material covered (but math is not subjective and profs should not make these assumptions, they should be explicit).  That may sound reasonable to an instructor, but theoretical problems can be solved many different ways and just because I'm in topology doesn't mean I shouldn't solve a problem with a proof from modern algebra.  The topology proof may be easier, but the modern algebra one shouldn't be marked incorrect just because it wasn't what the prof had in mind.  I've had profs mark many things like this incorrect, even if the proof was from the same course and just not the one they were thinking of.  This is partially because in order for them to know my proof was correct they would need to put a good deal of thought into understanding what I did in my proof, which would take ages to do for everyone.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes (often old timer profs) will not accept that my proof is correct, they are unable to articulate a reason that it is not correct, but claim it is wrong because it isn't how they solved it, or how their text did.  I have encountered many other cases where after speaking with the professor directly they would look at it seriously and return points (if I was in fact correct, otherwise they would show me where I went wrong, which is something I really appreciate).  Knowing one way works isn't good enough for me, I also want to know why the other way doesn't work.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you may get more helpful answers if you're more explicit about what things they think they know and do not.  Do I understand trigonometry, yep, you bet I do.  Can I solve a calculus problem without using references that is only made doable by hand by knowing a relatively obscure trigonometric transform, nope, no chance in hell I'm going to remember that in say my modern algebra class when I haven't had a calc class since my freshman year (I was done Differential Equations while in high school and took Partial Differential Equations freshman year.  Needless to say you don't remember these trig function when they randomly show up (in 1-2 proofs the whole semester in modern algebra, but they crush you out of the blue, and then occasionally out of nowhere in physics courses) 2-3 years later.  Of course you remember the basic transforms, but the more exotic ones, nada, and I see no reason for profs to expect students to have this knowledge after such a long time, but they expect it anyway and if you don't know, then they're all, &quot;but you said you took a course in trigonometry.&quot;  Yeah... three years ago you jerk.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know this may have seemed personal, but the idea is to try and give you a little perspective on the frustrations that even very bright students have in courses.  I know that a few years after finishing courses it is easily forgotten what it's like to have a full 18 credit course load with two majors (where one actually required more courses than was technically allowed by the university's policy) and being expected to have magically had time to re-memorize everything you did in every previous course in the subject dating back as far as five or six years to high school courses by the time you're graduating.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For all I know your students are idiots that really don't get that they don't know something... but keep all this in mind just in case they aren't.  Also, I know I said before to have office hours right after class, but also discuss what other time would be good with your students so that maybe once a week they will be at a different time for those with a course right after yours.  Mmmk, I think I hit all the major points there.  Please excuse the scattered train of thought, neither of my majors were English.  &lt;/p&gt;&#xA;" OwnerUserId="1367" LastActivityDate="2014-05-04T05:04:14.800" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2136" PostTypeId="2" ParentId="805" CreationDate="2014-05-04T05:14:22.607" Score="2" Body="&lt;p&gt;The reason I went to university was to gain experience dealing with time-pressure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exam questions rarely require too much thinking---they are there to test familiarity with concepts and the deftness with which students can apply techniques.&lt;/p&gt;&#xA;" OwnerUserId="1369" LastActivityDate="2014-05-04T05:14:22.607" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2137" PostTypeId="2" ParentId="2098" CreationDate="2014-05-04T12:18:31.287" Score="2" Body="&lt;p&gt;You asked for a productive way to fix the misconception. Here's one: &lt;em&gt;Nobody really knows it!&lt;/em&gt; Always incorporate enough content to reflect this. Try to construct a series of problems that moves from the very basic to the extremely tricky (think Putnam Exam level).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is, any course is just an introduction to a topic. Just because I teach Calculus doesn't mean I know it. This is the real misconception that we need to debunk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the &quot;stupid mistakes&quot;, just because you can throw a ball doesn't make you Nolan Ryan.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2014-05-05T00:58:25.977" LastActivityDate="2014-05-05T00:58:25.977" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2138" PostTypeId="2" ParentId="805" CreationDate="2014-05-04T14:01:00.923" Score="4" Body="&lt;p&gt;Giant disclaimer: I am a future math undergrad with experience in completely separate fields. So far I've just done high school and &quot;hobby&quot; math, i.e Numberphile and the like. This is also my very first post.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is the option of doing two exams, one for &quot;pass&quot; and one for showing greater depth. The first one being a greater number of easier questions with a larger questions/time quota, while the latter is just a few different problems to battle with that require greater insight. &#xA;I know this is used in a few engineering programs in Sweden, but I don't know how successful it's been. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another option is giving homework that gives credit in the upcoming exam. I'm also familiar with the x-amount of questions answered in A/B/C-difficulty version, where the student is not required (nor expected, in some cases) to answer everything, but at least gets the opportunity to show what level they're at. This is where I usually (as a student) get to show that I'm terrible at grinding out questions, but not too shabby in understanding difficult concepts. More than once I've answered the difficult once but failed the &quot;easy&quot; ones. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess they all have their pros and cons, but I think it's quite clear that the traditional time pressure test is not the only one, nor necessarily the best.&lt;/p&gt;&#xA;" OwnerUserId="1375" LastActivityDate="2014-05-04T14:01:00.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2139" PostTypeId="2" ParentId="1655" CreationDate="2014-05-04T15:03:04.553" Score="32" Body="&lt;p&gt;I agree with @kathleen, that being a young (and female) instructor can lead some students to be less respectful. In this case, we can use that to our advantage...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long ago, when I was less secure as a teacher, I would be so embarrassed and bothered by making a mistake that it would decrease my ability to teach well for the rest of that class period. I knew that I couldn't be sure of never making mistakes, so early on in my teaching career I came up with a way to make those mistakes a positive part of classroom culture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the beginning of the semester, I explain that I will make mistakes from time to time, and I want students to catch them, so they do not stay on the board, and get written into students' notes. Every mistake a student catches is a 'donut point'. When the class has caught me 30 times over the course of the semester, I will bring in donuts. I tell them that there is a second reason for this system. Students too often believe anything a math teacher says, and that's not a good way to learn math. They &lt;em&gt;should&lt;/em&gt; be questioning (in their minds) everything I say. Students like this system. Some try too hard to catch me, and I joke about them being awfully hungry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, when someone thinks there is a mistake, and I show there wasn't, they are usually apologetic. I say that's no problem, because they have been brave enough to trust their own reasoning, which is a good way to approach math. If the person were to continue to insist (this hasn't happened to me, at least not in many years), I'd ask the class how many followed my reasoning. If even 2 or 3 people admit to not following my reasoning (there are more not admitting it), I would find other ways to explain. If this student is the only one, I'd ask them to see me in my office for help.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-05-04T15:03:04.553" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2140" PostTypeId="2" ParentId="2121" CreationDate="2014-05-04T15:29:44.307" Score="4" Body="&lt;p&gt;For a more particular example, when we are finding the area (in calculus) under an even function (symmetrical across the y-axis) from x = -a to x = a, we can ask ourselves the easier question of finding the area from x = 0 to x = a, and multiplying that by two.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-05-04T15:29:44.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2141" PostTypeId="1" CreationDate="2014-05-04T15:53:49.630" Score="16" ViewCount="1353" Body="&lt;p&gt;A friend of mine recently tried a standards-based grading (SBG) approach for her Calculus II course. (You can read about Kate's experience on her &lt;a href=&quot;http://blogs.cofc.edu/owensks/2014/01/08/sbg-calculus2/&quot;&gt;blog&lt;/a&gt;.) I find this approach to evaluation very interesting and, after seeing Kate's success, I am eager to learn more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are plenty of resources for SBG at pre-college levels, but it seems that college level resources are somewhat scarce and hard to sort from the giant haystack of pre-college resources. I'm hoping we can build a list here or that someone can point me to a list compiled elsewhere. I am especially interested in seeing other accounts and examples of using SBG in college math courses, especially for calculus courses.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;SBG is based on a fine division of the course objectives into concepts, tasks, skills or other learning objectives, henceforth called &lt;em&gt;standards&lt;/em&gt; for simplicity. Students are evaluated for each standard on a coarse scale, typically 0-4, indicating their level of mastery of the standard in question. Final course grades are assigned according to what portion of the standards that each student has mastered at an appropriate level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Implementation details vary. In the course that my friend gave, students were regularly assessed and re-assessed for each standard using tests and quizzes to measure their current level of mastery as the course advanced. Students were also given some opportunities to ask for specific standards to be re-tested. In cases where a standard occurred on the final exam, the average of the latest course score and final exam score was used to compute the final score for that standard. For final course evaluations, letter grades A,B,C corresponded roughly to earning 4,3,2, respectively, on at least 85% of standards.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two features of this scheme that I really like are that the meaning of each grade is perfectly clear and students are constantly aware of their strengths and weaknesses as the course progresses.&lt;/p&gt;&#xA;" OwnerUserId="74" LastEditorUserId="74" LastEditDate="2014-05-04T23:40:55.480" LastActivityDate="2014-05-05T06:51:06.493" Title="Standards-based grading in calculus" Tags="|undergraduate-education|calculus|grading|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2142" PostTypeId="2" ParentId="2123" CreationDate="2014-05-04T17:45:50.130" Score="5" Body="&lt;p&gt;I see two factors:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Most good math authors tend to not be good entertainers and vice versa. I feel that the two talents are not correlated. But, to me, Colin Adams and his Knot Book (which I just used as a textbook) are a great example of how math can be informative and fun.&lt;/li&gt;&#xA;&lt;li&gt;Fun can get in the way of understanding. I've read papers or explanations of math that had a very joke-y attitude and it was very hard to keep track of what is fun and what matters.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2014-05-04T19:39:27.363" LastActivityDate="2014-05-04T19:39:27.363" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2143" PostTypeId="2" ParentId="2123" CreationDate="2014-05-04T20:21:13.287" Score="7" Body="&lt;p&gt;I know that for many people, illustrations (not necessarily cartoons/comics) can help them visualize what is being described, and can make the task of reading a mathematical exposition easier. But illustrations need to be well-matched to the content to be useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the best series of textbooks available to people who are learning on their own are those produced by &lt;a href=&quot;http://www.artofproblemsolving.com/index.php?mode=books&quot;&gt;Art of Problem Solving&lt;/a&gt;. Almost no illustration, but definitely based on understanding. I would like to know if the original poster would consider those dry, or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have definitely seen an author's attempt at humor get in the way, making their explanations less helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One author who uses illustrations (including comic strips) well is &lt;a href=&quot;http://www.amazon.com/Harold-R.-Jacobs/e/B001H6ENU4/ref=sr_ntt_srch_lnk_2?qid=1399234504&amp;amp;sr=1-2&quot;&gt;Harold Jacobs&lt;/a&gt;, whose textbooks are a joy to work from.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The textbooks I had to face in my college math work (back in the 70's and 80's) were so dense, and lacked not only illustrations, but any sense of the author's voice, any connection to anything outside the math topic at hand. I like making connections, and if I needed to read much (as opposed to just doing the exercises), I might have described them as dry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You asked why this happens. I think mathematicians are very interested in making logical arguments. They aren't always aware of the psychology related to learning from a text. If you tell us more, especially the level you're interested in, I can recommend delightful books.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-05-04T20:21:13.287" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2144" PostTypeId="2" ParentId="2079" CreationDate="2014-05-04T20:27:54.487" Score="4" Body="&lt;p&gt;Different people have different philosophies of education. You will want a program that reflects your own personal philosophy. I would recommend reading online discussions about math education, and find someone whose ideas move you. If that person is at an institution whose program you like, go there. Do not worry about &quot;best&quot;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-05-04T20:27:54.487" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2145" PostTypeId="1" AcceptedAnswerId="2277" CreationDate="2014-05-04T20:41:11.273" Score="17" ViewCount="817" Body="&lt;p&gt;&lt;a href=&quot;https://projecteuler.net&quot;&gt;Project Euler&lt;/a&gt; is a very popular self-challenge website where users complete various projects designed to test their number-theory intuition and programming skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been considering various ideas for a first year introduction to proofs course. One idea I've had is to choose project ideas from project Euler and run the class using the Moore method. Thus, students would work through the first large chunk of projects together, handing in proofs that their answer works and presenting their solutions periodically on the board.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pros:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Introduces mathematical programming and number theory&lt;/li&gt;&#xA;&lt;li&gt;Students can check their answers through the website&lt;/li&gt;&#xA;&lt;li&gt;The internet resists people cheating on Project Euler (mostly, there are exceptions).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Cons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students may be mad if the website says they are right and the teacher says their proof is incorrect.&lt;/li&gt;&#xA;&lt;li&gt;Students may have already seen most of the material if they previously worked through the project.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Building off the previous item, how could standard material like proof techniques and basic logic be incorporated into such a course?&#xA;  Does project Euler have a wide-enough selection of topics to be an effective introduction to rigorous thinking?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A familiarity with a programming language would be a prerequisite. This class would give students more experiences programming math, which would be helpful in preparing them for a variety of future jobs.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="11" LastEditDate="2014-05-12T21:52:17.283" LastActivityDate="2014-05-15T18:24:09.643" Title="How to build a class based on Project Euler?" Tags="|course-design|technology-in-education|project-euler|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2146" PostTypeId="2" ParentId="2123" CreationDate="2014-05-04T20:47:30.477" Score="10" Body="&lt;p&gt;It seems to me that the question rests on a false dichotomy: either (1) the book has cartoons or artwork, or (2) the book consists of non-exciting material which the author expects the reader to memorize by rote.&#xA;However, in mathematical writing I think that (1) and (2) are almost orthogonal.&#xA;(Diagrams are certainly useful in some fields of math, such as geometry, but I wouldn't call them cartoons or artwork and I'm not sure if that's what the OP meant.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is possible that some readers are &lt;em&gt;only&lt;/em&gt; excited by cartoons and artwork.  However, these readers cannot be expected to learn any math.  So naturally, mathematical writing is geared to other readers, ones for whom it is at least &lt;em&gt;possible&lt;/em&gt; to be excited by the mathematical ideas themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly some authors are more successful than others in writing mathematics in an exciting and engaging way.  However, including pictures is neither necessary nor sufficient for this, except with very young readers for whom pictures may be necessary.  (Even so, I would call the useful pictures &quot;diagrams&quot; and maintain that cartoons and artwork would mostly serve to distract the reader from the mathematical ideas being presented.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If, due to deficiencies of the author or the teacher, a mathematical idea is presented so poorly that rote memorization is the only way for the student to &quot;learn&quot; it (and I wouldn't even call this learning) I don't think that adding cartoons and artwork to the text would be likely to make in improvement.&lt;/p&gt;&#xA;" OwnerUserId="870" LastActivityDate="2014-05-04T20:47:30.477" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2147" PostTypeId="2" ParentId="1655" CreationDate="2014-05-04T22:52:59.227" Score="-6" Body="&lt;p&gt;This happened to me this year briefly. &amp;quot;I have literally a degree in this. I have done it professionally now, for something like 5 years. When I say it's this way, it's absolutely this way.&amp;quot; Shut them up immediately.&lt;/p&gt;&#xA;&lt;p&gt;Edit:&#xA;Since more details were requested, I will follow this up. You have three problems:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;They don't respect you&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They are disrupting your class&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And they are being rude TO you.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The most immediate issue is the classroom disruption. It is evident from your comment that you are not in charge of the situation here. This may be an unusual situation, or it may be that you were never in charge and this student is making it substantially worse. In either case, the correct thing to do is to move on without them. Put them into a position where they have to accept your decision to move on, or brazenly violate your control of the class. State you are moving on, AND THEN DO IT. You are giving them permission to disrupt your class by doing anything else. Don't consent to it. The other students in the class are well aware the student is being a jerk - the social pressure exerted on them when it is apparent you know what you are doing and they don't is enormous, so if they insist on doing the fight publicly, you can win it. They WILL shut up.&lt;/p&gt;&#xA;&lt;p&gt;Next up, there is the issue that they don't respect you/your abilities. You can address this directly as I did in my pithy answer, or you can do it indirectly. This student is evidently confident about their math abilities - you are a grad student. You want to see them realize how out of their depth they are? Put some real analysis on the board. Or better yet, some complex analysis. Nothing like a zeta function to terrify a student. If they are the keen and excited student you describe them as, this is the perfect way to deal with the issue. (Edit: to be clear, I suggest you do this after class, in private)&lt;/p&gt;&#xA;&lt;p&gt;Last, if all this fails, you have the issue that they are being rude to you, and apparently your efforts to control them in class or gain their respect have failed. At this point you need to deal with this directly through the university bureaucracy. This is the worst case, but it will deal with the problem. You have a student who is refusing to let you teach, and you have every right to have them removed. This isn't elementary school, they don't have a legal right to this education, if they don't like it they can get out. Just the threat of this is sufficient in the once in a VERY rare while it is needed. I've TAd for a decade now, and I've only had to threaten this once, and certainly I've never needed to carry it out.&lt;/p&gt;&#xA;" OwnerUserId="1381" LastEditorUserId="11" LastEditDate="2020-10-19T17:55:57.737" LastActivityDate="2020-10-19T17:55:57.737" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2148" PostTypeId="2" ParentId="2145" CreationDate="2014-05-04T22:54:47.450" Score="5" Body="&lt;p&gt;I don't think it is a disadvantage if the students have seen the material before. What you are interested in at the end of the course is that the students are familiar with it; if it was before or during the course is not that relevant. Or ask them beforehand what problems they have seen/solved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And presumably you are able to convince them of your assesment that their proof/development is wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd be more worried that the problems turn out too hard for the class, you'd need to screen them beforehand. That means solving (at least most of) the ones suggested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also give other problems, open ones, and encourage discussion e.g. on &lt;a href=&quot;http://math.stackoverflow.com&quot; rel=&quot;noreferrer&quot;&gt;MSE&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-04T22:54:47.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2149" PostTypeId="2" ParentId="2060" CreationDate="2014-05-05T02:15:47.383" Score="3" Body="&lt;p&gt;Calculus is seldom helpful for biology majors, if &quot;helpful&quot; means useful in a utilitarian, professional sense. The vast majority of biology majors are going into allied health fields: they intend to be doctors, pharmacists, physical therapists, vets, optometrists, and dentists. These professions are not like engineering, in which calculus is used from day to day. Here in California, the UC system decided ca. 1997 to start requiring biology majors to take calculus-based physics. The motivation was pretty transparent: they had too many biology majors (the major was &quot;impacted&quot;), and they wanted to get rid of some. This is similar to the fact that in 19th century Britain, if you wanted to be a military officer, you had to pass a test on Greek and Latin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does this mean that future military officers have nothing to gain from learning ancient Greek, or that future dentists have nothing to gain from taking calculus? Absolutely not. It simply means that for the future dentist, learning calculus is one possible ingredient in that quaint notion of a general education. It's a way of gaining broad knowledge about the world and getting experience in varied intellectual pursuits and ways of thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For comparison, it may be helpful to ask the similar question of whether biology coursework is helpful to biology majors. A lot of it clearly is not, if helpful is used in the sense of day-to-day professional utility. For example, biology majors learn about the reproduction of ferns and club mosses, which is likely to be of very little practical utility to an optometrist.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-05-05T03:17:55.093" LastActivityDate="2014-05-05T03:17:55.093" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2150" PostTypeId="1" AcceptedAnswerId="2172" CreationDate="2014-05-05T02:26:12.660" Score="4" ViewCount="625" Body="&lt;p&gt;At some point very early on in a freshman calc course, we use proof by induction to show that the derivative of $x^n$ is $nx^{n-1}$ if $n$ is a natural number. For the small minority of our students who are math majors, it's desirable that they then get a chance, as soon as possible, to practice the technique themselves. Are there any good exercises that require no more knowledge of differential calculus than the chain rule and product rule, and first derivatives? The examples I was able to come up with involved the nth derivative of some function (e.g., $1/(1-x)$ or $b^x$), but in the plan I'm following, higher derivatives come a little later in the course. &lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2016-09-28T14:16:53.430" Title="Good exercises in proof by induction, very early in freshman calculus?" Tags="|calculus|proofs|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2151" PostTypeId="1" AcceptedAnswerId="2160" CreationDate="2014-05-05T05:02:21.823" Score="34" ViewCount="6049" Body="&lt;p&gt;First, I am not a professor, but I was a teaching assistent for a couple of courses. One time I took over a few sections for a friend who was also a TA. The course was 'math for chemists' (I think it was the second quarter or third quarter of the course).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was explaining how to get a linear approximation for a differential equation and draw a phase plane from that (I hope I'm using the correct English terms here...). So you have:&#xA;$$ \frac{dx}{dt} = f(x, y) $$&#xA;$$ \frac{dy}{dt} = g(x, y) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You seek for the fixed points where $f(x, y) = 0$ and $g(x, y) = 0$ at the same time, then take the linear approximation at that point. When I explained this (of course I did a more thorough explanation than these two sentences...) I got very little feedback from the students. I felt like the students were not sure how to handle the linear approximation so I pointed out that it is analogous to taking the linear term of a Taylor series, but in two variables instead of one. When I asked if the students understood this, I got no reaction at all. This made me very uncomfortable: it was clear that not everybody fully understood the topic, but at the same time I had no idea what to explain, for from my point of view, the explanation was very clear. I'm more used to teaching math in a 1-1 setting, in which people can't really ignore your questions, so I didn't know how to handle this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I made the remark 'it's better to ask for clarification now than to discover you don't understand the material during the exam', and thought about asking a specific student (so it would be harder to ignore my question) but I didn't do this in the end (most of the students in that section seemed pretty shy and I didn't want to make them uncomfortable).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any other (better?) tricks or strategies you have used or you would use to avoid these situations?&lt;/p&gt;&#xA;" OwnerUserId="915" LastActivityDate="2014-05-08T16:13:00.960" Title="What to do when you get &quot;the empty stare&quot;?" Tags="|undergraduate-education|tutoring|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2152" PostTypeId="1" CreationDate="2014-05-05T05:14:08.443" Score="21" ViewCount="1915" Body="&lt;p&gt;Depressingly many of the physical &quot;applications&quot; of Taylor series that I can find in textbooks and online are actually just applications of linear approximation, since they only take the constant and linear term of a Taylor series.  For instance, this how you get the Newtonian limit of the kinetic energy of special relativity, the $1/r^3$ behavior of an electric dipole far away, and the standard approximation to the period of a pendulum.  Since the whole point of a Taylor series is that we can go &lt;em&gt;on&lt;/em&gt; from the linear approximation to get quadratic and higher approximations, I find this unsatisfying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what are some good physical applications of the &lt;strong&gt;higher order&lt;/strong&gt; terms in a Taylor expansion?  (I'm looking for something more than just &quot;we can get a better approximation than the linear one by including more terms&quot; &amp;mdash; ideally, a situation where the higher terms give some new &lt;em&gt;qualitative&lt;/em&gt; insight.)&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="61" LastEditDate="2014-05-06T10:43:16.593" LastActivityDate="2024-03-06T01:38:05.840" Title="Physical applications of higher terms of Taylor series" Tags="|calculus|concept-motivation|physical-sciences|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2153" PostTypeId="2" ParentId="2151" CreationDate="2014-05-05T06:07:26.213" Score="16" Body="&lt;p&gt;When I think of the what caused my students (or myself, back when I sat in lectures) to get to the point of an empty stare, &lt;strong&gt;it's that they were so confused that they didn't even know what to ask&lt;/strong&gt;. They know they don't understand, but they lost the thread a while ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what can you do? Two things:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make sure students feel comfortable asking questions, so they will ask questions before getting so utterly confused they can only stare blankly and hope that the lecture will end soon. To make sure your students ask questions: acknowledge, by name, good questions--&quot;Good question, Jane&quot;--and be gentle when students ask questions that reveal their ignorance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make sure your students trust you enough to keeping working even when they're confused. Math is difficult, and some topics won't be understood hours, days, or even weeks after a lecture. Tell your students that confusion is often a step on the road from ignorance to knowledge. Encourage your students to work and play with the mathematics until it is understood.&lt;/p&gt;&#xA;" OwnerUserId="182" LastActivityDate="2014-05-05T06:07:26.213" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2154" PostTypeId="2" ParentId="1082" CreationDate="2014-05-05T06:47:57.183" Score="2" Body="&lt;p&gt;Until you know what caused the course to get off-track, it's hard to know how to fix it. What follows are a few things to help you determine the cause of the trouble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often, universities have a program to help faculty and graduate students teach effectively. If your university has one, contact them and ask for help. They do things like observe your lectures and/or survey your students. Typically, this feedback isn't shared with your department, so you can learn from it without worrying about getting in trouble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask a colleague if you can sit in on one of their classes (it's best if it's the same course or one quite similar). Make sure you ask someone who has a good reputation as a teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask a colleague to sit in on one of your classes. Afterwards, buy them a beer and ask if they have any thoughts on how you might improve.&lt;/p&gt;&#xA;" OwnerUserId="182" LastActivityDate="2014-05-05T06:47:57.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2156" PostTypeId="2" ParentId="2141" CreationDate="2014-05-05T06:51:06.493" Score="5" Body="&lt;p&gt;This is in response to the bit:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I am especially interested in seeing other accounts and examples of using SBG in college math courses&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've done SBG (aka &quot;&lt;em&gt;criterion based assessment&lt;/em&gt;&quot;) in an undergraduate mathematics course.  I heartily recommend it.  I've been trying to write up my experience with the course that I tried it in.  Here's the part on SBG (warning: this is in draft form):&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h1&gt;Criterion Based Assessment&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;I have never been a fan of numerical assessment in exams. (This might have something to do with the fact that my undergraduate university - Oxford - used no less than &lt;em&gt;four&lt;/em&gt; different schemes for devising a final grade).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My reason for this is simple. Learning is not a continuous process; whilst it often proceeds in small increments there are definite leaps from one level to the next. It is these leaps that we should be focussing on measuring: has a student achieved the next level, or are they still at the previous one? Within those levels, there is not much to separate students.&#xA;Grading a paper purely numerically paints a false picture. It says that learning is continuous, and that the student who got 68 points is better than the student who got 65 points. We then draw graphs showing a nice normal distribution and select cut-off points for the different grades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we take something discrete, namely the amount the student has learnt, force it into something continuous, their grade, and finally make it discrete again. As any topologist could tell, discretising a continuous function is a Bad Idea. Suppose we decide on 70 as a grade boundary. What about the student who got 69? Are they really qualitatively worse than the one who got 70? It was only one point! But by that argument, all students should get the same grade since each possible total of points differs by only one point from the next one and therefore should get the same grade as it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most likely, the grade boundaries are not as solid as that. We look carefully at students on the boundaries (well, we ought to; whether we do or not is another matter). But this is actually worse. Not only are we acknowledging that the original system is incorrect, we are fixing it in an ad hoc way that is completely opaque to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Enter &lt;em&gt;Criterion Based Assessment&lt;/em&gt;. The core of this is to avoid the false assumption of learning being a continuous function. Instead, the grades are aligned with specific stages of learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The exam is viewed in a different light. Instead of a slow accumulation of points, each unit is graded according to the rule &quot;Is this evidence of a particular skill?&quot;. At the end, the question becomes &quot;Has this student shown sufficient evidence to justify this grade?&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When grading, my imaginary scenario is the following. I imagine that I award a student a certain grade. Then I imagine that sometime after the lecturer for another course has marched into my office demanding to know why I awarded that grade (since clearly the student is as thick as two short planks). The exam is then my evidence for that grade. If I cannot use the exam to defend the grade, the student does not get it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This has a useful side effect. It so happens that I sometimes get asked to write references for students who have taken this course. However, it is a medium to large course - about 100 students - so there are very few that I can say too much about. But as my grades are knit to specific achievements, I can list those achievements and say that the student has demonstrated that ability.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this course, the levels of achievement were the following.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Memorisation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was the student's ability to remember things they had been told: such as definitions and statements of results.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Utility.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was the student's ability to use things from the course in the manner in which they are intended.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Adaptation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was the student's ability to adapt what they had been told to new situations, or to join together ideas to build a bridge.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These three levels corresponded to grades E, C, and A. Not every part of every question could provide evidence for all three levels. In particular, one question simply asked for specific statements and so could only provide evidence for an E grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus at the end of grading an exam, the question was &quot;Of those questions which could provide evidence for a grade, were enough done to that level?&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The intermediate grades (D and B) were very definitely low versions of the next grade up. Thus the qualitative leap was between E and D, and C and B. The distinction between D and C and between B and A was quantitative. So a C-grade student was very definitely C-grade.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In addition to the above, let me link to the &lt;a href=&quot;http://mathsnotes.math.ntnu.no/mathsnotes/show/scheme+of+work+2011&quot; rel=&quot;noreferrer&quot;&gt;scheme of work&lt;/a&gt; for the course where you can see what the specific learning outcomes were (together with how they translate in to grades).  Exploring that site will tell you more about the course if that's useful to you.  There are also links to exam papers (of both types).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it is also worth pointing out a couple of additional advantages.  The exam was a doddle to write (comparatively), and the number of queries/complaints that I got about grades in that year was &lt;em&gt;far fewer&lt;/em&gt; than any other year (with the same course).  I have no proof of causation, but it's a very strong correlation!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, you say&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There are plenty of resources for SBG at pre-college levels&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;so use them!  I have the great benefit of being married to a(n excellent) teacher.  Every time I hear about some innovation in university level education, I get the same line from her: &quot;We've been doing that for years, it's called &lt;em&gt;XYZ&lt;/em&gt;.&quot;.  Then I get on my high horse and say that of course it's different in university because of &lt;em&gt;ABC&lt;/em&gt;.  Finally, when I actually try it then I realise that she was right all along and at heart, it really is the same.&lt;/p&gt;&#xA;" OwnerUserId="112" LastActivityDate="2014-05-05T06:51:06.493" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2157" PostTypeId="1" CreationDate="2014-05-05T07:19:15.493" Score="23" ViewCount="2122" Body="&lt;p&gt;One useful trick in mathematics is to prove something stronger instead of the question asked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works well in induction proofs (because strengthening the claim also strengthens the induction basis):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Example: Prove $\frac1{1\cdot 2} + \frac1{2\cdot 3} + \dots +\frac1{(n-1)\cdot n} &amp;lt; 1$. This admits a simple induction proof if one proves the exact formula instead.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But there are also other examples&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Example: Proving that inscribed angles in a circle are equal becomes easier when we try to show that all of them are half the central angle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course, there are also lots of research examples that involve the trivialization of results as soon as new parameters are introduced or the whole problem is generalized to the &quot;right&quot; setting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, my question is the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are some good examples for the principle &quot;Prove something stronger instead of the original statement&quot; that would work well with first-year university students?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some calculus is ok, as long as it is introductory content. But generally, the more basic, the better, because the point is to illustrate the principle.&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2023-11-19T09:01:49.160" Title="What are some good low-prerequisite examples for the heuristic advice &quot;If you cannot prove it, prove something stronger.&quot;?" Tags="|examples|proofs|" AnswerCount="8" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2159" PostTypeId="2" ParentId="2121" CreationDate="2014-05-05T07:41:07.210" Score="7" Body="&lt;p&gt;Some examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It's much easier to develop a symmetric plane or boat than to calculate all the forces regarding an asymmetric hull.&lt;/li&gt;&#xA;&lt;li&gt;Car keys being symmetric it's great usability feature: saves time and frustration.&lt;/li&gt;&#xA;&lt;li&gt;Pots are symmetric, because they are easier to produce and nicer to use.&lt;/li&gt;&#xA;&lt;li&gt;Wheels wouldn't work if not for symmetry.&lt;/li&gt;&#xA;&lt;li&gt;Games are often symmetric to achieve a form of fairness.&lt;/li&gt;&#xA;&lt;li&gt;Humans and animals in general are symmetric too (from outside, but the asymmetries inside are not so big either), but I have no idea way (somehow such a shape must be more fit).&lt;/li&gt;&#xA;&lt;li&gt;High-level symmetry is recognizable and feels deeply intentional, this can be used to great effects in art (i.e. not only for aesthetics, but also emphasis or symbolism, etc.); examples include &#xA;&lt;ul&gt;&#xA;&lt;li&gt;direct use like &lt;a href=&quot;http://en.wikipedia.org/wiki/Inversion_%28music%29&quot;&gt;inversion&lt;/a&gt; or &lt;a href=&quot;http://en.wikipedia.org/wiki/Permutation_%28music%29&quot;&gt;retrograde&lt;/a&gt; in music, e.g. in fugues by J. S. Bach,&lt;/li&gt;&#xA;&lt;li&gt;less direct use like the same scene from various perspectives, e.g. in &lt;a href=&quot;http://en.wikipedia.org/wiki/Another_World_%28M._C._Escher%29&quot;&gt;Another World&lt;/a&gt; or &lt;a href=&quot;http://www.escherinhetpaleis.nl/wp-content/uploads/2013/02/Boven-en-Onder.jpg&quot;&gt;Above and Below&lt;/a&gt; by M. C. Escher,&lt;/li&gt;&#xA;&lt;li&gt;even less direct like the same situation happening to pair of characters in a novel, but the second time with their roles reversed (taken to the extremes in the 7th Voyage in the &lt;a href=&quot;http://en.wikipedia.org/wiki/The_Star_Diaries&quot;&gt;Star Diaries&lt;/a&gt; by S. Lem).&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-05-05T09:10:33.740" LastActivityDate="2014-05-05T09:10:33.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2160" PostTypeId="2" ParentId="2151" CreationDate="2014-05-05T07:59:07.157" Score="26" Body="&lt;p&gt;It's perhaps worth noting that there are (at least) two distinct reasons why your students might offer no reply when you ask them if they understand what you've just explained:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;They might not understand what you've told them, or even &lt;strong&gt;understand enough to know what it is that they don't understand&lt;/strong&gt;.  This typically happens when the students lack (or have failed to properly internalize) some prerequisite knowledge that you're assuming they have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if your students have &lt;em&gt;never seen a differential equation before&lt;/em&gt;, and have no idea what the notation even means, asking whether they've understood your explanation of phase plane analysis and Jacobians is, most likely, going to be met by blank stares.  They may even assume that, since they don't know those things that you're clearly assuming they should know, the class is clearly not meant for them, and they should just keep quiet and not interrupt the other students' learning with stupid questions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Conversely, the other possible reason for silence is that your students &lt;strong&gt;already know everything you've been telling them&lt;/strong&gt;, e.g. because you're (intentionally or not) repeating something that was covered in an earlier course.  They &lt;em&gt;could&lt;/em&gt; answer your questions (at least with &quot;yes, I know all this&quot;), but they won't, because they assume that the questions are really meant for some other students who &lt;em&gt;don't&lt;/em&gt; know the subject already, or that you're only asking to make sure you can safely move to the next topic (and that you'll do so if nobody speaks up).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In your case, I do suspect that the first explanation is more likely, but you should keep in mind that both are, in fact, possible.  It's also quite possible for &lt;em&gt;both&lt;/em&gt; reasons to occur simultaneously in the same class: some of your students could be silent because they have no idea what you're talking about, while some others might be silent because they already know all of it and don't want to interfere while you explain it to the rest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can also be highly cultural.  Here in Finland, for example, I've heard several lecturers coming from abroad (especially from the U.S.) complain about getting &quot;nothing but blank stares&quot; from the class, because Finnish culture is traditionally a &lt;em&gt;lot&lt;/em&gt; less outspoken than, say, in the U.S., and people coming from other cultures may not immediately recognize the subtle body language cues that can, sometimes, signal the difference between &quot;I'm not asking questions because I want you to move on&quot; and &quot;I'm not asking questions because I have no idea what to ask&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;OK, so what to do about it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, the situation you describe seems like a perfect opportunity to &lt;strong&gt;take a few minutes to explain&lt;/strong&gt; that you &lt;em&gt;really do&lt;/em&gt; rely on feedback from your students to know whether you're going way too slow or way too fast (or both), and that you really want them to speak up even if (or &lt;em&gt;especially&lt;/em&gt; if) they have nothing to say but &quot;I don't get any of this&quot; or &quot;I already know all of this&quot;.  Try to create a friendly and relaxed atmosphere, let your students know that it's &lt;em&gt;always&lt;/em&gt; OK to ask question or speak up, and, when they do, make sure to respond positively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, if you're still getting no response by asking the entire class, don't be afraid to &lt;strong&gt;pick a random student&lt;/strong&gt; and address your questions to them personally.  As uncomfortable as it might, initially, feel to the student who ends up in the &quot;hot seat&quot;, it at least ensures that they can't just &quot;let someone else answer&quot;, and so gives you a single direct data point about that student's level of understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, since you say that you feel more comfortable teaching in a one-to-one setting, you can always just pick one student out and &lt;strong&gt;work it out with that student until they get it&lt;/strong&gt;.  The odds are that, by doing this, you're also helping all the &lt;em&gt;other&lt;/em&gt; students who may have the same or similar difficulties.  Besides, by taking time during class to specifically engage one student in dialogue, you're showing the other students that they can also ask specific questions without worrying about &quot;wasting time&quot; or disrupting your lesson plan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/2151/what-to-do-when-you-get-the-empty-stare#comment4456_2153&quot;&gt;Jsor's suggestion&lt;/a&gt; of &lt;strong&gt;picking a &quot;stooge&quot;&lt;/strong&gt; to sit in class and speak up when the other students won't is also excellent, and something I would've suggested myself.  If you're a professor, it can be a TA who sits in class; if you're the TA yourself, it can be any student you trust to do the job.  If necessary, ask a friend who's done the same class recently to join in and ask basic questions if your students won't.  I've &lt;em&gt;been&lt;/em&gt; that stooge before, and I can testify that it really does work.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Finally, and somewhat tangentially, it occurs to me that you might need to simply &lt;strong&gt;lower your expectations&lt;/strong&gt; a bit.  You're TA-ing a course on &quot;math for chemists&quot;, presumably aimed at undergrad chemistry majors with no prior math background since high school.  I don't know about your chemistry students, but back when I did my minor in chemistry, even using the quadratic formula to solve second-order equations was considered &quot;advanced math&quot; &amp;mdash; i.e. something that the students would not be generally required or assumed to know &amp;mdash; in first or second year chemistry courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, courses like &quot;math for chemists&quot; exist precisely to provide some of those missing math skills needed for certain topics in chemistry, but the point is that you &lt;em&gt;really&lt;/em&gt; can't assume in such a course that the students know anything past the minimum high school math curriculum &amp;mdash; and they may have forgotten much of that.  It &lt;em&gt;probably&lt;/em&gt; doesn't include differential equations, and I'm pretty sure that it won't include Taylor series.  You're going to have to start from the basics &amp;mdash; equations, variables, constants, basic algebra &amp;mdash; and work up from there, brutally distilling each concept down to the bare minimum necessary to work up to the more advanced topics you're really supposed to be teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or course, as a TA, that's &lt;em&gt;primarily&lt;/em&gt; your professor's responsibility, but &lt;em&gt;your&lt;/em&gt; job is to fill in the missing gaps for each student on an individual level.  If you've got a good textbook (or a good set of lecture notes) for the course that presents all the preliminary material in a clear and concise form, I'd suggest reading it until you know most of it by heart. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you &lt;em&gt;don't&lt;/em&gt; have one, try to &lt;em&gt;find&lt;/em&gt; one; even if it's not official material for the course, having a good introductory textbook aimed at students like yours to fall back on is worth its weight in gold.  It means that, when a student asks you &quot;what's a differential equation?&quot;, you can just paraphrase what the book says about them, instead of having to make up your own five-minute explanation of differential calculus on the spot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Of course, it should go without saying, but you should not follow any book slavishly.  Rather, use the book as a &quot;cheat sheet&quot; for your own teaching, as and when needed.)&lt;/p&gt;&#xA;" OwnerUserId="526" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-05T08:10:24.397" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2161" PostTypeId="2" ParentId="2150" CreationDate="2014-05-05T11:54:21.277" Score="1" Body="&lt;p&gt;If it is very early in a freshman calculus class, there are no good exercises in proof by induction:  they would distract from the focus of the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they haven't seen induction, I would prove this derivative formula without it, using the factorization of $y^n-x^n$.  E.g.:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(x^3)'&#xA;= \lim_{y\rightarrow x} \frac{y^3-x^3}{y-x}&#xA;= \lim_{y\rightarrow x}\; (y^2+xy+x^2)&#xA;=3x^2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(x^n)'&#xA;= \lim_{y\rightarrow x} \frac{y^n-x^n}{y-x}&#xA;= \lim_{y\rightarrow x}\; (y^{n-1}+xy^{n-2}+\cdots+x^{n-1})&#xA;=nx^{n-1}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would flesh out the first line, do a similar example like $x^5$, and then flesh out the second line similarly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The expression with &quot;$\cdots$&quot; gets at the right ideas, and introducing $\sum$-notation or induction instead would be distracting.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-05-05T15:31:03.133" LastActivityDate="2014-05-05T15:31:03.133" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2162" PostTypeId="1" CreationDate="2014-05-05T13:58:03.987" Score="21" ViewCount="1274" Body="&lt;p&gt;Both education researchers and mathematicians discuss the challenge of (epsilon, delta) type definitions in real analysis and the student reception of them. My impression has been that mathematicians often hold an upbeat opinion on the success of student reception of this, whereas education researchers often stress difficulties and their &quot;baffling&quot; and &quot;inhibitive&quot; effect (see below).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Paul Dawkins recently expressed a typical educational perspective on this: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;2.3. Student difficulties with real analysis definitions. The concepts of limit and continuity have posed well-documented&#xA;  difficulties for students both at the calculus and analysis level of&#xA;  instructions. Researchers identified difficulties stemming from a&#xA;  number of issues: the language of limits, multiple quantification in&#xA;  the formal definition, implicit dependencies among quantities in the&#xA;  definition, and persistent notions pertaining to the existence of&#xA;  infinitesimal quantities. Limits and continuity are often couched as&#xA;  formalizations of approaching and connectedness respectively.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;However, the standard, formal definitions display much more subtlety and complexity. That complexity often baffles students who cannot&#xA;  perceive the necessity for so many moving parts. Thus learning the&#xA;  concepts and formal definitions in real analysis are fraught both with&#xA;  need to acquire proficiency with conceptual tools such as&#xA;  quantification and to help students perceive conceptual necessity for&#xA;  these tools. This means students often cannot coordinate their concept&#xA;  image with the concept definition, inhibiting their acculturation to&#xA;  advanced mathematical practice, which emphasizes concept definitions.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[See &lt;a href=&quot;http://dx.doi.org/10.1016/j.jmathb.2013.10.002&quot; rel=&quot;noreferrer&quot;&gt;http://dx.doi.org/10.1016/j.jmathb.2013.10.002&lt;/a&gt; for the entire article;&#xA;  the online article provides links to references for several of the comments above.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To summarize, in the field of education, researchers decidedly have &lt;em&gt;not&lt;/em&gt; come to the conclusion that epsilon, delta definitions are either &quot;simple&quot;, &quot;clear&quot;, or &quot;common sense&quot;. Meanwhile, mathematicians often express contrary sentiments. Two examples are given below. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...one cannot teach the concept of limit without using the epsilon-delta definition. Teaching such ideas intuitively does not&#xA;  make it easier for the student it makes it harder to understand.&#xA;  Bertrand Russell has called the rigorous definition of limit and&#xA;  convergence the greatest achievement of the human intellect in 2000&#xA;  years! The Greeks were puzzled by paradoxes involving motion; now they&#xA;  all become clear, because we have complete understanding of limits and&#xA;  convergence. Without the proper definition, things are difficult. With&#xA;  the definition, they are simple and clear. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[See Kleinfeld, Margaret;&#xA;  Calculus: Reformed or Deformed? Amer. Math. Monthly 103 (1996), no. 3,&#xA;  230-232.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I always tell my calculus students that mathematics is not esoteric: It is common sense. (Even the notorious epsilon, delta definition of&#xA;  limit is common sense, and moreover is central to the important&#xA;  practical problems of approximation and estimation.)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[See Bishop,&#xA;  Errett; Book Review: Elementary calculus. Bull. Amer. Math. Soc. 83&#xA;  (1977), no. 2, 205--208.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When I compare the upbeat assessment common in the mathematics community and the somber assessments common in the education community, sometimes I wonder whether they are talking about the same thing. How does one bridge the gap between the two assessments? Are they perhaps dealing with distinct student populations? Are there perhaps education studies providing more upbeat assessments than Dawkins' article would suggest? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a parallel thread with a bounty offered at &lt;a href=&quot;https://math.stackexchange.com/questions/683513/evaluating-the-reception-of-epsilon-delta-definitions&quot;&gt;https://math.stackexchange.com/questions/683513/evaluating-the-reception-of-epsilon-delta-definitions&lt;/a&gt; so you can respond there (or here if you prefer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 1. Several approaches have been proposed to account for this difference of perception between the education community and the math community: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a) sample bias: mathematicians tend to base their appraisal of the effectiveness of these definitions in terms of the most active students in their classes, which are often the best students; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) student/professor gap: mathematicians base their appraisal on their own scientific appreciation of these definitions as the &quot;right&quot; ones, arrived at after a considerable investment of time and removed from the original experience of actually learning those definitions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(c) misguided educators: educators are not sufficiently concerned with rigor and cause horrible confusion to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of these approaches sound plausible, but it would be instructive to have field research in support of these approaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We recently published &lt;a href=&quot;http://dx.doi.org/10.5642/jhummath.201701.07&quot; rel=&quot;noreferrer&quot;&gt;an article&lt;/a&gt; reporting the result of student polling concerning the comparative educational merits of epsilon-delta definitions and infinitesimal definitions of key concepts like continuity and convergence, with students favoring the infinitesimal definitions by large margins.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-04-23T11:56:04.167" LastActivityDate="2019-08-21T22:56:06.337" Title="Evaluating the reception of (epsilon, delta) definitions" Tags="|undergraduate-education|mathematical-pedagogy|calculus|mathematical-analysis|infinitesimals|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2163" PostTypeId="2" ParentId="2151" CreationDate="2014-05-05T14:36:19.253" Score="9" Body="&lt;p&gt;Tutoring is complicated and hard! Here is how I conceptualize it, as a program.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;function Tutor(Problem) {&#xA;  print('It is okay to not know how to do this problem.')&#xA;  result = askStudentToSolve(Problem)&#xA;  if (result == 'no idea'):&#xA;    newProblem = createSimplerVersion(Problem)&#xA;    Tutor(newProblem)       // Notice this can create a recursive loop of problems!&#xA;  else if (result == 'correct first step'):&#xA;    recognizeSuccess()&#xA;    guideStudentThrough(Problem)&#xA;  else if (result == 'success'):&#xA;    recognizeSuccess()&#xA;}&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;First, make sure that the students know it is okay to not understand, and it is okay to tell you this. Then, present the problem and ask them to get started. If they cannot even get started, &lt;strong&gt;create a fundamentally simpler version of the problem&lt;/strong&gt;, either by reducing the complexity of the algebra, reducing the number of variables, or making the problem seemingly trivial. Creating a simpler version of a problem is, in my opinion, the most important skill for a tutor to develop.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Present them with the simpler problem -- on a blank sheet of paper with plenty of space for them to write. Repeat the process. (Make sure it is okay to say &quot;I don't know;&quot; if they don't know, then create a simpler version, etc). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end you may have created 4 problems -- the simplest one being something trivially easy, but they will solve that one. Then you will have them complete a simple version of the problem. Get out the next-easiest problem and see if they can do it now, with the benefit of the easier version being in their head. They will gradually work their way up to the main problem at hand.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Implementation Notes:&#xA;&#xA;I'd say the createSimplerVersion() function is the hardest &#xA;one to write, while many implementations of Tutor unfortunately &#xA;only have an implementation of guideStudentThrough(), totally &#xA;missing the recognizeSuccess() and the main if/else loop.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-05-05T14:36:19.253" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2164" PostTypeId="1" AcceptedAnswerId="3896" CreationDate="2014-05-05T14:56:53.420" Score="52" ViewCount="12515" Body="&lt;p&gt;In most secondary and early undergraduate courses, students purchase expensive and carefully-written textbooks. These textbooks contain, roughly, three things:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Exercises and Answers&lt;/li&gt;&#xA;&lt;li&gt;Reference Material&lt;/li&gt;&#xA;&lt;li&gt;Exposition&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Students are mostly adept at using the exercises and answers, and they often refer to the textbook for formula sheets or equations. However, they are mostly unable to utilize the expository text in the book -- even the top students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently created an extra-credit assignment guiding students through the process of critically reading a section of a math textbook, and I'd like to incorporate this more firmly in my courses. This leads to my question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teachers who have successfully or unsuccessfully done this in their classes already: What was your approach to teaching students how to read their math textbook, and what happened?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="143" LastEditDate="2014-06-23T12:35:29.030" LastActivityDate="2022-03-01T19:23:46.280" Title="How can we help students learn how to read their textbook?" Tags="|undergraduate-education|secondary-education|general-pedagogy|transferrable-skills|" AnswerCount="15" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2165" PostTypeId="1" AcceptedAnswerId="2383" CreationDate="2014-05-05T15:14:23.167" Score="10" ViewCount="660" Body="&lt;p&gt;What are some good references for high-school (or even middle-school) mathematics, written from an advanced point of view, especially texts written for prospective teachers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some references were given in a comment for an earlier question, but I cannot find that one now ...&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="262" LastEditDate="2014-05-05T17:04:59.030" LastActivityDate="2015-06-22T05:15:44.470" Title="References for &quot;high-school mathematics from an advanced point of view&quot;" Tags="|secondary-education|reference-request|textbooks|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2166" PostTypeId="2" ParentId="2162" CreationDate="2014-05-05T16:11:31.930" Score="5" Body="&lt;p&gt;As I said on &lt;a href=&quot;https://mathoverflow.net/questions/158145/assessing-effectiveness-of-epsilon-delta-definitions&quot;&gt;MathOverflow&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would attribute this to &lt;strong&gt;sample bias&lt;/strong&gt; -- which is &quot;distinct student populations&quot; of a sort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematicians who teach epsilon-delta definitions rarely study student reception of those definitions in a rigorous way. If they did, they would probably be publishing in education journals, not math journals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math teachers evaluating the reception of these ideas probably make &lt;strong&gt;informal evaluations, focusing on the most obvious students&lt;/strong&gt;: those who participated actively in class, those who stayed in touch with their teachers, those who continued with more mathematics. Those students are substantially more likely to appreciate the epsilon-delta definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Education researchers are less likely to suffer from this bias, though surely they suffer from others.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-05-05T16:11:31.930" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2167" PostTypeId="2" ParentId="2165" CreationDate="2014-05-05T16:20:20.907" Score="7" Body="&lt;p&gt;Right now the best-known book of this sort is &quot;Mathematics for High School Teachers -- An Advanced Perspective&quot; by Zalman Usiskin.  (I say &quot;right now&quot; because I have grandiose plans of writing such a textbook myself, but not until after I get tenure, i.e. not for at least 3 years.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-05-05T16:20:20.907" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2168" PostTypeId="2" ParentId="2151" CreationDate="2014-05-05T16:38:19.300" Score="12" Body="&lt;p&gt;As I imagine your interaction with the students, each of you had the following mental picture of what is going on.  This is obviously a big assumption about the nature of the class and of your own personality, but the gap I describe is definitely there regardless of the details.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;You: A differential equation (which is synonymous with a system thereof) is a general gadget that can have multiple constituent scalar equations involving arbitrary anonymous functions.  As functions (in this case differentiable ones) they have lots of properties and methods, such as Taylor series, at various points of their domain.  All of these methods are available as tools for modifying, analyzing, or simplifying the gadget of a DE containing them, and there is a relatively conceptual heuristic (the one encapsulated by the precise concept of &quot;linear approximation&quot;) that allows you to say how these alterations relate to the original.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Them: A differential equation is something like $x'' + 3x' + 2x = 0$.  Worse, it could be $x'' + 3x' + 2x = \sin(t)$, and at the absolute outside, maybe something like $x' + t^2x = 1$, with a variable coefficient (but only in the first order).  A &lt;em&gt;system&lt;/em&gt; of differential equations is something like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\begin{align}x' &amp;amp;= x + y \\ y' &amp;amp;= 2x - 3y\end{align}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and it is meant to be solved by a process involving linear algebra, which is a lot of work and has very specific steps.  If you were to change any of the numbers, the steps would be completely different, though the process is the same.  The solutions to the two equations (original and changed) are pretty much unrelated.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So you see how they could become confused: your explanation has &lt;em&gt;nothing&lt;/em&gt; to do with their experience.  It doesn't &lt;em&gt;mean&lt;/em&gt; anything to them to &quot;expand a DE in Taylor series&quot;, they don't even think of functions $f(x,y)$ and $g(x,y)$ appearing in a nonlinear equation (maybe in their examples they've seen $x(1 - y)$), and the very concept of solving a &lt;em&gt;different&lt;/em&gt; equation to analyze the first is fundamentally illogical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to connect with them, you need to be gradual, gentle, and totally explicit.  What they really want to know (and you know this because you threatened them with it) is how to do this kind of problem on the test; so, show them that.  Do a really simple example with 100% detail and explain what each step is.  Do another one.  That will satisfy almost everyone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems like you want to give them a deeper understanding: that's admirable but you can't make the horse drink.  In fact, since you are not the professor, you can't even lead them to the water: as the TA you really can't change the tone of the lessons.  If they weren't taught the phase plane method conceptually, then you aren't going to get them to care and they will be angry if you make their understanding of the &quot;real&quot; material dependent on an abstract concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Supposing it &lt;em&gt;were&lt;/em&gt; your job to teach them the ideas you want them to learn, you would &lt;em&gt;still&lt;/em&gt; have to set them up.  This depends a lot on the class: for example, I last taught ODEs to an honors class.  With them, I could assume some readiness to take me on faith until the payoff.  But even then, it's not a good idea to build a huge conceptual structure before showing them the application.  In my class, this &lt;em&gt;was&lt;/em&gt; the application, so that's okay.  But you are teaching a service class: math for scientists; definitely applied, probably purely practical.  In this class, no piece of theory counts as an application, so you have to show them a real computation first.  Then, maybe, you can begin the process of abstraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, the explanation you tried to give presupposes a pretty strong math background, a fair level of mathematical maturity and intuition, and is in itself worth about two days of lecturing when you are not the lecturer and probably tried to give it in only a few minutes.  You got blank stares because you went over their heads.&lt;/p&gt;&#xA;" OwnerUserId="513" LastActivityDate="2014-05-05T16:38:19.300" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2169" PostTypeId="2" ParentId="2109" CreationDate="2014-05-05T16:47:03.347" Score="6" Body="&lt;p&gt;I have many ways I have used technology in my math class to promote mathematical thinking. Here are five examples.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Share a &lt;a href=&quot;http://www.geogebratube.org/student/m53240?mobile=true&quot; rel=&quot;noreferrer&quot;&gt;dynamic Geogebra applet at the front of the room&lt;/a&gt;, and ask students to share things they notice, and share things they wonder, and ask the person at the front to change the sliders. As students notice things, other students notice other things, and pretty soon all of the important features of the graph fall out of their collaborative thinking. Imagine trying to do this activity (noticing and wondering about different forms of a linear function) without the technology!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Instead of giving students tutorial videos, or asking them to create tutorial videos, ask students to create video word problems, ones which involve the current mathematics they are learning, but with a challenging problem solving narrative. &lt;a href=&quot;http://davidwees.com/content/turning-math-word-problems-math-video-problems&quot; rel=&quot;noreferrer&quot;&gt;Here are some examples of the final product&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;With kindergarten students, I showed them briefly how to use Google's Blockly. They then used the simple programming language to draw shapes, and I asked students to attempt challenges like &quot;draw a square&quot; or &quot;draw a stair case&quot;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I created a &quot;&lt;a href=&quot;http://davidwees.com/graphgame/&quot; rel=&quot;noreferrer&quot;&gt;Graph Game&lt;/a&gt;&quot; for students to use. The idea is that students are attempting to recreate the distance-time graph by sliding the little stick-man across the screen. It turns out that this forces students to confront some of the most common misconceptions related to what graphs actually mean.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We did a project once with my 9th grade class, which was to look at the number of digits that repeat in any given fraction (or the number of digits before the decimal expansion terminates) and find a relationship between the numerator and the denominator. It turns out that a calculator display is too limiting to do this project, and in fact, most programming languages are not up to the task using their default operations, so &lt;a href=&quot;http://davidwees.com/divider/&quot; rel=&quot;noreferrer&quot;&gt;I created this calculator that just does division&lt;/a&gt;, up to an arbitrary length. In this case the technology enables the problem solving activity since actually doing the division would be much too time-consuming.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-05-05T16:47:03.347" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2170" PostTypeId="2" ParentId="2157" CreationDate="2014-05-05T17:27:22.937" Score="8" Body="&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; You might consider Klain and Rota's (1997) proof of Buffon's &lt;em&gt;noodle&lt;/em&gt; problem as a somewhat general example that can be used to resolve (the better known) Buffon's &lt;em&gt;needle&lt;/em&gt; problem, and also to prove Barbier's Theorem. See more on its &lt;a href=&quot;http://en.wikipedia.org/wiki/Buffon%27s_noodle&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipage&lt;/a&gt;; I came across this example on an old MO post &lt;a href=&quot;https://mathoverflow.net/a/5382/22971&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Here is a nice problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Twenty-three people, each with integral weight, decide to play football, separating into two teams of $11$ people, plus a referee. To keep things fair, the teams chosen must have equal total weight. It turns out that no matter who is chosen to be the referee, this can always be done. Prove that the $23$ people must all have the same weight.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The problem as stated above is $\#3.4.31$ in &quot;The Art and Craft of Problem Solving&quot; (2006, 2e, p. 107) by Paul Zeitz. However, solving it is probably easier when tackling the stronger case for $2n + 1$ people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;N.B.&lt;/strong&gt; Abstracting away from actual person-weights and extending from positive integers to rational numbers is reasonably straightforward. The result still holds when extended all the way to complex numbers, but this result is tougher to show - and no longer at the level about which you have asked. For a bit of history around this problem, see my earlier MO post &lt;a href=&quot;https://mathoverflow.net/questions/105400/extending-an-assignment-property-from-q-to-r-or-c&quot;&gt;here&lt;/a&gt; or the blitz summary &lt;a href=&quot;https://mathoverflow.net/a/165083/22971&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-05-15T15:37:11.893" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2171" PostTypeId="2" ParentId="2157" CreationDate="2014-05-05T17:41:20.767" Score="23" Body="&lt;p&gt;For a very basic example, how about proving that 59549121058965346178 can be expressed as a product of primes? It is much easier to prove the stronger result that every positive integer can be expressed as a product of primes.&lt;/p&gt;&#xA;&lt;p&gt;As a clarification, among professional mathematicians, the phrase &amp;quot;a product of primes&amp;quot; includes the case &amp;quot;a product of no primes&amp;quot; (which is one) or &amp;quot;a product of one prime&amp;quot; (which is the prime itself) as well as &amp;quot;a product of two or more primes&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="76" LastEditorUserId="2083" LastEditDate="2023-11-19T09:01:49.160" LastActivityDate="2023-11-19T09:01:49.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2172" PostTypeId="2" ParentId="2150" CreationDate="2014-05-05T19:02:21.730" Score="11" Body="&lt;p&gt;I personally never bothered with mathematical induction in a first calculus course (except for perhaps a very few exceptions), but I did point out how the sum rule for derivatives, $(f+g)' = f' + g',$ can be reapplied successively to get the sum rule for derivatives that textbooks actually require in exercises but almost never explicitly tell students about. For example,&#xA;$$(f + g + h)' \; = \; \left( (f + g) \, + \, h \right)' \; = \; (f+g)' \, + h' \; = \; (f'+g') \, + \, h' \; = \; f' + g' + h' $$&#xA;However, I have previously collected some induction examples &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=5965460&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=5966861&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, and &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=5968462&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. Below I've LaTexed some of the examples that might be of use in your situation. Also, in &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=5487150&quot; rel=&quot;nofollow&quot;&gt;this 16 January 2007 sci.math post&lt;/a&gt; is a bibliography of 72 books and papers on mathematical induction, mostly papers from elementary journals that are not reviewed by Mathematical Reviews or Zentralblatt MATH (and thus not easily found except by what I did, which was to look through every issue of several journal titles at various university libraries, which took me several years).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1.&lt;/strong&gt; Prove that the sum rule holds for arbitrary finite sums:&#xA;$$\frac{d}{dx} \left( f_1 + f_2 + f_3 + \ldots + f_n \right) \;\; = \;\; \frac{d}{dx}\left(f_{1}\right) + \frac{d}{dx}\left(f_{2}\right) + \frac{d}{dx}\left(f_{3}\right) + \ldots + \frac{d}{dx}\left(f_{n}\right)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2.&lt;/strong&gt; Prove that the $n$th derivative of the product of two functions follows the pattern for expanding the $n$th power of a sum (i.e. binomial expansion for positive integer powers):&#xA;$$(fg)' \; = \; f'g + fg'$$&#xA;$$(fg)'' \; = \; f''g + 2f'g' + fg''$$&#xA;$$(fg)''' \; = \; f'''g + 3f''g' + 3f'g'' + fg'''$$&#xA;Let the $0$th derivative of a function be the function itself in order to make the correspondence complete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This also works for the $n$th derivative of a product of $m$ functions (corresponds to $n$th power of an $m$-omial), but this is probably too elaborate for your needs. Anyway, the version for $m = 2$ (what I have above) is proved almost exactly like you prove the binomial expansion using induction in a high school or college precalculus course (which I realize most people skip, including myself, except when I taught honors level precalculus in a high school).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3.&lt;/strong&gt; Using logarithms and L'Hopital's rule, you can easily show:&#xA;$$\lim_{x \rightarrow 0^{+}} x^{x} \;\; = \;\; 1 $$&#xA;$$\lim_{x \rightarrow 0^{+}} x^{x^{x}} \;\; = \;\; 0 $$&#xA;$$\lim_{x \rightarrow 0^{+}} x^{x^{x^{x}}} \;\; = \;\; 1 $$&#xA;$$\lim_{x \rightarrow 0^{+}} x^{x^{x^{x^{x}}}} \;\; = \;\; 0 $$&#xA;Using induction, you can show that this pattern continues. For an odd number of $x$'s the expression approaches $0$ as $x$ approaches $0$ from the right, and for an even number of $x$'s the expression approaches $1$ as $x$ approaches $0$ from the right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is something that a calculator exploration will suggest, by the way. In fact, the very first day I ever used a graphing calculator (and this also includes any computer algebra system, since I had never used one of those until several years after I first used a graphing calculator) I graphed $y = x^{x},$ then $y = x^{x^x},$ then $y = x^{x^{x^x}},$ and so on, and noticed this pattern. I even skipped ahead and tried graphing functions such as $y= x^{{x}^{{x}^{{\cdot}^{{\cdot}^{{\cdot}^{x}}}}}}$ $(8\; x\text{'s})$ and $y = x^{{x}^{{x}^{{\cdot}^{{\cdot}^{{\cdot}^{x}}}}}}$ $(15\; x\text{'s}).$ I did this on a HP-28S which I had purchased that day, sometime back in Fall 1988 or Winter 1988-89 (I don't remember more precisely now).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4.&lt;/strong&gt; Let $x$ be a real number. Define $x\,\#\,0 = 1$ and, for&#xA;positive integers $n,$ define $x\,\#\,n \; = \; x(x-1)(x-2)\cdots(x-n+1).$ Then using induction one can prove that the forward difference operator applied to $x\,\#\,n\;$ is $\;n\cdot[x\,\#\,(n-1)].$ &lt;strong&gt;Note:&lt;/strong&gt; The &quot;forward difference operator&quot; applied to&#xA;the function $f(n),\;$ where $f:{\mathbb N} \rightarrow {\mathbb R},\;$ gives the result $f(n+1) - f(n).$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5.&lt;/strong&gt; Find an error in the following &quot;proof&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We prove by induction that $\frac{d}{dx}(x^n) = 0$ for each $n = 0,$ $1,$ $2,\, \ldots $ Note that $\frac{d}{dx}(x^0)=0,$ since the derivative of a constant is zero. Now assume $\frac{d}{dx}(x^k)=0$ for each $k \leq n.$ (We'll be using what's sometimes called &quot;strong induction&quot; or &quot;course of values induction&quot;.) Then $\frac{d}{dx}(x^{n+1})=0,$ as shown below:&#xA;$$ \frac{d}{dx}\left( x^{n+1} \right) \;\; = \;\; \frac{d}{dx}\left( x^{n} \cdot x \right) \;\; = \;\; \frac{d}{dx}\left(x^{n}\right) \cdot x \; + \; x^{n} \cdot \frac{d}{dx} (x) \;\; = \;\; 0 \cdot x \; + \; x^{n} \cdot 0 \;\; = \;\; 0  $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;6.&lt;/strong&gt; Let $(d/dx)^n$ be $n$-many differentiations with respect to $x.$&#xA;$$\frac{d^n}{dx^n}\left( \ln x \right) \;\; = \;\; (-1)^{n-1}(n-1)! \cdot x^{-n} $$&#xA;$$\frac{d^n}{dx^n}\left( a^{b^x} \right) \;\; = \;\; b^{n} (\ln a)^{n} a^{bx} $$&#xA;$$\frac{d^n}{dx^n}\left( \sin (ax+b) \right) \;\; = \;\; a^{n} \sin \left( ax+b + \frac{n\pi}{2} \right) $$&#xA;$$\frac{d^n}{dx^n}\left( \cos (ax+b) \right) \;\; = \;\; a^{n} \cos \left( ax+b + \frac{n\pi}{2} \right) $$&#xA;$$\frac{d^n}{dx^n}\left( e^{x}\cos x \right) \;\; = \;\; \left(\sqrt{2}\right)^{n}e^{x} \cos \left(x + \frac{n\pi}{4} \right) $$&#xA;$$\frac{d^n}{dx^n}\left( \sqrt{x+1} \right) \;\; = \;\; (-1)^{n-1} \cdot \frac{(1)(3)(5)\cdots (2n-3)}{2^{n}(x+1)^{n - \frac{1}{2}}} $$&#xA;$$\frac{d^n}{dx^n}\left( x^2 - 1 \right)^{-1} \;\; = \;\; (-1)^{n} \cdot \frac{n!}{2} \left[ (x-1)^{-n-1} \; - \; (x+1)^{-n-1} \right] $$&#xA;$$\frac{d^n}{dx^n}\left( \frac{3x-4}{2x^2 + 3x -2 } \right) \;\; = \;\; (-1)^{n} \cdot n! \left[ \frac{2^{n+1}}{(2x-3)^{n+1}} \; - \; \frac{3^{n+1}}{(3x+2)^{n+1}} \right] $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the last two, develop a formula for the $n$th derivative of $(ax + b)^{-1}$ and then use partial fractions (and additivity of the $n$th derivative operation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From what I've read in old texts (&gt; 100 years old), these kinds of results were often useful in calculating Taylor coefficients. In particular, the formulas often allowed one to come up with recursion formulas for the $n$th derivative evaluated at zero, which allowed for more rapid calculation of the coefficients than multiplying known power series expansions together.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="729" LastEditDate="2016-09-28T14:16:53.430" LastActivityDate="2016-09-28T14:16:53.430" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2173" PostTypeId="2" ParentId="2039" CreationDate="2014-05-05T20:00:29.957" Score="3" Body="&lt;p&gt;I'm in a project with 27 different schools from around New York city, all of which have geometry classes, and as far as I can tell, the only use of DGS I have seen is teacher use for demonstration purposes. There are teachers who would use it with students, but the process of getting students able to access it is too excessive with their current technology structures (either a computer lab, or a mobile laptop cart).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in every private school I worked in, we not only had access to DGS, we used it, probably every teacher in the department at some point. In two of these schools students had their own laptops, and in the 3rd school, my class was small enough that using 12 computers in the computer lab was not that big a deal, and so we just went when we needed the software.&lt;/p&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-05-05T20:00:29.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2174" PostTypeId="2" ParentId="2157" CreationDate="2014-05-05T20:11:16.583" Score="16" Body="&lt;p&gt;Consider the definite integral&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{0}^{1}\frac{x^{10} - 1}{\ln x} dx$$&lt;/span&gt;&#xA;This can be evaluated fairly easily by considering instead the more general integral&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{0}^{1}\frac{x^{b} - 1}{\ln x} dx,$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$b \geq 0$&lt;/span&gt; is an arbitrary parameter, and using a method called &amp;quot;differentiating the integral&amp;quot; or &amp;quot;differentiating under the integral sign&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$F(b) = \int_{0}^{1} \frac{x^{b} - 1}{\ln x} dx.$&lt;/span&gt; Ignoring the matter of interchanging the order of integration with differentiation (this is justified in older advanced calculus texts), we have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$F'(b) \;\; = \;\; \frac{d}{db} \int_{0}^{1}\frac{x^{b} - 1}{\ln x} dx \;\; = \;\; \int_{0}^{1}\frac{d}{db} \left( \frac{x^{b} - 1}{\ln x} \right) dx \;\; = \;\; \int_{0}^{1} x^{b} dx \;\; = \;\; \frac{1}{b+1}$$&lt;/span&gt;&#xA;Integrating both sides of &lt;span class=&quot;math-container&quot;&gt;$F'(b) = \frac{1}{b+1}$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$F(b) = \ln (b+1) + C$&lt;/span&gt; for some constant &lt;span class=&quot;math-container&quot;&gt;$C.$&lt;/span&gt; We can find &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; by using a value of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; for which we know the value of &lt;span class=&quot;math-container&quot;&gt;$F.$&lt;/span&gt; Clearly, &lt;span class=&quot;math-container&quot;&gt;$F(0) = 0.$&lt;/span&gt; This implies that &lt;span class=&quot;math-container&quot;&gt;$C = 0,$&lt;/span&gt; and so &lt;span class=&quot;math-container&quot;&gt;$F(b) = \ln(b+1).$&lt;/span&gt; Hence, the first integral I gave has the value &lt;span class=&quot;math-container&quot;&gt;$F(10) = \ln {11}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[[&lt;/strong&gt; &lt;em&gt;(added 4 years later)&lt;/em&gt; The same example with &lt;span class=&quot;math-container&quot;&gt;$b = 7$&lt;/span&gt; is an answer to a similar question in Mathematics Stack Exchange --- &lt;a href=&quot;https://math.stackexchange.com/questions/899109/problems-that-become-easier-in-a-more-general-form/900398#900398&quot;&gt;Problems that become easier in a more general form&lt;/a&gt;. This other question has several answers different from any of those given here, and thus it is worth looking at for more examples pertaining to the present question. &lt;strong&gt;]]&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can find this method discussed in older (say, before 1945) advanced calculus texts. This is also the method Feynman talks about in his book &lt;strong&gt;Surely You're Joking Mr. Feynman&lt;/strong&gt; (excerpt below). I used the example above on take home tests in the late 1990s when I was teaching at a residential &amp;quot;math-science high school academy&amp;quot;. The problem was supplied with a generous hint, and I also included the Feynman quote below. &lt;strong&gt;[[&lt;/strong&gt; &lt;em&gt;(Added 4 years later)&lt;/em&gt; See problem #5 on &lt;a href=&quot;http://mathforum.org/kb/servlet/JiveServlet/download/206-2048374-6998850-589649/LSMSA%20multi%20test%204b.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this take home test&lt;/a&gt; for the last time I assigned this problem. &lt;strong&gt;]]&lt;/strong&gt; Later (8 April 2000) I posted this in sci.math (see &lt;a href=&quot;https://groups.google.com/g/sci.math/c/j9o0OHIcsug/m/gUHLEF1xwYQJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Excerpt from Richard Feynman, &lt;strong&gt;Surely You're Joking Mr. Feynman&lt;/strong&gt; (1985), pp. 86-87. (Text enclosed in brackets, [...], are NOT from the original. These are additions I've included.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So every physics class, I paid no attention to what was going on with Pascal's Law, or whatever they were doing. I was up in the back with this book: &amp;quot;Advanced Calculus&amp;quot;, by Woods. Bader [Feynman's High School Physics teacher, who loaned Feynman his copy of Wood's book] knew I had studied &amp;quot;Calculus for the Practical Man&amp;quot; a little bit, so he gave me the real works--it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions--all kinds of wonderful stuff that I didn't know anything about.  That book also showed how to differentiate parameters under the integral sign--it's a certain operation. It turns out that [it's] not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2022-01-15T07:35:09.987" LastActivityDate="2022-01-15T07:35:09.987" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="2175" PostTypeId="2" ParentId="2074" CreationDate="2014-05-05T21:17:27.650" Score="10" Body="&lt;p&gt;I teach at the college level in the US, where geometry is a standard part of the high school curriculum, so my students are all supposed to have had previous exposure to it. However, probably only a certain percentage (50%?) of students here take a geometry course that requires them to write proofs.  I assume that in many cases in the US, the high school geometry teacher simply isn't willing to grade a stack of proofs from five classes, each with an enrollment of 40.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What frightens me is the results I get when I ask students to write even a trivial proof. For example, I introduce the vector cross product and give both its characterization in terms of components and its geometrical characterization. I then assign a homework problem in which the student is asked to prove that $\mathbf{A}\times(k\mathbf{B})=k(\mathbf{A}\times\mathbf{B})$. The most common response (among students who write an answer at all) is to make up an example and demonstrate that the property holds for that example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If students think this is a valid mode of reasoning, then I suppose they believe that all presidents of the United States are African-American, since we have an example that proves it. This is not a defect in their mathematical preparation, it's a defect in their critical thinking skills that could perhaps have been &lt;em&gt;addressed&lt;/em&gt; through their mathematical training. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are the results in an educational system that &lt;em&gt;does&lt;/em&gt;, at least in theory, require proof-writing in high school. I shudder to imagine how much worse it would be in a system where there had not even been any attempt to teach proof-writing. When Abraham Lincoln was a country lawyer in Illinois, traveling on horseback to county seats and sleeping in boarding houses, he always carried a copy of Euclid with him, which he would study at night by lamplight as a model of logical argumentation. I wonder how Lincoln would have turned out if he'd never been exposed to Euclid.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-05T21:17:27.650" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2176" PostTypeId="2" ParentId="2157" CreationDate="2014-05-05T23:24:51.133" Score="17" Body="&lt;p&gt;I think it is easier to prove Cauchy-Schwarz in an arbitrary inner product space than for dot product on $\mathbb{R}^n$, because trying to prove it in general limits the selection of tools you have available:  you only have inner product space axioms.  In $\mathbb{R}^n$, you might try all kinds of explicit horrible calculations to try to get at the inequality.  This illustrates in general why proving a stronger statement is oftentimes easier:  by working towards a stronger result you limit the search space of possible proofs.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-05-05T23:24:51.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2177" PostTypeId="2" ParentId="2152" CreationDate="2014-05-05T23:37:59.557" Score="12" Body="&lt;p&gt;First f all, check out this great &lt;a href=&quot;http://journals.aps.org/prstper/pdf/10.1103/PhysRevSTPER.9.020110&quot; rel=&quot;noreferrer&quot;&gt;paper&lt;/a&gt;. It has some interesting examples, especially in Appendix A.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A typical example is relativistic mechanics: the usual first order approximation yields the unsatisfactory Newtonian mechanics (see &lt;a href=&quot;http://en.wikipedia.org/wiki/Relativistic_mechanics#Classical_limit&quot; rel=&quot;noreferrer&quot;&gt;for example here&lt;/a&gt;), which is in many applications not enough to explain what is happening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;ADDED:&lt;/strong&gt; Since OP asked for an example, calculations in special relativity usually boil down to the use of the binomial series, hence they are &lt;em&gt;mathematically&lt;/em&gt; suitable for a calculus class. Whether people understand the underlying &lt;em&gt;physics&lt;/em&gt; is a different question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Section 15.5 in these &lt;a href=&quot;http://www-physics.ucsd.edu/students/courses/fall2010/physics200a/LECTURES/200_COURSE.pdf&quot; rel=&quot;noreferrer&quot;&gt;great notes&lt;/a&gt; explains how in the relativistic harmonic oscillator the period depends on the amplitude (Section 15.5.1, page 311), Formula (15.132).&lt;/p&gt;&#xA;" OwnerUserId="61" LastEditorUserId="61" LastEditDate="2014-05-06T07:43:25.783" LastActivityDate="2014-05-06T07:43:25.783" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2178" PostTypeId="2" ParentId="2152" CreationDate="2014-05-05T23:41:24.637" Score="13" Body="&lt;p&gt;Here's one that I just thought of, by modifying a problem in the ODEs section of my textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question: &lt;em&gt;In the presence of air resistance, does a thrown ball take longer to go up or to come down?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We assume that the force of air resistance is proportional to velocity, with constant of proportionality $p$.  Thus, we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ F = m a = - p v - m g. $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since $a=v'$, this is a differential equation for $v$.  For simplicity, let's divide out by $m$ and write $q=p/m$, so it becomes&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ v' = - q v - g. $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The differential equation is separable, so we can solve it to get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ v = \left(v_0 + \frac{g}{q}\right) e^{-q t} - \frac{g}{q}. $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Integrating again, we get the height as a function of time:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ y = \left(\frac{v_0}{q} + \frac{g}{q^2}\right)(1-e^{-q t}) - \frac{g t}{q}. $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can solve $v = 0$ to find the time $t_{\mathrm{max}}$ at which the ball reaches its maximum height, but it's not so easy to solve $y=0$ to find the time at which it reaches the ground.  The book answers the question by using a clever argument to conclude that $y(2t_{\mathrm{max}})&amp;gt;0$.  But we can also approximate $y$ by a power series in the small parameter $q$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's substitute the &lt;em&gt;second-degree&lt;/em&gt; Taylor polynomial for $e^{-q t}$, namely&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$1 - q t + \frac{1}{2} q^2 t^2,$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;into $v$.  We get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ v \approx \left(v_0 + \frac{g}{q}\right) \left(1 - q t + \frac{1}{2} q^2 t^2\right) - \frac{g}{q} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplying this out, we get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ v\approx v_0 + \frac{g}{q} - (v_0 q + g) t + \frac{1}{2} (v_0 q^2 + g q) t^2 - \frac{g}{q}. $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Canceling the $\frac{g}{q}$s and discarding the term involving $q^2$ (since $q$ is small), we obtain&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ v\approx v_0 + (v_0 q + g) t + \frac{1}{2} g q t^2 . $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;to first order in $q$.  Note that all the $q$s in the denominators canceled out, and this is a &lt;em&gt;linear&lt;/em&gt; approximation to $v$ as a function of $q$.  In principle, we could have derived it by simply taking the derivative of $v$ with respect to $q$ at $q=0$.  However, it's not at all obvious from the formula for $v$ that it's even differentiable at $q=0$!  (It's not even necessarily obvious that it has a limit as $q\to 0$, but of course the physical considerations imply that it must.)   The power series expansion is much nicer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can also integrate $v$ to get the height&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y \approx v_0 t + \frac{1}{2} (v_0 q + g) t^2 + \frac{1}{6} g q t^3. $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we can answer the original question.  Putting $v=0$ and solving for $t$ with the quadratic formula, we get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$t_{\mathrm{max}} \approx \frac{1}{q} + \frac{v_0}{g} \pm \frac{1}{q}\left(1+\frac{v_0^2 q^2}{g^2}\right)^{1/2}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the linear Taylor polynomial for the binomial series, we get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$t_{\mathrm{max}} \approx \frac{1}{q} + \frac{v_0}{g} \pm \frac{1}{q}\left(1+\frac{v_0^2 q^2}{2g^2}\right).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We must use the minus sign to cancel out the impossible $1/q$s, yielding&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$t_{\mathrm{max}} \approx \frac{v_0}{g} - \frac{v_0^2 q}{2 g}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the first term, $\frac{v_0}{g}$, is the obvious value in the no-air-resistance case $v = v_0 - g t$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can also set $y=0$ and solve for $t$ to find the time when the ball lands.  The solution $t=0$ (when it was thrown) factors out and we can use the quadratic formula again:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ t_{\mathrm{lands}} \approx \frac{3}{2q} + \frac{3v_0}{2g} \pm \frac{3}{2q}\left(1 - \frac{2qv_0}{3g} + \frac{v_0^2 q^2}{g^2}\right)^{1/2}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the &lt;em&gt;second-degree&lt;/em&gt; Taylor polynomial for the binomial series this time (since there is a power of $q$ rather than just $q^2$ inside the square root) but discarding the resulting $q^4$ term, we get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ t_{\mathrm{lands}} \approx \frac{3}{2q} + \frac{3v_0}{2g} \pm \frac{3}{2q}\left(1 - \frac{qv_0}{3g} + \frac{v_0^2 q^2}{2g^2} - \frac{q^2 v_0^2}{18 g^2}\right).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again we must take the minus sign to cancel the impossible $\frac{3}{2q}$, and we get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ t_{\mathrm{lands}} \approx \frac{2 v_0}{g} - \frac{2 v_0^2 q}{3g^2}. $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observe that $t_{\mathrm{lands}} &amp;gt; 2 t_{\mathrm{max}}$.&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="38" LastEditDate="2014-05-06T00:23:45.937" LastActivityDate="2014-05-06T00:23:45.937" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2179" PostTypeId="2" ParentId="2157" CreationDate="2014-05-05T23:41:41.530" Score="4" Body="&lt;p&gt;The proof that it is possible to cover a chessboard with L-shape triominos leaving a space free at the center is a classic. It is proven by induction on the size $2^n \times 2^n$ leaving an arbitrary square free. See e.g. &lt;a href=&quot;http://www.cut-the-knot.org/Curriculum/Geometry/Tromino.shtml&quot; rel=&quot;nofollow&quot;&gt;Cut the knot&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-05T23:41:41.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2180" PostTypeId="2" ParentId="1914" CreationDate="2014-05-06T00:28:13.947" Score="2" Body="&lt;p&gt;Try this special variation on the game of Nim:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Put a pile of 42 stones on the table (although if you don't have 42, any multiple of six will do). Explain to your student that each turn, each of you will remove either 1, 3, or 5 stones, and the person who removes the last stone from a pile wins. Then, explain that the person who moves second has a winning strategy, and offer them a choice of who goes first. Most likely, they'll make you go first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, challenge one of your students to win &lt;em&gt;exactly&lt;/em&gt; six of seven games of this type (perhaps offering some sort of prize as a reward). You could also do four out of five, two out of three, as long as they have to try to lose at least one game. They'll probably figure out some time during the first six games that they can force a win by taking 6 stones minus however many you took, especially if they've seen this sort of thing before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But when they try to &lt;em&gt;lose&lt;/em&gt; the seventh game, they'll find it's impossible.  Can you guess why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every turn when you take stones, you'll leave an odd number, and every turn when they take stones, they'll leave an even number. There's no way to break this pattern within the defined rules, and a consequence of this is that the person who moves second &lt;em&gt;always&lt;/em&gt; wins (since 0 is an even number), no matter whether he follows any sort of strategy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This introduces them to the idea of invariants. It's hard to know the exact level of theory you're looking for, but I think this is a good example purely from a discovery point of view, and doing something counterintuitive (being unable to &lt;em&gt;lose&lt;/em&gt; a game even when you want to).&lt;/p&gt;&#xA;" OwnerUserId="856" LastEditorUserId="856" LastEditDate="2014-05-06T00:38:52.210" LastActivityDate="2014-05-06T00:38:52.210" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2181" PostTypeId="2" ParentId="2162" CreationDate="2014-05-06T01:57:06.683" Score="8" Body="&lt;p&gt;If I may add my 2¢ as a &lt;em&gt;former student&lt;/em&gt;, in no way involved in teaching this stuff, my own initial confusion came from discussions on &quot;moving towards $x_0$&quot; and &quot;for very small ...&quot; and other such. When I came to grasp $\epsilon$ - $\delta$ as a &lt;em&gt;static&lt;/em&gt; situation, no &quot;moving,&quot; no &quot;very small&quot; required, things became clear. I had an exceptionally gifted teacher (he was the dean of Math here, and taught first year calculus out of love for the subject). He very carefully avoided the &quot;moving&quot; and &quot;very small&quot; imagery, and I believe that helped enormously. Even some earnest discussion of what to do if $\epsilon = 1000$ was thrown in.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-06T01:57:06.683" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2182" PostTypeId="2" ParentId="1260" CreationDate="2014-05-06T02:22:36.217" Score="2" Body="&lt;p&gt;One puzzle I rather like comes from a website (forget where) that gives a problem about four cards. Each card contains a letter on one side, and a number on another. Also, each card &lt;em&gt;ostensibly&lt;/em&gt; has the property that &lt;strong&gt;if one side contains a vowel, then the other side contains an even number&lt;/strong&gt;. Then, you're given four cards, with the following faces showing:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;7&lt;/li&gt;&#xA;&lt;li&gt;2&lt;/li&gt;&#xA;&lt;li&gt;E&lt;/li&gt;&#xA;&lt;li&gt;S&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and asked which cards would you need to turn over to be sure that that card followed the bolded property. It turns out that you need to flip 7 and E (because those are the only ones that could possibly have a vowel on one side and an odd number on another side), but most people who don't know this beforehand would get something else.&lt;/p&gt;&#xA;" OwnerUserId="856" LastActivityDate="2014-05-06T02:22:36.217" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2183" PostTypeId="2" ParentId="2152" CreationDate="2014-05-06T03:00:20.250" Score="7" Body="&lt;p&gt;I don't know if this is the sort of thing you're thinking about -- it seems to me like a too-obvious example, but it seems to fit your question pretty well.  Suppose you have a particle moving along some path $r(t)$, and suppose we expand $r(t)$ in a Taylor series as $r(t) = a_0 + a_1 t + (1/2) a_2 t^2 + (1/6) a_3 t^3 + ....$  Then $a_0$ tells us the initial location of the particle, $a_1$ its initial velocity, $a_2$ its initial acceleration, $a_3$ its initial jerk, $a_4$ the initial &quot;jounce&quot; or &quot;snap&quot; (beyond that the derivatives don't have standard names).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A second example, one that maybe isn't &quot;physical&quot; enough but also might give insight into the higher-order terms:  Compound interest.  If a bank account balance is growing according to $B(t)=A e^{rt}$, and we expand the RHS as a Taylor series, the constant term tells you the initial balance, and the linear term tells you how the balance would grow if we were only considering &lt;em&gt;simple&lt;/em&gt; interest; the higher-order terms take into account the effects of &lt;em&gt;compounding&lt;/em&gt;.  (You can adapt this example to deal with any kind of exponential growth or decay.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-05-06T03:00:20.250" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2184" PostTypeId="2" ParentId="2152" CreationDate="2014-05-06T03:29:19.460" Score="10" Body="&lt;p&gt;Much of quantum mechanics is only known in a perturbative framework. Essentially, the basic objects of interest are series. Terms are calculated by Feynman diagrams of increasingly  complex diagrams for higher order corrections. So, the series description is pragmatically fundamental. I suppose in principle, non-pertubative solutions exist, however, if no one can hack the math then for a physical purposes (literally in my answer) the series generated by Feynman diagrams represents the best view we have of quantum field theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, low order applications are interesting. We use $F=mg$ to approximate $F = \frac{GmM}{(R_E+h)^2}$ at $h=0$. In fact, $F=mg$ is just the constant term in the Taylor series for gravity. I submit to you this is still interesting, and, in view of the final I just graded, still sufficiently challenging for many undergraduates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applications where you use all the terms in a series are perhaps nicely found in math itself. For example, the alternating harmonic series convergence to $\ln(2)$ as seen from the expansion of $\ln(1+x)$. This is a nice example which uses all the terms in the series. Many examples of this nature are found in every textbook, they make formidable homework problems when taken out of context.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-06T03:29:19.460" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2185" PostTypeId="1" AcceptedAnswerId="2245" CreationDate="2014-05-06T04:18:20.997" Score="13" ViewCount="3302" Body="&lt;p&gt;This is very similar to a question I asked at MSE a while back ago but I think it is far more appropriate here. &lt;a href=&quot;https://math.stackexchange.com/questions/632139/examples-of-open-ended-calculus-class-project-ideas&quot;&gt;https://math.stackexchange.com/questions/632139/examples-of-open-ended-calculus-class-project-ideas&lt;/a&gt;. However, I'd like to throw in a twist. I want to emphasize writing about mathematics (or applied mathematics), so I want them to write a paper. In order for this to work, I'd like the open ended project ideas to be less like harder/more involved HW questions than what I got on that site, and more like open ended projects where the students really have some freedom in where their research takes them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone tried requiring a final report/paper for a calculus class? If so what did you ask them? and how did it go over in general? Would you do it again?&lt;/p&gt;&#xA;" OwnerUserId="1387" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-05-12T07:27:58.207" Title="What are some good final project prompts for a calculus class (where they have to write a &quot;paper&quot;)?" Tags="|calculus|writing|term-papers|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2186" PostTypeId="1" AcceptedAnswerId="2192" CreationDate="2014-05-06T04:40:53.143" Score="11" ViewCount="267" Body="&lt;p&gt;All students seem to accept that they should take notes during a lecture. However, they don't think about why they are doing this. Most students think that the reason to take notes is to have a reference for later, but we know that the real reason to take notes is to &lt;strong&gt;make your brain process the material&lt;/strong&gt;. As a result, many students frantically write down computations instead of thinking about what is going on -- and reserve the &quot;thinking about what is going on&quot; until later!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, students would not write down every little thing said by the instructor, but would take note of important facts and also write down many of their own thoughts and questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What methods &lt;strong&gt;have you actually tried&lt;/strong&gt; in your courses to help students learn how to take effective math notes? I'd welcome success stories and otherwise.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="143" LastEditDate="2014-06-15T11:01:54.777" LastActivityDate="2014-06-15T11:01:54.777" Title="How can we help students learn to effectively take notes?" Tags="|general-pedagogy|transferrable-skills|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2187" PostTypeId="2" ParentId="1026" CreationDate="2014-05-06T04:55:53.793" Score="2" Body="&lt;p&gt;I let this be up to the student. I tell them, &quot;these things will be provided on the exam, ... then proceed to give them a list of a few formulas, .... If it's not on that list and it's in the relevant chapters in the book or was presented in class, you will need to be able to derive it or have it memorized.&quot; I try to keep memorization to a minimum. I am often pleasantly surprised at how many students actually choose to derive some of the formulas. If I told them strictly to have memorized that formula, perhaps they would have gotten less out of the course.&lt;/p&gt;&#xA;" OwnerUserId="1387" LastEditorUserId="1387" LastEditDate="2014-05-06T19:19:17.973" LastActivityDate="2014-05-06T19:19:17.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2188" PostTypeId="1" AcceptedAnswerId="2212" CreationDate="2014-05-06T05:12:31.913" Score="11" ViewCount="312" Body="&lt;p&gt;So I have given quizzes in two different ways&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Quizzes will be a problem taken &quot;verbatim&quot; from an example in this weeks reading, prior to the topic being presented in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Quizzes consist of easy questions, made up by the instructor, that are used to make sure you are following the basic concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience quizzes of type (2) are liked by the students more, but quizzes of type (1) might help the students learn more, by encouraging them to keep on top of their readings. It's a small sample, but when I did the first option my teaching evaluations were lower but the students actually scored higher on a &quot;common final exam&quot; compared to students with different instructors. How have you used quizzes? Which ways do you think are the most effective.&lt;/p&gt;&#xA;" OwnerUserId="1387" LastEditorUserId="1387" LastEditDate="2014-05-06T18:28:56.687" LastActivityDate="2014-05-07T14:17:39.397" Title="How useful are quizzes in undergraduate math courses?" Tags="|exams|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2189" PostTypeId="2" ParentId="2164" CreationDate="2014-05-06T07:00:46.530" Score="40" Body="&lt;p&gt;One thing you might do is contrast reading mathematics textbooks with reading novels. I have seen this done at the start of a textbook draft for a course on the Kuratowski closure operators (&lt;a href=&quot;https://matheducators.stackexchange.com/a/910/262&quot;&gt;MESE sketch&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The material is not generally for distribution, but here is a brief excerpt from the start of the book:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/gtDGk.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see that Topology was mentioned in a comment above; &lt;a href=&quot;https://www.dropbox.com/s/v0x7w1110tg7v3t/topology%20text.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; are some notes I wrote up early on for students going through Munkres' text. Most of these questions come from just a couple of pages in the text (pp. 76-77) and my goal was to emphasize early on that reading a book like this passively would not suffice. I did not have a control group, but several students &lt;em&gt;did&lt;/em&gt; make an honest attempt at answering these questions, which I expect was helpful to them. Probably it would have been better if I had continued to ask questions like these (or had the students formulate their own questions in a similar vein).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this addresses some of the &lt;strong&gt;approaches&lt;/strong&gt;, but I do not think I can say much more about &lt;strong&gt;what happened&lt;/strong&gt; since these reminders were not provided in an on-going way. (Even among prospective mathematics teachers, it is quite surprising how often someone will remark with exasperation that they must rely on definitions to answer a question. Consider a problem like: &lt;em&gt;Prove that all cyclic groups are abelian.&lt;/em&gt; My experience is that students who have trouble with this problem are either unaware that they must know the words &lt;em&gt;cyclic&lt;/em&gt;, &lt;em&gt;group&lt;/em&gt;, and &lt;em&gt;abelian&lt;/em&gt; in order to make sense of the question; or they are not sure how to tackle a &lt;em&gt;prove all&lt;/em&gt; problem. I consistently expect more from graduate students, but it is likely that sometimes more is necessary than early remarks on textbook reading that are complemented by further explanations when problems arise.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-06T07:00:46.530" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2190" PostTypeId="2" ParentId="2188" CreationDate="2014-05-06T09:51:12.623" Score="3" Body="&lt;p&gt;Giving the same problem as in the text would tend to have them work all problems, which is nice. But it can also lead to trading in solved problems, learned by rote. Most &quot;solution manuals&quot; are readily available (even when they should only be distributed to bona-fide instructors).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Better make up a problem by varying one of the standard text problems (change some of the constants, ask to go a step further, ...).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-06T09:51:12.623" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2191" PostTypeId="2" ParentId="2164" CreationDate="2014-05-06T10:00:41.557" Score="7" Body="&lt;p&gt;Give (modest) extra credit for reading the text critically (very useful to fix your own lecture notes). Give homework on the lines of &quot;give an alternative proof&quot; or make them invent and work through some examples. It might be useful to collect wrong proofs (where the conclusion is right and where it is wrong) to exercise finding (and patching) holes, and finding counterexamples.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-06T10:00:41.557" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2192" PostTypeId="2" ParentId="2186" CreationDate="2014-05-06T11:23:58.137" Score="11" Body="&lt;p&gt;&lt;em&gt;It's a skill you can teach fairly explicitly&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach 16-18 year old students A-level maths in classes (not lectures) in the UK. Over the course of two years I gradually switch from telling them exactly what to write, where and when to letting them decide. They do occasionally still ask near the end, but that's OK, and we can discuss it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is comparable to teaching them any long term skill - at first I make every decision for them, then involve them increasingly in the decision making, and finally I'm only needed for occasions when they're unsure again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;~&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;As a student your needs may vary from what is expected&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the other side of the fence, as an undergraduate I found some lecturers had magic dust in that I understood them very clearly when they were in the room but when they went away I was lost. It took me a long time to figure out that the magic dust was the explanations they were saying out loud but not writing on the board. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I wrote things down that helped explain, I found I understood even when I was away from the lecturer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conclusion: students and teachers may actually underestimate how much needs to be written down, and it depends on the combination of subject, student and lecturer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;~&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Guide them to experienced choices&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's no easy way for students to see which details they will need to write down. Experience is the best teacher, but you can help them acquire that over time rather than they suddenly switch from all guidance to no guidance.&lt;/p&gt;&#xA;" OwnerUserId="1340" LastActivityDate="2014-05-06T11:23:58.137" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2193" PostTypeId="1" AcceptedAnswerId="2199" CreationDate="2014-05-06T11:33:47.953" Score="5" ViewCount="82" Body="&lt;p&gt;There is always a risk that the teaching of translation is neglected. Meaning: there is a process to go through to get from a word problem to a collection of variables, equations and constants. Very often, it’s neglected to cover this key step – resulting in students who are extremely adept with manipulating numbers or formulas, but who are utterly unable to apply their skills outside the math classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having a class with first-year (university) math students of varying ability levels is not really a good position to start out with, as there tend to be gaps in prerequisite knowledge (or each student's feelings towards their own ability to learn math)… and teachers must decide how to meet the needs of the individual students in their classrooms, without ignoring the rest of the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know there is no &lt;em&gt;“ideal”&lt;/em&gt; solution to this as it depends on individual situations, but what approach(es) would be recommendable to handle/balance such mixed-knowledge classes? Are there any references or papers that academically analyzed this issue and present potential solutions to the problem?&lt;/p&gt;&#xA;" OwnerUserId="35" LastActivityDate="2014-05-06T16:42:44.090" Title="Optimal management of mixed-knowledge classes" Tags="|reference-request|classroom-management|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2194" PostTypeId="2" ParentId="2162" CreationDate="2014-05-06T13:32:31.237" Score="9" Body="&lt;p&gt;The essence of your question is why do mathematicians and math educators come to different conclusions about teaching the epsilon-delta definition. May I submit that we have different primary allegiances:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;math educators&lt;/strong&gt;: do &lt;strong&gt;all&lt;/strong&gt; the students understand what is being taught? Typical response to difficult situation, take the technical and replace it with a heuristic so more of the student population &quot;gets&quot; it. Cost: no students learn the true definition. Benefit: many students learn some approximate version of the true, hopefully they return to it later and seek deeper truth in later coursework.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;mathematicians&lt;/strong&gt;: is the content correct? Typical response to student difficulty, face it head-on, refuse to replace true definition with a heuristic. Cost: many students do not learn the definition. Benefit: those who learn the definition correctly do not need to learn the material needlessly again in a later class, hopefully those who learned nothing get more next time they retake the course.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Of course, I'm kidding. The truth is some amalgam of these. Moreover, there is actually no demarcation between math educators and mathematicians. In fact, there are many individuals who are in both camps. I try to incorporate both modes because I don't want to be rude to either camp of students. I want to reach as much of the population I see in my classes as possible. However, one size never fits all. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-06T13:32:31.237" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2195" PostTypeId="2" ParentId="2151" CreationDate="2014-05-06T14:09:25.123" Score="19" Body="&lt;p&gt;It's hard to admit you don't understand if you think you're the only one in the class that's lost (rarely actually the case, but individuals get self-conscious).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I employed a system where if I wanted to check perceived understanding, I would do an &quot;understanding check&quot;.  Students would hold up 1-4 fingers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 - I'm so lost I can't even ask a good question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 - I'm confused, but I kinda get it a little.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3 - I pretty much get it, but there are some fuzzy bits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4 - I understand it well enough to teach it to someone else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quick straw poll, then I could say something like, &quot;Looks like I saw a lot of 2's--can anyone ask a specific question about what I didn't explain well?&quot; which both removed the fear of being 'the dumb one' while shifting the blame back to me.  &lt;/p&gt;&#xA;" OwnerUserId="1393" LastActivityDate="2014-05-06T14:09:25.123" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2196" PostTypeId="1" AcceptedAnswerId="2197" CreationDate="2014-05-06T14:53:23.670" Score="17" ViewCount="3150" Body="&lt;p&gt;Next semester I'm teaching a course where I want to grade a significant chunk of the homework for completeness without checking whether the answers are right, and separately post the answers and encourage students to check the answers themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is largely for homework early in our coverage of a topic, so part of the goal is to make the homework less stressful, and indicate that we don't expect them to understand the topic completely at this stage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My concern is that, especially in the middle of the semester when they get busy, students will be tempted to just write out nonsense knowing we don't check the answers.  (Students sometimes do this when we &lt;em&gt;do&lt;/em&gt; check the answers.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know other people have tried similar things, so I'm curious about methods that worked for getting the students to take the homework seriously.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-03-06T16:02:01.430" Title="Motivating students to take homework seriously without grades" Tags="|undergraduate-education|homework|" AnswerCount="7" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2197" PostTypeId="2" ParentId="2196" CreationDate="2014-05-06T15:52:37.420" Score="13" Body="&lt;p&gt;We tried something that worked nicely (though not for all students).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the beginning of the course we announced that there will be a quiz during the semester, and the questions in the quiz will be taken from their HW assignments. This way they had to check the answers to these assignments. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The downside is, as we noticed while checking the quizzes, that some students just memorized the answers without understanding the most basic ideas.  &lt;/p&gt;&#xA;" OwnerUserId="1392" LastActivityDate="2014-05-06T15:52:37.420" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2198" PostTypeId="1" CreationDate="2014-05-06T16:33:58.393" Score="9" ViewCount="590" Body="&lt;p&gt;I just finished teaching Calculus I this semester at my university, and I have several students that failed the course despite working hard.  The culprit in all of the cases was inadequate background knowledge coming into the course.  My university does not offer any lower level courses, so I'd like to come up with helpful suggestions for them, so that they can adequately prepare this summer to be able to retake Calculus I in the fall with a good chance of being able to pass the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, one option would be to take a college algebra or pre-calculus course at a local college near their hometown.  But if that option is not available to the students, can you recommend any resources for self-study that the students can use?  These students have a weak background, so they will need a lot of specific guidance on what topics to study, as well as explanations for all of the concepts they are studying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you in advance for any suggestions you can offer.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2014-05-06T23:34:04.753" Title="Support materials for student who has failed Calculus I" Tags="|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2199" PostTypeId="2" ParentId="2193" CreationDate="2014-05-06T16:42:44.090" Score="4" Body="&lt;p&gt;There are a great series of papers by or involving &lt;a href=&quot;http://www.dylanwiliam.org/&quot; rel=&quot;nofollow&quot;&gt;Dylan William&lt;/a&gt; including &lt;a href=&quot;http://www.kappanmagazine.org/content/92/1/81.abstract&quot; rel=&quot;nofollow&quot;&gt;Inside the black box&lt;/a&gt;, &lt;em&gt;Mathematics inside the black box&lt;/em&gt; and &lt;em&gt;Assessment for learning: why, what and how?&lt;/em&gt;, in which he examines strategies that actually have the largest measurable impact in education. The title &quot;Assessment for Learning&quot; has been rather abused since then; to quote him, &quot;if your approach to assessment for learning involves a spreadsheet in any way, you're doing it wrong!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Much of the evidence comes from a wide age range, but I don't believe people become fundamentally and radically different at age 18, for example.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In one of his papers he references a study that examined the effects of peer teaching on different ability groups. (Sorry I can't find which paper, nor the study he references, but the example was memorably compelling.) They felt that peer teaching was beneficial for students who were doing badly but held back students who would otherwise get further ahead and excel. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They categorised the students into strong, medium and weak, then had them work in mixed ability pairings with explicit peer teaching tasks over the course of weeks, and reassessed at the end. The weaker students did indeed improve ahead of the control group as expected, but the surprising result was that the stronger students improved compared to the control group by a greater margin - &lt;strong&gt;peer teaching actually helps the helper even more than it helps the one being helped!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps this isn't surprising considering that teaching forces you to verbalise things clearly, answer difficult questions, reiterate fundamental material etc, and generally puts you in a position where you strongly want to have the answers ready.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;~&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion&lt;/strong&gt;&lt;br&gt;&#xA;&lt;em&gt;We conclude then that the best way of dealing with mixed ability groups, for both ends of the spectrum, is to pair them up in deliberately mixed-ability pairs and give them joint-responsibility rich collaborative tasks, and explicitly teach things like coaching over telling.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We've found this to be very effective indeed in practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;~&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Aside: If you're stuck for ways to ensure collaboration instead of copying during class, you can do it in class with Dr Spencer Kagan's &quot;Kagan Strategies&quot;. On the face of it, you could worry that such an approach would be off-putting to serious older intellectual students, but our experience at our 16-18 college in the UK is that the older, brighter and more mathematically confident they are, the less they mind joining in with what appear at first to be silly activities. (The least prepared to take part are the groups of younger remedial students.) The key is to explain why they're doing the wacky thing and to show them an example before they start to break the ice. We've tweaked a few, but largely find that the original is best for promoting learning, even if it seems wacky at first. Certainly it's important to do this from the start of the year - students will accept almost anything when they're new but moan like crazy half way through term if you change the way they learn.]&lt;/p&gt;&#xA;" OwnerUserId="1340" LastActivityDate="2014-05-06T16:42:44.090" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2200" PostTypeId="2" ParentId="747" CreationDate="2014-05-06T16:47:03.930" Score="2" Body="&lt;p&gt;In Alabama, all public school jobs for Elementary, Middle, and High Schools are posted on Teach in Alabama located on the Alabama State Department of Education website. There is also a website/organization designed to certify people interested in teaching who already have a degree in an area of study called Teach for America, at teachforamerica.org. For other teaching avenues, such a private schools. Those are usually posted on the schools website, or can be found on indeed.com. &lt;/p&gt;&#xA;" OwnerUserId="839" LastActivityDate="2014-05-06T16:47:03.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2201" PostTypeId="2" ParentId="2196" CreationDate="2014-05-06T16:48:12.067" Score="11" Body="&lt;p&gt;The following answer is just my opinion and might not address your particular situation.  To oversimplify a bit, most students either&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;want to do the homework for their own reasons (enlightenment, fun, or preparation for the exams,) or&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;only do the homework in order to receive credit.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;To get group (1) to take the homework seriously, you don't need to do anything at all. So let's just consider group (2).&#xA;If you want them to take it seriously, then you need to grade it carefully and only give credit for work that demonstrates understanding.  If you grade based on &quot;completion&quot; then they will (rightly) understand this to mean that a nonsense answer is as good as a correct answer.  A nonsense answer is easier to write, so the choice will be clear to them.  Also, they will think &quot;if the instructor can't be bothered to grade properly, then why should I bother to write a proper answer?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if you want more students to take the homework seriously, but you don't want to grade more carefully, you will need to make more students from group (2) move over to group (1).  The straightforward way to do this would be to make the homework more enlightening, more fun, or more relevant to the exams.  You can make sure the homework is relevant to the exams when you write the exams, so let's think about how to make it more enlightening and fun.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a hard problem in general, but one easy thing you can do is to simply give fewer problems that are less tedious.  This makes it feasible for more of the students to turn in their own work, rather than giving up and writing nonsense or copying the answers from another source.  It is hard enough for a student to understand one new concept per week, so the instructor should focus on making that happen rather than assigning dozens of exercises with the unrealistic expectation that the student will understand them all simply by virtue of being forced by the grading scheme to &quot;do the work.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarize, a simple way to get the students to take the homework more seriously may be, paradoxically, to make it easier.  (The downside is that you will have to select the problems more carefully and make sure the students understand that they need to understand &lt;em&gt;all&lt;/em&gt; of the problems in order to be prepared for the exam.)&lt;/p&gt;&#xA;" OwnerUserId="870" LastActivityDate="2014-05-06T16:48:12.067" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2203" PostTypeId="2" ParentId="2152" CreationDate="2014-05-06T17:39:39.740" Score="4" Body="&lt;p&gt;While this isn't a 'mainstream' topic, I feel like homotopy analysis methods are worth a glance here. The idea is to approximate some non-linear equation by a linear one, take a continuous connection between them, solve the linear problem, and 'slide' the solution over to the non-linear case. If $L\{u(t)\}=0$ is the linear operator and $N\{u(t)\}$ is the non-linear operator, we introduce a new parameter $q$ and consider the equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(1-q)L\{u(t;q)\}+qN\{u(t;q)\}=0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where the solution now depends on $q$. $q=0$ is the linear case and $q=1$ is the full non-linear equation you're trying to solve. What we do is expand the dependence on $q$ as a Taylor series $u(x;q)=\sum u_n(x)q^n$. This leads to a recursion relation for $u_n$ in terms of $u_{n-1}$ that can be solved iteratively. Fiddling around with parameters to ensure that the radius of convergence includes $q=1$ then gives you the non-linear solution as an infinite series. While not physics &lt;em&gt;per se&lt;/em&gt;, it has major applications when dealing with nonlinear waves and similar things. &lt;a href=&quot;http://en.wikipedia.org/wiki/Homotopy_analysis_method&quot; rel=&quot;nofollow&quot;&gt;The wikipedia page&lt;/a&gt; says it better than I can. It might not be appropriate to go through it in detail (it &lt;em&gt;could&lt;/em&gt; be done with only an ODE course, but it would probably take as much time to understand as a major topic of the curriculum), but it's a good example of where arbitrarily high terms in the Taylor series become important. It's not just a formality; the first or second order approximations are rarely of any meaningful accuracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another major example (though I again apologize for it not being purely physics) would be in numerical approximations, where Taylor series are bread and butter. Being able to expand Taylor series is fundamental in deriving many methods for numerically solving ODEs, and is crucial for being able to estimate bounds and convergence orders. If you only take linear approximations you'll have a horrible time of everything.&lt;/p&gt;&#xA;" OwnerUserId="522" LastEditorUserId="522" LastEditDate="2014-05-07T01:47:46.400" LastActivityDate="2014-05-07T01:47:46.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2204" PostTypeId="2" ParentId="2196" CreationDate="2014-05-06T18:03:58.677" Score="9" Body="&lt;p&gt;This won't directly help you, but the way I was motivated to do my homework as an undergraduate was that I had one-on-two tutorials every week. If you haven't done the homework and your tutorial partner has, then you can't really hide in room of three people. There was no formal sanction for missing a small amount of homework unless you count sarcasm. I didn't always do it (or do it all), but I certainly didn't skip it lightly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can't hold tutorials (I confidently assume), but I think you should be looking for other ways to prevent the students from hiding. Busy or lazy people will drop your apparently-optional homework unless there are consequences for dropping it, so you're looking for a way to persuade the students to make an honest effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you've noted, grading for completeness without regard for content doesn't do this, since there's no difference between a wrong answer and &quot;Lorem Ipsum&quot;. But you could for example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Make clear that you will randomly sample homework and fail anyone you catch gaming the system. Of course this is &quot;unfair&quot; in the sense that someone might still take a risk and avoid getting sampled.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Include some &lt;em&gt;really easy&lt;/em&gt; questions that can be answered pretty much directly from the course material, and check that they get those correct. You probably need this to be non-obvious or else it can still be gamed by those who've at least attended lectures. But the inclusion of the easy questions should at least make things less stressful and ensure that even if they feel they don't understand everything they do at least understand &lt;em&gt;something&lt;/em&gt; properly. Up to you whether you count it a win if someone looks at all the questions but only bothers answering the ones they think are the easy ones.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Grade the homework for effort instead of completeness (which requires finding someone to actually read it and distinguish nonsense from failed effort).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Create an indirect means by which the homework is obviously necessary in order to pass the course. Ludolila already suggested an exam involving a subset of homework questions, and you can prevent people memorising the answers by setting an exam involving questions very similar to the homework but not quite identical. Students could then be persuaded to view homework as exam preparation, and that it's easier to do it than skip it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Set normal graded homework on the easiest parts of the material, plus advanced questions that are explicitly optional since the students aren't required to have mastered the material yet. However, stuff that's optional at the start of the course could be introduced into the normal questions in later weeks. Maybe you could even figure a way to set questions in week N+1 that are easy for anyone who has seen the sample solutions to the questions from week N, which motivates the students to check the answers. That might still result in students never attempting the optional questions, though. Just reading the solutions is a poor substitute for trying to do it yourself and only reading the solution after failing.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="321" LastEditorUserId="321" LastEditDate="2014-05-06T18:15:31.303" LastActivityDate="2014-05-06T18:15:31.303" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2205" PostTypeId="2" ParentId="1689" CreationDate="2014-05-06T19:15:26.990" Score="16" Body="&lt;p&gt;I was always honest:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You won't use this.  If you don't like math by now, it's probably not going to be useful to you in your career or life, or ever again once you graduate.  If you're like most people, you're going to completely forget this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's not why we're doing it.  Ever watch an athlete workout?  Maybe do a bench press?  Regardless of the sport he or she plays, how many times do they have to lie on their back during a game and heave a heavy weight over their chest?  They're not doing bench presses because that's a specific skill they need:  they're doing it to get stronger so they'll be better at what they &lt;em&gt;do&lt;/em&gt; need to do.  You're learning this advanced math to make your brain stronger and to be a smarter, more analytical person.  &lt;/p&gt;&#xA;" OwnerUserId="1393" LastEditorUserId="1393" LastEditDate="2018-10-30T12:21:28.813" LastActivityDate="2018-10-30T12:21:28.813" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2206" PostTypeId="1" CreationDate="2014-05-06T19:43:30.487" Score="21" ViewCount="3764" Body="&lt;p&gt;As a sidetrack in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2164/how-can-we-help-students-learn-how-to-read-their-textbook&quot;&gt;this question&lt;/a&gt; it came up that it is important to have students read texts (in particular proofs) critically. As examples it is nice to have correct proofs at hand (presumably in the textbook/lecture notes), but also a variety of others:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Proofs that are just plain wrong, like the nice &quot;proof by induction&quot; that all integers are larger than 10, just by omitting the base case&lt;/li&gt;&#xA;&lt;li&gt;Proofs that are wrong, but the result is actually right (missing cases, hidden assumptions, ...)&lt;/li&gt;&#xA;&lt;li&gt;Proofs that are right, but could be tightened up/simplified&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is in part motivated by the classic by Kernighan and Plaugher &lt;a href=&quot;http://en.wikipedia.org/wiki/The_Elements_of_Programming_Style&quot; rel=&quot;noreferrer&quot;&gt;&quot;The Elements of Programming Style&quot;&lt;/a&gt;, where the authors show bad program snippets culled from textbooks and other published sources, dissect them and show how they should be written right, and why. I'm aware of Edward Barbeau's &quot;Mathematical Fallacies, Flaws, and Flimflam&quot; (MAA, 2000), but that targets more bad computation than bad proving.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It it helps narrowing down, I'm mostly interested in combinatorics and discrete math. But approachable proofs, accessible to the relative layman (think students with Calculus I under their belt, often just taking Calculus II) in any area are quite welcome.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-05-07T00:01:07.670" Title="Example &quot;bad proofs&quot;?" Tags="|mathematical-pedagogy|proofs|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2207" PostTypeId="2" ParentId="2196" CreationDate="2014-05-06T22:01:13.110" Score="5" Body="&lt;p&gt;In a (small, graduate) course we got weekly homework. Four questions, only one (&quot;selected at random&quot;) was graded (rigurously), and the resulting homework grade was a minor part of the final grade. Partial answers (mostly hints) were published.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Worked nicely. Hard part is to come up with four equally interesting, and equivalent in work required, questions each week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last term (at the suggestion of one of my TAs) we started having tutorial sessions with three questions. One to be solved as an example by the TA, but active participation by the class is expected; another one to be solved in open discussion by the class, with hints from the TA as warranted; the third one is written up individually and handed in, to be graded as 0 (no work/completely irrelevant), 1 (looks related to the topic), and exceptionally 2 (very well done). This is a 5% of the final grade. Grades (and percentage of passing) soared. We are repeating the exercise this term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both alternatives cut down on the grading work, and (given good problems) do engage the students. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-06T22:01:13.110" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2208" PostTypeId="2" ParentId="2206" CreationDate="2014-05-06T22:03:39.293" Score="14" Body="&lt;p&gt;Students are, of course, the best source of &quot;bad proofs&quot;. And while one can pull from assignments/homework, I find the best &quot;bad proofs&quot; are the ones students give on competition style problems. To that end, I usually photocopy our students' MAA Team Contest and Putnam responses before mailing them in. Then I can type up those responses and use them in our Intro to Proofs course.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-05-06T22:03:39.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2209" PostTypeId="2" ParentId="2198" CreationDate="2014-05-06T23:34:04.753" Score="7" Body="&lt;p&gt;YouTube is a good source of tutorial videos, particularly &lt;a href=&quot;https://www.youtube.com/user/patrickJMT/&quot;&gt;PatrickJMT's channel&lt;/a&gt;. I would not steer the student toward Khan Academy, though that name might be the first one to come to mind when you think about math videos -- too much cookbook, not enough conceptual understanding that a student who failed Calc 1 needs. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-05-06T23:34:04.753" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2210" PostTypeId="2" ParentId="2206" CreationDate="2014-05-07T00:01:07.670" Score="9" Body="&lt;p&gt;It may not be as thorough as you're looking for, but there's a brief section in my &lt;a href=&quot;http://www.math.vt.edu/people/day/ProofsBook/IPaMV.pdf&quot;&gt;Intro to Proofs&lt;/a&gt; book about spoofs in Chapter 1 B with some examples/problems.&lt;/p&gt;&#xA;" OwnerUserId="1405" LastActivityDate="2014-05-07T00:01:07.670" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2211" PostTypeId="1" AcceptedAnswerId="2218" CreationDate="2014-05-07T13:45:49.597" Score="5" ViewCount="405" Body="&lt;p&gt;In my class today as I was checking homework of trigonometric operations, I called one of the students and gave her one of the problems of book which was $\frac{1}{(1-\cot (\theta ))}+\frac{1}{(\cot (\theta )-1)}$. The question was to simplify that problem so I told her that the question is in bit complex form so then she should make it simpler. In the process I told to take common (-) from the 2nd term of $\cot(\theta)-1$ so then it becomes simple to take LCM and solve the problem. As it seems she seems to have problem on algebra. So it seems like I have to start from the beginning but I do not know from where to start. Are there any systematic ways to make her understand all those things easily? A specific way to make understand and able her like student to solve those kind of problems?&lt;/p&gt;&#xA;" OwnerUserId="1172" LastEditorUserId="64" LastEditDate="2014-05-10T15:57:37.747" LastActivityDate="2014-05-17T20:31:20.877" Title="A student's problem on solving simple trigonometric operations" Tags="|algebra|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2212" PostTypeId="2" ParentId="2188" CreationDate="2014-05-07T14:17:39.397" Score="3" Body="&lt;p&gt;I have implemented a variety of different quiz schemes over the years:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I give one quiz paired with each test. This is often too long for class, but I give it anyway as a form of test review. This was very popular with the students, but it eats a lot of time. It also forces me to be done with tested material at least 2 meetings before the Test itself. It was not always possible to get them feedback in this mode since the quiz was often about 3 days before the actual test.&lt;/li&gt;&#xA;&lt;li&gt;I gave a short quiz at the start of every lecture where they were able to copy down the solution to one of the assigned textbook problems. The trouble with this was trading back and forth 40 assignments gets old in a hurry with 30-40 students. The downside, athletes always have an excuse and somebody's going to get married etc... there is a big time stamp for interfering with folks schedules here. So, some strategy to deal with missed quizzes equitably must be found. My solution was to just award them the grade of the quiz before or after the missed quiz. In addition, vonbrand is correct, we did have a problem with a student just copying down solutions from the online solutions manual brought to us by google. I will say, the student was good and showed understanding on tests, so perhaps that was research more then outright cheating. It is a problem though, so to do this right you really need to write your own quizzes.&lt;/li&gt;&#xA;&lt;li&gt;I give a weekly quiz. This I usually do in a junior level course. Here I have a weekly homework (which I write most of the problems for to avoid the google problem ) and the other day of the Tuesday/Thursday schedule is a quiz based on the homework and/or lectures from the previous week.&lt;/li&gt;&#xA;&lt;li&gt;just give tests and homework (no quizzes) this I do either when my schedule is too cramped to deal with quizzes, or when it is a senior or special topics type course.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Only option 2 is really intended to get them working in the text ahead of lecture. I think this confused them a bit because of the perceived &lt;strong&gt;unfairness&lt;/strong&gt; of the method: how can I test them on what I haven't even taught yet? Thus, to ward off this (understandable) complaint you must be careful to only give quizzes which are answered on the basis of a merely cursory understanding of the material as it would arise from a first reading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long story short, I'm tired of attempting 2. at the moment. My current strategy to get them to read for the course is to put an explicit problem on each homework which has them affirm they have read certain sections of the book and my notes. I think this is a bit more polite and communicates my expectations for them repeated in the semester. I can write a sentence in the syllabus about reading before class, but it's better to remind them often of their duty as a student; to &lt;strong&gt;study&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-07T14:17:39.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2213" PostTypeId="1" CreationDate="2014-05-07T14:51:34.657" Score="23" ViewCount="1611" Body="&lt;p&gt;This is a rather broad (and perhaps too philosophical) question about undergraduate and graduate mathematics education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gowers, in his article &quot;&lt;a href=&quot;https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf&quot;&gt;the two cultures of mathematics&lt;/a&gt;&quot;, observes differences between the problem solving and theory building aesthetics in mathematics. To slightly perturb an analogy of Alexander Grothendieck, one might view the problem solving culture as focusing more on a &quot;&lt;a href=&quot;http://en.wikipedia.org/wiki/Yin_and_yang&quot;&gt;Yang&lt;/a&gt;&quot; or &quot;attacking&quot; mathematical aesthetic, and the theory building culture as focusing on a more &quot;&lt;a href=&quot;http://en.wikipedia.org/wiki/Yin_and_yang&quot;&gt;Yin&lt;/a&gt;&quot; or &quot;yielding&quot; aesthetic. Of course, I have taken liberties in my perturbation of this analogy, and the distinction is probably best understood by reading Gowers's article linked to above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the whole of mathematics involves an interplay of the &quot;Yin&quot; and &quot;Yang&quot; of the two cultures, and every mathematician is at any moment looking at things more as a problem solver than as a theory builder or the other way around. Subjects emphasizing the different styles mostly take care of sorting mathematicians into appropriate communities for their tastes, but I can't help but wonder if there are better ways to recognize the diversity of mathematical tastes in my classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some students simply love to read and learn about mathematics, and gain a very strong base of ideas to think about, and there is a danger of these students becoming depressed by seeing a problem solver work incisively on a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some students are excellent at problem solving, but become depressed when looking at classmates who have a broader knowledge base. Many such clever students are turned off by routine mathematics classwork aimed at building up their knowledge base.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above perceptions extend rather far into one's career. Once, a problem-solver friend of mine was shocked when he overheard a discussion between a few mathematicians (one of whom had a Fields Medal) and noticed that the participants didn't have a solid grasp of a certain definition they were talking about. (I think it might be common among some theory-building types to try to tease new perceptions from an ill-defined understanding of a definition.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, a former graduate student of an IMO Gold Medalist who is now primarily a theory-builder told me stories about the disdain his advisor had for results that were &quot;just combinatorics&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What concretely can we do to better embrace the diversity of mathematical tastes in our departments? What can we concretely do as mathematics educators to encourage communication between the two cultures?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(EDIT: &lt;a href=&quot;http://pzacad.pitzer.edu/~dbachman/RUME_XVI_Linked_Schedule/rume16_submission_3.pdf&quot;&gt;Here&lt;/a&gt; is an example of the sort of thing I'm hoping to see, ideas for concrete pedagogical practices that encourage either &quot;culture&quot; or perhaps how to concretely achieve a balance.)&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2014-05-09T17:45:56.747" LastActivityDate="2014-05-09T17:45:56.747" Title="Teaching and &quot;The Two Cultures&quot;" Tags="|undergraduate-education|graduate-education|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2214" PostTypeId="2" ParentId="2196" CreationDate="2014-05-07T16:21:13.840" Score="3" Body="&lt;p&gt;The purpose of homework is to demonstrate understanding. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what about offering your students the chance to come up and solve/explain a question demonstrating their understanding to the rest of the class, and if they do so successfully they get full homework grade without having to do the homework assignment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Its a win for all parties involved.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You get to go through individual questions in class, like you originally planned&lt;/li&gt;&#xA;&lt;li&gt;You'll have less homework to grade, since kids who understand the stuff won't be submitting any.&lt;/li&gt;&#xA;&lt;li&gt;Students get an opportunity to demonstrate their understanding during class time, and don't have to waste their free time solving the the same problem repeatedly just to demonstrate to you that they understand it.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1416" LastActivityDate="2014-05-07T16:21:13.840" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2215" PostTypeId="1" CreationDate="2014-05-07T16:25:18.393" Score="5" ViewCount="1107" Body="&lt;p&gt;This question is motivated by the &quot;shaking hands&quot; method I first saw in this &lt;a href=&quot;https://matheducators.stackexchange.com/a/1732/533/&quot; title=&quot;answer&quot;&gt;response&lt;/a&gt;. I'm constantly learning about little things like this that some students find helpful even if I don't. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm curious to know what tricks, mnemonics, etc. you know that might be useful to low level students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some other examples: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Singing the quadratic formula to the tune of &quot;Pop Goes the Weasel&quot;&lt;/li&gt;&#xA;&lt;li&gt;Thinking of the x-y axis as a &quot;ground-wall&quot; axis when graphing. &lt;/li&gt;&#xA;&lt;li&gt;&quot;low d high, high d low, all over low squared&quot; for the quotient rule. &lt;/li&gt;&#xA;&lt;li&gt;FOIL&lt;/li&gt;&#xA;&lt;li&gt;&quot;Cross Multiply&quot;&lt;/li&gt;&#xA;&lt;li&gt;PEMDAS&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A caveat: I don't necessarily agree with the use (abuse) of all these. I cringe when students shout &quot;cross multiply!&quot; at the sight of a fraction without taking the time to think about the problem. But I've learned that some students need help in different ways and &lt;em&gt;some&lt;/em&gt; don't abuse these tricks. So perhaps the mention of these would be best left for office hours, but that's a different story. For now I would just like to see what you've used. &lt;/p&gt;&#xA;" OwnerUserId="533" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-19T22:43:32.493" Title="What math &quot;tricks&quot; or methods do you know for low level students?" Tags="|undergraduate-education|secondary-education|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2216" PostTypeId="2" ParentId="2151" CreationDate="2014-05-07T16:33:38.717" Score="1" Body="&lt;p&gt;I didn't see anyone mention this specifically so here's my input. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've sometimes found it difficult to get students to answer certain questions in class. Either they're lost, tired, bored, etc. Also, when you pick a specific individual from the class you might not get an accurate representation of what the class knows on average. That student might know the answer perfectly, might know nothing, or might not fully explain themselves in hopes of sitting down faster. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest taking a step back and asking a slightly easier but more specific question and having the students write their answer on their sheet of paper. I usually tell them to answer on the corner of the page of to circle their work that way I can quickly browse over a few students' work and get a feel for what the class knows. This seems to work well and I think the reason is because they don't have to answer in front of their classmates. &lt;/p&gt;&#xA;" OwnerUserId="533" LastActivityDate="2014-05-07T16:33:38.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2217" PostTypeId="2" ParentId="2215" CreationDate="2014-05-07T18:27:53.873" Score="7" Body="&lt;p&gt;Instead of FOIL, I have had luck with the phrase &quot;&lt;strong&gt;every times each&lt;/strong&gt;.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(x + 5 + \sqrt2)(x + 2)$ doesn't need to be harder than $(x+5)(x+2)$, but if all the students know is FOIL, then you have to explain the process as something like &quot;FOIIIL&quot; which is comical and terrible.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2014-05-07T19:00:59.943" LastActivityDate="2014-05-07T19:00:59.943" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="2218" PostTypeId="2" ParentId="2211" CreationDate="2014-05-07T19:00:29.480" Score="10" Body="&lt;p&gt;Yes. Repeatedly break the problem down by removing one layer of complexity at a time until you discover a problem she can complete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic layer of this problem is fractions. Ask her if she can add these fractions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\frac13 + \frac17$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The next layer of this problem is fractions within fractions. Ask if she can simplify the following expression:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\frac1{\frac13 + 1}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The next layer of this problem is rewriting trigonometric functions. For example, can she complete this problem?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Simplify $\tan x \cdot \cos x$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You will probably find that she is able to complete at most one, if any, of these subproblems. Asking these simpler problems is very productive. Ideally you will find one or two of these skills that she does not have. Then you can get out a blank sheet of paper, and together, cover it with examples of that one skill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the time you've touched all the skills, she should be able to complete that problem. And she should &lt;strong&gt;leave with a sheet of paper for each skill she practiced that day&lt;/strong&gt;. Even if it starts with adding fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also my answer to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2151/what-to-do-when-you-get-the-empty-stare/2163#2163&quot;&gt;what to do when you get the empty stare&lt;/a&gt; .&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-07T19:00:29.480" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2219" PostTypeId="2" ParentId="2215" CreationDate="2014-05-07T19:12:03.347" Score="4" Body="&lt;p&gt;$X^3 - 1 = (X - 1)(X^2 + X + 1)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rule is SOAP - Same, Opposite, Always Positive&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$X^3 + 1 = (X + 1)(X^2 - X + 1)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me offer the broader context - the assignment to factor $X^3 - 1$ is first solved via Benjamin's method, as he comments below. Then $X^3 + 1$ is also solved this way. The answers are compared and the general rule is created. When confronted with, say, $X^3 - 8$ in the context of a larger problem to solve or under the time pressure of a standardized test, the student might save valuable time by knowing such tricks. I assume that any &lt;em&gt;tricks&lt;/em&gt; the OP would gather would be presented in that spirit. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-05-09T10:54:59.387" LastActivityDate="2014-05-09T10:54:59.387" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2220" PostTypeId="1" CreationDate="2014-05-07T18:44:06.233" Score="31" ViewCount="2274" Body="&lt;p&gt;Depending on the university, there are always slight differences in the syllabus and the structure of the standard material undergraduate students learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I also noticed that undergraduate mathematics differs more greatly if one compares for example Germany (my home country) with the U.S. – I do not know this for sure, but I’ve heard rumours that in the U.S., it is common to start with some non-rigorous calculus (not introducing $ε$-$δ$-definitions for continuity, no formal definition of limits etc.) whereas here in Germany is seems to be common to start with more rigorous courses on linear algebra and analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This just now got me interested in how the standard syllabus differs by country.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;How does the mathematical education at university look like in your country?&lt;/strong&gt;&#xA;  What is the standard syllabus/what are the standard courses offered &lt;em&gt;in the first or first two years&lt;/em&gt; and in which order do (undergraduate) students undergo them/learn which material?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A quick google got me &lt;a href=&quot;http://mymathforum.com/new-users/1396-mathematics-education-different-countries.html&quot;&gt;this discussion&lt;/a&gt; which covers only few countries, but has the sort of answer I’m interested in. Maybe we can get something similar going? Each answer a country?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The question got migrated from MO.)&lt;/p&gt;&#xA;" OwnerUserId="666" OwnerDisplayName="k.stm" LastEditorUserId="666" LastEditDate="2014-05-08T05:33:34.503" LastActivityDate="2014-11-21T23:01:00.857" Title="Mathematical education by country" Tags="|undergraduate-education|curriculum|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2221" PostTypeId="2" ParentId="2215" CreationDate="2014-05-07T21:17:20.567" Score="4" Body="&lt;p&gt;This was supposed to be a comment, but it became too long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you want your students to understand the material, I think it's good practice to avoid remembering procedures (I avoid the term 'trick' because I want to distinguish between two types on tricks: IMO, there is a good and a bad kind) as much as possible. I think they are harmful for understanding of the topic. A much more useful skill is to derive formulas on the spot. I have to admit that there are examples in which it is just not feasible to do this (for example, I &lt;em&gt;would&lt;/em&gt; advise to just remember the quotient rule as it's too complex to derive on the spot every time, but I think it's also good to be able to derive it, so that forgetting a rule is no big deal).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to recap, I think it's better to teach techniques which &lt;em&gt;enable&lt;/em&gt; students to come up with formulas. Remembering a technique (normally, I'd refer to a technique as a trick, rather than a formula or abbreviation) usually works way better than remembering a formula (because there is usually some intuition behind the technique, and our brains just are better at remembering ideas than they are at remembering raw formulas).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can (hopefully) show what I mean, by an example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider two students that have learned how to find roots of a quadratic equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student A has learned to write $x^2 + bx + c = 0$ in the form $(x-a)^2 = d$, so that you can apply a squareroot on both sides, then solve for x to get: $ x = a \pm \sqrt{d} $. Ideally, he would understand that we translate the graph of the polynomial (by doing a substitution for x) to get rid of the $bx$ term, get an equation that is easy to solve, and then translate/substitute back again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student B has learned $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now suppose they learn about cubic polynomials and their roots. Using some clever geometric tricks, it is possible to solve $x^3 + px + q = 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student A's knowledge of the technique now enables him to see a similar possibility here. IF you have $x^3 + ux^2 + vx + w = 0$, and you can find an $a$ such that the quadratic term will vanish (just like the first-order term vanished in the quadratic equation), you can write &lt;em&gt;any&lt;/em&gt; cubic equation in the simpler form. So student A may realize that the solution to the simpler cubic equations is enough to solve for &lt;em&gt;all&lt;/em&gt; cubic equations.&lt;/p&gt;&#xA;" OwnerUserId="915" LastActivityDate="2014-05-07T21:17:20.567" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2222" PostTypeId="2" ParentId="2157" CreationDate="2014-05-07T22:13:08.620" Score="3" Body="&lt;p&gt;An example I just remembered is to compute e.g.&#xA;$$&#xA;\sum_k \binom{n}{k}^2 = \sum_k \binom{n}{k} \binom{n}{n - k}&#xA;$$&#xA;Notice that the variable $n$ appears three times, in such cases it can turn out much easier to separate them:&#xA;$$&#xA;\sum_k \binom{r}{k} \binom{s}{n - k}&#xA;$$&#xA;So a heuristic is to uncouple. Sure, it could be that keeping them the same allows simplifications that aren't applicable if separate, that's why this isn't sure-fire.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And this same example also shows the heuristic of rewriting in an equivalent form (second binomial coefficient in the original).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-07T22:13:08.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2223" PostTypeId="2" ParentId="2220" CreationDate="2014-05-08T13:26:33.827" Score="12" Body="&lt;p&gt;I studied and teach in Finland, but got some exposure to the US system as a graduate student. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our system is similar to what the OP described. The first year analysis uses $\varepsilon-\delta$. Much to the students dismay I may add.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My educated guesses as to the reasons for differences in the US/Finnish practices. I am prepared to be wrong about them, as I may hold mistaken beliefs about the US system.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In Finland the kids have their 19th or 20th birthday the year they start college (depending on whether you do conscript duty fresh out of high school or not). In the US it's more like 17-18, so they are younger meaning that you can usually expect a bit less (whether that holds in practice is open to discussion). May be the same is true in Germany?&lt;/li&gt;&#xA;&lt;li&gt;In these parts the freshmen apply and are accepted to a math or physics or law or medicine programs right away, in the US you have to wait a year or so to pick a major (law and medical schools are typically for graduate level only), and the freshman year curriculum is comprehensive as opposed to specialized (IIRC freshmen need to take one course in math, one in English, one in social sciences, all depending on the University - at Notre Dame one selective course in theology was compulsory to ALL the undergraduates). Therefore the set of students taking freshman calculus in the US is more diverse. Here we only have math, physics and engineering students sitting in that class. And it is easier to justify that those students actually should learn their $\varepsilon$s. In the US you may have kids in pre-med and such, who will probably use some math tools later on, but don't need to necessarily understand the limitations of those tools.&lt;/li&gt;&#xA;&lt;li&gt;And also here, in spite of the above points, we still get some pressure from other departments to offer &quot;tools only&quot; analysis (i.e. calculus) courses, to chemistry, computer science and engineering majors in particular. The physics department is not happy with the speed of progress in math department courses either. Ideally they would like the freshmen to learn multivariable calculus, differential equations and Laplace/Fourier transforms before Xmas. Also the local sorry excuse of high school math no longer prepares students to the world of definitions and proofs, so we may need to convert to US style calculus soon as well. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Sorry, that last bullet turned into a rant - I need to blow off some esteem every now and then. Reminiscing the good ole days when middle school here had proof based Euclidean geometry as a compulsory topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answering the question. Locally the freshmen take&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Analysis I - II: elementary functions, limits (1st sequences, then functions, then continuity, then derivatives) in the Fall, and integration and series in the Spring. Presumable a lot of this is review of high school (chain rule, improper integrals and power series are new), but now we bring in the epsilons.&lt;/li&gt;&#xA;&lt;li&gt;Linear algebra: ${\bf R}^n$, linear systems of equations, linear independence, determinants and matrices in the Fall, and&lt;/li&gt;&#xA;&lt;li&gt;First course in abstract algebra in the Spring: congruences, intro to elementary number theory, a very quick intro to groups, rings, fields, polynomials and abstract vector spaces. This is in two parts and many postpone the latter part to their second year.&lt;/li&gt;&#xA;&lt;li&gt;Ordinary differential equations (the basic stuff).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the second year it becomes more diverse, except that multivariable calculus in some form is a nearly universal choice. Number theory and combinatorics are popular choices as are first courses in metric spaces and complex analysis.&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2014-05-08T13:41:42.787" LastActivityDate="2014-05-08T13:41:42.787" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2224" PostTypeId="2" ParentId="2220" CreationDate="2014-05-08T14:49:10.867" Score="10" Body="&lt;p&gt;Jyrki Lahtonen is correct that in the US students do not specialize into a major subject until (usually) the end of their second year at University.  For that reason, in each of the first two years at University students typically take only two math courses -- one in each semester -- so by the time they begin their third year, they have only had four math classes.  The default sequence of these classes is, at most institutions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1st semester, year 1:  Calculus I (differentiation, some integration; see below)&lt;/li&gt;&#xA;&lt;li&gt;2nd semester, year 1:  Calculus II (integration, sequences &amp;amp; series)&lt;/li&gt;&#xA;&lt;li&gt;1st semester, year 2:  Calculus III (multivariable calculus)&lt;/li&gt;&#xA;&lt;li&gt;2nd semester, year 2:  Differential equations&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I think it is correct to say that most US institutions treat the notions of limit somewhat informally in this two-year sequence; the $\epsilon-\delta$ definition may be briefly presented but it is not emphasized, and overall the course is not proof-focused.  After a student chooses a mathematics major, he or she will take an &quot;Analysis&quot; course that covers the content of Calc I-II again but with a focus on rigorous definitions and proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I referred to this above as the default sequence, there are some variations in this scheme that are so common as to be almost standardized:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Many high school students -- and certainly the ones who are most likely to become Mathematics majors (but remember, that decision is not made until the end of the second year) -- will take a Calculus course in high school, and a large proportion of those students will take one of the two (optional) Advanced Placement exams in Calculus.  Depending on the outcome of that exam, the University may grant them college credit for one or both of the first two semester courses.  In that case, students &lt;em&gt;may&lt;/em&gt; begin right away with the higher-level courses; on the other hand many students choose to forgo their AP credit and instead take Calc I again, reasoning that it is likely to be an &quot;easy A&quot;.&lt;/li&gt;&#xA;&lt;li&gt;Some Universities offer in the first two years an alternative &quot;Honors Calculus&quot; track that gives more attention to rigorous definitions and proof.  (My alma mater the &lt;a href=&quot;http://math.lsa.umich.edu/&quot;&gt;University of Michigan&lt;/a&gt; has &lt;a href=&quot;http://www.lsa.umich.edu/math/undergrad/coursesforfreshmen/infinhonorssequences&quot;&gt;&lt;em&gt;four different&lt;/em&gt; Honors Calculus sequences&lt;/a&gt; with varying degrees of rigor.)  Depending on University policy and student enthusiasm, a student with AP credits for the first year might decide to forgo those credits and take an Honors Calculus sequence, not because it is going to be an &quot;easy A&quot; but because they want an intellectual challenge and to lay the foundation for more advanced coursework.&lt;/li&gt;&#xA;&lt;li&gt;The fourth semester Diff Eq course may be offered in several different versions, depending on whether the student is heading toward a major in pure mathematics or theoretical physics (in which case the course will be more theoretical and, typically, will include some introductory linear algebra material) or something in the applied sciences or engineering (in which numerical methods might get more emphasis).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;After the student completes this two-year sequence of four prerequisite courses -- typically at the beginning of the third year, to coincide with declaring a major, but possibly sooner if the student uses AP credit to skip one or more courses -- then a student will begin with the &quot;core courses&quot; of the math major:  Linear Algebra, Abstract Algebra, an advanced Geometry or Topology course, and so on.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-05-08T14:49:10.867" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2225" PostTypeId="2" ParentId="2151" CreationDate="2014-05-08T16:13:00.960" Score="2" Body="&lt;p&gt;Was going to add this as a comment but...I think you have to create a culture of questioning right from the beginning.  You might be dealing with student who come from places where asking questions is not expected.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three things I say on the first day:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&quot;Think of me as a resource, like a book or the internet.  I can't help you if you don't tell me what you don't understand.&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;Your most powerful weapon in this class is this:&quot; (point to raised hand)&lt;/li&gt;&#xA;&lt;li&gt;&quot;If there's one thing I know as a teacher, it's if you have a question it's more than likely someone else does too.&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you've said all of these things from the beginning instead of waiting until the class reaches the impasse then it's easier to say things like, &quot;OK, I bet at least two people have a question about what I just said,&quot; or &quot;Everyone who truly understood what I just said, raise your hand.&quot;  Gives you the chance to receive more feedback.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for putting yourself in the &quot;beginner's mind,&quot; they don't know, and they don't know what they don't know.  You can't go slow enough.  If you have to repeat something again from the beginning then do it.&lt;/p&gt;&#xA;" OwnerUserId="953" LastActivityDate="2014-05-08T16:13:00.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2226" PostTypeId="1" AcceptedAnswerId="2230" CreationDate="2014-05-08T16:28:34.653" Score="18" ViewCount="2393" Body="&lt;p&gt;I'm asking this question as a student, wondering what various pros/cons to the given formula for oral exams could be.&#xA;Let me give some context first. I am a first year mathematics student at a university. The are two approaches available to students for every subject: There is a &quot;basic&quot; group and an &quot;advanced group&quot; and we were able to choose which one we wanted to attend at the beginning of both semesters with the possibility of moving from one group to another if we found the level inadequate for 2 weeks since the semester's start.&#xA;The difference between groups is really large, to the point where the only people left in the advanced groups  are ones who have either studied a significant amount of university-level maths in their free time in high school.&#xA;So, in short, attending the advanced groups is completely voluntary and the courses can't be really treated as ordinary introductory ones, because while they might theoretically assume any background knowledge, they do require quite some mathematical maturity which is bound to come with the backgorund knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After this long rant we arrive at the actual question. In these advanced groups, students are required to know all the theorems together with complete proofs of all the theorems/propositions presented during the entire semester in the given course. Learning all that is obviously not an easy task - not only is that a lot of material, it is also significantly harder than expected from an ordinary first year course. What do you think about such a formula? Is it too much? Or is it actually a good thing to expect this much from students who are highly above &quot;average&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think I did somewhat benefit from learning all the proofs in the first semester, but on the other hand the task seems like a bit unreasonable, since even professors use notes when presenting proofs during lectures.&lt;/p&gt;&#xA;" OwnerUserId="1421" LastActivityDate="2018-02-17T23:42:34.003" Title="Requiring students to know all the proofs on an oral exam" Tags="|undergraduate-education|exams|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2227" PostTypeId="2" ParentId="2226" CreationDate="2014-05-08T17:00:54.780" Score="6" Body="&lt;p&gt;This is a good way, in my opinion the best way, to teach somebody that actually wants to study mathematics, especially if they should have the goal of becoming a mathematician. &#xA;I use the word teach above, since while the question is about the exam, this style of exam also impacts the way students (should) learn the material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can see nothing wrong with it. In various systems it is also not at all unusual, and not limited to some &quot;elite&quot; groups or institutions. &#xA;Obvioulsy, in order to judge if some course is hard or not one would have to know more. But, just the general layout is not that unusual.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason why I consider this as a good way is that it makes it necessary for students  to really engage with the &lt;em&gt;theoretical&lt;/em&gt; core of the material, and not just learn how to do some excercices as they often are used to from secondary education. This is often a  fundamentally different approach, and the first year of university seems like a good point for this shift of style.  &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-05-08T18:02:43.897" LastActivityDate="2014-05-08T18:02:43.897" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2228" PostTypeId="2" ParentId="2226" CreationDate="2014-05-08T17:18:21.827" Score="8" Body="&lt;p&gt;I'd certainly expect a student to be able to remember the more important proofs, and remember enough of more run-of-the-mill results to be able to come up with a proof (sketch) on short notice. But I'd also expect quite a bit more, in terms of being able to use them to solve (somewhat) similar problems that weren't covered in class. Given time constraints on exams, I'd probably leave out most of the &quot;state and prove theorem XYZ&quot; type questions. Some memorization is OK, but the focus should be elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then again, this is first term, and I'm spoiled by more advanced students anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[In the above you can freely replace &quot;proof&quot; with whatever analogue may be applicable to some other subject area, be it Computer Programming or Italian Renaissance Poetry.]&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-08T17:18:21.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2230" PostTypeId="2" ParentId="2226" CreationDate="2014-05-08T19:32:41.357" Score="13" Body="&lt;p&gt;This will depend in some part on the course you are taking. If it is an introduction to proofs class, then probably it is reasonable to hold such an expectation. For higher level courses, some of the details that go into a proof can make memorizing a lot of theorems difficult. (Concrete example: I think the proof that permutation groups can be partitioned into &quot;even&quot; and &quot;odd&quot; elements is the sort that, if there were a lot of these to be memorized for an Abstract Algebra final, would constitute asking too much of the student.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An important difference in how advanced vs. novice students think about proofs is related to how they &quot;chunk&quot; the information. An advanced undergraduate who has finished Calculus in high school might be taking a first course on Real Analysis by semester two of their first year; in such a case, consider the &lt;strong&gt;theorem&lt;/strong&gt; that says every closed and bounded infinite subset of $\mathbb{R}$ contains a limit point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I first saw this proof, I was very caught up in the details; I could not chunk it very well, having no real familiarity with this style of argument. Today, I certainly would not memorize how it is proved verbatim. Instead, I would think: Put subset into an interval (boundedness); divide in half and choose left half if it contains infinitely many points, otherwise choose right half (infinite); repeat indefinitely. This zooms in on a limit point [essentially by giving its binary representation if the interval is $[0,1]$, where &quot;choosing left&quot; gives a $0$ and &quot;choosing right&quot; gives a $1$] and that limit point is in the set (closed).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even in the paragraph above, I have written more details than I keep in my head. My &lt;em&gt;actual&lt;/em&gt; thinking is something like: divide in half repeatedly and use the three givens (boundedness, infinite, closed).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To keep many proofs (especially ones that are not being written out formally, but rather explained in an oral examination where they will probably give you hints if you get stuck) in one's working memory is a doable task. But it does, in your own language, &quot;require quite some mathematical maturity.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt; This task could be unreasonable if the proofs are overly involved and too many; but, more generally, the ideal for such an examination is that students can grasp the key concepts and reconstruct proofs or proof sketches without needing to store word-for-word arguments in their working memory.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-05-08T19:38:54.300" LastActivityDate="2014-05-08T19:38:54.300" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2231" PostTypeId="2" ParentId="2220" CreationDate="2014-05-08T19:46:39.620" Score="14" Body="&lt;p&gt;If this is of any interest here is the case of Romania -Bourbaki's heaven (or hell):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A year-long course of Analysis: starting with the Peano axioms, construction of Integer, Rational and Real Numbers (as the completion of $\mathbb{Q}$) sequences and series, general topology, differentiation and integration (Riemann) of function of one variable&lt;/li&gt;&#xA;&lt;li&gt;A year-long course of Algebra: basic set theory, general group theory and concrete examples (groups of permutations and groups of matrices), rings, vector spaces and linear algebra up to the Jordan normal form (only over $\mathbb{C}$)&lt;/li&gt;&#xA;&lt;li&gt;A year-long course in Geometry: Linear Algebra (again) and euclidean spaces , Affine and Euclidean (Analytical and not Axiomatic) Geometry and a little of&#xA;Projective Geometry (ideally start with the axioms and construct the field of&#xA;coordinates)&lt;/li&gt;&#xA;&lt;li&gt;A semester-long course on Logic and Set Theory -some ZF axiomatics, calculus with predicates, deductive theories (whatever this means)&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Three semester-long courses in computer programming and Graphs&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Students are supposed to have a strong background in Analysis from high-school&#xA;($\epsilon-\delta$ definition of continuity, the definition of the derivative as a limit, the definition of the integral as the limit of Riemann sums).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Clarification:&lt;/strong&gt; This is for the Pure Mathematics. For other majors requiring&#xA;Mathematics (Physics, Engineering, etc.) the programme is a condensed version of the above plus some ODEs, function of a complex argument and Numerics.&lt;/p&gt;&#xA;" OwnerUserId="330" LastEditorUserId="330" LastEditDate="2014-05-08T19:53:39.223" LastActivityDate="2014-05-08T19:53:39.223" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2232" PostTypeId="2" ParentId="2213" CreationDate="2014-05-08T23:28:12.970" Score="16" Body="&lt;p&gt;Perhaps a bit of a rant...:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As long as we condone &quot;mathematics&quot; being treated as a competitive sport, in whatever mode, it becomes a vehicle for ego and machismo. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Math contests&quot; stir passions, but by appealing to some unfortunate human weaknesses. Similarly, conducting mathematics courses in a fashion allowing or encouraging &quot;competition&quot;, as opposed to cooperation to move toward better collective understanding, is merely a milder form of &quot;contest&quot;. If one wants to claim that this is a necessary evil, then it is inevitable that the &quot;losers&quot; (in a given sub-contest) will &quot;feel bad&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perpetuating the supposed &quot;two cultures&quot; or &quot;birds and frogs&quot; talk makes things worse exactly by suggesting that there is conflict between the two alleged &quot;types&quot;, and that supposedly people tend to one or the other extreme. This is akin to the fake distinction between &quot;pure&quot; and &quot;applied&quot; math, whose most inimical feature is creating labels to which people can declare allegiance, and then be polarized from the &quot;opposing&quot; camp.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, the problem is the perpetuation of the description, and perpetuation of talking about mathematical activity in quasi-violent or quasi-sexualized/genderized terms, and in allowing mathematics to lose its own meaning, but become a vehicle for various eternal human-weakness/conflict issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would take issue with the premise that routine classwork is the way to &quot;get understanding&quot;, so, then, kids who are impatient with it are entirely reasonable. It is entirely reasonable to try to make the subject &quot;more meaningful&quot; than the standard curriculum model may give a student. Both &quot;broader understanding&quot; and &quot;problem-solving&quot; allow more exercise of judgement, more genuine intellectual activity, than the standard model which emphasizes obedience and suspension of critical judgement.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-05-08T23:28:12.970" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2233" PostTypeId="2" ParentId="2213" CreationDate="2014-05-09T00:05:28.253" Score="8" Body="&lt;p&gt;This is somewhat tangential to your thrust, but&#xA;I do believe that there is a distinction between&amp;mdash;and a continuum between&amp;mdash;the hedgehog and fox mentalities described in Isaiah Berlin's famous essay.&#xA;I know mathematicians who essentially have no focus other than solving one problem after&#xA;another, where &quot;solving one problem&quot; may require an investment of as much as five years.&#xA;Others dig in the same hole for their entire careers, repeatedly finding&#xA;new tangents and avenues to explore, but constantly returning to some overarching question.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Berlin, Isaiah. &lt;em&gt;The hedgehog and the fox: An essay on Tolstoy's view of history&lt;/em&gt;. Simon and Schuster, 1966.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If one accepts a spread along this dimension as fundamental, then the avenue&#xA;to reach students is via&#xA;problems wherever one sits on the continuum. But &quot;problems&quot; here is decidedly not&#xA;&quot;contest problems,&quot; but real conundrums, open, unsolved, perhaps amorphous problems.&#xA;Fortunately, with effort, accessible problems can be found on the&#xA;frontiers of mathematics reachable&#xA;by college students, and more rarely, even accessible to high school students.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-05-09T00:05:28.253" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2234" PostTypeId="2" ParentId="2215" CreationDate="2014-05-09T05:28:53.903" Score="3" Body="&lt;p&gt;In trigonometry, a well-known mnemonic is &lt;a href=&quot;http://en.wikipedia.org/wiki/Sohcahtoa#Mnemonics&quot; rel=&quot;nofollow noreferrer&quot;&gt;SOHCAHTOA&lt;/a&gt; but this only works for acute angles in right triangles.&lt;/p&gt;&#xA;&lt;p&gt;For the three basic trigonometric functions of any angle (with an angle &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; in standard position, vertex at the origin, terminal side intersecting the circle centered at the origin with radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; at the point &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;), I teach my students to relate &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with horizontal motion, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; with vertical direction; and &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; with a &amp;quot;road&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Thus, to remember that &lt;span class=&quot;math-container&quot;&gt;$\cos\theta=\frac xr$&lt;/span&gt;, think of a car (a &lt;a href=&quot;http://en.wikipedia.org/wiki/Convertible&quot; rel=&quot;nofollow noreferrer&quot;&gt;convertible&lt;/a&gt; with the top down) moving &lt;em&gt;over&lt;/em&gt; a road.  (In the Filipino language, the word that we use for car is &lt;em&gt;kotse&lt;/em&gt;, which is quite similar to &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;To remember that &lt;span class=&quot;math-container&quot;&gt;$\sin\theta=\frac yr$&lt;/span&gt;, think of looking up while you're &lt;em&gt;over&lt;/em&gt; the road.  You will see the sun (which is quite similar to &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;To remember that &lt;span class=&quot;math-container&quot;&gt;$\tan\theta=\frac yx$&lt;/span&gt;, think of staying in your car and looking up.  You will get a sun tan (&lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt;) because the sun (&lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt;) is&#xA;over the convertible (&lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-05-09T05:28:53.903" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2236" PostTypeId="1" AcceptedAnswerId="2237" CreationDate="2014-05-10T01:18:58.917" Score="7" ViewCount="2318" Body="&lt;p&gt;Can anyone comment knowledgeably on this intriguingly titled book?&#xA;I have received publisher's promotional materials (as no doubt have&#xA;many of you), but I know nothing about it.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jason Padgett. &lt;em&gt;Struck by Genius: How a Brain Injury Made Me a Mathematical Marvel&lt;/em&gt;.&#xA;  (&lt;a href=&quot;http://www.struckbygenius.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book link&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/5Vl99.jpg&quot; alt=&quot;Jason Event Horizon&quot;&gt; &lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="262" LastEditDate="2014-05-10T20:21:48.573" LastActivityDate="2019-04-25T21:44:54.860" Title="Jason Padgett's &quot;Struck by Genius&quot;" Tags="|creativity|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2237" PostTypeId="2" ParentId="2236" CreationDate="2014-05-10T01:48:27.090" Score="23" Body="&lt;p&gt;&lt;strong&gt;Edit 2:&lt;/strong&gt; This story continues to reappear periodically in mainstream media; this month (April 2019) it was featured in a BBC Feature story &lt;a href=&quot;http://www.bbc.com/future/story/20190411-the-violent-attack-that-turned-a-man-into-a-maths-genius&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. The article title is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The violent attack that turned a man into a maths genius&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To reiterate my earlier answer: The pictures created by Padgett are (perhaps) interesting, but the media's intention to depict him as a &amp;quot;maths genius&amp;quot; is severely misguided and, in particular, demonstrates a widespread and continued lack of mathematical literacy.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit 1:&lt;/strong&gt; In case anyone was concerned, Channing Tatum will star in a &lt;strong&gt;&lt;a href=&quot;http://www.hollywoodreporter.com/news/channing-tatum-tackling-struck-by-732971&quot; rel=&quot;noreferrer&quot;&gt;movie&lt;/a&gt;&lt;/strong&gt; version of this book. Also, for those interested, Padgett's personal site can be found &lt;strong&gt;&lt;a href=&quot;http://fineartamerica.com/profiles/jason-padgett.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;You can find more information about him &lt;a href=&quot;http://www.huffingtonpost.com/2014/05/06/brain-injury-jason-padgett-math-genius_n_5273609.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. He has not produced any great (&amp;quot;genius&amp;quot;) mathematical work, but he has taken up an interest in drawing interesting geometrical pictures. In particular, it appears that part of his brain was altered in a way that causes him to perceive fractal-like patterns (other parts of the brain now compensate for the damaged part). More generally, though, this is an excellent example of the lack of mathematical literacy on the part of the media.&lt;/p&gt;&#xA;&lt;p&gt;An excerpt about this individual from &lt;a href=&quot;http://nypost.com/2014/04/20/how-a-brain-injury-turned-a-college-dropout-into-a-genius/&quot; rel=&quot;noreferrer&quot;&gt;an article&lt;/a&gt; in the NY Post:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;During one of his meditations, he came to the conclusion that “circles don’t exist.” ... “It was like a bomb went off in my mind. In a matter of minutes, I was no longer just a receiver of geometric imagery or a researcher; I was a theorist,” he writes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is somewhat like when &lt;em&gt;Time&lt;/em&gt; magazine wrote up a child who had learned some basic Calculus as having the potential to disprove Einstein's theory of relativity. A particularly unimpressive account of this story can be found in an LA Times opinion piece &lt;a href=&quot;http://opinion.latimes.com/opinionla/2011/03/where-are-the-literary-boy-geniuses.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. This latter write-up is very short, but I find the latter half of it to be especially offensive:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I always have the same question when I read one of these stories: Where are the verbal prodigies? &lt;strong&gt;A 12-year-old who can disprove relativity is pretty special&lt;/strong&gt;, but so would be a 12-year-old who wrote a sublime sonnet or the great American novel. Yet you almost never see that sort of wunderkind on TV.&lt;/p&gt;&#xA;&lt;p&gt;My theory: Producing great literature requires experience and reflection, not just an IQ of 170.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I added the emphasized text above because I find it quite unpalatable. &amp;quot;Pretty special&amp;quot;?! Moreover, the implicit idea that &amp;quot;an IQ of 170&amp;quot; is somehow what it takes to &amp;quot;disprove relativity&amp;quot; is preposterous.&lt;/p&gt;&#xA;&lt;p&gt;Though there is no real story here, I thought I might direct you to an interesting source, given that you tagged the question with &lt;em&gt;creativity&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Weisberg, R. W. (2006). &lt;em&gt;Creativity: Understanding innovation in problem solving, science, invention, and the arts&lt;/em&gt;. John Wiley &amp;amp; Sons.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this book, Weisberg argues that (1) creative thinking is a type of ordinary thinking, and (2) one can effectively identify &lt;em&gt;creative thinking&lt;/em&gt; with &lt;em&gt;problem solving&lt;/em&gt;. He is the sort of author who writes in a scholarly way about creativity, and about whom much can be gleaned from his previous book titles (e.g., &lt;em&gt;Creativity: Genius and Other Myths&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Finally: If you want more information about mathematical creativity (again, based on your tag here), then another place to look is an earlier MESE response of mine &lt;a href=&quot;https://matheducators.stackexchange.com/a/906/262&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-04-25T21:44:54.860" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="2238" PostTypeId="2" ParentId="2220" CreationDate="2014-05-10T06:41:48.117" Score="9" Body="&lt;p&gt;The first two years in the mathematics program in my university in Brazil goes as follows:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;First semester:&lt;/strong&gt; Calculus I, comprised of a review of real functions and inequalities (triangle inequality and &lt;span class=&quot;math-container&quot;&gt;$|x| - |y| \leq |x-y|$&lt;/span&gt;, limits of one variable functions, continuity (all with &lt;span class=&quot;math-container&quot;&gt;$\varepsilon-\delta$&lt;/span&gt; arguments), definition of derivatives, computations with derivatives, important theorems (Weierstrass's theorem, mean value theorem, intermediate value theorem, rolle's theorem), chain rule, drawing a sketch of a function's graph using calculus, applications of derivatives, antiderivatives, definite integrals, all integration techniques (substitutions, partial fractions, integration by parts, trigonometric substitutions), improper integrals, applications of integration; Analytic geometry comprised of vectors in two and three dimensions, basis, lines in two and there dimensions, parametric representations of figures, conic sections, matrix operations, Gaussian elimination (I don't remember much of this one by head).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Second semester:&lt;/strong&gt; Multivariable calculus (Calculus II), comprised of functions of two variables, limits of functions of two variables, partial derivatives, differentiability, directional derivatives, gradient of a function, directional derivatives as dot product of the gradient of a function with a direction, applications of partial derivatives, optimization using the Hessian criterion and Lagrange multipliers, double and triple integrals in cartesian, cylindrical, spherical and general coordinates using the Jacobian, line integrals, conservative vector fields, fundamental theorem of line integrals, Green's, Gauss's and Stokes's theorem in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;; Linear algebra, comprised of vector spaces over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt;, subspaces, bases and dimension, coordinates, computations, linear transformations, isomorphisms, transformations as matrices, linear functionals (sometimes), eigenvalues and eigenvectors, inner products, unitary and normal operators.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Third semester:&lt;/strong&gt; Calculus III, comprised of ordinary differential equations using integration factors, constant coefficients, Cauchy-Euler's equation, Riccati's equation, Bernoulli's equation, Euler's method, method of undetermined coefficients, variation of parameters, Wronskians, Laplace's transform definition, application to solving ODEs, Dirac's impulse function, unit step function, convolutions, sequences, numerical series, convergence of numerical series, Taylor series, solving ODEs using Taylor series, Frobenius's series method of solving ODEs, function series, Fourier series, solving the heat and wave equation using Fourier series.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Fourth semester:&lt;/strong&gt; Complex calculus, comprised of complex functions of one complex variable, definition of limit and continuity using &lt;span class=&quot;math-container&quot;&gt;$\varepsilon-\delta$&lt;/span&gt;, differentiability of complex functions, Cauchy-Riemann equations in cartesian and in polar equations, definition of harmonic functions, analyticity, the various Cauchy theorems of complex analysis, Jordan curves, rectifiable curves, complex sequences, complex series, Laurent series, computation of said series, complex integrals, computation of real integrals using residues, conformal mappings; Analysis I, comprised of the usual analysis topics: basic set theory, natural, integer, rational and real numbers, sequences and series, real line topology, limits of functions, continuity of functions, differentiability and important theorems cited above; and Advanced Linear Algebra (graduate course), comprised of a review most of the previous linear algebra course now in general fields, a more thorough investigation of eigenvalues and eigenvectors, dealing with invariant subspaces, primary decomposition theorem, cyclic decomposition theorem and other forms, special emphasis on Jordan's form, a reinvestigation of inner products, dealing with all types of operators, sometimes the min-max theorem, and tensor algebra, the universal diagram, symmetric tensors, exterior product, and the rest of tensor constructions.&lt;/p&gt;&#xA;&lt;p&gt;I apologize for the space, I tried to be thorough. There are other simultaneous courses non-math related I left out, like basic physics, experimental physics and the like.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="47" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-05-10T06:41:48.117" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2239" PostTypeId="1" AcceptedAnswerId="2244" CreationDate="2014-05-10T07:03:19.990" Score="4" ViewCount="486" Body="&lt;p&gt;Do math teachers, mathbook writers AND math textbook writers generally target students who show a math aptitude initially? Are students who show lack of interest weeded out? I remember seeing the movie STAND and DELIVER with Edward James Olmos (forgive the spelling), where he was a high school teacher and he wanted to teach calculus to his grade 12 students.  The faculty was against him but he managed to make his students proficient in Calculus; A LOT of them initially disliked it. Why can't calculus be taught in grade 12 or 11?&lt;/p&gt;&#xA;" OwnerUserId="1359" LastEditorUserId="47" LastEditDate="2014-05-10T07:53:16.967" LastActivityDate="2014-05-21T11:27:15.743" Title="Do teachers and books target more apt students?" Tags="|undergraduate-education|calculus|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2240" PostTypeId="2" ParentId="2239" CreationDate="2014-05-10T08:08:39.790" Score="4" Body="&lt;p&gt;No, they aren't. That said, some subjects are genuinely difficult such that even the clearest writer cannot simplify concepts further. You cannot distinguish your readers in a text. Authors attempt to narrow the prerequisites for learning their area to their bare minimum, which may not account for the true amount of background knowledge and/or mathematical maturity (often invoked) necessary for dealing and struggling with the naturally presented difficulties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for your anecdote regarding calculus in grades 11 and 12, I am not from the USA but I believe this stands: it's not about age but about readiness and willingness to work through the concepts. I feel that whoever learns calculus earlier than usual is better prepared and nurtured to do so than most people, not that they are more capable than anyone else.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-05-10T08:08:39.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2241" PostTypeId="2" ParentId="2239" CreationDate="2014-05-10T11:53:55.837" Score="2" Body="&lt;p&gt;The text itself isn't the issue. In my opinion, student's aptitude fits a bell curve fairly well, and the struggle for teachers is to keep that lowest 1/6 passing, while not losing the top 1/6 to boredom. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my school, the advanced track has calculus as an offering for seniors (12th grade). I also see seniors who are struggling with algebra, of basic trig. &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2211/a-students-problem-on-solving-simple-trigonometric-operations&quot;&gt;A student&amp;#39;s problem on solving simple trigonometric operations&lt;/a&gt; is an example of this. The text and the teacher are expecting students proficient at basic manipulation, yet, the teacher will likely need to pause to give this a review.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An answer to that question (Chris) suggests that mastery isn't the right expectation. My opinion is that his response is correct and forms the basis of the full high school curriculum. i.e. that a certain fraction of class time spent on review reduces the amount of new material that can be delivered.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-10T12:08:52.953" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2242" PostTypeId="2" ParentId="2239" CreationDate="2014-05-10T12:57:53.807" Score="3" Body="&lt;p&gt;It is just that someone talented/very interested will be able to learn even from an awful book or teacher, and that teaching those is a lot more fun than dealing with the bottom fifth of the class. Given &lt;a href=&quot;http://en.wikipedia.org/wiki/Sturgeon%27s_law&quot; rel=&quot;nofollow&quot;&gt;Sturgeon's revelation&lt;/a&gt;, and teachers being just human beings explains your observation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't believe what Hollywood shows you. A modicum of knowledge in a given area (or just a bit of common sense) will show innumerable howlers in your random film. I'm not saying it &lt;em&gt;can't&lt;/em&gt; happen; but if it was reasonably common, it won't make a good story...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-10T12:57:53.807" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2243" PostTypeId="2" ParentId="2239" CreationDate="2014-05-10T13:34:43.620" Score="6" Body="&lt;p&gt;It's not necessarily the case that people are trying intentionally to target a certain type of student. However there is evidence that certain traditional methods of mathematics instruction do cater to certain kinds of people and it is not as simple as selecting for the more apt students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the process of learning, students form ideas about what mathematics is and who they are in relation to it. Many get an impoverished view of mathematics because of an approach that focuses on certainty, having a single answer, rigid procedural action (so-called &quot;machine agency&quot;). Some decide that this inaccurate view of math is just what they're looking for for themselves and their future. Others want a future in which they feel they can express themselves in their work and use their interpretive, meaning-making ability. Not surprisingly, those people think math is not for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, it turns out you're not getting the most &quot;apt&quot; students, and it's not necessarily &lt;em&gt;intentional targeting&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more details, and references to that evidence, see this other MESE answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/1455/how-to-handle-the-situation-when-you-made-a-stupid-mistake-in-front-of-the-class/1461#1461&quot;&gt;How to handle the situation when you made a stupid mistake in front of the class?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-10T13:34:43.620" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2244" PostTypeId="2" ParentId="2239" CreationDate="2014-05-10T17:04:23.900" Score="-1" Body="&lt;p&gt;Yes. In many places, students who are more capable get more resources allocated to help them excel.  The textbooks do not discriminate between different students, but the teachers can, and quite often do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not unheard of for 9th graders to take Advanced Placement Calculus BC.  It's just a matter of how ready they are to take the course.  I don't think that anyone is really neglected, it's just that the &quot;smarter&quot; kids get put into more advanced classes, which get more resources, allowing them to do even better.  It comes down to ability more than anything.&lt;/p&gt;&#xA;" OwnerUserId="1431" LastActivityDate="2014-05-10T17:04:23.900" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2245" PostTypeId="2" ParentId="2185" CreationDate="2014-05-10T20:03:27.077" Score="7" Body="&lt;p&gt;I have offered papers for a variety of classes. For most students, the idea of writing a math paper is completely foreign to them. As a mathematics educators, we should encourage the learning of all of the mathematics out there, and I have taken this past opportunity to offer historical papers. I think it would otherwise be difficult for mathematics students to write a mathematically rigorous paper on the mathematics that they are just learning now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on the level of students, I have required different lengths of papers. Below is a full copy of the assignment I have offered to have them write on, though I leave open the option for them to find other topics.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Please write a (minimum) five-page (double spaced, 1 in margins all around, &#xA;Times New Roman 12-point font, APA formatting) paper emailed to me by 11:59PM,&#xA;May 16. List all your sources. Each of these topics below has more than &#xA;enough material to fill five pages, but there are lots of other topics that &#xA;can also be written about. You must confirm your topic with me before you &#xA;begin.&#xA;&#xA;One possible outline:&#xA;&#xA;•   Introduction: Give a general overview of the subject&#xA;•   Body: Explain the topic. Explain why it is important. How does it related&#xA; to anything in the world today?  What other fields is it connected to?&#xA; Expand on one (or more) particular idea, and go over it in detail.&#xA;•   Conclusion: Explain what you learned doing the research. Was there &#xA;anything that was familiar you came across during your research?  List all&#xA;sources used. What was the most interesting thing in the research to you?&#xA;&#xA;&#xA;Examples of Extra Credit Topics:&#xA;&#xA;The number e, its history, and uses&#xA;The number π, its history, and uses&#xA;The Fibonacci numbers&#xA;Magic Squares&#xA;Prime numbers&#xA;Numeral systems&#xA;Complex Numbers&#xA;Perfect Numbers&#xA;The Mandelbrot Set&#xA;Inductive Proof&#xA;Pigeon Hole Principle&#xA;Modular Arithmetic&#xA;Computus&#xA;Adrien-Marie Legendre&#xA;Carl Friedrich Gauss&#xA;Joseph Louis Lagrange&#xA;Augustin Louis Cauchy&#xA;Jean Baptiste Joseph Fourier&#xA;Bernhard Riemann&#xA;Peter Gustav Lejeune Dirichlet&#xA;Archimedes&#xA;Euclid&#xA;Rene Descartes&#xA;Leonhard Euler&#xA;Georg Cantor&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Outcomes&lt;/strong&gt;: It has been a few years since I incorporated this in a class, but as I recall, most students chose not to write a paper. Many expressed that they thought it was impossible to write 3-4 pages on a mathematics topic. There were some who chose to write the papers though. Those who did, saw how much was out there, they were amazed at how much they had to cut out. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Did Students Learn Things?&lt;/strong&gt; This is difficult to quantify from my assignment. The main goal I was trying to accomplish was to have my students learn that there is a lot of other math out there; that it's not just pushing numbers around. I would say that those students who took the assignment seriously did see that there was at least &lt;em&gt;some&lt;/em&gt; other aspect of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What would I do differently?&lt;/strong&gt; After first giving this assignment, I found that many students were drawn to write about the well-known mathematicians. After reading 3 papers on Newton that provided nothing new for me to learn about him, I eliminated him, along with a few others perhaps (I think Einstein may have been on my original list).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Would I do this again?&lt;/strong&gt; I just gave this to a remedial class the other day. When I receive the papers, I will post on more results. I will continue to offer this as an extra credit assignment for appropriate classes until I find something more valuable.&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="526" LastEditDate="2014-05-12T07:27:58.207" LastActivityDate="2014-05-12T07:27:58.207" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2246" PostTypeId="1" AcceptedAnswerId="2270" CreationDate="2014-05-11T00:07:59.783" Score="25" ViewCount="1718" Body="&lt;p&gt;There is a well known &lt;a href=&quot;http://www.physics.orst.edu/bridge/papers/CMJdifferentials.pdf&quot; rel=&quot;noreferrer&quot;&gt;essay by Dray and Manogue&lt;/a&gt; which argues that differentials should be brought back into freshman calculus, and that we shouldn't worry too much about choosing a specific way of formalizing them, or about giving students a &lt;em&gt;formal&lt;/em&gt; answer to the question of &quot;what are differentials?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find myself in almost complete agreement with Dray and Manogue, and I feel especially that differentials are the natural way to approach related rates and implicit differentiation. Since differentials are used universally in the sciences and engineering, I think freshman calc students should be exposed to them. I would like to throw in just a taste of differentials when doing related rates and implicit differentiation, without going whole hog and using them throughout the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I will admit to quite a bit of apprehension about actually going ahead with such an approach. I suspect that, for example, I would get some push-back from colleagues who believe that differentials are inherently inconsistent; that students would go to tutors for help, and that the tutors wouldn't understand the approach; and that the kind of casualness about foundational aspects advocated by Dray and Manogue would be interpreted as carelessness or incompetence on my part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone implemented anything like what Dray and Manogue advocate, and if so, what were their practical experiences like?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/12/a-calculus-book-that-uses-differentials&quot;&gt;A calculus book that uses differentials?&lt;/a&gt; [I was surprised to find that Keisler's treatment of implicit differentiation looks exactly like a standard one, without differentials or infinitesimals.]&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="1385" LastEditDate="2017-04-21T14:57:14.223" LastActivityDate="2023-03-06T23:42:00.483" Title="Practical experience with teaching differentials in freshman calc?" Tags="|calculus|infinitesimals|" AnswerCount="6" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2247" PostTypeId="2" ParentId="2185" CreationDate="2014-05-11T14:58:12.560" Score="4" Body="&lt;p&gt;Not my own experience, so I can't comment on how it went (and therefore this doesn't really answer the question), but Gavin LaRose at the University of Michigan has &lt;a href=&quot;http://www.math.lsa.umich.edu/~glarose/courseinfo/calc/calcprojects.html&quot; rel=&quot;nofollow&quot;&gt;a page of Calculus writing prompts on his website&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-05-11T14:58:12.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2248" PostTypeId="2" ParentId="2185" CreationDate="2014-05-11T17:19:21.417" Score="1" Body="&lt;p&gt;Perhaps this type of question could work, supplemented with some structuring guidelines...?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Where and how is calculus used in &lt;em&gt;epidemiology&lt;/em&gt; today?&#xA;  Please try to find explicit examples, and explain the use of calculus in these examples.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And then replace &quot;epidemiology&quot; with: {genetics, exoplanet search, drug delivery, atmospheric pollution monitoring, oceanography, stockmarket predictions, global warming, ...}.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-05-11T17:19:21.417" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2249" PostTypeId="1" AcceptedAnswerId="2252" CreationDate="2014-05-11T17:20:37.173" Score="6" ViewCount="552" Body="&lt;p&gt;Today on the process of explaining how to study Mathematics to my students, I said that Mathematics is just like a language. To learn a language, first we have to know its alphabet, grammar, vocabulary, synatx and all the other things. Likewise, Mathematics is a kind of language with its own vocabulary, symbols, terms, definitions and theorems. To be successful in Mathematics, we must be able to to develop a decent vocabulary, use it as often as we can and speak and write in the language of Math. With that I emphasized on writing definitions, theorems and properties which I think will surely help them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think I am correct on those things above but I wanted to tell them more but I did not know what things to add. Do you have any suggestion? Probably on how student should see Math as? Or how should they can learn other theories that are not on their curriculum? &lt;/p&gt;&#xA;" OwnerUserId="1172" LastEditorUserId="47" LastEditDate="2014-05-11T20:11:58.813" LastActivityDate="2014-05-11T23:39:25.357" Title="Explaining how to study mathematics" Tags="|secondary-education|student-motivation|" AnswerCount="3" CommentCount="6" ClosedDate="2014-05-12T00:15:48.023" ContentLicense="CC BY-SA 3.0" />
  <row Id="2250" PostTypeId="2" ParentId="2249" CreationDate="2014-05-11T18:07:59.867" Score="4" Body="&lt;p&gt;As an outsider (I am a user of mathematics in my field -economics- and not a mathematician), I most certainly agree that &quot;Mathematics is a language&quot; (I think &lt;a href=&quot;http://en.wikipedia.org/wiki/Willard_Gibbs&quot; rel=&quot;nofollow&quot;&gt;J. Willard Gibbs&lt;/a&gt; is credited with the saying). And a very useful and powerful one. &#xA;But then I have to point out that I disagree with the OP sentence &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To learn a language, first we have to know its alphabet, grammar,&#xA;  vocabulary, synatx and all the other things&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;-especially with the use of the word &lt;em&gt;first&lt;/em&gt; : Not always does a person gets to learn a language in this &quot;linear&quot; (or would that be &quot;sequential&quot;?), and safe, way.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for students to actually see Mathematics as a language, I believe that they must first &lt;em&gt;translate&lt;/em&gt; something in the language of Mathematics, from a language that they already know (the joy of speaking a language per se comes to be appreciated usually later - James Joyce's &lt;em&gt;Ulysses&lt;/em&gt; has as a prerequisite a deep knowledge of the English language in order to fully enjoy its beauty).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would think along these lines, and give them situations from the real world to translate into mathematics (in an abstract and symbolic manner of course).&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2014-05-11T18:07:59.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2251" PostTypeId="2" ParentId="2249" CreationDate="2014-05-11T18:21:56.047" Score="2" Body="&lt;p&gt;I'd only add that as a toddler you didn't get a helping of grammar, and a dictionary for vocabulary, and Strunk and White for style roundup. You learned step by step, going from simple words and phrases to ever more complex (and interesting!) utterances. Much of what you'll have to learn and do initially will be the equivalent of toddler talk, but it is the way to reading and writing novels later on.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-11T18:21:56.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2252" PostTypeId="2" ParentId="2249" CreationDate="2014-05-11T23:33:38.377" Score="5" Body="&lt;p&gt;&lt;strong&gt;TL;DR:&lt;/strong&gt; In my opinion this is &lt;em&gt;not&lt;/em&gt; how it works. How to study math: &lt;em&gt;Don't just read it; fight it!&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;You claim that mathematics is just like a language. Mathematics &lt;em&gt;is&lt;/em&gt; a language (among being other things), so &lt;em&gt;technically&lt;/em&gt; you are correct. However, it is a language unlike any other, so &quot;just like a language&quot; does not apply (or I misunderstood your post). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many dissimilarities, and they show when one attempts to use standard natural language processing methods on math papers. To name a few, math is highly contextual, the notation is non-linear, deeply nested and the ambiguities reach a whole new level of quality (sentences like &quot;I saw her duck&quot; or &quot;I see a man on the hill with a telescope&quot; seem like a piece of cake in comparison). However, there is one key difference: &lt;em&gt;math frequently redefines things&lt;/em&gt;. Just compare a few papers, and you see the same notation used to denote so many various concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know these are not the problems that &lt;em&gt;humans&lt;/em&gt; have with math, I just wanted to argue why math is not just like any language. In fact, people doesn't have any problems with music notation, but similar arguments would work there as well; curiously notations for both music and math are &lt;em&gt;semantic&lt;/em&gt; in nature, for example there are only slight difference between countries, like &quot;h&quot; vs. &quot;b&quot; notes or &quot;$\leq$&quot; vs. &quot;$\leqslant$&quot; symbols. Finally, I'm well aware that the above mostly concerns not the actual language, but its notation, that is, realization of the language on some medium; nevertheless, the connection between the language and its notation is so deep that the two can hardly be separated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet, let's suppose that math is like a language, perhaps what is important to learning is common to both math and other languages. You continue with &quot;To learn a language, first we have to know its alphabet, grammar, vocabulary, synatx and all the other things.&quot; However, is it how you learned to speak? Or was it more like a number of people (e.g. parents) would say to you some incomprehensible sounds and then after some time you would respond with &quot;ga ga ga&quot; and then they would smile, and slowly there would be some progress and after several years you would be able to express basic ideas?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel that I learned a lot of math from my parents, but they &lt;em&gt;never&lt;/em&gt; introduced anything formally. Math was then essentially pattern recognition, predictions and hypotheses testing, and most importantly, a great deal of fun. Of course, I learned the vocabulary, tools, notations, and other concepts, but it was more by assimilation than explicit tutoring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give more recent example, every time I see students hear the formal definition of a &lt;a href=&quot;http://en.wikipedia.org/wiki/Deterministic_finite_automaton&quot; rel=&quot;noreferrer&quot;&gt;finite state machine&lt;/a&gt; there are some horrified looks. In fact, &quot;let XYZ be a $n$-tuple&quot; became a running joke at our university. On the other hand, if students are shown how something works first (in computer science this is easy, because you can &lt;em&gt;run&lt;/em&gt; programs, algorithms, show visualizations and many of those are intuitive once you see it work), then they have much easier time digesting the theory. It's much more effective to give several cases of recursive functions, simulate them on some small inputs, and then let the students play and figure things out by themselves, than, for example, introduce all the necessary ingredients step by step and then combine it all via formulas and induction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is even more true if you consider how mathematical theories developed across the centuries. The grammar which you want to introduce so early was not even there when most math which we know now was formed. For example, the foundations are recent developments and are not yet even fixed. In my opinion the evolution of concepts of integrals and derivatives much more resembles a child attempts to talk, than orderly introduction that is given in university courses. Yes, there is a significant difference between evolution of a concept and teaching it as we currently know it (we don't need to copy all the failed approaches, do we?). Still, I doubt that starting with the limit formula is a good way to learn derivatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admit, that this only &lt;em&gt;anecdotal&lt;/em&gt; evidence and not significant in any way. The more so, because of my bias, i.e. how I learned math in my early years. Yet, after reading pieces like the following paragraph by Paul Zeitz, one can only wonder...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;A question of culture&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;When I was a grad student at UC Berkeley (in the late 1980s), it was understood, among my American classmates, that the Eastern Europeans were simply &lt;em&gt;better&lt;/em&gt;. &#xA;  They weren't genetically superior; indeed, many of my American classmates, myself included, were themselves descended from Eastern European immigrants.&#xA;  And we knew that we weren't stupid.&#xA;  Many of us had excelled at mathematical olympiads, even at the international level.&#xA;  But at Berkeley, the Eastern Europeans -- students and faculty alike -- were known for their &lt;em&gt;intensity&lt;/em&gt;. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;All joking aside, we fledgling mathematicians understood that the single important thing was not raw intelligence or knowledge. What mattered was &lt;em&gt;passion&lt;/em&gt;. The way to become successful in mathematics, like almost every endeavor, is to care about it, to love it, to obsess over it. And in this, Eastern Europeans had a clear superiority, a &lt;em&gt;cultural&lt;/em&gt; advantage. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;One of the many laudable practives around the world, one that resonates particularly well with the American mathematical community is the Eastern European notion of a &lt;em&gt;mathematical circle&lt;/em&gt;. Most math circles have several features that distinguish them from simple math clubs.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;First, they are &lt;em&gt;vertically integrated&lt;/em&gt;: young students may interact with older students, college students, graduate students, industrial mathematicians, professors, even world-class researchers, all in the same room. The circle is not so much a classroom as a gathering of young initiates with elder tribespeople, who pass down folklore.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Second, the &quot;curriculum,&quot; such as it is, is dominated by &lt;em&gt;problems&lt;/em&gt;, rather than specific mathematical topics. A problem, in contrast to an exercise is a mathematical question that one doesn't know how, at least initially, to approach. [...] [Problems] do not come with algorithms attached. By their very nature, they require &lt;em&gt;investigation&lt;/em&gt;, which is both an art and a science, demanding technical skill along with psychological strategies, what I call the &quot;three Cs&quot; of concentration, confidence, and creativity. Math circles teach students these skills, not with formal instruction, but by &lt;em&gt;doing math&lt;/em&gt; and observing others doing math. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;To conclude:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have no real idea how to study, teach or teach studying math (I only know what has worked for me before). However, I feel that going in order through &quot;alphabet, grammar, vocabulary, syntax and all the other things&quot; is not a good idea. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I was pressed to give some advice, I would recommend &quot;&lt;a href=&quot;http://lesswrong.com/lw/k4n/a_brief_summary_of_effective_study_methods/&quot; rel=&quot;noreferrer&quot;&gt;A brief summary of effective study methods&lt;/a&gt;&quot; by Arran Stirton and then quote Paul R. Halmos: &lt;em&gt;Don't just read it; fight it!&lt;/em&gt; (the picture comes from the &lt;a href=&quot;http://www.abstrusegoose.com/&quot; rel=&quot;noreferrer&quot;&gt;Abstruse Goose&lt;/a&gt; webcomic).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://abstrusegoose.com/353&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yfsJB.png&quot; alt=&quot;How to study math&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-05-11T23:39:25.357" LastActivityDate="2014-05-11T23:39:25.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2253" PostTypeId="1" AcceptedAnswerId="2254" CreationDate="2014-05-12T00:23:39.203" Score="29" ViewCount="4952" Body="&lt;p&gt;I've seen a variety of approaches to the day we hand back our exams.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Show every score listed in order, so every student knows exactly how they compare to others&lt;/li&gt;&#xA;&lt;li&gt;Show an aggregated histogram, for example saying how many students got each grade&lt;/li&gt;&#xA;&lt;li&gt;Give only descriptive statistics (mean, median)&lt;/li&gt;&#xA;&lt;li&gt;Say almost nothing about the class's scores&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;How can we give information about the distribution of scores that can be productive and useful for students? I'd specifically like to hear about &lt;strong&gt;approaches you have tried,&lt;/strong&gt; and critical discussion or anecdotes about &lt;strong&gt;how students gained or did not gain from the information&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please assume that the class is &lt;strong&gt;not&lt;/strong&gt; graded on a strict curve where a certain percentage of students will get an A; in that case it's obvious that we have to be open about the distribution.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-05-15T11:31:35.103" Title="When handing back exams, what should we tell our students about the distribution of exam scores?" Tags="|exams|struggling-students|" AnswerCount="9" CommentCount="19" ContentLicense="CC BY-SA 3.0" />
  <row Id="2254" PostTypeId="2" ParentId="2253" CreationDate="2014-05-12T01:25:18.210" Score="13" Body="&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Favorite options &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;approximate&quot; grade breakdown (no mean, median, or standard deviation, so there is no comparison to other students, this doesn't necessarily mean fixed grading)&lt;/li&gt;&#xA;&lt;li&gt;no information at all (note at many institutions this might actually be bad. Please see the answer to a &lt;a href=&quot;https://matheducators.stackexchange.com/a/2267/11&quot;&gt;related question&lt;/a&gt; by AndrewC for a full explanation about how this could work)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Least favorite&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;full distribution (can create a competitive environment plus privacy concerns) &lt;/li&gt;&#xA;&lt;li&gt;limited statistics (special care has to be taken such that the ones you provide are not  misleading, and again comparisons are being made to some central tendency, see Chris Cunningham's answer for a full explanation) &lt;/li&gt;&#xA;&lt;li&gt;no information at all (can potentially lead to grade anxiety if the students are at an institution with a grade obsessed culture - although this can work well in the right setting)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Explanation + Past Experience:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have done three things is the past. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I gave just the median and standard deviation&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I gave a histogram, with rough rough grade cut offs. See (3)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I gave the median and &quot;rough grade cutoffs.&quot; I explicitly warned that these grade cutoffs are only guidelines for approximately where you stand in the class and not your actual grade on the test. For example I might give the following cut offs &quot;A: &gt;88, B: &gt;75, C: &gt;60, D: &gt;40&quot;. I then say &quot;These are not your actual grades, only approximate grades based on, roughly, how much I would expect an A, B, C, and D student to have learned the  material.&quot; In addition, I explicitly say &quot;The entire class will be curved, at the end, so note that there are ways that the average of your 'approximate letter grades' on these exams could potentially be lower or higher than your final grade (but this is rarely more than by a +/-). This is a very rough curve for you to gauge where you are at. No +s or -s are mentioned because this would give a false sense of accuracy.&quot; I also put this warning in the syllabus.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I found that students much preferred options (2 and 3 about equally) to option (1) because two and three give them a better idea of what your expectations are, and it also relieves some of their anxiety about not knowing what their grade is at. If you go this route, definitely don't forget to put the warning in your syllabus. This protects you with documentation in case a student were to complain to the administration about the average of their exam grades being slightly higher than their final grade. However, I have never had a student complain about this; they understand that the grade cutoffs are innacurate and are only presented as a service to them and that if they are close to a border line, they understand that they could be sitting at the lower grade. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The histogram is an option that could potentially add information to the grade cut-offs but while it does add information I think it has serious drawbacks.  The reason why I prefer &quot;approximate but slightly inaccurate grade breakdowns&quot; to the histogram method is that I don't want to create a competitive environment. I like to frame the test results in language that emphasizes what I expected them to learn, rather than a class rank (even though these things are highly correlated). I did sense some added competitiveness when using the histogram (although it wasn't a huge effect), so I think I'm not going to do it in the future. Note, several academics often mention the competitive nature of math classes as a reason for the lack of women and other underrepresented groups in mathematics. Because of this, in the future, I am no longer going to tell students the median. I would, especially, not recommend giving out the full distribution, because in addition to the reason above, it might be possible to figure out who scored what with some investigative work on the students end. I also think the full distribution would convey a false sense of certainty. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One might ask, well given the last paragraph why don't you know the grade breakdown in advance (i.e. fixed grading). The answer to this is, when writing a new test every year, with new, sometimes creative problems, I never know all the possible mistakes students will make until afterwards. Weighing the mistakes I see and looking at the distribution, I then identify the &quot;approximate cut offs&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, I'd really like to try not providing any information. I think if you create an atmosphere where grades are completely de-emphasized, and reinforce that the student's holistic performance will be used to determine a grade at the end of the course, this could work very well. If a student is anxious about where they sit in terms of grades, they could ultimately request a meeting with you to discuss what they need to work on and what they are doing really well. Unfortunately, the bureaucracy at my large University, and the general grade anxiety in a class of mostly freshman who earned 4.0+ GPAs in high school, makes creating this environment difficult. If you do go this route, you also need to explain it clearly, and in detail on your syllabus, and keep some form of documentation of student performance so that you can justify all your grades.&lt;/p&gt;&#xA;" OwnerUserId="1387" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-14T18:01:12.530" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2255" PostTypeId="2" ParentId="2253" CreationDate="2014-05-12T04:04:37.820" Score="8" Body="&lt;p&gt;I tell the students the mean, standard deviation, and maximum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mean and standard deviation, because they are the most statistically useful. (The median would be useful too, but it's unclear how it interacts with the standard deviation, and sharing both the mean and median might be confusing to some students.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The maximum, because I think the students with good scores will be curious about whether their scores are the highest, and also in order to confirm or deny students' suspicions that the exam was impossibly difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't include the minimum, or a histogram, because it might make the worst students feel bad.  Also, including too much information would encourage students to obsess over scores even more than they already do.&lt;/p&gt;&#xA;" OwnerUserId="870" LastActivityDate="2014-05-12T04:04:37.820" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2256" PostTypeId="2" ParentId="2246" CreationDate="2014-05-12T08:14:03.040" Score="12" Body="&lt;p&gt;A great quote from Dray and Manogue:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;. . . many mathematicians think in terms of infinitesimal quantities: apparently,&#xA;however, real mathematicians would never allow themselves to write down such&#xA;thinking, at least not in front of the children. —Bill McCallum [16]&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think in this type of discussion it is important to separate carefully between two issues: (a) what is the theoretical justification for using differentials/infinitesimals in the classroom, and (b) what is the actual situation on the ground, i.e. are teachers successful in imparting understanding of basic notions of the calculus such as slopes/derivatives, areas/integrals, seeking minima/maxima, basic theorems such as the extreme value theorem, etc. using such an approach?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as (b) is concerned, I can report based on classroom experience with Keisler' book last term that the approach is successful. Not only was it successful but the students petitioned their department to make sure the class is taught exactly the same way next year. The true infinitesimal calculus course has now been taught for 4 years to over 400 freshmen, with testable success; see &lt;a href=&quot;http://u.cs.biu.ac.il/~katzmik/infinitesimals.html#17h&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as (a) is concerned, of course we know today that infinitesimals are consistent. It is even more important to make the point that traditional calculus courses do not typically establish the foundations of the subject such as constructing the real numbers. Some foundational material has to be taken as given, and is taken up again in more advanced analysis or algebra courses.  Similarly, in the approach using calculus certain facts need to be taken as given and Keisler does a marvellous job of motivating whatever facts that are taken as given in this approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is worth mentioning the work by Kathlene Sullivan from the mid 1970s who performed a controlled experiment in the Chicago area, with some groups following the traditional approach, and others using Keisler's book. The groups using infinitesimals came away from the course with a slightly better understanding of the basic calculus notions, according to the conclusions of the study.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-04-23T08:13:32.370" LastActivityDate="2017-04-23T08:13:32.370" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="2257" PostTypeId="2" ParentId="2246" CreationDate="2014-05-12T12:25:40.923" Score="4" Body="&lt;p&gt;I think the best time to really claim that differentials make sense is during solids of revolution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trying to convince students that $\int f(x) \textrm{dx}$ is the sum of infinitely many rectangles only works for the top students. The students don't actually &lt;em&gt;use&lt;/em&gt; this knowledge when solving problems this early in calculus, so I think it is best skipped at that point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, drawing a thin cylinder and claiming that its volume is $\pi r^2 \textrm{dx}$ when oriented one way and $\pi r^2 \textrm{dy}$ when oriented the other way is golden. Shell method? The thickness of the shell is &quot;a little bit of x&quot; (dx) when oriented one way, and dy the other way. The students get to practice using this while doing problems and so this is (in my opinion) the right time for the concept to be stressed.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-05-12T12:25:40.923" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2258" PostTypeId="2" ParentId="2246" CreationDate="2014-05-12T14:41:59.890" Score="17" Body="&lt;p&gt;A faculty member in my department is a former student of Dray's and 2-3 years ago showed me the differentials approach. He and I both use it quite extensively in our first semester calculus courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the pros and cons (in no particular order) and comments on them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pro:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) It's generally awesome. Students in my class struggle a lot less with chain rule, implicit differentiation, related rates, optimization, u-substitution, and integration by parts. (I could go on about how awesome it is, but that's not part of the original question.) In fact, the students that struggle the most are the ones who are retaking it (either again in college or just because they took it in high school) and won't embrace the differentials method. They usually end up making what I assume are the same mistakes they made the first time they took the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) In Calculus II, it makes even more sense. Integration techniques make more sense because $dx$ or $du$ are not just marking the end of the integral, they actually mean something. For the geometrical computations you can talk generally and then have students apply it to the given situation. For example, instead of two formulas for volumes by cylindrical shells, I give them one: $\int 2\pi r h dr$. Then it's up to them to figure out what $r$, $h$, and $dr$ represent geometrically. Likewise, it makes introducing parametric and polar curves easier since students already are comfortable with equations that aren't $y=f(x)$. Also, it greatly opens up the applications of integration section. Work = (Force)*(distance) now becomes $dW = s \cdot dF$ (or $dW = F \cdot ds$ if appropriate, &quot;What is the small quantity?&quot;, &quot;Which are you assuming to be constant on your slice?&quot;), which is really how advanced physics/engineering classes think about such relations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pros/Cons (depending on how you look at it):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) It makes calculus extremely algebraic to the point that you really don't have to understand much of the concepts going on. As long as you know what algebraic fact you need/are using (e.g. solving for $dA/dt$ or setting $dA=0$), it's just algebra. Of course, a lot of students are weak in their algebra skills...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Transition to Calculus II: If they have me again, no big deal. If they have someone else, I would guess they might stutter a bit in the beginning. But I usually use $d/dx$ at least once or twice around the end of the course and only once in the past 3 years have I had a student ask me what that meant. When I explained it meant, &quot;Take a d, then divide by dx&quot; they gave an &quot;oh, duh&quot; response.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Students mix up $3x^2$ and $3x^2dx$, i.e. they sometimes will write a function when they mean a differential or vice versa. If they write a differential and mean a function, it usually isn't too big a deal, since if they evaluate the function they then drop the $dx$. If they write a function and needed a differential, then the error is a problem because they typically need to solve for a ratio of differentials or something and they don't have the right pieces to do so. All in all, I see this as a small trade-off.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Students will get confused when they look at the book. I had a lot of problems when I first started using differentials because students would look to the book to do the homework. Now I have notes that are NOT my examples from lecture, but the examples in the book re-worked using differentials. I've had far less problems since. I'm hoping in an upcoming sabbatical to rewrite an open-source Calc book to put it into a differentials flavor...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) Colleagues' opinions. Well, as I mentioned above, maybe I'm lucky that there's two of us out of the 6-7 Calc I instructors we have at our university. The others either think it's fine (as long as they don't have to do it) or are interested in using bits and pieces (but won't commit to going all out). Nobody challenges the rigor of it because, honestly, given our student population, there's little rigor in our Calc I anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6) Tutors. Usually I warn the tutors who haven't had me what they might see when students come in. Our tutors are typically junior or senior math majors. When I show them the differential approach they take to it very quickly (after only a couple of examples) and usually wonder why everybody doesn't do it that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So that's my (more than) 2 cents. I could go on and on, but I think that answers most of your questions.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-05-12T14:41:59.890" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2259" PostTypeId="2" ParentId="2253" CreationDate="2014-05-12T14:51:16.090" Score="12" Body="&lt;p&gt;I've personally stopped giving the mean entirely, because it almost always gives an incorrect perception to students. Since the distribution is almost always skewed by very low-scoring outliers, the mean incorrectly tells more than half of the class that they are above-average.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd argue that giving a measure of central tendency at all sends a fundamentally incorrect message: &quot;Here is the target value. Your target is to be about as good as the most mediocre student in your class this semester.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's never my message.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-05-12T14:51:16.090" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2260" PostTypeId="2" ParentId="2253" CreationDate="2014-05-12T16:02:21.120" Score="7" Body="&lt;p&gt;Some people use a fixed grading scale, while others determine ad hoc cut-offs for an A, a B, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use a fixed grading scale: 80%=A, 69%=B, 58%=C. Since the scale is fixed, there is no reason that a particular student needs to know how the other students are doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For people who use ad hoc cut-offs, I can see why it makes sense to tell students what the grade distribution is, since there is an element of arbitrariness in setting the cut-offs, and they deserve to know what the cut-offs were based on. But to me this just raises the question of why it's valid to set ad hoc cut-offs, especially in a subject like math that has objectively right and wrong answers. If students are supposed to be able to differentiate expressions involving transcendental functions using the chain rule, then that should be the standard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Giving out grade distributions suggests to the student that (1) the teacher wasn't really sure what standards were reasonable to apply; (2) the day when you get back the exam is a day to be spent in complaining and negotiating about grading standards; and (3) as Chris Cunningham remarks, it suggests that the goal is to achieve the same marginal level of competence as the average student in the class.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-12T16:02:21.120" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2261" PostTypeId="2" ParentId="2253" CreationDate="2014-05-12T16:43:43.903" Score="2" Body="&lt;p&gt;As an adult student, I have taken some fairly difficult courses - two advanced chemistry courses come to mind.  That professor, who was great, would give you the score &quot;as is,&quot; but then tell you how exactly how many points the curve would give you if there was one, along with the &quot;central score&quot; in his model.  No one could ever get him to fully divulge his process, but he did let slip that he discarded outliers - he probably used a truncated mean or trimean.  On non-curved tests I think he gave the median.  We all thought this method was fine, but it really all depends on the class environment you create.  This was a highly interactive environment where none of us were really competitive, and in fact would openly share our scores with each other.  The students that made an 'F' on one test would be making a 'B' on the next because of the extra &quot;love&quot; they would receive.  Most of us were in our mid-to-late 20's with full time jobs, so we didn't have the time or energy for pettiness.  We preferred to learn and help each other out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, we would receive our scores with some form of truncated mean and the extra points that slid the distribution over to where he wanted on curved tests.          &lt;/p&gt;&#xA;" OwnerUserId="1438" LastActivityDate="2014-05-12T16:43:43.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2262" PostTypeId="1" CreationDate="2014-05-12T16:53:02.007" Score="30" ViewCount="28015" Body="&lt;p&gt;There seem to be two approaches to calculus education:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Early transcendentals: introduce polynomials, rational functions, exponentials, logarithms, and trigonometric functions at the beginning of the course and use them as examples when developing differential calculus&lt;/li&gt;&#xA;&lt;li&gt;Late transcendentals: develop differential calculus using only polynomials and rational functions as examples and introduce the rest afterwards.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Most American universities seem to use the early transcendentals approach, but I have never seen any empirical evidence that it is superior (e.g. higher pass rates, success rates in courses which depend on calculus).  So my question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there any empirical research comparing the outcomes of the early transcendentals and late transcendentals approaches to teaching calculus?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am hoping someone has done actual controlled experiments, but I would also be interested to hear the opinions of people who have considerable experience with both approaches.&lt;/p&gt;&#xA;" OwnerUserId="1439" LastActivityDate="2024-03-30T01:55:53.483" Title="Early vs. late transcendentals" Tags="|calculus|" AnswerCount="7" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2263" PostTypeId="2" ParentId="2262" CreationDate="2014-05-12T18:19:39.973" Score="4" Body="&lt;p&gt;I'm not sure, if my experience is considerable, but I'd like to point it out:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The late &quot;transcendentals approach&quot; has the huge drawback, that students often miss out the mere algebraic properties (like linearity) of these functions. And in all but the simplest calculus examples involving transcendental functions, these algebraic properties are important as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The early &quot;transcendentals approach&quot; has the smaller drawback, that all steps of calculus need more time as they need to be applied to all function classes. &lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-05-12T18:19:39.973" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2264" PostTypeId="2" ParentId="2069" CreationDate="2014-05-12T19:24:05.057" Score="3" Body="&lt;p&gt;In regards to early education and specifically the math of the CCSS, I have a link. This is a new blog post that ultimately tries to wade in by outlining several OTHER blog posts that all make good points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the main points being offered is that you have to separate the STANDARDS themselves from the question of whether they are being implemented well, whether there is the professional development and knowledge amongst teachers to use them the way they are meant to be used, whether the textbooks that are &quot;aligned&quot; are actually well done, etc etc etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As many are stating, there is plenty of good that can come from the standards, but there are many other things at play. They can be implemented poorly and actually make things worse... or at least not any better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://earlymath.erikson.edu/4-blog-posts-help-understand-common-core-math-standards-debate/&quot; rel=&quot;nofollow&quot;&gt;4 Blog Posts to Help Understand the Common Core Math Standards Debate&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1446" LastActivityDate="2014-05-12T19:24:05.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2265" PostTypeId="1" AcceptedAnswerId="3895" CreationDate="2014-05-12T21:51:24.370" Score="5" ViewCount="189" Body="&lt;p&gt;After reading &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2145/how-to-build-a-class-based-on-project-euler&quot;&gt;How to build a class based on Project Euler?&lt;/a&gt; I was inspired to suggest some work on Project Euler over the summer as part of an ad-hoc math/programming club. It looks like I will be able to put together a small audience for the project and need to start organizing soon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm looking for any speculation on how well this might go, &lt;strong&gt;from people who have some experience with Project Euler problems&lt;/strong&gt; (I have no such experience). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine five to six students meeting with pizza and a goal to complete a few Euler problems in a collaborative setting. What are some pitfalls or possible issues I should avoid? Is there really any room for collaboration at all? The students are very good and are unafraid of programming.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-07-02T01:45:36.800" Title="Project Euler for a Casual Collaborative Student Group" Tags="|technology-in-education|project-euler|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2266" PostTypeId="2" ParentId="2262" CreationDate="2014-05-12T23:55:06.077" Score="13" Body="&lt;p&gt;Disclaimer: I don't have any empirical studies, but do have substantial experience of the UK's hybrid between early and late transcendental functions in the 16-18 A-level Mathematics qualification.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;How calculus is introduced in the UK&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;(There are a number of different examining boards with different specifications, but the split between first year &quot;AS&quot; mathematics and second year &quot;A2&quot; mathematics is specified by the UK government, and recently (2014) the January exam session has been abolished, so that all students sit all of the year's papers in the summer at the end of the academic year.)&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is just the part of the content I feel is most relevant to your question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first year, students learn how to&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;differentiate and integrate polynomials &lt;/li&gt;&#xA;&lt;li&gt;deal with negative and fractional powers, &lt;/li&gt;&#xA;&lt;li&gt;use derivatives to find and classify stationary points, find equations of tangents and normals&lt;/li&gt;&#xA;&lt;li&gt;solve polynomial differential equations of the form dy/dx=p(x) with a simple boundary condition&lt;/li&gt;&#xA;&lt;li&gt;use calculus in the context of simple rate-of-change and maximisation problems&lt;/li&gt;&#xA;&lt;li&gt;use logs to base 10, 2, a etc to solve equations and manipulate expressions (no calculus)&lt;/li&gt;&#xA;&lt;li&gt;solving trigonometrical problems involving sin, cos, tan (no calculus)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the second year, students learn how to &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;use $e^x$ and ln x algebraically, as for logs to other bases in the previous year&lt;/li&gt;&#xA;&lt;li&gt;differentiate and integrate sin, cos, tan, $e^x$, ln x&lt;/li&gt;&#xA;&lt;li&gt;use the chain rule, product rule, quotient rule&lt;/li&gt;&#xA;&lt;li&gt;integrate by parts, by substitution and by inspection (typically fractions resulting in ln (v(x)))&lt;/li&gt;&#xA;&lt;li&gt;use cot, sec, cosec and associated trigonometrical identities including to facilitate calculus&lt;/li&gt;&#xA;&lt;li&gt;solve seperable first order differential equations&lt;/li&gt;&#xA;&lt;li&gt;use calculus with transendental functions to model growth and decay and rates of change&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note that in the first year, treatment of the derivative as a limit is encouraged by the specification but not assessed. An additional A-level qualification called Further Mathematics is available that includes the study of limits, but it is taken by a self-selecting minority of students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Individual schools and examining boards are free to stage this as they please within the year, some covering all of first year differentiation in the first term followed by all of first year integration in the second term, others splitting the terms as calculus with positive integer powers then calculus with fractional and negative powers.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Advantages and Disadvantages I have come across for this two year arrangement&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Restriction to polynomials in the first year&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The restriction to polynomials in the first year helps students become familiar with them, and is very helpful at first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However there are significant problems in the second year with over-generalisation of the method, particularly when x appears as a power. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Admittedly, over-generalisation is common elsewhere with linearity/the distributive law being a student favourite, seemingly unbounded in the contexts in which it can be misapplied (logs, squares, square roots, 3(xy) etc etc).] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we were to introduce how to differentiate sin, cos and $e^x$ in the first year, students might be less inclined to over-generalise, but there's no guarantee that that's the case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Introduction of transendental functions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Introducing these is in my view necessary to motivate the product rule and integration by parts, otherwise in the presence of polynomials alone it becomes a rather bizzare way of obtaining the correct result. It also provides a much richer and diverse set of examples to use for all the calculus techniques of the second year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I certainly use $x^2$ and polynomial products as very early examples of the product rule, because it's then clear to students that multiplying the derivative of two factors gives the wrong answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I believe it's important to use these functions as much as possible and in as many different combinations from as early as possible (we do so at our college as soon as we start second year work), precisely because they behave very differently to each other and to polynomials, so that the students have a better understanding of calculus which they would otherwise perceive as primarily about powers and coefficients.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Summary&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Were I at liberty to alter the split, I would introduce sin, cos and $e^x$ alongside polynomials and fractional/negative powers in the first year, including in simple differential equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe early diversity is helpful in &lt;em&gt;correctly&lt;/em&gt; generalising concepts from examples to principles, and the longer they are exposed to these beautiful and interesting functions the better they will be able to deal with them.&lt;/p&gt;&#xA;" OwnerUserId="1340" LastActivityDate="2014-05-12T23:55:06.077" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2267" PostTypeId="2" ParentId="712" CreationDate="2014-05-13T01:51:07.643" Score="66" Body="&lt;h2&gt;The evidence says no&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;What research I'm aware of is all about how giving any overall data about their own performance is actively harmful in promoting further learning. They learn considerably more from instruction about what to do differently in the absence of a grade or numerical score. To repeat; individual scores discourage learning! (Explanations for this correlation are about strong students relaxing and weak students giving up, all students concentrating on an emotional rather than academic response and ignoring any pointers.) The data says don't grade if you want more learning, instead instruct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience the earlier you do this correction/instruction-only feedback, the better. I've been delighted with the change in attitude its early use has resulted in, with students preferring to re-test than give up, and much higher pass rates (our course is assessed by a national externally marked test). Later in the course I've been able to mark tests with numbers, particularly near exams, but the foundational practice builds a culture of improvement post-testing rather than finality. &lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Note: To improve learning, provide instructions for how to improve work, without a mark&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Warning: the research shows that any grading nullifies correctional/improvement feedback - don't bother doing both. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clearly separate research findings from my own experience: research says the presence of any summative grade/percentage/score nullifies the effect of any formative advice/feedback/instruction, and that such feedback is far more effective at improving performance. My experience says that building an improvement culture early on by providing advice-only feedback with resits for underperformers can have a lasting impact on approach to learning, which allows them to ride out our numerical common test scores with their &quot;not there yet, let's fix it&quot; approach. We got the culture right first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to tell them &quot;You are not an arrow! If an arrow misses its target, it lies helplessly on the floor. You are a human - you have legs and a brain! If you missed the target, keep going. If you fell on your face, get up!&quot;&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Alternatives to grading&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;You absolutely need to check they've done the work and made a reasonable attempt at it. As some of our students pointed out in a candid after-the-exams conversation, &quot;If we think there's a &lt;em&gt;chance&lt;/em&gt; you won't check we did the work, we'll take that gamble, every time.&quot;&#xA;This checking only takes seconds per student. The good news is Dylan William, an advocate of using feedback to improve learning says feedback should be more work for the student than for you, but what are some alternatives to marking and grading?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are loads of ideas, but here are three.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Comment-only marking: a medal and a mission&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't mark everything, point out a medal and a mission: something they did well, one thing to improve. If I came in your class and listed everything you did wrong as an educator you would be dejected and find your list of failings to rectify overwhelming. If instead I praise something and then tell you the most important thing to improve, you can handle that, and may well actually do it! Students are the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Next you should...&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Quickly&lt;/em&gt; read through and assess where you think the student is at (eg strong &amp;amp; accurate, OK but error prone, clueless), and have a follow-up task or resource, but just set a few questions based on their ability: strong and accurate students get harder problems, clueless students get basic technique drill. If all students get the same number of follow up questions no-one feels penalised for what they did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Peer assessment&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Done well with thorough in-class training this can teach students to understand how they will be graded in the terminal exams, but more generally it spreads the mathematical authority around so that it's &quot;is this valid mathematics?&quot; as opposed to &quot;did I write what the teacher wanted?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disadvantage: it's hard to train students to give good feedback that's not binary or numerical, and as I say, the number often drowns the improvement message in emotional response.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;What is and isn't worth grading?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;In my view, homework is for learning - I expect students to seek help (phone a friend, ask the internet, watch a video, read a book) and find out how to answer questions if they don't know. As a result their performance on homework is a very poor indicator of what they can do unaided, so formally grading it is misleading for you and them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tests under test conditions are a better indicator of what you can do in an exam, and giving good feedback about how to improve (i.e. not a number, letter and judgement, but something they can work on) is important and relevant, so I don't object to marking tests, I'm just very aware that summative numerical feedback or grading can encourage students to stop learning, and have found success through avoiding it in the first part of the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's no reason to suppose I won't find holding out for longer even more effective in the future. (I won't hold out indefinitely, since one of my aims is that students understand how they will be assessed, and peer marking is the best way of teaching that. It's hard not to get numerical feedback out of a mark scheme!)&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;References/further reading&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.nctm.org/news/content.aspx?id=11474&quot;&gt;NCTM:Five “Key Strategies” for Effective Formative Assessment&lt;/a&gt; says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The research on feedback shows that much of the feedback that students receive has, at best, no impact on learning and can actually be counterproductive. Kluger and DeNisi (1996) reviewed more than three thousand research reports on the effects of feedback in schools, colleges, and workplaces and found that only 131 studies were scientifically rigorous. In 50 of these studies, feedback actually made people’s performance worse than it would have been without feedback. The principal feature of these studies was that feedback was, in the psychological jargon, “ego-involving.” In other words, the feedback focused attention on the person rather than on the quality of the work——for example, by giving scores, grades, or other forms of report that encouraged comparison with others. The studies where feedback was most effective were those in which the feedback told participants not just what to do to improve but also how to go about it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A nice introduction to comment only marking on a good blog &lt;a href=&quot;http://marksmathsstuff.wordpress.com/2012/07/15/stop-giving-grades-comment-only-marking/&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;'Inside the Black Box: Raising Standards Through Classroom Assessment' (Paul Black and Dylan Wiliam, Kings College, London, 1998).&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;People seem to reference &lt;em&gt;Butler, R. (1988). 'Enhancing and undermining intrinsic motivation: The effects of task-involving and ego-involving evaluation on interest and performance.' British Journal of Educational Psychology, 58,1-14.&lt;/em&gt; a lot but I haven't read it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Search terms: Assessment for Learning (AfL), comment-only marking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://www.journeytoexcellence.org.uk/resourcesandcpd/research/summaries/rsassessment.asp&quot;&gt;here&lt;/a&gt; for an intro to AfL generally.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Where's the evidence?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;(In response to comments challenging my statements about the existence or content of evidence.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a number of educational researchers who advocate what is referred to as evidence based education. They have high standards for what constitutes a good study (you can read their selection criteria in their work) and review a large number of studies, and concentrate on effect size. Dylan William is a good example as he specialises to mathematics a great deal, and I recall his description of the comment-only evidence but have read too many papers, talks etc by him to locate which one(s) make the reference. John Hattie is another evidence-based education expert, and I'll quote him for those seeking reassurance beyond my experience in class and my recollection of the literature I've read.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://www.columbia.edu/~mvp19/ETF/Feedback.pdf&quot;&gt;The Power of Feedback&lt;/a&gt; (Review of Educational Research March 2007, Vol. 77, No. 1, pp. 81-112 DOI: 10.3102/00346543029848)&#xA;John Hattie introduces some shorthand:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Thus, there is a distinction between feedback about the task (FT), about the processing of the task (FP), about self-regulation (FR), and about the self as a person (FS). We argue that FS is the least effective, FR and FP are powerful in terms of deep processing and mastery of tasks, and FT is powerful when the task information subsequently is useful for improving strategy processing or enhancing self-regulation (which it too rarely does).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;before going on to talk about where the evidence for comment-only marking, against grades, and the nullification of the effectiveness of comments by the presence of grades can be found:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The effectiveness of marks or written comments has also been investigated. There is considerable evidence that providing written comments (specific FT) is more effective than providing grades (Black &amp;amp; Wiliam, 1998; Crooks, 1988). In one of the early and influential studies, Page (1958) found that feedback in the form of short written comments rather than grades alone significantly improved the test performance of students in 74 classrooms (see also Cardelle &amp;amp; Como, 1981; Elawar &amp;amp; Como, 1985; McLaughlin, 1974). R. Butler (1987) demonstrated that grades can increase involvement, but they do not affect performance (relative to a no-FT condition). She also showed (R. Butler, 1988) that feedback through comments alone led to learning gains, whereas marks alone or comments accompanied by marks or giving praise did not. She claimed that such results called in question the whole classroom culture of marks, grades, gold stars, merit awards, competition rather than personal improvement. As will become a theme later in this article, feedback that mixes FS with FT is less effective than FT by itself.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The paper includes a considerable bibliography including of course the studies referenced in this extract, and if you'd like to delve further into the evidence than references you should download the paper and start investigating. &lt;/p&gt;&#xA;" OwnerUserId="1340" LastEditorUserId="1340" LastEditDate="2014-05-13T23:17:33.977" LastActivityDate="2014-05-13T23:17:33.977" CommentCount="24" ContentLicense="CC BY-SA 3.0" />
  <row Id="2268" PostTypeId="1" CreationDate="2014-05-13T09:26:05.873" Score="17" ViewCount="1077" Body="&lt;p&gt;I originally asked &lt;a href=&quot;https://mathoverflow.net/questions/165982/where-can-i-find-resources-for-creating-a-mathematics-bridge-course&quot;&gt;&lt;strong&gt;this&lt;/strong&gt;&lt;/a&gt; over at MathOverflow, but it was suggested I might also find some good answers over here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My university department is in the very early stages of developing a &quot;bridge course&quot; or &quot;introduction to proofs&quot; course, motivated by our lower-level courses not currently doing a good job of preparing our majors to go on to more advanced topics, which in this case means primarily abstract algebra and real analysis. The various service needs of those lower courses precludes making them more mathematically rigorous, so we feel the need to add something else. I know that other universities have instituted such courses, but it's not so easy to find information about them through brute force web searches. Therefore, I'm hoping to gather some information more directly. Ideally, I'd love to get some links to course syllabi and some suggestions for textbook choices, but I'd also very much welcome any other general resources concerning such courses, including any input concerning what's worked well (or what hasn't) at other universities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that right now the faculty preference for such a course would be for it to focus more on getting the students practice with actual advanced material than on simply being a litany of propositional logic and proof techniques, but, again, any information about what's worked elsewhere would be a big help to us.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="1456" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2019-08-09T06:48:15.333" Title="Where can I find resources for creating a mathematics “bridge course”?" Tags="|undergraduate-education|course-design|textbooks|" AnswerCount="6" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2269" PostTypeId="2" ParentId="2268" CreationDate="2014-05-13T11:07:47.963" Score="4" Body="&lt;p&gt;Going for the subject more than the question itself, I like Hammack's &lt;a href=&quot;http://www.people.vcu.edu/~rhammack/BookOfProof&quot; rel=&quot;nofollow&quot;&gt;&quot;Book of Proof&quot;&lt;/a&gt; for it's organization and explanation of how to prove stuff. Knuth, Larrabee and Roberts' &quot;Mathematical Writing&quot; (MAA 1989) (a partial version is &lt;a href=&quot;http://www-cs-faculty.stanford.edu/~knuth/klr.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;) is a must. Perhaps Aigner and Ziegler's &quot;Proofs from THE BOOK&quot; (Springer, third edition 2003) would be a good complement.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-13T11:07:47.963" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2270" PostTypeId="2" ParentId="2246" CreationDate="2014-05-13T11:11:12.937" Score="6" Body="&lt;p&gt;There is a hybrid approach which is possible:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;teach $\frac{df}{dx}$ as traditionally is done. Motivate geometrically by secant lines and physically my instantaneous velocity. Emphasize that $x \mapsto \frac{df}{dx}$ is a new &lt;strong&gt;function&lt;/strong&gt; defined pointwise by the derivative. &lt;/li&gt;&#xA;&lt;li&gt;introduce differentials as they naturally appear in related rates problems. For example, if $A = lw$ and $l$ and $w$ are both functions of time then the chain-rule and product-rule for functions of &lt;strong&gt;time&lt;/strong&gt; naturally yield:&#xA;$$ \frac{dA}{dt} = \frac{dl}{dt}w+l\frac{dw}{dt} \qquad (\ \star \ ) .$$&#xA;As a good approximation for small durations of time $\triangle t$ we have $\frac{dA}{dt}\triangle t = \triangle A$. This approximation can be explicitly understood as it connects with the limit definition given earlier and I try to use $\triangle t$ to denote a finite change in time. Generally, $\triangle f$ denotes some &lt;strong&gt;finite&lt;/strong&gt; change. On the other hand, if all the variables involved are time-dependent and we consider a process where the duration of time $\triangle t$ is very small then I say as a heuristic we can think of $\triangle t \rightarrow dt$. Moreover, we may multiply by $dt$ to obtain the relation between the &lt;strong&gt;infinitesimal changes&lt;/strong&gt; in $A,l,w$ for my token example:&#xA;$$ dA = wdl+ldw $$&#xA;My interpretation for them goes something like this: when we write $dA = wdl+ldw$ we intend to approximate the change in $A$ as it related to the change and values of $l,w$. This relation is a short-hand for the related-rates equation $\star$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then, later in integration applications, I advocate the method once more. I call it the infinitesimal method. I tell them it is a notational convenience which is justified by more cumbersome expressions which make problems harder to work. Yes, we could set up an explicit Riemann sum for area, volume or work problems but it is far more efficient to calculate formally $dA,dV$ or $dW$ and express each in terms of one variable which parametrizes the situation. I include in my notes discussions of how we go back and forth from the infinitesimal to the currently-traditional function-based framework. The technique of $u$-substitution is far easier to grasp in the formal manipulations of the method of differentials. However, if you want to prove anything, the method of differentials can put you in a bad place. That said, I think there is room for both approaches. The question is what is the task at hand? If the task is foundations and geometric intuition then I tend to think differentials are not so helpful. However, if the task is problem-solving, especially involving integration as a continuous sum, then differentials are quite appropriate. The students can understand them as a convenient notation for a limit-based formulation. There is no need to choose, we can have the best of both worlds.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-13T11:11:12.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2271" PostTypeId="2" ParentId="2268" CreationDate="2014-05-13T12:42:10.853" Score="8" Body="&lt;p&gt;I taught such a course about 5 years ago. In principle you are ready to attempt such a course after calculus I, however, for reasons of crowd-control etc.. we make calculus II the prerequisite. Honestly, this course could follow college algebra for the right student and that would be ideal in a more perfect world. You can see my meager course notes at:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.supermath.info/math200.pdf&quot;&gt;http://www.supermath.info/math200.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and my course webpage &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.supermath.info/math200s09.html&quot;&gt;http://www.supermath.info/math200s09.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm certain there are better resources out there. This much I would like to see stressed in such a course:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;methods of proof. How to write math both formally and informally. (with and without the shorthands $\forall, \in, ...$&lt;/li&gt;&#xA;&lt;li&gt;properties of functions&lt;/li&gt;&#xA;&lt;li&gt;definition of equivalence relation, partitions of sets&lt;/li&gt;&#xA;&lt;li&gt;math in the modern age, what's the big picture? math as an active, creative enterprise&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For our particular program, I think it is also a good place to work out the particulars of modular arithmetic, but, in another institution perhaps the $\epsilon,\delta$ proofs would be a good fit here. In any event, enjoy your task, it's a great problem to solve!&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-13T12:42:10.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2272" PostTypeId="2" ParentId="2268" CreationDate="2014-05-13T14:22:27.093" Score="6" Body="&lt;p&gt;Highly recommended: Robert Talbert's blog, chronicling his adventures in an inverted intro-to-proof course: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://chronicle.com/blognetwork/castingoutnines/2013/01/22/inside-the-inverted-proofs-class-why-i-did-it/&quot; rel=&quot;noreferrer&quot;&gt;blog&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=UuETUEJo0rk&amp;amp;list=PL2419488168AE7001&quot; rel=&quot;noreferrer&quot;&gt;videos&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Extremely well-written, &lt;a href=&quot;http://scholarworks.gvsu.edu/books/7/&quot; rel=&quot;noreferrer&quot;&gt;free textbook&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2014-05-13T14:22:27.093" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2273" PostTypeId="1" AcceptedAnswerId="2290" CreationDate="2014-05-13T14:25:36.430" Score="6" ViewCount="188" Body="&lt;p&gt;Should material covered in the class period prior to a midterm be included on the exam?  My usual instinct is no---students can't (and shouldn't) be expected to have absorbed material until they've had a chance to think about it and work some problems.  However it's (understandably) difficult for students to concentrate on learning new material when they're about to be tested on something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I have the time, I like to use a review class as a buffer, but I don't always have the time in the course schedule.  What are good ways to handle material in the class period prior to a midterm?&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-05-14T14:40:28.963" Title="Should midterms exams include the most recently covered material?" Tags="|exams|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2274" PostTypeId="2" ParentId="2268" CreationDate="2014-05-13T16:59:30.557" Score="3" Body="&lt;p&gt;When I was a starting graduate student (note: in computer science), I found the following text book very helpful in filling in the gaps in my mathematical knowledge that you describe (well, the first part you describe: abstract algebra and proofs -- not real analysis).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Bloch, Ethan. Proofs and Fundamentals: A First Course in Abstract Mathematics, Springer 2011.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is organized into 8 chapters (the last two of which are &quot;extras&quot;).  The first six are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Informal Logic&lt;/li&gt;&#xA;&lt;li&gt;Strategies for Proofs&lt;/li&gt;&#xA;&lt;li&gt;Sets&lt;/li&gt;&#xA;&lt;li&gt;Functions&lt;/li&gt;&#xA;&lt;li&gt;Relations&lt;/li&gt;&#xA;&lt;li&gt;Finite Sets and Infinite Sets&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The preface section entitled &quot;To The Student&quot; contains a great summary of the author's goal:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This book is designed to bridge the large conceptual gap between computational courses such as calculus, usually taken by first- and second-year college students, and more theoretical courses such as linear algebra, abstract algebra and real analysis, which feature rigorous definitions and proofs of a type not usually found in calculus and lower-level course.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He goes on for another page or so, essentially describing a student with my background.  The preface also includes a similar section entitled &quot;To The Instructor&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is an opposing set of pedagogical imperatives when teach a transition course of the kind for which this text is designed:  On the one hand, students often need assistance making the transition from computational mathematics to abstract mathematics, and as such it is important not to jump straight into water that is too deep.  On the other hand, the only way to learn to write rigorous proofs is to write rigorous proofs; shielding students from rigor of the type mathematicians use will only ensure that they will not learn how to do mathematics properly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The entire preface (4 or 5 pages) contains a good description of the design and goals of the text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oh, and the short set of notes compiled from Knuth's seminar on Mathematical Writing mentioned in vonbrand's answer was very helpful too (but it is about &lt;em&gt;writing&lt;/em&gt; mathematics, not necessarily about mathematics itself).&lt;/p&gt;&#xA;" OwnerUserId="1460" LastActivityDate="2014-05-13T16:59:30.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2275" PostTypeId="2" ParentId="2273" CreationDate="2014-05-13T17:30:11.183" Score="3" Body="&lt;p&gt;I use the last class before each exam as a rapid review (more a bullet-point reminder of the subject covered), and solving any questions students might have. As questions merit, I expand on them and sometimes solve related/similar problems.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-13T17:30:11.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2276" PostTypeId="2" ParentId="1234" CreationDate="2014-05-13T18:09:22.077" Score="1" Body="&lt;p&gt;Logic 2010 (&lt;a href=&quot;http://logiclx.humnet.ucla.edu/&quot; rel=&quot;nofollow&quot;&gt;http://logiclx.humnet.ucla.edu/&lt;/a&gt;) seems to be what I am looking for, though I am not sure because I have not seen it in use.  I am going to a school where it is used next year, so I will ask my classmates to see it and then I can give more details.&lt;/p&gt;&#xA;" OwnerUserId="612" LastActivityDate="2014-05-13T18:09:22.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2277" PostTypeId="2" ParentId="2145" CreationDate="2014-05-13T18:30:08.857" Score="5" Body="&lt;p&gt;I am just now ending a semester in which I allowed students to do Project Euler problems for credit. There was a slush of math-y things to do that would garner credit, so the problems weren't required. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had a persistent problem with cheating, which is unusual at my university. There are a number of people (jerks!) that have posted their solutions in any reasonable programming language. Google &quot;euler python 38&quot; and you find several solutions to Problem 38, with Python code. Some have a detailed discussion of the problem and methods, which has some value, but several just have code. I consider such activity to be overtly evil and directly counter-productive to our future as a species.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I love the problems, and wish you success. With guidance and discussion, perhaps you will have a better experience.&lt;/p&gt;&#xA;" OwnerUserId="87" LastActivityDate="2014-05-13T18:30:08.857" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2278" PostTypeId="1" AcceptedAnswerId="2287" CreationDate="2014-05-13T18:34:26.500" Score="4" ViewCount="186" Body="&lt;p&gt;I'm asking this question mostly looking for advice as a student, but I also thought this could be an interesting discussion subject for this SE comunnity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A word of explanation first: I'm a first year undergraduate student of mathematics in Poland, and the &quot;program&quot; is organised so that before the start of the 2nd year one chooses in what &quot;aspect&quot; of mathematics they want to specialise. There's stuff like theoretical, applied, mathematics in economy and some other things I'm not really considering. &#xA;Based on the specialisation chosen, you're going to be attending a certain set of classes until the end of the undergraduate stage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, theoretical mathematics is definitely the most interesting one for me, but at this university it's also by far the hardest specialisation. In previous years, it got about ~12 of the best students total, out of over 100 in other specialisations and the classes one takes after choosing are indeed very demanding.&#xA;The question is, how beneficial would it really be to choose something this demanding. The way I see it, it certainly has a lot of benefits, attending difficult classes with the best students in the given year would certainly be very educating. On the other hand, it'd be very time-consuming, leaving little place for one to keep learning additional things, outside of the standard curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How much does a student actually benefit from attending classes at a significantly higher level than normal, at the expense of their freedom in choosing what branches of mathematics they could study on their own(which would be possible if an easier and less time-consuming specialisation was chosen)?&lt;/p&gt;&#xA;" OwnerUserId="1421" LastActivityDate="2014-05-14T12:54:00.097" Title="Benefits and cons of choosing a very demanding specialization" Tags="|undergraduate-education|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2279" PostTypeId="1" AcceptedAnswerId="2286" CreationDate="2014-05-13T19:13:26.700" Score="32" ViewCount="46661" Body="&lt;p&gt;I came across many video tutorials on youtube regarding mathematics.&#xA;I found &lt;a href=&quot;https://www.youtube.com/watch?v=uD4lJlfPn-0&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; video amazingly simple to understand for students.&#xA;I want to know about the tool/software used for it or similar videos.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Screenshots:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/unCr2.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/GhrOg.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Youtube Video: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;copy &lt;code&gt;https://www.youtube.com/watch?v=uD4lJlfPn-0&lt;/code&gt; or &lt;a href=&quot;https://www.youtube.com/watch?v=uD4lJlfPn-0&quot; rel=&quot;noreferrer&quot;&gt;click here&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;please feel free to provide your suggestions which may have better options&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="1461" LastEditorUserId="1892" LastEditDate="2014-07-10T00:47:04.197" LastActivityDate="2023-10-12T21:42:17.093" Title="Software to create video tutorial of mathematics topics" Tags="|technology-in-education|tutoring|software|" AnswerCount="7" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2280" PostTypeId="2" ParentId="2253" CreationDate="2014-05-13T19:30:42.377" Score="4" Body="&lt;p&gt;Using Webworks allows for some more meaningful data distribution. I have given students a scatterplot showing (# HW problems solved, # exam score), with an obviously positive correlation. Then I point out the lower right portion (students who did a lot of HW and did poorly on exam) and the upper left portion (students who did little HW, but did well on exam) are nearly empty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have no data that this actually helped the students evolve their behavior, but at least I feel like I'm sending a useful message.&lt;/p&gt;&#xA;" OwnerUserId="87" LastActivityDate="2014-05-13T19:30:42.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2281" PostTypeId="1" AcceptedAnswerId="2297" CreationDate="2014-05-13T20:49:06.907" Score="6" ViewCount="502" Body="&lt;p&gt;I see two options in teaching mathematics to undergraduate students in social sciences. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;strong&gt;I can write examples and solve them in class, and ask the students to solve them while I do.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This encourages students to participate and to solve questions.  It obliges them to try. However this option needs more time and effort to control the class environment. Also I have to answer students' simple and/or needless questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;I can put homework problems on the internet and ask students to solve them at home&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This saves me time, though I cannot be sure whether the students attempt the questions or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which method do you prefer?&lt;/p&gt;&#xA;" OwnerUserId="79" LastEditorDisplayName="user173" LastEditDate="2014-05-14T21:00:57.150" LastActivityDate="2014-05-14T21:00:57.150" Title="Encourage Students to solve problems at class" Tags="|undergraduate-education|student-motivation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2282" PostTypeId="2" ParentId="2281" CreationDate="2014-05-13T21:08:26.720" Score="3" Body="&lt;p&gt;You have to make sure &lt;em&gt;they&lt;/em&gt; solve the problems. By suggestion of a TA, we have tutoring sessions once a week with three problems: one solved by the TA, requesting input from the class; one solved by the class, discussion between them encouraged (and the TA at hand for questions and ordering the discussion), and one to be done individually and handed in, graded as &quot;some relation to the subject&quot;/&quot;not relevant&quot;. Those grades make 5% of the course to encourage participation. Did it once, seems to have had a very positive impact.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-13T21:08:26.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2283" PostTypeId="2" ParentId="2278" CreationDate="2014-05-13T21:15:38.957" Score="2" Body="&lt;p&gt;Very hard to answer, as this depends sensibly on your interests and aptitudes. If you are &lt;em&gt;really&lt;/em&gt; interested in the &quot;demanding&quot; track, by all means try it. If you don't, you could end up regretting it all your life. It can't be that you can't switch in the middle if you find out it isn't for you. And I doubt any courses you take that aren't needed in the end will be completely useless...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-13T21:15:38.957" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2284" PostTypeId="2" ParentId="2253" CreationDate="2014-05-13T21:20:41.883" Score="5" Body="&lt;p&gt;I say absolutely nothing about the statistics for the exam unless they specifically ask. All that info is available to them on Blackboard anyway. (Well, the mean is there, and I think the min and max are there too.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, I prefer to talk about what the results of the exam &lt;em&gt;tell me&lt;/em&gt;. Saying that the average was a 72 and the standard deviation was a 20 will not convey much information to students (unless it's a statistics class). But this might: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Based on the results, it looks like we are doing OK but have some work to do. Specifically there was a pretty wide range of scores, so some people are doing very well while others are doing very poorly. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If I do a item-by-item breakdown, I will also focus on overall themes, like this: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Most of us did fine on the basic computations. But on the conceptual multiple choice items, the scores were not as good. This tells me we need to work more on conceptual understanding, so expect more clicker questions in upcoming classes and less time spent on basic examples. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Pretty generic, but importantly it &lt;strong&gt;gets the focus off numerical scores&lt;/strong&gt; and therefore, hopefully, downplays the point-scoring mentality that students are so prone to fall into. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally speaking I want to view exams as assessments in the true sense of the word -- where &quot;assess&quot; literally means to &quot;sit down beside&quot;. My role as the instructor is to take the raw results of the exam, process those into descriptions of strengths and weaknesses for students (both corporately as a class and individually) and finally into actionable information that tells me what to do in class and tells students what to do in class and at home. That takes a lot more work than just saying the average, but it's important and IMO the right thing to do. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-05-13T21:20:41.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2285" PostTypeId="2" ParentId="712" CreationDate="2014-05-13T22:45:26.030" Score="5" Body="&lt;p&gt;There seems to be a huge literature on this sort of thing, but most of it seems to be about younger students, whereas this question is about college students taking calculus. For a summary of the research on children, see &lt;a href=&quot;http://www.nctm.org/news/content.aspx?id=13814&quot; rel=&quot;noreferrer&quot;&gt;this NCTM report&lt;/a&gt;. The research seems to be politicized and inconclusive, and there seem to be many mutually contradictory findings, which may have been influenced by styles and fads in education, e.g., many math and science teachers re-emphasized homework in the Sputnik era.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.johnsasser.com/pdf/article01.pdf&quot; rel=&quot;noreferrer&quot;&gt;This paper by Sasser&lt;/a&gt; describes a study on college students in a remedial math class. Students were randomly assigned to two groups, who received identical instruction in the same room at the same time. One group was assigned graded homework and one was not. They were evaluated using the same pre- and post-test. The study showed a small but significant improvement in performance with graded homework. The paper also has some discussion of the previous literature on the topic.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-13T22:45:26.030" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2286" PostTypeId="2" ParentId="2279" CreationDate="2014-05-13T22:58:43.647" Score="20" Body="&lt;p&gt;I've created roughly 40 videos for an online course and a flipped course in biology, so I've got some experience making videos. I researched the maker of your videos, and he deliberately does NOT reveal his process. So I don't &lt;em&gt;know&lt;/em&gt; if this is his method, but I'm comfortable that this would produce something very similar. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Supplies:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Computer (I use a Windows 7 machine, but Mac should work) with a decent mic&lt;/li&gt;&#xA;&lt;li&gt;Presentation software (I use PowerPoint) that has a plain &quot;graph paper&quot; background option&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Wacom-style tablet, about $200&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Screencast software like &lt;a href=&quot;http://www.techsmith.com/camtasia.html&quot;&gt;Camtasia Studio&lt;/a&gt;, about $200&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Calculator emulator software. TI makes a $160 one, but &lt;a href=&quot;http://www.studentcalculators.co.uk/acatalog/copy_of_software.html&quot;&gt;this one&lt;/a&gt; looks like the one your YouTube guy uses.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then, his workflow is to build up a powerpoint show with some drawings already made and set to appear when he clicks. He starts the screen cast, narrates, and draws in where needed. He can alt-tab the calculator onto the screen when he wants it (or set it as a picture-in-picture? I've never used one). &#xA;After completing the recording, some writing can be speeded up as long as you aren't talking during that part. Export, upload to YouTube.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All you need is money, time, some drawing skill, and the ability to be clear and concise and brisk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have I missed any steps?&lt;/p&gt;&#xA;" OwnerUserId="1207" LastEditorUserId="1207" LastEditDate="2014-05-14T14:53:14.920" LastActivityDate="2014-05-14T14:53:14.920" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2287" PostTypeId="2" ParentId="2278" CreationDate="2014-05-14T12:54:00.097" Score="3" Body="&lt;p&gt;Warning: personal preference-based answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tend to do much better at challenging things that I enjoy than easy things I find uninteresting, and as an undergraduate/postgraduate student I wouldn't have dreamed of picking something dull because it was easy - I would have been seriously demotivated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I'd rather spend three hours writing a program that works something out automatically than two hours working it out myself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My job is difficult and stressful, but it's fun, and there's no way I would swap it for something routine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;University is a great time of life to devote yourself wholeheartedly to learning; later on you will not be so free. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You're in a dilemma between doing the interesting thing as the main thing or when you have spare time. I'd choose it as the main thing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The more you can find out about what each route will be like, the better you will be placed to make this decision.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Caveats:&lt;/strong&gt; I'm not you, you might have different priorities in life; for example, I find many people prioritise money higher than I do. You have to make your own decision - it's a mistake to not work out what you believe is best for you.&lt;/p&gt;&#xA;" OwnerUserId="1340" LastActivityDate="2014-05-14T12:54:00.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2288" PostTypeId="1" AcceptedAnswerId="2291" CreationDate="2014-05-14T13:57:05.713" Score="25" ViewCount="4707" Body="&lt;p&gt;It's time to write exams, and when writing in committee we often discover differences in usage between various instructors.  Here's an example I noticed today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the proper verb to use in a question of the form: &quot;____ the following integrals&quot;?  I've seen &lt;em&gt;evaluate&lt;/em&gt;, &lt;em&gt;compute&lt;/em&gt;, &lt;em&gt;solve&lt;/em&gt;, and others.&lt;/p&gt;&#xA;" OwnerUserId="446" LastEditorUserId="77" LastEditDate="2015-09-03T05:17:45.423" LastActivityDate="2015-09-03T05:17:45.423" Title="What is the proper verb for &quot;doing&quot; an integral?" Tags="|calculus|test-design|terminology|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2289" PostTypeId="2" ParentId="2288" CreationDate="2014-05-14T14:22:54.677" Score="15" Body="&lt;p&gt;I would avoid the verb &lt;em&gt;solve&lt;/em&gt; as I reserve this for things like equations, inequalities and problems. An integral is equal to a number or a function, so verbs like &lt;em&gt;find&lt;/em&gt;, &lt;em&gt;evaluate&lt;/em&gt; etc are more appropriate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd use &lt;em&gt;compute&lt;/em&gt; only for numerical integration methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;evaluate&lt;/em&gt; and &lt;em&gt;find&lt;/em&gt; are the two verbs that are used in textbooks and exams that I've come across. I think &lt;em&gt;evaluate&lt;/em&gt; is better for a definite integral because the result is a number, and &lt;em&gt;find&lt;/em&gt; is OK for both, eg &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Find $\int \sin x e^{\cos x}dx$&lt;/em&gt;&lt;br&gt;&#xA;or&lt;br&gt;&#xA;&lt;em&gt;Find the value of $\int _1^\infty \frac 1 {x^2} dx$&lt;/em&gt;&lt;br&gt;&#xA;or&lt;br&gt;&#xA;&lt;em&gt;Evaluate $\int _2^5 x(2x+1)\sin x dx$&lt;/em&gt;  &lt;/p&gt;&#xA;" OwnerUserId="1340" LastEditorUserId="1340" LastEditDate="2014-05-14T15:15:00.627" LastActivityDate="2014-05-14T15:15:00.627" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2290" PostTypeId="2" ParentId="2273" CreationDate="2014-05-14T14:40:28.963" Score="2" Body="&lt;p&gt;One option is to cover the most recent material in a very superficial level, with very explicit instructions in the text of the question rather than problem solving, so they don't need to be proficient, being clear in advance that that's what will happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could even go to the extent of giving them a sample question saying it's the question on the midterm with some numbers or a function changed, but this still risks focussing their learning too sharply on the new content, which is bad news in the long term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;~ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can possibly spare the time, the review class is beneficial since it allows the students to focus on mastery across the content instead of being distracted by trying to figure out the new stuff at whatever level. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They may be prepared to work faster with you afterwards in compensation if you make it clear that you sacrificed the time to help them out with the midterm - you can explain it suits both them and you, since they want good credit and you both want good proficiency.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(One option is to negotiate with the class - the deal is they have to do an extra preparatory piece of work between the midterms and the next class rather than have the weekend or whatever off, to cover material you've sacrificed to help them out. The problem with this is you may have to negotiate this quite some time before they're thinking seriously about revising, since you can't edit the midterm at short notice!)&lt;/p&gt;&#xA;" OwnerUserId="1340" LastActivityDate="2014-05-14T14:40:28.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2291" PostTypeId="2" ParentId="2288" CreationDate="2014-05-14T15:10:02.097" Score="27" Body="&lt;p&gt;I'm no native English speaker, but you can tackle that question from the mathematical point of view as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best verb depends on how you view the nature of definite and indefinite integrals.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Operators/Functionals&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Indefinite integrals are operators mapping functions to a set of functions or function (as representative of the equivalence class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definite integrals are operators mapping functions to numbers for given intervalls or operators mapping numbers (as one boundary) to numbers for given function and other boundary. In that case, it's numbervalued function of numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consequently, you should use &lt;code&gt;evaluate&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Items of a special calculus named &quot;calculus&quot;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The theory &lt;a href=&quot;http://en.wikipedia.org/wiki/Calculus&quot;&gt;&quot;calculus&quot;&lt;/a&gt; is a special &lt;a href=&quot;http://en.wikipedia.org/wiki/Calculus_%28disambiguation%29&quot;&gt;calculus&lt;/a&gt;. Consequently, you should use &lt;code&gt;calculate&lt;/code&gt;, even for indefinite integrals.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Problems&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;If you see integrals as problems, which non-computable integrals or integrals difficult to compute are indeed, then you should use &lt;code&gt;find&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Differential equations&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Regard the differential equation $\frac{\mathrm{d}F(x)}{\mathrm{d}x}=f(x)$ to be solved for $F(x)$. Its solution is $F(x)=\int f(x)\,\mathrm{d}x$ (or $\{F(x)\}=\int f(x)\,\mathrm{d}x$).&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;As solution&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;If you regard the integral as that solution, you should use the verb associated to solutions of algebraic equations: &lt;code&gt;find&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;As equation itself&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;If you regard the integral as the equation itself, you should use the verb associated to (algebraic) equations: &lt;code&gt;solve&lt;/code&gt;.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="439" LastEditDate="2014-05-15T20:11:06.560" LastActivityDate="2014-05-15T20:11:06.560" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2292" PostTypeId="2" ParentId="2288" CreationDate="2014-05-14T15:36:28.113" Score="11" Body="&lt;p&gt;I usually phrase it as &quot;Determine $\int x^2\ dx$&quot; or &quot;Determine $\int_1^3 x^2\ dx$&quot;. This way&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) it doesn't tip them off to what type of answer they should arrive at, and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) it allows them to read the symbol $\int$ as either &quot;integral&quot; or (better in my opinion) &quot;antiderivative&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For completeness, in some problems I write &quot;Set-up, but do not evaluate, the integral that computes some geometrical/physical thing.&quot; But when I actually want them to perform the integration (which is another phrase I use colloquially but never as instructions) and arrive at an answer, I write &quot;Use integration to find the geometrical/physical thing.&quot;&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-05-14T15:36:28.113" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2293" PostTypeId="1" AcceptedAnswerId="2294" CreationDate="2014-05-14T15:37:17.997" Score="4" ViewCount="3642" Body="&lt;p&gt;I was talking to some of my students yesterday about their calculus courses, and several mentioned that they use the web site CalcChat.com. The site's home page says, &quot;Step-by-step solutions to odd-numbered exercises.&quot; One student joked that she spent so much time on calcchat that she'd set it as the homepage of her browser. Can anyone provide any insight into how this site works? From casually clicking around on it, it looks like it will give students solutions to odd-numbered problems from a whole bunch of current math books, and they don't have to pay or register an account. There also appear to be provisions for talking to a tutor, although when I visited this morning, it said, &quot;calc chat assistant: offline.&quot; What is their business model?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is the site basically offering the equivalent of a student solutions manual, which the publisher would like to charge $100 for? Do the solutions appear to be the original work of the people who run the site, or are they cut and pasted from the publishers' solutions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone offer any insights into how their students use the site, for good or for evil? I guess the common practice among textbook authors is to put answers to odd problems in the back of the book. Do many people here assign the odd problems for credit and check that their students have actually worked out the answers? If so, do they find themselves reading the same CalcChat solution over and over? In your experience, how does this compare or compete in terms of education with printed student solutions manuals from the publisher? How does it compare with Wolfram/Mathematica or other symbolic math software that can show the steps it used to get the solution?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/535/students-use-wolframalpha-can-we-change-calculus-instruction-to-exploit-it-whil?rq=1&quot;&gt;Students use WolframAlpha. Can we change calculus instruction to exploit it while discouraging &amp;#39;cheating&amp;#39;?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-14T16:15:49.340" Title="How does CalcChat work, and how can students who use it be encouraged to do so constructively?" Tags="|undergraduate-education|calculus|technology-in-education|homework|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2294" PostTypeId="2" ParentId="2293" CreationDate="2014-05-14T16:15:49.340" Score="4" Body="&lt;p&gt;The site only offers solutions to Ron Larson textbooks, and is run and maintained by Larson Texts, Inc.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is their business model?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;By providing an engaging, socially-connected, friendly and helpful website they aim to make Ron Larson textbooks much more popular. They are &lt;a href=&quot;http://calcchat.com/advertise.php&quot; rel=&quot;nofollow&quot;&gt;offering advertising&lt;/a&gt;, but they don't seem to have gotten much yet.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There also appear to be provisions for talking to a tutor (calc chat assistant), they don't have to pay or register an account&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They also have a cheery and positive twitter and facebook presence. It's important to them that students like the site and find it helpful, and even more so that teachers &lt;a href=&quot;http://calcchat.com/teacher_benefits.php&quot; rel=&quot;nofollow&quot;&gt;feel they get a free TA&lt;/a&gt;. I strongly suspect that the chat assistants and twitter operatives are undergraduates and not Rod Larson himself! It'll be entertaining work for a able mathematics students who can work from home, socially interact whilst solving easy problems - you probably don't need to pay as much as some places to get employees for this due to the pleasant nature of the work.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do the solutions appear to be the original work of the people who run the site, or are they cut and pasted from the publishers' solutions?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They &lt;em&gt;are&lt;/em&gt; the (single) publisher's solutions, updated to the social networking-aware internet accessibility age.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is the site basically offering the equivalent of a student solutions manual, which the publisher would like to charge $100 for? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes. It has just been published in a different format and with additional support - the content will have been generated previously.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They perhaps feel that the quantity of physical sales of the solutions manual was low enough that investing most of that money and more in welcoming, friendly and helpful free-access would drive enough extra sales to increase profits. Alternatively, Ron Larson is just a good guy who wants to help students learn mathematics as much as possible and can afford the hit.&lt;/p&gt;&#xA;" OwnerUserId="1340" LastActivityDate="2014-05-14T16:15:49.340" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2295" PostTypeId="5" CreationDate="2014-05-14T16:17:17.487" Score="0" Body="&lt;p&gt;&quot;Classroom interaction&quot; is the focus on teacher-student or student-student interactions. These  include discussion and analysis of traditional patterns of interaction (like IRE - inquiry-response-evaluation) but may also include the role of technology in mediating the interaction of participants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For clarity, and to contrast with &quot;classroom management,&quot; classroom interaction is not a concern about maintaining order in the classroom, but rather about what is contained in specific interactions between classroom participants, whether they be teacher-driven or student-driven.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="39" LastEditDate="2014-06-11T23:18:01.780" LastActivityDate="2014-06-11T23:18:01.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2296" PostTypeId="4" CreationDate="2014-05-14T16:17:17.487" Score="0" Body="A focus on the quality, content, or consequences of patterns of action that occur among classroom participants." OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-05-14T18:01:40.510" LastActivityDate="2014-05-14T18:01:40.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2297" PostTypeId="2" ParentId="2281" CreationDate="2014-05-14T18:18:33.223" Score="1" Body="&lt;p&gt;I believe that the two options are almost never mutually exclusive. I expect students to do HW problems on their own &lt;strong&gt;and&lt;/strong&gt; solve problems in lecture. However, to choose the best option for your class, the key first step is to ask yourself &quot;Why am I asking the students to solve problems at all?&quot; There are tons of benefits to having students solve problems, and you should be able to answer this questions with the reasons that are most important to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some potential reasons (there are countless others)&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You want feedback as to how much they are understanding the material to allow you to adjust your how you proceed with the class accordingly. This is pretty broad, it can mean how do I proceed in the next 10 minutes of lecture, or how do I proceed 10 lectures from now.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You want to give students direct feedback on their problem solving skills, attitudes, and techniques, in real time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You want them to sharpen their reasoning, and problem solving skills through practice.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You want them to improve their understanding of the material through practice&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You want them to be able to apply their knowledge of the material to new situations&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;When I give problems in lecture, they tend to be of a different flavor. I am specifically targeting reasons (1-2). I use both hard and easy problems to address (1). For example, If I do a basic computation on the board I might give the class 30 sec to start the problem (or finish the problem) on paper before I go over it. This gives the struggling students a chance to figure out what they are confused about, but also doesn't give the stronger students enough time to get board and distracted. The more time you pause for student self reflection the more questions they ask.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I also ask questions that probe for deeper understanding. For example you can ask a tricky true or false question and poll the class, via clickers or via hand raising. This gives them a chance to take a intuitive &quot;guess&quot; at the answer, and then &quot;solve&quot; the problem afterwards. Often students jump into problems haphazardly, and this technique reinforces that students should be thinking about what they expect is true or false before they do the problem. It helps improve their &quot;mathematical intuition.&quot; I like to pick questions that are hard enough that at least half the class gets it wrong (to minimize embarrassment choosing a wrong answer). After polling the class, I pair up students that think the statement is true and the ones that think it is false. Other times I pair up trues with trues and falses with falses; interestingly, this often leads to people changing their mind as well. In each case, I poll the class again. Then we discuss the intricacies behind the question. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example problem is a great way to start a lecture. You can tell how well the students understood the reading or HW this way, and adjust your lecture on the fly based on their response. This will proactively prevent the blank stare you get when you say &quot;Any questions?&quot; 10 minutes into your lecture. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tend to leave basic computation questions for HW or for a student lead TA tutoring/discussion section, and use lecture for applied and theoretical questions. In the HW I focus on 4, but also have problems targeting 3 and 5. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;summary: In lecture I tend to focus on two types of problems (1) the ones that give me feedback in real time, which I then use to adjust the pace, content and style of my lecture, and (2) the ones that allow me to modify student behavior and attitudes towards mathematics. I leave problems that are mainly for practice for time outside of lecture, and also include problems that reinforce the student habits I emphasize in class.&lt;/p&gt;&#xA;" OwnerUserId="1387" LastEditorUserId="1387" LastEditDate="2014-05-14T18:51:47.433" LastActivityDate="2014-05-14T18:51:47.433" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2298" PostTypeId="2" ParentId="2253" CreationDate="2014-05-14T22:13:50.170" Score="4" Body="&lt;p&gt;In addition to other good answers: on one hand, ideally students should not care how they compare against the other people who happen to be in the class, but should care about whether they've made the progress that was the presumed goal of the course. Education! Scholarship! Under that hypothesis, and under the assumption that the instructor is experienced in interpreting test performance, there would not need to be any &quot;curving&quot;, much less reporting on how other people did, which is irrelevant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In reality, in my experience, both positive and negative effects can be achieved by letting on some (privacy-preserving...) statistics about a given class's performance. People who feel they're doing badly but are outperforming many others can be cheered up. People who think they're doing fine but are underperforming... can be implicitly admonished. People who don't want to over-exert themselves can game the system. People who somehow get the idea that the instructor is unreasonably harsh may see (or not) that their standards are deficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apart from the obvious distinction between required/weeder courses and electives, whenever I have to assign &quot;meaningful grades&quot; (meaning giving signals about adequate command of material, and whether requisite thresh-holds are met, or not), I try to obtain the best of all worlds by declaring that there are absolute thresh-holds, based on much past experience (so that everyone might get an &quot;A&quot;, or a &quot;C&quot;, or...), while also reporting rough statistics on actual scores... with commentary on my happiness with most peoples' good work... and/or disappointment with the general level of effort, thus, supposedly, squelching the subliminally-game-the-system-and-themselves reactions to rationalize far too much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, there is a fraction of the population that worries too much, suffers from impostor syndrome, etc. Another fraction is confident that they're being short-changed. Another fraction is oblivious. Only roughly correlated with these traits are work ethics... :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-05-14T22:13:50.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2299" PostTypeId="2" ParentId="2288" CreationDate="2014-05-14T22:39:44.927" Score="4" Body="&lt;p&gt;From a general test writing standpoint:&#xA;If you and the students both know that the problems are to be solved by integrating a formula, why not &quot;Integrate the following...&quot;  If they have to figure out whether to use integration or some other calculus then that won't work.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practical applications, such as HPLC analysis, the usual instruction is calculate the amount of substance X by integrating the peak area in the chromatogram in comparison to a standard chromatogram&lt;/p&gt;&#xA;" OwnerUserId="1471" LastActivityDate="2014-05-14T22:39:44.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2300" PostTypeId="2" ParentId="2268" CreationDate="2014-05-15T01:19:23.117" Score="9" Body="&lt;p&gt;(I mentioned this in a reply to another comment, but it might be hard to see.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two years ago I redesigned our proofs course using a flipped classroom format, and all the materials for that are free and available online: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The book, by my colleague Ted Sundstrom: &lt;a href=&quot;http://scholarworks.gvsu.edu/books/7/&quot;&gt;http://scholarworks.gvsu.edu/books/7/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;A bunch of videos to go along with the book: &lt;a href=&quot;http://www.youtube.com/playlist?list=PL2419488168AE7001&quot;&gt;http://www.youtube.com/playlist?list=PL2419488168AE7001&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;All my class materials including class activities, syllabus, exams, and proof portfolio problems: &lt;a href=&quot;http://github.com/RobertTalbert/proof&quot;&gt;http://github.com/RobertTalbert/proof&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;David Steinberg also linked to one of my blog posts about the design and rationale of that class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think there's a lot of room for improvement in the course, but I was pretty pleased with the way it worked out. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing to remember about proofs courses is that they are going to be hard to pull off no matter what you do. Students hit a wall in these courses because they force them to leave behind some immature notions about what mathematics is -- it's no longer about getting the right answer to simple computations in the shortest amount of time. There's a cultural and professional barrier that students have to get over, and many don't make it. The research literature is all over the place regarding the proper handling of these courses. It's a major course design and teaching challenge, but the successes IMO make it well worth the effort. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-05-15T01:19:23.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2301" PostTypeId="1" CreationDate="2014-05-15T03:13:21.173" Score="14" ViewCount="624" Body="&lt;p&gt;When I have TAed calculus courses and led discussions, while I did not find myself grossly inept at interacting with students, there was a bit to be desired with making discussion interactive. I am probably average or below average with interpersonal skills. My evaluations always came back with the lowest scores on engaging with the class--asking and answering questions. I have never had my students work in groups either. It takes a bit of social courage to involve the students. What are strategies for getting students actively involved in discussion?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: With respect to some of the answers and comments, things that are not an issue include mastery of the material, preparation, good diction and clarity of words, and otherwise lecturing forms of communication. One-on-one tutoring or office hours with small groups I carry out successfully, though I am not too personal and mostly I just respond to questions with explanations or do certain problems they are interested in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have the impression that engagement is much better than simply lecturing. I am asking for an explanation of how engagement improves the quality of learning calculus and the specific steps and tasks I can take that I can learn as a skill, not as something that comes naturally.&lt;/p&gt;&#xA;" OwnerUserId="244" LastEditorUserId="244" LastEditDate="2014-05-15T22:45:30.050" LastActivityDate="2015-06-09T07:02:01.770" Title="Remedying poor interpersonal skills" Tags="|classroom-management|class-participation|classroom-interaction|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2302" PostTypeId="2" ParentId="2301" CreationDate="2014-05-15T07:58:54.547" Score="3" Body="&lt;p&gt;While I do believe much of this is natural talent, there is lots you can learn, by looking at how more experienced colleagues do it, asking around ( e. g. here :-) , ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much of the problem  might be just nervousness/anxiety. Work on that: Get &lt;em&gt;very&lt;/em&gt; familiar with the subject matter, see if you can (also) work in situations that are less threatening to you (tutoring one-on-one, work with small groups).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have a few problems prepared, from easy to somewhat hard, ask the class to solve them, after doing an example. Step aside! Let &lt;em&gt;them&lt;/em&gt; do most of the work. Ask for volunteers to write down the evolving solution.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-15T07:58:54.547" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2303" PostTypeId="2" ParentId="2220" CreationDate="2014-05-15T08:36:10.727" Score="6" Body="&lt;p&gt;Here in South Africa, I'm not sure how other universities work, but I'm in the second year of a maths degree at UCT.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's worth noting that in South Africa we do 12 years of school and students entering university are usually 18. The level of mathematics education in our high schools is pretty shocking, with nothing further than power rule being covered, unless you do AP maths and then you do some integration, although I'm not sure of the details since I did the CIE system. Also, a bachelors degree is 3 years and a fourth &quot;honours&quot; year has to be completed before moving on to your masters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first year of university we take a whole year course, which starts off with a review of high school maths and then moves on to limits, treated informally, and then moving on to calculus. By the end of the first semester, integration by parts has been covered; I'm not sure where this falls in relation to Calc I or II because we don't have such notions. Some methods of proof have also been covered (induction and contradiction). In second semester we do some applications of integration (applications of derivatives are in the first semester) and then move on to complex numbers, basic linear algebra, and basically how functions of multiple variables work in preparation for multivariate calculus. Also solving some basic DEs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maths majors are also expected to take a &quot;fundamentals of mathematics&quot; course at some point in their degree. It's a first year course but many, such as myself, choose to continue to take it in second year, or even third. In first semester the course covers set theory and logic, in second semester it covers group theory and basic analysis along with some other miscellaneous topics (complex numbers, combinatorics) in preparation for second year maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In second year first semester the course splits up into modules. Each module consists of 30 lectures which span 45 minutes each.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Semester 1: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Linear Algebra&lt;/li&gt;&#xA;&lt;li&gt;Multivariate Calculus (up to divergence theorem)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Semester 2: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;Introductory Algebra&quot;&lt;/li&gt;&#xA;&lt;li&gt;Real Analysis&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Students can also take a module on differential equations, but for maths majors that's not recommended as it would mean playing catch-up in third year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Year 3:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Semester 1: Students pick two of the following modules:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Metric Spaces&lt;/li&gt;&#xA;&lt;li&gt;Logic &amp;amp; Computation&lt;/li&gt;&#xA;&lt;li&gt;Algebra&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Semester 2: Students pick two of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Complex Analysis&lt;/li&gt;&#xA;&lt;li&gt;&quot;Topics in Algebra&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;Topics in Analysis&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There is also an optional research project which can be taken in third year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's the pure maths track. We also have applied maths modules which can be taken concurrently and cover nonlinear dynamics, ODEs, numerical analysis, boundary value problems (second year), and general relativity, fluid dynamics, methods of mathematical physics,  &quot;advanced numerical methods&quot;,  and methods of functions of complex variables (third year).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the honours programme (fourth year) there's a broad range of modules to choose from and a research project must be completed in that area before the end of the year.&lt;/p&gt;&#xA;" OwnerUserId="1476" LastEditorUserId="267" LastEditDate="2014-11-21T23:01:00.857" LastActivityDate="2014-11-21T23:01:00.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2304" PostTypeId="2" ParentId="2253" CreationDate="2014-05-15T11:31:35.103" Score="3" Body="&lt;p&gt;When handing back exams, I give the complete distribution, listing every score from highest to lowest, with names removed so that privacy is protected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not only that, but I also give the distribution of the total scores for all graded assignments, again ranked from highest to lowest and with names removed, so that students see their overall letter grade in the class at every moment in time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The day I started doing this, student complaints about grades ceased immediately. Now, my students never complain that the test was too hard, because they can see that there is a healthy number of high grades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students are also now fully aware of how their letter grades are computed, and again there are no complaints.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student evaluations improved when I instituted this policy. Grade grubbing stopped.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students find this information on their class standing to be valuable. The good students try harder to stay at the top. The struggling students know where they stand and what they need to do to improve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some students see that they are so far behind that they need to drop the course. In most cases, I agree that they should drop the course, and try again next semester now that they realize how hard they need to work to get a good grade. For many freshmen students, this is the most valuable lesson they learn, that this is no longer high school and that they need to work harder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will never again keep the complete grade distribution secret. Students deserve and benefit from this information. It should be freely available.&lt;/p&gt;&#xA;" OwnerUserId="306" LastActivityDate="2014-05-15T11:31:35.103" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2305" PostTypeId="2" ParentId="364" CreationDate="2014-05-15T12:05:51.210" Score="5" Body="&lt;p&gt;I notice Munkres has already been mentioned, but, I include it again:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Munkres is quite in depth, as I recall there are almost 100 pages devoted to the study of integration and his discussion of $n$-volumes is nice.&lt;/li&gt;&#xA;&lt;li&gt;Edward's Advanced Calculus (available as Dover reprint) also has over 100 pages devoted to multivariate integration. It is not overly abstract as the manifolds considered therein are all either parameterized or level-sets of $\mathbb{R}^n$. There is a proof of the generalized Stokes Theorem in that text.&lt;/li&gt;&#xA;&lt;li&gt;Hubbard and Hubbard's text is probably worth a look&lt;/li&gt;&#xA;&lt;li&gt;Advanced Calculus by James J. Callahan looks to be a fairly careful and thorough treatment of low-dimensional cases. &lt;/li&gt;&#xA;&lt;li&gt;Susan Colley's Vector Calculus text has a very accessible introduction to the calculus of differential forms. There are not too many calculus III texts which actually show how to integrate a $k$-form over a $k$-dimensional subspace, this one does. It is more or less at the level of an honors version of third semester calculus.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;All in all, I doubt you'll be happy with any of these. I always find something is missing, or something is included I'd rather hide. So, perhaps you can write some notes on this for all of us to enjoy.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-15T12:05:51.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2306" PostTypeId="1" CreationDate="2014-05-15T12:18:00.727" Score="6" ViewCount="2208" Body="&lt;p&gt;When teaching trigonometric identities, I found the students had trouble proving them. All the students taking turns to ask almost all the questions of an exercise embarrased me somewhat.  In order to make it easier, I told them that these questions are asked not to test your algebraic ability, but to test how you use the reciprocal, quotient and Pythagorean relations we studied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What should I mention to make it easier for them to solve those problems?  Are there fundamental things we should consider while proving any kind of identities?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for an example:&#xA;$\frac{\cos \left(\theta \right)}{1+\sin \left(\theta \right)}+\tan \left(\theta \right)=\sec \left(\theta \right)$&lt;/p&gt;&#xA;" OwnerUserId="1172" LastEditorUserId="64" LastEditDate="2014-05-16T16:33:09.087" LastActivityDate="2014-07-16T11:49:30.083" Title="Proving trigonometric identities" Tags="|secondary-education|trigonometry|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2307" PostTypeId="2" ParentId="2306" CreationDate="2014-05-15T14:28:32.527" Score="7" Body="&lt;p&gt;Trigonometric identities test students' algebraic ability.  I don't think it helps to tell students otherwise or tell them that that's not the point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you'd rather give trigonometry problems that don't rely so heavily on algebraic ability, you can try numerical ones.  Plot $\sin(x)-x$:  for what values of $x$ is this less than 0.1?  Plot some temperature data for your city on a reasonable axis:  what simple function has the same maximum and minimum in roughly the same places?  If the earth is a sphere and the equator has a circumference of 40,000 km, what is the circumference of the Arctic circle, at latitude 66 degrees?&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-05-15T14:40:49.847" LastActivityDate="2014-05-15T14:40:49.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2308" PostTypeId="2" ParentId="2301" CreationDate="2014-05-15T15:42:14.237" Score="7" Body="&lt;p&gt;I'm not a natural extrovert, but I've found that this is less about social skills than about establishing clear expectations. For example, if I put up a question on the projector that says &quot;Discussion question: ...&quot; and tell the class we're going to discuss it, there's no real uncertainty about what is expected. On the other hand, when I observe other teachers teaching, sometimes I see things like the teacher asking a question, pausing for about 500 milliseconds, and then answering his own question. This sends the opposite message: the expectation is that the students will &lt;em&gt;not&lt;/em&gt; participate. If your expectations aren't clear, then students will revert to the default, which is to sit passively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another important point when seeking active student participation is that the most common mistake for me has always been to give them tasks that were too hard. Tasks that seem trivial to me are not trivial for them, because the subject is new to them. You can always start with an easy task and then follow up with a harder one.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="123" LastEditDate="2014-05-15T17:52:25.537" LastActivityDate="2014-05-15T17:52:25.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2309" PostTypeId="2" ParentId="2301" CreationDate="2014-05-15T16:02:53.553" Score="4" Body="&lt;p&gt;It seems to me that there are two things that I would want in a teacher who is looking to improve their interpersonal skills: (1) ability to master the subject (2) ability to lead a flowing discussion. The first is not the question that I think you're really referring to, so I'll talk about the latter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have had the discussion with a few teachers about the idea that teaching is a form of stage acting. There are many similarities between teaching and being on stage. An excerpt taken from a colleague:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Students are known to tout their favorite teachers as &quot;funny.” Taking improv classes would presumably loosen me up a little at the board and make me more confident about my stage presence, thereby helping to facilitate the occasional joke or witty aside that could make for moments of shared humor between me and the class. Now, some years and many improv classes later, I have come to recognize that the art of improvising and the art of teaching share fundamental themes. In fact, I believe that they can be seen as crafts that share some fundamental rules.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To keep a class (or audience) engaged, you want to be able to present the material (plot) in a way that keeps them involved. If you're a TA at a college, many colleges have Improv acting classes or groups that perform. You may want to see what's available at your school or in town to help you become better at keeping your students engaged.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-05-15T16:02:53.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2310" PostTypeId="2" ParentId="2145" CreationDate="2014-05-15T18:24:09.643" Score="4" Body="&lt;p&gt;We're using Project Euler in a grad class for inservice teachers on Number Theory. I like the problems - intuitive and accessible. They'll allow us to develop some programming skills, which actually supports conceptual development in number theory. The forums which you can see after solving the problem allow them to see alternative approaches to the problem, some substantive discussion and programming tips in a variety of languages. Because of possibilities like what Kevin saw, I wouldn't want to grade on number of problems done. Instead, for us, it's one of the sources that students can use for mathematical writing. So I think that might fit into your proof writing course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems like it would also dovetail with my colleague Ted Sundstrom's freely available text for intro to proof writing. &lt;a href=&quot;https://sites.google.com/site/tedsundstrom/&quot; rel=&quot;nofollow&quot;&gt;https://sites.google.com/site/tedsundstrom/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12" LastActivityDate="2014-05-15T18:24:09.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2311" PostTypeId="2" ParentId="2306" CreationDate="2014-05-15T18:54:41.710" Score="6" Body="&lt;p&gt;It seems as though your students can't cross the barrier of trying out algebraic manipulations. This is a general problem, that needs to be addressed some time and rather early than late, but at the moment it helps, if you give them an enumerated list of manipulations to apply, that covers most easy examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Replace all trigonometric functions by $\sin$ and $\cos$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Apply addition theorems until only simple arguments ($x$, $y$, $z$, …) appear.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Remove all roots and fractions by reverse operations. (should already be a known heuristic step to them)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Factorize to get factors of the type $\sin^2x +\cos^2x$ and apply Trigonometric Pythagorean.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use other standard algebraic tools to proove identity.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This algorithm might create really long terms, but should work in most easy examples. If exercised some, students might observe occurences of shortcuts, that were left out (Trigonometric Pythagorean, Addition Theorems, standard algebraic tools, …). They should then apply these shortcuts whereever possible. After some more exercise, they might get the idea, that there is no enumerated list that always yields the shortest, easiest and correct result, but only a set of possible manipulations that can be applied and the rest is trial-and-error with some experience.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-05-15T18:54:41.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2312" PostTypeId="2" ParentId="2265" CreationDate="2014-05-15T19:06:08.267" Score="3" Body="&lt;p&gt;You probably know how much math/programming skills the students would bring along with the pizza.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that in mind, you should select and solve a few problems matching this skillset so that you can lend a hand or provide a bit of technical advice in case of need.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If all else fails, make good use of the &quot;google euler 38 python&quot; problem mentioned in the other question and teach the students how to read, understand, critique foreign code.&lt;/p&gt;&#xA;" OwnerUserId="1479" LastActivityDate="2014-05-15T19:06:08.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2313" PostTypeId="5" CreationDate="2014-05-15T19:58:55.410" Score="0" Body="&lt;p&gt;Questions related to &lt;a href=&quot;http://projecteuler.net&quot; rel=&quot;nofollow&quot;&gt;Project Euler&lt;/a&gt; in the context of Mathematical Education&lt;/p&gt;&#xA;" OwnerUserId="1479" LastEditorUserId="1479" LastEditDate="2014-05-15T22:49:46.850" LastActivityDate="2014-05-15T22:49:46.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2314" PostTypeId="4" CreationDate="2014-05-15T19:58:55.410" Score="0" Body="Project Euler in the context of Mathematical Education" OwnerUserId="1479" LastEditorUserId="1479" LastEditDate="2014-05-15T22:49:41.960" LastActivityDate="2014-05-15T22:49:41.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2315" PostTypeId="2" ParentId="2301" CreationDate="2014-05-15T21:48:31.607" Score="7" Body="&lt;p&gt;I've taught about 100 biology discussions and get good interaction evaluations; you can decide if this advice applies to you and calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am making some assumptions about your teaching environment:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You have 10 - 30 students in the room&lt;/li&gt;&#xA;&lt;li&gt;Students had lecture elsewhere, so you don't need to lecture&lt;/li&gt;&#xA;&lt;li&gt;Nobody is telling you what you have to do in discussion (e.g., cover homework problems)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If these are true, then your main job is to provide an environment for students to learn calculus. This is best done by students practicing problem-solving while you are available to help them over sticky points.  Here are some simple mechanisms that you can use to hugely increase your &quot;engaging with the class&quot; scores:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Learn their names&lt;/strong&gt;. I do this by bringing my camera on the first day, and passing out index cards, and having them write their first name on the card, and then they hold the card under their chin while I take the picture. I can sort these by discussion, and either print thumbnails or just flip through them on my phone or in Picasa. I can learn most of the names of a class of 100 in the first two weeks. Plus, students LOVE that you learn their names. They think the photo thing is kind of silly, but they LOVE that you know their name. The cool thing is they &lt;em&gt;think&lt;/em&gt; you know the names of everyone, even if you only really know 60%.&lt;/li&gt;&#xA;&lt;li&gt;Pick 3-5 good problems (from book or homework or whatever), and have a worksheet for them to &lt;strong&gt;download the night before, print, and bring to class&lt;/strong&gt;. These should be as much like exam problems as possible.&lt;/li&gt;&#xA;&lt;li&gt;Put students in &lt;strong&gt;groups of three&lt;/strong&gt;, and actually make them turn chairs and face each other. Tell them to start Problem #1, and you'll come around and help. Students can just work with people near them, but I'll sometimes randomize the groups with playing cards so they don't sit with friends.&lt;/li&gt;&#xA;&lt;li&gt;Walk around the room, and be friendly. Ask questions that encourage the group to try to answer you. Ask, &lt;strong&gt;&quot;What did you try first? How did that work? What part got you stuck?&quot;&lt;/strong&gt;  If you find that a lot of students are stuck on the same principle, go up to the board and say, &quot;hey everyone, let me help with this little part here&quot; and talk for 5 min. Then shut up, and let them work.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Don't get trapped&lt;/strong&gt; by one group. Encourage them to ask a neighboring group, and keep moving. &lt;/li&gt;&#xA;&lt;li&gt;Stop every 15 min or so, and ask them, &quot;&lt;strong&gt;what answer does your group have&lt;/strong&gt; for #X?&quot; This puts less pressure on the individual. You can have a random-order class list so you call on everyone equally. But give them time to work first, so they can answer. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;If you are talking less than 20% of the time&lt;/strong&gt; and students are on-topic talking for more than 50%, that is a good, engaging discussion.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1207" LastActivityDate="2014-05-15T21:48:31.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2316" PostTypeId="1" AcceptedAnswerId="2320" CreationDate="2014-05-15T23:12:37.540" Score="4" ViewCount="172" Body="&lt;p&gt;I am a graduate teaching assistant at my university. This will be the first time I teach a course in the summer, 4 days a week, for an hour and a half each day. I have a general question. I hope your experience can help me!&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What is the typical workload like, compared to teaching a class during the school year?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerDisplayName="user718" LastEditorUserId="143" LastEditDate="2014-05-17T14:44:04.723" LastActivityDate="2014-05-17T14:44:04.723" Title="Teaching in the summer at a university" Tags="|undergraduate-education|summer|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2317" PostTypeId="1" CreationDate="2014-03-20T20:42:21.187" Score="8" ViewCount="247" Body="&lt;p&gt;Based on Calculus II calendars everywhere, apparently polar area integrals are something we expect freshmen to fully grasp after one single lecture.  (Or even less: the one single lecture is often split between polar integrals and polar derivatives.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea behind polar integrals, leading to the formula $\int_{\alpha}^{\beta}\frac12 f(\theta)^2\,d\theta$, is beautiful and nice.  But then we expect freshmen to have no trouble figuring out the correct $\alpha$ and $\beta$ to use.  In my experience, finding the right $\alpha$ and $\beta$ in some run-of-the-mill polar area exercises is harder for many students than everything else in Calculus II, combined!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have never, after years of teaching, been able to effectively teach how to find $\alpha$ and $\beta$.  I could do an okay job teaching it if you gave me a week to devote to it, but usually only one lecture or less can be spared for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how do you teach polar integrals, especially how to find the correct bounds, to freshmen in a single lecture?&lt;/p&gt;&#xA;" OwnerDisplayName="Anonymous" LastEditorUserId="143" LastEditDate="2014-05-17T14:39:50.200" LastActivityDate="2014-05-20T08:50:20.577" Title="How to teach polar integrals" Tags="|undergraduate-education|calculus|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2318" PostTypeId="1" AcceptedAnswerId="2319" CreationDate="2014-05-16T04:18:44.527" Score="10" ViewCount="652" Body="&lt;p&gt;I am teaching calculus III in the upcoming semester. The course is fairly standard, just a brief run-down:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Test 1&lt;/strong&gt;: covers vectors and coordinate systems as well as the calculus of space curves including the Frenet frame, curvature and torsion.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Test 2&lt;/strong&gt;: covers functions of several variables, multivariate limits, partial derivatives, chain-rules,min/max theorems, Lagrange multipliers&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Test 3&lt;/strong&gt;: covers multivariate integration, double and triple Cartesian integrals, Jacobians and the change of variables theorem, possibly Greene's Theorem&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Test 4&lt;/strong&gt;: vector calculus. Theorems about conservative vector fields, line and surface integrals, curl, divergence, Gauss and Stokes' Theorems&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I'm using Salas and Hille's text. I've recently noticed that Maxima is fairly accessible and the Sage cloud is very user friendly in some sense, but, perhaps a bit slow. I'd like to do some animations and manipulations (sliders). My question is somewhat broad:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; how can I get students to investigate calculation and visualization via Sage ( or Maxima etc... open source software ) without overwhelming them with work? I don't intend to test on the software aspect, so these assignments are mainly intended to help them crack calculationally intense homework (think torsion) and to visualize geometric data. Ideally, I want to craft 4 computer projects which give them a template and task to modify the template to solve some problem. Perhaps such assignments exist already, I've seen nice work in the help documentation linked to Sage, but I'm trying to find something with a bit more structure. Perhaps a good tutorial.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Secondary question: should I use the cloud or is it better to go for local installation of Sage? Does anyone have experience with either option as it applies to Sage or Maxima.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-07-03T22:26:46.550" Title="Open Source Math Software in Multivariate Calculus" Tags="|technology-in-education|vector-calculus|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2319" PostTypeId="2" ParentId="2318" CreationDate="2014-05-16T13:01:50.943" Score="6" Body="&lt;p&gt;I have some experience teaching students to use Maxima for evaluating integrals. I did it in a classroom with almost as many computers as students. Maxima was installed on all the computers locally, so they didn't have to install Maxima themselves, which would have been somewhat of a barrier. It worked fine, and they seemed to enjoy it, but Maxima has various usability problems which they needed help with. Most college-age people today have no experience with a terminal-style user interface. Some specific things that they needed my help with:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Use log(x) for ln x, exp(x) for e^x.&lt;/li&gt;&#xA;&lt;li&gt;Use integrate(...), not integral(...).&lt;/li&gt;&#xA;&lt;li&gt;For integrals involving symbolic constants, Maxima may ask if the constant is positive, negative, or zero. They need to type a semicolon after their response.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have only a little experience with Sage, but it was pretty negative. When I tried it, the web browser interface was much too slow to be usable. If you download Sage and install it locally, it's a huge download. Sage seems to be sort of a hairball of other math packages, and the interface to the other packages often doesn't seem to be very good. The tutorial shows you how to do lots of fancy things using examples from abstract algebra, but doesn't tell you ordinary, useful things, like how to integrate $x^a$.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-16T13:01:50.943" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2320" PostTypeId="2" ParentId="2316" CreationDate="2014-05-16T13:33:34.533" Score="5" Body="&lt;p&gt;This is around twice the workload (for the teacher and the students!) than during regular semesters. This means that stuff that has to &quot;sink in&quot; won't have time to do so. You need to take that into account. Talk to people who have taught this course (during the year, in summer), see if you can filch some notes (or at lest an outline).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-16T13:33:34.533" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2321" PostTypeId="2" ParentId="2317" CreationDate="2014-05-16T16:11:26.617" Score="8" Body="&lt;p&gt;When I have taught polar representations of regions I used to use a physical pointer (a meter stick when at the board, a pen or pencil when working on paper) to &quot;sweep out&quot; the region like the hand of a clock.  For instance, in the case of $r = \cos(\theta)$, I would start by pointing from the origin along the positive x-axis.  I would say:  &quot;When $\theta$ is zero, the radius is 1.&quot;  As the pointer sweeps counter-clockwise I would say:  &quot;As $\theta$ increases, you can see the radius is dropping, until&quot; (now the pointer is pointing directly upwards) &quot;now we are at 90°, or $\pi/2$ radians, and the radius is zero.&quot;  Continuing to sweep counter-clockwise:  &quot;Now $\theta$ is larger than 90° and the radius is &lt;em&gt;negative&lt;/em&gt;.  See how the points on the graph are on the ray pointing in the opposite direction from the pointer?&quot;  Continuing to sweep:  &quot;We are approaching $\theta$ = 180° or $\pi$ radians, and the radius is approaching $-1$.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;End by summarizing:  &quot;The pointer had to sweep out just 180°, and the function traced out an entire circle.  So we integrate $\theta$ from 0 to $\pi$.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of comments on this approach:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;First of all you notice that I switch back and forth between degrees and radians.  I think that is an important support for students at this level.  Usually degrees are second nature to them, but radians are still unnatural.&lt;/li&gt;&#xA;&lt;li&gt;The reason the &quot;sweeping out&quot; approach is helpful is because it coordinates the visual notion of &quot;angle&quot; (which is hard to read directly off the graph) with the experiential notion of &quot;time&quot;.   &lt;/li&gt;&#xA;&lt;li&gt;Of course the same curve could also be parametrized over the interval $-\pi/2 \leq \theta \leq \pi/2$, which has the advantage that only positive values of $r$ are involved.  You could use the &quot;sweeping out&quot; approach to use this parametrization, too:  After reading $\pi/2$ you say something like &quot;So we've gone from 0 to 90 degrees, and so far we have swept out the top half of the circle.  Now let's reset back to 0 and rotate in the other direction.  See how as we go from 0 to -90 degrees we sweep out the bottom half of the circle?  So the entire circle can be swept out if we go from -90 to 90 degrees.&quot;&lt;/li&gt;&#xA;&lt;li&gt;Having said that, I think there is pedagogical value in trying to acclimate students to the idea that a point can have coordinates $(r, \theta)$ where $r$ is negative, and what that means.&lt;/li&gt;&#xA;&lt;li&gt;You can also draw several radial lines inside the bounded region as &quot;traces&quot; of the pointer that remain in the diagram after the moving demonstration is complete.  That makes it possible for students to look back at their notes later and reconstruct the process.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In the diagram below, the blue lines represent the pointer directions; the red dotted lines represent the negative values of $r$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/RcQDg.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here it is as an animated GIF; in this one, the blue rays represent the pointer direction and the red dotted segments represent the values of $r$ (whether positive or negative).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/TjXIo.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2014-05-16T16:25:50.047" LastActivityDate="2014-05-16T16:25:50.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2322" PostTypeId="1" AcceptedAnswerId="2328" CreationDate="2014-05-16T16:41:27.033" Score="12" ViewCount="689" Body="&lt;p&gt;In a freshman calculus text (Larson), I was surprised to find a definition of differentials as finite differences on the tangent line, and even more surprised to learn later that this definition apparently dates back to Leibniz. My main source on the history is Boyer, The History of the Calculus and its Conceptual Development, 1939, which is &lt;a href=&quot;https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment&quot; rel=&quot;nofollow noreferrer&quot;&gt;available online&lt;/a&gt;. Boyer characterizes Leibniz's definition as follows (p. 210):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the first published account of the calculus, Leibniz gave a&#xA;  singularly satisfactory definition of his first-order differentials.&#xA;  He said that the differential dx of the abscissa x is an arbitrary&#xA;  quantity, and that the differential dy of the ordinate y is defined as&#xA;  the quantity which is to dx as the ratio of the ordinate to the&#xA;  subtangent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Boyer then goes on to describe the fact that this approach didn't work for higher-order differentials, so that Leibniz ended up using a mixture of the above definition and a framework involving infinitesimals. In Larson's language, dx is &quot;any nonzero real number.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yesterday I interrogated a few students who had learned calc from the Larson text. I asked them something like this: You do integrals using substitutions, and during that process, you end up writing things like du=2dx. What kind of things are du and dx? I got blank stares, so I asked if they were numbers, small numbers, ...? The reaction I got was basically that they had learned a technique of integration called u-substitution, and that there was a set of procedures for doing that, which entailed manipulating symbols in this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is Larson's approach common pedagogy? In this approach, how does one get past the issues with higher-order differentials -- do these issues make this approach a dead end? Should this be understood as some kind of watered-down presentation of differential forms?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing that seems odd to me is that Larson writes things like $\Delta y\approx dy\approx 0.06$. I think the vast majority of scientists and engineers would consider this mathematically illiterate or a naive mistake, so it seems like Larson's approach has the disadvantage of being incompatible with the way the rest of the world talks about differentials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/2246/practical-experience-with-teaching-differentials-in-freshman-calc&quot;&gt;Practical experience with teaching differentials in freshman calc?&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/12/a-calculus-book-that-uses-differentials&quot;&gt;A calculus book that uses differentials?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-02-11T13:53:36.813" Title="Is it a dead end to define differentials as finite differences on the tangent line?" Tags="|calculus|textbooks|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2323" PostTypeId="2" ParentId="2306" CreationDate="2014-05-16T18:26:27.437" Score="5" Body="&lt;p&gt;I work as a math aide, students (grade 9-12) come to see me when they need help. I recently had a number of student come in, all looking for help on this exact issue. It's not teacher specific, three teachers' classes are in sync, the same chapter in the Trig book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your example is exactly an example of one that I needed to address. The fact that tan(θ) is sin(θ)/cos(θ) was the first thing I tried to point out. That was fine, but then it took more coaxing to move forward even a bit. &#xA;The process of multiplying the denominators to create a common denominator (and then multiplying the numerators to even things out, of course, before adding) was a struggle. I was relying on the assumption that these algebraic manipulations would come easily, but this wasn't the case. It took me solving a proof from beginning to end, then spelling out the steps for the next problem. Finally, on the 3rd or 4th problem, the student would work completely independently. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a number of distinct skills required to get through this type of exercise, and if I were in front of a classroom, I'd do my best to evaluate what knowledge gaps the class had. If the lower 2/3 of the class needs a quick arithmetic review (add 1/3 + 1/6, or 4/5 + 5/8) it might be worth it. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-05-16T19:33:01.340" LastActivityDate="2014-05-16T19:33:01.340" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2324" PostTypeId="1" CreationDate="2014-05-16T20:52:44.933" Score="18" ViewCount="751" Body="&lt;p&gt;I teach a graduate class in algorithms. Students take this class primarily as a breadth requirement in grad school, and they are a mix of MS and Ph.D students in computer science. Most students taking the class have a CS background, but a few don't: instead, they have enough math background that I deem it permissible for them to take the class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The class itself spends roughly a third of the time covering/reviewing what would be standard material in undergrad algorithms (basic algorithm design principles and methods for analyzing algorithms) and a third of time with more advanced material that is still core algorithms (flows, complexity theory, randomness) and a third with more advanced material (approximation algorithms, and advanced topics). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main emphasis in the class (and the reason why I'm posting the question here) is an understanding how to reason formally about algorithm (as opposed to learning recipes, which is common in undergrad algorithms). Students are expected to write proofs, and understand how to apply imperfect design strategies to specific problems and generate provably correct algorithms. Students also do assignments and projects that involve programming, but this is not a central part of the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As is probably not surprising, understanding what constitutes a proof is a major issue for students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Over the years, the class size has grown: partly because of the growth of the program and partly (I hope!) because of interest in the material. I started off some years ago with under 20 students, and now I'm upto 80 and counting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has forced many changes in the structure of my teaching. But I'm still in a &quot;standard&quot; lecture-style model, where I deliver lectures and assign homeworks. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'm thinking about doing is 'flipping' the class around to do something more interactive, especially to help students understand how to go about reasoning formally about algorithms. But at this size I'm not sure how possible this is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So finally, my question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;are there good pointers on ways to flip a &lt;strong&gt;large grad&lt;/strong&gt; class in a mathematical topic where the focus is on helping students assimilate algorithm design principles and understand how to reason about them ? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="236" LastActivityDate="2014-05-17T01:06:32.343" Title="Flipping (or not) a large graduate class" Tags="|graduate-education|computer-science|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2325" PostTypeId="2" ParentId="2324" CreationDate="2014-05-17T00:04:09.167" Score="5" Body="&lt;p&gt;You might take this in stages. One of our Chemistry professors, Kevin Shea, is using &lt;a href=&quot;https://learningcatalytics.com/&quot; rel=&quot;noreferrer&quot;&gt;Learning Catalytics&lt;/a&gt; in his 75-student Organic Chemistry class, a comparable enrollment and complexity of material to your situation. This enables him to pose a question at the very start of the class that was supposed to be explored the night before, and receive overlaid and comprehensive feedback in the first 5 minutes of the class, that gives him an indication of where everyone stands.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an example from his class, where he can conclude immediately that most of them&#xA;&quot;get it.&quot;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/KVvZ1.png&quot; alt=&quot;KevinExample&quot;&gt;&#xA;&lt;hr /&gt;&#xA;This is a useful step beyond clickers, which have limited question range.&#xA;Students may use any device&amp;mdash;phone, tablet, laptop&amp;mdash;that can access the web,&#xA;and the instructor sees the results, both in a aggregated format as above, and also individually, all pretty much instantaneously.&#xA;Your challenge would be to be concoct questions that get to the heart of&#xA;the algorithmic correctness and performance issues that are your focus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would not attempt to flip the entire semester's classes, but rather do it in stages.&#xA;It works for some material, but not for others. It takes some time to figure out&#xA;what works and what does not.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-05-17T00:04:09.167" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2326" PostTypeId="2" ParentId="2322" CreationDate="2014-05-17T00:11:36.793" Score="5" Body="&lt;p&gt;Differentials are not infinitesmals!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Larson's definition is not a watered down presentation of differential forms:  it is exactly an explanation of what $df$ is as a differential form when $f:\mathbb{R} \to \mathbb{R}$.  In particular $df\big|_p(h) = f'(p)h$.  We have the fundamental approximation that $f(p+h) \approx f(p)+df\big|_p(h)$.   I suppose you could rewrite this approximation as $\Delta f \approx df(\Delta x)$, if you understand that this applies at each basepoint, and that $\Delta f$ is the change in $f$ resulting from incrementing $x$ by $\Delta x$. So it would probably be better to write something like $\Delta y \approx dy(\Delta x) \approx 0.6$.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2014-05-17T00:23:28.797" LastActivityDate="2014-05-17T00:23:28.797" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2327" PostTypeId="2" ParentId="2324" CreationDate="2014-05-17T01:06:32.343" Score="8" Body="&lt;p&gt;It's definitely possible to flip the class you're mentioning here. Certainly the fact it's a graduate class makes it a good candidate for flipping since those students can and should be responsible for more of the learning. As for the size, there are two main areas you'll need to think carefully about: the logistics of the pre-class work, and what you will do during class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of flipping a class involves getting the &quot;lecture&quot; content out to students in a medium that can be taken in and studied &lt;em&gt;prior&lt;/em&gt; to class, to prepare for the class activities. If you are putting your materials online, or if you are just using a standard textbook, then pretty much the size of the class is irrelevant. However, you may also want students to &lt;em&gt;do&lt;/em&gt; something before class and not only read the textbook. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example students in my calculus classes have reading assignments and videos to watch, followed by simple exercises that instantiate a number of learning objectives I've laid out for them. (&lt;a href=&quot;https://github.com/RobertTalbert/calculus/blob/master/GuidedPractice/gp2-3.md&quot;&gt;Here is an example.&lt;/a&gt;) So this raises the issue of logistics. How will students submit the work that they do on those pre-class activities? For me, I had students submit their work using a Google Form, which just looks like an ordinary web form which then dumps the results into a spreadsheet, which I then scan prior to class for any big trends or misconceptions. Whichever way you go, you'll want the means of submitting work something that can be automated and scaled up if possible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the in class work, here are some ideas. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Clicker questions. Joe O'Rourke already hit on this, and he mentioned Learning Catalytics of which I am a user and big fan. Clicker questions are awesome for getting at conceptual understanding which can be very important in proof-based classes. I would highly recommend skimming through Derek Bruff's book on classroom response systems to get a lot of good ideas on how to use clickers or web-based classroom response systems. &lt;/li&gt;&#xA;&lt;li&gt;Proof evaluation tasks. This is where you give students a mathematical statement and an argument, possibly flawed, and ask them which category the argument falls into: (1) the statement is correct and the argument is a valid proof; (2) the statement is correct but the proof has a flaw; or (3) the statement is not correct. You can do proof evaluation as a class discussion activity, or a think-pair-share, or using clickers, or some combination. &lt;/li&gt;&#xA;&lt;li&gt;Proof writing workshops. This is where students practice their proof writing live and in class, collaborating with each other and giving feedback on the proofs they've constructed. Writing courses do this all the time. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Really, you can use the class time any way you want because by flipping, you've moved the lectures and basic content outside of class and made a ton of room for activities. And the active learning is really the magic ingredient -- the flipping is done solely for the purposes of launching the active learning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally I would say Joe's advice to take it slowly and incrementally is sound. Maybe you could reserve one day a week for the &quot;writers workshop&quot; I mentioned above, for example. Or, start the class with no flipping at all, then gradually increase the amount students to prior to class until by midterms the entire class is flipped. But certainly there's no need to go all in with the flipping unless you're totally comfortable with it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It sounds like an awesome class, and good luck with it. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-05-17T01:06:32.343" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2328" PostTypeId="2" ParentId="2322" CreationDate="2014-05-17T04:01:35.487" Score="6" Body="&lt;p&gt;The senior faculty at my institution were also unhappy with the presentation of the &quot;differential&quot; in the style you describe. I gather they were unhappy with the appearance of mixing $dx$ with $\triangle x$ and the lack of notation to indicate the base point of the approximation. That said, it is often the case that instructors slavishly follow the text so I would wager the pedagogy is quite common.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My approach in first semester calculus is simply to emphasize the linearization of the function at a point. In particular, I define $L_f^a(x) = f(a)+f'(a)(x-a)$ and we learn that $y= L_f^a(x)$ is the tangent line. I try to drive home the concept that this gives us the best affine approximation to the function near $x=a$. Or, equivalently, $L_f^a(x)-f(a) = f'(a)(x-a)$ gives us the best approximation to the &lt;strong&gt;change&lt;/strong&gt; in the function near the point. Of course, you can see how this notation is a bit of a drag so it's helpful to invent some simple notation $\triangle x = x-a$ and $\triangle y = f(x)-f(a)$ thus $\triangle y = f'(a)\triangle x$. I see no need for $dx$ or $dy$ here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me discuss the differential notation(s). If the differential of $f: \mathbb{R} \rightarrow \mathbb{R}$ is denoted $df_a:\mathbb{R} \rightarrow \mathbb{R}$ then $df_a(h) = f'(a)h$. In Edward's Advanced Calculus text he distinguishes the differential $df_a$ and the derivative $f'(a)$. This terminology continues for mappings $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ where $dF_a$ is a linear transformation whereas $F'(a)$ is the Jacobian matrix. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is clear enough that $df_a$ is a linear form on $\mathbb{R}$. However, for the student who goes on to deeper things, if $f: M \rightarrow N$ then $df_a: T_aM \rightarrow T_{f(a)}N$ thus (returning to the case $M=N=\mathbb{R}$) $df_a(h)$ does not even make sense. Unless of course, we identify $\frac{d}{dt}|_a = 1$ hence the vector $h\frac{d}{dt}|_a = h$. Naturally, this is the identification which is made. Here I assume that we take the set of derivations at $p \in M$ as the tangent space to the smooth manifold $M$ at $p$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice, as we transition to discussing differential forms which take tangent vectors in their domain to return real values we typically have an isomorphism or two which are oft used without explicit mention. In my current formulation of advanced calculus I spend about half the semester with traditional column-vector based differentials then the later half of the course uses the more abstract differential forms which &lt;em&gt;eat&lt;/em&gt; derivations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your comment about Leibniz getting confused about higher derivatives reminded me of my own confusion when I approached a question about higher derivatives of $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$. I don't know if we really appreciate this enough: higher derivatives calculus I style are a luxury. A fortunate accident that we can identify linear forms on $\mathbb{R}$ with real numbers themselves; $\alpha: \mathbb{R} \rightarrow \mathbb{R}$ linear is naturally identified with $b \in \mathbb{R}$ for which $\alpha(x)=bx$. In the same way, we replace $df_a$ with $f'(a)$. These naturally correspond. Moreover, the function defined by $x \rightarrow f'(x)$ is easy enough to differentiate again as to define $f'': \mathbb{R} \rightarrow \mathbb{R}$ in the same point-wise fashion. However, as you think about the deeper definition, the differential is basic. The second derivative must study the change in the map $x \rightarrow df_x$. This is a whole other ball game. Now, the output is not a number, it is a mapping. I'll not try to explain the resolution to this here, but, beware, this is something hidden by our reluctance to properly embrace the centrality of the linearization idea in calculus. But, don't take my word for it. Let me quote a master:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dieudonne said it best: this is the introduction to his chapter on differentiation in &lt;em&gt;Modern Analysis&lt;/em&gt; Chapter VIII.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The subject matter of this Chapter is nothing else but the elementary theorems of Calculus, which however are presented in a way which will probably be new to most students. That presentation, which throughout adheres strictly to our general ''geometric'' outlook on Analysis, aims at keeping as close as possible to the fundamental idea of Calculus, namely the ''local'' approximation of functions by &lt;em&gt;linear&lt;/em&gt; functions.  In the classical teaching of Calculus, the idea is immediately obscured by the accidental fact that, on a one-dimensional vector space, there is a one-to-one correspondence between &lt;em&gt;linear&lt;/em&gt; forms and numbers, and therefore the derivative at a point is defined as a  &lt;em&gt;number&lt;/em&gt; instead of a &lt;em&gt;linear form&lt;/em&gt;. This slavish subservience to the shibboleth of numerical interpretation at any cost becomes much worse when dealing with functions of several variables...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Dieudonne's  then spends the next half page continuing this thought with explicit examples of how this custom of our calculus presentation injures the conceptual generalization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarize, using $dx$ and $dy$ for linearization may be customary, but it makes it all the more difficult to cleanly articulate linearization. The focus ought to be more on the best linear approximation concept as opposed to a slick-looking formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(to be clear, I'm not advocating Dieudonne for first semester calculus)&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-17T04:01:35.487" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2329" PostTypeId="1" CreationDate="2014-05-17T09:01:05.187" Score="12" ViewCount="2508" Body="&lt;p&gt;I have recently come across &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/052171916X&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conceptual Mathematics: A First Introduction to Categories&lt;/a&gt; by Lawvere &amp;amp; Schanuel. It is a gentle introduction to Category Theory and strikes me as a potential alternative text for a bridging course to abstract mathematics. Has anyone here used it as such and what were the pros and cons you experienced in doing so?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2268/where-can-i-find-resources-for-creating-a-mathematics-bridge-course&quot;&gt;Where can I find resources for creating a mathematics “bridge course”?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Another text I've encountered is David Spivak's &lt;a href=&quot;http://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientists-spring-2013/textbook/MIT18_S996S13_textbook.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Category Theory for Scientists&lt;/a&gt;. It is also a gentle introduction to Category Theory, but with an emphasis on modeling scientific ideas.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-10-30T18:19:44.733" Title="Conceptual Mathematics by Lawvere &amp; Schanuel as text for bridging course?" Tags="|undergraduate-education|course-design|textbooks|category-theory|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2330" PostTypeId="2" ParentId="2288" CreationDate="2014-05-17T13:16:18.997" Score="7" Body="&lt;p&gt;The proper verb is whatever verb your textbook uses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This helped me one time when I wrote &quot;Evaluate this integral&quot;, and one student approximated the integral using rectangles. I was able to show the student that every problem I assigned from the textbook that required a symbolic answer said &quot;Evaluate this integral&quot; whereas every problem that required a numerical approximation said &quot;Approximate this integral&quot;. The student did not argue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a good policy for any exam problem, not just integrals. I always use the same phrasing on exams that my textbook uses.&lt;/p&gt;&#xA;" OwnerUserId="306" LastActivityDate="2014-05-17T13:16:18.997" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2331" PostTypeId="2" ParentId="2215" CreationDate="2014-05-17T13:20:17.263" Score="7" Body="&lt;p&gt;Tina Cardone has been gathering examples of tricks like this in an crowd-sourced online book, available here: &lt;a href=&quot;http://nixthetricks.com&quot;&gt;http://nixthetricks.com&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend reading her curated book. She has many, many of these tricks described, and the reasons why you should never use any of them.&lt;/p&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-05-17T13:20:17.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2332" PostTypeId="2" ParentId="1311" CreationDate="2014-05-17T20:06:42.403" Score="3" Body="&lt;p&gt;From the students' point of view, the main problem with being asked is most often &quot;If I mess up, I get a bad grade&quot;. So I tell my students (1st session and then repeatedly for weeeeeks) when and what exactly will be relevant for grading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often this amounts to the somewhat flippant &quot;Make as many errors in our sessions and in your homework as you can, so that you don't have to make them in written exams&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once they start believing, things get a lot more interactive.&lt;/p&gt;&#xA;" OwnerUserId="1479" LastActivityDate="2014-05-17T20:06:42.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2333" PostTypeId="2" ParentId="2211" CreationDate="2014-05-17T20:31:20.877" Score="4" Body="&lt;p&gt;I tend to channel Polya in such situation and try to guide by asking questions like: &quot;How do you add two fractions?&quot;, &quot;Are the denominators completely different or can we find similarites (which we could exploit)?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Colors usually help in discovering the structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{1}{(\color{green}1\color{black}-\color{red}\cot (\theta )\color{black})}+\frac{1}{(\color{red}\cot (\theta )\color{black}-\color{green}1\color{black})}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;How can we get from 'green minus red' to 'red minus green'?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's similar to what Chris Cunningham suggested, but I try to stay with the given problem as long as possible. Task switching is easy for the teacher, since he knows that &quot;all this is the same&quot;, but often a nontrivial job for a struggling student.&lt;/p&gt;&#xA;" OwnerUserId="1479" LastActivityDate="2014-05-17T20:31:20.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2334" PostTypeId="2" ParentId="2329" CreationDate="2014-05-18T00:04:54.380" Score="15" Body="&lt;p&gt;I find &lt;em&gt;Conceptual Mathematics&lt;/em&gt; creative, illuminating, and thought-provoking.  Subobject classifiers for high school students!  However, I've never taught from it and I don't think it's well-suited to the goals of a typical bridging course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the goal of the course is teaching good proof techniques, I would focus on things like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do you use &quot;Let x&quot; to begin a proof of $\forall x$ or $\exists x$?&lt;/li&gt;&#xA;&lt;li&gt;If you want to prove $P\rightarrow Q$, does it make sense to investigate $P\ \&amp;amp;\  \neg Q$ or $Q\ \&amp;amp;\ \neg P$?&lt;/li&gt;&#xA;&lt;li&gt;What is the difference between $\forall x\exists y$ and $\exists y\forall x$?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The students in Lawvere and Schanuel's dialogues remind me of the students in Proofs And Refutations, by Imre Lakatos -- nominally naive, actually not likely to be tripped up by any of the above questions -- and therefore more mathematically sophisticated than most students that would be taking a bridging course.  I'd stick with the suggestions from the other question.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-05-18T00:04:54.380" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2335" PostTypeId="2" ParentId="2329" CreationDate="2014-05-18T01:24:03.873" Score="10" Body="&lt;p&gt;My instinctive reaction is that a &quot;category error&quot; is being made here (in the philosophical sense, not the mathematical sense of category).  Namely, category theory is an abstraction of (standard, undergraduate level) abstract algebra, which is itself an abstraction of the sort of very concrete mathematical manipulations most students have seen up to that point.  In most undergraduate curricula I am familiar with, the sort of transition course you describe comes just before abstract algebra and gives students needed familiarity with (i) reading and writing proofs, (ii) very basic mathematical logic, and (iii) experience with the next level of abstraction in mathematics (i.e., rather arbitrary sets, functions and relations).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have taught the above bridge course twice at the University of Georgia.  Admittedly there is a class of undergraduates who do not take this course, so it is somehow the opposite of an honors course.  Nevertheless I think the students there are representative of the sort of math majors one meets in many American universities.  In this course I spend more than two weeks on mathematical induction, and the abstraction of induction as a statement about subsets of the natural numbers is very challenging for the students.  Proving things like: if $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are functions, then: if $g \circ f$ is injective, $f$ is injective; if $g \circ f$ is surjective, $g$ is surjective; $g \circ f$ surjective need not imply $f$ surjective, and so forth -- is something that my students can learn eventually but certainly need to be taught.  Each time I taught the course I ended up doing very little with cardinalities of infinite sets: this material occurs at the very end of the syllabus, and by the time I got there I felt it would be more worthwhile for the majority of the students if I spent the last part of the course reinforcing the basic techniques learned, rather than tacking on a new concept that (unfortunately, to be sure) does not get used much in the rest of their undergraduate career.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should admit that I do not own the book of Lawvere and Schanuel.  I looked at some of it on amazon just now, and it does look to be quite carefully written and unusually friendly.  At a preliminary glance it looks plausible and even intriguing to use this text for &lt;em&gt;some other undergraduate course&lt;/em&gt;.  However, to use it for a transitions course would involve increasing the level of abstraction in such a course and therefore seems to be less appropriate for (at least the standard versions of) that course than for other courses.  Using this text would involve abandoning most of the traditional content of a transitions course and, for the clientele to which the traditional content is pitched, that would be a loss.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't want to be &lt;em&gt;too&lt;/em&gt; discouraging though: a sufficiently talented, knowledgeable and enthusiastic instructor can make almost anything work.  In fact my first undergraduate introduction to abstract algebra began with five weeks of category theory.  Before we learned about groups, we learned about monoids and the free monoid functor (called the &quot;James construction&quot;: I have not gone back to try to track down its provenance).  Before we studied monoids we studied sets and mapping from the perspective of universal mapping properties, e.g. we learned the &lt;strong&gt;Quotient Principle&lt;/strong&gt; for when one map factors through another (in the category of sets).  The latter at least turned out to be extremely useful.  Overall the course at the time looked eccentric, and doing something more traditional would probably have worked even better, but it did work, because the instructor -- the still-present, great &lt;strong&gt;Arunas Liulevicius&lt;/strong&gt; -- had so much insight, enthusiasm and charm.  It also worked because the students were very talented and enthusiastic: this was the &quot;honors algebra course&quot; at the University of Chicago.  So you can make things work that sound like they shouldn't, sometimes.  &lt;/p&gt;&#xA;" OwnerUserId="176" LastActivityDate="2014-05-18T01:24:03.873" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2336" PostTypeId="1" CreationDate="2014-05-18T06:53:06.043" Score="2" ViewCount="260" Body="&lt;p&gt;I have just started teaching mathematics up to secondary level. I don't have much idea as to how to handle the class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to make students learn well, how can we divide the time in order to put a mix of lecture and specially the group work?&lt;/p&gt;&#xA;" OwnerUserId="1172" LastEditorUserId="5220" LastEditDate="2020-06-03T15:28:32.907" LastActivityDate="2020-06-03T15:38:18.183" Title="Good ideas on structuring a math class?" Tags="|mathematical-pedagogy|course-design|secondary-education|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2337" PostTypeId="2" ParentId="469" CreationDate="2014-05-18T09:42:08.897" Score="6" Body="&lt;p&gt;In the &quot;emergency&quot; scenario, many of the problems can be avoided, if you start off with having the class instruct you on &quot;how things are done here&quot;. Students will be delighted to reverse the direction of teaching, and get you up to speed on what happened when. Warn them that they have to watch you carefully, as you might slip up now and then ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For &quot;planned&quot; scenarios, you should have collected the relevant information before entering the classroom so that you have some common ground with the students.&lt;/p&gt;&#xA;" OwnerUserId="1479" LastActivityDate="2014-05-18T09:42:08.897" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2338" PostTypeId="1" CreationDate="2014-05-18T12:14:16.060" Score="49" ViewCount="5691" Body="&lt;p&gt;I am very uncomfortable with indefinite integrals, as I have a hard time giving them a precise sense that matches the way they are written and the usual meaning of other symbols.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, when one writes&#xA;$$ \int \sin(x) \,\mathrm{d}x = -\cos(x) + k$$&#xA;then the status of both $x$ and $k$ is pretty unclear (which quantifier in front of each of these variables?) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I personally know how to translate this sequence of symbols into a proper mathematical sentence, but for students it seems utterly difficult to give a precise meaning to this, in particular at the stage when we try to explain the distinction between a function and its value at a point, or when we consider functions of several variables. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, this kind of notation tend to reinforce the student's habit to see mathematical notation as a kind of voodoo formulas that can be manipulated using certain incantations: no one probably knows what the incantation mean, but using the wrong incantation is forbidden for some reason (maybe it will summon an efreet?). On the contrary, I would like to show them the &lt;em&gt;meaning&lt;/em&gt; behind everything we teach them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this reason, I try to never use indefinite integrals, relying instead on moving bounds, e.g.:&#xA;$$ \forall a,x \quad \int_a^x \sin(t) \,\mathrm{d}t = -\cos(x)+\cos(a).$$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt; what possible issues are there in avoiding completely&#xA;  indefinite integrals? Is there any pedagogical advantage to using&#xA;  them? Is there a third way to go?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; let me add another issue with the notation&#xA;$$ \int \sin(x) \,\mathrm{d}x = -\cos(x) + k$$&#xA;In the right-hand side, $x$ is implicitly a variable (as opposed to the &lt;em&gt;parameter&lt;/em&gt; $k$), but on the left-hand side it is both a global variable and a local (mute) variable of integration. Given the (already somewhat weird) role we give to the integration variable in definite integrals, this is a source of confusion that bothers me a lot. Does anyone even imagine writing something like&#xA;$$ \sum_n n^3= \frac{n^2(n+1)^2}4+k?$$&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="262" LastEditDate="2015-10-24T20:26:08.377" LastActivityDate="2023-11-04T13:01:44.110" Title="Should we avoid indefinite integrals?" Tags="|calculus|concept-motivation|integration|" AnswerCount="14" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="2339" PostTypeId="2" ParentId="2338" CreationDate="2014-05-18T15:42:34.483" Score="36" Body="&lt;p&gt;No, it is a bad idea to avoid indefinite integrals, the reason being simply that your students will encounter them elsewhere, and therefore need to be familiar with them. Calculus is a service course. The purpose of the course is to make science and engineering majors fluent in the language of calculus as used in their fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than always using moving bounds, why not just tell your students that when we write&#xA;$$ \int \sin(x) \,\mathrm{d}x = -\cos(x) + k$$&#xA;we're really describing a &lt;em&gt;set&lt;/em&gt; of functions on each side of the equals sign, with an implied quantifier over $k$ on the right? In the US educational system, students are introduced to the notion of a &quot;solution set&quot; very early on, so this should be natural to them.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-18T15:42:34.483" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2340" PostTypeId="2" ParentId="2338" CreationDate="2014-05-18T16:20:51.710" Score="58" Body="&lt;p&gt;On quizzes, homeworks, and tests, I repeatedly ask questions like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find &lt;strong&gt;three different&lt;/strong&gt; functions that have derivative equal to $x^2 + x$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Forcing them to do antiderivatives and deal with the quantifier on the +C without staring at the notation helps some of them separate the +C from the voodoo magic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do a similar thing in college algebra classes to deal with unpleasant quantifiers:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find &lt;strong&gt;three different&lt;/strong&gt; polynomials with variable x that have roots at $x=2$ and $x=3$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-05-18T16:20:51.710" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2341" PostTypeId="2" ParentId="2338" CreationDate="2014-05-18T18:27:35.563" Score="7" Body="&lt;p&gt;The textbook we use (a fairly standard one in the US, Thomas) is actually pretty careful about this: it says that an indefinite integral is a &lt;em&gt;collection&lt;/em&gt; of functions, namely, all the antiderivatives.  The resulting possibilities (&quot;an antiderivative&quot; versus &quot;the indefinite integral&quot;) are a bit confusing for students just learning the topic, especially since &quot;$\cos x+k$&quot; could mean either &quot;the indefinite integral, i.e. all the possible functions at once&quot; or &quot;a particular antiderivative with $k$ some constant&quot;, but I've found that Chris Cunningham's suggestion---emphasizing that this represents multiple functions by having them explicitly instantiate multiple cases---helps.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="376" LastEditDate="2016-04-14T17:21:32.020" LastActivityDate="2016-04-14T17:21:32.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2342" PostTypeId="1" AcceptedAnswerId="2344" CreationDate="2014-05-18T19:44:21.470" Score="13" ViewCount="1896" Body="&lt;p&gt;I'm teaching geometry to grade 8 and 9 students, and I find that they often mistake &lt;em&gt;theorems&lt;/em&gt; with their &lt;em&gt;converses&lt;/em&gt;. For example, when I give them a 3, 4, 5 triangle and ask them to decide whether or not it is a right triangle (using the converse of the Pythagorean Theorem), many of them begin incorrectly with $a^2+b^2=c^2$, thereby assuming what they're trying to prove.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to clarify &lt;em&gt;converses&lt;/em&gt; for my students, but I have little time to spare. I'd rather not take my grade 8's into propositional logic, though I know that's an option.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How can I meaningfully teach my students converses without going too deeply into propositional logic?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I can see, there are at least 3 possible &quot;depths&quot; for teaching the idea of a converse. Ideally I'd like to teach my students at a level somewhere between 2 and 3 below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Avoid using the word &lt;em&gt;converse&lt;/em&gt; altogether and just explain all theorems separately and carefully.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Explain that &lt;em&gt;converse&lt;/em&gt; means sort of the same thing as an &lt;em&gt;opposite&lt;/em&gt;, like in the same way I used the word opposite to help my students understand that $+$ and $-$ are &lt;em&gt;inverse operations&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Teach a bit of propositional logic, then explain that the &lt;em&gt;converse&lt;/em&gt; of $A \implies B$ is $B \implies A$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, here's a list of some of the theorems with which my students make mistakes related to the converse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Pythagorean Theorem $\iff$ converse of Pythagoras&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parallel lines $\iff$ Angle relationships formed by transversals&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Type of triangle/quadrilateral $\iff$ properties of triangle/quadrilateral&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Segment connecting midpoints of a triangle $\iff$ consequences of the midpoint theorem&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2014-05-28T03:22:43.173" Title="How do I Introduce Converse Theorems in Class?" Tags="|mathematical-pedagogy|secondary-education|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2343" PostTypeId="2" ParentId="2338" CreationDate="2014-05-18T20:34:14.440" Score="27" Body="&lt;p&gt;I go a step further than Thomas (see Henry Towsner's answer). In my view,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int f(x) \ dx = \{ F(x) \ | \ F'(x)=f(x) \} $$&lt;/span&gt;&#xA;On a connected domain, it is true that &lt;span class=&quot;math-container&quot;&gt;$F'(x)=G'(x)$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$F(x)-G(x)=c$&lt;/span&gt; hence, given an integrand which is continuous (or piecewise continuous, insert your favorite weakened set of functions here) we may write:&#xA;&lt;span class=&quot;math-container&quot;&gt;$ \int f(x) \ dx = \{ F(x)+c \ | \ c \in \mathbb{R} \}. $&lt;/span&gt;&#xA;Then, I tell the students that nobody wants to write this all the time so we drop the &lt;span class=&quot;math-container&quot;&gt;$\{ \}$&lt;/span&gt; and simply summarize it with a slogan: &lt;strong&gt;the indefinite integral is the most general antiderivative&lt;/strong&gt;. This really means it is the set of all functions which form antiderivatives of the integrand. Moreover, I warn them, for this reason the usual rules of equality do not apply. In fact, &lt;span class=&quot;math-container&quot;&gt;$\int x \, dx = x^2/2+c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\int x \, dx = x^2/2+42+c$&lt;/span&gt; are the same answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Truth is, we are working on equivalence classes of functions as we study indefinite integration as the notion of equality has properly been replaced with congruence. Moreover, if we take function space and quotient by the subspace of constant functions then for some connected domain the indefinite integral and derivative operator are inverse operations. This I do not tell first semester calculus students, however, in a good semester of linear algebra I think it makes a nice quotient space discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, the question remains, why on earth should we use the same symbol &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\int f(x) \, dx$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\int_{a}^{b} f(x) \, dx$&lt;/span&gt;? These are radically different objects. The indefinite integral is a set of functions whereas the definite integral is a number. The answer is the &lt;a href=&quot;//en.wikipedia.org/wiki/Fundamental_theorem_of_calculus&quot; rel=&quot;noreferrer&quot;&gt;FTC&lt;/a&gt;. That said, I think it important to make a point of emphasizing just how surprising it is that these two ideas have any connection at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit added 6/4/19&lt;/strong&gt; I see what Michael is saying in the comments about the error of me confusing a function with its value. The thing is, the notation &lt;span class=&quot;math-container&quot;&gt;$\int f(x) \, dx$&lt;/span&gt; already indicates a variable for the functions in play. I'm not at peace with an answer which has &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; on the LHS but not the RHS (say &lt;span class=&quot;math-container&quot;&gt;$\int f(x) \, dx = \{ F \ | \ F' = f \}$&lt;/span&gt;. If we are to go this route to talk about functions rather than their values then I'd probably adopt the notation &lt;span class=&quot;math-container&quot;&gt;$\int f$&lt;/span&gt; for the indefinite integral of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and keeping with the legalism of my current answer here I'd write:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int f = \{ F \ | \ F' = f \} $$&lt;/span&gt;&#xA;I think I prefer my answer with its abuse, but I see why others would rather engage in the subtlety which Michael points towards.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2019-06-04T15:22:19.370" LastActivityDate="2019-06-04T15:22:19.370" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="2344" PostTypeId="2" ParentId="2342" CreationDate="2014-05-18T23:38:51.663" Score="13" Body="&lt;p&gt;Teaching converses at the level of secondary Geometry is complicated by the fact that the vast majority of the theorems you want your students to learn are in fact &lt;em&gt;biconditional&lt;/em&gt;, so that for each theorem they know, the odds are that its converse is a theorem, too, and therefore the distinction you want to make seems (to students) like hair-splitting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The distinction between a theorem and its converse only seems meaningful when one of them is &lt;em&gt;true&lt;/em&gt; and the other one is &lt;em&gt;false&lt;/em&gt;.  Here are two examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If quadrilateral $ABCD$ is a rectangle, then it has congruent diagonals.&lt;/li&gt;&#xA;&lt;li&gt;If quadrilateral $ABCD$ is a rhombus, then it has perpendicular diagonals.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;One approach to teaching about converses is to begin with a proposition that is true in only one direction, illustrate it with examples, and then pose the converse question.  Odds are a lot of students in the class will think it is true.  Challenge the class to find a counterexample.  Assuming one is found, use that discovery -- the fact that &quot;If $A$ then $B$&quot; is true, but &quot;IF $B$ then $A$&quot; is false -- as an opportunity to introduce the word &quot;converse&quot;.  You can also use the same opportunity to ask them what they think about &quot;If $B$ is false, then $A$ is false&quot; (contrapositive) and &quot;If $A$ is false, then $B$ is false&quot; (inverse).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again I think it's important to have this conversation in a setting where the proposition under consideration is true only in one direction.  But you can mix it up:  Give them a proposition that is &lt;em&gt;false&lt;/em&gt; and ask them if its converse / inverse / contrapositive is &lt;em&gt;true&lt;/em&gt;.  Keep stressing that there is no automatic connection between the two directions of a proposition:  they might both be true, or both be false, or either one could be true and the other false.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Then&lt;/em&gt; move into some examples that are biconditional, such as the ones you listed in your question:  The Pythagorean Theorem and its converse, the various parallel lines/angles properties, etc.  The fact that these are biconditional seems much more significant if you already know that isn't always the case.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-05-18T23:38:51.663" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2345" PostTypeId="2" ParentId="2342" CreationDate="2014-05-19T01:10:33.260" Score="4" Body="&lt;p&gt;Converses may be clearer with quantifiers than with propositional logic.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The converse of &quot;all As are B&quot; is &quot;all Bs are A&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The converse of &quot;all basketball players are over 5' tall&quot; is &quot;all people over 5' tall play basketball&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The converse of &quot;All right triangles have $a^2+b^2=c^2$&quot; is &quot;all&#xA;triangles with $a^2+b^2=c^2$ are right.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Analyzing this with implication may not be necessary.  Euclid had the converse to the Pythagorean theorem without any propositional logic.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-05-19T01:10:33.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2346" PostTypeId="2" ParentId="2329" CreationDate="2014-05-19T02:20:17.033" Score="11" Body="&lt;p&gt;In case you have not yet seen it, I thought I would draw your attention to (what is currently) the most recent issue of the American Mathematical Monthly, and, in particular, the article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Leinster, T. Rethinking Set Theory. &lt;em&gt;The American Mathematical Monthly, 121&lt;/em&gt;(5), pp. 403-415.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An arXiv version can be found &lt;a href=&quot;http://arxiv.org/pdf/1212.6543.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The abstract says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematicians manipulate sets with confidence almost every day, rarely making mistakes. Few of us, however, could accurately quote what are often referred to as 'the' axioms of set theory. This suggests that we all carry around with us, perhaps subconsciously, a reliable body of operating principles for manipulating sets. What if we were to take some of those principles and adopt them as our axioms instead? The message of this article is that this can be done, in a simple, practical way (due to Lawvere). The resulting axioms are ten thoroughly mundane statements about sets.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In particular, the ten axioms (stated informally) are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/rjE1h.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leinster makes it a point to dispel a few misconceptions about Lawvere's presentation, namely, (1) that an underlying goal is to replace set theory with category theory; (2) that the axiomatization requires greater mathematical maturity than other systems (e.g., ZFC); and (3) that there might be some inherent circularity by mentioning both 'categories' and 'sets'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Respectively, he comments: this is literally set theory; it is not that complex; and the word 'category' is not used.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The primary motivation for this paper is that most working mathematicians use ZFC set theory without really paying attention to the axioms. As the author somewhat humorously remarks:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Dg76o.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article does mention some pros and cons of using the text to teach &quot;axiomatic set theory,&quot; but perhaps they could transfer to a bridge course:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/CM0wS.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The citations above are: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(5) F. W. Lawvere, R. Rosebrugh, Sets for Mathematics. Cambridge University Press, Cambridge, 2003.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(18) T. Trimble, ETCS: building joins and coproducts (2008). Retrieved online from &lt;a href=&quot;http://ncatlab.org/nlab/show/Trimble+on+ETCS+III&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ncatlab.org/nlab/show/Trimble+on+ETCS+III&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Still:&lt;/strong&gt; If you are interested in trying this text for a bridging course, then maybe using Leinster's presentation would be of help. (Or reading it over, making it available for students, etc.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As a final, offhand comment about bridging courses: You can find more on the &lt;em&gt;history&lt;/em&gt; of such courses at my MO response &lt;a href=&quot;https://mathoverflow.net/a/151910/22971&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-05-19T03:30:36.207" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2347" PostTypeId="2" ParentId="2338" CreationDate="2014-05-19T14:28:26.197" Score="7" Body="&lt;p&gt;You can emphasize definite integrals, by treating them first -- before the indefinite integrals and even before the derivative.  Apostol's calculus text was famous for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After enough definite integrals, students may both understand and appreciate the indefinite integrals better.  This order corresponds to the history also:  the earliest texts we have which look like calculus are calculations of areas by Archimedes.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-05-19T14:28:26.197" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2348" PostTypeId="1" AcceptedAnswerId="2354" CreationDate="2014-05-19T15:08:43.597" Score="23" ViewCount="3786" Body="&lt;p&gt;When chatting with a new acquaintance for a few minutes, our profession often comes into the conversation and then takes over. We're all familiar with this line at grocery stores, airplanes, trains, and more.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;MATH?&lt;/strong&gt; Oh, I was never a math person.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The conversation then continues on a predictable trajectory. The interesting thing I have noticed is: during conversations like this, our new acquaintance is actually willing to consider our opinion. There is usually time to give them about one new sentence about mathematics or mathematics education to think about. This means these conversations are actually a unique educational opportunity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leads to my question: &lt;strong&gt;When you have a willing audience of one, and you are able to transmit exactly one idea about mathematics or mathematics education, what do you choose?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note my goal is not to steer the conversation away from math; they want to talk about it and want to listen a tiny bit. How can we make this an effective learning experience?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-05-19T22:14:16.177" Title="How can we handle grocery store/bus stop conversations better?" Tags="|mathematics-in-daily-life|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2349" PostTypeId="2" ParentId="2220" CreationDate="2014-05-19T15:19:46.573" Score="7" Body="&lt;p&gt;&lt;strong&gt;Italy&lt;/strong&gt;&#xA;(the situation is not completely standard and may vary a little from one university to another):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Analisi I (calculus)&lt;/strong&gt; &#xA;Real functions of one variable + real line topology. Usually split in two: 1st semester  $\varepsilon-\delta$ limits, derivatives, 2nd semester integration+ series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Geometry I&lt;/strong&gt;&#xA;1st semester: Linear algebra (vector spaces, matrices, linear systems, eigenvectors)&#xA;Applications to affine geometry (time permitting classifcation of conics and quadrics).&#xA;2nd semester: Euclidean spaces (i.e. scalar products), orthonormal basis, dual space,&#xA;norms, applications to Euclidean geometry (problems concerning metric issues between lines adn planes in 3d space)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Algebra&lt;/strong&gt;&#xA;Basics of abstract algebra: Sets, functions, polynomials, groups (+actions, symmetric group, finite cyclic groups), rings, UFD. May include elements of Logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Physics I&lt;/strong&gt;&#xA;Elementary mechanics, some thermodynamics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Probability Theory&lt;/strong&gt; (1 semester - not everywhere)&#xA;Basic probability theory&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is for math majors, of course.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2014-05-19T15:19:46.573" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2350" PostTypeId="2" ParentId="2348" CreationDate="2014-05-19T15:36:02.200" Score="3" Body="&lt;p&gt;I might continue with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I'm glad I liked it and learned it well.&#xA;  I got 20 years of jobs based on my mathematical skills.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then I can talk about math in whichever job seems right for the context.  That might be health care, defense, academia, software, finance, consulting....I would emphasize that math has opened a lot of doors for me.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-05-19T15:43:54.107" LastActivityDate="2014-05-19T15:43:54.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2351" PostTypeId="2" ParentId="2338" CreationDate="2014-05-19T15:42:22.740" Score="5" Body="&lt;p&gt;I think it's funny that you/your students have a problem with the indefinite integral but don't mind the mysterious placeholder $dx$ at the end of a definite integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my Calculus course, I take a differentials approach (see &lt;a href=&quot;https://matheducators.stackexchange.com/a/2258/507&quot;&gt;here&lt;/a&gt;), so in that sense $\int$ is the inverse operator of $d$. However, since $d$ is a many-to-one operation, the $+k$ is required in the output of $\int$ in the same way that $\pm$ is required when undoing a square. When I first introduce indefinite integrals (which is after definite integrals which motivate the notational symbols), I'll do an easy example like $$\int 3x^2\ dx = \int d(x^3) = x^3+k$$ (which most of my students get pretty quickly, especially when compared to $\sqrt{9}=\sqrt{(\pm 3)^2} = \pm 3$). I'll also do a more advanced example: $$\int \frac{x}{x^2+3}\ dx = \int \frac{\frac{1}{2}d(x^2)}{x^2+3} = \frac{1}{2}\int\frac{d(x^2+3)}{x^2+3} = \frac{1}{2}\int d(\ln(x^2+3)) = \frac{1}{2}\ln(x^2+3)+k$$ (which all of them need to think about before understanding).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, my point is treat $\int$ as the inverse operator of $d$ and draw upon the analogy of square roots to explain the $+k$.&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-19T22:18:04.883" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2352" PostTypeId="2" ParentId="2348" CreationDate="2014-05-19T15:51:14.290" Score="9" Body="&lt;p&gt;I might say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A lot of kids learn math from teachers who don't make it interesting.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here in the US, kids usually get their first instruction in math (aside from learning to count) in public schools, from teachers with multiple-subject credentials. Some of these folks are wonderful at teaching math, but many of them limped through their own math coursework without understanding it very well, and hated math themselves. If a kid goes through 1st through 5th grades and has four teachers who are fantastic at teaching math and one who is terrible, often that kid ends up hating math. All it takes is a bad experience during one year, and the kid generalizes: I can't do math past this level because I'm not smart enough, and math past this level is too hard and scary and doesn't connect to real life.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-19T15:51:14.290" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2353" PostTypeId="2" ParentId="2338" CreationDate="2014-05-19T16:06:58.577" Score="5" Body="&lt;p&gt;Have you considered the following notation:&#xA;$$ \quad \int^x \sin(t) \,\mathrm{d}t = -\cos(x)+k$$&#xA;emphazing that the integral is a function of $x$ ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Abbreviations for integration and derivation can then be written in shorthand with $\quad \int^x$ and $d/dx$&lt;/p&gt;&#xA;" OwnerUserId="1498" LastActivityDate="2014-05-19T16:06:58.577" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2354" PostTypeId="2" ParentId="2348" CreationDate="2014-05-19T16:38:04.933" Score="9" Body="&lt;p&gt;My go-to response is: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I used to be &quot;not a math person&quot; as well, but I stuck with it, and eventually I learned what math was all about and I fell in love. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My response may make me seem like a bit creepy -- after all who &quot;falls in love&quot; with math but creepy people? -- but it has the benefit of inviting follow-up questions that are positive in nature. &quot;Oh really? Did you have a good teacher? What &lt;em&gt;is&lt;/em&gt; math all about anyway?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also has the benefit of being true. Math was my worst subject, by far, in school until my senior year of high school, when I had a teacher who gave me the time and space in class to explore Calculus on my own. Later still in college, the light bulb went off for me when I was taking abstract algebra and I learned about isomorphisms and it all just came together for me. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what I'm saying here is that I'd communicate the idea that &quot;not math people&quot; can become math people given the right combination of instruction, persistence, and inspiration -- and that a person can have a deeply personal relationship with this subject given the chance. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-05-19T16:38:04.933" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2355" PostTypeId="2" ParentId="2348" CreationDate="2014-05-19T16:39:53.530" Score="5" Body="&lt;p&gt;For the particular situations you mention, I might continue&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;at the grocery store:&lt;/em&gt;  Did your teachers ever talk with you about fitting&#xA;  oranges in a box?   And how every orange ends up touching 12 others?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;at the bus stop:&lt;/em&gt;  Did you ever watch a single point on a bus tire,&#xA;  to see its path as the bus moves?  It looks like this &lt;em&gt;(draw a cycloid in the air)&lt;/em&gt;,&#xA;  and the curve turned out to be useful for clock-makers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Math can be everywhere.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-05-19T16:39:53.530" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2356" PostTypeId="2" ParentId="2348" CreationDate="2014-05-19T16:52:42.187" Score="17" Body="&lt;p&gt;This happened to me two days ago! The &quot;What is it that you teach then?&quot; conversation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I first said everyone's different and has different brains, and that a bunch of us who did PhDs together decided being a PhD mathematician was a mild form of mental disease, with a lot of common symptoms (that I won't need to list to anyone who's spent a lot of time observing people who teach postgraduate mathematics), and told her to ask my wife about my maths friends at our wedding, remembered by the families for ever.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The wall&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I explained we all hit a wall in maths at some stage, (the wall being where your innate talent becomes insufficient to get on without a significant increase in effort, reached at very different ages by different people).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With a good teacher, you learn to climb over the wall and start finding wall climbing enjoyable. If the teacher at the time doesn't get you over your personal wall, then because it's so hierarchical (I didn't use that word) and jargon-laden, it's easy to get left behind and conclude maths is impossible and leave demoralised. I went on to explain how I've managed to get my students this year much better at (metaphorical) wall climbing, and they're the best bunch in terms of mock exam results I've had for a long long time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;(Aside: a lot of this is to do with &lt;a href=&quot;https://matheducators.stackexchange.com/a/2267/1340&quot;&gt;eliminating judgement from feedback and replacing it with instruction&lt;/a&gt;, in conjunction with opportunities to improve. The opportunities to improve have always been there, but the culture of not-there-&lt;em&gt;yet&lt;/em&gt;, but here's what to change is the cultural difference that has resulted in the behavioural difference.)&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She went on to tell me about her personal wall at work (tracheotomy re-insertions) and decided she wanted to do something about it - planning to start observing her colleague doing it and get accredited to do it. Apparently she's felt for more than a decade she ought to be able to do this, but decided to actually do something about it when hearing that it's possible for students who find mathematics hard to get past that! I don't know whether she'll follow through on that, but I'm delighted that some of what I said must have meant something to her.&lt;/p&gt;&#xA;" OwnerUserId="1340" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-19T20:19:51.467" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2357" PostTypeId="2" ParentId="2306" CreationDate="2014-05-19T17:26:49.070" Score="3" Body="&lt;p&gt;A trig &quot;identity&quot; is true if and only if the algebraic ``identity'' obtained by replacing $$\sin(k\theta) \mapsto \frac{z^k-z^{-k}}{2i}, \cos(k\theta) \mapsto \frac{z^k + z^{-k}}{2}$$&#xA;is true. We already teach them algebra, and trig identities are no more than algebra (with imaginary numbers). Add in $$d\theta \mapsto \frac{dz}{i z}$$ and the method of partial fractions, and you can do all of the trig integrals, too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trig identities such as the one in the OP are an anachronism. Since we don't sail ships by hand anymore, it is pretty rare for anyone to need to be facile with trigonometry past the right-triangle and unit circle interpretations of $\sin(\theta),\cos(\theta)$, and their immediate consequences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And while I'm on the subject of anachronisms, $$\sin( \tfrac{\pi}{4}) = \frac{1}{\sqrt 2}.$$&#xA;It's not $\frac{\sqrt 2}{2}$. Of course they are equal, but which expression is simpler? The only justification for the old $\sqrt{2}/2$ (how ugly is &lt;em&gt;that&lt;/em&gt;?) is if you use a table of square roots to handle conversions to decimal form. Our parents did that, and we learned how to do it, but I don't see a point to teaching our children how to use tables.&lt;/p&gt;&#xA;" OwnerUserId="87" LastActivityDate="2014-05-19T17:26:49.070" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2358" PostTypeId="1" CreationDate="2014-05-19T22:09:17.633" Score="13" ViewCount="3476" Body="&lt;p&gt;Does anyone have any good reasons to offer for introducing definite integrals before indefinite integrals, or the other way around? Definite integrals first because they have a clearer geometric interpretation? Indefinite integrals first because they can be defined without introducing the Riemann sum? Definite integrals first because they make it easier to understand the logic behind the Leibniz notation?&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-20T12:13:38.507" Title="Which first, definite or indefinite integrals?" Tags="|calculus|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2359" PostTypeId="2" ParentId="2348" CreationDate="2014-05-19T22:14:16.177" Score="1" Body="&lt;p&gt;I have yet to find the solution for this problem which works for myself. However, I heard a great story at a donut shop in Brooklyn a few years ago. I think this works for a bus stop or train station, whatever. This was told to a youngster by an elderly gentleman who was enjoying some conversation about pop culture whilst the the youngster and his friends were rather rowdy. First, he did not judge the boy. Second, he just started talking to him. I'll recount the story briefly:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;coffee man: &lt;em&gt;so, what's your favorite subject in school?&lt;/em&gt; &lt;/li&gt;&#xA;&lt;li&gt;boy: &lt;em&gt;math.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;coffee man: &lt;em&gt;really, I've got a question for you, think you can answer it?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;boy: &lt;em&gt;sure.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;coffee man: &lt;em&gt;all right, you ready?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;boy: &lt;em&gt;sure.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;coffee man: &lt;em&gt;all right, so there's a subway car and it has 4 people on it, you got it?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;boy: &lt;em&gt;yes.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;coffee man: &lt;em&gt;the subway stops at the next station and 5 people get on and 1 gets off, get it?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;boy: &lt;em&gt;yes.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;coffee man: &lt;em&gt;the subway stops again 7 people get on and 4 get off, get it?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;boy: &lt;em&gt;yes.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;coffee man: &lt;em&gt;the subway stops again 4 people get on and 1 get off, get it?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;boy: &lt;em&gt;yes.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;coffee man: &lt;em&gt;the subway stops again 11 people get on and 5 get off, get it?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;boy: &lt;em&gt;yes.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;coffee man: *the subway stops again 3 people get on and 1 get off, get it&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and this goes on for some time...&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;coffee man: how many stops did the subway car make?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The boy left and the rowdy behavior abated. I've tried this a few times on calculus etc... nobody ever gets it. So, perhaps one fun thing to do:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;find a good math joke to tell them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-19T22:14:16.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2360" PostTypeId="2" ParentId="2215" CreationDate="2014-05-19T22:43:32.493" Score="1" Body="&lt;p&gt;I'm not sure if this qualifies, but there is literature on &quot;tricks&quot; for arithmetic. Many of the, what you call &quot;low level&quot;, students that I have encountered could benefit from an additionally multiplication or division trick or two (or more). Two such books that I know of are &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471575631&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rapid Math Tricks and Tips&lt;/a&gt; by Edward H. Julius and &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B0007FDYKQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;High-Speed Math&lt;/a&gt; by Lester Meyers. The former is almost exclusively arithmetic tricks, whereas the latter contains some applied mathematics as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not recommending teaching all of these ideas, but perhaps integrating a few of these types of tricks might help free up some of the working memory of your struggling students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also linking to this MESE question: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1649/tricks-for-computing-things-in-your-head/1683#1683&quot;&gt;Tricks for computing things in your head&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-05-19T22:43:32.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2361" PostTypeId="2" ParentId="2358" CreationDate="2014-05-19T22:58:50.123" Score="5" Body="&lt;p&gt;A remark that has little to do with my answer: I believe Apostol's calculus does definite integration before differentiation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The application of indefinite integration to initial value problems is interesting. For example, generically, we can ask: what position function $x(t)$ gives rise to a particular velocity function $v(t)$. So, we're looking for a function $x(t)$ which when differentiated gives us $v(t)$. This is just antidifferentiation. Although, I'm fairly sure it's wrong to say given $v(t)$ find $V(t)$. I think such problems are interesting and accessible before any discussion of the Riemann integral has been given. If you choose to also discuss the indefinite integral here the very nice thing you get to say: &quot;this $c$ we add is to account for all the possible initial conditions.&quot; This seeds intuition for later work on differential equations in my view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a course management perspective, these give me a chance to get the students to study antiderivatives before we cover the Fundamental Theorem of Calculus (FTC). This means, as I'm teaching FTC there is a greater chance they understand what I mean when I say &quot;antiderivative&quot;. I suspect they are more confused about $\sum$ notation and limits of sequences by the time I get to FTC. It's ironic that we worry about these minor points and yet feel free to drop sequential limits on them when so many have not seen a sequence in years. Backgrounds vary I suppose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something I've worked out for the sake of showing them what not to do is an indefinite integral from an explicit calculation of the Riemann integral with varying bound. It's not too much trouble to show:&#xA;$$ \int_{0}^x dt = x \qquad \int_{0}^{x} t \, dt = \frac{1}{2}x^2 $$&#xA;from the definition. You can derive these from the known formulas for sums. Then, I ask them, can you work out $\int_{0}^{x} \sin (t) \, dt$? There is a class of students who needs to realize &lt;em&gt;why&lt;/em&gt; antidifferentiation is necessarily educated guessing. This discussion helps with that. Annoyingly, I can't seen to find the explicit examples in my posted notes, if you're interested let me know, I'll find them. For more on how I present antiderivatives etc... you might look at: &lt;a href=&quot;http://www.supermath.info/OldschoolCalculusII.pdf&quot; rel=&quot;noreferrer&quot;&gt;My Oldschool Calculus Notes circa 2010&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;( I am happy to email you LaTeX source if you want it, however, it has some institution specific comments and my horrible grammar, I'm working on an edit this summer if my student comes through as I hope...)&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-19T22:58:50.123" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2362" PostTypeId="2" ParentId="2342" CreationDate="2014-05-19T23:15:25.073" Score="3" Body="&lt;p&gt;I don't think they're confusing statements with their converses, I think that, like my own students, they don't initially know that to prove B you can't start with B, but must instead conclude with B. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is different from the converse - they didn't attempt to use Pythagoras to show that the triangle had sides 3,4,5, they just used Pythagoras because it was mentioned in the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Their directionless past&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It happens partly as a result of the directionless world they come from - before being introduced to proof, there are just things you write on a page - they factorise stuff, they multiply stuff out, they square stuff, they square root stuff. The rules of mathematics don't have a great deal of direction to them before they meet proof. So far, they think of some maths and they write it down, and they get ticks unless they made a mistake. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You asked them to get to a destination from a start, and they write the destination down first - this is because they didn't see that as cheating since all their questions so far only gave them stuff they were allowed to use whenever they like in their answer. Any feedback they've had so far on what order to write things in has seemed presentational rather than fundamental.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A sense of direction takes time to acquire&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This takes time to absorb and learn, so in initial lessons I give non-maths examples and then follow up strictly for about a year - &quot;That's cheating - you can't start with what you're trying to show!&quot;. (For fans of football (soccer) or rugby, you can refer to it as offside or a forward pass, but sadly I don't know enough to make a US analogy.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think there's a short route to this any more than there is to understanding that you can't distribute anything over everything. It takes practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Examples I've used to explain&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Innocent until proven guilty&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I pick on the most innocent, pleasant and helpful student that I get on with in the class (suppose she's called Lauren), and say&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my erasers went missing! Lauren stole it! Let me prove it to you: Lauren was the one who took it, but she's clever, so she wouldn't have done it when there was anyone else in the room, so clearly she'd have been in the room on her own, and that was when the eraser went missing. Now let's look at the facts we've established: Lauren was in the room, on her own, at the time when the eraser went missing. Before she was in the room on her own, the eraser was there, and after that it wasn't, and no-one else was there in between. We can only come to one conclusion from this evidence: it &lt;em&gt;was&lt;/em&gt; Lauren!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I explain that the court would reject my evidence since it was based on Lauren's assumed guilt. We can only say Lauren's guilty after we've used evidence to back that up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The trigonometrical identity 5 = 9&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5 = 9, so subtracting 7 we get&lt;br&gt;&#xA;-2 = 2, then squaring both sides we get&lt;br&gt;&#xA;4 = 4. Tick! True!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I use that in response to &quot;proofs&quot; of trigonometrical identities that conclude $3 \sec^2 x = 3 \sec^2 x$, true.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They pick up the rules of the game eventually through practice and being corrected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;When I &lt;em&gt;am&lt;/em&gt; explaining converses&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it's raining, there are clouds in the sky.&lt;br&gt;&#xA;Contrapositive: If there are no clouds in the sky, it isn't raining.&lt;br&gt;&#xA;Converse: If there are clouds in the sky it is raining.&lt;/p&gt;&#xA;" OwnerUserId="1340" LastActivityDate="2014-05-19T23:15:25.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2363" PostTypeId="2" ParentId="2342" CreationDate="2014-05-19T23:33:07.320" Score="4" Body="&lt;p&gt;As you will be unsurprised to learn, this is a common error for students learning about proofs. Two authors who write about this topic are A and J Selden; here is a citation and excerpt (pdf 472/598):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Selden, A., &amp;amp; Selden, J. (1987, July). Errors and misconceptions in college level theorem proving. In &lt;em&gt;Proceedings of the second international seminar on misconceptions and educational strategies in science and mathematics&lt;/em&gt; (Vol. 3, pp. 457-470). &lt;a href=&quot;http://files.eric.ed.gov/fulltext/ED293686.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Jl6C3.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;strong&gt;Side-note:&lt;/strong&gt; It seems that &quot;if it rains&quot; is a common example! I see AndrewC also includes the statement: &lt;em&gt;If it's raining, there are clouds in the sky&lt;/em&gt;. See also my example regarding the statement &lt;em&gt;I'll go unless it rains&lt;/em&gt; as posted on MESE &lt;a href=&quot;https://matheducators.stackexchange.com/a/1018/262&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have heard, in the past, instructors for proof-writing courses emphasize the last point about mathematical definitions using the &lt;em&gt;if&lt;/em&gt; convention when they really mean &lt;em&gt;if and only if&lt;/em&gt;. Moreover, this if vs. iff problem is mentioned by mweiss, who notes that many of the &quot;if, then&quot; propositions in eighth grade geometry are, in fact, biconditional. &lt;strong&gt;So:&lt;/strong&gt; We have here one suggested answer, which is that you emphasize when the word &quot;if&quot; is used to mean &quot;if, then&quot; and when it is used to mean &quot;if and only if&quot;; perhaps you give examples in which you ask students more often to prove a statement, and then to figure out whether or not it is biconditional. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Toscho's comment gives a nice example: &lt;em&gt;If the polygon $S$ is a square, then the polygon $S$ is a rectangle.&lt;/em&gt; Prove the statement. Is the converse true, too? Why or why not?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at your three points, I have now gone beyond 1 (by using the word &quot;converse&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You mention that you would like to teach somewhere between 2 and 3. I would actually &lt;em&gt;object&lt;/em&gt; to the idea of 2, insofar as the word &quot;opposite&quot; can have different meanings. In this particular context, you give the example of $+$ and $-$ as being &quot;opposites&quot; because they are &quot;inverses.&quot; Of course, in propositional logic, the implication $P \implies Q$ has both a &quot;converse&quot; (the subject of your question here) &lt;em&gt;and&lt;/em&gt; an &quot;inverse.&quot; Which one is the &quot;opposite&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are other subtle issues with figuring out the converse of a proposition. For an example from group theory, see:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hazzan, O., &amp;amp; Leron, U. (1996). Students' use and misuse of mathematical theorems: The case of Lagrange's theorem. &lt;em&gt;For the Learning of Mathematics, 16&lt;/em&gt;, 23-26. &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.111.1435&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;in which the authors write:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/jnh6v.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The phenomenon of experts vs. novices relating to mathematical statements/problems at different levels (i.e., on a structural vs. superficial level, respectively) is recurrent. For another example of this and related citations, I refer you to my MESE response &lt;a href=&quot;https://matheducators.stackexchange.com/a/1256/262&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for level 3: I think this is the best choice. I am quite confident that you can intersperse real-life examples among geometric examples and still keep everything digestable for eighth grade students. (Incidentally, when I took geometry as a ninth grader, we &lt;em&gt;began&lt;/em&gt; by covering propositional logic: implications, negations, and/or, truth tables, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt; Teach students that &quot;if $P$, then $Q$&quot; statements have as their &lt;em&gt;converse&lt;/em&gt; &quot;if $Q$, then $P$&quot;. Oftentimes, a true &quot;if, then&quot; statement run into by students will have its converse be true as well; however, this is not always the case. To help students understand this point, introduce the word &lt;em&gt;converse&lt;/em&gt; and repeatedly ask about &quot;if, then&quot; statements both &lt;em&gt;what&lt;/em&gt; their converse would be, and &lt;em&gt;whether&lt;/em&gt; the converse is true or false.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Finally:&lt;/strong&gt; An oft-incorporated technique in proof-writing classes is to show students examples of incorrect proofs. (See, e.g., Brendan Sullivan's MESE response &lt;a href=&quot;https://matheducators.stackexchange.com/a/336/262&quot;&gt;here&lt;/a&gt;.) If you are observing a particular error frequently (e.g., students using a proposition's converse erroneously) then try writing up sample &quot;proofs&quot; with this mistake and ask students to critique them.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-23T23:15:48.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2364" PostTypeId="2" ParentId="2358" CreationDate="2014-05-20T00:24:26.860" Score="15" Body="&lt;p&gt;In my opinion what is important is that one defer use of the term &lt;em&gt;indefinite integral&lt;/em&gt; until after one has seen definite integrals and the Fundamental Theorem of Calculus.  If you hear about &quot;indefinite integrals&quot; first, then it is only natural to &#xA;cognitively overlay definite integrals on top of indefinite integrals: it's an indefinite integral plus only a little more: $F(x)(+C) \mapsto F(b) - F(a)$. In other words, this seems to abet what is perhaps the single most frustrating cognitive process of calculus students: the tendency to regard the Fundamental Theorem of Calculus as a definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To help with this I try to use the term &lt;em&gt;antiderivative&lt;/em&gt; instead of &quot;indefinite integral&quot; as much as possible.  At some point I slip up because after all we are using the same integral sign for both: I &lt;em&gt;almost&lt;/em&gt; wish we didn't.  (Using almost identical notation for two &lt;em&gt;a priori&lt;/em&gt; incredibly different things which in the setting of the FTC become almost the same is one of the brilliant notational innovations of calculus.  It's up there with the in/famous $\frac{dy}{dx}$.)  I also begin discussion of antiderivatives while doing differential calculus.  Whenever you define an interesting operation in mathematics, it is an easy sell to consider the corresponding &lt;strong&gt;inverse problem&lt;/strong&gt;: what inputs, if any, lead to a given output?  For me, the uniqueness of antiderivatives up to a constant is one of the major applications of the Mean Value Theorem.  Generally at that time I mention the problem of &lt;em&gt;existence&lt;/em&gt; of antiderivatives of any continuous function and allude to the fact that to solve this problem we need &lt;em&gt;integration&lt;/em&gt;, by which I mean area functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(For many years I thought that you really did &quot;need&quot; integration to prove the existence of antiderivatives of continuous functions.  Rather recently I learned that there are proofs of this which do not use integration whatsoever.  The first person to give such a proof was apparently...Henri Lebesgue!  As usual there is a nice MONTHLY article about this...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also start with the notation $\int_a^b f$ for &quot;definite integrals&quot;.  I tack on the $dx$ only after doing FTC and introducing integration by change of variables (or, as I was taught in high school and still think of it in the one variable context: &quot;u-substitution&quot;), at which point I try to explain the sense in which the $dx$ is &quot;just&quot; a brilliant Leibnizian device to streamline the change of variables process.  (May the proponents of nonstandard analysis forgive me: I do not say anything about infinitesimals in my freshman calculus courses.)&lt;/p&gt;&#xA;" OwnerUserId="176" LastEditorUserId="176" LastEditDate="2014-05-20T06:39:29.873" LastActivityDate="2014-05-20T06:39:29.873" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2365" PostTypeId="2" ParentId="2317" CreationDate="2014-05-20T07:33:20.487" Score="2" Body="&lt;p&gt;A lot depends on what your students have seen earlier. It is, indeed, nearly impossible to do a good job in a single 2hr lecture, if you need to cover both differentiation and integration in polar coordinates in one go. The good ole spiral approach (bad pun intended) could work better. I describe how I do it, but unfortunately I don't know if this is any better. Alas I cannot cite any comparative studies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remedy: draw pictures! Lots of them! Insist that the students do so, too. It's nearly impossible to do well here if you don't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;IMO the basic problem is that students are not used to thinking in terms of polar coordinates, and polar coordinate curves, at all. In school they at least got to plot several lines, parabolas, trig functions, rational functions and such (YMMV). Thus they know what to expect when plotting $y=f(x)$ for some standard $f$. Not so in the polar world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After having banged my head against that wall for a year two I was in a position to partially redesign the 1st year calculus courses (read: write my own lecture notes). In these parts we do elementary functions, sequences, continuity and derivatives in the Fall, and the integration and series in the Spring. Except that my predecessor had done indefinite integrals at the end of the Fall semester. I moved that to the Spring and instead did differentiation on parametrized curves, including polar ones. I only set modest goals. While I did the obligatory example of tangents on a logarithmic spiral I largely only expected students to be able to plot in polar. A typical exam question is to determine which of given six polar curves come from which functions. They do well with four-leaved clovers and such, but even good students may err in noticing the difference between a left-handed and a right-handed spiral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, come Spring term and integration at least the good and intermediate students usually do well. Weaker brothers and sisters still think in terms of $xy$-plots out of inertia. Because they never drew those pictures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which brings us back to the pictures. Even relatively good students fail to draw enough of them. So the test question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find the area of the region bounded by the polar curves $r=\cos\varphi$ and $r=\sin\varphi$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;does not go as well as I had hoped. Way too often they only tried to solve the equation $\cos\varphi=\sin\varphi$. If only they had plotted it first and started thinking second.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit: Did I say that I think you need to make them plot curves in polar coordinates? Many of them! And you should give them examples showing how it is done. Also those that aren't in the textbook/lecture notes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Caveat: I'm not positive that the idea of introducing the students to polar coordinate curves at the end of a semester long course is best. Come December some of them are exhausted by the epsilons, and may not be receptive to learn about something they have never seen earlier. &lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2014-05-20T08:50:20.577" LastActivityDate="2014-05-20T08:50:20.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2366" PostTypeId="2" ParentId="352" CreationDate="2014-05-20T08:22:39.020" Score="18" Body="&lt;p&gt;I'm teaching a course of math for future primary school teachers and discussed with them a little bit this point.&lt;/p&gt;&#xA;&lt;p&gt;In my opinion teaching multiplication table is necessary up to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; and completely useless afterwards.&lt;/p&gt;&#xA;&lt;p&gt;The point is that the usual multiplication algorithm is devised exactly with this purpose in mind: you can multiply every pair of numbers as long as you know how to multiply two numbers which are both less than 10. So if you know the multiplication table up to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; by heart you know everything you need to multiply (and in fact also to divide) &amp;quot;everything&amp;quot;. I think this is an useful conceptual point. I do not mean you necessarily say something of this kind to first grade pupils but that you can, in such way, make them understand allows the power of certain algorithms.&lt;/p&gt;&#xA;" OwnerUserId="465" LastEditorUserId="12691" LastEditDate="2022-12-10T13:29:22.927" LastActivityDate="2022-12-10T13:29:22.927" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2367" PostTypeId="2" ParentId="352" CreationDate="2014-05-20T10:58:46.210" Score="9" Body="&lt;p&gt;The point of learning multiplication tables is not to get the product of two numbers. It is to be able to quickly recognize that a larger number can be factored into smaller parts that are easier to handle.&lt;/p&gt;&#xA;" OwnerUserId="1479" LastActivityDate="2014-05-20T10:58:46.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2368" PostTypeId="2" ParentId="2358" CreationDate="2014-05-20T12:13:38.507" Score="4" Body="&lt;p&gt;Both history and applications suggest doing definite integrals first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The applications of calculus to areas, volumes, moments, probabilities and expectations are all primarily definite integrals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The history of things which look like calculus begins with Archimedes's calculations of areas, which are again definite integrals.  And the things that developed first are usually the most obvious and most natural starting points.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do enough definite integrals numerically, it's easy to appreciate the utility of indefinite integrals afterwards.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-05-20T12:13:38.507" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2369" PostTypeId="1" AcceptedAnswerId="2372" CreationDate="2014-05-20T22:26:12.013" Score="2" ViewCount="2066" Body="&lt;p&gt;Let's say that a given course has $3$ exams that are worth $25\%, 35\%$ and $40\%$ each. When building and correcting the exam, do you give/take out points so that the total sums to $25,35$ or $40$ respectively, or do you score all of them on $100$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reasons for a $25,35$ and $40$ scoring system that I can think of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students know right away how much points they have cumulated;&lt;/li&gt;&#xA;&lt;li&gt;They know how much points each mistake is worth;&lt;/li&gt;&#xA;&lt;li&gt;If the teacher judges that some mistake should be penalized for $1\%$ of the grade each time, then it is always $1$ point out of your total, whatever that total might be.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Reasons for a all on $100\%$ scoring system that I can think of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A minus $2$ points might have less chance to get contested than a minus $0.5$.&lt;/li&gt;&#xA;&lt;li&gt;Students are more used to $100\%$ scale, and thus can better compare.&lt;/li&gt;&#xA;&lt;li&gt;This gives you the opportunity to give variable weights to your exams, should you need it for some reason, or allows for dynamic weighting (Say the best exam of the class or individually is worth more)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What do you guys use and why?&lt;/p&gt;&#xA;" OwnerUserId="805" LastActivityDate="2014-05-29T21:20:10.710" Title="On what total do you grade your exams?" Tags="|course-design|exams|grading|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2370" PostTypeId="2" ParentId="2369" CreationDate="2014-05-21T00:53:47.547" Score="1" Body="&lt;p&gt;I find it easier to asign points when the maximal grade is the same all around, be it an exam, quiz, or homework. It makes grades directly comparable too. To see what it means in terms of the final grade is just a simple proportion away (or stick into the appropiate formula, whatever).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-21T00:53:47.547" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2371" PostTypeId="2" ParentId="2369" CreationDate="2014-05-21T02:50:48.323" Score="3" Body="&lt;p&gt;My friend has a course design she likes. She knows the course will be worth 200 points total, and she knows exactly how many points to give to the two quizzes (10 ea), the midterm (50), the final (90), the clickers (10), and the homework (30). This works well for her, except she has to have an exact number of questions on every exam and homework, and she can't adjust assignments on a whim.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach the same course, and I like to keep things nimble. I have three midterms (10%, 25%, 10%) and a final (40%). I have regular online quizzes (10%) and random in-class assignments (5%). If I find my students are slow readers, I can reduce the number of questions on the next exam. If I suddenly want to have students turn in an index card with their group answer during a class, I can collect another assignment without worry. I can add a quiz retake whenever I want. I can make a quiz long or short. This works well for me, and I rarely get students asking me to figure out their grade for them. I will say that our course management system does the math for both me and the students, which helps the students track their progress. You may wish to stick to simple points if you don't have a good gradebook system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarize, do whichever suits your style, but always be happy to talk to your students about their grades.&lt;/p&gt;&#xA;" OwnerUserId="1207" LastActivityDate="2014-05-21T02:50:48.323" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2372" PostTypeId="2" ParentId="2369" CreationDate="2014-05-21T07:37:16.137" Score="8" Body="&lt;p&gt;&lt;strong&gt;You &lt;em&gt;should not&lt;/em&gt; design your exams, tests, whatever to have a special number of total points&lt;/strong&gt; in each exam, test, whatever. Reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It is mathematically totally irrelevant (as long as it isn't 0).&lt;/li&gt;&#xA;&lt;li&gt;We're talking about math courses:&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The students should be aware, that it's totally irrelevant.&lt;/li&gt;&#xA;&lt;li&gt;The students should be able to calculate their current standing on their own.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;It disturbs the evaluating in an exam, test, whatever.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Let me explan the last point:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume, that there are 23 items of equal importance in the exam. If you want to give 50 points in total, then each item will be worth approximately 2.17 points. If a task consists of 5 of these items and a student has got 3 of them, then his performance in this task is worth approximately 6.52 out of 10.87. This is just disturbing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The most natural way of assigning points&lt;/strong&gt; is to give each item, that the student should produce according to your expected solution, 1 point. If an item should be worth more or less than a standard item, use natural or easy fraction (0.5,0.25) weights.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the total number of points is 23 and the student got 3 out of 5 points in task (A). At the end, you can weigh the total number of points according to your plan for the whole course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That way, you support the didactic value of your exams, tests, whatever rather than their selective value.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-05-21T07:37:16.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2373" PostTypeId="1" CreationDate="2014-05-21T11:23:27.450" Score="13" ViewCount="1397" Body="&lt;p&gt;Let's be clear from the start - I'm NOT a maths educator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a reasonable aptitude for maths: I've passed my high-school level exams in the subject, taken a life science degree and gone on to work in computing. In my experience I can understand moderately complex mathematical techniques (such as calculus, or probability) when they're explained to me by a patient teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, my ability to recall mathematical concepts once learned is &lt;em&gt;appalling&lt;/em&gt;. I can literally spend an hour learning something and have forgotten how to answer questions on the subject within another hour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't have this experience with any other subject. Most things I read about, learn and can recall a decent amount without significant effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I've gotten older my struggle with remembering how to do things in maths has become more and more burdensome: I need it for aspects of my job, and now, as my daughter approaches high-school age, I need it to help her with her learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: I find mathematical concepts interesting, and I must possesses a reasonable level of capability. Is there anything I can do to help my ability to remember and apply mathematical concepts?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Examples were asked for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The worst example was several years ago when I joined a work study group. We were tasked with sitting through a lecture a week of Standford's Machine Learning course (&lt;a href=&quot;https://www.coursera.org/course/ml&quot;&gt;https://www.coursera.org/course/ml&lt;/a&gt;). Most of my colleagues were in a similar position to me having come from a numerate but not entirely mathematical academic background and didn't practice maths on a regular basis. Yet they were able to comprehend the course much better than me. Worse, they'd spend the study session explaining to me what I'd missed, and I'd feel like I understood, but then when I watched the lecture again that evening, it was all gibberish again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more recent - and less taxing - example was binomial distribution. I wanted to calculated the chance of a particular set of outcomes after some repeated trials so I looked up how to do it, and went through the maths, digging a little deeper into the algebra behind it before performing the calculation. A week later I wanted to do it again, and couldn't remember the first thing about how.&lt;/p&gt;&#xA;" OwnerUserId="833" LastEditorUserId="-1" LastEditDate="2014-05-21T23:39:58.303" LastActivityDate="2014-05-22T08:12:38.620" Title="Are there technique to help with poor recall?" Tags="|self-learning|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2374" PostTypeId="2" ParentId="2239" CreationDate="2014-05-21T11:27:15.743" Score="0" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Do teachers and books target more apt students?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am shocked at people either saying or implying that the answer is no.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have discussed this topic several times with my former fellow students and to me it's quite obvious the answer is yes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In first grade everyone, except for a few rare exceptions, learns the material. This is the kind of success rate any normal academic school should aim at.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In university, only a handful of people learn the material properly (and not just passing). How can people claim that the courses aren't out of the average students reach? They are so by definition. Arguments such as &quot;students just don't study&quot; aren't good because students don't study in first grade either, yet they learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With respect to books, the same happens. Take a regular mathematics book about a subject you don't know and you (unless you are one of &lt;em&gt;those&lt;/em&gt; handful of people) will struggle. Take a non-regular mathematics book like &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521675995&quot; rel=&quot;nofollow&quot;&gt;How to Prove It: A Structured Approach&lt;/a&gt; and things are very different. You can actually read this book like a novel. The level of detail is actually adequate for someone who doesn't know anything about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Picture yourself teaching a group of thirty people how to turn your water heater on. Would you really believe that you're teaching was adequate if only  two or three people really learned it and ten somehow managed to 'barely' turn it on while the others simply couldn't? The difference in difficulty between turning a water heater on and learning calculus is of no concern here, results are what matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Throughout my rationale I basically assumed that students should be learning without studying at home. I do not commit to this view, but if you reject it, I still maintain my position. If one removes people who don't study at home, there will still be a huge amount of people failing. (I can't support this claim with evidence, this is just something I observed during my undergraduate degree).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a more personal note, I had good grades throughout my BSc I had good grades, but I had to work hard to get them, so did my most of my friends who had good grades and I know about lots of people who worked hard and didn't get good grades. Wouldn't you expect, if you pick one of the best students, that he/she would be someone who learns easily? This isn't so...&lt;/p&gt;&#xA;" OwnerUserId="287" LastActivityDate="2014-05-21T11:27:15.743" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2375" PostTypeId="2" ParentId="2373" CreationDate="2014-05-21T14:11:06.670" Score="12" Body="&lt;p&gt;You may have been able to remember a lot of math for a short time, but you ma have lacked understanding. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you talk to people who excelled in school mathematics, they might be perplexed by your plight, or they may think you just didn't stick with it long enough. They might mistakenly tell you that you're just not good at math. However, traditional approaches that many of us experienced in school often especially privileged people with good memories or an ability to see patterns quickly, which might leave many of them to wonder why everyone does not come out of math education with the same insights they did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The phenomenon your talking about is addressed in the Learning Principle of the NCTM Principles and Standards for School Mathematics. That is to say, the phenomenon you are describing is a familiar thing to mathematics education researchers, and is something often seen when an approach to mathematics focuses on rote learning and memorization. Stylianides and Stylianides (2007) wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An important point set forth by the LP [Learning Principle] is that&#xA;  memorization of facts or procedures without  understanding often&#xA;  results in fragile learning. This remark corresponds to research which&#xA;  has  shown that mastery of facts and rote performance of procedures&#xA;  are not sufficient in thinking  mathematically (Schoenfeld, 1988),&#xA;  getting the right answers does not necessarily imply  mathematical&#xA;  proficiency (Erlwanger, 1973), and learning computational formulas is&#xA;  a poor  substitute for developing understanding of the underlying&#xA;  concepts (Pollatsek et al., 1981).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The learning principle, stated by the NCTM, can be found &lt;a href=&quot;http://www.nctm.org/standards/content.aspx?id=26807&quot;&gt;here&lt;/a&gt;. In short:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Research has solidly established the importance of conceptual&#xA;  understanding in becoming proficient in a subject. When students&#xA;  understand mathematics, they are able to use their knowledge flexibly.&#xA;  They combine factual knowledge, procedural facility, and conceptual&#xA;  understanding in powerful ways.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are a few models for understanding this phenomenon. One is that your view of math becomes more simple over time as you understand concepts that draw different aspects of mathematics together. It's a process of abstraction in which you have to remember fewer things, because abstraction compresses knowledge (that's one way to look at it). Another way to look at it is that greater understanding is not just about knowing more things, but it's about the connections among those things that you know. Without these connections, the things you have learned would remain in isolation, unuseful, neglected, and eventually (possibly) forgotten. A representation of this idea is in diSessa's (1988) Knowledge in Pieces, in which he talks about how people can have intuitive physics knowledge, but not a way of bringing that knowledge together into what we would see as understanding. Imagining these things in a sort of network, expert knowledge might look like a very efficient set of connections among ideas, tailored to the types of situations most useful to someone in a particular field (whether it be math or science).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;OK, but how do you become more expert? The Learning Principle is meant to address that by promoting the notion that teaching approaches should be mindful of how student understanding is not a given (or a goal, even) in some approaches to teaching mathematics. It tells us that avoiding the fragility you describe requires a conscious effort to make understanding an educational outcome, not just a hopeful result of the application of memory or a practiced facility with procedures. So, seek to learn from people who take this view and the results will be a less fragile knowledge of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One notable example, is Jo Boaler, who just happens to have an online course in which she addresses some of the issue of learning mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://class.stanford.edu/courses/Education/EDUC115N/How_to_Learn_Math/about&quot;&gt;How to Learn Math&lt;/a&gt; is her attempt to educate teachers and parents about the ongoing process of learning mathematics. It's a good start if you want to understand what you or your daughter need to do to learn for understanding, and what teachers can and should be doing (based on the last couple of decades of research). You should also get some idea of next steps to address areas of mathematics you're interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully, this has given you a way of understanding that your experience is not all that unusual, and &lt;em&gt;can&lt;/em&gt; be addressed. But how you learn is important in how you will be able to retain and use your knowledge.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Works Cited&lt;/p&gt;&#xA;&#xA;&lt;p&gt;diSessa, A. (1988). Knowledge in pieces. In G. Forman &amp;amp; P. B. Purfall (Eds.), Constructivism in the computer age (pp. 49–70). Hillsdale, NJ: Erlbaum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stylianides, A. J., &amp;amp; Stylianides, G. J. (2007). &lt;a href=&quot;http://www.math.umt.edu/tmme/vol4no1/TMMEv4n1a8.pdf&quot;&gt;Learning Mathematics with Understanding: A Critical Consideration of the Learning Principle in the Principles and Standards for School Mathematics&lt;/a&gt;. Montana Mathematics Enthusiast, 4(1).&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-05-21T14:11:06.670" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2376" PostTypeId="1" AcceptedAnswerId="2377" CreationDate="2014-05-21T14:31:40.510" Score="18" ViewCount="11559" Body="&lt;p&gt;In my class, what often happens is that some smart students do problems faster than other students. And they are actually going well ahead of my course plan. While I am teaching to other students a certain chapter, they are doing the next chapter to it. And in the meantime, while I am guiding the weak students, they ask me the question which they find difficult to do from the next section. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I kind of find that thing disturbing. They are calling me all the time and I am unable to give my time to other students. What should I do in this situation? Should I tell them to stop going ahead of me? If I tell them so, what other things can I tell them to do while I am teaching the weak students?&lt;/p&gt;&#xA;" OwnerUserId="1172" LastEditorUserId="143" LastEditDate="2014-05-23T20:42:47.777" LastActivityDate="2019-05-07T03:38:58.010" Title="How to deal with fast students without neglecting weaker ones" Tags="|classroom-management|gifted-students|general-pedagogy|" AnswerCount="7" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2377" PostTypeId="2" ParentId="2376" CreationDate="2014-05-21T14:44:00.370" Score="18" Body="&lt;p&gt;I don't think it's fair to tell the students to not read ahead. In fact, I'd encourage it. What you &lt;strong&gt;should&lt;/strong&gt; make clear is that the material in the current section takes precedence, and if somebody else has a question on the current material you'll sadly have to help those people before the questions on the next section.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a weaker student is raising their hand, even if you're in the middle of explaining the more advanced topic, simply ask the weaker student if it's about the current material. If it is, just say &quot;sorry, I have to get this question first, hold on to your thought.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they're really taking a &lt;strong&gt;lot&lt;/strong&gt; of time and they really do want questions answered, that is a good use of office hours and emails. Of course, you still have to put limits and make it clear that you won't answer 2000 emails a week, and that at office hours students on the current material still take precedence, but that's no big deal. In my experience, students react to this rather gracefully and office hours frequently will have a cadre of &quot;regulars&quot; asking advanced questions who stay quiet or even help out when a student with a basic or relevant question comes in.&lt;/p&gt;&#xA;" OwnerUserId="521" LastEditorUserId="521" LastEditDate="2014-05-22T11:05:49.913" LastActivityDate="2014-05-22T11:05:49.913" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2378" PostTypeId="2" ParentId="2373" CreationDate="2014-05-21T16:07:51.293" Score="8" Body="&lt;p&gt;Following points have always worked for me:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Learn the concepts, not the formulas.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Practice. If you really want something to stick in your brain, you have to convince the brain, you need it. The clue is not how many times you repeat something, but how many times you try to recall it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Only do one manageable theoretical step at a time and let yourself time to exercise it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Write down notes in a consistent manner to help your memory and do not hesitate to use the notebook, if your brain doesn't want to remember. The worst paper is still better than the best memory.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do not let non-understood or misunderstood topics aside. Take your time to disassemble them alone, or with help of someone who knows the topic better. Unless you try something really crazy, you can always find someone in the neighborhood, or online to help you further.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Read your notes at loud, when you need to reread them. Acoustic feedback will provide additional association layer to the information network in your head.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1520" LastEditorUserId="143" LastEditDate="2014-05-22T08:12:38.620" LastActivityDate="2014-05-22T08:12:38.620" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2379" PostTypeId="2" ParentId="2376" CreationDate="2014-05-21T16:32:33.207" Score="10" Body="&lt;p&gt;I use a technique I learned from my pilates teacher: in class activities, have different options for different ability levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have worksheets in almost all of my lessons.  I almost never cover all the problems in class (solutions are published later).  Some of the problems are there to challenge those who breeze through the easy problems, while I can spend time working with those still struggling with the easy problems.  &lt;/p&gt;&#xA;" OwnerUserId="446" LastActivityDate="2014-05-21T16:32:33.207" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2380" PostTypeId="2" ParentId="2373" CreationDate="2014-05-21T17:15:38.907" Score="5" Body="&lt;p&gt;This is an opinion.  Based on JPBurke's answer, I'd say that (as with everything memory-related) the problem is one of &lt;a href=&quot;http://www.sciencedaily.com/releases/2007/03/070314134812.htm&quot;&gt;associative memory&lt;/a&gt; (to give one reference that comes up quickly in Google).  Their answer suggests &lt;em&gt;understanding&lt;/em&gt; as a possible association, but this is a vague and insufficient concept if the comments are anything to go by.  My experience is that the relevant quantity is &lt;em&gt;interest&lt;/em&gt;, which helps the understanding remain active in my mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interest is not just appeal, but also a form of obsession or fascination.  It implies an urge to constantly revisit the subject outside of scheduled study sessions, turn it over and question it, rebuilding its foundations as you forget the received ones.  Being interested is more than &lt;em&gt;having an interest&lt;/em&gt;, as in the sense of career preparation, and more than just a pleasant reaction to the facts as received.  It's also more than &lt;em&gt;understanding&lt;/em&gt; in the sense that teachers often test in their students: the ability to reproduce known responses to standard challenges (even conceptual challenges: the interaction &quot;Why is $x = 1$ not a solution of $\sqrt x = x - 2$?&quot;, &quot;Because you have to check both solutions after squaring an equation&quot; reflects some competence with algebra, but misses the big picture).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anything you learn you have to teach yourself.  You may have heard it from a book first but that's only factual memorization, and unless you have an eidetic memory those facts won't be accessible after a short while.  To teach yourself, you need to think of questions and answer them.  I define &lt;em&gt;interest&lt;/em&gt; as &quot;having questions&quot;; this facilitates memory because the questions come from a place that existed before the material you're trying to absorb, and the answers connect that material with that place.  That's associative memory, and it's how the brain works.  It also works through repetition, but as you've noticed, learning based only on repetition is fragile.  Instead, I mean the kind of repetition I mentioned above: compulsively revisiting your understanding.  Keep it fresh in your mind, remind yourself of what you have already worked out, and the brain will (possibly literally) carve out a niche for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, this is my opinion, my experience, based on a superficial understanding of brain science that may as well be mysticism.  But there are a lot of facts I remember from classes I took more than ten years ago because I was inspired to dig into them, and also several classes that are complete blanks because nothing in them made me care.  You should try to learn not because you need to know, but because you want to be the kind of person who knows.&lt;/p&gt;&#xA;" OwnerUserId="513" LastActivityDate="2014-05-21T17:15:38.907" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2381" PostTypeId="2" ParentId="2376" CreationDate="2014-05-21T18:03:59.017" Score="8" Body="&lt;p&gt;My experience is that students that are ahead, and most importantly, asking questions too - are more likely to be in the position to be both independent learners, and able to take on peer-to-peer tasks related to teaching and learning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leads to a few options which might include:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Use the advanced students to help the slower students too, thus increasing the overall speed that the class is mastering new concepts.&lt;/li&gt;&#xA;&lt;li&gt;Require the more advanced students to ask a peer for help and then have them confirm the the answer reached was correct with you; if they're unable to reach an answer, clearly you'd help too.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1511" LastActivityDate="2014-05-21T18:03:59.017" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2382" PostTypeId="1" CreationDate="2014-05-21T21:09:02.297" Score="7" ViewCount="138" Body="&lt;p&gt;There is a wide variety of summer mathematics workbooks available for students in elementary and middle school (see &lt;a href=&quot;http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Daps&amp;amp;field-keywords=summer%20math%20workbook&quot;&gt;here&lt;/a&gt;). With summer coming up, these books seem to be available everywhere. Thus my questions are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Is there any evidence that suggests summer workbooks are beneficial?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;If so, which ones are the best?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1515" LastEditorUserId="1515" LastEditDate="2014-05-21T21:21:54.987" LastActivityDate="2016-02-04T00:53:20.273" Title="Summer Mathematics Workbooks for Elementary and Middle School" Tags="|self-learning|summer|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2383" PostTypeId="2" ParentId="2165" CreationDate="2014-05-21T21:44:27.557" Score="5" Body="&lt;p&gt;A good (arguably excellent) overview of pre-calculus mathematics can be found in &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1592441300&quot; rel=&quot;nofollow&quot;&gt;Precalculus Mathematics in a Nutshell&lt;/a&gt;&lt;/em&gt;, written by George F. Simmons. It is an excellent but terse resource. More thorough accounts for each topic can be found elsewhere if desired.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For algebra and trigonometry, you may wish to consider&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Israel Gelfand's &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0817636773&quot; rel=&quot;nofollow&quot;&gt;Algebra&lt;/a&gt;&lt;/em&gt; and &#xA;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0817639144&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Trigonometry&lt;/em&gt;&lt;/a&gt;,&lt;/li&gt;&#xA;&lt;li&gt;Serge Lang's &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387967877&quot; rel=&quot;nofollow&quot;&gt;Basic Mathematics&lt;/a&gt;&lt;/em&gt; (this is another good overview, but it's strongest in algebra).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For geometry,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Martin Isaac's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0821847945&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Geometry for College Students&lt;/em&gt;&lt;/a&gt;,&lt;/li&gt;&#xA;&lt;li&gt;HSM Coxeter's &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0883856190&quot; rel=&quot;nofollow&quot;&gt;Geometry Revisited&lt;/a&gt;&lt;/em&gt;,&lt;/li&gt;&#xA;&lt;li&gt;Gerard Venema's &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0136020585&quot; rel=&quot;nofollow&quot;&gt;Foundations of Geometry&lt;/a&gt;&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There is no Calculus in Simmon's text mentioned above. However, many excellent treatments can be found elsewhere. These include,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Michael Spivak's &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0914098918&quot; rel=&quot;nofollow&quot;&gt;Calculus&lt;/a&gt;&lt;/em&gt;,&lt;/li&gt;&#xA;&lt;li&gt;Tom Apostol's &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471000051&quot; rel=&quot;nofollow&quot;&gt;Calculus, Vol 1&lt;/a&gt;&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="52" LastEditorUserId="52" LastEditDate="2014-05-21T22:54:43.967" LastActivityDate="2014-05-21T22:54:43.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2385" PostTypeId="2" ParentId="2376" CreationDate="2014-05-21T23:09:05.757" Score="5" Body="&lt;p&gt;I would never advise you to tell your students not to read ahead, they are amazing for having the motivation to do that!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on the number of students I think there are a number of possible solutions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1-5) if the &quot;ahead&quot; students are relatively balanced in their abilities it might be valuable to form them into a group. Encourage them to work with each other to solve through problems.  This leaves more time for you, ensures that you don't have to re-explain concepts, and offers them work companions that are more frequently available.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6-10) I would probably try to form groups of 3-5 where skills are similarly matched (or if students have a desire to work with each other that should be fine, they are ahead of the course content after all)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;10) I would suggest splitting your class or seriously considering how the material is being covered.  With a large fraction of students significantly ahead of the course content they should probably be placed in a higher level course (or the other students should be placed in a lower level).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1179" LastActivityDate="2014-05-21T23:09:05.757" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2386" PostTypeId="1" CreationDate="2014-05-21T23:23:23.513" Score="15" ViewCount="14329" Body="&lt;p&gt;What is the perfectly (maybe unrealistically) ideal undergraduate sequence for a undergraduate majoring in pure mathematics who takes 2-3 mathematics courses per semester assuming a strong AP Calculus BC background and wishes to go on to graduate school for a PhD in mathematics?&lt;/p&gt;&#xA;" OwnerUserId="1516" LastActivityDate="2017-10-29T00:55:53.003" Title="Ideal Undergraduate Sequence" Tags="|undergraduate-education|curriculum|" AnswerCount="6" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2387" PostTypeId="2" ParentId="2382" CreationDate="2014-05-21T23:51:48.577" Score="4" Body="&lt;p&gt;What exactly do you want to accomplish? If it is helping a child who did poorly, I'd guess the most important ingredient is a patient tutor (probably with the same texts used during the year). If they are interested in looking deeper, perhaps a text book isn't ideal. If it is skipping ahead, better let them have their well-deserved vacation.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-21T23:51:48.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2388" PostTypeId="2" ParentId="2386" CreationDate="2014-05-22T01:09:25.063" Score="4" Body="&lt;p&gt;Here's my opinion/wish. For some of the classes the chronological order doesn't matter. For some it probably does and/or I might not have the best order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Semester 1:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus I (Differential Calculus),&#xA;Discrete Math,&#xA;Programming course&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Semester 2:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus II (Integral Calculus, Taylor Series),&#xA;Linear Algebra,&#xA;Physics I (statics, motion, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Semester 3:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multi variable Calculus,&#xA;Foundations of Math (Logic, proof techniques, set theory),&#xA;Physics II (E&amp;amp;M, special relativity, maybe basic quantum)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Semester 4:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Differential Equations,&#xA;Number theory,&#xA;Probability,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Semester 5:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Theory of Complex Variables,&#xA;Topology,&#xA;Statistics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Semester 6:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chaos/Dynamics,&#xA;Graph theory,&#xA;The thing I'm forgetting&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Semester 7:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analysis I (Sequences, more rigorous treatment of calculus),&#xA;Abstract Algebra (Group theory),&#xA;Independent research project (first half)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Semester 8:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analysis II (Sequences of functions, basic measure theory),&#xA;Abstract Algebra II (Rings),&#xA;Independent research project (second half)&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-05-22T01:09:25.063" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2389" PostTypeId="2" ParentId="2386" CreationDate="2014-05-22T02:06:51.920" Score="12" Body="&lt;p&gt;I wonder if we (or some formal international body) should rethink the entire curriculum? &lt;em&gt;Aeryk&lt;/em&gt;'s very logical and reasonable curriculum is focused totally on content, as opposed to what we want the students to learn along the way. In other words, it is not only sequence that matters!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am particularly concerned about three aspects that disappear in content lists:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the notion of what constitutes a &lt;strong&gt;&lt;em&gt;proof&lt;/em&gt;&lt;/strong&gt;.  &lt;/li&gt;&#xA;&lt;li&gt;the high-level view&#xA;of mathematics, with an active &lt;strong&gt;&lt;em&gt;frontier&lt;/em&gt;&lt;/strong&gt; being pushed daily, sometimes&#xA;even by undergraduates.&lt;/li&gt;&#xA;&lt;li&gt;This is likely already covered in any&#xA;carefully considered curriculum, but the &lt;strong&gt;&lt;em&gt;historical context&lt;/em&gt;&lt;/strong&gt; in which&#xA;mathematics emerges adds &lt;em&gt;so&lt;/em&gt; much to understanding the content&#xA;material.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="511" LastEditorUserId="5220" LastEditDate="2017-10-29T00:55:53.003" LastActivityDate="2017-10-29T00:55:53.003" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2390" PostTypeId="2" ParentId="2386" CreationDate="2014-05-22T06:03:03.487" Score="3" Body="&lt;p&gt;Ideally, well, in retrospect, call multiple disciplinary studies bluff and use their flexibility to eliminate the fluff. All the sudden, we have time for 4-5 technical courses a semester. A hard path, but, if we stay on it where does it take us? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 1:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;calculus III, introduction to proofs course, elementary number theory, discrete math&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 2:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;ODEs, basic matrix theory course, complex analysis, physics I&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 3:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;PDEs, algebra I, real analysis, linear algebra, physics II&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 4:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;basic measure theory, algebra II, multivariate analysis, elementary differential geometry&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 5:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;algebra III, point-set topology, numerical methods, variational calculus&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 6:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;algebra IV, algebraic topology, operations research, history of math with a focus on nineteenth century math which is still being developed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 7:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;algebra V, manifolds ala John Lee's text, combinatorics, classical mechanics based on variational approach&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 8:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;geometry of differential forms as in Shigeyuki Morita's text, analytic number theory, chaos and dynamics, electromagnetism with vector calculus approach&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I had one semester which was like the ones I list above. Once upon a time, I took a semester with 5 pure courses in math and physics. It was a pure joy. However, reality intervened and after that, the need to satiate graduation requirements diluted the joy with courses from other schools. The sequence I list above is slanted towards a particular direction of differential geometry or perhaps mathematical physics. If you were otherwise inclined, you would want to take more number theory or programming much earlier. By algebra I,II, III, IV, V I intend a elementary treatment from something like Gallian followed by a few semesters digging through Dummit and Foote. Also, if you could find a few foundations courses in set theory or categories it would be nice at some point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Honestly, it really depends on what you want to do. Also, it depends strongly on what can be done at your school if you go the traditional route. The path of independent study is worth investigating if your school leaves you wanting more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like it or not, summer research is a good move for the sake of seeing outside your school. Moreover, earlier is better, some programs are looking for students without too much course work. Those could be a good gateway. It is not easy to get into these so plan to do some research about where and what you want to attempt. Personally, I ignored an opportunity I had to go with a professor to Los Alamos and work in mathematical physics. Who knows, if I had been less of an idiot and taken the opportunity I might still be in physics. But, life is good, just saying, don't pass up such opportunities lightly.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-22T06:03:03.487" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2391" PostTypeId="2" ParentId="2386" CreationDate="2014-05-22T06:45:12.673" Score="10" Body="&lt;p&gt;Taking inspiration from @Aeryk, here is my (as realistic as possible) dream proposal for a good student at a good university but &lt;strong&gt;without access to graduate level classes&lt;/strong&gt; (such as at a LAC):&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 1&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mathematical Problem Solving Seminar I (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521675995&quot; rel=&quot;noreferrer&quot;&gt;Velleman&lt;/a&gt;; &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387961712&quot; rel=&quot;noreferrer&quot;&gt;Larson&lt;/a&gt;; &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0914098918&quot; rel=&quot;noreferrer&quot;&gt;Spivak&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Multivariable Calculus (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387964053&quot; rel=&quot;noreferrer&quot;&gt;Lang&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Either the most advanced physics or computer science course you can take (don't bother with physics if it isn't calculus based)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 2&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Elementary Number Theory (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/048646931X&quot; rel=&quot;noreferrer&quot;&gt;Dudley&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Linear Algebra I (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0980232716&quot; rel=&quot;noreferrer&quot;&gt;Strang&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Ordinary Differential Equations (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0470458313&quot; rel=&quot;noreferrer&quot;&gt;Boyce-DiPrima&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 3&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Real Analysis I (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/007054235X&quot; rel=&quot;noreferrer&quot;&gt;Rudin&lt;/a&gt; 1-7)&lt;/li&gt;&#xA;&lt;li&gt;Linear Algebra II (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387982582&quot; rel=&quot;noreferrer&quot;&gt;Axler&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Probability (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/032179477X&quot; rel=&quot;noreferrer&quot;&gt;Ross&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 4&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Real Analysis II (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0201315963&quot; rel=&quot;noreferrer&quot;&gt;Munkres&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Combinatorics (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521457610&quot; rel=&quot;noreferrer&quot;&gt;Cameron&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Introduction to Topology (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0131816292&quot; rel=&quot;noreferrer&quot;&gt;Munkres&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 5&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Complex Analysis (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/3540939822&quot; rel=&quot;noreferrer&quot;&gt;Freitag-Busam&lt;/a&gt; I-IV)&lt;/li&gt;&#xA;&lt;li&gt;Algebra I (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471010901&quot; rel=&quot;noreferrer&quot;&gt;Herstein&lt;/a&gt; 1-3)&lt;/li&gt;&#xA;&lt;li&gt;Independent Study&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 6&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Lebesgue Theory (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0763717088&quot; rel=&quot;noreferrer&quot;&gt;Jones&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Algebra II (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471010901&quot; rel=&quot;noreferrer&quot;&gt;Herstein&lt;/a&gt; 4-6)&lt;/li&gt;&#xA;&lt;li&gt;Number Theory II (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/038797329X&quot; rel=&quot;noreferrer&quot;&gt;Ireland-Rosen&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 7&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Algebraic Topology (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521795400&quot; rel=&quot;noreferrer&quot;&gt;Hatcher&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Independent Study&lt;/li&gt;&#xA;&lt;li&gt;Independent Research&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester 8&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mathematical Logic (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1441906142&quot; rel=&quot;noreferrer&quot;&gt;Manin&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Mathematical Problem Solving Seminar II (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387008926&quot; rel=&quot;noreferrer&quot;&gt;Souza-Silva&lt;/a&gt;; &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/9814304964&quot; rel=&quot;noreferrer&quot;&gt;Ta-Tsien&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Independent Research&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In the last two or three semesters, one may wish to take a fourth math class that piques ones interest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are just my initial thoughts -- please offer any criticism and suggestions in the comments.&lt;/p&gt;&#xA;" OwnerUserId="52" LastEditorUserId="52" LastEditDate="2014-05-25T06:05:10.557" LastActivityDate="2014-05-25T06:05:10.557" CommentCount="8" CommunityOwnedDate="2014-05-22T22:24:33.847" ContentLicense="CC BY-SA 3.0" />
  <row Id="2393" PostTypeId="2" ParentId="2386" CreationDate="2014-05-22T15:27:37.357" Score="8" Body="&lt;p&gt;Oh, this is quite straightforward. First, go to a school with quarters rather than semesters: this gives you 50% more classes to take. Then, take every upper-division class and anything marked as &quot;honors&quot;, in order of prerequisites and interest. Take graduate classes if you run out. &lt;/p&gt;&#xA;" OwnerUserId="513" LastActivityDate="2014-05-22T15:27:37.357" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2394" PostTypeId="2" ParentId="2386" CreationDate="2014-05-22T16:31:03.540" Score="4" Body="&lt;p&gt;For whatever it's worth, below is roughly the schedule followed by the &lt;strong&gt;5 or 6 top math students&lt;/strong&gt; over a 4-year period where I was an undergraduate in the late 1970s. Two went to Berkeley, one went to Princeton, one (who was a top 30 Putnam scorer) left math for riches in computer science, and one went to MIT for a Ph.D. in mechanical engineering. Incidentally, I know of at least one person from this time period who DID NOT follow this path (took advanced calculus in 3rd year, maybe two graduate level classes in 4th year) who is currently a full professor at a Group II university (I just checked).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm only listing math courses below, but most took several computer science and physics courses. Two of them, maybe three, took pretty much all of the undergraduate theoretical physics courses (2-semester sequence in classical mechanics, 2-semester sequence in electrodynamics, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/048648596X&quot; rel=&quot;nofollow&quot;&gt;Saxon level&lt;/a&gt; quantum mechanics, solid state physics, optics) along with two or three graduate physics courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In some cases the text varied in a predictable way depending on who taught the course. For example, Professor X would use Do Carmo in years n, n + 2, n + 4, etc. and Professor Y would use O'Neill in years n+1, n+3, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1st Year (Fall, Spring)&lt;/strong&gt;&#xA;Apostol's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471000078&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Calculus, Volume 2&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2nd year (Fall)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hoffman/Kunze's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0135367972&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Linear Algebra&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taylor/Mann's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471025666&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Advanced Calculus&lt;/strong&gt;&lt;/a&gt; (first few chapters and the last part on sequences, series, improper integrals, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2nd Year (Spring)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Standard cookbook ODE course [some may have skipped this]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fleming's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387902066&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Functions of Several Variables&lt;/strong&gt;&lt;/a&gt; [Special pull-out reading course for top 2 or 3 students each year taking the second semester of the Taylor/Mann Advanced Calculus sequence.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3rd and 4th Years (order taken varied; only one or two took all of these)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Royden's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/013143747X&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Real Analysis&lt;/strong&gt;&lt;/a&gt; [offered in Fall]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do Carmo's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0132125897&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Differential Geometry of Curves and Surfaces&lt;/strong&gt;&lt;/a&gt; OR O'Neill's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0120887355&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Elementary Differential Geometry&lt;/strong&gt;&lt;/a&gt; [offered in Fall]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Munkres' &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0131816292&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Topology&lt;/strong&gt;&lt;/a&gt; (entire book covered except for Stone-Cech compactification, some of the metrization results, and dimension theory) AND much of Massey's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387902716&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Algebraic Topology: An Introduction&lt;/strong&gt;&lt;/a&gt; [offered in Spring]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hirsch/Smale's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0123495504&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Differential Equations, Dynamical Systems, and Linear Algebra&lt;/strong&gt;&lt;/a&gt; [offered in Fall]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lang's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/038795385X&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Algebra&lt;/strong&gt;&lt;/a&gt; OR Hungerford's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387905189&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Algebra&lt;/strong&gt;&lt;/a&gt; [2-semester Fall-Spring sequence]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Warner's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387908943&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Foundations of Differentiable Manifolds and Lie Groups&lt;/strong&gt;&lt;/a&gt; OR Spivak's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0914098705&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;A Comprehensive Introduction to Differential Geometry (Volume 1)&lt;/strong&gt;&lt;/a&gt; [offered in Spring]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides courses from those above, these students took one or two other &quot;special topics&quot; graduate courses that were offered at irregular times, such as calculus of variations and ergodic theory.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-05-22T16:31:03.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2395" PostTypeId="1" AcceptedAnswerId="2716" CreationDate="2014-05-22T17:01:37.313" Score="8" ViewCount="272" Body="&lt;p&gt;I'm looking for a BYOD (Bring Your Own Device) student response system (like Socrative, Top Hat, Learning Catalytics, Pingo, etc.) with the following three properties:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It is free for both the instructor and students.&lt;/li&gt;&#xA;&lt;li&gt;It can display mathematical formulas.&lt;/li&gt;&#xA;&lt;li&gt;Students register and sign in individually, so that it can be used to track attendance/participation or give graded quizzes.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Is there such a thing?&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="261" LastEditDate="2014-05-30T20:49:48.537" LastActivityDate="2014-06-17T09:53:00.667" Title="A free BYOD response system that registers students?" Tags="|technology-in-education|software|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2396" PostTypeId="2" ParentId="2395" CreationDate="2014-05-22T18:13:31.307" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://moodle.org&quot; rel=&quot;nofollow&quot;&gt;Moodle&lt;/a&gt; is free (open source), it can be coerced to use MathJax, and can give quizzes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We use it here, but personally just as a way to communicating with the students and collecting homework. I haven't used its features for reporting grades back, nor have I looked at its facilities for giving automated quizzes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure that the work involved in creating a database of questions and answers by topic, all more or less equally dificult, will be formidable. Unless it is a really massive class, repeated more or less the same each year (also with &lt;em&gt;lots&lt;/em&gt; of help comming up with problems, checking them, and entering them into the system) it's not worth it. Then again, the larger the class, the larger the required database of questions so there aren't too many repeats...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-22T18:13:31.307" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2397" PostTypeId="1" AcceptedAnswerId="2399" CreationDate="2014-05-23T00:45:39.183" Score="19" ViewCount="2069" Body="&lt;p&gt;When I was self-studying as an undergraduate, I would spend up to two weeks working on a single problem or trying to understand a proof in Rudin's Principles of Mathematical Analysis. I realize now that I wasted a lot of time and could have learned a lot more if I had spent less time struggling and asked other people for help/looked up with answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Currently I only try to spend about 30 minutes to an hour on a single problem before asking for help. Is this reasonable?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How much time should one spend working on a single question? And by question, I mean ones that you would find in upper level undergraduate and beginning graduate level textbooks. &lt;/p&gt;&#xA;" OwnerUserId="1523" LastEditorUserId="143" LastEditDate="2014-09-21T10:21:16.063" LastActivityDate="2014-09-21T10:21:16.063" Title="How much time to spend on a single question?" Tags="|undergraduate-education|homework|general-pedagogy|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2398" PostTypeId="2" ParentId="2395" CreationDate="2014-05-23T00:57:26.810" Score="3" Body="&lt;p&gt;I think the answer right now is &quot;no&quot;. I've either played with or actually used all (I think) of the currently-available CMS's and many of them have two out of the three features you used -- e.g. Learning Catalytics does well with math and handles individual students, but isn't free -- but not all three, not yet anyway. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, I would love to see the open source community tackle this as a project. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2014-05-23T00:57:26.810" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2399" PostTypeId="2" ParentId="2397" CreationDate="2014-05-23T01:32:44.273" Score="18" Body="&lt;p&gt;The key is not just that you spend time on a problem - it is normal to struggle with problems at the level you have described - but that you do so in a productive way. In fact, the notion of &lt;em&gt;productive struggle&lt;/em&gt; is present in the literature on Mathematics Education; see, for example, my earlier MESE response &lt;a href=&quot;https://matheducators.stackexchange.com/a/878/262&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting source to consult is the Schoenfeld article &quot;What's All the Fuss About Metacognition?&quot; available on google books &lt;a href=&quot;http://books.google.com/books?id=1oP2aqI9LBsC&amp;amp;lpg=PA189&amp;amp;dq=what%27s%20all%20the%20fuss%20about%20metacognition&amp;amp;pg=PA189#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. Unfortunately, the images are not present in the link given; below is a particular diagram from the article, &lt;em&gt;Stages. time spent on each stage, sequencing, and management activity in the Expert Group (of Problem Solvers)&lt;/em&gt;, and the text corresponding to it:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/PQJLQ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point I wish to draw your attention to is the second feature discussed above. Namely, that expert problem solvers &lt;em&gt;ask themselves questions about their own thinking&lt;/em&gt; while struggling with a problem. This metacognitive feature (&lt;em&gt;metacognition&lt;/em&gt;, by the way, refers roughly to &lt;em&gt;thinking about thinking&lt;/em&gt;) is often enough to &lt;em&gt;distinguish&lt;/em&gt; expert and novice problem solvers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If these ideas are interesting to you more generally, then I suggest looking up Schoenfeld's (1985) &lt;em&gt;Mathematical Problem Solving&lt;/em&gt;. The area that you are asking about would be found under the discussion of &lt;em&gt;control&lt;/em&gt;, though the sense that 30 minutes is enough time to spend on a single problem would be quite relevant to his discussion of &lt;em&gt;beliefs and belief systems&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find a bit more discussion around these sources in an earlier MO response of mine &lt;a href=&quot;https://mathoverflow.net/a/135192/22971&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt; If you are looking for an actual &lt;em&gt;number&lt;/em&gt; of minutes that you should spend on an individual problem, then I cannot answer this in any sort of generality. Instead, I suggest engaging with the problem as long as you can carry out an idea &lt;strong&gt;or&lt;/strong&gt; recognize an idea is not working and then try to think up a new approach. (In the Mathematics Education literature, sometimes a &lt;em&gt;problem&lt;/em&gt; is distinguished from an &lt;em&gt;exercise&lt;/em&gt; by defining the former as: a question for which the method of solution is unknown at the outset.) Sometimes a single sitting is not the best way to do this; instead, you might walk away from the problem and come back later when you are ready to try some new method. There is a literature on this idea of taking a break from a problem that you are trying to solve (Wallas (1926) calls this the &lt;em&gt;incubation&lt;/em&gt; period) and much more to say in this direction, but let me conclude my response here at the risk of going off too far on a tangent.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-05-23T01:39:01.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2400" PostTypeId="2" ParentId="1691" CreationDate="2014-05-23T02:28:12.207" Score="3" Body="&lt;p&gt;(Community wiki.) I think a quite relevant answer to this can be found at MSE, by Willie Wong, &lt;a href=&quot;https://math.stackexchange.com/a/21540&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his answer, WW mentions as a source early work by Hadamard, which was more recently written up as &lt;em&gt;Lectures on Cauchy's problem in linear partial differential equations&lt;/em&gt;; this latter text is available, for free and in full, at the archives &lt;a href=&quot;https://archive.org/details/lecturesoncauchy00hadauoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, I re-iterate from my comment above that the trichotomies post on MathOverflow might be of interest. In particular, see Alexandre Eremenko's answer &lt;a href=&quot;https://mathoverflow.net/a/120629/22971&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-05-23T02:28:12.207" CommentCount="0" CommunityOwnedDate="2014-05-23T02:28:12.207" ContentLicense="CC BY-SA 3.0" />
  <row Id="2401" PostTypeId="2" ParentId="2336" CreationDate="2014-05-23T04:45:53.333" Score="3" Body="&lt;p&gt;In my 75-min algebra 2 class, I use this schedule daily: 5 min for warmup, 35 min for individual presentations to the class of homework solutions, up to 10 min for direct instruction if needed from me, 20 min for classwork in groups, and 5 for some quick exit exercise that I collect as a check on understanding. It's been working well for me; I got the basic structure from Lee Mahavier, a very experienced and successful teacher of secondary ELL's in Georgia. &lt;/p&gt;&#xA;" OwnerUserId="1525" LastActivityDate="2014-05-23T04:45:53.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2402" PostTypeId="2" ParentId="2006" CreationDate="2014-05-23T05:34:07.487" Score="4" Body="&lt;p&gt;Fawcett did a year-long experiment teaching high school geometry with the explicit aim of, not only teaching the content of geometry, but also transferring to students a new life-long behavior of considering propositions in all life arenas in terms of definitions, undefined terms and so on. Harold P. Fawcett, Ph.D. The Nature of Proof: The thirteenth annual yearbook of the National Council of Teachers of Mathematics, Teachers College, New York, 1938. 146pp. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://cut-the-knot.org/ctk/NatureOfProof.shtml&quot; rel=&quot;nofollow&quot;&gt;Cut-the-knot&lt;/a&gt; has a write up of the book, and I have written a few &lt;a href=&quot;http://dangoldner.wordpress.com/2012/07/08/the-greatest-course-ever/&quot; rel=&quot;nofollow&quot;&gt;notes&lt;/a&gt; about it too.&lt;/p&gt;&#xA;" OwnerUserId="1525" LastEditorUserId="1525" LastEditDate="2014-05-23T05:43:20.120" LastActivityDate="2014-05-23T05:43:20.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2403" PostTypeId="2" ParentId="397" CreationDate="2014-05-23T05:41:00.937" Score="6" Body="&lt;p&gt;An excellent and comprehensive reference on this question: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0470952121&quot; rel=&quot;nofollow&quot;&gt;Student Writing in the Quantitative Disciplines: A Guide for College Faculty Paperback&lt;/a&gt; by Patrick Bahls. It helped me design a lot of exercises I use in my high-school classes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, the best tip I've received for giving students helpful direction and feedback in a time-efficient way is to develop your use of rubrics. See this &lt;a href=&quot;http://dangoldner.wordpress.com/2012/08/25/how-an-ela-teacher-saved-my-life/&quot; rel=&quot;nofollow&quot;&gt;short post&lt;/a&gt; for details. &lt;/p&gt;&#xA;" OwnerUserId="1525" LastEditorUserId="1525" LastEditDate="2014-05-24T02:48:16.137" LastActivityDate="2014-05-24T02:48:16.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2404" PostTypeId="2" ParentId="2220" CreationDate="2014-05-23T12:46:24.770" Score="6" Body="&lt;p&gt;Czech Republic&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bachelor Study:&#xA;1st year&#xA;Mathematical analysis 1 (8 hours a week): Logic, Sets, Continuity, differentiation, Taylor series.&#xA;Algebra 1 (6 hours a week): Groups, Fields, Vector spaces, Bases, Linear mappings, Matrices, equations, determinants.&#xA;Foundations of programming I(2 hours)&#xA;Mathematical analysis II.(8 hours a week): Newton-Leibniz integral. Riemann Integral, techniques of integration.&#xA;Algebra II(6 hours a week): Matrix decompositions, linear forms, bilinear and quadratic forms. Multilinear algebra.&#xA;Foundations of programming II(2 hours)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2nd year:&#xA;Mathematical analysis III: Calculus on vector spaces, Gateux derivative, Frechet derivative, derivatives of higher order. Lagrange multipliers. Fourier series and transform.&#xA;Geometry: Manifolds, basics of topology, connections, curvature, Riemannian manifolds, Lie Derivative&#xA;Mathematical analysis IV: Measure theory, Lebesgue integral, Fubini theorem, manifolds, differential forms, chains, Stokes theorem.&#xA;Numerical mathematics: Basic interpolation, aproximations, SVD...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3rd year: &#xA;Topology: Definition, open and closed sets, factor spaces, Tychonoff theorem, Nets, Filters.&#xA;Ordinary differential Equations: clasification, methods of solution, stability os solutions, manifold of solutions.&#xA;Functional analysis 1: Topological vector spaces, Hahn-Banach theorem and other great theorems, functionals, operators.&#xA;Functional analysis 2: F-spaces, Frechet spaces, convex analysis, Hilbert spaces.&#xA;Diff. Geometry 1: Manifolds, connections, Covariant derivative, geodesics.&#xA;Diff. Geometry 2: Lie groups and algebras&#xA;Partial differential equations 1: Classification of PDE, classical theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Graduate:&#xA;Partial differential equations II: Distributions, LAx - Milgram lemma, Sobolev spaces, FEM.&#xA;Algebraic topology I: Category theory, fundamental groups, simplices&#xA;Algebraic topology II: Homology and cohomology&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And much more&lt;/p&gt;&#xA;" OwnerUserId="1528" LastActivityDate="2014-05-23T12:46:24.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2405" PostTypeId="2" ParentId="397" CreationDate="2014-05-23T15:00:15.637" Score="9" Body="&lt;p&gt;These are all good answers, however, let me add two general points. These points are more about the general idea of this question than the guiding example of calculus II.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Good writing comes from good reading. Generally, we need to encourage our students to make a continual habit of reading technical and nontechnical mathematical treatise. Ideally, some of this is not directly connected to completing a homework or getting a grade in some course. The larger goal, become a scholar in math. From the technical side, we read to learn more about what is out there and in the process, see how we express our mathematical ideas. From the non-technical side, popular books help give you ideas about how we can repackage our ideas to be understood by people who do not have the luxury of our mathematical lexicon.&lt;/li&gt;&#xA;&lt;li&gt;Good writing does not come from online homework and/or emporium style mathematics. If you don't know already, an &quot;emporium&quot; is a context where the students are forced to go to the emporium to watch instructional videos, read e-texts, e-tutorials etc... then complete the online homework while in house. There is little or at least almost no note-taking in this format of instruction. Moreover, there is almost no written homework. In principle, free of this format, you might take the opportunity to force them to write grammatically complete sentences. However, without a major reformat to include a human grading system, it is not possible to grade written work in this format of instruction. In particular, it is antithetical to the popular efficiency of the model to burden it with extra human interaction. So, generically, work to keep human grading wherever possible. The convenience of e-grading is seductive, but, it robs us of instructional opportunities. In short, if the students &quot;do&quot; the assignment they may be able to &quot;do&quot; the math relative to some metric, but, they might not be able to write even a few simple sentences explaining what they are doing. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-23T15:00:15.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2406" PostTypeId="1" CreationDate="2014-05-23T15:38:37.723" Score="12" ViewCount="258" Body="&lt;p&gt;Currently we are using &lt;a href=&quot;http://moodle.org&quot;&gt;Moodle&lt;/a&gt; to manage our courses here. It has meant structuring the publication and turn in of homework, and has helped in distributing lecture notes and shorter documents to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Moodle has also functionality like quizzes and a question bank. What are experiences with those? How hard is it to write quizzes (particularly considering the plethora of weird notations we are so fond of)? I have quite a collection of problems (old exams, homework, filched from MSE, ...), mostly written in LaTeX. Any tools to help migrate (if it even makes sense)? How effective is the reuse of questions from one term to the other, does sharing material among colleagues work smoothly in your experience?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking a bit further, do you think such quizzes encourage cheating too much? It is quite simple to get somebody else to answer for me, so perhaps this should be used more for self-assessment than grading?&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="1892" LastEditDate="2014-07-09T19:41:08.393" LastActivityDate="2014-07-09T19:41:08.393" Title="Effective use of Moodle (or similar)" Tags="|undergraduate-education|technology-in-education|general-pedagogy|software|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2407" PostTypeId="5" CreationDate="2014-05-23T20:51:29.507" Score="0" Body="" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-05-23T20:51:29.507" LastActivityDate="2014-05-23T20:51:29.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2408" PostTypeId="4" CreationDate="2014-05-23T20:51:29.507" Score="0" Body="for questions on general considerations and problems of teaching mathematics, i.e., issues specific to teaching mathematics yet relevant to various contexts and courses." OwnerUserId="143" LastEditorUserId="39" LastEditDate="2014-06-11T23:17:27.187" LastActivityDate="2014-06-11T23:17:27.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2409" PostTypeId="5" CreationDate="2014-05-23T20:52:11.047" Score="0" Body="" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-05-23T20:52:11.047" LastActivityDate="2014-05-23T20:52:11.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2410" PostTypeId="4" CreationDate="2014-05-23T20:52:11.047" Score="0" Body="For questions touching upon pedagogical considerations and problems of a general nature, i.e., not specific to mathematics. Please note that some aspect of the question should still be related to mathematics.&#xD;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-05-23T20:52:11.047" LastActivityDate="2014-05-23T20:52:11.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2411" PostTypeId="2" ParentId="2406" CreationDate="2014-05-23T22:45:11.360" Score="8" Body="&lt;p&gt;For already TeXed documents, I have generally attached them to threads in the discussion forum. As an example, I did this with a proof sketch of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cantor-Bernstein-Schroeder Theorem&lt;/a&gt; using a .docx attachment. (Indeed, one of the prospective teachers in the class gave a chocolate-worthy answer!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/dioPp.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, I find that the discussion forum through Moodle (technique 2 &lt;a href=&quot;https://matheducators.stackexchange.com/a/1695/262&quot;&gt;here&lt;/a&gt;) is quite nice to allow students to post, and, sometimes, to respond to one another. Here is an image of some of the posts from when I was the TA for prospective teachers taking a course on Mathematical Problem Solving:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/7Sosa.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I blanked out the names of participants other than myself who started threads; as you can see, the vast majority were started by students in the course, and sometimes they led to replies from me or from other students. I can also see on Moodle when students last logged in, which gives an idea of whether or not they are using the site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, it gives me a nice way to edit responses from other participants, such as when a student links to outside work rather than engaging with the problem personally. For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/wrAy1.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;As for quizzes:&lt;/strong&gt; I'm afraid that I hold the (pessimistic? unrealistic?) view that such online assessments allow too easily for cheating. Perhaps if you can convince the students that the quizzes are &lt;em&gt;formative&lt;/em&gt; and will not affect their grade in any way, then the risk of cheating can be lowered. In this case, I can see Moodle being a valuable tool, but I have restricted my use to the discussion fora; a choice, I might add, that lowered the individual emails I received/sent dramatically, and facilitated much better the sharing of ideas in a group-setting.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-05-23T22:45:11.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2412" PostTypeId="2" ParentId="298" CreationDate="2014-05-23T23:04:41.797" Score="3" Body="&lt;p&gt;FYI the (free, open source) textbook &quot;&lt;a href=&quot;http://scholarworks.gvsu.edu/books/7/&quot; rel=&quot;nofollow&quot;&gt;Mathematical Reasoning: Writing and Proof&lt;/a&gt;&quot; contains many exercises of this form: proofs with a variety of different kinds of errors in them.&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2014-05-23T23:04:41.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2413" PostTypeId="2" ParentId="2342" CreationDate="2014-05-24T02:00:24.377" Score="1" Body="&lt;p&gt;One of my colleagues was fond of using this kind of proposition to make students understand that the converse is not always true: &quot;If you live in Manhattan, then you live in New York.&quot; It is easily translated in an inclusion diagram. It is also easy to see that the inverse (&quot;If you don't live in New York, then you don't live in New York&quot;) is true.&lt;/p&gt;&#xA;" OwnerUserId="503" LastActivityDate="2014-05-24T02:00:24.377" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2414" PostTypeId="1" CreationDate="2014-05-24T05:43:06.617" Score="8" ViewCount="213" Body="&lt;p&gt;I am considering using &lt;a href=&quot;http://www.maplesoft.com/products/mapleta/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maple T.A.&lt;/a&gt; as a tool for formative assessment (and possibly at some stage, summative assessment) for courses such as calculus and linear algebra. What are your experiences and tips on using Maple T.A. effectively? Is there evidence that using Maple T.A. or similiar tools leads to more comprehensive understanding of the material and/or higher grades?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2406/effective-use-of-moodle-or-similar&quot;&gt;Effective use of Moodle (or similar)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: I recognize that there is a risk that using a software system for formative assessment might not facilitate developing good mathematical writing skills (see &lt;a href=&quot;https://matheducators.stackexchange.com/a/2405/376&quot;&gt;James S. Cook's answer&lt;/a&gt; to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/397/how-can-we-help-students-learn-to-write-about-their-mathematics&quot;&gt;How can we help students learn to write about their mathematics?&lt;/a&gt;). Suggestions on how to deal with this issue either in Maple T.A. or alongside it are also welcome.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-07-09T19:36:55.840" Title="Effective use of Maple T.A" Tags="|undergraduate-education|technology-in-education|reference-request|software|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2415" PostTypeId="1" CreationDate="2014-05-24T07:52:28.733" Score="10" ViewCount="903" Body="&lt;p&gt;A few months ago, I asked a question on &lt;a href=&quot;https://math.stackexchange.com/questions/634690/teaching-engineers-mathematical-thinking-skills&quot;&gt;teaching engineers mathematical thinking skills&lt;/a&gt; over at MSE. I also asked it a little later at &lt;a href=&quot;https://mathematicsteachingcommunity.math.uga.edu/index.php/918/mathematical-thinking-skills-for-engineers&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Mathematics Teaching Community&lt;/a&gt;, but traffic on that site is very low. (Update: the site is currently unavailable, but I will leave the link up in case it comes back online.) Now that we have MESE, I would like to ask it here, after first having checked on &lt;a href=&quot;https://matheducators.meta.stackexchange.com/questions/356/asking-a-question-on-teaching-engineers-mathematical-thinking-skills&quot;&gt;Meta&lt;/a&gt; whether doing so would be considered acceptable.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Preamble&lt;/strong&gt;: In my experience, many introductory engineering mathematics textbooks these days tend to skip proofs and discuss logic only in the context of digital electronics. On the other hand, I can imagine that engineering students (and others) could benefit from developing basic skills in mathematical thinking* &lt;em&gt;beyond&lt;/em&gt; the cookbook approach. See, for instance, Keith Devlin's &lt;a href=&quot;https://www.coursera.org/learn/mathematical-thinking&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Mathematical Thinking&lt;/a&gt; course. I hasten to add that many engineering mathematics textbooks do have strong points, such as numerous examples of mathematics applied to solving engineering problems. For the record, I currently use &lt;a href=&quot;http://www.pearsoned.co.uk/bookshop/detail.asp?item=100000000277020&quot; rel=&quot;nofollow noreferrer&quot;&gt;Engineering Mathematics: A Foundation for Electronic, Electrical, Communications and System Engineers&lt;/a&gt; by Croft et al. The material covered includes derivatives, integrals, complex numbers, matrices, differential equations, Laplace transforms and Fourier series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;: Can you suggest engineering mathematics textbooks that do contain material on mathematical thinking and/or share your experience(s) in teaching introductory mathematics to engineering students where you went beyond the cookbook approach?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;$\text{*}$ I realize that I haven't defined &lt;em&gt;mathematical thinking&lt;/em&gt;. Keith Devlin addresses this in his blog entry &lt;a href=&quot;http://devlinsangle.blogspot.nl/2012/08/what-is-mathematical-thinking.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;What &lt;em&gt;is&lt;/em&gt; mathematical thinking?&lt;/a&gt; In particular, he writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns. Moreover, it involves adopting the identity of a mathematical thinker.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also Terry Tao's &lt;a href=&quot;http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;There's more to mathematics than rigour and proofs&lt;/a&gt; and the anonymous answer to the question &lt;a href=&quot;http://www.quora.com/Mathematics/What-is-it-like-to-understand-advanced-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;What is it like to understand advanced mathematics?&lt;/a&gt; However, please note that I am focusing on introductory courses and basic skills in this question.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Related question: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/899/is-there-any-difference-between-teaching-calculus-for-math-and-engineering-stude&quot;&gt;Is there any difference between teaching calculus for math and engineering students?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-09-24T12:45:21.197" Title="Mathematical thinking skills for engineering students" Tags="|undergraduate-education|textbooks|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2416" PostTypeId="2" ParentId="2121" CreationDate="2014-05-24T11:37:19.093" Score="2" Body="&lt;p&gt;An example of a rather natural and rich question one can start to investigate at lower grades is that of designing a dice: how many ways are there to assign the values from 1 to 6 to the faces of a dice? As a first subquestion, one can ask whether there are even two different ways? One can also give pupils two different dices and ask whether they are &quot;the same&quot;; then the cubical symmetry makes a lot of sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is how to sit people on a round table; in many circumstances, one &quot;quotient out&quot; by circular symmetry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another one is the first hand the game of go: many &quot;different&quot; hands have in fact just the same effect by symmetry. It is also interesting to think about what happens after a few plays: when one approaches a corner stone lying on a diagonal, there are two version of most possible move which are locally equivalent because of diagonal symmetry, but global symmetry has usually broke at this point. Then the player has to take into account the distant surroundings, which have a very subtle effect. Many beginners spontaneously consider the situation as symmetric (showing how natural the notion is), while it is not quite so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More geometrically, high symmetry implies lower diversity in shapes. For example, among quadrilaterals squares have the most symmetries and are entirely described (up to isometry) by their side-length; one needs much more quantities to describe a general quadrilateral, or even a parallelogram.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-05-24T11:37:19.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2417" PostTypeId="2" ParentId="2121" CreationDate="2014-05-24T15:34:32.493" Score="1" Body="&lt;p&gt;You can use symmetry to cut snow flakes, stars or ornaments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know the english names, but &lt;a href=&quot;http://dekoherz.blogspot.com/2012/12/bine-brandles-bunte-welt-keksdosen-mit.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://basteln.ucoz.de/index/ornamentschnitt/0-68&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; are examples of these stars. The use rotational symmetry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With translational symmetry you can create a long ornament of angel figures for example. (I don't find links, sorry.)&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-05-24T15:34:32.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2418" PostTypeId="2" ParentId="397" CreationDate="2014-05-24T18:28:25.647" Score="10" Body="&lt;p&gt;I taught a course in the US which was designated as &amp;quot;Writing in the Major&amp;quot;.  Once I got used to the idea, I found it invigorating: both to prepare and to teach.&lt;/p&gt;&#xA;&lt;p&gt;I've managed to find a copy of the handout that I gave to the students.  Unfortunately, I've lost the source file but I'll see what &lt;code&gt;ps2ascii&lt;/code&gt; can do (yes, it was in the days when PDF was too &amp;quot;cutting edge&amp;quot; to use).&lt;/p&gt;&#xA;&lt;p&gt;I think that the key parts that made it work were the peer review and the detailed rubric.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Project on Fourier Series&lt;/h2&gt;&#xA;&lt;h3&gt;1 Overview&lt;/h3&gt;&#xA;&lt;p&gt;The aim of this assignment is to give you practice writing mathematical prose that is both readable&#xA;and precise.&lt;/p&gt;&#xA;&lt;p&gt;The subject of the project is Fourier series. The central part of the project is to prove Fejer's&#xA;Theorem. This theorem deals with the question of uniform convergence of the Fourier series of&#xA;a function. The statement of the theorem can be found below, together with some details of the&#xA;mathematics involved.&lt;/p&gt;&#xA;&lt;p&gt;There are several rounds of revision for this assignment. The first is to give a copy of your&#xA;assignment to another student in the class. Part of your grade will be the quality of feedback you&#xA;give to your partner in this phase. The papers will be then turned in a first time to me, at which&#xA;point comments and suggestions for improvement will be made. After the papers are returned, you&#xA;will be required to resubmit edited or rewritten versions. Editing is an integral part of the writing&#xA;process.&lt;/p&gt;&#xA;&lt;h3&gt;Important dates:&lt;/h3&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Tuesday 11th of May. First draft given to peer in class for review.&lt;/li&gt;&#xA;&lt;li&gt;Thursday 13th of May. Comments returned to peer.&lt;/li&gt;&#xA;&lt;li&gt;Tuesday 18th of May. Complete first draft due to me.&lt;/li&gt;&#xA;&lt;li&gt;Tuesday 25th of May. Assignments returned with comments.&lt;/li&gt;&#xA;&lt;li&gt;Tuesday 1st of June. Final rewritten versions due.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Note: When submitting the final version, you must also submit your first draft, your peer review,&#xA;your partner's review of your paper, and the tutor's comments on your first draft. Therefore, it is&#xA;vital that you keep copies of everything.&lt;/p&gt;&#xA;&lt;h3&gt;2 Parts of the paper&lt;/h3&gt;&#xA;&lt;p&gt;Your paper should consist of the following sections:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;An Introduction: an ideal introduction gives a clear, concise, and accurate overview of the&#xA;paper, presenting enough information to interest a reader in the rest of the work and persuade&#xA;them of its importance without presenting so much that the reader is overwhelmed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Proof: an ideal section containing the proof of a major theorem proves the theorem in a&#xA;correct, logical way; it provides any information that the reader is unlikely to already know (or&#xA;should be reminded of); and is laid out in a manner such that the reader always knows which&#xA;step of the proof is being done, how it fits into the whole proof, and its relative importance&#xA;in the proof.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An Application: an ideal section about an application explains the application concisely including any context that may be useful for the reader to know; it demonstrates clearly the&#xA;relevance of the application to the main topic; and does all this with just enough detail to&#xA;convince the reader without so much that the reader is confused.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;h3&gt;3 Guidelines for Writing&lt;/h3&gt;&#xA;&lt;p&gt;Your aim should be to make your paper understandable. You should imagine that another student&#xA;in the class has asked for your help on this topic.&lt;/p&gt;&#xA;&lt;p&gt;Use full sentences. The only sentence fragments that are acceptable are headings such as &amp;quot;Theorem&amp;quot; and &amp;quot;Proof&amp;quot;. You should not use symbols such as [symbols for therefore, implies, because, and so forth] nor should you use&#xA;abbreviations such as &amp;quot;s.t.&amp;quot; or &amp;quot;iff&amp;quot;. These detract from the clarity. It should be possible for someone with basic mathematical knowledge to read your paper aloud without hesitating over symbols&#xA;or poorly constructed sentences.&lt;/p&gt;&#xA;&lt;p&gt;Different parts of the paper have different purposes and these lead to different styles. In the&#xA;first part, the primary aim is to interest a person in your paper, to give them an overview, but not&#xA;to burden them with minutiae. In the second part, the aim is to present the theorem and its proof.&#xA;In this part, accuracy is paramount but it is not possible to be accurate without being clear. In the&#xA;third part - the application - clarity is again foremost.&lt;/p&gt;&#xA;&lt;p&gt;When proving the theorem, although the proof is in the book you should not regard this as a&#xA;copying exercise. The style adopted in the book is not the best style for this type of paper.&lt;/p&gt;&#xA;&lt;p&gt;[Section 4 was some mathematical background on Fejer's theorem.]&lt;/p&gt;&#xA;&lt;h3&gt;5 Rubric&lt;/h3&gt;&#xA;&lt;p&gt;The paper will be graded according to the scheme below. There are five categories: introduction,&#xA;proof, application, overall presentation, and review and revision. In each section, there are four&#xA;marks available, making a total of twenty. The specifics for each mark are laid out below. This&#xA;scale is incremental. That means that to get, say, two marks on the introduction you have to not&#xA;only satisfy the criteria laid out for two marks but also the criteria for one mark.&lt;/p&gt;&#xA;&lt;h3&gt;Mark Criteria&lt;/h3&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Introduction&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Introduction is present.&lt;/li&gt;&#xA;&lt;li&gt;Aims of the paper are stated. Broad area of mathematics to which material belongs&#xA;is stated.&lt;/li&gt;&#xA;&lt;li&gt;Aims of the paper are clearly stated. Concise background information is explained.&#xA;Relevance of subject matter is stated.&lt;/li&gt;&#xA;&lt;li&gt;A clear, concise, accurate introduction is present.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Proof of Theorem&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Methodological steps have been taken toward proving the result.&lt;/li&gt;&#xA;&lt;li&gt;Theorem is correctly proven.&lt;/li&gt;&#xA;&lt;li&gt;Relevant definitions are stated. Theorem is proved in a clear way with correct differentiation to the steps.&lt;/li&gt;&#xA;&lt;li&gt;A complete, precise, concise and detailed proof is given. Where relevant, especially&#xA;elegant or informative steps are highlighted.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Application&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;An application of the subject matter is given.&lt;/li&gt;&#xA;&lt;li&gt;The application is set in its proper context.&lt;/li&gt;&#xA;&lt;li&gt;The relevance of the subject matter to the application has been explained.&lt;/li&gt;&#xA;&lt;li&gt;The importance of this work to the application is evaluated.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Overall Presentation&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Paper is legible with adequate spelling, punctuation, and grammar.&lt;/li&gt;&#xA;&lt;li&gt;Paper is neatly and clearly presented with good spelling, punctuation, and grammar.&lt;/li&gt;&#xA;&lt;li&gt;Secondary sources are cited correctly.&lt;/li&gt;&#xA;&lt;li&gt;Segué between sections is smooth and logical.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Review and Revision&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;First draft, peer review, partner's peer review, and tutor's review are present.&lt;/li&gt;&#xA;&lt;li&gt;First draft has all sections, peer review comments on all sections of partner's first&#xA;draft (even if said section is missing).&lt;/li&gt;&#xA;&lt;li&gt;Final version shows improvement in accordance with partner's review, where relevant.&#xA;Peer review is conducted according to this rubric. Tutor's recommendations have been&#xA;implemented.&lt;/li&gt;&#xA;&lt;li&gt;Peer review gives practical guidelines for improvement in partner's paper.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="112" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-05-24T18:28:25.647" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2419" PostTypeId="2" ParentId="55" CreationDate="2014-05-24T22:10:28.913" Score="8" Body="&lt;p&gt;As Willie Wong suggests in the comments to your question, it could be possible to use Alan MacDonald's textbook &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1453854932&quot;&gt;Linear and Geometric Algebra&lt;/a&gt; as an alternative to a more conventional linear algebra course. See also MacDonald's webpage on the book at &lt;a href=&quot;http://faculty.luther.edu/~macdonal/laga/&quot;&gt;http://faculty.luther.edu/~macdonal/laga/&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have the freedom to teach linear algebra MacDonald-style &lt;em&gt;before&lt;/em&gt; multivariable/vector calculus then you can follow up with &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1480132454&quot;&gt;Vector and Geometric Calculus&lt;/a&gt;. Once more, see also MacDonald's webpage on the book at &lt;a href=&quot;http://faculty.luther.edu/~macdonal/vagc/&quot;&gt;http://faculty.luther.edu/~macdonal/vagc/&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The OP mentions undergraduates in &lt;em&gt;engineering, physics and mathematics&lt;/em&gt;. There's also  computer science. Geometric algebra has applications to computer graphics, computer vision and robotics. (Admittedly, the latter two topics, especially robotics, are also studied in engineering.) See, for instance, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0123749425&quot;&gt;Geometric Algebra for Computer Science&lt;/a&gt; by Dorst, Fontijne &amp;amp; Mann. The webpage for the book is: &lt;a href=&quot;http://www.geometricalgebra.net/&quot;&gt;http://www.geometricalgebra.net/&lt;/a&gt;. It would probably go particularly well after MacDonald's aforementioned &lt;em&gt;Linear and Geometric Algebra&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I own all three of the above books, but I have not used them yet in a course. One of the challenges is overcoming the inertia of standard approaches, especially when they work well for most practical purposes. A new approach may need to be demonstrably superior before having a chance of ousting the incumbent. &lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2014-05-24T22:10:28.913" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2420" PostTypeId="2" ParentId="2120" CreationDate="2014-05-25T06:23:41.370" Score="7" Body="&lt;p&gt;(Sorry if my answer is as offensive as responding to a question about Coca-Cola with data about Pepsi!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are certainly some lesson plans around looking at statistics for &lt;strong&gt;M&amp;amp;M colors&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, see Supplemental Activity S3.3 from COMAP's &quot;Mathematics: Modeling our World&quot; &lt;a href=&quot;http://www.comap.com/highschool/projects/mmow/assets/DOSWIN/Course3/Unit3/Resources/U3SUPPL.PDF&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. COMAP (COnsortium for Mathematics and its APplications) is my general go-to for modeling related activities; for example, see the discussion of a different sugary item in my &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/1558/262&quot;&gt;MESE post&lt;/a&gt;&lt;/strong&gt; on cake-cutting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another brief &lt;a href=&quot;http://www.rci.rutgers.edu/~mmm431/quant_methods_F13/MnMs_web_exercise.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;write-up&lt;/strong&gt;&lt;/a&gt; that I saw online recommended two other sources. One of these sources includes a classroom activity and has 'teaching' as a keyword; it is entitled: &lt;a href=&quot;http://www.rsscse-edu.org.uk/tsj/wp-content/uploads/2011/03/johnson4.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Testing Colour Proportions of M&amp;amp;Ms&lt;/strong&gt;&lt;/a&gt;. The other source, &lt;a href=&quot;http://psoup.math.wisc.edu/pub/mm/fricker.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Mysterious Case of the Blue M&amp;amp;M's&lt;/strong&gt;&lt;/a&gt;, recommends the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/JuPrh.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article is from 1996, so I'm not sure if this &quot;teaching device&quot; is still available (though I suppose you could call and ask). You might also look up the (Landwehr et al, 1987) source mentioned at the end. Googling did not lead me to that particular one-sheet guide, though I did find yet another &lt;strong&gt;&lt;a href=&quot;http://courses.washington.edu/stats315/newpage1.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;M&amp;amp;M project&lt;/a&gt;&lt;/strong&gt; on the subject described online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, you might download &lt;a href=&quot;https://www.stat.auckland.ac.nz/~iase/publications/dissertations/00.Meletiou.Dissertation.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;this thesis&lt;/strong&gt;&lt;/a&gt; and search for mentions of M&amp;amp;Ms (e.g., p. 33, p. 243).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In response to your comment (&quot;There are quite a few articles on using them as a method in the classroom, but I'm more interested in whether they are a useful model&quot;) I would say there are enough different items online to put something of interest together. &lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-05-25T06:23:41.370" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2421" PostTypeId="1" CreationDate="2014-05-25T14:34:23.150" Score="7" ViewCount="667" Body="&lt;p&gt;Mathematician Ian Stewart writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To criticize mathematics for its abstraction is to miss the point&#xA;  entirely. Abstraction is what makes mathematics work. If you&#xA;  concentrate too closely on too limited an application of a&#xA;  mathematical idea, you rob the mathematician of his most important&#xA;  tools: analogy, generality and simplicity. Mathematics is the ultimate&#xA;  in technology transfer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ian Stewart, &lt;em&gt;Does God Play Dice?&lt;/em&gt;, 2nd ed., 1997, Penguin Books Ltd&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;How do you emphasize the benefits of mathematical abstraction in your teaching, while not losing sight of applications? Do you have any prime examples?&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2014-05-25T15:51:31.600" LastActivityDate="2014-05-26T01:25:01.407" Title="Emphasizing benefits of mathematical abstraction" Tags="|undergraduate-education|secondary-education|examples|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2422" PostTypeId="2" ParentId="2421" CreationDate="2014-05-25T15:25:58.957" Score="2" Body="&lt;p&gt;The only way is to show them with concrete (!) examples that abstraction is useful. Perhaps the easiest way is to introduce (a bit) of abstract algebra, showing uses of group theory for instance.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-25T15:25:58.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2423" PostTypeId="1" CreationDate="2014-05-25T16:28:42.347" Score="12" ViewCount="6712" Body="&lt;p&gt;Suppose your students know already the geometric definition of $\sin$ $\cos$ and $\tan$ for angles between $0^{\circ}$ and $90^{\circ}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I motivate the definition of $\sin$, $\cos$ and $\tan$ for angles between $0^{\circ}$ and $360^{\circ}$ using the unit circle? What would you answer if a student asks: What is the reason that it is defined this way?&lt;/p&gt;&#xA;" OwnerUserId="1203" LastEditorUserId="64" LastEditDate="2014-05-25T23:44:08.830" LastActivityDate="2014-05-27T16:57:19.517" Title="How to motivate the geometric definition of trigonometric functions on the unit circle" Tags="|concept-motivation|trigonometry|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2424" PostTypeId="2" ParentId="2423" CreationDate="2014-05-25T16:47:13.310" Score="9" Body="&lt;p&gt;Suppose, in a given coordinate system with the origin at a given point, an object is $r$ units away from the given point at an angle (compass heading) of θ degrees -- find its coordinates.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example: radar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many other applications, too, but geometrically, the point is to solve the problem as a function of the angle (and distance).  It turns out the dependence on the angle may be reduced to the unit circle, one of the few easy things about trigonometric functions.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2014-05-25T16:47:13.310" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2425" PostTypeId="2" ParentId="2421" CreationDate="2014-05-25T18:00:38.300" Score="5" Body="&lt;h1&gt;The analogy of thin lenses and resistors&lt;/h1&gt;&#xA;&lt;h2&gt;Forumulas for thin lenses&lt;/h2&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{g}+\frac{1}{b}=\frac{1}{f}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$b+g=d$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(&lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is the distance object-lense. &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is the distance image-lense. &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is the focal width. &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is the distance object-image.)&lt;/p&gt;&#xA;&lt;h2&gt;Formulas for Resistors&lt;/h2&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{R_1}+\frac{1}{R_2}=\frac{1}{R_{\text{parallel}}}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$R_1+R_2=R_{\text{series}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Because the mathematics of both are abstractly equal, mathematical solutions to equal problems apply equally to both.&lt;/p&gt;&#xA;&lt;h2&gt;Example of equal problem&lt;/h2&gt;&#xA;&lt;p&gt;Find two resistances such that their total parallel resistance and their total series resistance has given values!&lt;/p&gt;&#xA;&lt;p&gt;Find the position of a lense between fixed object and visor, such that the image on the the visor is sharp!&lt;/p&gt;&#xA;&lt;p&gt;The solution to both can be found by solving the abstract system of equations&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{x}+\frac{1}{y}=\frac{1}{a}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x+y=b$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which happens to be transformable to&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x+y=b$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x\cdot y=a\cdot b=c$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which happens to have been solved by Vieta already.&lt;/p&gt;&#xA;&lt;h2&gt;Realization&lt;/h2&gt;&#xA;&lt;p&gt;No, all this just hasn't simply &lt;em&gt;happened&lt;/em&gt;. It is result of using abstract math.&lt;/p&gt;&#xA;&lt;h3&gt;Side remark&lt;/h3&gt;&#xA;&lt;p&gt;The mathematical solutions to the above problem are equal, but their physical interpretations might not be. Resistances may not be negative, but optical distances may be. Negative distances do not create real images, but if you replace the visor by an eye, you might see the virtual image.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-05-25T18:00:38.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2426" PostTypeId="2" ParentId="2421" CreationDate="2014-05-25T19:52:47.107" Score="3" Body="&lt;p&gt;When I teach physics, I vacillate between two modes.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;examples with explicit numerical data&lt;/li&gt;&#xA;&lt;li&gt;examples which are purely symbolic&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A wise student will notice that the examples without numerical data actually reveal more physical content. The numerical examples are needed to help them see how to work our their own numerical homework problems. This is actually an example of abstraction at work. The use of algebra variables is abstraction, we replace a word with a letter and use formal manipulation of symbols, calculus, and vectors to work out physical consequences which would not be obvious without the abstraction. All of this happens within physics, which, I think I can reasonably claim is &lt;em&gt;applied&lt;/em&gt;. At some point or another, usually several times in a given semester, I will make the following point:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To solve a problem with numbers is to solve &lt;strong&gt;that&lt;/strong&gt; problem. To solve a problem symbolically is to solve &lt;strong&gt;all&lt;/strong&gt; such problems. So, when faced with the question of putting numbers into variables, ask yourself the question: &quot;do I want to solve this problem, or, for almost the same effort, do I want to solve all such problems?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As to mathematics, abstraction is the heart of mathematics. So, how can I avoid abstraction if I am to teach math? The examples are endless.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-05-25T19:52:47.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2427" PostTypeId="2" ParentId="2423" CreationDate="2014-05-25T23:44:08.903" Score="9" Body="&lt;p&gt;Unfortunately I don't have any way to draw pictures on this computer, otherwise I could give a more helpful answer, but I have thought about this and I would start with a diagram a little like the following from wikipedia - but initially only with cos, sin and tan (and maybe cotangent): &lt;img src=&quot;https://i.stack.imgur.com/s3ybv.png&quot; alt=&quot;Wikipedia diagram of trig functions on a circle&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would start by motivating sine. Depending on how concrete the students need things to be (and some need more than others) I would imagine the unit radius line as (say) a crane or robot arm. The sine of the angle is then the height of the arm above the horizontal. This is true for all angles even ones outside the 0 - 360 degree range. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the arm goes below the horizontal, the sine is (obviously) negative because you are going below the line. It is easy to see that sine is positive from 0 to 180 and then negative from 180 to 360.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cosine is then the distance projected onto the horizontal by the line (eg the length of shadow cast by the crane or how far the robot arm stretches at that angle). Again this is true for all angles. You can see it is periodic and will become negative as the angle passes 90.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tangent is less obvious but it should be clear that it grows without bound as the angle approaches 90 (and you can draw some pretty long tangents to show this). The range -90 to 90 is fairly clear. Obviously you can extend the definition, but the intuition requires a little more work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sec and cosec are likely to be for more advanced students but you can see how they have their own definition as well as being multiplicative inverses of cosine and sine. Sin^2 + cos^2 = 1 is hopefully obvious if you know your pythagoras (and this is a way of re-using that formula and reinforcing it), other identities can be drawn out geometrically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I suspect that this works much better for some students than others. Some people really don't have or like to use a pictorial way of thinking about something, but I think this is useful. &lt;/p&gt;&#xA;" OwnerUserId="1536" LastActivityDate="2014-05-25T23:44:08.903" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2428" PostTypeId="1" CreationDate="2014-05-25T23:48:24.197" Score="39" ViewCount="7125" Body="&lt;p&gt;I am due to teach Calculus II in the fall at an American state university. Our calculus sequence is somewhat slow, due to the fact that many of our students come with limited backgrounds. Most of our typical syllabus consists of the following.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Techniques of integration.&lt;/em&gt; We teach the students to memorize and apply complicated techniques for evaluating obscure integrals --- all of which Wolfram Alpha &lt;a href=&quot;http://www.wolframalpha.com/input/?i=integral%20of%20sin%5E4%28x%29%2acos%5E2%28x%29&quot;&gt;can&lt;/a&gt; &lt;a href=&quot;http://www.wolframalpha.com/input/?i=integral+of+1%2F%28x%5E4+-+1%29&quot;&gt;do&lt;/a&gt; in a fraction of a second.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why do they work? Well, that requires a lot of mathematical maturity to understand, and for the most part we don't even &lt;em&gt;try&lt;/em&gt;. For example, Stewart's calculus book makes &lt;em&gt;no&lt;/em&gt; attempt to explain why partial fractions work, not even a handwavy one. Just &quot;A theorem in algebra guarantees that it is always possible to do this,&quot; and the rules are presented for memorization, in all of their complicated glory.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Sequences and series&lt;/em&gt;. This chapter is a huge, long slog in every calculus book I've ever seen. Stewart's book starts with lots of scary theorems (&quot;Every bounded monotonic sequence is convergent&quot;, proved with epsilon-delta), so that by the time you write $1 + \frac{1}{2} + \frac{1}{4} + \cdots = 2$ on the board, or even $.9999\dots = 1$, the students are made to feel that it secretly means something complicated and technical which they half-admit they don't really understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From there it gets even worse. Typical problem in Stewart: Does $\sum_{n = 1}^{\infty} \frac{n + 5}{\sqrt[3]{n^7 + n^2}}$ converge or diverge? Well, there is a rule for that -- one of very many -- and for the comparison and integral tests you can at least give a fairly convincing simple explanation for why they work. But I shudder to ponder how many of our students understand that the Latin word &lt;em&gt;converge&lt;/em&gt; means &lt;em&gt;equals a number&lt;/em&gt; -- this fact is obscured because &lt;em&gt;usually there is no good way to evaluate what this number is&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the meantime -- where do these series come from? The students will have seen few or none, and typically applications are de-emphasized in the book (and also feel a bit artificial or canned). The textbook does not provide the material with which to convince the student that such a series is a natural thing to write down, except if one is looking for convenient fodder for homework or an exam. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Now, I could revise the material to do it &quot;my way&quot;, but if I really did it my way, I'd end up teaching something radically different from your average Calculus II course. (And my department expects me to stay fairly close to the book.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm feeling unprepared to make the case to my students that this course is interesting, useful, or important. What can be done with this material?&lt;/p&gt;&#xA;" OwnerUserId="1535" LastEditorUserId="2083" LastEditDate="2024-03-02T20:42:23.803" LastActivityDate="2024-03-03T20:39:44.920" Title="How to make Calculus II seem motivated, interesting, and useful?" Tags="|undergraduate-education|calculus|usa|" AnswerCount="14" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="2429" PostTypeId="1" CreationDate="2014-05-25T23:59:48.387" Score="17" ViewCount="3571" Body="&lt;p&gt;I will be teaching a year-long undergraduate introduction to abstract algebra in the fall, and I am quite looking forward to it! I need to choose a textbook, and I don't have personal experience with any that I think will be suitable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that popular books are those written by Gallian, Fraleigh, and Beachy and Blair, among others. I can read the reviews on Amazon, and I can presumably also obtain copies of these and other books. (I have a copy of Gallian's, and it looks quite nice.) But browsing the books and the Amazon reviews gives me only a partial ability to understand what the differences will be in the long term from the students' perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the substantial differences between these or other recommended books -- especially those which won't be immediately evident from reading their tables of contents or otherwise skimming the books?&lt;/p&gt;&#xA;" OwnerUserId="1535" LastEditorUserId="143" LastEditDate="2014-05-26T10:02:32.727" LastActivityDate="2020-09-20T15:55:18.780" Title="What are the differences between popular undergraduate abstract algebra books?" Tags="|undergraduate-education|textbooks|abstract-algebra|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2430" PostTypeId="2" ParentId="220" CreationDate="2014-05-26T00:05:02.013" Score="16" Body="&lt;p&gt;Prove that the &lt;a href=&quot;http://en.wikipedia.org/wiki/Tower_of_hanoi&quot; rel=&quot;noreferrer&quot;&gt;Tower of Hanoi&lt;/a&gt; puzzle can be solved for any number of disks. Indeed, that it can be solved in $2^n - 1$ moves for $n$ disks.&lt;/p&gt;&#xA;" OwnerUserId="1535" LastActivityDate="2014-05-26T00:05:02.013" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2431" PostTypeId="2" ParentId="2428" CreationDate="2014-05-26T00:16:20.393" Score="22" Body="&lt;p&gt;While it doesn't motivate much of the dense and technical morass that is Calculus II, I've made it a little more fun (for me anyway) by thinking about the course as trying to understand the infinite.  Our version of the course also has improper integrals (clearly about infinity) and power series (a polynomial of infinite degree).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tend to drop in some facts about infinity throughout the semester.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-05-26T00:16:20.393" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2432" PostTypeId="2" ParentId="2421" CreationDate="2014-05-26T01:19:51.243" Score="4" Body="&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt; Promote abstractions through generalizations they come with.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Let me start with a digression...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Abstractions and generalizations&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Abstractions and generalizations are two different things, but they commonly appear together. One funny way to think about them is to regard abstraction as a kind of weird existential quantifier, while generalization would be a universal one. To give an example, for a statement:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{Any group of order $3$ is cyclic.}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;an abstraction could be&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{There exists a prime number $p$ such that any group of order $p$ is cyclic.}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;generalization only would be&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{Any group of order $2$, $3$ or $5$ is cyclic.}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;finally using both might give&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{For any prime number $p$, any group of order $p$ is cyclic.}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the &quot;universal quantifier&quot; in the last statement is apparent, it is not so obvious in the generalization-only example; but it is there:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{For any number $n \in \{2,3,5\}$, any group of order $n$ is cyclic.}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The takeaway is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Abstractions and generalizations may appear separately.&lt;/li&gt;&#xA;&lt;li&gt;But, the most useful setting is where there are both.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Natural&quot; generalization (i.e. one which is handled uniformly, does not involve cases, etc.) implies some abstraction (perhaps stupid, but still).&lt;/li&gt;&#xA;&lt;li&gt;Non-trivial abstraction (i.e. which has more than one instance) imply some generalization (also, might be stupid).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why to use abstractions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two main reasons to use abstractions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To blur out unnecessary details.&lt;/li&gt;&#xA;&lt;li&gt;To prepare for generalizations.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;While the first one is more obvious, it is rare that students will approach problems of complexity so high that would warrant use of abstractions. Most of topics are usually given in small, digestible chunks, the abstractions being often already overlaid on the content. It seems a lost cause: either the problem would be easy enough to be able to keep track of all the details, or it would be so complex that the weaker students would be lost before it even started. There are cases like random variables where abstraction visibly makes things simpler, but is it &lt;em&gt;necessary&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here there comes the justification of that lengthy introduction. The easiest way to show why abstractions are useful, in my opinion, is through the generalizations they provide. In other words, it is impractical to use multiple different cases, and any proof that handles various objects in a uniform way has to use some form of abstraction. Fortunately there are accessible examples of this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Examples of abstractions at work&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The best example I know in computer science comes from &lt;a href=&quot;http://blog.sigfpe.com/2008/11/approach-to-algorithm-parallelisation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this post&lt;/a&gt; by Dan Piponi. It is &lt;em&gt;not&lt;/em&gt; an elementary problem, but I've used it several times to great effect. &lt;em&gt;I recommend it very much!&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Basic example in mathematics might be in calculating $$\int_{-\pi}^{\pi} \sin^3 t \cdot \cos t\ \mathrm{d}t.$$ One can do it by parts, but it's easier to abstract $\sin$ and $\cos$ as &lt;a href=&quot;http://en.wikipedia.org/wiki/Even_and_odd_functions&quot; rel=&quot;nofollow noreferrer&quot;&gt;odd and even&lt;/a&gt; functions and conclude that the integral is zero almost immediately.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similar example might be with proving that &lt;a href=&quot;https://math.stackexchange.com/a/444835/26306&quot;&gt;$(x-a)^3 + (x-b)^3 + (x-c)^3 = 0$ has only one real root&lt;/a&gt; by observing that $x \mapsto x^3$ is strictly increasing.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Induction is a great source of examples of generalizations. For example, to prove that $$4 \mid 5^{2014}-1$$ we generalize to $4 \mid 5^n - 1$ and argue by $5^{n+1} - 1 = 5\cdot (5^n-1) + 4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is even a nicer example with covering the chessboard with L-shaped trominos, e.g. see &lt;a href=&quot;http://www.cut-the-knot.org/Curriculum/Games/TrominoPuzzle.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Information theory provides a general abstract approach by disregarding anything but the flow of bits. For example to prove that you need at least $5$ comparisons to sort $4$-element array it's best to calculate $\lceil\log_2(4!)\rceil = 5$, that is, any algorithm that distinguishes each of $4!$ permutations has to make at least $5$ decisions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Yet another example might be calculating $S_n = \sum_{k=0}^{n} k\cdot 2^k$; the trick is to make $2$ a variable, i.e. $S'_n = \sum_{k=0}^{n} k\cdot z^k$ to get $S'_n = \frac{z(nz^{n+1}-(n+1)z^n+1)}{(z-1)^2}$ and go back to $S_n = 2(2^nn-2^n+1)$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let the final example be as follows: let $P : \mathbb{N} \to \mathbb{N}$ be a function constructed by combining $\mathrm{gcd}$, $\mathbb{lcm}$, constants and input, e.g. $$P_1(x) = \gcd\Big(\gcd(5,x),\mathrm{lcm}(x,3)\Big),$$ how to check if $P(x) = n$ has any solutions for some given $n \in \mathbb{N}$? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observe that $\langle\mathbb{N},\gcd,\mathrm{lcm}\rangle$ is a lattice, and so it is enough to test whether $P(n) = n$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Abstraction is a very practical thing, the problem is that in mathematics the complexity level in which it shines enough is to high for basic courses. For example, this threshold is much lower in computer science and being able to find good abstractions is widely considered a characteristic of a good programmer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, even in math one doesn't need to sacrifice practicality to talk about abstractions, because the generalizations they are tied with are enough powerful to warrant their (abstractions) use. Such examples might be a bit contrived, nevertheless, seem persuasive enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-05-26T01:25:01.407" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2433" PostTypeId="2" ParentId="2423" CreationDate="2014-05-26T03:55:32.347" Score="1" Body="&lt;p&gt;You could say something like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Similar triangles all have the same trigonometric values, since they are all defined as ratios of side lengths.  We want to think about how sine and cosine vary with the angle.  In principle, to think about how $\sin(30^{\circ})$ compares to $\sin(31^\circ)$, for instance, I have to think up a right triangle with angle $30^\circ$, and another with angle $31^\circ$.  If I make one of these triangles very large, and the other very small, it will be difficult to compare the ratio opposite/hypotenuse for each triangle.  A nice solution is to fix the hypotenuse.  In fact, you can fix it to be of length $1$, so that the length of the opposite to the angle is the sine. Then thinking about how to compare these values becomes simpler, since I have &quot;standardized&quot; how I will compare them.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now draw several triangles which all have the same hypotenuse, with the angle based at the origin, and the angles all sharing the positive horizontal axis in common.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;What sort of figure am I drawing with all of these points?&quot;  [A circle]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Right, so the abscissa of this point corresponds to the cosine of this angle, and the ordinate to the sine.  Using our definition of sine and cosine in terms of right triangles, it does not make sense to speak about the sine of an angle larger than $90^\circ$.  But this new picture shows us how to extend the definitions of these functions in a natural way.&quot;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-05-26T03:55:32.347" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2434" PostTypeId="2" ParentId="2423" CreationDate="2014-05-26T05:10:29.447" Score="4" Body="&lt;p&gt;To use Law of Sines and Law of Cosines for triangles that have no right angle, we need the trig functions to be defined for angles greater than 90 degrees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To model phenomena that are periodic, we need trig functions that are defined for all real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definitions based on circles (unit or not) achieve these goals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also like the video I show as a lead-in for our work on y=sin x, &lt;a href=&quot;https://www.youtube.com/watch?v=0FjddxnLMmY&quot; rel=&quot;nofollow&quot;&gt;showing a human-powered ferris wheel&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-05-26T05:10:29.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2435" PostTypeId="2" ParentId="2428" CreationDate="2014-05-26T05:20:59.067" Score="15" Body="&lt;p&gt;Don't you also do volumes of rotation, and polar, parametric etc?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The volumes and work problems are the most down-to-earth problems in the course I teach. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I end the infinite series unit with Maclaurin series, and their ability to make integrals of things like e^(-x^2) solvable, which has a solid connection with statistics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the techniques of integration unit, I have them do an area project, using Simpson's rule, so they get a sense of how messy real-life applications may be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definitely, try to help your students understand why each technique works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find Calc I much more satisfying, but I enjoyed Calc II this past semester. I am still searching for ways to make it work better for the students.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-05-26T05:20:59.067" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2436" PostTypeId="2" ParentId="2428" CreationDate="2014-05-26T11:23:29.713" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Every bounded monotonic sequence is convergent&quot;, proved with epsilon-delta&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Snore&lt;/em&gt;. Seriously, that won't motivate anyone. But if you show me how it applies to real-world problems that I actually &lt;strong&gt;want&lt;/strong&gt; to solve....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I took calculus in University and even got a surprisingly good mark in the class. The one and only thing I can recall is I should use 2&amp;pi;r for circumference - the professor (who was great, and Welsh. That's important here) would mark &amp;pi;d wrong. Why do I remember this point? This is the only thing covered that had applications outside of physics labs, where we used differentials to work out absolute error. In 1985 that was a pencil-grinding slog that crashed the calculator half the time, and no one really cared.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you pointed out, Wolfram Alpha can do the grunt work instantly and produces pretty graphs. Skim over the mechanics (which I won't need any more than long division), connect input directly to output, and then explain why it works.&lt;/p&gt;&#xA;" OwnerUserId="1540" LastActivityDate="2014-05-26T11:23:29.713" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2437" PostTypeId="2" ParentId="2428" CreationDate="2014-05-26T12:58:21.647" Score="6" Body="&lt;p&gt;I'd try to get more &lt;em&gt;understanding&lt;/em&gt; than &lt;em&gt;rote regurgitation of formulas&lt;/em&gt; out of them. As you note, there are computer algebra systems around (I'm a fan of &lt;a href=&quot;http://maxima.sourceforge.net&quot; rel=&quot;noreferrer&quot;&gt;maxima&lt;/a&gt;, it even runs on my cellphone) that can handle the routine effortlessly. It would be nice to have the same at hand for everybody, and as an open source one maxima fits that bill. Or insist on some web-based system, like Wolfram alpha.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of having them compute nasty integrals by hand, let the CAS handle that. Do compute all sort of weird volumes of rotation and such, but get symbolic/numeric results (and ask for that in homework). Ask for the set up, and have them attach the commands and output of the CAS, and then state the result. In exams ask for how to set up the computation, how to use e.g. partial fractions on a sum or integral (not &quot;give me the coefficients&quot;, but &quot;to compute this, do you split ...&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For sequences and series, I'm afraid there isn't much that can be done. But freeing up time by leaving out the drudgery should help a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take care to coordinate with colleages working in parallel, check what exactly following courses require. In particular, if some CAS is standard there, or there are &quot;student editions&quot; available through the bookstore, use them.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-26T12:58:21.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2438" PostTypeId="2" ParentId="2429" CreationDate="2014-05-26T13:04:29.263" Score="3" Body="&lt;p&gt;I've looked at Pinter's &quot;A Book of Abstract Algebra&quot; (Dover, 2010), and I like it a lot (haven't taught algebra, mind you, just a bit of use in Discrete Math classes). It is a bit slow, but gives plenty of concrete examples of application of the theory. Without that, abstract algebra will seem just mindless (and pointless) pushing meaningless symbols around.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="376" LastEditDate="2016-10-18T18:23:42.070" LastActivityDate="2016-10-18T18:23:42.070" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2439" PostTypeId="2" ParentId="2428" CreationDate="2014-05-26T15:09:50.447" Score="-1" Body="&lt;p&gt;Try using smooth infinitesimal analysis and point out that if we leave the powers of epsilon in we get finite difference calculus. Also look at &lt;a href=&quot;https://mathoverflow.net/questions/77175/taking-zooming-in-on-a-point-of-a-graph-seriously&quot;&gt;this&lt;/a&gt;. It seems to me that limit theory only applies exactly to some early results of integral calculus (eg the quadrature of the hyperbola); and apart from that it's an approximation for which nilpotent infinitesimals are equivalent.&lt;/p&gt;&#xA;" OwnerUserId="1544" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-05-26T15:15:27.737" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2440" PostTypeId="2" ParentId="2428" CreationDate="2014-05-26T15:40:16.037" Score="4" Body="&lt;p&gt;From my experience tutoring, what has worked is first presenting a real world problem that needs solved then show how Calculus can be used to solve it.  Position Calculus as a tool that can solve real problems and show how to translate problems to math and back again.  Math is the tool that makes Science possible.  Science without math is simply observation!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also helps to reinforce that math is exact and a chain of steps that has to be done correctly otherwise it only takes one break in the chain to miss a problem.  Don't let one misunderstanding of one formula break the chain!  Students have to memorize formulas and understand the steps exactly.  It takes practice to perfect understanding; and patience with yourself to go through the perfection process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always say they are lucky as every problem presented in class has a correct answer, they just have to find it.  Thinking a problem cannot be solved is simply not true, they just need to find the secret on how to solve it.&lt;/p&gt;&#xA;" OwnerUserId="1545" LastEditorUserId="1147" LastEditDate="2017-08-18T04:03:59.560" LastActivityDate="2017-08-18T04:03:59.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2441" PostTypeId="2" ParentId="2429" CreationDate="2014-05-26T20:45:39.500" Score="5" Body="&lt;p&gt;An alternative approach is Childs' &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387745270&quot;&gt;A Concrete Introduction to Higher Algebra&lt;/a&gt; (3rd ed., Springer 2008). It starts with some basic number theory, followed by rings and fields. Groups don't make an appearance until later. It's worth a look if you want to give your course a number theoretic flavor with applications and don't mind de-emphasizing groups somewhat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Personally, though, I quite like group theory and its many applications, so I don't know if I'd follow Childs' approach. That said, I'm happy to have his book on my bookshelf as a resource.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2014-05-26T20:54:28.703" LastActivityDate="2014-05-26T20:54:28.703" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2442" PostTypeId="2" ParentId="2428" CreationDate="2014-05-27T00:09:43.433" Score="11" Body="&lt;p&gt;The real problem is that the course has to serve the needs of math majors, engineering/physical science majors, and life science majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For math majors, 100% of the material in a traditional freshman calc course is relevant, and no apologies are required. It doesn't matter that computer algebra systems can do hard integrals; we need to keep on educating human beings who have the skills to &lt;em&gt;write&lt;/em&gt; a computer algebra system. These students need to understand the foundations of the subject, which are nontrivial and took hundreds of years for the world's best minds to work out after the invention of the calculus by Newton and Leibniz. For example, people like Euler used to seriously manipulate series like $\sum_{n=0}^\infty (-1)^n n!$, because the notion of convergence wasn't obvious and hadn't been invented. Lagrange tried to build calculus on Taylor series as a foundation, in order to avoid limits and infinitesimals; people at that time didn't understand that not all functions were analytic. For a student who's a math major, the only serious problem with this type of course is that the curriculum typically ignores the rehabilitation of infinitesimals in the 20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For physical science and engineering majors, series, especially Taylor series, are important, practical, easy to motivate, and will be used by them for the rest of their working lives. Methods of integration, on the other hand, are less likely to be useful, although there may be times in their upper-division coursework where their teachers will want to use these ideas without having to apologize, and in contexts, e.g., proofs of general facts, where a CAS isn't the right tool. For example, integration by parts is used in the standard derivation of the Euler-Lagrange equations of motion from the principle of least action; because this is being proved in general, not for some specific example, you need to understand integration by parts in order to understand the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For nearly all life science majors the second semester of calculus is a ridiculous waste of time. It's required of them at some more selective schools because majors like biology are &quot;impacted,&quot; so the biology department is looking for a way to weed students out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no &lt;em&gt;good&lt;/em&gt; reason why all students who take calculus should take the same two-semester freshman calc course. At the community college where I teach, we offer four different flavors of freshman physics. The math department could do the same. The most probable reason that freshman calc is one-size-fits-all is that other departments want it as a &quot;weeder&quot; course.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-05-27T17:06:26.863" LastActivityDate="2014-05-27T17:06:26.863" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="2443" PostTypeId="1" CreationDate="2014-05-27T10:41:32.190" Score="11" ViewCount="5585" Body="&lt;p&gt;Suppose you have been told to give a multiple choice exam, and to nullify the effects of random guessing by penalizing incorrect answers.  Suppose there are $N$ available choices, of which exactly one is correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One common approach is to give $P$ points for the correct answer, $p = -P\, /\, (N-1)$ points for an incorrect answer, and 0 points for not answering or choosing more than one answer.&lt;sup&gt;*&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now suppose we have some questions with 4 available choices, and some true/false questions with only 2 available choices. &lt;strong&gt;How should we assign the corresponding correct answer scores $P_2$ and $P_4$&lt;/strong&gt; (and the respective penalties for incorrect answers $p_2$ and $p_4$) &lt;strong&gt;to make the questions &quot;worth the same&quot;?&lt;/strong&gt;&lt;sup&gt;**&lt;/sup&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For convenience we can normalise $P_4 = 3$ the corresponding penalty $p_4 = -1$:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Set $P_2=1, p_2 = -1$, so the penalties are the same for incorrect answers. &lt;/li&gt;&#xA;&lt;li&gt;Set $P_2=2, p_2 = -2$, so the differences are the same between correct and incorrect answers.&lt;/li&gt;&#xA;&lt;li&gt;Set $P_2=3, p_2 = -3$, so the differences are the same between answering correctly and &lt;em&gt;not&lt;/em&gt; answering.&lt;/li&gt;&#xA;&lt;li&gt;Set $P_2=\sqrt{3}, p_2 = -\sqrt{3}$ so the variances are the same for random guesses.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Is there a compelling reason to prefer any of the above options, or something else altogether? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;*&lt;/sup&gt; To prevent hurting the students' feelings by giving them a negative score on a test, we can equivalently give 0 points for incorrect answer and P/N points for null/multiple answers. Whether we normalize the score around 0 or P/N is inconsequential for the discussion here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;**&lt;/sup&gt; Without penalties (for example, in the case of free-response questions) the answer is simple: just make the maximum number of points awarded for each question be the same. The question here can be re-interpreted as: does it make more sense to count the weight of each question relative to the theoretical minimum score or the theoretical average score, or something else altogether? &lt;/p&gt;&#xA;" OwnerUserId="125" LastEditorUserId="125" LastEditDate="2014-05-28T14:01:28.730" LastActivityDate="2014-05-29T15:24:42.947" Title="Grading scale: how to handle multiple choice questions with different number of choices" Tags="|exams|grading|test-design|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2444" PostTypeId="1" CreationDate="2014-05-27T11:05:21.780" Score="16" ViewCount="7506" Body="&lt;p&gt;As it stands all of my lectures are presented by me writing on pre-printed acetate sheets (printed with my notes with gaps for pictures, examples, etc.) which are then projected onto a screen using the acetate projector.&lt;/p&gt;&#xA;&lt;p&gt;While I am happy enough with this I am not very happy with the amount of acetate sheets that I am using... perhaps of the order of 1000 per year which must cost in the hundreds rather than the tens.&lt;/p&gt;&#xA;&lt;p&gt;The obvious solution would be to reuse the sheets but there are a number of problems with this. Firstly cleaning the ink off is messy and cannot be done after about a month. Secondly, and far more importantly, I like to edit and update my notes from year to year so I need new acetate sheets every year.&lt;/p&gt;&#xA;&lt;p&gt;Therefore I have been thinking of employing a new system where I have a screen displaying the same notes (in pdf), that I can write on with a 'stylus' pen such that the file and the writing is projected from the overhead onto the white screen.&lt;/p&gt;&#xA;&lt;p&gt;The first advantage to this would be that there would be no more need for acetates and the second would be that I could save the files including the original content as well as the scribblings.&lt;/p&gt;&#xA;&lt;p&gt;I recall my PhD supervisor employing such a system and I emailed him&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;What was that device you had where you could write directly onto a screen and then have it projected (and presumably save the file with the writing).&lt;/p&gt;&#xA;&lt;p&gt;I am going through a small acetate rainforest every semester and it is starting to prick my conscience.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;He replied with&#xA;&amp;quot; I use a tablet PC, i.e. a laptop running some version of Windows with touch sensitive screen, and use the programme Windows Journal. There are plenty of other versions of such programmes now I believe, but I've been using the same set-up for 7 years or so(!), so haven't kept pace with the changes... Having said that, [a colleague] recently bought an iPad and a stylus for this which seems to let him do much the same thing, and this would be a cheaper option than a tablet PC; I imagine other tablets of the iPad variety, i.e. Android devices etc., probably can be equipped with styli. What you probably need to work out is how you aim to connect your device to whatever projection facilities you have available.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I assume that the projection facilities that I have are adequate. Can anybody make a recommendation or have any other advice or feedback? I suspect that I need a more accurate stylus than my PhD supervisor uses/used.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;POST ANDREW STACEY COMMENT EDIT:&lt;/strong&gt; Thank you very much for the great advice Andrew however having looked at your slides I don't think the IPad is going to do the trick. To show what I need my solution to do I have attached the following slides from my notes (sorry that they are sideways!). I think my 'screen' that I write on will have to be A4.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/MFycZ.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/34uKb.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/9hHBL.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;SECOND EDIT:&lt;/strong&gt; What do folks know about a system using &amp;quot;a graphics tablet and a laptop&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;THIRD EDIT:&lt;/strong&gt; I now use, both on- and offline, paper together with an Epsom document camera.&lt;/p&gt;&#xA;" OwnerUserId="584" LastEditorUserId="584" LastEditDate="2020-10-06T16:04:24.397" LastActivityDate="2020-10-06T16:04:24.397" Title="Advice for Tablet/Stylus for Mathematics Teaching" Tags="|technology-in-education|software|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="2445" PostTypeId="2" ParentId="2443" CreationDate="2014-05-27T12:52:12.080" Score="-1" Body="&lt;p&gt;The &lt;em&gt;worth&lt;/em&gt; of a question is $P$. Score and penalty for an answer are proportional to $P$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This worth should not be dependant on the number of possible answers, but on the didactic worth you see in that question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you have a question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Compute 2+3.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;with the possible answers: &lt;code&gt;1,3,5,7&lt;/code&gt; or &lt;code&gt;5,7&lt;/code&gt;. Why should it's worth depend on the number of possible answers given? What, if you give 100 possible answers? Should the worth be 50 times that of a question with only two possible answers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An extreme example would be a test with one question with 100 possible answers and 100 (or 50?) questions with 2 possible answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Valueing these questions to either of your proposed versions will render some possible outcome unfair.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-05-27T12:52:12.080" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2446" PostTypeId="2" ParentId="220" CreationDate="2014-05-27T14:26:33.220" Score="4" Body="&lt;p&gt;Prove that the &lt;a href=&quot;http://en.wikipedia.org/wiki/Change-making_problem#Greedy_method&quot; rel=&quot;nofollow&quot;&gt;greedy algorithm for the change-making problem&lt;/a&gt; with Fibonacci denominations produces an optimal solution (uses the minimum number of coins possible).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This problem is not as easy as it looks. Note that there may be more than one optimal solution. It would also be instructive to try and minimize the number of times the induction axiom is used. (But no, I won't suggest that one should also try to &lt;em&gt;prove&lt;/em&gt; this minimum. =P)&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2014-05-27T14:26:33.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2447" PostTypeId="2" ParentId="2443" CreationDate="2014-05-27T16:02:43.620" Score="1" Body="&lt;p&gt;I think a well-written test should not have the need for that kind of penalty. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the test were a full problem solve, the student would be encouraged to &quot;show all work.&quot; and odds are, they'd simply do their best. At least they'd feel that if they got an incorrect answer, the teacher might give partial credit. I've seen 10 pt questions get 9 pts as it was clear the error was careless, and not a lack of understanding. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of multiple choice, the negative points for a wrong answer creates a different layer of test anxiety as the student will have to ponder each question to be 'darn sure' they are right, and perhaps leave questions blank because they aren't 100% certain. Can a student really be expected to calculate their odds for each question, to decide how right they feel about an answer? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me add a bit to my own objection. Say there are 4 choices for each question. I am going to get 80% correct. I then have 20 more questions which I have no clue on. Guessing should give me 5% more in undeserved points. So wrong guesses should count as -1/3 to negate that extra 5% from my guessing. In reality, depending on the nature of the questions, a bright student should be able to narrow down the answer to the best 2 of 4, in which case the penalty factor should be higher, -1, so the  wrong guesses cancel the 50% that were guessed right. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on the level of the course, I can spend a bit of effort coaching a student to beat a multiple choice test with the same strategy I teach them for checking their own answers on the 'long' tests. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-05-28T01:40:26.807" LastActivityDate="2014-05-28T01:40:26.807" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="2448" PostTypeId="2" ParentId="2423" CreationDate="2014-05-27T16:57:19.517" Score="1" Body="&lt;p&gt;A force $F$ is applied to a railroad car, at an angle $\theta$ above the horizontal. Find the component of the force parallel to the tracks, which is the only component that can accelerate the car.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer: $F\cos\theta$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now what do you expect to happen physically if $\theta=-10$ degrees? What if $\theta=100$ degrees? Are these really different problems mathematically, requiring qualitatively different solutions?&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-27T16:57:19.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2449" PostTypeId="2" ParentId="220" CreationDate="2014-05-27T17:02:42.153" Score="12" Body="&lt;p&gt;&lt;strong&gt;Base problem&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some positive integers are written on the board. At each step one of the integers $n$ is erased and replaced with any number of positive integers that are all less than $n$. Of course, if the erased integer is $1$ then no new positive integers can be written on the board. Prove that no matter how this procedure is carried out the board must eventually become empty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This problem is just the right difficulty and can ensure that students really understand what induction is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;General problem&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prove that the following game always terminates regardless of the initial configuration and the choices (of node, subtree and number of copies) throughout the game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Start with any finite tree rooted at $r$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) At each step:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; (2.1) Select any node $x$ and a child node $y$ of $x$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; (2.2) Remove the subtree rooted at $y$ (and of course the edge $(x,y)$ as well)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; (2.3) If $x$ is not $r$, attach finitely many copies of the subtree rooted at $x$ to its parent node&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the following diagram shows one step, where the chosen subtree to be removed is shown in red and the subtree that would be copied is shown in green.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/qjaak.png&quot; alt=&quot;&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The base problem is equivalent to the special case of this general problem where the initial tree has depth at most $2$. This generalization is a harder problem that is an extension of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Hydra_game&quot; rel=&quot;noreferrer&quot;&gt;Hydra Game&lt;/a&gt;. A hint is to define a total ordering on trees:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For any finite trees $S,T$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; $S \le T$ iff:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; Either:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; &amp;emsp; $S$ is a leaf node&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; Or:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; &amp;emsp; $S,T$ are both not leaf nodes&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; &amp;emsp; The subtrees of $S$ when ordered forms a lexicographically smaller sequence than that of $T$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then one must show that at each step the whole tree decreases according to the ordering, and also that any decreasing sequence of trees must terminate. That last part is the hardest. Basically, induct on the tree depth first, which then allows inducting on the biggest subtree, and then induct on the number of copies of the biggest subtree.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this proof shows that with the right definitions and inductions it is quite intuitive, and there is no need to introduce ordinals.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2014-05-27T17:02:42.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2450" PostTypeId="2" ParentId="2429" CreationDate="2014-05-27T17:12:26.737" Score="4" Body="&lt;p&gt;An unusual choice could be &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/3319167200&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra&lt;/a&gt; by Cox, Little and O'Shea. It may not necessarily be a popular choice for a first course in abstract algebra, but I get the impression that it does get some mileage as an undergraduate introduction to aspects of algebraic geometry. The formal prerequisites are a course in linear algebra and a course involving doing proofs (in some situations, these &amp;quot;two&amp;quot; courses could be one and the same). Note that the book does &lt;em&gt;not&lt;/em&gt; require prior knowledge of abstract algebra and the authors suggest that it could be used for a first course in the subject. Naturally, the emphasis is on fields and rings, rather than groups.&lt;/p&gt;&#xA;&lt;p&gt;One potentially-attractive aspect of the book is the way in which it combines algebra, geometry and algorithms. There's plenty of material for a whole year, some nice applications to robotics and automatic geometric theorem proving, and an appendix containing suggested projects. You could also link up with other subjects such as geometric combinatorics (including polytopes) and algebraic statistics, although the latter could carry one too far afield for a first course. For the former, see Rekha Thomas' &lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=STML-33&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lectures on Geometric Combinatorics&lt;/a&gt;; for the latter, see for instance Seth Sullivant's &lt;a href=&quot;https://bookstore.ams.org/gsm-194/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebraic Statistics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2020-09-20T15:55:18.780" LastActivityDate="2020-09-20T15:55:18.780" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2451" PostTypeId="1" CreationDate="2014-05-27T17:32:42.573" Score="2" ViewCount="129" Body="&lt;p&gt;Particularly, I'm looking for an undergraduate text with an excellent explanation of $\delta$-$\varepsilon$ proofs, and many example questions related thereto.&lt;/p&gt;&#xA;" OwnerUserId="1553" LastEditorUserId="1553" LastEditDate="2014-05-27T21:15:49.983" LastActivityDate="2014-05-27T21:15:49.983" Title="Searching for an analysis textbook" Tags="|undergraduate-education|textbooks|mathematical-analysis|" AnswerCount="0" CommentCount="10" ClosedDate="2014-05-28T09:10:19.707" ContentLicense="CC BY-SA 3.0" />
  <row Id="2452" PostTypeId="1" CreationDate="2014-05-27T20:46:59.467" Score="12" ViewCount="285" Body="&lt;p&gt;Next year, I will be teaching a very challenging second year &quot;multivariable calculus and calculus on manifolds&quot; course, and will assign a large number of difficult problem sets. I am thinking of assigning students to go back to problems where they had good but not perfect solutions and rewrite an elegant, well presented solution. There are two goals here:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Pedagogically, to force students to really think through and present their ideas well, and to avoid the feeling that each problem set is passed through and ignored.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Logistically, to save myself the time of writing lengthy solutions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Have some of you tried this? How did you make the task fair, when some problems are much harder to write than others? What were the logistical practicalities: How long did students have to produce solutions, and how were they evaluated?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be clear about the sort of student population we are talking about, this is the top 20 or so kids at Michigan, who have already done a similarly challenging single variable course and should be anticipating a high workload. For those who know Michigan, this is 395-396; I believe the Chicago and Harvard equivalents are 20700-20800-20900 and 55b (but in a year, not the term that 55b tries to squeeze into.)&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="143" LastEditDate="2014-05-27T20:55:00.630" LastActivityDate="2014-06-05T00:05:47.437" Title="Assigning students to write solution sets" Tags="|undergraduate-education|homework|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2453" PostTypeId="2" ParentId="2443" CreationDate="2014-05-27T22:19:35.473" Score="2" Body="&lt;p&gt;I believe option number 2 is the correct one, given the circumstances that you describe.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;At its base, you want each question on this multiple-choice test, regardless of how many possible answers it has, to be worth $1$ point if the student has demonstrated he knows the answer to the question, and $0$ points otherwise. This is the only fundamental way to make each question &quot;worth&quot; the same. Anything else is just clever rescaling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most common way to formulate this is to give $1$ point for a correct answer, and $0$ points for an incorrect answer. However, complicating this is the fact that a student might get the correct answer purely through guessing. So, you decide to give them an &quot;incentive&quot; (really just removing the edge from the former choice) to not answer if they have no idea, by giving them $1/n$ points, the expected value from guessing randomly, for leaving the question blank. In your scenario, this is $1/4$ point for a question with four choices, and $1/2$ point for a true-false question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you scale this up by a factor of $4$ to make the point values all integers, you get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{matrix}&#xA;P_4 = 4 &amp;amp; O_4 = 1 &amp;amp; p_4 = 0\\&#xA;P_2 = 4 &amp;amp; O_2 = 2 &amp;amp; p_2 = 0&#xA;\end{matrix}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if you subtract enough from each row to make $O_n$ equal to $0$ in all cases, you get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{matrix}&#xA;P_4 = 3 &amp;amp; O_4 = 0 &amp;amp; p_4 = -1\\&#xA;P_2 = 2 &amp;amp; O_2 = 0 &amp;amp; p_2 = -2&#xA;\end{matrix}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which is exactly your second option.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another, less common way to formulate this is to give $1$ point for a correct answer, and $0$ points to the expected value of random guessing. This also fits the description, assuming that you equate random guessing to &quot;not knowing anything about the question&quot;. If this is the case, then you'll want a matrix where the following is true:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{matrix}&#xA;P_4 = 1 &amp;amp; O_4 = 0 &amp;amp; 3p_4 + P_4 = 0\\&#xA;P_2 = 1 &amp;amp; O_2 = 0 &amp;amp; p_2 + P_2 = 0&#xA;\end{matrix}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solve those two equations on the right and we get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{matrix}&#xA;P_4 = 1 &amp;amp; O_4 = 0 &amp;amp; p_4 = -1/3\\&#xA;P_2 = 1 &amp;amp; O_2 = 0 &amp;amp; p_2 = -1&#xA;\end{matrix}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And scale it up by three to make the point values integers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{matrix}&#xA;P_4 = 3 &amp;amp; O_4 = 0 &amp;amp; p_4 = -1\\&#xA;P_2 = 3 &amp;amp; O_2 = 0 &amp;amp; p_2 = -3&#xA;\end{matrix}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and we get your third option.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we've narrowed our choices down to your options 2 and 3 because the first two don't account for the model of &quot;$1$ point for knowing, $0$ points for not knowing&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point, I'd go with option 2 for two reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It follows the more conventional formulation of $1$ point for a correct answer and $0$ points for an incorrect answer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It feels more aesthetically pleasing. Imposing a penalty of $-3$ points for an incorrect answer to a true-and-false feels like way too much gravity to put onto a true-false question.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Note that the AP exams, before they dropped penalties altogether, went with option 3, taking away $1/4$ point for incorrect answers to questions with $5$ choices and $1/3$ point for incorrect answers to question with $4$ choices. However, there's not much difference between $1/4$ and $1/3$, while there is a huge difference between $-1/3$ and $-1$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;That being said, I really do need to criticize the usage of penalties for incorrect answers on multiple-choice exams here. I completely fail to understand how getting an answer incorrect somehow demonstrates that the student knows &lt;em&gt;less than nothing&lt;/em&gt; about the question, or conversely how leaving the answer blank means that the student knows any more about the question than how to take the test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may be (and most people are) doing this to compensate for the fact that getting an answer correct does &lt;em&gt;not&lt;/em&gt; necessarily demonstrate that the student knows the answer to the question. But that is a flaw inherent to multiple-choice tests. You can't get rid of that just by making leaving an answer blank an equal alternative (which is what the expected-value penalty really does).&lt;/p&gt;&#xA;" OwnerUserId="856" LastEditorUserId="856" LastEditDate="2014-05-28T16:10:22.093" LastActivityDate="2014-05-28T16:10:22.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2454" PostTypeId="2" ParentId="220" CreationDate="2014-05-27T23:49:25.490" Score="8" Body="&lt;p&gt;Three connected induction proofs:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) For a really easy one: Every tree of $n$ arcs has $n+1$ nodes.&#xA;($0$ arcs $\implies$ $1$ node. Remove a leaf node and its incident arc; induct.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) A bit more difficult: Every polygon triangulation can be $3$-colored.&#xA;Assume every polygon has a triangulation. See below.&#xA;Argue that the dual graph is a tree.&#xA;Remove a leaf node by clipping off an &quot;ear tip.&quot;&#xA;$3$-color the remainder and return the ear triangle, coloring its tip different from&#xA;its two neighbors.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/DbNcf.png&quot; alt=&quot;3-coloring&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(3) Every polygon has a triangulation. Prove there exists an interior diagonal. Induct on the two halves.&#xA;Proved, e.g., in &lt;a href=&quot;http://cs.smith.edu/~orourke/DCG/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Discrete and Computational Geometry&lt;/em&gt;&lt;/a&gt;, p.4.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-05-27T23:49:25.490" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2455" PostTypeId="2" ParentId="2342" CreationDate="2014-05-28T03:22:43.173" Score="1" Body="&lt;p&gt;Perhaps the best way is to understand any (well-formed) statement as being either true or false. Each statement may be about certain objects or an absolute statement independent of any object. Then it is quite clear that some statements are true and some are false, with no necessary link between them. A theorem of the form (If P, then Q) can be understood to mean that we somehow know that whenever the statement P is true, the statement Q is true too. Examples are important, and can even lay the foundation of first-order logic without ever talking about propositions and logical symbols, not to say &quot;converses&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask your students to take out 3 pencils each and place them on the table. Write the following three statements on the board:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(P) The first and second are parallel&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Q) The second and third are parallel&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(R) The first and third are parallel&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask them if there are any situations where:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(0) None of them are true&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Exactly one of them is true&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Exactly two of them are true&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) All of them are true&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After they experiment, ask them to tell you the answers and the constructions for the possible scenarios. They won't be able to arrange the pencils to make (2) true. Ask them if perhaps they did not try hard enough. When they are convinced that it is impossible, ask them why? Probably some of them will realize that when P and Q are both true, R is also true (which we know as $P \wedge Q \Rightarrow R$). But since (1) can happen, it is possible that R is true while P and Q are both false. Also, notice that this very fact is the same as saying that (2) implies (3). But certainly (3) implies (2), and hence we have (2) and (3) being equivalent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Emphasize that mathematics is about figuring out what is true based on what we know to be true already.&lt;/strong&gt; In the above example, if for a certain arrangement of 3 pencils we know that ( P and Q ) is true (someone we trust told us so), we can figure out that R is also true &lt;strong&gt;without having to look at the pencils&lt;/strong&gt;. But if we only know that R is true, we cannot claim that ( P and Q ) is true, because we don't have enough information to rule out the possibility that the first and third pencils are parallel but the second is slanted to both of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This same example will also be useful to explain equivalence relations and equivalence classes in the near future. It is also related to the fact that you need at least $n$ equations to determine the values of $n$ unknown real numbers, which in this case are rotations with respect to the first pencil. A lot can be discovered with pencils!&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2014-05-28T03:22:43.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2456" PostTypeId="2" ParentId="2443" CreationDate="2014-05-28T08:18:08.077" Score="1" Body="&lt;p&gt;To me, it seems desirable to make each question have the same weight in determining a student's final grade.   That implies that which rubric is best depends on the way you assign grades for the course.  Based on your first footnote, I assume that a student's grade for the course depends on the difference between his total score and the total scores of other students.  Based on that, it seems that option #2 is the most reasonable -- it will make each question have equal weight in determining the student's grade.&lt;/p&gt;&#xA;" OwnerUserId="1559" LastActivityDate="2014-05-28T08:18:08.077" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2457" PostTypeId="1" CreationDate="2014-05-28T11:46:03.043" Score="13" ViewCount="2765" Body="&lt;p&gt;This is perhaps the question that some thought &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2443/grading-scale-how-to-handle-multiple-choice-questions-with-different-number-of&quot;&gt;Grading scale: how to handle multiple choice questions with different number of choices&lt;/a&gt; really was.  I'm of the opinion that multiple choice questions should not be used for &lt;a href=&quot;http://en.wikipedia.org/wiki/Summative_assessment&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;summative&lt;/em&gt; assessment&lt;/a&gt;.  However, I have no actual evidence to back up that view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has there been actual research on this, specifically with regards to mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know if the use of multiple choice questions is different in mathematics to other subjects by dint of sheer ignorance in how other subjects are examined.  I do know that often multiple choice questions in mathematics are not simply &quot;Do you happen to know the answer?&quot; questions where the route to the answer is &lt;em&gt;irrelevant&lt;/em&gt; but the goal is more that by seeing the choice of answer one can deduce the route that they took and so there is no need to see the details (this assumes that they took a route and did not simply guess).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A sub-issue in this is the numerous schemes to discourage guessing.  I'd happily also learn about any research as to the effectiveness of these.  Again, my opinion is that they do not correct for the failings of the use of multiple choice questions but, again, I have no research-based evidence for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me conclude by re-emphasising that I am asking about &lt;strong&gt;summative&lt;/strong&gt; assessment and not &lt;a href=&quot;http://en.wikipedia.org/wiki/Formative_assessment&quot; rel=&quot;nofollow noreferrer&quot;&gt;formative assessment&lt;/a&gt; and about actual research.  I'm not interested in answers that are purely anecdotal or opinion-based (I'll happily hear those in another venue, though).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In thinking about MattF's comment about &lt;em&gt;correlation&lt;/em&gt;, let me try to focus it more precisely.  I have no doubt that multiple choice test scores are &lt;em&gt;correlated&lt;/em&gt; with every other type of test.  That doesn't speak to their &lt;em&gt;efficacy&lt;/em&gt; and &lt;em&gt;fairness&lt;/em&gt; though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the following two types of question:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Multiple choice, where the options include &quot;Some of the above&quot; and &quot;All of the above&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;No partial credit, but credit only given for an answer with reasoning.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In both, it is an &quot;all or nothing&quot;.  I've posited the extra options for the &quot;multiple choice&quot; variant to make it so that it isn't just a closed list to choose between and so to &lt;em&gt;know&lt;/em&gt; the answer then a student ought to have worked it out.  In the first, a student can guess.  In the second, they can't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question, then, could be phrased as: how much effect does the fact that students can guess an answer have on a multiple choice test's ability to report students' abilities when compared with a &quot;no partial credit&quot; test?&lt;/p&gt;&#xA;" OwnerUserId="112" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-08-28T18:18:08.227" Title="What research is there on the effectiveness of multiple choice questions for summative assessment?" Tags="|education-research|assessment|" AnswerCount="3" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="2458" PostTypeId="1" AcceptedAnswerId="2481" CreationDate="2014-05-28T15:44:16.470" Score="11" ViewCount="598" Body="&lt;p&gt;When studying linear algebra in mathematics (I mean, for the people studying mathematics) there are many ways of approaching it, depending of your &lt;em&gt;needs&lt;/em&gt;, however supposedly every mathematician should have about the same basic concepts, which I think I have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been asked to help this kid, who's studying engineering, and is having a lot of trouble with linear algebra, nonetheless I'm a little nervous because I don't know how to help him well. Some friends tell me that is practically the same as pure mathematics except without proofs and that is just doing excercices. But what does that mean? Getting really good with Gram–Schmidt without understanding where it came from? Do they study dual spaces? What is the best way to help without making a mess in his mind?&lt;/p&gt;&#xA;" OwnerUserId="99" LastEditorUserId="376" LastEditDate="2016-03-14T11:04:23.257" LastActivityDate="2016-03-14T11:04:23.257" Title="Linear algebra for engineers" Tags="|undergraduate-education|tutoring|linear-algebra|engineering-mathematics|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2459" PostTypeId="1" CreationDate="2014-05-28T18:05:21.200" Score="12" ViewCount="1252" Body="&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt; I'm teaching mathematics at a high school with emphasis on mathematics and natural sciences. Usually, pupils at our school have to buy a calculator (TI-89) and we work with those during class and sometimes they are allowed to use them for exams too. Recently, it has been proposed to work with tablets instead of conventional calculators. There has been a poll at the most recent meeting of the maths teachers giving the following result:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For the next school year, each maths teacher decides on his own&#xA;  whether his students buy a conventional calculator or a tablet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I myself think that conventional calculators are a lot easier to use, don't provide too much or too little information (as opposed to e.g. wolframalpha for students at that stage) and also at our local universities' exams, no tablets are allowed but only calculators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Do you think high school students should have to stick to conventional calculators or what clear advantages are there in switching to tablets instead?&lt;/p&gt;&#xA;" OwnerUserId="493" LastActivityDate="2015-02-26T21:14:01.610" Title="Should my high school students use tablets or conventional calculators?" Tags="|technology-in-education|aids|calculators|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2460" PostTypeId="2" ParentId="2459" CreationDate="2014-05-28T20:24:51.697" Score="16" Body="&lt;p&gt;My answer would be neither. A TI-89 is \$80, which is a lot of money for many families, and the functionality it provides beyond that of a \$5 calculator is hardly ever needed. I don't own a graphing calculator myself, so I can't see forcing my students to buy one. A tablet is even more money and even more overkill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this is a public high school in the US, isn't it illegal to force students to pay for either of these things? IIRC there have been lawsuits recently over schools' attempts to force families to buy supplies like kleenex, pencils, etc. Many state constitutions mandate free and compulsory K-12 education.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-28T20:24:51.697" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2461" PostTypeId="1" AcceptedAnswerId="2482" CreationDate="2014-05-28T21:40:49.223" Score="5" ViewCount="265" Body="&lt;p&gt;I'd like to explain the &lt;a href=&quot;http://en.wikipedia.org/wiki/MapReduce&quot; rel=&quot;noreferrer&quot;&gt;MapReduce&lt;/a&gt; programming model to a group of students. I thought about a little group work, where each student acts as a &amp;quot;Map-process&amp;quot; and has to perform a little computation. The results are then aggregated by other &amp;quot;Reduce&amp;quot; students to gain the overall result.&lt;/p&gt;&#xA;&lt;p&gt;For comparison, I'd have the best student calculate on his own. In the end, I'd like to compare the time needed for the calculation by the single student and the rest.&lt;/p&gt;&#xA;&lt;p&gt;The missing part for this group exercise is the type of calculation the students would have to do. Ideally, this would be something which a group can do much faster than a single student. Overall, the computation should not be too complicated and take more than one or two minutes.&lt;/p&gt;&#xA;&lt;p&gt;I thought about calculating the average temperature over the last 200 years. In this example, 20 &amp;quot;Map&amp;quot; students would calculate the average temperature for a decade, followed by a &amp;quot;Reduce&amp;quot; student calculating the overall average.&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure if the average temperature calculation is a bit too simple. Are there any calculations which are more suitable for such an exercise?&lt;/p&gt;&#xA;&lt;p&gt;The students are college students, so their maths levels should be like a high-school graduate.&lt;/p&gt;&#xA;" OwnerUserId="1564" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-05-29T20:45:41.837" Title="Good example for manual execution of MapReduce" Tags="|technology-in-education|interactive-teaching|class-participation|software|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2462" PostTypeId="1" CreationDate="2014-05-28T22:28:32.980" Score="16" ViewCount="1533" Body="&lt;p&gt;I'm looking for fun games for children (from 4 to 12 years) used in learning math concepts. They can be offline or online games.&lt;/p&gt;&#xA;" OwnerUserId="1565" LastEditorUserId="1892" LastEditDate="2014-07-11T11:06:04.840" LastActivityDate="2019-02-19T03:21:49.653" Title="Fun games for children" Tags="|technology-in-education|primary-education|preschool-education|games|" AnswerCount="6" CommentCount="8" ClosedDate="2019-02-20T15:20:46.050" ContentLicense="CC BY-SA 3.0" />
  <row Id="2463" PostTypeId="2" ParentId="2462" CreationDate="2014-05-28T22:57:52.180" Score="7" Body="&lt;p&gt;I think &lt;a href=&quot;http://setgame.com&quot;&gt;Set&lt;/a&gt; is a great commercial game.  There's a daily instance of the game that can be played at the same website I just linked to.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-05-28T22:57:52.180" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2465" PostTypeId="2" ParentId="2462" CreationDate="2014-05-29T00:15:12.960" Score="8" Body="&lt;p&gt;Another great game is &lt;a href=&quot;http://www.thinkfun.com/rushhourjr&quot;&gt;Rush hour&lt;/a&gt;.  It requires the important but in my opinion underemphasized skills of nonverbal problem solving, working backwards and trying all possible options.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-05-29T00:15:12.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2466" PostTypeId="1" AcceptedAnswerId="2469" CreationDate="2014-05-29T01:19:56.200" Score="8" ViewCount="729" Body="&lt;p&gt;The title pretty much explains the question. I've never taken a course in economics, so it's hard for me to judge what are the most important applications, the most interesting ones, or the ones that show the calculus techniques to best advantage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I have so far:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The marginal rate of substitution is minus the derivative of the indifference curve.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Various business-y applications of optimization, including economic order quantity.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Laffer-curve argument related to Rolle's theorem.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Free-market equilibrium occurs where the supply and demand curves intersect (intermediate value theorem).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Consumer's and producer's surplus as applications of the definite integral.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-30T20:45:35.833" Title="&quot;Greatest hits&quot; of applications of freshman calculus to economics?" Tags="|calculus|applications|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2467" PostTypeId="2" ParentId="2466" CreationDate="2014-05-29T01:58:46.297" Score="6" Body="&lt;p&gt;I'm fond of the &lt;a href=&quot;http://math.scu.edu/~ffarris/MonthlyFinal.pdf&quot;&gt;Gini index&lt;/a&gt;, a useful and interesting measure of the &quot;fairness&quot; of income distribution and requires the ability to integrate.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-05-29T01:58:46.297" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2468" PostTypeId="2" ParentId="2466" CreationDate="2014-05-29T09:48:21.957" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Expected_utility_hypothesis&quot;&gt;Expected utility&lt;/a&gt; is a key concept, a paradigm for rationality, which uses calculus in several ways:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Expected utility is an integral, $E[U]=\int U(x)\,p(x)\,dx$, where $U$ is utility and $p$ is probability.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The second derivative $U''(x)$ is typically required to be negative for risk aversion&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Expected utility is to be maximized (e.g. by varying the amount of risky assets in a portfolio)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-05-29T09:48:21.957" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2469" PostTypeId="2" ParentId="2466" CreationDate="2014-05-29T12:56:37.203" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Present_value&quot; rel=&quot;nofollow&quot;&gt;Present&lt;/a&gt; and &lt;a href=&quot;http://en.wikipedia.org/wiki/Future_value&quot; rel=&quot;nofollow&quot;&gt;future value&lt;/a&gt; of a continuous income stream $f(t)$, continuously compounded at a rate $r$, over $0 \le t \le T$:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Total income: $TI = \int_0^T f(t) \; dt$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Present value of the stream: $PV = \int_0^T f(t) e^{-rt} \; dt$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Future value over $0 \le t \le T$: $FT = e^{rT} \int_0^T f(t) e^{-rt} \; dt = e^{rT} PV$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="498" LastActivityDate="2014-05-29T12:56:37.203" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2470" PostTypeId="1" CreationDate="2014-05-29T13:54:35.387" Score="6" ViewCount="946" Body="&lt;p&gt;Why is factorization of integers important in a first number theory course at an undergraduate level? Where is factorization used in real life? Are there examples which have a real impact? I am looking for examples which will motivate students. &lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="11" LastEditDate="2014-05-29T15:09:04.600" LastActivityDate="2014-06-01T21:01:38.177" Title="Why teach factorization of integers?" Tags="|undergraduate-education|student-motivation|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2471" PostTypeId="1" AcceptedAnswerId="2472" CreationDate="2014-05-29T14:26:55.377" Score="11" ViewCount="1138" Body="&lt;p&gt;I asked my students the following question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt; Express &lt;span class=&quot;math-container&quot;&gt;$\cos(\pi+x)$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A:&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$-\cos(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Students:&lt;/strong&gt; Yeah, but where is the &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; part? If I got this in an exam then I'd think I was wrong...yada-yada-yada....&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Their issue is with the use of the word &amp;quot;and&amp;quot;. Logically speaking, it should be an &amp;quot;or&amp;quot;, but that would confuse them even more! Therefore, I was wondering if anyone has either,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A better way of phrasing this, and similar, question(s).&lt;/li&gt;&#xA;&lt;li&gt;A decent explanation of why they needn't look for the &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; part.&lt;/li&gt;&#xA;&lt;li&gt;A persuasive argument as to why I should bite the dust and just use &amp;quot;or&amp;quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I tried explaining to them that they could write it as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$-\cos(x)+0\sin(x)$&lt;/span&gt;&amp;quot;, but this didn't seem to help.&lt;/p&gt;&#xA;" OwnerUserId="169" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-09-06T22:15:29.560" Title="The word &quot;and&quot; rather than &quot;or&quot;" Tags="|mathematical-pedagogy|terminology|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2472" PostTypeId="2" ParentId="2471" CreationDate="2014-05-29T14:34:00.213" Score="14" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Express $\cos(x + \pi)$ in terms of $\sin x$ or $\cos x$ (possibly both).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the &quot;or&quot; is logically more correct and the parenthetical recalls/stresses/clarifies this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I am in favor of redundancy and details for such things.&lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-05-29T14:34:00.213" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2473" PostTypeId="2" ParentId="2443" CreationDate="2014-05-29T15:24:42.947" Score="4" Body="&lt;p&gt;Let's consider an extreme case: An exam with two questions. Questions 1 has 2 options (only one is correct). Question 2 has 1000 options (only one is correct). Let $c_i&amp;gt;0$ be the points awarded for a correct answer to problem $i$ and $w_i&amp;gt;0$ the points subtracted for a wrong answer. Without loss of generality, we can scale $c_1=1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then there are some desirable aspects we wish to achieve:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Random guessing for each problem has an expected value of 0 points. For this to occur we have to satisfy:&#xA;\begin{eqnarray*}&#xA;0.5c_1-0.5w_1 &amp;amp;=&amp;amp;0\\&#xA;0.001c_2-0.999w_2 &amp;amp;=&amp;amp;0&#xA;\end{eqnarray*}&#xA;Then $w_1=c_1=1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Each question carries the same relative weight in damaging/helping the overall score.&#xA;For this to occur we have to satisfy:&#xA;\begin{eqnarray*}&#xA;c_1+w_2 = w_1+c_2&#xA;\end{eqnarray*}&#xA;Since then missing exactly one question results in the same score. But, in light of $c_1=w_1$ above, this reduces to&#xA;\begin{eqnarray*}&#xA;w_2 =c_2&#xA;\end{eqnarray*}&#xA;This, of course, contradicts the null expected value condition. So we can't do (1) and (2) at the same time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two possible solutions/alterations:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1') Random guessing throughout the exam has an expected value of 0 points for the whole exam. For (1') and (2) to occur we have to satisfy:&#xA;\begin{eqnarray*}&#xA;0.5c_1-0.5w_1+0.001c_2-0.999w_2&amp;amp;=&amp;amp;0\\&#xA;c_1+w_2 &amp;amp;=&amp;amp; w_1+c_2&#xA;\end{eqnarray*}&#xA;With $c_1=1$, this reduces to&#xA;\begin{eqnarray*}&#xA;-0.5w_1+0.001c_2-0.999w_2&amp;amp;=&amp;amp;-0.5\\&#xA;w_1+c_2- w_2 &amp;amp;=&amp;amp;1&#xA;\end{eqnarray*}&#xA;This has solution set $w_1=1-\frac{998}{1499}c_2$ and $w_2=\frac{501}{1499}c_2$. To keep $w_1&amp;gt;0$, then $c_2$ should be chosen less than $\frac{1499}{998}$. Maybe $c_2=1$ makes sense, in which case $w_1=w_2=\frac{501}{1499}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2') The other possible solution is not to weigh each question equally (e.g. weigh easy ones less or something). Suppose we want Question 2 to weigh twice as much as Question 1 (I mean, c'mon, 1000 choices makes it hard to find just the right one.) Then (1) and (2') give:&#xA;\begin{eqnarray*}&#xA;0.5c_1-0.5w_1 &amp;amp;=&amp;amp;0\\&#xA;0.001c_2-0.999w_2 &amp;amp;=&amp;amp;0\\&#xA;2(c_1+w_2) &amp;amp;=&amp;amp; w_1+c_2&#xA;\end{eqnarray*}&#xA;Setting $c_1=1$ and solving gives: $w_1=1$, $c_2=\frac{999}{997}$, $w_2 = \frac{1}{997}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's about the best analysis I can think of. Of course, the above numbers can be modified for your case of 4 choices; I just think better in extremes. I'm guessing if you start to have more than two questions, the systems of equations get bigger but stay linear and solvable if you choose one of the suggested alterations.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-05-29T15:24:42.947" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2474" PostTypeId="2" ParentId="2470" CreationDate="2014-05-29T15:25:16.110" Score="1" Body="&lt;p&gt;Comparison of number systems: Why is the decimal system &lt;em&gt;mathematically&lt;/em&gt; worse than the duodecimal system? Why is the sexigesimal system so useful, that it's still used even if nearly everything else (outside of the US, Liberia and Burma) is decimal?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is the factoring of these numbers.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="439" LastEditDate="2014-05-30T09:37:11.770" LastActivityDate="2014-05-30T09:37:11.770" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2475" PostTypeId="2" ParentId="2471" CreationDate="2014-05-29T15:47:59.210" Score="3" Body="&lt;h1&gt;Better Phrasing&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;First of all, you should use $\sin(x)$ and $\cos(x)$ instead of $\sin$ and $\cos$ as mweiss explains in his comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, &lt;code&gt;and&lt;/code&gt; and &lt;code&gt;or&lt;/code&gt; are logical operators. They can naturally be interpreted as set operations $\cap$ and $\cup$. What you want is neither $\{\sin\}\cup\{\cos\}$ nor $\{\sin\}\cap\{\cos\}$, but rather something like $\mathrm{span}\{\sin,\cos\}$ where the underlying field is unclear, or the language of algebraic expressions using $\sin(x)$ and $\cos(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it's clear to you, what you mean and your students will understand that, than use that explicitely for example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Express $\cos(\pi+x)$ as a linear combination of $\sin(x)$ and $\cos(x)$ in the vector space of real valued functions over the field of real numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If it's unclear to you or your students won't understand it, than the most explicit proposal is surely that of Steven Gubkin:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Express $\cos(\pi+x)$ as a (algebraic) combination $\sin(x)$ and $\cos(x)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;h1&gt;Looking for the $\sin$-part&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;This is something, that should've been taught in high school. You can simply give several trivial transformations using $\sin(x)$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$-\cos(x)+0\sin(x)\quad,\quad -\cos(x)\cdot\sin(x):\sin(x)\quad,\quad -\cos(x)+\sin(x)-\sin(x)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, if someone is looking for a $\sin$-part, he may do so, but he needs not necessarily find one. Besides, why bother?&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Using &lt;code&gt;or&lt;/code&gt; instead&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;You shouldn't bite the dust. &lt;code&gt;or&lt;/code&gt; might be interpreted as &lt;code&gt;either…or&lt;/code&gt; or as not allowing a combination of $\sin(x)$ and $\cos(x)$.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="42" LastEditDate="2014-05-29T16:43:35.847" LastActivityDate="2014-05-29T16:43:35.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2476" PostTypeId="2" ParentId="2471" CreationDate="2014-05-29T16:15:00.450" Score="2" Body="&lt;p&gt;Q: Simplify $\cos(\pi+x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back in the days I made it a point that the students should be able to deal with &quot;soft&quot; instructions and &quot;high-level&quot; instructions. One of my favorite 40 min tests is: &quot;Discuss the function ...&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clarification: the instruction &quot;discuss the function&quot; makes sense to the students at that point because by then they should have learned that it means working out&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;domain and value &lt;/li&gt;&#xA;&lt;li&gt;symmetry&lt;/li&gt;&#xA;&lt;li&gt;asymptotes&lt;/li&gt;&#xA;&lt;li&gt;derivatives&lt;/li&gt;&#xA;&lt;li&gt;zeroes of function and derivatives with multiplicity and effect on the function's graph&lt;/li&gt;&#xA;&lt;li&gt;draw the graph in the &quot;interesting&quot; region with suitable elements discovered earlier&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1479" LastEditorUserId="1479" LastEditDate="2014-05-30T12:00:42.683" LastActivityDate="2014-05-30T12:00:42.683" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2477" PostTypeId="1" CreationDate="2014-05-29T16:16:04.690" Score="11" ViewCount="641" Body="&lt;p&gt;When talking with students it frequently happens that they misunderstand what you meant. The common example is the amount of rigor that one would consider &amp;quot;a proof&amp;quot;, but there are other things, like&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;smooth meaning &lt;span class=&quot;math-container&quot;&gt;$C^2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$C^\infty$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;ring meaning a ring with or without multiplicative identity,&lt;/li&gt;&#xA;&lt;li&gt;representing points as row or column vectors,&lt;/li&gt;&#xA;&lt;li&gt;defining Fourier transform with different constants,&lt;/li&gt;&#xA;&lt;li&gt;zero being a natural number or not,&lt;/li&gt;&#xA;&lt;li&gt;using term &amp;quot;graph&amp;quot; for undirected graphs only,&lt;/li&gt;&#xA;&lt;li&gt;tree meaning rooted binary tree,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;or for more soft examples&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;participation of the students in lectures,&lt;/li&gt;&#xA;&lt;li&gt;checking homework or some specific way of grading it,&lt;/li&gt;&#xA;&lt;li&gt;relation between exam problems and the problems covered in the class,&lt;/li&gt;&#xA;&lt;li&gt;the way the instructor answers questions,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;and others. Of course, it's best to address all the uncertain points (e.g. at the first lecture or in the notes), however, the problem is it's not always possible, to name a few reasons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;there is not enough resources (time/space/attention span),&lt;/li&gt;&#xA;&lt;li&gt;some things seem too obvious,&lt;/li&gt;&#xA;&lt;li&gt;one my find him/herself unprepared (for whatever reason),&lt;/li&gt;&#xA;&lt;li&gt;one doesn't know the audience,&lt;/li&gt;&#xA;&lt;li&gt;one doesn't have enough experience,&lt;/li&gt;&#xA;&lt;li&gt;some point was addressed, but the student missed or forgot it.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Naturally, it doesn't apply only to students, but this is frequent and may have some specifics (e.g., I'm not saying it's good, but it is possible to force some convention).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;General question:&lt;/strong&gt; How to best address these issues?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;More specific questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;What approach to use, when there is no enough time or space to explain?&lt;/em&gt; For example, if I want to fit all the homework problems on one sheet (for example to avoid tl;dr reaction).&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;How to notice misunderstanding when it happens?&lt;/em&gt; E.g. it's hard to distinguish between non-understanding and misunderstanding.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;How to fix the problem when spotted?&lt;/em&gt; For example, I may have noticed that the student uses a different definition, but we are in the middle of reasoning and making a digression may be undesirable. Another example, the student solved an effectively different (perhaps easier) problem.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;How can I make the students be aware of it and careful in this aspect?&lt;/em&gt; E.g. to make them ask for clarification instead of assuming some particular version.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; There's a very nice post about definitions by Jeff Stuart &lt;a href=&quot;https://web.archive.org/web/20160315230252/http://mathforum.org/kb/message.jspa?messageID=5947620&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, thanks to @Dave L Renfro for sharing it.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="6053" LastEditDate="2022-09-16T10:23:13.963" LastActivityDate="2022-09-16T10:23:13.963" Title="Common phrases having different meaning" Tags="|undergraduate-education|students-mistakes|teacher-preparation|classroom-interaction|terminology|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="2478" PostTypeId="2" ParentId="2470" CreationDate="2014-05-29T16:38:01.640" Score="8" Body="&lt;p&gt;Luckily there's a &lt;a href=&quot;http://www.ams.org/bookstore-getitem?item=STML-68&quot; rel=&quot;nofollow&quot;&gt;book&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;The Joy of Factoring&lt;/em&gt;.&#xA;  Samuel S. Wagstaff, Jr.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;that answers this exact question -- see the review on the linked page.  In addition to cryptography, the book motivates an interest in factoring via repunits, repeating decimal fractions, perfect numbers, Gauss's interest in factoring and the Cunningham project.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="511" LastEditDate="2014-05-30T03:01:58.967" LastActivityDate="2014-05-30T03:01:58.967" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2479" PostTypeId="2" ParentId="2470" CreationDate="2014-05-29T16:39:37.817" Score="6" Body="&lt;p&gt;There is (at least) one significant reason behind the factorization: the relevant intuition. For example&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;it allows you to think of natural numbers as vectors (e.g. $\gcd$ and $\mathrm{lcm}$ become component-wise $\min$ and $\max$);&lt;/li&gt;&#xA;&lt;li&gt;it allows you to think of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Chinese_remainder_theorem&quot;&gt;Chinese remainder theorem&lt;/a&gt; as base-reencoding from prime-based number system.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are also other good reasons, like the problem's importance in cryptography or as basic example of &lt;a href=&quot;http://en.wikipedia.org/wiki/Unique_factorization_domain&quot;&gt;unique factorization domain&lt;/a&gt; or &lt;a href=&quot;http://en.wikipedia.org/wiki/Principal_ideal_domain&quot;&gt;principal ideal domain&lt;/a&gt;. Still, the intuitions and mental tools factorization provides seem enough of a reason by itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-05-29T16:39:37.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2480" PostTypeId="2" ParentId="2477" CreationDate="2014-05-29T16:57:52.373" Score="4" Body="&lt;p&gt;When I teach some basic graph theory, I explicitly mention (and include in the notes) that certain definitions and terminology vary from author to author, in some cases giving the alternatives. In fact, the field is notorious for this. Of course, there is a risk that students will use one of the alternatives instead of my preferred definition, but my experience so far is that this doesn't happen extremely often. I try to be alert to this possibility and point it out when it occurs, such as with the question: Which definition did you use?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I try to develop my own awareness of clarity issues that may arise and inform students about them, preferably early on in a course or when the relevant topic is introduced, but sometimes on a need-to-know basis. &lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2014-05-29T16:57:52.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2481" PostTypeId="2" ParentId="2458" CreationDate="2014-05-29T17:23:19.713" Score="3" Body="&lt;p&gt;Dave Renfro's advice [below] in his comment to your question is eminently sensible. Familiarize yourself with this particular course and what is expected of the student so you can help him/her best.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also add that many engineering mathematics courses, but certainly not all, have less emphasis on proofs than courses on the same topic intended for mathematics majors.  Correspondingly, there may be more emphasis on applications to/in engineering. However, to be sure, it would be wise to check.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Renfro -- I recommend looking at the web page for the course, or if there is no web page, try to find what textbook is being used. Also, faculty who have previously taught the course might have old homework assignments, tests, handouts, etc. for the course still on their faculty web pages. Sometimes these are not visible from the faculty member's web page, but you can (sometimes) still find them by googling things like the professor's name along with &quot;linear algebra&quot; and the course number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2016-03-14T11:01:39.120" LastActivityDate="2016-03-14T11:01:39.120" CommentCount="0" CommunityOwnedDate="2014-06-02T19:42:39.910" ContentLicense="CC BY-SA 3.0" />
  <row Id="2482" PostTypeId="2" ParentId="2461" CreationDate="2014-05-29T18:21:35.473" Score="3" Body="&lt;p&gt;To craft the problem in a way that you can do it much faster with a group, you need something parallelisable. To give some examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Calculating some aggregate like the average, or maximum. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make it simple you could ask them to find the longest word in the first $n$ pages of their math book. It might be an introductory task to let them know what to expect.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;Checking if a number is prime. I doubt this needs additional explanation.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sorting, quick-sort is easier, but merge-sort has more involved reduce part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For quick-sort you could assign each mapper an interval, and then they are to filter the array and sort it, the reducer would concatenate the arrays in appropriate order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For merge-sort you could give each mapper a chunk to sort and then the reducer would merge all the partial results.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finding common friends, you can read more about it &lt;a href=&quot;http://stevekrenzel.com/finding-friends-with-mapreduce&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given an array of integers, find the interval of largest sum. To parallelise it, represent any number $x$ as 4-tuple $(x,0,x,0)$ and combine the tuples with&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align} &amp;amp;(a_1,b_1,c_1,d_1) \otimes (a_2,b_2,c_2,d_2) = \\ &amp;amp;\hspace{20pt}\Big(a_1+a_2, \max(a_1+b_2,b_1), \max(c_1+a_2,c_2), \max(c_1+b_2,d_1,d_2)\Big),\end{align}&#xA;the result will be the maximum of $c$ and $d$. The above formula is an expanded multiplication of $$\left[\begin{array}{c}a&amp;amp;-\infty&amp;amp;b\\c&amp;amp;0&amp;amp;d\\-\infty&amp;amp;-\infty&amp;amp;0\end{array}\right] \quad\text{starting with}\quad\left[\begin{array}{c}x&amp;amp;-\infty&amp;amp;0\\x&amp;amp;0&amp;amp;0\\-\infty&amp;amp;-\infty&amp;amp;0\end{array}\right]$$ in $(\max,+)$ ring and then reading of the middle of multiplication with $[0,-\infty,0]^T$. You can read more about it &lt;a href=&quot;http://blog.sigfpe.com/2008/11/approach-to-algorithm-parallelisation.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-05-29T20:45:41.837" LastActivityDate="2014-05-29T20:45:41.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2483" PostTypeId="2" ParentId="2462" CreationDate="2014-05-29T19:58:36.350" Score="3" Body="&lt;p&gt;The NCTM website has many mathematical games: &lt;a href=&quot;http://illuminations.nctm.org/&quot; rel=&quot;nofollow&quot;&gt;http://illuminations.nctm.org/&lt;/a&gt;. The site lists over 20 games in the &quot;pre-K-2&quot; grade category, though some of them are a bit advanced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For offline games, if you spend a little time searching, you can find some really good ideas on Pinterest. Here are a few boards which are worth checking out:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.pinterest.com/MandyMG2/math-activities-for-prek/&quot; rel=&quot;nofollow&quot;&gt;http://www.pinterest.com/MandyMG2/math-activities-for-prek/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.pinterest.com/ppetty/math-games/&quot; rel=&quot;nofollow&quot;&gt;http://www.pinterest.com/ppetty/math-games/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.pinterest.com/LearnAsWeGo/math-games-and-activities/&quot; rel=&quot;nofollow&quot;&gt;http://www.pinterest.com/LearnAsWeGo/math-games-and-activities/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2014-05-29T19:58:36.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2484" PostTypeId="1" AcceptedAnswerId="2580" CreationDate="2014-05-29T20:40:30.663" Score="8" ViewCount="777" Body="&lt;p&gt;It's been quite a while since I was tutoring a high school student and even longer since not a gifted one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this time, something was amiss. I have asked him to show me how he does some exercise, and then another and the only thing I wanted to do was to shout:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;You are doing it &lt;em&gt;wrong&lt;/em&gt;!&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I hope I didn't let it show and tried to work it out in small steps. However, it was like he didn't wanted to learn, he just wanted to get the problem set done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One mistake he would frequently do, is to mess the signs in trigonometric reductions (e.g. $\cos(90^\circ + x)$ would be $\sin x$ instead of $-\sin x$). In fact, he memorized all the formulas, but couldn't recall them properly. I've tried to teach him how to recover formulas from graphs of $\sin$ and $\cos$, but the response was along the lines of &quot;I don't need to understand it&quot; and &quot;I would like to do it without thinking&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, there was a pair of tasks 1) prove that $X(\alpha) = Y(\alpha)$ and  2) calculate $X(30^\circ)$, where $Y$ was simple and $X$ was not. However, in the class they covered such calculations, so he evaluated the $X$ directly. Yet, pointing out the simpler way didn't changed his approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example could be an expression with non-round numbers, where I suggested to hide them behind $\alpha$ and its variations (esp. since such substitution made the necessary formulas and reductions easily visible). Nevertheless, he tirelessly rewrote them all the way through calculations and clung to their concreteness like to some kind of lifeline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware that an experienced tutor would do much better than I did, but that's not what I'm after. What shook me most strongly was this strange quality of his approach to mathematics that would produce a strong feeling of dissatisfaction (similar to the one you have when seeing something ugly). What I would like to ask about is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How can I change such a student's approach to math?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems like a Catch-22. He sees math as horrendous, but won't learn otherwise, because he doesn't want to deal with it. On the other hand, when he does, it's so wrong that no wonder he sees math this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a change I wanted to show him something engaging, e.g. how math allows us to build nice things using 3D animation software, but he would not make a connection. It was just a few cool pics useful only as long as they can be used to raise his status (e.g. his friends think them awesome).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have no illusions on whether I could teach him to enjoy math. But, is it possible to make him not hurt mathematics? Right now, I admit defeat.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-05-29T21:05:35.773" LastActivityDate="2014-06-04T19:31:35.727" Title="Changing students' approach to math" Tags="|mathematical-pedagogy|secondary-education|student-motivation|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2485" PostTypeId="2" ParentId="2369" CreationDate="2014-05-29T21:20:10.710" Score="0" Body="&lt;p&gt;It sounds like this is a grade weight question. When weighing grades, like saying a test is worth 25%, it doesn't matter how many questions you put on the test/exam. There are two ways that I know of the weigh grades, points and percents. They are basically the same method, just represented differently. With points, you have a total number of points that you allot per assignment and the total points earned is divided by the total points available is the average. The percent method is a method in which you do not necessarily know the total number of points that will be allotted by the end of the course, however, you do know how many assignments you will give and what percentage you think they should be worth. Then you take the scores of the assignments in that category, average them, and then multiply them by the percent. Then you add up all the averages*percents and get the overall grade for the course. This will prevent the need for determining how many questions are needed to make sure you have the same number as your desired percent. It sounds like you are trying to combine the two methods. If you want to do a point method with 100 points total, then you need to be careful with how many points each question are worth. You can give fractions of points for questions if you feel the need to give more questions than a 1 point system will allow, also.&lt;/p&gt;&#xA;" OwnerUserId="839" LastActivityDate="2014-05-29T21:20:10.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2486" PostTypeId="1" AcceptedAnswerId="2488" CreationDate="2014-05-29T21:31:56.033" Score="20" ViewCount="1529" Body="&lt;p&gt;I'm currently teaching a second-semester calculus course in which a significant percentage of my students (over half) either tested into the course just out of high school, took a much easier first semester at a nearby community college, or simply got through the preceding course by the skin of their teeth.  For many of them, it has been over a year since their previous math course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems as though there is almost no recollection of limits, differential calculus, or the basics of integration that were included in the first semester material.  They seem to be able to keep up with the new topics as long as they are independent of the previous coursework, but whenever I ask them to recall old material, things start to get dramatic.  For example, a recent lecture on numerical integration (e.g. Simpson's rule) contained a discussion of error bounds, which involved maximizing a function on an interval.  This quickly dissolved into a review of finding extrema while I responded to student questions, and by the time I was done, we had no more time to talk about integration formulas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like the natural thing to do would be to simply &lt;em&gt;start over&lt;/em&gt;, i.e. give a comprehensive review of the entire first semester as quickly as possible before moving on.  However, this is a summer course, so it is accelerated, and there is no time for this.  Things are already behind schedule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I facilitate a reasonable understanding of the curriculum under these conditions?&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2014-05-31T20:47:05.757" LastActivityDate="2014-05-31T20:47:05.757" Title="What to do when a majority of students have insufficient mastery of the prerequisite material?" Tags="|undergraduate-education|mathematical-pedagogy|calculus|prerequisites|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2487" PostTypeId="2" ParentId="2486" CreationDate="2014-05-29T22:10:27.927" Score="3" Body="&lt;p&gt;The easiest method would be to issue an addendum to the syllabus that highlighted specific chapters of the Calculus I material that would be helpful as prerequisites to the new material.  Since you indicated in the comments that you are using a separate textbook, this doesn't seem as feasible, since it's tough to ask students to shell out another $150 on a book that they may not use heavily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, I would not worry about the students with advanced placement at all.  Given that they've already demonstrated at least some independence in learning the AP material, they should be able to dig up the resources that they need to supplement the current text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it were me, I would start the class with a worked-through Calc I &quot;warm up&quot; problem, but quickly finish it so that the lecture doesn't get diverted into a refresher course. For example, a quick optimization problem before the numerical integration lecture would force the students back into the proper mindset and prompt some recall of the older material, I think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you did elect to go back through for a firehose-style review of the material, giving a Calc I problems &quot;quiz&quot; as an assessment first would let you gauge where you need to focus your energy.  Given the fact that most students take a summer course under various circumstances (early start, remediation, etc.), there may be some students that are just so far behind they need to reconsider their options anyway.   &lt;/p&gt;&#xA;" OwnerUserId="16" LastActivityDate="2014-05-29T22:10:27.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2488" PostTypeId="2" ParentId="2486" CreationDate="2014-05-29T23:03:28.973" Score="15" Body="&lt;p&gt;I don't think there is any way to get students caught up without expending some effort. &#xA;If you &lt;em&gt;are&lt;/em&gt; willing to expend the effort, you can create a simple &quot;you should remember&quot; pre-class activity for days that require it, and preserve your class time for new concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Steps:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Look through your upcoming lectures (even just a few in advance), and determine which ones require a Calc 1 skill.&lt;/li&gt;&#xA;&lt;li&gt;Track down the &lt;a href=&quot;https://www.khanacademy.org/math&quot;&gt;Khan Academy video&lt;/a&gt; that correspond to the Calc 1 skill for that lecture, and post it on your class website.&lt;/li&gt;&#xA;&lt;li&gt;Post 2 simple questions for students to work that they need to show you when they walk into class.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There are some pretty awesome videos out there to re-learn things -- frankly it's a good life skill for students to grasp. &lt;/p&gt;&#xA;" OwnerUserId="1207" LastActivityDate="2014-05-29T23:03:28.973" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2489" PostTypeId="2" ParentId="2470" CreationDate="2014-05-30T00:35:29.267" Score="0" Body="&lt;p&gt;Factoring into primes is a very fundamental operation in number theory. It pops up all over the place. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-30T00:35:29.267" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2490" PostTypeId="2" ParentId="2470" CreationDate="2014-05-30T01:59:18.177" Score="4" Body="&lt;p&gt;As far as &quot;real life&quot; goes, it's fairly boring, but I honestly think the best use is simply number sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not &lt;strong&gt;complete&lt;/strong&gt; prime factorization that's important, IMO it's not super important to your average undergrad that $288 = 3^2 \cdot 2^5$, but the general process of $288 = 2 \cdot 144 = 2 \cdot 12 \cdot 12$ will make it much easier to multiply 3-4 digit numbers in your head. Or the inverse, simply that &quot;oh, 288 is about 12 times bigger than 24&quot;. While that specific number may not be useful, the general method is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, depending on your subfield of mathematics that's not necessarily an overwhelmingly useful skill, but it's useful for one-the-fly calculations in &quot;real life&quot;. As much as we like to make fun of them, &quot;if Sally has 12 apples...&quot; problems do come up occasionally in planning meetings and the like, and the ability to get a quick intuitive breakdown of numbers can speed things up.&lt;/p&gt;&#xA;" OwnerUserId="521" LastActivityDate="2014-05-30T01:59:18.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2491" PostTypeId="2" ParentId="2470" CreationDate="2014-05-30T04:50:46.030" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Where is factorization used in real life?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is used when we simplify fractions.  For example, to simplify $\frac{15}{51}$ to $\frac{5}{17}$ one must know that $15=3\times 5$ and $51=3\times 17$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is used when we need to know how many objects having a common value are needed to get a given value.  For example, to find out how many $16$-US-fluid-ounce cups of a softdrink are needed to fill up a $1$-US-gallon container, you need to know that $1$ US gallon = $128$ US fluid ounces = $8\times 16$ US fluid ounces.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-05-30T04:50:46.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2492" PostTypeId="1" CreationDate="2014-05-30T07:24:36.567" Score="9" ViewCount="1859" Body="&lt;p&gt;I'm teaching a section of Calculus 2 (integration techniques, arc length, surface area, improper integrals, parametric &amp;amp; polar functions, sequences, and series ) next semester and would like to assign a problem set near the end of the course to each student relevant to his or her major.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It can be expected that a handful of students will be majoring in physics. These students are going to be mostly first years, so they will have little exposure to physics outside what they learned in high school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have the lightest of physics backgrounds so I could use some advice in developing the problem set.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What are some interesting applications of the material taught in Calculus 2 to physics that require little in the way of physics prerequisites?&lt;/li&gt;&#xA;&lt;li&gt;Are there any good resources I may want to look into for ideas?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;: I asked this question on physics.SE but it was closed and users recommended I ask it on matheducators.SE.&lt;/p&gt;&#xA;" OwnerUserId="1576" LastActivityDate="2014-06-02T01:30:59.693" Title="Applications of Calculus 2 to Physics" Tags="|undergraduate-education|calculus|problem-design|physical-sciences|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2493" PostTypeId="2" ParentId="2471" CreationDate="2014-05-30T08:07:12.527" Score="0" Body="&lt;p&gt;How about this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Express $\sin(\pi+x)$ in terms of $\sin(x)$, $\cos(x)$ or a (linear) combination of these.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Depending on your audience and on how much information you want to convey, you may choose to include or omit the word “linear”.&lt;/p&gt;&#xA;" OwnerUserId="1248" LastEditorUserId="1248" LastEditDate="2014-05-30T09:16:33.350" LastActivityDate="2014-05-30T09:16:33.350" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2494" PostTypeId="2" ParentId="2484" CreationDate="2014-05-30T14:13:19.600" Score="4" Body="&lt;p&gt;&lt;em&gt;What does he want?&lt;/em&gt;  The question says more about what he doesn't want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe he wants to pass an exam to avoid math for a few years, and wants to improve his chance of passing.  Maybe he wants to run a business, and would be interested in business algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Starting from his goals can make the relevant activities more appealing for him.  That may get him in a frame of mind to appreciate other approaches to math.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-05-30T14:13:19.600" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2495" PostTypeId="2" ParentId="2492" CreationDate="2014-05-30T15:14:25.357" Score="4" Body="&lt;p&gt;In special relativity, we have $\gamma=(1-v^2)^{-1/2}$, where $v$ is the velocity in units of the speed of light. Relativistic momentum for a particle with $m\ne0$ is $p=m\gamma v$, and kinetic energy is $K=m(\gamma-1)$ (in units where $c=1$). (a) Expand $p(v)$ in a Taylor series and show that the lowest-order nonvanishing term recovers the nonrelativistic limit. (b) Do the same for $K$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Polar coordinates can be used to calculate things like the moment of inertia of a disk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The magnetic field of a long, straight wire is of the form $B\propto 1/r$. The energy density of the field (energy per unit volume) is proportional to $B^2$. Show that the improper integral diverges logarithmically at both $r\rightarrow0$ and $r\rightarrow\infty$. (Physically, the wire can't have zero radius, and the distant field isn't realistic because we need a complete circuit.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an object close to a concave mirror, the object's distance $u$ from the mirror and the image's distance $v$ from the mirror are related by $1/f=1/u-1/v$, where $f$ is a constant (the mirror's focal length). The magnification is $M=v/u$. What happens to the magnification in the limit as $u\rightarrow0$? (This gives an indeterminate form.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If an atom of a radioactive isotope is observed not to have decayed at $t=0$, then the probability distribution for the time at which it will decay is of the form $ke^{-bt}$. Use normalization to relate $k$ to $b$.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-30T15:14:25.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2496" PostTypeId="2" ParentId="2492" CreationDate="2014-05-30T15:21:54.190" Score="1" Body="&lt;p&gt;The &quot;classic&quot; Calculus 2 application to Physics are the &lt;a href=&quot;https://math.stackexchange.com/questions/47621/deriving-the-derivative-of-the-normal-vector-in-the-frenet-formula&quot;&gt;Frenet formulas,&lt;/a&gt; which incorporate tangents, normals and binormals. &lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-05-30T22:05:56.560" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2497" PostTypeId="1" AcceptedAnswerId="2499" CreationDate="2014-05-30T09:46:23.113" Score="33" ViewCount="2222" Body="&lt;p&gt;I am trying to teach a teenage person math, but he doesn't seem to be able to grasp the concept of negative numbers and $0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again and again he finds $-4$ greater than $-3$ because he has spent several years seeing $4$ greater than $3$. Similarly he has never experienced adding $0$ to stuff and stuff to $0$ and I am unable to make him understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please tell me a few methods which I can try with him to make him understand the concepts&lt;/p&gt;&#xA;" OwnerUserId="1586" OwnerDisplayName="rijul gupta" LastEditorUserId="1589" LastEditDate="2014-05-30T18:27:09.047" LastActivityDate="2015-01-25T00:36:35.103" Title="How to teach someone that $-3&gt;-4$?" Tags="|mathematical-pedagogy|concept-motivation|negative-numbers|" AnswerCount="18" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2498" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T10:02:51.220" Score="28" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Again and again he finds $-4$ greater than $-3$.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ask him who is &lt;strong&gt;richer&lt;/strong&gt;, he who has a &lt;strong&gt;smaller debt&lt;/strong&gt; $($like $3$ rupees$)$, or he who has a &lt;strong&gt;bigger debt&lt;/strong&gt; $($like $4$ rupees$)$, assuming both persons have no money, just debts.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;He has spent several years seeing $4$ greater than $3$.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A &lt;strong&gt;debt&lt;/strong&gt; of $4$ rupees is indeed &lt;strong&gt;bigger&lt;/strong&gt; than one of $3$ rupees. But the one that &lt;strong&gt;owes less&lt;/strong&gt; is &lt;em&gt;richer&lt;/em&gt; than the one who &lt;strong&gt;owes more&lt;/strong&gt;. There's a difference between &lt;em&gt;owing&lt;/em&gt; and &lt;em&gt;owning&lt;/em&gt;. :-)&lt;/p&gt;&#xA;" OwnerUserId="4653" OwnerDisplayName="Lucian" LastActivityDate="2014-05-30T10:02:51.220" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="2499" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T10:23:50.930" Score="33" Body="&lt;p&gt;Draw a number line and label all the integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tell him that adding $x&amp;gt;0$ is moving $x$ units to the right and subtracting $x&amp;gt;0$ is moving $x$ units to the left.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tell him that adding $0$ is not moving at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tell him that adding $x&amp;lt;0$ is moving $-x$ units to the left and subtracting $x&amp;lt;0$ is moving $-x$ units to the right.&lt;/p&gt;&#xA;" OwnerUserId="1589" OwnerDisplayName="Jasper Loy" LastEditorUserId="1589" LastEditDate="2014-05-30T18:18:42.140" LastActivityDate="2014-05-30T18:18:42.140" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2500" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T10:42:37.627" Score="4" Body="&lt;p&gt;I would begin with the definition of the &lt;strong&gt;opposite of a number&lt;/strong&gt; as each of the numbers in a pair:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;such as $1$ and $-1$ or $-\frac{1}{2}$ and $\frac{1}{2}$, also called additive inverse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then introduce the &lt;strong&gt;property of opposites&lt;/strong&gt;: for every real number $a$ there is a unique real number $-a$ such that $$a+(-a)=0.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the above equation you can ask him to substitute the familiar positive numbers in for $a$. This will then provide the motivation for the natural conclusion that a negative number is what you add to a positive number to obtain zero, as the property of opposites states. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you feel ambitious, you could use the property of opposites as a special case to motivate the definition of subtraction. &lt;/p&gt;&#xA;" OwnerUserId="216" OwnerDisplayName="skullpatrol" LastEditorUserId="216" LastEditDate="2015-01-25T00:36:35.103" LastActivityDate="2015-01-25T00:36:35.103" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2501" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T11:29:58.483" Score="3" Body="&lt;p&gt;Negative numbers are strange beasts, since they're completely artificial constructs. Perhaps we have forgotten how difficult they are to comprehend, since we learned them so long ago?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Negatives are useful for &lt;em&gt;modelling&lt;/em&gt; phenomena like opposite directions, debts, etc. but those phenomena themselves contain no negatives; speed is an absolute quantity regardless of direction, debts are sets of rules for moving around absolute quantities, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They're also useful for scales, like temperature, but we can only ever measure absolute quantities (eg. the length of mercury in a thermometer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an excellent analogy to be made with Complex numbers, since they are also artificial constructions which cannot be measured; when they're used to model real phenomena, the solutions always turn out to be Real.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not recommending you teach negative numbers by analogy with complex numbers, but that you see which methods of teaching complex numbers work, and adapt them to help you teach negatives. Complex numbers are usually taught to teenagers, eg. advanced high school or introductory University courses, so I imagine that the difficulties experienced may be similar (eg. at a young age, we may just accept the rules at face value, rather than questioning them like a teenager would).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the concept of zero, one way might be to represent numbers as sets of things &lt;em&gt;in containers&lt;/em&gt;. &quot;Here is 1 apple, here are 2 apples, together they make 3 apples&quot; works very well, but there's nothing to point at for &quot;0 apples&quot;. By putting the apples in boxes, you can use an empty box to represent &quot;0 apples&quot;.&lt;/p&gt;&#xA;" OwnerUserId="15287" OwnerDisplayName="Warbo" LastActivityDate="2014-05-30T11:29:58.483" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2502" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T12:54:01.737" Score="-1" Body="&lt;p&gt;There is an intuitive way to do this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The solution is the common ground between addition, subtraction, multiplication and division. Mathematics has to be thought in a certain order, all &quot;lower&quot; concepts have to be accepted by the learner first before you advance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-4 + 3 does not make sense unless you introduce commutativity.&#xA;-4 + 3 is equivalent to 3 - 4 which is easy to do equals -1.&#xA;But wait how does he know that we can go below zero?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(-2) * 6 does not make sense unless you introduce distributivity.&#xA;(0 - 2) * 6 is equivalent to (0 * 6) - (2 * 6) = 0 - 12 = -12.&#xA;But wait, how does he know 0 multiplied by anything is 0&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-2 + 4 + 1 - 3 does not really make any sense as negative numbers are weird, how can you have -4 apples. Teach him to put a zero before the -2 so it becomes 0 - 2 + 4 + 1 - 3.&#xA;But wait, how can you subtract from zero?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So to cut a long story short, here is the way to teach basic math. It must be precisely taught in this order:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Teach addition of positive numbers big and small (1 to 99999999). Make sure he can add two and triple digits. Behind his back choose only addition of numbers that when added at any level do not go over 10 AND don't have any zeros in them. For example 123 + 451 = 574, 1+3 or 5+4&#xA;*Do not choose any exercises that go over 10 for example 5+6 to 399 + 3. If he asks, tell him you'll show him later something extraordinary :D or give him candy.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;wait a while and have him do about 500 additions this way and write them down.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Teach the number 0 as when added to another number the result is the same number. 0+1, 0+7 and so on and all the other numbers at step 1. NOW add 0 to all the other numbers you left out at step 1. 499+0 567+0 and so on.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;wait a while and have him do about 100 additions this way and write them down.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;NOW add numbers like 489+344 and teach him to &quot;carry&quot; the 1 when 8+4 is 12 and he writes a 2 and adds the one to the higher order and so on. He knows how to add zero to things so you can put 560 or 870 to the mix.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;wait a while and have him do about 500 additions this way (it might seem like a lot but he can do 50 per day and it will take him 10 days of practice)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's it you are done with addition!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice how I used the fact he doesn't know math to my advantage. The &quot;trick&quot; to teaching anything is &quot;not to fail&quot;, only teach cases that enforce the rules he already knows, avoid the exceptions. Only teach special cases when all the necessary building blocks are there for HIM to draw the conclusion on his own. WAIT for him to do that, sometimes he catches on really fast, sometimes it will take a little bump in the right direction.&lt;/p&gt;&#xA;" OwnerDisplayName="unmircea" LastActivityDate="2014-05-30T12:54:01.737" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2503" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T14:28:22.403" Score="2" Body="&lt;p&gt;Maybe you should introduce him to the concept of distance away from zero. The negative is only a direction (on the previously suggested scale/timeline). Maybe try a few analogies to help him make the learning relevant, or even suggest that he already understands the concepts in practise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It sounds like he is struggling with conceptual maths, but the actuality is that he will have used these principles at some stage in his life. Your task here is to help him recognise this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mentored a young chap when I was at university that had the same challenges, and we eventually got around most of his problems by making the discipline of maths a less threatening beast, with the added benefit of him accepting that maths was not a bunch of irrelevant theories or contrived questions, but something he was already using and would benefit from understanding better. (For the record, he went on to exceed everybody's expectations in his exams by 3 grades)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps, and best of luck.&lt;/p&gt;&#xA;" OwnerUserId="1596" OwnerDisplayName="Paul Merryfellow" LastActivityDate="2014-05-30T14:28:22.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2504" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T14:31:30.163" Score="4" Body="&lt;p&gt;Try to introduce this with his own age. Let's see he's 8 years old, ask him what was his age 4 years ago, then he will say 4 and after you ask him what was his age 3 years ago and he will say 5 so then it will maybe be easier for him to understand that -4 is smaller that -3 because 3 years ago he was older than 4 years ago. It's a simple solution that can work, but all child are differents. Hope this help. Concepts are way more easier to learn than facts.&lt;/p&gt;&#xA;" OwnerDisplayName="pcharb" LastActivityDate="2014-05-30T14:31:30.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2505" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T14:31:49.397" Score="25" Body="&lt;p&gt;Apologies, this should be a comment on the answer provided by @Jasper Loy but I don't have enough rep on this site. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just wanted to add that in my experience, struggling students have an easier time grasping negative numbers when the number line is oriented vertically rather than horizontally. I think we as humans naturally make the 'up=greater, down=less' association; while the 'right=greater, left=less' association may seem natural to those of us with a stronger math background, it usually isn't natural for students like yours. &lt;/p&gt;&#xA;" OwnerUserId="1585" OwnerDisplayName="JVL" LastActivityDate="2014-05-30T14:31:49.397" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2506" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T16:43:19.893" Score="6" Body="&lt;p&gt;Perhaps teaching about other negative numbers would be easier, like &lt;span class=&quot;math-container&quot;&gt;$-1 &amp;gt; -2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-100 &amp;gt; -200$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Some possible approaches:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose you have &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; apples. If I took from you &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, in which case would you have more?&lt;/p&gt;&#xA;&lt;p&gt;Now suppose you have no apples and again I take &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; from you (that means, should you get a hold of some apple, you have to give it to me, until I have &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; respectively). In which case would you have more?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some elevators have levels numbered via negative numbers (often basement or car parks). You can ask then, which level is on top of the other &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;. If there is no such lift around, you could talk about stair steps or ladder rungs that leads to basement.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sometimes the temperature in the Arctic drops below &lt;span class=&quot;math-container&quot;&gt;$0^\circ$&lt;/span&gt;. When it would be warmer, at &lt;span class=&quot;math-container&quot;&gt;$-30^\circ$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-40^\circ$&lt;/span&gt;? You can also turn it around, i.e. when it is &lt;em&gt;colder&lt;/em&gt;, at &lt;span class=&quot;math-container&quot;&gt;$30^\circ$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$40^\circ$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If the kid is interested in geography, you could talk about depressions (i.e. places with elevations below sea level). Which one is higher, &lt;a href=&quot;http://en.wikipedia.org/wiki/Lake_Asal_(Djibouti)&quot; rel=&quot;noreferrer&quot;&gt;Lake Asal in Djibouti&lt;/a&gt; or the &lt;a href=&quot;http://en.wikipedia.org/wiki/Dead_Sea&quot; rel=&quot;noreferrer&quot;&gt;Dead Sea&lt;/a&gt;? You could also talk about some other forms like the &lt;a href=&quot;http://en.wikipedia.org/wiki/Mariana_Trench&quot; rel=&quot;noreferrer&quot;&gt;Mariana Trench&lt;/a&gt;, such curiosities may spark some interest.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I hope this helps &lt;span class=&quot;math-container&quot;&gt;$\ddot\smile$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-05-30T18:14:26.393" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2507" PostTypeId="1" AcceptedAnswerId="2529" CreationDate="2014-05-30T18:48:09.143" Score="5" ViewCount="366" Body="&lt;p&gt;As a math educator, do you distinguish between the concepts of &quot;learning&quot; and &quot;development?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If so, what is the distinction for you?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would you say the consequences of this distinction are to your practice as a math educator?&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-05-30T19:57:08.693" LastActivityDate="2014-06-01T16:35:33.680" Title="Learning vs. Development in Practice as a Math Educator" Tags="|psychology|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2508" PostTypeId="1" AcceptedAnswerId="2509" CreationDate="2014-05-30T19:01:24.790" Score="4" ViewCount="108" Body="&lt;p&gt;I recently started tutoring a $9^{\text{th}}$ grader who had managed to rotely memorize some processes (thats how he got this far).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, I have someone that is trying to learn basic concepts (that we use a positional number system for example and what that means) at the same time as learning things that rely on that (scientific notation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it standard practice in these cases to jump right to expressing numbers as $\sum\limits^{n}_{i}{a_i10^i}$ (in words of course) and how that can be generalized, or should I in essence do the 3rd? grade part first, and then do scientific notation normally?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; This is just one example. There are many others, and I am asking for a general rule for what to do in these cases.&lt;/p&gt;&#xA;" OwnerUserId="1591" LastEditorUserId="143" LastEditDate="2014-05-30T21:21:26.653" LastActivityDate="2014-05-30T21:21:26.653" Title="Using formal underpinnings to teach basic concepts" Tags="|secondary-education|tutoring|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2509" PostTypeId="2" ParentId="2508" CreationDate="2014-05-30T20:36:32.300" Score="3" Body="&lt;p&gt;I'd go for basic concepts, explained (somewhat) formally. Specially if they do know how to &quot;do&quot; problems, it might be enlightening to see &quot;why&quot; it works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Be careful not to overdo it. What I find easy, even obvious, is often a deep mistery to students. Ditto, what I find interesting and fascinating in its possibilities is often seen as utterly pointless. Putting yourself into their shoes is &lt;em&gt;hard&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sites like &lt;a href=&quot;http://www.cut-the-knot.org&quot; rel=&quot;nofollow&quot;&gt;Cut the knot&lt;/a&gt; have lots of discussion on serious mathematical problems in form of puzzles, many accompanied by interactive applets that allow you to experiment. This I can recommend highly, I'm sure others can give references to other (not so dry/serious) sites. Don't worry if they aren't learning why e.g. long division works, there is plenty of time for that later (if at all required).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-30T20:36:32.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2510" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T20:37:13.587" Score="1" Body="&lt;p&gt;Show the student a globe.  Perth, Australia is -32 degrees latitude (aka 32 degrees south where north is defined as &quot;positive&quot;); the capital city of Canberra is -35 degrees.  Perth has a higher (more northern) latitude, while the &lt;em&gt;magnitude&lt;/em&gt; of Canberra's latitude is larger. &lt;/p&gt;&#xA;" OwnerUserId="650" LastActivityDate="2014-05-30T20:37:13.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2511" PostTypeId="2" ParentId="2497" CreationDate="2014-05-30T20:38:12.030" Score="1" Body="&lt;p&gt;If I add 10 to -3, I get 7.  If I add 10 to -4 I get 6.  6 is less than 7, so -4 must have been less than -3.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-05-30T20:38:12.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2512" PostTypeId="2" ParentId="2466" CreationDate="2014-05-30T20:45:35.833" Score="2" Body="&lt;p&gt;By far, the &quot;freshman calculus&quot; Superstar in economics is &lt;strong&gt;differentiation&lt;/strong&gt;, because it is pervasive in all branches of the discipline, and at all levels, undergraduate post-graduate, PhD, professional academic.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And this is because, the core &lt;em&gt;behavioral&lt;/em&gt; assumption in economics at the micro-level is that economic agents (persons, companies organizations, whatever), are all the time &lt;em&gt;consciously&lt;/em&gt; and &lt;em&gt;purposefully&lt;/em&gt; trying to &lt;em&gt;optimize&lt;/em&gt; some measure, metric, whatever: utility, profit, value (maximization), cost, risk, loss (minimization)... in the world of economics, there is no such thing as &quot;sit back and enjoy life&quot;, you are always on the look out (our agents, that is, inside our models), to ameliorate your position, in whatever way this &quot;position&quot; is represented -this is how you &lt;em&gt;do&lt;/em&gt; enjoy life.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we take derivatives as we breath, since, apart for special cases where the objective function may be constructed to be a bit weird, our functions are always differentiable -and even we have a model in discrete time, we find ways to do things similar to differentiation w.r.t time...  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I guess, what everybody knows about Economics is that the discipline has an obsession with the concept of &quot;equilibrium&quot;, right? Well, watch us &lt;em&gt;differentiate the equilibiurm condition&lt;/em&gt; of a model to obtain the relation between two variables of interest at the equilibrium point -and we are so used to it, some don't even know this is called the implicit function theorem.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2014-05-30T20:45:35.833" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2513" PostTypeId="2" ParentId="2477" CreationDate="2014-05-30T20:46:18.543" Score="0" Body="&lt;p&gt;Define your terms, if needed point out there are alternative definitions/notations (and perhaps point some of them out). When using them, be careful to go back to first definitions whenever it makes sense.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-30T20:46:18.543" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2514" PostTypeId="2" ParentId="2486" CreationDate="2014-05-30T22:02:38.970" Score="1" Body="&lt;p&gt;I have two suggestions... &#xA;1) for the students that are having the most difficult time, you might ask them to get a copy of &quot;Hurricane Calculus&quot;.  This book is immensely helpful in simplifying calculus concepts quickly and succinctly.  It is also less intimidating to students because it reads less like a text book and more like a how-to guide.  You can get it for a buck, used on Amazon.&lt;br&gt;&#xA;2) In my experience, when you have a group of mixed ability students and a short time line, it is best to get the students working together.  I do break-out groups and have them report out... usually in something like a game format or non-graded group quiz.  This works even better when you can get the groups heterogeneous so that each group has students who get it and who don't get it.  The biggest lesson I've learned about education in general is that the more you get students talking to each other about the content and sharing with the class, the quicker they learn, since students generally communicate better with their peers than with their teachers.&lt;/p&gt;&#xA;" OwnerUserId="1595" LastActivityDate="2014-05-30T22:02:38.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2515" PostTypeId="2" ParentId="2458" CreationDate="2014-05-30T22:20:01.153" Score="4" Body="&lt;p&gt;I learned linear algebra (decades ago) as an &quot;adjunct&quot; to &quot;engineering mathematics.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The focus of most engineering students is &quot;systems of equations.&quot; As long as you remember that, you're fine. So engineers are most focused on those parts of linear algebra that provide solutions to systems of equations, including lower echelon row reduction and eigenvalue problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just avoid the more abstract issues, such as &quot;kernels&quot; of spaces, where they'll get lost.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-05-30T22:20:01.153" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2516" PostTypeId="1" CreationDate="2014-05-30T22:30:09.687" Score="9" ViewCount="540" Body="&lt;p&gt;Sometimes it happens that a person doesn't want to accept your argument, because he claims not all the inferences are valid. There's a famous example of Lewis Carroll, namely &lt;a href=&quot;http://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles&quot;&gt;What the Tortoise Said to Achilles&lt;/a&gt;, where nonacceptance of a basic inference rule leads to infinite regression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes, it's the other way around, where a student doesn't want to acknowledge your proof. It might be more subtle, like with the non-constructive proof that an irrational raised to the power of another irrational can be rational, or it could be less subtle, e.g. when pupil doesn't concur that $$(\forall \varepsilon &amp;gt; 0\;. \ |x| &amp;lt; \varepsilon) \implies x = 0$$ for $x \in \mathbb{R}$ (I remember one guy that agreed on $x \neq 0 \implies |x| \not&amp;lt;\frac12|x| &amp;gt; 0$, but wouldn't accept the statement above).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give a different kind of example, consider the following. For propositions $P(x)$ and $Q(x)$ we could infer that $\exists x. P(x) \land Q(x)$ from $\exists x. P(x)$ and $\exists x.\ Q(x)$. This is not true in general, but there are cases where it works, e.g. if $P$ and $Q$ are independent (this could be formalized as, for any considered model $\mathcal{M}$ and element $x \in \mathcal{M}$ we have $\mathcal{M},P(x) \not\models Q(x)$ and $\mathcal{M},P(x) \not\models \neg Q(x)$, etc.). While a scrutinizing approach would be to work out all the details, one might want the students to apply a more intuitive, higher-level perspective; in my opinion, such exercises are helpful to develop a deeper understanding. Yet, some don't even want to start participating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This happens also in non-math fields, like psychology or biology where papers on non-widely-accepted theories (like &lt;a href=&quot;http://en.wikipedia.org/wiki/Group_selection&quot;&gt;group selection&lt;/a&gt;) are ridiculed, for example, because the audience doesn't handle well the concept of &quot;assumption&quot;, or because the author insists on something bizarre without considering it might be false.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; How can we resolve conflicts that stem from disagreements on the fundamentals?&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="160" LastEditDate="2014-05-31T00:59:53.827" LastActivityDate="2014-06-04T17:23:01.420" Title="What to do if there is a disagreement on fundamentals, e.g. axioms or inference rules?" Tags="|undergraduate-education|graduate-education|mathematical-pedagogy|general-pedagogy|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2517" PostTypeId="2" ParentId="2516" CreationDate="2014-05-30T22:56:35.013" Score="5" Body="&lt;p&gt;A small partial-answer: relativize it, as you can easily anticipate. Be up-front and explicit about logical principles and such. Someone might still (reasonably!) object, but if you conscientiously acknowledge your assumptions, the issue changes from &quot;is this right?&quot; to the relative one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did once try to give a seminar for advanced undergrads about ideas of modern mathematics, and the philosophy students gave me endless trouble, ... :) ... in the sense that I felt (at the time, much younger...) that we couldn't get off the ground for all the quibbling. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, I think it is a bad thing to squelch intellectually honest queries or skepticism. Heaven forbid a calculus student gets access to Bishop Berkeley's essay? :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Making the context (more) explicit&quot; is often the answer to such questions.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-05-30T22:56:35.013" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2518" PostTypeId="2" ParentId="2516" CreationDate="2014-05-30T22:58:04.267" Score="3" Body="&lt;p&gt;If there is a disagreement, then have an argument!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Argument&quot; in the sense of &quot;reasoned discussion&quot; - it is not an accident that the word is used commonly in mathematics. Even if you are sure that you are objectively &quot;right&quot;, you may learn something  by taking the student's perspective more seriously. They are clearly wrong in &lt;em&gt;your&lt;/em&gt; sense, but perhaps they are correct in &lt;em&gt;their&lt;/em&gt; sense, and are just having trouble communicating their ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, mathematics is about proof. If you cannot prove something to someone's satisfaction, then it probably means that you should re-evaluate your understanding of that concept. These issues are often more subtle than you would expect.&lt;/p&gt;&#xA;" OwnerUserId="1598" LastActivityDate="2014-05-30T22:58:04.267" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2519" PostTypeId="2" ParentId="2492" CreationDate="2014-05-31T00:05:35.150" Score="2" Body="&lt;p&gt;I would say that Taylor series come up fairly often in physics (both elementary and not):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;E.g., the force due to gravity is usually given by $F=mg$ but this is merely an approximation.  See exercise 4 &lt;a href=&quot;http://math.kennesaw.edu/~plaval/math2202/series_apps.pdf&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; for example.&lt;/li&gt;&#xA;&lt;li&gt;E.g., the motion of simple pendulum can be described via a Taylor series.  See &lt;a href=&quot;http://www.physics.udel.edu/~jim/PHYS211_14S/Class%20Notes/Class_04.pdf&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;E.g., the first order approximation to the binomial $(1+x)^n$ has applications to electrostatics.  See &lt;a href=&quot;http://www.lajpe.org/sep09/5_LAJPE_272_Mungan.pdf&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The $n$-body problem has a formulation which can be solved via Taylor series.&lt;/li&gt;&#xA;&lt;li&gt;Position given by $$x(t)=x_0+v_0(t-t_0)+a/2(t-t_0)^2$$ is really just a power series approximation for position.&lt;/li&gt;&#xA;&lt;li&gt;They also have various applications to Statistical Mechanics.  See &lt;a href=&quot;http://arxiv.org/pdf/1112.4231v2.pdf&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-05-31T00:05:35.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2520" PostTypeId="2" ParentId="2516" CreationDate="2014-05-31T00:45:12.257" Score="1" Body="&lt;p&gt;At least when trying to explain mathematical ideas to my friends, one way that I try to give them an intuitive feel for why something is true is to ask them to provide counterexamples. If the &quot;counterexamples&quot; are invalid I try to explain why. This does not constitute a valid proof, but at the very least it change their &quot;gut feeling&quot; for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: One person refused to believe that $0.\overline{9}=1$. I asked him the following: &quot;In between any two distinct real numbers, there is a third distinct real number, eg. via taking the average. Find me a number between $0.\overline{9}$ and $1$.&quot; After some reflection, he acquiesced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is rather similar, I believe, to your example of $\forall \varepsilon, |x|&amp;lt;\varepsilon\implies x=0$. &quot;Find me $x$ such that this statement is false.&quot;&lt;/p&gt;&#xA;" OwnerUserId="1573" LastActivityDate="2014-05-31T00:45:12.257" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2521" PostTypeId="2" ParentId="2477" CreationDate="2014-05-31T01:07:16.287" Score="3" Body="&lt;p&gt;My website lists a good many math words with two widely used meanings.  It is at [dead link].&#xA;Search the index for &quot;two meanings&quot; and you will find a bunch of them.  --Charles Wells&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit many years later: It may still be possible to find some things at an archive of this site; try perhaps:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20180831121859/http://www.abstractmath.org/MM/MMGlossary.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://web.archive.org/web/20180831121859/http://www.abstractmath.org/MM/MMGlossary.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20180831121859/http://www.abstractmath.org/MM/MMIndex.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://web.archive.org/web/20180831121859/http://www.abstractmath.org/MM/MMIndex.htm&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1599" LastEditorUserId="11" LastEditDate="2019-12-02T06:57:13.993" LastActivityDate="2019-12-02T06:57:13.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2522" PostTypeId="2" ParentId="852" CreationDate="2014-05-31T02:54:39.507" Score="4" Body="&lt;p&gt;One way of handling this situation that I haven't seen above is an examination of their knowledge of the subject. The first day of class, review your syllabus and typical basic class procedures, and give an optional examination that could be a mock final for the class, if they get (whatever grade) they get an A in the class. Students are free to leave at this point if they don't wish to take the exam. This does a couple of things, the people that leave this examination are people who you will not have this problem with. (because if they really thought they knew it, they would have stayed). It also lets you pick out which students (if any) in the class might actually know the material, and saves them waisted time in a class they understand. The kids who take this exam and don't pass, will get their examination back the second class and take a private blow to their ego that doesn't offend them, or make them think they made a stupid mistake. &lt;/p&gt;&#xA;" OwnerUserId="1600" LastActivityDate="2014-05-31T02:54:39.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2523" PostTypeId="2" ParentId="2497" CreationDate="2014-05-31T03:07:57.790" Score="4" Body="&lt;p&gt;Just with respect to the concept of negative numbers being greater than or equal to each other - maybe try talking about it in terms of things on earth, in the air, and under the ground.  Things &lt;strong&gt;underground are negative&lt;/strong&gt;, things &lt;strong&gt;on the surface of earth are zero&lt;/strong&gt;, and things &lt;strong&gt;in the air are positive&lt;/strong&gt;.  The higher up you are the greater you are.  In this system, 1 foot under ground is higher up (more positive, greater, etc.) than 2 feet under ground.  Three feet under ground is even lower, and so on.&lt;/p&gt;&#xA;" OwnerUserId="1601" LastActivityDate="2014-05-31T03:07:57.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2524" PostTypeId="2" ParentId="2516" CreationDate="2014-05-31T09:20:59.880" Score="4" Body="&lt;p&gt;You could ask the student to come up with his own rules and axioms from which he can prove whatever theorems he thinks are valid. Usually when they really get down to doing that systematically and formally, they will understand the rules and axioms better anyway, and then they can precisely pinpoint what is it that they don't agree with. Otherwise, often they won't be able to give justification for their disagreement, and arguments would just go in (periodic) circles.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2014-05-31T09:20:59.880" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2525" PostTypeId="2" ParentId="2492" CreationDate="2014-05-31T11:55:58.597" Score="0" Body="&lt;p&gt;I'd probably stay away (or use very simple examples) if I didn't really understand the physics behind the application. As several of the other answers show, many applications involve approximations or the simple models give formulas that fail due to being non-realistic in some way. You are asking how to explain use of a tool, if you don't know the application, you can't really use the tool on it.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-05-31T11:55:58.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2526" PostTypeId="2" ParentId="2497" CreationDate="2014-05-31T17:00:39.200" Score="10" Body="&lt;p&gt;I want to post this answer just to give show any future prospectors what &lt;strong&gt;did&lt;/strong&gt; work for me in this particular case! and also because it's tooo long for a comment (pardon my over emphasis)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mixed the concepts of number line first horizontal, but then as he found it difficult I showed him a vertical number line as suggested by @JVL, along with pictures of water level where I marked a portion in middle as 0L.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also demonstrated the fundamentals of richness in debt as suggested by @Lucian, since the child understood the concept of debt before hand, as he has had such experiences (probably because of being from an economically backward background, or maybe just like that) It was quite easy for him to grasp that he was richer when he was in a debt of Rs 3 rather than when he was in a debt of Rs 4 (I live in India, that's why the rupees)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have chosen the answer by @JasperLoy because number line was my first approach, and I had given him quite a few problems to work on and he had already grasped quite a bit just by thinking which numbers lied at which side. I had also taught him addition on number line before posting this question and he seemed to be doing better now, even though there were numerous conceptual mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also love to declare that I believe I have finally taken him across this initial strata of basic number line and can soon take him on a fast track (he's quite intelligent, just lacks formal education).&lt;/p&gt;&#xA;" OwnerUserId="1586" LastActivityDate="2014-05-31T17:00:39.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2527" PostTypeId="2" ParentId="852" CreationDate="2014-05-31T17:13:07.410" Score="6" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;they feel the need to share it with the rest of the class interrupting my lecture.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is the problem (1) that they're interrupting, (2) that their interruptions are excessive or not of interest to the rest of the class, (3) that their comments are incorrect, or (4) they the way they do it is rude? I wouldn't consider #1 a problem at all; there is little excuse in 2014 for making students sit through a noninteractive presentation that they could just as easily have watched on youtube.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some suggestions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Avoid reactions that might give the impression to other students that their participation would also be unwelcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't shut down these students in a way that they will interpret as an attempt to embarrass them in front of their peers, because teenagers are extremely sensitive to this kind of thing, and even if you're right, it can be damaging to the student-teacher relationship. However, if the problem is rudeness or excessiveness, address that. &quot;I think the issue you've raised is an interesting one, and I can tell that you're not yet satisfied with my answer. However, we've spent about 10 minutes on this topic. Class time is limited, and we need to be sensitive to the fact these other folks in the room may not be as interested in this question as we are. Let's continue this discussion after class or in my office hours.&quot;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-05-31T17:13:07.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2528" PostTypeId="1" AcceptedAnswerId="2535" CreationDate="2014-05-31T23:36:38.097" Score="13" ViewCount="938" Body="&lt;p&gt;There's an interesting read: &lt;a href=&quot;http://grantwiggins.wordpress.com/2014/04/23/conceptual-understanding-in-mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conceptual Understanding in Mathematics&lt;/a&gt; by Grant Wiggins. In that text the author proposes &quot;a test for conceptual understanding&quot; which should be given &quot;to 10th, 11th, and 12th graders who have passed all traditional math courses through algebra and geometry.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would have thought that as a person with higher math education, I should have enough of &quot;conceptual understanding&quot; of high school math$^\dagger$. Yet, the following question, number 12 in the author's list, stumped me.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;12)   “In geometry, we begin with undefined terms.” Here’s what’s odd, though: every Geometry textbook always draw points, lines, and planes in exactly the same familiar and obvious way – as if we CAN define them, at least visually. So: define “undefined term” and explain why it doesn’t mean that points and lines have to be drawn the way we draw them; nor does it mean, on the other hand, that math chaos will ensue if there are no definitions or familiar images for the basic elements.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What is this about? What is undefined and what we &quot;CAN&quot; define visually? In what way the answerer is expected to define undefined terms? How to interpret the last sentence about chaos in mathematics? I don't even...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would greatly appreciate your help in deciphering author's intentions.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;$\dagger$ &lt;sub&gt;Note that my skills aren't rusty, I'm an active researcher and as time of writing in &lt;a href=&quot;https://math.stackexchange.com/tags/geometry/topusers&quot;&gt;top 20&lt;/a&gt; geometry answerers at math.stackexchange; it's not much, but ought to be enough to answer high school geometry question, right?&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-09-04T16:29:34.673" Title="Wiggins' question #12" Tags="|secondary-education|concept-motivation|geometry|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2529" PostTypeId="2" ParentId="2507" CreationDate="2014-05-31T23:37:31.270" Score="6" Body="&lt;p&gt;You have asked three questions, and, in mostly answering the second, my response is already quite long. For now, let me not broach the third question about &lt;em&gt;consequences of the distinction in practice&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Yes&lt;/strong&gt;, I distinguish between the two concepts. As for how I distinguish between them:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me note first that a &lt;em&gt;rigorous&lt;/em&gt; answer would probably come from within some realm of developmental psychology. The main sources I would start with would be Piaget (who might have said he wasn't working in developmental psychology but rather in &quot;genetic epistemology&quot;; but let us ignore such an objection) and Vygotsky. For the former, one might start by looking at his stages of (cognitive) &lt;strong&gt;development&lt;/strong&gt;, and how &lt;strong&gt;learning&lt;/strong&gt; fits in. As an example introduction, see &lt;a href=&quot;http://epltt.coe.uga.edu/index.php?title=Piaget%27s_Stages&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. For the latter, one might start by looking at his theory around the zone of proximal &lt;strong&gt;development&lt;/strong&gt;, and, again, how &lt;strong&gt;learning&lt;/strong&gt; fits in. As an example introduction, see &lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Zone_of_proximal_development&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;. (This is not directed at the OP, for whom I expect this background information is already known.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, a rigorous answer might involve asking you first to define what you mean by 'learning' and 'development'; of course, if there were clear-cut definitions, then I expect (1) you would be able to determine if/how they are distinguished without posting here, and (2) probably the main responses would be support for or objections to your definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My sense, then, is that your question effectively boils down to putting forth one's own definitions of these broad terms, whereby the distinguishing features of the two will be elucidated. To this end, I can only answer in a relatively &lt;em&gt;non-rigorous&lt;/em&gt; way, for they are very big words indeed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me direct your attention to a different psychologist, &lt;a href=&quot;http://en.wikipedia.org/wiki/Howard_Gruber&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Howard Gruber&lt;/strong&gt;&lt;/a&gt;, a contemporary of Piaget's whose work was focused on creativity as inspired by his own lengthy case study on Charles Darwin. (You can find a bit more about Gruber in my earlier &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/906/262&quot;&gt;MESE response&lt;/a&gt;&lt;/strong&gt; about creativity.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roughly speaking, Gruber conceives of the individual as being an evolving system composed of loosely coupled subsystems of knowledge, affect, and purpose; furthermore, he considers how these different subsystems interact with one another &lt;em&gt;over time&lt;/em&gt;, and how they relate to the projects in which one is engaged (which Gruber refers to as one's &lt;em&gt;network of enterprises&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So:&lt;/strong&gt; I will define (hence distinguish between) &lt;strong&gt;learning&lt;/strong&gt; and &lt;strong&gt;development&lt;/strong&gt; as follows. First, I extend the three subsystems (without filling in the newly introduced theoretical lacunae) by allowing each to include some sort of meta- component as well. This means &lt;em&gt;knowledge&lt;/em&gt; becomes &lt;em&gt;knowledge&lt;/em&gt; and a tower of types of &lt;em&gt;meta-knowledge&lt;/em&gt;; speaking messily again, something like &lt;em&gt;cognition&lt;/em&gt; and &lt;em&gt;metacognition&lt;/em&gt;. (For more on metacognition as broached by Schoenfeld, see my &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/2399/262&quot;&gt;MESE response&lt;/a&gt;&lt;/strong&gt; about time spent on a question.) In the same vein, &lt;em&gt;affect&lt;/em&gt; extends from something like &lt;em&gt;feelings&lt;/em&gt; to include also &lt;em&gt;feelings about feelings&lt;/em&gt; and so forth; and, with more subtleties hence less discussion here, &lt;em&gt;purpose&lt;/em&gt; and a tower of types of &lt;em&gt;meta-purpose&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I consider &lt;strong&gt;learning&lt;/strong&gt; to be what happens within a single one of these three subsystems (probably some would limit this to knowledge and its meta- component, which I'll denote &lt;em&gt;knowledge+&lt;/em&gt; as the combined unity; but let us go one step further and allow learning to occur in any one of the three). Heuristically, one can think about them as &lt;strong&gt;learning about the world&lt;/strong&gt; (knowledge+), &lt;strong&gt;learning about the self&lt;/strong&gt; (affect+), and &lt;strong&gt;learning about how the self fits into the world&lt;/strong&gt; (purpose+).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Meanwhile, I consider &lt;strong&gt;development&lt;/strong&gt; to be what happens as these three different subsystems interact with one another over time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there established precedent for conceiving of &lt;strong&gt;learning&lt;/strong&gt; as an &lt;em&gt;intra&lt;/em&gt;-system process and &lt;strong&gt;development&lt;/strong&gt; as an &lt;em&gt;inter&lt;/em&gt;-system process? I'm not quite sure; but this is my non-rigorous answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, as somewhat of a side-note, let me end by quoting from the final lines of Gruber's &lt;em&gt;Darwin On Man&lt;/em&gt; (I highly recommend this final chapter, entitled &quot;Creative Thought: The Work of Purposeful Beings,&quot; and ensure you that the excerpt does not ruin anything). Gruber writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In his explorations of the world, the individual finds out what needs doing. In his attempts to do some of it, he finds out what he can do and what he cannot. He also comes to see what he need not do. From the intersection of these possibilities there emerges a new imperative, his sense of what he must do. How &quot;It needs&quot; and &quot;I can&quot; give birth to &quot;I must&quot; remains enigmatic (p. 257).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-06-01T16:35:33.680" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2530" PostTypeId="2" ParentId="2528" CreationDate="2014-06-01T00:05:54.053" Score="9" Body="&lt;p&gt;I think you might find some value in Raymond Wilder's classic textbook &lt;em&gt;Introduction to the Foundation of Mathematics&lt;/em&gt;. A free copy is archived &lt;strong&gt;&lt;a href=&quot;https://archive.org/details/IntroductionToTheFoundationsOfMathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;. Here is a short excerpt from p. 10 of Wilder's text:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/9Dxhg.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, you could broach this using the traditional &lt;em&gt;point&lt;/em&gt; and &lt;em&gt;line&lt;/em&gt; (or &lt;em&gt;blob&lt;/em&gt; and &lt;em&gt;streak&lt;/em&gt;) model. But here is an alternative way that uses something like &lt;a href=&quot;http://en.wikipedia.org/wiki/Set_%28game%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;SET&lt;/strong&gt;&lt;/a&gt; cards (limited to a single shape and filling):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/hcZ2d.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here the model is not the standard one we think of for plane geometry, at least in its presentation above, yet still satisfies the mathematical axioms without anything chaotic ensuing. Note further that I have &quot;defined the undefined terms,&quot; namely, I have given definitions for both &quot;point&quot; (which consists of some number - one, two, or three - of colored dots) and &quot;line&quot; (this is inspired by the game of SET).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the author just wants you to think about how the axioms of plane geometry are generally interpreted by, e.g., secondary school textbooks in the same way, despite the &lt;em&gt;undefined terms&lt;/em&gt; having nothing that forces a point to be thought of as a single dot, and so forth. If you are to construct a different model that satisfies these axioms, even if it looks different, it still ought not to lead to any chaos in the sense of contradictions. This is because the axioms are consistent insofar as they are satisfiable.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;As a side-note:&lt;/strong&gt; By satisfying the third Axiom above, the model presented here gives an indication as to how one answers the &quot;fun question&quot; I posed on ncr's &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/2463/262&quot;&gt;MESE answer&lt;/a&gt;&lt;/strong&gt; of SET as a game for learning mathematical concepts. In particular, each pair of SET cards uniquely determines the third card that forms a &lt;em&gt;set&lt;/em&gt;. The full game has four characteristics (shape, color, filling, number) and three choices for each, hence the full deck has $3^4 = 81$ total cards. The total number of sets, then, can be found as follows: You have $81$ choices for the first card, $80$ choices for the second, and $1$ for the third; but this over-counts by $3!$. So the &lt;strong&gt;total&lt;/strong&gt; is $81\cdot80/6 = 1,080$ possible sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Another note:&lt;/strong&gt; See &lt;a href=&quot;http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/set.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;this paper&lt;/strong&gt;&lt;/a&gt; (Davis and Maclagan) for other SET-math connections.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-09-03T20:53:10.550" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2531" PostTypeId="1" CreationDate="2014-06-01T01:23:39.733" Score="15" ViewCount="1272" Body="&lt;p&gt;Has anyone attempted to introduce, or has data on such endeavor, Lebesgue integration before Riemann? I've seen many discussions about how the Riemann integral is obsolete and that it is presented because it appeals to intuition*, contrary to Lebesgue's theory, which is considerably harder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside, it does not have to be Lebesgue's integration theory exclusively. I've seen people advocating for the gauge integral as well, but I don't know much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Citation needed. I don't think I know &lt;em&gt;anyone&lt;/em&gt; that understood the geometric appeal of the Riemann integral when meeting it for the first time.&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="143" LastEditDate="2014-06-01T12:04:46.970" LastActivityDate="2014-06-01T12:04:46.970" Title="Introducing the Lebesgue integral before Riemann's" Tags="|undergraduate-education|calculus|curriculum|mathematical-analysis|integration|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2532" PostTypeId="1" CreationDate="2014-06-01T01:34:06.353" Score="11" ViewCount="2135" Body="&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Why was advanced calculus removed as the first proof-based course in favor of real analysis in most curriculums? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I regularly see in advanced calculus books either that: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;its purpose is, along with teaching certain knowledge, to introduce undergraduates to their first proof-based course after a usual computation-based calculus sequence (mostly older books); or &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;it has been superseded by traditional real analysis courses. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What motivated this? I feel much geometric thinking was lost in the process, as most advanced calculus usually started with vector calculus.&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="262" LastEditDate="2014-06-01T01:46:25.820" LastActivityDate="2015-03-27T00:24:25.273" Title="Advanced Calculus vs. Analysis for a first proof-based course" Tags="|undergraduate-education|mathematical-pedagogy|curriculum|mathematical-analysis|vector-calculus|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2533" PostTypeId="2" ParentId="2531" CreationDate="2014-06-01T02:10:42.997" Score="1" Body="&lt;p&gt;I believe you answer youself: Riemann's integral (while intuitively quite appealing) isn't easy to get to grip with. Lebesgue's definition is much harder to aprehend, and solves problems with Riemann's integral that have to be explained it that terms. So you would be introducing a more complex definition, while taking out the justification to define it. For &quot;everyday engineering use&quot; Riemann's integral is more than enough, most of the times integrals of interest are expressible through antiderivatives.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-01T02:10:42.997" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2534" PostTypeId="2" ParentId="2531" CreationDate="2014-06-01T03:35:50.833" Score="9" Body="&lt;p&gt;In the US, you'd be hard-pressed to find any student seeing the Lebesgue integral before having ever seen the Riemann integral.  Every calculus book I've seen defines the integral as the Riemann integral.  That said, when I was a graduate student, I came across this &lt;a href=&quot;http://www.cambridge.org/US/academic/subjects/mathematics/abstract-analysis/lebesgue-integration-and-measure&quot;&gt;book&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Lebesgue Integration and Measure&lt;/em&gt;&#xA;  by Alan J. Weir&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;that, instead of connecting the Lebesgue integral to the Riemann integral, connects it directly to the problem of finding areas.  This might not be what you want, but it's the only textbook with the Lebesgue integral that I know of that is both written for undergraduates and doesn't have a chapter or section explicitly on the Riemann integral.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quick internet search tells me the math department at the &lt;a href=&quot;http://maths.york.ac.uk/www/Home&quot;&gt;University of York&lt;/a&gt; uses this text and in its standard sequence doesn't appear to make reference to the Riemann integral.  You might consider contacting someone there.  &lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-01T03:35:50.833" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2535" PostTypeId="2" ParentId="2528" CreationDate="2014-06-01T04:43:58.023" Score="5" Body="&lt;p&gt;Here's &lt;strong&gt;Hilbert&lt;/strong&gt;, &lt;a href=&quot;http://www.gutenberg.org/ebooks/17384&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Foundations of Geometry&lt;/em&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let us consider three distinct systems of things.&#xA;  The things composing the first system, we will&#xA;  call &lt;em&gt;points&lt;/em&gt;...; those of the second, we will call &lt;em&gt;straight&#xA;  lines&lt;/em&gt;...;&#xA;  and those of the third system, we will call &lt;em&gt;planes&lt;/em&gt;....We think of these points, straight lines, and planes&#xA;  as having certain mutual relations....The complete and exact description of these relations&#xA;  follows as a consequence of the &lt;em&gt;axioms of geometry&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The things called points, lines, and planes are just things, which are assumed to exist.  No further definition is given.  However from the assumptions made about the existence of relations and the properties these have as well as a few other assumptions, Hilbert constructs geometry.  &lt;strong&gt;Euclid&lt;/strong&gt; on the other hand gives definitions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;A point is that which has not even one part.&lt;/li&gt;&#xA;  &lt;li&gt;A line is breadthless length.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Euclid defines point in terms of part and line in terms of dimensions.  But part, breadthless, and length themselves are not defined.  If one were to define those terms, they would be in terms of other undefined terms (unless one were to make a circular definition).  The modern way is to start with things -- that is to say, a set (or class) of objects.  The elements of the set are considered undefined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Measurement as a basis for geometry is introduced [G.D.Birkhoff, A set of postulates for plane geometry (based on scale and protractor), &lt;em&gt;Ann. of Math.&lt;/em&gt; &lt;strong&gt;33&lt;/strong&gt;&#xA;(1932), 329–345].  Birkhoff further influenced the teaching of geometry by producing a textbook &lt;em&gt;Basic Geometry&lt;/em&gt; with R. Beatly.  This approach was later adopted for high school in the US (see, for instance, Moise and Downs, &lt;em&gt;Geometry&lt;/em&gt;, 1964).  &lt;strong&gt;Birkhoff&lt;/strong&gt; uses, without definition, the term undefined:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Undefined elements and relations.&lt;/strong&gt;&#xA;  The undefined elements are (a) &lt;em&gt;points&lt;/em&gt;,...and (b) certain classes of points called (straight) &lt;em&gt;lines&lt;/em&gt;.&#xA;  The undefined relations are (c) &lt;em&gt;distance&lt;/em&gt; between any two points,...&#xA;   (d) &lt;em&gt;angle formed by three ordered points&lt;/em&gt;....&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To respond to a question by the OP in a comment to another question, the points and lines are assumed to be given (or supplied) but there is no definition of them.  What they are arises out of the axioms that are assumed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example given in the comment raises the question whether the $x$ is undefined in $f(x) = x^2$, which seems a different issue if I've understood the point correctly.  Ignoring the modern definition of $f$ as a set, one could say that $f(x) = x^2$ is short for &quot;let $f$ be the function defined by the property that for all $x \in {\bf R}$, $f(x) = x^2$. The variable $x$ represents an element from the set ${\bf R}$.  As a matter of logic, a variable is indefinite, and I think that is different that being defined or undefined.  A closer analogy would be to say that in the definition of $f$, the set ${\bf R}$ is undefined.  That is an approach some analysis texts: a set ${\bf R}$ is assumed to exist and further assumed to have certain properties.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;P.S.&lt;/em&gt;&#xA;Some quibbles about Wiggins question:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What does Wiggins mean by define, as in &quot;define 'undefined term'&quot;?  Whatever is meant by define, undefined means it was not defined.  This is not really a serious objection, but the wording, at least in the excerpt, might be confusing.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What does Wiggins mean by draw?  This seems more serious (see also below).  I know very well what it means in art class.  Mathematically, I am uncertain what can be meant by &quot;the way we draw them.&quot;  There are several problems with the way I draw them.  I've not drawn a mathematical line ever.  Two points do not determine a straight line, because there is always some error; if I draw a line twice through the same two points, I get two different lines.  Likewise, extending a straight errs, when I do it.  But I &lt;em&gt;imagine&lt;/em&gt; two points determine a line and I can &lt;em&gt;imagine&lt;/em&gt; my figures are perfect.  I dismiss my error and do not think my lack of perfection disproves the axioms.  Maybe Wiggins is after that sort of explanation, and that mathematically lines need not be drawn that way (in which, say, lines have a thickness or are not &quot;straight,&quot; whatever that means).  I was taught in H.S. that we cannot draw a geometric point or line, which is only a slightly different explanation.  It also sounds like Wiggins might be opening the door to models of non-Euclidean geometry.  The might be that we do not have to imagine that straight lines follow our intuition of straightness, but that we should deduce properties only from the axioms.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;P.P.S.&lt;/em&gt;&#xA;For Proclus (&lt;em&gt;Comm. Eucl.&lt;/em&gt;, 400s CE), &quot;leading geometry out of Calypso’s arms, so to speak, to more perfect intellectual insight and emancipating it from the pictures projected in imagination&quot; would be the &quot;perfect culmination&quot; of studying Euclid.  This shows that the concern that images and what we imagine about them might lead us astray in our reasoning goes back to ancient times.  Proclus' take is somewhat opposed to the perspective in Wiggins' question.  Wiggins seems to think his readers will find comfort in familiar images and definitions for every thing.  His readers might fear not having them.  The term &quot;chaos&quot; might be a pun as it tends to imply formlessness, which would be an ungeometrical state.  I think the question is asking for an explanation of why mathematics is not invalid just because of undefined terms and &quot;straight&quot; do not have to be what we imagine as straight.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;P.P.P.S.&lt;/em&gt;&#xA;As a side note, in both Euclid and Hilbert, but not Birkhoff, a line is not a set of points, which unnerves most college students.  Some find it liberating, though.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2014-06-02T10:49:28.687" LastActivityDate="2014-06-02T10:49:28.687" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2536" PostTypeId="1" AcceptedAnswerId="2537" CreationDate="2014-06-01T08:46:48.360" Score="17" ViewCount="2051" Body="&lt;p&gt;I am a new calculus teacher in a high school for gifted students. I am the youngest teacher, I am not in my home country, and this country particularly values age and experience, so I have little room to discuss syllabi and homework assignments. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The level of maths is quite high (senior students deal with vector calculus and Stokes' theorem) and the class requires a large amount of weekly homework (to give an idea: about 15 exercises for each section in Stewart's Calculus, about 3 sections a week). Deadlines for these homework assignments are very strict (new assignment every week, none accepted after the deadline).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that practice is important, especially in calculus, but this leads to unintended outcomes:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Grading takes a lot of time, so many teachers don't grade seriously (little feedback; just checking that students did the homework) and instead distribute a solutions printout from the textbook to the students.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students don't spend a lot of time thinking about or engaged with the problems; they just rephrase the solutions. Good students will try to understand the solutions, but other will not really care.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even if I don't give them the solutions printout, then they will find the answers online.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Many students don't enjoy calculus and few of them would take an optional class related to calculus (say analysis or differential geometry). Freshmen are really afraid of their future calculus class.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I have seen this with a couple of students: they believe that they will be successful if they copy the solutions for all homework problems (including non-assigned ones) from the textbook (at least 60 per section!). This has led to very poor results. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What can I do? I like teaching in the school, but the general philosophy of education in this country is: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If a student has unsatisfying grades, give him more homework and extra classes. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So far, I have just decided to relax the deadlines, and not to care too much about late homework, but the outcome is still not satisfying. I am also not comfortable with having different rules for homework than other classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there an efficient way to manage a huge load of homework?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What are the alternatives to the &quot;practice, practice, practice&quot; method to gain good skills in computations?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The ideal solution would be to completely change the homework list in accordance with the philosophy of &quot;less but better,&quot; but my colleagues will be hurt if I change the school's habits. How can I discuss this problem with them without hurting their sense of hierarchy?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thank you in advance, and sorry if my questions are imprecise. To sum up, it could be changed to &quot;What are the most efficient methods to teach Calculus, with respect to the ratio (computation skills)/(homework load)?&quot;&lt;/p&gt;&#xA;" OwnerUserId="1538" LastEditorDisplayName="user507" LastEditDate="2014-06-01T17:21:09.283" LastActivityDate="2017-12-30T16:49:36.163" Title="Overload of Calculus homework assignments" Tags="|calculus|student-motivation|homework|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2537" PostTypeId="2" ParentId="2536" CreationDate="2014-06-01T14:31:30.763" Score="11" Body="&lt;p&gt;One way of reducing the load of grading is to use an online answer checker. There are several that are free and open-source, including &lt;a href=&quot;http://webwork.maa.org/&quot; rel=&quot;noreferrer&quot;&gt;Webwork&lt;/a&gt;, &lt;a href=&quot;http://www.lon-capa.org/&quot; rel=&quot;noreferrer&quot;&gt;Lon-Capa&lt;/a&gt;, and &lt;a href=&quot;http://www.lightandmatter.com/spotter/spotter.html&quot; rel=&quot;noreferrer&quot;&gt;Spotter&lt;/a&gt; (which I wrote). Webwork seems very popular among math teachers, and there is a huge body of problems set up in Webwork.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only work you really have to grade by hand is work that involves writing, critical thinking, and higher-level reasoning. If you assign a lot of that and feel that it's too much to grade, then one option is to grade a random sample of the problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The overarching issue is that at some point you're going to have to make a decision as to whether your perceived lack of academic freedom is a real lack of academic freedom, and how to deal with that. It's possible that not all of your colleagues are as rigid or conservative as you imagine.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-06-01T14:31:30.763" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2538" PostTypeId="2" ParentId="2306" CreationDate="2014-06-01T16:19:30.733" Score="1" Body="&lt;p&gt;As a response to Kevin's answer, and for those who might better understand what the OP is required to teach as part of a high school trig class. OP did not explicitly it, but often, the instruction given is to manipulate the left side of the equation until it's clearly equal to the right side, therefore no multiplying &lt;em&gt;both&lt;/em&gt; sides, cross multiplying, etc. Some teachers ask students to use a two column proof style, justifying each manipulation along the way. This is how I walked students through the OP's example. The first transition from (1) to (2) was the one that took the most discussion even though it seems the most obvious (to me) thing to do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) $\frac{\cos \left(\theta \right)}{1+\sin \left(\theta \right)}+\tan \left(\theta \right)=\sec \left(\theta \right)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) $\frac{\cos (\theta )}{1+\sin (\theta )}+\frac{\sin (\theta )}{\cos (\theta )}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) $\frac{\cos (\theta )}{1+\sin (\theta )}\left(\frac{\cos \theta }{\cos \theta }\right)+\frac{\sin (\theta )}{\cos (\theta )}\left(\frac{1+\sin (\theta )}{1+\sin (\theta )}\right)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) $\frac{\left(\cos ^2\theta +\sin ^2\theta +\sin \theta \right)}{\left(1+\sin (\theta )\right)\cos \theta }$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(5)  $\frac{\left(1+\sin (\theta )\right)}{\left(1+\sin (\theta )\right)\cos \theta }$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(6) $\frac{1}{\cos \theta }$&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-06-01T16:19:30.733" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2539" PostTypeId="1" AcceptedAnswerId="2541" CreationDate="2014-06-01T17:01:19.917" Score="5" ViewCount="657" Body="&lt;p&gt;Today I explained to my students how the value of $\sin(90°)$ is 1 by first drawing a right angled triangle and taking $90°$ as a reference angle, and told them that now $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Then I used pythagorean relation to show that $\cos(90°)$ is 0, i.e $\sin^2(\theta)=1-\cos^2(\theta)$. And finally I used quotient relation to show how $\tan(90°)$ is infinity. &#xA;The students didn't seem much convinced by my methods as they were asking why we couldn't find the value of $\cos(90°)$ and $\tan(90°)$ the same way we did to find the value of $\sin(90°)$. &#xA;Is there any other ideas to show the value of $\sin(90°)$, $\cos(90°)$ and $\tan(90°)$ understandble to them?&lt;/p&gt;&#xA;" OwnerUserId="1172" LastEditorUserId="123" LastEditDate="2014-06-03T00:41:51.743" LastActivityDate="2014-06-04T02:37:06.467" Title="The values of trigonometric ratios" Tags="|secondary-education|trigonometry|mathematical-pedagogy|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2540" PostTypeId="2" ParentId="2528" CreationDate="2014-06-01T17:03:02.657" Score="0" Body="&lt;p&gt;This question is simply testing if the students&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;sees a difference between mathematical objects and their relations (drawing &lt;em&gt;undefined&lt;/em&gt; objects)&lt;/li&gt;&#xA;&lt;li&gt;can work with the relations regardless of the mathematical objects (question of &lt;em&gt;undefined&lt;/em&gt;)&lt;/li&gt;&#xA;&lt;li&gt;realizes, that Euclidean Geometry is not the only theory satisfying geometric axioms (points and lines don't have to be drawn that way)&lt;/li&gt;&#xA;&lt;li&gt;knows, that working only with relations will only yield relational results. Concrete results will only be achieved if working with mathematical objects or feeding these to the achieved relational results. (math &lt;em&gt;chaos&lt;/em&gt; will ensue without concrete objects)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-06-01T17:03:02.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2541" PostTypeId="2" ParentId="2539" CreationDate="2014-06-01T17:12:21.230" Score="8" Body="&lt;p&gt;Yes:  Use the definitions of these functions on the unit circle.  All of these facts then become geometrically apparent.  These concepts do not even really apply to right triangles, since you would need a triangle with 2 right angles, which is not possible.  &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-06-01T17:12:21.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2542" PostTypeId="2" ParentId="2536" CreationDate="2014-06-01T19:22:22.417" Score="8" Body="&lt;p&gt;In addition to having an online homework system if possible, you might also consider asking the students present a few problems in class. This would make the students at the very least care more about the homework and the subsequent class discussion would help you identify some of the difficulties with the assignment. Often this helps a great deal with the computational practice many calc students seem to need. Doing algebra in front of your classmates will certainly inspire to get every step right and will ensure that you are able to explain the validity of each step. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Online homework combined with presentations seems ideal. Lots of practice, instantaneous feedback on all problems, detailed feedback on several problems. &lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-01T19:22:22.417" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2544" PostTypeId="2" ParentId="2532" CreationDate="2014-06-01T22:40:33.083" Score="12" Body="&lt;p&gt;This is somewhat of a hypothesis rather than a definitive answer, but one reason why vector calculus may no longer be the first proof based course at many colleges is because vector calculus involves teaching a lot of new material*. Real analysis, is mostly (at least in the first quarter/semester), material they have already seen before in calculus - minus all of the abstraction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In math there are often two axes for the content being taught. (axis - 1) the abstractness and (axis - 2) the familiarity. The goal of introducing people to proofs is to venture down the &quot;abstractness&quot; axis that is rarely explored in high school and lower level college classes. To really focus on this abstraction it is best to do so with content that is familiar (i.e. 1-dimensional calculus). This way the students can really focus on the proof writing and the abstraction because they already have some intuitive (calculation based) understanding of the concepts from calculus. However, if you were to introduce rigorous proof writing in vector calculus you would have to teach a lot of new material along the way. This distracts from teaching the basics of proof writing; students can easily get overwhelmed if they have to do both at once. In addition when a student doesn't understand something, there is the added confusion of &quot;do I not understand how to do the basic vector calculus calculations?&quot; or &quot;do I not understand the 'intuitive idea'?&quot; or &quot;do I not understand the proof&quot; or all of the above. Often the struggling students can't even answer this question correctly, which makes it nearly impossible for them to overcome their difficulties without some major intervention (because they don't know what to work on). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I stole this idea from professor Steve Strogatz (the author of the somewhat recent &lt;a href=&quot;http://opinionator.blogs.nytimes.com/category/steven-strogatz/&quot;&gt;math column&lt;/a&gt; in the NY times), from a &lt;a href=&quot;http://www.cornell.edu/video/steven-strogatz-doing-math-in-public&quot;&gt;talk&lt;/a&gt; he gave about what makes a &quot;pop-math article&quot; popular. He pretty much said, take something really familiar to the audience and dive into it a bit more abstractly, or talk about something unfamiliar but in the language of a concrete example that the audience can easily follow. If he wrote something that was both unfamiliar and abstract, he lost his audience. I think this general comment holds for students transitioning to higher math. Eventually, once they are comfortable with abstract thinking, you can then introduce to them new material abstractly, but not until they master proof writing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My guess is that the real answer is going to vary a lot from school to school and have to do with how they sequence their other courses, but I feel like the above guess is a pretty good one. &lt;strong&gt;Note that there are also obvious pitfalls to this method&lt;/strong&gt;, especially for advanced students, who really could benefit from going abstract and unfamiliar at once. However, I'd argue that such students are most likely already familiar with the abstract, through their own mathematical investigations. This is why often big universities have two mathematics major tracks, one geared towards prepping for graduate school and one for future high school math teachers and others who are interested in math but aren't ready for the really abstract stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Note I am taking liberty here and changing the question to &quot;why isn't intro to proof writing emphasized in vector calculus as frequently as it used to be?&quot; based on several people's comments on how &quot;Advanced Calculus&quot; is ill-defined. Proof based vector calculus seems to be the type of class Fantini is referring to as Advanced Calculus.&lt;/p&gt;&#xA;" OwnerUserId="1387" LastEditorUserId="1387" LastEditDate="2014-06-04T18:16:20.007" LastActivityDate="2014-06-04T18:16:20.007" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="2545" PostTypeId="2" ParentId="1817" CreationDate="2014-06-01T23:25:19.950" Score="6" Body="&lt;p&gt;The Association for Women in Mathematics has a great annual essay contest. Here is a &lt;a href=&quot;https://sites.google.com/site/awmmath/programs/essay-contest/Past-Winning-Essays&quot; rel=&quot;noreferrer&quot;&gt;link&lt;/a&gt;.  There are some great essays and the subjects run the gamut from industry to academia to the arts. &lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-01T23:25:19.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2548" PostTypeId="2" ParentId="2492" CreationDate="2014-06-02T01:30:59.693" Score="4" Body="&lt;p&gt;What I try to emphasize is that &lt;em&gt;any&lt;/em&gt; formula they're familiar with requires integral calculus once some of the quantities vary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basic example: (distance)=(rate)*(time). How far do you go if your rate is a function of time?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Medium example: (force)=(mass)*(acceleration). What is the end velocity of a rocket that burns mass (i.e. fuel) to exert constant force?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Advanced example: (Work)=(Force)*(distance). What if you have a cube of butter sliding on a hot pan? The force of friction is proportional to the mass, but the mass is decreasing because it's melting. The more you push it, the faster it melts (since the butter can spread away easier). So how much work is done in pushing a pad of butter around until it melts in your pancake griddle?&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-06-02T01:30:59.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2549" PostTypeId="2" ParentId="2536" CreationDate="2014-06-02T03:20:00.863" Score="2" Body="&lt;p&gt;If you are worried about &lt;em&gt;student&lt;/em&gt; overload (I would), try to select few problems that show different aspects for homework. Try to make sure they learn, not just practice. Complement with a selection of lots of practice problems, to be done voluntarily (and not graded). It does help if solutions are available, or even better if they are checked on-line.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-02T03:20:00.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2550" PostTypeId="2" ParentId="18" CreationDate="2014-06-02T14:05:00.910" Score="8" Body="&lt;p&gt;In many of the classes I've taught, I've allowed students to have a sheet of notes (e.g., a course on differential equations where some of the recipes they're asked to apply can be easily mixed up) but with the following two features:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I talk a lot about how, like others mention above, if they're going to rely on what they've written down during the exam, they probably won't have time to finish the exam.  I sell it more as a way for them to organize their own understanding of the material.&lt;/li&gt;&#xA;&lt;li&gt;I ask them to turn it in to me a few days before the exam and I return it to them in class, on the day of the exam.  This helps them not only to start the studying process earlier, but also helps them to not rely on it in the last few days before the exam.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-02T14:05:00.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2551" PostTypeId="1" AcceptedAnswerId="2562" CreationDate="2014-06-02T14:13:24.037" Score="57" ViewCount="17002" Body="&lt;p&gt;The kid I am teaching math (subtraction for large numbers right now) just said this is all too easily done by a calculator, why don't we use it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, I did tell him that you can only learn more stuff by learning this, but this seems like an extremely wrong answer. I am lucky that he is interested in learning, otherwise for a kid who does not have a keen interest, this may have been a devastating answer, because the kid may think learning this only brings more learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what should be the correct way to teach someone that it's better not to use the calculator?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question was just for small children, but I believe the concept can be carried forward to higher standards to the limit when computational mathematics is a necessity. &lt;/p&gt;&#xA;" OwnerUserId="1586" LastEditorUserId="851" LastEditDate="2014-06-07T10:50:11.787" LastActivityDate="2022-08-08T18:16:58.320" Title="Student: Why not use a calculator?" Tags="|technology-in-education|primary-education|calculators|" AnswerCount="15" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2552" PostTypeId="1" CreationDate="2014-06-02T14:38:13.690" Score="10" ViewCount="180" Body="&lt;p&gt;Consider the following two problems:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(A) Square 2 has sides that are longer than those of square 1 by a factor of 1.1. Compare the areas of the squares.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(B) Square 2 has sides that are longer than those of square 1 by 10%. Compare the areas of the squares.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find that many college students majoring in the sciences get A right, but for B they compute $0.10^2=0.01$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is going on here? Does K-12 education in the US not ever teach students to convert a fractional difference to a ratio? (In the lower grades, it seems like the word &quot;ratio&quot; is for some reason coupled to notation like &quot;2:3,&quot; and some students never realize that this connects to fractions.) Does the symbol % just trigger a sequence of operations in the student's brain, distracting from the larger context? Obviously the people who get $0.01$ aren't thinking about whether their answer makes sense. A contributing issue may be that they get little experience with problems in which the &lt;em&gt;answer&lt;/em&gt; is a ratio.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2022-07-17T10:52:04.017" Title="Failure to distinguish ratios from fractional or percent differences" Tags="|proportional-reasoning|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2553" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T14:42:10.790" Score="13" Body="&lt;p&gt;All of math is a logical progression. One should master a type of calculation before turning it over to a calculator or computer. &#xA;Can I multiply 4386x934754 by hand? Of course I can. And it was important to learn to do this in 4th (?) grade, but soon after, no need for long multiication or division. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every day, I'm intrigued at the disparity between the students who rely on the calculator for the simplest of problems vs those who take pride in doing it in their heads. I can't quite claim cause and effect, but in my opinion, it's rare to find a heavy reliance on the calculator and a high mastery of the subject going hand in hand. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's missing from the conversation so far is exactly when it becomes appropriate to use the calculator. What scares me is when a student's answer is grossly wrong and he doesn't have the skill to see that for example, a rock wont take 1000 seconds to fall from the tallest building on earth. Or that when you get a result of .84 for sin(1), you really meant to hit &lt;em&gt;2nd sin&lt;/em&gt; asking for inv sine to give you 90 degrees. And the calculator should be on degrees, not radians. A certain level of mastery should be required on a process before letting the calculator take over. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above translates to &quot;tell the student it's important to understand how to do math at each level before trusting a calculator's answer.&quot; &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-06-04T12:08:31.027" LastActivityDate="2014-06-04T12:08:31.027" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2554" PostTypeId="2" ParentId="798" CreationDate="2014-06-02T15:00:51.100" Score="4" Body="&lt;p&gt;I have some experience using discovery-based learning at the college level, with gen ed students. I liked it, and I think it succeeded better than other modes of instruction in building deep understanding and critical thinking skills. But there were several significant disadvantages:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Some students didn't buy in to it, and some students had trouble with the intellectual challenges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) The rate at which we could cover material was much lower than with traditional instruction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) At some point you need the students to consolidate what they've figured out. The method I used was to have them maintain a file of notes online (in a system similar to google docs), which I corrected, commented on, and graded once or twice a week. This worked, but it was a huge amount of work for me. The role of a textbook becomes awkward in a course like this. Students deserve some kind of organized source of information that isn't just their own notes.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-06-02T15:00:51.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2555" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T15:02:35.080" Score="31" Body="&lt;p&gt;I think the issue fundamentally is about a student's comfort with symbols and notation.  A student who relies on a calculator thinks of it as a magical oracle that returns an answer to the question being asked.  So, when that same student is asked to solve even the simplest of algebra problems, since the calculator can't answer that problem, the student is lost.  And since they're not comfortable with symbols, they don't know how to start.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, when someone learns subtraction by hand they should simultaneously learn that to check the answer they do the addition by hand from the bottom up.  That way, the idea of addition and subtraction being opposites is cemented from the beginning.  That way when a student is asked to solve $x+3=5$ they know the should subtract to get rid of the three.  And by &lt;em&gt;know&lt;/em&gt; I mean at some deeper level than &quot;my teacher told me so&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-02T15:02:35.080" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2556" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T15:47:08.010" Score="9" Body="&lt;p&gt;I don't think there is one &quot;right&quot; answer to this.  But one possible answer (assuming that we agree you are right!) is that you want to be able to know whether you made a mistake, or at least to recognize when things are suspicious.  There's no auto-correct for subtraction on a calculator, or computer.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I usually use this argument when it comes to differentiation and integration, but the same principle applies.  Naturally, this reason won't obtain in every situation, and of course you don't want to do 100-digit subtraction by hand - the calculator will be more accurate than you are, assuming you type it in right (also a necessary condition for doing it correctly by hand).  Yet, the better you understand what the tool is doing, the better you will use the tool.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2014-06-02T15:47:08.010" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2557" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T16:00:27.080" Score="6" Body="&lt;p&gt;When doing a problem like 8x6, how many students do you witness adding eight sixes together?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's been my experience that most students will instead memorize a series of reference points and skip the repetitive addition.  The question then is, what is gained by spending months memorizing the results of an equation (for instance, times tables) as apposed to spending a few minutes learning how to work with a calculator?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I empathize with the argument against calculators.  It's much harder to prove somebody knows what multiplication means when they can blindly punch a few numbers and get the same numeric results, but maybe that just means we have to update our tests to match current technology.&lt;/p&gt;&#xA;" OwnerUserId="1610" LastActivityDate="2014-06-02T16:00:27.080" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2558" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T16:18:49.873" Score="5" Body="&lt;p&gt;When you solve problems without a calculator, you are training your mind to solve that class of problems.  It warps the way you think about numbers, and forces you to abstract problems to solve them more easily and quickly, until one day they become trivial to solve in your head.  Rinse and repeat for every new concept you learn in math.  A calculator robs you of this experience.  Learning about math isn't about solving a singular trivial problem, it's about being able to solve the next problem faster.&lt;/p&gt;&#xA;" OwnerUserId="1611" LastActivityDate="2014-06-02T16:18:49.873" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2559" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T16:52:48.170" Score="4" Body="&lt;p&gt;Given that the time alloted to math in school is finite, I'd be glad if the children learned more &lt;em&gt;meaning&lt;/em&gt; and &lt;em&gt;how&lt;/em&gt; and &lt;em&gt;why&lt;/em&gt; to sum/multiply/integrate than mindless application of rules by rote. It is certainly much harder to teach/test...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-02T16:52:48.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2560" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T16:53:41.827" Score="-1" Body="&lt;p&gt;Teaching math isn't all about teaching numbers, it's about teaching a step-by-step approach to solving larger problems.  8x6 = 48 is one thing (is this math or just retrieving memorized information?), but think about bigger problems later on with &quot;solve for x&quot; problems in algebra.  PEMDAS is a good example: step 1, step 2, step 3, [...], you're done.  Problem solving is what you're really teaching.  We apply this same line of thinking to other subjects, like mechanics.  If you open the hood of a car, you see lots of complex machinery.  However, if you can follow directions, you can fix all sorts of things on your own.  Math teaches us problem solving skills early on, unlike most other subjects.&lt;/p&gt;&#xA;" OwnerUserId="1612" LastActivityDate="2014-06-02T16:53:41.827" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2562" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T18:15:47.853" Score="60" Body="&lt;p&gt;To the student who wants to know why he or she should learn addition/multiplication/other mathy thing when they can use a calculator or computer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;It is a matter of independence.&lt;/strong&gt;  You will not always have access to your preferred tool.  Being able to compare prices in a store without shamefully pulling out a calculator will boost your confidence that you can make do in an increasingly numerical world.  The older you are, the more you will appreciate this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;You will be able to ballpark-verify outcomes.&lt;/strong&gt;  If you want 49 times 51, and you mistype and your calculator says 100 instead of something closer to 2500, you will know this is wrong.  A less numerically literate person is resigned to accept the calculator's word as law, which fill help a fat lot when it says ERR.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;You can prevent being cheated.&lt;/strong&gt; &lt;sup&gt;(*)&lt;/sup&gt;  People will try to take advantage of you in many ways, not the least being sleight of hand with numbers.  Being skilled enough to quickly do these calculations in your head will help you verify or counter numbers fronted by others, and can prevent nasty surprises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;(*)&lt;/sup&gt; &lt;sub&gt;This is close in meaning to independence, but I find it helpful to stress both self-sufficiency in everyday situations and being able to recognize fraudulence.&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;You are not asked to disavow your calculator.&lt;/strong&gt;  It is not evil or corrupting.  Do fall back on it when a calculation takes too much time or you just want a second opinion, as we do too.  But you will be a more rounded, self-sufficient individual when you are able to do head calculations or even paper calculations.&lt;/p&gt;&#xA;" OwnerUserId="1193" LastActivityDate="2014-06-02T18:15:47.853" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2564" PostTypeId="2" ParentId="2552" CreationDate="2014-06-02T19:56:16.480" Score="7" Body="&lt;p&gt;Some of this comes down to the recipes the students overlearn and teachers (myself included) try to make too efficient. For example, when I tutored kids who arein high school, they were taught &quot;something multiplied by a factor x&quot; as multiplying the something by x, so they'd almost never mess up a problem like the first one.  On the other hand, when, in the second problem they see &quot;are longer than those by 10%&quot; they use their recipe to translate the percent to .1 and then think to square it because they can't think of anything else to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something that is absent from many of these recipes is to draw some kind of visual representation of the problem. If they draw a square, extend the sides, and then try to find the area, I'd be surprised if the second kind of mistake would ever happen. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I don't have firm research to back up these two observations, I have had many students I've tutored talk about these kinds of problems in terms of recipes and I have also helped them overcome their struggles by having them draw pictures.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-02T19:56:16.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2565" PostTypeId="2" ParentId="2459" CreationDate="2014-06-02T20:09:55.287" Score="1" Body="&lt;p&gt;I'd go with a graphing calculator.  I can't imagine a scenario where a tablet would be more useful in math class than a graphing calculator.  Besides, I think my TI-84 Plus made it through a full year of precalculus class, which involved a lot of calculator use.  I'd like to see a tablet make it through a full year in a similar setting.&lt;/p&gt;&#xA;" OwnerUserId="1613" LastActivityDate="2014-06-02T20:09:55.287" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2566" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T20:52:37.413" Score="5" Body="&lt;p&gt;I think you have to separate the issue of whether the student understands numbers at all from the use of calculators. I once had the rather thankless job of trying to teach continuing studies physics to someone that was convinced that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;0.5 + 0.5 = 0.10  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their answer when I pointed out to them this wasn't right - get out their calculator !!  Needless to say, trying to educate this student didn't go so well. The person had never understood numbers at all, and still in their 30's didn't. It's not so much about can we use a calculator - but ensuring that the student doesn't get through by mindlessly pressing buttons and copying answers off the screen whilst having no clue whatsoever what they are doing. Unfortunately, over reliance on the use of calculators  can lead to this sort of total failure of mathematical education. Once the student has masters numbers decimals and fractions, then in my opinion if they want to use calculators - let them. &lt;/p&gt;&#xA;" OwnerUserId="1615" LastActivityDate="2014-06-02T20:52:37.413" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2567" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T22:21:43.003" Score="5" Body="&lt;p&gt;I would argue the correct way to answer this is is the same answer that one might use in teaching the foundations of any discipline. Everything starts with the basics - eventually yes, it will be easier to use the calculator, but if you never understand the basic way of thinking that all math relies on - then you will never be able to do &lt;em&gt;anything&lt;/em&gt; with math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Present the student with the concept that in any job, any personal endeavor they ever undertake, they may likely use &lt;em&gt;some&lt;/em&gt; kind of math, be it algebra or basic addition/subtraction/etc. In some cases, a calculator may or may not be at hand (these days, some form of calculation device or app will likely be easily available in the developed world). Someone who has some basic grounding in mathematical concepts - a comfort with numbers - is less likely to get cheated when buying a car, a house, a microwave, a video game, and is going to be better able to cope with computers, which operate entirely on a mathematical basis.&lt;/p&gt;&#xA;" OwnerUserId="1619" LastActivityDate="2014-06-02T22:21:43.003" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2568" PostTypeId="2" ParentId="2551" CreationDate="2014-06-02T23:07:38.250" Score="17" Body="&lt;p&gt;People who ask why calculators don't resolve the (very low-level) issues well enough are quite accurate in their skepticism, I think, which makes it hard to present an &quot;absolutist&quot; defense of alternatives. For that matter, seriously (!), why is it ok to use Hindu-Arabic numerals and the weird associated algorithms, rather than &quot;honest&quot; manipulations of hash-marks or pebbles? (As a kid, I was required to do a bit of arithmetic in Roman numerals... ah, a lost art!?!) As a small kid, I &lt;em&gt;did&lt;/em&gt; do quite a lot of seat-of-the-pants arithmetic by visualizing dots or marks, e.g., multiplication of small integers, while resisting memorizing multiplication tables, and even resisting, for a time, the algorithms. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, schools don't explain &lt;em&gt;why&lt;/em&gt; these algorithms work, and, thus, could be claimed to exactly &lt;em&gt;fail&lt;/em&gt; to teach &quot;understanding&quot;. What they teach is execution of algorithms with a certain interface.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point is that the defense of learning how to manipulate the standard algorithms with Hindu-Arabic numerals cannot be &lt;em&gt;mathematical&lt;/em&gt;, cannot be absolutist, but can only refer to particular contexts, _if_any_, in which there is some immediate, practical advantage. Not fake in-principle advantages, not marooned-on-desert-island advantages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, it is hard to defend a claim that the standard algorithms with Hindu-Arabic numerals give &quot;understanding&quot; more operational than facility with a calculator. Although I once would have claimed so, I capitulated at a point when each month's bank statement had 50 checks, and my (on the whole very quick and accurate) mental or on-paper standard calculations simply had too many chances to fail... while my wife's (for professional reasons) extreme facility doing arithmetic on a calculator did truly afford a much lower error-rate. For 20 years, I have not checkbook-balanced by hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, since I am essentially always near a computer, if I need to do any computation (numerical or otherwise) beyond a certain very, very low complexity, I fire up Python or Sage or...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The immediate defenses of learning the standard things are just two, and somewhat flimsy: first, many older people will judge one on this, and this (dubious) expertise will be used as a filter/weeder (as in the &quot;do you want to be a manual laborer?&quot; defense), and, second, for a person interested in science/technology, much older literature (and other context) assumes familiarity with such stuff, as &lt;em&gt;implied&lt;/em&gt; context. The latter reason to pay attention is less invidious than the former.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In reality, I (as a professional mathematician) think there's scant reason to pretend, especially to small kids, that there are absolutist reasons to acquire facility in execution of visibly peculiar algorithms in manipulation of Hindu-Arabic numerals... even though &quot;all the old people&quot; know these things. Any sensible kid will believe anything else you say less after any kind of rant that attempts to claim that they shouldn't depend on computers, much less calculators. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar issue arises in having the first-year calculus be so caricatured that many different software packages, many &quot;free/open&quot;, can do it all far better. That is, what the heck should we convert these courses to... rather than be foot-draggers and insist on a highly antique, stylized, unconnected-to-reality drill that few of the participants take seriously?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, the &lt;em&gt;real&lt;/em&gt; problem, to my perception, is that as machines can do more and more algorithmic things, the &quot;easy&quot; (algorithmic) stuff that once was a fine human province is no longer available. Only more difficult things are left... and it becomes ever more difficult to pretend that we'll insist that the whole population learn lots of things that machines cannot do. An awkward situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summary: I'd recommend &lt;em&gt;not&lt;/em&gt; trying to defend any prohibition of calculators. Rather, &lt;em&gt;grant&lt;/em&gt; calculators, and then you can ask the next level of questions. When the kid gets access to and learns how to use symbol-manipulation software, then those issues, too, are taken for granted, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, I know, your local curricular constraints... Still, I do often find that clearing one's head by thinking what would make sense if there were not random exogenous constraints is, first, soothing, and, second, does put the necessary compromises in perspective. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-06-02T23:07:38.250" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="2570" PostTypeId="2" ParentId="2551" CreationDate="2014-06-03T12:41:02.110" Score="21" Body="&lt;p&gt;The reason we teach strategies for multi-digit operations is (in a large part) because students are learning to manipulate the symbol system we use to represent numbers, and they need to see that there is meaning behind these strings of digits. That understanding carries through not just for subtraction, but for anywhere they'll use multi-digit numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To put it another way: practicing subtraction is an opportunity to learn about how these number representations are made up (their structure). They're learning how that structure yields to &lt;em&gt;their&lt;/em&gt; desires as mathematicians. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is precisely why teaching subtraction as a specific algorithm fails. Subtraction as a &quot;standard algorithm&quot; actually isn't any better than using a calculator. Both substitute a mechanism (rote steps or calculator clicking) for understanding something about the actual mathematics behind multi-digit numerical representations in a base 10 number system and its implications for performing an operation. Standard multi-digit subtraction simply makes the student into a kind of computer that follows subtraction steps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if you're teaching the student ways of breaking down multi-digit numbers to make subtraction easier (understanding the base 10 number system, decomposing, manipulating, and composing a large number, and using it to your advantage) then there is a lot of actual mathematics going on, and decisions based on mathematical knowledge. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short (and this might be obvious), the answer depends on what you're teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's say that in your classroom a student sees this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align*}&#xA;2367 \\&#xA;-1989\\&#xA;\hline&#xA;\end{align*}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... and it would not be unusual for them to respond with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;I can make this problem simple. 1989 is really close to 2000. If I&#xA;  add one, I get 1990. I can add ten to that and get 2000. Ten plus one&#xA;  is eleven. I think of 2367 as my starting place. If I'm trying to&#xA;  subtract from it, I can add any number to it, as long as I remember to&#xA;  subtract that out again, too. So I can write the problem like this:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;\begin{align*} 2367 + 11\\ -1989 + 11\\ \hline \end{align*}&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;... and that's really this:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;\begin{align*} 2378\\ -2000\\ \hline \end{align*}&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;...which is simple. The answer is 378.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It's hard to argue against the view that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This student is thinking mathematically, not just following an algorithm.&lt;/li&gt;&#xA;&lt;li&gt;This student did some problem solving, involving decisions based on number sense and understanding of the place value system.&lt;/li&gt;&#xA;&lt;li&gt;This student got some practice with arithmetic.&lt;/li&gt;&#xA;&lt;li&gt;This is very different from what a calculator does, and difficult to say you could replace this reasoning with a calculator.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of course, maybe your student doesn't just say this right away. Maybe there is some work going on, different representations like this along the way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/KfLKP.png&quot; alt=&quot;Getting from 1989 to 2000 in two decimal hops&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... it doesn't really matter as long as it's good mathematical reasoning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is: &lt;strong&gt;if the students are reasoning mathematically, that's something a calculator cannot do for them&lt;/strong&gt;. In that case, it is a lot easier to both see and argue that the calculator is not appropriate for the activity. &#xA;Other times and in other cases in math class, there are a lot of ways it is appropriate to use or explore with a calculator. But if the only thing they're doing is something that a calculator &lt;em&gt;can&lt;/em&gt; do for them, and that's why it's hard to see why they couldn't just substitute the calculator in, that's not a problem of explaining to the student why not to use a calculator. That's a problem with what the student is being asked to do.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-06-03T12:41:02.110" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2571" PostTypeId="2" ParentId="2539" CreationDate="2014-06-03T15:54:31.180" Score="1" Body="&lt;p&gt;A mix of your approach, Steven's answer, and also the graphed Sine, Cosine, and Tangent functions. To Joel's point, the graph will clearly show tan θ going to either plus or minus infinity at 90∘ depending on which side you approach from. While sine and cosine run into issues on right triangles, the curves are easy to understand and have no discontinuities.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-06-03T15:54:31.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2572" PostTypeId="2" ParentId="2539" CreationDate="2014-06-03T16:16:19.390" Score="2" Body="&lt;p&gt;I don't know if, in good faith, you can really assert than $\sin(90^\circ)=1$ without your students having a very firm grasp of the unit circle or at least some kind of limiting process.  But, then, if the students have a firm grasp of the unit circle, your argument for why $\cos(90^\circ)=0$ &lt;em&gt;should&lt;/em&gt; be seen as unnecessarily complicated.  And then, as mentioned above, you shouldn't say that $\tan(90^\circ)$ is infinite but you should really say that it's undefined (careful, though, if a curious student enters $\tan(\pi/2)$ into a certain free online calculator, they'll get something like $1.63317787 \times 10^{16}$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you could do to accomplish what it seems you want to accomplish (i.e., to calculate values of sine and cosine using identities) you could have them do a bunch of consistency checks in the form of exercises such as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Using the double angle prove that $\sin(90^\circ)=1$&lt;/li&gt;&#xA;&lt;li&gt;Using the Pythagorean identity, prove $\cos(90^\circ)=0$.&lt;/li&gt;&#xA;&lt;li&gt;By evaluating $\tan(x)$ using the definition at $x=89^\circ,\, 89.9^\circ,\, 89.99^\circ,\dots$ and at $x=91^\circ,\, 90.1^\circ,\,90.01^\circ,\dots$ try to explain a little more about they sense in which $\tan(90^\circ)$ is undefined.  Now try to explain it using the unit circle.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="1537" LastEditDate="2014-06-04T02:37:06.467" LastActivityDate="2014-06-04T02:37:06.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2573" PostTypeId="2" ParentId="2459" CreationDate="2014-06-03T17:16:20.710" Score="9" Body="&lt;p&gt;I think there are (at least) three different issues here, which do not benefit from being confounded with each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, yes, there is the literal issue of cost. Requiring tablets as opposed to inexpensive calculators, if paid for by parents, would be an example of a &quot;regressive tax&quot;, and a bad thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, first-part-b, while a calculator would not be perceived as good for much &quot;outside of school&quot;, a tablet &lt;em&gt;does&lt;/em&gt; have many other uses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second issue, ignoring cost, is whether this is an instance of closing the barn door after the horse is out of the barn. That is, relatively inexpensive computers are widely available. Phones are ubiquitous. Kids are familiar with many aspects of &lt;em&gt;that&lt;/em&gt; interface, while less so with calculators. Why not try to play into existing competence? Further, making &quot;math class&quot; somehow deliberately and ostentatiously disconnected from the rest of the day and context doesn't help our cause. I loathe the idea of math class as an environment with artificial strictures, artificial rules, and denial of the obvious ambient culture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third, at a more technical level, why not teach the low end of genuine, long-term-useful computer systems? The incidental hazard that we can't stop them from finding and using software that trivializes the tasks we might want to give them is as much an indicator that we need to redesign those tasks, rather than that we have to work hard to constrain the environment. &quot;Teacher as rule-enforcer&quot; is not the happiest role.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-06-03T17:16:20.710" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2574" PostTypeId="2" ParentId="2551" CreationDate="2014-06-03T17:33:10.710" Score="11" Body="&lt;p&gt;For young students especially, there is intrinsic value in all 'physical' work done with numbers, and that intrinsic value is the steady accumulation of number sense. I don't mean that kids should be endlessly crunching numbers, but a balance should be met.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students who are moved towards calculators at the first instant that they're more convenient often end up on a slippery slope. I've seen high school students punch expressions like $\frac{4*3}{2}$ through a calculator, which betrays severe innumeracy, and is an enormous waste of time besides. If this step occurred in a more serious 'problem', then the time of punching these calculations into their calculator is also likely to have been enough to interrupt their working memory and distract their train of thought from the more abstract (and interesting) thing that they were supposed to be working on. This is an enormous problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's possible to explain the difference between 'problems' and 'exercises' even to young children, and if there is good rapport, it's possible that they can be convinced to trust your judgement that prescribed exercises are for their benefit.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2014-06-03T18:03:02.467" LastActivityDate="2014-06-03T18:03:02.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2575" PostTypeId="1" CreationDate="2014-06-03T18:24:46.773" Score="11" ViewCount="445" Body="&lt;p&gt;I have seen remedial courses before a first undergraduate calculus course that consist of a mix of algebra, trigonometry and coordinate geometry. However, have there been any experiments with remedial courses focusing more on developing problem-solving skills and/or the basics of what one would usually see in a transition-to-higher-mathematics course? If so, were such experiments successful in improving performance in calculus?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Note that while this is a general question, my specific interest at the moment is in assisting engineering students just beginning their undergraduate studies. Some of them only just scraped through in secondary school mathematics, while others have first completed a technician-style course that is light on mathematics.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2015-12-25T16:30:44.853" LastActivityDate="2015-12-25T16:30:44.853" Title="Alternative remedial courses before calculus" Tags="|undergraduate-education|calculus|course-design|reference-request|remedial-courses|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2576" PostTypeId="2" ParentId="2575" CreationDate="2014-06-03T19:48:03.547" Score="5" Body="&lt;p&gt;First, it certainly shouldn't be called or considered &quot;remedial&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think a course on mathematical modeling would be best.  You could include a little bit of statistics, a little bit of rudimentary calculus and a little bit of scientific computing/programming.  That should be enough to get some people hooked, it should be different from what they've seen previously and it won't be in any sense remedial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I imagine something in between&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mathematics 112 at Bucknell University and&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://catalog.macalester.edu/preview_course_nopop.php?catoid=10&amp;amp;coid=58855&quot; rel=&quot;nofollow&quot;&gt;Mathematics 135&lt;/a&gt; at Macalester College&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Anecdotally, I understand that Math 135 at Mac has greatly increased the number of students taking mathematics courses.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorDisplayName="user173" LastEditDate="2014-06-04T12:08:30.393" LastActivityDate="2014-06-04T12:08:30.393" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2577" PostTypeId="2" ParentId="2444" CreationDate="2014-06-03T22:03:47.557" Score="1" Body="&lt;p&gt;A &quot;convertible tablet&quot; notebook computer might fit with what you're looking for—I've used an older version of &lt;a href=&quot;http://shop.lenovo.com/SEUILibrary/controller/e/web/LenovoPortal/en_US/catalog.workflow%3acategory.details?current-catalog-id=12F0696583E04D86B9B79B0FEC01C087&amp;amp;current-category-id=89ECFD45CA0654775864CDBAD2606AC7&amp;amp;tabname=Features&quot; rel=&quot;nofollow&quot;&gt;this ThinkPad from Lenovo&lt;/a&gt; to teach (I had been using an overhead and transparencies and switched to a convertible tablet and projector, using Windows Journal) and to give presentations/workshops.  Basically, the idea would be to get a computer that runs a full version of Windows and has a built-in pen digitizer (more accurate than the stylus add-ons for touch-based screens).  I think there might be a Microsoft Surface Pro that has pen input as well, but I haven't kept up with the technology as much over the past few years.&lt;/p&gt;&#xA;" OwnerUserId="482" LastEditorUserId="482" LastEditDate="2014-06-05T14:39:37.963" LastActivityDate="2014-06-05T14:39:37.963" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2578" PostTypeId="2" ParentId="2552" CreationDate="2014-06-04T02:56:37.700" Score="1" Body="&lt;p&gt;Not sure if this is what's going on with your class, but I've found that when many students have the same incorrect answer that you can't explain, it's because they were all copying from each other and the original was never very good.  At least, for homework I've seen that quite a few times.  You may want to consider giving students similar questions with different numbers, which helps curtail this problem.&lt;/p&gt;&#xA;" OwnerUserId="1595" LastActivityDate="2014-06-04T02:56:37.700" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2579" PostTypeId="2" ParentId="2516" CreationDate="2014-06-04T17:12:54.077" Score="0" Body="&lt;p&gt;Don't claim that the axioms are true, nor that the rules of inference are valid.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Just&lt;/em&gt; claim that &lt;em&gt;if&lt;/em&gt; the axioms qualify as true, as well as rules of inference qualify as valid, &lt;em&gt;then&lt;/em&gt; the rest follows.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consequently, it follows that everything can get proved from modus ponendo ponens (which ends up residing in the meta-meta-language here).  If the student wants to try and deny modus ponendo ponens, then tell him or her to get to work on developing propositional calculi without them (it has gotten done before, and though I haven't seen it myself, it seems to involve a lot of work).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, say I write the following &quot;push-back&quot; proof in (a subsystem of) Lukasiewicz infinite-valued logic (in Polish notation):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;axiom 1 CCpqCCqrCpr&lt;/p&gt;&#xA;&#xA;&lt;p&gt;axiom 2 CpCqp&lt;/p&gt;&#xA;&#xA;&lt;p&gt;axiom 3 CCNpNqCqp&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;  1 p/Np, q/CNqNp, r/Cpq&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;4 CCNpCNqNpCCCNqNpCpqCNpCpq&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;  2 p/Np, q/Nq&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;5 CNpCNqNp&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;  3 p/q, q/p&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;6 CCNqNpCpq&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;  detachment 4, 5&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;7 CCCNqNpCpqCNpCpq&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;  detachment 7, 6&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;8 CNpCpq&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I say that theorem 8 CNpCpq implies that from the negation of a proposition p, and the proposition p, by detachment we can infer any proposition at all in infinite-valued Lukasiewicz logic.  But, the student objects, because as is well-known, because the principle of contradiction fails in infinite-valued Lukasiewicz logic.  Consequently, I would respond by saying that if those are permissible axioms, if the rule of inference of substitution comes as permissible, if the rule of formal detachment: {C$\alpha$$\beta$, $\alpha$} $\vdash$ $\beta$ comes as permissible, and if soundness holds, then the above suffices for my argument (and then I'd probably explain that it's CKpNpq that fails in Lukasiewicz infinite-valued logic, but that's another issue). &lt;/p&gt;&#xA;" OwnerUserId="1356" LastEditorUserId="1356" LastEditDate="2014-06-04T17:23:01.420" LastActivityDate="2014-06-04T17:23:01.420" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2580" PostTypeId="2" ParentId="2484" CreationDate="2014-06-04T18:27:39.283" Score="13" Body="&lt;p&gt;&lt;em&gt;(This is a very, very long answer because I want to highlight both the likely mindset of such a pupil, and possible approaches to win them over.  Also, please check &lt;a href=&quot;https://matheducators.stackexchange.com/questions/895/student-poisoned-experience-with-math&quot;&gt;Student Poisoned Experience with Math&lt;/a&gt; in case it may help you.)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;h1&gt;Understanding the Mindset&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Another example could be an expression with non-round numbers, where I suggested to hide them behind α and its variations (esp. since such substitution made the necessary formulas and reductions easily visible). Nevertheless, he tirelessly rewrote them all the way through calculations and &lt;strong&gt;clung to their concreteness like to some kind of lifeline&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You sound like you're teaching a younger me.  I feel sorry for you.&lt;/p&gt;&#xA;&lt;p&gt;This was exactly what I did.  I would reduce the problem, not to its simplest abstract form from which a solution becomes more apparent, but &lt;em&gt;to the form conceptually most recognisable to me&lt;/em&gt;, which would lead me around and often down a blind alley.  And when you would say that there is an easier way to do it, and you'd even show it, I'd just ignore it because I didn't understand what you were saying, and I wanted to get there on my own steam, dammit.&lt;/p&gt;&#xA;&lt;p&gt;In short, you have a pupil who is disenchanted, depressed, and close to belligerent.&lt;/p&gt;&#xA;&lt;p&gt;I feel that, as tutors, we tend to make a mistake.  We see someone take the long way around and we offer a short-cut.  We approach the subject from the perspective of a master, with insights of simplicity and foresight.  By imparting our knowledge, we're hoping something will stick, and that their performance will improve.&lt;/p&gt;&#xA;&lt;p&gt;The perspective of our pupils, however, is that of a frog trying to cross a busy highway: at some point, they just make the plunge, race screaming and flailing for the other side, and hope not to get hit by a truck.  They can't see the short-cut because they don't understand the short-cut, even when it's pointed out with neon signs.&lt;/p&gt;&#xA;&lt;p&gt;And that's the really hardest part here, I think: debugging the misconceptions op your pupil.  Stepping out of our frame of reference, and getting into theirs.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;I don't need to understand it&amp;quot; and &amp;quot;I would like to do it without thinking&amp;quot;. &lt;sup&gt;(*)&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.youtube.com/watch?feature=player_detailpage&amp;amp;v=pIwToj3p3vM#t=62&quot; rel=&quot;noreferrer&quot;&gt;How's that working out for you, partner?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Help him out with the rules by putting them on display.  For trigonometry, show the unit circle, and put an example in each quadrant.  It'll help him visualize and internalize the relations between the trigonometric functions.  At some point, remove the unit circle--but let him draw it himself, if he so chooses.&lt;/p&gt;&#xA;&lt;p&gt;It will also help him realise that these 'rules' are not axiomatic, independent laws, but properties and relations; tools rather than obstacles.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;(*)&lt;/sup&gt; &lt;sub&gt;Objects follow the path of least physical resistance.  People follow the path of least intellectual resistance.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h1&gt;Changing the Approach&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How can I change such a student's approach to math?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Sneak the math onto him.  Really.  Seriously.&lt;/p&gt;&#xA;&lt;p&gt;My brother had the same problem tutoring me in physics.  I was horrible.  I didn't get anything right.  I hated it.  I wanted to burn my course book after smothering it in the blood of a sacrificial goat.&lt;/p&gt;&#xA;&lt;p&gt;We had been covering potential and kinetic energy, and then he decided we needed a better exercise.  So we were going to calculate the horsepower of &lt;em&gt;throwing a horse off the Niagara falls.&lt;/em&gt;  Because the exercise was so far removed from what I personally experienced as the boring, horrible physics, I did not have the instant mindblock I would otherwise have &lt;sup&gt;(*)&lt;/sup&gt;.  Suddenly, &lt;strong&gt;physics was a means to an end&lt;/strong&gt;, and I had the (admittedly morbid) drive to find the answer.&lt;/p&gt;&#xA;&lt;p&gt;A logic and philosophy professor at our university had this approach to teaching where he would start with anecdotes and slowly, sneakily slide into his courses.  By the time students 'caught on' that he was back to teaching, he brought up another anecdote to reel them in.&lt;/p&gt;&#xA;&lt;p&gt;Our probability and statistics professor would take topics from recent news for exercises and show either why the news report was highly misleading, or what the implications of something would be: error margins on polling, probabilities on game shows, casino games, poker, stuff like that.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;(*)&lt;/sup&gt; &lt;sub&gt;We didn't actually throw a horse off a cliff.  My brother wasn't too big on empirical physics.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For a change I wanted to show him something engaging, e.g. how math allows us to build nice things using 3D animation software, but he would not make a connection. It was just a few cool pics useful only as long as they can be used to raise his status (e.g. his friends think them awesome).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;3D animations can be cool and flashy, but they don't immediately, intuitively link back to math for a layperson.&lt;/p&gt;&#xA;&lt;p&gt;Stick closer to home.  Does he like video games?  Maybe he played Call of Duty, and heard of the &lt;a href=&quot;http://www.youtube.com/watch?v=9cf6wfVyb1Q&quot; rel=&quot;noreferrer&quot; title=&quot;Call of Duty 4 - One Shot, One Kill&quot;&gt;Coriolis effect&lt;/a&gt;.  Or maybe he played Realm of the Mad God; that uses Voronoi diagrams for world generation.  Maybe he's addicted to his cell- or smartphone; introduce satellite orbits or talk about GPS systems.  Math itself is difficult to connect directly, so you may have to go through physics.&lt;/p&gt;&#xA;&lt;p&gt;You're right: you can't make him go from loathe to love in an instant.  Even now, I don't love mathematics, but I am no longer terrified of it, and I regularly use it as a powerful tool of abstraction.&lt;/p&gt;&#xA;" OwnerUserId="1193" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-06-04T18:27:39.283" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2581" PostTypeId="2" ParentId="2484" CreationDate="2014-06-04T19:31:35.727" Score="6" Body="&lt;ol&gt;&#xA;&lt;li&gt;JvR's answer is very good.&lt;/li&gt;&#xA;&lt;li&gt;I'd like to point out a missing aspect:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;h1&gt;Emotion&lt;/h1&gt;&#xA;&lt;p&gt;Your student has developped highly negative emotions concerning math. He has lost motivation and is in despair concerning math. He feels, he'll never understand math also he knows, that it's important. He does &lt;em&gt;not&lt;/em&gt; understand, what he's doing at the moment, but he also doesn't want to do something new because of two reasons:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Learning something new has the risk of forgetting or loosing the little he thinks he knows or understand. (That's wrong, but can't see it in the right way.)&lt;/li&gt;&#xA;&lt;li&gt;Trying to learn something new has the risk of failure. That'd increase the negative feelings towards math.&lt;/li&gt;&#xA;&lt;li&gt;The new stuff is gonna be harder than the current stuff. As he doesn't even understand the current stuff well, he won't understand the next stuff at all.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;You cannot win against these emotions. You have to reduce them.&lt;/p&gt;&#xA;&lt;h2&gt;Bring in good emotions&lt;/h2&gt;&#xA;&lt;p&gt;Forget about the current subject, if you hamper with it, give him the recipes he wants, but don't try to change him.&lt;/p&gt;&#xA;&lt;p&gt;Take the next subject or an alternate subject to teach him something new, that &lt;em&gt;really works for him&lt;/em&gt;. He needs success with math. The way of solving, you choose to teach him, need not be the &amp;quot;right&amp;quot; way. He need not understand it fully. It only must work and bring him from a given task to its solution.&lt;/p&gt;&#xA;&lt;h2&gt;Let him stay with that successful way for a while&lt;/h2&gt;&#xA;&lt;p&gt;He must gain confidence in math. He must gain confidence in you. He must gain confidence in himself being able to do math. He only has this one working way. Don't take that away from him. His school teacher is already doing that regularly.&lt;/p&gt;&#xA;&lt;p&gt;When he's so confident, that he thinks himself, something else -- some development might be due, then you can show him better ways. He'll trust you then.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-06-04T19:31:35.727" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2582" PostTypeId="2" ParentId="2452" CreationDate="2014-06-04T21:55:49.010" Score="6" Body="&lt;p&gt;As a person who still has a fresh memory of being assigned problems, I would like to point out several possible problems/reasons why this might not be the best approach. I'm not sure to what extent this answers the question.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The most rewarding thing about solving problems is the thrill of figuring out a solution to a problem you previously did not know how to solve. The rest, in my experience, is mostly mundane (writing things down, doing explicit computation, etc.). What you will be asking them is to do the boring work without the exciting bit. This probably won't be very motivating. [Maybe consider just giving a new assignment, somehow related to one that a lot of people got wrong?]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This opens an opportunity for strategically making mistakes (depending on your student's ethics, it may be fair from their point of view.). They will have an incentive to take a perfect solution, introduce artificially some mistakes, hand it in, then have you ask them to produce an improved version, and hand in the original. (This saves them the bother of producing one extra assignment, and lets them avoid a situation when they have to improve a problem which they find difficult). &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As you yourself mention, this is inherently 'unfair', since the students with few/easy mistakes will have an easier assignment than the rest. (Unless you're willing to balance it on a case-by-case basis, which will involve a &lt;em&gt;lot&lt;/em&gt; of work on your part.) Even when you balance things out (somehow), the class will likely not feel that the system is fair, because the system won't be transparent/simple enough. [Maybe consider using another scheme with similar effect, but which feels like it is just working in the student's favour? E.g. allow them to resubmit a solution to a problem, but only under the conditions that (a) new solution is very well written (b) old solution was at least OK; the new mark then replaces the old one. Make the grading scheme such that it makes sense for them to fight for those few extra points. Or a prize (extra credit) to whoever produces the best solution.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you give them comments on their previous work, they may end up parroting what you told them. This is bad because they can hand in correctly looking things without really understanding what is going on. Even worse, they may easily produce results which clearly show that they do not understand the problem, but which are still technically correct/difficult to grade down. Also, you'd be doing part of their work for them.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1645" LastActivityDate="2014-06-04T21:55:49.010" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2583" PostTypeId="2" ParentId="2551" CreationDate="2014-06-04T22:55:58.613" Score="4" Body="&lt;p&gt;One point that seems worth emphasizing is that learning to perform an operation (even a mechanical and uninspiring one, like long addition/division) helps you to understand better how the the objects you operate on really work. Being able to perform addition is about a lot more than being able to magic up the sum of two numbers. It also involves knowing what would happen &lt;em&gt;if&lt;/em&gt; you were to perform addition, without actually doing it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you want to do something a little more exciting than add integers base 10. Suppose, for example, that you are learning about a different number system, and you want to add numbers base 3. The standard techniques for adding two numbers carry through without much modification. The approach &quot;use a calculator&quot; fails miserably. Suppose you want to write a computer program adding two very long integers (i.e. program the very calculator!). You have to know how &lt;em&gt;you&lt;/em&gt; would do things, in the first place. This may fall under the heading of &quot;learning stuff to learn more stuff&quot;, but may be useful for people who like computers, or are mature enough to learn about different bases. (I would be so bold as to suggest that mentioning different bases would be very helpful to some students in understanding the &quot;standard&quot; base.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As another example, consider a situation when your input is not just a bunch of random digits, but is something more structured. For a trivial example, say you are adding $1230000000000000$ to $4560000000000000$. It may be obvious that the answer is $5790000000000000$ (i.e. the zeros stay where they are), but if it's not, then imagining you are doing long addition is a sufficient argument. A less trivial example is adding $1$ to $999999999$, or multiplying $111111111$ by $11111111$. Or you can derive the criterion for divisibility by $9$ by imagining you are performing long division, without ever using the word &quot;modulo&quot;. Or adding two very &quot;sparse&quot; numbers. Or... etc, etc. To rephrase this: if you are operating on numbers that have some regularity, this can help you if you are doing things by hand. It won't if you are using a machine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, there are also a lot of problems where you are supposed to find long numbers where some digits are unknown and a single equation is given (One example I just found is to find a solution to: SEND+MORE=MONEY, where each letter is a digit; there are also simpler ones.) The only way to approach these is to know about long addition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, if you are adding two numbers, but are only interested in the digit at a given position, for whatever reason, then long addition allows you to find this digit with minimal effort. Again, calculators do not provide this. &lt;/p&gt;&#xA;" OwnerUserId="1645" LastActivityDate="2014-06-04T22:55:58.613" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2584" PostTypeId="2" ParentId="2452" CreationDate="2014-06-05T00:05:47.437" Score="3" Body="&lt;p&gt;I see a few other ways to accomplish your stated goals:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If time permits, have students present solutions.  This is an even more transparent way to see what they were thinking and how well they understand their own solution.&lt;/li&gt;&#xA;&lt;li&gt;Having students present at least a few problems and maybe asking a scribe to write down what the presenters say will save you from having to write solutions to those problems.&lt;/li&gt;&#xA;&lt;li&gt;I don't know that students want elegant and streamlined solutions -- at least I know that as a student I'd rather have seen the big ideas and insights of the argument rather than the details.  The details I would try to figure out on my own.  If you were to write solutions that were a little more skeletal, I think the students would be satisfied.  Something you could do if you're not comfortable with only a skeletal solution is to have a skeletal solution, put it on a computer and then record your voice talking over it.  That way you could say out loud all the elegance that takes a long time to write.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-05T00:05:47.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2585" PostTypeId="1" AcceptedAnswerId="2609" CreationDate="2014-06-05T01:59:57.353" Score="7" ViewCount="2548" Body="&lt;p&gt;I was presented with a power point slide by a friend about math education and one of his slides talked about &quot;the seven benchmark numbers&quot;. He said that: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The seven benchmark numbers to develop a &quot;complete&quot; number sense are: $0, \frac{1}{10}, \frac{1}{2}, 1, 10, 12,$ and $100$. These numbers form the foundation of the mathematics curriculum in primary and secondary education.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, when pressed to do so, my friend was unable to explain why these numbers were &quot;benchmarks.&quot; Does anyone know what he may be referring to or, better yet, does anyone know where he gets this information from?&lt;/p&gt;&#xA;" OwnerUserId="1647" LastEditorUserId="11" LastEditDate="2014-06-05T03:52:55.527" LastActivityDate="2014-06-11T04:00:32.140" Title="The seven benchmark numbers?" Tags="|secondary-education|reference-request|primary-education|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2586" PostTypeId="2" ParentId="1042" CreationDate="2014-06-05T03:53:50.600" Score="3" Body="&lt;p&gt;As a professional mathematician, I certainly appreciate the projects listed above, and this might not be exactly the kind of answer you're looking for.  The projects below will definitely help students understand a particular concept (dimension and proof-writing, respectively) but will not directly help them solve any problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I used to teach high school, I would usually try to develop projects that connected mathematics to other disciplines.  I would focus especially on the humanities and arts, since students whose interests leaned in those disciplines were often the most challenging to motivate and engage.  Two projects with which I had some success:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In a geometry class, I had students read &lt;em&gt;Flatland&lt;/em&gt; and then asked that they come up with a creative project based on what they read.  One student turned it into an analysis of women's rights in Victorian England; another painted, on a thin strip of wood, what an inhabitant of Flatland might see; another made the argument that cubist paintings were a 2-d rendering of what a 4-d person might see looking at a 3-d person and then made some paintings in that style.&lt;/li&gt;&#xA;&lt;li&gt;Also in the same geometry class, I tried to make an analogy between proof-writing (even though this was only ten years ago, we were still teaching proofs at this school) and editorial writing.  What are your givens/the editorial writer's biases and assumptions?  What are you trying to prove/the editorial writer trying to say?  Can you/the editorial writer justify each step in your argument?  I then had the students take an editorial in a local newspaper and attempt to decompose it into a two column proof.  &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-05T03:53:50.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2587" PostTypeId="2" ParentId="1294" CreationDate="2014-06-05T04:47:27.753" Score="1" Body="&lt;p&gt;How about &lt;a href=&quot;http://en.wikipedia.org/wiki/Arithmetico-geometric_sequence&quot; rel=&quot;nofollow&quot;&gt;arithmetico-geometric series&lt;/a&gt;?  I.e., a series of the form&#xA;$$&#xA;\sum_{n=1}^\infty P(n) a^n&#xA;$$&#xA;where $P(n)$ is a polynomial and $a\in \mathbb{R}$ is the &lt;em&gt;ratio&lt;/em&gt; of the series.  For example, find &#xA;$$&#xA;\sum_{n=1}^\infty n \left(\tfrac12\right)^{n+1}.&#xA;$$&#xA;A proof-without-words and one with words (although they're in Spanish) can be found &lt;a href=&quot;http://gaussianos.com/demostracion-sin-palabras-sobre-la-suma-de-una-serie-numerica/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-05T04:47:27.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2588" PostTypeId="1" CreationDate="2014-06-05T09:55:15.937" Score="9" ViewCount="886" Body="&lt;p&gt;I am first year masters student. This past year, I did a first year lab with three different sections. I received the TA evaluations of the said lab and got not-so-great reviews to my surprise. I got around a 3.8-3.9 for two of them and one of them was about a 4.5 out of 5. And I also got 3 written comments. All three comments all complained about how I grade horribly, and I am disorganized. They compared the grade they got with the grades their friends  from a different sections with  different TAs for similar solutions (they claimed they worked with their friends). One claimed that his friend had marks in the 90s while he had marks in the 70s. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like I am a fair grader. I grade based on the given grading scheme. I take away marks when students make mistakes in concepts or methods (not 2+2=5 errors). I judge how many marks to take off based on how bad the error is. I don't believe in giving students free marks or spoon feeding them since they don't pay attention to my comments (from experience) unless I take away marks. I have noticed assignments left around by other TAs in the lab rooms and seen that some TAs mark very leniently. Some mark lazily, where even if the student has the wrong method/answer, they get almost full marks, and some just act lenient because they say they feel for the students. During proofs, and multiple-step problems, I make sure to go through the steps to point out the mistakes in the concepts and make sure they're getting the full knowledge they deserve. However, I find myself feel horrible because its unfair if they see their friends ending up with much better grades than them for the same amount of work. Should I be more lenient at this stage? This is a first year course, so I can see why they would be weaker in the concepts, however, I have graded for a third year course and realized that students don't know the basics they should have learnt in the first year courses just because concepts are not reinforced to them from the very beginning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if students come to me about the grades, I either explain why they lost marks or if they have a justifiable explanation, I give them back some marks. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the lab part, I think I need to work on speaking with more confidence and such but in general, I think I know how to improve on it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is your approach to grading? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This year, I am teaching another first year course, and I find myself taking off points from people who use horrible notation, such as omitting limits when taking them, plugging in numbers into half the equation and leaving the variables intact in the rest, and such. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point I am just questioning on whether spending more of my time on checking their steps is worth it if I am just getting hated for it when there are other TAs getting by on being lazy about it and getting better reviews. I want the best for the students but they don't seem to appreciate my methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry for the long post! Any advice is much appreciated! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cheers!&lt;/p&gt;&#xA;" OwnerDisplayName="Merry" LastEditorUserId="123" LastEditDate="2014-06-05T22:19:49.463" LastActivityDate="2014-06-08T01:35:01.660" Title="How should I modify my grading as a TA?" Tags="|grading|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2589" PostTypeId="2" ParentId="2588" CreationDate="2014-06-05T11:28:16.390" Score="0" Body="&lt;p&gt;The best strategy for consistent grading (and being able to explain later) is to solve in detail, and &lt;em&gt;add&lt;/em&gt; points for things done right. Publish the solution and grading criteria. If there are several sections, coordinate with the other TAs to use consistent grading, for example by rotating the problem sets or having every one solve (and define grading) for one problem, or even grade one each.  For programming/laboratory assignments, perhaps set up a general schema (and publish it beforehand!) for grading, with some overall criteria like orderliness, use of the tools, clear result. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-05T11:28:16.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2590" PostTypeId="1" AcceptedAnswerId="2592" CreationDate="2014-06-05T14:09:03.340" Score="6" ViewCount="709" Body="&lt;p&gt;I'm referring here to (math or other subjects) textbooks which include an exposition, main text and examples, references and exercises in each of the chapters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my experience, in most cases, it isn't feasible to really work through a chapter in a single reading. I keep finding myself getting tired (and confused) with the texts and jumping forward to see more complete examples and to try my hand at the exercises. And I keep wondering, why won't anyone make textbooks with more manageable chapters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why not cut the subject matter into small chapters, as in popular fiction, or as in Khan Academy tutorials?&#xA;Is anyone here familiar with any research measuring the effect of chapter length on comprehension or any other parameters?&lt;/p&gt;&#xA;" OwnerUserId="688" LastActivityDate="2014-06-05T14:42:29.050" Title="Why are textbook chapters so long?" Tags="|textbooks|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2591" PostTypeId="2" ParentId="2588" CreationDate="2014-06-05T14:10:11.587" Score="6" Body="&lt;p&gt;In the future prior to the course starting, all the TAs and the lead instructor should meet to discuss the grading scheme for HW. It's a good idea to set a very rough approximate median for HW. This way at the end HW averages aren't too different. I generally like to grade two or 3 HW problems in detail, and give a set number of points for &quot;honest attempts&quot; for the other problems. I set up a score sheet for my graders when I have multiple people grading that details exactly what a 4/4,3/4,2/4,1/4 and 0/4 HW problem looks like and include examples. When I do that the graders become much more consistent. Also I like a 3 or 4 point scale for HW problems because it usually means graders all choose the same number, once you start assigning 5 or more points you leave a lot up to interpretation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4 - perfect, or arithmetic error (note many sign errors are conceptual)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3 - minor conceptual errors (you know what the student meant and it was correct) but their solution was unclear, or misused words in a way that made it unintentionally wrong&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 - major conceptual errors. They are clearly going about this problem completely incorrectly but they do show some evidence of going to lecture and reading the chapter&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 - states something like &quot;this is where I am stuck on this problem ...&quot;, if this is done very well it can be worth a 2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0 - intentional BS or writing down an answer with no work&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd then give them specific examples of past problem solutions that are archetypes of these 4 grades. It is completely reasonable to ask your instructor or have the TAs collectively come up with such a scheme. The 1 point for honest description of difficulties encourages academic honesty and gives you a lot of feedback. It saves the graders a lot of work from having to read BS answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I was a bit vague about what setting an &quot;approximate&quot; median, prior to the start of the course, means. It doesn't mean make all TAs hit an &quot;exact&quot; median score. What it means is have all TAs have an &quot;approximate&quot; median in mind while grading, it is only used to guide the TAs who are grading way too easily or harshly. You would tell your TAs something like this, &quot;In the past, 95% percent of sections in Course X, when I have been the lead instructor, have had HW medians between 71 and 79 percent, please come talk to me if you think your HW score distribution is very different from this.&quot; &lt;/p&gt;&#xA;" OwnerUserId="1387" LastEditorUserId="1387" LastEditDate="2014-06-08T01:35:01.660" LastActivityDate="2014-06-08T01:35:01.660" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2592" PostTypeId="2" ParentId="2590" CreationDate="2014-06-05T14:42:29.050" Score="4" Body="&lt;p&gt;This phenomenon is actually debatable. The length of the individual &quot;lessons&quot; in a textbook is one of the indications of its quality, because the smallest division in the book is the natural material for a single class: it is hard to teach half a section because the internal organization does not support it. A textbook with sections containing too much material to cover in (say) one hour is poorly designed as a classroom reference. So in that sense, what you describe is answerable with &quot;a better book wouldn't do that&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often, in the highly polished mega-textbook franchises (e.g. Stewart's &lt;em&gt;Calculus&lt;/em&gt;), there is bloat with different contents: examples. They try to give one for every kind of problem, since their goal is not for students to appreciate the theory, but to do the standard exercises fluently. These, again, can't all be done in class, but it doesn't mean the book is poorly designed: rather, it provides insurance against confusion. Though it is somewhat condescending.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether this is what you are seeing, or whether you simply need to skip to the exercises without reading the whole section first because of your personal learning style, I can't say.&lt;/p&gt;&#xA;" OwnerUserId="513" LastActivityDate="2014-06-05T14:42:29.050" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2594" PostTypeId="2" ParentId="2588" CreationDate="2014-06-05T16:22:36.563" Score="1" Body="&lt;p&gt;When I worked in the lab, we usually either tried to have the same person grade everyone for a given lab, or at least round-robin who grades which students on different assignments, so each student got an equal taste of the stickler/gimme guy.&lt;/p&gt;&#xA;" OwnerUserId="1611" LastActivityDate="2014-06-05T16:22:36.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2595" PostTypeId="1" CreationDate="2014-06-05T18:42:24.370" Score="12" ViewCount="6384" Body="&lt;p&gt;I taught Calculus 2 at my institution the past two semesters and several students have left comments in their course evaluations that advocate grading homework problems based on whether they were completed, not whether they were correct. For instance, when asked, &quot;What changes to the course would you recommend?&quot;, one student wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;grade homework on completeness and not correctness because we are putting in the effort even if we don't necessarily understand it yet&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think they have a good point here. I view homework as required practice of the course material. Furthermore, I've realized that grading every problem carefully for correctness is prohibitively time-consuming for me!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to implement a new grading scheme next year wherein homework problems will be assigned and I will give a grade based on whether they were done, and add some helpful comments about what students should work on. Ideally, everyone will get just about full credit on this component of their grade. (Perhaps someone will skip an assignment during a busy week, but if someone actually puts in effort, they will get credit.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I'm not sure how to synthesize this new idea with the overall grading scheme. Previously, I made homework assignments were 15-20% of a student's final grade. It feels strange, though, to essentially make this a &quot;gimme&quot; portion of their grade just for doing the problems. But if I lower this to 10%, what should I do ... Have another in-class exam? That eats up class meeting time. Have regular quizzes? That also takes up some time, and should I grade those carefully on correctness, even though the students are not used to that? Should I have a once-a-month &quot;take home exam&quot; that amounts to being a difficult homework assignment of sorts?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Essentially, my question boils down to this: &lt;strong&gt;I fully intend to take these students' suggestion and assign regular homework problems to be graded solely on completion, and not correctness. I am curious about how to modify the rest of my grading scheme so that the students' final grades are still accurate and fair, and without too much extra class time taken away.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in personal suggestions/anecdotes here, as well as any education research (if there is any). I am particularly interested if you have made a similar change and can explain some observed differences in the two schemes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note: There is a &lt;a href=&quot;https://matheducators.stackexchange.com/a/2267/80&quot;&gt;great answer here&lt;/a&gt; to the question, &quot;Is it worth grading calculus homework?&quot;. My question is not the same; I have already decided to implement this &quot;grade for completion and add suggestive comments&quot; method, partly based on that answer I linked to. I am curious about adjusting other components of the course to account for this decision.)&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-06-05T22:53:57.233" Title="Grading homework problems on completion vs. correctness, and overall grading" Tags="|undergraduate-education|calculus|course-design|grading|assessment|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2596" PostTypeId="2" ParentId="2595" CreationDate="2014-06-05T19:07:53.173" Score="6" Body="&lt;p&gt;My standard breakdown for calculus courses is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;10% total for homework outside of class (on a 0,1,2 scale: 0 means they didn't turn it in, 1 means they turned in a partially complete assignment and 2 means they turned in a complete one) &lt;/li&gt;&#xA;&lt;li&gt;15% total for two or three (depending on the size of the class) problem presentations on which I call pairs of students randomly to present a homework and they are given a few &quot;I didn't get that one&quot; kind of chances during the course of the semester&lt;/li&gt;&#xA;&lt;li&gt;15% for each of three midterms that are 90% based on homework or simple extensions of the homework and whose remaining 10% are somewhat different that what they've seen (e.g., more theoretical, repeated application of the same method)&lt;/li&gt;&#xA;&lt;li&gt;30% for a final exam&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the end, with this kind of break down, the median grade for a typical calculus course of mine is a B-, which is where my department would like it to be.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-05T19:07:53.173" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2597" PostTypeId="1" CreationDate="2014-06-05T19:44:51.710" Score="8" ViewCount="369" Body="&lt;p&gt;I teach at a smallish (~2000 undergrads) liberal arts college. The math department is small and we have only about 8-10 majors per year. As far as I know, the Putnam exam has not been offered here for at least a few years, due to lack of interest. But a student recently asked me about it, saying he knew of at least one other student interested in doing it. To &lt;a href=&quot;http://math.scu.edu/putnam/registration.html&quot;&gt;register&lt;/a&gt;, we need at least 3 students to take the exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: Here is the website for the &lt;a href=&quot;http://math.scu.edu/putnam/&quot;&gt;William Lowell Putnam Competition&lt;/a&gt;. And here is an archive of &lt;a href=&quot;http://kskedlaya.org/putnam-archive/&quot;&gt;past years' problems and solutions&lt;/a&gt;. In the USA, this is the premier mathematics competition for undergraduates. It is notoriously difficult, and graded very stringently; quite often the &lt;em&gt;median&lt;/em&gt; score is 0.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1&lt;/strong&gt;: How do I drum up interest in studying for and, ultimately, taking the exam? Seeing that participation has been nonexistent, is there anything special I can do to make students not only aware of, but also enthusiastic about it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2&lt;/strong&gt;: Is there anything I can do to help coach these students? Of course, going over old problems can be effective; what else do you suggest? Keep in mind that these students might not be willing to put in that much extra time outside of coursework. How can I maximize their enjoyment/benefits?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested in experiences with this kind of environment, a small liberal arts college, and especially if you started this &quot;from scratch&quot;, as it were. However, I'm interested in any more general experiences you might have in working with students to study for this exam.&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-06-05T20:17:42.157" LastActivityDate="2014-06-07T06:09:12.487" Title="Developing and fostering a Putnam group" Tags="|undergraduate-education|student-motivation|math-contests|liberal-arts|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2598" PostTypeId="5" CreationDate="2014-06-05T19:48:24.120" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-06-05T19:48:24.120" LastActivityDate="2014-06-05T19:48:24.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2599" PostTypeId="4" CreationDate="2014-06-05T19:48:24.120" Score="0" Body="For questions about math contests, such as the Putnam competition, Math Olympiad, American Math Competition, etc. This might include questions about designing problems, or working with students to succeed at these contests." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-06-05T21:00:26.780" LastActivityDate="2014-06-05T21:00:26.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2600" PostTypeId="5" CreationDate="2014-06-05T19:50:24.810" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-06-05T19:50:24.810" LastActivityDate="2014-06-05T19:50:24.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2601" PostTypeId="4" CreationDate="2014-06-05T19:50:24.810" Score="0" Body="For questions about math education at a liberal arts college. In the USA, this is an undergraduate institution with departments in humanities, arts, and sciences; that is, one that is not focused on a particular area. Students might be required to take a breadth of courses in a variety of subjects." OwnerUserId="80" LastEditorUserId="80" LastEditDate="2014-06-06T04:56:32.190" LastActivityDate="2014-06-06T04:56:32.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2602" PostTypeId="2" ParentId="2595" CreationDate="2014-06-05T21:00:02.147" Score="4" Body="&lt;p&gt;You should ask yourself what the homework is for (check they do work, check they can &lt;em&gt;do&lt;/em&gt; the problems correctly, ...), and grade according to that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my Discrete Math class I give many (some 6 to 8) homework problems, which check if they know how to work carefully on problems related to what is seen in class. This is 30% of the grade. With the TAs, we select 3 problems each week. One is to be solved by the TA, asking for input from the class; one is solved by the class, perhaps asking the TA for guidance; and one is solved individually, graded as turned in or not. This is 5%. And a midterm (30%) and a final (35%).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In homework they have time, and can consult external sources and use whatever tools they want, grading is according to work done carefully and completely. What I want is that they understand what they turn in, even if they got it off MSE, so a &quot;random&quot; selection of students get their grade from explaining to the TA what they turned in, not by grading what was turned in. The sessions with the TAs are geared towards not having them cram the day before the exams (which has shown time and again leads to a masacre), the exams are meant so see if they understand the material (apply it in situations at most mildly ouside what was seen in class/homework, &lt;em&gt;do not&lt;/em&gt; ask for careful development but e.g. just explain how to set up or solve a problem, or why something is done a specific way).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-05T21:00:02.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2603" PostTypeId="1" CreationDate="2014-06-05T21:38:22.970" Score="4" ViewCount="165" Body="&lt;p&gt;For work, I tend to have to explain quantitative ideas to people with only some informal computer science training. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yesterday, my friend wanted to draw a map on his computer and was concerned we could locate specific latitude and longitude on his computer screen.  This image is 256 pixels across and 256 pixels wide. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/sumyx.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here a really close-up view of San Juan, Puerto Rico.  According to &lt;a href=&quot;https://wiki.openstreetmap.org/wiki/Zoom_levels&quot; rel=&quot;nofollow noreferrer&quot;&gt;Openstreetmap&lt;/a&gt; the scale is &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Level    Degree  Area             m / pixel  ~Scale&#xA;13       0.044   village or town  19.093     1:70,000&#xA;14       0.022                     9.547     1:35,000*&#xA;15       0.011   street            4.773     1:15,000&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Since all maps are stored as squares tiles of various sizes, he has to identify the tiles he will need and place them together himself.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then if he wants to identify specific landmarks - he needs to compute the proportion between the pixels and the lat-long using information he can find in this page on the various &lt;a href=&quot;https://wiki.openstreetmap.org/wiki/Zoom_levels&quot; rel=&quot;nofollow noreferrer&quot;&gt;zoom levels&lt;/a&gt; or other resource.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;&#xA;Even though it is easy to point things out on a map, this is because computers do the quantitative reasoning for us.  This is about as simple as I can make the work-flow for my friend -- and I didn't do nearly as good a job yesterday.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These concepts are not obvious and it took me a while to learn about map projections and re-learn them when using them on a computer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;math&quot; involved was hardly traditional... I think we do one or two multiplcation problems in the entire process.  Instead we deal with issues like &lt;strong&gt;scale&lt;/strong&gt; and &lt;strong&gt;proportion&lt;/strong&gt; maybe &lt;strong&gt;angle&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Worst of all, I didn't get to explain to him that in &lt;a href=&quot;https://en.wikipedia.org/wiki/Mercator_projection&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mercator Projection&lt;/a&gt;, column 4 - meters per pixel - actually changes with the latitude on the map &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://wiki.openstreetmap.org/w/images/thumb/8/87/Tissot_mercator.png/400px-Tissot_mercator.png&quot; width=&quot;300&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What level would this be considered?  More importantly, this computation is a sequence of steps I will have to explain over and over - how could I explain this better?&lt;/p&gt;&#xA;" OwnerUserId="1654" LastEditorUserId="-1" LastEditDate="2017-03-10T09:42:41.960" LastActivityDate="2014-06-06T12:39:46.663" Title="Explaining the algebra behind a problem in computer geography" Tags="|secondary-education|geometry|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2604" PostTypeId="2" ParentId="2595" CreationDate="2014-06-05T22:03:35.913" Score="10" Body="&lt;p&gt;There's a compromise between &quot;correctness&quot; and &quot;completion&quot; called &quot;Standards-based grading&quot;.  Here's a few links about it with various people who have tried using it for Calculus: &lt;a href=&quot;http://blogs.cofc.edu/owensks/2014/01/08/sbg-calculus2/&quot;&gt;http://blogs.cofc.edu/owensks/2014/01/08/sbg-calculus2/&lt;/a&gt;, &lt;a href=&quot;http://alwaysformative.blogspot.com/p/standards-based-grading-implementation.html&quot;&gt;http://alwaysformative.blogspot.com/p/standards-based-grading-implementation.html&lt;/a&gt;, &lt;a href=&quot;http://speced.fivetowns.net/lcs/content/Standards%20Based.pdf&quot;&gt;http://speced.fivetowns.net/lcs/content/Standards%20Based.pdf&lt;/a&gt;, &lt;a href=&quot;http://www.computer.org/csdl/proceedings/fie/2012/1353/00/06462211-abs.html&quot;&gt;http://www.computer.org/csdl/proceedings/fie/2012/1353/00/06462211-abs.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can implement it in a variety of ways, but the general idea is that associate all problems you give in the course with one or more &quot;standard(s)&quot; that you expect students who pass to achieve, and you give students (effectively) infinite attempts to &quot;pass&quot; each standard.  In general, this means that, for students who don't need the extra practice, they can do very few problems, but for the students who need significant practice, they can keep on trying until they're solid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often, courses that use SBG do it on exams too, but you could probably only do it on the homework if you wanted.  And you can still give students significant credit for attempting the problems, but this way, still making the homework a &quot;gimme&quot;--they still have to try, but only until they get it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One clear concern here is the overhead of keeping all this information (since you now need to record which standards students are passing, possibly review more [but smaller!] batches of hw, etc.).  In general, I think the benefits outweigh the extra overhead though, particularly, because there should be (a) fewer grading overall and (b) more significant learning overall.&lt;/p&gt;&#xA;" OwnerUserId="93" LastActivityDate="2014-06-05T22:03:35.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2605" PostTypeId="2" ParentId="2588" CreationDate="2014-06-05T22:10:52.613" Score="2" Body="&lt;p&gt;Once I got quite good evaluation from a lab class but for my grading approach, which received exactly the same comments as you, e.g. that I like to cut points and things which I consider bad weren't even mentioned in other groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is despite the fact that we had &lt;em&gt;really detailed rules&lt;/em&gt; on how to grade each type of mistake. Not only each author of the problem prepared a list of common mistakes, but he would grade his group first, also skim over other groups solutions and then adjust the rules. Finally the rest would grade their groups, and yet there would be an update if someone found an error without an appropriate rule. It was &lt;em&gt;almost&lt;/em&gt; as one person grading it all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, there was huge inconsistency between graders. I was curious and went over other groups submissions and what I found was that &lt;strong&gt;the scores depended on how lazy or meticulous the grader was&lt;/strong&gt;. It was easy to tell from the number and level of detail of comments the TAs left.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The actual problem was that we would start from some top possible score and cut points for errors. Lazy graders would find less mistakes while scrupulous TAs would find a lot. Hence, the inconsistency. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think I did a good job in spite of comments about grading, and in my opinion &lt;em&gt;so did you&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, imagine that if you would give positive points for all nice things you found, the result would be quite different. Unfortunately, it's easier to find mistakes rather than correct pieces. One solution could be to include both kinds of points, but then there might be pressure to consider the solution good unless one finds a mistake, which would turn back into previous, flawed approach. Personally I have no idea how to fix this, without some person doing much more work than the others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concluding, don't worry too much about it and keep up the eagerness you have (it's great that you found this site and asked about your problem).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-06-05T22:10:52.613" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2606" PostTypeId="2" ParentId="2595" CreationDate="2014-06-05T22:53:57.233" Score="3" Body="&lt;p&gt;I've had professors do a hybrid approach. A small set of problems would be graded and returned with detailed comments on errors. This was to help us know if we were understanding the material well enough for the exams. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A larger set of problems would be turned in and graded mainly on completeness. This was to encourage a lot of practice, which, despite everyone's best intentions, just doesn't happen unless the work has to be turned in. We knew in advance which set was which, and we received solutions after-the-fact for both sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your main question, homework was still 15-20% of the grade. Each graded answer was a larger portion, but I didn't hear any of my fellow students complain about that. (Although, perhaps the ones who didn't do well on the homework didn't have the math skills to figure that out. &lt;em&gt;evil grin&lt;/em&gt;)&lt;/p&gt;&#xA;" OwnerUserId="908" LastActivityDate="2014-06-05T22:53:57.233" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2607" PostTypeId="2" ParentId="2471" CreationDate="2014-06-06T00:31:22.133" Score="10" Body="&lt;p&gt;Here's what happened: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;You asked a question using standard phrasing.&lt;/li&gt;&#xA;&lt;li&gt;The student found it surprising that the phrasing allowed the answer given.&lt;/li&gt;&#xA;&lt;li&gt;The student was worried about this and blamed the phrasing of the question.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Two things I think it's important that should happen:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The student needs to understand what is meant by this standard terminology, and &lt;/li&gt;&#xA;&lt;li&gt;The student will also in the future have to get more used to mathematics allowing special cases that are covered by general cases without explicitly mentioning it, for example:&#xA;&lt;ul&gt;&#xA;&lt;li&gt;if we say something's a parallelogram, we allow it to be a square, &lt;/li&gt;&#xA;&lt;li&gt;if we ask for complex roots of quadratics we allow real ones, &lt;/li&gt;&#xA;&lt;li&gt;if we have a function of x and y we allow it to be 3y&lt;/li&gt;&#xA;&lt;li&gt;if we express cos(x+a) in terms of cos(x) and sin(x) we allow it to be sin(x)&lt;/li&gt;&#xA;&lt;li&gt;0 is a number, 4 is complex, $1 \le 200$ and each set is a subset of itself.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The phrase &quot;express cos(x+a) in terms of cos(x) and sin(x)&quot; means, &lt;em&gt;to mathematicians&lt;/em&gt;, find an equation of the form cos(x+a)=f(cos(x),sin(x)), and experience tells the mathematician that f will be a linear function. I feel strongly that students should be taught the meaning of standard mathematical terminology and that edge cases are valid. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rephrasing the question does &lt;em&gt;not&lt;/em&gt; help them with either of the two things that need to happen, and you were correct to point that 0 is a valid coefficient, something it is &lt;strong&gt;very&lt;/strong&gt; important that they come to terms with.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Historically, the solution of quadratic equations was hampered by the fact that $x^2+x=6$, $x^2=x+6$, $x^2=6$ and $x^2=5x$ were seen as four completely separate types of quadratic equation. Only once you're prepared to accept negative and zero numbers, and write them all as $ax^2+bx+c$ is a general solution even expressible. It's very important to mathematical development to allow expressions to cover cases as generally as possible, and to resist the very understandable human trait to always treat edge cases separately.&lt;/p&gt;&#xA;" OwnerUserId="1340" LastActivityDate="2014-06-06T00:31:22.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2608" PostTypeId="2" ParentId="1649" CreationDate="2014-06-06T00:58:51.923" Score="6" Body="&lt;p&gt;Because of an exercise we ran every year, I accidentally memorised $\log_{10}2=0.301$, and I could find many logarithms quickly in my head using the log laws. One holiday I decided to take it a little further:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you memorise the base 10 logarithms of 2, 3 and 7, you can quickly deduce the logs of all the other digits in your head and amaze people with calculators by working out logs to three decimal places of arbitrarily large or small numbers with one significant figure. I'll use log to mean $\log_{10}$ throughout:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Memorised:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;log 2 = 0.301   &lt;/li&gt;&#xA;&lt;li&gt;log 3 = 0.477   &lt;/li&gt;&#xA;&lt;li&gt;log 7 = 0.845&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Examples you can do in your head - students are entertained by the straining looks on your face and a correct answer with three decimal places: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;log 50 000 = log 100 000 - log 2 = 5 - 0.301 = 4.699&lt;/li&gt;&#xA;&lt;li&gt;log 80 000 000 000 = 3 log 2 + log 10 000 000 000 = 0.903 + 10 = 10.903&lt;/li&gt;&#xA;&lt;li&gt;log 0.00006 = log 2 + log 3 + log 0.00001 = 0.301 + 0.477 - 5 = 0.777 - 5 = -4.223&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can say &quot;I can only do one significant figure&quot; and they'll accept it, but the bigger and more apparently random the number thus feels, the more amazed they are that you get it right, even though the number of zeros is the trivial bit of the calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do this as the introduction to my first lesson on logarithms. Some day I'll memorise some logs of larger primes and go to 2s.f..&lt;/p&gt;&#xA;" OwnerUserId="1340" LastEditorUserId="1340" LastEditDate="2015-10-18T19:40:28.780" LastActivityDate="2015-10-18T19:40:28.780" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2609" PostTypeId="2" ParentId="2585" CreationDate="2014-06-06T01:37:37.033" Score="9" Body="&lt;p&gt;A decent volume on elementary mathematics is &lt;a href=&quot;http://books.google.com/books?id=n8DjRAAACAAJ&amp;amp;dq=Mathematics%20for%20elementary%20teachers%20beckmann&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=0hKRU_X9AcjPsATCuIGAAw&amp;amp;ved=0CDAQ6AEwAA&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics for elementary teachers&lt;/a&gt; (Beckmann, 2010). The book is intended to help strengthen teachers knowledge of the mathematics behind the ideas in elementary curricula (especially reform curricula, I think). As such, it's often a good place to check for things like this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Benchmarks (also called &quot;landmarks&quot;) are introduced in the context of comparing fractions. When students are trying to determine which fraction is larger, $\frac{4}{9}$ or $\frac{3}{5}$, one strategy suggested is for students to reason about their relationship to some other number, like the fraction $\frac{1}{2}$ :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When we compared $\frac{4}{9}$ and $\frac{3}{5}$ by comparing both fractions with $\frac{1}{2}$, we used $\frac{1}{2}$ as a &lt;strong&gt;benchmark (or landmark)&lt;/strong&gt;. The fractions $\frac{1}{2}$, $\frac{1}{4}$, $\frac{3}{4}$, $\frac{1}{3}$, and $1$ are good to use as benchmarks. (p. 73)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It's clear from this text that the numbers are somewhat arbitrary; there's not meant to be a definitive list of benchmark numbers. Students would choose a fraction benchmark that helps them compare.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't say whether others use benchmarks the same way (a quick look at some other books I have within arms reach doesn't show up the term). However, the use &lt;strong&gt;here&lt;/strong&gt; is clear: a benchmark number is a number useful in reasoning about a problem. In this case, the benchmark is used as a reference point for comparison of fractions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The intent is to encourage reasoning rather than procedure. There are algorithms some students are taught to use for fraction comparison, which allow them to replace mathematical reasoning with a couple of memorized steps and some arithmetic. But reasoning allows them to practice conjecture, work through coming up with a justification for their answer, and eventually have a way to defend their answer other than &quot;this is what the procedure produced.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should think any useful number used in reasoning could be called a benchmark. For example, in my response to another question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2551/student-why-not-use-calculator/2570#2570&quot;&gt;(seen here)&lt;/a&gt;, I wrote about student reasoning that transforms a subtrahend into the number $2000$. In that case, $2000$ is useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another type of mathematical reasoning that might benefit from a benchmark is estimation. Numbers can be replaced by nearby benchmarks that make for quicker calculation, if the aim is to just ballpark an answer (an often quite useful strategy for many real world applications). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, &lt;strong&gt;I don't think there is support for a definitive list of benchmarks&lt;/strong&gt;. The ones Dr. Beckmann provides are suggestions (&quot;good to use&quot;) but the real test is whether they are useful to the thinker in the midst of their mathematical reasoning.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Works Cited:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beckmann, S. (2010). Mathematics for elementary teachers. New York: Pearson Addison-Wesley.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-06-06T01:37:37.033" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2610" PostTypeId="1" CreationDate="2014-06-06T04:06:03.603" Score="9" ViewCount="309" Body="&lt;p&gt;I teach at a University in a town of roughly 5700 people with a few towns of comparable size within twenty minutes drive.  I am very much interested in starting a Math Circle in the area and am looking for advice from folks who have successfully had math circles in comparable geographic areas.  In particular, I'd appreciate advice on the following topics:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;em&gt;Location:&lt;/em&gt;  Should I host the math circle on my University's campus or somewhere in the community (e.g., public library or community center)?  Should its location rotate between the various towns or should it stay in the same place?&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Audience:&lt;/em&gt;  Is it easier to attract students older than 14?  Between 10 and 14?  Younger than 10?  Are there any examples of math circles that simultaneously cater to both older and younger students?  At the same time in the same room?  Or on alternating weeks?&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Publicity:&lt;/em&gt;  What's the best way to announce the existence of the math circle?  Local newspapers, directly talking to local school teachers?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;And I'd certainly welcome any advice on topics I haven't mentioned.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="1537" LastEditDate="2014-06-06T16:39:02.057" LastActivityDate="2014-06-20T05:06:05.973" Title="Advice on starting a math circle in a small rural University town" Tags="|math-circle|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2611" PostTypeId="2" ParentId="2610" CreationDate="2014-06-06T05:37:30.110" Score="4" Body="&lt;p&gt;One set location is definitely better. And I'd recommend starting with one circle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe Circle in a Box says that high school age students are busier with their other activities and less likely to come. Math clubs for younger kids (under 10) often feature more activity-based sessions. I did a &quot;math salon&quot; at my home for a few years. New people came each time. It was parents and kids together. I'd rather work with a math circle, but haven't had the time and energy to start something big. I'm offering a math circle at the community college I work at. Last semester I got 5 students. This fall, I'll ask other teachers to mention it to their students, and maybe I'll get a few more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.mathcircles.org/&quot; rel=&quot;nofollow&quot;&gt;www.mathcircles.org&lt;/a&gt; may answer more of your questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you'd like more specific help, you can email me at mathanthologyeditor@gmail.com. (If the answers here don't seem complete, I can invite a few people to come and give their expertise.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bob and Ellen Kaplan wrote &lt;em&gt;Out of the Labyrinth: Setting Mathematics Free&lt;/em&gt;, an excellent book on math circles. That's what got me involved.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2014-06-20T05:06:05.973" LastActivityDate="2014-06-20T05:06:05.973" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2612" PostTypeId="1" CreationDate="2014-06-06T09:13:29.797" Score="9" ViewCount="2222" Body="&lt;p&gt;I noticed the discussion of whether the teaching of &lt;a href=&quot;https://mathoverflow.net/questions/34125/is-galois-theory-necessary-in-a-basic-graduate-algebra-course&quot;&gt;Galois Theory is necessary&lt;/a&gt; on MathOverflow. Here at &lt;a href=&quot;http://www.lse.ac.uk&quot; rel=&quot;nofollow noreferrer&quot;&gt;LSE&lt;/a&gt;, everything we teach in mathematics should have some application to the social side of life.&lt;/p&gt;&#xA;&lt;p&gt;Granted, there are physics applications. But are there any more social or economic applications?&lt;/p&gt;&#xA;&lt;p&gt;Really there is a hole in what they learn in algebra here, and that is Galois Theory. But there would be an objection to teaching it if it didn't have some connection to our primary mission.&lt;/p&gt;&#xA;" OwnerDisplayName="Robert Simon" LastEditorUserId="376" LastEditDate="2021-02-28T11:58:06.980" LastActivityDate="2021-02-28T17:21:22.103" Title="Galois Theory: necessary?" Tags="|undergraduate-education|curriculum|graduate-education|abstract-algebra|applications|" AnswerCount="7" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="2613" PostTypeId="2" ParentId="2612" CreationDate="2014-06-06T10:42:56.743" Score="1" Body="&lt;p&gt;A theory about the kind of knowledge that can be accessed through our human conception of an explicit solution seems to me to have applications in philosophy (and neuroscience). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also consider differential Galois theory: although physicists derived the «equations of the world» there are fundamental obstructions to solve them «explicitly». This non-integrability property relates to what we humans call «chaos» (this property can be proved mathematically in some cases) and questions our belief that we master the world through our technology. The latter statement is clearly arguable, but it underlines the fact that despite our ability to give a precise, deterministic model of reality, we are in general no closer to predict the evolution of a system. I'm thinking of &lt;em&gt;e.g.&lt;/em&gt; weather forecast (Lorentz's system is proved both chaotic and non-integrable), but I surmise micro-economics is somehow modelled by non-integrable equations (though I confess my total lack of knowledge in that respect).&lt;/p&gt;&#xA;" OwnerDisplayName="user1655" LastEditorDisplayName="user1655" LastEditDate="2014-06-06T12:34:19.170" LastActivityDate="2014-06-06T12:34:19.170" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2614" PostTypeId="2" ParentId="2610" CreationDate="2014-06-06T11:13:26.270" Score="0" Body="&lt;p&gt;Use technology. You don't have to restrict yourself to a limited geographic area that way. Welcome anyone interested, limiting yourself leads nowhere. Make sure everyone has an important task (if they want), feeling useful is an important part of being a member.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-06T11:13:26.270" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2617" PostTypeId="1" CreationDate="2014-06-06T13:55:47.930" Score="9" ViewCount="559" Body="&lt;h1&gt;Context&lt;/h1&gt;&#xA;&lt;p&gt;For the last few years I have been pondering about the use and disuse of calculators in secondary school level mathematics. Now, after coming across a 'hot network question' &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2551/student-why-not-use-calculator&quot;&gt;regarding a student enquiring why calculator shouldn't be used&lt;/a&gt; and the extend to which Math Educators seemed to feel obligated to defend it. On the other hand, for &lt;em&gt;most&lt;/em&gt; (&lt;strong&gt;not&lt;/strong&gt; all students) I am inclined to argue that it's more valuable to teach them 6 years of using calculators and things like WolframAlpha, r/statistics and programming than it is to teach so much paperwork (thus allowing also harder questions to be solved). Now, that's just my opinion and one I believe I can defend, however not the crux of this question.&lt;/p&gt;&#xA;&lt;h1&gt;Question&lt;/h1&gt;&#xA;&lt;p&gt;What I am wondering about is whether there are any curricula that do allow calculator use (including better calculators such as the TI98) on &lt;em&gt;all&lt;/em&gt; papers/examinations and are build around extensive technology use (doesn't matter if it it's from some far away random country).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h1&gt;What I do know&lt;/h1&gt;&#xA;&lt;p&gt;The two I know best: the IB program and the VWO program both allow the use of calculators on part of the exam, however contain questions explicitly designed to be unsolvable with a calculator or have entire sections where a calculator may not be used. Additionally both limit the 'power' of the calculator significantly (simply put, nothing above the TI94).&lt;/p&gt;&#xA;" OwnerUserId="860" LastEditorUserId="860" LastEditDate="2021-03-02T14:05:49.487" LastActivityDate="2022-08-08T17:09:16.807" Title="Is there a calculator centered secondary school curriculum somewhere?" Tags="|secondary-education|technology-in-education|calculators|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="2618" PostTypeId="2" ParentId="2612" CreationDate="2014-06-06T17:12:20.243" Score="4" Body="&lt;p&gt;Does Galois theory really have no connection with the mission? Consider,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;LSE was founded 1895 by four idealistic young Fabians for the &quot;betterment of society&quot;. It adopted rerum cognoscere causas, which means to know the causes of things,  as its motto. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You could argue, to really understand the structure of factoring polynomials you must study Galois theory. It answers the insolvability of the quintic. Moreover, applications to encryption etc.. must implicitly abound in that Galois extensions seem to be everywhere in number theory (I say this as an admitted outsider!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For future teachers to know the causes of things they teach, Galois theory has great relevance. Surely factoring polynomials is an essential part of &quot;real world&quot; mathematics. In addition, I would argue that understanding the interplay between extending the number system and solving equations is quite relevant to future math teachers. Galois theory is just another chapter in that saga. The need for understanding Galois theory both in content and historical context is even more evident for the teachers of future math teachers. Is Galois theory a required math education topic? Probably not. But, if you want to be an innovator, you could study the effect of including it in your curriculum. How does it aid their future work? I'm not certain this is relevant to your students, but perhaps you could adapt the idea I sketch here. That sort of education/innovation study seems to be something you could sell at LSE.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-06-06T17:12:20.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2620" PostTypeId="2" ParentId="1649" CreationDate="2014-06-06T22:22:33.887" Score="7" Body="&lt;p&gt;I devised a trick in 8th grade for converting repeating decimals to fractions.  They were teaching a very long drawn-out process.  My trick basically does the same thing but for some reason they wouldn't let me use it on the test!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:  $0.36298298298\overline{298} \ldots$ (0.36298 with the 298 repeating).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explanation: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) take the complete part of the decimal before the first repetition occurs ($36298$)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) subtract the non-repeating part ($36$)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) the denominator will have as many $9$s as there are repeating digits, followed by as many $0$s as there are non-repeating digits ($99900$)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solution = $(36298 - 36) / (99900) = 36363/99900$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Done!&lt;/p&gt;&#xA;" OwnerUserId="1595" LastEditorUserId="511" LastEditDate="2014-09-04T10:56:11.730" LastActivityDate="2014-09-04T10:56:11.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2621" PostTypeId="2" ParentId="2610" CreationDate="2014-06-07T01:23:35.567" Score="7" Body="&lt;p&gt;You absolutely need to read Circle in a Box, Sam Vandervelde, as soon as possible. It is pretty short and it goes through a ton of surprisingly useful specifics instead of waxing poetic about the ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the moment, the PDF can be found here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.mathcircles.org/GettingStartedForNewOrganizers_WhatIsAMathCircle_CircleInABox&quot;&gt;http://www.mathcircles.org/GettingStartedForNewOrganizers_WhatIsAMathCircle_CircleInABox&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2014-06-07T01:30:16.400" LastActivityDate="2014-06-07T01:30:16.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2622" PostTypeId="2" ParentId="2597" CreationDate="2014-06-07T03:06:17.433" Score="7" Body="&lt;p&gt;There are a lot of resources for problem solving online.&lt;/p&gt;&#xA;&lt;p&gt;One place to look is MIT's OpenCourseware; see their &lt;strong&gt;&lt;a href=&quot;http://ocw.mit.edu/courses/mathematics/18-s34-problem-solving-seminar-fall-2007/index.htm&quot; rel=&quot;noreferrer&quot;&gt;Problem Solving Seminar&lt;/a&gt;&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Course Description&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This course, which is geared toward Freshmen, is an undergraduate seminar on mathematical problem solving. It is intended for students who enjoy solving challenging mathematical problems and who are interested in learning various techniques and background information useful for problem solving. Students in this course are expected to compete in a nationwide mathematics &lt;a href=&quot;http://math.scu.edu/putnam/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;contest&lt;/strong&gt;&lt;/a&gt; for undergraduates.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You might also be interested in some of the Art of Problem Solving (AoPS) &lt;strong&gt;&lt;a href=&quot;http://www.artofproblemsolving.com/Resources/&quot; rel=&quot;noreferrer&quot;&gt;resources&lt;/a&gt;&lt;/strong&gt;. The site also has recommended books for a variety of mathematics contests; see &lt;strong&gt;&lt;a href=&quot;http://www.artofproblemsolving.com/Store/index.php&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Ideally you would find someone with experience doing MathCounts, IMOs, or (since you are in the Boston area) &lt;strong&gt;&lt;a href=&quot;http://www.promys.org/&quot; rel=&quot;noreferrer&quot;&gt;PROMYS&lt;/a&gt;&lt;/strong&gt;. You might look specifically into contacting the people with &lt;strong&gt;&lt;a href=&quot;http://www.promys.org/pft/about&quot; rel=&quot;noreferrer&quot;&gt;PROMYS for Teachers&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If none of your prospective participants has experience with problem solving competitions: I will second Dave L Renfro's comment about focusing on the first couple of problems in each session, i.e., A1, A2, B1, and B2. It is quite possible for a mathematics major who has no experience with these competitions to rack up points on these problems: usually the scoring would be 1 (something of relevance), 2 (partial solution), 9 (almost a full solution), or 10 (a full solution). With a fair bit of practice, first time test takers should be expected to score a 10 or 11 on the examination (and, as you said, even scoring a single point can be enough to surpass the median).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A word of caution:&lt;/strong&gt; More than helping students to do well, I would emphasize that a poor performance on the Putnam &lt;em&gt;does not&lt;/em&gt; entail an inability to do mathematics (or even be a professional mathematician). Like all things, doing well requires practice, and students who have not competed in MathCounts, IMOs, etc in the past are unlikely to score very high in this competition. Moreover, you write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Keep in mind that these students might not be willing to put in that much extra time outside of coursework.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If the students are unwilling to put in a lot of time &lt;em&gt;and&lt;/em&gt; are not experienced in this type of problem solving, then they may well be shut out.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Final suggestions:&lt;/strong&gt; Free food is one way to entice students. Another method is to see if some sort of prize (even if it is just a certificate) can be awarded to the student who scores the highest at your institution on the Putnam. Note that the highest scorer might &lt;em&gt;not&lt;/em&gt; be a mathematics major: See if those with other majors - e.g., Computer Science, Economics, Physics - have any interest in participating (especially with the free food). Separately, I know of at least one other small, MA liberal arts college that offers its own internal problem solving award: see more on Amherst College's Walker Prize Examination &lt;strong&gt;&lt;a href=&quot;https://www.amherst.edu/academiclife/departments/mathematics/prizes_and_awards&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt; and links (without solutions) to tests from &lt;strong&gt;&lt;a href=&quot;https://www.amherst.edu/system/files/media/walker2012_0.pdf&quot; rel=&quot;noreferrer&quot;&gt;2009-2012&lt;/a&gt;&lt;/strong&gt; and &lt;strong&gt;&lt;a href=&quot;https://www.amherst.edu/system/files/media/Walker13.pdf&quot; rel=&quot;noreferrer&quot;&gt;2013&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;And if you &lt;em&gt;do&lt;/em&gt; manage to get (non-graduating) students interested enough to take it: Try and see if they will help out with the preparations the next year. Doing well on this competition and maintaining student interest are both matters of school culture in addition to &lt;em&gt;some&lt;/em&gt; raw talent and &lt;em&gt;a lot of&lt;/em&gt; practice.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-06-07T06:09:12.487" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2623" PostTypeId="1" CreationDate="2014-06-07T12:14:32.153" Score="51" ViewCount="10806" Body="&lt;p&gt;I am interested in how blind people learn mathematics at any level, but particularly before college. Math is often taught using a lot of visualization; how does this work with blind people? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My interest in this is a little round-about. I have nonverbal learning disabilities (NLD) and am writing a book about it. I am good at math (I work as a statistician) but very poor at visualization. Most people with NLD are quite poor at math (ranging from difficulties at the very earliest levels to problems with upper-elementary school &quot;word problems&quot; to problems with geometry (high school level). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My idea is that most people learn math visually, but I learn it largely non-visually. But NLD is relatively little known. So, I thought that the methods used to teach blind people might be useful for others who have visual difficulties, even if we can see. &lt;/p&gt;&#xA;" OwnerUserId="1663" LastEditorUserId="1663" LastEditDate="2014-06-07T15:17:01.873" LastActivityDate="2020-01-16T23:28:08.317" Title="How do blind people learn mathematics?" Tags="|secondary-education|primary-education|mathematical-pedagogy|special-needs-education|" AnswerCount="3" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="2624" PostTypeId="2" ParentId="2610" CreationDate="2014-06-07T12:32:46.193" Score="3" Body="&lt;p&gt;We'd welcome you at our Math Circle summer teacher training Institute, held at Notre Dame July 6th-12th. E-mail us for details.&lt;/p&gt;&#xA;" OwnerUserId="1664" LastActivityDate="2014-06-07T12:32:46.193" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2625" PostTypeId="1" AcceptedAnswerId="2629" CreationDate="2014-06-08T02:23:00.287" Score="17" ViewCount="782" Body="&lt;p&gt;I have been reading this site for a while, and was glad to find an entire tag devoted to &quot;concept motivation,&quot; which is currently my area of interest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, my particular focus has not been addressed. Skimming through standard textbooks used in today's high school and (early) undergraduate coursework, I get the sense that there is a fundamental problem with motivation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Problem:&lt;/strong&gt; &lt;em&gt;the &quot;motivations&quot; often feel contrived and somewhat artificial - e.g., a standard multivariate calculus text explains that the chain rule still works for general functions $f:\mathbb{R}^m\to\mathbb{R}^n$, as long as matrix multiplication procedures are obeyed; this, the text enthusiastically declares, is why matrix multiplication is defined as it is!&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Um... don't think so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a long time I believed that the solution, as an educator, was to strive for better understanding of the material, so that concepts would lose their artificial &quot;school feel&quot; and take on more of a &quot;someone-in-the-real-world-actually-came-up-with-this feel&quot;. But I am now convinced that I was wrong, and that taking a &lt;em&gt;historical&lt;/em&gt; approach is in fact the right way to go (&lt;em&gt;edit: I am not saying that comprehension is bad, just that it cannot explain the real &quot;why&quot; behind ideas&lt;/em&gt;). I have learned a surprising amount about concepts in mathematics, simply by researching why the came about, how they were discovered, what they were originally used for, etc. Hence the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What research is there regarding the effectiveness of incorporating historical perspective (in the sense of &lt;em&gt;mathematical context&lt;/em&gt;, not names and dates) into the motivation of concepts? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would also be interested in any textbooks which may have been written from such perspectives. Histories of maths, obviously, are not relevant, as they are focused entirely on history and less on attempting to explain or illuminate concepts. I am more looking for a text which develops a subject in modern notation, terminology and rigour, while doing so as the subject came about historically - e.g., such a calculus text would not start with limits, which came last, but would rather start with the original problems considered by early practitioners.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I apologize for the somewhat verbose query; please comment if anything could be clarified!&lt;/p&gt;&#xA;" OwnerUserId="1598" LastEditorUserId="262" LastEditDate="2014-06-08T17:45:35.010" LastActivityDate="2016-02-19T13:45:02.907" Title="Historically Motivating Concepts" Tags="|undergraduate-education|secondary-education|concept-motivation|history|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="2626" PostTypeId="2" ParentId="2625" CreationDate="2014-06-08T02:48:50.950" Score="7" Body="&lt;p&gt;While I agree that the &quot;school feel&quot; you mention is a definite turn-off (or should be) for any thinking being, the historical development is in many cases very confusing in itself. Consider the development of calculus (as e.g. Dunham's &quot;The Calculus Gallery: Masterpieces from Newton to Lebesgue&quot; (Princeton University Press, 2008) masterfully shows it), includes Leibnitz' tortuous derivation and use of integration by parts. For us today it is a trite result of the rule for differentiating a product. Archimedes' genius was required to derive results that today are simple exercises for calculus beginners, using wildly different methods. Groups where originally conceived only as sets of permutations of objects. I'm sure there are lots of other examples of very illogical historic development, superseeded by better/cleaner/simpler techniques.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sadly for true historians, I believe that in math (and other sciences) it is best to come up with some cleaned up development of the concepts. It is still very true that tracing back to the &lt;em&gt;problems&lt;/em&gt; solved, and their importance, is required. Tracing the &lt;em&gt;methods&lt;/em&gt; used, I'm not so sure.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-08T02:48:50.950" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2628" PostTypeId="2" ParentId="2625" CreationDate="2014-06-08T17:37:53.577" Score="8" Body="&lt;p&gt;One source you might look to is the following Master's thesis:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dickey, G. (2004). A Historical Approach to Teaching the British Columbia Mathematics Eight Course.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In particular, see the literature review (and its references) beginning on page six. It begins:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/v4bfW.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not recommending the actual thesis, which I have not read, but there is lots in the second chapter that could be of interest if you track down the cited works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; You can find a copy uploaded &lt;strong&gt;&lt;a href=&quot;https://www.dropbox.com/s/lva8zk6c08dhflw/history.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;. (For more on &lt;em&gt;Linking to research&lt;/em&gt; see &lt;strong&gt;&lt;a href=&quot;https://matheducators.meta.stackexchange.com/questions/89/linking-to-research/91#91&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-03-16T16:41:53.880" LastActivityDate="2014-06-08T19:59:06.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2629" PostTypeId="2" ParentId="2625" CreationDate="2014-06-08T18:48:45.547" Score="6" Body="&lt;p&gt;I'm going to point you in the direction of Realistic Mathematics Education (RME). My reason for doing so is this: you're clear that the important part of your question is the mathematical context and motivation that arise from the types of problems people faced historically when mathematics was developed (to solve those problems). You already recognize that this idea of incorporating history into mathematics can be interpreted broadly. So the question (to me)becomes, how have people envisioned something similar to what you are suggesting, which may be useful to you as a starting point to narrow your exploration? I would also say that there is the possibly to introduce you to the mathematics education research concepts (theory) that have led people to learning approaches similar to what you describe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of RME, researchers were not motivated by the idea that history provided a sensible (to experts) structure of the content. In fact, Tall (referenced in the Gravemaijer and Doorman (1999) article I provide below) specifically warns against the expert tendency to structure based on pre-existing understanding. Your reference to limit makes me think you may already be familiar with Tall's argument, because this is precisely the example he uses to characterize some pitfalls of a traditional approach to calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of expert sense-making, researchers here are motivated by what will be allow students to pursue their own sense-making, and in the process make this mathematics real to them. It does not necessarily mean real-world situations, but rather an approach that is realistically (from a student perspective) motivated. To that end, Gravemeijer and Doorman (1999) don't actually describe their approach as being motivated by the inclusion of history. Instead, they look for elements of history that they find relevant to instructional design. For example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Looking  at  the  history  of calculus  from  a modeling  perspective,  we see a &#xA;  development  of calculus  that  starts  with  modeling  problems  about  velocity &#xA;  and  distance.  Initially  these  problems  are  tackled  with  discrete  approxima- &#xA;  tions,  inscribed  by  discrete  graphs.  Later,  similar  graphs  -  initially  discrete &#xA;  and later  continuous  -  form the  basis for  more formal  calculus.  From &#xA;  a  modeling perspective,  we  could say  that  graphs  of  discrete  functions &#xA;  come to the fore as models  of situations,  in  which velocity and  distance &#xA;  vary,  while  these  graphs  later  develop  into  models  for formal  mathematical &#xA;  reasoning  about  calculus. (p. 122)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They go on to discuss the possibility of structuring student activities based on an approach that includes student modelling activity. later posing problems for investigation within the narrative of Galileo's work. But the focus is how participation in the modelling activities gives students &lt;em&gt;mathematizing&lt;/em&gt; opportunities. In the RME approach, this mathematizing is the organization of subject matter (whether it be real world information, or previously understood mathematical objects and processes) into new mathematical objects (A central RME idea of Freudenthal's (1971)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Gravemeijer and Doorman (1999) point out, the intended outcome is reification of the important understandings that underlie the content. For more details about Freudenthal's RME ideas, you could look at &lt;a href=&quot;http://p4mriunismuh.files.wordpress.com/2010/08/revisiting-mathematics-education.pdf&quot;&gt;the China lectures&lt;/a&gt; (Freudenthal, 1991) in which he looked back on RME and the mathematics education research work of the Freudenthal Institute. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Gravemeijer and Doorman paper is well-referenced, and you may find a lot further detail and research on the effect of this approach within &lt;a href=&quot;http://scholar.google.com/scholar?cites=5538273704739635679&amp;amp;as_sdt=40000005&amp;amp;sciodt=0,22&amp;amp;hl=en&quot;&gt;the list of studies that reference it (via. Google Scholar)&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Works Cited&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Freudenthal, H. (1971). Geometry between the Devil and the Deep Sea. &lt;em&gt;Educational Studies in Mathematics&lt;/em&gt;, 3(3/4), 413–435. doi:10.2307/3482035&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Freudenthal, H. (1991). Revisiting Mathematics Education: China Lectures. Retrieved from &lt;a href=&quot;http://www.eric.ed.gov/ERICWebPortal/recordDetail?accno=ED341576&quot;&gt;http://www.eric.ed.gov/ERICWebPortal/recordDetail?accno=ED341576&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.jstor.org/stable/3483163&quot;&gt;Gravemeijer, K., &amp;amp; Doorman, M. (1999). Context Problems in Realistic Mathematics Education: A Calculus Course as an Example. &lt;em&gt;Educational Studies in Mathematics&lt;/em&gt;, 39(1/3), 111–129.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-06-08T18:48:45.547" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2630" PostTypeId="1" AcceptedAnswerId="2636" CreationDate="2014-06-08T15:05:07.097" Score="1" ViewCount="290" Body="&lt;p&gt;Are there criteria, that allow one to decide, whether something, which isn't a consquence of a theorie's axioms, but is exploitet in a proof, is a further axiom and not an assumption?    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A concrete example is the statement, that in euclidean geometry, to every line and a point, that is not on that line, there exist a unique parallel which contains the point.&lt;br&gt;&#xA;What is it, that qualifies the statement as an axiom?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example would be the assumption, that the points, that are fed into a computational-geometry algorithm, are in &quot;general position&quot;; why isn't that an axiom.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I don't doubt that every mathematician knows the difference, but what if you were asked to explain it to math &quot;newbies&quot;?&lt;/p&gt;&#xA;" OwnerUserId="1681" OwnerDisplayName="Manfred Weis" LastEditorUserId="143" LastEditDate="2014-06-09T11:30:25.047" LastActivityDate="2014-06-09T19:47:07.117" Title="How to tell the Axioms from the Assumptions" Tags="|mathematical-pedagogy|axioms-foundations|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2631" PostTypeId="2" ParentId="2630" CreationDate="2014-06-09T00:00:34.327" Score="0" Body="&lt;p&gt;Axioms are general assumptions, they are taken to hold always.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The specific case you mention, &quot;points in general position&quot; is more making explicit what wasn't said: Unless I'm told that the points, say, lie on a circle, I have to take them to lie anywhere. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-09T00:00:34.327" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2632" PostTypeId="2" ParentId="2617" CreationDate="2014-06-09T02:37:22.177" Score="3" Body="&lt;p&gt;This doesn't answer the part of your question about calculator use, but having worked for Jim Kaput I feel the need to say that yes, there have been curricula designed for extensive technology use. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One such combination of technology and curriculum is the SimCalc research, which produced SimCalc MathWorlds and the curriculum that goes with it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One version of SimCalc made use of calculators (but as programmable handheld graphing and motion simulation devices rather than as calculating devices. &lt;a href=&quot;http://education.ti.com/en/us/software/details/en/286EBE56A56E43F5820664AEBEF0F485/simcalc_8384&quot; rel=&quot;nofollow&quot;&gt;That software can be found here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.kaputcenter.umassd.edu/products/curriculum_new/&quot; rel=&quot;nofollow&quot;&gt;The Algebra 1 and Algebra 2 curriculum, which can also be used with the Windows/Mac version of the software, can be found here.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What this combination of curriculum and technology were used for is summarized here:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The mathematics content of SimCalc is the mathematics of change and&#xA;  variation (Mathematical Sciences Education Board and National Research&#xA;  Council 1990), which is a core strand within US school mathematics&#xA;  (National Council of Teachers of Mathematics 2000) that is&#xA;  representationally demanding and studied at many levels (Kaput 1994).&#xA;  We have systematically tested SimCalc across a series of mathematical&#xA;  topics and have begun to produce a set of curriculum materials that&#xA;  build on research and experimentation. Such materials exploit the&#xA;  power of both parallel software and the new, classroom-based&#xA;  communication infrastructure, to facilitate rich inquiry-based&#xA;  classrooms, fueled by group based or whole-class collaborative efforts. &lt;strong&gt;(Hegedus &amp;amp; Penuel, 2008)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=Rx08SIvYMxkC&amp;amp;pg=PA354&amp;amp;lpg=PA354&amp;amp;dq=simcalc,%20its%20con&amp;amp;source=bl&amp;amp;ots=47bTOrTczE&amp;amp;sig=Inu5zEngP2c4x3w1Go4aiX7EWwc&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=3xeVU-fiEtPRsATl6oHQCQ&amp;amp;ved=0CCAQ6AEwAA#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;A recent book also summarizes the research contributions of nearly two decades of SimCalc project work.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bonus:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was one of Kaput's views that developing educational software was pointless without developing curriculum along side of it. And that it was important to the future of education to think critically about the inclusion of technology. There is a difference between a technology that someone decides to use for math education and a technology that is designed for mathematics education. We shouldn't be surprised when a technology is plopped in the classroom and fails to make a difference. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tying it back to calculators, I say &quot;plopped&quot; intentionally, because some references to calculators in classrooms do little to talk about what students might do with them that they couldn't do before. There are clearly two important aspects of technology use in mathematics classrooms that are missing when the conversation is simply about whether or not we use the technology. 1) How is that technology designed to transform a student's capabilities, opportunities, thinking, and participation in the mathematics classroom? 2) How is that technology used to transform the mathematics classroom (capabilities, opportunities, thinking, and participation), whether or not we are using it for its intended design? And when I refer to transformation, we must consider what the use of technology is replacing (what activity of the students, or other aspect of the classroom, has changed).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We should probably ask at least these questions of any technology used in math classrooms, including calculators. And if I'm going to call for that critical eye, I feel the need to point out that only part of the many possible answers to the &quot;problem of calculator use in math classrooms&quot; lies within the technology itself. It may be sensible to reject calculators if mathematics instruction equates to giving students practice doing arithmetic. To use a crude metaphor, if the point of a particular driving lesson is to teach a student shifting gears, it doesn't make sense to do that in a car with automatic transmission. However, if the point of the driving lesson is to teach a student to parallel park, or to drive to a friend's house, you may take whichever car you like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something to think about, anyhow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;EDIT: Adding a description of SimCalc Activities to clarify the use of this technology. I realize that has made this answer long. The answer proper is above this point. Below here is only for interest in SimCalc, to clarify what this sort of technology means for the mathematics classroom. Works cited are at the very bottom.&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I described in the comments, SimCalc MathWorlds is an environment for mathematics education, meaning that it contains tools used to manipulate linked mathematical representations. Students use these tools for various purposes in the context of the curriculum activities. The educational aspect of the environment comes from the students use of tools acting on mathematical representations, interpreting mathematical representations, having discussions about their own contributions and the contributions of other students, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Early research on SimCalc (late 1990's) focused on what could be done in classrooms with the linked mathematical representations. Later, classroom networks were created using TI-Navigator, and ultimately school wifi between laptops (running on the mobile laptop labs some schools were buying). The connected aspect of the software now allowed students to work briefly in private before submitting a mathematical contribution to the group discussion -- all student work would be aggregated by the teacher and projected in a public display. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often, students were given parameters that were to be used in their contributions which caused &quot;families&quot; of related functions to appear on the public display when all the student work was seen simultaneously. Students and the teacher could then discuss the patterns and what mathematical meaning could be made from them. &quot;Why are these lines all parallel?&quot; (to use a relatively simple example)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.kaputcenter.umassd.edu/projects/simcalc/cc1/materials/sackraces/&quot; rel=&quot;nofollow&quot;&gt;Some specific activities using SimCalc MathWolrds are described on this page.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My description of two of these activities:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The background story of these activities is that students are involved in an exciting race (a potato sack race is sometimes used). The technology is going to allow them to edit functions that represent their own performance in the race. The results are displayed on their handheld device (or laptop, depending on which software they're using) as a plot on a position vs. time graph. They also have an &quot;executable representation&quot; which is essentially a simple animated environment where a character will act out the race, as defined by the function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[The software itself has a number of ways to edit and represent functions and information related to functions (graphs, tables, &quot;y=&quot; symbolic representation, &quot;number sentence&quot;, animation world, and something called &quot;marks&quot;.)  Many activities depend on creative use of restricted access to these representations or how they are edited. For example, you might be able to see a character moving in the animation but you are not given his position graph. This makes sense if you are trying to determine for yourself what the function that defines his motion would look like on a graph.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first &quot;race,&quot; students are shown the teacher's function (Sack A), which starts at 0 and goes 3 ft/sec for 5 seconds. Each student will be in Sack B on their own devices. Each student has been assigned some number and they are told:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Your job is to write a linear function y = mX + b, where b = &quot;your NUMBER&quot; so that you will go at the same speed for the same amount of Time, but starting at your NUMBER. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They have an input mode that allows them to edit the y = mX + b function representation. They're given &quot;b&quot; explicitly, but have to determine start time, end time, and &quot;m&quot; from the description (or their observations) of Sack A. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once they have created their functions (position as a function of time) and seen their graphs, the functions are collected by the teacher and displayed together. Since they had a bunch of different &quot;b&quot; values, you can see why this activity is called &quot;staggered start, staggered finish.&quot; At this point, the teacher leads a whole class discussion which should include the meaning of the variation students see in the functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, students are asked to go back and create a new race. This time, they'll edit the function graphs directly, adding &quot;pieces&quot; to create a piecewise-defined position function. Their objective is to create an exciting race that begins at position=0, time=0. It must also end in a tie with a constant velocity racer who finds himself at position=12, time=10 seconds. However, along with their own version of the race they're asked to write a narrative about what happens &lt;em&gt;during&lt;/em&gt; the race. Of course, their own races will have stops and starts, backwards motion, extreme speed, extreme slowness, etc. They're essentially given free rein. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typically, students come up with an outlandish and personally interesting narrative. After the different funcitons are collected by the teacher, they are displayed and invariably are a sort of mess of non-systematic variation. Except that they all begin and end at the same point. Now students are asked to give their narrative of how their own race went. The discussion can focus on a number of things: &quot;How much mathematical language does a student use in his or her narrative?&quot; &quot;Is the narrative descriptive enough that we can actually pick out the student from among all the class' contributions?&quot; &quot;What does the shape of the graph mean at various points?&quot; &quot;Are student stories consistent with the graph shape?&quot; &quot;How does using more mathematical language contribute to other students being able to find your motion from among all other students' motions?&quot; There are many other interesting questions that arise, but these are some of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We've found, also, that students can get into mathematical arguments about the meaning of the graphs, how to modify them, the best way to describe them, etc. Rather than the teacher mediating such disagreements, there can be productive use of the representations, with students trying to convince their peers. The personal contribution aspect, I believe, is part of what draws students in to identifying with the mathematical representation they have had placed in the public space. And having a stake contributes to the motivation to argue for how that representation should be interpreted, including mathematically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students sometimes violate the rules of the activity, creating outliers. These are generally not disruptive, as they can form the basis for a discussion about the parameters of the activity and why it is impossible to hide the outlier among the rest of the class' contributions. Representations can also be edited or hidden by the teacher, if necessary. Students can even be allowed to update their contribution on the fly, from their seat, while the class looks on. All changes in the public display (or access to the ability to change) are controlled by the teacher. The teacher can even pause the activity, which locks out all screens immediately with a message directing attention back to the teacher.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Works Cited:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hegedus, S. J., &amp;amp; Penuel, W. R. (2008). Studying new forms of participation and identity in mathematics classrooms with integrated communication and representational infrastructures. &lt;em&gt;Educational Studies in Mathematics&lt;/em&gt;, 68(2), 171–183. doi:10.1007/s10649-008-9120-x&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-06-09T09:32:36.990" LastActivityDate="2014-06-09T09:32:36.990" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2633" PostTypeId="2" ParentId="2617" CreationDate="2014-06-09T03:16:27.043" Score="2" Body="&lt;p&gt;As Steven Gubkin didn't come back to post his answer from the comment at least I will write it out into a short answer:&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;&lt;a href=&quot;http://www.computerbasedmath.org/&quot; rel=&quot;nofollow&quot;&gt;ComputerBasedMath.org&lt;/a&gt;&lt;/h1&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The importance of math to jobs, society, and thinking has exploded over the last few decades. Meanwhile, math education is in worldwide crisis—diverging more and more from what's required by countries, industry, further education... and students.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;math ≠ calculating: Computers are key to bridging this chasm: only when they do the calculating is math applicable to hard questions across many contexts. Real-life math has been transformed by computer-based calculation; now mainstream math education needs this fundamental change too.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They work with this &lt;strong&gt;very&lt;/strong&gt; nice concept where they split up the default math process into four steps:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Define Questions&lt;/li&gt;&#xA;&lt;li&gt;Translate to Math&lt;/li&gt;&#xA;&lt;li&gt;Looooots of paperwork or Compute Answers&lt;/li&gt;&#xA;&lt;li&gt;Interprent&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;And their argument is that step 3 is taking extremely much time in the current system, whilst in reality all steps deserve a lot of attention.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Their progress&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Right now they don't have an entire curriculum developed, but they have developed a framework and sample materials and are campaigning for policy makers to build upon this framework and collaborate on creating a fully functional curriculum. &lt;/p&gt;&#xA;" OwnerUserId="860" LastActivityDate="2014-06-09T03:16:27.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2634" PostTypeId="2" ParentId="2630" CreationDate="2014-06-09T13:20:31.073" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Of course, I don't doubt that every mathematician knows the difference, but what if you were asked to explain it to math &quot;newbies&quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are some ways to approach this for math newbies, one more 'to the bone' than the other.  Which explanation you choose depends on the expected level of math proficiency (or tolerance).&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Axioms are atomic, indivisible building blocks.  They are the basic rules with which we build a system.  Because axioms &lt;strong&gt;create&lt;/strong&gt; the system, the system itself cannot meaningfully prove them.  That would be like proving blue is prettier than red by assuming blue is prettier than red.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Axioms are the what-if genesis of a story.  What if there's only one line parallel to another through a given point?  What kind of story (mathematical system) does that imply?  Like in a story, there is no benefit in trying to prove the genesis: the Harry Potter series starts with &quot;there are wizards;&quot; it's axiomatic to the story.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Axioms are like types of Lego blocks: all of the tall 2x2 blocks are an axiom, and all of the flat 1x4 are an axiom, and so on.  With these types of blocks, you can build structures (theorems).  A proof of such a theorem is like the blueprint of a structure.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there criteria, that allow one to decide, whether something, which isn't a consquence of a theorie's axioms, but is exploitet in a proof, is a further axiom and not an assumption?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The difference between an axiom and an assumption is contextual and fudgeable, and not really set in stone.  Assumptions that are considered axiomatic in one field of study may be contestable in other fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Loosely speaking, an assumption is a statement you take to be true with no proof that it is so, so it could be false.  An axiom is the same, but it is essential to your system that it &lt;em&gt;not be false&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1193" LastActivityDate="2014-06-09T13:20:31.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2635" PostTypeId="1" AcceptedAnswerId="2638" CreationDate="2014-06-09T17:49:44.133" Score="15" ViewCount="202" Body="&lt;p&gt;&lt;strong&gt;How do we get students to see their own explanations as valuable to their learning?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, talking can help a student uncover misconceptions that might otherwise remain unexplored until much later.  Students can also benefit from examining the misconceptions of their peers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The act of participating reinforces how valuable students' contributions are.  With feedback from an instructor, they can also see how mathematical their contributions are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, students often express a belief that a good math class is one where the teacher or book gives good explanations, rather than one that requires students to explain.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What type of activity in the classroom have you seen that helps students express their mathematical thinking, and helps show it as valuable?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorDisplayName="user173" LastEditDate="2014-06-09T22:26:41.373" LastActivityDate="2014-06-10T03:56:51.383" Title="Strategies for encouraging student discussion, explaining, and argumentation about mathematics" Tags="|class-participation|classroom-interaction|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2636" PostTypeId="2" ParentId="2630" CreationDate="2014-06-09T19:47:07.117" Score="3" Body="&lt;p&gt;I am not completely sure I understand the distinction you are making between &quot;axioms&quot; and &quot;assumptions&quot;, so this response may be missing the mark, but here is my interpretation of the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the following example of a theorem in, say, ring theory:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $u$ be an invertible element in a ring $R$.  Then...  (something something something)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now the assertion that $u$ is an invertible element is an assumption of the theorem; it is the hypothesis that logically entails a conclusion.  Not all elements in a ring have multiplicative inverses, but for the sake of this theorem (whatever it is) we are thinking about an element that &lt;em&gt;does&lt;/em&gt; have a multiplicative inverse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, &lt;em&gt;every&lt;/em&gt; element of a ring has an &lt;em&gt;additive&lt;/em&gt; inverse; that is built into the &lt;em&gt;axioms&lt;/em&gt; of a ring, and as soon as we say we are thinking about a ring those axioms are implicitly invoked.  So the fact that $-u$ exists is not an assumption, it is an axiom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this is the kind of distinction you have in mind, then I would say that the difference between axioms and assumptions has to do with whether the assumptions are local and temporary -- the conditions under which a certain conclusion holds -- or whether they are global and stable -- part of the conditions that define the sort of object we are thinking about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is of course some gray area in between.  If we are studying general rings, then asserting that $R$ is a commutative ring might be a (local) assumption for a particular theorem or a particular set of theorems.  But if we say in that specialized theory for a while, it might eventually become part of the background context and appear more like an axiom of the sub-theory.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-06-09T19:47:07.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2637" PostTypeId="1" CreationDate="2014-06-09T20:03:29.387" Score="8" ViewCount="358" Body="&lt;p&gt;I'll be giving a guest lecture in a grade 12 calculus class.&#xA;I'm going to give them an introduction to machine learning, and go through some of the math involved in training an &lt;a href=&quot;http://en.wikipedia.org/wiki/ADALINE&quot; rel=&quot;nofollow&quot;&gt;adaline&lt;/a&gt;. It would be helpful to have some simple examples of problems that could be solved with such a technique.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Give an example, either practical or fun, of a binary decision that depends on one or two real valued variables.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For example, an email filter might mark as spam a message which contains the words &lt;code&gt;nigerian&lt;/code&gt; and &lt;code&gt;prince&lt;/code&gt;.&lt;/p&gt;&#xA;" OwnerUserId="538" LastEditorUserId="538" LastEditDate="2014-06-10T03:17:35.850" LastActivityDate="2014-06-14T18:30:45.770" Title="What is a good teaching example of machine learning?" Tags="|calculus|secondary-education|examples|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2638" PostTypeId="2" ParentId="2635" CreationDate="2014-06-10T03:56:51.383" Score="9" Body="&lt;p&gt;I will try, though I am a biologist.&#xA;The main difference between your question and the one linked in the comments seems to be that you want students to realize that articulating mathematics leads to better learning of mathematics. Not just being willing to answer questions, but talking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would recommend a technique that can be done with either clickers or a paper &quot;quiz.&quot; I'll present it as a clicker technique, but the same things can be done with a short 3-5 question quiz.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Pose a somewhat difficult clicker problem, expected accuracy at perhaps around 60%. &lt;/li&gt;&#xA;&lt;li&gt;Students work independently and submit answer via clicker.&lt;/li&gt;&#xA;&lt;li&gt;Students are then encouraged to discuss in pairs or small groups of three. They are allowed to resubmit a new answer.&lt;/li&gt;&#xA;&lt;li&gt;Students are then given an isomorphic question (same concept, slightly different presentation), and asked to independently solve and submit again.&lt;/li&gt;&#xA;&lt;li&gt;Instructor points out that student answers overall have improved- indicating that talking about mathematics helps organize thinking and uncovers confusions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A paper in Science using this technique shows that students who miss the first question but are given the opportunity to discuss and then answer an isomorphic question improve more than students who do not discuss (reference below, open-source).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You would still have to do the sell, of course. Students tend to avoid expending extra calories when possible and will hope to avoid serious thinking during class. But with practice (and exam questions similar to the clicker questions), motivation will rise. It's more fun to be an instructor too - students talking! about the subject!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Smith, M. K., Wood, W. B., Adams, W. K., Wieman, C., Knight, J. K., Guild, N., &amp;amp; Su, T. T. (2009). Why peer discussion improves student performance on in-class concept questions. Science, 323(5910), 122-124.&lt;/p&gt;&#xA;" OwnerUserId="1207" LastActivityDate="2014-06-10T03:56:51.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2639" PostTypeId="2" ParentId="2637" CreationDate="2014-06-10T07:30:17.700" Score="4" Body="&lt;p&gt;I'm not sure that neural network is the best example of machine learning, or that spam filters are good example of neural network usage. In particular, a decision that depends on &quot;one or two real valued variables&quot; seem neither practical nor fun. You can easily create a more sophisticated network, but there are still some issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Discrete things like &quot;contains words $X$&quot; cause problems because you would need to have a separate input for each (common) word indicating if it is in the message. If you insist, you can find some info about spam filter networks &lt;a href=&quot;https://stackoverflow.com/q/770238/2260841&quot;&gt;in this question&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also neural networks are good only in some range of applications, and so it happens that humans are also quite good in number of them (guess why...), so that neural network machine learning doesn't seem that impressive (e.g. &lt;a href=&quot;http://en.wikipedia.org/wiki/Random_forest&quot; rel=&quot;nofollow noreferrer&quot;&gt;random forests&lt;/a&gt; give nice results and are easy to understand, while &lt;a href=&quot;http://en.wikipedia.org/wiki/Bayesian_classifier&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bayes-based classifiers&lt;/a&gt; may have some educational advantage).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, I would recommend anything based on pattern matching, e.g.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;approximating a function, a visualization how the network learns is often engaging, there are tons of examples online;&lt;/li&gt;&#xA;&lt;li&gt;character recognition (simple OCR), the simplest feature set is take a random set of short lines, if you would like to use something closer to human visual system, you can use &lt;a href=&quot;http://en.wikipedia.org/wiki/Gabor_filter&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gabor stimuli&lt;/a&gt;;&lt;/li&gt;&#xA;&lt;li&gt;voice recognition, I've never did this, but it seems possible, you could record the voices of your students and then train the network to recognize them, you could also try to guess the gender of the speaker (e.g. based on the histogram).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also, to indicate how the neural networks works, you could do some weird things, like guessing the color of the hair based on voice. Then explain that it's reasonable to suspect there is no correlation between the two things in general (esp. if some girls use hair dye), but the classifier learned the voices of the people in class and approximately who has which color, it would fail on someone out of the training set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-05-23T12:41:54.857" LastActivityDate="2014-06-10T07:35:22.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2640" PostTypeId="2" ParentId="2637" CreationDate="2014-06-10T10:12:52.667" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://class.coursera.org/ml-005&quot; rel=&quot;nofollow&quot;&gt;Coursera's Machine Learning Course by Andrew Ng&lt;/a&gt; used two examples to motivate the training of neural networks:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Programming of Logic gates, such as AND, OR, XNOR and so on - the latter requires two layers. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Number/character recognition. This was demonstrated with a video from Yann LeCuns NCR for AT&amp;amp;T from the 1990's as well as a programming exercise.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="108" LastActivityDate="2014-06-10T10:12:52.667" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2641" PostTypeId="1" AcceptedAnswerId="2642" CreationDate="2014-06-10T16:30:03.807" Score="5" ViewCount="1058" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been studying undergraduate philosophy for the last two years. It seemed to me that philosophers could (and I suspect that many do) use mathematical methods, other than logic, to  investigate some philosophical matters more efficiently. On that thought, I decided to pursue a double major in mathematics and philosophy, doing so involves completing a remedial elementary-algebra course during the summer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The Problem&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Success in this course seems to involve quick and correct usage of the problem solving techniques taught in class. Accordingly, at the end of each class, the instructor assigns about 70 practice problems whose solutions involve correctly using one of the two to three methods taught in that class. Usually, I can easily solve all but a few of the problems. When I find problems that I can't solve, I contemplate why my thinking produced an incorrect solution, and adjust my understanding accordingly. However, the overall repetitiveness of the practice problems bores me so much that I often don't bother to solve all of them. I'd enjoy the homework more if it comprised questions like 'Using principles A, B, and C, prove and explain why method Y is the most efficient method of solving problems of form X' (which I do when I find a problem that I can't easily answer). Regardless, in order to study more math, I need to do well in this class, and doing well in the class involves doing well on the exams, which involves quickly and correctly using the problem solving techniques taught in class, and being able to do that involves a lot of repetitive problem solving, which I can't motivate myself to do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The Question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would motivate me to know that success in future math courses depends more on understanding principles and deducing their implications than on rote calculation.&#xA;Is that the case? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How else could a person motivate themselves to do these problems?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you. &lt;/p&gt;&#xA;" OwnerUserId="961" LastEditorUserId="961" LastEditDate="2014-06-10T18:10:16.953" LastActivityDate="2014-06-23T19:36:05.180" Title="Motivation for practicing algebra" Tags="|student-motivation|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2642" PostTypeId="2" ParentId="2641" CreationDate="2014-06-10T17:38:33.473" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;It would motivate me to know that success in future math courses depends more on understanding principles and deducing their implications than on rote calculation. Is that the case? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Yes&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Honestly I don't even know what more I can say about this.  Rote calculation is the lowest-level (and most maligned) form of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately if you are starting out with remedial algebra then you have a long, tedious slog ahead of you before you get to the interesting stuff, and you might want to take matters into your own hands and learn some &quot;real&quot; math independently while simultaneously pursuing your entry-level course work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I also think you should make sure you speak to a college advisor about your double-major plans...  If you are in your third year at a university and just starting out with elementary remedial algebra, it may not be possible to complete the graduation requirements for a mathematics major in fewer than four more years.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe start by reading Hardy's &lt;em&gt;A Mathematician's Apology&lt;/em&gt; to get a feeling for what the subject is all about.  You might also enjoy the book &lt;em&gt;Gödel, Escher, Bach&lt;/em&gt; by Douglas Hoftstadter.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-06-10T17:38:33.473" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2643" PostTypeId="2" ParentId="2641" CreationDate="2014-06-10T19:19:15.900" Score="4" Body="&lt;p&gt;As you are following a definitely non-standard track, you should schedule an in-depth talk with your advisor, and with somebody in such a capacity at the math department. Be prepared, have a reasoned outline of where you want to go and why. Take a look at the math courses offered, set up a sequence of courses to take and be prepared to defend it (and also to change it if the suggestions to do so convince you). Do it soon, changing direction in mid-study is painful; the more time goes by the worse.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-10T19:19:15.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2644" PostTypeId="2" ParentId="2585" CreationDate="2014-06-11T04:00:32.140" Score="2" Body="&lt;p&gt;I can't back this up, but here's a thought as a mathematician and father of school aged kids (in order that the benchmarks arise):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1: Represents the whole idea of what a number is. Once you get 1, you just have to memorize 2, 3,..., 9.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0: Represents understanding that nothing is a quantity/number, too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;10: At first &quot;10&quot; is just another symbol for a number like &quot;7&quot;. But if you really get that it's a 1 and a 0, then the symbols 11,...,99 become immediately understandable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;100: Understanding &quot;ten&quot; is one thing. The next step is understanding that there must be a new name for ten 10s. Once you get &quot;hundred&quot;, then &quot;thousand&quot;, &quot;ten thousand&quot;, &quot;million&quot;, etc. become memorization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1/2: Being able to truly understand 1/2 means you get what fractions are. I know students really struggle with fractions, but it all starts with 1/2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1/10: Once you get fractions, the question of decimal representation is natural. So I'm guessing 1/10 should really mean understanding 0.1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;12: A bit of an odd ball on the list. My guess is one of two possibilities: It's important because most students memorize multiplication tables to 12x12, or because in English, &quot;twelve&quot; is the last number whose name tells you nothing about its decimal representation, e.g. Maybe it should've been called &quot;seconteen&quot;.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-06-11T04:00:32.140" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2645" PostTypeId="1" AcceptedAnswerId="2646" CreationDate="2014-06-11T05:00:31.453" Score="13" ViewCount="1343" Body="&lt;p&gt;I am currently tutoring a student for the &lt;a href=&quot;http://www.nysedregents.org/integratedalgebra/&quot;&gt;Integrated Algebra regents&lt;/a&gt;, and he's really doing great.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's his father that bothers me.&lt;br&gt;&#xA;He does do some math (he's an accountant or actuary of some sort), but doesn't have the time to help his son. Whatever. But whenever I meet him, he makes an argument that I should not be focusing so much on the multiple choice, but more on the free-form questions at the end. &lt;em&gt;(Every. Single. Time.)&lt;/em&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;FWIW, I think he's wrong for a few reasons: the way I teach, I teach math over test method (so I'm really teaching both whenever I tutor); the argument that they're worth more is really false (30 multiple choice * 2 points each = 60 points, which &lt;a href=&quot;http://www.nysedregents.org/integratedalgebra/613/ialg62013-ccrev.pdf&quot;&gt;puts us in the 80's&lt;/a&gt;); and I'm really just grading my student's practice tests, on which (taking the last one for an example) he gets &lt;em&gt;all&lt;/em&gt; the free-form questions right, and really needs the most help on the multiple choice; on the free-form if you make a mistake, you can get partial credit, unlike multiple choice; and a few other reasons.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have given the father most of these reasons at one point or another, in a rational and respectful manner (he is, after all, the one paying me;), but I would like some advice on how to tell him to let me do the job he hired me to do without repeated, unsolicited and unhelpful input from him.&lt;hr&gt;&#xA;&lt;em&gt;Bonus: I'm not too closed-minded to think that my way is the only way. Does my employer have any saving grace &lt;strong&gt;specifically&lt;/strong&gt; in his arguments that more focus should be placed on the final sections of this test?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="1685" LastEditorUserId="1685" LastEditDate="2014-06-20T19:34:29.793" LastActivityDate="2014-06-20T19:34:29.793" Title="Tutor: How to deal with meddling parent?" Tags="|tutoring|testing|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="2646" PostTypeId="2" ParentId="2645" CreationDate="2014-06-11T06:11:21.620" Score="12" Body="&lt;p&gt;It might help to review a written summary of the student's progress, e.g.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\begin{array}{c|cc}&#xA; &amp;amp; \text{Multiple-Choice} &amp;amp; \text{Free-Form} \\&#xA;\hline&#xA;\text{Session #1} &amp;amp; 50\% &amp;amp; 100\% \\&#xA;\text{Session #5} &amp;amp; 75\% &amp;amp; 100\% \\&#xA;\end{array}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the father sees that and prefers that you focus on the free-form questions, that is his prerogative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However: the free-form problems are intended to test a broader range of skills.  So this pattern of getting more free-form than multiple-choice problems right seems odd, and calls for explanation.  What would the numbers look like if you graded the free-form answers more harshly?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My guess is that the father believes that the free-form work will help his son in contexts beyond this test.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-06-11T06:11:21.620" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2647" PostTypeId="2" ParentId="2645" CreationDate="2014-06-11T11:40:33.650" Score="8" Body="&lt;p&gt;I believe there is a serious issue of talking past each other here. As MattF. hints at, I'm sure the parent has his reasons, which you should ask him about. You can't know if he is right or not without knowing what he thinks, and even less convince him he is wrong (if he in fact is). As you tell it, you have explained your reasons, but it seems they weren't understood either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps he is looking not for training for a specific set of tests, but more broad learning?&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-11T11:40:33.650" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2648" PostTypeId="1" CreationDate="2014-06-11T16:14:29.787" Score="8" ViewCount="725" Body="&lt;p&gt;I was helping my kid study for a precalculus examination and looking at her old tests from the year, and came across a question about vectors. Below is the typed up version of the question and her answer; an image from the actual test can be found at the end.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;6.&lt;/strong&gt; A vector with tail at the origin has a magnitude of &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and a direction angle of &lt;span class=&quot;math-container&quot;&gt;$\pi/3$&lt;/span&gt;. When trying to find the coordinates of the head, Nick says, &amp;quot;That's easy. It's just &lt;span class=&quot;math-container&quot;&gt;$r \cos \theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r \sin \theta$&lt;/span&gt;; we've been doing this forever now!&amp;quot; Olathe counters, &amp;quot;No, we should use the idea of components to find out how much our vector goes along each axis.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Who's correct here? Give a thorough, mathematically sound, explanation!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The student writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nick is correct. As long as we have the magnitude and directional angle, and know that the tail is located at the origin, the method he proposed is suitable. (&lt;em&gt;Crossed out: Olathe is correct. Even though we have the magnitude and directional &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; we can't&lt;/em&gt;) Components are unnecessary b/c we know the tail is at the origin, we know how far out the vector goes, and we know the &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; it makes with the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm not sure how thorough this needed to be, but I agree with the kid.  Or I agree that both approaches are equivalent - maybe that was the point.  Would any high school math educators care to weigh in?&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Cw437.png&quot; alt=&quot;Image&quot; /&gt;&lt;/p&gt;&#xA;" OwnerUserId="1690" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-06-16T12:31:04.167" Title="Vector problems in high school precalculus" Tags="|grading|precalculus|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2649" PostTypeId="2" ParentId="2648" CreationDate="2014-06-11T16:42:14.807" Score="7" Body="&lt;p&gt;I feel that the student demonstrated the relevant understanding here, but my guess is that the teacher was looking for some acknowledgement of the validity of Olathe's point of view and/or the equivalence of the two points of view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This type of question is tricky to grade for precisely this reason. It seems to me the question was designed to elicit discussion of the relationship between the two points of view, and points were deducted because the answer didn't do this. This is hazy judgement territory though because (IMHO) it was reasonable for the student to take the language of the question at face value and opine about whose point of view was better, especially since they referenced the key fact that the tail is at the origin which makes Nick's method valid.&lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2014-06-11T16:42:14.807" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2650" PostTypeId="2" ParentId="2648" CreationDate="2014-06-11T17:22:40.640" Score="2" Body="&lt;p&gt;The analysis of Nick's approach is ok, although the first part is just a shortened repetition of the information given in the task.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question indeed asks for an analysis of both approaches. A comparison of both approaches isn't necessary, although an equivalence argument can be used instead of the second analysis.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-06-11T17:22:40.640" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2651" PostTypeId="1" AcceptedAnswerId="2657" CreationDate="2014-06-11T17:30:33.717" Score="6" ViewCount="308" Body="&lt;p&gt;Assume you have decided to call on students to present HW solutions in your course. For educators who have done this before, how do you implement this and why? There are a few problems I can imagine an instructor having&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;making sure every student presents at least once (or a similar number of times compared to other students)&lt;/li&gt;&#xA;&lt;li&gt;having it appear random so that students don't correctly guess when they are going to be called.&lt;/li&gt;&#xA;&lt;li&gt;making sure students that have presented don't lose motivation to do future assignments because they think they won't be called on again (either because they guessed your algorithm correctly, or because of some incorrect gamblers fallacy logic)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So what do you do to avoid these problems? I have never called on students before, but am considering it next semester. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; this is not a discussion on whether someone &lt;strong&gt;ought&lt;/strong&gt; to call on students. There are obviously major drawbacks and advantages to this approach.&lt;/p&gt;&#xA;" OwnerUserId="1387" LastEditorUserId="143" LastEditDate="2014-06-14T12:45:57.810" LastActivityDate="2014-06-14T12:45:57.810" Title="How to randomly call on students to present HW? (without students losing motivation , and high probability everyone presents)" Tags="|homework|classroom-management|grading|class-participation|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2652" PostTypeId="2" ParentId="2651" CreationDate="2014-06-11T20:28:48.900" Score="3" Body="&lt;p&gt;Doing this requires an impossibly large number of homework sets or a large proportion presenting each time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I do is to have everybody turn in homework, and each time a &quot;random&quot; selection (of around five) gets their grades by explaining what they turned in to the TAs, not by what they turned in. I'm not interested if they got the answers off MSE or Wikipedia, I want them to understand their answers. TAs tend to select suspected cheaters...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-11T20:28:48.900" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2653" PostTypeId="1" AcceptedAnswerId="2659" CreationDate="2014-06-11T22:19:18.157" Score="15" ViewCount="1251" Body="&lt;p&gt;I am considering a new HW policy that I haven't seen elsewhere. Each week the student must turn in $n$ problems, of her choosing. There will be no HW score beyond documenting whether the student is turning in honest attempts at these $n$ problems. (&lt;s&gt;my general thought would be to make each HW contain $n$ challenging problems, a few medium problems, and $n-1$ easy problems&lt;/s&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student chooses to do so, they circle one of their solutions and the &quot;grader&quot; reviews it and gives the student very detailed feedback, including comments on the correctness and clarity of each step, the writing, and one take home message they should work on. No numerical grade would be assigned here, it is purely for feedback purposes, and it is optional (although there is no reason not to choose this option).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is my desired outcome&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;students do the problems that will allow them to learn the most. Struggling students can work on the easier problems and get feedback on them before moving to harder ones.&lt;/li&gt;&#xA;&lt;li&gt;stronger students can turn in the hardest $n$ problems and choose to get feedback on the one they found most challenging (but still had a solution for), without wasting time on the problems they found trivial.&lt;/li&gt;&#xA;&lt;li&gt;the lack of a numerical grade tied to the HW (beyond completion) seems to increase learning. See the answer to: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/712/is-it-worth-grading-calculus-homework/2267#2267&quot;&gt;Is it worth grading calculus homework?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;by choosing the problems that they will receive feedback on, they are learning how to identify when they aren't sure about a solution and when they are confident in it. Being able to identify you don't know something or when you think your argument is shaky is one of the most important skills in mathematics.&lt;/li&gt;&#xA;&lt;li&gt;students will see the value in HW as a learning tool rather than something they need to do just to get a grade.&lt;/li&gt;&#xA;&lt;li&gt;students are learning to own and take pride in their struggle. By circling the question they most want feedback on, I think it reinforces them to constantly be trying to learn more and not focus on the things they already know well. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What I want to avoid (possible drawbacks)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;non-struggling students doing easy problems just to quickly turn in the assignment. They end up learning less than in the traditional HW setting and do poorly on the tests because they haven't gotten feedback on problems of the appropriate difficulty level.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The goal is to make HW purely a learning experience that everyone can get the most out of but to also move away from a grade obsessed classroom culture. What are the advantages and disadvantages of this HW policy? and what changed to overcome some of the disadvantages?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; another way of breaking down the questions (instead of easy/hard) is to use standards based grading as in this question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2141/standards-based-grading-in-calculus&quot;&gt;Standards-based grading in calculus&lt;/a&gt;. So I'd mark a question as &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;computation - applying formulas and rules used in this class,&lt;/li&gt;&#xA;&lt;li&gt;application - abstracting a concrete problem and figuring out which mathematical tools are most useful to solve it, &lt;/li&gt;&#xA;&lt;li&gt;adaptation - changing a formula or rule to fit a new, but similar type of problem &lt;/li&gt;&#xA;&lt;li&gt;theory - demonstrating a deep, theoretical, understanding of a concept presented in this class&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And I might have a requirement that they must do problems from of at least 3/4 types and not tell them which ones are the hardest.&lt;/p&gt;&#xA;" OwnerUserId="1387" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2022-03-13T14:52:50.143" Title="What are the pros and cons of having students choose which HW problems are looked at?" Tags="|homework|grading|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2654" PostTypeId="2" ParentId="2653" CreationDate="2014-06-11T23:03:56.977" Score="8" Body="&lt;p&gt;Knowing (a little) of how student's brains work, you'll get $n - 1$ &quot;easy&quot; problem attempts + 1 &quot;medium&quot; problem attempt each time, period.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have to force their hand a bit. Perhaps give extra credit for turning in harder problems or some such. Or make it a requirement that $x$ number of hard problems get turned in for getting a homework grade (where $x$ is, say, half the number of programmed homework sets).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You want to give weaker students extra oportunities/homework. But you have to consider that students that are weak in your class are probably also struggling elsewhere, so just piling more work on them won't help. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-11T23:03:56.977" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2655" PostTypeId="2" ParentId="2612" CreationDate="2014-06-12T01:42:21.683" Score="8" Body="&lt;p&gt;Galois Theory is the place where insights from one field (structure of groups) impacts another field (study of solutions of polynomial equations). I think it's the only time undergraduate students study such a phenomenon- certainly it's a classical and profound example of the interconnectedness of ideas.&lt;/p&gt;&#xA;&lt;p&gt;I would argue on that basis that Galois Theory is in fact be the ideal course at LSE. You see, the motto of LSE is &amp;quot;rerum cognoscere causas&amp;quot;, meaning &amp;quot;to know the causes of things&amp;quot;. In all other undergraduate math courses, the causes are &amp;quot;near&amp;quot;, &amp;quot;right under our noses&amp;quot;. But not in Galois Theory. In Galois Theory, to find the `cause' one has to look deeper! And that's the whole raison d'etre for LSE, is it not?&lt;/p&gt;&#xA;&lt;p&gt;So I would not argue for Galois Theory on the basis of some exotic social science or economic application that might or might not exist, but rather on the basis of it being the very embodiment of the LSE motto!&lt;/p&gt;&#xA;" OwnerUserId="185" LastEditorUserId="470" LastEditDate="2021-02-28T17:21:22.103" LastActivityDate="2021-02-28T17:21:22.103" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2656" PostTypeId="2" ParentId="1294" CreationDate="2014-06-12T02:21:47.873" Score="1" Body="&lt;p&gt;If you're looking for examples you can show to students without any background in computer science, consider the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;For geometric series, family trees. Each person has two parents, and each parent has two parents, and each grandparent also has two parents, and so on. This is an example of sums of powers of two, which is a geometric series.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For arithmetic series, acceleration. Every second, the number of centimetres per second an object moves increases by a constant amount.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="856" LastActivityDate="2014-06-12T02:21:47.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2657" PostTypeId="2" ParentId="2651" CreationDate="2014-06-12T03:21:47.593" Score="4" Body="&lt;p&gt;I do the following in my Pre-Calc and first-semester Calculus course:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assignments: A week's worth, so usually 3 sections with 8-15 problems per (depending on difficulty).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Homework Day: About once a week, students are called randomly to present* their solution on the board to an assigned problem. About third to a half of the problems are pre-selected (by me, unknown to students) to be done this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;* Here &quot;presents&quot; means writes a full complete solution on a section of the board that can be understood without any additional verbal exposition (though I sometimes ask clarifying questions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Random Method (Concept): Conceptually the idea is that at the beginning of each Homework Day, every student's name is added into a hat. Names are drawn one by one and we go down a sublist of assigned problems. For students whose names are not called that period, their name remains in the hat so that next time (after adding everyone's name again) they will be in there twice, and so on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Random Method (Implementation): I do all the randomization using Mathematica to manipulate a list of student names and a list of problem numbers. This is then printed off and I just read down my list. The &quot;hat&quot; is a Mathematica file that I update throughout the semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Adjustments: Sometimes if I have a lot of left-over names already in &quot;the hat&quot; I won't add everyone's name again. But I don't tell the students this. If I have more problems then students (which happens in smaller classes) then I'll add names twice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Results: Every student has a possibility of getting called on at least once, so all are motivated to be ready. Also, every time a student doesn't get called on, they are more likely to get called on next time. I have had students go 3 or 4 times without getting called and then for the next couple Homework Days they end up presenting multiple times each day. The longer they go without presenting, the more ready they realize they need to be. In a semester with 13 homework days and ~25 students, by the end everybody has presented about the same number of times, maybe +/-2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grading: Since not everyone presents the same amount, points are given out of the number of times they were called and then this percentage entered for the homework grade. Also, since I really want students to show work so we can learn from it, the rubric is as follows: 4 pts for a solution (even if wrong), 3 pts for a solution to a different assigned problem (with permission, even if wrong), 1 pt for being there, 0 pts if absent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Extra Credit: Some homework problems are hard and students need to pass. If 4 students in a row pass and/or ask to do a different problem (for the deduction), then the hard problem becomes extra credit and I ask for a volunteer. This is no longer random.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additional Notes: This is way better than grading papers and it seems to benefit students more or by the same (but certainly not less). It's good for classes were the concepts are really new (e.g. Calc I, Pre-Calc, even Number Theory), but I wouldn't recommend it for classes where the problems can be long, tedious, and/or computationally heavy (Calc II, Diff EQ, Linear Algebra).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-06-12T03:21:47.593" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2658" PostTypeId="2" ParentId="2612" CreationDate="2014-06-12T11:19:32.473" Score="2" Body="&lt;p&gt;I'd start by asking:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What would a class on Field Theory look like, if designed for students&#xA;  concentrating in Mathematics and Economics?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(Note that at LSE, the only concentrations with math in the name are&#xA;  &quot;Business Mathematics and Statistics&quot;, &quot;Mathematics and Economics&quot;, or &quot;Mathematics with Economics&quot;.) &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As prerequisite we assume a first &lt;a href=&quot;http://www.lse.ac.uk/resources/calendar/courseGuides/MA/2013_MA211.htm&quot; rel=&quot;nofollow&quot;&gt;course&lt;/a&gt; in Algebra and Number Theory.  From there we can see options in terms of the mathematics subject classification, e.g.:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;12D10 (location of zeroes)&lt;/li&gt;&#xA;&lt;li&gt;12F10 (Separable extensions, Galois theory)&lt;/li&gt;&#xA;&lt;li&gt;12L05 (decidability)&lt;/li&gt;&#xA;&lt;li&gt;12Y05 (computational aspects)&lt;/li&gt;&#xA;&lt;li&gt;14P10 (semialgebraic sets)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The class could start with definitions and examples.  The route to Galois theory would then focus on finite extensions of $\mathbf{Q}$, but that route is not necessary.  Another route would focus on $\mathbf{R}[x]$, and real algebraic computation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The highlights of that syllabus could include:  Sturm's algorithm for finding real roots.  Tarski's method for deciding the existence of points with algebraic properties.  An application to the economic problem of optimizing a quadratic utility function under algebraic constraints.  &lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-06-12T11:48:53.167" LastActivityDate="2014-06-12T11:48:53.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2659" PostTypeId="2" ParentId="2653" CreationDate="2014-06-12T16:13:31.260" Score="9" Body="&lt;p&gt;&lt;em&gt;Question 1&lt;/em&gt;: Will you tell your students which are the easy problems and which are the hard ones? &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If yes, you get the &lt;a href=&quot;https://matheducators.stackexchange.com/a/2654/125&quot;&gt;vonbrand scenario&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;If not, how will the &quot;strong students&quot; know which are the challenging questions to work on? &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Question 2&lt;/em&gt;: how much mathematical maturity do you expect in these students? &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are talking about junior or senior level college students in a subject they are doing their major in (so computer scientists in an algorithm class, or pure math majors in a topology class), then your scheme of having students figure out which questions to ask for help on may work. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But if you are talking about more introductory or &quot;service&quot; classes, my experience&lt;sup&gt;1&lt;/sup&gt; suggests that oftentimes the students do not know what they don't know. They &lt;em&gt;may&lt;/em&gt; dismiss a question as easy and solved, and be completely wrong about their solution. The policy of only one question being read for each set may be indirectly punishing the students for something that is not really any fault of their own. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Question 3&lt;/em&gt;: Will you supplement the homework assignments with detailed solutions? &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If yes, why bother asking the students to circle a question to be read? They can just compare against the model solution. (And the TA will thank you for it, even if it is the TA's job to write up the solution sheets.)&lt;/li&gt;&#xA;&lt;li&gt;If not, how will students learn if they have more than 1 question they don't know how to do? &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Some suggested modifications:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Don't make the problem sets too long. Keep $n$ small such that students at least have time to try every problem. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Don't tell them which are the hard problems. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Grader grades each question turned in on the scale of 0 to 2, where 0 is effectively blank/little effort shown, 2 is mostly correct, and 1 is good try but some mistakes. Grader is not require to comment. At the end, the grader totals up the scores, any sum bigger than $k$ for some $k$ depending on $n$ gets a pass on the homework. Your description of the desired outcome suggests that $k \approx n$ is a good choice. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Publish solutions promptly after the due date of the homework. (Ideally written by the person who will be fielding questions about the official solutions.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have someone (grader? you?) field any and all &lt;em&gt;specific&lt;/em&gt; questions about the solutions, after they are published. (But note, not necessarily after the students get their work back from the grader.) (And note, don't answer questions of the type: &quot;Can you show me?&quot; Make sure that they read the solutions and formulate a clear question first.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Why?&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;By not telling the students which are the easy problems, they will most likely, on average, have to try working on a few of the hard problems before they discover it is beyond their understanding or ability. This will at least show them what they don't know. (And knowing that is a good thing.) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;By grading them on a scale of 0,1, 2, you (and the grader) are given the opportunity to indicate to them that something they thought they know is perhaps something they don't know very well. (Clears up misconceptions.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;By publishing the solutions and letting the students work through them before asking questions, and with the possible additional feedback from the grading, the students actually develop the mathematical maturity to appreciate, without your help, what they know and what they don't know. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; FWIW, in my previous job homework is not graded, and I discuss the students' works with them in a small group (two or three students at a time) setting. In my current institution homework is also not graded, while we have weekly exercise sessions during which the students can work on the homework and ask for help from TAs. &lt;/p&gt;&#xA;" OwnerUserId="125" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-06-12T16:13:31.260" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2660" PostTypeId="2" ParentId="2653" CreationDate="2014-06-12T21:37:54.243" Score="4" Body="&lt;p&gt;This is more appropriate for a comment, but I lack sufficient rep to post it as such.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your concern that non-struggling students will do easy problems in order to quickly finish the assignment is valid. One way around this that I have seen work well is to assign each problem a number of points based on difficulty and require a fixed number of points worth of problems be turned in for each homework set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The advantage is that you can tune your point distribution each week, based on the previous week's results, until you have students consistently working on problems appropriate to their level. &lt;/p&gt;&#xA;" OwnerUserId="1154" LastActivityDate="2014-06-12T21:37:54.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2661" PostTypeId="2" ParentId="2612" CreationDate="2014-06-14T00:42:00.833" Score="5" Body="&lt;p&gt;I'm going to claim that there is no need to teach Galois theory to students at LSE.  Of course there's a need for Galois Theory in the world (in the same way that literature, music and painting are necessary) but I can't see why a student in applicable mathematics would choose this over areas like: operations research, statistics, game theory, econometrics, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the question is if there are any applications of Galois theory in the &quot;real world&quot;.  I can say that Koblitz curves are used in elliptic curve cryptography (see the NIST recommendations &lt;a href=&quot;http://csrc.nist.gov/groups/ST/toolkit/documents/dss/NISTReCur.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;) and an efficient way to add a point on this curve to itself, uses an element of the Galois group (this element is known as Frobenius).&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-14T00:42:00.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2662" PostTypeId="2" ParentId="1325" CreationDate="2014-06-14T01:07:46.557" Score="5" Body="&lt;p&gt;Years ago, there was a news story that coffee caused cancer. It was great, my opportunity to quickly tell everyone I ran into that I'd bet a year's pay this was a strong, but false, correlation. It was pretty obvious to me that for whatever reason, the coffee population had a higher smoking rate than the non-coffee drinkers. It took some time, but that was exactly what created the initial conclusion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, TV watching has been correlated with a sedentary lifestyle. A fair correlation, but that shouldn't discourage one from mounting a TV on the wall in front of their treadmill. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-06-14T01:07:46.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2663" PostTypeId="2" ParentId="1325" CreationDate="2014-06-14T02:43:14.157" Score="5" Body="&lt;p&gt;Folks often think that an event having probability zero and being impossible are the same thing.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-14T02:43:14.157" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2664" PostTypeId="2" ParentId="2612" CreationDate="2014-06-14T03:22:50.420" Score="1" Body="&lt;p&gt;One possible application to consider/work toward is &lt;a href=&quot;http://books.google.com/books/about/Algebraic_Statistics.html?id=3xYJLWR0K-4C&quot; rel=&quot;nofollow&quot;&gt;Algebraic Statistics&lt;/a&gt;. It's graduate level statistics, but maybe a focused approach can get undergraduates there in a semester or two.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-06-14T03:22:50.420" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2666" PostTypeId="2" ParentId="1848" CreationDate="2014-06-14T15:17:35.097" Score="3" Body="&lt;p&gt;Following up on Mike Miller's comment that &quot;many incredible textbooks have incredible exercises,&quot; you might want to take a look at Pugh's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387952977&quot; rel=&quot;nofollow&quot;&gt;Real Mathematical Analysis&lt;/a&gt; for its excellent selection of problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also also like to recommend Steele's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/052154677X&quot; rel=&quot;nofollow&quot;&gt;The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities&lt;/a&gt; as a book both entertaining and instructive in inequalities and problem-solving techniques.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2014-06-14T15:17:35.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2667" PostTypeId="2" ParentId="2637" CreationDate="2014-06-14T18:30:45.770" Score="3" Body="&lt;p&gt;Let me reiterate &lt;em&gt;Roland&lt;/em&gt;'s and &lt;em&gt;dtidarek&lt;/em&gt;'s suggestion to use character recognition,&#xA;and specifically numeral recognition, as a teaching example.&#xA;There are three advantages to this example:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It is easy to understand. &lt;/li&gt;&#xA;&lt;li&gt;It is the basis of zipcode recognition: It is being used every day, implemented in hardware used by the US Postoffice.&lt;/li&gt;&#xA;&lt;li&gt;There are simulators all over the web to explore performance. E.g., &lt;a href=&quot;http://www.cs.toronto.edu/~hinton/adi/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hinton's&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Here is a seminal paper on the topic that has subsequently been cited more than 1000 times:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;LeCun, Yann, Bernhard Boser, John S. Denker, Donnie Henderson, Richard E. Howard, Wayne Hubbard, and Lawrence D. Jackel. &quot;Backpropagation applied to handwritten zip code recognition.&quot; &lt;em&gt;Neural Computation&lt;/em&gt; 1, no. 4 (1989): 541-551.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Vxlc8.png&quot; alt=&quot;Fig1&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;sub&gt;(Part of) Fig. 1.&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-06-14T18:30:45.770" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2668" PostTypeId="2" ParentId="1848" CreationDate="2014-06-14T20:13:20.447" Score="2" Body="&lt;p&gt;I like these books:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.springer.com/mathematics/book/978-0-387-25765-5&quot; rel=&quot;nofollow&quot;&gt;Putnam and Beyond&lt;/a&gt; by Gelca, and Andreescu.  Brief introductions to mathematical topics from all over the curriculum and a number of competition-style problems to try.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.math.brown.edu/~jhs/frint.html&quot; rel=&quot;nofollow&quot;&gt;A Friendly Introduction to Number Theory&lt;/a&gt; by Silverman.  The open-ended way in which the problems in this textbook are phrased, are suggestive of the kinds of questions that a researcher might has.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.pearsonhighered.com/educator/academic/product/1,4096,0201199122,00.html?type=FEA&quot; rel=&quot;nofollow&quot;&gt;Discrete and Combinatorial Mathematics&lt;/a&gt; by Grimaldi.  This is a textbook that must have more than 1000 problems, many of which are quite good, challenging and give students access to a more general theory than what is being covered in the book.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-14T20:13:20.447" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2669" PostTypeId="2" ParentId="49" CreationDate="2014-06-14T20:59:53.137" Score="4" Body="&lt;p&gt;Eric Mazur's influential book Peer Instruction advocates a number of teaching techniques, one of which is to use short, easy, multiple-choice tests at the beginning of each meeting to enforce a requirement that students read the book before coming to class. The point is that if you don't do this, students will never read the book, and then you'll be forced to spoon-feed them every topic from scratch in lecture. Even weaker students can get some kind of feel for what's going on by reading. This allows you to spend class time on the topics that almost &lt;em&gt;everyone&lt;/em&gt; has trouble with, because they're just plain hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even so, this problem is probably inevitable to a certain extent in a freshman calc course. It's traditionally structured as a one-size-fits-all course for everyone from biology majors to math majors. Furthermore, you will inevitably end up with students who, for example, took AP Calculus in high school but for one reason or another are taking calc again in college.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-06-14T20:59:53.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2670" PostTypeId="2" ParentId="1204" CreationDate="2014-06-14T22:26:58.693" Score="1" Body="&lt;p&gt;The cross product is an important element of &quot;determinants,&quot; which is a topic normally covered in linear algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As such, it belongs in most courses of linear algebra. Except one that is &quot;light&quot; on determinants (and heavier in vector spaces and mappings).&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-06-14T22:26:58.693" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2672" PostTypeId="2" ParentId="1691" CreationDate="2014-06-14T22:38:09.737" Score="-1" Body="&lt;p&gt;Second order partial differential equations are characterized as hyperbolic, parabolic, or elliptic based on their &lt;a href=&quot;http://www.leb.eei.uni-erlangen.de/winterakademie/2009/report/content/course02/pdf/0202.pdf&quot; rel=&quot;nofollow&quot;&gt;discriminant function.&lt;/a&gt; So are conic sections.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-06-14T22:38:09.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2673" PostTypeId="2" ParentId="377" CreationDate="2014-06-14T22:42:47.837" Score="1" Body="&lt;p&gt;I would characterize the &quot;New Math&quot; as a &quot;top down&quot; approach to math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea was to have young students see the &quot;big picture.&quot; The danger is that they fail to &quot;see the trees for the forest&quot; (the reverse of the usual malady).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was a grade/junior high school student during the &quot;new math&quot; debate of the late 1960s.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-06-14T22:42:47.837" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2674" PostTypeId="2" ParentId="22" CreationDate="2014-06-14T22:54:10.857" Score="5" Body="&lt;p&gt;One nice family of examples is brought to us from &lt;strong&gt;thermodynamics&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the ideal gas law states:&#xA;$$ PV = nRT $$&#xA;where $P,V,n,T$ are generally variables which describe the pressure, volume, number of particles and temperature of a gas. We calculate the total differential:&#xA;$$ VdP+PdV= RTdn+RndT $$&#xA;This &lt;em&gt;equation of differentials&lt;/em&gt; contains a multitude of possible interpretations. One beautiful truth, at the outset, this total differential contains all those choices. In the absence of further constraints, any one of the four variables here can be taken as the dependent variable which is given by a function of the three remaining variables. The possibility of taking a given variable as dependent simply amounts to the algebraic freedom to solve for the differential of that variable. For example,&#xA;$$ dP = \frac{1}{V}(-PdV+RTdn+RndT) \ \ (\star)$$&#xA;Clearly $V=0$ is troubling, we can only expect to find $P = P(V,n,T)$ in some set where $V \neq 0$. Of course, this is not really an impediment here as negative volumes are hard to find. In any event, we can read the partial derivatives of $P$ with respect to $V,n,T$ directly from $\star$,&#xA;$$ \left(\frac{\partial P}{\partial V}\right)_{n,T} = \frac{-P}{V}, \ \ \&#xA;\left(\frac{\partial P}{\partial n}\right)_{V,T} = \frac{RT}{V}, \ \ \&#xA;\left(\frac{\partial P}{\partial T}\right)_{n,V} = \frac{Rn}{V} $$&#xA;The notation above is perhaps redundant given the context, however, in general in applications where there are multiple interpretations possible for a set of variables it is nice to be able to explicitly indicate which variables are taken to be independent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Generalizing a bit:&lt;/strong&gt;&#xA;This example is not hard enough to really appreciate the method of differentials as it explains partial derivatives for sets of variables. Essentially the same analysis is possible for several equations which govern some set of variables. In general, the meaning of something like $\frac{\partial w}{\partial x}$ is ambiguous. The value depends on the set of variables which is taken as independent as well as the defining equations (constraints) for the problem. What justifies this formal method? It's the &lt;strong&gt;implicit function theorem&lt;/strong&gt;. Thus, if you play with these differentials a bit and gain a better sense of how the Jacobian can be used to linearize a mapping then the implicit function theorem is not at all abstract. It's simply the condition needed to solve a given linearization for certain variables in terms of the remaining variables. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can be more greedy, the contraction mapping principle paired with a Newton's method type argument even gives a sequence of functions which converges to the implicit solution in a particular manner. So, the intuition I sketch above is just the first step in understanding this deeper result. Edwards' Advanced Calculus is once source for the implicit and inverse mapping theorems paired with the contraction mapping based solution sequence results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a better answer to give here. So many of the interesting theorems ultimately rest on the implicit function theorem. In some sense, those theorems are the real triumph of the implicit function theorem. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-06-14T22:54:10.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2675" PostTypeId="2" ParentId="1817" CreationDate="2014-06-14T23:11:02.710" Score="1" Body="&lt;p&gt;The &quot;classic&quot; female mathematician of the past century was Marie curie.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She was also a great chemist and physicist.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-06-14T23:11:02.710" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2676" PostTypeId="2" ParentId="1587" CreationDate="2014-06-14T23:16:34.437" Score="2" Body="&lt;p&gt;High school algebra courses have to cater to both serious and &quot;nonserious&quot; math students (the latter are taking their last course in math). As such, they are not in a position to match college levels of rigor, even for the same class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's only beginning with (senior year) calculus, where all the students are &quot;serious,&quot; that math is taught a true college level.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-06-14T23:16:34.437" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2677" PostTypeId="2" ParentId="1680" CreationDate="2014-06-14T23:24:07.547" Score="0" Body="&lt;p&gt;&quot;Multivariable analysis&quot; is the key to &quot;advanced calculus.&quot; The latter includes (but is not limited to):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Differential equations, especially if they include partial differential equations&lt;/li&gt;&#xA;&lt;li&gt;Vector and matrix calculus&lt;/li&gt;&#xA;&lt;li&gt;Real and complex analysis&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-06-14T23:24:07.547" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2678" PostTypeId="2" ParentId="1587" CreationDate="2014-06-15T02:14:11.157" Score="5" Body="&lt;p&gt;The actual level of rigor is going to depend on the school and the teacher, and has very little to do with the level of rigor claimed in the official course outline or in the textbook. Colleges are required by accrediting bodies to use college-level textbooks, but accrediting bodies can't tell whether teachers are actually applying college-level standards.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It depends on the country we're talking about, but in the US, the material you're talking about is stuff that students are supposed to be exposed to for the first time at an age somewhere between 13 and 15. At this age, they aren't adults intellectually. At age 13, many kids are barely able to read, so they need a textbook pitched to a very low reading level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reality is that remedial math courses at the college level have extremely low success rates. This fact has very little to do with the level of rigor of the courses and a lot more to do with the fact that the students who place into these courses are students who have already built up a record of failure in math. Past failure correlates with present failure.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-06-15T02:14:11.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2679" PostTypeId="1" AcceptedAnswerId="2681" CreationDate="2014-06-15T03:18:43.563" Score="47" ViewCount="19778" Body="&lt;p&gt;Talking to some friends, I was asked to explain the answer to the &lt;a href=&quot;http://en.wikipedia.org/wiki/Monty_Hall_problem&quot; rel=&quot;noreferrer&quot;&gt;Monty Hall problem&lt;/a&gt; (see also &lt;a href=&quot;http://xkcd.com/1282/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;;) .... they were having some trouble because whoever explained it to them didn't do a very good job.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;#Humblebrag&lt;/sub&gt; I was able to explain it in under 30 seconds to most people (winning on staying is 1/3; winning on switching is 2/3 because it's based on losing the first pick, which is a 2/3); but some people just didn't understand....&lt;strong&gt;is there any sure-fire method of teaching the answer to people?&lt;/strong&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;&#xA;Note: This question is not being asked about a math class per se, but is probably equally applicable to the classroom setting as it is to a group of friends.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;&#xA;A lot of the new answers don't understand what this question is asking. Pay attention to the bold text above, and realize that I understand the solution to the problem, but had some trouble explaining it to some friends....the focus of this question is &lt;em&gt;how to explain the answer&lt;/em&gt;, &lt;strong&gt;not&lt;/strong&gt; &lt;em&gt;what is the answer&lt;/em&gt;.&lt;br&gt;&#xA;A question focusing just on the answer would probably fit in a lot better on &lt;a href=&quot;https://math.stackexchange.com/&quot;&gt;Mathematics&lt;/a&gt; or &lt;a href=&quot;https://mathoverflow.net/&quot;&gt;MathOverflow&lt;/a&gt;; possibly &lt;a href=&quot;https://stats.stackexchange.com/&quot;&gt;Cross Validated&lt;/a&gt; (statistics).&lt;br&gt;&#xA;Thanks! :)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After posting where to find other questions regarding directly to the problem itself, I found &lt;a href=&quot;https://stats.stackexchange.com/q/373&quot;&gt;this question&lt;/a&gt; on Cross Validated -- it appears to be very, very related to my question.&lt;br&gt;&#xA;&lt;em&gt;(The letter copied in this &lt;a href=&quot;https://stats.stackexchange.com/a/23674&quot;&gt;answer&lt;/a&gt; to that question is hilarious, but very telling....)&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="1685" LastEditorUserId="2083" LastEditDate="2019-01-22T07:36:02.607" LastActivityDate="2022-01-01T14:01:56.120" Title="How to explain Monty Hall problem when they just don't get it" Tags="|mathematical-pedagogy|proofs|math-puzzle|probability|" AnswerCount="24" CommentCount="31" ContentLicense="CC BY-SA 4.0" />
  <row Id="2680" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T05:10:26.980" Score="13" Body="&lt;p&gt;I found the most helpful way to think about the problem is to expand it to a larger number of doors. For example, if you have them select from 100 doors, where 99 are losers and 1 is a winner. Then after the initial selection is made, eliminate 98 doors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The crux of this explanation is from one of Polya's heuristics, as explained below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The initial answer remains above, as I feel it is a quick answer to the question, thought it might not satisfy everyone. I answered almost immediately after the question was posed, and in the time since this edit, many more thorough answers exist on the board that should satisfy such readers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do wish to provide an answer to the comment by @Tutor who asked &quot;what Polya's heuristics are&quot;. George Polya is a well-renowned 20th century mathematician whose seminal work is &lt;a href=&quot;http://en.wikipedia.org/wiki/How_to_Solve_It&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to Solve it&lt;/a&gt;. In it, he discusses his heuristics (techniques) for solving (perhaps difficult) problems. Under the principle of &quot;Devise a Plan&quot; he discusses the idea of &lt;em&gt;solving a different problem&lt;/em&gt; if you are having difficulty solving the one posed. Sub-heuristics of this are: (1) solve a simpler problem, (2) solve a special problem, (3) solve a more general problem. The statement of a new problem with 100 doors could fall under any one of these categories.&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="2083" LastEditDate="2019-01-22T07:42:09.883" LastActivityDate="2019-01-22T07:42:09.883" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="2681" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T05:33:14.027" Score="28" Body="&lt;p&gt;Your explanation, by the way, is very elegant.  As an experienced mathematician, I see immediately that it cuts right to the heart of the matter and admits no ambiguity.  Unfortunately, this is precisely the quality that makes it unconvincing to others; the main confounding aspect of Monty Hall is that it ruthlessly exploits an intuitive misunderstanding of probability, and that misunderstanding remains even when one hears the correct explanation.  People don't learn by being told what's true if that truth is in conflict with a previously more strongly believed falsehood; you have to directly confront and discredit their preconceptions before they will be able to appreciate the slick answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possible misconception is that, after one door is eliminated and two remain, many people will feel that as an &quot;either/or&quot; choice, each door has probability of success 1/2, whereas in fact the distribution is not uniform.  You might test the possibility of this being the problem by posing other, simpler questions about non-uniform probabilities, for instance: play a game where I flip a coin and give it to you if it comes up heads.  What's the probability after two flips that you have 0, 1, or 2 coins?  If they know the right answer, ask them to explain why, and if not, work it out with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This can become Monty-Hall-like if you change the game once they understand.  I flip both in secret and announce that there were &quot;more than zero&quot; heads, without saying exactly how many.  Now what are the probabilities?  This is very close to one of the branches of the actual Monty Hall problem)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of the coins setup where the &quot;doors&quot; are dynamically generated rather than static and the implication that the game is repeatable is more obvious, it may conform better to the intuition.  The language a problem is stated in is very important for it being intuitive: people have different opinions about different but isomorphic versions of a single problem.  I don't claim that everyone will like coins either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would avoid an explanation that rests upon the other person successfully adopting a mathematical mindset.  For instance, the &quot;100 doors&quot; version immediately displays the problem but the concept of specializing from a generalization, or solving a related problem, especially one with extreme behavior, is not always natural to people.&lt;/p&gt;&#xA;" OwnerUserId="513" LastActivityDate="2014-06-15T05:33:14.027" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2682" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T07:26:35.320" Score="55" Body="&lt;p&gt;For some reason, the 'extend it to 100 doors and eliminate 98' explanation doesn't make it any clearer for me. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than talk about probabilities as fractions, I explain it this way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;If you picked the car (without knowing it) on the first choice, you'll lose it by switching, whereas if you didn't pick the car, you'll gain it by switching.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(stop here and make sure they get that.  Then...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Suppose you do the experiment 100 times*, and always stick with your initial choice.  About 33 of those times, you picked the car on the first choice.  So if you always stick, you'll end up with the car 33 times out of 100.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, about 66 of the times, you didn't pick the car on the first choice, but if you switch, you'll switch to the car.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore if you always switch, you'll get the car about 66 times out of 100, whereas if you always stick, you'll get the car about 33 times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it's twice as likely to get the car if you switch, as if you stick.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;*or 99 if they're picky&lt;/p&gt;&#xA;" OwnerUserId="980" LastActivityDate="2014-06-15T07:26:35.320" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2683" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T08:35:33.747" Score="3" Body="&lt;p&gt;The problem is most of the time something in the line of that a switch isn't believed to be helpful, because nothing has changed in the eyes of the person you are explaining the problem to: they consider the chances to be the same as before. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explaining that &lt;strong&gt;something has changed&lt;/strong&gt; can be the best step to take first: make sure they understand that they are getting &lt;strong&gt;more information&lt;/strong&gt; about the system. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ofcourse, then you go on, that because of this information, the chances have been changed. Then you can explain that the information is different based on the case you were right (1/3, no real new information) and the case you were wrong (2/3, new information is he tells you were the winner is).&lt;/p&gt;&#xA;" OwnerUserId="1700" LastActivityDate="2014-06-15T08:35:33.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2684" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T08:54:01.333" Score="20" Body="&lt;p&gt;There isn't any sure-fire method of explaining &lt;em&gt;anything&lt;/em&gt;, and especially in math. But specifically in the case of the Monty Hall problem it has been proven by extensive experience that many individuals with otherwise above average intellectual capacities exhibit an exceptional tenacity in refusing to accept the (otherwise) widely agreed upon solution; don't waste your time on repeating this experience, unless you should find pleasure in doing so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It should also be noted that for experts in probability theory there is a lot of nitpicking to be done about the exact hypotheses that need to be formulated in order to get a completely well defined probabilistic problem with the purported solution as correct answer (the Wikipedia page on the subject illustrates this). So if the resistance you get is of the type: &quot;it is not so clear cut, it might depend&quot;, then very possibly they could actually be right, unless you took all the required precautions in explaining the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Supposing you are trying to convince one person (otherwise it may become complicated) who has honest misunderstanding (as opposed to for instance a problem of pride in admitting to have been wrong) I could suggest the following approach. First make sure you have the same understanding of the problem formulation; otherwise any reasoning is pointless. Then if the other person agrees this should be a well defined probability problem with an unambiguous answer (again pointless if not) let him/her propose a method that should be able to determine that answer; it does not have to be efficient, but it should be acceptable independent of your beliefs about the answer. Since you are (convinced to be) right, you can afford to be generous; of course you should take care to refuse obviously incorrect models (like anything where the winning location is determined/changes &lt;em&gt;after&lt;/em&gt; the game starts). Some people would agree to be convinced by an experiment/simulation, but not everybody; you cannot impose this method. (Also if you do agree to this, be sure to fix &lt;em&gt;beforehand&lt;/em&gt; a number of trials sufficient for you to accept the risk of being accidentally &quot;shown&quot; wrong; asking for more trials after such an eventuality would make your credibility zero.) Another option would be to make a tree of &lt;em&gt;all&lt;/em&gt; possibilities, with agreed upon probabilities for descending into each branch. (Many arguments start &quot;without loss of generality let the first door be chosen&quot;. Don't do such a thing; while it can be justified, you cannot expect your adversary to accept the WLOG). Then compute the probabilities for each leaf, and add up. Or some other method still that could be proposed. Just make sure you agree about the method and its validity before applying it. Then just work it out, and accept the result.&lt;/p&gt;&#xA;" OwnerUserId="682" LastEditorUserId="682" LastEditDate="2014-06-15T10:52:15.883" LastActivityDate="2014-06-15T10:52:15.883" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2685" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T10:05:42.007" Score="9" Body="&lt;p&gt;I never understood it until my Maths lecturer explained it to me. Unfortunately I can't remember the exact way he explained it, but I'll try my best to remember. I really like your own explanation, so there is a chance that the following won't work. In that case, you may have to resort to empirical experiment. Another observation from my own experience is that sometimes I get too &quot;mathsy&quot; too quickly, and people don't follow my reasoning. Perhaps you need just need to slow down a bit!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try this explanation. Get three cards to represent the doors. Use an ace as the winning door. Now, take two of the cards, and give one to your friend. There is a 2:3 chance that you have the ace, because you have two of the three cards, and there is a 1:3 chance that your friend has the ace, because he has one of the three cards. Now, peek under your cards, and discard one. Ask your friend if they want to swap for the card that you kept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is essentially identical to what happens in the problem, but it feels different because &lt;em&gt;you&lt;/em&gt; are choosing two of the cards. You can highlight that you had a 2:3 chance of having the ace, because you had 2 of the cards. This may appeal to their intuition.&lt;/p&gt;&#xA;" OwnerUserId="683" LastActivityDate="2014-06-15T10:05:42.007" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2686" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T11:17:57.563" Score="6" Body="&lt;p&gt;Speaking from my own experience, a stage I had to go through before understanding any explanations of the logic and math involved was indignation at being baffled by a cheap fairground trick that probably goes back centuries. Also the eventual recognition that what put me, the punter, at a disadvantage to the operator was the fact that s/he is working to a strategy and is therefore prepared to lose a game every so often (1 out of 3 in this case), whereas I'm limiting myself to only thinking I've got to win the present game each time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another motivational prepping step I'd recommend is a highly entertaining &lt;a href=&quot;https://www.youtube.com/watch?v=o_djTy3G0pg&quot; rel=&quot;nofollow&quot;&gt;presentation&lt;/a&gt; (available on YouTube) by the great tele-mathematician Marcus du Sautoy playing the game with someone who sportingly agrees to be the fall guy who never switches and so loses heavily. This brings out the compelling practicalities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of points about the explanation stage. A clear diagram with your three doors and a list of the three possibilities of what's behind them:  Car Goat Goat; Goat Car Goat; Goat Goat Car. If my first choice of door happens to be the correct one, and I switch, then sure as hell I'm going to lose on that occasion. Don't mince words about that, but use it to show how it means that if instead my first choice happens to be the &lt;em&gt;wrong&lt;/em&gt; one, which is going to be 2 times in 3, then switching will reverse those odds into my favour. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I agree with others that I've never understood how considering lots and lots of doors is helpful)&lt;/p&gt;&#xA;" OwnerUserId="1702" LastEditorUserId="1685" LastEditDate="2014-06-15T20:03:33.310" LastActivityDate="2014-06-15T20:03:33.310" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2687" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T11:43:36.383" Score="19" Body="&lt;p&gt;Perhaps it's not the explanation that's the problem. I suggest you have &lt;em&gt;them&lt;/em&gt; explain to &lt;em&gt;you&lt;/em&gt; their understanding of the problem. Listening to their justification might reveal why your explanation is not gaining traction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even in the cases where people are saying they now agree with you, you don't necessarily know that they understand the problem. It's possible that all you did was get them to agree with you. People often just agree with another person who seems to have a better grasp of math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if they explain it to you, you'll have a much better idea of their understanding. No matter whether they agreed or disagreed with your conclusions.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-06-15T11:43:36.383" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2688" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T15:03:02.327" Score="4" Body="&lt;p&gt;The best way I came up with is to &lt;strong&gt;draw all 9 possible outcomes&lt;/strong&gt;.&lt;br&gt;&#xA;Prize is in A, you pick A;&lt;br&gt;&#xA;Prize is in B, you pick A;&lt;br&gt;&#xA;Prize is in C, you pick A;&lt;br&gt;&#xA;Prize is in A, you pick B;&lt;br&gt;&#xA;and so on.&lt;br&gt;&#xA;Each outcome is equally probable and &lt;strong&gt;6 of those will make you win&lt;/strong&gt;.&lt;br&gt;&#xA;I don't think it get's any easier than this.&lt;/p&gt;&#xA;" OwnerUserId="1703" LastActivityDate="2014-06-15T15:03:02.327" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2689" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T16:21:40.580" Score="7" Body="&lt;p&gt;While your explanation is correct, and easily understood by people like us, it's a bit too terse (a quality we like) for a lay person to understand. You can simply expand on it. And most importantly ask the audience questions along the way. Every explanatory sentence should have a question that goes a long with it. The key is for you to identify where the hangup is. I like to think about the problem in terms of the chosen door rather than the switching door. So this is how I might go about asking the questions.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask them before removing a door &quot;what is the probability that the door they chose has the car?&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then say, &quot;I remove a door. Since I am not evil, I am not going to remove the door with the car.&quot; Then ask &quot;Why does the probability that the car is behind the door you initially chose not change?&quot; If they don't get this right, work out the cases, as others have mentioned (continuing to ask questions along the way).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then say, &quot;Since there is 1/3 probability that the car is behind your door, what must be the probability the car is behind the other door?&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Explanation at each stage may be required, but this socratic method seems to work well. I deployed something similar to a bunch of small children at the NY county fair. By making the argument composed only of leading questions, its a way of making sure they are following at every step. Of course you can ask them different questions. The key is to make it socratic.&lt;/p&gt;&#xA;" OwnerUserId="1387" LastEditorUserId="1387" LastEditDate="2014-06-15T16:31:13.653" LastActivityDate="2014-06-15T16:31:13.653" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2690" PostTypeId="1" CreationDate="2014-06-15T16:53:12.940" Score="5" ViewCount="112" Body="&lt;p&gt;The pumping lemmata (for regular languages and for context free languages) are used to prove languages non-regular/non-context free by contradiction. But such proofs are often horribly botched by students. Any insights on how to explain how to use them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For reference, the pumping lemma for regular language (the context free one is similar): If $L$ is a regular language, then there is a constant $N \in \mathbb{N}$ such that all strings $\sigma \in L$ that are longer than $N$ ca be written $\sigma = \alpha \beta \gamma$, where the length of $\alpha \beta$ is at most $N$, $\beta$ isn't empty, and for all $k \in \mathbb{N}_0$ you have $\alpha \beta^k \gamma \in L$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typical mistakes are, roughly in decreasing order of occurrence:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Picking some nice $N$&lt;/li&gt;&#xA;&lt;li&gt;Selecting a particular split&lt;/li&gt;&#xA;&lt;li&gt;Not making sure the selected $\sigma$ is long enough&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Other mistakes, not fatal ones, are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Making sure $\sigma$ is (almost) exactly of length $N$&lt;/li&gt;&#xA;&lt;li&gt;Selecting awkward values of $k$ to derive a contradiction&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-16T11:22:35.693" Title="How to teach application of pumping lemma (automata theory)?" Tags="|proofs|students-mistakes|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2691" PostTypeId="1" CreationDate="2014-06-15T17:03:35.583" Score="8" ViewCount="521" Body="&lt;p&gt;The setting is undergraduate students in Computer Science, a course in Discrete Mathematics (first proof-oriented course they take, they had a mostly computation oriented first course in calculus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As examples I need nice proofs using cases, where the cases aren't automatically apparent (i.e., this mostly rules out proofs with absolute values). The restriction is also that not much more than high-school algebra (and perhaps a dash of common sense) is required.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bonus points for proofs that aren't obviously &quot;mathematical&quot; (i.e., not algebra, no formulas).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-07-09T06:31:57.180" Title="Nice examples of proofs by cases?" Tags="|undergraduate-education|proofs|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2692" PostTypeId="1" CreationDate="2014-06-15T17:07:01.357" Score="2" ViewCount="633" Body="&lt;p&gt;The setting is undergraduate students in Computer Science, a course in Discrete Mathematics (first proof-oriented course they take, they had a mostly computation oriented first course in calculus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As examples I need nice proofs using contradiction. The typical proof that $\sqrt{2}$ is irrational is standard fare, but they seem not to &quot;see&quot; the contradiction clearly, so any hints at making that one clearer would be appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The restriction is also that not much more than high-school algebra (and perhaps a dash of common sense) is required.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bonus points for proofs that aren't obviously &quot;mathematical&quot; (i.e., no algebra, no formulas).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-15T22:11:42.673" Title="Nice examples of proof by contradiction?" Tags="|undergraduate-education|proofs|" AnswerCount="2" CommentCount="2" ClosedDate="2014-06-16T13:02:03.510" ContentLicense="CC BY-SA 3.0" />
  <row Id="2693" PostTypeId="2" ParentId="2691" CreationDate="2014-06-15T17:26:25.507" Score="2" Body="&lt;p&gt;Suppose any two people at a party have either met or not. If every pair of people in a group has met, we’ll call the group a club. If every pair of people in a group has not met, we’ll call it a group of strangers.  Prove: Any six people at the party includes a club of 3 people or a group of 3 strangers.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-15T17:26:25.507" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2694" PostTypeId="1" CreationDate="2014-06-15T17:31:56.453" Score="8" ViewCount="160" Body="&lt;p&gt;As a mathematics instructor, what is the most valuable (to you) thing that classroom experience taught you about mathematics education? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be specific, I'm looking for answers not about mathematics itself, or student discipline unrelated to mathematics, but about the learning, teaching, or reasoning of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[To split some hairs, there is knowledge of mathematics that falls into this category; the difference between quotative and partitive division is a good example of pedagogical content knowledge. It's something most math users don't need to know about, but it helps teachers understand student conceptions of division. If you learned something like that through classroom experience, it would be a great answer and probably a great story as well.]&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-06-15T17:42:34.760" LastActivityDate="2014-06-23T01:03:13.277" Title="Valuable Classroom Experience" Tags="|teacher-development|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2695" PostTypeId="2" ParentId="2692" CreationDate="2014-06-15T17:33:00.727" Score="3" Body="&lt;p&gt;Here's one that I use in an analogous course I teach:  A lossless compression algorithm that makes some files smaller must make some (other) files larger.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-06-15T17:33:00.727" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2696" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T17:35:43.180" Score="5" Body="&lt;p&gt;The big problem with the &quot;Monty Hall&quot; problem is that there are many problems that sound superficially the same, but have different solutions. The terms of the game have to be stated &lt;em&gt;very precisely&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, Marily vos Savant's statement of the problem as it is quoted in the Wikipedia article is imprecise. It doesn't state whether the game host &lt;em&gt;must&lt;/em&gt; or &lt;em&gt;may&lt;/em&gt; open another door, whether he &lt;em&gt;must&lt;/em&gt; open a door with a goat or whether he is allowed to open a door with the car, and if he has some freedom of choice, whether he is trying to help you or not. Or whether he is even allowed to &quot;cheat&quot; and swap the contents of the two unopened doors sometimes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you listened to the explanation why changing doors is so good, but they just changed the rules - after &lt;em&gt;you&lt;/em&gt; choose the first door, if you chose a goat the game host will open the door and say &quot;tough luck, you lost&quot;. If you chose the door with a car, the game host will open another door with a goat, so if you swap, you lose. So you see that without stating the rules precisely, you can't give the correct answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;100 doors&quot; explanation doesn't help. So you are the lucky guy that by a one in hundred chance picked the car, and you are shown 98 doors with goats. If you swap, you lose. On the other hand, people who picked a goat will be shown fewer than 98 doors (once in a while, someone will be shown 98 doors, or the game would be given away). &lt;/p&gt;&#xA;" OwnerUserId="697" LastActivityDate="2014-06-15T17:35:43.180" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2697" PostTypeId="5" CreationDate="2014-06-15T17:36:12.627" Score="0" Body="&lt;p&gt;Teacher development is about how we, as mathematics educators, learn and develop in the profession. It can refer to gaining specific mathematics content knowledge related to teaching mathematics (pedagogical content knowledge), knowledge of students and how they learn mathematics, specific teaching practices that are instrumental in the mathematics classroom, etc.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-06-15T20:02:44.363" LastActivityDate="2014-06-15T20:02:44.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2698" PostTypeId="4" CreationDate="2014-06-15T17:36:12.627" Score="0" Body="Developing knowledge and abilities as a teacher of mathematics" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-06-15T20:02:54.433" LastActivityDate="2014-06-15T20:02:54.433" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2699" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T19:16:46.537" Score="2" Body="&lt;p&gt;Short and sweet summation of the method in the question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's more likely that your first choice will be wrong than it will be right. Therefore, if you are given the opportunity to switch, it's more likely that switching will be right than it will be wrong.&lt;/p&gt;&#xA;" OwnerUserId="1704" LastEditorUserId="2083" LastEditDate="2019-01-22T12:11:30.343" LastActivityDate="2019-01-22T12:11:30.343" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="2700" PostTypeId="2" ParentId="1028" CreationDate="2014-06-15T19:22:28.197" Score="0" Body="&lt;p&gt;A tennis instructor once wrote that if a student insisted that s/he had mastered a technique even tough s/he had not, the instructor would hit shots to &quot;pick on&quot; the weakness in the technique, and thereby demonstrate that student actually had a lesser mastery of that &quot;material&quot; than of others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the teacher would give quizzes on the supposedly &quot;earlier&quot; material and demonstrate that the students actually score lower on it than more recent material.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-06-15T19:22:28.197" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2701" PostTypeId="2" ParentId="1403" CreationDate="2014-06-15T19:40:35.213" Score="2" Body="&lt;p&gt;In my personal experience, &quot;mathematical maturity&quot; began when I began understanding how mathematical concepts &lt;strong&gt;interact&lt;/strong&gt; with each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Anyone&quot; can memorize concepts. Understanding &quot;how they fit together&quot; is the real sign of maturity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So tell students that they have &quot;mathematical maturity&quot; when they can understand the interrelationships between key mathematical concepts, and not just the concepts themselves.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2014-06-15T20:29:40.740" LastActivityDate="2014-06-15T20:29:40.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2702" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T20:03:15.120" Score="6" Body="&lt;p&gt;I find that most people who THINK they understand the Monty Hall Problem, actually don't. For about 5 years I was one of them, until a further insight made me understand it better. More of that at the end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I came up with a super-exaggerated version that seems to give people pause for thought at least.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the UK, our lottery has a probability of 14,000,00:1 for the jackpot. Suppose I buy all 14,000,000 tickets one Saturday and let you choose 1 ticket with your favourite numbers, the rest I put in my shed. I lock you in a room until the draw with no access to the lottery result and I sit in my shed and watch the lottery draw. After the draw, I walk out of my shed with 1 ticket in my hand and set light to the entire shed and the rest of the tickets. I unlock the door of your room and announce, “Good news! One of us has the winning ticket! Would you like to swap with me?” Why would you think that your ticket has suddenly become a 50/50 chance of being the winner? If you believe that, imagine we did the very same thing NEXT week and the week after. Why has your lottery ticket suddenly acquired a 50/50 chance of winning every week – yet never wins it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The crucial thing I alluded to at the top is that Monty HAS TO KNOW where the car is before any of the normal interpretations become valid. The “100 doors” scenario illustrates the point very well here. You pick door one and Monty opens door 2, then door 3; all the way to door 36. He bypasses door 37 and moves to 38, 39; all the way to 100. He then asks if you want to swap door one for door 37. Clearly the car is behind door 37 (well, 99 times out of a 100).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, imagine you watch the program for 3 months before you are a participant and every week Monty keeps opening doors to find that the car is behind: door 17, door 81, door 53 – each week he keeps stumbling upon the car BEFORE the point he would give the contestant the chance to swap. When you go on the show,  you choose door 1, then Monty proceeds to open each door from 2 to 99 and does not find the car. When he offers to swap door 100 with you, there really is a 50/50 chance that you have the car already and there is no advantage in swapping.&lt;/p&gt;&#xA;" OwnerUserId="1705" LastActivityDate="2014-06-15T20:03:15.120" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2703" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T20:51:33.603" Score="9" Body="&lt;p&gt;The best sure-fire method for teaching this to people who don't want to learn it is to set up a Monty Hall style game for small stakes of real money.  When they start losing 2/3 of the time, they will become more receptive to your explanation.&lt;/p&gt;&#xA;" OwnerUserId="1707" LastActivityDate="2014-06-15T20:51:33.603" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2705" PostTypeId="2" ParentId="2692" CreationDate="2014-06-15T22:02:56.827" Score="1" Body="&lt;p&gt;If you don't mind probability-related problems, this one works pretty well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose $A$, and $B$ are sets in $S$.  Then $A \cap B$, $A \cap B^C$, $A^C \cap B$ and $A^C \cap B^C$ partition $S$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proof by contradiction is in showing that the members of the partition are disjoint.  Suppose, for example that $A \cap B$ and $A \cap B^C$ are not disjoint.  Then there is some element $x \in S$ that belongs to both sets.  But $x$ must be an element of both $B$ and $B^C$.  The only such set is the null set.  Therefore, $A \cap B$ and $A \cap B^C$ are disjoint.  Note now that any pair of partition elements have at least one set and its complement, and so the partition members are disjoint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That they jointly cover $S$ is obvious, for any $x \in S$ must belong to $A$ or $A^C$ by definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can also be done with Boolean (set) algebra, but in teaching Bayes' rule I find that this argument is more persuasive.&lt;/p&gt;&#xA;" OwnerUserId="1708" LastActivityDate="2014-06-15T22:02:56.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2707" PostTypeId="2" ParentId="2679" CreationDate="2014-06-15T23:40:10.680" Score="2" Body="&lt;p&gt;Switching turns a winning guess into a losing guess, and a losing guess into a winning one. Since your guess is twice as likely to be losing as it is to be winning, you do better to switch. &lt;/p&gt;&#xA;" OwnerUserId="1711" LastActivityDate="2014-06-15T23:40:10.680" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2708" PostTypeId="2" ParentId="2679" CreationDate="2014-06-16T01:18:32.087" Score="-2" Body="&lt;p&gt;&lt;strong&gt;TL;DR There are still 3 doors even though one of them has been opened.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fact 1. After your initial choice there is a 2/3 chance that the prize is behind one of the other two doors.  Your chosen door has a 1/3 chance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fact 2. When the empty door is shown to you, there is &lt;strong&gt;still&lt;/strong&gt; a 2/3 chance that the prize is behind one of those two doors.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fact 3. Except now you know that one of those two doors is empty, so you choose the door that was not shown.  That one door has a 2/3 chance that the prize is behind it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conclusion: The unknown door has a 2/3 chance of having the prize, but your chosen door only has a 1/3 chance. So always switch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mistake that most people make is thinking that it is a now a two door problem and therefore has a 50-50 chance.  It is still a 3 door problem and can be easily solved with basic logic.&lt;/p&gt;&#xA;" OwnerUserId="1713" LastActivityDate="2014-06-16T01:18:32.087" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2709" PostTypeId="2" ParentId="2694" CreationDate="2014-06-16T03:20:54.907" Score="3" Body="&lt;p&gt;It was quite a shock to me when I found out I had the expert's curse (can't understand the problems newbies have because at their root is something &lt;em&gt;so&lt;/em&gt; familiar that you can't imagine somebody doesn't know it). Now I know I'm cursed, dunno if it has done my victims much good...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-16T03:20:54.907" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2710" PostTypeId="2" ParentId="2679" CreationDate="2014-06-16T04:38:51.110" Score="14" Body="&lt;p&gt;I've had best luck by simplifying the problem to asking whether you want to pick one door or two doors. Everyone understands they'd be better off picking two doors. At that point I tell them to just ignore the fact that Monty showed them what was behind one of the doors they didn't pick. They're still picking either one door or two doors.&lt;/p&gt;&#xA;&lt;p&gt;Those more sophisticated, but still confused  after listening, will object that this ignores the additional information gained from Monty opening one of the doors. But that's a red herring. There's always a losing door in the two you didn't pick, so Monty is not revealing any new information.&lt;/p&gt;&#xA;" OwnerUserId="1716" LastEditorDisplayName="user155" LastEditDate="2022-01-01T14:01:56.120" LastActivityDate="2022-01-01T14:01:56.120" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="2711" PostTypeId="2" ParentId="2679" CreationDate="2014-06-16T05:40:08.593" Score="6" Body="&lt;p&gt;Here's my take on explaining it. (And it usually involves a drawing for me :-) )&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming the host opens a door the odds of making the right choice without switching the door is 1/3 because it was made before having the knowledge the host provided.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you decide to make the switch here is a breakdown of the probability of losing VS winning:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/1S54Y.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="1717" LastActivityDate="2014-06-16T05:40:08.593" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2713" PostTypeId="2" ParentId="2690" CreationDate="2014-06-16T11:22:35.693" Score="2" Body="&lt;p&gt;In my opinion the main issue is that the pumping lemma is often presented as a black box, many students ignorant of the basic idea behind it. Because of this I prefer to use the &quot;pumping lemma&quot; phrase as a concise way of saying all the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A finite automaton has finite number of states, so for any (sub)word of length greater than $|A|$ some state has to repeat in the automaton run.&lt;/li&gt;&#xA;&lt;li&gt;If some state happens more than once in a single word, then there exists a non-empty substring $\beta$ that leads from that state to itself.&lt;/li&gt;&#xA;&lt;li&gt;Independent of how many times we repeat $\beta$, the rest of the automaton run stays unchanged.&lt;/li&gt;&#xA;&lt;li&gt;In particular, the word's acceptance doesn't change.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Each bullet above is intuitive and doesn't pose any problems, even for weaker students. I've never seen anyone to err while having such an operational intuition in mind (I'm not saying that it doesn't happen, only that it is rarer). Yet, there are people who don't understand it this way despite having seen the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is why I would prefer to stop presenting it as a lemma we use to prove non-regularity (and make some students disregard the proof), but rather each time prove non-regularity from scratch and then call such approach in general a proof &quot;via pumping lemma&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-06-16T11:22:35.693" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2714" PostTypeId="1" AcceptedAnswerId="2722" CreationDate="2014-06-16T11:44:56.467" Score="7" ViewCount="227" Body="&lt;p&gt;I'm teaching discrete math to computer scientists.  This is mostly their first proof-based course.  We cover a bit of number theory, and as I hate loose ends, I cover a bit of Möbius inversion. (It is required later to justify some transfer formulas for the symbolic method in combinatorics and elsewhere.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My problem is that I haven't been able to find clear applications of arithmetic sums (sums over the divisors of $n$) more than Euclid/Euler on perfect numbers. Even less convincing uses of Möbius inversion.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="263" LastEditDate="2014-06-17T00:37:15.497" LastActivityDate="2014-06-19T19:53:45.337" Title="Understandable, interesting uses of Möbius inversion?" Tags="|undergraduate-education|examples|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2716" PostTypeId="2" ParentId="2395" CreationDate="2014-06-16T12:32:07.113" Score="3" Body="&lt;p&gt;The program &lt;a href=&quot;http://www.letsgeddit.com/&quot; rel=&quot;nofollow&quot;&gt;geddit&lt;/a&gt; might be what you are searching for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding your requirements&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It is free for both the instructor and students.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Yes&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;It can display mathematical formulas.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It can show arbitrary images, so this is &lt;strong&gt;partly covered&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Students register and sign in individually, so that it can be used to track attendance/participation or give graded quizzes.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Yes&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="607" LastEditorUserId="1685" LastEditDate="2014-06-17T09:53:00.667" LastActivityDate="2014-06-17T09:53:00.667" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2717" PostTypeId="2" ParentId="1691" CreationDate="2014-06-16T12:37:55.847" Score="9" Body="&lt;p&gt;Benjamin Dickman &lt;a href=&quot;https://matheducators.stackexchange.com/a/2400/125&quot;&gt;already linked to&lt;/a&gt; my answer on Math.SE, but I didn't talk too much about exactly the motivation of the characterisation for second order equations. So let me write a little bit about that here. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;First, I want to emphasize again that the classification is &lt;strong&gt;incomplete&lt;/strong&gt;: the usual scheme does not include all possible equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us start by looking at the principal part of the equation (that is the coefficient matrix concerning the second order derivatives) $a^{ij}\partial^2_{ij}$. We know/assume that $a^{ij}$ is a symmetric $n\times n$ matrix. By &lt;a href=&quot;http://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sylvester's law&lt;/a&gt;, we can classify $a^{ij}$ by its positive, zero, and negative indices of inertia $n_+, n_0,n_-$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We pause here to note that $n_0$ agrees with the nullity of $a^{ij}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recall here the usual definitions (up to swapping $n_+$ and $n_-$ by an overall sign change):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;elliptic: $n_+ = n$, $n_0 = n_- = 0$&lt;/li&gt;&#xA;&lt;li&gt;parabolic: $n_+ = n-1$, $n_0 = 1$, $n_- = 0$. &lt;/li&gt;&#xA;&lt;li&gt;hyperbolic: $n_+ = n-1$, $n_0 = 0$, $n_- = 1$. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note that higher order nullity is already ruled out, and also are the cases where $n_0 = 0$ and $n_- = 2, 3,\ldots, n-2$. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So why do we focus on just these three? &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The &lt;em&gt;parabolic&lt;/em&gt; case is a very good model for diffusive phenomenon; this was already noted by Fourier who gave a physical derivation of his heat equation.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;hyperbolic&lt;/em&gt; case is a very good model for wave propagation, and also to some extent dispersive phenomenon; this was noted by d'Alembert who gave a physical derivation of the wave equation. &lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;elliptic&lt;/em&gt; case is important physically as elliptic equations arise naturally when one considers solutions to parabolic/hyperbolic equations which are &lt;em&gt;stationary in time&lt;/em&gt;. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So one motivation of why the literature have largely focused on developing these three types of equations, and not on the other second order PDEs, is that from very early on the &lt;em&gt;physical relevance&lt;/em&gt; of these three types of equations are understood. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I should note here that in the 20th century another second order linear PDE, which by virtue of it having complex valued coefficients does not fit into the existing classification scheme, also rose to prominence, and that is Schrodinger's equation from quantum mechanics. Again, the interest in the equation is driven by physics.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Why do we classify the equations in these three types? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is simply our human desire to organize knowledge. It turns out that the qualitative features of elliptic, parabolic, and hyperbolic PDEs allow us to write down separate, somewhat comprehensive theories for each of the types. I include some high-lights here:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;For both hyperbolic and elliptic types, but notably &lt;em&gt;not&lt;/em&gt; for parabolic type equations (let us assume constant coefficients), we can write down solutions to the Cauchy problem in the case of real analytic initial data using the theorem of &lt;a href=&quot;http://en.wikipedia.org/wiki/Cauchy-Kowalevski_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy-Kowalevski&lt;/a&gt;. &#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For hyperbolic types, this solution to the initial data problem turns out to be &quot;stable&quot;, in the sense that for initial data that has finite regularity we can also find solutions of finite regularity. For the elliptic case this stability is lost. &lt;/li&gt;&#xA;&lt;li&gt;For elliptic equations, the more natural question to ask is not an initial value problem, but a boundary value problem on a bounded domain $\Omega$. Note that whereas for the initial value problem we usually prescribe both the value of the function and its transversal derivative at the initial time, for the boundary value only one out of the two is prescribed. This is related to the next point. &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;For both the elliptic and parabolic types, but notably &lt;em&gt;not&lt;/em&gt; for the hyperbolic type equations, we have the notion of &lt;a href=&quot;http://en.wikipedia.org/wiki/Maximum_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;maximum principle&lt;/a&gt; for solutions. &lt;/li&gt;&#xA;&lt;li&gt;Solutions to the elliptic and parabolic type equations enjoy &lt;em&gt;smoothing properties&lt;/em&gt;, while solutions to the hyperbolic type equations have &lt;em&gt;propagation of singularities&lt;/em&gt;. Examples:&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In the elliptic case: Let $\Omega$ be a suitably regular, bounded domain. Then for any continuous function $f$ specified on $\partial\Omega$ we can solve $\triangle u = 0$ in $\Omega$ with $u = f$ on $\partial\Omega$. Furthermore, we know that strictly in the interior of $\Omega$, $u$ is real analytic (even in the case where $f$ cannot extended to a $C^1$ function). &lt;/li&gt;&#xA;&lt;li&gt;In the parabolic case: the &lt;a href=&quot;http://en.wikipedia.org/wiki/Heat_kernel&quot; rel=&quot;nofollow noreferrer&quot;&gt;heat kernel&lt;/a&gt; evaluated with $y = 0$ is formally a solution to the heat equation whose initial value is the Dirac-$\delta$. Note that at all times $t &amp;gt; 0$ the solution is $C^\infty$. &lt;/li&gt;&#xA;&lt;li&gt;In other words, solutions to elliptic and parabolic problems are often smoother than their boundary/initial values. &lt;/li&gt;&#xA;&lt;li&gt;For hyperbolic equations, singularities in the initial value are propagated into the future, along characteristic curves of the equation. For example, consider the one dimensional wave equation $\partial_t^2 u = \partial_x^2 u$, with initial data &#xA;$$ u(0,x) = \begin{cases} x^3 &amp;amp; x \geq 0 \\ 0 &amp;amp; x &amp;lt; 0\end{cases}, \quad u_t(0,x) = \begin{cases} - 3x^2 &amp;amp; x \geq 0 \\ 0 &amp;amp; x &amp;lt; 0 \end{cases}$$&#xA;the data here is in $C^2 \times C^1(\mathbb{R})$. The solution to the equation turns out to be&#xA;$$ u(t,x) = \begin{cases} (x-t)^3 &amp;amp; x \geq t \\ 0 &amp;amp; x &amp;lt; t\end{cases} $$&#xA;and is only twice continuously differentiable, with its third derivative discontinuous along the characteristic line $x = t$. &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Concerning the initial value problem, the parabolic type equations only have existence forward in time, while the hyperbolic type equations have existence both forward and backward in time. &lt;/li&gt;&#xA;&lt;li&gt;Somewhat more technically: with elliptic problems it is often very convenient to work in spaces of functions which are Holder continuous, with occasional call to use Lebesgue/Sobolev spaces of various Lebesgue exponents. Similarly for parabolic problems. For hyperbolic equations in spatial dimension $&amp;gt;1$, pretty much the only good tool is via $L^2$-based Sobolev spaces. &lt;/li&gt;&#xA;&lt;li&gt;Hyperbolic, and not &lt;em&gt;elliptic&lt;/em&gt; or &lt;em&gt;parabolic&lt;/em&gt; (at least in the linear case, for nonlinear equations there are some exceptions) enjoy the property of &quot;finite speed of propagation&quot;. (In the case of the linear wave equation $\partial_t^2 u = c^2 \triangle u$ this says, in particular, that if the initial data is zero outside of a ball of radius $R$, the solution at time $T$ is zero outside the ball of radius $R + c|T|$.) This is noticeably false for the initial value problem for the heat equation. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Lastly, note that if your equation has constant coefficients, then through a linear change of coordinates you can bring your equation into the form of a wave/heat/Laplace equation if it is classified as hyperbolic/parabolic/elliptic. So you can also motivate the general definition thus as saying that a hyperbolic equation looks, in a small neighbourhood, like the wave equation plus some small perturbations etc. This has the advantage of leading to a discussion of the method of freezing coefficients for elliptic estimates, say. &lt;/p&gt;&#xA;" OwnerUserId="125" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-06-16T12:37:55.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2718" PostTypeId="1" CreationDate="2014-06-16T15:03:17.220" Score="7" ViewCount="823" Body="&lt;p&gt;Assume you don't know the result of $(-1)\cdot 3$. You might try calculating&#xA;$$&#xA;\begin{align*}&#xA;3\cdot 3=9\\ 2\cdot 3=6\\1\cdot 3=3\\0\cdot 3 = 0&#xA;\end{align*}&#xA;$$&#xA;thus you may argue $$(-1)\cdot 3=-3$$ just by recognition of the pattern (first factor decreases by 1, result decreases by 3).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such argumentations may work, especially when you &lt;strong&gt;define&lt;/strong&gt; something new (like multiplication on negative numbers). However, things may go wrong as cases of $0^0$ show where you could define it as $0$ according to $0^n$ or as $1$ according to the $n^0$-sequence.&#xA;I would also say that $\int 1/x=ln(x)$ does not fit the pattern of $\int x^z=\frac{x^{z+1}}{z+1}$ for $z \neq -1$. (You might want to add a constant to be exact.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt; Do you have more of such examples?&lt;/p&gt;&#xA;" OwnerUserId="199" LastEditorUserId="112" LastEditDate="2014-06-16T15:08:26.430" LastActivityDate="2014-06-18T08:51:31.317" Title="Examples for reasoning by analogy going wrong" Tags="|examples|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2719" PostTypeId="2" ParentId="224" CreationDate="2014-06-16T16:00:04.890" Score="12" Body="&lt;p&gt;I'm very happy to see this discussion here, because all of you are saying exactly the things that led to the project I and my collaborator (Ken Monks, Univ. Scranton) are working on, &lt;a href=&quot;http://lurch.sf.net&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lurch&lt;/a&gt;.  It's free, open-source, and cross-platform, so there's no barrier to trying it out any time.&lt;/p&gt;&#xA;&lt;p&gt;It was mentioned briefly in one of the comments above, but it's so directly related to the issues in this thread that I want to bring it up to the level of a full answer, especially since one of the other answers asserts that no such thing exists (ack!) but says he'd love to try it if it did (great!).&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The OP asks for software related to &lt;em&gt;intro-to-proof courses&lt;/em&gt;, which are exactly the specialties Lurch focuses on, although it can do more than just those.&lt;/li&gt;&#xA;&lt;li&gt;He and some commenters mentioned the potential benefits of the &lt;em&gt;short feedback loop&lt;/em&gt;, which not only is one of Lurch's strengths, but informal classroom testing confirms its value.&lt;/li&gt;&#xA;&lt;li&gt;The OP and one answerer also hit the nail on the head regarding the difficulty of formal syntax muddying the potential value of most proof assistants; it is for this reason that Lurch was designed to be a &lt;em&gt;normal math word processor&lt;/em&gt; (normal math notation and typeset math built in, no funny ASCII).  See the video on the Lurch homepage, linked to above.&lt;/li&gt;&#xA;&lt;li&gt;Finally, the OP asked only for projects that have &lt;em&gt;actually been used&lt;/em&gt;, not just ideas; Lurch is such a project.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The discussion between the OP and @vonbrand re: many, tedious steps in formal proofs is also one of our main concerns.  Lurch can handle varying levels of formality and style, and gets better at it as new versions are developed.  Specific examples of what I'm talking about, with many other details, can be found in &lt;a href=&quot;http://ceur-ws.org/Vol-1010/paper-04.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Lurch overview whitepaper here&lt;/a&gt;.  (That whitepaper does not cover the latest feature of math typesetting, but the video mentioned above does.  The software also comes with a built-in tutorial that covers all its features.)  In fact, the OP's original question, about experiences using it in class, is answered in detail, for Lurch, in that paper, for two separate courses at two different institutions.&lt;/p&gt;&#xA;" OwnerUserId="1732" LastEditorUserId="1732" LastEditDate="2020-09-14T12:21:53.203" LastActivityDate="2020-09-14T12:21:53.203" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="2721" PostTypeId="2" ParentId="2718" CreationDate="2014-06-16T18:37:07.373" Score="4" Body="&lt;p&gt;The polynomial $n^2 + n + 41$ gives only prime values for $0 \le n \le 40$, while for 40 it is $41 \cdot 41$.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-16T18:37:07.373" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2722" PostTypeId="2" ParentId="2714" CreationDate="2014-06-16T21:21:58.340" Score="11" Body="&lt;p&gt;How many binary sequences are there that have period $n$?  The sequence $\ldots110110110\ldots,$ for example, has period 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we wanted to compute the number of binary sequences with period $6.$  There are $2^6$ binary strings of length $6,$ but not all of them, when infinitely repeated in both directions, produce a sequence with period $6$.  Among the strings of length $6,$ we find examples such as $111111,$ $101010,$ and $110110,$ which produce sequences with periods $1,$ $2,$ and $3,$ rather than $6.$  We can break the set of strings of length $6$ down as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;there are $2$ strings that give period $1$, namely $111111$ and $000000;$&lt;/li&gt;&#xA;&lt;li&gt;there are $2$ strings that give period $2$, namely $101010$ and $010101,$ the second of which is a cyclic permutation of the first;&lt;/li&gt;&#xA;&lt;li&gt;there are $6$ strings that give period $3$, namely $110110$ and its two cyclic permutations, and $100100$ and its two cyclic permutations;&lt;/li&gt;&#xA;&lt;li&gt;there are $54$ strings that give period $6,$ including $111110$ and its five cyclic permutations, $111100$ and its five cyclic permutations, $111010$ and its five cyclic permutations, and so on.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The answer to our question is therefore $9:$ there are nine classes of strings that give period $6$, each consisting of six strings that are cyclic permutations of each other, and therefore $9$ sequences of period $6.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's how we could have used Möbius inversion to compute this number.  Let $M(n)$ be the number of binary sequences with period $n.$  Each such binary sequence could have been produced by any of $n$ strings, which are all cyclic permutations of each other.  From this observation, we have&#xA;$$2^n=\sum_{d\mid n}d M(d).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applying the Möbius inversion formula to this equation, we get&#xA;$$nM(n)=\sum_{d\mid n}\mu(d)2^{n/d}$$&#xA;or &#xA;$$M(n)=\frac{1}{n}\sum_{d\mid n}\mu(d)2^{n/d}$$&#xA;Applying this to $n=6$, we get&#xA;$$M(6)=\frac{1}{6}(\mu(1)2^6+\mu(2)2^3+\mu(3)2^2+\mu(6)2^1)=\frac{1}{6}(64-8-4+2)=9.$$&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2014-06-16T21:21:58.340" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2723" PostTypeId="1" CreationDate="2014-06-16T22:56:56.137" Score="7" ViewCount="250" Body="&lt;p&gt;Recently I asked the following identity on Math.StackExchange knowing it had several proofs:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \binom{n}{k}^2 \geq \binom{n}{k-1}\binom{n}{k+1} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://math.stackexchange.com/questions/834786/log-concavity-of-binomial-coefficients-binomnk2-geq-binomnk-1-bin&quot;&gt;here&lt;/a&gt;. They quickly give the one-liner:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In your initial inequality divide the left hand side by the right hand side and simplify.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is kind of a proof.  It works on math.SE since many people know combinatorics there. &lt;br&gt; Probably they meant:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Using the identity $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ and cross multiplying, the identity is equivalent to: $$ k!^2 (n-k)!^2 \leq (k-1)!(k+1)! (n-k-1)!  (n-k+1)! $$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Let's try proving: $k!^2 \geq  (k-1)!(k+1)!$ Dividing both sides by $k!^2$ we get $1 \leq \frac{k+1}{k-1}$ which is true.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A similar proof works for the factors of $(n-k)!$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Overall, people had no idea where I was coming from.  I had been reading a Richard Stanley article about &lt;a href=&quot;http://math.mit.edu/~rstan/pubs/pubfiles/72.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;log-concave sequence of numbers&lt;/a&gt; and I was hoping to digest it for my class.  The logic made sense to me:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Instead of clearing denominators every single time (which maybe neither exciting nor informative), why not exploit the that $a_n^2 \geq a_{n-1}a_{n+1}$ for this particular sequence of numbers?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Eventually two other proofs surfaced using other branches of math, but I still don't feel like I got any point across.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, how do I motivate even this one proof?  Is this equation &lt;em&gt;really&lt;/em&gt; that obvious that it only deserves one sentence?&lt;/p&gt;&#xA;" OwnerUserId="1654" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-08-17T23:03:36.767" Title="Motivating combinatorics identity to different audiences: $ \binom{n}{k}^2 \geq \binom{n}{k-1}\binom{n}{k+1} $" Tags="|undergraduate-education|computer-science|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2724" PostTypeId="2" ParentId="2691" CreationDate="2014-06-17T00:07:27.917" Score="1" Body="&lt;p&gt;I recommend the &lt;a href=&quot;http://en.wikipedia.org/wiki/Hall%27s_marriage_theorem&quot; rel=&quot;nofollow&quot;&gt;Hall's theorem&lt;/a&gt; which has a nice inductive proof by cases (some people like to call it &quot;either it's trivial or it's trivial&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The formal wording is like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $G$ be a bipartite graph with bipartition $U \uplus V$ and denote by $\mathcal{N}(S)$ the set of neighbors of vertices in $S$. Then, there is a $U$-saturating matching in $G$ if and only if $$\forall S \subseteq U.\ |\mathcal{N}(S)| \geq |S| .$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, this can be rephrased in more friendly terms, for example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose there are two groups of girls and boys respectively, both with the same count. For any boy and girl (i.e. any potential pair) they would like to dance together or not. There is a perfect pairing (everybody has a dance partner) if and only if for any subset of boys there is at least the same number of girls willing to dance with them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The proof proceeds by induction on the size of the set $U$ and the inductive step has two cases:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\forall \varnothing \neq S\subsetneq U.\ |\mathcal{N}(S)| \color{red}{&amp;gt;} |S|.$ In this case we match any pair and the pairing for the rest follows from inductive hypothesis.&lt;/li&gt;&#xA;&lt;li&gt;$\exists \varnothing \neq S \subsetneq U.\ |\mathcal{N}(S)| \color{red}{=} |S|.$ In this case let $S$ be such a set which is minimal with regard to inclusion. We split vertices of $G$ into $G_1$ and $G_2$ so that $U(G_1) = S$, $V(G_1) = \mathcal{N}(S)$ and by induction hypothesis each part has its own matching.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-06-17T00:07:27.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2725" PostTypeId="2" ParentId="2723" CreationDate="2014-06-17T00:34:45.247" Score="0" Body="&lt;p&gt;It depends on the audience... I (and you, clearly) don't need more; my second year students (struggling with basic factorial identities and such) need a lot more handholding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question then is how to do the handholding. I have found that it is helpful to state what proof technique will be used (even if &quot;obvious&quot; from the proof itself), and write out most steps, even if rather simple algebra. If on the blackboard, you can skimp steps, but be prepared to fill in when asked (and you end up spending more time that way). This helps when fine-tuning.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-17T00:34:45.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2726" PostTypeId="2" ParentId="2645" CreationDate="2014-06-17T00:52:14.607" Score="3" Body="&lt;p&gt;The real answer is you should discuss this openly with the father.  Rather than viewing this as him questioning your method, think of you two as a team, dedicated to the goal of educating the child.  Teams can freely discuss strategy with one another.  He brought you in for your teaching skill, but he may have some insight as well - after all, it is his kid, he should know pretty well how he learns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is pure speculation, but my guess is that the father feels that learning how to answer multiple choice questions about math is not the same as learning math, and that the free-form questions are, at least in some sense, closer.  Hypothetically, if one knows how to write out full, complete solutions to every problem, one should also be able to get the answer on multiple choice without any issues.  (There is also a peripheral benefit to this in that the kid will do better in future mathematics, after the test is over.)  He may not realize that you are teaching math over test method, and you reassurance that this is the case could be exactly what is needed to appease him.&lt;/p&gt;&#xA;" OwnerUserId="263" LastActivityDate="2014-06-17T00:52:14.607" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2727" PostTypeId="2" ParentId="978" CreationDate="2014-06-17T02:12:51.137" Score="34" Body="&lt;p&gt;I have lots of experience tutoring students like that. The main problem is that they are convinced that they can't be good in math, so your task is more that of a psychologist than a math instructor. Also it is very common to see that the ground problem is a deficiency in the very basics. I have seen people that somehow make it almost all the way to university without knowing the multiplication tables, and I wouldn't be surprised if that was part of her problem. How do they manage to get that far is beyond me, but it happens more than you would think, and I have seen it both in Mexico (my home country) and in Germany.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is my list of suggestions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Don't argue with her or try to prove her wrong. That will only make things worse and will reinforce her feeling of not being able to do it. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Whenever she makes a mistake try something like &quot;Ok, maybe we can try this a different way&quot;, trying to imply that it was maybe you the one that didn't explain things that clear. It will lower her defenses and rejection to what you explain after that. Use your best poker face.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to diagnose exactly where the problem(s) is/are. If she was able to do things right once, she will be able to learn it, so try to forget the prejudice of her not being able to do it. If you think that, you shouldn't be teaching her. Whenever I take a new student, I am always convinced I can make him or her learn whatever it is they need to learn, and then I can do it. In over 20 years of tutoring, I have never found someone I can positively say is incapable of learning math, and I've had some real hard cases.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another very important distinction that needs to be made is there are two types of things to learn with mat: procedural, which need to be memorized and logic which need to be understood. Try to divide every exercise in these two components, and make sure that once the logic is understood, the student realizes that the rest is memory training, and that there is nothing more to be understood. More often than not, people have the feeling there must be something hidden beyond what they understood; something really complicated and it is always useful to explicitly dispel this feeling. Usually after an exercise I try to explicitly point out what was the objective of the exercise, or do it beforehand, depending on whether I use the exercise to reach a conclusion or to reinforce a previous explanation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to find other holes in her basics. Usually this repeated failure to learn a simple procedure stems from a whole weak foundation, so make sure to do a summary of the basic fraction operations, making sure she can add, subtract, multiply and divide properly, both fractions and real numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This I would say is probably the most important step. Whenever she does things right, always point it out in a positive manner, like &quot;See? There is nothing here you can not do&quot;, or &quot;it wasn't as hard as it looked, or was it?&quot;. Usually after a few sessions of working with a student, I start pointing out he or she can be among the first in their class, and this usually works wonders on their motivation. Of course I have to first convince myself, but in over 20 years of tutoring has consistently worked.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another very important aspect, which unfortunately in your case might not be that easy, is working also on the parents. You have to convince them also that their kid is capable of being among the first, because more often than not, this idea of &quot;being bad for math&quot; is believed to be inherited and most parents constantly reinforce on their children that they can not do it. In my experience once people find out they can be good in math, they start liking it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This is probably the most difficult. Once you find a student with whom you will be having to work on the basics for a few long sessions, it is tempting to feel frustrated and bored. However, I always see it as a challenge to try to figure out which tools does the student actually have and design a strategy to teach, only with those tools, whatever that has to be taught. I find really fascinating to witness the process people follow while learning new things, and I consider it an honor to be allowed to be part of it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also as important is finding out if she sleeps enough or not too much. Altered sleeping patterns are usually a sign of depression, and they affect people's cognitive abilities. Lack of sleep makes people forgetful, and it could be responsible for some of her problems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Practice. Keep in mind that to fix something in your memory, you have to repeat it a few times during the first week, then review it after 3 or 4 weeks, and then review it after 4 or 5 months. After this point, you will most likely not forget it. She is probably not practicing what you explain to her, so next time it looks as if she had a factory reset.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to keep track of her attention span by constantly asking questions. Many people with math problems have very short attention spans and it is hard to keep them focused.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Expose openly to her things that you don't know and make her see it is not such a big deal not to know something. That might make her feel comfortable admitting the same to you. Sometimes even making a few mistakes on purpose, or admitting it openly and without embarrassment when you make one helps to make the student feel comfortable with admitting their mistakes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The last point would be not thinking that you are enlightening her. Teaching is team work, as it is learning with a teacher. Convince her that you are a team and that her success is your success. Whenever you have to tell her something hard to digest try to talk in plural. As an example, instead of saying &quot;You need to learn scientific notation&quot;, you might choose a different wording like &quot;we need to review scientific notation&quot;. Keep in mind that the wording you choose says more about what you think than what you would like to reveal. To counteract this problem, you have to first convince yourself of her capabilities and then convince her.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given her defensive behavior, one way of finding out whether she knows the multiplication tables is pretending you are having problems to remember the multiplication at hand (as if you were trying to remember it) and see how long she takes to complete it. If she doesn't seem willing to help, you can say something like &quot;help me&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also try to find something she is good at and ask her to try to teach you or explain you something. That will put you in her mind at a similar level to hers and might reduce her level of defensiveness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;13 . One extremely important point I had forgotten. Stress the fact that most geniuses have changed the world by asking themselves questions no one did before. Of course it is the answers what made them famous, but as Voltaire once said: &quot;Judge a man by his questions rather than by his answers&quot;. So when she asks something that reflects insight, always point out that it was a very good question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck.&#xA;DP&lt;/p&gt;&#xA;" OwnerUserId="1742" LastEditorUserId="1742" LastEditDate="2014-06-18T14:46:06.667" LastActivityDate="2014-06-18T14:46:06.667" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2728" PostTypeId="2" ParentId="2645" CreationDate="2014-06-17T05:27:26.460" Score="4" Body="&lt;p&gt;You really need to have a conversation with the parent and &lt;strong&gt;listen more than talk&lt;/strong&gt;. Make the parent feel heard, and compromise a little (note this can be a 1% to 99% percent compromise). Ask him why he feels the way he feels. You likely agree with some aspect of his reasoning. See how you can incorporate the things you agree with. The student will ultimately do better if there isn't this lingering (somewhat unrelated to learning) argument between parent and tutor. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While your first priority should always be the student's best interest, being a tutor is different than being a teacher. If your tutoring sessions are one-on-one and you are paid by the parent, then the parent is your client (in a business sense) and hence you have to make a big effort to make the parent happy in addition to maximizing the student's learning. &lt;/p&gt;&#xA;" OwnerUserId="1387" LastActivityDate="2014-06-17T05:27:26.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2729" PostTypeId="2" ParentId="2718" CreationDate="2014-06-17T08:32:13.000" Score="5" Body="&lt;p&gt;Given $ax + b = 0$ with $a \neq 0$, I can find a formula for $x$. In particular, we have $x = -b/a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, for $ax^2 + bx + c = 0$ with $a \neq 0$, I can also find a formula for $x$. In particular, the quadratic formula gives all possible values for $x$ (there can be up to $2$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the cubic case, one can again come up with an analogous formula, which gives all possible values for $x$ (there can be up to $3$). Probably you see where this is going...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the quartic case, we have a formula (i.e., using the four operations and radicals) to identify the different possible values of $x$ (here, indeed, there can be up to $4$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By analogy, one would really expect there to be a &quot;quintic formula.&quot; But there's not (in a strong sense, i.e., not just because neither you nor I can think of one), and the &lt;em&gt;reasoning by analogy goes wrong&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Meanwhile, the idea of &quot;possible values&quot; needs to be made precise, since you might end up allowing for complex values of $x$; this notion does generalize nicely, i.e., a polynomial with degree $n$ can have up to $n$ distinct complex roots.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the &quot;reasoning by analogy gone wrong&quot; described above is frequently mentioned early on in a class on, say, Galois Theory to motivate its study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(And speaking of classical problems: Similar comments could be made about trisecting an angle...)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Below is an example that is a bit simpler to explain in full.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume you do not know if a two digit number is prime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Here is a primality test:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, memorize the twelve by twelve times table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, check if the given number is even (look at units digit), a multiple of three (check if the digital sum is a multiple of three), or a multiple of five (look at units digit).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third, if the number does not appear in the times table and has no obvious factors of 2, 3, or 5, then it must be &lt;strong&gt;prime&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, this approach to finding primes does not work so well for larger numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, it misses a two digit number: 91, the product of 7 and 13.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Quite a number of students will misidentify 91 as prime; try presenting it in a list of numbers to a student, and ask him or her to circle all the primes given some reasonable time constraint.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, for numbers through 100 one needs to check prime factors up to its square root, i.e., 2, 3, 5, and 7. But the tricks for 2, 3, and 5 are very simple, and those of us who memorized twelve by twelve tables get the multiples of 7 up through 84 (hence the missed 91, whereas 98 is even).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, I think there is a nice example in here somewhere, and I think it is related to the sense that multiplication tables contain &quot;all the products&quot; in a stronger sense than they really do. For example, the ten by ten table is missing all primes from 11 on. As for composites, it is missing plenty, starting at 22; it just so happens that the primality test described above is effective for all two-digit numbers besides 91.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-06-18T08:51:31.317" LastActivityDate="2014-06-18T08:51:31.317" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2730" PostTypeId="1" CreationDate="2014-06-17T14:13:02.943" Score="13" ViewCount="722" Body="&lt;p&gt;Different properties of different types of orders including partial, total, scattered and well-orders are a part of any graduate/undergraduate set theory course. I am looking for interesting &quot;non-mathematical&quot; examples of these orders in physics, biology, sociology, politics, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are some interesting &quot;non-mathematical&quot; examples of orders and other mathematical objects of a same nature? Any useful reference is welcome. &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="77" LastEditorDisplayName="user230" LastEditDate="2017-12-26T16:00:17.647" LastActivityDate="2017-12-26T16:00:17.647" Title="Non-Mathematical Examples of Orders" Tags="|reference-request|course-design|examples|" AnswerCount="7" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2731" PostTypeId="2" ParentId="2730" CreationDate="2014-06-17T14:24:46.407" Score="4" Body="&lt;p&gt;The classic example of a non-mathematical total ordering is people's heights. Each person is either taller or shorter than another. And if Alice is taller than Bob, who is taller than Carol, then Alice must be taller than Carol as well. Although measuring their heights converts this example into a numerical comparison, it's not strictly &quot;mathematical&quot; in nature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of a partial ordering might be a corporate hierarchy. A CEO commands multiple managers, each of whom supervises multiple employees. Although a manager is strictly superior to his own employees, he may or may not have any relation to the employees that other managers supervise.&lt;/p&gt;&#xA;" OwnerUserId="856" LastActivityDate="2014-06-17T14:24:46.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2732" PostTypeId="2" ParentId="2730" CreationDate="2014-06-17T14:52:42.633" Score="6" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Partial_order#Formal_definition&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; names genealogical descendancy as a partial order.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-06-17T14:52:42.633" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2733" PostTypeId="1" CreationDate="2014-06-17T14:55:16.043" Score="17" ViewCount="1542" Body="&lt;p&gt;This is a question I originally posted on Math Stackexchange.&#xA;I've just seen a very good discussion of Monty Hall problem, and someone mentioned the &quot;siblings&quot; paradox. I've had some success explaining the Monty Hall, but ran into some tough resistance in the siblings paradox. A summary of the arguments I faced is below.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So here is the problem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a well-known paradox (mentioned, among other sources, by Martin Gardner).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A) You know that I have two children, and the oldest is a daughter. What is the probability that both are daughters? (let's ignore the possibility of identical twins, which may muddle the question; and, for simplicity, assume that probability that a child is a daughter is exactly 1/2). This is very easy to answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B) Now, suppose you know that at least one child is a daughter (but you don't know if it's the oldest one or not). What is the probability that both are daughters?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The common way to answer is: you got extra information that the number of daughters, let's call it D, is at least one.&#xA;Thus, $P(D=2 | D \ge 1)= P(D=2,D \ge 1)/P(D \ge 1)=(1/4)/(3/4)=1/3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way to answer (B) is to consider all possible outcomes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The paradox is that the answers in (A) and (B) are different!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, a student told me that this is a wrong way to look at it. How do you know that at least one child is a daughter? Perhaps you overheard me saying &quot;my daughter ... (just had a birthday or something)&quot;. But in this case you were more likely to overhear in the first place, if I had had two daughters, and not one. This messes up the probabilities calculated above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To put it another way: let's generate the list of random 2-child families. Let's randomly &quot;overhear&quot; the gender of one random child in each family on the list. [This is what I foolishly suggested the student do!] Ignore sons and concentrate on daughters. Calculate which proportion of the families we picked had two daughters. Then your answer is 1/2 again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you describe a &lt;strong&gt;realistic&lt;/strong&gt; situation in which you can come up with the information &quot;at least one girl&quot;, and obtain the &quot;right&quot; answer of 1/3?&lt;/p&gt;&#xA;" OwnerUserId="1750" LastActivityDate="2016-02-25T19:48:12.050" Title="Explaining the &quot;siblings&quot; paradox" Tags="|math-puzzle|probability|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2734" PostTypeId="2" ParentId="978" CreationDate="2014-06-17T16:53:09.083" Score="23" Body="&lt;p&gt;This is more a comment than an answer, but it's too long to make a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When dealing with students who are couched in defensive self-confidence issues, you can often disarm them by asking earnest, simple questions about their own work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Watching someone 'simplify' $\frac{63}{3}$ to $\frac{61}{1}$ is an excellent teachable moment, but I'm afraid that you've taken a wrong turn. When seeing a move like this (I see such moves often) it's a great time to drop everything involving the context of the current exercise and ask some questions to reveal the person's understanding of their own actions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a script for this particular situation.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You've simplified $\frac{63}{3}$ to $\frac{61}{1}$ here. I'd like to think about that. Tell me - ballpark - what is 63 divided by 3?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It's not uncommon to get a blank stare at this point. Don't let it linger very long - it's demeaning. Follow up quickly with a reminder that the notation $\frac{a}{b}$ refers to the same thing as $a \div b$, and form the present question into a canonical division example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If you and I and Fred were sharing 63 dollars equally, how much would we each get?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Press for some approximate answer. You'll almost certainly get something between 15 and 30, which is good enough to proceed.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;OK, so $\frac{63}{3}$ is roughly $x$, right?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A little less than half of students will jump through the rest of the logic at this point and realize that they've made an error. If they do, congratulate them on sussing it out, and remind them that these sorts of estimatory checks on work in progress catch lots of errors. Go back to their work, and ask them to justify the simplification. They'll either fail to say anything, or claim it's a memorized rule. Remind them that it's very rarely a good idea in mathematics to exercise procedures from memory, but rather they should aim to take steps whose meaning they understand perfectly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the other half, keep going.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Alright then, so what is 61 divided by 1?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Blank stare is possible again at this point. It's absolutely critical that you handle this with no hint of incredulity, or with any whiff of disdain for their abilities. Follow it the same way as before:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Division by one is almost a trick question - if I have 61 dollars and I don't share it with anyone, how much am I left with?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some conversation here is required to identify the $1$ in the denominator with the count of people sharing in the money. Talk through $\frac{6}{6}$, $\frac{6}{3}$, $\frac{6}{2}$, and $\frac{6}{1}$, until you are convinced that there is enough mutual agreement on the meaning and interpretation of the division notation, and what division means in the first place.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Alright, so $\frac{63}{3}$ is about $x$, but $\frac{61}{1}$ is $61$, and you've replaced one with the other from this line to that line. Is this a problem?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;We're into the hard part now. About half of the students at this point will recognize that a mistake must have occurred, and that it's not OK to replace the $\frac{63}{3}$ with $\frac{61}{1}$, since they represent different values. For these students, proceed as above - ask for their justification of the original simplification, yadda yadda. The other students are going to need serious work, but I'm going to cap my reply here before it really gets out of hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main point is to concentrate on the results rather than the procedure. Your post indicates to me that you concentrated on trying to convince your friend that she couldn't cancel the denominator with only one digit in the numerator. This is very common even amongst math educators, but that's a shame! Instead, work with her to develop the equipment necessary to decide whether procedures are valid. Students never stop learning new incorrect procedures until they can read and think about the results that they are producing. It's a difference between bailing out a leaky boat and plugging the hole altogether.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2014-06-19T10:54:33.763" LastActivityDate="2014-06-19T10:54:33.763" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2735" PostTypeId="2" ParentId="2733" CreationDate="2014-06-17T16:58:49.680" Score="9" Body="&lt;p&gt;I think there are a few independent questions here that can be considered teaching mathematics (I will not broach the subject of how to solve the problem).  First, there is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can you state this problem unambiguously without using artificial language?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Indeed, as I see it, the reason this problem sounds paradoxical is that the phrase &quot;one of them is a daughter&quot; is used slightly differently in English than in mathematics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In English, to say that &quot;you know that one of his children is a daughter&quot; can mean &lt;em&gt;either&lt;/em&gt; that you know the formal statement $\exists i (\text{$x_i$ is a daughter})$ (where the children are $x_1$ and $x_2$) &lt;em&gt;or&lt;/em&gt; that you know its instantiation $\text{$x_i$ is a daughter}$ for one value of $i$.  In fact, the latter is preferred, since this sentence seems to default to talking about &quot;one of his children&quot; as a &lt;em&gt;place in the family&lt;/em&gt; filled by a person, rather than the person themselves.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In math, the phase defaults to the former statement (quantified), though because of the linguistic ambiguity, if the latter is meant it is often written in the modified form &quot;you know that a particular one of his children is a daughter&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, there is no common modified form that indicates that the quantified statement is the one intended, which leads to lawyerly language in attempting to specify it, and this makes the problem sound unnatural.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second problem is really annoying, in my experience.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can you present an everyday model for the precise statement of this problem, averting any possible objection along the lines of &quot;who would do that&quot; or &quot;that would never happen&quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is always the danger of trying to meet someone halfway from mathematical abstraction to familiar reality: if they care, it's never far enough, and if they don't care, then there was no point in abandoning abstraction in the first place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;s&gt;Here is a realistic scenario that seems to work: you see a man in his early thirties waiting outside a schoolyard at the end of the school day, obviously to pick up his child.  You chat, and he comments that his wife had to take &quot;our other kid&quot; to the doctor today.  Soon, a girl comes out and meets him, and they depart.  What's the probability that their &quot;other kid&quot; is a girl?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this covers all bases.  The point of the age and setting being specified is to create an image in the mind of the listener that nothing can be deduced about the relative likelihood of meeting a particular child in this place. A man of that age probably has two kids in a similar school year, so you can't tell by looking whether you're seeing the older or the younger one (kids always get sick at that age!).  The implication is that both would have been there but for illness.  The language is neutral, and it doesn't seem to me that there is any greater chance of a child of a particular gender or (in that range) age getting sick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This relies heavily on interpretation, but then, as discussed above, that's the whole point and problem.&lt;/s&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a scenario that &lt;em&gt;actually&lt;/em&gt; models the precise statement.  It is quite difficult to naturally talk indirectly about an unspecified child!&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You meet a pregnant woman that you don't know and, as you talk, you politely inquire about her family.  She tells you it's her third child, and jokes that although the house will be a lot louder, at least it won't be as bad as with three boys.  You guess this means that the newborn will be a girl, but she says that she didn't want to learn the gender in advance, so doesn't know.  What's the probability that her current two children are both girls?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Looking past the fluff, I believe this story says exactly and &lt;em&gt;no more than&lt;/em&gt; that she has two children, and one of them is a girl.&lt;/p&gt;&#xA;" OwnerUserId="513" LastEditorUserId="513" LastEditDate="2014-06-18T16:50:44.987" LastActivityDate="2014-06-18T16:50:44.987" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2736" PostTypeId="1" AcceptedAnswerId="2748" CreationDate="2014-06-17T17:58:52.960" Score="5" ViewCount="698" Body="&lt;p&gt;I have been teaching my student BODMAS and giving him problems like &#xA;$$4\times4-6+8\div4+10\div2+5-4\times2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While solving the problems he asked me that, &quot;So small answers for such big questions?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I sadly had no answer other than &quot;yes&quot;, but I sense that this might be interpreted as unfruitful in some students if not particularly mine, so is there any justification for this problem ?&lt;/p&gt;&#xA;" OwnerUserId="1586" LastEditorUserId="1685" LastEditDate="2014-06-22T03:29:02.493" LastActivityDate="2014-06-23T00:58:00.217" Title="How to justify single digit answers to long problems?" Tags="|mathematical-pedagogy|" AnswerCount="8" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2737" PostTypeId="2" ParentId="2736" CreationDate="2014-06-17T18:18:46.927" Score="1" Body="&lt;p&gt;I'd offer him another example, namely take 2 * (any 20 digit number) and show the result. But. When you divide the result by the original number you get 2. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, a string of +, *, and / can land you anywhere, a single digit, to many digits. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When looking particularly at * and /, it's not too tough to explain these are inverse functions, and Y*4/4=Y, so if you average equal numbers of multiplications and divisions, and the numbers you are using, you'll land close to where you started. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-06-17T18:18:46.927" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2738" PostTypeId="2" ParentId="2730" CreationDate="2014-06-17T18:27:51.503" Score="7" Body="&lt;p&gt;Lexicographical (alphabetical) order is an example of total order, and it can be helpful to introduce a total order in  $\mathbb{R}^n$&lt;/p&gt;&#xA;" OwnerUserId="1754" LastEditorUserId="1754" LastEditDate="2014-06-17T21:49:52.633" LastActivityDate="2014-06-17T21:49:52.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2739" PostTypeId="2" ParentId="2730" CreationDate="2014-06-17T19:10:15.440" Score="11" Body="&lt;p&gt;Preference relations in economics. e.g. I prefer Big Macs to Whoppers (denoted $\mathrm{Big Mac} \succ \mathrm{Whopper}$) and that induces a lattice on the space $\{\mathrm{BigMac},\mathrm{Whopper},\mathrm{Baconator}\}$. This idea has applications in choice theory, matching, and game/auction theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The preference-ordering framework is notably used to prove &lt;a href=&quot;http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem&quot;&gt;Arrow's Impossibility Theorem&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1756" LastActivityDate="2014-06-17T19:10:15.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2740" PostTypeId="2" ParentId="2679" CreationDate="2014-06-17T20:45:18.513" Score="0" Body="&lt;p&gt;Originally, the chances are 1/3 that the car is behind Door 1, 1/3 that the car is behind Door 2, and 1/3 that the car is behind Door three. If you choose Door 1, your chances of winning are now 1/3. More to the point, the chances are 2/3 that the car is behind Doors 2-and-3 together. This fact will hold true for the rest of the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Monte Hall opens Door 3 and shows a goat, meaning that there are ZERO chances of the car behind Door 3. But the chances are still 2/3 that the car is behind Doors 2-and-3 together. What's left of this &quot;together?&quot; Door 2. ALL of these chances are now behind Door 2, which &quot;inherited&quot; Door 3's chances. So the increase in your chances of winning by &quot;switching,&quot; to Door 2, are 2/3 to 1/3, or 2 to 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example: Suppose there was an election with Candidates A, B, and C, each with 10 votes. Suppose candidate C (Door 3) drops out and says, don't vote for me, vote for B over A (and his supporters follow him faithfully so he gets zero votes). What happens next? Candidate B gets candidate C's ten votes plus his own ten for 20, versus ten votes for candidate A.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2014-06-18T01:27:47.870" LastActivityDate="2014-06-18T01:27:47.870" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2742" PostTypeId="2" ParentId="1330" CreationDate="2014-06-17T21:19:20.790" Score="1" Body="&lt;p&gt;I would use &lt;strong&gt;polar coordinates.&lt;/strong&gt; &quot;Everyone&quot; is familiar with working in Cartesian coordinates. Not everyone knows how to translate these into polar coordinates. But the fact of the matter is that polar coordinates make (easily) accessible a number of applications and &quot;spaces&quot; that would be difficult to manage using Cartesian coordinates.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-06-17T21:19:20.790" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2743" PostTypeId="2" ParentId="2730" CreationDate="2014-06-17T22:01:19.437" Score="1" Body="&lt;p&gt;Anything that can be modelled by a directed acyclic graph is an order if extended suitably (transitive closure). So relations like &quot;is the boss of,&quot; &quot;branches off from&quot; in a tree, &quot;descends from&quot; in genealogy (but be sure to consider both mother and father to make for more interesting examples).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-17T22:01:19.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2744" PostTypeId="2" ParentId="2730" CreationDate="2014-06-18T01:13:43.997" Score="8" Body="&lt;p&gt;Some examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pieces of clothing, shoes, etc., for example you put socks before shoes.&lt;/li&gt;&#xA;&lt;li&gt;Building towers using toy blocks, e.g. you need to use some blocks to form a base, then put some elements in the middle and so on.&lt;/li&gt;&#xA;&lt;li&gt;Any tree-like structure induces an ordering, for example genealogical tree, or even an ordering of leaves of some real tree if there's no &lt;a href=&quot;http://en.wikipedia.org/wiki/Inosculation&quot;&gt;inosculation&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Subjects to learn, i.e. how more advanced subjects list other as prerequisites.&lt;/li&gt;&#xA;&lt;li&gt;Cooking, for example to bake a cake you need the dough and some topping which are prepared separately (each using its own steps) and joined later. Some recipes are really complicated!&lt;/li&gt;&#xA;&lt;li&gt;Any class of things ordered by its features, e.g. cars (speed, acceleration), buildings (height), mobile devices (screen size, processor, camera). Yet, this is not very non-mathematical.&lt;/li&gt;&#xA;&lt;li&gt;As for orders without lower bound, I would recommend human stupidity (and with its multitude of flavors, it might be even a partial order).&lt;/li&gt;&#xA;&lt;li&gt;As for total orders, you could try students by the time they can sit straight without making any noise.&lt;/li&gt;&#xA;&lt;li&gt;For more exotic orderings, one could use the &lt;a href=&quot;http://en.wikipedia.org/wiki/Dershowitz%E2%80%93Manna_ordering&quot;&gt;Dershowitz-Manna ordering&lt;/a&gt; to get nice results:&#xA;&lt;ul&gt;&#xA;&lt;li&gt;let's assume that we partially order chess pieces by relative strength: $$\text{Pawn} &amp;lt; \text{Knight},\text{Bishop} &amp;lt; \text{Rook} &amp;lt; \text{Queen}$$&lt;/li&gt;&#xA;&lt;li&gt;we can order some two (multi)sets $A$ and $B$ of chess pieces $A &amp;lt; B$ by&#xA;$$\text{if $A$ includes more pieces of some kind, then $B$ contains } \\\text{more pieces of some other kind that dominates the first.}$$&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Added two more orders.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2014-06-18T08:28:50.093" LastActivityDate="2014-06-18T08:28:50.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2745" PostTypeId="2" ParentId="2736" CreationDate="2014-06-18T07:14:13.600" Score="2" Body="&lt;p&gt;The point of problems like these is not that a long expression can be reduced to a short one. Instead, the idea is that expressions need to be well-defined, i.e., non-ambiguous. To attain such a goal, one introduces the notion of an &quot;order of operations&quot;; for you, this means BODMAS, while those in the U.S. are probably more familiar with PEMDAS.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to get this point across would be to present a single expression and talk about different ways you could interpret it (i.e., depending on the order in which you carry out the various operations). You could then enter the expression into a calculator several times. Note that the calculator gives you the same answer every time, and ask the student if he thinks the calculator will ever give a different answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It won't. Why? Because the calculator is programmed to follow a particular order of operations; that way, two people with the same computation to carry out will not somehow end up with different results (assuming that they don't make any errors of their own). Just as there is an order of operations implemented in the calculator, there is one to which we subscribe, as well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the reasoning is the same: to eliminate ambiguity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might go further and ask him about different values an expression can take on depending on where you put the brackets (or, as those in the U.S. would say, parentheses). Even for a single operation: Does it matter where the brackets go? This can lead naturally to a discussion of binary operations for which associativity holds. For example, $a+(b+c) = (a+b)+c$ for any $a,b,c \in \mathbb{Z}$; that is to say, for the set of integers, addition is an associative binary operation. But what happens when the operation is subtraction rather than addition? Similarly, $a\times(b\times c) = (a \times b)\times c$ for any $a,b,c \in \mathbb{Z}$; that is to say, for the set of integers, multiplication is an associative binary operation. But what about division?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to the student's astute observation: Expressions can be written in many ways. For example, the relatively &quot;short&quot; expression $3^{3^{3}}$ has only three numbers, but is equal to $7,625,597,484,987$. (If you raise it to $3$ once more, then you get an obscenely large number.) Meanwhile, you could write $1 - 1$ a bunch of times with a $+$ in between each of them; the sum would then be $0$. So here we could have a &quot;long&quot; expression that, when evaluated, gives a short answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The salient point is that a well-defined expression for your purpose refers to exactly one number. There are many equivalent ways to write a single expression; some short, some long. In fact, our decimal notation is really a way of abbreviating, so that, e.g., $365 = (3\times 100) + (6 \times 10) + (5 \times 1)$. In fact, I could have written the previous expression without brackets, given that BODMAS makes it unambiguous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;As a final note:&lt;/strong&gt; Students are &lt;em&gt;often&lt;/em&gt; asked to &quot;simplify&quot; expressions. The idea of such exercises is to remove ambiguity, and, the hope is, it will be clear for any particular class and teacher what constitutes a correct simplification. For example, many teachers ask that students &quot;rationalize denominators&quot; and re-write something like $\displaystyle \frac{1}{\sqrt{2}}$ as $\displaystyle \frac{\sqrt{2}}{2}$, thereby ensuring the denominator is an integer. Generalizing far beyond your student's query, one might ask whether this notion of simplifying is &lt;em&gt;itself&lt;/em&gt; unambiguous. You can find more on this question in a MathOverflow post &lt;strong&gt;&lt;a href=&quot;https://mathoverflow.net/questions/126519/is-there-a-mathematical-definition-of-simplify&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;, though I mention this more for the curious reader than as a matter of responding to your student.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-06-18T07:14:13.600" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2746" PostTypeId="2" ParentId="2679" CreationDate="2014-06-18T08:10:00.020" Score="1" Body="&lt;p&gt;Once I set up a bet, using a bunch of (future) lotto drawing. Lotto in Italy uses 90 numbers, and traditionally those from 1 to 30 are marked 1, those from 31 to 60 are marked 2, nd those from 61 to 90 are marked X. I would have taken the role of Monty Hall, and I explained in advance the algorithm which I will use to show a result (among 1, 2, X) which did not appear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exposing the algorithm makes people think that Monty Hall cannot usually choose a random door (or a random value, in my example). At least for me, in this way is easier to understand that for the opponent too the choices are not casual.&lt;/p&gt;&#xA;" OwnerUserId="26" LastActivityDate="2014-06-18T08:10:00.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2747" PostTypeId="2" ParentId="2736" CreationDate="2014-06-18T09:54:28.473" Score="4" Body="&lt;p&gt;Tell him that it is a common occurrence (esp. in science) that the complexity of the problem is a whole different matter than the complexity of its formulation. The issue here is with our perception, i.e. what seems easy or complex to us.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example &lt;a href=&quot;http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem&quot; rel=&quot;nofollow&quot;&gt;Fermat's last theorem&lt;/a&gt; can be worded very concisely, while its proof is very long and difficult. Similarly, a detailed answer to a short question like &quot;why a plane can fly&quot; involves a lot of physics, aerodynamics, flow equations and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand some apparently long and tedious calculations might have simple answers, for example if some big number is even, or when the final result is to be multiplied by $0$, or calculating integrals of odd functions or limits with multiple insignificant terms, etc. It also happens outside math, e.g. in a restaurant when the waiter takes an order, the menu is actually a part of the question, yet the answer is probably relatively short.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discrepancy between complexity of the problem and its formulation is rather common, so a more interesting question would be where our intuition comes from (i.e. why do we have it if it is wrong so often)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have no idea, but if I were to speculate:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;because of too little experience, with more experience we get better intuition (i.e. Fermat's last theorem wouldn't seem simple to an experienced mathematician);&lt;/li&gt;&#xA;&lt;li&gt;in ancient times $\text{bigger} = \text{stronger}$ was actually almost always true;&lt;/li&gt;&#xA;&lt;li&gt;human mind doesn't handle scale and complexities well, we mostly use thumb-rules that fail when any higher complexity is involved;&lt;/li&gt;&#xA;&lt;li&gt;in school students rarely face problems of high complexity, tasks are picked by the teacher and longer formulation often does mean more work.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-06-18T09:54:28.473" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2748" PostTypeId="2" ParentId="2736" CreationDate="2014-06-18T10:41:25.130" Score="10" Body="&lt;p&gt;A string of random multiplications, divisions, additions, and subtractions, even one involving single-digit numbers, isn't likely to have a nice, small answer.  Ask your student to create some problems of his own and solve them.  This will teach him two things:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;that nice answers aren't typical,&lt;/li&gt;&#xA;&lt;li&gt;that the teachers who created these problems actually took a great deal of care to ensure that the answers came out to nice numbers.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then tell him that the reason they did that was so he could focus on learning order of operations without getting distracted by lots of hairy arithmetic.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2014-06-18T10:41:25.130" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2749" PostTypeId="1" CreationDate="2014-06-18T11:16:08.753" Score="9" ViewCount="635" Body="&lt;p&gt;It seems that a lot of high school seniors contemplating college are too naive about borrowing and repaying a loan to tell whether a given loan choice is rational or irrational.  Or so we are being told.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would think that the prospect of taking out a major loan, and one from which there is no bankruptcy protection, would be an ideal situation to make a subject like compound interest into a study of real life, practical math.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal would be two-fold.  First, there's an immediate goal of producing millions of college applicants who are capable of &quot;doing the math&quot; when they are offered a loan and need to decide whether to sign or not to sign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second goal is to expose them to some math that goes beyond simple linear proportion problems that they have been dealing with for several years in grade school, and to motivate the lesson with a real-life application.&lt;/p&gt;&#xA;" OwnerUserId="1707" LastActivityDate="2014-06-18T17:44:35.560" Title="What is the best way to teach compound interest in high school?" Tags="|secondary-education|course-design|usa|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2750" PostTypeId="2" ParentId="2749" CreationDate="2014-06-18T11:45:53.773" Score="2" Body="&lt;p&gt;Combine it with computational modelling / simulation / software!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most students can't see the connection between sequence or function definitions, proportions and the total money. You need to make it more visual.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Use a simple programming language or high level modelling software to compute iterations of the total value depending on simple models in advertisements/leaflets.&lt;/li&gt;&#xA;&lt;li&gt;Use plots of functions of different loan cost models / interest models. Let them compare and analyse, under what conditions what model is better.&lt;/li&gt;&#xA;&lt;li&gt;Create graphical presentations of case overviews: a medical student and an arts student at bank A. The medical student at bank A or bank B.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-06-18T11:45:53.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2751" PostTypeId="2" ParentId="2736" CreationDate="2014-06-18T12:00:04.660" Score="-1" Body="&lt;p&gt;This question is clearly not mathematically motivated. The student might accept a mathematical answer on a cognitive level, but it will not touch or him on the aesthetic or emotional level where this question comes from.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So give some non-mathematical analogy, for example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Sports&lt;/strong&gt; Let's compare two games: basketball and association football. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Basketball with equally good teams nearly every move, effort and engagement will result in points. The more effort, engagement or moves, the more points for either team. So at the end of a game you may have 93 to 98 points (or whatever). In short: many moves and large results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In association football with equally good teams, most moves, effort and engagement will not result in points. They have no obvious impact at all. Only a very small number of moves will result in goals and consequently points. So at the end of a game, you may have 1 to 0 points. In short: many moves and small results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math is so flexible, it can be anything: many numbers/operations and large results or many numbers/operations and small results.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-06-18T12:00:04.660" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2752" PostTypeId="2" ParentId="2733" CreationDate="2014-06-18T12:09:20.730" Score="5" Body="&lt;p&gt;I see three issues:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the limitations of the English language,&lt;/li&gt;&#xA;&lt;li&gt;information leakage,&lt;/li&gt;&#xA;&lt;li&gt;sampling.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Issue 1 was addressed by Ryan Reich.  In brief, there is no natural English phrase corresponding to the mathematician's &quot;I have two children, at least one of whom is female.&quot;  Real people just don't talk that way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Issue 2: in real life, when people talk about their children, they reveal information about them.  Any story that allows one to conclude that a specific individual child is female requires one to assign a probability of $1/2$ to the other child being female.  This includes Ryan Reich's example, so I have to side with PA6OTA on that one.  The child in the schoolyard is a specific individual, as is the child visiting the doctor.  Each independently has a $50\%$ chance of being female.  Learning that that the one in the schoolyard is female does not influence the probability that the one at the doctor is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any story one might make up is likely to be isomorphic to Gardner's older child / younger child problem.  You sit next to a stranger on the bus and start talking about your families.  She tells you she has two children but doesn't specify female or male.  Later in the conversation she mentions that there was a birthday party for one of the children yesterday, saying &quot;she got lots of presents.&quot;  Now you know that one of the children is female.  But you know more than that: you know that the child whose birthday it was yesterday is female.  (Maybe both had birthdays yesterday?  Real people would have mentioned that, but even if not, see below.)  Even if your traveling companion's reference to &quot;she&quot; was disconnected from any story and gave you no specific information whatsoever about her child, real people will be thinking of a particular child when they use the pronoun &quot;she&quot;.  This allows you to conclude that a particular child&amp;mdash;the one your companion was thinking of when she used the word &quot;she&quot;&amp;mdash;is female.  The probability that the other child&amp;mdash;the one she wasn't thinking of&amp;mdash;is female is therefore $1/2.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Returning to the birthday example, it is possible, using mathematician-speak, to partially, but not completely specify an individual.  This is the &quot;Tuesday&quot; problem: a mathematician says to you &quot;I have two children, at least one of whom is a girl born on Tuesday.&quot;  In some ways this is like the problem with answer $1/3:$ we know that at least one of the children is a girl, but not which specific child.  We do know something else however, namely that a girl was born on Tuesday.  Having been born on Tuesday is not enough, in contrast to my earlier examples, to specify her as a unique individual.  After all, this is a mathematician speaking, and the other child might also have been born on Tuesday.  On the other hand, Tuesday is only one of seven days of the week, so the probability that the other child was born on Tuesday is small.  So it is likely, but not certain, that this information uniquely specifies the child.  If you do the calculation, you find that this partial specification is enough to drive the probability that the other child is a girl from $1/3$ almost, but not quite, to $1/2.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is what I referred to as the information leakage issue: in any real example, there is likely to be $100\%$ information leakage: enough information will leak to specify the female child as a unique individual.  But as the Tuesday example shows, even partial leakage is enough to change the probability away from $1/3.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Issue 3 is the one identified by your student.  Any real-life story is likely to involve implicitly sampling from a population, and you need to have a decent model of the sampling procedure to do any calculations.  I actually had never thought about this issue in quite this form.  Your student is extremely perceptive and obviously thought very hard about the problem.  This is the kind of student we like to have!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My conclusion: get away from stories involving individuals, and make the sampling procedure clear.  Here's my attempt:  you have an internship with the census bureau and your division is doing an analysis of two-child families.  One of your colleagues has a hypothesis about two-boy families, and takes all of the files for those families to her office in order to perform an analysis.  You pick a random file from those remaining in the filing cabinet.  What is the probability that you pick a two-girl family?&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2014-06-18T12:09:20.730" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2753" PostTypeId="2" ParentId="2749" CreationDate="2014-06-18T13:35:52.743" Score="5" Body="&lt;p&gt;Instead of a computer simulation, just give them a couple problems that require them to calculate compound interest for different principal amounts and interest rates, then have &lt;em&gt;them&lt;/em&gt; plot the results. Use realistic numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, give them a more extended problem in which they have an annual income, annual expenses, and a compounded-interest loan. Ask them to figure out how much they can borrow given a certain income, or how much income they need to finance a given loan, or how much they can spend given a loan and their income. Maybe give them a choice between no college, community college, and 4-year college, generate starting post-grad starting salaries that correspond to real data, and have them figure out whether the loans are worth it.&lt;/p&gt;&#xA;" OwnerUserId="1756" LastActivityDate="2014-06-18T13:35:52.743" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2754" PostTypeId="1" CreationDate="2014-06-18T14:30:26.670" Score="4" ViewCount="18745" Body="&lt;p&gt;If you want to convert for example $2\,\mathrm{\frac{m}{s}}$ to $\mathrm{\frac{km}{h}}$ you have just to multiply it by $3.6$ and get $2\,\mathrm{\frac{m}{s}} = 7.2\,\mathrm{\frac{km}{h}}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally I memorize this by knowing the factor $3.6$ (makes sense since the hour has $3600$ seconds and $1\, \mathrm{km}$ has $1000$ meters) and that the value before the $\mathrm{\frac{km}{h}}$ has to be higher. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a good way how can students memorize that the $\mathrm{\frac{km}{h}}$ has to be higher? I often see that the divide by 3.6 instead of multiplying or vice versa. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is there some simple intuitve reason, you can easily memorize why the $\mathrm{\frac{km}{h}}$ has to be higher?&lt;/li&gt;&#xA;&lt;li&gt;Is there any real life situation you can intuitively see that the $\mathrm{\frac{km}{h}}$ has to be higher?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am not looking for a verbal memnomic but more for a conceptual one. I am also not looking for a formal derivation, which is simple and most of my studends can do it, if they have to (but which clearly is too long just for resonable memorization).&lt;/p&gt;&#xA;" OwnerUserId="1203" LastActivityDate="2021-12-29T19:34:28.930" Title="Best way to memorize the conversion between m/s and km/h" Tags="|physical-sciences|" AnswerCount="7" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2755" PostTypeId="2" ParentId="2736" CreationDate="2014-06-18T14:54:10.683" Score="1" Body="&lt;p&gt;I like Will Orrick's answer in this thread. I'd like to propose a possible investigation for students to help them answer their own question of why  &quot;So small answers for such big questions?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Begin with $2 + 2 = 4$ and ask:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it possible to make this expression on the left any length and still have the result come to 4?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Have a student or group of students add pairs (and/or triplets and/or singletons) of operations that return the result to 4 every time. You'll end up with a special case (in which each pair brings it back to 4) but perhaps your student or students, as a result of this investigation, will have some of their own conjectures about whether there is any necessary relationship between the length of the expression on the left and the result on the right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more variety in your pairs of operations, you could specify a range of single digit results, or $0...5$ or whatever. Instead of $4$.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-06-18T14:54:10.683" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2756" PostTypeId="2" ParentId="2749" CreationDate="2014-06-18T17:44:35.560" Score="5" Body="&lt;p&gt;I would start by making them produce an amortization table for a compound interest loan &quot;by hand&quot; (in a spreadsheet program like Excel).   You can have them do a standard amortization table, and then ask the students to look at what happens when (due to some income-based repayment program or other) they are paying less than the computed payment, or worse, less than the total interest due each month, and what happens if they pay an extra $100 per month above what the computed payment for a 10-year amortization is.  I think working these examples out manually really drives home how interest works, and how much making reduced payments hurts you, and how much making extra payments saves you.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And yes, then have them compare various scenarios, like whether it is better to take a loan at 6.8% that does accrues interest while they are in college, versus one at 8.0% that does not.  (The answer to that depends in part on how many years of college they have left, and how quickly they expect to be able to pay off the loan afterwards.)  So that can be a more open-ended question where you ask them to state and justify assumptions, or compare the result for different assumptions, or even figure out where the cutoffs are for when one is better than the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Although I believe federal student loans are actually simple and not compound interest.)&lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-06-18T17:44:35.560" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2757" PostTypeId="2" ParentId="2733" CreationDate="2014-06-18T18:07:23.310" Score="3" Body="&lt;p&gt;As with the Monty Hall problem, it is important to know &lt;em&gt;exactly&lt;/em&gt; what information you have. Let's say I offer a parent a present of 100 dollars if they have two girls, and 10 dollars if they have at least one girl. If that parent answers &quot;I have at least one daughter&quot;, the probability for having two daughters is practically zero (because a parent with two daughters would have said &quot;two girls&quot; to get the 100 dollars, and not the ten dollars). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's say I have prepared cards saying &quot;I have two boys&quot;, &quot;I have a boy and a girl&quot;, &quot;I have two girls&quot;, &quot;I have at least one boy&quot;, &quot;I have at least one girl&quot;. I ask the parent to pick cards at random until one with a true statement comes up. The card says &quot;I have at least one girl&quot;. With two girls, the chance of giving this answer is 50%. With a boy and a girl, the chance of giving this answer is 33.33%. A bit of maths shows the chance of having two girls is (1/4) / (1/4 + 1/6) = 60%. &lt;/p&gt;&#xA;" OwnerUserId="1723" LastEditorUserId="1723" LastEditDate="2014-07-02T16:29:36.847" LastActivityDate="2014-07-02T16:29:36.847" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2758" PostTypeId="2" ParentId="2736" CreationDate="2014-06-18T18:48:09.867" Score="1" Body="&lt;p&gt;Ask your student if he would rather have a long answer. If he says &quot;yes,&quot; ask him why, and that should end the whole business.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, you can point out that the question is actually quite short. It asks the student to simplify a single expression. One task, one answer. If the answer were &quot;longer,&quot; it wouldn't be a good answer. Outside of the classroom, people ask questions for the explicit purpose of taking a long idea and making it short. That's why people hire other people to do math in the first place, because they don't have time to sort through the long stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A related alternative is to explain that much of the point of mathematics is to find understandable order in complexity. Ask your student whether the long form or the simplified form is easier to understand and work with.&lt;/p&gt;&#xA;" OwnerUserId="1756" LastActivityDate="2014-06-18T18:48:09.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2759" PostTypeId="1" CreationDate="2014-06-18T18:50:02.083" Score="3" ViewCount="107" Body="&lt;p&gt;This question is not directly related to Mathematics.However, I am hoping to find some guidance from educators here. Some faculty members are required to perform service to the community. Providing free education that improve societal value is one of the many ways to deliver the service. However, we recently are faced with a difficult situation in this channel, and I hope faculty here who had experience in teaching outside the school or social science researchers working in underprivileged groups will be able to point us to the right direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are a small group of software developers and teachers who give free computer lessons to pretty much anyone who is interested. So far we have taught 32 students and they are all working as programmers in various places. However the dropout rate has been 8 to 1 so far. But lately the big problem is finding new students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We don't charge money, there are no contracts to sign and we even provide required software and books free of charge. Sometimes we even supply the computers. The classes are online last about a hour a day. Still, even with sharing success stories and having former students themselves talk to new students, we are finding it hard to find new students who would be interested in taking the classes. I personally know people who would rather complain about the unfairness of life than take a job working as a paid intern somewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: What can we do to make free learning more appealing to people who are not in a school environment?&lt;/p&gt;&#xA;" OwnerUserId="1768" LastActivityDate="2014-06-18T18:50:02.083" Title="How do I encourage learning to an underprivileged group" Tags="|course-design|online-instruction|" AnswerCount="0" CommentCount="4" ClosedDate="2014-06-19T07:10:59.820" ContentLicense="CC BY-SA 3.0" />
  <row Id="2761" PostTypeId="2" ParentId="2723" CreationDate="2014-06-18T21:53:51.977" Score="1" Body="&lt;p&gt;Does this help?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(k-1)!(k+1)! = (\frac{k!}{k})(k!(k+1)) = k!^2\frac{k+1}{k} \ge k!^2\frac{k}{k}=k!^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and for $(n-k)!$ define $a=(n-k)$ and use the same trick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way would be to note that ${n\choose{k-1}} = \frac{1}{k}{n\choose{k}}$ and that ${n\choose{k+1}} = k{n\choose{k}}$&lt;/p&gt;&#xA;" OwnerUserId="1773" LastEditorUserId="1773" LastEditDate="2014-06-18T22:04:09.657" LastActivityDate="2014-06-18T22:04:09.657" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2762" PostTypeId="2" ParentId="2754" CreationDate="2014-06-18T22:52:23.983" Score="13" Body="&lt;p&gt;Better don't learn such conversion factors, learn how to derive such on the fly. This helps also in the case you want to change inches per second into feet per hour, or cubic feet per hour into cubic centimeters per second.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-18T22:52:23.983" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2763" PostTypeId="1" CreationDate="2014-06-19T02:00:27.870" Score="42" ViewCount="7182" Body="&lt;p&gt;I've had this discussion with a couple of friends. I argued that teaching multiplication as repeated addition isn't a good idea because it doesn't help children differentiate between the two operations and see them as independent. My friends argued that it would otherwise be too hard and that multiplication as repeated addition yields a more intuitive approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there research on what, if it exists, is a good method for teaching multiplication and keeping it as an independent operation of addition?&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="47" LastEditDate="2014-06-19T21:02:41.877" LastActivityDate="2021-12-28T13:09:50.740" Title="Is it advisable to avoid teaching &quot;multiplication as repeated addition&quot;?" Tags="|mathematical-pedagogy|primary-education|" AnswerCount="12" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="2764" PostTypeId="2" ParentId="2763" CreationDate="2014-06-19T02:11:08.533" Score="10" Body="&lt;p&gt;Why do you feel the description of multiplying as repeated addition is problematic? Hopefully it's just a brief stage, but an important one. &#xA;I'd even suggest that this is exactly the method we all use when multiplying larger numbers in our heads. I don't know 21 times 21, but can easily multiply 21 times 20, and then just add another 21. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In response to JPBurke's comment, I suggest a viewing of Vi Hart's &lt;a href=&quot;https://www.youtube.com/watch?v=N-7tcTIrers&amp;amp;feature=kp&quot; rel=&quot;noreferrer&quot;&gt;How I feel about logarithms&lt;/a&gt;, in which she starts off suggesting that for integer math, even adding 3 and 5 is &lt;em&gt;really&lt;/em&gt; a series of repeated &lt;em&gt;plus ones&lt;/em&gt;. She then goes on to support my view that &lt;em&gt;multiplication is adding&lt;/em&gt;, at least for integers. Respectfully, I understand comments to be brief requests for clarification or short remarks. Unfortunately, they have a minute long edit time, and I intended to return to add the link once I found it. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-06-19T12:15:54.837" LastActivityDate="2014-06-19T12:15:54.837" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2765" PostTypeId="2" ParentId="2763" CreationDate="2014-06-19T06:40:02.107" Score="42" Body="&lt;p&gt;There's a lot in this brief question, and I would like to try to give a brief answer, so I'm going to pick and choose what I respond to (and others might choose different things). Here are the parts of this question I see:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is teaching multiplication as repeated addition problematic?&lt;/li&gt;&#xA;&lt;li&gt;Is the problem with teaching multiplication as repeated addition that it makes it difficult to differentiate the two operations (or see them as independent)?&lt;/li&gt;&#xA;&lt;li&gt;Is multiplication too hard to teach without using repeated addition?&lt;/li&gt;&#xA;&lt;li&gt;Is multiplication as repeated addition more intuitive?&lt;/li&gt;&#xA;&lt;li&gt;Is there research on teaching methods for multiplication that do not use repeated addition?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Not all these are questions you are asking directly, but they are implicit (such as when your friends assert that repeated addition is more intuitive -- I have problematized it).&lt;/p&gt;&#xA;&lt;p&gt;We could also well ask &amp;quot;Is multiplication repeated addition?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Because if we're going to wonder whether we should teach multiplication &lt;em&gt;as&lt;/em&gt; repeated addition, we should probably at least consider whether it &lt;em&gt;is&lt;/em&gt; repeated addition. If it &lt;em&gt;is&lt;/em&gt; repeated addition, then this isn't a problem to teach it that way. If it &lt;em&gt;isn't&lt;/em&gt; repeated addition, then why would we teach it as repeated addition? Now, repeated addition may be a strategy students use in certain situations. But that's different from teaching multiplication as repeated addition.&lt;/p&gt;&#xA;&lt;p&gt;OK, I'm going to draw heavily from &lt;a href=&quot;http://www.jstor.org/discover/10.2307/749486?uid=3739696&amp;amp;uid=377752083&amp;amp;uid=2&amp;amp;uid=3&amp;amp;uid=3739256&amp;amp;uid=60&amp;amp;sid=21104173713927&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simon and Blume (1994)&lt;/a&gt; because it's such an interesting paper for many reasons and it references other resources that address multiplicative reasoning, and this will allow me to be a little lazy. The article is actually about elementary teachers understanding of area as a product of linear measures.&lt;/p&gt;&#xA;&lt;p&gt;In this paper, Simon and Blume (1994) on page 474 reference earlier works by Kaput and by Schwartz which points out that multiplicative reasoning can result in the production of intensive quantities (that is, a quantity that is not counted or measured directly and is invariant with the scale of the system). This has implications in understanding proportion later. We see intensive quantities in division. Miles traveled divided by hours elapsed produces an entirely new quantity (speed).&lt;/p&gt;&#xA;&lt;p&gt;The paper suggests &amp;quot;intensive&amp;quot; vs. &amp;quot;extensive&amp;quot; quantities may help us understand not just whether a student is solving a multiplication problem, but how sophisticated their approach is. This idea comes from Thompson (1994) who observed students solving problems relating to speed without conceiving of speed as an extrinsic quantity, revealing something about the sophistication of their conceptual understanding.&lt;/p&gt;&#xA;&lt;p&gt;All that is simply to point out that there is some complexity to how learners may think about multiplication, and it is worth paying attention to not only because it is part of how multiplication makes sense.&lt;/p&gt;&#xA;&lt;p&gt;The property of multiplicative reasoning to produce a quantity that is different from the factors in the problem can be considered a referent-shifting aspect of multiplication. Simon and Blume explain:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Schwartz&#xA;argues&#xA;that&#xA;multiplication and division  are &amp;quot;referent  transforming&amp;quot; operations,&#xA;because&#xA;they&#xA;take two&#xA;quantities&#xA;with different referents as&#xA;input&#xA;and&#xA;output&#xA;a third&#xA;quantity&#xA;whose  referent is  different from  either of&#xA;the  first  two&#xA;(in  Example&#xA;1: number  of  cookies  x  number  of  dollars/cookie&#xA;=  number  of&#xA;dollars).&#xA;Schwartz and&#xA;Kaput&#xA;further&#xA;point&#xA;out  that  the  notion&#xA;of&#xA;multiplication as&#xA;repeated&#xA;addition  is&#xA;problematic&#xA;because addition  is  referent&#xA;preserving,&#xA;whereas&#xA;multiplication&#xA;is  referent&#xA;transforming.&#xA;Repetitions&#xA;of  addition therefore cannot&#xA;yield&#xA;the  referent that  is&#xA;appropriate for  the&#xA;product&#xA;in  a&#xA;multiplicative  situation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, yes to #1 -- there is reason to believe that repeated addition is a problematic way to teach multiplication: it may obscure the referent-transforming aspect of multiplication. To #2, you may want to investigate the idea of independence further, but clearly there are other reasons that repeated addition has a different meaning than multiplication. In some sense: yes; failing to make clear the meaning of multiplication is a lost opportunity to separate the operations from one another.&lt;/p&gt;&#xA;&lt;p&gt;I'll address #4 in a limited way by saying that we don't necessarily want to reinforce student intuitions. So the question of whether it is more intuitive or not may be moot. Is it helpful to our students in achieving the ability to reason multiplicatively?&lt;/p&gt;&#xA;&lt;p&gt;I will address #3 and #5 together by directing you to a couple of other resources, but also by giving you an example.&lt;/p&gt;&#xA;&lt;p&gt;Kouba and Franklin (1995) give a brief overview of their view of the research on introducing students to multiplication and division. Their conclusion is that students need a varied conceptual basis for multiplication and division. They include an example of students using objects they can touch in the process of scalar multiplication (which they may accomplish by using repeated addition as one of their own strategies).&lt;/p&gt;&#xA;&lt;p&gt;However, as a teacher, how can we help students conceptualize multiplication in a way that is consistent with Kaput's and Schwartz's observation that multiplication involves a referent transformation? I look to an example in &lt;em&gt;Mathematics for elementary teachers&lt;/em&gt; (Beckmann, 2010).&lt;/p&gt;&#xA;&lt;p&gt;Dr. Beckmann gives a number of different examples for modeling multiplication, but in one example she uses an array of soft drink cans to show how the idea of cans-in-groups and then number of groups can be put in a useful representational structure. She points out that the rows or the columns can be used as the groups. I would also add that this lets us see that this is not just repeated addition of cans; this representation really does show referent transformation: &lt;code&gt;cans per row * rows = cans&lt;/code&gt; also &lt;code&gt;cans per column * columns = cans&lt;/code&gt;. This is conceptually different from saying &amp;quot;what's three times five cans?&amp;quot; One obvious difference is that the numbers have meaning in the array model, when we talk about them as a number of groups, or a size of a group.&lt;/p&gt;&#xA;&lt;p&gt;And, if you want to discuss repeated addition, this model allows us to show why, in this case, repeated addition gives us the correct answer for the multiplication problem.&lt;/p&gt;&#xA;&lt;p&gt;Repeated addition can be something you do, but it can be separate from a conception of what multiplication is.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;In summary&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;There is reason to question the teaching of multiplication &lt;em&gt;as&lt;/em&gt; repeated addition&lt;/strong&gt; on the basis of the other meanings and understandings of multiplication we want for our students. This is discussed in some of the research that highlight the differences between the reasoning that repeated addition produces and multiplicative reasoning.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Students need a varied conceptual basis&lt;/strong&gt; to form an understanding of multiplication; examples of these can be found in the resources cited, along with representations that support them.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Student use of repeated addition is one strategy&lt;/strong&gt;. I gave an example of how a representation could possibly be used to connect this strategy to another conceptual basis for multiplication.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Cited:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Beckmann, S. (2010). Mathematics for elementary teachers. New York: Pearson Addison-Wesley.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.ccsu.edu/mitchell/math412tcmmultdivsensemakingmeaning.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kouba, V. L., &amp;amp; Franklin, K. (1995). &lt;strong&gt;Research into Practice: Multiplication and Division: Sense Making and Meaning&lt;/strong&gt;. &lt;em&gt;Teaching Children Mathematics&lt;/em&gt;, 1(9), 574–77.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.jstor.org/discover/10.2307/749486?uid=3739696&amp;amp;uid=377752083&amp;amp;uid=2&amp;amp;uid=3&amp;amp;uid=3739256&amp;amp;uid=60&amp;amp;sid=21104173713927&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simon, M. A., &amp;amp; Blume, G. W. (1994). &lt;strong&gt;Building and understanding multiplicative relationships: A study of prospective elementary teachers&lt;/strong&gt;. &lt;em&gt;Journal for Research in Mathematics Education&lt;/em&gt;, 472–494.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://pat-thompson.net/PDFversions/1994ConceptSpeedRate.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thompson, P. W. (1994). &lt;strong&gt;The development of the concept of speed and its relationship to concepts of rate&lt;/strong&gt;. &lt;em&gt;The Development of Multiplicative Reasoning in the Learning of Mathematics&lt;/em&gt;, 179–234.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="6423" LastEditDate="2021-12-28T13:09:50.740" LastActivityDate="2021-12-28T13:09:50.740" CommentCount="22" ContentLicense="CC BY-SA 4.0" />
  <row Id="2766" PostTypeId="1" AcceptedAnswerId="2770" CreationDate="2014-06-19T10:40:40.253" Score="5" ViewCount="146" Body="&lt;p&gt;I was discussing this with a colleague, who showed some papers on &quot;interactive engagement&quot; in physics education. Another keyword there was &quot;force concept inventory&quot;, and the realted (newer) &quot;calculus concept inventory&quot;. Any references to good empirical papers?&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="123" LastEditDate="2014-06-20T00:41:47.407" LastActivityDate="2014-06-20T00:41:47.407" Title="Empirical papers on &quot;interactive engagement&quot; in calculus education?" Tags="|undergraduate-education|reference-request|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2767" PostTypeId="2" ParentId="2754" CreationDate="2014-06-19T13:48:43.957" Score="7" Body="&lt;p&gt;To answer the question &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there any real life situation where you can intuitively see that the km/h number is higher?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and seeing that you are based in Germany:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.bahn.com/i/view/USA/en/trains/overview/ice.shtml&quot; rel=&quot;noreferrer&quot;&gt;ICE&lt;/a&gt; goes up to 300 km/h, but nobody suggests that it comes anywhere near the &lt;a href=&quot;http://en.wikipedia.org/wiki/Speed_of_sound&quot; rel=&quot;noreferrer&quot;&gt;speed of sound&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This, of course, presupposes that one knows the rough speed of high speed trains in km/h, and also the rough speed of sound in m/s (whose value at sea level around 20 centigrade is about 340). &lt;/p&gt;&#xA;" OwnerUserId="125" LastActivityDate="2014-06-19T13:48:43.957" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2768" PostTypeId="2" ParentId="2733" CreationDate="2014-06-19T14:11:41.840" Score="2" Body="&lt;p&gt;Given two genders (daughters and sons), and two birth slots, there are four permutations of gender and birth order. They are, 1) DD, 2) DS, 3) SD, 4) SS. They all have equal chances of occurring, after &quot;rounding.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you know that at least &lt;strong&gt;one&lt;/strong&gt; of the children is a daughter, the three allowable permutations are, 1) DD, 2) DS, and 3) SD. The fourth permutation, SS, has been eliminated from consideration. &quot;Both daughters&quot; is the first permutation, one out of three allowable ones, and the chances of that are one third. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you know that the &lt;strong&gt;oldest&lt;/strong&gt; child is a daughter, the allowable permutations are &quot;only,&quot; &#xA;1) DD and 2) DS. (The third permutation has been eliminated, because this is a more restrictive condition than the previous one.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The chances of DD, or both daughters, is one out of two in this restricted space. &lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2014-06-20T22:20:25.767" LastActivityDate="2014-06-20T22:20:25.767" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2769" PostTypeId="1" AcceptedAnswerId="2771" CreationDate="2014-06-19T15:58:43.770" Score="14" ViewCount="3462" Body="&lt;p&gt;I'm not a math teacher but am actively involved in teaching my children mathematics (elementary age).  I learned the traditional &quot;long&quot; approach to multiplication, but the school systems now emphasize (or perhaps require) the use of the partial products method.  And, I was also recently introduced to the term &quot;grid&quot; method.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know if these other newer methods of teaching multiplication have good evidence to show improved learning outcomes?  (I consider them &quot;newer&quot; only in the sense that they were not part of my elementary education which was a looooong time ago.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit:  I would be very interested to know whether there is a &quot;systematic review&quot; or, preferably, a &quot;meta-analysis&quot; on this topic.  From an empirical standpoint, is there a consensus to support one method or another on any particular learning outcome?&lt;/p&gt;&#xA;" OwnerUserId="1734" LastEditorUserId="143" LastEditDate="2014-06-21T19:43:45.883" LastActivityDate="2015-09-23T13:42:13.237" Title="Traditional &quot;long&quot; method of multiplication versus grid and partial products -- evidence of better outcomes?" Tags="|primary-education|mathematical-pedagogy|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2770" PostTypeId="2" ParentId="2766" CreationDate="2014-06-19T16:23:02.287" Score="3" Body="&lt;p&gt;The following web sites seem to be good starting places:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://math.arizona.edu/~lomen/conceptests.html&quot; rel=&quot;nofollow&quot;&gt;http://math.arizona.edu/~lomen/conceptests.html&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.math.cornell.edu/~GoodQuestions/index.html&quot; rel=&quot;nofollow&quot;&gt;http://www.math.cornell.edu/~GoodQuestions/index.html&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The &quot;news&quot; link from the Cornell site has a link labeled &quot;Preliminary Project Results (password needed)&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;People:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Jerome Epstein&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;David O. Lomen&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Scott Pilzer&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maria K. Robinson&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maria Terrell&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Papers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Epstein, &quot;Development and validation of the calculus concept inventory&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Epstein, &quot;The calculus concept inventory -- measurement of the effect of teaching methodology in mathematics,&quot; Notices of the AMS 60 (2013) 1018&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pilzer, &quot;Peer instruction in physics and mathematics,&quot; Primus XI (2001) 185&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lomen and Robinson, &quot;Using conceptests in single and multivariable calculus&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thomas and Lozano, &quot;Analyzing calculus concept inventory gains in introductory calculus&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of these papers can be found online by googling.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-06-19T16:33:42.073" LastActivityDate="2014-06-19T16:33:42.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2771" PostTypeId="2" ParentId="2769" CreationDate="2014-06-19T17:02:00.997" Score="14" Body="&lt;p&gt;You will probably not find precisely what you're looking for if you want to compare one algorithm to another for effectiveness. The reason is, the problem that was identified in the 70's (Erlwanger, 1973) was not about the unsuitability of the algorithm itself. It could better be thought of as a problem of focusing on algorithm. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Erlwanger (1973) demonstrated that mastery of the content did not imply understanding. So, researchers began to consider whether teaching for understanding was possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there evidence that a shift to teaching for understanding &quot;worked?&quot; Yes, in the sense that students were able to achieve computational fluency &lt;em&gt;not&lt;/em&gt; at the expense of conceptual learning and problem solving. An example of a study that demonstrated this is this longitudinal study of helping teachers teach for understanding (Fennema et al., 1996).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But rather than write a lit review on teaching for understanding, I'll point you to Schoenfeld's chapter (2008) in &lt;em&gt;Powerful Learning&lt;/em&gt;. He does an excellent job explaining mathematics learning as a sense-making endeavor, and includes quite a bit of a bibliography to the research that counts as evidence for the approach. It's a great resource, I think, for parents who want to understand some things that have changed about mathematics education as a result of real research.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another great resource is &lt;a href=&quot;https://class.stanford.edu/courses/Education/EDUC115N/How_to_Learn_Math/about&quot;&gt;Jo Boaler's How To Learn Math online EdX course&lt;/a&gt;. She mentions her book (What's math got to do with it?) as well. But I believe the EdX course is free. She's a researcher and educator who has done influential work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not what you were looking for, but &lt;a href=&quot;http://www.edutopia.org/inquiry-project-learning-research&quot;&gt;here is an article that is a snippet of the Powerful Learning book&lt;/a&gt;, by some of the authors. It doesn't have references, but it does discuss why learning for understanding is a thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Reorganizing into points:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Newer approaches to mathematics education that value conceptual understanding are often going to involve more sense-making on the part of the students (ideally). So algorithms take a back seat, meaning that there isn't a competition between algorithms. &lt;/li&gt;&#xA;&lt;li&gt;However, there is a reluctance to teach an algorithm that backgrounds sense-making because a student may not be motivated to work toward conceptual understanding if they know they can just use the standard algorithm. Analogy: algorithms are technology, like calculators. If you knew the calculator (or algorithm) was going to give you the answer, and it was ever-present, you might feel the urge skip the conceptual work and just use the calculator.&lt;/li&gt;&#xA;&lt;li&gt;There is evidence that a lack of conceptual understanding could hide in procedural fluency. This was discovered in the 70's.&lt;/li&gt;&#xA;&lt;li&gt;Use of alternative algorithms may help students not because the alternative algorithm is better. Other reasons are paramount.For instance, an approach may allow teachers to see student mathematical reasoning and intervene (formative assessment). It may allow flexibility in approach, changing an activity from procedural practice to actual problem solving based on number sense and other mathematical knowledge.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I hope this is helpful.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cited&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.wou.edu/~girodm/library/benny.pdf&quot;&gt;Erlwanger, S. H. (1973). &lt;strong&gt;Benny’s conception of rules and answers in IPI mathematics&lt;/strong&gt;. &lt;em&gt;Journal of Children’s Mathematical Behavior&lt;/em&gt;, 1(2), 7–26.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.jstor.org/discover/10.2307/749875?uid=3739696&amp;amp;uid=377752083&amp;amp;uid=2&amp;amp;uid=3&amp;amp;uid=3739256&amp;amp;uid=60&amp;amp;sid=21104175950457&quot;&gt;Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., &amp;amp; Empson, S. B. (1996). &lt;strong&gt;A Longitudinal Study of Learning to Use Children’s Thinking in Mathematics Instruction&lt;/strong&gt;. &lt;em&gt;Journal for Research in Mathematics Education&lt;/em&gt;, 27(4), 403–434.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=ZQ-wKAAACAAJ&amp;amp;dq=powerful%20learning&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=4xOjU7qXMcXesASApoHoDw&amp;amp;ved=0CEoQ6AEwCA&quot;&gt;Schoenfeld, A. H. (2008). &lt;strong&gt;Mathematics for understanding&lt;/strong&gt;. In L. Darling-Hammond, B. Barron, P. D. Pearson, A. H. Schoenfeld, E. K. Stage, T. D. Zimmerman, … J. L. Tilson, &lt;em&gt;Powerful Learning: What We Know About Teaching for Understanding&lt;/em&gt; (1st ed., pp. 113–150). San Francisco, CA: Jossey-Bass.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-06-19T17:02:00.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2772" PostTypeId="2" ParentId="2769" CreationDate="2014-06-19T18:05:35.630" Score="1" Body="&lt;p&gt;Whatever is the &quot;best&quot; way of doing it (or teaching it) is mostly irrelevant for your case. Just go with the way your children are taught at school. Teaching them another method will just confuse them, get them into trouble when they use the &quot;better&quot; method in class/exams, and create useless arguments with their classmates over the matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[If you ask me, I'll get them a cheap calculator and teach them what to &lt;em&gt;do&lt;/em&gt; with the operations, instead of drilling them in algorithms they'll hardly ever use in earnest in later life. But that's just me, not your neighborly school board.]&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-19T18:05:35.630" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2773" PostTypeId="2" ParentId="2763" CreationDate="2014-06-19T18:10:40.107" Score="9" Body="&lt;p&gt;If we're talking about &lt;strong&gt;young children&lt;/strong&gt; (elementary school) being taught multiplication for the first time, there is &lt;em&gt;no alternative&lt;/em&gt; to presenting it as repeated addition. Let's assume they have addition down. &lt;strong&gt;They're just not going to be able to grok any other concept when just being introduced to it, given their limited toolkit.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once they have down the concept of &quot;2 x 3 is 2 + 2 + 2&quot;, and can generalize it to &quot;m x n is m + m + ...&quot; (for specific integers), you can start expanding to multiplying multidigit integers (first, rote digit-by-digit and add subresults, and then the concept of shift-multiply digit-add). At some point they'll be ready to handle negative numbers and decimal numbers and even fractions, but not right off the bat.&lt;/p&gt;&#xA;" OwnerUserId="1781" LastActivityDate="2014-06-19T18:10:40.107" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2774" PostTypeId="2" ParentId="2754" CreationDate="2014-06-19T19:17:54.443" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;It's important to be able to do every conversion of complex units knowing base unit conversions. (Competence)&lt;/li&gt;&#xA;&lt;li&gt;It's very practical in physics courses to have the conversion of km/h and m/s memorized.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;That said, let's go to answer your question:&lt;/p&gt;&#xA;&lt;h2&gt;Smaller/Larger&lt;/h2&gt;&#xA;&lt;p&gt;As a side effect of the above 1., students should already know, that the bigger the unit the smaller the number before when doing conversions. For km/h and m/s it's &lt;em&gt;intuitively seeming&lt;/em&gt; to be the other way around: In km/h the number is larger than in m/s, although km/h contains the larger units than m/s. Do an example with them. From this moment on, the students will remember: for these units it's the other way around.&lt;/p&gt;&#xA;&lt;h2&gt;Number&lt;/h2&gt;&#xA;&lt;p&gt;For the number: It will stick after a small number of computations. Explicitely do exercises with speeds of 1 m/s, 10 m/s, 100 m/s, 1000 m/s. The students will 3.6 so often, they will just remember it.&lt;/p&gt;&#xA;&lt;h2&gt;Side Remark&lt;/h2&gt;&#xA;&lt;p&gt;I guess, the above only works in countries using the metric system. Imperial units have so many different conversion numbers, that 3.6 doesn't seem any special and won't stick around. In the metric system, the only conversions are 10,100,1000 except for times. And as velocities relate to times, we have another association that helps remembering 3.6.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-06-19T19:17:54.443" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2775" PostTypeId="2" ParentId="2714" CreationDate="2014-06-19T19:53:45.337" Score="1" Body="&lt;p&gt;In the end I adopted the following beautiful result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A string can be written as repetitions of its &lt;em&gt;root&lt;/em&gt;, the smallest substring that it is a repeat of. I.e., &quot;abababab&quot; has &quot;ab&quot; as root. If a string is its own root, it is called &lt;em&gt;primitive&lt;/em&gt;. Call $p_n$ the number of primitive strings of length $n$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the strings are written over an alphabet of $s$ symbols, there are $s^n$ strings in all. But they are all repetitions of some primitive root, whose length divides $n$:&#xA;$$&#xA;s^n = \sum_{d \mid n} p_d&#xA;$$&#xA;Möbius inversion gives:&#xA;$$&#xA;p_n = \sum_{d \mid n} \mu(n / d) s^d&#xA;$$&#xA;For example, with $s = 2$, $n = 4$ this gives $p_4 = 12$. This can also be computed by noting that primitive strings of length $1$ over $\{a, b\}$ are just $a$ and $b$ (giving rise to $aaaa$ and $bbbb$), of length $2$ the only ones are $ab$ and $ba$ (giving rise to $abab$ and $baba$). All other length $4$ strings are primitive, i.e., there are $2^4 - 4 = 12$ of them.   &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-19T19:53:45.337" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2776" PostTypeId="2" ParentId="416" CreationDate="2014-06-19T19:58:56.427" Score="4" Body="&lt;p&gt;Raising a number to a natural-number power, and raising a number to a real-number power, are different operations whose results &lt;em&gt;mostly&lt;/em&gt; coincide.  In any abstract algebraic ring, raising any element $x$ to a natural number power $n$ is equivalent to multiplying the multiplicative identity of that ring by $x$, $n$ times.  If $n$ is zero, the result in any ring will always the multiplicative identity, regardless of $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that raising the additive identity (&quot;zero&quot; for most kinds of numbers) to any natural-number power other than zero will always yield the additive identity (&quot;zero&quot;) and raising it to the zero power will yield the multiplicative identity (&quot;one&quot;), this doesn't represent a &quot;discontinuity&quot; since the whole numbers represent a set of discrete values rather than a continuum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Raising things to powers that are not natural numbers makes things more complicated.  The function $x^y$ is continuous for all $x$ and $y$ when $x&amp;gt;=0$, except when $x$ and $y$ are both zero; the discontinuity at (0,0) means that many rules which are true for any other values $x$ and $y$ in the domain cannot hold true there.  When using real-number exponents, it is simpler and more practical to say that the rules applicable to such exponents apply in all cases where a value is defined, but say that the value of $0^0$ is undefined, than it would be to define $0^0$ as equal to 1 but say that many of the rules of exponentiation don't hold there.&lt;/p&gt;&#xA;" OwnerUserId="1740" LastActivityDate="2014-06-19T19:58:56.427" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2777" PostTypeId="2" ParentId="2763" CreationDate="2014-06-19T20:55:03.377" Score="21" Body="&lt;p&gt;First, an answer to your title question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it advisable to avoid teaching “multiplication as repeated addition”?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;No&lt;/strong&gt;, one should not &lt;em&gt;avoid&lt;/em&gt; teaching multiplication in this way. There are other ways to teach it, carrying their own advantages and disadvantages, but such is the way that multiplication is defined for the natural numbers, and this (also building off of number chanting by $2$s and $3$s that occur even in pre-K) indicates that it is one of the ways in which multiplication should be taught. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;However&lt;/strong&gt;, it is often advisable to discuss concepts in multiple ways, and I do not find multiplication to be an exception. Thus, being that this is a site for Mathematics Educators, and given that I have thought a fair bit about multiplication tables (see, e.g., &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/1750/262&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;), I will sketch briefly an alternative way of teaching multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us begin with a concrete object: a $10 \times 10$ array of squares. We now proceed as follows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pick a square. Designate this square as the bottom-right corner of a rectangle whose top-left corner is the top-left of our entire array. Count the number of discrete squares that make up the rectangle, and write the total inside of the chosen square.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if we choose the square that is $4$ to the right and $6$ down, then we form a rectangle of squares with $4$ total columns and $6$ total rows; counting up all of the squares in this rectangle, we find a total of $24$, and so we write $24$ in that particular square.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do this for all of the individual squares. In our $10 \times 10$ example, that will be a total of $100$ individual squares, and, at the end, we will have filled out what many will recognize as the $10 \times 10$ times table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;strong&gt;Side problem:&lt;/strong&gt; For each pair $i, j \leq 10$, consider the total number of $i \times j$ rectangles that can be formed using the individual squares in our $10 \times 10$ array; demonstrate that a &quot;times table&quot; can be formed in this way, as well. For example: There is only $1$ way to form a $10 \times 10$ rectangle, i.e., the entire array; write $1$ in the top-left corner. At the other end of the spectrum, there are $100$ different $1 \times 1$ rectangles possible, i.e., each individual square; write $100$ in the bottom-right corner. The $i \times i$ rectangles will fill out the NW/SE main diagonal, whereas the $i \times j$ and $j \times i$ for $i \neq j$ rectangles come in pairs, just as, e.g., $4 \times 6$ and $6 \times 4$ appear symmetrically about the main diagonal.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some of the pedagogical advantages to this approach?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is easy to get into: Draw an array of squares (perhaps a $5 \times 5$ one for younger children) and just start filling them in by drawing the corresponding rectangles described above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the tedious nature of filling out so many squares, it encourages learners to look for short-cuts, possibly with scaffolding from the instructor. &quot;Counting by $3$s&quot; or noticing that the $i \times j$ and $j \times i$ rectangles have the same number of squares can both be incorporated. The former allows for the exploration of multiplication as repeated addition, though it is not explicitly defined as such; the latter can be used to demonstrate the commutative property of multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, if we are to take the individual squares as &quot;unit squares,&quot; we are essentially using an area model for multiplication. When we move on to, say, positive rational numbers, our model extends naturally to discuss what, e.g., $1/2 \times 1/3$ is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also natural ways to interpret the distributive law, which I leave as an exercise to the reader. In fact, I leave other methods of exploring this model for multiplication to the reader, too, as my answer length has grown somewhat unwieldy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;As a final remark:&lt;/strong&gt; Though the concern of the initial question stems from a desire to generalize to non-integers (non-naturals, even), I would like to post a cautionary note about not being too hasty to leave the standard times table. The link I posted earlier on in this post gives a number of examples of nonstandard problems that can be asked about the $10 \times 10$ table, which I think has been viewed quite unfairly, historically speaking, as epitomizing rote learning, whereas it can serve instead as a fecund area for mathematical exploration.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-06-19T20:55:03.377" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2778" PostTypeId="1" CreationDate="2014-06-20T03:26:00.020" Score="5" ViewCount="232" Body="&lt;p&gt;I read the answers to &lt;a href=&quot;https://matheducators.stackexchange.com/q/416/128&quot;&gt;What are argument one can give to students on the definition $0^0$?&lt;/a&gt; and I noticed some people are mildly bothered that $0^x$ is discontinuous if we define $0^0=1$. However, I think is fairly obvious that if $f(x)=0^x$ then $\text{domain}(f)$ does &lt;strong&gt;not&lt;/strong&gt; contain negative numbers. Surely, if $x &amp;lt;0$ then &#xA;$$0^x = 0^{-(-x)} = \frac{1}{0^{-x}}=\frac{1}{0}$$&#xA;usually an unwelcome expression. My thought here, $0^x$ is not a proper exponential function because the natural domain of the expression $0^x$ is at largest $[0, \infty)$ and, given the convention $0^0=1$, is continuous on $(0,\infty)$. In other words, it is the constant function $g(x)=0$ restricted to $(0,\infty)$. On the other hand, exponential functions with $a&amp;gt;0$ and $a \neq 1$ all have nice properties: they're injective functions with $\text{range}(a^x) = (0, \infty)$ and $\text{domain}(a^x) = (-\infty, \infty)$. Of course, $a=1$ is somewhat boring $a^x=1^x=1$ hence we just have the constant function with an awkward formula. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ok, enough about the happy place. What then about $f(x)=a^x$ where $a&amp;lt;0$. What is the natural domain for this formula? This leads to my question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do you mention the problem of understanding $f(x)=a^x$ for $a&amp;lt;0$ in your teaching?&lt;br&gt;&#xA;  Is it wise to omit discussion of $a \leq 0$ ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't have a firm opinion on this, I usually find myself talking about the bizarre behavior of such a formula sometime in the first week of calculus. A secondary question, does appreciating the weirdness of negative base &quot;exponential functions&quot; help explain why we shouldn't have much hope for $0^x$ to behave nicely? I put quotes here because I think the definition of an exponential function properly assumes the base is a positive number which is not one.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-06-21T01:44:26.247" Title="Functions with formula like an exponential function" Tags="|undergraduate-education|secondary-education|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2779" PostTypeId="2" ParentId="2778" CreationDate="2014-06-20T04:15:30.747" Score="5" Body="&lt;p&gt;I think I do the same as what you described, but in pre-calc. I try to get the students to tell me what different bases would do. I show them how bizarre a graph of (-2)^x would be, and then we dismiss negative numbers as possible bases.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-06-20T04:15:30.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2780" PostTypeId="2" ParentId="2769" CreationDate="2014-06-20T04:28:28.473" Score="3" Body="&lt;p&gt;I tutored a homeschooled boy who loves math. He would do multiplication in his head, and didn't know how to write it down. I was not tutoring in any conventional way, and never tried to get him to write it down. But when we were working through &lt;em&gt;Introduction to Number Theory&lt;/em&gt;, by Matthew Crawford (an AOPS book), he had a bit of trouble doing multi-digit base 8 multiplication in his head. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After watching him think for almost a minute one day, I asked him to describe what steps he was taking in his head. It turned out he was going left to right (which works better for most people than right to left, when doing things in one's head), and using the top number's digits times the bottom number - everything was backwards from the conventions. It would have worked, but there was too much. I wrote it down for him a few times, and then he knew how to write what was in his head.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His method and the conventional method are equally good algorithms. My opinion, if you're not trying to all do it the same in a classroom, is that the method the students likes best is the best method for them.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-06-20T04:28:28.473" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2781" PostTypeId="2" ParentId="2694" CreationDate="2014-06-20T04:57:49.447" Score="4" Body="&lt;p&gt;I have loved the zen of teaching. Accept where a student is, and figure out how to help them move forward. I remember years ago being a bit bored teaching algebra for the nth time. And then it got less boring, because I was able to focus more on the students and less on the content. Or rather, my focus was on how the students were seeing the content. Don't know if this is detailed enough to be useful...&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-06-20T04:57:49.447" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2782" PostTypeId="2" ParentId="2778" CreationDate="2014-06-20T10:33:56.330" Score="2" Body="&lt;p&gt;When you cover exponential functions, you should always discuss negative bases and base 0. But using power laws, it's relatively easy to see, that $(-2)^x$ is problematic for non-integer $x$, and $0^x$ is problematic for negative $x$. If you use &lt;a href=&quot;http://de.wikipedia.org/wiki/Permanenzprinzip&quot; rel=&quot;nofollow&quot;&gt;permanence sequences&lt;/a&gt;, you can also easily see, that $0^x$ is also problematic for $x=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These discussions are an important repetition of power laws and an important step in understanding functions (and sequences) in general. So &lt;strong&gt;no&lt;/strong&gt;, it is not wise to omit non-positive bases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And no, the weirdness of negative bases is no indicator for problems with 0. Root functions behave perfectly fine for radicand 0, only negative radicands are problematic. Exponential functions are applications of multiplication and the 0 (not negative numbers) are very special to multiplication.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-06-20T10:33:56.330" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2783" PostTypeId="2" ParentId="2769" CreationDate="2014-06-20T13:33:16.337" Score="4" Body="&lt;p&gt;JP's answer brings up some disturbing issues. &quot;a student may not be motivated to work toward conceptual understanding if they know they can just use the standard algorithm.&quot; 100% true, and tough to navigate around. Disturbing, in my opinion, because many countries are pushing the use of standardized testing which in my opinion, ignores that conceptual understanding JP highlights. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the times table should be memorized up to 12 X 12. Then the 'old' method of lining up the numbers under each other and multiplying by hand. The &lt;a href=&quot;http://www.calculatorsoup.com/calculators/math/latticemultiplication.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;lattice method&lt;/a&gt;, shown below, is an interesting way to multiply by hand, but avoid some of the confusion of carrying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/hxFHj.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are multiplying 327 by 586 here, and it should be obvious that this is just doing single digit math, with the lattice helping to avoid any carry, as each unit place has two cells. &#xA;My daughter was taught this method in 7th grade (age 12/13) and dismissed it as unnecessary, the old way worked fine. &#xA;I was tutoring a high school (age 14/15) freshman who was having multiplication issues, and pulled out this method. His eyes lit up. Regretting my inability to use a Vulcan Mind Meld, I came to accept that this alternate method could, and should be used as a secondary way to explain what we'd call simple multiplying of numbers beyond 12 X 12. My time with the students is always constrained, I suspect the lattice helped clarify the 'carry.' At least that's what I hoped.&#xA;To JP's point, the level of analysis would be tough to produce. My anecdotal evidence suggests that it's a worthwhile tool to offer students who are struggling with multiplication, but not useful as a replacement for the standard method we all learned. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-06-20T13:57:09.377" LastActivityDate="2014-06-20T13:57:09.377" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2784" PostTypeId="2" ParentId="2763" CreationDate="2014-06-20T13:48:31.523" Score="4" Body="&lt;p&gt;Teach it whatever way the students can understand the concept you are getting at &lt;em&gt;right now&lt;/em&gt;, and be explicit that it is a &lt;em&gt;model&lt;/em&gt; of multiplication -- that multiplication itself is this other thing they will learn more about later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Models change as new ideas are encountered. It is a normal part of the process of teaching: to provide a simple model sufficient for a trivial example case initially, and then blast that model and replace it with a more detailed one as the student advances. Its the only way we are capable of learning anything complex anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are teaching young children, multiplication usually involves only positive integers. In this case thinking of multiplication as iterative counting is sufficient. Obviously, this model falls apart entirely once you get into more interesting cases of multiplication and functional composition -- but that's not the point. The point is to teach at the level of abstraction that is appropriate to the student's level and the problem at hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your students are bright and analytically mature you can teach them about abstraction itself and how this &lt;em&gt;particular&lt;/em&gt; model of multiplication is a necessary simplification. But do &lt;em&gt;not&lt;/em&gt;, for example, dive into foundational mathematics with 2nd graders in an effort to get them to grasp the principle of abstract reduction. After all, that defeats the whole purpose of abstract reduction!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit; observation:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After reading some more of the comments and (long) posts on this topic, it is clear to me that preliminary mathematics educators and electrical engineering / comp-sci educators do not talk much. There are strong parallels between teaching a first-year engineering student something hopelessly complex by way of successively accurate models of the same thing and teaching primary school students something hopelessly complex by the same method. Peano arithmetic (where addition can be thought of as successive &quot;pile swapping by ones&quot;) provides a foundation for deeper concepts of addition of things that aren't unitary in the same way that teaching multiplication by &quot;successive adding&quot; provides a foundation for deeper concepts of multiplication of entities higher up on the numerical type order. Fast forward and you see the same thing in, for example, SICP where thinking of a language interpreter as a direct implementation of Lambda calculus provides a foundation for thinking of machines in a stateful (and therefore explosively complex) way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I write this observation because while I see engineers embracing the concept of teaching by abstraction, I see basic math educators arguing over whether its a good idea or not to do this. I find these to be remarkably parallel cases with strikingly opposite interpretations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Second edit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A talented member of the Erlang community, Anthony Ramine, linked &lt;a href=&quot;http://worrydream.com/AlligatorEggs/&quot; rel=&quot;nofollow&quot;&gt;http://worrydream.com/AlligatorEggs/&lt;/a&gt; , a light introduction to the untyped lambda calculus today. That led me to &lt;a href=&quot;http://worrydream.com/#!/SomeThoughtsOnTeaching&quot; rel=&quot;nofollow&quot;&gt;http://worrydream.com/#!/SomeThoughtsOnTeaching&lt;/a&gt; . I think both are highly relevant to the discussion, because the issue here is not multiplication, it is models of pedagogy and their underlying motivations.&lt;/p&gt;&#xA;" OwnerUserId="1786" LastEditorUserId="1786" LastEditDate="2014-06-21T11:50:21.910" LastActivityDate="2014-06-21T11:50:21.910" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="2785" PostTypeId="2" ParentId="2763" CreationDate="2014-06-20T15:33:12.537" Score="4" Body="&lt;p&gt;I try to avoid (and advocate that teachers try to avoid) lying to students. Stating explicitly that &quot;Multiplication IS repeated addition&quot; (MIRA) is a lie and an easily avoided one. First, the area model can be presented with sufficiently small integers that counting isn't necessary - subitizing (instant recognition of &quot;how many&quot; without counting - suffices. For most humans that's somewhere in the 5- 9 range, making areas up to about 8 feasible candidates. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When getting into the relationship between counting and addition, do we say &quot;addition is 'fast counting' or 'repeated counting'? Probably not. We can think about addition that way, but it's a lot more and different, even for integers, which is evident to most people when the addends are at all &quot;large.&quot; And while &quot;repeated addition&quot; may get us the correct product for a given multiplication problem, it simply isn't true that multiplication is the same thing. It's a different operation in a fundamentally different way than subtraction is a different operation from addition. One of many reasons multiplication is qualitatively different from addition is suggested by the fact that we need to worry about units and/or common denominators in order to add; no such things need concern us with multiplication. If that doesn't puzzle you, you likely shouldn't teach mathematics at any level. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To avoid lying to kids, it's reasonable to state things like &quot;You can think about multiplication as repeated addition, but it's both different from and more than that.&quot; Or perhaps, &quot;One way to think about multiplication is as repeated addition, but that only gives part of the story.&quot; I'm sure we can come up with other reasonable and TRUE things to say. But why would we want to tell children, &quot;addition always makes things larger&quot;? It's simply not true. Same for the other operations. You simply cannot truthfully claim that any of the four basic arithmetic operations ALWAYS yields a larger or smaller result than the initial input. Period. So why claim it? So doing only serves to confuse many kids down the line. More modest claims can be made honestly. It's also a lie to aver that &quot;You can't subtract a larger number from a smaller one.&quot; It's true that you can't do that with strictly non-negative numbers. But even some young elementary students are aware of the existence of negative numbers and some actually have a degree of understanding of and/or facility with them. Why lie? It's legitimate to tell someone that &quot;With the numbers we're currently using, there is no answer to 7 - 9.&quot; But doing that already suggests that there might be numbers within which the answer to that problem exists, and kids can begin thinking about what that could be on their own. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being doctrinaire on these issues, insisting upon a narrow and false claim because &quot;kids can't handle the truth&quot; or &quot;that's the way I learned&quot; or &quot;that's how it's always been done&quot; simply doesn't make sense. We are supposed to be a species capable of growth and change, of learning from past mistakes, etc., but when it comes to teaching elementary mathematics, at least in this country, we resort to almost blind religious defenses of outright lies. Frankly, that's disappointing and disturbing. &lt;/p&gt;&#xA;" OwnerUserId="1785" LastActivityDate="2014-06-20T15:33:12.537" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2786" PostTypeId="1" CreationDate="2014-06-20T16:07:08.077" Score="21" ViewCount="777" Body="&lt;p&gt;I just finished a PhD in math at a top department, but not one that placed much emphasis on graduate student teaching.  Grad students here teach only as TAs, and the training is minimal.  I got great student evaluations, but I realize this was due more to holding lots of office hours and making study guides than to any real pedagogical brilliance on my part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next year I'll start a postdoc at a similarly research-oriented place, and I'd like to be a more effective instructor.  The question is: where do I start?  What are some good survey books or articles that would bring me (partially) up to speed on best practices for teaching university-level math?  My formal knowledge of good teaching strategies is essentially nil.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll have the chance to teach a range of undergraduate courses.  I realistically won't have the ability (or longevity, or experience...) to make sweeping changes in e.g. the way the calculus classes are taught, but I'd like to do the best I can within the current framework.  I may also have the chance to design a one-off mid-level undergrad course from scratch, so something touching on course development would be helpful.&lt;/p&gt;&#xA;" OwnerUserId="1788" LastEditorUserId="1788" LastEditDate="2014-06-20T16:34:49.840" LastActivityDate="2019-08-03T22:35:46.240" Title="Becoming a better instructor: where to start?" Tags="|undergraduate-education|reference-request|teacher-development|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2787" PostTypeId="2" ParentId="2786" CreationDate="2014-06-20T17:28:59.043" Score="8" Body="&lt;p&gt;One thing to watch out for is learning to understand what the student's problems with the subject are. Once you identify that, fixing is typically easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take care when designing exams and homework. Ask yourself what you want to accomplish (what should be measured, what you want them to learn). It is important to think of grading when designing exams, don't ask for overlong derivations (they won't have the time to do more than a very few, and much material won't be examined that way; someone will make the most absurd error off the bat, and force the grader to follow a convoluted path through a completely different jungle).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Collect material from previous terms as a (rough) guide. There are plenty of lecture notes, past exams and homework on a wide range of subjects on the 'web, see if some are to your liking. Even if the policy is that exams (and solutions) aren't published, assume they &lt;em&gt;are&lt;/em&gt; available to interested parties. Ask around on what they had before, what follows, where they'll use the material later on. Talk to people who have taught the subject before (or will do so in paralell), ask former students and TAs on the rough (and smooth) spots, request input on them. Yes, that will probably make you look somewhat goofy at the start, but then again, you'll be the new kid on the block.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Look out for trouble (yours, the class', individual students), and make sure it gets handled early. Too often some minor problem grows into unmanageable proportions as term end looms near.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specific advice: It is often useful to start with a short summary (perhaps list the most important definitions and results from last class for reference). End with a recapitulation of what was done, explaining again what is important and what was just scaffolding or examples.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-20T17:28:59.043" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2788" PostTypeId="2" ParentId="2786" CreationDate="2014-06-20T19:16:09.980" Score="7" Body="&lt;p&gt;I recommend the book &lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=htm/2&quot; rel=&quot;noreferrer&quot;&gt;How to Teach Mathematics by Steven G. Krantz&lt;/a&gt;. He covers almost all of what you are asking from low level courses, advanced courses, large and small. The second edition also contains many essays on reflections on teaching that I found enlightening.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-06-20T19:16:09.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2789" PostTypeId="2" ParentId="2786" CreationDate="2014-06-20T21:29:21.190" Score="6" Body="&lt;p&gt;After teaching college courses (full disclosure - I'm a biologist) for about 15 years, my main teaching discovery is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teachers overestimate how much students learn from a lecture.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This occurs because a) teachers themselves were part of the select group that learned great from lectures, or they wouldn't have survived the flaming hoops of academia, and because b) teachers learn a BUNCH by preparing a lecture. But math &lt;em&gt;students&lt;/em&gt; learn best by doing math problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are entering the field during a time when pre-class video assignments and online quizzes are becoming part of the math curriculum. This presents a great opportunity to lecture less and spend at least 10 minutes of every class time having students &lt;strong&gt;work in groups&lt;/strong&gt; on their first homework problems. I promise that if you clear time in class to tell students to work in pairs on a slightly-difficult problem and walk around and ask, &quot;How's it going? What have you figured out so far?&quot; you will be AMAZED at where students get stuck. And once you learn the sticky parts, you will have a great time trying different techniques to help students understand them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some reading on math education, try:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/users/942/robert-talbert&quot;&gt;Robert Talbert&lt;/a&gt;'s Casting Out Nines &lt;a href=&quot;http://chronicle.com/blognetwork/castingoutnines/&quot; rel=&quot;nofollow noreferrer&quot;&gt;blog&lt;/a&gt; on Chronicle of Higher Ed&lt;/li&gt;&#xA;&lt;li&gt;Dan Meyer's &lt;a href=&quot;http://blog.mrmeyer.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;blog&lt;/a&gt; on math education (high school focus, but still relevant to college)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(perhaps &quot;real&quot; math educators can recommend additional resources)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For general reading on being a college instructor, try &lt;a href=&quot;http://amzn.com/0470401044&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching at its Best&lt;/a&gt; by Linda Nilson.&lt;/p&gt;&#xA;" OwnerUserId="1207" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-06-20T21:29:21.190" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2790" PostTypeId="1" CreationDate="2014-06-20T22:06:46.197" Score="23" ViewCount="3383" Body="&lt;p&gt;This provocative essay&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Igor Rivin.&#xA;  &quot;Some Thoughts on the Teaching&#xA;  of Mathematics—Ten Years Later.&quot;&#xA;  &lt;em&gt;Notices of the AMS&lt;/em&gt;, Jun/Jul 2014.&#xA;  (&lt;a href=&quot;http://www.ams.org/notices/201406/rnoti-p597.pdf&quot;&gt;PDF download link&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;suggests that a discussion of Igor's &quot;principle&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A computer program IS a proof&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;might be worthwhile. I myself am very interested in conveying the notion of&#xA;what constitutes a &lt;strong&gt;&lt;em&gt;proof&lt;/em&gt;&lt;/strong&gt; as early as possible in the educational system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So given that most students have little understanding of what constitutes a proof&#xA;by the time they reach college in the US (13th grade), what might be a good way to&#xA;introduce that type of rigor earlier via a programming language, which effectively&#xA;IS a proof, to quote Igor?&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="4925" LastEditDate="2015-03-29T00:41:11.187" LastActivityDate="2015-03-31T01:53:31.850" Title="&quot;A computer program IS a proof&quot;: Introducing rigor via programming" Tags="|secondary-education|proofs|computer-science|" AnswerCount="9" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2791" PostTypeId="2" ParentId="2790" CreationDate="2014-06-20T22:49:00.527" Score="12" Body="&lt;p&gt;(By the way, I am in considerable agreement with Igor Rivin's premises and observations in that essay, and many of the in-principle conclusions, but might tend to doubt that the course of the juggernaut that is American culture and the concommitant educational system can be altered by the actions of a few quasi-rogue mathematicians in universities...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, first, my comment/reaction is that it's maybe not that a &lt;em&gt;program&lt;/em&gt; is a proof, but the demonstration of correctness of an algorithm certainly is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For context in regard to that article: I, too, have been frustrated by the intellectual state of undergrads, from perceptions of calculus, and I. Rivin's example of linear algebra, and even worse with &quot;intro to abstract algebra&quot;. The latter was supposedly aimed at future high school math teachers, future actuaries, and math majors not going to grad school in math. After several experiments, it became clear that a majority of students could not only &lt;em&gt;not&lt;/em&gt; write proofs or logical arguments or even sensible arguments, but could not reliably compose coherent paragraphs about very mundane actions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Motivated by this situation (and other things), I cobbled together a &quot;crypto&quot; course and an &quot;error-correcting codes&quot; course to allow discussion of abstract algebra (&lt;em&gt;and&lt;/em&gt; other important, basic mathematics along the way: combinatorical probability, a little linear algebra, finite fields, ...) with practical applications, and &lt;em&gt;algorithm&lt;/em&gt;-oriented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I still did ask students to narrate their execution of algorithms (with human-scale data). I had thought this would be a fair-if-minor challenge/goal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was to a considerable degree amazed that even very many &quot;good&quot; students could not, even with prompting, say what they had done, or why, even in very simple computations. &quot;It's obvious.&quot; I tried to get them to write as though they were explaining something over the phone to someone who had no idea, but this seemed ... impossible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, as artifact of the educational system in the U.S., people cannot reliably explain a process or algorithm. Observation suggests that a decade or so of &lt;em&gt;not&lt;/em&gt; being required to explain things, but only to follows rules, does not beget narrative skill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, I strongly hesitate to describe what even mathematicians would like from students as &quot;proof&quot;. &quot;Explanation&quot;, as a less stylized, more colloquial label, might be less polarizing and more explanatory to the general population.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But should kids be required to write programs? Maybe in pseudo-code. While I do understand Scheme and other dialects of Lisp and some other languages, I'd hesitate to insist on the &quot;purity&quot; of Scheme, as opposed to Python, for example. Not to mention the opportunities for killing time wrangling with syntax that is wildly different from English. No real point in that, unless we are wanting to inadvertently return to the (popular!) idea of mathematics as a discipline of adherence to semi-inexplicable &quot;rules&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was also dismayed to learn that many, many students believe that computers can do things that are even in principle impossible for humans, not merely on grounds of time or memory... but that they (computers) can compute trig functions and detect primes and do all sorts of things that no human could conceivably do. There is scant understanding that the atomic steps are perhaps even smaller for computers than for human-scale operations. Computers are magic?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, it would be good for U.S. students to learn that &quot;proof&quot; might mean, for example, explaining why an algorithm (whether executed by them or by a computer) produces useful outputs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then there're probabilistic algorithms (very relevant in crypto and in coding, thus to the internet, thus to everyday life...), which reliably disturbed many of the math majors... :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-06-20T22:49:00.527" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="2792" PostTypeId="2" ParentId="2754" CreationDate="2014-06-20T22:52:59.073" Score="3" Body="&lt;p&gt;I have faced this teaching problem before, but I am not very experienced a teacher.&lt;br&gt;&#xA;This is what I think right now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than try to get them to memorize both the factor (3,6) and the &quot;side&quot; (multiply from m/s to km/h) it might be better to have a standard conversion that they think about each time (till it becomes automatic)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;If I walk 1 meter each second, I walk 60 meters per minute.&lt;br&gt;&#xA;There are 60 minutes in a hour. I walk 60*60=3600 meters in an hour.&lt;br&gt;&#xA;We have about 3 kilometers. Actually, 3,6.&lt;br&gt;&#xA;When I walk 1 meter per second, I walk 3,6 kilometers in an hour&quot;&lt;/p&gt;&#xA;" OwnerUserId="1662" LastActivityDate="2014-06-20T22:52:59.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2793" PostTypeId="2" ParentId="2790" CreationDate="2014-06-21T00:14:03.567" Score="7" Body="&lt;p&gt;The first poster said that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;it's maybe not that a &lt;em&gt;program&lt;/em&gt; is a proof, but the demonstration of correctness of an algorithm certainly is.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would modify this slightly (although I agree with the content of that claim). I think the phrase, in the context of the article, can be written as &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Writing a program&lt;/strong&gt; is equivalent to a proof.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In what sense? In the sense that getting a computer to correctly execute a function requires a total understanding, down to the last detail, of how that function works. In other words, making a computer find the GCD of two numbers has as its necessary and sufficient burden (besides fluency in the computer language, obviously) an understanding of &lt;em&gt;how&lt;/em&gt; a method for doing so works (where &quot;how&quot; should convey both the &quot;what&quot; and the &quot;why&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So from an educational standpoint, making students write programs to execute mathematical tasks is an excellent way to ensure that they will understand both &lt;em&gt;how&lt;/em&gt; the method works (they wrote the code) and &lt;em&gt;why&lt;/em&gt; the method works (they planned/problem-solved to come up with the idea behind the code). The &lt;em&gt;why&lt;/em&gt; part constitutes the &quot;proof&quot; of the method - they could, in theory, give an adequate demonstration of the theorem if it was demanded of them.&lt;/p&gt;&#xA;" OwnerUserId="1598" LastActivityDate="2014-06-21T00:14:03.567" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="3793" PostTypeId="2" ParentId="2790" CreationDate="2014-06-21T00:57:54.197" Score="14" Body="&lt;p&gt;From the article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For example, the student who wrote the program in the section “Source&#xA;  Code for a Modular Arithmetic Package in Scheme” had to have&#xA;  complete understanding of the Chinese Remainder Theorem, and his&#xA;  program is the Chinese Remainder Theorem in that, given some&#xA;  quantities satisfying the hypotheses of the theorem, it never fails to&#xA;  produce a quantity satisfying the conclusion.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;This correspondence between program and proof (though far from&#xA;  perfect) allows us to make mathematics hands on...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the principle in effect is only saying that &quot;an algorithm to construct something of which a theorem states the existence, is a proof of the theorem&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that as mathematicians we would dispute that principle, and further dispute its out-of-context generalisation to the statement &quot;a program is a proof&quot;, because it will lead to incomplete attempted proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in terms of recognising a proof when you see one, a construction can certainly &quot;be&quot; an existence proof once you add the proof that the thing constructed has the properties stated by the theorem. So there's an aid to comprehension and possibly a useful rule of thumb here, it's just not actually &lt;em&gt;true&lt;/em&gt; since, as the author says, there's no perfect correspondence between the program and the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The imperfection of the correspondence can be illustrated by presenting the Python code:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;n = int(input())&#xA;while n != 1:&#xA;    if n % 2 == 0:&#xA;        n /= 2&#xA;    else:&#xA;        n = 3 * n + 1&#xA;    print n&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;I propose that this clearly is &lt;em&gt;not&lt;/em&gt; a proof of the Collatz Conjecture, please do not burst into rapturous applause ;-) Writing it didn't require any deep understanding of the conjecture, that gets me towards a proof. If the conjecture is true, then I've written a program that &quot;given some quantity satisfying the hypotheses of the theorem, never fails to produce a quantity satisfying the conclusion&quot;, where the quantity it produces is a sequence ending in 1. Actually resource limits can bite, and I haven't checked whether the author of the article addresses that. The code I wrote is still not (on its own) a proof even that the conjecture is true for n smaller than some manageable bound.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the conjecture is false then the program fails to halt on a counter-example, so is certainly not a disproof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, under normal circumstances we cannot call things &quot;proofs&quot; that don't even tell us whether the theorem is true or not! With modification my code could be used to produce a (ridiculously long) proof for small bounds, simply by generating all the sequences required. That leads to an argument about computer proofs that probably isn't really relevant to the thrust of Rivin's points, but I don't think that by accepting a computer-aided proof we have to accept that the program itself &lt;em&gt;is&lt;/em&gt; the whole proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it's pretty typical to write programs that &lt;em&gt;if the proposition is true&lt;/em&gt; can be fairly easily seen to produce the required quantity. For another example, one can and usually does write a program to factorise an integer, whose design and proof of correctness both rely on the fundamental theorem of arithmetic. Claiming the resulting program to be a proof of the theorem would be wholly circular.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proving that my program produces the required quantity &lt;em&gt;would&lt;/em&gt; prove the Collatz Conjecture, but a program on its own is not in general a proof that its outputs have particular desired properties. So by all means, programming might help introduce the notion of mathematical rigour if it's easier to ask and answer &quot;how do we know this program does what it's supposed to?&quot; than it is to ask and answer &quot;what is a proof?&quot;. A proof is still needed, but the act of programming might make it much more clear what needs to be proved, and thus easier to write a proof and easier to see what a proof is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my example, it probably doesn't help, if only because computing the Collatz sequence is so simple that it's no easier to understand the question with the help of a computer. It's easier to explore what some sequences actually look like, that's about it. In other examples it might well help a lot, and perhaps the principle can encourage you to design programming tasks in a series of steps that ensure once the task is completed, the student has seen what they need to see in order to appreciate the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose that Euclidean geometry can and has provided the same introduction to rigour that programming can, since there too one can invent or be presented with a process that's observed to produce a certain result, and then challenged to relate this process to a proof of a proposition.&lt;/p&gt;&#xA;" OwnerUserId="321" LastEditorUserId="321" LastEditDate="2014-06-22T14:40:14.830" LastActivityDate="2014-06-22T14:40:14.830" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3794" PostTypeId="2" ParentId="2790" CreationDate="2014-06-21T01:44:44.267" Score="5" Body="&lt;p&gt;I agree that the emphasis on calculation instead of understanding is a crime, and that skills in &quot;liberal arts&quot; areas (writing down a coherent description of a procedure, be it a recipe for jambalaya or pseudocode to compute the greatest common divisor) are extremely important, perhaps even more in technical areas (which tend to attract people who shy away from them) than otherwise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, equating &quot;programming&quot; with &quot;proof&quot; is a complete misunderstanding of both. Yes, in the hands of people like Dijkstra a program is a proof. Or perhaps better, Dijkstra's programs did include proofs, or had proofs written in parallel. Knuth's literary programming points in the same direction, write the program and the proof hand in hand, have the program typeset with the explanation for reading/checking by people, and compiled for running. But this is so rarely practiced that it gets lost in the noise. Most programming is (disturbingly) done by the &quot;patch until it seems to work&quot; technique. It isn't for nothing that &lt;a href=&quot;http://en.wikiquote.org/wiki/Programming&quot;&gt;Weinberg's Second Law&lt;/a&gt; states: &lt;em&gt;If builders built buildings the way programmers wrote programs, then the first woodpecker that came along would destroy civilization.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-21T01:44:44.267" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3795" PostTypeId="2" ParentId="2790" CreationDate="2014-06-21T05:51:36.207" Score="7" Body="&lt;p&gt;Another thing worth noting is often times computers can only provide &quot;approximations&quot; of pure mathematical concepts and objects, simply due to the nature of the technology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in C and many other languages, a statement such as:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;(1e200 + 1.0) == (1e200)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Will evaluate to &lt;code&gt;true&lt;/code&gt;, simply due to precision loss in floating-point numbers. Did we just prove that $10^{200}+1=10^{200}$? We could easily construct a program that &quot;proves&quot; inductively that $10^{200}+n=10^{200}\&amp;gt;\forall\&amp;gt;{n}\in\mathbb{Z}$ (although we may reach limits of representable integer values), and another that proves the same thing not to be true by contradiction. We can &quot;prove&quot; that for any sufficiently large $n$, that $n+1=n$. And the only way to get to the bottom of it is to examine what is happening at the low level of bits and bytes and storage formats and digital arithmetic - all of which distract from the problem at hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, irrational and non-terminating numbers cannot be easily represented. For one common floating-point representation, for example, the non-terminating rational number:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;1.0 / 3.0&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Is actually:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;0.3333333432674408000&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;There are plenty of other examples with numbers - integers wrapping around depending on the size of the variables used to store them, etc. I'm sticking to arithmetic because it's an easy example but there are much more subtle problems - memory limitations, incorrect usage of tools, design choices that affect results, bugs in underlying libraries, and also inherent limitations in using a discrete digital system to approximate continuous models.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So while it's possible sometimes to represent high level concepts in a program and have that program closely (or exactly) resemble a proper proof, many times it is not necessarily a direct proof of correctness because all of the lower level issues related to the actual technology can easily get in the way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much of this depends on the choice of programming language and other tools used as well. There are many different programming languages. For example, functional languages like SML and Haskell were essentially designed with induction in mind; programs written in those languages are often direct inductive proofs of correctness - whereas other languages can get bogged down in the technical details of the underlying system and subtle things sometimes nondeterministic, can ruin the success of a proof. In a language like Haskell, you can concentrate on the proof and not get bogged down in details.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Computers are useful tools just as calculators are useful tools, and programs can be used to aid constructions of proofs, but by no means are proofs in themselves. More often, you would use a proper proof to prove that the algorithms in the &lt;em&gt;program&lt;/em&gt; were correct. There is simply too much &quot;noise&quot; -- other things to worry about -- involved in a computer program that take away from the strength of using it as a proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wouldn't overlook the importance of programming, however. While programs aren't always themselves appropriate for proofs, teaching certain languages to a student (going back to the SML example) can, at the very minimum, teach a student a whole new way of thinking, and those skills can then be extended to constructing mathematical proofs. Learning a program language that is focused on induction most certainly changes the way a person looks at certain problems, and has the potential to add a few more great tools to somebody's mental tool set. Also some of these languages are great illustrative tools - for example, a calculator that solves algebraic equations and prints the steps is a good self-learning tool; just as an inductive proof written in Haskell opens itself up to examination and experimentation with instant and clear feedback.&lt;/p&gt;&#xA;" OwnerUserId="680" LastEditorUserId="680" LastEditDate="2014-06-21T15:27:03.420" LastActivityDate="2014-06-21T15:27:03.420" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3796" PostTypeId="2" ParentId="351" CreationDate="2014-06-21T11:42:31.293" Score="3" Body="&lt;p&gt;In general, I am in favor of publishing old exams for similar reasons to those given in other answers. However, some caveats are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If the course material has changed, questions from old exams may be misleading. Likewise, if the course instructor has changed and the new instructor has a different style of asking questions or likes to emphasize different aspects of the material.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students may be overly reliant on previous exam questions when studying and not make sufficient use of other resources to master the material.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If full answers are supplied, some students might decide to memorize them, in the hope/expectation of encountering similiar questions in the exam.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="376" LastActivityDate="2014-06-21T11:42:31.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3797" PostTypeId="1" CreationDate="2014-06-21T13:17:30.283" Score="3" ViewCount="238" Body="&lt;p&gt;Most students with aspirations of a pursuing a PhD in pure mathematics at a top grad school takes the &quot;standard&quot; curriculum as an undergrad which includes 2x algebra, 2x analysis, 1x complex analysis, 1x topology, 1x linear algebra, 1x probability, and 1x combinatorics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a rising junior who needs to figure out what to do over the next two years and knows he wants to go to grad school but doesn't have the faintest clue in what particular field:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are the most important &quot;nonstandard&quot; undergraduate math courses for pure math grad school?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: I asked this question on math.se but it was put on hold there. Hopefully this question is better suited for this site.&lt;/p&gt;&#xA;" OwnerUserId="1795" LastActivityDate="2014-06-21T13:17:30.283" Title="Most important nonstandard math courses" Tags="|undergraduate-education|curriculum|" AnswerCount="0" CommentCount="7" FavoriteCount="0" ClosedDate="2014-06-22T00:11:34.500" ContentLicense="CC BY-SA 3.0" />
  <row Id="3798" PostTypeId="2" ParentId="2497" CreationDate="2014-06-21T23:18:01.660" Score="5" Body="&lt;p&gt;I work with students with disabilities and often have to teach them about negative numbers.   I tend to use an elevator analogy that I borrowed from a math methods class I took years ago; I create a fictional building that has 5 levels above ground, a ground floor (0), and then an underground parking structure with a P1, P2, P3, P4.    Then I tell them we're going to call the floors below ground, -1, -2, -3, and -4.   Using a diagram of the elevator shaft, I can have them do both addition (moving up a certain number of floors) and subtraction (moving down a certain number of floors).   When they add and subtract negative numbers, you end up moving in the opposite direction - so adding -5 means to move down 5, and subtracting negative 3 means to move up 3.    Using something like this, it becomes clear that -4 is less than -3, since you have the floors in the order they would appear in the building...  &lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-21T23:18:01.660" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3799" PostTypeId="1" AcceptedAnswerId="3801" CreationDate="2014-06-22T03:06:21.410" Score="13" ViewCount="3135" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Related but different: &lt;a href=&quot;https://matheducators.stackexchange.com/q/41/1685&quot;&gt;What is a good handwriting font for mathematics?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A (former) student of mine (no longer! Thankfully passed his Trigonometry regents last week:) had a problem with his handwriting -- I am not exaggerating when I say that at times, he would not be able to read what he wrote 30 seconds before.&lt;br&gt;&#xA;The student is in 11th grade, around 16-17 years old.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Needless to say, this was a problem for a number of reasons:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Occasionally, as I watched him solve a problem, he would misinterpret work he had done on a problem earlier, thereby finding a wrong answer, when he really understood the material and did all the right work (what a waste!)&lt;/li&gt;&#xA;&lt;li&gt;While not verified, this probably also caused problems with teachers grading tests -- I would imagine that sometimes they might not understand what he wrote, and taking off points either for wrong answer (as in above example) or for lack of work.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What should a Mathematics Educator (either teacher or tutor) tell a student with this problem to do to fix this?&lt;/p&gt;&#xA;" OwnerUserId="1685" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-06-23T00:50:13.257" Title="What to do about a student's poor handwriting" Tags="|general-pedagogy|transferrable-skills|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3800" PostTypeId="2" ParentId="3799" CreationDate="2014-06-22T09:42:25.370" Score="3" Body="&lt;p&gt;It might be some kind of disability, in which case training better handwriting might be of no (or little) use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps an option is to use a computer to type text/math? I'm at the point where writing LaTeX is almost as fast as handwriting (but that is after decades of daily use). Or use some tablet with handwriting-recognizing software. Some of my students use that successfully to take notes in class.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-22T09:42:25.370" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="3801" PostTypeId="2" ParentId="3799" CreationDate="2014-06-22T13:52:26.347" Score="10" Body="&lt;p&gt;There are many reasons for handwriting problems, and many of these reasons may not be &lt;em&gt;remedied&lt;/em&gt; with mathematics education methods and approaches. For example, &lt;a href=&quot;http://www.nha-handwriting.org.uk/publications/selected-articles/handwriting-problems-in-children-with-aspergers&quot;&gt;Asperger's Syndrome has been associated with difficulty in handwriting&lt;/a&gt;, and it is possible that you would not be aware of this student being on the spectrum. While not a disability, Asperger's demonstrates that there is a possibility we will see students with unbalanced levels of ability. It's especially surprising to see a student exhibit mathematical understanding, and then struggle working a problem because of writing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Practice&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the problem is simply that the student just has not had enough practice with handwriting and has not had a chance to become good at it, the answer will clearly be: practice. Or help from someone who specializes in writing (not a mathematics educator in either case). However, the older the student, the less likely it would seem that lack of practice would be the issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Speed&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the problem is something along the lines of what I suggested above (or many similar issues outside of a lack of practice), it is possible that time pressure is a contributing factor. As noted in studies on Asperger's-related handwriting issues, speed is a factor in student writing, and speed demands increase as students progress in age. So, one possible remedy is to recognize that writing is going to be slower than what you would expect, and accommodate that by changing expectations and allowing more time. Stress clarity, not speed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Accomodations&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To address the school problem, if it actually has been a problem for this student, students can often get accommodations to allow for mismatched skill levels that would interfere with assessment. That may only occur with a diagnosis; the school administration can guide you here. Local ways of doing things differ (all over the world, of course). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We might also more generally talk about the problems that ELL (English Language Learners) have with mathematics education. It's another case in which a skill outside of the mathematics education realm may interfere with either learning or assessment. And there are many people concerned with this problem. &lt;a href=&quot;http://ell.stanford.edu/sites/default/files/math_learnmore_files/2.Principles%20for%20Math%20Instruction%208-14-13.pdf&quot;&gt;If we look at some of what researchers consider there&lt;/a&gt;, what can we take away? One thing is that, as math educators, our job is not necessarily to make all students do things the same way. Developing mathematical skills and knowledge may mean working with a student's strengths (like the ability to discuss mathematics) rather than a focus on deficits (which you may not be able to address in math tutoring or classrooms).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A short answer here: workarounds. Possibly more structure in the writing and less actual writing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Structure&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What form would that structure take? You specifically mention a difficulty going back and interpreting handwriting that was done 30 seconds ago. Assuming the student is doing fine &lt;em&gt;while writing&lt;/em&gt;, what may help here is the ability to recognize the higher-value portion of the writing and pay special attention to that so that the student &lt;em&gt;can&lt;/em&gt; come back later if needed. For example, an important result, or a simplified expression that will be of use later? Re-write that patiently and put a box around it. This may be more effective than just having the student slow down everywhere -- just learn to recognize what part of the writing is important, slow down there, and put a box around it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aside from the research I referenced in the links, I personally have struggled with handwriting issues (not diagnosed with anything in particular) and organizational problems; this is how I deal with them. Focusing on what's important, taking more time and care with the most important things, and adopting some kind of structure that supports my efforts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;In summary:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While we may not be able to remedy handwriting problems in math education, we can address the problem somewhat by understanding how handwriting, and the particular abilities of a student, interact with their mathematics activity. We can accommodate, or we can provide supports that help them to use their ability most effectively. And we can teach them skills and habits so that they can support themselves in the future.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-06-22T20:24:37.503" LastActivityDate="2014-06-22T20:24:37.503" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3802" PostTypeId="2" ParentId="2497" CreationDate="2014-06-22T16:19:22.737" Score="2" Body="&lt;p&gt;This:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.tcm.com/mediaroom/video/241887/Stand-and-Deliver-Movie-Clip-Fill-the-Hole-.html&quot; rel=&quot;nofollow&quot;&gt;http://www.tcm.com/mediaroom/video/241887/Stand-and-Deliver-Movie-Clip-Fill-the-Hole-.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Seriously. Maybe don't phrase it quite the same way, but the &quot;fill in the hole&quot; analogy always resonated with me.&lt;/p&gt;&#xA;" OwnerUserId="1756" LastActivityDate="2014-06-22T16:19:22.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3804" PostTypeId="1" AcceptedAnswerId="3805" CreationDate="2014-06-22T22:39:08.423" Score="2" ViewCount="363" Body="&lt;p&gt;I am writing a book and for the sake of simplicity I want to do something as follows.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Coef((-3x^2 +5x -1)(x^2 +1), 2) = -3 -1 = -4&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;where the first argument represents polynomial expression and the second one represents order of coefficient we want to output.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a conventional function notation that takes a polynomial and order and yields the coefficient corresponding to the order? In other words, is there &lt;code&gt;Coef&lt;/code&gt;-like notation defined by mathematicians?&lt;/p&gt;&#xA;" OwnerUserId="153" LastEditorUserId="143" LastEditDate="2014-10-07T09:55:32.243" LastActivityDate="2021-03-20T13:36:49.713" Title="Is there a conventional function notation that takes a polynomial and order and yields the coefficient corresponding to the order?" Tags="|notation|functions|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3805" PostTypeId="2" ParentId="3804" CreationDate="2014-06-22T23:16:19.747" Score="7" Body="&lt;p&gt;A notation common when dealing with generating functions (where this operation is crucial) is to write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;[z^n] A(z) = a_n&#xA;$$&lt;/span&gt;&#xA;When &lt;span class=&quot;math-container&quot;&gt;$A(z) = \sum_{k \ge 0} a_k z^k$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But also see e.g. Knuth &amp;quot;Bracket notation for the 'coefficient of' operator&amp;quot; in &lt;em&gt;A classical mind: Essays in honour of C. A. R. Hoare&lt;/em&gt;, pages 247-258 (Prentice Hall 1994), also &lt;a href=&quot;https://arxiv.org/abs/math/9402216&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2021-03-20T13:36:49.713" LastActivityDate="2021-03-20T13:36:49.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3806" PostTypeId="2" ParentId="2786" CreationDate="2014-06-23T00:11:23.983" Score="4" Body="&lt;p&gt;I read in an article the other day that said rather cynically that a sign of a good classroom is one where students are allowed to say more than one sentence about mathematics without being interrupted by the instructor.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been reading a lot of blogs written by university-level mathematicians who are interested in teaching, and the latest buzzword seems to be &quot;inquiry-based learning&quot; - basically, forcing students to develop and prove things on their own rather then merely working rote exercises after someone else has done all the real mathematical work.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;IBL comes out of something called the Moore Method, named after a math teacher who would basically give his students a list of things to be proved and the class consisted entirely of students making presentations of their proofs on the chalkboard and writing up those proofs for critique by classmates.   Here's a site that gives background information on the Moore Method:  &lt;a href=&quot;http://legacyrlmoore.org/&quot; rel=&quot;noreferrer&quot;&gt;http://legacyrlmoore.org/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's an organization that's based around those principles that offers workshops, grants, and a curriculum library for University-level instructors:  &lt;a href=&quot;http://www.inquirybasedlearning.org/&quot; rel=&quot;noreferrer&quot;&gt;http://www.inquirybasedlearning.org/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, if you want students to learn mathematics - maximize the amount of time that students spent doing mathematics in your class rather than WATCHING you do mathematics.   The students need to be the ones thinking mathematically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, a lot of what you might face in teaching mathematics in higher education parallel the issues in teaching mathematics in K-12.   So I'd suggest seeking out the education department on your campus and to work collaboratively with them on ideas that you might have.&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-23T00:11:23.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3807" PostTypeId="1" AcceptedAnswerId="3812" CreationDate="2014-06-23T00:36:13.087" Score="11" ViewCount="886" Body="&lt;p&gt;I've been looking for possible journals to publish an article about undergraduate mathematics pedagogy.   A lot of math journals apparently don't publish articles about pedagogy, or require pedagogical articles to be illuminating some new aspect of math content.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far, here's what i've found:&#xA;&lt;a href=&quot;http://www.tandfonline.com/toc/upri20/current&quot;&gt;http://www.tandfonline.com/toc/upri20/current&lt;/a&gt;&#xA;&lt;a href=&quot;http://sigmaa.maa.org/rume/journals.html&quot;&gt;http://sigmaa.maa.org/rume/journals.html&lt;/a&gt;&#xA;&lt;a href=&quot;http://www.editorialmanager.com/rund/default.asp&quot;&gt;http://www.editorialmanager.com/rund/default.asp&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For those of  you who do research in undergraduate mathematics education - where have you published things?  For those of you who teach undergraduate mathematics, where do you go to get new ideas about teaching?&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-23T17:57:55.043" Title="What are some places to publish pedagogical articles about teaching undergraduate mathematics?" Tags="|undergraduate-education|mathematical-pedagogy|publishing|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3808" PostTypeId="2" ParentId="3799" CreationDate="2014-06-23T00:50:13.257" Score="3" Body="&lt;p&gt;I had a lot of difficulty with illegible handwriting as a child (the lowest grade I've ever gotten in a class in my life was a D- in penmanship).   One thing that was helpful to me was trying different types of pencils - I mean, there's a lot more out there than just your basic #2 pencil.   I think I tried a #4, and that worked pretty well for me, but eventually I settled on mechnical pencils.   Now, strangely enough, I use pens, since they contact the paper differently than pencils and I like to be able to see a record of my mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a special education teacher now as my day job, and at least for our special education students we can refer them to occupational therapy to get help with handwriting.   In the United States, any teacher or parent can initiate a special education evaluation, which would involve a comprehensive look to figure out whether there's some sort of underlying disability.   If assistive technology was needed, too, the special education process would be a great way to document the school's responsibility to pay for it and to be able to get a specialist to do a formal assistive technology assessment.   See here for some guidelines about what assistive technology assessment looks like: &lt;a href=&quot;http://www.ocali.org/project/guidelines_for_assistive_technology_assessment&quot; rel=&quot;nofollow&quot;&gt;http://www.ocali.org/project/guidelines_for_assistive_technology_assessment&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student has health insurance, perhaps the insurance company would pay for occupational therapy with a referral from their doctor.  Actually, having them see a doctor to rule out any kind of medical condition might be a good idea - perhaps they have something causing unsteady hands, for example?   Or perhaps they have an eyesight problem that is causing them to not be able to clearly see what they are writing?&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-23T00:50:13.257" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3809" PostTypeId="2" ParentId="2736" CreationDate="2014-06-23T00:58:00.217" Score="0" Body="&lt;p&gt;My background is in computer programming, so in that sense the mathematical operations, say:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4×4−6&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Get transformed by a compiler into stack operations:&#xA;Put 4 on the stack.&#xA;Put 4 on the stack.&#xA;Multiply the top two things on the stack.&#xA;Store the new thing on the stack.&#xA;Place a 6 on the stack.&#xA;Subtract the last two things on the stack.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when all is said and done, you'd have one thing left on the stack, your answer.  I find myself wondering what your student would make of an RPN calculator.  Or at least diagramming out everything that's happening in order to solve this problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note too, that since all the operations are binary - each time you do an operation it takes two numbers and only gives you back one.   Perhaps this distinction between a binary and unary operation would be useful to consider.   Like the negation operation, -5 - takes a 5 and negates it.   Whereas 3+5 takes two things and gives you back one.   So if you have nothing but binary operators on integers, in the end you're just going to be left with a single answer.&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-23T00:58:00.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3810" PostTypeId="2" ParentId="2694" CreationDate="2014-06-23T01:03:13.277" Score="1" Body="&lt;p&gt;One thing that hit me this past summer while teaching Algebra 2 was that exponential functions and geometric series were basically the same thing, exponential functions being over the reals and geometric series being over the integers.   I had spent years studying mathematics and math education, and never made the connections between those two topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&#xA;1,2,4,8,16&#xA;Is a geometric series, with a ratio of 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But you could also write it as an exponential function, with each term being 2^(n-1).  &lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-23T01:03:13.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3811" PostTypeId="2" ParentId="2641" CreationDate="2014-06-23T01:17:57.890" Score="2" Body="&lt;p&gt;Most schools have a placement examination, in which you can test out of algebra 1, geometry, algebra 2, pre-calculus, etc. and start in Calculus.   Assuming you're fairly mathematically astute and self-motivated, you could probably teach yourself the mathematics up through Calculus and then test out of those prerequisite courses.   Perhaps get a few friends who are good at mathematics that you can have on-call (preferably that have different schedules so someone's always around to ask) and come up with a pacing schedule for yourself and see how far you can get on your own...   You have the freedom this way to choose your own textbook, set your own pace, and ask people you truly genuinely like and enjoy talking to for help when you need it...&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-23T01:17:57.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3812" PostTypeId="2" ParentId="3807" CreationDate="2014-06-23T01:20:26.520" Score="5" Body="&lt;p&gt;The links you've already provided (especially the second one) are quite good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The top journal, I'd say, is the &lt;em&gt;Journal for Research in Mathematics Education (JRME)&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on the content of what you wish to publish, it may be that a journal about education (not specific to mathematics) or one about psychology or cognitive studies (again, not specific to mathematics education) would be viable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One option would be to look at where prominent mathematics educators have published; for example, you can find some of A Schoenfeld's recent publications &lt;a href=&quot;http://gse.berkeley.edu/people/alan-h-schoenfeld#user_user_full_group_research&quot;&gt;here&lt;/a&gt;; some of D Ball's publications &lt;a href=&quot;http://www-personal.umich.edu/~dball/background/DBall_cv_may2014.pdf&quot;&gt;here&lt;/a&gt; (CV pdf); or, more generally, by searching for individuals on Google Scholar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A lesser known option is the &lt;em&gt;Journal of Mathematics Education at Teachers College (JMETC)&lt;/em&gt;. The next issue has as its focus post-secondary education, so it may be a good fit for an article on undergraduate mathematics pedagogy. A link to the &lt;em&gt;JMETC&lt;/em&gt; page can be found &lt;a href=&quot;http://journals.tc-library.org/index.php/matheducation/&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I might note that, despite its relative newness, the &lt;em&gt;JMETC&lt;/em&gt; has featured some well-known names in mathematics education: H Burkhardt, HP Ginsburg, C Greenes, J Kilpatrick, HO Pollak, A Schoenfeld, E Silver, WF Tate, U Treisman, and HH Wu. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have listed some of these individuals' names, since they are generally good people to search for in order to see where else they have published recently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As for my own reading: I really like &lt;em&gt;Math Horizons&lt;/em&gt; and the &lt;em&gt;American Mathematical Monthly&lt;/em&gt;; I think the &lt;em&gt;Mathematics Teacher&lt;/em&gt; is a good education-specific choice, too.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-06-23T01:20:26.520" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3813" PostTypeId="1" AcceptedAnswerId="3820" CreationDate="2014-06-23T02:57:00.613" Score="9" ViewCount="1072" Body="&lt;p&gt;Students are exposed to sets very early in their education, so my first inclination is that this would be the best method to give children in the early primary grades an introduction to probability and statistics.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I struggle with the notion of how to best promote the ideas of chance, etc. to students that wouldn't understand, for example, that a smaller set of two items out of a set of four would represent the chances of the smaller set being drawn are about half.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are fractions and percents -- I know these early concepts are two sides of the same coin, pun intended, but aren't always taught concurrently -- necessary prerequisites for learning probabilistic and statistical concepts, or are there ways to provide intuition (for example with math based on the analog clock) into these lofty ideas without them? &lt;/p&gt;&#xA;" OwnerUserId="16" LastActivityDate="2015-07-22T20:04:10.537" Title="What is a logical way to introduce probability and statistics to students that don't know fractions or percentages yet?" Tags="|primary-education|statistics|probability|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="3814" PostTypeId="2" ParentId="3807" CreationDate="2014-06-23T12:14:00.510" Score="5" Body="&lt;p&gt;As for who will publish undergraduate math research, I suggest that &lt;a href=&quot;http://www.springer.com/education+&amp;amp;+language/mathematics+education/journal/10649&quot; rel=&quot;nofollow&quot;&gt;Educational Studies in Mathematics&lt;/a&gt; is influential in math education research, and will publish research at the undergraduate level. A member of my dissertation committee recently published a study of the concept of limit in an undergraduate mathematics classroom there. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure what journals people are most likely to go to to inform their undergraduate mathematics instructor practice, but here are some practitioner journals that focus on undergraduate mathematics teaching:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.maa.org/publications/periodicals/college-mathematics-journal/the-college-mathematics-journal&quot; rel=&quot;nofollow&quot;&gt;The College Mathematics Journal&lt;/a&gt; an MAA publication (Apparently they have some rules about articles that are purely pedagogical.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.nctm.org/publications/toc.aspx?jrnl=mt&quot; rel=&quot;nofollow&quot;&gt;Mathematics Teacher&lt;/a&gt; extends into undergraduate mathematics&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-06-23T17:57:55.043" LastActivityDate="2014-06-23T17:57:55.043" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3815" PostTypeId="2" ParentId="3813" CreationDate="2014-06-23T12:45:59.027" Score="0" Body="&lt;p&gt;Seeing that our mathematically mature students (after a few rounds of calculus and some discrete math with proofs) struggle with this, and that rather simple problems like Monty and siblings lead to heated discussions here and on MSE, I'd forget about it all.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-23T12:45:59.027" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3816" PostTypeId="2" ParentId="3813" CreationDate="2014-06-23T12:54:26.897" Score="2" Body="&lt;p&gt;You need to reduce the nature of fractions and percentages. Some ideas:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Reduce the nature of ratios, Laplace probabilities and relative frequency to &lt;code&gt;x out of y&lt;/code&gt;. Representations of this can be:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;symbolic: as the sentence „$x$ out of $y$“ or even the fraction symbol $\frac{x}{y}$.&lt;/li&gt;&#xA;&lt;li&gt;iconic: Venn-Diagrams encircling the whole set and the event set.&lt;/li&gt;&#xA;&lt;li&gt;enactive: a bowel representing the whole set and glass inside representing the event set, colored wheels of fortune&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Only compare different fractions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;enactively: put them as beads onto rubber bands with equal distance and draw the rubber bands until equally long. (Beware, that the last bead has a free distance as well.)&lt;/li&gt;&#xA;&lt;li&gt;iconic/symbolic: if you have a fraction lower than $\frac{1}{2}$ and one larger than $\frac{1}{2}$, you can compare them via $\frac{1}{2}$: Are more results inside the event set or outside the event set?&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-06-23T12:54:26.897" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3817" PostTypeId="2" ParentId="3813" CreationDate="2014-06-23T12:55:40.773" Score="7" Body="&lt;p&gt;I'm going to suggest what you're considering is very difficult. And also, I don't think it's something I'd want to do. (But we'll do it below anyway, for fun!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fractions are an early concept that involves a multiplicative relationship between things. In the case of fractions, one of those things is a whole. Students may have some understandings of fraction concepts without understanding the multiplicative relationship part. They can view fraction as a number (because a fraction is also a number). But this is an early opportunity for students to become acquainted with the idea of mathematics involving relationships.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the question is, what sort of understanding of relative &lt;em&gt;anything&lt;/em&gt; (chances or whatever) is possible without other foundational understanding about relationships between numbers (beyond differences)? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Freeing myself to speculate:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would be an example goal understanding of probability be, which does not involve any sort of multiplicative comparison? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it possible with some knowledge of quantitative comparison to understand how changes in a system do not change the relative chance (where &quot;relative&quot; just means that one is a better chance) of two things? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would try to build off children's understanding of fairness, I guess. (Fairness, as in equal sharing, underlies fractions, but it is not an understanding of fractions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance: There's ten cracker jack boxes to share. A toy is only in one. We could demonstrate that if I take a lot more boxes, I'm probably going to get the toy and you are not. It doesn't necessarily matter how often you get the toy with fewer (let's say &quot;2&quot;) boxes. Kids understand &quot;fair.&quot; And this is not fair. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other uses of fair might be related to probability. Instead of equal sharing of objects, equal sharing of chance is underlying the notion of random sampling. If it is possible to demonstrate that the outcome can be different every time, yet everyone was treated in an equal way (fairness) that random sampling is a type of fairness that does not depend on the outcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously I'm focusing on concepts here. It is difficult to do anything numerical without the understandings our exercise constrained us from using. But it is a bit of a fun exercise to think about.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-06-23T12:55:40.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3818" PostTypeId="2" ParentId="2164" CreationDate="2014-06-23T15:33:28.027" Score="13" Body="&lt;p&gt;When I've taught mathematics, getting students to even attempt to read the textbook has always been a challenge.   One of the things that I always start with is having students type sentences into this letter scrambler:&#xA;&lt;a href=&quot;http://www.douglastwitchell.com/scrambled_words.php&quot;&gt;http://www.douglastwitchell.com/scrambled_words.php&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and see that English text still makes sense if you scramble it up.  I then have them putting in a section from a math textbook, and see how it doesn't make sense, particularly with the many different types of data in a textbook - words, numbers, symbols, tables.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I explain to them that a very different kind of reading is needed with mathematics.   This isn't just about reading formulas or reading text.  Linda Takami, in her dissertation, observed:  &quot;“In mathematics, it is necessary to read from left to right, right to left, top to bottom, bottom to top, and diagonally. In addition, tables, graphs, charts, symbols, and illustrations are a part of mathematics reading and are critical to comprehension of mathematics concepts ” (2009, p. x).   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These tables, graphs, and charts are often presented in a different spot than the text that refers to them, and so students encounter them and then just skip over them.   Takami suggests that a teacher might say something like: “you’re going to be asked to write an equation from the data on these graphs” in order to help create a purpose for reading.   (If you have proquest dissertations at your school, you can find the full dissertation here: &lt;a href=&quot;http://search.proquest.com/docview/305018920&quot;&gt;http://search.proquest.com/docview/305018920&lt;/a&gt;).   When I taught Algebra 2, I generally had to scaffold the reading of the text - walk students through it the first few times, and gradually get them to the point where they could read it on their own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roni Jo Draper ( &lt;a href=&quot;http://www.ed.sc.edu/raisse/pdf/MathArticles/SchoolMathematicsReform,Constructivism,andLiteracy.pdf&quot;&gt;http://www.ed.sc.edu/raisse/pdf/MathArticles/SchoolMathematicsReform,Constructivism,andLiteracy.pdf&lt;/a&gt; ) had similar ideas about having a purpose for reading.   She says that students should have a purpose for reading - some question they'd like to answer, some problem they've been unable to solve.   Then they can read the section of the textbook looking for that answer.   Much like an English class, students should also stop and summarize and make predictions as they read a math text.   Again, I found that I had to explicitly teach these skills, rather than just talk to the students about them, and gradually get it to the point where they could do this on their own.&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-23T15:33:28.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3819" PostTypeId="2" ParentId="2641" CreationDate="2014-06-23T19:36:05.180" Score="0" Body="&lt;p&gt;Unfortunately, in order to get to the heavy stuff, the slog is necessary. I think the worst mistake I have seen students make is rushing through math courses, doing just what they need to in order to get to the upper level stuff they need (which they then struggle with because they just skimmed through the boring stuff). When you get to calculus, you will use lots of your algebra and trig principles. In linear algebra, you will use an immense amount of algebra. In statistics, you will use calculus and algebra. And in real analysis and other proof classes, you will use a ton of algebra, though in more abstract terms (without actual numbers often). I would advise to continue thinking about exactly how the solutions are derived and why, because that will help when you get to your proof-based courses and make you stronger with proving theorems. Proofs are 60% philosophy and 40% mathematical theory.&lt;/p&gt;&#xA;" OwnerUserId="1813" LastActivityDate="2014-06-23T19:36:05.180" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3820" PostTypeId="2" ParentId="3813" CreationDate="2014-06-23T20:05:49.493" Score="6" Body="&lt;p&gt;I definitely think you can get the intuitions behind probability to elementary age kids, and doing so might even be a way to introduce the important notions of equivalent fractions / related rates / etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take a six-sided die and mark 1 on one side, 2 on two sides, 3 on three sides.  Show the kids the die.  Ask them, if you roll this die a bunch of times, which number do you think will come up most often?  Ask them how much more often they think 2 will show up than 1? Some of them should have the intuition that it will be twice as often. Ask how much more often they think 3 will come up than 1.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have them do an experiment and create a bar chart with their results.  Make them do enough rolls to see the expected trend.  Ask them why everyone's bar chart is not exactly the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tell them that since there are six sides on the die, and one of them is marked &quot;1&quot;, every time you roll the die, there is one chance out of six that it will land on the 1.  Ask them how many chances out of six there are that the die will land on a 2.  And how many chances out of six are there that the die will land on a 3.  Ask them, if you roll the die 6 times, are you guaranteed to get 1 once, 2 twice and 3 three times?  Since they have done the experiment they know this is not so.  Ask them why they think it would or would not be guaranteed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Propose different games to be played with this die, and ask the students if they are fair or not.  For example, we roll the die once, and if it comes up 1 you win, and if it comes up 2 I win, and if it comes up 3, we roll again.  Ask: Is that fair?  What would be fair?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing you could do (to get at equivalences and scaling) is ask the kids, if I had a 12-sided die, how should I mark the sides if I want the chances of getting 1, 2 or 3 to be the same as it was for the 6-sided die that we were just using.  In other words, how could I label the die so that if you rolled that die a large number of times, your bar charts would look similar to these ones you all made with the 6-sided die.  This will be tricky for them because the chances are now &quot;out of 12&quot; instead of &quot;out of 6&quot;.  Is 1 chance out of 6 the same as 1 chance out of 12?  If not, what would be the same?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also try doing basic probability with a spinner.  Take a game spinner on a background broken into one half and two quarters.  Label the spot that covers half of the spinner area A, and the two quarters B and C.  Ask the kids, if they spin the spinner a lot of times, which letter will come up most often.  Ask them to compare how often they expect to get A versus B, and how often they expect to get C versus B.  If you can come up with decent spinners for them to play with, you can have them do that experiment too.  Ask them if they can give you a &quot;chances out of&quot; statement for how often B will come up.  If they are stuck, try dividing A's region in half with a dotted line, so they can see 4 equal sized (and thus equally likely regions) and then ask them if they can come up with a &quot;chances out of&quot; statement.  If they are still stuck, point out that there are 4 equally likely regions on the spinner, just like there were 6 equally likely sides on the die.  So they need to figure out how many chances out of four each letter has of being chosen on the spinner.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to try introducing the notion of equivalence and scaling up with the spinners, you could then divide all the segments in half again, so there are now 8 segments on the spinner.  Now B looks like it has 2 chances out of 8, and before it looked like 1 chance out of 4.  Are those the same thing?  Can they explain why they think so?   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they are getting good at it, then you could try some &quot;or&quot; questions, like on the die, what are the chances of getting a 1 or a 2 on any given roll? (3 chances out of 6)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't see as easy a way of doing &quot;and&quot; questions, like what is the chance of getting a 1 the first time you roll the die and then a 2 the second time.  Making the chart of all 36 combinations, especially when some look identical, will not go well in the primary grades, IMO.  If you do want to get into that, get two different colored dice, normally numbered 1 through 6, and ask what is the chance of getting a 1 on the red die, and a 2 on the blue die.  And then help them enumerate all 36 combinations.  But honestly, I think at that level, you're better off working with slightly older kids who understand multiplication and fractions.&lt;/p&gt;&#xA;" OwnerUserId="817" LastEditorUserId="817" LastEditDate="2014-06-24T00:20:58.977" LastActivityDate="2014-06-24T00:20:58.977" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3821" PostTypeId="2" ParentId="2415" CreationDate="2014-06-24T00:41:11.137" Score="2" Body="&lt;p&gt;This is not explicitly an Engineering Mathematics textbook, but take a look at &lt;a href=&quot;http://www.artofproblemsolving.com/Store/viewitem.php?item=calculus&quot; rel=&quot;nofollow&quot;&gt;The Art of Problem Solving's Calculus text&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The AoPS textbooks are intended for students with a strong interest in mathematical competitions.  As such, they include a lot of &quot;non-routine&quot; problems taken from past problem solving competitions, or based on such problems.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a &quot;Calculus for Engineers&quot; course, I'd probably stick with a more traditional textbook and supplement with materials from the AoPS text.&lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-06-24T00:41:11.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3822" PostTypeId="2" ParentId="2415" CreationDate="2014-06-24T00:41:45.230" Score="4" Body="&lt;p&gt;I come at this question both having been a computer science student in undergraduate but also having studied mathematics education in graduate school.   In my computer science program, they spent very little time actually teaching us about the process of solving problems - what little we learned about this came from trial and error.   We had a lot of really mathematically intense classes, but almost no training in mathematical thinking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We did have a class on techniques of proof, which was called &quot;discrete math&quot; and included combinatorics and things like series, sequences, and recursion, and recurrence relations as well.   So in looking at your question, I looked around and found an open textbook for a discrete math class for engineers: &lt;a href=&quot;http://faculty.uml.edu/klevasseur/ADS2_zips/ADS_Fundamentals_V2-0.pdf&quot; rel=&quot;nofollow&quot;&gt;http://faculty.uml.edu/klevasseur/ADS2_zips/ADS_Fundamentals_V2-0.pdf&lt;/a&gt;   - you can probably find similar things in other textbooks on discrete math - that tends to be where formal mathematical thinking is taught to engineers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By contrast, in my math education program, we spent most of the program focusing on mathematical problem solving and looking at the process and heuristics of how people solve mathematical problems.   So the other thing I searched for was about teaching &quot;problem solving&quot; and &quot;mathematical problem solving&quot; to engineers.   Here's the two things that seemed relevant:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A chapter on teaching problem solving to engineering students from a book on teaching engineering:  &lt;a href=&quot;https://engineering.purdue.edu/ChE/AboutUs/Publications/TeachingEng/chapter5.pdf&quot; rel=&quot;nofollow&quot;&gt;https://engineering.purdue.edu/ChE/AboutUs/Publications/TeachingEng/chapter5.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A study of how some engineering students used mathematical problem solving in their capstone courses along with a lot of theoretical reflection:&#xA;&lt;a href=&quot;http://monicacardella.com/mc/wp-content/uploads/2012/06/cardella2005_ASEE_engineeringstudentsmath.pdf&quot; rel=&quot;nofollow&quot;&gt;http://monicacardella.com/mc/wp-content/uploads/2012/06/cardella2005_ASEE_engineeringstudentsmath.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-24T00:41:45.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3823" PostTypeId="1" CreationDate="2014-06-24T01:28:13.287" Score="2" ViewCount="256" Body="&lt;p&gt;I love deriving formulas. I am pretty good at Math. Maybe because I have a little asperger syndrome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most people don't learn Math that way. Why not?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently taught peano axioms to my daughter. She seems to be able to grasp it a little, then I'll teach the basic again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just want Math to be very easy for her like it is for me.&lt;/p&gt;&#xA;" OwnerUserId="1814" LastEditorUserId="1814" LastEditDate="2014-06-27T10:04:21.493" LastActivityDate="2014-06-27T10:04:21.493" Title="Is derivation based Math a good way to teach my 5 years old daughter addition?" Tags="|axioms-foundations|" AnswerCount="1" CommentCount="5" ClosedDate="2014-06-24T18:08:07.483" ContentLicense="CC BY-SA 3.0" />
  <row Id="3824" PostTypeId="2" ParentId="2790" CreationDate="2014-06-24T03:49:49.990" Score="9" Body="&lt;p&gt;I'm surprised not to see any mention yet of constructive type theory and computer proof assistants, in which a computer program &lt;em&gt;is literally&lt;/em&gt; a proof and vice versa.  For instance, if you wrote a program to compute the result of the Collatz function in a verified and termination-checked programming language like Coq or Agda, then it &lt;em&gt;would&lt;/em&gt; be a proof of the Collatz conjecture.  Conversely, a proof of the existence of an object in constructive type theory is, literally, an algorithm to compute that object.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think computers have many roles to play in mathematics education.  The coding of algorithms certainly has a place in education, but it seems to me that a more promising way to use computers to teach logical reasoning is to use a computer program that actually &lt;em&gt;understands&lt;/em&gt; logical reasoning.  Hence why I asked &lt;a href=&quot;https://matheducators.stackexchange.com/questions/224/teaching-logic-with-a-proof-assistant&quot;&gt;this question&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-06-24T03:49:49.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3826" PostTypeId="2" ParentId="3823" CreationDate="2014-06-24T13:16:21.680" Score="-5" Body="&lt;p&gt;My mother used to be a primary school teacher (now retired). And one of the things she complained a lot about was parents teaching children the subjects &lt;em&gt;before&lt;/em&gt; they came up at school, and specially parents that used &lt;em&gt;different&lt;/em&gt; approaches than those taught. Children used to do things &quot;some other way&quot; get confused, and are often a distraction on the other pupils.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please don't do this. Find out what she sees at school, and how, and help her understand that. See what other interests she has (perhaps plants or animals in the garden? going to the zoo and learning more about exotic animals? palenteology, and learning about the animals which used to live where you are?) and see to share that with her (as long as it doesn't interfere with what is done at school, it should be fine).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Yes, I'm assuming she attends school, or will do so soon(ish). At home tutoring is another kettle of fish...]&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-24T13:16:21.680" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="3827" PostTypeId="1" CreationDate="2014-06-24T16:38:11.223" Score="11" ViewCount="434" Body="&lt;p&gt;&lt;a href=&quot;https://chess.stackexchange.com/questions/5199/does-playing-tournament-chess-help-one-in-the-field-of-chess-related-math&quot;&gt;https://chess.stackexchange.com/questions/5199/does-playing-tournament-chess-help-one-in-the-field-of-chess-related-math&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I often wonder whether the visualization of the various types of chess-related graphs is helped along by playing competitively.  This would seem to fly in the face of a recent peer-reviewed study in the sense that there would be a high-level benefit (ie teaches skills that can be transferred to other domains) to chess if this line of thinking is correct, namely in the field of chess-related math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://people.brunel.ac.uk/~hsstffg/preprints/chess_and_education.PDF&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://people.brunel.ac.uk/~hsstffg/preprints/chess_and_education.PDF&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, the study seems to be very thorough and is one of the few peer-reviewed articles available on the subject.  One would expect if there were some high-level gain to playing competitively then the test that shows this would be repeatable.  However, there has been mixed results, as the above article claims.  Perhaps the field of chess-related math is only chess-related in that the only skill that transfers is knowing how the pieces move, without any regard for strategy. What do you think?&lt;/p&gt;&#xA;" OwnerUserId="1816" LastEditorUserId="-1" LastEditDate="2017-04-13T12:48:53.500" LastActivityDate="2015-08-07T02:10:22.487" Title="Does playing tournament chess result in high-level mathematical gains, and not merely a skill whose benefits can be gained through any other activity?" Tags="|education-research|transferrable-skills|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3828" PostTypeId="2" ParentId="2164" CreationDate="2014-06-24T16:52:51.897" Score="11" Body="&lt;p&gt;Reading intellectually challenging material simply requires practice, so all you can really do to help your students learn this skill is to force them to practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students have an immature approach to education, and therefore tend to do only what we give them direct incentives for doing. I give incentives such as easy, multiple-choice reading quizzes that happen during the first 5 minutes of every class. Without such an incentive, I'm sure most students would never attempt read the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They also will take their cues from your behavior. If your classes consist of nothing but straight lecturing in which the material is presented from scratch, then they will see that you didn't expect them to read. I use lecture for active learning and for working on concepts that I know are hard for everyone -- not for spoon-feeding them simple facts and definitions of terms that they were responsible for getting from the reading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These techniques work well for me, and nearly all my students do the reading. They're not original ideas of mine. You can find them described in by Eric Mazur's book Peer Instruction: A User's Manual.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-06-24T16:59:11.413" LastActivityDate="2014-06-24T16:59:11.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3829" PostTypeId="2" ParentId="2457" CreationDate="2014-06-24T19:27:47.063" Score="4" Body="&lt;p&gt;I don't know of research on the use of multiple choice questions for summative assessment specifically in mathematics. However, what's important is the approach to construction of multiple choice questions, if they are to overcome challenges like the ones you list. Following course design basics from sources like &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0131950843&quot; rel=&quot;nofollow&quot;&gt;Wiggins &amp;amp; McTighe&lt;/a&gt;, what we know about assessments are that they are most effective when they are aligned with your learning goals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With this in mind, I did come across &lt;a href=&quot;http://reference.sabinet.co.za/sa_epublication_article/amesap_n69_a1&quot; rel=&quot;nofollow&quot;&gt;one article&lt;/a&gt; that looks promising, but at the moment I only have access to the abstract, so I can't guarantee its value. (&lt;strong&gt;Edit:&lt;/strong&gt; The article is available in full &lt;a href=&quot;http://www.amesa.org.za/amesap_n69_a1.pdf&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.) From the abstract it appears the authors are taking a somewhat nuanced look at assessment in mathematics, and considering connections between the types of things you might want to assess, and how different assessment tools (in particular MCQs) can be of value.&lt;/p&gt;&#xA;" OwnerUserId="1817" LastEditorUserId="262" LastEditDate="2014-06-24T21:42:09.233" LastActivityDate="2014-06-24T21:42:09.233" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3830" PostTypeId="2" ParentId="294" CreationDate="2014-06-25T00:03:30.323" Score="5" Body="&lt;p&gt;Eli Maor's book e: Story of a number was what first got me interested in mathematics when I read it back in high school.   He presents the history of e, and of topics such as logarithms and calculus in a narrative that is engaging and math that is both simple enough that I could follow as a high school student but complex enough that I still come back to it to unpack its depth over a decade later.    ( &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691141347&quot; rel=&quot;noreferrer&quot;&gt;http://www.amazon.com/Story-Number-Princeton-Science-Library/dp/0691141347&lt;/a&gt; )&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He also has a great book on infinity, To Infinity and Beyond, which covers infinity in lots of different contexts and situations, spanning across the mathematical disciplines.   I used this as a textbook for a high school seminar I taught, but I think it's really more on the beginning college level - the high school students found it pretty challenging  ( &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691025118&quot; rel=&quot;noreferrer&quot;&gt;http://www.amazon.com/Infinity-Beyond-Cultural-History-Infinite/dp/0691025118&lt;/a&gt; ).   It looks like he has some more recent books on the Pythagorean Theorem, Trigonometry, and Geometry, too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also if the book on infinity piques your interest, there's another great book I found that comes at it more from the story of Cantor and set theory:  Amir Aczel's The Mystery of an Aleph.  This one has some good ties to infinity and religion, as well as a great historical component.   It also really gives a good layman's introduction to the paradoxes of set theory and the continuum hypothesis.    &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0743422996&quot; rel=&quot;noreferrer&quot;&gt;http://www.amazon.com/Mystery-Aleph-Mathematics-Kabbalah-Infinity/dp/0743422996&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, also on the topic of math history, Theoni Pappas has a great book, Mathematical Scandals which is an easy read and highly engaging:  &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/188455010X&quot; rel=&quot;noreferrer&quot;&gt;http://www.amazon.com/Mathematical-Scandals-Theoni-Pappas/dp/188455010X&lt;/a&gt; - this is one that I'd probably give to someone who said &quot;they weren't a math person&quot; as the other three I recommended are chock full of formulas and diagrams and lots of &quot;mathy&quot; content :-)&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-25T00:03:30.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3831" PostTypeId="2" ParentId="238" CreationDate="2014-06-25T00:08:29.640" Score="4" Body="&lt;p&gt;My favorite has to be The Greedy Triangle, the story of a triangle that's unsatisfied with its appearance and decides to add another side.  I won't spoil the story for you, but you can imagine what happens when it gets too many sides - a nice preview of Calculus and limits.    Plus, how many children's books end up in Shape-Fiction and Self-Acceptance-Fiction as the Library of Congress Subject Headings?  :-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0545042208&quot; rel=&quot;nofollow&quot;&gt;http://www.amazon.com/The-Greedy-Triangle-Scholastic-Bookshelf/dp/0545042208&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, sort of a meta-answer to this question:  The NCTM (National Council of Teachers of Mathematics) and NCTE (National Council of Teachers of English) collaborated on a nice book on linking literature and mathematics for kids that not only gives tons of recommended books, but goes over criteria for selecting a good book and gives suggestions for how to use them in the classroom:  &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0814133487&quot; rel=&quot;nofollow&quot;&gt;http://www.amazon.com/New-Visions-Linking-Literature-Mathematics/dp/0814133487&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-25T00:08:29.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3832" PostTypeId="2" ParentId="699" CreationDate="2014-06-25T00:11:56.237" Score="2" Body="&lt;p&gt;Back in 1995, Theoni Pappas wrote this wonderful little book of mathematical quotes that I really liked at the time.   I know that today probably most of these are on the web, but I find something to be said for having a tangible book on a shelf:&#xA;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1884550045&quot; rel=&quot;nofollow&quot;&gt;http://www.amazon.com/Music-Reason-Experience-Mathematics-Quotations/dp/1884550045&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-25T00:11:56.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3833" PostTypeId="2" ParentId="1760" CreationDate="2014-06-25T00:14:54.370" Score="3" Body="&lt;p&gt;Edelsbrunner's new book,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;A short course in computational geometry and topology&lt;/em&gt;. Springer, 2014.&#xA;  (&lt;a href=&quot;http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-3-319-05956-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer link&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;might serve as a course skeleton.&#xA;It is indeed short (110 pages), and written in his&#xA;laconic but precise style. And beautifully illustrated.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/8Xl4N.png&quot; alt=&quot;HEFig1.2&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-06-25T00:14:54.370" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3834" PostTypeId="2" ParentId="689" CreationDate="2014-06-25T00:28:20.790" Score="2" Body="&lt;p&gt;Los Alamos National Laboratory wrote a nice lesson plan in the mid-90s around an embellished version of the Hilbert Hotel story, including the alignment to the 1989 NCTM standards:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hotel Infinity  (you'll need to click around, as the interface is very 90s...)&#xA;&lt;a href=&quot;http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html&quot; rel=&quot;nofollow&quot;&gt;http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you prefer a youtube video (which didn't exist in the 90s when LANL wrote their lesson plan), here's a 4 part version of the tale sketched out in pen.  Here's part one: &lt;a href=&quot;https://www.youtube.com/watch?v=ZZ7ipq9T_bw&quot; rel=&quot;nofollow&quot;&gt;https://www.youtube.com/watch?v=ZZ7ipq9T_bw&lt;/a&gt; and here's the video author's website:  &lt;a href=&quot;http://www.hotel-infinity.com/&quot; rel=&quot;nofollow&quot;&gt;http://www.hotel-infinity.com/&lt;/a&gt;   &lt;/p&gt;&#xA;" OwnerUserId="1798" LastEditorUserId="1798" LastEditDate="2014-06-25T01:30:37.747" LastActivityDate="2014-06-25T01:30:37.747" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3835" PostTypeId="2" ParentId="294" CreationDate="2014-06-25T01:32:43.780" Score="3" Body="&lt;p&gt;I love the connection between mathematics and physical models&#xA;that serves as the theme in&#xA;Bryant and Sangwin's delightful 2008 book &lt;a href=&quot;https://press.princeton.edu/titles/8624.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How Round Is Your Circle&lt;/em&gt;&lt;/a&gt;:&#xA;&lt;br /&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/6HFit.jpg&quot; alt=&quot;Constant width bodies&quot;&gt;&#xA;&lt;br /&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/lFsSs.png&quot; width=&quot;320&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-12-29T01:01:14.397" LastActivityDate="2018-12-29T01:01:14.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3836" PostTypeId="2" ParentId="2457" CreationDate="2014-06-25T04:15:52.413" Score="5" Body="&lt;p&gt;I've read your question a few times, plus the comments. As best I can determine, your gut feeling is that constructed response exams should be a much better indicator of student understanding than multiple choice exams in math, because a) students cannot guess and b) partial learning can be recognized by analyzing the worked problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, &lt;strong&gt;you want to see a comparison of student performance on multiple choice versus short-answer questions testing the same concepts within the same exam.&lt;/strong&gt; If the average student's performance is similar on the two parts of the exam, this would disprove your hypothesis that short-answer is better. Is this correct? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question is a month old, but I'll assume you as OP think this does state your question and I will continue. I've listed some research articles below. The short answer is that when the question stems are matched, there is little difference in student strategy or student performance. It is possible that very carefully written CR exams might produce more information about student learning, but it is also likely that this careful question-writing hardly ever happens. My 20-min review indicates there is &lt;strong&gt;surprisingly little evidence that CR exams are better than multiple choice&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lukhele, R., Thissen, D., &amp;amp; Wainer, H. (1994). On the Relative Value of Multiple‐Choice, Constructed Response, and Examinee‐Selected Items on Two Achievement Tests. Journal of Educational Measurement, 31(3), 234-250.  They found student answers for AP Chem and AP History did not differ significantly between MC and constructed response.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Katz, I. R., Friedman, D. E., Bennett, R. E., &amp;amp; Berger, A. E. (1996). &lt;a href=&quot;http://www.ets.org/Media/Research/pdf/RR-96-20.pdf&quot; rel=&quot;noreferrer&quot;&gt;Differences in Strategies Used to Solve Stem-Equivalent Constructed-Response and Multiple-Choice SAT&lt;/a&gt; [R]-Mathematics Items. RESEARCH REPORT-EDUCATIONAL TESTING SERVICE PRINCETON RR. Found that matched-stem problems triggered similar student problem-solving and similar results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shepard, L. A. (2008). Commentary on the National Mathematics Advisory Panel recommendations on assessment. Educational Researcher, 37(9), 602-609. This is a short critique of research that matches MC vs CR question stems. They argue that CR allows for &lt;em&gt;different&lt;/em&gt; sorts of testing and this should be taken into account.&lt;/p&gt;&#xA;" OwnerUserId="1207" LastActivityDate="2014-06-25T04:15:52.413" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3837" PostTypeId="2" ParentId="294" CreationDate="2014-06-25T19:46:37.653" Score="12" Body="&lt;p&gt;For a recent suggestion, check &lt;strong&gt;How Not to Be Wrong&lt;/strong&gt; by &lt;a href=&quot;http://en.wikipedia.org/wiki/Jordan_Ellenberg&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Jordan Ellenberg&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/w3Ldi.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lying in the &quot;simple and profound&quot; quadrant, the book also gives deserved attention to &lt;a href=&quot;http://en.wikipedia.org/wiki/Marquis_de_Condorcet&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Condorcet&lt;/strong&gt;&lt;/a&gt;, in addition to providing a very readable book for a wide audience. Rather than my saying more, let me direct you to some recent reviews/responses:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;http://www.latimes.com/books/jacketcopy/la-et-jc-jordan-ellenberg-how-not-to-be-wrong-20140617-story.html#page=1&quot; rel=&quot;noreferrer&quot;&gt;LA Times&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;http://www.salon.com/2014/06/08/how_not_to_be_wrong_what_the_literary_world_can_learn_from_math/&quot; rel=&quot;noreferrer&quot;&gt;Salon&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;http://blogs.scientificamerican.com/roots-of-unity/2014/05/31/how-not-to-be-wrong-book-review/&quot; rel=&quot;noreferrer&quot;&gt;Scientific American&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;http://online.wsj.com/articles/book-review-how-not-to-be-wrong-by-jordan-ellenberg-1402696154&quot; rel=&quot;noreferrer&quot;&gt;WSJ&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/8khQJ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For another author who demonstrates an ability in both the mathematical and literary world, I must recommend &lt;a href=&quot;http://en.wikipedia.org/wiki/Alan_Lightman&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Alan Lightman&lt;/strong&gt;&lt;/a&gt;, and, specifically, &lt;strong&gt;Einstein's Dreams&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/HKBtT.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Incidentally, for better or worse, free copies of the latter are easily found online.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-06-25T19:46:37.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3838" PostTypeId="1" CreationDate="2014-06-25T21:13:55.833" Score="21" ViewCount="847" Body="&lt;p&gt;In a store last weekend, I saw a simple solar-powered calculator for $2. It had four functions plus a square root key. I'm considering the possibility of buying 50 such calculators and handing them out for use during exams in my freshman calc course, as the only calculators allowed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Advantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;No difficulties with determining whether a device is a calculator or a phone, is a graphing calculator, has internet or text-messaging, can do symbolic math, can store notes, ...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In some cases it's sensible to check one's results on calc problems numerically. For instance, I give homework where they differentiate some function and then check themselves numerically by calculating a delta-y/delta x. It would be nice to allow them to use calculators on exams so that they could do the same thing.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prevents an arms race for students trying to use fancier calculators. Saves students money by eliminating a perceived incentive to buy an expensive calculator that actually has low added utility.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Possible disadvantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;A hassle to maintain the sack full of calculators, replace broken ones, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;May mask incompetence in basic arithmetic skills.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Problems would have to be carefully designed so that if a numerical check is required, it stays within the capabilities of the calculators (not so good for transcendentals).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Any problem with a calculator (broken key, ...) becomes my fault.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Has anyone tried this? Is it a good idea?&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-02-05T15:51:33.760" Title="Handing out $2 calculators for tests?" Tags="|exams|testing|calculators|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="3839" PostTypeId="1" AcceptedAnswerId="3841" CreationDate="2014-06-25T21:36:02.860" Score="37" ViewCount="1958" Body="&lt;p&gt;I am designing a course for the upcoming fall semester, and I am tossing around an idea in my head. While planning which topics to cover each week and how to set the pacing of the course, I figured I might as well plan ahead for all of the homework problems I'd like to assign. And if I've already done that before the semester starts ... well, &lt;strong&gt;Why not make every homework assignment I plan on giving to the students during the semester available on the first day?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I plan on giving one assignment per week, due on Fridays. Each assignment will have a specific due date, and students will be required to turn in those designated problems by that date (but I will, of course, accept early submissions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Potential benefits for students:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They can see the &quot;story arc&quot; for the course, beyond just a list of topics in the syllabus (which might have terminology they don't understand anyway)&lt;/li&gt;&#xA;&lt;li&gt;They might be more motivated to engage with the course, seeing the kinds of problems they'll be able to solve later on&lt;/li&gt;&#xA;&lt;li&gt;They can get started on problems ahead of time, alleviating time pressure during future busy times of the semester&lt;/li&gt;&#xA;&lt;li&gt;They can set their own pace, rather than waiting for each week's problems to be &quot;released&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Potential downsides for students:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Seeing the difficult problems posed later on, or just seeing the sheer number of problems, might demotivate some daunted students&lt;/li&gt;&#xA;&lt;li&gt;They might feel like standards will be higher, knowing that every problem has been available for so long&lt;/li&gt;&#xA;&lt;li&gt;Some students might get distracted from the current topic of the course by working on later problems too far ahead of time&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Potential benefits for me, as the instructor:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;While introducing a topic in class, I can point to specific problems they can now solve with this knowledge (instead of vaguely saying, &quot;You'll see this later on, trust me&quot;)&lt;/li&gt;&#xA;&lt;li&gt;I anticipate more office hours visitors as the students work on the problems ahead of time and realize they need help&lt;/li&gt;&#xA;&lt;li&gt;I'll have a set schedule and can use this as an impetus to keep the desired pace of the course going throughout the semester&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Potential downsides for me:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Having such a rigid schedule might be too restrictive, and small tweaks to the pacing/content of the course will be difficult to implement. (For instance, I would feel uncomfortable adjusting an assignment if some students had already started working on those problems.)&lt;/li&gt;&#xA;&lt;li&gt;Might this lead to more student collusion?&lt;/li&gt;&#xA;&lt;li&gt;It might be difficult to keep students on task if they're generally disinterested in the current topic, for whatever reason&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Main questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Is this worthwhile?&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Do you have personal experience with doing this? How did it go? Do you have any suggestions for how to do this well?&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;If you'd recommend against it, why? What are the major downsides that outweigh the benefits?&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Even if you don't have direct experience, do you have any items to add to the lists of benefits/downsides that might help me make this decision?&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(For reference, I'm planning this for an introduction to combinatorics and graph theory, a 3000-level course, so mostly 3rd and 4th year students, and almost surely entirely math majors. If your answer takes this into consideration, even better, but this is not a requirement for a good answer!)&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2022-11-05T07:04:12.473" Title="Are there any benefits to having an entire course's homework problems available from day one?" Tags="|undergraduate-education|course-design|homework|problem-design|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3840" PostTypeId="2" ParentId="3838" CreationDate="2014-06-26T01:18:34.657" Score="11" Body="&lt;p&gt;I haven't tried this, but here are some comments and suggestions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although one might expect students to be able to do basic arithmetic (and thus not need calculators at all), if your exam involves numbers with many significant figures or with square roots, it would make sense to let them use a basic calculator (since you are teaching freshman calculus and not, say, grade school arithmetic).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider who will shoulder the expenses, that is, who will own the calculators.  If you use school money to buy the calculators, you will be held accountable if they are lost, stolen, or damaged.  If you sell the calculators to the students, this might be seen as an unethical practice, even if it is clear that you did not profit from the sales.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The following suggestions assume that either you or the school owns the calculators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assign each student a unique number, label the calculators with these numbers, and assign each calculator to the corresponding student.  This will make it easy to identify who is responsible if the calculator is stolen or damaged.  Make it clear from the start what the penalties are for these cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it would be better if you were the one to keep the calculators when they are not in use.  If you let the students take the calculators home it becomes more difficult to hold them accountable for the calculators (for example, a student might claim someone else stole his/her calculator when in fact he/she was the one to steal it).&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-06-26T01:18:34.657" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3841" PostTypeId="2" ParentId="3839" CreationDate="2014-06-26T02:21:14.067" Score="23" Body="&lt;p&gt;I always organize my courses with the totality of the course set from the outset whenever I can. I see this as being closely tied to your question concerning homeworks assigned.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The benefits I see:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Keeps me and my students on track&lt;/strong&gt;. The semester invariably gets busy, it's nice to have a go-to place where everything is set from the outset. I can always tell the students to go look at my &quot;course planner&quot;. This is technically distinguished from the syllabus which I am bound by various university regulations to clutter with useless information no one cares about. My course planner is usually like 2 pages at most with a spreadsheet with the day by day for the class including both reading from the text, my notes and upcoming assignments.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;When test time comes&lt;/strong&gt;, I've already told them where the test gets to and what they should have done to prepare. If they happen to be absent it doesn't matter. It's all in writing and they have no excuse (well, no me-based excuse let's say)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Organization of the course&lt;/strong&gt;. If you organize the whole course in one sitting there will probably be more continuity in exposition than there might be otherwise. At least, that is the case for me. For others, it might not matter.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Allows students to work ahead&lt;/strong&gt; this is a somewhat rare event. But, in principle, if a student was like me, they might want more homework to attack if they finish what is due soon. Or, if they have some family reunion or honeymoon etc... they can work ahead to not destroy their semester. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Unless you're making your own homework&lt;/strong&gt; solutions are available online already. It doesn't matter if you give them a week or 12 weeks when the solution-pdf is just a google search away. Anecdotally, my colleagues who make no course planner have the same trouble with textbook based problems. For problems I write, the trouble is much more with bad group work. I encourage working together, but, I insist they write their own solution. I am often troubled by the group apathy.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In many course delivery systems there is this idea the course unfolds as it progresses. As a student this would have been quite annoying. I mean, am I an adult? Can I work towards a clearly defined goal, or do I need to be tricked into climbing a hill half as high before I climb the rest? The students will respect your honesty and see the hard work and care you've placed in the course from the outset. It has value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Words of caution: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;do include the word &quot;tentative&quot; somewhere prominent. &lt;/li&gt;&#xA;&lt;li&gt;use your best judgement when test time comes. Students rarely complain about pushing a test back a week and covering new material. Yet, if you plan to cover new material before the Test officially this is greeted differently. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Here are links to a few of my past plans:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.supermath.info/math321s2014planner.pdf&quot;&gt;Linear Algebra&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.supermath.info/math334PlannerSpring2014.pdf&quot;&gt;Differential Equations&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.supermath.info/Physics231S2014planner.pdf&quot;&gt;Physics I&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, for a non-example, since this was something I updated as the course progressed:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.supermath.info/CoursePlannerInProgress.pdf&quot;&gt;Elementary Differential Geometry&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2014-06-26T02:26:32.080" LastActivityDate="2014-06-26T02:26:32.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3842" PostTypeId="1" CreationDate="2014-06-24T16:07:44.067" Score="5" ViewCount="613" Body="&lt;p&gt;I am interested in coming up with a few lectures from first-semester calculus that I can incorporate Mathematica into in a natural way.  I have already written some code for a lecture on the $\delta\mbox{-}\epsilon$ definition of a limit, and I am looking for suggestions about things other people have done in their classrooms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also be interested in knowing if anyone has done examples that &quot;break&quot; Mathematica.  That is, examples that require human ingenuity that a computer can't simply compute the answer to. &lt;/p&gt;&#xA;" OwnerDisplayName="Trevor" LastEditorUserId="143" LastEditDate="2014-06-26T17:10:16.630" LastActivityDate="2014-06-27T12:53:41.893" Title="Use Mathematica to teach calculus" Tags="|calculus|course-design|technology-in-education|software|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="3843" PostTypeId="2" ParentId="3839" CreationDate="2014-06-26T05:02:21.830" Score="7" Body="&lt;p&gt;It seems you've done a good analysis of the student &amp;lt;--&gt; course  and the instructor &amp;lt;--&gt; course relationships with regards to your proposal.  One factor you may not have considered is the student &amp;lt;--&gt; instructor relationship.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Posting homework ahead of time in this way will make students feel that you are a careful planner, but also inflexible and rigid.  You will be seen as less creating a learning experience for them and more as a conduit for knowledge, such as a talking textbook.  I see such a policy as being mildly negative in this regard.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since in the U.S. there are typically student evaluations in which the students share their satisfaction with the instructor and course, such a policy in my view would have a small but negative impact on your career.  I would not recommend it; instead I suggest using the creativity and enthusiasm for teaching that you clearly have, to find ways to better engage and delight the students.&lt;/p&gt;&#xA;" OwnerUserId="270" LastActivityDate="2014-06-26T05:02:21.830" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3844" PostTypeId="2" ParentId="3838" CreationDate="2014-06-26T05:21:21.257" Score="7" Body="&lt;p&gt;As of this academic year, for many courses, our students are restricted to a basic relatively-inexpensive scientific calculator during tests/exams. Students are required to buy such a calculator themselves, but we also keep a box of calculators owned by us handy for emergencies. A student can borrow a calculator for the duration of a test and then return it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the reason for the restriction was to level the playing field, as the occasional student was found using a calculator with advanced or unknown capabilities such as symbolic mathematics or a device running in a foreign language. Permitting a scientific calculator means that students do have access to transcendental functions.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2014-06-26T05:21:21.257" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3845" PostTypeId="2" ParentId="294" CreationDate="2014-06-26T07:18:53.327" Score="3" Body="&lt;p&gt;When I was younger I  was fascinated by Jorge Luis Borges' ' Library of Babel'. It is a story about combinatorics, countability and the hugeness of finite numbers.&lt;/p&gt;&#xA;" OwnerUserId="901" LastActivityDate="2014-06-26T07:18:53.327" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3846" PostTypeId="1" CreationDate="2014-06-26T08:46:26.617" Score="4" ViewCount="113" Body="&lt;p&gt;In some situations one need to design some compact courses for undergraduate/graduate students. I would like to know how to do it without any loss of generality, main ideas and any other important parameters of a usual course? Is there any difference between compact course design for graduate and undergraduate students? In fact I am searching for the main parameters which I should bring in my talks and those subjects which I should avoid. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By a compact course I mean the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Very Short: 1 lecture&lt;/li&gt;&#xA;&lt;li&gt;Short: 3 lectures&lt;/li&gt;&#xA;&lt;li&gt;Medium: 6 lectures&lt;/li&gt;&#xA;&lt;li&gt;Long: 12 Lectures  &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="64" LastEditDate="2014-06-27T17:08:05.257" LastActivityDate="2014-06-27T17:08:05.257" Title="How to design a compact course without loss of main benefits of a usual course?" Tags="|course-design|general-pedagogy|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3847" PostTypeId="2" ParentId="3842" CreationDate="2014-06-26T10:59:59.590" Score="9" Body="&lt;p&gt;For several years I taught I course (which no longer exists) making heavy use of Maple.&#xA;All the material is at &lt;a href=&quot;http://neil-strickland.staff.shef.ac.uk/courses/MAS100/&quot;&gt;http://neil-strickland.staff.shef.ac.uk/courses/MAS100/&lt;/a&gt;.  Obviously technical details would change if you wanted to use Mathematica instead, but the general approach might still be useful.&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2014-06-26T10:59:59.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3848" PostTypeId="1" CreationDate="2014-06-26T11:49:09.833" Score="11" ViewCount="799" Body="&lt;p&gt;My class is mostly are in business and politics and most have made the choice never to look at math again.  Is there still any chance I can motivate exact notions like $\sqrt{2}$ in a manner that is relevant to them? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One counterargument might be that we can use the decimal system and we usually only need 2 points after the decimal anyway...&lt;/p&gt;&#xA;" OwnerUserId="1654" LastActivityDate="2014-08-25T09:26:56.677" Title="is it possible to motivate square roots e.g. $\sqrt{2}$ in business math?" Tags="|secondary-education|concept-motivation|algebra|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="3849" PostTypeId="2" ParentId="3848" CreationDate="2014-06-26T12:18:08.883" Score="-3" Body="&lt;p&gt;Unless the concept of square root (or any other operation that gives irrational numbers, what I'm guessing you are really after) comes up naturally, it is probably hopeless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Areas where such come up are when you compute e.g. interest rates (given a present value and a schedule of payments, what is the interest rate). Sure, you can compute approximate values, but the exact value will most of the time turn out irrational.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry, I don't know enough of this area (even less what your class has seen) to come up with more examples.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-06-26T12:18:08.883" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3850" PostTypeId="2" ParentId="3848" CreationDate="2014-06-26T12:38:38.233" Score="7" Body="&lt;p&gt;I don't know about the square root of 2, per se, but the nth root is an important concept. e.g. If a stock goes up 40% one year, and down 40% the next, the average gain is 0%, yet the compound growth is 1.4 * .6 = .86 and the square root of this number, about .927 shows an average compound loss of 7.3% per year. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-06-26T14:39:02.073" LastActivityDate="2014-06-26T14:39:02.073" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3851" PostTypeId="2" ParentId="3848" CreationDate="2014-06-26T17:12:20.100" Score="1" Body="&lt;p&gt;Our money is growing such that it is compounded once per year.&lt;/p&gt;&#xA;&lt;p&gt;If we double our money in two years, what is the yearly interest rate?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(1+r)^2 = 2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$r = \sqrt2 - 1$&lt;/span&gt; which is about 41%. It's going well!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-06-26T17:12:20.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3852" PostTypeId="2" ParentId="3842" CreationDate="2014-06-26T17:44:07.287" Score="0" Body="&lt;p&gt;Here's some courseware that uses a custom version of Mathematica to teach Calculus, but it looks like it costs about $99/student.   On the other hand, that's probably cheaper than buying Mathematica:  &lt;a href=&quot;http://www.makingmath.com/&quot; rel=&quot;nofollow&quot;&gt;http://www.makingmath.com/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-26T17:44:07.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3853" PostTypeId="2" ParentId="3848" CreationDate="2014-06-26T17:49:20.110" Score="13" Body="&lt;p&gt;There seem to be two different issues mixed together in your question, and I think they have different answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first issue is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a way to motivate the &lt;em&gt;concept of&lt;/em&gt; square roots, for example, $\sqrt{2}$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As the other answerers have pointed out, square roots do arise quite naturally in many financial applications.  If some quantity (say, an investment account balance) grows in successive periods of time by $r_1\%,  r_2\%, \ldots r_n\%$ then the average rate of growth can be computed by defining the &lt;em&gt;growth factors&lt;/em&gt; $k_i=1+\frac{r_i}{100}$, computing their &lt;em&gt;geometric mean&lt;/em&gt; $k=\sqrt[n]{k_1 k_2 k_3 \cdots k_n}$, and interpreting $k-1$ as a percent.  (This also works if some of the $r_i$ are negative, i.e. in the case of loss rather than growth.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second issue that I think seems to be implied by your question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a way to motivate the &lt;em&gt;concept of irrationality&lt;/em&gt;, and in particular the virtue of using a symbolic representation of an exact quantity like $\sqrt{2}$ rather than a rational approximation?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here I think you are on shakier ground.  I think the &lt;em&gt;idea&lt;/em&gt; of irrationality -- and in particular, proving that $\sqrt{2}$ cannot be expressed as a ratio of two integers -- is one of the greatest and most beautiful discoveries of pure logic in human history, and I think it's a crying shame that it is not known to all educated people.  But even I have to concede that for practical purposes, there is no real harm in using (let's say) a 12-digit approximation to $\sqrt{2}$ rather than its true value.  I do think JoeTaxpayer's comment above regarding precision beyond the 2nd decimal place is definitely on-target:  Students should see that truncating everything to the nearest penny at each stage of a calculation results in an error that accumulates dramatically over time.  (See also:  the plots of the movies &lt;em&gt;Superman III&lt;/em&gt; and &lt;em&gt;Office Space&lt;/em&gt;.)  And this could, I suppose, be used to justify as a heuristic that it is better to work with exact expressions like $\sqrt{2}$ throughout a calculation and only produce an approximate value at the very last stage.  But to be honest this is a little bit untrue, because (as I noted) using a 12-digit decimal approximation is going to produce a final result almost certainly every bit as reliable as what you would get if you resisted using approximations until the very last moment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edited to add:&lt;/strong&gt;&#xA;Can't believe that neither I nor anybody else has mentioned it yet, but one can hardly discuss compound interest without talking about continuous compounding, which naturally introduces the irrational number $e$ into the conversation.  But here again the issues raised above apply:  While the &lt;em&gt;relevance of the irrational number&lt;/em&gt; is undeniable, the &lt;em&gt;fact that it is irrational&lt;/em&gt; seems more like a curiosity, and it is hard to argue with a straight face that there is some real harm in using a rational approximation, as long as that approximation is &quot;good enough&quot;.  Here too there is probably some good substantive discussion to be had about how many digits is &quot;enough&quot; to get reliable results; 2 digits is definitely not enough, 8 is probably more than enough, and rounding off too much at intermediate stages of a calculation produces an error that can accumulate substantially by the time you reach the end.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2014-06-26T21:12:05.067" LastActivityDate="2014-06-26T21:12:05.067" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3855" PostTypeId="2" ParentId="3842" CreationDate="2014-06-27T12:53:41.893" Score="1" Body="&lt;p&gt;Here is an example that &quot;breaks&quot; Mathematica, for now:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_{-\infty}^\infty \text{erf}(x+1)\,e^{-x^2} dx = \sqrt{\pi}\ \text{erf}(\,1/\sqrt{2}\,)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematica doesn't find this answer, even though you can find the exact result with human ingenuity, or with the right table of integrals.  However....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have submitted this comment to the Mathematica developers, and I expect that they will teach Mathematica some techniques for integrating this soon enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, this is not an example at the level of a first-semester calculus course.  Mathematica has been trained to solve all the problems in various textbooks, so I doubt you'll find a simple example of a break. &lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-06-27T12:53:41.893" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3856" PostTypeId="2" ParentId="3846" CreationDate="2014-06-27T14:30:14.437" Score="4" Body="&lt;p&gt;The purpose of a course is normally determined when the course is planned and designed, not when it is executed by an instructor.  If this course is part of a college curriculum, then there is normally a planning process where this purpose is clearly stated and approved by a variety of stakeholders.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are engaged in this planning yourself, then the questions I would ask are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;What are the most important topics that you want students to retain?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What are some non-essential but nice topics you'd like to include?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What, if anything, will completion of this course certify them to do?  Take some other course, perhaps?  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What prerequisites do you need to impose to enable student success?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What resources will the students and instructor have available?  Books, handouts, computers, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Who are the students and who is the instructor?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="270" LastActivityDate="2014-06-27T14:30:14.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3857" PostTypeId="1" CreationDate="2014-06-27T14:32:02.893" Score="6" ViewCount="859" Body="&lt;p&gt;Sometimes a math teacher becomes sick or depressed like any other human and is not able to be prepared for lecture. Sometimes a math teacher spends too much time on a research project and does not prepare the course material appropriately. These are just a few reasons which can make somebody unprepared for a lecture course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; How can a math teacher manage a class when unprepared? What tricks can help an unprepared math teacher teach a nice lecture course in a way that is useful for the students?&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="143" LastEditDate="2014-06-27T23:19:38.810" LastActivityDate="2014-06-27T23:19:38.810" Title="How can a math teacher manage a class when unprepared?" Tags="|course-design|classroom-management|general-pedagogy|teacher-preparation|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3858" PostTypeId="1" AcceptedAnswerId="3870" CreationDate="2014-06-27T16:57:26.423" Score="17" ViewCount="12442" Body="&lt;p&gt;Many students who take courses in mathematics go on to pursue &quot;non-mathematical&quot; careers. I'm wondering, in particular, about those who go on to study Law, and how mathematics is (or can be made) relevant to them in the classroom. I have not specified a particular age of the students, but if you wish to restrict to (middle school, secondary school, college/university) in your answer, then that would be fine.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are concrete ways in which one's mathematics education can contribute to a career in law? (If your answer is, &quot;It doesn't,&quot; then an explanation of this would be fine, too!) Connections between mathematics education at any level and its impact on careers in law would be most welcome.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For the motivation, see &lt;a href=&quot;https://math.meta.stackexchange.com/q/11664&quot;&gt;this meta question&lt;/a&gt;; moreover, &lt;a href=&quot;https://math.meta.stackexchange.com/a/13726/79539&quot;&gt;this user&lt;/a&gt; recommended I post here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another interesting place to look is Law and Mathematics Professor Peter Rosenthal's &lt;a href=&quot;http://www.maa.org/careers/career-profiles/law/peter-rosenthal&quot; rel=&quot;noreferrer&quot;&gt;MAA article&lt;/a&gt;; for example, consider the following excerpt:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Law sometimes makes a pretense of being logical, but it is only a pretense.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user507" LastEditDate="2020-12-20T17:29:48.310" LastActivityDate="2023-10-24T13:45:02.503" Title="How is education in mathematics relevant to law?" Tags="|student-motivation|" AnswerCount="14" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="3859" PostTypeId="2" ParentId="3857" CreationDate="2014-06-27T17:35:28.890" Score="7" Body="&lt;p&gt;As JoeTaxpayer suggested, my comment is more of an answer than a comment, so I'm copying my comment here. Maybe this will get others to answer, although I realize that appearing to admit that something like this has ever happened can reflect poorly on one's teaching. Personally, this almost never happened to me, if at all. I had more of a problem with having too much that I wanted to cover, and I often resorted to turning the things I didn't get to (or decided I shouldn't try to cover) into supplementary extra credit projects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can make it into a &quot;review day&quot; where you ask students to suggest old topics they want you to go over in more detail or you can make it into a &quot;problem day&quot; where you and/or the students present solutions to old homework problems. Or you could simply switch the order of some topics and discuss a later topic you are more comfortable with that doesn't depend on anything new (i.e. do implicit differentiation, which is quite mechanical, if you didn't get a chance to properly prepare for solving some applied max/min problems). &lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-06-27T17:35:28.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3861" PostTypeId="2" ParentId="2754" CreationDate="2014-06-27T19:02:41.903" Score="7" Body="&lt;p&gt;It might help to make vivid what this conversion means. 10 km/h is a reasonable running speed. A soccer field is 105 m long. In one second, can you run a few meters, or can you run a third of a soccer field? (American football fans might have a better sense for this, as the number of yards traveled by a runner is key to the game, but I assume that soccer is more relevant in Germany.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, 100 km/h is a slow highway driver. Does a driver travel 1/4 of a soccer field, or 3 and a half soccer fields, in a second?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For both of these computations, although I would guess the right way, I could imagine someone guessing wrong. But once someone knows the right answer, I imagine they would remember it, as running/driving down a playing field is more memorable than the numbers $3.6$ and $0.27$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way: My dad is a lawyer. At one point, I was reading one of his law textbooks, and it recommended memorizing 60 mph = 90 ft/sec in order to catch witnesses who give impossible descriptions of car accidents. So there is someone in the world who finds this worth memorizing, although it feels like a pretty specialized case.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2014-06-28T01:54:59.507" LastActivityDate="2014-06-28T01:54:59.507" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3862" PostTypeId="2" ParentId="3857" CreationDate="2014-06-27T19:55:51.970" Score="2" Body="&lt;p&gt;How about letting students work in groups of 2 or 3 people to understand the next block of theory from their textbooks themselves? I am a huge believer in learning to understand mathematical notation, because this will make almost any new topic easier to learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach a supplementary statistics course and the professor supplied a set of lecture notes that students will be allowed to use during the exam. Many of them are struggling to understand the theoretical mathematical notation of the content, but for the exam they should understand what all these formulas, assumptions, etc. actually mean. Therefore, I let them help each other read little sections of the lecture notes. Basically, they're supposed to translate the formulas from mathematical notation into actual sentences. Afterwards, I give a brief explanation of what the formulas mean. I think this should work even without a lot of preparation since you know the concepts already (also, the group work phase will give you time to prepare a brief explanation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This probably won't work if your students are too young to even have formulas in their textbooks or if the topics are so hard that the session will end in hopeless confusion without your guidance).&#xA;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="1824" LastActivityDate="2014-06-27T19:55:51.970" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3863" PostTypeId="1" AcceptedAnswerId="3867" CreationDate="2014-06-27T21:37:03.057" Score="17" ViewCount="1311" Body="&lt;p&gt;I'm tutoring a very bright middle school student in a difficult geometry class. He understands most of the material, but struggles with some of the hardest problems and proofs. I find that when he goes through his homework, he gets stuck on the very hard problems and refuses to move on to another more than once, preventing him from finishing in time. I think his desire to solve every problem is good, but I am unwilling to give too much help with his homework, so in these problems I've given all the help I'm willing to give and he sits for hours trying to figure them out. I have tried to convince him he would be well-advised to move on and come back to the problem with a fresh mind, but he has a problem with skipping &quot;too much&quot; (more then one problem) for fear he won't get back to the skipped problems. How can I convince him to give the problem a rest for now and work on something else?&lt;/p&gt;&#xA;" OwnerUserId="1380" LastActivityDate="2014-07-18T23:41:49.140" Title="Convincing a student it's OK to move on to another problem" Tags="|tutoring|" AnswerCount="6" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3864" PostTypeId="2" ParentId="3863" CreationDate="2014-06-27T21:42:59.603" Score="7" Body="&lt;p&gt;Consider pointing out how much each of the homework questions are worth. If his motivation is intrinsic, then how much will you learn from trying and failing to do this problem rather than doing the easier ones you already know how to do? If extrinsic, then note how much the problems are worth.&#xA;Another thing I would point out is that if you do all the easier questions, you will have gained a better understanding of the material, and be better prepared to tackle the harder question. One of the most important things I've ever been told in mathematics is to &quot;try some small numerical examples.&quot; It is annoying, but it improves intuition with proofs, and ultimately saves time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if one skips a bunch of problems, and doesn't have time to go back to them, then one wouldn't have had time to finish them all without skipping in the first place. Skipping doesn't cost as much time as it appears it will because often homework is designed to improve understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="1825" LastActivityDate="2014-06-27T21:42:59.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3865" PostTypeId="1" CreationDate="2014-06-28T01:18:17.653" Score="13" ViewCount="1478" Body="&lt;p&gt;Does anyone know of any course called Mathematical Knowledge for Teaching?  Where is this course taught?  If it's not a course, is there any workshop about this?  Or possible a book? It's not knowing how to do math but how to teach math (focus on conceptual understanding and problem solving rather than merely procedural skills) especially in elementary/middle school.&lt;/p&gt;&#xA;" OwnerUserId="1828" LastActivityDate="2015-12-05T15:06:41.573" Title="Mathematical Knowledge for Teaching" Tags="|undergraduate-education|course-design|primary-education|graduate-education|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3866" PostTypeId="2" ParentId="3858" CreationDate="2014-06-28T01:51:34.997" Score="12" Body="&lt;p&gt;If you mean Pure math, then a firm grasp of logic and logical arguments to form proofs may be incredibly helpful in building a good case. It would also help in finding logical flaws in opposing briefs. &#xA;From an applied math view, being able to check the stats and whatnot that a contract is based on seems reasonable (like in insurance).&#xA;Finally, for both types of math majors, you are likely to have friends in the CS, physics, and other science departments. Those people will always need patent lawyers to make them some money. Having a firm grasp of how their invention functions should make life a lot easier when trying to determine if they even have a case when you have noon-literal infringement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above is just some brainstorming. Hopefully it is helpful.&lt;/p&gt;&#xA;" OwnerUserId="1825" LastActivityDate="2014-06-28T01:51:34.997" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3867" PostTypeId="2" ParentId="3863" CreationDate="2014-06-28T02:17:50.927" Score="11" Body="&lt;p&gt;Nothing ironclad, but I have a couple of ideas. I hope they're of use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's admirable that he has persistence in solving problems. This is, of course, an important mathematical practice identified by many (not to mention codified in &lt;a href=&quot;http://www.corestandards.org/Math/Practice/#CCSS.Math.Practice.MP1&quot;&gt;CCSS.MATH.PRACTICE.MP1&lt;/a&gt;: &quot;Make sense of problems and persevere in solving them.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most often teachers of mathematics must work to instill students with a sense that problem solving takes time, to counter a common view that &quot;smart people&quot; or &quot;people who are good at math&quot; are people who &quot;get the answer,&quot; and quickly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But persevering is only part of the story. It's important to help students develop an ability to learn with and from others (Martin, 2007). It's important to understand that collaboration is an appropriate activity when you're learning, and it's important to reflect on progress to know when one's own activity in solving a problem is productive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Reflecting on progress (metacognition)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know of no formulaic way to teach this, I only know that it can be helpful to get students to think not only about the problem, but about their own thinking about the problem. To take a slightly different example, there has been research asking students to track certain emotions during problem solving to gather data on their experience of problem solving (Lesh, 2010). Reflecting on this data helped some students realize that the feeling of discomfort that accompanied a difficult problem often came just before a necessary change in thinking. This changed the way they thought about (and experienced) those feelings in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not suggesting you turn reflection into a modelling activity (well, not &lt;em&gt;necessarily&lt;/em&gt;), but one part of the approach taken in that study was a periodic approach to reflection. In other words, after a certain amount of time, the students were asked to record their emotions. In the case of OP's student, I suggest establishing a certain amount of time after which he or she reflects on whether the struggle is productive. You can probably come up with a good list of indicators of progress. Here are a few to start:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What have we ruled out since the last check?&lt;/li&gt;&#xA;&lt;li&gt;What have we learned since the last check?&lt;/li&gt;&#xA;&lt;li&gt;Can we list one or more avenues we haven't ruled out yet?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In any case, with a decent list, you can help a student discover for him or herself just how productive the struggle is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Collaboration&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Learning appropriate collaboration is important (Martin, 2007). We rarely are completely alone in our investigations; our social resources include talking to other people, consulting books, searching the Internet, etc. Depending on the task, some of these are appropriate, and some are not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it's almost always appropriate to at least talk about the problem with a peer who has not solved the problem yet (or a teacher/tutor who can listen and prompt appropriately rather than lecture). Compared to either struggling unproductively or looking up an answer, discussing the problem has distinct advantages. First, it can help a student expose thinking about the problem. Second, collaborating ideas means the possibility of having to argue and justify. Third, mathematical sense-making sometimes benefits from a change of perspective, which another set of eyes can help with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are skills that students often do not experience in a classroom, so if they can get experience engaging in them outside of the classroom, that is a kind of productivity beyond finding a solution. Though, of course, it can help get the student on a more productive path toward solution.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cited:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lesh, R. (2010). Tools, researchable issues &amp;amp; conjectures for investigating what it means to understand statistics (or other topics) meaningfully. &lt;em&gt;Journal of Mathematical Modelling and Application&lt;/em&gt;, 2(1), 16.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Martin, T. S. (Ed.). (2007). &lt;em&gt;Mathematics teaching today&lt;/em&gt; (2nd ed.). Reston, VA: National Council of Teachers of Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-06-28T02:17:50.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3868" PostTypeId="2" ParentId="3865" CreationDate="2014-06-28T02:20:50.570" Score="12" Body="&lt;p&gt;(Perhaps this should be a comment, but it's pretty much an answer.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematical Knowledge for Teaching (MKT) is based on the more general term Pedagogical Content Knowledge (PCK) due to Lee Shulman. For MKT, see the work of DL Ball. &lt;strong&gt;&lt;a href=&quot;http://scholar.google.com/scholar?q=deborah%20ball%20%22mathematical%20knowledge%20for%20teaching%22&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=0%2C22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Deborah Ball is currently at the University of Michigan; you can find courses and related research projects on MKT there. For example, see &lt;a href=&quot;http://www-personal.umich.edu/~dball/projects/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;this section&lt;/strong&gt;&lt;/a&gt; of her site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an image to help summarize the MKT model (see &lt;a href=&quot;http://www.uni.edu/ctlm/content/mathematical-knowledge-teaching&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; for more):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Vee2b.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-06-28T02:20:50.570" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3869" PostTypeId="2" ParentId="3865" CreationDate="2014-06-28T02:36:32.540" Score="7" Body="&lt;p&gt;Benjamin gave the appropriate answer about what is MKT (and the famous diagram known to my friends as &quot;the egg&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm hoping to address the part of your question in which you are looking for a course that might help you explore the idea of &quot;how to teach math (focus on conceptual understanding and problem solving rather than merely procedural skills).&quot; This sort of view of math falls very much within Jo Boaler's research. And you're lucky, because she has a &lt;a href=&quot;https://class.stanford.edu/courses/Education/EDUC115N/How_to_Learn_Math/about&quot;&gt;free online course How To Learn Math&lt;/a&gt; running in a couple of weeks, open for registration now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The summer version of the course is for educators, and the school year version is for anyone wanting to learn more about their own math learning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be clear, this is not a course about MKT. It's a course that addresses math &lt;em&gt;teaching and learning&lt;/em&gt; from someone who values a problem solving approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=Ien-86bXCrI&quot;&gt;Here's a video with Prof. Boaler&lt;/a&gt; talking a little bit about her approach.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-06-28T02:36:32.540" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3870" PostTypeId="2" ParentId="3858" CreationDate="2014-06-28T13:17:37.480" Score="31" Body="&lt;p&gt;A 16 year old once told me that he has no need for mathematics since he wanted to become a lawyer. I told him that he wouldn't make a good lawyer if he jumps to such conclusions without first collecting all available evidence. So what evidence is there? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a book in front of me: Mathematics, Physics and Finance for the Legal Profession by Ashley Saunders Lipson, 2011. Chapters include Logic and Set Theory, Probability, Statistics, Graphs and Diagrams, Classical Physics, Modern Physics and Accounting and Finance. I have only read a little bit, yet already I have been pleasently surprised: The use of trig in land deeds; the logic in legal interpretations similar to the interpretations of Euclidean and non-Euclidean geometry; and an interesting question, can you patent a mathematical formula?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I teach probability, I always discuss the &lt;a href=&quot;http://en.wikipedia.org/wiki/Sally_Clark&quot;&gt;Sally Clark case&lt;/a&gt;. It highlights what can happen when no-one in the case, whether it be defense or prosecution, judge or jury, has the ability to appreciate something as basic as independence of events. It makes you think!&lt;/p&gt;&#xA;" OwnerUserId="685" LastEditorUserId="3237" LastEditDate="2015-05-11T14:53:21.650" LastActivityDate="2015-05-11T14:53:21.650" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3871" PostTypeId="2" ParentId="3863" CreationDate="2014-06-28T14:08:43.630" Score="1" Body="&lt;p&gt;Geometry can be very frustrating for bright students who excel with the linear nature of Algebra. That said, it is incumbent upon you to see what is stumping him. What is it about the &quot;hard&quot; problem that is causing him to be stumped about what to do next? Is there a pattern of difficulty? With Geometry, it's &quot;what does what we know, tell us?&quot; When he gets stuck, maybe it's time to list what we know and where each piece of the knowledge leads us. Also, working backwards can help. &#xA;Perseverance is a great thing and I don't think we should encourage a student to give up. Certainly the problem has a solution and he should continue to work towards it. The problem here is not that he doesn't know when to quit but rather he doesn't know how to solve the problem.&lt;/p&gt;&#xA;" OwnerUserId="1831" LastActivityDate="2014-06-28T14:08:43.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3872" PostTypeId="2" ParentId="3858" CreationDate="2014-06-28T23:10:05.207" Score="4" Body="&lt;p&gt;Just as mathematical definitions and axiom systems and so on do not (and maybe cannot) genuinely capture &quot;mathematical reality&quot;, (as a non-jurist) it seems that law does not and cannot capture &quot;justice&quot; (or &quot;morality&quot;, etc).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of mathematics, to my perception there is &lt;em&gt;eventually&lt;/em&gt; more flexibility, once one is beyond the typical undergrad and beginning graduate curriculum. That is, the business is not nearly as rule-bound as undergrad curricula often are presented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Mercifully) I have no experience in law or the courtroom... but it would be my impression that judges and lawyers are far more bound by the random approximations to justice that are the laws on the books. Yes, of course, they are intended to approximate real justice and fair play, but this seems impossible to completely accomplish.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The upshot is that the somewhat rule-based undergrad math curriculum is perhaps a good abstracted version of the &quot;game&quot; part of law. Not many other undergrad curricula amount to adherence to possibly unfathomable, unbending rules, but, more often appeal (reasonably) to common sense. Not a bad thing! But, in the popular U.S. culture, appeals to &quot;common sense&quot; are often merely veiled appeals to previous prejudices or popular beliefs, rather than any sort of genuine seat-of-the-pants. (Perhaps certain higher-level video games may be a significant exception...) Thus, in the past, few undergrads have experience in manipulation of &quot;rules&quot;, making deductions from the rules (without addressing the truth or falsity of the rules themselves), and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Capsulization: where else to in-effect learn about modus ponens? :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The beyond-the-rules research parts of mathematics don't seem to have much real relation...)&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2016-01-01T15:25:36.050" LastActivityDate="2016-01-01T15:25:36.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3873" PostTypeId="1" AcceptedAnswerId="3876" CreationDate="2014-06-29T03:24:52.883" Score="11" ViewCount="984" Body="&lt;p&gt;I was taught in 9th grade the two column proof, and it wasn't until 11th (when I saw some number theory) that I realized what a poor method that is. However, it is certainly effective in getting simple statements proven. Since improving one's ability to write proofs is always helpful, and because helping others is fun, I was wondering if there is a better way to teach/learn proofs. Repetition works for induction, and contradiction in many cases, but how does one get good at coming up with the non-obvious approaches. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="1825" LastEditorUserId="143" LastEditDate="2014-06-29T09:26:57.863" LastActivityDate="2022-10-04T00:08:51.833" Title="How to teach Proofs" Tags="|secondary-education|proofs|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3874" PostTypeId="2" ParentId="3863" CreationDate="2014-06-29T03:31:06.143" Score="2" Body="&lt;p&gt;I have a rule, which is called the 90% rule. Love it or hate it, it works for people in employment, as well as for students, and right now, this student, is not going to do well as a student, or in life unless they learn the 90% rule. Teaching them my 90% rule, will prepare them for exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 90% rule, is that they need to aim to solve as much as they can within a fixed time frame. Its better to answer, and solve 90% of an exam, and miss out on 10% due to time constraints, then to focus on the 10%, and fail the exam miserably. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would set fixed study times, 90mins max. At the end of 90mins, mark them on what they have solved so far. Tell them they have failed. After the 90mins, let them work on whatever, but everyday, dont help them for that 90minute period, and emphasize their mark, and if they failed or not and sure enough they will change their tactics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In real life, I have seen people get fired for not-following the 90% rule. A process worker got canned, as accepted too many &quot;difficult&quot; repair jobs, following the 90% rule, he would have done the quick and easy ones, and after reaching minimum target levels could have done a few from the &quot;too hard&quot; basket. Fixing 10-20 complicated repair jobs when 60-80 they can do 60-80 repairs for the easy ones, makes them look slow, incompetent, and ultimately fired. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The value of people that can work on something until they solve it is worth almost zero, if you have nothing to show for an entires day work, and if anything, makes it clear to employers that you are not suitable, or smart enough for the job. In real world terms, doing 90% of your tasks quickly, and the remaining 10% in whatever time you have left, will let you keep your job.&lt;/p&gt;&#xA;" OwnerUserId="1835" LastActivityDate="2014-06-29T03:31:06.143" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3875" PostTypeId="2" ParentId="3863" CreationDate="2014-06-29T05:19:35.923" Score="4" Body="&lt;p&gt;In my experience (both as a student and as a tutor), the best way to deal with question on both tests and homework is a triage-style system:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Look at all the questions, and take care of those that you know you can answer quickly; whatever you can't skip and come back to it ( = &lt;em&gt;take the low-hanging fruit first&lt;/em&gt;).  &lt;/li&gt;&#xA;&lt;li&gt;On the second pass, don't answer anything, but take note of (and&#xA;maybe mark on the test/homework sheet) those answers that you known&#xA;that you can answer for sure, but would take a little more effort (these&#xA;are the ones that you should attack first on the third pass), and&#xA;those that you're not certain that you can answer.&lt;/li&gt;&#xA;&lt;li&gt;Third pass: answer those questions you marked in the second pass.&lt;/li&gt;&#xA;&lt;li&gt;Then, if there's any time left over, you can deal with those monster&#xA;questions that eat up so much time.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In my own experience as a high-school student, there is a possibility of a near-infinite loop between the third and fourth steps.....that's not a problem, as long as time is being spent productively, i.e. thinking about the questions as opposed to just looking at them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I hadn't even thought of until JPBurke mentioned it in a &lt;a href=&quot;https://matheducators.stackexchange.com/questions/3863/convincing-a-student-its-ok-to-move-on-to-another-problem/3875?noredirect=1#comment7636_3875&quot;&gt;comment&lt;/a&gt; was that this method will help in a different way, namely that by looking at all the questions first, a student may be reminded of things that were known and forgotten, and has a great ability to help answer problems that were skipped the first or second time around.&lt;/p&gt;&#xA;" OwnerUserId="1685" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-07-18T23:41:49.140" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3876" PostTypeId="2" ParentId="3873" CreationDate="2014-06-29T09:25:47.253" Score="7" Body="&lt;p&gt;Coursera has a periodically taught course by a rather famous Stanford professor about mathematical thinking that you might of interest the next time it's offered:  &lt;a href=&quot;https://www.coursera.org/course/maththink&quot;&gt;https://www.coursera.org/course/maththink&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also suggest broadening your inquiry from &quot;how to do proofs&quot; to &quot;how to solve mathematical problems and then demonstrate that your answer is correct&quot;   In that sense, books such as Thinking Mathematically: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0273728911&quot;&gt;http://www.amazon.com/Thinking-Mathematically-2nd-Edition-Mason/dp/0273728911&lt;/a&gt;  (although the 1st edition is fine, and a lot cheaper) would be an excellent resource.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good proof provides insight into the mathematical or logical structure underlying what you're proving.   A mediocre proof merely demonstrates that it works all the time, without giving any new insight into what's going on.   So I'd urge you to rely less on proofs, say, by induction and more on careful and thoughtful reasoning about patterns and why they must hold true for all cases.   Mason will be a great start in doing that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, a nice (if a bit heavy) article that a colleague just recommended to me about the importance of proofs that explain what's going on mathematically:&#xA;Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-06-29T09:25:47.253" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3877" PostTypeId="2" ParentId="3873" CreationDate="2014-06-29T17:42:10.300" Score="3" Body="&lt;p&gt;There are resources on writing proofs, e.g. Hammack's &lt;a href=&quot;https://web.archive.org/web/20171219230825/http://mathforum.org/library/drmath/view/54290.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Book of proof&amp;quot;&lt;/a&gt; and many others. For mathematical writing style, look for e.g. Knuth, Larrabee, and Roberts' &amp;quot;Mathematical Writing&amp;quot; (MAA, 1989, a (somewhat mutilated) version &lt;a href=&quot;http://www-cs-faculty.stanford.edu/%7Eknuth/klr.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). I'm sure a search for lecture notes on intermediate mathematics courses will turn lots of hits.&lt;/p&gt;&#xA;&lt;p&gt;In my experience, for beginners it is important to structure proofs in familiar ways (announce the overall strategy, do e.g. induction in styzed ways, don't skip steps), thus I can understand the idea of column proofs. But more interesting proofs aren't always easy to classify. For an astonishing collection of nice proofs see Aigner and Ziegler's &amp;quot;Proofs from THE BOOK&amp;quot; (Springer 2012, 4th edition). Most don't require calculus at all.&lt;/p&gt;&#xA;&lt;p&gt;If you want them get interested (or want something to give more restless students) Dunham's &amp;quot;Journey though genius&amp;quot; (Penguin, 1990) or &amp;quot;The mathematical universe&amp;quot; (Wiley, 1997) would be good starting points.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="6053" LastEditDate="2022-10-04T00:08:51.833" LastActivityDate="2022-10-04T00:08:51.833" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3878" PostTypeId="2" ParentId="3863" CreationDate="2014-06-29T19:22:40.950" Score="1" Body="&lt;p&gt;While we've gotten good test-taking skills offered, I think they miss an important part of the question.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;he has a problem with skipping &quot;too much&quot; (more then one problem) for&#xA;  fear he won't get back to the skipped problems. How can I convince him&#xA;  to give the problem a rest for now and work on something else?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Two things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1st. He has an (irrational) fear. We know he can skip the problem and come back but he can't do that. Even if he moves on to another problem, he'll still be thinking about it while he's trying to do the next. If he skips two problems then it only compounds the situation. You can not reason someone out of an irrational fear. They can only be  convinced by experience that there is nothing to fear. Have him skip a problem on purpose (say #3) and then come back to it later. I like the idea of ranking the problems by difficulty but he's still going to run into the blockage problem eventually.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2nd. The value judgement that he needs to move on. The problem isn't that he won't move on, it's that he doesn't know what to do. Organizing his knowledge so he can see the connections between the different parts might help. I often had students in geometry who would get stumped because they didn't &quot;see it&quot;. I would try to get them to break the problem down. What do we know? What do we want? These two lines are parallel. What are all the thing we know from that? These two angles are congruent. What are all the things that tells us? Maybe use index cards to record theorems and properties. Pull out the ones that apply to that problem. You mentioned proofs. Can he do a flow-chart or paragraph proof or does it have to be two-columns? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry, but it just frosts me that as educators we would encourage him to move on.&lt;/p&gt;&#xA;" OwnerUserId="1831" LastActivityDate="2014-06-29T19:22:40.950" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3879" PostTypeId="2" ParentId="2006" CreationDate="2014-06-30T02:41:17.480" Score="0" Body="&lt;p&gt;&quot;Ethical and Professional Dilemmas for Educators&quot; are the focus of this &lt;a href=&quot;http://www.ctteam.org/df/resources/Module5_Manual.pdf&quot; rel=&quot;nofollow&quot;&gt;document&lt;/a&gt; from Connecticut.  It presents and analyzes several examples, e.g. on p. 36:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ms. H teaches ninth grade English. She gives the class directions for&#xA;  a writing assignment that is to be completed by the end of the period.&#xA;  Josh isn’t paying attention and missed the directions.... He asks for&#xA;  the directions to be repeated. The other students in the class begin&#xA;  to make fun of Josh....&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-06-30T02:41:17.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3881" PostTypeId="2" ParentId="3873" CreationDate="2014-06-30T17:10:03.613" Score="-1" Body="&lt;p&gt;I tried to learn (teach myself) proofs by grouping them into categories. For instance: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Proof by contradiction: Assume the converse and show that it leads to contradicting the hypotheses, therefore the original must be true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Proof by induction: This is basically a recursive argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Proof by common terms: Prove A= B by showing that they both &quot;reduce&quot; to a common term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Construction and verication (c &amp;amp; v for short). Construct the formula, and verify that it has the desired properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) Proof by completion: A is &quot;too weak&quot; to imply B, but A does imply B when you add a further condition, C.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And there's always the Polya admonition: &quot;Invert!&quot; Jacobi Inversion formula, LaPlace Transforms, deMorgan's Laws, etc.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-06-30T17:10:03.613" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3882" PostTypeId="2" ParentId="2006" CreationDate="2014-06-30T19:06:39.587" Score="1" Body="&lt;p&gt;One of the best discussion of this in the Math Ed literature is Alan Schoenfeld's wonderful (1988) article &quot;When Good Teaching Leads to Bad Results: The Disasters of 'Well-Taught' Mathematics Classes.&quot;  From the abstract:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This article describes a case study in mathematics instruction, focusing on the development of mathematical understandings that took place in a 10-grade geometry class. Two pictures of the instruction and its results emerged from the study. On the one hand, almost everything that took place in the classroom went as intended -- both in terms of the curriculum and in terms of the quality of the instruction. The class was well managed and well taught, and the students did well on standard performance measures. Seen from this perspective, the class was quite successful. Yet from another perspective, the class was an important and illustrative failure. There were significant ways in which, from the mathematician's point of view, having taken the course may have done the students as much harm as good.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &quot;harm&quot; Schoenfeld refers to has to do with what I would call &lt;em&gt;mathematical dispositions&lt;/em&gt;, certain attitudes about what mathematics is that are out of sync with the values of mathematicians.  These dispositions are very difficult to measure empirically, and there does not even seem to be a consensus that they are important to measure.  So while the class in his case study was successful in terms of &quot;measurable outcomes&quot;, that success seems to have come at a cost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schoenfeld presents his case study as a cautionary tale -- the moral seems to be &quot;beware, just because everything looks good on conventional measures does not mean that all is well&quot; -- but a more cynical reading of his evidence &lt;em&gt;could&lt;/em&gt; be that these &quot;disasters&quot; are an &lt;em&gt;inevitable consequence&lt;/em&gt; of teaching students to perform well on conventional measures; in other words, the cost of teaching &quot;well&quot; (as it is commonly assessed) might be that we have to distort the discipline along the way.  Schoenfeld does not go there, and I don't really think that's the case, but sometimes we all have our dark days when it seems like there is a trade-off that cannot be avoided.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-06-30T19:06:39.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3883" PostTypeId="2" ParentId="3873" CreationDate="2014-06-30T19:09:33.707" Score="4" Body="&lt;p&gt;This question is interesting because it points to what many of us believe is the problem with &quot;introduction to proof&quot; courses: Just because proofs have a common underlying logical architecture doesn't mean that new proofs don't require genuinely creative new ideas that are specific to the mathematical questions they address. An intro to proof course generally fails to give adequate weight to the very important fact that the idea for a new proof is most likely to come from the careful consideration of a very specific difficulty of a very specific problem. It's hard not to feel that such courses, by definition, don't correctly teach the heart of proof &lt;em&gt;because&lt;/em&gt; they are general. Truly new proofs arise from careful thought that is spurred by a mathematician's encounter with a specific problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The OPs question reminds me of an image (I believe found in Derrida) used to describe the activity of structuralists as attempting to reproduce a beetle using only the beetle's shed shell. Introductory proof courses may give a false impression that reproducing the beetle from its shell is possible, and that there is a &quot;method&quot; for genuinely creative mathematical work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I may be wrong, but it seems that the way to find genuinely new proofs is to develop expert knowledge about a problem in a field over a very long time, until what is needed eventually becomes clear. This is the same thing that is needed for the development of a useful tool in any other field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How does one come up with the idea of a hammer, screwdriver or wrench? How does one come up with the idea for a turbine engine? How about a microprocessor? All of these require various degrees of deep familiarity with the contour of some problem combined with hard work, knowledge of one's field and a little bit of luck!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not saying there aren't healthy methods for getting better at mathematical research, but I am saying that there is no shortcut. I've asked a closely &lt;a href=&quot;https://math.stackexchange.com/questions/8532/what-is-your-method&quot;&gt;related question that might be interesting to you&lt;/a&gt;. It also might be worth looking at Terry Tao's blog.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the best advice on this topic came in a short remark of &lt;a href=&quot;http://en.wikipedia.org/wiki/Yitang_Zhang&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yitang (Tom) Zhang&lt;/a&gt;: &quot;Just keep thinking&quot;.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-07-01T10:47:01.580" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3884" PostTypeId="2" ParentId="3827" CreationDate="2014-06-30T19:15:58.720" Score="-1" Body="&lt;p&gt;Playing tournament level chess results in high-level, transferrable, cognitive skills. These skills lie partially in the field of maths. For example a Knight's Tour is a graph theory problem. Many other aspects of chess lie in the fields of discrete geometry and game theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But all these aspects never cover any of these theories or sub-theories in a whole. And they mostly don't cover their basics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consequently, it only &lt;strong&gt;helps&lt;/strong&gt; understanding or doing math better, but it cannot convey math on its own. The ultimate abstraction even in elementary math goes much further than the abstraction in chess: just remember that chess is finite.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-06-30T19:15:58.720" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3885" PostTypeId="2" ParentId="3858" CreationDate="2014-06-30T19:21:41.267" Score="4" Body="&lt;p&gt;There are &lt;em&gt;many&lt;/em&gt; math-related aspects of pure and applied law. Just look at the other answers. But the simplest example is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Necessity and Sufficiency&lt;/strong&gt;&#xA;Take any example of a legal text mentioning &lt;code&gt;conditions&lt;/code&gt; and ask a lawyer, if this means a &lt;code&gt;necessary condition&lt;/code&gt; or a &lt;code&gt;sufficient condition&lt;/code&gt;. You'll understand by his reaction.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-06-30T19:21:41.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3886" PostTypeId="1" AcceptedAnswerId="3892" CreationDate="2014-06-30T21:58:29.370" Score="23" ViewCount="2749" Body="&lt;p&gt;I'm asking this question from the point of view of an introductory non-rigorous calculus instructor. Calculus textbooks have different approaches about how to define $e$ and $\ln$. For example, my current textbook defines $e$ as the number such that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) $\int_1^e  \frac1x \, \mathrm{d}x= 1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've also seen $e$ defined as the number such that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) $\frac{\mathrm{d}}{\mathrm{d}x} e^x |_{x = 0} = 1$, or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) $e := \lim_{n \to \infty} (1 + \frac1n)^n$, or &lt;/p&gt;&#xA;&#xA;&lt;p&gt;d) $e := \sum_{n=0}^\infty \frac{1}{n!}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course no matter where you start, you can derive the others, but the distinction between which facts are definitions and which are derived feels very artificial, especially to the students! The questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do you prefer one of these approaches, and if so, why?&lt;/li&gt;&#xA;&lt;li&gt;Is it a good idea to explain the (arbitrarily) chosen definition and the subsequent derivations to students? I often wonder:  why not just say $e$ is an important number that is about 2.718, and talk about its properties?  Maybe the appropriate place for these worries is an analysis course.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2014-07-01T10:28:14.387" LastActivityDate="2014-10-29T00:09:54.170" Title="The definition of natural log and e" Tags="|undergraduate-education|secondary-education|calculus|definitions|" AnswerCount="9" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3887" PostTypeId="2" ParentId="3886" CreationDate="2014-06-30T22:45:45.330" Score="10" Body="&lt;p&gt;When I teach calculus I, my initial working definition for $e$ is that it is the real number &lt;strong&gt;implicitly&lt;/strong&gt; defined by:&#xA;$$ \lim_{h \rightarrow 0} \frac{e^h-1}{h}=1 $$&#xA;Naturally, this allows me to derive that $\frac{d}{dx} e^x = e^x$ as&#xA;$$ \lim_{h \rightarrow 0} \frac{e^{x}e^h-e^x}{h}=e^x\lim_{h \rightarrow 0} \frac{e^{h}-1}{h}=e^x. $$&#xA;Of course, we know this. At this point, I usually say something like, you know, if we wanted, we could have defined $e$ to be the number for which $\frac{d}{dx}e^x = e^x$. That is an interesting equation, it says the slope of the tangent line to $y=e^x$ is the same as the value of the function at the point of tangency. In the world of all possible exponential functions the base $e$ is very special.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In what follows below, this is the sequence I make the comments, but, you could rearrange them depending on what sequence of topics you follow.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Later, when I discuss the second derivative test and give an introduction to Taylor polynomials I show $e^x = 1+x+ \frac{1}{2}x^2 + \cdots$. This allows me to explicitly calculate $e$ with just a little arithmetic. Often at this point, I admit that this is our third definition and I know of at least 5 more or less distinct definitions. That said, the power series definition is near and dear to my heart as it is the method I use to define exponentials in supermath and other formal systems... (this I usually abstain from saying)&lt;/li&gt;&#xA;&lt;li&gt;Later, when I treat indeterminant powers I show $e = \lim_{n \rightarrow \infty} \left( 1+\frac{1}{n}\right)^n$. At this point, if I have my wits about me, I will foolishly attempt to explain how this ties into continuous compounding of interest. Indeed, continuous multiplication is how the exponential is thought of in physics and a few other places. So, it's worthwhile to mention this.&lt;/li&gt;&#xA;&lt;li&gt;Later, after we've covered FTC part I, I point out how $\frac{d}{dx} \int_1^x \frac{dt}{t} = \frac{1}{x}$ suggests that $\int_1^x \frac{dt}{t} = \ln (x)$ as we have already shown that $\frac{d}{dx} \ln (x) = \frac{1}{x}$ via implicit differentiation. Then, I'll show how to derive the various laws of logs as consequences of differential and integral calculus. I then confess that there are other instructors who take the integral as the definition of the log and so the exponential is introduced as the inverse function of the log. However, I tell them that was not our approach since I thought they all could use a whole semester to practice logs and exponential functions. Of course, my approach also has holes ( I don't prove laws of exponents, I just assume we know them and $a^x$ is somehow reasonable for $a&amp;gt;0$, admittedly, there is more substructure here, but I haven't found a class with sufficient curiosity to push the issue)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Naturally, there are other ideas to plan. Intuition for exponential functions is worth cultivating as it is one of those constructions which appears in a multitude of abstract fields of mathematics. I also like this topic because it gives us one topic which spans the entire course of calculus both integral and differential. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-06-30T22:45:45.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3888" PostTypeId="2" ParentId="3886" CreationDate="2014-07-01T01:08:04.743" Score="15" Body="&lt;p&gt;I think a lot depends on context -- what course you are teaching and what the characteristics of that course are.  If by &quot;Freshman college calculus&quot; you mean what I think you mean -- namely, a non-rigorous course that treats a lot of things using intuitive and heuristic arguments -- then I wonder if &quot;define&quot; is even the right verb to use.  Students come into such a class having already encountered $e$; they may not feel like they really understand it well, but they should know it is a number about equal to 2.718, and they know it is used to describe exponential growth and decay, and that it has something to do with &quot;continuous compounding&quot;.  They have some familiarity with $f(x)=\ln x$, although the motivation for using $e$ as the base of a logarithm may seem far from &quot;natural&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If that is the background of the students coming in, then I would not &lt;em&gt;define&lt;/em&gt; $e$.  Rather, I would do (have done) something roughly like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We want to find the derivative of functions of the form $y=a^x$, where $a$ is some positive real number.  So we compute  $$ \lim_{h \to 0}  \frac{a^{x+h}-a^x}{h} = a^x \lim_{h \to 0} \frac{a^h-1}{h}$$&#xA;and we notice that the limit does not depend on $x$ at all; it's just some constant that depends only on the choice of $a$.  Let's denote that constant (temporarily) by the symbol $K_a$.  Then we have&#xA;$$y=a^x \implies y'=K_a a^x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a really interesting (and surprising) formula.  When we take the derivative of a power function, the exponent decreases; but when we take the derivative of an exponential function, we end up with &lt;em&gt;the same exponential function&lt;/em&gt;, times some constant that depends only on the base of the exponent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what is that constant?  Well, students can do some numerical approximation with a graphing calculator or a spreadsheet:  Choose a single specific value for $a$ and make a table of values of $\frac{a^h-1}{h}$ for smaller and smaller values of $h$ to estimate the corresponding value of $K_a$.  Compare results with other students to compile a table of values of $K_a$ for different values of $a$.  Graph them.  Any ideas what the relationship is?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point I reveal the &quot;amazing fact&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For any choice of $a$, the constant $K_a$ turns out to be exactly $\ln a$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Students can, at the very least, convince themselves that this claim is consistent with what they found above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now as a consequence of this amazing fact, we have that &lt;em&gt;if a happens to be equal to e, then the exponential function is its own derivative&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now here is where my teaching sequence may differ from others:  I make no attempt to &lt;em&gt;prove&lt;/em&gt; this amazing fact -- to do so you would have to already have a rigorous definition of $\ln a$ available, and in this type of course we are dealing with students who have no such definition to work with.  Nor do I take the amazing fact as a &lt;em&gt;definition&lt;/em&gt; of $\ln a$; one could certainly do that, but then you would have to prove that this definition syncs up in some way with what they already know about $\ln$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, I take the position that $\ln$ and $e$ are basically already &quot;out there&quot; in their background knowledge, not really defined but available to them to use.  They know some properties about them, but none of them are &quot;primitive&quot;, nor are any of them &quot;derived&quot;.  And I am adding one more property to that constellation, making it seem plausible through numerical examples but not trying to do anything more rigorous than that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of that seems to me to make sense in a 1st-year, non-rigorous college Calculus course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this were a Precalc course?  Then we'd be having a completely different conversation.  We'd be talking about compound interest, and what it means to compound interest twice per year, or three times per year, or $N$ times per year.  We'd see the effective growth rate $(1 + 1/N)^N$ appear naturally in that context, and we'd explore what happens when $N$ gets very large.  Eventually we'd be convinced (but would not prove) that as $N$ gets very large this value seems to converge to some number about equal to $2.718$, and we'd call that number $e$, and then talk about how whenever something is growing (or decaying) continuously (rather than at discrete intervals) it makes sense to express that growth (or decay) in terms of $e$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this were a rigorous (i.e. proof-based) Analysis course?  The sequence of topics in such a course likely has convergence of sequences (&quot;limit as $N \to \infty$&quot;) before either limit at a point (which necessarily precedes derivatives) and also before convergent series (which necessarily precedes integrals), so the first opportunity to define $e$ would be as $\lim_{N \to \infty} (1 + \frac{1}{N})^N$.  I would start there (making sure to motivate it by connecting it to what students remember about compound interest and other contexts of continuous growth).  Then later, when we are covering other types of limits, use that definition to prove both that $\lim_{h \to 0} \frac{e^h-1}{h} = 1$.  Then go on from there, revisiting the topic when we cover derivatives, series and integrals, each time proving the &quot;new&quot; property in terms of the previously established ones.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2014-07-01T17:46:25.710" LastActivityDate="2014-07-01T17:46:25.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3889" PostTypeId="2" ParentId="3886" CreationDate="2014-07-01T01:21:18.223" Score="2" Body="&lt;p&gt;I prefer d), which fits with the all-important definition&#xA; $$\exp(x):=\sum_{n=0}^\infty \frac{x^n}{n!}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a straightforward definition.  It makes obvious sense.  The proof that the sum converges is about as easy as it can be.  By contrast:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definition a), via the integral of $1/x$ is indirect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definition b), via the derivative of $e^x$, only makes sense if you've already defined both differentiation and exponentiation over arbitrary real bases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definition c), the limit of $(1+1/n)^n$, is both a more difficult limit than the sum, and not as central a property as the others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I prefer d).&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-07-01T01:21:18.223" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3890" PostTypeId="2" ParentId="3886" CreationDate="2014-07-01T09:35:14.197" Score="4" Body="&lt;p&gt;My personal oppinion is that the number e itself has virtually no significance and one should rather focus on the exponential function.&#xA;Depending on which course you are teaching, it can show up in various ways. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are giving an introduction to analysis, defining limits, series, etc. than c) and d) are very nice, but of course the exponential series&#xA;$$&#xA;\exp(x)= \sum_{n=0}^\infty \frac{x^n}{n!}&#xA;$$&#xA;is much more important than only its value at $x=1$. It can be highlighted that this is a very important if not &lt;em&gt;the&lt;/em&gt; stereotype of a power series, one can discuss radius of convergence, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the focus is on functions and derivatives or differential equations, one may of course ask for a function that solves&#xA;$$&#xA;f'=f, f(0)=1&#xA;$$&#xA;to again find $\exp$. (Notice that your version of b) is not a definition, e.g. $f(x)=x$ also fulfils $f'(0)=1$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My favourite version for lower grades is to introduce exponentiation by integers in the obvious way: multiple addition -&gt; multiplication with integers and similarly multiple multiplication -&gt; exponentiation. Then one can introduce $\log_b$ as the inverse of $b^x$, explain the calculus of exponentials and logarithms, in particular &#xA;$$&#xA;b^x =c^{xlog_c(b)}&#xA;$$&#xA;which highlights the &lt;em&gt;insignificance&lt;/em&gt; of the basis and mention the preferred choices of $c=10$ for convenience and $c=e$. This leads then naturally to the question why $c=e$ is actually preferred for e.g. implementation in calculators, which could bring you back to the exponential series. (If pupils are actually interested: &lt;a href=&quot;https://math.stackexchange.com/questions/18445/fastest-way-to-calculate-ex-upto-arbitrary-number-of-decimals&quot;&gt;https://math.stackexchange.com/questions/18445/fastest-way-to-calculate-ex-upto-arbitrary-number-of-decimals&lt;/a&gt; )&lt;/p&gt;&#xA;" OwnerUserId="1849" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-07-01T09:35:14.197" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3891" PostTypeId="2" ParentId="3886" CreationDate="2014-07-01T09:48:17.087" Score="7" Body="&lt;p&gt;A textbook I have used first defines $$\ln x =\int_1^x  \frac 1 t \, \mathrm{d}t,$$ and then defines the number $e$ as in your Definition a).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, the exponential function $e^x$ is defined as the inverse of the function $\ln x$. (They show that this definition of $e^x$ coincides with the usual one from repeated multiplication when $x$ is an integer.) &#xA;Finally, for $a&amp;gt;0$, they define exponentiation with real exponent as $$a^x=e^{(\ln a)x}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems a bit backwards, but I can't find anything wrong with it. One advantage could be that formal properties like $$a^x a^y=a^{x+y}$$ can be proven more directly than the way it's done in the Real Analysis textbooks I have seen.&lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2014-07-01T09:48:17.087" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3892" PostTypeId="2" ParentId="3886" CreationDate="2014-07-01T14:58:20.950" Score="21" Body="&lt;p&gt;I think it is a good thing to talk about how there are some concepts where there are choices for where you start when definining them. It happens in linear algebra too, with the definition of linear dependence. You need to talk about how there is this web of connected properties, and it depends on what you're trying to achieve as to where you start. I do recommend prefacing any discussion of it with a general discussion on logs and exponentials using familar numbers, so they get a feel for how they ought to work first. This can even be done without graphs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to which definition of e I prefer...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, it's the functions ln(x) and $e^x$ that are the important thing, rather than the number $e$ itself per se. The number $e$ just crops up as a useful number to unify things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I quite like the one where ln(x) is defined to be $\int_1^x \frac{1}{t} dt$, and e is the number for x that makes this area come out to 1. I have several reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a real need for this function to have an integral because no power of $x$ differentiates to give $\frac{1}{x}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You can approximate e by using upper and lower sums of an easy function like 1/x.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The area truly exists because you can draw it there on the page, whereas it's a bit dicey as to whether it's reasonable to differentiate a function like $a^x$ when you don't know what it even means when $x$ is irrational!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Defining e using the area of a shape we can draw makes it feel a bit like $\pi$, which can be defined as the area of a circle and approximated by polygonal areas, and is one of the other famous irrationals we are familar with.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; For reference, here is the full sequence of definitions as I understand it (since it has been hotly debated in the comments):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I understood it, it goes like this: for positive real $a$ you still define $a^1:=a$, $a^n:=a^{n−1}\times a$ for $n\in \mathbb{N}$, $a^0:=1$, $a^{−n}:=\frac{1}{a^n}$ for $n\in\mathbb{N}$, $a^\frac{1}{n}:=\sqrt[n]{a}$ for $n\in\mathbb{N}$ and $a^\frac{m}{n}:=(a^\frac{1}{n})^m$ for $n\in\mathbb{N}$ and $m\in\mathbb{Z}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can still prove all the familiar properties of powers from these definitions, and  that $\frac{d}{dx}x^r=rx^{r−1}$ for any rational $r$ using methods of first principles, without using logs at all. The issue is how you define $a^x$ for irrational $x$, because it needs a definition before you can differentiate $a^x$ or indeed define an inverse for the function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we want, can implicitly assume it does have a consistent definition which works for all the other properties we observe about powers, and investigate the behaviour of this function and its inverse and their derivatives, calling the inverse $\log_a (x)$, but realising that there is a hole in our definition of it and a hole in our arguments about derivatives until we know what $a^x$ means for irrational $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we define $\ln(x) := \int_1^x \frac{1}{t}dt$ which immediately gives us that $\frac{d}{dx} \ln(x) = \frac{1}{x}$. From this definition we are able to prove all the familiar log laws for $\ln$. For example, to prove that $\ln(x^r) = r\ln(x)$ we can do the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the chain rule, $\frac{d}{dx} \ln(x^r) = \frac{1}{x^r}\cdot rx^{r-1} = \frac{r}{x}$. However $\frac{d}{dx} r\ln(x) = \frac{r}{x}$. Since these two functions have the same derivative, they differ by a constant and so $\ln(x^r) = r\ln(x) +C$ for some $C$. Substituting $x=1$ gives $\ln(1) = r\ln(1) + C$, so $C=0$ and $\ln(x^r) = r\ln(x)$. Note that this only works for $r$ rational, because we still don't know what $x^r$ means for $r$ irrational.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We define $\exp$ to be the inverse of $\ln$, and using the properties of $\ln$ we are able to prove that it behaves the same as the powers laws we know. For example, we can prove that $\exp(a+b) = \exp(a)\times\exp(b)$ and $\exp(rx) = \exp(x)^r$. However this last one is only currently proved for rational $r$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We define the particular number $e$ as the number such that $ln(e)=1$ or equivalently $e := exp(1)$, because we notice that $\exp(x)$ is the same as $e^x$ for all rational $x$ (and we know what $e^x$ is for rational $x$ because we defined what rational powers meant already without reference to $\exp$ or $\ln$). So now we &lt;em&gt;define&lt;/em&gt; $e^x := \exp(x)$ for irrational $x$ and we &lt;em&gt;define&lt;/em&gt; $a^x := \exp(\ln(a)\times x)$ for irrational $x$. We don't need to define $a^x$ for rational $x$ because it already has a definition, but we notice that even for rational $x$ this new calculation gives the correct result. We also notice that now $\exp(x) = e^x$ for all real $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, we can prove how derivatives of power functions work, and we can define $\log_a$ as the inverse of the function sending $x$ to $a^x$. And finally we observe that $\log_e(x)$ is the same as $\ln(x)$ for all real $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems like a roundabout way of doing things, but if we define $a^x$ for $x$ irrational as the limit of a sequence, then all our proofs have to involve how limits of sequences interact with these functions and their derivatives and they are therefore actually harder. &lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-07-03T23:38:25.850" LastActivityDate="2014-07-03T23:38:25.850" CommentCount="19" ContentLicense="CC BY-SA 3.0" />
  <row Id="3893" PostTypeId="2" ParentId="3886" CreationDate="2014-07-01T19:24:15.460" Score="4" Body="&lt;p&gt;Consider the question: What is the form of the function $f:\mathbb{R}\rightarrow\mathbb{R}$ that has the following two properties?&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$f'(x)=f(x)$&lt;/li&gt;&#xA;&lt;li&gt;$f(0)=1$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The unique answer is of course $f(x)=e^x$ however the proof is longer than people generally expect if they haven't considered this before, (for instance, we don't assume a priori that the solution is in the form of an exponential) but it consists of a series of steps that are mostly within the grasp of freshman calculus students. The only step that may perhaps have to be put off until later in the course is proof of absolute convergence. Here it is in outline form:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Derive the Maclaurin series $f(x)=\sum_{n=0}^\infty\frac{x^n}{n!}$.&lt;/li&gt;&#xA;&lt;li&gt;Prove absolute convergence (can be put off until later.)&lt;/li&gt;&#xA;&lt;li&gt;Derive $f(x+y)=f(x)f(y)$ for all $x,y\in\mathbb{R}$ by multiplying out $\sum_{n=0}^\infty\frac{(x+y)^n}{n!}$ and invoking the binomial theorem.&lt;/li&gt;&#xA;&lt;li&gt;Prove $f(xy)=f(y)^x$ for all $y\in\mathbb{R}$, as $x$ varies over successively larger sets $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$.&lt;/li&gt;&#xA;&lt;li&gt;Define $e\equiv f(1)$. Thus $f(x)=e^x$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This may seem like a lot of overhead for a seemingly elementary result, but I have asked this to a number of math grads and asked it &lt;a href=&quot;https://math.stackexchange.com/q/486731/92427&quot;&gt;here&lt;/a&gt;. There doesn't seem to be a substantially shorter proof that doesn't take for granted a known result about $e$ or the natural logarithm.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do you prefer one of these approaches, and if so, why?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Not speaking as a professional educator, but as an occasional tutor, I prefer the above $f'(x)=f(x)$ approach as I feel it is both the most &quot;natural&quot; motivation, as well as one which is most directly applicable to real-life applications (e.g. Newton's law of cooling, etc.). The other results as listed in the question follow fairly readily from this approach. It is also a good introduction to more formalized/advanced proofs.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it a good idea to explain the (arbitrarily) chosen definition and the subsequent derivations to students? I often wonder: why not just say e is an important number that is about 2.718, and talk about its properties? Maybe the appropriate place for these worries is an analysis course.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One can begin to compile a list of all the places where $e$ shows up. E.g. question to college freshmen: Did you know that in order to create mp3 files, one must use $e$ in the fourier transform to encode the sound as frequencies? (get as in-depth on this point as desired.) Another: It is one of the most important numbers in mathematics. The one equation which relates the 5 most important constants is: $$ e^{i\pi} + 1 = 0 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This list can go on and on...&lt;/p&gt;&#xA;" OwnerUserId="1857" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-07-01T19:42:48.993" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3894" PostTypeId="2" ParentId="2262" CreationDate="2014-07-02T00:48:39.637" Score="5" Body="&lt;p&gt;Many places have Calculus I as a co-requisite for University Physics I. The physics instructors like students in thier class to be familiar with derivatives of exponential functions before the end of the semester, hence a need for early transcendentals. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2014-07-02T00:48:39.637" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3895" PostTypeId="2" ParentId="2265" CreationDate="2014-07-02T01:45:36.800" Score="0" Body="&lt;p&gt;Now that my club has been running for a few weeks, I feel qualified to answer my own question. Here are some points I suggest:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Create one account&lt;/strong&gt; for the whole group. This allows students to be more invested in each others' solutions and creates a feeling of shared progress. It also makes it more natural for students to split up on different problems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Decide in advance &lt;strong&gt;which programming language&lt;/strong&gt; you will foist on students who have little or no programming experience. Seasoned programmers will use whatever they are comfortable with, but it is completely possible for smart people with no formal programming training to jump into programming with these problems. I of course recommend Python, but if you go with JavaScript you don't even need to install anything (they have a web browser)! To complete the first problem, all they need is a modulus function, a loop, and an if statement. Be ready to walk through those three pieces with newbies.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When someone completes a problem, announce the success to everyone and &lt;strong&gt;get everyone to cheer&lt;/strong&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Don't try to hide the fact that the solutions are posted online. The students know this is likely true. This is a great time to &lt;strong&gt;lead by example&lt;/strong&gt;: that looking up the solutions would be extremely silly.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Let students choose&lt;/strong&gt; terrible strategies for their programs. If the strategy is headed for a complete dead-end, then save them -- but if it will work in a terrible way, let them complete it the terrible way! It feels awesome when they get the answer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-07-02T01:45:36.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3896" PostTypeId="2" ParentId="2164" CreationDate="2014-07-02T01:52:16.443" Score="5" Body="&lt;p&gt;Here is what I gave out as an extra credit assignment to students who were interested in how to get more out of their textbook. Students mostly got the last problem correct, so the results were that &lt;strong&gt;the people who did this voluntarily really got something out of it&lt;/strong&gt;. The handout was 5 pages long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The last problem was as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How many pages of the book have we made it through?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(answer: 5) &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How many pages of our own math writing did it take to make it this far?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(answer: 5)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When you have to read a math book in the future, what is an&#xA;important tool you will absolutely need in order to do so&#xA;successfully?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;*(answer: your own paper and pencil!)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I copy-pasted the activity below although it is out of a google document instead of being properly TeX'd, so you probably won't be able to use it as is (for example, the integral signs didn't make it, nor the exponents or fractions). Still, it's here if you're interested.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The purpose of this exercise is to help you think about how to read a math book. Reading a math book is much different from reading other books!&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Step 1: Open your book to section 4.5. Begin reading the “Pattern&#xA;  Recognition” part. Read page 292 -- everything before the first&#xA;  example. Don’t worry too much about the details of the “theorem” for&#xA;  now, but then answer the following question:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Each differentiation rule has an integration rule that tells you how to go backwards. Which differentiation rule goes with “integration&#xA;  by substitution?”&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Step 2: Read Example 1. Then complete this question:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Find the derivative of 13(x2 + 1)3 + C. Why is this answer relevant to example 1?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Step 3. Read Example 2. Then complete this question:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Find the derivative of sin 5x  + C. Why is this answer relevant to example 2?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Step 4. Look at the “Exploration” box at the bottom of the page,&#xA;  after Examples 1 and 2.&#xA;  4. Complete questions a through e on this page. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;a. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;b.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;c.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;d. (multiply by ½ and also  by 2, as your first step).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;e. (multiply by &lt;strong&gt;__  and also  by _&lt;/strong&gt; as your first step).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Step 5. Read Example 3. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What does this have to do with the things you did in parts d and e&#xA;  on the previous page?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;There is a sentence after Example 3 that says “After all, if it&#xA;  were legitimate to move variable quantities outside the integral sign,&#xA;  you could move the entire integrand out and simplify the whole&#xA;  process. But the result would be incorrect.”&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Use this wrong method (factor everything out of the integral) to find&#xA;  x2 dx. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What is the actual correct answer for x2 dx? &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Step 6. Read the paragraph and formula under the heading “Change of&#xA;  Variables.” &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It says in here that if u = g(x), then du = g’(x) dx. That last&#xA;  equation looks weird. What happens if you divide both sides of it by&#xA;  dx? Does that make more sense?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If u = x2, what is du?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Step 7. Read Example 4. &#xA;  Check their answer to see if it is right (how do we even do that?)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Step 8. Read Example 5.&#xA;  Check their answer to see if it is right (how do we even do that?)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Following the examples, solve x3x - 1 dx. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Check your answer to see if it is right (how do we even do that)?&#xA;   Step 9. Read Example 6. It is getting crazy now.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Remember that sin2 x means (sin x)2, and sin3 x means (sin x)3. Check their answer by taking the derivative.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Step 10. How to read a math book.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;11a. How many pages of the book have&#xA;  we made it through?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;How many pages of our own math writing did it take to make it&#xA;  this far?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;When you have to read a math book in the future, what is an&#xA;  important tool you will absolutely need in order to do so&#xA;  successfully?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-07-02T01:52:16.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3897" PostTypeId="2" ParentId="380" CreationDate="2014-07-02T02:18:27.520" Score="1" Body="&lt;p&gt;I find two things helpful:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Explicitly tell students how you make the decision of what rule to use. There are features of the problem that sing &quot;chain rule&quot; to you, so explicitly tell them what those features are. Also include in this information a list of things to try in order of most likely to produce a helpful result.&#xA;It's a bit like a doctor in the emergency room of a hospital. The patient presents with various symptoms, which they learn how to associate with various medical conditions. Those possible conditions are also sorted in their minds into an order of likelihood and they start with the most likely first.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give them a &quot;mantra&quot; to state. Something they can repeat to themselves to help them focus. The one I use with students is &quot;I would know what to do if THIS part was x.&quot; This helps them choose the chain rule AND focus on which part needs to be differentiated when. &#xA;I should point out what I do for the rest of the procedure: I say &quot;I'll pretend it IS x to start with. But it's NOT x, so I have to multiply by the derivative.&quot; I cover up the part that is pretending to be x until I differentiate it, and I also use the &quot;arrows&quot; approach mentioned by Dave L Renfro to help them focus.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-07-02T02:18:27.520" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3898" PostTypeId="2" ParentId="3813" CreationDate="2014-07-02T09:50:07.117" Score="1" Body="&lt;p&gt;Children have some intuitive notions about chance and probability.  Here is an example:  When my daughters were about 6 and 7 years old, I tried this experiment.  We had two urns  (well, shoe boxes, important they were identical), in one 8 black balls, 2 white, in the other numbers switched.  Then we put the urns in a sack, mixed them and draw one.  From the drawn urn we draw one ball. It was black.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I asked: from which urn do you think we got the ball?  Thy thought about 1-2 seconds, then answered simultaneously: The urn with most black balls! ---Why do you think so? --- because then it is easier to draw black! they said, in chorus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So thats it, more probable is the same thing as &quot;easier&quot;.  No need fro fractions!&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2014-07-02T09:50:07.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3899" PostTypeId="1" AcceptedAnswerId="3902" CreationDate="2014-07-02T11:52:04.467" Score="14" ViewCount="699" Body="&lt;p&gt;This is a detail, but getting the details right means a lot when you design your course. For the purposes of this question &lt;em&gt;please assume calculators are not allowed in the course&lt;/em&gt;, which is true in many college calculus courses and is a debate I do not wish to have here.&lt;/p&gt;&#xA;&lt;p&gt;I've tried things along the spectrum, from&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Essentially never simplify&lt;/strong&gt;. Simplifying doesn't matter unless it is useful. Is your answer &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\sqrt{256}}$&lt;/span&gt;? Fine, circle it. Finding the square root of 256 is not the point of this class and in the real world you won't need to know it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;to&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Essentially always simplify&lt;/strong&gt;. If you don't simplify, it betrays a lack of knowledge about what is going on. Circling &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\sqrt{256}}$&lt;/span&gt; is embarrassing and means you haven't completed the problem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I find it difficult to provide a well-defined policy inbetween the two extremes, but I don't like either extreme:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;If simplification is not required, then students have no external incentive to simplify something like&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{\sqrt4 - 5}{3}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;but they realize that if they simplify it incorrectly, they will lose points. The answer is -1, and the students should be able to turn that into -1. A &amp;quot;no-simplification-required&amp;quot; policy deirectly incentivizes students to circle something like that as their final answer.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;On the other hand, if simplification is required, then we are spending time and effort on computational issues that really don't matter in the end, and drawing a line of what needs to be simplified is difficult. Is &lt;span class=&quot;math-container&quot;&gt;$\sec^2 x - 1$&lt;/span&gt; simplified enough? What about &lt;span class=&quot;math-container&quot;&gt;$\sin(\frac{x}2)\cos(\frac{x}2)$&lt;/span&gt;? What about &lt;span class=&quot;math-container&quot;&gt;$\frac{x - 1}{\sqrt{x} - 1}$&lt;/span&gt;? Do I really want them worrying about any of this? I'd rather they spend more time on the content of the course.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Have you found a good simplification policy for college courses without calculators?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-07-07T16:32:21.003" Title="What is a good &quot;simplification policy&quot; for a college course with no calculators?" Tags="|undergraduate-education|calculus|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3900" PostTypeId="2" ParentId="2246" CreationDate="2014-07-02T12:07:35.563" Score="4" Body="&lt;p&gt;I run the Maths Learning Centre at my University and I came up with the idea of using differentials on the spot when a student asked me to explain how the chain rule works. The problem is that the chain rule just doesn't relate directly to slopes drawn on a graph, so I needed to come up with a different approach. Following the tendency of the Economics lecturers to use differentials I started there. Later in a revision seminar for economics students, I used it as a way to unify their thinking about ordinary and partial differentiation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not using pure differentials as described in the article but a bit of a mixture. I've been saying that we are looking for a way to describe how big dy is relative to dx. So I write &quot;dy = (?)*dx&quot;. Then I say that the name for the ? is the derivative and it's represented by &quot;dy/dx&quot;. Later, with partial derivatives (say with z dependent on x and y) we have dz = (?) dx + (??) dy and the name for (?) is $\partial z/\partial x$ and the name for (??) is $\partial z / \partial y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did make it clear that dy/dx is ALSO the formula for a function in its own right, which is where I introduce the f' notation, saying it's &lt;em&gt;another&lt;/em&gt; function related to f.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, this is all connected to a pictorial approach that is &lt;em&gt;not&lt;/em&gt; the graph of the function. I draw a horizontal number line for x and a separate horizontal number line for y. Then I talk about how when you start at a point on the x-line, the function produces a matching point on the y-line. I draw a few pairs of matching points. Then I pick one of them and draw a little arrow to represent dx. Then I draw a matching little (but slightly bigger) arrow for dy on the second line. And I point to these two arrows and discuss how big one is relative to the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the two-variable function, I draw the x-line and the y-line next to each other, and below in the middle I draw the z-line. Then the dx arrow and the dy arrow each produce a separate arrow on the z-line, which are arranged head-to-tail to make the whole dz arrow.&#xA;&lt;img src=&quot;https://i.stack.imgur.com/In44U.png&quot; alt=&quot;relationship between dx, dy and dz&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students I have used this with one-on-one have seemed to really like this, especially for the two-variable case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the future I would like to actually make a table of x's, y's, dx's and dy's so that they can see how the relationship between dx and dy can be different for different x's and therefore be a formula rather than a single number.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-07-02T22:27:47.063" LastActivityDate="2014-07-02T22:27:47.063" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3901" PostTypeId="2" ParentId="3848" CreationDate="2014-07-02T14:17:24.873" Score="2" Body="&lt;p&gt;First I have to make an assumption: that when you say &quot;they are into business&quot; you do &lt;em&gt;not&lt;/em&gt; mean &quot;professionals&quot; in general -because if they are, say, engineers building bridges or spacecrafts, they should and most probably do care a lot about whatever mathematical toll can give them increased precision. So my assumption is that &quot;into business&quot; means &quot;into business administration&quot; (I won't bother discussing the &quot;politics&quot; part).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, from my experience as a &quot;business person&quot;, I suggest (for precision's shake) that you separate &quot;business people&quot; (let's call them BP's), into those that are in the Financial Services Sector, and those that are not.&lt;br&gt;&#xA;The first group has as a usual level of precision four decimal digits, the &quot;basis points&quot;: interest rates are usually quoted as percentages with two decimal digits: $6.23$% $= 0.0623$. Of course, in very large contracts this can go further, but I am stating the usual everyday practice.&lt;br&gt;&#xA;The second group, well, they haven't decided yet: business reports, charts, &quot;contribution margins&quot; etc are quite often expressed also as above... &lt;em&gt;but almost nobody will talk, communicate, think, negotiate or decide at that level of precision&lt;/em&gt;. 2 decimal digits (i.e. whole percentages and nothing more, $6% = 0.06$), is the usual level here... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;...for percentages. because when you go into &lt;em&gt;levels&lt;/em&gt; of magnitudes, then you loose at least 3 zeros &lt;em&gt;before&lt;/em&gt; the dot: The amount $100.450$ USD is &quot;a hundred thousand&quot;, while if the business is in the tens of millions, well, we talk using tens of millions (only accountants resist this, and this is why BP's call them derogatorily &quot;bean counters&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this apparent avoidance of precision, perhaps surprisingly, means that BP's should &lt;em&gt;adore&lt;/em&gt; things like $\sqrt 2$. Why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, why are they &lt;em&gt;not&lt;/em&gt; after precision in the first place? Because, there is always the trade-off between the gains from precision and the cost of attaining it.This cost is measured in &lt;em&gt;time units&lt;/em&gt;: time required to &lt;em&gt;think&lt;/em&gt; precisely, time required to &lt;em&gt;compute&lt;/em&gt; precisely...   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;...and BP's are obsessed (rightfully) with efficient time management. So they will take &lt;em&gt;anything&lt;/em&gt; that cuts the time to do &lt;em&gt;anything&lt;/em&gt;: $\sqrt 2$ requires less button pushes compared to its rational approximation -and so it increases precision &lt;em&gt;at a time gain&lt;/em&gt;: that's a dream come true (I am not joking). BP's will still &lt;em&gt;think&lt;/em&gt; of $\sqrt 2$ as &quot;about $1.5$&quot; -but when it comes to actual computing, $1.5$ takes three button hits, while $\sqrt 2$ takes only two.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it appears that if you want to push this agenda, your chance is to make the BP's see the reality that, with all these computers and calculators around, they can think in whatever level of precision they like -but that it is &lt;em&gt;in their time-management interest&lt;/em&gt; to &lt;em&gt;compute&lt;/em&gt; using these strange looking mathematical concepts and tools...while avoiding also the possibility of fatal compounding approximation error: a dream come true.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2014-07-02T14:17:24.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3902" PostTypeId="2" ParentId="3899" CreationDate="2014-07-02T14:31:36.223" Score="5" Body="&lt;p&gt;If you use online homework, then when it comes to symbolic answers you don't have much wiggle room. These systems generally work by generating random numbers as the values of the variables that appear in an expression, evaluating the expression, and comparing the numerical outputs. (I believe the folks developing the LON-CAPA system started out this way and then augmented it with some CAS strategies.) The software doesn't care whether the answer is entered as $x-2y$, $-2y+x$, or $x-2y+0$. It's not smart enough to know whether the answer is simplified or not. The notion of simplification is in a certain sense &lt;a href=&quot;https://mathoverflow.net/questions/126519/is-there-a-mathematical-definition-of-simplify&quot;&gt;not definable by algorithms&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For problems with numerical answers that are based on finite-precision given data, answers should be fully evaluated with the correct number of significant figures. Example question: A square has a diagonal of length 0.76 cm. Find the length of one of its sides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Correct answer: $0.54$ cm&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incorrect answer 1: $(0.76)/\sqrt2$ cm. This answer shows that the student has not understood the point of preserving exact expressions like $\sqrt2$ rather than evaluating them as decimal approximations. Applying that habit in this context is wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incorrect answer 2: $(0.38)\sqrt2$ cm. Another habit applied in a context where it doesn't make sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incorrect answer 3: $0.53740$ cm. Ths student doesn't understand that the precision of the final result depends on the precision of the input data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rule of fully evaluating the numerical result in this sort of context can and should be implemented in whatever online homework software you're using.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my favorite types of problems to assign is of the form: (a) find a symbolic expression for $x$ in terms of $y$, $z$, ...; (b) interpret the equation's dependence on the variables. For example, the student might be asked to check whether increasing $z$ has the effect of increasing $x$ or decreasing it, and then to compare with the expected behavior based on the real-world situation being described. E.g., if $z$ is the price of a good and $x$ is the quantity sold, and our expression is $x=1/z$, then we check the mathematical behavior, which is that as $z$ gets bigger, $x$ gets smaller, and compare with the expected behavior. In tasks like these, simplification is often necessary as a preliminary step before the student can reason successfully about the behavior of the result. The requirement of simplification is sort of self-enforcing, because the student can't complete the problem without doing it.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-07-02T14:31:36.223" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3903" PostTypeId="2" ParentId="3873" CreationDate="2014-07-02T15:50:26.357" Score="3" Body="&lt;p&gt;Proofs are what make mathematics special as a subject. The sciences have theories which change with time. Mathematics has theorems that don't change with time. However, I think &quot;teaching proofs&quot; is over emphasized. Showing &quot;samples&quot; of proofs that illustrate different approaches to proving is much more important than being able to &quot;prove theorems&quot; as an output skill for a &quot;course.&quot; Students should see proofs by contradiction, using parity arguments, using the pigeon hole principle, using induction, proofs without words, etc. Teaching students about using mathematical tools and ideas is much more important than teaching proofs it seems to me. No one is likely to be able to find a proof of the 4-color theorem using mathematical induction. However, Carsten Thomassen showed that plane graphs are 5-choosable, using mathematical induction in a clever way. (The issues is having lists of colors where the vertices can be colored using any color on the list for that vertex but the usual rule that two vertices joined by an edge can't get the same colors holds). (Thomassen, Carsten (1994), &quot;Every planar graph is 5-choosable&quot;, Journal of Combinatorial Theory, Series B 62: 180–181) The idea of list colorings is more important to show students than getting them to master how to do induction proofs of statements that are already known to be true. Making interesting/important conjectures is very hard, in many ways, harder than proofs.  &lt;/p&gt;&#xA;" OwnerUserId="1862" LastActivityDate="2014-07-02T15:50:26.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3904" PostTypeId="2" ParentId="3899" CreationDate="2014-07-02T15:55:02.360" Score="9" Body="&lt;p&gt;As simplification is &lt;a href=&quot;https://mathoverflow.net/questions/126519/is-there-a-mathematical-definition-of-simplify&quot;&gt;not definable&lt;/a&gt; (from Ben Cromwell's answer), there cannot be a general rule for simplification that works for all examples under all circumstances.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, &lt;strong&gt;use specific rules for each task/question&lt;/strong&gt;. Your example of $\frac{\sqrt{4}-5}{3}$ &lt;em&gt;might&lt;/em&gt; best be changed to &lt;em&gt;for example&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;multiple choice, if just checking the given answers for correctness isn't easier or not the competence you want to check&lt;/li&gt;&#xA;&lt;li&gt;comparison, under similar conditions like multiple choice&lt;/li&gt;&#xA;&lt;li&gt;explicitly ask for decimal number/integers/fractions (if computations are comparably easy)&lt;/li&gt;&#xA;&lt;li&gt;practical application, where uncomputed answers are useless or not customary.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And beware: &lt;strong&gt;algebraic simplification may also veil structure&lt;/strong&gt;. If I teach students the base values of trigonometric functions, I give them&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\sin(0°)=\frac{1}{2}\sqrt{0}$$&#xA;$$\sin(30°)=\frac{1}{2}\sqrt{1}$$&#xA;$$\sin(45°)=\frac{1}{2}\sqrt{2}$$&#xA;$$\sin(60°)=\frac{1}{2}\sqrt{3}$$&#xA;$$\sin(90°)=\frac{1}{2}\sqrt{4}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you simplify these terms algebraically, you &lt;em&gt;may&lt;/em&gt; lose structure. &lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-07-05T18:49:48.720" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="3905" PostTypeId="2" ParentId="2790" CreationDate="2014-07-02T16:23:41.037" Score="3" Body="&lt;p&gt;Here's what could realistically happen: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, someone states a theorem T, which may be true or not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, I write a program. I claim &quot;if this program has output X, then the theorem T is true. If this program has output Y, then the theorem T is false. If the program produces output Z or doesn't produce any output at all in the time that I'm willing to wait, then I need to try harder&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third, I run the program, which may output X or Y. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fourth, the important step, if the program output was X or Y, is that I need to prove my claim from Step 2. So a computer program is not a proof. A computer program together with a proof what the output will be if the theorem is true or false, that's a proof. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is quite easy for example to write a program that &lt;em&gt;might&lt;/em&gt; demonstrate that the Collatz Conjecture is false (by finding an x where the Collatz sequence starting with x ends in a cycle); I have no idea how to even start writing a program that &lt;em&gt;might&lt;/em&gt; demonstrate that the Collatz Conjecture is true. And I have seen many programs written by beginners where a proof would fail because of bugs in the program. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is the minor problem that program + proof + output still require that the compiler compiled the program correctly, and that the hardware didn't make any changes to the calculation because of some cosmic rays and so on, but then any &quot;normal&quot; mathematical proof could contain errors that somehow everybody missed.  &lt;/p&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2014-07-02T16:23:41.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3906" PostTypeId="2" ParentId="2164" CreationDate="2014-07-02T16:35:25.573" Score="17" Body="&lt;p&gt;For reading individual proofs, my colleagues and I have had some success in research studies with self-explanation training adapted for mathematics students.  We haven't extended this work to whole sections of books yet, but it was encouraging to find that students could improve their reading after light-touch training - they did better in subsequent proof comprehension tests, and they showed reading behaviours more like those of professional mathematicians (as measured using eye-movement data).  The research paper is at &lt;a href=&quot;http://www.jstor.org/stable/10.5951/jresematheduc.45.1.0062&quot;&gt;http://www.jstor.org/stable/10.5951/jresematheduc.45.1.0062&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1863" LastActivityDate="2014-07-02T16:35:25.573" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3907" PostTypeId="2" ParentId="3873" CreationDate="2014-07-02T16:50:56.637" Score="5" Body="&lt;p&gt;Three more books that might be of interest:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Allenby, R.B.J.T. Numbers and Proofs&lt;/li&gt;&#xA;&lt;li&gt;Solow, D. How to Read and Do Proofs;&lt;/li&gt;&#xA;&lt;li&gt;Velleman, D.J. How to Prove it: A Structured Approach.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I think it's also worth bearing in mind that students who have learned earlier mathematics primarily as a set of procedures might not be in the right frame of mind to appreciate what proofs are for and why we bother with them.  I wrote some chapters for such students (and for those with different experiences) in my book 'How to Study for a Mathematics Degree' (the US version is called 'How to Study as a Mathematics Major').  The book is based on research in mathematics education and psychology and it won't suit everyone, but it aims to explain how proof-based mathematics differs from earlier work, and how a student might go about tackling it.&lt;/p&gt;&#xA;" OwnerUserId="1863" LastActivityDate="2014-07-02T16:50:56.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3908" PostTypeId="2" ParentId="3858" CreationDate="2014-07-02T17:25:21.583" Score="9" Body="&lt;p&gt;There is a whole book devoted to game theory and law: Game Theory and the Law by D. G. Baird, R. Gertner, and R. Picker, Harvard U. Press, 1994.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although much of Game Theory was developed by mathematicians such as John Von Neumann, Lloyd Shapley, John Nash, and Alvin Roth, people who currently do research in game theory are more likely to teach in economics, operations research, or political science departments rather than mathematics departments, even though they have degrees in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastEditorUserId="1865" LastEditDate="2014-07-02T20:21:40.817" LastActivityDate="2014-07-02T20:21:40.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3909" PostTypeId="2" ParentId="3899" CreationDate="2014-07-02T19:54:17.767" Score="3" Body="&lt;p&gt;Simplifying $\frac{x-1}{\sqrt x - 1}$ can be misleading insofar as while it's mostly equivalent to $\sqrt x + 1$, the unsimplified expression is undefined at $x = 1$ but not the latter. If you get the fraction expression in evaluating the problem, it's likely that the missing point on the curve is part of the solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as was mentioned earlier, the whole problem of simplification is not algorithmically deterministic and the choice, e.g., of $\cos 2x$ vs $2\cos^2 x -1$ is one that must be determined based on the context.&lt;/p&gt;&#xA;" OwnerUserId="1866" LastActivityDate="2014-07-02T19:54:17.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3910" PostTypeId="2" ParentId="12" CreationDate="2014-07-03T08:37:20.050" Score="9" Body="&lt;p&gt;Was Silvanus Thompsons lovely &quot;&lt;a href=&quot;http://en.wikipedia.org/wiki/Calculus_Made_Easy&quot; rel=&quot;noreferrer&quot;&gt;Calculus made easy&lt;/a&gt;&quot; mentioned already? It's a classic (100 years old) freely available on &lt;a href=&quot;http://www.gutenberg.org/ebooks/33283&quot; rel=&quot;noreferrer&quot;&gt;gutenberg.com&lt;/a&gt;. Some opinions of it can be found on &lt;a href=&quot;https://mathoverflow.net/questions/58193/leibnizian-calculus-textbook/58196#58196&quot;&gt;mathoverflow&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It doesn't go very far so it might need to be supplemented with another text, but I believe it does a great job at teaching the physical and geometrical intuition on differentials. It seems that it's closer to synthetic differential calculus than to non-standard analysis in the way it treats infinitesimals.&lt;/p&gt;&#xA;" OwnerUserId="590" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-07-03T08:37:20.050" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3911" PostTypeId="2" ParentId="61" CreationDate="2014-07-03T12:13:32.920" Score="3" Body="&lt;p&gt;&lt;strong&gt;&lt;em&gt;Geometric models&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students who have trouble imagining often need to see it with their own eyes in order to imagine. So a model is often the best approach to support their imagination, and it's not too infeasible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Using playdough&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can buy commercial playdough, or make your own if you want a really big blob. (Recipe below)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to draw an x and y axis on a piece of paper, and then put the playdough on top in some sort of shape. To mark points on the surface, you can poke a tiny hole with a sharp pencil. A sharp pencil can also be used to scratch lines in the surface to make level curves. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you pick a line and slice it with a knife, you can show the exposed face and note that it makes an ordinary curve, which has slopes of its own. Marking an axis on the paper along the knife-cut helps to point out that direcitonal derivatives require a new axis with points marked by a unit vector.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Slicing parallel to the x-axis or y-axis helps you point out the meaning of the ordinary partial derivatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are in a lecture theatre with a document camera, you can very successfully do this on the document camera -- just remember to lift and tilt it every so often so they can see the three-dimensional nature of it. If you have tutorials with a smallish number of students, you can get them to stand around one table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PLAYDOUGH RECIPE (from McKenzie's Cream of Tartar box):&lt;br&gt;&#xA;Ingredients:&lt;br&gt;&#xA;2 cups plain flour, 1 cup cooking salt, 4 tbsp cream of tartar powder, 2 tbsp cooking oil, 2 cups water, food colouring&lt;br&gt;&#xA;Instructions:&lt;br&gt;&#xA;1. mix the dry ingredients in a saucepan.&lt;br&gt;&#xA;2. add the wet ingredients including the colour and mix well.&lt;br&gt;&#xA;3. cook over medium heat stirring continuously until the mixture congeals.&lt;br&gt;&#xA;4. tip out and allow to cool.&lt;br&gt;&#xA;5. knead until smooth.&lt;br&gt;&#xA;6. store in a plastic bag in an airtight container.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Using a sand-box&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't tried it myself yet, but I think a clear container with slightly wet sand could do a similar job to playdough, though it is messier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cardboard for tangent planes&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When talking about tangent planes, students often have real trouble imagining them. You need to have a proper model of the plane. Paper is too floppy and so doesn't work. You need something thicker. Any cardboard should do. You can draw vectors on the plane to represent the x-slope and the y-slope. A playing card would be a good thing to use -- you could even use pins to attach several of them to the playdough at the points they are tangent to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Water levels for level curves&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can get your hands on a big clear container and a heavy object, then you can explain level curves using water. Put the object into the container and start pouring water slowly. The edge of the water describes a level curve on the surface. I reckon it would be even more striking with coloured water, especially if the object was a very different colour, though I haven't tried this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Potato printing for level curves&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take a potato (or other vegetable/object) and cut it. Dip the end in paint and press it down on a piece of paper. The shape left is the shape of the level curve. Slicing it again at another height and redoing it can show how the curve changes as you move up the potato. Changing colours as you go can highlight a sequence of level curves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Slightly less geometric models&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The temperature or smell analogy&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of trying to &lt;em&gt;visualise&lt;/em&gt; the function, you can appeal to the students' other senses. I often use the analogy that at each point in a room, there is a temperature -- each point &quot;feels&quot; hot or cold. A gradient in a particular direction is how quickly the temperature changes when moving that direction. For example, the temperature will increase when you move towards the heater. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another alternative is smell. You can imagine a smell being stronger in one place in a room than in another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A visual version in 2D is to use colour -- possibly red-to-blue gradients to match with the temperature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These three reinforce the idea that the function doesn't have to represent a height, and that you can think of it like the function value &quot;lives&quot; at each point -- if you inspect the point itself you can make it reveal its function value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Colour-coded level curves&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way of thinking about a function is that at each point there is a value &lt;em&gt;right there&lt;/em&gt; for the function. If you colour in all the points with the same function value, you will make curves. If you use a gradient of colour for changing values (say blue-to-red), students can see the speed at which colour is changing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you space your curves evenly by function-value, then you can pick any point and any direction and draw a line, noticing how far you have to travel to get to the next function value. This is a representation of the rate of change. When lines are close together you pass by a lot of them very quickly so the function value is changing quickly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is important to colour the lines so that you can tell whether it's going up or down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Before-and-after&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to think of functions is as input-output machines. A two-variable function takes a point in 2D as input and produces a number as an output -- which can be interpreted as a line. So you can actually draw two separate pictures -- a plane for the input and a line for the output.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can mark points in the plane, and then mark the matching points on the line. Colour-coding can help to identify which point produces which output. If you move an input point along a line, you can investigate how quickly the output point moves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geogebra should allow you to do this by having two graphics windows open at once. Make a point A in one window. In the second window turn of the x-axis and make a point B = (0, x(A) + y(A)^2) [or whatever function]. Then try moving the point A and see what happens to the point B. See a quickly-made example &lt;a href=&quot;http://www.adelaide.edu.au/mathslearning/resources/maths1b/two_var_function_before_after.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-07-25T06:51:35.517" LastActivityDate="2014-07-25T06:51:35.517" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3912" PostTypeId="2" ParentId="654" CreationDate="2014-07-03T13:59:24.453" Score="3" Body="&lt;p&gt;Others have mentioned several advantages/disadvantages and personal experience of using slides. I will add some comments that are not strongly mentioned already:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Maths that is already written down does not make any sense&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You know yourself that when you read text, you have the tendency to read the words and skim over the maths. The same happens when reading slides. Also, maths is very dense in terms of meaning, so students will have trouble figuring out which bit you are referring to if they look away for even a second. Finally, students learning new notation need to know how it is pronounced so that they can talk about it, and so they know how their speech will translate into notation when they write. I see a lot of students who point to notation in the slides and ask &quot;what does this symbol say?&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if you choose to use slides you need to:&#xA;* Read all the maths aloud in full.&lt;br&gt;&#xA;* Preferably highlight any part of the screen/slide that you are talking about at any given moment. (If you are recording your lecture you need to do this in such a way that the recording device can capture your highlighting/pointing.)&lt;br&gt;&#xA;* Consider seriously the option of pausing the slides momentarily to write some maths live, even if it will appear on the slide shortly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Slides are not good for the students' personal study&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though slides are effective at presenting a whole class, they are not the best thing for personal study. Well-designed slides tend to have only a small amount of information per slide, and to be arranged specifically for the format of accompanying a spoken component. Without the speech, they are often quite useless. Students visit my Maths Learning Centre with their slides dutifully printed out, but they cannot find information in them, and they often become frustrated when a maths example spreads across several slides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to give your students printed material as well (and you probably should if you have slides), you should probably prepare material specifically for the purpose. I suggest reorganising your content into a collection of one-or-two page summaries, or books of worked examples. Students seem to really appreciate this sort of thing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;If your content is not linear, consider using Prezi&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I do use slides, I use Prezi most of the time (though of course I make print resources specially and use handwriting on the document camera/blackboard too when necessary). In case you don't know, Prezi is like an infinite whiteboard, and instead of flipping through pages, you tell it to move along a path and zoom in on things in succession.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find Prezi has several advantages.&lt;br&gt;&#xA;* I don't have to put in multiple slides if I keep coming back to the same point -- I can just keep coming back to the same place in the Prezi.&lt;br&gt;&#xA;* It's easier to skip to a previous part of the presentation if someone asks because you can simply zoom out and find it visually.&lt;br&gt;&#xA;* I can put in a full worked example from a pdf and zoom in on different parts of it to analyse what is written there.&lt;br&gt;&#xA;* I can also organise the content spatially to reinforce connections between things. This  is particularly useful for maths because many things have multiple connections in multiple directions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prezi does have a disadvantage in that it is not well-suited to simply printing the &quot;slides&quot;, but as I said, that's not necessarily the best plan anyway.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-07-03T14:48:55.207" LastActivityDate="2014-07-03T14:48:55.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3913" PostTypeId="2" ParentId="3886" CreationDate="2014-07-03T14:30:42.720" Score="2" Body="&lt;p&gt;When I taught high school Calculus, I talked about how e is a transcendental number and explained that it went on forever, never repeating, and that any finite length number sequence you search for could be found inside it.  I had a student give me their date of birth and then looked it up in the pi and e search engine and found it in both pi and e (with 2 billion digits the odds of a 6 digit number being found it in is rather high):&#xA;&lt;a href=&quot;http://www.subidiom.com/pi/&quot; rel=&quot;nofollow&quot;&gt;http://www.subidiom.com/pi/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which rather impressed them.   I then had them consider whether e is found inside pi or pi inside of e - and that rather boggled their minds.   I also presented the difference between an algebraic and trascendental irrational number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also had them consider e ^ pi and pi ^ e and whether those algebraic and transcendental.  Also that e ^ pi is transcendental, but we don't know about pi ^ e.   (Which is strange, because it feels like it should be, but no one has been able to prove it according to most sources I've looked at...)   Here's some other fun transcendental numbers:  &lt;a href=&quot;http://sprott.physics.wisc.edu/pickover/trans.html&quot; rel=&quot;nofollow&quot;&gt;http://sprott.physics.wisc.edu/pickover/trans.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That seemed to be &quot;good enough&quot; for high school calculus - they were convinced that it was a number similar to pi and that it was important enough to have its own button on their calculator.   And they could see that it worked similar to other logarithm bases to solve exponential functions - like, you could use ln instead of log and still get the same answers, and in fact you could use any base to solve the types of things we were doing.   I didn't really rigorously prove it's derivative, but using the graphing calculator they could see rather readily that e^x and ln x are inverse functions and that the derivative of ln x was 1/x.   (Most graphing calculators let you graph, say, the 1st derivative of a function, using a numerical approach).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think if I had gone on to teach second semester calculus I might have used the area under the curve definition, or perhaps the limit of the compound interest formula.   But the big thing, I think, is what you hope to get from the definitions - like, are there particular results you're hoping to prove that need a certain definition?   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If not, maybe it's okay to have a bit of mystery around e and to merely pique students interest in learning more about it.  Like, I'd rather have students going &quot;oh wow, e is really fascinating - let's go read a book on it&quot; (say, Eli Maor's excellent Story of E) then &quot;I know a formal definition of e, but it put me to sleep.&quot;   &lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-07-03T14:30:42.720" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3915" PostTypeId="2" ParentId="2246" CreationDate="2014-07-04T12:44:24.273" Score="9" Body="&lt;p&gt;I had the chance to read Dray &amp;amp; Minogue's online stuff shortly before I was first assigned to teach a Calculus course, an Applied Calculus course intended for biology and economics majors.  As it was a terminal math course, I felt justified in taking an unusual approach; as it was applied, I didn't worry about rigour (as long as I knew that it could be done).  So I used differentials.  It seemed to go all right.&lt;/p&gt;&#xA;&lt;p&gt;That was three years ago.  Since then, I've also taught the regular three-semester-equivalent Calculus series for physical science/engineering and mathematics majors.  Sometimes I teach the whole sequence in a row, sometimes I only teach Caclulus 3 (multivariable).  I have used differentials every time.  I particularly love it for multivariable calculus, where I also introduce differential forms.&lt;/p&gt;&#xA;&lt;p&gt;It is generally not too hard to link this to what's in the book (which is chosen at the departmental level).  After all, differentials (and differential forms in multivariable integrals) appear in the text; it's just that they only appear in certain combinations.  So I can pick out a subexpression from the expressions in the book and explain that it has a meaning in its own right.  Sometimes I write some new exercises, but mostly I just show them another way to do the exercises in the book.  Sometimes they do it my way, sometimes the book's way; it's all good.&lt;/p&gt;&#xA;&lt;p&gt;In the applied course, I do differentials before derivatives, which I do before limits.  We're not pretending to define things rigorously, so there is no harm in this, and this seems to be in order of increasing difficulty.  I actually introduce rigorous definitions (even ones that are not in the textbook), but at the end, in the context of approximation methods (which is actually how they first appeared historically); I mention that such-and-such actually provides a rigorous definition, which they take as one of my remarks important for their well-rounded liberal education that won't be on the test, and we move on.&lt;/p&gt;&#xA;&lt;p&gt;In the regular course, I feel obligated to give the rigorous definitions a more prominent place, since they are explicitly part of the syllabus.  This means an approach as in Aeryk's answer: limits, then derivatives, then differentials.  I pretty much agree with everything that Aeryk has said, right down the line.  But I will add that differentials are especially awesome in multivariable calculus, where I can introduce partial derivatives as in David Butler's answer.  Often it is easier to work with the differentials directly, and never mind the individual partial derivatives that appear within them.&lt;/p&gt;&#xA;&lt;p&gt;I must caution against conflating different kinds of differentials and infinitesimals.  In particular, Dray &amp;amp; Minogue's approach is NOT the same as Keisler's approach using nonstandard analysis.  (Although a nice thing about Keisler's book is that it allows you to do derivatives before limits, rigorously even.)  Dray &amp;amp; Minogue's (and hence my) differentials are the linear differentials from differential geometry (and in differential forms), not the nonstandard hyperreal numbers from nonstandard analysis (nor the nilpotent infinitesimals from synthetic differential geometry, for that matter).  To distinguish these, consider: if ‘d’ indicates a finitesimal change, then dy/dx = f'(x) is an approximation; if it indicates a nonstandard infinitesimal, then dy/dx = f'(x) is an adequality (equality of standard parts); if it indicates a linear differential, then dy/dx = f'(x) is an equality, period.  (All on the assumption that y = f(x), of course.)&lt;/p&gt;&#xA;&lt;p&gt;I also want to stress the triviality of the Chain Rule with differentials.  I don't mean that you can prove it by cancelling du in dy/dx = dy/du du/dx; it is a real theorem that requires a real proof.  (I once read a review of Keisler's book that praised Keisler for being able to prove the Chain Rule with a similar cancellation, but if you read that part of the book, it wasn't this trivial.)  But when working with differentials, it becomes a theoretical result, not a tool for practical calculation.  The equation dy/dx = dy/du du/dx is a triviality, but it is not the Chain Rule; or rather, it is the Chain Rule only if you change the meaning of the symbols from one place to another, an unfair trick to play on students.  The real Chain Rule is d(f(u)) = f'(u) du.  It tells you how, if you can differentiate [take the derivative of] a function, you can apply that function to any differentiable expression and differentiate [take the differential of] the resulting expression.&lt;/p&gt;&#xA;&lt;p&gt;An example will show the power of this approach.  Suppose that you establish, say by the usual argument involving sum-angle formulas and special limits, that the derivative of the sin function is the cosine function.  Then you apply the Chain Rule —once!— to conclude that d(sin u) = cos u du for any differentiable expression u; this fact is called the Sine Rule.  Now when faced with such expressions as sin(3x²), sin(5x - sin x), sin(2x + 3y), etc, you do NOT use the Chain Rule; you use the Sine Rule.  So in the end, instead of having a rule for each algebraic operation, a derivative for each special function, and a Chain Rule that requires you to analyse expressions as composites, you have a rule for each algebraic operation and a rule for each special function, and every differentiation is done by working your way from the outside in, applying the appropriate rule to whatever operation comes next in the expression as it is written down.  (This is at least part of what Aeryk means by saying that Calculus becomes algebraic.)  The Chain Rule is only needed explicitly if you are dealing with a new or unknown function.&lt;/p&gt;&#xA;&lt;p&gt;As you can see with sin(2x + 3y) above, differentials work seamlessly with any number of variables.  Students have done Algebra with multiple variables, and they can just as easily do Calculus with multiple variables.  Some things, such as optimization, graphing, and integration, are legitimately more complicated with multiple variables, but the basic operation of differentiating [taking the differential of] an expression is not.  This is why related rates and implicit differentiation comes so smoothly, but it also means that you can talk about partial derivatives right away too.  I make sure to do this, even though it is again an aside that's not on the test, because Calculus 3 is not required for everybody who is going to use partial derivatives, after all.&lt;/p&gt;&#xA;&lt;p&gt;You can see material for my courses at &lt;a href=&quot;http://tobybartels.name/courses/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://tobybartels.name/courses/&lt;/a&gt;.  Generally the later material is more polished.  Update: For the Calculus 1/2/3 sequence specifically, you can find the most up-to-date presentation at &lt;a href=&quot;http://tobybartels.name/calcbook/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://tobybartels.name/calcbook/&lt;/a&gt;.  (It's been several years now since I've taught the applied course.)&lt;/p&gt;&#xA;" OwnerUserId="408" LastEditorUserId="408" LastEditDate="2023-03-06T23:42:00.483" LastActivityDate="2023-03-06T23:42:00.483" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="3916" PostTypeId="1" AcceptedAnswerId="11167" CreationDate="2014-07-04T21:14:43.433" Score="5" ViewCount="153" Body="&lt;p&gt;The &lt;a href=&quot;http://www.stockmarketgame.org/expteacher.html&quot;&gt;Stock Market Game&lt;/a&gt; has been a staple in mathematics and economics curricula for quite some time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a number of different skill sets that can be taught to middle school students (grades 6-8) using such a project, which I think would include basic statistics (tables of data, mean, median, percents), basic accounting principles (balance sheets), data visualization (scatter plots, bar charts), pre-algebra/basic algebra (plotting, linear fitting, very basic extrapolation), etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which other skills at the middle school level, particular dealing with such matters as &quot;number sense&quot;, can be taught using the Stock Market Game (or its generic equivalent)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The site itself seems to have some lessons, but they are only open to those teams that are actively registered for the contest, as far as I could tell.    &lt;/p&gt;&#xA;" OwnerUserId="16" LastActivityDate="2016-06-23T22:20:01.287" Title="What other skill sets for middle school students can be introduced during the Stock Market Game?" Tags="|secondary-education|applied-mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3917" PostTypeId="2" ParentId="3899" CreationDate="2014-07-04T21:53:02.937" Score="2" Body="&lt;p&gt;I'm not sure if you're asking specifically about simplifying radicals, or if you're asking a more general question, but for your radical examples, I would think that the Algebra I notion of simplifying radicals would serve here.  You need to simplify radicals so that they (a) have no radical in the denominator of a rational expression, and (b) have no factors that can be pulled out of the radical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{1}{\sqrt{256}}$ leaves a perfect square under the square root sign (and has a radical in the denominator) so it's not simplified.  That should be simplified to $\frac{1}{16}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\sqrt{512}$ leaves a perfect square factor under the square root sign so it's not simplified.  That should be simplified to $16\sqrt{2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{1}{\sqrt{512}}$ leaves a perfect square under the square root sign (and has a radical in the denominator) so it's not simplified.  That should be simplified to $\frac{1}{16\sqrt{2}}$ which must be further simplified by multiplying by $\frac{\sqrt{2}}{\sqrt{2}}$ to give $\frac{\sqrt{2}}{32}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would expect anyone in a college math course to be able to do this without breaking a sweat and without a calculator, as long as the numerical values you're giving them to work with are &quot;reasonable&quot;.  If you're giving them problems where they end up with things like $\sqrt{703423}$ to simplify without a calculator, that's not very nice.&lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-07-04T21:53:02.937" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3918" PostTypeId="2" ParentId="2279" CreationDate="2014-07-04T23:09:40.237" Score="9" Body="&lt;p&gt;Here are some tools that can be used for creating high quality Maths video lessons:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hardware&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Computer with tablet, or just a tablet computer (e.g. Microsoft Surface Pro 3).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;External microphone of a reasonably high quality. I use the Blue Yeti which seems to be a good compromise between quality and price. Previously I used a Logitech Headset which I found frustrating because it would pick up unwanted sounds, and generally made things too quiet.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Software&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Some sort of screen capturing software. Most people seem to think that &lt;a href=&quot;http://www.techsmith.com/camtasia.html&quot; rel=&quot;noreferrer&quot;&gt;Camtasia Studio&lt;/a&gt; is the best for recording and subsequent editing. If you are creating shorter videos and don't require editing there are free options around.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A program which allows you to write and draw. My preference here is OneNote as it seems to use the pressure-sensitivity of tablet computers and make handwriting look neat.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Optionally, a program to record the audio separately like &lt;a href=&quot;http://audacity.sourceforge.net/&quot; rel=&quot;noreferrer&quot;&gt;Audacity&lt;/a&gt;, and you then synchronise your recorded script with the video. This method requires decent editing software, like Camtasia Studio, mentioned above.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mathematics specific software&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://desmos.com/calculator/&quot; rel=&quot;noreferrer&quot;&gt;Desmos&lt;/a&gt; is an excellent web-based program that allows you to create graphs and animate them. This is a great way to create animations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A calculator-emulator (possibly just web-based). I've found typing 'calculator' into Google and using the built-in one is sufficient for my purposes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT: Some sample videos created with setup described above&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://vimeo.com/humberstone/dilation&quot; rel=&quot;noreferrer&quot;&gt;A video on dilation/reflection&lt;/a&gt; with animations created using Desmos and handwritten portions created in OneNote.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://vimeo.com/humberstone/unitcircle&quot; rel=&quot;noreferrer&quot;&gt;A video on the unit circle&lt;/a&gt; again with animations created using Desmos.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://vimeo.com/humberstone/sincosgraphs&quot; rel=&quot;noreferrer&quot;&gt;A video on sin and cos graphs&lt;/a&gt; showing the use of multiple video tracks, one with the Unit Circle showing and one for the graph itself. This is the first one I recorded using the Blue Yeti microphone.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://vimeo.com/humberstone/factorising&quot; rel=&quot;noreferrer&quot;&gt;An earlier video on factorising quadratics&lt;/a&gt; created just using Camtasia and OneNote. This was before I realised that better quality videos are created by recording the audio separately from the video. (The physical constraints of handwriting means that such videos feel a lot slower, whereas in other videos I generally speed up my writing to about double speed.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1873" LastEditorUserId="1873" LastEditDate="2014-07-19T02:26:54.863" LastActivityDate="2014-07-19T02:26:54.863" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3920" PostTypeId="1" CreationDate="2014-07-05T23:18:09.160" Score="8" ViewCount="26408" Body="&lt;p&gt;Are there any good examples for high school studends where: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Interquartile range is &quot;better&quot; to describe &quot;spread&quot; in an (empirical) statistical distribution of data&lt;/li&gt;&#xA;&lt;li&gt;standard deviation is a &quot;better&quot; measure and why it is used instead of interquartile range in many applications&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1203" LastEditorUserId="39" LastEditDate="2014-07-06T06:20:00.123" LastActivityDate="2014-07-31T02:07:11.480" Title="Teaching the difference between standard deviation and interquartile range" Tags="|examples|statistics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3921" PostTypeId="2" ParentId="654" CreationDate="2014-07-06T15:16:02.960" Score="2" Body="&lt;p&gt;I use a tablet PC in class - it happens to be a Surface, which has been surprisingly robust and easy to use.  I find if I use slides, I click WAY too quickly through them.  So I write the notes on my tablet (I use Windows Journal) and then post the notes at the end of class.  This way, students don't panic about getting everything I write and can focus more on understanding.  The blackboards I use for specific examples or for equations, comments, etc. that I want available throughout the lecture so I am not scrolling back to review previous comments (I hate that as an audience member.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think you need to find a medium that fits your teaching sytle - experiment but don't assume that if it works for someone else, it will work for you.  And it could be different for each class -- what works for one might not work for another.&lt;/p&gt;&#xA;" OwnerUserId="1876" LastActivityDate="2014-07-06T15:16:02.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3922" PostTypeId="2" ParentId="3920" CreationDate="2014-07-06T17:46:44.890" Score="9" Body="&lt;p&gt;&lt;strong&gt;A simple example for the IQR&lt;/strong&gt; is to consider the following two data sets:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A = {1,1,1,1,1,1,1} and B = {1,1,1,1,1,1,100000000}. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;IRQ for both is 0, but SD is &lt;em&gt;very&lt;/em&gt; different. You can argue about which is really better, but this example very nicely illustrates that the &lt;em&gt;IQR tells you where the middle 50% of the data is located&lt;/em&gt; while the &lt;em&gt;SD tells you about the spread of the data&lt;/em&gt;. It also shows that the IQR is very resistant to outliers (and to some degree skew) while the SD is not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although maybe not directly relevant to the beginning student, the standard deviation plays a central role in statistics for two reasons: it is a key factor in the central limit theorem (which explains to students why increasing the sample size of a data set should give us a &quot;better&quot; estimate for the mean) and also because it gives us the variance (which is fundamental for such methods as ANOVA, linear regression, etc). &lt;em&gt;However, one use of the standard deviation that is very important for beginning students to understand is that for the normal distribution, we actually think of the SD as a measuring stick that helps us tell how likely an even is to occur (i.e. the all important &quot;68–95–99.7 rule&quot;).&lt;/em&gt; &lt;strong&gt;An example I give to my students wrt SD&lt;/strong&gt; is to ask them to consider 2 hypothetical job offers: one is in city A, located in rural area where the salary ~ N(30K,5K) and the other is in city B, a large city where salary ~N(60K,10K). Suppose that salary offer from city A is 45K and the salary offer from city B is 60K, which should you choose? Here simply looking at how far the salary offers from the means in units of SD, tells us that city A is the better offer. &lt;/p&gt;&#xA;" OwnerUserId="1877" LastEditorUserId="1877" LastEditDate="2014-07-06T18:45:35.960" LastActivityDate="2014-07-06T18:45:35.960" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3923" PostTypeId="2" ParentId="2164" CreationDate="2014-07-06T19:03:55.090" Score="6" Body="&lt;p&gt;A friend of mine who taught college math used to give pop quizzes at the beginning of his topology class on the material which was assigned to have been read for that day's class. The questions were of a type that if a person had made a decent effort to read the material he/she could do well on the quizzes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not sure if &lt;em&gt;I&lt;/em&gt; would recommend doing this, but he claimed it worked well for him.&lt;/p&gt;&#xA;" OwnerUserId="1877" LastActivityDate="2014-07-06T19:03:55.090" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3924" PostTypeId="2" ParentId="3920" CreationDate="2014-07-07T00:53:08.580" Score="3" Body="&lt;p&gt;&lt;strong&gt;Heights and weights are roughly normal, so standard deviation&lt;/strong&gt; is more standard for them.  In those cases it's easy to translate from IQR to standard deviation by a factor of 1.35, so it's better to use the more standard number.  By contrast:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Economic data is rarely normal, so interquartile range&lt;/strong&gt; is often more useful in that area.  This is why interquartile ranges are more commonly quoted for both incomes and real estate values.  Consider:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;What is the interquartile range of New York City incomes?&lt;/em&gt;  We can calculate it reasonably well from a reasonable sample.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;What is the standard deviation of New York City incomes?&lt;/em&gt;  We'd need to know details about the highest-earning people like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do we count David Koch as a resident, even if he spends less than half the year in the city?&lt;/li&gt;&#xA;&lt;li&gt;Did Michael Bloomberg claim the NYC mayoral mansion as his primary residence for tax purposes, and how did that affect the deducibility of mortgage interest on his other properties?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The standard deviation of income depends so much on these details that it won't be as useful as the interquartile range.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-07-07T00:53:08.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3925" PostTypeId="1" CreationDate="2014-07-07T10:48:55.183" Score="7" ViewCount="297" Body="&lt;p&gt;For the purposes of this question, assume that you have a student who is aiming for a B in an introductory calculus or college algebra course but plans to take many more math classes after this. Most students in my courses match this description.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At midterm, this student will look at all his or her grades to determine which classes need more attention. If the student has a high B in the calculus or college algebra course at midterm, they are very likely to refocus their efforts on chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The end result of this is that they end up with a very low B or high C, then do &lt;strong&gt;very poorly in the subsequent class&lt;/strong&gt; (C or D) due to this refocus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can a math educator combat this phenomenon to help their students do well and continue to do well in later courses?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-07-07T19:06:19.990" Title="How can we prevent students who are &quot;doing okay&quot; from slipping?" Tags="|undergraduate-education|student-motivation|grading|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3926" PostTypeId="2" ParentId="3925" CreationDate="2014-07-07T12:11:42.110" Score="4" Body="&lt;p&gt;I think the idea would be to pry into that motivation that these student have for taking &quot;many more math classes&quot; and work some of that into your lectures.  Are the students going to become majors?  If so, present some more advance proofs and allow students to complete them for &quot;extra credit&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students are going to apply the math (and are mostly non-majors), bring some concrete examples into your class.  Show them, for example, how they can use your calculus course in calculating rates of reaction in chemistry class.  See &lt;a href=&quot;https://chemistry.stackexchange.com/questions/9242/what-are-some-good-examples-of-rate-equations-for-a-math-class&quot;&gt;this Chem.SE question&lt;/a&gt; for some additional ideas.  They will be motivated to do more work in your class &lt;em&gt;and&lt;/em&gt; in chemistry, and there will be less competition for their attention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realize you're just using chemistry for an example here, so for the non-majors, get some idea of what kind of applications they are looking for.  Do a quick SurveyMonkey (or equivalent) to get some idea of their plans and goals.  Give them quiz credit for taking it (or donuts or something).  This will keep their eyes on their future, but will also help them stay grounded in what may seem like a forest of abstract material at times.&lt;/p&gt;&#xA;" OwnerUserId="16" LastEditorUserId="-1" LastEditDate="2017-04-13T12:57:06.603" LastActivityDate="2014-07-07T19:06:19.990" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3927" PostTypeId="1" AcceptedAnswerId="3931" CreationDate="2014-07-07T14:11:01.673" Score="6" ViewCount="1897" Body="&lt;p&gt;I'm struggling to find any good &quot;problem generation&quot; software or sites. I want a program that will do the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Generate typical problems at all sorts of levels (from toddler age through to teen, say).&lt;/li&gt;&#xA;&lt;li&gt;Do this reasonably &lt;strong&gt;intelligently&lt;/strong&gt;, meaning, for instance, solutions for $x$ would actually exist when they're asked for. Think of hand-constructed problems from text books that are designed to be reasonable.&lt;/li&gt;&#xA;&lt;li&gt;Print out a long list of desired problems as a PDF file for printing purposes, &lt;strong&gt;on paper&lt;/strong&gt;. (I'm afraid I very much extremely do not like &quot;online&quot; or &quot;onscreen&quot; math problems, online / app &quot;math games&quot; and similar!)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There are any number of, say, web sites, that rather badly generate &quot;poorly thought out&quot; random problems. I want a system that will generate lists of &lt;strong&gt;good&lt;/strong&gt; problems to be done on paper. Does anyone know of an app/program/website that does this? Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="1880" LastEditorUserId="80" LastEditDate="2014-07-07T19:58:46.183" LastActivityDate="2014-07-09T10:08:29.260" Title="Arithmetic / Algebra problem generation software?" Tags="|textbooks|algebra|problem-design|software|aids|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3928" PostTypeId="2" ParentId="2164" CreationDate="2014-07-07T16:20:29.237" Score="9" Body="&lt;p&gt;I do something at least mildly annoying: I make 10% of their class grade be showing up each week with the examples from the sections of the book we will cover that week written out by hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, if in week 3 I will be covering sections 4.1, 4.2, and 4.3, then at the beginning of class that week they show up with a stack of paper on which they have copied all the examples in the text from sections 4.1, 4.2, and 4.3 by hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a lot of work, but not an undoable amount for most textbooks, and for most students it's the first time they actually read the text. I get lots of positive feedback about this, despite how tedious it sounds.&lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-07-07T16:20:29.237" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3929" PostTypeId="2" ParentId="3899" CreationDate="2014-07-07T16:32:21.003" Score="4" Body="&lt;p&gt;First some general thoughts: I could never understand the obsession with no radicals in the denominator.Those situations where it is worth one's time to &quot;simplify&quot; such a thing seem remarkably rare. I feel similarly about many other &quot;simplifications.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, demanding &quot;simplification&quot; in all cases seems very often to lead to rather irrational beliefs -- I can't tell how many students have told me that you can't divide by $\sqrt{2}$, for example. I can only surmise they feel this way because they had someone who blew a gasket every time they tried to leave a $\sqrt{2}$ in the denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So that's that. For solutions, I ask (often) for an estimate at the end of the problem. That is, if I think students might try to leave their answer in the form $\frac{\sqrt{4}-3}{5}$, then I append and extra part to the problem that asks something like &quot;is your answer closer to -2 or to 1?&quot; Simplification then happens naturally and for the right reason -- so you can get a better idea of where your answer lies. &lt;/p&gt;&#xA;" OwnerDisplayName="user614" LastActivityDate="2014-07-07T16:32:21.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3930" PostTypeId="1" CreationDate="2014-07-07T18:27:05.053" Score="6" ViewCount="207" Body="&lt;p&gt;I'm doing some research about cooperative learning in vector calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems like what cooperative learning in calculus is referred to varies over time.  In 1987, there was an MAA book, Calculus for a New Century: A Pump, Not a Filter that used the term &quot;calculus reform&quot;.  In the early 90s there was a lot of talk about a &quot;laboratory approach.&quot;  I found a 1991 MAA book, the Laboratory Approach to Teaching Calculus, a 1993 MAA book Learning by Discovery: A Lab Manual for Calculus.   Then in the mid to late 90s people were referring to &quot;cooperative learning.&quot;   Then there's a 1995 MAA book called A Practical Guide to Cooperative Learning in Collegiate Mathematics and a 1997 MAA book called Readings in Cooperative Learning for Undergraduate Mathematics.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More specifically to vector calculus, Dray and Manogue (&lt;a href=&quot;http://k12s.phast.umass.edu/stemtec/pathways/Proceedings/Papers/Dray-p.pdf&quot; rel=&quot;nofollow&quot;&gt;http://k12s.phast.umass.edu/stemtec/pathways/Proceedings/Papers/Dray-p.pdf&lt;/a&gt;) write about using cooperative groups to bridge the gap between vector calculus and physics.   But they don't seem to cite any of the work from the 90s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So then the questions I have are - Who else besides Dray and Manogue are using cooperative groups in vector calculus (or in calculus in general)?  What are those that are using cooperative groups in calculus currently calling it?  (Is there a new term besides &quot;calculus reform&quot;, &quot;laboratory,&quot; or &quot;cooperative learning&quot;?)   How does whatever is being done in the present connect to the work in the 80s and the 90s?    &lt;/p&gt;&#xA;" OwnerUserId="1798" LastEditorUserId="376" LastEditDate="2016-08-11T11:04:09.917" LastActivityDate="2016-08-11T11:04:09.917" Title="How is cooperative learning being used in vector calculus, and what are the origins of this work?" Tags="|curriculum|mathematical-pedagogy|vector-calculus|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3931" PostTypeId="2" ParentId="3927" CreationDate="2014-07-07T19:34:53.727" Score="4" Body="&lt;p&gt;It looks like there's a lot of moderately useful, or partially coded things out there but nothing that's really comprehensive and does fully what you want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of the websites that generate PDFs, this one looked reasonably customizable - each of their worksheets had a number of different settings to control what ended up on it:  &lt;a href=&quot;http://www.math-aids.com/&quot; rel=&quot;nofollow&quot;&gt;http://www.math-aids.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's some source code for a math worksheet generator (under a CC-NC-SA license) that might be a good starting point for coding your own.   Apparently the author had a coworker with a kid who could do math worksheets faster than the coworker could write them, so he made a program to generate them:&#xA;&lt;a href=&quot;https://github.com/davmillar/lasagnadothtml&quot; rel=&quot;nofollow&quot;&gt;https://github.com/davmillar/lasagnadothtml&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't actually find the webapp anywhere, but this grad student claims to have written a worksheet generator webapp, so you could maybe talk to him about what he was working on:&#xA;&lt;a href=&quot;http://math.berkeley.edu/~alvin/resume-tech.pdf&quot; rel=&quot;nofollow&quot;&gt;http://math.berkeley.edu/~alvin/resume-tech.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here's a prototype math worksheet generator from Microsoft, although it is no longer supported by Microsoft.  You would give it a particular type of problem and it would generate more of the same:&#xA;&lt;a href=&quot;http://www.pil-network.com/Resources/Tools/Details/852875ce-b376-4b49-8f79-41c5cd75b067&quot; rel=&quot;nofollow&quot;&gt;http://www.pil-network.com/Resources/Tools/Details/852875ce-b376-4b49-8f79-41c5cd75b067&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-07-07T19:34:53.727" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3932" PostTypeId="2" ParentId="3927" CreationDate="2014-07-07T20:37:18.707" Score="9" Body="&lt;p&gt;There are a number of resources I like out there.  I tend to use a variety of different sites for different needs -- I don't think there is one that &quot;does it all&quot; but here are some of my favorites.  Most of them &lt;em&gt;are&lt;/em&gt; pretty random, though the first few below that have lots of customization options tend to make very usable worksheets, and for some applications the others are great as well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://homeschoolmath.blogspot.com/search/label/worksheets&quot;&gt;http://homeschoolmath.blogspot.com/search/label/worksheets&lt;/a&gt; has a number of links to individual worksheet generators, each of which has a number of useful parameters you can set to customize the worksheets to your needs.  For example, her generator for basic operations worksheets is here: &lt;a href=&quot;http://www.homeschoolmath.net/worksheets/basic-operations-worksheets.php&quot;&gt;http://www.homeschoolmath.net/worksheets/basic-operations-worksheets.php&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://themathworksheetsite.com/&quot;&gt;http://themathworksheetsite.com/&lt;/a&gt; offers several very nice worksheet generators that you can use for free, plus others that are available by paid subscription.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math-aids.com/&quot;&gt;http://www.math-aids.com/&lt;/a&gt; offers worksheets at all levels, with a good amount of customization.  It says through grade 12, but I don't see much beyond Algebra 1 / Geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.kidzone.ws/math/index.htm&quot;&gt;http://www.kidzone.ws/math/index.htm&lt;/a&gt; offers some pre-made and some generated worksheets on topics for elementary school mathematics.  They're cute with a little graphic added and laid out nicely.  They also offer some word problems that you can re-generate with different numbers plugged in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.worksheetworks.com/math.html&quot;&gt;http://www.worksheetworks.com/math.html&lt;/a&gt; offers generators for a number of topics.  Some of the topics, like Geometry and Probability also have some well designed problem solving worksheets available (one well-designed problem, not auto-generated).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.kutasoftware.com/free.html&quot;&gt;http://www.kutasoftware.com/free.html&lt;/a&gt; offers free sample worksheets for high school level courses and sells software (that I haven't tried) for creating more.  The free samples are pretty basic, but the features of the paid software sound interesting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another algebra worksheet generator: &lt;a href=&quot;http://www.math.com/students/worksheet/algebra_sp.htm&quot;&gt;http://www.math.com/students/worksheet/algebra_sp.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This one has pre-made worksheets for high school math through trig, but they are not generated &lt;a href=&quot;http://www.mathworksheetsgo.com/&quot;&gt;http://www.mathworksheetsgo.com/&lt;/a&gt;  This one is also pre-made worksheets, but there are several instances of each sheet with different numbers.  &lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-07-07T20:37:18.707" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3933" PostTypeId="1" AcceptedAnswerId="3934" CreationDate="2014-07-08T02:59:35.233" Score="15" ViewCount="1974" Body="&lt;h2&gt;Pre-algebra&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;If the student is taking this branch of mathematics, they are expected to show their work because they're expected to solve specific problems in a certain way. Ex, when they're solving for a variable they're supposed to manually find the value of x by isolating it rather than entering the left and right hand of the equal sign in the calculator (in slope-intersect form) and finding the intersection point. If they only get the correct answer without showing work, we give them 2 points. 8 more points if work is shown.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Algebra&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Students in algebra are expected to know how to get the correct answer. We don't care how they do it, as long as it's the correct answer. Full credit is given for the correct answer, basically.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Are we being too &quot;harsh&quot; on our pre-algebra students? Should we let them get full credit for the correct answer or should we make sure they're ready for algebra with our current system?&lt;/p&gt;&#xA;" OwnerUserId="1885" LastEditorUserId="52" LastEditDate="2014-07-23T09:13:27.540" LastActivityDate="2016-12-01T23:37:31.863" Title="Should students get full credit for getting the correct answer (without work)?" Tags="|primary-education|algebra|" AnswerCount="12" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="3934" PostTypeId="2" ParentId="3933" CreationDate="2014-07-08T09:59:34.283" Score="25" Body="&lt;p&gt;Personally, I'd say &quot;no.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students who don't write out the steps in their algebra classes appear in calculus thinking that they should be able to write down all the answers without any intermediate steps. I even have some students in Calculus 2 who think that there is some kind of value in not writing down the steps. None of these students can complete any calculus problems without errors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd prefer that in all math classes, student submitted solutions &lt;strong&gt;should be solutions&lt;/strong&gt;: I should be able to read them and be &lt;em&gt;convinced&lt;/em&gt; that the final answer is correct. &quot;x = 7&quot; out of nowhere isn't convincing anyone, and I don't see it as a useful submission.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the earlier this standard becomes expected, the better.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-07-08T09:59:34.283" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3935" PostTypeId="2" ParentId="3933" CreationDate="2014-07-08T11:08:42.883" Score="2" Body="&lt;p&gt;I'll agree with what's been said, and say no.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Showing the work means showing that you've grasped the concept itself. Being able to produce the answer is more often than not the easy part. Often times, the answer is immediately apparent when reading the question, and one might be tempted to &quot;show off&quot; and simply state that &lt;strong&gt;I can&lt;/strong&gt; tell you the answer without doing any work, and hope that the reader will be impressed by this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It &lt;em&gt;is&lt;/em&gt; impressive. It means that you've gained a very intuitive understanding of the concept. But with that, you should be able to tell the reader exactly why it is so intuitive and obvious. That's the &lt;strong&gt;really&lt;/strong&gt; impressive part, and therefore should be the thing that awards full credit.&lt;/p&gt;&#xA;" OwnerUserId="796" LastActivityDate="2014-07-08T11:08:42.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3936" PostTypeId="2" ParentId="2262" CreationDate="2014-07-08T13:20:47.530" Score="8" Body="&lt;p&gt;&lt;strong&gt;Literature:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, I have never seen a study that compares these two approaches to teaching Calculus (nor did a cursory search through ProQuest and Google Scholar produce anything of import). I expect a reason for this is that your terminology is nonstandard. The &lt;em&gt;late transcendentals&lt;/em&gt; approach is somewhat akin to what might be done in a (U.S.) course on pre-Calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did see at least one study that might be of interest to you in this regard; essentially, it introduces ideas around finding &quot;turning points&quot; for polynomials and rational functions, and later on connects these ideas to finding derivatives for the polynomials.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Cherkas, B. (2003). Finding polynomial and rational function turning points in precalculus. &lt;em&gt;International journal of computers for mathematical learning, 8&lt;/em&gt;(2), 215-234.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Excerpt:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In today’s typical precalculus course, students are informed that a polynomial&#xA;  of degree $n$ has at most $n−1$ turning points. Representative graphs&#xA;  of polynomial functions are also given to help students recognize turning&#xA;  points. Yet it is only for quadratic functions that students are asked to find&#xA;  turning points (vertices), using the technique of completing the square. The&#xA;  general topic of finding turning points is left to calculus, where students&#xA;  learn to compute derivatives, set them equal to zero, and solve the resulting&#xA;  equations. In the case of polynomial and rational functions, however, it&#xA;  is unnecessary to wait for calculus. The purpose of this snapshot is to&#xA;  describe procedures that rely on the Fundamental Theorem of Algebra,&#xA;  spatial reasoning, and solving equations to find turning points for any&#xA;  polynomial or rational function with real coefficients.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The article continues a bit later on:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To gain insight into what happens behind the scenes when using this&#xA;  method, we first examine general cubic functions, obtaining a complete&#xA;  classification for the number and location of turning points in terms of the&#xA;  cubic’s coefficients. With the understanding gained from this investigation,&#xA;  we link this algebraic approach for finding turning points to derivatives&#xA;  of polynomials. We then illustrate these techniques with the assistance of&#xA;  &lt;em&gt;Mathematica&lt;/em&gt; to find all turning points for specific polynomials of degrees&#xA;  $4$ and $5$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Major drawbacks of the study are, first, that it incorporates technology as a major part (which is not something you were asking about), and, second, the &lt;em&gt;Mathematica&lt;/em&gt; approach that is discussed for a laboratory portion of the course was not explicitly connected to the classroom-based learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, the author might agree with your skepticism about the different ways Calculus is initially broached, though perhaps he would suggest directing some of the &lt;em&gt;late&lt;/em&gt; approach's material to what would be learned prior to a first course in Calculus. As he writes in the discussion section:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In any precalculus study of turning points, it is natural to include the&#xA;  related concepts of local and global maxima and minima. These visually rich ideas add to the overloading of calculus with concepts that can be&#xA;  introduced earlier in a revised precalculus curriculum. Presenting these&#xA;  concepts in advance of differential calculus introduces a spiral effect to&#xA;  the learning curve: students first see the concepts in the limited perspective&#xA;  of precalculus and subsequently revisit them in the more general context&#xA;  of calculus. Introducing optimization problems in precalculus that are&#xA;  modeled by polynomial or rational functions is part of this spiral approach.&#xA;  It also provides students with realistic precursors of the specific cognitive&#xA;  demands they will encounter in calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My own thoughts:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One difficulty in comparing the approaches you have described is that it is not clear (to me) what &lt;em&gt;success&lt;/em&gt; would look like. Your example suggestions are &quot;higher pass rates&quot; and &quot;success rates in courses which depend on calculus.&quot; For the former suggestion: I think it is clear (perhaps with some thought) that this will not be a good metric. For the latter suggestion: Calculus students are remarkably diverse with regard to their goals. Are they intending to follow differential Calculus with integral Calculus? Are they intending to take Physics or Engineering courses? Are they satisfying pre-med requirements? Are they satisfying general education requirements? Etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Allow me to go out on a limb and suggest that the &lt;em&gt;fundamental&lt;/em&gt; (the word choice is no accident...) goal of learning Calculus is to make sense of the Fundamental Theorem of Calculus. In particular, I think the goal must be for students to see the relation between differentiation and integration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For polynomials (say, with coefficients in $\mathbb{R}$ or $\mathbb{Q}$), they have the nice property that differentiating and integrating is not so tough: The derivative of such a polynomial is a polynomial; the antiderivative of such a polynomial gives a family of polynomials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But you are suggesting &lt;em&gt;rational functions&lt;/em&gt;, too, and now we will have a problem. In fairness, you are asking about an approach to teaching &quot;differential Calculus&quot;; but I think it is nice to have an earlier familiarity with the more commonly appearing functions so that, when the time for finding antiderivatives comes around, some of the cognitive load can be reduced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question of &lt;em&gt;which&lt;/em&gt; functions should be covered is a difficult one, particularly with regard to finding antiderivatives. Even after going through the full Calculus sequence, a student might rightfully ask: What is the antiderivative of $\sin(x)/x$? What is the antiderivative of $x\tan x$? By a theorem of Liouville, these antiderivatives cannot be expressed in elementary functions; cf. &lt;a href=&quot;https://math.stackexchange.com/q/694615/37122&quot;&gt;&lt;strong&gt;former&lt;/strong&gt;&lt;/a&gt; and &lt;a href=&quot;https://mathoverflow.net/q/108598/22971&quot;&gt;&lt;strong&gt;latter&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And so there will be picking and choosing when it comes to which functions are to be introduced, and to which derivatives or antiderivatives are to be taken. Given the nature of the AP examinations and the subsequent courses in Calculus in most mathematics departments, I think it wise to introduce at least the functions mentioned in what you call the &lt;em&gt;early&lt;/em&gt; approach. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For students who do not intend to take the AP examination (or an equivalent one) and departments that can organize themselves around this nonstandard approach - perhaps they have a different idea of the fundamental goal of learning Calculus? - it would be interesting to see whether lightening the cognitive load early on will pay dividends later, or whether introducing bits and pieces earlier on will make their re-appearance later more palatable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In most situations, I would advocate sticking to the &lt;em&gt;early&lt;/em&gt; approach. With regard to theory: It would be interesting to compare the two (perhaps such studies exist and I am just unaware of the terminology!). With regard to practice: For a mathematics educator who is interested in the &lt;em&gt;late&lt;/em&gt; approach, perhaps its inclusion in a pre-Calculus course would make for a better fit.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-07-08T13:20:47.530" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3937" PostTypeId="1" CreationDate="2014-07-08T14:09:55.717" Score="6" ViewCount="174" Body="&lt;p&gt;I am looking for:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;an undergraduate research program&lt;/li&gt;&#xA;&lt;li&gt;in mathematics/theoretical physics&lt;/li&gt;&#xA;&lt;li&gt;offered online (e.g. via skype or something)&lt;/li&gt;&#xA;&lt;li&gt;given by a good institution&lt;/li&gt;&#xA;&lt;li&gt;which can be followed while attending another university&lt;/li&gt;&#xA;&lt;li&gt;which provides material for further study relevant to research&lt;/li&gt;&#xA;&lt;li&gt;which lasts for a minimum of two months&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The last two criteria are not strictly necessary.  Does such a program exist?&lt;/p&gt;&#xA;" OwnerUserId="1888" LastEditorDisplayName="user173" LastEditDate="2014-07-08T15:41:33.407" LastActivityDate="2014-07-08T15:41:33.407" Title="Undergraduate maths research" Tags="|undergraduate-education|education-research|undergraduate-research|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3938" PostTypeId="1" CreationDate="2014-07-08T17:48:43.713" Score="15" ViewCount="1312" Body="&lt;p&gt;This is a soft question perhaps not well suited for the format of the site but I'm interested to hear opinions from this community on this topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;K-12 mathematics textbooks (understandably) divide their content into chapters. My concern is that the level of compartmentalization from chapter to chapter is so extreme that it hinders the development of real mathematical thinking, and instead fosters the development of what I call flow-chart mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most discouraging example I know comes from the texts used in grades 7 through 9 in my own Atlantic Canadian province. Each book contains a chapter dedicated to &quot;Real Numbers&quot; where students learn to do arithmetic with fractions. In each book this chapter is followed by another called &quot;Measurement&quot;, where children work on problems to do with perimeter, (surface) area, and volume. The thing that's shocking to me is that no problems in the Measurement chapter involve objects with fractionally labelled lengths!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe this is an error, and I sometimes try to address it when working with kids this age. After establishing the prerequisite skills, I'll draw a rectangle that's $4$m by $\frac{2}{3}$m and ask them to calculate the area. A surprising number of students who are capable of moving from $4 * \frac{2}{3}$ to $\frac{8}{3}$, and who are also able to explain to me how and why the area of a rectangle is equal to its length times its width, still have substantial difficulty putting these skills together in order to find the area of this given rectangle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that this reveals that they don't understand that the multiplication of fractions that they've learned in the previous chapter is the same operation as the familiar multiplication of integers. Rather, they've learned a flow-chart to follow when they come across $\frac{a}{b}*\frac{c}{d}$. This, I think, is a huge problem!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to hear arguments for and against this level of compartmentalization, and if possible, examples of situations where it's particularly beneficial or harmful.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2015-11-25T22:50:44.397" Title="Cost and benefits of compartmentalization in k-12 curriculum" Tags="|secondary-education|textbooks|curriculum|primary-education|mathematical-pedagogy|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3939" PostTypeId="1" AcceptedAnswerId="3940" CreationDate="2014-07-08T20:37:18.717" Score="5" ViewCount="7856" Body="&lt;p&gt;Many people on this site refer to undergraduate courses called &quot;Calculus I&quot; or &quot;Calculus II&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my country (Australia) there is no standard naming convention for university maths courses, and courses with the same name at different universities have different topics. For example, at our University the first-year courses are called &quot;Mathematics 1A&quot; and &quot;Mathematics 1B&quot; and contain both linear algebra and calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a standard list of topics in courses with these names in the USA, and if so, what are these lists of topics?&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-07-10T19:32:59.697" Title="Naming of calculus courses" Tags="|undergraduate-education|curriculum|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3940" PostTypeId="2" ParentId="3939" CreationDate="2014-07-08T21:47:58.900" Score="8" Body="&lt;p&gt;You can get a good idea of American standards by looking at the most popular calculus textbooks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An Amazon search for &quot;&lt;a href=&quot;http://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&amp;amp;field-keywords=calculus%20textbook&quot;&gt;calculus textbook&lt;/a&gt;&quot; suggests Larson &amp;amp; Edwards and Stewart as the most popular.  They largely agree on the chapter titles and their ordering.  If we divide each book into three parts, the agreement between parts is nearly complete.  Here's what we get, which corresponds to my experience of these courses:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Calculus I&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Preparation for Calculus&lt;/li&gt;&#xA;&lt;li&gt;Limits&lt;/li&gt;&#xA;&lt;li&gt;Derivatives&lt;/li&gt;&#xA;&lt;li&gt;Applications of differentiation&lt;/li&gt;&#xA;&lt;li&gt;Integrals&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Calculus II&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Applications of integration&lt;/li&gt;&#xA;&lt;li&gt;Transcendental functions&lt;/li&gt;&#xA;&lt;li&gt;Techniques of integration&lt;/li&gt;&#xA;&lt;li&gt;Differential equations&lt;/li&gt;&#xA;&lt;li&gt;Polar coordinates and parametric equations&lt;/li&gt;&#xA;&lt;li&gt;Infinite sequences and series&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Calculus III&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Vectors and the geometry of space&lt;/li&gt;&#xA;&lt;li&gt;Vector functions&lt;/li&gt;&#xA;&lt;li&gt;Partial derivatives&lt;/li&gt;&#xA;&lt;li&gt;Multiple integrals&lt;/li&gt;&#xA;&lt;li&gt;Vector analysis&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-07-08T21:47:58.900" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3941" PostTypeId="2" ParentId="2691" CreationDate="2014-07-08T23:01:08.147" Score="4" Body="&lt;p&gt;Suppose that $x_1,\dots, x_5$ are numbers such that $x_1\le x_2\le x_3\le x_4\le x_5$ and $x_1+x_2+x_3+x_4+x_5=50$. Prove that $x_1+x_2\le 20$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof: We consider the cases $x_2\le 10 $ and $x_2&amp;gt;10$. If $x_2\le 10$ then the conclusion is obvious. If $x_2&amp;gt;10$ then $x_3+x_4+x_5&amp;gt;30$ so $x_1+x_2&amp;lt;20$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not so easy to prove this without cases.&lt;/p&gt;&#xA;" OwnerUserId="1889" LastActivityDate="2014-07-08T23:01:08.147" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3942" PostTypeId="2" ParentId="2691" CreationDate="2014-07-09T06:31:57.180" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;How about the proof $f(x)=2x^2+x+1$ has no zero in $\mathbb{Z}/3\mathbb{Z}$. Using modular arithmetic with $n=3$,&#xA;$$ f(0)=1, \ \ \ f(1)=2+1+1=1, \ \ \ f(2) = 2(4)+2+1=2.$$&#xA;thus, $f(x)=2x^2+x+1$ has no zero in $\mathbb{Z}/3\mathbb{Z}$. &lt;/li&gt;&#xA;&lt;li&gt;Another stub of an idea I had: Truth table arguments involve multiple cases. The proof technique is perhaps not so exciting, so perhaps you could combine the truth table proof with the method for finding a minimal expression which represents the logic. In particular, show them &lt;a href=&quot;http://en.wikipedia.org/wiki/Karnaugh_map&quot; rel=&quot;nofollow&quot;&gt;Karnaugh Maps&lt;/a&gt;. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-07-09T06:31:57.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3943" PostTypeId="2" ParentId="3933" CreationDate="2014-07-09T09:58:31.813" Score="2" Body="&lt;p&gt;It depends on the mark allocation but in my opinion I would say no. I've seen before that a learner somehow arrives at the correct answer in homework but none of the work is correct. At the end of the day we are assessing a number of things, not just the answer. A learner must demonstrate the appropriate skills and knowledge when answering questions. &lt;/p&gt;&#xA;" OwnerUserId="1891" LastActivityDate="2014-07-09T09:58:31.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3944" PostTypeId="2" ParentId="3927" CreationDate="2014-07-09T10:08:29.260" Score="1" Body="&lt;p&gt;Here in South Africa we have a very useful program called MathsBuddy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;MathsBuddy South Africa is a high quality, independent online maths tutoring programme based on the South African curriculum for Grades 1 to 12. MathsBuddy:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;was developed by experienced teachers,&lt;/li&gt;&#xA;&lt;li&gt;features over 1,800 animated and narrated (full audio-visual) maths lessons and more than 65 000 interactive questions,&lt;/li&gt;&#xA;&lt;li&gt;includes Advanced Programme Maths (AP Maths), &lt;/li&gt;&#xA;&lt;li&gt;offers relevant content for international curricula,&lt;/li&gt;&#xA;&lt;li&gt;available on supported tablets and other mobile devices (Android and Apple platforms),&lt;/li&gt;&#xA;&lt;li&gt;offers a speed skills game to enhance arithmetic speed and accuracy.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I know that MathsBuddy also has platforms in other countries so perhaps they have one for your country? Go to www.mathsbuddy.co.za where they have 150 free lessons that you can try. I currently have 100 of my learners registered on MathsBuddy and it is by far the most helpful program I have worked with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I encourage all math teachers from all grades to go take a look at this website. It's easy to use, inexpensive (R150 a year here) and your kids will love it.&lt;/p&gt;&#xA;" OwnerUserId="1891" LastActivityDate="2014-07-09T10:08:29.260" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3945" PostTypeId="1" CreationDate="2014-07-09T12:26:56.110" Score="12" ViewCount="1212" Body="&lt;p&gt;My name is Tammy and I am a high school math teacher in South Africa. In 2012, SA was ranked last IN THE WORLD wrt to math and in 2013 we were ranked second last. Despite the fact that  R253.8 billion (about $23.6 billion) was spent on education this year, we still have mud schools, ghost schools, a lack of textbooks, training and even the basics like water, electricity, desks and sometimes even teachers. Heck, two months ago a little girl died at school when she fell into the long drop (school couldn't afford normal plumbing).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I ask you this, what can be done to improve the standard of mathematics in my country. Can you think of any initiatives or ways of helping learners?&lt;/p&gt;&#xA;" OwnerUserId="1891" LastEditorUserId="2065" LastEditDate="2014-08-29T12:19:23.250" LastActivityDate="2014-08-29T12:19:23.250" Title="What can be done to improve the standard of mathematics in South Africa?" Tags="|student-motivation|africa|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3946" PostTypeId="1" AcceptedAnswerId="3947" CreationDate="2014-07-09T14:48:53.487" Score="4" ViewCount="102" Body="&lt;p&gt;Forgive me if this is off topic, but I've been looking for an online resource that I can use to polish up and improve my math skills. I've looked at math.com and purplemath.com but they don't have much past algebra. I want to know if there is a site that contains &quot;complete&quot; coursework for Trigonometry, Calculus, and possibly Statistics. &lt;/p&gt;&#xA;" OwnerUserId="1893" LastEditorUserId="47" LastEditDate="2014-07-09T17:05:15.890" LastActivityDate="2014-07-13T18:56:05.590" Title="Looking for an online mathematics practice resource" Tags="|undergraduate-education|reference-request|self-learning|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3947" PostTypeId="2" ParentId="3946" CreationDate="2014-07-09T17:02:39.507" Score="1" Body="&lt;p&gt;I believe &lt;a href=&quot;http://www.khanacademy.org/&quot; rel=&quot;nofollow&quot;&gt;Khan Academy&lt;/a&gt; is what you're looking for. As far as introductory mathematics university courses go, it has plenty of resources. Certainly it covers Trigonometry, Geometry and Calculus well enough.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-07-09T17:02:39.507" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3948" PostTypeId="2" ParentId="1817" CreationDate="2014-07-09T19:45:39.170" Score="7" Body="&lt;p&gt;&lt;strong&gt;Mary Somerville&lt;/strong&gt; (1780-1872) was a self-taught mathematician and an expert on theoretical astronomy. The &lt;a href=&quot;http://en.wikipedia.org/wiki/Dictionary_of_National_Biography&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;Dictionary of National Biography&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt; (London, 1897) described her as 'the most remarkable woman of her generation'. See this &lt;a href=&quot;http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1996JBAA..106..201B&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; about her.&lt;/p&gt;&#xA;" OwnerUserId="1892" LastActivityDate="2014-07-09T19:45:39.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3949" PostTypeId="2" ParentId="3938" CreationDate="2014-07-10T00:42:59.267" Score="10" Body="&lt;p&gt;The compartmentalization drives me crazy.  I'll have students who can use the distributive law perfectly when we're on the &quot;distributive law&quot; chapter in the book.  But if I later give them a problem to solve 37 x 40 + 37 x 48 + 37 x 12 in another context, they will do the 3 multiplications out by hand and never notice that that could have been simplified to 37 x 100 using the distributive law.  Likewise, if I give them that problem they will ask me &quot;Do we have to use order of operations?&quot; like that's some special thing you only do when you're explicitly asked to.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the middle school level, I try to bring in problems from sources like MATHCOUNTS to expose the kids both to problem solving strategies, but also to using the math they know even when no one told them which topic was going to be exercised.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I try to emphasize (with limited success) that the things we learn are &lt;em&gt;tools&lt;/em&gt; that they can use for all sorts of problems going forward, and not just tricks they have to learn to pass the chapter test.  And I do &lt;strong&gt;try&lt;/strong&gt; to back that up by bringing them problems that use prior chapters' (and years') skill sets.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I can't cite you any research or anything, but my experience tells me that this compartmentalization is bad for students, and contributes to misconceptions as to what mathematics is really all about!&lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-07-10T00:42:59.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3950" PostTypeId="1" CreationDate="2014-07-10T09:33:49.217" Score="5" ViewCount="5771" Body="&lt;p&gt;Can you suggest me some &lt;strong&gt;&lt;em&gt;REALLY&lt;/em&gt;&lt;/strong&gt; hard books on calculus and analysis. By hard I don't mean difficult in explanations, but with &lt;strong&gt;&lt;em&gt;extremely&lt;/em&gt;&lt;/strong&gt; challenging exercises (all worked out if possible) and useful insights and tricks. Also, I would like you to share similarly defined books (or handouts) about other disciplines in an undergraduate course (that is to say, the subjects mentioned &lt;a href=&quot;https://www.ocf.berkeley.edu/~abhishek/chicmath.htm&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="1888" LastEditorUserId="77" LastEditDate="2017-12-13T15:53:42.347" LastActivityDate="2022-10-31T12:32:53.063" Title="Extremely &quot;hard&quot; books (or handouts) for undergrad studies" Tags="|undergraduate-education|reference-request|reference-books|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3951" PostTypeId="2" ParentId="780" CreationDate="2014-07-10T10:13:52.357" Score="0" Body="&lt;p&gt;The Tower of Hanoi is a classic puzzle which has a lot of mathematical aspects: &#xA;recurrence relations, combinatorics, binary number system... It can be analyzed using Graph Theory... Also the classical Peg Solitaire can be analyzed using Group Theory.&lt;/p&gt;&#xA;" OwnerUserId="1892" LastActivityDate="2014-07-10T10:13:52.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3952" PostTypeId="1" CreationDate="2014-07-10T10:18:11.633" Score="19" ViewCount="1844" Body="&lt;p&gt;Most Calculus courses/textbooks I have seen teach the different topics in that order: limits of functions, continuity, differentiability, integration. Then, depending on the teacher/textbook's preference, sequences, series, power series, differential equations, curves are taught in various orders.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to discuss the following variation: what are the pros and cons of teaching first limits of sequences (I assume that generalities about series - definitions $a_n=f(n)$ or $a_n=f(a_{n-1})$, increasing, decreasing sequences,... - is part of a preCalculus class but it could be included here) and then limits of functions, continuity, differentiability, integration,...?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pros&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I think many students have difficulties to understand all the various $\epsilon-\delta$ definitions of limits of functions. For a function $f(x)$, you need to understand the different meanings of the locution &quot;is close to&quot; for the input $x$ (close to a point $a$, from the right $a^+$, close from the left, &quot;close to $\infty$&quot;=&quot;large enough&quot; and &quot;close to $-\infty$&quot;=&quot;negatively large enough&quot;), as well as for the output $f(x)$. I found that some students are confused by all these possibilities. For a sequence $\{a_n\}$, everything simplifies for the input since $n$ can only tend to $\infty$. Thus, studying sequences first allows to slow down the pace of learning $\epsilon-\delta$ definitions by considering only the output first.&lt;/li&gt;&#xA;&lt;li&gt;I find that sequences and the Sequential Characterization of Limits (one of my favorite theorem :D) are very convenient to prove that a limit of a function does not exist. For example, showing that $\sin$ has no limit at $\infty$ is a piece of cake with the SCL. It can also be used for functions of several variables and I think it is simpler to write why a given function has no limit with the SCL than by considering limits along a curve.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cons&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The only disadvantage I see with this order of topics is that you probably need to spend time reviewing sequences before starting series. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My question: are there any other advantages and disadvantages in starting a Calculus course by studying limits of series?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Subsidiary question: any reference for a textbook following this particular order?&lt;/p&gt;&#xA;" OwnerUserId="1538" LastActivityDate="2018-07-24T14:46:09.513" Title="Teaching limits of sequences before limits of functions in Calculus?" Tags="|calculus|sequences|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3953" PostTypeId="2" ParentId="3952" CreationDate="2014-07-10T10:52:50.803" Score="2" Body="&lt;p&gt;According to me, another disadvantage of this approach is that limits of functions have a visual aspect because of the graphic representation of functions which can help to understand concepts.&lt;/p&gt;&#xA;" OwnerUserId="1892" LastActivityDate="2014-07-10T10:52:50.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3954" PostTypeId="1" CreationDate="2014-07-10T12:29:22.057" Score="7" ViewCount="259" Body="&lt;p&gt;The purpose of this question is to ask for your experiences using PISA released questions at classroom. How do you use them to improve the mathematical competencies of your students? How do you implement assessment of mathematical competencies when working with this kind of questions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PISA is a program of the OECD. It is an acronym for 'Programme for International Student Assessment'. It is a triennial international survey which aims to evaluate education systems worldwide by testing the skills and knowledge of 15-year-old students. It develops tests which are not directly linked to the school curriculum. The tests are designed to assess to what extent students at the end of compulsory education, can apply their knowledge to real-life situations and be equipped for full participation in society. Fifteen-year-old students from randomly selected schools worldwide take tests in these subjects: reading, mathematics and science. There are also optional assessments (optional for the country which decides to participate in PISA) in Problem Solving and Financial Literacy. The students and their school principals also answer questionnaires to provide information about the students' backgrounds, schools and learning experiences and about the broader school system and learning environment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find PISA released questions and more information in this link:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.oecd.org/pisa/pisaproducts/pisa-test-questions.htm&quot; rel=&quot;noreferrer&quot;&gt;http://www.oecd.org/pisa/pisaproducts/pisa-test-questions.htm&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="1892" LastEditorUserId="1892" LastEditDate="2014-07-11T15:28:50.223" LastActivityDate="2014-07-21T04:00:04.617" Title="How can PISA released questions be used to improve the mathematical competencies of students?" Tags="|secondary-education|assessment|pisa|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3955" PostTypeId="2" ParentId="3945" CreationDate="2014-07-10T13:14:11.027" Score="6" Body="&lt;p&gt;At the recommendation, I've converted my comment into an answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It'll be hard without adequate government support, which is quite the political issue as it deals with funding. I'm inclined to say that an individual teacher cannot affect much without strong support. The only thing you could do is work on teacher training with your peers, but poverty (sadly) has been shown to be a massive disadvantage in education. You could start looking into works of Servaas van der Berg, one that I found being &lt;a href=&quot;http://unesco.atlasproject.eu/unesco/file/7ccb1dde-ab76-4631-94de-2f5f7088c0cf/c8c7fe00-c770-11e1-9b21-0800200c9a66/181754e.pdf&quot;&gt;Poverty and Education&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-07-10T13:14:11.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3956" PostTypeId="2" ParentId="3938" CreationDate="2014-07-10T13:22:01.203" Score="4" Body="&lt;p&gt;I agree with you: this is a huge problem and to avoid it, mathematics should be taught in a way completely different that the compartmentalized approach given by textbooks. But, unfortunately, this is the way that our current society wants mathematics to be taught. At least in my country is that way: not only textbooks have this completely compartmentalized structure, but official curriculum tends to be much more compartmentalized every day. Mathematics is a subject with an increasing social pressure in such a way that, if you try to implement some kind of innovative way to teach math to get rid of such compartmentalization (for instance, a problem solving approach to introduce new mathematical contents), you will have lots of parents complaining that you are not doing real mathematics at your classroom.&lt;/p&gt;&#xA;" OwnerUserId="1892" LastActivityDate="2014-07-10T13:22:01.203" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3957" PostTypeId="2" ParentId="3950" CreationDate="2014-07-10T15:36:18.120" Score="4" Body="&lt;p&gt;For Calculus, I suggest: &lt;a href=&quot;http://www.artofproblemsolving.com/Store/viewitem.php?item=calculus&quot; rel=&quot;nofollow&quot;&gt;http://www.artofproblemsolving.com/Store/viewitem.php?item=calculus&lt;/a&gt; (which has an available solutions manual as well).  Many of the problems in the book are collected contest problems.  Also each chapter has a challenge problems section after the review problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beyond that, although not limited to a particular discipline like a textbook, I recommend the Putnam Competition problems and solutions.&lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-07-10T15:36:18.120" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="3958" PostTypeId="2" ParentId="3950" CreationDate="2014-07-10T15:49:41.413" Score="3" Body="&lt;p&gt;Richard Courant's Differential and Integral Calculus has both unusual &quot;content&quot; and exercises. Generally, speaking Calculus books from the turn of the century had mechanistic problems which were quite difficult. Many of these books survived into the 1960's including the Calculus book by Granville, Smith and Longley which appeared in 1904 with Granville as the only author.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-07-10T15:49:41.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3959" PostTypeId="2" ParentId="3950" CreationDate="2014-07-10T16:09:55.017" Score="13" Body="&lt;p&gt;My pick would be Volume 1 of Richard Courant and Fritz John's &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/354065058X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Introduction to Calculus and Analysis&lt;/strong&gt;&lt;/a&gt;. For discussions of how Courant/John compares with Spivak, Apostol, and other books, see the math StackExchange question &lt;a href=&quot;https://math.stackexchange.com/questions/79865/difficulty-level-of-courants-book&quot;&gt;Difficulty level of Courant's book&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Volume 1 of Courant/John does not have any solutions to the exercises, but complete solutions to all the problems were separately published by A. A. Blank:&lt;/p&gt;&#xA;&lt;p&gt;Albert Abraham Blank, &lt;a href=&quot;https://math.stackexchange.com/questions/375964/whats-the-name-of-the-pamphlet-in-company-with-introduction-to-calculus-and-ana&quot;&gt;&lt;strong&gt;Problems in Calculus and Analysis&lt;/strong&gt;&lt;/a&gt;, John Wiley and Sons, 1966, x + 264 pages. &lt;a href=&quot;https://archive.org/details/ProblemsInCalculusAndAnalysisAlbertBlank&quot; rel=&quot;nofollow noreferrer&quot;&gt;archive.org copy&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You might also want to look at the list of honors calculus books posted in the following math StackExchange question:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/731087/joseph-kitchens-calculus-reference&quot;&gt;Joseph Kitchen's Calculus (reference)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;(ADDED NEXT DAY)&lt;/em&gt; Regarding Joseph Malkevitch's answer, yesterday I was thinking of mentioning some books from roughly the 1880s to 1890s, when textbooks in algebra through calculus were, on average, pitched at the highest level, but I didn't get around to it. Throughout the 1800s there was a gradual overall trend towards more difficult texts until the end of the 1800s, at which time there were several &amp;quot;downward adjustments&amp;quot; (I'm mostly thinking of the U.S. and England) due to various reforms and the increasing percentages of students going to college. (This is discussed in Châteauneuf &lt;strong&gt;[1]&lt;/strong&gt; for those wishing a reference.)&lt;/p&gt;&#xA;&lt;p&gt;Off-hand, I can think of three authors from this time who wrote calculus treatises at a fairly high level (in the algebraic-manipulative-mechanical sense, not in the modern rigorous sense), and with the beauty of the internet now-a-days it only takes me a few minutes to track down freely available digital copies of their books. See &lt;strong&gt;[2]&lt;/strong&gt; through &lt;strong&gt;[8]&lt;/strong&gt; below. I think you'll find plenty of &amp;quot;extremely challenging exercises&amp;quot; in these books, and many will be on topics you are probably not familar with. The books by Edwards are probably the most extreme in this respect, and over the years I've read book reviews in old journals that pretty much say this (while also being very critical of Edwards' lack of rigor, especially in reviews written after the mid 1890s).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Amy Olive Châteauneuf, &lt;strong&gt;Changes in the Content of Elementary Algebra Since the Beginning of the High School Movement as Revealed by the Textbooks of the Period&lt;/strong&gt;, Ph.D. Dissertation (under John Harrison Minnick), University of Pennsylvania, 1929, x + 191 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Also published by Westbrook Publishing Company in 1929, and reviewed by Lao Genevra Simons in &lt;a href=&quot;http://www.jstor.org/stable/27951306&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Mathematics Teacher&lt;/strong&gt; 24 #1 (January 1931), 58-59&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Joseph Edwards, &lt;a href=&quot;http://archive.org/details/cu31924001130354&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;An Elementary Treatise on the Differential Calculus&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Joseph Edwards, &lt;a href=&quot;http://archive.org/details/treatiseonintegr01edwauoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A Treatise on the Integral Calculus, Volume 1&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Joseph Edwards, &lt;a href=&quot;http://archive.org/details/treatiseonintegr02edwauoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A Treatise on the Integral Calculus, Volume 2&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[5]&lt;/strong&gt; Isaac Todhunter, &lt;a href=&quot;https://archive.org/details/atreatiseondiff05todhgoog/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A Treatise on the Differential Calculus&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[6]&lt;/strong&gt; Isaac Todhunter, &lt;a href=&quot;http://archive.org/details/atreatiseoninte00todhgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A Treatise on the Integral Calculus and Its Applications&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[7]&lt;/strong&gt; Benjamin Williamson, &lt;a href=&quot;http://archive.org/details/anelementarytre19willgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;An Elementary Treatise on the Differential Calculus&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[8]&lt;/strong&gt; Benjamin Williamson, &lt;a href=&quot;http://archive.org/details/cu31924031264769&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;An Elementary Treatise on the Integral Calculus&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, solutions manuals to older texts (in English) were often called &lt;em&gt;Keys&lt;/em&gt;, and I'm sure some of the texts above have keys (typically prepared by someone other than the text's author), which I'll leave you to search for if you're interested. I did happen to come upon the following Key while looking up the books above, so I'll include it:&lt;/p&gt;&#xA;&lt;p&gt;Hunter, &lt;a href=&quot;http://archive.org/details/cu31924058531959&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Key to Todhunter's differential calculus&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2022-10-24T09:37:09.103" LastActivityDate="2022-10-24T09:37:09.103" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="3960" PostTypeId="2" ParentId="3945" CreationDate="2014-07-10T16:11:01.923" Score="0" Body="&lt;p&gt;It is unfortunate that issues related to where one stands in international comparisons drives mathematics education in many countries, including the United States. In my experience emphasizing mathematical applications that affect one's students and doing mathematical modeling make a big difference. Examples might include how vaccination programs improve both short term health and life expectancy. How operations research ideas improve the function of government and businesses - improving municipal services (trash collection, scheduling of workerss at medical centers, etc.)&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-07-10T16:11:01.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3961" PostTypeId="1" AcceptedAnswerId="3963" CreationDate="2014-07-10T19:14:05.190" Score="14" ViewCount="1538" Body="&lt;p&gt;When I was a student (in the 1970s) I was taught linear algebra  as an &quot;adjunct&quot; to &quot;engineering mathematics&quot; such as differential equations. That was during my sophomore year, which seems a bit late, but those were the days when &quot;calculus&quot; was all the rage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays, the application emphasis is on information technology, which stores information in matrices, strings, and vectors. Given this emphasis should linear algebra be taught earlier, concurrently with computer science? Would that mean making it &quot;simultaneous&quot; with first year calculus or even pre-calculus?&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-07-13T18:47:38.820" Title="When is a good time to teach linear algebra?" Tags="|linear-algebra|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3962" PostTypeId="2" ParentId="3939" CreationDate="2014-07-10T19:32:59.697" Score="1" Body="&lt;p&gt;In the United States, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Calculus I&lt;/strong&gt; typically covers &lt;strong&gt;differential&lt;/strong&gt; calculus (in one variable), plus related topics such as limits.&lt;br&gt;&#xA;  &lt;strong&gt;Calculus II&lt;/strong&gt; typically covers &lt;strong&gt;integral&lt;/strong&gt; calculus in one variable. &lt;br&gt;&#xA;  &lt;strong&gt;Calculus III&lt;/strong&gt; is the term for  &lt;strong&gt;multivariate&lt;/strong&gt; calculus, and is an introduction to vector calculus.&lt;br&gt;&#xA;  &lt;strong&gt;Advanced Calculus&lt;/strong&gt; is advanced &lt;strong&gt;vector&lt;/strong&gt; calculus, usually incorporates elements of real analysis, and includes topics in differentiation such as the implicit and inverse function theorems, and theorems of comparable importance in integration. &lt;br&gt;&#xA;  &lt;strong&gt;Linear Algebra&lt;/strong&gt; is considered a separate topic from calculus, and is mostly taught separately. Many calculus books will have one or two chapters of linear algebra (mostly matrix theory), to &quot;support&quot; their teaching of calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-07-10T19:32:59.697" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3963" PostTypeId="2" ParentId="3961" CreationDate="2014-07-11T02:01:10.167" Score="10" Body="&lt;p&gt;My personal opinion. Anytime is a good time to teach linear algebra. I'd like to see at least two courses in the curriculum. Ok, more accurately:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;elementary matrix theory:&lt;/strong&gt; solving systems, matrix math, determinants, routine computations of all sorts. Very limited proof emphasis. Here we are merely rounding out a computational core to give better tools to later calculus and other courses. This could be taught before calculus, in parallel with precalculus if need be. I would hope to include some appreciation of spanning and linear independence here. We'd like to use these terms intelligently in other courses before they hit linear algebra. Applications abound in this course. For example, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0321385179&quot;&gt;Lay's Linear Algebra&lt;/a&gt; would be ideal.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;linear algebra:&lt;/strong&gt; focus on proofs and abstraction. Includes abstract vector spaces, subspace and basis theorems. Either this course comes after the introduction to proofs course and gets to the Jordan form, or, the topics covered are narrowed considerably and the course is used as the introduction to formal proofs course. For the non-introductory to proofs version, you might use &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0130084514&quot;&gt;Insel, Spence and Friedberg&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There are good reasons for making this demarcation in courses. The elementary matrix theory course is really worthwhile for almost every technical major and if it was earlier in the degree plans then many sections could be offered at a larger school making scheduling easy for the student. On the other hand, linear algebra proper is really just a course for math majors and it is an absolute drudgery to try to teach proofs to engineers. I speak with considerable experience.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-07-11T02:01:10.167" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3964" PostTypeId="1" CreationDate="2014-07-11T03:03:22.770" Score="32" ViewCount="2001" Body="&lt;p&gt;Why are students taught to represent one and a half as $1 \frac{1}{2}$ rather than $1 + \frac{1}{2}$? This mode of expression seems standard at least throughout North America. I believe that it is bad pedagogy for a couple of distinct reasons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, doing it the 'correct' way would give students a great deal of personal experience equating the English word 'and' with the mathematical symbol '+'. This is a good thing, since it fosters the notion that mathematical statements (or in this case expressions) have tangible meaning. Students who understand what a half is do well with understanding what 3 and a half is, and I expect that representing it as $3 + \frac{1}{2}$ can do a great deal to cement the 'true meaning' of addition in their understanding. Consider that the previous experience of students at this age is dominated by calculating 8+4 either counting 9-10-11-12 or by rote, neither of which is all that connected to the physical reality of addition (this much &lt;strong&gt;and also&lt;/strong&gt; that much).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, students will reach a point where they are expected to abide by the convention that $ab$ represents $a * b$. Strong students will do OK with this other than a few early mistakes during an adjustment period. But students struggling in math, especially those experiencing phobia or anxiety around the subject, will have little chance but to understand this shift in notation as yet another in an unending string of indications that what's expected from them in math class is entirely arbitrary, changes from one teacher to another, and is some sort of arcane magic. The horrible thing is that in this case they're correct to interpret it this way!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have to be missing something. What advantages does the current scheme provide?&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2017-05-15T19:40:10.470" Title="What is the rationale for the absent (+) in mixed fractions?" Tags="|primary-education|mathematical-pedagogy|notation|fractions|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3966" PostTypeId="2" ParentId="3964" CreationDate="2014-07-11T07:44:31.360" Score="16" Body="&lt;p&gt;One reason is that mathematics was not handed down by the gods fully formed and unambiguous. It is a human construction over a very long time and mathematical notation even more so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any time a notation doesn't &quot;work&quot; for all possible contexts, it's an opportunity for us to talk about this human side of mathematics, and about the pros and cons of notation conventions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to why the missing +, I think it is so that you feel that $1\frac{1}{2}$ is an actual number in its own right, which is harder to perceive with the plus. Also, we write a lot of other numbers in shorthand: 35 and not  30+5, for example, so it feels similar to other representational conventions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, perhaps some of the reasons you mentioned are why most mathematicians avoid mixed numbers in their own work, preferring improper fractions (not that I advocate this for children - it's much easier to get a feel for the size of $1\frac{1}{2}$ as opposed to $\frac{3}{2}$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: The history section of the wikipedia article on fractions &lt;a href=&quot;http://en.wikipedia.org/wiki/Fraction_%28mathematics%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Fraction_%28mathematics%29&lt;/a&gt; has some interesting notes about the history of mixed number notation.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2017-05-15T19:40:10.470" LastActivityDate="2017-05-15T19:40:10.470" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3968" PostTypeId="5" CreationDate="2014-07-11T15:23:47.667" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-07-11T15:23:47.667" LastActivityDate="2014-07-11T15:23:47.667" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3969" PostTypeId="4" CreationDate="2014-07-11T15:23:47.667" Score="0" Body="For questions about the use of games as a tool to learn mathematics." OwnerUserId="1892" LastEditorUserId="1892" LastEditDate="2014-07-11T23:28:21.180" LastActivityDate="2014-07-11T23:28:21.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3970" PostTypeId="5" CreationDate="2014-07-11T15:29:22.713" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-07-11T15:29:22.713" LastActivityDate="2014-07-11T15:29:22.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3971" PostTypeId="4" CreationDate="2014-07-11T15:29:22.713" Score="0" Body="For questions related to the 'Programme for International Student Assessment' (PISA) of the OECD. &#xD;&#xA;" OwnerUserId="1892" LastEditorUserId="1892" LastEditDate="2014-07-11T23:28:15.913" LastActivityDate="2014-07-11T23:28:15.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3972" PostTypeId="2" ParentId="3954" CreationDate="2014-07-12T15:12:06.197" Score="3" Body="&lt;p&gt;Although I am not a big fan of basing what we do in American mathematics education as a consequence of how America performs on international tests, assessments of any kind do give one &quot;feedback.&quot; The purpose of PISA is to see to what extend the mathematics that students learn can be used in their interaction with problems that come up in &quot;daily life.&quot; The issue is not can you solve this quadratic equation or factor that polynomial but in essence mathematical modeling skills. At whatever level one teaches - K-12 or college, I think discussing mathematical modeling and the applicability of mathematics is very important. I don't think one has to go to PISA problems here but can draw on the vast array of mathematical questions that encourage modeling. I particular like urban optimization problems (pot hole inspection routes; routes for meals on wheels; scheduling problems) and fairness questions (elections, bankruptcy, apportionment, cost sharing).&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-07-12T15:12:06.197" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3973" PostTypeId="2" ParentId="3964" CreationDate="2014-07-12T16:42:14.907" Score="6" Body="&lt;p&gt;The university math department where I work runs a math competition for high school students each year. One year there was a problem with answer $7/2$. May students put $3\frac{1}{2}$ down as the answer. Professors who were grading all marked this as incorrect--not because they abhor mixed numbers, but because they naturally parsed it as $3\cdot\frac{1}{2}=\frac{3}{2}$. Later when some students questioned the grading, we all had an aha! moment.  &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2014-07-12T16:42:14.907" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3974" PostTypeId="2" ParentId="3964" CreationDate="2014-07-12T18:52:58.553" Score="13" Body="&lt;p&gt;&lt;strong&gt;Manipulation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mixed fraction form makes subtraction and negative values easier to parse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$2\frac{1}{2} - 1\frac{1}{4}$ is substantially easier to read, write, and understand at this stage than either $2+\frac{1}{2} - (1+\frac{1}{4})$ or $2+\frac{1}{2} - 1-\frac{1}{4}$. Students at this age have not encountered the distributive property (or even brackets?), and they may have trouble attaching the '$-$' to the $\frac{1}{4}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The simple expression of a negative value are similarly challenging.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think that this is a good enough reason (in particular, I'm not sure if I'm a big fan of doing manipulations with mixed fractions at all) but it's a rationale that occurred to me.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2014-07-12T18:52:58.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3975" PostTypeId="2" ParentId="3933" CreationDate="2014-07-13T07:27:37.227" Score="3" Body="&lt;p&gt;Great Question. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In school, and in math especially the goal is to model systems. Wether the system be large, or complex. Super applicable or abstract. Our goal as educators is to teach students how to think deeply about the concepts, and to apply the skill sets. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, mandating showing work allows me a new dimension of teaching. It demonstrates an understanding of the conceptual, and of being able to follow a modeling process. For you as a teacher, it will allow you to differentiate your teaching to meet the needs of students who are lacking understanding or follow through in a particular area. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Showing work is equally helpful for the instructor and student. For the student, it forces them to compartmentalize their work in a framework that they can communicate to others. It also allows them to track mistakes, organize methods for problem solving, and build a framework that is applicable in all areas of life. For the teacher is allows you insight into the needs of the student, and a map for how your students are thinking differently. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And on the practical note, habit building is crucial. And in any field in industry these students will need to be able to document their work in a way that successfully communicates how they've solved the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm an Undergraduate researcher, and this is something one of my favorite advisors told me, when I didn't think I needed to be thorough in the way I documented my findings. &lt;em&gt;&quot;If you can tell someone the answer, but you can't explain the process, or teach anyone to get there. More often than not your solution falls on deaf ears&quot;&lt;/em&gt; Essentially. Communication is &lt;strong&gt;key&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best Mathematician I've ever met, Dave Prince, always told me, &lt;em&gt;&quot;What works, is Work!&quot;&lt;/em&gt;. So if you don't show it then it doesn't work. We would give 20% for correct solution, and 80% for process. &lt;/p&gt;&#xA;" OwnerUserId="1902" LastActivityDate="2014-07-13T07:27:37.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3976" PostTypeId="2" ParentId="3950" CreationDate="2014-07-13T16:11:09.917" Score="6" Body="&lt;p&gt;I learned calculus (by self-study) from the book by Granville, Longley, and Smith that Joseph Malkevitch mentioned in his answer. Later I  had to work considerably harder on problems from Polya and Szegö's &quot;Aufgaben und Lehrsätze aus der Analysis&quot;, which I believe now exists in English also.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; The English translation is entitled &lt;strong&gt;Problems and Theorems in Analysis&lt;/strong&gt; (links to &lt;strong&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/3540636404&quot; rel=&quot;noreferrer&quot;&gt;one&lt;/a&gt;&lt;/strong&gt; and &lt;strong&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/3540636862&quot; rel=&quot;noreferrer&quot;&gt;two&lt;/a&gt;&lt;/strong&gt;).&lt;/p&gt;&#xA;" OwnerUserId="326" LastEditorUserId="262" LastEditDate="2014-07-14T12:01:54.133" LastActivityDate="2014-07-14T12:01:54.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3977" PostTypeId="1" AcceptedAnswerId="4004" CreationDate="2014-07-13T16:46:44.760" Score="7" ViewCount="637" Body="&lt;p&gt;This question is related to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2074/is-euclid-dead-or-should-euclidean-geometry-be-taught-to-high-school-students&quot;&gt;Is Euclid dead? or Should Euclidean geometry be taught to high school students?&lt;/a&gt;, but I am not asking about whether Euclidean geometry should be taught at all, but whether alternate systems should be taught first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;non-Euclidean geometries such as Taxicab geometry have always had a bit of a &quot;cult following&quot; among mainstream mathematics educators, since the notions of &lt;a href=&quot;http://taxicabgeometry.net/angles_trig/angles.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;angles&lt;/a&gt; can be presented in a completely different fashion (though traditional Taxicab geometry uses Euclidean angles), and of course, the notion of a &lt;a href=&quot;http://en.wikipedia.org/wiki/Taxicab_geometry#Circles_in_Taxicab_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;square&quot;&lt;/a&gt; circle is a bit of a paradigm shift for most people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since these generalizations may be helpful in creating a more abstract notion of space and definitions of metrics, etc., would it be effective to teach these concepts before the Euclidean ones, or has this been tried before and there have been deleterious effects on students' ability to learn the Euclidean principles?&lt;/p&gt;&#xA;" OwnerUserId="16" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-01-05T15:24:05.530" Title="Will presenting non-Euclidean geometries to students before Euclidean geometry give them a better intuition about shapes on the plane?" Tags="|secondary-education|geometry|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3978" PostTypeId="2" ParentId="3945" CreationDate="2014-07-13T18:35:50.663" Score="1" Body="&lt;p&gt;As a Peace Corps Volunteer who served in South Africa, I can say that there are a whole lot of things the South African education system needs to do before it can be said to be adequate. I'll show some restraint and just suggest one thing: the foundations of mathematics can't be neglected. Especially the way South African curriculum is designed, every year builds on the previous year, so if learners are neglected one year, it becomes very hard for them to catch up. The worst-case scenario is that the teacher never goes to class until the learners are preparing for matric, and then they have to somehow learn 12 years of mathematics in one or two years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, stop having departmental and union meetings in time meant for class. That's just common sense. I know that individual teachers can't just refuse to go to a meeting, but if you're not going to be in class, be honest about it in your timetable and your planning.&lt;/p&gt;&#xA;" OwnerUserId="1903" LastActivityDate="2014-07-13T18:35:50.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3979" PostTypeId="2" ParentId="3977" CreationDate="2014-07-13T18:41:58.297" Score="4" Body="&lt;p&gt;You can certainly show the geometry e.g. on a sphere. Going any futher would be very counterproductive: most (as in &quot;almost all&quot;) everyday geometry is Euclidean. Trigonometry (and much of &quot;initial calculus&quot;) requires/assumes Euclidean geometry. The list goes on.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-07-13T18:41:58.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3980" PostTypeId="2" ParentId="3961" CreationDate="2014-07-13T18:47:38.820" Score="2" Body="&lt;p&gt;The way &quot;programs store data in arrays&quot; has next to nothing in common with linear algebra. Don't mix them up, the superficial similarities are confusing enough as is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a nice look at what linear algebra is really about check out Treil's &lt;a href=&quot;http://www.math.brown.edu/~treil/papers/LADW/LADW.html&quot; rel=&quot;nofollow&quot;&gt;&quot;Linear Algebra done Wrong&quot;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-07-13T18:47:38.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3981" PostTypeId="2" ParentId="3946" CreationDate="2014-07-13T18:56:05.590" Score="0" Body="&lt;p&gt;You might also consider &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;MSE&lt;/a&gt; for lots of problems (but perhaps most harder than what you are looking for). Look at the relevant tags. Search for math olympiad preparation material, and for problems and solutions in such. Again, probably much harder than what you are interested in for a first look,&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-07-13T18:56:05.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3982" PostTypeId="1" CreationDate="2014-07-13T23:16:49.390" Score="5" ViewCount="234" Body="&lt;p&gt;I'm going to be teaching small (&amp;lt; 10 students) but very mixed group of middle schoolers (US system, ages 11-13) who have had varying amounts of pre-algebra and algebra instruction thus far.  I need to give a placement test so that I know who has mastered which topics, to plan appropriately.  I will probably then split the group into two subgroups, and also have enrichment activities available for students who have &quot;tested out&quot; of a topic the rest of their group needs to cover.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is, they will show up on the first day of school rusty and not remembering much of what they learned last year.  How can I quickly &quot;de-rust&quot; them to the point where I can get reasonable results on a placement test, preferably within one or two 45-minute sessions (with homework in between if helpful).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm thinking the homework might be a sample of problems like the ones they will see on the placement test, that they can hopefully remind themselves how to do. These are mostly above-average math students, but mostly lacking a parent who remembers algebra who could help remind them of what to do.  So, if I could come up with (or find pre-made online) some kind of quick reference of pre-algebra and algebra topics (preferably not more than a couple of pages) that I could give them as a reference to refresh memories, that would probably be helpful.&lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-07-14T05:16:02.710" Title="How do I get my middle schoolers warmed back up at the start of the year prior to giving a placement test" Tags="|secondary-education|assessment|review|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3983" PostTypeId="1" AcceptedAnswerId="3999" CreationDate="2014-07-14T00:31:39.373" Score="13" ViewCount="3117" Body="&lt;p&gt;How can we break down the complexity of eigenvalues/vectors to something that is more intuitive for students. I feel like the proofy way isn't a good intuitive representation of the mechanism that eigenvalues/vectors represent. What are the best reasons as to why a student might need to understand eigenvalues, and the tangible real world applications for eigenvalues, and eigenvectors?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teaching this to all ages, high school, through college. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Can assume students have a foundation in calculus (differentiation ~ multivariable)&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="1902" LastEditorUserId="1902" LastEditDate="2014-07-16T16:46:08.060" LastActivityDate="2022-05-18T14:12:47.267" Title="What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?" Tags="|mathematical-pedagogy|linear-algebra|eigenvalues|" AnswerCount="14" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="3984" PostTypeId="2" ParentId="3982" CreationDate="2014-07-14T03:25:30.327" Score="1" Body="&lt;p&gt;Presumably you have access to a class list over the summer?  I would send the students a series of review problems in the mail once a week that form some sort of scavenger hunt.  You can include a stamped postcard so they can send the answers back to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answers to the review problems could correspond to letters of the alphabet (so a problem with the solution x = 1 would give the letter &quot;A&quot; as a &quot;clue&quot;, the next week's problem could have the answer y = 3, which would yield the letter &quot;C&quot;.  By the end of the summer, the students would collect all the letters and solve the puzzle, which could be something like &quot;ICE CREAM PARTY&quot;.  This can then be their reward after the examination.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's no guarantee the students will do the work, but the more motivated students will want to know the answer.  Even if the students don't complete all the problems, they would still get some review out of it.&lt;/p&gt;&#xA;" OwnerUserId="16" LastActivityDate="2014-07-14T03:25:30.327" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3985" PostTypeId="2" ParentId="2462" CreationDate="2014-07-14T04:55:16.150" Score="1" Body="&lt;p&gt;This is a very large age range to be covering.  I'm assuming children will be playing in groups that have a more limited ages range (so 4,5,6 year old children together or something of that nature).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on the attention span of the children, something like math bingo can help them remember multiplication facts (or addition facts for the younger kids).  Classic games like Quizmo have a readily available set of clue cards.  I think that brand only exists in multiplication/division and decimals/place value, but you could certainly make up your own cards to fit the range of numbers you need.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dice games are also a lot of fun and can teach a range of skills depending on the age of the participants.  Something like Yahtzee and variants can be used for counting and pairing for the younger kids and skills such as combinations and permutations for the older set (the 10-12 year olds).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, one particular game for 4-12 year olds might be difficult to find, but you can certainly vary some old favorites by creating new wrinkles in the rules of the game.&lt;/p&gt;&#xA;" OwnerUserId="16" LastActivityDate="2014-07-14T04:55:16.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3986" PostTypeId="2" ParentId="3982" CreationDate="2014-07-14T05:07:25.693" Score="2" Body="&lt;p&gt;I recommend devoting class time to helping the students review last year's content. I wouldn't recommend just telling them stuff, but getting them to identify things that they need to know themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First I would get them to try to write down things they learned last year. Give them say 15 mins to write down everything they could think of individually without talking. The focus is on getting as much stuff down on the paper, and handwriting isn't important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After they've written down as much as they can remember, get them in groups to compare notes. I recommend giving them guidance here or they might just argue. Tell them that at this stage they're looking for a list of topic headings (eg &quot;negative numbers&quot;, &quot;areas&quot;). Maybe another 15 mins on that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally get each group to share and collate it together to get a list of topics for the whole class. You can fill in any topics that are missing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For homework, they can do the same &quot;try to remember as much as you can&quot; for each of those topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the next class, get them into groups again to talk about each topic. The point here is to remember the main facts/rules from that topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can give them a certain amount of time for each topic, then summarise these at the board. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This process ought to give you a feel for where the students are with regard to previous learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For homework the second day, they can try some example problems in preparation for the test.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-07-14T05:16:02.710" LastActivityDate="2014-07-14T05:16:02.710" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3987" PostTypeId="1" CreationDate="2014-07-14T09:32:26.410" Score="5" ViewCount="269" Body="&lt;p&gt;P vs NP is a million dollar millenium problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Essentially it boils down to, &lt;em&gt;If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem?&lt;/em&gt; This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Per a comment, it became clear that I should add some extra clarity! :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In what ways can we intuitively teach HS-college students how to understand and attack a problem like this. Not P vs NP in particular but rather, a truly difficult, if not unsolvable problem. I think finding ways to to reconcile these types of problems, is important to developing really fine tuned critical thinking and analysis skills. With that, I do want them to have an intuition around P vs NP in this particular case, to set the stage for a very difficult problem. And I do want them to try out ways to approach difficult problems&lt;/p&gt;&#xA;" OwnerUserId="1902" LastEditorUserId="1902" LastEditDate="2014-07-14T16:28:07.753" LastActivityDate="2014-07-16T14:12:03.947" Title="What is the best way to intuitively explain, understand and approach P vs NP" Tags="|student-motivation|intuition|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3989" PostTypeId="1" AcceptedAnswerId="3993" CreationDate="2014-07-14T16:01:24.490" Score="9" ViewCount="3567" Body="&lt;p&gt;Single variable calculus is typically (and reasonably) taught over a whole year, with the first semester being devoted to &quot;differential&quot; calculus, and the second semester being devoted to &quot;integral&quot; calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my own experience, &quot;multivariable&quot; calculus is taught in one semester. That is a course with vector calculus, partial derivatives, gradient and the chain rule. Then onto line integrals, multiple integrals, and Green's Gauss' and Stokes' theorems. That's a lot for one semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would it make sense to break &quot;multivariable up into a two semester course, differential and integral, as with single variable calculus?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That would be a &quot;differential&quot; course with vector calculus, partial derivatives, gradient and the chain rule, Taylor series, constrained optimization, matrices, mappings, and determinants, plus the Jacobian in the first semester. And an &quot;integral&quot; course with improper integration, power series, change of variables, line integrals, multiple integrals, Fourier integrals and Green's, Gauss' and Stokes' theorems in the second semester.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2014-07-14T17:17:25.230" LastActivityDate="2022-11-14T22:08:29.050" Title="Should multivariable calculus be a two- rather than one- semester course?" Tags="|calculus|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3990" PostTypeId="2" ParentId="2060" CreationDate="2014-07-14T16:06:16.580" Score="4" Body="&lt;p&gt;One important application of calculus in biology is called the &lt;a href=&quot;http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation&quot; rel=&quot;nofollow&quot;&gt;predator-prey model&lt;/a&gt;, which determines the equilibrium numbers of predator and prey animals in an ecosystem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's actually an application of &quot;differential equations&quot; but you will need calculus to &quot;get there.&quot;&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-07-14T16:06:16.580" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3991" PostTypeId="2" ParentId="3983" CreationDate="2014-07-14T16:10:06.677" Score="1" Body="&lt;p&gt;Eigenvectors are vectors that map onto themselves. (Eigen= &quot;oneself&quot; in German.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To accomplish this, they are defined by vectors of eigenvalues that &quot;solve&quot; the so-called &quot;characteristic equation&quot; (which defines them) for the value zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This equation is derived from: $A v = \lambda v$, where A is the &quot;transformation matrix, v is the eigenvector, and $\lambda$ is a vector to be solved for. If you subtract $\lambda v$ from both sides, you get zero on the right hand side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By factoring out the v, and through matrix operations on the left hand side, you get a polynomial equation that can be solved for $\lambda$.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2014-07-14T16:20:07.450" LastActivityDate="2014-07-14T16:20:07.450" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3992" PostTypeId="2" ParentId="1149" CreationDate="2014-07-14T16:24:59.180" Score="1" Body="&lt;p&gt;Expansion by &quot;cofactors&quot; is really a form of &quot;cancellation.&quot; That is, you try to get to a point where all the terms but one in a row or column is zero, which equation &quot;trivializes&quot; the exercise. Of course, when you add or subtract columns or rows to produce this result, you end up &quot;compensating&quot; the other terms in the array.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the whole operation is one of &quot;compensated&quot; cancellation to determine the volume of the matrix.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-07-14T16:24:59.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3993" PostTypeId="2" ParentId="3989" CreationDate="2014-07-14T16:49:42.957" Score="6" Body="&lt;p&gt;I believe it is true that teaching the material in more time would be better. However, there are a lot of useful things for students to do with their time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, there will be a lot of pressure from your Engineering faculty against your idea. They might propose the opposite of your plan! -- more calculus material squeezed into fewer credit hours. For example, &lt;a href=&quot;http://www.math.illinois.edu/UndergraduateProgram/freshmen-calculus.html&quot; rel=&quot;noreferrer&quot;&gt;at the University of Illinois&lt;/a&gt;, the standard calculus sequence is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;5 credits: Math 220 (Differential Calculus and many Integrals and Applications) &lt;/li&gt;&#xA;&lt;li&gt;3 credits: Math 231 (Integration Techniques, Sequences &amp;amp; Series)&lt;/li&gt;&#xA;&lt;li&gt;4 credits: Math 241 (Multivariable as you described)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;However, this 15-credit sequence was becoming too burdensome to fit into Engineering degrees, so a special accelerated intro calculus course was created to make the engineering calculus sequence only &lt;strong&gt;14 credits&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;They are not evil!!&lt;/strong&gt; and their intentions are extremely good -- the engineering people have a lot of things that they want their students to take, and it seems reasonable to them to compress first-semester calculus a bit in order to make room for another credit hour of other valuable things. Many of their students have AP credit anyway, and many of their students are fully able to succeed in the accelerated intro course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally I agree with you, that something like 18 credits for all this material might be better than 15. However, the math instructor's entirely legitimate desire to teach more credits of math will come into direct conflict with everyone else's entirely legitimate desire to teach more credits of everything else. This will be the primary thing that will stop you from expanding multivariable calculus into more semesters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possibility might be to teach a two-semester sequence that combines multivariable calculus and the generally-boring linear algebra class into a nice soup. This kind of approach might avoid much of the counterpush by other interests.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-07-14T16:49:42.957" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3994" PostTypeId="1" AcceptedAnswerId="3998" CreationDate="2014-07-14T17:11:33.980" Score="14" ViewCount="1635" Body="&lt;p&gt;Starting with &quot;precalculus,&quot; students learn trigonometric functions. After they've spent a semester or more learning how to differentiate and integrate these trigonometric functions in calculus, students are then introduced to &quot;hyperbolic&quot; functions, but mainly &quot;in passing.&quot; (At least, that was my experience.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While there is some mention of hyperbolic functions at somewhat higher levels, such as multivariate calculus and linear algebra, by far the greater emphasis is on trigonometric functions. Hyperbolic functions don't get (more or less) equal time until about complex variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any pedagogical reason for this, or is it just the relative frequency of (low level) applications that produces this pattern?&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2022-05-09T15:01:04.377" Title="Why are hyperbolic functions given &quot;short shrift&quot; at &quot;low&quot; levels of math?" Tags="|functions|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3996" PostTypeId="1" CreationDate="2014-07-14T17:26:46.793" Score="10" ViewCount="124" Body="&lt;p&gt;Systems and electrical engineering graduate students often take a course on stochastic systems (a.k.a. &quot;Probabilistic Systems Analysis&quot;).  A typical course will present such topics as multivariable probability distributions, auto- and cross-correlations (and Fourier transformations of them), Ergodicity, stationarity, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students wanting a more advanced treatment after completing the above often gravitate towards a more rigorous Stochastic Processes course in the math department, which normally smacks them over the head with the concept of Measure Theory, which throws them for a loop, even if some have taken a course in advanced engineering mathematics.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the best way to bridge the gap between what is a more applied treatment of these probabilistic concepts into the more abstract notions, particularly for students that may not have had a graduate-level Real Analysis course?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related questions: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/178/best-textbooks-to-introduce-measure-theory-and-lebesgue-integration/317#317&quot;&gt;Best textbooks to introduce measure theory and Lebesgue integration?&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/2415/mathematical-thinking-skills-for-engineering-students&quot;&gt;Mathematical thinking skills for engineering students&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-07-14T17:26:46.793" Title="How can instructors bridge the gap between an engineering course in stochastic systems and a more rigorous Stochastic Processes course?" Tags="|probability|engineering-mathematics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3997" PostTypeId="2" ParentId="3989" CreationDate="2014-07-14T18:40:29.770" Score="1" Body="&lt;p&gt;At my University for engineering Calc I is two or three weeks of differential calculus while Calc II is multivariable and then we have a differential equations course. Not all engineers are required to take differential equations but most (possibly all) engineering majors have to take linear algebra. It used to be that Calc I was the weed out course now Calc II is. Either way adding more semesters to this sequence would lead to problems with prerequisites for major related classes that require these maths. So I guess my answer is that in the context of an engineering school slowing down this sequence will lead to a domino effect slowing down major matriculation. Anyone taking multi variable calculus should be taking it with an eye toward it being specifically useful for some other course or further math study and as a result it is a probably a class taken only by those with an aptitude for math. &lt;/p&gt;&#xA;" OwnerUserId="1906" LastActivityDate="2014-07-14T18:40:29.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3998" PostTypeId="2" ParentId="3994" CreationDate="2014-07-14T19:23:10.943" Score="15" Body="&lt;p&gt;There are a few reasons I can think of, not all of them strictly pedagogical.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The trigonometric ratios (as opposed to functions) have a direct relationship to measurement of triangles, which are an everyday concept, and so the introduction of these functions via triangles is a natural thing to do during high school. Hyperbolic trig functions do not relate so directly to everyday geometry and so are not as easy to introduce so early.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The fundamental definition of the trig functions involves the geometry of a circle, which is familiar to students, whereas the hyperbolic trig functions relate to the geometry of a hyperbola, which for many students is not familiar at all!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The graphs of the trig functions have repeating shapes with nice maxima/minima that allow for intense investigation of shifts and dilations. The graphs of the hyperbolic trig functions have no major distinguishing features other than their even/odd-ness and one important point (minimum for cosh, inflexion point for sinh and tanh).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The definition of the hyperbolic trig functions using the exponential function is not at all obviously related to the geometry of the hyperbola, and many teachers/lecturers do not even know the connection. I myself only learned this year where the &amp;quot;angle&amp;quot; is in the hyperbola geometry. I still don't know why hanging ropes make cosh curves.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(Inverse) Hyperbolic trig functions seem to be mainly used to provide integrals of functions that inverse trig functions can also be used for, so I think many people choose to ignore them because they know it can be done with ordinary trig functions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A couple of the major connections between hyperbolic and ordinary trig functions involve concepts that the students are just not familiar with until they have done more maths -- Taylor series and complex numbers, for example.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So that's several reasons why they are not introduced earlier.&lt;/p&gt;&#xA;&lt;p&gt;However, I do wish more time was spent on them at the point they are introduced, though, regardless of when that introduction happens. In my experience, students find them quite difficult to assimilate when they are introduced with a wave of the hand and then promptly forgotten about. A little discussion of the connections between them and the trig functions, and a proper investigation of their trig identities and how much easier they make some integrals would make them a bit easier to swallow!&lt;/p&gt;&#xA;&lt;p&gt;PS: Here is a paper about the history of them: &lt;a href=&quot;http://www.jstor.org/discover/10.2307/3219227&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jstor.org/discover/10.2307/3219227&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;" OwnerUserId="1853" LastEditorUserId="9551" LastEditDate="2022-05-09T15:01:04.377" LastActivityDate="2022-05-09T15:01:04.377" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="3999" PostTypeId="2" ParentId="3983" CreationDate="2014-07-14T22:31:55.107" Score="9" Body="&lt;p&gt;Here's an example I use for myself. I don't teach this topic in regular class but I have used this example in private conversations with advanced students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Think of an object (perhaps a globe) that is stretched on one or more directions then rotated in various ways and perhaps reflected. We can show that at least one line through the object is either still pointing the same direction or pointing the opposite direction. The vector for this direction is an eigenvector. The amount of stretching in that direction is the eigenvalue for that eigenvector. If the direction is opposite the original direction, the Eigenvalue is negative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works since one-directional stretching, rotating, and reflecting are linear functions, and 3-dimensional space requires at least one real eigenvalue.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2014-07-14T22:31:55.107" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4000" PostTypeId="2" ParentId="3983" CreationDate="2014-07-15T01:49:00.097" Score="4" Body="&lt;p&gt;To be perfectly honest, the applications of eigenvalues tends to be much more complex than the actual theory of finding eigenvalues itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first time I found a practical application of them was in Differential Equations, particularly solving the problems of&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y' = Ax$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;whereas $A$ is a constant matrix $y$ is a vector function of $x$ which itself is a vector.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To some high school/very bright middle school students that have seen calculus this may be worth exploring but obviously this example isn't the best application for a wide-spread audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another perhaps more primitive understanding is &quot;eigenvalues&quot; and &quot;eigenvectors&quot; are curious objects that satisfy the equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$Ax = \lambda x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which in itself is an interesting problem (for reasons mentioned above particularly @Rory Daulton and her stretching/rotating analogy)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And it just so happens these things can also be used to do a hell of a lot more for example factoring matrices, etc...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See here: &lt;a href=&quot;http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Practical_implications.5B1.5D&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Practical_implications.5B1.5D&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1909" LastActivityDate="2014-07-15T01:49:00.097" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4001" PostTypeId="2" ParentId="3994" CreationDate="2014-07-15T02:02:09.953" Score="9" Body="&lt;p&gt;I am of the personal opinion that until complex numbers and manipulation of infinite series is introduced that it is pointless to introduce hyperbolic trig functions as I had learned them in that order and fared just fine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said the rest of the math worlds seems to disagree insisting on learning hyperbolic-trig indentities and integrals before showing most students what these things actually are. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason they aren't seen too much though and given the 'fair treatment' is because it's not pedagogically useful to focus too much on them though since what ends up happening is that students need to be taught a plethora of trigonometric-style tricks without much (if any) intuition or proof when they are first introduced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By studying Trig first, followed by taylor series, and complex exponentials one much more naturally arrives at the theory of hyperbolic-trig and has an algebraic means of understanding exactly why they behave the way they do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For people that insist on geometric intuition: the diagram below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Hyperbolic_function#mediaviewer/File:Hyperbolic_functions-2.svg&quot; rel=&quot;noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Hyperbolic_function#mediaviewer/File:Hyperbolic_functions-2.svg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the use of some heavy algebra (and knowledge of tools such as arc-length formulas) can help them derive all the trigonometric-style tricks &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would think this is a useful method too to learn them but have never seen this done, EVER, in a class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I think the best order of learning is: differential and integral calculus for simple algebraic functions followed by taylor expansions, then followed by an introduction to trigonometry whereas as soon as some of the basics have been learned, students are more than well equipped to derive taylor expansions for the trigonometric functions, determine their complex exponential formulas, etc, etc... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly hyperbolic trig, if anything should be something students derive on their own attempting to solve their associated differential equations.&lt;/p&gt;&#xA;" OwnerUserId="1909" LastActivityDate="2014-07-15T02:02:09.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4002" PostTypeId="2" ParentId="2060" CreationDate="2014-07-15T06:59:33.127" Score="2" Body="&lt;p&gt;There is at least one very good reason to know calculus as a Biologist. There was a certain paper published, I don't know the details, but could probably look it up, by a biologist to a bio journal detailing how to calculate the area under a curve by using this amazing approximation using rectangles and trapezoids. This of course was peer reviewed and hailed as a major advance for some part of bio that constantly needed to do this. The story goes on to say that the biologist knew that this came for math somewhere, but so many other biologists wanted to use the technique and needed something to cite, so he published the paper. However, the issue remains: Biologists, didn't know basic integration. &#xA;I'm sure you could find this story online. I am not sure if it is valid, but I find it likely to be in part true at the very least. So being a respectable scientist is a good enough reason to learn something like calculus.&lt;/p&gt;&#xA;" OwnerUserId="1825" LastActivityDate="2014-07-15T06:59:33.127" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4003" PostTypeId="1" AcceptedAnswerId="4007" CreationDate="2014-07-15T19:31:26.637" Score="8" ViewCount="3885" Body="&lt;p&gt;I have developed a small Android app. The app is a pocket calculator for addition and subtraction, in which the calculation is explained via both video and audio. If it is unclear please have a quick look at the &lt;a href=&quot;http://www.youtube.com/watch?v=0iJXksW-Gdg&quot; rel=&quot;nofollow&quot;&gt;video&lt;/a&gt; I made that explains it.&lt;br&gt;&#xA;Of course I understand subtraction using the American method, but I am having a hard time explaining the idea of borrowing over more then one digit. For example 1000-1, already in the first step, I need to borrow a 1 from all the other digits. For me that is a recursion, but how can I explain that in my app? My target users are school kids and their parents.&lt;/p&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="1537" LastEditDate="2014-07-19T02:28:34.860" LastActivityDate="2014-07-19T02:28:34.860" Title="How to explain 1000 - 1 using the american method" Tags="|primary-education|tutoring|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="4004" PostTypeId="2" ParentId="3977" CreationDate="2014-07-15T20:00:24.923" Score="6" Body="&lt;p&gt;When many people use the term &quot;non-Euclidean&quot; geometry they are often referring to the &quot;classical&quot; non-Euclidean geometries - elliptic geometry and Bolyai-Lobachevsky geometry. The question asks about intuition about shapes in the (Euclidean) plane. So my answer to the question if studying the classical non-Euclidean geometries helps with intuition about Euclidean shape is &quot;no.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I think that notation of distances other than Euclidean distance should be looked at much earlier than has often been the case. Most students arrive in college with the idea that the only &quot;distance&quot; is Euclidean distance. In fact, in some ways learning about Euclidean distance is &quot;harder&quot; than learning about taxicab distance because for many students square roots are a rather subtle issue. Discussing in K-12 or college about the different ways to measure distance is very &quot;liberating.&quot; The points that are unit distance from me in an open field depends on the distance function I use, and &quot;shape&quot; of circles depends on distance. It also would not do any harm to discuss Hamming distance and show how certain spell checkers use Hamming distance to help correct spelling errors.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastEditorUserId="376" LastEditDate="2017-01-05T15:24:05.530" LastActivityDate="2017-01-05T15:24:05.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4005" PostTypeId="2" ParentId="4003" CreationDate="2014-07-16T03:53:15.473" Score="4" Body="&lt;p&gt;I'll take a stab at it. What does $1000$ mean? It is a decimal representation:&#xA;$$ 1000 = 1 \cdot 10^3+0 \cdot 10^2+0 \cdot 10^1+0 \cdot 1$$&#xA;This is hard to work with, however, if we notice $1000=999+1$ then we have:&#xA;$$ 1000 = (9 \cdot 10^2+9 \cdot 10^1+9 \cdot 1)+1$$&#xA;Furthermore, if we are willing to violate the purist's form of the decimal representation then we can just as well write (this is &quot;carrying the one&quot;):&#xA;$$ 1000 = 9 \cdot 10^2+9 \cdot 10^1+10 \cdot 1$$&#xA;Finally, to subtract $1$ we can subtract&#xA;$$ 1 = 0\cdot 10^2+0 \cdot 10^1+1 \cdot 1$$&#xA;from $1000$ as follows:&#xA;$$&#xA;\begin{align}&#xA; 1000 &amp;amp;= 9 \cdot 10^2+9 \cdot 10^1+10 \cdot 1 \\&#xA;1 &amp;amp;= 0\cdot 10^2+0 \cdot 10^1+1 \cdot 1 \\ \hline&#xA;1000-1 &amp;amp;= 9 \cdot 10^2+9 \cdot 10^1+9 \cdot 1&#xA;\end{align}&#xA;$$&#xA;Which gives $1000-1=999$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The procedure of carrying can always be understood in the way I indicate here. If we have to subtract too much from a particular digit then we can borrow from higher digits by taking one of the first larger digit and breaking it up for the smaller digits. Of course, there are more cases I have not discussed...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe you can see how to twist my work here into something for your app?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A second view:&lt;/strong&gt; my wife had this comment. How to explain to children? My way is wrong. Instead, when faced with the question $1000-1$ the story goes as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;We want to subtract $1$ from the one's place in $1000$ but the one's place has nothing so it has to borrow $1$ from the ten's place which is $10$ of the one's place.&lt;/li&gt;&#xA;&lt;li&gt;Unfortunately, the ten's place also has nothing so it has to borrow $1$ from the hundred's place which counts as $10$ in the ten's place. But, the ten's place already gave $1$ to the one's place so it only has $9$.&lt;/li&gt;&#xA;&lt;li&gt;Yet again, the hundred's place has nothing so it has to borrow $1$ from the thousand's place which gives $10$ hundreds. Again, the hundred's place already gave $1$ to the ten's place so only $9$ remains.&lt;/li&gt;&#xA;&lt;li&gt;Finally, the thousands place only had $1$ and since it gave it up the hundreds there is nothing more to do.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Perhaps we could think of this as a series of banks from which each level can only borrow from the level immediately above.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2014-07-16T16:21:00.037" LastActivityDate="2014-07-16T16:21:00.037" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4006" PostTypeId="2" ParentId="3977" CreationDate="2014-07-16T04:11:55.217" Score="3" Body="&lt;p&gt;If you would like to look at a treatment of axiomatic geometry which begins with the abstract, featureless space then places structure (Euclidean, Projective, Hyperbolic) then you might want a copy of &lt;a href=&quot;http://www.xulonpress.com/bookstore/bookdetail.php?PB_ISBN=9781625093127&quot; rel=&quot;nofollow&quot;&gt;Fundamentals of MODERN GEOMETRY for College Students by&#xA;Honorè P. Mavinga &lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had the opportunity to sit in some of his lectures. His approach is really clean and, while it may require some patience your students do not possess, at the end of it you can clearly understand the distinctions between the different axiomatic geometries as they arise by adding structure to the same blank template: affine space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal inclination, given students who care etc.. I very much like the idea of the text by McCleary.&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521424801&quot; rel=&quot;nofollow&quot;&gt;Geometry from a Differentiable Viewpoint by John McCleary&lt;/a&gt;. Part A is axiomatic classical geometry. Part B is classical differential geometry of curves and surfaces, the usual topics including Gauss Bonnet. I think it might be very nice to teach a course where a few weeks concisely covered the axiomatic problems then the remainder of the course shows how the geometry of curved surfaces gives not just one noneuclidean geometry, rather, boundless choices. There are various institutional pressures which make the birth of this course very unlikely, but, perhaps somewhere there exists an institution whose local conditions (students, needed prerequisites...)  would encourage such a course...&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-07-16T04:11:55.217" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4007" PostTypeId="2" ParentId="4003" CreationDate="2014-07-16T04:12:48.543" Score="5" Body="&lt;p&gt;The first three digits of $1000$ represent $100$ tens.  If you regroup that as $99$ tens and $10$ units,then you can subtract $1$ unit from the $10$ units and get $99$ tens and $9$ units, i.e. $999$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edited:&lt;/strong&gt; I'm going to substantially expand on my answer above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think most school-aged children are taught to execute the algorithm by &quot;borrowing&quot; or &quot;regrouping&quot; from one digit to another, one at a time:  from thousands to hundreds to tens to ones.  I've illustrated this with the GIF below.&#xA;&lt;img src=&quot;https://i.stack.imgur.com/540YC.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the problem becomes much simpler if, instead, you aggregate the first three digits together (as I described in my response originally).  Then the process looks like this:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Nxz5p.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2014-07-16T20:27:36.950" LastActivityDate="2014-07-16T20:27:36.950" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4009" PostTypeId="2" ParentId="3987" CreationDate="2014-07-16T14:12:03.947" Score="3" Body="&lt;p&gt;One way might be to assign optional readings from a select group of authors who popularize these problems and have students give a brief report.  Two that come to mind are &quot;The Golden Ticket&quot; by Lance Fortnow (for P vs NP) or &quot;The Music of the Primes&quot; by Marcos du Sautoy (for the Riemann  Hypothesis), or the &quot;Millennium Problems&quot; by Keith Devlin (the 7 prize problems of the Clay Math Institute).  It may also help to assign readings for difficult problems which have already been solved (Fermat's Last Theorem, Poincare Conjecture, Godel's Incompleteness Theorem)  to see how mathematics is more than high school algebra, calculus, etc.  but is really a collaborative effort which has spanned centuries.  I hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="1914" LastActivityDate="2014-07-16T14:12:03.947" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4010" PostTypeId="2" ParentId="3983" CreationDate="2014-07-16T22:13:43.783" Score="5" Body="&lt;p&gt;Although the following quantum-ish-physics-y &quot;explanation&quot; begs-the-question in several ways, it is genuine, and may convey something to students: given a linear operator (a.k.a. &quot;matrix&quot;), an eigenvector is &quot;a pure state&quot; (of what, we don't quite ask), meaning that the operator acts on it in an especially simple fashion. In good situations, a general &quot;state&quot; is a &quot;superposition&quot; (a.k.a. &quot;linear combination&quot;) of pure states.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although one might dismiss such half-explanations as too vague, the &quot;hook&quot; of &quot;quantum ...&quot; can instill belief that the thing is serious and &quot;big-time&quot;.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-07-16T22:13:43.783" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4011" PostTypeId="1" AcceptedAnswerId="4050" CreationDate="2014-07-16T23:31:24.803" Score="21" ViewCount="5485" Body="&lt;p&gt;I believe it is the case that, between spaces,&#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/Homeomorphic&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;homeomorphism&lt;/em&gt;&lt;/a&gt;&#xA;is stronger than&#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/Homotopy#Homotopy_equivalence&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;homotopy equivalence&lt;/em&gt;&lt;/a&gt;&#xA;which is stronger than having &#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/Simplicial_homology&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;isomorphic homology groups&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example,&#xA;the annulus and the circle are not homeomorphic&#xA;but they have the same homotopy type.&#xA;For the comparison between homotopy and homology, I&#xA;am less clear, but it was addressed in an&#xA;&lt;a href=&quot;https://math.stackexchange.com/q/99302/237&quot;&gt;MSE question&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;. What is the best way to explain the differences among&#xA;  the three concepts to students, beyond simply pointing to the definitions?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am interested in approaches that would clearly convey the intuitive&#xA;differences and yet be responsive to the technical differences.&#xA;The audience is undergraduates learning topology.&#xA;I am primarily interested in simplicial complexes, but perhaps it is&#xA;not necessary to specialize to get across the concepts.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2020-03-06T22:33:42.843" Title="Pedagogical challenge: Homeomorphic vs. Homotopy equivalent vs. Homologous?" Tags="|undergraduate-education|intuition|topology|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4012" PostTypeId="2" ParentId="4011" CreationDate="2014-07-17T02:01:38.517" Score="8" Body="&lt;p&gt;I am not entirely sure on the best way to convey the difference between homotopy equivalence and isomorphic homology groups (or even isomorphic homotopy groups, though on CW-complexes I guess this isn't as big of a concern), except by way of examples. I remember my algebraic topology exam had an explicit example of spaces with all the same homology and fundamental groups. I'm pretty sure one was $S^1 \vee S^2 \vee S^3$, and the other was $S^2 \times S^1$. However the proof these were not homotopy equivalent required showing through a guided exercise that for homotopy equivalent nice spaces the equivalence lifts to the universal covers, which in this case are clearly different by considering the third homology group. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For homotopy and homeomorphism, my favorite exercise I've ever seen was identifying which letters of the standard print alphabet are homotopy equivalent and homeomorphic. It gives you a real idea for the types of topological manipulation the two equivalencies permit.&lt;/p&gt;&#xA;" OwnerUserId="1915" LastEditorUserId="1915" LastEditDate="2014-07-17T04:21:15.057" LastActivityDate="2014-07-17T04:21:15.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4013" PostTypeId="2" ParentId="3983" CreationDate="2014-07-17T02:08:29.223" Score="5" Body="&lt;p&gt;I think a good motivation is the idea of dynamical systems and stability a la Markov chains. If we have a system which can be modelled by taking a vector of data $v(0)$ and then some matrix $A$ we have $v(t) = Av(t-1)$. Observe that such a system is in some sense stable, and will undergo consistent exponential growth if it is a eigenvector. Even more, one can actually data and see that often the maximal or minimal eigenvalue will dominate. This can be motivated by supply/demand/competition in a marketplace, discrete models of populations with predator prey populations, or many other things. &lt;/p&gt;&#xA;" OwnerUserId="1915" LastActivityDate="2014-07-17T02:08:29.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4014" PostTypeId="1" AcceptedAnswerId="4016" CreationDate="2014-07-17T07:46:40.537" Score="2" ViewCount="157" Body="&lt;p&gt;I am really passionate about theoretical and quantitative biology and I would like to build my future career around this topic.&#xA;I've just got my bachelor's degree in biology (ecology) but scince I've had only one math class through the years, my math skills are very poor, so I have problems in understanding even simpe models. &#xA;My main interests are population dynamics, adaptive dynamics and game theoretical models of social conflicts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to improve my skills, but I have just no idea where/how to start. What kinds of math should I try to focus on learning to get where I want to go?&lt;/p&gt;&#xA;" OwnerUserId="1917" LastEditorUserId="1333" LastEditDate="2014-07-17T19:21:39.260" LastActivityDate="2014-07-17T19:21:39.260" Title="Advice on studying mathematical biology" Tags="|reference-request|self-learning|reference-books|applied-mathematics|biology|" AnswerCount="1" CommentCount="2" ClosedDate="2014-07-17T21:03:41.177" ContentLicense="CC BY-SA 3.0" />
  <row Id="4015" PostTypeId="2" ParentId="4011" CreationDate="2014-07-17T12:37:59.217" Score="5" Body="&lt;p&gt;(Moved from comment...) Examples in the plane: a twice-punctured annulus shows the difference between homotopy and homology, because the fundamental group (free on two generators) has a much smaller abelianization. That is, for $s,t$ two loops, one around one hole, the other around the other, $sts^{-1}t^{-1}$ is non-zero in homotopy, but is homologous to $0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, two circles attached at a single point (equivalently, a figure-eight) has the homotopy type of a twice-punctured disk, but is visibly of lower dimension.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-07-17T12:37:59.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4016" PostTypeId="2" ParentId="4014" CreationDate="2014-07-17T14:05:45.153" Score="1" Body="&lt;p&gt;Basically, you want a course of study in applied, as opposed to pure, mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such courses would include probability and statistics. Also &quot;engineering mathemetics&quot; such as differential equations, which studies the interaction of various parties in an ecosystem.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-07-17T14:05:45.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4017" PostTypeId="2" ParentId="3945" CreationDate="2014-07-17T15:13:33.850" Score="3" Body="&lt;p&gt;I believe your question refers to the &lt;a href=&quot;http://mybroadband.co.za/news/general/103307-sa-has-worst-mathematics-science-education-in-the-world-wef.html&quot; rel=&quot;nofollow&quot;&gt;World Economic Formum rankings&lt;/a&gt;. As &lt;a href=&quot;http://mybroadband.co.za/news/government/103677-truth-about-sas-mathematics-science-education-quality.html&quot; rel=&quot;nofollow&quot;&gt;the government&lt;/a&gt; and &lt;a href=&quot;http://mg.co.za/article/2014-06-13-theres-madness-in-wef-methods&quot; rel=&quot;nofollow&quot;&gt;Mail &amp;amp; Guardian&lt;/a&gt; noted in their responses, the WEF rankings are based on outside perception, not student achievement. Nevertheless, student achievement makes it clear that South Africa has among the worst education systems in the world. Although the matric pass rate for mathematics has increased recently, some have argued that the standard has &lt;a href=&quot;http://mybroadband.co.za/news/government/94380-matric-results-a-fraud-experts.html&quot; rel=&quot;nofollow&quot;&gt;decreased&lt;/a&gt;. The most frequently quoted stat is that the &lt;a href=&quot;http://www.education.gov.za/Curriculum/AnnualNationalAssessment/ANATest2013Grade9/tabid/1027/Default.aspx&quot; rel=&quot;nofollow&quot;&gt;2013 grade 9 maths&lt;/a&gt; average on the &lt;a href=&quot;http://www.education.gov.za/LinkClick.aspx?fileticket=Aiw7HW8ccic%3D&amp;amp;tabid=424&amp;amp;mid=1831&quot; rel=&quot;nofollow&quot;&gt;Annual National Assessment (ANA) was only 14%.&lt;/a&gt; Unfortunately, South Africa has not participated in &lt;a href=&quot;http://en.wikipedia.org/wiki/Programme_for_International_Student_Assessment&quot; rel=&quot;nofollow&quot;&gt;PISA&lt;/a&gt; for many years, but the &lt;a href=&quot;http://en.wikipedia.org/wiki/Trends_in_International_Mathematics_and_Science_Study&quot; rel=&quot;nofollow&quot;&gt;TIMS&lt;/a&gt; indicates that South Africa's grade 9's scored &lt;a href=&quot;http://www.hsrc.ac.za/uploads/pageContent/2929/TIMSSHighlights2012Dec7final.pdf&quot; rel=&quot;nofollow&quot;&gt;next to last out of forty two&lt;/a&gt; countries, despite the fact that in nearly all other countries the test was given to grade 8 students!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To understand South Africa's education problem it's important to go back and acknowledge that South Africa has made significant progress since the end of Apartheid. Under &lt;a href=&quot;http://en.wikipedia.org/wiki/Bantu_Education_Act,_1953&quot; rel=&quot;nofollow&quot;&gt;Bantu education&lt;/a&gt;, for example, black South Africans were denied any meaningful amount of mathematics education. H.F. Verwoerd famously said (back in 1953):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is no place for [the Bantu] in the European community above the level of certain forms of labour ... What is the use of teaching the Bantu child mathematics when it cannot use it in practice?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It's difficult to quantify the lingering effect of Bantu education now that today's students were all born after the end of apartheid in 1994, but I'd say offhand that South African maths education is recovering from Bantu education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is very clear that since 1994, South Africa's education system has been well funded. For example, in &lt;a href=&quot;http://africacheck.org/reports/is-sas-education-system-the-worst-in-africa-not-according-to-the-data/&quot; rel=&quot;nofollow&quot;&gt;2007&lt;/a&gt;, South Africa spent USD 1 225 per pupil, whereas Kenya (with higher maths and reading scores) spent only USD 258. Though there have been complaints about corruption (such as the &lt;a href=&quot;http://www.dailymaverick.co.za/article/2013-10-23-limpopo-textbook-debacle-oh-what-a-tangled-web/#.U8fMjY2SxT4&quot; rel=&quot;nofollow&quot;&gt;Limpopo textbook fiasco&lt;/a&gt; ) and wastefulness, but &lt;a href=&quot;http://www.southafrica.info/business/economy/policies/budget2014c.htm#.U8fKK42SxT4&quot; rel=&quot;nofollow&quot;&gt;60% of South Africa's schools are free&lt;/a&gt;, and this percent is growing. South Africa's 250 000 teachers are paid reasonably well, especially considering South Africa's very high unemployment rate and low cost of living.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, here are some suggestions as to how South African mathematics education can improve:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;strong&gt;Give teachers better training&lt;/strong&gt;. In a test of maths teachers' content knowledge, a researcher from the University of Stellenbosch &lt;a href=&quot;http://www.citypress.co.za/news/grade-6-pupils-beat-teachers-maths/&quot; rel=&quot;nofollow&quot;&gt;found that 5% of grade 6 pupils at disadvantaged schools know more than 20% of their teachers&lt;/a&gt;. The same report goes on to say that only 32% of grade 6 teachers have adequate content knowledge. &lt;a href=&quot;http://www.timeslive.co.za/opinion/columnists/2013/05/09/teachers-must-be-tested&quot; rel=&quot;nofollow&quot;&gt;Jonathan Jansen&lt;/a&gt; has made a compelling argument that teachers - rather than students - should write comprehensive national tests to see if they are fit for teaching. &lt;a href=&quot;http://www.education.gov.za/LinkClick.aspx?fileticket=RlQ3WgihTOA%3D&amp;amp;tabid=348&amp;amp;mid=1018&quot; rel=&quot;nofollow&quot;&gt;The department of education has made some effort to train teachers the recent adoption of the CAPS&lt;/a&gt; curriculum, and more training should be done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before I move on to other suggestions, I must note that this is probably the most practical thing you and I can do to improve mathematics education in South Africa. I can't think of any practical way I can solve issues 2 and 3 without becoming high up in government or education, but there are &lt;em&gt;many&lt;/em&gt; ways to promote better maths teaching. As a second-year maths teacher outside of Pietermaritzburg, I'm currently in charge of an outreach programme at a disadvantaged school where my high school students teach maths and reading to primary school students. Another teacher in my department spends much of her assisting other maths teachers by doing demonstration lessons, teaching teachers, and supporting teachers who want to improve their results. I know of others who are encouraging schools to adopt technology and helping teachers use computer labs effectively. Others are gifted at networking between schools, fundraising, running after school programmes, etc. Anyone (including those who dislike maths) can be involved in this important work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;Improve relations between the teachers union (SADTU) and the department of education&lt;/strong&gt;. Recently in &lt;a href=&quot;http://www.news24.com/SouthAfrica/News/Sadtu-plans-KZN-mass-action-20140604&quot; rel=&quot;nofollow&quot;&gt;KZN&lt;/a&gt;, SADTU has complained about low and missing teacher pay, while reports have surfaced that &lt;a href=&quot;http://www.news24.com/SouthAfrica/News/More-unions-implicated-in-teaching-posts-scam-20140514&quot; rel=&quot;nofollow&quot;&gt;principal positions have been sold for R30 000&lt;/a&gt;. In all this, the &lt;a href=&quot;http://www.news24.com/SouthAfrica/News/SADTU-PRESIDENT-EXPELLED-NTOLA-20140519&quot; rel=&quot;nofollow&quot;&gt;SADTU president was recently expelled&lt;/a&gt;. Whatever side one takes, it's very clear that there is a lot of labor unrest in South Africa. The story of principal positions being sold is especially disheartening considering the vital role principals play in developing good teachers and removing incompetent ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &lt;strong&gt;Increase standards&lt;/strong&gt;. Though the department of education has proudly announced that matric results are improving, there's little doubt that the standard of maths education has declined in recent years, especially for students taking maths literacy or receiving receive a 30% &quot;pass&quot;. &lt;a href=&quot;http://africacheck.org/reports/why-the-matric-pass-rate-is-not-a-reliable-benchmark-of-quality-education/&quot; rel=&quot;nofollow&quot;&gt;This report&lt;/a&gt; sums up the main criticisms well, including a high dropout rate, low pass rate, easier subjects, and high variance.&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="867" LastEditDate="2014-07-18T11:46:03.733" LastActivityDate="2014-07-18T11:46:03.733" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4018" PostTypeId="1" CreationDate="2014-07-17T16:10:19.113" Score="10" ViewCount="628" Body="&lt;p&gt;I am teaching (for the first time) a 2nd course in linear algebra. The students will have had a beginning course in linear algebra and a beginning course in abstract algebra. I am considering Hoffman and Kunze (which I had in an undergraduate course) or Curtis. Would you please tell me about your experiences with this course and/or these books or others? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="1919" LastEditorUserId="1608" LastEditDate="2019-01-15T15:48:18.403" LastActivityDate="2020-08-25T17:49:26.057" Title="Textbook for 2nd linear algebra course" Tags="|textbooks|linear-algebra|" AnswerCount="2" CommentCount="8" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4019" PostTypeId="1" CreationDate="2014-07-17T15:48:28.270" Score="3" ViewCount="50" Body="&lt;p&gt;I stumbled across this site while searching for Hoffman and Kunze. There was a discussion about using HK for a beginning linear algebra course. I am teaching (for the first time) a 2nd course in linear algebra. The students will have had a beginning course in linear algebra and a beginning course in abstract algebra. I am considering HK (which I had in an undergraduate course) or Curtis.  Would you please tell me about your experiences with this course and/or these books or others? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="1919" OwnerDisplayName="Angela" LastActivityDate="2014-07-17T17:04:37.697" Title="Text book for 2nd Linear Algebra course" Tags="|linear-algebra|" AnswerCount="0" CommentCount="3" ClosedDate="2014-07-17T18:16:01.720" ContentLicense="CC BY-SA 3.0" />
  <row Id="4020" PostTypeId="2" ParentId="33" CreationDate="2014-07-17T18:29:07.213" Score="2" Body="&lt;p&gt;Here is a github page that details connections between &lt;strong&gt;all&lt;/strong&gt; standards: &lt;strong&gt;&lt;a href=&quot;http://jeffbaumes.github.io/standards/&quot; rel=&quot;nofollow&quot;&gt;CCSSM Flow View&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, you can hover over the standards for &lt;strong&gt;K&lt;/strong&gt; and see how they relate to others. Moreover, you can click through to see the description of the standard, its prerequisites, and its dependent strands.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Site description:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hover over a standard to see what it depends on (blue) and what depends on it (red). The colors fade as the dependency chain to the standard becomes longer. Click on a standard for more detail on the standard and its prerequisites and dependent standards. The dependencies are derived from a &lt;a href=&quot;https://docs.google.com/spreadsheet/ccc?key=0AtYMTchBa3Y2dEpIUHpudlRfbWs0eGY1TTJ1ZE01UXc#gid=0&quot; rel=&quot;nofollow&quot;&gt;shared spreadsheet&lt;/a&gt;, referenced from a &lt;a href=&quot;http://dl.dropbox.com/u/11459286/ccssmgraph.pdf&quot; rel=&quot;nofollow&quot;&gt;document&lt;/a&gt; by Jason Zimba. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Hopefully this helps clarify both what is to be known for CCSSM, and also how the covered material will be built upon in subsequent years.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-07-17T18:29:07.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4021" PostTypeId="2" ParentId="18" CreationDate="2014-07-17T20:03:48.977" Score="6" Body="&lt;p&gt;I' m only going to answer one of the questions:&lt;br /&gt;&#xA;&lt;strong&gt;What are the disadvantages of of allowing aids during tests/exams?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;First a point of terminology: Here in Australia, a sheet of notes you are allowed to take into your exam is called a &amp;quot;cheat sheet&amp;quot;, and in Asia it's known as a &amp;quot;help sheet&amp;quot;. In the USA it seems to be most often called a &amp;quot;crib sheet&amp;quot; or &amp;quot;crib notes&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Anyway I read quite a bit about this a couple of years ago, and you can see a short literature review in my conference paper &lt;a href=&quot;https://www.merga.net.au/Public/Public/Publications/Annual_Conference_Proceedings/2011_MERGA_CP.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Student experiences of making and using cheat sheets in mathematical exams&lt;/em&gt;, David Butler &amp;amp; Nicholas Crouch&lt;/a&gt;. To summarise: some people say it improves grades, while others find no significant difference; some think it encourages a deep approach, while others think it encourages dependency; most agree it helps students feel less stressed, whereas my research indicates it actually increases stress for some students.&lt;/p&gt;&#xA;&lt;p&gt;The strongest conclusion is that different students will react differently to the opportunity to make or use a cheat sheet. From the surveys in my own research, we learned that students have widely varied expectations for what a cheat sheet is capable of, and indeed what an exam is for, and this coloured their view of how useful it was for study and stress reduction. If you choose to allow them, then you need to give students guidance in how to use them to study for understanding, how to study generally, and manage their expectations of the exam. For example, do emphasise that a lot of the exam is &lt;em&gt;not&lt;/em&gt; about memory and there will often be several questions that are not the same as anything they've seen before (if that's what your exam is like of course). I have a video of a seminar for students on this topic, &lt;a href=&quot;https://www.youtube.com/watch?v=-6GqeqabP8k&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;SEMINAR: Writing and Using a &amp;quot;Cheat Sheet&amp;quot;&lt;/em&gt;&lt;/a&gt;: &lt;a href=&quot;https://www.youtube.com/channel/UC01Nr5bWVlQeZdxlx_BPN_Q&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maths Learning Centre UofA&lt;/a&gt;. Please excuse the strange clicking in the sound, I'll redo it with better sound quality when I get time.&lt;/p&gt;&#xA;&lt;p&gt;I would recommend if you allow cheat sheets, to make sure that students are aware that they don't &lt;em&gt;have&lt;/em&gt; to make one. Indeed, I would actually &lt;em&gt;also&lt;/em&gt; provide a formula sheet with the exam and let them know what is on the formula sheet. They can use the formula sheet as a basis for making their cheat sheet, or only include extra information on top of it, or not make a cheat sheet at all. Giving them this choice helps them to have more power over their own study process.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorDisplayName="user7990" LastEditDate="2020-08-29T10:39:02.030" LastActivityDate="2020-08-29T10:39:02.030" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="4022" PostTypeId="2" ParentId="49" CreationDate="2014-07-17T20:18:23.590" Score="3" Body="&lt;p&gt;Some ideas:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Talk about why you find things interesting and about the cool stuff, even if it is at a high level. I run a maths learning centre, so the majority of students I talk to are struggling. However I still talk about the higher-level stuff in there and the &quot;weaker&quot; students say they like to hear people talk about those things even if they don't necessarily understand. They key is to do it because you think it's cool/interesting/fascinating/exciting. The student who doesn't get that bit needs to still catch your enthusiasm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Be extremely clear with your explanations. This tempers the above point so that the students with less experience don't get lost. However it is worth noting that some concepts are hard for almost everyone, and there is nothing like a quick learner to get distressed when they don't understand! So be very very clear about all the steps involved in things and give lots of thinking strategies. I find the clever ones appreciate you being clear, even if they've heard it before. You just might say something they've never thought of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arrange for lots of time to work on problems in class time. If you have tutorials/workshops as part of your program, then set them up where students solve problems in groups, and give them a range of problems to choose from. This will benefit everyone, but the quick ones will either self-select into groups where they do the harder problems, or they will take it upon themselves to help teach the other students. This will be a win-win for everyone. You do need to make sure there is a rule that they need to make sure everyone in the group is ok to move on before moving on!&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-07-18T06:54:47.680" LastActivityDate="2014-07-18T06:54:47.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4023" PostTypeId="2" ParentId="57" CreationDate="2014-07-17T20:37:05.240" Score="4" Body="&lt;p&gt;Get them to work on problems in small groups (4 to 6 students) and walk around watching/listening and giving advice. This works best if they are working at a white/black board. If you don't have enough board spaces, dark whiteboard markers work well on windows, and butcher's paper is ok at a pinch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to make your expectations clear on what is supposed to happen. I also tell them that no-one is allowed to move on without helping the others to understand. (EDIT based on recieved comment: Perhaps a better thing is to tell them they need to check if everyone is ok with moving on before moving on, rather than &lt;em&gt;requiring&lt;/em&gt; that people help others to understand.) I also tell them it's always ok to ask someone to how/why they did something, as long as they're polite about it. When you are watching/listening, look for problem-solving strategies you can help them with, misconceptions you can fix, and also good things you can point out to them to increase confidence. If everyone is stuck at the same place, tell them to leave that one for later and you'll do it at the board at the end of the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end of the class, do some presenting to the whole class, perhaps doing a problem they all struggled with. Another thing to do in the whole-class time is ask them to volunteer things they learned today. This sort of &quot;cognitive closure&quot; is good for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One practical advantage of doing group work is that once you've set up how it works, students can start working even before the other students arrive, in some cases even before &lt;em&gt;you&lt;/em&gt; arrive, so you don't have to waste time waiting for people to turn up. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of learning, you get to actually see what their understanding is and have a hope of fixing misconceptions. This can't happen very well in any whole-class setting. Imagine the situation where you &quot;got through&quot; the wole amount of content but half the class lost you at the first example. All the rest of the examples are wasted. This way what they work on is more easily targeted at what they need. Also, students at different ability-levels can engage at different levels, sometimes by self-selecting into different groups, sometimes by the stronger students acting as teachers for the others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, it's more fun for everyone!&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-07-17T23:24:30.983" LastActivityDate="2014-07-17T23:24:30.983" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4024" PostTypeId="2" ParentId="59" CreationDate="2014-07-17T20:50:52.717" Score="7" Body="&lt;p&gt;I gave a response over at this question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18/formula-sheets-and-books-during-tests-and-exams&quot;&gt;Formula sheets and books during tests and exams&lt;/a&gt; that might be useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, the education literature seems to be divided over the usefulness of open book exams, and there appears to be very little on the maths education front, with much more from medicine or literature-focussed courses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From personal experience, most students I talk to hate open-book exams because they don't know how to study for them effectively and are worried during the exam that they will miss something important in their answers that they could have found in their book. At least with a fully closed-book exam they know where they stand. Therefore, if you choose to do it, you need to train students in the skill of doing an exam with open-book. You should probably give them lots of advice about exams generally, regardless of the type of exam you do!&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-07-17T20:50:52.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4025" PostTypeId="1" AcceptedAnswerId="4029" CreationDate="2014-07-19T05:13:07.377" Score="14" ViewCount="1063" Body="&lt;p&gt;Could an effective and 'comprehensive' course on advanced math be taught through a series of fun comic books, say a fun and adventurous series of stories each exploring advanced math principles somehow entangled in the story. I know of a fair amount of advanced math books with some cartoons but I don't think they get much respect. Could one teach a PhD level cource in math with lots of cartoons?&lt;/p&gt;&#xA;" OwnerUserId="1359" LastEditorUserId="759" LastEditDate="2014-07-21T09:54:45.483" LastActivityDate="2022-12-27T14:08:00.350" Title="Teaching advanced math using books with cartoons" Tags="|undergraduate-education|course-design|graduate-education|mathematical-pedagogy|" AnswerCount="8" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="4026" PostTypeId="1" CreationDate="2014-07-19T13:50:48.720" Score="3" ViewCount="782" Body="&lt;p&gt;I would like a suggestion on the 'deepest' books in&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus and analysis (something along the lines of Rudin's)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Linear algebra&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Abstract algebra&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometry (and topology); (even something along the lines of New horizons in geometry is  acceptable).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Numerical analysis&lt;/p&gt;&#xA;&#xA;&lt;p&gt;General physics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematical physics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probability and statistics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By 'deepest' I mean the ones whose reading can be &quot;useful&quot; for you to help build some kind of mathematical maturity and gain some useful insights. In other words, I'm not asking merely for books which are good for learning a subject (there are lots, I guess), but for some books whose reading can be formative for your mathematical thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you very much.&lt;/p&gt;&#xA;" OwnerUserId="1888" LastActivityDate="2014-07-19T17:20:03.063" Title="''Deep'' maths books" Tags="|mathematical-pedagogy|reference-request|reference-books|" AnswerCount="2" CommentCount="7" ClosedDate="2021-02-04T23:08:53.583" ContentLicense="CC BY-SA 3.0" />
  <row Id="4027" PostTypeId="1" CreationDate="2014-07-19T15:56:42.880" Score="14" ViewCount="1631" Body="&lt;p&gt;As math majors and math educators we take for granted the importance of proofs and being precise.  However with I have found that non-math majors are content with &lt;em&gt;anything that looks reasonably quantitative, whether the logic is correct or not&lt;/em&gt;.  I get two types of responses why it is a waste of time.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the statement is obviously true and so we don't have to analyze&lt;/li&gt;&#xA;&lt;li&gt;the computation is too difficult therefor we should just -- &lt;em&gt;insert superstitious mumbo jumbo here&lt;/em&gt; -- &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In math courses, a typical examples are showing that $-1\times a = -a$ or that $\displaystyle \lim_{n \to \infty} \tfrac{1}{n} = 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Outside of math, words like &lt;strong&gt;good&lt;/strong&gt; or &lt;strong&gt;better&lt;/strong&gt; are usually signs of trouble since they mean a different thing to just about everybody.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I am trying to discuss a computation, I am trying to motivate the idea there can be several choices when we are trying to be precise.  Then we have to pick the best one and sometimes we get it wrong on the first try.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do I explain the notions of proof and mathematical proof and why it is necessary?  Maybe it's not?&lt;/p&gt;&#xA;" OwnerUserId="1654" LastActivityDate="2014-07-20T16:36:46.987" Title="Why do we prove things we already know?" Tags="|proofs|mathematics-in-daily-life|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4028" PostTypeId="2" ParentId="4026" CreationDate="2014-07-19T16:13:38.107" Score="0" Body="&lt;p&gt;Take a look at &lt;a href=&quot;https://math.stackexchange.com/questions/417167/mathematical-books&quot;&gt;https://math.stackexchange.com/questions/417167/mathematical-books&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math.StackExchange and MathOverflow have a quite a few &quot;book-list&quot; questions&lt;/p&gt;&#xA;" OwnerUserId="1654" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-07-19T16:13:38.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4029" PostTypeId="2" ParentId="4025" CreationDate="2014-07-19T16:19:09.407" Score="13" Body="&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20140122022649/https://forbiddenplanet.com/picks/50-best-graphic-novels/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graphic novels&lt;/a&gt; are an underappreciated means of pedagogy.  Please look at:  &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0817636889&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galois' Dream&lt;/a&gt; by Michio Kuga  It teaches:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Group Theory&lt;/li&gt;&#xA;&lt;li&gt;Differential Equations&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To first-year undergraduates from a course at University of Tokyo.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3v3j5.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3v3j5.jpg&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;sub&gt;(source: &lt;a href=&quot;https://images.springer.com/sgw/books/medium/9780817636883.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;springer.com&lt;/a&gt;)&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;p&gt;Graphics should certainly make the material more engaging, but I suspect difficulty is getting an artist and mathematician to collaborate on the same project.&lt;/p&gt;&#xA;" OwnerUserId="1654" LastEditorUserId="6053" LastEditDate="2022-12-27T14:08:00.350" LastActivityDate="2022-12-27T14:08:00.350" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="4030" PostTypeId="2" ParentId="4026" CreationDate="2014-07-19T17:20:03.063" Score="7" Body="&lt;p&gt;I think of myself as a &quot;geometer&quot; so let me try to offer suggestions for books that I take to be entries to geometry at different levels. Let me first mention that geometry is a huge field and I tried to point this out with my &quot;essay&quot; Geometry in Utopia but I have not updated  it in quite some time:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.york.cuny.edu/~malk/utopia.html&quot;&gt;https://www.york.cuny.edu/~malk/utopia.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is especially appealing about geometry is that it is so quick starting. One can get to the &quot;research frontier&quot; much more quickly than would be the case for many parts of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of these books are surveys rather than texts. Once one finds out what areas of geometry one wants to learn more about there are many ways sources to learn more. I want to emphasize that while some of these books are aimed at high school teachers or students that does not mean they can' t be read with profit by graduate students and researchers. Similarly, the books at the graduate school level can also be read by others profitably.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;High School:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Coxeter, Harold Scott Macdonald, and Samuel L. Greitzer. Geometry revisited. Washington, DC: Mathematical Association of America, 1967.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Craine, T. and R. Rubenstein (eds.) Understanding Geometry for a Changing World, NCTM, Reston, 2009.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Krause, Eugene F. Taxicab geometry: An adventure in non-Euclidean geometry. Courier Dover Publications, 2012.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;O'Rourke, J., How To Fold It, Cambridge University Press, NY, 2011&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Undergraduate&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Berger, Marcel, and Lester J. Senechal. Geometry revealed: a Jacob's ladder to modern higher geometry. Springer, 2010.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hartshorne, Robin. Geometry: Euclid and beyond. Springer, 2000.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pritchard, Chris, ed. The changing shape of geometry: celebrating a century of geometry and geometry teaching. Cambridge University Press, 2003.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Senechal, Marjorie, and George Fleck. &quot;Shaping Space—A Polyhedral Approach.&quot; AMC 10 (1988): 12. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Graduate&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Goodman, Jacob E., and Joseph O'Rourke, eds. Handbook of discrete and computational geometry. CRC press, 2010.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grunbaum, Branko, Victor Klee, Micha A. Perles, and Geoffrey Colin Shephard. Convex polytopes. New York: Interscience, 1967.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grünbaum, Branko, and Geoffrey Colin Shephard. Tilings and patterns. Freeman, 1987.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-07-19T17:20:03.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4031" PostTypeId="2" ParentId="3964" CreationDate="2014-07-19T17:45:03.137" Score="4" Body="&lt;p&gt;I was educated in different country, even so I was taught to write $1\!\frac12$ not $1 + \frac12$. I think it is less clear when $1 + \frac12$ is used.&lt;/p&gt;&#xA;" OwnerUserId="1922" LastEditorUserId="11" LastEditDate="2014-07-21T02:51:39.087" LastActivityDate="2014-07-21T02:51:39.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4032" PostTypeId="2" ParentId="4027" CreationDate="2014-07-19T18:14:29.620" Score="9" Body="&lt;p&gt;You've touched on why it's problematic for educators to only talk about proving in the context of formal proofs. Students need to be accustomed to mathematical reasoning and justification well before they ever see a formal proof. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the reason that &lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot;&gt;Common Core State Standards Mathematical Practice 3&lt;/a&gt; exists:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;CCSS.MATH.PRACTICE.MP3&lt;/strong&gt; Construct viable arguments and critique the&#xA;  reasoning of others. Mathematically proficient students understand and&#xA;  use stated assumptions, definitions, and previously established&#xA;  results in constructing arguments. They make conjectures and build a&#xA;  logical progression of statements to explore the truth of their&#xA;  conjectures. They are able to analyze situations by breaking them into&#xA;  cases, and can recognize and use counterexamples. They justify their&#xA;  conclusions, communicate them to others, and respond to the arguments&#xA;  of others. They reason inductively about data, making plausible&#xA;  arguments that take into account the context from which the data&#xA;  arose. Mathematically proficient students are also able to compare the&#xA;  effectiveness of two plausible arguments, distinguish correct logic or&#xA;  reasoning from that which is flawed, and—if there is a flaw in an&#xA;  argument—explain what it is. Elementary students can construct&#xA;  arguments using concrete referents such as objects, drawings,&#xA;  diagrams, and actions. Such arguments can make sense and be correct,&#xA;  even though they are not generalized or made formal until later&#xA;  grades. Later, students learn to determine domains to which an&#xA;  argument applies. Students at all grades can listen or read the&#xA;  arguments of others, decide whether they make sense, and ask useful&#xA;  questions to clarify or improve the arguments.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can see important foundations of formal proofs in this mathematical practice. Practices are meant to be developed throughout the mathematics curruculum, across all grade bands (at whatever level is appropriate for the students). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ways that some of these practices can be addressed are found in many places. For teachers with students before secondary school I suggest looking at &lt;a href=&quot;http://www.nctm.org/catalog/product.aspx?ID=13794&quot;&gt;Developing Essential Understanding of Mathematical Reasoning for Teaching Mathematics in Grades Pre-K-8&lt;/a&gt;. Central ideas of this book are conjecturing and generalizing, along with justifying and refuting, and how they can be a part of the mathematical activity of students in the classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're getting the idea that I am not suggesting a quick fix, you've got the right idea. I really think that students should come out of their public school education realizing that mathematics is something other than calculation -- that the mathematical practices themselves are ways of thinking that help you understand and act in the world more effectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my ideal solution is to be adamant that the mathematical practices are a focus of school mathematics. That way students will have a chance to know why conjecturing, generalizing, justifying, and refuting are important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Okay, but how do you convince people in &quot;everyday life?&quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is another book in the Essential Understandings series that deals with proof and proving for secondary school age students: &lt;a href=&quot;http://www.nctm.org/catalog/product.aspx?ID=13803&quot;&gt;Developing Essential Understanding of Proof and Proving for Teaching Mathematics in Grades 9-12&lt;/a&gt;. This is not a book on &quot;how to do a proof&quot; but, rather, how and why proofs and proving are a part of mathematics and life. I wouldn't say it addresses your question directly (although, perhaps if you consider everyday people as similar to secondary school students, there is some utility here). Even so, I like these ideas from the book:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Big Idea 4: Not all arguments are proofs.&lt;/strong&gt; &lt;em&gt;A proof is not an argument based on authority, perception, popular consensus, intuition,&#xA;  probability, or examples.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A good deal of the rest of the book talks about what proof &lt;em&gt;is&lt;/em&gt;, but this idea about what it isn't may be especially helpful when explaining to people that proof has a unique place in mathematical reasoning that is not fulfilled by these other sorts of arguments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why do we need this aspect of mathematics? &lt;em&gt;The role of proof&lt;/em&gt; is discussed in &lt;strong&gt;Big Idea 5&lt;/strong&gt;'s subcategories.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;to verify the truth or falsehood of a statement&lt;/li&gt;&#xA;&lt;li&gt;to provide insight as to why a statement is true&lt;/li&gt;&#xA;&lt;li&gt;as an entry point into the development of a new idea&lt;/li&gt;&#xA;&lt;li&gt;as a structure for communicating mathematical knowledge&lt;/li&gt;&#xA;&lt;li&gt;as an impetus for the use of precise mathematical language&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As an example of how that last one can be demonstrated, I worked on a math ed technology research project that used software to collect a whole classroom's piecewise-defined position vs. time functions and graph them on one Cartesian space. Students created narratives to go along with their functions (the context was an &quot;exciting race&quot; in which all the participants run for the exact same amount of time, have the same start and end position, but all have different ways of completing the race). One student is chosen to read his narrative, and then the whole class tries to pick out his graph from all the other ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally, it is difficult to chose one such graph out of the 30 or so student functions. And, usually, it is because their narratives lack mathematically precise language. So this becomes an opportunity to show how precise mathematical descriptions communicate much more than a loose narrative, but also how those mathematical statements can be used to justify the conjecture that &quot;Sam's race is the red graph.&quot; &quot;Why?&quot; &quot;Because he ran 6 meters/second for 10 seconds, then stopped for three seconds, and that's what the red graph represents.&quot; &quot;How do you know the red graph represents that?&quot; Etc. It is much harder to make a convincing argument without the mathematical language and the precision it brings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my advice is to come up with examples that fit into the categories listed above in the role of proof. Cater the examples to what your audience cares about. But if it's students, there are some decent resources. Look specifically to resources that focus on mathematical practices (especially CCSS MP3).&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-07-19T18:14:29.620" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4033" PostTypeId="2" ParentId="4027" CreationDate="2014-07-20T12:58:34.020" Score="2" Body="&lt;p&gt;Addressing the relatively narrow issue of the question itself, the first proof of a theorem, especially an important theorem, is not always the &quot;easiest&quot; or &quot;most insightful.&quot; Also different proofs provide different insights. A good example is Euler's Polyhedral Formula, and the associated graph theory version about planar graphs. Euler's attempts to prove&lt;/p&gt;&#xA;&#xA;&lt;p&gt;V(vertices) + F(faces) - E(edges) = 2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for (convex) 3-dimensional polyhedra was not correct (but can be fixed). David Eppstein has collected 20 proofs of this result which show different aspects of this remarkable theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ics.uci.edu/~eppstein/junkyard/euler/&quot; rel=&quot;nofollow&quot;&gt;http://www.ics.uci.edu/~eppstein/junkyard/euler/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From an educational point of view since the &quot;polyhedral formula&quot; is not true for graphs embedded on the torus, a proof at some point must use some aspect of the space into which the graph is embedded - in the plane this is the Jordan Curve Theorem. &lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-07-20T12:58:34.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4034" PostTypeId="2" ParentId="4027" CreationDate="2014-07-20T16:36:46.987" Score="3" Body="&lt;p&gt;I would say that the important point is to determine, for each public and course, &lt;strong&gt;which axioms are relevant&lt;/strong&gt; and try to convince (often this is very easy) student that the axioms can reasonably be accepted. Then everything that you want to teach and that is not an axiom should be proved, simply because if the proof was not interesting enough for the given curriculum you would have chosen the result as an axiom. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now with this I have not really answered your question, but I am not so far: with this point of view, &lt;em&gt;you&lt;/em&gt; know precisely why you want to show any given proof to your students, so its usually easier to justify them. Also, it makes teachers quite free to admit some things without guilt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We should not forget that we &lt;em&gt;never&lt;/em&gt; prove everything, and it takes a long time even for math majors to be able to prove every basic theorem from the classical axioms (ZFC say), if it happens at all. The difference between proof-oriented courses and math-as-a-tool courses is simply the deepness of the axioms compared to the most sophisticated results of the course. In some case, all results can be axioms, and the only proofs are mere application of assumed theorems. This is ok as long as it fits the goals of the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me finally consider a couple of examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, take $\lim \frac1n = 0$. If you are teaching biology majors who needs to be able to interpret asymptotic behavior of simple models, then you do not need to prove it; but you want your students to know it, so you will take it as an axiom. If you teach math majors a course one of whose goals is to teach epsilon-delta proofs, then you will rather assume a set of axioms for real numbers, define rigorously limits, and prove that $\lim \frac1n = 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second example: for a first or second year course in analysis for math majors, I prefer not to develop an integration theory (Riemann's one is cumbersome, and Lebesgue is out of the question; there are other possibilities but that's not my point here). I rather assume as axioms that there exist a theory of integration that takes a piecewise continuous function on a segment and yield a number; and that it is positive, linear, and satisfy the fundamental theorem of analysis. Then I develop the rest of the theory (e.g. integration by part, etc.) from these basic principles. It emphasizes the basic properties, contains some model proofs, and concentrates on what we do daily. For a later course, I think it is important to come back to integration and explain a way to construct integrals from deeper first principles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As students get more mathematically mature, we should both teach more sophisticated things &lt;em&gt;and&lt;/em&gt; go deeper into the foundations. Hopefully, at each step we prove things that do not sound too obvious, or only after students had time to realize how nontrivial foundations of mathematics are.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-07-20T16:36:46.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4035" PostTypeId="2" ParentId="3983" CreationDate="2014-07-20T18:38:38.837" Score="1" Body="&lt;p&gt;For students of all ages, I would go the “dynamical system road” &lt;strong&gt;with pictures&lt;/strong&gt; (never tried it though, mostly by lack of opportunity).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Introduce a problem that is in fact a simple recursive vector-valued sequence (such as the evolution of a population with youngsters and olds, in which at each step a given proportion of youngster grows old while the other die, and every old gets a given number of young children and then die), and then show pictures of the possible evolutions given different starting points (you may want to choose another model to allow for negative values).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typically, if you choose an hyperbolic $2\times 2$ matrix (i.e. two positive eigenvalues, one $&amp;lt;1$ and the other $&amp;gt;1$), you will have interesting pictures. Some very particular trajectories are going to zero or diverge linearly, with most trajectories starting close to the first type then close to the second type.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After showing pictures for different starting values, and then for different systems, you can easily introduce eigenvectors at the observed special directions, eigenvalues as rate of dilation/contraction, etc. It also gives a nice illustration of why decomposing a vector on a non-canonical basis is meaningful. Student will have a mental picture, which they usually lack even after many years of mathematical studies.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="3272" LastEditDate="2015-08-25T09:09:31.193" LastActivityDate="2015-08-25T09:09:31.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4036" PostTypeId="1" CreationDate="2014-07-20T21:00:53.540" Score="12" ViewCount="3111" Body="&lt;p&gt;&lt;strong&gt;How can I help my 12 year old daughter strengthen her math skills?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My strategy up until a year or so ago had been relaxed. I subscribe to the idea that the best motivation for learning is the intrinsic kind. Unfortunately it's become clear that my daughters intrinsic motivation to learn math is close to zero. Waiting and hoping that she'll fall in love with the subject seems like an irresponsible gamble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My new strategy since a few months back is to force her to spend more time working on math examples, on her own and with me. So far it's hard to tell if this is helping. One definite result is more antagonism between the two of us. She's frustrated because she can't spend as much time on Netflix and I'm frustrated because she rushes through the examples getting half of them wrong (if I give her a set number of examples to solve) or drags her feet (if I tell her to work a fixed amount of time).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I made her play an iPad game that teaches basic algebra and that worked pretty well. This didn't made the work more enjoyable to her (she still hated it), but it took away the appeal of rushing it or dragging her feet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there similar gamified approaches with examples to last her all the way through high school? Pointers to any iPad apps or websites like that are very welcome. Please note that English isn't her first language, so anything language-independent gets a bonus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also open to other, non-technical, approaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Some updates and further thoughts.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your questions: I'm not homeschooling. Grade wise she's passing but not much more, which has me worried that she'll be in for a real uphill struggle in the coming years if she does not get a stronger foundation now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was her age I loved math and I loved programming so much so that I ended up becoming a programmer. For the longest time I thought that any person could love any subject given the right impetus. First time I started doubting this was when I spoke to one of my instructors at University. He was involved in a project teaching philosophy to kids. Being a parent himself he tried to interest his own kids. One of his kids was totally fascinated by the subject and his other kid just found it boring. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today, as a father of three, I find this very unsurprising. Children, even within the same family, with the same upbringing, are very different and have very different interests. My five year old is showing more interest in math than my 12 year old did at that age. My three year old is has an unfearing, adventurous side that is completely absent in her older siblings. People are different. Unfortunately the educational system doesn't really care about that. In three years my 12 year old will be forced to do equations and trigonometry, and failing to master those subjects will limit her choices for higher education,  even if she decides to major in history or some other subject that doesn't actually require any math beyond basic addition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;How would your daughter like to learn origami?&quot; &lt;a href=&quot;https://matheducators.stackexchange.com/users/724/opal-e&quot;&gt;Opal E&lt;/a&gt; asks. Well a couple of years ago she asked me to show her how to fold a paper plane. So I took out a paper and started showing her: &quot;You fold along the middle. Then you bring this edge down to the middle line.&quot; Right about there she stopped me and blurted out in frustration: &quot;Why are you jabbering about lines? Just show me how to do the damned plane!&quot; I don't think modular origami is going to be a hit with her.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/users/16/jonsca&quot;&gt;jonsca&lt;/a&gt; suggests that I leverage her interest in Netflix by introducing her to the inner workings of recommendation engines. I'll give it a try but I'm not too hopeful about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of answers suggest introducing an element of play. Right now talking about math with her is more like pulling teeth. Going from that to a state of playfulness is hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel that the answers given here fail to see that some kids are just not fascinated by math, no matter how many trippy fractals you show them. How do we, as parents or educators, help these kids attain the mathematical knowledge they need to function in society?&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;A year later&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;It's been more than a year and things are looking very different. My daughter still isn't in love with math but I think that something more important has happened. She has acquired a learning mindset. Last year she was already beginning to feel, as I think a lot of children do, that she was inherently not good at math. She'd adopted a worldview in which people are either smart or dumb. The smart ones are good at math and that's the end of it. I've been working a lot on explaining and demonstrating to her that pretty much anyone can learn pretty much anything. We can't all win a Nobel prize, but we can all master the curriculum of basic education. Just like we can all learn to swim or to drive a car. I think that I've managed to take her out of her old conviction. She's now spending a lot more time with all of her schoolwork. I don't think that she'll ever love math the way I did as a kid, but a lot of the resentment that she expressed is gone, and I think that most that resentment was a consequence of the world view she held. If you think that only dumb people struggle with math then every piece of homework you can't solve without effort will be telling you that you're dumb. Who wouldn't try to do as little of that as possible?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All the time we're spending together with her schoolwork has also brought the two of us a lot closer, at least that's my feeling. But don't ask her :)&lt;/p&gt;&#xA;" OwnerUserId="1925" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-12-09T11:29:19.017" Title="Helping a reluctant 12 year old" Tags="|technology-in-education|mathematical-pedagogy|games|" AnswerCount="8" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="4037" PostTypeId="2" ParentId="4036" CreationDate="2014-07-21T00:06:53.570" Score="15" Body="&lt;p&gt;The best thing you can do for your daughter is to talk to her and especially have her talk back to you. You noted that she doesn't like doing exercises. If you want her to like math, and she doesn't like doing exercises, more exercises are not going to help her like math. She is already intrinsically motivated to do certain things. People learn when they are faced with a problem they are motivated to solve; unfortunately, many problems of 12 year olds do not have mathematical solutions. So what you have to do is construct situations that are desirable to be in, in which mathematical thinking can be a part of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have a good relationship with her, her desire to interact with you (which may wane a little into teenagerhood) is an advantage you need to use. Instead of suggesting games on the iPad, or doing exercises in isolation, I'm going to suggest more interaction between the two of you in ways that show her other aspects of mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Games like &lt;a href=&quot;http://www.nctm.org/uploadedFiles/Conferences/Annual_Meetings/pdfs_for_jumpstarts/SarahNSarahMiddle.pdf&quot; rel=&quot;nofollow&quot;&gt;the ones here&lt;/a&gt; involve mathematical thinking and involve some of the content areas that she is encountering in school at her age. And playing them with her has the advantage of you helping her to see mathematical patterns. Or, as she notices things, you can help her mathematize what she is seeing (by talking about ways to generalize, use the scholastic langauge out of the books she's using in her math classes, etc.). If you are working with her, that also solves the language problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your question specifically was &quot;How can I help my 12 year old daughter strengthen her math skills?&quot; -- you didn't specify which skills, so I am focusing on thinking mathematically in general rather than any specific problems. Especially because your description of her reaction to working exercises does not suggest &quot;more exercises&quot; is the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is: spend a bunch of time with her regularly talking, but have that talk involve math. Play games that involve mathematical thinking. Also, listen for the things she likes to do and think of how any aspect of mathematical thinking can be a part of them. Instead of working &lt;em&gt;more&lt;/em&gt; problems and worrying about whether she got them right or wrong, have her talk more (to you) about the few problems she is doing. Make sure she is doing most of the talking. Teach her to explain her reasoning in the problems. When she gets a problem wrong, focus on getting her to explain what she did and what she thought, and how she justified her thinking. This can help her to be more reflective as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may not make her like working exercises any more. I can't say. But it may help her have a way you can show her that she and math can have a better relationship than what she may be starting to believe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addressing your further thoughts&lt;/strong&gt; (&lt;em&gt;which may or may not be worth anyone's time, since they address issues related to, but apart from, the original question. This reflects the addition's slightly wider focus extending to thoughts about people in general.&lt;/em&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You express dissatisfaction with the answers given on the basis that they will be hard to follow through on (specifically, that playfulness in math education with your daughter may be hard). It could be said that this sort of difficulty is at the heart of math education - that teaching can be a very difficult endeavor. Judging by most media reporting I see about teachers, this is a profound and uncommon realization you (may have) arrived at.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You also note that people are different. You're uncovering truths about education here that I think are very important. When you hear people refer to a past of education when traditional methods were dominant and everything just worked better, this difference you refer to is what is responsible. In other words, we had a system that catered to very specific types of learners (i.e. the ones who were likely to succeed almost no matter what). And so, we had a filtering function. And the students who succeeded believed that the education that resulted in their success was the best type. And that others who did not succeed just had some sort of failing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I referred to you talking to your daughter, and more importantly getting her to talk, &lt;em&gt;I am mentioning precisely the only thing in the world that can address your observation that people are different&lt;/em&gt;. Even knowing that people are different we only learn how they are different by listening to them. Where you have generalized about the education system to be uncaring of student differences, I will grant you that this is a better description of more traditional approaches to mathematics instruction. Research has brought upon realizations that understanding student thinking is (or can be) an important key to improving teaching (Fennema et al., 1996). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many people have taken this realization and tried to understand how this changes things like teacher preparation and instruction inside (and outside) the classroom. And that has often involved getting students' ideas to be a part of instruction more and more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps it seems backwards. Why should students be talking more -- don't we want them to learn what &lt;em&gt;we&lt;/em&gt; think is right? But, ignoring that misconception of learning as putting our ideas in kids' heads, if we want to address your concern about individuality of people, we need to understand those differences. We need to understand those students. And that means more listening and more interacting. How else do you respond to differences?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To narrow this down back to your case, none of us here can tell you what motivates your daughter or what is best for her in her specific situation. That she doesn't want to do exercises &lt;em&gt;does not uniquely identify her&lt;/em&gt;. The responses given are not only notable for the playful aspects. They have also applauded and encouraged your own interaction with her in different activities. Bot parts of that are important: your involvement (talking and listening) and trying different activities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whereas &quot;play&quot; may indicate games to you, I find a more helpful (to your situation) use of the word in the writing of Bruner. Bruner (1976) noted that an important function of play in development is the combination of behaviors that might not otherwise be tried. Clearly, this can result in something being discovered. In your daughter's case, the hoped-for discovery is some activity that is unexpectedly engaging to her but also one you can help make mathematically meaningful by your presence. That last part is a special challenge of math teaching. But choosing the right activities can help with that (and you've gotten some suggestions in the responses here). Play, to Bruner, also served to provide a lower-stakes environment in which mistakes are acceptable. Since it is clear that your daughter becomes frustrated with exercises, some of the respondents here are probably thinking in line with this when they suggest playful engagement. It reduces frustration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;None of this changes my answer, and I think much of it is in line with he other responses you're getting. I hope it helps you understand a little better &lt;em&gt;why&lt;/em&gt; you are getting these sorts of responses. And also to understand that nobody really thinks that helping your daughter suddenly becomes easy with any of these suggestions.  Consider them part of a large bank of things to try because, as you noted, people are different. That makes your daughter uncharted territory. And you are the one there with the most motivation (and time, relatively speaking) to chart it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cited:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bruner, J. S. (1976). Nature and uses of immaturity. In J. S. Bruner, A. Jolly, &amp;amp; K. Sylva (Eds.), Play: Its role in development and evolution (pp. 28–64). New York: Basic Books, Inc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., &amp;amp; Empson, S. B. (1996). &lt;a href=&quot;http://www.jstor.org/discover/10.2307/749875?uid=3739696&amp;amp;uid=377752093&amp;amp;uid=2134&amp;amp;uid=2&amp;amp;uid=70&amp;amp;uid=3&amp;amp;uid=377752083&amp;amp;uid=3739256&amp;amp;uid=60&amp;amp;sid=21104374716007&quot; rel=&quot;nofollow&quot;&gt;A Longitudinal Study of Learning to Use Children’s Thinking in Mathematics Instruction.&lt;/a&gt; Journal for Research in Mathematics Education, 27(4), 403–434.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-07-23T01:52:46.647" LastActivityDate="2014-07-23T01:52:46.647" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4038" PostTypeId="2" ParentId="4025" CreationDate="2014-07-21T02:29:46.387" Score="3" Body="&lt;p&gt;Be sure to look Logicomix: An Epic Search for Truth, by Doxiadis. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2014-07-21T02:29:46.387" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4039" PostTypeId="2" ParentId="3954" CreationDate="2014-07-21T04:00:04.617" Score="2" Body="&lt;p&gt;PISA evaluates the students' abilities to see real-life situations in a mathematical point of view. My country had a terrible performance at the last assessments, and those results can be used to impulse a model that focuses on implementation of problems (complementary, but to a greater extent than the algebra exercises), and suggestions to improve reasoning for mathematical translation of real-life situations.&lt;/p&gt;&#xA;" OwnerUserId="1926" LastActivityDate="2014-07-21T04:00:04.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4040" PostTypeId="2" ParentId="3933" CreationDate="2014-07-21T04:16:47.083" Score="2" Body="&lt;p&gt;Professors worldwide should use the procedure as the basis of grading. Three reasons.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It allows the examiner to catch potential misunderstandings/mistakes, and it gives him/her the opportunity to signal them to the student.&lt;/li&gt;&#xA;&lt;li&gt;It rewards students in a way such that the work is more valuable than simple fraud (it can be easy to copy a classmate's answer, but it degrades the whole methodology, turning it into something similar to a multiple selection exam).&lt;/li&gt;&#xA;&lt;li&gt;As mathematics builds up, topics from first courses start getting trivial, and can be forgotten by students. Sometimes, a small error generated by this can ruin the answer, and deprive a student from a good grade, even if the actual course's theory is correct in the solution. Grading by procedure allows the student to receive a good (if not the whole) grade, because he has demonstrated that he/she understands and applies the actual course theory.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1926" LastActivityDate="2014-07-21T04:16:47.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4041" PostTypeId="1" AcceptedAnswerId="4043" CreationDate="2014-07-21T10:38:57.973" Score="8" ViewCount="287" Body="&lt;p&gt;In everyday life, you don't encounter many people who express that they feel a connection between mathematics and fun. Not where I live, in any case. This is true, despite that fact that many people who have achieved some math knowledge relish making use of that knowledge (and find joy in new mathematical realizations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder whether it is possible for someone who is an avid recreational reader to pick up a book, become engaged in it, and learn something about mathematics despite not seeing him or herself as a lover of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book(s) would need to be engaging, obviously. They may or may not ostensibly be about mathematics. Possibly one of more &quot;big ideas&quot; or important concepts would be covered in a way that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;The reader gains a new understanding of what math, itself, is.&lt;/em&gt; Perhaps they only encountered math as arithmetic and calculation in school and never clicked with algebra beyond some symbol manipulation. Through this book they encounter math as a way to see aspects of the world that otherwise remain unseen.&lt;/li&gt;&#xA;&lt;li&gt;OR: &lt;em&gt;The reader revisits some aspect of school math and sees it in a different light.&lt;/em&gt; They never imagined it would actually be &lt;em&gt;useful&lt;/em&gt; or &lt;em&gt;interesting&lt;/em&gt; somewhere.&lt;/li&gt;&#xA;&lt;li&gt;OR: &lt;em&gt;The reader actually learns some mathematics they can use.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My questions to you are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Do you think it's possible or probable&lt;/strong&gt; to recommend a book of this type to a non math-lover and have them later tell you &quot;I really enjoyed that book. And it turned out a good portion of it was about math.&quot;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;What specific math content or aspect of math&lt;/strong&gt; do you think would make a good subject for a book of this type?&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Do you know of any books that already exist&lt;/strong&gt; which have been or could become popular, and are as I described above?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Feel free to answer any or all of the questions.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-07-21T18:38:22.550" LastActivityDate="2014-07-22T09:41:08.903" Title="Reading Math for Fun - Could it be for Anyone?" Tags="|mathematics-in-daily-life|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4042" PostTypeId="2" ParentId="4041" CreationDate="2014-07-21T10:38:57.973" Score="2" Body="&lt;p&gt;This doesn't completely satisfy the criteria, but the book &lt;a href=&quot;http://books.google.com/books/about/Super_Crunchers.html?id=1O8-B8y7BkoC&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Super Crunchers: Why Thinking-by-numbers is the New Way to be Smart&lt;/em&gt; by Ian Ayers&lt;/a&gt; argues for why standard deviation should be a more widely used concept in everyday life. Therefore, this book could be thought of as a book that both motivates the learning of standard deviation as a concept (through the case it makes of what it can be used for in--for example--news reporting) and gives an informative introduction to the concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this book is essentially all about quantitative analysis of data (obviously mathy!). So a person would need to already be interested in that, or in the specific applications discussed (such as wine vintage pricing). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While there may be better examples, I like how this book identifies a mathematical concept and tries to make it relevant to the reader. The author may not have specifically intended it this way, but it could be seen as &quot;here's why you're going to like knowing about standard deviation, and you'll wish people used it more.&quot;&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-07-21T10:38:57.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4043" PostTypeId="2" ParentId="4041" CreationDate="2014-07-21T15:04:47.330" Score="5" Body="&lt;p&gt;A couple books spring to my mind that fit the criteria:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Alice's Adventures in Wonderland&lt;/em&gt; by Lewis Carroll is a striking example in that although it might be weak in heavily structured mathematics, it uses modular arithmetic, non-euclidean geometry, and logic amongst other fields of math to tell the story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Godel, Escher, Bach: An Eternal Golden Braid&lt;/em&gt; by Douglas Hofstadter is a clearly mathematical book covering many topic in math, but I could see that it is daunting to many readers: not just in length, but sometimes needing to spend a minute to think about each passage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my favorites is &lt;em&gt;Flatland: A Romance of Many Dimensions&lt;/em&gt; by Edwin A. Abbott. He discusses math of several dimensions along with a critique of Victorian society.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other particular example that I know of but have not looked over is &lt;em&gt;Group Theory in the&#xA;Bedroom, and Other Mathematical Diversions&lt;/em&gt; by Brian Hayes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As probably noted by these examples, it seems that geometry and logic lends itself nicely to popular books. Character arguments can be framed in a logical sense which is frequently used in the above books, while geometry is pretty clear in that when writing, you can place your mind in a non-euclidean space quite nicely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that many people, unless they already like math, might not be as appreciative of the mathematics as you might expect. They might read it for the story and not realize the mathematical concepts that it is using, critiquing, or explaining.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-07-21T15:04:47.330" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4044" PostTypeId="2" ParentId="3933" CreationDate="2014-07-21T15:32:32.593" Score="4" Body="&lt;p&gt;My last undergraduate exam (many years ago...) was for the course &quot;Microeconomic Theory II&quot; - not a 4th-year course, but we had freedom to schedule. I was aiming to get an overall &quot;Excellent&quot; grade in my BA (= above 8.5/10 as grades are measured in my country). Moreover, this was my last exam, so I wanted it to be a triumph (vanity is never far away...). We knew that the exam would require drawing diagrams, but also performing basic algebraic calculations (finding the extremum of a function -cost minimization, utility maximization, etc). So I went in with pencil, ruler, eraser, sharpener, millimetre paper to draw the diagrams, and glue-tape to glue them on the exam papers. But no calculator - we were allowed calculators only in Statistics, to perform linear regression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And indeed, there were the diagrams, and I was excited to draw them in such a high-quality (and flashy) manner, and of course, there was the &quot;find the extremum&quot; part. I did -and the end result was not a nice looking, round number. I got suspicious: we knew the professors were usually giving numerical exercises with nice round solutions - as a gift to the students, and perhaps a little easier (to the eye) to grade afterwards. So I re-checked the whole calculations &lt;em&gt;twice&lt;/em&gt; (so in all, I did them three times). I could not find anything wrong, so I thought, &quot;hey, this time, no nice and round solution&quot;. Apart from this little worry, I was pretty sure I had answered everything perfectly (i.e. completely and correctly).&lt;br&gt;&#xA;After the exam ended I realized what I had done, three times in a row: I had &quot;divided a multiplication&quot;: there was in front of my eyes something like &quot;$3 \times 4$&quot; ($=12$) and I have repeatedly calculated it as $3/4$ (=$0.75$). Down goes your triumphant final exam...  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I got a 10/10. I am not saying that this was a fair grade (because there were maybe other students that performed at the same level as me without making the silly mistake), but obviously the professor saw that it was a silly mistake, and decided to ignore it, (impressed, perhaps, by the unexpectedly executed diagrams). But, my point is, that the only reason he &lt;em&gt;could&lt;/em&gt; see it as a silly mistake, was because I had written down all the steps leading to the solution.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So while the correctness of the end-result is important, it does not really convey anything about what the student knows, if it is presented alone (and leaving also aside issues of cheating, etc). It is the arrival method that gives the instructor something to evaluate (the journey and not the destination, as it is said in other contexts...). So I would agree that &quot;answers out of nowhere&quot; should get a &quot;nowhere&quot; grade.&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2014-07-21T15:56:04.213" LastActivityDate="2014-07-21T15:56:04.213" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4045" PostTypeId="1" CreationDate="2014-07-21T16:12:52.583" Score="10" ViewCount="7377" Body="&lt;p&gt;Why are perfect numbers important?&#xA;What is the best way of introducing these numbers to a first course on number theory?&#xA;I could not find any application apart from the relation to Mersenne primes. Are there any other applications of perfect numbers?&lt;/p&gt;&#xA;" OwnerUserId="1587" LastEditorUserId="143" LastEditDate="2014-07-21T16:33:44.457" LastActivityDate="2017-03-10T05:20:18.863" Title="Why do we need perfect numbers?" Tags="|undergraduate-education|concept-motivation|interactive-teaching|number-theory|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4046" PostTypeId="2" ParentId="4045" CreationDate="2014-07-21T17:04:57.423" Score="15" Body="&lt;p&gt;I think it turns out that &quot;perfect&quot; numbers do not interact much with other parts of number theory. Some of these very old, elementary, very ad-hoc definitions of special classes of integers have proven (and will prove) to interact interestingly with other ideas, but some seem not to. It's not easy for a beginner to guess the significance or subtlety of one of these classes of integers from the innocuous elementary-sounding definitions, usually. But tradition itself lends some significance to such notions. And there is always the possibility that the future discloses some surprising connections. And sometimes the sheer long-standing irresolution of a question gives it considerable cachet, even without profound consequences.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-07-21T17:04:57.423" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4047" PostTypeId="2" ParentId="4045" CreationDate="2014-07-21T20:03:40.120" Score="2" Body="&lt;p&gt;We &quot;need&quot; the natural numbers to count, we need the real numbers to measure and we need the complex numbers to guarantee that every quadratic equation has solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whenever we have some collection of objects and we can create a new definition that makes it possible to make distinctions between the objects (some are the &quot;same&quot; and some are &quot;different&quot;) the potential exists to get new insights into the objects involved, and to other related objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, we can talk about the integers that are prime, those that are squares, cubes, the sum of two squares, etc. Some of these definitions create wonderful new theoretical and applied playgrounds. The primes are a good example. After noticing the primes then one can show the prime factorization theorem. Primes can be used to design a cryptographical system  (RSA) powerful enough to protect many financial transactions at the current time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perfect numbers create a &quot;playground&quot; for the interested. One of my undergraduate professors, Leo Zippin, made the observation to me that if some mathematician can create a &quot;nifty&quot; bit of theoretical mathematics, eventually some other mathematician will find a &quot;nifty&quot; use for those ideas. &lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-07-21T20:03:40.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4048" PostTypeId="2" ParentId="4036" CreationDate="2014-07-22T01:50:34.707" Score="5" Body="&lt;p&gt;In all honesty, and I say this not to be harsh, but it sounds like what you are doing is going to avert her to mathematics completely, and this is not desirable for obvious reasons.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As &lt;a href=&quot;https://matheducators.stackexchange.com/a/4037/&quot;&gt;the other answer&lt;/a&gt; touches on, numeracy, which is a skill that is most useful to students, regardless of their future vocations, is something that you should be emphasizing, if anything.  It's helpful to have an intuition about &quot;Exit polls indicate 54% of people (margin of error 3%) approved of the town referendum&quot; and whether she'll buy enough paint/carpet to cover a room, etc., rather than being able to solve for &quot;X&quot;.  If she likes Netflix so much, discuss with her the process of movie recommendations, as this was a super-hot topic in machine learning a few years ago.  This will give her an opportunity to apply mathematics reasoning to something she's seen and is important to her, and she's able to learn about correlation and regression in the process (I realize the process of recommendations requires a lot more than that, but it's a start).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, she may end up gravitating towards &quot;formal&quot; math in the long run anyway if what she is truly interested in requires some mathematics.  Many students (young and old) like to build things and get introduced into subjects like engineering through a &quot;product design&quot; sequence in high school.  These students soon realize that to get anywhere with the engineering theory, they need to strengthen their mathematics skills, and the association becomes more natural, rather than the regimented approach that many instructors and parents are using.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I applaud your involvement in your daughter's education, and I think your philosophy is great, but it is my humble opinion that it may not serve her best interests in the long run.&lt;/p&gt;&#xA;" OwnerUserId="16" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-07-22T01:50:34.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4049" PostTypeId="2" ParentId="4041" CreationDate="2014-07-22T09:41:08.903" Score="1" Body="&lt;p&gt;A book that may satisfy your conditions is &lt;strong&gt;&lt;em&gt;&lt;a href=&quot;http://books.google.es/books?id=EnycZwEACAAJ&quot; rel=&quot;nofollow&quot;&gt;The Cartoon Guide to Calculus&lt;/a&gt;&lt;/em&gt;&lt;/strong&gt; by Larry Gonick, well at least I have heard that some people that have love it. In fact, it is an introductory book on calculus which treats the topics of functions, limits, derivatives, and integrals.  &lt;/p&gt;&#xA;" OwnerUserId="1892" LastActivityDate="2014-07-22T09:41:08.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4050" PostTypeId="2" ParentId="4011" CreationDate="2014-07-22T13:14:53.267" Score="15" Body="&lt;p&gt;&lt;strong&gt;Homotopy equivalence v. Homeomorphism.&lt;/strong&gt;&#xA;I believe an accessible difference between homotopy equivalence and homeomorphism is that one preserves an intuitive (though hard-to-define) topological invariant, while the other almost never does: the invariant of dimension.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, any $\mathbb R^n$ is homotopy equivalent to a point. Similarly, $\mathbb R^n - \{0\}$ is homotopy equivalent to the $(n-1)$-sphere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back when I was learning this stuff, this was one of the examples where I learned quickly that homotopy equivalence does not preserve rigid invariants like dimension. And if, like me, some of your students are not so used to the &quot;flexibility&quot; or &quot;wiggly-ness&quot; of homotopical invariants, this might be taken as a sign that homology seems like a less-ambitious invariant of detecting when spaces are different. For instance, if the space of our universe had a single hole in it, homology would not be able to tell the difference between the whole of our universe and the surface of the earth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside, it is probably easiest for a student to visualize a homeomorphism, while it takes some getting used to to learn about homotopy equivalence. I believe most useful examples of homotopy equivalences are strong deformation retracts, though a notable non-example is in proving the contractibility of EG for a group G.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Preserving homology groups.&lt;/strong&gt;&#xA;This is a great place to hammer home the notion that invariants aren't just groups that simply emerge from having a space: Most useful invariants are functorial, in that maps of spaces induce maps of invariants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be useful to talk about the Whitehead-Hurewicz theorem (as a &quot;Coming Attraction&quot; for their future mathematics), which tells you that if two simply-connected, reasonable spaces admit a &lt;em&gt;map inducing&lt;/em&gt; isomorphisms on their homology groups, then those two spaces are homotopy equivalent via that map. This might also encourage your students to stay away from simply-connected spaces if you're looking for these kinds of examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Regardless, a possibly accessible example of two simply-connected spaces with isomorphic homology groups, that do not admit a homotopy equivalence between them, is $CP^2$ and $S^2 \vee S^4$. This may be a bit advanced for undergraduates, as $CP^2$ is not always covered. Also, the proof that they aren't homotopy equivalent usually utilizes cohomology, which is also very advanced. Undergraduates sometimes enjoy examples even if they can't be proven, though.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Homology groups v. Homotopy equivalence&lt;/strong&gt;&#xA;This is hard, because proving two things aren't homotopy equivalences usually requires some invariant like cohomology or homotopy groups (if the spaces begin with isomorphic homology groups).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regardless, a good example might be $S^2$ wedged with $2g$ circles, compared to the surface of genus $g$ (for $g \geq 2$). The fundamental groups of these differ so they can't be homotopy equivalent, but they have isomorphic homology groups.&lt;/p&gt;&#xA;" OwnerUserId="1935" LastActivityDate="2014-07-22T13:14:53.267" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4052" PostTypeId="2" ParentId="4036" CreationDate="2014-07-22T19:39:13.297" Score="7" Body="&lt;p&gt;I will echo the sentiment of other commenters here and say that the way you are approaching seems like it will squelch, rather than foster, any interest in mathematics your daughter may have. I will share my personal experience with my father, an engineer, and his father, an engineering professor, in an attempt to help you find ways to help your daughter have better mathematical reasoning and possibly begin to enjoy math. At the same time, this may require you to rethink your opinion of what &quot;math&quot; is, and to divorce any opinion that calculation is math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is, in my opinion, in play. Not just forcing her to play, but playing yourself. I always saw my father messing around on his computer. I think he had a mandelbrot set generator when I was little. He'd show me how clicking in different places led only to more complexity, and say they used math to generate it. I spent an hour clicking different places, going back up to the top and starting over. Later on he showed me the snowflake fractal (Koch's snowflake). I learned how to draw it myself. One day he asked me how long the snowflake's edge was. We spent some time discussing its edge length and area. Discussing fractals can be beneficial if your daughter struggles with fractions and fraction addition. You don't have to go to limits, but just try to figure out how long the snowflake's edge is with iteration 1, 2, 3. My dad was not just asking me questions, but asking himself questions and sharing them with me, stuff like &quot;I read somewhere that flower petals follow the fibonacci sequence, but I don't understand why. I'm curious. Want to join me counting flower petals?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll make a plug here for geometric games and trinkets. My dad had little puzzle toys, you know the topological ones, where you had to get a ring off of the rod, or free the rope from the wooden shape. My grandfather had 3d puzzles where we tried to arrange the blocks into a cube. My grandfather had a illusion mirror (I'm not sure what they're called) that had two parabolic mirrors and formed an illusion of whatever was inside it at the top.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tessellation blocks are fun to make patterns with, especially magnets. Give her some to decorate her wall with (but don't tell her they have anything to do with math), possibly get a magnet board on her wall. She might have some fun creating new patterns on her own, and it will give her geometric intuition. Any geometric game or puzzle can help--tangrams, tetris... &lt;a href=&quot;http://www.sciencedaily.com/releases/2009/09/090901082851.htm&quot;&gt;There is evidence tetris can improve cognitive function&lt;/a&gt; and I would not be surprised if other geometric games can do so as well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How would your daughter like to learn origami? It was a favorite pastime of mine. Make her some different origami figures, share them with her, particularly ones that move -- then find her a book and let her have at it. Preferably one that includes modular origami at the end. Kirigami is another good variant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, build something with her. As a kid my grandfather built a trebuchet. Figure out something she'd like to make and have her design it. Help her out. If you are learning at the same time, that's fine -- she needs a role model for learning, not just one telling her the answers. She needs to see that it's okay to not know the answers, and see how you find them. She needs to see you asking questions and being curious. She may very well see math as ver prescriptive and oppressive, and I hope that if you do some of the things I've listed here, she might begin to find more joy in it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: In any sense, rote calculation will be miserable for her, and I'm merely aiming to suggest some things she can do, that might be fun/interesting, that would also enhance her math ability. Another option would be to try and find some &lt;a href=&quot;http://jfmueller.faculty.noctrl.edu/toolbox/tasks.htm&quot;&gt;authentic tasks&lt;/a&gt; at her level. People don't learn unless they're motivated to learn.  &lt;a href=&quot;http://jfmueller.faculty.noctrl.edu/toolbox/examples/tasks_middle_math.htm&quot;&gt;Here are some examples for math.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2014-07-22T22:59:51.507" LastActivityDate="2014-07-22T22:59:51.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4053" PostTypeId="1" AcceptedAnswerId="4355" CreationDate="2014-07-22T20:33:42.480" Score="8" ViewCount="491" Body="&lt;p&gt;Is it easier to remember something if it is expressed in a funny and/or fascinating way rather than by learning through repetitious exercises that hopefully instill the necessary understanding ? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;limit&lt;/em&gt; concept is very hard for novices, the concept of the infinitesimal was argued about for a long time by experts before it was finally included in textbooks. I don't think repetitious often uninspiring exercises could cover the 'subtleties' of limits or differentials or any related concepts that often require $very$ creative approaches to understanding them. Yet maybe a very creative even artistic approach could make such ideas 'reachable'.&lt;/p&gt;&#xA;" OwnerUserId="1359" LastEditorUserId="1385" LastEditDate="2017-04-21T14:57:08.753" LastActivityDate="2017-04-21T14:57:08.753" Title="Learning math through fun rather than rote learning" Tags="|undergraduate-education|mathematical-pedagogy|infinitesimals|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="4054" PostTypeId="2" ParentId="4036" CreationDate="2014-07-23T03:08:57.423" Score="3" Body="&lt;p&gt;First, I think it's great that you want to get your daughter to like math. However, this seems both ambitious and (based on your descriptions) possibly misguided. It seems like your main stated worry is her possible future struggles with mathematics, rather than her possible future love of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would you do if your child were struggling with a subject such as English* or Science, or another subject where early struggles can indicate future struggles? You don't have to love Shakespeare to get an A in English, but you may have to work harder on your essays and your reading skills. Along these lines, your emphasis on &quot;math&quot; (and trying too hard to encourage a love of it) may actually be causing you and your daughter to lose track of a more general fact: If a struggle has to be overcome, sometimes you have to overcome it, whether you like it or whether you prefer Netflix. This is not a problem unique to math: Most students need to work hard to get good grades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the antagonism from your daughter: It's unfortunate that these kinds of conflicts can cause antagonism between parents and children, but it happens, and maybe someday she'll realize why you chose possible antagonism over letting her flounder in mathematics. That said, you can still try approaches that don't antagonize her too much. Even small changes like, when she wants to just &quot;know the answer,&quot; telling her the answer might signal that you are willing to listen and work with her. (This is an imagined scenario raised as an example; I know you didn't mention such an interaction in your original post.) Different students gain confidence in different ways, so it'll be important for you and her to explore the ways in which she can gain this comfort with mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, at 12, she may be socially in-tune enough to realize that the games you propose are simply attempts to get her to do math. Depending on her personality, a certain cynicism may turn her off from these games. If so, you may be better off telling her straight that she simply needs to do better at math in school. Or that you care about her future, and while you know she doesn't like it, she has to do it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to say one more thing about teaching math, because I know a lot of mathematically inclined people become caught up in teaching the love of math rather than focusing on the real need for math. It is easy to hang onto the idea of the student who learns to love math under your tutelage, but the majority of students benefit simply from feeling confident or comfortable about math. While we should encourage enthusiasm, we shouldn't create the illusion that mathematics requires enthusiasm, nor the illusion that it is a realm for only those who love it. Mathematics is also a real necessity, and whether you like it or not, you need to do it. Many of us view math as play--but for most people, it isn't; especially when they begin to dislike it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This brings me to your question: &quot;How do we, as parents or educators, help these kids attain the mathematical knowledge they need to function in society?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I have no specific answers aside from what I've already said, I should say I was very much influenced by Lisa Delpit's &quot;Other People's Children,&quot; where she talks about the same issue from the perspective of teaching English to underprivileged students. (She focuses also on students in cities, and in Native American Reservations, in the United States.) Her book influenced my educational philosophy greatly, and I recommend it for anybody interested in education in the United States.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*I apologize for using English as an example, but I don't know where your daughter is growing up, or what the equivalent &quot;native language&quot; class is in your schools. I also apologize if my answer seems to center on societal values that aren't helpful to you--I've worked mostly with inner city students in the United States, so my thought process is probably biased by that experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, in a completely different direction: Depending on your daughter's personality, a book by &lt;a href=&quot;http://www.danicamckellar.com/math-books/&quot; rel=&quot;nofollow&quot;&gt;Danica McKellar&lt;/a&gt; may help. &lt;/p&gt;&#xA;" OwnerUserId="1935" LastEditorUserId="1935" LastEditDate="2014-07-23T03:21:36.273" LastActivityDate="2014-07-23T03:21:36.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4055" PostTypeId="1" AcceptedAnswerId="4065" CreationDate="2014-07-23T03:14:08.183" Score="14" ViewCount="346" Body="&lt;p&gt;I've seen many courses labelled as 'mathematical physics', but I'm interested in knowing about the opposite: 'physical mathematics'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've noticed that some areas of mathematics which I found extremely abstract and not interesting were directly inspired by or popularized by physics: functional analysis, differential geometry, symplectic geometry and contact structures, partial and ordinary differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've noticed, however, that students often dislike interrupting a mathematical course to introduce some motivating physics. So my question is,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What topics could be included into a 3-credit hour, one semester graduate course dedicated to describing physical systems and the math which results from them? Specifically, is there a set of topics that would form a somewhat cohesive whole while allowing enough time to get to interesting math?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One specific example I've thought of is doing the first few chapters of Dirac's Principles of Quantum Mechanics to teach functional analysis from a physics viewpoint and combining this with the first three chapters of Wald's General Relativity to teach differential geometry from a physics viewpoint. However, these topics are very different, and their is probably enough material in those combined chapters to teach several courses. What topics would be interconnected but not contain too much material?&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2014-07-29T04:19:03.177" Title="Physics-based mathematical courses" Tags="|course-design|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4056" PostTypeId="1" CreationDate="2014-07-23T03:48:41.780" Score="12" ViewCount="964" Body="&lt;p&gt;Once in a discussion a colleague told me that he thinks:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is better to use women to teach maths to little children including preschoolers and children in elementary school. Also it is better to use men to teach math courses in high school and university. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that in some sense he divided students by two categories, children and adults.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have no idea about truth or falsity of his claim. Although I believe that in some cases and for special communities using a male/female math teacher could have a positive or negative influence on students.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; Is there any research to show that there is a meaningful relevance between gender of a math teacher and quality of teaching maths (for different levels of students) in any sense?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; Is above claim really true?   &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="262" LastEditDate="2014-07-23T22:13:42.393" LastActivityDate="2016-08-19T10:14:29.707" Title="Are women better math teachers for little children?" Tags="|secondary-education|reference-request|primary-education|teachers|gender|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4057" PostTypeId="2" ParentId="4036" CreationDate="2014-07-23T05:27:39.377" Score="3" Body="&lt;p&gt;I'll reiterate the suggestion to play games with your daughter and give her games to play, rather than requiring her to do exercises.  If your daughter has a limited amount of &quot;screen time&quot; (for Netflix, video games, TV, etc.) you might allow her to earn extra for any time she spends on one of these games (which would also not count against her limit, since they are &quot;educational&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some suggestions, in addition to the excellent games linked by JPBurke:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sudoku -- you can find these all over the place, and many kids who don't like math do like solving these.  If she likes Sudoku, expand her horizons to KenKen and Kakuro which require some arithmetic.  A lot of my students who &quot;don't like math&quot; enjoy working on KenKen and Sudoku puzzles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;KenKen is similar to Sudoku, but requires numbers in certain outlined regions to add up to a certain sum (in the simplest version) or to adhere to other similar restrictions.&lt;br&gt;&#xA;KenKen online: &lt;a href=&quot;http://www.kenken.com/play_now&quot; rel=&quot;nofollow&quot;&gt;http://www.kenken.com/play_now&lt;/a&gt;&lt;br&gt;&#xA;They have a mobile app too: &lt;a href=&quot;http://www.kenken.com/store/mobile&quot; rel=&quot;nofollow&quot;&gt;http://www.kenken.com/store/mobile&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kakuro is another kind of Japanese number puzzle -- these are like crosswords but you have to make certain sums, using each digit only once per entry.  Two places to download Kakuro for printing:&#xA;&lt;a href=&quot;http://akidsheart.com/math/prints/kakurop/index.html&quot; rel=&quot;nofollow&quot;&gt;http://akidsheart.com/math/prints/kakurop/index.html&lt;/a&gt;&lt;br&gt;&#xA;&lt;a href=&quot;http://krazydad.com/kakuro/&quot; rel=&quot;nofollow&quot;&gt;http://krazydad.com/kakuro/&lt;/a&gt;&lt;br&gt;&#xA;play online here: &lt;a href=&quot;http://www.youplay.com/games/view/kakuro/&quot; rel=&quot;nofollow&quot;&gt;http://www.youplay.com/games/view/kakuro/&lt;/a&gt; (among other places)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't have an iPod or iPad so I haven't tried this one but it looks excellent:&#xA;&lt;a href=&quot;https://itunes.apple.com/us/app/wuzzit-trouble/id600190128?ls=1&amp;amp;mt=8&quot; rel=&quot;nofollow&quot;&gt;https://itunes.apple.com/us/app/wuzzit-trouble/id600190128?ls=1&amp;amp;mt=8&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a great game that will help with fractions:&#xA;&lt;a href=&quot;http://illuminations.nctm.org/ActivityDetail.aspx?ID=178&quot; rel=&quot;nofollow&quot;&gt;http://illuminations.nctm.org/ActivityDetail.aspx?ID=178&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This one is good and quick for thinking about numbers in terms of their factors:&#xA;&lt;a href=&quot;http://illuminations.nctm.org/Activity.aspx?id=4134&quot; rel=&quot;nofollow&quot;&gt;http://illuminations.nctm.org/Activity.aspx?id=4134&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The SET daily puzzle is great for finding patterns:&#xA;&lt;a href=&quot;http://www.setgame.com/set/daily_puzzle&quot; rel=&quot;nofollow&quot;&gt;http://www.setgame.com/set/daily_puzzle&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope that helps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One bonus game: The Logical Journey of the Zoombinis is an excellent game for developing logic and problem solving skills.  Unfortunately, it's an old PC game that doesn't run well on modern computers.  But apparently TERC is planning to put out a mobile version sometime this year.  &lt;a href=&quot;https://external-wiki.terc.edu/display/ZOOM/Home&quot; rel=&quot;nofollow&quot;&gt;https://external-wiki.terc.edu/display/ZOOM/Home&lt;/a&gt; If that happens, it's definitely worth getting.&lt;/p&gt;&#xA;" OwnerUserId="817" LastActivityDate="2014-07-23T05:27:39.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4058" PostTypeId="2" ParentId="4056" CreationDate="2014-07-23T10:27:43.390" Score="20" Body="&lt;p&gt;Most of the research on gender and math education is focused on student gender differences. However, a few references can be found that focus on the what differences there may be based on the gender of the teacher. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that appears to be common among some of these studies appears to be that perception of student performance varies based on gender of teacher and student. For instance, Ehrenberg, Goldhaber, and Brewer (1995) found that white female teachers evaluated their white female students more highly than did white male teachers. Yet they did not find learning differences. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A different analysis of the same data (Dee, 2006) appears to agree that perception differences of student performance exist. And the analysis found performance distances as well. However, the analysis also found that there were confounding circumstances in that same data set that led the author to exclude math classes from his conclusion that the gender of the teacher does matter in school. Dee referred to differences that (he determines) were probably attributable female teachers being assigned more difficult students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, there is a perception in line with what your colleague suggested, and it appears to influence the outcomes in this particular data set, namely the National Education Longitudinal Survey (NELS) from 1988. Teachers were assigned the less advanced math students, resulting in a finding of lower achievement for students of female teachers of math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other studies have noted: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1348/000709999157527/abstract;jsessionid=3FAF8A628BAE333C195383BC66200995.f03t02?deniedAccessCustomisedMessage=&amp;amp;userIsAuthenticated=false&quot;&gt;differences in perception of student performance based on teacher gender&lt;/a&gt;; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/0013188990410106#.U8-MvvldV8E&quot;&gt;differences in teachers assessment of student achievement based on student gender, but inconclusive links between belief and teacher gender&lt;/a&gt;; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://link.springer.com/article/10.1023/A:1014892408105&quot;&gt;some evidence that female teachers, in some secondary school mathematics classrooms, interact more with male students&lt;/a&gt;;&lt;/li&gt;&#xA;&lt;li&gt;But not much to make a case that there is a clear link between teacher gender and performance.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I think we can safely say that this perception you refer to exists, but the &lt;em&gt;basis&lt;/em&gt; for it is dubious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder what reputable evidence people would refer to if challenged in this assumption. And, of course, anyone making such a claim should be challenged to produce some sort of evidence. Based on a look at what's out there, I can say that I would not want to be the one having to defend a claim like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cited:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dee, Thomas S. (2006). The Why Chromosome. Education Next. Retrieved from &lt;a href=&quot;http://educationnext.org/the-why-chromosome/&quot;&gt;http://educationnext.org/the-why-chromosome/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ehrenberg, R. G., Goldhaber, D. D., &amp;amp; Brewer, D. J. (1995). Do teachers’ race, gender, and ethnicity matter? Evidence from the National Education Longitudinal Study of 1988 [Electronic version]. Industrial and Labor Relations Review, 48(3), 547-561.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-07-23T10:27:43.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4059" PostTypeId="2" ParentId="3964" CreationDate="2014-07-23T14:26:55.607" Score="8" Body="&lt;p&gt;That's the way non-mathematicians write it.  Surely you have seen road signs, showing distances, with entries like&#xA;$$&#xA;1\,\frac{1}{2}\;\text{mi}&#xA;$$&#xA;at least if you are in the US....  I wonder, in Europe would they write it that way, too? or maybe, for example,&#xA;$$&#xA;5.5\;\text{km}&#xA;$$&#xA;???&lt;br&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/He710.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2015-01-23T22:15:12.267" LastActivityDate="2015-01-23T22:15:12.267" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="4060" PostTypeId="1" AcceptedAnswerId="4345" CreationDate="2014-07-24T00:04:58.647" Score="8" ViewCount="296" Body="&lt;p&gt;I teach college students and wanted to make my classes more engaging. I found this recent article interesting/useful &lt;a href=&quot;http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html&quot;&gt;http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html&lt;/a&gt;? My question is besides Lampert's book are there other books that discuss &lt;strong&gt;math&lt;/strong&gt; teaching techniques that promote class participation?&lt;/p&gt;&#xA;" OwnerUserId="1941" LastActivityDate="2014-09-08T12:49:33.953" Title="Teaching math in an engaging way" Tags="|class-participation|reference-books|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4061" PostTypeId="2" ParentId="4045" CreationDate="2014-07-24T02:13:04.110" Score="8" Body="&lt;p&gt;Mathematically, even perfect numbers give a good number theory example to the general idea of classification, i.e. all even perfects have a specific form. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use perfect numbers in my number theory class for two or three pedagogical reasons: with some trial and error (and the help of some computational software), I have the students essentially discover Euclid's formula. There's a lot of good conjecturing and testing along the way. We then look at the proof which is one of the few that are long enough to be nontrivial from the student view, yet easy enough to follow that they can actually understand it (especially after the preliminary explorations). When it's all done, I talk a little about odd perfect numbers: no one knows of any nor can prove they don't exists. This ties into a common thread in my upper level courses: math is still alive, still being discovered, still full of the unknown.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this being said, I surely wouldn't look down on a teacher or student who knew nothing about perfect numbers and didn't have them covered in class.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-07-24T02:13:04.110" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4062" PostTypeId="2" ParentId="4056" CreationDate="2014-07-24T02:35:48.320" Score="12" Body="&lt;p&gt;A study I have read indicates that &lt;a href=&quot;http://www.pnas.org/content/107/5/1860.long&quot;&gt;female teachers' math anxiety (negatively) affects girls' math achievement&lt;/a&gt; at the elementary level, which would run counter to your claim that women are &quot;better&quot; for math teachers at the elementary level. (However, as a woman myself, this to me means that we need to do better at countering math anxiety in our female elementary teachers!). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As others mentioned, this effect could possibly be due to contributing to stereotype threat. As far as I know, there is no study that indicates that women are objectively better at teaching elementary math and men objectively better at teaching upper-level math, or vice versa, or really much of a comparison at all. My anecdotal experience would suggest otherwise, as well (my best math teachers/professors were female), but while some slight overall correlation might exist between math teaching ability and gender I doubt we currently have a means to indicate whether it is due to some innate gender difference as opposed to social and cultural phenomena. The idea of innate vs. constructed gender differences is an ongoing and heated discussion.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2014-07-24T02:35:48.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4063" PostTypeId="2" ParentId="4060" CreationDate="2014-07-24T12:09:11.650" Score="10" Body="&lt;p&gt;One recommendation that I have also given in response to similar questions, check out Jo Boaler's edX course: &lt;a href=&quot;http://online.stanford.edu/course/how-to-learn-math-for-teachers-and-parents-s14&quot;&gt;How to Learn Math for Parents and Teachers&lt;/a&gt;. It's currently in progress, but I'm sure it will run again. (I don't know if you can join while it's going on).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the book &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0470276673&quot;&gt;Powerful Learning: What We Know About Teaching for Understanding&lt;/a&gt; has a chapter by Alan Schoenfeld on an approach to mathematics learning that involves inquiry. This book is also good for references to the research that supports the teaching of mathematics with a focus on understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some thoughts on making engagement and participation relevant to mathematics learning:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try to think carefully about both engagement and participation as they apply to mathematics education. Sometimes people consider students engaged if they look interested. I consider engagement in a math classroom not to be interest, but to be &quot;students are engaged in explaining, demonstrating, or arguing about mathematics.&quot; I might also add &quot;listening, because they are about to do one of those other things&quot; but that is not observable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You'll see workshops about how to get students engaged by using candy, or how a particular technology increases student engagement and participation because suddenly students appear interested or they are doing more than sitting in their chairs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes: activity and apparent interest are good things. But when we're talking about student engagement and participation, I recommend you ask an additional question: what mathematics is a student engaged in? And how are they participating in some mathematical practice? When a person tells you an engaging app is teaching kids math, ask them: what kind of mathematical thinking is going on? The 2048 game involves numbers. And it's certainly engaging. But apart from matching symbols (you can replace all the numbers with pictures and still play the game without ever thinking of doubling or powers of 2), it's not a mathematical game. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To circle back around to your question, if the classroom expectations are that students must argue and explain using mathematics, and they understand how (or how to make) their contributions are both valuable and mathematical (even when they are struggling), the classroom will be engaging. The students may need to become accustomed to this new way of being a student in a math classroom, but that's mathematical engagement In My Humble Opinion. &lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-07-24T12:09:11.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4064" PostTypeId="1" CreationDate="2014-07-24T13:44:45.557" Score="6" ViewCount="3422" Body="&lt;p&gt;There are far too many mathematicians for me to name here, you only need to wiki a mathematician of note to find out he is Jewish. Are they taught to think logically and analytically from a young age? What are the possible explanations for this? Have their been any studies on this? Might there be any aspects of Jewish culture or pedagogy that would give rise to their success in mathematics? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P. S. If someone could edit the tags that would be appreciated, as I'm not entirely sure they're the correct ones.&lt;/p&gt;&#xA;" OwnerUserId="1677" LastEditorUserId="1677" LastEditDate="2014-07-25T13:12:30.477" LastActivityDate="2014-07-27T04:28:28.457" Title="What is it about the education of the Jewish people that has allowed them to produce such remarkable Mathematicians?" Tags="|mathematical-pedagogy|education-research|cultural-differences|religious-institution|" AnswerCount="4" CommentCount="7" ClosedDate="2014-07-27T04:11:51.113" ContentLicense="CC BY-SA 3.0" />
  <row Id="4065" PostTypeId="2" ParentId="4055" CreationDate="2014-07-24T14:10:37.787" Score="3" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Virasoro_algebra&quot; rel=&quot;nofollow&quot;&gt;Virasoro Algebra&lt;/a&gt; arises in the study of the quantum mechanics of strings. This is one of the fundamental examples which leads to the study of vertex operator algebras. When I saw the derivation in a string theory course it wasn't too lengthy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another topic would be exotic four dimensional manifolds. &lt;a href=&quot;http://en.wikipedia.org/wiki/Donaldson_theory&quot; rel=&quot;nofollow&quot;&gt;Donaldson Invariants&lt;/a&gt; arose from the study of &lt;a href=&quot;http://en.wikipedia.org/wiki/Instanton#Mathematics&quot; rel=&quot;nofollow&quot;&gt;instantons&lt;/a&gt;. The physics of instantons was used to discover the existence of four dimensional manifolds. I gather that the &lt;a href=&quot;http://en.wikipedia.org/wiki/Seiberg%E2%80%93Witten_theory&quot; rel=&quot;nofollow&quot;&gt;Seiberg-Witten&lt;/a&gt; theory has supplanted some of the Donaldson theory and it would also appear to fit the bill of physics being used to do mathematics. I believe that is the theme of Witten's work which impacts mathematics; some insight in quantum field theory has been used to &lt;em&gt;do&lt;/em&gt; math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a lower level, since geometry is one of the oldest forms of physics, you can argue that physics is used whenever we resolve a proof by geometric intuition. For example, the proof of Rolle's Theorem; &quot;what goes up must come down, so somewhere it had to stop going up and come down&quot;. That would be an example of using physics to prove mathematics. I say &quot;prove&quot; in an very relaxed sense of the word!&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-07-24T14:10:37.787" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4066" PostTypeId="2" ParentId="4064" CreationDate="2014-07-24T17:13:12.327" Score="7" Body="&lt;p&gt;My answer is just my educated guess and so it is probably flawed. But I think besides mathematicians, you could also include writers or other scientists, think Freud or  Oppenheimer for example. Steven Gubkin in his comments already pointed the major points. I just wanted to add the practice of Judaism is centered on reading and studying the Torah but since the Torah is not clear cut with regard many aspects (oral law) - a great part of studying centers on reading the &lt;strong&gt;debates&lt;/strong&gt; amongst the old rabbis of the Talmud. That is: most of the religious law emerged as result of debates using logical arguments on what is the correct interpretation of some text. The analysis and dissection of the text, creates a culture that gives incentives to debates and arguments based on logic. This culture goes beyond the religious practitioners. Freud, again as an example, was somewhat anti-religious: after his wedding, he forbade his wife to light candles on Friday evenings. Perhaps you could ask the question at &lt;a href=&quot;https://judaism.stackexchange.com/&quot;&gt;Mi Yodeya&lt;/a&gt; at Stack Exchange.&lt;/p&gt;&#xA;" OwnerUserId="1941" LastEditorUserId="-1" LastEditDate="2017-04-13T12:42:08.043" LastActivityDate="2014-07-27T03:17:38.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4067" PostTypeId="2" ParentId="4064" CreationDate="2014-07-24T20:45:54.957" Score="12" Body="&lt;p&gt;This question seems like a big opportunity for casual social science conjecturing which may or may not be productive. I hope I can clarify a couple of things in my response.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to your supporting statement, your question is &quot;what are the possible explanations for why so many mathematicians you look up on Wikipedia are Jewish?&quot; For the purposes of this answer I'm going to assume we are talking about one monolithic group with genetics and culture in common so that I can speak hypothetically (while acknowledging that others have challenges to that aspect of the question, and that they are at least worth thinking about).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One explanation &lt;em&gt;could be&lt;/em&gt; that the group &lt;em&gt;is actually&lt;/em&gt; better at math. To just ask &lt;em&gt;why&lt;/em&gt; this group is better at math, however, would be to &lt;em&gt;assume&lt;/em&gt; they are better at math (so I won't do that). It is not actually demonstrated by the wiki search mentioned. If there is more support for this sort of explanation--for instance, a difference in the comparison of means as part of some quantitative study--we could search for an explanation of why those means are different, and look to culture, perhaps. But we are not actually at that point, unless someone provides such evidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another explanation we could conjecture about is that there might be similar means in the Jewish population and other populations, but there might be a greater variance in the data. Meaning that good Jewish mathematicians become more accomplished and thus are among the most famous mathematicians. And, if the means are similar, there would be similarly exceptional low achievers in the same population (or more slightly less accomplished mathematics users-- depending on the shape of the curve). We would look for a different sort of explanation if this is the quantitative difference we noticed in comparison of Jewish and other groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There could be cultural reasons that some accomplished Jewish mathematicians are better supported for some reason. &lt;em&gt;Or&lt;/em&gt; that they have happened to be in groups at opportune places or times (enough to make really important contributions, but not in great enough numbers to change the mean over a reasonable sample of Jewish people). Yes, smart people &quot;math better&quot; but they do it even betterer in groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conjectures are great starting points for research questions. But outside of using them for that purpose (to propose research) or having actual research and discussing how to extend from that research, I think we enter a realm of imagination rather than scientific enlightenment. One benefit of good social science is that assumptions must be made explicit. In casual conjecturing you will see assumptions pop up like dandelions and participants trample over them toward a possible explanation that catches their eye. I think I would be a lousy research assistant if I didn't point out that outside of scientific research, we often see a compelling conclusion driving whatever justification can connect things. And this is (at the very least) unfair to the groups of people subjected to our conjectures. &lt;em&gt;Discretion and caution&lt;/em&gt; are called for. Even if we &lt;em&gt;think&lt;/em&gt; we're saying something positive.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-07-24T20:45:54.957" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4068" PostTypeId="2" ParentId="4064" CreationDate="2014-07-25T03:21:16.690" Score="7" Body="&lt;p&gt;If you're looking up German mathematicians of note of the 1870s to the 1930s, there are some historical reasons for a high fraction of Jews:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;strong&gt;German academic anti-Semitism&lt;/strong&gt; may have pushed Jews away from higher-prestige fields like experimental physics.  So Jews with physical and mathematical interests may have focused on their mathematical interests or opportunities as a result.  (See Jungnickel and McCormmach, Intellectual Mastery of Nature, &lt;a href=&quot;http://books.google.com/books?id=7eggQO_Eey4C&amp;amp;pg=PA286&amp;amp;dq=%22intellectual%20mastery%20of%20nature%22%20jewish&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=LMLRU-e4AYSdyAT7k4L4CA&amp;amp;ved=0CB4Q6AEwAA#v=onepage&amp;amp;q=%22intellectual%20mastery%20of%20nature%22%20jewish&amp;amp;f=false&quot;&gt;pp. 286-287&lt;/a&gt;:  &quot;Among academic fields in Germany, theoretical physics had an unusually high proportion of Jews, who found more opportunity there than in the more desirable field of experimental physics&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;Felix Klein&lt;/strong&gt; believed that Jewish mathematicians tended to have particular habits of thought, and promoted those habits of thought as part of a diverse approach to mathematics.  He had enough influence over hiring in German math departments for these views to have an effect.  (See &lt;a href=&quot;http://www.jstor.org/stable/231607&quot;&gt;David Rowe&lt;/a&gt;, &quot;'Jewish mathematics' at Gottingen in the era of Felix Klein&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &lt;strong&gt;The German mathematical emigration of the 1930's&lt;/strong&gt; may have led to German Jewish mathematicians becoming better-known that their non-Jewish counterparts.  (See Sanford Segal on Mathematicians Under the Nazis, or his entry on Helmut Hasse in the Dictionary of Scientific Biography.)  The Jews who emigrated to the U.S. may have became more integrated into the US-centric mathematical world of the later 20th century than the non-Jews who did not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whatever our beliefs about the value of Jews in various fields of math or physics, similar beliefs among various Germans had theirs, and that was enough to cause historical effects.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-07-25T03:21:16.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4069" PostTypeId="2" ParentId="3964" CreationDate="2014-07-25T12:06:29.977" Score="7" Body="&lt;p&gt;This is a typical case of emphasis trade off. Omitting the $+$ sign wants to show that this is a single number (result) rather than an arithmetic expression to be processed further. I don't see any problem with it as long as the mentioned confusion with multiplication can be avoided in the context. Numbers written in the mixed form can be compared to each other more easily than pure fractions by mortals like me.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2014-07-25T12:06:29.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4070" PostTypeId="1" AcceptedAnswerId="4075" CreationDate="2014-07-25T13:53:17.077" Score="11" ViewCount="826" Body="&lt;p&gt;I know Euclid's proof of there being infinite number of primes. I want to let my brother (age 15) arrive at that proof by himself. He knows definition of a prime number (number divisible only by 1 and itself). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First when I asked him how many prime numbers are there, he said there must be an infinite number of them. Why? Because it's not possible to know which one is the largest. Why? ... blank.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I told him that since larger a number becomes, more numbers are there below it. So chances of a larger number $N$ being divisible by at least one of the $N-2$ numbers increases as $N-2$ larges with $N$. He agreed with this logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I asked him if he could generate a prime number larger than any two given given primes (and a third constant) using some simple function of them. I was trying to make him arrive at $\text{prime}_1 \times \text{prime}_2 + 1$ formula. I gave him two primes $2$ and $5$ and asked to think of some examples. He summed them -&gt; $7$ is a prime. Then I asked him to use $1$ also, he said $251$ is a prime. Then he got bored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What approach can I use to intuitively arrive at the proof?&lt;/p&gt;&#xA;" OwnerUserId="1945" LastEditorUserId="143" LastEditDate="2015-12-02T23:54:27.277" LastActivityDate="2015-12-02T23:54:27.277" Title="How to arrive at infinitude of primes proof?" Tags="|proofs|number-theory|prime-numbers|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4071" PostTypeId="1" CreationDate="2014-07-25T15:21:33.430" Score="17" ViewCount="929" Body="&lt;p&gt;In general, I think (and I am told by students) that I am good as a tutor. However, I would like to become more rigorously familiar with the discovery-based and inquiry-based learning applied to mathematics. Can you suggest some reference material please?&lt;/p&gt;&#xA;" OwnerUserId="1888" LastEditorUserId="39" LastEditDate="2015-01-26T14:46:29.950" LastActivityDate="2015-04-28T19:24:54.883" Title="Discovery-based and inquiry-based learning" Tags="|reference-request|education-research|teacher-development|inquiry-based-learning|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4072" PostTypeId="2" ParentId="4070" CreationDate="2014-07-25T15:42:44.330" Score="6" Body="&lt;p&gt;I think using actual small primes actually detracts from finding the solution.  If you are thinking about finding a number that is not divisible by 2,3, or 5, it is easy to come up with one (11) without having to use a formula. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, I would get him thinking about primes more symbolically.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given 3 primes, $p_1, p_2, p_3$ what is their LCM?  What is their GCD?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the LCM and GCD of $p_1^2, p_1p_2, p_1p_3$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are all the factors of $p_1p_2p_3$?  (for this one, you  might need to do an example with small primes first, but the idea is for him to eventually &quot;see it&quot; using the variables)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this all makes sense, then I would say he would be primed (no pun intended) to think about the problem of &quot;can you think of an expression using $p_1, p_2, p_3$ that is guaranteed not to be divisible by any of $p_1, p_2$ or  $p_3$?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I think most of use have to have a number of clever proofs shown to us before we are ready to discover them on our own.&lt;/p&gt;&#xA;" OwnerUserId="817" LastEditorUserId="817" LastEditDate="2014-07-25T16:54:25.920" LastActivityDate="2014-07-25T16:54:25.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4073" PostTypeId="2" ParentId="4060" CreationDate="2014-07-25T16:05:57.420" Score="2" Body="&lt;p&gt;This &quot;comment&quot; addresses a narrow aspect of your post, the article in the New York Times (July 23, 2014). This article was very disappointing. It confuses arithmetic (and implicitly basic algebra) and mathematics which is a much bigger subject than arithmetic (and algebra). Arithmetic and &quot;realistic&quot; applications of arithmetic are very important but how about discussing with elementary school students the mathematics behind how some spell checkers work (Hamming distance and/or edit distance)?. If we teach &quot;content&quot; that is not optimal for the K-12 audience  (and there is no wide agreement about what content goals here should be despite CCSS-M &quot;hype&quot;) then no matter how well we teach it there is a high price to pay. Oversimplifying very greatly (smile, there is a CCSS-M that deals with modeling) we have designed a curriculum with too much attention to the needs of future STEM majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might find this article of mine of interest here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=EvuQdO3h-DQC&amp;amp;pg=PA89&amp;amp;lpg=PA89&amp;amp;dq=malkevitch+public+perceptions&amp;amp;source=bl&amp;amp;ots=gIU3iCOsC8&amp;amp;sig=8hRG0ROR-J43c2SC0PWvI-IXj90&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=qXbSU97wGMSZyATouIDQBQ&amp;amp;ved=0CDgQ6AEwAw#v=onepage&amp;amp;q=malkevitch%20public%20perceptions&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;http://books.google.com/books?id=EvuQdO3h-DQC&amp;amp;pg=PA89&amp;amp;lpg=PA89&amp;amp;dq=malkevitch+public+perceptions&amp;amp;source=bl&amp;amp;ots=gIU3iCOsC8&amp;amp;sig=8hRG0ROR-J43c2SC0PWvI-IXj90&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=qXbSU97wGMSZyATouIDQBQ&amp;amp;ved=0CDgQ6AEwAw#v=onepage&amp;amp;q=malkevitch%20public%20perceptions&amp;amp;f=false&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or it can be downloaded from this site, which also contains other background articles about some of these issues:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.education.umd.edu/MathEd/conference/briefbook.html&quot; rel=&quot;nofollow&quot;&gt;http://www.education.umd.edu/MathEd/conference/briefbook.html&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-07-25T16:05:57.420" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4074" PostTypeId="1" AcceptedAnswerId="4391" CreationDate="2014-07-25T17:28:41.583" Score="-4" ViewCount="275" Body="&lt;p&gt;Has learning through doing repetitive exercises and mechanical non-creative exercises  been researched and analysed sufficiently for College and University level courses? Have there been surveys and polls for High school and University Level students about whether they appreciate these repetitive and uninspiring exercises for learning and if it helps them? Have any College level students indicated Repetitive and uninspiring often mechanical learning is tedious or monotonous and doesn't really explain that much?&lt;/p&gt;&#xA;" OwnerUserId="1359" LastEditorUserId="376" LastEditDate="2014-10-04T12:26:22.413" LastActivityDate="2014-10-04T12:26:22.413" Title="About Rote learning" Tags="|undergraduate-education|mathematical-pedagogy|" AnswerCount="1" CommentCount="9" ClosedDate="2014-10-05T02:49:50.530" ContentLicense="CC BY-SA 3.0" />
  <row Id="4075" PostTypeId="2" ParentId="4070" CreationDate="2014-07-25T18:04:16.880" Score="14" Body="&lt;p&gt;It might not be possible to get your brother to arrive at the proof himself, no matter how much you scaffold it. (&quot;You can lead a horse to water&quot; and all that.) If he's into maths and appreciates a good proof, you might get the desired enthusiasm by just showing him the proof!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, here's an idea. Forget primes for a moment, and think about divisibility. If you multiply several numbers together (where all are greater than 1), and then add 1, is the result divisible by any of them? No. Why not? Because you'll always have that 1 you added as the remainder!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the challenge for your brother. You've got &quot;all of the primes&quot;. We know that every number greater than 1 is either prime, or is divisible by (a unique list of) primes. But using what we just did, is it possible to make a number that can't be divided by any of our prime numbers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid's proof: multiply &quot;all of the primes&quot; together and add 1. So either the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic&quot;&gt;fundamental theorem of arithmetic&lt;/a&gt; is wrong (oh horror!), or our list of &quot;all the primes&quot; must be missing at least one prime number. And since this goes for &lt;em&gt;any&lt;/em&gt; finite list that claims to contain &quot;all the primes&quot;, there must be infinitely many primes. QED.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(He does know the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic&quot;&gt;unique factorisation theorem&lt;/a&gt;, right? I'm just checking, because you pretty much need to be aware of it to tackle Euclid's proof.)&lt;/p&gt;&#xA;" OwnerUserId="1946" LastEditorUserId="1946" LastEditDate="2015-11-24T09:35:04.843" LastActivityDate="2015-11-24T09:35:04.843" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4076" PostTypeId="2" ParentId="4071" CreationDate="2014-07-25T19:32:42.067" Score="5" Body="&lt;p&gt;A simple search give you plenty of reference materials. Here is &lt;a href=&quot;http://www.inquirybasedlearning.org/&quot;&gt;The Academy of Inquiry Based Learning&lt;/a&gt;.  And here is &lt;a href=&quot;http://www.jiblm.org/&quot;&gt;The Journal of Inquiry-Based Learning in Mathematics&lt;/a&gt;. Enjoy exploring and experiencing. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2014-07-25T19:32:42.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4077" PostTypeId="2" ParentId="4053" CreationDate="2014-07-25T20:43:34.230" Score="6" Body="&lt;p&gt;First of all, I am not happy with putting rote learning against fun, since one of them is more cognitive and the other more affective. Moreover, fun or not, there is something called &quot;&lt;strong&gt;cognitive obstacle&lt;/strong&gt;&quot; which basically implies that it doesn't matter whether you enjoy learning what you are supposed to learn or not, there are certain fundamental  obstacles that occur for you. Regarding the concept of your interest, the most famous obstacle is the gap between &quot;&lt;strong&gt;potential infinity&lt;/strong&gt;&quot; and &quot;&lt;strong&gt;actual infinity&lt;/strong&gt;&quot; .   &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2014-07-25T20:43:34.230" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4078" PostTypeId="1" AcceptedAnswerId="4082" CreationDate="2014-07-25T21:11:36.707" Score="4" ViewCount="331" Body="&lt;p&gt;I will be working as a TA in Calculus 1 this semester. In Norway, this means leading a study group with about 20 people (most of the days, I guess around five or six will show up), writing up the exercise-numbers on the blackboard, and answer any questions they have about them. More than likely, I will get students without an interest in mathematics (biology students, chemistry, geology etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students have no interest in learning the rigorous and (subjectively) interesting parts of the subject, should I still force these concepts upon them? Many of them will be forced into taking more advanced subjects later (Linear Algebra, Calc 2). However, the nature of Calc 1 allows you to get a C, or maybe even a B, by simply cramming methods. I am having a hard time making up my mind about this, because it is very likely that my students wishes (and maybe what's best for them in terms of grades) will be in strict conflict with my philosophy as someone who hopes to become a good expositor of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The best example may be the formal definition of the limit. This can be taught well, as in, &quot;this is &lt;em&gt;why&lt;/em&gt; it works&quot;, or by simply reducing it to a humanized algorithm, which will often give you a correct value of delta.)&lt;/p&gt;&#xA;" OwnerUserId="1710" LastEditorUserId="1710" LastEditDate="2014-07-25T21:50:31.810" LastActivityDate="2014-07-26T04:17:47.850" Title="Should I go out of my way and give non-interested students a rigorous understanding of the subject?" Tags="|undergraduate-education|calculus|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4079" PostTypeId="2" ParentId="4071" CreationDate="2014-07-25T21:51:33.090" Score="6" Body="&lt;p&gt;Because you used the education-research and reference-request tags, I will give my usual recommendation of &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0470276673&quot;&gt;Powerful Learning: What We Know About Teaching for Understanding&lt;/a&gt; for the reason that Schoenfeld's chapter is well-referenced and written for practitioners. The benefit here is that it is a good introduction to methods of &quot;teaching for understanding&quot; while providing some research background which includes research done on &quot;discovery&quot; and inquiry methods. Because it has a number of influential references in it, you can build a decent reading list from it (and look to Schoenfeld's contextualization).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can supplement reading very recent journals that more specifically target Inquiry-based learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: (Chris noticed the blurb on Amazon says the book is for K-2, but that is incorrect. The book pertains to K-12, as mentioned on the book jacket.)&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-07-26T01:41:00.323" LastActivityDate="2014-07-26T01:41:00.323" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4080" PostTypeId="2" ParentId="4078" CreationDate="2014-07-25T22:19:06.527" Score="2" Body="&lt;p&gt;I imagine a scenario like this: you see an error and ask them how they arrived at what they have written. They can't explain it. You then ask them whether they want an explanation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose a student might say &quot;just tell me the answer&quot; but that's treating you like a glorified answer key. How will they get the answer on the class' assessment? But their reasoning may be different, even if they did decide, in some way, to come to a tutor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think your focus should be on understanding. But, then, that's my philosophy. When you're a tutor, you're working for someone and they may have different ideas. Your colleagues can support you in your philosophy, but some people are hired (for example) to do test-prep work and would not get rehired if they did not use cramming approaches that are (let's be honest) near useless for understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While someone else may have an insight from their experience that will help you, I'm going to say that this may be very much a personal decision because of the employment ramifications. If I am understanding correctly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A somewhat analogous situation from public school: a school administration in public school may mandate a certain curriculum pr pedagogy. Teachers very often face philosophical differences. They may well ask &quot;should I follow what my math coordinator and district want me to do, or should I go outside of the acceptable leeway I've been given?&quot; &lt;em&gt;If&lt;/em&gt; one is in a situation with personal consequences, then it's a personal decision. Especially after other measures (like petitioning the administration) have already been tried.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-07-25T22:19:06.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4081" PostTypeId="2" ParentId="4070" CreationDate="2014-07-26T03:24:48.277" Score="13" Body="&lt;p&gt;I don't really answer the question but: why do you want your brother to come to the answer &lt;strong&gt;right now&lt;/strong&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, your brother understands that, to prove that the set of prime numbers is infinite, you can assume it is finite. And, from this pool of prime numbers, you can construct a new one that is not in the family. Why don't you let your brother play with that idea for a short time (a couple of days or weeks)? Present the question to him as a challenge and tease him regularly to be sure he keep thinking about it. Except at school or for career-related reasons, there are no deadline in mathematics to solve a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your brother is genuinely interested by maths, he will probably enjoy this challenge. If he is not, or his motivation gets down, you may motivate him with a reward. A small reward, since you probably don't want him to work for a reward. So, the reward should not be money or gift, but maybe an intellectual reward: explaining about the twin conjecture &quot;You came up with a proof. Great! Do you know about twin primes? &lt;em&gt;Bla bla bla&lt;/em&gt;&quot;, or a magic trick involving prime numbers,...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, please keep in mind this is not a big deal if he finally does not arrive to the complete proof. Finding a proof is fine, but searching and finally enjoying the beauty of someone else's brillant reasoning is great too! How many people could have came by themself to &lt;em&gt;Cantor's diagonal argument&lt;/em&gt;? Isn't it a great proof that most mathematics enthusiasts enjoy?&lt;/p&gt;&#xA;" OwnerUserId="1538" LastEditorUserId="1538" LastEditDate="2014-07-26T03:58:01.847" LastActivityDate="2014-07-26T03:58:01.847" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4082" PostTypeId="2" ParentId="4078" CreationDate="2014-07-26T04:17:47.850" Score="5" Body="&lt;p&gt;This is a very narrow answer, but I feel that if you construct the questions cleverly, you can teach the neat concepts conceptually, even to previously &quot;uninterested&quot; students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For epsilon-delta, I've had some success with questions like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Suppose someone is trying to prove that $\lim_{x \to 3} x^2 = 12$, and tries $\epsilon = 8$. Find a value of $\delta$ that works with&#xA;  this $\epsilon$.&lt;/li&gt;&#xA;  &lt;li&gt;What is an example of an $\epsilon$ that will not work when trying to show that $\lim_{x \to 3} x^2 = 12$?&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They almost can't help but gain some understanding of why it works if you ask this kind of question.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-07-26T04:17:47.850" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4083" PostTypeId="1" CreationDate="2014-07-26T05:07:48.403" Score="19" ViewCount="1117" Body="&lt;p&gt;When we ask students to prove a particular result in a math class, students often reply with examples. For example, if I state: &lt;em&gt;if a number is even its square will be even&lt;/em&gt;, and ask the students to prove it, they will reply with an example (such as, &quot;The square of two is four, and both are even&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that the problem lies in the wording (which we cannot change). &lt;em&gt;Proof&lt;/em&gt; in a social context means supporting your arguments with evidence, which is different from the mathematical definition of the word.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can we handle such problems in class?&lt;/p&gt;&#xA;" OwnerDisplayName="Madhu" LastEditorUserId="39" LastEditDate="2014-07-28T19:24:12.267" LastActivityDate="2014-09-13T16:19:13.483" Title="&quot;Proof&quot; meaning in maths and society" Tags="|proofs|concept-motivation|mathematical-pedagogy|terminology|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4084" PostTypeId="2" ParentId="4064" CreationDate="2014-07-26T10:57:08.293" Score="2" Body="&lt;p&gt;This is a really nice question. Although it is discussed by OP, Steven Gubkin and Sergio Parreiras implicitly, I would like to emphasize on &lt;strong&gt;cultural root&lt;/strong&gt; of this phenomena again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Religion&lt;/strong&gt; is the most important part of Jewish culture. Some mysterious parts of this special religious culture are &lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Jewish_mysticism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jewish mysticism&lt;/a&gt;&lt;/strong&gt; and &lt;a href=&quot;http://en.wikipedia.org/wiki/Kabbalah:_Primary_texts&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Kabbalah&lt;/strong&gt;&lt;/a&gt; which play a vital role in &lt;em&gt;old&lt;/em&gt; tradition of Jewish mathematics. In fact one can see numbers all around Torah. A famous example is &lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Book_of_Numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book of Numbers&lt;/a&gt;&lt;/strong&gt; which is full of strange numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a consequence of this Jewish philosophical idea that &lt;em&gt;God created world in the language of numbers (maths) and then coded its secrets somewhere in the Torah&lt;/em&gt;, one may think about discovering these secrets using numbers (maths). This idea  in Jewish culture gives a &lt;em&gt;divine&lt;/em&gt; importance to maths. In this point of view mathematics is a tool to unfold all secrets of the universe and those who believe in this idea analyze every phenomena mathematically to find the &lt;em&gt;signs&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to find more on this topic see &lt;a href=&quot;http://en.wikipedia.org/wiki/Pi_%28film%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Pi movie&lt;/strong&gt;&lt;/a&gt; and &lt;a href=&quot;https://movies.stackexchange.com/questions/15230/why-is-aronofskys-pi-not-pi&quot;&gt;&lt;strong&gt;my question&lt;/strong&gt;&lt;/a&gt; in movies.stackexchange.com&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="-1" LastEditorDisplayName="user230" LastEditDate="2017-04-13T12:54:04.933" LastActivityDate="2014-07-27T04:28:28.457" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4085" PostTypeId="2" ParentId="4083" CreationDate="2014-07-26T15:36:03.117" Score="8" Body="&lt;p&gt;The mention of &quot;evidence&quot; in addition addition to &quot;proof&quot; is a good way to start. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can explain that the fact that the square of 2 is also even is evidence for the assertion/conjecture/hypothesis/claim made. The fact that the square of 6 is also even is further evidence for it. &#xA;And so on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But no matter how much evidence of this form one generates there might always rest some possibility, some doubt, that the general claim could be false. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, one could recall that in court one common standard is to say that for something to be considered proved there needs to be &lt;a href=&quot;http://en.wikipedia.org/wiki/Reasonable_doubt&quot;&gt;evidence beyond a reasonable doubt&lt;/a&gt;; but for other situations one might only need to show a &quot;Preponderance of the evidence&quot; or still something else. See the Wikipedia site &lt;a href=&quot;http://en.wikipedia.org/wiki/Legal_burden_of_proof&quot;&gt;Legal burden of proof&lt;/a&gt; for various concepts.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, one could say in mathematics the standards of proof are very strict and there needs to be evidence beyond any doubt whatsoever for something to be considered as proved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;To sum it up:&lt;/strong&gt; also in everyday usage there is (or at least there should be) a distinction between &quot;evidence&quot; and &quot;proof,&quot; and this is a good opportunity to recall this. Which quantity of supporting evidence is considered as a proof differs depending on context, and in mathematics is especially high.  &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-07-26T16:33:20.553" LastActivityDate="2014-07-26T16:33:20.553" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4086" PostTypeId="1" CreationDate="2014-07-26T17:11:58.333" Score="4" ViewCount="145" Body="&lt;p&gt;After having met yet another person confused by indefinite integrals today, I've finally decided to ask the community.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you think it makes sense to teach indefinite integrals? My opinion is that only definite integration should be taught since it is the only one that makes formal sense to me. Of course indefinite integrals can be used by people who know what they are doing, but it doesn't justify the introduction of this notion from the very beginning to the layest of the laymen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to argue as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;One often read/hears $\int..dx$ is the inverse of differentiation, its the anti-derivative. While one can of course make some sense of it, of course everybody knows that differentiation is an irreversible operation where information is lost, so there is no true inverse of that operation. For me the usage of &quot;anti-&quot; in the sense of &quot;almost-anti-&quot; is one source of confusion.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In my opinion $\int f(x) dx$ should not be seen as a function, written like that, for my taste, I would say that it's not well-defined as a function. If it is a function, of what variable? Certainly not of $x$. It would make slightly more sense to write $\int^t f(x)dx$ as now at least one can use this form differentiation. But still, as a function it is not completely unambiguous. Of course, there are applications where this additive unsertainty (which can be infinity) does not play a role, but again this is of no concern for people who are just being taught what integrals are. I&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The only sensible use of writing $\int f(x)dx$ that I can imagine is as a sort of abbreviation in the sense &quot;you know what boundaries you have to insert, so let's just skip it&quot;. It is like writing sums without giving the boundaries: $\sum f(n)$, which I would generally avoid to do, unless everyone knows what is meant.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Given that I see school text books full of indefinite integrals from the beginning and that the search on math.stackexchanges for &quot;indefinite integral&quot; gives &gt;1000 results, where sometimes calculations of this sort &lt;a href=&quot;https://math.stackexchange.com/questions/591718/indefinite-integrals&quot;&gt;(Link)&lt;/a&gt; are carried out with the result that $\int\frac{dx}{x}=\ln(x)$ without anyone complaining about the notation which is at most sketchy, and finally that searching wikipedia for &quot;indefinite integral&quot; automatically redirects to &quot;anti-derivative&quot;, I would like to ask, what do you think about using indefinite integrals in mathematics? Should school children be exposed to it? Should it be taught?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S.: this question has also been posted on SE: &lt;a href=&quot;https://math.stackexchange.com/questions/878851/how-useful-useless-is-the-indefinite-integral&quot;&gt;(Link)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1948" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-07-26T17:11:58.333" Title="How useful/useless is the indefinite integral" Tags="|integration|" AnswerCount="0" CommentCount="3" ClosedDate="2014-07-27T16:25:35.013" ContentLicense="CC BY-SA 3.0" />
  <row Id="4087" PostTypeId="1" CreationDate="2014-07-27T04:59:05.713" Score="7" ViewCount="400" Body="&lt;p&gt;Nowadays almost everybody uses such devices almost every time! Particularly during my lectures in the class I can see some students who are working with their iPads and mobile phones. In such a situation I usually think about two possible situation and two possible reactions as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Good Case:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The student is searching in the internet for more information about what I am teaching.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In this case &lt;em&gt;Good Saint Georg&lt;/em&gt;'s reaction could be:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Waiting for a possibly good question by that student which is an opportunity to introduce new aspects of the teaching material.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bad Case:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The student is doing something unrelated to what I am teaching.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In this case &lt;em&gt;Evil Saint Georg&lt;/em&gt;'s reaction could be:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Declaring loudly to all students that &quot;&lt;em&gt;Shut down your devices or get out of my class!&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What is a wise strategy about students iPads and mobile phones in class during my lectures? Should I remove the problem by banning all students to bring any devices to my classes or there are more soft ways to manage this situation?     &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="143" LastEditDate="2014-07-28T19:45:57.767" LastActivityDate="2024-03-19T15:29:24.777" Title="Students iPads and mobile phones in class: good or distracting?" Tags="|technology-in-education|classroom-management|general-pedagogy|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4088" PostTypeId="2" ParentId="4087" CreationDate="2014-07-27T11:25:19.587" Score="1" Body="&lt;p&gt;&quot;While I am lecturing&quot; is generally not the time that mobile devices at their best use in a mathematics classroom - at least if we're talking about purposes that are specifically mathematical. For the students to actually be engaging in some sort of mathematics, they should be focused on whatever they're trying to do with the devices, talking to each other about what they're doing on the devices, or the like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the device to be put to good mathematical use, you should be able to answer the question: &quot;what sort of mathematical thinking was the student doing while using the device? What is evidence that suggests this thinking was going on?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question applies at any level (obviously the mathematics will be different) but a good example is some pilot research I was involved with using Sketch Explorer on the iPad with classes of fourth grade students. Groups of students engaged in a number of geometry activities together on the iPads, argued about different approaches to the problems they faced, and then reported their group's strategies back to the whole classroom for further discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a use that is not specifically or demonstrably mathematical, I often used my iPad to take notes in my graduate classes and colloquia. I would create concept maps, for example, that structured my notes of the lecture or classroom discussion or the colloquium speaker's presentation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A wise strategy that has something to do so mathematics would be to understand how technology can be part of the mathematical environment of your classroom. That's going to take more than an answer on stackexchange. The final question as you have worded it, however, is a very general question about classroom management and conduct, and I don't see it as having much to do with mathematics education specifically. Above, I have tried to link it to a little of what I know about technology use in mathematics education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since providing references is inconvenient for me at the moment, you may consider this an opinion answer.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-07-27T11:25:19.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4089" PostTypeId="1" CreationDate="2014-07-27T19:04:48.977" Score="12" ViewCount="457" Body="&lt;p&gt;Today's NY Times Magazine has an article titled &lt;a href=&quot;http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html&quot;&gt;&quot;Why do Americans stink at math?&quot;&lt;/a&gt; It gives a very positive depiction of a set of methods advocated by Magdelene Lampert in her book &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0300099479&quot;&gt;Teaching Problems and the Problems of Teaching&lt;/a&gt; (which I haven't read). As described in the article, what Lampert advocates is changing from the traditional &quot;I, We, You&quot; method (demonstrate how to solve a problem, then do one with the students, then have them try one individually) to what the author of the article describes as &quot;You, Y'all, We.&quot; In this method, the teacher starts class by posing a single &quot;problem of the day.&quot; Students first try to tackle it individually, then get together in groups, and at the end the whole class discusses it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lampert seems to specialize in elementary education, especially grade 5-ish. Is there anyone here who has tried this method at the college level, or who knows of an account of whether it's been tried, or whether it is or is not likely to be appropriate?&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-09-17T19:00:49.897" Title="Use of methods similar to Magdalene Lampert's at the college level?" Tags="|interactive-teaching|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4090" PostTypeId="2" ParentId="4087" CreationDate="2014-07-27T20:22:29.690" Score="1" Body="&lt;p&gt;The ubiquitousness of cell phones has made it incredibly challenging to hold students' attention throughout class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I feel students should be held accountable and responsible for their own learning. After all, why should other students' learning be disrupted while another student is texting or browsing the internet or whatever quietly?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student asks me to go over something I just got done talking about (and they were on their phone), I have no quarrels with telling them they should have paid attention.&lt;/p&gt;&#xA;" OwnerUserId="625" LastActivityDate="2014-07-27T20:22:29.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4091" PostTypeId="1" CreationDate="2014-07-28T00:03:32.920" Score="20" ViewCount="1031" Body="&lt;p&gt;In a set theoretic point of view all mathematical objects are sets. We &quot;call&quot; some of them as &lt;em&gt;numbers&lt;/em&gt; (e.g. sets in $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $Ord$, $Card$) but what is rigorous definition of a &lt;em&gt;number&lt;/em&gt;? In fact this question is not as simple as it seems. There is a long sequence of logical and philosophical papers related to this subject which unfolds the complexity of the notion of number. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes in my set theory courses when I refer to members of $Ord$ and $Card$ as ordinal and cardinal &quot;numbers&quot;, some students ask me if they are really numbers or not? In such a situation I usually say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What do you mean by a &quot;number&quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They usually say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We mean something like natural, integer, rational, real and imaginary numbers?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then I ask:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why do you think those objects are more natural numbers than infinitary numbers? Which properties of $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ makes them a set of numbers? Why do you believe that $i$ is a number but $\omega_{\omega+1}+2$ is not? Are you sure that you don't refer to members of $\mathbb{C}$ as &quot;numbers&quot; just because you &quot;heard&quot; this word from your teachers before and this is just a relative social common belief?! &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When we reach this point in the discussion, I emphasize on complexity of the notion of number and refer my students to philosophical papers as a project. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes I think it is good if I know what a usual school/university math teacher who is not familiar with logic and philosophy of maths professionally, says when his/her students ask him/her about the notion of number and the reason which makes members of $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ numbers. Surely most of students ideas about numbers is inspired by ideas of their teachers who are not necessarily logicians and it is important to understand students (possibly false) common beliefs about numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are the most common beliefs about notion of number amongst non-logician math teachers? What is a &quot;number&quot; in their point of view?  &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-08-02T00:18:32.603" LastActivityDate="2021-03-08T00:24:55.263" Title="What is a number?" Tags="|mathematical-pedagogy|logic|set-theory|real-numbers|philosophy-of-maths|" AnswerCount="6" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="4092" PostTypeId="2" ParentId="4091" CreationDate="2014-07-28T11:29:22.567" Score="14" Body="&lt;p&gt;I think the level of the student is very important to this question. &lt;strong&gt;If the student has never had an abstract math course&lt;/strong&gt; (like my students), then the lack of a definition of &quot;number&quot; is a great way to introduce the idea of abstract algebra. They are very happy to initially believe a definition like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A number is anything that you can add and multiply, such that addition and multiplication work and follow the right rules.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course I haven't defined &quot;number&quot; at all -- I've defined something else entirely -- but this is a great opening that allows you to discuss new objects. $\mathbf{Z}/12$ is a pretty easy starting point. You can explore $\mathbf{Z}/12$ and find all kinds of strange properties, but still see that addition and multiplication play nicely together, so it should still count as a set of &quot;numbers.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you get to the point: &lt;strong&gt;you can use the definition of a &quot;number&quot; as something pliable that is open to opinion&lt;/strong&gt;. For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\mathbf{Z}/12$ has zero-divisors (don't say this or define it or even give the name &quot;zero-divisor,&quot; just look around and find some). Does that feel illegal enough that these should not count as numbers? Or are they just strange numbers?&lt;/li&gt;&#xA;&lt;li&gt;$\mathbf{Z}/12$ does not permit division. Does that feel illegal enough that these should not count as numbers?&lt;/li&gt;&#xA;&lt;li&gt;$\mathbf{Z}/12$ does not allow you to add things like $\frac34$ to a number. Does that feel illegal enough that these should not count as numbers?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The resolution, which is that in abstract algebra all these things have names (&quot;ring,&quot; &quot;field,&quot; &quot;$\mathbf{R}-algebra$,&quot; etc) is a stupid punchline that I don't even give them. The idea of abstract algebra is cooler than the names. Leaving the punchline as &quot;the definition of number is open to opinion&quot; is way, way better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best students will be really bothered by this open-ended piece of mathematics, and come back the next week and want to know if we can make $\infty$ is a number, or if we can make $i^i$ a number, of if 3:00pm $\cdot \frac34$ could be a number, and then we get to talk about ordinals, or surreals, or extensions, or parallel parking, or whatever else is in that direction.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-07-28T11:29:22.567" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4093" PostTypeId="1" AcceptedAnswerId="4104" CreationDate="2014-07-28T12:22:58.030" Score="13" ViewCount="1159" Body="&lt;p&gt;My mathematics education was essentially rote, you learned the formulas and applied them almost algorithmically to the problems you were presented with; the teacher dictated a method and you followed it. This conveyed a dry and static picture of mathematics with no room for creativity and where curiosity was strait-jacketed and questions weren't encouraged. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel when I approach problems now the seeds of this have come to bear. I think very linearly; I see a certain method I can apply to a problem and I keep attacking the problem with the same method till either I give up or the problem beats me. It is almost like tunnel vision, I see things very narrowly. The saying everything looks like a nail when you have a hammer comes to mind. I feel this is particularly prevalent when I'm tackling Olympiad problems, I feel certain students have a fount of creativity they can dip into, mines in contrast seems depleted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I assume a good pedagogy would engender a broad, deep and intuitive understanding of mathematics and would enable you to think more creatively and draw on broader areas of mathematics when attempting to tackle a problem and would further allow you to think spontaneously when something out of the norm is encountered. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;How does one concretely cultivate this? In essence what I am asking is:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How does one think creatively when it comes to problem solving in mathematics? Can I teach myself to think creatively?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there any books or resources that are particularly helpful with regards to this?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What are the pre-requisites to creative thinking? Does one have to be deeply proficient in a subject and have a good grasp of its concept before they can be creative/inventive? Is the creativity I perceive in others just subject proficiency?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Or does one just have to be &quot;efficient&quot; (as opposed to proficient) to be creative? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Can creativity be learnt? Or are people inherently creative: either you have it or don't? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there any ways to correct the tunnel vision I described, that will enable me to think more open and broadly and just generally shake off the ill effects of a poor education? (I guess this is what I attribute my inadequacies to).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1677" LastEditorUserId="1677" LastEditDate="2014-08-15T12:59:30.593" LastActivityDate="2014-08-15T12:59:30.593" Title="Moving From Rote Learning To Creative Thinking" Tags="|mathematical-pedagogy|reference-request|creativity|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="4094" PostTypeId="2" ParentId="4087" CreationDate="2014-07-28T15:30:49.823" Score="4" Body="&lt;p&gt;This depends strongly on the age-group. My course notes for grad-level courses are on-line as PDFs, and many of my students use their iPads to mark on the PDFs during my lectures... the latter talking more informally about the material, highlighting certain details, generally trying to complement the inevitably more formal tone of the PDF'd notes.&lt;/p&gt;&#xA;&lt;p&gt;No, I cannot be sure they're not &lt;em&gt;also&lt;/em&gt; checking email or messages. But it's infeasible to separate one from the other without being over-bearing, I think.&lt;/p&gt;&#xA;&lt;p&gt;Certainly note-taking is ever-more feasible on iPads, and also on laptops, so how could we really demand &amp;quot;paper&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Attempting to distinguish &amp;quot;phones&amp;quot; from &amp;quot;tablets&amp;quot; is ever harder (cf. &amp;quot;iPad mini&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;The thing to attempt to require is &amp;quot;civility&amp;quot; (of students toward the teacher and toward their peers)... although younger students may not understand what this entails.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: Ten years later, 100% of the students in my grad-level math course take notes on iPads or equivalent. No one takes notes on paper. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2024-03-19T15:29:24.777" LastActivityDate="2024-03-19T15:29:24.777" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="4095" PostTypeId="2" ParentId="4045" CreationDate="2014-07-29T04:04:33.453" Score="3" Body="&lt;p&gt;I don't know of a practical application of perfect numbers per se (not saying they don't exist), but its historical status dating back to the Greeks makes them interesting for pedagogical purposes. The Greeks (the Pythagoreans in particular) put mystical significance on special sequences such as the primes, the figurate numbers (notably the triangular numbers), and of course the perfect numbers. If you find that injecting anecdotes into your teaching helps you grab the attention of your students, then perfect numbers are a well-documented example of the ongoing human fascination with number theory since antiquity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a very detailed history of perfect numbers, check out &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Perfect_numbers.html&quot; rel=&quot;nofollow&quot;&gt;http://www-history.mcs.st-and.ac.uk/HistTopics/Perfect_numbers.html&lt;/a&gt;. (In fact entire MacTutor History of Mathematics Archive is quite fascinating, and makes for quite nice leisure reading.)&lt;/p&gt;&#xA;" OwnerUserId="663" LastActivityDate="2014-07-29T04:04:33.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4096" PostTypeId="2" ParentId="4055" CreationDate="2014-07-29T04:19:03.177" Score="2" Body="&lt;p&gt;I think you might appreciate an example of what has been tried and can be done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently took a one-semester seminar-type course in mathematical general relativity, servicing both mathematics and physics students, and I would say I learned quite a lot. We met 80 minutes per week with the students (undergrads) preparing and giving the lectures with guidance from the organizer (faculty). The first half of the course was devoted to developing the language of Riemannian manifolds, mostly rigorously but tiptoeing past the more abstract definitions. The second half focused on general relativity; we were able to formulate the Einstein field equations and discussed the Schwarzchild solution toward the end, and a quick description of Penrose diagrams and what black holes are. Given more time (say as a proper course meeting twice a week) we would probably have been able to talk about geometric analysis (developing the language of analysis along the way).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Courses like this can be quite successful. I think quantum mechanics is a great subject for mathematicians and physicists to learn from together; statistical mechanics is probably another, and I think fluid dynamics could be done (maybe mathematically from the dynamical systems perspective, or perhaps the analysis of Navier-Stokes).&lt;/p&gt;&#xA;" OwnerUserId="663" LastActivityDate="2014-07-29T04:19:03.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4097" PostTypeId="1" AcceptedAnswerId="4380" CreationDate="2014-07-29T06:18:24.290" Score="8" ViewCount="775" Body="&lt;p&gt;I am currently in a course titled &quot;teaching reading in a content area.&quot; While there are plenty of examples of different strategies that can be used in different subject areas, there are little suggestions for mathematics, due in part I am guessing to our book author's unfamiliarity with the subject. A little is discussed on how to approach textbook reading with your students (involving pointing out the structure and function of different aspects of the book, which is of course important), but I am feeling very wanting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are some of the ways I know that reading comprehension can affect math class performance. In an answer I would be hoping for ways to address any or all of these. Additionally, mentioning other ways that reading comprehension can affect math ability would be something I'm interested in as well. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Straightforward: Not knowing how to read the textbook (or not knowing &lt;em&gt;when to&lt;/em&gt; read the textbook) affects a student's independent learning ability. This applies to all classes, but I'd argue particularly math, since it seems that students expect teachers to talk them through everything and if they don't catch something, it's out of sight out of mind. (This comes from a lot of tutoring: many students don't bring their textbooks to the tutoring center I am at and those who do seemed stunned that the formulas are in the book)&lt;/li&gt;&#xA;&lt;li&gt;Content (mathematical) vocabulary. This is something I believe most of us already focus on since definitions are at the heart of mathematics. However, we all know a student can memorize a definition without having understanding of the concept behind it. Perhaps this can be summed up as &quot;making meaning of mathematical ideas.&quot; &lt;/li&gt;&#xA;&lt;li&gt;Comprehension of &quot;word problems&quot; and turning &quot;words&quot; into mathematical diagrams/formulas-- if a student cannot decode their word problems, they cannot do them. This was evident in a remedial geometry course I taught for the first time last year, where it took me until mid-course to realize that some of my students' reading skills affected their ability to learn math (it was the first course I ever taught). The question I posed was, nearly verbatim: &quot;A church has a 6&quot; stair leading to its entrance. They want to build a wheelchair ramp to the stair. The maximum legal angle for wheelchair ramps is 6 degrees from the ground. How far out horizontally from the stair will the ramp reach? Make sure to draw the stair and the ramp.&quot; One particularly struggling student drew a stair, but did not even draw a ramp. Does this signify defeat or lack of comprehension? &lt;/li&gt;&#xA;&lt;li&gt;Comprehension of causative statements. The &quot;If...then&quot; relationship tends to be a challenge to effectively convey as evidenced by even upper-division math students struggling in proofs courses with it. While this may in part be due to difficulty in logical reasoning skills, this may also be tied to lack of clear exposure to &quot;If...then&quot; and &quot;cause...effect&quot; instruction in language learning. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To recap: What are ways that reading comprehension (at the lower, decoding level AND the higher, reading for meaning level) can be addressed meaningfully in a math classroom? What are other ways that poor reading comprehension can affect mathematical performance?&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2018-12-05T23:22:29.613" Title="How to address struggling readers in the mathematics classroom?" Tags="|secondary-education|problem-design|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4098" PostTypeId="2" ParentId="4093" CreationDate="2014-07-29T06:46:58.873" Score="6" Body="&lt;p&gt;There are studies indicating that &lt;a href=&quot;http://www.colorado.edu/physics/EducationIssues/T&amp;amp;LPhys/PDFs/Schoenfeld_metacognition.pdf&quot;&gt;metacognition training can improve problem attack.&lt;/a&gt; In the linked article by Schoenfield (which I may have gotten from here, I'm not sure), he discusses the differences in how experienced mathematicians approach a problem compared to inexperienced students, characterizing each approach by the stages &quot;Read, Analyze, Explore, Plan, Implement, Verify.&quot; An inexperienced problem-solver will stay in the &quot;Explore&quot; stage, but with some training, can be taught to use planning, implementation, and verification to improve their problem solving skills. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another process which is highly cited is &lt;a href=&quot;http://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=X3xsgXjTGgoC&amp;amp;oi=fnd&amp;amp;pg=PP2&amp;amp;dq=polya+problem+solving&amp;amp;ots=t6Jw1HpGs7&amp;amp;sig=oqxD5EBfYrnspY0TInTkzoEwg9k#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;George Polya's Problem Solving Process&lt;/a&gt; where he describes four stages he claims form the problem-solving process: Understanding the Problem, Devising a Plan, Carrying out the Plan, and Looking Back. While clearly a bit behind where you are since you are doing Olympiad-level mathematics, it may come in handy to someone else looking for problem-solving strategies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also personally think that &lt;strong&gt;metacognitive skills&lt;/strong&gt; can vastly improve problem-solving performance, but also &lt;strong&gt;exposure to similar problems&lt;/strong&gt; can as well. Sometimes there is a &quot;trick&quot; that you wouldn't normally think of except you saw it used once in a similar problem. The evaluation of the integral $\int_0^{\infty} e^{-x^2}$ comes to mind. I also recall some problems on the Putnam which would have been infinitely easier had a particular result from, say, graph theory come to mind (For the record, I have little experience in competitive mathematics and my participation was purely for fun). In that sense, &lt;strong&gt;familiarity with background information,&lt;/strong&gt; too, is important. It would take someone with considerable mathematical prowess to be able to derive new theorems in time to complete a competitive mathematics exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The three factors I have mentioned probably only scratch the surface, but all three are things that can be worked on and improved and do not necessarily require innate ability.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2014-07-29T06:46:58.873" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4099" PostTypeId="2" ParentId="4091" CreationDate="2014-07-29T12:48:10.240" Score="4" Body="&lt;p&gt;I think this question is important. I'd love to see an actual answer to it and cannot upvote it enough. I don't have an answer, but would like to share some intuitions/speculation. I think the following is what teachers have absorbed as the notion of number. I do not think most teachers could articulate this.&lt;/p&gt;&#xA;&lt;p&gt;Our early exposure to numbers comes about usually by naming groups of objects. The action of &amp;quot;counting&amp;quot; as pointing to objects in succession and saying number words allows us to move from the objects counted to the process of counting itself. The number words are then &amp;quot;reified&amp;quot; into objects themselves. One may regard this as a primitive &amp;quot;language game&amp;quot; in the sense of Wittgenstein. From this point of view a (natural) number encodes an action...something we do when faced with a set of objects where we can carry out a physical/linguistic process. It seems important that the stability or consistency of the linguistic pattern with the physical process be abstracted into language. When we abstract the process into language it becomes number.&lt;/p&gt;&#xA;&lt;p&gt;Other simple physical/linguistic processes like grouping objects yield &amp;quot;addition&amp;quot; and &amp;quot;subtraction&amp;quot; language games. Again, working with a few stones is good enough to &amp;quot;understand&amp;quot; addition and subtraction, but bootstrapping using the brain's language capability to form a &amp;quot;pattern language&amp;quot; that encodes and extrapolates the observed regularity in the physical objects yields a first concept of number.&lt;/p&gt;&#xA;&lt;p&gt;From this point of view, numbers are abstracted from processes found in language games. One might say that numbers, since tied with processes, are connected with the movement of time in some way. A good text on abstract algebra defined that subject as concerned with &amp;quot;general notions of number&amp;quot; as another answer to this question does.&lt;/p&gt;&#xA;&lt;p&gt;Of course there is a significant leap from the above naive experience with number and the axiomatic approach to number, which employs quantifiers to axiomatize the abstracted properties even further. Of course we know what happens if we try to completely logicize basic number: Whitehead and Russel's 379 page proof of 1+1=2.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="216" LastEditDate="2021-03-08T00:24:55.263" LastActivityDate="2021-03-08T00:24:55.263" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="4100" PostTypeId="2" ParentId="4091" CreationDate="2014-07-29T14:15:21.220" Score="6" Body="&lt;p&gt;Whatever teachers may think about the nature of numbers, the foundations of &quot;arithmetic&quot; and the nature and concept of number in particular are very subtle. For a recent and sophisticated look at the issues, see: John Horton Conway, On Numbers and Games, second edition, A.K. Peters, 2001. Conway gives his approach to the surreal numbers, relates these to what has come before, and puts what he did a broader setting.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-07-29T14:15:21.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4101" PostTypeId="2" ParentId="4097" CreationDate="2014-07-29T18:54:50.907" Score="2" Body="&lt;p&gt;I am unclear at what &quot;grade level&quot; you are dealing with, however, independent of grade level you may be interested in the relatively new series of books published by Princeton University Press - The Best Writing on Mathematics: Year X, where books for 2010, 2011, 2012, and 2013 have appeared so far. The Forewords of these books have been written by William Thurston (now deceased), Freeman Dyson, David Mumford, and Roger Penrose, respectively. While I may quibble with the presence of some individual articles that have appeared in these volumes, on the whole these are very varied and interesting materials, over a broad swath of different aspects of mathematics and mathematics education. The reading levels for the different articles also varies a fair amount but there is a lot to like in these books.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-07-29T18:54:50.907" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4103" PostTypeId="2" ParentId="926" CreationDate="2014-07-29T23:01:39.867" Score="15" Body="&lt;p&gt;I guess you have already got plenty to read for &quot;Where does it come from?&quot; part of you question. Thus I just shortly introduce my favorite strategy that I use for &quot;What can we do about it?&quot; part, when there is a case to work on: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Encourage your students to find conditions that linearity assumption indeed works!&lt;/strong&gt;   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For examples, for which values of $a$ and $b$, $(a+b)^2=a^2+b^2$. Quite often, it turns to a nice challenge and rewarding in the direction you need. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2014-07-29T23:01:39.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4104" PostTypeId="2" ParentId="4093" CreationDate="2014-07-30T10:16:14.437" Score="8" Body="&lt;p&gt;The question is very broad, as you surely realize, since you broke it up into six sub-questions (which are also very broad!). In order to attempt an answer, I will have to send you on a bit of a scavenger hunt:&lt;/p&gt;&#xA;&lt;p&gt;First, see my answer &lt;a href=&quot;https://matheducators.stackexchange.com/a/906/262&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; and note that I have a literature review from my doctoral dissertation, which was entitled &lt;em&gt;Conceptions of Creativity in Elementary School Mathematical Problem Posing&lt;/em&gt;. Practice with formulating your own problems is one way in which you can foster creativity; the literature review contains many references you can follow up on with regard to creativity, mathematical problem posing, or both. Moreover, you can find further remarks of mine about problem posing &lt;a href=&quot;https://matheducators.stackexchange.com/a/1382/262&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Another potential way to cultivate creativity is to impose &lt;em&gt;constraints&lt;/em&gt; on your work; to this end, check PD Stokes' (non-mathematical) book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0826178456&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Creativity from Constraints&lt;/strong&gt;&lt;/a&gt; and see if you find it helpful.&lt;/p&gt;&#xA;&lt;p&gt;Other good sources are some of the work by B Sriraman and his collaborators; most recently, I received a recommendation for a book by P Borwein et al that I have yet to check out, &lt;strong&gt;&lt;a href=&quot;http://maa-store.hostedbywebstore.com/dp/B00KAEYWM8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematicians on Creativity&lt;/a&gt;&lt;/strong&gt;, which is described as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This book aims to shine a light on some of the issues of mathematical creativity. It is neither a philosophical treatise nor the presentation of experimental results, but a compilation of reflections from top-caliber working mathematicians. In their own words, they discuss the art and practice of their work. This approach highlights creative components of the field, illustrates the dramatic variation by individual, and hopes to express the vibrancy of creative minds at work.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you like creative people's recollections, then a non-mathematical source (though it includes a nice piece by Feynman) is F Barron's &lt;a href=&quot;http://www.amazon.ca/Creators-Creating-Frank-Barron/dp/0874778549&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Creators on Creating&lt;/strong&gt;&lt;/a&gt;. Of course, one cannot always take self-reported stories at face value; a discussion of this matter and how storytelling relates to creativity can be found in work by MC Bateson.&lt;/p&gt;&#xA;&lt;p&gt;More generally, if you are going to go with a single textbook to gain a broad sense about creativity and the different ways in which it is conceived of (especially as related to &lt;em&gt;problem solving&lt;/em&gt;) see Weisberg's text mentioned in my answer &lt;a href=&quot;https://matheducators.stackexchange.com/a/2237/262&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. Again, this is not explicitly about mathematics - in fact, there is a dearth of work on STEM creativity! - but you may well find tidibits of interest. (And there are at least some mentions of the hard sciences, e.g., genetics, biology.)&lt;/p&gt;&#xA;&lt;p&gt;As far as whether people are inherently creative: &lt;strong&gt;No&lt;/strong&gt;, I do not believe so, though any question about the construct will likely depend on  the definition (or conception) you have in mind. One creativity theorist of whom I am particularly fond is Howard Gruber; he effectively defines creativity as &lt;em&gt;purposeful work&lt;/em&gt;, and this is not something we are born able to churn out. Purpose is something that is developed over time, and Gruber works to &amp;quot;demystify creativity&amp;quot; as many others who use case studies are inclined towards. For a couple more of my answers that mention Gruber and his work, see &lt;a href=&quot;https://matheducators.stackexchange.com/a/2529/262&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; (less relevant) or &lt;a href=&quot;https://matheducators.stackexchange.com/a/1377/262&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Though following the links above (and links therein!) will already entail quite a bit of reading, I thought I would voice one pet peeve of mine: Many studies in the mainstream media that are purportedly about creativity use the Torrance Tests of Creative Thinking (&lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Torrance_Tests_of_Creative_Thinking&quot; rel=&quot;nofollow noreferrer&quot;&gt;TTCT&lt;/a&gt;&lt;/strong&gt;) as their main instrument. However, the TTCT really measures what is today known as &lt;strong&gt;divergent thinking&lt;/strong&gt; (DT) and, for those who research creativity, this is no longer thought to be the same as creativity. In fact, such an identification should have been put to rest in the 1970s (!).&lt;/p&gt;&#xA;&lt;p&gt;Even (or especially) major researchers of DT - such as Runco - will tell you this. For example, see (or at least glean the essence from the title) his &lt;strong&gt;&lt;a href=&quot;http://creativitytestingservices.com/satchel/pdfs/2008%20Runco,%20M.%20A.%20%282008%29.%20Commentary-%20Divergent%20thinking%20is%20not%20synonymous%20with%20creativity.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Commentary: Divergent Thinking Is Not Synonymous With Creativity&lt;/a&gt;&lt;/strong&gt; (pdf). One must be wary of studies that use TTCT-like assessments to measure DT, and then form strong (i.e., unfounded) conclusions about creativity.&lt;/p&gt;&#xA;&lt;p&gt;If you are interested in other ways to assess creativity: I am somewhat fond of TM Amabile's &lt;strong&gt;Consensual Assessment Technique&lt;/strong&gt; (CAT) for which you can find a strong overview article by Baer and McKool &lt;strong&gt;&lt;a href=&quot;http://users.rider.edu/%7Ebaer/BaerMcKool.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt; (pdf) that has a fair bit in the context of higher education. Insofar as the above-mentioned TTCT is concerned, note that the authors write in their introduction:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If creativity is to be assessed in college settings in a meaningful way, divergent-thinking tests like the Torrance Tests of Creative Thinking and other commonly used creativity tests are inadequate because they fail to meet even the loosest standards of validity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Not all of what is written in the mainstream media about creativity is as unpalatable as some of the TTCT based work. A recent piece by Brooks in The New York Times called &lt;a href=&quot;http://www.nytimes.com/2014/07/08/opinion/david-brooks-the-creative-climate.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Creative Climate&lt;/strong&gt;&lt;/a&gt; includes the following interesting snippet:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Today we live in a distinct sort of creative environment. People don’t so much live in the contradiction between competing worldviews. We live in a period of disillusion and distrust of institutions.&lt;/p&gt;&#xA;&lt;p&gt;This has created two reactions. Some monads withdraw back into the purity of their own subcultures. But others push themselves into the rotting institutions they want to reinvent. If you are looking for people who are going to be creative in the current climate, I’d look for people who are disillusioned with politics even as they go into it; who are disenchanted with contemporary worship, even as they join the church; who are disgusted by finance even as they work in finance. These people believe in the goals of their systems but detest how they function. They contain the anxious contradictions between disillusionment and hope.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I find the above excerpt quite interesting for a few reasons. One, it relates to some of the other literature on creativity: the anxious confrontation alluded to in the last sentence reminds me of R May's work in existential psychology, and especially &lt;a href=&quot;http://moe.machighway.com/%7Ecliffor1/Site/EXSupplementalReadings_files/23692564-ROLLO-MAY-Tthe-Courage-to-Create.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Courage to Create&lt;/strong&gt;&lt;/a&gt; (pdf). (The word &lt;em&gt;monads&lt;/em&gt; also comes up in Darwin's work; cf. Gruber's book &lt;a href=&quot;http://www.amazon.ca/Darwin-Man-Psychological-Scientific-Creativity/dp/0226310078/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Darwin on Man&lt;/strong&gt;&lt;/a&gt;.) Two, the notion of distrusting institutions is an important theme in Latour's &lt;a href=&quot;http://www.amazon.ca/Inquiry-into-Modes-Existence-Anthropology/dp/0674724992/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;An Inquiry into Modes of Existence&lt;/strong&gt;&lt;/a&gt;, which is part of an on-going collaborative project of potential interest to those who study creativity. Its &lt;strong&gt;&lt;a href=&quot;http://www.modesofexistence.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;corresponding site&lt;/a&gt;&lt;/strong&gt;, for which registration is free, contains the entire book; it also contains a video of a mathematician (darij grinberg) writing a question and uploading it to MathOverflow (the specific question can be seen &lt;strong&gt;&lt;a href=&quot;https://mathoverflow.net/q/138148/22971&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Perhaps it would be best to curtail myself here; my apologies for the lengthy nature of this response.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-07-31T01:30:05.150" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4105" PostTypeId="1" CreationDate="2014-07-30T13:43:23.663" Score="14" ViewCount="750" Body="&lt;p&gt;As mathematicians and mathematics educators we very often see the &lt;a href=&quot;http://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect&quot; rel=&quot;noreferrer&quot;&gt;Dunning-Kruger effect&lt;/a&gt; in action. Our calculus students are certain that they are masters of Calculus because they took the AP exam. To be fair, we ourselves are not immune: I often think I know something better than I do and end up choosing a more ambitious research project than I should.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A way to combat this effect in research is to check whether you can lecture clearly on a topic without preparation and fill in all of the details. In fact, an excellent research colleague of mine actually seems to write his papers this way: he lectures on his topic many times over and over again and then once he seems to have everything in his mind and can reproduce exactly how it all fits together he goes and writes the paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course it is possible to write mathematical papers without having everything about one's project on the tip of one's tongue, but the effort to do so may produce better work since the effort to compress and organize one's ideas in memory is reflected in the resulting writing. In fact, it is likely that such careful organization of one's ideas may lead to better and more surprising results, since having the entire framework of a problem or theory in mind allows the subconscious to have access to that data.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;My question is the following:&lt;/strong&gt; What does the mathematics education literature say about the effect of such memory effort on student performance in coursework? Does knowing the solution to a problem inside and out translate to better performance on other problems that are not obviously analogous?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although this question seems a bit strange, I think it is very important to address. Implicitly, Inquiry-Based Learning Courses focus in depth on a few problems in order to encourage this sort of deep familiarity, whereas traditional instruction often requires working on many problems in order to build skill and identify analogy. Discovery methods, though, may be weak in providing the memory framework on which to hang one's discovered results. One might remediate this by requiring students in an Inquiry-Based Learning course to learn a &quot;working knowledge outline&quot; around which to think. Data should drive such decisions, though. Hence this question... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, graduate students studying for comprehensive exams often spread themselves too thin, I think, as a result of being too &quot;expansive&quot; with their efforts rather than instead learning some central arguments deeply. I'd like to know if there is evidence that this is a valid observation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; One may view the question as a subquestion of the following about a more general &quot;theory&quot; of mathematics education: If mathematics is about mathematical analogy, where a mathematical analogy requires the wholes to be analogous, the parts to be analogous and the way these fit together to be analogous (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691025096&quot; rel=&quot;noreferrer&quot;&gt;as discussed in Polya&lt;/a&gt;), then must the structures between which the analogy is drawn be committed to memory for their most effective comparison?&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="262" LastEditDate="2015-05-04T22:25:31.723" LastActivityDate="2015-05-04T22:25:31.723" Title="The interplay of memory and mathematical performance" Tags="|undergraduate-education|secondary-education|primary-education|graduate-education|problem-solving|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4106" PostTypeId="2" ParentId="4071" CreationDate="2014-07-30T14:45:24.577" Score="4" Body="&lt;p&gt;This is only a brief answer, but I hope that it's a useful counterpoint to the others which present a very positive view of discovery learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one reference that I have found most useful for discovery- and inquiry-based learning is the paper &lt;a href=&quot;http://www.cogtech.usc.edu/publications/kirschner_Sweller_Clark.pdf&quot; rel=&quot;nofollow&quot;&gt;Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching&lt;/a&gt;. Essentially discovery learning works less well than guided instruction for most groups of learners and this has been shown in empirical studies. The reason it is less effective is that discovery approaches overload learners' limited working memory by requiring them to search for a solution, whilst more guided forms of instruction reduce this load.&lt;/p&gt;&#xA;" OwnerUserId="1958" LastActivityDate="2014-07-30T14:45:24.577" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4107" PostTypeId="2" ParentId="4105" CreationDate="2014-07-30T14:53:59.897" Score="5" Body="&lt;p&gt;The strongest students often tell me that they like taking my courses because they learn so much in doing the work for my course. I lecture and give traditional, human graded, homework. One thing I do which I see being done less in other courses is to ask some difficult open-ended problems along-side the &quot;standard working knowledge&quot; component. If I had a recitation day, I would love to give &quot;modular&quot; projects which tell a story with a sequence of 4 or 5 problems which move from basics to abstraction. My brother has done this with considerable success in modern algebra courses. That said, I don't have that sort of time in the higher math courses so I use homework to encourage them to delve &lt;strong&gt;deeper than my tests&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I assign homework problems for a given course I usually try to diversify the problem sets: roughly:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;routine problems, to encourage the building of rudimentary computational skill,&lt;/li&gt;&#xA;&lt;li&gt;global concept problems, to encourage understanding of how the theory for the course weaves together,&lt;/li&gt;&#xA;&lt;li&gt;common danger problems, to pose paradoxes to test if the students understand the need for preconditions to a given theorem,&lt;/li&gt;&#xA;&lt;li&gt;open ended problems, these lead a student in a particular direction then ask them to get the rest of the way themselves, or, more disturbingly, if there even is a way to find.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For the weak students, if they complete item 1 and some of 2 and 3 they can usually pass. Those make up about 80-90 percent of the problems. On the other hand, for the stronger students, completing all the problems is a significant achievement. After the homework the test is comparably easier. In my best efforts, the problem sets encourage both the adventurous discovery learning as well as the standard mode of learning efficient, well-known techniques. In other words, I insist everyone acquires the &quot;working knowledge outline&quot; and I give guided-opportunities for some open-ended questions. In short, if the students do the work, they have seen the content of the course from several different angles. All of this in homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, a couple years ago, I roughly asked a student if there was some way to transfer the cross-product to a particular set of matrices. At the time, the student just had calculus III behind him and was a week into linear algebra. This was part of my sometimes ambitious advanced calculus course. The student responded by solving the problem in the next day. He learned multiplication of matrices to solve the problem and &lt;strong&gt;discovered&lt;/strong&gt; the commutator bracket for matrices. Of course, the student was excited about his solution. It gave me something quantitative and special to write for his recommendation letters. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience with that student is special. In large lecture sections populated by mainly degree-seeking students it would not go well to ask such questions. I tone it down quite a bit for calculus or physics. But, in courses where the students choose to take the course I try to include something where the student can take ownership of a problem or topic either by the problems assigned, or sometimes a term-paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this said, I mainly &lt;strong&gt;lecture&lt;/strong&gt; during the class meeting time. I leave the discovery time for the home or dorm.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2014-07-30T16:39:07.060" LastActivityDate="2014-07-30T16:39:07.060" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4108" PostTypeId="2" ParentId="4105" CreationDate="2014-07-30T16:21:18.437" Score="9" Body="&lt;p&gt;Anecdotally, based on self-observation and observation of many faculty and grad students: &quot;if it's not in your head in some form, you can't think about it&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A funny point here is that it seems not strictly necessary to &quot;completely understand&quot; something, if one can keep it in one's mind. Indeed, I don't see how to make the transition from not-understanding to understanding without having the thing in one's head already at the not-understanding phase. Nevertheless, it appears to me that many people have great difficulty &quot;remembering&quot; something they don't already understand, or at least &quot;feel comfortable with&quot;. This is obviously an obstacle to gaining understanding... if, in effect, one can't bear to contemplate the thing one wishes to understand?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, indeed, one cannot make analogies to things one has truly forgotten, or never known.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The more one can keep in one's head, while remaining able to manipulate the things, the better. But/and I don't think this at all requires &quot;complete understanding&quot;, but, rather, something more like &quot;awareness&quot;. And it would be misguided, I think, to try to force-memorize things, since the resultant clutter and fragility often makes &quot;manipulation&quot; difficult.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-07-30T16:21:18.437" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4110" PostTypeId="1" AcceptedAnswerId="4113" CreationDate="2014-07-30T22:21:31.820" Score="8" ViewCount="287" Body="&lt;p&gt;As I apply for several other colleges who are hiring part time math teachers, I find myself wondering about this question as it is asked on ALL college instruction applications. Some of the questions that have came about in the adjunct faculty math applications are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Describe your experience working with diverse socio-economic students&lt;/li&gt;&#xA;&lt;li&gt;Ability and willingness to work with students from a wide range of socioeconomic backgrounds.&lt;/li&gt;&#xA;&lt;li&gt;Ability to motivate and teach students with diverse cultural, socio-economic and ethnic backgrounds, preparation, learning styles, and disabilities, using instructional methods that reflect cultural sensitivity and interdisciplinary approaches to subject matter.&#xA;Incorporating materials and activities that reflect the role of mathematics in students’ lives.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;My question is, how do you all attack such a question? Here's my response for this question:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As Jaime Escalante puts it, “Mathematics is the great equalizer.” I have had the privilege in working with students from a culturally diverse community: African Americans, Native Americans, Asian Americans, Hispanic Americans, mixed-race students, students with disabilities, students with limited English proficiency, AB 540 students, students from low income communities, students of particular religious groups, and students with different sexual orientations. Math never cares what your social background is, it does not care what language you speak, nor will it ever discriminate.  Mathematics is accessible for all students. This will be their language and by helping them build this academic language content, they will be better prepared for the challenges they will encounter in college and therefore be more successful. So no matter what background they come from, Math will provide an equal opportunity for all my college students to reach their goals. Even though Math is the equalizer people will engage in Math from different cultural perspectives therefore it is imperative that a Mathematics professor uses cultural distinctions to access the universality of Mathematics. Math origins are spread throughout the world and I would bring in these distinct and unique cultural vignettes to make representatives from different areas produce and acknowledge the contribution of others so that they respect and acknowledge the contribution to the field.&lt;/p&gt;&#xA;" OwnerUserId="712" LastEditorUserId="8571" LastEditDate="2020-08-27T12:26:39.473" LastActivityDate="2020-08-27T12:26:39.473" Title="Answering the Diversity Question for Mathematics Instructor Applications" Tags="|mathematical-pedagogy|cultural-differences|employment|philosophy-of-maths|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="4111" PostTypeId="2" ParentId="3920" CreationDate="2014-07-31T00:23:36.590" Score="0" Body="&lt;p&gt;What makes one better or worse may depend on many things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately (though this is not especially enlightening) - if you want to look at the kind of thing IQR measures, IQR is good at that, while if you want to look at the kind of thing standard deviation measures, standard deviation is good at that. The different ways they respond to data helps determine where they're more useful (in particular, sd is much more impacted by large outliers, so tends to be more valuable in situations where the data doesn't have big tails)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to give a slightly different take that may or may not be of some help. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both IQR and standard deviation can be thought of as measures of a kind of &quot;typical distance between data points&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the IQR is effectively the distance between the median of the top half of the data and the median of the bottom half of the data, and in that sense is a kind of 'typical distance'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The variance (up to a Bessel correction factor in the sample case) is half the average squared distance between pairs of points; the standard deviation is thereby a root-mean-square distance between pairs of points, divided by $\sqrt{2}$ (or times $\sqrt{\frac{n}{2(n-1)}}$ in samples when applying a Bessel correction in the sample case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The different way those 'typical distance' measures are affected by observations in different parts of the data is useful; it may be worth exploring an empirical influence function (without &lt;em&gt;necessarily&lt;/em&gt; naming it, the concept is intuitive enough).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These lines of discussion may not be suitable for every level of student, but it's a take on two common measures of spread that may be easier to motivate that some others, and has at least one advantage: it reduces the size of that 1.35 &quot;asymptotic conversion factor&quot; at the normal to the perhaps less surprising 0.954 (or 1.048, depending on which way you go) - that's a good deal closer to 1, and it's smaller for the measure less affected by very large observations. &lt;/p&gt;&#xA;" OwnerUserId="755" LastEditorUserId="755" LastEditDate="2014-07-31T02:07:11.480" LastActivityDate="2014-07-31T02:07:11.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4112" PostTypeId="2" ParentId="3983" CreationDate="2014-07-31T09:06:24.173" Score="0" Body="&lt;p&gt;Say you have 3 cities A, B, C and for each of them you know  the percentage of people each year remain in the city or move to one of the two other cities. Given the population distribution at time 0 how will people distribute in the long run?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say you have 3 car rental X,Y,Z and for each of them you know the percentage of rented car that are returned in the same place or to one of the two others. Given the car distribution at time 0 how will car distribute in the long run?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What if in the first model I add the percentage of dying and newborns? An in the second one I add lost cars and new cars?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are simple real life examples (where of course the 3 can be easily changed into N) where eiganvalues and eigenvectors come into play into a Markov chain form, as commented before. The question you consider is easily motivated (I'm running the car rental and I want to know if in 2-3 years I will have to pay someone to bring back cars to their original disposition or not) and the answer easily understood. &lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2014-07-31T09:06:24.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4113" PostTypeId="2" ParentId="4110" CreationDate="2014-07-31T10:20:05.590" Score="7" Body="&lt;p&gt;My take for what it is worth. I see at least two very strong statements you make. I don't see strong statements as necessarily problematic; sometimes it is very important to take a stand. But stands often have to be defended.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first very strong statement essentially summarized in your sentence:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Math never cares what your social background is, it does not care what&#xA;  language you speak, nor will it ever discriminate.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which (unless I am mistaken) relies on the assertion that mathematics is something discovered rather than constructed. Since not everyone shares this view, you may encounter opposition to this in your job search. I won't give a counter-argument, but if you get into a discussion of mathematics as objective, someone might ask you how the choices that make up the department's curriculum were discovered rather than culturally constructed (considering that this shapes what the students will be studying). If you say that don't see the choices of what to study as part of the objectivity of mathematics and leave it at that, you may be considered to have missed the point of the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You also make a strong statement that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;[...] Math &lt;em&gt;will provide&lt;/em&gt; an equal opportunity for all my college&#xA;  students to reach their goals.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[emphasis mine] Consider how you will answer challenges to whether math can actually provide the opportunity you promise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen very intelligent people I respect take very different and compelling views on diversity questions in mathematics, which tells me that it is a difficult set of questions. And some of it may come down to beliefs and values.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-07-31T10:20:05.590" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4114" PostTypeId="1" CreationDate="2014-07-31T19:00:06.850" Score="3" ViewCount="876" Body="&lt;p&gt;Based on &lt;a href=&quot;http://boingboing.net/author/maggie_koerth_baker&quot; rel=&quot;nofollow&quot;&gt;Maggie Koerth-Baker&lt;/a&gt;'s article, &quot;&lt;em&gt;&lt;a href=&quot;http://boingboing.net/2012/08/07/what-do-christian-fundamentali.html&quot; rel=&quot;nofollow&quot;&gt;What do Christian fundamentalists have against set theory?&lt;/a&gt;&lt;/em&gt;&quot;, it seems there are some parts of culture which put some restrictions on math education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, there were and there are several types of such limitations on the empirical sciences and philosophy. For example, in several countries, some parts of research on genetic engineering, particularly those which are related to cloning humans and animals, are not legal for &lt;em&gt;ethical&lt;/em&gt; reasons.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My question is about current or historical examples of such cultural restrictions on research in particular fields of mathematics or mathematics in general. Please introduce references for each example.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-08-01T03:42:52.640" LastActivityDate="2014-08-03T06:26:21.970" Title="Examples of cultural limitations on math education" Tags="|reference-request|general-pedagogy|history|set-theory|culture|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4115" PostTypeId="1" AcceptedAnswerId="4187" CreationDate="2014-07-31T19:43:47.140" Score="4" ViewCount="311" Body="&lt;p&gt;Mathematics has many applications in industry. Here I would like to focus on a special part of these industries and the role of mathematics in its development.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Consider the countries which invested heavily in mathematics for military purposes. How did they format their math education programs in light of their warlike or expansionist nature? Is there information on particular methods or books they have used to effectively increase the country's mathematics education and knowledge? Was their focus exclusively on applied mathematics or was pure mathematics as important, possibly having a higher place?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-08-13T06:28:05.787" LastActivityDate="2014-08-13T16:25:35.590" Title="Mathematics of War" Tags="|mathematical-pedagogy|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="4116" PostTypeId="2" ParentId="4114" CreationDate="2014-07-31T20:11:06.607" Score="10" Body="&lt;p&gt;A quintessential example of a cultural clash over math education in US schools, I would say, is the entire Math Wars, an ongoing struggle over standards and pedagogy in K-12 schools. It is a struggle of curriculum reformers trying to bring research-based innovations of mathematics thinking, learning, and teaching over the last three decades into schools facing opposition from some parents and mathematicians who prefer a traditionalist approach relying on outdated (or, in some cases, just frankly unscientific) understandings of a whole host of issues surrounding education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a thorough introduction, I recommend Schoenfeld's (2004) writing on the subject. It's 10 years old, and some people speak about it as though it is in the past (many people have tried to declare &quot;math wars&quot; as over) but I think some of the arguments over the Common Core State Standards have made it clear just how much Math Wars are still a reality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question as titled and asked is a little muddled (education is missing from the wording of the question in yellow, though prominent in the title). This addresses the (I think) intended question of an example of culture placing a restriction on mathematics education. In short: research findings conflict with traditional cultural values, resulting in some people rejecting the science and adhering to tradition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cited:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schoenfeld, A. H. (2004). &lt;a href=&quot;http://www.math.cornell.edu/~henderson/courses/EdMath-F04/MathWars.pdf&quot;&gt;The Math Wars&lt;/a&gt;. Educational Policy, 18(1), 253–286. doi:10.1177/0895904803260042&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-07-31T20:11:06.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4117" PostTypeId="2" ParentId="4114" CreationDate="2014-07-31T20:23:29.553" Score="7" Body="&lt;p&gt;Perhaps this historical example fits the bill:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Khovanova, T., &amp;amp; Radul, A. (2012). Killer problems. &lt;em&gt;American Mathematical Monthly, 119&lt;/em&gt;(10), 815-823.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The piece was published earlier on the arXiv as &lt;a href=&quot;http://arxiv.org/abs/1110.1556&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Jewish Problems&lt;/strong&gt;&lt;/a&gt;. Here is the abstract:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This is a special collection of problems that were given to select applicants during oral entrance exams to the math department of Moscow State University. These problems were designed to prevent Jews and other undesirables from getting a passing grade. Among problems that were used by the department to blackball unwanted candidate students, these problems are distinguished by having a simple solution that is difficult to find. Using problems with a simple solution protected the administration from extra complaints and appeals. This collection therefore has mathematical as well as historical value. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-08-03T06:26:21.970" LastActivityDate="2014-08-03T06:26:21.970" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="4118" PostTypeId="1" AcceptedAnswerId="7747" CreationDate="2014-07-31T21:56:42.503" Score="15" ViewCount="1542" Body="&lt;p&gt;I've recently dipped my toes into the world of number theory; and I've bought a book that to me is quite unconventional: R. P. Burn, &lt;em&gt;A Pathway into Number Theory&lt;/em&gt;. I've yet to put the book through its paces, but it seems agreeable enough to me. The book is unique in that it poses a sequence of questions to you in the hope that you'll be able to answer them and by thus doing so, begin to discover the fundamentals of number theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a style of learning that I find agreeable as the knowledge I gain this way is assimilated and retained better. I like being able to discover for myself however most times I don't have the necessary direction (I am self-studying) but that's where this textbook comes in. I feel that this book in the process of nudging you in the right direction also helps you think more like a mathematician (from my very limited experience with it). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I enjoy books that give you a &quot;pathway&quot;, although I guess this is the aim of all textbooks. Is it possible for anyone to recommend texts that take a similar aided discovery/inquiry based approach? &lt;/p&gt;&#xA;" OwnerUserId="1677" LastEditorUserId="1677" LastEditDate="2017-01-16T21:27:37.557" LastActivityDate="2017-01-17T22:22:14.557" Title="Recommendations for inquiry based/aided discovery textbooks" Tags="|reference-request|textbooks|reference-books|inquiry-based-learning|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4119" PostTypeId="1" CreationDate="2014-07-31T22:27:56.000" Score="12" ViewCount="2215" Body="&lt;p&gt;There are many examples of artistic works which have some mathematical aspects. A high school or undergraduate math teacher can use them as interesting examples in his/her teaching. e.g. In a combinatorics course when a teacher introduces the notion of a magic square, he/she can add examples form paintings and literature as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(1)&lt;/strong&gt; &lt;a href=&quot;http://en.wikipedia.org/wiki/Albrecht_D%C3%BCrer&quot; rel=&quot;noreferrer&quot;&gt;Albrecht Dürer&lt;/a&gt;'s magic square in his famous painting &lt;a href=&quot;http://en.wikipedia.org/wiki/Melencolia_I&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Melancolia I&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/xbDob.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(2)&lt;/strong&gt; &lt;a href=&quot;http://en.wikipedia.org/wiki/Goethe&quot; rel=&quot;noreferrer&quot;&gt;Goethe&lt;/a&gt;'s description of a magic square in &lt;a href=&quot;http://en.wikipedia.org/wiki/Goethe%27s_Faust&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Faust&lt;/em&gt;&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\begin{array}{l|l}&#xA;\text{Du musst verstehen!} &amp;amp; \text{You shall see, then! } \\&#xA;\text{Aus Eins mach Zehn,} &amp;amp; \text{From one make ten!} \\&#xA;\text{Und Zwei lass gehen,} &amp;amp; \text{Let two go again,} \\&#xA;\text{Und Drei mach gleich} &amp;amp; \text{Make three even,} \\&#xA;\text{So bist du reich.} &amp;amp; \text{You’re rich again.} \\&#xA;\text{Verlier die Vier!} &amp;amp; \text{Take away four!     } \\&#xA;\text{Aus Fünf und Sechs,} &amp;amp; \text{From five and six,} \\&#xA;\text{So sagt die Hex,} &amp;amp; \text{So says the Witch,} \\&#xA;\text{Mach Sieben und Acht,} &amp;amp; \text{Make seven and eight,} \\&#xA;\text{So ist's vollbracht:} &amp;amp; \text{So it’s full weight:} \\&#xA;\text{Und Neun ist Eins,} &amp;amp; \text{And nine is one,  } \\&#xA;\text{Und Zehn ist keins.} &amp;amp; \text{And ten is none.} \\&#xA;\text{Das ist das Hexen-Einmaleins!} &amp;amp; \text{This is the Witch’s one-times-one!} \\&#xA;\end{array}$$&#xA;(See &lt;a href=&quot;http://www.poetryintranslation.com/PITBR/German/FaustIScenesIVtoVI.htm&quot; rel=&quot;noreferrer&quot;&gt;Faust&lt;/a&gt;, line 2540.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the above phrases are describing a magic square as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/HOR81.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are interesting examples of using mathematical notions in arts which are useful to introduce these notions to students at high school/undergraduate level? &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-08-02T01:00:10.043" LastActivityDate="2017-06-25T20:12:36.487" Title="Examples of Artistic Works with Mathematical Aspects" Tags="|mathematical-pedagogy|arts|" AnswerCount="9" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4120" PostTypeId="2" ParentId="258" CreationDate="2014-07-31T22:52:43.893" Score="9" Body="&lt;p&gt;As coordinator of a Maths Learning Centre, where many many students have actually seen some calculus at school but still struggle with calculus at University, these are my top few:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The notation f(x)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quite a few students think that $f(x)$ means $f \times (x)$. This is particularly true with trigonometric functions, where many of them actually say &quot;sine times ex&quot; when they read $\sin(x)$ aloud. I teach them it's pronounced  &quot;sine of ex&quot;, but for some that doesn't help because they were taught at school that &quot;of&quot; always means &quot;times&quot; (as in &quot;10% of 15&quot;). Constantly reinforcing that $f(x)$ is the object produced when the action $f$ is performed upon the object $x$ -- or &quot;the result of f acted upon x&quot; -- would make life better for many of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if students do realise that $f(x)$ is not about multiplication, many of them do not know at a deep level that $f(x)$ is actually a number. They see it as a call to action, rather than the name of the &lt;em&gt;result&lt;/em&gt; of this call to action. While it may be &lt;em&gt;possible&lt;/em&gt; to calculate the value of $f(3)$ and it might come out to the answer $4$, say, you don't necessarily &lt;em&gt;need&lt;/em&gt; to calculate it, and you can use the symbol &quot;$f(3)$&quot; every time you mean &quot;$4$&quot; if you want. Basically, $f(3)$ is literally able to stand in the place of whatever value it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may seem like a small issue, but it is a problem when students attempt to differentiate $\ln(2)$ to give $\frac12$ (for example, given $y = e^{\ln(2)x}$, they write $\frac{dy}{dx} = e^{\ln(2)x}(\frac12 x + \ln(2))$ ). I keep reinforcing to them that $\ln(2)$ is just a number. It's somewhere between 0 and 1, but I don't have to calculate it to know it's a number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The fundamental meaning of an x-y equation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I really wish students knew that the fundamental meaning of an x-y equation is to tell if a point is part of set in 2D space. That is, when we see something like $y = 3x+1$, it is telling us a rule for how x and y must be related in order for the point $(x,y)$ to be on a line. If we sub in the point and it works, then it's on the line, and if it doesn't work, then it's not on the line. The question of &quot;Is the point (1,4) on the line with equation $y = 3x+1$?&quot; stumps many of my students, even if they can tell me what the slope and intercepts are without even thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another point I'd like them to realise is that any equation such that all the points on the line satisfy it and all the points not on the line don't satisfy it, is a possible equation of the line. For example, $y - 3x = 1$ and $x = \frac{y-1}{3}$ or even $(y - 3x +1)^2 = 0$ are perfectly reasonable equations of the same line. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically my point is that I would like students to know that an equation does not have to be explicitly tied to a function, and even if it is, it can still be used to tell if points are part of a set. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The connection between algebra and geometry&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The flip-side of the above point is that equations &lt;em&gt;do&lt;/em&gt; represent geometrical objects, and you can glean information from the equation to tell you about the geometry of the object. I guess what I'd like them to realise is that writing an equation in a certain way will tell you certain information about the object, and especially that different ways of writing the equation will tell you different information. These ideas are often known separately for different equations (eg quadratic equations where completeing the square tells you the vertex but factorising tells you the x-intercepts), but I'd love them to have the general concept reinforced for everything so that they realise it's actually one idea worked out in different situations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Knowing what the graphs of various functions look like&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My students self-identify this one as their own major problem. They just don't have an instinct for what the graph of a function looks like without having to get a computer/graphics calculator draw it for them. If you ask them &quot;what does the graph of a cubic look like?&quot;, they will reach for their graphics calculator, when all you wanted them to do was draw a &quot;swoosh&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would love it if they simply &quot;knew&quot; what various graphs looked like. A bit of a stimulus-response thing where if they see either of the forms of the equation of a line, they'll think &quot;line&quot;, or if they see $y = \text{quadratic in }x$ they'll automatically think of an up-curve or a down-curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realise this may be at odds with the previous point, but I think you can achieve both. Plus it really is an issue for them when their university lecturers use these functions as their basic examples, and the students simply can't imagine them!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Skill at algebraic manipulation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, this really is a prerequisite that gets in the way for a lot of people. It's not as bad a problem as you might think, because often learning algebra &quot;in passing&quot; while doing calculus is actually more successful than the students' previous algebra instruction. However, it would be nice if they knew how to expand/factorise/manipulate fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This last one of manipulating fractions particularly comes up when they are doing differentiation by first principles with limits, and when they are using the quotient rule, and many students show that they simply do not know how to add fractions, even of the number variety. (Though there are some students who can do the algebraic ones and not the number ones.)&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-07-31T23:25:05.863" LastActivityDate="2014-07-31T23:25:05.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4121" PostTypeId="2" ParentId="4118" CreationDate="2014-07-31T22:55:26.440" Score="8" Body="&lt;p&gt;Check out &lt;a href=&quot;http://jiblm.org&quot;&gt;http://jiblm.org&lt;/a&gt;. There are lots of scripts here, some better than others. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A nice book in this style is &lt;a href=&quot;http://www.maa.org/publications/ebooks/distilling-ideas&quot;&gt;&quot;Distilling Ideas&quot; by Brain Katz and Michael Starbird&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also recommend the following method: Take a reputable text on a topic, and try to prove all the theorems for yourself. If you get stuck for a long time, take a quick peek to get unstuck…or find a new thing to assume by looking at the proof. You'll find that if you do this honestly that you can find out which ideas are novel and which are routine. I learned this approach from a Halmos student, and it really is enjoyable to do. The proofs in the book are just regarded as the answer key.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have fun! &lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2014-07-31T22:55:26.440" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4122" PostTypeId="1" AcceptedAnswerId="4131" CreationDate="2014-07-31T23:34:53.510" Score="10" ViewCount="459" Body="&lt;p&gt;I am writing a paper directed at a physics-education journal and I want to briefly refer to the phenomenon of students “overassociating” (in lack of a better term) mathematical concepts with each other which have similar notations, names or properties, but are not similar in all respects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the multiplication of real numbers, scalar multiplication and the cross product are all called &lt;em&gt;multiplication&lt;/em&gt; and partially share their symbols, because they have similar properties. I am interested in the phenomenon that a student treats, e.g., a scalar multiplication like a normal multiplication where this is not allowed and divides by a vector.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is the &lt;a href=&quot;https://matheducators.stackexchange.com/q/926/75&quot;&gt;misassumption of universal linearity&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As giving the above explanation or something similar would be too much, I am looking for one of the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;An established term for such an association going too far, if one exists.&lt;/li&gt;&#xA;&lt;li&gt;A general reference from mathematics-education literature about this phenomenon.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="75" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-08-02T18:13:38.343" Title="Term and reference for the problem of students “overassociating” concepts with each other" Tags="|reference-request|students-mistakes|terminology|" AnswerCount="2" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="4123" PostTypeId="2" ParentId="4119" CreationDate="2014-07-31T23:36:46.873" Score="1" Body="&lt;p&gt;I haven't looked at many of &lt;a href=&quot;http://www.math.dartmouth.edu/~mqed/eBookshelf/art/index.php&quot; rel=&quot;nofollow&quot;&gt;these&lt;/a&gt; in a while, but there are several links to the math behind patterns, perspective, etc. &lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-07-31T23:36:46.873" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4124" PostTypeId="2" ParentId="4119" CreationDate="2014-07-31T23:48:41.720" Score="11" Body="&lt;p&gt;Henderson &amp;amp; Taimina (2006) point out that models in hyperbolic geometry are aesthetically compelling, and that the artist M. C. Escher made us of this in a representation of infinity within his works:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Repeating patterns on the sphere have an aesthetic appeal through&#xA;  their simplicity and finiteness. However, in these various hyperbolic&#xA;  models, the patterns have an aesthetic appeal for us because of their&#xA;  connections with infinity--there are infinitely many such patterns and&#xA;  each also draws us to the infinity at the edge of the disc, leaving&#xA;  sufficient space for our imagination.&#xA;  (p. 69)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/DWxbn.jpg&quot; alt=&quot;M.C. Escher&amp;#39;s Circle Limit III (based on the Poincaré disc model)&quot;&gt;&#xA;M.C. Escher's Circle Limit III &lt;em&gt;(based on the Poincaré disc model)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Henderson, D. W., &amp;amp; Taimina, D. (2006). Experiencing meanings in geometry. In N. Sinclair, D. Pimm, &amp;amp; W. Higginson (Eds.), Mathematics and the aesthetic: New approaches to an ancient affinity (pp. 58–83). Springer Verlag.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-07-31T23:48:41.720" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4125" PostTypeId="2" ParentId="4119" CreationDate="2014-08-01T01:01:21.000" Score="8" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Salvador_dali&quot; rel=&quot;noreferrer&quot;&gt;Salvador Dali&lt;/a&gt;'s 1954 &lt;a href=&quot;https://en.wikipedia.org/wiki/Crucifixion_%28Corpus_Hypercubus%29&quot; rel=&quot;noreferrer&quot;&gt;Crucifixion (Corpus Hypercubus)&lt;/a&gt; &quot;deviates from traditional portrayals of the Crucifixion by depicting Christ on the polyhedron net of a hypercube [...].&quot;  It could be used to introduce the idea of a fourth spatial dimension.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/S69NJ.jpg&quot; alt=&quot;Crucifixion (Corpus Hypercubus)&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the Wikipedia entry:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Instead of painting Christ on a wooden cross, Dalí depicts him upon the net of a hypercube, also known as a tesseract. The unfolding of a tesseract into eight cubes is analogous to unfolding the sides of a cube into six squares. The use of a hypercube for the cross has been interpreted as a geometric symbol for the transcendental nature of God. Just as God exists in a space that is incomprehensible to humans, the hypercube exists in four spatial dimensions, which is equally inaccessible to the mind. The net of the hypercube is a three-dimensional representation of it, similar to how Christ is a human form of God that is more relatable to people.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-08-01T01:01:21.000" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4126" PostTypeId="2" ParentId="4119" CreationDate="2014-08-01T03:46:49.447" Score="12" Body="&lt;p&gt;Some crazy mathematicians have managed to calculate the missing centre of the Escher artwork below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://escherdroste.math.leidenuniv.nl/&quot; rel=&quot;noreferrer&quot;&gt;Go to their website to learn all about it.&lt;/a&gt;&#xA;They have some pretty cool animations of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, &lt;a href=&quot;http://im-possible.info/english/articles/escher_printgallery/&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt; gives a pretty good overview of the project.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://web.cortland.edu/jubrani/Escher%20-%20Sheila%20Haak.pdf&quot; rel=&quot;noreferrer&quot;&gt;Here is a link&lt;/a&gt; to a paper about Escher and symmetry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/7iXj3.jpg&quot; alt=&quot;Original&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/VZ9dH.jpg&quot; alt=&quot;Filled in&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="683" LastEditorUserId="683" LastEditDate="2014-08-01T03:55:16.877" LastActivityDate="2014-08-01T03:55:16.877" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4127" PostTypeId="1" CreationDate="2014-08-01T04:40:01.910" Score="4" ViewCount="376" Body="&lt;p&gt;The sentence in the title is inspired by a sentence on the gate of hell in &lt;a href=&quot;http://en.wikipedia.org/wiki/Dante_Alighieri&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Dante Alighieri&lt;/em&gt;&lt;/a&gt;'s masterwork &lt;a href=&quot;http://en.wikipedia.org/wiki/Divine_Comedy&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Divine Comedy&lt;/em&gt;&lt;/a&gt; (&lt;a href=&quot;http://en.wikipedia.org/wiki/Inferno_%28Dante%29&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Part I: Inferno&lt;/em&gt;&lt;/a&gt;). I usually write it on my office door after exams when I think there are some requests of failed students for a remedial exam. It illustrates that I don't accept such requests. Recently I thought it is not bad to be more flexible about this situation.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What is the best policy about requests of students for remedial exams (or more time for their homework)? Are there situations that a teacher should consider such remedial opportunities? Should I replace the sentence on my office door with &quot;&lt;em&gt;Never say never!&lt;/em&gt;&quot;?   &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastActivityDate="2014-08-04T14:29:36.797" Title="Abandon all hope, ye who enter my office!" Tags="|course-design|exams|homework|exercises|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="4128" PostTypeId="2" ParentId="4105" CreationDate="2014-08-01T05:24:52.737" Score="6" Body="&lt;p&gt;In light of your edited &quot;more general&quot; question, I thought I would make a few remarks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, an early treatment of the subject of reasoning by analogy can be found in work on Gestalt psychology. One of the better known authors/books in this area (where mathematical examples are plentiful) is Max Wertheimer/&lt;em&gt;Productive Thinking&lt;/em&gt;. For a somewhat more recent consideration of this book (by the author's son, Michael Wertheimer) see &lt;strong&gt;&lt;a href=&quot;http://files.eric.ed.gov/fulltext/ED406635.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;, where &lt;em&gt;memory&lt;/em&gt; is broached by contrasting &quot;reproductive thinking&quot; and &quot;productive thinking.&quot; In particular, see the following excerpt:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/5IsTs.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The younger M Wertheimer continues later on in the same piece:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/eN9nz.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This latter excerpt bifurcates nicely into two areas of potential interest. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;First&lt;/strong&gt;, there is the discussion of the relation to &quot;insight&quot; or &quot;illumination&quot; (terminology originally due to Wallas' (1926) &lt;em&gt;The Art of Thought&lt;/em&gt;, I believe) which suggests there is a fair amount of information that can be found in the literature on &lt;strong&gt;creativity&lt;/strong&gt;. For a survey of literature related to creativity, see my earlier MESE response &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/4104/262&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;. With regard to analogies, in particular, see (for example) &lt;em&gt;The Encyclopedia of Creativity&lt;/em&gt;; a relevant bit related to Wertheimer is &lt;strong&gt;&lt;a href=&quot;http://books.google.com/books?id=1fsW-k5wypkC&amp;amp;lpg=RA1-PA444&amp;amp;ots=X3IkObUHcW&amp;amp;dq=productive%20thinking%20wertheimer%20analogy&amp;amp;pg=RA1-PA444#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;, which references a section specifically about analogies, in the same book, &lt;strong&gt;&lt;a href=&quot;http://books.google.com/books?id=1fsW-k5wypkC&amp;amp;lpg=RA1-PA444&amp;amp;ots=X3IkObUHcW&amp;amp;dq=productive%20thinking%20wertheimer%20analogy&amp;amp;pg=PA71#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;. (See also Fauconnier and Turner on &lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Conceptual_blending&quot; rel=&quot;nofollow noreferrer&quot;&gt;conceptual blending&lt;/a&gt;&lt;/strong&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Second&lt;/strong&gt;, the excerpt above mentions &lt;em&gt;anagram tasks&lt;/em&gt;. In keeping with your mention of Polya, I thought I would record here an observation that I have not seen made so explicitly in the literature on mathematics education: Jacques Hadamard, in his investigation of mathematical insight, finds that almost no mathematicians are thinking in words. The one true exception, to whom Hadamard devotes a couple of pages, is George Polya. (See the latter half of my MESE response &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/1801/262&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;.) Unlike other mathematicians with whom Hadamard conversed, Polya finds analogies with words, letters, and turns of phrase to be critical to his thinking. You may recall from &lt;em&gt;How to solve it&lt;/em&gt; that Polya (1945) gives an example asking how to rearrange (anagram!) the letters &lt;em&gt;DRY OX TAIL IN REAR&lt;/em&gt;. His suggestion, as best I recall, is first to separate the letters into the vowels and consonants (I won't spoil the solution here). This is but a single instance related to the more general distinguishing feature noted by Hadamard. In this respect, I find it of note that the study of problem solving within Mathematics Education has sprung forth from an opus by a mathematician whose modality of thought is quite different from that of his peers. Might this conceptual provenance have led to meaningful ramifications later on? I am not sure, and leave the reader with this question as food for thought...&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-08-01T05:24:52.737" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4129" PostTypeId="2" ParentId="270" CreationDate="2014-08-01T09:01:23.587" Score="1" Body="&lt;p&gt;I have several arguments for having some hand-calculations in the classroom, but with some caveats:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, as a pure mathematician, whose entire research was doing things by hand entirely in symbols, I can confidently say that it was important to me! Also, when I enter various applied mathematician's or statistician's  or scientist's offices at my university I see whiteboards and scraps of paper with symbols scrawled on them every time. I have watched even those whose work mostly involves computers constantly switching back and forth from the computer to hand-caclulations. So clearly hand-calculations are still important for the working scientist. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often there is a decent amount of algebraic thinking that needs to be done in order to make it possible to turn your problem into something the computer can solve. Without some training in this, you may never be able to even enter the correct input. (Of course, this also means it is probably a good idea to introduce calculators/computers early so that people can learn to use them in conjunction with their algebraic manipulations, rather than leaving it till later to learn that skill.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, symbolic thinking is a style of thinking that is actually quite useful, and allows people to do things they could not otherwise do. Sometimes an in-depth study of the symbols and rules allows you to know what is possible and what is not possible and what can be made possible. Deferring all of that to a machine can rob us of coming up with a way that is different and makes more sense to us, rather than the way that makes sense to the person who programmed the computer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, sometimes our approach to using symbols and calculations does not highlight these thinking skills. Techniques of integration &lt;em&gt;can&lt;/em&gt; be a study of how integrals are connected to each other and &lt;em&gt;can&lt;/em&gt; be a study of the difficulties that integration poses compared to differentiation, and it &lt;em&gt;can&lt;/em&gt; be a study in how to earn how to choose from a set of strategies in a field where multiple approaches may work or none may work. However, sadly, it usually just feels like a lot of &quot;here's another fancy trick&quot;. So in terms of structuring a course, I'd suggest making sure your emphasis is on the higher thinking if you do choose to do calculations (or have no choice).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, something that is never mentioned in these discussions is self-efficacy. There is something highly satisfying about knowing that you can do something yourself without the need of technology or assistance. My daughter learned to crochet for no purpose but to know that she could. I personally love doing integrals simply because I can. The nursing students I help love to be able to calculate their drug caclulations by hand once they know how, because it gives them power over their calculations. This personal power is an important aspect of doing things by hand. Note you don't necessarily need absolute fluency to have this feeling, but it does take &lt;em&gt;some&lt;/em&gt; practice.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-08-01T09:01:23.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4130" PostTypeId="2" ParentId="4119" CreationDate="2014-08-01T12:27:57.677" Score="3" Body="&lt;p&gt;MAA has a special interest group devoted to mathematics and the arts. Here is the link to its web page which includes galleries and resources:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://sigmaa.maa.org/arts/exhibits.html&quot; rel=&quot;nofollow&quot;&gt;http://sigmaa.maa.org/arts/exhibits.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;AMS sponsors an exhibit of mathematical art at the Joint Mathematical Meetings and here is a link for that and related things:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/mathimagery/&quot; rel=&quot;nofollow&quot;&gt;http://www.ams.org/mathimagery/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also an annual conference dealing with connections between mathematics and the arts which generates lots of materials both of research and pedagogical interest:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://bridgesmathart.org/&quot; rel=&quot;nofollow&quot;&gt;http://bridgesmathart.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-08-01T12:27:57.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4131" PostTypeId="2" ParentId="4122" CreationDate="2014-08-01T15:41:47.017" Score="6" Body="&lt;p&gt;I am not aware of standard terminology for this. Perhaps you have a chance to introduce some!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I propose &lt;strong&gt;&quot;concept overload&quot;&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This suggests the underlying principle, that students have one concept in mind but are piling too many &quot;things&quot; onto it. This could be because they truly are similar concepts (like a limit of a function vs. a limit of a sequence), or because they share a name or notation (like the &quot;multiplication&quot; examples you mention).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your article, you could briefly explain this idea early on, define the terminology (whichever you choose), and continue to use it throughout.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also think this is a good idea because of the similarity to &lt;a href=&quot;https://en.wikipedia.org/wiki/Function_overloading&quot;&gt;&quot;function overloading&quot; in programming&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[It] allows creating several methods with the same name which differ from each other in the type of the input and the output of the function. It is simply defined as the ability of one function to perform different tasks.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-08-01T15:41:47.017" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4132" PostTypeId="2" ParentId="4083" CreationDate="2014-08-01T16:30:50.213" Score="6" Body="&lt;p&gt;I think it would be helpful to simply reiterate the differences between a &quot;proof&quot;, in the real-world sense of the word, and a &lt;strong&gt;mathematical proof&lt;/strong&gt;. Indeed, it would be best to keep adjoining the &quot;mathematical&quot; part to &quot;proof&quot; whenever you ask your students to write a proof. This will continually remind them that they are working on a specific kind of argument style with rigorous standards.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another idea is to provide students with written arguments (they could either created by you, or come from past students' assignments) and have them critique them. Have some that are correct proofs of correct results, some that are false proofs of correct results, and some that are false proofs of false results. Have the students read them and discuss them, for both correctness and clarity. By getting them to think and &lt;em&gt;talk&lt;/em&gt; about different arguments, they will get a better understanding of what constitutes a &lt;em&gt;mathematical&lt;/em&gt; argument, as opposed to a regular ol' argument.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-08-01T16:30:50.213" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4133" PostTypeId="2" ParentId="186" CreationDate="2014-08-02T00:25:48.340" Score="5" Body="&lt;p&gt;I very much liked how the real numbers were introduced to me at university. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The professor introduced the usual axioms of arithmetic. Then defined continuous functions, Cauchy sequences, limits, and so on with all kinds of proofs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then started proving the Intermediate Value Theorem. (If f (x) is continuous on [a, b], and f (a) &amp;lt; 0, f (b) &gt; 0, then there is an x in [a, b] such that f(x) = 0). So he takes the set S of reals in [a, b] where f (x) &amp;lt;= 0, S is not empty and has an upper bound, so we take x as the least upper bound of S and f (x) must be 0. Pause. The audience does not like the proof. The audience slowly realises that there is something wrong with that proof. How do we know that S has a least upper bound? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So he sneaked in the Axiom of Completeness - actually used it literally in the proof, word by word as you would find it defined in a text book. And that's the missing axiom that turns the axioms of arithmetic into the axioms of real numbers. Introduced in a way where it totally made sense. If you put that axiom at the start of real numbers, there is no understanding. In the middle of the proof for the Intermediate Value Theorem, it is absolutely obvious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;BTW. I like proving the existence of real numbers by equating them with equivalence classes of Cauchy sequences. &lt;/p&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2014-08-02T00:25:48.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4134" PostTypeId="1" CreationDate="2014-08-02T02:50:55.870" Score="6" ViewCount="361" Body="&lt;p&gt;Some day when I was a young school boy, our math teacher gave us a strange question in an exam. As far as I can remember that question was something like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $a$ be such that $a^2+a+1=0$ then prove $a$ has the property $P$. (I can't remember that property correctly).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;At that time we were not familiar with the notion of imaginary numbers so our world of numbers was restricted to $\mathbb{R}$. As $a^2+a+1=0$ has no root in real numbers I told my teacher that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Your assumption is false! Are you sure that this is not a typo? Did you mean $a^2+a-1=0$?! If not, the question is trivial because such an $a$ doesn't exist at all!&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He told me (with a mysterious smile!):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;There is nothing wrong with my question. Just try to &lt;strong&gt;imagine&lt;/strong&gt; existence of what doesn't exist and go forward in the world of imagination!&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;At that moment I didn't understand his comment properly. I simply wrote: &quot;The assumption is false so the theorem is true!&quot; in my answer to the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After a while when I learned more about imaginary numbers, I understood the creative intuition behind such a strange question that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Some contradictory assumptions can lead us to an expanded intuition using expanding our world with new notions and objects. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Long after that I attended some lectures about &lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Field_with_one_element&quot;&gt;Theory of Fields with One Element&lt;/a&gt;&lt;/strong&gt; and the same story repeated. Assuming existence of what doesn't exist and trying to understand its properties! Later I tried to introduce such a phenomena in mathematical thinking to my students. I usually say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Proving truth of a theorem by proving falsity of its assumption is &lt;strong&gt;not fair&lt;/strong&gt;! Try to find a &lt;strong&gt;fair&lt;/strong&gt; proof!&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; When a contradictory assumption like existence of an one element field or existence of a number $a$ with $a^2+a+1=0$ can lead us to a richer theory and deeper intuition? What are characteristics of such &lt;em&gt;useful&lt;/em&gt; contradictory assumptions? How to teach a math student to distinguish between &lt;em&gt;useful&lt;/em&gt; and &lt;em&gt;useless&lt;/em&gt; contradictory assumptions? How to teach them to deal with &lt;em&gt;useful&lt;/em&gt; contradictory beings and use them to expand their understanding of the nature of the subject?   &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastActivityDate="2014-08-11T16:05:10.303" Title="Assuming existence of what doesn't exist!" Tags="|mathematical-pedagogy|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4135" PostTypeId="1" CreationDate="2014-08-02T05:20:50.387" Score="3" ViewCount="313" Body="&lt;p&gt;In mathematics we usually can generalize a particular notion in many different ways. Some of these generalizations could be contradictory. When I teach maths/logic to my students I usually encourage them to think about possible generalizations. Sometimes their generalizations are very strange. In such circumstances I usually discuss with them on &quot;truth&quot;, &quot;usefulness&quot; or &quot;naturality&quot; of such generalizations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are our philosophical, mathematical or meta-mathematical &lt;strong&gt;criterion&lt;/strong&gt; to decide on the most &lt;strong&gt;natural&lt;/strong&gt; or most &lt;strong&gt;useful&lt;/strong&gt; generalization of a particular mathematical notion? Are the most natural and the most useful generalizations of a notion necessarily same? Which is the &lt;strong&gt;best&lt;/strong&gt; generalization?    &lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastActivityDate="2018-03-16T16:35:43.823" Title="What is the true generalization of a notion?" Tags="|general-pedagogy|mathematical-pedagogy|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4136" PostTypeId="2" ParentId="4134" CreationDate="2014-08-02T11:49:23.557" Score="7" Body="&lt;p&gt;As you wrote, there is no real number $a$ satisfying $a^2+a+1=0$, and this can be proven using the axioms of the real numbers. As you also wrote, there actually is such an $a$ in the complex numbers. However, $a$ and the complex numbers can exist because the complex numbers &lt;strong&gt;do not obey all the axioms of the real numbers&lt;/strong&gt;. To allow the complex numbers you must remove at least one axiom. (In fact, we remove all the ordering axioms.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So your teacher was not fair in his assignment. The proof that such an $a$ would have property $P$ relies on a somewhat diminished set of axioms, and your teacher did not say so. A particularly bright student could answer, &quot;If we remove these particular axioms then...&quot;. A less bright student could still answer by simply forgetting about those axioms or not realizing the contradictions that would ensue by leaving them in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your question, you could &quot;teach a math student to distinguish between useful and useless contradictory assumptions&quot; by asking them to rigorously show an argument for the contradiction, to clearly specify the axioms used in the argument, and to present a reduced set of axioms that would invalidate that argument. For extra credit, the student could present different, new axioms that would show that such an $a$ exists and would not contradict the other axioms. This of course would be very difficult, but I was given questions similar to this in my &quot;math camp&quot; many years ago. I must admit that I was not good enough to answer such questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't point out the different axioms between the real numbers and the complex numbers or other field extensions, you could just confuse the better students. If you are asking for the best balance between rigor and creativity,... that question has consumed better mathematicians than I.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2014-08-02T16:49:41.857" LastActivityDate="2014-08-02T16:49:41.857" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="4137" PostTypeId="2" ParentId="3983" CreationDate="2014-08-02T14:06:56.683" Score="0" Body="&lt;p&gt;See these two posts: &lt;a href=&quot;http://blogs.mathworks.com/cleve/2013/07/08/eigshow-week-1/?s_tid=Blog_Cleve_Category&quot; rel=&quot;nofollow&quot;&gt;eigshow: week 1&lt;/a&gt;, &lt;a href=&quot;http://blogs.mathworks.com/cleve/2013/07/22/eigshow-week-2/?s_tid=Blog_Cleve_Category&quot; rel=&quot;nofollow&quot;&gt;eigshow: week 2&lt;/a&gt;  written by C. Moler about illustrating eigenvalue in MATLAB software.&lt;/p&gt;&#xA;" OwnerUserId="1878" LastActivityDate="2014-08-02T14:06:56.683" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4138" PostTypeId="2" ParentId="4122" CreationDate="2014-08-02T18:13:38.343" Score="0" Body="&lt;p&gt;When a student draws improper analogies between two concepts after wrongly associating one with the other, that would be &quot;false association.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept is similar to &quot;false friends&quot; in foreign languages, where two similar- sounding/looking words actually have different meanings.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-08-02T18:13:38.343" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4139" PostTypeId="2" ParentId="4135" CreationDate="2014-08-03T01:54:53.923" Score="0" Body="&lt;p&gt;Mathematics grows not only through the process of generalization but also through the process of specialization. If one has a definition and one can generalize or specialize that definition so that where before one could not tell X from Y in the original case but the &quot;refined&quot; definition allows one to tell X from Y mathematical progress can be made.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A favorite example of mine of this type is the idea of a polygon - usually thought of as something in the plane which is non-self-intersecting but &quot;closed.&quot;. The essence of being a polygon was the &quot;flatness&quot; of its edges and that there are only a finite number of edges. So when one invents the idea of convexity one gets lots of new developments. One can talk about regularity of polygons whether or not they are convex. One can allow vertices of polygons to coincide (as Branko Grünbaum does). Many types of polygons are now studied: star shaped, rectilinear, those with infinitely many edges, closed, open, non-planar, knotted, etc. The resulting world of polygons is so rich that geometers will be kept busy for a long time investigating &quot;named&quot; kinds of polygons, and almost certainly there will be new types to catch our attention. &lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-08-03T01:54:53.923" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4140" PostTypeId="2" ParentId="186" CreationDate="2014-08-03T04:14:54.857" Score="8" Body="&lt;p&gt;Whether you construct the reals or just axiomatize them, at some point you need to talk about the completeness property or completeness axiom. The students should ultimately learn several versions of this (Cauchy completeness, nested intervals, least upper bounds). I've found the following version, not listed in previous answers (unless I overlooked something) to be relatively easy to formulate, without a lot of prior definitions, and thus useful as a first version to show the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose $X$ and $Y$ are two sets of real numbers, and suppose $x\leq y$ for all $x\in X$ and all $y\in Y$. Then there is a real number $z$ such that $x\leq z$ for all $x\in X$ and $z\leq y$ for all $y\in Y$.  &lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-08-03T04:14:54.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4141" PostTypeId="2" ParentId="75" CreationDate="2014-08-03T16:29:42.687" Score="8" Body="&lt;p&gt;Of the common teaching methods used in STEM, evidence from educational research strongly suggests that lecturing is the worst. A 2013 meta-analysis[Freeman] compared the effectiveness of lecturing with that of active engagement techniques. The metrics used were success rates and scores on standardized tests or other assessments. In all STEM subjects and by both types of measures, active engagement gave better results, with the difference being statistically significant at the 95% level in nearly all cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The research is not necessarily fine-grained enough to detect whether lecturing or active engagement might be better in one specific situation but not in another. However, there is some evidence that opposes certain hypotheses to the effect that lecturing would sometimes be better. Some of the most detailed evidence comes from physics,[Hake,Mazur] and that's what I'll summarize below.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The evidence does not seem to support the hypothesis that better students are better served by lecturing. In the survey by Hake, students were given a standardized test before and after instruction, and a normalized gain score was calculated for each student by dividing the increase in the student's score with instruction by the maximum increase that would have been possible. Students at highly selective schools often had high pretest scores, but such students still had poor normalized gains when they received instruction by lecturing.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The evidence does not seem to support any belief that lecturing can be a good technique when done by a highly skilled lecturer. The large survey by Hake found no cases in which lecturing resulted in high normalized gain scores.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The evidence does not seem to support the claim that problem-solving and other technical skills suffer due to active learning. Mazur shows evidence that active learning improves over-all performance, not just conceptual performance.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Freeman et al., &quot;Active learning increases student performance in science, engineering, and mathematics,&quot; &lt;a href=&quot;http://www.pnas.org/content/early/2014/05/08/1319030111&quot;&gt;http://www.pnas.org/content/early/2014/05/08/1319030111&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hake, &quot;Interactive Engagement Versus Traditional Methods: a Six-Thousand Student Survey of Mechanics Test Data for Introductory Physics Courses,&quot; Am. J. of Phys, 66 (1997) 64&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mazur, Peer Instruction: A User's Manual, 1996, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0135654416&quot;&gt;http://www.amazon.com/Peer-Instruction-A-Users-Manual/dp/0135654416/ref&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-08-03T16:29:42.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4142" PostTypeId="1" AcceptedAnswerId="4143" CreationDate="2014-08-03T19:18:32.600" Score="20" ViewCount="613" Body="&lt;p&gt;As I grade my final exams, I sometimes find calculus 1 students who will pass with a C but lack critical skills like the chain rule. This can be for a number of reasons, for instance:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They understand the concepts but cannot do the computations.&lt;/li&gt;&#xA;&lt;li&gt;They are able to memorize processes well, but they missed this process.&lt;/li&gt;&#xA;&lt;li&gt;They understand geometric intuition but their algebra skills are lacking.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;When I pass these students with a C, I am doing them a disservice -- without the ability to do the chain rule, or the product rule, or whatever other basic skill they need, this student will definitely be doomed in the next calculus course. However, I don't want to increase the weight on these &quot;easy&quot; basic concepts, since they are critical but they are also just a small part of the material we cover.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can we structure the grading system in our courses so that students who pass the course definitely have the basic skills required for the next course, without the bad plan of massively increasing the grading weight on these concepts?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-08-05T13:24:45.183" Title="Requiring basic skills to pass a course without increasing their grading weight" Tags="|calculus|course-design|assessment|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="4143" PostTypeId="2" ParentId="4142" CreationDate="2014-08-03T22:33:20.947" Score="8" Body="&lt;p&gt;The notion of &quot;gateway&quot; (exam or other thresh-hold) is not universal, and generates some conflicts, but addresses the issue head-on. That is, throughout a course (perhaps at the beginning, but this is in some ways the most harsh and least hopeful/optimistic) separate &quot;exams&quot; are given that test exactly and only the absolutely indispensible skills. Not only is everyone expected to get nearly everything right, but, if they don't, they are required to keep doing it until they do succeed. In the harshest version, the very registration for the course is not allowed without passing gateway exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can make successful completion of all the gateway exams some sort of prerequisite for taking the final, or for getting a passing grade, or... hence the name &quot;gateway&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This makes considerable sense to me, but I would anticipate considerable backlash from students who're too accustomed to Xtreme Mediocrity (really, semi-incompetence) being ajudged good-enough to be allowed to pass on...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't have quite a bit of authority over the grading system, or are somehow vulnerable to skepticism about your teaching ideas, this could cause you a lot of trouble...&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2014-08-03T22:43:50.370" LastActivityDate="2014-08-03T22:43:50.370" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4144" PostTypeId="2" ParentId="4134" CreationDate="2014-08-04T03:16:58.783" Score="10" Body="&lt;p&gt;Whenever we state a theorem in the form &quot;If $P$ then $Q$&quot; we are usually omitting a whole slew of tacit additional hypotheses, namely those that specify what kind of theory we are talking about.  What we really mean (but don't bother saying because it is understood from context) is &quot;If $P$ [in the context where $\{ H_1, H_2, \dots H_n \}$  ] then $Q$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the statement &quot;If $a$ is a number such that $a^2 + a + 1 = 0$, then $Q$&quot; really means &quot;If $a$ is a number [where &quot;number&quot; means a member of $\mathbb{R}$, which satisfies the axioms of a complete ordered field] such that $a^2 + a + 1 = 0$, then $Q$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you say &quot;The theorem is true because the assumption is false!&quot; what you mean is that under the hypotheses $\{ H_1, H_2, \dots H_n \}$, $P$ is false, and therefore &lt;em&gt;under those implicit hypotheses&lt;/em&gt; $P \implies Q$ is vacuously true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sense that this proof is &quot;unfair&quot; stems from the fact that it takes no notice whatsoever of $Q$ and therefore the logical relationship between $P$ and $Q$ -- which is presumably at the heart of the question's intent -- is completely unexamined.  What you really want students to do is consider the possibility that under some (not fully specified) &lt;em&gt;weakening&lt;/em&gt; of the hypotheses -- say $ \{ H_1', \dots H_m' \}$ --  $P$ might be possible after all, and to then show if that were the case, $Q$ would be true, too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that any question that simply asks &quot;Assume $P$ and prove $Q$&quot; without directly addressing the fact that $P$ is false (under the prevailing set of implicit assumptions that govern the rest of what is going on in the course) is &lt;strong&gt;a badly worded question&lt;/strong&gt; and one that invites an &quot;unfair&quot; proof.  If you want students to consider that a change of the &quot;ground rules&quot; could allow for things other than those normally possible, you should say so directly, rather than be mysterious about it.  Here is one way you might word such a question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is easy to prove that there is no real number $a$ such that $a^2 + a + 1 = 0$.  (Before continuing with this paragraph, write such a proof.)  But suppose we introduce a new kind of number for which $a^2 + a + 1 = 0$ &lt;em&gt;is&lt;/em&gt; true.  Such an $a$ would not be a real number; it would have to be some sort of special, non-real number.  Without worrying about what exactly such an $a$ would need to be, just assume that it exists.  Prove that, under this assumption, &lt;em&gt;every&lt;/em&gt; quadratic equation has a solution that can be written in the form $r + sa$, where $r$ and $s$ are ordinary real numbers of the type that we are familiar with, and $a$ is the &quot;special&quot; number satisfying $a^2 + a + 1 = 0$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Admittedly, this is wordy, and some students might balk at the complexity of the question.  But there is nothing wrong with making students read a paragraph of text when doing mathematics!  In fact it probably is one of the most under-appreciated tools in a mathematician's skillset, and one that we often do a poor job of teaching our students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With a question like the above, I don't think you would get any students who simply say &quot;There is no such $a$ so the theorem is vacuously true.&quot;  You have forestalled that response by &lt;em&gt;acknowledging up front&lt;/em&gt; that no such $a$ exists under the normal hypotheses of what a &quot;number&quot; is, and asked students to entertain the possibility of expanding the number system by introducing a new kind of number that would be a solution.  Any student who tried to just say &quot;The hypothesis is false, so the theorem is true&quot; is clearly missing the entire point of the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that doing this stops rather short of Rory Daulton's suggestion of asking students to actually &lt;em&gt;identify new hypotheses&lt;/em&gt; under which $P$ could be true.  That might be asking too much of students, and is also sort of missing the point of the question, which presumably is &lt;em&gt;not&lt;/em&gt; supposed to be about whether and/or how $P$ could be true, but instead about &lt;strong&gt;the relationship between $P$ and $Q$&lt;/strong&gt;.  The formulation I propose simply stipulates that &lt;em&gt;somehow&lt;/em&gt; $P$ can happen; just accept that as a working hypothesis and get to work on proving $Q$.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2014-08-11T16:05:10.303" LastActivityDate="2014-08-11T16:05:10.303" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4146" PostTypeId="2" ParentId="4091" CreationDate="2014-08-04T09:09:00.327" Score="3" Body="&lt;p&gt;I did a search on this question as well as the three existing answers for the term &quot;applications&quot; and did not find any occurrences. The natural numbers $\mathbb N$ do not cause students any particular difficulties of motivation. Arguably the main point to emphasize in introducing the successive enlargements $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{R^\ast}\ldots{}$ is their usefulness in applications. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first two extensions are helpful in solving the simplest linear equations. If we wish to solve harder problems in algebra and geometry (finding diagonal of a square, area of a circle) it is convenient to extend further to the reals. To do calculus, it is convenient to include infinite numbers and infinitesimals, so we extend to the hyperreals. Complex numbers are of course easily motivated in terms of roots of polynomials, quaternions in terms of spacial rotations, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without applications it is difficult to motivate today's students. Specifically I wonder how useful it would be to tell them about transfinite ordinals which are after all not necessarily the most useful way of modeling infinity as far as concrete applications are concerned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Responding to the comment by Matt, one of my favorite applications of the extenstion $\mathbb{R}\subset\mathbb{R^\ast}$ is Cauchy's definition of continuity of $f$, namely an infinitesimal increment $\alpha$ always leads to infinitesimal change $f(x+\alpha)-f(x)$. In my teaching experience students have an easier time relating to this than to the four-quantifier definition used in the epsilon-delta approach.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2014-12-09T10:05:03.720" LastActivityDate="2014-12-09T10:05:03.720" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4148" PostTypeId="2" ParentId="4127" CreationDate="2014-08-04T14:29:36.797" Score="13" Body="&lt;p&gt;I wouldn't put any such sentence on your door, just one in the syllabus explaining your make-up exam policy which should be determined at the beginning of the semester. You don't want to come off as mean, intimidating, or otherwise to students who might seriously have an issue where a make-up exam would be justifiable (illness, death of family, etc); not that you should want to come off that way to any student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would focus on a welcoming atmosphere for students, where they can discuss their exam if they are concerned about it, but if they ask for a make-up exam (as they will, no matter what you have written) you can just refer the the syllabus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for if you should offer a make-up or remedial exam, I would discuss this with your peers and see what the common departmental standard are. I usually lean towards no make-up or remedial exams unless they are on academic leave and I get notified by the administration or my coordinator.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-08-04T14:29:36.797" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4149" PostTypeId="1" CreationDate="2014-08-05T06:00:20.920" Score="5" ViewCount="78" Body="&lt;p&gt;This coming Fall semester, I will have to organize a course to prepare high school students for the maths interview they will have before entering to university. This is my first time teaching this, so I am concerned about the organization of such a class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question: How such a class should be organized compared to a normal class?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Informations that may help to answer:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The class will be only on the math part of the interview. Questions about motivation to apply to the university will be handle in the English course.&lt;/li&gt;&#xA;&lt;li&gt;The range covered by the interview is PreCalculus and single variable Calculus (roughly, AP examination curriculum). All students already covered this material in the previous years.&lt;/li&gt;&#xA;&lt;li&gt;Small group of students (about 10).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Any advice on references or textbooks is also welcomed.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: Format of the interview: the interview is about 15-20 minutes long. It starts with a precise theoritical question like &quot;State the Fundamental Theorem of Calculus&quot;, followed by a vague (in the sense: no single correct answer) theoretical question like &quot;Why is it useful?&quot;, &quot;Give an application&quot; or &quot;Which can of problem can you solve using this?&quot;. Finally, a problem has to be solved by the applicant. The focus is on the reasoning and methodical thinking of the applicant, its reaction to suggestion more than finding the exact numerical solution. There is no preparation time for the interview.&lt;/p&gt;&#xA;" OwnerUserId="1538" LastEditorUserId="1538" LastEditDate="2014-08-05T15:56:53.780" LastActivityDate="2014-08-05T15:56:53.780" Title="Organization of an interview preparation course" Tags="|course-design|classroom-management|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4150" PostTypeId="2" ParentId="4142" CreationDate="2014-08-05T06:55:42.187" Score="4" Body="&lt;p&gt;In my view, the method to make sure they leave calculus I with all the required skills is to simply weave them into all the questions. I warn them, if you don't know the basic computations cold then it will hurt your grade. By the time of the final, it ought not be a struggle to do chain rules and such. If it still is a burden to them mentally then it's still time for them to study calculus I. This is my draconian view. Actually, since they don't tend to study before the next course (especially if they need to), it is the kind thing to keep them back before they find themselves trapped in the much more harsh environment of calculus II.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At each stage of the course I try to test them on everything they've learned up to that point. By the time we reach the final, they should see it coming. Of course, you have to be sensitive to the difficulty of the topic, look for the right places to layer concepts.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-08-05T06:55:42.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4151" PostTypeId="1" CreationDate="2014-08-05T09:08:00.173" Score="8" ViewCount="167" Body="&lt;p&gt;Since it's currently summer break, and I've a bit more time than normal, I've been organizing my old notes. I seem to have an almost unwieldy amount of old handouts and tests from classes previously taught. I'm hesitant to get rid of these, as they may provide useful for some future course. Because I adjunct at a few different colleges, with slightly different course content, I find that a lot of these materials are useful for multiple classes. I've begun organizing them according to specific content, but I'm curious, does anyone have an organizational style that they have found useful?&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="143" LastEditDate="2014-08-05T10:25:59.807" LastActivityDate="2014-08-08T00:16:43.987" Title="Organizing Old Notes" Tags="|teacher-preparation|organization|lecture-notes|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4152" PostTypeId="2" ParentId="4091" CreationDate="2014-08-05T13:21:43.177" Score="2" Body="&lt;p&gt;'Number' is just a word. It's meaning strongly depends on the audience and the information you want to give. For a small child a number is something she can show on her fingers. Growing up with number, it is something used to count things (N), measure dimensions (R), describe positions (C) and so on. You decide where to stop. For me, the number is the smallest piece of information you can reason about in the given context.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2014-08-05T13:21:43.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4153" PostTypeId="2" ParentId="4142" CreationDate="2014-08-05T13:24:45.183" Score="3" Body="&lt;p&gt;I have the same concern as you have, and even more so I guess: in the French system, not only various subjects of a given course can compensate one another, but student can even pass into the next year by compensating a bad grade in their majors by good enough grades at their minors (which are numerous in first years).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that a passing grade should mean that the student is ready for the next stage, and it makes little sense to average over different skills and pieces of knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One suggestion I read somewhere in this site is the following: make an exam divided in three parts, C B and A. Part C is made of basic questions (10 say) that are expected to turn out almost all alright. Part B is made of slightly more advanced questions (6 say), corresponding to what you expect a reasonably good student to achieve, and part A is made of advanced questions (4 say) that needs more independent thinking. Then the grading goes like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;you grade part C; if less than say 9 (maybe 8) answers out of 10 are correct, then the student fails and you stop grading. Otherwise, the student passes, and then:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;you grade part B; if less than say 4 (maybe 3) answers out of 6 are correct, then the student has a C and you stop grading. Otherwise:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;you grade part A; the student has an A if one (maybe 2) answer out of 4 is correct, otherwise he or she has a B.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One of the main difficulty would be to have student truly understand and accept this grading system. I have not used such a system myself (but we grade from 0 to 20, which is less prone to this method).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A lot of adjustments can be made of course; for example above in part A, the important part is to be able to solve one difficult problem among several; this is most suited for graduate courses, as in research you succeed by finding a problem you can solve rather than by being able to solve whatever problem you encounter.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-08-05T13:24:45.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4154" PostTypeId="2" ParentId="4151" CreationDate="2014-08-06T01:19:57.530" Score="4" Body="&lt;p&gt;While my data organizing problems are usually more related to research, they still often involve notes, papers, and sheets of data collection interview items that are much like handouts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One solution that I have used (although I am still perfecting it) is to scan everything and get rid of the paper copies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once I have electronic copies of things, a number of organizational methods are possible. The most obvious is a hierarchical organization, which can be accomplished in any computer's file system folder structure. As a bonus, if you have some sort of cloud storage, store or mirror your file structure there so you can browse your scans any time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The limitation of a hierarchical system is that you can only put a scan into one folder. That handout may have been useful for multiple projects or classes. So, a system that allows tagging is better if you need to be able to find one handout but you would like to remember that you used it in your most recent class (because you don't remember what class you first used it in). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tags allow you to create a new tag for every use (or class). Then you can (and should) tag the handout for every use you have considered. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: lets say you decide to use Evernote to store all your scanned notes. It allows tagging. You have a handout you particularly like. You put it into Evernote and tag it with a subject matter tag. But then you also create a tag for the class you used it in. Finally, you add the &quot;handout&quot; tag that you give all your handouts. Now you can find the thing by scanning the list of handouts, or by remembering one of the classes you used it in, or by remembering the subject matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see that my main concern is finding stuff later. Information can pile up, and it's nice to have things like this easily retrievable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://evernote.com/evernote/guide/&quot; rel=&quot;nofollow&quot;&gt;Here's the getting started guide for Evernote.&lt;/a&gt; If you're going to scan your notes, that can be tedious. But you did say you had some time.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-08-06T01:19:57.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4155" PostTypeId="2" ParentId="1572" CreationDate="2014-08-06T11:56:28.193" Score="32" Body="&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/X2ccP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A recent Times article titled &lt;a href=&quot;http://www.nytimes.com/2014/08/04/your-money/americans-are-bad-at-math-but-its-not-too-late-to-fix.html&quot; rel=&quot;noreferrer&quot;&gt;Americans Are Bad at Math, but It’s Not Too Late to Fix&lt;/a&gt; offered an example - &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A&amp;amp;W's &quot;Third Pounder hamburger failed to catch on because &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;During focus groups, the company discovered that customers believed&#xA;  they were getting less meat. Because the “3” in ⅓ was smaller than “4”&#xA;  in ¼, “customers believed they were being overcharged.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If this is not classic innumeracy, I don't know what is. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorDisplayName="user507" LastEditDate="2014-08-06T18:11:25.340" LastActivityDate="2014-08-06T18:11:25.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4156" PostTypeId="2" ParentId="4151" CreationDate="2014-08-06T12:46:00.290" Score="0" Body="&lt;p&gt;I use schoology.com to keep my notes and such.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's great because it goes where I go plus it's free for both students and teachers.&lt;/p&gt;&#xA;" OwnerUserId="625" LastActivityDate="2014-08-06T12:46:00.290" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4157" PostTypeId="2" ParentId="75" CreationDate="2014-08-06T23:37:31.457" Score="9" Body="&lt;p&gt;This is a bit tongue-in-cheek, but it is good to lecture &lt;em&gt;after&lt;/em&gt; students think they have mastered something via an active learning experience, provided the lecture is the result of long effort to find an efficient or beautiful path to a result. A lecture should be given with the disclaimer that such efficient and beautiful paths are the result of a long distillation, and should not be demoralizing to those who have struggled hard to find a result for themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for this is reasonably obvious: Discovery learning does not necessarily provide the neatest way through new material, but it is very effective for uncovering difficulties and essential points. After struggling with material deeply, an elegant proof that deals neatly with a horrible difficulty will be highly memorable and will help structure the experience for future use.&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2014-08-06T23:37:31.457" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4158" PostTypeId="1" CreationDate="2014-08-07T02:47:50.720" Score="3" ViewCount="182" Body="&lt;p&gt;Students questions and answers reflect their thoughts. It is very important to teach them how to ask nice questions and how to answer them. In order to improve these abilities I would like to commence a Q&amp;amp;A contest in one of my undergraduate courses (&amp;quot;&lt;em&gt;Foundations of Maths&lt;/em&gt;&amp;quot;) but I am not sure about all possible aspects of such a competition.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; What are possible disadvantages of a math Q&amp;amp;A contest in an undergraduate course? Could it be more useful for graduates? What sort of problems could happen during such a process?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; How to organize a Q&amp;amp;A contest effectively? What sort of algorithm should I follow to manage this additional event during semester?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 3:&lt;/strong&gt; How to judge that which question/answer is better? Are there any clear criterion or I should trust on my feeling?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 4:&lt;/strong&gt; What is a good prize for the winner?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Please feel free to add your personal experiences if you had something similar in your courses before.&lt;/p&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-10-14T06:59:11.337" Title="How can a Q&amp;A contest help a math teacher to improve his/her students' understanding?" Tags="|course-design|mathematical-pedagogy|math-contests|student-question|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4159" PostTypeId="2" ParentId="4091" CreationDate="2014-08-07T14:38:54.523" Score="2" Body="&lt;p&gt;How about:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The Book of Numbers&lt;/em&gt;&#xA;by John H. Conway, Richard Guy&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2014-08-07T14:38:54.523" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4160" PostTypeId="1" AcceptedAnswerId="4162" CreationDate="2014-08-07T20:19:03.583" Score="11" ViewCount="419" Body="&lt;p&gt;I am a guest here, having responded to a general invitation extended to the &lt;a href=&quot;https://stats.stackexchange.com/questions&quot;&gt;Cross Validated&lt;/a&gt; community, to possibly contribute answers whenever some question related to Statistics comes up in this site.&#xA;I do not teach Mathematics, but I do occasionally teach Statistics. And one of the less obvious and most difficult aspects to teach, is the difference between Descriptive and Inferential Statistics.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me of course, it seems pretty clear: Descriptive Statistics just summarize some characteristics of a specific data set. Inferential Statistics is our attempt to draw inferences about something &quot;larger&quot; than the data set available. How do we manage that?  &lt;em&gt;By making a whole new set of assumptions&lt;/em&gt;. And what do we do after? &lt;em&gt;We use the exact same results we derived from Descriptive Statistics&lt;/em&gt; -but which now lead us to totally different conclusions in nature and in scope.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here lies the problem: these additional assumptions are simply sewn alongside the &lt;em&gt;tools&lt;/em&gt; of Descriptive Statistics. And students get uneasy: in the previous (Descriptive Statistics) class, this was just the &quot;average of the data set&quot;, &quot;a centrality measure&quot;. How on earth the exact same number, calculated in the exact same way, has now become &quot;an estimate of the population mean&quot; that moreover has been derived through the interaction of the data set with a &lt;em&gt;function of random variables&lt;/em&gt;,(the estimator), a function that is (say), &quot;unbiased and asymptotically consistent&quot;?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is not whether the concepts themselves need work and mental effort to understand. The problem is that this &quot;switch of vision&quot; of the same thing (the data set), from a &quot;vector of numbers&quot; to a &quot;set of realizations of distinct random variables that belong to the same (statistical) population, forming a sample of this population&quot; is so big, that, the consequent use of the exact same tools and results seems in total disharmony: surely, such a big step in the set up should lead to some brand new tools also... and it does, but mostly &quot;later on&quot;, while the tools of Descriptive Statistics remain prominent, with maybe minor modifications (like bias-correction).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bigger problem, is that the true problem takes time to show: students may play along, some may even like this new stochastic and probabilistic world, -but I keep getting the feeling that deep down, they feel that all the theoretical apparatus of Inferential Statistics is just an ingenious way to make something out of nothing (or too much out of too little), since after all, we keep adding the values in the data set and we divide by their number...  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since &quot;too much out of too little&quot; is demonstrably &lt;em&gt;not&lt;/em&gt; the case (if the general public knew how many procedures they consider as deterministically controlled, are in reality driven by statistical algorithms, I suspect they would had a serious panic attack), I believe it is important to find ways to deal with this.&lt;br&gt;&#xA;One way would be to &lt;em&gt;start&lt;/em&gt; with Inferential Statistics, and get rid of the notion that &quot;Descriptive Statistics are a good introduction and familiarization step&quot; (I just argued that they are not).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question(s)?&lt;/strong&gt; &lt;strong&gt;1)&lt;/strong&gt; To those that teach Statistics, what are your experiences and how do you deal with the passage from Descriptive to Inferential Statistics?&lt;br&gt;&#xA;And to everybody,&lt;br&gt;&#xA; &lt;strong&gt;2)&lt;/strong&gt; what are some other fields in Mathematics where such &quot;structural breaks&quot; happen, i.e. where objects and concepts already taught, acquire a &lt;em&gt;totally different meaning&lt;/em&gt; by activating a new set of assumptions? And how do you teach that?&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="-1" LastEditDate="2017-04-13T12:44:19.443" LastActivityDate="2014-08-09T22:22:26.840" Title="Passage from Descriptive to Inferential Statistics - analogies with other Math-fields?" Tags="|concept-motivation|mathematical-pedagogy|statistics|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4161" PostTypeId="2" ParentId="4151" CreationDate="2014-08-08T00:16:43.987" Score="1" Body="&lt;p&gt;I find that the most efficient way to store things is to scan them in. You will hopefully find that the photocopier at your school can batch-scan and email the copies to you, or put them on a USB.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that I give them very descriptive names according to a naming scheme, so that they easy to search for. By storing them in my dropbox or google-drive folders, they easy to access on my iPad etc. as well as my laptop.&lt;/p&gt;&#xA;" OwnerUserId="1984" LastActivityDate="2014-08-08T00:16:43.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4162" PostTypeId="2" ParentId="4160" CreationDate="2014-08-08T00:19:54.443" Score="7" Body="&lt;p&gt;&lt;strong&gt;On the (false) distinction between descriptive and inferential statistics&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my view, it is rare that your only purpose is simply to describe the data you have. Even a simple graph is usually used to make a tentative hypothesis about the relationship between variables or the distribution of a single variable, and this is really the beginnings of inferential statistics. Also, presenting means in a research paper is usually to forward your argument about differences in populations. In that sense the mean is already a representation of the population and not technically only a representation of your sample.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if you really want to investigate the data you currently have, then summary stats are not usually fit for the purpose - you are often much more interested in identifying outliers and investigating them further. For example you might notice a negative skew in your class results, but then you would naturally be interested in those students in that left tail and how you might help them, so you end up looking at the individual data points. Even if you do calculate a summary, such as the pass rate, then your thoughts naturally turn to how you can make the pass rate higher in future students, and now you're thinking about a population beyond your current class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even the usual measures of central tendency are problematic. Many students I talk to express concern that they remove so much of the data they are trying to describe, and their natural instinct is instead to talk about clumps in the data or the percentage in a certain zone. It seems to me that the choice of the mean is actually guided by the estimation if a parameter, rather than actually being  natural way for people to summarise data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, in practice, when a statistician does analyse data with the view to make conclusions about the population, they &lt;em&gt;always&lt;/em&gt; draw graphs and calculate summaries. But we have a hard time convincing students that they should do this. I wonder if the dichotomy we set up between descriptive statistics and inferential statistics encourages students not to draw graphs. It seems they have learned that graphs only describe the data you have and they think they therefore cannot help in the process of making inferences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Big ideas of variability, population and distribution&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It distresses me that even though we know that stats is built on the big ideas of variability, population and distribution, we tend to forget about them when teaching certain parts of the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even when teaching probability, we often focus on specific events, whereas in fact the only reason any one event has a probability is because it is in comparison to all the other events that could have been.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1111/j.1751-5823.2007.00029.x/full#ss4&quot; rel=&quot;nofollow&quot;&gt;This article&lt;/a&gt; reviews research into learning statistics, and the section I have linked to is the part about those big ideas and is well worth a read.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My experience teaching all stats as inference&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To orient you, I am coordinator of a Maths Learning Centre at a university, so I talk to many students from many disciplines about their stats. I also do guest lectures on stats within courses (such as the med students during their research course).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taking the previous discussion into account, I decided that when helping students understand statistics,  I would try to remove the distinction between descriptive and inferential stats. I tell them that the main purpose of data collection and analysis is actually to make conclusions about populations, and that descriptive statistics are the essential first step. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I talk about distributions very early on, usually when they are studying descriptive stats. I introduce the big idea that any number you might measure or calculate has a distribution that describes all the possibilities and how likely they are. Your data is always a sample of the data you could have had, and the mean is just one mean that could have happened. Yet it has the tendency to be close to that central clump of the distribution, which is why it's pretty good as a measure of centre if your view is towards inference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find that recognising at the outset that almost all stats has a view to inference matches best with the reasons why they are forced to learn stats (generally it's a research methods course), and also helps to unify what they are learning in their stars course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept of &quot;all things have a distribution&quot; helps to get them to understand why the calculations for test statistics need to be different, because we need to create something with a distribution we actually know. It also helps them to pull away from the data itself to imagine the bigger picture of the distribution/population it came from.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-08-08T06:51:03.927" LastActivityDate="2014-08-08T06:51:03.927" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4163" PostTypeId="2" ParentId="4160" CreationDate="2014-08-08T01:20:02.930" Score="3" Body="&lt;p&gt;I've encountered something similar in &lt;strong&gt;logic&lt;/strong&gt;, where symbols can be interpreted either syntactically ($\vdash$) or semantically ($\models$).  I've had the same feeling that logic students were only playing along about appreciating the difference between the two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;statistical results connecting descriptive and inferential statistics&lt;/strong&gt; look like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The sample mean is an unbiased estimate of the population mean.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The sample variance times $N/(N-1)$ is an unbiased estimate of&#xA;  the population variance.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Fortunately, these results are easy to prove.  The numerical factor helps show the difference clearly.  There are lots of other nearby results to consider.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;logical results connecting syntax and semantics&lt;/strong&gt; look like Godel's completeness theorem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A set of sentences is consistent by these inference rules&#xA;  iff it has a model with these semantics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;These are harder to prove.  I suspect that most logic teachers would have a hard time coming up with different hypotheses and nearby theorems.  So it's harder to demonstrate understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps this can reframe your pedagogical issues?  $\vdash$ vs $\models$ might be even harder for students and teachers than $s^2$ vs $\sigma^2$ and $N$ vs $N-1$.  &lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-08-08T01:33:36.550" LastActivityDate="2014-08-08T01:33:36.550" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4164" PostTypeId="2" ParentId="1572" CreationDate="2014-08-09T01:10:57.957" Score="8" Body="&lt;p&gt;Let me mention a bit of (somewhat) good news in this area. Some years ago, Wal-Mart sold cardboard shipping boxes, labeled as 14&quot; by 14&quot; by 14&quot;. They were also labeled in metric units; since an inch is 2.54 centimeters, the dimensions were printed as 35.56 cm. I used to make fun of that --- dimensions of a shipping box accurate to a tenth of a millimeter. The good news is that now those boxes, still 14&quot; on a side, are labeled 35.5 cm.  It still seems to claim an unwarranted level of precision, but not as absurdly unwarranted as before.  &lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-08-09T01:10:57.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4165" PostTypeId="1" AcceptedAnswerId="4166" CreationDate="2014-08-09T03:18:38.903" Score="23" ViewCount="6558" Body="&lt;p&gt;&lt;strong&gt;I'm curious the views of those who teach calculus.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you know the &lt;strong&gt;continuity of a function&lt;/strong&gt; at a point is defined in terms of the &lt;strong&gt;limit&lt;/strong&gt; in the typical course. I'd like to ask a pair of questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Consider $f(x)=1/x$. Is $f$ continuous ?&lt;/li&gt;&#xA;&lt;li&gt;Let $\text{dom}(g) = (-\infty, 0 ) \cup (0, \infty)$ where $g(x)=1/x$ for each $x \in \text{dom}(g)$. Is $g$ continuous?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I don't want to say much more initially as I'd rather not influence the answers at the outset. The audience I have in mind is a mixed math majors / science majors calculus course at the university and time is not a large issue.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="376" LastEditDate="2022-02-26T09:11:49.633" LastActivityDate="2023-05-10T17:50:16.763" Title="Is the reciprocal function continuous?" Tags="|undergraduate-education|calculus|" AnswerCount="7" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4166" PostTypeId="2" ParentId="4165" CreationDate="2014-08-09T04:03:45.743" Score="21" Body="&lt;p&gt;I think the point of the question is that although there is no mathematical ambiguity, in different circumstances one would give different answers.  So:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I. From the perspective of a freshman calculus textbook: well, I know my enemy.  The textbook answer to (1) is &quot;A function is 'continuous' if it is continuous at $c$ for every real number $c$.  A function is continuous at $c$ if it is defined at $c$, if $\lim_{x \rightarrow c} f(x) = L$ exists and $L = f(c)$.  Since $f(x)$ is not defined at $0$, it is not continuous there.&quot;  Yikes.  To be fair, most calculus textbooks would classify the discontinuity at $0$ as not being removable.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel ever so slightly worried that I've unfairly caricatured this addle-pated view of continuity.  I know that freshman calculus textbooks would give the answer &quot;No, because it is not defined at $0$&quot; to (1).  As written, it sounds like they would give a similar answer to the continuity of, say, $f(x) = \sqrt{x-1}$, and I think that they may not be as legalistic as I implied above that a function can only be called continuous if it is defined at all real numbers: sometimes being defined on a closed interval is regarded as sufficient.  I think question (2) outfoxes the freshman calculus text: even by asking it you show understanding that the continuity of a function depends upon what domain you take.  The calculus texts are not clear on this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;II. From the perspective of a mathematician: (1) I would ask, &quot;Well, what is the domain and codomain of the function?  If the function is meant to take real or complex values, then it is not defined at $0$.  As a function from $\mathbb{R} \setminus \{0\}$ to $\mathbb{R}$ or from $\mathbb{C} \setminus \{0\}$ to $\mathbb{C}$, it is continuous (and more...).  The function cannot be extended continuously to $0$, e.g. because it is unbounded in every neighborhood of $0$.  However, it extends continuously [and more...] to a function on the Riemann sphere: $f(0) = \infty$.&quot; &#xA;(2) Yes, of course $g$ is continuous [and more...].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;III. If a freshman calculus student asked me this in class: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) &quot;There are some tricky issues in continuity for functions which are not defined at isolated points.  Our intuitive notion of continuity is that the graph is a nice unbroken curve.  [Draws the graph of f] As you can see, in this case the graph is not nice at $0$.  Now in fact the function is not even defined at $0$.  Whether a function is continuous at a point where it is not defined is a bit legalistic!  More pertinently, sometimes functions which are not defined at an isolated point &lt;em&gt;can&lt;/em&gt; be defined at the point so as to be continuous there.  This happens exactly when the limit as you approach the point exists and is called a 'removable discontinuity'.  However, in this case, $\lim_{x \rightarrow 0^{-}} f(x) = -\infty$ and $\lim_{x \rightarrow 0^{+}} f(x) = \infty$ so it's pretty bad: each one-sided limit is infinite, but the signs are different, so we can't even say that the overall limit is $\pm \infty$ [even if we could, since $\pm \infty$ are not real numbers, the limit would not exist].  The upshot is that there is no &quot;good&quot; way to define $f(x) = \frac{1}{x}$ at $0$.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) &quot;Yes, the function is definitely continuous on the given domain, as you can see from the graph.  Beware though: the given domain is not an &lt;strong&gt;interval&lt;/strong&gt; but rather a union of two non-overlapping intervals.  Be very careful with this: most of the big theorems in calculus concern continuous functions &lt;em&gt;defined on an interval&lt;/em&gt;.  When the domain is 'more than one interval', some funny things can happen.  [Depending on where we are in freshman calculus...] For instance the intermediate value theorem fails here: the function takes negative values and positive values but never takes the value $0$.  Also one antiderivative of the function is $\log |x|$, but not every antiderivative is of the form $\log |x| + C$!  [Maybe say more about this if someone seems interested; it is a tricky point.]  It may be more useful to think of the function $g$ as really being 'two different functions', one defined for negative values and one defined for positive values.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;b&gt;Added&lt;/b&gt;: The most general antiderivative of $g$ seems like a tricky point indeed, since more than one person has asked about it in the comments.  It may be worth explaining the answer.  The uniqueness of antiderivatives up to a constant is equivalent to the &lt;strong&gt;Zero Velocity Theorem&lt;/strong&gt;: if $F$ is differentiable and $F' = 0$ (identically) then $F = C$ is constant.  This in turn is a consequence of the Mean Value Theorem, and the hypothesis that the domain be an interval is critical: more generally any &lt;strong&gt;locally constant&lt;/strong&gt; function has derivative zero, and if the domain has $N$ connected components, the vector space of locally constant functions has dimension $N$.  In this case the domain of $g$ consists of two disjoint intervals, so the most general antiderivative is $G(x) = $&#xA;$\log(x) + C_1$, $x &amp;gt; 0$&lt;br&gt;&#xA;        $\log(-x) + C_2$, $x &amp;lt;0$&lt;br&gt;&#xA;for any two $C_1,C_2 \in \mathbb{R}$.  &lt;/p&gt;&#xA;" OwnerUserId="176" LastEditorUserId="176" LastEditDate="2014-08-10T21:34:33.300" LastActivityDate="2014-08-10T21:34:33.300" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4167" PostTypeId="2" ParentId="4165" CreationDate="2014-08-09T04:07:12.573" Score="5" Body="&lt;p&gt;This is my experience as a calculus TA for the last 4 years, and I have had no active role in setting the curriculums I'm talking about. The &quot;continuity of a function&quot; may be defined and explained thoroughly in an advanced honors rigorous calculus class, but in the trenches that are the first year calculus at most first year universities it does not play a large role. There are essentially 3 options for defining continuity for these students.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A function is continuous if its graph can be drawn without picking up the pen.&lt;/strong&gt; This is the most popular method (in what I've observed) , and it leads to the fallacy you are hinting at. Still, this definition (whether flawed or not) is understandable to calculus students and is of the kind of thinking that mathematicians use.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\lim_{x\rightarrow a} f(x)=f(a)$. This definition is in every calculus text and is of course entirely correct. The issue is that this definiton has the giant black box of a &quot;limit&quot;, which many students simply will not understand unless they are forced  to (through HW or lectures). At least in my university the pressure to skip over these more difficult and harder to test aspects of calculus in favor of a larger more impressive-looking curriculum guarantees that they will not ever get experience in things like limits.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall\epsilon\ \exists\delta$ &lt;strong&gt;s.t&lt;/strong&gt; $|f(x)-f(x_0)|&amp;lt;\epsilon\ \forall|x-x_0|&amp;lt;\delta$. Again if an effort is made to teach this kind of a definition, it can be taught to first years. There simply isn't time for that in the curriculums I have seen. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The goal of most calculus curriculums I have been party to is to get the students calculating integrals and derivatives of elementary functions. What is done after that is not necessairly constant, but almost all of these classes are in a sprint to get to this material. Generally what I think the very best students in such a class understand that continuity of a function allows us to apply the fundamental theorem of calculus to that function and perhaps the first definition I've given. There is a huge expanse of material to cover in a couple of semesters and foundations generally get short shrift. Saying that, without some power to change the curriculum, I cannot expect students to try and understand difficult concepts when they will  be given tests that completely ignore these concepts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: If this seems as an indirect answer it somewhat is. I understand what continuity means if it were said in a research seminar, but I choose to reinforce definition 1 to calculus students then the others as I think it is much more likely to be understood.&lt;/p&gt;&#xA;" OwnerUserId="1274" LastActivityDate="2014-08-09T04:07:12.573" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4168" PostTypeId="2" ParentId="4165" CreationDate="2014-08-09T12:07:07.237" Score="5" Body="&lt;p&gt;I cover three different concepts of continuity in my high-school calculus class (separate from advanced concepts such as uniform continuity). This vocabulary is based on the AP textbook &quot;Calculus: Graphical, Numerical, Algebraic&quot; by Ross L. Finney et al., pages 75 and 77.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A function $y=f(x)$ is &lt;strong&gt;continuous at an interior point&lt;/strong&gt; $c$ of its domain if&#xA;$$\mathop {\lim }\limits_{x \to c} f(x) = f(c)$$&#xA;And similarly for an endpoint of an interval, using a one-sided limit. A function is not continuous at any point not in its domain. Hence your reciprocal function is continuous at every value of $x$ other than $x=0$, where it is discontinuous.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A function is &lt;strong&gt;continuous on an interval&lt;/strong&gt; if and only if it is continuous at every point of the interval. Your reciprocal function is continuous on every interval not containing $x=0$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;strong&gt;continuous function&lt;/strong&gt; is one that is continuous at every point in its domain. Hence, the answer to your first question is: for $f(x)=\frac1x$, $f$ is a continuous function. And so is your function $g$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These definitions are not perfect but they are indeed helpful in my teaching. I can use theorems about continuous functions without being concerned about points not in the domain.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2014-08-09T12:07:07.237" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4169" PostTypeId="2" ParentId="1572" CreationDate="2014-08-09T20:27:05.317" Score="7" Body="&lt;p&gt;A general answer of &quot;every time the unit pricing makes no sense.&quot; The specific example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Xbrcj.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The product is identical. Presale, 12 for 2.29, 24 for 3.99. Fair enough, a bit of a discount to buy twice the number of pencils. Now, the unit price for the 24 pack is 50% more than the unit price of the 12 pack. I wonder how many of the large pack they'll sell this summer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All too often I've seen a 1lb package of a grocery item sell for 1.99, and the 2lb package of the identical item, 3.99. But this is the first I've seen the side by side unit price so off. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-08-10T12:42:57.280" LastActivityDate="2014-08-10T12:42:57.280" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4170" PostTypeId="2" ParentId="4160" CreationDate="2014-08-09T22:22:26.840" Score="3" Body="&lt;p&gt;You might find this article in the Journal of Mathematical Behavior of interest: Conceptual issues in understanding the inner logic of statistical inference: Insights from two teaching experiments, by Luis A. Saldanha, Patrick W. Thompson, pages 1-30, Volume 35, September, 2014.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-08-09T22:22:26.840" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4172" PostTypeId="1" CreationDate="2014-08-11T12:57:18.093" Score="5" ViewCount="746" Body="&lt;p&gt;The &lt;a href=&quot;http://www.promys.org/program/counselors&quot; rel=&quot;nofollow&quot;&gt;Information for Counselors&lt;/a&gt; page at Promys is not very sapecific: what does one need to do to apply? On which basis are candidates selected?  Does anyone have any information from past years (I'm willing to apply next year)?&lt;/p&gt;&#xA;" OwnerUserId="1994" LastEditorUserId="143" LastEditDate="2014-08-18T07:37:18.833" LastActivityDate="2014-08-18T07:37:18.833" Title="What does the selection process for counselors at PROMYS involve?" Tags="|undergraduate-education|undergraduate-research|summer|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4173" PostTypeId="2" ParentId="4158" CreationDate="2014-08-11T19:04:09.350" Score="1" Body="&lt;p&gt;Let's start with question 2, because it's easier on me as the answerer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Over in the mathematics and economics world, we like markets.  I'd envision market system for Q&amp;amp;A.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each student receives n units of currency (let's call them something sexy like MATH_Bux!) into a trust.  They can only spend MATH_Bux! on answers.  At the end of the quarter, student's MATH_Bux! earned for answers (those in the trust are not counted) are logged as a distribution and you scale 5% of their grade based on the MATH_Bux! distribution.  5% sounds reasonable to me as a student.  This also sounds a lot like an answer to question 4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On some sort of forum or something (I would use the internet for this), students may post questions from class (or independent study?) and answers.  Students bid on which answer they like the best with their MATH_Bux!.  I'd allocate myself, as the instructor, some fixed daily allotment of MATH_Bux! that I always spend within one week of accrual, to guarantee a certain level of activity (maybe do this with the students too, but then that punishes the student with an absurdly good idea in week 9 that is hailed as a genius by her classmates).  I'd also post both questions and answers as a teacher, but never reward my MATH_Bux! to answers of questions I ask (to avoid bias problems).  Hmmmm, this sounds like a good use of a sub-reddit.  And it also sounds a lot like an answer to question 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer question 1, well, &lt;em&gt;everything&lt;/em&gt; is better for graduates.  They have more background and more dedication, so they'll take better advantage of something like this.  But one of the reasons they're better at things like this is because &lt;em&gt;they should've already done them as undergrads&lt;/em&gt; so I would be pretty optimistic for your undergrads.  After all, if you've identified this as an important educational area, until a better way to teach it comes up this is all there is.  I'd expect problems with participation (which is why I suggested a slight grade incentive), point trading (which actually still incentivizes some discussion), and concerns over quality (what I've experienced in Q&amp;amp;A classes, for which I generally recommend large numbers of examples).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And invariably someone will think MATH_Bux! is a silly name, and want dataPointz or something that isn't nearly as cool.  Don't cave on that, MATH_Bux! is an awesome name and I'm sure everyone will learn to love it.&lt;/p&gt;&#xA;" OwnerUserId="1996" LastEditorUserId="1996" LastEditDate="2014-08-11T19:17:15.457" LastActivityDate="2014-08-11T19:17:15.457" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4174" PostTypeId="1" AcceptedAnswerId="4177" CreationDate="2014-08-11T21:50:21.270" Score="8" ViewCount="947" Body="&lt;p&gt;I'm developing a course that focuses on the transistion from arithmetic to algebraic thinking, particularly in grades 5-8.  We will do this through focus on the common core.  I'm also putting together a collection of suggested readings from the math education literature.  I would be interested to hear your suggestions for suggested readings.&lt;/p&gt;&#xA;" OwnerUserId="1346" LastActivityDate="2015-01-08T00:42:33.930" Title="Literature on development of algebraic thinking" Tags="|algebra|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4175" PostTypeId="2" ParentId="4174" CreationDate="2014-08-11T22:07:01.057" Score="3" Body="&lt;p&gt;Although these two NCTM books cover K-12 (there are items specifically directed at middle school level) they have ideas related to how to develop algebraic thinking: The Ideas of Algebra, K12, 1988 Yearbook, A. Coxford and A. Shulte, editors, and Developing Mathematical Reasoning in Grades K-12, 1999 Yearbook, Lee Stiff and Francis Curcio, editors. &lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-08-11T22:07:01.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4176" PostTypeId="2" ParentId="4165" CreationDate="2014-08-11T22:16:08.950" Score="8" Body="&lt;p&gt;In this situation, I'd claim there's the &quot;prior error&quot; of seemingly requiring a boolean answer, yes-or-no. Surely we're not so much interested in pranking the students by giving them a function whose domain is not topologically connected, which &lt;em&gt;is&lt;/em&gt; continuous on each connected component, but not &quot;continuous&quot; in an obvious, intuitive sense &quot;across the gap&quot;. The situation is clear, because everyone in Calc I knows the graph of this function. What is not clear is the semantics of a formal definition, whose pitfalls will not have caught the students' attention... and maybe don't merit it, besides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, &quot;$f(x)=1/x$ has a problem at $x=0$, but otherwise is fine&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&quot;What about $1/x^2$?&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, I'd propose not over-burdening a boolean &quot;continuity-or-not&quot; with this situation in Calc I. It's not so much a mathematical issue as a semantic one, which oughtn't be a primary point, in my opinion.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-08-11T22:16:08.950" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4177" PostTypeId="2" ParentId="4174" CreationDate="2014-08-12T00:38:45.893" Score="6" Body="&lt;p&gt;Early algebra research necessarily deals with the development of algebraic reasoning and questions like &quot;what is algebra&quot; and &quot;what counts as algebraic thinking and reasoning?&quot; And my own readings on early algebra have helped me to focus on what about algebraic thinking are students developing, apart from the manipulation of symbols. This is where, I think, some of the early algebra research may have more general interest, since some of the authors deal with what students are capable of that is connected to algebra well before they are doing what we are familiar with in an Algebra 1 course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A book collecting some different research on Early Algebra is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0805854738&quot;&gt;Kaput, J. J., Carraher, D. W., &amp;amp; Blanton, M. L. (2007). Algebra in the early grades.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a review of it in JRME &lt;a href=&quot;http://www.jstor.org/discover/10.2307/20720131?uid=3739696&amp;amp;uid=377752083&amp;amp;uid=38070&amp;amp;uid=2&amp;amp;uid=3&amp;amp;uid=364742151&amp;amp;uid=67&amp;amp;uid=62&amp;amp;uid=38069&amp;amp;uid=3739256&amp;amp;uid=60&amp;amp;sid=21104579409983&quot;&gt;(Chazan &amp;amp; Edwards, 2010)&lt;/a&gt;, and &lt;a href=&quot;https://www.education.umd.edu/MathEd/info/DChazanScans/2010JRME03-203.pdf&quot;&gt;someone is sharing the review here as a PDF (in case you don't have access)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the review:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In addition to these direct assaults on the rhetorical challenge posed by early &#xA;  algebra, one of the strengths of the volume is the richness of the presentations aimed &#xA;  at helping readers appreciate how early algebra is not the sort of activity that a &#xA;  reader might associate with a high school Algebra 1 course. In particular, the seven &#xA;  chapters in Part II offer theoretical arguments for children’s capacity to learn &#xA;  algebra early and empirical studies of children’s algebraic reasoning in a variety of &#xA;  different elementary-level classroom and curricular contexts. These chapters &#xA;  provide illustrations of what early algebra curriculum and instruction look like, as &#xA;  well as existence proofs of children’s capacity to do this work. (p. 204)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Even though you're not talking about algebrafying early mathematics, and depending on what sort of things you want students to consider and even argue about, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0805854738&quot;&gt;this book&lt;/a&gt; (along with other early algebra research) may provide perspectives that spark thoughtful discussions and reflections.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-08-12T00:38:45.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4179" PostTypeId="2" ParentId="2429" CreationDate="2014-08-12T09:48:19.803" Score="3" Body="&lt;p&gt;My favorite is &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/8120339320&quot; rel=&quot;nofollow&quot;&gt;Abstract Algebra by I.H Sheth&lt;/a&gt;, printed in &lt;a href=&quot;http://phindia.com&quot; rel=&quot;nofollow&quot;&gt;india&lt;/a&gt;!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do not have to be directly and immediately discuss on groups theory, this book will be useful. It covers more additional requirements comparing to Fraleigh's book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;IMO, the Fraleigh problem is its proof approach! They are not fast and clean as well as others. However it moves slow: step by step and will be a good choice for Intermediate students. though, it covers extra topics for BA.&lt;/p&gt;&#xA;" OwnerUserId="1998" LastActivityDate="2014-08-12T09:48:19.803" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4180" PostTypeId="1" CreationDate="2014-08-12T11:48:46.760" Score="8" ViewCount="327" Body="&lt;p&gt;Last semester I taught Linear Algebra using the standard textbook of Lay. Online one can find nice &lt;a href=&quot;http://www.laylinalgebra.com/free_site/tm_index.html&quot;&gt;&quot;class handouts&quot;&lt;/a&gt; that serve very well as lecture notes for students to follow along with during class. There are blanks to fill in the bodies of examples and theorems, so students can interact with the notes while not having to try to write down everything said during lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students responded very positively to these notes, so I'd like to try to craft a similar set of short Calculus lecture notes. I don't want to reinvent the wheel, though, and I'd like to be able to borrow from some &lt;em&gt;good&lt;/em&gt; guided lecture notes while writing mine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Does anyone know of a particularly good set of Calculus I and II &lt;a href=&quot;http://www.interventioncentral.org/academic-interventions/study-organization/guided-notes-increasing-student-engagement-during-lecture-&quot;&gt;guided lecture notes&lt;/a&gt; analogous to the Lay notes found above?&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2014-08-12T16:35:53.690" LastActivityDate="2014-08-12T16:35:53.690" Title="Guided Lecture Notes for Calculus" Tags="|calculus|reference-request|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4182" PostTypeId="1" CreationDate="2014-08-12T16:37:05.013" Score="8" ViewCount="318" Body="&lt;p&gt;Fractions are a well-recognised issue in maths learning, with all sorts of complex issues involved. One particular aspect of this is difficulty recognising fractions as numbers which describe the size of things, especially fractions greater than 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once saw a talk or read a paper (which I am unable to now find) suggesting that we can help to deal with this right from the start by introducing children to mixed numbers with halves and quarters (such as $1 \frac12$) before we introduce them to general proper fractions (such as $\frac37$). The argument was that if children can recognise half as an amount, then they can easily recognise one and a half and two and a half as amounts too, and even add them together consistently. This would mean they could understand these simple mixed numbers before learning about other fractions, and then, when they do learn about other fractions, all the mixed numbers can be inroduced at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is twofold:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Does anyone have a reference for this idea?&lt;/li&gt;&#xA;&lt;li&gt;Has anyone actually tried it in practice who can describe how it worked (or did not work)?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-03-29T06:25:49.177" Title="Teaching &quot;and a half&quot; early, possibly before general proper fractions" Tags="|reference-request|primary-education|fractions|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4183" PostTypeId="2" ParentId="2429" CreationDate="2014-08-12T19:50:13.770" Score="4" Body="&lt;p&gt;Perhaps implicit in this question is the issue of the content design of abstract /modern algebra courses at colleges and universities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some abstract algebra courses begin with groups, and go on to rings and fields. Some aspects of group theory are helpful in thinking about issues related to integral domains, rings, and fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, for schools that train a lot of high school and middle school teachers, the integers are an example of an integral domain in the K-12  curriculum, so perhaps one should start an abstract algebra course with integral domains then go on to fields and groups? If one teaches something about modular arithmetic in K-12 perhaps having teachers who realize there is a concept such as a zero-divisor is useful and important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Zero_divisor&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Zero_divisor&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book one chooses may reflect the difference in approach to the content design of abstract algebra that is picked for the course at your school.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-08-12T19:50:13.770" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4184" PostTypeId="1" CreationDate="2014-08-13T05:51:51.693" Score="2" ViewCount="183" Body="&lt;p&gt;I have some personal criterion to choose my graduate students including PhDs. Here I would like to know what my colleagues' criterion are to choose their math PhD/MSc students. I hope this may help me to note some subtle points which are important but less well-known about choosing a math PhD student. In fact I would like to know more on those useful abilities which a math PhD student should have.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are nice criterion to choose a good PhD student in pure mathematics?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user230" LastEditorDisplayName="user230" LastEditDate="2014-08-21T02:50:17.040" LastActivityDate="2014-08-21T02:50:17.040" Title="How to choose a good PhD student in pure mathematics?" Tags="|graduate-education|" AnswerCount="0" CommentCount="4" ClosedDate="2014-08-18T20:21:03.950" ContentLicense="CC BY-SA 3.0" />
  <row Id="4186" PostTypeId="1" CreationDate="2014-08-13T14:30:26.100" Score="6" ViewCount="565" Body="&lt;p&gt;I am looking for toys or props that illustrate concepts from abstract algebra. The best known example is of course the &lt;a href=&quot;http://en.wikipedia.org/wiki/Rubik&amp;#39;s_Cube&quot; rel=&quot;noreferrer&quot;&gt;Rubik's Cube&lt;/a&gt;; another great example is the &lt;a href=&quot;http://en.wikipedia.org/wiki/Fifteen_puzzle&quot; rel=&quot;noreferrer&quot;&gt;Fifteen Puzzle&lt;/a&gt;, especially if like Sam Loyd you offer $1,000 to any student who manages to switch the 14 and the 15.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There should be many, many more. For example it should be possible to construct a toy that physically illustrates that $S_8$ (say) is generated by a transposition and an $8$-cycle, and I vaguely even remember having something like this when I was a kid, but I have no idea where to get one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where can one get such toys or props? I am especially interested in actual, physical toys which I can buy somewhere and use during my abstract algebra course this fall.&lt;/p&gt;&#xA;" OwnerUserId="1535" LastActivityDate="2014-08-13T17:27:22.567" Title="What toys and props illustrate concepts from abstract algebra?" Tags="|abstract-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4187" PostTypeId="2" ParentId="4115" CreationDate="2014-08-13T16:25:35.590" Score="3" Body="&lt;p&gt;I can see several ways of addressing the question, although I'm afraid that a comprehensive answer would require a number of chapters and quite a bit of scholarship. I also think this question's premises raise interesting issues on their own: are arguments underlying mathematics education programs ever so explicitly about militarism, or are they always made on a basis of diverse views of utility and value of mathematics knowledge? How easy or difficult is it to tie national trends and concerns to the very specific materials (books) mentioned in the question, especially considering how some nations (notably, the U.S.) have historically and intentionally decoupled local education from national policy? Is mathematics education, the bulk of which exists in K-12 classrooms (where policy has most of its focus), thought of as a dichotomy of applied vs. pure mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For my part, I will touch on one small aspect of the question: &lt;strong&gt;a cycle in the justification for mathematics education in the school curricula of the U.S.A. that sometimes involves concerns of war&lt;/strong&gt;. The preparation for or looming of war has been a part of education policy in the USA. Some in mathematics education research and policy research have noted that there is a cycle of recurring themes in the choices made to determine the place of mathematics education within the American school curriculum (Garrett &amp;amp; Davis Jr, 2003). War concerns are one kind of concern that has acted as a type of critical event in this set of themes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Drawing directly from, but paraphrasing, Garrett and Davis (2003), the themes fit together like this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Math in school curriculum is seen to come under attack (Why do we need so much math!? and/or Isn't this math useless for many of these students? -- Shouldn't most of them just learn arithmetic?)&lt;/li&gt;&#xA;&lt;li&gt;People (university professors and others outside of schools) come to the defense of mathematics in the K-12 curriculum.&lt;/li&gt;&#xA;&lt;li&gt;Everyone notices some looming threat, or some threatening event occurs (such as war). Arguments turn to using this threat to justify the role of mathematics.&lt;/li&gt;&#xA;&lt;li&gt;For a time, people accept the role of mathematics in the curriculum.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(see pp. 493-494)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As part of the cyclical nature of this set of themes, Garrett and Davis noted that mathematicians and math educators have failed to make a deep argument for the role of mathematics in the curriculum that is lasting. Consequently, we will all see the same arguments again in the future when memory of the critical event fades. So these events have acted as a ready justification for mathematics in the curriculum, but the strength of their emotional impact has obviated the need for a deeper logical argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A number of the chapters in the Stanic and Kilpatric (2003) volumes deal with the different motivations behind mathematics education policy and related reforms, including intervention from notable military voices who were concerned about mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem described by Garrett and Davis can be thought of as not only a failure to give a deeper argument for mathematics education in the curriculum, but a failure to confront the fact that these curricular choices are value-laden, and that education reform is values driven (Stanic &amp;amp; Kilpatrick, 1992). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking to the question of &lt;em&gt;what&lt;/em&gt; mathematics to teach (e.g. where applied vs. pure might be one dimension considered) and whether or not it is driven by war, Stanic and Kilpatrick (1992) wrote that we are divided by some of the same faith (and arrogance) that has led us to promote mathematics in the first place. Their paper concluded:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The mathematics education community has traditionally acted from and&#xA;  defended a fundamental faith in the importance of mathematics for&#xA;  everyone. According to this faith, more people should study more&#xA;  mathematics in school, for their own sakes as well as the sake of our&#xA;  society. Ironically, the same faith that established and sustained the&#xA;  community gets in the way of seeing and dealing with the important&#xA;  differences that divide it and that hamper curriculum reform.&#xA;  Competing visions – that is, competing answers to the questions of&#xA;  what we should teach, why we should teach one thing rather than&#xA;  another, and who should have access to what knowledge – can be&#xA;  healthy, but only if the are recognized and dealt with. It is naïve,&#xA;  moreover, to assume that wide-ranging reform in school mathematics&#xA;  will result from any effort that focuses only on schools and is not&#xA;  somehow linked to reform of the wider society. (p. 414)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Often we seem so focused on these kinds of differences within the broader field of those who promote mathematics education, we lose the ability to get traction with reform until a looming national concern, such as war or the surprising advancement of a feared enemy (e.g. the U.S reaction to the launch of Sputnik) creates an impetus to stray from &quot;basics&quot; into more research-based approaches (e.g. the boom in exploratory approaches to mathematics education in the U.S. collectively referred to as &quot;New Math&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;As an aside&lt;/strong&gt;, I think that war and other impactful events that have reforms associated them garner a disproportionate amount of attention, distorting our view of history. For instance, our &quot;back to basics&quot; movement of the 1970's was a reaction to the (not entirely justified) perceptions of New Math-era reforms, especially based on misunderstandings of their effects and scope. However, during the &quot;back to basics&quot; era, researchers remarked on impoverished middle school curriculum, an over-focus on processes, and a public's perception of mathematics as limited to arithmetic (Fey &amp;amp; Graeber, 2003).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;As a second aside&lt;/strong&gt;, we might ask whether actual war or threat of war are even needed when war metaphor can be used to generate a sense of urgency. This linguistic theme in calls for reform is notably present in the oft-cited report &lt;em&gt;A Nation at Risk:&#xA;The Imperative for Educational Reform&lt;/em&gt; (National Commission on Excellence in Education, 1983). As you have undoubtedly heard before, they summed up the decade of &quot;back to basics&quot; mathematics reforms with this call to action:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If an unfriendly foreign power had attempted to impose on America the&#xA;  mediocre educational performance that exists today, we might well have&#xA;  viewed it as an act of war. As it stands, we have allowed this to&#xA;  happen to ourselves. We have even squandered the gains in student&#xA;  achievement made in the wake of the Sputnik challenge. Moreover, we&#xA;  have dismantled essential support systems which helped make those&#xA;  gains possible. We have, in effect, been committing an act of&#xA;  unthinking, unilateral educational disarmament. (p. 6)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We might ask whether it is really productive (i.e. aligned with our goals and values) to create, or at least attempt to create a sort of hysterical urgency to drive education reform. Or, at least, what problems it creates for us in the longer term if we agree that there is something fundamentally important about the understanding to be gained via school mathematics apart from its utility in defense and offense.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cited:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fey, J. T., &amp;amp; Graeber, A. O. (2003). From the new math to the Agenda for Action. In G. Stanic &amp;amp; J. Kilpatrick, A history of school mathematics (Vol. 1, pp. 521–558).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Garrett, A. W., &amp;amp; Davis Jr, O. (2003). A time of uncertainty and change: School mathematics from World War II until the new math. In G. M. A. Stanic &amp;amp; J. Kilpatrick (Eds.), A history of school mathematics (Vol. 1, pp. 493–520). Reston, VA: National Council of Teachers of Mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform. Washington, D.C.: U.S. Government Printing Office.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stanic, G., &amp;amp; Kilpatrick, J. (1992). Mathematics curriculum reform in the United States: A historical perspective. International Journal of Educational Research, 17(5), 407–417.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stanic, G. M. A., &amp;amp; Kilpatrick, J. (Eds.). (2003). A history of school mathematics. National Council of Teachers of Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-08-13T16:25:35.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4188" PostTypeId="2" ParentId="4186" CreationDate="2014-08-13T17:27:22.567" Score="3" Body="&lt;p&gt;I have the answer for you: &lt;strong&gt;Reptangles: turtles that snap&lt;/strong&gt;. These are little plastic turtles which allow connections in a bunch of different ways. Like legos, but more possible angles. The end result is you can assemble shapes with all sorts of symmetries. You also have color to help illustrate your point. Here's a picture:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Qim6A.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The instructions have a fair amount of symmetry discussion built-in, I have the feeling a math person designed these, but it's just a hunch. My 6 year old also loves to play with them and I got him some at home just as a toy (for now, I mean I don't expect abstract algebra until middle school ;)&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-08-13T17:27:22.567" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4189" PostTypeId="2" ParentId="4165" CreationDate="2014-08-13T17:47:44.517" Score="6" Body="&lt;p&gt;I thought it might be interesting to look at a variety of different Calculus textbooks to see how they handle the terminology.  Here is an unscientific survey of the half-dozen different Calc textbooks I have on my bookshelf right now:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Stewart, Single Variable Calculus, 4e (both Early Transcendentals and Late Transcendentals versions) and 5e&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stewart defines the phrase &lt;em&gt;continuous at a number a&lt;/em&gt; in the usual way; the text also defines &lt;em&gt;continuous from the right at a&lt;/em&gt; and &lt;em&gt;continuous from the left at a&lt;/em&gt;.  Armed with these definitions, the text defines &lt;em&gt;continuous on an interval&lt;/em&gt;, with one-sided limits used at the endpoints of the interval if appropriate in context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With respect to the specific question of the OP, Stewart's Theorem 5 states that &quot;Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The phrase &quot;a continuous function&quot; (without qualifiers like &quot;at a point&quot; or &quot;on an interval&quot;) is used in the exposition, but never in the definitions, theorems, proofs, or exercises. The phrase is used solely as a kind of informal shorthand that makes it possible to refer to several different types of continuity at once, for example in statements like &quot;a sum or product of continuous functions is continuous&quot;.  A question like &quot;Is the reciprocal function continuous?&quot; would not be found in this text; however, &quot;Is the reciprocal function continuous on its domain?&quot; and &quot;At what points is the reciprocal function discontinuous?&quot; would be legitimate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Gillett, Introduction to Calculus and Analytic Geometry, 4e&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like Stewart, Gillett defines &quot;continuous at $x_0$&quot;.  This text also defines &quot;continuous in S&quot; (where S is a subset of the domain).  (Interestingly the definition does not impose any restrictions on what types of subsets may be considered, but every example seems to be a countable union of intervals.)  The phrase &quot;continuous in its domain&quot; comes next, and Gillett has as Example 6 that &quot;The reciprocal function is continuous in its domain.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the terminology &quot;a continuous function&quot;, Gillett has the following remark immediately following Example 6:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When a function is continuous in its domain, many writers simply call it &lt;strong&gt;continuous&lt;/strong&gt;, using the global term (unqualified)…  This is not unreasonable… but it can be misleading.  To say that f(x) = 1/x is continuous (without any qualification) is to risk overlooking the infinite jump at x = 0.  It also sounds paradoxical to say &quot;f(x) is continuous&quot; in one breath and &quot;f(x) is discontinuous at x=0&quot; in the next.  That is why we prefer the language &quot;continuous in its domain.&quot;  It leaves the door open for further remarks.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So Gillett explicitly rejects the phrase &quot;a continuous function&quot; without specifying a point or a subset of its domain, and justifies this using the reciprocal function as its example of why that usage would be poor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Moise, Calculus:  Part I.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This text (from 1966), like the more modern examples, starts by defining &quot;continuous at $x_0$&quot;.  Unlike Stewart and Gillett, Moise continues by defining &quot;If $f$ is continuous at every point $x_0$ of its domain, then we say that $f$ is &lt;em&gt;continuous&lt;/em&gt;.&quot;  So it uses precisely the terminology that Gillett critiques.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example of how continuous functions can be combined to produce more continuous functions, Moise considers the example of $f(x) = x^2 + 1$ and $h(x) = x^2 - 1$.  The text states that $f+h$ and $fh$ are both &quot;continuous&quot; (note the absence of any qualifier) and then says that $f/h$ is continuous except at $1$ and $-1$.  The example continues, &quot;Of course, at $x=1$ and $x = -1$ it is not just continuity that breaks down: the quotient function is not even defined at these points, because the denominator of $f/h$ becomes $0$.&quot;  Oddly enough, Moise stops short of saying that $f/h$ is a &quot;continuous function&quot;, despite the fact that it would be completely in keeping with his own usage conventions to do so.  Presumably that is because, as Gillett noted in his remark quoted above, it would seem weird to say that &quot;$f/h$ is continuous&quot; and then also say that &quot;$f/h$ is discontinuous at $1$ and $-1$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Zill (1993), Calculus (3e)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like all the others, Zill begins by defining &quot;continuity at a point&quot;.  Zill states that that rational functions are discontinuous at points where they are not defined.  The text&#xA;defines continuity on an interval (with &quot;continuous from the right/left&quot; used at end points as appropriate).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With respect to the usage &quot;continuous function&quot;, Zill says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Functions that are continuous on $(-\infty, \infty)$ are said to be&#xA;  &quot;continuous everywhere&quot; or simply &quot;continuous&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So Zill reserves the phrase &quot;continuous function&quot; for use only with functions whose domain is all of $\mathbb{R}$.&#xA;On the other hand (and somewhat in contradiction to this explicit definition),  &quot;continuous function&quot; is used throughout the exposition (but never in the theorems!) as a shorthand, much in the same way that Stewart does.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So, to summarize, I'm going to go out on a limb here and, based on the single example of the 1966 textbook, suggest that the use of the phrase &quot;continuous function&quot; to mean &quot;function that is continuous at each point of its domain&quot; might be an artifact of an earlier generation of textbook authors.  More recent texts either avoid giving the phrase any technical meaning at all, using it instead as an informal shorthand, or (in the case of Zill) define it to mean &quot;continuous on all of $\mathbb{R}$&quot;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-08-13T17:47:44.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4190" PostTypeId="2" ParentId="4172" CreationDate="2014-08-14T00:57:08.733" Score="1" Body="&lt;p&gt;See the &lt;a href=&quot;http://www.promys.org/program/faq&quot; rel=&quot;nofollow&quot;&gt;FAQ&lt;/a&gt;:  &quot;Are counselors always former PROMYS students?  Not always, but often.&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they're like most American summer programs, they prefer to hire alumni.  When there aren't enough good alums, they'll hire people that some alum or staff member knows personally.  Strangers are usually a last resort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if you're asking here, you're not in a great position....but you have plenty of time before next summer to make friends with someone from the program.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-08-14T00:57:08.733" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4191" PostTypeId="2" ParentId="1572" CreationDate="2014-08-14T03:26:37.123" Score="4" Body="&lt;p&gt;Paulos defines innumeracy as &quot;an inability to deal comfortably with the fundamental notions of number and chance&quot; (Paulos, 1988, p. 3). I read (and enjoyed) his book back when I was primarily a software developer and didn't know enough to ask why he wrote the definition that way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first example he gives of innumeracy is a newscaster concluding that the weekend had a 100% chance of rain if both Saturday and Sunday had a 50% chance. But he also found that a fellow viewer of the weather report found no problem with the conclusion. Today I would ask Paulos: &quot;What evidence do you have of the level of comfort felt by the participants in your example?&quot; And if he told me &quot;That's not what I meant&quot; I would ask why he wrote it that way if he didn't mean it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beside the point? It is not. Definitions are important in our understanding of mathematics as well as in our understanding people (and education). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book is really mostly about the types of errors similar to this weathercaster's. In this case, conclusions that don't follow mathematically from the available data. Some people collect examples of this sort of thing. I credit user &lt;a href=&quot;https://matheducators.stackexchange.com/users/498/michael-e2&quot;&gt;Michael-E2&lt;/a&gt; for bringing this collection to my attention: &lt;a href=&quot;http://test.causeweb.org/wiki/chance/index.php/Collected_Forsooths&quot; rel=&quot;nofollow noreferrer&quot;&gt;Collected Forsooths&lt;/a&gt; of the newsletter of Royal Statistics Society. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there is a problem here. So little context is given with these examples that it is impossible to determine whether the errors are truly the result of mathematical errors, some different sort of error, or no error at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How bad a problem is this? For education purposes, I say it is no problem at all, depending on the lesson and the point you are trying to make. In my view, mathematics in the world is a sense-making endeavor. An education gives your students a greater capability to make sense of more of the world, allowing the world to actually make more sense to them. And that includes the things people say -- also including the occasional nonsensical-sounding mathematical claim.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How useful is it to be given a list of known mathematical errors? It is certainly of some use. But is not it also of good use to be given a possible error and to have to argue for what it &lt;em&gt;could&lt;/em&gt; mean? In some cases, these errors may not reflect realty, but they could have some meaning. How do we use our mathematical understanding to make sense of the situation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At some point, someone said something; at another point, someone used mathematical knowledge to determine &quot;that was a problematic statement.&quot; How do we choose sides? Can we narrow the choices of what the statement &lt;em&gt;could&lt;/em&gt; mean? What do we rely on? What arguments can we make?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this would be a valuable educational activity: give students statements and have them argue about the mathematical reasoning used. And what they can make sense of. And how they support their own view of the situation. These are types of mathematical reasoning that appear in standards documents, but also are what they will have to rely on if they use math to make sense of their world. And of course, what they will do to convince themselves and others of what they know.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-01-28T23:15:44.203" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4192" PostTypeId="2" ParentId="1817" CreationDate="2014-08-14T10:22:36.040" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Christine_Ladd-Franklin&quot; rel=&quot;nofollow&quot;&gt;Christine Ladd-Franklin&lt;/a&gt; was involved in the early development of a community of American Mathematical research. She was the first female student and Johns Hopkins (though unofficially enrolled) and produced results in algebraic logic.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-08-14T10:22:36.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4193" PostTypeId="2" ParentId="4182" CreationDate="2014-08-14T10:30:50.347" Score="2" Body="&lt;p&gt;I have tried this with my own son (whom my wife and I are homeschooling), and at least for him, the approach seems to be working. What we have basically done is introduced fraction concepts as they are naturally needed and discussed them like we might discuss any other number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a conversation I had with him recently about fractions.&#xA;&lt;a href=&quot;http://maththinking.org/2014/07/29/talking-about-fractions/&quot; rel=&quot;nofollow&quot;&gt;http://maththinking.org/2014/07/29/talking-about-fractions/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice the lack of symbols in my description of our conversation. This is because we have not yet introduced any symbols when thinking about fractions. My hypothesis is that introducing symbols based on division to represent fractions is pretty meaningless until children have a better understanding of division, so hence my delay in that respect.&lt;/p&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-08-14T10:30:50.347" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4194" PostTypeId="2" ParentId="4093" CreationDate="2014-08-14T10:34:53.010" Score="4" Body="&lt;p&gt;If you are looking for a book you can use to study an effective practice of teaching mathematics in a way that is definitely not intended to leave students with mathematical tunnel vision, can I recommend Teaching Problems and the Problems of Teaching by Magdalene Lampert?&lt;/p&gt;&#xA;" OwnerUserId="254" LastActivityDate="2014-08-14T10:34:53.010" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4196" PostTypeId="2" ParentId="1817" CreationDate="2014-08-15T01:18:41.443" Score="40" Body="&lt;p&gt;&lt;a href=&quot;http://www.cbc.ca/news/technology/maryam-mirzakhani-becomes-1st-woman-to-win-fields-medal-1.2734950&quot;&gt;Maryam Mirzakhani&lt;/a&gt;, who just won the Fields Medal, and also was the first Iranian student to win a gold medal in the IMO in 1995 with a perfect score.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My colleague Mohammad Javaheri was on Iran's IMO team with her in 1995. He told us the other day that after Maryam won the gold, when the rest of the team went up to congratulate her she said &quot;next, the Fields Medal&quot;. Nineteen years later, she did it!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I'm not sure what else to say that would be interesting to high school students...as moduli spaces of Riemann surfaces may not be so interesting to them...)&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2014-08-15T18:25:01.330" LastActivityDate="2014-08-15T18:25:01.330" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4198" PostTypeId="1" AcceptedAnswerId="4201" CreationDate="2014-08-15T14:53:07.607" Score="3" ViewCount="412" Body="&lt;p&gt;Significant digits are in the curriculum for both my Chemistry and my Physics class, and all my Calculus students have taken at least Chemistry. It also pops up in the school's Precalculus class (which I do not now teach). Although significant digits are not in the Calculus curriculum I get many questions about them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should I continue my present practice of ignoring sig digs in my Calculus class (and stating the desired precision of approximate answers in quizzes and tests) or should I include them in some way?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I am not asking if sig digs should be used in science classes: that is a given for now.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2014-08-16T04:13:21.617" Title="Significant digits in high-school Calculus?" Tags="|secondary-education|calculus|mathematical-pedagogy|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4199" PostTypeId="2" ParentId="294" CreationDate="2014-08-15T17:18:59.440" Score="10" Body="&lt;p&gt;I highly recommend Paul Lockhart's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0674057554&quot;&gt;Measurement&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1677" LastActivityDate="2014-08-15T17:18:59.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4200" PostTypeId="1" CreationDate="2014-08-15T17:41:08.190" Score="3" ViewCount="295" Body="&lt;p&gt;I am currently a &quot;self-studier&quot;, so the manner in which I learn is I have a textbook and I  work through it in a relatively linear fashion. As I am introduced to new concepts I ask questions (sometimes I go down the &quot;rabbit hole&quot;, but I feel I've got a good handle on this now) and I attempt to answer these myself and if all else fails I ask Maths SE. Most importantly, only after I've understood a certain topic do I then use Anki to make flashcards to ensure I retain most of what I've learned. Learning like this is what seems to work for me, however I feel like  this could be optimised.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I have a few questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When exposed to a new topic what kind of questions should one be asking oneself? What are the most incisive kinds of questions that cut to the core of a matter? How do I ask the right questions?&lt;/li&gt;&#xA;&lt;li&gt;Is there a proven effective way of learning? How does one learn something inside-out?&lt;/li&gt;&#xA;&lt;li&gt;How does one learn to think really analytically?&lt;/li&gt;&#xA;&lt;li&gt;What is the best way to make connections with things you've learned previously? I feel that with Anki (a spaced repetition system) that I make all these flashcard decks full of bite-sized bits of knowledge that I can recall on demand, but just because you can recall something on cue doesn't mean you can use that knowledge when a certain context demands it. How does one make strong connections between all these disparate bits of knowledge? I want to build a strong &quot;network&quot; of knowledge if you get what I mean. &lt;/li&gt;&#xA;&lt;li&gt;What study-habits should you instill in yourself if you want be an effective learner?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;P.S. If this is too opinion-based is it possible to place this in community wiki or something?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1677" LastEditorUserId="1677" LastEditDate="2014-08-15T20:07:39.423" LastActivityDate="2014-08-15T20:07:39.423" Title="Learning a topic deeply" Tags="|undergraduate-education|general-pedagogy|self-learning|" AnswerCount="0" CommentCount="5" ClosedDate="2014-08-16T03:02:17.407" ContentLicense="CC BY-SA 3.0" />
  <row Id="4201" PostTypeId="2" ParentId="4198" CreationDate="2014-08-16T01:26:59.640" Score="6" Body="&lt;p&gt;No. If you are going to have questions that have decimal/numerical answers, it makes sense to decide on some particular place value for students to round to for the sake of uniformity, but basing this off of significant digits reflects a misunderstanding about what significant digits are used for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sciences use significant digits because problems are posed in terms of experimental givens, where the amount of listed digits loosely implies confidence in measurement, for example a 1 kg vs 1.00 kg mass. Significant digit rules make sure that the answer implies the same level of measurement accuracy as the given quantities. This is not normally considered in math classes because the given quantities are assumed to be mathematically exact/ideal, since they are just &quot;invented&quot; for the purpose of the exercise and not measured.&lt;/p&gt;&#xA;" OwnerUserId="2011" LastActivityDate="2014-08-16T01:26:59.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4202" PostTypeId="2" ParentId="4198" CreationDate="2014-08-16T04:13:21.617" Score="0" Body="&lt;p&gt;Most of the methods of calculus are at first directed towards describing the exact formulas for differentiation and integration. In these cases I don't really see the point of sigfigs. If you were interested in such things, maybe talk about error in a slightly more sophisticated way by describing how the derivative of a function can be used to propagate error in input. The is, we have data $x$ with error $\pm \delta$ then we can approximate error in a function $f$ by $f(x) \pm |f'(x)|\delta$ of course, it should be emphasized that this is not the only source of error (error may be in the model/function not the data) nor is it perfect (for example it is absurd to claim error at a max is $0$.&lt;/p&gt;&#xA;" OwnerUserId="1915" LastActivityDate="2014-08-16T04:13:21.617" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4203" PostTypeId="1" CreationDate="2014-08-16T19:28:31.997" Score="3" ViewCount="249" Body="&lt;p&gt;I am a new math major, and I will be starting in the third year of the program (transfer from engineering) this September. I need to get comfortable taking limits of functions, differentiating them, integrating them, etc. I used to be able to do this quite well in the past, but I must admit that my knowledge was fairly mechanical. I didn't &lt;em&gt;really&lt;/em&gt; understand stuff deep down (even if I knew the epsilon-delta definition of a limit...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am tempted to use Apostol's book to review, rather than a standard calculus book (Stewart, boring!). This is because I feel like I might no longer be satisfied by the more mechanical understanding that a book like Stewart might present, having been exposed (somewhat) to books by people like Courant or Gowers. When starting to read Apostol though, after he mentions &lt;em&gt;Foundations of Analysis&lt;/em&gt; by Landau, I can't help but wonder how interesting that book might be...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Might going with Stewart indeed be the best idea, since I am just trying to reawaken deadened muscle memory? Maybe the deeper understanding can be left for the process of continuous education, while I have to realize that my immediate goal is to get muscle memory back up for school? Or, is there a short and sweet single variable calculus document, that is still insightful, that I might benefit from during my review? Perhaps that's hoping for a silver bullet...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some wisdom and advice from teachers would be very much appreciated. Although, I am afraid the advice might just be: &quot;this is a stupid question, just sit down and do your work even if it bores you, princess&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I thought of a good example that illustrates my concerns. If I were to go through Stewart's book for limits, I am going really be left behind the task of memorizing the limit rules, and then I'll have to apply them to various problems. On the other hand, if I go through Apostol's book, I'll know where and why each rule exists (is it an axiom, or for instance, is something like the limit sum rule (the limit of a sum is the sum of the limits) the consequence of some deeper properties of limits...?&lt;/p&gt;&#xA;" OwnerUserId="2013" LastEditorUserId="2013" LastEditDate="2014-08-16T21:04:03.677" LastActivityDate="2014-08-17T15:02:16.233" Title="Reviewing single variable limits and calculus: how to go about it?" Tags="|undergraduate-education|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4204" PostTypeId="1" CreationDate="2014-08-17T11:01:19.940" Score="6" ViewCount="235" Body="&lt;p&gt;For the incoming semester, my school gave me an amazing new classroom, where all walls are covered by white boards. Even windows are covered by movable boards. It is great to have that much space and I will use lateral boards to keep information such as classroom life (homeworks, examinations,...), and keep important theorems, definitions or figures all lecture long. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure what to do with the board in the students'back. Since their chairs have wheels, I could use it normally but I don't feel it will be convenient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My idea is to leave this board to the students and let them use it to present a math related project. The two main constraints are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Only this board can be used to present the project. In particular, it should be self-contained, and the students should assume they would not be present to answer to the audience's question. The aim is that any passer-by could enter into the classroom and learn something from the project (maybe work on it - see below).&lt;/li&gt;&#xA;&lt;li&gt;The project will evolve during the semester, the students improving or modifying it during small classbreak, or during their lunch break,... The idea is that the project does not take too much of the students time. Also, I planned to take photos of their work regularly.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My question: Do you have examples of this kind of project? Or resources on how to organize it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specific constraints related to this precise class (but I am interested in answers in another setting, addressing the two main constraints above):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;They are high school students. They will start the study precalculus, but already know quite a lot about functions and trigonometry. A project involving multivariable calculus would also be interesting (for another class).&lt;/li&gt;&#xA;&lt;li&gt;It is a special high school for science students. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My idea so far would be a timeline about Asian history of mathematics (Asian because I am a European teacher in Asia, and I don't know anything about this part of the history of mathematics). &lt;/p&gt;&#xA;" OwnerUserId="1538" LastActivityDate="2014-09-19T09:01:08.933" Title="A one semester-long visual mathematical project" Tags="|class-participation|board-use|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4205" PostTypeId="2" ParentId="4203" CreationDate="2014-08-17T15:02:16.233" Score="3" Body="&lt;p&gt;I honestly think Rudin's book, &quot;Principle of Mathematical Analysis&quot;, is a bit too dense for your needs.My classic recommendation would be &quot;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0914098918&quot; rel=&quot;nofollow&quot;&gt;Calculus&lt;/a&gt;&quot;, by Spivak.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I've come across a couple of great books which make the subject even more bearable: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521684242&quot; rel=&quot;nofollow&quot;&gt;A First Course in Mathematical Analysis&lt;/a&gt;&quot;, by Brannan;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This book is wonderful for self-study: it's very comprehensive, has meaningful examples and is rigorous enough for your needs. It's a good prequel to a more serious course in real analysis.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/8847008751&quot; rel=&quot;nofollow&quot;&gt;Mathematical Analysis I&lt;/a&gt;&quot;, by Canuto and Tabacco.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another great book for self-study. To some extent it's more rigorous than Brannan's book. It has lots of examples and solved exercises. (You can have a look at some extra theorems at the &lt;a href=&quot;http://calvino.polito.it/canuto-tabacco/analysis_1/top.htm&quot; rel=&quot;nofollow&quot;&gt;book's website&lt;/a&gt;). There's also a sequel to this book by the same authors, which deals with multivariable calculus: &quot;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/8847017831&quot; rel=&quot;nofollow&quot;&gt;Mathematical Analysis II&lt;/a&gt;&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1461496373&quot; rel=&quot;nofollow&quot;&gt;Real Analysis for the Undergraduate: With an Invitation to Functional Analysis&lt;/a&gt;&quot;, by Pons&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book by Pons is the most rigorous of my suggestions and will be very handy if/when you take real analysis. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;PS: I'm not a teacher. My suggestions are based on the fact that you're moving from engineering to math.&lt;/p&gt;&#xA;" OwnerUserId="1997" LastActivityDate="2014-08-17T15:02:16.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4206" PostTypeId="2" ParentId="294" CreationDate="2014-08-17T21:04:36.587" Score="1" Body="&lt;p&gt;You might be interested in the expansive answers that were generated on math.stackexchange by the questions &lt;a href=&quot;https://math.stackexchange.com/questions/828458/book-ref-request-starting-from-a-mathematically-amorphous-problem-and-comb/889732#889732&quot;&gt;Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/questions/828648/book-series-like-ams-student-mathematical-library&quot;&gt;Book series like AMS' Student Mathematical Library?&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2013" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-08-17T21:04:36.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4207" PostTypeId="2" ParentId="4172" CreationDate="2014-08-18T04:29:10.380" Score="4" Body="&lt;p&gt;There will be an application form on the website during the application season.  It gets removed during the rest of the year to make sure nobody wastes time filling it out when there's no hiring going on, but this ought to be clarified on the website.  (I'll look into fixing this.  Full disclosure: I've been involved with PROMYS in different ways over the years, but I've never played any role in hiring counselors.  This answer is not official or on behalf of PROMYS.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The counselors are chosen based primarily on their mathematical knowledge, talent, experience, and enthusiasm.  This is judged partly from the application form and partly from letters of recommendation.  Of course other factors can play a role.  For example, if you come across as a jerk, you'll be less likely to be hired.  However, you can't make up for a weak mathematics background by being really good at working with teenagers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many counselors are former PROMYS students, but many others had no connection with the program before applying.  (I don't have statistics handy.)  Alum applications are often easier to judge since they already have track records at the program.  Whether this is an advantage or not depends on the track record, of course, and PROMYS wouldn't hire a less qualified alum in preference to other applicants.&lt;/p&gt;&#xA;" OwnerUserId="2018" LastActivityDate="2014-08-18T04:29:10.380" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4208" PostTypeId="1" AcceptedAnswerId="4210" CreationDate="2014-08-18T12:28:24.030" Score="12" ViewCount="496" Body="&lt;p&gt;Is there any game or gamified application for teaching/learning mathematics covering a &lt;strong&gt;high school/secondary school&lt;/strong&gt; curriculum?&lt;/p&gt;&#xA;" OwnerUserId="2020" LastEditorUserId="262" LastEditDate="2015-04-05T15:11:02.797" LastActivityDate="2015-04-05T15:11:02.797" Title="Game or gamified application for teaching/learning secondary school mathematics" Tags="|secondary-education|technology-in-education|interactive-teaching|games|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4209" PostTypeId="1" AcceptedAnswerId="4211" CreationDate="2014-08-18T14:32:54.127" Score="9" ViewCount="704" Body="&lt;p&gt;When I was growing up during the 1960s, &quot;set builder notation&quot; constituted a large part of what was then the &quot;new math.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; When and why did &quot;set theory&quot; become popular in math education? Was it because of the earlier application of sets (and related topics such as topology) to real analysis?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(At heart, the question is what (if any) connection was there between the use of set theory in New Math at the elementary level, versus the expansion of real analysis to include set theory by professional math researchers?)&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="262" LastEditDate="2015-07-10T19:26:37.640" LastActivityDate="2017-07-10T02:12:12.877" Title="What caused the (relatively) recent popularity of set theory?" Tags="|reference-request|curriculum|set-theory|new-math|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4210" PostTypeId="2" ParentId="4208" CreationDate="2014-08-18T16:35:31.150" Score="9" Body="&lt;p&gt;This is one major goal of &lt;a href=&quot;https://www.expii.com/&quot; rel=&quot;nofollow&quot;&gt;Expii&lt;/a&gt;, a crowd-sourced work in progress.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some other related projects I've seen:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Alcumus from the Art of Problem Solving (AoPS): &lt;a href=&quot;http://www.artofproblemsolving.com/Alcumus/index.php&quot; rel=&quot;nofollow&quot;&gt;http://www.artofproblemsolving.com/Alcumus/index.php&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Brilliant: &lt;a href=&quot;http://brilliant.org/&quot; rel=&quot;nofollow&quot;&gt;http://brilliant.org/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Clever Math: &lt;a href=&quot;https://www.clevermath.org/&quot; rel=&quot;nofollow&quot;&gt;https://www.clevermath.org/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Curriki: &lt;a href=&quot;http://www.curriki.org/&quot; rel=&quot;nofollow&quot;&gt;http://www.curriki.org/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;DragonBox App: &lt;a href=&quot;http://www.dragonboxapp.com/&quot; rel=&quot;nofollow&quot;&gt;http://www.dragonboxapp.com/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Euclid: The Game: &lt;a href=&quot;http://euclidthegame.com/Tutorial/&quot; rel=&quot;nofollow&quot;&gt;http://euclidthegame.com/Tutorial/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Established MOOCs like Khan Academy, Coursera, edX, etc.&lt;/li&gt;&#xA;&lt;li&gt;Oppia: &lt;a href=&quot;https://www.oppia.org/&quot; rel=&quot;nofollow&quot;&gt;https://www.oppia.org/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Terry Tao's linear equation games: &lt;a href=&quot;http://terrytao.wordpress.com/2012/04/18/new-version-of-algebra-game/&quot; rel=&quot;nofollow&quot;&gt;http://terrytao.wordpress.com/2012/04/18/new-version-of-algebra-game/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;From what I can tell, Khan Academy currently seems like the &quot;most comprehensive&quot; answer to your question, but I don't know if it's what you're looking for.&lt;/p&gt;&#xA;" OwnerUserId="2017" LastEditorUserId="2017" LastEditDate="2015-04-03T19:30:09.863" LastActivityDate="2015-04-03T19:30:09.863" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4211" PostTypeId="2" ParentId="4209" CreationDate="2014-08-19T00:12:43.213" Score="9" Body="&lt;p&gt;Whenever &quot;the new math&quot; comes up, I must point out that there was no one &quot;new math&quot; curriculum and much of that post WWII exploration into math curriculum and pedagogy had different aims and philosophies (Davis, 2003). (This doesn't directly pertain to your question, but the actual use of new instructional methods and curricular content during this period was also not widespread or deeply penetrating into the educational system of the USA.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in order to get a really good answer when talking about what was going on in the post WWII era, it may help to talk about a specific curriculum. Because, while some of the projects doing research at the time may have incorporated set theory, they may have done so for different reasons and there may not be one unifying philosophy. There possibly 30 or so projects that could be associated in some way with that era.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Angela Lynn Evans Walmsley (2003) sought to chronicle that time of reform and focus on the most prominent projects of the so-called &quot;New Math movement,&quot; and she has this to say about the use of set theory:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Also, the language of sets and abstraction became part of the &quot;new&#xA;  mathematics&quot;. Both set theory and abstract algebra became known&#xA;  synonymously with &quot;new mathematics,&quot; but neither was a large part of&#xA;  the curriculum. (p. 75)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Later in that same book, the author makes reference to Morris Kline, an outspoken opponent of curriculum reforms who specifically attacked the requirement of using precise language and the inclusion of set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own conjecture is that the prominence of set theory in the minds of people when they consider &quot;the new math&quot; may be, in part, attributable to the critics who focused on set theory as either too abstract, or as substituting for the arithmetic that was the focus of a lot of traditional mathematics education approaches. Possibly, people saw it as something associated with the professional mathematicians who, in part, drove some of the ideas behind new (but not necessarily research-based) approaches to mathematics education. For whatever reason, its close association does appear to be disproportionate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for why it is there in the first place, the question would really need to be more specific. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My general way of addressing the question (without a reference to back it up) would be that a structuralist approach to mathematics influenced the thinking of some mathematicians who wanted to see students build up their knowledge from axioms. Mathematicians might have been paying close attention to the writings of Bourbaki at the time, but to get reasons of set theory's inclusion in a curriculum, it would really be better to focus on a specific curriculum or set of related curricular materials.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Works Cited:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Davis, R. B. (2003). Changing school mathematics. In G. M. A. Stanic &amp;amp; J. Kilpatrick (Eds.), A history of school mathematics (Vol. 1, pp. 623–646). Reston, VA: National Council of Teachers of Mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Walmsley, A. L. E. (2003). A history of the “New Mathematics” movement and its relationship with current mathematical reform. University Press of America.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-08-19T00:12:43.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4212" PostTypeId="2" ParentId="4209" CreationDate="2014-08-19T02:28:47.813" Score="9" Body="&lt;p&gt;&lt;strong&gt;My very incomplete understanding:&lt;/strong&gt; I know from Henry Pollak (first-hand, but mentioned briefly &lt;a href=&quot;http://blogs.tc.columbia.edu/mst/2014/02/05/25th-year-faculty-anniversary-dr-henry-pollak-mathematics-education/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; and in an article below) that one of the driving forces behind SMSG (with whom &lt;em&gt;New Math&lt;/em&gt; is often associated) was Ed Begle. It should be noted that Begle's influence on mathematics education moving forward was also nontrivial: He served as advisor (along with Polya) to &lt;strong&gt;&lt;a href=&quot;http://www.genealogy.math.ndsu.nodak.edu/id.php?id=51364&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeremy Kilpatrick&lt;/a&gt;&lt;/strong&gt;, who is perhaps one of the three most prominent names in mathematics education (the other two, in my opinion, being Schoenfeld and Ball; &lt;strong&gt;&lt;a href=&quot;http://www.genealogy.math.ndsu.nodak.edu/id.php?id=45584&quot; rel=&quot;nofollow noreferrer&quot;&gt;Uri Treisman&lt;/a&gt;&lt;/strong&gt; is also pretty well known, and his mentor/advisor was the logician Leon Henkin, who was a student of Alonzo Church... but I digress).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to the main-story: You can see from &lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Edward_G._Begle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Begle's wikipage&lt;/a&gt;&lt;/strong&gt; that he served as the director of SMSG (School Mathematics Study Group) and that his background was in topology: Begle earned his doctorate under (algebraic) topologist &lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Solomon_Lefschetz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solomon Lefschetz&lt;/a&gt;&lt;/strong&gt;, and he was also deeply influenced by the work of Raymond Wilder, as mentioned in &lt;strong&gt;&lt;a href=&quot;http://www.tau.ac.il/~corry/publications/articles/pdf/Moore%20-%20New%20Math.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;&lt;/strong&gt; (pdf) from &lt;em&gt;The International Journal for the History of Mathematics Education&lt;/em&gt;. You can find my citation of Wilder's classic text on &lt;em&gt;Set Theory&lt;/em&gt; in an earlier MESE response &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/2530/262&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;. So, with minimal digging, we have already uncovered a fair amount of potential sources for set theoretical ideas to make their way into curricula associated with New Math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Recap:&lt;/strong&gt; &quot;New Math&quot; would have been influenced by, among other mathematicians, SMSG's director, Ed Begle, who was fond of the sort of notation you allude to when you ask about set theory. However, I'd like to return briefly to Pollak and another article in &lt;em&gt;The International Journal for the History of Mathematics Education&lt;/em&gt;, available for free &lt;strong&gt;&lt;a href=&quot;http://www.comap.com/historyjournal/pdf/Vol2No2InterviewHenryPollak.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt; (pdf). In particular, allow me to highlight the following two excerpts:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/AskQ6.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;interviewee&quot; above is Henry Pollak, and you can find in the article the foot-noted items. The real item of import is in the subsequent paragraph, and the point made that it was not so much about using &lt;em&gt;set theory&lt;/em&gt; but rather about using &quot;set ideas&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/hV4bc.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This conception of the ideas in &quot;New Math&quot; provides, perhaps, some support for JPBurke's conjecture about an over-focus on &quot;set theory&quot; in NM as overly abstract by its critics, whereas - at least in the case of Pollak, who worked with SMSG - the real motivation may have come from a more &quot;applied&quot; setting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This post really does not do the &quot;full story&quot; justice, but I thought it might shed a bit of light on the large question, and maybe encourage others to dig even further...) &lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-08-19T02:33:48.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4213" PostTypeId="2" ParentId="380" CreationDate="2014-08-19T10:13:38.100" Score="-2" Body="&lt;p&gt;It seems as though the real problems, your students have, are not the chain rule but analysing algebraic expressions and doing derivatives based on this analysis on the one hand and coping with unknown objects (like &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;) on the the other hand.&lt;/p&gt;&#xA;&lt;p&gt;For the first problem, your consequence shouldn't be to go back to simple derivatives, but to algebraic expressions. Let them analyse the expressions to be derived. You can support this by using colors either for the different layers or for the different operations/derivation rule prototypes. Another support are &lt;a href=&quot;http://wikis.zum.de/zum/Klammergebirge&quot; rel=&quot;nofollow noreferrer&quot;&gt;Klammergebirge (parentheses mountain ranges)&lt;/a&gt;. You can do this with LEGO or magnets on a black board.&lt;/p&gt;&#xA;&lt;p&gt;Then, let them give you the derivatives of every single layer first, and mark them in the respective color/at the respective level and finally put them back together.&lt;/p&gt;&#xA;&lt;p&gt;Yes, this will look silly to your students, but you can show them, that what they need to learn, is something very basic, very elementary, very abstract.&lt;/p&gt;&#xA;&lt;p&gt;For the second problem, give them &lt;em&gt;exotic&lt;/em&gt; functions (maybe like &lt;span class=&quot;math-container&quot;&gt;$\operatorname{arcoth}$&lt;/span&gt;), they don't know the derivative of, and use them in easy examples (like &lt;span class=&quot;math-container&quot;&gt;$f(x)=\operatorname{arcoth}(x^2+1)$&lt;/span&gt;. Then give them problems like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f'(x)=\sin(x)/x$&lt;/span&gt;. Find &lt;span class=&quot;math-container&quot;&gt;$(f(x)f'(x))'$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Force them, to treat &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; as a mathematical object and not as a task/action/process. (See the other answers' reference to APOS.)&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="6053" LastEditDate="2022-09-27T16:59:37.980" LastActivityDate="2022-09-27T16:59:37.980" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="4214" PostTypeId="1" AcceptedAnswerId="4216" CreationDate="2014-08-20T03:24:41.647" Score="6" ViewCount="431" Body="&lt;p&gt;I'm currently working as a teacher assistant. My job now is to orientate differential calculus' students and try to solve any questions that may rise in the development of the course and exercises. They're instructed to solve a particular set of problems; in that set you can find the following question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there infinitely many irrational numbers?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They're asked to answer either &lt;em&gt;yes&lt;/em&gt; or &lt;em&gt;no&lt;/em&gt; and to justify their answer. I can't seem to find an easy way of explaining that there is indeed an infinite amount of irrational numbers. To do so, I only imagine talking about Cantor's diagonal argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Resuming, my question's this: &lt;strong&gt;is there an easy, naive way of explaining that there's an infinite amount of irrationals?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="1551" LastEditorUserId="1551" LastEditDate="2014-08-20T03:31:09.890" LastActivityDate="2014-08-20T04:54:04.440" Title="Explaining there's an infinite amount of irrationals to differential calculus freshmen" Tags="|undergraduate-education|calculus|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4215" PostTypeId="2" ParentId="4204" CreationDate="2014-08-20T04:53:17.513" Score="2" Body="&lt;p&gt;A study of the Platonic solids can be a nice visual project. A challenge to the students could be to find cartesian coordinates that correspond to the vertices of the these 5 solids. The octahedron and cube would be quite easy. Finding the vertices of the dodecahedron and icosohedron with trigonometry is challenging and fun. Another Platonic solid project would be finding radius from center to face center, edge center and vertice. Also finding dihedral angles. &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486614808&quot; rel=&quot;nofollow&quot;&gt;Coxeter's Regular Polytopes&lt;/a&gt; is an inexpensive Dover book and would be helpful for such a project.&lt;/p&gt;&#xA;" OwnerUserId="603" LastActivityDate="2014-08-20T04:53:17.513" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4216" PostTypeId="2" ParentId="4214" CreationDate="2014-08-20T04:54:04.440" Score="15" Body="&lt;p&gt;(Migrated from comments at request of OP.) The question is a bit difficult to answer without knowing how the topics (e.g., infinitude, irrational number) have been defined. Nevertheless, since the question does not broach countability and is strictly about explaining the infinitude of irrational numbers, one might proceed as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Show (or recall) that the sum and difference of any two rationals is rational.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Observe that the sum of any irrational and any rational is &lt;em&gt;irrational&lt;/em&gt;. To prove this, suppose $a$ is irrational and $b$ is rational. Now: If $a+b$ were rational, then we could write $a = (a+b) - b$, i.e., we could write $a$ as the difference of rational numbers, hence $a$ would need to be rational; a contradiction. Thus, it must be the case that $a+b$ is irrational.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Show (or recall) that $\sqrt{2}$ is irrational.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Using #2, observe that $\sqrt{2} + q$ is irrational for all $q \in \mathbb{Q}$. This gives &quot;infinitely many irrational numbers&quot; as desired. If preferable, you could even modify and use $\sqrt{2} + n$ for all $n \in \mathbb{N}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;But since this is MESE (not MSE): Ask students what they have already tried. Have they seen the proof that $\sqrt{2}$ is irrational? If so, has anyone suggested $\sqrt{p}$ is irrational for all primes $p$? Maybe someone has even gone and proved such a thing. Has someone suggested $\sqrt{n}$ is irrational for all natural numbers $n$? If so, where does such an argument break down? (And why can it be rescued by looking at, e.g., the square roots of primes?) If someone has tried the trick suggested above: Can they prove the different building blocks? Can they state anything meaningful about the sum of two irrationals? Can they solve related problems (e.g., given an irrational number and a rational number, prove there is always an irrational number strictly between them)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, if the irrationals have been defined in some other way, then your explanation may differ. For example, if one defines them as the non-repeating decimals, then can they prove $0.1234567891011\ldots$ is irrational? If they can prove &lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Champernowne_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;&lt;/strong&gt; does not repeat, then can the proof be adapted to show the same about $1.1234567891011\ldots$? What about $2.1234567891011\ldots$? And so forth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You write that your job is to help the students in this Calculus course to solve the problems, and that this particular one showed up. I am not convinced that showing them &quot;an easy way&quot; to answer this problem is the best approach with regard to helping them further down the road. Let them &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/878/262&quot;&gt;struggle&lt;/a&gt;&lt;/strong&gt; with the problem; let them &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/2399/262&quot;&gt;spend some time&lt;/a&gt;&lt;/strong&gt; with the problem; demand to see &lt;em&gt;some&lt;/em&gt; evidence that they have given the problem thought before you start to scaffold and assist them in figuring out a solution!&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-08-20T04:54:04.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4218" PostTypeId="2" ParentId="55" CreationDate="2014-08-20T07:19:49.483" Score="5" Body="&lt;p&gt;My own introduction to this topic was Alan MacDonald's early paper, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://faculty.luther.edu/~macdonal/GAGC/GAGC.html&quot; rel=&quot;noreferrer&quot;&gt;A Survey of&#xA;Geometric Algebra&#xA;and&#xA;Geometric Calculus&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Capable Advanced level Mathematics and Physics pupils could be directed towards this freely available paper and in particular to Section 3.1.6. Electromagnetism. When they see the simplification (unification) offered by the multivector approach some may realize that Geometric Calculus could, in fact, be a new paradigm in mathematics. They might then seek out undergraduate courses offering the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I believe that the two books, abbreviated LAGA and VAGC, will eventually become standard texts and that the first sight of Geometry (and Algebra), for even the youngest students, will be Geometric Algebra (GA).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Returning to the main question though...&#xA;The entry point in the curriculum could be whenever the concept of complex numbers is usually introduced in mathematics. I would avoid any usage of the word &quot;abstract&quot; as in the phrase &quot;applied abstract algebra&quot;. The whole ethos of GA is its power in the unification of physics.&lt;/p&gt;&#xA;" OwnerUserId="2025" LastActivityDate="2014-08-20T07:19:49.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4219" PostTypeId="2" ParentId="2429" CreationDate="2014-08-20T15:48:04.747" Score="5" Body="&lt;p&gt;I can speak to the books by Fraleigh and Beachy/Blair, since I've taken courses where each was the primary text. (Sadly, I got out of university teaching before I could teach undergrad algebra myself.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally enjoyed Fraleigh's approach much more than Beachy/Blair's; however, as @mini mentioned above, a nontrivial portion of the teaching happens using problems. If I recall correctly, Fraleigh's recommendations to the instructor include spending the first third of every class session with students at the board presenting their solutions to problems. Later sections of the text also refer to results which were to have been proved in problem sets. So it may feel awkward to teach using this book if your classroom format differs significantly from the recommended format.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On other matters, I found Fraleigh's prose far more readable and clear than Beachy/Blair's, and I valued the fact that his definitions were more general -- e.g. not assuming that all rings have 1, which allows Fraleigh better parallels between subgroups/subrings and normal subgroups/ideals (if all rings have 1, then ideals aren't subrings).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, Beachy/Blair is structured more as a journey to one particular big goal (Galois theory). Fraleigh is more of an exploration with some particular highlights. Since it's mostly number theorists who actually need/use Galois theory these days, I thought the &quot;exploration&quot; approach was preferable.&lt;/p&gt;&#xA;" OwnerUserId="1403" LastActivityDate="2014-08-20T15:48:04.747" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4220" PostTypeId="2" ParentId="294" CreationDate="2014-08-20T18:29:19.897" Score="4" Body="&lt;p&gt;&lt;em&gt;How to Lie with Statistics&lt;/em&gt;, by Darrell Huff. Pictures by Irving Geis. &lt;a href=&quot;http://www.amazon.com/exec/obidos/ISBN=0393310728/&quot; rel=&quot;nofollow&quot;&gt;http://www.amazon.com/exec/obidos/ISBN=0393310728/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The World of Mathematics&lt;/em&gt;.  I have Volume Four, which was edited by James R. Newman in 1956.  Most of the articles are 10 - 30 pages long -- long enough to take &quot;what if&quot; questions seriously.  There are no dry mathematical derivations.  Instead, there are articles like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Alan Turing's explanation of the Turing Test.  (Can a man pretend to be a woman?  Can a computer do better than a human at figuring out if a man is pretending to be a woman?  When you think about it, it is really quite bizarre, especially for the 1950s.)&lt;/li&gt;&#xA;&lt;li&gt;&quot;Mathematics in Warfare&quot; -- How basic numerology can help win wars.&lt;/li&gt;&#xA;&lt;li&gt;&quot;How to Hunt a Submarine&quot; -- By the folks who figured out how to win the Battle of the Atlantic in World War II.&lt;/li&gt;&#xA;&lt;li&gt;Excerpts from &quot;Flatland&quot;, Jonathan Swift, Lewis Carroll, and others.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2014-12-14T09:14:07.790" LastActivityDate="2014-12-14T09:14:07.790" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4221" PostTypeId="1" AcceptedAnswerId="4239" CreationDate="2014-08-20T23:35:35.870" Score="2" ViewCount="447" Body="&lt;p&gt;For a math circle activity on Euler Characteristic, I'd like the students to be able to triangulate some surfaces and count the number of vertices, edges, and faces of the result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a lot of ways to get a nice big physical sphere that the students can draw on... but what about other surfaces? Have any of you found a nice, preferably cheap way to do this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be clear, I am demanding that the end result be a physical object here, and am simply not interested in software that will do this for us. Thanks!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-09-01T15:53:48.697" Title="How can I demonstrate triangulations of surfaces with real hands-on objects?" Tags="|topology|manipulatives|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4222" PostTypeId="1" AcceptedAnswerId="4233" CreationDate="2014-08-20T23:52:27.693" Score="12" ViewCount="625" Body="&lt;p&gt;In the question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/737/why-do-students-have-problems-with-showing-that-somehing-is-well-defined-how-ca&quot;&gt;Why do students have problems with showing that something is well-defined? How can this be improved?&lt;/a&gt;, it was suggested that perhaps students have never seen something that is &lt;strong&gt;not&lt;/strong&gt; well-defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree with this and would like to hear what you think is a &lt;strong&gt;good starting example for a not-well-defined object&lt;/strong&gt;. I think a good starting example should require as little background knowledge as possible while avoiding the pitfalls of the following examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Too fixable&lt;/em&gt;: You could claim that &quot;the antiderivative of a function&quot; is not well-defined, since there are lots of them. But students will easily protest that the antiderivative can just be something with a +C on it, which seems to clear up the issue. This doesn't help anyone who doesn't already understand the idea.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Too obvious&lt;/em&gt;: You could say that &quot;the denominator of a rational function&quot; is not well-defined (or similar things in the link above), but the students see it immediately. This gives the wrong impression; in fact checking well-definedness carefully is critical and you can almost never see it immediately.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-08-23T19:16:36.707" Title="What is a good prototypical example of a construction that is not well-defined?" Tags="|undergraduate-education|examples|students-mistakes|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4223" PostTypeId="1" AcceptedAnswerId="4224" CreationDate="2014-08-20T23:58:30.157" Score="16" ViewCount="89754" Body="&lt;p&gt;I think this is a nontrivial question, because to avoid embarrassment or singling-out, I cannot ask students about:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;their age&lt;/li&gt;&#xA;&lt;li&gt;anything about their body (height, weight)&lt;/li&gt;&#xA;&lt;li&gt;anything involving money&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Ideally I would like some nice questions with numerical answers where we will get non-embarrassing outliers, so one student can verify that yes, he is the reason the mean of this data set is so crazy -- without feeling awkward or ashamed or really having any emotional attachment to it whatsoever.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2016-05-27T19:38:53.133" Title="What are good survey questions for statistics students to ask each other in class?" Tags="|examples|statistics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4224" PostTypeId="2" ParentId="4223" CreationDate="2014-08-21T02:05:39.063" Score="8" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If you already have the class roster, you can look at the students' names and see if you can find any &lt;strong&gt;outliers in measures associated with their names&lt;/strong&gt;. Numbers of letters, longest run of consonants, total number of consonants, largest difference between letters in the first and last names, etc. It would depend on your data set.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Number of items they are carrying&lt;/strong&gt; on their person (not counting items of clothing, but counting anything in pockets or purses, but not book bags). There may be outliers, especially within subgroups. And differences in the means of subgroups. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Get to the classroom early and note the &lt;strong&gt;time each student arrives for class&lt;/strong&gt;. You can do this manually, or with a digital camera that will timestamp pictures. As people may not want their pictures taken, you could assign an index to each student or group of students as they enter (a number on a slip of paper). All you need do is photograph the slip of paper. They can hold onto the paper, so they can be associated with the time. Or, write the index on the chalkboard and photograph that, and ask them to remember the index they are assigned. Or, completely disassociate the person from the index and just record the times to keep the discussion focused on the data rather than the people.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Where were they when they last got into a car? Find it on Google Maps, ask for directions from there to your campus, and record &lt;strong&gt;the distance Google Maps gives you&lt;/strong&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The results of a game involving some trivial-to-explain but variably difficult task&lt;/strong&gt;. For instance, the number of words they can think of in 30 seconds that begin with one letter and end with another. Have them choose the letters from a bag filled with Scrabble tiles. Or a simple script written in any scripting language (or even Excel script) that assigns the beginning and ending letter. Have them pair up so that one person just says as many words as he or she can think of and the other counts and is in charge of time. Each person has a different pair of letters, making it possible for some people to get a very difficult combination. Tweak the parameters here as desired.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Alternately, you could grab some data from the &lt;a href=&quot;http://lib.stat.cmu.edu/DASL/&quot;&gt;DASL Data and Story Library&lt;/a&gt;. :&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;You can search based on the method of analysis you're interested in working with, but this also relates to points of interest in the different sets. For instance, go to &quot;stories by method&quot; and then choose &quot;Outlier&quot; -- &lt;a href=&quot;http://lib.stat.cmu.edu/cgi-bin/dasl.cgi?query=Outlier&amp;amp;submit=Search%21&amp;amp;metaname=methods&amp;amp;sort=swishrank&quot;&gt;it will give you data stories with outliers&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Somewhat related:&lt;/strong&gt; I stumbled upon this resource when I was preparing to present Tinkerplots as a technology for visually exploring sets of data. &lt;a href=&quot;http://blog.keycurriculum.com/who-survived-the-titanic/&quot;&gt;A quick example of this use is in this blog post and video, regarding some interesting attributes of the set of passengers of the Titanic.&lt;/a&gt; The question being explored is &lt;em&gt;&quot;Who survived the TItanic?&quot;&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-21T02:31:22.437" LastActivityDate="2014-08-21T02:31:22.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4225" PostTypeId="1" CreationDate="2014-08-21T03:32:16.737" Score="4" ViewCount="233" Body="&lt;p&gt;Historically, in set theory we use two different notations to refer set theoretically same objects $\aleph_{\alpha}$ and $\omega_{\alpha}$. The folklore justification of this dual notation is that we use $\aleph_{\alpha}$ and $\omega_{\alpha}$ when we are dealing with cardinal and ordinal arithmetic respectively in order to avoid any confusion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method of using notations seems strange because one can define several structures (arithmetics) on a given set/class (in this case we are working in the proper class $Card$) and then using these different structures we can look at its objects from essentially different &lt;em&gt;point of views&lt;/em&gt;. One should note that this difference comes from the &lt;em&gt;structure&lt;/em&gt; not the &lt;em&gt;objects&lt;/em&gt; in these structural contexts.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In very few set theory texts I saw an alternative approach which uses a common notation for both ordinal and cardinal numbers and emphasizes on their different arithmetics by means of using different notations for cardinal and ordinal arithmetic. For example: $\omega_1 +\omega_0$ means ordinal sum and $\omega_1\oplus\omega_0$ means cardinal sum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example from ordinary mathematics consider three different summation operators $+$, $\oplus$, $\boxplus$ on the set of natural numbers $\mathbb{N}$. In the first approach we should use three different notations $\mathsf{n}$, $\mathfrak{n}$, $\mathcal{n}$ to refer natural numbers with respect to the context and in the second approach we just need to use different symbols $+$, $\oplus$, $\boxplus$ on $\mathbb{N}$ and show each natural number in a uniform way. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Which approach is better to teach? What would be (dis)advantages of each method in my particular example in set theory?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user230" LastActivityDate="2014-08-21T14:16:42.250" Title="On using different notations for the same objects" Tags="|notation|set-theory|history|mathematical-pedagogy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4226" PostTypeId="1" CreationDate="2014-08-21T04:08:45.860" Score="1" ViewCount="429" Body="&lt;p&gt;Many musicians believe that if one wants to become a professional pianist he/she should begin playing piano when he/she is a child.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; Is this true about mathematics? In other words, is it possible for a person without any academic math education at university level to become a professional mathematician after his mid-life (about 40 years old)? Is there any limitation in age for entering the realm of research level maths? What about legal academic limitations?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; If the answer of question 1 is negative and there is no limitation, what are good suggestions for such a person to make his/her entrance onto stage of professional mathematics easier and more successful? What are good examples of such persons?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 3:&lt;/strong&gt; Is the first paragraph about professional pianists really true?! I heard this from one of my friends who is a conductor.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="-1" LastEditorDisplayName="user230" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-08-25T15:49:39.653" Title="Is there any age limitation to become a professional mathematician?" Tags="|mathematical-pedagogy|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="4227" PostTypeId="2" ParentId="4225" CreationDate="2014-08-21T14:16:42.250" Score="2" Body="&lt;p&gt;In my experience, I use what ever notation is most simple, which doesn't get mixed up with other notation being used concurrently, and follows whichever text you choose to use for the course. If we were to not reuse notation, I think mathematics itself would collapse under the weight of it's own alphabet. One thing to always do is to define each operation and if there are other notations that can be used, then list those as well with emphasis on the one you choose to use. I strongly encourage people to avoid being pedantic with their notation, I think it distracts from our goals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, I can't think of a situation where you would add a cardinal number to an ordinal number, so using regular addition (+) for both situations should be sufficient. Another example is finite direct products and finite direct sums which in this sense the same, but in linear algebra can get confused with inner direct products as they use the same notation ($\oplus$).&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-08-21T14:16:42.250" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4228" PostTypeId="2" ParentId="4222" CreationDate="2014-08-21T15:28:19.573" Score="11" Body="&lt;p&gt;Supposedly defining a ring homomorphism $f:\mathbb Z/m \to \mathbb Z/n$ by $f(k)=k$ is a popular, traditional attempt to define a map ... initially $\mathbb Z\to \mathbb Z/n$, that may be imagined to &quot;factor through&quot; the quotient $\mathbb Z/m$, but which does not without assumptions on $m,n$. Yet, in my experience, students look at the &quot;formula&quot; for it, and cannot see the difficulty, since the formula has such a straightforward appearance. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-08-21T15:28:19.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4229" PostTypeId="2" ParentId="4222" CreationDate="2014-08-21T20:09:10.247" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/737/why-do-students-have-problems-with-showing-that-somehing-is-well-defined-how-ca&quot;&gt;The linked question&lt;/a&gt; has &lt;a href=&quot;https://matheducators.stackexchange.com/questions/737/why-do-students-have-problems-with-showing-that-somehing-is-well-defined-how-ca/742#comment2705_742&quot;&gt;a decent example&lt;/a&gt;, from mweiss: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Well-defined&quot; only becomes a meaningful concept if you have&#xA;  experience with cases in which something is not well-defined. Here is&#xA;  a simple case in which something seems entirely reasonable: Let $m,n$&#xA;  be two integers and $[m],[n]$ their equivalence classes mod $p$.&#xA;  Define $[m]^{[n]}=[m^n]$. Seems reasonable, especially because of the&#xA;  way we define the other arithmetic operations mod $p$. But as soon as&#xA;  you try to calculate particular examples you realize the definition is&#xA;  broken; different representatives of $[n]$ yield different results.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-08-21T22:06:41.913" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4231" PostTypeId="2" ParentId="4226" CreationDate="2014-08-22T07:17:49.330" Score="4" Body="&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; No, there is no limitation. In some countries, such limitations would be illegal due to anti-discrimination laws, as maths are never safety-relevant. If it's possible, depends on the person itself and cannot be answered in general on a pure age basis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; There is but one suggestion: Study math at the university of your choice and supplement it with private studies at home.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 3:&lt;/strong&gt; No idea. That question doesn't belong here.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-08-22T07:17:49.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4232" PostTypeId="2" ParentId="4221" CreationDate="2014-08-22T07:29:56.080" Score="3" Body="&lt;p&gt;Sphere: No problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Torus: Swim Ring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Double Torus: &lt;a href=&quot;http://en.wikipedia.org/wiki/Rock-climbing_equipment#Figure_eight&quot; rel=&quot;nofollow&quot;&gt;Figure Eight (rock climbing)&lt;/a&gt;, but could be to small for your purposes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$n$-Torus: Craft them yourself fronm Polystyrene and cover them with paper to draw upon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Non-orientable, bounded surfaces: Hmm, not possible, if you really want to draw triangles upon them.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-08-22T07:29:56.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4233" PostTypeId="2" ParentId="4222" CreationDate="2014-08-22T09:28:00.693" Score="10" Body="&lt;p&gt;The naive &quot;addition&quot; on rational numbers, denote it $\oplus$, where you simply add numerator with numerator and denominator with denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:&#xA;$\tfrac 1 3 \oplus \tfrac 3 4 = \tfrac 4 7$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is less obviously non-well-defined than the other examples involving fractions.&lt;/p&gt;&#xA;" OwnerUserId="39" LastEditorUserId="39" LastEditDate="2014-08-22T17:51:54.593" LastActivityDate="2014-08-22T17:51:54.593" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4234" PostTypeId="2" ParentId="4222" CreationDate="2014-08-22T17:59:01.730" Score="5" Body="&lt;p&gt;Here is a troubling one that requires very little background:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given $x \in \mathbf{R}$ and $q \in \mathbf{Q}$, define $x^q$ by:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Write q as a fraction $\frac{a}{b}$.&lt;/li&gt;&#xA;&lt;li&gt;Evaluate $\sqrt[b]{x^a}$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Then, for example, we compute something unpleasant like &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$i = \sqrt{-1} = (-1)^{1/2} = (-1)^{2/4} = \sqrt[4]{(-1)^2} = 1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or even&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$-1 = \sqrt[3]{-1} = (-1)^{1/3} = (-1)^{2/6} = \sqrt[6]{(-1)^2} = 1$&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-08-22T17:59:01.730" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4235" PostTypeId="1" CreationDate="2014-08-23T00:53:27.350" Score="11" ViewCount="220" Body="&lt;p&gt;Teaching some courses in mathematics requires drawing too many shapes or figures (e.g. Geometry, some parts of calculus, etc.) but I am not good at drawing shapes. Of course I can use slides, but I want to use only blackboard.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; What to do when a math course needs too many shapes/figures and I'm not good at drawing and also I don't want to use slides at all?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; How to improve my ability in drawing mathematical shapes? Is there any reference?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user230" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-09-13T14:05:09.793" Title="What to do when a math course needs too many shapes/figures and I'm not good at drawing?" Tags="|reference-request|classroom-management|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4237" PostTypeId="2" ParentId="4222" CreationDate="2014-08-23T01:52:39.470" Score="5" Body="&lt;p&gt;You can't get much more prototypical than Euclid. &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html&quot;&gt;Book I, proposition 1&lt;/a&gt; assumes that the two circles intersect, but there is no postulate that guarantees that they intersect. From a modern point of view, the hidden assumption is that you're working in a space like $\mathbb{R}^2$, with completeness, rather than, say, $\mathbb{Q}^2$.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-08-23T01:52:39.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4239" PostTypeId="2" ParentId="4221" CreationDate="2014-08-23T13:42:45.450" Score="0" Body="&lt;p&gt;Instead of finding a pre-built object and drawing triangles on it, we've settled on just building the objects up from scratch using &lt;a href=&quot;http://en.wikipedia.org/wiki/Zome&quot; rel=&quot;nofollow&quot;&gt;Zometools&lt;/a&gt;, which is a more engaging plan anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a small problem where when you build a torus out of these toys, it's hard to tell which &quot;faces&quot; are faces and which ones are not, so the solution is to stick your arm through the &quot;hole&quot; in the torus to make sure you don't count the faces that are not really there.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-08-23T13:42:45.450" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4240" PostTypeId="5" CreationDate="2014-08-23T16:27:37.280" Score="0" Body="&lt;p&gt;Questions related to the use of various kinds of software for learning and teaching mathematics. Such software may be involved with mathematical exploration, or mathematical representation, communicating mathematical ideas among instructors and students, or supporting writing in mathematics for the purposes of education. However, if the software is meaningfully used in mathematics education, it is not limited to the above characteristics.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:27:37.280" LastActivityDate="2014-08-23T16:27:37.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4241" PostTypeId="4" CreationDate="2014-08-23T16:27:37.280" Score="0" Body="Questions related to the use of various kinds of software for learning and teaching mathematics." OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:27:37.280" LastActivityDate="2014-08-23T16:27:37.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4242" PostTypeId="5" CreationDate="2014-08-23T16:32:31.210" Score="0" Body="&lt;p&gt;Questions about education research are related to the various social scientific research work being done in areas related to mathematics education. This includes research on student thinking, development of mathematical thinking, teacher knowledge, classrooms, school institutions, policy, curriculum development, attitudes, beliefs, identity, social interactions, culture, symbols and representation, and many other areas that help us address the phenomena surrounding mathematics education.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="39" LastEditDate="2016-01-13T01:41:55.077" LastActivityDate="2016-01-13T01:41:55.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4243" PostTypeId="4" CreationDate="2014-08-23T16:32:31.210" Score="0" Body="Questions about education research are related to the various social scientific research work being done in areas related to mathematics education. " OwnerUserId="759" LastEditorUserId="39" LastEditDate="2016-01-13T01:42:03.450" LastActivityDate="2016-01-13T01:42:03.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4244" PostTypeId="5" CreationDate="2014-08-23T16:33:31.137" Score="0" Body="&lt;p&gt;Questions about how someone learns on their own, outside of traditional classroom environments.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:33:31.137" LastActivityDate="2014-08-23T16:33:31.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4245" PostTypeId="4" CreationDate="2014-08-23T16:33:31.137" Score="0" Body="Questions about how someone learns on their his or her own, outside of traditional classroom environments." OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:33:31.137" LastActivityDate="2014-08-23T16:33:31.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4246" PostTypeId="5" CreationDate="2014-08-23T16:35:24.883" Score="0" Body="&lt;p&gt;Questions related to teaching students how to approach small scale mathematical questions that are often pursued for recreation purposes and require novel (to the participant) and compelling problem solving approaches.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="5321" LastEditDate="2016-03-03T06:33:23.943" LastActivityDate="2016-03-03T06:33:23.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4247" PostTypeId="4" CreationDate="2014-08-23T16:35:24.883" Score="0" Body="Questions related to small scale mathematical questions that are often pursued for recreation purposes and require novel (to the participant) and compelling problem solving approaches." OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:35:24.883" LastActivityDate="2014-08-23T16:35:24.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4248" PostTypeId="5" CreationDate="2014-08-23T16:37:00.107" Score="0" Body="" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:37:00.107" LastActivityDate="2014-08-23T16:37:00.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4249" PostTypeId="4" CreationDate="2014-08-23T16:37:00.107" Score="0" Body="Questions about books that can be used by instructors, administrators, or students as resources in mathematic-educational situations." OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:37:00.107" LastActivityDate="2014-08-23T16:37:00.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4250" PostTypeId="5" CreationDate="2014-08-23T16:38:18.600" Score="0" Body="" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:38:18.600" LastActivityDate="2014-08-23T16:38:18.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4251" PostTypeId="4" CreationDate="2014-08-23T16:38:18.600" Score="0" Body="Questions referring to math education related activities that take place outside of classrooms, specifically over the summer for students that are between grades or years of an educational program." OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:38:18.600" LastActivityDate="2014-08-23T16:38:18.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4252" PostTypeId="5" CreationDate="2014-08-23T16:40:59.210" Score="0" Body="&lt;p&gt;Questions about the process of preparing pre-service teachers for duties related to classroom instruction in mathematics. This may be about the details of programs, comparisons of programs, particular approaches, etc.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="39" LastEditDate="2014-09-22T20:43:54.643" LastActivityDate="2014-09-22T20:43:54.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4253" PostTypeId="4" CreationDate="2014-08-23T16:40:59.210" Score="0" Body="Questions about the process of preparing pre-service (or supporting in-service) teachers for duties related to classroom instruction in mathematics." OwnerUserId="759" LastEditorUserId="39" LastEditDate="2014-09-22T20:43:42.900" LastActivityDate="2014-09-22T20:43:42.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4254" PostTypeId="5" CreationDate="2014-08-23T16:42:02.447" Score="0" Body="" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:42:02.447" LastActivityDate="2014-08-23T16:42:02.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4255" PostTypeId="4" CreationDate="2014-08-23T16:42:02.447" Score="0" Body="Questions about the use of various handheld calculating devices and their role or effect in mathematics education." OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-08-23T16:42:02.447" LastActivityDate="2014-08-23T16:42:02.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4256" PostTypeId="2" ParentId="4222" CreationDate="2014-08-23T19:16:36.707" Score="2" Body="&lt;p&gt;The simple example I use is an operation on rational numbers, call it $\star$ where&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\frac{a}{b}\star\frac{c}{d} := \frac{\max(a,c)}{\max(b,d)}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This then motivates the need usual proof that one needs to check that $+$ and $\cdot$ are still well defined modulo an ideal.&lt;/p&gt;&#xA;" OwnerUserId="2035" LastActivityDate="2014-08-23T19:16:36.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4257" PostTypeId="2" ParentId="4226" CreationDate="2014-08-23T21:50:15.880" Score="2" Body="&lt;p&gt;Q2: Regardless of the age, the way to enter professional mathematics is to get a Ph.D. in mathematics.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q1: Mathematics has been done as a second career.  I know one case where the first career was in the Merchant Marine, and another case where the first career was in the U.S. Army.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2014-08-23T21:50:15.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4258" PostTypeId="1" CreationDate="2014-08-23T22:59:19.683" Score="5" ViewCount="81" Body="&lt;p&gt;How can a high school maths teacher find opportunities to teach online?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This could be for students from any country.&lt;/p&gt;&#xA;" OwnerUserId="2036" LastEditorDisplayName="user173" LastEditDate="2014-08-24T00:46:18.487" LastActivityDate="2014-09-13T15:28:59.183" Title="Teaching online when living outside america" Tags="|tutoring|teachers|" AnswerCount="1" CommentCount="1" ClosedDate="2015-03-05T10:04:56.627" ContentLicense="CC BY-SA 3.0" />
  <row Id="4259" PostTypeId="2" ParentId="4221" CreationDate="2014-08-24T00:41:48.220" Score="2" Body="&lt;p&gt;I recommend pretzels (large, unsalted) and bagels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you triangulate with &lt;a href=&quot;http://www.amazon.com/s/field-keywords=food+markers&quot; rel=&quot;nofollow&quot;&gt;food markers&lt;/a&gt;, you can eat the results.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-08-24T00:41:48.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4260" PostTypeId="1" CreationDate="2014-08-24T04:05:00.360" Score="3" ViewCount="230" Body="&lt;p&gt;I found some derivatives and integrals involving the absolute value functions are expressable in terms of the absolute value functions, for examples:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\dfrac{d}{dx}|x|$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=\lim\limits_{\Delta x\to 0}\dfrac{|x+\Delta x|-|x|}{\Delta x}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=\lim\limits_{\Delta x\to 0}\dfrac{(|x+\Delta x|-|x|)(|x+\Delta x|+|x|)}{\Delta x(|x+\Delta x|+|x|)}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=\lim\limits_{\Delta x\to 0}\dfrac{|x+\Delta x|^2-|x|^2}{\Delta x(|x+\Delta x|+|x|)}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=\lim\limits_{\Delta x\to 0}\dfrac{(x+\Delta x)^2-x^2}{\Delta x(|x+\Delta x|+|x|)}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=\lim\limits_{\Delta x\to 0}\dfrac{x^2+2x\Delta x+(\Delta x)^2-x^2}{\Delta x(|x+\Delta x|+|x|)}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=\lim\limits_{\Delta x\to 0}\dfrac{2x+\Delta x}{|x+\Delta x|+|x|}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=\dfrac{x}{|x|}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=\dfrac{x|x|}{|x|^2}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=\dfrac{x|x|}{x^2}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=\dfrac{|x|}{x}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\int|x|~dx$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=x|x|-\int x~d(|x|)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=x|x|-\int x\times\dfrac{|x|}{x}dx$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=x|x|-\int|x|~dx$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\therefore\int|x|~dx=\dfrac{x|x|}{2}+C$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why does current mathematics education often not to teach like these?&lt;/p&gt;&#xA;" OwnerUserId="152" LastActivityDate="2014-08-26T00:37:37.057" Title="About teaching the derivatives and integrals involving the absolute value functions in the current mathematics education" Tags="|secondary-education|calculus|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4261" PostTypeId="2" ParentId="1479" CreationDate="2014-08-24T08:06:22.890" Score="3" Body="&lt;p&gt;FWIW, I have created a website of public domain instructional materials in Mathematics and Language Arts. I began writing these materials long before the web came into existence. I sent a copy of everything to the National Science Foundation. They eventually sent me a kudos letter for my work. You can find my website, where a copy of the the NSF kudos letter is also located, by googling for &quot;public domain materials&quot; and my name: &quot;Mike Jones&quot;. (or, use the address www.public-domain-materials.com)&lt;/p&gt;&#xA;" OwnerUserId="2039" LastActivityDate="2014-08-24T08:06:22.890" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4262" PostTypeId="2" ParentId="4260" CreationDate="2014-08-24T12:04:29.900" Score="4" Body="&lt;p&gt;I do not know the full reason why these are not taught, but &lt;em&gt;one&lt;/em&gt; reason is that there are many ways to write your solutions. For example we also have:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\dfrac{d}{dx}|x| = \operatorname{sgn} (x) = \frac {1} {\operatorname{sgn} (x)}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int|x|~dx = \frac {\operatorname{sgn} (x) \cdot x^2} 2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\operatorname{sgn} (x)$ is the signum function: +1 for $x&amp;gt;0$, $-1$ for $x&amp;lt;0$. Usually, $\operatorname{sgn} (0)$ is $0$, but in the first solution in the first line above it would need to be undefined. (It could be zero or undefined in the second solution in the first line, since zero would make the expression undefined.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pesky little details like that make using the absolute value formulae somewhat problematic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I do give an exercise each year where I have my students come up with formulae for $\dfrac{d}{dx}|x|$ and $\int|x|~dx$. They rarely come up with proper ones, and I end by giving them the formulae that you proposed.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2014-08-24T12:38:30.960" LastActivityDate="2014-08-24T12:38:30.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4263" PostTypeId="2" ParentId="348" CreationDate="2014-08-24T22:37:40.820" Score="9" Body="&lt;p&gt;I will answer you from my own experience: I am a lecturer in a Maths Learning Centre at a University. We provide a drop-in centre, and also revision lectures and other resources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Students' problems with lecturers skipping the &quot;simple&quot; calculations&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students often talk to me about how frustrating it is when a lecturer skips steps, or at least when the lecturer does not indicate that they have skipped steps. This is not just the struggling students, but the ones who are very clever with most of the rest of the course. They stare at it for 20 minutes trying to figure out what happened, when what really happened was some expanding and simplifying. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have other students who got through their high school maths, or worse &lt;em&gt;didn't&lt;/em&gt; and the university let them do the course anyway (economics students are the classic example), for whom it is precisely the basics that they don't know. In particular, they have a shaky hold on the rules for algebraic manipulation of powers and fractions. Now we can work with them with the general rules, and give them exercises to do, but no matter how much you do this, they still need to see these operations in context with the new stuff. They don't know it well enough to embed it in something else, or to have the instinct of when an algebratic operation might be useful. They need to see full examples with the old stuff fully worked out so they can see where it fits with the new stuff. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even some very high achievers have this issue -- there is some particular algebraic trick they just have never happened to see, and which only appears as useful while doing something new, such as simplifying the result of a quotient rule. These students will be able to see it as algebra and transfer it to other areas. (Others might need to be told that this is a possibility.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are in danger here of thinking that algebra is the only problem, but for many students it's actually the arithmetic that's the problem! Don't forget that many of your students actually have very poor arithmetic skills, having done all their maths on a calculator from a young age. They can often do a lot of high-level maths stuff, but have very few strategies to draw on with the low-level stuff. Students often tell me they really appreciate me doing the most basic whole-number calculations because they learn tricks they had never seen before. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These ideas are not restricted to strugglers with prerequisite knowledge, but at many points throughout the course. Often we teach X in one week, then assume they have it down pat next week and skip all the parts involving X when we teach Y. But of course, they are still trying to assimilate X. The classic example is eigenvalues. Many of my students simply don't have good strategies for calculating determinants, so when they try to find the characteristic polynomial, they struggle. Showing the whole process of actually expanding it out / using row operations cements the previous learning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, on a purely practical level, the students will have to be able to do all these calculations actually by themselves at some point (ie the exam). There is a danger that if they have never seen an example of doing the whole thing including all the little bits, that they will never realise that it actually is all necessary. They will think they know how the process works, but struggle to actually do it themselves, often failing simply because of the calculations rather than the understanding. To never show the whole thing is to give a false picture of your expectations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Choosing when to do full calculations&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you have probably guessed, my general rule is to do full calculations whenever you can. Choosing examples where the numbers are easy to work with is one strategy, but you must include harder ones too at some point! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your purpose is to show the structure of the new solution, I recommend going back through the solution you have done and highlighting the overall structure, noting when you were using theory and when you were doing calculations. I find drawing boxes around the different parts of the solution really helps students to see the parts. So does actually putting headings in that say what you are doing eg: &quot;calculate determinant&quot;, &quot;solve equation&quot;. Indeed, if they learn to put in helpful signposts like this in their own working then you will find it easier to mark!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For processes that do involve a lot of calculation, these headings can allow you to skip parts of the working without it seeming like magic. The notes say what to do and the students are not staring at it wondering what you did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, if you have the ability to record your lectures, those who are a bit slower can catch up on the details later.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-08-24T22:37:40.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4264" PostTypeId="2" ParentId="348" CreationDate="2014-08-24T22:45:00.810" Score="4" Body="&lt;p&gt;I teach high-school Calculus. When I teach a new concept, I almost always give a specific example, then I choose a student at random and ask him or her how to begin solving the problem. After he/she gives one step, I write that down then ask another random student for the next step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this way I see whether or not this kind of problem is truly easy for the students. I then adjust my teaching. This means I do not have a standard pace, but then the students really do understand what is being taught.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2014-08-24T22:45:00.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4265" PostTypeId="2" ParentId="4260" CreationDate="2014-08-24T23:48:30.150" Score="1" Body="&lt;p&gt;The expressions in the question are good for minimizing the number of symbols.  But:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$|x|' = &#xA;\begin{cases}&#xA;+1,  &amp;amp; \text{if $x&amp;gt;0$} \\&#xA;-1, &amp;amp; \text{if $x&amp;lt;0$}  \\&#xA;\end{cases};&#xA;\ \ \ &#xA;\int_0^a|x|\,dx = &#xA;\begin{cases}&#xA;\ \ \ a^2\,/\,2,  &amp;amp; \text{if $a\geqslant 0$} \\&#xA;-a^2\,/\,2, &amp;amp; \text{if $a&amp;lt;0$}  \\&#xA;\end{cases}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;have the pedagogical advantages of being easier to read and quicker to understand.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-08-25T03:38:09.340" LastActivityDate="2014-08-25T03:38:09.340" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4266" PostTypeId="2" ParentId="4260" CreationDate="2014-08-25T02:04:56.700" Score="1" Body="&lt;p&gt;There is something really pretty about the formula $\int |x| dx = \frac{1}{2}x|x|+c$. It extends across $x=0$. This means we don't have to break into cases in an explicit calculation. For example,&#xA;$$ \int_{-3}^2 |x| \, dx = \frac{1}{2}x|x|\bigg{|}^2_{-3} = \frac{1}{2}[4+3]= \frac{7}{2}.$$&#xA;This is half the computation in comparison to the technique which breaks into cases. Ok, admittedly, the $x=0$ bounds are fairly easy to calculate, but, still, I think this is interesting. Moreover, I am fairly sure I can use this observation to craft nasty problems which become far more difficult if one thinks in cases. To my taste, this formula is interesting precisely because it hides cases without bringing any new unknown functions into play.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the derivative, I usually emphasize $|x| = \sqrt{x^2}$ for $x \neq 0$ so we can derive the derivative by chain-rule:&#xA;$$ \frac{d}{dx} (|x|) = \frac{d}{dx}\sqrt{x^2} = \frac{1}{2\sqrt{x^2}}(2x) = \frac{x}{|x|}.$$&#xA;The function $f(x)=x|x|$ is interesting as it is an example of a once, but not twice, differentiable function. In fact, this function is usually mentioned in the study of the Wronskian as $\{ x^2, x|x| \}$ has a identically vanishing Wronskian on $\mathbb{R}$ yet $f(x)=x^2$ and $g(x)=x|x|$ are not linearly independent functions on $\mathbb{R}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, I think these are worth talking about and they're not that hard. So, if someone was deliberately avoiding them I would hope it was just a quirk of the textbook in play etc.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-08-25T02:04:56.700" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4267" PostTypeId="2" ParentId="4260" CreationDate="2014-08-25T03:43:12.260" Score="1" Body="&lt;p&gt;I personally prefer writing $|x|$ as $\sqrt{x^2}$ and then using the chain rule:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\dfrac{d}{dx} |x|$ = $\dfrac{d}{dx} \sqrt{x^2}$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$= \frac12(x^2)^{-1/2}\cdot 2x$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ = \frac{x}{\sqrt{x^2}}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ = \frac{x}{|x|}$&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-08-25T03:43:12.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4268" PostTypeId="2" ParentId="4221" CreationDate="2014-08-25T05:13:13.397" Score="1" Body="&lt;p&gt;For spheres, balloons would work well. Everyone could have a balloon and a marker and draw their own triangulations. You can imagine the awesome squeaking noise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An advantage of this would be that people could try to draw really funky triangulations to test if the Euler characteristic actually worked. I feel starting with spheres (ie balloons) and trying lots of different triangulations would really convince people that it really works. Only then is it really worthwhile to try other shapes, in my opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to the other shapes, play-dough is a possibility, but it's probably a bit too floppy for the purpose. Taking someone else's answer, you could use &lt;em&gt;real&lt;/em&gt; dough baked into bread of various shapes, but of course not easy to give one to every student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of making them smooth, you could make them out of cubes glued together. Cubes don't take too much effort to make out of paper and tape and with enough of them the students could all make their own, triangulate them and group them based on their characteristic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has an advantage that you could actually start with a single cube and show it comes to the same characteristic as the sphere. Also, students could easily do it at home if you gave them a cube net they could print. Making it possible for students to investigate by themselves is always a good thing. (This is one reason why I prefer not to use commercially-produced products but things you can find in your home.)&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-08-25T05:20:30.667" LastActivityDate="2014-08-25T05:20:30.667" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4269" PostTypeId="2" ParentId="3848" CreationDate="2014-08-25T09:26:56.677" Score="1" Body="&lt;p&gt;It (the square root of 2) is the one and only aspect ratio that results in the same aspect ratio of the two halves when cut in half (width-wise). This is the basis of &quot;metric-size&quot; paper, in particular, its most common form: A4. Establishing this fact about aspect ratio is a trivial algebraic exercise.&lt;/p&gt;&#xA;" OwnerUserId="2039" LastActivityDate="2014-08-25T09:26:56.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4270" PostTypeId="2" ParentId="358" CreationDate="2014-08-25T10:25:27.327" Score="1" Body="&lt;p&gt;&lt;strong&gt;Why the &quot;find things that add and multiply&quot; is taught before the quadratic formula&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method of &quot;think of $p$ and $q$ so that $b=p+q$ and $c = pq$&quot; is used in the service of factorising the polynomial equation into $(x+p)(x+q) = 0$. The concept that you can factorise and then set each part equal to zero is the fundamental idea of this method, and is in fact a fundamental idea to solving many equations, not just polynomial equations. As such, it's a part of a larger more general method that the quadratic formula  is not part of. Often this section of the maths curriculum follows a section where you tell them about noticing a common factor (eg in $2x^2 + 4x = 0$ becomes $2x(x+2) = 0$), and is trying to match with this earlier idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quadratic formula is usually introduced as a &quot;last resort&quot; when all other methods have failed. Personally, I have a sense of power when I think of a way to factorise it, rather than &quot;resorting&quot; to the quadratic formula. I conjecture that many people feel that using factorisation is more &quot;righteous&quot; than using the formula, in the same way that they feel guilty asking a computer to solve an equation for them. While you can argue on the correctness of this feeling of guilt, you can't deny it's there for a lot of people, and many of these people are teaching the students. It is probably worth noting at this point that we don't teach a &quot;linear formula&quot;, that solves $ax + b =0$ by doing $x = -b/a$, because there are other more general methods that work just fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another reason is that the concept of writing a polynomial as a product of factors is an important idea even if you are not trying to solve equations. As algebraic objects, polynomials have products and factors the way that integers do, and factoring them is a way of reducing them to their &quot;lowest form&quot;. The quadratic formula is a way of solving an &lt;em&gt;equation&lt;/em&gt; and is rarely used as a way to find a factorised form. Of course, there is nothing stopping you doing so -- I do find it odd that it is rare to see someone saying that they found $x = p, q$ by the quadratic formula and therefore the factorised form is $(x-p)(x-q)$!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Factors are also a way to strengthen the connection between a polynomial and its graph. Noting that you can tell the x-intercepts of a function from its formula when it is in factored form is a powerful thing. It's cool that numbers pulled out of the structure of an equation written on the page tell you about what the graph is doing. Interestingly, completing the square also strengthens this connection, because when you rewrite $y = ax^2 + bx + c$ as $y = a(x-d)^2 + e$, then you can immediately read off the location of the vertex $(d,e)$. Unfortunatley, I find it rare that students know that completing the square allows you to do this trick. (Note that we do often tell them a bit about this with linear graphs, citing the $y=mx+c$ and the $ax + by = k$ forms of a line that tell you different information.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why not teach (synthetic) polynomial division&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I never learned synthetic division, and when I saw my South Australian friends do it when I got to Uni I found the whole thing a bit mystifying. (I may have been biased, since I already knew how to do ordinary polynomial division it seemed like a waste of time to learn another method when I had one that worked well for me, especially considering how few polynomials I really needed to divide even in my first year of a maths degree.) When I look at the first few hits of a Google search for &quot;synthetic polynomial division&quot;, the explanations there are not particularly illuminating. It is not at all obvious how synthetic division might be used to extend understanding from quadratics to higher-degree polynomials, except for dividing by linear factors until you get a quadratic, which ordinary polynomial division does too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So perhaps the question is really why is &lt;em&gt;polynomial division&lt;/em&gt; (synthetic or otherwise) not taught earlier? I wish I knew. I think it would be great if the process of dividing polynomials was taught earlier. As algebraic objects it makes sense that if you can multiply polynomials then you would want to divide them too, and also it has a nice lead-in to rational functions. As you say, it would allow extension of quadratics to other functions.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-08-25T10:25:27.327" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4272" PostTypeId="2" ParentId="4226" CreationDate="2014-08-25T15:49:39.653" Score="1" Body="&lt;p&gt;I am familiar with the situation in Russia and in the US, and your country may have different practices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your biggest impediment is that agility of one's brain, and one's ability to learn, really depends on age: it is easier to learn stuff when you are young; that's just biology of the brain. You know this, and other people know this. So while there may be no legal limitations on age, and most civilized countries would explicitly forbid age discrimination, the admission committees in most grad schools would likely screen you out and prefer younger candidates. Remember: the fame of the program depends on both what the faculty publish and on where the graduates are placed, and if they perceive that you will not be placed well five or seven years down the road when you are fifty (and everybody's thinking on the job market would be, &quot;I have plenty of 50-year old professors with three dozen publications and 20 years of teaching experience in my department, why should we bother with a 50-year old Ph.D. graduate who will retire in 10 years?&quot;), they may not take you upfront.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have not defined what you mean by &quot;professional mathematician&quot;; there's a spectrum from teaching math in high school to working full time on math research at &lt;a href=&quot;http://www.cims.nyu.edu/&quot; rel=&quot;nofollow&quot;&gt;Courant Institute&lt;/a&gt;. You would probably be able to get up to Master's if you can pay for your education out of your pocket. But even getting admitted into a Ph.D. would probably require that the department knows you in and out, and knows your dedication to learning math -- which means you will be limited to the school where you can get your Bachelor's and/or Master's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note, in Russia, there is usually little way for you to get into a Ph.D. (or rather the 3-year research-only program called &quot;candidate of science&quot; in Prussian manner) unless it is right after your first college degree, especially in mathematics where agility of the brain is the primary issue. 90% of the post-graduate students would go to the candidate program back-to-back after their basic degrees (which were 5-6 year programs called &quot;specialist&quot;, which are easily mapped to the Western Master's degree), and the remaining 10% were one or two years out of college, and would have done some work in a related area. For that candidate program, you'd have to take formal entrance exams in the subject (not unlike subject GRE), in English, and some sort of philosophy of science (unlike the American process based on application, standardized scores and recommendation letters). You will likely fail the subject exam unless you just had an intense math training in the immediately preceding years, and retained most of the math you've just learned.&lt;/p&gt;&#xA;" OwnerUserId="742" LastActivityDate="2014-08-25T15:49:39.653" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4273" PostTypeId="2" ParentId="358" CreationDate="2014-08-25T21:15:47.150" Score="0" Body="&lt;p&gt;The power of learning synthetic division is not that it speeds up being able to do divisions by (x-a) but as a conceptual and/or practical tool for evaluating a polynomial function f(x) at x = a. f(a) is the last &quot;term&quot; of the process. Often in computer science classes it is pointed out that when one wants to compute f(a) for a polynomial of high degree one does not want to compute high powers of a large number but use &quot;synthetic&quot; division. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am old enough that I learned this in a Theory of Equations course but this lovely idea is not usually taught in the linear algebra courses that replaced Theory of Equations. &lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-08-25T21:15:47.150" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4274" PostTypeId="2" ParentId="460" CreationDate="2014-08-25T23:07:18.540" Score="6" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I stopped being a high-school teacher and returned to uni to do my PhD, I had exactly this problem when I started running tutorials for first-year maths. We tutors were given a set of problems and were instructed to present solutions to them at the board. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even having been a schoolteacher, and having some skills in engaging students in whole-class presentations, it was like pulling teeth a lot of the time, and for many students it was boring. Indeed, for me it was boring. Also, I had the fear that some students were not actually understanding how to do these things themselves, since they were simply &lt;em&gt;watching&lt;/em&gt; someone else do them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This fear is confirmed now as coordinator of a Maths Learning Centre, where I see people who have been in lectures and lecture-style-tutorials, who just don't know how to start, and who have fairly basic misunderstandings that no-one has yet noticed. Also, even though these students have seen a quite high-quality presentation already, they &lt;em&gt;still&lt;/em&gt; needed to have it explained individually anyway.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I decided after a semester I would ignore my instructions and do something different...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What I did in tutorials&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I organised the students into groups of about five, and gave each group a white/black-board space to work on. I divided the front board in the room into two or three sections, brought in a rolling whiteboard with two sides, and gave them whiteboard markers to use on the windows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I asked them to work through the example problems together on the board. I told them the reason was so that they had experience trying things with someone present to give advice, because everything looks easy when someone else is doing it. I also said that it was always ok to ask the person using the board a question at any point about what they were doing. I usually directed them to have a go at a particular question first, knowing that it would be most useful for them this week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that I walked around the room listening to what they were saying and giving advice. If all the groups were stuck on the same thing, I'd tell them to skip that one and I would do it at the board in front of the whole class later. Sometimes I would notice a serious misunderstanding in a lot of people and stop them there and then to do a whole-class session at the big board.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd usually wrap up the session with a whole-class part (maybe 10-15 mins) where I went through one thing they all struggled with, or where I asked for volunteers to tell me something they learned today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How it turned out&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I certainly had a much better time teaching these tutorials than the old style! I didn't have to wait around for people to arrive to start the class, because they could get going on problems straight away, and I had more time with the individual students. I was able to learn all their names because I talked to them individually. As to the students, they seemed to enjoy themselves in the tutes, and they did seem to get a lot out of it. In whole-class discussions, they were much more responsive than in classes where I hadn't done group-work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 2008 I had about a hundred students across all my tutorials, so I made sure I did evaluations with them to see how they perceived my teaching in this style of tute. I had above 95% agreement with all the measures, with 100% agreement for &quot;This person shows enthusiasm for encouraging student learning&quot;, &quot;This person shows concern for students&quot; and &quot;All things considered, how would you rate the effectiveness of this person as a university teacher.&quot; In the free-response comments, about a half of the students mentioned the group-work as the best aspect of my teaching. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is important to mention that the remainder of the free response comments said that the best aspect were my clear explanations and the way I talked about the details of problem-solving when I gave examples. So you can't rely totally on the format to get good results -- you have to actually be good at explaining stuff and focus on the skills they need when you do!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some did say that they would like more whole-class explanations of lecture content. This comment from previous semesters is one of the reasons I started doing a whole-class session at the end of every tutorial. However I still beleive that the groupwork is more important for this type of class. In the end it's the lecturer's job to teach the content clearly and you as a tutor can't be responsible for &quot;fixing&quot; that. Plus, with this format you can actually tell what is in the students' heads and so know where to pitch your explanations anwway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Note I gave a similar answer to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/57/how-do-you-handle-a-wide-ability-range-when-delivering-a-50-min-tutorial-with-lo&quot;&gt;this question&lt;/a&gt;, but this one includes more detail on why I chose to do it in the first place and the various ways I knew it worked.] &lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-08-25T23:07:18.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4275" PostTypeId="2" ParentId="4260" CreationDate="2014-08-26T00:37:37.057" Score="0" Body="&lt;p&gt;It's just a matter of priorities. First, many or most of the students are only taking calculus as a filter/weeder course (certainly in the U.S.), so the specific content doesn't matter _at_all_. Second, even if a student will be using some elementary mathematics (basic calculus) later, it is my impression/experience that mostly the questions of such integrals and derivatives will not arise immediately.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In more substantive situations (some parts of physics, some parts of mathematical analysis) where such a question would be &quot;live&quot;, there would be a very pointed need to address very-nearby questions about &quot;generalized functions&quot;, such as Dirac's delta and its derivatives... which naturally arise in such context, ... unless one artificially constrains the situation to a point of uselessness and dysfunction. That is, at the point that one choose to go down that road, there is much more that immediately merits attention... which is not part of the traditional Calc I syllabus in the U.S. ... for better or worse.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-08-26T00:37:37.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4277" PostTypeId="2" ParentId="3945" CreationDate="2014-08-26T08:12:33.717" Score="1" Body="&lt;p&gt;For me, the key to solving the problem (and I think it is generally agreed that there is a problem with maths education) is to understand the causes. I recently did a quick and dirty bit of research about the perceived causes of the problems. Many of the respondents were teachers but I also had some others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most people responded that the biggest causes are poor teaching and negative societal attitudes towards mathematics. The fact that the language of learning and teaching in South Africa (LoLT) is not the mother tongue is also seen as big cause. I have published my findings on a &lt;a href=&quot;http://fasmedaimssec.wordpress.com/the-south-african-context/what-are-the-causes-of-problems-in-maths-education-in-south-africa/&quot; rel=&quot;nofollow&quot;&gt;blog&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware that this doesn't answer the question, but as the responses above show, there probably isn't an easy answer to the question! However, understanding the context better is surely highly important. 'The context' is enormously complex, not least because of the great differences between the best schools and the worst schools.&lt;/p&gt;&#xA;" OwnerUserId="2048" LastEditorDisplayName="user173" LastEditDate="2014-08-26T11:31:04.153" LastActivityDate="2014-08-26T11:31:04.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4278" PostTypeId="2" ParentId="484" CreationDate="2014-08-26T10:27:57.947" Score="-1" Body="&lt;p&gt;To be completely honest, sometimes I wonder why we include limits in the curriculum at all, at least at the place we do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The traditional place they are found is just before derivatives, which makes mathematical sense, since the definition of derivative is a limit. However, almost all of the students I talk to individually (several hundred a year), keep the ideas of limits and derivatives completely separate in their minds, at best treating limits as an annoying thing they did before they got to the real business of differentiating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would much prefer that differentiation was taught as a process for finding a formula for a rate of change of a function at particular points -- a new function where when you sub in the point, it tells you the rate of change. Through much observation our forefathers realised there were all these neat relationships for various functions and combinations thereof, and also that it could be used to solve all sorts of useful problems. In essence you teach differentiation straight up as something that you do to functions. I have no evidence to suggest this is the best approach, but I have much empirical evidence that this is how students already think about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not necessarily suggesting to completely remove limits and the definition of derivative from the curriculum (though it probably wouldn't hurt your non-maths majors too much to do so). What I am suggesting is to get them familiar with the mechanics of how they behave and the usefulness of them before getting into the foundations of mathematical rigour it's built on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can imagine, after you've gotten them smoothly doing derivatives, that you could have a section of the course where the explicit aim is to lay the mathematical foundation for derivatives and other concepts of function. This is where limits belong -- they belong to the general discussion of continuity and differentiability. From the raw meaning of function, what does it mean to say a function is continuous? What does it mean to say a function has a derivative? What sorts of things can you generally say about functions and their derivatives and when and where they exist? (That is, theorems like the Mean Value Theorem.) These concepts only really make sense to ask about when you have already got a good understanding of the behaviour of functions, and indeed they will probably make more sense because you can use their already-existing intuition of what ought to happen to support their learning of what &lt;em&gt;does&lt;/em&gt; happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the purpose was to actually discuss the mathematical foundations -- after having built something that is worthy of having a foundation -- then I don't see a problem using epsilons and deltas. It would seem fitting if we were going to be rigorous to really do it properly. You could choose how quickly you switched from that to the limit laws based on the level of your students, of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would of course be important that any time you did ask them to prove anything, that you were clear on what they were and were not allowed to use -- for example, the epsilon-delta definition or the limit laws; differentiating using the rules or with limits?&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-08-26T10:27:57.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4279" PostTypeId="1" AcceptedAnswerId="4282" CreationDate="2014-08-26T14:04:56.563" Score="10" ViewCount="712" Body="&lt;p&gt;In helping a family member who is studying calculus, I was asked about the meaning of the following, which is straight out of a calculus text book (Varberg and Purcell)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$v'(t) = \frac{{\rm d}\,v}{{\rm d}t}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've also seen the same thing elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I haven't done much calculus since my undergrad, but I've done a lot of logic and C.S.; so much, in fact, that I am now unable to read a formula without fretting about the type and binding of each variable. To my (perhaps hypercritical) eye this looks plain wrong: on the, left $v$ is being treated as a function. On the right, $v$ is being treated as a (dependent) real variable. To me, it seems the equation should be written as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$v'(t) = \frac{{\rm d}\,v(t)}{{\rm d}t}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\dot{v} = \frac{{\rm d}\,v}{{\rm d}t}$ ,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;depending on whether you want to treat $v$ as a function or as a variable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question: Is the text book being sloppy about notation, or is there some defensible interpretation of $v'(t) = \frac{{\rm d}\,v}{{\rm d}t}$ that I'm missing?&lt;/p&gt;&#xA;" OwnerUserId="527" LastEditorUserId="527" LastEditDate="2014-08-27T17:01:41.873" LastActivityDate="2014-08-29T23:08:20.213" Title="Is this example of Leibniz notation sloppy?" Tags="|calculus|notation|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4280" PostTypeId="1" CreationDate="2014-08-26T15:39:40.963" Score="3" ViewCount="384" Body="&lt;p&gt;I have the common misconception in my business calculus classes that the Average Rate of Change, say from $x=1$ to $x=5$, is the statistical average of the rates on the four unit intervals $1$ to $2$, $2$ to $3$, etc. So when introducing/reviewing Average Rate of Change, I'm tempted to say &quot;This is not an Average like you learn about in your stats class.&quot; Except, in reality, it is. From the perspective at the end of the course, the difference quotient is the result of finding the average value of the derivative. And the idea of the average value of a continuous function is very much like the averages they compute in a stats class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So my question is: How can I tell them not to compute a statistical average without lying to them about the relation between average rate of change and continuous statistics?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2014-08-26T15:45:19.990" LastActivityDate="2023-05-05T17:41:54.667" Title="Average Rate of Change isn't/is Statistics" Tags="|calculus|mathematical-pedagogy|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4282" PostTypeId="2" ParentId="4279" CreationDate="2014-08-27T00:45:12.500" Score="4" Body="&lt;p&gt;&lt;strong&gt;Tl;dr:&lt;/strong&gt; They are all equal and this is a matter of preference. The notations mean the same concept, each with its respective advantages and disadvantages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's notation $\dot{v}$ is commonly used in physics and will always mean you are considering a function of time, denoted by the parameter $t$. This notation has the advantage that you avoid confusion regarding with respect to what is the function being differentiated. The disadvantage is that this is one of the cases of frozen notation: don't dare writing $\dot{v}$ for differentiation with respect to anything else but $t$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibniz's notation $\frac{dv}{dt}$ is very flexible. The advantages of it are that it already tells you what is the function differentiated and with respect to what. It also adapts easily to all cases, as this is not frozen notation. The disadvantage is that there is the slight abuse you have noticed. Correctly we should write $\frac{dv(t)}{dt}$, but $\frac{dv}{dt}$ conveys the idea pretty well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lagrange's notation $v'$ is the most confusing of all, in my opinion. Is not entirely clear what you are differentiating with respect to. Sure, if you have a one variable function it works, but I've seen people write $v'$ when $v$ is a function of space and it meant differentiation with respect to time, without mentioning it. If abused it can lead to great trouble. The chain rule in this notation states that $(f(g(x))' = f'(g(x)) g'(x)$, but what if you omit the parameters? It becomes $(f \circ g)' = f' g'$, which is horrible (it may even confuse students with the product rule).&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-08-27T00:45:12.500" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4283" PostTypeId="2" ParentId="4280" CreationDate="2014-08-27T07:32:52.233" Score="6" Body="&lt;p&gt;$$\frac{1}{4}\int_1^5 f'&#xA;= \frac{1}{4}\left(&#xA;\int_1^2 f'&#xA;+ \int_2^3 f'&#xA;+ \int_3^4 f'&#xA;+ \int_4^5 f'&#xA;\right)$$&#xA;The LHS is an average rate of change.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The RHS is a statistical average of four average rates of change.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This requires some differentiability, but I would not treat this as a misconception!&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-08-27T07:32:52.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4284" PostTypeId="1" AcceptedAnswerId="4299" CreationDate="2014-08-27T23:25:17.517" Score="26" ViewCount="4253" Body="&lt;p&gt;Calculus instructors are mostly very careful to explain that $\frac{\mathrm{d}y}{\mathrm{d}x}$ is not a fraction, and multiplying both sides of an equation by $\mathrm{d}x$ is nonsense, wrong, or evil. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I really want to be able to say that &quot;$y$ is the sum of all the little pieces of $y$,&quot; i.e. that $y = \int \mathrm{d}y$. And I'd really like to say that when I integrate $\frac{\mathrm{d}y}{\mathrm{d}x} \mathrm{d}x$, it's the same as integrating $\mathrm{d}y$. And when I separate variables, I really like splitting up the $\frac{\mathrm{d}y}{\mathrm{d}x}$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This comes to the question: what is a good example of the harm that we will cause if we allow students to think $\frac{\mathrm{d}y}{\mathrm{d}x}$ is a fraction? Are we worried that students will think that $\mathrm{d}y$ and $\mathrm{d}x$ are numbers?&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="5220" LastEditDate="2018-03-01T16:58:28.187" LastActivityDate="2018-03-12T08:03:43.187" Title="Why are we so careful in saying that dy/dx is not a fraction?" Tags="|calculus|mathematical-pedagogy|" AnswerCount="7" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="4285" PostTypeId="2" ParentId="4284" CreationDate="2014-08-28T00:19:39.847" Score="24" Body="&lt;p&gt;A big issue I've seen is in understanding the multivariable chain rule. I've had many students over the years argue that&#xA;$$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}$$&#xA;implies&#xA;$$\frac{\partial f}{\partial t} = 2\frac{\partial f}{\partial t}, \text{ so } \frac{\partial f}{\partial t}=0$$&#xA;since the $\partial x$ and $\partial y$ cancel.&lt;/p&gt;&#xA;" OwnerUserId="18" LastActivityDate="2014-08-28T00:19:39.847" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4288" PostTypeId="2" ParentId="4284" CreationDate="2014-08-28T03:08:55.227" Score="21" Body="&lt;p&gt;When Leibniz invented the notation, he considered $dx$ and $dy$ to be infinitesimal numbers and $dy/dx$ to be their quotient. Non-standard analysis (NSA) has essentially vindicated that view. One has to distinguish between the quotient $dy/dx$ and its standard part, which is the derivative. In fact, Leibniz made this distinction [Blaszczyk 2012], although it was not carefully observed by later users of his notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.wisc.edu/~keisler/calc.html&quot;&gt;Keisler&lt;/a&gt; is a very nice, free freshman calc book that develops calculus from the NSA point of view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was a period ca. 1820-1965 when this stuff was not clearly understood, and it was believed, incorrectly, that limits were the only possible way to give calculus a rigorous foundation. Even during this period, scientists and engineers never stopped using infinitesimals. Freshman calc students need to be literate in the practices of science and engineering, so a disservice is done to them when the discredited belief is inculcated upon them that these practices are somehow wrong, ignorant, or logically suspect.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This comes to the question: what is a good example of the harm that we will cause if we allow students to think $\dfrac{dy}{dx}$ is a fraction? Are we worried that students will think that $dy$ and $dx$ are numbers?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is no such harm. Students should of course be told that $dx$ and $dy$ are not &lt;em&gt;real&lt;/em&gt; numbers. In old-school terminology, they're infinitesimal numbers. In the termninology of NSA, they're hyperreal numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Blaszczyk et al., &quot;Ten Misconceptions from the History of Analysis and Their Debunking,&quot; &lt;a href=&quot;http://arxiv.org/abs/1202.4153&quot;&gt;http://arxiv.org/abs/1202.4153&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-08-28T03:19:17.243" LastActivityDate="2014-08-28T03:19:17.243" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4289" PostTypeId="1" AcceptedAnswerId="4321" CreationDate="2014-08-28T07:09:55.923" Score="11" ViewCount="333" Body="&lt;p&gt;In elementary maths education literature, they distinguish multiple concepts that rational numbers are used to represent: fractions, quotients, ratios, rates, and possibly more. These words seem to be used to refer to the concept, rather than to the number itself per se. For example, when they say &quot;fractions&quot; they are referring to part-whole relationships, and the numbers themselves are rational numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most high-school and university teachers don't distinguish between all of these though. When we say &quot;fractions&quot;, we normally mean simply numbers produced via division.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I myself am not completely sure on the definitions and usages of these words in an elementary education context. For example, I don't know if it would be considered bad to say &quot;Find the rate of change of volume with time and express your answer as a fraction.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is twofold:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In the elementary maths education context, what are the definitions of these words, and how are they used?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is it important or useful to distinguish these words in this way for maths learners, or are they simply words we use to describe the concepts to ourselves in papers etc? - for example, should the word fraction always have a part-whole concept attached to it even when we use it with learners?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-08-28T08:33:08.343" LastActivityDate="2014-09-03T00:22:24.670" Title="Using terminology for the different concepts of rational number" Tags="|primary-education|fractions|terminology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4291" PostTypeId="2" ParentId="4284" CreationDate="2014-08-28T15:06:47.207" Score="1" Body="&lt;p&gt;I think the problem is understanding the line between intuition and rigor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Riemann rearrangement theorem? Intuitive operation (you can just move around terms right?) unintuitive result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar for things like $1+2+3+\cdots = -1/12$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So with derivatives, it's like to think they are &lt;em&gt;like&lt;/em&gt; fractions since $dx$ and $dy$ are &lt;em&gt;like&lt;/em&gt; infinitescimals (and per Ben Crowell's answer even that viewpoint can be rigorized), but numerous theorems, such as the chain rule, are required to vindicate this view. Perhaps show a proof of the chain rule &lt;em&gt;guided&lt;/em&gt; by our understanding of them as fractions, but remember to emphasize the technicalities it takes to underpin this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also note that the hypotheses are essential. These rules don't work when things aren't differentiable, for instance.&lt;/p&gt;&#xA;" OwnerUserId="1988" LastActivityDate="2014-08-28T15:06:47.207" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4292" PostTypeId="2" ParentId="1589" CreationDate="2014-08-28T17:59:59.380" Score="5" Body="&lt;p&gt;$\def\RR{\mathbb{R}}\def\CC{\mathbb{C}}$ Here is a proof via ODE. I imagine it's been found before, but I hadn't seen it before.&#xA;The essential idea is that, if $z: \RR \to \CC$ is an inverse function to $f$ then we should have $z'(t) = 1/f'(z(t))$ (formula for the derivative of the inverse function, in an exotic setting). Therefore, we can solve this ODE and get an inverse function, and we can evaluate the inverse function at $0$ to find a root of $f$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $f(z) = f_n z^n + \cdots + f_1 z + f_0$ be the polynomial which we wish to prove has a root. We first make a simplifying assumption:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simplifying Assumption: We may assume that there is no point $w$ where $f(w) \in \mathbb{R}$ and $f'(w)=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof: Let $w_1$, $w_2$, ..., $w_k$ be the zeroes of $f'(z)$, there are finitely many of them by the easy part of the FTA. If $f(w_i)=0$ for any $i$, we are done. If not, replace $f(z)$ by $e^{- i \alpha} f(z)$ for some $\alpha$ not equal to the arguments of any $f(w_i)$. $\square$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With this assumption, we will show &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Theorem: There is a smooth curve $t \mapsto x(t) + i  y(t)$, parametrized by $\RR$, so that $f(x(t)+i y(t))=t$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, the point $x(0)+i y(0)$ is a zero of $f$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lemma 1: There is a constant $c&amp;gt;0$ so that $|f'(z)| &amp;gt; c$ for $z \in f^{-1}(\RR)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof: Choose $R$ large enough that $n |f_n| S^{n-1} &amp;gt; \sum_{k=0}^{n-1} k |f_k| S^{k-1} + 1$ for $S&amp;gt;R$. Then $|f'(z)| &amp;gt; 1$ for $|z|&amp;gt;R$.&#xA;The set $f^{-1}(\RR) \cap \{ |z| \leq R \}$ is closed and bounded, hence compact, so $|f'(z)|$ has a minimal value $b$ on that set, and by the simplifying assumption $b&amp;gt;0$. &#xA;Take $c = \min(b,1)$. $\square$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to set up an ODE, we also need an initial value:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lemma 2: There is a point $z_0 \in \CC$ where $f(z_0) \in \RR$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof: Choose $R$ large enough that $|f_n| R^n &amp;gt; 2 \sum_{k=0}^{n-1} |f_k| R^k$. Let $f_n = |f_n| e^{i \alpha}$. &#xA;Then $\mathrm{Im} f(R e^{i (-\alpha-\pi/2)/n})&amp;lt;0$ and $\mathrm{Im} f(R e^{i (-\alpha+\pi/2)/n})&amp;gt;0$. So, by the intermediate value theorem, there is some $\theta$ with $f(R e^{i \theta}) \in \RR$. $\square$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Write $t_0$ for $f(z_0)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the ODE $z'(t) = 1/f'(z(t))$ on $\RR \times \CC$, with initial value $z(t_0) = z_0$. Then this ODE is solvable for all $t \in \RR$, and $f(z(t)) = t$ for all $t$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We first check that any solution to the ODE obeys $f(z(t))=t$. The proof is simply to differentiate the left hand side: $z'(t) f'(z(t)) = \frac{1}{f'(z(t))} f'(z(t)) = 1$ and $f(z(t_0)) = t_0$, so $f(z(t)) = t$. However, this assumes that students are comfortable differentiating complex valued functions of real arguments by the usual rules. So I'd first do the quick but questionable way and then I'd write it out: Put $f(x(t)+iy(t)) = \sum (g_k + i h_k) (x(t)+iy(t))^k$ and carefully take the derivative with respect to $t$, seeing that you get $z'(t) f'(z(t))$ at the end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, we must check that solutions to the ODE extend to all $\RR$. Suppose, for the sake of contradiction, that there is a solution on $[t_0, q)$ but not beyond $q$; a similar argument works to show the ODE is solvable for all negative time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the previous computation, $f(z(t)) = t$ for $t \in [t_0,q)$, so in particular $z(t) \in f^{-1}(\RR)$. By Lemma 1, the right hand side of the ODE is bounded above by $1/c$ for $t \in (p,q)$, so $\lim_{t \to q^{-}} z(t)$ exists; call this limit $w$.  By continuity of $f$, we have $f(w) = f \left( \lim_{t \to q^{-}} z(t) \right) = \lim_{t \to q^{-}} f(z(t)) = \lim_{t \to q^{-}} t = q$.  In particular, by our Simplifying Assumption, $f'(w) \neq 0$. So we can solve the ODE again with initial condition $z(q) = w$. This gives a solution which agrees with the previous $z$ on a neighborhood to the left of $q$ (by uniqueness of solutions to ODEs), and is defined to the right of $q$, a contradiction. &lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2017-01-20T18:30:45.297" LastActivityDate="2017-01-20T18:30:45.297" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4293" PostTypeId="2" ParentId="2457" CreationDate="2014-08-28T18:18:08.227" Score="0" Body="&lt;p&gt;I prefer giving multiple choice exams.  I can give frequent feedback to students, since the grading is easier. Below is a study the indicates no difference between multiple choice and short answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pubmed/24274234&quot; rel=&quot;nofollow&quot;&gt;http://www.ncbi.nlm.nih.gov/pubmed/24274234&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chris&lt;/p&gt;&#xA;" OwnerUserId="2058" LastActivityDate="2014-08-28T18:18:08.227" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4294" PostTypeId="2" ParentId="3945" CreationDate="2014-08-29T09:31:30.170" Score="0" Body="&lt;p&gt;I think to improve the standard of mathematics in poor countries is not about the money, because that way of thinking is creating only an excuse and feel of disempowerment. People needs to work together to improve the quality of life by motivating each other and come up with new ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a teacher, you should propose new ideas to the management, e.g.:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;ask management what can be done to improve the quality of the school and are there any possibilities of additional funds, if not, check how the existing costs can be improved to save some budget,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;to get some additional funding, school management can consider to contact the right non-profit organisations or you can use some on-line sites to raise money (e.g. &lt;a href=&quot;http://www.justgiving.com/&quot; rel=&quot;nofollow&quot;&gt;JustGiving&lt;/a&gt;, &lt;a href=&quot;http://www.thebiggive.org.uk/&quot; rel=&quot;nofollow&quot;&gt;TheBigGive&lt;/a&gt;, etc.),&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;if school can't afford normal plumbing and there are some drop holes, use the hazard warning signs (on the internet you can find some free printable hazard warning signs) - good organisation and fresh ideas with minimal cost can save people's lives (if school doesn't have printers, just use pen and paper),&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;mud schools and ghost schools can be depressing, but the most important is the outcome and what people learns,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;if school needs some work (e.g. plumbing, cleaning, small refurbishment) try to find some volunteers to help, involve the people and parents to make the quality of schools better,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a lack of textbooks can be solved by printing the materials or using some cheap mobiles, tablets or other devices which can be shared as well (&lt;a href=&quot;http://mygaming.co.za/news/gadgets/51029-cheapest-tablets-in-sa.html&quot; rel=&quot;nofollow&quot;&gt;the cheapest tablets&lt;/a&gt; are from around 700 rands, cheapest Android smartphones from 1400 rands and second-hand should be much cheaper) and you may use some free apps or math text books,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;for water supply issues, contact the right non-profit organisations and see what can be done (see: &lt;a href=&quot;http://thewaterproject.org/water-in-crisis-south-africa&quot; rel=&quot;nofollow&quot;&gt;The Water Crisis in SA&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Water_supply_and_sanitation_in_South_Africa&quot; rel=&quot;nofollow&quot;&gt;Water supply and sanitation in South Africa&lt;/a&gt;),&lt;/li&gt;&#xA;&lt;li&gt;the electricity in South Africa is available almost everywhere, but if there is lack of it or school can't afford it, try to contact the right non-profit organisations and ask them for help,&lt;/li&gt;&#xA;&lt;li&gt;some desks and other school equipment, check for non-profit charities and adverts, some people giving some stuff for free, or post the advert explaining the situation,&lt;/li&gt;&#xA;&lt;li&gt;if school really can't do anything to improve the quality, consider to relocate to better area or change your school.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2065" LastActivityDate="2014-08-29T09:31:30.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4295" PostTypeId="1" CreationDate="2014-08-29T11:11:45.567" Score="6" ViewCount="430" Body="&lt;p&gt;Let me start by saying that there isn't necessarily a very good selection of answers to this question. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am teaching some very basic mathematics (see below for module content) to a group of Computing Students who will not fail this class. I am trying to teach the material as well as I can. For example, all of our arithmetic of fractions will be born out of the definition that $\frac{1}{n}$ is a number that when multiplied by $n\neq 0$ gives one... and one is the number that when multiplied by any number gives the same number: $1\times m=m$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I am happy with that side of things. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;I am just wondering is there any point in adding in applications to&#xA;computing (specifically computing)? Are there any good examples?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I do not have a computing background and am happy explaining to them why they should be adept at all of this stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three good examples I have are units of data for conversion of units, a crude way of drawing lines on a pixel screen for coordinate geometry and Moore's Law for curve fitting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Module Outline&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Fundamentals of Arithmetic with Applications ---&#xA;The arithmetic of fractions. Decimal notation and calculations. Instruction on how to use a calculator. Ratio and proportion. Percentages. Tax calculations, simple and compound interest. Mensuration to include problems involving basic trigonometry. Approximation, error estimation, absolute, relative and relative percentage error. The calculation of statistical measures of location and dispersion to include arithmetic mean, median, mode, range, quartiles and standard deviation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basic Algebra ---&#xA;The laws of algebra expressed literally and illustrated both numerically and geometrically. Algebraic manipulation and simplification to include the factorisation of reducible quadratics. Transposition of formulae. Function notation with particular emphasis on functions of one variable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indices and Logarithms ---&#xA;Indices with a discussion of scientific notation and orders of magnitude. Conversion of units. Logarithms and their use in the solution of indicial (exponential) equations. Discussion of the number e and natural logarithms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Graphs ---&#xA;Graphs of quantities which are in direction proportion and indirect proportion. Graphs of simple linear, exponential and logarithmic functions. Reduction of non-linear relations to linear form to allow for the estimation of parameters.&lt;/p&gt;&#xA;" OwnerUserId="584" LastEditorUserId="584" LastEditDate="2014-09-02T10:23:50.430" LastActivityDate="2015-01-29T19:25:16.957" Title="Examples of Applications of Basic Mathematics to Computing" Tags="|undergraduate-education|examples|concept-motivation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4298" PostTypeId="2" ParentId="4280" CreationDate="2014-08-29T14:08:44.313" Score="4" Body="&lt;p&gt;From the point of view of an outsider like me, what the OP describes is the following: there is a function $f()$ and we define the Average Rate of Change of this function between points $a&amp;lt;b$ as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ARC_f = \frac {f(b) - f(a)}{b-a} \tag{1}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the OP considers as an example the interval $[1,5]$, for which we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ARC_f(1,5) = \frac {f(5) - f(1)}{5-1} = \frac 14 [f(5) - f(1)]\tag{2}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He then speaks of the &quot;misconception&quot; that students believe that they can recover $ARC_f(1,5)$ by calculating the arithmetic average of the Average Rates of Change in the intermediate unit-length intervals, namely&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ARC_f(1,5) = ?? \frac 14 \left(\frac {f(2) - f(1)}{2-1}+\frac {f(3) - f(2)}{3-2}+\frac {f(4) - f(3)}{4-3}+\frac {f(5) - f(4)}{5-4}\right) \tag{3}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is obvious that the right-hand-side of $(3)$ is identical to the right-hand side of $(2)$, and so there is no misconception at all, the students are right, &lt;em&gt;as long as&lt;/em&gt; we are talking about an interval that can be exactly decomposed into unit-length intervals, and that we take the arithmetic average of the ARCs related to these unit-length intervals.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rigor? I am more than certain that all involved have in their minds a function that is defined everywhere in $[a,b]$, and any other regularity condition that might be needed -after all, this is &lt;em&gt;business&lt;/em&gt; calculus, as the OP informs us.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2014-08-29T14:08:44.313" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4299" PostTypeId="2" ParentId="4284" CreationDate="2014-08-29T16:39:46.817" Score="12" Body="&lt;p&gt;If $y$ is a highly oscillating function of $x$, it can become quite unclear exactly what is going on with domains of integration if one tries to write out in precise terms the kind of &quot;integration identity&quot; you would like to express (in terms of definite integration).  My high school physics teacher used exactly the kind of reasoning your are suggesting (with integrating $(dy/dx)dx = dy$ to get $y$) as a way of suggesting that the Fundamental Theorem of Calculus is almost a triviality, or a matter of defining notation (without carefully thinking through why the notation is well-posed, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since we know that the Fundamental Theorem of Calculus is a real theorem, the danger of putting too much emphasis on the fraction notation is that it deludes students into thinking that FTC has no content, or is some kind of triviality. For most students the goal can't be to make them appreciate the subtleties of definitions and proofs of theorems, but it would be good to at least make sure they understand that the fractional formalism gives &quot;right answers&quot; because it is well-designed notation masking the content of some real theorems (a fact which is hard for them to accept if they think the derivatives literally are fractions). That isn't to say one should completely divorce the intuition behind the notation from the properties one wants it to satisfy, of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, the fractional notation &lt;em&gt;does&lt;/em&gt; work really well (for Chain Rule, u-substitution, derivative of inverse functions, etc.), so I wouldn't belabor the point after making one attempt with an example of non-invertible $y$ that the possibility to compute &quot;as if&quot; $y$ were a variable in its own right is an amazing feature of Leibniz' notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, I think the situation with the multivariable chain rule mentioned in one of the other answers is a very good illustration of why too much blind faith in the fractional notation without carefully thinking through why it is working will lead students into traps when they move on to multivariable calculus (and the reality is that most students cannot be expected to think very carefully about the notation when they haven't mastered or been taught the precise definitions). &lt;/p&gt;&#xA;" OwnerUserId="2069" LastActivityDate="2014-08-29T16:39:46.817" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4300" PostTypeId="1" CreationDate="2014-08-29T20:11:17.193" Score="11" ViewCount="479" Body="&lt;p&gt;I'm teaching undergraduate group theory [again] this term. I've been increasingly dissatisfied with the approach in various books (I've used Fraleigh and looked at others). They're all reasonable books, but I always feel as though much of the material is unmotivated and kind of introduced as &quot;well, here's the next topic&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I started this year by taking some material from the start of &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/088385757X&quot;&gt;Visual Group Theory&lt;/a&gt;, showing how object symmetries lead naturally to the definition of a group via arrows in Cayley diagrams. While I like that book, in the end I feel as though it concentrates too much on the visual, stretching it beyond its natural usefulness, and doesn't present enough theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Over the past couple of days, I've been thinking about how to continue with the class. The idea of groups arising from symmetries leads naturally to the idea of group actions. So I have in my mind the following sequence of topics and arguments, which constructs most of the basic results using group actions as the basic concept; I'd like comments (mathematic, pedagogical, stylistic, or whatever):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Introduce group isomorphisms via Cayley diagrams. Discuss homomorphisms and basic properties of kernel and image.&lt;/li&gt;&#xA;&lt;li&gt;Introduce group actions as natural extensions of symmetries, prove Cayley embedding theorem, correspondence of actions and homomorphisms $G\to \Omega_X$.&lt;/li&gt;&#xA;&lt;li&gt;Discuss conjugation action of $G$ on itself; note that conjugation takes subgroups to [isomorphic] subgroups, note that subgroups that are taken to themselves by every element of $G$ should be worth study; those are [defined to be] normal subgroups.&lt;/li&gt;&#xA;&lt;li&gt;Prove that $gHg^{-1} = H \Leftrightarrow gH = Hg$&lt;/li&gt;&#xA;&lt;li&gt;Note that kernels are example of normal subgroups.&lt;/li&gt;&#xA;&lt;li&gt;Prove that all normal subgroups can be realized as kernels as follows: Given $K\triangleleft G$, we want a map $\varphi$ from $G$ to some other group whose kernel is $K$. That means that if $\varphi(g_1) = \varphi(g_2)$, then $g_1^{-1}g_2\in K$ so that $g_2 = g_1k$. So all elements of the set $gK$ must be taken to the same element of the target group. So lets define the codomain of $\varphi$ to be $\{gK\ |\ g\in G\}$. Then show that there is a group structure on this set exactly when $K$ is normal, and that you get a homomorphism. (Perhaps this all sounds obvious, but to me it completely reverses the normal order of events, which is to introduce normal subgroups as those subgroups whose cosets form a group --- not necessarily an obvious thing to want).&lt;/li&gt;&#xA;&lt;li&gt;Discuss quotient groups using the above.&lt;/li&gt;&#xA;&lt;li&gt;Use the conjugation action to prove the conjugacy class equation.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;At this point I think the students will have seen most of the basic results, assuming that Lagrange and Cauchy and some portion of the abelian group structure theorem are part of the above discussions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: what have I done so far?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Develop diagrams of symmetries of a few standard objects --- a rectangle, an equilateral triangle, and a tetrahedron --- using a minimal set of generators. Here the emphasis is on the symmetry space of the object, not on the operations that induce the rigid motions.&lt;/li&gt;&#xA;&lt;li&gt;Point out that while a tetrahedron, a dodecagonal pyramid, and a hexagonal plate each have 12 symmetries, the &quot;relations&quot; between the symmetries are not the same in each case. So for example the tetrahedron has 7 axes of symmetry, the pyramid one, and the plate 7, but the orders of the symmetries are different in the various cases.&lt;/li&gt;&#xA;&lt;li&gt;Conclude that to understand symmetries, it's not enough to see how many there are - we have to understand their relationship. Use that to motivate switching focus from the vertices of the graph (the orientations of the objects in space) to the operations themselves; regard the operations as elements, understand what properties they have, and use that to define a group.&lt;/li&gt;&#xA;&lt;li&gt;The next section, which I haven't taught yet, is just for basic familiarity: notion of subgroup (using standard arguments --- it's just too hard and not informative to do this visually), extended exploration of cyclic groups, extended exploration of permutation groups.&#xA;After all of that, I hope to be ready for the material in the first part of this post.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="174" LastEditorUserId="174" LastEditDate="2014-08-29T21:36:33.277" LastActivityDate="2014-08-29T22:33:38.460" Title="Group theory via actions" Tags="|undergraduate-education|mathematical-pedagogy|abstract-algebra|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="4301" PostTypeId="1" CreationDate="2014-08-29T22:29:23.227" Score="13" ViewCount="307" Body="&lt;p&gt;I am teaching an independent-reading course from Simon's &quot;Representations of finite and compact groups&quot;.  I chose this book based on fond memories from a previous reading course in which I had participated as assistant, but then discovered that it has no exercises.  At least for the early material, I'm not finding it easy to create relevant exercises myself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What books, or online resources for other courses, have relevant exercises for this book?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously it's best if they are specifically intended as companions to Simon's book, but I am interested in exercises for other first courses in representation theory as well.  The book's point of view is intended to lead up to the representation theory of compact Lie groups.  More specifically, the table of contents is available at &lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=GSM-10&quot;&gt;http://www.ams.org/bookstore-getitem/item=GSM-10&lt;/a&gt;.  (The first 3 chapters listed there are spurious, and appear to be a typo.)&lt;/p&gt;&#xA;" OwnerUserId="2070" LastEditorDisplayName="user173" LastEditDate="2014-08-30T08:40:45.470" LastActivityDate="2014-08-30T08:40:45.470" Title="Exercises to go with Simon's &quot;Representations of finite and compact groups&quot;" Tags="|reference-request|exercises|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4302" PostTypeId="2" ParentId="4300" CreationDate="2014-08-29T22:33:38.460" Score="3" Body="&lt;p&gt;I have only ever skimmed it, and it may suffer from the same sort of over-specialisation as Carter's book, but I have always liked the idea of teaching a course from (if only to learn the specialised material myself!) Joyner's &quot;Adventures in group theory&quot; (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0801890136&quot; rel=&quot;nofollow&quot;&gt;http://www.amazon.com/Adventures-Group-Theory-Merlins-Mathematical/dp/0801890136&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="2070" LastActivityDate="2014-08-29T22:33:38.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4303" PostTypeId="2" ParentId="298" CreationDate="2014-08-29T22:48:45.263" Score="1" Body="&lt;p&gt;The other answers seem mostly to be about giving this sort of question in a proof-based course, but, in the spirit of your question (which seemed to be about more introductory courses), I have used this methodology with (I think) great success in Business Calculus, Calculus for Engineers, and Differential Equations (for Engineers).  Basically, whenever I am preparing an exam question on which I find myself thinking &quot;they'll all make the following important mistake&quot;, I convert it into a question of the form:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Your friend did the following ….&#xA;(30%) What mistake did your friend make?&#xA;(70%) What is the correct solution?&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;I haven't done any formal testing to see what impact it has on grades or, more importantly, understanding, but at the very least it forces them to recognise the mistake (rather than just making it, and getting no chance to remedy it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another, perhaps more pointed, tactic, which was suggested to me by a colleague, is to give students the chance to correct &lt;em&gt;their own&lt;/em&gt; mistakes (submitted a week after an exam as part of that week's homework)—but to require them to explain what they did wrong and why, rather than just to give a correct answer.  I like this in principle, but found it extremely hard to judge fairly whether something was a 'real' explanation.&lt;/p&gt;&#xA;" OwnerUserId="2070" LastActivityDate="2014-08-29T22:48:45.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4304" PostTypeId="2" ParentId="1598" CreationDate="2014-08-29T22:57:44.927" Score="5" Body="&lt;p&gt;When I am teaching a course for the second (or third or fourth or …) time, I pick problems that I know that students will get wrong, or wrong approaches that I know that they will take, and discuss them in class.  After some student feedback indicated that this was confusing, I introduced the tactic of switching to red when I would write an intentionally wrong statement on the board.  I also say that I will be doing this and that it will be signalled by the change to red, and I try to give a brief explanation of why I think that it is important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I enjoy this approach, and wish in retrospect that my own teachers had implemented something like it—I believe that a trained mathematician's first two questions about any new definition are &quot;What is an example?&quot; and &quot;What is a non-example?&quot;, and suspect that any scientist will ask similar questions.  However, feedback about it, from students at all undergraduate levels, has ranged from neutral (no comment) to viciously negative.  Perhaps the most common comment is &quot;We don't want to learn the &lt;em&gt;wrong&lt;/em&gt; way to do something.&quot;  I do not remember that I have ever received a positive comment specifically mentioning this method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that this feedback is based on incorrect perceptions of what a mathematics class should be, and further that the long-term benefit will outweigh the negative student perceptions, so I have continued to implement the method.  However, it is maybe worth knowing in advance the extent of the push-back that you might receive!  (I should mention that I have not taught an undergraduate course targeted at math majors since I implemented this policy, so it may be that I am just seeing the perspective of non-majors on this approach.)&lt;/p&gt;&#xA;" OwnerUserId="2070" LastActivityDate="2014-08-29T22:57:44.927" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4305" PostTypeId="2" ParentId="4279" CreationDate="2014-08-29T23:08:20.213" Score="5" Body="&lt;p&gt;I disagree with your descriptions of the meaning of the various uses of $v$; in fact, I would flip them!  I &lt;em&gt;think&lt;/em&gt; that what I say below describes the common mathematical perspective, but it could be that I am wrong; for example, Mark Fantini above (&lt;a href=&quot;https://matheducators.stackexchange.com/a/4282/2070&quot;&gt;https://matheducators.stackexchange.com/a/4282/2070&lt;/a&gt;) seems to disagree.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematical convention (which differs from some computer-science conventions), a 'bare' name $v$ refers to a function, whereas an 'argumented' name $v(t)$ refers to its value.  It is very, very common to abuse this, and so to refer to &quot;the function $\sin(x)$&quot; rather than &quot;the function sin&quot; (which looks terrible, or perhaps theological) or &quot;the function $x \mapsto \sin(x)$&quot;, especially in contexts where the function has no name (like &quot;the function $x^2$&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, I would argue that combining convention with precision (which is not necessarily desireable!) would dictate writing $v' = \frac{\text dv}{\text dt}$ (or $\dot v$ if you prefer it in place of $v'$), or, if you really want, the cumbersome and fairly ugly notation $v'(t) = \frac{\text dv}{\text dt}(t)$.  (Another common approach to indicating a value to be plugged into a derivative is to write something like $\left.\frac{\text dv}{\text dt}\right|_{t = 7}$ for $v'(7)$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find an approach that rarely mentions explicit arguments particularly valuable in a differential-equations course, where, for example, we might have a height $y$ depending on a position $x$, which in turn depends on a time $t$—in which context it is much easier to get to the bottom of the physical meaning if one writes just $\frac{\text dy}{\text dx}$ or $\frac{\text dy}{\text dt}$ rather than, say, $\frac{\text dy(x)}{\text dx}$ or $\frac{\text dy(x(t))}{\text dt}$.&lt;/p&gt;&#xA;" OwnerUserId="2070" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-08-29T23:08:20.213" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4306" PostTypeId="2" ParentId="348" CreationDate="2014-08-29T23:25:16.177" Score="3" Body="&lt;p&gt;When I want to present a problem (&quot;if a man can row so fast and walk so fast, then what is the quickest path from A to B?&quot;) in a lower-level math class that is important, but that requires detailed calculation, I generally display a set of slides for it, &lt;em&gt;which I then make available on the class web page&lt;/em&gt;.  (This is similar to, but I think also meaningfully different from, Matt F. (&lt;a href=&quot;https://matheducators.stackexchange.com/a/881/2070&quot;&gt;https://matheducators.stackexchange.com/a/881/2070&lt;/a&gt;)'s answer.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This allows me to present every detail, but removes the problems of requiring the students to wait for me to write the details, and of requiring me to wait for them to copy those details.  It also allows for limited interactivity, because I can pause at crucial steps and allow students to fill in details.  (It would be fun to have 'choose-your-own-adventure' slides to see the result of expected common mistakes!)&lt;/p&gt;&#xA;" OwnerUserId="2070" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-08-29T23:25:16.177" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4307" PostTypeId="2" ParentId="4284" CreationDate="2014-08-29T23:37:19.733" Score="0" Body="&lt;p&gt;I am not sure that it has the same rigorous foundation as Santiago Canez's &lt;a href=&quot;https://matheducators.stackexchange.com/a/4285&quot;&gt;answer&lt;/a&gt;, but I think that, if one wanted to argue for the mathematical legitimacy of the notation $\frac{\text dy}{\text dx}$, then one could view it (pointwise) as a quotient of elements of a cotangent space $T^*$ that is 1-dimensional, for which therefore the scalar multiplication $\mathbb R \times (T^* \setminus \{0\}) \to T^*$ may be 'inverted' to give a division $T^* \times (T^* \setminus \{0\}) \to \mathbb R$.  In this sense, it seems to me not so different from the common device of writing $Q - P$ for the &lt;em&gt;vector&lt;/em&gt; $\vec{PQ}$ (&lt;em&gt;i.e.&lt;/em&gt;, viewing Euclidean space as an affine space under $\mathbb R^n$).  Of course, such a viewpoint should never be explicitly mentioned in the calculus classroom; but I remember that it was actually advanced in a Monthly article that, unfortunately, I can no longer find.&lt;/p&gt;&#xA;" OwnerUserId="2070" LastEditorUserId="2070" LastEditDate="2018-03-10T23:20:44.533" LastActivityDate="2018-03-10T23:20:44.533" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4309" PostTypeId="1" AcceptedAnswerId="4316" CreationDate="2014-08-31T21:32:08.120" Score="11" ViewCount="610" Body="&lt;p&gt;I've begun a &quot;Question of the Day&quot; (QOTD) contest in my grade 8 class where I post a problem solving question on the wall each day and students are challenged to answer it in their own time. I need more ideas for good questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question: Where (either online or in print) can I go to find a long list of short and simple problem solving questions for my students?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are my criteria for QOTD questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Non-routine questions for grade 7-9 students, most often based upon content in the syllabus. Questions can range widely from easy to difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;Very important&lt;/strong&gt;: Questions should be &lt;strong&gt;short&lt;/strong&gt; and &lt;strong&gt;easily understood&lt;/strong&gt; so that they fit on an A4 paper with large font and I don't have to spend time explaining what the question means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Questions should have clear, concise solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some examples of questions I've given in the past that have worked reasonably well:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;What is the measure of an interior angle of a regular decagon?&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;How many pairs of parallel edges does a rectangular prism have?&lt;/em&gt; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;If I flip a coin 3 times in a row, what is the probability of getting heads all 3 times?&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Find the next number in the pattern: $1, 1, 2, 3, 5, 8, 13, 21, ...$&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="867" LastActivityDate="2022-10-18T01:53:00.797" Title="Grade 8 &quot;Question of the Day&quot; Ideas" Tags="|secondary-education|problem-design|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4310" PostTypeId="2" ParentId="1589" CreationDate="2014-09-01T01:51:14.967" Score="6" Body="&lt;p&gt;In this &lt;a href=&quot;https://www.youtube.com/watch?v=shEk8sz1oOw&amp;amp;feature=youtu.be&amp;amp;t=9m32s&quot;&gt;video&lt;/a&gt;, David Eisenbud (a math prof at Berkeley) explains a proof of the fundamental theorem of algebra that is very enlightening / beautiful, and almost makes the result seem obvious.&lt;/p&gt;&#xA;" OwnerUserId="2049" LastActivityDate="2014-09-01T01:51:14.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4312" PostTypeId="2" ParentId="4309" CreationDate="2014-09-01T13:25:34.333" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the most number of confined regions you can form by drawing an equilateral triangle&#xA;  on top of a square?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Ga9Dw.jpg&quot; alt=&quot;SqTri&quot;&gt;&#xA;&lt;hr /&gt;&#xA;One can concoct many such problems, and some can be quite difficult, especially in 3D.&#xA;&quot;Confined region&quot; might need some explication.&#xA;You could change to count the number of triangular regions, the number of edge-edge points of intersection, etc.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-09-01T13:25:34.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4313" PostTypeId="2" ParentId="4221" CreationDate="2014-09-01T15:53:48.697" Score="4" Body="&lt;p&gt;I have used &lt;em&gt;&lt;a href=&quot;http://www.polydron.co.uk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;polydrons&lt;/a&gt;&lt;/em&gt; with 5th-grade students&#xA;through to college students:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/fXJnY.jpg&quot; alt=&quot;stellated&quot;&gt;&#xA;&lt;hr /&gt;&#xA;A store in Massachusetts used to sell them,&#xA;but recently I've had to purchase&#xA;them from England.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-09-01T15:53:48.697" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4315" PostTypeId="2" ParentId="4309" CreationDate="2014-09-02T10:42:17.573" Score="4" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;This website has a host of great problems of various levels of difficulty, and it's actually called &amp;quot;problem of the week&amp;quot;. Note that many of them are higher than the level of your students, but many are ok. &lt;a href=&quot;https://web.archive.org/web/20171228030032/http://mathforum.org/wagon/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Henry Dudeney's books are very cute and are available on project Gutenberg. They were published in the 19-teens and have a certain charm about them. We have particularly enjoyed the Canterbury Puzzles at our puzzle club here at Adelaide Uni: &lt;a href=&quot;http://www.gutenberg.org/files/16713/16713-h/16713-h.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.gutenberg.org/files/16713/16713-h/16713-h.htm&lt;/a&gt; and &lt;a href=&quot;http://www.gutenberg.org/files/27635/27635-h/27635-h.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.gutenberg.org/files/27635/27635-h/27635-h.htm&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Mathematics Task Centre has activities that are actually &amp;quot;tip of the iceberg&amp;quot; problems that have nice closed problems to begin but a lot of extension after. They can often be adapted to puzzles. &lt;a href=&quot;http://mathematicscentre.com/taskcentre/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://mathematicscentre.com/taskcentre/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;On top of these, many things that come up in the normal curriculum can turn easily into problems that are quite interesting. The advantage of these is that they can create several puzzles along the same theme. Some examples:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Draw the diagonals of some a regular hexagon or other shape -- how many triangles are formed? how many right-angled triangles are formed?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Construct a strange-looking shape out of bits of various shapes and ask them to find the area and perimeter. Good ones are those that require you to take away bits (eg a square with triangles removed).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give them a property of numbers and ask them how many numbers from 1 to 100 (or 1 to 1000) have the propery. How many numbers are there such that they are both a square and triangular number? How many have digits that sum to 6? How many are more than the sum of their factors? How many are multiples of 7 but not 8? How many can be written as the sum of two consecutive numbers? This can lead into all sorts of interesting discussion about numbers.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give them a long calculation that will simplify if you do some rearranging first.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One final comment: avoid questions that are simply annoying. There is nothing worse than seeing the answer and groaning because it's a bad pun, or thinking &amp;quot;I suppose that makes sense, but where the hell did that inspiration come from?&amp;quot; You don't want to give the impression that problems can only be done with divine inspiration or are about guessing the mind of the problem-poser. Examples of such problems are &amp;quot;what's the next number in the sequence?&amp;quot; problems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="6053" LastEditDate="2022-10-16T20:16:24.237" LastActivityDate="2022-10-16T20:16:24.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4316" PostTypeId="2" ParentId="4309" CreationDate="2014-09-02T14:11:41.763" Score="7" Body="&lt;p&gt;A decent source of questions is the AMC 8 math competitions that are held yearly. The competition consists of 25 questions in roughly increasing order of difficulty. There are also 10th and 12th grade versions of the competition (titled AMC 10 and AMC 12, respectively). Here is a link to &lt;a href=&quot;http://www.artofproblemsolving.com/Wiki/index.php/AMC_8_Problems_and_Solutions&quot;&gt;aops&lt;/a&gt; which presents the questions and solutions by year.&lt;/p&gt;&#xA;" OwnerUserId="1647" LastActivityDate="2014-09-02T14:11:41.763" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4317" PostTypeId="1" CreationDate="2014-09-02T16:18:55.940" Score="7" ViewCount="148" Body="&lt;p&gt;Others have already &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1266/students-using-ambiguous-notation&quot;&gt;here pointed out&lt;/a&gt; that students can struggle with notation in mathematics. I can often think that the lack of proper notation gets in the way of solving a problem correctly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, I would like a online &lt;em&gt;free&lt;/em&gt; reference/resource about proper mathematics notation that I can direct my students to. I appreciate any suggestions.&lt;/p&gt;&#xA;" OwnerUserId="53" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-09-02T19:34:05.807" Title="Resource about notation for students" Tags="|reference-request|notation|online-instruction|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4318" PostTypeId="2" ParentId="4289" CreationDate="2014-09-02T19:16:59.197" Score="3" Body="&lt;p&gt;I cannot imagine that there would ever be a sufficient consensus on the meanings of the various words. And even if many teachers agree on the &quot;correct&quot; meaning and usage of every fraction-related word, students of such teachers should be able to cope with teachers and students with different views &amp;mdash; including the chance of no view at all. And there is, of course, the additional problem that the verbiage does not have exact counterparts in other languages. This is to say that if such distinction is to be made, I doubt it can be meaningfully made beyond very elementary math education. Everyone will eventually be exposed to sloppy language, and I see no reason why this should not start in school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there is a point to making some distinctions, though. For example, pupils should understand that cutting a pie in six equal pieces and eating two of them is a different action than cutting it in three equal pieces and eating one, although $\frac26=\frac13$. (I see this so that the fraction stands for a number, not the process behind it.) I would find it beneficial that pupils are given suitable words to express this, although I'm not sure if this is too confusing to too many. The gain from this distinction is not a huge one: after all, both action lead to the same amount of pie eaten, which is reflected by the equality of the two fractions. Instead of making the distinction by giving exact definitions and sticking to them, it might be better (also for the sake of developing one's skill in explaining and arguing) to discuss this issue in free words.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As summary, let me answer your questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm not a native speaker of English, so I have no idea.&#xA;(This may actually be true of a large percentage of the users of this site.)&#xA;I do not recall any such distinction made in my elementary education (in Finnish).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think it can be helpful to try to use maximally consistent terminology, but pupils should eventually be able to communicate these concepts with the outside world, where the usage of these words is sloppy.&#xA;I find it much more important that a pupil can explain in their own words something similar to my pie example.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2014-09-02T19:16:59.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4319" PostTypeId="2" ParentId="4317" CreationDate="2014-09-02T19:34:05.807" Score="2" Body="&lt;p&gt;I am not aware of the kind of notation reference manual you ask for.&#xA;Although such a thing could be helpful, I doubt that alone will solve the problem.&#xA;Something that might help is &lt;em&gt;active discussion of notation with the students&lt;/em&gt;.&#xA;I do not recall seeing a teacher engage in such discussion in all my years of taking math classes (including everything from elementary school to grad school).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I doubt that it would be of no benefit at all to occasionally take a piece of notation and discuss in which possible ways it could be interpreted.&#xA;Does the notation serve its purpose if the class can easily come up with several interpretations?&#xA;Do students recognize bad notation when they see it?&#xA;(I have understood that recognizing words and non-words is a fairly good test of language proficiency; why not apply this test to mathematics?)&#xA;If students are somehow forced to try to interpret their own (or their friends') notation and actively compare what they are trying to say with what they have written down, I believe something would go across.&#xA;A notation guidebook might &lt;em&gt;support&lt;/em&gt; this process well.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2014-09-02T19:34:05.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4321" PostTypeId="2" ParentId="4289" CreationDate="2014-09-03T00:22:24.670" Score="10" Body="&lt;p&gt;You can find a discussion of these considerations in:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Chapin, S., &amp;amp; Johnson, A. (2006). &lt;em&gt;Math matters: Grade K-8 understanding the math you teach&lt;/em&gt;. Sausalito: Math Solution Publications.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Specifically, pp. 99 - 131, Chapter 5, Fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me just quote from &lt;strong&gt;Teaching Fractions&lt;/strong&gt; on p. 131:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fractional numbers are a rich part of mathematics. However, many students find them difficult to understand. To help students learn about and use fractions, it is important to introduce the multiple meanings of fraction and to emphasize sense making in all mathematical activities. Instruction in the early grades should focus on the part-whole interpretation of fraction but include all other interpretations as well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To give you an idea of the multiple meanings, I will also quote from the start of the chapter on pp. 99-100:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Interestingly, fractions have multiple meanings and interpretations. Educators generally agree that there are five main interpretations: fractions as parts of wholes or parts of sets; fractions as the result of dividing two numbers; fractions as the ratio of two quantities; fractions as operators; and fractions as measures (Behr, Harel, Post, and Lesh 1992; Kieren 1988; Lamon 1999).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The book then goes through each of these five meanings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Alternatively:&lt;/strong&gt; The National Council of Teachers of Mathematics (NCTM) has released a relevant book:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Barnett-Clarke, C., Fisher, W., Marks, R., &amp;amp; Ross, S. (2010). &lt;em&gt;Developing Essential Understanding of Rational Numbers for Teaching Mathematics in Grades 3-5&lt;/em&gt;. Reston, VA: National Council of Teachers of Mathematics. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The entire book would be a good read if you are interested in these subjects. I will just quote the &quot;Essential Understanding&quot; definition given on p. 13:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The rational numbers are a set of numbers that include the whole numbers and integers as well as numbers that can be written as the quotient of two integers, $a \div b$, where $b$ is not zero.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But, of course, this single definition does not do the book justice, as it goes on to provide multiple interpretations and explore the ramifications of each.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To &lt;strong&gt;summarize&lt;/strong&gt; my answer to your twofold question:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In the context of elementary school mathematics, there are many definitions; see either of the sources provided above as well as the works they cite.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To re-quote from the first excerpt: &quot;it is important to introduce the multiple meanings of fraction and to emphasize sense making in all mathematical activities.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="262" LastActivityDate="2014-09-03T00:22:24.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4322" PostTypeId="2" ParentId="4295" CreationDate="2014-09-03T01:20:05.033" Score="4" Body="&lt;p&gt;Since you mentioned Moore's Law...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could make a nice unit on exponential explosion.&#xA;You could tell the story of the Persian prince and grains of wheat,&#xA;nicely described in &lt;a href=&quot;http://en.wikipedia.org/wiki/Wheat_and_chessboard_problem&quot; rel=&quot;nofollow&quot;&gt;this Wikipedia page&lt;/a&gt;.&#xA;Even better, look at Chapter 2 (&quot;The Persian Chessboard&quot;) of Carl Sagan's book, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Sagan, Carl. &lt;em&gt;Billions &amp;amp; Billions: Thoughts on Life and Death at the Brink of the Millennium&lt;/em&gt;. Random House LLC, 2011.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can find PDFs of the book on the web, e.g., &lt;a href=&quot;https://www.andrew.cmu.edu/user/jksadegh/A%20Good%20Atheist%20Secularist%20Skeptical%20Book%20Collection/carl%20sagan%20-%20billions%20and%20billions.pdf&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-09-03T01:20:05.033" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4323" PostTypeId="2" ParentId="4135" CreationDate="2014-09-03T15:44:14.513" Score="3" Body="&lt;p&gt;I can't help thinking of the idea of a &lt;a href=&quot;https://en.wikipedia.org/wiki/Unification_(computer_science)#Syntactic_unification_of_first-order_terms&quot; rel=&quot;nofollow noreferrer&quot;&gt;most general unifier&lt;/a&gt;.  Indeed, maybe the analogy isn't entirely specious; mgu's provide a satisfactory first-order theory of unification, and most of the examples one would think of for difficult notions to generalise are second order.  Well, that's just philosophical babbling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps more usefully, this kind of question has been considered before.  An old Monthly article that I enjoyed discussed how a search for generalisation could lead to mathematical discovery:  &lt;a href=&quot;http://www.jstor.org/stable/2974557&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stolarsky - Searching for common generalizations: The case of hyperbolic functions&lt;/a&gt;.  Although I haven't read it, I discovered in searching that Zeilberger wrote a similarly themed article:  &lt;a href=&quot;http://www.jstor.org/stable/2975373&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zeilberger - The method of undetermined generalization and specialization&lt;/a&gt;.  Other authors have also considered the process of generalisation in its own right; for example, &lt;a href=&quot;http://www.jstor.org/stable/2303955&quot; rel=&quot;nofollow noreferrer&quot;&gt;Blumberg - On the technique of generalization&lt;/a&gt;, and, of course, perhaps most famously, Pólya (&lt;a href=&quot;http://www.jstor.org/stable/2305589&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pólya - Generalization, specialization, analogy&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;None of these answers the question, and I think that it is more or less obvious that there is no answer; but each of them seems to have a useful perspective to add to the pile of thoughts.&lt;/p&gt;&#xA;" OwnerUserId="2070" LastEditorUserId="2070" LastEditDate="2018-03-16T16:35:43.823" LastActivityDate="2018-03-16T16:35:43.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4324" PostTypeId="1" CreationDate="2014-09-03T17:48:00.507" Score="6" ViewCount="284" Body="&lt;p&gt;I have recently ended up in a position where I can supervise bachelor's and master's theses, and there has been some discussion in my research group of organizing thesis supervision within the group.&#xA;I would like to hear your opinions on getting this work started.&#xA;I want to focus here on bachelor's theses, which are the first longer (more than five pages) mathematical writings that students have to write &amp;mdash; if the corresponding thesis in your university is called something else, don't let that bother.&#xA;Comments and answers regarding all pre-PhD university theses are welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what is supposed to happen:&#xA;When a student thinks that the time is ripe for writing such a thesis, he contacts the BSc thesis coordinator.&#xA;He tells the coordinator what kind of mathematics he most likes or which courses he enjoyed the most, and possibly describes his attitude towards some possible advisors.&#xA;Then the coordinator sends the student to speak with a suitable advisor.&#xA;The advisor presents possible topics (related to their own research or other expertise) and together with the student they pick a thesis topic and the actual thesis work commences.&#xA;The exact topic may of course change during the process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to make sure that my possible future advisees find a topic they actually like (enough) and that the challenge is of suitable level, so that they find the motivation to finish what they start.&#xA;My concern here is getting a good start, not the thesis work itself.&#xA;I would like to hear comments and experiences regarding the following questions (which are listed in no particular order):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If I meet a student for the first time, is he likely to be so overwhelmed and excited that he will not say his true opinions?&#xA;If yes, how to prepare the student so that the meeting could be more useful?&#xA;(This preparation could be an email conversation or something on my homepage.)&lt;/li&gt;&#xA;&lt;li&gt;Is it useful to list possible thesis topics on my (or my group's) homepage?&#xA;Do students actually read such pages?&#xA;Has using such lists brought students to your door?&lt;/li&gt;&#xA;&lt;li&gt;Would it be helpful to compile written presentations of possible topics instead of describing them orally and on the blackboard?&#xA;Would it be better to give such presentations at my homepage instead of showing them when a student comes knocking on my door?&#xA;(My group is planning to make a poster advertising our field of expertise, but the poster can only contain coarse ideas and possible titles.)&lt;/li&gt;&#xA;&lt;li&gt;What are the most important things about a thesis topic from a student's point of view?&#xA;How does this depend the level of the student?&#xA;Related to the previous question, what should a presentation/advertisement of a topic include?&#xA;(Topics are not supposed to be hardcore research problems at this stage, but hopefully something that gives a flavor of actual research. The exact difficulty level of the thesis will be adjusted to fit the student, and the topic proposals are therefore somewhat open-ended.)&lt;/li&gt;&#xA;&lt;li&gt;Is it better to first present several different topics and let the student choose or offer one topic at a time until the student accepts one?&#xA;(I might first ask if the student has proposals of his own, but I cannot assume that such exist.)&lt;/li&gt;&#xA;&lt;li&gt;What are the most important things to ensure that the thesis work starts well and the student commits himself to it?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If there is an important question that belongs to this category but which I have overlooked, let me know.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2014-09-03T18:47:23.777" LastActivityDate="2014-09-04T02:07:03.920" Title="How to propose subjects for bachelor's theses?" Tags="|undergraduate-education|graduate-education|undergraduate-research|advising|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4325" PostTypeId="2" ParentId="4284" CreationDate="2014-09-03T23:36:04.787" Score="5" Body="&lt;p&gt;On the same line of &lt;a href=&quot;https://matheducators.stackexchange.com/a/4285&quot;&gt;Santiago Canez's answer&lt;/a&gt;, an identity that appears all over thermodynamics is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{\partial x}{\partial y}\bigg|_z\frac{\partial y}{\partial z}\bigg|_x\frac{\partial z}{\partial x}\bigg|_y=-1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(where there exists $f(x,y,z)=0$)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first approach of many people is to apply the chain rule, thereby not understanding where the minus sign came from.&lt;/p&gt;&#xA;" OwnerUserId="2080" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-09-03T23:36:04.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4326" PostTypeId="1" CreationDate="2014-09-04T00:36:01.273" Score="11" ViewCount="4795" Body="&lt;p&gt;Motivating what is often called &quot;Calculus 2&quot; can be hard, which is probably why there are &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2492/applications-of-calculus-2-to-physics&quot;&gt;multiple&lt;/a&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2152/physical-applications-of-higher-terms-of-taylor-series?rq=1&quot;&gt;other&lt;/a&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2428/how-to-make-calculus-ii-seem-motivated-interesting-and-useful?rq=1&quot;&gt;attempts&lt;/a&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/questions/995/unusual-applications-of-integration&quot;&gt;at&lt;/a&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1245/for-calculus-students-what-should-be-the-intuition-or-motivation-behind-series?lq=1&quot;&gt;motivating&lt;/a&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2466/greatest-hits-of-applications-of-freshman-calculus-to-economics?rq=1&quot;&gt;it&lt;/a&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1294/what-are-some-fun-nonstandard-examples-of-arithmetic-geometric-series?lq=1&quot;&gt;here&lt;/a&gt;. I have just begun teaching such a course, beginning with the bag-of-tricks for performing integration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We will be spending a large amount of time on trig substitution, by which I mean evaluating integrals like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \int \frac{1}{\sqrt{a^2 - x^2}} \mathrm d x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;by substituting $x = a \sin \theta$, and similar integrals of that type. This was similarly emphasized when I first learned calculus years ago, but though I can defend many aspects of a &quot;calculus 2&quot; course in terms of use or intellectual interest, I don't have a repository of interesting or motivating reasons why we spend so much time on this subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In principle, these types of problems might come out of physical phenomena, as these are very Pythagoreanesque.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I wonder:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are &lt;strong&gt;specific&lt;/strong&gt; applications and motivations for learning trig substitution?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have started with some simple applications - but I'm really hoping for more.&lt;/p&gt;&#xA;" OwnerUserId="20" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2018-01-29T08:17:47.437" Title="Direct applications and motivation of trig substitution for beginning calculus students" Tags="|calculus|concept-motivation|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4327" PostTypeId="2" ParentId="4326" CreationDate="2014-09-04T00:36:01.273" Score="3" Body="&lt;p&gt;One possible motivation originates in a very simple problem. Suppose a boat located at the origin is tied to a buoy located at $(0, l)$ (and where the rope starts out taut). If the boat travels upwards on the $y$-axis, the buoy will travel in a path described by a function $f(x)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the buoy is always pulled in the direction of the boat, the line from the boat to the buoy is a tangent to $f(x)$, and so we can write down the differential equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ f'(x) = \frac{-\sqrt{l^2 - x^2}}{x},$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which is clearly separable and clearly leads to a classical trig substitution problem. $\diamondsuit$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, any object pulling other free-moving objects with rigid materials of fixed length lead to isomorphic questions. &lt;/p&gt;&#xA;" OwnerUserId="20" LastActivityDate="2014-09-04T00:36:01.273" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4328" PostTypeId="2" ParentId="4326" CreationDate="2014-09-04T00:37:03.720" Score="3" Body="&lt;p&gt;Although beyond the level of most students the first time they're learning trig substitution, a variety of problems asking about the electric field around uniformly charged 1-dimensional lines in space end up yielding integrals that should be done with trig substitution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, a uniformly charged rod of length $L$ lying with one end at the origin and the other at the point $(0,l)$ produces an electric field at the point $(x,y)$ given by &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_{-x}^{l-x} \frac{y}{4\pi \epsilon_0(t^2 + y^2)^{3/2}} \mathrm{d}t.$$&lt;/p&gt;&#xA;" OwnerUserId="20" LastActivityDate="2014-09-04T00:37:03.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4329" PostTypeId="2" ParentId="4326" CreationDate="2014-09-04T00:38:12.617" Score="7" Body="&lt;p&gt;There are some areas that are naturally calculated with trig substitution, and which appear somewhat naturally. For example,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Finding the area of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ can be done with trig substitution.&lt;/li&gt;&#xA;&lt;li&gt;Given two intersecting circles in the plane, finding the area enclosed within one circle but outside the other can be done with trig substitution. This shape is a &lt;a href=&quot;http://en.wikipedia.org/wiki/Lune_(mathematics)&quot;&gt;lune&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Finding the area within a circle and under a specified height leads to a trig sub integral.&lt;/li&gt;&#xA;&lt;li&gt;Equivalently, finding the amount of soda in a cylindrical can lying on its side given the height of the soda leads to a trig sub.&lt;/li&gt;&#xA;&lt;li&gt;A torus can be constructed by rotating a circle around the origin as a volume of revolution. Finding the area of this torus leads to a trig sub.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In general, finding areas of regions bounded by quadratic algebraic curves might often lead to integrals that can be tackled with trig substitution.&lt;/p&gt;&#xA;" OwnerUserId="20" LastActivityDate="2014-09-04T00:38:12.617" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4330" PostTypeId="2" ParentId="4326" CreationDate="2014-09-04T00:40:49.153" Score="6" Body="&lt;p&gt;Although it's not particularly specific, given a rational function $F(\sin(x), \cos(x))$, the substitution $z = \tan\frac{x}{2}$ will transform the integral $\displaystyle \int F(\sin x, \cos x) \mathrm d x$ into an integral of a rational function of $z$, which can in turn (and in principle) always be integrated with partial fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These come up when calculating the average angular velocity of the output shaft of a &lt;a href=&quot;http://en.wikipedia.org/wiki/Universal_joint&quot; rel=&quot;noreferrer&quot;&gt;universal joint&lt;/a&gt; when the two shafts are not aligned (as in the picture on wikipedia).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, I do not have intuition for coming up with interesting questions yielding rational functions of $\sin x$ and $\cos x$ (even though I have actually used this before!), so I cannot list more.&lt;/p&gt;&#xA;" OwnerUserId="20" LastActivityDate="2014-09-04T00:40:49.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4331" PostTypeId="2" ParentId="4324" CreationDate="2014-09-04T02:07:03.920" Score="3" Body="&lt;p&gt;&lt;strong&gt;Computations&lt;/strong&gt; are often the core of good bachelor's and master's theses:  &lt;strong&gt;do a non-trivial computation, and then explain the result&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can tailor this to their area of interest; most of the real work is in understanding the subject well enough to do the computation in the first place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found and read a few such theses in the past couple years when doing research online.  E.g.:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Would you get more money investing in a mutual fund with higher&#xA;management fees, or in an ETF with higher transaction costs?&lt;/li&gt;&#xA;&lt;li&gt;How do&#xA;various measures of skewness, or measures of risk-adjusted&#xA;performance compare?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These projects don't require creativity:  they require surveying the literature, and systematically doing calculations on that basis.  It's within the range of a thesis project, provides a learning experience that benefits the student, and can result in a useful contribution.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-09-04T02:07:03.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4332" PostTypeId="2" ParentId="4326" CreationDate="2014-09-04T14:24:11.057" Score="23" Body="&lt;p&gt;In my opinion, trig substitution is presented in a terrible fashion in every calculus book I have ever seen. &quot;If you see $\sqrt{a^2 - x^2}$, substitute $x = a \sin \theta$, and then use such-and-such trig identity, blah, blah, blah...&quot; Yet another unmotivated rule to memorize.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always present trig substitution as follows: If you see any algebraic expression that looks like the Pythagorean theorem (i.e., $\sqrt{a^2 - x^2}$, $\sqrt{x^2 - a^2}$, or $\sqrt{x^2 + a^2}$), then &lt;strong&gt;&lt;em&gt;DRAW A TRIANGLE&lt;/em&gt;&lt;/strong&gt; such that the algebraic expression is one of the side lengths, choose an angle $\theta$, use the triangle to write out the values of trigonometric functions involving $\theta$, and substitute.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This takes three unmotivated &quot;rules&quot; off the table, and gives the students a different rule in their place: &lt;em&gt;when in doubt, appeal to geometry.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="1535" LastActivityDate="2014-09-04T14:24:11.057" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4333" PostTypeId="1" AcceptedAnswerId="4344" CreationDate="2014-09-04T14:59:56.753" Score="28" ViewCount="3187" Body="&lt;p&gt;I'd be happy even with an explanation in the simplest case: an explanation of why expressions of the form $\frac{ax + b}{(x - c)(x - d)}$ with $c \neq d$ can always be rewritten in the form $\frac{A}{x - c} + \frac{B}{x - d}$. I'm following Stewart's book, and his explanation, in its entirety, is &quot;A theorem in algebra guarantees that it is always possible to do this&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In some sense, that &lt;em&gt;is&lt;/em&gt; enough, because the thing to do is to &lt;em&gt;try&lt;/em&gt; to rewrite it as $\frac{A}{x - c} + \frac{B}{x - d}$, solve for $A$ and $B$, and the &quot;theorem in algebra&quot; says that you will always be able to. But can one be more convincing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My students will not have seen linear algebra, so that explicit discussion of linear independence will be unhelpful, even though this is the proper language in which to formulate a proof. They will also not have seen number theory, so that comparisons with modular arithmetic and the Chinese Remainder Theorem are also likely to be unhelpful.&lt;/p&gt;&#xA;" OwnerUserId="1535" LastActivityDate="2018-03-12T06:49:24.297" Title="Is there a simple explanation for calculus classes of why partial fractions work?" Tags="|calculus|" AnswerCount="8" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4334" PostTypeId="2" ParentId="4333" CreationDate="2014-09-04T16:06:21.120" Score="11" Body="&lt;p&gt;It is a matter of taste, but I find the following approach easier to understand for beginners (and thus more convincing) than trying to rewrite the fraction in a good form and solving for coefficients.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be good to start with stating that if $ax+b$ can be written in the form $A(x-d)+B(x-c)$, then $\frac{ax + b}{(x - c)(x - d)}$ can be written in the form $\frac{A}{x - c} + \frac{B}{x - d}$; don't expect that all students will think of this unless told.&#xA;It is quite simple to explain that this is possible if for all $a,b$ there are $A,B$ so that $A+B=a$ and $Ad+Bc=-b$.&#xA;This last statement can be tricky to prove (if one does not have more advanced tools and there is not much time to spend on it), but little experimentation will certainly make it plausible.&#xA;Proving the last claim by hand can be given as an extra exercise for those that are interested in order not to disturb the flow of the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to organize the above argument so that the problem is reduced to a more tractable one in a couple of steps and the first step is (hopefully) the simplest one to grasp.&#xA;I assume your goal is not to give a full proof, but to give a justification for why it works so that all students can understand at least a little.&#xA;If the problem is linked to some other problems (formally equivalent does not mean equivalent in all students' minds), it suffices that the students believe that at least one of them can be solved; students will probably disagree with each other about the simplest formulation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trying to give a fuller proof before appropriate machinery is available might &amp;mdash; depending on the exact situation &amp;mdash; be a waste of time.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2014-09-04T16:06:21.120" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4335" PostTypeId="1" AcceptedAnswerId="4337" CreationDate="2014-09-04T17:20:41.703" Score="10" ViewCount="196" Body="&lt;p&gt;I have a differential equations student who said the following (I'm paraphrasing of course):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A long time ago, my math teacher told me that when I finished differential equations, I would be able to read Einstein's paper on relativity. I'm really excited for that.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I went and looked at a paper of Einstein's, but the one I found didn't seem like a good idea. I'd like to provide a suitable substitute, and I really like this form of motivation: just finish this series of classes, and you will understand [famous paper or idea].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have you ever taken an undergraduate through a famous text in this way? Can differential equations students really look forward to anything like this? I'm not really asking specifically about Einstein.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2014-09-04T21:01:13.463" LastActivityDate="2014-09-05T03:34:25.523" Title="Can we motivate undergraduates by saying they will be able to read famous papers?" Tags="|undergraduate-education|reference-request|student-motivation|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4336" PostTypeId="2" ParentId="4333" CreationDate="2014-09-05T02:18:46.650" Score="3" Body="&lt;p&gt;Given they have no algebra background, the simplest thing is just to do the algebra.&#xA;$$&#xA; \frac{ax + b}{(x - c)(x - d)} = \frac{A}{x-c}+\frac{B}{x-d}&#xA;$$&#xA;Multiplying by $(x-c)(x-d)$ yields (for $x \neq c,d$)&#xA;$$ ax+b= A(x-d)+B(x-c) = (A+B)x-(Ad+Bc)  $$&#xA;From which we read $a=A+B$ and $b = -Ad-Bc$. Multiplying by $d$ we obtain $ad=Ad+Bd$ and adding equations yields $b+ad=Bd-Bc=B(d-c)$ and as $d \neq c$ we find $B = \frac{b+ad}{d-c}$. It follows that $A=a-B = a-\frac{b+ad}{d-c} = \frac{a(d-c)-b-ad}{d-c} = \frac{-ac-b}{d-c} = -\frac{ac+b}{d-c}$. Now, I will reverse engineer the algebra:&#xA;\begin{align}&#xA;\frac{-\frac{ac+b}{d-c}}{x-c}+\frac{\frac{b+ad}{d-c}}{x-d} &#xA;&amp;amp;= \frac{1}{d-c} \left[\frac{-(ac+b)}{x-c}\cdot\frac{x-d}{x-d}+\frac{b+ad}{x-d}\cdot\frac{x-c}{x-c} \right] \\&#xA;&amp;amp;= \frac{1}{d-c} \left[\frac{-(ac+b)(x-d)+(b+ad)(x-c)}{(x-c)(x-d)}\right] \\&#xA;&amp;amp;= \frac{1}{d-c} \left[\frac{(d-c)ax-bx+bx-acd+acd+b(d-c)}{(x-c)(x-d)}\right] \\&#xA;&amp;amp;= \frac{ax + b}{(x - c)(x - d)}.&#xA;\end{align}&#xA;Ok, so, you can start with the last expression, multiply by $1$ and add $0$ until you obtain the partial fractions decomposition $\displaystyle \frac{-\frac{ac+b}{d-c}}{x-c}+\frac{\frac{b+ad}{d-c}}{x-d} $. If the student thirsts for formulas, here they are:&#xA;$$ A = -\frac{ac+b}{d-c} \qquad \&amp;amp; \qquad B=\frac{b+ad}{d-c}. $$&#xA;Of course, for specific examples, it is far easier to simply work out the algebra. That said, I think what I have written above can be formalized into a proof. You just have to add zero and multiply by $1$ creatively. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-09-05T02:18:46.650" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="4337" PostTypeId="2" ParentId="4335" CreationDate="2014-09-05T03:34:25.523" Score="11" Body="&lt;p&gt;Surely some undergraduates will respond to the motivation of reading famous papers, but I'd usually recommend motivating them with ideas or applications instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;strong&gt;What they expect in the paper may not be in the paper.&lt;/strong&gt;  Your student probably wants to see $E=mc^2$ from the time Einstein understood this.  They may be disappointed to learn that Einstein's closest paper from 1905 &lt;a href=&quot;http://www.itp.kit.edu/~ertl/Hauptseminar_SS13/papers/einstein_ist_die_traegheit_eines_koerpers_von_seinem_Energiegehalt_abhaengig.pdf&quot;&gt;said only&lt;/a&gt; &quot;&lt;em&gt;Gibt ein Korper die Energie L in Form von Strahlung ab, so verkleinert such seine Masse um&lt;/em&gt; $L\,/\,V^2$&quot;, &lt;a href=&quot;http://www.doublexscience.org/did-einstein-write-his-most-famous-equation-does-it-matter/&quot;&gt;or&lt;/a&gt;&#xA;&quot;if a body gives off the energy $L$ in the form of radiation, its mass diminishes by $L\,/\,V^2$.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;Many famous papers are hard to read.&lt;/strong&gt;  Often the first presentation of an idea is messy. Often after an idea is recognized as noteworthy, people start talking about the idea in a different context, so that the original context seems odd and difficult to understand.  What was common knowledge at the time the paper was written may not be common knowledge now, or may not be known to the student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &lt;strong&gt;Even if they succeed in reading the paper, they may not be able to answer basic questions&lt;/strong&gt; like:  What was new in this paper?  What similar things were known beforehand?  Answering those questions requires studying the historical context, which is hard.  Most profs would have difficulty providing accurate historical context for most famous mathematical papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However:  if they want to read some historical papers in an area (rather than one particularly famous paper), there are lots of sourcebooks with accessible selections and helpful introductions.  I recommend those sourcebooks for studying the history of math, rather than fixating on one famous paper.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-09-05T03:34:25.523" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4338" PostTypeId="1" CreationDate="2014-09-05T04:33:16.907" Score="5" ViewCount="66" Body="&lt;p&gt;I'm curious what strategies people have found effective for teaching operations with integers, specifically addition and subtraction. I've used number lines and zero pairs (drawings and two-colored counters). Is there a specific order that would be better when introducing multiple strategies?&lt;/p&gt;&#xA;" OwnerUserId="2085" LastActivityDate="2014-09-05T14:16:16.293" Title="Integer Operations" Tags="|secondary-education|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4339" PostTypeId="1" CreationDate="2014-09-05T10:15:28.150" Score="12" ViewCount="614" Body="&lt;p&gt;Thought a little background is required. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The place I teach students learn Calculus in school, not the theory but only the computational aspect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to teach Calculus to first year Engineering students using Apostol but there is huge resistance from my own Department. The arguments range from Engineers don't need this, they won't understand theory and its useless for them, they should focus more on concepts and visualization (geometric intuition). I am feeling a bit defenseless and cornered as I never been part of discussions where people discussed pedagogy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All I know, I spent 15 odd years in an Engineering Institute and 95% percent of my friends are Engineers and they mostly know/did their maths courses rather well; which was a proper course, as in they did proofs, knew about the real number system (lub axiom) , Sequences (Bolzano Weierstrass) , Continuity, Roll's Theorem and MVT, Taylor's Expansion all, Fundamental Theorem of Calculus and they did all these in their first semester along with proofs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here I am facing a scenario where Cauchy Sequence is not to be touched on as if it is going to make students head explode. Taylor's Expansion is not to be talked about, forget Taylor's Series I am not allowed to teach Series. The problem is the resistance comes from my own head/department and not the Engineering Department. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am totally confused and do not know how can I argue my case. Apparently all these is part of Real Analysis and not Calculus and also Proofs are not required, how does one argue against this ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not complaining at all, I am asking for advise regarding how to present my case in a compelling and convincing way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regards&#xA;Vb &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Added in Response to comment by Douglas Zare &lt;/p&gt;&#xA;&#xA;&lt;p&gt;those with different preparation would probably go through a preparatory course I suppose. What I mean to what extent do the Engineering students need to know Calculus to do well in their Career. What is the basic minimum. Where they begin is a different issue, what's the least they must &amp;amp; should know ?&lt;/p&gt;&#xA;" OwnerUserId="2131" OwnerDisplayName="Vagabond" LastActivityDate="2022-09-24T11:33:40.267" Title="What Should be there in a Single Variable Calculus course for Engineers?" Tags="|engineering-mathematics|calculus|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4340" PostTypeId="2" ParentId="4339" CreationDate="2014-09-05T12:00:22.370" Score="2" Body="&lt;p&gt;For one thing, you can argue that students need to know separation of variables&#xA;&lt;a href=&quot;http://en.m.wikipedia.org/wiki/Separation_of_variables&quot; rel=&quot;nofollow&quot;&gt;http://en.m.wikipedia.org/wiki/Separation_of_variables&lt;/a&gt;&#xA;and therefore need to know Series.&lt;/p&gt;&#xA;" OwnerUserId="8865" OwnerDisplayName="Bjørn Kjos-Hanssen" LastActivityDate="2014-09-05T12:00:22.370" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4341" PostTypeId="2" ParentId="4339" CreationDate="2014-09-05T13:16:57.497" Score="7" Body="&lt;p&gt;I strongly agree with Douglas Zare that it is highly dependent on what incoming students are prepared for in terms of their mathematical background beforehand.  As far what is the minimum calculus knowledge that engineers need, this too is dependent on what field of engineering they go into.  I know several engineers that use no more than basic trigonometry in what they do and do not need to know much (if any)  calculus, though that will certainly not be true for all engineers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the US (where I teach), many schools make a distinction between a calculus course and a real analysis course.  A calculus course is essentially a problem-solving course, in which students are introduced to basic facts related to the operations of differentiation and integration, with a few supplementary facts about underlying notions such as the real number line and limits, and then expected to be able to solve a wide range of problems that, in my view at least, often range from essential calculations that everyone should be able to do to esoteric trivia that has remained in the curriculum due to inertia.  On the other hand, a real analysis course should be a careful introduction to the real number line and functions of a real variable, based on rigorous proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typically, engineering students are required to take the sequence of calculus courses, but not required to take an analysis course.  We generally push the stronger students to continue with the analysis sequence, and their &quot;reward&quot; for doing so is the opportunity to add a second major in mathematics, which has a non-trivial amount of employment value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This system suggests that analysis is not required in a &lt;em&gt;bare minimum&lt;/em&gt; of what a prospective engineer needs to know, but that it is a valuable asset that some engineering firms will value (either directly for the subject knowledge obtained or indirectly as a signaling mechanism to indicate an ability for abstract thought).&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2014-09-05T13:16:57.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4342" PostTypeId="2" ParentId="4338" CreationDate="2014-09-05T14:16:16.293" Score="1" Body="&lt;p&gt;The age matters. I use money (including debt), temperature, and sometimes height (above or below ground) with my community college students. &lt;a href=&quot;http://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=1&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CCAQFjAA&amp;amp;url=http%3A%2F%2Fwww.jamestanton.com%2Fwp-content%2Fuploads%2F2012%2F03%2FCurriculum-Newsletter_June-2013.pdf&amp;amp;ei=RcUJVIaGJYi6ogS9lIDYDg&amp;amp;usg=AFQjCNE2Zis30tndkZYZ7FwQvCM_oiqUig&amp;amp;sig2=2EmBComyrZex_fJaV-4HqQ&amp;amp;bvm=bv.74649129,d.cGU&quot; rel=&quot;nofollow&quot;&gt;James Tanton&lt;/a&gt; has a metaphor with sand piles and holes in the sand. &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-09-05T14:16:16.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4343" PostTypeId="2" ParentId="4309" CreationDate="2014-09-05T14:28:58.950" Score="2" Body="&lt;p&gt;I like &lt;em&gt;&lt;strong&gt;&lt;a href=&quot;https://www.alibris.com/search/books/isbn/9781911093701?invid=17260790817&amp;amp;utm_source=Google&amp;amp;utm_medium=cpc&amp;amp;utm_campaign=NMPi&amp;amp;gclid=CjwKCAjw-rOaBhA9EiwAUkLV4j5-FFbbpltPeGkZCsrfgHLLWcg-CTClPzKWltMNsfe2i-xR5xQYHhoChZkQAvD_BwE&amp;amp;gclsrc=aw.ds&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometry Bites&lt;/a&gt;&lt;/strong&gt;&lt;/em&gt;, by Ed Southall and Vincent Pantaloni. (There's also &lt;em&gt;&lt;strong&gt;More Geometry Bites&lt;/strong&gt;&lt;/em&gt;.) Some of the problems are too hard, but many are fine for this age. I bought an online copy, so I could easily use them in my online classes.&lt;/p&gt;&#xA;&lt;p&gt;Along the same lines, &lt;a href=&quot;https://drive.google.com/file/d/1hVP8tLURVDphmHsphz5BQLVzHCeTts29/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;Catriona Shearer's puzzles&lt;/a&gt; are often way too hard, but the easiest ones are lovely. She also posts on twitter as Cshearer41.&lt;/p&gt;&#xA;&lt;p&gt;Also, you might like &lt;em&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0821887459&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Circle Diaries, Year 1: Complete Curriculum for Grades 5 to 7&lt;/a&gt;&lt;/em&gt;, by Anna Burago.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-10-18T01:53:00.797" LastActivityDate="2022-10-18T01:53:00.797" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="4344" PostTypeId="2" ParentId="4333" CreationDate="2014-09-05T14:42:13.960" Score="18" Body="&lt;p&gt;Here is what I might do, if I were teaching a class which taught some abstract proof skill (but not a true proof class) or if I was happy reaching only the top half of a standard Calculus class here at Michigan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we want to expand&#xA;$$\frac{f(x)}{(x-r_1)(x-r_2)(x-r_3)(x-r_4)}$$&#xA;where all the $r_i$ are distinct, and $\deg f \leq 3$. I claim that&#xA;$$\begin{array}{rcl}&#xA;\frac{f(x)}{(x-r_1)(x-r_2) \cdots (x-r_4)} &amp;amp;=&amp;amp;\phantom{+} \frac{f(r_1)}{(r_1-r_2)(r_1-r_3)(r_1-r_4)} \frac{1}{x-r_1}\\&#xA;&amp;amp; &amp;amp;+ \frac{f(r_2)}{(r_2-r_1)(r_2-r_3)(r_2-r_4)} \frac{1}{x-r_2}\\&#xA;&amp;amp; &amp;amp;+ \frac{f(r_3)}{(r_3-r_1)(r_3-r_2)(r_3-r_4)} \frac{1}{x-r_3}\\&#xA;&amp;amp; &amp;amp;+ \frac{f(r_4)}{(r_4-r_1)(r_4-r_2)(r_4-r_3)} \frac{1}{x-r_4}\\&#xA;\end{array}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, you might wonder how on earth I found that formula, but let's first take it as handed down by magic and show that it works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clearing denominators, the claim is that&#xA;$$\begin{array}{rcl}&#xA;f(x) &amp;amp;=&amp;amp; \phantom{+}\frac{f(r_1)}{(r_1-r_2)(r_1-r_3)(r_1-r_4)} (x-r_2)(x-r_3)(x-r_4)\\&#xA;&amp;amp; &amp;amp; +\frac{f(r_2)}{(r_2-r_1)(r_2-r_3)(r_2-r_4)} (x-r_1)(x-r_3)(x-r_4)\\&#xA;&amp;amp; &amp;amp; +\frac{f(r_3)}{(r_3-r_1)(r_3-r_2)(r_3-r_4)} (x-r_1)(x-r_2)(x-r_4)\\&#xA;&amp;amp; &amp;amp; +\frac{f(r_4)}{(r_4-r_1)(r_4-r_2)(r_4-r_3)}(x-r_1)(x-r_2)(x-r_3)\\&#xA;\end{array}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now remember: $f$ and the $r$'s are all constants that we know. In the last problem we did, they were $8x-13$, $1$ and $2$ (or whatever). The supposed claim is that this is a true equation for all $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that both sides of this equation are polynomials in $x$. What is the degree of that polynomial? Right, $3$, or possibly less if the coefficients of $x^3$ cancel on the right hand side. A degree $3$ polynomial is determined by $4$ values, just like a linear polynomial is determined by $2$ values. Which $4$ values might we plug in for $x$? Let's try the $r_i$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pause for at least $10$ seconds. Do the computation. Pause for at least $10$ seconds again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Okay, how did I find this formula? This approach is much more worth memorizing than the formula is. Suppose that there were a formula like&#xA;$$\frac{f(x)}{(x-r_1)(x-r_2)(x-r_3)(x-r_4)} = \frac{c_1}{x-r_1} + \frac{c_2}{x-r_2} + \frac{c_3}{x-r_3} + \frac{c_4}{x-r_4}.$$&#xA;So the $c_i$ are the constants we want to find: in the last problem, they turned out to be $3$ and $5$ (or whatever). Here is the trick: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Multiply both sides by $x-r_i$ and take the limit as $x \to r_i$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Pause for at least $10$ seconds. Do the limit. Pause for at least $10$ seconds again.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As a general note, I've found that sum and product notation, and using variables for discrete quantities like the degree of a polynomial, are huge stumbling blocks. I pretty much always use specific values of $n$, like $4$, whenever working with non-math majors.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2014-09-16T16:30:04.510" LastActivityDate="2014-09-16T16:30:04.510" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4345" PostTypeId="2" ParentId="4060" CreationDate="2014-09-05T15:11:16.933" Score="3" Body="&lt;p&gt;I have made my classes more participatory over the years. I made great strides after I began reading math teacher blogs. Participating in math circles, and leading them, has also helped. Even in discussion, if we keep asking students if they agree with another student's proposed step, and keep asking why something works or is useful, we can get a lot more mathematical thinking than with a conventional lecture.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-09-05T15:11:16.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4346" PostTypeId="2" ParentId="4036" CreationDate="2014-09-05T15:24:51.510" Score="1" Body="&lt;p&gt;Entice, don't require. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Play around with things at the edges of math. Might she like programming in &lt;a href=&quot;http://scratch.mit.edu/&quot; rel=&quot;nofollow&quot;&gt;Scratch&lt;/a&gt;? Might she like the book &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B009OZN7IO&quot; rel=&quot;nofollow&quot;&gt;The Number Devil&lt;/a&gt;&lt;/em&gt;? What about chess and go? Does she like you to read to her? Read her &lt;em&gt;&lt;a href=&quot;http://www.betterworldbooks.com/the-man-who-counted-id-9780393309348.aspx&quot; rel=&quot;nofollow&quot;&gt;The Man Who Counted&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-09-05T15:24:51.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4347" PostTypeId="1" CreationDate="2014-09-05T16:28:38.733" Score="13" ViewCount="1702" Body="&lt;p&gt;All of the calculus textbooks I've used (teaching at community colleges) start with the first chapter covering limits. (Perhaps after a review chapter.) I think this order is wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, Newton and Leibniz thought in terms of fluxions or infinitesimals. It took mathematicians 150 years to develop the logical (epsilon-delta) machinery of limits that puts calculus on a sound logical foundation. (And it turns out that this is not the only way to do so. Non-standard analysis uses infinitesimals in a logically rigorous way.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The big ideas of calculus are derivatives for rate of change, and integration for areas and volumes. I think the course makes more sense for students if we start with &quot;slopes of curvy lines&quot; and velocity. So I'm bringing in outside material to help them explore the basic concept of the derivative from many perspectives. (Our textbook only has two sections that do this. Most only have one. I spend three weeks on it. Much of my material comes from Boelkins' &lt;em&gt;&lt;a href=&quot;http://faculty.gvsu.edu/boelkinm/Home/Download.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Active Calculus&lt;/a&gt;&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I give the limit definition of the derivative, but I say that for now we'll think of the limit as meaning that we get infinitely close. After we are done with simple derivatives, I go back to chapter one's limit exercises before doing trig derivatives, because we hit some unusual limits there (e.g. sinx/x).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to know why textbooks cover limits first. Is it just because the mathematicians writing them are stuck at a higher level, and want to first establish the validity of what will be done later? I don't think that's good pedagogy. I think calculus is beautiful and powerful, and that I can convey this better by waiting to introduce limits after students see why they need them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why cover limits first?&lt;/strong&gt; &lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2017-04-26T13:41:26.617" LastActivityDate="2017-05-03T23:34:16.330" Title="When should we get into limits in introductory calculus courses?" Tags="|mathematical-pedagogy|calculus|infinitesimals|limits|" AnswerCount="7" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="4348" PostTypeId="2" ParentId="4347" CreationDate="2014-09-05T17:27:36.983" Score="9" Body="&lt;p&gt;My currently preferred approach is to start the course with an introductory lecture explaining the difference between average velocity over a time interval (something we can always in principle measure using a stopwatch) and instantaneous velocity at a given instant of time (which we do not have a means to measure).  This motivates the question of what exactly instantaneous velocity is, using the intuition that most of us have that the notion makes sense.  (Of course, that in and of itself is a philosophical issue and one that I want to avoid in the course for the obvious reason of limited time!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I explain that one can &lt;em&gt;approximate&lt;/em&gt; instantaneous velocity by measuring the average velocity over smaller and smaller intervals of time centered around the given instant of time.  (In fact, I never actually use a centered time interval, but rather use the formulation with the instant of time as one endpoint of the time interval, so that the formulas I write down look like what the definition of derivative will be when we define it officially later on.)  Then I say that the instantaneous velocity is the number that we would get &lt;em&gt;in the limit&lt;/em&gt; where the time interval became closer and closer to $0$.  And I emphasize that, of course, we can never actually make a measurement with a time interval of duration $0$ (both physically because we can't start and stop our stopwatch at the exact same instant of time and mathematically because using the same starting and ending time in the formula for average velocity yields the undefined expression &quot;$0/0$&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This serves to motivate the introduction of limits, and in fact, in the courses I teach, typically this is more or less the definition of a limit.  I use the heuristic and imprecise version of $\lim\limits_{x \rightarrow a} f(x) = L$ means that as $x$ gets closer and closer to $a$ without actually reaching $a$, the values $f(x)$ get closer and closer to $L$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, this seems to strike the right balance of motivating limits and discussing how to compute them in the types of cases that will come up throughout the semester, while still being able to quickly get to derivatives so that most of the semester is spent on the two fundamental notions of derivative and integral and, ultimately, the link between the two provided by the Fundamental Theorem of Calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I strongly prefer to follow pretty closely to the textbook's development in a course such as introductory calculus, where I feel that students are often not sufficiently mature to handle the discrepancy between the textbook presentation and the presentation given in lecture.  And they are not equipped to handle the inevitable situation where the textbook has referred to concept X in Section A which appears before Section B and I want to cover the ideas in Section B before covering concept X, but too much of the presentation and choice of problems refer to or are based on understanding concept X.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for why I prefer to discuss limits before derivatives and integrals (neglecting the obvious cultural bias that that is the way I learned the subject), since I cannot speak for textbook publishers, to me there are two big pictures to take away from calculus.  One is the study of change (derivatives) and accumulation (integrals) and the fact that the two problems are intimately related (Fund. Thm. of Calc.).  The other is that derivatives and integrals, key ideas for understanding our dynamic natural world, are best understood computationally as the limiting process of simpler calculations that only require basic arithmetic (difference quotients for derivatives, Riemann sums for integrals).  It should be noted that the subjective value judgment indicated by the use of the word &quot;best&quot; is not universally agreed upon.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, these two big pictures should both be presented and stressed to the students.  I find it easiest, logically and pedagogically, to present derivatives and then later integrals to model independent processes and stress the point of view that these are both limiting processes.  As such, I need to have discussed what a limit is.  After defining integrals as limits of Riemann sums, I point out that the fact that integration, like differentiation, is a limiting process is one of two reasons why the two topics are both covered in a common course.  And then I foreshadow what is to come by alluding to the fact that there is a deeper and more important link between these two concepts, which I explain a few lectures later when we are ready to discuss the Fundamental Theorem of Calculus.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2014-09-08T12:34:53.747" LastActivityDate="2014-09-08T12:34:53.747" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4349" PostTypeId="2" ParentId="4333" CreationDate="2014-09-05T20:26:25.687" Score="6" Body="&lt;p&gt;I have always found it instructive and fun to address this by thinking graphically. Students in Calculus II will know the graphs of the rational functions $f(x)=\frac{ax+b}{(x-c)(x-d)}$ and $g(x)=\frac{A}{x-c}+\frac{B}{x-d}$ have the same vertical asymptotes and same horizontal asymptote.  Varying $a$ and $b$ moves the $x$ and $y$ intercepts of $y=f(x)$. It is also clear that varying $A$ and $B$ moves the $x$ and $y$ intercepts of $y=g(x)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fixing $c$ and $d$, we have a two-parameter family of curves $y=f(x)$ and similarly for $y=g(x)$. The key point here is that two families are identical--we can find $A$ and $B$ so that the $g$-curve matches any given $f$-curve. It is straightforward to illustrate this in class with animations generated by software--Geometers Sketchpad or Maple for example. As you vary the two parameters of one family, the curves fill out the curves from the other family. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not a proof, but rather a conceptual understanding that calculus students can appreciate. The proof that Stewart alludes to really is straightforward and mundane, and is covered in a precalculus course--it reduces to the existence of a solution to two linear equations in two unknowns.  &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2014-09-05T20:26:25.687" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4351" PostTypeId="1" AcceptedAnswerId="4354" CreationDate="2014-09-06T01:54:15.003" Score="100" ViewCount="18978" Body="&lt;p&gt;In the comments to David Speyer's answer &lt;a href=&quot;https://matheducators.stackexchange.com/a/4344/117&quot;&gt;here&lt;/a&gt;, he points out that &quot;the distinction between 'if there is a formula, it is this one' and 'this formula works' is subtle.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any simple, natural examples where there is a &quot;unique candidate for a thing satisfying properties&quot;, but that candidate does not actually work?  This is really a particular kind of reductio ad absurdum, but somehow different from most.  Most reductio ad absurdum arguments do not produce a &quot;reasonable looking&quot; candidate for the nonexistent object, if you know what I mean. &lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2024-01-31T16:06:41.547" Title="Unique candidate that fails" Tags="|undergraduate-education|mathematical-pedagogy|" AnswerCount="20" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="4352" PostTypeId="2" ParentId="4347" CreationDate="2014-09-06T02:56:56.170" Score="11" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I'd like to know why textbooks cover limits first.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't think there's any big mystery as to why commercial textbooks tend to be similar. It's a market mechanism known as the network effect, the same mechanism that makes Microsoft Windows so popular. Once people start to see something as a standard, anything different becomes non-viable in the marketplace. Stephen J. Gould wrote a nice essay about this in the field of biology; he uses the example of how biology texts propagate the meme of  Lamarckian and Darwinian explanations of the giraffe's neck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see that there is no objective reason for doing limits first, we just have to look at some older textbooks. Hutton 1807 uses fluxions. Davies 1836 uses limits, but also connects with infinitesimals.  Granville 1904 uses limits. Thompson 1910 uses infinitesimals. Different authors did different things, which is how it should be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also some modern texts that don't start with limits and then move on to derivatives: Marsden 1981, Keisler 1976. At the risk of seeming self-promoting, &lt;a href=&quot;http://www.lightandmatter.com/fund/&quot;&gt;here's a link&lt;/a&gt; to a book that I'm a coauthor of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There may have been a time ca. 1850-1965 when limits were believed to be the only possible rigorous foundation for calculus. That era ended when NSA essentially vindicated Leibnizian infinitesimals.[Blaszczyk 2012]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other than the network effect, a second and more cynical explanation would be that a course in calculus is used today as a filter in order to get rid of some of the students seeking a valuable credential. This is the phenomenon of credential creep. For example, I teach quite a few students who want to be physical therapists, and many DPT programs require them to take a year of calculus. Calculus is fundamentally an easy subject, but it can be made much harder by emphasizing epsilontics and by requiring these folks to learn a bag of tricks for integration. I'm not a cynic at heart, but I do have a very hard time coming up with any other explanation for why we require future physical therapists to learn how to do integrals using trig substitutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Blaszczyk, Katz, and Sherry, &quot;Ten Misconceptions from the History of Analysis and Their Debunking,&quot; 2012, &lt;a href=&quot;http://arxiv.org/abs/1202.4153&quot;&gt;http://arxiv.org/abs/1202.4153&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Charles Davies, Elements of the Differential and Integral Calculus (1836), &lt;a href=&quot;https://archive.org/details/elementsdiffere03davigoog&quot;&gt;https://archive.org/details/elementsdiffere03davigoog&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Granville, Elements of the Differential and Integral Calculus (1904), &lt;a href=&quot;https://archive.org/details/elementsdiffere01smitgoog&quot;&gt;https://archive.org/details/elementsdiffere01smitgoog&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Charles Hutton, A Course of Mathematics: In Two Volumes. For the Use of Academies as Well as Private Tuition (1807), &lt;a href=&quot;https://archive.org/details/acoursemathemat02huttgoog&quot;&gt;https://archive.org/details/acoursemathemat02huttgoog&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keisler, Elementary Calculus: An Infinitesimal Approach, 1976, &lt;a href=&quot;http://www.math.wisc.edu/~keisler/calc.html&quot;&gt;http://www.math.wisc.edu/~keisler/calc.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Marsden, Calculus Unlimited, 1981, &lt;a href=&quot;http://www.cds.caltech.edu/~marsden/books/Calculus_Unlimited.html&quot;&gt;http://www.cds.caltech.edu/~marsden/books/Calculus_Unlimited.html&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thompson, Calculus Made Easy, 1910, &lt;a href=&quot;http://www.gutenberg.org/ebooks/33283&quot;&gt;http://www.gutenberg.org/ebooks/33283&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-09-06T03:34:35.497" LastActivityDate="2014-09-06T03:34:35.497" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4353" PostTypeId="2" ParentId="4351" CreationDate="2014-09-06T07:24:31.617" Score="8" Body="&lt;p&gt;How about having three distinct points on a plane, so there should exist a &lt;em&gt;unique&lt;/em&gt; parabola that passes through all of them, but it turns out that the points are collinear so the parabola &lt;em&gt;does not exist&lt;/em&gt; after all?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Comments&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You probably mean something like &quot;having three distinct points on the xy-plane, there should exist a unique parabola with vertical axis that passes through all of them&quot;. The coordinate-free presentation allows parabolas in so many directions that three points do not determine the shape. –  Matt F. Sep 6 '14 at 11:36   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on your point of view, the parabola does exist, but it has infinite focal length. A straight line can be so many things, it is an arc of oh so many infinitely sized curves :-) –  steveverrill Sep 6 '14 at 13:41    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;@MattF. FWIW by Bezout's theorem, two parabolas intersect in 4 points, so 5 points determine a parabola in the plane (which are not necessarily vertically aligned). –  Steven Gubkin Sep 6 '14 at 14:48&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2015-02-22T00:15:20.240" LastActivityDate="2015-02-22T00:15:20.240" CommentCount="0" CommunityOwnedDate="2015-02-22T00:15:20.240" ContentLicense="CC BY-SA 3.0" />
  <row Id="4354" PostTypeId="2" ParentId="4351" CreationDate="2014-09-06T10:08:14.037" Score="88" Body="&lt;p&gt;For a positive real number $x$ consider &#xA;$$x^{x^{x^{\dots}}}$$&#xA;or formally (the limit of) the sequence $a_n= x^{a_{n-1}}$ (and $a_0= 1$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Determine the value $x$ (if it exists) such that &#xA;$$x^{x^{x^{\dots}}}=4.$$&#xA;Uniqueness can be proved like this: since for such an $x$ we ought to have $x^4 = 4$ we get that $4^{1/4} = \sqrt{2}$ is the unique &lt;em&gt;candidate&lt;/em&gt; for a solution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this value does not work. A way to appreciate this (though not strictly a proof of it) is to do the same for $2$ instead of $4$, getting also $x=\sqrt{2}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\dots}}}$$ would have to be both $2$ and $4$, which is absurd, so (at least) for one of the two there is no solution. (It is indeed $4$ where there is no solution.) &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-09-06T10:08:14.037" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="4355" PostTypeId="2" ParentId="4053" CreationDate="2014-09-06T17:21:34.737" Score="6" Body="&lt;p&gt;What you are referring to, I believe, is students who are actively engaged with, thinking about, and emotionally connected to the material rather than being told to passively absorb it. This distinction between Active and Passive learning modes is a hot topic in education research these days. I don't know about mathematics specifically, but physics departments are whole-heartedly embracing active strategies, lead by Eric Mazur at Harvard. You can check out his publications and videos for great examples of how to apply active strategies to traditionally &quot;dry&quot; material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quick google search shows that there is current research into active learning in mathematics:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://iteachem.net/wp-content/uploads/2014/05/Freeman-S-Proc-Natl-Acad-Sci-USA-2014-Active-learning-increases-student-performance-in-science-engineering-and-mathematics.pdf&quot; rel=&quot;noreferrer&quot;&gt;Active learning increases student performance in&#xA;science, engineering, and mathematics, 2014&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.sagepub.com/booksProdDesc.nav?prodId=Book229224&quot; rel=&quot;noreferrer&quot;&gt;Active Learning in the Mathematics Classroom, Grades 5-8, 2007&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2094" LastEditorUserId="2094" LastEditDate="2014-09-06T18:45:03.283" LastActivityDate="2014-09-06T18:45:03.283" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4356" PostTypeId="2" ParentId="4347" CreationDate="2014-09-06T19:01:42.057" Score="2" Body="&lt;p&gt;I am not advocating any order but one reason to start with limits first is that the concept of &lt;strong&gt;continuity&lt;/strong&gt; is extremely important and useful and limits are the natural way to introduce it. &lt;/p&gt;&#xA;" OwnerUserId="1941" LastActivityDate="2014-09-06T19:01:42.057" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="4357" PostTypeId="2" ParentId="4347" CreationDate="2014-09-07T10:56:57.853" Score="3" Body="&lt;p&gt;I think this book hasn't been mentioned yet:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1418441244&quot; rel=&quot;nofollow&quot;&gt;John C. Sparks: Calculus without Limit - Almost&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Disclaimer: I have it on my bookshelf but have to admit that I haven't read it yet, so I can't really comment on whether it's good or not.]&lt;/p&gt;&#xA;" OwnerUserId="2089" LastActivityDate="2014-09-07T10:56:57.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4358" PostTypeId="2" ParentId="4351" CreationDate="2014-09-07T12:06:36.727" Score="40" Body="&lt;p&gt;If there were a linear formula for $\int_0^1 f(x) dx$ in terms of $f(0)$ and $f(1)$, it would be $\int_0^1 f(x) dx = \frac{1}{2}(f(0) + f(1))$. Proof: Suppose $\int_0^1 f(x) dx = af(0)+bf(1)$. Taking $f(x) = 1$, we deduce $a+b=1$; taking $f(x) = x$ we deduce $b=1/2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there is no such formula, but this computation does hint at something useful: If all you know is $f(0)$ and $f(1)$, then $\frac{1}{2}(f(0) + f(1))$ is a reasonable guess. This leads to the trapezoidal rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you were teaching &lt;a href=&quot;http://en.wikipedia.org/wiki/Simpson%27s_rule&quot;&gt;Simpson's rule&lt;/a&gt; or &lt;a href=&quot;http://en.wikipedia.org/wiki/Gaussian_quadrature&quot;&gt;Gaussian quadrature&lt;/a&gt;, you could similarly ask &quot;Can we get a rule which works for a few more functions by incorporating $f(1/2)$?&quot; or &quot;Can we get a rule which woks for even more functions by using $f(a)$ and $f(b)$ instead of $f(0)$ and $f(1)$, for some well chosen $a$ and $b$?&quot;&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2014-09-07T12:06:36.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4359" PostTypeId="2" ParentId="4351" CreationDate="2014-09-07T13:50:53.663" Score="114" Body="&lt;p&gt;The obvious example I immediately thought of is that, if the divergent geometric series $$1 + 2 + 4 + 8 + \dotsb = \sum_{k=0}^\infty 2^k$$ converged, it would converge to $-1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Proof:&lt;/em&gt; If the series converged to some number $x = 1 + 2 + 4 + 8 + \dotsb$, then clearly this number $x$ would have to satisfy $2x = 2 + 4 + 8 + \dotsb = x - 1$.  Solving $2x = x-1$ for $x$ yields $x = -1$, Q.E.D.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Like many other such examples of &quot;unique false solutions&quot;, this one also hints at deeper things.  For example:&lt;/p&gt;&#xA;&#xA;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Whenever the &lt;a href=&quot;//en.wikipedia.org/wiki/Geometric_series&quot;&gt;geometric series&lt;/a&gt; $1 + x + x^2 + x^3 + \dotsc$ converges, its limit equals $\frac{1}{1-x}$.  This expression is defined for all values $x \ne 1$, even those for which the original series does &lt;em&gt;not&lt;/em&gt; converge; for $x = 2$, it equals $-1$.  This is, essentially, an example of &lt;a href=&quot;//en.wikipedia.org/wiki/Analytic_continuation&quot;&gt;analytic continuation&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In the system of &lt;a href=&quot;//en.wikipedia.org/wiki/P-adic_number&quot;&gt;2-adic numbers&lt;/a&gt;, which have a different notion of distance, and thus of convergence, than the ordinary real numbers, the series $1 + 2 + 4 + 8 + \dotsb$ does converge.  The number it converges to is $-1$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In fact, the proof above is sufficient to show that &lt;em&gt;any&lt;/em&gt; &lt;a href=&quot;//en.wikipedia.org/wiki/Divergent_series#Properties_of_summation_methods&quot;&gt;stable and linear summation method&lt;/a&gt; that produces a finite sum for this series must yield the answer $-1$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another way of looking at this is in terms of &lt;a href=&quot;//en.wikipedia.org/wiki/Iterated_function&quot;&gt;iterated functions&lt;/a&gt;.  Specifically, the partial sums $a_n = \sum_{k=0}^n 2^k$ satisfy the recurrence relation $a_{n+1} = f(a_n) = 2a_n + 1$.   The unique fixed point of the map $f$ is $f(-1) = -1$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Yet another, and possibly the most natural, way is to think of the series as converging to $-1$ &lt;em&gt;when run backwards&lt;/em&gt;.  Specifically, the sequence of partial sums $a_n = \sum_{k=0}^n 2^k = 2^n-1$ can be naturally extended to negative values of $n$.  (Indeed, the same can be done for any geometric series.)  And, clearly, $\displaystyle \lim_{n\to-\infty} a_n = -1$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, if you naïvely sum the powers of two on a computer using signed $n$-bit arithmetic, you will normally get the answer $-1$.  For a &lt;a href=&quot;//ideone.com/CRAzZv&quot;&gt;demonstration&lt;/a&gt;, try the following Java code:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;int sum = 0, term = 1;       // these are 32-bit signed integers&#xA;while (sum + term != sum) {&#xA;    sum = sum + term;&#xA;    term = term * 2;&#xA;}&#xA;System.out.println(sum);     // this prints &quot;-1&quot;&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The reason this happens is because most computers handle signed integers using the &lt;a href=&quot;//en.wikipedia.org/wiki/Two%27s_complement&quot;&gt;two's complement&lt;/a&gt; representation, in which a bitstring consisting of all 1 bits represents the number &amp;minus;1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, the reason &lt;em&gt;why&lt;/em&gt; this representation is so commonly used is because, in a sense, it is the natural one &amp;mdash; in particular, it allows basic arithmetic operations like signed addition and multiplication to be implemented using the exact same code / circuitry as their unsigned counterparts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The underlying mathematical reason is that both the usual representation of unsigned $n$-bit integers and the two's complement representation of signed integers correspond to &lt;a href=&quot;//en.wikipedia.org/wiki/Modular_arithmetic&quot;&gt;arithmetic modulo $2^n$&lt;/a&gt;, just with different representatives chosen for the equivalence classes.  The bitstring that in signed arithmetic represents $-1$ corresponds to the unsigned number $2^n-1$ $=$ $1 + 2 + 4 + \dotsc + 2^{n-1}$, which is equivalent to $-1$ modulo $2^n$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For more information and links, see &lt;a href=&quot;//en.wikipedia.org/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B%AF&quot;&gt;the Wikipedia article on this series&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="526" LastEditorUserId="526" LastEditDate="2015-10-23T09:47:35.370" LastActivityDate="2015-10-23T09:47:35.370" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4360" PostTypeId="1" CreationDate="2014-09-07T16:06:04.297" Score="9" ViewCount="349" Body="&lt;p&gt;There's a new &lt;a href=&quot;http://www.bbc.com/news/technology-29063614&quot; rel=&quot;nofollow&quot;&gt;story&lt;/a&gt; about an app that &quot;teaches UK pupils 18 months of maths in six weeks&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming this result is true (and such claims need a strong justification), what could be the cause of this app working so well, and of the huge difference?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And do any other places show such benefits for technology? &lt;/p&gt;&#xA;" OwnerUserId="2101" LastEditorDisplayName="user173" LastEditDate="2014-09-09T12:02:00.470" LastActivityDate="2014-09-09T18:26:42.937" Title="New app teaches math really fast. How does it work?" Tags="|secondary-education|technology-in-education|primary-education|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4361" PostTypeId="2" ParentId="1817" CreationDate="2014-09-07T20:58:38.747" Score="8" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Ingrid_Daubechies&quot;&gt;Ingrid Daubechies&lt;/a&gt;, known for &lt;a href=&quot;http://en.wikipedia.org/wiki/Daubechies_wavelet&quot;&gt;Daubechies wavelets&lt;/a&gt; and former president of the &lt;a href=&quot;http://en.wikipedia.org/wiki/International_Mathematical_Union&quot;&gt;IMU&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Ruth_Moufang&quot;&gt;Ruth Moufang&lt;/a&gt;, known for &lt;a href=&quot;http://en.wikipedia.org/wiki/Moufang_loop&quot;&gt;Moufang loops&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2089" LastActivityDate="2014-09-07T20:58:38.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4362" PostTypeId="2" ParentId="4060" CreationDate="2014-09-08T12:49:33.953" Score="4" Body="&lt;p&gt;Since there have been a number of questions along these lines, the following observation may be useful:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a mathematician, I've often read summary background material in a very high powered groundbreaking paper that makes me wonder why, if the only persons who will read the paper are specialists, it is important to include such material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may be related to the fact that thinking about mathematics effectively (at least in the &quot;algebraic&quot; subjects) requires that the practitioner hold a reasonably well-developed and compressed picture of the whole of a project in mind at one time. The brisk and highly compressed &quot;review&quot; serves to activate the memory of the expert so that the requisite facts are more easily in reach while thinking about the new details.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This said, writing a very &quot;jokey&quot; or &quot;entertaining&quot; mathematical text while keeping the level of compression high may become very difficult. The reason is that certain difficult-to-process sentences might compress a precise intuition of the author. The ability to remember the sentence becomes the ability to recall the details associated to a construction. A good joke or entertaining real-world application may help with this, but more often than not such information blocks the ability to remember what is truly essential for processing an argument or calculation! It feels (to me) like trying to memorize something in a room full of loud noises like people yelling and blaring music. (the yelling and blaring music being analogous to the joke or application.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyhow, I think that at the root this may be why many of the very entertaining and witty mathematicians I know don't opt to be entertaining while writing mathematics texts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way to think about this: Imagine this were a golf forum and someone posted the analogous question &quot;Why don't we teach golf in a more engaging way?&quot; The fact is, golf (and math) is hard and the only way to get better at it is to practice and concentrate very hard on what you are doing. Hoopla while teaching math (although we all involve some...just ask my students) is sort of like talking during someone's swing. Golf (and Math) is its own reward, when you have very occasionally succeeded in doing something difficult with a beautiful result. There's a good chance this would never be seen if we changed the teaching of the game to make it easier or more engaging to learn...which is what most entertaining approaches inadvertently do.&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2014-09-08T12:49:33.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4363" PostTypeId="2" ParentId="4351" CreationDate="2014-09-08T16:48:13.443" Score="2" Body="&lt;p&gt;The enigmatic &lt;a href=&quot;https://en.wikipedia.org/wiki/Field_with_one_element&quot; rel=&quot;nofollow&quot;&gt;&quot;field with one element&quot;&lt;/a&gt; comes to mind. This is a non-existent object that behaves like a finite field, but which has only one element. There is a rich history of deep mathematics that grapples with it.  &lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="688" LastEditDate="2015-02-21T12:24:56.957" LastActivityDate="2015-02-21T12:24:56.957" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4364" PostTypeId="2" ParentId="4360" CreationDate="2014-09-09T11:48:14.963" Score="8" Body="&lt;p&gt;I read the article. Note that, after reading the article:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;We don't know what they assessed&lt;/li&gt;&#xA;&lt;li&gt;We don't know how they assessed it&lt;/li&gt;&#xA;&lt;li&gt;We don't know anything about the comparison groups, including what they are comparing the treatment group &lt;em&gt;to&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;We don't know anything about what the app does. What subject matter? How is it represented? What do students do with it?&lt;/li&gt;&#xA;&lt;li&gt;Their definition of rigorous is &quot;demands a score of 10 out of 10.&quot; That's rigor? What's it have to do with learning?&lt;/li&gt;&#xA;&lt;li&gt;How did they research their treatment? Where did they publish?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If I had to guess wildly, they're giving students some sort of repeated practice in some algorithms procedures or math facts, and then assessing those facts and procedures directly. If this is the case, their results would not be too surprising. But that's a wild guess on very little information. Looking into it a little further, the apps seem to focus on counting to 10, counting to 20, and the times tables for 2s, 5s, and 10s. That sounds like something fairly reasonable to teach to someone in 6 weeks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without actual research, we're left to speculate as I have done above. You can take my comments above as addressing your &quot;what could be the causes&quot; question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, the causes &lt;em&gt;could&lt;/em&gt; be: 1) comparison to a situation that was previously lacking in time actually spent on learning. 2) Math content that relies on memorization and repetition rather than understanding. We already know that certain things, when drilled repeatedly, can result in gains performing tasks or recalling facts. If they're essentially creating a mechanized and more engaging form of drilling, and testing exactly those things, then you will see gains. &lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-09-09T18:26:42.937" LastActivityDate="2014-09-09T18:26:42.937" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4365" PostTypeId="2" ParentId="487" CreationDate="2014-09-09T23:26:59.260" Score="8" Body="&lt;p&gt;&lt;strong&gt;Solutions that could not have been concieved by the average student&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is important to realise that many more solutions can't be concieved by the average student than we usually realise. Even solutions we think are quite routine can seem like strokes of genius to students new to the area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students often say to me, &quot;I can follow the lecturer's proof/solution, but I have no idea how I would come up with something like that myself.&quot; And they really can't. In first year it's integrals and proofs about subspaces they struggle with. In second year it's &quot;basic&quot; proofs about limits using the $\epsilon$-$\delta$ definition. The problem is that our insights are based on a fairly deep understanding of the area and experience with lots of these proofs which has given us a familarity with the sorts of things that might help. The students do not have this experience, so they can't imagine even things that to us seem basic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What the purpose of showing a solution is&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think before we answer the question, we should probably consider what the purpose of showing a solution is at all. Is it to convince the student of the truth of a statement? Is it to show them how to go about coming up with solutions? Is it to show them the patterns of argumentation that are acceptable in your discipline? Is it to gain an appreciation for the clever method of solution? Or some combination of these? Each of these purposes will give you a slightly different approach to presenting solutions, though they will overlap.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it is students presenting the solutions, what is the purpose of that? Is it so that they get better at their communication skills? Are the students they present to supposed to have come to an understanding already, or are they supposed to use what the presenting student does to understand? Is it the correctness of the solution that is being assessed or the effectiveness of the presentation? Is the effectiveness judged by how the other students understand or on the structure of the solution itself? You need to consider these things so you can advise students on what sorts of things you want them to do when they present, so you can choose what sort of feedback you give them, and so you can decide when to intervene.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In most courses, the ultimate goal is usually to get students to solve problems on their own. Also, from my above observations about students, they need a lot of support to learn to do this. Thereforre, in my opinion the main reason for showing any solution is actually to help the students learn how to solve problems themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Some suggestions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In light of this, I have some suggestions for things you can do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, for all solutions, no matter how routine they seem, talk about the thought processes that help make the decision of what to do. Compare the given information to the goal, look up definitions of terms, make reference to other solutions that are similar, keep looking to the goal to see if you can do something to get closer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes, you should try something that doesn't work, followed by some reasoning as to how you could tell in advance next time that it wouldn't work. You can also list some different things to try and make a quick assessment of them to choose the one that will work, rather than actually trying them out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For students presenting, you can tell them the purpose is to describe how they came up with the solution, and you can make sure you ask them how they came up with the idea when they do present. You can also ask other students if they had a different solution. (I remember a tutorial when one of my students used two blackboards to do a proof, and when I asked the shocked audience if they had another way, another student came up and showed the intended three-line proof. It was a great lesson in different approaches for all of us.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For things that really are strokes of genius, even for us who are experienced, there does come a point where you can say that people worked on this for quite some time and after a lot of false starts they came up with THIS idea, which is really a very brilliant inspiration you couldn't predict. There are still some morals for the students' own problem-solving though: they should persevere when problem-solving because you might have to make some false starts before hitting upon the right idea; more experience with solutions in a particular area make it more likely to come up with a helpful idea; going outwards to a bigger domain (like going up to 3D) can sometimes help. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The important thing for me is to be clear that you can learn about problem-solving from solutions to problems, even when it is a stroke of genius.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-09-09T23:26:59.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4366" PostTypeId="1" CreationDate="2014-09-10T19:20:59.820" Score="10" ViewCount="190" Body="&lt;p&gt;I am math teacher in secondary school and I have a few thousands of mathematical exercises in many Microsoft Word files.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to organize them in folders and keep track of new exercises to update my content, but it was hard working with many different type of exercices. I was thinking to create a web app for personal use where I could add all the exercises and also implement a query system to retrieve only those I want to create exams papers with specific difficulty or specific chapter. But, I had a problem with exercises with tables, bullets and not just simple text and math symbols.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for a software which help you to organize and categorize all your exercises and make your life as a teacher easier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I am looking a software to organize the exercises and not the documents. For example, in the web app I tried to create, I had to import the exercises one by one including meta-tags like difficulty, tags, chapters, images etc. With that way, I could run a query to retrieve all the exercises with specific tag or difficulty.&lt;/p&gt;&#xA;" OwnerUserId="2061" LastEditorUserId="2061" LastEditDate="2014-09-11T09:55:41.637" LastActivityDate="2014-10-13T17:56:58.543" Title="Software to organize and track your exercises" Tags="|software|organization|exercises|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4367" PostTypeId="1" AcceptedAnswerId="4369" CreationDate="2014-09-10T21:44:39.983" Score="18" ViewCount="4510" Body="&lt;p&gt;I already tried the direct approach, starting with &quot;this is how it works&quot;. That turned out ok but took too long and was boring for all of us.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My second attempt was using the twofingered alien. This worked better but needs improvement. The problem was that most kids in 5th grade are not aware of how the 10 based number system works. They just use it. I ended up having to explain the 10 based system in the middle of my binary alien story. That kind of interrupted the flow. But explaining the 10 based system before starting binary takes away most of the &quot;suprises&quot; of the aliens counting with just two fingers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;edit: I should probably add that I'm a math teacher in Germany and that binary numbers are on the agenda at this grade.&lt;/p&gt;&#xA;" OwnerUserId="2110" LastEditorUserId="2110" LastEditDate="2014-09-10T22:25:48.487" LastActivityDate="2021-03-20T03:27:18.187" Title="How to teach binary numbers to 5th graders?" Tags="|primary-education|mathematical-pedagogy|concept-motivation|" AnswerCount="14" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="4368" PostTypeId="1" AcceptedAnswerId="4375" CreationDate="2014-09-11T00:42:42.390" Score="6" ViewCount="171" Body="&lt;p&gt;Neither of the two most evident sources,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;MathWorld: &quot;&lt;a href=&quot;http://mathworld.wolfram.com/FinslerMetric.html&quot; rel=&quot;nofollow&quot;&gt;Finsler&#xA;Metric&lt;/a&gt;.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Wikipedia: &quot;&lt;a href=&quot;http://en.wikipedia.org/wiki/Finsler_manifold&quot; rel=&quot;nofollow&quot;&gt;Finsler&#xA;Manifolds&lt;/a&gt;.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;seems to provide me with the high-level intuition that I could convey&#xA;to students in ~10 minutes. And that is because I myself am not &#xA;sufficently schooled in&#xA;Finsler geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Either a pointer to a better source, our your own take, would be welcomed. Thanks!&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="143" LastEditDate="2014-09-11T18:51:57.847" LastActivityDate="2014-09-12T04:31:59.943" Title="Intuitive explantion: What is a Finsler metric?" Tags="|concept-motivation|geometry|intuition|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4369" PostTypeId="2" ParentId="4367" CreationDate="2014-09-11T11:47:24.223" Score="18" Body="&lt;p&gt;A sketch of one idea. I think it's probably better spread over a couple of days.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Day one:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start them counting, from zero, out loud to you. Write the numbers on the board as they go. Zero (0), one (1), two (2), ... , ten (10). Stop here. Prompt a discussion about what happened - how is the most recent number different than all of the previous numbers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I expect (admittedly I don't have a lot of experience with this age group) that they'll be able to come up with the fact that it's now a two digit number. Press them to explain why we weren't able to express ten as a single digit. With some guidance, I think it's possible for them to come up with something that approximates 'because we don't have any more numbers'. Go into that - explain that the digits 0-9 are symbolic in nature, and that we use them to represent quantities. Since we have ten of them, we're able to express ten different values with single digits, and when we want to express other values we have to use the digits in combination.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start over now, except ask the class to imagine that there were only 5 numeric symbols in the alphabet (0, 1, 2, 3, 4) and that 5 through 9 never existed. Again, count up from zero to four on the board, but put it to the class how five might be represented. This may be a struggle since lots of them will want very badly to write '5' - potential for some humor here though as you can remind them that '5' doesn't exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully you're able to get the suggestion of '10' (5) out of the class, and then you're able to proceed with '11' (6), '12' (7), on up to '14' (9). If you're able to get them to '20' for ten, then you're set - they get it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Day two:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Run through the base 5 counting exercise again. 1,2,3,4,10,11,12,13,14,20,21, etc. If things go well enough here then it's time to get into generalized discussion about how this sort of counting works in comparison with 'normal' counting. Ask if there's anything special about counting with 10 digits, or with 5 digits, or with any particular number of digits. I expect that they'll say no.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Move to binary now - tell the kids that there are only two number symbols (0 and 1), and get them to count aloud and instruct you on how to represent the numbers. They'll probably get through 0, 1, 10, 11 without much trouble, but you can help them move from here to 100 if they're stuck by asking about the transitions from 9 to 10 and 99 to 100 in 'normal' numbers. If things are going OK at this point, it might be appropriate to talk about computers, yadda yadda.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Thoughts...&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My original thought after reading this question was that binary for elementary students is an awful idea, and representative of the bad habit of 'back-porting' desired skills into inappropriate age groups. (We need computer technicians? Computers use binary! Teaching binary to kids will help!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But done carefully, it can potentially speak to one of the great joys in math - the whole idea is that you can make suppositions and then wonder about their implications. Suppose 5-9 don't exist? Absurd, but OK. Now how does counting work again?? Uh, did I understand counting in the first place? It's confusing in the way that mathematicians (of any age) love to be confused.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it's best to take the intermediary step here (base 5) because it's substantially less of a jump than going straight to binary. In base five, things proceed in the natural fashion most of the time - when incrementing the ones digit - and only the 'rollover' conditions are different. Binary rolls over with every other increment, and requires a carrying rollover very early on, which is really a difficult thing. By learning base 5 first, you make it easier for them to understand the fundamental &lt;em&gt;validity&lt;/em&gt; of different numbering systems, which makes it easier to process the ultimate extreme of using binary.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="-1" LastEditDate="2018-07-31T13:10:08.113" LastActivityDate="2018-07-31T13:10:08.113" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="4370" PostTypeId="2" ParentId="4367" CreationDate="2014-09-11T13:52:57.573" Score="4" Body="&lt;p&gt;A colleague of mine has pretty good success with a sitting and standing activity. The basic idea is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Line up 3 to 5 chairs in front of the class with a person sitting in each chair. The rules are: You sit down or stand up (i.e. change position) if the person on your left sits down and the person all the way to the left can sit and stand freely. The goal is to find the number of moves the person all the way to the left has to make to get the person all the way to the right to stand up.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-09-11T13:52:57.573" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4371" PostTypeId="2" ParentId="4351" CreationDate="2014-09-11T14:08:57.613" Score="10" Body="&lt;p&gt;Suppose we want a genuine tensor product $X\otimes Y$ of Hilbert spaces $X,Y$, meaning a Hilbert space (or, more generously, some locally convex topological vector space) $Z$ and continuous bilinear $b:X\times Y\to Z$ so that, for every continuous bilinear $B:X\times Y\to V$ (with $V$ a Hilbert space, most cautiously) there is a unique continuous linear $Z\to V$ making the obvious diagram commute.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;obvious&quot; candidate is to give the algebraic tensor product of $X,Y$ the inner product&#xA;$$&#xA;\langle x\otimes y,\;x'\otimes y'\rangle\;=\; \langle x,x'\rangle_X\cdot &#xA;\langle y,y'\rangle_Y&#xA;$$&#xA;and complete to form a Hilbert space. The obvious map of $X\times Y$ to that Hilbert space is continuous. However, if both $X,Y$ are infinite-dimensional, then this construction fails to have the desired universal property. (Whence the terminology &quot;projective&quot; and &quot;injective&quot; tensor products, in the literature, meaning TVS's which have one half or the other, but not both, the properties of a genuine tensor product.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-09-11T14:08:57.613" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4372" PostTypeId="2" ParentId="4367" CreationDate="2014-09-11T17:58:43.437" Score="3" Body="&lt;p&gt;Esmaya, I have done math circles around base 8 and base 3. I made a little story for base 8 called &lt;a href=&quot;https://drive.google.com/file/d/0B4Lou9CsLnQxbVkyZlZYOUotZ2c/edit?usp=sharing&quot; rel=&quot;nofollow&quot;&gt;Eight Fingers&lt;/a&gt;. Your students might have fun illustrating it. (The 'code' at the end is actually binary. If you have time for that sitting and standing activity, they would go together well.) &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-09-11T17:58:43.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4373" PostTypeId="2" ParentId="4367" CreationDate="2014-09-11T21:19:41.630" Score="7" Body="&lt;p&gt;I've never done this, but maybe something like the following will work. Since they don't really understand base $10$ (indeed, one can consider that being the end goal of doing the binary stuff), you don't want to get all tangled up in binary position notation and the like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start by looking at the numbers you get when you begin with $1$ and successively double:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$1, \;\; 2, \;\; 4, \;\; 8, \;\; 16, \;\; 32, \;\; 64, \;\; 128, \;\; 256, \;\; 512, \;\; \ldots$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leave this list on the blackboard or in some other place where the students can see it in what follows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then tell the students (in an age-appropriate manner) that even though this process removes relatively more and more numbers from a listing of all positive integers, the numbers that are left have a really neat property. Namely, &lt;strong&gt;every positive integer&lt;/strong&gt; can be written as a sum of these numbers where each of these numbers is either used just once or not at all. Moreover, there is only one way to do this for each positive integer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, $13 = 8 + 4 + 1$ and $23 = 16 + 4 + 2 + 1.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then pick a number, say $53,$ and show students how they can find the powers of $2$ that work. First, find the largest power of $2$ that doesn't &quot;overshoot&quot; the integer. In this case it's $32.$ Now subtract $32$ from $53$ to see what's left: $53 - 32 = 21.$ Now find the largest power of $2$ that doesn't &quot;overshoot&quot; the part that's left. This will be $16.$ Next, subtract $16$ from $21$ to get $5,$ and so on in the manner we all know about. At the end, tell the students that we'll record the result using a shorthand code. In the case of $53,$ the shorthand code is $110101$ because (reading the $0$'s and $1$'s from right to left) we used $32,$ we used $16,$ we didn't use $8,$ we used $4,$ we didn't use $2,$ and we used $1.$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can mention that the shorthand code always starts with a $1$ (because there will always be a largest power of $2$ we can use), but after that anything can happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps then give students two or three examples to try among themselves (small groups ought to be great for this). After most students seem to understand how to carry this out, you can then consider the reverse problem where someone gives you a code for the number and you figure out what the number is, such as $10010$ being $2 + 16 = 18$ and $110011$ being $1 + 2 + 16 + 32 = 51.$ Point out that when deciphering the code for a number, it's easiest to work through the digits from right to left (opposite the direction we used when finding the codes).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far there is nothing about positional notation or how this is base $2$ and we're used to working with base $10,$ but at least this will get across some aspects of binary notation without delving into abstract positional notation system stuff (which will likely fly right over their heads if you start there).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also be able to work in &lt;a href=&quot;http://en.wikipedia.org/wiki/Twenty_Questions&quot;&gt;the game of $20$ questions&lt;/a&gt;, explaining (carefully!) how the number of digits in the code tells you how many yes/no questions are needed.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2014-09-11T21:24:57.190" LastActivityDate="2014-09-11T21:24:57.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4374" PostTypeId="2" ParentId="2376" CreationDate="2014-09-12T01:58:40.933" Score="6" Body="&lt;p&gt;I agree with Mathew's opinion above about including challenging problems in worksheets for those who usually manage to breeze through the normal curriculum. Like the posts above, I don't encourage asking students not to read ahead because as a student myself, I may sometimes feel interested to learn in advance because of the nature of the topics (some might interest me) or perhaps I just want to learn more (in order to read deeper in some articles that I may have encountered elsewhere requiring the knowledge of the next topic for example). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a student, I like having challenging problems (perhaps labelled as Optional Questions) in addition to the normal worksheet. I would recommend non-routine kind of problems that gives students the space to thinking beyond what they are usually exposed to in classroom. Perhaps one good site to search for such question will be &lt;a href=&quot;https://brilliant.org/&quot; rel=&quot;noreferrer&quot;&gt;Brilliant.org&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another idea I have would be to give out articles of applications/fun facts/games/research/history (whatever's of a suitable level) of the topic at hand to students who finish their individual component/work faster. This might engage them in the remaining time of the lesson without disrupting the other students who might still be doing their work, and might also ignite new interests for the topic! :) &lt;/p&gt;&#xA;" OwnerUserId="1921" LastEditorUserId="1921" LastEditDate="2014-09-12T03:10:24.310" LastActivityDate="2014-09-12T03:10:24.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4375" PostTypeId="2" ParentId="4368" CreationDate="2014-09-12T04:31:59.943" Score="3" Body="&lt;p&gt;The easiest examples of Finsler metrics are manifolds where the length of a curve is defined by&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int \sqrt[\large{4}]{\dot{x}^4 + \dot{y}^4} dt.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Distances are then defined by the corresponding infimum.  With this in mind the definitions are easier.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-09-12T04:31:59.943" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4376" PostTypeId="1" CreationDate="2014-09-12T14:17:45.887" Score="12" ViewCount="9098" Body="&lt;p&gt;Some years ago I used mobile phone or internet rates (for example, with basic fees and a given charge per minute or by data volume) to introduce and motivate the study of linear functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, today (at least in Germany) most people have flat rates (or some semi- flat rates where the data speed is throttled after a given GB volume is reached). So mobile phone and internet rates are not ideal examples for students anymore. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Are there other examples that students (in my case, they are ~15 years old, most of them boys who are also interested in technology and science) may find motivating and that also have the didactic clarity and variability which some years ago the mobile phone rate examples had? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(I thought about something like electricity tariffs, but I have the feeling that they would find it boring and not related to their real lives.)&lt;/p&gt;&#xA;" OwnerUserId="1203" LastEditorUserId="262" LastEditDate="2014-10-15T01:51:52.093" LastActivityDate="2014-10-22T17:39:59.027" Title="Real life examples to motivate the study of linear functions" Tags="|secondary-education|reference-request|concept-motivation|functions|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4377" PostTypeId="2" ParentId="4376" CreationDate="2014-09-12T16:26:36.350" Score="4" Body="&lt;p&gt;Maybe encourage them to come up with some examples?  Affine maps are so ubiquitous that it seems hard to motivate them:  it is like trying to motivate the air.  Any time some quantity is changing at a constant rate with respect to some other quantity, you have a linear function.  It is probably more fun for the students to try to come up with examples than listen to your examples.  For a sequence of progressively more demanding examples, maybe the following will do:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Assuming 1 pound of bodyweight is 3500 kcal, model effect on bodyweight of various daily caloric deficits and surpluses compared to daily energy requirements.  Applications to bodybuilding.  This can be coupled with &quot;real world&quot; diet planning.  How many grams of which foods would you need to include...etc&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Starting amount of money and weekly paycheck?  What is the relationship between the functions when you take day, week, or month as unit of time?  When you take dollar, euro, or pound as unit of money?  What happens to the graph of these functions? Try to get them to think of this in terms of composition of linear functions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Linear interpolation of data.  Have a bunch of discrete points, maybe stock market data, and approximate the continuous data by connecting the dots with straight lines.  How can you &quot;automate&quot; this process? (This will motivate the &quot;point slope&quot; form of a line).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could even throw some lightweight &quot;calculus&quot; at them, by given them discrete data for velocity of some object, have them linearly interpolate, and then find area under these piecewise linear curves using geometry to recover an approximation of distance traveled.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-09-12T16:26:36.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4378" PostTypeId="2" ParentId="4376" CreationDate="2014-09-12T18:51:14.177" Score="8" Body="&lt;p&gt;In his blog post, &quot;&lt;a href=&quot;http://blog.mathpl.us/?p=101&quot;&gt;An Argument Against the Real World&lt;/a&gt;&quot;, Mr. K argues that fictional (even mythical) problems can work better than 'real-world' problems. I think he has a great idea.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-09-12T18:51:14.177" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4379" PostTypeId="1" CreationDate="2014-09-12T20:42:21.497" Score="22" ViewCount="6090" Body="&lt;p&gt;I have reposted this question on &lt;a href=&quot;https://softwarerecs.stackexchange.com/q/11882/2323&quot;&gt;Software Recommendations&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm looking for a web app for tutoring math remotely to high school and junior high kids that ideally has the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Build in graphing.  The exemplar for this is the application Geogebra, which not only allows graphs, but also has a table view, sliders, and animation, and is scriptable.  Indeed, a networked multi-user large canvas version of geogebra would meet most of my requirements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Built in drawing tools.  At a minimum:  freehand, line, polygon, text, and eraser.  Drawing tools should be vector based and not pixel based, so they can be easily moved and edited. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Editing tools:  It should be possible to change stroke, fill, fontsize, font color, location, rotation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additional desirable tools:  Marque select, group, layer, lock, bezier, arc, fill&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Desirable Stationery:  Various kinds of lined/graph paper including graph paper with index lines (heavier lines) Polar coordinates, log, log-log etc.  This could be done as a PDF or GIF image layer, or best, would be as a fill you could position in a rectangle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multipage canvas.  Application should not be limited to a single screen. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Persistent sessions.  I want to come back tomorrow and see it how I (or my partner) left it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiple simultaneous editing.  I should be able to edit one thing&#xA;while another user is editing elsewhere.  This may mean that the&#xA;other user may be on a different page from me.  As an adjunct, a way&#xA;of seeing what changes were made while I wasn't looking at a page&#xA;would be useful -- deleted items in pink, and new ones in green&#xA;maybe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Handwriting recognition for math.  This one is tough.   MyScribe MathPad looks good, but it's online demo only does a single equation at a time.  webFluidMath has one approach, but it's not really ready for prime time. It took me 6 attempts to get ax^2 +bx + c = 0, and I never was able to do a 2x2 matrix.  In principle I could create the equation in one app and paste it in, but this makes editing difficult, and gets in the way of teaching the derivation of a theorm or equation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A keyboard approach is an acceptable alternative.  The best of these I've found so far is Lyx, which has a combination of keyboard and mouse selection. MathML requires lots of extra () to be added.  FrameMaker's equation editor got one thing correct with the use of a space character to exit one level of nesting.  Lyx does this too.  I do &lt;em&gt;NOT&lt;/em&gt;  want to type full TeX.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this reason I'm not considering various TeX enabled chat room programs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Handwriting recognition is desirable, but not required.  I will live with my scrawls if need be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compatible with iPad.  Apple limits script functionality on their platform.  Notability is has a decent approach on the ipad.  You can draw, you can bring in PDFs and annotate, add images, and draw on them etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that none of the applications mentioned (Geogebra, Mathpad, webFluidMath, Lyx, Notability) is collaborative software in the sense that multiple people can use it in real time, seeing the changes that the other party has made with only minor delays.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The use of 'networked' here means Internet -- The two parties will NOT be on the same LAN.  I do not care if it is application based, or web based, except that if the former, it has to run on iPad, Windows &gt;=XP, and Mac &gt;= 10.6 (Snow Leopard)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;TeamViewer is one example of a desktop sharing app.  This is one approach, but only one user can control the mouse/keyboard at a time, and whatever app you are using is limited to the present screen.  Teamviewer is clunky to use on a network with significant latency (over 100 ms)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definitions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Real Time Interactive: (RTI)&lt;/em&gt;  All parties can write at the same time, on different parts of the canvas, possibly different screens, using different tools. Delay between updates measured in seconds.&#xA;&lt;em&gt;Turn Based Interactive: (TBI)&lt;/em&gt;  Only one person can draw at a time.  All people see the same screen/zoom/view.  TeamViewer works like this.&#xA;&lt;em&gt;Shareable:&lt;/em&gt;  One person can create a document, publish it to some form of cloud repository, then other people can edit it.  Google docs works  like this most of the time.&#xA;&lt;em&gt;Presentation&lt;/em&gt; Other people have read only access, possibly in real time, possibly saved as a recorded session.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Persistent&lt;/em&gt;  A session can be stopped then later resumed even if there is a period when no one is using the document.&#xA;&lt;em&gt;Transient&lt;/em&gt;  Session vanishes when creator or last client quits the application.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Pixel based graphics (PBG)&lt;/em&gt;  Tools change the color of pixels on the screen.  Generally impossible to edit.&#xA;*Vector Graphics (VG): *  Entities have individual existence, so you can move parts around after creating them, and change attributes such as line width, size, stroke color and fill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Existing product limitations:  Most of these evaluations were done quickly, and stopped once I ran into 2-3 items not on my wish list.  Some of these may be useful for other purposes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;iDroo&lt;/em&gt;  VG, RTI, limited drawing set.  No math.  Can drag and drop images onto the canvas, or into a column on the edge, where they are stored as thumbnails. Active development.  Most usable whiteboard  of the lot.  Best for tutoring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;VMT&lt;/em&gt;  This is the virtual math team project at Drexel university.  Their Java app can run sharable whiteboard or geogebra in a tabbed interface.  Whiteboard is clunkly and somewhat limited.  GeoGebra has a HTML5 Canvas version in Beta.  Worth Watching.  Geogebra tabs can be pre-loaded with a Geogebra worksheet.  Best for instructing a class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Baiboard&lt;/em&gt; is one collaborative app that has possibilities.  It's sketch + PDF annotate with the ability to save snapshots at any time.  This is a good example of the collaboration aspect. The drawing tools are weak.  Only available on iPad and Mac, and the Mac version is much more limited.  Persistent, RTI&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Groupboard:&lt;/em&gt;  Limited VG: Objects can be moved, but not modified, RTI. Math is limited to pasting from a limited symbol set.  &lt;a href=&quot;http://www.groupboard.com/demo/math.shtml&quot; rel=&quot;noreferrer&quot;&gt;http://www.groupboard.com/demo/math.shtml&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Groupworld:&lt;/em&gt;  This is from the same people as Groupworld.  Same problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Board 800:&lt;/em&gt;   Limited VG: Objects can be moved, but not modified, RTI. Limited drawing set. No math.  Multi page.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Tutorsbox:&lt;/em&gt;  VG, RTI.  Objects can be moved but not modified.  Limited tools.  Line, circle, square. Function grapher.  Has wysiwyg math editor, but it operates in a modal window, which makes deriving something tricky -- you cannot see the previous line. &lt;a href=&quot;https://tutorsbox.com/en/&quot; rel=&quot;noreferrer&quot;&gt;https://tutorsbox.com/en/&lt;/a&gt; Plans start at $9/month&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Show me:&lt;/em&gt; Pixel based.  No editing. P.o.S.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;iDroo:&lt;/em&gt;  VG, RTI.  Limited drawing set. Pencil, line, rectangle, elipse, text. No color. No math.  Objects can be moved and rotated.  Marque select to move multiple objects.  &lt;a href=&quot;http://www.idroo.com/&quot; rel=&quot;noreferrer&quot;&gt;http://www.idroo.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;RocketBoard&lt;/em&gt; &#xA;  This is an actual white board sharing app:  You write on a whiteboard using standard markers, and the app corrects for perspective, and adjusts contrast.  Slick way to lecture.  BUT...&#xA;* No way to import material, or graph paper.&#xA;* No way for me to annotate another person's work, or for them to copy/paste from my work except as a bit map.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;AWW A Web Whiteboard:&lt;/em&gt; Pixel based, no erase except clear, 4 colors.  P.o.S.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Scribblar:  $9/month.  Free plan doesn't have many features.  Untested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have asked for recommendations elsewhere on StackExchange but have not received good results. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar posts:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://webapps.stackexchange.com/questions/8761/is-there-a-web-application-that-allows-tandem-drawing-painting&quot;&gt;https://webapps.stackexchange.com/questions/8761/is-there-a-web-application-that-allows-tandem-drawing-painting&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://apple.stackexchange.com/questions/89304/math-via-skype-on-a-mac&quot;&gt;https://apple.stackexchange.com/questions/89304/math-via-skype-on-a-mac&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://apple.stackexchange.com/questions/98874/software-recommendation-for-fast-entry-of-equations&quot;&gt;https://apple.stackexchange.com/questions/98874/software-recommendation-for-fast-entry-of-equations&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://webapps.stackexchange.com/questions/13532/tool-for-teaching-maths-online&quot;&gt;https://webapps.stackexchange.com/questions/13532/tool-for-teaching-maths-online&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2114" LastEditorUserId="-1" LastEditDate="2017-04-13T12:57:24.690" LastActivityDate="2020-09-06T02:47:00.353" Title="Real time interactive whiteboard for tutoring math" Tags="|secondary-education|tutoring|software|online-instruction|" AnswerCount="5" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="4380" PostTypeId="2" ParentId="4097" CreationDate="2014-09-12T21:59:39.627" Score="4" Body="&lt;p&gt;You can't build on sand.  If they don't have the reading skills, you need to teach that first.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(There is a strong temptation to go to the prior teacher of reading and beat them about the head and ears for their incompetence, along with the administrators of the school for passing kids that aren't ready, but this, while satisfying, seldom gets results, and can result in your search for a new job.  This, may be a solution.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest giving a short course in decoding word problems.  One phrase at a time.  Where possible illustrate the problem with an undimensioned example of what they are supposed to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with simple examples. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The shadow of a 50 foot flagpole is 110 feet from the base of the pole.  How many degrees above the horizon is the sun?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or even simpler:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;John has 6 times as many apples as Chris.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;John has 6 more apples than Chris.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;John has 6 fewer apples than Chris.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;John has half as many apples as Chris does.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;John's speed is 1/3 more than Chris's speed.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also:  Go the other way.  Draw the picture on the board, and have them generate a word problem.  (You may also find that your students do not understand drawings.  That too is a learned skill.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concentrate on the phrases.  There is a rather odd phrasing in word problems in math books, with passive case being used a lot, and far more complex sentence structure than commonly used in speech.   Do easy ones first, then do ones taken from achievement tests and standardized tests to get them familiar with the (sometimes arcane) wording of those instruments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may find yourself skipping a lot of word problems while you make up their reading deficiencies.  Do not worry about doing word problems at a couple years below the current nominal grade level.  At this point, you are looking for the translation from words to math.  Let them be easy math problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One step at a time.&lt;/p&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-12T21:59:39.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4381" PostTypeId="2" ParentId="3938" CreationDate="2014-09-12T22:17:46.503" Score="3" Body="&lt;p&gt;Decompartmentalize.  Start the new chapter as written, then connect it back to things you did before.  Also introduct things that you can't do yet with what they know so that there is mystery and anticipation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math is about patterns.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A lot of math is taught just as symbol shuffling.  Stress visualization.  E.g.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw 3 rectangles that are labeled 47 x 37  etc.  This works great if you have a presentation screen and can move figures around and rotate them.  Show them that they line up end to end with a common 37' length that they can factor the 37 out.  Do this with different combinations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then show it with one of the 37's divided into a 20 and a 17.  So now you apply the factoring twice. in different ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leave them with a packing problem where you have a whole list of rectangles to pack into another rectangle.&lt;/p&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-12T22:17:46.503" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4382" PostTypeId="1" CreationDate="2014-09-12T22:51:41.383" Score="13" ViewCount="693" Body="&lt;p&gt;Substituting $x=1$ into $px^{p-1}$, why do so many students get $p^{p-1}$? I saw this four or fives times in my office hours this week as students worked on the same problem. Not a single student spontaneously got this step right. Some write $p(1)^{p-1}$ as an intermediate step. I doubt that any of them would have had trouble evaluating $3x^2$ at $x=1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to get them to articulate why they would think this, and none of them could. It could be that they don't understand that exponentiation has higher priority than multiplication, but I doubt they would have evaluated $3x^2$ to be 9.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems like the symbolic exponent makes their brains turn off. Many students couldn't evaluate $1^{p-1}$ by itself, even after I asked them what $1^{87}$ was, and they knew it was 1.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2022-11-13T19:41:50.710" Title="Substituting $x=1$ into $px^{p-1}$, why do so many students get $p^{p-1}$?" Tags="|algebra|students-mistakes|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4383" PostTypeId="2" ParentId="4235" CreationDate="2014-09-12T23:21:23.173" Score="3" Body="&lt;p&gt;Several strategies:&#xA;1.  Practice.  Copy illustrations out of books.  Start simple. Mix your practice:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Draw small copies -- about the same size as the one in the book.&lt;/li&gt;&#xA;&lt;li&gt;Draw full page copies.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Draw on the black/white board&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;If you have access to a projector you can do some impressive illustrations of geometry and calculus using Geogebra.  Geogebra has a lot of teaching posibiltiies if your kids have computer access.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You can also used shared applications that echo your iPad to a projector.  Some of these allow you to drag in presaved PDF's or images into the screen.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use a grid of dots as a guide.  I covered one whiteboard with a tiny + written in permanent sharpie on 2 inch spacing.  Took a couple hours to do.  But that made for slick graph sketching.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-12T23:21:23.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4384" PostTypeId="2" ParentId="737" CreationDate="2014-09-12T23:26:55.133" Score="8" Body="&lt;p&gt;Part of the problem is that we often talk and write as if something is well-defined and then check well-definedness afterward.  For example, we might write $f(x)=\dots$ and afterward check that &quot;$\dots$&quot; really does define a function. Unfortunately, in the (usually very brief) time between writing $f(x)$ and checking that it's well-defined, we have probably confused some students. We've used a notation $f(x)$ that (usually) presupposes that $f$ is a function, without yet having verified the presupposition.  Strictly speaking, we should perhaps have used a notation for relations (see David Speyer's answer), like $R(x,\dots)$ or $(x,\dots)\in R$, then verified uniqueness, and only afterward used function notation.  But that's not what we usually do.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For people who already understand the issue, it's no problem that we use function notation before checking its legitimacy, but for students who are just learning (or trying to learn) these things, it can cause confusion.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-09-12T23:26:55.133" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4385" PostTypeId="2" ParentId="4235" CreationDate="2014-09-12T23:33:44.937" Score="4" Body="&lt;p&gt;I think the solution for your problem depends very strongly on what sort of course you're teaching.  If you &lt;em&gt;really&lt;/em&gt; need accurate drawings, then I can't help you; my own drawings are usually pretty bad.  But in many circumstances, the drawing need not be very accurate; it just needs to convey the right idea.  In that case, I'd suggest doing the best you can while drawing and then alerting the class to the (perhaps obvious) fact that it's not very accurate.  I've often drawn some sort of closed convex curve on the board and told the class &quot;Pretend this is a circle.&quot;  &lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-09-12T23:33:44.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4386" PostTypeId="2" ParentId="4382" CreationDate="2014-09-13T00:38:27.903" Score="5" Body="&lt;p&gt;I don't think the proper question is why they do this. A better question is: How can we get them to keep their brains turned on while doing math? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try to ask a student in this situation a series of questions: What is 1^87? You say that's 1? Why? (Good reason given.) Now what is 1^n? [If that is too hard, make some questions that get at their trouble. Maybe just ask 1^15, 1^20, possibly until they roll their eyes and say it's always 1. Oh? What does n represent here?  ... What wrong answer did they give? Can you figure out where it comes from? That will help you decide what questions to ask them.] If that's not too hard, now you get to ask about 1^(n-1). Is n easier for them than p? That would be interesting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think what I'm suggesting would be called Socratic dialogue. Maybe we need some youtube videos of this sort of thing...&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-09-13T00:38:27.903" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4387" PostTypeId="2" ParentId="4382" CreationDate="2014-09-13T01:59:47.857" Score="13" Body="&lt;p&gt;I think this is what goes on in their head:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$3x^2$ is very familiar to them, and it is clearly $3 \times (x^2)$. It helps that $3$ is a number and $x$ is a variable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$px^{p-1}$ is not so familiar, and unless they are paying complete attention (using their &lt;a href=&quot;https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow#Two_systems&quot;&gt;System 2&lt;/a&gt;), their lazy System 1 automatically associates it with the nearest familiar expression, namely those of the form $31^2$ (in $px^{p-1}$, all are variables, and in $31^2$, all are numbers $-$ this probably makes them associated). But $31^2$ is, of course, $(31)^2$, and not $3 \times 1^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you &lt;em&gt;ask them&lt;/em&gt; about it, however, they are forced to pay attention, and System 2 makes sure they think it through, so the mistake cannot be replicated. And as System 1 works automatically and subconsciously, they do not even understand what they were thinking before.&lt;/p&gt;&#xA;" OwnerUserId="1606" LastActivityDate="2014-09-13T01:59:47.857" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4388" PostTypeId="2" ParentId="4382" CreationDate="2014-09-13T05:12:03.543" Score="11" Body="&lt;p&gt;It may not be helpful to think of it as their brains turning off. Instead, perhaps their brains have never turned &lt;strong&gt;on&lt;/strong&gt;!They may never have thought about what the operations there mean, and have been working on a stimulus-response basis where they see certain symbols and respond in certain ways. A combination of symbols that doesn't fit neatly into one of the categories they are familiar with will be responded to a fairly random way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My instinct is that their expected response to an expression with a pronumeral in it is to give a new expression which still has a pronumeral in it, matching with their experience of expanding and/or simplifying. The answer being a plain number like 1 is not something they expect to happen and so it doesn't even occur to them as an option. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, their experience with pronumerals in the exponent is probably very limited, and I would hazard a guess that in all of those experiences, there was still a pronumeral in the exponent after simplification. So the pull to keeping the pronumeral in the exponent is probably strong. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They are probably not consciously seeing a pronumeral as a representation of a possible number. They see numbers and pronumerals as completely different entities with different rules of reckoning. All they have in their heads are rules for manipulating things. They don't realise that a good strategy to assess their understanding is to try it out with various numbers to see if it makes sense for each case, because they don't see it as a description of something that works for multiple different numbers -- it's just symbols. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To go further on this line, in one of your comments to another answer you say&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... $p1^{p-1} = p^{p-1}$. It seems like there must be some reason that&#xA;  nearly all my students think this same thing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is probably not true that the students think of these as the same thing at all. They probably don't even think of $3x^2$ and $3xx$, or $3 \times 1^2$ and $3$ as the same thing. All they see is manipulations that are or are not allowed and have expectations for what might or might not happen as a result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In light of this discussion, you may need to really get them to talk about what they think expressions mean. Trying the expressions out with various different values of both x and p might help them to see that the expression is a description of how those numbers are combining to make an answer. I mention using different x's to highlight that 1 is a special case. Talking about how even if you don't know p, you can be sure the answer is 1 and how this would not be the case for most other x's would be helpful where the answer is expressable as a number, as opposed to not being able to do any manipulation.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-09-13T05:12:03.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4389" PostTypeId="2" ParentId="4235" CreationDate="2014-09-13T13:19:39.327" Score="2" Body="&lt;p&gt;This is not an answer, and will only make us envious.&#xA;I think that one reason &lt;a href=&quot;http://www.math.upenn.edu/~ghrist/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Ghrist&lt;/a&gt;&#xA;is such an amazingly effective lecturer is that he has a remarkable drawing talent:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/4ul4C.png&quot; alt=&quot;GhristTopology&quot;&gt;&#xA;&lt;hr /&gt;&#xA;See also his &lt;em&gt;&lt;a href=&quot;https://play.google.com/store/books/details?id=HbbeRHUozJcC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Funny Little Calculus Text&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-09-13T13:19:39.327" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4390" PostTypeId="2" ParentId="4258" CreationDate="2014-09-13T15:28:59.183" Score="1" Body="&lt;p&gt;Teaching Math online is non-trivial.&lt;br&gt;&#xA;1.  You need both some form of blackboard/whiteboard link that updates in near real time.  (Delay of no more than a second or two.)&#xA;2.  You need a good audio link for the students to hear you speak.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far so good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need a way for a student to ask a question, so the audio link has to be selectable and two way.  You need a way for the student to indicate that she/he wants to ask a question.  This can be done with chat, but also really needs a way for the student to use a mouse on a particular point on the screen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally you do not want the entire class to be able to edit the screen, although having kids do a problem on their own page then review, is a good way to do things.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Whiteboard should have provision for bringing in backgrounds.  Much easier to graph on a sheet of electronic graph paper.  Drawing on a grid is generally easier.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;You need a reasonably fast link to your students.  Interactive whiteboards start to get clunky at delays of over 150 ms or so.  This is fairly easy to achieve on a single continent, but overseas doesn't work well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to speak the same language as your students.  More than that, you need to speak with the same accent. Accent problems are not helped by the distortions introduced in digital links.  I have difficulty with English as spoken by most people from India.  I had a Chinese teaching assistant in calculus that was so bad that I stopped going to the help sessions.  Impossible accent and very limited vocabulary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way to advertise is on the internet.  Use the online classifieds such as craigslist, kijiji, backpage and place an ad in the services section.  I strongly recommend just doing one region at a time.  It will be far easier to teach if all your students are using the same textbook.&lt;/p&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-13T15:28:59.183" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4391" PostTypeId="2" ParentId="4074" CreationDate="2014-09-13T15:35:57.647" Score="-3" Body="&lt;p&gt;Has it been studied?  I don't know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does it work?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, no.  I have taught kids that came from asia where their math was done by rote.&#xA;Their arithmetic skills were impressive, but when presented with a large number of new ideas they had trouble seeing how the pieces fit together, and were often at loose ends when faced with a new variation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a place for rote learning?  Yes.  Doing many problems.  This helps build facility with algorithms.  &lt;/p&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-13T15:35:57.647" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4392" PostTypeId="2" ParentId="4083" CreationDate="2014-09-13T16:19:13.483" Score="4" Body="&lt;p&gt;Here's a few things that may help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Borrow a concept from science -- the notion of falsifiability.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There repeated verification of a theory is just that -- verification but not proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In science we cannot prove something correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may find help in the phrase &quot;formal proof&quot; or in the word &quot;derivation&quot;  Introduce them to conjectures too such as Goldbachs.  By using a different label, you can help separate the concepts in their head.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have a discussion about 'what is truth?'  Show that in everyday life we require different levels of evidence and different kinds of proof.  I am more likely to believe stuff I see in the Washington Post rather than in the National Enquirer.  NBC news has a better rep than Fox.  When a friend tells you that Susan is sleeping with Mike, you require less evidence than you would if that same friend tells you that Challenger has exploded. (Happened to me.  Took him 10 minutes to convince me that he wasn't kidding.)  At this point Mathematical Proof becomes yet another type of proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is material that should be covered in a course on logic, but alas, we don't generally teach logic anymore.  &lt;/p&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-13T16:19:13.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4393" PostTypeId="2" ParentId="1860" CreationDate="2014-09-13T16:48:01.340" Score="2" Body="&lt;p&gt;I like the historical example.  Computationally division is lots slower than multiplication somewhat depending on the numbers. (A quick check shows 8-20 times slower) A good compiler would optimise dividing by 2 with a shift operation.  I don't know if generally dividing by a long decimal is either more time consuming or less accurate than dividing by a small integer.  These obviously don't matter to someone using a calculator, but could well matter when doing the billions of operations needed to update a screen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition I think there is merit in having canonical forms.  Why do you ask students to reduce 3/6 to 1/2.  Both are valid.  Why not leave an an answer as 31+12-16 instead of 27?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly checking agreement between two polynomials starts getting messy if they are in arbitrary order. (and if there are multiple variables I'm not aware of any canonical form for the answer)&lt;/p&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-13T16:48:01.340" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4394" PostTypeId="2" ParentId="4366" CreationDate="2014-09-13T16:58:12.530" Score="1" Body="&lt;p&gt;Word?  Yuck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately you need a document management system, each exercise has to become it's own file, although you can cheat and put multiple similar exercises in the same file, then each exercise is assigned keywords.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could do this with screen snapshot software and a good photo cataloging package.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Adjust word settings so that the full path to the file shows on screen.&lt;/li&gt;&#xA;&lt;li&gt;Take a snap of each exercise.&lt;/li&gt;&#xA;&lt;li&gt;Catalog the resultant images.  You can uses faceted keywords to make later search easier.  E.g. Diff:Hard allows you to find the hard problems without finding the ones that otherwise use the word hard.  Top:DifEq could stand for topic differential equations.  Cor:Math211 could stand for what math course(s) you used it for, and so on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-13T16:58:12.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4396" PostTypeId="2" ParentId="1042" CreationDate="2014-09-14T21:21:04.967" Score="2" Body="&lt;p&gt;You might also want to read Euclidlab - &lt;a href=&quot;http://euclidlab.org/unsolved&quot; rel=&quot;nofollow&quot;&gt;http://euclidlab.org/unsolved&lt;/a&gt;. They have many interesting projects, which should appeal to high-schoolers (I am a high-schooler myself and am thus speaking from ``experience''). In fact, I am planning on joining Camp Euclid, which other high-school students might find interesting (it costs a lot, though).&lt;/p&gt;&#xA;" OwnerDisplayName="user207" LastEditorDisplayName="user207" LastEditDate="2014-09-15T22:27:30.503" LastActivityDate="2014-09-15T22:27:30.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4397" PostTypeId="1" CreationDate="2014-09-14T21:33:52.763" Score="10" ViewCount="1255" Body="&lt;p&gt;I'm a 25 years old guy and I wanna learn math. I haven't had a good relationship with them until now. My background is that I know how to add, subtract and multiply. In theory I also know how to divide, but some times I'm messing it up or get stuck with what the next step is supposed to be. I blame calculators for that. I know how to solve basic equation and I used to know how to solve quadratic equations too, but last time I solved one was 7 years ago. I'm rumbling, I know. Basically I'm trying to say is that I don't know what I know about math, but I could say that I'm at a very basic level.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to learn math because about a year ago, I was watching the Linear Algebra lectures from MIT on YouTube. It was amazing! I could feel my mind working to its limits to understand all this new information! I was feeling challenged for the first time in my life and my brain working at 110% capacity to cope. But I tried to learn too much too quickly (because linear algebra was a prerequisite for a course in coursera, about to start in one week) and I failed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I want to start from the beginning and take this slowly step by step. But I also would like a challenge. For example I would like to tackle proofs of theorems without learning them. Just something (a lecture, a book..) giving me the tools to prove something and then make me discover it, following the steps of those who first discovered the theorem. I want to exercise my creativity, my abstract thought and my problem solving. I want to feel my mind expanding! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where do I start? What books or online material would you advice me to read(or watch)? Please make sure that you give me a list for the next years, so I don't have to come back here and ask the same question once I'm done with the basics.. And how much time would you say I have to give to studying? I'm working and I'm trying to write a novel, so I don't think I can study for let's say 5 hours per day, every day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well that's it! I know it was not the best worded question in the world, but I hope it is sufficient to give you all the data you need to put me in the right road.&lt;/p&gt;&#xA;" OwnerUserId="2119" LastEditorUserId="2122" LastEditDate="2014-09-15T09:57:49.080" LastActivityDate="2014-09-16T00:35:58.920" Title="I wanna learn math!" Tags="|self-learning|" AnswerCount="5" CommentCount="3" ClosedDate="2014-09-16T01:49:15.813" ContentLicense="CC BY-SA 3.0" />
  <row Id="4398" PostTypeId="2" ParentId="4397" CreationDate="2014-09-14T23:10:10.207" Score="1" Body="&lt;p&gt;&lt;a href=&quot;http://www.khanacademy.org&quot; rel=&quot;nofollow&quot;&gt;Khan Academy&lt;/a&gt; has a nice overview of primary, secondary, and some university mathematics, presented in a way that is clear and easy to self assess. I would encourage you to start there, and then when you find a topic you're especially interested in, go on and explore it independently.&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2014-09-14T23:10:10.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4399" PostTypeId="2" ParentId="4397" CreationDate="2014-09-15T00:36:04.903" Score="6" Body="&lt;p&gt;It seems you have discovered your new great passion for math. I suggest continuing to feed this passion. In addition to Khan Academy, you may also find &lt;a href=&quot;http://www.cut-the-knot.org/&quot;&gt;cut the knot&lt;/a&gt; a good place for interesting problems online. Keep searching online and find the sites that are most suited to your interests.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also suggest visiting your local library or bookstore. You should be able to find reading material to keep you interested for many lifetimes.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-09-15T00:36:04.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4400" PostTypeId="2" ParentId="4397" CreationDate="2014-09-15T02:26:06.237" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://www.jamestanton.com/&quot; rel=&quot;nofollow&quot;&gt;James Tanton&lt;/a&gt; has some very interesting videos. He treats math as an adventurous exploration. (While Khan helps people slog through the topics in their math courses.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd also recommend Keith Devlin's &lt;a href=&quot;https://www.coursera.org/course/maththink&quot; rel=&quot;nofollow&quot;&gt;Introduction to Mathematical Thinking&lt;/a&gt;, on coursera.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have fun!&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-09-15T02:26:06.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4401" PostTypeId="2" ParentId="4397" CreationDate="2014-09-15T02:42:53.347" Score="2" Body="&lt;p&gt;If you want to &lt;em&gt;learn&lt;/em&gt; math, then you have to &lt;em&gt;do&lt;/em&gt; math.  (As the other answers suggest, one way to do this is through Khan Academy or something similar to it.)  This will take a lot of time and effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you simply want to &lt;em&gt;appreciate&lt;/em&gt; math, then I suggest you read books and articles by writers who popularize mathematics:  &lt;a href=&quot;http://en.wikipedia.org/wiki/Martin_Gardner&quot; rel=&quot;nofollow&quot;&gt;Martin Gardner&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Ian_Stewart_%28mathematician%29&quot; rel=&quot;nofollow&quot;&gt;Ian Stewart&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Keith_Devlin&quot; rel=&quot;nofollow&quot;&gt;Keith Devlin&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Danica_McKellar&quot; rel=&quot;nofollow&quot;&gt;Danica McKellar&lt;/a&gt;, and so on.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-09-15T02:42:53.347" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4402" PostTypeId="1" AcceptedAnswerId="4411" CreationDate="2014-09-15T20:56:15.693" Score="3" ViewCount="159" Body="&lt;p&gt;I am teaching Mathematical Economics for undergraduate students. Students have taken Mathematics I and II during their first year and they must take Mathematical Economics when they are third year students. &#xA;I have two options for teaching: first, I can teach Mathematics Rules and proofs (such as derivatives and integrals), or, as a second option I can teach only Economical applications. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I take the first option I must pay more time for reteaching basics that they had seen during first year, but I think they may relearn the basics of Mathematics, and Mathematical Economics too. On the other hand, with the second option I can focus on Economical usages of mathematics only and offer more application examples. But in this case students may lose their interest in math and maybe start to fear Mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which option is better for my students?&lt;/p&gt;&#xA;" OwnerUserId="79" LastEditorUserId="2083" LastEditDate="2014-09-16T07:23:47.253" LastActivityDate="2014-09-17T20:08:10.603" Title="Teaching Mathematics rules or its applications" Tags="|undergraduate-education|applied-mathematics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4403" PostTypeId="2" ParentId="4397" CreationDate="2014-09-15T23:23:30.883" Score="10" Body="&lt;p&gt;It's great to hear about your enthusiasm to learn math!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No fewer than three answers have suggested Khan Academy. Khan Academy sucks. (In my not-so-humble opinion.) The Khan Academy videos basically crank through calculations, without encouraging any deeper understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It sounds like you have bits and pieces of algebra. Next, you probably want to check an algebra textbook out of the library and sort the topics out into three categories: (1) stuff you already know, (2) stuff you need to know, (3) stuff you don't need to know, at least for now. A &lt;em&gt;lot&lt;/em&gt; will fall into category 3, e.g., factoring polynomials, which is a topic much beloved by math teachers but seldom necessary outside of an algebra class. If you're not sure whether something fits in category 3, make a note of it, remember that such a thing exists, and if you need it later, come back to it. Once you've identified category 2, solve some problems that have answers in the back of the book, so you can check yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you're solid in algebra, the world is your oyster. You can pretty much pick up any math book and dig in. A cool, old book on calculus is &lt;a href=&quot;http://www.gutenberg.org/ebooks/33283&quot;&gt;Calculus Made Easy&lt;/a&gt; by Silvanus P. Thompson.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="64" LastEditDate="2014-09-16T00:35:58.920" LastActivityDate="2014-09-16T00:35:58.920" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4404" PostTypeId="5" CreationDate="2014-09-16T05:28:02.410" Score="0" Body="&lt;p&gt;For questions related to applications of mathematics and the teaching thereof.  Topics include science and engineering, economics, statistics, computer science, operations research, optimization, etc.&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2014-09-16T14:11:06.653" LastActivityDate="2014-09-16T14:11:06.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4405" PostTypeId="4" CreationDate="2014-09-16T05:28:02.410" Score="0" Body="For questions related to applications of mathematics and the teaching thereof." OwnerUserId="263" LastEditorUserId="263" LastEditDate="2014-09-16T14:11:16.050" LastActivityDate="2014-09-16T14:11:16.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4406" PostTypeId="2" ParentId="4333" CreationDate="2014-09-16T13:31:08.407" Score="4" Body="&lt;p&gt;As in @user52817's graphically-oriented answer, I think it is fair to secretly borrow some ideas nowadays labelled &quot;complex analysis&quot; to help explain &lt;em&gt;why&lt;/em&gt; this should work: that is, while thinking in terms of the &lt;em&gt;residue&lt;/em&gt; at a simple pole $x_1$, one can explain it (perhaps quasi-geometrically...) as giving the &quot;asymptotics&quot; of the rational function as $x\to x_1$. That is, for low-degree numerator, $f(x)/(x-x_1)\ldots(x-x_n)\sim {1\over x-x_1}\cdot {f(x_1)\over (x_1-x_2)\ldots(x_1-x_n)}$ near $x_1$, and so on. Subtracting all these &quot;polar parts&quot; (or whatever one chooses to call them for calculus students) leaves a rational function that has no poles, and does goes to $0$ for large $x$, so has to be $0$ (a tiny prototype for Liouville's theorem, perhaps).&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-09-16T13:31:08.407" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4407" PostTypeId="1" CreationDate="2014-09-16T18:20:40.807" Score="11" ViewCount="465" Body="&lt;p&gt;For many years, I've been an instructor for lower level undergraduate math classes (precalculus through calculus III).  During that time, I've noticed that the vast majority of problems I assigned were mostly computations rather than proofs.  I gained a lot of expereince teaching these classes and gained a sense of how to grade computationally focused problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, now I'm teaching an advanced mathematics course (linear algebra) with both computation and proofs as significant components.  I feel that proofs should be graded somewhat differently from how computations are graded, but I can't seem to put my finger on how.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some useful strategies for grading proofs and how do they compare to (or differ from)  grading computational problems?&lt;/p&gt;&#xA;" OwnerUserId="1315" LastActivityDate="2014-09-17T05:09:50.587" Title="Grading Computations vs. Grading Proofs: Is there a difference?" Tags="|proofs|grading|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4408" PostTypeId="1" AcceptedAnswerId="17378" CreationDate="2014-09-16T20:40:56.963" Score="15" ViewCount="897" Body="&lt;p&gt;When I was an undergraduate, someone presented to me a proof of the Fundamental Theorem of Calculus using entirely vegetables. I found this incredibly fun at the time, but I can't remember who presented it to me and my internet searching has not been successful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proof involved pinning various vegetables to a board and using their locations as variable names. I hope to find a reference for this -- obviously it's a bit of a longshot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was at the University of Michigan and it probably happened around 2004. But I have no reason to believe it was invented at that location or that time.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2019-10-30T16:46:32.100" Title="The Fundamental Theorem of Calculus and Vegetables" Tags="|calculus|reference-request|visual-proof|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="4409" PostTypeId="1" AcceptedAnswerId="4419" CreationDate="2014-09-17T01:49:56.100" Score="23" ViewCount="1395" Body="&lt;p&gt;What are some examples of math history that can be mentioned in calculus classes, either to liven things up or to provide additional perspective / insight on the material being learned?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, when discussing complex numbers, one might tell the story about how Hamilton discovered the quaternions.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The great breakthrough in quaternions finally came on Monday 16&#xA;  October 1843 in Dublin, when Hamilton was on his way to the Royal&#xA;  Irish Academy where he was going to preside at a council meeting. As&#xA;  he walked along the towpath of the Royal Canal with his wife, the&#xA;  concepts behind quaternions were taking shape in his mind. When the&#xA;  answer dawned on him, Hamilton could not resist the urge to carve the&#xA;  formula for the quaternions, \begin{equation} i^2 = j^2 = k^2 = ijk =&#xA; -1, \end{equation}&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;into the stone of Brougham Bridge as he paused on it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What are your favorite math history stories to tell when teaching calculus?  Are there any historical math problems, such as the Brachistochrone problem, that could be incorporated into a calculus class?&lt;/p&gt;&#xA;" OwnerUserId="2049" LastActivityDate="2018-03-12T14:17:48.283" Title="Historical tidbits to liven up calculus classes" Tags="|calculus|history|" AnswerCount="13" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4410" PostTypeId="2" ParentId="4407" CreationDate="2014-09-17T04:14:44.797" Score="8" Body="&lt;p&gt;I can't speak to linear algebra, but I did teach standard Euclidian geometry where it was almost all proofs.  Many of the exercises were tantamount to the most recent proof, but drawn in a different way.  Made for a horrendous marking load.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically I treated much like I would an essay.  (I also taught lit in this school...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would do the proof myself.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My normal score for a problem was 1 point per line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would look at the student proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often there were several ways to go about it, and there was a certain amount of changes in order possible.  All the stuff present and in the right order -- Full marks.  Missing stuff -1 per line.  Out of order -1/2  Correct step, but incorrect reference to justifying theorem -1/2 Clever alternative use +2.  Clever use that was shorter too +3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would start by grading the 3 smartest kids in the class.  That would catch my own mistakes.&lt;/p&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-17T04:14:44.797" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4411" PostTypeId="2" ParentId="4402" CreationDate="2014-09-17T04:18:21.900" Score="2" Body="&lt;p&gt;Forget the first option.  If they didn't pick it up the first time, they won't this time either.  The second option is far more likely to induce either fear or boredom with Math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your mission here is to make math useful to them.  Teach the applications.  Review the background material as needed for those applications.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside I would expect an economics course would have far more use for linear algebra and statistics than in calculus.&lt;/p&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-17T04:18:21.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4412" PostTypeId="2" ParentId="4089" CreationDate="2014-09-17T04:41:02.107" Score="-2" Body="&lt;p&gt;Frankly I expect it to fail as an overall technique.  There is a place for this sort of thing, but typically in any group one person does it, one might comment, the rest watch.  And that is true where the group is two or a class.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The place for this is to keep the class interesting.  Show where math comes into play in unexpected places.  I would be willing to use 10% of class time for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A huge amount of math is chunking.  At any given level, you use the stuff you learned last week with some difficulty, the stuff you learned last year with more ease.  At a college level you shouldn't have to think about factoring or rationalizing a complex denominator -- you've done enough problems that it is automatic.  You don't want to have to think about all the levels of the problem.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math is a lot like sport that way.  You practice one action until you do it right.  Then you build that action into a more complex action.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math is dull for many people.  Our choice of what we teach is partially at fault.  The way we teach it is partly at fault.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(In the day to day life of a Registed Nurse how likely is she to need to do a definite integral or solve a system of two equations, two unknowns?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most useful math course I've taken?  Euclidian Geometry.  The number of times I've used concepts from EG to do row layout on my tree farm, cutting diagrams for plywood....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are a meteorologist, then yes, you have to understand first and second partial differentials.  But 90% of college students need little more math than that required for bookkeeping and estimation.&lt;/p&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2014-09-17T04:41:02.107" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4413" PostTypeId="2" ParentId="4407" CreationDate="2014-09-17T05:09:50.587" Score="5" Body="&lt;p&gt;When teaching Linear for engineers, I grade the proofs very similarly to the computations; the proofs are simple enough that there is only one reasonable way to prove it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I write out a proof, break it in to 2-5 core ideas, and assign point values in a rubric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grading proofs in more complicated classes is very different, to me. It is not linear, but when grading the intro proofs class (and any class where serious mathematical students might have to write proper proofs for the first time), I placed a lot of emphasis on rigor, rather than problem solving. Students need to learn how to formally argue, which is a big mental shift for many of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I have only been teaching for a few years.&lt;/p&gt;&#xA;" OwnerUserId="1963" LastActivityDate="2014-09-17T05:09:50.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4414" PostTypeId="1" AcceptedAnswerId="4415" CreationDate="2014-09-17T12:29:48.217" Score="12" ViewCount="546" Body="&lt;p&gt;There is a abbreviation used in high school mathematics that is almost never seen outside of it: &lt;span class=&quot;math-container&quot;&gt;$\operatorname{cis}(\theta) = \cos(\theta) + i \sin(\theta)$&lt;/span&gt;, where &lt;strong&gt;cis&lt;/strong&gt; stands for &lt;b&gt;c&lt;/b&gt;osine + &lt;b&gt;i  s&lt;/b&gt;ine.&lt;br&gt;&#xA;As soon as students get into university Euler's formula &#xA;&lt;span class=&quot;math-container&quot;&gt;$e^{i\theta} = \cos(\theta) + i \sin(\theta)$&lt;/span&gt; is almost exclusively used as it makes a lot of things easier and more obvious. The reason Euler's formula is not used in high school is probably that it's not easy to prove or to develop an intuition for at that level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: does anyone know the origins and history of the abbreviation &lt;strong&gt;cis&lt;/strong&gt; ?&lt;/p&gt;&#xA;" OwnerUserId="395" LastEditorUserId="205" LastEditDate="2020-01-16T00:30:38.280" LastActivityDate="2020-01-16T00:30:38.280" Title="The origins of $\operatorname{cis}(\theta)$" Tags="|trigonometry|history|complex-numbers|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4415" PostTypeId="2" ParentId="4414" CreationDate="2014-09-17T14:21:00.030" Score="11" Body="&lt;p&gt;The notation exists since a long time. It was used already by  Irving Stringham in &lt;a href=&quot;http://www.archive.org/details/uniplanaralgebra00stri&quot;&gt;'Uniplanar Algebra (1893).'&lt;/a&gt; This is claimed to be the earliest use on &lt;a href=&quot;http://jeff560.tripod.com/trigonometry.html&quot;&gt;http://jeff560.tripod.com/trigonometry.html&lt;/a&gt; giving Cajori vol. 2, page 133 as reference. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this book, Uniplanar Algebra, the notation is used first, as far as I can see, in chapter III (The algebra of complex quantities). It is introduced without much ado as a shorthand notation. It is said it should be read as &quot;sector of $\theta$&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cajori also mentions another early usage (thanks to Dag Oskar Madsen for the information!), namely,  Harkness and Morley &lt;a href=&quot;http://www.archive.org/details/introductiontot00morlgoog&quot;&gt;&quot;Introduction to the theory of analytic functions (1898)&quot;&lt;/a&gt;. Also there it is introduced  as shorthand, and the link to the exponential function that will be seen &lt;em&gt;later&lt;/em&gt; is mentioned right away. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems the motivation is, as speculated in the question, to have some notation at hand when the notation via the exponential function is not yet available in a justified or motivated way. &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-09-17T15:17:59.910" LastActivityDate="2014-09-17T15:17:59.910" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4416" PostTypeId="2" ParentId="4089" CreationDate="2014-09-17T15:16:27.493" Score="1" Body="&lt;p&gt;At the undergraduate level, this kind of teaching does is often called &quot;Inquiry-Based Learning&quot;.  The University of Michigan has a center devoted to IBL mathematics at &lt;a href=&quot;http://www.math.lsa.umich.edu/ibl/&quot; rel=&quot;nofollow&quot;&gt;http://www.math.lsa.umich.edu/ibl/&lt;/a&gt;.  At that link you can find resources on how IBL is implemented, and some program assessment information.  IBL is closely related to the &lt;a href=&quot;http://legacyrlmoore.org/method.html&quot; rel=&quot;nofollow&quot;&gt;&quot;Moore Method&quot;&lt;/a&gt;, although that phrase seems to mean different things to different people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My impression (and I do not have evidence to back this up) is that when IBL is taught well it is far superior to more traditional methods, but it is really, really hard to teach IBL well.  In fact Lampert's book is mostly about how hard it is to do it well, even at the elementary level, and the types of problems (in the sense of conflicts between competing imperatives) this type of teaching creates for teachers -- hence the title of her book.  Sherwood Botsford in his answer identifies one of those problems (the possibility that some students will freeload off of the efforts of a small minority of productive students).  Lampert deals with this problem (and many others) extensively.  I think it is a fair summary of her work to say that she does not claim to have any easy solutions to these problems; her goal was to explore and illuminate the complexity of the work.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2014-09-17T19:00:49.897" LastActivityDate="2014-09-17T19:00:49.897" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4417" PostTypeId="2" ParentId="4409" CreationDate="2014-09-17T19:40:23.490" Score="5" Body="&lt;p&gt;There are SO many.  See e.g. &lt;a href=&quot;http://fredrickey.info/hm/CalcNotes/default.htm&quot; rel=&quot;noreferrer&quot;&gt;Fred Rickey's website&lt;/a&gt;.  My two favorites are &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The story of Hobbes' response to Torricelli's computation of the volume of the so-called Gabriel's Horn (see e.g. Calculus Gems for this), and &lt;/li&gt;&#xA;&lt;li&gt;Leibniz' wrong product rule, which I had a handout from Rickey on, but which I can't find now - perhaps it's linked &lt;a href=&quot;http://fredrickey.info/talks.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; somewhere.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2014-09-17T19:40:23.490" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4418" PostTypeId="2" ParentId="4402" CreationDate="2014-09-17T20:08:10.603" Score="1" Body="&lt;p&gt;Why not try to do some of both: introduce a mathematical idea, and then spend time carefully developing economic applications. I would be inclined to spend the semester alternating between two weeks on theory and two weeks of applications.  &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2014-09-17T20:08:10.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4419" PostTypeId="2" ParentId="4409" CreationDate="2014-09-17T20:29:43.783" Score="26" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What are some examples of math history that can be mentioned in calculus classes, either to liven things up or to provide additional perspective / insight on the material being learned?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You mention two different goals here. Personally, I have used anecdotes about and discussion of historical events/people in calculus courses to...&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;provide some context for course material within the broader history of human intellectual development.&lt;/li&gt;&#xA;&lt;li&gt;provide some context for course material within the history and development of mathematics and/or physics and/or the sciences and/or problem-solving.&lt;/li&gt;&#xA;&lt;li&gt;make a particular idea more memorable or enlightening by tying it to a specific story or discussion.&lt;/li&gt;&#xA;&lt;li&gt;make class time more engaging /interesting/entertaining for the students.&lt;/li&gt;&#xA;&lt;li&gt;show students that math is not a static body of knowledge, but rather an ever-changing social structure that happens to have a lot of &quot;facts&quot; in it.&lt;/li&gt;&#xA;&lt;li&gt;satisfy myself personally, to be able to say, &quot;I told them about $X$!&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It's rare that any particular anecdote/discussion will satisfy more than three of these goals; usually, I'm just satisfying a couple, depending on the audience. And occasionally, yes, on the spur of the moment, I might achieve only #4 or #6 by mentioning something that comes to mind that I hadn't planned on saying during class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the list that follows, I'll point out which particular goals (from my list) that the anecdote/discussion might achieve, based on my past experiences talking about these ideas in class. I'll also give a context for when the anecdote/discussion might be relevant in a calculus course.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Leibniz/Newton and derivative notation.&lt;/strong&gt; There has been discussion here about &lt;a href=&quot;https://matheducators.stackexchange.com/questions/562/should-students-be-asked-to-use-more-than-one-notation-for-the-derivative-in-an?rq=1&quot;&gt;whether students should be required to use multiple notations&lt;/a&gt;. I find it worthwhile to point out the variety of notations for the derivative of a function, and describe where they come from, who wanted to use them, and why. Leibniz's philosophy made frequent use of &lt;em&gt;monads&lt;/em&gt;, directly analogous to &lt;em&gt;infinitesimals&lt;/em&gt;. His notation conveys his idea that a derivative is related to infinitesimal change in a function. Newton was developing calculus to be used in solving physical problems, with functions of &lt;em&gt;time&lt;/em&gt;. His notation lends itself nicely to those problems. Lagrange's notation is more concise but not necessarily as clear. Euler's notation generalizes to other things but is a bit overblown for just Calc 1. In a vector calculus course, you could even engage students in a discussion about the &quot;operator&quot; notations of divergence and the gradient, and point to overarching concepts of functional analysis. &lt;em&gt;In the context of an introductory calculus course, I think this hits goals #2,3,5.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;&quot;Leibniz got the product rule wrong! Har har!&quot;&lt;/strong&gt; I cannot find the reference at the moment, but I distinctly remember a marginal note in the Stewart calc text. When it introduces the product rule, it has a short aside about how Leibniz first incorrectly guessed, &quot;as you might, dear reader&quot;, that $(fg)'=f'g'$. I am not at all against the idea of pointing out how mathematicians have corrected themselves, individually and collectively, over time. But I remember feeling like this was an overt jab at Leibniz. Nowhere in the text does it mention anyone else's mistakes, nor does it say that Leibniz fixed it and deserves much credit for developing the calculus, nor does it describe how Newton and his cronies browbeat the others into accepting his as the primary/premiere publication. But anyway ... I now like to point out this example and show that, yes, humans get things wrong, but it's all about our progress in figuring things out, not just &quot;getting the right answer&quot; right away. &lt;em&gt;In an intro calc course, I think this hits goals #1,2,3,5.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Should &quot;L'Hopital's Rule&quot; be &quot;Bernoulli's Rule&quot;?&lt;/strong&gt; We train students to recognize when and how to use L'Hopital's Rule. But very few of them know that L'Hopital was paying the Bernoullis for personal tutoring, and he assumed that anything they developed in their lessons was ripe for publishing under his own name. This can lead to an interesting discussion of intellectual property and the &quot;body&quot; of mathematical knowledge, as well as serve as a way to point out the historical influence of the Bernoulli family throughout mathematics. &lt;em&gt;In an intro calc course, I think this can hit all of the listed goals.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Euler and the Basel Problem.&lt;/strong&gt; This is one of the historical touchstones in evaluating infinite sums exactly, and it's also the event that birthed Euler onto the mathematical landscape. Moreover, there are new proofs of this fact published all the time (a handful in the last year or two, at least) over 250 years later. A calc 2 course can follow some of the big ideas in Euler's proofs (he had three, actually), and, in the past, I have set aside 30 minutes of class time to outline one of the proofs. This story establishes Euler as the genius that he was, and shows that some results never really &quot;get old&quot;. &lt;em&gt;Personally, I've mostly used this for goal #6, but it also addresses #3,4.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Nicole Oresme's proof (14th century) that the harmonic series diverges.&lt;/strong&gt; This is, in fact, the &quot;standard&quot; proof, showing that blocks of terms add to $&amp;gt;1/2$, so the whole series is $&amp;gt;1/2+1/2+\dots$. But hardly anyone points out that this proof is due to Oresme in the mid-1300s and was lost for the next few centuries during the Dark Ages (when not much science or math was really going on, at least in the Western world) and was &quot;rediscovered&quot; by Pietro Mengoli around when he posed the Basel Problem. &lt;em&gt;In a calc 2 course, this hits goals #1,3,5.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;&lt;a href=&quot;http://www.reddit.com/r/math/comments/eero3/1_write_scientific_paper_on_using_the_trapezoidal/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;The reinvention of the trapezoid rule&quot;&lt;/a&gt; in 1994.&lt;/strong&gt; This incident brought about some vitriol between the paper's author and those who wrote letters to the associated journal and, as I've noticed, has inspired some not-so-nice discussion about how most scientists &quot;don't know how to do math&quot; or are &quot;uninterested&quot; in spending the time to learn the math of their discipline (I mean, just look at the comments on that reddit thread I linked to). I'd recommend avoiding those discussions. Instead, I like to bring up this example when we study numerical approximation of integrals and say that some ideas are just so important and fundamental that humans will keep &quot;rediscovering&quot; them, apparently. (And yes, you can follow this up to say, &quot;It behooves you, future scientists, to learn whatever you can so that you don't spend time reinventing the wheel,&quot; if you'd like.) &lt;em&gt;In a calc 1/2 class, this hits goals #1,2,3.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Weierstrass' Function and &quot;the lamentable scourge of continuous functions with no derivatives.&quot;&lt;/strong&gt; This story is more worthwhile in a class with math-oriented students, potential future majors who will take Real Analysis. (I'd avoid anything more than a precursory mention with a class of non-math majors.) But when you bring up the idea of &quot;differentiable implies continuous&quot; but not the converse, you can point to Hermite's quote from 1893 -- &quot;&quot;I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives.&quot; -- as well as to Weierestrass' pathological function. You could even use this to preview some notions about infinite cardinalities and say how &quot;almost every&quot; $f:\mathbb{R}\to\mathbb{R}$ is not even continuous. &lt;em&gt;In an honors calc class, this can hit goals #2,3,5.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-09-17T20:29:43.783" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4420" PostTypeId="1" CreationDate="2014-09-17T20:51:01.357" Score="13" ViewCount="1122" Body="&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do you have any examples from pop culture (say, a joke, or an episode of a show, or a song lyric, etc.) that utilize and/or effectively illustrate a mathematical concept?&lt;/li&gt;&#xA;&lt;li&gt;Have you used any examples in class, and have they proven effective in making a concept easier to grasp at first and/or more memorable in the long term?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This comes to mind because I recently used a bit from a standup comedian to illustrate how widely-used and easily-understandable the &lt;em&gt;Pigeonhole Principle&lt;/em&gt; is, even though most people don't really know it by that name (or any, for that matter). &lt;em&gt;Here is the joke: &lt;a href=&quot;http://youtu.be/L8yP_3ZEMdg?t=29m33s&quot; rel=&quot;nofollow noreferrer&quot;&gt;video link&lt;/a&gt;, 29:33 to 29:55.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My lucky number is four billion. That doesn't come in handy when you're gambling. &quot;C'mon 4 billion! @#%$, seven. I'm gonna need more dice: four billion divided by six, at least...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I used this example to show my students that we are comfortable with the use of the Pigeonhole Principle already (i.e. best case scenario is we roll a bunch of 6s to get as high as 4 billion, but we might need more dice), that the underlying idea is pervasive and understandable. I think this sets a nice tone for learning and applying a new concept: &lt;em&gt;this is kinda new to you, mathematically speaking, but you are already familiar with it, conceptually&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would consider examples like this one to be good answers to this question, and I welcome a list of such examples. (Consider posting multiple examples as separate answers so we can vote accordingly on each one.) &lt;strong&gt;Essential property of a good example:&lt;/strong&gt; it should &lt;em&gt;describe/exemplify&lt;/em&gt; the concept, not merely serve as a &lt;em&gt;reference&lt;/em&gt; to the concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wouldn't necessarily consider as a good answer an example like that famous episode of Futurama, &lt;a href=&quot;http://en.wikipedia.org/wiki/The_Prisoner_of_Benda&quot; rel=&quot;nofollow noreferrer&quot;&gt;the one with the &quot;mind swap&quot; permutation problem&lt;/a&gt;. It doesn't illustrate a particular concept, nor does it make anything more enlightening, nor does it show that a particular idea is already familiar to people. It is certainly interesting and very &quot;mathy&quot;, but it's not exactly what I'm looking for. (That said, this could be an interesting example for, say, an undergrad discrete math course that happens to be studying permutations, so maybe it could be a good example. I suppose it depends on context.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Meta: I wanted to add a &quot;big-list&quot; tag to this, but I see that &lt;a href=&quot;https://matheducators.meta.stackexchange.com/questions/74/should-we-have-a-big-list-tag/&quot;&gt;the relevant discussion on meta.MESE&lt;/a&gt; doesn't have an apparent resolution. I'm fine if this Q ends up getting deleted because we decide that &quot;big-list&quot; is a bad idea, but I'm also fine with this starting up the discussion again :-) ]&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2017-03-16T16:41:53.880" LastActivityDate="2014-09-19T18:07:38.930" Title="Using joke / song / film / pop culture to exhibit a new mathematical concept" Tags="|undergraduate-education|secondary-education|examples|concept-motivation|mathematics-in-daily-life|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4421" PostTypeId="2" ParentId="4420" CreationDate="2014-09-18T01:05:31.713" Score="4" Body="&lt;p&gt;From &lt;a href=&quot;http://en.wikipedia.org/wiki/Numbers_(TV_series)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Numb3rs&lt;/a&gt;,&#xA;an American TV show (which I've never seen).&#xA;Several friends notified me when it touched upon &lt;em&gt;art gallery theorems&lt;/em&gt;&#xA;(e.g., see &lt;a href=&quot;https://matheducators.stackexchange.com/a/1099/511&quot;&gt;this earlier MEd posting&lt;/a&gt;):&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/WIZjr.png&quot; alt=&quot;Numb3rs&quot;&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/wIKbM.png&quot; alt=&quot;Quote&quot;&gt;&#xA;&lt;hr /&gt;&#xA;From the Numb3rs Teacher Activity Guide:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/mN41h.png&quot; alt=&quot;ActivityBook&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-09-18T01:05:31.713" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4422" PostTypeId="2" ParentId="780" CreationDate="2014-09-18T18:16:06.583" Score="0" Body="&lt;p&gt;How about a game based on the Traveling Salesperson Problem from graph theory?  Put pedometers on each of your participants.  Pick several local landmarks, provide those locations to your competitors, and see who can visit -- by walking -- all the landmarks in the least amount of distance, as measured by the pedometers.  &lt;/p&gt;&#xA;" OwnerUserId="2125" LastActivityDate="2014-09-18T18:16:06.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4423" PostTypeId="2" ParentId="3938" CreationDate="2014-09-18T20:44:12.357" Score="5" Body="&lt;p&gt;The opposite of compartmentalization as the question describes it may be &lt;em&gt;connections&lt;/em&gt;. &lt;em&gt;Connections&lt;/em&gt; is a &lt;a href=&quot;http://www.nctm.org/standards/content.aspx?id=322&quot; rel=&quot;nofollow&quot;&gt;process standard&lt;/a&gt; of the U.S. National Council of Teachers of Mathematics (NCTM), and they have written widely about it, including &lt;a href=&quot;http://www.nctm.org/about/content.aspx?id=39616&quot; rel=&quot;nofollow&quot;&gt;this article&lt;/a&gt; which includes a story about a breakthrough moment in class when a student connected past work with factors and multiples to current work with area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.nctm.org/uploadedFiles/Math_Standards/12752_exec_pssm.pdf&quot; rel=&quot;nofollow&quot;&gt;Elsewhere&lt;/a&gt; from the NCTM website:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematics is not a collection of separate strands or standards, even though it is often partitioned and presented in this manner. Rather, mathematics is an integrated field of study. When students connect mathematical ideas, their understanding is deeper and more lasting, and they come to view mathematics as a coherent whole. They see mathematical connections in the rich interplay among mathematical topics, in contexts that relate mathematics to other subjects, and in their own interests and experience. Through instruction that emphasizes the interrelatedness of mathematical ideas, students learn not only mathematics but also about the utility of mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="867" LastEditorUserId="867" LastEditDate="2014-10-12T17:46:18.467" LastActivityDate="2014-10-12T17:46:18.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4424" PostTypeId="1" CreationDate="2014-09-18T21:15:15.703" Score="12" ViewCount="289" Body="&lt;p&gt;My high school students' textbooks have a full answer key in the back. Nearly all of my students make at least one of these mistakes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Not knowing that the answers are in the back of the book. Hence they miss out on opportunities that their classmates have to assess their own work as they do homework.&lt;/li&gt;&#xA;&lt;li&gt;Checking the back of the book too early and too often. Hence they try to make their answer fit the back of the book without paying much attention to the question content.&lt;/li&gt;&#xA;&lt;li&gt;Plagiarising from the back of the book, either with or without an attempt to show some working out.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What guidelines can I give my students so that they can use the answer key in the back of their books correctly?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: I am not seriously concerned about my students improving their grade by copying the back of their homework book, since we write more than enough quizzes and tests. However, I want guidelines which can help &lt;em&gt;students&lt;/em&gt; understand how they should and should not go about learning, and help &lt;em&gt;me&lt;/em&gt; respond correctly when students are out of line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another note: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/6/how-to-assign-homework-when-answers-are-freely-available-or-attainable-online&quot;&gt;This other question&lt;/a&gt; is similar, but I'm interested in secondary education and in teaching my students how to use the help given rater than circumvent it.&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-01-01T22:38:09.463" Title="Coaching Students to Use the Back of the Book Correctly" Tags="|secondary-education|general-pedagogy|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4425" PostTypeId="2" ParentId="4420" CreationDate="2014-09-18T23:53:34.423" Score="2" Body="&lt;p&gt;This may be known already but &quot;The Simpsons&quot; has many (advanced) math and physics references. Here is a relevant &lt;a href=&quot;http://homepage.smc.edu/nestler_andrew/SimpsonsMath.htm&quot; rel=&quot;nofollow&quot;&gt;link&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1647" LastEditorUserId="1647" LastEditDate="2014-09-18T23:59:08.503" LastActivityDate="2014-09-18T23:59:08.503" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4426" PostTypeId="2" ParentId="4420" CreationDate="2014-09-19T01:02:47.060" Score="1" Body="&lt;p&gt;I think the video &quot;Jokes with Einstein&quot; about crossing an elephant and a banana is just right for a course such as Calculus III when you introduce cross products. Here is a link&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.youtube.com/watch?v=r651684kxKg&quot; rel=&quot;nofollow&quot;&gt;http://www.youtube.com/watch?v=r651684kxKg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can play it four times consecutively in class and students laugh each time. It does not seem to get old.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2014-09-19T01:02:47.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4427" PostTypeId="1" CreationDate="2014-09-19T05:34:23.963" Score="10" ViewCount="1288" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/41/what-is-a-good-handwriting-font-for-mathematics&quot;&gt;What is a good handwriting font for mathematics?&lt;/a&gt; addresses handwriting on a board. How do you clearly distinguish calligraphic letters, such as &lt;span class=&quot;math-container&quot;&gt;$\mathcal{ABCDEFG}$&lt;/span&gt;, or script letters, such as &lt;span class=&quot;math-container&quot;&gt;$\scr{ABCDEFG}$&lt;/span&gt;, from regular letters &lt;span class=&quot;math-container&quot;&gt;$ABCDEFG$&lt;/span&gt; in handwritten mathematics?&lt;/p&gt;&#xA;&lt;p&gt;Edit:&#xA;Since posting this question, I have discovered &lt;a href=&quot;https://math.stackexchange.com/questions/236303/how-to-write-mathcal-letters-by-hand/236396#236396&quot;&gt;How to write \mathcal letters by hand?&lt;/a&gt; on MSE and Andrew Stacey's article: &lt;a href=&quot;http://loopspace.mathforge.org/CountingOnMyFingers/Calligraphy/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Old Pappus' Book of Mathematical Calligraphy&lt;/a&gt;, although his main example is how to write a Fraktur &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{g}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2022-01-29T15:16:22.900" LastActivityDate="2022-01-29T22:07:51.177" Title="Distinguishing calligraphic or script letters from regular letters in handwritten mathematics" Tags="|board-use|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="4428" PostTypeId="2" ParentId="4427" CreationDate="2014-09-19T08:41:52.070" Score="8" Body="&lt;p&gt;I always give this piece of advice to my students: only use letters that you can distinguish yourself.&#xA;(Some people really don't do this unless told so.)&#xA;When you teach, apply the same more strictly: only use letters that most students can distinguish.&#xA;This includes avoiding using $v$ and $\nu$ in the same context.&#xA;If you run out of letters, use something else: overlines, underlines, accents...&#xA;Or just use different letters for different things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I make my curly letters more curly than the regular ones, but this distinction is not always easy.&#xA;If my students (or I myself) seem to have hard time following, I change my notation.&#xA;Script letters should never be absolutely necessary.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2014-09-19T08:41:52.070" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4429" PostTypeId="2" ParentId="4204" CreationDate="2014-09-19T09:01:08.933" Score="2" Body="&lt;p&gt;How about a gallery of &lt;a href=&quot;https://en.wikipedia.org/wiki/Triangle_center&quot; rel=&quot;nofollow&quot;&gt;triangle centers&lt;/a&gt;?  From the Wikipedia page:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles. For example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of them has the property that it is invariant under similarity. In other words, it will always occupy the same position (relative to the vertices) under the operations of rotation, reflection, and dilation. Consequently, this invariance is a necessary property for any point being considered as a triangle center.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As of 3 July 2014, Clark Kimberling's &lt;a href=&quot;http://faculty.evansville.edu/ck6/encyclopedia/ETC.html&quot; rel=&quot;nofollow&quot;&gt;Encyclopedia of Triangle Centers&lt;/a&gt; contains an annotated list of 5,875 triangle centers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of particular interest are the &lt;em&gt;1st Liu point&lt;/em&gt; and the &lt;em&gt;2nd Liu point&lt;/em&gt; (&lt;a href=&quot;http://faculty.evansville.edu/ck6/encyclopedia/ETCPart3.html&quot; rel=&quot;nofollow&quot;&gt;X(3598) and X(3599)&lt;/a&gt;, respectively) which &quot;were discovered by Kang-Ying Liu of St. Andrew's Priory School, Honolulu, Hawaii, during 2010&quot; while she was still in high school.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-09-19T09:01:08.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4430" PostTypeId="2" ParentId="4424" CreationDate="2014-09-19T13:08:00.320" Score="2" Body="&lt;p&gt;I would start by addressing the issue openly on the first day of class.  This solves issue #1.  I assume your evaluation method is primarily based on students solving problems for which they do not have access to the solutions (e.g. almost any exam format fits this characterization).  I would explain that in order to be able to complete that task successfully, they have to practice that very task.  Looking at the answers first for guidance or -- much worse -- simply copying the answers will not improve their skill level and consequently does nothing to prepare them for exams.  (In theory, the primary concern for the students in the course should be increasing their skill level, but in practice, the primary concern for students in the course who engage in #2 or #3 is, of course, their grade.)  Emphasize that the answers are meant as a diagnostic tool to help them instantly confirm whether or not they have mastered the given problem.  I like to emphasize that even if they produced an answer on their own and the back of the book confirms that their answer is correct, they have not mastered the material if they cannot explain the thought process that underlies the method they used to obtain the answer.  Of course, many students are not interested in putting forth the effort to achieve this level of mastery (whether it be disinterest / laziness or simply an accumulation of lack of understanding in earlier grades that makes such a level of mastery difficult for them to attain).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You also have to accept that no matter how clearly you communicate the proper use of solutions, some students are simply going to ignore your advice and will only learn &quot;the hard way&quot; after some failure.  Keep in mind that failing and learning from failure is our natural way of learning, while trusting the directions of someone else to avoid a failure we've never experienced before is not natural.  But if you address the issue in advance, then students who ignore your advice may have an epiphany later on when they do not do as well as they wanted to.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2014-09-19T13:08:00.320" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4431" PostTypeId="2" ParentId="4420" CreationDate="2014-09-19T13:15:29.797" Score="2" Body="&lt;p&gt;There are many examples from pop culture which “utilize and/or effectively illustrate” game theoretical concepts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find some examples &lt;a href=&quot;http://www.gametheory.net/popular/film.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concepts are such as e.g. Prisoner’s Dillemma, Game of Chicken, common knowledge, etc.&lt;/p&gt;&#xA;" OwnerUserId="2127" LastActivityDate="2014-09-19T13:15:29.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4432" PostTypeId="2" ParentId="4427" CreationDate="2014-09-19T21:41:17.323" Score="6" Body="&lt;p&gt;If you have clear handwriting on the board (if you don't, forget about calligraphic letters and use diacritics instead), you can &amp;quot;invent&amp;quot; your own calligraphic set of letters for the blackboard. It just has to be clearly distinguishable from your normal one, e.g. like this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;look at the top left corner of the capital letter&lt;/li&gt;&#xA;&lt;li&gt;if it is a corner (B,D,E,F,P,R), make it an arc instead&lt;/li&gt;&#xA;&lt;li&gt;if it is a line cap (H,I,J,K,L,T,W,Z), add an arc to it&lt;/li&gt;&#xA;&lt;li&gt;if it is an arc (C,G,O,Q,S), make it a corner instead&lt;/li&gt;&#xA;&lt;li&gt;if it is skipped (A), make it a corner instead&lt;/li&gt;&#xA;&lt;li&gt;for X and Y make the line from the top left corner a downward arc&lt;/li&gt;&#xA;&lt;li&gt;add a swoosh to the U (or don't use it, because it too easily gets confused with V, Y and the normal variants)&lt;/li&gt;&#xA;&lt;li&gt;make both lines of the V inward arcs (or don't use it)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, except for V and U (and X and Y) the letters are easily distinguishable and the rules are fairly simple.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="376" LastEditDate="2022-01-29T15:20:10.563" LastActivityDate="2022-01-29T15:20:10.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4433" PostTypeId="2" ParentId="780" CreationDate="2014-09-20T03:44:27.670" Score="1" Body="&lt;p&gt;There are very simple graph theoretic games like  &lt;a href=&quot;http://www.mathteacherscircle.org/assets/session-materials/crisscross.pdf&quot; rel=&quot;nofollow&quot;&gt;Criss-Cross&lt;/a&gt;, invented by Sam Vandervelde as far as I know, and harder slightly related games like Sprouts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well-known games like SET are actually games on 4-dimensional geometry; defining lines and planes and so on can be a lot of fun.  You can emphasize the geometry a bit with some of the games like Planet in  &lt;a href=&quot;http://www.maa.org/sites/default/files/pdf/pubs/SetsPlanetsAndComets.pdf&quot; rel=&quot;nofollow&quot;&gt;Brian Conrey et al&lt;/a&gt;.  There's also some great geometry in the game of Spot It! and you can easily ask questions about projective planes.&lt;/p&gt;&#xA;" OwnerUserId="2130" LastActivityDate="2014-09-20T03:44:27.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4434" PostTypeId="2" ParentId="4367" CreationDate="2014-09-20T03:57:16.927" Score="4" Body="&lt;p&gt;I often invent a land where the money comes in units of 1, 2, 4, 8, 16, ... and then ask how to pay for certain amounts, using exact change, with the fewest possible coins.  It's nice that the greedy algorithm of always using the largest possible coin always works!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I make the analogy starting with $101 meaning one hundred-dollar bill, no tens, and one one to lead them toward using binary notation to represent the way that they're paying the amount of money that they need to pay.&lt;/p&gt;&#xA;" OwnerUserId="2130" LastActivityDate="2014-09-20T03:57:16.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4435" PostTypeId="1" AcceptedAnswerId="4446" CreationDate="2014-09-20T04:37:09.880" Score="11" ViewCount="195" Body="&lt;p&gt;AMATYC has a draft position statement on the academic preparation of mathematics faculty at two-year colleges that will be voted on soon, if it hasn't happened already. The draft is attachment L in &lt;a href=&quot;http://c.ymcdn.com/sites/www.amatyc.org/resource/resmgr/board_minutes_and_motions/sbm_2013_minutes_final.pdf&quot;&gt;these minutes&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The draft states in part:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;All full-time mathematics instructors at two-year colleges should begin their careers with at least a master's degree in mathematics or in a related field with at least 30 semester hours (45 quarter hours) in graduate level mathematics and have mathematics teaching experience at the secondary or collegiate level. The teaching experience may be fulfilled through a program of supervised teaching as a graduate student. Just as a strong knowledge of Calculus has always been a core standard, Statistics has become equally important, and some background in this area is desirable. Course work in pedagogy and the community college is desirable. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Does this position preclude a master's degree in math education &lt;em&gt;without&lt;/em&gt; 30 hours of graduate level mathematics courses, but perhaps &lt;em&gt;with&lt;/em&gt; a lot of undergraduate math and teaching experience? The ambiguity for me lies in the  &quot;at least a master's degree in mathematics&quot;. Between the fields of mathematics and math education, neither is greater than the other. So I assume the &quot;at least&quot; refers to the tiers of master's and doctorate degrees. And at this point of the statement I want to know if the field of mathematics education is included in &quot;mathematics&quot;, or if it has to be amongst the &quot;related fields&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be asking this community for its interpretation, but opinion-based questions are not for Stack Exchanges. Instead I am asking if anyone in this community knows the intent of AMATYC.&lt;/p&gt;&#xA;" OwnerUserId="767" LastActivityDate="2014-09-21T21:08:17.410" Title="Regarding AMATYC's draft position statement on the academic preparation of mathematics faculty at two-year colleges" Tags="|teacher-preparation|community-colleges|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4436" PostTypeId="1" CreationDate="2014-09-20T05:52:10.940" Score="5" ViewCount="575" Body="&lt;p&gt;What is a good source for the history of infinite series?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, why do we learn them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are they really useful on their own, or are just tools / stepping stone for studying series of functions later?&lt;/p&gt;&#xA;" OwnerUserId="2131" LastEditorUserId="376" LastEditDate="2014-10-13T13:15:35.703" LastActivityDate="2014-10-14T02:11:00.823" Title="History Of Infinite Series" Tags="|history|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4437" PostTypeId="1" AcceptedAnswerId="4442" CreationDate="2014-09-20T10:25:20.370" Score="9" ViewCount="772" Body="&lt;p&gt;In which areas of mathematics is it traditional to use calligraphic letters, such as &lt;span class=&quot;math-container&quot;&gt;$\cal{ABCDEFG}$&lt;/span&gt;, or script letters, such as &lt;span class=&quot;math-container&quot;&gt;$\scr{ABCDEDG}$&lt;/span&gt;, and is there a pedagogical advantage to doing so?&lt;/p&gt;&#xA;&lt;p&gt;(Note that, in line with a comment by Joel Reyes Noche, I am using &amp;quot;calligraphic&amp;quot; and &amp;quot;script&amp;quot; in the sense of the LaTeX commands &lt;code&gt;\cal&lt;/code&gt; and &lt;code&gt;\scr&lt;/code&gt;, respectively. However, when I originally wrote the question, I used &amp;quot;script&amp;quot; more broadly to refer to both of them, not necessarily in LaTeX, but possibly also handwritten. This broader usage can be seen in the answers.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2022-01-29T15:09:07.350" LastActivityDate="2022-01-30T20:14:49.863" Title="Advantages of using calligraphic or script letters" Tags="|mathematical-pedagogy|board-use|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="4438" PostTypeId="1" CreationDate="2014-09-20T16:50:27.640" Score="9" ViewCount="433" Body="&lt;p&gt;I'm teaching mathematics at a high school and I am offered the opportunity to get extra lessons, namely: I can provide an optional course about some topic of mathematics. The optional course will take two lessons (each 45 minutes) a week and last a whole semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mathematical skills of the pupils will vary a lot: Usually, they will have covered one or two the three topics &quot;vector geometry&quot;, &quot;differentiation&quot; or &quot;integration&quot;. I myself am mostly interested in linear algebra and theoretical physics (quantum physics).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm having a hard time to come up with something interesting I know a lot about to teach them. Obviously, I can also read a bit about something not in my area, because I doubt I can show them any proper theoretical physics because a) it won't count as a maths course and b) not all of them know integration and none know complex numbers yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have a suggestion what I could do? A friend suggested to do number theory but I'm not sure if that is appealing to high school pupils (I never cared for it) plus I still don't know anything about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;//edit: It should also not be something that is taught in the standard curriculum, e.g. not complex numbers.&lt;/p&gt;&#xA;" OwnerUserId="493" LastEditorUserId="493" LastEditDate="2014-09-20T19:16:42.930" LastActivityDate="2016-06-24T00:48:18.163" Title="What topic could I cover in an optional course at high school taking two lessons a week during a whole semester?" Tags="|secondary-education|course-design|" AnswerCount="6" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4439" PostTypeId="2" ParentId="4438" CreationDate="2014-09-20T17:37:33.153" Score="7" Body="&lt;p&gt;We had something like that when I was in high school. It was 40 minutes a week and we studied Axiomatic Geometry. I benefited a lot from this course! Now, looking back, I think I understand why. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, we were already familiar with the basic theorems of geometry so it wasn't intimidating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, it was a great opportunity to learn some history of mathematics, to see where and how it began.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third, it really prepared (some of) us for the university courses we later took. So when I got to the university, I was already familiar with axiomatic approaches, definitions, lemmas, theorems, the concept of formal proof (from the axioms) etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, it was fun! &lt;/p&gt;&#xA;" OwnerUserId="1392" LastActivityDate="2014-09-20T17:37:33.153" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4440" PostTypeId="2" ParentId="4438" CreationDate="2014-09-20T18:04:07.173" Score="3" Body="&lt;ul&gt;&#xA;&lt;li&gt;Treat it like a math circle, where you propose an 'accessible&#xA;mystery.' &lt;/li&gt;&#xA;&lt;li&gt;You don't need to care about how much content you 'cover', so you can&#xA;focus on the students' mathematical thinking, and help them to&#xA;uncover mathematical concepts.  &lt;/li&gt;&#xA;&lt;li&gt;Pick a topic you like, so you'll have passion for it.&lt;/li&gt;&#xA;&lt;li&gt;Be aware that your students will struggle with actually thinking mathematically, instead of the usual method of following procedures they've been shown. (Sometimes 'good students' struggle more, because they're used to it coming easily.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Possible topics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Non-euclidean geometry might be cool. I don't think you'd need much research to be able to mentor the group well in this.&lt;/li&gt;&#xA;&lt;li&gt;If you do want to address complex numbers (unclear to me from your post), a good starting question might be &quot;What is i to the i?&quot;&lt;/li&gt;&#xA;&lt;li&gt;Fold a long strip in half repeatedly, creating a dragon curve. Describe it mathematically. Go on from there...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Check out this site for more ideas: &lt;a href=&quot;http://www.mathcircles.org/content/math-circle-problem-collection&quot; rel=&quot;nofollow&quot;&gt;http://www.mathcircles.org/content/math-circle-problem-collection&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-09-20T18:04:07.173" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4441" PostTypeId="2" ParentId="4438" CreationDate="2014-09-20T18:16:53.157" Score="4" Body="&lt;p&gt;I highly recommend introducing the students to a survey of discrete mathematics: sets, logic, counting techniques, probability, graphs, all that good stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most likely the students will not have encountered such a topic before, and will be pleased with the break from the monotonous calculations of algebra and calculus. Even the students who are not so hot about mathematics will benefit, since the aforementioned topics are usually covered in a course in university along the lines of &lt;em&gt;Mathematics for the Liberal Arts.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="2135" LastActivityDate="2014-09-20T18:16:53.157" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4442" PostTypeId="2" ParentId="4437" CreationDate="2014-09-20T18:42:26.383" Score="9" Body="&lt;p&gt;When using categories it is common (though not universal) to denote the categories by script letters; see &lt;a href=&quot;http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf&quot;&gt;http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf&lt;/a&gt; and &lt;a href=&quot;http://wwwhome.ewi.utwente.nl/~fokkinga/mmf92b.pdf&quot;&gt;http://wwwhome.ewi.utwente.nl/~fokkinga/mmf92b.pdf&lt;/a&gt; (page 8) for two examples. [The examples are not chosen very specifically, they just came up early on in a search for examples.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example where a script letter is common is for denoting the powerset of a set $S$, as $ \mathcal{P}(S)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the pedagogical advantage, I would say that it can be useful if one has several 'layers' or 'types of things' to distinguish them by different fonts. For example, it is common to denote sets by upper-case and their elements by lower-case letters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there is a third layer it can make sense to ressort to a third font, and thus use script letters. &#xA;Like, categories  $\mathcal{C}, \mathcal{D}$ have objects $C_1, C_2$ and $D_1, D_2$, respectively; and let $f$ and $g$ denote a morphism of $C_1, C_2$ and $D_1, D_2$, respectively. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question now splits a bit as there are at least two separate questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Does it make sense to use several fonts in this way? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is script a good choice for a font?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For 1. I would say that it can help to structure things and to have a permanent reminder what is what. It is (with different fonts) also very wide spread, for example, Greek letters for angles. Yet, one should also not overdo this. There is a risk that students become so fixated on some specific notation that they are completely off track when then confronted with a different set of notational conventions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For 2. I would also say, yes, in some cases. What is good about the font is that it is not too different but still distinguishable. By contrast, Fraktur (as somewhat frequently used for ideals or the symmetric group and related things) can be difficult to read and write. For instance, I am under the impression that for not few students the notations $\mathfrak{S}_n$ and especially $\mathfrak{A}_n$ for the symmetric and alternating groups are not at all intuitive, since they do not really &quot;see&quot; an &quot;S&quot; and an &quot;A&quot; in a different font but just some unknown symbols. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is not so good is that it can be a bit 'heavy' or 'voluminous' (for lack of better words). So, it seems better to use it for things that do not occur that frequently and/or are rather 'high' in the hierarchy, like, element $a$, in set $A$, in a collection of sets $\mathcal{A}$. &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-09-20T18:42:26.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4443" PostTypeId="2" ParentId="4435" CreationDate="2014-09-20T23:06:27.500" Score="4" Body="&lt;p&gt;A graduate degree does not necessarily correlate closely with knowledge of the subject. It's common for people with a master's in math education to be hired at a community college despite simply not knowing calculus. Something like half of people interviewing for a tenure-track position will flub a question that probes their knowledge of first-semester calculus. On the other hand, I have a bachelor's in math and physics and a PhD in physics, zero graduate work in math -- but I teach calculus as well as physics. Of course I'm biased, but I consider myself pretty competent to teach calculus; if anything, the challenge is to bring myself down to the level of students who write PEMDAS in the margin of their paper in order to remember the order of operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We see the same lack of close correlation when we interview people for physics jobs. We've had applicants with graduate degrees in physics, even PhDs, who nevertheless demonstrated elementary misconceptions about Newton's third law (e.g., that it involves equilibrium), or who couldn't use Gauss's law to find the field of a point charge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Community colleges are odd beasts. We do a lot of different things, including transfer-level and non-transfer courses, remedial courses, vocational courses, and courses that exist simply to serve the community. If someone with a master's in math ed is going to teach remedial math for the next 30 years, and will be happy doing that, it's not necessarily a bad thing. Possibly someone with a PhD would go nuts doing that. But it would be a bad thing IMO if a department became packed with too many faculty with master's degrees in math ed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;California has quite a few rural community colleges that are very small, and I suspect that the FSA rules are as loose as they are partly so that those schools can hire one person to teach more than one subject.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-09-21T00:02:56.387" LastActivityDate="2014-09-21T00:02:56.387" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4444" PostTypeId="2" ParentId="4351" CreationDate="2014-09-21T16:24:33.927" Score="62" Body="&lt;p&gt;Here is a more elementary example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes, &lt;a href=&quot;http://en.wikipedia.org/wiki/Extraneous_and_missing_solutions&quot;&gt;extraneous solutions&lt;/a&gt; in algebra are of the type you describe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, to solve the equation &#xA;$$\sqrt {2x-1}=-x, \quad (x \in \mathbb R)$$ &#xA;we might take the square on both sides and get the equation&#xA;$$2x-1=x^2,$$&#xA;which has a unique solution $x=1$. Substituting back in the original equation we see that our unique candidate is actually not a valid solution.&lt;/p&gt;&#xA;" OwnerUserId="39" LastEditorUserId="39" LastEditDate="2014-09-21T18:49:18.233" LastActivityDate="2014-09-21T18:49:18.233" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4445" PostTypeId="2" ParentId="4435" CreationDate="2014-09-21T18:11:02.847" Score="3" Body="&lt;p&gt;I'd say you should look also at the &quot;Minimal Qualifications&quot; section, which seems to be very very lenient toward the amount of graduate mathematics taken:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;All full- and part-time mathematics instructors at two-year colleges should possess at least a master's degree in mathematics or in a related field with at least 18 semester hours (27 quarter hours) in graduate-level courses strongly related to mathematics, and at least six of which are graduate-level mathematics. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the end, they are only requiring two graduate mathematics courses, with the rest of the graduate courses allowed to be &quot;strongly related to mathematics.&quot; I think that's incredibly low, and certainly not restrictive of math education degrees.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-09-21T18:11:02.847" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4446" PostTypeId="2" ParentId="4435" CreationDate="2014-09-21T21:08:17.410" Score="7" Body="&lt;p&gt;After writing an email, I received a response from Ernie Danforth, the NE regional vice president of AMATYC which answer my question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Keep in mind these are GUIDELINES, they are not hard and fast rules.&lt;/p&gt;&#xA;&lt;p&gt;No mathematics education courses do not count as mathematics preparation, although they are still considered valuable.&lt;/p&gt;&#xA;&lt;p&gt;You are correct that under the guidelines; neither bachelor's degrees or master's degrees in mathematics education are considered Standard Preparation (incidentally this has not changed since the guidelines were first adopted in 1993).&lt;/p&gt;&#xA;&lt;p&gt;I am not sure what you consider a large amount of work experience.  It can be teaching a few courses as a graduate assistant, but more than just running tutoring sessions.&lt;/p&gt;&#xA;&lt;p&gt;And, yes, the expectation of the standard preparation does include a good deal of graduate level mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Does this answer your question?  If not, let me know and I will try to clarify further.  I will also be running the session to review the guidelines in Nashville.&lt;/p&gt;&#xA;&lt;p&gt;Ernie Danforth&#xA;NE Region Vice President&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="767" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-09-21T21:08:17.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4448" PostTypeId="1" AcceptedAnswerId="6058" CreationDate="2014-09-22T07:10:00.760" Score="14" ViewCount="2642" Body="&lt;p&gt;What are some ideas for math topics to teach to gifted second and third grade math circle students?  Let's assume the math circle meets for one hour every week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example activity is learning about various types of ciphers that can be used to send secret messages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know I can look at math circle archives online (and I have), but I think it'd be useful to compile a list of topics here.  I'm curious to see what suggestions this community comes up with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Archives for the Los Angeles Math Circle can be found &lt;a href=&quot;http://www.math.ucla.edu/~radko/circles/archive.shtml?&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, a little background on math circles from Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematical enrichment activities in the United States have been&#xA;  around for at least thirty years, in the form of residential summer&#xA;  programs, math contests, and local school-based programs. The concept&#xA;  of a math circle, on the other hand, with its emphasis on convening&#xA;  professional mathematicians and secondary school students on a regular&#xA;  basis to solve problems, has appeared only within the past twelve&#xA;  years. This form of mathematical outreach made its way to the U.S.&#xA;  most directly from Russia and Bulgaria, where it has been a fixture of&#xA;  their mathematical culture for decades. (The first ones appeared in&#xA;  Russia during the 1930s; they have existed in Bulgaria for a century.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2049" LastEditorUserId="2049" LastEditDate="2014-09-25T01:24:33.467" LastActivityDate="2015-02-01T08:02:06.953" Title="Math activities for gifted second and third grade math circle students" Tags="|math-circle|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4449" PostTypeId="2" ParentId="4438" CreationDate="2014-09-22T13:05:21.857" Score="3" Body="&lt;p&gt;A fun topic that could be covered at the high school and would introduce students to more advanced ways of thinking is Combinatorial Game Theory.  You could use Volume 1 of Berlekamp, Conway, and Guy's &lt;em&gt;Winning Ways for your Mathematical Plays&lt;/em&gt; as the source for your material.  You can let the students discover strategies for various games, and depending on your judgment, you can also consider discussing how one &lt;em&gt;proves&lt;/em&gt; that the optimal strategies are just that.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2014-09-22T13:05:21.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4450" PostTypeId="2" ParentId="4409" CreationDate="2014-09-22T13:30:55.343" Score="2" Body="&lt;p&gt;I love the answers already here, but just wanted to add: I have always rather enjoyed the &quot;May we not call them the Ghosts of departed Quantities?&quot; quote of George Berkeley re: infinitesimals. I found it fascinating that infinitesimals/limits had been pretty controversial at one point; since many students also struggle with these notions this has seemed to hearten them as well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See more here: &lt;a href=&quot;http://en.wikipedia.org/wiki/The_Analyst&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/The_Analyst&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="126" LastActivityDate="2014-09-22T13:30:55.343" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4451" PostTypeId="2" ParentId="4438" CreationDate="2014-09-22T13:45:37.263" Score="2" Body="&lt;p&gt;As a topologist myself, I am somewhat biased, but given such an opportunity I would teach a course on low-dimensional topology, using a book such as &lt;a href=&quot;http://people.albion.edu/ram/TopologyNow!/&quot; rel=&quot;nofollow&quot;&gt;Topology Now!&lt;/a&gt; or &lt;a href=&quot;http://math.umaine.edu/~franzosa/ITPA.htm&quot; rel=&quot;nofollow&quot;&gt;Introduction to Topology: Pure and Applied&lt;/a&gt;. If you are willing to do something a bit more focussed, one could use something like &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0821836781&quot; rel=&quot;nofollow&quot;&gt;The Knot Book&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Topology Now! in particular does not require a whole lot of background, and whatever background is needed could be covered as an aside during one of the weeks (since it's an optional course it presumably doesn't matter too much how far you get into the chosen book.). The reason I recommend this, apart from the aforementioned potential bias towards my own field, is that topology/knot theory are really quite different from the types of math HS students usually see; as a result, such a course would reveal to them some unexpected places where math plays a role, and also possibly level the playing field a bit for your students who, as you said, might have different mathematical backgrounds. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a short review of two of the books I mentioned: &lt;a href=&quot;http://www.jstor.org/stable/40391111&quot; rel=&quot;nofollow&quot;&gt;http://www.jstor.org/stable/40391111&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="126" LastActivityDate="2014-09-22T13:45:37.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4452" PostTypeId="2" ParentId="487" CreationDate="2014-09-22T14:14:11.533" Score="4" Body="&lt;p&gt;In the case of Desargues' theorem, and many others, the solution is not so unexpected when you know the history. Desargues was one of the inventors of &lt;a href=&quot;http://en.wikipedia.org/wiki/Projective_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;projective geometry&lt;/a&gt;, whose original goal was to develop a mathematical theory of perspective drawing. He also worked as an architect, where he prepared accurate perspective drawings from blueprints. If you've ever taken an art class where they teach you how to do perspective drawing, they'll teach you to take each family of parallel lines in your space and choose some vanishing point somewhere on your easel to draw them passing through. If you have many families of parallel lines in coplanar directions, than these &quot;points at infinity&quot; will lie on a &quot;line at infinity&quot;. Check out this &lt;a href=&quot;https://www.youtube.com/watch?v=2gAbvquhbmw&quot; rel=&quot;nofollow noreferrer&quot;&gt;YouTube drawing instruction video&lt;/a&gt; for the sort of art lesson I am talking about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, here is Escher's &lt;i&gt;Inside St. Peters&lt;/i&gt; (1935). See how all the vertical lines pass through one point down below the edge of the paper, all the north-south floor tiles converge at a second point and all of the east-west floor tiles converge at a third.&#xA;(I just realized this wasn't the best example to choose: These three points at infinity aren't colinear because the three families of parallel lines don't lie in coplanar directions. See around 8:40 in the above linked video for a better example..)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/T551t.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not so strange that some one who was intimately familiar with translating between figures in three space and geometry of lines on the page would look at a figure and see it as a sketch of a three dimensional figure. Specifically, seeing $(\overline{AB}, \overline{ab})$, $(\overline{AC}, \overline{ac})$ and $(\overline{BC}, \overline{bc})$ as pairs of parallel lines in coplanar directions, and the line on which $\overline{AB} \cap \overline{ab}$, $\overline{AC} \cap \overline{ac}$, $\overline{BC} \cap \overline{bc}$ lie as the line at infinity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this particular case, I wonder if more is true: I wonder if Desargues invented the problem in order to show off how powerful projective geometry is! One of the easiest ways to solve a problem is to start with the solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can't always take the time to pull back the curtain and talk about motivation like this, but I've found that students really appreciate it when you can.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2014-09-22T14:29:15.920" LastActivityDate="2014-09-22T14:29:15.920" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4453" PostTypeId="1" CreationDate="2014-09-22T19:02:00.817" Score="6" ViewCount="556" Body="&lt;p&gt;I am looking for a book on differential equations (ordinary/partial) of a particular kind. I figure out I am terrible at solving differential equations. This a vast field and unfortunately I don't really have time to study that in depth now. And the terrible thing is that I forget everything very quickly. The dream book right now would be :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I encounter a differential equation to solve.&lt;/li&gt;&#xA;&lt;li&gt;Most of time, the resolution is simple but still, I fail.&lt;/li&gt;&#xA;&lt;li&gt;I open the book where tens of typical equations are solved, possibly with different techniques. Techniques should be emphasized more than theory.&lt;/li&gt;&#xA;&lt;li&gt;When I forget one resolution, I can easily find back in the book how to tackle it again.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This way I am hoping learn and get used progressively to the most common equations I am facing in my work (wave propagation, diffusion, reaction rates...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you for your recommendation.&lt;/p&gt;&#xA;" OwnerUserId="2140" LastEditorUserId="2140" LastEditDate="2014-09-22T19:13:02.053" LastActivityDate="2015-12-23T06:37:43.033" Title="Differential equations book" Tags="|textbooks|reference-books|differential-equations|" AnswerCount="2" CommentCount="7" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4454" PostTypeId="2" ParentId="715" CreationDate="2014-09-23T23:43:42.727" Score="14" Body="&lt;p&gt;I've always been surprised by this way of measuring the area of an ellipse. Instead of the classical use of the semi-major and semi-minor axes, you can actually do something similar to what you do with a parallelogram:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw two parallel lines that just touch the ellipse on the outside. We'll say these lines are horizontal. We'll call the distance between these lines h. Halfway between them, measure across your ellipse horizontally. Let's call half this width r. Then the area of the ellipse is $\pi r (\frac12 h)$. It's particularly impressive when $h=2r$ because then the area is just $\pi r^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works because your ellipse is a shear of another ellipse with the same height and horizontal radius. In essence because of the 2D version of Cavalieri's principle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/XWiM1.png&quot; alt=&quot;circle and ellipse&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-09-23T23:43:42.727" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4455" PostTypeId="2" ParentId="6" CreationDate="2014-09-24T03:53:17.310" Score="6" Body="&lt;p&gt;The question as posed asks &quot;What are possible, good ways of overcoming the availability of free and proficient homework assistance?”  I find myself wondering what the purpose of the homework is. Since this was not clear to me from the question, I am going to take as given that the purpose of homework (as the rest of the activities undertaken by students and teacher related to the course) is to promote learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If so, I think the specific exercises included are not just “the wrong ones” (in that their answers are easily locatable), but the wrong &lt;strong&gt;kind&lt;/strong&gt;. So what is the right kind? First of all, there should be (as Taalman calls it) a &quot;Problem Zero&quot; (&lt;a href=&quot;http://www.maa.org/problem-zero&quot; rel=&quot;noreferrer&quot;&gt;http://www.maa.org/problem-zero&lt;/a&gt;), where students are asked to articulate the key ideas. This is likely to elicit more thinking on their part, require more work, and gives the instructor more to go on to ensure more learning in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, there are many simple adjustments possible to the exercises you have. One is the “give them the answers and ask them to justify” that Michael Pershan proposed above (it seems with limited acclamation). Specific examples include asking them to explain whether their method would work in another case, or articulate the limits of how far it generalizes; comparing worked solutions (which can be harvested from prior student work) to identify either connections or contrasts in reasoning or diagnose errors; or writing their own problems and explaining how they are related.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the answers are already findable, clearly the student generating the answers (again) does not have some inherent value. The desired output of the course is for them to learn.&lt;/p&gt;&#xA;" OwnerUserId="2144" LastActivityDate="2014-09-24T03:53:17.310" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4456" PostTypeId="1" AcceptedAnswerId="4485" CreationDate="2014-09-24T14:44:57.200" Score="11" ViewCount="250" Body="&lt;p&gt;&lt;em&gt;I asked &lt;a href=&quot;https://math.stackexchange.com/questions/940182/the-use-of-sound-in-mathematics&quot;&gt;this question&lt;/a&gt; on Mathematics Stack Exchange first. It was well-received, but put on hold as off-topic. This site seems more appropriate. Here it is verbatim.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I'm not sure that this question is appropriate here. There's a good chance it's too opinion-based. If that's the case, I'm sorry.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was sat in a research seminar recently and wondered whether it'd be tacky if the speaker made the mathematics in his presentation somehow audible (in a non-trivial sense).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been tempted to write a programme or something ever since, that'd attach all the bells &amp;amp; whistles to a given symbol-heavy proof. I picture something like a beamer presentation where each relevant slide makes a noise once it's called, so that any patterns in the equations (or whatever) really leap out at the audience. If used sparingly and with &lt;a href=&quot;http://scratch.mit.edu/projects/1954944/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;taste&lt;/em&gt;&lt;/a&gt;, I suppose it could be quite effective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\color{red}{\text{The }}\color{green}{\text{use }}\color{blue}{\text{of }}\color{magenta}{\text{colour }}$, $\Large{\text{font sizes}}$, &lt;em&gt;italics&lt;/em&gt;, etc., are certainly effective on me, so why not sound?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In studying Mathematics alone, too, especially with dry, dense proofs, I have tried making small noises. It is a bit silly, yeah, but the odd &quot;pop&quot; for certain substitutions can be really handy. If anything, it makes things less dry . . .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here's my question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How best does one use sound in the study and/or presentation of Mathematics?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Again, I'm sorry if this is too opinion-based.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;NB:&lt;/strong&gt; I suppose what I'm looking for is suggestions for its use, ideally drawing upon experience and research. I don't know (yet) how to decide which answer (if any) would be considered the best. I'm sorry.&lt;/p&gt;&#xA;" OwnerUserId="396" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-10-03T04:05:59.203" Title="The Use of Sound in Mathematics" Tags="|self-learning|presentation|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4457" PostTypeId="2" ParentId="1316" CreationDate="2014-09-24T14:55:58.330" Score="9" Body="&lt;p&gt;Perhaps the experience of just such a student may help you? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a student, when I chose my second-year maths courses, I compared the list of topics that were in second-year courses to the topics I studied in first-year courses and I decided what I wanted to do based on the things I enjoyed most in first year. It seemed to me that the courses labelled as &quot;applied maths&quot; contained mostly things I did &lt;em&gt;not&lt;/em&gt; enjoy in first year maths. That is, heavily computer-based things, numerical things and differential equations. So I chose not to study any of those courses and did statistics and pure maths instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It wasn't until much much later that I thought about &lt;em&gt;why&lt;/em&gt; I didn't enjoy those topics in first year. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, the maths-on-computers thing: In first year, this was mainly just frustrating. I would regularly get error messages for missing what seemed like small things, and I got told off by the teachers for not using the &quot;most efficient&quot; methods in my programs even though there didn't seem to be any instructions in how to make them more efficient. It seemed like a wasted effort when most of the problems we did were just as easy for me to do by hand.  Moreover, my way of understanding things is very much visual and geometrical and the computer programming was very much text-based and just didn't feel the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much later, I discovered that I could use the computer to simulate things and create lots of data that allowed me to investigate things I could never have achieved by hand. For example, I could simulate a board game I was designing to see what it did, or decide if a hypothesis test in stats really did seem to have 80% power. This power of the computer was not really apparent to me in first year. I also discovered the pleasure in having to make a program do something and checking and rechecking and editing it until it did what I needed it to well. This pleasure of a job well done was also lost on me in first year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, the differential equations: In first year, differential equations was very much a procedural sort of topic. It was all very much of the form: this particular situation goes with this particular solution because it just does, so get good at recognising it. This went against everything I felt about understanding maths. I felt that there ought to be unifying principles and big picture ideas. I also liked the problem-solving aspect of other bits of maths where you had a small amount of tools and you worked out yourself how they fit together to show/prove/find something and there were often different methods of doing it, all of which were kind of clever. I disliked it when people said that maths was a list of problems and how to solve each one, yet this was precisely my experience of differential equations in first year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much much later, when helping students to study the higher-level courses, I saw more methods and how they fit together. I saw how quite deep bits of theory were brought to bear on these problems, such as treating functions as a vector space, or representing functions as infinite series. I also saw how very clever some of these methods were, and was able to stand in awe of the cool problem-solving aspect of the discipline. I wished then that more had been said about this side of it when I was in first year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the numerical methods stuff: In first year the extent of this was calculating integrals by the trapezoidal and Simpson's rules. We brushed over this stuff very quickly and only did it on functions where we knew how to calculate it another way anyway. I certainly never had a feeling of a bigger picture. Also, they taught us these crazy error formulas which were just made me think  &quot;where the hell did this 2880 come from?&quot; and &quot;surely if I know the fourth derivative of a function I can do better than using an approximation?&quot;. Finally, it was just a lot of hard work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much later, I realised that there were unifying principles in these methods (such as the area being related to the average value, so it's reasonable to average some values). I also realised how supremely clever it was that you &lt;em&gt;could&lt;/em&gt; figure out an error formula, and I wished I knew more about where those sorts of formulas came from. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is one final aspect that is worth mentioning: the applied-maths-ish topics were the ones I just didn't understand so well, probably because they didn't fit with my natural way of thinking. So another reason I didn't enjoy them was simply because I didn't enjoy the sensation of feeling confused and they were a lot more like hard work and a lot less like play!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to sum up: it wasn't so much that I didn't like the applications, but that the style of teaching, learning and problem-solving in the applied maths topics in first year was something that I didn't enjoy. Later, I realised that there were aspects of it that I could enjoy. (Though to be completely honest, when I watch people doing it, it often still doesn't look like fun to me.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And what can you take away from this as a teacher? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well perhaps you should ask them about the topics they have seen as applied maths in the past, and whether they enjoyed it or not. And indeed, ask them about what particular aspects of other maths topics they &lt;em&gt;did&lt;/em&gt; enjoy. Then you might have a different sort of picture of why they feel grumpy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also try to include elements in what you're doing that make it feel more like the other maths topics. Talk about the big picture structure. Talk about the clever maths problem-solving aspects of it (as opposed to the solving real-world issues). Talk about the pleasure of a job well done with the computing stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, you can make it really really really clear. There is nothing like simply not understanding that gets students looking around for reasons not to do something!&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-09-24T14:55:58.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4458" PostTypeId="1" CreationDate="2014-09-24T17:21:59.930" Score="7" ViewCount="370" Body="&lt;p&gt;I have a question where we are given a statement and asked to prove whether or not the statement is true. If the statement is true, then we must prove it; otherwise we must provide a counterexample to prove the statement incorrect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the statement is false, and the student provides a counter-example; should the student also provide an explanation of why the counter-example disproves the statement? &lt;/p&gt;&#xA;" OwnerUserId="2145" LastActivityDate="2014-09-25T02:20:42.967" Title="Explanation of Counter-example?" Tags="|undergraduate-education|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4459" PostTypeId="2" ParentId="1316" CreationDate="2014-09-24T17:47:15.683" Score="2" Body="&lt;p&gt;First, rather than &quot;pure&quot; and applied mathematics, I prefer &quot;theoretical&quot; and applied mathematics. What is &quot;impure&quot; mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, many students are put off by the &quot;word problems&quot; they see in K-12 since if these problems are typical of the &quot;power&quot; of applying mathematics, then they are skeptical. (If Mary is twice as old as her brother was when ..... How much does Mary's dog weigh?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience the best way to expose students to applications of mathematics in lower grades is to show them how powerful graph theory is as a tool for getting insight into the world (garbage collection and mail delivery routes, routing packets on the internet, and software that provides driving routes from A to B, etc.) and examples that show the role of mathematics in the technologies that are so much part of their worlds. Thus, the cell phone in your pocket (or pocketbook) is made possible by error correction and data compression ideas, frequency assignment algorithms, as well as GPS technology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sadly, the recently adopted CCSS-M do little to change the perceptions of mathematics that many students get, and I think are likely to make these perceptions worse.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-09-24T17:47:15.683" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4460" PostTypeId="2" ParentId="4458" CreationDate="2014-09-24T18:26:01.957" Score="9" Body="&lt;p&gt;I believe this depends on how clearly the counterexample is stated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider this claim: $f:\mathbb{R}\to\mathbb{R}$ is continuous $\implies f$ is differentiable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine a student, let's call him Karl, who says&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Take any function of the form $\sum_{n=0} ^\infty a^n \cos(b^n \pi x)$, where $0&amp;lt;a&amp;lt;1$, $b$ is a positive odd integer, and $ab &amp;gt; 1+\frac{3}{2} \pi$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Would you give him credit for this counterexample? Even presuming that the difficulty level of your course and the talents of your students are such that you did, indeed, expect them to come up with this result, I don't feel like this answer is sufficient. Obviously, Karl must have understood &lt;em&gt;something&lt;/em&gt; about the exercise to be able to come up with this answer (never mind the great ingenuity required) but his written argument fails to demonstrate that knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Contrast that with a disproof of this claim: For any function $f:A\to B$ and any sets $S,T\subseteq A$, $f[S]\subseteq f[T]\implies S\subseteq T$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the kind of statement you might consider in an introductory analysis course. I'd expect a student to find an easy counterexample, say by defining $A=\{\heartsuit,\spadesuit\}$ and $B=\{\star\}$ and setting $f(\heartsuit)=f(\spadesuit)=\star$ and $S=\{\heartsuit\}$ and $T=\{\spadesuit\}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would I expect the student to have to show all the details? Do they really need to explicitly say, &quot;Observe that $S,T\subseteq A$ and $f[S]=f[T]$ and yet $S\not\subseteq T$&quot;? Well, that depends on your standards. If you're really training them to write clearly and explicitly (concision be darned), then you might require them to show these details. If you're just checking to see that they can &lt;em&gt;generate&lt;/em&gt; such examples, then merely stating the example would suffice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Main answer:&lt;/strong&gt; It depends on your standards for such a written argument, which in turn should depend on what you expect the student to be able to understand and demonstrate.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2014-09-24T18:26:01.957" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4461" PostTypeId="2" ParentId="4458" CreationDate="2014-09-24T23:42:00.027" Score="4" Body="&lt;p&gt;Certainly the student should be aware of the expectations beforehand. To this end, I think there are at least two approaches. One is to specify on individual questions whether counterexamples should be explained, and the other is to establish (preferably from the outset, i.e., setting &quot;norms&quot; at the beginning of a course) the expectation that reasoning/justification for &lt;em&gt;all answers&lt;/em&gt; will be provided.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, I think explaining counterexamples is important. Let me give just two reasons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Consider the following problem: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is every whole number divisible by 6 and 8 also divisible by 6 $\times$ 8 = 48? If so, prove your answer; if not, please provide a counterexample. &lt;strong&gt;Answer:&lt;/strong&gt; 24.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what does such a student understand? Perhaps she just remembered 24 is divisible by both and realized it is not divisible by 48. Or perhaps she realizes that the issue is that the two divisors in this proposed rule are not relatively prime, and could correct it by using, e.g., 3 and 16 instead. Or, even more seriously, perhaps she has the deeper understanding, but accidentally writes 32. In this last case, the understanding is the deepest, but it would be tough to justify giving the response any credit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Asking that the counterexample be justified ameliorates this potential problem. Personally, I would expect more than just an explanation of the particular counterexample, e.g., more than:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;No, and 24 is a counterexample: It is divisible by 6 because 24 = 4 $\times$ 6, and it is also divisible by 8 because 24 = 3 $\times$ 8. But it is &lt;em&gt;not&lt;/em&gt; divisible by 48, because 24/48 is not a whole number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would prefer to see an answer that mentions being &quot;relatively prime&quot; or something equivalent, and perhaps even one that gives an example of two numbers that &lt;em&gt;would&lt;/em&gt; yield a divisibility rule for 48.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) More subtle and perhaps not related to the specific question you have in mind (if there is one): Note that (speaking slightly messily) counterexamples can settle &lt;em&gt;for all&lt;/em&gt; questions, but not &lt;em&gt;there exists&lt;/em&gt; questions. The previous example is the assertion that &lt;em&gt;all&lt;/em&gt; whole numbers (etc). If this turns out to be false, then it can be settled with a counterexample. If it turns out to be true, then it will need a further proof or justification. On the other hand, a &lt;em&gt;there exists&lt;/em&gt; statement that is true might be settled with a single example. If it turns out to be false, then it will need a further proof or justification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This latter point about proving/disproving by (counter)example in the context of &lt;em&gt;for all&lt;/em&gt; vs. &lt;em&gt;there exists&lt;/em&gt; assertions may seem a bit pedantic, but it is a place in which many students who are just beginning to understand ideas around &quot;proof&quot; can struggle quite a bit.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-09-25T02:20:42.967" LastActivityDate="2014-09-25T02:20:42.967" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4462" PostTypeId="1" CreationDate="2014-09-25T11:32:09.413" Score="2" ViewCount="197" Body="&lt;p&gt;I'm 26 and hold a 2.2 degree in Biomedical Science from a British university. I took my A Levels in Biology and Chemistry but only got an AS Level in Maths and Art. My work background is that I have been a carer for family members for many years. I would love to pursue Maths teaching but I am unsure whether I would be accepted onto the programmes and whether I should complete an A Level in Maths with a view to apply for PGCE or Schools Direct Route in 2015.&lt;/p&gt;&#xA;" OwnerUserId="2148" LastActivityDate="2014-09-28T01:19:42.130" Title="Can I be a Maths Teacher?" Tags="|graduate-education|" AnswerCount="1" CommentCount="9" ClosedDate="2014-09-30T04:28:09.227" ContentLicense="CC BY-SA 3.0" />
  <row Id="4463" PostTypeId="1" AcceptedAnswerId="4468" CreationDate="2014-09-25T13:32:17.473" Score="19" ViewCount="621" Body="&lt;p&gt;I am working as a teaching assistant in a course that first year mathematics students take. Some students who have maths as a minor also take it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I run what is called a supervising session in the official translation - guidance session would be more literal translation. I help students who try to solve exercises similar to the ones they are to complete for the exercise sessions. There is no effect on grading or any other incentive besides learning for students to come to the guidance sessions. There are maybe 10 to 20 students (in maybe 6 to 13 groups, though some groups are singletons), and one tutor. I will visit the student groups, moving through them in order. Sometimes I will write a sample proof or discuss something several groups have problems with  on the blackboard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often the students have several problems. Consider the following, which is very close to a real example, though most attempts are not quite this weak: The task is to prove that if $a \geq 4$ and $b \geq 2$, then $a+b \geq 6$.&#xA;The student has written: $a \geq 4$, $b \geq 2$, $a+b \geq 6$. Antithesis: $a &amp;lt; 4$. After this the student has done calculations, some of them faulty, to get a contradiction with $a+b \geq 6$. The calculations are simply lines of inequalities without words or implication arrows joining them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the mistakes: Assumptions and the claim are not clearly separated. The antithesis is wrong. There are mistakes in the calculation. The writing is poor, as it only consists of formulae.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;When encountering a student's proof attempt (or calculation) with several mistakes, should I focus on correcting one mistake at a time, or should I point out all of them?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I fear that a barrage of corrections will only confuse the students. On the other hand, correcting only something may not be enough for them to complete an exercise, and I may not guess what issue or which issues are worst ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any literature on this, or any best practices?&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2014-09-26T11:13:35.733" Title="Should I correct only one mistake at a time?" Tags="|undergraduate-education|tutoring|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4464" PostTypeId="2" ParentId="4463" CreationDate="2014-09-25T14:59:15.410" Score="12" Body="&lt;p&gt;I strongly believe that the best tutoring skill to have is to identify one issue they need to fix, then on the spot, &lt;strong&gt;craft a new, simpler question&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you give them the right simplified question, they will make one error -- and as you point out, it is easy to correct one error. Once they feel confident, you can give them a new question in which they will make a new error. Then you can help fix that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The hard part, of course, is being able to make up questions on the spot that are targeted at the student's exact problem.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For your example, I would probably start with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Solve 3 - 4x &amp;lt; 8.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They will make an error with the inequality signs. Fix it. Then follow up with&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Prove that if x &amp;lt; 5, then 1 - 3x &gt; -14.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They will not connect their claims with implication arrows or be organized. Fix it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then keep going.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-09-25T14:59:15.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4465" PostTypeId="2" ParentId="4456" CreationDate="2014-09-25T15:52:02.203" Score="3" Body="&lt;p&gt;This is not a direct answer to your interesting question, but rather just&#xA;one instance where I found it useful to use sound to convey aspects&#xA;of an algorithm's operation.&#xA;The algorithm walks through the space of polygonizations of a point set,&#xA;and I used the polygon perimeter to control the pitch of a &quot;Crystal&quot; MIDI instrument.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/AikWI.png&quot; alt=&quot;Exponential&quot;&gt;&#xA;&lt;hr /&gt;&#xA;If your browser enables Quicktime movies, you can see/hear it at this link:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://cs.smith.edu/~orourke/Twangs/Exponential.k8.n100.web.mov&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quicktime .mov download&#xA;(0.6MB)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://cs.smith.edu/~orourke/papers.php#263&quot; rel=&quot;nofollow noreferrer&quot;&gt;The paper&lt;/a&gt; and video were well-received when I presented at a conference.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-09-25T17:23:51.850" LastActivityDate="2014-09-25T17:23:51.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4466" PostTypeId="2" ParentId="4448" CreationDate="2014-09-25T21:19:35.777" Score="4" Body="&lt;p&gt;Have you seen &lt;a href=&quot;http://talkingsticklearningcenter.org/blog/&quot; rel=&quot;nofollow&quot;&gt;Rodi Steinig's blog&lt;/a&gt;? She does lots of very cool math topics. Her stuff may not fit your audience, though. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't yet worked with &lt;em&gt;groups&lt;/em&gt; of gifted kids, but I'd start out with an intriguing topic, and see where it took us. You might like &lt;a href=&quot;http://www.geometer.org/mathcircles/tangle.pdf&quot; rel=&quot;nofollow&quot;&gt;rational tangles&lt;/a&gt;, or one of the many other topics on &lt;a href=&quot;http://www.geometer.org/mathcircles/&quot; rel=&quot;nofollow&quot;&gt;Tom Davis' page&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-09-25T21:19:35.777" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4467" PostTypeId="2" ParentId="6" CreationDate="2014-09-26T06:02:55.960" Score="8" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Unfortunately, writing my own exercises (which I currently do) doesn't prevent or penalize the use of WolframAlpha or online math fora.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wolfram Alpha will do great if you have a long list of problems that are all posed in the same formulaic manner, and that all consist of nothing but calculation. 1. $d(x^7)/dx=?$ 2. $d(x^7+x^6)/dx=?$ etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a whole different story if you make the problems you're assigning not so formulaic -- even trivially so.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Evaluate $d(r^j)/dr$, where $j$ is a contstant.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;To you and me, this is obviously a trivial variation on problem 1, but to a student who is trying to use WA without understanding anything, it's a complete mystery, and they won't even know how to put it in a format that WA can accept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;WA can't do word problems. WA can't do proofs. Ask your students to check whether the answer has the right units, or to check that the answer makes sense in a special case, or to describe why the answer increases when one of the parameters decreases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic point is that WA can't &lt;em&gt;think&lt;/em&gt;. Ask your students to think. A computer can't do that.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-09-26T06:08:16.910" LastActivityDate="2014-09-26T06:08:16.910" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4468" PostTypeId="2" ParentId="4463" CreationDate="2014-09-26T11:13:35.733" Score="9" Body="&lt;p&gt;I believe your intuition &quot;that a barrage of corrections will only confuse the students” is on point. Even if you &quot;may not guess what issue or which issues are worst ones”, you (who have a big-picture view of the ideas, how they relate, and where they are going) probably will have a fighting chance of diagnosing the issue (especially if you actively keep these things in mind and try to make sense of their thinking).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While &quot;correcting only something may not be enough for them to complete an exercise”, that is not really the goal; the exercise, their response, and your corrections or feedback are tools to get to improved understanding on their part. A good reference on this is Dylan Wiliam’s &lt;em&gt;Embedded Formative Assessment&lt;/em&gt; (&lt;a href=&quot;http://www.solution-tree.com/embedded-formative-assessment.html&quot;&gt;http://www.solution-tree.com/embedded-formative-assessment.html&lt;/a&gt;); one of the main points he makes is that &lt;strong&gt;feedback should be more work for the recipient than the donor&lt;/strong&gt;. New simpler questions are a great example of putting this into practice. Comparing responses — to a similar question, or better yet the same question they have already worked -- is another way, that helps give the learner a “nose for quality.” &lt;/p&gt;&#xA;" OwnerUserId="2144" LastActivityDate="2014-09-26T11:13:35.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4469" PostTypeId="2" ParentId="4462" CreationDate="2014-09-27T09:23:48.937" Score="1" Body="&lt;p&gt;Mathematics and teaching it is not a suitable job for everybody. Math is a love and if you haven't got a love for mathematics it would be boring for you and your students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have huge interest to work with mathematical subjects, you would be the happiest person in your circumstance. Elsewhere it is not good idea to make it as your job. &lt;/p&gt;&#xA;" OwnerUserId="79" LastEditorUserId="1853" LastEditDate="2014-09-28T01:19:42.130" LastActivityDate="2014-09-28T01:19:42.130" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4470" PostTypeId="1" CreationDate="2014-09-29T13:17:28.537" Score="12" ViewCount="572" Body="&lt;p&gt;Here's a link to the full paper, colloquially known as &lt;a href=&quot;http://www.genealogy.math.ndsu.nodak.edu/id.php?id=37815&quot; rel=&quot;nofollow&quot;&gt;Lockhart&lt;/a&gt;'s Lament:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Link:&lt;/strong&gt; &lt;a href=&quot;https://www.maa.org/external_archive/devlin/LockhartsLament.pdf&quot; rel=&quot;nofollow&quot;&gt;https://www.maa.org/external_archive/devlin/LockhartsLament.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the context of K-8 learning materials that take into account the concerns discussed in Paul Lockhart's essay &lt;strong&gt;A Mathematician's Lament&lt;/strong&gt;, I wonder if anyone knows of syllabi, curricula, or worksheets that I can use to supplement my child's &quot;Core&quot; program in school, and what experiences (if any) others have with using said materials.&lt;/p&gt;&#xA;" OwnerUserId="2154" LastEditorUserId="262" LastEditDate="2014-10-01T04:08:29.467" LastActivityDate="2014-10-01T04:08:29.467" Title="Teaching K-8 math in the style of &quot;A Mathematician’s Lament&quot;" Tags="|reference-request|primary-education|mathematical-pedagogy|curriculum|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4471" PostTypeId="2" ParentId="4470" CreationDate="2014-09-29T19:30:17.300" Score="13" Body="&lt;p&gt;In &lt;em&gt;A Mathematician's Lament&lt;/em&gt;, Paul Lockhart describes what is wrong with math education in schools. To get a good idea of how he would do things differently, you should read his book titled &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0674057554&quot; rel=&quot;nofollow&quot;&gt;Measurement&lt;/a&gt;&lt;/em&gt;. The ideas in it are for slightly older students, but the approach is what you're looking for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend that you try to work through &lt;em&gt;Measurement&lt;/em&gt; yourself, and see what questions pop up for you that you can share with your child. If you are comfortable with math yourself, you might want to start a math circle. If you are not that comfortable, and you live close to one, check out a &lt;a href=&quot;http://juliarobinsonmathfestival.org/&quot; rel=&quot;nofollow&quot;&gt;Julia Robinson Mathematics Festival&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also find my book, &lt;em&gt;&lt;a href=&quot;http://www.moebiusnoodles.com/playingwithmath/&quot; rel=&quot;nofollow&quot;&gt;Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers&lt;/a&gt;&lt;/em&gt;, useful. It should be out in November.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="262" LastEditDate="2014-10-01T03:59:51.783" LastActivityDate="2014-10-01T03:59:51.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4473" PostTypeId="1" AcceptedAnswerId="4474" CreationDate="2014-09-30T14:43:15.023" Score="23" ViewCount="2171" Body="&lt;p&gt;We often use the word &lt;em&gt;roots&lt;/em&gt; when referring to the solutions of an equation. For instance, when we have a polynomial $P(x)$, we call its zeros the &lt;em&gt;roots&lt;/em&gt; of $P(x)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some polynomials we can relate the zeros with a root function of some kind, say &#xA;$$&#xA;x^2-4=0,&#xA;$$&#xA;we can take the square roots of $4$ to get the solutions. However with some functions, like trigonometrics, there doesn't seem to be a clear relation with the typical root functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask because in an assignment I gave, students where asked to find the roots of a rational function.  I found out that many students had never heard the word &lt;em&gt;roots&lt;/em&gt; used in a &quot;find the zeros&quot; context. Many of them took the square roots of the function and tried solving some equations that made no sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When looking online, I could not find any solid reference to the first uses of the word in that context.&lt;/p&gt;&#xA;" OwnerUserId="805" LastEditorUserId="1706" LastEditDate="2014-10-01T00:39:07.023" LastActivityDate="2022-10-18T19:45:14.943" Title="Where does the word &quot;roots&quot; come from when talking about zeros" Tags="|terminology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4474" PostTypeId="2" ParentId="4473" CreationDate="2014-09-30T19:04:04.913" Score="21" Body="&lt;p&gt;Doctor Peterson in &lt;a href=&quot;https://web.archive.org/web/20170829212727/http://mathforum.org/library/drmath/view/71575.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Math Forum&lt;/a&gt; refers to the following sources.&lt;/p&gt;&#xA;&lt;p&gt;From D. E. Smith, &lt;em&gt;History Of Mathematics Vol II&lt;/em&gt; (1925), footnote page 393:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Arabs also used &lt;em&gt;jidr&lt;/em&gt; (&lt;em&gt;dyizr&lt;/em&gt;, root), whence the Latin &lt;em&gt;radix&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A longer quote from W. W. R. Ball, &lt;em&gt;A Short Account of the History of Mathematics&lt;/em&gt; (4th edition, 1908):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The algebra of Alkarismi holds a most important place in the history&#xA;of mathematics, for we may say that the subsequent Arab and the&#xA;early medieval works on algebra were founded on it, [...].&#xA;The unknown quantity is&#xA;termed either &amp;quot;the thing&amp;quot; or &amp;quot;the root&amp;quot; (that is, of a plant), and from&#xA;the latter phrase our use of the word root as applied to the solution&#xA;of an equation is derived. The square of the unknown is called &amp;quot;the&#xA;power&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="39" LastEditorUserId="6053" LastEditDate="2022-10-18T19:45:14.943" LastActivityDate="2022-10-18T19:45:14.943" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="4475" PostTypeId="1" AcceptedAnswerId="4476" CreationDate="2014-10-01T12:04:47.147" Score="35" ViewCount="2103" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Metonymy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Metonymy&lt;/a&gt; is a figure of speech where a word or another expression is used to mean something other than its literal meaning.&#xA;This phenomenon is not restricted to the &quot;usual human languages&quot; (such as English or Finnish) but also appears in the symbolic language of mathematics.&#xA;I am not interested in drawing boundaries between English and mathematics in a mathematical presentation; I am interested in metonymy in all mathematical contexts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples of metonyms:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When we discuss a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, we often actually write or say &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, although &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is the value of the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. That is, the value of the function at a &quot;generic point&quot; is used as a metonym for the function itself. (An extended example of this metonymy is the statement &quot;The Fourier transform of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\hat f(\xi)$&lt;/span&gt;&quot;.)&lt;/li&gt;&#xA;&lt;li&gt;A continuous function &lt;span class=&quot;math-container&quot;&gt;$\gamma:I\to\mathbb R^n$&lt;/span&gt; from a real interval &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is a curve. &quot;The curve &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;&quot; occasionally also refers to the image &lt;span class=&quot;math-container&quot;&gt;$\gamma(I)$&lt;/span&gt;, the trace of the curve, by metonymy. Different authors use different conventions here, but the word &quot;curve&quot; is used in various different meanings.&lt;/li&gt;&#xA;&lt;li&gt;A group is a set together with a (certain kind of) binary operation. The set alone is often used to denote the group by metonymy, the operation being understood implicitly.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Metonyms are always about using the same expression (word, symbol&amp;hellip;) in several different meanings.&#xA;Metonymy is often convenient, but it can also be a source of confusion if it is not introduced explicitly.&#xA;Sloppy notation and abuse of notation are typically metonymous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How should I take metonymy into account when teaching mathematics?&#xA;Should I try to explicitly state whenever this linguistic phenomenon is present?&#xA;In my experience this is rarely done, and I have a feeling that some students have difficulties because they do not recognize when an expression is used metonymously.&lt;/li&gt;&#xA;&lt;li&gt;What are the most important metonyms that I should make my students aware of?&#xA;(I am thinking about university level education, but answers on any level are welcome.)&lt;/li&gt;&#xA;&lt;li&gt;Are there some metonyms that I should avoid or instruct my students to avoid?&#xA;Do students commonly have misconceptions of a metonymous origin?&#xA;This is the kind of problem that I had in mind:&#xA;If a banana costs &lt;span class=&quot;math-container&quot;&gt;$3\$$&lt;/span&gt; and an apple &lt;span class=&quot;math-container&quot;&gt;$2\$$&lt;/span&gt;, a student may be tempted to write &lt;span class=&quot;math-container&quot;&gt;$\text{banana}=3\$$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\text{apple}=2\$$&lt;/span&gt; and conclude &lt;span class=&quot;math-container&quot;&gt;$2\times\text{banana}=3\times\text{apple}$&lt;/span&gt; which is not true.&#xA;This is because of using the symbol &quot;&lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;&quot; metonymously, not actually referring to two things being equal.&#xA;If one introduces a price function &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and lets &lt;span class=&quot;math-container&quot;&gt;$p(\text{apple})=2\$$&lt;/span&gt; and so on, then &lt;span class=&quot;math-container&quot;&gt;$p(2\times\text{banana})=p(3\times\text{apple})$&lt;/span&gt; is a reasonable statement &amp;mdash; and a true one if the price function is linear.&lt;/li&gt;&#xA;&lt;li&gt;To what extent should I delay exposition to metonyms?&#xA;Metonymy is a part of standard mathematical communication, and mathematical educations should prepare students for it.&#xA;But this does not necessarily mean that I should express myself metonymously from day one when teaching a new topic.&lt;/li&gt;&#xA;&lt;li&gt;Can metonymy be used as a tool to help learn mathematics?&#xA;That is, are there useful metonyms that I should introduce?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2020-01-07T11:54:11.877" LastActivityDate="2020-01-07T11:54:11.877" Title="Metonymy in mathematics" Tags="|undergraduate-education|students-mistakes|notation|terminology|language-use|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="4476" PostTypeId="2" ParentId="4475" CreationDate="2014-10-01T15:08:49.443" Score="21" Body="&lt;p&gt;Metonymy and its relatives, metaphor, polysemy, synecdoche occur all over the place in mathematical writing, and sometimes cause students problems and sometimes don't, because those thought processes are basic to our understanding of everything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Where mathematics comes from&lt;/em&gt;, by George Lakoff and Rafael E. Núñez. Basic Books, 2000.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example&lt;/em&gt;, by Michelle J. Zandieh and Jessica Knapp.&#xA;The Journal of Mathematical Behavior, Volume 25, Issue 1, Pages 1-17.  This is owned by (boo hiss) Elsevier who charges an exorbitant amount to get a copy, so I have not read it.  I have not checked if it is on Jstor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://ejournals.library.ualberta.ca/index.php/complicity/article/viewFile/8835/7155&quot;&gt;Your Metaphor, or My Metonymy?&lt;/a&gt;, by John Mason, 2010.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.sunysb.edu/~tony/whatsnew/column/metonymy-1199/metonymy1.html&quot;&gt;Metonymy and metaphor in mathematics&lt;/a&gt;, by Tony Phillips, 1999.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cwru.edu/artsci/math/wells/pub/pdf/hyperhbk.pdf&quot;&gt;Handbook of mathematical discourse&lt;/a&gt;, by Charles Wells.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.abstractmath.org/MM/MMImagesMetaphors.htm&quot;&gt;Images and metaphors&lt;/a&gt;, article in &lt;a href=&quot;http://www.abstractmath.org/&quot;&gt;abstractmath.org&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://abstractmath.org/MM/MMMoreLanguage.htm#incompletenotation&quot;&gt;Incomplete notation&lt;/a&gt;, section of an article in abstractmath.org.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://abstractmath.org/MM/MMMoreLanguage.htm#embedding&quot;&gt;Embedding&lt;/a&gt;, section of an article in abstractmath.org.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my favorite examples of this sort of thing is when a mathematician says something like &quot;x^2-1 &lt;strong&gt;vanishes&lt;/strong&gt; at x=1&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1599" LastActivityDate="2014-10-01T15:08:49.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4477" PostTypeId="1" AcceptedAnswerId="4479" CreationDate="2014-10-01T18:42:34.083" Score="23" ViewCount="3715" Body="&lt;p&gt;Our college is switching to an Early Transcendentals calculus text, and this seems like a good time to consider which text we are using in general. Larson, Stewart, Thomas, Briggs/Cochran, etc are all on the table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the books we have no experience with have the burden of proof. If we don't form a strong enough opinion about any of them, we will certainly decide to just stay with the one we have been using.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is an example of a good reason to change entirely from one author to another?&lt;/strong&gt; This change would cause inconveniences for faculty and students alike. I don't currently have any preferences that are strong enough to outweigh this textbook inertia, and I'd like to know how other faculty have approached the problem. &lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-03-12T13:32:08.273" Title="What is a good reason to change calculus texts?" Tags="|undergraduate-education|calculus|textbooks|" AnswerCount="8" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4478" PostTypeId="2" ParentId="4477" CreationDate="2014-10-01T22:42:15.953" Score="5" Body="&lt;p&gt;We make good use of WeBWorK in many of our mathematics courses. WeBWorK is an open source online homework system supported by the MAA. My university uses a textbook rental system. When the stock of calculus textbooks got low a year or two ago, a significant factor in deciding on a new calculus textbook was the quality and quantity of questions in the WeBWorK National Problem Library that were well aligned with the textbook. Stewart Early Trancendentals was a good choice. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2014-10-01T22:42:15.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4479" PostTypeId="2" ParentId="4477" CreationDate="2014-10-01T22:48:11.733" Score="28" Body="&lt;p&gt;I would like to encourage consideration of a free textbook. The conventional textbooks are outrageously expensive. (&lt;em&gt;Actually, if your department insists on one of the choices you mentioned, I'd want to go with the cheapest.&lt;/em&gt;) Students are suffering from massive amounts of debt these days, with no guarantees of good jobs with which to pay off the debt. Expensive textbooks are an added burden.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;em&gt;If your department does decide to look into free and/or low-cost textbooks, choosing among those is a different question.&lt;/em&gt;)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="-1" LastEditDate="2014-10-03T01:32:05.197" LastActivityDate="2014-10-03T01:32:05.197" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4480" PostTypeId="2" ParentId="4477" CreationDate="2014-10-02T00:13:16.517" Score="25" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What is a good reason to change calculus texts?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Reason #1: the text you're currently using costs money.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reason #2: your current text uses an online homework system such as MyMathLab that costs money, rather than a free one like WeBWorK.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some good free texts:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Robbin and Angenent - &lt;a href=&quot;http://www.math.wisc.edu/~angenent/Free-Lecture-Notes/&quot; rel=&quot;noreferrer&quot;&gt;http://www.math.wisc.edu/~angenent/Free-Lecture-Notes/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boelkins - &lt;a href=&quot;http://faculty.gvsu.edu/boelkinm/Home/Open_Calculus.html&quot; rel=&quot;noreferrer&quot;&gt;http://faculty.gvsu.edu/boelkinm/Home/Open_Calculus.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keisler - &lt;a href=&quot;http://www.math.wisc.edu/~keisler/calc.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.math.wisc.edu/~keisler/calc.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm currently using my own spin-off of Robbin and Angenent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Picking a text based on cost may seem like putting the cart before the horse, but in fact the commercial texts are all very much alike, and although the free ones show a little more individuality, what you're still getting is basically a freshman calc book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's also worth carefully considering the assumption that the department must adopt a single text. The only good reason for this is economic: students need to be able to sell back their books, you want there to be a used market, and the bookstore may want the option of holding on to unsold copies rather than having to ship them back. But none of these economic reasons holds if you're using a free text. For example, nothing bad happens if your department decides to recommend to all faculty that they use either Larson (commercial) or a free text of their choice. A simple desire for uniformity is &lt;em&gt;not&lt;/em&gt; a valid reason to force everyone to use the same book. This is an academic freedom issue. If someone can't be trusted to pick their own book, then that person isn't qualified to teach the course in the first place.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-10-03T02:14:01.133" LastActivityDate="2014-10-03T02:14:01.133" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4481" PostTypeId="2" ParentId="4351" CreationDate="2014-10-02T02:20:32.100" Score="34" Body="&lt;p&gt;A very simple example (if you know calculus) is finding the local minimum of a downward-opening parabola, like $y = -x^2$. The unique candidate comes from setting the derivative equal to zero, but then you still have to check the second derivative to see whether it is, in fact, a minimum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is also a very physically relevant example, because it comes up pretty much any time you are trying to see whether a physical system has a stable equilibrium. Checking that the minimum is really a minimum makes the difference between the system settling into a stable configuration and blowing itself apart because it's unstable.&lt;/p&gt;&#xA;" OwnerUserId="463" LastActivityDate="2014-10-02T02:20:32.100" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4482" PostTypeId="1" AcceptedAnswerId="17366" CreationDate="2014-10-02T12:16:02.220" Score="22" ViewCount="1403" Body="&lt;p&gt;As I have been &lt;a href=&quot;https://matheducators.stackexchange.com/questions/4477/what-is-a-good-reason-to-change-calculus-texts&quot;&gt;recently informed&lt;/a&gt;, it is a good idea to consider free calculus textbooks for college and university courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this feels risky to me, because:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I don't know anyone who is using one, so I don't have a core &quot;users group&quot; to chat with and I don't have any assurance of quality control.&lt;/li&gt;&#xA;&lt;li&gt;I want to be sure my course will transfer to other institutions (at a community college this concern is paramount).&lt;/li&gt;&#xA;&lt;li&gt;The distribution of the textbook sounds tricky; what about students who want a hard copy to read?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So, with these concerns in mind, &lt;strong&gt;which free calculus textbooks are the best&lt;/strong&gt;? I know this is an opinion question at its core, but to make it more objective, my ideal free textbook would have:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A users group, message board, or Github page, that allows discussion and shows a history of the authors fixing errors or issues with the text.&lt;/li&gt;&#xA;&lt;li&gt;A group of respectable colleges that have adopted the book for their calculus courses.&lt;/li&gt;&#xA;&lt;li&gt;A &quot;printable version&quot; of the textbook or some optional hard-copy version available.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of course I am open to criticism of my assumptions, and I am also open to hearing about any other considerations I should have. &lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2023-02-07T05:32:38.747" Title="How can I choose a free calculus textbook?" Tags="|undergraduate-education|calculus|textbooks|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4483" PostTypeId="1" AcceptedAnswerId="4489" CreationDate="2014-10-02T22:34:00.713" Score="12" ViewCount="428" Body="&lt;p&gt;I am soon going to explain to my students what the alternating group is. The definition is subtle.... one must prove that the notation of an &quot;even permutation&quot; is well defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There seem to be two proofs of this. One writes down a polynomial (a square root of an appropriately defined discriminant) for which the alternating group preserves the sign. The other is a fairly straightforward induction argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't think of any way of describing the alternating group that avoids giving the impression that it is something complicated and confusing. (And Gallian, Beachy and Blair, Saracino, Herstein etc. don't seem to offer too much beyond what I already described.) Is there something simple I can say?&lt;/p&gt;&#xA;" OwnerUserId="1535" LastEditorUserId="143" LastEditDate="2014-10-03T12:42:24.443" LastActivityDate="2014-10-10T14:19:29.350" Title="How can one convincingly present the alternating group?" Tags="|concept-motivation|abstract-algebra|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4484" PostTypeId="1" AcceptedAnswerId="4491" CreationDate="2014-10-03T03:41:31.717" Score="17" ViewCount="1141" Body="&lt;p&gt;This question is long but I am asking for educated opinions on a question of math education and for this reason I'd like it not to be closed on the grounds that it invites subjective discussion.  Educational practice is not a subject on which there is a single right answer, but there is such a thing as a well-reasoned answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've taught a lot of ODEs, either as a dedicated course or as a unit within a linear algebra course.  I never actually took one myself, though I did read Coddington's book &lt;em&gt;Ordinary Differential Equations&lt;/em&gt;, and I would call myself an algebraist, though not the kind that hates analysis (I enjoy teaching it at this level, actually).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My feeling is that the traditional ODEs course is not very good, but it is nearly universally agreed upon by textbooks.  It looks like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;First-order equations of all kinds, both linear and nonlinear, with techniques such as separation of variables, integrating factors, substitutions, and exact equations (so, more integrating factors).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Second-order equations, linear and constant-coefficient only, with an ad-hoc derivation of the characteristic equation and informal justification of undetermined coefficients and variation of parameters; the Wronskian appears here as a trick.  Big presentation of the &quot;three cases&quot; for the characteristic equation.  Then maybe variable-coefficient reduction of order.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mention of higher-order equations, with all of the above recapitulated halfheartedly.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some kind of discussion of linear systems, complicated by the fact that linear algebra is often not a prerequisite for the course so must be taught on the spot (or simply avoided, in which case this topic is incomplete).  The three cases are discussed &lt;em&gt;again&lt;/em&gt;.  The matrix exponential may be introduced, and this entails another discussion of the three cases.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is often a single day on the existence and uniqueness theorems at some point, perhaps here, but since the proof is serious analysis, it's hard to make this sound like anything but a footnote.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is also often a small unit on numerical methods, which doesn't get to anything serious.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Infinite series, possibly even including Frobenius series, are given a lot of attention.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Laplace transforms get just as much attention, often, though typically only applied to inhomogeneous constant-coefficient equations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nonlinear systems are somewhere in there, with a discussion of the phase plane and equilibrium points.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;That's a lot of topics and some of them are a lot of work, and I have often gotten the impression that students find the subject entirely disunified.  Once, I think they didn't even understand that the &quot;three cases&quot; for linear equations were actually the same but in different contexts, rather than somehow being a different technique for second-, third-, ...order equations, and systems.  The same problem exists in a larger sense: almost all of the techniques overlap and students can confused about whether they even solve the equations in the same way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my question is: what is the best way to actually structure this course so it teaches &lt;em&gt;an idea&lt;/em&gt;, rather than &quot;how to solve some differential equations&quot;?  For the record, my own preference is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Start with a few days of first-order linear equations presented in a way as to presage the general method used for systems, and avoid special-purpose tricks.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then, teach (or review) linear algebra and constant-coefficient systems at the same time, with emphasis on using a (generalized) eigenbasis to find the fundamental solutions.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Introduce the matrix exponential as a unified method to solve an initial value problem directly, and from here, on to inhomogeneous systems, variation of parameters, and generalized undetermined coefficients.  The Wronskian shows up here completely naturally.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Now&lt;/em&gt;, talk about higher-order equations and the reduction to systems of first-order equations, and quickly derive everything about them from the preceding.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Talk about how there aren't such highly effective techniques for most differential equations and use that to motivate the existence and uniqueness theorems.  They should be explained somewhat, since the proof technique (Picard iteration) motivates both series and also the step methods for numerical solutions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Now you're free to talk about special-purpose tricks.  I like to go back and do exact first-order equations and integrating factors, because that's the context in which the latter really make sense.  I &lt;em&gt;don't&lt;/em&gt; bother with reduction of order, or substitution, because they seem to have no serious purpose.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Since I'm on nonlinear equations, I move onto nonlinear systems and don't skimp on the trajectory sketching: it's as much a solution method as writing down an infinite series.  The connection with linear systems is obviously to be emphasized strongly.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Logically, the next thing must be numerical solutions, since it is a fail-safe for getting some kind of a solution.  It would be nice if there were a way to pull computer work into this.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't like Laplace transforms because they are only helpful for finding explicit solutions for linear equations, but those are already solved.  The subject can't be taught in a way consistent with its real significance.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't like power series because they are a ton of work for dubious benefit.  This is a shame, because Frobenius series are very interesting.  There's probably time to do this subject anyway, if you drop Laplace transforms.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This obviously reflects one person's biases about the subject.  I feel that it unifies the whole thing around linear equations and makes the abstraction of the existence and uniqueness theorems seem worthwhile.  It compartmentalizes the strange methods in a way that makes it clear when to use them.  It does decrease the &quot;methods&quot; aspect of the course, and I did do this in an honors class, so maybe that's undesirable in general.  But what does the traditional structure have going for it?&lt;/p&gt;&#xA;" OwnerUserId="513" LastActivityDate="2017-06-07T01:26:00.560" Title="What is the right way to order the topics in a first ODEs course?" Tags="|mathematical-pedagogy|differential-equations|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4485" PostTypeId="2" ParentId="4456" CreationDate="2014-10-03T04:05:59.203" Score="5" Body="&lt;p&gt;&lt;strong&gt;Sound effects!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I substituted once for a friend in her ODE class. A week later I met her in the Department's lounge and she complained with a smile in her face that I had ruined her class because the students wanted the &quot;bloop&quot; professor back. My task had been to illustrate exponential growth, which I did with reference to a bacterium in a Petri dish. After 20 minutes, it goes &quot;bloop&quot; and you have 2 bacteria; after 20 more minutes they go &quot;bloop, bloop&quot; and you have 4...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea is to add a layer of realism to your gesturing, making the concept more palpable. Alberto Verjovsy has a wonderful explanation of how geodesics look in a torus. You see... (he says) you have this torus (he pretends to hold one) and it has a geodesic with rational angle, so it goes around a few times (fingertip flies around) and comes back to the original point, so the geodesic closes. But if the angle turns irrational..... (he wiggles his fingers around the whole surface of the torus) the geodesic never closes and is in fact dense. Sounds nice, BUT while wiggling his fingers, he gently blows air on his fingers, making a rustling noise between his teeth, and it sounds exactly as when you blow the pages of a book to separate them. The effect is really spectacular, and completes the visual presentation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My classrooms have whiteboards, which I hate because the markers are always dry. Nevertheless, the squeak that you get when removing the cap is quite useful :) Say I try to explain how a simple linear transformation stretches things differently in different directions. I take the marker in my hand with thumb and index holding the cap; then, as I slowly open my arms wide to represent the stretching, I twist the cap to get as much squeak as I can. It really sounds as if poor $\mathbb{R}^2$ is stretching and is about to rip.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my classes, things go 'bloop' and 'ping' and 'boom'. It helps students, and I could not stop it even if required; it is already second nature.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2014-10-03T04:05:59.203" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4486" PostTypeId="2" ParentId="4484" CreationDate="2014-10-03T04:47:49.603" Score="5" Body="&lt;p&gt;This is not a complete answer, just a few pointers of what has worked for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Different students react differently to different &quot;methods&quot; of solving linear equations. Should one learn to find the homogeneous solution first, or do the exponential-of-the-unmotivated-integral, or...? Other solvable families have the same problem. Moreover, I want to get to the fun part (read dynamical) part of the course. If we spend time teaching methods of solution, the students get burned by mid-semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my first ODE class I did a daring thing. On the first day of class the students received a 100 equations homework sheet. These 20 are linear; solve them. These 20 are separable; solve them. These are homogeneous, these are exact... The instructions were to use any method in any book that they liked. They could work together and ask me questions. The result was better than I had hoped for. After a week they could all solve equations on the spot (each using the recipes they liked), and we could get started with the really interesting portions of the course. Basically they did not think that &quot;solution methods&quot; was a laborious part of the course, but rather an quick application of Calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also suggest that you take a look at the concepts of fences and funnels introduced by Hubbard and West. There is a nice article in a 1994 issue of the MAA's College Math Journal which was devoted to DE teaching. This is an awesome way to motivate a geometric understanding of DEs, and helps explain existence/uniqueness and other concepts.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="2172" LastEditDate="2014-10-03T14:36:13.443" LastActivityDate="2014-10-03T14:36:13.443" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4487" PostTypeId="2" ParentId="4484" CreationDate="2014-10-03T04:58:51.770" Score="6" Body="&lt;p&gt;I agree that the standard textbook approach is not too exciting from a pure math perspective. I would say the order of the topics is largely based on an attempt to keep the subject from getting too abstract too soon. Also, I would wager the applications which are solved by the methods also benefit from the parsing of topics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the first order problems (even nonlinear) represent interesting physics problems perhaps involving friction. These we could not solve before the clever first order techniques are given. There are fairly simple physical problems which necessitate the use of the Ricatti equation or Bernoulli equation. Really, pretty much every weird first order problem is tied to some obscure example in physics, biology or perhaps chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, the second order problems have much direct physical relevance. Harmonic oscillators are everywhere. The trichotomy of solutions have disparate and interesting application to the separate applications in question. Mostly I mean RLC-circuits and damped spring mass oscillators. Furthermore, the &quot;forcing function&quot; is an honest to goodness force at this juncture. So, discussions of superposition are really direct discussions of how the sum of the forces cause a sum of responses which eventually lead to the response of the net-force.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I whole-heartedly agree the recapitulation of the constant coefficient case at higher order does not seem to be a good economy of presentation. Moreover, applications at higher order are harder to come by... this is probably a lot of the reason this material gets less air time in many courses. No application = no interest. I don't agree, I just observe this for the purpose of understanding. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me just briefly explain how I structure the course:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;week 1: terminology ODE vs. PDE solution, implicit vs. explicit. First order ODEs, I look at the three main methods: separation of variables, integrating factor technique, exact equations.&lt;/li&gt;&#xA;&lt;li&gt;week 2: geometry of DEqns, maybe play with some pplane, applications&lt;/li&gt;&#xA;&lt;li&gt;week 3: uniqueness, existence (by now they appreciate the question)&lt;/li&gt;&#xA;&lt;li&gt;week 4: theory of higher order ODEs, focus on structure of solution sets, Wronskian&lt;/li&gt;&#xA;&lt;li&gt;week 5: develop theory of smooth differential operators. Build solution set from kernels of your basic factors. (generalized e-vectors abound, but, no matrix exponential in sight). Discussion of complexification of problems is given.&lt;/li&gt;&#xA;&lt;li&gt;week 6: solve $n$-th order ODEs. All of them. We own them now.&lt;/li&gt;&#xA;&lt;li&gt;week 7: method of annihilators solves easy nonhomogenenous problems. Then I tackle the $n$-th order variation of parameters. Make a point to emphasize it is a general method and we get to make nice connections with the theory from week 4.&lt;/li&gt;&#xA;&lt;li&gt;week 8: systems of ODEs introduced, theory from week 4 in some sense repeated, but it is a bit different, see eigenvector solutions&lt;/li&gt;&#xA;&lt;li&gt;week 9: complex eigenvectors, again see complexification idea in play. Of course, e-vectors are not enough so we discover matrix exponential and I show them the basics of how to find real Jordan form ( I don't tell them this directly usually) for arbitrary solutions of $x'=Ax$. I make a point at the end of it to emphasize the use of our method to linearize more general problems which are not merely linear.&lt;/li&gt;&#xA;&lt;li&gt;week 10, 11, 12 series, singular points, Frobenius. This is hard, but they actually do pretty well. They're much worse at finding eigenvectors. (go figure)&lt;/li&gt;&#xA;&lt;li&gt;week 13, 14 Laplace Transforms. Here we finally find a good method to treat discontinuity. And, we get a glimpse at the dark art of distributions. Interesting connections with Green's functions can be made here and the general theory of linear systems can give joy to the purest of mathematicians. It's just hard to find in the current books.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I recommend Rabenstein. Or, for a pale echo, see my notes: &lt;a href=&quot;http://www.supermath.info/DEqns2014.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.supermath.info/DEqns2014.pdf&lt;/a&gt; Keep in mind, my main goal is computation. I do not have an audience which appreciates the difference between $\Leftrightarrow$ and $\Rightarrow$. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-10-03T04:58:51.770" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4488" PostTypeId="2" ParentId="4482" CreationDate="2014-10-03T05:58:01.663" Score="4" Body="&lt;p&gt;For the most part, what you're going to look for in a free text will probably be the same as what you'd have looked for in a commercial text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're interested in the ability to customize a text, then the good news is that many free texts make this easy to do. Look at the license and the format that the text is available in. For example, Robbin and Angenent is available in LaTeX format and is under the GFDL license, which means you can modify it however you like and redistribute it to your students.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A &quot;printable version&quot; of the textbook or some optional hard-copy version available.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If the license is permissive, then you can do this without depending for permission on the authors, either through your campus bookstore or through a company like lulu.com.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you use nontraditional teaching methods, then you may want a book that supports those methods. Boelkins is specifically designed for active learning.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-10-03T05:58:01.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4489" PostTypeId="2" ParentId="4483" CreationDate="2014-10-03T09:33:09.920" Score="10" Body="&lt;p&gt;It seems to me that the first thing to do is to define the signature. It is a very important morphism, and you need to have a bunch of example of morphisms to present to your students anyway. Only then is the alternating group really relevant, and you get all properties you want from what you will have done with the signature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I am telling you to pass all the difficulties from the introduction of the alternating group to the introduction of the signature. But it seems that this way is much more natural, hence should be easier to convey convincingly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, following the comment of PVAL, you can first show the important property that every permutation is a product transpositions, and ask whether the needed number of transposition is well defined. Of course it is not, point this out. Then what one could try is to associate to each permutation the minimal number of transposition needed to write it, but then it easily turns out that this map does not define a morphism (but it is linked to a lot of interesting math, from geometry of groups to work of Helfgott and others on the diameter of finite groups far arbitrary set of generators). Then, when you show the students that looking at the &lt;em&gt;parity&lt;/em&gt; of the number of transpositions generating an element yields a morphism, you are in a very meaningful context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are other ways to introduce signature, for example looking at the number of pairs $(i,j)$ with $i&amp;lt;j$ whose order is changed by the permutation. The above one seems best to me right now, but if anyone can suggest other in comments I would be happy to have more choice when I have to teach that myself.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-10-03T09:33:09.920" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4490" PostTypeId="1" AcceptedAnswerId="4492" CreationDate="2014-10-03T20:16:18.330" Score="8" ViewCount="430" Body="&lt;p&gt;I'm making a video right now about the unit circle definitions of the basic trig functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've done sine and cosine, and am now talking about the tangent function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As most of you know, it can be rewritten as in the title, but off the top of my head, I can't remember when this might be genuinely useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certain derivatives and integrals come to mind. Do you have any other examples?&lt;/p&gt;&#xA;" OwnerUserId="796" LastActivityDate="2014-10-04T22:24:41.937" Title="Examples of when $\tan(x) = \frac{\sin(x)}{\cos(x)}$ is useful" Tags="|algebra|trigonometry|functions|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4491" PostTypeId="2" ParentId="4484" CreationDate="2014-10-03T21:17:17.717" Score="8" Body="&lt;p&gt;For a thoughtful view on the subject, see Gian-Carlo Rota, &quot;&lt;a href=&quot;https://web.archive.org/web/20130517105634/http://www.ega-math.narod.ru/Tasks/GCRota.htm&quot; rel=&quot;noreferrer&quot;&gt;Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations&lt;/a&gt;.&quot; The tone of the linked text may be off-putting to some, but the ideas within are a good starting point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I share many of his frustrations and agree with many of his conclusions. I've pasted the &quot;ten lessons&quot; below; please see the link for details on a justification of each. According to the website, Rota delivered the invited address on this topic at the meeting of the Mathematical Association of America at Simmons College on April 24, 1997.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Lesson 1: Most of the material now taught in an introductory differential equations course is hopelessly obsolete&lt;/li&gt;&#xA;&lt;li&gt;Lesson 2: Reduce to a minimum the discussion of first order differential equations at the beginning of the course&lt;/li&gt;&#xA;&lt;li&gt;Lesson 3: Linear differential equations with constant coefficients are the bottom line&lt;/li&gt;&#xA;&lt;li&gt;Lesson 4: Teach changes of variables&lt;/li&gt;&#xA;&lt;li&gt;Lesson 5: Forget about existence and uniqueness of solutions&lt;/li&gt;&#xA;&lt;li&gt;Lesson 6: Linear systems with constant coefficients are the meat and potatoes of the course&lt;/li&gt;&#xA;&lt;li&gt;Lesson 7: Stay away from differentials&lt;/li&gt;&#xA;&lt;li&gt;Lesson 8: Avoid word problems&lt;/li&gt;&#xA;&lt;li&gt;Lesson 9: Motivate the Laplace transform&lt;/li&gt;&#xA;&lt;li&gt;Lesson 10: Teach concepts, not tricks&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In particular, I've reduced my discussion of existence and uniqueness a little more each time I cover it, and I do believe that next time I will not present it at all.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-10-03T21:17:17.717" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4492" PostTypeId="2" ParentId="4490" CreationDate="2014-10-03T22:07:14.310" Score="7" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;You are simplifying the equation for a function and you get down to something like $f(x) = \cos x\tan x $. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You want to know where tan(x) is discontinuous without memorizing it (assuming you already know where cos(x) is zero).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastEditorUserId="759" LastEditDate="2014-10-04T11:50:36.590" LastActivityDate="2014-10-04T11:50:36.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4493" PostTypeId="2" ParentId="4490" CreationDate="2014-10-03T22:16:38.750" Score="5" Body="&lt;p&gt;My previous answer (which was basically &quot;That is the definition of tangent!&quot;) was not well received. I admit that that answer was made out of exasperation with the question. I have written a longer response as to why I feel the way I do, with some mathematical and historical context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You say you are defining these functions based on the unit circle.  If this is the case, then you are probably defining $\cos(\theta)$ to be the $x$-coordinate, and $\sin(\theta)$ to be the $y$-coordinate of the point obtained by taking a piece of string of length $\theta$, fixing one end at $(1,0)$, and winding it it counterclockwise around the circle.  In this case, if you define tangent to be $\frac{y}{x}$ this is also defining tangent to be $\frac{\sin(\theta)}{\cos(\theta)}$.  The two should be so closely identified that these two expressions &quot;say&quot; exactly the same thing to you.  Because of this I am confused by the question:  it appears that the definition of tangent is $\frac{\sin(\theta)}{\cos(\theta)}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The unit circle approach takes some care to make rigorous.  You at least need to be able to define arclength to make it work.  For this reason, most introductions to analysis which I have seen either define sine and cosine in terms of their power series representations, or as solutions to their defining differential equations.  In either content, the tangent function would again be defined by the ratio of the two.  Trying to define tangent &quot;on its own&quot;, without first defining sine and cosine, is difficult because it is not an entire function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a certain sense, I am suspicious of the attention given to the other trig functions.  We no longer pay homage to haversine or exsecant, yet we still talk about tangent.  Why? Fundamentally the reason that tangent, secant, exsecant, etc even have names of their own is that they were needed in some computations quite commonly (say, while sailing), and it was easier to compile tables of values for these &quot;special functions&quot; than to calculate them by hand given only tables for sine and cosine.  I do not know the historical reason that only 6 of these functions survived into the common usage, but from a mathematical perspective I feel like only sine and cosine really &quot;deserve&quot; names in the current era.  It would be as if we had a new operation &quot;inc sum&quot;, denoted $\oplus$ with $a \oplus b = a+b+1$.  A useful operation, to be sure, but not useful enough to warrant its own distinct mental real estate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also think that the habit of questioning where each and every mathematical fact is &quot;useful&quot; is misguided.  Asking about the &quot;usefulness&quot; of an identity which is at most one mental step away from the definition is not a healthy attitude with which to approach mathematics.  When someone is learning mathematics, they should &lt;strong&gt;always&lt;/strong&gt; be exploring 3 or 4 steps away from what is written.  This is just part of building a mental model, gaining &quot;familiarity with the landscape&quot;.  Playing with identities to understand the basic relationships between various functions should be appreciated for its own sake.  &lt;strong&gt;Individual mathematical facts are not useful&lt;/strong&gt;:  a mathematical outlook is useful, a mathematical framework is useful.  This particular fact is just one part of building up the mathematical framework of trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not to say all questions of the form &quot;Examples of when this identity is useful&quot; are not good.  For example, I think that the equality expressed by &quot;completing the square&quot; on an arbitrary quadratic is &quot;just an identity&quot;, but it has a huge story behind it, and how it connects to the larger story of understanding quadratics.  So asking &quot;how is this useful&quot;, can prompt an introduction to a whole framework of thinking.  Similar comments apply to many identities, like the fundamental theorems of calculus, or Cauchy's formula, etc.  $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ on the other hand, is on the same level as $(x+1)(x-1) = x^2-1$ (or even lower on the totem pole):  it is just an immediate consequence of definitions.  One should not have to justify exploring it. It is not a pivotal part of any story, so it should just be one of those things to observe in passing and appreciate for what it is.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit:  perhaps the following analogy is useful.  If you give someone a blanket, they might ask &quot;What is the use of this particular thread?&quot;.  The answer to this question for any thread is that, all by itself, it is useless.  When you weave all of the threads together you get an object which is incredibly useful.  This is the way with mathematics as well.  Mathematics, when properly understood, is a powerful framework for understanding many diverse problems.  $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ is like one of those threads.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2014-10-04T18:31:41.557" LastActivityDate="2014-10-04T18:31:41.557" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="4494" PostTypeId="2" ParentId="4483" CreationDate="2014-10-03T22:21:47.087" Score="4" Body="&lt;p&gt;I'd second @BenoitKloeckner's suggestions about the larger discussion... a sort of dialectic, explaining why, although it's almost obvious that parity-of-number-of-2-cycles is well-defined, this is in fact the crux of the matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As @BenoitKloeckner says, indeed, the (minimal) length (in terms of the adjacent-transposition generators for $S_n$) is the number of pairs $i&amp;lt;j$ whose order is reversed by the permutation. It may be a little unclear how to see that this behaves as hoped under composition. One handy book-keeping device is (as Steve Gubkin noted) to let $S_n$ act by permutations on the coordinates of $\mathbb R^n$ (or any characteristic-zero field, but $\mathbb R$ is more familiar) and take sign-of-determinant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: also, a sort of shadow of the previous is to let $S_n$ permute the variables in a Vandermonde determinant... to define &quot;sgn&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, the work to prove that determinants exist is a superset of the work to prove that the sgn character of $S_n$ exists. Some people would say that the $S_n$ case is the &quot;field with one element&quot; case of the other. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2014-10-03T22:32:49.373" LastActivityDate="2014-10-03T22:32:49.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4495" PostTypeId="2" ParentId="4409" CreationDate="2014-10-04T02:38:57.270" Score="2" Body="&lt;p&gt;The Maya had a place-value number system with a &quot;numeral&quot; to represent zero long before those in the other hemisphere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is evidence that &quot;Pascal's Triangle&quot; was known and used in China long before Pascal lived.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is likely the ancient Egyptians knew an equivalent to the &quot;Pythagorean Theorem.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why do we refer many of these important results only be the name associated with a European who discovered or invented or was the first to publish them?&lt;/p&gt;&#xA;" OwnerUserId="2177" LastActivityDate="2014-10-04T02:38:57.270" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4496" PostTypeId="2" ParentId="4409" CreationDate="2014-10-04T02:51:28.533" Score="2" Body="&lt;p&gt;The story that DeCarte came up with the idea of rectangular coordinates by wondering if he could describe the position and path of a bug on the ceiling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story that Gauss was acting up at school, and his teacher tried to punish him by asking him to find the sum of the integers from 1 to 100 thinking it would keep him busy for a LONG time. It only took him a few minutes, and the only thing he wrote on his slate was the result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we can talk about which of these types of stories are fact, fiction, or myth and who gets to write the history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oh, one more.  Sonia Kovalevsky learned calculus at a young age because her father wall-papered he bedroom with the pages of a calculus text.&lt;/p&gt;&#xA;" OwnerUserId="2177" LastActivityDate="2014-10-04T02:51:28.533" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4497" PostTypeId="1" CreationDate="2014-10-04T17:58:30.273" Score="7" ViewCount="1107" Body="&lt;p&gt;I am studying Electrical Engineering and I want to specialize in signal processing.&#xA;However, I have to study complex analysis first (I am an undergraduate, so I lack some terminology). In your opinion: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) What's the difference between studying complex analysis in pure and applied contexts (topics I mean, and how much should I dig down toward purity)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Do I need real analysis, and what topics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Please suggest books for both pure and applied classes of either real or complex analysis.&lt;/p&gt;&#xA;" OwnerUserId="2181" LastEditorUserId="376" LastEditDate="2020-12-30T10:39:47.497" LastActivityDate="2020-12-30T10:39:47.497" Title="Complex analysis (Applied versus pure)" Tags="|undergraduate-education|mathematical-pedagogy|mathematical-analysis|applied-mathematics|engineering-mathematics|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4498" PostTypeId="2" ParentId="4490" CreationDate="2014-10-04T19:34:46.523" Score="5" Body="&lt;p&gt;One reason the tangent function is useful is that it yields the slope of a line,&#xA;$\tan \theta$ where $\theta$ is the angle of a line tangent to a&#xA;curve at a point (and note the connection of the &lt;em&gt;tangent&lt;/em&gt; function to a line &lt;em&gt;tangent&lt;/em&gt; to&#xA;a curve):&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/VRQMg.jpg&quot; alt=&quot;TangentSlope&quot;&gt;&#xA;&lt;hr /&gt;&#xA;If you identify $y$ with $\sin \theta$ and $x$ with $\cos \theta$, &#xA;the connection to slope could be explained.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More pragmatically, you can use the tangent to estimate the height $y$ of a building:&#xA;If you are standing about $x$ units from the building, and you need to look up $\theta$&#xA;to the top, then $y = x \tan \theta$. Again, this could be explained in terms of $\sin \theta$ and $\cos \theta$.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-10-04T22:24:41.937" LastActivityDate="2014-10-04T22:24:41.937" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4499" PostTypeId="2" ParentId="4497" CreationDate="2014-10-04T19:45:14.727" Score="2" Body="&lt;p&gt;1) &quot;Pure&quot; complex analysis is filled with theory and proofs, and is more like real analysis. &quot;Engineering&quot; complex analysis is more about problems, and more like calculus. I recommend the latter for you, an engineering student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) You need real analysis for the former.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) A good book for &quot;pure&quot; is &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0070542341&quot; rel=&quot;nofollow&quot;&gt;Rudin&lt;/a&gt; A good book for applied is Kreyszig,&quot; &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0470458364&quot; rel=&quot;nofollow&quot;&gt;Advanced Engineering Mathematics, Chapters 11-17.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-10-04T19:45:14.727" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4501" PostTypeId="2" ParentId="4497" CreationDate="2014-10-06T01:47:45.560" Score="6" Body="&lt;p&gt;This may not be a direct hit, but since you mentioned &quot;emphasis on visualization,&quot; may I suggest you investigate Tristan Needham's &lt;em&gt;Visual Complex Analysis&lt;/em&gt;. E.g., see &lt;a href=&quot;https://math.stackexchange.com/a/85244/237&quot;&gt;this MSE answer&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;img src=&quot;https://i.stack.imgur.com/HpbKO.png&quot; alt=&quot;Figure 13&quot;&gt;&lt;hr /&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-10-06T01:47:45.560" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4502" PostTypeId="2" ParentId="59" CreationDate="2014-10-06T11:14:03.777" Score="4" Body="&lt;p&gt;The biggest advantage of open-book exams is that they are closer to real world problems. People can look things up in the real world. So why should they not do that during exams? However, the big disadvantage is that students, because they are not used to it, have the common misconception that an open-book (or even open-internet) exam means that &quot;I can just look everything up on the fly&quot;, which is bad, but is just that: A misconception.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make sure to explicitly communicate how you expect them to prepare for this exam. Consider even to ask and guide them on their exam preparations. Give them a recipe for exam success (while, of course, avoiding &quot;teaching to the test&quot;). Also, make sure not to make the open-book exam terribly hard. If you transform a closed-book exam into an open-book exam, but the degree of difficulty stays the same, you have a slight chance of making your students' lives just a little bit easier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;exam taker&quot; students might complain, so maybe compensate with &quot;bonus&quot; problems or other goodies (possibly, outside of exams) that compensate people who invest more time, and are actually interested in learning this stuff. What about rewarding students helping other students (outside of exams)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said that, let us re-consider what an exam is and does: Exams exams are short-time, high-stakes, high-pressure turning points in a young person's career. They generate large amounts of stress and &lt;strong&gt;disgust&lt;/strong&gt; in the average young (and old) person's mind, and there is tons of research showing that they barely help the student in understanding things, solving any problems or achieving any other form of goal. Exams do two things:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Exams generate one (and only one) kind of &lt;em&gt;extrinsic motivation&lt;/em&gt; in students. We call that kind of motivation &quot;pressure&quot; or &quot;stress&quot;. In the world of &quot;carrots and sticks&quot;, its the stick.&lt;/li&gt;&#xA;&lt;li&gt;They generate a grade. Institutions (and thus teachers) are required to generate a student's evaluation, usually in the form of a low-dimensional vector of &quot;grades&quot;. As a result, generating &quot;good grades&quot; ends up being the only goal for institutions, and often distracts them from what should be their real goal: To help the students.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Do you think, that open- or closed-book exam can really help students? Regarding your exams, is it more important to &quot;get it right the first time&quot; (because else their grades, and thus their future career and chances to get scholarships will deteriorate), or do you focus on students learning from the exams and the feedback you give them on the exams?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you give students a chance to generate good grades, if they have the passion and put in the effort, even if they are not good &quot;exam takers&quot;? Do you think that the quality of being a &quot;good exam taker&quot; correlates with a person's ability to succeed in or contribute to society? What about those who are not good at exams? Do we know anything about those students?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here are some suggestions: Have you tried non-one-shot (i.e. repeatable) exams? If that's too much effort (or you don't have the infrastructure to make it effortless), did you consider take-home exams? What about project work? Please don't rule it out just because students are bad project managers. No one has taught them yet. Of course, good project work also needs the right kind of structure. The same issues apply to all approaches that diverge from the traditional &quot;lecture, homework, exam&quot; cycle of death - It's hard. Hopefully, we will be able to come up with better approaches that we can also share more easily, soon enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: Maybe you want to take a look at &lt;a href=&quot;http://chronicle.com/blognetwork/castingoutnines/&quot; rel=&quot;nofollow&quot;&gt;Robert Talbert's awesome blog on new ideas in teaching Mathematics, flipping the classroom and a lot more&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="2190" LastEditorUserId="2190" LastEditDate="2014-10-06T13:38:46.863" LastActivityDate="2014-10-06T13:38:46.863" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4503" PostTypeId="1" CreationDate="2014-10-06T12:38:36.987" Score="13" ViewCount="716" Body="&lt;p&gt;&lt;strong&gt;Main question: How do I, in a medium- to large-sized undergraduate class setting, appropriately and effectively discourage students from relying too heavily on calculators?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There have been several questions here about calculator usage (both in class and during exams) and its effects on student learning and understanding:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/3838/handing-out-2-calculators-for-tests&quot;&gt;Handing out $2 calculators for tests&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/1976/how-does-a-reliance-on-calculators-affect-student-performance&quot;&gt;How does a reliance on calculators affect student performance?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/894/a-study-comparing-effects-of-calculator-usage-on-later-math-skills&quot;&gt;A study comparing effects of calculator usage on later math skills?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I don't want to rehash a discussion of whether or not calculator usage is effective for student learning, whether or not they should be allowed on exams, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the following posted question is very close to what I want to ask here: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2551/student-why-not-use-a-calculator&quot;&gt;Student: Why not use a calculator?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That question has attracted a variety of thoughtful and, I believe, effective answers. However, because of the setup of the question and its &quot;primary-education&quot; tag, the answers are aimed at promoting effective mathematical thinking in young children, and most answers presume the educator is working one-on-one (or can at least speak to the student directly and hear their responses). Here is a summary of the points made by the top, accepted answer there:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(i) It is a matter of independence. (ii) You will be able to ballpark-verify outcomes. (iii) You can prevent being cheated. (iv) You are not asked to disavow your calculator. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here are my questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;How can I adapt some of these techniques to, say, a 20- to 50-student basic algebra or pre-calculus course in an undergraduate college?&lt;/strong&gt; I cringe every time I put some practice problems on the board and ask students to work on them, and immediately hear the sliding and locking of calculator covers and the rattling of keys, especially when I &lt;em&gt;explicitly designed&lt;/em&gt; the questions to avoid needing any difficult arithmetic. Is this just a knee-jerk reaction on my part that I need to curb?&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Is it worthwhile to have a discussion with the class at large about this, or would it only be effective one-on-one anyway?&lt;/strong&gt; That is to say, should I devote 5 or more minutes of class time to talking about calculator usage, or would it just be a waste of time? Should I only address this topic individually when I see students acting on their &quot;addiction&quot;?&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Is there a succinct and effective quote/saying/analogy that you have used to point out calculator overreliance?&lt;/strong&gt; I want to say something like, &quot;Why take the elevator to go up 3 stairs?&quot;, but this isn't quite the right sense, and I don't want to be insensitive. Do you have any better suggestions?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Ultimately, I'm wondering what to do with adult-aged students who likely should have some basic arithmetic skills, and surely do, but still seem to turn to their devices whenever any numbers appear. While they're in my courses and I have the opportunity to say something appropriate and effective about this issue ... what should I do/say?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I would understand if this question is too similar to the one I linked above and is subsequently closed.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good answer here might include&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;an effective saying or anecdote or presentation you tell to students about this, or&lt;/li&gt;&#xA;&lt;li&gt;a homework/exam policy that you have found to effectively dissuade students from unnecessarily reaching for a calculator, or&lt;/li&gt;&#xA;&lt;li&gt;a persuasive argument as to why I should just not care about this issue, that the students are &quot;on their own&quot; now,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;or some combination thereof.&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-10-07T23:08:07.597" Title="Appropriate ways/sayings to discourage undergraduate students' overreliance on calculators" Tags="|undergraduate-education|technology-in-education|algebra|calculators|dyscalculia|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4504" PostTypeId="2" ParentId="4503" CreationDate="2014-10-06T12:52:49.553" Score="8" Body="&lt;p&gt;I think one reason students reach for their calculators so readily is out of fear.  They are afraid that they will get the answer wrong, and would rather rely on an oracle which &quot;always gives the right answer&quot;.  Fundamentally, I think this fear results from the threat of punishment for getting the answer wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is so ingrained that it is hard to change.  One thing which might be effective (I have no studies, or even anecdotal evidence, which supports this) is to encourage students to avoid calculator use on formative assessment activities which carry no weight for their grade.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-10-06T12:52:49.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4505" PostTypeId="1" CreationDate="2014-10-06T15:11:24.463" Score="18" ViewCount="922" Body="&lt;p&gt;First off sorry if this question isn't suited to matheducators.stackexchange I just thought here would be a good place to ask seeing as you would probably have experience with students. Anyway here is my question and other related information:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question simply is &quot;What is the best method to make sure you retain what you have learnt?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Okay so I've tried learning mathematics up to where I should be at in the past. Every time though I end up stopping for some reason, whether that be health reasons, family reasons etc. sometimes not even for a long time, such as for a couple days/week. Once this happens the information seems to just drop out of my brain no matter and I have to start again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am only trying to currently get up to year 12 level for here in South Australia so I will be caught back up and ready to continue my university study and move onto Discrete Mathematics among other things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway I've tried the practice, practice, practice approach a while back where I spent probably 6+ months like 5 hours a day on Khan Academy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To add to the problem I am currently an external student so teaching other people for example maybe difficult as I hear teaching is the best way to learn. Finding other people to teach that will listen would be difficult even as an internal student. I probably wouldn't even feel confident enough to teach due to a lack of understanding myself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway I'm not giving up yet, still looking for a method that might work. Sure you might argue why waste time asking this question, go do problems and practice more but as I said that doesn't seem to work exactly. I don't have this problem with programming for example so I don't know what is wrong. Plus once I'm back studying at university I doubt I will have enough time to dedicate to just doing problems everyday to make sure the information stays fresh in my mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So yeah any help is appreciated (hopefully I'm posting in the correct stackexchange site...), also hopefully not too specific to my case or something too, which I just realised might be a problem...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="2192" LastActivityDate="2015-03-17T12:24:03.077" Title="How can I retain the mathematics that I've supposedly learnt?" Tags="|secondary-education|mathematical-pedagogy|education-research|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4506" PostTypeId="2" ParentId="4505" CreationDate="2014-10-06T17:06:18.273" Score="6" Body="&lt;p&gt;Keep doing math and thinking about problems, even if you can only devote a little bit of time each day!  I'm no expert in neuroscience of the brain, but my understanding is that a lot of learning constitutes forming connections (synapses) between different neurons in our brain; the more we think about and learn about a particular topic, the stronger (more interconnected) the network of related neurons becomes.  Conversely, as we stop thinking and learning about a given topic, the more likely it is that the connections between the associated neurons will weaken and our brain may eventually use that neural space for other knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if some of that attempt at neuroscience is incorrect or overly simplified, it is an observable fact that people retain knowledge, especially mathematical knowledge, much more if they are constantly practicing it.  So one thing to do is to try to find some time each day, even if it is only a small amount of time, to think about math and try problems.  Even if it is only 30 minutes out of your busy schedule, it will make a difference.  The other solution, as a complement or replacement for finding available time to work on math at a desk, is to think about math problems in your head when you are walking somewhere, in the shower, or eating a meal.  Even if you cannot fully engage your thoughts because of lack of immediate access to paper and pencil, you'll be consolidating what you do know and keep your brain's neural network developing its mathematical connections.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2014-10-06T17:06:18.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4507" PostTypeId="2" ParentId="59" CreationDate="2014-10-06T18:42:47.920" Score="4" Body="&lt;p&gt;One minor point: I would not permit books and forbid other print resources (student's prepared notes, reference cards, other text books…) That strikes me as too hard a rule for proctors to enforce, because it is hard to see what a student is flipping through.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2014-10-06T18:42:47.920" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4508" PostTypeId="1" AcceptedAnswerId="4509" CreationDate="2014-10-06T19:06:45.583" Score="15" ViewCount="1640" Body="&lt;p&gt;I've recently started learning linear algebra on my own. I always try to prove the theorems I encounter by myself, without looking at the book (only to check if my proof is correct), because I found out that this way I can remember both the statements and the proofs much more easily. Moreover, it's much more engaging to me than just passively reading a text, and since I don't have time limits (I'm a high school student but I don't take any math classes) it's the technique I always adopt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I've spent three whole weeks trying to prove that all the bases of a finite-dimensional vector space have the same number of vectors, without much success. I thought that professional mathematicians have to deal with this all the time, since they spend months or years just to prove a single result, and it would be a good training if I'll become one. So I just moved on to something else, hoping to get back to it when I had become more experienced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now it's been over a week that I'm struggling to prove that the rank of the rows of a matrix equal to the rank of the columns. I'm getting extremely frustrated because I've run out of ideas and keep using the same ones over and over, or I just get a bunch of absolutely useless ones. But I keep thinking that maybe if I persist I'll eventually find the solution, although it's difficult to concentrate being bored and frustrated. Still, I'd feel worthless and incompetent if I just looked at the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What should I do? Is it useless to keep trying for such a long time? &lt;/p&gt;&#xA;" OwnerUserId="2194" LastEditorUserId="376" LastEditDate="2019-04-10T07:48:26.813" LastActivityDate="2019-04-10T08:01:36.803" Title="Proving theorems on one's own: how long should one persist?" Tags="|proofs|linear-algebra|self-learning|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="4509" PostTypeId="2" ParentId="4508" CreationDate="2014-10-06T19:36:05.800" Score="11" Body="&lt;p&gt;I think it's very commendable to try proving things yourself first; even a failed attempt has value. However, it's also important to learn from others' proofs, so don't be afraid to sneak a peek at the full proof, if only the first line or two to give you a hint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may also find it worthwhile to get hold of a book on proof techniques, such as Velleman's &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0521675995&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to Prove It&lt;/a&gt;. You might be missing some important strategies, such as proof by contradiction. Also, you might consider looking at Chapter 3 of Alcock &amp;amp; Simpson's &lt;a href=&quot;https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/8846/1/Alcock_Simpson_book.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ideas from Mathematics Education&lt;/a&gt; on semantic and syntactic reasoning strategies. It's available for free from Alcock's &lt;a href=&quot;https://laraalcock.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt;. The book is written for mathematicians, but I think there are some eye-openers in there for students too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Caveat&lt;/strong&gt;: books/courses on proof techniques can equip you with an array of useful tools, approaches and examples, but they are merely a starting point, not a universal remedy. Take the time to become familiar with the particulars, peculiarities and main ideas of the area you are studying. Sometimes you'll find that certain methods and conceptual approaches are used repeatedly and profitably.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: regarding how much time to spend and the idea of &lt;em&gt;productive struggle&lt;/em&gt;, please see &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2397/how-much-time-to-spend-on-a-single-question&quot;&gt;How much time to spend on a single question?&lt;/a&gt;, especially Benjamin Dickman's answer.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2019-04-10T08:01:36.803" LastActivityDate="2019-04-10T08:01:36.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4510" PostTypeId="1" AcceptedAnswerId="4512" CreationDate="2014-10-06T22:44:01.277" Score="5" ViewCount="438" Body="&lt;p&gt;In Spain, there's a famous website called &lt;a href=&quot;http://herramientas.educa.madrid.org/wiris/&quot; rel=&quot;nofollow&quot;&gt;Wiris&lt;/a&gt; that is a multi-platform website with the ability to create and calculate some of the most basic mathematical operations, but also you can save your project and continue after a while!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering if there is any other website or program similar to this, to create your own project with formulas and variables and, after only inputting the necessary data, the program calculates everything you want!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was trying by creating autofilling sheets in Microsoft Word or LaTeX but they don't convince me (also in LaTeX I couldn't)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, any suggestions? &lt;/p&gt;&#xA;" OwnerUserId="2195" LastEditorUserId="5321" LastEditDate="2015-11-23T18:50:49.533" LastActivityDate="2015-11-23T18:50:49.533" Title="Program or Website similar to Wiris?" Tags="|specific-software|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4512" PostTypeId="2" ParentId="4510" CreationDate="2014-10-07T11:33:53.563" Score="7" Body="&lt;p&gt;Having taken a quick look, what you are looking for seems to be a &lt;strong&gt;computer algebra system&lt;/strong&gt; with a workbook, into which you can save your progress. The most well-known is of course &lt;a href=&quot;http://www.wolfram.com/mathematica/&quot;&gt;Mathematica&lt;/a&gt;, but depending on what you want to do some people prefer &lt;a href=&quot;http://www.mathworks.ch/products/matlab/&quot;&gt;MatLab&lt;/a&gt; or &lt;a href=&quot;http://www.maplesoft.com/products/maple/&quot;&gt;Maple&lt;/a&gt;. For a pretty big list you can consult the &lt;a href=&quot;http://en.wikipedia.org/wiki/List_of_computer_algebra_systems&quot;&gt;Wikipedia listing of CAS&lt;/a&gt;. Here I note a few free (as in beer, and sometimes as in speech) programs or websites. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.gnu.org/software/octave/&quot;&gt;GNU Octave&lt;/a&gt; is designed as a free software replacement for MatLab, and covers many similar functionality, and uses similar syntax. &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.sagemath.org/&quot;&gt;Sage&lt;/a&gt; is a bundle of free mathematics softwares with a more-or-less unified interface. They have currently an online service that is in beta called &lt;a href=&quot;https://cloud.sagemath.com/&quot;&gt;SageMathCloud&lt;/a&gt; which seems to be very similar to Wiris in terms of functionality. &lt;/li&gt;&#xA;&lt;li&gt;One can also make use of &lt;a href=&quot;http://sympy.org/en/index.html&quot;&gt;SymPy&lt;/a&gt; together with an &lt;a href=&quot;http://ipython.org/notebook.html&quot;&gt;IPython Notebook&lt;/a&gt; to reproduce somewhat the Wiris experience. &lt;/li&gt;&#xA;&lt;li&gt;(A little bit of a shameless plug: I &lt;a href=&quot;http://williewong.wordpress.com/2012/07/27/ipython-notebook-take-2/&quot;&gt;wrote some code&lt;/a&gt; sometime ago for helping with algebraic computations using IPython Notebook. There is &lt;em&gt;no&lt;/em&gt; built-in CAS capabilities in my code; that is to say all math must be done in your brain. It is just a convenient replacement for paper and pen when I want to be more organized about writing things down.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://maxima.sourceforge.net/&quot;&gt;Maxima&lt;/a&gt; can be used with various frontends (I personally prefer &lt;a href=&quot;http://andrejv.github.io/wxmaxima/&quot;&gt;wxMaxima&lt;/a&gt;) to get the similar experience of having a workbook which you can save. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="125" LastActivityDate="2014-10-07T11:33:53.563" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4513" PostTypeId="5" CreationDate="2014-10-07T13:24:40.927" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-10-07T13:24:40.927" LastActivityDate="2014-10-07T13:24:40.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4514" PostTypeId="4" CreationDate="2014-10-07T13:24:40.927" Score="0" Body="Use of language in communicating mathematics. Includes all kinds of languages: spoken, written, symbolic etc. There is a separate tag for terminology, which is a narrow subfield of language use." OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2014-10-07T14:54:43.637" LastActivityDate="2014-10-07T14:54:43.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4515" PostTypeId="2" ParentId="4503" CreationDate="2014-10-07T19:34:26.027" Score="10" Body="&lt;p&gt;Modeling mental math strategies as part of your teaching is important. I make a show of the fact that I'm not pulling out a calculator. I also show them how I think through certain calculations: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;36*50? That's half of 36 times twice the 50, or 18 times 100.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[This semester I have a student who I noticed was counting on her fingers. She came to my office and told me how dumb she felt. I talked to her about accepting where she's at right now, and how to move forward. (I also told her I thought she must be pretty good at some parts of math, seeing as how she's done pretty well even with this deficit.) I showed her some online games I thought might help her. And I encouraged her to notice when she's pulling out her calculator for something simple, so she can think about how to improve her basic arithmetic skills.]&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-10-07T19:34:26.027" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4516" PostTypeId="2" ParentId="4505" CreationDate="2014-10-07T22:55:07.017" Score="3" Body="&lt;p&gt;If you try to memorize things, you will lose them. If you try to understand and make connections, you can retain what you learn much longer. Perhaps finding puzzles related to what you're trying to learn. If you give some examples of topics you've learned and lost, I can give examples of different ways to approach those topics.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-10-07T22:55:07.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4517" PostTypeId="2" ParentId="4503" CreationDate="2014-10-07T23:08:07.597" Score="5" Body="&lt;p&gt;Show them instances where it would be easier to calculate mentally than it would be to use a calculator.  Some examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$1+1$&lt;/li&gt;&#xA;&lt;li&gt;$123456789+0$&lt;/li&gt;&#xA;&lt;li&gt;$11111+11111+11111+11111+11111$&lt;/li&gt;&#xA;&lt;li&gt;$123456789-123456789$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Show them instances where a calculator would give an incorrect answer.  For example, it is easy to see that $\cot 90^\circ$ is zero if one understands the concepts.  But if one relies on a calculator by, say, trying to get the reciprocal of $\tan 90^\circ$, then a calculator error results leading the student to think that the answer does not exist (when in fact it does).&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-10-07T23:08:07.597" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4518" PostTypeId="1" AcceptedAnswerId="4521" CreationDate="2014-10-08T14:09:25.513" Score="1" ViewCount="3358" Body="&lt;p&gt;What topics should I read before studying mathematical analysis? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to have a solid foundation in terminology, notation and concepts in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please suggest titles for books.&lt;/p&gt;&#xA;" OwnerUserId="2181" LastEditorUserId="376" LastEditDate="2014-11-15T17:41:57.473" LastActivityDate="2014-11-24T20:34:04.230" Title="Prerequisites of mathematical analysis" Tags="|reference-request|mathematical-analysis|" AnswerCount="4" CommentCount="2" ClosedDate="2014-12-01T07:17:28.400" ContentLicense="CC BY-SA 3.0" />
  <row Id="4520" PostTypeId="2" ParentId="4505" CreationDate="2014-10-08T14:17:54.463" Score="5" Body="&lt;p&gt;There are many ways to increase the chances that you will retain what you learn. I list here a few that I know:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Review regularly&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The more often you review what you learn, the more likely you will retain it even longer. By review, I mean stopping to look back over what you have learned to remind yourself of the major ideas and how they are connected. You may want to skim over what you've done, or think about it, or say it to yourself, or write a short summary, or draw a picture. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;People recommended to me when I was a student that you should review what you learned at the end of every study session, at the end of each day, at the end of each week, and at the end of each month. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, when you have been studying for an hour and you want to stop, before you leave, try to summarise what you learned. At the end of each day, plan a short study session where you look back over the whole day and summarise what you learned. At the end of each week, plan a study session where you look back over the whole week and summarise what you learned. Similarly at the end of each month.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, when I was a student I used to use my daily train journey home to quickly jot down in a diary what I learned today. On the weekend, I would have a session where I looked back through the week's diary and thought about the week as a whole. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your particular situation this has an advantage that if your study is interrupted at any moment, then you have been reviewing as you go anyway, so you are more likely to retain across a gap in time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Explicitly make connections&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are more likely to be able to recall things you learned if they are connected to other things in your mind already. Imagine your mind as a map with towns on it an roads between them. If you want to get to town A, the more roads there are between any town and any other town, the more chance you have of hitting the right road to town A eventually. But if town A is only connected to one other town, you have much less chance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the sensation of understanding happens when new ideas are connected to existing ideas in your mind (literally the brain cells connecting to each other).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you should explicitly try to make connections. Constantly ask yourself &quot;How is this similar to other things I have learned?&quot;, &quot;How is this different to other things that look similar?&quot;, &quot;Where does this idea come up?&quot;. Another strategy is to draw the connections on paper, in a diagram like a mind-map. Don't just try to connect the ideas within one topic, but try to connect to other topics too. A memorable example of this for myself was when I explicitly made connections between algorithms in polynomial arithmetic to algorithms in whole-number arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above paragraph would seem to suggest that the only connections are to other ideas in maths, but the more connections there are to &lt;em&gt;anything&lt;/em&gt; the better chance you will have of understanding and retaining. So ask yourself if the things you are learning remind you of other tasks you know how to perform, or life events you've been part of, or movies you once watched, or people you know, or foods you ate etc. Personally for me the movies I have watched often give me great connections to draw upon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Use multiple senses&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The connections mentioned earlier are not just to ideas, but also to sensory input. Each of your senses is stored in different places in your brain and is associated in your memory with different experiences and ideas. So a way of increasing learning retention is to use more than one sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you are summarising the day, write a list, and also say your list aloud. Construct models out of play-dough or paper. Associate a sound-effect to a particular numerical operation. Write different types of ideas in different colours.  Draw pictures with your finger on the back of your hand so you can &lt;em&gt;feel&lt;/em&gt; them. When you explain an idea to yourself, use hand-gestures as well as words.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hand gestures are particularly important -- there is research to suggest that the movement of your body and in particular your hands plays an important role in all learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Practice using the ideas and choosing which one to apply&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As others already mentioned, using the ideas to do something will help you retain them. So seek out problems to do. The best problems are the ones that involve more than one idea, or involve a bit of playing around to get somewhere. These ones will force you to make more connections and really use what you have learned to do something new. Puzzles are another good thing to do because they often use ideas in unexpected ways. Don't forget to review the things you learned while doing problems at the end of a problem-solving session!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While it is obviously good to practice each separate skill or type of problem separately, it is also important to have practice at choosing what skill to use or what type of problem the problem is. Many excercises are in blocks where all the problems are about the same thing or require the same skill (such as those on Khan academy). You also need to seek out exercises which mix together different types of problems and skills. Choosing one from each block in a textbook is a good start.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-10-08T20:24:56.133" LastActivityDate="2014-10-08T20:24:56.133" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4521" PostTypeId="2" ParentId="4518" CreationDate="2014-10-08T14:21:56.700" Score="1" Body="&lt;p&gt;To begin your analysis journey, perhaps a book like &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1441928669&quot; rel=&quot;nofollow&quot;&gt;Abbott&lt;/a&gt; to introduce you to the basic topics and some basic proofs for typical analysis classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a rigorous analysis course (think &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/007054235X&quot; rel=&quot;nofollow&quot;&gt;Baby Rudin&lt;/a&gt;), I believe that a good grasp of the concept of proof and set theory is required. The little bit of familiarity with topology (particularly that of the topology of $\mathbb{R}$) that you need will be covered concurrently in many analysis books. I've not used the book, but I hear &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521675995&quot; rel=&quot;nofollow&quot;&gt;How to Prove it&lt;/a&gt; is a good introduction to the transition from solving problems to proving theorems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To move up to real analysis (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0070542341&quot; rel=&quot;nofollow&quot;&gt;Big Rudin&lt;/a&gt;, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/013143747X&quot; rel=&quot;nofollow&quot;&gt;Royden&lt;/a&gt;, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471317160&quot; rel=&quot;nofollow&quot;&gt;etc&lt;/a&gt;), the basics covered in Baby Rudin should be sufficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For topology, I find that &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0131816292&quot; rel=&quot;nofollow&quot;&gt;Munkres&lt;/a&gt; is a great read and has many wonderful problems. Again, confidence with proof tactics is required.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are essentially the books I used in classes progressing through analysis.&lt;/p&gt;&#xA;" OwnerUserId="201" LastEditorUserId="201" LastEditDate="2014-10-09T00:12:12.640" LastActivityDate="2014-10-09T00:12:12.640" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="4522" PostTypeId="1" AcceptedAnswerId="5553" CreationDate="2014-10-08T15:35:38.403" Score="9" ViewCount="391" Body="&lt;p&gt;I am currently tutoring a few students in an entry level physics course and had some trouble recently when it comes to helping them with problem solving. The students I am helping don't have many issues when the problems are very straightforward. They understand what formulas to use and what each variable in the formulas represents but they seem to struggle when a problem isn't completely straight forward. I'll give a small example as to exactly what I'm refering to. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students have the following formula for displacement: $$\ d = v_i t + \frac{1}{2}a t^2.$$ The students had no trouble with any of cookie cutter the hw problems that would ask them to find the displacement after some time $t$ of an object that started from rest ($v_i =0$) and had some acceleration $a$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Inevitably on their test the students received a problem similar to: Two cars $A$ and $B$ start from rest exactly $88\,\mathrm{m}$ away from each other and accelerate towards each other with accelerations of $A_a$ and $B_a$ respectively. How far has car $A$ travelled when it collides with car $B$? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students all found this problem nearly impossible to solve and really didn't even know where to begin. Most tried something like $$ 88\,\mathrm{m} = 0t_1 + \frac{1}{2}A_a t_1^2 ,$$ and $$88\,\mathrm{m} = 0t_2 + \frac{1}{2} B_a t_2^2$$ but then didn't know how to go any further. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The trick here is that when they collide the total distance covered is $88 \text{ m}$ and the cars have both been traveling for the exact same amount of time so the equation that needs to be solved is $$\left[0t + \frac{1}{2}A_a t^2\right] + \left[0t + \frac{1}{2} B_a t^2\right] = 88\,\mathrm{m}.$$ For me this was easy to see but when I tried to explain this to the students they really did not understand the reasoning behind it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is just one example but the same is true in many other cases. How do I teach them the critical thinking skills necessary to solve questions and not just apply formulas to specific situations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also feel free to add the appropriate tags. I found none that seemd to fit.&lt;/p&gt;&#xA;" OwnerUserId="2200" LastEditorUserId="1961" LastEditDate="2014-10-14T11:57:26.150" LastActivityDate="2014-10-14T11:57:26.150" Title="Teaching Critical Thinking Skills" Tags="|creativity|critical-thinking|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="4523" PostTypeId="1" AcceptedAnswerId="4524" CreationDate="2014-10-08T16:59:10.043" Score="9" ViewCount="866" Body="&lt;p&gt;I'm teaching first semester algebra, the majority of the students share the rest of the classes. Sometimes I want to take examples from other courses so they can see that everything intertwines, but it has happend too many time that whenever I try to do that they say that they haven't study that with the corresponding teacher for whatever reason. To give a for instance, yesterday I wanted to show them that sequences could be seen as functions, but they told me that they haven't study anything about them! which is a little frightening because at this point in the semester they are suppose to know something. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When that happened at the beggining I opted to just go over what I wanted to show them in particular very quickly, but as we progress I can tell that they are falling behing way too much and fast that is also affecting my own course. At first I thought that they were exaggerating, they even told me that they geometry teacher hates them, but now the &lt;em&gt;good students&lt;/em&gt; are failing as well. What should I do? or more over, what can be done?&lt;/p&gt;&#xA;" OwnerUserId="99" LastEditorUserId="99" LastEditDate="2014-10-09T18:40:41.397" LastActivityDate="2014-10-09T18:40:41.397" Title="What to do when all the other teachers fail?" Tags="|undergraduate-education|large-lecture|struggling-students|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4524" PostTypeId="2" ParentId="4523" CreationDate="2014-10-08T17:18:24.983" Score="12" Body="&lt;p&gt;If you want to rely on things convered in other course, one thing I think you should try to do is to get information from the &lt;em&gt;teachers&lt;/em&gt; of these other courses what they have discussed; if possible, in detail, so that you know exact terminology and notation they use. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience (both as teacher and student!), students can be (for various reasons) &lt;em&gt;quite&lt;/em&gt; unreliable in conveying  what they have seen in another course and (should) know, especially when asked in the middle of a large lecture, possibly using slightly different terminology.&lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-10-08T17:18:24.983" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4525" PostTypeId="1" CreationDate="2014-10-08T19:52:57.047" Score="2" ViewCount="105" Body="&lt;p&gt;What is the mathematics needed to delve in signal processing? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know if it correct to dig toward purism downwards or stay at the applied level. Specifically, in complex analysis I find the $\varepsilon-\delta$ definition and limits. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know whether am going to use them for next concepts in the book or not. I found that the word neighborhood has specific meanings in topology and analysis. do I need some pure math?&lt;/p&gt;&#xA;" OwnerUserId="2181" LastEditorUserId="47" LastEditDate="2014-10-08T20:43:09.993" LastActivityDate="2014-10-09T14:00:05.910" Title="Mathematics necessary for signal processing" Tags="|reference-request|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4526" PostTypeId="1" CreationDate="2014-10-09T00:04:01.817" Score="4" ViewCount="292" Body="&lt;p&gt;In an off-topic discussion,&#xA;I tried to explain to a student why a &quot;ray-gun&quot; that (somehow!) shoots a ray that followed&#xA;a spiral path would be much more difficult to aim at a particular target&#xA;(point in $\mathbb{R}^3$)&#xA;than a gun that shoots a ray along a straight ray.&#xA;(&lt;em&gt;Added&lt;/em&gt;: The spiral has a uniform radius and uniform spacing along its axis.)&#xA;My attempted explanation failed to get at the essence, and consequently failed to convince.&#xA;Perhaps the ambiguity is that the notion of what it might mean to &quot;aim&quot; such a gun&#xA;is not clear.&#xA;But even with precise knowledge of the spiral's geometry, I feel that under&#xA;any interpretation of &quot;aim,&quot; it is still difficult.&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/5Mdky.jpg&quot; alt=&quot;SpiralGun&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;. Assuming I am not wrong that it &lt;em&gt;is&lt;/em&gt; more difficult,&#xA;  can you think of a convincing way to explain this?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-10-09T00:28:28.763" LastActivityDate="2019-07-21T10:45:39.917" Title="Explanation challenge: Why is a spiral ray-gun difficult to aim?" Tags="|mathematical-pedagogy|geometry|" AnswerCount="3" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="4527" PostTypeId="1" AcceptedAnswerId="4547" CreationDate="2014-10-09T01:49:22.163" Score="11" ViewCount="2995" Body="&lt;p&gt;In hopes of helping my students practice and deepen their understanding of math knowledge, I wanted to bring up the topic of a test correction analysis - I didn't find much on this topic when I searched for it here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here's a screenshot of a test correction analysis that I provide to my current Intermediate Algebra college students who didn't do well on an exam:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/hXNLg.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One reason for allowing students to do a test correction analysis and get points for doing it (half credit least) is to improve student retention. Another one is to reduce fear. Students can get a passing score if they complete an error analysis to an &quot;F&quot; exam grade. This college serves a low income community and the retention level of this Math class is very low because students lose hope of passing the course. Instructors start with 35 students and end up with 18 or 19 in the end. I have sort of remedied a cure for this by reducing the fear of failing for my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I talk about fear, I mean that my goal as a teacher is to instill in my students that success in this life comes to those who are willing to persevere with a task even when they are not having fun, that instant gratification is not always possible, but that the long term benefits&#xA;and rewards are well worth the investment of time and energy I am asking them to&#xA;make. With this in mind, if they sacrifice some&#xA;of their own time, then they deserve to be rewarded with the opportunity to&#xA;improve their grade by doing this test correction. What is most important to me&#xA;is that they learn the concepts and material so that they can apply them to&#xA;problems, even if I have to wait an extra day or two. One reward for their&#xA;sacrifice is a better grade in my class. The ultimate reward is knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rules for test correction:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Must be completed within 1 week and at the Math Tutoring Center. Proof of math tutoring must be verified (I can check if they went to tutoring on my online portal). If no tutoring services were attended, I cannot accept the test correction (this is what I mean by them sacrificing their own time to attend Math center).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) If you failed an exam, the highest possible score you can get is a passing grade (70%) after an error analysis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) If you did not fail an exam, only 1/4 credit will be added to your exam grade per wrong problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My main question is: would this forum consider this an effective tool? I would appreciate some feedback. I started with 35 students and still am at 35 students (which is a miracle). I am also providing one day out of the whole semester for any student to retake an exam (different version) for a better grade. I have been told that I'm lowering the standard. But my rebuttal to that is that I am making learning more accessible. Your thoughts? &lt;/p&gt;&#xA;" OwnerUserId="2198" LastEditorUserId="2198" LastEditDate="2014-10-09T02:52:04.510" LastActivityDate="2024-01-02T17:04:47.043" Title="Test Correction Analysis" Tags="|mathematical-pedagogy|testing|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4528" PostTypeId="2" ParentId="4522" CreationDate="2014-10-09T01:53:24.383" Score="7" Body="&lt;p&gt;There is quite a bit of evidence that critical thinking can be taught (which I found surprising). College students' gains in critical thinking skills over the course of their education are correlated with high standards set by instructors and greater time spent studying. So I think the basic answer is that you have to assign students tasks that require critical thinking, and grade them on those tasks. It's not something you can do overnight, and the gains are not large, but the gains do exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some references:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Academically Adrift: Limited Learning on College Campuses, Arum and Roksa, 2011, summarized in &lt;a href=&quot;http://www.newyorker.com/arts/critics/atlarge/2011/06/06/110606crat_atlarge_menand&quot; rel=&quot;nofollow&quot;&gt;http://www.newyorker.com/arts/critics/atlarge/2011/06/06/110606crat_atlarge_menand&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pascarella and Terenzini, &quot;How Robust Are the Findings of 'Academically Adrift?'&quot; &lt;a href=&quot;http://www.education.uiowa.edu/docs/default-source/crue-publications/AcadAdriftChangeArticleFINAL.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.education.uiowa.edu/docs/default-source/crue-publications/AcadAdriftChangeArticleFINAL.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-10-10T00:14:31.460" LastActivityDate="2014-10-10T00:14:31.460" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4529" PostTypeId="2" ParentId="4527" CreationDate="2014-10-09T03:01:16.403" Score="8" Body="&lt;p&gt;I would not give exam points for corrections. I would worry that the tutor might do too much of the work, and the student might still not understand.&lt;/p&gt;&#xA;&lt;p&gt;What I do is tell the students they must correct their problem, explain what went wrong in their process, and do two others like it. They must do this for each problem they got wrong. Then they can do a test retake (alternate version, and get full credit for their new exam score.&lt;/p&gt;&#xA;&lt;p&gt;I do not believe I have lowered the standards.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2024-01-02T17:04:47.043" LastActivityDate="2024-01-02T17:04:47.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4530" PostTypeId="2" ParentId="4518" CreationDate="2014-10-09T05:12:24.543" Score="3" Body="&lt;p&gt;This is my first answer here, so I hope I'm not breaking any inside rule in this answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your question is too vague in my opinion: Mathematical Analysis is a very wide area of mathematics and is extremely hard to answer it without knowing a bit about your background.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you already have taken a calculus course you probably can start studying real analysis and Rudin is a timeless classic, although I like very much &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0201086468&quot; rel=&quot;nofollow&quot; title=&quot;Real Analysis: An Introduction, by AJ White&quot;&gt;Real Analysis: An Introduction, by AJ White&lt;/a&gt;, with its well crafted exercises that will leave a lot of work to do on your own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no formal pre-requisite, but it will help a lot if you have some experience with one-variable calculus, series and sequences of real numbers.&lt;/p&gt;&#xA;" OwnerUserId="2206" LastActivityDate="2014-10-09T05:12:24.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4531" PostTypeId="2" ParentId="4518" CreationDate="2014-10-09T10:58:01.293" Score="1" Body="&lt;p&gt;In theory, there is very little pre-requisite to study introductory level real analysis (sometimes it's called advanced calculus, although this term is also seen applied to multivariable/vector calculus). Things as fundamental as number system (specifically real number) will be rigorously defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Practically, I think having the following will help a lot.&lt;br&gt;&#xA;1. Some exposure to math proof in other courses (e.g. linear algebra, discrete math, etc.)&lt;br&gt;&#xA;2. Non-rigorous version of calculus&lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For textbook, I also recommend Pugh's Real Mathematical Analysis. &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/144192941X&quot; rel=&quot;nofollow&quot;&gt;http://www.amazon.com/Mathematical-Analysis-Undergraduate-Texts-Mathematics/dp/144192941X&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user2139" LastEditorDisplayName="user2139" LastEditDate="2014-10-10T01:17:45.770" LastActivityDate="2014-10-10T01:17:45.770" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4532" PostTypeId="2" ParentId="4525" CreationDate="2014-10-09T14:00:05.910" Score="1" Body="&lt;p&gt;An important mathematical tool for signal processing is &lt;a href=&quot;http://en.wikipedia.org/wiki/Differential_equation&quot; rel=&quot;nofollow&quot;&gt;differential equations&lt;/a&gt;. Other necessary forms of math derive from this, including difference equations, transform theory, linear algebra, functional analysis, etc.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-10-09T14:00:05.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4533" PostTypeId="2" ParentId="4523" CreationDate="2014-10-09T15:07:21.313" Score="15" Body="&lt;p&gt;Here is an example from an algebra class I teach.&lt;/p&gt;&#xA;&lt;p&gt;We cover the quadratic formula, completing the square, the multiple ways to find the vertex, and try to connect it all together with the effects on the graph of the functions. In practice problems from the online homework, the students see things like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Solve &lt;span class=&quot;math-container&quot;&gt;$x^2 - 9x + 18 = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, when I ask them to do the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Solve &lt;span class=&quot;math-container&quot;&gt;$x^2 - 9x = -18$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They invariably respond that they &lt;strong&gt;have not seen that kind of problem before&lt;/strong&gt;. This is a very robust and repeatable experiment.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;What I am saying is that your students have probably seen what you are talking about before, but you may be surprised how few connections they are making between material. Instead many math students (and some teachers) think that mathematics is all about formula-plugging and pattern-matching.&lt;/p&gt;&#xA;&lt;p&gt;Asking the other teacher for the problem sets the students have seen on sequences will probably give you a lot of information.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As a further example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1a) Find the y-intercept of the following line: &lt;span class=&quot;math-container&quot;&gt;$2x + 5y = 10$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;1b) Find the x-intercept of the following line: &lt;span class=&quot;math-container&quot;&gt;$2x + 5y = 10$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;College algebra students will almost always think that 1a is easy and 1b is confusing. If you are wondering if they understand intercepts and you give them only (1b), you would probably think their other teacher had never even mentioned intercepts, when actually the problem is a lack of conceptual understanding and sense-making.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-10-09T15:07:21.313" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4535" PostTypeId="1" AcceptedAnswerId="4536" CreationDate="2014-10-09T21:32:04.440" Score="14" ViewCount="439" Body="&lt;p&gt;This is my first question and I hope it's appropriate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often in the process of teaching a subject I start with examples of a phenomenon, exhibiting similar properties between the examples and building an abstract concept of something. Turns out that the concept underlying the technical definition is much more powerful than the definition itself and becomes a hindrance when the theory is not strong enough to support the idea in its full generality. Students come up with examples compatible with the idea but not the technical definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, when teaching functions I discuss that for each street in a city there is a name to it (here in Brazil our streets are rarely numbered but have the name/date of important events or people). For each name we can compute the length of the word and therefore we can associate with each street the length of its name. This illustrates what I would later name function composition. This goes on for sometime until I define function as a relation of two sets satisfying certain properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student could ask me if associating a set with its size is an example of a function and I answer: &quot;Although the idea is correct, we can't do that because our function would have no domain, because there is no such thing as the set of all sets.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student is often disappointed because concepts he thought he understood (set as the extension of a property, like Frege's original idea, and a function as I taught) are not achievable by the mathematical entity called function and set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I deal with similar situations in a encouraging way for the students, while mathematically rigorous and concrete?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I apologize for eventual mistakes in the text as English is not my native language.)&lt;/p&gt;&#xA;" OwnerUserId="2206" LastEditorUserId="47" LastEditDate="2014-10-09T22:01:51.703" LastActivityDate="2017-02-27T23:35:44.227" Title="Explaining subjects whose justification requires demanding technical content" Tags="|student-motivation|examples|concept-motivation|definitions|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4536" PostTypeId="2" ParentId="4535" CreationDate="2014-10-09T21:55:24.090" Score="10" Body="&lt;p&gt;You could reply something like this: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Yes&lt;/em&gt;, associating each subset of the integers with its size would be an example of a function. And, in fact this does not just work for the integers but for all the subsets of any set. But note that to do it really for all sets would cause some problems as &quot;the set of all sets&quot; is a problematic notion. However, this does not really concern us here and it goes a bit beyond our current scope, but we can discuss it after the class if you like.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What I feel is key is to give some clear positive feedback, whence the &quot;yes&quot; at the start &#xA;(leaving it out the reply could come over as &quot;While something similar is true, what you said was &lt;em&gt;wrong&lt;/em&gt;.&quot;)&#xA;It was a good idea, and this should be made clear. But I feel it is also important not to say something false. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally, in such a situation I &lt;em&gt;try to turn what the student said into something that is right&lt;/em&gt;, and only mention the need for a correction or addition in passing or indirectly.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-10-09T22:35:06.090" LastActivityDate="2014-10-09T22:35:06.090" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4537" PostTypeId="1" CreationDate="2014-10-09T23:37:23.300" Score="7" ViewCount="177" Body="&lt;p&gt;How do modern primary-school math curricula like Everyday Math, Bridges, TERC, etc. approach the &quot;problem&quot; of providing enough challenge to young students who are very comfortable with math and who learn math concepts very quickly relative to peers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm asking because my 2nd-grader (in California) tells me that her Bridges math class is &quot;too easy&quot; and from what I've seen I admittedly agree with her.  The class is working on concepts she &quot;gets&quot; already like basic multiplication.  She wants more challenge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do primary school teachers using these new curricula deal with this problem? How do they keep the fast learners engaged given the new curricula's emphasis on group learning and avoiding &quot;tracking&quot; fast learners in separate &quot;gifted&quot; groups (like was common when I was in primary school long ago) ?  If you're an elementary teacher, how do you challenge a kid who's already working 1+ years ahead of grade level and who feels like the class is moving too slowly?&lt;/p&gt;&#xA;" OwnerUserId="2208" LastActivityDate="2014-10-10T21:40:16.953" Title="Modern primary-school math curricula (e.g. Everyday Math, Bridges) for accelerated learners" Tags="|primary-education|curriculum|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4538" PostTypeId="2" ParentId="4522" CreationDate="2014-10-10T06:40:33.590" Score="2" Body="&lt;p&gt;&quot;Do like a sports commentator: just say what's happening. As a sport commentator is talking to the public, whatever they are looking to the live-event or listening at the radio, they will understand just because he is saying exactly what's happening and nothing else. I think if you explain to your students, you're probably telling them your interpretation of the facts, so they have first to interpret your interpretation, then maybe understand. Teach as simple as you can!&quot;   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EXPLANATION: by &quot;do like a sports commentator&quot; I mean: tell to the public (your students) step by step what is happening, so that the public can understand everything from the event. After the event, an amount of sports experts meet each other in a tv-show and discuss about the sport match. They do comments about it, they suppose how could it be in a different way, they say which were the mistakes.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now transfer this situation into your class. The problem has to be solved. The teacher or tutor, explains to the students what is the problem asking for. Then the theacher gives his solution, and says step by step how he achieved it.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that, he asks to the students to give their solutions.  At this point, begins a discussion about the problem and its different ways to solve it. Of course every sentence should be motivated and explained as easy as possible, to avoid any misunderstandings caused by a to personal interpretation.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this process could encourage in the students a sort of &quot;critical thinking&quot;. &lt;/p&gt;&#xA;" OwnerUserId="2211" LastEditorUserId="2211" LastEditDate="2014-10-12T20:37:49.973" LastActivityDate="2014-10-12T20:37:49.973" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4539" PostTypeId="2" ParentId="4483" CreationDate="2014-10-10T13:58:10.950" Score="5" Body="&lt;p&gt;I basically agree with Benoit Kloeckner's answer. If you wanted to give a bit more context to the motivation, you could assign a sequence of exercises like the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $S_4$ act on $k[w, x,y,z]$ by permuting the variables.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;What is the stabilizer of $w$? How big is the orbit of $w$?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is the stabilizer of $w+x+2y+3z$? How big is the orbit of $w+x+2y+3z$?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is the stabilizer of $w+x+2y+2z$? How big is the orbit of $w+x+2y+2z$?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is the stabilizer of $wx+yz$? How big is the orbit of $wx+yz$?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Then, in class, start looking at the orbit of $(w-x)(w-y)(w-z)(x-y)(x-z)(y-z)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any intro group theory course has to do a sequence of basic exercises on group actions and orbits, and this isn't worse than any other. Plus, it will be excellent preparation if any of them take Galois theory!&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2014-10-10T14:19:29.350" LastActivityDate="2014-10-10T14:19:29.350" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4540" PostTypeId="2" ParentId="4025" CreationDate="2014-10-10T15:37:42.010" Score="8" Body="&lt;p&gt;I wanted to add:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1593274130&quot;&gt;The Manga Guide to Linear Algebra&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was published in 2012, and, is apparently one of an ongoing series of such texts. Now, as they currently stand it's not a stand-alone textbook, but it might be a fun supplemental for some courses. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-10-10T15:37:42.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4541" PostTypeId="2" ParentId="4025" CreationDate="2014-10-10T16:16:22.923" Score="2" Body="&lt;p&gt;The book &lt;a href=&quot;http://www.americanas.com.br/produto/6791770/livro-teoria-das-categorias-para-ciencia-da-computacao&quot; rel=&quot;nofollow&quot;&gt;Teoria das Categorias para Ciência da Computação&lt;/a&gt; has some cartoons involving a cat. The book is about category theory, it is meant for computer science students and written in Portuguese.&lt;/p&gt;&#xA;" OwnerUserId="2212" LastActivityDate="2014-10-10T16:16:22.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4543" PostTypeId="2" ParentId="4025" CreationDate="2014-10-10T20:53:18.647" Score="6" Body="&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; There was a relevant talk just last week (with some names that may be worth following up on) entitled &lt;a href=&quot;http://www.tc.columbia.edu/news.htm?articleID=9868&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Visual Learning and Teaching&lt;/strong&gt;&lt;/a&gt; (and headlined with: &lt;em&gt;Cartoons Can Teach Science&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I see that the top answer is about Galois Theory. A number of years ago, I thought it might be nice to try and create some supplementary materials (in the &quot;cartoon&quot; or storybook spirit) for a course on Abstract Algebra II. I only made a few PPT slides, but I've uploaded them &lt;strong&gt;&lt;a href=&quot;https://www.dropbox.com/s/04tpvudtn4q88js/Galois%20Storybook.pptx&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A sample slide:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Dplpw.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should re-caveat that these slides are not only old, but also unedited; in particular, there are some things that I would change today (e.g., my &lt;em&gt;field&lt;/em&gt; definition is really for a &lt;em&gt;commutative field&lt;/em&gt;, and the examples of &quot;non-fractions&quot; in this context should really be algebraic over $\mathbb{Q}$, meaning that $\pi$ and $e$ should be removed).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regard to using the slides: Anyone is &lt;strong&gt;free&lt;/strong&gt; to modify these notes, though I would be interested to see anything that could be squeezed out of them! My overarching wish was to start with something conceptually small, such as the single number $i = \sqrt{-1}$, and (perhaps at the expense of rigor) build outwards towards something more general, such as Galois Theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, I have only a vague proof of concept, which is enough to convince &lt;em&gt;me&lt;/em&gt; that the answer to the question here is &lt;em&gt;Yes&lt;/em&gt;; however, I do not have any plans (personally) to develop the slides linked above.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2015-03-05T18:26:18.007" LastActivityDate="2015-03-05T18:26:18.007" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4544" PostTypeId="2" ParentId="4537" CreationDate="2014-10-10T21:40:16.953" Score="2" Body="&lt;p&gt;Sometimes, people interpret a student's facility with calculation as understanding all the concepts that the curriculum is designed to address. That's not necessarily the case.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://investigations.terc.edu/library/implementing/qa-1ed/working_with_gift_students.cfm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Barbara Bissell wrote&lt;/a&gt; about an experience working with a whole class of gifted students.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the Horse Problem, a man buys a horse for \$60. Then sells it for \$70. Later he buys it back for \$80 and sells it again for \$90. The question to the students is &amp;quot;How much money does the man make or lose?&amp;quot; We spent the full hour working on the problem, arguing about it and finally acting it out.&lt;/p&gt;&#xA;&lt;p&gt;Now, what was so hard about that problem? Clearly, the arithmetic is easy. It must have been the reasoning that stumped these kids.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, it is not easy to know whether the curriculum &lt;em&gt;truly&lt;/em&gt; is easy. (It makes sense to read the whole piece this is excerpted from, for a more complete picture.)&lt;/p&gt;&#xA;&lt;p&gt;My recommendation is to talk to the teacher. And when you do, instead of characterizing the curriculum as too easy, ask the teacher what the goals are in her classroom and how to help your daughter get the most out of the class in terms of the goals.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-10-10T21:40:16.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4546" PostTypeId="2" ParentId="4526" CreationDate="2014-10-10T23:59:08.807" Score="1" Body="&lt;p&gt;To riff off of the ideas I develop in the comments:  A unit disk starts off in the $xy$ plane centered at the origin.  It is spinning at $\omega$ revolutions per minute, and traveling straight up along the $z$ axis at a rate of $S$ units per minute.  A target is located at the point $(1,0,L)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You obtain a different &quot;ray gun path&quot; by placing a mark at a different place on the boundary of the disk, and letting it trace out a path.  Only one of these paths will pass through the target.  To a human eye, all of these paths look qualitatively very similar:  If your ray gun fired one beam versus another I doubt you would notice.  It stands to reason that it would be very hard to hit your target, since only one out of an uncountable number of choices which are identical looking to you hits the target.  Even if you give some width to the beam, say $\delta$, your chance of hitting the target should be somewhere around $\frac{\delta}{2\pi}$, which is not too great.  You could, of course, calculate where your beam should start in this case at $(\cos(2\omega L\pi/S),-\sin(2\omega L\pi/S))$, but that is not the sort of thing which would be easy to &quot;eyeball&quot;.  On the other hand, humans are quite good at aiming things along straight lines, or parabolic arcs.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-10-10T23:59:08.807" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4547" PostTypeId="2" ParentId="4527" CreationDate="2014-10-11T05:09:14.597" Score="5" Body="&lt;p&gt;What are you worried about?  1) Are you worried about passing a certain percentage of students?  2) Are you worried that students may cheat to achieve this passing grade?  3) Are you worried that students do not learn your material?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With little doubt, you will answer #3, but is that the truth? (only you can answer that question--you will not answer it truthfully here).  Passing a percentage of students is easy, &lt;em&gt;and&lt;/em&gt;, should be the ultimate goal.  The reason it should be an ultimate goal is that you should have a majority of your class &lt;em&gt;understand&lt;/em&gt; the concepts presented--thus passing a majority of your class &lt;em&gt;should&lt;/em&gt; be easy.  But it's &quot;easy&quot; because you can make up whatever grades you like such that you pass a majority of your class.  Being concerned about whether or not your students learn the material is much harder...but that's a different, and unanswerable question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't see a problem with offering corrections to a test (and even giving &lt;em&gt;FULL&lt;/em&gt; credit for the corrections--if done properly).  Let's assume it's completely passive: they simply turn in corrections and you give them half credit.  One way or another, they had to go out of their way to get the solutions, whether by a tutor or not, they had to at least write down their solutions--I think half credit here is fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I taught calculus for a year and I allowed corrections (it was for half credit, but I would have been fine with full credit the way I did it), but to get the full credit you would have to have to come to me and explain your solutions (even if I had already given the solution).  That doesn't mean you turn in a paper assignment with the solution, it means you come to me (during office or tutoring hours) and you explain to me each problem you got wrong.  During that time, I ask you why your solution is valid and I probe the student to understand whether or not they actually understand their mistakes (or their solution).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This type of probing will tell you whether or not the student understands the material and thus whether or not you should change their grade accordingly.&lt;/p&gt;&#xA;" OwnerUserId="2204" LastEditorUserId="2204" LastEditDate="2014-10-11T05:23:24.763" LastActivityDate="2014-10-11T05:23:24.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4548" PostTypeId="2" ParentId="4535" CreationDate="2014-10-11T05:37:17.663" Score="3" Body="&lt;p&gt;A function is a &quot;weird&quot; thing.  You must emphasize this first, if you are to explain functions.  My example is a function is an oven: you take milk, baking soda, flour, etc. and you make biscuits.  The function is that you take those &lt;em&gt;input&lt;/em&gt; variables (milk, baking soda, flour, etc,) and do something with them and out comes a biscuit (something that is &lt;em&gt;not&lt;/em&gt; the sum of its parts)!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A problem I see of many high school curricula, is that a function is taught as an &lt;em&gt;explicit&lt;/em&gt; function, i.e. $y = f(x) =ax + b$ where you simply put in the value of $x$ and you get the function's value.  Even &lt;em&gt;worse&lt;/em&gt;, I see that a function is taught as $y = f(x)$--which, while it serves the purpose of the objectives, is &lt;em&gt;not&lt;/em&gt; what a function &lt;em&gt;is&lt;/em&gt;!&lt;/p&gt;&#xA;" OwnerUserId="2204" LastActivityDate="2014-10-11T05:37:17.663" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4549" PostTypeId="2" ParentId="4025" CreationDate="2014-10-11T05:39:57.680" Score="2" Body="&lt;p&gt;&quot;Indra's Pearls&quot; (by Mumford, Series, and Wright) is just a wonderful book describing the mathematics behind Schottky groups. It is accessible to undergraduate students, and has fun projects for computer visualization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the nicest features of the book is the cartoons drawn by Larry Gonick. His character, Dr. Stickler, cuts and glues rubber material to construct manifolds in an entertaining and very intuitive way.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2014-10-11T05:39:57.680" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4550" PostTypeId="2" ParentId="4526" CreationDate="2014-10-11T06:30:50.953" Score="1" Body="&lt;p&gt;First we need a definition of &quot;aim&quot; that is interesting. If the gun had a laser sight, you would just wiggle until the red spot falls on the target (assuming, of course, that the laser also travels in a spiral). But that is just going in circles :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So let's say that we are given 3D coordinates for gun and target, and our goal is to compute a direction vector such that when the gun is 'aimed' in that direction and fired, the target is hit. Thus, we have to determine the shot's axis; the line $L$ through the center of the spiral. But we also have to account for the initial direction of the ray: the muzzle can be placed on any point around a circle of radius $r$ orthogonal to $L$. It seems that &quot;aiming&quot; is more difficult due to our having one more parameter to determine. One would have to find the relative direction $\theta$ from $L$ to the target as seen from the gun, and then determine the initial position of the muzzle, given the spiral parameters and the distance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the line $L$ is not unique. In fact, the is a circle's worth of lines around the target all of which give hits, provided the initial direction is chosen correctly. Once you work out the formula (which I won't do since this is just a conceptual discussion) you can hit the target in infinitely many different skew directions!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, I would say that it is indeed more difficult to aim a spiral ray gun because the formula is more complicated than a straight line equation. This is an anticlimactic answer, but after trying to set up a reasonable model, it is the answer that matches my intuition.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2014-10-11T06:30:50.953" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4551" PostTypeId="1" AcceptedAnswerId="4553" CreationDate="2014-10-11T14:53:27.300" Score="8" ViewCount="544" Body="&lt;p&gt;I'm teaching in China, and I have been told that I will be leading the students in mathematics competitions.  I feel very ill prepared for this.  I never did mathematics competitions in either high school nor university. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There will be at least five different competitions, likely more.  The first is &lt;a href=&quot;https://cemc.math.uwaterloo.ca/contests/past_contests.html&quot;&gt;Waterloo:CEMC&lt;/a&gt;.  Also, the student's parents send their kids to compete in competitions outside of the school's scheduled events.  So, on top of coaching the 90 grade 10 (US system) students plus, grade 11 and 12; I am also to help a couple of students to do well at the &lt;a href=&quot;https://www.math.duke.edu/dumu/mathmeet/&quot;&gt;Duke University Math Meet&lt;/a&gt;, and some various other mathematics competitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question has two parts: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How to coach these students when the problems themselves are challenging for myself. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The bright spot, is that I have a decent budget to order books. Please, suggest titles of books to place on my order sheet and any other resources that would benefit my coaching and broaden my and the students knowledge base for mathematics competitions.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2182" LastActivityDate="2019-09-03T15:22:57.533" Title="Good resources for coaching for mathematics competitions (Highschool Level)" Tags="|reference-request|math-contests|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4552" PostTypeId="2" ParentId="4551" CreationDate="2014-10-11T16:08:01.480" Score="4" Body="&lt;p&gt;You could have them work through problems on &lt;a href=&quot;http://www.artofproblemsolving.com/Wiki/index.php/AMC_Problems_and_Solutions&quot; rel=&quot;nofollow&quot;&gt;aops&lt;/a&gt;. It contains AMC questions as well as video lectures on the problems. Furthermore it has &lt;a href=&quot;http://www.artofproblemsolving.com/Store/index.php&quot; rel=&quot;nofollow&quot;&gt;books&lt;/a&gt; and other resources you may be interested in.&lt;/p&gt;&#xA;" OwnerUserId="1647" LastActivityDate="2014-10-11T16:08:01.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4553" PostTypeId="2" ParentId="4551" CreationDate="2014-10-11T19:45:35.003" Score="6" Body="&lt;p&gt;I highly recommend Paul Zeitz, &lt;em&gt;&lt;a href=&quot;http://www.betterworldbooks.com/the-art-and-craft-of-problem-solving-id-9780471789017.aspx&quot;&gt;The Art and Craft of Problem Solving&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The books coauthored by Richard Rusczak on problem solving are probably excellent, though I haven't yet looked at them myself: &lt;a href=&quot;http://www.artofproblemsolving.com/Store/contests.php&quot;&gt;&lt;em&gt;The Art of Problem Solving, Volumes 1 and 2&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then there are &lt;a href=&quot;http://www.ams.org/bookstore?s3=msri&amp;amp;quicksearch.x=0&amp;amp;quicksearch.y=0&amp;amp;pg1=&amp;amp;s1=&amp;amp;op1=AND&amp;amp;co1=AND&amp;amp;pg2=&amp;amp;s2=&amp;amp;op2=AND&amp;amp;co2=AND&amp;amp;op3=AND&amp;amp;co3=AND&amp;amp;pg3=ALLF&amp;amp;fn=100&amp;amp;d=BOOK&amp;amp;p=1&amp;amp;u=&amp;amp;r=0&amp;amp;l=100&amp;amp;f=S&amp;amp;subj=mathed&quot;&gt;the math circle books&lt;/a&gt;, many of them geared to competition preparation...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also find James Tanton's book, &lt;a href=&quot;http://www.jamestanton.com/?page_id=22&quot;&gt;&lt;em&gt;Solve This!&lt;/em&gt;&lt;/a&gt;, very useful as you learn to coach math teams.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-10-11T19:45:35.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5553" PostTypeId="2" ParentId="4522" CreationDate="2014-10-11T22:19:42.103" Score="9" Body="&lt;p&gt;I know I shouldn't add another answer, but I think my other answer went off on a tangent that didn't really address your question.  I did not initially read it carefully enough to realize you were tutoring students, not teaching a class.  I really don't think the label of &quot;Teaching Critical Thinking Skills&quot; is either relevant or pertinent--teaching is teaching, if we want to label &quot;teaching&quot; as &quot;teaching critical thinking skills&quot; then I think we are defining want teaching is--if you actually want to &quot;teach&quot; critical thinking, then this is a far more broad question which is not at all specific to mathematical instruction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When tutoring students, you have a very good opportunity to give them deep and meaningful instruction (much more so than in a classroom setting).  It's easy to just give them the solution, but obviously not very helpful (and unlikely to further the students learning).  Having said that, the end goal &lt;em&gt;must&lt;/em&gt; be to have the student leave with a solution, preferably one they arrive at as close to on their own as possible.  Different students will come to you with different abilities and for some students, you may have to end up explaining the entire solution (but this should be a last resort).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Asking questions is vital.  You need to nudge the student but not push them over a cliff (i.e. give them the entire solution).  I will use your problem as an example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Step 1:&lt;/strong&gt;  Ask the student whether or not they drew a picture.  If no, then tell them to draw one, if yes, have them show you their picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Step 2:&lt;/strong&gt;  Analyze their picture.  Is the picture correct?  Perhaps they drew the cars going in the same direction (as opposed to driving towards each other).  You want to avoid cuing the student; so do not automatically accept a correct picture (assume it's wrong no matter what they draw).  Ask them to explain their picture.  Does the picture agree with the setup of the question?  Why did they draw car $A$ on the left and car $B$ on the right (or vice versa)?  Is that correct?  Does the problem specify this?  Does it matter?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Step 3:&lt;/strong&gt; Label the picture.  What is given in the question?  Can we introduce variables (unknowns) that we may or may not be able to solve for?  Assuming the picture is now correct, hopefully the student will, at a minimum, label the initial distance between the cars as $88$ m.  If this is all they label, then point out the distance between the starting point of one of the cars and the collision point.  Ask them if they can find this distance.  No?  Then let's just label it was an unknown.  If you do that for one car, hopefully they will see you can do that for &lt;em&gt;both&lt;/em&gt; cars.  If not, you will have to point that one out too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Step 4:&lt;/strong&gt; Point out relationships we can deduce from the picture.  Is there anything we can say about the two distances?  Do we know &lt;em&gt;anything&lt;/em&gt; about them?  If they are baffled, ask does the initial distance of $88$ m change?  They may say yes, it changes as the cars get closer together.  If they go this route, then let them continue.  Explain that they now need a second picture with the cars closer together.  Do we have more or &lt;em&gt;less&lt;/em&gt; information in this second picture?  Do we know how far apart cars are now?  Can we figure it out?  How much time as elapsed from the first picture to the second?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Step 5:&lt;/strong&gt; Start solving the problem.  With any luck, you stopped around the 2nd or 3rd step and the student did the rest, but you may still have to go this far.  If the student is still confused, you will need to start asking about time.  OK, we have the two distances labeled, how long did this take?  Can we come up with a relationship between the distance and time?  If that doesn't work, you may have to ask what if I told you that car A traveled its distance in 2 seconds?  Could you find the distance then?  If car A traveled &lt;em&gt;its&lt;/em&gt; distance in 2 seconds, then how long should it take car B to travel &lt;em&gt;its&lt;/em&gt; distance?  Hopefully the student will answer 2 seconds.  So what can we say about the times it takes car A to travel its distance and the time it takes car B to travel its distance?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student is still baffled, you may have to just tell them the two times must be the same.  Now ask, what if the time &lt;em&gt;was&lt;/em&gt; 2 seconds?  What would the two distances be?  Is that correct?  How do you know it's not correct?  Hopefully the student will realize that the two distances (using $t = 2$ seconds) do not add up to the full $88$ meters or they add up to more (and hopefully $t = 2$ seconds is not coincidentally the correct time--if so choose a time you know won't be correct).  Can we find the correct time?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they are still unsure, ask them again, how you knew $t = 2$ seconds was incorrect.  Have them write out the equations without evaluating (i.e. $d_A = \frac{1}{2}A_a(2)^2$ and $d_B = \frac{1}{2}B_a(2)^2$).  Ask why these two distances are incorrect (hopefully they already answered and thus have the answer ready).  Because they do not add up to the full distance.  So if $2$ seconds &lt;em&gt;was&lt;/em&gt; the correct time, what equation would you expect to be true.  If they are unsure, tell them to write it out in full, you may have to simply tell them to write down $\frac{1}{2}A_a(2)^2 + \frac{1}{2}B_a(2)^2$.  Ask them what this equation should equal.  They now should essentially have the equation.  Can you find the time &lt;em&gt;now&lt;/em&gt;?  If they are &lt;em&gt;still&lt;/em&gt; baffled, ask them what do we do when we have an unknown?  We create a variable for it.  At this point, if they are still confused, you may have to flat out tell them to substitute $t$ where they placed $2$ (remind them that $2$ was the time we guessed).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you have two problems: 1) we need to solve for time, so you are now no longer a physics tutor, but an Algebra I tutor and 2) you need to ask them, after &lt;em&gt;all&lt;/em&gt; of this work, what was the question asking for?  Did it ask for the time?  No?  It asked for the distance car A traveled?  If we know the time, can we find this value?  Hopefully they realize they can, but if not, remind them that we &lt;em&gt;have&lt;/em&gt; an expression for $d_A$ already, we just need to plug in our value of $t$ into it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that this explanation was very specific to your example problem, but the strategy is essentially the same: 1) Draw a picture, 2) analyze the picture, 3) label the picture, 4) find relationships, and 5) solve the problem.  The absolute &lt;em&gt;hardest&lt;/em&gt; part is trying to avoid cuing the students' responses.  For instance, don't ask: &quot;does [insert kinematic equation] help us?&quot;  If you &lt;em&gt;do&lt;/em&gt; have to resort to this, then you should go through them all and have the student explain why each is either useful or not applicable.  Don't automatically accept correct answers and don't automatically reject incorrect answers.  Let them continue with the wrong way until you hit a point where you can no longer solve, then help the student backtrack to find where the mistake is.  Your goal as a tutor, should be to do as &lt;em&gt;little&lt;/em&gt; as possible.  The more time you spend with a particular student, the more you will learn what they need from you (duh).&lt;/p&gt;&#xA;" OwnerUserId="2204" LastEditorUserId="2204" LastEditDate="2014-10-13T04:04:06.027" LastActivityDate="2014-10-13T04:04:06.027" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5554" PostTypeId="1" CreationDate="2014-10-12T04:36:00.460" Score="7" ViewCount="211" Body="&lt;p&gt;I am looking for the conceptual/visualisation way of tackling the algebra problems . I found that Singapore Maths caters this need. What are the good sources for Singapore Maths - any online classes for Singapore Maths. My aim is to mentally solve any word problems - algebra so that I score high marks in the competitive exams like GRE. Also, This way I can crack the algebra problems visually rather than x, y and z.&lt;/p&gt;&#xA;" OwnerUserId="3214" LastActivityDate="2015-07-14T23:05:53.827" Title="What are the good sources for Singapore Mathematics?" Tags="|algebra|linear-algebra|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5555" PostTypeId="2" ParentId="36" CreationDate="2014-10-12T05:40:22.257" Score="3" Body="&lt;p&gt;For an easy-to-use software that is both simple and capable of ploting beautiful 3d graphs you can try downloading my &lt;a href=&quot;http://www.runiter.com/graphing-calculator/&quot; rel=&quot;nofollow&quot;&gt;Graphing Calculator 3D&lt;/a&gt; software.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It supports explicit and implicit equations, parametric equations, cartesian, polar, cylindrical and spherical coordinates.&lt;/p&gt;&#xA;" OwnerUserId="2214" LastEditorUserId="2214" LastEditDate="2015-10-15T14:13:04.833" LastActivityDate="2015-10-15T14:13:04.833" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5556" PostTypeId="2" ParentId="4551" CreationDate="2014-10-12T15:14:37.653" Score="6" Body="&lt;p&gt;The (university) course I took on problem solving and mathematics competitions had several problem sets with competition-level problems that were &lt;strong&gt;sorted into problem-solving strategies&lt;/strong&gt;. For example, during week 3, we solved problems using the Extremal Principle and got familiar with it. In another week we solved Pigeonhole Principle problems. You might get some mileage out of such problems, depending on the level of the students.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Edit, years later: The materials now appear to be available at&#xA;&lt;a href=&quot;https://sites.lsa.umich.edu/hderksen/wp-content/uploads/sites/614/2018/05/ProblemSolving-1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://sites.lsa.umich.edu/hderksen/wp-content/uploads/sites/614/2018/05/ProblemSolving-1.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The course was taught by Harm Derksen at the University of Michigan. The textbook used for the course was Loren C. Larson, Problem-Solving through Problems, Problem books in Mathematics, Springer, 1983 -- although I have no memory of the textbook, only of the excellent problem sets.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2019-09-03T15:22:57.533" LastActivityDate="2019-09-03T15:22:57.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5557" PostTypeId="1" CreationDate="2014-10-12T16:11:14.900" Score="7" ViewCount="4665" Body="&lt;p&gt;I am looking to go back to school for a Mathematics degree (Bachelor). I want to do it for pure knowledge, and not a degree (but I do want to kill 2 birds with one stone) - many areas I'm interested in require a thorough understanding of math. I have very basic calculus/linear algebra (from 1st year Engineering) understanding which is often not enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions are as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there severe limitations for doing distance education in mathematics? I have access to people with thorough understanding of higher mathematics who can readily help me if I'm completely stuck.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have you or anyone you know completed distance education degree in mathematics (level of the degree does not matter)? Could you describe some of the experience?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do you know of any universities (especially in Europe, but not necessarily) that offer distance education in mathematics (Bachelor)?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I do not need to be actively taught. Just a way to submit work is fine.&#xA;Another point to mention is that I would prefer cheaper education alternative. As much as I would like the freedom of not being tied down to a city, it stops making sense to pay $6000 for online education when I can just go to a local university and not attend all lectures. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3216" LastEditorUserId="3216" LastEditDate="2014-10-12T19:26:20.283" LastActivityDate="2014-10-15T10:37:38.613" Title="Distance Bachelor's degree in mathematics in Europe" Tags="|undergraduate-education|online-instruction|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5558" PostTypeId="2" ParentId="5557" CreationDate="2014-10-12T19:35:01.813" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://www.open.ac.uk/&quot; rel=&quot;nofollow&quot;&gt;The Open University&lt;/a&gt; offers distance learning and has &lt;a href=&quot;http://www.mathematics.open.ac.uk/study&quot; rel=&quot;nofollow&quot;&gt;undergraduate and postgraduate programmes in mathematics&lt;/a&gt;. &#xA;Unfortunately, I do not know anything beyond this about it, but the linked to pages should lead to all kind of information. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For your first question of severe limitations, I would say, no, mathematics is rather well-suited for this, as there is no lab-work or anything like this needed. Especially since you have access to people that can help you, you should be fine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be hard for some people to learn in relative isolation and independence, but this is a general issue not specific to mathematics.  &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-10-12T19:35:01.813" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5559" PostTypeId="1" CreationDate="2014-10-12T19:39:14.283" Score="-5" ViewCount="857" Body="&lt;p&gt;It is now well known that a random coin toss has 1/6000 probability of landing on its edge. So the out-dated model that a coin toss always land on either heads or tails with probability 1/2 is wrong. Further, it has also been revealed that the physical coin toss process is not random, but deterministic. So the coin toss should be described as a phenomenon in Newtonian dynamical systems, and the &quot;randomness&quot; comes from the sensitivity of initial data, so it is not really random at all. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore it clearly &lt;em&gt;does not&lt;/em&gt; fit into the definition of &lt;a href=&quot;http://en.wikipedia.org/wiki/Classical_definition_of_probability&quot; rel=&quot;nofollow&quot;&gt;classical probability&lt;/a&gt;. Because if we are talking about the coin toss as a random event with 1/2 for either heads or tails, then this is taking conditional probability excluding the case of landing on its edge already. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the heads and tails had a pedagogy advantage for being easy to understand, I believe this is not an appropriate example that should appear in the textbook. And we all know there are a lot of good examples in quantum mechanics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is, why we still mistaken coin toss to be an example of classical probability? To me this is almost the same as leaving the mistake to the next generation while science has proved it is not true. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://journals.aps.org/pre/abstract/10.1103/PhysRevE.48.2547#abstract&quot; rel=&quot;nofollow&quot;&gt;http://journals.aps.org/pre/abstract/10.1103/PhysRevE.48.2547#abstract&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="tuzhexiu" LastEditorUserId="143" LastEditDate="2014-10-12T20:38:42.697" LastActivityDate="2014-10-14T13:14:13.917" Title="Why we mistaken coin toss to be an example of classical probability?" Tags="|mathematical-pedagogy|curriculum|probability|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5560" PostTypeId="2" ParentId="5559" CreationDate="2014-10-12T19:55:26.293" Score="17" Body="&lt;p&gt;When mathematics is applied to the real world, it is always only an approximation.&#xA;Usual probabilistic model for the experiment of coin tossing is a good approximation.&#xA;Good and useful. &#xA;So there is nothing wrong with teaching this example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arguing like you do, we should not teach classical mechanics, because it is only an approximation to reality, and teach only quantum mechanics and (general) relativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a wrong approach. Because classical mechanics works perfectly in many practical problems. And probabilistic model works very well when applied to coin tossing, card games,&#xA;etc.&lt;/p&gt;&#xA;" OwnerUserId="20727" OwnerDisplayName="Alexandre Eremenko" LastActivityDate="2014-10-12T19:55:26.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5561" PostTypeId="2" ParentId="5559" CreationDate="2014-10-12T20:12:35.833" Score="3" Body="&lt;p&gt;Stating that it is &quot;well known&quot; that the probability of a coin landing on its side is 1/6000 is incorrect. In fact - probability is determined by a variety of factors. In this case, the coin landing on its side would depend on the nature of the surface upon which it was landing most notably, as well as the nature of the coin's surfaces.&#xA;In fact, taking all factors into account probability is actually very predictable. If you were able to model the majority of forces and masses acting on the coin as it was tossed, you would (to a limit) essentially be able to predict the outcome with a very high accuracy. Such a thing could be done with a computer.&#xA;In fact, probability is more the percentage occurrence of an event within a group of large events, rather than something that can be applied to one-time occurrences. For example, last week's lottery numbers had an immensely small chance of occurring. Some would say that if the lottery was played once that the chances of those numbers occurring would make the event a miracle almost. However, since we have the lottery every week, this become normal - obviously those numbers might occur some week (it just happened to be last week).&#xA;Your comments about quantum mechanics are interesting: read the last part of this blog post.&#xA;&lt;a href=&quot;http://lifephysics.blogspot.co.uk/2014/03/computers-don-always-do-what-they-told.html&quot; rel=&quot;nofollow&quot;&gt;http://lifephysics.blogspot.co.uk/2014/03/computers-don-always-do-what-they-told.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3217" OwnerDisplayName="Resquiens" LastEditorUserId="3217" LastEditDate="2014-10-12T20:26:44.603" LastActivityDate="2014-10-12T20:26:44.603" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5562" PostTypeId="2" ParentId="5557" CreationDate="2014-10-12T22:05:34.703" Score="2" Body="&lt;p&gt;Have you looked at &lt;a href=&quot;http://www.snhu.edu/&quot; rel=&quot;nofollow&quot;&gt;http://www.snhu.edu/&lt;/a&gt;  (Southern New Hampshire University), which offers online eduction at the undergraduate and graduate level.  It's not in Europe, though.   &lt;/p&gt;&#xA;" OwnerUserId="3219" LastEditorUserId="143" LastEditDate="2014-10-13T06:08:03.663" LastActivityDate="2014-10-13T06:08:03.663" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5563" PostTypeId="2" ParentId="5559" CreationDate="2014-10-13T00:00:41.287" Score="10" Body="&lt;p&gt;One often sees the phrase &quot;a &lt;em&gt;fair&lt;/em&gt; coin.&quot; So, if you wish, this is a theoretical object which does not exist in the real world but the name is evocative and useful. In real life a (real) coin flip is used to decide some things (football). One may object to an element of chance in a contest of skill. Perhaps the side winning the toss gets an unfair advantage. But no-one worries that the side calling the toss gets an unfair advantage. In the world of theory no-one outside a middle school science fair is tossing actual coins. Then a fair coin is a useful name for an imaginary binary object which behaves exactly like the best first order approximation behavior of real coins (or at least what we would like it to be).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you see as the problem?&lt;/p&gt;&#xA;" OwnerUserId="3220" LastActivityDate="2014-10-13T00:00:41.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5564" PostTypeId="2" ParentId="5557" CreationDate="2014-10-13T00:20:07.177" Score="2" Body="&lt;p&gt;I think what you seek is (a) not yet available, but (b) will likely be available eventually.&#xA;I base the latter on the existence of the &lt;a href=&quot;https://www.edx.org/xseries&quot; rel=&quot;nofollow noreferrer&quot;&gt;X-series&lt;/a&gt;,&#xA;a sequence of edX courses that leads to an XSeries Certificate (&quot;&lt;a href=&quot;https://www.edx.org/press/mitx-introduces-xseries-course-sequence&quot; rel=&quot;nofollow noreferrer&quot;&gt;EdX introduces new ID verification service&lt;/a&gt;&quot;). You can earn a certificate&#xA;in Aerodynamics, Astrophysics, Foundations of Computer Science, and other topics. But not (yet) Mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one of the Astrophysics courses, in progress just this month:&#xA;&lt;a href=&quot;https://www.edx.org/course/anux/anux-anu-astro3x-violent-universe-3021#.VDsaRdR4r7c&quot; rel=&quot;nofollow noreferrer&quot;&gt;ANU-ASTRO3x The Violent Universe&lt;/a&gt;, taught by Brian Schmidt &amp;amp; Paul Francis.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Sda2L.png&quot; alt=&quot;ViolentUniverse&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-10-13T11:33:34.567" LastActivityDate="2014-10-13T11:33:34.567" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5565" PostTypeId="1" CreationDate="2014-10-13T02:20:32.350" Score="4" ViewCount="230" Body="&lt;p&gt;My wife and I will be homeschooling our son and twin daughters. We're interested in adopting an honors-level mathematics curriculum for secondary students that culminated in calculus AB in 12th grade. Which textbooks on the market (or not) would you math brains recommend for honors level courses in Algebra I, Proof-Heavy Geometry, Algebra II/Trig, and Intro Math Analysis/Precalc. My wife and I both have taken calculus through differential equations and are in science related fields. We'd be able to take a formal approach to teaching the math. Any suggestions?  &lt;/p&gt;&#xA;" OwnerDisplayName="Ryan Cavanaugh" LastEditorUserId="143" LastEditDate="2014-10-13T20:41:32.593" LastActivityDate="2014-10-13T20:41:32.593" Title="Honors Level High School Coursework Curriculum?" Tags="|secondary-education|textbooks|homeschooling|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5566" PostTypeId="2" ParentId="5565" CreationDate="2014-10-13T14:12:55.260" Score="2" Body="&lt;p&gt;Check into the courses offered by &lt;a href=&quot;http://www.artofproblemsolving.com/&quot; rel=&quot;nofollow&quot;&gt;Art of Problem Solving&lt;/a&gt;. They get good reviews from the homeschoolers I am in contact with. Although their description does not mention honors-level or AB, they train students for competitions, and their materials include the rigor you desire.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-10-13T14:12:55.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5567" PostTypeId="2" ParentId="333" CreationDate="2014-10-13T23:25:17.863" Score="4" Body="&lt;p&gt;I mostly teach physics, not math, so I get a lot of experience in my lab classes where I see students manipulating numerical data. In particular, I often see them do numerical calculations that are long enough  so that rounding errors really could accumulate. (I doubt that math teachers very often see their students do a really lengthy numerical calculation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thing that stands out to me most from this experience is that most students at this level (freshman college STEM majors) show an almost complete inability to reason &lt;em&gt;about&lt;/em&gt; how a calculation works. This extends to the most incredibly simple concepts, such as the idea that when you change the input to a calculation, the output will also change.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, I will often see students write down intermediate steps in their lab notebooks, recording all the digits from their calculators. As an imaginary example, suppose that they've just immersed a cube in water and measured its volume to be 124 cubic centimeters. They next use this to calculate the length of the cube's edge, which is going to be a step in a longer calculation. They write it down as 4.98663095223864 cm. When I ask them why they're doing that, here are some typical answers I get:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;That's what my calculator gave me.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In chemistry you're not supposed to round off. Is it different in physics?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Answers like these show that they're not reasoning at all about how the calculation works. They expect to be told an arbitrary rule to follow, which will ensure them a good grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For these reasons, I agree very much with Sue VanHattum. When something like this comes up, its value lies in the fact that we can use it to guide students toward thinking more deeply about what would make sense. Teaching the student a more statistically desirable procedure should not be the point, unless the student is ready to make sense of the procedure. Sometimes we can say things to our students that will nudge them into thinking:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;When I was a kid, my family got our first electronic calculator, and it had a 6-digit display. I see that you iPhone calculated that cube root with 14 digits of precision, and you wrote them all down in your lab notebook. If you got a newer phone and it displayed 100 digits, would you write all of those down?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This other lab group measured 123 cm$^3$ for the volume of the cube. If we took the cube root of that number instead of your 124 cm$^3$, which of those digits in your result do you think would change?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As a practical matter, for people who are more numerate, I don't think it makes a lot of sense to follow any particular rule for rounding fives. In almost all real-world applications, the answer is that if rounding a 5 up or down is enough to make a difference in your life, then you're rounding too much; you should retain more digits of precision.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-10-13T23:25:17.863" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5568" PostTypeId="1" CreationDate="2014-10-13T23:54:47.217" Score="9" ViewCount="322" Body="&lt;p&gt;There is a set of related topics in a freshman calc course that includes the completeness axiom for the reals, the intermediate value theorem, extreme value theorem, Rolle's theorem, and mean value theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How can one interest a freshman calc student in these topics, when they almost certainly have zero interest in foundational issues?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This material seems to me to have a fairly tenuous connection to the mainstream of a freshman calc course. I'm not an expert in the history of mathematics, but it seems that Dedekind cuts and such were not invented until about 1860. That leaves a period of about 150 years during which the world's greatest mathematicians happily did calculus without feeling a need for these foundational issues to be addressed. They would probably have felt that a result like the extreme value theorem was geometrically obvious. (Some of the theorems were stated earlier, e.g., Rolle's theorem dates to 1691, but I don't see how Rolle could have given a meaningful proof 150 years before the arithmetization of the reals.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further adding to the difficulties in selling this to a class of students are the facts that:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;These theorems are not particularly useful in solving problems (because in most problems where they were needed, the theorems' claims would have been intuitively obvious anyway).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In a freshman calculus course we don't really attempt to present this material in detail. E.g., textbooks tend to omit most of the proofs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(Just to put this in perspective, I recently asked a class of students who had all already had a year of calculus to describe in general terms how the derivative was defined -- I wasn't asking them to regurgitate the definition, just to describe its flavor. Despite a lot of poking, prodding, and hinting, not a single student volunteered that it was defined using a limit.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best idea I've been able to come up with so far is just to give the class a series of tasks in which they come up with counterexamples to the various theorems when one of the assumptions is omitted, e.g., give an example of a function that is continuous everywhere in its domain but that never attains a maximum value. This might at least help to convince them that the results make sense and have a geometrically visualizable interpretation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://mathoverflow.net/questions/184358/what-did-rolle-prove-when-he-proved-rolles-theorem&quot;&gt;https://mathoverflow.net/questions/184358/what-did-rolle-prove-when-he-proved-rolles-theorem&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-10-14T09:03:11.417" Title="Selling completeness, extreme value theorem, etc.?" Tags="|calculus|proofs|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5569" PostTypeId="2" ParentId="4436" CreationDate="2014-10-14T02:11:00.823" Score="3" Body="&lt;p&gt;Of course infinite series are useful.  Consider three early examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Euclid (~300 BC) found the geometric series $\sum 1/r^n$ useful, and contemporary applications would have included Archimedes's quadrature of the parabola and for Zeno's paradoxes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Oresme (~1350) found the harmonic series $\sum 1/n$ useful, as an example of a divergent sum.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mercator (1668) found $\sum \pm\, x^n/n$ as a useful series for calculating logarithms $\log(1+x)$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are plenty of historical sources on any of those three topics, though not necessarily connected to each other.  Also I wouldn't focus on the history too much, since contemporary applications to compound interest may be a better hook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, people used infinite series well before there were Taylor series or series of functions in much generality.  The same series, with updated interpretations, still provide good reasons to learn the topic today.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2014-10-14T02:11:00.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5570" PostTypeId="2" ParentId="4158" CreationDate="2014-10-14T06:59:11.337" Score="2" Body="&lt;p&gt;You can organize it as a low-tech version of StackExchange!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Collect questions (anonymously) and present them to the class, letting them upvote the questions they feel are relevant. Then collect answers (anonymously) and present them for evaluation. Students should upvote those answers they feel help them understand better. The reason anonymity is important is to avoid friends helping each other, which certainly will tend to occur if you use the &quot;reputation points&quot; to award bonus credit at the end of the semester.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2014-10-14T06:59:11.337" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5571" PostTypeId="2" ParentId="5568" CreationDate="2014-10-14T08:52:23.613" Score="12" Body="&lt;p&gt;In the preamble to your question you gave several great arguments &lt;em&gt;against&lt;/em&gt; dwelling on foundational issues in a freshman calculus class, and indeed I think it is inappropriate to focus too much on these issues.  In my view, the point of a calculus class is to teach students how to use a particular set of tools to solve problems, and we should give them time to understand how to use the tools before expecting them to appreciate why the tools work.  It's better to teach someone how to ride a bike before teaching them how to build one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, you were asking &lt;em&gt;how&lt;/em&gt;, not &lt;em&gt;if&lt;/em&gt;, you should sell foundational concepts to your students.  My suggestion is to integrate them into your examples as much as possible.  If you go over the intermediate value theorem in one class toward the beginning of the semester and then never revisit it, the theorem will appear as an esoteric curiosity that they can safely dismiss.  But suppose that later on you do an example such as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f(x) = e^x + 5x^2 + 3x$.  Is there a point on the graph of $f$ where the tangent line has slope $2$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This comes down to solving the equation $f'(x) = e^x + 10x + 3 = 2$.  This can't be done algebraically, but you can observe that $f'(0) = 4 &amp;gt; 2$ and $f'(-1) = e^{-1} - 7 &amp;lt; 2$, so indeed $f'(x) = 2$ for some $x$ between $-1$ and $0$ by the intermediate value theorem.  This makes the intermediate value theorem feel more like a technique than just an idle fact.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With a little thought, you can find ways to sneak other theorems into your examples.  For instance, after you've done a bit of integration you can give the following example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find the average value of $f(x) = x \sin x$ on the interval $[-\pi, \pi]$.  Does $f$ attain its average at some point in the interval?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After you find that the average is $1$, you can remark that the equation $x \sin x = 1$ can't be solved algebraically, and a direct application of the intermediate value theorem is no help since $f(-\pi) = f(\pi) = 0$.  But $f$ attains its minimum and maximum values at points $x_{min}$ and $x_{max}$ by the extreme value theorem, and you can observe that&#xA;$$f(x_{min}) \leq \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\, dx \leq f(x_{max})$$&#xA;So by the intermediate value theorem $f$ has to attain its average.  (Notice that this actually sneaks the proof of the mean value theorem for integrals into an example.  The word &quot;proof&quot; shuts off students' brains, but anything you say after the word &quot;example&quot; commands their full attention.)&lt;/p&gt;&#xA;" OwnerUserId="1439" LastEditorUserId="1439" LastEditDate="2014-10-14T09:03:11.417" LastActivityDate="2014-10-14T09:03:11.417" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5572" PostTypeId="2" ParentId="5559" CreationDate="2014-10-14T13:14:13.917" Score="3" Body="&lt;p&gt;Probabilistic models are used with great effect in many deterministic settings, and also in settings where we freely admit that the model is wrong (e.g. because events of small probability are ignored!).  In fact this is the most common scenario!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's easy to miss the point of modelling.  &quot;All models are wrong, but some are useful&quot;.  Logic, math, probability are models.  We study them because they're &lt;strong&gt;useful&lt;/strong&gt;.  The definition of &quot;useful&quot; here is situation-specific and open to debate!  (There are other reasons to study abstract models too, e.g. for fun!)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway I say your question is ill posed, because it isn't really a &quot;mistake&quot; to model a coin flip using a Bernoulli(1/2) random variable.  On the other hand, I think it's good you're thinking about the limitations of your models and not just blindly assuming they're &quot;right&quot;!&lt;/p&gt;&#xA;" OwnerUserId="3226" LastActivityDate="2014-10-14T13:14:13.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5573" PostTypeId="2" ParentId="4510" CreationDate="2014-10-14T17:36:21.433" Score="6" Body="&lt;p&gt;this is Ramon from WIRIS team.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are working on a Javascript version of WIRIS CAS. Still in beta.&#xA;&lt;a href=&quot;http://www.wiris.com/calc/&quot;&gt;www.wiris.com/calc/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3227" LastActivityDate="2014-10-14T17:36:21.433" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5574" PostTypeId="2" ParentId="4376" CreationDate="2014-10-14T19:37:53.573" Score="0" Body="&lt;p&gt;&lt;strong&gt;Real world situation:&lt;/strong&gt; Weekly (or monthly) pocket money.&lt;br&gt;&#xA;&lt;strong&gt;Constant term:&lt;/strong&gt; agreed upon disbursement from parents.&lt;br&gt;&#xA;&lt;strong&gt;Variable/function argument:&lt;/strong&gt; number of nagging episodes per week (or total length thereof, for a truly continuous argument).&lt;br&gt;&#xA;&lt;strong&gt;Argument coefficient&lt;/strong&gt;: extra money per unit of variable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They can even arrive at the concept of non-linear functions this way.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if you want to turn it statistical, you can include an &quot;error term&quot; describing the &quot;visiting grandparents&quot; disturbance. &lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2014-10-14T19:37:53.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5575" PostTypeId="1" CreationDate="2014-10-14T20:53:04.377" Score="7" ViewCount="64" Body="&lt;p&gt;Is there anyone that is experienced in teaching Informal Inferential Reasoning in primary school? Mind to share your experience?&lt;/p&gt;&#xA;" OwnerUserId="3228" LastActivityDate="2014-10-14T20:53:04.377" Title="Statistics for Primary School: Informal Inferential Reasoning" Tags="|primary-education|statistics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5576" PostTypeId="1" CreationDate="2014-10-15T03:08:10.840" Score="3" ViewCount="210" Body="&lt;p&gt;I just read somewhere the question, &quot;What was wrong with the old way?&quot;  That was followed by the statement that &quot;Clearly the old way did not work&quot;. This was, I think, based on years of following what was going on with students and math.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have my own experience from which to draw (my children did not need my help).  I learned what subtraction was in my second grade classroom by placing 10 articles on a table, removing 3 of those 10 articles, and counting the 7 articles remaining on the table.  10 take away 3 equals 7.  I was able to generalize that experience to cover what I was doing every time subtraction was required by a math problem.  I just saw an example of &quot;You paid \$20.00 for a \$3.18 purchase.&quot;  Then it was said that most people go through some mental exercises to find out the change.  What would be wrong with \$20.00 - \$3.18 = \$16.82.  My first job, at age 14, was at S. S. Kresge Co.  I counted back that change with &quot;\$3.18 (handed 2 pennies) \$3.20 (handed 1 nickel) \$3.25 (handed 3 quarters - 1 at a time) \$3.50, \$3.75, \$4.00 (handed a dollar bill) \$5.00 (handed a five dollar bill) \$10.00 (handed a ten dollar bill) \$20.00.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have observed that there are few who can count back change that way these days.  What happened?  If the power fails and the cash register doesn't do the math, they seem to be lost! &lt;/p&gt;&#xA;" OwnerUserId="3231" LastEditorUserId="143" LastEditDate="2014-10-15T16:56:03.780" LastActivityDate="2014-10-15T16:56:03.780" Title="Common Core Subtraction" Tags="|primary-education|common-core|" AnswerCount="1" CommentCount="2" ClosedDate="2014-10-17T13:29:54.867" ContentLicense="CC BY-SA 3.0" />
  <row Id="5577" PostTypeId="2" ParentId="4376" CreationDate="2014-10-15T03:28:36.663" Score="0" Body="&lt;p&gt;I personally tend to avoid examples that make use of money, but not for the reasons outlined by Mr. K (my reasons are more about the kids who don't have money).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How about some simple things like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;how much your glass of water goes up each time you add an ice cube&lt;/li&gt;&#xA;&lt;li&gt;assuming that the public transportation system in your area has stops that are the same time apart (say 5 minutes) and that you start at stop 10, what stop are you on 20 minutes later?  x minutes later?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2014-10-15T03:28:36.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5580" PostTypeId="2" ParentId="5557" CreationDate="2014-10-15T10:37:38.613" Score="7" Body="&lt;p&gt;A few years back I followed a distance master degree in maths at UPMC (Université Pierre et Marie Curie, in Paris). At the time the university sent course documents and exercices by post every week, solutions to exercices had to be sent back, and students had to be physically present for the final exams. Subscription was cheap (a few hundreds euros I think). I don't know to what extent all this has changed since then; see here for the Bachelor degree: &lt;a href=&quot;http://www.telesciences6.upmc.fr/fr/licencemath.html&quot;&gt;http://www.telesciences6.upmc.fr/fr/licencemath.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did not get my master degree in the end. Reasons are: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I was already in the process of getting a master degree in engineering, so pressure was low to get another master degree. And I think a bit of self-pressure is required with any distance solution, because no one but you is there to push you on a daily basis. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Not only pressure but also emulation. Being part of the same boat with other students is important to discuss ideas, problems, wrong solutions, etc. and sometimes to do something else. I was lacking interactions with peers. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some courses such as stochastic processes were entirely new and strange to me, and I had much difficulties to understand them without a teacher. At the same time the temptation was high to spend more time on courses I liked such as groups and representation theory. It means that a lot of self-discipline is required, more than for regular courses where the agenda &quot;forces&quot; you to tackle with all topics. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For maths you need to do exercises to make progress, and you need to experience it to know how much it is true. The few exercises that were asked were maybe not enough, and I did not have the motivation to make additional exercise sessions by myself. The danger is that reading maths (i.e. reading the printed document courses) is something very pleasant I think, but this is not very rewarding if not completed with taking notes, doing exercises, asking questions, etc. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I guess the development of MOOCs nowadays can moderate several of the points mentioned above; I have followed a few MOOC math courses recently and I have been quite enthusiastic about them - but this is certainly another topic by itself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some years later I started a PhD in parallel to my regular professional activity. I worked on my PhD during evenings, week-ends, and vacations. I did it for me, not for carrier purposes. Concretely I was rarely in the lab, I used to see and email my advisor and a few other researchers from time to time, I attended a few conferences only, so this can probably be called a distance PhD degree. I succeeded in this enterprise, but I think the context was fundamentally different from that of the master degree: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;this was research, not education, so I did not really need to learn things by heart or to force myself to work on topics I disliked; I naturally followed the path of maths I felt comfortable with, and this is where I found results: magic! &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;research is not a linear process, so I think your discoveries depend more on a maturing process than on the total time you spend working on it. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Hope this can help. &lt;/p&gt;&#xA;" OwnerUserId="3233" LastActivityDate="2014-10-15T10:37:38.613" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5581" PostTypeId="2" ParentId="5576" CreationDate="2014-10-15T16:21:29.713" Score="3" Body="&lt;p&gt;There are a lot of sources of information on the history of mathematics education reform, and such sources fill many books. What you may be interested in is knowing that, basically, our understanding of human learning has changed over time. To hugely oversimplify, much of learning that older generations are familiar with is based on ideas of learning linked to behaviorist psychology. A focus on the basics of arithmetic and an idea that practice makes perfect is a very old, traditional approach to math education. Since then we've learned that there are a lot of problems with this approach. Among them, they create a curriculum largely devoid of creativity, problem solving, context, reasoning, etc. But also, they don't reflect the understanding of how students come to adopt new ways of mathematical thinking, and how they make connections to their existing body of knowledge. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are political wars over this sort of thing. I'm not sure there's a great single treatment of this issue that I can link to, but this one is not too bad:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://mathematicallysane.com/reform-mathematics-vs-the-basics/&quot; rel=&quot;nofollow&quot;&gt;http://mathematicallysane.com/reform-mathematics-vs-the-basics/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It describes some of the conflict in mathematics education reform. It's very limited. I only provide it to help you understand that nothing actually &lt;em&gt;happened&lt;/em&gt; in the sense I think you mean. But it might be a starting place for you if you're truly interested in understanding how curriculum and instruction in mathematics has had to change. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A better resource is the following, very large, pair of books: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0873534719&quot; rel=&quot;nofollow&quot;&gt;A History of School Mathematics. Volumes 1 &amp;amp; 2&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In reality, a lot of things have happened in the policy world. Part of the difficulty for teachers is that so much policy change (as political winds shift) has left them suspicious of any new policy. How long (they must think to themselves) before a new administration comes in and reverses all these directives? Research has marched forward, but politics does not have that sort of cumulative thrust.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-10-15T16:21:29.713" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5582" PostTypeId="1" CreationDate="2014-10-15T17:05:32.450" Score="3" ViewCount="7235" Body="&lt;p&gt;In my Business Calculus class we cover finding the production level that minimizes average cost. I get what average cost is measuring, but all the texts I've looked at don't really mention why we care to minimize it. Maximum profit relates to marginal cost, so points of minimum average cost are not even necessarily giving maximum profit (unless by coincidence (marginal cost)=(marginal revenue)=(average cost)). In what situations do we look at the problem and see that the answer is where the average cost is minimized?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best I can come up with is this: A nonprofit is making medicine kits that it wants to sell at the lowest price. Since they don't want to make a profit, the price will be average cost. So finding the production level that minimizes average cost will allow them to sell at the lowest price.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems highly contrived. Any other good set-ups/questions that are answered by minimizing the average cost?&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-10-16T02:15:22.813" Title="Application of Minimizing Average Cost" Tags="|calculus|applied-mathematics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5583" PostTypeId="2" ParentId="5582" CreationDate="2014-10-15T21:30:29.390" Score="5" Body="&lt;p&gt;Cost is fairly easy to calculate, so finding the minimum average cost is also easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Profit depends partly on revenue, which is much harder to predict. You can't know just how customers will react to each possible price that you charge. Your product may become a fad, or it may not. So how can you get a formula that gives you revenue, much less profit?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Minimum average cost is not the same as maximum profit, but it probably will not be terribly far from it. You can use it to easily approximate what you really want. The alternative is to get a worse approximation with more effort. At least, all that could happen fairly often.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another consideration: a planner needs to take many things into account. The CEO may want to know both how to get minimum average cost and how to get maximum profit. A good CEO should know the proper weight to give each fact when making his decisions.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2014-10-16T00:51:01.043" LastActivityDate="2014-10-16T00:51:01.043" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5584" PostTypeId="2" ParentId="5582" CreationDate="2014-10-15T22:06:21.183" Score="2" Body="&lt;p&gt;When prices can change quickly while it is difficult to adjust the output,&#xA;then to produce at minimum average cost could be a good idea;&#xA;especially, if  avoiding to make a loss is important.&lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-10-15T22:06:21.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5585" PostTypeId="2" ParentId="5582" CreationDate="2014-10-16T02:15:22.813" Score="1" Body="&lt;p&gt;We are considering Total Cost $(C)$as a function of output, $q$, $C= C(q)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Average Cost is defined as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$AC \equiv \frac {C(q)}{q}, q&amp;gt;0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finding the minimum average cost production level $q^*$requires to set $AC$'s derivative equal to zero&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac {dAC}{dq} = 0 \Rightarrow \frac{C'(q)q-C(q)}{q^2} = 0 \Rightarrow C'(q)q-C(q) = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\Rightarrow q^*: C'(q^*)=\frac{C(q^*)}{q^*}$$&#xA;(and you want the cost function to be convex, $C''&gt;0 in order for this to be a minimum).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But $C'(q^*)$ is the &lt;em&gt;marginal&lt;/em&gt; cost function calculated at $q^*$, so&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ q^*: MC(q^*)=AC(q^*)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, at $q^*$, average and marginal cost coincide.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, we have to remember that in Economic Theory the concept of &quot;profit&quot; has nothing really to do with how the word is understood in every day business, or say, in Accounting. &quot;Profit&quot; is above and beyond some measure of &quot;normal returns to capital&quot;, while Accounting profits, the &quot;bottom line&quot;, have not subtracted from revenues capital returns fully (they have subtracted the returns to foreign-borrowed- capital, but not the returns to own capital). So roughly speaking, Economic Theory sees Accounting profits as &quot;own capital returns&quot; -and will characterize a portion of them as &quot;profits&quot; &lt;em&gt;only if&lt;/em&gt; said portion exceeds the &quot;normal rate of own capital return&quot;, however this is measured.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why this digression? Because in competitive markets the argument is that firms are more or less &quot;price-takers&quot; (they may &quot;set the prices&quot; but they have to eventually adjust them due to competition), meaning that the revenue function is linear in the price, and so marginal revenue does not depend on the quantity produced by a single company. Moreover, competition will drive the price down enough so that &lt;em&gt;economic&lt;/em&gt; profits (as defined above) are extinguished: but this means that total revenue will equal total cost, and so marginal revenue (which equals average revenue here), will equal &lt;em&gt;average&lt;/em&gt; cost. Since the company equates marginal revenue with marginal cost, this leads us to the conclusion/prediction that what we observe is companies producing at a point where all three are equal -and marginal cost is equal to average cost when the firms produce at $q^*$ level.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the practical side of things, producing at minimum average cost, &lt;em&gt;maximizes your chances&lt;/em&gt; to make, after all, a profit, in the &lt;em&gt;economics&lt;/em&gt; sense of the word. Note that the above imply that when doing cost calculations, in &quot;costs&quot; we should include the &quot;opportunity cost of own capital&quot;, which is another way to talk about the &quot;normal&quot; returns to capital.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And yes, when market imperfections, market power, market asymmetries, market externalities, are taken into account, the above picture becomes an approximate benchmark case -and it is then when things get really interesting.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2014-10-16T02:15:22.813" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5586" PostTypeId="1" CreationDate="2014-10-17T12:42:43.707" Score="13" ViewCount="2625" Body="&lt;p&gt;I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation and limits as its the first time I am studying it. Here are the lessons that are required to study for the first semester:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Limit of a function at a point&lt;/li&gt;&#xA;&lt;li&gt;Limits Theorems&lt;/li&gt;&#xA;&lt;li&gt;Limits of fractional functions&lt;/li&gt;&#xA;&lt;li&gt;Limits of Trigonometric functions&lt;/li&gt;&#xA;&lt;li&gt;Limits at Infinity&lt;/li&gt;&#xA;&lt;li&gt;Continuity at a point&lt;/li&gt;&#xA;&lt;li&gt;Continuity on an interval&lt;/li&gt;&#xA;&lt;li&gt;Rate of Change&lt;/li&gt;&#xA;&lt;li&gt;First derivative&lt;/li&gt;&#xA;&lt;li&gt;Continuity and differentiation&lt;/li&gt;&#xA;&lt;li&gt;Differentiation Rules&lt;/li&gt;&#xA;&lt;li&gt;Derivatives of Higher Order&lt;/li&gt;&#xA;&lt;li&gt;The chain rule&lt;/li&gt;&#xA;&lt;li&gt;Implicit differentiation&lt;/li&gt;&#xA;&lt;li&gt;Geometric applications of differentiation&lt;/li&gt;&#xA;&lt;li&gt;Physical applications of differentiation&lt;/li&gt;&#xA;&lt;li&gt;Related Rates&lt;/li&gt;&#xA;&lt;li&gt;Increasing and Decreasing functions&lt;/li&gt;&#xA;&lt;li&gt;Extreme Values.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Limits are relatively easy. However, related rates and extreme values are disgustingly difficult, is there any way to make those two lessons easy and routine? Something like a book filled with questions on those two or something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="3243" LastEditorUserId="-1" LastEditDate="2014-10-17T21:47:25.667" LastActivityDate="2017-07-06T01:15:22.713" Title="How can you be perfect at maths (highschool)?" Tags="|calculus|secondary-education|" AnswerCount="7" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5587" PostTypeId="2" ParentId="5586" CreationDate="2014-10-17T14:10:52.657" Score="8" Body="&lt;p&gt;Welcome to the site!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's great that you're motivated and want to get a perfect grade in your studies! There is no actual formula for getting a perfect grade. One goal among all the goals of your math educators is to deepen your understanding of the mathematics you are studying in class and in whatever books your reading. There are things you can do to address this as well. I will suggest one. Because you seem so motivated, I think this is something that could work for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you haven't already done this, consider forming a study group with other students in your class. It doesn't matter so much whether they are more or less advanced students than you. A mix may actually be good. The aim, primarily, is to gather a few like-minded people who are motivated to think about mathematics together. It's important that you're comfortable talking together about ideas, and especially putting forward suggestions that may not be &quot;correct.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can form such a group (even maybe just 2 or 3 people), have meetings around a number of goals and questions such as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Discuss the ideas in a chapter (or chapter section) you all have agreed to read.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What was challenging about it?&lt;/li&gt;&#xA;&lt;li&gt;What seemed obvious about it?&lt;/li&gt;&#xA;&lt;li&gt;What are you still unclear about?&lt;/li&gt;&#xA;&lt;li&gt;Where might it be useful?&lt;/li&gt;&#xA;&lt;li&gt;Did anything seem especially interesting? Exciting? Pointless? Annoying? If so, why?&lt;/li&gt;&#xA;&lt;li&gt;Do you feel there is a better way to do something presented in the chapter? Why is this other way better? &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Find online sources of information related to things you've already read in the book chapters.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Are there differences in how these concepts are presented?&lt;/li&gt;&#xA;&lt;li&gt;Did anyone find a presentation they found easier to understand? How does that explanation relate to what you read in your textbook?&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Solve problems you've found in other books (or online)&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What different approaches occur to you? &lt;/li&gt;&#xA;&lt;li&gt;What are the relative advantages/disadvantages of various approaches?&lt;/li&gt;&#xA;&lt;li&gt;What is surprising about these problems?&lt;/li&gt;&#xA;&lt;li&gt;Are there problems you just can't solve? Do people get hung up in the same place? Consider asking a teacher if there is some approach that will help you solve this problem. &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Brainstorm applications for the mathematics you're learning. &#xA;&lt;ul&gt;&#xA;&lt;li&gt;What contexts do the textbooks and other sources suggest?&lt;/li&gt;&#xA;&lt;li&gt;Where is this mathematics generally used?&lt;/li&gt;&#xA;&lt;li&gt;Search online to find what applications the math may relate to. What did different people find? Does it intersect with any of your interests?&lt;/li&gt;&#xA;&lt;li&gt;Is the math you're learning part of a trajectory leading to some advanced math you know you will be encountering int he future? &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can come up with your own ways of discussing the mathematics you're learning, I'm sure. Obviously, your group could mostly be about discussing and comparing approaches to solutions, or even working problems collaboratively. The point is that there can be a lot to talk about related to your mathematics learning, so a group like this has very few limits. And these discussions could be very productive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One word of caution: be very mindful of your teacher's rules about homework collaboration. There's nothing wrong with taking on extra problems and working them collaboratively, but your teacher may have good reasons to establish very specific rules on the homework you've been individually assigned. Just be clear on those rules and you'll be fine.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-10-18T19:00:47.497" LastActivityDate="2014-10-18T19:00:47.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5588" PostTypeId="2" ParentId="5586" CreationDate="2014-10-17T16:06:01.220" Score="19" Body="&lt;p&gt;The best way to ensure a good grade is to make sure you &lt;em&gt;deeply understand&lt;/em&gt; the topics you are supposed to learn.&#xA;It is of course important to remember the routine solution methods, but you should also be able to tell intuitively and at a glance &lt;em&gt;why&lt;/em&gt; these methods work and &lt;em&gt;where&lt;/em&gt; any given method is applicable.&#xA;You of course need to remember some key results, but you should also be able to justify those results &amp;mdash; or even better, give a (sketch of a) proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that if you understand the topic well, you can quickly and reliably reconstruct all necessary information.&#xA;If you remember the topic as a whole, it does not matter if you forget some little details.&#xA;I have a PhD in mathematics and I still occasionally forget elementary things, but I can fill the gaps.&#xA;For example, if you remember the differentiation rule of quotients but you are not sure about the signs, test it with some simple functions &amp;mdash; the sign in the general case must be the same as in any example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teachers often focus on telling what is true (differentiation rules, ways to calculate limits), but I strongly recommend learning also what is not true.&#xA;For example, if $\lim_{x\to\infty}f(x)=0$ and $\lim_{x\to\infty}g(x)=\infty$, do we necessarily have $0&amp;lt;\lim_{x\to\infty}f(x)g(x)&amp;lt;\infty$?&#xA;If you are aware of some common &quot;false rules&quot; that are easy to believe, you can recognize when you have made a mistake.&#xA;When solving a problem, try to make sure that you understand what you are doing at all times and test your claims in special cases if you are unsure.&#xA;(The last sentence may sound trivial, but many students seem not to do this.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will make mistakes and you will forget things.&#xA;We all do.&#xA;If you want to make yourself good, try to make yourself robust &amp;mdash; so that if you forget something, you can reconstruct it based on something else, and if you make a mistake, you can recognize it yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far I have answered a question like this: &quot;What kind of a student will almost surely get perfect grades?&quot;&#xA;Another important question is: &quot;How does one become such a student?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For one thing, you should know what you want to become.&#xA;If you really want to understand mathematics well, examine your own skills.&#xA;Ask yourself what are the most important ideas, results and methods in higher order differentiation.&#xA;If you cannot answer with confidence and give a couple of examples demonstrating these ideas, you need to work more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For another thing, do not limit your scope to the present course if possible.&#xA;The big picture you create for yourself shouldn't be only about the course at hand, but mathematics as a whole.&#xA;I would even suggest not trying to remember which course a given topic was covered in and which course you are having at the moment.&#xA;The borders between different courses are somewhat artificial and you don't need to respect them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if you have the extra time, look what is coming ahead: find a follow-up course that builds on your current course and take a look at its book.&#xA;When I was in high school (or the closest equivalent in Finland), other students thought that I didn't have to work at all because I understood quickly and could solve problems quite intuitively.&#xA;The reason was that I was working ahead of them: I had already read the book of the next course, and that gave me plenty of context and motivation for the present topic and I could focus on building a solid big picture.&#xA;I was working hard, but I was working on something different than others.&#xA;It often happens that you properly understand something only when you have applied it in something else; no one masters the last thing they have learned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As JPBurke suggests, working in a group also helps.&#xA;But a group is not strictly necessary if you can't find equally motivated friends or suitable ways to collaborate.&#xA;What you do need is someone to ask from if you don't understand something on your own.&#xA;It can be a fellow student, a teacher, an older sibling or anyone willing to help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realize that this answer gives somewhat grandiose goals.&#xA;A perfect understanding is too much to ask for, but I do suggest putting goals in this direction.&#xA;For me playful interest and idle curiosity in mathematics is what kept and still keeps me going; there is no need to be serious in order to become good.&#xA;The most valuable thing you can have when trying to get good grades is a passion to understand.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2017-07-05T07:23:10.753" LastActivityDate="2017-07-05T07:23:10.753" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5589" PostTypeId="1" AcceptedAnswerId="5607" CreationDate="2014-10-17T17:28:16.620" Score="9" ViewCount="531" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I sometimes tutor high school students and I have come across various problem types that are best represented by the following two problems.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;They are unable to keep track of the correct direction of inequality symbols. That is, if I ask then to write &amp;quot;5 is bigger than 4&amp;quot; they will guess between &lt;span class=&quot;math-container&quot;&gt;$5 &amp;gt;4$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$5&amp;lt;4$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They are unable to set up a division problem. That is, if I ask them to &amp;quot;write 10 divided by 2&amp;quot;, they are tempted to often write &lt;span class=&quot;math-container&quot;&gt;$10\overline{)2}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{10}$&lt;/span&gt;. However, if I ask them &amp;quot;how many times does &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; go into 10,&amp;quot; they are able to provide the correct answer, but they cannot perform long division.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Keeping in mind that I have no professional training as an educator, how can I help students keep these two ideas straight?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Here is what I have tried&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I vaguely recall a math teacher from elementary school telling me that the inequality sign is like the mouth of an alligator and it likes to eat the bigger number. The students have no problem recognizing which number is bigger but for some reason this explanation doesn't help. Since the student doesn't seem to have a problem with &amp;quot;solving&amp;quot; an inequality problem with algebraic manipulation, i.e, a problem like &lt;span class=&quot;math-container&quot;&gt;$x+7&amp;lt;2x$&lt;/span&gt;, their teacher has told them (and the rest of the class) that there will be no &amp;quot;word problems on their tests or quizzes. That is, an inequality will always be set up for them to solve. Should I not worry too much about whether or not a student can properly write an inequality and be satisfied that they are able to perform just the algebraic steps?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For the division problem, their teacher told them to just type it in the calculator. That is, &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; divided by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; translates directly to &lt;span class=&quot;math-container&quot;&gt;$10÷2$&lt;/span&gt; on a calculator. Should I also not worry about this?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Further considerations&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I work at a tutoring center so sometimes the kids will come in with their own homework and sometimes they will work on assignments we give them. The reason why I feel that the students should learn the two concepts above is because their teacher next year (or the year after) may have a different policy for setting up inequalities, etc. Furthermore, the students are often faced with other problems like the two listed so maybe helping the student work through them will help them sort out other ones like it. Note that I only have an hour a week (sometimes 2 hours) with the students.&lt;/p&gt;&#xA;" OwnerUserId="1647" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-12-16T07:02:29.043" Title="Helping high school students remember inequalities and division" Tags="|secondary-education|tutoring|notation|" AnswerCount="6" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5590" PostTypeId="2" ParentId="5589" CreationDate="2014-10-17T18:24:34.813" Score="5" Body="&lt;p&gt;For the larger and smaller: I do not find the aligator explanation intuitive. If this is a mouth it could be just as well the larger eats the smaller, which would give the wrong thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I was told long ago was it is like a beak of a bird, and the larger pecks on the smaller.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in all these stories there is a risk of confusion. In my opinion the symbol is rather intuitive in itself. Where it is wide(r) open there is the larger thing, where it is closer together there is the smaller thing. &#xA;To me it  makes sense, you give more space for the larger thing. &#xA;If you want more plasticity you could say, the equality signed got widened up where the larger thing is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I should perhaps add as a disclaimer that I have little to no experience teaching children that age.) &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-10-17T18:24:34.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5591" PostTypeId="2" ParentId="5586" CreationDate="2014-10-17T19:26:46.673" Score="10" Body="&lt;p&gt;In college I had a friend who was also a Math major, and had been in the Marine Corp. His advice was a little coarse and echoed his military background: &quot;Work problems until you puke, then wipe off the puke and work some more&quot;.  I followed this advice and was an A student in college.  I found that in Calculus in particular, the difficulties I had were due to my weakness in algebra and trigonometry. The equations in Calculus problems will almost always simplify.  Hope this helps!&lt;/p&gt;&#xA;" OwnerUserId="3245" LastActivityDate="2014-10-17T19:26:46.673" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5592" PostTypeId="2" ParentId="36" CreationDate="2014-10-17T20:11:53.193" Score="2" Body="&lt;p&gt;A new player on the scene is called Desmos. It's oddly... pretty. It's also super simple to use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The website is here: &lt;a href=&quot;https://www.desmos.com/&quot; rel=&quot;nofollow&quot;&gt;Desmos&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sliders can be added rather easily. If you're in the calculator and want to do transformations for example, you might type:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$y=sin(nx)$ -- it will ask you if you want to add a slider for n. This is doable for any number of sliders (so, $y = a(x-h)^2+k$ ); you can restrict the values to integers or to certain ranges, etc. Polar graphing is also available. It's easily explorable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the benefit of this is that anyone can use it on any computer, no installing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, the graphs can be imported to other documents or e-mailed easily if a student is trying something and needs to show you their graph when not in your presence! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anyone else has used this and has comments, suggestions, or negative experiences I know that I'd love to hear them! &lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2014-10-17T20:11:53.193" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5593" PostTypeId="2" ParentId="5589" CreationDate="2014-10-17T20:48:21.437" Score="1" Body="&lt;p&gt;the larger side of the greater sign always goes to the larger quantity, so since 5&gt;4, then when you see x&gt;y, x is greater than y.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for division, read the division symbol as &quot;over&quot;, so &quot;ten over 2&quot; &lt;/p&gt;&#xA;" OwnerUserId="2154" LastActivityDate="2014-10-17T20:48:21.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5594" PostTypeId="2" ParentId="5589" CreationDate="2014-10-17T20:49:48.237" Score="1" Body="&lt;p&gt;The direction of the inequality symbols should be intuitive enough.  The big end of the symbol corresponds to the side with the larger quantity.  The small pointy end indicates the smaller side.  Perhaps the mistakes are attributable to dyslexia rather than confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For division, it may help to remind them of more familiar fractions like $\frac{1}{2}$, which is 1 divided by 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The long division setup is less intuitive, though.  I'm not sure that there is a good explanation, other than that that is how the convention is, and it takes practice to learn the algorithm.&lt;/p&gt;&#xA;" OwnerUserId="996" LastActivityDate="2014-10-17T20:49:48.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5595" PostTypeId="2" ParentId="5586" CreationDate="2014-10-17T20:55:47.757" Score="3" Body="&lt;p&gt;The advice that JPBurke and Joonas Ilmavirta have given is excellent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to be perfect, you need to check your work (as explained at the end of this answer).  You also need to know that some problems do not have answers -- and that the correct answer may be to point out why.  Furthermore, real-world problems have limited precision (or accuracy) of the input data.  Understand significant figures, and when &quot;close enough&quot; really is &quot;good enough&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regard to &lt;strong&gt;calculus&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The Fundamental Theorem of Calculus and the Reynolds Transport Theorem are the two most important concepts in calculus.  Yes, they are even more important than limits or the definition of a derivative.  This is because they let you sanity-check your work in real life, even if you only have estimates available.&lt;/li&gt;&#xA;&lt;li&gt;The Fundamental Theorem of Calculus proves that integration and differentiation are inverse processes.  You can use integration to check your derivatives, and &lt;em&gt;vice versa&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The Reynolds Transport Theorem is a detailed version of &quot;what goes in, either stays in, or comes out.&quot;  It is the basis for all Conservation Law problems, such as almost all practical problems in physics and engineering.  Learn this Theorem, and you will have a much easier time in Physics, Quantitative Chemistry, Statics, Mechanics, Electrodynamics, Fluid Mechanics, and Thermodynamics.  This is because almost all of the problems in all of these subjects use the same math -- special cases of the Reynolds Transport Theorem.&lt;/li&gt;&#xA;&lt;li&gt;Try to solve most problems without using a calculator.&lt;/li&gt;&#xA;&lt;li&gt;Follow a good process for solving (and checking) problems, as discussed below.&lt;/li&gt;&#xA;&lt;li&gt;For every problem (that is not a proof), graph the problem (either on paper, or in your head).  Mark on the graph where the function crosses the x-axis, where it crosses the y-axis, where it has minimum(s) and maximum(s), and where it has inflection points.  Know where the slope is positive, zero, or negative.  Doing this will seem like a lot of work at first, but it will give you an intuitive sense for the shape of the graph.&lt;/li&gt;&#xA;&lt;li&gt;You will learn some important things from the graph practice I just mentioned.  The slope of a function is the function's derivative.  If the slope is zero, the graph is either at a minimum, a maximum, or an inflection point.  Where the second derivative is zero, the graph is probably at an inflection point.&lt;/li&gt;&#xA;&lt;li&gt;Pay attention to symmetry.  Is the function even or odd?  (In other words, is it symmetric with respect to x = 0?)  Unshifted even-powered polynomials and unshifted cosines are even; unshifted odd-powered polynomials and unshifted sines are odd.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;When you are doing things with &lt;strong&gt;vector&lt;/strong&gt;s, the following &lt;strong&gt;notation&lt;/strong&gt;s work well:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are allowed to choose your notation, use x, y, and z's &quot;hat notation&quot; to indicate the directions of unit vectors.  For example, &quot;y-hat&quot; is a letter &quot;y&quot; with a circumflex accent, and is written &quot;ŷ&quot;.  This is less confusing than using the i, j, and k &quot;hat notation&quot; (such as &quot;ĵ&quot;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Either use angle bracket notation for vectors (such as &amp;lt;x,y,z&amp;gt;) or explicitly add the vectors (such as x x-hat + y ŷ + z ẑ, and use the proper symbol instead of x-hat).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Unfortunately, you cannot use the hat notation to indicate unit vectors in Quantum Mechanics, because Quantum Mechanics uses hats to indicate &quot;multiplication&quot; by an operator.  For example, &quot;ŷ&quot; in Quantum Mechanics means &quot;multiply by y&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;With regard to &lt;strong&gt;solving word-problems&lt;/strong&gt; in general:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Try to draw a picture.&lt;/li&gt;&#xA;&lt;li&gt;If the problem uses units, keep the units with the associated numbers.  For example, if t = 2 seconds, never abbreviate this as t = 2.&lt;/li&gt;&#xA;&lt;li&gt;Label what you know.  Label where each variable is zero.  Label which direction it is increasing in.&lt;/li&gt;&#xA;&lt;li&gt;Label what you are trying to find.&lt;/li&gt;&#xA;&lt;li&gt;Write out your variables and parameters.  For example, &quot;t = time since rocket was fired.&quot;&lt;/li&gt;&#xA;&lt;li&gt;Write out the relevant formulas.  Count up your knowns and unknowns.  For each unknown you have, you need another independent equation if you are to determine the unknown's value.  Make sure to include both &quot;boundary conditions&quot; and &quot;field conditions&quot;.&lt;/li&gt;&#xA;&lt;li&gt;Systematically solve the equations using algebra and/or calculus.  Make sure that you either do not divide by zero, or that you only divide by zero as part of a limit.  (Derivatives and L'Hopital's rule use limits to perform &quot;valid&quot; divisions by zero.)  Make sure that you bifurcate the problem as necessary.  For example, when you factor a polynomial to find its roots, you will have a separate subproblem for each factor.&lt;/li&gt;&#xA;&lt;li&gt;Only after you have found a formula for your answer should you compute a numerical answer.  Make sure to write out the units as you compute the numerical answer.&lt;/li&gt;&#xA;&lt;li&gt;Check whether each solution is valid.  For example, if you are finding positive solution(s), explicitly rule out any negative solution(s) you find.  Also, check whether the units make sense.  For example, if the answer is supposed to be in meters per second per second, but your answer is in meters per second, then you have probably made a mistake.  If necessary, use unit conversion factors.  (For example, multiply a parameter by (1000 ms / s) if the parameter is in seconds, but the answer needs to be in milliseconds.)&lt;/li&gt;&#xA;&lt;li&gt;Clearly label your answer, and make a note about what the answer means.  For example, &quot;t = 4 s.  The rocket reaches maximum altitude 4 seconds after it is launched.&quot;  I was taught to circle this answer in a cloud.&lt;/li&gt;&#xA;&lt;li&gt;Sanity check your answer.  If the rocket was supposed to go to the moon, 4 seconds seems awfully short.&lt;/li&gt;&#xA;&lt;li&gt;Check-By-Substitution (CBS).  Plug your answer into the original formulas (making sure to keep the units).  Reduce both sides of each equation until you can either confirm that the answer is (a) correct answer, or that it is incorrect.  Mark the initial and intermediate equals-signs with question marks.  If the CBS works, mark the final equals-sign with a check mark.  If the CBS fails, use an inequality sign, and look for a mistake.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2014-12-24T20:13:41.840" LastActivityDate="2014-12-24T20:13:41.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5596" PostTypeId="5" CreationDate="2014-10-18T12:41:34.737" Score="0" Body="&lt;p&gt;In mathematics, a limit is the value that a function or sequence &quot;approaches&quot; as the input or index approaches some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.&lt;/p&gt;&#xA;" OwnerUserId="3246" LastEditorUserId="3246" LastEditDate="2014-10-18T13:33:13.233" LastActivityDate="2014-10-18T13:33:13.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5597" PostTypeId="4" CreationDate="2014-10-18T12:41:34.737" Score="0" Body="For question regarding the properties and evaluation of limits." OwnerUserId="3246" LastEditorUserId="3246" LastEditDate="2014-10-18T13:33:20.783" LastActivityDate="2014-10-18T13:33:20.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5598" PostTypeId="2" ParentId="5586" CreationDate="2014-10-18T14:08:10.593" Score="1" Body="&lt;p&gt;Solve problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's literally the only way to get better in math. Do more problems. Don't do the same problem with different numbers, do harder and harder problems until you are mentally exhausted, take a break, and delve in again.&lt;/p&gt;&#xA;" OwnerUserId="3250" LastActivityDate="2014-10-18T14:08:10.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5599" PostTypeId="1" AcceptedAnswerId="5604" CreationDate="2014-10-18T14:12:58.590" Score="14" ViewCount="356" Body="&lt;p&gt;Once my professor told me that sometimes questions that she gets from her students turn out to be precious inputs for deepening her own mathematical understanding or for doing some new research in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did this happen to some of you too? That is, did questions of students help you to understand some mathematical concept in a better or more profound way, or did they even give you input for your own mathematical research? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If so, please, mention these questions and explain in what way they helped you. &lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="143" LastEditDate="2014-10-20T10:33:22.467" LastActivityDate="2014-10-20T10:33:22.467" Title="What are questions from students that improved the teachers understanding of mathematics?" Tags="|student-question|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5600" PostTypeId="2" ParentId="4482" CreationDate="2014-10-18T15:33:36.997" Score="14" Body="&lt;p&gt;We started here: &lt;a href=&quot;http://aimath.org/textbooks/approved-textbooks/&quot;&gt;http://aimath.org/textbooks/approved-textbooks/&lt;/a&gt; as suggested by David Steinberg. Their selection criteria seem very good, and as a result many good candidates appear at that page, but many have very subpar image quality and would feel like a &quot;budget book&quot; to our students and administration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We've settled on Guichard's Calculus, since it was the only one on the list, in our opinion, that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Has an early transcendentals version&lt;/li&gt;&#xA;&lt;li&gt;Goes all the way through Stokes' Theorem&lt;/li&gt;&#xA;&lt;li&gt;Has a nicely navigable PDF&lt;/li&gt;&#xA;&lt;li&gt;Has good image quality (not obviously scanned from an old book)&lt;/li&gt;&#xA;&lt;li&gt;Has online resources -- WebWork already implemented to some degree&lt;/li&gt;&#xA;&lt;li&gt;Has a clear history of refinement and improvement over time (the MAA review mentions a section of the book that should be there -- the newest edition has that section added!).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;All the other options we looked at had deficiencies in one of the above areas. I am still sad that the Google Group does not seem to be active, and many of the sections in the textbook do not have enough exercises, but for the first year, at least, we plan to have a required purchase of something like a Schaum's Outline at the bookstore to provide supplemental practice problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks all!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-10-18T15:33:36.997" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5601" PostTypeId="2" ParentId="5589" CreationDate="2014-10-18T16:07:41.740" Score="2" Body="&lt;p&gt;[I agree with all of the previous answers on the inequality symbol: Big side, big number/value. It seems intuitive to me. My guess is that those stories are what messed students up in the first place.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have trouble with keeping the symbols straight for division myself. (I have never been diagnosed with dyslexia, but I believe I am mildly dyslexic.) To remind myself of the proper orders of symbols, I always use what I see as the simplest example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6 ÷ 2 = 6/2 (picture this vertically) = 2 goes into 6 (I can't write this symbol) = 3. This grounds me, and I can move on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I'm working with a student, I might start by asking &quot;How many twos in six?&quot; And then ask them how they'd write that as a problem. Most of my students feel sure of the answers to all three problems (6 ÷ 2, 6/2, and 2 goes into 6), and that helps them see the structure. I sounds like yours aren't. Go as basic as you can, and discuss the meanings with them. If a student wrote 2/6 instead of 6/2, I'd ask them to tell me what 2/6 means, and try to get at the fact that its value is less than one.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-10-18T16:07:41.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5602" PostTypeId="2" ParentId="5599" CreationDate="2014-10-18T17:52:14.903" Score="7" Body="&lt;p&gt;I am not a research-level mathematician, but this absolutely happens to me. There is so much in math that we can think more deeply about. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just last week, a student came into my office to get help on a derivative problem that used the quotient 'rule'.  He was getting zero in the numerator. I showed him his mistake, and said &quot;You won't get zero in the numerator.&quot; He said he thought it could happen on some problems. I said &quot;Well, let's prove it. Here's my conjecture: If y = f(x)/g(x), where f(x) does not equal g(x), and neither is a constant, then y' will not equal zero.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Embarrassingly, I was stuck then. I was trying to think of it algebraically, and was getting nowhere. The way to think about it (which I realized while discussing it with another teacher) is conceptually. If y is not a constant, then y' cannot equal zero. That's all there is to it. Is my understanding deeper now? My understanding of derivative was fine, but my approach to proof was too constrained. I hope I've learned something. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pam Sorooshian, an unschooling advocate, has said that, &quot;Once people memorize a technique or a 'fact' they have the feeling that they 'know it' and they stop questioning it or wondering about it.&quot; I think this probably happened to even the best mathematicians among us, when we were young. How much of elementary math do you take for granted? Can you get a deeper understanding of it, and still stay connected to a child's perspective? What bits of math do you 'know' to be true even though you've never personally proved them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For years in my algebra courses, I had students using the Pythagorean theorem. It finally occurred to me that I didn't really know why it was true. (And I only knew &lt;em&gt;that&lt;/em&gt; it was true because of all the history that references it, not because of any logical necessity.) This was not one of the times that a student question provoked me to dig deeper, but it could have been. Ever since then, I always get my students to work through the most visual proof I know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not remembering all of the student questions that have made me think, but I know it has happened repeatedly. It's always a delight to me when I have to say that I don't know and I'll think about it. I never would have imagined that I could keep learning more about basic algebra, just about every time I teach it. Calculus too. And that is part of the joy of teaching for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope others (with better memories) will offer more questions students have asked, that got them thinking more deeply or more carefully.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2014-10-19T19:02:46.873" LastActivityDate="2014-10-19T19:02:46.873" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5603" PostTypeId="2" ParentId="4551" CreationDate="2014-10-18T20:53:22.173" Score="4" Body="&lt;p&gt;I'd like to highly recommend &lt;a href=&quot;https://en.wikipedia.org/wiki/How_to_Solve_It&quot; rel=&quot;nofollow&quot;&gt;George Pólya's &quot;How to Solve It&quot;&lt;/a&gt;. This 1945 book contains great theory and advice on how to approach hard problems in math.&lt;/p&gt;&#xA;" OwnerUserId="688" LastActivityDate="2014-10-18T20:53:22.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5604" PostTypeId="2" ParentId="5599" CreationDate="2014-10-19T11:55:04.903" Score="11" Body="&lt;p&gt;At my university, the usual multivariable calculus class is split into two semesters: the first does geometry in $\mathbb{R}^3$ and then derivatives (gradients, Lagrange multipliers, etc.) while the second focuses on integration (Green's theorem, Stokes' theorem etc.)  In the second semester there are usually about three free weeks or so at the end of the course for supplementary topics, and I decided to give my students a primer on the calculus of variations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two of the examples that I discussed were the isoperimetric inequality (the curve of a given length which encloses the maximal area is a circle) and geodesics on a sphere.  One of my students tried to make a connection between these two examples and asked a good question: can one use the calculus of variations to solve the isoperimetric problem on the sphere?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The standard variational proof of the planar isoperimetric inequality uses Green's theorem, so it is natural to try to use Stokes' theorem on the sphere.  I quickly realized that this can't work, and after a little research learned that one should use the Gauss-Bonnet theorem instead.  This led me into the literature on isoperimetric inequalities on Riemannian manifolds and got me thinking about research questions involving connections between isoperimetric problems and Dirac operators.  I never got any results, but the questions are still in the back of my mind.&lt;/p&gt;&#xA;" OwnerUserId="1439" LastActivityDate="2014-10-19T11:55:04.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5605" PostTypeId="1" CreationDate="2014-10-19T17:42:51.980" Score="8" ViewCount="3462" Body="&lt;p&gt;I am looking for a good linear/matrix algebra textbook, suitable for self-study, that covers topics relevant to statistics and machine learning. I have access to &lt;code&gt;Gentle's &quot;Matrix Algebra&quot;&lt;/code&gt;, but have found it to be too dry and more of a reference book for a practitioner who's already studied the subject before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the books I'm considering are:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Seber, &quot;A Matrix Handbook for Statisticians&quot;&#xA;Searle, &quot;Matrix Algebra Useful for Statistics&quot;&#xA;Harville, &quot;Matrix Algebra from a Statistician's Perspective&quot;&#xA;Gruber, &quot;Matrix Algebra for Linear Models&quot; (just came out)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;and the more generally oriented:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Strang, &quot;An Introduction to Linear Algebra&quot;&#xA;Strang, &quot;Linear Algebra and Its Applications&quot;&#xA;Axler, &quot;Linear Algebra Done Right&quot;&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The trouble is, all these texts have excellent reviews on Amazon, but so did Gentle's text and it doesn't really suit my purposes.&lt;/p&gt;&#xA;" OwnerUserId="660" LastEditorUserId="376" LastEditDate="2019-01-09T14:21:27.300" LastActivityDate="2019-01-09T23:46:47.540" Title="Self-study Linear Algebra textbook for Machine Learning and Statistics" Tags="|statistics|linear-algebra|self-learning|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="5606" PostTypeId="1" AcceptedAnswerId="5609" CreationDate="2014-10-20T11:39:07.747" Score="5" ViewCount="22387" Body="&lt;p&gt;Given that:- (1) a : b = 5 : 24, and (2) b : c = 16 : 7&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In getting the combined ratio a : b : c, there are two approaches:-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Method 1&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1.1 Rewrite (1) as a : b = 5*16 : 24*16 = 80 : 384&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1.2 Rewrite (2) as b : c = 16*24 : 7 * 24 = 384 : 168&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1.3 Then a : b : c = … = 80 : 384 : 168 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1.4 Reduce the finding by the HCF of (80 : 384 : 168) and get 10 : 48 : 21.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Method 2&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2.1 Find the LCM of (24, 16) = 48.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2.2 Rewrite (1) as a : b = 5*(48/24) : 48 = 10 : 48;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2.3 Rewrite (2) as b : c = 48 : 7*(48/16) = 48 : 21.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2.4 The result is a : b : c = 10 : 48 : 21&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;What do you (as a teacher) think by considering more easier to teach? more logical?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the view of a learner, which is more readily acceptable? more simpler to learn (or understand)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other considerations, which is more systematic? more easier to program?&#xA;. . . &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no need to answer this question in a hurry. If you have students of this level, show them both approaches and then ask for their feedback. You probably will be surprised.&lt;/p&gt;&#xA;" OwnerUserId="3258" LastActivityDate="2017-11-25T22:51:09.590" Title="Which method of combining ratios is ‘better’?" Tags="|interactive-teaching|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5607" PostTypeId="2" ParentId="5589" CreationDate="2014-10-20T17:23:00.370" Score="3" Body="&lt;p&gt;This is somewhere between a comment and an answer, but it's too long for a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When my students have similar difficulties, I usually want to know what they're thinking about as they do the problem, because there are (at least) two things that might be going on that call for very different approaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's possible that the students have a heuristic in mind (perhaps the alligator story or something similar) but consistently get it mixed up.  (&quot;Which side eats which one again?&quot;)  In that case the right approach is probably to find a more useful heuristic; finding the right one may depend on the student, and other answers have some good suggestions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the second possibility (and some of your post suggests this to me) is that they aren't getting to the point of really trying to answer the question at all.  They may have been told some rule for remembering the order, but faced with the question &quot;is 5&amp;lt;4 or 5&gt;4?&quot;, they guess based on a gut feeling, and never try to apply the rule.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(At least with my students, I've noticed that once they've decided a problem should be straightforward, they're very reluctant to approach it slowly, even when they're consciously aware that they're struggling with it.  So having decided that they're supposed to automatically know whether 5&amp;lt;4 or 5&gt;4, and realizing that they don't, they feel lost and make a guess.  Backing off---&quot;I don't know this, but I know how to find it out&quot;---isn't something they try.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your question suggests that they might not think it's important for them to get the problems you're talking about, which might contribute: having realized that they don't immediately know the answer, they may feel like it's not important enough to really try to get it right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the second case, if you decide it's worth pressing them to get it right (and on behalf of their future teachers, &lt;em&gt;someone&lt;/em&gt; will one day appreciate your efforts), the issue isn't teaching them a rule, it's teaching them to recognize when they need to stop and consciously apply the rule they do know.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-10-20T17:23:00.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5609" PostTypeId="2" ParentId="5606" CreationDate="2014-10-20T22:55:57.923" Score="4" Body="&lt;p&gt;I've never seen that particular type of problem, but the issue you're referring to comes up similarly in adding fractions, a much more common problem type. Given the problem: Find 5/24 + 7/16, we could ask which is better: (1) multiplying the first fraction, top and bottom, by 16, and the second one, top and bottom, by 24, or (2) Realizing that the LCD is 48, and proceeding to get a common denominator of 48 (instead of the huge and unwieldy 384). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know which students prefer. I would expect to work with them on easier problems where both methods make sense (have fractions with denominators of 4 and 6, work the problem both ways), and discuss the pros and cons with the students. I think I would try to slowly ratchet up the difficulty, hoping to get students to agree that method 2 is more sensible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, many students would argue that their calculator can do this more effectively than we can, whichever method we're trying to use.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-10-20T22:55:57.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5610" PostTypeId="2" ParentId="5606" CreationDate="2014-10-20T23:01:07.397" Score="4" Body="&lt;p&gt;JPBurke is correct that the best way is whatever way &amp;quot;that your student can explain &lt;em&gt;why&lt;/em&gt; it works.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The format of this question may make it difficult for some students to understand.&lt;/p&gt;&#xA;&lt;p&gt;Here are some basic rules that are needed to solve problems like this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You can multiply both sides of a ratio by the same number, and the ratio will still be the same.&lt;/li&gt;&#xA;&lt;li&gt;You can divide both sides of a ratio by the same number, and the ratio will still be the same.&lt;/li&gt;&#xA;&lt;li&gt;Except -- you cannot divide by zero.&lt;/li&gt;&#xA;&lt;li&gt;When in doubt, go step-by-step.&lt;/li&gt;&#xA;&lt;li&gt;You can always check your work (Check-By-Substitution, or CBS).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And here are some follow-up lessons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;All ratios are really fractions in disguise.&lt;/li&gt;&#xA;&lt;li&gt;All fractions are really division problems in disguise.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;When I saw the problem, I immediately thought of a third way to solve it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a : b =  5 : 24, and&lt;/p&gt;&#xA;&lt;p&gt;b : c = 16 : 7&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Reading them out loud, like an old-fashioned SAT analogy problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;a&amp;quot; is to &amp;quot;b&amp;quot; as 5 is to 24.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;b&amp;quot; is to &amp;quot;c&amp;quot; as 16 is to 7.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Why abbreviate the variable names to just one letter?  It is easier to understand a problem if the variables describe things that are understandable.  And if the variables have units, keep the units with the quantities.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For every 5 apples, there are 24 bananas.&lt;/p&gt;&#xA;&lt;p&gt;For every 16 bananas, there are 7 cucumbers.&lt;/p&gt;&#xA;&lt;p&gt;For every 24 bananas, there are how many cucumbers?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Notice that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;24 = 16 * (3/2).&lt;/p&gt;&#xA;&lt;p&gt;For every 16 * (3/2) bananas, there are 7 * (3/2) cucumbers.&lt;/p&gt;&#xA;&lt;p&gt;For every 24 bananas, there are 10 1/2 cucumbers.&lt;/p&gt;&#xA;&lt;p&gt;For every 5 apples, there are 24 bananas, and 10 1/2 cucumbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Multiplying through by 2 gets rid of the fraction:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For every 10 apples, there are 48 bananas, and 21 cucumbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The greatest common denominator of 10, 48, and 21 is 1, so we do not need to simplify any further.  The answer is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a : b : c = 10 : 48 : 21&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;a&amp;quot; is to &amp;quot;b&amp;quot; is to &amp;quot;c&amp;quot; as 10 is to 48 is to 21.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Check-By-Substitution:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is 10 : 48 the same ratio as 5 : 24?&lt;/p&gt;&#xA;&lt;p&gt;Yes.  2 * 5 : 2 * 24 is the same ratio as 5 : 24.&lt;/p&gt;&#xA;&lt;p&gt;Is 48 : 21 the same ratio as 16 : 7?&lt;/p&gt;&#xA;&lt;p&gt;Yes.  3 * 16 : 3 * 7 is the same ratio as 16 : 7.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-10-21T00:27:02.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5611" PostTypeId="1" CreationDate="2014-10-21T09:17:11.630" Score="3" ViewCount="247" Body="&lt;p&gt;Like many teachers, I see the students reach for the calculators to solve problems - now graphics calculators are indeed very useful for modelling in topics such as optimisation.&lt;br&gt;&#xA;However, I notice students also using the calculators to calculate basic times tables parts of their working out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When not using the calculator, many students struggled with these basic skills.  Particularly with the 7 and 13 (and higher) times tables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my question is &lt;strong&gt;how to reinforce mental times table skills within a calculus (optimisation) context?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user3264" LastActivityDate="2014-10-22T21:25:56.217" Title="How to reinforce mental times tables skills in a senior calculus (optimisation) class?" Tags="|calculus|calculators|optimization|" AnswerCount="2" CommentCount="9" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5612" PostTypeId="1" AcceptedAnswerId="6133" CreationDate="2014-10-21T17:15:26.963" Score="10" ViewCount="485" Body="&lt;p&gt;I teach algebra at a community college.  I assign homework which counts for 2 points.  After going over the HW, I give the students a quiz (worth 20 points) with problems similar to the HW.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I spend about 30 minutes at the start of class going over the homework.  I have tried a few different things and I don't really think either of the methods really works great.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Method A--I ask students to work in pairs and talk to one another about the homework.  Meanwhile, I circulate and answer individual questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Method B--I ask students which problems they would like to go over.  I write the problem numbers on the board and then ask for volunteers to write the problem on the board.  I then go one by one talking through the problems, while directing my comments to the person who asked about the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any thoughts on how to make this &quot;going over homework&quot; time better?&lt;/p&gt;&#xA;" OwnerUserId="3265" LastEditorUserId="1147" LastEditDate="2014-12-23T17:57:55.287" LastActivityDate="2018-07-07T18:14:21.753" Title="Effective Method for Reviewing Homework in Class" Tags="|homework|community-colleges|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5613" PostTypeId="2" ParentId="839" CreationDate="2014-10-21T21:15:23.990" Score="1" Body="&lt;p&gt;The other answers are great and I agree with them completely. This one comes with a lot of caveats:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;b&gt;IF&lt;/b&gt; you know the student well and have a good relationship with him or her &lt;b&gt;AND&lt;/b&gt; the student is a pretty self confident person who is generally doing well in the class, you can say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;That was a pretty silly question. Thank you very much for asking it. I&#xA;  really want to hear about the basic things that my students are&#xA;  confused about, no matter how silly. Remember, if you are confused, so&#xA;  are a half dozen of your fellow students. Please ask!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then answer the question.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2014-10-21T21:15:23.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5614" PostTypeId="1" CreationDate="2014-10-22T05:20:17.007" Score="29" ViewCount="28878" Body="&lt;p&gt;Consider a real function $f(x)$ and imagine its graph in the plane. Then the graph of $f(x+2)$ is simply the graph of $f$ shifted to the &lt;em&gt;left&lt;/em&gt; 2 units while the graph of $f(x-2)$ is that of $f$ shifted to the &lt;em&gt;right&lt;/em&gt; 2 units. Also, the graph of $f(2x)$ is the graph of $f$ &lt;em&gt;shrunk towards&lt;/em&gt; the $y$-axis by a factor of $1/2$ and the graph of $f(\frac{1}{2}x)$ is that of $f$ &lt;em&gt;stretched outwards&lt;/em&gt; from the $y$-axis by a factor of $2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These transformations operate in the opposite way than one would initially think insomuchas that the transformations that operate on the whole function (such as $f(x)+2$, $2f(x)$, and $\frac{1}{2}f(x)$) act exactly as one would expect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only after unraveling through the definition of evaluation and comparing points, do we see that transformations applied to the inside of a function operate in this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Presenting graph transformations has been a hurdle for me in teaching algebra, but it's limited to this one point. And the only way I've come up with teaching it is giving a 'opposite-than-you-would-think' rule to keep in mind, but I want something better to tell my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know it's because transformations on the inside are going to require you to perform the inverse operation to get the same point that $f$ was considering, but putting that thought process in words that my student's can understand and grok has been a challenge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I explain these transformations better?&lt;/p&gt;&#xA;" OwnerDisplayName="user1916" LastActivityDate="2023-11-12T13:33:24.497" Title="How does one explain that transformations 'inside' a function operate in the opposite direction than intuition suggests?" Tags="|algebra|" AnswerCount="17" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5615" PostTypeId="2" ParentId="5612" CreationDate="2014-10-22T06:37:46.803" Score="1" Body="&lt;p&gt;For method A, you should not wait for people to ask questions. Glance at what they have done, and if there is a mistake or unclear parts, ask about them. If everything looks okay, you can still ask how does the student know that this or that manipulation is legal, for example. Then ask them to write down their reasoning, not only the calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also combine the methods - have people discuss in pairs or groups first, and then have someone do it on blackboard (with enhanced confidence or understanding from the group work). Or use method A, but when there is a common issue or problem, have someone do the exercise on board or do it by yourself. Students might be more interested when they have first discussed the issue and noticed a problem.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2014-10-22T06:37:46.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5616" PostTypeId="2" ParentId="5614" CreationDate="2014-10-22T08:43:56.670" Score="4" Body="&lt;p&gt;As a mathematician I do not explicitly learn these by heart. Rather, I use a simple function (say, a linear one) and test how it behaves under the transformation. Repeating this often enough leads to memorisation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest teaching the students how to test if their guess or intuition is correct. It is a powerful tool in cases where the student roughly remembers what should happen - the function should be shifted to left or right, say - but is not quite clear on the details.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2014-10-22T08:43:56.670" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5617" PostTypeId="2" ParentId="5614" CreationDate="2014-10-22T10:36:22.123" Score="5" Body="&lt;p&gt;I handle this by giving the form as&#xA;$$y-d=f(x-c)$$&#xA;or, if expansion/contraction is included,&#xA;$$y-d=a \cdot f[b(x-c)]$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then it is more clear that a positive $d$ (which is right next to the $y$) means a movement in the positive $y$ direction and a positive $c$ (which is right next to the $x$) gives movement in the positive $x$ direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I changed to this form recently, but so far this has helped. Putting the complications in the equation form does seem to work better than putting the complications in the interpretation of the form. I do explain to the class why my base equation form is more complicated than the one they see in the textbook, and so far that has been accepted.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2014-10-22T10:36:22.123" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5618" PostTypeId="2" ParentId="5614" CreationDate="2014-10-22T10:58:32.317" Score="24" Body="&lt;p&gt;The way I help students understand this is to flip the idea on its head and say to them that replacing either variable with something &lt;em&gt;always&lt;/em&gt; does the opposite to the graph, whether you replace the x or the y. The full story goes something like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first thing it is worth pointing out to the students is that when you graph a function, there are in fact &lt;em&gt;two&lt;/em&gt; variables involved. You aren't graphing $f(x)$, you are in fact graphing $y=f(x)$. This equation is a rule for telling whether a point is part of the graph or not: you sub the coordinates $(x,y)$ of the point into the equation and if it's true then the point is on the graph and if it's not true then it's not on the graph. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pictorially you can imagine forming the graph by choosing a large number of random points, testing to see if they're on the graph or not, and colouring them in black if they are and white if they're not. This is not the normal way of making graphs, but it does help to point out the usefulness of the equation for telling if a point is on the graph, and it will make the shifting/stretching make more sense too. (Note it also helps to picture the meaning of equations that &lt;em&gt;aren't&lt;/em&gt; functions like $x^2+y^2=1$, and later to get the idea of general sets which are defined by equations, like subspaces.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once this idea has been introduced, you can describe what happens to graphs when you replace $x$ by $x-2$. If you have a point $(a,b)$ and you sub it into the new equation $y=f(x-2)$ and it works, then the point $(a,b)$ is on the new graph. But of course, this is the same as subbing the point $(a-2,b)$ into the old equation $y=f(x)$. So a point is on the new graph when the point two steps back is on the old graph. That is, the old graph is -2 steps from the new graph, which means of course that the new graph is +2 steps from the old graph. That's a key place to point out the backwardsness: the -2 tells you where the old graph is relative to the new graph, not the other way around, which is why it seems backwards.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(You already mentioned this idea &quot;unraveling through the definition of evaluation and comparing points&quot;, but I find it works best in the 2D coordinate algebra rather than the 1D of just the x's.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key point I like to make with students is that this process &lt;em&gt;works just as well if you replace the $y$ with something&lt;/em&gt;. So if you replace the $y$ with $y-2$ then a point $(a,b)$ will be on the new graph of $y-2 = f(x)$ when the point $(a,b-2)$ is on the old graph of $y=f(x)$. So a point is on the new graph when the point two steps down is on the old graph. This means that the new graph is two steps up from the old graph.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So replacing the $x$ or the $y$ tells you where the old graph is relative to the new graph, which means you do the opposite to move the old to the new. There is no difference at all except for which axis. The reason it &lt;em&gt;looks&lt;/em&gt; like there's a different rule for inside/oustide is because the numbers are in the wrong place! The graph with equation $y = f(x)+2$ is actually the graph with equation $y - 2 = f(x)$, which has had $y$ replaced with $y-2$ and so the old graph is two steps down. But of course shifting the 2 to the other side makes it a +. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This particular description helps quite a few of my students, because they can combine it all into &lt;em&gt;one&lt;/em&gt; rule rather than two separate rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you had only a couple of minutes to say something pithy they could hold onto, I'd suggest something like: &quot;The numbers next to the $x$ (or $y$) tell you where the &lt;em&gt;old&lt;/em&gt; graph is relative to the &lt;em&gt;new&lt;/em&gt; one. So it seems backwards when you have the old and you're finding the new. This works just as well with $x$ or the $y$ only on the other axis. Numbers on the outside of the function can be moved to be next to the $y$ and then they work the same way as numbers inside the function.&quot;&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-10-22T10:58:32.317" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5619" PostTypeId="2" ParentId="5611" CreationDate="2014-10-22T13:28:44.323" Score="2" Body="&lt;p&gt;One thing that has been very helpful for me is reinforcing how valuable quick mental arithmetic can be. I recently taught about some great mental math shortcuts for squaring any reasonably sized number ending in 5, and using the difference of squares to quickly multiply any two numbers that are centered around an &quot;easy&quot; square (and now 5s are in that list as well as 10s). Upperclassmen in particular have tended to be fascinated by these &quot;unlocks&quot; - and I use them to illustrate how important the memorization of times tables is (up through 12s, at least, which was always good enough for me). At the Calculus level specifically, it might be worthwhile to assign some basic derivatives with higher exponents, and make them do it in class without the use of a calculator. Even though we normally only run into variables with relatively low exponents (especially at the HS level), this can be a good way to sneak in some times tables drilling without deviating from your curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since my students respect my expertise on the subject, I try to communicate to them the mental techniques I use when doing a problem in my head. Every teacher has their own stable of tricks; have you shared yours with your students? They seem to take notice of that, and when I tell them they shouldn't be using a calculator for a simple problem, they usually agree with me. Once they feel that way, the next step is teaching them how we made it easy for ourselves - so, you'll have to remember the methods you used that worked in your own mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another technique that has worked for me, if you can spend some time on it in class, is to hammer the memorization in as many different ways as you can: Flashcards, timed drilled written problems, any sort of game involving rapid mental calculation, etc. I know it's not technically your job in a calculus course to review elementary ideas, but a single day spent on this can be fun for students if done right, and will end up benefiting them much more than another day of over-relying on the calculator. I use the analogy with my students that the calculator is like a crutch, and you won't build your mind up to be strong enough if you lean too heavily on it.&lt;/p&gt;&#xA;" OwnerUserId="3269" LastActivityDate="2014-10-22T13:28:44.323" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5620" PostTypeId="2" ParentId="5611" CreationDate="2014-10-22T14:31:58.470" Score="4" Body="&lt;p&gt;Memorization of multiplication tables is a skill which used to be valuable, but is much less so nowadays. I do not actually see a valid reason to force students to learn multiplication tables, other than it is a minor convenience to be able to multiply numbers in one's head. Bear in mind there are always a couple of iPhone's sitting around that can do much more than multiply numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students behave rationally, given the environment they live in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why don't you force them to learn the periodic table of elements? Do you think they should know that? Should they know how many years it takes for 100 dollars to double at 1% interest rate? Should they know the speed of light? Memorization is as obsolete as being able to skin a rabbit. If and when the civilization goes to ruins, people will quickly master the old skill, no worries about that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lest I be accused of violating a policy: you cannot force students to learn multiplication tables directly. You &lt;em&gt;may&lt;/em&gt; be able to motivate them though, if you can somehow demostrate that it is advantageous to them to know this stuff. For instance, you could have a set of problems which do not ask for exact answers, but for &lt;em&gt;approximate&lt;/em&gt; answers calculated &lt;em&gt;mentally&lt;/em&gt; (no paper) in a very &lt;em&gt;short&lt;/em&gt; time. You can try arguing that's an advantage in real life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addendum:&lt;/strong&gt; I asked my 9-year old daughter why she thought it was useful to learn the multiplication table (which she did last year). She said right away &lt;em&gt;&quot;because then you can answer when the teachers asks you&quot;&lt;/em&gt;. (I find this answer very sad by the way). I pressed on, I wanted to know where outside school multiplication was useful. She said something like &quot;then you can tell how many coins you need if you need three by six coins&quot;. I think she was just trying to get rid of me. Can your daugher do better? Can you? I can't.&lt;/p&gt;&#xA;" OwnerUserId="165" LastEditorUserId="165" LastEditDate="2014-10-22T21:25:56.217" LastActivityDate="2014-10-22T21:25:56.217" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5621" PostTypeId="2" ParentId="5614" CreationDate="2014-10-22T16:34:15.890" Score="9" Body="&lt;p&gt;In my opinion, there are three main reasons that students are ill equipped to handle function transformations:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;They think about transformations as operations on expressions:  &amp;quot;If I add &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in this &lt;strong&gt;expression&lt;/strong&gt;, then the graph will shift this way&amp;quot;.  In fact &lt;strong&gt;function transformation is about the relationship between two functions&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Relatedly, although two different functions are involved in a transformation, usually only one is named. For example, in your question above, you talk about &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=f(2x)$&lt;/span&gt;.  There is no explicit mention of the function &lt;span class=&quot;math-container&quot;&gt;$g(x)=f(2x)$&lt;/span&gt;.  The function they are trying to graph is never explicitly written as a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, which is a big problem, since before transformations, this is all they have seen.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think the quantifiers are all mixed up in students heads.  When we write &amp;quot;the graph of &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt;&amp;quot; we mean &amp;quot;All the points &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; which satisfy &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt;&amp;quot;, but to the student the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&amp;quot; in &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt; and the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&amp;quot; in &lt;span class=&quot;math-container&quot;&gt;$y=(x+3)^2$&lt;/span&gt; look like &amp;quot;the same &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;When I teach, I usually do so through structured sequences of exercises, with discussion between to cement the &amp;quot;moral of the story&amp;quot; which we are supposed to be learning from the exercises.  So here is my sequence of questions, which I believe avoids these three pitfalls.  I would make sure students do more than one question of each type before moving on.  We discuss each one, and make sure we can verbalize everything.  Also note that I will only discuss horizontal shift type questions below, but at each stage I would be using each type of transformation, not just horizontal shifts.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt;  If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; are functions, and we know that &lt;span class=&quot;math-container&quot;&gt;$(4,5)$&lt;/span&gt; is on the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, and we know &lt;span class=&quot;math-container&quot;&gt;$f(x) = g(x+2)$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, what point can we determine is on the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;? (Note that the universal quantification of the equation is made explicit!)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solution 1:&lt;/strong&gt; Since &lt;span class=&quot;math-container&quot;&gt;$f(x)=g(x+2)$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, it is in particular true for &lt;span class=&quot;math-container&quot;&gt;$x=4$&lt;/span&gt;.  So &lt;span class=&quot;math-container&quot;&gt;$f(4)=g(4+2)$&lt;/span&gt;.  &lt;span class=&quot;math-container&quot;&gt;$(4,5)$&lt;/span&gt; was on the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$f(4)=5$&lt;/span&gt;.  Thus &lt;span class=&quot;math-container&quot;&gt;$5=g(6)$&lt;/span&gt;.  So &lt;span class=&quot;math-container&quot;&gt;$(6,5)$&lt;/span&gt; is on the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;.  &lt;strong&gt;Note that this does exhibit the &amp;quot;expected&amp;quot; behavior&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Moral 1:&lt;/strong&gt; Did you notice that the point on &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; had the same ordinate as the point on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, but its abscissa was &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; greater than the point of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is was related to?  Think about why.  Will this relationship always be true?  Try to verbalize that.  &amp;quot;For each point on the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, there is another point on the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; with the same ordinate, but whose abscissa is two greater&amp;quot;.  Get good at moving from the symbolic expression to the verbal expression.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; If this is the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; (GRAPH), and we know &lt;span class=&quot;math-container&quot;&gt;$f(x)=g(x+2)$&lt;/span&gt;, what does the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; look like?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solution 2:&lt;/strong&gt; Apply insight from first question to determine that &amp;quot;For each point on the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, there is another point on the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; with the same ordinate, but whose abscissa is two greater&amp;quot;.  Use this realization on several points.  Determine that, graphically, each point on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is related to a point on &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; which is &amp;quot;two units to the right&amp;quot;.  Draw the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;.  Get good at thinking through this sort of relationship quickly.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 3:&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; are functions, and we know that &lt;span class=&quot;math-container&quot;&gt;$(4,5)$&lt;/span&gt; is on the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, and we know &lt;span class=&quot;math-container&quot;&gt;$f(x+2) = g(x)$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, what point can we determine is on the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solution 3:&lt;/strong&gt; We only know that &lt;span class=&quot;math-container&quot;&gt;$f(4)=5$&lt;/span&gt;.  So to use the equation relating &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, we will need to choose an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; which results in &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; receiving the input &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;.  So we need &lt;span class=&quot;math-container&quot;&gt;$x+2=4$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt;.  Putting this into the equation, we have &lt;span class=&quot;math-container&quot;&gt;$f(2+2)=g(2)$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$g(2)=5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Moral 3:&lt;/strong&gt; This problem was harder because we had to think carefully about what value &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; needed to be so that we were applying &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to something we know.  Compare this problem to the last kind of problem.  They are basically the same:  we have a relationship between two functions, and a point on one, and we need to find the point on the other.  In this case &amp;quot;For every point on &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; there is a corresponding point on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; whose ordinate is the same, but whose abscissa is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; less&amp;quot;, or equivalently &amp;quot;For every point on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; there is a corresponding point on &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; whose ordinate is the same, but whose abscissa is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; more&amp;quot;.  Either verbalization is cool.  In any case, we have a point on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, and so the corresponding point on &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; must have abscissa &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; less.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 4:&lt;/strong&gt; If we know &lt;span class=&quot;math-container&quot;&gt;$f(x+2) = g(x)$&lt;/span&gt;, and this is the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, what is the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solution 4:&lt;/strong&gt; By the verbal description of the relationship, we see that the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; shifted &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; units to the left.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 5:&lt;/strong&gt;  If &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$g(x) = f(x+3)$&lt;/span&gt;, what is a formula for &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 6:&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x) = (x-2)^3$&lt;/span&gt;, can you write an equation expressing the relationship between &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solution 6:&lt;/strong&gt; At this point, the student can write either &lt;span class=&quot;math-container&quot;&gt;$g(x)=f(x-2)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$g(x+2)=f(x)$&lt;/span&gt;, although the first form is more common. If both forms come up, talk about why they are both valid, and both express the same idea.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 7:&lt;/strong&gt; You know the graph of &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^3$&lt;/span&gt;.  Now graph &lt;span class=&quot;math-container&quot;&gt;$g(x)=(x-2)^3$&lt;/span&gt; by first finding an equation expressing the relationship between &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, and then using that relationship to relate points on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to points on &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For completeness, here is what I would expect as a solution to the&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt; &amp;quot;Graph &lt;span class=&quot;math-container&quot;&gt;$f(x) = (x-3)^2+1$&lt;/span&gt; by finding a sequence of related functions&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Ideal student response&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Plan of attack:  I know the graph of &lt;span class=&quot;math-container&quot;&gt;$g_0(x) = x^2$&lt;/span&gt;.  I will relate &lt;span class=&quot;math-container&quot;&gt;$g_0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$g_1(x) = (x-3)^2$&lt;/span&gt;, and then relate &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The relationship between &lt;span class=&quot;math-container&quot;&gt;$g_0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$g_1(x) = g_0(x-3)$&lt;/span&gt;.  This says that for each point on the graph of &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; there is another point on the graph of &lt;span class=&quot;math-container&quot;&gt;$g_0$&lt;/span&gt; whose ordinate is the same, but whose abscissa is &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; less.  So the graph of &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; is the graph of &lt;span class=&quot;math-container&quot;&gt;$g_0$&lt;/span&gt; shifted &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; units to the right.&lt;/p&gt;&#xA;&lt;p&gt;[Illustration with graph of &lt;span class=&quot;math-container&quot;&gt;$g_0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt;, with a few key points and their relationship shown.  A generic &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x-3$&lt;/span&gt;&amp;quot; are marked to show that &lt;span class=&quot;math-container&quot;&gt;$g_1(x) = g_0(x-3)$&lt;/span&gt;]&lt;/p&gt;&#xA;&lt;p&gt;We also know &lt;span class=&quot;math-container&quot;&gt;$f(x) = g_1(x)+1$&lt;/span&gt;.  So to each point on &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; there is a point of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; with the same abscissa, but with ordinate one greater.  So the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is the graph of &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; shifted up by one.&lt;/p&gt;&#xA;&lt;p&gt;[Illustration with graph of &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, with a generic &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; marked to show &lt;span class=&quot;math-container&quot;&gt;$f(x)=g_1(x)+1$&lt;/span&gt;].&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I will note that I have only tried this approach as a tutor, not as a teacher.  I have probably tutored &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; students this way, and everyone has become a master of function graphing.  Over time, much of this scaffolding can drop away, and the student can start graphing &amp;quot;on sight&amp;quot;, but this is appropriate &lt;a href=&quot;http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;post rigorous&lt;/a&gt; thinking, not mere procedural memorization.  This is also the most powerful way I know to instill students with the knowledge that &amp;quot;Functions are gadgets which take inputs and return outputs, they are not just expressions&amp;quot;.  Internalizing this seems to be essential to this approach, which is an added &amp;quot;side benefit&amp;quot; (or the main point depending on your perspective).&lt;/p&gt;&#xA;&lt;p&gt;I have almost always paired these lessons with completing the square.  Having first learned transformations, now graphing any quadratic of the form &lt;span class=&quot;math-container&quot;&gt;$f(x) = a(x-h)^2+k$&lt;/span&gt; is easy.  Completing the square allows us to put quadratics in that form, so we can graph arbitrary quadratics without resorting to plotting points!  This is powerful, and I think appreciated by students.  Coincidentally it also makes solving quadratic equations much easier...&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2023-11-02T10:20:59.903" LastActivityDate="2023-11-02T10:20:59.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5622" PostTypeId="2" ParentId="4376" CreationDate="2014-10-22T17:39:59.027" Score="1" Body="&lt;p&gt;Patterning tasks are a great way to introduce linear equations. Something like the Pool Border Task (&lt;a href=&quot;http://www.lausd.net/lausd/offices/iss/Math/HS/AlgebraConcept/SURROUND_THE_POOL.pdf&quot; rel=&quot;nofollow&quot;&gt;here's a sample lesson plan&lt;/a&gt;; I use a stripped-down version) builds a lot of student buy-in for small group work, and naturally leads students to discover a linear relationship for themselves. I've had a good deal of success with this task and the discussion afterward, even for students who typically have a low output. You can easily invent your own version of such a task, but I find this one is simple enough and yet rich enough to make for a good day in the classroom.&lt;/p&gt;&#xA;" OwnerUserId="3269" LastActivityDate="2014-10-22T17:39:59.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5623" PostTypeId="2" ParentId="5614" CreationDate="2014-10-22T19:47:58.143" Score="9" Body="&lt;p&gt;I think a key tool here is &lt;strong&gt;tables of values&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My approach has always been to give students functions like &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)=f(x-1)$&lt;/span&gt; and ask them to fill out tables of values for some range of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, say &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; from -4 to 4 in steps of 1. When you step back and look at the two tables, the abstraction starts to lift.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next present students with a graph something like the one below. Say it is the graph of &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt;, and ask them to make tables of values for &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=f(x-1)$&lt;/span&gt; using the coordinates of the corners. Now they have to figure out the implied domain for &lt;span class=&quot;math-container&quot;&gt;$y=f(x-1)$&lt;/span&gt;. Finally have students graph &lt;span class=&quot;math-container&quot;&gt;$y=f(x-1)$&lt;/span&gt; using their table of value. When they get this done, the abstraction and mystery of the horizontal shifting should go away. Of course it will come back later, but that's another battle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Vfnlf.jpg&quot; alt=&quot;sample graph&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="6053" LastEditDate="2019-04-01T11:46:41.200" LastActivityDate="2019-04-01T11:46:41.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5624" PostTypeId="2" ParentId="5614" CreationDate="2014-10-23T01:40:07.810" Score="4" Body="&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/0sIER.jpg&quot; alt=&quot;FncComp&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To obtain the $x$ used in $g(x)$, you need to subtract $2$ from the&#xA;argument to $f(x+2)$. &lt;/li&gt;&#xA;&lt;li&gt;To obtain the $x$ used in $g(x)$, you need to&#xA;multiply the argument to $f(2x)$ by $\frac{1}{2}$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(Credit to Steven Gubkin for the main idea.)&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-10-23T01:40:07.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5625" PostTypeId="2" ParentId="5612" CreationDate="2014-10-23T02:45:26.823" Score="0" Body="&lt;p&gt;When reviewing homework, if you gather a list of problems from students and then do them by popularity (simple raise of hands vote). This should help with getting to the problems that the most students have issue with and keep the majority's attention. Additionally, if you get down with a few problems with only one or two votes, then that might be an indication that you can move on with the lecture. I like to present the problems myself instead of having them come to the board as that seems like grade-school type of attitude. I also tend to go quickly through the jist of the problem or ask the class which part of the problem they are having issue with. For all you know, they can get everything up to the end where they are just messing up punching in numbers to a calculator. They are adults and should be responsible enough to keep up with this type of presentation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can then swing the lecture a bit by kind-of flipping the classroom. Perhaps instead of having them present the homework, work on review handouts in class and have them present the solutions to those. This de-emphasizes the &quot;I must have the correct answer for a solution to get a good grade&quot; drive that the homework has and encourages a bit more of a creative and build a more intuitive approach to the problems for them. In the long run, this might help mitigate the homework review in the beginning.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-10-23T02:45:26.823" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5626" PostTypeId="1" CreationDate="2014-10-23T04:53:14.570" Score="7" ViewCount="10301" Body="&lt;p&gt;The typical explanation for modular arithmetic is calling it by another name, &quot;clock-arithmetic&quot;, and comparing it to the way the hour value of clocks &quot;resets&quot; every time it has passed midnight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is &lt;strong&gt;great&lt;/strong&gt; illustration in Europe. However, in America, notation for time presents the following problems:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The numerical hour resets after 12 hours, rather than after a full day&lt;/li&gt;&#xA;&lt;li&gt;You have to deal with AM/PM&lt;/li&gt;&#xA;&lt;li&gt;Time goes from 1 to 12, rather than 0 to 23&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For these reasons, I fear using the same illustration with American students would cause more confusion than clarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a simple illustration that would show modular arithmetic, that would be more familiar to American students?&lt;/p&gt;&#xA;" OwnerUserId="1266" LastActivityDate="2019-03-28T19:47:18.517" Title="What real-example of modulo-arithmetic would work for American students?" Tags="|examples|mathematics-in-daily-life|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5627" PostTypeId="2" ParentId="5626" CreationDate="2014-10-23T05:02:56.860" Score="3" Body="&lt;p&gt;How about the mechanical &lt;a href=&quot;http://en.wikipedia.org/wiki/Odometer&quot; rel=&quot;nofollow&quot;&gt;odometer&lt;/a&gt; of a car (or any other mechanical measuring device that exhibits rollover---restarting to zero after reaching the maximum reading)?&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2014-10-23T05:02:56.860" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5629" PostTypeId="2" ParentId="5614" CreationDate="2014-10-23T09:04:07.520" Score="0" Body="&lt;p&gt;This is a complicated question.  It's complicated because we do not define our axes very well, yet we assume the following: $+x$ goes to the right and $+y$ goes up, therefore $-x$ goes to the left and $-y$ goes down.  There is &lt;em&gt;absolutely no&lt;/em&gt; rational for this other than convention (and then you can argue about how people think--although I might argue that left-handed people are wired to think &quot;to the left&quot; is &quot;positive&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to talk about directions in a graph, then you better define your axes &lt;em&gt;very well&lt;/em&gt; (and no teacher that I've ever met ever does--mainly because it will likely confuse students--but a good teacher &lt;em&gt;should&lt;/em&gt; confuse their students, i.e. make them think).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that in mind, we can come up with &lt;em&gt;two&lt;/em&gt; different scenarios where $f(x+\Delta x)$ shifts either to the right &lt;em&gt;or&lt;/em&gt; to the left!  Likewise we can come up with axes which show that $f(x) + \Delta f$ shifts either up &lt;em&gt;or&lt;/em&gt; down!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1 Assume &quot;to the right&quot; is positive&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then how do we shift $f(x + \Delta x)$?  We initially get the graph $f(x)$, what &lt;em&gt;value&lt;/em&gt; of $x$ gives $x + \Delta x = 0$ (the original $f(x)$)?  We can solve: $x + \Delta x = 0 \rightarrow x = -\Delta x$.  This suggests that we should move the $x = 0$ point of $f(x)$ to the $x = -\Delta x$ point, which is to the left, since positive is to the right, negative must mean to the left!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2 Assume &quot;to the left&quot; is positive&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then how do we shift $f(x + \Delta x)$?  We initially get the graph $f(x)$, what &lt;em&gt;value&lt;/em&gt; of $x$ gives $x + \Delta x = 0$ (the original $f(x)$)?  We can solve: $x + \Delta x = 0 \rightarrow x = -\Delta x$.  This suggests that we should move the $x = 0$ point of $f(x)$ to the $x = -\Delta x$ point, which is to the right, since positive is to the left, negative must mean to the right!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I copy and pasted my response to &lt;em&gt;both&lt;/em&gt; of the above--that's because &quot;to the right&quot; or &quot;to the left&quot; are arbitrary choices that can be switched by a simple exchange of variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an illustration:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/EqR64.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="2204" LastActivityDate="2014-10-23T09:04:07.520" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5630" PostTypeId="1" AcceptedAnswerId="8266" CreationDate="2014-10-23T11:38:52.967" Score="11" ViewCount="483" Body="&lt;p&gt;Recently I found &lt;a href=&quot;https://photomath.net/&quot; rel=&quot;noreferrer&quot;&gt;this device&lt;/a&gt; advertised which uses any mobile phone's built-in camera to solve equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the site -&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;PhotoMath reads and solves mathematical expressions by using the&#xA;  camera of your mobile device in real time. It makes math easy and&#xA;  simple by educating users how to solve math problems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/4yr4z.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What impact will it have on how Algebra will be taught in the future? In particular will maintaining skills in solving equations become a thing of the past?&lt;/p&gt;&#xA;" OwnerUserId="216" LastEditorUserId="64" LastEditDate="2014-10-26T19:50:19.377" LastActivityDate="2015-06-12T18:59:05.013" Title="What is the impact of such a device on math education?" Tags="|technology-in-education|algebra|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5631" PostTypeId="2" ParentId="5626" CreationDate="2014-10-23T12:55:30.330" Score="4" Body="&lt;p&gt;I like the application of angle-measure. You have a few different systems to choose. You can discuss radians mod $2\pi$ or degrees mod $360^o$. But, I suppose you could do some silly thing and invent pizza slice angles where $8$ slices gives you a full circle. So, you can either go $7$ slices or $-1$ slices to get to the same cut. These would be equal mod $8$ slices since $7-(-1)=8 \equiv 0$ (mod 8 slice).&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-10-23T12:55:30.330" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5632" PostTypeId="1" AcceptedAnswerId="5682" CreationDate="2014-10-23T15:29:59.197" Score="6" ViewCount="420" Body="&lt;p&gt;The result $$\zeta(2) = \frac{\pi^2}{6},$$ tends to amaze young students because of its beauty. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, although in literature there are many proofs of this result, I find that none is suitable for someone who didn't take some advanced classes in calculus. Moreover, &lt;em&gt;as far as I know&lt;/em&gt;, they do not offer an &lt;em&gt;intuitive&lt;/em&gt; explanation of why this result &lt;em&gt;should&lt;/em&gt; be true. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the key intuition -- that is, the picture --  behind the result? Are there any &lt;em&gt;visual&lt;/em&gt; of anyway &lt;em&gt;intuitive&lt;/em&gt; proofs (*) of the statement that can be used to convey a &lt;em&gt;better&lt;/em&gt; (or, at least, different) understanding of it (to people who haven't taken advanced calculus yet)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(*) Note that &lt;em&gt;full&lt;/em&gt; rigour is not compulsory for the scope of this question (it is very appreciated by me personally, though).&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="143" LastEditDate="2014-10-28T17:38:33.097" LastActivityDate="2015-08-03T18:54:16.323" Title="Intuition behind $\zeta(2) = \frac{\pi^2}{6}$" Tags="|mathematical-pedagogy|proofs|intuition|number-theory|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5633" PostTypeId="1" CreationDate="2014-10-23T18:20:52.287" Score="10" ViewCount="314" Body="&lt;p&gt;How the distinction between problems (find/describe such values of &lt;em&gt;x&lt;/em&gt; that… ) and universal truths (identities) is taught to secondary-school students and higher?&#xA;Especially in English-speaking countries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ī mean the following thing (sorry if Ī present an obvious stuff in a confused way): if we have a relation, equality or else, and a proposition with this relation at the top and some variable(s), then we have two distinct useful cases, although notation might be the same. In the first case the proposition &lt;em&gt;is not necessarily true&lt;/em&gt; and we have to find values of the variable where it is true. In the second case the proposition &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1056/standard-word-for-a-formula-that-is-always-true&quot;&gt;is necessarily true (law/identity/“theorem”)&lt;/a&gt; and variables become bound by universal quantification. Universal truths can be used in subsequent equivalent transformations of expressions, equations, inequalities, or something alike. Students &lt;em&gt;must learn to distinguish these cases,&lt;/em&gt; haven’t they? &lt;strong&gt;Examples:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Binary &lt;strong&gt;relations:&lt;/strong&gt;         “=”  (equals)     “&gt;” (greater than)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;   &lt;strong&gt;Problems:&lt;/strong&gt;   “$x^2 - x - 1 = 0$” (&lt;strong&gt;equation&lt;/strong&gt;)   “$x^2 - x - 1 &amp;gt; 0$”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Universal &lt;strong&gt;truths:&lt;/strong&gt;   “$(a+b)(a-b) = a^2 - b^2$”    “$\exp(x) &amp;gt; 0$”&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&#xA;How may be used: “α² = δ² ⇒ α = δ ∨ α = −δ ”      exp(α)⋅ℓ &gt; 0 ⇔ ℓ &gt; 0&#xA;          (transformed equation)       (transformed inequality (problem))&#xA;          “(α + 1)(α − 1) = α² − 1 ”       cosh α &gt; 0&#xA;          (transformed expression)      (derived inequality (truth))&#xA;&lt;/pre&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Why am Ī preoccupied with it? A year ago, the only person happened to &lt;a href=&quot;https://en.wikipedia.org/wiki/Talk:Equation#Identities&quot; rel=&quot;nofollow noreferrer&quot;&gt;cooperate with me about precise equation–identity distinction in English Wikipedia&lt;/a&gt; was a mathematician from France; was a coincidence he wasn’t taught in English? Today Ī argued on this topic (against two presumedly native English speakers) at a neighbouring site and become frustrated. Could an English speaker be correct in insisting that something like &lt;em&gt;a&lt;/em&gt;&lt;sup&gt;&lt;em&gt;n&lt;/em&gt;+1&lt;/sup&gt; = &lt;em&gt;a&lt;/em&gt; × &lt;em&gt;a&lt;/em&gt;&lt;sup&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sup&gt; is an “&lt;strong&gt;equation&lt;/strong&gt;” and calling the “=” symbol in it “equation sign”?&lt;/p&gt;&#xA;" OwnerUserId="3272" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-10-30T08:10:10.377" Title="Distinction between problems (such as equations), and universal truths" Tags="|secondary-education|algebra|logic|language-use|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5634" PostTypeId="2" ParentId="5633" CreationDate="2014-10-23T19:13:25.433" Score="6" Body="&lt;p&gt;I am comfortable saying &quot;Solve the equation $x+2$=4&quot; and also saying &quot;Using the equation $(a+b)(a-b)=a^2-b^2$, we see that...&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On other other hand I would only ever speak of solving an equation, not an identity, and I would be willing to use the word &quot;identity&quot; in my second example above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think I would say that an identity is a kind of equation, one which is universally quantified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, one can avoid confusion jut by saying what you mean, as in &quot;Solve the equation $x+2=4$&quot;, or &quot;For all numbers $a$ and $b$, $(a+b)(a-b)=a^2-b^2$&quot;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-10-23T19:13:25.433" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="5635" PostTypeId="1" AcceptedAnswerId="5718" CreationDate="2014-10-23T21:23:28.230" Score="20" ViewCount="2853" Body="&lt;p&gt;I am a student and I would love to become a research mathematician one day.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So I would like to ask you---experts in mathematics but also in&#xA;education---what are some influential (&lt;span class=&quot;math-container&quot;&gt;$\star$&lt;/span&gt;) books that  I (and&#xA;everyone in my situation) should (&lt;span class=&quot;math-container&quot;&gt;$\star \star$&lt;/span&gt;) read.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\bigstar$&lt;/span&gt; By that, I mean: &lt;em&gt;&lt;strong&gt;(1)&lt;/strong&gt;&lt;/em&gt; books that you think may help to develop positively the way one thinks about mathematics (in general&#xA;or some areas); &lt;em&gt;&lt;strong&gt;(2)&lt;/strong&gt;&lt;/em&gt; books that help developing intellectual and&#xA;mathematical maturity; &lt;em&gt;&lt;strong&gt;(3)&lt;/strong&gt;&lt;/em&gt; books that are in any way&#xA;inspirational; &lt;em&gt;&lt;strong&gt;(4)&lt;/strong&gt;&lt;/em&gt; books that are useful because they offer&#xA;&lt;em&gt;(generally) unknown&lt;/em&gt; way to look at things in maths.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; These books do not have to be strictly mathematical (that is, textbooks).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\bigstar \bigstar$&lt;/span&gt; I know that question is rather subjective and quite broad, but I &lt;em&gt;do&lt;/em&gt; think that it may be helpful for this site and I honestly&#xA;hope that you will share your experiences, which may be extremely&#xA;precious for many people. I welcome any suggestion to edit, improve&#xA;(possibly retag) the question.&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2023-09-13T17:49:18.290" Title="Books that every aspirant mathematician should read" Tags="|undergraduate-education|mathematical-pedagogy|general-pedagogy|reference-books|" AnswerCount="11" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5636" PostTypeId="1" AcceptedAnswerId="5642" CreationDate="2014-10-23T21:40:43.983" Score="5" ViewCount="508" Body="&lt;p&gt;Are there any books that are substantially based on a geometric approach to explain topics in number theory (elementary and more advanced)? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If so, is such approach -- judging from your teaching (or studying) experience -- fruitful and should it be encouraged in students?&lt;/p&gt;&#xA;" OwnerUserId="3251" LastActivityDate="2020-10-22T08:38:42.997" Title="Teaching number theory: geometric approach" Tags="|undergraduate-education|mathematical-pedagogy|number-theory|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5637" PostTypeId="2" ParentId="4551" CreationDate="2014-10-24T00:30:30.423" Score="1" Body="&lt;p&gt;&quot;The problems themselves are challenging for myself.&quot; That's not a bad situation to be in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Work through the problems and figure out why they are challenging to you, Then teach the students what you have learned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chinese students are well trained and disciplined by their parents. They will pretty much follow where you, their teacher, leads them. Pretend that you are training for one of these contests, then pass on your knowledge to the students. Use the materials you feel would be most helpful to you.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2014-12-12T20:02:33.120" LastActivityDate="2014-12-12T20:02:33.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5638" PostTypeId="2" ParentId="5614" CreationDate="2014-10-24T07:49:19.883" Score="8" Body="&lt;p&gt;&lt;em&gt;[Disclaimer: not field tested nor backed up by research.]&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How about teaching the students that any transformations inside the function is changing the domain? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine $\mathbb{R}$ being an infinitely long rubber band. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The difference between $f(x)$ and $f(x-2)$ is that we &lt;strong&gt;hold the graph of the function fixed&lt;/strong&gt; while we &lt;strong&gt;slide the $x$ axis as a whole two units to the left&lt;/strong&gt;. &lt;/li&gt;&#xA;&lt;li&gt;The difference between $f(x)$ and $f(2x)$ is that we &lt;strong&gt;hold the graph of the function fixed&lt;/strong&gt; while we &lt;strong&gt;stretch the $x$ axis by a factor of 2&lt;/strong&gt;. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The picture in mind can be &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Hold graph steady&lt;/li&gt;&#xA;&lt;li&gt;Change the axis by translating or stretching.&lt;/li&gt;&#xA;&lt;li&gt;Put graph back down over the axis. &lt;/li&gt;&#xA;&lt;li&gt;Let go of the axis, now with a &quot;boing&quot; it bounces back to its original place carrying the graph with it. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you are good with computer animations this can be pretty easily illustrated. If not you can try to actually get a translucent sheet of rubber and some student to help you hold it down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do it this way, be sure to emphasize the semantic difference between $2f(x)$ and $f(2x)$. In the first case the 2 is acting on the function. In the second case the 2 is acting on the domain. It will help if you introduce a third situation for the graph as $\{ y = f(x)\}$ versus $\{2y = f(x)\}$. In this case you have that the 2 acting on the codomain and so you stretch the $y$-axis (in exactly the same way as the changes to the $x$ axis). Whereas when the 2 acts on the function you change the function value, not the axis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(All these are somewhat similar to, but also different from, the distinction between &lt;a href=&quot;http://en.wikipedia.org/wiki/Active_and_passive_transformation&quot; rel=&quot;noreferrer&quot;&gt;active and passive transformations&lt;/a&gt;. )&lt;/p&gt;&#xA;" OwnerUserId="125" LastActivityDate="2014-10-24T07:49:19.883" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5639" PostTypeId="2" ParentId="5614" CreationDate="2014-10-24T12:22:15.863" Score="7" Body="&lt;p&gt;How about starting with a number line?&lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$X=0$&lt;/span&gt; is just the point &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(X-2)=0$&lt;/span&gt; solves to &lt;span class=&quot;math-container&quot;&gt;$X=2$&lt;/span&gt;, i.e. the point &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; on the line. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;so&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$X-A$&lt;/span&gt; is a positive shift of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; units on the line. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="6053" LastEditDate="2019-04-01T09:34:07.617" LastActivityDate="2019-04-01T09:34:07.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5640" PostTypeId="2" ParentId="5636" CreationDate="2014-10-24T13:38:42.967" Score="4" Body="&lt;p&gt;I have not taught number theory, but I do keep an eye out for books with a geometric or visual approach. One that I've come across is Allen Hatcher's draft text &lt;em&gt;Topology of Numbers&lt;/em&gt;, available from &lt;a href=&quot;http://www.math.cornell.edu/%7Ehatcher/TN/TNpage.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.cornell.edu/~hatcher/TN/TNpage.html&lt;/a&gt;. Something to note is that by &lt;em&gt;topology&lt;/em&gt;, the author means &amp;quot;the spatial arrangement and interlinking of the components of a system&amp;quot; in this book. Perhaps it's what you're looking for.&lt;/p&gt;&#xA;&lt;p&gt;(Allen Hatcher is also the author of &lt;a href=&quot;http://www.math.cornell.edu/%7Ehatcher/AT/ATpage.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebraic Topology&lt;/a&gt;, in which &lt;em&gt;topology&lt;/em&gt; is used in its traditional mathematical sense.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2020-09-19T16:31:14.290" LastActivityDate="2020-09-19T16:31:14.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5641" PostTypeId="2" ParentId="5635" CreationDate="2014-10-24T19:10:17.507" Score="8" Body="&lt;p&gt;Two very good books, inspiring autobiographies:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;strong&gt;&quot;A mathematician grappling with his century&quot;&lt;/strong&gt;, by &lt;em&gt;Laurent Schwartz&lt;/em&gt;. Another great french mathematician. This book is even better, in my opinion than the one of André Weil, &quot;The Apprenticeship of a Mathematician&quot;, which is already very interesting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;&quot;I Want to Be a Mathematician: An Automathography&quot;&lt;/strong&gt;, by &lt;em&gt;Paul R. Halmos&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3239" LastEditorUserId="3239" LastEditDate="2014-12-12T03:56:02.013" LastActivityDate="2014-12-12T03:56:02.013" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5642" PostTypeId="2" ParentId="5636" CreationDate="2014-10-25T01:15:56.040" Score="2" Body="&lt;p&gt;This is likely not exactly what you seek, but it might be interesting nonetheless:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;The Geometry of Numbers&lt;/em&gt;, C. D. Olds, Anneli Lax, Giuliana P. Davidoff, 2000.&#xA;  (&lt;em&gt;&lt;a href=&quot;http://www.maa.org/publications/ebooks/the-geometry-of-numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMS link&lt;/a&gt;&lt;/em&gt;).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;This is a self-contained introduction to the geometry of numbers, beginning with easily understood questions about lattice points on lines, circles and inside simple polygons in the plane. A minimum of mathematical expertise is required beyond an acquaintance with elementary geometry. The authors gradually lead up to the theorems of Minkowski and others who succeeded him. On the way the reader will see how this powerful approach gives improved approximations to irrational numbers by rationals, simplifies arguments on ways of representing integers as sums of squares, and provides a natural tool for attacking problems involving dense packings of spheres.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Za8zU.jpg&quot; alt=&quot;BookCover&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-10-25T01:15:56.040" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5643" PostTypeId="2" ParentId="5635" CreationDate="2014-10-25T12:33:48.343" Score="9" Body="&lt;p&gt;I recommend &lt;strong&gt;How to Solve It&lt;/strong&gt; by George Polya. I merely glanced at it in high school and it helped me to work on math problems. More recently I bought it and now read the whole thing annually, and it now governs how I help my students to solve problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This book definitely meets your first three criteria. Although it tries to be broader than just mathematics, it inspires you to both find answers to problems and to develop proofs that are both sound and intuitive. It definitely helps in mathematical maturity and it inspires. The fourth point is debatable. The basic techniques on &lt;strong&gt;How to Solve It&lt;/strong&gt; are no longer &quot;unknown&quot; since so many other books were inspired by it. But the depth of Polya's work, based on his own significant mathematics work, goes beyond the other books. The book has a great balance of accessibility for all and depth for the mathematician.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2014-10-25T12:33:48.343" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5644" PostTypeId="1" AcceptedAnswerId="5686" CreationDate="2014-10-25T14:24:55.573" Score="11" ViewCount="2539" Body="&lt;p&gt;I've heard from a friend of mine that Dieudonné's &lt;em&gt;Foundations of Modern Analysis&lt;/em&gt; is &quot;painful reading&quot; and &quot;a little outdated&quot;; however, my teacher actually suggested it to me, describing it as a &quot;wonderful text&quot;. So I would like to ask here if anyone has gone through it and can describe the general approach followed by the author, and (more importantly) if it is pedagogically and mathematically worthwhile to study analysis using it.&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="376" LastEditDate="2014-10-26T13:32:51.337" LastActivityDate="2017-08-18T14:35:24.663" Title="Pedagogy, mathematics and Dieudonné's Foundations of Modern Analysis" Tags="|mathematical-pedagogy|textbooks|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5645" PostTypeId="1" CreationDate="2014-10-25T14:32:43.077" Score="13" ViewCount="559" Body="&lt;p&gt;In Jennifer Roberts' article &lt;a href=&quot;http://harvardmagazine.com/2013/11/the-power-of-patience&quot;&gt;The Power of Patience: Teaching students the value of deceleration and immersive attention&lt;/a&gt; she talks about intentionally slowing down to contemplate deeply a work of art. To my mind, the study of mathematics also benefits from slow and careful thought in a distraction-free environment, in addition to spending time on interactive discussion and group work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has research been carried out on immersive attention or sustained concentration when learning mathematics? If so, what were the results and are there any useful suggestions for teachers and students, both inside and outside of the classroom? This could include lesson pacing, variation of activities and selection of assignments, for instance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, does anyone have experiences to share of the process and results of setting students an exercise that involved  contemplation of a mathematical object or theorem for a seemingly-excessive length of time a la Roberts' example in the above-mentioned article?&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2014-10-26T17:35:14.383" LastActivityDate="2016-08-10T15:05:20.247" Title="Immersive attention when learning mathematics" Tags="|reference-request|mathematical-pedagogy|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5648" PostTypeId="1" AcceptedAnswerId="5663" CreationDate="2014-10-25T22:02:25.910" Score="36" ViewCount="10070" Body="&lt;p&gt;I was having a discussion with a friend and fellow mathematics teacher the other day when the topic of dividing by zero came up. She is the department head and had this in a questionnaire she gave to prospective teachers at her school. She was shocked by the number of incorrect responses provided by mathematics teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leads me to my question: &quot;How do/would you explain why division by zero does not produce a result.&quot; I would imagine that answers would vary depending on the age/level of the student. &lt;strong&gt;Please indicate for which level you think your explanation would be appropriate.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit - 10/28/14&lt;/strong&gt;&#xA;&lt;em&gt;In response to many of the comments that address the fact that it &quot;does produce a result&quot; or &quot;is defined&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discussion was initially set in, and meant to address division with the real numbers. I recognize that an answer of &quot;undefined&quot; is a &quot;result,&quot; but it is not a result in the image set in this scenario. Extending the reals or defining division in some other less-well known (and by &lt;em&gt;less-well known&lt;/em&gt; I mean by the non-mathematical world) is certainly a valuable thing for mathematicians to be able to do. I feel like the discussion ultimately breaks down into one in Group Theory with different sets and operators. Perhaps this question should have initially been split as such:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How do/would you explain why division by zero does not produce a result for Elementary age children? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How do/would you explain why division by zero is undefined for Algebra/Calculus students?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How do/would you explain division by zero to the advanced mathematics student?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This, too, may not be partitioned enough. That is why I have additionally edited my original question by highlighting the last sentence!&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="267" LastEditDate="2014-11-19T05:23:20.710" LastActivityDate="2022-07-12T15:32:40.483" Title="Dividing by zero" Tags="|concept-motivation|division|arithmetic-operations|" AnswerCount="17" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="5649" PostTypeId="2" ParentId="5648" CreationDate="2014-10-26T00:07:02.857" Score="7" Body="&lt;p&gt;Here is my answer to high school underclassmen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we let $\frac10=n$ for any $n$, we then get&#xA;$$\frac10 \cdot 0 = n \cdot 0$$&#xA;$$1=0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works for any non-zero value divided by zero. Allowing any value here leads to a contradiction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we let $\frac00=n$ we get&#xA;$$\frac00 \cdot 0 = n \cdot 0$$&#xA;$$0=0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works for any value of $n$, so which value of $n$ should we choose? We do not have a good choice, so we make no choice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admit the last argument is weaker than the first, but better arguments require more time or a higher level of math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;ADDED LATER&lt;/strong&gt; (in response to comments)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above is my short answer. I have a longer answer, which I developed as a response to the related Algebra 2 question &quot;Why can we take the square root of a negative number but not divide by zero?&quot; Here is more (but not all) from that longer answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My short answer assumes we are talking about the real number system and also assumes that we keep all the usual properties of that number system. Jared's comments are valid but are applicable to another number system, such as the projective extended real number system (where we add the value $\infty$, let $1/0=\infty$, and say that $0 \cdot \infty$ is undefined or &quot;indeterminate&quot;). The main problem with such extended systems is that they lose some fundamental properties of the real numbers. Jared's comments assume that multiplication is not closed, disallowing $0 \cdot \infty$, which is much more drastic and un-intuitive than disallowing $1/0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In brief, I was keeping things simple and sticking to the real number system.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2014-10-26T11:05:03.133" LastActivityDate="2014-10-26T11:05:03.133" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5650" PostTypeId="2" ParentId="5648" CreationDate="2014-10-26T00:17:14.240" Score="12" Body="&lt;p&gt;Dividing $1$ disk into $\frac{1}{n}$-ths ($\frac{1}{3}, \ldots$), leads to &#xA;$\frac{1}{1/n} = n$ pieces ($3,\ldots$):&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/apRmu.jpg&quot; alt=&quot;PieCharts&quot;&gt;&#xA;&lt;hr /&gt;&#xA;As $\frac{1}{n}$ approaches $0$, the number of pieces $n$ grows without bound.&#xA;The result upon division by $0$, $1/0$, should be this limit. But there is no limit.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-10-26T00:17:14.240" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="5651" PostTypeId="2" ParentId="5636" CreationDate="2014-10-26T01:38:10.403" Score="2" Body="&lt;p&gt;This &lt;a href=&quot;http://illustratedtheoryofnumbers.wordpress.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt; when it's finally done might be exactly what you're looking for. I find the sample pages pretty compelling.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit: The book has since been published. It's &lt;a href=&quot;http://illustratedtheoryofnumbers.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Illustrated Theory of Numbers&lt;/a&gt; by Weissman.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="376" LastEditDate="2020-09-10T15:22:01.630" LastActivityDate="2020-09-10T15:22:01.630" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5652" PostTypeId="2" ParentId="5648" CreationDate="2014-10-26T04:21:16.890" Score="0" Body="&lt;p&gt;The answer should be that if you divide a finite value by zero then it can be either $\pm \infty$ (it's not clear if it's positive or negative without further context).  This doesn't mean that the value doesn't clearly exist--you can say something about it if you know it's infinite.  On the other hand, if you try to divide $\frac{0}{0}$ then the answer is indeterminate--it could range from $\pm \infty$ to some finite value (including $0$) to the limit does not exist at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In answer to your second question about age/level of appropriateness.  I would say that at the level of Algebra, dividing by zero is always undefined.  The real level depends on whether or not you've introduced the idea of limits (which could be as early as pre-calculus and certainly by calculus).  You cannot talk about dividing by zero without limits.&lt;/p&gt;&#xA;" OwnerUserId="2204" LastEditorUserId="2204" LastEditDate="2014-10-26T04:27:44.727" LastActivityDate="2014-10-26T04:27:44.727" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5653" PostTypeId="1" CreationDate="2014-10-26T05:02:36.913" Score="3" ViewCount="30639" Body="&lt;p&gt;On a 4th grade test the question was asked: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;17 is a multiple of only two numbers, 1 and 17. Tell why this&#xA;  statement is true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If it is true then every number must be a multiple of 1 since 1 is a factor of every number.  Right?&lt;/p&gt;&#xA;" OwnerUserId="3280" LastEditorUserId="64" LastEditDate="2014-10-26T16:09:34.253" LastActivityDate="2014-10-28T19:28:43.297" Title="Is every number a multiple of one?" Tags="|assessment|" AnswerCount="2" CommentCount="1" ClosedDate="2014-10-29T11:30:28.043" ContentLicense="CC BY-SA 3.0" />
  <row Id="5658" PostTypeId="2" ParentId="5648" CreationDate="2014-10-26T07:42:37.813" Score="10" Body="&lt;p&gt;This answer is intended for the second category of students:&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;How do/would you explain why division by zero is undefined for Algebra students?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Begin by introducing the reciprocal of a real number as that number that satisfies the property of reciprocals:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a⋅\dfrac 1{a} = 1$$&lt;/span&gt;&#xA;Note here that &lt;span class=&quot;math-container&quot;&gt;$\dfrac1a$&lt;/span&gt; must be thought of as the result of dividing  &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; in the same way as &lt;span class=&quot;math-container&quot;&gt;$\dfrac a{1}$&lt;/span&gt; is thought of as equal to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, and emphasize that &lt;span class=&quot;math-container&quot;&gt;$a≠0$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; times any number is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Next state the definition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Definition of Division&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;For every real number &lt;em&gt;a&lt;/em&gt; and every &lt;em&gt;nonzero&lt;/em&gt; real number &lt;em&gt;b&lt;/em&gt;, the &lt;strong&gt;quotient&lt;/strong&gt;&#xA;a&lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt;b,&lt;/p&gt;&#xA;&lt;p&gt;or &lt;span class=&quot;math-container&quot;&gt;$\dfrac{a}{b}$&lt;/span&gt;, is defined by:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a\div b=a \cdot \frac{1}{b}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Dividing by zero would mean multiplying by the reciprocal of 0.&lt;/p&gt;&#xA;&lt;p&gt;But 0 has no reciprocal (because 0 times &lt;em&gt;any&lt;/em&gt; number is 0, &lt;em&gt;not&lt;/em&gt; 1.)&lt;/p&gt;&#xA;&lt;p&gt;Therefore, &lt;em&gt;division by 0 has no meaning in the set of real numbers.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;To show that &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; cannot satisfy the property of reciprocals we must prove the multiplicative property of &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Multiplicative property of 0&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Prove:&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is any real number,&#xA;then &lt;span class=&quot;math-container&quot;&gt;$a\cdot 0 = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$0\cdot a = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Proof:&lt;/p&gt;&#xA;&lt;p&gt;Statement _________________________Reason&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$0 = 0 + 0$&lt;/span&gt; ___________________1. Identity property of addition&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a\cdot0 = a(0 + 0)$&lt;/span&gt; ______________2. Multiplication property of equality&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a\cdot0 = a\cdot0 + a\cdot0$&lt;/span&gt; ___________3. Distributive property of mult. with respect to add.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But &lt;span class=&quot;math-container&quot;&gt;$a\cdot0 = a\cdot0 + 0$&lt;/span&gt; __________4. Identity property of addition&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\therefore$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$a\cdot0 + a\cdot0 = a\cdot0 + 0$&lt;/span&gt; _____5. Transitive property of equality&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a\cdot0 = 0$&lt;/span&gt; ____________________6. Subtraction property of equality&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$0\cdot a = 0$&lt;/span&gt; ____________________7. Commutative property of multiplication&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Therefore, 0 times &lt;em&gt;any&lt;/em&gt; number is 0, &lt;em&gt;not&lt;/em&gt; 1.&lt;/p&gt;&#xA;&lt;p&gt;(Source: Algebra: Structure and Method Book 1)&lt;/p&gt;&#xA;&lt;p&gt;The two cases are sometimes presented as follows:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Dividing a nonzero number by zero, &lt;em&gt;violates&lt;/em&gt; the &lt;strong&gt;multiplicative property of zero&lt;/strong&gt; and therefore the properties of the real numbers upon which it is proven, as shown above.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Dividing zero by zero, which  does not violate the &lt;strong&gt;multiplicative property of zero&lt;/strong&gt;, but multiplication by zero is an operation that results in zero for every real number.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The following argument is presented in an older edition of the above source:&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$\dfrac{a}{0}$&lt;/span&gt; = c, then &lt;span class=&quot;math-container&quot;&gt;$a = 0\cdot c$&lt;/span&gt;.&#xA;But &lt;span class=&quot;math-container&quot;&gt;$0\cdot c = 0$&lt;/span&gt;.&#xA;Hence, if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is not equal to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;,&#xA;no value of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; can make the statement &lt;span class=&quot;math-container&quot;&gt;$a = 0\cdot c$&lt;/span&gt; true,&#xA;while if &lt;span class=&quot;math-container&quot;&gt;$a = 0$&lt;/span&gt;,&#xA;every value of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; will make the statement true.&lt;/p&gt;&#xA;&lt;p&gt;Thus, &lt;span class=&quot;math-container&quot;&gt;$\dfrac{a}{0}$&lt;/span&gt; either has &lt;em&gt;no&lt;/em&gt; value or is &lt;em&gt;indefinite&lt;/em&gt; in value.&lt;/p&gt;&#xA;&lt;p&gt;This separation into two cases, one of which results in &lt;strong&gt;no value&lt;/strong&gt; satisfying the multiplicative property of zero and the other resulting in an &lt;strong&gt;indefinite value&lt;/strong&gt; satisfying it, gives the impression that &lt;span class=&quot;math-container&quot;&gt;$\dfrac{0}{0}$&lt;/span&gt; is allowed.&lt;/p&gt;&#xA;&lt;p&gt;The direct argument &lt;em&gt;starts&lt;/em&gt; with the equation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$a⋅\dfrac 1{a}= 1$$&lt;/span&gt;&#xA;and emphasises that &lt;span class=&quot;math-container&quot;&gt;$a≠0$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; times any number is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thus, the product of &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;strong&gt;no real number&lt;/strong&gt; equals &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This further reinforces the importance of the central idea that &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; has no number, called its reciprocal, that when multiplied by it equals &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; has no reciprocal, division by &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is not defined.&#xA;I've found a similar detailed explanation &lt;a href=&quot;http://ee.usc.edu/stochastic-nets/docs/divide-by-zero.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="216" LastEditorUserId="216" LastEditDate="2020-10-02T00:05:42.240" LastActivityDate="2020-10-02T00:05:42.240" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="5659" PostTypeId="2" ParentId="5648" CreationDate="2014-10-26T12:58:12.913" Score="29" Body="&lt;p&gt;Have the students tell &lt;em&gt;you&lt;/em&gt; that division by zero is a non-sequitur. This is possible at any age where division is understood at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Teacher&lt;/em&gt;: If there are eight cookies and four children, how many cookies does each child get?&#xA;&lt;br&gt; &#xA;&lt;em&gt;Student&lt;/em&gt;: Uh, two.&#xA;&lt;br&gt;&#xA;&lt;em&gt;Teacher&lt;/em&gt;: Yes! This is a division problem. $\frac{8}{4} = 2$. Now, if there are 8 cookies shared by only two children, how many cookies would each child get?&#xA;&lt;br&gt;&#xA;&lt;em&gt;Student&lt;/em&gt;: Four.&#xA;&lt;br&gt;&#xA;&lt;em&gt;Teacher&lt;/em&gt;: Yes, right. Now tell me the division statement for that problem.&#xA;&lt;br&gt;&#xA;&lt;em&gt;Student&lt;/em&gt;: Um, 8 divided by 2 is 4.&#xA;&lt;br&gt;&#xA;&lt;em&gt;Teacher&lt;/em&gt;: Good! Now, suppose we have 8 cookies and there are no children. How many cookies does each child get?&#xA;&lt;br&gt;&#xA;&lt;em&gt;Student&lt;/em&gt;: That question doesn't make any sense.&#xA;&lt;br&gt;&#xA;&lt;em&gt;Teacher&lt;/em&gt;: Why not?&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;Things will vary considerably here depending on individual students, because it's a bit difficult to put your finger on the concept that it's impossible/meaningless to assign characteristics to things which don't exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real point though is that when considering the physical and logical meaning of division, the absurdity of division by zero becomes obvious. Students have trouble getting here because they often don't have a strong sense of the physical and logical meaning of the operations that they are working with, but instead consider division as just one more button on the calculator, and just one more hoop to jump in math class.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2014-10-26T12:58:12.913" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5660" PostTypeId="2" ParentId="5653" CreationDate="2014-10-26T15:03:20.583" Score="8" Body="&lt;p&gt;Welcome to the site, Donna. I hope you find my answer helpful to your situation. Please let us know if there's another aspect of the situation you'd like to think about with our help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Test question: &quot;17 is a multiple of only two numbers, 1 and 17. Tell why this statement is true.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think they are asking the student to show that 17 is not a multiple of any other numbers. To do so, one can show that dividing by 2,3, ... always leaves a remainder. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you are asking whether it is right to conclude that &quot;every number must be a multiple of 1 since 1 is a factor of every number&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, every whole number is a multiple of 1. We say that b is a multiple of a when a*n = b (where n is a whole number).   Since 1*b = b, for any number b, all numbers are multiples of 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It sounds like you also want to double-check your understanding of the two words, 'factor' and 'multiple'. If b is a multiple of a, then a is a factor of b. The two terms describe the same situation from different perspectives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this helpful to you?&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-10-26T15:03:20.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5661" PostTypeId="2" ParentId="5648" CreationDate="2014-10-26T17:48:47.237" Score="4" Body="&lt;p&gt;Division itself is defined like this: The result of division (the quotient) of &quot;&lt;em&gt;b&lt;/em&gt; divided by &lt;em&gt;a&lt;/em&gt;&quot; is defined as the number &lt;em&gt;x&lt;/em&gt; in&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;a&lt;/em&gt;⋅&lt;em&gt;x&lt;/em&gt; = &lt;em&gt;b&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, let's divide 8 by 4:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We look for the number &lt;em&gt;x&lt;/em&gt; with: 4⋅&lt;em&gt;x&lt;/em&gt; = 8&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We find &lt;em&gt;x&lt;/em&gt; = 2. So 8 divided by 4 equals 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if we want to divide for example 8 by zero (!) we have to look for &lt;em&gt;x&lt;/em&gt; with&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0⋅&lt;em&gt;x&lt;/em&gt; = 8.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, you would agree it makes sense to say&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0⋅&lt;em&gt;x&lt;/em&gt; = 0 for any &lt;em&gt;x&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, there is no &lt;em&gt;x&lt;/em&gt; that satisfies 0⋅&lt;em&gt;x&lt;/em&gt; = 8 and 8/0 can't be found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you divide by numbers closer and closer to zero&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0.1⋅&lt;em&gt;x&lt;/em&gt; = 8&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0.00001⋅&lt;em&gt;x&lt;/em&gt; = 8&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0.0000001⋅&lt;em&gt;x&lt;/em&gt; = 8&lt;/p&gt;&#xA;&#xA;&lt;p&gt;etc. you see that &lt;em&gt;x&lt;/em&gt; becomes greater and greater. So it could make sense to say that 8/0 = infinity. You could define it that way if it made sense to any calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's try another thing and divide zero by zero. That means: let's find the &lt;em&gt;x&lt;/em&gt; with&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0⋅&lt;em&gt;x&lt;/em&gt; = 0.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will agree this is true for any number. So 0/0 is any number if you wanted it to be this way. But that's disappointing because we want division to always have exactly one result. But once again, you could define it that way if it made sense to any of your calculations.&lt;/p&gt;&#xA;" OwnerUserId="1674" LastEditorUserId="3272" LastEditDate="2014-10-28T15:05:02.593" LastActivityDate="2014-10-28T15:05:02.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5662" PostTypeId="2" ParentId="4025" CreationDate="2014-10-26T18:38:56.167" Score="7" Body="&lt;p&gt;The Cartoon Guide to Calculus by Larry Gonick should be mentioned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0061689092&quot;&gt;http://www.amazon.com/The-Cartoon-Guide-Calculus-Guides/dp/0061689092&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="607" LastActivityDate="2014-10-26T18:38:56.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5663" PostTypeId="2" ParentId="5648" CreationDate="2014-10-26T19:33:04.303" Score="40" Body="&lt;p&gt;&lt;strong&gt;You asked:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;How do/would you explain why division by zero does not produce a&#xA;  result.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Any such explanation that is not rooted in student understanding would be talking to ourselves, not to students. Therefore both &lt;em&gt;meaning&lt;/em&gt; and &lt;em&gt;student understanding&lt;/em&gt; are important. Otherwise, what's the point? So I have grounded my response there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Young students (elementary level) have different understandings of division situations, prior to generalizing to the understanding of division that more advanced learners have formed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For such students there are two meanings of division: the &lt;strong&gt;partitive&lt;/strong&gt; (or &quot;&lt;em&gt;fair sharing&lt;/em&gt;&quot;) meaning and the &lt;strong&gt;quotative&lt;/strong&gt; (or &quot;&lt;em&gt;measurement&lt;/em&gt;&quot;) meaning of division.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To discuss division by zero to these students (or any students) you have to be aware of their understanding of division. In the case of students who see quotative and partitive situations differently, if you do not address both then you are not giving them an explanation (or allowing a discussion) that covers all of the situations that we would like them to see as division.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How to address it?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For &lt;em&gt;partitive division&lt;/em&gt;, you could ask them to think about:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The box has 10 cookies. If 10 friends are at a party, how many cookies&#xA;  does each friend get, if everything is to be fair? [After some&#xA;  discussion] Now, what if there are no friends -- no people at the&#xA;  party. What does partitioning mean if you're partitioning cookies&#xA;  among no people? Is there even a party? How do we make sense of this?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That may be a puzzler to them, and worth thinking about. However, the &lt;em&gt;quotative or measurement division&lt;/em&gt; example may be even more fruitful.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Willy Wonka has determined that a serving of gum is 2 pieces. If you&#xA;  arrive in class with a bag of 10 gumballs, how many students in class&#xA;  can you give a full serving to? [After consideration of this problem&#xA;  that verifies understanding] OK, now the Surgeon General doesn't think&#xA;  sugary gum should be in anyone's diet, so she has determined that 0&#xA;  pieces of gum is a serving. With your bag of 10 gumballs, how many&#xA;  people can you distribute servings of gum to now?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.learner.org/courses/learningmath/number/session4/part_a/division.html&quot;&gt;For more on quotative and partitive division, including seeing students work with these understandings, see the Annenberg Learner site.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;EDIT:&lt;/strong&gt; There is no reason to skip to the answer for them. It's a worthwhile problem for them to consider on their own and if the object is to get them to think about it the surest way to prevent it is to simply tell them that it is defined a certain way. That authoritative approach is likely to remove the need and impetus for them to actually consider, if indeed they are curious about it.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;My recommendation at this point, then, would be to continue to mostly give them questions instead of answers. &quot;Is it possible to give &lt;strong&gt;everyone&lt;/strong&gt; a 0 serving of gum? Or is it that you can give &lt;strong&gt;nobody&lt;/strong&gt; a 0 serving of gum?&quot; &quot;Does it even make sense?&quot;&lt;/em&gt; &lt;em&gt;&quot;How should we handle this?&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;You can always tell them later that it is undefined, and have a discussion about that. Having puzzled over it for a time (and possibly frustrated themselves productively), that revelation might actually make sense to them.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="759" LastEditDate="2014-10-26T19:47:35.357" LastActivityDate="2014-10-26T19:47:35.357" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5664" PostTypeId="2" ParentId="5648" CreationDate="2014-10-26T20:48:03.820" Score="33" Body="&lt;p&gt;From the viewpoint of a mathematician and a formalist, you're asking the wrong question: the right question is not &quot;why does division not produce a result&quot;, but &quot;why do we define division so that it's not defined for division by zero?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arithmetic operations aren't &quot;god-given&quot; or properties of &quot;nature&quot;: we define arithmetic so that we can describe things and solve problems. In the context of arithmetic like what students learn in primary school, the point can be roughly summarized as we want to solve equations $ax=b$ for $x$ (or any equivalent 'visualization' of this idea), and define division so that $x = b/a$ is &lt;em&gt;the&lt;/em&gt; solution. This doesn't make sense when $a=0$, so we define division so that we forbid zero denominators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is worth noting that in other contexts, division by zero is not only allowed, but it would be silly to forbid it aside from the case of $0/0$ (and there are even a handful of contexts where even &lt;em&gt;that&lt;/em&gt; should be defined): e.g. in one-dimensional projective geometry, or in complex analysis. In both cases, for $x \neq 0$, the value of $x/0$ would be the &quot;point at infinity&quot;.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2014-10-26T20:48:03.820" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5665" PostTypeId="1" CreationDate="2014-10-26T20:51:33.460" Score="8" ViewCount="1537" Body="&lt;p&gt;Let's say we're asked to solve the following system of equations for the unknowns $x$ and $y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x+y=1 \qquad \text{(1)}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x-y=3 \qquad \text{(2)}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My approach, which I'll refer to as serial elimination of variables (SEV for convenience) would be to solve one of the equations for one of the unknowns, substitute that in for that variable in the other equation, solve this equation for the other variable, and then back-substitute to get the other unknown. E.g., $x=3+y$, $(3+y)+y=1$, $y=-1$, $x=2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of my students seem to have been taught a different procedure, which I'll call SEE for setting equations equal. They would solve both equations for a particular variable, set them equal to each other, solve for that variable, and then back-substitute. So faced with system (1,2), they would do $x=1-y$, $x=3+y$, $1-y=3+y$, $y=-1$, $x=2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SEE seems correct but slower. I can think of examples where it doesn't work, e.g.,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$e^x+xy=1 \qquad \text{(3)}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\sin y+xy=3 \qquad \text{(4)}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can't even get going with (3,4) by the SEE method, whereas with SEV you can at least reduce it to a single equation in a single variable, and maybe look for a solution by graphing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also see the following over and over. Say we have the system (5,6):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x+y=0 \qquad \text{(5)}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x-y=7 \qquad \text{(6)}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My students proceed by writing $x-y-7=0$, then setting $x+y=x-y-7$ (setting 0 equal to 0). This hasn't helped, and they don't know what to do next. This seems to suggest that SEE has a disadvantage, which is that it obscures what one is trying to do (eliminate a variable), and this obscurity causes problems for students in applying it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is SEE seemingly so popular? Does it have some advantage that I'm not aware of?&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-10-28T20:55:23.180" Title="Is there an advantage to teaching solution of simultaneous equations by setting equations equal to each other?" Tags="|algebra|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5666" PostTypeId="2" ParentId="5665" CreationDate="2014-10-27T00:08:40.233" Score="1" Body="&lt;p&gt;There are multiple ways to solve these &quot;2 equations with 2 unknowns.&quot; For your (1) / (2) I'd say they lend themselves to being added up, to 2x=4. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the variables had coefficients, multiplying through to get the same coefficient on either both X or both Y then let you subtract. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is also a good time to introduce matrix arithmetic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students will use what they are familiar with and comfortable to use repeatedly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a method won't work for a certain type of problem you introduce, that's when you offer the reason why and explain the new (to them) method.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-10-27T00:08:40.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5667" PostTypeId="2" ParentId="5648" CreationDate="2014-10-27T00:12:45.637" Score="5" Body="&lt;p&gt;Division is multiplication, backwards. These two equations are exactly equivalent, by definition:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a / c = b$$&#xA;$$a = b \times c$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's easy to understand division by zero if we look at the equivalent multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a / 0 = b$$&#xA;$$a = b \times 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For any real number $b$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a = b \times 0 = 0$$&#xA;$$a = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two cases with division by zero:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If $a = 0$, then $a/0 = b$ is unconstrained, any real number $b$ satisfies the equation. You can discard such an equation which does not constraint the result.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If $a \ne 0$ then $a / 0 = b$ is contradictory. There is no real number $b$ which satisfies that equation. This is still useful to know; &quot;there is no answer&quot; can be a sort of meta-answer.  For example if trying to solve a system of equations of static forces, &quot;there is no answer&quot; might mean you need to consider a different design for your bridge!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There is no need to consider advanced concepts such as limits in order to fully understand division.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In short, $a / 0 = b$ is true if and only if $a = 0$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you see such an equation $a / 0 = b$, you may simplify it to $a = 0$.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$a / 0 = b \iff a = b \times 0 \iff a = 0$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3287" LastEditorUserId="3287" LastEditDate="2014-12-05T00:53:16.727" LastActivityDate="2014-12-05T00:53:16.727" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5668" PostTypeId="2" ParentId="5633" CreationDate="2014-10-27T07:25:36.417" Score="1" Body="&lt;p&gt;In my experience (being taught in a US school), there is no explicit distinction between an identity and an equation.  The reason being is that to make the distinction you have to introduce first order logic which is not generally appropriate for secondary (high school) mathematics education.  Students can usually ascertain an identity (a formula) from an equation to be solved so this is not usually an issue.  The students that have trouble using formulas or have trouble solving equations are unlikely to benefit from a more detailed explanation (the detailed explanation will likely only further confuse them and make the topic of mathematics far less tangible and far more abstract).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Examples:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Quadratic equation.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;We are generally taught that given $ax^2 + bx + c = 0$ such that $a \neq 0$ that $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.  This is an identity.  It's a &lt;em&gt;first order&lt;/em&gt; equation.  Formally it would state the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\forall x \in \mathbb{C}.\forall a, b, c\in \mathbb{R}:\left( \left(a\neq 0\wedge ax^2 + bx + c = 0\right) \rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\right)&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Solving an equation:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Solve the following equation: $10x^2 + 6x + 8 = 0$ wher $x$ must be a real number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The very first thing to do when solving an equation is to determine whether or not it &lt;em&gt;can&lt;/em&gt; be solved.  This too is a first order logic equation.  The question (which must be answered) is whether or not the following is true:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\exists x\in \mathbb{R}: 10x^2 + 6x + 8 = 0&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this particular instance, the answer would be that the above is false therefore there are &lt;em&gt;no&lt;/em&gt; solutions.  But how do you know that?  How do you prove that?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a different example: Solve the following: $x^2 + 6x + 8 = 0$.  Again, we must first &lt;em&gt;decide&lt;/em&gt; whether or not the following first order equation is true or false:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\exists x\in \mathbb{R}: x^2 + 6x + 8 = 0&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case it is true (how do we know?).  Once we know the equation &lt;em&gt;can&lt;/em&gt; be satisfied, we next attempt to find the particular solutions.  This is not easy and in fact, &quot;impossible&quot; in general.  Here is an example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Find real values of $x$ that satisfy the following equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;e^x = x + 2&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It can be &lt;em&gt;proved&lt;/em&gt; that a solution exists:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;f(x) = e^x - x - 2 \rightarrow \text{find } f(x) = 0 \\&#xA;f'(x) = e^x - 1 \rightarrow f'(x) = 0 \rightarrow e^x = 1 \rightarrow x = \ln(1) = 0&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the derivative changes from negative to positive at $x = 0$, this represents a relative minimum (and since this is the &lt;em&gt;only&lt;/em&gt; extrema this is the global minimum).  $f(0) = 1 - 0 - 2 = -1$.  Further since $\lim_{x\rightarrow-\infty} e^x = 0$ and $\lim_{x\rightarrow-\infty} = -\infty$ (thus subtracting negative infinity creates a positive infinity) and that $e^x$ dominates $-x$ such that $\lim_{x\rightarrow\infty} e^x - x = +\infty$, we &lt;em&gt;know&lt;/em&gt; that this function, $f(x)$, goes towards $+\infty$ on either side, there &lt;em&gt;must&lt;/em&gt; be two spots where $f(x) = 0$ and therefore there are &lt;em&gt;exactly&lt;/em&gt; two real solutions to $e^x = x + 2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet even though we &lt;em&gt;know&lt;/em&gt; the equation to be satisfiable, it is not clear how (or analytically possible) to solve for those two values.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when solving equations there are two separate issues that we usually do &lt;em&gt;not&lt;/em&gt; identify in secondary education (largely because it's beyond the scope).  The first is, whether or not there &lt;em&gt;is&lt;/em&gt; a solution and the second is how to find the solution once you can show that one exists--note that the first step becomes tedious to the point of being prohibitive towards learning for the vast majority of problems encountered in elementary to high school level mathematics courses.&lt;/p&gt;&#xA;" OwnerUserId="2204" LastEditorUserId="2204" LastEditDate="2014-10-30T08:10:10.377" LastActivityDate="2014-10-30T08:10:10.377" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5669" PostTypeId="1" AcceptedAnswerId="5687" CreationDate="2014-10-27T14:27:57.130" Score="4" ViewCount="295" Body="&lt;p&gt;&lt;em&gt;Note:&lt;/em&gt; This question is ment to extend the scope of some related questions of mine. I would appreciate very much any suggestion to improve the way the question is posed.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I would like to ask what is -- according to your experience as students beforehand and educators then -- a sensible approach to develop undergraduate students' mathematical intuition (*) (that is visual thinking, etc.). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As the question is very broad, collections of references to research papers in the field of mathematical education and to fitting textbooks are very welcome ad accepted as helpful answers (at least by me).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(*) in general, but in particular as regards the following fields:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Analysis;&lt;/li&gt;&#xA;&lt;li&gt;Mathematical/theoretical physics/ dynamical systems;&lt;/li&gt;&#xA;&lt;li&gt;Linear Algebra; &lt;/li&gt;&#xA;&lt;li&gt;Geometry;&lt;/li&gt;&#xA;&lt;li&gt;Abstract Algebra;&lt;/li&gt;&#xA;&lt;li&gt;Number Theory;&lt;/li&gt;&#xA;&lt;li&gt;Probability/Combinatorics.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3251" LastActivityDate="2014-11-01T21:55:57.303" Title="Promoting intuition (for undergraduate students): visual thinking, geometic approaches, etc. in the classroom" Tags="|mathematical-pedagogy|reference-request|textbooks|intuition|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5670" PostTypeId="1" CreationDate="2014-10-27T15:08:55.557" Score="8" ViewCount="294" Body="&lt;p&gt;&lt;strong&gt;Context:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some time ago there was a post on a brief study conducted by Alexis Wiggins (she was shadowing two students for two days), you can find it &lt;a href=&quot;https://grantwiggins.wordpress.com/2014/10/10/a-veteran-teacher-turned-coach-shadows-2-students-for-2-days-a-sobering-lesson-learned/&quot;&gt;here&lt;/a&gt;, which &lt;a href=&quot;https://grantwiggins.wordpress.com/2014/10/19/a-ps-to-the-guest-post-on-shadowing-hs-students-and-the-author-revealed/&quot;&gt;got quite an attention&lt;/a&gt;. One interesting thing is that, as the author of the blog have put it, it is &quot;not news&quot;. To be more precise, let me quote a paragraph:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Not news.&lt;/strong&gt; Yet, as a number of commenters pointed out – and I agree – the passive and sitting life of a HS student is not news. Our surveys document this, and I have written about my own observations of the boredom I see. Ted Sizer’s Coalition of Essential Schools (for which I worked as the 1st director of research 30 years ago) began life due to &lt;em&gt;Horace’s Compromise&lt;/em&gt; and &lt;em&gt;The Shopping Mall HS&lt;/em&gt; which both documented the problem (along with massive data from John Goodlad’s &lt;em&gt;A Place Called School&lt;/em&gt;). All of which was preceded by Postman &amp;amp; Weingartner, Kozol, Silbermann and Friedenberg in the 60s.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[Grant Wiggins, &lt;a href=&quot;https://grantwiggins.wordpress.com/2014/10/19/a-ps-to-the-guest-post-on-shadowing-hs-students-and-the-author-revealed&quot;&gt;A PS to the guest post on shadowing HS students (and the author revealed)&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But despite the fact that it is not news, Alexis, &quot;a veteran HS teacher who just became a Coach&quot; states&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I have made a terrible mistake. I waited fourteen years to do something that I should have done my first year of teaching: shadow a student for a day.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and later in the article follows with a number of changes she would immediately implement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question:&lt;/strong&gt; What else we miss?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suspect that a veteran teacher would know something about the students' daily routine. From the story she told, it is clear it was not enough (at least for her). As someone with much less experience I'm concerned about other things that we (as teachers and educators) are unaware of that hurt our students the most. What I'm looking for, is a collection of posts, statements, research results, that would help getting the fundamentals right (I regard the students' inability to follow the lesson quite fundamental). The scenario I'm counting for is that we crowd-source such a list here, each of us contributing some important bits (the perspectives vary, collaboration might result in a more balanced collection).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A follow up:&lt;/strong&gt; Is missing such things very bad, and if it is, how we can prevent missing such things? (Perhaps that should be another question, I will separate it if there is enough interest in it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt; I'm aware that this question is more about teaching in general than teaching mathematics, but I don't know any other SE site that fits better (and there surely are some math-related issues too, I just don't know how they compare).&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-10-30T11:02:58.797" Title="What else we miss?" Tags="|secondary-education|reference-request|teachers|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5671" PostTypeId="2" ParentId="5669" CreationDate="2014-10-27T16:39:00.533" Score="2" Body="&lt;p&gt;I've been using origami and pop-up card design as vehicles to teach geometry (and algebra and combinatorics and ...).&#xA;Here is one example; pardon the self promotion:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;From Pop-Up Cards to Coffee-Cup Caustics: The Knight's Visor,&quot; 2012.&#xA;  (&lt;a href=&quot;http://arxiv.org/abs/1206.1312&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv link&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/sqrpT.jpg&quot; alt=&quot;KnightsVisor1&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/c7Yem.png&quot; width=&quot;360&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-10-28T00:57:08.197" LastActivityDate="2014-10-28T00:57:08.197" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5672" PostTypeId="2" ParentId="5665" CreationDate="2014-10-27T16:46:11.693" Score="2" Body="&lt;p&gt;From my experience as a user of mathematics, the &quot;SEE&quot; method, making the one-side of two (or more equations) identical and then equating the other sides, is helpful in various instances, for example, when a &quot;non-interesting&quot;, or an auxiliary, variable is present.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume we are optimizing under equality constraints, say,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\max f(x,y)  \;\;s.t. \;\;h(x)+g(y) = a$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The langrangean here is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\Lambda = f(x,y) +\lambda [a-h(x)-g(y)]$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first-order conditions require&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac {\partial \Lambda}{\partial x} = \frac {\partial f(x,y)}{\partial x} -\lambda h'(x) = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac {\partial \Lambda}{\partial x} = \frac {\partial f(x,y)}{\partial y} -\lambda g'(x) = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we don't really care about $\lambda$ (and in many cases we don't) then the next step is to solve for it,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac {\partial f(x,y)}{\partial x}\frac 1{h'(x)}= \lambda $$&#xA;$$\frac {\partial f(x,y)}{\partial y}\frac 1{g'(x)}= \lambda $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and equate the left-hand sides&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac {\partial f(x,y)}{\partial x}\frac 1{h'(x)}= \frac {\partial f(x,y)}{\partial y}\frac 1{g'(x)} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;eliminating from sight the auxiliary variable, the multiplier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another case where the SEE method, as a general approach, can be useful to &quot;have in mind&quot;, is when one wants to show that two expressions are equal. For example in mathematical statistics, we consider the probability limit of two functions of random variables. In order to &lt;em&gt;check&lt;/em&gt; that they are equal, we calculate both separately, and conclude that the left-hand sides are equal if the right-hand sides are equal - we don't equate them directly and try to see whether the equality is true or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that the above possibly relate to other levels of mathematics, and the &quot;SEE&quot; method may not help in all cases of solving systems of equations, but I think it is useful for the students to be familiar with this approach also from early on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the &quot;SEV&quot; approach, sometimes it produces complicated-looking intermediate steps and I have the feeling it makes plain calculation errors more probable (this does not render it &quot;inferior&quot; of course).&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2014-10-27T18:37:56.333" LastActivityDate="2014-10-27T18:37:56.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5674" PostTypeId="2" ParentId="5648" CreationDate="2014-10-27T19:37:04.393" Score="-1" Body="&lt;p&gt;The term &quot;division&quot; is actually applied to a few different operations in different context, depending upon whether the operands and intended result are whole numbers, integers, or real numbers.  Although whole numbers are often thought of as simply being a superset of the integers, integers, whole numbers, and real numbers have some fundamental differences in nature which can be important when performing operations like multiplication or exponentiation, but become most apparent when performing division.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fundamentally, the meaning of whole numbers is defined in terms of multiplying &lt;em&gt;any kind of thing with a defined additive identity and associative addition operator&lt;/em&gt; by them.  $ONE$ is the thing which, when multiplied by any x, will simply yield X.  $TWO$ is the thing which, when multiplied by any x, will yield x+x.  $ZERO$ is the thing which, when multiplied by any x, will yield $ZERO$; it may also be added to anything without affecting it.  Note that whole numbers don't require &quot;mathematical&quot; multiplication.  If a person has one rock and adds another rock, the person has &quot;two&quot; rocks.  An important feature of whole numbers is that multiplying $ZERO$ by &lt;em&gt;anything&lt;/em&gt; will yield $ZERO$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Real numbers behave a lot like integers, but with a twist: although multiplying real-number zero by any finite number will yield real-number zero, some &quot;problematic values&quot; which would cancel out when multiplied by (whole-number) $ZERO$ do not cancel when multiplied by real-number zero but may yield anything--either a problematic value or a normal-seeming number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this becomes important when trying to define division.  Whole division entails finding the largest whole number which, when multiplied by the divisor, will yield something which is defined as being no &quot;larger&quot; than the dividend according to some defined means of comparison.  The difference between the dividend and the result of that multiplication is called the remainder.  Note that when dividing by any form of zero or additive identity, there is no defined quotient, since there is no limit as to how large a whole number one could multiply by the divisor without yielding a result larger than the dividend.  Nonetheless, the equation $dividend = quotient * divisor + remainder$ holds just as well when dividing by zero as when dividing by anything else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Division of an arbitrary thing by a whole number seeks to find a value which, when added to the dividend type's additive identity the number of times specified by the divisor, will yield the dividend &lt;em&gt;precisely, without a remainder&lt;/em&gt;.  If the whole number divisor is $ZERO$, however, the operation becomes one of finding a value which, when multiplied by $ZERO$, will yield the dividend.  Unless the dividend is itself $ZERO$ (as opposed to merely being an additive identity like real-number zero), no such value will exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Division of a real number by another real number seeks to find a real number which, when multiplied by the divisor, will yield the dividend without a remainder.  If the divisor is real-number zero and the dividend isn't, there won't be any real number which could satisfy the constraints, but the division may yield a &quot;problematic value&quot; which, when multiplied by real-number zero, might arbitrarily yield the dividend.  Note that while the result of multiplying such a problematic divisor by a real number is not well-defined, the result of multiplying by $ZERO$ is well-defined--it yields $ZERO$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The three styles of division have slightly different behavior when dividing by zero; the kinds of inferences one may draw in cases involving &quot;division by zero&quot; will thus differ.  When performing whole division by zero or any additive identity, the &quot;division equation&quot; will be satisfied by making the remainder equal the dividend.  Division of anything other than $ZERO$ by $ZERO$ is simply undefined.  Division of a real number by zero may yield a &quot;problematic value&quot;.  In no situation is division by zero &quot;nicely defined&quot;, but the exact meanings differ.&lt;/p&gt;&#xA;" OwnerUserId="1740" LastActivityDate="2014-10-27T19:37:04.393" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="5676" PostTypeId="1" AcceptedAnswerId="7962" CreationDate="2014-10-27T22:25:21.190" Score="8" ViewCount="553" Body="&lt;p&gt;Consider, in a basic statistics class, the difference between the following two examples:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A six-sided die is rolled twenty times. It comes up a &quot;two&quot; eight times, which seems unfair. The owner of the die claims that it is fair, namely that each face of the die is equally likely to come up. Complete a hypothesis test to assess this claim.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;versus&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A store manager assigns shifts to six different employees, including you. Twenty workdays into the job, you notice that you have been assigned the &quot;bad shift&quot; eight times, which seems unfair. The manager claims that he assigns shifts randomly, namely that each employee has an equal chance of getting assigned the &quot;bad shift.&quot; Complete a hypothesis test to assess this claim.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The second one puts statistics directly into the real world, and seems (to me) to be more engaging.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides shift assignments from your boss, &lt;strong&gt;what other topics from students' lives are yield interesting statistical tests of claims?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although it is not entirely accurate, for the purposes of this question, let's assume that the mean age of my students is 20 years, with a standard deviation of 1 year.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-01-28T19:03:17.797" Title="Hypothesis Tests in Students' Lives" Tags="|examples|statistics|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5677" PostTypeId="2" ParentId="5665" CreationDate="2014-10-27T22:44:24.130" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Why is SEE seemingly so popular?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The technique I have observed is not just to solve both equations for &lt;em&gt;a&lt;/em&gt; particular variable, but to solve both equations &lt;em&gt;for y&lt;/em&gt;. There are logical reasons why they might do this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most students think of y as a function of x, even if it is not presented that way. Therefore it is natural for them to rewrite both as &quot;y =&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When written as &quot;y=&quot;, the students imagine two curves on an x-y plane. At the points where they meet, the curves have the same height and therefore the y for one curve is equal to the y for the other curve. Hence they equate the two y's which they know should tell them the x at which the two curves meet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, the above sentence is not wholly accurate because many students will not actually equate the two y's and do the algebra. Instead they will enter both equations into their graphics calculator and ask the calculator to find the intersection of the graphs. This is another explanation for the &quot;y=&quot; approach: the only way to get the calculator to graph the curve is to rearrange it into this format. They're used to this and so it's what they do first, even if they are forced to use algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly, the maths/science/engineering students at my university learn the &quot;adding multiples of equations together&quot; method, and will use it most of the time if presented with raw equations. But if the context where they see the equations is strongly graphical (eg drawing convex sets), they will revert to the &quot;convert to y= and use the graphics calculator&quot; approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even more interestingly, the economics students learn multiple approaches, but their lecturers actually use different ones depending on context, rather than sticking to one. If they have supply/demand curves (whose variables are price P and quandity Q) they will always convert them both to Q= and equate them, despite having shown the students the week before how to use another method for solving simultaneous equations. I think this is because the pull towards the logical story is strong: the equilibrium point is where the quantity supplied is equal to the quantity demanded so equating the two quantities makes sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think there are two reasons why it is so popular:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It matches logically to a particular interpretation of the meaning of the equations. That is, at the point where the two curves meet, one of the variables is equal, so you should &lt;em&gt;set&lt;/em&gt; one of the variables equal.&lt;/li&gt;&#xA;&lt;li&gt;It matches with other approaches they have used already, and they are just making it look more like those other situations so they can use the same approach.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does it have some advantage that I'm not aware of?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I can see one advantage over your SEV method, which is that there is less scope for errors. You would be aware in yourself how careful you have to be when subbing one expression into another, especially when there are minus signs involved! The SEE method dosn't require this sort of care at the equating stage because you just copy what you have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I don't think there is an advantage to only showing them one method. Really you should show multiple methods and a way of choosing which one will work for them today. Almost no method will be foolproof. Personally if I see two linear equations I'll multiply them by things until I can add/subtract to make a variable vanish. But if I see nonlinear equations I'm more likely to make one &quot;x=&quot; or &quot;y=&quot; and sub one into the other. Sure I could conceivably use the same approach for both but I find the first approach easier for linear equations.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-10-27T22:44:24.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5678" PostTypeId="2" ParentId="5676" CreationDate="2014-10-28T02:44:03.093" Score="6" Body="&lt;p&gt;Students in the class could google their birth dates before class and come to class knowing the day of the week on which they were born (MTWRF). In class you then test the hypothesis that days of birth are equally distributed. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2014-10-28T02:44:03.093" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5679" PostTypeId="1" CreationDate="2014-10-28T04:48:38.713" Score="3" ViewCount="159" Body="&lt;p&gt;I am a teaching assistant at a school. My job is mostly to help them solve problems given by their teachers. Students are at high school level. I assume the curriculum In Brazil is similar to the one in US, except that we don't have the possibility of taking AP Calculus/Physics type of courses. You don't get to pick your courses, everyone is subject to everything at the same level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope the answers are applicable despite of what specific things they are dealing with, but as a small breakdown we have&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Last year high school students are dealing with polynomials in algebra;&lt;/li&gt;&#xA;&lt;li&gt;Second year high school students are learning matrices;&lt;/li&gt;&#xA;&lt;li&gt;First year high school students are learning functions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;They all have geometry in a separate class, creating two math classes, but I don't know at what stage each of them are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see that they study hard and do many exercises, yet they lack basic understanding and their knowledge is almost completely procedural. As a remedy for this I've recommended them some of the strategies proposed by Polya, with a strong emphasis in understanding what are the terms involved and what is asked. Consequently, I'm constantly hammering the point that they know the definitions and how to apply them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this sound? I know this is learned behavior and that it's uncommon to proceed from definitions, but I think this approach will result in better outcomes than they have now.&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="47" LastEditDate="2014-10-28T23:12:52.363" LastActivityDate="2014-10-29T17:47:03.943" Title="Focusing on definitions for understanding in secondary education" Tags="|secondary-education|mathematical-pedagogy|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5680" PostTypeId="2" ParentId="5653" CreationDate="2014-10-28T05:03:35.400" Score="2" Body="&lt;p&gt;Yes, every number and every thing is a multiple of one.  2 is. 5 is. 0.1 is.  Potato salad is.  Seriously, one times potato salad is still potato salad.  Multiplying by one does nothing and you can do nothing to anything.  And this has almost nothing to do with answering the test question.  It just complicates the way it has to be asked.  Answering the test question goes like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because 17 is a PRIME number.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word in the test question to obsess on here isn't multiple, or factor, it's ONLY.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;BTW, the test question, as quoted, is actually false.  It needs to be corrected to read: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;17 is a multiple of only two &lt;strong&gt;whole&lt;/strong&gt; numbers, 1 and 17. Tell why this statement is true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Because there are an infinite number of numbers that can be multiplied together to give you 17: 1.7 x 10, sqrt(17) x sqrt(17), (17/2) x 2, etc. But there are only two whole numbers.  That is why 17 is called a prime number.  Any number that has &lt;strong&gt;only&lt;/strong&gt; two whole number multiples is a prime number.&lt;/p&gt;&#xA;" OwnerUserId="3305" LastEditorUserId="3305" LastEditDate="2014-10-28T19:28:43.297" LastActivityDate="2014-10-28T19:28:43.297" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5681" PostTypeId="2" ParentId="5632" CreationDate="2014-10-28T13:59:29.533" Score="5" Body="&lt;p&gt;Euler's original heuristic may &quot;fly&quot; with people who don't know about calculus: just as polynomials (e.g., with all real roots) are essentially products of linear factors $x-\alpha$ where $\alpha$ runs through the roots, one might imagine that $\sin \pi z$ (using radian measure) is a products of something like $1-{z\over n}$ for $n$ non-zero integer, and also a factor of $z$ for the zero at $0$. We can get the constant right by looking at the limit as $z\to 0$. Then it is plausible that the next-order term is the sum of inverses of roots... which would give the desired formula.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-10-28T13:59:29.533" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5682" PostTypeId="2" ParentId="5632" CreationDate="2014-10-28T14:59:21.087" Score="6" Body="&lt;p&gt;You probably already know that $1/\zeta(2)$ is the probability of two &quot;random&quot; integers being relatively prime. Since this constant is related to number theory, you should expect any proof of the Basel formula to be hard work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the best proof from the point of view of transparency is obtained by evaluating&#xA;$$\int_0^1\int_0^1 \frac{1}{1-xy}{\rm d}x{\rm d}y$$&#xA;in two different ways; one gives $\zeta(2)$, and the other $\pi^2/6$. A version appears in&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Aigner, Martin; Ziegler, Günter (2009). Proofs from THE BOOK (4th ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-642-00855-9.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But a shorter version appears in&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Apostol, Tom M., A proof that Euler missed: evaluating $\zeta (2)$ the easy way, The Mathematical Intelligencer, vol 5, #3, 1983, 59--60.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It has the advantage of pointing out that the second evaluation of the double integral is achieved via a very appealing (read tricky) change of variables; namely rotating the coordinate axes by $\pi/4$.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="2070" LastEditDate="2015-08-03T17:25:16.223" LastActivityDate="2015-08-03T17:25:16.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5684" PostTypeId="2" ParentId="416" CreationDate="2014-10-28T18:20:27.677" Score="6" Body="&lt;p&gt;Answers of @dtldarek and @vonbrand are excellent, but Ī’m ready to add my few cents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider &lt;em&gt;k&lt;/em&gt; random independent events of &lt;strong&gt;zero&lt;/strong&gt; probability each (for example, Ī shoot for the Moon). Which probability has “all successes” event? Of course, 0&lt;sup&gt;&lt;em&gt;k&lt;/em&gt;&lt;/sup&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have no problems to realize that for &lt;em&gt;k&lt;/em&gt; = 1, 2, … “all successes” is a zero-probability event. But what for &lt;em&gt;k&lt;/em&gt; = 0? Ī didn’t attempted any shoots. Ī didn’t fail any time. Hence, Ī won with probability 1.&lt;/p&gt;&#xA;" OwnerUserId="3272" LastActivityDate="2014-10-28T18:20:27.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5685" PostTypeId="1" CreationDate="2014-10-28T19:58:56.050" Score="6" ViewCount="149" Body="&lt;p&gt;I ask &quot;At what stage during an undergrad degree should a student move beyond just 'doing' the equations?&quot; because I am taking some Year 1 maths papers in 2015. I am worried I won't be able to actually see the deeper meaning and reasoning behind the equations. I may be able to plug numbers into the formula and remember the procedures to follow, but I feel this is different to really knowing what is going on and why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked a 3rd year student and he said during years 1 and 2 that I shouldn't get too worried about the deeper meaning, rather the teachers/lecturers are wanting to see that I can do the work and arrive at the correct answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is anyone able to shed some light on this and how they see it, or share their experiences?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not fresh out of high school, rather an adult trying to get an education later in life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;thanks&lt;/p&gt;&#xA;" OwnerUserId="2003" LastEditorUserId="143" LastEditDate="2014-10-28T20:25:50.610" LastActivityDate="2014-10-29T11:31:02.713" Title="At what stage during an undergrad degree should a student move beyond just 'doing' the equations?" Tags="|undergraduate-education|intuition|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5686" PostTypeId="2" ParentId="5644" CreationDate="2014-10-28T20:00:42.477" Score="13" Body="&lt;p&gt;That is the book that had the most deleterious effect on me in my life! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the courses  in (advanced) calculus I had as an undergraduate were very old-fashioned (emphasis on esoteric criteria for convergence like Raabe's test, nonsensical definition of differentials, ..), a young teaching assistant told me to read Dieudonné's book.&lt;br&gt;&#xA;I tried to, but the result was a disaster: I could formally understand the definition of the differential of a map between Banach spaces but I didn't learn that the differential of  a map $\mathbb R^2\to \mathbb R^2$ can be represented by a $2$ by $2$ matrix.&lt;br&gt;&#xA;One of the first examples in the chapter on metric spaces  is $\mathbb Q$ with a $p$-adic distance ...&lt;br&gt;&#xA;In the first (!) set of exercises one is  requested to prove that in an ultrametric space, given  two non-disjoint balls, one is included in the other.&lt;br&gt;&#xA;I was completely discouraged, since I couldn't do the exercises, and thought that I would never understand mathematics, but fortunately I stumbled on Buck's &lt;em&gt;Advanced Calculus&lt;/em&gt; and fell in love with differntial calculus in several variables.     &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my take on &lt;em&gt;Foundations of Modern Analysis&lt;/em&gt; is that it is inadequate for learning said foundations  and that the ratio abstraction/content is unreasonably high.&lt;br&gt;&#xA;Also it contains zero pictures, which I find intolerable.&lt;br&gt;&#xA;I would recommend instead Lang's excellent, richly and intelligently illustrated, books on Calculus, Advanced Calculus and Analysis  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Warning: the book and the author are not to be confused!&lt;/strong&gt;&lt;br&gt;&#xA;I knew Dieudonné, admired and loved him.&lt;br&gt;&#xA;He was one of the great mathematicians of the twentieth century, the main and most energetic contributor to Bourbaki, the collaborator of Grothendieck  who  would not have  written EGA without Dieudonné, the inventor of paracompact spaces and partitions of unity, the inventor of determinants over skew-fields, etc...&lt;br&gt;&#xA;Even more importantly, he was selfless, utterly  honest and, despite his important position in  French academic life, was ready to very humbly put himself at the disposal of young mathematicians like Grothendieck whose talent  he helped develop.&lt;br&gt;&#xA;I miss the mathematician and the gentleman.&lt;/p&gt;&#xA;" OwnerUserId="3313" LastActivityDate="2014-10-28T20:00:42.477" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5687" PostTypeId="2" ParentId="5669" CreationDate="2014-10-28T20:48:22.357" Score="2" Body="&lt;p&gt;For Linear Algebra, try this classic MAA volume:  Resources for Teaching Linear Algebra.   It gives a lot of examples of different ways teachers have worked to make linear algebra more concrete, applications oriented, visual, technology-based, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://eric.ed.gov/?id=ED407244&quot; rel=&quot;nofollow&quot;&gt;http://eric.ed.gov/?id=ED407244&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For Vector Calculus, I particularly liked the problems and curricula from the Vector Calculus Bridge Project.  They apparently have disabled the download link, though, and you'll have to email them to get a copy of their labs and instructor's guide.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://math.oregonstate.edu/bridge/&quot; rel=&quot;nofollow&quot;&gt;http://math.oregonstate.edu/bridge/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-10-28T20:48:22.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5688" PostTypeId="2" ParentId="5665" CreationDate="2014-10-28T20:55:23.180" Score="0" Body="&lt;p&gt;I think if I were teaching this I'd probably start with the graphical approach, like, have them put both in slope-intercept y=mx+b form and then find the point of intersection using a graphing calculator, and then from there teach one of those two methods to find the answer.   I don't know that your method is necessarily &quot;better&quot; then the other method.   Generally I would want students to develop their own methods for things, to have as many different methods as they can in their &quot;toolbox,&quot; and to learn how to choose methods that they like to use to solve problems they encounter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also find myself wondering if the context would change how students approach this.  Like, what happens when you give different problems that require intersecting lines or systems of equations in multiple variables to solve?  Does the nature of the problem you give change how students would approach it, absent you giving direct instruction favoring one method over another...?&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-10-28T20:55:23.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5689" PostTypeId="2" ParentId="3886" CreationDate="2014-10-29T00:09:54.170" Score="5" Body="&lt;p&gt;I have few remarks which may be enlightening. I have not seen most of these spelled out anywhere for some reason.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not prefer one definition over the others:  in fact I think this is a very nice place to try to get students to compare and contrast various definitions, and to derive each definition from the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are my observations:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$e^x = [e^{\frac{x}{n}}]^n$.  For very large values of $n$,  $e^{\frac{x}{n}} \approx 1+\frac{x}{n}$ by the tangent line approximation to $y=e^t$ at $t=0$.  So we obtain $e^x \approx (1+\frac{x}{n})^n$.  This approximation should get better and better as $n$ increases without bound, motivating the possibility that $e^x = \displaystyle\lim_{n \to \infty} (1+\frac{x}{n})^n$.  Proving this is a nice challenge.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another way to $(1+\frac{x}{n})^n$ is Euler's Method.  Apply Euler's method to the diff EQ $f'(x)=f(x)$ with $f(0)=1$ to the interval $[0,x]$, with $n$ subintervals. We obtain:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;$\begin{align*}&#xA;f\left(\frac{x}{n}\right) &amp;amp;\approx 1+\frac{x}{n}\\&#xA;f\left(\frac{2}{n}\right) &amp;amp;\approx f\left(\frac{x}{n}\right)+\frac{x}{n}f\left(\frac{x}{n}\right) = \left(1+\frac{x}{n}\right)f\left(\frac{x}{n}\right) \approx \left(1+\frac{x}{n}\right)^2\\&#xA;&amp;amp;\vdots\\&#xA;f(x) &amp;amp;\approx \left(1+\frac{x}{n}\right)^n&#xA;\end{align*}$&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Assuming existence and uniqueness of ODE's is all that is needed to derive $e^{a+b} = e^ae^b$ and $e^{ax} = (e^x)^a$.  For the first, observe that $f_1(x) = e^{a+x}$ and $f_2(x) = e^ae^x$ both satisfy the DE $y'=y$ with initial condition $y'(0)=e^a$.  For the second note that $f_1(x) = e^{ax}$ and $f_2(x) = (e^x)^a$ both satisfy the DE $y'=ay$ with initial condition $y'(0)=1$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You may also enjoy &lt;a href=&quot;https://math.stackexchange.com/questions/498339/demystify-integration-of-int-frac1x-mathrm-dx/498790#498790&quot;&gt;this post&lt;/a&gt; where I spell out how solving equations like $2^x=3$ is connected to finding areas under $\frac{1}{t}$, and how this is all linked with the number $e$.  This story can also be turned into a very nice activity, where the students discover all of these connections themselves.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-10-29T00:09:54.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5691" PostTypeId="2" ParentId="3983" CreationDate="2014-10-29T02:56:42.027" Score="1" Body="&lt;p&gt;As illustrated by other answers, Markov processes and linear ODE systems are the foremost applications. I like to mess a little with students explaining that the &quot;only&quot; thing that Google machines do all day is compute one large eigenvector over and over agin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But to get a feeling for why are eigenvectors useful, think simply of base changes: Let's say there is no issue with multiplicity, so our transformation $A:\mathbb{R}^n \longrightarrow \mathbb{R}^n$ has $n$ distinct eigenvectors. Then the basis of eigenvectors is the most natural one since it is invariant (up to scale). How does an object transform under $A$? It stretches by eigenvalue #$1$ in the first direction, by eigenvalue #$2$ in the second direction, and so on. It is the convenience of this description that makes eigenstuff useful...&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2014-10-29T02:56:42.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5692" PostTypeId="1" CreationDate="2014-10-29T08:24:53.167" Score="8" ViewCount="467" Body="&lt;p&gt;I've been searching for a repository or a database with math problems (K-12 levels) written in &lt;code&gt;LaTeX&lt;/code&gt;, but with no luck. Neither in Spanish (my student's language) nor in English. But I bet this is something someone must have done before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are things like &lt;a href=&quot;http://www.mathmaster.org/worksheet/&quot;&gt;mathmaster.org&lt;/a&gt; where you can download pdf worksheets. Indeed, &lt;code&gt;mathmaster&lt;/code&gt; uses &lt;code&gt;LaTex&lt;/code&gt; as a backend, but users can just download the final pdf (apart from this technical detail, mathmaster is a great resource).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyone knows about such LaTeX repository? Does it exist?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if there is no LaTeX... Is there any open source repo of any kind?&lt;/p&gt;&#xA;" OwnerUserId="3260" LastActivityDate="2015-10-08T18:42:51.580" Title="Is there a repository with K-12 math exercises and problems written in LaTeX (English or Spanish)?" Tags="|textbooks|homework|problem-design|exercises|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5693" PostTypeId="2" ParentId="5685" CreationDate="2014-10-29T11:31:02.713" Score="3" Body="&lt;p&gt;My experience has been that this can happen as early as you like. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach College Algebra, Calculus, and Differential Equations at a community college. In all of these courses, it is possible to see connections and ask &quot;why&quot; questions. I truly believe that if you start asking why and finding out what is going on, it is significantly easier to learn and retain the material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even as early as the identity $x^0 = 1$, it is very productive to ask why!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your professors will almost certainly have office hours available. Appearing in their office to ask these questions is a very good idea.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-10-29T11:31:02.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5694" PostTypeId="2" ParentId="5679" CreationDate="2014-10-29T14:58:29.653" Score="2" Body="&lt;p&gt;What you call &lt;code&gt;proceed from definitions&lt;/code&gt; is quite similar to the teaching practices of the  &lt;a href=&quot;http://en.wikipedia.org/wiki/New_Math&quot; rel=&quot;nofollow&quot;&gt;New Math&lt;/a&gt; or Modern Mathematics (UK) during the 1960s. Definitions, corollaries, theorems, etc. seem to us, as teachers, the perfect method to learn pure mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, not every student in your class will want to become a mathematician or an engineer. Even those who show now interest in Mathematics, don't have the necessary background to learn in a significant way. Just because they are young or do not have the required cognitive level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;Polya strategies&lt;/code&gt; are fine. But just giving them to the students doesn't make a miracle. I bet you don't think in Polya when solving a problem. Nor your students. Yes, they can learn the rules and the steps, but most of them will fail to apply them in a new problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can we handle it? Here are a couple of proposals. And I say just two, because there are a &lt;code&gt;bunch of movements&lt;/code&gt; (not just Polya) which try to answer this question:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Obvious: one learn to solve problems by solving problems. &lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As we have a limited period for teaching (3-4 hours a week?), we should work efficiently. This means to choose the proper problems. And, better than problems, situations with a rich context. How do we know if problem, a collection of problems, or a situation is good as a learning experience?: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Looking at how many different concepts and other math objetcs (properties, arguments, etc.) does it articulate, the more, the better.&lt;/li&gt;&#xA;&lt;li&gt;Looking at how many different ways do a concept or a math object appear. For example, algebraically, graphically, verbally, etc. Again, the more, the better.&lt;/li&gt;&#xA;&lt;li&gt;By using contexts that may be interesting for our students. Or real-life contexts.&lt;/li&gt;&#xA;&lt;li&gt;By using manipulatives (even if they are in high school). The thing is to make the situation appear as real as possible. &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="3260" LastEditorUserId="3260" LastEditDate="2014-10-29T17:47:03.943" LastActivityDate="2014-10-29T17:47:03.943" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5695" PostTypeId="1" AcceptedAnswerId="5704" CreationDate="2014-10-29T17:49:53.037" Score="2" ViewCount="252" Body="&lt;p&gt;I have been telling my students to try to solve the problems on their as much as they can. As I tell them, it will help them be better at solving and understanding problems of Mathematics. This thing I learned from Khan Academy actually. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But students seem to have problems on this aspect. To me it seems like they don't know what to think. Before asking them to solve the exercises, I first do some problems on the board. As I do that, the students now can solve only the problems that was similar to the ones I did on the board. When questions come with a little tweak, they blank out. And then I ask them to think harder but they become blank. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would like to ask you how to tell them to think, let's say, mathematically in this context? Secondly, how should I encourage them to try to solve the problems on their own?&lt;/p&gt;&#xA;" OwnerUserId="1172" LastEditorUserId="143" LastEditDate="2014-10-29T19:05:47.867" LastActivityDate="2014-10-31T16:36:09.637" Title="Solving problems" Tags="|secondary-education|mathematical-pedagogy|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5696" PostTypeId="2" ParentId="5676" CreationDate="2014-10-30T02:32:52.377" Score="2" Body="&lt;p&gt;The problem is, in almost all &lt;em&gt;recurring&lt;/em&gt; human interactions, almost nothing exhibits &quot;random&quot; (i.e. equiprobable) behavior, especially at the micro-level (and at the macro-level it is more likely to observe behavior represented by other distributions rather than the Uniform one -Normal and Pareto come easily to mind). So even though the distinction &quot;fair/not fair&quot; is very powerful and indeed engaging compared to say &quot;test that &quot;so and so&quot; follows or not a Pareto distribution&quot;, it will be rather hard to find many convincing real world &lt;em&gt;examples&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my Statistics professors when I was an undergraduate narrated to us how he had an argument with his wife about whether two different and long urban itineraries from their house to some relative they would visit often, were the same in terms of travel time (using car)... and he conducted an &quot;&lt;strong&gt;equality of means&lt;/strong&gt;&quot; hypothesis test to check it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the &quot;average&quot; is a fundamental statistical concept which is also intuitively known to everybody even before he hears the word &quot;Statistics&quot; for the first time, perhaps building examples and task-assignments around &quot;equality of means&quot; could be engaging and educationally beneficial. It would also have the students  &lt;em&gt;collecting the data themselves&lt;/em&gt; which is &lt;em&gt;very&lt;/em&gt; useful as an experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, there may be the impression that Sally &quot;talks more on her cell phone&quot; than her friend Sue. But is this statistically validated? Record number of phonecalls and duration of each call (both are provided by the cell phones), for say 7 consecutive days (to include possible daily seasonality over a week), and conduct &quot;equality of means&quot; test for number of phone calls, and another for duration of calls. This specific example serves also to bring in the concept of independence - we expect that Sally and Sue talk to each other over the cell phones, being friends - so the two samples will include common observations. Should they? Should we exclude them? Why? etc&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2014-10-31T02:39:04.623" LastActivityDate="2014-10-31T02:39:04.623" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5698" PostTypeId="2" ParentId="1896" CreationDate="2014-10-30T14:26:51.367" Score="3" Body="&lt;p&gt;Obviously, and its ilk, do not contain much information. They can be replaced by referring to the tool that is used to prove or notice the obvious fact.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example: By triangle inequality, by definition (of something specific where necessary), by calculation, by integrating by parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also: By lengthy calculation, by very clever choice of test function, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here the statement communicates something about the difficulty and the tools used, thus making it easier to immediately verify something, or alternatively telling why one should not be discouraged at not immediately seeing it.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2014-10-30T14:26:51.367" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5700" PostTypeId="1" AcceptedAnswerId="5707" CreationDate="2014-10-31T02:14:27.747" Score="21" ViewCount="1398" Body="&lt;p&gt;I was helping out in a learning support class today and we were working through some absolute value problems when something like $|x + 4| - 5 = 10$  came up and both students I was working with split the problem up into &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$|x + 4| - 5 = 10$ and&#xA;$|x + 4| - 5 = -10$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;when I tried to explain that this was not correct, I couldn't seem to find a way to put it concisely.  I ended up just telling them, &quot;make sure that the absolute value is by itself&quot; which  I feel like doesn't really explain the reasoning and the students didn't seem to be able to connect it with what we had been doing previously with absolute values and inequalities.  Any suggestions on a better way to explain this?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2014-12-04T16:40:26.023" LastActivityDate="2015-12-21T23:22:29.843" Title="Good way to explain why an absolute value in an equation does not automatically mean to make the other side +/-" Tags="|secondary-education|algebra|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5701" PostTypeId="2" ParentId="5700" CreationDate="2014-10-31T04:00:54.433" Score="12" Body="&lt;p&gt;I find it helpful to teach the concept that $|x - a|$ is exactly the &lt;em&gt;distance of $x$ from $a$&lt;/em&gt;. This also generalizes easily to higher dimensions, where $|x - a|$ will be the norm of the vector $x - a$ (no matter which norm/metric you use). While teaching this, explain it using the number line, with a circle centred at $a$. Of course, since the only possible points are &lt;em&gt;on&lt;/em&gt; the number line, you have to consider only the points where the circle intersects the line, which are at equal distance from $a$ and on either side of it. That is, if $|x - a| = d$, then the points (limits of $x$) will be at distance $d$ from $a$, and these will clearly be $a + d$ and $a - d$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once this concept is understood, $|x + 4| - 5 = 10$ becomes: &quot;The distance of $x$ from $-4$, less $5$, is $10$, so the distance of $x$ from $-4$ is $15$, thus $x$ is either $-4 + 15 = 11$, or $-4 - 15 = -19$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One other advantage of this approach is that it is also just as easy to solve &lt;em&gt;inequalities&lt;/em&gt; involving $|\cdot|$. There is no need to remember complicated [for students] rules about changing the inequality with the sign.&lt;/p&gt;&#xA;" OwnerUserId="1606" LastActivityDate="2014-10-31T04:00:54.433" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5702" PostTypeId="1" CreationDate="2014-10-31T09:06:56.720" Score="7" ViewCount="309" Body="&lt;p&gt;&lt;em&gt;I &lt;a href=&quot;https://math.stackexchange.com/q/989000/74479&quot;&gt;asked&lt;/a&gt; this question in Math.S.E. but got no answer!&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm going to introduce my middle school students to the distributive law in arithmetic, in a meaningful way so that they understand its importance and value. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I need some interesting examples and problems like the one below. Do you know any such example?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My example:&lt;/strong&gt; In this figure, the solid arcs make a path. How long is it? Find the answer in less than one minute! No calculator allowed! $\pi$=3.14$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/NwTW1.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-11-02T16:31:11.690" Title="Appreciating the Distributive Law" Tags="|secondary-education|algebra|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5703" PostTypeId="2" ParentId="5702" CreationDate="2014-10-31T11:36:00.847" Score="6" Body="&lt;p&gt;People who are competent at mental arithmetic routinely use the distributive law when doing multiplication. If asked to evaluate $38\cdot14$, I would immediately break it up as $38\cdot10+38\cdot4=380+38\cdot2\cdot2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I think it's more important to focus on the &lt;em&gt;meaning&lt;/em&gt; of the distributive law rather than on its applications. The meaning is of course that if I have $a$ of something in one hand and $b$ in the other, than I have $a+b$ in total, and this applies even if that &quot;something&quot; is $x$ of some object, thus we conclude that $ax+bx=(a+b)x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe you're worried that once this is explained, students will feel it's so obvious that it couldn't possibly be of any use. Even in the mental arithmetic example I gave above, you don't have to be explicitly aware of the symbolic law $ax+bx=(a+b)x$ to apply that kind of reasoning, most people do it without thinking. You only really appreciate the value of the law when you do algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think most students should be able to solve equations of the form $ax=b$ even before taking algebra. Until you've taken algebra, though, an equation like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$7(5x+6)+2x=3x+18(5x+3)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;seems utterly intractable. What's remarkable is that through simple application of distributivity, the above reduces to the form $ax=b$ and can then immediately be solved.&lt;/p&gt;&#xA;" OwnerUserId="115" LastEditorUserId="115" LastEditDate="2014-10-31T12:03:13.160" LastActivityDate="2014-10-31T12:03:13.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5704" PostTypeId="2" ParentId="5695" CreationDate="2014-10-31T15:18:47.437" Score="5" Body="&lt;p&gt;Perhaps they go blank when you ask them to think because it's been so long since they've tried to think in math class. Sadly, many (most?) students just try to memorize procedures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You'll need to find problems that suck them in, so they really &lt;em&gt;want&lt;/em&gt; to think. Also, you'll need to give them some strategies for problem-solving. &lt;em&gt;How to Solve It&lt;/em&gt;, by George Polya has lots of great ideas. (Here's &lt;a href=&quot;http://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=1&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CCUQFjAA&amp;amp;url=http%3A%2F%2Fmath.berkeley.edu%2F~gmelvin%2Fpolya.pdf&amp;amp;ei=caJTVL_YJqL7igKIoYDYBQ&amp;amp;usg=AFQjCNFKYGQCAcVuQxPWXBcBA3hcB5Yuxg&amp;amp;sig2=zKZ4fqZiPo85cFOfdqCxJA&amp;amp;bvm=bv.78677474,d.cGE&quot; rel=&quot;noreferrer&quot;&gt;an online copy&lt;/a&gt; of the one-page outline he devised.) You can also find some excellent material at Avery Pickford's blog, by searching on &quot;&lt;a href=&quot;http://www.withoutgeometry.com/search?q=habits%20of%20mind&quot; rel=&quot;noreferrer&quot;&gt;habits of mind&lt;/a&gt;.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Brainstorming helps, which means working in groups is much more effective than working solo. But it takes a lot of savvy to facilitate good groupwork. One principle for what types of problems are best used in this situation is that they have multiple solution paths, and a somewhat open answer. This sort of problem is often called a &quot;&lt;a href=&quot;http://www.risps.co.uk/&quot; rel=&quot;noreferrer&quot;&gt;rich task&lt;/a&gt;.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're teaching high school, there has been work done on &quot;&lt;a href=&quot;http://mathmamawrites.blogspot.com/2011/07/complex-instruction.html&quot; rel=&quot;noreferrer&quot;&gt;complex instruction&lt;/a&gt;,&quot; which refers to a suite of strategies used to get students thinking, while using lots of groupwork. I've found this work useful, although I am not comfortable using all their strategies with my college students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One more note on your students' responses to your attempts so far: Students have taken math for many years by the time they reach you, and have expectations of math class. When you violate their expectations, there will be pushback. According to James Stigler, author of &lt;em&gt;The Teaching Gap&lt;/em&gt;, much of it is deeply imprinted in us, as what school &lt;em&gt;is&lt;/em&gt;. He writes, “The scripts for teaching in each country appear to rest on a relatively small and tacit set of core beliefs about the nature of the subject, about how students learn, and about the role that a teacher should play in the classroom.”&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-10-31T15:18:47.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5705" PostTypeId="2" ParentId="5695" CreationDate="2014-10-31T16:36:09.637" Score="2" Body="&lt;p&gt;I think that you may try this procedure. Go to the classroom. Teach them the material, if possible with a bit of historical and some amusing facts. But don't solve any problem. Tell the students to tell you to give a problem that they thinks is difficult. If someone gives you a problem, don't work it out fully on the board. You first ask him/her to tell you how far he/she has progressed. Then your job will simply be to involve all others of the class to the problem. If the student's approach is on the right path, just help him a bit and leave the rest upto him/her. If he/she is on the wrong track, produce an example which contradicts the student's procedure and say him/her to explain that using his/her theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example if the student says that he proved that the interchange of limits in any case is always possible. You may then ask him to evaluate the limits $\displaystyle\lim_{x \to 0}\displaystyle\lim_{y \to 0}\dfrac{x^2}{x^2+y^2}$ and $\displaystyle\lim_{y \to 0}\displaystyle\lim_{x \to 0}\dfrac{x^2}{x^2+y^2}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This practice will help the students to develop their own thinking skills. If the students see that the teacher is taking special attention to those who can independently solve problems, they will (as is expected in general), try to put more effort in independently solving problems.      &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that in this respect the approach of &lt;a href=&quot;https://www.google.co.in/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=2&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CCUQFjAB&amp;amp;url=https%3A%2F%2Farchive.org%2Fdetails%2FTarasovCalculus&amp;amp;ei=1bdTVMHYDdGD8gW2yIHIAw&amp;amp;usg=AFQjCNGrbRYglcQHYGpktgkhLXnJ8hGhuw&amp;amp;bvm=bv.78677474,d.dGc&quot; rel=&quot;nofollow&quot;&gt;this book&lt;/a&gt; will help you a lot.&lt;/p&gt;&#xA;" OwnerDisplayName="user3319" LastActivityDate="2014-10-31T16:36:09.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5706" PostTypeId="1" AcceptedAnswerId="5744" CreationDate="2014-10-31T20:48:46.340" Score="8" ViewCount="1202" Body="&lt;p&gt;$5^x=326$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A book titled &quot;algebra and trigonometry&quot; by Paul Foerster, offers this problem. It came before the logarithm chapter and specifically said not to solve with any log manipulation. It ended with the requirement that the result be expressed to 3 digits. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am posting here and not the Math.SE site as I'm less interested in the solution, per se, than the method used to encourage the students to look at this in a way that makes sense for them to solve. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The time was limited, and with so many questions, I skipped this one, offering the hint that $5^3=125$ and $5^4=625$ but at the time couldn't offer much more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After sleeping on it, I have a method that uses the square key, and nothing else beyond 4 basic functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be clear - the question - &quot;how to encourage students to solve a problem of this type, given the constraints?&quot;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="1147" LastEditDate="2014-11-02T01:31:22.270" LastActivityDate="2022-10-24T23:51:19.293" Title="Method of Solving $5^x=326$ (Logs not allowed)" Tags="|inverse|exponential|logarithm|" AnswerCount="9" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5707" PostTypeId="2" ParentId="5700" CreationDate="2014-10-31T22:31:10.540" Score="31" Body="&lt;p&gt;One of the reasons your students are putting the $\pm$ on the 10 is probably because someone told them that when you remove a pair of absolute value signs that the $\pm$ goes on the number after the $=$. They are simply doing what they thought they were told to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I try to avoid telling them this sort of thing. I like to tell them is that $|x|$ is equal to $+x$ or $-x$ depending on the value of $x$. In short, $|x| = \pm x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may seem like it's not any different, but the key point I stress is that &lt;em&gt;the $|x|$ itself and all by itself is equal to $\pm x$ regardless of whether it's in an equation or not&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this basically means that when you remove the absolute value signs, the plus or minus attaches itself to whatever had the absolute value signs on it before. So the solution is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;|x+4| - 5 = 10\\&#xA;\pm(x+4) - 5 = 10\\&#xA;+(x+4) - 5 = 10 \quad \text{or} \quad -(x+4) - 5 = 10\\&#xA;x+4 - 5 = 10 \quad \text{or} \quad -x - 4 - 5 = 10\\&#xA;x - 1 = 10 \quad \text{or} \quad -x - 9 = 10\\&#xA;x = 11 \quad \text{or} \quad  -x = 19\\&#xA;x = 11 \quad \text{or} \quad x = -19&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I do this even for the simple absolute value equations like $|x| = 3$, beginning with $\pm x = 3$, even though it introduces an extra step. I do this because I know in advance that it will prevent problems later, and also it reinforces that it is all to do with the meaning of the absolute value signs, not the equation per se. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This approach works quite well when there are two sets of absolute values in an equation such as $|x+2| + |x-2| = 5$, because you get $\pm(x+2) \pm (x-2) = 5$ so you can much more easily see that there are in fact four options.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something else it is worth stressing is that they should always check their solution to see if it works in the original equation. Because $|x| = \pm x$ depending on the value of $x$ and we haven't been careful to specify which values of $x$ make it $+$ and which make it $-$ in our solution, sometimes we get extra solutions that don't exist, because the value of $x$ that we find doesn't match with when it's the $+$ or the $-$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Graphing the function $y = |x+4| - 5$ and finding where it hits the horizontal line $y=10$ is also a good activity to help them see visually what the answer ought to be, but I find that while it helps them see that their answer is wrong, it doesn't help them see how to go about solving the equation on paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: of course the above equation-solving approach is not what I would actually use, because I always tell students they should try to deal with the things that are furthest from the $x$ first. So from their experience solving ordinary equations, they should deal with the $-5$ first like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;|x + 4| - 5 = 10\\&#xA;|x+4| = 15\\&#xA;\pm(x+4) = 15\\&#xA;+(x+4) = 15 \quad \text{or} \quad -(x+4) = 15\\&#xA;x + 4 = 15 \quad \text{or} \quad x + 4 = -15\\&#xA;x = 11 \quad \text{or} \quad  x = -19&#xA;$$&#xA;However, I still put the $\pm$ on the thing that had the absolute value signs though, because this transfers to other situations.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-11-02T23:12:39.893" LastActivityDate="2014-11-02T23:12:39.893" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5708" PostTypeId="2" ParentId="5706" CreationDate="2014-10-31T22:52:59.813" Score="10" Body="&lt;p&gt;One could solve this by &quot;guess and check&quot;, together with the knowledge that $5^x$ is increasing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So start with your observation that $3 &amp;lt; x &amp;lt; 4$.  Test $3.1, 3.2, 3.3, ...$ and observe that $3.5&amp;lt;x&amp;lt;3.6$.  Repeat the process for the next two decimal places.  I am not sure if this is what they intended.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another method which could be used if they had access to a spreadsheet is to make a table of values for $(1.001)^n$. Then do a reverse lookup for $5$ and $326$.  You will find that $(1.001)^{1610} \approx 5$ and $(1.001)^{5790} \approx 326$.  So we are trying to solve $[(1.001)^{1610}]^x = 1.001^{5790}$, which amounts to solving the linear equation $1610x=5790$, so $x \approx \frac{5790}{1610} \approx 3.596$ which is pretty good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This last method is motivated by &quot;finding a common base&quot;.  If you cannot find a common base, you can always &lt;em&gt;approximately&lt;/em&gt; find a common base by using a number slightly bigger than, but very close to, $1$.  This in turn motivates the natural logarithm, and can be used to see the connection between the natural logarithm and the function $y=\frac{1}{x}$, as in &lt;a href=&quot;https://math.stackexchange.com/questions/498339/demystify-integration-of-int-frac1x-mathrm-dx/498790#498790&quot;&gt;this post&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-10-31T22:52:59.813" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5709" PostTypeId="2" ParentId="5706" CreationDate="2014-11-01T03:41:06.997" Score="24" Body="&lt;p&gt;From the perspective of mathematics education, I must ask:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why do you &lt;em&gt;want&lt;/em&gt; to encourage students to solve such a problem without logarithms?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is a connection between square roots and logarithms (e.g., &lt;strong&gt;&lt;a href=&quot;https://web.archive.org/web/20190302045628/http://mathforum.org/library/drmath/view/66630.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;) but I consider the mathematics that underlies it quite opaque without understanding &lt;span class=&quot;math-container&quot;&gt;$x\mapsto \log(x)$&lt;/span&gt; beforehand. Personally, my inclination is &lt;em&gt;not&lt;/em&gt; to encourage students to solve a problem like this under the given constraints; the best approach I can think of is the already mentioned one using a combination of guess-and-check and assuming continuity plus knowledge of the intermediate value theorem.&lt;/p&gt;&#xA;&lt;p&gt;For the sake of completeness, the &lt;em&gt;mathematics&lt;/em&gt; described in the aforelinked suggests computing:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{n \rightarrow \infty}\frac{326^{(1/2^n)}-1}{5^{1/2^n} - 1} = 3.5956\ldots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which can be carried out to reasonable precision by clicking the square root key many, many times.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps this is the method that you already had in mind; I include it here for the mathematically curious reader, though I do not see how to satisfy the ensuing curiosity as to &lt;em&gt;why that works&lt;/em&gt; without bringing logarithms (and limits) into the discussion.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="6053" LastEditDate="2022-10-22T10:13:49.080" LastActivityDate="2022-10-22T10:13:49.080" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="5710" PostTypeId="2" ParentId="5706" CreationDate="2014-11-01T05:28:58.403" Score="2" Body="&lt;p&gt;There is only one way (that I know of) to solve this without knowledge of Calculus (and a calculator--that is without tables--although my solution will rely on the computational power of a calculator).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, I &lt;em&gt;do not&lt;/em&gt; see the question as appropriate for the level of algebra &lt;em&gt;or&lt;/em&gt; trigonometry (I certainly don't see the link to trigonometry).  Frankly, I don't see this problem appropriate for any class other than a scientific computations class (where you attempt to approximate solutions).  As has been expressed, the &quot;guess and test&quot; method is not something I recommend &lt;em&gt;at all&lt;/em&gt;!  Guess and test literally means, guess &lt;em&gt;anything&lt;/em&gt; and see if it works--there is little rhyme or reason to the guesses.  A computer science class (or calculus class) on approximating solutions can shed some light on approximation methods, but this is &lt;em&gt;far&lt;/em&gt; beyond the scope of an algebra or trigonometry class.  As is, &quot;guess and test&quot; at the level of algebra or geometry is basically guess an integer, if it's not right, guess another integer.  That is sort of what I will employ.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bisection Method&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You must first bound the solution.  Just start guessing integers: $5^2 = 25$, $5^3 = 125$. $5^4 = 625$, there we have bounded the solution, it's between $3$ and $4$.  Assuming $5^x$ is monotonic (which it is) then the solution &lt;em&gt;must&lt;/em&gt; be between $3$ and $4$.  So choose something exactly in between: $3.5$ (which actually &lt;em&gt;does&lt;/em&gt; require a calculator):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;5^{3.5} \approx 279.5085 &amp;lt; 326&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is less than the target solution of $326$, so the &lt;em&gt;actual&lt;/em&gt; solution must be between $3.5$ and $4$.  Keep going:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;5^{3.75} \approx 417.96269061 &amp;gt; 326 \\&#xA;5^{3.625} \approx 341.795441065 &amp;gt; 326 \\&#xA;5^{3.5625} \approx 309.086929645 &amp;lt; 326 \\&#xA;5^{3.59375} \approx 325.030003916 &amp;lt; 326 \\&#xA;5^{3.609375} \approx 333.307325974 &amp;gt; 326 \\&#xA;5^{3.6015625} \approx 329.142646077 &amp;gt; 326 \\&#xA;5^{3.59765625} \approx 327.079861109 &amp;gt; 326 \\&#xA;5^{3.595703125} \approx 326.053321617 &amp;gt; 326 \\&#xA;5^{3.5947265625} \approx 325.541260675 &amp;lt; 326 \\&#xA;5^{3.59521484375} \approx 325.797190544 &amp;lt; 326 \\&#xA;5^{3.59545898438} \approx 325.925230923 &amp;lt; 326&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since we &lt;em&gt;finally&lt;/em&gt; got to values that have the same three last digits, we arrive at $5^x = 326 \rightarrow x \approx 3.595$.&lt;/p&gt;&#xA;" OwnerUserId="2204" LastActivityDate="2014-11-01T05:28:58.403" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5711" PostTypeId="1" AcceptedAnswerId="5712" CreationDate="2014-11-01T06:07:09.043" Score="13" ViewCount="714" Body="&lt;p&gt;I am teaching abroad to Chinese non-native English speakers with a large variance of language skills.&lt;/p&gt;&#xA;&lt;p&gt;I teach both pre-calculus and AP calculus (AB &amp;amp; BC). For both of those classes I define the new terms.  I use word problems, have them read from the textbook, take notes in English during class, have the students speak English in class, and teach the class in English.&lt;/p&gt;&#xA;&lt;p&gt;I am asking this question because my CP (Center Principle) told me I need to embed more English into my lesson, so the students learn English.&lt;/p&gt;&#xA;&lt;p&gt;What are some techniques to embed English into a mathematics lesson, and what skills and lessons have you learned from teaching mathematics in an ESL environment?&lt;/p&gt;&#xA;" OwnerUserId="2182" LastEditorUserId="19777" LastEditDate="2023-05-11T16:23:01.763" LastActivityDate="2023-05-11T16:23:01.763" Title="How to embed English into Calculus lessons" Tags="|secondary-education|general-pedagogy|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5712" PostTypeId="2" ParentId="5711" CreationDate="2014-11-01T08:10:59.817" Score="8" Body="&lt;p&gt;I teach in an ESL environment where students come from a variety of countries. I try to be particularly sensitive to differences in notation/convention in mathematics, which seem to come up most often in the notation for the decimal point, what sign to use for multiplication and how logarithms are written.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have also learned to take my time and avoid using too much idiomatic or low-frequency vocabulary. I usually state difficult points twice, often reformulated slightly the second time, to give students a second chance to understand. Of course, this applies to a non-ESL environment too, but it is especially important when students are struggling to follow the subtleties of mathematics in a second or foreign language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I do not attempt to turn my mathematics lessons into English lessons, apart from the occasional remark. I do, however, mention when mathematical English usage differs from everyday English usage, a classic example being the word &quot;series.&quot; I would also say that getting students to work in small groups to solve problems will provide practice in speaking English. You can also set written assignments requiring students to write in full English sentences and not just the symbolic language of mathematics. For the latter, Kevin Houston's &lt;a href=&quot;http://www1.maths.leeds.ac.uk/~khouston/pdf/htwm.pdf&quot; rel=&quot;nofollow&quot;&gt;How to Write Mathematics&lt;/a&gt; may come in handy.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2014-11-01T19:14:15.717" LastActivityDate="2014-11-01T19:14:15.717" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5713" PostTypeId="2" ParentId="5711" CreationDate="2014-11-01T21:27:43.917" Score="2" Body="&lt;p&gt;Well, apart from what you are already doing, there are not too many things you can do. You teach mathematics in English during Math class. And your students learn the grammar, general vocabulary, etc. during English class. And this is something your principal should understand. Depending on the level of your students, it will be also allowed to use their native language to explain some misunderstandings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, you could satisfy your principal if using, for example, short video fragments taken from films, TV series, Youtube, etc. Take a look to these sites, they are just a sample of what you can find:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.qedcat.com/moviemath/index.html&quot; rel=&quot;nofollow&quot;&gt;Mathematical Movie Database&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://numb3rs.wolfram.com&quot; rel=&quot;nofollow&quot;&gt;The Math Behind Numb3rs&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://mathsci2.appstate.edu/~sjg/simpsonsmath/&quot; rel=&quot;nofollow&quot;&gt;Simpsons Math&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main idea is to create a problem or situation with a real-life context, or easily imaginated by the students. That is where the fragments help. They provide a visual anchor where develop further mathematics. Sometimes an explicit problem will also be set from an scene (it's a classic the movie &lt;em&gt;Die Hard: With a Vengeance&lt;/em&gt;, where some problems are proposed to Bruce Willis). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By this way you will not only satisfy your principal, but also improving your teaching practice. &lt;/p&gt;&#xA;" OwnerUserId="3260" LastActivityDate="2014-11-01T21:27:43.917" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5714" PostTypeId="2" ParentId="5635" CreationDate="2014-11-01T22:21:17.977" Score="3" Body="&lt;p&gt;&lt;strong&gt;&quot;A History of School Mathematics&quot;&lt;/strong&gt; edited by George M. A. Stanic and Jeremy Kilpatrick is a must for any mathematics educator. I understand that you plan on becoming a research mathematician, but even if you do achieve your ultimate goal, at some point you will probably also be in front of the classroom as a teacher. The book is more a collection of essays on topics that are organized in a chronological and cohesive way, and really only focuses on North American schools, but it is still valuable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Plus, you can buy it &lt;a href=&quot;http://www.nctm.org/catalog/product.aspx?ID=12057&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; for relatively cheap (about a nickel a page).&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-11-01T22:21:17.977" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5715" PostTypeId="1" AcceptedAnswerId="5716" CreationDate="2014-11-01T23:06:36.307" Score="10" ViewCount="311" Body="&lt;p&gt;What is a good way to explain the slightly different kinds of continuity to students? I have in mind these kinds of continuity:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A function is continuous at a point. (This also has two sub-kinds: continuity at an interior point and at a left or right endpoint of the domain.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A function is continuous in an interval.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A function is continuous. (I.e. it is continuous at every point in its domain.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My students get these ideas mixed up, no matter how hard I try to distinguish the ideas and the terminology. Do you have a good way to teach these concepts?&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="4872" LastEditDate="2015-06-14T07:28:21.540" LastActivityDate="2015-06-14T14:05:26.990" Title="What is a good way to explain the slightly different kinds of continuity?" Tags="|secondary-education|calculus|functions|definitions|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5716" PostTypeId="2" ParentId="5715" CreationDate="2014-11-02T00:28:48.917" Score="4" Body="&lt;p&gt;In my Calculus classes I really like using the function $f(x) = \frac{1-x^2}{1-x}$ as a running example. First I use it to exemplify how the definition of a function must include information about its domain. For instance, $f$ is not quite the same as the function $1+x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later, I se $f$ to talk about continuity, because it is a good example of a function that is not continuous at exactly one point, even though the limit exists. Then we talk about how you can extend a function by continuity, thus recovering $1+x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, we use $f$ to visualize the derivative of $x^2$ at $x=1$. The point I make is that the quotient definition of derivative &lt;strong&gt;needs&lt;/strong&gt; to use a limit, because the quotient itself is not defined. Usually (meaning for quite a few students) the understanding of the continuity issue becomes ingrained as a collateral effect of understanding the definition of derivative. And that kind of is the point in a Calculus class, right?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As an aside, I also use $f$ in Complex Analysis, now as a function of a complex variable, to explain the subtle notion of a removable singularity. Without a concrete example, they usually miss that there is meat to this definition.)&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2014-11-02T00:28:48.917" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5717" PostTypeId="2" ParentId="5706" CreationDate="2014-11-02T00:43:33.607" Score="5" Body="&lt;p&gt;This is my method of solving. It uses the square function, and division. One then must accumulate the powers of 1/2, i.e 1/2,1/4,1/8, etc that are noted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/8CK96.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Method - &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A) Divide by the base, in this case 5, and determine the exponent has a 3 prior to the decimal. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;B) Take the result and square it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;C) If you can divide  the result by 5, a binary 1 is noted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;D) If you can't divide by 5 (i.e. the number is less than 5) square it, and note a binary 0.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Repeat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since $2^{10} \approx 1000$ 10 bits will offer 3 significant decimal digits. 10 square functions are needed, but, on average, only 5 divides, so 15 operations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The chart above shows a binary .100110000111, which is easily converted to decimal using reciprocal of 2,4,8, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;No calculus, no logs, and since the original 4 function calculator allowed X= ('times equal') to square the number shown, this can be calculated using nothing but a 4 function calculator. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I purposely didn't post on MSE, because I was more interested in the pedagogical aspect of such an approach. In my opinion, this method lends itself to a segue to logarithms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's no guessing, and the calculation to achieve N digits of accuracy is both known and in my opinion, reasonable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the calculations aren't clear, please comment. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-02T00:43:33.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5718" PostTypeId="2" ParentId="5635" CreationDate="2014-11-02T01:48:21.333" Score="9" Body="&lt;p&gt;Here goes nothing. I have many other suggestions that are not listed; there is always one more example of a great math book:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;The enjoyment of Mathematics&quot; H. Rademacher, O. Toeplitz.&lt;/em&gt; You can't beat this classic collection of serious (and beautiful) results with elementary proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;A mathematician's miscellany&quot; J. Littlewood.&lt;/em&gt; This is much more advanced, but it is a wonderful collection of math, anecdotes, and insights into the life of a professional mathematician. Sometimes it has a very pedantic tone, but always, always profoundly human.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Development of mathematics in the 19th century&quot; F. Klein.&lt;/em&gt; You can browse and browse and always extract a bit more insight into what those mythical giants were thinking about when they created modern mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Anything by M. Gardner.&lt;/em&gt; Our enterprise is a serious one, but its recreational side is a major source of inspiration and ideas. The older books are better in the sense that they portray better the network of curious minds that kept in contact through Gardner in the primitive pre-internet ages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Goedel, Escher and Bach&quot; D. Hofstadter &lt;strong&gt;plus&lt;/strong&gt; anything by R. Smullyan.&lt;/em&gt; Someone said that Goedel's Theorem is the one result from 20th century Mathematics that every mathematician should know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Anything by Milnor.&lt;/em&gt; I have said it &lt;a href=&quot;https://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/116427#116427&quot;&gt;elsewhere&lt;/a&gt;. If you want to learn to write better, absorb the style of the masters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Specific topics:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Three-dimensional geometry and topology vol. 1&quot; W. Thurston.&lt;/em&gt; Just open at random and swim into an ocean of ideas. The original Thurston notes are already available in the web. They are very different from the book, so you can swim into two connected oceans.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Complex analysis: an introduction&quot; L. Ahlfors.&lt;/em&gt; Although the language feels now a little old, you can open almost any page and find jewels of exposition. I am always surprised by the incredible precision in the choice of proofs, exercises, topics, and topic order. It is an incredibly methodic book and very thorough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Differential equations, dynamical systems and linear algebra&quot; M. Hirsch, S. Smale.&lt;/em&gt; I was told to study by hunting for the many, many, typos. The newest edition has been cleaned, updated, and expanded by Devaney, but it remains a great book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Ordinary differential equations&quot; V. Arnold.&lt;/em&gt; Even if this is not your thing, you should read at least the first three chapters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Introduction to commutative algebra&quot; M. Atiyah, I. MacDonald.&lt;/em&gt; A short book, and very readable (at least the first few chapters).&lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-11-02T01:48:21.333" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5719" PostTypeId="2" ParentId="5676" CreationDate="2014-11-02T03:30:15.990" Score="7" Body="&lt;p&gt;An example which I like to use in class is &lt;a href=&quot;http://en.wikipedia.org/wiki/Paul_the_Octopus&quot;&gt;Paul the Octopus&lt;/a&gt; (predicted 11/13 games in 2010 World Cup).&#xA;Sports are useful sources of statistical examples in general, but I particularly like this one because it illustrates how using hypothesis testing in real life can lead to nonsense.&lt;/p&gt;&#xA;" OwnerUserId="814" LastActivityDate="2014-11-02T03:30:15.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5720" PostTypeId="2" ParentId="5711" CreationDate="2014-11-02T14:47:48.637" Score="2" Body="&lt;p&gt;One way to teach English along with calculus is to assign so-called &quot;word problems.&quot; (E.g. if g is the acceleration due to gravity, how do you integrate/differentiate to get Newton's Law.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a situation where students have to &quot;translate&quot; from English to math. Then they have to do the math.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-11-02T14:47:48.637" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5721" PostTypeId="2" ParentId="5702" CreationDate="2014-11-02T16:05:06.570" Score="5" Body="&lt;p&gt;Nothing calls for students to respect the law of distribution more than provide them with arithmetic problems that conduct their lives in school. You can simply build many of the issues that seem to make the distribution more feasible to perform routine procedure, such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6 × 15 = 6 × ( 10+5 ) = 60 + 30 = 90&lt;/p&gt;&#xA;&#xA;&lt;p&gt;also 24 × 98 = 24 × ( 100 - 2 ) = 2400- 48 =2352&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or 13 × 103 = 13 × ( 100 + 3)= 1300 + 39 = 1339&lt;/p&gt;&#xA;" OwnerUserId="3306" LastEditorUserId="3306" LastEditDate="2014-11-02T16:31:11.690" LastActivityDate="2014-11-02T16:31:11.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5722" PostTypeId="2" ParentId="5706" CreationDate="2014-11-02T17:48:03.997" Score="6" Body="&lt;p&gt;This method does not give 3 digits precision, so is not really an answer to the original question -- but if all one is looking for is a quick-and-dirty approximation, here is a strategy that is very elementary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, let's change the problem to one that will be easier to solve:  $5^x=325$.  Since we are only looking for a reasonable approximation to the original problem, this problem is just as good as the original one (in the sense that an approximate solution to the new problem will also be approximately good for the original problem) (and yes, I am being very vague about what &quot;approximately good&quot; means).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now why is this easier to solve?  To start, notice that $325 = 25 \cdot 13 = 5^2 13$.  So the problem of solving $5^x=325$ reduces to solving $5^t=13$; if we can solve that, then $x=2+t$ is the value we need.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Okay, we now have reduced the problem to finding a decent approximate solution to $5^x=13$.  By &quot;decent approximate solution&quot; I mean that I would like to find a rational number whose numerator and denominator are not too large.  One easy way to do this without using logarithms or any functions beyond a basic 4-function calculator is to write the (integer) powers of 5 and 13 in parallel lists and look for numbers in the two lists that are sort-of close to each other:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Powers of 5:  $5, 25, 125, 625, 3125, 15625, 78125, \textbf{390625}, 1953125 \dots$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Powers of 13:  $13, 169, 2197, 28561, \textbf{371293}, 4826809, 62748517, \dots$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now inspecting this list we find that $5^8 \approx 13^5$ (boldfaced in the lists above).  That is to say, they differ from each other by about 5%.  So we have $5^{8/5} \approx 13$.  Thus to a reasonable approximation the solution to $5^t=13$ is about $1.6$, and therefore the solution to $5^x=325$ is about $3.6$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How good is this solution?  Well, $5^{3.6} \approx 328.3$ (notice that this can't be done on a basic 4-function calculator), which is less than 1% larger than the target value of $326$.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2014-11-02T22:51:12.413" LastActivityDate="2014-11-02T22:51:12.413" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5725" PostTypeId="1" AcceptedAnswerId="5726" CreationDate="2014-11-02T21:20:01.853" Score="13" ViewCount="2425" Body="&lt;p&gt;For the students of grade 9: Is $a^0 = 1$ for a non zero real a, considered a theorem or an axiom?&lt;/p&gt;&#xA;" OwnerUserId="3306" LastEditorUserId="267" LastEditDate="2014-11-04T08:27:46.850" LastActivityDate="2016-08-12T06:54:46.390" Title="Is $a^0 = 1$ for a nonzero, real number $a$, a theorem or an axiom?" Tags="|secondary-education|algebra|" AnswerCount="11" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5726" PostTypeId="2" ParentId="5725" CreationDate="2014-11-02T21:27:41.190" Score="11" Body="&lt;p&gt;It is more a 'corollary' or a 'property' derived from the basic power rules. Once you have shown how products and divisions of powers with the same base work, $a^0=1$ is just a consequence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{a^m}{a^n}=a^{m-n}$$ Then if we set $m=n$, we obtain $$\frac{a^m}{a^m}=a^{m-m}=a^0$$ and as  $\frac{a^m}{a^m}=1$, we can state that $a^0=1$.&lt;/p&gt;&#xA;" OwnerUserId="3260" LastEditorUserId="6724" LastEditDate="2016-08-12T06:54:46.390" LastActivityDate="2016-08-12T06:54:46.390" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5727" PostTypeId="2" ParentId="5725" CreationDate="2014-11-02T22:26:16.543" Score="13" Body="&lt;p&gt;In grade 9, you should teach that as a result of the usual properties, as Pablo B. wrote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a^0=a^{n-n}=a^n/a^n=1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(whatever $a^n$ is: it just needs to be non-zero).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In higher grades, once the students have learned mathematical induction, you would make it an axiom, together with&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a^{n+1}=a \cdot a^n$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Those two axioms define $a^n$ for all non-negative integers $n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, for your students, your equation starts as a theorem then later becomes an axiom. Of course, many or most secondary students never get that far.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2014-11-02T22:26:16.543" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5728" PostTypeId="2" ParentId="5725" CreationDate="2014-11-03T08:57:22.123" Score="1" Body="&lt;p&gt;It is more of a definition than an axiom. I.e. it is part of the definition of the power function which would proceed by defining the function first for whole numbers inductively starting from $a^0 = 1, a \ne 0$ and $a^{n+1} = a \cdot a^n$ , then you would define it for negative integers, rational numbers, reals, complex in the well known ways. There are alternatives such as defining $e^x$ as a power series and then using $a^x = e^{(\ln{a})x}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not usual to define functions like this axiomatically but you could do it if you wanted to teach the axiomatic method. In that case it would be better to start with axioms $a^1 = a$ and $a^x a^y = a^{x+y}$ This would be sufficient to derive the power function rules for rational numbers. You would need to assume an axiom of continuity to extend to the reals. If you did this you would have to justify the assertion that the axioms are consistent. This is why it is better to use a definition rather than axioms. Axioms should be kept to a minimum and used only for the essential elements of the mathematical system you are working in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case this would be more than you wanted to explain to most grade 9 students. It is better to just call them definitions. You can motivate them with statements like $a^0 = a^n/a^n$ to help memory and understanding, but you should not confuse definitions with axioms.&lt;/p&gt;&#xA;" OwnerUserId="3330" LastEditorUserId="3330" LastEditDate="2014-11-03T09:08:23.730" LastActivityDate="2014-11-03T09:08:23.730" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5729" PostTypeId="2" ParentId="5725" CreationDate="2014-11-03T08:58:15.360" Score="4" Body="&lt;p&gt;Neither, exponentiation is &lt;strong&gt;defined&lt;/strong&gt; as:&#xA;$$a^0:=1 \quad (a\not= 0)$$ $$a^{n+1}:=a \cdot a^n$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, it holds by its &lt;strong&gt;Definition&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify things here:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;em&gt;Definition&lt;/em&gt; is a precise description of the meaning of a mathematical term. In this case it is not much more than an abbreviation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An &lt;em&gt;Axiom&lt;/em&gt; is a mathematical statement that is assumed to be true without proof &amp;ndash; often it is not possible to prove it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;em&gt;Theorem&lt;/em&gt; is a mathematical statement that is proved.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;em&gt;Corollary&lt;/em&gt; of a &lt;em&gt;Theorem&lt;/em&gt; is a statement which can be easily proved by the theorem.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3331" LastEditorUserId="3331" LastEditDate="2014-11-03T10:12:56.857" LastActivityDate="2014-11-03T10:12:56.857" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5730" PostTypeId="1" CreationDate="2014-11-03T10:05:45.983" Score="20" ViewCount="4085" Body="&lt;p&gt;I asked this question in a class of 13-year-old students: &lt;em&gt;Bag 1 contains 2 black marbles and 1 white one. Bag 2 contains 20 black marbles and 10 white ones. Which bag is more likely to yield a black marble? Or maybe the chances are equal?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We were discussing this problem informally and intuitively (the probability as a ratio is not introduced yet). One of the students conjectured that when the ratios of number of blacks to the number whites are equal in two bags then &lt;strong&gt;the bag with greater difference between blacks and whites&lt;/strong&gt; is more likely to yield a black one (a known misconception, as mentioned &lt;a href=&quot;http://books.google.com/books?id=d0ybYaoKmegC&amp;amp;pg=PA153&amp;amp;dq=choices%20that%20Green%20consistently%20identified%20across%20similarly%20structured%20items.&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=lVJXVP3nBofBObL4gYgD&amp;amp;ved=0CB8Q6AEwAA#v=onepage&amp;amp;q=choices%20that%20Green%20consistently%20identified%20across%20similarly%20structured%20items.&amp;amp;f=false&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I convince him that his answer is wrong? I simulated the situation with &lt;em&gt;Mathematica&lt;/em&gt; for 100,000 draws, for many times. However it made him doubt his conjecture just a tiny bit!&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2018-05-11T22:05:16.447" Title="Probability Misconception: Two Bags with Black and White Marbles" Tags="|statistics|probability|" AnswerCount="14" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5731" PostTypeId="2" ParentId="5725" CreationDate="2014-11-03T10:28:46.753" Score="30" Body="&lt;p&gt;This is what really happens during &lt;em&gt;education&lt;/em&gt; (not in the mind of a mathematician): The teacher introduces $4^9$ as just a way to abbreviate $4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\times 4$. So, the $a^n \times a^m=a^{m+n}$ and othere &lt;em&gt;rules&lt;/em&gt; can be shown to be true, simply via thinking to the meaning of this abbreviation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, someday, $a^0$ should be introduced; a total nonsense regarding to the abbreviation mentioned above! An approach (I thikng the best one) in this point is as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Giving motivation:&lt;/strong&gt; Explaining that accepting $n$ to be $0$ in $a^n$ helps us in our mathematical practice. We can say it's nonsense or &lt;strong&gt;give&lt;/strong&gt; it a meaning. Which one is better?&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Recognizing different ways to &lt;em&gt;define&lt;/em&gt; $a^0$&lt;/strong&gt; It's not anyone outside the classroom to tell us what to define $a^0$! We can define it anything we want, but we will pay the price for bad choices!&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;&lt;em&gt;Good&lt;/em&gt; definition for $a^0$&lt;/strong&gt; What is the value for $a^0$ which won't make us to add new &lt;em&gt;rules&lt;/em&gt; to our rulebox? What is the value which is consistent with what we know about exponents and makes the least trouble? It's $a^0=1$!&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2014-11-03T10:28:46.753" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5734" PostTypeId="2" ParentId="5676" CreationDate="2014-11-03T11:42:05.667" Score="3" Body="&lt;p&gt;On the softer side:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: Murder Trial&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a murder trial, the null hypothesis (innocent until proven guilty) is given by:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The defendant did not murder the deceased ($H_0$)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The alternate hypothesis is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The defendant did murder the deceased ($H_A$)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Implicit in $H_0$ is a range of possibilities that are consistent with $H_0$. It is up to the prosecuting council to produce evidence which lies outside this range of possibilities (e.g. the probability of the defendant being innocent AND owning the gun is small). If the prosecuting council succeed in producing such evidence (similar to a sample mean outside the confidence interval), the jury reject $H_0$ and the man is convicted. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A Type I error here is convicting an innocent man — which is considered worse than a Type II: acquitting a guilty man.&lt;/p&gt;&#xA;" OwnerUserId="584" LastActivityDate="2014-11-03T11:42:05.667" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5735" PostTypeId="2" ParentId="5730" CreationDate="2014-11-03T11:52:10.007" Score="1" Body="&lt;p&gt;Nice question! Two ideas:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Compare &lt;em&gt;A&lt;/em&gt;: {2 black, 1 white, difference = 1}, to &#xA;&lt;em&gt;B&lt;/em&gt;: {100 black, 98 white, difference = 2}. &#xA;Should &lt;em&gt;B&lt;/em&gt; have a greater chance than &lt;em&gt;A&lt;/em&gt; of drawing black?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Compare &lt;em&gt;A&lt;/em&gt;: {2 black, 1 white, difference = 1}, to &#xA;&lt;em&gt;C&lt;/em&gt;: {1 black, 0 white, difference = 1}. &#xA;Should &lt;em&gt;A&lt;/em&gt; and &lt;em&gt;C&lt;/em&gt; have the same chance of drawing black?&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-11-03T11:52:10.007" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5736" PostTypeId="1" CreationDate="2014-11-03T12:13:58.930" Score="6" ViewCount="493" Body="&lt;p&gt;I was looking at this previous question on writing lecture notes he says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Most students think that the reason to take notes is to have a reference for later, but we know that the real reason to take notes is to make your brain process the material. As a result, many students frantically write down computations instead of thinking about what is going on -- and reserve the &quot;thinking about what is going on&quot; until later!&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;From&lt;/strong&gt;: &lt;em&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/2186/how-can-we-help-students-learn-to-effectively-take-notes&quot;&gt;How can we help students learn to effectively take notes?&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This may have been true when I was a student.  Now as a adult, there's a chance I'd have wanted to use my notes for future reference or even to distribute to others - if my ability to write had improved any since high school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As we progress through school, our knolwlege has beceome more specialized and there's a chance I am the only person in the wrong who is knowledgeable on a given topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Theorem-Proof format works well for serious students  Yet, I have seen many examples of lectures notes which are logically clear but very painful to read visually and logically disorganized.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't feel like writing a textbook, I am just looking for strategies for laying out materials on the page, and building narrative or logical sequences from the material that I have been given.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I don't even know if it's possible to translate the principles of good writing in other disciplines to a technical subject like math.  Are there examples of really clear, easy to read lecture notes?&lt;/p&gt;&#xA;" OwnerUserId="1654" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-11-03T14:04:26.193" Title="How can I make my notes more useful for future reference?" Tags="|general-pedagogy|lecture-notes|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5737" PostTypeId="2" ParentId="5725" CreationDate="2014-11-03T13:11:32.740" Score="1" Body="&lt;p&gt;$a^b$ is a notation for the multiplicative operand equivalent to $b$ repeated multiplications by $a$, the same way that $a*b$ is a notation for the additive operand equivalent to $b$ repeated additions of $a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a*0 = 0$ for all $a$, since adding no $a$s, ie, doing no addition, is the same as adding zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a^0=1$ for all $a$ (including $0$), since multiplying by no $a$s, ie, doing no multiplication, is the same as multiplying by one.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2014-11-03T18:03:33.270" LastActivityDate="2014-11-03T18:03:33.270" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5738" PostTypeId="2" ParentId="5730" CreationDate="2014-11-03T13:29:29.420" Score="9" Body="&lt;p&gt;Ask whether they think putting the marbles in a bag into any particular arrangement would affect the outcome. If they're ok with this, have them consider the arrangement of &lt;em&gt;Bag 2&lt;/em&gt; where the marbles are arranged in 10 &lt;strong&gt;wbb&lt;/strong&gt; groups. They're likely to agree that the hand now has 10 equivalent groups to pick from, and that each group yields the same odds of getting a black marble as &lt;em&gt;bag 1&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a really neat problem, by the way.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2014-11-03T13:29:29.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5739" PostTypeId="2" ParentId="5635" CreationDate="2014-11-03T13:33:06.520" Score="4" Body="&lt;p&gt;Assuming that you have still not started college, here is a list that I had read in my school days and pre college days which influenced me a lot and for a large part were the reason I choose to become a mathematician:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Men of Mathematics by E. T. Bell; the best biographies on mathematicians that I have ever read, this two volume book is still in print and is still a classic. You should definitely read this.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is Mathematics? by Corant and Robbins; written by two master mathematicians, this book contains everything that you have ever encountered in math and much more. Ideally suited for any young student eager to see the world beyond the school curriculum.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Dr. Riemann's Zeros by Karl Sabbagh; a book on a famous problem which discusses many connections with different branches of math and physics. It will definitely show you that math is not a disjoint union of different subjects but is woven in a thinly knit fabric which encompasses all.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A Mathematician's Apology by G. H. Hardy; a long essay on what it is like to do math and be a mathematician. It makes for a very enjoyable reading and is recommended to see some less known sides of a mathematician.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="3326" LastEditorUserId="22834" LastEditDate="2023-09-13T17:49:18.290" LastActivityDate="2023-09-13T17:49:18.290" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5740" PostTypeId="2" ParentId="5736" CreationDate="2014-11-03T14:04:26.193" Score="1" Body="&lt;p&gt;I try to keep my work reusable, and from all things (which work or not, YMMV) I would stress one:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$${\Huge\text{context}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many issues with searchability, readability, complexity, etc., but whatever you do, it is much easier to reuse your work when you know the context in which it had been done or used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(All the arguments I would add were either obvious or pointless or unnecessary in some other way, so I cut them down.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope it helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2014-11-03T14:04:26.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5741" PostTypeId="2" ParentId="5730" CreationDate="2014-11-03T15:56:46.177" Score="7" Body="&lt;p&gt;One suggestion that might be worth considering is asking the students the same question but for a bag with no white balls. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you have a bag with 1 black and no white balls ( probability of picking black = 100% and difference between white and black = 1)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now suppose you have a bag with 10 black and no white balls ( probability of picking black = 100% and difference between white and black = 10) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second case obviously has a greater difference as your student suggested might be a determining factor but clearly the probability of picking a black ball in this scenario is the same as when the difference was = 1.&lt;/p&gt;&#xA;" OwnerUserId="2200" LastEditorUserId="11" LastEditDate="2014-11-04T16:12:54.377" LastActivityDate="2014-11-04T16:12:54.377" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5742" PostTypeId="2" ParentId="5635" CreationDate="2014-11-03T16:00:21.863" Score="7" Body="&lt;p&gt;As a counterpoint to Polya: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486201074&quot;&gt;The psychology of invention in the mathematical field&lt;/a&gt; by Hadamard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As much as we all love Polya, Hadamard's study indicates that mathematicians don't often think in language...with Polya as a notable exception.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also recommend the &lt;a href=&quot;http://press.princeton.edu/titles/8350.html&quot;&gt;Princeton Companion to Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2014-11-03T16:00:21.863" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5743" PostTypeId="2" ParentId="5730" CreationDate="2014-11-03T16:14:35.013" Score="5" Body="&lt;p&gt;Describe hitting bag 1 with a hammer three times. &#xA;Stipulate that each blow shatters one of the three marbles into ten equal-sized pieces. &#xA;Ask them whether they think the probability is different now.&lt;/p&gt;&#xA;" OwnerUserId="3338" LastActivityDate="2014-11-03T16:14:35.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5744" PostTypeId="2" ParentId="5706" CreationDate="2014-11-03T16:27:59.557" Score="14" Body="&lt;p&gt;First, I'll answer the question posed by Benjamin Dickman: Solving problems with limitations is good practice for working with algebraic structures that do not have analogous functions. For example, solving $5^x \equiv 326 \mod 331$ is a situation where the log button on your calculator isn't going to help at all. (Neither is the bisection method.) So you have to be able to solve it using techniques that can translate to the system you're working in. It's good to force students to practice with these constraints so that (1) they appreciate the readily available techniques of $\mathbb{R}$ and (2) prepare for future topics where many algebraic systems are limited to $+$, $-$, $\times$, $\div$, and integer exponentiation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, I'll answer the original question: You motivate them by showing them examples of what I just mentioned above. For example, a significant portion of cryptography depends on the fact that logs are hard/impossible in discrete systems. If you can introduce modular arithmetic, that would certainly help motivate the problem. But if you've done any matrix algebra you can see that (without diagonalization and power series), problems there need to be solved with $+$, $-$, $\times$, matrix inversion, and integer exponentiation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last, I'll solve the problem: (This is a method known as Shank's Algorithm. It uses the fact that continued fractions provide the quickest and best approximations to irrational values.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We know that $5^3 &amp;lt; 326 &amp;lt; 5^4$. So let $x=3+x_1$ for some $0&amp;lt;x_1&amp;lt;1$. Now divide by $5^3$ to get $5^{x_1} = 2.608$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since $2.608^1 &amp;lt; 5 &amp;lt; 2.608^2$, then $x_1 = \frac{1}{1+x_2}$ for some $0&amp;lt;x_2&amp;lt;1$. Divide again $5/2.608^1 = 1.91717791...$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since $1.91717791 &amp;lt; 2.608 &amp;lt; 1.91717791^2$, then $x_2 = \frac{1}{1+x_3}$ for some $0&amp;lt;x_3&amp;lt;1$. Divide: $2.608/1.91717791=1.36033280...$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since $1.36033280^2 &amp;lt; 1.91717791 &amp;lt; 1.36033280^3$, then $x_3 = \frac{1}{2+x_4}$ for some $0&amp;lt;x_4&amp;lt;1$. Divide: $1.91717791/1.36033280^2 = 1.03602939...$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since $1.03602939^8 &amp;lt; 1.36033280 &amp;lt; 1.03602939^9$, then $x_4 = \frac{1}{8+x_5}$ for some $0 &amp;lt; x_5&amp;lt;1$. Divide: $1.36033280/1.03602939^8 =1.02486949...$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And so on. At any point you can just set $x_i = 0$ and stop the process. Doing so gives the following sequence of rational numbers: $3,4,\frac{7}{2},\frac{18}{5},\frac{151}{42},\frac{169}{47},\frac{489}{136},...$. The theory of continued fractions says that the error in these approximating fractions is bounded by 1 over the square of the next denominator. Since $\frac{1}{47^2} &amp;lt; .0005$, then $3+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{2+\dfrac{1}{8}}}} =\frac{151}{42} = 3.595...$ solves the problem.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-11-03T16:27:59.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5745" PostTypeId="2" ParentId="5730" CreationDate="2014-11-03T20:29:51.453" Score="25" Body="&lt;p&gt;Firstly, don't forget that your student has thought hard to come up with his answer and to be told it is wrong may be taken as invalidating his effort, or even insulting his intelligence. This might be at least a small part of his resistance to accepting your response.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would have recommended starting by asking him to explain more about his thinking, and withholding judgement for a little longer. Sounding like you are genuinely interested in learning more about his ideas will help (actually being interested will help more, of course ;) ). Another helpful thing to ask him is how he might go about checking if his intuition is right, rather than presenting him with your simulation first up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The misconception might be caused by thinking about repeated trials &lt;em&gt;without&lt;/em&gt; replacement. He may be thinking about taking a handful of marbles, or one after the other. In that case, with the second bag, it's possible to pull out many black marbles in a row, but in the first it's not. Also in the first bag, you only pull out two before you're guaranteed a white, whereas in the second you have to pull out 20 before being guaranteed a white. Even with formal probability, it actually is more likely to pull out two black in a row from the second bag! If this seems to be the issue, my suggestion is to ask him what would happen if you only had one chance to pull out one marble. Would it still be more likely in the second bag then?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A computer simulation may not be convincing to him, simply because it's a computer and therefore not real. It would be much more real if he did it with actual bags of actual marbles (or beads or m&amp;amp;ms or whatever). This is more likely to be the suggestion he comes up with himself if you ask how he could test his understanding. To really drive home the the &quot;only one chance&quot; idea, I would get everyone in the class to give it a go once. Even better, give everyone their own bag and all pull out one marble simultaneously. Then, if you do do the simulation later, you can say you're simulating ten thousand different people all having their own one and only chance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only other suggestion I have is to investigate it not just for 1:2 versus 10:20 but also for 2:4, 3:6, 4:8, 5:10...The weight of evidence from all of these will be more convincing, and highlight more strongly that it's just the ratio and not the difference that is the important thing.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-11-03T20:29:51.453" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5746" PostTypeId="2" ParentId="5730" CreationDate="2014-11-03T23:32:57.473" Score="2" Body="&lt;p&gt;I think you tell him to write down and remember his answer to this question, then you move on to teaching probability and how to calculate it. Some time later when probability is understood ask him to calculate the probability of drawing a black ball from each bag, then see if he wants to change his answer, hopefully he will. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This will give him the satisfaction of knowing that he has learnt and understood something useful that was not clear to him before. It also allows him to correct his own misconception which is more positive than trying to explain to him why he is wrong. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that you can ask them if he draws two balls which bag is more likely to yield one white and one black ball.&lt;/p&gt;&#xA;" OwnerUserId="3330" LastActivityDate="2014-11-03T23:32:57.473" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5747" PostTypeId="1" CreationDate="2014-11-04T01:41:55.083" Score="1" ViewCount="114" Body="&lt;p&gt;If you were starting to teach now (as I am), what advice you wish you had gotten?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Things as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;hopes you wish you did/didn't have&lt;/li&gt;&#xA;&lt;li&gt;specific knowledge you wish you were more knowledgeable&lt;/li&gt;&#xA;&lt;li&gt;books/references you wish you knew&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;EDIT: I know this is a bit generic, but here is a bit more of information about my situation. I will be teaching high school level math. Here in my school we teach everything, so advice on any subject is welcome.&lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="1890" LastEditDate="2014-11-04T12:30:20.467" LastActivityDate="2014-11-04T12:30:20.467" Title="Advice for a starting teacher" Tags="|teacher-development|" AnswerCount="1" CommentCount="5" ClosedDate="2014-11-04T13:40:13.763" ContentLicense="CC BY-SA 3.0" />
  <row Id="5749" PostTypeId="2" ParentId="5730" CreationDate="2014-11-04T07:15:19.317" Score="3" Body="&lt;p&gt;13 year olds? Marbles? The answer is easy (as was suggested by David Butler) -- do the physical experimental test of the probabilities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And take the opportunity to expand your teaching beyond the symbols of mathematics and into &lt;strong&gt;physical&lt;/strong&gt; experimentation: teach the students how to keep a lab journal with the experiment, experimenter, results, additional factors, etc all recorded, signed, and dated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As someone with an engineering degree, I appreciate both the pure value of mathematics and its application to the physical world. I humbly suggest that your classes would become even more interesting if you seize opportunities to apply math to the physical world whenever the opportunities present themselves. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible source of professional collaboration would be your friendly local physics teacher. She or he might have some fun physical experiments that you can then use to explore the mathematics of the experiment. (Or do the math first and then the experiment.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also discuss how the great mathematicians / scientists of the past would work on both the symbolic and experimental--get your history teachers involved too!&lt;/p&gt;&#xA;" OwnerUserId="3343" LastActivityDate="2014-11-04T07:15:19.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5750" PostTypeId="2" ParentId="5747" CreationDate="2014-11-04T08:46:49.157" Score="5" Body="&lt;p&gt;I will tell you as a previous teacher my hope that I wish I didn't have was that I was going to teach my students math and that they would understand because I was going to teach them the &quot;how&quot;.  This is an unrealistic goal (particularly for a beginning teacher).  This pessimistic view will probably be met with down votes, but you need to hear it as a beginning teacher.  You should &lt;em&gt;lower&lt;/em&gt; your expectations.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if you &lt;em&gt;are&lt;/em&gt; a great teacher, it's unlikely that you will be successful for your first few years.  You need to lower your expectations.  You need to both lower your expectations for what you think you can teach and, &lt;em&gt;especially&lt;/em&gt;, lower your expectations of what your students already know.  You are likely going to have to reteach many topics that you think are prerequisites to your course (whether it be geometry, pre-cal, algebra, etc.).  You &lt;em&gt;cannot&lt;/em&gt; teach assuming your students already understand/know the requisite skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming this is for high school/junior high education, the thing I wish I had been more knowledgeable about was classroom management.  This is &lt;em&gt;far&lt;/em&gt; more important that your knowledge of mathematical topics.  If you assume that you are going to keep your students' attention by explaining things in the most perfect way then you are seriously mistaken.  High school and junior high students are &lt;em&gt;not&lt;/em&gt; in your classroom to learn about math, they are in the classroom because they are forced to be there and I guarantee that a majority of your students &lt;em&gt;don't&lt;/em&gt; want to be there.&lt;/p&gt;&#xA;" OwnerUserId="2204" LastActivityDate="2014-11-04T08:46:49.157" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5751" PostTypeId="2" ParentId="5730" CreationDate="2014-11-04T16:09:42.627" Score="7" Body="&lt;p&gt;I wrote a long answer to this, which then made me realise that I think this needs the rule. [Restart:]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few pointers first:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Don't number the bags if you have numbers of marbles: big/small, black/brown, Peter/Paul, A/B.&lt;/li&gt;&#xA;&lt;li&gt;Don't actually use bags when you want to see what is going on: bowls, or for class experiments transparent vases.&lt;/li&gt;&#xA;&lt;li&gt;Marbles are fine, since they are familiar yet insignificant to them, but in most sets they're not black and white. (For the class, use ping pong balls in e.g. orange and white.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In private discussion (Class discussion is more about what the right answer should be.), I guess he'd bring the marbles and you the bowls and together you study it from the very small scale up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, and this is what I referred to above, I think this is going to need a systematic approach, and that means deriving that you count all balls, and count the part of them that you're after - probability ratio. You can make it special with &quot;I haven't told this in class yet ...&quot;, but I think that to effectively scale between 1 and 2, 2 and 4, and 10 and 20 and make it stick, you'll have to go that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the completely different alternative you have the runway coloured in black sections and white sections and the pilot of the biplane losing his shawl (etc.) somewhere over them. Just as random as a draw, but without the confusion of lack of order. (Coloured fence, horses on the merry-go-round, any random selection of a sectioned surface would work.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And don't worry. You're a good teacher, as you know when to ask for help. So be assured that you will get the subject across to him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;  &lt;/p&gt;&#xA;" OwnerUserId="3347" LastEditorUserId="122" LastEditDate="2017-04-20T08:54:59.603" LastActivityDate="2017-04-20T08:54:59.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5752" PostTypeId="1" CreationDate="2014-11-04T16:14:33.373" Score="8" ViewCount="437" Body="&lt;p&gt;In general the secondary student may not ask why is $\sin 90^\circ = 1$ because they can see the answer in the graph of the sine wave. However the students in grade 8 are not familiar with the graph of sine, so how the teacher could find a simple way to convince the student?? All what they have is the sine definition as a ratio between the lenght of the opposite side to the angle and the length of the side opposite to the right angle.&lt;/p&gt;&#xA;" OwnerUserId="3306" LastEditorUserId="3246" LastEditDate="2014-11-04T18:05:45.340" LastActivityDate="2014-11-08T19:37:21.827" Title="How to convince the students of grade 8 that $\sin 90^\circ =1$? ( calculator not allowed )" Tags="|trigonometry|precalculus|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5753" PostTypeId="2" ParentId="5752" CreationDate="2014-11-04T17:48:54.753" Score="19" Body="&lt;p&gt;Maybe draw this picture?&#xA;Make it clear that the green hypothenuse is fixed in length,&#xA;but the red altitude is growing and approaching that hypothenuse in length&#xA;as the angle approaches $90^\circ$.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/CjRq4.jpg&quot; alt=&quot;Sin90&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-11-04T17:48:54.753" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5755" PostTypeId="2" ParentId="5752" CreationDate="2014-11-04T18:20:41.590" Score="12" Body="&lt;p&gt;By their definition (using a right triangle), $\sin 90^\circ$ is undefined, since a triangle cannot have two right angles. You need the definition based on a circle ($\sin \theta = y/r$) to have a value for $\sin 90^\circ$.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="3246" LastEditDate="2014-11-04T18:50:20.160" LastActivityDate="2014-11-04T18:50:20.160" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5756" PostTypeId="1" AcceptedAnswerId="5760" CreationDate="2014-11-04T19:40:46.897" Score="11" ViewCount="2005" Body="&lt;p&gt;According to your experience as students and professors, what are (and why) the courses that should be part of a math &lt;em&gt;undergraduate&lt;/em&gt; degree, but that are missing in most institutions? &lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="143" LastEditDate="2014-11-08T10:26:21.240" LastActivityDate="2014-11-15T18:52:20.397" Title="Topics that should be in an undergraduate math programme" Tags="|undergraduate-education|mathematical-pedagogy|curriculum|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5757" PostTypeId="2" ParentId="5725" CreationDate="2014-11-04T19:52:56.887" Score="2" Body="&lt;p&gt;I will say something a bit different than other people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, there is the question about what the correct mathematical definition is. How do you define $a^x$ for a real number $x$ (and non-negative $a$). To do this one could &lt;em&gt;define&lt;/em&gt; $a^x$ first for natural numbers. Then we can extend to the non-zero integers in the obvious way. To define the expression at a real number you can then take limits in the usual way. If you do all this, then $a^0 =1$ for all $a &amp;gt; 0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, there is the question on how you should teach this in your 9th grade class. Often we have to teach students have to do stuff that we can't really define rigorously. My favourite example of this is sets. We learn about sets early on, but we never really define what a set is until (most often) we reach graduate school and learn about set theory. So I think you have the same problem with the example of $a^x$. So then you will hear different opinions on how to convince the students that $a^0 = 1$ makes sense. My preferred way would simply be to first define $a^n$ for a natural number the obvious way and then just say that we define $a^0 =1$. You can then point out that with this definition the usual nice rules (for example $a^ma^n = a^{m+n}$) hold. But I be careful what you say and how you say it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third, what ever you do don't lie to the students. Don't tell them that it is an axiom. This will just confuse them when they later on possibly have to do real work with axioms.&lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-11-04T19:52:56.887" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5758" PostTypeId="2" ParentId="5752" CreationDate="2014-11-04T19:58:50.007" Score="3" Body="&lt;p&gt;I would simply define $\sin(x)$ from the unit circle. I would use a right-angled triangle exactly because (as others have point out already) this definition doesn't work for $90^\circ$. Note that is also seem to work well for negative values.&lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2014-11-04T19:58:50.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5759" PostTypeId="2" ParentId="5756" CreationDate="2014-11-04T21:19:39.700" Score="5" Body="&lt;p&gt;A course in number theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Number theory is one of the &quot;big four&quot; topics in mathematics (with algebra, geometry, and calculus/analysis). It has a long history, equal to that of geometry. It is foundational to algebra and calculus, inspiring and giving many examples to those topics. It is the first application of set theory to the &quot;big four.&quot; It has strong links to non-big-four topics such as set theory, probability, statistics, and combinatorics. It is simple enough to be understood by elementary school students but deep enough to provide many of our famous unsolved problems (Goldbach conjecture, twin-prime conjecture, etc.). It forms many puzzles that can be understood by non-mathematicians. It inspired the most famous current unsolved problem, the Riemann hypothesis. Finally, it has important real-world applications in topics such as cryptography.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was not able to take a number theory class when I was a math undergrad at Cornell University. A professor tried to get a class together but could not get the minimum number of students. My main exposure to number theory was in &quot;math camp&quot; during a high school summer, and it was the only required topic since it has practically no prerequisites but is studied in university.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2014-11-04T21:37:50.150" LastActivityDate="2014-11-04T21:37:50.150" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5760" PostTypeId="2" ParentId="5756" CreationDate="2014-11-05T00:13:00.303" Score="7" Body="&lt;p&gt;Before I answer your question, I want to preface my answer with the statement that, what courses are available at any one particular university are contingent on the faculty. If there is nobody appropriate to teach a branch of mathematics, then it is difficult to run such a course!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Discrete Mathematics&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In America at least, I feel as though there is such a heavy emphasis on the continuous track of mathematics. The discrete track, I feel, is underrepresented. Courses such as &lt;strong&gt;Graph Theory&lt;/strong&gt;, &lt;strong&gt;Game Theory&lt;/strong&gt;, and &lt;strong&gt;Combinatorics&lt;/strong&gt; are classes I have taken but gained so much value from, and are more difficult to come by. I've been blessed by attending two schools (undergraduate and graduate) that have had a wide variety of courses to choose from. Even then, these courses were frequently only offered every few semesters. There are a wide number of sub-genres of topics that could be developed into a full class, if the right person were available to create the course.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I want to answer another question that seems relevant: &lt;em&gt;What courses should be a part of a mathematics education degree program?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have long yearned for a course that focuses specifically on the methodology of a particular commonly taught mathematics course. A course on &lt;strong&gt;teaching high school algebra&lt;/strong&gt;, &lt;strong&gt;teaching high school geometry&lt;/strong&gt;, &lt;strong&gt;teaching college group theory&lt;/strong&gt; or &lt;strong&gt;teaching college analysis&lt;/strong&gt; would do wonders, I think. To my knowledge, no such courses exist.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-11-05T00:13:00.303" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5761" PostTypeId="2" ParentId="82" CreationDate="2014-11-05T00:28:26.510" Score="10" Body="&lt;p&gt;In my experience, there are a few key things that students misunderstand around the central limit theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, many students don't actually have a concept of distribution in the first place. They need the idea that anything you could record about an object/person/group has various possibilities for what the outcome could be, some of which might be more likely than others. The description of what the possible outcomes are and how likely they all are is called the distribution. This description is sometimes a table of probabilities, sometimes a formula, sometimes a graph; and sometimes, the pattern of probabilities is so common and so useful it has a name, and just the name is enough to tell a statistician everything they need to know. The normal distribution is one of these. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, students don't have any distributions in their heads that &lt;em&gt;aren't&lt;/em&gt; normal! When they think of a measurement with a mean, the exercises they've done so far have required them to draw a normal distribution anyway. So it's hardly surprising or useful to them that the mean of the means is normal because they think, well isn't everything anyway? Therefore, it's probably useful to show some distributions of things that aren't normal so that they appreciate what it might mean for it to &lt;em&gt;be&lt;/em&gt; normal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thirdly, and most importantly, many students don't have a concept that the sample mean has a distribution at all! To them the sample mean is one value they calculate so how could it have a distribution? It is important to stress that &lt;em&gt;anything&lt;/em&gt; that can be recorded &lt;em&gt;or calculated&lt;/em&gt; about an object &lt;em&gt;or group of objects&lt;/em&gt; has a distribution. That is, there is a description of all the outcomes that &lt;em&gt;could possibly be&lt;/em&gt; and &lt;em&gt;how likely they are&lt;/em&gt;.  This point is almost never made explicitly about groups of objects, and I feel it's a big gap in most stats teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to how to deal with these things, well, here is what I would do. I have only actually tried some of these things as presentations rather than as class activities, so if you do try them I'd love to hear how they worked out!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, I would list several things you could record/calculate about a group of objects and show graphs of the distributions of these things: the mean height of a group of 6 people, the standard deviation of the IQ scores of 10 people, the median of the prices of 100 houses, the maximum of the potassium level in 5 bananas, etc. The point you are trying to make is that any measurement you could calculate has a distribution, and this distribution might or might not be something recognisable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If at all possible, I would get everyone to do their own little sample. It could be as mundane as getting everyone to roll a die 5 times and record the result, or as interesting as getting them to measure the length of 5 snake lollies. If you can't get them to do it themselves, you could always provide everyone with their own 5 measurements, but I think it would help if it was an everyday thing they could measure themselves if they wanted to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then get everyone to calculate various things from their sample and record the distributions of them on a big graph on the board. I'd recommend doing a couple of crazy calculations, like &quot;the smallest, plus the biggest, minus two of the middle one&quot;, just to really reinfoce that &lt;em&gt;any&lt;/em&gt; calculation has a distribution. This should give them a real feel for how different calculations produce different distributions. It should be clear that the mean has a particularly nice distribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you could do it with a bigger sample size, maybe this time with just the mean. After having done this twice, it's probably ok to proceed by just showing them the results that you would have gotten rather than getting them to do it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you can show how the distribution of the possible means is nicely clumped in the centre and more so as the sample size increases. And you can say that it's not just clumped in the centre, there is a specific well-known distribution that describes this shape -- the normal distribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A final comment: the fact that the distribution of sample means is approximately normal is actually not the point, of course. The point is that it's really &lt;em&gt;useful&lt;/em&gt; to know that it will come out this way because then you can use everything that mathematicians know about normal distributions to say something useful. Doing all the preceding is probably a waste of time without some discussion of why you care so much about it!&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-11-05T00:28:26.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5762" PostTypeId="2" ParentId="5730" CreationDate="2014-11-05T01:00:56.473" Score="3" Body="&lt;p&gt;How many marbles are you going to draw?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My initial answer was like the student's answer. Since you implied by the question that there was a difference, my thought pattern went like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you draw one marble the odds are equal. (1 in 3)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you draw two marbles (without replacement), the small bag ahs a 100% chance of yielding a black marble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To convince the student I would start by acknowledging the question is a bit ambiguous (it never hurts to give his ego a shot). ... Then try the question over with a bag with the same number of marbles of each color (say 5 of each). Ask the student &quot;If I draw one marble from this bag, is it more likely to be black or white?&quot; When he says there is equal chance, double the number of marbles in the bag but keep them 50-50 and ask again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If he sees that the proportion is the important thing here, he might see it for the other case as well.&lt;/p&gt;&#xA;" OwnerUserId="3352" LastActivityDate="2014-11-05T01:00:56.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5763" PostTypeId="2" ParentId="5730" CreationDate="2014-11-05T02:48:30.300" Score="8" Body="&lt;p&gt;DavidButlerUofA's answer hits the spot on pedagogical methodology. However, the actual question is about establishing the correct intuition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that said, a good way to falsify the wrong intuition is to &lt;em&gt;enlarge&lt;/em&gt; the bag to contain millions of marbles; or rather, to shrink the marbles until they are black and white grains of dust. With 2/3 of the dust grains being black, I bet the student will sense that the probability of extracting one black grain is 2/3. &quot;Clearly&quot; the numerical difference (which is in the millions) is not relevant.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="2172" LastEditDate="2014-11-06T02:25:54.403" LastActivityDate="2014-11-06T02:25:54.403" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5764" PostTypeId="2" ParentId="238" CreationDate="2014-11-05T03:29:06.690" Score="4" Body="&lt;p&gt;In addition to your excellent list on your own site, I would add:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The Grapes of Math&lt;/em&gt; by Greg Tang and illustrated by Harry Briggs&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each page has a picture puzzle described in rhyme, where the goal is to count how many of something there is (in particular on one page you are counting grapes). However the focus on each page is that if you look at it the right way you don't have to count them all individually. On one page you can count by fives, on another you can notice it's a square but with a couple missing. It's so cool that with just simple counting you can introduce problem-solving.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Math Curse&lt;/em&gt; by John Scieszka and illustrated by Lane Smith&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Describes the journey of a girl who wakes up one morning with a Math Curse, where everything she sees becomes a maths problem. I like it because it is precisely how my mathematical mind works -- everything &lt;em&gt;is&lt;/em&gt; a maths problem! But it also gently pokes fun at the strange way that maths problems in school are written, which children can notice for themselves. Also, it has a good message that you an solve problems if you think about them in the right way.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-11-05T03:29:06.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5765" PostTypeId="2" ParentId="5725" CreationDate="2014-11-05T10:52:44.527" Score="2" Body="&lt;p&gt;To me it is clearly a definition. Maybe, in addition to Behzad's post have one more idea:&#xA;What if $a=0$? Then $a^n=0$ for positive values of $n$. So $3^0, 2^0, 1^0$ all equal $1$ wheras $0^3, 0^2, 0^1$ equal $0$. Which value should $0^0$ have? Both choices, $0$ and $1$ have good reasons.&#xA;I think this is a good starting point to learn, that mathematics is made by humans (by us!) and we may decide (define) some things. &lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2014-11-05T10:52:44.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5766" PostTypeId="1" CreationDate="2014-11-05T11:51:45.677" Score="7" ViewCount="473" Body="&lt;p&gt;Especially (but not only) in the case of induction proofs, it happens that a stronger claim $B$ is easier to prove than the intended claim $A$ since the induction hypothesis gives you more information. I am trying to come up with exercises for beginner students that help to demonstrate this point (and also interested in the general phenomenon). Do you know any good examples (preferably elementary ones) where strengthening a claim makes the proof easier?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an example of what I mean (Problem 16 from chapter 7 of Engel's `Problem solving strategies'):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Show that $\frac{1}{2}\frac{3}{4}...\frac{2n-1}{2n}\leq\frac{1}{\sqrt{3n}}$ for $n\geq 1$. This is much harder than proving the stronger statement that $\frac{1}{2}\frac{3}{4}...\frac{2n-1}{2n}\leq\frac{1}{\sqrt{3n+1}}$ for $n\geq 1$, which is a straightforward induction.&lt;/p&gt;&#xA;" OwnerUserId="3354" LastActivityDate="2014-11-05T14:51:34.163" Title="Examples where it easier to prove more than less" Tags="|proofs|problem-design|" AnswerCount="2" CommentCount="1" ClosedDate="2014-11-06T17:42:30.853" ContentLicense="CC BY-SA 3.0" />
  <row Id="5767" PostTypeId="2" ParentId="5766" CreationDate="2014-11-05T12:13:40.710" Score="4" Body="&lt;p&gt;This has been asked many times before in MathOverflow&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathoverflow.net/q/21214/12357&quot;&gt;Particular problem solved by solving a more general problem.&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathoverflow.net/q/31699/12357&quot;&gt;Strengthening the Induction Hypothesis&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathoverflow.net/q/40005/12357&quot;&gt;Generalizing a problem to make it easier&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2014-11-05T12:13:40.710" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5768" PostTypeId="1" AcceptedAnswerId="5773" CreationDate="2014-11-05T12:42:18.747" Score="15" ViewCount="529" Body="&lt;p&gt;Writing mathematics is an important activity of the mathematician. In trying to write one's mathematics, one finds ways to balance intuition and rigor and to efficiently communicate concepts and ideas along with the results of calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In requiring students to write mathematics in full sentences, we often require that they provide justification for what they do. We get a more complete picture of what the students are thinking, and check blind and mindless computation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have recently told my students that mathematical notation is a very powerful shorthand that one must earn the use of by demonstrating that one knows why one is doing what one is doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, in reality we all compute when working on a problem: we all make formal substitutions according to rules appropriate to a symbolic system. We are not always mindful, in language, of the validity of these substitutions as we are doing them. Arguably, a different part of our brain is engaged when we are carrying out computations than when we are writing them up. In short, we all (I imagine rather often) behave like formalists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more serious objection to writing while one works is that mathematical notation &lt;em&gt;is&lt;/em&gt; a very powerful shorthand. Using symbolic substitutions provides a great deal of leverage that writing everything down misses. Just think of basic arithmetic with place value!!!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, attempts to reduce mathematics to formal logic a la Whitehead and Russell surround the issue.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Has there been any study of the dichotomy between the mental processes of writing mathematics and calculation?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Writing and calculating just feel so different, I wonder if someone has studied this.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="376" LastEditDate="2016-08-17T06:25:18.710" LastActivityDate="2016-08-17T06:25:18.710" Title="Calculation versus writing in mathematics" Tags="|mathematical-pedagogy|reference-request|writing|psychology|philosophy-of-maths|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5769" PostTypeId="1" CreationDate="2014-11-05T13:29:20.120" Score="5" ViewCount="228" Body="&lt;p&gt;Say we have a property $P$ defined on the  natural nubers. Usually students are taught that to  pove $P(n)$ is true for all $n\in\mathbb N$ you have to do the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;make a basis&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and use either of the following arguments:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;if $P(k)$ is true then  $P(k+1)$ is true&lt;/li&gt;&#xA;&lt;li&gt;if $P(n)$ is true $\forall n\leq k$ then $P(k+1)$ is true.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But you can also use slightly different methods. In particular I would like to look at a situation like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You prove that $P(k)\implies P(k+3)$ and combine this with three (or less) induction bases.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;(or less)&lt;/em&gt; part could be used, if something is only true for all numbers $2\bmod3$ for instance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The number $3$ can be any other natural number  of course.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also asked this question &lt;a href=&quot;https://math.stackexchange.com/q/947065/145141&quot;&gt;here, on MSE&lt;/a&gt;, where it already has two answers (please check these first). However I thought this site would perhaps be more appropriate and thus lead to more response. So does anyone know  problems that would be &lt;strong&gt;easier&lt;/strong&gt; (less time consuming) to solve with a method like this than with the ordinary methods. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think problems like these would be very nice examples to help students understand the power of induction.&lt;/p&gt;&#xA;" OwnerUserId="3246" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-01-08T09:01:03.060" Title="On problems which can be proved easier if we use a different induction step" Tags="|proofs|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5770" PostTypeId="2" ParentId="5769" CreationDate="2014-11-05T13:39:20.307" Score="1" Body="&lt;p&gt;For example: to prove that $8|n^2-1$ for odd numbers $n$, we can go about like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$8|1^2-1=0$&lt;/li&gt;&#xA;&lt;li&gt;Suppose $8|k^2-1$, for an odd $k$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now we have \begin{align}(k+2)^2-1=&amp;amp;k^2+4k+4-1\\=&amp;amp;k^2-1+4(k+1)\end{align} which concludes our proof since $k+1$ is even by assumtion.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This is my own very poor example, which saves you about a few seconds. I'm sure others will provide much more usefull contributions. &lt;/p&gt;&#xA;" OwnerUserId="3246" LastActivityDate="2014-11-05T13:39:20.307" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5771" PostTypeId="5" CreationDate="2014-11-05T14:18:16.547" Score="0" Body="&lt;p&gt;Division is the reversed operation of multiplication. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If division is defined within the context, then $a$ divided by $b$ is commonly, interchangeably denoted by any of the following expressions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\dfrac ab$&lt;/li&gt;&#xA;&lt;li&gt;$a/b$&lt;/li&gt;&#xA;&lt;li&gt;$a\div b$&lt;/li&gt;&#xA;&lt;li&gt;$ab^{-1}$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If defined, $\dfrac ab$ should be the unique element such that: $$\frac ab\cdot b=b\cdot\frac ab=a.$$&lt;/p&gt;&#xA;" OwnerUserId="3246" LastEditorUserId="39" LastEditDate="2014-11-30T18:08:56.440" LastActivityDate="2014-11-30T18:08:56.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5772" PostTypeId="4" CreationDate="2014-11-05T14:18:16.547" Score="0" Body="For questions regarding the arithmetical operation of division. For question specifically concerning the use of fractions, please consider the (fractions) tag." OwnerUserId="3246" LastEditorUserId="39" LastEditDate="2014-11-30T18:09:09.120" LastActivityDate="2014-11-30T18:09:09.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5773" PostTypeId="2" ParentId="5768" CreationDate="2014-11-05T14:27:30.113" Score="13" Body="&lt;p&gt;What do we understand as &lt;em&gt;mental processes&lt;/em&gt;? All of us (Math teachers) dream of &lt;em&gt;entering&lt;/em&gt; the brain of our students, see what's happening and adjusting some connections...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the thing is that their brains are a kind of black box for us. So the only way we actually have to be sure they have catched some mathematical concept (or argument, or property, etc.) is to check if they manage to use all of the possible semiotic representations of such objects. What you call &lt;strong&gt;writing&lt;/strong&gt; is just our verbal language, a semiotic register. On the other hand, numerical calculations, algebra, graphic representations, etc. are different semiotic registers. The more registers used, the better. All of this is one of the principles explained in &lt;a href=&quot;http://www.researchgate.net/publication/237210212_The_Onto-Semiotic_Approach_to_Research_in_Mathematics/file/e0b4952a5f5fc658d4.pdf&quot;&gt;The onto-semiotic approach to research in mathematics education (Godino, Batanero &amp;amp; Font, 2007)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, answering to your question, I'd say that instead of a dicotomy, writing and calculation are complementary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's also interesting &lt;a href=&quot;http://math.unipa.it/~grim/semiotic.pdf&quot;&gt;F. Spagnolo's approach&lt;/a&gt; (Semiotic and hermeneutic can help us to interpret teaching/learning?). In that article, he shows how to analyze a didactic situation from the semiotic point of view. Which registers are employed and when to express the ideas? The natural tongue? A symbolic representation? Algebraic language?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both references will lead you to more bibliography.&lt;/p&gt;&#xA;" OwnerUserId="3260" LastActivityDate="2014-11-05T14:27:30.113" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5774" PostTypeId="2" ParentId="5766" CreationDate="2014-11-05T14:51:34.163" Score="3" Body="&lt;p&gt;Proving Cauchy-Schwarz for a general inner product is easier than proving it for any particular inner product.  For example, it is quite hard to prove for the dot product on $\mathbb{R}^n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason it is easier to prove it in general is that the definition of an inner product high light only the essential properties needed for the proof, so the &quot;space&quot; of possible proofs is constrained.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is on of the general reasons why it is easier to prove something harder:  there is a more limited number of possible approaches to the proof.  What is harder is to &lt;strong&gt;discover&lt;/strong&gt; a hard theorem.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-11-05T14:51:34.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5775" PostTypeId="2" ParentId="5626" CreationDate="2014-11-05T17:37:14.157" Score="8" Body="&lt;p&gt;What about using it to calculate the day of the week for some future date?  I'm just spitballing here but something like...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Today is Tuesday.  Jacob knows that his math test is going to be in&#xA;  17 days.  What day of the week will his math test be on?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This type of problem is very accessible and can be solved without explicitly using modular arithmetic.  I would imagine this to be a good warm up problem just to get students thinking about &quot;wrapping around&quot; and similar concepts.  After discussing, you could then ask&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Today is a Friday.  Janet knows that her mother's birthday is in 241&#xA;  days.  What day of the week will her mother's birthday fall on?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which is easy with modular arithmetic, but would be tedious to do week by week.  I feel like these are good, contextual examples that might actually be useful for students every once in awhile.  Also, the intuitive way of solving these problems, where you would divide by 7, find the remainder, and add that to the current day, is modular arithmetic, which may make it stick more readily.  Also, the modular base is a small number, 7, which I have found helps students to understand mod operations easier.&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2014-11-05T17:37:14.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5776" PostTypeId="2" ParentId="5626" CreationDate="2014-11-05T17:47:13.527" Score="8" Body="&lt;p&gt;I've always preferred the module 60 argument in seconds per minute. This is closer to the actual modulo arithmetic than the 12 or 24 hour clock. We might say things like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;30 minutes and 14 seconds&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;but would never say&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;30 minutes and 60 seconds&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;we would say &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;31 minutes (and 0 seconds)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This same idea also works with minutes per hour.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-11-05T17:47:13.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5777" PostTypeId="2" ParentId="5626" CreationDate="2014-11-05T18:08:48.620" Score="7" Body="&lt;p&gt;I think you could build a lively unit around percussion timelines:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Perhaps the most &#xA;  quintessential timeline is what most people familiar with rockabilly music dub the &#xA;  &lt;em&gt;Bo Diddley Beat&lt;/em&gt;, &#xA;  and salsa dancers call the &lt;em&gt;clave son&lt;/em&gt;, which they attribute to much older Cuban and Latin American  music.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/B9DhC.png&quot; alt=&quot;ToussaintFig2&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Godfried T. Toussaint, &quot;The rhythm that conquered the world: What makes a &quot;good&quot; rhythm good,&quot; &lt;em&gt;Percussive Notes&lt;/em&gt;, November Issue, 2011, pp. 52-59. (&lt;a href=&quot;http://cgm.cs.mcgill.ca/~godfried/rhythm-and-mathematics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;web link&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-03-19T19:13:23.950" LastActivityDate="2019-03-19T19:13:23.950" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5778" PostTypeId="1" AcceptedAnswerId="5992" CreationDate="2014-11-05T18:48:34.013" Score="9" ViewCount="2339" Body="&lt;p&gt;I am currently student teaching, and the main class that I am focusing on is a secondary geometry class.  I am currently following my classroom mentors curriculum sequence which looks something like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Intro to Geometry (points, lines, planes, shape basics)&lt;/li&gt;&#xA;&lt;li&gt;Logic (basic propositional logic to introduce the idea of proofs)&lt;/li&gt;&#xA;&lt;li&gt;Triangles (properties, congruence, special right triangles, centers) &lt;strong&gt;where we are now&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;Quadrilaterals/Polygons&lt;/li&gt;&#xA;&lt;li&gt;Circles&lt;/li&gt;&#xA;&lt;li&gt;Symmetry&lt;/li&gt;&#xA;&lt;li&gt;Transformations&lt;/li&gt;&#xA;&lt;li&gt;Constructions&lt;/li&gt;&#xA;&lt;li&gt;Trigonometry&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;which I feel like doesn't have a super good flow.  We are jumping around in the textbook and there is no real intuitive flow between subjects.  As it is already a quarter of the way through the school year, I know there is only so much I can do to adjust this sequencing but I would like to improve this for when I am teaching geometry in the future.  Does anyone have suggestions for a better way to sequence my secondary geometry curriculum?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2014-12-09T04:33:35.277" Title="Secondary Geometry Curriculum Sequencing?" Tags="|secondary-education|curriculum|geometry|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5779" PostTypeId="2" ParentId="5752" CreationDate="2014-11-05T20:31:29.227" Score="-2" Body="&lt;p&gt;Can't we think to find $\sin 90^\circ $ directly as the ratio between the side opposite to the angle and the side opposite to the right angle ( the basic defenition ) ?? in this privet case the side opposite to the right angle is the same side ! so $\sin 90^\circ = 1$.&#xA;Notice the diagram.&lt;img src=&quot;https://i.stack.imgur.com/APvXy.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="3306" LastActivityDate="2014-11-05T20:31:29.227" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5780" PostTypeId="2" ParentId="5635" CreationDate="2014-11-05T21:48:41.697" Score="6" Body="&lt;p&gt;You've asked for a book, but the most inspiring piece of writing about mathematics I've ever read is William Thurston's &lt;a href=&quot;http://arxiv.org/abs/math/9404236&quot;&gt;On proof and progress in mathematics&lt;/a&gt;. The title is roughly self-explanatory, but particularly helpful to aspiring teachers of mathematics are his insights on the way the subject is &lt;em&gt;communicated&lt;/em&gt;. &lt;/p&gt;&#xA;" OwnerDisplayName="user37" LastActivityDate="2014-11-05T21:48:41.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5781" PostTypeId="2" ParentId="5778" CreationDate="2014-11-06T00:38:03.077" Score="3" Body="&lt;p&gt;I cannot answer your (good) question, but if I may proselytize a bit&#xA;re your &quot;where we are now&quot;:&#xA;The challenge of cutting out a (generic) triangle from the center of a piece&#xA;of paper with one straight scissors cut leads to an almost physical&#xA;proof that the three angle bisectors of a triangle meet in a common point:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/fopPD.png&quot; alt=&quot;HTFI_Fig5.7&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;sup&gt;(Figure from &lt;a href=&quot;http://howtofoldit.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book&lt;/a&gt;.)&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;I distribute paper and scissors (8th-grade geometry in the U.S.)&#xA;and run through examples leading to this gem.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-11-06T01:48:31.230" LastActivityDate="2014-11-06T01:48:31.230" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5783" PostTypeId="1" AcceptedAnswerId="5788" CreationDate="2014-11-06T07:42:36.973" Score="19" ViewCount="2567" Body="&lt;p&gt;In Brazil's curriculum students are taught matrices in high school. Here, however, there is no linear algebra or pre-calculus, therefore matrices end up being just tables with lots of &quot;arbitrary&quot; definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a way to motivate the definitions without teaching them things not on the curriculum?&lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="47" LastEditDate="2014-11-06T07:54:40.510" LastActivityDate="2019-01-21T20:02:41.523" Title="Motivating the study of matrices" Tags="|student-motivation|concept-motivation|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5784" PostTypeId="5" CreationDate="2014-11-06T11:39:36.147" Score="0" Body="&lt;p&gt;Terminology refers to the use of terms, which can be words or word groups. The precise meaning of a term is called a &lt;em&gt;definition&lt;/em&gt; and the two should not be confused. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example: the terms &lt;em&gt;division ring&lt;/em&gt; and &lt;em&gt;skew field&lt;/em&gt; are two terms which have the same definition. &lt;/p&gt;&#xA;" OwnerUserId="3246" LastEditorUserId="77" LastEditDate="2019-12-31T03:33:10.970" LastActivityDate="2019-12-31T03:33:10.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5785" PostTypeId="4" CreationDate="2014-11-06T11:39:36.147" Score="0" Body="How words are used in mathematics or mathematics education" OwnerUserId="3246" LastEditorUserId="77" LastEditDate="2019-12-31T03:33:10.970" LastActivityDate="2019-12-31T03:33:10.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5786" PostTypeId="2" ParentId="5783" CreationDate="2014-11-06T13:04:48.107" Score="3" Body="&lt;p&gt;Short of changing the curriculum :-), maybe you can use &lt;a href=&quot;http://mathworld.wolfram.com/MagicSquare.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;magic squares&lt;/a&gt;, or even&#xA;Sudoku,&#xA;to help motivate.&#xA;See, e.g., &quot;&lt;a href=&quot;http://blog.themathmom.com/2011/05/from-magic-squares-to-sudoku.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;From Magic Squares to Sudoku&lt;/a&gt;&quot;. Here is Dürer's $4 \times 4$ magic square:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/q31bV.jpg&quot; alt=&quot;DurerSquar&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-11-06T13:04:48.107" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5787" PostTypeId="2" ParentId="5783" CreationDate="2014-11-06T14:19:16.923" Score="4" Body="&lt;p&gt;Maybe you can try solving sets of linear equations (2 × 2 or 3 × 3) as a motivation example for matrices and their basic manipulations.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastEditorUserId="3272" LastEditDate="2015-08-24T13:06:08.790" LastActivityDate="2015-08-24T13:06:08.790" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5788" PostTypeId="2" ParentId="5783" CreationDate="2014-11-06T14:46:16.720" Score="22" Body="&lt;p&gt;Even without explicitly introducing the language of &quot;linear maps&quot;, &quot;vectors&quot;, and so on, you can still develop matrices as a shorthand for such maps, thought of as exchange rates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Machine A can make 3 sprogs and 2 sprakets a day.  Machine B can make 1 sprog and 3 sprakets a day.  We summarize this data in a table of values:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{bmatrix} 3 &amp;amp; 1 \\ 2 &amp;amp; 3 \end{bmatrix}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where the first column represents machine A and the second column machine B, the first row sprogs, and the second row sprakets.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now If my company buys $7$ machine As and $3$ machine Bs, what is my production capacity?  This is a reasonable question for a 3rd grader, probably.  All the matrix does is provide a structure for solving such a problem systematically, as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{bmatrix} 3 &amp;amp; 1 \\ 2 &amp;amp; 3 \end{bmatrix} \begin{bmatrix} 7 \\ 3 \end{bmatrix}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplying a matrix by a vector should be defined to make the interpretation above valid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now say each sprog sells for $5$ and each spraket sells for $4$ dollars.  Then we can get a new table telling us how much money machine A makes and how much money machine B makes.  Again, this is a &quot;third grade&quot; problem, and we just introduce new notation for the problem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{bmatrix} 5 &amp;amp; 4\end{bmatrix}\begin{bmatrix} 3 &amp;amp; 1 \\ 2 &amp;amp; 3 \end{bmatrix}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example might be that each sprog lets another company build 2 cars and 3 buses, while a spraket lets them build 4 cars and 2 buses.  So ultimately, to figure out how machine A and B relate to cars and buses, you can perform:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{bmatrix} 2 &amp;amp; 4 \\ 3 &amp;amp; 2 \end{bmatrix}\begin{bmatrix} 3 &amp;amp; 1 \\ 2 &amp;amp; 3 \end{bmatrix}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At each stage you should be carefully tracking the meaning of these objects.  The columns each represent one of a set of inputs, and each row represents an output.  The entry at the $i^{th}$ column and $j^{th}$ row is the quantity of output $j$ produced by $1$ input $i$.  Matrix multiplication corresponds to figuring out a new table if you are given two tables where the outputs of one are the inputs of the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is to say, I suggest teaching the linear algebra in a particularly down to earth way, which is unlikely to intimidate students (or other teachers!) with words like &quot;linear transformation&quot; or &quot;vector space&quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-11-06T14:46:16.720" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5789" PostTypeId="1" CreationDate="2014-11-06T19:00:30.560" Score="4" ViewCount="192" Body="&lt;p&gt;I have a few students that need lots of individual attention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you have the other students do during this time?&#xA;Review problems?  Challenge problems?  start on their homework?  play a math game?  Ideally, they would come see me during office hours, but I teach at a community college, and it just doesn't happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Currently, I teach Intro Algebra and I have about 30 students in the class.  I run a lecture-discuss-solve problems classroom. What do I mean by this? I set up a topic by lecturing for a few minutes. I then ask them to discuss some aspect of it. Then they share out ideas with the whole class, and I lecture/they discuss a bit more. Then we work on problems which are similar to the problem discussed, but may include slight differences. I offer &quot;basic&quot; and &quot;medium&quot; problems, to offer some differentiation. I sometimes offer a &quot;challenge&quot; or open-ended problem, to satisfy the stronger students. It mostly works........but I was just looking for more ideas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like I am meeting the needs of the A, B, and C students, but I don't meet the needs of the really struggling student.&lt;/p&gt;&#xA;" OwnerUserId="3265" LastEditorUserId="3265" LastEditDate="2014-11-07T20:06:15.450" LastActivityDate="2014-11-07T20:06:15.450" Title="Spending time with students individually" Tags="|undergraduate-education|classroom-management|community-colleges|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5790" PostTypeId="1" AcceptedAnswerId="5793" CreationDate="2014-11-06T22:27:20.110" Score="8" ViewCount="5508" Body="&lt;p&gt;From a pedagogical as well as strictly mathematical perspective, which one of Lang's &lt;em&gt;Linear algebra&lt;/em&gt; and &lt;em&gt;Introduction to linear algebra&lt;/em&gt; would you recommend to an undergraduate with not much experience with linear algebra, but a fairly good grip of some abstract algebra and real analysis, who wants to gain a rigorous and precise knowledge of the topic? Are these two books really different? How do they compare with each other?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Note that other recommendations are also always welcome.&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="376" LastEditDate="2014-11-07T15:10:02.217" LastActivityDate="2014-11-10T08:27:59.863" Title="Lang's Linear algebra or Introduction to linear algebra for an undergraduate" Tags="|undergraduate-education|textbooks|linear-algebra|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5791" PostTypeId="2" ParentId="5789" CreationDate="2014-11-07T00:09:02.447" Score="4" Body="&lt;p&gt;As this question still seems vague to me, (I'm curious as to how your classroom is currently structured), I can provide only anecdotal evidence of what I do with my Community College Algebra class. First, some background on the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found difficulty teaching this course, since I often receive students with a wide range of abilities. Some come in with a strong sense of algebra, but need only minor tutelage to pass the exam; others come in struggling with the very basics of arithmetic; most fall somewhere in between. Additionally, whether or not a student passes hinges entirely on the final exam: If they pass the exam, they pass the class; if they fail the exam, they fail the class (with little else that I can do).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have structured the class in a different way the past two semesters. Since I am getting students with varying degrees of ability, I do not teach the curriculum in an orderly manner, and because the class is so heavily dependent on the results of the final exam, I make this my primary focus. I have created 8 cycles, with each cycle mimicking different sections on the test. Cycle 1 consists of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Quiz 1&lt;/strong&gt;: Handed out first, without any discussion. I let them struggle with the material and after allowing some time for individual work, I suggest they &quot;cheat&quot; by going online, asking their classmates, or looking through their notes. At this time I also begin to circulate the classroom. This allows me to help individuals and small groups more intimately on their specific problems. After some amount of time has passed, I have individual students volunteer to put their solutions on the board. I then go through their solutions providing my &quot;expert&quot; opinion and critique their work. This is as close to a &quot;Lecture style&quot; that I get. This quiz has 5 questions. The quiz is not graded.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Worksheet 1A&lt;/strong&gt;: Twice the amount of questions, with two of each type. I let them work in whatever way they prefer going around and working with groups or individuals again. Usually, pockets of students form who like to work together, but I encourage students who I see have done good work to share it with other groups that may be struggling. After some time, questions are presented on the board for critiquing again. The worksheet is not graded.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Board Set 1&lt;/strong&gt;: I &quot;lecture&quot; for 5 similar problems; again providing my &quot;expert&quot; analysis. Questions are encouraged and we dissect each problem.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Quiz 1B&lt;/strong&gt;: 5 questions, 20 minutes, graded. When quizzes are collected, I put the answers on the board.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Worksheet 1C&lt;/strong&gt;: 10 questions, due the next class period. If the worksheet is handed in and it is higher than the Quiz 1B grade, it will replace the Quiz 1B grade in my gradebook. I do not allow extensions, but I let them use whatever resources they can. Students will know how they did on the previous quiz since the answers were posted on the board immediately after they handed it in.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Repeat the cycle for 5 new questions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, there are also online homework assignments that are associated with each cycle during the timeframe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This entire process allows me a lot of individual time. It lays a lot of the onus on the students to self-motivate, but this is an issue I had in the past before. My feeling is that students who are &lt;em&gt;really&lt;/em&gt; ready to learn and dedicate time to the material, will do so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I make it a point to focus heavily on factoring. While the questions that I pull are usually from old exams, many of the issues that my students have fall under the category of &quot;not really understanding the prerequisite material&quot;. Most students struggle with basic factoring facts, so I spend a lot of time, early, on factor trees and finding the factors of integers and variables. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, I emphasize that the laws of exponents involve &lt;em&gt;seeing one operation&lt;/em&gt; (e.g., multiplication), and &lt;em&gt;performing a different operation&lt;/em&gt; (e.g., addition). I often create a concept map of hierarchy pairing multiplication and division together, but also associating each of those to addition and subtraction, respectively. This extends to the concept of associating &lt;em&gt;raising powers to higher powers&lt;/em&gt; and &lt;em&gt;multiplication of exponents&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;How might you incorporate these ideas to your classroom?&lt;/em&gt; The structure of quizzes and worksheets allows me to work with individual students who are struggling more closely. It also promotes group learning and creates an overall sense of community for the classroom. Emphasizing this style of teaching, and moving away from the traditional lecture styles that many of have learned mathematics from, is valuable for struggling students. Perhaps if you could incorporate some of the organizational styles that I have described above, you can find more time for these students who require more of your attention.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-11-07T00:09:02.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5792" PostTypeId="2" ParentId="5789" CreationDate="2014-11-07T00:11:46.893" Score="2" Body="&lt;p&gt;I teach basic mathematics among algebra in a Danish High school. My students are also very diverse, with some having a big interest in math and others still lacking understanding of basic concepts. I have 28 students in a class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tend to give the good students some challenging puzzle, which build upon the subject we are working with. I write this up on the black board named as a bonus exercise. The students then know that it is not mandatory to do it and that the exercise can be extremely difficult.&#xA;Sometimes too difficult for even the best students to solve, which leaves them puzzled after class, eager to find a solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like this setup because, it keeps both the strong and the weak students engaged and challenged. But without the strong students moving too much ahead in the curriculum, instead they gain a deeper understanding of the subjects. &lt;/p&gt;&#xA;" OwnerUserId="3359" LastActivityDate="2014-11-07T00:11:46.893" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5793" PostTypeId="2" ParentId="5790" CreationDate="2014-11-07T02:40:31.320" Score="3" Body="&lt;p&gt;In my opinion, it is difficult to surpass &lt;a href=&quot;http://www-math.mit.edu/~gs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gilbert Strang's&lt;/a&gt; &lt;a href=&quot;http://math.mit.edu/linearalgebra/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Introduction to Linear Algebra&lt;/em&gt;&lt;/a&gt;, 4th Ed.:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/dfZND.png&quot; alt=&quot;Fig4.3&quot;&gt;&#xA;&lt;hr /&gt;&#xA;Plus there are &lt;a href=&quot;http://web.mit.edu/18.06/www/&quot; rel=&quot;nofollow noreferrer&quot;&gt;video lectures available&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-11-10T01:05:34.400" LastActivityDate="2014-11-10T01:05:34.400" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5794" PostTypeId="1" CreationDate="2014-11-07T05:53:01.627" Score="32" ViewCount="6304" Body="&lt;p&gt;Actually, there is no algebraic problem to show that $-(-x) = x$. This proof can be build upon the concept of the addition of the opposite like this:&#xA;$- x + x = - x + [- ( - x) ]$, and thus by eliminating $- x$ from both sides we obtain $x = - ( - x )$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder how we can introduce the concept that $-(-x)=x$ and, the closely related and in some sense more general, $(-a)(-b)=ab$ through an example, without using abstract analysis.&lt;/p&gt;&#xA;" OwnerUserId="3306" LastEditorUserId="3237" LastEditDate="2015-05-05T18:02:58.727" LastActivityDate="2020-02-17T19:26:34.587" Title="How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?" Tags="|undergraduate-education|algebra|negative-numbers|" AnswerCount="20" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5795" PostTypeId="2" ParentId="5794" CreationDate="2014-11-07T09:10:09.797" Score="0" Body="&lt;p&gt;Not sure this is what you are looking for, but multiplying a positive by a negative is very straightforward, i.e. $+8\cdot-1$.  If we take eight negative ones then we get: $-1 + -1+ -1+ -1+ -1+ -1+ -1+ -1 = -8$.  However what is &lt;em&gt;not&lt;/em&gt; straightforward is what does it mean to have $-1\cdot 8$.  Here we can argue that $-1\cdot8 = 8\cdot-1$ (because we &lt;em&gt;assume&lt;/em&gt; multiplication is commutative) and thus make sense of it.  However, this doesn't really help when we multiply a negative by a negative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We really need a new definition when we multiply by a negative number.  I think first you need to define what a negative number is: $-8 = 0 - 8$.  When we multiply by a negative, we aren't &lt;em&gt;adding&lt;/em&gt;, rather we are &lt;em&gt;subtracting&lt;/em&gt;!  So when we say $-1\cdot8$ we are subtracting one &quot;8&quot; value: $0 - 8 = -8$.  If we do $-2\cdot3$ we are &lt;em&gt;subtracting&lt;/em&gt; two $3$'s: $0 - 3 - 3 = -6$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in that regard, if we do $-1\cdot-1$ then we should &lt;em&gt;subtract&lt;/em&gt; one negative one: $-1\cdot-1 = 0 - -1 = +1$.  Likewise if we do $-3\cdot-4$ then we should &lt;em&gt;subtract&lt;/em&gt; three negative fours: $0 - -4 - -4 - -4 = 0 - -12 = +12$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's how I would make sense of multiplying by a negative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Can we prove that $0--x = x$?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well obviously from a mathematical standpoint it's easy to prove but it &lt;em&gt;also&lt;/em&gt; easy to understand from an intuitive standpoint (explained later).  Mathematically, subtraction is the inverse to addition just as division is the inverse of multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we have: $x + y = z$ therefore the inverse is $x = z - y$.  Thus if we have $0 - -x = a$, then  $0 = a + -x$.  If we recognize that adding a negative is the same as &lt;em&gt;subtracting&lt;/em&gt; the positive (which is basically the definition of negative numbers) then we realize that $0 = a - x$.  Now we recognize that the &lt;em&gt;inverse&lt;/em&gt; of this is that $a = 0 + x$ which equals $x$ because $0$ is the additive identity (i.e. $0 + z = z$), so we arrive at $0 - -x = a = x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Intuitive Explanation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What can negative numbers mean intuitively?  My example would be debt.  If you loan me $x$ dollars then I am in debt to you by $x$ dollars--my balance can be said to be &lt;em&gt;negative&lt;/em&gt;, that is $-x$ represents that I &lt;em&gt;owe&lt;/em&gt; you $x$ dollars.  On the &lt;em&gt;other&lt;/em&gt; hand, if I loan &lt;em&gt;you&lt;/em&gt; $y$ dollars, then my balance can be said to be &lt;em&gt;positive&lt;/em&gt; that is &lt;em&gt;you&lt;/em&gt; owe &lt;em&gt;me&lt;/em&gt; $y$ dollars ($y$ dollars is the amount I &quot;hold&quot;--assuming you will eventually pay me back).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you loaned me $x$ dollars, then my current balance is $-x$.  If you decide that you are going to &lt;em&gt;forgive&lt;/em&gt; that loan, i.e. you are going to subtract $x$ dollars of debt from me, then my &lt;em&gt;new&lt;/em&gt; balance would be $-x - -x$.  Why subtraction?  Because you are &lt;em&gt;subtracting&lt;/em&gt; debt from me--you are &lt;em&gt;taking away&lt;/em&gt; debt (which is a negative entity).  By forgiving my debt, you would make my &lt;em&gt;new&lt;/em&gt; balance $0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's say that you &lt;em&gt;give&lt;/em&gt; me $x$ dollars (&lt;em&gt;not&lt;/em&gt; loan me $x$ dollars).  How can you do this if all there is is loans and loan forgiveness?  It's simple: instead of loaning me $x$ dollars you &lt;em&gt;forgive&lt;/em&gt; me $x$ dollars!  This means you subtract, from my initial balance of $0$, $-x$ dollars (since a loan means I go negative).  By simply &lt;em&gt;giving&lt;/em&gt; me $x$ dollars (with no expectation of repayment) you subtract the debt of $-x$ dollars from me: $0 - -x = x \leftarrow$ you &lt;em&gt;gave&lt;/em&gt; me $x$ dollars (with no expectation of repayment).&lt;/p&gt;&#xA;" OwnerUserId="2204" LastEditorUserId="2204" LastEditDate="2014-11-07T09:48:09.950" LastActivityDate="2014-11-07T09:48:09.950" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5796" PostTypeId="2" ParentId="5794" CreationDate="2014-11-07T14:05:05.830" Score="5" Body="&lt;p&gt;You could have also proven this algebraically, by doing this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To show that $-(-x)=x$ is the same as showing $-(-x)-x=0$. We then simply say&#xA;  $$-(-x)-x=-1(-x+x)=0  \tag{QED}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or without using the distributive law, it could go something like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If the students already believe that $-1\times-1=1$, then $$--a=-1\times-1\times a=1\times a=a\tag{QED}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;${   }$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As for an example: I suppose you can appriciate something intuitive. How I was taught, was with the example of a big bowl of water. It goes like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have a man (/women/machine/etc...) standing by a big bowl of water. With him the man has two piles of blocks. Red blocks and blue blocks. &lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Now the red blocks are hot. They have a positive temperature.  &lt;/li&gt;&#xA;  &lt;li&gt;The blue blocks are cold. They have a negative temperature. &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;At the begining, the temperature can be whatever you like. It will now probably speak to the students'  imagination that&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;adding a red block will increase the temperature of the water. So adding something positive leads to an increase. &lt;/li&gt;&#xA;  &lt;li&gt;adding a blue block will cause a decrease in temperature. So adding something negative leads to a decrease.&lt;/li&gt;&#xA;  &lt;li&gt;removing a red block will make the water colder. So subtracting  something positive leads to a decrease.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;And now for your case:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;removing a blue block will make the water warmer. So substracting something negative will lead to an increase.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;The magnitudes of the decreases and increases depend on the sizes of the blocks of course. I'll take it you won't find that very hard to explain. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So there we have it. An algebraic example and an intuitive example.&lt;/p&gt;&#xA;" OwnerUserId="3246" LastEditorUserId="3246" LastEditDate="2016-06-09T09:44:48.417" LastActivityDate="2016-06-09T09:44:48.417" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5797" PostTypeId="1" AcceptedAnswerId="5804" CreationDate="2014-11-07T14:08:22.750" Score="12" ViewCount="424" Body="&lt;p&gt;Many students, teachers and parents view problems as confrontational. Many students develop a self concept in mathematics based on failed attempts to easily win such confrontations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leads me to wonder if it is possible to consider a terrible heresy: Is it possible to imagine mathematics without problems?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As ludicrous as this initial idea seems, it may actually be possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is generally believed that a problem is good if it generates a lot of ideas, i.e. if there is a lot of &quot;back and forth&quot; between the problem and a person trying to solve it. Polya, in &quot;Induction and analogy in Mathematics&quot;, discusses the centrality of mathematical analogy in mathematics. Lakatos, in &quot;Proofs and refutations&quot;, proposes that the nature mathematics itself has the essential quality of a conversation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These facets suggest a connection of mathematical practice with the &lt;a href=&quot;http://en.wikipedia.org/wiki/Hermeneutic_circle&quot; rel=&quot;nofollow noreferrer&quot;&gt;hermeneutic circle&lt;/a&gt; and the work of &lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1111/1467-9752.00243/abstract&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hans Georg Gadamer&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What if &quot;conversation&quot; replaced &quot;problem&quot; in the mathematics classroom?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Has any connection between the work of Hans Georg Gadamer and mathematical practice appeared in the mathematics education literature? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This question is closely related to an answer posted to my other question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5768/calculation-versus-writing-in-mathematics&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-09-05T05:44:18.540" Title="Mathematics and the hermeneutic circle" Tags="|reference-request|mathematical-pedagogy|education-research|philosophy-of-maths|mathematical-discourse|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5798" PostTypeId="5" CreationDate="2014-11-07T14:28:27.120" Score="0" Body="&lt;p&gt;Negative numbers are usually only defined within the context of the real numbers $\mathbb R$ or within a subset of real numbers. In any case defining negative numbers requires a totally ordered set with a neutral element. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We usually assume that we are dealing with numbers in $\mathbb R$. In that case we say:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A negative number is any number smaller than $0$. The opposite of a negative number is a positive number i.e. a number that is greater than $0$. &#xA;The number $0$ is generally defined to be neither negative nor positive. Some people define  $0$ to be both  negative and positive. The former definition is however considered more usual. &lt;/p&gt;&#xA;" OwnerUserId="3246" LastEditorUserId="39" LastEditDate="2014-12-13T21:19:33.323" LastActivityDate="2014-12-13T21:19:33.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5799" PostTypeId="4" CreationDate="2014-11-07T14:28:27.120" Score="0" Body="For questions on teaching students about negative numbers. Both formally and intuitively. " OwnerUserId="3246" LastEditorUserId="3246" LastEditDate="2014-11-07T14:31:06.767" LastActivityDate="2014-11-07T14:31:06.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5800" PostTypeId="2" ParentId="5790" CreationDate="2014-11-07T14:51:55.103" Score="0" Body="&lt;p&gt;Caveat: this answer is predicated on the student having &quot;a fairly good grip of some abstract algebra and real analysis [and] who wants to gain a rigorous and precise knowledge of the topic [of linear algebra].&quot; Therefore, I would tend to recommend a book more in the vein of Lang's &lt;em&gt;Linear Algebra&lt;/em&gt; than his &lt;em&gt;Introduction to Linear Algebra&lt;/em&gt;, unless one has not seen any linear algebra at all. Usually, the difference between such books is that the introductory type focuses on vectors, matrices, computations and concrete examples, whereas the more advanced books take a more abstract approach, emphasizing structures, mappings and proofs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possible alternative to Lang's &lt;em&gt;Linear Algebra&lt;/em&gt; is Roman's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387728287&quot; rel=&quot;nofollow&quot;&gt;Advanced Linear Algebra&lt;/a&gt;. Formally, Roman assumes little more than some knowledge of matrices and determinants, but a healthy dose of mathematical maturity is recommended, as one would likely have obtained from courses on abstract algebra and real analysis. In fact, a prior course in abstract algebra would be particularly handy, as the basic concepts of groups and rings are introduced fairly rapidly in a kind of chapter zero, entitled &lt;em&gt;Preliminaries&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book is written very much in the definition-theorem-proof style, and may need supplementing with additional examples if the student in question has not taken a prior course on the level of Strang/Lay/Poole etc. It starts with vector spaces and linear transformations, but then moves on to isomorphism theorems and modules, so that should give a rough indication of the level; (it's in Springer's Graduate Texts in Mathematics series, but the dividing line between graduate and advanced undergraduate is fuzzy and depends on many factors). There's a nice selection of topics in the second half of the book, ranging from Hilbert spaces to the umbral calculus.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2014-11-10T08:27:59.863" LastActivityDate="2014-11-10T08:27:59.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5801" PostTypeId="2" ParentId="5794" CreationDate="2014-11-07T15:06:12.173" Score="41" Body="&lt;p&gt;In a comment, the OP has suggested that he actually wants a practical example convincing students that the product of two negative numbers is positive.  This is related to, but psychologically distinct from, the question asked.  In particular, students often have different mental models for multiplication (a binary operator) and negation (an unary operator).  Actually the problem is easy from the negation point of view:  negation just &quot;switches the sign&quot;, and switching the sign twice takes you back to where you started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a &quot;real world&quot; example which models multiplication of real numbers of any sign:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If Jill walks $x$ miles per hour north, and travels for $y$ hours, what is her displacement from her starting position?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $x=3$ and $y=2$, then she ends up $6$ miles north of where she started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $x=-3$ and $y=2$, then we have to interpret what walking $-3 \textrm{mph}$ north means.  It literally means that every $3$ hours, you will be $3$ miles &quot;less north&quot; than you were before.  This is equivalent to walking $3 mph$ south.  So she ends up 6 miles south of where she started, aka $-6$ miles north of where she started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $x=3$ and $y=-2$, then we have to interpret what walking $-2 \textrm{h}$ means.  It means that she has been walking at a constant rate, and we are interested in her position $2$ hours ago.  In $2$ hours she walks $6$ miles north, so $2$ hours ago she was $6$ miles south of where she started, aka $-6$ miles north of where she started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $x=-3$ and $y=-2$, then she is walking $3 \textrm{mph}$ south, and we are interested in her position $2$ hours ago.  She was $6$ miles north of where she is now $2$ hours ago.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2014-11-07T15:16:24.367" LastActivityDate="2014-11-07T15:16:24.367" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5802" PostTypeId="2" ParentId="5700" CreationDate="2014-11-07T23:02:38.333" Score="4" Body="&lt;p&gt;You should get the students to graph $y = |x + 4| -5$ and $y = 10$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's obvious where the solutions are then from that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first graph can be done by shifting the graph of y=$|x|$ to the left by 4 units then down by 4 units.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XndBc.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://i.stack.imgur.com/XndBc.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can then relate +(x+4) and -(x+4) to the different branches.&lt;/p&gt;&#xA;" OwnerUserId="3363" LastEditorUserId="3363" LastEditDate="2014-11-07T23:08:22.853" LastActivityDate="2014-11-07T23:08:22.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5803" PostTypeId="2" ParentId="5756" CreationDate="2014-11-08T02:03:32.140" Score="3" Body="&lt;p&gt;At large research schools, or moderately large state universities there is often an undergraduate or graduate course which deals with the following topics. However, at a small liberal arts school, too often one, two or all three of the following are either missing or under-emphasized:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;elementary topology&lt;/li&gt;&#xA;&lt;li&gt;differential geometry&lt;/li&gt;&#xA;&lt;li&gt;complex analysis&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Actually, topology is very common, but, I list it for my own reason. More important than all of these:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a second course in linear algebra including modules and cannonical forms, multilinear algebra and other fun things we dare not cover in the linear course for the masses.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-11-08T02:03:32.140" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5804" PostTypeId="2" ParentId="5797" CreationDate="2014-11-08T05:04:01.697" Score="7" Body="&lt;p&gt;First, a direct answer:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Yes&lt;/strong&gt;, there is work in mathematics education connected to Gadamer. One soure for such studies is articles by Brent Davis; more generally, &lt;a href=&quot;http://scholar.google.com/scholar?hl=en&amp;amp;q=%22mathematics%20education%22%20%22gadamer%22&amp;amp;btnG=&amp;amp;as_sdt=1%2C22&amp;amp;as_sdtp=&quot; rel=&quot;nofollow noreferrer&quot;&gt;google scholar&lt;/a&gt; yields such connections readily.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here are some examples of BA Davis' work in which Gadamer is cited:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Davis, B. (1997). Listening for Differences: An Evolving Conception of Mathematics Teaching.&#xA;&lt;em&gt;Journal for Research in Mathematics Education, 28&lt;/em&gt;(3), pp. 355-376. &lt;a href=&quot;http://www.jstor.org/stable/749785&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jstor.org/stable/749785&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Davis, B. (1995). Why Teach Mathematics? Mathematics Education and Enactivist Theory. &lt;em&gt;For the Learning of Mathematics, 15&lt;/em&gt;(2), pp. 2-9. &lt;a href=&quot;http://www.jstor.org/stable/40248172&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jstor.org/stable/40248172&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an article of his with no paywall, check &lt;a href=&quot;http://www.doingmathematics.com/uploads/5/0/2/0/5020119/from_telling_to_listening.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and, e.g,. foot-note 19 (pdf 11/19, p. 277).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, I am not totally clear on what you mean by &lt;em&gt;problem&lt;/em&gt;, but here are a couple of remarks about the role of &lt;em&gt;conversation&lt;/em&gt; in the teaching and learning of mathematics. (My post, though somewhat long, necessarily elides over heaps and mounds of literature!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A professor of Mathematics Education once remarked to me that the best problem solvers are those who are good at talking with themselves; this observation can be found more formally in Schoenfeld's (1985) work on &lt;em&gt;Mathematical Problem Solving&lt;/em&gt;, in which expert problem solvers distinguished themselves from novice problem solvers by virtue of self-checking / asking themselves if they were on the right track in the problem solving process. For more, see, e.g., my previous MESE response &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/2399/262&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, features such as &lt;em&gt;accruing mathematical vocabulary&lt;/em&gt; and &lt;em&gt;attending to precision when articulating one's (mathematical) thoughts&lt;/em&gt; are both consistent with the Common Core State Standards. For example, see the &lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Standards for Mathematical Practice&lt;/a&gt; (SMP) and earlier work from NCTM.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quoting from the immediately aforelinked (emphasis added):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, &lt;strong&gt;communication&lt;/strong&gt;, representation, and connections.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The corresponding SMP is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/0MOsA.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How this manifests in the classroom (or even in a two person conversation) will vary based on the participants. In a healthy mathematics classroom, there should be an ongoing conversation about the new ideas that are coming up. However, engaging in conversation of this nature requires significant background knowledge (both mathematical content and pedagogical) and a fair amount of bravery on the part of the instructor. After all, it is easy to find oneself on the receiving end of a question that is intractable, but not obviously so; it takes a certain deftness to engage with students in mathematical conversation while maintaining the wherewithal needed to move them in the general direction of understanding that a given course intends to develop.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the perspective of research in mathematics education, one difficulty presented by conversation-based learning is that it is difficult to identify qualitatively how ideas collaboratively emerge. I believe that some of the literature on distributed creativity (e.g., &lt;a href=&quot;http://scholar.google.com/scholar?q=collaborative%20emergence%20RK%20Sawyer&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=0%2C22&quot; rel=&quot;nofollow noreferrer&quot;&gt;work by RK Sawyer&lt;/a&gt;) may be relevant in this respect. On an individual basis, researchers can certainly evaluate student learning before and after units that use discourse-based learning (e.g., using standard pre- and post-tests). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are interested in how to implement conversation into the mathematics classroom, then the best recommendation that I have for K6 is the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Chapin, S. H., O'Connor, C., O'Connor, M. C., &amp;amp; Anderson, N. C. (2009). &lt;em&gt;Classroom discussions: Using math talk to help students learn, Grades K-6&lt;/em&gt;. Math Solutions. &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1935099019&quot; rel=&quot;nofollow noreferrer&quot;&gt;Amazon link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am confident that these ideas can be implemented in later years, too, and scaled all the way up to the level of research mathematicians conversing with one another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To close, I might also note that the use of conversation in teaching students mathematics &lt;em&gt;and&lt;/em&gt; gaining a better understanding of their (mathematical) understanding can be accomplished to great effect by using clinical interviews. For more on clinical interviews, the best text of which I am aware is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ginsburg, H. (1997). &lt;em&gt;Entering the child's mind: The clinical interview in psychological research and practice&lt;/em&gt;. Cambridge University Press.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As to case studies on individual students' understandings of mathematics, the classic piece you are likely to encounter is Erlwanger's article on Benny (&lt;a href=&quot;http://www.uky.edu/~mfi223/EDC670OtherReadings_files/ErlwangersBenny.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;link without paywall&lt;/a&gt;). More recently, I read a lovely write-up about a young child discussing how she knows number names; the citation is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Brizuela, B. (1997). Inventions and conventions: A story about capital numbers. &lt;em&gt;For the learning of Mathematics&lt;/em&gt;, 2-6.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-11-08T05:04:01.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5805" PostTypeId="1" AcceptedAnswerId="5811" CreationDate="2014-11-08T15:29:04.673" Score="19" ViewCount="10416" Body="&lt;p&gt;I am currently self-studying, so I learn using a textbook which I work through in a  linear fashion. As I am introduced to new concepts I ask myself questions and I attempt to answer these questions and if all else fails I ask Maths SE. Only after I've understood a certain topic do I then formulate all that i've learnt in a Q&amp;amp;A format: i.e. formulas, definitions, theorems, etc. so that it's ready to be inputted in &lt;a href=&quot;http://ankisrs.net/&quot;&gt;Anki&lt;/a&gt; (a spaced repetition system) that is used to create flashcards and schedule them optimally so that you don't forget them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anki is very helpful in ensuring that I retain most of what I've learned. My concern is that I'm essentially storing all that i've learnt in memory as bits of discrete knowledge that I can recall on demand. I feel like this removes the &quot;connections&quot; that are made between concepts; the knowledge  does not seem &quot;networked&quot;, just disparate bits of knowledge i can recall. I want to cultivate a strong &quot;network&quot; of knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my questions are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How can I utilise a spaced repetition system to help me learn Mathematics most effectively? &lt;/li&gt;&#xA;&lt;li&gt;What are the best ways of making strong connections between mathematical knowledge you've learned?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1677" LastEditorUserId="1677" LastEditDate="2014-11-08T18:42:59.057" LastActivityDate="2021-01-14T08:12:40.973" Title="Learning Mathematics with the aid of spaced repetition systems" Tags="|mathematical-pedagogy|technology-in-education|education-research|self-learning|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5806" PostTypeId="1" CreationDate="2014-11-08T17:05:19.090" Score="3" ViewCount="495" Body="&lt;p&gt;Many of my students' parents have asked me to suggest websites to help their children practice the common core math standards.  I work in a very rural school and many of my students are from lower socioeconomic backgrounds.  The ones I have seen have required a fee.  Can anyone suggest a common core math practice site that is free?&lt;/p&gt;&#xA;" OwnerUserId="3367" LastEditorUserId="47" LastEditDate="2014-11-08T17:30:18.607" LastActivityDate="2015-01-04T21:14:30.083" Title="Free websites to practice math for the common core standards" Tags="|technology-in-education|homework|self-learning|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5807" PostTypeId="2" ParentId="5730" CreationDate="2014-11-08T17:13:34.307" Score="5" Body="&lt;p&gt;As no one has mentioned this yet, I should note that the cumbersome question &quot;&lt;em&gt;Which bag is more likely to yield a black marble?&lt;/em&gt;&quot; is yet another linguistic convolution that is more than likely to yield an unnecessary distraction for the 13-year-old adolescent who is simply trying to grasp the crux of the matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No 13-year-old uses the world 'yield', as might be the wont of many an ancient mathematician. Use plain language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;One bag contains 3 balls - 2 black and 1 white.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A second bag contains 30 balls - 20 black and 10 white.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;You are blindfolded and pull one ball out of each bag.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;From which bag are you more likely to get a black ball?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, by the way, repeated simulations on &lt;em&gt;Mathematica&lt;/em&gt; is to any kid just another meaningless black box that won't convince him of anything. Indeed probably not just to kids. Mathematicians who couldn't believe the Monty Hall solution would not have been convinced by  &lt;em&gt;Mathematica&lt;/em&gt; simulations either. &lt;/p&gt;&#xA;" OwnerDisplayName="user378" LastActivityDate="2014-11-08T17:13:34.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5808" PostTypeId="2" ParentId="5794" CreationDate="2014-11-08T17:37:06.120" Score="4" Body="&lt;p&gt;First of all I want to thank a lot the moderator who volunteer to edit my contributions in this nice site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact I have a good practical example to illustrait that $- ( - x ) = x $, but I was a bit greedy to find an other example that meight be easier to accept for students minds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example is as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we have a monybox:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 - If we add 2 dollars a day, then after 5 days we will have increasing that equal to $2 × 5 = + 10 $ .. That was $+ × + = + $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 - If we pull 2 dollars a day, then after 5 days we will have decrease equal to $-2 × +5 = - 10 $ .. This was $ - × + = - $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3- If we add 2 dollars a day, then BEFORE 5 days we have a decrease = $ + 2 × -5 = -10 $ .. That was $+ × - = - $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4- Now suppose we PULL 2 dollars a day, then BEFORE 5 days we were had an increase = + 10&#xA;That was $ - 2 × -5 = + 10 $&lt;/p&gt;&#xA;" OwnerUserId="3306" LastActivityDate="2014-11-08T17:37:06.120" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5809" PostTypeId="2" ParentId="5790" CreationDate="2014-11-08T19:12:11.433" Score="1" Body="&lt;p&gt;Learning calculus and linear algebra for the first time as an undergraduate, I found Lang's books hard to understand, because they didn't go into enough detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reviewing those same subjects as a graduate student, I found that Lang's books provided a good synthesis for people who had a reasonable grasp of the mechanics, and needed to get the &quot;big picture.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a mathematically mature and relatively &quot;experienced&quot; student in the junior or senior year, I would recommend the advanced &quot;Linear Algebra&quot; book. The &quot;Introduction to Linear Algebra&quot; is &quot;kid's stuff&quot; for freshmen, or less advanced sophomores.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2014-11-10T01:29:52.247" LastActivityDate="2014-11-10T01:29:52.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5810" PostTypeId="2" ParentId="5752" CreationDate="2014-11-08T19:37:21.827" Score="0" Body="&lt;p&gt;The sine is the ratio of the &quot;opposite&quot; axis (relative to theta, the angle in question) compared to the hypotenuse. The hypotenuse is always longer because it is opposite the right angle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as theta approaches 90 degrees, the length of the opposite axis approaches the hypotenuse as limit (in fact, they both approach infinity).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conversely, as theta approaches 90 degrees, the third angle approaches zero (90 degrees minus theta), and the axis opposite this angle shrinks as theta angle does. T value of the axis approaches zero as the &quot;non-theta&quot; angle approaches zero.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-11-08T19:37:21.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5811" PostTypeId="2" ParentId="5805" CreationDate="2014-11-08T21:41:49.080" Score="6" Body="&lt;p&gt;Making connections between concepts can be done with something called (simply enough) a &lt;strong&gt;Concept Map&lt;/strong&gt;. (In fact, this is a key part of some current research that I'm involved in, and we plan to present at both the AMTE 2015 and NCTM 2015 conferences in Orlando and Boston, respectively.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concept mapping was developed by Joseph D. Novak in the 1970s. His book &lt;a href=&quot;http://books.google.com/books/about/Learning_How_to_Learn.html?id=8jkBcSDQPXcC&quot; rel=&quot;noreferrer&quot;&gt;Learning How to Learn&lt;/a&gt;, coauthored with D. Bob Gowin, talks of how you can create these concept maps yourself. Additionally, he talks of creating a knowledge Vee, which may also be another helpful visual to make connections between ideas. (Though, I must confess, I don't know much about the knowledge Vee apart from its existence.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that Concept Maps are not Mathematics specific, and can be applied to any number of fields, but Novak himself holds degrees in Science and Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-11-08T21:41:49.080" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5812" PostTypeId="1" CreationDate="2014-11-09T01:00:46.050" Score="18" ViewCount="656" Body="&lt;p&gt;A colleague of mine in a math department at another university is looking for a textbook on multivariable calculus that discusses applications of higher-dimensional integrals that feel contemporary rather than solidly traditional. In particular, he is looking for a path through the subject that culminates in something other than Stokes' theorem, since that is hard to get majors outside of math and physics excited about. The students who take the course in its current form are willing to work hard to pass the class, but they are &lt;em&gt;not&lt;/em&gt; math-oriented (e.g., they major in economics or biology) and they simply don't see an overlap between the standard big integral theorems of vector calculus and their own interests. (I once asked a statistics professor if he ever used Stokes' theorem and he said no.) It's not even necessary that the multivariable calculus be applied to a student's major, but at least to something that looks fresh and modern (computer graphics, forecasting of all kinds) and maybe even exhibits an awareness that the people who use it and don't live in a math department rely on computers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Edit: My colleague is looking for a new book as part of faculty discussions to change the syllabus of the department's multivariable calculus course.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any textbooks on the market that have a genuinely different approach to what multivariable calculus can be good for and present a series of interesting applications of multidimensional integrals? &lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="143" LastEditDate="2014-11-09T10:40:28.777" LastActivityDate="2015-08-07T15:20:44.243" Title="Textbook for multivariable calculus with interesting modern applications" Tags="|undergraduate-education|textbooks|vector-calculus|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5813" PostTypeId="1" CreationDate="2014-11-09T03:46:49.357" Score="7" ViewCount="409" Body="&lt;p&gt;I want to make the binary class fun for my students, and I would like to apply activities to make it easier.&lt;/p&gt;&#xA;" OwnerUserId="3366" LastEditorUserId="376" LastEditDate="2014-11-09T09:15:34.940" LastActivityDate="2014-12-24T06:21:56.030" Title="How to teach addition, subtraction, multiplication and division of binary numbers? Are there any activities that can be recommended?" Tags="|secondary-education|mathematical-pedagogy|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5814" PostTypeId="2" ParentId="5794" CreationDate="2014-11-09T10:56:29.577" Score="18" Body="&lt;p&gt;I would use the (real) number line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First introduce the concepts of positive numbers (distance to the right of zero) and negative numbers (distance to the left of zero).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then introduce the concept of negation.  Instead of teaching it as flipping with respect to the vertical line at zero, I recommend teaching it as the rotation of $180^\circ$ (counter-clockwise) about the point zero.  (I'll explain why later.)  This handles the case of why negating a negative number yields a positive number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then introduce the concept of addition (of positive numbers) as moving to the right, subtraction (of positive numbers) as moving to the left.  Note that the negation of a positive number yields the same result as subtracting a positive number from zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now introduce the concept of addition (of negative numbers) as moving to the left, that is, show that adding a negative number yields the same result as subtracting a positive number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, introduce the concept of subtraction (of negative numbers) as moving to the right, that is, show that subtracting a negative number yields the same result as adding a positive number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Comment&lt;/em&gt;:  I teach negation as rotation of $180^\circ$ (counter-clockwise) about zero.  This helps later on when the complex numbers are introduced.  With the imaginary axis (and the complex plane) introduced, I teach multiplication by $i$ as a rotation of $90^\circ$ (counter-clockwise) about zero.  This makes it easy to understand why $i^2=-1$.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2015-10-30T14:18:59.670" LastActivityDate="2015-10-30T14:18:59.670" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5815" PostTypeId="2" ParentId="2497" CreationDate="2014-11-09T19:47:37.337" Score="1" Body="&lt;p&gt;At the beginig of teaching the negative numbers, I prefere to use the temperture, becaues children need a non - abstract example. It will be easy for them to understand that temperture at $0$ is more than any negative temperture. Consequently $-3 &amp;gt; -4$ will be clear.&lt;/p&gt;&#xA;" OwnerUserId="3306" LastEditorUserId="3306" LastEditDate="2014-11-09T19:55:52.093" LastActivityDate="2014-11-09T19:55:52.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5816" PostTypeId="2" ParentId="5794" CreationDate="2014-11-09T21:05:54.783" Score="9" Body="&lt;p&gt;This is redundant with several nice answers, but let me try it anyway,&#xA;as (for me) the intuition is strong.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A gambler (or spendthrift) loses \$10 per day.&#xA;  If they have \$30 today, how much did they have 5 days ago?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So I'm using loses per day as one negative, and past time as another.&#xA;It is intuitively clear that the further in the past, and the greater&#xA;the loses per day, the larger the initial pot of money.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-11-09T21:05:54.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5817" PostTypeId="2" ParentId="5805" CreationDate="2014-11-09T23:22:04.160" Score="3" Body="&lt;p&gt;For the past eight years or so I have used spaced repetition to maintain proficiency in a number of areas. primarily it is language study, but I also use it to maintain a grasp on many scientific concepts, although I have not yet begun to focus entirely on math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that learning mathematics would be similar in principle to learning a language. When I am learning a new language, I have to focus on two areas: grammar and vocabulary.I have found that creating multiple flashcards that test my knowledge of a particular grammar point is very helpful at making sure that your knowledge forms a cohesive network, one flash card by itself does not seem to be sufficient for something as broadly &quot;connectable&quot; as grammar is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, when learning about a more difficult subject, I need to create flashcards that connect each of the pieces together. Imagine that your concept is a large dinosaur skeleton, and flashcards or the cables and glue that hold the statue together. For every connection, you need at least one flash card. While this might seem excessive, over the next 30 years reviewing that single flash card barely takes minutes of your cumulative time, so add many flashcards and delete the ones you find to not be useful as you review them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using spaced repetition seems to be the most efficient way to maintain synaptical connections in the brain that humans have at their disposal right now. You are definitely on the right track, you just need to learn to capture the connections of a more nuanced discipline in flashcard form. I highly recommend you read this article that is basically the 20 Commandments of flash card creation: &lt;a href=&quot;http://www.supermemo.com/articles/20rules.htm&quot; rel=&quot;nofollow&quot;&gt;http://www.supermemo.com/articles/20rules.htm&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3368" LastActivityDate="2014-11-09T23:22:04.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5818" PostTypeId="1" AcceptedAnswerId="5820" CreationDate="2014-11-10T00:37:58.740" Score="19" ViewCount="1541" Body="&lt;p&gt;Here is a statistics question I asked on a recent exam:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How high should the doorway be to allow 97% of men to fit through it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I got a very large number of answers like &quot;3 inches&quot; or &quot;0.0001 feet&quot;. This makes me sad. I'd like to know if these students notice that their answers are nonsense, and reward them if they have this level of sense-making.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I set up my grading scheme to reward people who &lt;strong&gt;notice&lt;/strong&gt; their answer is nonsensical? Is there any way to do this besides the side benefit that nonsense answers are a clue that the work is probably wrong?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-11-13T18:07:58.353" Title="How to award points for sense-making at the end of a problem?" Tags="|exams|grading|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5819" PostTypeId="2" ParentId="5818" CreationDate="2014-11-10T00:59:29.040" Score="11" Body="&lt;p&gt;In my grading scheme, any answer that is wrong, checkable, and not checked gets $1/4$ off the partial credit. (I give partial credit for work shown, depending on how much knowledge is shown, the pettiness of the mistake, and so on.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student gets a wrong answer, checks it and sees that it is wrong, &lt;em&gt;and notes this in his answer or work&lt;/em&gt;, I do not remove the $1/4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps I should add that I accept a &quot;sanity check&quot; as a check. It is only partial, as it does not catch a &quot;close but no cigar&quot; answer, but it is still a valid check.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2014-11-10T11:16:06.840" LastActivityDate="2014-11-10T11:16:06.840" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5820" PostTypeId="2" ParentId="5818" CreationDate="2014-11-10T01:31:35.863" Score="8" Body="&lt;p&gt;Introducing the slogan/motto of &quot;sanity check&quot; may be useful, especially if/when mathematics is being taught as something that is entirely compatible with the students' prior experience, general sense, and so on. Situations (all-too-often upper-division undergrad...) in which one's physical intuition is utterly disallowed and students are &quot;pranked&quot; on syntactic or semantic technicalities have many severe faults, not-the-least of which is conditioning to disregard &quot;sanity checks&quot; ... because, apparently, such things are irrelevant or disjoint from the enterprise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in fact, to exhort people that their physical intuition is basically reasonable, and to look at mathematics as an extension of that, making &quot;sanity check&quot; sensible, is a good thing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(A similar issue is &lt;em&gt;prioritization&lt;/em&gt; of concern for details. I.e., not all details are of equal importance, despite the &quot;logical&quot; principle that if any single thing is &quot;wrong&quot;, then the whole is wrong, etc. In real life, it's not that boolean...)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2014-11-10T01:31:35.863" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5821" PostTypeId="2" ParentId="5818" CreationDate="2014-11-10T02:22:25.117" Score="4" Body="&lt;p&gt;Might you be able to offer an initial &quot;guesstimate&quot; as a sanity check for their final answer? In this example, they are forced to consider the question absent any math. Their initial guess should be about 6ft, in my opinion. Once they solve the problem if they come up with even 5ft, they should realize something is wrong. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 3 inch answer is another issue, they are not applying any sanity check at all to their work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, I've seen a result of 900 for the number of seconds for a rock to fall to the ground off a building. When the student is forced to ponder whether this makes any sense, they agree, 15 minutes is too long. A calculator error was made, 9 seconds was correct. &#xA;In the end, problems that are real-life need a forced sanity check somewhere along the way. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-10T02:22:25.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5822" PostTypeId="1" CreationDate="2014-11-10T06:31:12.147" Score="8" ViewCount="408" Body="&lt;p&gt;As math teachers, sometimes we may wish that we could enter our students' minds to see how they think. In early years, while children are learning language, they build the concept of the word &lt;em&gt;greater&lt;/em&gt; in relation to the concept of volume, shown visually in the shape below: $1 &amp;gt; 3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/1esLa.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact if we rename the greater than sign to be &lt;em&gt;more than&lt;/em&gt;, then I believe it will be more harmonious with the language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, to avoid conflict with negative numbers, we could say &lt;em&gt;more positive&lt;/em&gt;. For example, read $ -2 &amp;gt; -5 $ as &quot;$-2$ is more positive than $-5$.&quot; On the line of integers it would be clearer with this language that $-2$ is closer to the positive numbers than $-5$. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My question is: &lt;strong&gt;How deep and meaningful could this language alteration change children's concept of the inequality?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3306" LastEditorUserId="267" LastEditDate="2014-11-10T06:45:22.533" LastActivityDate="2014-11-12T16:25:30.310" Title="Should we rename the greater than sign?" Tags="|undergraduate-education|negative-numbers|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5823" PostTypeId="2" ParentId="1609" CreationDate="2014-11-10T15:21:01.940" Score="5" Body="&lt;p&gt;I know this isn't a full answer, but I wanted to mention John Holt's excellent &lt;a href=&quot;https://en.wikipedia.org/wiki/How_Children_Fail&quot; rel=&quot;noreferrer&quot;&gt;&quot;How Children Fail&quot;&lt;/a&gt; which has many examples on children's failure to learn math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the issues mentioned in the book are of a more &quot;affective&quot; nature, but are still, very relevant to this discussion. In particular, it seems that many of the examples of &quot;failure&quot; in the book stem from the children not understanding the place of math in the real world and thus fail to develop intuitions about it. It's as if, to some, math is only arbitrary manipulation of symbols devoid of any meaning.&lt;/p&gt;&#xA;" OwnerUserId="688" LastActivityDate="2014-11-10T15:21:01.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5824" PostTypeId="2" ParentId="5822" CreationDate="2014-11-10T16:56:58.413" Score="7" Body="&lt;p&gt;Talking about your proposed language alteration with your students would make more of a difference than just using it. You could help them see that math is partly about defining relationships carefully, and that language, with all its richness, isn't so good at that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In calculus, when I ask where the derivative is greater, and we're looking at negative rates of change, I know there will be a problem in translation from words to math, so I point it out. (With positive slopes, greater means steeper, with negative slopes, greater ends up meaning less steep. It takes some getting used to.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Asking students how they would describe things is more important than changing the vocabulary ourselves. Take a student's term, and make it a class term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Of course, if you use non-standard terms it's important to remind students of the official term that goes with your non-standard term, so they don't go out thinking the rutabaga of 9 is 3. See &lt;em&gt;The Number Devil&lt;/em&gt; for more fun non-standard terms!)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-11-10T16:56:58.413" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5825" PostTypeId="1" CreationDate="2014-11-10T18:02:32.197" Score="10" ViewCount="457" Body="&lt;p&gt;Our district has changed its approach to placing students in grades 6 and 7 math classes. Students considered for placement above grade level must now take a test composed of problems drawn from the next-level CMP textbook: for instance, in the spring of grade 5 they take a test based on problems from the CMP 6 textbook. Students must score 95% or better on this test to be placed above grade level; otherwise they will be in CMP 6 in 6th grade. Teacher input and scores from standardized tests are not considered. I cannot find data supporting the use of this method. Do you have any comments on the usefulness of this approach to placement?&lt;/p&gt;&#xA;" OwnerUserId="3369" LastActivityDate="2015-11-15T03:04:36.727" Title="Placement tests for middle school math" Tags="|mathematical-pedagogy|exams|assessment|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5826" PostTypeId="2" ParentId="5818" CreationDate="2014-11-10T22:18:16.433" Score="10" Body="&lt;p&gt;Students tend to only pay attention to feedback that affects their grade. In your example, where the student calculates the height of the door to be 0.0001 inches, showing no sign of realizing that the answer is impossible, I would give a zero on the problem. If the student writes, &quot;Hmm...obviously this is wrong, but I've run out of time to track down the mistake,&quot; I would give appropriate partial credit.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-11-10T22:18:16.433" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5827" PostTypeId="2" ParentId="5822" CreationDate="2014-11-11T14:53:03.470" Score="4" Body="&lt;p&gt;The answer to the topic question: No.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Toscho should upgrade his comment to an answer. The name of the '&gt;' sign is used in too many places to be replaced without causing confusion elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there is a more important point I want to make: One thing can have more comparable properties as you have demonstrated and Toscho commented. Size of circle vs count of the circles. Height of a person vs her age, or weight. Make sure you define the property you want to compare first. Then you can get another comparative adjectives (3 yrs older, 1 inch taller etc.) to help you explain what the greater means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you decide to work with numbers later, keep only numbers! No circles, no distractions. You may want to show the numbers as moves along the number line. Greater than to the right, less than (or whatever you find suitable) to the left. Note that the movement has always direction, this way the sign (+/-) comes into the mind as necessity.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2014-11-11T14:53:03.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5828" PostTypeId="1" AcceptedAnswerId="5836" CreationDate="2014-11-11T16:37:36.863" Score="5" ViewCount="291" Body="&lt;p&gt;I notice that all of the analysis books that I've studied start from dealing with limits of sequences and only then move on to limits of functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does this kind of approach have any particular advantage from an educational point of view? Or maybe it follows the historical development of the subject?&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="47" LastEditDate="2014-11-11T18:36:40.510" LastActivityDate="2014-11-12T20:42:39.693" Title="Real analysis: why usually first limits of sequences and then limits of functions?" Tags="|undergraduate-education|mathematical-pedagogy|course-design|mathematical-analysis|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5829" PostTypeId="2" ParentId="5635" CreationDate="2014-11-11T20:32:46.020" Score="2" Body="&lt;p&gt;Another fine book which is, in a way, a modern counterpart to &lt;em&gt;What is Mathematics?&lt;/em&gt;, is &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486284247&quot; rel=&quot;nofollow&quot;&gt;Concepts of Modern Mathematics&lt;/a&gt; by Ian Stewart&lt;/p&gt;&#xA;&#xA;&lt;p&gt;who has also written many other great books (see e.g. the &lt;a href=&quot;http://www.amazon.com/Ian-Stewart/e/B000APQ9NM/ref=dp_byline_cont_pop_book_1&quot; rel=&quot;nofollow&quot;&gt;list&lt;/a&gt; at Amazon).&lt;/p&gt;&#xA;" OwnerUserId="365" LastActivityDate="2014-11-11T20:32:46.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5830" PostTypeId="2" ParentId="5828" CreationDate="2014-11-11T23:48:11.827" Score="1" Body="&lt;p&gt;&quot;Sequences&quot; refer to sequences of numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Numbers&quot; are easier to deal with than functions, hence limits of sequences, versus limits of functions. (For most people, anyway.)&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-11-11T23:48:11.827" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5831" PostTypeId="2" ParentId="5783" CreationDate="2014-11-11T23:56:26.127" Score="2" Body="&lt;p&gt;One of the favorite mathematical activities of many students is &quot;cancellation.&quot; An example is when you are asked to reduce a complicated fraction, and you can do so by &quot;factoring out&quot; a large number from both numerator and denominator, leaving &quot;lowest terms.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can convince students that matrices are really representative of systems of equations, they can have a lot of fun reducing the systems of equations to lowest terms. Using &quot;low echelon row reduction&quot; for instance, you can create a diagonal matrix, where one equation is a number of x, the equation above it relates to only x's and y's, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing to do is to show how matrices determine polyhedrals in space. And how the related determinants &quot;reduce&quot; to the volumes of these polyhedrals.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-11-11T23:56:26.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5832" PostTypeId="2" ParentId="5635" CreationDate="2014-11-12T03:29:06.530" Score="0" Body="&lt;p&gt;I rather admire the book by Sherman Stein (who has written lots of fine books) called Mathematics The Man-Made Universe. &lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-11-12T03:29:06.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5833" PostTypeId="2" ParentId="5822" CreationDate="2014-11-12T15:55:09.530" Score="2" Body="&lt;p&gt;It is a good habit, in speaking or writing, to get used to saying &quot;more positive&quot; or &quot;more negative&quot; in situations where confusion may arise. However, students should also be taught that $&amp;lt;$ has a completely standard meaning, so that they can read books and listen to people who are not careful about this.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2014-11-12T15:55:09.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5834" PostTypeId="2" ParentId="5805" CreationDate="2014-11-12T16:02:48.850" Score="6" Body="&lt;p&gt;I've been tempted to try to use spaced repetition in the past, as there is certainly some kind of linguistic component to learning mathematics. This always meets the paradox that many mathematicians work nonverbally, as pointed out in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486201074&quot; rel=&quot;nofollow&quot;&gt;Hadamard's study of the mental habits of mathematicians&lt;/a&gt;. Mathematics is accumulated less by mnemotechnics than by a slow and difficult struggle to use all of one's mental capacity to produce reasonably faithful mental models, as observed many places by &lt;a href=&quot;http://arxiv.org/pdf/math/0503081.pdf&quot; rel=&quot;nofollow&quot;&gt;William Thurston&lt;/a&gt;, e.g. as expressed in this quote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Mathematics is amazingly compressible: you may struggle a long time, step by&#xA;  step, to work through some process or idea from several approaches. But once you&#xA;  really understand it and have the mental perspective to see it as a whole, there&#xA;  is often a tremendous mental compression. You can file it away, recall it quickly&#xA;  and completely when you need it, and use it as just one step in some other mental&#xA;  process. The insight that goes with this compression is one of the real joys of&#xA;  mathematics&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What worries me about trying spaced repetition/mnemotechnics is that these try to get the compression/memory without the long struggle to form the mental structure. Such a process is precisely the opposite of trying to store things cleanly in order to remember them quickly and accurately. Thurston's quote suggests that memory of a topic is won by engaging very painstakingly for a long time with an extraordinary mess.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This sort of thing is even reported in interviews with more algebraic thinkers like &lt;a href=&quot;http://www.alainconnes.org/docs/Inteng.pdf&quot; rel=&quot;nofollow&quot;&gt;Alain Connes&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot; I always proceed in&#xA;  the following way : whatever the complexity of the problem, instead of trying it first on a&#xA;  piece of paper, with a pencil, I just go out for a walk, and try to have all the ingredients&#xA;  present in my mind, in order to start manipulating them mentally. Only after this exercise,&#xA;  am I able to see clearly, think about the various steps and begin to get a mental picture.&#xA;  This is a painful process which consists in gathering in your mind, in your memory, all&#xA;  the elements of the problem, in order to begin manipulating them. It is an exercise that I&#xA;  recommend - well, of course, different people function differently- if one wants to be able not&#xA;  to depend upon paper and pencil. Because with paper and pencil, you get tempted to start&#xA;  writing immediately, and if you haven’t thought long enough before, you will get nowhere.&#xA;  You will get discouraged before having had enough time to create in the linguistic part of&#xA;  the brain specific mental pictures that you can then manipulate, as usual, by zipping them,&#xA;  transforming them into something smaller, and then moving them around.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This compression is again won by struggle. I'd be wary of &quot;learning&quot; mathematics via ANY sort of mnemotechnics. It's probably best to struggle with things and work for their compression without looking for a shortcut! Of course, this is just my opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps there is one honest mnemotechnic mathematicians use: the simplest nontrivial example of a given phenomenon. Such an example is analogous to a model of a finite geometry, in that any such example encodes the whole theory and the example can be generated from a relatively small seed that is more compatible with our mental modules, perhaps computationally or visually. One gets the impression that such an example is more concrete since its interrelations are less disembodied, or could be generated from the simpler rules used to define the example.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; The following may be helpful regarding the use of spaced repetition. It might be helpful to internalize &quot;atomic facts&quot; with which to work (think arithmetic facts) but is not likely to be helpful for working with these facts in creative ways, (think fancy mental arithmetic a la Art Benjamin). This said, the following excellent quote of Paul Halmos about the mathematician's work points to a potentially useful application of spaced repetition:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For the professional pure mathematician, mathematics is the logical dovetailing of a carefully selected sparse set of assumptions with their surprising conclusions via a conceptually elegant proof.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From this standpoint, it may be useful to employ spaced repetition to quickly internalize the &quot;sparse set of assumptions&quot; in order to begin thinking about them. This is interesting in light of Alain Connes's quote above, as well.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2015-05-07T23:59:19.587" LastActivityDate="2015-05-07T23:59:19.587" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5835" PostTypeId="2" ParentId="5822" CreationDate="2014-11-12T16:25:30.310" Score="2" Body="&lt;p&gt;Determining how mathematics applies to a situation (how to mathematize a situation) often requires measurement. To determine whether one thing is greater than another thing, first those things must be identified as quantities. And then they need to be compared somehow; this can involve a measurement process that results in coming up with a number to represent the quantity in some appropriate system of units.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the OP example given, we could compare total diameters. We could compare individual diameters. We could compare total area or individual areas. We could compare hue. We could compare a measure of curvature. I'm sure there are a lot of other things to compare.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is, identifying quantities and measuring them is an important part of mathematical thinking that, among other things, helps to put mathematics in context. This should be a process students become used to engaging in, else their mathematical knowledge is trapped in specifically-worded problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, the real solution here is to get students to become aware of quantity and measurement and to ask questions if the quantities under comparison are not obvious. And perhaps specify their own basis for comparison if one is not handed to them by an authority figure.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-11-12T16:25:30.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5836" PostTypeId="2" ParentId="5828" CreationDate="2014-11-12T20:42:39.693" Score="4" Body="&lt;p&gt;One reason for this is that many beginning analysis books start with a rigorous development of the real numbers.  If the chosen foundation is based on Cauchy sequences of rational numbers, then limits of sequences are logically prior to limits of functions of real variables.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-11-12T20:42:39.693" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5837" PostTypeId="1" CreationDate="2014-11-12T20:58:04.963" Score="9" ViewCount="289" Body="&lt;p&gt;I think a significant start to my development as a mathematician was playing card games (mostly Euchre) with my parents in my youth. After a particular round, my father would tell me, &quot;Well, with your hand, you should have done this. It would have forced me to do that. And then you could've come back with this, thus winning the round.&quot; Basically, I feel like I was taught early on how to reason and make an argument in the context of a game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fast forward, next semester I'm teaching an &quot;intro to proofs&quot; course (not for the first time) and I'd really like to start the semester off with having the students play a game in pairs such that they can explain after each round who should have won. Unfortunately, most of the games I know and enjoy playing are too complicated or too long to achieve this efficiently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is: Does anyone know of a game that satisfies some or all of&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Easy to learn (so it doesn't take all class to explain it). &lt;/li&gt;&#xA;&lt;li&gt;Rounds are short (so multiple rounds can be played and students can get a sense of how it works). &lt;/li&gt;&#xA;&lt;li&gt;Easy to move back and forth in time (so it's easy to replay the game from various points, so probably only a handful of moves in each round). &lt;/li&gt;&#xA;&lt;li&gt;Sometimes, if the opponent plays a certain way, your choice of moves may be reduced to only one option.&lt;/li&gt;&#xA;&lt;li&gt;Conducive to basic logical analysis.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The only idea I've had so far is Tic-Tac-Toe, but I thought I'd ask to see if there's maybe a lesser-known game that also fits the bill.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-01-12T21:29:31.240" Title="A basic game to make arguments about" Tags="|proofs|games|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5838" PostTypeId="2" ParentId="5825" CreationDate="2014-11-12T21:12:24.243" Score="0" Body="&lt;p&gt;In America at least, there is a strong emphasis on &quot;social&quot; promotion, and keeping children with their &quot;peer group.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, they won't allow promotion to the &quot;second&quot; level above the current one, unless a student can get a 95% on a test &lt;em&gt;before&lt;/em&gt; going into the &quot;next&quot; one. This is a difficult (but not insurmountable) hurdle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, the American grade school systems want to make special provision only for students that are clearly &quot;gifted,&quot; not merely &quot;good.&quot;&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-11-12T21:12:24.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5839" PostTypeId="2" ParentId="5837" CreationDate="2014-11-12T21:40:05.150" Score="9" Body="&lt;p&gt;&lt;strong&gt;Sprouts&lt;/strong&gt; is a very excellent game that has a lot of scope for proving things, not just for strategy but also for other aspects of the game too. &lt;a href=&quot;http://en.wikipedia.org/wiki/Sprouts_%28game%29&quot;&gt;http://en.wikipedia.org/wiki/Sprouts_%28game%29&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The game begins with a number of dots drawn on the page (5 is good for a short game). On your turn you can join two dots or a dot to itself as long as any line you draw doesn't cross other existing lines or pass through other dots. Then you draw a new dot on the line you just drew. Also, if a dot has three paths exiting from it, then it can no longer be used. Note that this means any dot drawn on a line has only one more line that can go from it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The winner is the last person who draws a line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good strategies involve enclosing dots inside loops so that even though they have a free path you can't draw it without crossing another line.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-11-12T21:40:05.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5840" PostTypeId="2" ParentId="4367" CreationDate="2014-11-12T21:45:48.953" Score="4" Body="&lt;p&gt;I have a suggestion for getting a feel for how the number system works, which might even convince them that it is a useful thing to know how to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Step 1: A method of counting&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take a pile of small objects (beads, lollies, blocks, pieces of spirali pasta, Go counters -- whatever takes your fancy -- though anything that naturally joins together might have a slight advantage). Then say you are going to count how many there are by a new and creative method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First you'll group them into pairs. There may or may not be one left over but that's ok. Next you'll group your pairs into pairs, making groups of four. There may or may not be a pair left over but that's ok. Now you'll group your groups of four into pairs, making groups of 8.Now group the groups of 8 into pairs, making groups of 16.Now group the groups of 16 into pairs, making groups of 32.If you haven't chosen too many things in your pile it means that now you'll have to stop because there are no pairs of groups. Of course if there are more you'll just keep going on pairing up the biggest groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/piLS7.png&quot; alt=&quot;grouping in pairs to get a binary representation&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now how many have we got? Well let's start with the biggest group. We have one group of 32, one group of 8, one group of 2 and a single object. So that's 43 things. As a way of counting it was quite efficient because we weren't actually counting at any point, only pairing things up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this stage you can give everyone a pile of things and they can do the pairing up themselves and figure out how many objects they have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[You can also imagine this being done with the students themselves -- get them to pair up, then pair up the pairs etc. Of course some of them might be a bit miffed at being the ones left out at the end.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Step 2: A new way to represent numbers&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should be able to say at this point that in fact every number can be grouped like this because they can see that all their piles have been effectively split into pairs progressively. So that means it's a neat way of representing what number you have. But there's an even neater way to do it and here it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;List the sizes of the piles on the board/document camera/screen: 32, 16, 8, 4, 2, 1&#xA;Now underneath say you'll write how many of each pile you have. In our example you have 1 0 1 0 1 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask them to do this for their own piles and get some or all of them to write their numbers and their list of piles on the board.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this stage they should be able to see that every number has a representation as a string of 0's and 1's. Ask them if it's possible to get anything other than 0 or 1 (the answer is no, but you want them to be sure).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can say that this is the binary representation of a number. Each digit is how many groups of that size you get when you successively pair things up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[It may be worth doing this process for other sized groupings, like grouping into 10's, which they may be familiar with from earlier years.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you can give them various binary representations and ask them to tell you what number they represent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Step 3: Adding binary numbers&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now pair up your students and get them to add their numbers together and find out what the new binary representation is. See if they can do this without having to start from scratch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully they hit upon the idea that they can join any groups that are the same size into bigger groups, and continue this process until there are no more. They'll probably do this organically with no structure, and after discussion you can practice by always starting with the smallest piles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can see how this works on paper with the binary representation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this method could work quite well to get them started on binary numbers, though including some other activities involving counting in binary with the fingers of your hand (as others have described) wouldn't go astray.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-11-12T21:45:48.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5841" PostTypeId="2" ParentId="4367" CreationDate="2014-11-12T22:33:13.350" Score="2" Body="&lt;p&gt;Give them some motivation to learn it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three years ago I was meeting 3 fifth-graders once a week for half an hour of math.  Mostly we did puzzles and games.  One time I gave them the rules of Nim; we played it in various combinations, and (naturally) I could always win.  I even played all 3 of them simultaneously and won all the games.  So by then they were asking how I did it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some later week (reminding them about Nim) we discussed using bases other than 10, and converting back and forth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then a third week I told them the strategy for Nim, based on writing the numbers in base 2.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2014-11-12T22:33:13.350" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5842" PostTypeId="2" ParentId="5837" CreationDate="2014-11-13T01:57:32.690" Score="5" Body="&lt;p&gt;I think you will like &lt;a href=&quot;http://en.m.wikipedia.org/wiki/Eleusis_(card_game)&quot;&gt;Eleusis&lt;/a&gt;, a card game where one player invents a rule for valid sequences of cards, and the rest of the players &quot;experiment&quot; by proposing a sequence and being told whether it is acceptable or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The game rules are a bit involved, but there is a version called &quot;Eleusis express&quot; that may be suitable. Quoting the Wikipedia link above, &lt;em&gt;&quot;In 2006, John Golden developed a streamlined version of the game, intended to assist elementary school teachers in explaining the scientific method to students.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2014-11-13T01:57:32.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5843" PostTypeId="2" ParentId="5778" CreationDate="2014-11-13T15:57:22.023" Score="4" Body="&lt;p&gt;Trying to &quot;linearize&quot; the mathematics in a course is among the hardest problems facing curriculum designers and teachers. This is especially true for geometry, where if one does not use an axiomatic approach it is quite hard to both select good topics and the order to teach them. In a high school geometry class the typical student will not go on to become a STEM major in college so, it seems to me, highly desirable to organize what is taught to show students the appeal, applicability, and power of using geometrical ideas. Logic and proofs though important for mathematics majors is rather dry to say the least for many other students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I teach geometry for undergraduate majors I often start with with Euler's traversability theorem for graphs which has the charm of being a lovely result and highly applicable. (e.g. snow removal, garbage collection) I try to use the theory of graphs as a unifying backbone for the geometry I teach. I realize this approach works better for a survey of geometry in a college level course than what can be done in lower grades given the poor choice of geometry in the K-12 curriculum.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-11-13T15:57:22.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5844" PostTypeId="2" ParentId="4367" CreationDate="2014-11-13T16:08:15.647" Score="2" Body="&lt;p&gt;It may be helpful when introducing binary numbers to begin with showing the difference between counting and arithmetic in binary and binary codes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Binary_code&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Binary_code&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may help clarify the complex issue of symbols and what these symbols are used for. By the 5th grade students have a fair amount of experience with the decimal system. Decimal codes are less common than binary codes and so when one moves to showing students how one can count and do arithmetic in other systems than decimal it may be useful that binary codes be part of what one does. Furthermore there are lots of lovely applications of binary codes (Huffman codes, for example).&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-11-13T16:08:15.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5845" PostTypeId="2" ParentId="5818" CreationDate="2014-11-13T18:07:58.353" Score="1" Body="&lt;p&gt;In accounting and finance work (my field) &quot;order of magnitude&quot; (thousands, millions, billions), is nearly as important as the actual number itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, I would give partial credit for answers in the correct order of magnitude, and zero for answers outside it. In this example, people should know that the correct order of magnitude is 70-80 inches, or even &quot;a large multiple of 10 inches, but less than 100,&quot; even if they can't do the calculations for means and standard deviations, etc. Or a somewhat acceptable answer might be &quot;mean, plus about two standard deviations,&quot; either in those terms, or using numerical values if given, or calculable.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-11-13T18:07:58.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5846" PostTypeId="2" ParentId="4367" CreationDate="2014-11-13T20:09:36.643" Score="2" Body="&lt;p&gt;I'm posting this answer to complement @JoshuaZucker's one, whose idea is basically the same I'm introducing. Instead of using money, use &lt;strong&gt;weights&lt;/strong&gt; and... the &lt;strong&gt;magic cards&lt;/strong&gt; (they'll love them).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we have a set of weights. One weight of 1kg, one of 2kg, one of 4kg, one of 8kg and one of 16 kg. Then, I propose the students to make a table specifying which weights are required to weigh 1kg, 2kg, 3kg, 4kg, 5kg, ... 31kg. For example, to weigh 7kg you need the weights of 4kg and 3kg. And so on. Actually, when they'll finish, they'll have a complete binary representation of numbers from 1 to 31.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just for amusement (motivation) and reinforce it, they can prepare the so called &lt;a href=&quot;http://www.mathmaniacs.org/lessons/01-binary/Magic_Trick/&quot; rel=&quot;nofollow&quot;&gt;magic cards&lt;/a&gt;. You could even start the lesson showing the &lt;em&gt;power&lt;/em&gt; of this cards. Show the cards with the PC and tell a student to think a number. Then tell him/her what number it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Magic cards are just tables in which you put all the weights you can weigh with each one of your original weight set. The first card includes all numbers that require the 1kg weight. The second, that require the 2kg weight. The third, that require the 4kg weight. And so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once the students are done with this activity, the best option is to continue as @DavidButlerUofA explains, just to make them realize how counting systems work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then tell them how would they count when having eight fingers instead of ten!&lt;/p&gt;&#xA;" OwnerUserId="3260" LastActivityDate="2014-11-13T20:09:36.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5847" PostTypeId="1" CreationDate="2014-11-14T17:58:31.990" Score="7" ViewCount="1285" Body="&lt;p&gt;If no calculator is allowed, and we want to find the square root of a square number if it is  large and analyzing to prime factors is hard, how can one proceed? For example, what to do if the number is a square of a large prime number like 5329?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are methods to perform this and what are  pedagogical reasons to prefer one over the other?&lt;/p&gt;&#xA;" OwnerUserId="3306" LastEditorUserId="143" LastEditDate="2014-11-15T13:24:48.913" LastActivityDate="2014-11-18T05:10:08.840" Title="Is there any other procedure to find the square root?" Tags="|undergraduate-education|arithmetic-operations|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5848" PostTypeId="2" ParentId="5847" CreationDate="2014-11-14T18:19:49.070" Score="9" Body="&lt;p&gt;There is an algorithm to calculate square roots with paper and pencil (like the long division algorithm). See &lt;a href=&quot;http://www.homeschoolmath.net/teaching/square-root-algorithm.php&quot;&gt;How to calculate a square root without a calculator&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="729" LastActivityDate="2014-11-14T18:19:49.070" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5849" PostTypeId="2" ParentId="5847" CreationDate="2014-11-14T18:20:01.697" Score="14" Body="&lt;p&gt;You can use (basically) Newton's method. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take $x_0&amp;gt;0$, and define $$x_{k+1} = \frac{1}{2} (x_k + \frac{n}{x_k}).$$ &#xA;The seqeunce converges to $\sqrt{n}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;And&lt;/em&gt; if one starts with $x_0= n$ one has, as soon as $|x_{k+1} - x_k|&amp;lt; 1$, that $\lfloor x_{k+1} \rfloor =  \lfloor\sqrt{n} \rfloor$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the above is not very feasible for computing by hand, and there is better variant, if you just want to do this for perfect squares or test if something is a perfect square, or more generally compute $\lfloor\sqrt{n} \rfloor$. (It seems this is the case for you.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can do &#xA;$$x_{k+1} = \left \lfloor \frac{1}{2} (x_k +  \lfloor \frac{n}{x_k} \rfloor)  \right \rfloor$$&#xA;&lt;em&gt;only doing arithemtic with  integers&lt;/em&gt; (including euclidean division).&#xA;This will converge to $\lfloor\sqrt{n} \rfloor$ in a finite number of iterations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://en.wikipedia.org/wiki/Integer_square_root&quot;&gt;Integer Square Root&lt;/a&gt; on Wikipedia for further references.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this could be an interesting method to teach, as one can provide a good geometric motivation for it.  &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-11-14T18:20:01.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5850" PostTypeId="2" ParentId="5847" CreationDate="2014-11-15T02:17:51.857" Score="3" Body="&lt;p&gt;A fancy way, for those who know a little calculus, is to linearize $f(x)=\sqrt{x}$ at some point near the number you want to square root.  This can only be done when you can think of a square rootable number that isn't too far away, so it's not going to work if you're trying to square root a super huge number, but if you can think of one, it's very doable by hand, and very quick.  It also has the added benefit of giving you a rational approximation, if that's something you want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, to compute $\sqrt{2}$, we first have to come up with some other number that is reasonably close to $2$ which we &lt;em&gt;do&lt;/em&gt; know how to square root. The first thing that comes to mind is $9/4$ (though there are many other choices that are even more accurate, e.g. such as $49/25$).  We linearize $\sqrt{x}$ at $9/4$: $$\left.\frac{d}{dx}\sqrt{x}\right|_{x=9/4}=\left.\frac{1}{2\sqrt{x}}\right|_{x=9/4}=\frac{1}{2\sqrt{9/4}}=\frac{1}{3}$$&#xA;$$\mathcal{L}_{9/4}(x)=f^\prime\left(\frac{1}{3}\right)\left(x-\frac{9}{4}\right)+f\left(\frac{9}{4}\right)=\frac{1}{3}\left(x-\frac{9}{4}\right)+\frac{3}{2}$$&#xA;Now, we plug $2$ in there.&#xA;$$\mathcal{L}_{9/4}(2)=\frac{1}{3}\left(2-\frac{9}{4}\right)+\frac{3}{2}=\frac{17}{12}$$&#xA;From here, if you want decimals, you can simply use long division.  Numerically, $17/12$ comes out to be $1.41667$, while the actual value of $\sqrt{2}$ is $1.41421\ldots$.  That's pretty close, and likely close enough to do the job for many on the fly calculations.&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2014-11-15T02:23:47.170" LastActivityDate="2014-11-15T02:23:47.170" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5851" PostTypeId="1" CreationDate="2014-11-15T02:45:54.710" Score="8" ViewCount="269" Body="&lt;p&gt;One of my classes has about 15 people in it. Everybody else is going okay except for precisely one student. If it was 2, that'd be fine because I can pair them up. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, with this student, I am fairly sure that English isn't their first language, so they struggle to keep up with my explanations. However, I can't slow down for the sake of one student otherwise the rest of the students feel frustrated (and I have observed this when I sometimes go through a question individually with that student).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: &lt;strong&gt;What do you normally do when you have a class with just the one struggling student? Or what strategies do you suggest?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="3383" LastActivityDate="2014-11-19T00:26:37.387" Title="How do you teach a class with one struggling student?" Tags="|classroom-management|struggling-students|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5852" PostTypeId="1" CreationDate="2014-11-15T04:38:27.570" Score="19" ViewCount="442" Body="&lt;p&gt;I recall in one of my classes earlier this year that I had a student who always 'hid' their working out from me when I walked around to see how well they're going with the question. The usual practice is for me to do a few worked examples, and then for me to give them time to try similar questions themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During this time, I walk around the class to see how they are going and explain things individually if needed. Sometimes I call attention to the class to address a common problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot help this student because they are clearly struggling and are too embarrassed to let me see their working out. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What strategies do you suggest in this kind of scenario?&lt;/strong&gt; I would like to pull them out after class to have a quiet chat with them about it, but I feel that they'd be too intimidated even though I try to be as friendly as possible!&lt;/p&gt;&#xA;" OwnerUserId="3383" LastEditorUserId="143" LastEditDate="2014-11-15T09:31:54.597" LastActivityDate="2015-11-18T03:30:52.383" Title="What should I do if I have a student 'hiding' their working out?" Tags="|general-pedagogy|struggling-students|classroom-interaction|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5853" PostTypeId="2" ParentId="5852" CreationDate="2014-11-15T07:08:02.930" Score="2" Body="&lt;p&gt;I often have the same practice than you, but when I walk around I particularly go to student hiding their work out and ask them to show me what they tried. I ask this casually, but making it clear that it is mandatory and not optional to show me their work. I have had no trouble that way, but I may have different students than you have.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-11-15T07:08:02.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5854" PostTypeId="2" ParentId="5837" CreationDate="2014-11-15T07:39:05.620" Score="5" Body="&lt;p&gt;The most basic one would possibly be the following classical game-theoretic example (also appearing in the movie &quot;Last Year at Marienbad&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The game starts with a number of matches on the table (replaceable by any collection of similar object if matches are unsuitable for class, of course). The two players take rounds; at each round the player picks $1$, $2$ or $3$ matches from the table. The player picking the last match loses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The interest of this game is that it is easy to analyse it completely,  determining which player has a winning strategy given the number of matches. It is usually played with 13 matches, in which case the second player has a winning strategy. This produces the following dialogue in &quot;Last year at Marienbad&quot; (approximately remembered and translated):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A: &quot;This is a game I always win.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B: If you can't loose, it is not a game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A: I can loose, but I always win. I'll let you play first&quot;&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-11-15T07:39:05.620" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5855" PostTypeId="2" ParentId="5847" CreationDate="2014-11-15T12:37:51.647" Score="3" Body="&lt;p&gt;Use log tables and the identity $x^\alpha = e^{\alpha \log x}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From an educational point of view, that gives evidence of the usefulness of logarithms (you can calculate any power of $x$ using just a log table and multiplication) and teaches about transforming between problems.  It also allows all roots to be calculated by the same method.&lt;/p&gt;&#xA;" OwnerUserId="1285" LastActivityDate="2014-11-15T12:37:51.647" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5856" PostTypeId="2" ParentId="4518" CreationDate="2014-11-15T17:38:47.703" Score="1" Body="&lt;p&gt;Since you ask what to read &lt;em&gt;before&lt;/em&gt; studying mathematical analysis (emphasis mine), might I suggest Lara Alcock's recent book: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0198723539&quot; rel=&quot;nofollow&quot;&gt;How to Think About Analysis&lt;/a&gt; (2014, OUP). She has written a very accessible and helpful guide to introductory analysis, replete with study advice and how to overcome common points of difficulty. It would go well prior to or alongside a textbook such as Abbott or Bryant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I was also interested to read about &quot;Self-Explanation Training&quot; on pp. 39-43 as a way to aid understanding proofs. See &lt;a href=&quot;http://setmath.lboro.ac.uk&quot; rel=&quot;nofollow&quot;&gt;http://setmath.lboro.ac.uk&lt;/a&gt; and &lt;a href=&quot;http://homepages.lboro.ac.uk/~mamji/files/JRME_SelfExpl_Paper.pdf&quot; rel=&quot;nofollow&quot;&gt;http://homepages.lboro.ac.uk/~mamji/files/JRME_SelfExpl_Paper.pdf&lt;/a&gt; for further details.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2014-11-24T20:34:04.230" LastActivityDate="2014-11-24T20:34:04.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5857" PostTypeId="2" ParentId="5756" CreationDate="2014-11-15T18:52:20.397" Score="2" Body="&lt;p&gt;A course in &lt;a href=&quot;http://en.wikipedia.org/wiki/Experimental_mathematics&quot; rel=&quot;nofollow&quot;&gt;experimental mathematics&lt;/a&gt; could be valuable so that students are introduced to making conjectures and testing their plausibility. This would give them a taste of an aspect of research mathematics and might help counter the notion that all the mathematics there is has already been discovered. Appropriate subject matter, depending on level and interests, could include combinatorics, dynamical systems, discrete geometry and/or number theory.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2014-11-15T18:52:20.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5858" PostTypeId="1" CreationDate="2014-11-16T10:41:46.090" Score="6" ViewCount="420" Body="&lt;p&gt;In my undergraduate logic courses I introduce several types of logics to my students including propositional, first order, second order, intuitionistic and fuzzy logics and it usually happens that somebody asks: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Which logic is better? Which logic is correct? Which one is more useful in mathematics, business, philosophy, etc?&lt;/em&gt;  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't like to tell my students that your question is meaningless because even amongst professional logicians, mathematicians and philosophers, some of them prefer some logics more than others.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What is a nice answer to above question?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user230" LastActivityDate="2014-11-17T14:22:08.297" Title="What is a good answer to the question &quot;Which logic is better?&quot;" Tags="|course-design|mathematical-pedagogy|logic|student-question|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5859" PostTypeId="2" ParentId="5858" CreationDate="2014-11-16T16:51:52.037" Score="5" Body="&lt;p&gt;It is meaningful to ask which logic is more useful in mathematics and business.  E.g.:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Learning classical first-order logic can help you program, most dramatically in &lt;a href=&quot;http://en.wikipedia.org/wiki/SQL&quot; rel=&quot;nofollow&quot;&gt;SQL&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'd rather not confuse undergraduates with the other logics you mention.  I would advise them:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Instead of learning fuzzy logic, learn probability directly.&lt;/li&gt;&#xA;&lt;li&gt;Instead of learning second-order logic, learn multi-sorted first-order logic.&lt;/li&gt;&#xA;&lt;li&gt;Instead of learning intuitionist logic, learn rhetoric, to be alert&#xA;to how people use dichotomies.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So I would deflect the questions of &quot;better&quot; or &quot;correct&quot;, and stick with &quot;useful&quot;, which gets clearer answers.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-11-17T01:25:28.957" LastActivityDate="2014-11-17T01:25:28.957" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5860" PostTypeId="2" ParentId="5858" CreationDate="2014-11-16T19:36:31.317" Score="7" Body="&lt;p&gt;The following is written as if I were giving my own best answer to a student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's probably an accident of history that Aristotle defined classical logic in a specific way, and that classical logic has been almost exclusively used as the foundation of mathematics ever since. Common sense tells us that Aristotelian logic is an oversimplification of how we actually reason about the real world. For example, Aristotelian logic claims that every proposition is either true or false, but the following are statements that many reasonable people would say are neither true nor false:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Green is a nicer color than black.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Harry Potter had a great-great-great-great grandfather named Joe.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Based on the equal protection clause of the U.S. constitution, it's unconstitutional to deny gay people the right to get married.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;My terrier is an intuitive Buddhist.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The vast majority of the literature of mathematics assumes classical logic, so for professional mathematicians in most fields, strong knowledge of classical logic is both necessary and sufficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But people didn't just invent fuzzy logic or multi-valued logic for no reason. They invented them because those logics were more suitable for what they wanted to do.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-11-16T19:36:31.317" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5861" PostTypeId="2" ParentId="5847" CreationDate="2014-11-17T00:43:00.950" Score="2" Body="&lt;p&gt;An intuitive approach to find the square root of A is to draw a square with the number (A) in the middle as the area, and the sides labeled x and y. We know that x times y equal A, and x is the square root of A when x = y.  Begin with a rough estimate of x and calculate y as A/x. Now replace x with the average of the original x and y, and repeat.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, let's find the square root of 200. As a rough guess, let's begin with x = 15. Solve for y = 200/15 = 13.333. The new value for x, call it x1 = (15 + 13.333)/2 = 14.167. Now the new value y1 = 200/14.167 = 14.117. The square root of 200 is between 14.167 and 14.117. The average is 14.142. If we use x2 = 14.142, then y2 = 200/14.142 = 14.142, verifying that we have the square root of 200 with five significant digits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This approach has the advantage of showing what a square root is visually, which can give a problem more meaning. The process makes intuitive sense, and it does not require mathematics beyond division.&lt;/p&gt;&#xA;" OwnerUserId="3391" LastEditorUserId="3391" LastEditDate="2014-11-18T05:10:08.840" LastActivityDate="2014-11-18T05:10:08.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5862" PostTypeId="2" ParentId="5847" CreationDate="2014-11-17T07:19:23.807" Score="1" Body="&lt;p&gt;Here is the method I used to solve the original poster's problem in about three seconds:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Square root (5329):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Steps 1a-d estimate the answer, to one significant figure, with the correct order of magnitude:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1a)  Start with an &lt;strong&gt;estimated answer&lt;/strong&gt; of 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1b)  Find the order of magnitude of the answer.  Make a &lt;strong&gt;copy of the argument&lt;/strong&gt; (5329).  Shift the decimal of the &lt;strong&gt;copy of the argument&lt;/strong&gt; by two places at a time, to get a number between 1 and 100.  Shift the decimal of the &lt;strong&gt;estimated answer&lt;/strong&gt; by one place at a time, in the opposite direction.  For example, 53.29 and 10.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1c)  Assume that the student knows the squares of the integers between 0 and 10 by heart.  What is the &lt;strong&gt;closest such square&lt;/strong&gt;?  For example, 49.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1d)  Multiply the &lt;strong&gt;estimated answer&lt;/strong&gt; by the &lt;strong&gt;square root&lt;/strong&gt; of the &lt;strong&gt;closest square&lt;/strong&gt;.  For example, square root (49) = 7, so the new &lt;strong&gt;estimated answer&lt;/strong&gt; is 70.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Step 2 checks the estimate:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2)  Square the current &lt;strong&gt;estimated answer&lt;/strong&gt;.  For example, 70 * 70 = 4900.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Steps 3a-d refine the estimate using  uses (a+b)^2 = a^2 + 2*a*b + b^2.  This is a special case of Newton's method:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3a)  Subtract the value found in step 2 from the original argument, to yield a &lt;strong&gt;remainder&lt;/strong&gt;.  For example, 5329 - 4900 = 429.  Keep track of the sign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3b)  Make a copy of the &lt;strong&gt;estimated answer&lt;/strong&gt;.  Multiply the copy by 2, to yield a &lt;strong&gt;double estimate&lt;/strong&gt;.  For example, 70 * 2 = 140.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3c)  Divide the &lt;strong&gt;remainder&lt;/strong&gt; by the &lt;strong&gt;double estimate&lt;/strong&gt;, to yield an &lt;strong&gt;adjustment&lt;/strong&gt;.  Round off as convenient.  Keep track of the sign.  For example, 429 / 140 ~ 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3d)  Add the &lt;strong&gt;adjustment&lt;/strong&gt; to the &lt;strong&gt;estimate&lt;/strong&gt; (not the double estimate).  We kept track of the sign, so that we could correctly choose whether to add or subtract.  For example, 70 + 3 = 73.  For subsequent calculations, this value replaces &lt;strong&gt;estimate&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Repeat steps 2 - 3 until the answer is close enough.  For example, 4900 + 420 + 3*3 = 5329, so the square root of 5329 is 73.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some advantages of this method:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I can do it in my head.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It only requires keeping track of a few numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It quickly produces an estimated answer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is a special case of Newton's method.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Each time through the loop, the student can round off the adjustment to a convenient decimal or fraction.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Every iteration, the student checks their work.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Three worked examples:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;       Argu-   Estimated   Found                Double     Adjust-  &#xA;Step   ment      Answer    So Far  Remainder    Estimate    ment     Notes  &#xA; 1a    5329        1  &#xA; 1b               10                                                 53.29  &#xA; 1c                                                                  7*7=49  &#xA; 1d               70&#xA; 2     5329       70       4900  &#xA; 3a    5329       70       4900       429  &#xA; 3b    5329       70       4900       429        140  &#xA; 3c    5329       70       4900       429        140          3&#xA; 2     5329       73       5329                                 73*73=4900+420+9&#xA;Done!&#xA;&#xA;       Argu-   Estimated   Found                Double     Adjust-  &#xA;Step   ment      Answer    So Far  Remainder    Estimate    ment     Notes  &#xA; 1a     739        1  &#xA; 1b               10                                                  7.39  &#xA; 1c                                                                  3*3= 9   &#xA; 1d               30  &#xA; 2      739       30        900&#xA; 3      739       30        900       -161        60        -3&#xA; 2-3    739       27        729         10        54         0.2&#xA; 2-3    739       27.2      739.84      -0.84     54.4      -0.015&#xA; 2-3    739       27.185    739.024225  -0.024225 54.37     -0.0005&#xA; 2-3    739       27.1845   738.99704025          54.369     0.00005&#xA; 2      739       27.18455  738.9997587025&#xA; ...Stop when satisfied...  &#xA;&#xA;       Argu-   Estimated   Found                Double     Adjust-  &#xA;Step   ment      Answer    So Far  Remainder    Estimate    ment     Notes  &#xA; 1a    4285        1  &#xA; 1b               10                                                 42.85  &#xA; 1c                                                                  6*6=36   &#xA; 1d               60  &#xA; 2-3   4285       60       3600       685       120         5&#xA; 2-3   4285       65       4225        60       130         0.4&#xA; 2-3   4285       65.4     4277.16      7.84    130.8       0.06&#xA; 2-3   4285       65.46    4285.0116   -0.0116  130.92     -0.0001&#xA; 2-3   4285       65.4599  4284.99850801        130.9198    0.00001&#xA; 2     4285       65.45991 4284.9998171081&#xA; ...Stop when satisfied...  &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2014-11-17T22:10:54.357" LastActivityDate="2014-11-17T22:10:54.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5863" PostTypeId="2" ParentId="5852" CreationDate="2014-11-17T12:52:28.277" Score="6" Body="&lt;p&gt;When dealing with students, especially more quiet and private ones, the most important thing is to build some sense of trust before trying to breach things that make them uncomfortable.  If you just outright go up to them and tell them to show their work to you because you are the teacher it will only make them more uncomfortable and less likely to want to learn.  You might get to see their work but it is not worth the trade-off by making them uncomfortable.  I would definitely talk to them and just let them know that you aren't doing this to embarrass them or snoop on them, you are just doing it to get a better sense of how they are doing with the content and that it is completely ok if they aren't getting it, you just need to know so that you know what to focus on during the lesson.  Don't do this in front of the other students and don't make a big deal when you ask them to stay after class and definitely don't make it a power struggle (&quot;I'm the teacher so you need to show me this&quot;, even if that is the case).  If they are still resistant, ask them if they are comfortable giving you their work after class to look over.  I find that some students who don't like to let you look at their work during class are ok with letting you see their work when they aren't present, this way you can still get a sense of how they are doing.  I hope this helps, such a difficult problem to tackle! &lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2014-11-17T15:54:57.920" LastActivityDate="2014-11-17T15:54:57.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5864" PostTypeId="2" ParentId="5858" CreationDate="2014-11-17T14:22:08.297" Score="0" Body="&lt;p&gt;Logic is a (formal) tool for reasoning. There is not better, or worse one. Only more or less suitable for given problem. To ask which one is better is similar to ask which tool is better: hammer or tweezers, or which language is better japan, or french?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given a particular task, you may compare suitability of logic variations and results obtained using one, or another.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2014-11-17T14:22:08.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5865" PostTypeId="1" CreationDate="2014-11-17T16:29:46.380" Score="9" ViewCount="242" Body="&lt;p&gt;I had asked the title question in an answer &lt;a href=&quot;https://matheducators.stackexchange.com/a/5760/267&quot;&gt;here&lt;/a&gt;, but it didn't get any recognition, so I'm posing it formally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key word in the question is &lt;em&gt;should&lt;/em&gt;. In my Mathematics Education degree education, I have had the option to take many content classes, but few methods classes. &lt;strong&gt;As current teachers, if you were to design a degree program, what classes would you ideally offer?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the construct of this hypothetical question:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A Masters (not dually earned with undergraduate) degree consists of 12 courses, and most students go on to become secondary school teachers;&lt;/li&gt;&#xA;&lt;li&gt;A PhD or EdD degree consists of 25 courses, and many students go on to become post-secondary teachers or researchers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A (perhaps) similarly relevant post can be found here: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/247/what-are-the-differences-between-graduate-and-undergraduate-classes-relevant-to&quot;&gt;What are the differences between graduate and undergraduate classes, relevant to course design and teaching?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-11-19T05:35:06.697" Title="What courses should be a part of a Mathematics Education degree program?" Tags="|course-design|graduate-education|teacher-preparation|teacher-training|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5866" PostTypeId="1" AcceptedAnswerId="5873" CreationDate="2014-11-17T18:47:29.693" Score="10" ViewCount="3517" Body="&lt;p&gt;My kids Cindy (5 1/2, in Kindergarden) and Jamie (4, in pre-K) are taking some math enrichment classes. I have been telling and teaching them to count items and do single-digit arithmetic in their head, hoping to train their brains rather than working the problems mechanically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below is the verbatim evaluation I received from their tutors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I fully believe that people (not just kids!) should be able to do arithmetic in their head, I am wondering if I should wait a few more years.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Cindy is very good at math, especially being able to visualize an&#xA;  answer. However, I'd like to get her to a point where she can&#xA;  calculate the answer before guessing so that she has the absolute&#xA;  correct answer. We went over simple addition after this set and I had&#xA;  her practice counting on her fingers. She did extremely well with this&#xA;  and got the hang of it right away.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;James was required to add more objects to a given amount. He did very&#xA;  well but it didn't seem as if he had any strategy to solve each&#xA;  problem. When I asked him why he chose a particular answer he replied,&#xA;  &quot;because you said that.&quot; We pertained using our finger to help him&#xA;  find the answer. It was hard to determine whether Jamie was doing the&#xA;  work on his head and couldn't verbalize it or if his guessed were&quot;&#xA;  lucky guesses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;" OwnerUserId="2154" LastActivityDate="2016-05-26T15:55:48.963" Title="Learning counting and addition: fingers or in their head?" Tags="|preschool-education|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5867" PostTypeId="1" AcceptedAnswerId="5871" CreationDate="2014-11-17T19:36:38.097" Score="10" ViewCount="317" Body="&lt;p&gt;Normally, I am comfortable when I have to explain a piece of mathematics to a group of people using pen and papers as supports. However, I find it quite difficult to give small talks and explanations using the blackboard as a support.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is mainly because the blackboard is huge, my handwriting has to be quite big (to let students see from their seats), and I have to stand at 50cm from the blackboard: as a result, I find it extremely difficult to have a complete picture of what I'm writing (to make one example, sometimes I cannot even keep track of the letters I've used to denote variables) and therefore to follow my own reasoning (whereas, on a piece of paper, I would have everything under better control).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do professors do to avoid this kind of loss of bearing when using such huge blackboards to give extended lectures? Could you give me some advice based on your experience as students and educators?&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="3251" LastEditDate="2014-11-18T00:47:31.693" LastActivityDate="2014-11-18T07:07:10.410" Title="Explaining mathematics, giving talks, and using the blackboard as a support" Tags="|teacher-preparation|presentation|teacher-development|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5868" PostTypeId="2" ParentId="5865" CreationDate="2014-11-17T21:03:17.830" Score="2" Body="&lt;p&gt;If I were to take my master's degree again, I would take:&#xA;1-2 courses in abstract algebra, 1-2 courses in analysis, 1-2 courses in topology,  1 class in statistics, and 1 class in combinatorics for a total of 5-7 classes to get a good solid background in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would then set up a year long apprenticeship with a master teacher.  I guess this would would count as 2-3 courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would take:&#xA;1-2 classes in classroom management and pedagogy (discussing and reflecting on the apprenticeship) and 1-2 classes that give a deeper look at the math content of high school math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think a high school math teacher should have a very solid understanding of math, but that most of the learning of how to teach would happen in the apprenticeship with the master teacher.&lt;/p&gt;&#xA;" OwnerUserId="3265" LastActivityDate="2014-11-17T21:03:17.830" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5869" PostTypeId="2" ParentId="5866" CreationDate="2014-11-17T22:41:00.113" Score="10" Body="&lt;p&gt;&lt;strong&gt;Edit (May 2016):&lt;/strong&gt; From &lt;em&gt;The Atlantic&lt;/em&gt; is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Boaler, J. &amp;amp; Chen, L. &quot;Why Kids Should Use Their Fingers in Math Class.&quot; Apr 2016. &lt;a href=&quot;http://www.theatlantic.com/education/archive/2016/04/why-kids-should-use-their-fingers-in-math-class/478053/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Evidence from brain science suggests that far from being “babyish,” the technique is essential for mathematical achievement.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(See the link for more!)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Counting is one of the most important activities to engage in with young children; it is also one of the key precursors for developing number sense later on. If you are looking to help young children with their mathematical development, then I would say that the single best choice is to &lt;strong&gt;count&lt;/strong&gt; with them (another important activity is to practice using words meaningfully, e.g., identifying different shapes). If you are wondering about how to help them count, the standard approach is to begin with the number chant (it is pretty tough till around 13, and then gets a bit easier in the later teens, and then goes rapidly from 20 on); next, move on to skip counting (by 2s, by 5s, by 10s); and counting in reverse order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to the question here, there is a lot written about the subject. For example, Gelman writes &lt;strong&gt;&lt;a href=&quot;http://ruccs.rutgers.edu/faculty/GnG/pdf_files/2000TheEpigenesisOfMathematical.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/3IxSM.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real reason to post such an excerpt is to indicate how complex a task this is!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find some good work on counting by &lt;strong&gt;&lt;a href=&quot;http://scholar.google.com/scholar?q=counting%20ginsburg%20baroody&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=0%2C22&quot; rel=&quot;nofollow noreferrer&quot;&gt;searching&lt;/a&gt;&lt;/strong&gt; for AJ Baroody and HP Ginsburg.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are not so keen to read through the literature yourself and would just like a quick answer, here is the best that I can do (though, I reiterate, counting is not so simple!): The ultimate goal is for children to be able to count without needing to use their fingers; however, in building up to that goal, it is totally fine for them to count using their fingers. Keep practicing the chanting, the skip counting, and then try counting groups of objects! It is fine for children to tap each object when they are counting them; later on, they might just point; later still, they may be able to count without needing their fingers; ultimately, the goal is to count without even having the objects in front of themselves. But the time that this takes can vary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note about &lt;em&gt;multiplication&lt;/em&gt; and finger counting: Later on, when students move on from basic facts about addition to &quot;basic facts&quot; about multiplication, i.e., multiplying pairs of whole numbers from 1 to 10, they can still use finger counting to help them: especially with the skip counting numbers of 2, 5, and 10. For example, one way to compute 2x8 is to hold up 8 fingers, and then count on them by 2s (2, 4, 6, 8, 10, 12, 14, 16), putting down a finger each time it has been used. Again, the ultimate goal is to move towards abstraction and not needing a concrete representation, but using fingers to count - for addition or for multiplication, which can be thought of at that age as repeated addition - can be very helpful.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps one of the stressful factors is not knowing what the &lt;em&gt;norms&lt;/em&gt; are for a given age (or stage). The language that Jamie is using looks totally age-appropriate; my bigger concern would be that his tutors are not quite sure how to elicit his reasoning. For example, the response of &lt;em&gt;because you said that&lt;/em&gt; looks very familiar if you have spoken with children about mathematics. One good way to try to understand their thinking is by using clinical interviews; a single particular technique is to say something like, &lt;em&gt;what if a little kid asked you to explain your thinking?&lt;/em&gt; Rather than delve deeply into techniques for using the clinical interview, I will give the same recommendation I included in earlier answers &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/843/262&quot;&gt;here&lt;/a&gt;&lt;/strong&gt; and &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/5804/262&quot;&gt;here&lt;/a&gt;&lt;/strong&gt; (see the very end of each):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ginsburg, H. P. (2009). Early mathematics education and how to do it. &lt;em&gt;Handbook of child development and early education: Research to practice&lt;/em&gt;, 423-428.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Finally, if you would like to know a bit more about norms for kids learning mathematics at an early age, I recommend the following book (or its 1989 reprinting):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ginsburg, H. (1977). &lt;em&gt;Children's arithmetic: The learning process&lt;/em&gt;. D. van Nostrand.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is an excerpt from Chapter 2, Learning to Count, to give you an idea around norms:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/v4pX3.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-05-26T15:55:48.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5870" PostTypeId="2" ParentId="5867" CreationDate="2014-11-18T06:58:16.737" Score="1" Body="&lt;p&gt;You may be able to produce smaller legible letters with a good &lt;a href=&quot;https://matheducators.stackexchange.com/questions/41/what-is-a-good-handwriting-font-for-mathematics&quot;&gt;font&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But mostly, I don't write down my reasoning on a blackboard.  For an extended presentation, I use a projector with prepared slides.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, blackboards are great for explanations and discussions in small groups.  Think of how you want the board to look at the end:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Start your writing uncomfortably close to the upper left corner, so it&#xA;flows across the board easily.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use multiple colors to highlight steps in the argument or construction or calculation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Put a big diagram or two in the center of the board as a focal point.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Good graphic design helps.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-11-18T06:58:16.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5871" PostTypeId="2" ParentId="5867" CreationDate="2014-11-18T07:07:10.410" Score="12" Body="&lt;p&gt;I struggled with this as a new lecturer, and I found a few ways to manage the process of using a big blackboard:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Observe people lecturing&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I first started lecturing, I visited the lectures of other staff to see how they did it. I realised that when I was a student I had seen many lectures, but I had never thought about the actions of the lecturers at the time. So I went to lectures to see what the lecturer did: I noted whether they had notes they referred to, and I noted how often they referred to them; I watched how often they turned to face the class to talk rather than write; I watched how often they stood back to observe their own work; and I tried to take note of anything else they did to do the job of lecturing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Write your notes in advance&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If at all possible, write your notes in advance. Then, if you get lost, you can look at your notes to see where you are up to. I found it helpful to lay the notes on the bench and turn the pages over as I moved through the lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that if your students are copying what you write, they themselves will flick back through what they have written to check where you are up to, so there is nothing wrong with you doing the same!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Regularly stand back to see where you are up to&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My PhD supervisor gave me a piece of advice which is very logical: when you are 50cm away from the board you can't see what's going on, so stand back every so often to look at the whole thing at once. You should do this before you rub out the board, and also any time you finish a particular paragraph or calculation. Also its especially important to do it whenever you feel you are getting lost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not just useful for you to keep your perspective, it's also useful for your students. They will appreciate the extra few seconds to write down what you've written and to assimilate the information. I know I did when I was a student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Signpost your writing&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you write on the board, you should use headings and underlining and other markers to tell you where thoughts begin and end and to tell the students (and yourself) where you've been and where you're going. These should be consistent across the time you are speaking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if you have a theorem, write &quot;THEOREM&quot; on the board and underline it. If you feel brave you can always use a special symbol like a triangle or picture next to it to visually indicate what's there. When you do an example, say &quot;EXAMPLE 1&quot; and underline it. It is amazing how helpful this is as a student to know that you are beginning or ending a new thought.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Stop to check where you have been&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you lecture a very helpful thing for the students is to stop to see where you have been. Not just in your own head but aloud. So in combination to stopping and looking at what you have written on the board, stop and talk through what you have done so far, including the things you have rubbed off. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example: &quot;We began by defining what a metric space is, which was a way of defining distance in a reasonable way. Then we looked at a few examples of metric spaces and metrics to get a feel for how this works. Then we talked about what it means for a set to be open, and we proved that an open ball is itself open. Now we're up to proving that the union of any collection of open balls is open.&quot; As you do this, you can flick through your pre-written notes. This is very helpful to your students, but will help you to keep focus as well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't have pre-written notes and are making the lecture up as you go (as sometimes happens), then this is even more important. I did a revision lecture for students last week where I made it up as I went (on metric spaces as it happens) and I stopped every time I got to the end of a thought to check where we had been. It felt much better to know in my own head where I was up to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd also recommend making space on the board to &lt;em&gt;write&lt;/em&gt; where you've been -- just the major headings along the side somewhere. Then when you look back at where you've been you have something to refer to and you can add to it as you go.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Don't use the blackboard&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And finally, I have to say I don't use the blackboard/whiteboard anymore. The lecture theatres in my uni all have document cameras so I use paper and pencil on the document camera. At a pinch I have used my tablet PC and written on a PDF on the screen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The advantage of this is that when you do go back to see where you've been you can flick through the actual notes you actually wrote (or scroll if you're on a computer). Not to mention being able to use colour more effectively!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All the other comments about stopping to see where you've been, and recapping the story so far and using signposts still apply to make it more effective, of course. And if you have no choice but to use a blackboard/whiteboard then they are the most helpful things I can suggest.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-11-18T07:07:10.410" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5872" PostTypeId="1" AcceptedAnswerId="5875" CreationDate="2014-11-18T07:41:37.467" Score="13" ViewCount="617" Body="&lt;p&gt;I am tutoring a high school level student (who is currently at the level of being introduced to anti derivatives) who has quite some trouble with grasping mathematics.&#xA;We have been making good progress but then with this concept I get to a hault: he understands that&#xA;$$\frac 22, \quad \frac 33, \quad \frac 44,\quad  \ldots$$&#xA;all clearly are equal to &lt;em&gt;one&lt;/em&gt;&#xA;but as soon as I replace the number with a letter,&#xA;$$\frac xx$$&#xA; he does not get it and starts guessing:   &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;nothing? x? zero? one?&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;What is a good approach to teaching this?&lt;/p&gt;&#xA;" OwnerUserId="1770" LastEditorUserId="64" LastEditDate="2019-03-31T01:34:57.967" LastActivityDate="2019-03-31T12:31:08.727" Title="How do I help my student understand concepts such as &quot;$x$ divided by $x$&quot;?" Tags="|mathematical-pedagogy|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="5873" PostTypeId="2" ParentId="5866" CreationDate="2014-11-18T13:58:29.703" Score="6" Body="&lt;p&gt;I was working with a student (high school level, algebra) and she blushed when she realized that she was counting on her fingers in front of me. She quickly told me her friends made fun of her.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;35 years ago, my classmates poked fun that I counted on my fingers. Then we took our SATs, and I aced the math portion, an 800. They stopped laughing.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a bit anti-calculator, for multiple reasons, but as far as fingers go, I think it's a great way to avoid simple errors, and after adding 5 to 7, counting 8,9,10,11,12 on your hand, it eventually sinks in. It's when student use the calculator to perform even simpler math, that I worry. The finger math has a time honored tradition going back to the abacus.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-18T13:58:29.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5874" PostTypeId="2" ParentId="5866" CreationDate="2014-11-18T14:04:53.860" Score="3" Body="&lt;p&gt;(Too long for a comment...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your question suggests a distinction between 'using ones brain' to do mathematics and 'performing a mechanical process' to do mathematics. This is an idea I always love to jump on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The truth is that much of mathematics &lt;em&gt;is&lt;/em&gt; mechanical (this is why computers can do it so well), and the machine involved in performing these mechanical processes is your children's brains.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Human brains obtain skills via long repetition, be it a motor skill (how many hours do violinists spend drilling the fingering for scales x y and z?) or an intellectual skill (in this case counting and the internalizing the concept of addition/subtraction). My own inclination would be to let them count until they abandon it of their own accord after it has become boring for them and/or their accumulated practice leaves them able to instantly know that five and three are eight.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2014-11-18T14:04:53.860" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5875" PostTypeId="2" ParentId="5872" CreationDate="2014-11-18T16:08:20.697" Score="17" Body="&lt;p&gt;I find that I can ask the following series of questions, which bridge the gap between &amp;quot;3&amp;quot; and &amp;quot;x.&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$\frac22$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$\frac33$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$\frac66$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$\frac{17}{17}$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is key -- 17 is a number that no one has any intuition about. If the student can deal with 17, they are already abstracting the idea as something that works with any number. If they get stuck before 17, then you know they are not successfully abstracting the idea at all, and you should go back to 5.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$\frac{179}{179}$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Continue like this -- be ridiculous if needed, use 1578302, until you sense that the student is slightly annoyed by your questions because they definitely get the pattern. Then I like to go with&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$\frac{\textrm{smiley face}}{\textrm{smiley face}}$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a concept they are struggling with, then the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; itself might be threatening and might cause a mental block. Don't use &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; itself. Use some other placeholder symbol. A picture of a bird from angry birds, if you want. They will probably be able to work with smiley face and get the answer because &lt;strong&gt;that is the same idea they have to use to work with 17&lt;/strong&gt;. No one is actually imagining a pile of 17 things, divided into 17 piles. If they can do &lt;span class=&quot;math-container&quot;&gt;$\frac{17}{17}$&lt;/span&gt;, then they already get it, you just have to show them they already get it.&lt;/p&gt;&#xA;&lt;p&gt;Then ask the big question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Does it matter what I write here? [pointing at the top and bottom]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You basically already made it to the punchline which is that &lt;span class=&quot;math-container&quot;&gt;$\frac{x}{x}$&lt;/span&gt; is 1. But saying that it is the same fact as &lt;span class=&quot;math-container&quot;&gt;$\frac{y}{y} = 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{z}{z} = 1$&lt;/span&gt; is a big help.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-11-18T16:08:20.697" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5876" PostTypeId="2" ParentId="5872" CreationDate="2014-11-18T16:36:35.097" Score="10" Body="&lt;p&gt;The problem may very well be that the student was expected to learn fractions and instead learned to manipulate symbols, but did not learn the meaning of a fraction. To reference part of  &lt;a href=&quot;https://matheducators.stackexchange.com/a/5875/759&quot;&gt;Chris Cunningham's example&lt;/a&gt;, it certainly is possible to convey to a student that if the symbol on the top matches the symbol on the bottom ( i.e. something like $\frac{\textrm{smiley face}}{\textrm{smiley face}}$ ) that this is a kind of special case (which can evaluate to $1$). But an understanding grounded in meaning rather than just symbols is more likely to connect with the students greater body of mathematical understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student doesn't see $\frac{3}{3}$ as &quot;three parts of a whole that is composed of three parts&quot; then this is an opportunity to address this understanding. (You could ask the student to describe what the $3$ in the denominator means, and the $3$ in the numerator, and see if they even grasp the underlying meaning of a fraction). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you've addressed the part-whole relationship meaning of fraction, which (even if neglected earlier in his education) should be comparatively easy to explain, since it is grounded in meaning, you can address the generalization of  $\frac{x}{x}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Assuming this represents a fraction, what does the $x$ in the denominator represent?&quot; &quot;Now, how many of $x$ pieces do we have in this so-called generalized fraction? How much of a whole does that make?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The obvious problem here is that you're asking for a numerical answer, using fractions as examples, but $\frac{x}{x}$ can represent non-fractions like $\frac{0}{0}$, which you have not ruled out. So, depending on his understanding, your student may well be right to hesitate and, perhaps, wonder what you're getting at.&lt;/p&gt;&#xA;" OwnerUserId="759" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-11-18T16:36:35.097" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5877" PostTypeId="1" AcceptedAnswerId="5883" CreationDate="2014-11-18T19:30:28.697" Score="30" ViewCount="2778" Body="&lt;p&gt;I've recently run across a series of problems that didn't reflect reality. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example -  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;An algebra problem with two teens on bicycles. The resulting times showed the bike was moving at 120MPH. &lt;/li&gt;&#xA;&lt;li&gt;A quadratic equation, &quot;The football follows a path of.....&quot; but the equation didn't reflect proper gravity, either in feet or meters, and the solution had the football hanging in the air for an absurd time. &lt;/li&gt;&#xA;&lt;li&gt;A trig problem, the answer being that a man's height, calculated from his shadow, was 8 feet. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Disclosure - I am a tutor in a High School math center. These questions were from homework, and in one case, a quiz I proctored. The student pointed out to me that the answer made no sense, but had checked it twice. I'm not supposed to coach or comment during quizzes or tests, but I saw the answer was right, and told the student so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As the title asked - Should word problems be reasonable? I often ask students to reflect on their answers to understand if it makes sense. These recent questions caught me off guard. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-10-07T20:14:02.013" Title="Should word problems be reasonable?" Tags="|algebra|trigonometry|word-problems|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5878" PostTypeId="2" ParentId="5877" CreationDate="2014-11-18T20:10:02.213" Score="17" Body="&lt;p&gt;The answers to a word problem should in my opinion  make sense (within reasonable limits).&#xA;The goal you mentioned that students should check their answers is one shared by many, as also witnessed by our recent question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5818/how-to-award-points-for-sense-making-at-the-end-of-a-problem&quot;&gt;How to award points for sense-making at the end of a problem?&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if it is not a given any more that the solution does indeed make sense, this is completely moot. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, there can be simplifying assumptions, a word problem cannot be an example of or exercise  in state-of-the-art mathematical modeling either. Yet, the results should at least not be glaringly implausible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one only wants to practice skills in manipulation and calculation, one should simply not do word problems but something else. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In brief, in my opinion, either do word problems that make some sense, or just don't do them. &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-11-18T20:10:02.213" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5879" PostTypeId="1" CreationDate="2014-11-18T21:09:31.287" Score="6" ViewCount="464" Body="&lt;p&gt;I'm preparing a task on the Cartesian product of two sets and I have run into the following confusion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that the Cartesian product is not a commutative operation. Generally speaking, AxB does not equal BxA unless A=B or A or B is the empty set. This is usually easy to explain to students because in the definition of a cartesian product, we define it as an &lt;em&gt;ordered pair&lt;/em&gt;, meaning order would matter. However, once we move on from this idea to explain what the product set represents, things get a bit fuzzy. For example, If we think of the 52 cards in a standard deck as a product set, we can define set A as the ranks and set B as the suits. How can we explain to students that while AxB and BxA both represent the 52 cards in a standard deck, the sets AxB and BxA are distinct and disjoint sets?&lt;/p&gt;&#xA;" OwnerUserId="1312" LastActivityDate="2014-11-18T22:14:16.967" Title="Cartesian product set" Tags="|algebra|set-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5880" PostTypeId="2" ParentId="5879" CreationDate="2014-11-18T21:53:21.420" Score="8" Body="&lt;p&gt;They are isomorphic. While $A\times B \neq B \times A$ for any arbitrary distinct sets $A$ and $B$, by defining the map $\phi:A\times B  \to B \times A$ by $\phi(a,b)=(b,a)$, we can show that algebraically (or set theoretically) that they behave the same. Just a quick check of the set theoretic parts:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\phi(a,b) = \phi(c,d)$ iff $a=c$ and $b=d$. Hence $\phi$ is injective.&lt;/li&gt;&#xA;&lt;li&gt;$\phi$ is clearly surjective.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One way to explain this to the students is to say that you just need to be consistent in the order of the sets throughout (i.e., if you choose $A \times B$, always use that ordering), but your initial choice doesn't matter. You can switch to the other ordering by just flipping the order of everything everywhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To show they are distinct, I would say say something along the lines that if $A=\{a,b,c\}$ and $B=\{1,2,3\}$, then $(a,1) \in A \times B$ but $(a,1)$ is not in $B \times A$.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-11-18T21:53:21.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5881" PostTypeId="2" ParentId="5879" CreationDate="2014-11-18T22:14:16.967" Score="6" Body="&lt;p&gt;I like Chris C's answer; I will offer another point of view. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the difficulty lies in the example that has been chosen: a deck of cards isn't naturally a cartesian product for exactly the reason that there is no natural ordering of rank and suit. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, you might look at a (simplified) menu, and consider all of the meals that consist of a main course and a dessert, and represent that as a cartesian product. Here, there is  natural ordering. (I'm sure there are better examples out there, this is just the first thing that came to mind.)&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2014-11-18T22:14:16.967" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5882" PostTypeId="2" ParentId="5851" CreationDate="2014-11-19T00:26:37.387" Score="5" Body="&lt;p&gt;Sometimes I think of a visibly struggling student as the proverbial canary in the mine, pointing out when something clearly isn't working in the way that I'm teaching something.   So my first question is: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are you &lt;strong&gt;sure&lt;/strong&gt; everyone else is with you?  How do you know that 14 students get it and 1 doesn't?   I'm thinking back to my own teaching, and that 1 student visibly confused probably means that 4-5 other students are confused too and just hiding it.   You don't necessarily have to slow down the entire lesson just for one student, but it could be helpful to check in with other students for understanding in those moments where you see the one student struggling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second thing I want to offer a bit of a different take on pairing students up than what you mention.  You mention pairing up the struggling student with another struggling student.  One method of pairing that's worked well in the research literature is called peer assisted learning strategies (PALS), where you are pairing up a struggling student with a student who's not necessarily at the top of the class, but maybe just a little bit about average in the class.   You can start by having the student who understands the concept tutoring the struggling student.  Then they can switch roles and you can have the struggling student tutor the student who understands it better, so that both have the opportunity to practice being the teacher and the student.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you wanted to do this as an entire class, you could divide the class in half, and pair the student at the bottom of the lower half with the student at the bottom of the upper half, and so on and so forth, so that the most struggling student is paired with the average student, and the student just at the average is paired with the top student.   You could assign the partners, and assign everyone to be A and B, and then that way a student wouldn't necessarily know they've been assigned as the &quot;less smart&quot; one in the pair.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The final thing I want you to think about is looking at having students occasionally work in groups on more complex problems that have multiple ways of reaching a solution and invite mathematical dialogue and collaboration.   These are known in the literature as groupworthy tasks, as in a task that's worthy to be worked on by a group of students.   Here's some examples of what these tasks look like at the elementary school level:  &lt;a href=&quot;http://cimath.org/tasks-for-download/&quot;&gt;http://cimath.org/tasks-for-download/&lt;/a&gt;  .  Here's another site that offers some more advanced tasks that might be a good starting point for a high school class: &lt;a href=&quot;http://nrich.maths.org/public/leg.php&quot;&gt;http://nrich.maths.org/public/leg.php&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-11-19T00:26:37.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5883" PostTypeId="2" ParentId="5877" CreationDate="2014-11-19T00:57:14.113" Score="29" Body="&lt;h3&gt;Absolutely!&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;In fact, in my opinion, the most important &quot;math skill&quot; that should be taught in conjunction with, and using, word problems is &lt;em&gt;checking whether the answers make sense&lt;/em&gt;.  This is an absolutely invaluable part of making &lt;em&gt;any&lt;/em&gt; practical use of mathematics, as opposed to just blindly applying formulas for the sake of passing an exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are several parts to this skill, which all should be explicitly taught and reinforced at every level, such as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//en.wikipedia.org/wiki/Dimensional_analysis&quot;&gt;&lt;strong&gt;Dimensional analysis&lt;/strong&gt;&lt;/a&gt;: A bicycle does not move at a speed of 15 kg/m&amp;sup2;.  Any such answer is simply nonsense, and students need to recognize it as such.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//en.wikipedia.org/wiki/Fermi_problem&quot;&gt;&lt;strong&gt;Fermi estimation&lt;/strong&gt;&lt;/a&gt;: The number of bicycles on Earth is probably of the same order of magnitude as the number of people, i.e. in the billions.  There are several ways to guesstimate this, but the important point is that any calculation resulting in, say, less than a thousand or more than a trillion bicycles is almost certainly nonsense.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Physical and logical constraints&lt;/strong&gt;: Little Timmy might own zero, one, two or even three bicycles.  He most likely does not own 0.3675 bicycles, and he &lt;em&gt;definitely&lt;/em&gt; does not own $\sqrt{-5}$ or infinitely many bicycles.  That would just be ridiculous.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Just plain common sense&lt;/strong&gt;: A bicycle does &lt;em&gt;not&lt;/em&gt; travel faster than sound (unless fired from a gun or dropped from orbit).  Any answer that implies this is nonsense. Any question that expects such an answer is equally nonsense.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Most children naturally learn these skills at an early age, but unfortunately, through the efforts of all too many early and middle school math teachers, a large fraction of them manage to forget them, or at least to acquire the regrettable belief that such sensible rules only apply &lt;em&gt;outside&lt;/em&gt; the classroom.  Thus, in order to give such students a practical level of everyday numeracy, and to help them see mathematics as a useful tool rather than just a pointless classroom exercise in memorization and rule-application, &lt;strong&gt;these skills need to be re-taught&lt;/strong&gt; as an explicit part of any &lt;em&gt;responsible&lt;/em&gt; math curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exams and homework exercises &amp;mdash; being, for better or worse, an unavoidable part of most math teaching methods &amp;mdash; also need to support these lessons, or &lt;em&gt;at least should not actively undermine them&lt;/em&gt;, as &quot;word problems&quot; like those quoted above do.  Word problems as such are a great tool for teaching practical problem solving, but only when properly used.  As a responsible teacher, you need to make sure that any problems you present have sensible answers, and to instill in your students the understanding that an answer is not complete until it has been verified, and that a nonsensical answer is &lt;em&gt;worse than wrong&lt;/em&gt; &amp;mdash; a wrong but plausible answer may still be worth partial credit, but a clearly nonsense answer, presented with no indication that the student understands that it must be wrong, is simply unexcusable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If &lt;em&gt;other&lt;/em&gt; teachers are undermining your teaching by giving your students nonsense questions like those quoted above, I'd suggest first having a friendly chat with those teachers about it, perhaps even pointing them to this thread.  If that doesn't help, I'd go ahead and tell your students that, while other classes might include word problems that don't make any sense, yours doesn't (or, at least, shouldn't) &amp;mdash; and that, even in other classes, they should still make as much use of the answer-checking skills you've taught them as possible.  (Make sure to note that, even for nonsense problems, double-checking the answer still helps them catch mistakes in exams.  That's always a good motivation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As tempting as it may be, though, you &lt;em&gt;may&lt;/em&gt; wish to refrain from suggesting to your students that, when they answer word problems from other classes, they should point out any nonsensical assumptions made in the problem statement in their answers.  Or maybe not &amp;mdash; it depends on the situation.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Addendum: As noted by neminem in the comments, one possible exception to the rule of keeping word problems reasonable are problems &lt;em&gt;deliberately&lt;/em&gt; meant to be funny and absurd, &lt;em&gt;and clearly signaled as such.&lt;/em&gt;  The last part is important &amp;mdash; students need to clearly recognize that certain parts of those particular problems may violate common sense.  One way of doing so, as noted in neminem's comment, is to pick an &lt;a href=&quot;//en.wikipedia.org/wiki/Inherently_funny_word&quot;&gt;inherently funny&lt;/a&gt; animal or other being or object, and make clear (via your lecture examples, and/or simply by openly stating so) that, in &lt;em&gt;your&lt;/em&gt; exercises, those &lt;em&gt;particular&lt;/em&gt; creatures or things may frequently violate laws of physics or common sense.  That gives your students an easy way to identify not-so-realistic exercises &amp;mdash; &quot;&lt;em&gt;that's another one with a llama, it's probably silly&lt;/em&gt;&quot; &amp;mdash; while also setting up a &lt;a href=&quot;//en.wikipedia.org/wiki/Running_gag&quot;&gt;running gag&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="526" LastEditorUserId="526" LastEditDate="2014-11-19T01:10:54.587" LastActivityDate="2014-11-19T01:10:54.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5884" PostTypeId="1" CreationDate="2014-11-19T05:21:30.820" Score="13" ViewCount="1647" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm currently teaching a Statistics course (for the first time), one in which I been trying to teach with a hands-on approach. I have had my students (all traditional college age, non-STEM majors) collect their own data and form their own hypotheses before doing any calculations for $\chi^2$, Regression Lines, and applications of the Central Limit Theorem. I'm having difficulty, while preparing for the upcoming unit on ANOVA testing, finding motivation for ANOVA, with its tedious calculations and cumbersome notation/naming of terms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that, while in class we often utilize Excel to help compute values for calculation-heavy situations, for the final exam, a departmental requirement is to have an ANOVA calculation question (with calculator only). While this example is relatively simple (3 groups of 5 observations each) it still results in a lot of calculation, and so I want to make sure my students understand the meaning behind the calculations as well as how to keep everything organized for when they will have to do it on the exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Has anyone with more Statistics teaching experience sought and found an effective way to teach ANOVA testing? Preferably, something that clearly describes the conceptual aspects as well as the computational ones. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="267" LastActivityDate="2015-03-22T16:58:57.897" Title="Looking for a creative way to teach ANOVA" Tags="|undergraduate-education|statistics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5885" PostTypeId="1" CreationDate="2014-11-19T09:34:14.930" Score="8" ViewCount="2330" Body="&lt;p&gt;A colleague of mine, teaches math to 6th graders. Some of his students are completely fluent in arithmetic, but can't solve word problems. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What practical ways exist to help them? &lt;/li&gt;&#xA;&lt;li&gt;Is there any detailed article of book dealing with this matter?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2018-09-27T00:11:40.807" Title="Unable to solve word problems, fluent in arithmetic" Tags="|word-problems|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5886" PostTypeId="1" CreationDate="2014-11-19T16:50:50.900" Score="4" ViewCount="1183" Body="&lt;p&gt;I have been planning ahead and it looks like i will have a solid three to four days before the winter break where we wont really be in a unit.  We will have just finished a unit and i do not want to start a new one right before such a long break. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So im curious, what are some fun activities or games in the geometry vein that you have done in the past that are appropriate for the high school level? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm also open to general critical thinking or reasoning activities, the kids love them and it is still developing their math skills, thank you!&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2014-11-27T01:22:58.047" Title="Fun Activities/Games Before a Break/Between Units for Secondary Geometry" Tags="|secondary-education|geometry|games|" AnswerCount="7" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5887" PostTypeId="2" ParentId="5886" CreationDate="2014-11-19T17:26:21.927" Score="3" Body="&lt;p&gt;You might use the amazing &quot;Fold &amp;amp; Cut&quot; Theorem, mentioned in this &lt;em&gt;MESE&lt;/em&gt; answer&#xA;&quot;&lt;a href=&quot;https://matheducators.stackexchange.com/a/5781/511&quot;&gt;Secondary Geometry Curriculum Sequencing&lt;/a&gt;.&quot;&#xA;Templates available at &lt;a href=&quot;http://howtofoldit.org/#Chapter5&quot; rel=&quot;nofollow noreferrer&quot;&gt;this web site&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another idea is to have your students discover the $11$ incongruent &quot;nets&quot; for (unfoldings&#xA;of) a cube (without knowing in advance that there are $11$):&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/2wFff.png&quot; alt=&quot;CubeNets&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;sup&gt;(Figure from &lt;a href=&quot;http://howtofoldit.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book&lt;/a&gt;.)&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;I hand out pre-grided paper, with cells about 5cm$\times$5cm, and scissors.&#xA;A class can be fruitfully engaged for quite some time trying to answer the question,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;. How many distinct (incongruent) &lt;a href=&quot;http://en.wikipedia.org/wiki/Net_(polyhedron)&quot; rel=&quot;nofollow noreferrer&quot;&gt;nets&lt;/a&gt; of the cube are there?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-11-27T01:22:58.047" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5888" PostTypeId="2" ParentId="5886" CreationDate="2014-11-19T17:36:32.567" Score="2" Body="&lt;p&gt;Another idea, albeit more arithmetic than geometry: Clueless sudoku:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/e9jJY.png&quot; alt=&quot;Henle&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Creating Clueless Puzzles.&quot;&#xA;  Gerard Butters,&#xA;  Frederick Henle,&#xA;  James Henle,&#xA;  Colleen McGaughey.&#xA;  (&lt;a href=&quot;http://www.math.smith.edu/~jhenle/clueless/creatingclueless.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is also an &lt;a href=&quot;http://www.math.smith.edu/~jhenle/clueless/puzzle.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;interactive website&lt;/a&gt; on which to solve these puzzles.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-11-19T18:06:28.580" LastActivityDate="2014-11-19T18:06:28.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5889" PostTypeId="2" ParentId="5886" CreationDate="2014-11-19T18:54:00.677" Score="3" Body="&lt;p&gt;Have them explore low dimensional topology by cutting up strips of paper: &lt;a href=&quot;https://www.youtube.com/watch?v=JNtKcK27x1s&quot; rel=&quot;nofollow&quot;&gt;Mobius Strip Activities&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-11-19T18:54:00.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5890" PostTypeId="2" ParentId="5885" CreationDate="2014-11-19T20:51:38.287" Score="1" Body="&lt;p&gt;I think that Benjamin Dickman made a useful comment. Also, you can talk with your pupils about their own experience. For example, you can ask them:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;You have 10 dollars. Then you buy ice cream with 4 dollars.  How many dollars do you have left?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You go to the market and buy 8 pencils for 50 cents each.  How much did you pay in total?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can build more examples that motivate your students to think and to solve the exercise. &lt;/p&gt;&#xA;" OwnerUserId="3306" LastEditorDisplayName="user173" LastEditDate="2014-11-20T02:07:00.843" LastActivityDate="2014-11-20T02:07:00.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5891" PostTypeId="2" ParentId="4036" CreationDate="2014-11-20T01:00:00.680" Score="1" Body="&lt;p&gt;Wow Mathdad!  You and I have a lot in common.  I also have a 12 year old girl that is (was?) struggling in math and I tried many of the things suggested here.  I'd say try them all and see if they work for you.  Many of the ideas suggested in the answers did not work for me. [I'm a huge fan of Danica McKellar and quickly bought her book for my daughter.  My daughter uses it as a coaster for her drink :) ].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know I'll get flamed for my suggestion, and I'm frankly curious to see how bad a parent folks will think I am.  The only thing that has worked for me (and it's an ongoing experiment) was to &quot;put my money where my mouth is&quot;.  I told her that I would pay money for A's and B's on report cards (twice a year in my daughter's school).  You have to figure out the denominations you are comfortable with but I went with 5 dollars for an A, 2.50 for a B, nothing for anything else and nothing for P.E., citizenship, etc. But here's the key point, I announced a special 25.00 bounty for an A in mathematics. 12.50 for a B.  When she questioned why other classes couldn't be higher since after all, &quot;English, history, science are all important&quot;, I gave my speech that other folks have different opinions and they may be right, but it's my money and I can spend it anyway I choose.  I then went on to reiterate that in my opinion, math is more important than all other subjects.  That with a strong basis in math, one can master any other subject.  I firmly believe this.  I of course also offer to help her relieve her old man of 25.00 by helping her with homework and studying for tests anytime she wants.  When she gets frustrated or isn't paying attention, I remind her that she is in fact trying to get 25.00 from me and should be a little nicer with me. The semester is almost over and her grade seems to have gone up from a 'D' to an 'A-'.  I'll be interested to see what the final grade is.  There is no way on earth that her grade could have gone from a D to an A if the teacher is equally weighting the tests (she had a D for close to half the semester).  I have come to the conclusion that either (1) the teacher is weighting recent tests more heavily, or (2) my daughter has informed her of the situation and offered to give her a cut if she raises the grade. :).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, my daughter's first language is English but she is a French immersion school and that does add to the complexity of every subject.  I sometimes have trouble helping her with her homework simply because my knowledge of French is comparatively limited. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dave&lt;/p&gt;&#xA;" OwnerUserId="3405" LastActivityDate="2014-11-20T01:00:00.680" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="5892" PostTypeId="5" CreationDate="2014-11-20T05:53:20.553" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-20T05:53:20.553" LastActivityDate="2014-11-20T05:53:20.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5893" PostTypeId="4" CreationDate="2014-11-20T05:53:20.553" Score="0" Body="For questions relating to teaching math at the community college or two year college." OwnerUserId="3265" LastEditorUserId="3265" LastEditDate="2014-11-20T10:34:32.093" LastActivityDate="2014-11-20T10:34:32.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5894" PostTypeId="1" AcceptedAnswerId="5927" CreationDate="2014-11-20T06:38:03.103" Score="12" ViewCount="476" Body="&lt;p&gt;Can anyone recommend a textbook for an introduction-to-proofs bridge course that discusses the rules for &quot;proving and using&quot; (aka &quot;introduction and elimination&quot;) each connective and quantifier, as in type theory or natural deduction?  All the books I've looked at so far either explain the connectives using truth tables, or don't explain them specifically at all.  They often discuss some of the rules (like &quot;direct proof&quot; = implies-intro and &quot;proof by cases&quot; = or-intro and &quot;constructive proof&quot; = exists-intro), but usually just in a list of &quot;proof methods&quot; without an organizing structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(To clarify, I'm &lt;em&gt;not&lt;/em&gt; looking for a textbook in formal logic.  I know that some people use textbooks in formal logic for bridge courses, but I don't think that would be appropriate for my students.  I want a textbook which introduces students to the idea of proof, to basic concepts in mathematics like induction, divisibility, and sets, and to other aspects of mathematics like mathematical writing and exploration / proof search.  There are lots of such textbooks, but I haven't found one yet that organizes the proof rules by their governing connectives as above.)&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2014-11-30T04:16:59.300" Title="Introduction-to-proof book using natural-deduction-style rules?" Tags="|textbooks|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="5896" PostTypeId="1" AcceptedAnswerId="5928" CreationDate="2014-11-20T19:07:50.280" Score="7" ViewCount="646" Body="&lt;p&gt;I'm teaching mathematics at my former high school and the next topic will be vector geometry. When I attended high school, I was only taught vector geometry and never learnt anything about matrices until university. I feel like my pupils would be missing out on something if I didn't teach them very basic linear algebra and I also have the permission to teach linear algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two possibilites: Treat vector geometry and linear algebra as two &quot;different&quot; topics (i.e. with other topics in between, such as integration, probability, etc.) or merge the two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I experienced the &quot;merged&quot; variant in university, but obviously following that course wouldn't make any sense for high schoolers. However, as I didn't learn linear algebra in high school, I don't really know how I should merge the material I already have from my time in school with new linear algebra. Whenever I try to come up with a plan, I end up putting all of vector geometry (scalar product, vector product, intersection problems) before and linear algebra (matrices, Gauss elimination, determinants, eigenvalues) after.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have a suggestion, how I could teach the two topics in a comprehensive &quot;linear algebra&quot; course for high schoolers?&lt;/p&gt;&#xA;" OwnerUserId="493" LastActivityDate="2014-11-26T21:58:25.553" Title="How should I teach linear algebra and vector geometry together at high school?" Tags="|linear-algebra|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5897" PostTypeId="2" ParentId="5885" CreationDate="2014-11-21T00:36:50.033" Score="12" Body="&lt;p&gt;Does your colleague have any information about these students' reading comprehension? The reason I ask is that, in my opinion, difficulty with word problems is not always a matter of taking the content of the word problem and reformulating it as equations.  Rather, it is often a matter of getting the content of the problem in the first place.  In other words, the reason some people can't convert a word problem into a mathematical formulation is that they don't understand the word problem.  In such cases, it is their reading skills rather than their mathematical skills that need to improve.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-11-21T00:36:50.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5898" PostTypeId="1" CreationDate="2014-11-21T04:50:13.597" Score="21" ViewCount="776" Body="&lt;p&gt;Is there any research supporting the idea that a single professor should teach the entire calculus sequence as opposed to splitting the duty amongst multiple professors? What are the pros and cons of having the same professor teach all the calculus classes?&lt;/p&gt;&#xA;" OwnerUserId="179" LastEditorUserId="262" LastEditDate="2014-11-21T22:40:13.457" LastActivityDate="2015-05-05T05:35:15.340" Title="Benefits of Having the Same Professor Teach the Entire Calculus Sequence" Tags="|undergraduate-education|calculus|teacher-development|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="5899" PostTypeId="1" CreationDate="2014-11-21T14:07:45.907" Score="5" ViewCount="252" Body="&lt;p&gt;How can I teach a $7^\text{th}$ grade student the concept of ratio and proportion &lt;em&gt;with the use of teaching aids ( such as models or Charts)&lt;/em&gt; . One of my friends asked me this question. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possible solution could be found using  of spring of length length 5 units (,say) and a scale  (marked with numbers from $1$ to $10$). I colour the spring to the length of 3 units . Then I stretch the whole spring upto $10(= 5 \times 2)$ units and see the coloured portion stretches upto $6 (=3 \times 2)$ units.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any other easier way (of course by using appropriate teaching aids) to grow the concept of ratio and proportion ? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance for your time.&lt;/p&gt;&#xA;" OwnerUserId="3411" LastEditorUserId="39" LastEditDate="2017-05-07T12:23:56.740" LastActivityDate="2017-05-07T12:23:56.740" Title="How can I teach a $7^\text{th}$ grade student the concept of ratio and proportion" Tags="|secondary-education|proportional-reasoning|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5900" PostTypeId="2" ParentId="5899" CreationDate="2014-11-21T14:45:13.100" Score="4" Body="&lt;p&gt;One nice model is mixing paint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make a color by mixing one part yellow to two parts red by volume.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can ask questions like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;If I add one more cup of yellow paint, how much red paint must I add to get the same color?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;If I double the amount of yellow paint, what will I need to do to the red paint to maintain the same color?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;If I add one tsp of yellow and one tsp of red paint, will the new color be 'yellower' or 'redder' than the old color&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students can actually play with mixing small quantities of paint this way.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-11-21T14:45:13.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5901" PostTypeId="1" CreationDate="2014-11-21T16:32:10.787" Score="14" ViewCount="4271" Body="&lt;p&gt;I teach the definite integral before the indefinite for a few reasons, one of which is that I want students to recognize that the definite integral means area (not anti-derivative). If we do indefinite integrals before the Fundamental Theorem of Calculus, the students are likely to think the FTC is not a big deal. (&quot;Yep, just find the anti-derivative, that's what we already learned to do when we see this tall skinny S.&quot;) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also the issue that the symbols make more sense in the definite integral. &#xA;$$\int_1^2 x^2\, dx$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can read the integral sign as a summation, so that we get &quot;add up an infinite number of infinitely skinny rectangles, from x=1 to x=2, with height x^2 times width dx.&quot; Each part of the symbol makes sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But with indefinite integrals, the tall skinny S does not mean summation, and the dx has no real meaning. I think it's a very unfortunate symbol choice. [See &lt;a href=&quot;https://matheducators.stackexchange.com/q/2338/60&quot;&gt;Should we avoid indefinite integrals?&lt;/a&gt; for more on this.] But, since we're stuck with it, I'd like to know if there is any explanation one can give for the symbols. I tell my students that the S part means &quot;find the anti-derivative&quot;, and the dx means &quot;and x is the variable&quot;. &lt;strong&gt;Is there a better way to describe the meaning of the dx (and maybe even the S) here?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-02-05T13:01:45.307" Title="Explaining the symbols in definite and indefinite integrals" Tags="|calculus|notation|" AnswerCount="6" CommentCount="7" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5902" PostTypeId="2" ParentId="5901" CreationDate="2014-11-21T16:39:31.427" Score="6" Body="&lt;p&gt;It sounds like what you want to do is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Do definite integrals first using $\int_a^bf(x)\ dx$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Then do antiderivatives using Newtonian notation: use capital letters to represent antiderivatives. So for instance, if $f(x)=2x$, then $F(x)=x^2+C$ so that $F'(x)=f(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Do the Fundamental Theorem of Calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Since the two ideas are related, now introduce the indefinite $\int f(x)\ dx$ as another way of saying antiderivative (which, by the way, is Leibniz's notation).&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2014-11-23T03:55:50.017" LastActivityDate="2014-11-23T03:55:50.017" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5903" PostTypeId="1" AcceptedAnswerId="5906" CreationDate="2014-11-21T17:01:52.680" Score="4" ViewCount="414" Body="&lt;p&gt;In a lecture, the teacher told us that the notion of imaginary number can be introduced using the geometric mean. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$i=\sqrt{-1}=\sqrt{(-1)\cdot1}$&#xA;which is the geometric mean of $-1$ and $1$.&#xA;We can place $1$ and $-1$ on the real line. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is how could you explain that the two geometric mean $i$ and $-i$ are placed on the imaginary line (--EDIT-- without using complex numbers, in order to introduce $i$ ans $-i$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;HINT : In a right triangle, the altitude from the hypotenuse to its 90° vertex splits the hypotenuse into two segments. The geometric mean of these segment lengths is the length of the altitude.&lt;/p&gt;&#xA;" OwnerUserId="3412" LastEditorUserId="3412" LastEditDate="2014-11-21T19:31:51.153" LastActivityDate="2014-11-21T19:31:51.153" Title="How to introduce the notion of imaginary number using the geometric mean?" Tags="|complex-numbers|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5904" PostTypeId="2" ParentId="5899" CreationDate="2014-11-21T17:48:28.373" Score="3" Body="&lt;p&gt;I like food math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students wanting 2 slices of pizza for lunch. This sets up a 2:1 ratio, pizza:students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can then move on to ravioli, at which point the ratio is 6:1 for medium ravioli. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next step is a recipe in which you have butter, eggs, sugar, flour, in a given ratio and you can change the number of results (the cookies/muffins/etc) and ask for how much of each ingredient is needed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above started with my daughter nearly a decade ago. We were making brown rice and had a 'recipe' that was 4 cups water, 2 cups rice. When she measured the rice and saw only 1-1/2 cups, I said &quot;I guess we can't make rice for dinner.&quot; She showed her knowledge of ratios, and said we'll just use 3 cups of water. Not bad for a 7 year old. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-21T17:48:28.373" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5905" PostTypeId="1" AcceptedAnswerId="5912" CreationDate="2014-11-21T18:07:01.410" Score="33" ViewCount="1092" Body="&lt;p&gt;At many colleges in the United States, incoming students are required to take placement tests in basic skills such as math and reading. Those who score below a cut-off are required to take remedial coursework. Is there solid evidence about the effectiveness of remedial math courses? I'm particularly interested in community colleges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only empirical study I was able to turn up was Calcagno 2008. The method they used was to examine data from Florida community colleges to look for evidence of a discontinuity in average educational outcomes as a function of placement test scores, at the cut-off. The reasoning is that a student who scores at the cut-off score $x$ has almost the same math skills as one who scores $x-1$ and is forced into remediation, but would have a worse educational outcome because of not getting remediation. The result seems to be that math students who get remediation don't do better in later math coursework, and aren't more successful in college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The subject seems to be controversial, however. Are there other studies that do support remediation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the community college where I teach, it wouldn't be surprising to me if remediation was uncorrelated with later success, simply because our curriculum has so much review built into it. For example, a student who starts at a low math level and gradually climbs up through our sequence will probably be led over and over again through reviews of the same topics, such as the properties of the reals or how to solve multiple equations in multiple unknowns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calcagno and Long, &quot;The Impact of Postsecondary Remediation Using a Regression Discontinuity Approach: Addressing Endogenous Sorting and Noncompliance,&quot; National Center for Postsecondary Education, 2008, &lt;a href=&quot;http://ies.ed.gov/director/conferences/08ies_conference/pdf/calcagno_long.pdf&quot;&gt;http://ies.ed.gov/director/conferences/08ies_conference/pdf/calcagno_long.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2015-12-03T23:14:08.963" LastActivityDate="2015-12-04T16:14:40.020" Title="What is the evidence about the effectiveness of remediation in math?" Tags="|undergraduate-education|community-colleges|remedial-courses|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5906" PostTypeId="2" ParentId="5903" CreationDate="2014-11-21T18:43:08.813" Score="4" Body="&lt;p&gt;Consider complex numbers in polar form $r e^{i \theta}$. Then multiplying by $re^{i \theta}$ stretches the original number by a factor of $r$ and rotates about the origin by an angle $\theta$. For geometric mean, consider a couple of cases:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Two real positive numbers $x=r$ and $y$. For both $\theta = 0$, and the geometric mean is $\sqrt{xy}$. No rotation is require to get from one to the other, only real stretching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Real $x&amp;gt;0$, $y&amp;lt;0$. Then $y = |y|e^{i \pi}$, so $\sqrt{xy} = (x\cdot |y|e^{i \pi})^{1/2} = \sqrt{x|y|}e^{i \pi/2} = i\sqrt{x|y|}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Arbitrary complex $x=r_xe^{i \theta_x}$, $y=r_ye^{i \theta_y}$. Then $\sqrt{xy} = \sqrt{r_xr_y}e^{i(\theta_x+\theta_y)/2}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, the geometric mean is the usual geometric mean of the lengths but in the direction halfway between them (i.e. at the arithmetic mean of the angles). Rotating halfway between positives and negatives gives you a purely imaginary answer.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-11-21T18:43:08.813" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5907" PostTypeId="2" ParentId="5901" CreationDate="2014-11-21T19:00:37.713" Score="1" Body="&lt;p&gt;The $dx$ means &quot;with respect to $x$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose water flows over a waterfall at a rate of $(3+\sin(t))x$, where $x$ is the distance from the riverbank and $t$ is the time of year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you want to know how much is flowing over the falls at once?  Or how much flows over one rock over a period of time?  You would set up different integrals:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int (3+\sin(t))\ x\ dx = (3+\sin(t))\ x^2/\,2 + C$$&#xA;$$\int (3+\sin(t))\ x\ dt = (3t-\cos(t))\ x + C\ \ \ $$&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-11-21T19:05:48.093" LastActivityDate="2014-11-21T19:05:48.093" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5908" PostTypeId="2" ParentId="5899" CreationDate="2014-11-21T21:31:18.603" Score="0" Body="&lt;p&gt;In order to achieve a deeper understanding of what &lt;code&gt;ratio&lt;/code&gt; and &lt;code&gt;proportion&lt;/code&gt; are, you can use short fragments from Peppa Pig's 4x02 episode, &lt;em&gt;The new house&lt;/em&gt;. Peppa's dad works as an architect and he designs a new house for his neighbours, the Wolf family. He gives &#xA;his drawing, which can be seen in an explicit way, to the construction workers. Then, they build the house.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After watching these fragments, you can think of a similar activity. Make your students to draw a design (with just straight lines and so) in their notebooks. Just to finish, tell them to build it with construction blocks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/9o8nS.jpg&quot; alt=&quot;Peppa Pig 4x02: ratio and proportion concepts&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="3260" LastActivityDate="2014-11-21T21:31:18.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5909" PostTypeId="2" ParentId="5901" CreationDate="2014-11-22T00:14:43.413" Score="7" Body="&lt;p&gt;If I were teaching the indefinite integral to students who already understand the definite integral and are comfortable with the usual notation for it, then I would introduce the indefinite integral as a definite integral with a variable upper limit, i.e.,&#xA;$$&#xA;F(t)=\int_a^tf(x)\,dx,&#xA;$$&#xA;where $a$ is a constant.  (Different choices for $a$ give different &quot;constants of integration&quot;.)  So far, the concept and the notation should look pretty reasonable.  But then questions arise about the usual notation, which looks like &#xA;$$&#xA;\int f(x)\,dx=F(x)+C.&#xA;$$&#xA;Why do we omit the limits of integration ($a$ and $t$ above), allowing the role of $a$ to be taken by the $C$ on the right side of the equation, and replacing $t$ as the argument of $F$ with the variable of integration $x$?  The use of $C$ can probably be explained thanks to the fundamental theorem of calculus.  The integrals with various $a$'s as lower limits are some of the antiderivatives of $f$; for some purposes we'll want all the antiderivatives of $f$, and the formulation with $C$ provides these.  As for the use of $x$ in place of $t$, I won't try to defend a convention that notationally identifies a bound variable (in the integral) with a free variable (in $F(x)$).  In such situations, I usually tell the class something like: &quot;I'm not responsible for this notation. It's so ancient that it was fixed even before I was born.  So I never had a chance to correct it.&quot;&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2014-11-22T00:14:43.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5910" PostTypeId="2" ParentId="5905" CreationDate="2014-11-22T17:27:02.753" Score="17" Body="&lt;p&gt;I don't know about the research you ask for - &lt;em&gt;supporting&lt;/em&gt; remediation. I am learning a lot lately about how problematic remediation is. There are two sorts of problems - placement and the remediation coursework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Placement tests can be harder than the tests students encounter at the end of their courses. I took the ACT/Compass placement test that our college uses, to see what it's like. Many of the questions were interesting to me. That translates to - way too hard for most students. The questions required real mathematical thinking, coming at topics from unusual directions. I asked afterward how many students place into calculus. They said almost none do. (Our department has control over the cut scores, but we probably set them too high long ago, and haven't dealt with that recently.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another problem with placement tests is the issue of &lt;a href=&quot;http://www.reducingstereotypethreat.org/&quot; rel=&quot;noreferrer&quot;&gt;stereotype threat&lt;/a&gt;, the tendency to feel threatened by the stereotypes about your people (gender, race, etc), and because of that feeling, to do worse on a test of ability. (See &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B00E86KDKY&quot; rel=&quot;noreferrer&quot;&gt;Whistling Vivaldi, by Claude Steele&lt;/a&gt; for a deeper understanding of how big a problem this is.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of these issues, my department is looking at changing placement decisions to a combination of GPA and grades from prior coursework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The short version: Many students who place into remedial courses don't really need to be there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But some students are clearly not ready for college level math courses. Are remedial courses the right place for them? Probably not. Any course that feels like a repetition of things a student first saw in elementary school is going to activate all sorts of emotional baggage. Also, taking so many courses means that at least a few drop out at each step along the way. At our (typical) community college, many of the students place into a pre-algebra course. If we did super-well and 90% passed, some would still not go on to the next course (they move, they need to take on more paid work and have no time, etc). Let's say we get 90% going to the next course. How many ever arrive at a first college-level course, even with these wildly optimistic success rates? (90% pass pre-alg)(90% go to next course)(90% pass beginning alg)(90% go on)(90% pass int alg) (90% go on) = .9^6 = 53% of these students make it to the college level course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When over 70% of students place into remedial courses, and we fail over half of them (because that 90% I used above was not realistic), we are allowing our courses to become a barrier to their success.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way around this that looks like it's working is &quot;acceleration&quot; programs that get students ready for the college level course they'll take. Many students need just one college level course, and choose statistics. We are piloting an Algebra for Statistics course that students at any level can take. Another college in our district has found that the students who take this course do better in statistics than students who took the traditional remedial sequence (and made it to stat). I put acceleration in quotes because it's not about going faster through material (which would be disastrous); instead, it's about getting to your goal faster, by learning some mathematical thinking skills, playing around with basic stat concepts, and coming into the stat course ready to think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it possible to develop a program that gets students ready for STEM courses, including calculus, with fewer semesters of coursework? I'm looking into that now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Resources:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.washingtonpost.com/blogs/answer-sheet/wp/2014/07/21/why-college-remediation-needs-to-be-overhauled/&quot; rel=&quot;noreferrer&quot;&gt;Washington Post: Why college remediation needs to be overhauled&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://cap.3csn.org/developing-pilots/pre-statistics-courses/&quot; rel=&quot;noreferrer&quot;&gt;California Acceleration Project&lt;/a&gt; &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2015-12-04T16:14:40.020" LastActivityDate="2015-12-04T16:14:40.020" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5911" PostTypeId="2" ParentId="5905" CreationDate="2014-11-22T18:19:54.990" Score="8" Body="&lt;p&gt;Remediation is more about the politics of mathematics education than about learning mathematics. When students don't use their mathematics they often lose it quickly. My own experience was that it was much better for students to take the courses that their high school record said they had taken the prerequisite courses for than to have them take &quot;remedial&quot; courses first. I reviewed the skills that experience often suggested students had troubles with when it was needed for a topic I was teaching. I think it is especially cynical to have students take remedial algebra courses if there is little need for algebra as part of their career plans.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.insidehighered.com/news/2014/08/21/college-level-statistics-trumps-remedial-algebra-cuny-study&quot;&gt;https://www.insidehighered.com/news/2014/08/21/college-level-statistics-trumps-remedial-algebra-cuny-study&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2014-11-22T18:19:54.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5912" PostTypeId="2" ParentId="5905" CreationDate="2014-11-22T19:15:20.513" Score="24" Body="&lt;p&gt;Sonnert and Sadler (2014) investigated what factors influence success in college calculus courses, including putting students into a pre-calculus course, which they described as &quot;often a review of material students learned (or were supposed to learn) in high school.&quot; This may correspond to what you are calling &quot;remedial.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In their results, they found students who took the pre-calc course showed no significant benefit over their peers in their outcomes in calculus courses. No matter how unprepared for calculus they were (pp.15-16).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This study examined a large sample (about 10,500) of U.S. calculus students. This is the most recent research of this type I am aware of. The literature review might be helpful as well. Dr. Sadler came to speak at the Kaput Center recently and he was very frank that there are many possible implications and questions that we might have as a result of this analysis, but he didn't entertain any sort of speculation. This gave me confidence in the results he did report.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cited:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sonnert, G., &amp;amp; Sadler, P. M. (2014). The impact of taking a college pre-calculus course on students’ college calculus performance. International Journal of Mathematical Education in Science and Technology, 45(8), 1188–1207. doi:10.1080/0020739X.2014.920532&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-11-22T19:15:20.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5913" PostTypeId="1" AcceptedAnswerId="5918" CreationDate="2014-11-22T19:56:19.113" Score="14" ViewCount="531" Body="&lt;p&gt;&lt;strong&gt;Briefly&lt;/strong&gt;: I am looking for research on the extent to which learning mathematics (let's say &quot;college algebra&quot; if we want to be specific) impacts problem solving skills, abstract reasoning, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Less briefly&lt;/strong&gt;: I teach a college algebra class in an american research university. Most of my students struggle with motivation, and want to know (for example) why they need to learn how to complete the square. My stock answer is to compare the classroom to a gym and to the instructor (me) as a personal trainer: the purpose of mathematics is then to strengthen their &quot;analytic-reasoning muscles.&quot; The students accept this answer for the most part, but I feel a little guilty in that I don't actually know of any research that supports the argument that learning mathematics improves abstract reasoning or problem solving skills. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Possibly unnecessary&lt;/strong&gt;: What do I mean by &quot;abstract reasoning / problem solving skills&quot;? That is a  good question. I guess what I want to know is whether or not learning to complete the square is helps students learn how to solve problems beyond those that ask them to complete the square. I suspect this depends on what I am actually doing when I say that I am teaching them to complete the square. Am I giving them a recipe for them to follow? Or am I giving them something more to engage with? &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2022-09-28T15:40:38.153" Title="Research on how mathematics skills transfer to other areas" Tags="|reference-request|student-motivation|education-research|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5914" PostTypeId="1" CreationDate="2014-11-22T20:32:52.160" Score="8" ViewCount="90" Body="&lt;p&gt;Motivated by &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5913/research-on-how-mathematics-skills-transfer-to-other-areas&quot;&gt;this other question&lt;/a&gt; I'm interested in getting to know the literature on mathematics skills transfer within itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All I know is what I've read in David Perkins's book &quot;Knowledge as Design&quot;, in which he touches on low road transfer and high road transfer but does not elaborate further and the book is not centered on mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are good books dealing with the subject?&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-11-22T20:32:52.160" Title="Literature on skill transfer" Tags="|reference-request|education-research|reference-books|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5915" PostTypeId="2" ParentId="5913" CreationDate="2014-11-22T21:12:41.130" Score="6" Body="&lt;p&gt;I think the evidence suggests arguing for college algebra differently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students do not develop these sorts of skills in college at all, according to the book &lt;em&gt;Academically Adrift&lt;/em&gt;.  A key &lt;a href=&quot;http://www.huffingtonpost.com/2011/01/18/45-of-students-dont-learn_n_810224.html&quot; rel=&quot;noreferrer&quot;&gt;result&lt;/a&gt; from the book is that &quot;45 percent of students show &lt;strong&gt;no significant improvement in the key measures of critical thinking&lt;/strong&gt;, complex reasoning and writing by the end of their sophomore years.&quot;  If the students are majoring in education, social work or business, it's &lt;a href=&quot;https://www.linkedin.com/pulse/article/20130509131339-17000124-congratulations-college-graduate-now-tell-us-what-did-you-learn&quot; rel=&quot;noreferrer&quot;&gt;especially likely&lt;/a&gt; that they're not developing these skills.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would sell college algebra &lt;strong&gt;without&lt;/strong&gt; reference to general problem-solving skills or abstract reasoning, and preferably without completing the square.  More directly, it helps to be able to answer:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How many widgets do we need to sell to make a profit?&lt;/li&gt;&#xA;&lt;li&gt;Given my scores so far and the exam weights, what score do I need on the final to get an A or a B?&lt;/li&gt;&#xA;&lt;li&gt;If my income tax rate is 30%, how much do I need to earn before tax to have $1,000 more after tax?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I would also want the class to focus on answering questions like these.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2014-11-24T01:19:30.223" LastActivityDate="2014-11-24T01:19:30.223" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5916" PostTypeId="1" CreationDate="2014-11-22T21:16:04.163" Score="14" ViewCount="378" Body="&lt;p&gt;I work as a teaching assistant in a high school and geometry gives most students headaches.&lt;/p&gt;&#xA;&lt;p&gt;I emphasize understanding the problem by constructing lines and using elementary properties to arrive at the solution instead of merely pulling a formula and plugging numbers. However, even when I've presented multiple ways to think about a problem they still do not attempt constructions to understand and/or solve it or similar ones.&lt;/p&gt;&#xA;&lt;p&gt;What are good ways to foster this kind of initiative in students dealing with elementary geometry?&lt;/p&gt;&#xA;&lt;p&gt;I mean constructions in the sense of drawing diagrams, constructing (by hand) auxiliary lines that shape the problem into something known (perpendiculars, etc).&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="2083" LastEditDate="2022-06-09T06:44:14.233" LastActivityDate="2022-06-09T06:47:38.690" Title="Helping students use constructions in geometry" Tags="|secondary-education|mathematical-pedagogy|geometry|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="5917" PostTypeId="2" ParentId="5872" CreationDate="2014-11-23T00:31:47.863" Score="8" Body="&lt;p&gt;It sounds to me like the student may well not understand that a fraction indicates a division.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, as others have already noted, this student may be performing all these computations purely formally without having any real notion of their significance.  This is not at all uncommon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once was teaching a first-year algebra class.  At one point it occurred to me, after much thrashing around, that the students perhaps didn't really understand what a fraction was.  I drew a number line on the board (probably from -4 to 4) and asked where 2/3 was on the line.  About half the class thought it was somewhere between 2 and 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These relationships -- fractions, division, points on a number line, ... are historically very deep.  It took the human race a really long time to see what was going on.  It's no surprise that our students have difficulty with this.  What's particularly galling, though, is that so many elementary school teachers are clueless about this, and that they aren't helped much by the ed courses they take.  This creates a formidable problem by the time students get to Junior High School, or even (gasp!) algebra.&lt;/p&gt;&#xA;" OwnerUserId="1214" LastActivityDate="2014-11-23T00:31:47.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5918" PostTypeId="2" ParentId="5913" CreationDate="2014-11-23T05:09:36.140" Score="9" Body="&lt;p&gt;The answer you are giving to your students overlaps with the so-called &quot;Mental Discipline&quot; theory (also &quot;Theory of Formal Discipline&quot;) for justifying mathematics education. As you search for research related to your favored justification, this theory may help you focus in on more relevant work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One researcher who has written about this is George Stanic. &lt;a href=&quot;http://www.flm-journal.org/Articles/6F9A84FC509C2D73F45435A001FB0.pdf&quot;&gt;You can see one of his articles on the subject here&lt;/a&gt;. But this comes up in a few places in the detailed and wide-ranging Stanick and Kilpatrick edited work on the history of mathematics education (Stanic &amp;amp; Kilpatrick, 2003).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are varied opinions on whether or not mathematics education disciplines your thinking in ways that extend beyond using math in the specific situations in which it is taught. Some consider it &quot;debunked.&quot; However, it is an idea that is still the subject of research today, &lt;a href=&quot;http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0069399&quot;&gt;as exemplified here&lt;/a&gt; (Attridge &amp;amp; Inglis, 2013).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Cited:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Attridge, N., &amp;amp; Inglis, M. (2013). Advanced Mathematical Study and the Development of Conditional Reasoning Skills. PLoS ONE, 8(7), e69399. doi:10.1371/journal.pone.0069399&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stanic, G. M. A., &amp;amp; Kilpatrick, J. (Eds.). (2003). A history of school mathematics. National Council of Teachers of Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="759" LastActivityDate="2014-11-23T05:09:36.140" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5919" PostTypeId="2" ParentId="5886" CreationDate="2014-11-23T08:40:53.023" Score="1" Body="&lt;p&gt;A teacher I once worked with ran a tesselation competition. The students had to create a tesselation similar to those of MC Escher. Points were awarded both for the artistic side (the tesselation pieces looking like something eg an animal, and for use of colour) and for the explanation of how the tesselating shape was created.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-11-23T08:40:53.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5920" PostTypeId="2" ParentId="5886" CreationDate="2014-11-24T00:20:20.563" Score="0" Body="&lt;p&gt;This is a supplement to @DavidButlerUofA's answer:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/s03x8.jpg&quot; width=&quot;360&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;sup&gt;(Image from &lt;a href=&quot;http://galleryhip.com/bear-tessellation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt;.)&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-11-24T00:20:20.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5921" PostTypeId="2" ParentId="5916" CreationDate="2014-11-24T00:43:00.240" Score="3" Body="&lt;p&gt;Almost all constructions in geometry at the level of your students boil down to using the rigidity of triangles (SSS).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Want to copy an angle?  Make a triangle with that angle as one of its three angles, and copy the triangle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Want to copy a length?  Make a triangle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my suggestion would be to focus on finding triangles creatively, even in problems where there are none to start with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer:  I have never taught a geometry class at this level.  I have only tutored a few students in these subjects.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-11-24T00:43:00.240" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5922" PostTypeId="2" ParentId="5886" CreationDate="2014-11-24T01:18:05.610" Score="2" Body="&lt;p&gt;Maybe have the students discover that every triangle can be dissected and rearranged&#xA;to form a rectangle:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/EomPy.png&quot; alt=&quot;Fig1.21&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;sup&gt;(Figure from &lt;a href=&quot;http://cs.smith.edu/~orourke/DCG/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this textbook&lt;/a&gt;.)&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Not easy to discover from scratch. Needs blank paper and lots of scissors.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-11-24T01:18:05.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5923" PostTypeId="2" ParentId="5813" CreationDate="2014-11-24T06:08:02.833" Score="3" Body="&lt;p&gt;One suggestion: You could you focus on binary operations on $\mathbb Z$, the integers, and then talk about the parity of your two inputs as compared with the outputs? This could lead nicely into classifying all even numbers as 0 mod 2 and all odd numbers as 1 mod 2.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-11-24T06:08:02.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5925" PostTypeId="2" ParentId="5916" CreationDate="2014-11-25T00:36:21.593" Score="11" Body="&lt;p&gt;This is likely to be tough, for two reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Students' previous experience in math has probably been exclusively devoted to carrying out algorithms.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Their initial attempts will probably end in failure and negative feedback.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I teach mainly physics, not math, but as an example along these lines, I assign freshman physics students this problem: given the radius $R$ of the earth and its period of rotation $T$, find the velocity of a point on the earth's surface lying at latitude $\theta$. Over and over again, students show up in my office hours wanting help on this problem, and they usually have not attempted a sketch. If I then ask them to make a diagram without giving additional guidance, they usually draw something in perspective. I then have to suggest that they do non-perspective drawings either from the side or the top, or possibly a cross-sectional view. They find this utterly alien, and they also usually have not had real-life experiences that would help, such as taking a shop or architecture class. They require extensive guidance, with reminders that, e.g., a length has to be measured from a definite point to a definite point, and that an angle has to be defined between two lines that cross. Or they draw a triangle with vertices at the north pole, center of the earth, and a point on the equator, and label one of its angles $\theta$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is a case where something that you and I consider easy and natural (drawing a diagram) is actually an extremely sophisticated skill that needs to be built up over time. A similar example would be order-of-magnitude estimates, such as the volume of a cow or the number of piano tuners in New York.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Keep requiring them to do it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Provide explicit guidance, possibly in the form of a list of guidelines. E.g., you can tell them not to make their diagrams tiny, or not to draw a diagram in such a way as to imply a symmetry that isn't known (e.g., drawing a triangle as isoceles when it's not known to be so).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Expecting them to do it without being forced may be unrealistic until they've had enough time to do it successfully a certain number of times and accept that it really is &quot;normal&quot; to do so.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2014-11-25T00:36:21.593" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5926" PostTypeId="2" ParentId="5916" CreationDate="2014-11-25T19:32:38.350" Score="4" Body="&lt;p&gt;Some suggestions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;give them problems not drawn to scale to force them to have to redraw the figure.  The act of redrawing helps to give a better understanding of the problem and demonstrates how important having a good picture is&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;give them diagrams with many intersecting lines and have an activity to see who find the most number of polygons amongst the lines.  This will help them to &amp;quot;see through the mess&amp;quot; and pick out different shapes.  Once they get the hang of this give them similar diagrams except with a few angles given and have them find the missing angles, again looking for shapes and line patterns they recognize&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;introduce them to various forms of geometric art (islamic tile design, tessellations, isometric drawing, perspective drawing etc) to give them a context to why drawing shapes can be fun/important&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;literally practice drawing shapes based off of a description (ex: &amp;quot;Draw a square with sides 4 inches&amp;quot;, &amp;quot;Draw a triangle with sides 3, 5, 7 inches&amp;quot;, &amp;quot;draw quadrilateral QUAD&amp;quot;) some of your students may just have no experience with drawing, especially geometric shapes, so good old fashion practice might help them to get more comfortable&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;give them problems that &amp;quot;require&amp;quot; drawing a picture,  i.e. it is very hard to reason about the problem without drawing a picture.  The &lt;a href=&quot;http://books.google.com/books/about/Discovering_Geometry.html?id=d9YTPwAACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discovering Geometry&lt;/a&gt; textbook has a lot of good problems where drawing a picture is crucial&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;model drawing a diagram for a student when working through a problem with them (ex: &amp;quot;so the way that I thought about this problem was to draw a picture, what do you think my picture looked like?&amp;quot;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These are all things I have done with my Geometry class and while they do not always draw a picture, they have definitely been drawing them more often since the beginning of the school year.&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="2083" LastEditDate="2022-06-09T06:47:38.690" LastActivityDate="2022-06-09T06:47:38.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5927" PostTypeId="2" ParentId="5894" CreationDate="2014-11-26T01:55:03.363" Score="8" Body="&lt;p&gt;The book &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521675995&quot;&gt;How to Prove It&lt;/a&gt; by Daniel Velleman has an entire chapter devoted to the different types of proofs and the strategies on how to approach each one. It's actually the first book my university uses in its &quot;Intro to Advanced Math&quot; course. Even though the book's language tends to be fairly casual and conversational, it does a great job of introducing true mathematics!&lt;/p&gt;&#xA;" OwnerUserId="2135" LastActivityDate="2014-11-26T01:55:03.363" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5928" PostTypeId="2" ParentId="5896" CreationDate="2014-11-26T21:58:25.553" Score="2" Body="&lt;p&gt;The CME project is working very hard to make this happen more in high schools. Although they haven't published their book yet (coming this winter!), they have an outline of the material on this website:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://mpi.edc.org/resources/linearalgebra&quot; rel=&quot;nofollow&quot;&gt;https://mpi.edc.org/resources/linearalgebra&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take a look! It's a good guide. &lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2014-11-26T21:58:25.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5929" PostTypeId="2" ParentId="5886" CreationDate="2014-11-26T23:45:42.843" Score="2" Body="&lt;p&gt;lots of great answers! figured i'd contribute for anyone who finds this in the future,  I did something similar to @JosephORourkes answer today and it went great!  I also have found that &lt;a href=&quot;https://en.wikipedia.org/wiki/Tangram&quot; rel=&quot;nofollow&quot;&gt;Tangrams&lt;/a&gt; are something that my students enjoy and they can even be played as a competitive game if they wanted by racing another student to get the correct configuration.  Opens up an interesting conversation on the tangram paradox, history, and westernization of the puzzle.  &lt;a href=&quot;http://math.about.com/od/tangrams/&quot; rel=&quot;nofollow&quot;&gt;Here&lt;/a&gt; are some tangram resources and many more printable puzzles can be found via google.  I have a few sets of nice plastic ones but in a pinch cutting up some old folders works just as well.&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2014-11-26T23:45:42.843" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5932" PostTypeId="1" CreationDate="2014-11-28T14:48:56.113" Score="7" ViewCount="172" Body="&lt;p&gt;I'm working with a number of students who have an interest in programming, and who represent a wide range of numeracy skills. I aim to exploit their interest in programming in order to 'trick' them into some mathematical thinking, with the specific hope that lessons can be learned which are relevant to their own mathematical skill set and academic troubles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a few exercises that I've had success with, which I will share as an answer later, but I'm sure that there will be some good suggestions.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2024-03-07T19:43:06.287" Title="Numerical / Computational problems suitable for grade 9-12 audience" Tags="|secondary-education|examples|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5933" PostTypeId="2" ParentId="5932" CreationDate="2014-11-28T15:28:36.313" Score="6" Body="&lt;p&gt;A few suggestions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Have them practice what I call &amp;quot;function deconstruction and reconstruction&amp;quot; in which they are given a mathematical function (conversion between Fahrenheit and Celcius, quadratic formula, etc) and have them code a separate function for each individual step of calculating the equation.  Then in their &lt;em&gt;main&lt;/em&gt; function (whichever language it may be) have them attempt to combine all of their functions so that the program actually performs the desired output.  It is also useful to have them play around with the order of function calls etc. to explore the order of operations and how it effects the output.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give them code (preferably one that calculates/evaluates something as per your learning goal) which has some redundancies in it and ask them to &amp;quot;optimize&amp;quot; it, i.e. take out the redundancies.  This will require them to not only understand the code and how it relates to the math, but then to analyze where the math is being extraneous.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give them the challenge to come up with as many ways as possible to calculate the Fibonacci sequence; a great way to get students thinking about sequences, recursive definitions, iteration, and multiple approaches to the same problem. Once they do it, you can have them seed their functions with different numbers to see how the sequences will differ.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give them code that calculates an equation/formula and have them &amp;quot;decode&amp;quot; the program and come up with what equation/formula it calculates; a great way to promote &amp;quot;thinking backwards&amp;quot; and also practice translating between code and maths.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have them diagram a program by showing functions/blocks as boxes that are linked with arrows representing function calls/input/output. At first this sounds not math related but I have found that it not only gives them a better understanding of the functional aspect of coding but it also lets them practice diagramming skills which are incredibly helpful when solving math problems, especially word problems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These are just a few that I have tried/thought about trying.  If I come up with more I will definitely edit my answer. I hope these help!&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="7365" LastEditDate="2024-03-07T19:43:06.287" LastActivityDate="2024-03-07T19:43:06.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5934" PostTypeId="2" ParentId="5932" CreationDate="2014-11-29T18:45:15.420" Score="2" Body="&lt;p&gt;(1) The &lt;a href=&quot;http://en.wikipedia.org/wiki/Collatz_conjecture&quot; rel=&quot;nofollow&quot;&gt;Collatz conjecture&lt;/a&gt;, a.k.a. the $3n+1$ problem, is easy to code, fascinating to explore, and amazing that it is an unsolved problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Generating the prime numbers upto $n$ is a good programming exercise, with&#xA;many possible solutions. Maybe they'll discover &lt;a href=&quot;http://en.wikipedia.org/wiki/Generating_primes#Prime_sieves&quot; rel=&quot;nofollow&quot;&gt;prime sieves&lt;/a&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) Generating strings of $2n$ balanced parentheses is more challenging, but can be explored&#xA;recursively, e.g., with &lt;a href=&quot;http://code.activestate.com/recipes/578458-generating-balanced-parenthesis/&quot; rel=&quot;nofollow&quot;&gt;this code&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;def pargen(left,right,string):&#xA;    if(left==0 and right==0):&#xA;        print string;&#xA;    if(left&amp;gt;0):&#xA;        pargen(left-1,right+1,string+'(');&#xA;    if(right&amp;gt;0):&#xA;        pargen(left,right-1,string+')');&#xA;pargen(3,0,'');&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Eventually they may discover the &lt;a href=&quot;http://en.wikipedia.org/wiki/Catalan_number&quot; rel=&quot;nofollow&quot;&gt;Catalan numbers&lt;/a&gt;.&#xA;E.g., there are $5$ strings with $n=3$ left parens.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-11-29T21:18:57.103" LastActivityDate="2014-11-29T21:18:57.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5937" PostTypeId="1" CreationDate="2014-11-30T23:13:16.070" Score="2" ViewCount="366" Body="&lt;p&gt;I am currently studying Calculus and there are some topics giving me some difficulty, such as the epsilon-delta definition of a limit. I searched the Internet and found that this epsilon-delta definition is not particularly important in Calculus study, but more so in Analysis, and since my focus is primarily in Calculus, I could skip it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is &lt;strong&gt;what other topics in Calculus books (differential and integral) could I skip?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit: 1/7/15&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I kept searching and found this &lt;a href=&quot;https://math.stackexchange.com/q/818700/131893&quot;&gt;question&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/a/827512/131893&quot;&gt;answer&lt;/a&gt; on Mathematics Stack Exchange that talks about what topics could be studied in Real analysis, Complex analysis, and differential equations. These topics are ordered which is helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My new request is for suggestions based on the links provided. I am unable to contact the writer of the posted answer. If you could also provide What books would be helpful for any suggestions, it would be greatly appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please keep in mind that I am a beginner and I'm looking for something that provides gradual increases in difficulty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="2181" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-01-07T08:51:57.387" Title="Self study question in calculus" Tags="|calculus|self-learning|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5938" PostTypeId="1" CreationDate="2014-12-01T16:52:02.390" Score="12" ViewCount="464" Body="&lt;p&gt;I'm currently teaching a small section of calculus (~35 students). As the semester comes to a close, I'm looking for an effective way to use one or two classes to help my students review for the final exam.  In the past, I've used the time as a question/answer period, but this doesn't seem to accomplish anything that couldn't be done in office hours.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have an effective way to use class time to review for a final exam?  Answers specific to calculus are most welcome, but I imagine there are good generic answers as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for your responses.&lt;/p&gt;&#xA;" OwnerUserId="3443" LastActivityDate="2014-12-02T22:15:59.583" Title="Effective Final Review" Tags="|calculus|review|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5939" PostTypeId="2" ParentId="5938" CreationDate="2014-12-01T18:09:31.683" Score="5" Body="&lt;p&gt;In my calculus class, I give a practice test that covers (almost) the same material as the test. I give some class time to do the practice test but they largely do it in their own time. In the class before the test I discuss the practice test. The students know this is the last time for them to ask any questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This system isn't perfect, but what is? It has worked pretty well for me.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2014-12-01T18:09:31.683" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5940" PostTypeId="1" CreationDate="2014-12-01T19:28:36.347" Score="10" ViewCount="209" Body="&lt;p&gt;In this semester I've realized that many of the problems (my) students have can be solved by reading better. The most recent example I've encountered was in the last exam; I asked them the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given the following set $X=\{a,b,c,d,e\}$, find how many subsets with $r$ elements are in $X$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What many of them did was to find how many subsets there are in $X$, regardless of the number of elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I figured that maybe if I made some reading exercises, they could actually worry about the maths and not lose points because of things like this. Is this something accurate/doable? where can I find good books for this?&lt;/p&gt;&#xA;" OwnerUserId="99" LastEditorUserId="11" LastEditDate="2014-12-02T15:40:22.813" LastActivityDate="2014-12-02T15:40:22.813" Title="How to read better?" Tags="|mathematical-pedagogy|student-motivation|assessment|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5941" PostTypeId="2" ParentId="5938" CreationDate="2014-12-01T22:11:29.430" Score="1" Body="&lt;p&gt;I will often give two review sessions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first session, I will reorganise the content of a particular section of the course into an order more useful for review. I tell the students that often we learn things in a certain order because it's the way our brain has to learn them, or because that's how the topic is built up axiomatically. However it's often more useful to &lt;em&gt;remember&lt;/em&gt; it in a different order. So I will talk through the ideas in the topic  not necessarily in chronological order, and highlight the connections between them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the second session, I will actually do a question and answer session. I take requests about anything in the course, including questions from past exams or assignments. [If there aren't past exams available I would recommend (like @celeriko) &lt;em&gt;giving&lt;/em&gt; them a practice exam to do in their own time.] I encourage students to email me questions they have beforehand, so I can prepare and also so I can give them at least an indication of at least one thing that will be covered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I'd like to deal with your comment: &quot;In the past, I've used the time as a question/answer period, but this doesn't seem to accomplish anything that couldn't be done in office hours.&quot;  There is at least one thing a group question/answer period can accomplish that can't be done in office hours -- students will be able to hear the answers to other students' questions! Some students can't think clearly enough about an issue to have a proper question to ask. These students can benefit from hearing the answers to questions other students ask. Even students who have a clear enough picture to have questions to ask will hear answers to questions they never thought to ask. (Of course it's still a good idea to have office hours as well.)&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-12-01T22:11:29.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5942" PostTypeId="2" ParentId="5938" CreationDate="2014-12-01T23:51:57.500" Score="5" Body="&lt;p&gt;I spend one entire class period with the students working in groups on a massive stack of problems. More problems than is really reasonable to complete in the time allotted. In this &quot;review mode,&quot; certain kinds of problems are more effective than usual.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One type: where the problem is easy once you identify the correct strategy, but the correct strategy is elusive. Often students struggle on exams because they can't distinguish two fundamentally distinct questions, and this is a good time to delve into the issue. For example &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1a. Find $\int_{-5}^5 25 - x^2 \, \mathrm{dx}$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1b. Find $\int_{-5}^5 \sqrt{25 - x^2} \, \mathrm{dx}$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1c. Find $\int_{-5}^5 x\sqrt{25 - x^2} \, \mathrm{dx}$ &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1d. Find $\int_{-5}^5 (\sin x)(25 - x^2) \, \mathrm{dx}$ &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Hint: (1d) is only reasonable to do in one way, and it doesn't require any computation. (1c) is actually doable by the same method, although there are other options.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Second type: can also use up your elegant &quot;trick&quot; questions on these review packets since they are legitimately a learning exercise, so none of the negative feelings of the trick question happen but you get all of the educational benefit. For example, in an algebra class, I can give this on a review day (but maybe not on an exam, although the ideas under the trick question will appear on the exam):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;2a. Solve for $x$: $x^2 = 81$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;2b. Solve for $x$: $\sqrt{x} = -3$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Hint: neither answer is simply &quot;$x = 9$&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-12-01T23:51:57.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5943" PostTypeId="2" ParentId="5937" CreationDate="2014-12-02T07:59:09.910" Score="2" Body="&lt;p&gt;I wouldn't recommend skipping anything. If you've found a calculus book with good reviews, it's likely that it's set up in such a way that it provides an organic progression between topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I somewhat disagree that the epsilon delta definition of limits should be skipped while learning calculus. While it's not immediately useful in solving limits (since there are other techniques), it does provide valuable insight into how we go about dealing with infinitesimal change.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This ends up providing you with some useful intuition when progressing further into differential/integral calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my bottom line is; learn limits the &quot;simplified&quot; way first. &lt;em&gt;Then&lt;/em&gt; learn epsilon delta. Learning the simplified way first might help you digest the latter more easily.&lt;/p&gt;&#xA;" OwnerUserId="796" LastActivityDate="2014-12-02T07:59:09.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5944" PostTypeId="1" CreationDate="2014-12-02T09:39:35.500" Score="5" ViewCount="176" Body="&lt;p&gt;Dip a wire frame into a bubble mix and you have a bubble forming a &lt;a href=&quot;http://en.wikipedia.org/wiki/Minimal_surface&quot; rel=&quot;nofollow&quot;&gt;minimal surface&lt;/a&gt; spanning the wire frame. These bubble surfaces have been used to demonstrate math and physics concepts like &lt;a href=&quot;http://en.wikipedia.org/wiki/Surface&quot; rel=&quot;nofollow&quot;&gt;topological surfaces&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Steiner_tree_problem&quot; rel=&quot;nofollow&quot;&gt;Steiner trees&lt;/a&gt; and of course minimal surfaces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to do a demonstration on making &lt;a href=&quot;http://en.wikipedia.org/wiki/Seifert_surface&quot; rel=&quot;nofollow&quot;&gt;Seifert surfaces&lt;/a&gt; using wireframe knots. How can I solidify such a surface? Ideally I'd like students to dip a wireframe knot in, take it out with the bubble surface, and then do something to the surface to solidify it. This will allow students to rotate it around to consolidate their spatial understanding of the surface, and refer back to it constantly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can it be &lt;strong&gt;frozen&lt;/strong&gt;? I've looked up freezing bubbles with dry ice but that causes the bubbles to burst. How about liquid nitrogen?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or could there be some &lt;strong&gt;special bubble mix&lt;/strong&gt; that allows me to do this kind of thing?&lt;/p&gt;&#xA;" OwnerUserId="3447" LastEditorUserId="3447" LastEditDate="2014-12-05T00:37:27.187" LastActivityDate="2014-12-06T10:36:39.807" Title="Solidifying Bubble Surfaces" Tags="|visual-proof|topology|manipulatives|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5945" PostTypeId="2" ParentId="5940" CreationDate="2014-12-02T14:10:27.180" Score="6" Body="&lt;p&gt;I'd like to enthusiastically reinforce celeriko's comment above. It seems that students often don't pick up on the fact that really &quot;juicing&quot; something basic is what allows for fluent thought about later mathematics. &lt;strong&gt;We should build assignments that aim to teach students to slow down and read carefully to process what is being said. We should encourage them to work out simple examples and to modify hypotheses, to find easier questions and harder ones.&lt;/strong&gt; I suggest that for assignments like these it should be made very clear that the point is not to get an answer but to build a solid mental picture that one can fluently think and talk about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a &lt;a href=&quot;http://rs.io/bill-thurston-reading-hard-things/&quot;&gt;quote&lt;/a&gt; from Bill Thurston: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I was really amazed by my first encounters with serious mathematics textbooks. I was very interested and impressed by the quality of the reasoning, but it was quite hard to stay alert and focused. After a few experiences of reading a few pages only to discover that I really had no idea what I’d just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;—Bill Thurston, from the foreword to Teichmüller Theory and Applications to Geometry, Topology, and Dynamics&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2014-12-02T14:10:27.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5946" PostTypeId="2" ParentId="5938" CreationDate="2014-12-02T22:15:59.583" Score="1" Body="&lt;p&gt;This may be unpopular, but what exactly is the purpose of giving a final? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On my tests, I do sections that spiral prior knowledge so students have to still study those throughout the semester. So by the time the semester is over, I've assessed them multiple times on skills and conceptual understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe final exams cause unnecessary stress not only on students, but on professors as well.&lt;/p&gt;&#xA;" OwnerUserId="625" LastActivityDate="2014-12-02T22:15:59.583" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5947" PostTypeId="2" ParentId="5648" CreationDate="2014-12-03T15:34:57.087" Score="7" Body="&lt;p&gt;Having recently covered this topic in a course for pre-service elementary school teachers, I thought I would write a bit about the somewhat subtle difficulties entailed in tackling this question. I am going to use language that may be at the level of undergraduate majors in mathematics or mathematics education, but I believe that the content can be scaled so that it can be implemented at the elementary level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answer is to some extent a re-hashing of other responses here; however, I am confident that multiple presentations of similar material can still be valuable in developing our own understandings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As the OP writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I feel like the discussion ultimately breaks down into one in Group Theory with different sets and operators.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In a related spirit, I will list below four possible difficulties (phrased as questions).&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;When we use a binary operator (in this case, division) in elementary school, what are the associated sets under discussion (and what do we need to know about binary operators)?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If our ultimate goal (in elementary school mathematics) is to come to a consensus that $a\div0$ is &lt;em&gt;undefined&lt;/em&gt;, then how will we explain what we mean by this word (undefined)?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If we are working with the rational numbers, then what background knowledge do students need?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What are the different &lt;em&gt;interpretations&lt;/em&gt; of division, and how can we use them to make sense of an expression such as $a\div0$?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In thinking about 1, we might recall earlier discussions of the binary operator of addition and the set of whole numbers. In particular, we can take as input any two whole numbers (let us say order matters, though we would show at some point that addition is commutative), and give as output a unique whole number. The &lt;em&gt;uniqueness&lt;/em&gt; here corresponds to the concept covered in an undergraduate mathematics course as &lt;em&gt;well-definedness&lt;/em&gt;. For early mathematics education, even this feature is not obvious for addition: Consider an interpretation of addition that involves combining and counting discrete objects, and note that (cf. &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/5869/262&quot;&gt;MESE 5866&lt;/a&gt;&lt;/strong&gt; on counting) &quot;children sometimes believe the same collection can be characterized by two or more numbers; yes it has 14, and it also has 15!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Subtraction presents problems of its own, because, unlike the above case, we may take as input two whole numbers in a particular order, but the output might &lt;em&gt;not&lt;/em&gt; be a whole number. For example, if we are given the whole numbers $3$ and $4$, then we find that $3-4$ is not a whole number. How does one deal with such a scenario? The mathematical temptation might be to extend to the integers; early on in one's mathematics education, though, the simplest subtraction problems (i.e., those that involve an action: &quot;take away/separate problems&quot;) are only posed if the minuend is greater than the subtrahend. We do not begin our discussion of subtraction by showing a child $3$ apples and asking that s/he take away $4$ apples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These brief remarks about subtraction relate back to 2; namely, if we are talking about subtraction strictly using whole numbers, then an expression that cannot be evaluated as a whole number is said to be &lt;em&gt;undefined&lt;/em&gt;. It is not that $3-4$ is &lt;em&gt;secretly negative one&lt;/em&gt;; rather, we have not defined what happens when the minuend is less than the subtrahend, and so any such expression is (theretofore) literally undefined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Briefly, with regard to 3: Doing justice to the (let us say positive) rational numbers is nontrivial, for it requires students to grasp the fractions (ordered pairs of a whole number and non-zero whole number) as well as their standard equivalence relation and the corresponding set of equivalence classes (cf. &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/1449/262&quot;&gt;MESE 1447&lt;/a&gt;&lt;/strong&gt;). Therefore, discussed below is only the case of why a whole number divided by zero is said to be &lt;em&gt;undefined&lt;/em&gt; in elementary school mathematics; similar reasoning can be extended when one wishes to discuss more generally the division of rational numbers, but is not pursued here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More precisely: Just as subtraction problems are initially presented only when the minuend is greater than the subtrahend, division problems for whole numbers are initially presented only when the remainder is zero. Sometimes a discussion of remainders is used as scaffolding in helping to deepen students' understanding of division; one must be careful, though, for the notation involved in this endeavor can be misleading. E.g., $4\div3 = 1R1$ and $3\div2 = 1R1$, but we would not wish to conclude that $4\div3 = 3\div2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to 4, there are essentially three different interpretations of division. These are sometimes referred to as: the partitive (equal sharing), quotative (measurement AKA repeated subtraction), and missing factors interpretations. The former two interpretations are both discussed in &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/5663/262&quot;&gt;JPB's answer&lt;/a&gt;&lt;/strong&gt;; the missing factors interpretation states that $a\div b = c$ &lt;em&gt;means&lt;/em&gt; there is a unique $c$ for which $a = b \times c$ (cf. &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/5648/dividing-by-zero#comment10401_5648&quot;&gt;my comment&lt;/a&gt;&lt;/strong&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer the OP's question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How do/would you explain why division by zero does not produce a result?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Recall that in the notation $a\div b = c$ we call $a$ the &lt;em&gt;dividend&lt;/em&gt;, $b$ the &lt;em&gt;divisor&lt;/em&gt;, and $c$ the &lt;em&gt;quotient&lt;/em&gt;. In both the equal sharing and measurement interpretations, the dividend refers to the total amount of objects. In the equal sharing interpretation, the divisor is the number of (equal sized) groups of objects, and the quotient is the number of objects in each group. In the measurement interpretation, these meanings switch: The divisor is the number of objects in each group, and the quotient is the number of (equal sized) groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To resolve a question such as $a\div0$, it may be wise to begin with a few other questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use each of the three interpretations to explain why $6\div2 = 3$. For the equal sharing and measurement interpretations, what might the corresponding pictures look like? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;strong&gt;Sketch&lt;/strong&gt;: In equal sharing, we could begin by drawing two circles to represent the meaning of the divisor; then we would alternate putting one dot in each group until all six were used up. At the end of this process, there would be three dots in each group, which tells us that the quotient - i.e., the number of objects in each group - is three. &lt;strong&gt;Alternatively&lt;/strong&gt;, using measurement, we could begin by drawing six dots to represent the dividend. Next, we note that the divisor tells us there are two dots in each group, so we begin to draw circles around the dots, two at a time. At the end of this process, there would be three groups of two dots, which tells us that the quotient - i.e., the number of equal sized groups - is three.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the missing factors interpretation, $6\div2 = 3$ means that $6 = 2 \times 3$, which is a true number sentence. Moreover, we ought to observe that $3$ is the &lt;em&gt;unique&lt;/em&gt; whole number that, when multiplied by $2$, gives $6$. A smaller number multiplied by $2$ will be less than $6$, and a bigger number multiplied by $2$ will be greater than $6$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From here, we can segue into a discussion of $0\div6$ and $6\div0$; then, finally, $0\div0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that you will find all three interpretations can be readily applied (as in the parenthetical example using $6\div2$ above) to each of the former two scenarios. The most difficult case to discuss is that of $0\div0$; again, pushing students to explain what is &lt;em&gt;meant&lt;/em&gt; in each case (as suggested by JPB) will allow them to understand why dividing by zero (in the context of whole numbers) does not make sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In such a discussion, I would emphasize that the output is supposed to be a unique whole number. Without this attention to binary operators and underlying sets, one difficulty you might run into is that students believe, e.g., $6\div0$ is undefined (there is no whole number that, when multiplied by zero, gives six) whereas $0\div 0$ is &quot;all numbers&quot; (because any number, when multiplied by zero, gives zero). I believe that the latter remark ought not to be viewed simply as a misunderstanding, but rather as an authentic effort to make sense of the mathematics. However, it asserts that the answer is a(n infinite set) of whole numbers, whereas we have required (by &lt;em&gt;definition&lt;/em&gt;) that any admissible answer be a single, unique whole number. Bearing this in mind, one should be able to convince elementary school students (better: have students convince themselves) why it does not make sense to divide a whole number by zero.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-12-03T15:34:57.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5948" PostTypeId="1" AcceptedAnswerId="5951" CreationDate="2014-12-03T15:53:54.770" Score="21" ViewCount="10143" Body="&lt;p&gt;How does one explain, clearly and simply, that we live in a three-dimensional world? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The explanation has to be understandable for a twelve year old child.&lt;/p&gt;&#xA;" OwnerUserId="3412" LastEditorUserId="262" LastEditDate="2014-12-04T08:31:02.223" LastActivityDate="2018-06-07T14:27:52.277" Title="How to explain that we live in a three-dimensional world?" Tags="|secondary-education|concept-motivation|geometry|physical-sciences|" AnswerCount="12" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="5949" PostTypeId="1" CreationDate="2014-12-03T16:01:58.767" Score="8" ViewCount="184" Body="&lt;p&gt;A &lt;em&gt;test statistic&lt;/em&gt; is a measure of significance of a data set with respect to a claim.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My statistics textbook is insistent that the students should be able to interpret the test statistic in two different ways:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Convert the test statistic to a P-value, then decide whether that P-value is sufficiently low.&lt;/li&gt;&#xA;&lt;li&gt;Find critical values, then decide whether the test statistic is more extreme than the critical values.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My understanding of how statistics is used &quot;in the real world&quot; is that researchers always convert their data and hypothesis into a P-value, then go from there. This is possible because we have technology that allows us to easily convert test statistics of all types into P-values.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am tempted to stop discussing critical values entirely. If the students have access to the technology to turn everything into a P-value, why should we make them think about critical values? Does this topic have educational value, or is it simply a relic of a less-technological time, when tables of statistical values were the only way to operate?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-12-04T03:55:05.147" Title="In introductory statistics hypothesis testing, why not always use P-values?" Tags="|mathematical-pedagogy|technology-in-education|statistics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5950" PostTypeId="2" ParentId="5948" CreationDate="2014-12-03T16:21:08.857" Score="8" Body="&lt;p&gt;Unfortunately, dimensionality is a very tough concept to truly understand, even for adults. I know that you are trying to find a clear and simple method, which what I have typed below is certainly not.  My hope is you can use the information below to get ideas on how to develop your own scaffolded method for your student.  What I would suggest would be a scaffolded approach by which you start with the very basics of dimensionality, showing how to &quot;construct&quot; higher dimensions using objects of a lower dimension, and progressing on once the student demonstrates they understand the previous dimension.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;0th Dimension&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Individual points exist as objects with 0 dimensions.  They do not have a size, they do not have a length/width/height.  They merely exist at one specific spot in space.  Points do not express dimensionality because they cannot be traversed or &quot;moved through&quot; in anyway.  &quot;Moving through&quot; a point would just lead to another, wholly different point. Make sure to enforce the fact that the &quot;points&quot; that we draw and talk about are not true points because the graphite/ink they are drawn with has (albeit very minuscule) length, width, and height&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To move from the 0th dimension to the 1st dimension, one need only draw any two points and connect them with a line.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1st Dimension&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;True lines (linear/straight NOT curved or bending) exist as objects with 1 dimension, which we can call &quot;length&quot;.  &quot;Moving through&quot; a line can exist in only one way, along the length of the line.  Trying to move somewhere not along the line will take you out of that one dimension because you will no longer be on the line.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To move from the 1st dimension to the 2nd dimension, one need only draw two lines and connect them with additional lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2nd Dimension&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Polygons, circles, shapes all exist as objects with 2 dimensions.  they still have the same first dimension as lines, &quot;length&quot;, but they now have a second dimension, which we can call &quot;width&quot;.  Having two dimensions now allows us to move in two distinct directions, &quot;left/right&quot; and &quot;up/down&quot;.  If you draw a shape on a piece of paper you can see that you can move anywhere inside that shape or along the edges, and still only be in two dimensions.  What you CANNOT do is raise off of the paper, because the shape does not exist above the paper, it only exists on the paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To move from the 2nd dimension to the 3rd dimension is a little tricker because it cannot be done on a single sheet of paper.  A simple way to get around this is to draw two squares on two separate pieces of paper and hold one above the other.  In the same way as before, we can draw two, 2D objects and &quot;connect&quot; them between the two pieces of paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3rd Dimension&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Pretty much everything that we interact with daily are objects that exist in 3 dimensions.  They contain the same first dimension as lines, &quot;length&quot;, the same second dimension as a shape, &quot;width&quot;, but now they have &lt;em&gt;risen off the paper&lt;/em&gt;, so to say, into a third dimension, which we can call &quot;height&quot;.  You can verify this by standing up and stepping to the left/right (1st Dimension), stepping forward/backward (2nd Dimension), and jumping up and down (3rd Dimension).  This can also be verified by picking up a book or a box and seeing that it, in fact, has three, distinct, dimensions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I hope this helps some, this is the way that dimensionality was explained to me when I was younger.  It also made it very intuitive for me to understand higher dimensions (4th, 5th, etc).  Another idea might be to have your student try to imagine life as a 2D object (a square on a piece of paper, etc) and have them think about what life would be like, ala the novel &lt;em&gt;Flatland&lt;/em&gt;, good luck!&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2014-12-03T16:27:00.967" LastActivityDate="2014-12-03T16:27:00.967" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5951" PostTypeId="2" ParentId="5948" CreationDate="2014-12-03T17:03:37.587" Score="9" Body="&lt;p&gt;As celeriko took the what I consider as what &quot;ambient&quot; space the object lies in approach (I.e., Whittney's embedding), I'll consider the degrees of freedom of movement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the normal directions you would have someone move when they are standing straight up. Have the child walk back and forth in a straight line. We can consider this the forward-backward freedom as the first dimension. Then have them shuffle sideways left or right without turning. We can consider this the left-right freedom as the second dimension. Put together, we get the usual ability to walk around on the surface of the planet (a 2-manifold!). But what if we jump in a rocket or dig through the planet? Or even if we just moved up or down in space? We can call that the up-down freedom giving us the usual third dimension. If you want to get crazy, have her stand still for 5 minutes (or like 30 seconds, I couldn't  keep still that long). While you are not moving, time slides forwards. While you can't go back in time (as far as we know!), many people consider this to be the fourth dimension!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It'll be fun and give you a chance to run around a bit while exploring dimensions. Also you can get into a little bit of coordinate systems (move forward 10 feet and left 3 feet!). &lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-12-03T17:03:37.587" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5952" PostTypeId="2" ParentId="5949" CreationDate="2014-12-03T17:17:17.063" Score="2" Body="&lt;p&gt;When I learned statistics at university, the people teaching me made direct reference to p-values even when we were using critical values. The process of using critical values was something like this (for a significance level of 0.05):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Calculate the test statistic, suppose it's called $t$.&lt;/li&gt;&#xA;&lt;li&gt;Look up the critical value of $t^\ast$ in the table. This is the test statistic that would have given me a p-value of exactly 0.05.&lt;/li&gt;&#xA;&lt;li&gt;My test statistic is beyond $t^\ast$, so my p-value must be less than 0.05. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The key part was that final step. We would never just say &quot;my test statistic is beyond $t^\ast$&quot;. We would always include the last part where we say &quot;so p-value &amp;lt; 0.05&quot;. We never only compared our test statistics to critical values without also referring to p-values.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some may argue that this is making it more confusing, because we are introducing an extra step in the reasoning. I disagree. I think it is better to have one more step in the reasoning than a whole separate approach which is listed as a separate approach. Indeed, students in my Maths Learning Centre are often confused by having two approaches and this explanation usually allows them to understand what's going on by unifying the two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, I was not technically taught to use the above approach at all, but instead to &lt;em&gt;find the p-value as accurately as you can&lt;/em&gt; and then make your decision. If you were using the F-table you could usually only narrow your p-value down to &quot;less than 0.05&quot; or &quot;more than 0.05&quot;. If you were using the t-table you could get a bit more accurate and get it to &quot;between 0.05 and 0.01&quot;. If you were using the z-table you could be more accurate again. And if you used software you could get it to 15 decimal place accuracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the &lt;em&gt;one&lt;/em&gt; approach I use is this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Calculate the test statistic.&lt;/li&gt;&#xA;&lt;li&gt;Find the p-value as accurately as you can. If using tables, the best you can usually do is compare to the critical values to get a range for the size of the p-value.&lt;/li&gt;&#xA;&lt;li&gt;Based on the size of the p-value relative to the significance level, make your decision.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It is worth noting that in some modern uses of statistics (eg testing for equivalence), the concept of p-value is difficult to tie down and so they talk about &quot;rejection regions&quot;. Basically, if the test statistic is in a certain region then you will reject the null hypothesis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In line with this, one course at my university does use two methods, but they use the &quot;p-value method&quot; and the &quot;rejection region method&quot;. Instead of getting students to refer to simply refer to the size of the test statistic relative to the critical value, they get the students to define the rejection region, and then say whether the test statistic is in it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally feel it is better for beginning students to only be expected to use one approach rather than two. It's possibly a good idea to mention the idea of rejection regions, but then always expect them to refer to p-values in your course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, you do still need the terminology of critical value for when you construct confidence intervals, but again I refer to the critical value for 0.05 as &quot;the test statistic that would have given you a p-value of exactly 0.05&quot;. So it's still p-values really.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-12-03T17:17:17.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5953" PostTypeId="2" ParentId="5948" CreationDate="2014-12-03T17:33:56.707" Score="3" Body="&lt;p&gt;I have no experience with this age group, so this suggestion might be bonkers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One reason it might be hard about dimensions is that there are so few examples (point, line, plane, 3-space is usually it), so it is hard to see exactly what property of these examples you are &quot;abstracting&quot; with the number &quot;dimension&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to rectify this dearth of examples, is to give a bunch of interesting &quot;nonspatial&quot; examples of spaces, and try to figure out their dimensions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The configuration space of a robot arm with two ball joints (4 dimensional, isomorphic to $S^2 \times S^2$), versus one with two fixed hinges (2 dimensional, isomorphic to $S^1 \times S^1$)&lt;/li&gt;&#xA;&lt;li&gt;The configuration space of a rod trapped in a sphere with indistinguishable ends (this is the projective plane!) (2 dimensional)&lt;/li&gt;&#xA;&lt;li&gt;A piece of shaded string, where the shade can run between $0$ (black) and $1$ (white).  To build the imagination here, you could talk about how every knot in $\mathbb{R}^4$ can be unknotted.  Anytime you want the string to &quot;pass through itslef&quot;, just adjust the shade. (shaded string is one dimensional, but is embedded in a 4 dimensional space)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I am sure you can come up with many more examples.  This should convey the concept of dimension, and also give a taste of &quot;real mathematics&quot;.  The idea to model a set of physical constraints as a high dimensional &quot;space&quot; can be really captivating!&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2014-12-03T17:33:56.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5954" PostTypeId="1" CreationDate="2014-12-03T19:11:04.787" Score="7" ViewCount="473" Body="&lt;p&gt;I'm currently teaching &quot;Real Analysis I&quot;. I took a midterm exam and one of the questions was to show that there is a unique positive number $r\in\Bbb{R}$ so that $r^2=17$. One of the students tried to answer this question with a similar method that we used in the class for a similar question about the existence of a unique positive number $r\in\Bbb{R}$ so that $r^2=2$. It's based on &quot;&lt;em&gt;Completeness Axiom&lt;/em&gt;&quot; and a set $S$ defined as $S=\{x\in\Bbb{R}:x\ge0 \quad and\quad x^2&amp;lt;17\}$, then trying to show that $(\sup{S})^2=17$. Now this specific student wrote a correct reasoning about $S$ and how it is true that $(\sup{S})^2=17$. But the problem is he didn't defined $S$ in his/her answer! What's your opinion? How should I mark this answer?&lt;/p&gt;&#xA;" OwnerUserId="3450" LastActivityDate="2014-12-18T03:44:42.597" Title="How to mark student's answer, when you realize that he/she just memorized the answer?" Tags="|undergraduate-education|grading|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5955" PostTypeId="2" ParentId="5948" CreationDate="2014-12-03T19:53:59.940" Score="4" Body="&lt;p&gt;You'll want to keep it simple and visual. Use easy, real-world examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say a dimension is similar to a &quot;direction you can move in&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a string you can only move left and right. It has one dimension. On a piece of paper, you can move left and right, but you can also move up and down--two dimensions. In a fish tank you can move up, down, left, right, but now also forward and backward--three dimensions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask the twelve year old to imagine a video game where you can only move left and right, up and down (a side scroller, like Mario). That would be what the world would be like if it was only two dimensional. It would be a cartoon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm seeing that my answer is very similar to Chris C's... Sorry. I'll post it anyway since I made it this far.&lt;/p&gt;&#xA;" OwnerUserId="3452" LastActivityDate="2014-12-03T19:53:59.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5956" PostTypeId="2" ParentId="5948" CreationDate="2014-12-03T20:48:32.543" Score="3" Body="&lt;p&gt;You could use basis vectors as a basis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 2D, because everyone talks about lower dimensions when talking about any dimensions, you can draw two lines at a right angle to each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, by only translating, but not rotating, that group of lines you can get to any other 2D point.&#xA;You may also be able to describe the vectors as &quot;rails&quot; that the object slides along to get to its point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, you can draw a third line coming from the same point. You can show that any two of the lines can get to the same point that the third gets to.&#xA;This allows you to talk about linear independence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With one line, though, there are many many points we can't get to.&#xA;So, the paper is 2D because we need two lines to describe points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you make a 3D set of axis, you can show that in the same way, moving along those lines can get them anywhere they need to go.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A fourth line is redundant, and two lines obviously misses a lot of points, so we can perceive 3D.&lt;/p&gt;&#xA;" OwnerUserId="3455" LastActivityDate="2014-12-03T20:48:32.543" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5957" PostTypeId="2" ParentId="5954" CreationDate="2014-12-04T02:29:57.360" Score="4" Body="&lt;p&gt;Are you completely sure they memorized the answer, or might they understand the original proof just forgot to include the definition of the set? Either way, there is some benefit for them memorizing the solution as I've found that some learning has happened, and they did put in enough effort to memorize it. Anecdotally, I have found proofs that have stumped me at first, but I know I should know their steps. I find that I almost always understand what occurs when I take another look later, either the same semester or even years later since I was more than likely lacking familiarity with the subject that time helps with. Sometimes it is more worthwhile to remember the proof in order to pass to other topics and re-approach them later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I would do is deduct the appropriate number of points you think that not including the set definition is worth. It is critical that they include it, so some deduction might be necessary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you think they just forgot to include it, inviting them into your office as brendansullivan07 suggested to see if they deserve the full points might be a good option, as long as you would do it for other students as well. I would use this method more of a &quot;I think you have the right idea, but just want to check&quot; rather than trying to catch they memorizing.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-12-04T02:29:57.360" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5958" PostTypeId="2" ParentId="5949" CreationDate="2014-12-04T03:55:05.147" Score="1" Body="&lt;p&gt;If we place value in understanding the &lt;em&gt;meaning&lt;/em&gt; of a p-value, it seems almost essential to discuss the test statistic, and hence the corresponding critical value. (The p-value is the probability that the test statistic is as &quot;extreme as it is&quot; assuming the null hypothesis, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you do a good job of explaining to a student the meaning of a p-value without ever mentioning the test statistic? Once you relate the p-value and the test statistic, referring to the critical value is a natural part of the picture. I think the approach of &quot;getting by&quot; without understanding the meaning of a p-value is something more typical of a social science methods course. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2014-12-04T03:55:05.147" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5959" PostTypeId="1" CreationDate="2014-12-04T06:42:09.553" Score="11" ViewCount="1019" Body="&lt;p&gt;High school students have a lot of trouble understanding word problems, and I don't know how to help them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's been too long since I've had trouble understanding usual word problems, so when I read the problem I automatically know what to do and how to do it. This, however, makes for terrible pedagogy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are good strategies to help them develop skill at solving these exercises?&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="47" LastEditDate="2014-12-04T07:16:25.327" LastActivityDate="2014-12-16T12:04:39.763" Title="Teaching how to read and translate word problems" Tags="|secondary-education|mathematical-pedagogy|advising|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5961" PostTypeId="1" AcceptedAnswerId="5967" CreationDate="2014-12-04T10:10:23.497" Score="12" ViewCount="1378" Body="&lt;p&gt;I am looking for some resources which tells someone how to mark a (1st year undergraduate level) maths exam paper. Ideally it would cover the basic stuff, like making an error at the top doesn't affect the marks at the bottom (so long as the question hasn't been overly simplified, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Google-ing has turned up nothing. I could try and write something myself, but time....so little time....... (Also, it would be nice to read something someone else wrote, just to see if how I was taught was in any way standard - such as crossed-ticks means &quot;correct according to an earlier error&quot;).&lt;/p&gt;&#xA;" OwnerUserId="169" LastActivityDate="2023-06-18T06:52:51.080" Title="Resources on how to mark a maths exam" Tags="|undergraduate-education|reference-request|exams|grading|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5962" PostTypeId="2" ParentId="5948" CreationDate="2014-12-04T14:06:57.967" Score="2" Body="&lt;p&gt;It might be helpful to explain that space is popularly thought as being three dimensional because it requires three components for a coordinate to locate any point from a mathematical perspective. Mathematics is a very popular tool for describing the universe in scientific terms, but while useful for study and application, our perception of the universe does not explain why the universe works the way it does. Sometimes the things that we believe must be revised to more closely coincide with reality, so we see that while we do not know precisely how reality works, over time we are coming up with mathematically better approximations to correspond with our observations and experiments.&lt;/p&gt;&#xA;" OwnerUserId="1618" LastActivityDate="2014-12-04T14:06:57.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5963" PostTypeId="2" ParentId="5948" CreationDate="2014-12-04T15:50:38.970" Score="1" Body="&lt;p&gt;Yes, there is an experiment that shows that we live in a 3D world, and not in 2D or 4D world.&#xA;The answer comes from Coulombs law. The force F follows the inverse square law. You can test that by doing an experiment. &#xA;You can prove that if our world had more or less dimensions that force would be impossible to follow the inverse square law. &#xA;For me this is the most straightforward way to convince someone that our world is 3D. It links a hypothesis (is it 3D) with an experiment. &#xA;The same holds for gravity since it also follows the inverse square law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will not go into the maths. &#xA;Im sure that if you search the internet with the words &quot;how would coulombs law, or gravity be in 4D&quot; you will find many things to find your answer. &#xA;The  main idea is that the concepts of flux and curl (basic in maxwells equations) are different in 2D 3D and 4D.&#xA;Those concepts are not that hard to explain in children because you can always show them with basic hydrodynamic examples (water flowing from a hose, or curls in water etc)&lt;/p&gt;&#xA;" OwnerUserId="3463" LastActivityDate="2014-12-04T15:50:38.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5964" PostTypeId="2" ParentId="5961" CreationDate="2014-12-04T16:07:11.823" Score="5" Body="&lt;p&gt;This answer is not a published resource, etc. but below are techniques that me and many of my colleagues use and it seems to work pretty well.  My suggestion is to first go through the test yourself, question by question and determine a &quot;rubric&quot; for each question, by determining what are likely to be the &quot;most popular&quot; methods of solving each problem and assigning point values to each correct step.  For example, if a question was &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find the roots of the following function:  $f(x) = x^2 - 9$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and it was worth 6 points your rubric might look something like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Method A&#xA;&lt;ul&gt;&#xA;&lt;li&gt;2 pts:  Correctly factoring (1 pt for each binomial) &lt;/li&gt;&#xA;&lt;li&gt;2 pts:  Setting factored binomials equal to zero (1 pt for each binomial)&lt;/li&gt;&#xA;&lt;li&gt;2 pts:  Finding the roots by solving each equation (1 pt for each root)&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Method B&#xA;&lt;ul&gt;&#xA;&lt;li&gt;2 pts:  Setting equation equal to zero&lt;/li&gt;&#xA;&lt;li&gt;1 pts:  Isolating the $x^2$ term&lt;/li&gt;&#xA;&lt;li&gt;3 pts:  Correctly taking the square root of both sides and having both +/- 3&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It will not always work out perfectly depending on the problem/methods used to solve the problem but it gets you started on being objective as possible.  What I also recommend is to have a list of &quot;sloppy/low-risk&quot; mistakes and point deductions for each.  By &quot;sloppy/low-risk&quot;, I mean if the student makes this mistake they will receive a deduction but as long as they continued the process correctly, they should not be further penalized.  Common mistakes I apply this rationale to are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;dropping a negative sign (-1 pt)&lt;/li&gt;&#xA;&lt;li&gt;error in rewriting between steps (-1 pt)&lt;/li&gt;&#xA;&lt;li&gt;transposing digits/terms (-1 pt)&lt;/li&gt;&#xA;&lt;li&gt;getting the correct answer somewhere along the way but reporting the wrong answer (-1 pt)&lt;/li&gt;&#xA;&lt;li&gt;not fully simplifying (-1 pt)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This will depend on the test material and the level of expectations you have for your students, but i feel like these are good guidelines.  Another suggestion is to grade problem by problem or page by page, i.e. grade the first problem/page of each students exam, then grade the second problem/page of each students exam, etc.  This helps to keep it fresh in your mind exactly where you took off points for each question because not everything will fit nicely into the rubric you make, making it easier to be more objective.  Once you finish a page, tally up the points gained/lost and write them in the bottom corner of each page, so it is easy to add them all up once the exams are marked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On your question about a standard way to mark papers, to my knowledge, there is no standard way established on how to mark papers. I have seen &quot;subtractive&quot; grading where only deductions are marked and anything not marked is assumed to be full credit.  I personally use &quot;additive&quot; grading where the student starts with a 0 and every point they earn is explicitly marked.  I have also seen teachers just mark right or wrong and put the percentage of credit earned next to any marked wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;hope this helps!  &lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2014-12-04T16:07:11.823" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5965" PostTypeId="2" ParentId="5959" CreationDate="2014-12-04T16:25:58.067" Score="6" Body="&lt;p&gt;I feel your angst, word problems are a consistent weakness across most grade/subject levels.  Learning to decode and solve word problems is a learned skill and so it needs to be practiced.  However, just giving students 50 word problems and telling them to solve them for homework wont help anyone. I'd suggest to focus on each of the following items individually, guiding students at first and then slowly letting them try on their own.  Once they begin to pick them up, you can start having them combine the different techniques, again with guidance at first and gradual release.   &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;decoding &quot;buzz words&quot; (sum, difference, product, each, etc.)&lt;/li&gt;&#xA;&lt;li&gt;underline/highlight important words/numbers&lt;/li&gt;&#xA;&lt;li&gt;crossing out extraneous information&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;draw a picture!&lt;/strong&gt; (one of the most helpful skills in life)&lt;/li&gt;&#xA;&lt;li&gt;translating sentences into equations/expressions&lt;/li&gt;&#xA;&lt;li&gt;underline/highlight what the final question of the word problem is&lt;/li&gt;&#xA;&lt;li&gt;reading word problems (it sounds trivial, but many of my students have trouble even finishing reading certain word problems)&lt;/li&gt;&#xA;&lt;li&gt;make a plan (outline steps of what you need to do, sentence by sentence)&lt;/li&gt;&#xA;&lt;li&gt;make a table (for certain problems, this can really help visualize what is going on)&lt;/li&gt;&#xA;&lt;li&gt;smart solving (if you just spent the previous week calculating slope, there is a good chance the word problem today will have something to do with slope)&lt;/li&gt;&#xA;&lt;li&gt;working backwards (asking yourself &quot;What do I know? What do I need to know? How can I know it?&quot;, one of the most difficult practices for students to develop)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Another area I would focus on, which is not necessarily used in the process of solving a WP, is the aspect of explaining an answer.  If a student can explain their answer with words it proves a certain level of understanding.  One idea is to give them a word problem, let them read through it, then present them with the work of someone else and see if they can explain what this other person did to solve the problem.  In a similar vein, you can give them a worked through solution and have THEM write the word problem.  This gives them a chance to be creative and also practice working backwards.  I hope this helps!  &lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2014-12-04T16:25:58.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5966" PostTypeId="2" ParentId="5948" CreationDate="2014-12-04T18:26:14.563" Score="2" Body="&lt;p&gt;Though not an answer to the original question, here's how you might explain that you live in a &lt;strong&gt;four&lt;/strong&gt; dimensional world, the fourth dimension being time. Maybe this intuition is helpful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you ask someone to meet you, you would never say, &quot;Meet me at the intersection of first and second street, on the third floor&quot;, without also specifying a time. Likewise, you would never specify a time without including a location to meet.&lt;/p&gt;&#xA;" OwnerUserId="1305" LastActivityDate="2014-12-04T18:26:14.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5967" PostTypeId="2" ParentId="5961" CreationDate="2014-12-04T18:41:49.823" Score="7" Body="&lt;p&gt;I don't think there's a &quot;universal&quot; rubric or way to grade. But the people at CollegeBoard (a.k.a AP) think a lot about fair grading rubrics and have developed descent guidelines on how to grade the open response questions on the AP exam. Many of their philosophies translate to regular course-work grading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a few of their resources:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;a href=&quot;http://apcentral.collegeboard.com/apc/public/repository/2011_Calculus_FR_Instruction_Commentary.pdf&quot; rel=&quot;noreferrer&quot;&gt;Commentary on Instructions&lt;/a&gt;&quot; (pdf)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specific scoring guidelines for past problems can be found &lt;a href=&quot;http://apcentral.collegeboard.com/apc/members/exam/exam_information/8031.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I've served as a reader for the exams, they also had general guidelines on how to handle decimal errors, etc. However, this information appears not to be public (or at least not easily found online) and frequently had to be tweaked to suite a given problem.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-12-04T18:41:49.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5968" PostTypeId="2" ParentId="5948" CreationDate="2014-12-04T19:17:54.707" Score="3" Body="&lt;p&gt;If you want to boggle minds, then you could start with some &lt;a href=&quot;http://www.math.cornell.edu/~dwh/papers/crochet/crochet.html&quot; rel=&quot;nofollow&quot;&gt;hyperbolic crochet&lt;/a&gt;, which is only flat in a 4th dimension, and then get your 12-year-olds to understand that the world that they (believe they) live in is simpler than that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or you could play &lt;a href=&quot;http://www.tcm.com/mediaroom/video/244843/Purple-Rose-of-Cairo-The-Movie-Clip-Tom-Baxter.html&quot; rel=&quot;nofollow&quot;&gt;the scene from Woody Allen's The Purple Rose of Cairo&lt;/a&gt;, where a character from a 2-dimensional film comes out of the screen into a 3-dimensional world, and get them to discuss what it would mean to live in world with a different number of dimensions.&lt;/p&gt;&#xA;" OwnerUserId="3464" LastActivityDate="2014-12-04T19:17:54.707" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5969" PostTypeId="2" ParentId="5948" CreationDate="2014-12-04T19:49:16.760" Score="5" Body="&lt;p&gt;All the above good answers aside, I think that this question misses the mark in so far as the real problem is to internalize a good notion of what 'dimension' means. In particular, someone that age who familiar with the science-fiction notion of 'dimensions' referring to parallel universes (eg, traveling between dimensions with a magical vortex or portal) would be in a very rough spot trying to reconcile this notion with the the suggestion that our own world is three dimensional.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try this out as a definition for your student(s):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The number of dimensions of a space is the number of pieces of information it takes to determine the location of something. For our world, you can (relatively) locate something by&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;p&gt;1) turning the head left or right a certain amount&#xA;&lt;p&gt;2) tilting the head up or down a certain amount&#xA;&lt;p&gt;3) naming the distance of the object&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, things can be up or down, left or right, and near or far. Each of those is a dimension in our physical space. Give someone any two of these and locating something is hopeless (or at least difficult/time consuming). Give them all three and they know exactly where it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that this spherical-coordinate, self-centered approach is probably easier for kids to grok in terms of 'number of dimensions' than the notion of us all floating around in a Cartesian 3-space.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2014-12-22T12:14:44.503" LastActivityDate="2014-12-22T12:14:44.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5970" PostTypeId="1" CreationDate="2014-12-04T21:02:38.387" Score="4" ViewCount="182" Body="&lt;p&gt;I am interested in answers to the title question without parentheses, but I found this method below rather interesting, and I am hoping to find it published somewhere, along with a teacher's guide with suggestions on how to present it.  (My current target audience consists of U.S. students aged 10 and younger.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one wants just powers of 2 (or of some number b), one just takes some ruled filler paper (I like college rules with the lines spaces something like 1/3 inch apart), and on (say) the left edge of the paper places 1,2,4, etc on each rule, and copies this labeling to the right edge of a different sheet.  One gets a primitive slide rule to multiply powers of 2 (or of b).  A logarithmic scale is just several of these power scales aligned and intertwined, and one needs a good value of (say) (log b / log 2) to get the intertwining right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the idea.  Take a second sheet of ruled paper and (using b=3 as an example) label the rules somewhere 3^0(=1),3,9,27,... successively.  Now tilt the paper so that the rule  labeled 3^0 intersects the rule on the first sheet labeled 1, with the point of intersection on the edge of the first paper.  Tilt the paper and adjust so that the first&#xA;rule stays fixed at the intersection, but now the second rule labeled 3 intersects the edge between (the rules labeled) 2 and 4.  Continue tweaking to get (the intersection points with the edge of the rules labelled) 9, 27, 81, and so on properly positioned (between the rules labeled with powers of 2).  When the last tweak is done, mark the edge of the first paper with ticks at the points where the rules labeled 3^i intersect the edge.  Now you have the start of a log scale that includes powers of 2 and 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can also add powers of 5 and 7 in an analogous fashion.  When one starts with about 28&#xA;powers of 2 and fills in powers of 3, 5, and 7 in this fashion, one can then transfer marks to the edge of the second ruled paper and start using this to compute marks for non prime powers, most notably 6 and 10.  Compute and add marks as desired.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My search so far reveals some material talking about the basic properties of the log scale and its use as a slide rule.  There is likely material on rational approximation of ratios of logs (and this is what is going on with tilting the second sheet, is coming up with a&#xA;physical measurement of such ratios).  However, I have not seen any thing which starts with&#xA;two sheets of nicely ruled paper and constructs a log scale or a slide rule.  Further, the above method does not require exact computation of powers: knowing that 2^19 is near 520,000 and that 3^12 is near 530,000 suffices for the precision expected with this method and tools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this could be useful in multiplication practice as well as an introduction to exponentiation and its inverse.  Alternately, it could serve as an example for those wanting to learn more about nomographs and other physical calculating devices.  However, it may be that I can't find it because it may be considered too challenging for my intended audience.  So in addition to references, I also ask others here what background/age range/maturity level would be wanted in order to show this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Ask Me About Course Design&quot; Paseman, 2014.12.04 &lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2014-12-05T01:40:59.940" Title="What is a good reference for (this way of) generating a logarithmic scale?" Tags="|reference-request|primary-education|aids|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5971" PostTypeId="2" ParentId="5970" CreationDate="2014-12-05T01:40:59.940" Score="3" Body="&lt;p&gt;This is only slightly an answer, but I've taken a different approach to (I think) the same goal, where I ask students to find log 2, log 3, log 4, and so on on their calculators, then build a slide rule out of it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd argue that most students will not appreciate the elegance of using only lined paper anyway, unless you are working with more mathematically mature students. So using the calculator, even though it abstractly defeats the purpose of the question by using circular reasoning, might still accomplish the same goals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, the goal of this exercise is simply to convince them that &lt;strong&gt;expressions like log 5 represent numbers&lt;/strong&gt; as opposed to arcane symbols. If they get one of the log rules ingrained in their head using the slide rule, that's an added bonus. I'm not sure what your objective is for your activity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've done this only with college freshmen.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2014-12-05T01:40:59.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5972" PostTypeId="1" CreationDate="2014-12-05T06:48:07.330" Score="20" ViewCount="1235" Body="&lt;p&gt;I would like to introduce the concept of a manifold to a high school student. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This person isn't familiar with the axioms of set theory nor topology. However, despite these deficits, I would like to get the general point across that manifolds are useful for studying gravity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of any resources that address the general idea of a manifold for someone with this level of knowledge? &lt;/p&gt;&#xA;" OwnerUserId="3471" LastActivityDate="2014-12-07T01:34:49.237" Title="Teaching manifolds to high schoolers" Tags="|geometry|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5973" PostTypeId="2" ParentId="5959" CreationDate="2014-12-05T06:52:38.967" Score="7" Body="&lt;p&gt;The answer provided by celeriko seems like it hits all of the main points. However, I have found in my experience teaching word problems with students that a few additional strategies are most effective for struggling students.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Rewriting the important mathematical &quot;givens&quot; in a problem. By having students rewrite these ideas, they're taking possession of these parts of the problem, and begin dissecting the problem from a &quot;word problem&quot; to a &quot;regular problem&quot;. It is also helpful for them to have these pieces easily accessible and to not have to dig back into the paragraph of the word problem to use them. Writing things down is also helpful for students who are struggling because it gives them something to do. A common complaint I have heard from students is not knowing how to get started. This gives them something to start doing.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Decoding &quot;buzz words&quot; as celeriko mentions is key. There are other less common buzz words that I think are even more helpful than the ones listed in this other answer. Some of these other are: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;IS:&lt;/strong&gt; This is usually translated into the mathematical symbol $=$ but is also used very often in common English grammar, so it is often glossed over.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;OF:&lt;/strong&gt; Similar to &lt;strong&gt;IS&lt;/strong&gt;, it needs to be emphasized in mathematics as multiplication.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;PER:&lt;/strong&gt; Division, or fractions, with the word or words prior belonging to the numerator and the word(s) after belonging to the denominator.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;PERCENT:&lt;/strong&gt; I discuss this with &lt;strong&gt;PER&lt;/strong&gt; and break down the word into its two roots. &lt;strong&gt;PER&lt;/strong&gt; meaning division, and &lt;strong&gt;CENT&lt;/strong&gt; standing for 100. There are plenty of examples you can use to connect &lt;strong&gt;CENT&lt;/strong&gt; to 100, such as cent or century.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;THAN:&lt;/strong&gt; This is an interesting one since it doesn't actually stand for any operation, but when paired with &lt;strong&gt;LESS&lt;/strong&gt;, like &quot;$5$ less than $9$&quot; actually commutes the order of the elements when translating the word problem into a mathematical expression.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Creating a dialog and discussing problems with students in class is also important. This promotes a common method that many mathematicians use to solve their own problem, which is creating a dialog with themselves (I believe Polya mentioned this, but I could be wrong). In having these discussions, hopefully your students can glean from you some of your thought processes, and you may also discover how you actually think about these problems to enable you to break them down for your students.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-12-05T06:52:38.967" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5974" PostTypeId="2" ParentId="5972" CreationDate="2014-12-05T07:23:41.460" Score="8" Body="&lt;p&gt;Introducing the concept of a manifold in general might be hard, but working with specific examples shouldn't be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea that each point $x$ of a n-manifold $M$ has local coordinates in $\mathbb{R}^n$ (for real manifolds) can be realized with the idea that there are projections from parts of a globe ($S^2$) onto the real plane ($\mathbb{R}^2$). This generally shows that (via stereographic projection, which shouldn't be too hard to visually show with a physical globe) the sphere is a 2-manifold. You can even get into the fact that there are many such projections (different coordinate systems on the manifold) by using different mapping types (&lt;a href=&quot;http://en.wikipedia.org/wiki/Map_projection&quot;&gt;the wiki page has many discussions on the subject&lt;/a&gt;) and even get into the fact that $S^2$ is not $\mathbb{R}^2$ by showing where several types of projections fail and distort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can then diverge into discussions of other common manifolds (ones embedded in $\mathbb{R}^3$ would be best so they can visualize them) such as the Torus, the Mobius band, or even lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That should cover the concept of &quot;what is a manifold&quot; with enough of a concept image that they might appreciate some of the idea of how they interact in the study of gravity. But alas, I am unfamiliar with the subject myself to give voice to that specific example. I would be unsurprised that it would be a challenge to give them any specific mathematics for it besides a &quot;gut feeling&quot; that manifolds are involved. Perhaps using rays of light as geodesics in this coordinate system on the gravity wells would be useful.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-12-05T07:23:41.460" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5975" PostTypeId="2" ParentId="2623" CreationDate="2014-12-05T07:32:33.137" Score="10" Body="&lt;p&gt;This may not directly answer your question, but perhaps it will help. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The brain is not born preprogrammed. It evolves during development (ie conception to adulthood) and gradually adapts to its environment. As such, it is really a misconception that the brain has particular areas that we are born with that must automatically carry out certain functions. Of course, it would be ludicrous to say that there aren't certain distinct regularities. But one must first appreciate just how adaptive brains are. Let me illustrate this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in all mammalian brains, there is a particularly important area called the thalamus. Destruction of this region results in a loss of consciousness. So it essentially controls the flow of sensory information into the various higher order regions. Researchers who studied the thalamus have run experiments with different mammals and found that, if you connect their visual organs to the traditional auditory regions during development, that brain area will actually grow into a visual processing center instead of an auditory one. In other words, it's no longer an auditory processor, but now a visual one. That's extraordinary. It means an area that one would think would be devoted completely to one sense (ie vision) can actually develop into a completely different kind of circuit (ie auditory processing). Thus, it is really inaccurate to say that a particular brain region by necessity has to correspond to a particular function. It all depends on how the system evolves from birth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because most humans grow up with fairly similar sensory inputs and overall wiring patterns, the average human brain tends to share similar patterns of functional circuitry. So it isn't a free for all. Nature gives our brains strong biases and incentives so that there's a tendency to prefer certain circuit relationships. That's why for most people, the auditory circuits don't get mixed up with visual. It's not a free for all, but a guided self-organizing process. So it works out that most of the time, but on occasion you do see irregularities (eg synesthesia). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, as for mathematical areas in particular, in humans, the brain has select areas that tend to participate in mathematical calculation and reasoning. For example, the angular gyrus is one such area. Here's a general article about the different regions: &lt;a href=&quot;http://newsoffice.mit.edu/1999/math-0512&quot; rel=&quot;noreferrer&quot;&gt;http://newsoffice.mit.edu/1999/math-0512&lt;/a&gt; You can read the original papers if you're interested. It's not hard to find general information about math and the brain. However, there are so many kinds of math and types of math thinking, they can't study all of them. So the best you can do is google around and hope someone has done experiments relating to your topic of interest. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me elaborate on this point further.  The brain doesn't just have a &quot;math center&quot;. Quite the contrary. Each type of math may have induce different kinds of firing patterns spread out in different regions. Geometry may induce different patterns than algebra. Algebra may be different from mere arithmetic. For instance, just because I can add 5 + 5 doesn't mean my brain understands the concepts of inverses, commutativity, associativity, etc. So don't assume that all math is processed the same way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, to make meaningful comments about how the brain regions that participate in a blind person, nothing substitutes for real data. I'm not familiar with this topic, so I don't know if there are any studies about math, blindness, and visualization in particular. But I do know that people who are blind from birth have very different wirings from those who go blind after having sight. People who go blind after having seen retain the ability for visual imagination. They can still dream visually and presumably do other things like spatial rotation. This is not true for those who are born blind. In all case though, most blind brains start to channel extra processing through other senses. For instance, they may start to show augmented spatial processing capacities in their auditory cortical areas. Same for haptic (ie touch). To get a sense for why this would happen, turn off your lights in your room at night and you'll immediately start to rely on these additional sense to give you cues. But ---interestingly --- unlike a blind brain, your brain is still getting visual data.  Well, a blind brain realizes that its not getting data and starts to adapt. It says, &quot;what a waste of neurons! I might as well use them for other stuff.&quot; So the circuits may shift some of the functions to other regions like auditory areas. This is similar to how some people develop savant capabilities after brain damage. You might google the term plasticity, although beware since it's an overused term and thus is only so useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps!!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case your main question was about the practical strategies people use, I'll add the following remarks:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since different kinds of math use different brain areas, you might expect people to display deficits associated with the particular senses they have trouble with. So if visualization is a problem, then geometry might be a weakness or harder to learn whereas algebra and numbers might come more easily. I have a friend who is terrible with visualization. Can't even hold up four fingers properly! But she can understand the concept of a vector algebraically. So if someone stinks at algebraic thinking, teach them vectors with arrows. Conversely, if a person can't visualize a vector, introduce it algebraically through axiomatic vector space theory. Math is cool because you can teach a concept many different ways and some of the coolest math isn't visual at all. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also know that business and communications people have argued for a while that its best to target all senses while teaching to an audience through what is called &quot;neuro-linguistic programming&quot; (NLP). You may find this literature provides you with ideas for how to convert a teaching method in one modality into another. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, there are many books on learning disabilities that involve systematic studies on how to help people with them learn best. You might consider consulting that type of literature. It's hit or miss but there are definitely intelligent and systematic people researching practical strategies for helping LDs of all kinds. &lt;/p&gt;&#xA;" OwnerUserId="3471" LastEditorUserId="3471" LastEditDate="2014-12-10T00:32:12.803" LastActivityDate="2014-12-10T00:32:12.803" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5976" PostTypeId="2" ParentId="5972" CreationDate="2014-12-05T15:33:12.793" Score="3" Body="&lt;p&gt;To obtain some heuristic idea about geometry and physics there are many entry-level, mostly non-technical, books. Many of these books are written to popularize string theory for the masses. Whatever your opinion on that matter, the fact remains that geometrically there is something to be gained from reading classics like &lt;strong&gt;Hyperspace&lt;/strong&gt; by Michio Kaku or &lt;strong&gt;Elegant Universe&lt;/strong&gt; by Brian Greene. Both of these make an effort to explain the interplay between General Relativity and the geometry of manifolds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, if they are really, really interested, they could learn something from Susskind:&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=JRZgW1YjCKk&amp;amp;index=10&amp;amp;list=PLpGHT1n4-mAvcXwzOIz3dHnGZaQP1LEib&quot; rel=&quot;nofollow&quot;&gt;Leonard Susskind's General Relativity Lectures&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-12-05T15:33:12.793" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5977" PostTypeId="1" CreationDate="2014-12-05T22:04:43.870" Score="7" ViewCount="470" Body="&lt;p&gt;I'm today trying to construct a lesson on theory and problems of group theory. A specification of this lesson would be to use as much visual representations as possible.&#xA;I have found some trivial ways to represent some basic concepts of group theory but I'm stuck with cartesian product.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I represent the cartesian product visually?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you for your time.&lt;/p&gt;&#xA;" OwnerUserId="3474" LastEditorUserId="1853" LastEditDate="2014-12-17T08:30:21.420" LastActivityDate="2014-12-17T17:09:33.957" Title="Visual representation of Cartesian Product of groups" Tags="|abstract-algebra|visual-proof|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5978" PostTypeId="2" ParentId="5944" CreationDate="2014-12-06T10:36:39.807" Score="1" Body="&lt;p&gt;Usual bubbles can be freezed. See &lt;a href=&quot;https://www.youtube.com/watch?v=cW8ZO7EBRkc&quot; rel=&quot;nofollow&quot;&gt;Bubbles freezing at -26°C&lt;/a&gt;  and &lt;a href=&quot;https://www.youtube.com/watch?v=ddST_7n9peg&quot; rel=&quot;nofollow&quot;&gt;Frozen Bubbles 01-24-11&lt;/a&gt;. According to WikiHow &lt;a href=&quot;http://www.wikihow.com/Make-a-Frozen-Bubble&quot; rel=&quot;nofollow&quot;&gt;no special bubble mix is required&lt;/a&gt;. Obvious idea: use very cold water to minimize the required time.&lt;/p&gt;&#xA;" OwnerUserId="729" LastActivityDate="2014-12-06T10:36:39.807" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5979" PostTypeId="2" ParentId="5972" CreationDate="2014-12-06T22:33:17.480" Score="7" Body="&lt;p&gt;If you try to convey the idea of what a manifold is, and if you do not want to restrict to surfaces (and curves), then at some point you will have to explain what an intrinsic manifold is (as opposed to a submanifold). Many mathematicians working in areas remote from differential geometry are happy with working with submanifolds when they need to, but they are familiar with $\mathbb{R}^n$, $n&amp;gt;3$; a general audience will not fall for that trick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, how to explain intrinsic manifold to an audience with less mathematical knowledge than our respected colleagues?  Chris C already suggested using the sphere as an example; I second him on this, except that: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;rather than stereographic projection, I would use longitude and latitude to show that a position on earth can be defined by two numbers, as it relates to known concepts;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;distortion of maps is a great subject, but it is a metric property, not a differential-topological one, so I would avoid it in a first approach,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;one can use earth to explain intrinsic manifold in a convincing way.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Below, I will first give details about 3. above, and after that I'll give a way to deal with higher dimensions (at least, with dimension 3).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of having your audience vizualizing Earth inside the Euclidean space, let them instead imagine people in very primitive time, before the roundness of earth was known, who are only aware of some relatively small region around them (hey! here is a chart! but don't tell them yet). Clearly, first neglecting hills, mountains and valleys, one would easily consider standing on a Euclidean plane, even without knowing formal Euclidean geometry. It is probably better to avoid discussing too much geometry, as it would lead us far from the core subject; but the following &quot;obvious&quot; property is useful to discuss: if one starts walking straight ahead and never stops, then one gets farther and farther away from its starting point, forever.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, we can appeal to the higher knowledge we share with our audience: the Earth is in fact a sphere! For the sake of the argument, let us forget about slight geometrical distortions as Himalaya and slight inconveniences as oceans, and let's assume Earth is a perfect round sphere. Then the obvious property above is of course wrong: if one walks straight for a sufficiently long time (80 days?), then one gets to its starting point again after a whole turn around the world. Here you are starting to touch the main point: there can be &quot;spaces&quot; (i.e. set of positions one can occupy) where locally everything looks like a plane, but that have different global properties. These spaces are exactly manifolds. Note that the neglected inconveniences above can be taken into account without harming the point too much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point, it is certainly a good idea to propose another example; the torus is the obvious one, but instead of talking about a donut, let's use the classical video game world: a rectangle where moving up at the upper edge makes you beam out at the bottom, and moving right at the right edge makes you bam out at the left, and vice-versa. This is well known to most, and is more intrinsic than the donut surface. Also, you can use the case when one does not see the whole world, but only the small piece that fits the screen, and where there is no feeling of beaming out at any point: one just go straight up and ends at one's starting point after some time. The edges and the beam stuff  are really only a convenience to represent the space on the map screen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ok, now it is pretty straightforward to engage your audience with a three-dimensional manifold: simply picture a huge but finite box in space, enclosing Earth, Sun, etc.,  such that moving through the upper face makes you beam out to the bottom face, etc. But of course, this edges and beam stuff are again only convenient ways to picture this space inside the one we are accustomed with; you have the three-dimensional torus. At this point you can say that more generally, a manifold is simply the global shape of a space (or surface) which is locally just as plain, boring old space (or plane); you could have said that at the very beginning, but it would not have been clear that other examples than plain, boring old space can exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you draw a higher-genus surface in $\mathbb{R}^3$ (these ones are difficult to picture intrinsically), you can easily convince your audience that a lot of different surfaces exist. But after the previous discussion, this leads you to let them imagine that the same should be true in 3D. They may very well be living in a universe that is not shaped as plain, boring old space! Maybe that a rocket sent in a given straight direction would come back to its starting point after some  (very long) time!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also gives you the opportunity to solve a longstanding dilemma. It is difficult to imagine that the universe is infinite, because infinite is difficult to grasp; but on the other hand, it is difficult to imagine that the universe has a boundary, because what would then be on the other side of the boundary? The above discussion, where all examples are closed manifolds, shows that the universe could in principle be finite and without boundary. Making ma able to truly understand that is probably one of the most precious gifts mathematics has given to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next step would be to engage with higher-dimensional spaces and manifolds, but this answer is really long already. I will therefore only point to a reference, which is tremendously clever to explain this: &quot;flatland&quot; by Edwin Abbott Abbott. I vaguely remember that his idea has also been reused in more modern texts.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-12-06T22:33:17.480" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5980" PostTypeId="2" ParentId="5972" CreationDate="2014-12-07T01:34:49.237" Score="2" Body="&lt;p&gt;An accessible book that I found inspiring as a HS student was &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486234002&quot; rel=&quot;nofollow&quot;&gt;Geometry, Relativity and the Fourth Dimension&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your student is especially interested in ideas in relativity, then the geometric concepts here are key, without getting bogged down in the technicalities of defining manifolds more generally.&lt;/p&gt;&#xA;" OwnerUserId="3478" LastActivityDate="2014-12-07T01:34:49.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5981" PostTypeId="1" AcceptedAnswerId="5982" CreationDate="2014-12-07T04:38:52.773" Score="24" ViewCount="2918" Body="&lt;p&gt;A few days ago I had my last discussion session on probability theory as a TA. In the end I asked students to ask me questions as this is the last class. One of the student asked me about the (real) definition of expectation; he said he is confused by &quot;simple facts&quot; like&#xA;$$&#xA;E(c)=c, c\in \mathbb{R}&#xA;$$&#xA;and why the expectation is linear. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Embarrassed that I did not explain this well enough earlier, I tried to explain to him expectation is a form of weighted sum, and for continuous case we are using some kind of probability measure coming from the density function. So the expectation is really some kind of integration. The student seemed to follow at this point. Then I tried to explain that the integration involved this way is not the same as the Riemann integration he learned in calculus classes, which is partly why the expectation of Cauchy distribution does not exist. I drew a graph and showed the Lebesgue integral can be viewed as a kind of &quot;horizontal&quot;-decomposition of the integration area. He asked me a very good question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;What is the benefit of using horizontal instead of vertical? Isn't that the same thing?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did not really know how to answer this appropriately in a short time. I showed him that the horizontal decomposition would involve general (measurable) sets, not rectangles. And I also intuitively defined the outer measure of a set using open boxes. As a practice I showed that $\mathbb{Q}\cap [0,1]$ has zero measure using this definition, and as a result the Dirichlet function has zero integral on the unit interval, while it is impossible to define the Riemann integral rigorously. However, I noticed that by this point he was more or less lost when I showed $m(\mathbb{Q}\cap [0,1])=0$. I told him that to understand it properly he needed to take a year of real analysis classes, and that he should consult a professor I know in my department. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to ask what is a good way to explain the ideas of the Lebesgue integral to a student like him next time without making him/her confused. I later learned that the student did not have a proper proof writing background (like he did not know what it means to be injective and surjective). Since these students constitute the majority of my my classes, I feel obliged to find a way to explain myself better without forcing them to pay a visit to my professor or read a serious textbook. &lt;/p&gt;&#xA;" OwnerUserId="3218" LastEditorUserId="3218" LastEditDate="2014-12-07T18:38:22.410" LastActivityDate="2022-06-14T21:43:02.303" Title="What is a good way to explain the Lebesgue integral to non-math majors?" Tags="|undergraduate-education|concept-motivation|probability|integration|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5982" PostTypeId="2" ParentId="5981" CreationDate="2014-12-07T15:17:44.613" Score="14" Body="&lt;p&gt;As you told the student, the easiest way is &lt;a href=&quot;http://en.wikipedia.org/wiki/Lebesgue_integration&quot;&gt;to regard the Lebesgue integral as beginning with a partition of the range&lt;/a&gt;, rather than the domain. Perhaps a more refined way to view this is that the partition, rather than the &quot;heights&quot; of the rectangles, can be used to encode the &quot;shape&quot; of function being integrated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way to encode the function in the partition is to build the parts using the natural sets $S_{\lambda}=\{t\in \mathbb{R}:f(t)&amp;gt; \lambda\}$. These sets behave well with respect to limits. (Of course to be honest you must talk about sigma algebras…but this is a trailer for the film!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since you are talking to a non-major, you could stop here and just say that following up with this we obtain natural and desirable properties that Riemann integrability is too strong to satisfy, like the Monotone Convergence Theorem. Also, you might note that the question of when a function is Riemann integrable is naturally answered using the resulting class of sets (f on [a,b] is Riemann integral if and only if it is bounded and continuous almost everywhere in terms of the Lebesgue measure). &lt;a href=&quot;http://en.wikipedia.org/wiki/Lebesgue_integration&quot;&gt;All of this is discussed on the Wikipedia page&lt;/a&gt;…so then you could point the student there!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, see the &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521711835&quot;&gt;excellent book by David Bressoud&lt;/a&gt; for an extremely detailed account of the history and motivation.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2014-12-08T00:36:07.377" LastActivityDate="2014-12-08T00:36:07.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5983" PostTypeId="1" AcceptedAnswerId="5984" CreationDate="2014-12-07T17:42:02.253" Score="8" ViewCount="1801" Body="&lt;p&gt;Is the 'start with the desired result and work backwards' problem solving method discouraged? I'm now in my 3rd university year and every physics problem has been solved using the 'given values, manipulate until you find the final answer' (let's call it 'working forwards', unless there is an actual name).&lt;/p&gt;&#xA;&lt;p&gt;I think working forwards is a good method for derivation questions, when formulas for the final answer are not known. However, for application problems this often makes the solution unclear until the last step, when it all comes together. In my limited experience working backwards always yields a more logical process. I'd like to know if there is a reason that it isn't used more frequently in solutions, at least when it comes to physics problems.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;To illustrate&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Given some initial values for variables, and the name of the desired variable, you could approach it in (at least) 2 ways:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Method 1&lt;/em&gt;: Take the variables with given values and manipulate them to find a formula for the desired variable. E.g.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Given a, b, c. Find d.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;One relation with a is x = f(a,b). One with c is y = f(c). It is also known that d = f(x,y). Substituting, we can now write d = f(a,b,c).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Method 2&lt;/em&gt;: Take formulas for the final variable, and manipulate other variables until you get it in terms of the given values. E.g.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Given a, b, c. Find d.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;One solution for d is d = f(x,y). It's also known that x = f(a,b) and y = f(c). So we can now write d = f(a,b,c).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3480" LastEditorUserId="2083" LastEditDate="2023-11-08T18:46:15.563" LastActivityDate="2023-11-08T18:46:15.563" Title="Disadvantages of working-backwards solutions" Tags="|undergraduate-education|physical-sciences|" AnswerCount="3" CommentCount="4" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5984" PostTypeId="2" ParentId="5983" CreationDate="2014-12-07T19:19:51.667" Score="5" Body="&lt;p&gt;In both my high school Physics and Calculus classes, I emphasize that problems can be approached both ways: forwards or backwards. A student should try one method, and if that doesn't work he or she should try the other method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the multiplicity of formulas at his command, a student may not be sure which information that can be derived from the known facts is useful. Working backwards may be more likely to stick to useful information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one problem that my Physics class almost always only succeeds with working backwards. Given a mass, with a distance and a time to raise it, find the needed power of a motor to raise the mass (assuming constant power). The way the students get is:&#xA;$$P=\frac Wt=\frac{Fd}t=\frac{mgd}t$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Working forward is difficult as they do not see the need to include $g$ (the free-fall acceleration due to gravity at the surface of the earth).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;It should also be noted that the classic book &lt;em&gt;How to Solve It&lt;/em&gt; by George Polya emphasizes the backward approach. Polya devotes little space to the forward approach, and some of that space concerns combining it with the backward approach. This shows that at least one master of problem solving considered the backward approach to be more important and useful.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;My own philosophy is to teach and show both approaches. Use whatever works!&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2014-12-07T19:19:51.667" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5985" PostTypeId="2" ParentId="5977" CreationDate="2014-12-07T23:50:24.133" Score="2" Body="&lt;p&gt;to illustrate $A \times B,$ draw two orthogonal axis and mark the points of $A$ on one and the points of $B$ on the other. the elements of $A \times B$ are the points where lines through points on the axes meet. or is this too obvious and adequate for the purposes.&lt;/p&gt;&#xA;" OwnerUserId="3441" LastActivityDate="2014-12-07T23:50:24.133" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5986" PostTypeId="1" CreationDate="2014-12-08T01:22:41.860" Score="10" ViewCount="2789" Body="&lt;p&gt;I have met kids and teenagers who I believe are capable of understanding the concept of a vector. However, I have only heard of a select number of schools who teach it to young people. Here's my main question: how early and how much can you teach young kids and teenagers about vectors? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, my mom (who is very smart) somehow managed not to learn about it until I taught it to her. And that floored me because I u see them everyday. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I learned about vectors intuitively in high school through free body diagrams. When I got to college, I was introduced to them formally in Tom Apostol's calculus. But a few years later, I'm stunned I didn't learn this earlier, like middle school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't understand why it isn't universally taught earlier. It seems like a concept that's very applicable and eases learning concepts in areas such as physics. And I know lots of kids who hate math because its boring. So they may never take vectors if it is taught so late in life. Thus a lot of people could benefit if you taught them earlier in their lives. Just say &quot;hey! Vectors are cool and important! So learn about them and you'll like math since they are very useful!&quot; And then show them what you can do with them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyways, I'm looking for a sort of Piaget/development type answer. Example: 5 years old you do adding arrows. 10 years old you introduce scaling in size through multiplication by numbers, etc.      &lt;/p&gt;&#xA;" OwnerUserId="3471" LastEditorUserId="122" LastEditDate="2017-04-19T16:10:08.620" LastActivityDate="2021-03-28T16:18:49.390" Title="How early and how much can you teach kids and teenagers about vectors?" Tags="|primary-education|curriculum|" AnswerCount="8" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5987" PostTypeId="2" ParentId="5983" CreationDate="2014-12-08T07:19:19.070" Score="-2" Body="&lt;p&gt;I recently had a conflict about this with a pupil. He had answered a question in which you were supposed to determine a parameter such that the equation that contained it had a specific solution. He guessed a value for the parameter and then confirmed it by calculating out the equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I took points off because it was possible, and part of the tested material, to derive a value for the parameter by reasoning forward in logical steps. Very often I am interested in testing in whether the student has used logical steps rather than guesswork. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the problem makes it possible to work backwards from the answer in logical steps  I have no problem with that strategy. If it involves luck or guesswork I would be against it.&lt;/p&gt;&#xA;" OwnerUserId="3483" LastEditorUserId="3483" LastEditDate="2014-12-08T07:44:53.223" LastActivityDate="2014-12-08T07:44:53.223" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5988" PostTypeId="2" ParentId="5778" CreationDate="2014-12-08T08:19:08.853" Score="1" Body="&lt;p&gt;I recommend that you use the sequence designed into the textbook. Whether it is inspired or not, it will help you prepare classes and your students organize their revision. It is much easier to study for a test on chapter 5 than to study pieces of five different chapters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am inclined to introduce logic/proofs material with triangles (classical construction proofs are very tangible) and then keep on hitting it with each new topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The curriculum I teach (secondary school in the Netherlands) breaks trig into Tangents, which turn up naturally after similar triangles and the rest (Sin, Cos etc) which only appear after we have handled quadratic equations. &lt;/p&gt;&#xA;" OwnerUserId="3483" LastActivityDate="2014-12-08T08:19:08.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5989" PostTypeId="1" CreationDate="2014-12-08T15:01:02.517" Score="22" ViewCount="1261" Body="&lt;p&gt;My colleague and I are currently teaching &quot;true infinitesimal calculus&quot; (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by Keisler &lt;a href=&quot;https://www.math.wisc.edu/~keisler/calc.html&quot;&gt;https://www.math.wisc.edu/~keisler/calc.html&lt;/a&gt;. Two of my colleagues in Belgium are similarly teaching TIC at two universities there. I am also aware of such teaching going on in France in the Strasbourg area, based on Edward Nelson's approach, though I don't have any details on that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Which schools, colleges, or universities teach true infinitesimal calculus?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A colleague in Italy has recently told me about a conference on using infinitesimals in teaching in Italian highschools. This NSA (nonstandard analysis) conference was apparently well attended (over 100 teachers showed up). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Geneva, there are two highschools that have been teaching calculus using ultrasmall numbers for the past 10 years. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anybody with more information about this (who to contact, what the current status of the proposal is, etc.) is hereby requested to provide such information here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 1. On Gerald's suggestion, also of interest would be any educational studies comparing the two approaches (the one using infinitesimals and the one using epsilon, delta). That is, in addition to the study &lt;em&gt;Sullivan, Kathleen; Mathematical Education: The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach. Amer. Math. Monthly  83  (1976),  no. 5, 370–375.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-04-20T13:33:35.097" LastActivityDate="2017-04-20T13:33:35.097" Title="Which universities teach true infinitesimal calculus?" Tags="|undergraduate-education|calculus|reference-request|infinitesimals|" AnswerCount="3" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="5990" PostTypeId="1" AcceptedAnswerId="6016" CreationDate="2014-12-08T21:19:10.637" Score="8" ViewCount="504" Body="&lt;p&gt;I'm looking into holding my basic statistics class in a computer lab in the future to avoid a lot of the by-hand computing that happens in statistics classes. For example, very few students get anything out of the calculation of standard deviations or correlation coefficients by hand. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm looking for &lt;strong&gt;experiences of other math educators who have taught in a computer lab&lt;/strong&gt; -- not speculation about whether this is a good idea. What were the unique challenges of teaching in a computer lab, and what were the most surprising benefits and drawbacks?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-04-25T14:41:07.153" Title="Statistics Class Held in a Computer Lab" Tags="|mathematical-pedagogy|technology-in-education|statistics|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5991" PostTypeId="2" ParentId="5990" CreationDate="2014-12-08T23:01:16.107" Score="5" Body="&lt;p&gt;I was a TA for a Probability class (which means that I had to do recitations and tutorials for a small group of students which are enrolled in a bigger class guided by a professor), my experience is that you need to be strict with students in the first few weeks, otherwise your students will end up in facebook or twiiter and not paying any attention to your class (I know because it happen to another TA of the same class). I usually started with 10 to 20 minutes of a lecture setting (that is, not using the computer but introducing the concepts first and then showing how the computer might help), after that I began by provide the students with a &quot;lab guide&quot; which contains an activity (which for this course, consisted in getting proficient with the R language) that took often an hour an a half to complete. My advice after this experience is that, you need to have a small group, and be ready to deal with students that are not tech-savy. The good things were that most of my students liked the material, and I provided guides to more material and activity guides that introduced new concepts for the next tutorial.  &lt;/p&gt;&#xA;" OwnerUserId="3253" LastActivityDate="2014-12-08T23:01:16.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5992" PostTypeId="2" ParentId="5778" CreationDate="2014-12-09T04:33:35.277" Score="11" Body="&lt;p&gt;While I am not sure if your mentor teacher is up for the discussion for this year, I will note that, in practice, it is important to continue to question the sequencing of topics in a particular course. Mathematics is organized logically, and when a school &lt;em&gt;year&lt;/em&gt; forces a chronology, the order of topics will be something you continue to revisit depending on the needs of your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before I address your concern for “intuitive flow” in sequencing a secondary geometry curriculum, I must acknowledge &lt;strong&gt;two constraints&lt;/strong&gt; that might limit one’s opportunities to make adjustments to the sequencing &lt;em&gt;puzzle&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If you are constrained by the school’s desire to make sure that &lt;strong&gt;students pass the state exam&lt;/strong&gt;, then your sequencing might value the topics in geometry that are “new” or “most tested” over those that you find interesting. These topics—including circles, special right triangles, triangle centers, and precise definitions of quadrilaterals—may not have been introduced in K-8 mathematics, and it may be your responsibility to address these at least haphazardly before your students tackle the state exam. For instance, the state math exam that my students will take in Massachusetts places a heavy emphasis on computations using algebra. Topics such as logic, trigonometry, constructions, and proofs are not tested at all. Hence, the sequencing that I was handed when I started teaching skipped over the topics in geometry that would not be covered on the exam.&lt;/li&gt;&#xA;&lt;li&gt;If you are constrained by the need for &lt;strong&gt;heavy Algebra I remediation&lt;/strong&gt; because your geometry class serves as a bridge to Algebra II, then your sequencing might value time spent reviewing algebra explicitly over development of proofs. The latter relies on precision of language and nuance that may not necessarily be meaningful for students who do not intend to continue to study mathematics for too long after your class. Along a similar vein, if you are constrained by the &lt;strong&gt;needs of students with learning, behavioral, or language challenges&lt;/strong&gt;, then your sequencing might focus on practical applications of geometry and skip proof-based geometry altogether.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Your need for “intuitive flow” indicates to me that you might not be too concerned about these two constraints during your student teaching, but keep in mind that these are realistic constraints that might affect how you make decisions in your own geometry class (or even in your mentor teacher's).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are &lt;strong&gt;three guiding principles&lt;/strong&gt; to how I have sequenced your topics below:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Encourage exploration and discovery.&lt;/strong&gt; The high school geometry class that I teach uses an inductive approach based on Michael Serra’s &lt;em&gt;&lt;a href=&quot;https://www.kendallhunt.com/discoveringgeometry/&quot;&gt;Discovering Geometry&lt;/a&gt;&lt;/em&gt;. Students perform investigations and keep a running list of conjectures throughout the year. Students test their conjectures and apply them to problems that may require some skills from Algebra I. As indicated in Serra’s text, constructions play an important part in the discovery process, so I introduce them early. Each subsequent topic is the answer to a problem posed at the end of a previous investigation. For instance, “What symmetries do translations, reflections, and rotations create?” might link a unit on rigid transformations to a unit on symmetry.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Build from students’ previous experiences with geometry.&lt;/strong&gt; The K-8 curriculum exposes students to a basic understanding of geometric shapes and vocabulary. When introducing the next topic, the narrative of the sequencing should acknowledge that students have had some exposure to the topic before they entered your class, perhaps with less precise language and less reasoning. While it is in your course that these ideas are refined and extended, helping students make these connections to prior knowledge provides access points to build discussion.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Save proofs for the end.&lt;/strong&gt; &lt;a href=&quot;http://www.michaelserra.net/weblog/2013/08/proof-in-discovering-geometry.html&quot;&gt;Serra explains using the van Hiele model&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The van Hiele research shows students aren’t ready for formal proof until they have had concrete experiences with geometric figures and have successfully mastered earlier levels of visual thinking. Research shows 70% of high school students enter geometry operating at level 0 or 1 on the van Hiele measure of geometric reasoning (Shaughnessy and Burger 1985; Senk 1989). Yet &quot;traditional&quot; geometry textbooks that begin with establishing postulates and proving theorems are expecting students to immediately begin their geometry experience at levels 3-4. When the teacher and the textbook are presenting geometry at van Hiele level 3 or higher, while the students are functioning at level 0 or 1, it should be no surprise that there is such a high failure rate in traditional geometry courses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In my class, students are still asked to provide reasoning for their answers throughout the year. However, it is not until the end of the course, after repeated experimentation with their conjectures, that students are introduced to “formal proof.” (Compare to Glencoe McGraw-Hill and Holt McDougal secondary geometry textbooks, which introduce proofs in Chapter 2.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;With these guiding principles and possible constraints in mind, the following is &lt;strong&gt;one way&lt;/strong&gt; to sequence the units in a secondary geometry course that may provide the “intuitive flow” that you are looking for. I have also included a few ideas for how the units may connect thematically and &lt;em&gt;added some units&lt;/em&gt;. (Note of practice: this particular sequencing is only somewhat in effect and still undergoing testing/revision as the year progresses!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Part I: Building Blocks&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Understandings of geometry from primary school are accessed, refined, and formalized through precise definitions and notation.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Introduction to Geometry &lt;/li&gt;&#xA;&lt;li&gt;Constructions &lt;/li&gt;&#xA;&lt;li&gt;Parallel Lines and&#xA;Transversals&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Part II: Basic Shapes and Measurement&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Triangles from primary school are further investigated to discover connections to polygons. Pythagorean Theorem serves as a foundation for distance in the coordinate plane. Primary conceptions of measurement are extended to area, perimeter, circles, and volume.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Triangle Properties and Centers&lt;/li&gt;&#xA;&lt;li&gt;Polygons and Quadrilaterals&lt;/li&gt;&#xA;&lt;li&gt;Pythagorean Theorem, Special Right Triangles&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Length, Area, Perimeter&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Circles&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Solids and Volume&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Part III: Transformations&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rudimentary understandings of “slide, flip, turn” become formally defined as translation, reflection, and rotation in the coordinate plane. These lead to an investigation of symmetry and tessellation. A contrast is drawn between isometries and dilation. The discussion of similarity extends to an introduction to trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;10&quot;&gt;&#xA;&lt;li&gt;Rigid Transformations&lt;/li&gt;&#xA;&lt;li&gt;Symmetry and Tessellations&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Dilations and Similarity&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Trigonometry&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Part IV: Proofs&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Logic and algebraic proofs are an entry point to an intuition about proofs in geometry. Circling back to rigid transformations, congruent polygons pave way for the need for triangle congruence shortcuts, which are then used to prove other conjectures investigated during the course. Finally, the year ends by connecting algebra and geometry through coordinate proofs.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;14&quot;&gt;&#xA;&lt;li&gt;Logic&lt;/li&gt;&#xA;&lt;li&gt;Triangle Congruence&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Proofs of Conjectures for Quadrilaterals and Circles&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Coordinate Proof&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3484" LastActivityDate="2014-12-09T04:33:35.277" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5993" PostTypeId="2" ParentId="5700" CreationDate="2014-12-09T12:48:05.733" Score="2" Body="&lt;p&gt;One of the new foci in the Common Core State Standards is on the role of translations and other form of transformations and how they affect points in a plane.   For example, see the Congruence standards in the high school standards:&#xA;&lt;a href=&quot;http://www.corestandards.org/Math/Content/HSG/CO/&quot; rel=&quot;nofollow&quot;&gt;http://www.corestandards.org/Math/Content/HSG/CO/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;|x+4|−5=10 and |x+4|−5=−10&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I taught algebra 2, I'd have students experiment with these first without using absolute value in order to get familiar with moving equations to the left, to the right, and up and down.    You can do this with lines, but i think that it works a lot nicer with parabolas.   Using graphing calculators at this point is fairly nice, as students are convinced a lot easier of things when they can see them instantly :-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try doing this with x^2 and then x^2 -5.   the -5 moves the parabola down 5 points on the y axis.   Then (x+4)^2 - 5 moves the parabola - okay, this part is a bit tricky, it moves it left.   My students were always totally baffled by this.   I would have them do a table:&#xA;x x^2 x+4 (x+4)^2&#xA;-6 36 -2 4&#xA;-5 25 -1 1&#xA;-4 16 0 0&#xA;-3 9  1 1&#xA;-2 4  2 4&#xA;-1 1  3 9&#xA;0  0  4 16&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, interestingly enough, when you add 4 before squaring, it means that -4 becomes 0, -3 becomes 1, -2 becomes 2, and -1 becomes 3.   Hence, the parabola moves left, not right!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your first one, |x+4| - 5 = 10 - we can think of as intersecting y= |x+4| -5 and y=10.   Start by graphing y = |x|.  Now move it down 5 points on the y axis.   Then move it - again, tricky! - 4 spaces left on the x axis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then graph y= 10 and find where it intersects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that showing this with other functions like parabolas helps make this a lot clearer.   As in, this is a type of operation you can do on functions, not just on absolute value equations.&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2014-12-09T12:48:05.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5994" PostTypeId="2" ParentId="5990" CreationDate="2014-12-10T00:44:37.713" Score="8" Body="&lt;p&gt;I see you teach university level courses, however, I think that my experience in a high school classroom may offer some insight. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach an introductory applied statistics class at my high school and I have made the transition this year to having students use of technology very frequently. Unlike you I have the luxury of having computer carts, so I get to choose when a student has a computer in front of them -- which has become more and more often. I would estimate that I allow students to use some form of technology to engage in statistical calculation at least 50% of class periods. If am not able to secure computers, I will allow students to use their cell phones to access various online statistics calculators.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the greatest benefit to this is that you can spend more time having students run through multiple examples and data sets. For instance, inquiry approaches to developing an understand of statistics are far more possible as computation is much quicker. Unlike in past years, by having user friendly technology (e.g. not a ti-83) I was able to cover the following introductory topics through investigation:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Outlier affect on measures of center&lt;/li&gt;&#xA;&lt;li&gt;Comparing data sets based on measures of center and spread&lt;/li&gt;&#xA;&lt;li&gt;Transformations on data sets and their affect on center and spread&lt;/li&gt;&#xA;&lt;li&gt;Correlation coefficient&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and I plan to use it extensively in my upcoming probability unit. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has also allowed me to introduce estimation as a routine activity to encourage students to become familiar with the mental math of statistics -- something all of us who are familiar with statistics do. I ask students to make estimates of some statistical measure and then check their answer. This has helped them interpret statistics more easily by having a more fluent sense of what these statistics imply about the underlying data. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Computers if used well can help student see into the heart of these measures by preventing the actual calculations from becoming an impediment to deeper understanding. That being said, the calculations themselves offer insight into the value and meaning of a particular statistic. The temptation I have found is that with technology prevalent in my classroom, I am much more likely to decide that it not worth it to spend time manual statistical calculations and graphs. But these calculations do have their worth if presented properly. For instance, I also decided not to requires students to use a statistical table for the normal distribution by allowing students to use an &lt;a href=&quot;http://davidmlane.com/hyperstat/z_table.html&quot;&gt;online calculator&lt;/a&gt;. Some might argue however that there is value in using these tables in the same way that using a slide rule or abacus helps you develop intuition about numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Technology can be just as procedural as manual calculation. As with all teaching, it largely comes down to how you use it.  &lt;/p&gt;&#xA;" OwnerUserId="3485" LastActivityDate="2014-12-10T00:44:37.713" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5995" PostTypeId="2" ParentId="5885" CreationDate="2014-12-10T00:56:34.153" Score="6" Body="&lt;p&gt;One of my colleagues introduced me to a strategy called &quot;3 Reads&quot; which he details in his blog &lt;a href=&quot;http://misteristhisright.com/2014/08/20/strategy-showcase-3-reads/&quot; rel=&quot;noreferrer&quot;&gt;Misteristhisright&lt;/a&gt;. This strategy has two major features:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;You remove the question from the problem. The problem only gives the context. This can be done with &lt;a href=&quot;http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/&quot; rel=&quot;noreferrer&quot;&gt;Dan Meyer's 3 act problems&lt;/a&gt; as well. &lt;/li&gt;&#xA;&lt;li&gt;You read the problem 3 different times in order to become familiar with it: first for understanding context, second for understanding quantities, and third to generate questions that you can ask about the context and quantities. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The goal of the strategy is to have students understand the context before they begin answering the question. In my experience students (as they are often taught) jump right to the question and then go back to understand which results primarily in students mashing numbers together hoping that they will result in something meaningful. &lt;/p&gt;&#xA;" OwnerUserId="3485" LastActivityDate="2014-12-10T00:56:34.153" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5996" PostTypeId="1" AcceptedAnswerId="5997" CreationDate="2014-12-10T02:21:26.797" Score="7" ViewCount="7075" Body="&lt;p&gt;I often find myself spending too much time coming up with systems of equations for exercises for my students to practice on (that come out to nice integers). For two variables this is trivial, albeit annoying, but for three, four, five, etc variables it becomes increasingly more arduous.  I was wondering if anyone knows of a program/website that will generate systems of equations with an arbitrary (up to at least four) number of variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, any resource on generating systems of equations would be awesome. I have a feeling, based on google searches, that this ideal tool may not exist so I am totally willing to code it myself, but a starting point would be nice.&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2020-09-23T19:59:54.493" Title="System of Equations Generator" Tags="|reference-request|algebra|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5997" PostTypeId="2" ParentId="5996" CreationDate="2014-12-10T03:37:49.070" Score="9" Body="&lt;p&gt;This can be done very easily, with nothing more complex than a spreadsheet:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Start by randomly generate your &lt;em&gt;solution&lt;/em&gt; to the system, using whatever &quot;niceness&quot; criteria you like.  In Excel, for example, use the function &lt;code&gt;RANDBETWEEN(-10,10)&lt;/code&gt; to generate a random integer between $-10$ and $10$. Call the four numbers so generated $x,y,z,w$.&lt;/li&gt;&#xA;&lt;li&gt;Now randomly generate the &lt;em&gt;coefficients&lt;/em&gt; of an equation, again using whatever &quot;niceness&quot; criteria you like.  If you are using four equations with four unknowns, you need to generate 4 coefficients.  Call them $a, b, c, d$.&lt;/li&gt;&#xA;&lt;li&gt;Now compute $ax+by+cz+dw$, call the total $N$, and write the equation $$ax+by+cz+dw=N$$ (replacing $a,b,c,d,N$ with their randomly-generated numerical values and leaving $x,y,z,w$ as variables).  You have now generated an equation.  If you don't like it -- for example if $N$ is an &quot;ugly&quot; number -- just refresh the values of $a,b,c,d$ and get a new one.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Iterate steps 2-3 until you have four equations you like.  You now have a system for which the &quot;nice&quot; values generated in Step 1 is a guaranteed solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course it is possible that this method will produce a degenerate system, but you can also test for that by having Excel calculate the determinant of the matrix of coefficients.  As long as it isn't $0$, your solution is unique.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-12-10T03:37:49.070" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5998" PostTypeId="1" CreationDate="2014-12-10T06:09:00.053" Score="13" ViewCount="633" Body="&lt;p&gt;I am not an expert on inquiry-based learning, but it seems to be a useful teaching technique to use in smaller classrooms. Since many college math courses are quite huge, I was wondering how inquiry-based learning can be implemented in this situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How can inquiry-based learning be utilized in a large classroom situation?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="228" LastEditorUserId="39" LastEditDate="2015-02-04T11:55:26.283" LastActivityDate="2015-02-05T16:58:10.107" Title="How can inquiry-based learning be used in a college classroom?" Tags="|undergraduate-education|mathematical-pedagogy|teacher-development|inquiry-based-learning|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5999" PostTypeId="1" AcceptedAnswerId="6019" CreationDate="2014-12-10T10:28:10.910" Score="10" ViewCount="4672" Body="&lt;p&gt;At least in Germany, nearly all teachers and textbooks use the notation&#xA;$$P(x,y)$$&#xA;for the point $P$ with coordinates $x$ and $y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own math professors at university always cried about this, as the notation should be&#xA;$$P=(x,y)$$&#xA;and I've come to understand the truth of this. Alas, it's difficult to convince other teachers of this rationally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where can I find ressources to prove my point, that at least at university level, the second notation is used and the first notation is at least deemed formally incorrect?&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2014-12-14T04:51:46.257" Title="Notation of points with coordinates" Tags="|geometry|notation|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="6000" PostTypeId="2" ParentId="5996" CreationDate="2014-12-10T13:10:24.787" Score="3" Body="&lt;p&gt;If you have access to software such as Maple, Mathematica, or Matlab, then you can figure out how to have it generate an nxn matrix A with integer entries and with determinant 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have such a matrix, you are essentially done. The system of equations is AX=B where B is a column matrix that can be arbitrary integers. The fact that the det(A)=1 guarantees a solution vector X that will be integers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have used this approach to write MapleTA and WeBWorK questions like you are trying to come up with. You always want to check the size of the entries of the solution vector X to make sure they are not too big. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2014-12-10T13:10:24.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6001" PostTypeId="1" CreationDate="2014-12-10T18:24:06.270" Score="9" ViewCount="204" Body="&lt;p&gt;In graduate school I once tried to rapidly learn mathematics by working through a text and collecting (and precisely remembering) where I had been stuck, and what parts of arguments or computations I did not follow. After writing precisely down where I was stuck, I immediately moved on. Of course, I then immediately got in trouble with supervisors who spotted the holes in my understanding. I have, from time to time, wondered whether sticking with this approach longer may have paid off.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that in this way, one can accumulate a lot of facts and intuitions about a lot of things. It may be argued that the sticking points might actually eventually be dissolved by frequent enough encounter with the same difficulty in a diverse variety of circumstances. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is, perhaps, expressed in the extreme viewpoint (probably not seriously) stated by Halmos that one should &quot;never read the proof of a theorem&quot;. The sentiment here being that one should instead think about what the theorem can do and eventually aim to prove the theorem for oneself. One envisions a mathematician spending time to collect theorems like mushrooms and using them to &quot;cook his own mathematics&quot; rather than poring over other people's proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: Is there evidence that some mathematicians effectively learn in the above fluid way described in my first paragraph? In particular, is it possible to develop deep insights by moving &quot;laterally&quot; and accumulating knowledge, cataloging where one is stuck as one moves along, but immediately moving on to further understand the landscape rather than remaining stuck for any significant period of time?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am asking this question as a &quot;devil's advocate&quot; position to the one I actually espouse, as I have expressed in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5805/learning-mathematics-with-the-aid-of-spaced-repetition-systems/5834#5834&quot;&gt;my answer here&lt;/a&gt;. I personally believe that depth is sacrificed by an approach like the above, and that it is too tempting to move on to something easier in the above strategy to develop anything substantial. On the other hand, such rapid lateral movement might better have a chance to uncover connections between different areas, and is likely to be much more fun than cracking one's skull on details. More interestingly, perhaps, does anyone teach this from this standpoint...especially to undergraduates? Undergraduates might enjoy a more pedestrian approach, and if there is evidence that significant gain can be acheived by treading more lightly, this might affect one's approach to undergraduate mathematics education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this is subjective and argumentative...or too metacognitive for the forum, I'll happily delete it.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-12-11T05:50:52.973" Title="Learning modulo sticking points, or &quot;fluid study&quot; in mathematics" Tags="|undergraduate-education|graduate-education|undergraduate-research|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6002" PostTypeId="2" ParentId="6001" CreationDate="2014-12-11T05:50:52.973" Score="1" Body="&lt;p&gt;I'll just address the issue of teaching and detail. I do think there is something to be gained in providing a big picture on a given topic. For the students who care, I hope to present them with some sort of roadmap to math beyond the course. By the nature of the discussion, there is no way they will be able to struggle through the details implicit within the big picture. For example, I would mention the Jordan form in linear algebra despite the fact I do not work out the gory details. Or, in calculus III, I will spend a little time at the end to alert them to the existence of these wonderful objects called differential forms. Are they ready to prove the generalized Stokes' Theorem on $p$-chains or whatever? No. Should I avoid it because of that? I say no. Whatever class I teach, I always look for opportunities to give brief pointers to avenues of future study. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, the bulk of my teaching is detail-oriented. I don't really grade the big-picture part. Furthermore, I tend to show more proofs than might be typical because I do believe that the details matter. Especially when you get to the point of generalization. How can you generalize an argument you never saw? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, I have seen students who seek a big-picture actively discouraged by certain professors. The desire to know more than you are currently able to prove is healthy in my view. So, this discouraging posture is an enigma in my view. I see a student who wants to know math. Good. Whatever skill they fall short in we should take the posture of encouraging them to attain that skill so they can own the new math rather than collect it as trivia. I think it's really important to give students a forum to express their excitement about math, we should never dismiss their interests. However, we should help them see what they need to unwrap before they can compress it down as Thurston says in your answer.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-12-11T05:50:52.973" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6003" PostTypeId="1" CreationDate="2014-12-11T15:53:01.477" Score="8" ViewCount="465" Body="&lt;p&gt;I'm writing (together with a colleague) a minicourse on mathematical analysis (currently we want to cover the Weierstrass theorem on functions on compact intervals, so the aim is to present only the material needed for that particular result; we want to enlarge the scope later).  I'm a bit stuck on the definition of a limit of a sequence.  Does anyone know any intuition or metaphor which would naturally motivate the epsilon-ish definition?  Ideally, after reading it, the student would think &quot;yeah, this is obviously how this should be defined.&quot;  I have an idea or two, but I do not like them too much, so I decided to ask here.&lt;/p&gt;&#xA;" OwnerUserId="704" LastActivityDate="2016-09-26T22:51:26.100" Title="Any metaphors/intuitions for a limit of a sequence?" Tags="|mathematical-analysis|intuition|sequences|limits|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6004" PostTypeId="2" ParentId="6003" CreationDate="2014-12-12T00:40:48.873" Score="-1" Body="&lt;p&gt;Maybe you could contrast $\sin(x)/x$ and $\sin(1/x)$ as $x \to 0$,&#xA;arguing that the former stays within an $\epsilon$-ball (vertical $y$-interval)&#xA;while it approaches $1$ (somewhat nonobviously) from the right/$+x$,&#xA;whereas the latter oscillates and does not stay within an $\epsilon$-ball,&#xA;and in fact covers the $y$ interval $[-1,1]$ as $x \to 0$...?&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/wRo3H.jpg&quot; alt=&quot;SinPlots&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-12-12T00:40:48.873" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6005" PostTypeId="2" ParentId="5998" CreationDate="2014-12-12T04:53:06.927" Score="7" Body="&lt;p&gt;IBL is a really wide umbrella term, nowadays at least.  I strongly suggest browsing the &lt;a href=&quot;http://www.inquirybasedlearning.org/&quot;&gt;Academy of Inquiry Based Learning&lt;/a&gt; website.  They have videos for many of the talks at the R.L. Moore conferences, such as &lt;a href=&quot;https://www.youtube.com/watch?v=nMTeafAW2gs&amp;amp;index=67&amp;amp;list=PL7HVaYibt1klxwJ8OQH0QJz5GhCUKSXAY&quot;&gt;this one that seems to be about &quot;larger&quot; classes&lt;/a&gt;.  The &lt;a href=&quot;http://www.maa.org/ebooks/notes/NTE75.html&quot;&gt;canonical text&lt;/a&gt; on &quot;Moore method&quot; may have some ideas for your larger classroom situation, but I think that you are right that there are not enough resources available.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at the physics literature may help, because a lot of the &quot;flipped&quot; classroom enthusiasm recently comes from that direction.  The videos from &lt;a href=&quot;http://ocw.mit.edu/courses/chemistry/5-95j-teaching-college-level-science-and-engineering-spring-2009/&quot;&gt;Sanjoy Mahajan's class&lt;/a&gt; on MIT's OpenCourseWare have some very interesting ideas that are closely related to IBL, for instance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I would say on a personal note that just trying ONE activity that has the students trying to grapple with their own 'construction' of something is worthwhile.  Surely you have one day that you can set aside for some activity that they can do in small groups that is 'inquiring'.  See how it goes, get feedback - and be honest that they are doing a service to your future students, to make 'lecture' feel less like, well, lecture.  I think many will respond to that.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2014-12-12T04:53:06.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6006" PostTypeId="2" ParentId="5986" CreationDate="2014-12-12T04:57:46.210" Score="2" Body="&lt;p&gt;Anyone who can understand that if you buy five apples and three oranges today, but two apples and one orange tomorrow, then you have seven apples and four oranges at the end, can do vectors.  Forget about the physics angle; naturally there is a lot of usefulness there, but this sort of thing is easy to do.  Indeed, matrices come up very naturally in such contexts as well - check out any 'business math' or 'finite math' text and there will be a number of very interesting, if elementary, examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why this isn't in the curriculum is a long historical question, probably having a lot to do with our curriculum's generally still being beholden to the Sputnik outlook.  That said, I wouldn't introduce the dot produce to a third-grader without a good reason!  But vectors are fine, especially if you call them something else like, say, &lt;em&gt;lists&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2014-12-12T04:57:46.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6007" PostTypeId="2" ParentId="6003" CreationDate="2014-12-12T10:27:33.633" Score="9" Body="&lt;p&gt;My feeling is that the $\epsilon$-$\delta$ formulation is already pretty close to what one should think about limits; that is, the language can be hard to grasp at first, but the idea is very literally expressed in that language and can also be expressed in words. I find this more convincing using a physics measurement approach (which is slightly more natural for limits of functions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, we think of numbers as measurements by a physical device, which can have arbitrary (non perfect) precision. We have a sequence of such measurements $(u_n)$; we will say that this sequence has limit $\ell$ if, however precise our device is, there is a time (rank) after which all our subsequent measurements are indistinguishable from $\ell$: our device cannot tell them apart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another approach would be game-theoretic &lt;strong&gt;[edit:&lt;/strong&gt; &lt;em&gt;as pointed out in the comment, this had been proposed by Terence Tao on MO; in any case I do not pretend than any of the ideas in this post are original ideas of mine&lt;/em&gt; &lt;strong&gt;]&lt;/strong&gt;. Player A wants to prove that $u_n$ has limit $\ell$, and player B wants to disprove it. Player B plays by proposing an error, as small as she wants but non-zero. Player A then answers with a rank, as large as she wants. If all subsequent numbers in the sequence are closer to $\ell$ than specified by the error chosen by B, than A wins the round, otherwise she looses. One says that $(u_n)$ has limit $\ell$ if A can win all rounds, whatever the strategy of B is.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2014-12-12T18:26:25.557" LastActivityDate="2014-12-12T18:26:25.557" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6008" PostTypeId="2" ParentId="6003" CreationDate="2014-12-12T17:06:06.123" Score="1" Body="&lt;p&gt;Maybe a topological approach would be better: The limit of a sequence is $L$ if and only if any open interval containing $L$ eventually contains the rest of the sequence. Of course, making this more rigorous in a metric space immediately gives the $N-\epsilon$ definition. But the intuition is that eventually the sequence never gets far away from $L$.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-12-12T17:06:06.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6009" PostTypeId="2" ParentId="5990" CreationDate="2014-12-12T17:14:29.477" Score="1" Body="&lt;p&gt;I am writing as someone who has been a statistics &lt;strong&gt;student&lt;/strong&gt; next to a computer lab. I think this was a great opportunity, because I could learn the statistics concepts in &amp;quot;class,&amp;quot; then walk over to the lab and program them into a computer as soon as &amp;quot;class&amp;quot; was over. I couldn't imagine a better transition from &amp;quot;theory to practice&amp;quot; in &lt;strong&gt;any&lt;/strong&gt; course than this one.&lt;/p&gt;&#xA;&lt;p&gt;And the experience put students on notice that these applications were &amp;quot;live,&amp;quot; not just out of a textbook.&lt;/p&gt;&#xA;&lt;p&gt;There were no drawbacks, because the course was taught in a real &amp;quot;classroom,&amp;quot; just within easy walking distance of computers. (This was during the 1970s, when there were no desktops or PCs.)&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2021-04-25T14:41:07.153" LastActivityDate="2021-04-25T14:41:07.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6010" PostTypeId="2" ParentId="6003" CreationDate="2014-12-12T19:39:29.133" Score="2" Body="&lt;p&gt;This is very similar to the answer Benoit Kloeckner gave above, but perhaps more informal (and hence more intuitive?).  I like to explain the limit of a sequence in the form of a dialogue between two people, &lt;em&gt;A&lt;/em&gt; and &lt;em&gt;B&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;A&lt;/em&gt;:  As this sequence of values continues, it gets closer and closer to 12.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;B&lt;/em&gt;:  How close?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;A&lt;/em&gt;:  What do you mean?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;B&lt;/em&gt;:  How close does it get to 12?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;A&lt;/em&gt;:  As close as you want.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;B&lt;/em&gt;:  Does it ever get so close that the values stay always between 11.9 and 12.1?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;A&lt;/em&gt;:  Yes.  You just have to go out past the 230th term in the sequence.  From that point on, the values are always between 11.9 and&#xA;  12.1&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;B&lt;/em&gt;:  Does it ever get so close that the values stay always between 11.99 and 12.01?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;A&lt;/em&gt;:  Sure.  Just wait until you get past the 12,042nd term in the sequence.  From that point on, the values are always between 11.99 and&#xA;  12.01&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;B&lt;/em&gt;:  Well, does it ever get so close that the values always stay between --&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;A&lt;/em&gt;:  Look, let me make it simple for you.  You pick &lt;em&gt;any&lt;/em&gt; tiny little window around 12 that you want, no matter how small, and I &lt;em&gt;guarantee&lt;/em&gt;&#xA;  you that I can find some term in the sequence beyond which the values&#xA;  always stay within that window.  Okay?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;B&lt;/em&gt;:  Got it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The $\epsilon - N$ definition of convergence is pretty much a straightforward transcription into formal language of &lt;em&gt;A&lt;/em&gt;'s last sentence in this dialogue.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-12-12T19:39:29.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6011" PostTypeId="2" ParentId="5986" CreationDate="2014-12-12T19:54:56.783" Score="8" Body="&lt;p&gt;You can teach kids about vectors in the eighth or ninth grade (ages 13-14). Basically, when they learn about Cartesian coordinates with x and y (that is, multiple) variables. Because (x,y) is just a vector, when measured from the origin.&lt;/p&gt;&#xA;&lt;p&gt;Students at that level are beginning to learn about lines &lt;strong&gt;on&lt;/strong&gt; planes. There's no reason why they can't learn to generalize these concepts in multivariate space, that is, to planes in hyperplanes. That will remove some of the artificial distinctions between e.g. &amp;quot;plane&amp;quot; and &amp;quot;solid&amp;quot; geometry. Besides, a knowledge of vectors is a good foundation for matrix theory in math, and for physics in general.&lt;/p&gt;&#xA;&lt;p&gt;FWIW, I believe that kids can, and perhaps should, be taught linear algebra late in high school (rather than freshman or sophomore year in college). &lt;a href=&quot;https://matheducators.stackexchange.com/questions/3961/when-is-a-good-time-to-teach-linear-algebra&quot;&gt;When is a good time to teach linear algebra?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2021-03-28T09:23:29.563" LastActivityDate="2021-03-28T09:23:29.563" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="6012" PostTypeId="2" ParentId="2099" CreationDate="2014-12-12T20:14:09.570" Score="3" Body="&lt;p&gt;For high school math students, there are two basic &quot;tracks.&quot; One leads to pre-calculus and calculus, etc. The other is called &quot;general math.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Statistics makes a good &quot;general math&quot; study for students who don't want to pursue calculus. That is because it deals with things that are found in everyday life, such as the &quot;averages&quot; and &quot;dispersions&quot; of things like height, weight, grades,and other key attributes for high school students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus-bound students could also benefit from the study of statistics (and probability), but they can wait to take the calculus first, and then learn about integrating conditional density functions in probability, or differentiating normal equations in statistics.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2014-12-12T20:14:09.570" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6013" PostTypeId="1" CreationDate="2014-12-12T20:28:32.203" Score="5" ViewCount="149" Body="&lt;p&gt;My 5-year-old son recently brought home (from kindergarten) a worksheet that he had done at school. 6 boxes were given. In one box were 3 kites, in another, 2 bears, in another 8 drums, and so on. Each box had a list of 3 numbers at the bottom to choose from. The instructions were:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“Count the objects in each set. Draw a circle around the right number.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As we all know, such exercises are common for young students, but I have always wondered whether such exercises were teaching something that would later have to be unlearned, namely, the false idea that members of a set are always all instances of the same thing. Anyway, at the very least we should ask when in their academic career students finally encounter the truth that the members of a set can be things of different type. 8th grade? senior year of high school? College? Only math majors in college after their first year?&lt;/p&gt;&#xA;" OwnerUserId="2039" LastActivityDate="2014-12-14T15:57:12.780" Title="When are non-homogeneous sets first encountered by students?" Tags="|mathematical-pedagogy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6014" PostTypeId="2" ParentId="5725" CreationDate="2014-12-13T00:01:29.163" Score="2" Body="&lt;p&gt;For students in the ninth grade, evaluate $3^1$, $3^2$, $3^3$ and ask them how they would get the next value, $3^4$ (multiply).  Then if we traverse this table in the opposite way, how do we get 'next' values?  If they can tell you how to get to 9 from 27 and how to get to 3 from 9, then perform that operation on $3^1$ to evaluate $3^0$, and similarly, $3^{-1}$.    &lt;/p&gt;&#xA;" OwnerUserId="3492" LastEditorUserId="5664" LastEditDate="2016-08-10T10:12:14.710" LastActivityDate="2016-08-10T10:12:14.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6015" PostTypeId="2" ParentId="5999" CreationDate="2014-12-13T03:03:57.710" Score="4" Body="&lt;p&gt;As mentioned in the comments already, there isn't one universally accepted correct notation.  And the prevalence of both notations throughout the literature would make it quite a Herculean task to &lt;em&gt;correct&lt;/em&gt; this notation in the writing of others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find aspects of both notations bothersome.  So in my own writing, I tend to avoid using either.  Instead, I write &lt;code&gt;let P have coordinates (1,2)&lt;/code&gt; or &lt;code&gt;P has coordinates (1,2)&lt;/code&gt; or &lt;code&gt;P is a point with coordinates (1,2)&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem I have with &lt;code&gt;P=(1,2)&lt;/code&gt; is, perhaps, a philosophical one (but being logician by training, I like to think philosophically).  To me, it seems that the plane exists prior to the coordinate system.  We are perfectly capable of talking about points and their relationships amongst themselves and other objects without taking recourse to any coordinate system.  (Read Euclide to see the extent to which he avoides ever even talking about numbers.  Indeed, coordinate systems are a rather modern development.)&#xA;After all, there are other coordinate systems, why identify a point with one particular choice of coordinate system.  Consider how, if we decide our &lt;em&gt;plane&lt;/em&gt; is inbedded in a larger 3 dimensional space (or even higher dimension), the point is still &lt;code&gt;P&lt;/code&gt;, but it's coordinates must change: let's say surreptitiously &lt;code&gt;P&lt;/code&gt; has coordinates &lt;code&gt;(1,0,2)&lt;/code&gt; in this higher dimension.  Given the transitive nature of equality, would we really want to say &lt;code&gt;(1,2)=(1,0,2)&lt;/code&gt;?  I think not.  But all notations have their short-comings: so perhaps in this context we're willing to disclaim that the transitivity of equality can be so cavalierly applied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A coordinate system helps us locate the point and differentiate one point from another via functions:  that is, a coordinate system allows us to analyze what's happening between points by reducing the analysis to that of what's happening between numbers and vectors etc.&#xA;And a great deal of mathematical thinking has gone in to showing that thinking about numbers and vectors will not lead us astray in regards to points and their ilk.  But I would caution that there's nothing obvious about how coordinates relate to each other being equivalent to how points relate to each other.  And in that sense &lt;code&gt;P=(1,2)&lt;/code&gt; suggests that this equivalence properties obvious where, in fact, it took a long time to be discovered and verified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say it's best to look at the context in which a particular notation is being used as respect that within a particular field one style of notation may be preferred over another.  Perhaps I overgeneralize, but I don't believe many mathematicians put too much thought into why they use a particular notation until they find the notation creates more problems then it solves.  But also, mathematicians are generally not worried about the deeper philosophy a particular notation suggests.&lt;/p&gt;&#xA;" OwnerUserId="3491" LastEditorUserId="3491" LastEditDate="2014-12-13T03:09:14.953" LastActivityDate="2014-12-13T03:09:14.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6016" PostTypeId="2" ParentId="5990" CreationDate="2014-12-13T09:06:46.457" Score="10" Body="&lt;p&gt;I have been an IT/maths teacher, so I have spent several years teaching in a computer lab, and watched others use the room, sometimes very effectively, sometimes not so much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have recently seen some excellent computer based courses and now can not imagine teaching statistics (or calculus or physics) without regular access to a computer lab. It totally transforms what you are able to do with students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teaching in a computer lab is an entirely different skill than most other kinds of teaching. I have seen very experienced and competent teachers totally lose a class, or at least a large portion of it, and not even know about it until things are totally out of control. Just as a lecturer learns to read a lecture hall to know what to do next, you will need to learn to read the computer lab. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make matters worse, there are also some computer lab layouts that make it extremely difficult to work with as they are often designed to fit in a ton of computers rather than accommodate effective teaching. You will need to figure out how to use the space effectively. Computer labs are not interchangeable like regular classrooms or lecture theatres.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For students that &quot;speak computer&quot; fluently, they can use it to learn the material much faster and more deeply. The best students will be able to play with the computers and discover things faster than you can teach them. These students can create challenges you need to be prepared for. Do you have more advanced stuff for students to experiment with after they finish the lab work in 15 minutes? If not, it is dead boring for the students, and potentially disruptive for the class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then there are those that will spend the hour looking for the Start button... if they make it past the login screen! Do you have good TAs or student helpers available for those who just don't get it? Losing a student is unfortunate in a lecture, but near fatal in a computer lab, as catching up to the flow of the class is much harder, and the motivation smaller as there is always something to distract them at a computer. Nothing replaces 1-1 help. Organising students into small groups is useful at high school, but I don't know how it would work in your situation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to have well designed activity sheets that students will be able to work through independently, and maybe have a manned tutorial session for students to catch up on what they missed in the lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make sure that the digital resources are well organised and easy to find, without diving through four levels of sub-folders. Make sure any printed resources match the current digital resources as much as possible - a wrong file name or hard to access sub-folder can throw the lesson for a surprisingly long time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you get to know how a computer lab works, you will get a feel for when to bring students' attention together for instruction, and when to allow it to run a little longer in tutorial mode. Don't be afraid to run it a little looser if you can see students are engaged and learning. The extra experimenting that they are doing may be more valuable to their understanding than cramming in the last bullet point of your planned class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It will take a while to get right, and you may be constrained by the lab setup and other resources available, but I think you will find the payoff will be worth it. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2014-12-13T09:14:19.890" LastActivityDate="2014-12-13T09:14:19.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6017" PostTypeId="1" CreationDate="2014-12-13T23:01:18.307" Score="7" ViewCount="4413" Body="&lt;p&gt;When starting compass and straightedge geometric constructions in my grade 8-9 maths classes, I usually begin by mentioning a little about Euclid and the fact that constructions have been done for thousands of years. The problem is that my students haven't heard about Euclid before, don't really understand how a geometric construction differs from a sketch drawing, and don't continue constructions beyond triangles and quadrilaterals in grade 10-12. Moreover, as far as I know there aren't any &quot;real world&quot; applications of geometric constructions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What hook can I use to engage students in learning about geometric constructions?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally the hook should be somewhat informative and very engaging, such as a story, video, article, starting problem, etc.&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2016-03-03T20:16:56.267" Title="How Can I Motivate Geometric Constructions?" Tags="|secondary-education|concept-motivation|geometry|constructions|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6018" PostTypeId="2" ParentId="6017" CreationDate="2014-12-14T01:06:19.603" Score="2" Body="&lt;p&gt;you can use paper folding, which does not take many resources, to motivate elementary constructions. i think there is a book by johnson, d. a. published by maw that cover many basic constructions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;students don need to have heard of euclid to enjoy and benefit from solving construction problems. if the teacher practices construction, then it would be easier to pass that joy along. when was the last time you used a pair of compasses?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;i have taught a week long class on polyhedral construction. at the beginning we will cover lots simple construction: construction simple angles, square roots and so on. eventually we learn how to construct a regular pentagon. student build all the platonic and archimedian solids not from cut outs but their o  &lt;/p&gt;&#xA;" OwnerUserId="3441" LastEditorUserId="3441" LastEditDate="2014-12-14T02:12:57.703" LastActivityDate="2014-12-14T02:12:57.703" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6019" PostTypeId="2" ParentId="5999" CreationDate="2014-12-14T04:51:46.257" Score="5" Body="&lt;p&gt;First, notation sometimes evolves, sometimes not.  I don't think $P(x,y)$ is bad, but it is somewhat old, I believe, although still in use but not quite &lt;em&gt;à la mode,&lt;/em&gt; somewhat like French these days.  :)  It's not as bad, say, as the notation $\sin^2 x$ or worse (for me), $\ln^2 x$, which I first learned meant $\ln \ln x$ but my students use for $(\ln x)^2$.  But what I have to say should not be taken as promoting the old notation.  As $P(x,y)$ is used less and less, fewer and fewer people will be familiar with it.  That in itself is argument against it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $E^2$ be a Euclidean plane.  A coordinate system on $E^2$ is a bijection $E^2 \leftrightarrow {\bf R}^2$ (with certain properties).  The point $P(x,y)$ is a notation represents a point $P \in E^2$ corresponding to the element $(x,y) \in {\bf R}^2$. The point $Q(-1,3)$ is similarly understood.  When this convention is well-known, it is not particularly hard to understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a couple of competing impulses in current mathematics.  One is that notation should clearly indicate a single object that depends only on the symbols out of which the notation is constructed.  I can trace this back to Frege, but it seemed to take hold in the 20th century.  The other is that notation should be efficient.  The desire for efficiency quite often leads to a violation of the first principle.  The notation $P(x,y)$ is certainly efficient, the parentheses being arguably extraneous; but since they are used to denote a coordinate pair, they might also be defended.  The implicit coordinate mapping is rarely important.  Usually when the notation $P(x,y)$ is used, it is in a simple geometric problem and there is only one coordinate mapping.  The context can be made clear by words, which is often a better way, in my opinion, than with abstruse and superfluous notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was young and impatient, I did not like $P(x,y)$.  The $P$ seemed extraneous.  In many cases the letter $P$ was not needed for the problem, but the author seemed to feel every point needed a letter attached to it.  Part of the problem was that I thought ${\bf R}^2$ was the same as the Euclidean plane.  If they are considered distinct, and they often are in modern synthetic geometry, then $P = (x,y)$ is certainly wrong, unless $(\_,\_)$ denotes a function ${\bf R}^2 \rightarrow E^2$, which normally it does not.  If the plane and ordered pairs are identified, as I was led to think in my early years, then the notation is okay.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The normal way is to define or posit a mapping $\phi \colon E^2 \rightarrow {\bf R}^2$ (by introducing coordinate axes), and let $P = \phi^{-1}(x,y)$.  But this seems cumbersome.  So it seems to me one can chose which is the lesser of two evils: the notation $P(x,y)$, which does not conform with current functional notation; and the notation $P = (x,y)$, which treats the plane as identical with ordered pairs of real numbers.  The current trend in usage is for the latter, of course.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2014-12-14T04:51:46.257" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6020" PostTypeId="2" ParentId="5725" CreationDate="2014-12-14T12:43:48.963" Score="0" Body="&lt;p&gt;I find it best to think of exponents as repeated ratios in a geometric progression. When defined in this way, it automatically solves the edge cases via definition rather than requiring an axiom or theorem. It also builds a good foundation for understanding how powers of different bases relate to each other, logarithms, binomial powers etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As such I think exponent should always be thought of as&#xA;$$ x^n = 1\times x \times x \times x ...$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Year 9 students should be able to recognise an easy geometric progression: $3, 6, 12,24,48$&#xA;$$24 = 3\times 2 \times 2\times 2= 3\times 2^2$$&#xA;$$12 = 3\times 2 \times 2= 3\times 2^1$$&#xA;$$6 = 3\times 2 = 3\times 2^1$$&#xA;So what was the first in the series?&#xA;$$3=3\times 2^0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leads to a general formula for the $n^{th}$ term in a geometric progression, starting from $a$ increasing with a ratio $b$ : $a\times b^n$. &#xA;$$a\times b^2 = a\times b\times b$$&#xA;$$a\times b^1 = a\times b$$&#xA;$$a\times b^0 = a$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So now we understand geometric progressions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only remaining question is, when $a$ is unwritten, what do we place in front of the repeated ratios? $1$ is the only choice:&#xA;$$b^2 = 1\times b^2 = 1\times b\times b$$&#xA;$$b^1 = 1\times b^1 = 1\times b$$&#xA;$$ b^0 = 1\times b^0 = 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To further demonstrate that the above is the standard way of understanding exponents, mathematicians now generally agree the following (though in the past there was more disagreement, so probably not for discussion in a 9th grade class!)&#xA;$$0^2=1\times 0\times 0$$&#xA;$$0^1=1\times 0$$&#xA;$$0^0=1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $0^0$ is not treated this way, all sorts of mathematical formula break, for instance the binomial theorem (the leading coefficients are the binomial coefficients from Pascal's triangle)&#xA;$$(1+2)^2=1\times 2^0 + 2\times 2^1 + 1\times 2^2=9$$&#xA;$$(1+2)^1=1\times 2^0 + 1\times 2^1=3$$&#xA;$$(1+2)^0=1\times 2^0 =1$$&#xA;$$(1+0)^2=1\times 0^0 + 2\times 0^1 + 1\times 0^2=1$$&#xA;$$(1+0)^1=1\times 0^0 + 1\times 0^1=1$$&#xA;$$(1+0)^0=1\times 0^0 =1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In conclusion, the leading &quot;$1\times$&quot; in the style of a geometric progression does not need to be used all the time for exponents, but defining exponents in this way does clarify the edge cases nicely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for your original question, I would say that $x^0 = 1$ is simply part of the definition of exponents. However, the fact that their has been significant disagreement about another edge case ($0^0$), with people sometimes arguing about &quot;what it is&quot; rather than &quot;what it should be defined as&quot; indicates that in practice the line between definition and axiom can be blurry. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2014-12-14T12:43:48.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6021" PostTypeId="2" ParentId="6017" CreationDate="2014-12-14T13:21:15.947" Score="5" Body="&lt;p&gt;As for real world applications, most of the buildings around you older than 30 years were designed by an architect that manually used geometric constructions for their drawings. I learned geometric constructions in my Tech Drawing class before I learned them in maths class. Of course, CAD programs do all of that in the background now, but the reason that Euclid chose the axioms that he did for his constructions was that it was about the real world. Otherwise, why include the parallel postulate?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So as for a story, how about using a compass and straightedge (no rulers if you want to be pure) to design a net of an interesting building (ie. not a cube). Starting from a single length to define the height of a wall, you could build a gabled square house, a hexagonal round(ish) house, or a geodesic dome (soccer ball) all from simple geometric constructions. You would end up with a little Euclidean village.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have studied symmetry, the structures could be decorated with various wallpaper and frieze patterns, maybe replicated with similar techniques. One of your students could invent a recursive/fractal wallpaper!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have the opportunity, using software like Geogebra can extend the scope of design in purely geometric terms. The infinite scalability from changing the first and only measurement is also amazing to watch. A revolutionary tool for any part of geometry if you have computers or tablets available. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though not using Euclidean constructions, I had some able students create nets for every Platonic and Archimedean solid (and samples from the two infinite series of prisms and skew prisms) in Geogebra, discovering Euler's theorem for plane networks and then using that to discover the more difficult combinations. A couple of the most complicated Archimedean solids were hard to lay out and they needed a helping hand, but it was a worthwhile activity, though it takes some time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Constructions can be very rich, especially when combined with other areas of mathematics. I hope your students have fun with whatever you choose. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2014-12-14T13:21:15.947" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6022" PostTypeId="2" ParentId="5986" CreationDate="2014-12-14T15:13:48.317" Score="7" Body="&lt;p&gt;As a solid example, I taught my kids vectors in primary and middle school.  I barely had to teach them, it just was common sense to them in the context. Keep in mind that they are good at math, but they are no more able and much less experienced than early High School students. I went the traditional route and did it via Cartesian coordinates and physics. Once you know coordinates, the concept of arrows, vector addition and vector multiplication just come naturally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I liked the example from kchrisman about adding apples and oranges for the youngest kids. That is seriously no harder than mid primary school to lean the notation, and late primary for scalar multiplication. Of course with no variables. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dot-product was harder conceptually, but fairly easy mechanically. In physics &quot;it just works&quot;, and that is all they needed. Using the apples and oranges example, doing a dot product with the price list for the final cost should be easy for middle schoolers. It is what it looks like on a supermarket docket - so it can't be that hard. Still fairly concrete without variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Vector variables for substitution can be introduced just after normal variables. It actually make more sense at the beginning for vectors, as you can use them just to save space writing down and make formulas clearer - while normal variables use the same room as the numbers they are replacing and are no clearer. Normal 1D variables only are useful with generalised functions, which is a more abstract concept than simple substitution in a particular context. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Along with vectors, my kids were able to use a computer algebra system, so once they had the vector concepts, using them was as easy as using single dimensions numbers. Because of this, they already have an intuitive concept of a mathematical &quot;group&quot; or &quot;field&quot; (though not technically accurate) as they use the same addition and scalar multiplication with vectors as the did with integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you know vectors, much of maths becomes more concrete and qualitatively easier, reducing the age at which students are able to grasp it. Solving simultaneous equations - what is the big deal about? Gaussian elimination is just scaling and adding vectors until you get the arrows you want. It is just like basic arithmetic (not quite - but you know what I mean). Maybe mid high school for 2 equations, later for 3 equations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was much easier learning vectors and dot products than learning the 'cos' function - the traditional equivalent. In fact I still haven't really taught them trig - vectors are just too natural and too powerful to bother for most applications. If you know vectors, Pythagorus, and ratios, you can do most things you ever want to do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Add the ability to transform radial coordinates into vectors, and you have most of the rest. You will need to know unit vectors which is a step up from dot product. But IMHO coordinates of an arrow is more concrete than the ratios of a triangle, early high school should be fine for teaching it. I know that it is really just trig, but it is trig without the yucky trig concepts, just the idea of different ways of writing a vector. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have unit vectors you can learn parallel and perpendicular vectors. Could probably learn at the same time as trig is currently taught. Just involves dot product times unit vector for parallel, and subtracting parallel for perpendicular. So technically just a small step from translating radial to Cartesian coordinates, but it requires some more abstract thinking that suits mid-high school. However, the vector method is much easier to use than trig in a computer algebra system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have recently taught them 2D cross product for areas (triangle and rhombus), and that did not come naturally at all, and it is not something they need yet. At least as abstract as trig, but the mechanics are much more down to earth. Direct, easy multiplication and addition rather than via trig ratios. For appropriate problems, it may save you from the sine rule or cosine rule, and anything that can achieve that for even a few problems has to be a good thing! Probably upper high school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to rotate vectors directly rather than through radial translation, I think complex arithmetic is probably easier than introducing transformation matrices. If the student has been using vectors through high school, complex arithmetic should be easier to understand, but probably best left to last year of high school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect the biggest problem is that most teachers, school administrators and parents don't know vectors, and it doesn't have the kind of hero worship that calculus has, so it is really hard to get into a curriculum. Combine with the current &quot;back to basics&quot; movement ... &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2014-12-14T15:13:48.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6023" PostTypeId="2" ParentId="6013" CreationDate="2014-12-14T15:57:12.780" Score="4" Body="&lt;p&gt;It is very early primary school that they learn that sets can hold instances of different things. If they are of the same thing, then they are not really sets - or rather they have cardinality 1 as sets can't have repetition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, It is a good that they learn early on that set should contain things of the same type. A set of 3 apples can be unioned (added) with a set of 2 oranges, but what would the result be? Not a set of apples or oranges, but a set of 5 pieces of fruit. And then add in a dog, and you have a set of 6 formerly alive objects. Add in a cloud...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The more varied the type of thing in the set, the less you can do with it. For instance, you could have a set of a person, a name, a number, and mathematical equation and the set of prime numbers, but I can't imagine what you would ever do with it. What would be the point of counting it (5 items)? What kind of set would you intersect it with?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Computer Science, arrays, lists, or sets are often typed to prevent randomly heterogenous sets. Tuples, records, or structs may contain heterogenous objects, but they can not be manipulated as sets.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2014-12-14T15:57:12.780" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6024" PostTypeId="2" ParentId="6003" CreationDate="2014-12-14T21:25:49.260" Score="3" Body="&lt;p&gt;The natural language description of a limit is usually &quot;the number that the values in the sequence get closer and closer to&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The phrase &quot;closer and closer&quot; strongly indicates a time element to sequences. That is, $n$ increasing is the same as time moving forwards. I imagine the values of the sequence as dots on a number line, with a new dot appearing every second.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/e6vv6.gif&quot; alt=&quot;sequence using time&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The picture in my head of a sequence converging on a limit is the dots appearing one at a time and them being literally closer to the limit. It wouldn't do to periodically have a dot appear that is far away from the limit, so we need to be sure that &lt;em&gt;all&lt;/em&gt; future dots will be as close as we are now. Just one or two future dots being further out would be fine, as long as at some future time all of them are closer. This has to work no matter how close we are - otherwise it's not &quot;and closer&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And there is the idea of the epsilon. It has a limit L if no matter the $\varepsilon$, there is a time such that all future values are at least that close to L. I imagine it as zooming in super close to L and then waiting to see if the dots start appearing in the zone I am looking at. If there is a time past which all future dots will be in view, then this $\varepsilon$ is good. We need this to work no matter how far we zoom in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/FczyL.gif&quot; alt=&quot;time sequence with epsilon zone&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess it comes about because of an attempt to pin down what it means to be &quot;closer&quot; and what it would look like if it &lt;em&gt;didn't&lt;/em&gt; get closer.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2014-12-14T23:01:28.563" LastActivityDate="2014-12-14T23:01:28.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6025" PostTypeId="2" ParentId="6017" CreationDate="2014-12-15T02:26:55.763" Score="2" Body="&lt;p&gt;I question whether engagement is what is preventing students from grasping construction. I would argue that constructions are very much like geometric proofs in that they rely heavily on abstraction and formalism for something that students are already somewhat familiar with. In this thread about &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5778/secondary-geometry-curriculum-sequencing&quot;&gt;geometry sequencing&lt;/a&gt;, a colleague of mine makes the case that proof should be taught at the end of the year. I agree with this. In order for students to write a geometric proof they need to be fluent in requisite geometric knowledge so that they can reason about these ideas and connect them. I believe that if we want a student to learning mathematical reasoning, they can't simultaneous learn the content. For example, if we want students to prove something about triangles, they should be able to fluently be able to do this with concrete numbers because generalizing. To this end, I think that students need to develop fluency in creating accurate sketches of what they are constructing before they can engage with construction. If I had a nickle for every angle bisector that was off by more than 10%, I would be a rich man. If students don't value precision in a concrete setting, they certainly won't be able to carry that value over to an abstract one. &lt;/p&gt;&#xA;" OwnerUserId="3485" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-12-15T02:26:55.763" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6026" PostTypeId="1" AcceptedAnswerId="6028" CreationDate="2014-12-13T08:56:14.293" Score="11" ViewCount="441" Body="&lt;p&gt;I'm in undergrad engineering, freshman year, in IT. I suffer anxiety when I have to do calculus. Trigonometry is just tortuous. Anyone got any tips for lessening that? I guess hard work is the best, but anything else? Anyone else had any personal experiences of the same sort?&lt;/p&gt;&#xA;" OwnerDisplayName="Suraj Zala" LastEditorUserId="3499" LastEditDate="2014-12-15T15:22:32.760" LastActivityDate="2014-12-24T19:31:43.433" Title="How to overcome mathematics-related anxiety?" Tags="|student-motivation|anxiety|" AnswerCount="4" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="6027" PostTypeId="2" ParentId="6026" CreationDate="2014-12-13T11:05:10.190" Score="3" Body="&lt;p&gt;Based on my own experience, the only way to overcome math anxiety is to realise that it's a beautiful subject that can be applied to an enormous amount of areas and applications, a subject that makes you a more powerful thinker and problem solver. Until you do that, you will always be sort of afraid of it. This is, of course, easier said than done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used to be afraid of math at some point in my life, mainly due to the fact that I had bad teachers. When I was in college, however, I had a few really good professors who somehow managed to convince me that math is amazing; I've been excited about the subject ever since.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest reading a couple of math-related books in your free time, such as &quot;Flatland: A Romance of Many Dimensions&quot; by Edwin Abbott Abbott, &quot;How to Think Like a Mathematician: A Companion to Undergraduate Mathematics&quot; by Dr Kevin Houston, &quot;One Two Three ... Infinity: Facts and Speculations of Science&quot; by George Gamow, and other related books. These will help you create a positive attitude towards math, which in the long run will allow you to look at it from a different perspective, i.e. not as a scary subject that you need to pass, but as an amazing area that is worth exploring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have this positive attitude, solving problems will become much easier and your hard work will be transformed into more effective work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck.&lt;/p&gt;&#xA;" OwnerDisplayName="Aleksandar Mitrevski" LastActivityDate="2014-12-13T11:24:03.300" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6028" PostTypeId="2" ParentId="6026" CreationDate="2014-12-13T20:35:55.470" Score="4" Body="&lt;p&gt;Maybe you could try doing homework with friends.  I used to have pretty bad programming anxiety and it got a lot better when I coded with other people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The caveat is you don't want to do problem sets with friends who are much better at math than you, because then you will just feel discouraged when they get all the answers before you do.&lt;/p&gt;&#xA;" OwnerUserId="3504" OwnerDisplayName="Ben Bitdiddle" LastActivityDate="2014-12-13T20:35:55.470" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6030" PostTypeId="2" ParentId="5589" CreationDate="2014-12-15T13:35:40.393" Score="1" Body="&lt;p&gt;I vaguely remember having this problem with inequalities (so this will be anecdotal). The problem I had with teacher is the way they would read it from &lt;strong&gt;left to right&lt;/strong&gt; then double check it by reading from &lt;strong&gt;right to left&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kids already know which number is bigger. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;strong&gt;just agree&lt;/strong&gt; on reading it from left to right. Let them use their own words if they like smaller or bigger. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you feel the need to reverse check (reading from right to left) then &lt;strong&gt;physically write the numbers&lt;/strong&gt; in reverse for example 5&gt;4 or 4&amp;lt;5. Do not perform reverse reading &lt;strong&gt;verbally&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Never ever &lt;strong&gt;verbally read&lt;/strong&gt; 5&gt;4 as &lt;strong&gt;4 is less than 5&lt;/strong&gt; (which is what my teachers used to do when they reverse check). I know its true (we can read is as 4 is less than 5) but I think the children &lt;strong&gt;do not rotate&lt;/strong&gt; the numbers and inequality symbols in their minds &lt;strong&gt;when we read it backwards verbally.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just agree on the left to right reading order, since the greater than sign (&gt;) and less than sign (&amp;lt;) are both &lt;strong&gt;tied to a left to right reading order&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;or read it like this (which still follows the &lt;strong&gt;consistent left to right&lt;/strong&gt; reading order):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;the number on the left is (__) than the number on the right&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(again anecdotal)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;easier with gif&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/fB5FL.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="3503" LastEditorUserId="3503" LastEditDate="2014-12-16T07:02:29.043" LastActivityDate="2014-12-16T07:02:29.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6031" PostTypeId="1" CreationDate="2014-12-15T16:33:26.857" Score="18" ViewCount="5922" Body="&lt;p&gt;so we are working through various methods of factoring quadratic equations and the students seem comfortable factoring basic quadratics such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x^2 - 7x + 12 = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;by finding the factors of $12$ that add up to $-7$.  I feel like this is intuitive because it is just reasoning through &quot;anti-distribution&quot; and the students have picked it up very well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, we have gotten to the point where the coefficient on the $x^2$ term is not $1$ such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3x^2 -14x - 5 = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and I feel like finding all of the factors and essentially guessing and checking is much less intuitive.  This is the way that I learned to factor these types of equations but it took a very long time for me to get comfortable and develop an intuition as to which factors to use. My feeling that this is a less than ideal instructional strategy is backed up by the fact that many of the students who are great with $a=1$ are totally lost when the $a$ term is changed.  what are some intuitive ways to teach how to factor an equation like this other than finding all of the factors and guessing and checking?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2014-12-15T16:43:08.410" LastActivityDate="2022-06-11T00:38:14.057" Title="Factoring quadratics where the coefficient on the $x^2$ term does not equal 1" Tags="|secondary-education|algebra|solving-polynomials|" AnswerCount="12" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6032" PostTypeId="2" ParentId="6031" CreationDate="2014-12-15T18:09:32.887" Score="7" Body="&lt;p&gt;Assuming the problem comes from the happy world of textbook problems... a typically successful method is to assume integer factorizations:&#xA;$$ (3x+a)(x+b) = 0$$&#xA;where $ab=-5$. In the world free of those complicated fractions, we have just $a= \pm 1 $ and $b= \mp 5$ to choose. So, our options are:&#xA;$$ (3x+1)(x-5) \qquad \&amp;amp; \qquad (3x-5)(x+1)$$&#xA;$$ (3x-1)(x+5) \qquad \&amp;amp; \qquad (3x+5)(x-1)$$&#xA;and multiplication reveals $(3x+1)(x-5)=0$ is the winner. In summary, guided guess and check. &lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2014-12-17T14:44:58.270" LastActivityDate="2014-12-17T14:44:58.270" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6033" PostTypeId="2" ParentId="6031" CreationDate="2014-12-15T18:25:34.077" Score="12" Body="&lt;p&gt;Teach them how to complete the square. This is probably the simplest systematic method for factoring quadratic polynomials, and it's also very geometrically intuitive (you can literally &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Completing_the_square.gif&quot;&gt;visualize it in terms of a square&lt;/a&gt;). Guessing and checking usually works for very simple examples — monic quadratic polynomials with small integer factors — but it's not effective for anything much more complicated than that. (I suspect your students would also have trouble factoring polynomials with non-integer roots in this way.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Completing the square is also essential for understanding the quadratic formula, which is derived using this technique. Since roots of a polynomial correspond to linear factors ($r$ is a root if and only if $x - r$ is a factor), this is the same thing in slightly different language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, one of the most important mathematical skills is being able to reduce more complicated problems to simpler ones. Given a quadratic polynomial $ax^2 + bx + c$, we can write this as $a(x^2 + \frac{b}{a} x + \frac{c}{a})$, and now study the &lt;em&gt;monic&lt;/em&gt; quadratic polynomial $x^2 + \frac{b}{a}x + \frac{c}{a}$, which has exactly the same roots as $ax^2 + bx + c$. So, if your students learn how to factor monic quadratics (by completing the square or by another method), then they can factor &lt;em&gt;all&lt;/em&gt; quadratics by reducing in this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can motivate completing the square as coming from a series of such reductions to simpler problems, starting from the easiest cases. An equation of the form $x^2 = c$ is easily solved, and $(x - d)^2 - c = 0$ isn't much harder. But $(x - d)^2 - c = x^2 - 2dx + d^2 - c$, so this is general enough to encompass all monic quadratics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Guessing and checking might lead to an ad hoc way of completing the square anyway. Factoring a polynomial $x^2 + rx + s$ means finding numbers whose sum is $r$ and product is $s$, so one might start looking around $\frac{r}{2}$ and work outwards until numbers with the appropriate product are found. For instance, starting with $x^2 + 8x + 12$, one could guess $4$ and $4$ (whose product is too big), then $3$ and $5$ (still too big), then $2$ and $6$ (which is correct). This corresponds to the fact that $x^2 + 8x + 12 = (x + 4)^2 - 2^2$, whose roots are $4 \pm 2$. (Of course, this needs to be understood more explicitly to handle something like $x^2 + 8x + 14$.)&lt;/p&gt;&#xA;" OwnerUserId="3505" LastEditorUserId="3505" LastEditDate="2014-12-16T04:09:36.607" LastActivityDate="2014-12-16T04:09:36.607" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6035" PostTypeId="2" ParentId="6031" CreationDate="2014-12-15T23:04:47.323" Score="13" Body="&lt;p&gt;Factoring non-monic quadratic polynomials can be done by factoring with respect to a particular constraint. More precisely, DL Renfro points to &lt;a href=&quot;https://people.richland.edu/james/misc/acmeth.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;the ac Method of Factoring&lt;/strong&gt;&lt;/a&gt; which can be summarized roughly as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given a quadratic $ax^2 + bx + c$, the polynomial can be factored iff there is a factor pair for $ac$ whose sum is $b$; here, I denote by &quot;factor pair&quot; a pair of &lt;em&gt;integers&lt;/em&gt; whose product is the target integer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The aforelinked works out a few examples, and also suggests pulling out common factors of $a,b,c$ beforehand (when relevant).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applying the approach to the OP's quadratic of $3x^2−14x−5$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observe that $3(-5) = -15$ has a factor pair, $(-15, 1)$, whose sum is $-15 + 1 = -14$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we know the answer is &lt;em&gt;yes&lt;/em&gt;, it can be factored; rather than providing the general justification, one can work it out as per the linked examples (I include an optional $1$ coefficient for clarity):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3x^2 - 14x - 5 = 3x^2 - 15x + 1x - 5 = 3x(x - 5) + 1(x - 5) = (3x+1)(x-5)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do note the role of the factor pair $(-15, 1)$ in re-writing the original expression; more generally, the factor pair for $ac$ indicates precisely how to re-write the expression in a manner that makes the polynomial factorization more transparent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will &quot;leave as an exercise&quot; why this approach works, but I would expect an explanation to be covered at some point in a relevant course (whether before or after the algorithm is presented may depend on constraints outside the scope of this question). &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Given the above discussion, I would like to make one additional comment:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Viewing the task at hand as consisting of &quot;factoring with respect to a particular constraint,&quot; it is important that students be able to factor integers more generally. But when do students have the opportunity to practice factoring integers? Unfortunately, the answer seems to be: Not often enough. To this end, here are some brief bits from a recent article about factoring (written in the context of working with pre-service teachers, but certainly applicable here). The citation is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Feldman, Z. (2014). &lt;a href=&quot;http://www.jstor.org/stable/10.5951/mathteacmiddscho.20.4.0230&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Rethinking Factors&lt;/em&gt;&lt;/a&gt;. &lt;em&gt;Mathematics Teaching in the Middle School, 20&lt;/em&gt;(4), 230-236.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;First, a bit on when factoring is (or should be) covered in the United States:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/sd2FH.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, and perhaps surprising to some, students can struggle to work with a number's (prime) factorization; again, from the same MTMS Nov. 2014 article:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/V4VCo.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without overgeneralizing, the risk indicated in my reading of this work is students (who are unlikely to encounter an actual &lt;em&gt;number theory&lt;/em&gt; course unless they are college mathematics majors) may lack a great deal of fluency in working with factoring integers. &lt;strong&gt;So&lt;/strong&gt;, before delving too deeply into topics around factoring polynomials, it may be worthwhile to (re)visit integer factorization to ensure that students are not lacking a more foundational component.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2014-12-15T23:10:45.577" LastActivityDate="2014-12-15T23:10:45.577" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6036" PostTypeId="2" ParentId="6031" CreationDate="2014-12-16T00:08:46.580" Score="26" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$3x^2 -14x - 5 = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Multiply through by A or here, 3&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$9x^2 -42x - 15 = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, use substitution, u=&lt;span class=&quot;math-container&quot;&gt;$3x$&lt;/span&gt; (3X is the square root of this first term, and by using the u substitution, we now have an 'a' of 1. )&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$u^2 -14u - 15 = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;factor to&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(u-15)(u+1)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Substitute back u=3x&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(3x-15)(3x+1) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;last, divide out that 3 we multiplied by -&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(x-5)(3x+1)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you look at the number of steps, you'll realize it's consistent, 5 steps every time. Compare this to the guess and bang head method, and you'll find this one is a bit better. Students who get this are happy to learn and use it. But the u substitution isn't for everyone. I use this with high schoolers at multiple levels who need to factor just as you asked, with a non-zero A.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-06-11T00:38:14.057" LastActivityDate="2022-06-11T00:38:14.057" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="6037" PostTypeId="2" ParentId="6031" CreationDate="2014-12-16T05:18:08.713" Score="7" Body="&lt;p&gt;The more little tricks and techniques we teach our students, the more they see math as an arcane toolbox of things to remember until the next exam and forget thereafter.&lt;/p&gt;&#xA;&lt;p&gt;Instead of teaching an N-step process for each problem type, whenever possible we should try to find a memorable, generalizable, useful concept that unites all similar problems. In this case, we should use the same idea to factor any difficult quadratic, including things like &lt;span class=&quot;math-container&quot;&gt;$x^2 - 7x + 1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x^2 + 9$&lt;/span&gt; or OP's &lt;span class=&quot;math-container&quot;&gt;$3x^2−14x−5$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You should only ever try to factor a quadratic as long as you find it interesting to keep trying. &lt;strong&gt;As soon as you are sick of factoring, just use the quadratic formula&lt;/strong&gt; to find the roots. It's okay if you give up immediately when &lt;span class=&quot;math-container&quot;&gt;$a \neq 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If the roots of a quadratic are &lt;span class=&quot;math-container&quot;&gt;$r_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r_2$&lt;/span&gt;, then the quadratic factors as &lt;span class=&quot;math-container&quot;&gt;$y = a(x - r_1)(x - r_2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't see any reason to learn factoring tricks or skills beyond this one. This method has several advantages:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It reinforces the fundamental intuition of algebra: that the factors of a polynomial line up with its zero-places.&lt;/li&gt;&#xA;&lt;li&gt;It works in many more situations and even generalizes to higher degree polynomials.&lt;/li&gt;&#xA;&lt;li&gt;It is a fundamentally geometric viewpoint, so you can draw pictures of it to illustrate.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2014-12-16T05:18:08.713" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="6038" PostTypeId="2" ParentId="5996" CreationDate="2014-12-16T05:22:24.447" Score="0" Body="&lt;p&gt;Just work backwards.  For instance let's say you want x=2, y=3, z=1 as your answers. So you need to do what amounts to row operations on those equations to generate a linear system.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can add them up into new arbitrary combinations or multiply any of your equations by a non-zero constant.  For instance,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x+y+z = 6$$&#xA;$$x + y = 5$$&#xA;$$2x +z = 5$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or I can add up Eqs 1 and 2,  2 and 3, and 1 and 3 to get a new system:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$2x+2y+z = 11$$&#xA;$$3x + y+z = 10$$&#xA;$$3x+y +2z = 11$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both of these system will have solutions (x,y,z) = (2,3,1).&lt;/p&gt;&#xA;" OwnerUserId="3506" LastActivityDate="2014-12-16T05:22:24.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6039" PostTypeId="2" ParentId="5959" CreationDate="2014-12-16T05:23:18.253" Score="4" Body="&lt;p&gt;Have you tried doing the reverse. Show them the solution and ask them to form a word problem (based on the solution). It would show you how students parse (break down) word problems in their minds. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start easy for example: x + 2 = 8 form a word problem using slices of pizza. The way the children form sentences will show you the words they use to define a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would also allow you to &lt;strong&gt;reword some problems using words that the children use,&lt;/strong&gt; since it looks more of a language problem than a math problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't be strict in the child's sentence construction, the point is you want an insight into their minds of how they form, break down, and express math word problems in their heads. You will probably see word problems looking like conversations instead of properly structured math word problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have a common ground you can later show them what a properly structured word problem looks like - with emphasis on buzzwords as suggested by Andrew Sanfratello - allowing you to compare buzzwords to the words your students use.&lt;/p&gt;&#xA;" OwnerUserId="3503" LastEditorUserId="3503" LastEditDate="2014-12-16T12:04:39.763" LastActivityDate="2014-12-16T12:04:39.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6041" PostTypeId="1" CreationDate="2014-12-16T17:29:59.693" Score="6" ViewCount="453" Body="&lt;p&gt;I've recently discovered that the following theorems require the axiom of choice to be proven:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;every surjective function has a right inverse. &lt;/li&gt;&#xA;&lt;li&gt;a real-valued function that is sequentially continuous at a point is necessarily continuous in the neighbourhood sense at that point. &lt;/li&gt;&#xA;&lt;li&gt;every vector space has a basis.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;When I revisited the proofs I was taught in first year, I was surprised that my lecturers had used the axiom of choice without declaring so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems strange that so much effort was dedicated to establishing that mathematics is a rigorous subject [indeed much time was spent on learning the field axioms, well-ordering axioms, Archimedean principle and (later) the completeness principle] but to ignore the axiom of choice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested if there are reasons for omitting to mention the axiom of choice. Are there pedagogical reasons? Is it deemed too complicated? Is it more contentious than the other axioms?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question also &lt;a href=&quot;https://matheducators.stackexchange.com/questions/6041/why-is-the-axiom-of-choice-not-taught-from-the-start-to-mathematics-undergraduat&quot;&gt;asked&lt;/a&gt; at Mathematics S. E..&lt;/p&gt;&#xA;" OwnerUserId="3407" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-12-28T01:30:30.957" Title="Why is the axiom of choice not taught from the start to mathematics undergraduates?" Tags="|undergraduate-education|axioms-foundations|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6042" PostTypeId="2" ParentId="426" CreationDate="2014-12-16T18:44:53.333" Score="4" Body="&lt;p&gt;The history of this topic in the US is long and complicated. It differs from state to state, and also between rural and urban areas. In rural areas before about 1850, teachers were in short supply; a prospective schoolteacher had to have a very basic education, and also would have to pass an oral exam administered by the local school board. In the second half of the 19th century, cities began to &quot;professionalize&quot; teaching. First there were &quot;normal departments&quot; within high schools, and later these became independent normal schools. Around 1900-1930, there started to be schools and colleges of education within universities, and eventually many of the more prestigious ones became graduate schools of education. As early as 1905, California required five years of college for prospective teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can read all this as simply a gradual raising of professional standards, and a lot of these changes did coincide with the great era of progressivism. On the other hand, there were certainly winners and losers in the process, and one can tell an economic story. Supply and demand fluctuated. For example, during World War I, many women left the teaching profession to take the better-paying jobs vacated by men who had been drafted. This led to a shortage of teachers, and teachers' pay went up. During the Depression, there was an oversupply of teachers, and credentialing was used by many districts to control the supply.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Economic liberals such as Milton Friedman have argued that the proliferation of licenses (for morticians, auctioneers, cosmetologists, ...) allows the profession to set up a cartel similar to a medieval guild, so as to artificially limit supply and drive up prices. But I'm not sure that this argument applies well to high school teachers. If it did, you would expect that the credentials would have been created at the behest of teachers' unions. Teachers unions have existed for a long time (e.g., the California Teachers' Association since 1863), but it was only relatively recently that they became the political behemoths that they are in a state like California, where CTA is described as the fourth and most powerful branch of the state government. It wasn't legal for teachers' unions to carry out collective bargaining until about 1960, which is many decades later than the creation of certificates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My best read on the history is that credentialing was driven by the institutional self-interest of the schools providing training to teachers, as these gradually evolved from normal departments of high schools to graduate schools of education. These institutions employed a lot of professors and administrators, and the livelihood of those people was crucially tied to requiring specialized education coursework for prospective teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taking California as an example of the current system, if you want to teach high school math, you need a single-subject teaching credential. This requires the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A bachelor's degree.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;College coursework in education, which is often taken while the prospective teacher is an undergrad.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Passing a standardized test of basic skills (CBEST).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Passing a standardized test showing competence in math (CSET).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The above requirements get you a five-year preliminary credential, after which you need to jump through more hoops (education courses or testing) to get a permanent credential.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I would assume that anyone well qualified to teach math would easily satisfy requirements 1, 3, and 4. But the evidence does not provide much support for the requirement of educations courses. A study by Goldhaber using a a value-added methodology finds that although there are huge differences in effectiveness between teachers, only about 3% that variation is explained by objectively measurable factors including education and experience. In defense of the people who originally set up the credentialing system, it should be pointed out that this evidence is very recent, and there was probably no reliable evidence for or against the effectiveness of education coursework at the time when the requirements were set up; Goldhaber discusses the earlier studies, which were flawed methodologically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Sources I read:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The dubious case for professional licensing,&quot; &lt;a href=&quot;http://economix.blogs.nytimes.com/2013/10/11/the-dubious-case-for-professional-licensing/?_r=0&quot; rel=&quot;nofollow&quot;&gt;http://economix.blogs.nytimes.com/2013/10/11/the-dubious-case-for-professional-licensing/?_r=0&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Angus, &quot;Professionalism and the public good: a brief history of teacher certification,&quot; available online by googling&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Goldhaber, &quot;The mystery of good teaching: surveying the evidence on student achievement and teachers' characteristics,&quot; &lt;a href=&quot;http://educationnext.org/the-mystery-of-good-teaching/&quot; rel=&quot;nofollow&quot;&gt;http://educationnext.org/the-mystery-of-good-teaching/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2014-12-16T18:49:58.227" LastActivityDate="2014-12-16T18:49:58.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6043" PostTypeId="2" ParentId="6041" CreationDate="2014-12-17T01:55:16.363" Score="5" Body="&lt;p&gt;It is at once a philosophical, mathematical, foundational, and pedagogical issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than expand on the first sentence, I recommend Gregory Moore's book which discusses some of the first three parts (and perhaps the fourth also).  As a practical&#xA;matter, the Axiom of Choice is a challenge to formulate precisely and formally in a first-order language, where as most of the other axioms are easy to state in such a language.&#xA;Also, in practice much mathematics outside mathematical logic and some parts of computer science only needs to acknowledge the use (if any) of the axiom; the application usually does not concern foundational issues such as weak systems of arithmetic or set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are trying to highlight or raise the awareness level of foundational issues, make&#xA;note of the Axiom of Choice and suggest a reference or two for those who are interested.  Spending more time on it in the context of almost any mathematics class is time lost to spending on the class material proper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Pedagogical Practice Has Few Grounds&quot; Paseman, 2014.12.16&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2014-12-17T01:55:16.363" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6044" PostTypeId="2" ParentId="5977" CreationDate="2014-12-17T06:35:38.443" Score="1" Body="&lt;p&gt;When I was originally taught groups, one concrete example was the symmetry transformations of an object, like an equilateral triangle ($S_3$) or a rectangle ($V_4$). Usually you label or colour-code the corners so you can tell the difference between them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One easy way to form a Cartesian product would be to have transformations of both object simultaneously. So an element might be &quot;flip the rectangle horizontally and turn the triangle $120^\circ$&quot;. You can imagine the students with an actual rectangle and triangle and doing one action with one hand and one with the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A useful thing to do would be to find the order of such an element. So keep doing both simultaneously until both shapes come back to the original configuration. The number of times is the order. They should very easily come up with the idea that the order will be the greatest common multiple of the orders of the two individual transformations.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-12-17T06:35:38.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6045" PostTypeId="1" AcceptedAnswerId="6047" CreationDate="2014-12-17T08:07:22.737" Score="8" ViewCount="955" Body="&lt;p&gt;I'm helping with an algorithms course next term. I've taught intro programming courses and seminars, but never intro algorithms. I've spoken with previous TAs and instructors, and one of the biggest problems they face is student assignments -- both on the writing and grading side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The course is more or less abstract (no concrete implementation assignments). While you can argue whether or not this is ideal, it's the hand I'm dealt. This means that pseudocode is heavily important -- in fact, it goes into linear programming quite a bit and handwaves a large number of details on things like &quot;actually efficiently finding a feasible point/local optimum&quot; for simplex and such.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leads to two major problems: students don't deal with a suitable level of abstraction. They tend to refuse to consider &quot;black boxes&quot; even if they're used in class. While wanting to know details isn't wrong, and if you want to use the algorithms in real work it can be a big deal, it can lead to actually doing the assignments being intractable. Especially when the efficient algorithms they should be black boxing for certain purposes are either complex, open problems, or something they shouldn't need to reroll (sorting algorithms after the sorting section is covered).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On my end, it also leads to grading hazards. I've seen past assignments, and they tend to be very dense, boilerplate ridden, implementation detail-considering chunks of copy+pasted Java or C++ code. Some of this is simply poor coding practice, and would be an issue even with pseudocode, but it's absolutely clear that pseudocode solutions would be shorter (often by several pages) and easier to read and understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, attempts to teach pseudocode tend to leave students confused. They (perhaps understandably) shy away from black boxes, think their Java code is self evident, or think &quot;real code&quot; is inherently better because it can be run. They don't embrace the terseness and magic-of-convenience of things like &quot;sort x ascending by length&quot; and instead rely on verbose comparator implementations. How can I effectively communicate the benefits of pseudocode in abstract algorithms and data structures contexts for both my and their sanity?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Quick disclaimer: while this is a real question, this is also partially to test how the community feels about erring on the side of inclusiveness on computer science and programming as mentioned in &lt;a href=&quot;https://matheducators.meta.stackexchange.com/questions/57/how-does-computer-science-education-fit-into-the-community-if-at-all&quot;&gt;this meta discussion&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="521" LastEditorUserId="-1" LastEditDate="2017-03-16T16:41:53.880" LastActivityDate="2015-01-01T21:28:18.450" Title="How to effectively teach pseudocode" Tags="|mathematical-pedagogy|computer-science|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6046" PostTypeId="2" ParentId="6031" CreationDate="2014-12-17T08:12:25.397" Score="4" Body="&lt;p&gt;Edit 2:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Andrew Sanfratello, in a new answer to this question, seems to have discovered that the method I describe below is called the &quot;Berry method&quot;(for reasons unknown). My explanation shows that it is actually a refinement of the substitution method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comparison of the &lt;a href=&quot;http://peacemanzach.blogspot.com/2011/05/alternate-factoring-methods.html&quot; rel=&quot;nofollow&quot;&gt;ac and berry methods:&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just taught this today. The curriculum requires that they solve these problems, but all the examples have at least one of a or c as prime. This doubles the problem space compared to a monic quadratic, but it is far from combinatorial explosion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Completing the square/quadratic formula are really over the top, &quot;ac&quot; method or substitution introduce another unnecessary layer and sometimes a very large &quot;ac&quot; to factor. In reality they had no problem at all simply extending the guessing method used for monic quadratics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Original:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I would tend to teach completing the square/quadratic formula, I have an updated alternative to the &quot;ac method&quot; which is a cut down version of the substitution method also presented as an answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was surprised how similar the &quot;ac method&quot; is to the substitution method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To factor:$ax^2 + bx + c$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As in the &quot;ac method&quot; find the factors $d,e$ of $ac$ that add to $b$. As in the &quot;ac method&quot; &lt;strong&gt;this step does not need to be written down.&lt;/strong&gt;&#xA;$$u^2 + bu + ac = (u+d)(u+e)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Different from the &quot;ac method&quot;, simply plug the found factors $d,e$ into the following equation&#xA;$$ax^2 + bx + c = (ax + d)(ax + e)\div a$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a,d$ and/or $a,e$ will have common factors that will cancel with the $\div a$ if needed to give the simplest integer factorisation if possible. If you simply want to find zeros, the $\div a$ can be ignored. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method works by $u=ax$ and multiplying the whole equation by $a$, but the student can actually safely ignore the u step. The only step that needs writing is the final step. I think the full u substitution is beautiful for understanding, but tedious for lots of problem solving. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method is of limited usefulness as it only works with integer factorisation. I am interested in what people think about this and whether there are any improvements to be made.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-02-24T10:50:14.960" LastActivityDate="2015-02-24T10:50:14.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6047" PostTypeId="2" ParentId="6045" CreationDate="2014-12-17T11:58:48.610" Score="9" Body="&lt;p&gt;Here are a few ideas, which may or may not fit into your course. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;To start with, would it be possible to build an explicit pseudo-library of black box pseudo-functions and pseudo-operators that they should feel comfortable using? After they have used this library for a few assignments, they may feel more comfortable creating their own black-boxes. I remember feeling lost when concurrently taking maths courses from different year levels, and it was hard to know what theorems were allowed to be used in proofs, so I proved too much using only basic theorems. It would have helped to have an explicit &quot;library&quot; of theorems from each year that I could use in my work.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If they think Java or C++ code is in any way self-evident, they have a lot of unlearning to do. Maybe they should be forced to write in a scripting language for the first 1-2 weeks of the course. You could have appropriate high level libraries available for the required tasks, so they are forced to write in abstractions. Pseudo-code would be the next step of abstraction that they move to.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For the first assignment, you could present pseudo-code as part of a rapid algorithm development process, enabling the rapid communication and evaluation of algorithms. They could have a project that actually enforces a pseudo-agile development process, with several iterations of algorithms and  maybe also enforcing pair or group work - emphasising the communication aspect of pseudo-code. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;After their pseudo-code rapid algorithm development process, they could even create a real program from the pseudo-code. This would help them see that pseudo-code is a valid part of real computer science, not just mathematics. (Not using mathematic pejoratively, but the reason many do programming rather than mathematics is that they like to make things and see things work). This would only be one program for the first assignment, just to &quot;sell&quot; pseudo-code as a useful language that leads to real results. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Anyway, I hope one of these is useful.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2014-12-17T11:58:48.610" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6048" PostTypeId="2" ParentId="5977" CreationDate="2014-12-17T13:59:44.133" Score="3" Body="&lt;p&gt;Perhaps this &#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=MnRcc9eINaQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;YouTube video on Visualizing Group Theory&lt;/a&gt;,&#xA;by Nathan Carter (Bentley Univ), may help. Here is a snapshot&#xA;illustrating $C_3 \times C_4$:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/GsM02.png&quot; alt=&quot;DirectProduct&quot;&gt;&#xA;&lt;hr /&gt;&#xA;Blue represents the cyclic group $C_4$ horizontally,&#xA;and red the cyclic group $C_3$ vertically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nathan is the author of the wonderful book, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Carter, Nathan. &lt;em&gt;Visual group theory&lt;/em&gt;. MAA, 2009. (&lt;a href=&quot;http://www.maa.org/publications/maa-reviews/visual-group-theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;MMA link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-12-17T17:09:33.957" LastActivityDate="2014-12-17T17:09:33.957" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6049" PostTypeId="2" ParentId="1834" CreationDate="2014-12-17T23:25:25.927" Score="4" Body="&lt;p&gt;here is another way of looking at solving the problem $$3x + 5 = 14$$&#xA;break it up into two simpler problems solve $u + 5 = 14$ for $u$ and solve $3x = 9$ for x; see if they can solve that. in fact, this is how the student ought to have arrived at the problem.&lt;/p&gt;&#xA;" OwnerUserId="3441" LastEditorUserId="3441" LastEditDate="2014-12-18T12:02:35.523" LastActivityDate="2014-12-18T12:02:35.523" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6050" PostTypeId="2" ParentId="5954" CreationDate="2014-12-18T03:44:42.597" Score="4" Body="&lt;p&gt;In my opinion, for the purpose of grading, you should not try to guess or otherwise ascertain the student's reason for writing an incorrect answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you should grade the answer as you would grade any other answer such that&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;there is a crucial piece missing, and&lt;/li&gt;&#xA;&lt;li&gt;you can fill in the piece, but someone reading the question and answer in isolation would not be able to fill it in routinely.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In my case, I give such answers low scores.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student can then draw his or her own conclusions about the advisability of attempting to regurgitate memorized answers.&lt;/p&gt;&#xA;" OwnerUserId="870" LastActivityDate="2014-12-18T03:44:42.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6051" PostTypeId="2" ParentId="5998" CreationDate="2014-12-18T19:42:04.783" Score="4" Body="&lt;p&gt;There are numerous little things you can do to instill an IBL atmosphere in your class; most of them can be found on the AIBL website mentioned earlier. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also recommend watching this TED talk:  &lt;a href=&quot;http://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en&quot; rel=&quot;nofollow&quot;&gt;http://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I love his examples of getting the students to be the ones to generate the questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main example that sticks with me is about predicting when a tank will be full. Textbooks typically will give you an abstract diagram of the tank and a flow rate then ask the question, &quot;When will the tank be full?&quot;  He found a tank and made a video of the tank getting filled at an excruciatingly slow pace. Playing the video for the class motivated the students to be the ones to ask the question, &quot;How long will it take for the tank to be full?&quot; followed almost immediately by, &quot;What do we need to know in order to figure this out?&quot; He also had the class in groups placing predictions on the board which became more and more accurate. Further, instead of having the predicted answer cleanly in the back of the book, they simply watched the video until the end to see the time stamp.&lt;/p&gt;&#xA;" OwnerUserId="3515" LastEditorUserId="3515" LastEditDate="2014-12-18T22:08:11.330" LastActivityDate="2014-12-18T22:08:11.330" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6052" PostTypeId="1" CreationDate="2014-12-18T21:45:41.393" Score="1" ViewCount="177" Body="&lt;p&gt;It isn't so hard to convince students of grade 8. 9 that $-2$ is greater than $-4$, but it still make some confusion with an axiom says that ( The whole is greater than part ), which is true in the positive numbers. For example if we consider the ( Whole) to be $4$ and the half to be ( The part ), then 4 is greater than its half which is 2. The question is how to convince students that the converse should happen in the negative numbers, i.e, the half of $-4$ ( $-2$) is greater than the ( Whole ) which is $-4$ ?!&lt;/p&gt;&#xA;" OwnerUserId="3306" LastEditorUserId="1853" LastEditDate="2014-12-18T23:37:55.483" LastActivityDate="2014-12-22T08:48:30.297" Title="How to preserve axioms from the positive numbers in the negative numbers?" Tags="|algebra|negative-numbers|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6053" PostTypeId="2" ParentId="6045" CreationDate="2014-12-19T02:32:13.217" Score="4" Body="&lt;p&gt;While I have never actually taught pseudocode, my experience as a math teacher and a computer science student might provide some insight. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the exam for an object oriented programming courses the prof asked us to write a piece of pseudocode, and was chagrined that most students (including myself) chose to write out the full C++ code, since he, like you, had been using pseudocode in class. Challenge lay in that he had used pseudocode, but he had never explicitly taught it. This is similar to the way that most college math classes use proof, but do not explicitly teach students how to write a proof. This reflects the tension between content (object oriented programming, real analysis) and skills (pseudocode, proof). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be helpful to discuss different representations of code (specification, pseudodcode, code) and ask students to &quot;translate&quot; between representations. To do this you might give students an example of pseudocode and ask them to translate it into a specification or actual code or you might give them code or a specification and translate it into pseudocode. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a more general comment, students learn all sorts of bad habits when the correct answer is valued more the process. This is endemic in computer science courses where high value is placed on having functioning code rather than well written code. Just because you can't run tests on code that does not compile, but that does not mean that the code is worthless in terms of the ideas it represents. If the classes don't value the process part of code writing, then the students certainly won't.&lt;/p&gt;&#xA;" OwnerUserId="3485" LastActivityDate="2014-12-19T02:32:13.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6055" PostTypeId="2" ParentId="6052" CreationDate="2014-12-19T20:11:13.443" Score="2" Body="&lt;p&gt;Based on the comment by @GeraldEdgar and others, I'd like to answer that &quot;The whole is greater than the part&quot; is just not a valid axiom about numbers (of any sort). This statement is relevant for collections (e.g. finite sets, or everyday physical objects), but it doesn't quite make sense regarding numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To provide a concrete example, a statement that 2 is a half of 4 has in its essence a very different meaning than saying that the set {Eve, Sarah} constitutes a half of {Adam, Eve, Abraham, Sarah}. Speaking about parts of numbers appears to me to be an abuse of notation that inevitably leads to confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps you could provide us with a context for the statement of this &quot;axiom&quot;?&lt;/p&gt;&#xA;" OwnerUserId="688" LastActivityDate="2014-12-19T20:11:13.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6057" PostTypeId="2" ParentId="5948" CreationDate="2014-12-20T02:32:58.740" Score="1" Body="&lt;p&gt;Use an android or ipad tablet to represent a portion of an infinite 2 dimensional plane you can turn or incline that plane but any object you draw can only exist only on that plane. Then place a book on top of that plane and you have a third dimension. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe 15 graphing papers on top of each other bottom graphing paper is 0 on z axis, next graphing paper is 1 on z axis etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For easy transition 3d to coordinate system - childrens blocks representing  dimensionless points - a 2x2 grid (in case you don't have enough blocks) with labelled coordinates for 2 dimensions, another grid on top, for the third dimension. The students can label a block's coordinate for more interactivity. You can easily choose a 0,0,0 coordinate if you need to explain negative coordinates.&lt;/p&gt;&#xA;" OwnerUserId="3503" LastActivityDate="2014-12-20T02:32:58.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6058" PostTypeId="2" ParentId="4448" CreationDate="2014-12-20T15:58:43.183" Score="6" Body="&lt;p&gt;I've created a gallery of math activities, aimed at learners and children, with descriptions and images. You can see it at &lt;a href=&quot;http://mariuskempe.net/Activities/&quot;&gt;http://mariuskempe.net/Activities/&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="369" LastActivityDate="2014-12-20T15:58:43.183" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6059" PostTypeId="2" ParentId="4448" CreationDate="2014-12-20T16:23:46.267" Score="3" Body="&lt;p&gt;some things that i did in my elementary math circle (we called it math enrichment)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Fibonnaci Numbers (there are a plethora of activities involving Fib. with a simple Google search)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://mathcraft.wonderhowto.com/how-to/create-parabolic-curves-using-straight-lines-0131301/&quot; rel=&quot;nofollow&quot;&gt;Curve Stitching&lt;/a&gt; (Link shows the basics. We took shoeboxes and stitched curves using thin string from one side of the box to another. Very cool!)&lt;/li&gt;&#xA;&lt;li&gt;Pascal's Triangle a la the way it was presented in &lt;a href=&quot;http://en.wikipedia.org/wiki/The_Number_Devil&quot; rel=&quot;nofollow&quot;&gt;The Number Devil&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.math.washington.edu/~mathcircle/mmc/mmc2010/PicoFermiBagel.pdf&quot; rel=&quot;nofollow&quot;&gt;Pico, Bagel, Fermi&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.setgame.com/set&quot; rel=&quot;nofollow&quot;&gt;SET Card Game&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Symmetry (mostly involving making snowflakes and other fun hands-on activities)&lt;/li&gt;&#xA;&lt;li&gt;Origami (relating it to geometry and symmetry)&lt;/li&gt;&#xA;&lt;li&gt;Basic Geometric Constructions (expose students to using compass and straightedge)&lt;/li&gt;&#xA;&lt;li&gt;&quot;Mathy&quot; board games such as Connect Four, Mancala, or Othello&lt;/li&gt;&#xA;&lt;li&gt;Infinite series (have a student walk half way across the room, then walk half way across the remaining part of the room, then half way and half way etc. and discuss why they will never actually reach the other wall)&lt;/li&gt;&#xA;&lt;li&gt;Mobius Strips&lt;/li&gt;&#xA;&lt;li&gt;&quot;Non-standard&quot; functions (factorial, absolute value, modular arithmetic)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this list helps, i know some of these topics seem way to advanced for elementary students but all of these can be introduced in very basic, and easy to understand terms.  Gifted students should have a natural predilection for math so if you can just spark their interest in any one of these topics they will be willing to push their thinking to the limits in order to learn more.  Again, these were all things that I did in my elementary math circle and i am now a secondary math teacher :)&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2014-12-20T16:23:46.267" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6060" PostTypeId="5" CreationDate="2014-12-20T16:38:37.710" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-12-20T16:38:37.710" LastActivityDate="2014-12-20T16:38:37.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6061" PostTypeId="4" CreationDate="2014-12-20T16:38:37.710" Score="0" Body="For questions about teaching mathematics online and online resources. " OwnerUserId="39" LastEditorUserId="39" LastEditDate="2014-12-20T19:08:14.377" LastActivityDate="2014-12-20T19:08:14.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6062" PostTypeId="1" CreationDate="2014-12-21T03:11:07.343" Score="26" ViewCount="1749" Body="&lt;p&gt;I learned to do math proofs in college. But recently I have begun studying more advanced math books and I've noticed some mathematicians frequently make assumptions that I don't. When I write proofs, I tend to state every detail possible. But mathematicians often omit details or repetition. I used to think this was annoying but having written more proofs I realize such omissions create concision and may aid in understanding as a result. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said this, I do not feel I have a good grasp on what makes for good mathematical writing. Does anyone have any good resources on this topic? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark 1: I have extensively studied general writing and verbal communication and very few of the principles used in explaining things (eg use of analogies or metaphors) work well for mathematics -- at least at higher levels. So math writing is clearly a distinct skill. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark 2:&#xA;A very trivial, simple example of such concision is that when we refer to $\mathbb{R}^{n}$, we mean a set of numbers for $n=1$ but a set of ordered pairs for $n&amp;gt;1$. Even I wouldn't state this, but for younger learners, I'm sure that distinction isn't known initially. I certainly didn't and had to learn it.  &lt;/p&gt;&#xA;" OwnerUserId="3471" LastEditorUserId="3471" LastEditDate="2014-12-21T14:49:47.757" LastActivityDate="2019-12-02T15:53:59.117" Title="How to write like a mathematician?" Tags="|writing|" AnswerCount="8" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6063" PostTypeId="2" ParentId="6062" CreationDate="2014-12-21T03:46:39.390" Score="3" Body="&lt;p&gt;The first example I know of that comes to mind when I read your question is &quot;Clean Writing in Mathematics&quot; found in Appendix 4 of &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387983791&quot; rel=&quot;nofollow&quot;&gt;Calculus: A Liberal Art&lt;/a&gt; by W. M. Priestley. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, writing mathematics is just like any other skill: The more you practice and the more you see others do it, the better you'll get. I find I take bits of pieces of styles from mathematicians that I read and admire most, thus creating my own unique style.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2014-12-21T03:46:39.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6064" PostTypeId="2" ParentId="6062" CreationDate="2014-12-21T04:15:45.980" Score="4" Body="&lt;p&gt;I strongly recommend Steven G. Krantz's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0821806351&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;A Primer of Mathematical Writing&lt;/em&gt;&lt;/a&gt;. I consult it regularly when I'm not doing recreational writing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are writing something for fun or for your own benefit alone, do whatever you think will help you think best.&lt;/p&gt;&#xA;" OwnerUserId="3525" LastActivityDate="2014-12-21T04:15:45.980" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6065" PostTypeId="2" ParentId="6062" CreationDate="2014-12-21T07:55:22.233" Score="5" Body="&lt;p&gt;Franco Vivaldi has a useful book on mathematical writing available from Springer: &lt;a href=&quot;http://www.springer.com/mathematics/book/978-1-4471-6526-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Writing&lt;/a&gt; (2014).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also say in response to Remark 1 that analogies and metaphors remain valuable in mathematical writing, as they can aid understanding, but they must be used with care. For instance, if a metaphor contradicts an established mathematical definition, then the definition wins.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2019-12-02T06:03:11.343" LastActivityDate="2019-12-02T06:03:11.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6066" PostTypeId="2" ParentId="41" CreationDate="2014-12-21T19:02:46.027" Score="14" Body="&lt;p&gt;This is my handwriting font for mathematics. (Excuse the shakey-ness, this was made using the stylus on my phablet.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/r7lkQ.jpg&quot; alt=&quot;&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It isn't as minimalist as some of yours, but I have never had anyone tell me they had trouble reading my writing. I make it clear which letters are which by using size and giving attention to negative space. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find that using big loops and serifs is really good for board work, where it can otherwise be difficult to distinguish capital letters. (Spencerian cursive is actually pretty good for board work too, just not for equations.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The prototypical gesture here is the J, from left to right to down. That is the leftmost part of many capitals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keeping track of where the bottom line of the word is also important, for example the tail on the lower case t is just below thst bottom line, and it is crossed exactly on the middle letter line. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My biggest problem is that my vs look like nus.&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2015-02-08T21:05:32.927" LastActivityDate="2015-02-08T21:05:32.927" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6067" PostTypeId="2" ParentId="6062" CreationDate="2014-12-22T08:19:17.453" Score="12" Body="&lt;p&gt;I'd suggest Halmos' &quot;&lt;a href=&quot;http://www.math.washington.edu/~lind/Resources/Halmos.pdf&quot; rel=&quot;noreferrer&quot;&gt;How to Write Mathematics&lt;/a&gt;&quot;&lt;/p&gt;&#xA;" OwnerUserId="3528" LastEditorDisplayName="user7990" LastEditDate="2019-12-02T14:57:39.420" LastActivityDate="2019-12-02T14:57:39.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6068" PostTypeId="2" ParentId="6052" CreationDate="2014-12-22T08:48:30.297" Score="1" Body="&lt;p&gt;Reflecting on this, I think the difficulty comes because there are 2 ways of interpreting integers which have different ordering functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first (and standard) interpretation is as a totally ordered set, from $^-\infty$ to $\infty$. From this perspective $^-4 &amp;lt; {^-2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second way (as used in high school physics) is to interpret integers as a 1 dimensional vector. The direction is given by $^+/_-$, and the magnitude is given by a totally ordered set from $0$ to $ \infty$. From this perspective, it would make sense to think of $^-4 &amp;gt; {^-2}$, as a velocity of $^-4$ is clearly bigger than a velocity of $^-2$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, to be mathematically consistent, interpretation 2 integers are only partially ordered, with no overall supremum, as $^+4$ and $^-4$ are not able to be compared. It may be best not to call directed natural numbers such as these integers, and refer to them as 1D vectors. However, they do conform to your whole/part intuition, and are routinely used according to this intuition, while still (incorrectly?) being called integers. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2014-12-22T08:48:30.297" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6069" PostTypeId="1" CreationDate="2014-12-22T09:35:58.637" Score="5" ViewCount="188" Body="&lt;p&gt;In this question I would like to ask you about material showing the uses (or occurrences) of mathematics in the everyday world. The aim is to encourage with it a group of young undergraduate mathematics students, and for this reason I am looking for:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Material rich in pictures, animations, interactive applets perhaps, which may attract the attention of those young students, and useful for conferences and other activities.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Pedagogical material: showing the power of mathematics perhaps without having studied the topic rigorously.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Material dealing with advanced mathematics: the real challenge is to show the importance of advanced mathematics, because basic mathematics examples are abundant and overused.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Of course, besides material, any idea or suggestion is welcomed. Perhaps it is also helpful to have a look at the same question I asked in Mathematics: &lt;a href=&quot;https://math.stackexchange.com/questions/1070873/sources-for-mathematics-outside-the-mathematics-world&quot;&gt;https://math.stackexchange.com/questions/1070873/sources-for-mathematics-outside-the-mathematics-world&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3529" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-06-26T23:25:49.793" Title="Sources for mathematics outside the mathematics world" Tags="|undergraduate-education|secondary-education|student-motivation|interactive-teaching|applications|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6070" PostTypeId="1" CreationDate="2014-12-22T17:09:05.237" Score="3" ViewCount="364" Body="&lt;p&gt;I finished high school 2 years ago and now I'm stuck in a university in Turkey. I am interested in learning precalculus, discrete mathematics, physics and chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question: I need to learn math and physics from the bottom, and I mean by from the bottom all the basic material, possibly starting in from precaluclus and up to calculus, etc. I would also like advice for anything related to buying books or searching online for material. Anything would really be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="3533" LastActivityDate="2014-12-22T21:03:24.770" Title="I want to learn math from the beginning" Tags="|mathematical-pedagogy|precalculus|" AnswerCount="1" CommentCount="4" ClosedDate="2014-12-23T07:40:11.963" ContentLicense="CC BY-SA 3.0" />
  <row Id="6071" PostTypeId="2" ParentId="6026" CreationDate="2014-12-22T17:37:52.823" Score="4" Body="&lt;p&gt;Sometimes our past experiences pile up, and create anxiety from the stories we tell ourselves about how bad we are at something, or how often we mess it up. One way to lessen anxiety is to do a guided visualization, telling ourselves more positive things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My students often say they blanked on a test, even though they felt like they really knew the material. So I created &lt;a href=&quot;http://mathmamawrites.blogspot.com/2009/09/math-relax-guided-visualization-for.html&quot; rel=&quot;nofollow&quot;&gt;a guided visualization for relaxing with math study and math tests&lt;/a&gt;. Check it out, and see if it helps any. (Though working with others may be more helpful in your situation.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-12-22T17:37:52.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6072" PostTypeId="2" ParentId="6070" CreationDate="2014-12-22T21:03:24.770" Score="2" Body="&lt;p&gt;Just to give you some sense of how broad is your question, here is a small sampling of Precalculus textbook cover images:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Q6vW7.jpg&quot; alt=&quot;PreCalcBooks&quot;&gt;&#xA;&lt;hr /&gt;&#xA;Perhaps start with this MSE question &quot;&lt;a href=&quot;https://math.stackexchange.com/q/264591/237&quot;&gt;what is the best book for Pre-Calculus?&lt;/a&gt;.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-12-22T21:03:24.770" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6073" PostTypeId="2" ParentId="6062" CreationDate="2014-12-22T21:10:39.637" Score="8" Body="&lt;p&gt;This is not the type of resource you request, but if you seek examples&#xA;of excellent expositions geared toward the general public:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The &lt;em&gt;Best Writing on Mathematics&lt;/em&gt; series (&lt;a href=&quot;https://press.princeton.edu/books/paperback/9780691198354/the-best-writing-on-mathematics-2019&quot; rel=&quot;nofollow noreferrer&quot;&gt;Princeton link&lt;/a&gt;). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now &lt;strike&gt;five&lt;/strike&gt; ten years: 2010, ..., 2019. All edited by Mircea Pitici.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/FUqAE.jpg&quot; width=&quot;150&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-12-02T15:53:59.117" LastActivityDate="2019-12-02T15:53:59.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6074" PostTypeId="1" AcceptedAnswerId="6112" CreationDate="2014-12-22T21:45:08.687" Score="9" ViewCount="330" Body="&lt;p&gt;My professor advised us to get either Hardy&amp;amp;Wright or Ireland&amp;amp;Rosen for our introductory number theory course. I would like to ask what are the differences between these textbooks in terms of pedagogical effectiveness; completeness; relevance; difficulty; material covered (also, some personal thoughts and advice would be really good). Are there other books which I should prefer to the ones mentioned above?&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="2083" LastEditDate="2021-07-29T04:56:57.103" LastActivityDate="2021-07-29T04:56:57.103" Title="Differences between Hardy&amp;Wright and Ireland&amp;Rosen for number theory course" Tags="|undergraduate-education|textbooks|number-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6075" PostTypeId="1" CreationDate="2014-12-22T22:25:30.037" Score="9" ViewCount="387" Body="&lt;p&gt;I have been strongly recommended to read the book &lt;a href=&quot;http://www.springer.com/mathematics/algebra/book/978-0-387-78214-0&quot;&gt;Naive Lie Theory&lt;/a&gt;. In the introduction one can read: &quot;This naive approach to Lie theory is originally due to von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to ask:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;first of all, what is Lie theory and is there some agreement that it should be a fundamental part of undergraduate education?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;could you elaborate a little on the book mentioned above? Is the naive approach somewhat &lt;em&gt;too&lt;/em&gt; naive? Should I prefer other references?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="3251" LastEditDate="2014-12-23T11:14:16.177" LastActivityDate="2014-12-25T13:24:27.193" Title="Lie Theory: significance and relevance to undergraduate education" Tags="|undergraduate-education|textbooks|algebra|curriculum|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6076" PostTypeId="1" CreationDate="2014-12-22T22:50:24.150" Score="6" ViewCount="313" Body="&lt;p&gt;I would like to ask if, in the research field of mathematical education, some work has been done to investigate the relationship between 1) and 2):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;mathematical education and student motivation&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the great quantity of &amp;quot;folkloristic&amp;quot; writing (so to say) on trivia and anecdotes about the mathematical community (e.g.  the ones collected in &lt;em&gt;Mathematical Apocrypha Redux&lt;/em&gt; by Steven Krantz).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;To be clear, I want to know if there are links between cultural stories of mathematics (which may have reached &amp;quot;apocryphal&amp;quot; status), and with student performance and/or motivation in mathematics in those cultures.&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="4791" LastEditDate="2021-01-06T13:44:29.120" LastActivityDate="2021-01-06T13:44:29.120" Title="Links between mathematical folklore and educational success" Tags="|undergraduate-education|mathematical-pedagogy|student-motivation|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6077" PostTypeId="1" AcceptedAnswerId="6080" CreationDate="2014-12-23T00:11:00.747" Score="14" ViewCount="1258" Body="&lt;p&gt;I've been asked by an high school student what are the $5$ major theorems in Lang's Linear Algebra (and therefore, by extension, in an undergraduate linear algebra course). Firstly, I bluntly said that the question holds a great deal of subjectivity and that I couldn't answer it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, coming to think of it, I realized that the question is indeed somewhat well-posed. Indeed, it could be natural to ask which of the theorems presented in a course are most useful for the prosecution of the study; and it requires some maturity to pick the key ideas that are most relevant for a subject as a whole.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I would like to ask what are the $5$ theorems that best epitomize an undergraduate linear algebra course and communicate the key ideas of the subject.&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="3251" LastEditDate="2014-12-23T14:27:39.943" LastActivityDate="2015-07-04T23:35:38.420" Title="Key theorems in undergraduate linear algebra" Tags="|undergraduate-education|curriculum|linear-algebra|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6078" PostTypeId="2" ParentId="6077" CreationDate="2014-12-23T02:15:13.750" Score="2" Body="&lt;p&gt;Here's my list, but I'm curious what others will say...&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Properties of the extended Echelon form.&lt;/li&gt;&#xA;&lt;li&gt;The Four subsets theorem.&lt;/li&gt;&#xA;&lt;li&gt;det(AB)=det(A)det(B).&lt;/li&gt;&#xA;&lt;li&gt;Every finite n dimensional complex vector space is isomorphic to C^n&lt;/li&gt;&#xA;&lt;li&gt;Every linear transformation is equivalent to matrix multiplication.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In case the names for 1 and 2 aren't standard, see &lt;a href=&quot;http://linear.ups.edu/html/section-FS.html&quot; rel=&quot;nofollow&quot;&gt;Theorems PEEF and FS here.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2014-12-23T02:15:13.750" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6079" PostTypeId="2" ParentId="6077" CreationDate="2014-12-23T04:12:00.217" Score="3" Body="&lt;p&gt;Let's see. If I had to pick, I'll keep it $\mathbb{R}$,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The linear correspondence theorem; any linear combination of columns present in $A$ is likewise found in the row reduced form of $A$. It follows that the solutions of $Av=b$ are easily read from $\text{rref}(A|b)$.&lt;/li&gt;&#xA;&lt;li&gt;A square matrix $A$ is invertible iff $Av=0$ has only $v=0$ solution iff $Null(A)= \{0 \}$ iff $Col(A) =\mathbb{R}^n$ iff $\text{det}(A) \neq 0$ iff $A$ does not have zero as an eigenvalue iff $Av=b$ has a solution for all $b$ iff $Row(A) = \mathbb{R}^{1 \times n}$ iff $rref(A)=I$.&lt;/li&gt;&#xA;&lt;li&gt;$L: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a linear transformation if and only if $L(v)=Av$ for $A \in \mathbb{R}^{ \ m \times n}$ and matrix multiplication is defined such that the matrix of the composite is the product of the composed transformations. Consequently matrix multiplication is associative, but not commutative, as it is essentially a covert notation for function composition (ok, I cheated, there are two or three here)&lt;/li&gt;&#xA;&lt;li&gt;The dimension of a finite dimensional vector space is well-defined; if there exists a finite minimal spanning set then the number of vectors in that set is an invariant for a given vector space.&lt;/li&gt;&#xA;&lt;li&gt;A matrix is diagonalizable iff it has an eigenbasis.(but, I'd really like to replace this with every matrix has a real Jordan form)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="128" LastActivityDate="2014-12-23T04:12:00.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6080" PostTypeId="2" ParentId="6077" CreationDate="2014-12-23T04:27:44.987" Score="4" Body="&lt;p&gt;It depends on your view of what is crucial to the topic. In no particular order, here's what I think is critical for a linear algebra course.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Rank-nullity Theorem: Very useful to show isomorphism&lt;/li&gt;&#xA;&lt;li&gt;Cayley-Hamilton Theorem&lt;/li&gt;&#xA;&lt;li&gt;Jordan Normal Form&lt;/li&gt;&#xA;&lt;li&gt;The Gram-Schmidt process&lt;/li&gt;&#xA;&lt;li&gt;Dual Spaces&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="201" LastActivityDate="2014-12-23T04:27:44.987" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6081" PostTypeId="1" CreationDate="2014-12-23T05:45:06.433" Score="5" ViewCount="233" Body="&lt;p&gt;Three months ago, we defined a programming project for some gifted middle school students: Writing code to generate simple fractals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two of them have finished writing Pascal programs already! The programs generate fractals using L-Systems, and do virtually everything  can be done &lt;a href=&quot;http://www.kevs3d.co.uk/dev/lsystems/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. We didn't think that just three month would be enough for them to finish this part of the project!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question: How can they continue their fractal generation project? What can be their next destination?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please keep in mind that they're 8th grade students, so know nothing about complex numbers, and their exposed to algebra just a year ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I &lt;a href=&quot;https://math.stackexchange.com/posts/1077561/edit&quot;&gt;posted&lt;/a&gt; this question in Math.SE. but got no answer.)&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-12-25T12:59:20.053" Title="Fractal Generation Project: How to Continue?" Tags="|secondary-education|gifted-students|fractals|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6082" PostTypeId="2" ParentId="6081" CreationDate="2014-12-23T12:44:09.080" Score="4" Body="&lt;p&gt;You might extend the rules so the &quot;turtle&quot; moves in 3 dimensions. &#xA;It is not much more difficult, but the results can be stunning.&#xA;(You will need software to display 3D graphics.)&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/PW2Cq.gif&quot; alt=&quot;3DPlant&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;(&lt;a href=&quot;http://www.nbb.cornell.edu/neurobio/land/OldStudentProjects/cs490-94to95/hwchen/&quot; rel=&quot;nofollow noreferrer&quot;&gt;HW Chen Cornell Univ link&lt;/a&gt;)&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/c08wS.jpg&quot; alt=&quot;Serbian3DLsystem&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;(&lt;a href=&quot;http://elearning.rcub.bg.ac.rs/moodle/course/view.php?id=151&amp;amp;aformatforumid=13&quot; rel=&quot;nofollow noreferrer&quot;&gt;Univ Belgrade link&lt;/a&gt;)&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/vQdFV.jpg&quot; alt=&quot;Hilbert&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;(&lt;a href=&quot;http://cyberxaos.blogspot.com/2006/04/3d-l-systems-at-last.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;cyberxaos link&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2014-12-25T12:59:20.053" LastActivityDate="2014-12-25T12:59:20.053" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6083" PostTypeId="1" CreationDate="2014-12-23T12:58:26.203" Score="20" ViewCount="8960" Body="&lt;p&gt;I am interested in comparing mathematics (and also physics) final examinations from all around the world for the highest secondary school qualification in the particular country. Especially I am interested in countries which ranked high in TIMSS/PISA. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the German Abitur (I am a mathematics and physics teacher in Germany), there are some web sites which collect problems and solutions of the Abitur exams of the last years for example: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://ne.lo-net2.de/selbstlernmaterial/m/abi/abiindex.html&quot;&gt;http://ne.lo-net2.de/selbstlernmaterial/m/abi/abiindex.html&lt;/a&gt; (for Mathematics for different federal states)&lt;br&gt;&#xA;&lt;a href=&quot;http://ne.lo-net2.de/selbstlernmaterial/p/abi/abiindex.html&quot;&gt;http://ne.lo-net2.de/selbstlernmaterial/p/abi/abiindex.html&lt;/a&gt; (for Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I am looking for similar links for the original problems (and possibly solutions) of the final examinations of other countries (or other examinations roughly equivalent to the German Abitur). Ideally it would be great to have a version which was translated to English (or German) but having the exam in the original language would be sufficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am also interested in comparing older (or even historical) exams of different countries if available (though I didn't find a good link for this even for German exams).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please add if Calculators, Graphical Calculators or Computer Algebra Systems are allowed. Same for formula sheets/booklets, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clearly one might find such resources using a search engine, but because of the different languages and school systems this is not so easy. Because of the international user base, this site seems to be a good place to ask this question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; It would be useful if you could add some background about the exam system of the particular country or state. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of Germany:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The German exams linked above are part of the German Abitur, which allows you to study every subject in university if you pass it (&quot;Allgemeine Hochschulreife&quot;). The Abitur is slightly different in each federal state, but you can roughly say this: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Main subjects of the math exams are calculus, vector geometry (and very little linear algebra) and elementary probability theory (depending on the year and the federal state one of the latter might be left out)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Over the last 10 years most states changed to a system, where students graduate after the 12th grade and everybody has to take a mathematics course in grades 11 and 12 with 4x 45 min lessons a week. Based on this course (and implicitly content of the lower grades) they have to write the exams at the end of the 12th grade. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Before that most states had a system where students graduated after the 13th grade and were allowed to choose between a typcically 3x45 min per week math course in 12 and 13 (&quot;Grundkurs&quot;) or a 5x45 min per week math course (&quot;Leistungskurs&quot;) with different exams at the end of the 13th year. In the Leistungskurs, typically some very basic real analysis was part of the curriculum (convergence of sequences, continuity (using sequences or $\epsilon$-$\delta$ test), some proof of theorems of calculus, proof by induction etc) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In the last 10 years many states introduced graphical calculators (&quot;GTR&quot;) or even computer algebra systems (&quot;CAS&quot;) which were allowed in the exams. Often there was a part without any calculator and a part with a GTR or CAS. Before that students had typically only a usual scientific calculator. Usually a table of formulas is available for the students in the exams. One might add that in the case of Baden-Württemberg the GTR and CAS are abandoned for the coming Abitur 2019 (on &quot;Allgemeinbildenen Gymnasien&quot; and 2017 on &quot;Beruflichen Gymnasien&quot;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1203" LastEditorUserId="1203" LastEditDate="2014-12-24T10:46:45.193" LastActivityDate="2022-12-06T08:54:04.810" Title="Worldwide collection of final examinations in mathematics" Tags="|secondary-education|exams|" AnswerCount="6" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6085" PostTypeId="2" ParentId="6083" CreationDate="2014-12-23T17:46:51.420" Score="3" Body="&lt;p&gt;Around 1997-1998 I became aware of some neat math problems on the internet. This was back in the day when you could pretty much find everything of mathematical interest on the internet with search engines and by knowing some places to visit where sites of interest were collected, although in this particular case I might have been led to the problems by a Russian mathematician (student of E. P. Dolzhenko) that I was in correspondence with. At the time I was teaching high school (a specialized math-science boarding high school for gifted students), and in one of the classes (multivariable calculus) I had only 5 or 6 students, all very strong (possibly all now have a graduate degree in a science field, and one has a Ph.D. in mathematics from Rice University), so we did these &quot;time-outs&quot; two or three times a week for the last 15 or 20 minutes of class to work on the problems that I had found on the internet. (Just now I had to go back to some old LaTeX class handouts to get the URLs, then use the Internet Archive to locate valid URLs for them.) The problems are in Russian, but many can be understood without being able to read Russian. It also helped in my case that one of my students was fluent in Russian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20001006124351/http://mech.math.msu.su/admission/var1995.html&quot; rel=&quot;nofollow&quot;&gt;1995 problems&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20000417123105/http://mech.math.msu.su/admission/var1996.html&quot; rel=&quot;nofollow&quot;&gt;1996 problems&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20000417160901/http://mech.math.msu.su/admission/var1997.html&quot; rel=&quot;nofollow&quot;&gt;1997 problems&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20001006174447/http://mech.math.msu.su/admission/var1998.html&quot; rel=&quot;nofollow&quot;&gt;1998 problems&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding mathematics problems on the Moscow entrance exams, much has been written about them lately. The following seem to be especially useful discussions of these problems:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alexander Shen, &lt;a href=&quot;http://www.3038.org/press/shen.pdf&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Entrance Examinations to the Mekh-mat&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;The Mathematical Intelligencer&lt;/strong&gt; 16 #4 (1994), 6-10.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tanya Khovanova and Alexey Radul, &lt;a href=&quot;http://arxiv.org/abs/1110.1556&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Jewish Problems&lt;/em&gt;&lt;/a&gt;, arXiv:1110.1556v2, 15 October 2011, 21 pages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jay Egenhoff, &lt;a href=&quot;http://www.math.umt.edu/tmme/vol11no3/Egenhoff_10.pdf&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Math as a Tool of Anti-Semitism&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;The Mathematics Enthusiast&lt;/strong&gt; 11 #3, 649-664.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tanya Khovanova's Math Blog, &lt;a href=&quot;http://blog.tanyakhovanova.com/2011/05/a-math-exams-hidden-agenda/&quot; rel=&quot;nofollow&quot;&gt;A Math Exam’s Hidden Agenda&lt;/a&gt;, 11 May 2011.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2014-12-23T17:46:51.420" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6086" PostTypeId="2" ParentId="416" CreationDate="2014-12-23T18:54:22.160" Score="2" Body="&lt;p&gt;Exponential notation is a shorthand for repeated multiplication in the same way that multiplication is a shorthand for repeated addition. Where multiplication returns an addend, exponentiation returns a multiplicand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In multiplication, $0x = 0$ since adding no '$x$'s is the same as doing no addition, which is the same as adding zero.&lt;br&gt;&lt;br&gt;&#xA;In exponentiation, $x^0 = 1$ since multiplying by no '$x$'s is the same as doing no multiplication, which is the same as multiplying by one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no need for special consideration in the case that $x=0$. Not multiplying by zero is the same as not multiplying by any other number.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2014-12-23T18:54:22.160" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6087" PostTypeId="2" ParentId="416" CreationDate="2014-12-23T19:19:12.543" Score="7" Body="&lt;p&gt;Others have already given good reasons for the usual convention that $0^0=1$.  A while ago, I explained at &lt;a href=&quot;https://math.stackexchange.com/questions/475337/&quot;&gt;https://math.stackexchange.com/questions/475337/&lt;/a&gt; why I consider  the most common alternative, namely to declare $0^0$ undefined, to be inadequately motivated.  Specifically, it serves only to protect calculus students from certain sloppy mistakes, and alternative protection is available, namely not to assume continuity at points of discontinuity.&lt;/p&gt;&#xA;" OwnerUserId="326" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2014-12-23T19:19:12.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6088" PostTypeId="2" ParentId="6083" CreationDate="2014-12-24T03:32:44.683" Score="4" Body="&lt;p&gt;For the United States (NYC, specifically): You can find the archive of Regents Examinations &lt;a href=&quot;http://www.jmap.org/JMAP_REGENTS_EXAM_ARCHIVES.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(See also the Regent Examinations &lt;a href=&quot;http://en.wikipedia.org/wiki/Regents_Examinations&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;wikipage&lt;/strong&gt;&lt;/a&gt; if you are unfamiliar with these assessments.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mathematics examinations are available from 1866 through today, though the archive is incomplete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One prominent researcher of the math regents is Eileen Donoghue, cf. &lt;a href=&quot;https://matheducators.stackexchange.com/a/1838/262&quot;&gt;&lt;strong&gt;MESE 1820&lt;/strong&gt;&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/a/806/262&quot;&gt;&lt;strong&gt;MESE 369&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an example of the start of the earliest test in the archive (1866):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/6hooo.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here is an example of the start of a recent test in the archive (2014):&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/HsG6U.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2014-12-24T03:32:44.683" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6089" PostTypeId="1" CreationDate="2014-12-24T09:50:40.383" Score="12" ViewCount="1678" Body="&lt;p&gt;I recently started giving maths lessons and it seems like I am at my wits end. My own background is: I'm a masters student in physics, already did several tutorials for younger students, especially for a course on mathematical methods in physics. Of course, some years ago when going to school myself, I was helping some of my friends with school maths, so teaching school maths is also not entirely new to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However the current situation really gives me a headache. The pupil is going to 8th grade, the current topics are simple systems of two linear equations in two variables and the intuitive methods of solving them. The problem which I and the pupil are facing is however another one: She really has a hard time doing simple calculations with numbers, multiplying, dividing, working with fractions. It is not a problem of understanding. Given a big amount of time she can do all the calculation using long division and multiplication. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However she is entirely missing what I would call &quot;intuition for numbers&quot;, i.e. she doesn't see if one number is a multiple of another (i.e. she doesn't notice that 8/4 can be simplified to 2), she doesn't know the most useful fraction to decimal fraction conversions (i.e. 1/4=0.25). She knows the 10x10 multiplication table in principle, but it takes a huge amount of time till she recalls the result. Sometimes she even needs some seconds to calculate 2x3=6.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can imagine that it is very hard teaching new topics in such a situation. A week ago we discussed the methods for solving systems of linear equations and I intentionally gave only examples with small coefficients and such that no fractions appeared at all. The result was that she understood the topic very well and could solve such exercises. However, if in an exam she encounters exercises with higher numbers or examples where fractions will pop up during the calculations, she again won't be able to solve them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And now I'm really stuck and don't know how to proceed. Make her learning the multiplication table by heart? But she already knows it, sort of. However, doing operations with numbers in her head just doesn't work properly. Returning to standard exercises from 6th and 7th grade? Takes a huge amount of time and doesn't really solve the actual problem since the exercises will be diluted with other concepts which are not of importance now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for any hint,&#xA;Michael&lt;/p&gt;&#xA;" OwnerUserId="3542" LastActivityDate="2019-10-10T02:06:53.273" Title="How to teach quick multiplication and division in head?" Tags="|fractions|multiplication-table|division|" AnswerCount="7" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6090" PostTypeId="2" ParentId="6083" CreationDate="2014-12-24T11:23:25.963" Score="7" Body="&lt;p&gt;The Sixth Term Examination Papers are national and public examinations taken by 18 year old school leavers (pre-university), intending to study mathematics at one of the more selective/over-subscribed UK universities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although there is a close association with Cambridge University - and are considered to be entrance papers to read mathematics there - several UK universities now ask for grades (although not particularly high) in mathematics STEP.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because STEP are public examinations open to any candidate, they are based on what is considered to be &quot;core&quot; syllabus concepts. So in some sense, the range of questions is quite narrow, but in practice the questions are usually much more demanding than standard A level, with each one drawing upon a number of different syllabus areas and often requiring the application of standard techniques in novel/interesting ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much information is available here &lt;a href=&quot;http://www.admissionstestingservice.org/for-test-takers/step/preparing-for-step/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and in a &lt;a href=&quot;http://www.maths.cam.ac.uk/undergrad/admissions/step/advpcm.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;study booklet&lt;/a&gt;, solutions and past papers are available.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Taken by 18 year old school leavers for university entrance (Abitur&#xA;equivalent) &lt;/li&gt;&#xA;&lt;li&gt;No calculator, books or other material is permitted, a formula sheet is provided.&lt;/li&gt;&#xA;&lt;li&gt;Examinations are three hours.&lt;/li&gt;&#xA;&lt;li&gt;Candidates attempt 6 questions, out of a possible 14 (or so). Questions can be taken from pure, statistics and mechanics.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For standard A Level mathematics, the UK consists of 7 (yes, 7) examination boards - but information is available &lt;a href=&quot;http://www.aqa.org.uk/subjects/mathematics/a-level/mathematics-6360&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Both STEP and A Level are directly comparable to the German Abitur.&#xA;&lt;img src=&quot;https://i.stack.imgur.com/4IthC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="3541" LastEditorUserId="3541" LastEditDate="2014-12-24T11:38:05.873" LastActivityDate="2014-12-24T11:38:05.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6091" PostTypeId="2" ParentId="6077" CreationDate="2014-12-24T14:59:02.113" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Understanding the connection between matrix multiplication and composition of linear maps.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The rank-nullity theorem&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Complex linear maps always have eigenvectors.  Maybe that we can always find a basis where the matrix is upper triangular?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Gram-Schmidt algorithm/theorem&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The real and complex spectral theorems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastEditorUserId="4930" LastEditDate="2015-07-04T23:35:38.420" LastActivityDate="2015-07-04T23:35:38.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6092" PostTypeId="2" ParentId="6089" CreationDate="2014-12-24T15:59:58.327" Score="5" Body="&lt;p&gt;In my experience, I feel this is best learned by just practicing.  I understand that it may seem remedial for whatever lessons you are giving (or perhaps it isn't) and that it can take away from time you need to teach other subjects. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I honestly do find that I have certain values (multiplication tables, special fractions) just &quot;memorized&quot;, but I still know how to calculate the values if I ever forget them (i.e., doing the long division for fractions).  Whenever I would teach about fractions I would always do the long division first and point out the ones that are really important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, sometimes fractions can be easier to remember when they are related to something used in everyday life.  In the U.S., a quarter is .25 or 25 cents.  Most students know the value of the quarter so it's just teaching them that $1 \over4$ is another way of saying a quarter.  2 quarters is .50 or 50 cents just like $2 \over 4$ is $1 \over2$ and things like that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course every student learns differently and other countries don't have the same denominations of coins as here in the US, but I was just providing that as an example. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something that a lot of students these days do have are smart phones and I have seen apps that can test quick calculations like these (it's sort of like the flashcards students used to use when I was younger).  Or, perhaps the students can just use actual flashcards with a friend or relative (or themselves).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;tl;dr:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe for the simple fractions and quick multiplication you mentioned, it all comes down to repetition and practice, but there are plenty of resources to get that practice if the student is willing to provide the time for it (10 minutes on the smartphone before bed doing quick math quizzes instead of candy crush could be good!)&lt;/p&gt;&#xA;" OwnerUserId="3522" LastActivityDate="2014-12-24T15:59:58.327" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6093" PostTypeId="2" ParentId="6081" CreationDate="2014-12-24T16:39:42.610" Score="2" Body="&lt;p&gt;If you want the project to be a mathematics project rather than only a programming project, then the next stage would be to do a proper maths investigation, since now they have a tool they can use to investigate the connection between the rules of their L-system and the resulting fractals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Get them to try changing the parameters in various ways and see what happens to the resulting fractal, then record this information in a systematic way. For example, they could keep the same rule and change the angle, then record any features of the fractal they notice that change as the angle changes, and what stays the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would get them to choose one of the things they found that was interesting and create a presentation about it. Perhaps they could video themselves doing the presentation so this too is a product they can be proud of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another option is for them to create a &quot;guess my rule&quot; competition, where they create fractals for each other and they have to figure out what rules created them. This would motivate them to try lots of different things to get a fractal worth entering into the competition for others, and it should also motivate them to investigate what features of a fractal correspond to what features of the rules of the L-system in order to guess others' rules. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again I would get them to do a presentation describing their investigation process and how they made the final decision of the rule -- a bit like a crime scene investigation.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2014-12-24T16:39:42.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6094" PostTypeId="2" ParentId="6089" CreationDate="2014-12-24T17:05:36.767" Score="2" Body="&lt;p&gt;Permit me to recommend this book as a source of fun topic ideas:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Benjamin, Arthur, and Michael Shermer. &lt;em&gt;Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks&lt;/em&gt;. Random House LLC, 2008.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And see &lt;a href=&quot;https://www.youtube.com/watch?v=1JW9BA57aR8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Art Benjamin's YouTube video&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/l3dAd.png&quot; alt=&quot;ArtBenjamin&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-12-24T17:05:36.767" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6095" PostTypeId="2" ParentId="6026" CreationDate="2014-12-24T19:31:43.433" Score="1" Body="&lt;p&gt;In my teaching years (34), many times, the anxious students would come up to me and say: &quot;I don't understand anything.&quot; My work was not mainly to give them an answer, but of helping them find out exactly what they didn't understand so that they could 1) focus on what they don't understand and 2) use what they did understand to advance their work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe you could make a list of what you find easy and what you find difficult. You could also add to every subject of this list if you understand the element or not. You may find out that there are subjects you find easy but don't understand; don't worry about that for the moment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Armed with this list, try to figure out how you could prepare yourself for an exam. You could also build an exam in which you think you'd have a perfect score. Also, try to identify in your list the subjects you think you could better understand by reading your textbook and/or doing more exercises. Then identify the subjects where you think a friend could help you out. And finally identify the subjects for which only your teacher could come to your aid and go see him or her with questions pertaining to those subjects. Prepare your questions in writing; read them over to be sure you'll be asking the questions that are important to you and that the teacher will understand you.&lt;/p&gt;&#xA;" OwnerUserId="503" LastActivityDate="2014-12-24T19:31:43.433" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6096" PostTypeId="2" ParentId="6089" CreationDate="2014-12-24T19:48:44.847" Score="3" Body="&lt;p&gt;In fourth grade, our daily math class included a three minute timed exercise.  Each student had a page of approximately 50 or 100 math problems.  All of the math problems were of one kind -- either adding 2-digit numbers, or subtracting 2-digit numbers, or multiplying numbers between 0 and 12, or dividing numbers between 0 and 144 by numbers between 1 and 12.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you should get a book of these worksheets for your student, and have her do one 3-minute exercise (on one worksheet) per day.  The worksheet itself won't improve her skills, but it will:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;give her a chance to practice&lt;/li&gt;&#xA;&lt;li&gt;let her see her improvement&lt;/li&gt;&#xA;&lt;li&gt;give her a goal of being able to do arithmetic quickly.&lt;/li&gt;&#xA;&lt;li&gt;learn to spot what she knows how to do, and do those problems first.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can also teach her some valid shortcuts and estimating techniques.  I would teach one new shortcut per lesson, and have her see how much that shortcut helps with her 3-minute timed exercise.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2014-12-25T01:34:48.770" LastActivityDate="2014-12-25T01:34:48.770" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6097" PostTypeId="2" ParentId="4448" CreationDate="2014-12-24T20:22:27.740" Score="2" Body="&lt;p&gt;&lt;strong&gt;How high can you count on your fingers?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Most people can count from 1 through 10.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I use roman numerals to count from 1 through 99.  (right fingers = Is; right thumb = V; left fingers = Xs; left thumb = L)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Basketball referees can count from 0 through 35.  (They use base 6.  Each hand is from 0 through 5, and the uniform numbers don't include any digits from 6 through 9.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some people can use binary to count from 0 through 1023.  (Warn them they'll get in trouble if they go around flashing the numbers 4, 128, or 132.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/1834#1861&quot;&gt;How to do Check-By-Substitution&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;As I have explained in other answers, you can teach them to check their work (especially on story problems).  This will give them a huge advantage going forward.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How to write out very big numbers&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;thousands, millions, billions, trillions... decillions&lt;/li&gt;&#xA;&lt;li&gt;1,000;  1,000,000;  1,000,000,000; 1,000,000,000,000...&lt;/li&gt;&#xA;&lt;li&gt;10^3, 10^6, 10^9, 10^12, ...&lt;/li&gt;&#xA;&lt;li&gt;kilo-, mega-, giga-, tera-, ...&lt;/li&gt;&#xA;&lt;li&gt;Ask the students to provide examples of real-world big numbers, and real-world words that use the prefixes, and figure out what they mean.&lt;/li&gt;&#xA;&lt;li&gt;1 googol (10^100)&lt;/li&gt;&#xA;&lt;li&gt;1 &lt;a href=&quot;https://en.wikipedia.org/wiki/Mole_%28unit%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;mole&lt;/a&gt; (the number of carbon-12 atoms in 12 grams of carbon-12.  Or the net number of electrons that go past a point in one second when a one Amp current is flowing.  About 6x10^23, or 602 sextillion.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How to write out very small numbers&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;thousandths, millionths, billionths, trillionths... decillionths&lt;/li&gt;&#xA;&lt;li&gt;0.001;  0.000 001;  0.000 000 001; 0.000 000 000 001...&lt;/li&gt;&#xA;&lt;li&gt;10^-3, 10^-6, 10^-9, 10^-12, ...&lt;/li&gt;&#xA;&lt;li&gt;milli-, micro-, nano-, pico-, ...&lt;/li&gt;&#xA;&lt;li&gt;Ask the students to provide examples of real-world small numbers, and real-world words that use the prefixes, and figure out what they mean.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is a negative number?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Tricks for setting up story problems&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Draw a picture&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/1834/how-do-i-teach-algebra#1840&quot;&gt;What is a variable?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Label what you know&lt;/li&gt;&#xA;&lt;li&gt;Keep your units&lt;/li&gt;&#xA;&lt;li&gt;Solve the problem&lt;/li&gt;&#xA;&lt;li&gt;Write out your answer in words&lt;/li&gt;&#xA;&lt;li&gt;Check-by-substitution&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;http://www.amazon.com/exec/obidos/ISBN=0393310728&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to Lie with Statistics&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Maps, Areas, and Volumes&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Measure the classroom's length, width, and height&lt;/li&gt;&#xA;&lt;li&gt;Calculate the classroom's floor area&lt;/li&gt;&#xA;&lt;li&gt;Calculate the classroom's volume&lt;/li&gt;&#xA;&lt;li&gt;Draw a &quot;map&quot; of the classroom&lt;/li&gt;&#xA;&lt;li&gt;Ditto for a desk (or other object)&lt;/li&gt;&#xA;&lt;li&gt;Use an atlas or big map to measure the distances between cities, or calculate the areas of states.  Look up the &quot;official&quot; answers, and see how close you were.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-02-01T08:02:06.953" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6098" PostTypeId="1" CreationDate="2014-12-24T23:13:16.173" Score="6" ViewCount="480" Body="&lt;p&gt;I see that most of the american universities have mathematical summer programs with lectures and mini-courses (note: &lt;strong&gt;I'm not talking about undergraduate &lt;em&gt;research&lt;/em&gt; programs in this question&lt;/strong&gt;) for university students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately most of such programs are only for US citizens. So I would like to ask what summer programs are there in the US, Europe, Asia, South America, or Africa that are accessible to international students (note: in particular, I'm European)?&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="3251" LastEditDate="2014-12-29T18:05:50.087" LastActivityDate="2014-12-29T18:05:50.087" Title="Summer program for 'international' undergraduate students of mathematics" Tags="|undergraduate-education|curriculum|summer|" AnswerCount="2" CommentCount="5" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6099" PostTypeId="2" ParentId="6075" CreationDate="2014-12-25T00:25:19.893" Score="4" Body="&lt;p&gt;&lt;strong&gt;is there some agreement that [Lie theory] should be a fundamental part of undergraduate education?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2014-12-25T00:25:19.893" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6100" PostTypeId="2" ParentId="4448" CreationDate="2014-12-25T03:10:17.883" Score="2" Body="&lt;p&gt;I've had some success using basic &lt;a href=&quot;http://en.wikipedia.org/wiki/Kirigami&quot; rel=&quot;nofollow noreferrer&quot;&gt;kirigami&lt;/a&gt; at that age, in particular, cutting&#xA;out snowflakes:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/wFMm2.png&quot; width=&quot;300&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;(&lt;a href=&quot;http://www.omiyageblogs.ca/2013/12/cut-fold-kirigami-snowflakes.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kirigami snowflakes link&lt;/a&gt;)&#xA;&lt;hr /&gt;&#xA;Memorably, I had one nearly mute kid who could cut out the most amazing designs,&#xA;but could not explain what he was doing.&#xA;It was as if kirigami was a language in which he was fluent, but he was decidedly &#xA;not fluent in English (his native and only language).&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2014-12-25T03:10:17.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6101" PostTypeId="2" ParentId="6089" CreationDate="2014-12-25T05:09:25.197" Score="3" Body="&lt;p&gt;I have 2 ideas for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, I have helped a student get number intuition by getting him to building factorisation lattices. For instance, 12 can be represented by a 2D, 2 node by 3 node grid/lattice:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;            12&#xA;&#xA;      6            4&#xA;&#xA;3           2&#xA;&#xA;      1&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;They get really pretty when you get 3D lattices (3 unique factors, like 60). It can help to draw the connecting lines in different colours, but it is essential that each factor has its own axis/direction. More than 3 unique factors is almost impossible to draw nicely, but is not needed until 210 (2*3*5*7). Having multiplicities of factors is not a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It gives students a good feel for numbers and how they interrelate, apart from good practice with division/multiplication. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;HOWEVER... (idea 2)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal experience is that mental arithmetic is overrated in importance for future academic achievement. If a student does not have good mental arithmetic by Year 8/9, most teachers I know would give them a calculator. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I do not like calculators!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculators often do not help visualisation, or allow for easy experimentation. But I have had amazing results available with a tablet using iPython with a web interface. Other Computer Algebra Systems may work just as well, but I have not tried them with students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This year I coached a year 8 student with poor mental arithmetic and very little algebra through a Physics course that used numerical integration techniques and vectors. How did he do it? He programmed it all in Python. Mental arithmetic never came into it. Algebra was never an issue, as he found computer manipulation of variables much more intuitive than algebraic manipulation. Now he is experimenting with fluid dynamics - beyond what I studied at university.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His times tables are still atrocious, his algebra is improving (through practice with Physics equations and DragonBox), but he is doing university level physics and applying it to complex problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The use of technology presents problems in exams, but appropriate guided use can give students confidence and experience that will eventually translate into maths intuition. It also provides the practical technology/maths skills that are of use outside a maths classroom, something traditional maths curricula struggle with. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2014-12-25T05:09:25.197" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6102" PostTypeId="2" ParentId="6075" CreationDate="2014-12-25T06:21:46.383" Score="2" Body="&lt;p&gt;First of all let me mention that I'm a physics student and therefore cannot tell you whether Lie theory is interesting for a mathematician. Also I don't know the specific book you are referring to. However, I can tell you why Lie groups and Lie algebras are one of the most essential pieces of maths that a theoretical physicists needs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lie groups and algebras appear throughout physics to mathematically implement the concept of symmetry. For example, physics should be invariant under translations in space and in time, spatial rotations and boosts. Spatial rotations are described by the rotation (&quot;special orthogonal&quot;) group SO(3), spacetime rotations, that is spatial rotations and boosts, are described by the bigger Lorentz group SO(3,1) and if you add the spacetime translations you end up with the Poincaré group. All of these groups are Lie groups. The Poincaré symmetry leads to the very way how you write down physical laws, using so-called Lorentz indices (covariant notation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In quantum field theory and elementary particle physics you encounter even more Lie groups describing so called internal symmetries. For example, the internal symmetry group of the standard model describing the strong and electroweak interaction is SU(3)xSU(2)xU(1) where SU(n) and U(n) are again Lie groups. Such symmetries help you to write down the theory and also work with it, i.e. do the spontaneous symmetry breaking using the Higgs mechanism which leads to the breaking of an initial internal symmetry group to a smaller one. The so called grand unified theories for example start with a big group like SU(5) and break it down to the standard model group SU(3)xSU(2)xU(1) using the Higgs mechanism.&lt;/p&gt;&#xA;" OwnerUserId="3542" LastEditorUserId="3542" LastEditDate="2014-12-25T07:34:20.793" LastActivityDate="2014-12-25T07:34:20.793" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6103" PostTypeId="1" CreationDate="2014-12-25T08:33:57.680" Score="3" ViewCount="314" Body="&lt;p&gt;In a &lt;a href=&quot;http://vserver1.cscs.lsa.umich.edu/~spage/pnas.pdf&quot; rel=&quot;nofollow&quot;&gt;2004 article&lt;/a&gt; entitled &lt;em&gt;Groups of diverse problem solvers can outperform groups of high-ability problem solvers&lt;/em&gt;, L. Hong and S. Page argued that &quot;diversity trumps ability&quot; and provided a mathematical theorem to support their conclusion. In his 2007 book, Page added that &quot;$\ldots$ the veracity of the diversity trumps ability claim is not a matter of dispute. It's true, just as $1+1=2$ is true&quot;, apparently referring to the said theorem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a &lt;a href=&quot;http://www.ams.org/notices/201409/rnoti-p1024.pdf&quot; rel=&quot;nofollow&quot;&gt;2014 article&lt;/a&gt; &lt;em&gt;Does diversity trump ability?&lt;/em&gt;, A. Thompson provided a counterexample to the theorem of Hong and Page. In a subsequent exchange of letters, Hong and Page pointed out that Thompson missed a hypothesis that bars Thompson's counterexample, and Thompson retorted that the hypothesis was not given in the 2004 article. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this surely boosts readership but it still leaves the purely mathematical and therefore educational question unanswered: &lt;em&gt;is there nontrivial mathematical content to the Hong-Page theorem (once all the hypotheses are in place, that is)?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2014-12-26T08:04:19.233" LastActivityDate="2014-12-26T08:04:19.233" Title="Is there mathematical substance to the claim that diversity trumps ability?" Tags="|undergraduate-education|secondary-education|" AnswerCount="0" CommentCount="6" ClosedDate="2014-12-27T00:32:17.493" ContentLicense="CC BY-SA 3.0" />
  <row Id="6104" PostTypeId="2" ParentId="6075" CreationDate="2014-12-25T09:37:40.950" Score="4" Body="&lt;p&gt;One refreshing aspect of Stillwell's book is that he takes the viewpoint not merely of von Neumann but also of Sophus Lie himself who thought of his algebras as groups of infinitesimal transformations, and Stillwell repeatedly returns to the intuitively appealing infinitesimal descriptions of the Lie algebra throughout the book.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the reason this is called the &lt;em&gt;naive approach&lt;/em&gt; is because infinitesimals used to be thought of as being naive.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2014-12-25T13:24:27.193" LastActivityDate="2014-12-25T13:24:27.193" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6105" PostTypeId="2" ParentId="6089" CreationDate="2014-12-25T14:54:07.683" Score="4" Body="&lt;p&gt;Permit me to direct you to read an answer to another question by another user &lt;a href=&quot;https://matheducators.stackexchange.com/a/6035/267&quot;&gt;Benjamin Dickman&lt;/a&gt; first, notably the second part of the answer that begins with the line &quot;Given the above discussion, I would like to make one additional comment&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot improve on the linked answer but I can connect it to your question. What you call an &quot;intuition for numbers&quot; and I might classify as &quot;number sense&quot;, I feel could be improved with an emphasis on delving into the prime factorizations of integers. An earlier answer by @Richard suggests making lattices of these factorizations (though I know them as &quot;trees&quot;). In my experience, many students who struggle with the same things that you say your student is struggling with have more success when working more with prime factorization.&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2014-12-25T14:54:07.683" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6106" PostTypeId="2" ParentId="2497" CreationDate="2014-12-25T16:42:15.957" Score="2" Body="&lt;p&gt;The explanation to be given to the student depends upon whether you are teaching him &lt;em&gt;Pure Mathematics&lt;/em&gt; or &lt;em&gt;Applied Mathematics&lt;/em&gt;. I'd like to write my answer from the &lt;em&gt;Pure Mathematical&lt;/em&gt; point of view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Mathematics a Number is a &lt;em&gt;Mathematical object&lt;/em&gt;, in a less strict language, a &lt;em&gt;symbol&lt;/em&gt;. E.g. the natural numbers $2$ and $3$, are symbols which represent the concept of being &lt;em&gt;two&lt;/em&gt; and being &lt;em&gt;three&lt;/em&gt;. The most common use of the natural numbers is to count things, e.g. $5$ eggs -- $7$ books. &lt;em&gt;Mathematics&lt;/em&gt; doesn't bother where these objects(natural numbers) are used in real life or not. &lt;em&gt;Mathematics&lt;/em&gt; studies the properties of these objects. Natural numbers $a,b$ and $c$ always follow the law:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{cases}&#xA;      \text{If}\ \ a+c=b+c \\ &#xA;      \ \ \ \ \ \ \ \ \ \ \ \  a=b \tag{Law - I}&#xA;\end{cases}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above mentioned law can be understood intuitively by recalling the concept of numerical equality. Let $a,b$ and $c$ represent the number of apples in three different groups. Now, if it is given that $a+c=b+c$, then the only way this can be true is that a and b represent the same number of apples. Law - I can never be proved, so we take it as an &lt;em&gt;axiom&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another law that these numbers follow is, $$\begin{cases}&#xA;      \text{If}\ \ a+c&amp;gt; or &amp;lt;b+c \\ &#xA;      \ \ \ \ \ \ \ \ \ \ \ \  a&amp;gt; or &amp;lt;b \tag{Law - II}&#xA;\end{cases}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, I'd explain the concept of subtraction. The symbol $a-b$ is defined by the equation,&#xA;$$(a-b)+b=a$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case $a=b$ or $a&amp;lt;b$, we don't have any natural number for $a-b$. Again &lt;em&gt;Mathematics&lt;/em&gt; does not care where the concept of subtraction is used in real life. It does not care if the $a-a$ or $a-b$( when $a&amp;lt;b$) have any use in real life or not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We had the law $1$, if $a+c=b+c$ then $a=b$. By the &lt;em&gt;principle of permanence&lt;/em&gt; we define the law to be true for, if $(r-r)+c=(p-q)+c$, then $(r-r)=(p-q)$--whatever the meaning of this symbolic equation is. This definition eventualy implies that any symbol $x-x$ doesn't depend upon the value of $x$. So we &lt;em&gt;choose&lt;/em&gt; the symbol $0$ for that expression. Also $a-a=b-b=x-x=0.$ By the definition of subtraction we have, $(b-b)+b=b$. This is equivalent to $(a-a)+b=b$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having defined $0$, I'd explain the negative numbers. A negative number $-b$ is a brief symbol for $0-b$. By the &lt;em&gt;principle of permanence&lt;/em&gt; we define all the laws of natural numbers to be true for the symbol $-b$, e.g. commutative, associative etc; whatever the meaning in real life may be assigned to this symbol, it will obey all those laws, even the law-1 and law-2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a+(-b)=a+(0-b)=a+0-b=(a+0)-b=a-b$. So $a-a=a+(-a)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider two numbers $r$ and $s$, such that $b&amp;gt;a$. We have,&lt;br&gt;&#xA;$$\ \ \ \ \ \ \ \ \ \ \ b&amp;gt;a$$&#xA;$$ \implies \ \ 0+b&amp;gt;0+a$$&#xA;$$ \implies (a-a)+b&amp;gt;(b-b)+a$$&#xA;$$ \implies (-a)+(a+b)&amp;gt;(-b)+(a+b)$$&#xA;$$ \implies -a&amp;gt;-b$$&#xA;Briefly, if $b&amp;gt;a$ then $-a&amp;gt;-b$. In particular for the &lt;em&gt;objects&lt;/em&gt; $3,4,-3$ and $-4$ we have $4&amp;gt;3$ so $-3&amp;gt;-4$. &lt;/p&gt;&#xA;" OwnerUserId="1642" LastActivityDate="2014-12-25T16:42:15.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6107" PostTypeId="1" CreationDate="2014-12-25T17:25:49.760" Score="26" ViewCount="3679" Body="&lt;p&gt;I'm going to introduce my students to the fundamental theorem of arithmetic (uniqueness of integer factorization to prime factors), and I don't want them to take the uniqueness for granted! To make my students understand that the uniqueness is not trivial by any means, I'm looking for a non-unique &lt;em&gt;factorization&lt;/em&gt; of integers. What &lt;em&gt;factorizations&lt;/em&gt; do you suggest?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a suggestion myself, but it's definitely not the best one: Suppose that Goldbach's conjecture is true. So every integer can be written as the sum of at most 3 primes, but this &lt;em&gt;factorization&lt;/em&gt; is not unique!&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="-1" LastEditDate="2014-12-27T07:32:47.260" LastActivityDate="2018-06-28T19:44:31.117" Title="A Non-Unique Factorization of Integers!" Tags="|secondary-education|prime-numbers|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6108" PostTypeId="2" ParentId="2497" CreationDate="2014-12-25T18:10:22.027" Score="1" Body="&lt;p&gt;Possible Geometrical method for this particular teenager:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming that he has no difficulty with the whole numbers and the non-negative number line, imagine a reflection about 0.  Let us call this operation *.  Then each positive whole number has a reflected image on the new extended number line.  For example, the reflected image of $3$ is $3^*$ and the reflected image of $4$ is $4^*$.  If he believes that $a&amp;gt;b$ if $a$ lies to the right of $b$ on the number line, then he should also believe that $3^*&amp;gt;4^*$.  As a bonus you get $(3^*)^*=3$.&lt;/p&gt;&#xA;" OwnerUserId="3546" LastActivityDate="2014-12-25T18:10:22.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6109" PostTypeId="2" ParentId="6107" CreationDate="2014-12-25T19:52:25.087" Score="11" Body="&lt;p&gt;You can &quot;factor&quot; every positive integer into a sum of distinct Fibonacci numbers. This is not unique. However, it is unique if you add the condition that no two summands are adjacent Fibonacci numbers.&lt;/p&gt;&#xA;" OwnerUserId="3547" LastActivityDate="2014-12-25T19:52:25.087" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6110" PostTypeId="2" ParentId="6107" CreationDate="2014-12-25T20:20:32.453" Score="22" Body="&lt;p&gt;Integer factorization itself isn't unique: for example, $6 \cdot 2 = 4 \cdot 3$. The remarkable theorem is that there is only one factorization (up to reordering and sign) in which all the factors are prime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, you could give an example where unique factorization into irreducibles genuinely fails. If we look at numbers of the form $m + n \sqrt{10}$, where $m$ and $n$ are integers, then we don't have unique factorization. For example, $2 \cdot 3 = 6 = (4 + \sqrt{10}) \cdot (4 - \sqrt{10})$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As a side note, the latter situation is important in the history of number theory. Dedekind's theory of &quot;ideal numbers&quot;, which led to much of modern algebraic number theory, was motivated by trying to recover some sort of unique factorization in such rings. Such failures of unique factorization are closely related to why Kummer was unable to prove Fermat's Last Theorem in full generality, and it's plausible that the error in Fermat's own claim of a proof was due to incorrectly assuming that cyclotomic fields always have unique factorization.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's another illustrative example: There isn't a good notion of &quot;prime factorization&quot; in the rational numbers or real numbers (or more generally, in any field): everything but zero is invertible, so we get &quot;factorizations&quot; like $5 = \frac{5}{1728} \cdot 1728$. This shows that some &lt;em&gt;failure of invertibility&lt;/em&gt; of multiplication is required for an interesting notion of factorization, which explains why we don't pay attention to scalar factors when factoring polynomials with real coefficients. Similarly, we can factor $6$ as $(-2) \cdot (-3)$, but we don't consider this to be a breakdown of unique prime factorization because $-1$ has a multiplicative inverse in the integers.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2014-12-25T20:20:32.453" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6111" PostTypeId="2" ParentId="6098" CreationDate="2014-12-25T21:20:10.263" Score="1" Body="&lt;p&gt;If you like mathematical physics. There is a summer school in Bogotá, Colombia. &lt;a href=&quot;http://matematicas.uniandes.edu.co/~cursillo_gr/escuela2015/index_en.php&quot; rel=&quot;nofollow&quot;&gt;http://matematicas.uniandes.edu.co/~cursillo_gr/escuela2015/index_en.php&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3253" LastActivityDate="2014-12-25T21:20:10.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6112" PostTypeId="2" ParentId="6074" CreationDate="2014-12-25T23:06:27.353" Score="8" Body="&lt;p&gt;Some caveats: I own both books, and have taken number theory courses up to graduate level. Also, your questions are clearly &lt;em&gt;somewhat&lt;/em&gt; subjective: what is difficult, relevant or complete for one person, depends on their previous exposure, inclination, course content, and ability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So...in my opinion.. both books are &lt;em&gt;classics&lt;/em&gt;, which means they have been around for some time, have been reprinted, corrected and reprinted again, and used by many students all over the world. HW is certainly an introductory text: you could give this to an able 16/18 year old for them to work through, from chapter 1. The treatment is comprehensive and is logically ordered. I found the notation comfortable and it more than covered my first course in number theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The IR book &lt;em&gt;covers slightly more ground in fewer pages&lt;/em&gt;. I think that last statement accurately sums up this book. My appreciation for this book &lt;em&gt;only&lt;/em&gt; grew once I had covered a number of other courses (on groups, rings and fields) - and matured somewhat in my mathematical thinking (for example there is excellent treatment of quadratic reciprocity). Indeed, IR's book is an excellent taster of algebraic number theory - but that is unlikely to feature in an introductory course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, both books are excellent. I own both. I mostly used HW for my introductory course. As a bridge to more advanced levels, IR's text is excellent if you already know you are interested in algebraic number theory.&lt;/p&gt;&#xA;" OwnerUserId="3541" LastEditorUserId="3541" LastEditDate="2014-12-29T22:10:03.730" LastActivityDate="2014-12-29T22:10:03.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6113" PostTypeId="2" ParentId="6107" CreationDate="2014-12-25T23:24:16.900" Score="35" Body="&lt;p&gt;A really nice example is factorization of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert_number&quot; rel=&quot;noreferrer&quot;&gt;Hilbert numbers&lt;/a&gt; - that is the numbers&#xA;$$1,\,5,\,9,\,13,\,17,\,\ldots,4n+1,\,\ldots.$$&#xA;Now, we can talk about factoring in this domain too - for instance, $45=5\times 9$ is a factorization of $45$. Moreover, $5$ and $9$ are both &quot;Hilbert Primes&quot; - they are not the product of two other Hilbert numbers - meaning that this factorization cannot be further reduced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This &lt;em&gt;looks&lt;/em&gt; very similar to factoring an integer. Except - surprise! - it's not unique:&#xA;$$1617=21\times 77=33\times 49$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I encountered this example in the book &lt;em&gt;In Code&lt;/em&gt; by Sarah Flannery)&lt;/p&gt;&#xA;" OwnerUserId="3229" LastEditorUserId="10105" LastEditDate="2018-06-27T05:29:17.560" LastActivityDate="2018-06-27T05:29:17.560" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="6114" PostTypeId="2" ParentId="6107" CreationDate="2014-12-25T23:37:10.780" Score="18" Body="&lt;p&gt;You can restrict to subsets of the integers to get nice examples of structures that do not have unique factorization. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, take the set of all natural numbers, or integers it does not really matter, that are congruent to $1$ modulo $4$. This is a semigroup with identity under multiplication, and every element is the product of irreducible elements. However, factorizations into irreducible elements are not unique in an essential way. For example, you have:&#xA;$$21 \cdot 21 = 9 \cdot 49$$&#xA;and $9,21,49$ are irreducible in that structure as their divisors are $3$ modulo $4$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This structure is sometimes called Hilbert semigroup as he used (it is said) this as instructional example. (Added: this is also mentioned in another answer that appeared while I was writing this.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another possibility would be to consider only the even integers (and $1$). Again you have a factorization into irreducible elements, but it is not unique;&#xA;$$42 \cdot 2 = 6 \cdot 14$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still another one would be to consider all integers greater than $2$ (and $0$) in this case with addition as operation. The only two irreducible elements are $2$ and $3$ and one has non-unique factorizations: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$2+2+2 = 3 + 3$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above is an example of a numerical semigroup. &#xA;If you do not like this is additive, you can instead consider polynomials that do not have a linear term. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure if this is relevant in your context, but if you want you could continue to use some of these examples in more advanced contexts. For instance, the Hilbert semigroup is contained in the (odd) natural numbers and that structure has unique factorization. This situation is analogous, in fact not just analogous but in a suitable sense littelraly the same, as one encounter when studying factorizations into ideals to restore unique factorization in algebraic number theory. &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2014-12-25T23:37:10.780" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6115" PostTypeId="2" ParentId="6062" CreationDate="2014-12-26T15:16:30.507" Score="5" Body="&lt;p&gt;This is a classic reference: &quot;&lt;strong&gt;&lt;em&gt;Handbook of writing for the mathematical sciences&lt;/em&gt;&lt;/strong&gt;&quot; by N. Higham. See &lt;a href=&quot;http://www.siam.org/books/ot63/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3541" LastEditorUserId="3541" LastEditDate="2014-12-26T15:28:33.880" LastActivityDate="2014-12-26T15:28:33.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6116" PostTypeId="2" ParentId="416" CreationDate="2014-12-27T00:21:59.290" Score="1" Body="&lt;p&gt;If working with high school students, I would suggest mentioning that all of the usual exponent rules hold. It could be a fun exercise to ask your students to interpret the numerical value of $0^0$ and to write a brief paragraph explaining their reasoning.&lt;/p&gt;&#xA;" OwnerUserId="3528" LastActivityDate="2014-12-27T00:21:59.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6117" PostTypeId="2" ParentId="5872" CreationDate="2014-12-27T01:10:51.630" Score="2" Body="&lt;p&gt;It's possible that they don't understand that x stands for a number, or that they don't understand that you are making a statement about ALL nonzero numbers, x. &#xA;I'd suggest starting like this:&#xA;&quot;Let's figure out for what values of $x$, $\frac{x}{x}=1$. When $x=3,$ is $\frac{x}{x}=1$? How do we check? Well, let's see. Is $\frac{3}{3}=1$?&quot; and lead them to a generalization.&lt;/p&gt;&#xA;" OwnerUserId="3528" LastActivityDate="2014-12-27T01:10:51.630" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6118" PostTypeId="2" ParentId="2763" CreationDate="2014-12-27T01:21:31.610" Score="2" Body="&lt;p&gt;I think that there is some merit to teaching properties of the integers &lt;em&gt;before&lt;/em&gt; extending them to rational numbers. This well help students understand important properties such as the remainders that come with whole-number division, prime decomposition, modular arithmetic, etc. More generally, it will help students understand the ring structure of the integers. Moreover, it may help exponentiation make more sense as repeated multiplication. In these contexts, multiplication &lt;em&gt;is&lt;/em&gt; repeated addition. See, for example, the work done by Stephen Campbell and Rina Zazkis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This isn't to say that students shouldn't also learn rational number operations, just as we teach students to extend division to the rational numbers. &lt;/p&gt;&#xA;" OwnerUserId="3528" LastEditorUserId="3528" LastEditDate="2014-12-27T04:27:31.877" LastActivityDate="2014-12-27T04:27:31.877" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6119" PostTypeId="2" ParentId="6083" CreationDate="2014-12-27T11:55:22.910" Score="3" Body="&lt;p&gt;Sorry, I do not have the ability to translate these; however, for the high school mathematics exam of China, &lt;a href=&quot;http://www.51test.net/show/4526162.html&quot; rel=&quot;nofollow&quot;&gt;this&lt;/a&gt; website works, search for 会考数学.  That means huikao (high school final exam) and shuxue (mathematics).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the physics use &lt;a href=&quot;http://www.51test.net/huikao/&quot; rel=&quot;nofollow&quot;&gt;this&lt;/a&gt; and from there you can choose which region of China and subject you would like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the college entrance exam: &lt;a href=&quot;http://www.233.com/gaokao/shiti/shuxue/shanghai/20140306/134138963-8.html&quot; rel=&quot;nofollow&quot;&gt;高考&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="2182" LastActivityDate="2014-12-27T11:55:22.910" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6120" PostTypeId="1" CreationDate="2014-12-27T19:19:16.930" Score="4" ViewCount="652" Body="&lt;p&gt;So far I wasn't able to find one single open source set of lecture notes or textbook &lt;strong&gt;&lt;em&gt;with $\LaTeX$ source code available&lt;/em&gt;&lt;/strong&gt; on &quot;introductory real analysis&quot; that could suit my needs. It seems to me that you can find either sloppy calculus books or books whose very first chapter is on Banach Spaces. Could you point out something in the middle: that is, some open source lecture notes that cover the same material covered by Rudin and Ross and treat the subject with rigour?&lt;/p&gt;&#xA;" OwnerUserId="3251" LastEditorUserId="3251" LastEditDate="2014-12-27T19:47:32.763" LastActivityDate="2015-01-10T19:17:57.657" Title="Open source lecture notes textbook in &quot;introductory real analysis&quot;" Tags="|undergraduate-education|textbooks|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6121" PostTypeId="2" ParentId="6120" CreationDate="2014-12-27T19:40:39.160" Score="2" Body="&lt;p&gt;You may be interested in &lt;a href=&quot;http://www.trillia.com/zakon-analysisI.html&quot; rel=&quot;nofollow&quot;&gt;Zakon's Analysis I&lt;/a&gt; and &lt;a href=&quot;http://www.trillia.com/zakon-analysisII.html&quot; rel=&quot;nofollow&quot;&gt;II&lt;/a&gt;. It is free for personal use, it is rigorous, it covers the same content as Rudin and Ross. I haven't personally studied it, but it looks good.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2014-12-27T19:40:39.160" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6122" PostTypeId="2" ParentId="6074" CreationDate="2014-12-28T00:52:17.570" Score="8" Body="&lt;p&gt;Ireland &amp;amp; Rosen is a graduate textbook which starts with general number theory and moves to algebraic number theory. Abstract Algebra is not strictly required, but students would benefit from a course in it. The first half of the textbook can be used, with some care, as an advanced undergraduate textbook.&lt;/p&gt;&#xA;&lt;p&gt;Hardy &amp;amp; Wright is an introductory number theory text. It's suitable for undergraduates or perhaps advanced high school seniors (first few chapters). The latter half of the book could be adapted for use in a graduate class, especially in combination with another book.&lt;/p&gt;&#xA;&lt;p&gt;Very subjective: Ireland &amp;amp; Rosen is more modern and better-organized. Hardy &amp;amp; Wright is more eclectic and perhaps more fun. Both are good textbooks, but I don't have enough experience to compare them broadly to others.&lt;/p&gt;&#xA;" OwnerUserId="3558" LastEditorUserId="3558" LastEditDate="2021-07-28T19:38:19.380" LastActivityDate="2021-07-28T19:38:19.380" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="6123" PostTypeId="2" ParentId="6041" CreationDate="2014-12-28T01:30:30.957" Score="3" Body="&lt;p&gt;Much of mathematics works perfectly well with so-called &quot;naive&quot; set theory. That is, working with sets without worrying too much exactly what sets are. This isn't philosophically that different from calculus courses where you work with real numbers without worrying so much what they are (equivalence classes of Cauchy sequences of rational numbers), or for a more advanced example, modern homotopy theory research which uses $(\infty, n)$-categories without being too bothered by exactly what these are. These are all things that you generally have some intuition for, and until that intuition fails, it's okay to rely on it at first pass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At some point, students (and mathematicians) need to understand the basics on which the theory is founded, but that isn't necessarily the beginning of their studies. Mathematics isn't just about going down to the foundations and checking that everything is rigorous; you also want to prove theorems once in a while, and sometimes hiding the foundational difficulties until later lets you get things done faster and more intuitively. This is no different from the fact that you weren't taught the Peano axioms of arithmetic in primary school, even though you were learning about the natural numbers. Sure, it wasn't completely rigorous, but finger-counting made it a whole lot easier to prove that $3+4=7$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The right time to discuss the axiom of choice in any depth seems to me to be at the same point that you begin discussing the other axioms of ZFC and set up an axiomatic foundation of set theory. There's nothing really special about choice; it's an axiom that almost every mathematician is fine with accepting or to whom it doesn't even matter. If you try to talk about the axiom of choice without any discussion of the ZF axioms, you're still using naive set theory, and you aren't avoiding any of the paradoxes that axiomatic set theory was made to circumvent. Giving special privilege to AC isn't any more rigorous than not discussing it at all, just more indecisive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That isn't to say that mentioning its use in passing does much harm; it may even be overall beneficial to mention such things. The majority of students won't care either way, but there may be a minority who are interested in other foundations besides ZFC, and that minority might be quite interested in the fact that every model of ZF¬C has vector spaces without bases. But spending significant lecture time on it in a course on e.g. linear algebra isn't a good use of the time. Interested students will likely study it on their own (now or later), and the rest of the class doesn't particularly need to know that the existence of bases for arbitrary vector spaces requires choice to prove the spectral theorem or put a matrix in Jordan normal form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, this &lt;em&gt;was&lt;/em&gt; in first-year undergraduate courses (and in some cases even in high school). For some other people I know, they didn't learn axiomatic set theory until graduate school. But for what it's worth, those people are now working mathematicians, while I'm a physicist; you can draw your own conclusions on that.&lt;/p&gt;&#xA;" OwnerUserId="3559" LastActivityDate="2014-12-28T01:30:30.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6124" PostTypeId="1" AcceptedAnswerId="6129" CreationDate="2014-12-28T12:07:19.793" Score="13" ViewCount="5802" Body="&lt;p&gt;I was browsing IEEE xplore the other day and found this gem called &lt;a href=&quot;http://ieeexplore.ieee.org/xpl/abstractAuthors.jsp?reload=true&amp;amp;arnumber=4320540&quot; rel=&quot;noreferrer&quot;&gt;&quot;What Mathematics Courses Should an Electrical Engineer Take? A Report on the National Study of Mathematics Requirements for Scientists and Engineers&quot;&lt;/a&gt; by G.H. Miller. dated 1970. I believe this paper is most likely aimed at undergrad electrical engineers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this paper he collected a survey from two groups: awards group and abstracts group. Awards group is a list of electrical engineers who won nationally or internationally recognized awards. Abstracts group were people who published profusely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal was to determine just which courses should ELECTRICAL engineers take during undergrad. Note again this was from the 70s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ViM4D.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The conclusion of the paper is as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Highly recommended: calculus sequence, vectors, elementary DE, intermediate ODE, advanced calculus, elementary complex variables, matrix theory, elementary probability&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moderately recommended: tensor analysis, advanced ODE, advanced PDE, calculus of variations, complex variables and machine computation, numerical analysis and integral transform&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and this quote: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There was little use for newer courses in modern mathematics such as&#xA;  group theory, lie lgebras, multilinear algebra, mathematical logic,&#xA;  game theory, and geometric algebra. Therefore, these courses shoud be&#xA;  given low priority.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This quote raises obvious questions. Group theory is the foundation of signal processing. Lie algebra is used widely in quantum mechanics which is important for semiconductor physics. Mathematical logic is core of modern programming and control theory. Geometric algebra has many application in both signal processing and control theory. So it appears highly likely that the paper is out of date.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three questions I have in mind hope some one can help me with (in no particular order)&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Are recent or more modern sources for undergrad electrical engineer to decide on which math courses to take?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As the work force becomes more segmented, it is highly likely that a working electrical engineer will never use much of the math taught in school. (how much math is needed in quality assurance anyhow?) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So this is more aimed towards research engineers. Are there any research electrical engineers who might wish to dispute some of the claims in the paper?&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;What courses outside of the high and moderate recommendations should an electrical engineer student consider and why?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3560" LastActivityDate="2017-07-16T16:23:23.770" Title="What math courses should be taught to undergrad electrical engineers: a 40 years update" Tags="|undergraduate-education|mathematical-pedagogy|education-research|undergraduate-research|applied-mathematics|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="6125" PostTypeId="1" CreationDate="2014-12-28T19:49:53.447" Score="10" ViewCount="1780" Body="&lt;p&gt;One of my favorite books is Flatland by Edwin Abbott, which is incredibly rich in geometric concepts especially relating to dimensionality.  I would really like to assign it to my secondary geometry class to read but I am struggling with a final project/assessment for them to do after they finish reading.  I would like to stay away from just answering questions about the reading/writing an essay.  Even if that was the route I went, I can't seem to come up with a solid essay topic or list of questions that go beyond regurgitation of what they read.  Does anyone have any suggestions on creative ideas for projects or assessments I could use with the novella Flatland?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-01-29T16:33:20.663" Title="Project/Assessment based on Flatland for Secondary Geometry Class" Tags="|secondary-education|geometry|assessment|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6126" PostTypeId="2" ParentId="6125" CreationDate="2014-12-28T19:56:40.200" Score="9" Body="&lt;p&gt;The book goes into detail giving the two-dimensional cross-sections of a sphere. You could ask the class to find the cross-sections of a cube or of another 3D object more complex than a sphere. It seems hard to find all possible cross-sections of a cube, but you could assign points for each possibility they do find. (It would be interesting to see how many students would find the regular hexagon.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also ask for other ways to represent a cube in two dimensions. Students could think of a square inside another square with segments joining the corners--a kind of perspective drawing. They could also try to find all the ways to &quot;flatten out&quot; the cube: cut along some edges to fold the cube down to the plane. The most common way results in the Latin cross, but there are other ways.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2014-12-28T19:56:40.200" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6127" PostTypeId="2" ParentId="294" CreationDate="2014-12-29T04:18:25.840" Score="5" Body="&lt;p&gt;For more mathematical philosophy try:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Godel, Escher Bach&quot; by Douglas Hofstadter&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a great book for exploring mindset when approaching natural mathematical problems. Many problems are entirely theoretical in nature, as in, they results that flow from axiomatic constructions. This book teaches that natural mathematics is as much about creativity of perception and the explicit recognition of the usefulness of abstractions as skill with mechanics and identities.&lt;/p&gt;&#xA;" OwnerUserId="3562" LastEditorUserId="3562" LastEditDate="2014-12-30T13:00:15.310" LastActivityDate="2014-12-30T13:00:15.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6128" PostTypeId="1" AcceptedAnswerId="6132" CreationDate="2014-12-29T07:06:24.840" Score="15" ViewCount="341" Body="&lt;p&gt;Beginning next semester I am teaching a course in proofs and mathematical problem solving at my local university. For some background, the university is primarily a commuter university and the students I will be teaching will be mostly math education, and computer science students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am teaching from the book &lt;em&gt;Introduction to Mathematical Thinking&lt;/em&gt; by Scott A. Vanstone and Will J. Gilbert. The departmental course outline suggests starting at Chapter 2 - which would mean beginning with topics such as divisibility, diophantine equations, integers in different bases and prime numbers before talking about logic, sets, proofs and methods of proofs (this stuff is given in chapter 1). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I prefer starting in chapter 1 I see that there may be some wisdom to starting with very concrete topics and moving foward. But if I do not explain proofs and logic from the start I will be unable to prove any of the small intermediate results that are given in chapter 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, I was thinking of opening the class with an excercise using missing entries of a sudoku puzzle and having the students explain what numbers go in the missing entries and writing and explanation of why they believe it. I want to use this exercise to introduce direct proof, proof by cases and proof by contradiction to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1)&lt;/strong&gt; I was wondering if someone has experienced teaching a course of this nature and has any recommendations on the suggested progression for the beginning of the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2)&lt;/strong&gt; Assuming I start with chapter 2, would it still be a good idea to start with the sudoku exercise - and if so, how would I naturally segue from it into the rest of the class.&lt;/p&gt;&#xA;" OwnerUserId="3563" LastActivityDate="2014-12-30T01:45:32.050" Title="Order of Topics in Introductory Proofs Class" Tags="|undergraduate-education|course-design|mathematical-pedagogy|curriculum|proofs|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6129" PostTypeId="2" ParentId="6124" CreationDate="2014-12-29T13:54:21.107" Score="8" Body="&lt;p&gt;In my experience teaching undergraduate engineering students, key topics include at least some calculus, linear algebra and differential equations. Exactly how much depends on the field/subfield of engineering. A sampling of other topics includes:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Boolean algebra (digital electronics)&lt;/li&gt;&#xA;&lt;li&gt;Graph theory and algorithms (networks)&lt;/li&gt;&#xA;&lt;li&gt;Probability and statistics (reliability engineering, quality assurance and communications)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The main thing I miss from a modern perspective in Miller's lists is discrete mathematics, which has increased in importance with the rise of computers in the last few decades. On a more specialized note, I would add discrete/computational geometry for its applications in areas such as robotics and sensor networks.&lt;/p&gt;&#xA;&lt;p&gt;Research engineers may need to draw on a much wider and deeper array of knowledge. An example would be Michael Robinson at American University (&lt;a href=&quot;http://www.drmichaelrobinson.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.drmichaelrobinson.net/&lt;/a&gt;), whose research includes signal processing, dynamics and applied topology. See, for instance, his recent book &lt;a href=&quot;http://www.drmichaelrobinson.net/publications.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Topological Signal Processing&lt;/a&gt;, which draws on ideas from sheaf theory, category theory and computational topology. Note that this book is published in Springer's &lt;a href=&quot;http://www.springer.com/series/8445&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Engineering&lt;/a&gt; series, which in the words of the publisher&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...presents new or heretofore little-known methods to support engineers in finding suitable answers to their questions, presenting those methods in such manner as to make them ideally comprehensible and applicable in practice.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;An undergraduate interested in pursuing such an area would conceivably be advised to take courses in topology and abstract algebra as preparation, both of which remain unorthodox choices for most engineers to the best of my knowledge.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit: The 2014-2015 ABET (Accreditation Board for Engineering and Technology) accreditation criteria can be found at &lt;a href=&quot;http://www.abet.org/uploadedFiles/Accreditation/Accreditation_Step_by_Step/Accreditation_Documents/Current/2014_-_2015/E001%2014-15%20EAC%20Criteria%203-13-14(2).pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.abet.org/uploadedFiles/Accreditation/Accreditation_Step_by_Step/Accreditation_Documents/Current/2014_-_2015/E001%2014-15%20EAC%20Criteria%203-13-14(2).pdf&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(Thanks Michael E2 for the link.)&lt;/p&gt;&#xA;&lt;p&gt;In particular, on page 11:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;These program criteria apply to engineering programs that include “electrical,” “electronic,” “computer,” “communications,” or similar modifiers in their titles.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Curriculum&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The structure of the curriculum must provide both breadth and depth across the range of engineering topics implied by the title of the program.&lt;/p&gt;&#xA;&lt;p&gt;The curriculum must include probability and statistics, including applications appropriate to the program name; mathematics through differential and integral calculus; sciences (defined as biological, chemical, or physical science); and engineering topics (including computing science) necessary to analyze and design complex electrical and electronic devices, software, and systems containing hardware and software components.&lt;/p&gt;&#xA;&lt;p&gt;The curriculum for programs containing the modifier “electrical” in the title must include advanced mathematics, such as differential equations, linear algebra, complex variables, and discrete mathematics.&lt;/p&gt;&#xA;&lt;p&gt;The curriculum for programs containing the modifier “computer” in the title must include discrete mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="376" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-01-02T17:33:01.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6130" PostTypeId="1" CreationDate="2014-12-29T15:00:35.990" Score="4" ViewCount="245" Body="&lt;p&gt;I am an engineering student trying to study mathematical analysis because it will help me in my post graduate studies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My problem is that when I searched the internet I found that some university sites recommend topology class while others do not. Also, some recommend a &quot;mathematical analysis&quot; text before going into real and complex analysis. The problem is in the diversity of titles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is, if I want to study real and complex analysis, what topics do I need and under what book titles are those topics given?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to emphasis that am searching in English so I ask that all book titles are according to an American or English syllabus. If you are of a different nationality, please tell me the order of studying real and complex analysis in your country.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt; The particular topics I want to study are functions of complex variables,  integration in the complex plane, series and residues, and conformal mapping.&lt;/p&gt;&#xA;" OwnerUserId="2181" LastEditorUserId="376" LastEditDate="2015-01-02T16:29:08.983" LastActivityDate="2015-01-03T03:28:06.437" Title="Mathematics curriculum and book titles to study mathematical analysis for post-grad studies" Tags="|mathematical-pedagogy|mathematical-analysis|functional-analysis|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6131" PostTypeId="2" ParentId="6098" CreationDate="2014-12-29T15:51:31.350" Score="1" Body="&lt;p&gt;As you're discovered, many such summer programs are limited to US citizens and permanent residents; my understanding is that this is due to the source of their funding. Nonetheless, some programs exist which are accessible to international students. Unfortunately, since the funding source is what limits participants to US citizens/permanent residents, international participants might have to forgo funding, or pay. Here are the few programs I know of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://pcmi.ias.edu/application-ugss&quot; rel=&quot;nofollow&quot;&gt;The Undergraduate Summer School at PCMI&lt;/a&gt;: From their website,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Undergraduates funded by PCMI may receive lodging, travel, meals and a&#xA;  stipend (stipends are awarded to U.S. Citizens or U.S. permanent&#xA;  residents only).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.edgeforwomen.org/summer-session/&quot; rel=&quot;nofollow&quot;&gt;The EDGE program&lt;/a&gt;: From their website, &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Applicants to the program should be women who: 1) are applying to PhD&#xA;  programs in the mathematical sciences, or 2) just completed their&#xA;  first year in a PhD program in the mathematical sciences.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Women from underrepresented minority groups are especially encouraged&#xA;  to apply. Final acceptance to the program is contingent upon&#xA;  enrollment in a PhD program in Fall 2015. Unfortunately, at this time&#xA;  we are only able to fund US citizens and permanent residents.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.math.cmu.edu/cna/summer_institute_upcoming.html&quot; rel=&quot;nofollow&quot;&gt;SUAMI at Carnegie-Mellon&lt;/a&gt;: From their website, &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;U.S. citizens or permanent residents who are selected into the&#xA;  program, will receive a stipend of $3,600, housing in the university&#xA;  dormitories, and allowances for food and travel.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Non U.S. citizens or non residents who are selected into the program,&#xA;  will be expected to pay a registration fee of 4,000. In addition,&#xA;  they are responsible for their own housing, travel and food.&#xA;  University housing will be available for approximately $2,000 for the&#xA;  duration of the Institute.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In addition, as mentioned in the comments (to the question) by James Cook, there might be sources of internal funding. If you are currently enrolled in a university, it is probably worthwhile to ask faculty members there. &lt;/p&gt;&#xA;" OwnerUserId="126" LastEditorUserId="126" LastEditDate="2014-12-29T15:57:31.937" LastActivityDate="2014-12-29T15:57:31.937" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6132" PostTypeId="2" ParentId="6128" CreationDate="2014-12-29T20:16:57.673" Score="5" Body="&lt;p&gt;I teach such a course to mathematics majors, a somewhat different audience, and I find there are always some who are little unclear about &quot;if-then&quot; vs. &quot;and&quot;, quantifiers, and free/bound variables.  I mean that in a practical sense -- they confuse themselves, &quot;accidentally&quot; as it were.  Theoretical instruction in formal logic helps only a few; they need examples to clarify formal logic, at which point formal logic might be useful to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach the formal stuff and proof structures and so on, but I have to keep in mind the pitfalls above that my students are facing.  I haven't thought about what examples can be shown with sudoku, but it seems a good thing to use, if you can create good examples with them.  (It will be fun for many of them, but don't be surprised if not all find it so.  It's a hard criterion, to please everybody.)  Now you're talking about proof by cases and proof by contradiction, but the pitfalls I mentioned seem more elementary to me.  They seem like they should be addressed before getting to cases and contradiction.  Probably that can be done with sudoku.  (I'll give an example of mine later.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My main point here is to be aware of what problems your student might face and to smooth the way for them, if only a little, before they stumble on them.  If this is your first time teaching the course, expect to learn a little of what you wish you had known about your students before you started.  I don't have any definite view about whether you should start in Chapter 1 or Chapter 2 -- I wouldn't even if I knew the book, which I do not at all (a preview is not available on amazon).  I think it could be done either way, especially if your colleagues are suggesting to start in Chapter 2.  The way I imagine that could be successful is that by using concrete examples in number theory you can illustrate the logical principles that may or may not be presented somewhat too abstractly in Chapter 1.  (There's no reason to think Chapter 1 has to be abstract.)  The other main thing is to be confident in your approach.  Don't do something that doesn't make sense to you just because someone else recommended it.  It might turn out to have been good advice, but if you don't understand it, you probably won't do a good job with it.  Sometimes the advice you get is incomplete: what is clear in the other person's mind is not entirely conveyed to you.  (I've had that experience.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;How I start.&lt;/em&gt;  All my students have had at least a year of calculus.  Many of them are still quite good at calculus.  So I use continuous functions and differentiable functions for examples.  I ask them to come up with functions that are continuous (C) or not at, say, $x=1$, or differentiable (D) or not.  I ask them how many combinations are possible and help them see all four, (C, D), (C, not D), etc.  Now not all four are in fact achievable.  Why?  Someone will say because a differentiable function is continuous.  Then we look at &quot;$f$ is D and $f$ is C&quot; and &quot;If $f$ is D, then $f$ is C.&quot;  Before addressing the implication, I make sure they all agree it is true (because it is a theorem, it has a proof).  I put the examples they come up with in a table with the logical values T/F in columns for C and D.  Then I go over the meaning of &quot;and&quot; and &quot;if-then&quot;.  The relation to the rest of the course, which you asked about, is this.  I use and reuse the examples when we talk about proving &quot;if-then&quot;, whenever confusion arises, and whenever I feel clarification is needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also use other examples.  Using a non-theorem is good for illustrating how to show an implication is false.  A simple one is &quot;if $x^2 &amp;gt; 4$, then $x &amp;gt; 2$&quot;, where $x$ represents an integer or a real number (but be sure to include negative numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if those examples are not appropriate for your students, my hope is that it might give you an idea how to use the sudoko puzzles.  Elementary number theory is also a good source of simple examples, which may be why it is the recommended starting point by the department.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2014-12-30T01:45:32.050" LastActivityDate="2014-12-30T01:45:32.050" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6133" PostTypeId="2" ParentId="5612" CreationDate="2014-12-29T22:33:02.230" Score="3" Body="&lt;p&gt;Method B is great, but I'd suggest that you require students put away their pens, pencils, and own homework assignments away while other students present the problems on the board. If you don't, the students will just either: copy down exactly what is on the board, or check to see if there solution matches, without actually listening to the presentation. Several times, I've seen a non-presenting student say something like &quot;that's not what i did, but it looks like it could be right,&quot;.&#xA;I'd suggest further signing a volunteer &quot;referee&quot;: a person designated to evaluate the solution of the presenter (of course others can make remarks as well). Require that the referee do more than just say &quot;yeah it's right&quot; or &quot;that's what I did&quot; but comment on the presentation and the reasoning.&lt;/p&gt;&#xA;" OwnerUserId="3528" LastActivityDate="2014-12-29T22:33:02.230" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6135" PostTypeId="1" AcceptedAnswerId="13349" CreationDate="2014-12-30T19:31:05.910" Score="20" ViewCount="915" Body="&lt;p&gt;I will soon have to teach Taylor's formulas (with Lagrange and Peano¹ remainder terms) to first-year students in math-related curricula (computer science, physics, engineering), and I am not terribly satisfied with the proofs I have seen. I do not care so much about having optimal hypotheses, so I am happy to derive Taylor-Peano for $C^k$ functions from Taylor-Lagrange. But the only proof I have found (in numerous places) of Taylor-Lagrange looks like a clever trick: one introduce the number $A$ such that the function defined from $f$ by&#xA;$$g(x) := f(b)-\Big(f(x) + (b-x) f'(x) + \dots + (b-x)^k \frac{f^{(k)}}{k!}&#xA;  +(b-x)^{k+1} \frac A{(k+1)!}\Big)$$&#xA;satisfies $g(a)=0$, and then applies Rolle' theorem to $g$.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do you know of a proof that carries more intuition about why the result is true? Alternatively, how would you explain the intuition, if any, behind this proof?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What bothers me is that the mean value theorem is deduced from Rolle's theorem in a very clear way; maybe this is linked with the statement itself being more intuitively explained. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added in edit 1.&lt;/strong&gt; Let me clarify what bothers me here: the mean value theorem is pretty clear in its statement, as it basically says that a car that drove 60 miles in 2 hours must have had instant speed 30 mph at some point. Moreover a simple drawing explains both the statement and the proof. Somehow, I feel there should be some sort of equivalent explanations, only slightly more involved, for the Taylor-Lagrange formula and its proof. An example of what bothers me is that I found no proof that explicitly involved acceleration for the $k=1$ case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added in edit 2.&lt;/strong&gt; Let me precise the background of my students: they are assumed to have the background of French scientific section of high school, plus one semester course on reals, sequences and functions (including derivatives, Rolle's theorem and the mean value theorem). My goal would be to show them proofs that they feel they could have discovered, if given time to think about them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Too many proofs look (to students) like aliens gave them to us; this can be either because we choose to teach shorter but less conceptual proofs, or because they where discovered after long and hard thinking by mathematicians with much more training and experience and intuition than they currently have.&#xA;Independently of why it is so, I think this matter of fact tends to reinforce their common belief that math is a collection of magical formulas that have no true meaning and can only be learned by heart (and beware what happens if you use the wrong formula!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;¹ This is called after Young in France.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2014-12-31T10:43:47.770" LastActivityDate="2017-12-26T02:12:04.570" Title="How to prove Taylor formulas?" Tags="|undergraduate-education|calculus|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6136" PostTypeId="2" ParentId="6135" CreationDate="2014-12-31T13:17:46.097" Score="11" Body="&lt;p&gt;I think that teaching for understanding is a noble cause and that teaching for procedural proficiency is a necessary part of learning mathematics - I am still making my mind up about where and when it is best to stress the motivation or the procedure in my own teaching and, certainly, there comes a point where it is better to be able to do something first and then understand it later. Taylor polynomials are one of those grey areas for me, so I will share my thoughts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we want to think about the Taylor polynomial of $f(x)$ about $0$ and we want to know how good the approximation is at some point $x &amp;gt; 0.$ In order to avoid using integration I think it is necessary to consider taking the Taylor polynomial at different points along the curve $(x,f(x))$ whilst keeping $x$ fixed and considering what happens to the remainder at $x.$ I will talk about the Taylor polynomial of degree $n=1$ first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the picture below, $A$ is the point where the Taylor polynomial has been taken, $B$ has $x$ coordinate $x$ and is the point where we are interested in the size of the remainder. The size of the remainder is represented by the vertical segment $BC$ and, clearly, it changes as the position of $A$ changes and is $0$ when $A=B$ because the value of a function always coincides with the value of the Taylor polynomial at the point where the polynomial is taken.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Eee7t.gif&quot; alt=&quot;A first degree Taylor polynomial of a typical curve&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see that a 'more curved curve' - i.e. one whose second derivative is a large number for $t \in [0,x]$- will impose a larger remainder than a less curved curve: this is the relationship between the case $n=1$ and 'acceleration' (in this case, curvature) you were thinking of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now for the details. Let&#xA;$$ f(x) = f(t) + f^{\prime}(t)(x-t) + R(t) $$&#xA;so that we have taken the degree $1$ Taylor polynomial of $f$ at $t$ and the remainder is $R(t).$ Think of $t$ as the $x$-coordinate of $A$ and $x$ as the $x$-coordinate of $B$ in the image. $R(t)$ is the length $BC.$ If we rearrange we get&#xA;$$ R(t) = f(x) - f(t) - f^{\prime}(t)(x-t) $$&#xA;and we are not surprised to find that $R(x) = 0$ because the remainder when $t=x$ should be $0$ since when $t = x, A= B$ in our picture. The thing we are interested to find is $R(0),$ and for a very curvy curve, $R(0)$ will be larger than for a less curvy curve. It is natural to ask: what is the rate of change of $R(t)$ as $t$ changes? Well it's pretty clear that...&#xA;$$R^{\prime}(t) = -f^{\prime\prime}(t)(x-t) $$&#xA;for each $t.$ And so we see, the remainder will certainly be effected by the curvature of $f.$ Now, to find $R(0),$ we could try to get a handle on&#xA;$$ \frac{R(x) - R(0)}{x-0} = -\frac{R(0)}{x} $$&#xA;The mean value theorem tells us that this is equal to $R^{\prime}(t)$ for some fixed $t \in (0,x),$ so we get&#xA;$$ -\frac{R(0)}{x} = R^{\prime}(t) = -f^{\prime\prime}(t)(x-t)$$&#xA;or&#xA;$$ R(0) = xf^{\prime\prime}(t)(x-t) $$ where $t$ is now a fixed number,&#xA;which is the Cauchy form of the remainder term when $n=1.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, when you want to know that value of something ($R(0),$ here) it is natural to find its value at a later point (for us, $R(x)$) and to try to work out information about how it has changed from the initial to the final position. Not surprisingly, this is the essence of the fundamental theorem of calculus:&#xA;$$ \begin{align*}&#xA;\text{Value at end} &amp;amp; = \text{Value at beginning} + \text{Amount of change} \\&#xA;g(b) &amp;amp; = g(a) + \int_{a}^{b} g^{\prime}(t) \, dt&#xA;\end{align*} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for higher values of $n,$ I think it is normal to lose some amount of intuition: 'the rate of change of the rate of change of the rate of change of...' just gets intractable, but we are used to taking such leaps into the abstract (as we start thinking about $\mathbb{R}^{n},$ for example, as $n$ increases our intuition is based more on algebra and comparisons to smaller $n$s than direct diagrams and tangible examples).&lt;/p&gt;&#xA;" OwnerUserId="901" LastEditorUserId="901" LastEditDate="2014-12-31T13:24:32.577" LastActivityDate="2014-12-31T13:24:32.577" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6137" PostTypeId="2" ParentId="6124" CreationDate="2014-12-31T17:55:47.503" Score="4" Body="&lt;p&gt;I believe Discrete Mathematics, Logic, and Calculus to Electrical Engineers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How much has to be studied in Calculus, and how the curriculum is to be arranged, however, is a tough question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The current educational system in my country teaches Electrical Engineers the following:&#xA;Year 1: Linear Algebra, Matrix Theory, Differentiation and Integration, Logic&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that at least this much is necessary to understand circuits, programming, and the physics curriculum (which includes basic forms of the Maxwell Equations)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Year 2: Partial Differentiation, Differential Equations, Discrete Mathematics, Probability and Statistics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this curriculum is set to  help them understand topics they study in Modern Physics, programming, and electronics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After the 2nd year, undergraduates start to specialize to the extent that it is no longer possible to provide one universal curriculum which all electric engineers must know. Engineers working on electronics will need to know different mathematical concepts than those working in communication, for example.&lt;/p&gt;&#xA;" OwnerUserId="3568" LastActivityDate="2014-12-31T17:55:47.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6139" PostTypeId="2" ParentId="369" CreationDate="2014-12-31T23:01:21.917" Score="1" Body="&lt;p&gt;There was an Algebra II course using SMSG, as I remember taking it in 1964-1965 at College Park High School, Concord, CA. I do not know what the text was named, but it followed the SMSG format.&lt;/p&gt;&#xA;" OwnerUserId="3570" LastActivityDate="2014-12-31T23:01:21.917" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6140" PostTypeId="2" ParentId="575" CreationDate="2015-01-01T20:15:06.517" Score="3" Body="&lt;p&gt;Regular expressions are the approach that is most readily understood by people who use computers, and shows a theoretical foundation for a common tool. That makes it an attractive approach if your audience is a more technical, computer-savvy audience, both because they'll be familiar with some of the basics and because it nicely illustrates how theory solves practical problems. You can then progress to relate regular expression to finiteness of state, either through NFAs or through quotients. Both require a significant leap in the level of abstractness, which pays off by providing a theoretical motivation for the concrete tool but does have a risk of losing students on the way. At an undergraduate level, I don't think you need to show both finite automata and quotients.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Starting with finite automata is good for a course about models of computation. If you've already introduced Turing machines and formal languages, you've already introduced finite automata, and it's rather natural to study them on their own and see what kinds of languages are recognizable. Regular expressions can be introduced progressively, first by proving that various constructions (concatenation, alternation, Kleene star, etc.) conserve the set of recognizable languages, then by proving or stating that all recognizable languages can be built from a small set of operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The syntactic monoid approach is only suitable for students with a solid mathematical background. It's not that it requires much prior knowledge, but if the set manipulations are to have any meaning, the students must be very experienced in reasoning formally about sets. If there's a long sludge through symbol pushing and applications only come afterwards, there's a high risk that the initial presentation will not be understood at all and will be wasted time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whichever formalism you start with, it's critical to tie regular expressions with a theoretical foundation. Even if you don't have time to do all the proofs, make sure that the equivalence between finite state and regular expressions is well-accepted. This is a nice example where some things that are hard to prove under one approach are trivial under the other approach. For example, I've often seen programmers ask how to negate a regular expression; that's trivial to do on an NFA, but not so much on a DFA, leading to the practical answer that it's always possible but the resulting regexp may be very large.&lt;/p&gt;&#xA;" OwnerUserId="170" LastActivityDate="2015-01-01T20:15:06.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6143" PostTypeId="2" ParentId="6045" CreationDate="2015-01-01T21:28:18.450" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;This leads to two major problems: students don't deal with a suitable level of abstraction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Finding the right level of abstraction for a problem is a major challenge. It takes enough knowledge of the subject not to get bogged down in details, and a lot of experience generally. You can't expect students to easily navigate between levels of abstraction. They can't know which details are actually important and which ones it's safe to skip over. Even professional mathematicians sometimes get it wrong, which is why some complex proofs take a while to review and to build confidence in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actual programming languages with formal rules provide a concrete anchor: if it isn't in the language, it needs to be defined. Mathematical language is also defined formally enough to express proofs, but for algorithms it has the same ambiguity as pseudocode: it doesn't distinguish between what is algorithmically trivial and what is easy mathematically but not constructively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to give very solid guidance as to what can be treated as a black box and what needs to be spelled out. Choosing a programming language has the advantage that it provides a formalism for black boxes: it's a function that students don't need to write by themselves.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;it can lead to actually doing the assignments being intractable. Especially when the efficient algorithms they should be black boxing for certain purposes are either complex, open problems, or something they shouldn't need to reroll (sorting algorithms after the sorting section is covered).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You need to make it clear to students what they may assume and what they have to write! If you want them to skip over some difficult problem, that absolutely needs to be made explicit. In the assignment, list out what students may assume. If you don't want students to reroll a sort algorithm, specify something like “assume that all classes have a &lt;code&gt;compare&lt;/code&gt; method and all arrays can be sorted by calling their &lt;code&gt;sort&lt;/code&gt; method”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your students aren't mature enough for pseudocode, I don't think you should try to force them into it. Do keep insisting on readable descriptions of algorithms, and do tell them that you don't care about all the punctuation as long as what they write is comprehensible to humans. Tell them to stick to basic language features and not care about code reusability, so no need to wrap code in classes, no templates, no need to catch exceptions, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some languages work better than others: Python can often work pretty well as pseudocode, Java or C++ markedly less so. But if the students are only comfortable with writing Java, an algorithms course isn't the place to teach them something else. Provide experience in figuring out the right level of abstraction, but don't make it a course requirement. Of course, do give them the option to use English, tell them that they're more likely to succeed if they don't get bogged down in programming micromanagement. But some students will have a hard time deviating from the programming language they know, and that's ok.&lt;/p&gt;&#xA;" OwnerUserId="170" LastActivityDate="2015-01-01T21:28:18.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6144" PostTypeId="2" ParentId="6130" CreationDate="2015-01-01T22:39:21.380" Score="4" Body="&lt;p&gt;If you want to study real and complex analysis, a classic text is W. Rudin's &quot;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0070542341&quot; rel=&quot;nofollow&quot;&gt;Real and complex analysis&lt;/a&gt;&quot;. In fact pretty much anything by Rudin is excellent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more accessible title of Rudin is &quot;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0070856133&quot; rel=&quot;nofollow&quot;&gt;Principles of Mathematical Analysis&lt;/a&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another classic complex analysis book is by &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0070006571&quot; rel=&quot;nofollow&quot;&gt;Alfors&lt;/a&gt;, although the title by &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521287634&quot; rel=&quot;nofollow&quot;&gt;Stewart and Tall&lt;/a&gt;, both called &quot;Complex analysis&quot;, might be more accessible (and slightly more modern in its approach).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are all introductory (1st year mathematics undergraduate) texts - for UK universities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rudin's &quot;Real and complex analysis&quot; does contain advanced material, and is probably not suitable as an introductory text for an engineering student. Worth a look, just in case.&lt;/p&gt;&#xA;" OwnerUserId="3541" LastEditorUserId="3541" LastEditDate="2015-01-02T01:12:47.733" LastActivityDate="2015-01-02T01:12:47.733" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6145" PostTypeId="2" ParentId="2099" CreationDate="2015-01-02T18:21:41.450" Score="4" Body="&lt;p&gt;The premise here is that we are limited by traditional curriculum of these courses. Continuous rates of change are essential to understanding the world. The machinery of calculus is not. Calculus, however is structured to focus on the later rather than the former. In fact I suspect that if you asked many calculus students basic conceptual questions about the nature of calculus, they would not do very well. Courses focus on what I will call &quot;calculytic techniques&quot; -- techniques used to compute derivatives and integrals. While for advanced students these can be fun puzzles, for many these are exercises in futility that seem to have no relation to the the concepts underlying calculus. Furthermore, I would argue that in an age of Ti-89's and Wolfram Alpha, we need to reevaluate how valuable teaching students these techniques actually is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would make the same argument for statistics. Statistics education can be equally befuddling for many students because it focuses on manual computation of statistics in a way that does not usually develop understandings of how to interpret statistics in context and how to understand the significance of that interpretation. Again, statistics curriculum is shaped by an age without technology, and we have to reevaluate it in the context of ubiquitous computational technology. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this way, teaching statistics also does not solve the problem posed by the original author -- that students learn math that will be valuable for them. &lt;strong&gt;Fundamentally, our mathematics curriculum (at all levels) does not reflect what we want students to be able to do -- transfer mathematical thinking from one topic to another, one class to another or to their lives filled with work, games, and puzzles that all require mathematical thinking.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In all of the our courses at the high school level, we focus on content (and computation) at the expense of teaching mathematical problem solving skills to most students. In the age of the common core, there is more pressure than ever to do both -- teach skills and content, but I do not believe that this is fully possible. Students cannot simultaneously engage in developing transferable problem solving skills around content they are not fluent with. That, however, is a post for another time. &lt;/p&gt;&#xA;" OwnerUserId="3485" LastEditorUserId="3485" LastEditDate="2015-01-04T15:38:33.117" LastActivityDate="2015-01-04T15:38:33.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6146" PostTypeId="1" AcceptedAnswerId="6151" CreationDate="2015-01-02T19:21:14.507" Score="31" ViewCount="1004" Body="&lt;p&gt;I volunteer with a group that provides tutoring to kids from grades nine through twelve. The included kids have been determined to be 'at risk of not graduating high school'. Of course, many of the students bring mathematics homework to the group, and they present a wide range of skill levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of the tutors mean well (it's hard not to in a volunteer position) but the majority of them are some combination of incapable, unwilling, or uncomfortable with working on mathematics with the students. A student looking for help with math is often met with phrases in the range&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;I'm not good at math&quot; &lt;/li&gt;&#xA;&lt;li&gt;&quot;I can't math&quot; (this one drives me insane)&lt;/li&gt;&#xA;&lt;li&gt;&quot;Everyone has trouble with math, hey?&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;That looks hard...&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The tutor then seeks out one of the 'math people' in order to pair them up, and quickly abandons the scene.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To my mind, this communicates to the students that the tutors believe some combination of the following &lt;em&gt;very bad things&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Only certain people can learn math.&lt;/li&gt;&#xA;&lt;li&gt;That they don't value math.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and subtly encourages them to adopt these beliefs themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can such interactions be steered toward something more productive? How can math phobic adults be convinced to participate in actively talking, learning, and participating in mathematics? How  one communicate to a well meaning volunteer that their actions are potentially damaging without alienating them or hurting their feelings?&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2015-01-04T09:07:10.007" Title="How to tactfully discourage casual, implicit disparagement of mathematics" Tags="|student-motivation|tutoring|teacher-motivation|culture|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6148" PostTypeId="2" ParentId="2763" CreationDate="2015-01-02T21:52:38.410" Score="12" Body="&lt;p&gt;We all want students to know how to compute and also to have the conceptual understanding of why their computational method works. I suggest that we can teach repeated addition &lt;strong&gt;as a computational strategy&lt;/strong&gt; when multiplying with whole numbers. But there is more to the question can computation. JPBurke breaks the question into multiple questions, and I won’t repeat the ground he covers. I’ll address a different aspect: “What do students miss when multiplication is taught as repeated addition?”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When students are taught multiplication as repeated addition, they are only being taught how to compute the answer. They are not taught the skills that are required for multiplicative reasoning, so they have great difficulty with ratio and proportion and rational numbers. Multiplicative reasoning requires a different way of thinking about units than additive reasoning. With addition, there is only one unit. With multiplication there are two units, and more importantly, a ratio relationship between the two units. This relationship can be thought of as a many-to-one correspondence, for example, 12 inches per foot, or $3 for every pound. Nunes and Bryant emphasize this ratio relationship as a basic element of multiplication in &lt;em&gt;Children Doing Mathematics&lt;/em&gt;. Michael Goldenberg goes into more detail on the Nunes and Bryant arguments on his blog.(I am allowed only two links, link below) Park and Nunes did a study testing repeated addition versus correspondence as a basis for understanding multiplication. In multiplicative reasoning tasks, the students who were taught multiplication through correspondence significantly outperformed the children taught multiplication through repeated addition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is another important difference in how students think about units with multiplication that Nunes and Bryant do not address. With additive reasoning, since there is only one unit, there is little attention paid to what is one whole. It is one, or what we could call the standard unit 1. With multiplicative reasoning, what is one whole or 100% can change with different elements in the problem situation. Davydov defines multiplication as counting with an intermediate, non-standard unit. For example, 3 x 7 can be considered as 3 units of 7, where seven is the new one whole. The &lt;em&gt;seven&lt;/em&gt; is considered an intermediate unit because we want the count in standard units: 3 &lt;em&gt;sevens&lt;/em&gt; is 21 &lt;em&gt;ones&lt;/em&gt;. Thinking of units this way helps students be more flexible in thinking about what is one whole, and multiplicative reasoning requires that flexibility and more; it often requires actively looking for what is the whole or 100%. One last example, comparing 20 to 4 multiplicatively, 4 is the whole and 20 is 5 wholes or 500% of 4. This is essentially the quotative meaning of division, in which what is one whole can change with every division expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplicative reasoning requires thinking of numbers not as individual fixed entities, but in relation to other numbers—that is, to see a correspondence of one number to another, or to see a number relative to one whole that is not the standard unit 1, but relative to a non-standard reference unit. When we teach multiplication, we should help students begin to think in terms of different units (scales) and the relationship between numbers or scales. Joan Moss says,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We know from extensive research that many people—adults, students, even teachers—find the rational-number system to be very difficult. Introduced in early elementary school, this number system requires that students reformulate their concept of number in a major way. They must go beyond whole-number ideas, in which a number expresses a fixed quantity, to understand numbers that are expressed in relationship to other numbers. These new proportional relationships are grounded in multiplicative reasoning that is quite different from the additive reasoning that characterizes whole numbers. (p.310)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When students are taught multiplication, we should begin to build the new knowledge that they need for multiplicative reasoning, and not just continue to reinforce additive reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cited&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Davydov, V.V. (1992). The psychological analysis of multiplication procedures. &lt;em&gt;Focus on Learning Problems in Mathematics&lt;/em&gt;, vol 14, pp. 3-67. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Goldenberg, M.P. (13 March 2010). Tereza Nunes and Peter Bryant dole out the multiplicative harshness, Retrieved from Rational Mathematics Education blog: &lt;a href=&quot;http://rationalmathed.blogspot.com/2010/03/terezinha-nunes-and-peter-bryant-dole.html&quot; rel=&quot;noreferrer&quot;&gt;http://rationalmathed.blogspot.com/2010/03/terezinha-nunes-and-peter-bryant-dole.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moss, J. (2005). Pipes, tubes, and beakers: New Approaches to teaching the rational number system, in &lt;em&gt;How Students Learn: History, Mathematics, and Science in the Classroom&lt;/em&gt;.  Donovan, M.S. &amp;amp; Bransford, J.D. (eds), Washington, DC: National Academy Press. &lt;a href=&quot;http://www.nap.edu/catalog/10126.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.nap.edu/catalog/10126.html&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nunes, T. &amp;amp; Bryant, P. (1996). &lt;em&gt;Children Doing Mathematics&lt;/em&gt;. Malden, MA: Blackwell Publishers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Park, J-H &amp;amp; Nunes, T. (2001). The development of the concept of multiplication. &lt;em&gt;Cognitive Development&lt;/em&gt;, 16, 763-773.&lt;/p&gt;&#xA;" OwnerUserId="3578" LastEditorUserId="3578" LastEditDate="2015-01-02T21:58:45.780" LastActivityDate="2015-01-02T21:58:45.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6149" PostTypeId="2" ParentId="2763" CreationDate="2015-01-02T22:15:29.090" Score="6" Body="&lt;p&gt;[note: links below in reference section]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let’s put this issue in the larger context of how we teach math. Phil Daro says Japanese teachers have the goal of teaching math concepts while American teachers have the goal of teaching how to compute answers. But all teachers will say that their goal is to teach students to understand math concepts as well as to compute correct answers. The evidence, such as the TIMSS video’s that Phil Daro reviewed, says that teachers believe they are teaching math concepts but are only teaching how to compute answers. They don’t in practice distinguish between conceptual knowledge and procedural knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why do we confuse a computational procedure with the concept? Why are we still stuck on the idea that if students can compute the answer to a problem then they understand the concepts involved? I can think of two reasons. One, that is how math has been taught. We think math involves memorizing computational procedures and computing the answers to problems, and we don’t understand much else. That is why we can multiply but we can’t explain what multiplication is. We don’t think of it as anything other than how to compute the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, carrying out procedures does involve concepts--procedural concepts, not concepts of the underlying mathematics. So the term “conceptual understanding” is ambiguous and easily interpreted as understanding how to use a computational procedure correctly. This is what Liping Ma describes in her book &lt;em&gt;Knowing and Teaching Elementary Mathematics&lt;/em&gt; on pages 36-37. Some of the teachers in her study had a strong belief in “teaching mathematics for understanding.” In their explanations and teaching practices, however, “no conceptual learning was evidenced at all.” What the teachers wanted the students to “understand” was the correct steps in carrying out the computational procedure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The larger issue, then, is to make a clear distinction between what Richard Skemp calls relational understanding and instrumental understanding. Both involve “concepts” and “understanding,” but mean different things. The distinction is an issue for every topic in math, not just multiplication. Phil Daro says if there was one thing he could change in our educational system, it would be to go beyond answer getting and teach relational understanding. For this to happen, research by James Spillane says that teachers need to see a different way of teaching. They need to make their teaching public and collaborate. Elizabeth Green’s book &lt;em&gt;Building a Better Teacher&lt;/em&gt; has many examples of the power of teacher collaboration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cited&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Daro, P. Against “Answer-Getting.” Video retrieved from Vimeo: &lt;a href=&quot;http://vimeo.com/79916037&quot; rel=&quot;noreferrer&quot;&gt;http://vimeo.com/79916037&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Green, E. (2014) &lt;em&gt;Building a better teacher: How teaching works (and how to teach it to everyone)&lt;/em&gt;. New York: W. W. Norton &amp;amp; Company.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ma, L. (1999). &lt;em&gt;Knowing and teaching elementary mathematics&lt;/em&gt;. Mahwah, NJ: Lawrence Erlbaum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Skemp, R. (1976). Relational Understanding and Instrumental Understanding. &lt;em&gt;Mathematics Teaching&lt;/em&gt;, 77:20-26. Also at &lt;a href=&quot;http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Spillane, J. P. (1999). External Reform Initiatives and Teachers' Efforts To Reconstruct Their Practice: The Mediating Role of Teachers' Zones of Enactment. &lt;em&gt;Journal of Curriculum Studies&lt;/em&gt;, 31(2): 143-75.&lt;/p&gt;&#xA;" OwnerUserId="3578" LastActivityDate="2015-01-02T22:15:29.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6150" PostTypeId="1" CreationDate="2015-01-02T23:20:44.350" Score="4" ViewCount="157" Body="&lt;p&gt;Recently, my junior high school (aka middle school) reorganized their curriculum. Instead of teaching middle school physics, chemistry, and biology in that order, they reversed it. The logic was that older children have more mathematical knowledge and therefore are better able to absorb concepts in the physical sciences later. This parallels how I was taught in high school but I found it surprising my middle school made this observation so late. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Upon further reflection I realized, given that my middle school teachers were not as educated as my high school teachers, I am not surprised it took longer for them to discover this order was better. Most of the middle school science teachers I had were not particularly talented at mathematics. But I feel if they had more mathematical knowledge, they would have realized this order was better in the first place. I personally found learning Newton's law in sixth grade excruciating. It was as though my teacher use the concept of energy carelessly without ever defining it. I had no idea what it was! Now I study quantum fields. So clearly the problem was that teachers were dodging around the mathematics because they didn't think we could handle it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My middle school's reorganization got me thinking: what if mathematics teachers had played an earlier role in helping to structure the science curriculum? For science teachers with less mathematical experience, suggestions from the math department could prove very useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question:&#xA;Do junior high math mathematics educators frequently establish dialogue with their science departments? How is this received by the science departments? Across junior high schools, is it often that the two departments work together to give kids the best math and science education possible? If not, why?  &lt;/p&gt;&#xA;" OwnerUserId="3471" LastEditorUserId="3471" LastEditDate="2015-01-03T00:24:35.637" LastActivityDate="2015-07-06T16:46:19.053" Title="Is there communication between junior high math and science departments?" Tags="|curriculum|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6151" PostTypeId="2" ParentId="6146" CreationDate="2015-01-03T01:36:59.727" Score="19" Body="&lt;p&gt;The tutors need training. Any volunteer job has job requirements, and one of the requirements of this job [for it to be done well] is being able to tutor anything that comes along. The training can mainly involve intriguing problems that will help the tutors enjoy math themselves. Then, if they get stuck on a student's problem, they can seek help in a positive way. &quot;Oh cool, I'm stuck. Let's talk to X, who might be able to point us in the right direction.&quot;&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-01-03T01:36:59.727" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="6153" PostTypeId="2" ParentId="6130" CreationDate="2015-01-03T03:11:15.010" Score="2" Body="&lt;p&gt;A would agree with Rusan Kax but like to add that another good introductory analysis book is by Tom Apsotol. Also, for beginners in analysis, I would say either start with Rudin's baby analysis or Apsotol. As for Complex analysis, you may want to learn some abstract algebra and/or number theory since many texts incorporate the use of Algebra. Books in complex analysis that I think are good for beginners are Graduate Text in Complex Analysis by Serge Lang and Complex Analysis by Alfors as well. If you are keen to learning some Algebra, I would suggest Dummit and Foote; however, it is not a small text.&lt;/p&gt;&#xA;" OwnerUserId="3581" LastEditorUserId="3581" LastEditDate="2015-01-03T03:28:06.437" LastActivityDate="2015-01-03T03:28:06.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6154" PostTypeId="2" ParentId="6146" CreationDate="2015-01-03T10:59:21.207" Score="2" Body="&lt;p&gt;I don't see that there is anything you can do other than teach the non-math volunteers math.  Your normal &quot;Joe&quot; isn't going to be able to do pre-cal homework.  The only real thing I would push is tell them to find someone who can help, but never, ever, under &lt;em&gt;any&lt;/em&gt; circumstances say things like &quot;Everyone has trouble with math, hey&quot; or anything else that pushes the notion that math is OK to be &quot;bad&quot; at.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The tutoring center should really be split up so that math tutors are the ones tutoring math and non-math people are never approached with math questions.  Even the honest &quot;I'm not good at math&quot; is dangerously close to reinforcing the idea that it's OK to be bad at math.  So those people really should &lt;em&gt;not&lt;/em&gt; be allowed to field any kind of math question.  If faced with something like that, they should say something like &quot;Let me find you someone who can better help you with that question.&quot;&lt;/p&gt;&#xA;" OwnerUserId="2204" LastActivityDate="2015-01-03T10:59:21.207" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6155" PostTypeId="2" ParentId="6135" CreationDate="2015-01-03T17:58:34.137" Score="1" Body="&lt;p&gt;You can start with powers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;f (x) = $x^0$ -&gt; f (x) = $x^0 / 0! * f (0)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;f (x) = $x^1$ -&gt; f (x) = $x^1 / 1! * f' (0)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;f (x) = $x^4$ -&gt; f (x) = $x^4 / 4! * f^{iv} (0)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd write it down without the x^0, ^1, 0! and 1! and fill them out later so that all the formulas are identical. Then we observe for example when f (x) = $x^3$, then the formula $x^4 / 4! * f^{iv} (0)$ gives 0. So we can add in all the other formulas like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;f (x) = $x^4$ -&gt; f (x) = $x^0 / 0! * f (0) + x^1 / 1! * f' (0) + ... + x^4 / 4! * f^{iv} (0) + ...$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the same formula applies for all powers. It also works for all polynomials. So now the obvious question is what happens for other functions? &lt;/p&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2015-01-03T17:58:34.137" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6156" PostTypeId="2" ParentId="5837" CreationDate="2015-01-03T20:40:56.777" Score="1" Body="&lt;p&gt;To some, chess is.  It's basic in the sense that it is perhaps the most popular game, and it provides many fruitful visual aids containing TONS of information!&lt;/p&gt;&#xA;" OwnerUserId="1816" LastActivityDate="2015-01-03T20:40:56.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6157" PostTypeId="1" AcceptedAnswerId="6182" CreationDate="2015-01-04T00:26:17.587" Score="6" ViewCount="346" Body="&lt;p&gt;My younger cousin (2nd grade USA) is advancing very quickly in his current math class.  The school does not have much to support him in moving ahead in the material.  What websites (preferably game or challenge based) are good for kids that want to practice and learn more math?&lt;/p&gt;&#xA;" OwnerUserId="3582" OwnerDisplayName="MicFin" LastEditorUserId="3582" LastEditDate="2015-01-04T00:44:33.093" LastActivityDate="2015-01-06T20:44:07.597" Title="Fun resources and games for advanced elementary school math students?" Tags="|primary-education|self-learning|games|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6158" PostTypeId="2" ParentId="6146" CreationDate="2015-01-04T00:39:07.953" Score="4" Body="&lt;p&gt;Most of my job tutoring high school math is giving my students confidence -- it's rare that a student of mine will actually have trouble with math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Switch at least part of your focus to &lt;strong&gt;building your students' confidence&lt;/strong&gt;:  &quot;hey, that wasn't so hard, was it?,&quot; &quot;See, you &lt;em&gt;can&lt;/em&gt; do this!&quot;  If you can't show their students how to do something, then you &lt;em&gt;should not&lt;/em&gt; make discouraging statements, and can say instead &quot;I don't know, let's find out!&quot;  At that point, you can either look up the answer, or ask someone else for help. (Also, as Sue &lt;a href=&quot;https://matheducators.stackexchange.com/a/6151/1685&quot;&gt;pointed out&lt;/a&gt;, the tutors should &lt;em&gt;definitely&lt;/em&gt; prepare themselves to help out with whatever comes up.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One last note -- if you &quot;can't math,&quot; then I'd add that you can't English, either.&lt;/p&gt;&#xA;" OwnerUserId="1685" LastEditorUserId="-1" LastEditorDisplayName="user173" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-01-04T08:21:31.557" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6159" PostTypeId="2" ParentId="6157" CreationDate="2015-01-04T00:44:13.390" Score="0" Body="&lt;p&gt;From what I know, Alexey Root has a series of wonderful chess books that aren't bad!  Yamie Chess also seems to be popular although I haven't touched any of its contents.  It was invented by former US Women's Champion Jennifer Shahade.&lt;/p&gt;&#xA;" OwnerUserId="1816" LastActivityDate="2015-01-04T00:44:13.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6160" PostTypeId="2" ParentId="6146" CreationDate="2015-01-04T09:07:10.007" Score="3" Body="&lt;p&gt;Most people can learn the math that is useful to them.  One problem here is that much of high school math isn't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not bothered by tutors saying:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;I'm not good at factoring&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;I can't figure out what axioms to put down in geometry problems&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;Everyone has trouble with trig identities, hey?&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;By contrast, tutors would be embarassed to say things like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;I'm not good at budgeting&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;I can't figure out how big an air conditioner I need&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;Everyone has trouble evaluating choices, hey?&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;...because when the tutors face practical issues they probably figure out how to do the math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when students get word problems, I recommend that tutors talk through the situations.  Algebra classes can have word problems on budgeting.  Geometry classes can have problems on air conditioners.  Statistics classes can have problems on evaluating choices.  In context, students and tutors both will see the point of going to the numbers, and figuring out what to do with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When students get problems on factoring and axioms and trigonometry, then I recommend an entirely different response:  &quot;This stuff is about manipulating formulas, and you may never have to do that after you leave high school.  But people have set it as a test anyway, and they will use it to determine whether to admit you to a college or offer you a job.  You can choose to avoid the challenge...and accept that some influential people will look down on you for it.  Or you can figure out how to meet the challenge, and see what opportunities it opens up.&quot;&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2015-01-04T09:07:10.007" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6161" PostTypeId="2" ParentId="5806" CreationDate="2015-01-04T09:16:25.330" Score="4" Body="&lt;p&gt;I think it is great that the parents are asking for resources. Generally, the more involved the parents are, the better. They may need to help the students on any website, at least in the beginning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As others have said, the Common Core is a set of standards. They are intended to make instruction more rigorous, which means to increase the depth of understanding. When students &lt;strong&gt;practice&lt;/strong&gt; alone, they are mostly reinforcing computational skills. I can recommend a couple of sites where some new learning may also take place.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.mathmammoth.com/worksheets/free.php&quot; rel=&quot;nofollow&quot;&gt;Maria Miller’s Freebies&lt;/a&gt; page has worksheets, video lessons, written lessons, assessment tests, and printable children’s activities, for elementary and middle school children. All are free; no registration required.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://thesingaporemaths.com/index.html&quot; rel=&quot;nofollow&quot;&gt;The Singapore Maths Teacher&lt;/a&gt; site gives step-by-step instructions on solving word problems primarily using the &lt;em&gt;block models&lt;/em&gt; that are taught in Singapore Math, but also using the following strategies: draw a picture, look for a pattern, guess and check, make a systematic list, logical reasoning, and work backwards. These problems are similar to those used in Singapore Math grades 3 to 6, and can be accessed without even registering.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;To increase understanding, students need to reflect on the math that they have done or are doing. This often occurs in small group or whole class discussions when students are trying to resolve differences of opinions, or solve a problem as a group, sharing, evaluating, and building on each others’ ideas. Unfortunately, group discussions don’t apply to students working alone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way that students can reflect on math when working alone is by writing. The teacher can give writing assignments that students can do as homework, or the parents can prompt the child to write about math. &lt;a href=&quot;http://www2.ups.edu/community/tofu/lev2/journaling/writemath.htm&quot; rel=&quot;nofollow&quot; &gt;Using Writing in Mathematics&lt;/a&gt; gives teachers and parents guidelines on getting students to write about math. Here the parent’s involvement is important. Can parents discuss the children’s writing with them? Do the parents &lt;em&gt;understand&lt;/em&gt; the math concepts enough to help the students refine their writing (without telling the students what to write)? Even if the parents do nothing to help the students evaluate their writing, the process of writing will be beneficial for the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parents helping students at home should know what math topics their children are learning, but I don’t think parents should be concerned about aligning with specific standards. Alignment with standards is technical, and too much focus on it could lead to a mechanical treatment of the standards. Parents should focus more on the intent of the Common Core. Do their children like math rather than dread it? Do they understand the concepts? Are they confident? Can they justify their answers with logical explanations, not just list the procedural steps used to calculate the answers? If the answers are “No” then the standards are not being met. Parents should be pleased with “Yes” answers to these questions for the content that the students cover in class.&lt;/p&gt;&#xA;" OwnerUserId="3578" LastEditorUserId="3578" LastEditDate="2015-01-04T21:14:30.083" LastActivityDate="2015-01-04T21:14:30.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6162" PostTypeId="2" ParentId="5806" CreationDate="2015-01-04T13:36:53.217" Score="-2" Body="&lt;p&gt;Another place to look around for resources would be&lt;br&gt;&#xA;&lt;a href=&quot;http://www.artofproblemsolving.com&quot; rel=&quot;nofollow&quot;&gt;The Art of Problem Solving&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-01-04T13:36:53.217" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6163" PostTypeId="1" CreationDate="2015-01-04T16:11:13.670" Score="35" ViewCount="11535" Body="&lt;p&gt;A colleague and I recently discussed what we call the &quot;Triangle Trap.&quot; High school geometry covers a very large unit reflecting the common core:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Classifying Triangles&lt;/li&gt;&#xA;&lt;li&gt;Triangle Angle Properties&lt;/li&gt;&#xA;&lt;li&gt;Constructing Triangles&lt;/li&gt;&#xA;&lt;li&gt;Properties of Triangles (e.g. Equilateral, isosceles)&lt;/li&gt;&#xA;&lt;li&gt;Triangle Similarity&lt;/li&gt;&#xA;&lt;li&gt;Triangle Congruence&lt;/li&gt;&#xA;&lt;li&gt;Triangle centers&lt;/li&gt;&#xA;&lt;li&gt;Pythagorean Theorem&lt;/li&gt;&#xA;&lt;li&gt;Special Right Triangles&lt;/li&gt;&#xA;&lt;li&gt;Right Triangle Trig&lt;/li&gt;&#xA;&lt;li&gt;Law of Sines and cosines&lt;/li&gt;&#xA;&lt;li&gt;Area of Triangles&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This represents almost a third of our year. And our students get triangle fever -- they get sick and tired of triangles, and honestly so do we. There are so many other exciting topics to delve into. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the value of placing this large of an emphasis on triangles in a secondary geometry context?&lt;/p&gt;&#xA;" OwnerUserId="3485" LastActivityDate="2020-04-05T00:22:39.710" Title="Why are triangles so prevalent in high school geometry?" Tags="|secondary-education|curriculum|geometry|" AnswerCount="17" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="6164" PostTypeId="2" ParentId="6163" CreationDate="2015-01-04T18:25:27.790" Score="15" Body="&lt;p&gt;This is not an answer, but I share what I take to be your skepticism. I think &lt;em&gt;polygons&lt;/em&gt; are a rich topic. For example, just understanding that every simple polygon can be triangulated would be an achievement. And it would lead to the often surprising conclusion that all $n$-gons, regardless of shape, have the same sum of internal angles: $(n-2)\cdot180^\circ$.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/JjHbZ.png&quot; alt=&quot;Fig.1.3&quot;&gt;&#xA;&lt;hr /&gt;&#xA;Another surprise: Not every simple polyhedron can be tetrahedralized.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="3237" LastEditDate="2015-01-05T17:33:49.490" LastActivityDate="2015-01-05T17:33:49.490" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6165" PostTypeId="2" ParentId="6163" CreationDate="2015-01-04T18:35:56.803" Score="4" Body="&lt;p&gt;Maybe it's not worth placing such a large emphasis on the topic.  The Common Core standards about triangles are &lt;a href=&quot;http://www.corestandards.org/Math/Content/HSG/SRT/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, about 11 out of the 43 standards for geometry.  I see at most one triangle center on that list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you move on to other topics like similarity and areas sooner, rather than including them in the triangles unit?  Then you can use triangles as a familiar example in the midst of a more interesting discussion.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2015-01-04T19:08:13.050" LastActivityDate="2015-01-04T19:08:13.050" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6166" PostTypeId="2" ParentId="6163" CreationDate="2015-01-04T19:05:33.107" Score="15" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt; Triangles helped me understand both unit circle and trig functions. Super cool and super useful. Didn't really use much of my other geometry. Don't drop triangles. If you do, you should have a good reason and better replacement.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am not a mathematics educator. But I did take junior high geometry that covered all those topics. Here's a student's perspective. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't see a topic on that list I regret studying. I always thought studying triangles in-depth made it easier to appreciate the role of triangles in defining the trig functions using the unit circle. I took Trigonometry freshman year of high school. If I had just learned SOHCAHTOA alone, I would have never appreciated the important the properties of right triangles versus nonright triangles. For me, my first look at trig was seeing how geometry could give me a new powerful mathematical tool (ie sin cos tan) through the geometry I has learned. Thus, understanding many properties of  triangles gave me a better appreciation for how trig functions are defined and why right triangles are useful for that. I thought trigonometry was beautiful in part because I was so familiar with triangles. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I feel my future math courses never used much of the geometry I learned in that course. I have never used SAS or SSS ever since then.  In fact, triangles were one of the few things I actually used after that course. Only many years later when I began studying differential geometry and non-Euclidean geometry did I really feel a curiosity to revisit the principles I learned in my geometry class. And I personally always was disappointed by this. Now that I reflect on it, most courses I took in high school I have used. Geometry would be an exception in that much of it I haven't used. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So perhaps if curriculums made more use of non-triangle concepts, it would be worthwhile to study stuff besides triangles. But, especially if you do physics or anything involving Fourier methods, you want to have a good grasp of trig and I feel triangles are a key piece of that. In other words, I wouldn't change how triangles are taught unless I see topics I know will be more useful that should replace them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I actually would have liked to study stuff besides triangles more in-depth. So I am not necessarily pro-triangle. But I am always trying to invest my time in things that are useful and triangles clearly were the most valuable part of my geometry course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 1: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of the above was, so to speak, in defense of triangles. But suppose there are better things to teach. Like what? If teachers want to teach less about triangles, I would be interested to hear a proposal for what should be taught in place of them. This proposal should keep in mind that secondary school isn't designed to train mathematicians but to provide children and young adults mathematical tools that will be useful for their professions. This ranges from finance to physics and beyond, so we aren't looking for merely the most interesting math but the most useful. So such a proposal should (1) specify what use the math would serve and (2) why this is more useful than studying triangles. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 2: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also think you could simply break up the triangles into separate units instead of bundling them together. This would allow you to teach other topics in between. For instance, split off together the trigonometric properties of triangles as well as the laws of sines and cosines. Make those a separate unit and do that unit later in the year. It should be workable I would imagine without having to drastically refresh the students on everything. &lt;/p&gt;&#xA;" OwnerUserId="3471" LastEditorUserId="3471" LastEditDate="2015-05-15T23:22:15.253" LastActivityDate="2015-05-15T23:22:15.253" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6167" PostTypeId="2" ParentId="6163" CreationDate="2015-01-04T19:39:06.507" Score="2" Body="&lt;p&gt;Geometry fan here! There's so much stuff about triangles to learn. For example we have at least &lt;a href=&quot;http://faculty.evansville.edu/ck6/encyclopedia/ETC.html&quot; rel=&quot;nofollow&quot;&gt;5000 triangle centers compiled by Clark Kimberling&lt;/a&gt; and still being updated. That said, there's so much stuff with triangles that doesn't work as well for every quadrilateral, hence why whatever section on quadrilaterals always divides it into parallelograms, trapezoids, rhombuses, rectangles, sqauares, kites, etc. There's so few that can be said about any general quadrilateral besides Newton Gauss line and Miquel point, but all these require the knowledge of circles and of power of a point, whose proof relies on similar triangles. If you wanted to study only cyclic quadrilaterals you need to study first circles (done in the final 1/4 of the year) which itself requires the knowledge of right triangles. &lt;/p&gt;&#xA;" OwnerUserId="3585" LastActivityDate="2015-01-04T19:39:06.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6168" PostTypeId="2" ParentId="5635" CreationDate="2015-01-04T20:57:44.003" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Aleksandrov, Aleksandr Danilovich, and Andreĭ Nikolaevich Kolmogorov, eds. &lt;em&gt;Mathematics: its content, methods and meaning&lt;/em&gt;. Courier Dover Publications, 1999.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/ZC1tG.jpg&quot; alt=&quot;Cover&quot;&gt;&#xA;&lt;hr /&gt;&#xA;This 1,103-page book was originally published in 1956. It is translated from the Russian and now reprinted by Dover (so it is relatively inexpensive).&#xA;I purchased it for A.D. Aleksandrov's contributions, but the list of chapter authors is a who's-who in Russian mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A NYTimes review said, &quot;What makes these volumes so readable as compared with usual mathematics textbooks is the emphasis here upon basic concepts and results rather than upon the intricate and wearying proofs that make such demands in conventional textbooks and courses.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one little result I learned from this book: &quot;&lt;a href=&quot;https://mathoverflow.net/q/77760/6094&quot;&gt;Why is the half-torus rigid?&lt;/a&gt;.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-01-04T20:57:44.003" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6169" PostTypeId="2" ParentId="6163" CreationDate="2015-01-05T00:13:23.110" Score="10" Body="&lt;p&gt;There is a school of thought that holds that triangles are so big in the high school curriculum at least partly for historical reasons. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, triangles are big in Euclid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, trigonometry was important to the military of the 19th century imperial powers in mapping and navigation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the circular functions are very important within mathematical analysis - you'll see a lot of them when you do calculus later in high school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we think of applications, geometry can also be approached through use of coordinate geometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which approach is more practical depends on the starting information. If you are a surveyor using a theodolite, trigonometry is the technique to use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, if you are writing software or using GPS, your starting point is coordinates not angles. In that case, coordinate geometry using scalar and wedge products has many benefits. However, this is a new situation; it takes a long time for the curriculum to adapt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd also note that the Pythagorean Theorem is not only famous, but perhaps the first theorem you see proved. Theorem proving is what mathematics is actually about.&lt;/p&gt;&#xA;" OwnerUserId="3587" LastActivityDate="2015-01-05T00:13:23.110" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6170" PostTypeId="2" ParentId="5937" CreationDate="2015-01-05T08:39:43.283" Score="3" Body="&lt;p&gt;I think it's important to keep in mind that reading a mathematics textbook is not the same as reading other texts. I direct you to the MESE answer that discusses this at length &lt;a href=&quot;https://matheducators.stackexchange.com/a/2189/267&quot;&gt;here&lt;/a&gt;, which may help you to determine what topics to skip.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, there is value in going over every topic in a textbook, but I doubt many people completely understand every single topic to their fullest degree the first time meeting a new field. Constant reviewing and rereading of material is the best way to make gains in comprehension.&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-01-05T08:39:43.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6171" PostTypeId="2" ParentId="6157" CreationDate="2015-01-05T11:22:58.223" Score="2" Body="&lt;p&gt;I would propose Kenken (&lt;a href=&quot;http://www.kenkenpuzzle.com&quot; rel=&quot;nofollow&quot;&gt;http://www.kenkenpuzzle.com&lt;/a&gt;). It gives you a feeling of numbers, that is: Which (prime) factors appear in which numbers.&lt;/p&gt;&#xA;" OwnerUserId="3542" LastActivityDate="2015-01-05T11:22:58.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6172" PostTypeId="2" ParentId="6163" CreationDate="2015-01-05T11:40:13.873" Score="22" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;TL;DR: It's not the triangles that are interesting; it's the&#xA;  mathematical concepts that can best be explained by using one of the&#xA;  most primitive geometrical shapes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The reason for intensive use of triangles goes beyond knowledge about triangles per se.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's the act of and steps in proving a theorem that's important to learn at this stage - start with some knowledge you &quot;know for sure&quot; (You probably don't want to use the word &quot;axiom&quot; at this stage), formulate logical deductions, arrive at something new you know must be true even if it seems counter-intuitive. This works much better with geometry than with algebra, because high-schoolers don't have ZFC as a foundation to build upon, but you can introduce Euklid's axioms easily, because they're so intuitive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, you can tackle many proofs in a purely geometrical way, or by transforming your geometrical representation into a coordinate system. This is easy for straight lines, but much harder for anything else, so triangles are, again, a good way to introduce the concept. This teaches your students that different fields of mathematics can profit from each other, and in some cases, something that might be hard to prove using geometry alone can be proven much easier in arithmetics. Or vice versa. Which shows to your students, as well, that sometimes there are several wildly different methods of proving something which, at the end, provide the same result.&lt;/p&gt;&#xA;" OwnerUserId="3590" LastActivityDate="2015-01-05T11:40:13.873" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6173" PostTypeId="2" ParentId="6163" CreationDate="2015-01-05T22:56:13.023" Score="2" Body="&lt;p&gt;This isn't an answer either, but in response to Joseph's answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not just polygons that can be triangulated, but any writer/director of films.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/EvoEC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a lithograph hanging in my home office. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-01-05T22:56:13.023" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6174" PostTypeId="2" ParentId="5837" CreationDate="2015-01-05T23:56:33.727" Score="3" Body="&lt;p&gt;You might also consider the game &lt;em&gt;Set&lt;/em&gt;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Set_(game)&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Set_(game)&lt;/a&gt; shows some of the basic games with the cards and some associated combinatorics. I also have some more materials for this game if you are interested.&lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2015-01-05T23:56:33.727" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6175" PostTypeId="2" ParentId="6163" CreationDate="2015-01-06T00:57:09.897" Score="2" Body="&lt;p&gt;I just happen to visit here. I am not a math-educator. It is my opinion that trigonometic functions - which comes from triangle - uses in the broad range area of applications. Even in the mathematics geometry and analysis requires them in order describe or prove something. For example, Fourier serirs and Fourier transforms are so important in the digital world which it is hard to survive without it today. &lt;/p&gt;&#xA;" OwnerUserId="3594" LastActivityDate="2015-01-06T00:57:09.897" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6176" PostTypeId="2" ParentId="4174" CreationDate="2015-01-06T09:22:37.873" Score="6" Body="&lt;p&gt;In case you haven't already seen these, I'll suggest articles by Luis Radford and Jean Schmittau. Both are influenced by Vygotsky, but take quite different approaches to algebra. Most of Radford's articles are available as pdfs on &lt;a href=&quot;http://luisradford.ca/luisradford/?page_id=13&quot; rel=&quot;nofollow&quot;&gt;his website&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Radford, L. (2010). Algebraic thinking from a cultural semiotic perspective. &lt;em&gt;Research in Mathematics Education&lt;/em&gt;, 12(1), 1-19.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;pdf at &lt;a href=&quot;http://www.luisradford.ca/pub/22_RME2010Algebraicthinkingfromaculturalsemioticperspective.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.luisradford.ca/pub/22_RME2010Algebraicthinkingfromaculturalsemioticperspective.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Abstract: In this article, I introduce a typology of forms of algebraic thinking. In the first part, I argue that the form and generality of algebraic thinking are characterised&#xA;  by the mathematical problem at hand and the embodied and other semiotic&#xA;  resources that are mobilised to tackle the problem in analytic ways. My claim is&#xA;  based not only on semiotic considerations but also on new theories of cognition&#xA;  that stress the fundamental role of the context, the body and the senses in the way&#xA;  in which we come to know. In the second part, I present some concrete examples&#xA;  from a longitudinal classroom research study through which the typology of&#xA;  forms of algebraic thinking is illustrated.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Schmittau, J. &amp;amp; Morris, A. (2004). The development of algebra in Davydov’s elementary curriculum, &lt;em&gt;The Mathematics Educator&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also available as a pdf at  &lt;a href=&quot;http://math.nie.edu.sg/ame/matheduc/tme/tmeV8_1/Schmittau.pdf&quot; rel=&quot;nofollow&quot;&gt;http://math.nie.edu.sg/ame/matheduc/tme/tmeV8_1/Schmittau.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Abstract: A comparison of the development of algebra in Davydov’s elementary mathematics curriculum with the approach to algebra advocated by the National Council of Teachers of Mathematics in the US reveals striking differences. Rather than developing algebra as a generalization of number, Davydov’s curriculum develops algebraic structure from the relationships between quantities such as length, area, volume, and weight. The arithmetic of the real numbers follows as a concrete application of these algebraic generalizations. The instructional approach, while similar to constructivist teaching methodology, emanates from a very different theoretical perspective, namely, the findings of Vygotsky and Luria that cognitive development is enabled by overcoming obstacles for which previous methods of solution prove inadequate. In a study in which the entire three-year elementary curriculum of Davydov was implemented in a US school setting, children using the curriculum developed the ability to solve algebraic problems normally not encountered until the secondary level in the US.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;UPDATE: I’d like to add a comment on the takeaway from the Schmittau and Morris article. The Davydov curriculum is very unconventional, and unlikely to be widely adopted in the United States. However, there are important insights that we could gain from it. I’ll highlight two of them. First, Schmittau points out in this and other articles that traditional instruction does not make as much use of what Vygotsky calls &lt;em&gt;psychological tools&lt;/em&gt; as Davydov’s curriculum. Psychological tools are graphics, tables, language and other ways of making ideas concrete. Psychological tools turn thoughts into objects that we can reflect upon, manipulate and consciously apply when appropriate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, Davydov’s curriculum develops a thorough understanding of concepts. When I first read the article, I thought, “Wow, they spend an excessive amount of time on one topic.” The students work on different aspects of a concept in a concrete way. Now I ask, ”Is this a good example of what we call rigorous instruction? Is this what the opposite of a ‘mile wide and inch deep’ curriculum looks like?” Concepts are developed deeply, in a logical progression. The zone of proximal development is moved forward so other higher level concepts can be taught. And it works. At first, students in the Davydov curriculum seem to be behind students in our traditional curriculum, as the Davydov curriculum builds a solid foundation. But they overtake students in traditional classes, seemingly performing at levels that we may consider to be developmentally impossible. Vygostsky says, “Learning leads development.” The secret may be a deep understanding of concepts that prepares students to move to the next level.&lt;/p&gt;&#xA;" OwnerUserId="3578" LastEditorUserId="3578" LastEditDate="2015-01-06T21:16:03.933" LastActivityDate="2015-01-06T21:16:03.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6178" PostTypeId="2" ParentId="5769" CreationDate="2015-01-06T15:51:42.490" Score="3" Body="&lt;p&gt;The first time I worked through Cauchy's proof of the AM-GM inequality I had to stop myself from shouting out loud: THIS IS OUTRAGEOUS! Less time consuming I don't know, but easier (in the sense of more intuitive) for me, yes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I first came across the problem in Spivak's excellent book Calculus (Chapter 2, Problem 22.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $a_{1},\ldots,a_{n}$ be a set of real numbers such that $a_{i}\geq 0$ for each $i.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Define&#xA;$$ \begin{align*}&#xA;A_{n} &amp;amp; = \frac{a_{1} + \ldots + a_{n}}{n} \\&#xA;G_{n} &amp;amp; = \left[a_{1}\ldots a_{n}\right]^{\frac{1}{n}}&#xA;\end{align*} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then $A_{n} \geq G_{n}.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proof&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We use induction on the set of integers $2^{k}$ for $k \in \mathbb{N}.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the case $k=1,$ we need to show $\frac{a_{1}+a_{2}}{2} \geq \sqrt{a_{1}a_{2}},$ which is done by considering the fact that $(\sqrt{a_{1}}-\sqrt{a_{2}})^{2} \geq 0.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume the proposition holds for $2^{k}.$ For $2^{k+1}$ we have&#xA;$$ \begin{align*}&#xA;		A_{2^{k+1}} &amp;amp; =  \frac{a_{1} + \ldots + a_{2^{k}} + a_{2^{k}+1} + \ldots + a_{2^{k+1}}}{2^{k+1}} \\&#xA;		&amp;amp; =  \frac{1}{2} \left[ \frac{a_{1} + \ldots + a_{2^{k}} } {2^{k}} + \frac{ a_{2^{k}+1} + \ldots + a_{2^{k+1}}}{2^{k}} \right] \\&#xA;		&amp;amp; =  \frac{1}{2}\left[ A + B \right] \\&#xA;		&amp;amp; \geq  \sqrt{AB} \\&#xA;		&amp;amp; \geq  [(a_{1}\ldots a_{2^{k}})^{\frac{1}{2^{k}}}( a_{2^{k}+1}\ldots a_{2^{k+1}})^{\frac{1}{2^{k}}}]^{\frac{1}{2}} &amp;amp; \text{(by inductive hypothesis)}\\&#xA;		&amp;amp; =  G_{2^{k+1}}&#xA;		\end{align*}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note we have even used the base step here!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we go back and think about general $n.$ Let $m$ be defined so that $2^{m} &amp;gt; n$ and let $a_{n+1}=\ldots=a_{2^{m}} = A_{n}.$ Then $$&#xA;		A_{2^{m}} = \frac{a_{1} + \ldots + a_{n} + (2^{m}-n)A_{n}}{2^{m}} = A_{n}&#xA;		$$&#xA;    so that&#xA;        $$&#xA;		(A_{n})^{2^{m}}  \geq  a_{1}\ldots a_{n}(A_{n})^{2^{m}-n}&#xA;		$$&#xA;    and rearranging easily proves the theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are certainly a few details missing here which you would have to fill out, and perhaps &lt;a href=&quot;http://www.artofproblemsolving.com/Wiki/index.php/Cauchy_Induction&quot; rel=&quot;nofollow&quot;&gt;this article&lt;/a&gt; is a bit clearer (a less slick proof), but the proof can definitely be broken down so that undergraduates can work through it as an exercise in several steps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; The reason I find this proof so appealing is that the step $P(2^{k}) \Rightarrow P(2^{k+1})$ is really reduced to basic algebra, whereas $P(n) \Rightarrow P(n+1)$ requires more lemmas. My conceptual focus when reading this proof is on the &lt;em&gt;idea&lt;/em&gt; of using a different induction step and how this simplifies the argument so much. In this sense, I find Cauchy's proof easier and more understandable even if it is not faster than others.&lt;/p&gt;&#xA;" OwnerUserId="901" LastEditorUserId="901" LastEditDate="2015-01-08T09:01:03.060" LastActivityDate="2015-01-08T09:01:03.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6179" PostTypeId="2" ParentId="6163" CreationDate="2015-01-06T17:55:34.150" Score="5" Body="&lt;p&gt;The reason triangles are so prevalent in geometry is because we work with a Cartesian coordinate system (XY planes/graphs). It's how we measure distances, by breaking them into two numbers. Three numbers if we're working with 3 dimensions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in order to reach something from (0,0) to (a,b) we need to have an X direction and a Y direction. The total distance is found out by the Pythagorean theorem, a third number. &#xA;What do three lengths make? A triangle!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometry is the math between these relationships: the x, the y, and the distance. Add in oscillations around a circle, and you have your sine and cosine functions representing the X and Y coordinates. Check out the Trigonometry wikipedia page and notice how many triangles there are. We have an X, a Y, and a distance.&lt;/p&gt;&#xA;" OwnerUserId="3603" LastActivityDate="2015-01-06T17:55:34.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6180" PostTypeId="2" ParentId="6157" CreationDate="2015-01-06T20:12:41.987" Score="2" Body="&lt;p&gt;This was a popular comment and is more appropriate as an answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the games on the Freudenthal Instituut website are very good in many ways, especially for discovery/inquiry learning. It's always good to follow up with some formalisation of the ideas afterwards. &lt;a href=&quot;http://www.fisme.science.uu.nl/publicaties/subsets/en&quot; rel=&quot;nofollow&quot;&gt;http://www.fisme.science.uu.nl/publicaties/subsets/en&lt;/a&gt; for the website. &lt;/p&gt;&#xA;" OwnerUserId="901" LastActivityDate="2015-01-06T20:12:41.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6181" PostTypeId="2" ParentId="5635" CreationDate="2015-01-06T20:37:03.800" Score="1" Body="&lt;p&gt;There are already some fantastic answers here. A few that I love, but haven't seen mentioned:&#xA;Mathematician's Delight- W W Sawyer: Anything by Sawyer is worth a look, especially for aspiring teachers, but this one is the easiest to get, and to get into.&lt;/p&gt;&#xA;&lt;p&gt;Everything and More- David Foster Wallace: One one the best writers of recent times tackles infinity, and it predictably makes for fascinating reading.&lt;/p&gt;&#xA;&lt;p&gt;The IMO Compendium- Djukic et al.: Incredibly thick, expensive and difficult, but there are enough great problems here (with, thankfully, worked solutions) to last you forever.&lt;/p&gt;&#xA;&lt;p&gt;The Road to Reality- Roger Penrose: A Physics book, but unlike Hawking, Penrose doesn't skimp on the Maths.&lt;/p&gt;&#xA;&lt;p&gt;The Art and Craft of Problem Solving- Paul Zeitz: Does what it says on the tin.&lt;/p&gt;&#xA;&lt;p&gt;Which Way did the Bicycle Go?- Stan Wagon et al.: My favourite book of problems, and he used to do a weekly question, archive &lt;a href=&quot;https://web.archive.org/web/20171228030032/http://mathforum.org/wagon/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3571" LastEditorUserId="6053" LastEditDate="2022-10-31T20:35:03.697" LastActivityDate="2022-10-31T20:35:03.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6182" PostTypeId="2" ParentId="6157" CreationDate="2015-01-06T20:44:07.597" Score="4" Body="&lt;p&gt;Britain has the NRich site, which has some great targeted problems, games and interactivities:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://nrich.maths.org/frontpage&quot; rel=&quot;nofollow&quot;&gt;http://nrich.maths.org/frontpage&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You'll be looking for the Upper Primary or Lower Secondary level.&lt;/p&gt;&#xA;" OwnerUserId="3571" LastActivityDate="2015-01-06T20:44:07.597" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7179" PostTypeId="2" ParentId="5648" CreationDate="2015-01-07T08:51:04.877" Score="3" Body="&lt;p&gt;I think it's important to explain students that formally it's not defined, but also give them motivation for &lt;em&gt;why&lt;/em&gt; it is not defined. Yes, definitions are in some sense arbitrary, but we have motivations for defining things certain ways, and I think it is good for high school students to grasp some of those motivations and intuitions behind certain definitions. A couple of ideas, and I apologize if I am repeating what has already been said:&#xA;1, Ask them how they &quot;check&quot; division with multiplication: &quot;why is 6 divided by 3 equal to 2? because 2 times 3 is equal to 6. Ok, so who has a proposal for what 6 divided by 0 is?&quot; And they will realize that there is nothing that they can multiply zero by to get 6. Naturally your students might ask about 0/0. This is a good chance for a class discussion. It is possible that one student may say zero, since zero times zero is zero, and then another point out that it could be 3, since 0 times 3 is equal to 0 as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2, I think it will be hard to come up with a satisfying reason on the spot for why 0/0 is undefined. I suppose you could do something about wanting to preserve continuity. I'd suggest having students come up with definitions of 0/0 and explain and defend their results - hopefully some students will show what rules about multiplication and division will have to be amended to account for the 0 case. For example, distributivity.&lt;/p&gt;&#xA;" OwnerUserId="3528" LastActivityDate="2015-01-07T08:51:04.877" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7180" PostTypeId="2" ParentId="4174" CreationDate="2015-01-07T08:54:27.577" Score="1" Body="&lt;p&gt;I'd suggest literature on students' understanding of the equals sign, e.g., &#xA;1. &quot;Concepts Associated with the Equality Symbol&quot; by Kieran. &lt;a href=&quot;http://link.springer.com/article/10.1007/BF00311062&quot; rel=&quot;nofollow&quot;&gt;http://link.springer.com/article/10.1007/BF00311062&lt;/a&gt;&#xA;2.A Longitudinal Examination of Middle School Students' Understanding of the Equal Sign and Equivalent Equations, Alibali et al.&#xA; &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/10986060701360902#.VK3Rx4rF_dc&quot; rel=&quot;nofollow&quot;&gt;http://www.tandfonline.com/doi/abs/10.1080/10986060701360902#.VK3Rx4rF_dc&lt;/a&gt;&#xA;3. &quot;From an operational to a relational conception of the equal sign,&quot; Molina&#xA;&lt;a href=&quot;http://www.researchgate.net/publication/46593005_From_an_operational_to_a_relational_conception_of_the_equal_sign._Thirds_graders_developing_algebraic_thinking&quot; rel=&quot;nofollow&quot;&gt;http://www.researchgate.net/publication/46593005_From_an_operational_to_a_relational_conception_of_the_equal_sign._Thirds_graders_developing_algebraic_thinking&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3528" LastEditorUserId="3528" LastEditDate="2015-01-08T00:42:33.930" LastActivityDate="2015-01-08T00:42:33.930" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7182" PostTypeId="1" CreationDate="2015-01-07T19:26:17.157" Score="5" ViewCount="186" Body="&lt;ul&gt;&#xA;&lt;li&gt;Andy has 4 apples, and then eats 2. How many does he have left?&lt;/li&gt;&#xA;&lt;li&gt;Beth drives for 3 hours at 80 km/h. How far did she go?&lt;/li&gt;&#xA;&lt;li&gt;Carl, Debbie and Earl earned $30 together shoveling driveways. How much does each of them get?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Is there a name for this sort of 'simplest possible' word problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any well-defined classification or hierarchy of such problems?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the third on my list requires the pre-processing step of recognizing group of &lt;strong&gt;three&lt;/strong&gt; entrepreneurs before doing the division, which adds a degree of difficulty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that such a classification system might be pedagogically useful, but I don't know whether one exists or is widely accepted.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="77" LastEditDate="2021-01-01T09:14:51.087" LastActivityDate="2021-01-01T09:14:51.087" Title="Is there a name for 'simple' two-input-one-output word problems?" Tags="|mathematical-pedagogy|primary-education|terminology|word-problems|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7183" PostTypeId="1" CreationDate="2015-01-07T20:24:10.317" Score="4" ViewCount="238" Body="&lt;p&gt;The task is to plan a one hour lesson to teach 'one-on-one' someone (a 16 year old girl), who is at an F grade in maths, how to solve a simple equation like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;0.24 = \frac{1}{x}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have done this successfully many times (I have a PhD in maths), though there must be an agreed method which you guys think is the most clear and penetrating for a young girl who understands fractions, variables, multiplication and very elementary techniques. Any advice?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If by the end, we could tackle problems like solving&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;4x-1 = 2x + 4(x+4) - 5&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;that would be a successful lesson I think. I'm looking for the optimal technique here, any discussion of why you think this works would be advantageous....&lt;/p&gt;&#xA;" OwnerUserId="4606" LastActivityDate="2015-01-08T04:16:31.853" Title="How to: Learn how to solve a simple equation...in one hour!" Tags="|algebra|teacher-preparation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7184" PostTypeId="2" ParentId="6163" CreationDate="2015-01-07T21:43:51.747" Score="3" Body="&lt;p&gt;My humble opinion;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that three factors might stand out for the prominence of trigonometry in mathematics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A (natural?) tendency to represent our 3D spacial experience in 2D; Up until a few centuries ago, the majority of our real word representations (paintings and drawings) have been two dimensional. I think that's around 50000 years of 2D doodling.&lt;/li&gt;&#xA;&lt;li&gt;The low cost of 2D communication; We only need sand and a stick  to discover/develop, record and transmit spacial ideas (I wonder how this relates to the previous point).&lt;/li&gt;&#xA;&lt;li&gt;The enormous power of trigonometry; Both in our 2D representations and 3D experience, the primordial concepts of location and distance seem to be perfectly captured (and perfectly translated between them) with trigonometry.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So I guess I see it as the cheapest and most effective way to communicate regarding the space we experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In regards to its teaching value, and taking into account my ignorance in education theory, the fact that all of you are tired of so much trigonometry convinces me that it is overdone. I would hope that the curriculum could be shortened (leaving out simple stuff that can be developed/researched when needed if the basics are known). Surely exposing other areas (such as non-euclidean geometries) could provide students a higher level insight that would compensate any reduction in trigonometry curricula?&lt;/p&gt;&#xA;" OwnerUserId="4607" LastEditorUserId="4607" LastEditDate="2015-01-08T19:16:24.410" LastActivityDate="2015-01-08T19:16:24.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7185" PostTypeId="2" ParentId="7183" CreationDate="2015-01-08T04:16:31.853" Score="6" Body="&lt;p&gt;It takes a few seconds to download &lt;em&gt;Dragon Box 12+&lt;/em&gt;, then your job is just about finished. It will take her a few hours to go through, but it will teach her well. It is also a game, so it neatly sidesteps any maths phobia that your students has acquired. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have used this program with students over the last few months. I have found that most of the Algebra you are looking at is taught in a way that requires very little tutor input, though I have actively reinforced good equation solving techniques in the last few levels. On the last level I have sometimes got them to repeat a problem to obtain the standard form of an answer that is not enforced by the program. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It does not cover polynomial factorisation or exponents, but it covers some fairly hairy problems up to that point. It also cleverly allows students to take common shortcuts in equation solving, but only after the &quot;correct&quot; way has been taught for a few levels (ie. subtracting from both sides vs. swapping the sign as you switch the side). It does not cover real numbers, but that should not be hard after she has learned how to do it with rationals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think I am a good tutor, but occasionally a new technology emerges that teaches something far better than I could, so I am happy to mostly sit back and let it do its job. Studies show that students are often happier to experiment and attempt more difficult problems with a computer tutor than a human. However, students often still require us humans to put things in context and make sure that their mental models are correct, so we are not out of a job yet!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another similar program that is not gamified and that covers simple exponents is &lt;em&gt;Algebra Touch&lt;/em&gt;, but &lt;em&gt;Dragon Box 12+&lt;/em&gt; (not to be confused with &lt;em&gt;Dragon Box 5+&lt;/em&gt;) is currently the gold standard in algebra tutoring. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-01-08T04:16:31.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7187" PostTypeId="1" CreationDate="2015-01-08T19:28:30.797" Score="11" ViewCount="390" Body="&lt;p&gt;I'm looking for some examples of proofs where it's easier to prove 'cyclical implications' $A\implies B\implies C\implies A$ than to prove $A\iff B$ directly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can think of some (relatively) advanced examples — but I'm looking for examples accessible to (ordinary) high-school students.&lt;/p&gt;&#xA;" OwnerUserId="4" LastEditorUserId="4" LastEditDate="2015-01-08T19:45:28.660" LastActivityDate="2015-01-27T17:06:39.200" Title="Examples of proofs that use a cycle of implications to prove equivalence" Tags="|secondary-education|proofs|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="7188" PostTypeId="1" AcceptedAnswerId="7190" CreationDate="2015-01-08T22:39:13.837" Score="21" ViewCount="4537" Body="&lt;p&gt;I base this question off where I got my motivation for math and science. Throughout several attempts in my junior years, I was able to design a perpetual motion machine, design a free energy device, prove Einstein $E=mc^2$ wrong. Obviously I was way, way off, and I eventually found that out by myself. However, the amount of interest in math and science and the amount I learned was greatly benefited. So I ask the question: should students be told they're wrong about an idea they have?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example I will give is a student &quot;proving&quot; $1=2$ by method of division by 0, such as the one described &lt;a href=&quot;http://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html&quot;&gt;here&lt;/a&gt;. I classify this in 4 ways:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;They should be told they're right. This makes the student feel smart and makes them have a higher interest in the subject. Eventually they will find out they are wrong. (This obviously being a lie.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They are told that they came up with good thoughts, not confirming or denying they are right. They may not receive as much interest, but they are still in the process of confirming if they are right and wrong, and if curious enough and believe they are right, perform their own study.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They are told they are wrong, but are not told why. Which still may present some curiosity, maybe enough to find it themselves. In this way they are not risking the trust between the teacher and student.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They are told they are wrong, and explained why. This way they are taught the process themselves, but may not receive as much interest from it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What's the most effective way for their education?&lt;/p&gt;&#xA;" OwnerDisplayName="user1244" LastEditorUserId="77" LastEditDate="2016-08-21T02:46:36.413" LastActivityDate="2016-08-21T06:47:59.693" Title="Should students be told they're wrong" Tags="|secondary-education|student-motivation|students-mistakes|" AnswerCount="7" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7189" PostTypeId="2" ParentId="7188" CreationDate="2015-01-08T23:24:24.987" Score="7" Body="&lt;p&gt;Let me start by saying that I believe what effectively educates one person will not necessarily be effective for someone else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that in mind, I would never personally use number 1 as I feel it is lying.  However if I knew the person well enough, I &lt;em&gt;might&lt;/em&gt; consider doing it if I feel they learn it better that way (you sound like you benefit from this from your perpetual machine example).  However, I think those situations would be quite rare.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another problem I have with 1 is that sometimes a person may not figure out they are wrong until it is too late!  Sometimes thinking something is correct when it is not can be harmless, but what if that person starts telling other people it is true?  Sometimes it is surprising how far people can get in life never being told something they believe to be true is wrong.  If this person ever became a teacher and started teaching what they believed is correct, when it is not, there could be problem.  I would feel it is best not to reassure them they are right when they are in fact incorrect.  I feel like this can become more than just a white lie.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, I would probably lean more towards a mixture of 3 and 4.  Tell the person they are incorrect but encourage them to try to find out why without giving away the answer (much like your 1=2 link does towards the end by asking:  &lt;strong&gt;See if you can figure out in which step the fallacy lies.&lt;/strong&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But again, a lot of people have a different learning style.  For situations like this, I would let the person they know that they are overlooking something within their assumptions, and encourage them to find out what it is.&lt;/p&gt;&#xA;" OwnerUserId="3522" LastActivityDate="2015-01-08T23:24:24.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7190" PostTypeId="2" ParentId="7188" CreationDate="2015-01-08T23:40:21.430" Score="51" Body="&lt;p&gt;All four of your options lead with &quot;They are told...&quot; Consider asking the student questions instead. At the very least, this shows interest, and they may end up catching their own mistakes as they try to explain to you what they had glossed over in their own heads.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I have the opportunity, I like to challenge my students to explain EVERY step of their work to me, including the steps that are correct. This builds their self-checking reflexes and their communication skills, and when they do catch a mistake, they don't feel like I did the work for them.&lt;/p&gt;&#xA;" OwnerUserId="5244" LastActivityDate="2015-01-08T23:40:21.430" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7191" PostTypeId="2" ParentId="7188" CreationDate="2015-01-09T01:49:39.550" Score="9" Body="&lt;p&gt;Presumably students who &quot;prove&quot; $1=2$ already know they're wrong.  They don't really believe that $1=2$ do they?  If a student brought me the &quot;proof&quot; you linked to, I'd first make sure they realized that $1\neq2$. Then, I'd suggest they check the &quot;proof&quot; by choosing a particular number, say $3$, for $a$ and $b$.  That way they can see exactly where the equations make the transition from true to false. I'd hope they can also see why that particular step was wrong, but if not then (and only then) I'd explain it.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2015-01-09T01:49:39.550" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7192" PostTypeId="1" CreationDate="2015-01-09T12:54:16.220" Score="12" ViewCount="601" Body="&lt;p&gt;I'm not sure how the concept of $i$ is taught in other places, but in our district the curriculum defines $i = \sqrt{-1}$, which is how it has been traditionally taught (for a while now) and also how our state standards align the concept of imaginary numbers. This definition is, in my opinion and the opinion of Math StackExchange (&lt;a href=&quot;https://math.stackexchange.com/questions/13801/whats-the-thing-with-sqrt-1-i&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/questions/1097134/why-is-sqrt-1-i-and-not-pm-i&quot;&gt;here&lt;/a&gt;), a poor definition for many reasons.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This definition excludes the completely legitimate possibility of $-i = \sqrt{-1}$, it causes difficulty when combining radicals through multiplication, and it is inconsistent with the definition of square root for positive numbers, such as $\sqrt{4} = \pm 2$.  It should instead be taught that $i^2 = -1$, which is a much better definition because it avoids all of these problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However this does not align with any of the curricula or standards that I am expected to teach to.  Any suggestions on how to get around this unfortunate situation?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-08-18T14:22:58.637" Title="Pedagogical quandary with the definition of $i$" Tags="|secondary-education|algebra|complex-numbers|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7193" PostTypeId="2" ParentId="7192" CreationDate="2015-01-09T15:19:11.050" Score="13" Body="&lt;p&gt;There are issues with how to define $i$ no matter which you choose, as the answers to the linked questions attest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To begin with, it is not actually true that $\sqrt 4 = \pm 2$. The square root function is defined (by mutual consent of the maths community) to only produce non-negative outputs, so $\sqrt 4 = 2$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two reasons for this. Firstly, a function can't produce two outputs simultaneously or it wouldn't be a function. Secondly, if it did produce two outputs then what on earth would be meant by the notation $\sqrt 2$? If $\sqrt 2$ was two numbers then so would $-\sqrt 2$ and indeed they would have to be the same two numbers so $\sqrt 2 = -\sqrt 2$ which would imply that $\sqrt 2 = 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So $\sqrt 4$ is in fact just 2. If you want to point out that there are two numbers that square to give 4 then you have to say something like &quot;The solutions to $x^2 = 4$ are $x = \pm 2$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in that sense, saying that $\sqrt{-1}=i$ does not at all preclude us from saying that the solutions to $x^2=-1$ are $x=\pm i$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What it does do is implicitly assume that there is a mathematically consistent way to choose which of the two possible solutions is the one meant when you do the square root of a nonpositive number. For example, if you can unambiguously say $\sqrt{-1} =i$, then there ought to be an unambiguous way to define which of the two possible solutions to $x^2 = 1-i$ is the single one meant by $\sqrt{1-i}$. There is a way, but it is not obvious until you talk about polar form, which comes much later!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Defining $i$ using $i^2 = -1$ is problematic too, because if $i$ was a variable then that equation has two solutions, so which of them is $i$? In fact either of them could be $i$ and there is no problem whichever one you choose, as long as you choose one and carry through with it for the rest of the maths you do. If each individual is consistent and doesn't switch between them willy-nilly it will be fine, and it won't matter if your $i$ is different to mine because switching all the $i$'s for $-i$ produces the same set of complex numbers and doesn't interfere with the operations on them. This is the idea of conjugation!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The resolution of all of this is the representation of complex numbers with points on a 2D plane. Mathematicians agree to choose $i$ to be the point anticlockwise from 1 rather than clockwise.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it would be remiss of you as a teacher not to mention some of this, but on the other hand it would be a bad idea to go into full detail straight up because that would confuse the students. Also, if locally the definition is $i = \sqrt{-1}$ then they need to know this to get the marks in their exam locally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would choose the approach of using the definition you have, but pointing out that there are some technical issues you're glossing over by using this definition. I would also tell them that it's possible to define it as a number such that $i^2=-1$, which also has some technical issues. You can possibly get back to these issues later if they are interested. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, if a task they are doing doesn't require them to explicitly mention the definition, then there is absolutely nothing wrong with them using the fact that $i^2 =-1$, which I find is easier than using square roots anyway. Indeed, many students I have taught don't realise it's ok to do this and end up having extra lines in their working and make more errors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One final thought: whenever there are multiple ways that people define things, it's probably a good idea to let your students know this, and which definition is the preferred one locally. (Another common example is the definition of the number $e$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This tells students that it is possible for there to be differences in definition. It also means that they are less likely to be confused by resources they find online which use a different definition.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-08-17T20:27:18.827" LastActivityDate="2015-08-17T20:27:18.827" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7194" PostTypeId="2" ParentId="6163" CreationDate="2015-01-09T16:00:28.030" Score="17" Body="&lt;p&gt;I see one important reason: &lt;strong&gt;triangles are simply the simplest non-trivial configurations of points&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One point configuration are boring: every two points are mapped one to the other by a translation. In other words, every point can be translated to the origin, which thus represents very well every single point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two-points configuration are only slightly more interesting: any pair of points can be moved by a translation and a rotation (thus preserving all geometric properties) to a pair $((0,0), (x,0))$ with $x&amp;gt;0$ being the distance between the two points. So the distance is the only meaningful geometric quantity for a pair of points. Now, if one allow for dilation, which preserve all dimensionless geometric quantities (e.g. ratios of lengths), then there are only two type of pairs: twice the same point, and two different points. Boring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three point configuration are incredibly rich: there exist a two-dimensional configuration space of pairwise non-similar triangle. Highly non-boring!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the same arguments make triangle interesting &lt;em&gt;in any dimension&lt;/em&gt;. As a side note, when one studies affine geometry without a Euclidean structure triangle become boring again. When one studies the projective line, triples of points are boring but quadruples lead to the cross-ratio, etc. So this point of view really make this high-school stuff fit into a much deeper and wider framework.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-01-09T16:00:28.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7195" PostTypeId="2" ParentId="7192" CreationDate="2015-01-09T16:27:52.777" Score="1" Body="&lt;p&gt;Celeriko,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll directly respond to your exact question first then rant on about my real feelings after :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are exactly right that this alleged definition which your standards is using is to say the least... unappealing... for all the reasons you mentioned. &#xA;If you teach your students $i^2=-1$ then you can automatically show them the case with the square root. It is highly unlikely that on some state test that they would ask for something like &quot;Define $i$,&quot; but if they did... your students would be giving a correct definition or would know to choose $i=\sqrt{-1}$ after working with you on this. Teach them the more logical thing, then point out how to pass the ridiculous test. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On an even more important note, really... complex numbers are points. It makes more geometric sense and the actual use of these numbers. They aren't &quot;imaginary,&quot; and their uses  are necessary for the modern world. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see a geometric, more logical approach that is completely within the realm of high schoolers, watch this wonderful mathematics professor's videos:&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=S7NDSPtoVP0&quot; rel=&quot;nofollow&quot;&gt;Complex Numbers (Wildberger)&lt;/a&gt;  and his lecture on the &lt;a href=&quot;https://www.youtube.com/watch?v=5Ui5W_BDa8s&quot; rel=&quot;nofollow&quot;&gt;history of complex numbers and algebra&lt;/a&gt; (more advanced but good info to have!)&lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2015-01-09T16:27:52.777" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7196" PostTypeId="2" ParentId="7188" CreationDate="2015-01-09T17:31:07.580" Score="6" Body="&lt;p&gt;It depends on the student and the problem at hand, though I'd never suggest one lie to them - when they discover the lie they will suspect everything else you've taught them, and possibly what others have taught.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In most cases I'd suggest that the right answer is to encourage and challenge them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was in High School a teacher indicated that a high school student shouldn't be capable of solving a particular problem.  Some overconfident students respond to that type of challenge - I did, and was rewarded with what I learned along the journey.  More important than the skill and knowledge gained, was the affirmation that I can accomplish difficult tasks if I set my mind to them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When a student presents something you already know to be wrong, rather than shooting them down, or praising them falsely, consider challenging them to take the next step. &lt;em&gt;&quot;That's an interesting design for a perpetual motion machine.  What is the calculated energy output?  How much of that energy will be used in friction in each of 29 bearings and sliding surfaces? Where do you propose to obtain single-pole magnets?&quot;&lt;/em&gt;  Rather than placing a road block on the path they are traveling, encourage them to continue down that path - they will gain more concrete knowledge with experience than they will by a quick, &quot;It can't work.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it's dangerous to pursue, or they would learn nothing new by taking something to its logical conclusion, then it might be worthwhile short-circuiting the process and engaging their mind in a more enlightening direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found, however, that &quot;Fail fast, fail often&quot; is not a bad way to learn things, and with good mentorship it can yield unexpected benefits in a learning environment.&lt;/p&gt;&#xA;" OwnerUserId="4617" LastActivityDate="2015-01-09T17:31:07.580" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7197" PostTypeId="2" ParentId="7192" CreationDate="2015-01-09T18:27:05.530" Score="18" Body="&lt;p&gt;A bit of personal opinion: Any time I find myself debating over a bit of pedantry while preparing course material, I ask myself, &quot;Is this something that &lt;em&gt;is really important to a student taking this course&lt;/em&gt;, or is it just interesting to &lt;em&gt;me&lt;/em&gt; because I'm a mathematician?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally would put the question about how to define $i$ in the &quot;only important to me&quot; category, at least if I was teaching a high school algebra or calculus (or similar level course). And, as a result, I would teach $i = \sqrt{-1}$. So I guess my answer to the question in the post is: I don't think the situation is unfortunate!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think this causes any difficulty, at least not any more than any other new concept. Yes, they will have to learn that $\sqrt{(-1)(-1)} \neq \sqrt{-1}\sqrt{-1}$, but I think they could be convinced pretty easily that this is no big deal! After all, this thing $\sqrt{-1}$ is new and weird. It's not one of the usual numbers we know and love, so why should we expect it to behave like one!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It sounds also like any other student taking this course in your district will likely be taught that $i = \sqrt{-1}$, so I would stick with this for the benefit of your students being able to communicate with other students. &lt;/p&gt;&#xA;" OwnerUserId="4618" LastActivityDate="2015-01-09T18:27:05.530" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7199" PostTypeId="1" CreationDate="2015-01-10T00:55:32.573" Score="20" ViewCount="801" Body="&lt;p&gt;I will be teaching a course in history of mathematics to juniors/seniors who are math and math education majors, many future school teachers. It should include highlights from antiquity to early 19-th century, but I'd like to supplement it with &quot;hands on&quot; experiences of what it was like doing mathematics in centuries past. Especially since it's a small class, 20-25 students. I don't mean &quot;hands on&quot; literally, like manipulatives, it could be group activities, individual tasks on paper, or even small exploratory projects, but with doing rather than just listening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What concerns me the most is balance, it is always hard to come up with activities that are meaningful but at the same time what is given in class is enough to reasonably expect students to work out things on their own, I don't want to get them discouraged. Unfortunately, our students have limited background and skills, they can be expected to know (algorithmic) linear algebra, calculus, differential equations, etc., but I expect them to struggle with any kinds of proofs, even in Euclidean geometry and elementary number theory, or with problems not directly analogous to examples. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking about breaking down some of the classical proofs/solutions into steps and letting them do some of the steps, with hints and nudges, but it is hard to do for the entire course on my own, or to gauge the balance. So I was hoping to find online resources or books that do something like this, or other things, I am open to suggestions. Unfortunately, although books like Eves's or Burton's list a lot of exercises and suggested activities, almost all of them seem to be much too advanced for semi-independent work. And what I saw online is mostly for grade school and/or historical nuggets embedded into other math courses, not a systematic history course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any advice and links/references are appreciated.    &lt;/p&gt;&#xA;" OwnerUserId="4621" LastActivityDate="2022-02-15T05:07:09.697" Title="Hands on activities for a college history of mathematics course" Tags="|undergraduate-education|mathematical-pedagogy|reference-request|history|" AnswerCount="8" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7200" PostTypeId="1" CreationDate="2015-01-10T09:01:17.370" Score="5" ViewCount="169" Body="&lt;p&gt;There are a lot of good particular examples of innumeracy at &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1572/examples-of-innumeracy&quot;&gt;this question&lt;/a&gt;.  But in addition what I would like is to be able to consistently bring &lt;em&gt;current&lt;/em&gt; examples in to class.  Does anyone maintain a blog that collects such examples as they occur?&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-01-10T17:52:30.017" Title="A blog or newsfeed for innumeracy in the media?" Tags="|examples|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7201" PostTypeId="2" ParentId="7199" CreationDate="2015-01-10T15:26:50.940" Score="12" Body="&lt;p&gt;I had my students in Math for Elementary Teachers doing arithmetic in the Babylonian, Mayan, and Egyptian systems. It's not beyond them at all, and it helps them understand place value more deeply.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't have recommendations for a text for the whole course, but you may find a lot of useful material in &lt;em&gt;Count Like an Egyptian&lt;/em&gt;. I hope to run a math circle with material from this book. It explains multiplication and division by Egyptian methods, and Egyptian fractions, which we'll be exploring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't finished it yet, so I just read the &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691160120&quot;&gt;reviews on Amazon&lt;/a&gt; to see what I might be missing. Apparently, the book is broader than I'd thought: &quot;Despite its apparently limiting title “Count Like an Egyptian”, this book delivers all that its subtitle “A Hands-On Introduction to Ancient Mathematics” promises. Besides presenting a comprehensive overview of the ancient Egyptian computing methods, Professor Reimer also introduces you to the Mesopotamian sexagesimal system, then compares these with the Roman numerals and Mayan counting glyphs, as well as with the modern decimal and even binary ways of expressing numbers.&quot;&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-01-10T15:26:50.940" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7202" PostTypeId="2" ParentId="7200" CreationDate="2015-01-10T17:52:30.017" Score="2" Body="&lt;p&gt;This one looks promising: &lt;a href=&quot;http://goodstatsbadstats.com/&quot; rel=&quot;nofollow&quot;&gt;http://goodstatsbadstats.com/&lt;/a&gt; But the last posting was in October.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-01-10T17:52:30.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7203" PostTypeId="2" ParentId="7199" CreationDate="2015-01-10T18:08:15.800" Score="5" Body="&lt;p&gt;You might use various proofs of the Pythagorean theorem. Several lend themselves to&#xA;experimenting with paper cutouts. And they can be connected to historical documents.&#xA;See the &lt;a href=&quot;http://www.cut-the-knot.org/pythagoras/&quot; rel=&quot;noreferrer&quot;&gt;Cut-the-Knot website&lt;/a&gt; for over 100 proofs, &#xA;including this ancient gem:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/oWoUC.png&quot; alt=&quot;Proof#2&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-01-10T18:08:15.800" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7204" PostTypeId="2" ParentId="6120" CreationDate="2015-01-10T19:17:57.657" Score="1" Body="&lt;p&gt;(Migrated from the comments.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below are two different textbooks that may fit the bill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, note that the OP has previously asked a more general question about textbooks &lt;strong&gt;&lt;a href=&quot;https://math.stackexchange.com/q/1027936&quot;&gt;here&lt;/a&gt;&lt;/strong&gt; on MSE. That earlier question included an answer with the Intro to Real Analysis book &lt;strong&gt;&lt;a href=&quot;http://www.jirka.org/ra/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt;, which has the LaTeX on the same page: scroll down to Source. The textbook is available as a free &lt;strong&gt;&lt;a href=&quot;http://www.jirka.org/ra/realanal.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pdf&lt;/a&gt;&lt;/strong&gt;, too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The text has been used at a bunch of quality institutions, from UW-Madison to UC-Berkeley, and it has a favorable &lt;strong&gt;&lt;a href=&quot;http://www.maa.org/publications/maa-reviews/basic-analysis-introduction-to-real-analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA review&lt;/a&gt;&lt;/strong&gt; from 2013 (of the December 2012 version of the text). Although it has been updated as recently as December 2014, the citation on google scholar is given as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lebl, J. (2011). Basic analysis-introduction to real analysis. &lt;em&gt;AMC, 10&lt;/em&gt;, 12.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For another free text (with available TeX) see &lt;strong&gt;&lt;a href=&quot;http://digitalcommons.trinity.edu/mono/7/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/strong&gt; for Trench's (December 2013) work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The recommended citation there is given as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Trench, William F., &quot;Introduction to Real Analysis&quot; (2013). &lt;em&gt;Books and Monographs&lt;/em&gt;. Book 7.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Its description:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Using a clear and informal approach, this book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Moreover, the page notes that: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This book was previously published by Pearson Education. This free edition is made available in the hope that it will be useful as a textbook or reference.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A citation for the earlier version is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Trench, W. F. (2003). &lt;em&gt;Introduction to real analysis&lt;/em&gt;. Upper Saddle River, NJ: Prentice Hall/Pearson Education.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-01-10T19:17:57.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7205" PostTypeId="2" ParentId="7188" CreationDate="2015-01-11T02:24:27.970" Score="14" Body="&lt;p&gt;This is obviously a subjective topic, but here's my take:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an educator, you should see yourself as a resource to your students. You have certain knowledge that they seek to obtain. You should &lt;strong&gt;never&lt;/strong&gt; withhold useful knowledge to a student if they explicitly ask for it. Rather than you attempting to &lt;em&gt;force&lt;/em&gt; their actions, it's more about how you react to their actions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, if a student approaches you and hands you his proof that 1 = 2, you should review it and perhaps say, &quot;I disagree with step 4&quot;. You should &lt;strong&gt;absolutely&lt;/strong&gt; say that you think there is an error in their work, otherwise they may have misconceptions that will lead to confusion down the road, or a lack of trust in your knowledge as a teacher, or a feeling that you don't care about their learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After you mention that you disagree with step 4, the student will then do one of a few things - they'll (a) review step 4, find their mistake, and that will be that. Or (b) they'll review step 4 and not be able to spot a mistake and ask for you to explain their mistake. Or (c), they won't bother to review their work and will directly ask you to explain their mistake to them. That's &lt;strong&gt;o.k.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In either scenario, you should react to the way in which they want to learn. If they specifically ask for you to explain their mistake, don't withhold your knowledge by forcing them to work it out for themselves - that might have the unintended effect of frustrating them, rather than encouraging them. In most cases, you're o.k. to reveal their error gradually though, so that you give them many &lt;em&gt;opportunities&lt;/em&gt; to prove themselves. So say, for example, &quot;Did you consider what would happen if b=0 in step 4?&quot; rather than just rewriting step 4 so that it's correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, they'll react in one of the ways covered above. And you should respond accordingly - reveal more information if they request it until they understand their error, all the while giving them &lt;em&gt;opportunities&lt;/em&gt; to make the leap themselves. But if they choose to give up and wish for you to explain their error to them in full, then I believe you should comply. It's either that, or they leave knowing that some part of their knowledge is wrong, without knowing exactly what or why. And you cannot expect for every person to be capable of solving each of their misunderstandings on their own. That is one reason for why we share our knowledge.&lt;/p&gt;&#xA;" OwnerUserId="4626" LastEditorUserId="4626" LastEditDate="2015-01-12T07:24:22.450" LastActivityDate="2015-01-12T07:24:22.450" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7206" PostTypeId="2" ParentId="2060" CreationDate="2015-01-12T00:41:30.690" Score="5" Body="&lt;p&gt;I know I am a little late to the party on this question, but when I read this question, I felt I could add some information of value. First, I am not a biologist, but I have taken a course in Mathematical Biology and Ecology where a wide range of topic were covered. Additionally, there are two good resources that show and discuss the math involved in Biology one is a two volume set. The books are &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387952233&quot; rel=&quot;nofollow&quot;&gt;Mathematical Biology I: An Introduction&lt;/a&gt; and &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387952284&quot; rel=&quot;nofollow&quot;&gt;Spatial Models and Biomedical Applications&lt;/a&gt; by JD Murray and &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0898715547&quot; rel=&quot;nofollow&quot;&gt;Mathematical Models in Biology&lt;/a&gt; by Leah Edelstein-Keshet. Another book I own which isn't entire Biology based but does have Biology in it is &lt;a href=&quot;http://www.westviewpress.com/?s=9780738204536&quot; rel=&quot;nofollow&quot;&gt;Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering&lt;/a&gt; By Steven Strogatz.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the topics may be mentioned in another post but I will still list them for completeness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Topics that require Calculus based mathematical maturity are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Continuous Population Models for Single Species&#xA;$$&#xA;    \frac{dN}{dt}=\text{birth}-\text{deaths}+\text{migration}&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;Discrete Population Models for a Single Species&#xA;$$&#xA;    N_{t+1}=N_tF(N_t)=f(N_t)&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;Models for Interacting Populations&#xA;\begin{align}&#xA;    \frac{dN}{dt}&amp;amp;=N(a-bP)\\&#xA;    \frac{dP}{dt}&amp;amp;=P(cN-d)&#xA;    \end{align}&lt;/li&gt;&#xA;&lt;li&gt;Reaction Kinetics&#xA;$$&#xA;    S+E\mathrel{\mathop{\rightleftharpoons}^{k_1}_{k_{-1}}} SE\to P+E&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;Biological Oscillators and Switches&#xA;$$&#xA;    \frac{d\mathbf{u}}{dt}=\mathbf{f}(\mathbf{u})&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;Perturbed and Coupled Oscillators and Black Holes (Not in space)&#xA;$$&#xA;    \frac{d\mathbf{u}}{dt}=\mathbf{f}(\mathbf{u},\lambda)&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;Dynamics of Infectious Diseases: SIR models&#xA;\begin{align}&#xA;    \frac{dS}{dt}&amp;amp;=-rSI\\&#xA;    \frac{dI}{dt}&amp;amp;=rSI-aI\\&#xA;    \frac{dR}{dt}&amp;amp;=aI&#xA;    \end{align}&lt;/li&gt;&#xA;&lt;li&gt;Reaction Diffusion, Chemotaxis, and Nonlocal Mechanisms&#xA;$$&#xA;    \frac{\partial}{\partial t}\int_Vc(\mathbf{x},t)dv=-\int_S\mathbf{J\cdot ds}+\int_Vfdv&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;Oscillators-Generated Wave Phenomena and Central Pattern Generators&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These next topics are a little more difficult and require knowledge of PDEs but an advanced undergrad could handle this&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Biological Waves: Single Species Models&#xA;$$&#xA;    \frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;The Use of Fractals&lt;/li&gt;&#xA;&lt;li&gt;Multiple-Species Waves&#xA;$$&#xA;    \frac{\partial\mathbf{u}}{\partial t}=\mathbf{f(u)}+D\nabla^2\mathbf{u}&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;Spatial Pattern Formation with Reaction Diffusion Systems&lt;/li&gt;&#xA;&lt;li&gt;Bacterial Patterns and Chemotaxis&#xA;$$&#xA;    \nabla\cdot\mathbf{J}_c=\nabla\cdot[\chi(n,c)\nabla c]&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;Mechanical Theory of Vascular Network Formations&#xA;$$&#xA;    \frac{\partial n}{\partial t}=-\nabla\cdot\frac{\partial\mathbf{u}}{\partial t} + \nabla\cdot\nabla\cdot(\mathbf{D(\epsilon)}n)&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;Epidermal Wound Healing&#xA;\begin{align}&#xA;    f(n)&amp;amp;=\lambda c_0\frac{n}{n_0}\frac{n_0^2+\alpha^2}{n^2+\alpha^2}\\&#xA;    f(n)&amp;amp;=\frac{\lambda c_0}{n_0}n&#xA;    \end{align}&lt;/li&gt;&#xA;&lt;li&gt;Neural Models of Pattern Formations&#xA;$$&#xA;    \frac{\partial n}{\partial t} = f(n)+\int_Dw(x-x')[n(x',t)-1]dx'&#xA;    $$&lt;/li&gt;&#xA;&lt;li&gt;Geographic Spread and Control of Epidemics&#xA;\begin{align}&#xA;    \frac{\partial S}{\partial t}&amp;amp;=-rIS+D\nabla^2S\\&#xA;    \frac{\partial I}{\partial t}&amp;amp;=rIS-aI+D\nabla^2I&#xA;    \end{align}&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3581" LastEditorUserId="3581" LastEditDate="2015-01-12T22:27:57.897" LastActivityDate="2015-01-12T22:27:57.897" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7207" PostTypeId="1" AcceptedAnswerId="10579" CreationDate="2015-01-12T06:42:37.727" Score="7" ViewCount="234" Body="&lt;p&gt;There is a concept called &lt;a href=&quot;https://en.wikipedia.org/wiki/Infographic&quot; rel=&quot;nofollow&quot;&gt;'Infographics'&lt;/a&gt; that uses lots of colourful diagrams and graphs and artwork to visually depict some set of concepts and tries to explain the ideas through pictures. I recommend these Infographics to any person trying to learn some complicated principle in Math, say. I was wondering if there are any easy-to-comprehend infographics for hard subjects like Category theory or Topology?&lt;/p&gt;&#xA;" OwnerUserId="4632" LastEditorUserId="376" LastEditDate="2016-02-17T21:54:26.597" LastActivityDate="2019-11-21T14:08:33.237" Title="About visual ways of teaching Math" Tags="|self-learning|visual-proof|arts|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7208" PostTypeId="1" CreationDate="2015-01-12T15:41:58.453" Score="7" ViewCount="5682" Body="&lt;p&gt;I'm going to teach exponentiation to my 7th grade math students, and I like the exponentiation rules to arise in a natural way (through real-world problems or &lt;strong&gt;meaningful&lt;/strong&gt; math problems). Do you have any such problem in mind?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2015-09-20T02:48:54.573" Title="Motivation for Exponent Rules" Tags="|secondary-education|powers|exponential|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7209" PostTypeId="2" ParentId="7208" CreationDate="2015-01-12T16:25:29.253" Score="5" Body="&lt;p&gt;You could use combinatorics:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How many words with 4 letters of 26 letter alphabet do exist? If you have the answer: How many passphrases of two word with 4 letters of the same alphabet do exist? How many passphrases of a 4 letter word and a 6 letter word do exist?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The real-world problem arises from the security of passwords (for Facebook for example). Another example is the range of telephone numbers, license plates, IP-addresses that sometimes happen to be too short for their application.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course you could also go into probability theory and use Bernoulli processes.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2015-01-12T16:25:29.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7210" PostTypeId="2" ParentId="7208" CreationDate="2015-01-12T17:53:15.120" Score="7" Body="&lt;p&gt;Here's something I used to do in college algebra and precalculus classes, from the mid 1980s to the mid 2000s, which has the additional advantage of being an example in which estimation is used. I think you can adapt this to your case. (Use metric system units if appropriate.) Of course, you'll want to go a lot slower than I do below, which is written for the teacher, not the student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; How many times would one theoretically have to successively fold a sheet of paper in order for the (theoretical) folded thickness to equal the distance from the Earth to the Moon?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thickness of $300$ sheets is about an inch. You can hold up a book to show how to come up with this. The moon's distance is about $240000$ miles $= 24 \times 10^4$ miles $= 24 \times 10^4 \times 5280$ feet, which is about $25 \times 5 \times 10^4 \times 10^3$ feet $= 125 \times 10^7$ feet $= 125 \times 10^7 \times 12$ inches, which is about $100 \times 10^7 \times 10 = 10^{10}$ inches $= 300 \times 10^{10}$ sheet-thicknesses $= 3 \times 10^{12}$ sheet-thicknesses, so we need to know how many doublings gives this last number. $10$ doublings gives us almost exactly $1000 = 10^3$ (only a $2$% error), so $20$ doublings gives $10^{6},$ $30$ doublings gives $10^{9},$ $40$ doublings gives $10^{12},$ and two more doublings after this (for a total of $42$ doublings) gives us $4 \times 10^{12} &amp;gt; 3 \times 10^{12}.$ You can also use $2^{10}$ is approximately $10^3$ (i.e. $10$ doublings has the effect of approximately multiplying by $1000)$ to estimate other situations with geometric growth when the doubling time (or half-life) is known.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also worth pursuing is how insensitive the final result is to the approximations used. For example, if the cummulative effect of all the estimates is a factor of $2,$ then the number of foldings is off by $1$ (i.e. it's between $41$ and $43).$ If the cummulative effect of all the estimates is a factor of $4,$ then the number of foldings is off by $2$ (i.e. it's between $40$ and $44).$&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2015-01-12T17:53:15.120" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7211" PostTypeId="2" ParentId="7208" CreationDate="2015-01-12T18:09:46.547" Score="9" Body="&lt;p&gt;might i suggest a different approach that I have found very helpful when teaching about exponentiation rather than real world examples?  I have found that until students understand &lt;em&gt;why&lt;/em&gt; a rule works (i.e. the derivation or something similar), they won't be able to understand &lt;em&gt;how&lt;/em&gt; to use it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of teaching the rules to the students, have them expand all exponents out by hand and combine like factors just like they have (hopefully) been doing and this should lead them to discover the rules for themselves.  If they ever forget a rule, they can just go back to how they discovered them, by expanding out exponents, and essentially &quot;derive&quot; the rule right there. so for example present them this problem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$4x^4y\cdot3x^5y^2$$&#xA;Which they can expand to&#xA;$$4x^4y\cdot3x^5y^2 = 4 \cdot x \cdot x \cdot x \cdot x \cdot y \cdot 3 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y $$&#xA;And simplifying yields&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$12x^9y^3$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same idea works exactly the same for each of the other exponent rules so for instance for the division/subtraction rule (whatever it is called:))&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{12x^5y^8}{2x^2y^3} = \frac{12 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y}{2 \cdot x\cdot x \cdot y \cdot y \cdot y} = 6x^3y^5$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By expanding out the exponents it is a lot easier to see what cancels and what can be combined.  Doing this all by hand is a huge pain and should motivate your students to try and figure out the rules as fast as possible.  Reinforcing what it actually means to raise something to an exponent by having them do this all manually, will certainly help down the line when the expressions/equations get more complicated.  &lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="60" LastEditDate="2015-01-15T14:40:06.150" LastActivityDate="2015-01-15T14:40:06.150" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7212" PostTypeId="2" ParentId="7207" CreationDate="2015-01-12T20:59:33.970" Score="2" Body="&lt;p&gt;For complex variables, there is the famous &quot;visual&quot; book  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tristan Needham, &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0198534469&quot; rel=&quot;nofollow&quot;&gt;Visual Complex Analysis&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-01-12T20:59:33.970" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7213" PostTypeId="2" ParentId="5837" CreationDate="2015-01-12T21:29:31.240" Score="1" Body="&lt;p&gt;There is a simple game that I like a lot as it illustrates in an interesting fashion the theorem that in deterministic two-players games without draws nor hidden information, either the first or the second player has a winning strategy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The game starts with a chocolate bar, of size $n$ by $m$ say, whose uppest rightmost piece is called the poisoned piece. At one's turn, one chooses a piece of chocolate and removes all pieces lying in the smallest lower-left semi-infinite rectangle containing the chosen piece (so , choosing the poisoned piece would remove all pieces, while choosing the lowest leftmost piece on the first move would only remove one piece). One says the player &quot;eats&quot; the removed pieces, and the player eating the poisoned piece looses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, by the above-mentioned theorem, either the first or second player has a winning strategy. Claim: for this game, we can prove that &lt;strong&gt;the first player has a winning strategy regardless of $n$ and $m$&lt;/strong&gt; (except for $n=m=1$, that is); but as far as I know, &lt;strong&gt;the winning strategy is unknown&lt;/strong&gt; except for very small values of $n$ or $m$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof of the claim: assume the second player has a winning strategy. Then if the first player first picks the lowest leftmost piece, the second player has a follow-up move consisting in picking some piece $P$. But then, for the first player to pick $P$ as its first move would be a winning strategy, as it leads to exactly the same position as the above scenario, except with player two to take her turn.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-01-12T21:29:31.240" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7214" PostTypeId="2" ParentId="6107" CreationDate="2015-01-12T23:12:43.047" Score="5" Body="&lt;p&gt;I know this might be perhaps be too advanced, but perhaps you can look at other &lt;em&gt;integer-like&lt;/em&gt; structures, (non unique factorization domains) like e.g. the classical counterexample $\mathbb Z[\sqrt{-5}]$ where $6= 2 \cdot 3= (1+\sqrt{-5})(1-\sqrt{-5})$ (which are all primes). Perhaps someone can come up with another example, but I fear this is the simplest one.&lt;/p&gt;&#xA;" OwnerUserId="1936" LastEditorUserId="511" LastEditDate="2015-01-12T23:54:47.293" LastActivityDate="2015-01-12T23:54:47.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7215" PostTypeId="2" ParentId="7192" CreationDate="2015-01-13T01:58:36.937" Score="4" Body="&lt;p&gt;Another way to get to the definition of $i$ is by starting with that the Complex numbers are an ordered pair of Real numbers $(a,b)$. Let $x=(a,b)$ and $y=(c,d)$. Then addition and multiplication is defined as&#xA;\begin{align}&#xA;x+y &amp;amp;= (a+c,b+d)\\&#xA;xy &amp;amp;= (ac-bd,ad+bc)&#xA;\end{align}&#xA;From the definitions above, the Complex numbers are Field which you can very with the students or just say so. Now we define $i=(0,1)$. &#xA;$$&#xA;i^2=(0,1)(0,1)=(-1,0)=-1&#xA;$$&#xA;Now we can show $(a,b)=a+bi$ to achieve the more common definition of the Complex numbers.&#xA;\begin{align}&#xA;a+bi &amp;amp;= (a,0)+(0,b)(0,1)\\&#xA;&amp;amp;= (a,0) + (0,b)\\&#xA;&amp;amp;= (a,b)&#xA;\end{align}&lt;/p&gt;&#xA;" OwnerUserId="3581" LastActivityDate="2015-01-13T01:58:36.937" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7216" PostTypeId="2" ParentId="7187" CreationDate="2015-01-13T03:40:42.213" Score="2" Body="&lt;h2&gt;Geometry&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The following two theorems came from &lt;a href=&quot;http://www.sonoma.edu/users/w/wilsonst/papers/Geometry/parallel/default.html&quot; rel=&quot;nofollow&quot;&gt;Sonoma&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a line intersects two lines then the following conditions are equivalent.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The alternate interior angles are the same size. &lt;/li&gt;&#xA;&lt;li&gt;The corresponding angles are the same size.&lt;/li&gt;&#xA;&lt;li&gt;The opposite interior angles are supplementary. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If two lines are crossed by a third, then the following conditions are equivalent.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The alternate interior angles are the same size. &lt;/li&gt;&#xA;&lt;li&gt;The corresponding angles are the same size.&lt;/li&gt;&#xA;&lt;li&gt;The opposite interior angles are supplementary. &lt;/li&gt;&#xA;&lt;li&gt;The two lines are parallel.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;Algebra&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I am not sure what to think of the difficult of this one but it could be explainable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $p\geq 2$ be an integer. Then the following conditions are equivalent.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$p$ is prime.&lt;/li&gt;&#xA;&lt;li&gt;For any $a\neq 0\in\mathbb{Z}_p$, the equation $ax=1$ has a solution in $\mathbb{Z}_p$.&lt;/li&gt;&#xA;&lt;li&gt;Whenever $ab=0$ in $\mathbb{Z}_p$, then $a=0$ or $b=0$ in $\mathbb{Z}_p$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;More Advanced Option&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;If the class is familiar with linear systems, the following theorem could be used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $\mathbf{A}$ be an $n\times n$ matrix. Then the following are equivalent. (Note you could make this for only $2\times 2$ matrices so it isn't over the head of the students.)&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$\mathbf{A}$ is row equivalent to $\mathbb{I}$&lt;/li&gt;&#xA;&lt;li&gt;$\mathbf{A}$ is a product of elementary matrices&lt;/li&gt;&#xA;&lt;li&gt;$\mathbf{A}$ is invertible&lt;/li&gt;&#xA;&lt;li&gt;The system $\mathbf{Ax}=\mathbf{0}$ has only the trivial solution&lt;/li&gt;&#xA;&lt;li&gt;For any $n$ dimensional vector $\mathbf{b}$, the system $\mathbf{Ax}=\mathbf{b}$ has a unique solutions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3581" LastActivityDate="2015-01-13T03:40:42.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7217" PostTypeId="2" ParentId="5872" CreationDate="2015-01-13T04:42:01.240" Score="1" Body="&lt;p&gt;A method I would try if they understand numbers but not symbols is: Let $x\in\mathbb{R}-\{0\}$ to avoid division by zero. Now we know that $x = x^1$ and $\frac{1}{x} = x^{-1}$ (if not known explain). When we take &#xA;$$&#xA;\frac{x}{x} = x^1x^{-1} = x^{1-1} = x^0 = 1&#xA;$$&#xA;where $x^1x^{-1} = x^{1-1}$ by the power rule and any nonzero real number to the zero power is $1$.&lt;/p&gt;&#xA;" OwnerUserId="3581" LastActivityDate="2015-01-13T04:42:01.240" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7218" PostTypeId="1" CreationDate="2015-01-13T10:55:56.260" Score="11" ViewCount="1739" Body="&lt;p&gt;From looking at undergraduate mathematics programmes it's quite apparent that mathematics degrees are demanding, one could even say the work load is gruelling. However I'm certain that there are things that one could do to prepare in advance for the rigours of such a degree. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'd like to know is what foundations must be in place so that the experience of learning mathematics at university is an enjoyable one. Enjoyable in the sense that if you're exposed to a new topic you aren't floundering and you can dive straight in and enjoy the exposition, without having to go backwards plugging in numerous gaps and addressing other deficiencies in your knowledge. I'm certain that a good grounding in pre-calculus mathematics and calculus are a prerequisite but aren't all that's needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are the things must one know to have a solid grounding in mathematics with the aim of studying mathematics at a higher level? What are the things you wish you knew at the beginning of your degree?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's assume it's a quite a demanding degree programme: MIT, Harvard, Cambridge, etc.&lt;/p&gt;&#xA;" OwnerUserId="1677" LastEditorUserId="1677" LastEditDate="2015-10-05T13:28:11.817" LastActivityDate="2015-10-05T16:08:07.393" Title="What things should one know in order to enjoy their undergraduate degree?" Tags="|undergraduate-education|secondary-education|mathematical-pedagogy|self-learning|" AnswerCount="8" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7219" PostTypeId="2" ParentId="7218" CreationDate="2015-01-13T11:06:11.770" Score="10" Body="&lt;p&gt;Don't denigrate &quot;pre-calculus mathematics and calculus.&quot; Many of the problems of students is a lack of a solid foundation in the lower mathematics, especially algebra. Make sure you can do the algebra &lt;em&gt;reliably&lt;/em&gt; and &lt;em&gt;quickly&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say the next most important knowledge is that of basic logic--not necessarily mathematical logic, which will be taught in college, but the basic ideas. Can you quickly tell which logical expressions are equivalent? Do you know the major proof strategies? That used to be emphasized in high school geometry, but the logic part of geometry has been almost completely dropped these days.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2015-01-13T11:06:11.770" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7220" PostTypeId="2" ParentId="7218" CreationDate="2015-01-13T11:37:49.600" Score="0" Body="&lt;p&gt;There is much that can be learned about the kind of mathematics one encounters at University from taking part in high school level mathematical competitions. The key difference between HS and university maths is increased rigour; less focus on methods, and more on explanations and justifications; and as you say, a wider diversity of topics. These new topics are not encountered in isolation, and require the same persistence, logical thinking and creativity that a student will learn by completing challenging and unfamiliar problems in even elementary maths.  &lt;/p&gt;&#xA;" OwnerUserId="3571" LastActivityDate="2015-01-13T11:37:49.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7221" PostTypeId="1" AcceptedAnswerId="7226" CreationDate="2015-01-13T15:20:23.517" Score="14" ViewCount="705" Body="&lt;p&gt;I am teaching a short &quot;topics in geometry&quot; course to future high school math teacher in France.&#xA;I plan to cover some spherical geometry. I will be treating the following topics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;volume of the ball and area of the sphere, &lt;/li&gt;&#xA;&lt;li&gt;length-minimizing property of great circles, &lt;/li&gt;&#xA;&lt;li&gt;intrinsic distance on the sphere &lt;/li&gt;&#xA;&lt;li&gt;Girard's formula for the area of spherical triangles. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;From this I would like to address some elementary questions of cartography, in particular I would like to prove the following theorem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem :&lt;/strong&gt; Given an open set $U$ of the sphere, there is no map from $U$ to the plane, which preserve distances (with $U$ endowed with the intrinsic distance on the sphere, and the plane endowed with the euclidean distance).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proof I know uses Gauss Egregium Theorem: a (smooth) map which preserves intrinsic distances between two surfaces will preserve the first fundamental form, and will also preserve the Gauss curvature (by the Egregium Theorem). Since the plane has curvature $0$ and the sphere has curvature $1$, this is impossible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that a more elementary proof should be available because the Girards formula actually expresses (via the local Gauss-Bonnet theorem) that the sphere has curvature $1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question :&lt;/strong&gt; Does there exists an elementary proof of the theorem above ? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I guess by elementary here I mean understandable by an average undergraduate student without prior exposition to differential geometry and little knowledge in multivariate calculus.)&lt;/p&gt;&#xA;" OwnerUserId="400" LastActivityDate="2015-01-15T16:05:00.700" Title="Is there an elementary way to explain that a map of the earth cannot preserve distances?" Tags="|undergraduate-education|geometry|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7222" PostTypeId="2" ParentId="7218" CreationDate="2015-01-13T15:29:15.850" Score="4" Body="&lt;p&gt;&quot;Enjoyable in the sense that if you're exposed to a new topic you aren't floundering and you can dive straight in and enjoy the exposition, without having to go backwards plugging in numerous gaps and addressing other deficiencies in your knowledge.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure that ever happens (it certainly hasn't for me).  Mathematics is an immensely rich and deeply interconnected topic, and learning a new topic always changes how you understand previous topics, which means going back and rethinking them and learning parts that you'd overlooked or hadn't appreciated before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(But math is enjoyable anyway!)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-01-13T15:29:15.850" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7223" PostTypeId="2" ParentId="7221" CreationDate="2015-01-13T15:32:18.023" Score="11" Body="&lt;p&gt;The accepted answer works well for the scope and goal of your course to give the students a solid understanding of &lt;em&gt;why&lt;/em&gt; the theorem is true without getting too indepth.  However, if they were to take this approach when &lt;em&gt;they are teaching in the future&lt;/em&gt; i think that it would likely be way more advanced than the scope of a secondary geometry class.  It may be useful to at least discuss (not necessarily actually perform) the following teaching strategy for them to implement when they have their own classrooms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;instead of trying to formally prove the theorem, it maybe more beneficial to their learning to have them explore the idea of &quot;flattening a sphere&quot; and come up with their own conjectures.  Especially for students who do not have a strong math background, as per your question, digging into the complexities of a proof may be overwhelming.  Instead, how about have them all bring in, or provide them with, various balls of different sizes (tennis ball, basketball, soccer ball, beach ball, etc), have them draw some shapes on the ball with markers and measure the shapes' dimensions (easily done with lengths of string), and then challenge them to both flatten the ball (via cutting it up, or any other method they may want to try) and preserve the dimensions of the shapes that they drew.  As we already know, this task is impossible but they should have a lot of fun trying and experimenting with different methods of flattening and will eventually conclude that it is impossible.  Especially for future high school math teachers, i think that this approach will provide a better learning experience than just going through a proof&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:   Edited my answer to reflect @BenCrowell astute comment that this method would be great to use to teach students but may be too elementary for a college level geometry course&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2015-01-15T16:05:00.700" LastActivityDate="2015-01-15T16:05:00.700" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7224" PostTypeId="2" ParentId="7218" CreationDate="2015-01-13T17:46:47.187" Score="5" Body="&lt;p&gt;I think this is a difficult question to answer usefully because the notions of &quot;fun/enjoy/...&quot; vary so wildly among individuals. There are also extremely disparate impulses about delayed gratification, long-term versus short-term this-and-that...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, especially after the questioner's edit that indicates that the context should be &quot;high-end&quot; programs:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, I think there is considerable utility in realizing that some difficulties are genuinely mathematical, and some are essentially completely artifactual (even if traditional!) created by curricula or instructors. Unless we go so far as to believe that mathematics itself is altogether nothing more than an artifact, which I think is too extreme, it can be important to learn to distinguish genuine complication from contrived. This is not to claim that contrived issues needn't be fun or entertaining or profitable. Indeed, doing well in contest-math, if one has the inclination and the capacity, is effective PR and can be an ego-boost. Doing well in homework assignments or exams can have similar benefits. However, I think one should not lose sight of the fact that such things are mostly quite deliberately contrived by other people as challenges to others, without regard to deeper meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This distinction might seem pointless... but I have seen many cases of failure to appreciate the distinction between genuine and contrived lead to vast wastes of time and effort in misguided attempts to defend against sniping and prank questions. Sure, prepping for contests may warrant such, but at a certain point one should realize that it is not only essentially impossible to pre-emptively defend against deliberate prank-questions, but it is &lt;em&gt;very&lt;/em&gt; &lt;em&gt;expensive&lt;/em&gt;... and displaces much more worthwhile work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I claim that this argues for moderation in line-by-line reading of texts, for moderation in doing exercises at the end of the chapter, and for moderation in thinking mathematics is only legitimate to the extent that it is formalized.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather, most mathematics &lt;em&gt;is&lt;/em&gt; based in physical intuition (if of a refined sort), and one can come very close to &quot;correct answers&quot;, at least if one is allowed to work &lt;em&gt;iteratively&lt;/em&gt; (which also seems disallowed in most formalized settings), without necessarily playing the game of formal arguments. I'm not trying to argue against formal arguments per-se, but only that formality as an alleged minimal criterion for mathematical correctness ought not be a foregone conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mention this sort of thing as a push-back against the (understandably convenient) formality of most texts and most curricula...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an action item, I strongly recommend working to get an &lt;em&gt;overview&lt;/em&gt; of the mathematics that is reflected in the standard curriculum, since that larger picture often explains the quirks of small details, thus avoiding any seeming need to &quot;memorize&quot; formal things whose motivations would only arise later. That is, read &lt;em&gt;lightly&lt;/em&gt; through textbooks... just to see what happens later, how things re-appear or don't, etc. &lt;em&gt;Especially&lt;/em&gt; beneficial is the bit of psychological familiarity this confers, in advance of appearance of new ideas in classes. Many people greatly underestimate the role that a sense of prior familiarity can play in &quot;comfort&quot;. Discomfort is obviously an inhibitor, for most people...&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2015-01-13T18:00:57.850" LastActivityDate="2015-01-13T18:00:57.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7225" PostTypeId="2" ParentId="7218" CreationDate="2015-01-13T18:02:00.970" Score="7" Body="&lt;p&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt; You assume a &quot;quite a demanding degree programme&quot;. In turn I will assume &quot;quite good student&quot;, that is, a person who is able to handle such studies within reasonable time frame.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I would place the following three things as most important for enjoying higher mathematical education:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;curiosity:&lt;/strong&gt; it's pointless if you don't really want to know the things you are learning;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;patience:&lt;/strong&gt; math can be frustrating, yet the delayed gratification makes doing it worth it, just make sure you don't waver too much in-between;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;be able do do hard (mathematical) work when it's required:&lt;/strong&gt; if drawing 20 complicated diagrams is what it takes to understand some concept, just do it.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In my opinion if these &lt;em&gt;foundations&lt;/em&gt; are in place, then the student does not have anything to worry about. However, if I would have to list some practical mathematical skills, then there are two:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;quantifiers manipulations:&lt;/strong&gt; in my opinion there is a big difference between high school and university-level definitions, where the latter are much more quantifier-heavy; being able to easily understand them seems important.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;reliable substitution:&lt;/strong&gt; substitution is used everywhere and doing it reliably greatly speeds up some derivations.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope that helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2015-01-13T18:02:00.970" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7226" PostTypeId="2" ParentId="7221" CreationDate="2015-01-13T20:44:30.230" Score="15" Body="&lt;p&gt;&lt;strong&gt;A simple proof by counter-example:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1)  Draw a map that preserves distances between (0°E 0°N), (0°E 90°N = north pole), and (90°E 0°N).  If you start with a sphere, these points are 90° apart from each other.  A distance-preserving map will be an equilateral triangle, possibly with bulging sides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2)  Try to add the point (0°E 90°S = south pole).  A distance-preserving map would need to simultaneously place it 90° from the two equator points, and 180° from the north pole.  But with two equilateral triangles, it will only be 90° * sqrt(3) away from the north pole.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, even for a very simple example, there is no way to make a distance-preserving flat map for just these four points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A more general proof:&lt;/strong&gt;&#xA;As Matt F. suggests, we can do the same exercise for a small square around a point, instead of for an entire hemisphere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1)  We are trying to prove that we cannot make a flat, uninterrupted map of a sphere that preserves all distances.  Formally, &quot;Given an open set U of the sphere, there is no map from U to the plane, which preserve distances (with U endowed with the intrinsic distance on the sphere, and the plane endowed with the euclidean distance).&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;This is an example of two important points for math teachers to teach their students:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;In proofs, start by clearly stating what you are assuming, and what you intend to prove.  In story problems, start by clearly indicating what you know, and what you are trying to find.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;You might need to translate the detailed &quot;formal&quot; language of mathematics into simpler language that other people can understand.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;2)  Let  be an angle between 0° and 180°.  (In the previous proof,  was 90°.)  For ease of drawing, 15° or 30° is convenient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;This is an example of some important points for math teachers to teach their students:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;At the start of your problem, write out the names of your variables.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Angle variables typically do not show a degree symbol (°).&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;There are two common units for angles:  360° for a full circle = 2 radians for a full circle.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Consistently use the units that are most convenient for a problem.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;If necessary, use unit conversions (such as multiplying by a fraction that equals 1) to convert between units.  For example, (2 radians) / 360° and 360° / (2 radians) are both fractions that equal 1.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;3) To keep the math simple, this proof implicitly assumes a unit sphere (with radius = 1).  The proof can be extended to a sphere with arbitrary radius R by multiplying all distances on the surface of the sphere and the surface of the flat map by R.  Alternatively, if the sphere represents the Earth, the student can assume that 1° = 60 nautical miles, or that 90° = 10,000 kilometers.  Nautical miles are &lt;strong&gt;not&lt;/strong&gt; the same as the more common &quot;statute miles&quot;. Nautical miles and kilometers were originally defined so that these statements were true when measuring from the north pole to the equator via Paris.  The error due to the Earth being an oblate spheroid instead of a perfect sphere is less than one percent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;This is an example of some more important points for math teachers to teach their students:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;When setting up a problem, sometimes it helps to make an approximation.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;It is good to know what those approximations are, and estimate how much they could affect the result.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;It is often easier to start with a special case, and then add extra variables as needed to find a more general answer.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;4)  We are going to draw a map around an arbitrary point on the sphere.  To make the math easier, we are going to make this point be (0°E 0°N).  But the proof applies to any point on the sphere, not just (0°E 0°N).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;This is an example of three more important points for math teachers to teach their students:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;If possible, draw a picture of the problem.  Label your variables and coordinate system.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;You can choose any coordinate system that is both convenient and consistent.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;On a sufficiently symmetrical object (like a sphere), it does not matter where you start.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;5)  On the sphere, label the point (0°E 0°N).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6)  On a flat map, label the point (0°E 0°N) at the origin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7)  On the sphere, label the four points ( E 0°N), (0°E  N), ( W 0°N), and (0°E  S).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;8)  On the flat map, label the four points ( E 0°N), (0°E  N), ( W 0°N), and (0°E  S).  Note that these four points are all a distance  from the origin.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;9) The four points (marked in Step 8) should form a square, because -- if it were possible to preserve distances -- any two adjacent corners should be the same distance from each other as any other two adjacent corners.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;10)  On the flat map, the length of a side of the square is (√2).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;11)  Inside the sphere, the straight-line distance as the heat-resistant rock-tunnelling gopher travels between opposite corners of the square is 2sin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;12)  Inside the sphere, the straight-line distance as the heat-resistant rock-tunnelling gopher travels between adjacent corners of the square is (√2)sin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;13)  Along the surface of the sphere, the distance between adjacent corners of the square is 2arcsin( sin / (√2) ).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;14) Using Taylor series expansions, we can show that (for angles measured in radians):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2arcsin( sin / (√2) ) &lt;a href=&quot;http://www.wolframalpha.com/input/?i=2arcsin%28sin%28x%29%2F%E2%88%9A2%29&quot;&gt;≈ (√2) - (^3)/6(√2) - 7(^5)/240(√2) + O(^7)&lt;/a&gt; ≠ (√2) (unless  = 0°).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;15) The flat map cannot have all of the distances match the distances on the sphere. &lt;strong&gt;QED.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;This is another important point for math teachers to teach their students:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Clearly label the solution to the problem.  For many story problems, this is best done by writing out a sentence describing the answer, and circling the answer in a cloud.  For many proofs, this is done by stating &quot;QED&quot;, which means &quot;This is what I set out to prove.&quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;16) Fortunately, for a map of a small area, the approximation is close.  For angles measured in radians, in the limit as →0 radians, sin = .&#xA;In the limit as →0°, 2arcsin( sin / (√2) ) ≈ 2arcsin(  / (√2) ) ≈ 2(  / (√2) ) ≈ (√2)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a sanity check:  We have confirmed that a flat map approximately preserves distances for a small enough portion of a sphere.  This is consistent with our real world experience that we can make useful flat maps of small parts of the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;This is an example of two more important points for math teachers to teach their students:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;At the end of a problem, do a sanity check to confirm that the answer is reasonable.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;If possible, perform a check-by-substitution, to confirm that the answer matches the information that you knew at the beginning of the problem.&lt;/em&gt; The previous limit is both a sanity check, and a partial check-by-substitution.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;17) As a further partial check-by-substitution, when  = 180°, 2arcsin( sin / (√2) ) = 2arcsin( 0 / (√2) ) = 2arcsin( 0 ) = 0°.  We can verify this by &quot;pushing&quot; the square halfway around the sphere to where the points of the square join up again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;18) And as one last partial check-by-substitution, we will look at the special case in the previous proof.  When  = 90°, 2arcsin( sin / (√2) ) = 2arcsin( 1 / (√2) ) = 2(45°) = 90° ≠ (√2).&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-01-14T22:17:06.510" LastActivityDate="2015-01-14T22:17:06.510" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7227" PostTypeId="2" ParentId="7218" CreationDate="2015-01-13T22:56:04.603" Score="5" Body="&lt;p&gt;&lt;strong&gt;Caveat:&lt;/strong&gt; From a UK university perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the first thing to say, as perhaps other answers have said, is that almost all students at some time wrestle with one concept/theorem, or another. After all, mathematics &lt;em&gt;is&lt;/em&gt; doing...and &lt;em&gt;doing&lt;/em&gt; takes energy and effort. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, for many mathematicians and students alike, the challenge of mathematical understanding forms part of the enjoyment - or even &lt;em&gt;is&lt;/em&gt; the enjoyment. An integral part of university mathematics is to make the transition from learning the tools of mathematics to applying those tools to solve new mathematical problems. So learning to enjoy mathematical challenge is important (if not a prerequisite for a university mathematics course!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if you do not venture down the pure mathematics / analysis route, it usually forms the foundation of university mathematics - and having attended a demanding course from one of the institutions you mentioned, I can say that the largest knowledge gap (for many students) seemed to be around basic set theory, including the notion of mathematical proof, and algebra and geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have no affiliation to either author of these books. I did, however, work through &lt;em&gt;similar&lt;/em&gt; books before the start of my university course. I own these two books now, and would recommend them as pre-university reading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0989472108&quot;&gt;Sets and Proofs&lt;/a&gt;, and in algebra and geometry &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521890497&quot;&gt;Algebra and Geometry&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being familiar with the concepts presented in these two books would certainly ease the transition from high-school/secondary school to university mathematics.&lt;/p&gt;&#xA;" OwnerUserId="3541" LastActivityDate="2015-01-13T22:56:04.603" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7228" PostTypeId="2" ParentId="5872" CreationDate="2015-01-14T10:03:23.983" Score="1" Body="&lt;p&gt;It seems your student didn’t understand fractions, nor that the expression could be interpreted as division, and especially had trouble thinking in general terms. You asked for “a good approach to teaching this.” You got an inductive approach to generalize, answers focused on the meaning of fractions and division, and one on powers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an approach to generalization based on meaning, not inductive reasoning. It is from Davydov’s implementation of Vygotsky’s theory, references below. The approach is to start with unspecified quantities that are &lt;em&gt;concrete&lt;/em&gt; &lt;strong&gt;and&lt;/strong&gt; &lt;em&gt;general&lt;/em&gt;, such as two line segments or two strips of wood not of equal length. Use the shorter segment as a basic unit to measure the longer segment. How many of the shorter unit fits into the longer segment? Say that the longer one is exactly 3 times the shorter. Then the answer is 3. Label the shorter line as “U” for the basic unit, and the longer as “Q” for the quantity measured. The action of measuring can be expressed as Q/U, which is measurement division: how many U’s in Q? Then for these segments we can write Q/U=3. (First graders actually get this, although it is developed in many steps.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then make Q  the same length as U. How many U’s in Q--clearly only one. Now Q/U=1. Since Q=U in this case, we don’t need two different letters. We can write it as U/U=1. How many units in one unit? It is one unit. Now go from the general to the specific to enrich and clarify the meaning. If U is 5 then 5/5=1. There is one 5 in 5, and so on with other numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used a length of 3 at first because it is easier to understand the meaning of the measurement if the units are not the same. After establishing the measurement and division expression, then move to equal lengths. I wouldn’t think that a high school student would have a problem understanding this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Vygotsky’s approach is to start with the general and “ascend to the specific.” The emphasis is on understanding relationships, then doing computation. For example, the additive relationships of A + B = C, C – B = A, and C – A = B are thoroughly examined in concrete situations, then symbols, before problems are given. The initial tasks are to understand the relationships. Then problems are created by making one of the three elements the unknown.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;References&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moxhay, P. (2008, September). “Assessing the scientific concept of number in primary school children.” Paper presented at the International Society for Cultural and Activity Research conference, San Diego.&#xA;&lt;a href=&quot;http://lchc.ucsd.edu/MCA/Mail/xmcamail.2008_10.dir/att-0111/Moxhay_ISCAR_2008.pdf&quot; rel=&quot;nofollow&quot;&gt;http://lchc.ucsd.edu/MCA/Mail/xmcamail.2008_10.dir/att-0111/Moxhay_ISCAR_2008.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schmittau, J. and Morris, A. (2004). “The Development of Algebra in the Elementary Mathematics Curriculum of V.V. Davydov” in The Mathematics Educator, Vol.8, No.1, 60 – 87.&#xA;&lt;a href=&quot;http://math.nie.edu.sg/ame/matheduc/tme/tmeV8_1/Schmittau.pdf&quot; rel=&quot;nofollow&quot;&gt;http://math.nie.edu.sg/ame/matheduc/tme/tmeV8_1/Schmittau.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3578" LastActivityDate="2015-01-14T10:03:23.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7229" PostTypeId="2" ParentId="7218" CreationDate="2015-01-14T11:43:04.977" Score="0" Body="&lt;h3&gt;The Greek Alphabet&lt;/h3&gt;&#xA;&lt;p&gt;A lot of mathematics professors prefer using greek symbols over the roman alphabet (a-z).  Learn to recognize the names and both the small and capital symbols, and practice writing them.&lt;/p&gt;&#xA;&lt;p&gt;One of my first lectures in Maths at Cambridge, UK used small xi - &lt;strong&gt;ξ&lt;/strong&gt; and small zeta &lt;strong&gt;ζ&lt;/strong&gt; for preference rather than x and y; challenging lectures are not a good place for a first encounter with these!&lt;/p&gt;&#xA;&lt;h3&gt;Matrix calculations&lt;/h3&gt;&#xA;&lt;p&gt;The other non-obvious thing is matrix calculations (how to multiply matrices etc).  Since the algorithm for doing this is straightforward, a fast course will touch on it lightly and then move on to the complex things that you actually use it for. However, most students require a lot of practice before they can do this reliably.&lt;/p&gt;&#xA;&lt;h3&gt;Calculus and other Things You Still Won't Have Memorized&lt;/h3&gt;&#xA;&lt;p&gt;Experienced mathematicians will 'just know' all of the formulae they may want to know.  Don't flounder around trying to keep up - that cheat sheet you made for revising?  The one with all of the common calculus derivatives &amp;amp; integrals, trigonometric formule etc?  Keep it and take it with you!&lt;/p&gt;&#xA;" OwnerUserId="4638" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-01-14T11:43:04.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7230" PostTypeId="2" ParentId="7221" CreationDate="2015-01-14T11:45:38.760" Score="4" Body="&lt;p&gt;Try a distance preserving flattening of an orange skin. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-01-14T11:45:38.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7231" PostTypeId="2" ParentId="369" CreationDate="2015-01-14T20:21:22.070" Score="12" Body="&lt;p&gt;The textbooks we used for Algebra 2 at Miramonte High School, Orinda, California, in 1967 - 1968 must have been SMSG Units 17 - 18, entitled &lt;em&gt;Intermediate Mathematics, Part I and II&lt;/em&gt;. I recognize the content as being that of the PDF Documents (scan) available at ERIC corresponding to the same titles.&#xA;Ref.:&#xA;&lt;a href=&quot;http://eric.ed.gov/?id=ED135625&quot;&gt;http://eric.ed.gov/?id=ED135625&lt;/a&gt;&#xA;&lt;a href=&quot;http://eric.ed.gov/?q=%22Allen+Frank+B.%22&amp;amp;id=ED135626&quot;&gt;http://eric.ed.gov/?q=%22Allen+Frank+B.%22&amp;amp;id=ED135626&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4641" LastActivityDate="2015-01-14T20:21:22.070" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7232" PostTypeId="2" ParentId="7221" CreationDate="2015-01-15T13:47:51.100" Score="7" Body="&lt;p&gt;On the unit sphere, construct a circle of radius $r$, i.e., the set of all points that lie at arc length $r$ from a given point. The circumference of this circle is $C=2\pi \sin r$, which is strictly less than the Euclidean value $2\pi r$. If there were a distance-preserving map, then it would have to preserve both $r$ and $C$, but we've just shown that if it preserves $r$, it can't preserve $C$.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2015-01-15T13:47:51.100" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7233" PostTypeId="2" ParentId="1760" CreationDate="2015-01-15T17:45:32.097" Score="8" Body="&lt;p&gt;Wow, thanks for the recent shout-out. I hope this is the right place for me to add a few references that might be useful and haven't already appeared in the answers.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Rob Ghrist has just written a fantastic new book called &lt;a href=&quot;http://www.math.upenn.edu/~ghrist/notes.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elementary Applied Topology&lt;/a&gt; which provides a soaring and current overview of the field. There are no exercises (yet!) but the figures are exquisite, and if you want a big-picture survey of which aspects of algebraic topology have been applied to which problems, it is hard to beat this book right now.&lt;/li&gt;&#xA;&lt;li&gt;Another text that really, really helped me out when I was in the late engineering undergrad stage is the very concrete and grounded &lt;a href=&quot;http://www.springer.com/mathematics/geometry/book/978-0-387-40853-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Computational Homology&lt;/a&gt; by Tomasz Kaczynski, Konstantin Mischaikow and Marian Mrozek. &lt;a href=&quot;http://www.math.rutgers.edu/~mischaik/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Konstantin&lt;/a&gt; taught a wonderful undergrad course out of this book at Georgia Tech in 2004, so it can definitely be used in your context. The book is replete with exercises that students can get their hands around, but sadly it was released in the pre-persistence era and hence does not get into Vietoris-Rips complexes, etc. On the other hand, the introduction is masterfully written and will appeal to the undergrad scientist/engineer audience.&lt;/li&gt;&#xA;&lt;li&gt;I wrote a recent &lt;a href=&quot;http://people.maths.ox.ac.uk/nanda/source/PersBio-Final.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;survey article&lt;/a&gt; with Radmila Sazdanovic that deconstructs much of the topological data analysis pipeline: from point clouds to simplicial complexes to persistence diagrams to applications. The paper has a slight biological bias towards the end but the beginning is fairly domain-agnostic.&lt;/li&gt;&#xA;&lt;li&gt;And certainly, don't miss the course schedule of Paul Bendich's 2012 Duke class on topology and applications, available &lt;a href=&quot;http://www.paulbendich.com/412fall2012.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. I find that it is remarkably well laid-out!&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4644" LastEditorUserId="376" LastEditDate="2020-06-09T13:22:16.700" LastActivityDate="2020-06-09T13:22:16.700" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7234" PostTypeId="1" CreationDate="2015-01-15T21:12:20.187" Score="20" ViewCount="2128" Body="&lt;p&gt;&lt;a href=&quot;http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf&quot;&gt;The National Mathematics Advisory Panel final report&lt;/a&gt; states that algebra is the gateway to higher math, to a college degree, and higher earnings from employment. It also states that success in algebra is dependent on understanding multiplicative reasoning, including proficiency with rational numbers and proportional relationships. Furthermore, there is ample evidence that difficulty with multiplicative reasoning is pervasive and failure rates in algebra classes are unacceptably high. One of the report's recommendation is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percent, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected. (p. 18)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If improvement in the teaching of fractions is necessary for improvement in student achievement in algebra, then isn’t multiplicative reasoning the gatekeeper to higher math rather than algebra?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of any research that sheds light on this question? For example, math placements exams for entering college students could be analyzed for proficiency with multiplicative reasoning and with algebra, and related to math courses taken in college. The same could be done for 8th grade and 12th grade standardized math tests or other tests like the PISA, and related to college attendance as well as math courses taken. The difficulty, I expect, would be access to all the necessary the data for individual students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a &lt;a href=&quot;http://kristiankiili.com/2014/09/18/rational-numbers/&quot;&gt;blog post&lt;/a&gt;, krikii says “Research has shown that the extent to which elementary school students master rational numbers is a strong predictor of future success in mathematics.” There is ample research showing poor performance on rational number tasks, but what research is there that uses rational number knowledge &lt;em&gt;to predict&lt;/em&gt; future success in math?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way to research this is to ask, &quot;When do students start to dislike math and not want to take any more math courses?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;UPDATE: littleO’s comment and Tom Au’s answer got me thinking about defining more clearly what I mean by “gatekeeper to higher math.” Students are having difficulty in making the change from additive reasoning to multiplicative reasoning, and in making the change from working with numbers to working with variables. Which adjustment in thinking is a greater separator between those who continue on to higher math and those who don’t? An important research question would be, “What percent of those who are not proficient in algebra are proficient in multiplicative reasoning?” &lt;/p&gt;&#xA;" OwnerUserId="3578" LastEditorUserId="3578" LastEditDate="2015-01-16T18:43:50.693" LastActivityDate="2015-04-12T12:22:09.607" Title="Is algebra really the gatekeeper to higher math, or is it multiplicative reasoning?" Tags="|algebra|education-research|fractions|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="7235" PostTypeId="2" ParentId="7234" CreationDate="2015-01-16T07:04:03.953" Score="3" Body="&lt;p&gt;Why do kids hate maths? In my view, performance anxiety and  the abstract nonsense that often passes for algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: these are not a main stream views as I believe curriculum change is necessary. This is intended to be thought provoking rather than &quot;the answer&quot; and is primarily based on personal teaching experience, and seeing alternative teaching techniques in practice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Timed tests&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is research that has been mentioned before (I'll look for it, but if someone remembers, please add it) that timed tests common in maths education leads to anxiety which makes kids hate maths. They are also expected to have high levels of accuracy, so that it is more about avoiding mistakes than constructing an answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students need to avoid mistakes at a rapid pace (often a question every 10 seconds), in an academic version of dodge ball, but a dodge ball that goes on your record and is reported to parents. If you do badly you may need to have make up academic dodge ball ordeals which are repeated until you are good enough at it. I feel stressed just imagining it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my teaching experience, I've seen a very mathematically gifted kid who hated maths because the teacher's measure of maths success was speed tests in times tables. Recent &quot;back to basics&quot; movements have just made it worse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Algebra&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a former student of pure maths, I find the next section difficult to write.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest that rational numbers, algebra, and analytic calculus encourage a thinking model that is largely at odds with decimal numbers, algorithms, and numeric methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[edited] Though I often hear the sentiment that mathematics is all about proof (algebraic thinking) this is not in line with many mathematicians' and engineers' experience with mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason that so many people hate school maths is that it is largely dominated by algebraic thinking models that most students find unintuitive and irrelevant. In other words, to them it is just abstract nonsense. &#xA;[end edit]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is no surprise that comfort with algebraic thinking models, which have largely dominated high school curricula, is highly correlated with success in current curricula (we needed a survey to find this out?). It does not investigate whether this style of mathematics deserves its current dominant position in curricula. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Decimals are often taught as a special case of rational numbers, but I have heard arguments that suggests students would be better served by dropping rational numbers from curricula until high school algebra. I am not sure if there is research on this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At lest schools currently teach decimals, unlike much other practical mathematics. Most mathematics curricula do not even include algorithms. It is mostly relegated to an optional subset of technology, along with cooking and fashion design. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though numeric calculus is easier to understand and totally dominates real world applications of calculus, current curricula only briefly include it as an intro to analytic calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my own experience, replacing algebra and calculus with algorithms and numeric methods allowed a middle school student to complete much of a calculus based Physics course. This student had virtually no algebra beyond substitution (the basis of algorithms) and he used memorised heuristics for common patterns of equation rearrangement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Multiplicative Reasoning&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience of teaching how to combine ratios and proportions, I have caught myself teaching formal algebraic methods to students that can do it intuitively. I eventually realised that by enforcing the formal algebraic methods, I was working against and possibly undermining students' intuition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally wonder if the cause of some problems in multiplicative reasoning is caused by current teaching methods undermining intuition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to fit problems into the formal algebraic language we are used to, we are forced to express problems in an unintuitive way that most students rightly react against, in an attempt to preserve their often algorithmic mathematical intuition. Similarly, generations of programmers have mostly rejected mathematically correct &quot;functional&quot; programming and algebraic type theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Papert, Resnick, and others at MIT have researched the use of algorithmic style thinking to allow students to develop abstract thinking and mathematical skills, however, it has been many years since I have researched this, so if anyone has any references, please add them. I don't think that they looked much at multiplicative reasoning or number specific skills. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I have also been an IT teacher, I can confirm that teaching algorithmic mathematics allows students to develop mathematical skills much more quickly and without the cognitive dissonance often associated with teaching algebra. Students naturally solve mathematical problems that they may not even recognise as &quot;maths&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry I don't have links to solid research yet, but it has been a long time since I researched it, and hopefully others can fill in my blanks. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-01-16T23:32:36.347" LastActivityDate="2015-01-16T23:32:36.347" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="7236" PostTypeId="2" ParentId="7234" CreationDate="2015-01-16T15:14:27.683" Score="3" Body="&lt;p&gt;I would say that algebra is the gateway to higher &quot;pure&quot; math, while multiplicative reasoning is more like the gateway to higher applied math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplicative reasoning is about algorithms and numerical methods that &quot;engineers&quot; use to solve problems on a day-to-day basis. In that regard, it is the &quot;gateway&quot; for these kinds of workers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But algebra is the gateway to higher pure math, precisely because it forces students to stop thinking in terms of numbers, and start thinking in terms of symbols. These &quot;symbols&quot; are found more and more at progressively higher levels of pure math, which gives algebra its importance.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2015-01-16T15:14:27.683" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7237" PostTypeId="1" AcceptedAnswerId="7240" CreationDate="2015-01-16T16:20:51.890" Score="18" ViewCount="1102" Body="&lt;p&gt;this came up in class yesterday and I feel like my explanation could have been more clear/rigorous.  The students were given the task of finding the zeros of the following equation $$6x^2 = 12x$$ and one of the students did $$\frac{6x^2}{6x}=\frac{12x}{6x}$$ $$x = 2$$ which is a valid solution but this method eliminates the other solution of $$ x = 0$$  When the student brought it up, I explained to the student that if $x = 0, 2$ and we divide by $6x$ there is a possibility that we would be dividing by 0 which is undefined.  The student, very reasonably, responded &quot;Well, obviously I didn't know that zero was an answer when I was doing the problem&quot;.  The student understands why we can't divide by 0 but is still struggling with how that connects to dividing by $x$.  I went on to explain that by dividing by $x$ you are &quot;dropping a solution&quot; because the problem, which was quadratic, is now linear.  Again, this didn't seem to click with the student.  Does anyone have maybe an axiom/law/theorem that I can show the student to give a rigorous reason as to why you can't just divide by $x$?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2021-03-17T13:04:53.037" Title="Rationale for not dividing both sides of an equation by $x$ (ex: $6x^2 = 12x$)" Tags="|secondary-education|algebra|" AnswerCount="8" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="7238" PostTypeId="2" ParentId="7237" CreationDate="2015-01-16T17:00:48.087" Score="19" Body="&lt;p&gt;I think your student pointed out the key issue: &quot;Well, obviously I didn't know that zero was an answer when I was doing the problem&quot;.  That's exactly right: at the time the student was dividing, they didn't know whether or not x is 0, which means they didn't know whether or not it made sense to divide by x.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you know you can't divide by 0, you know not to divide by things which &lt;em&gt;might&lt;/em&gt; be 0, because you don't know whether or not it's vaild.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-01-16T17:00:48.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7239" PostTypeId="2" ParentId="7237" CreationDate="2015-01-16T17:35:27.453" Score="10" Body="&lt;p&gt;Most experienced mathematicians will immediately identify $x = 0$ as a solution to this equation, and then do the same division (factoring) as your student in order to obtain the other solution at $x=2$. The only difference is that the experienced mathematician has already done due diligence with respect to the possibility of erasing a root/dividing by zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am all for dividing both sides by $6x$ here, with the caveat that when dividing by an unknown quantity you are responsible for first checking whether that unknown quantity is (or might be) zero. Your student is most of the way there - all that's left is to push the conclusion C from information A and B which they already seem to have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A) I don't know what $x$ is (or $6x$, as is the case here)&lt;br&gt;&#xA;B) I'm 'not allowed' to divide by zero&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;C) I don't know whether or not I'm allowed to divide by $x$ (because it might be zero)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are useful things to think about in this neighborhood as well. A similar situation where the variable-including-divisor being equal to zero represents the only solution, then we end up producing an absurdity. Eg,&#xA;$$6x = 4x$$&#xA;$$\frac{6x}{x} = \frac{4x}{x}$$&#xA;$$6 = 4$$&#xA;If we trust our teacher, or the textbook, to have provided us with a logically sound equation in the first place, then the absurdity reached here tells us that we've broken some rules in our working with it. Since the only thing we did was to divide by $x$, we can guess that dividing by $x$ introduced the absurdity. Since we know that dividing both sides of an equation by a quantity is fair game for any value other zero, we now know that $x=0$, etc...&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2015-01-16T17:52:45.983" LastActivityDate="2015-01-16T17:52:45.983" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7240" PostTypeId="2" ParentId="7237" CreationDate="2015-01-16T17:48:45.983" Score="40" Body="&lt;p&gt;Just before dividing, you can reason &quot;Either $x=0$ or I can divide by $x$.&quot; This creates two separate cases to be analyzed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works for dividing by anything. You want to divide by $\sin(x)$? You need to make two cases: $\sin(x) \neq 0$ and $\sin(x)=0$. And then analyze each independently.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-01-16T17:48:45.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7241" PostTypeId="2" ParentId="7237" CreationDate="2015-01-16T18:13:36.273" Score="7" Body="&lt;p&gt;I think it is important to emphasize Mark Fantini's remark:  factoring is different from dividing, and factoring is the way to get a complete solution.  I would also suggest that the&#xA;problem be rearranged and then factored to give 6x(x-2)=0, or even 2*3*x*(x-2)=0.  Then&#xA;one can see that one of 2 , 3, x, or x-2 must be zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When one is comfortable with using the factoring method, then one can consider &quot;shortcuts&quot; by jumping from the given equation to saying &quot;Suppose 6x is not 0: then divide...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Shortcuts Can Cut One Short&quot; Paseman, 2015.01.16&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-01-16T18:13:36.273" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7242" PostTypeId="2" ParentId="7237" CreationDate="2015-01-17T05:01:14.760" Score="2" Body="&lt;p&gt;Frankly, I think your reasoning is the best.  You initially have a quadratic, thus you should &lt;em&gt;expect&lt;/em&gt; to find two solutions (although you may not).  By dividing by $x$, you get only a single solution: so what is the other?  You clearly cannot find it by dividing by $x$, but you &lt;em&gt;can&lt;/em&gt; find it by either factoring or using the quadratic equation (which should be discouraged in this case).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Really this is about pattern recognition.  The student should recognize when they have:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;ax^2 + bx = 0&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That they should factor out $x$ to get: $x(ax + b) = 0$ and then set each factor to zero to solve: $x = 0$ and $ax + b = 0 \rightarrow x = -\frac{b}{a}$.  The second part is recognizing that when you have something like $ax^2 + bx + c = dx^2 + ex + f$ that they should solve by setting the equation to zero, collecting like terms, and then attempting to factor (using the quadratic equation as a last resort).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they can understand factoring and that &lt;em&gt;this&lt;/em&gt; is the correct approach to solving such equations then they should have less trouble generalizing to higher degree polynomials (i.e. ones that &lt;em&gt;don't&lt;/em&gt; have a nice formula like the quadratic formula).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rationale for this is 1) the fact that polynomial equations, in general, have multiple solutions (so you should either find them or justify why there are fewer) and 2) we &lt;em&gt;really&lt;/em&gt; only &quot;know&quot; how to solve linear equations so we must somehow convert our problem into a set of linear equations to be solved and the best way to do that is to attempt to factor (but explain that this is &lt;em&gt;not&lt;/em&gt; always possible/&quot;easy&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Granted, dividing by $x$ is one way to get &lt;em&gt;a&lt;/em&gt; linear equation, but since we see that it's quadratic we should find &lt;em&gt;two&lt;/em&gt; linear equations...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;...which could be redundant like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;x^2 = 2x - 1 \\&#xA;x^2 - 2x + 1 = 0 \\&#xA;(x - 1)^2 = 0 \\&#xA;(x - 1)(x - 1) = 0 \\&#xA;x - 1 = 0 \\&#xA;x - 1 = 0&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We got our two linear equations, but they were identical/gave the same solution.&lt;/p&gt;&#xA;" OwnerUserId="2204" LastActivityDate="2015-01-17T05:01:14.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7243" PostTypeId="1" AcceptedAnswerId="7254" CreationDate="2015-01-17T12:55:31.573" Score="27" ViewCount="2629" Body="&lt;p&gt;Many students have a fairly good grasp of the topics they are learning but fall down because they miss fatal errors in their work. Some don't check for errors at all, while many simply can't find them. Indeed, many of them get very frustrated when they &lt;em&gt;know&lt;/em&gt; they are wrong somewhere (perhaps because they checked at the back of the book) but can't find where.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The skill of checking your own work is a very useful skill, but as with any skill it is easier to get good at with some specific strategies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am specifically interested in problems involving some sort of calculation such as integration, solving equations, finding a basis for a subspace. (Proofs are an entirely different thing and have different sorts of &quot;errors&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What strategies can be taught to help students find and correct their own errors, and how can they be taught to students?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="77" LastEditDate="2016-08-21T02:36:19.670" LastActivityDate="2017-05-30T11:52:45.300" Title="Teaching students to find and correct their own errors" Tags="|undergraduate-education|secondary-education|students-mistakes|critical-thinking|" AnswerCount="9" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7244" PostTypeId="2" ParentId="7243" CreationDate="2015-01-17T14:55:19.797" Score="6" Body="&lt;p&gt;I teach high-school calculus, and many questions can be checked by a graphing calculator. So my first strategy is to teach the use of the calculator and using it to check answers when possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, often a student will see that his final answer disagrees with the calculator but he does not know which of his steps introduced an error.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found that I need two more strategies to avoid this. First, I encourage my students to check some of their steps if the final answer seems wrong. By using a &quot;binary search&quot; they can narrow the problem to just one step and sometimes even to a part of one step. They can then focus their attention to figure out the exact error. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me explain the search for errors more. The best steps to check are the key ones. (For example, in finding a normal line to a point given by its $x$ coordinate on the graph of a given function, the steps to check are the $y$ coordinate of the point, the derivative of the function, the slope of the graph at the point, the slope of the perpendicular line, and finally the equation of the normal line.) If there are no obvious key steps, the student should check a step near the middle of his calculations. If that step checks, ignore the steps in the first half and check the step halfway into the second half; if it does not check, ignore the second half and check the step halfway into the first half. At each stage decide which half of the remaining steps hold the error and cut that group into half. (For example, if there are 16 steps then one possible series of checks is 8 which doesn't check, 4 which does, 6 which doesn't, and 5 which doesn't. There then must be an error in step 5.) This halving strategy is similar to the &quot;binary search&quot; used in computer science. When a step is pinpointed, sometimes it can be broken down into smaller steps which can be checked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another strategy is to teach a mild distrust of the calculator. Sometimes the answer is correct and the calculator is wrong. I teach them several reasons the calculator can be wrong. The student may have mistyped the expression (the most common problem). The student may misunderstand the proper syntax ($xsin(x)$ is not interpreted as $x\cdot \sin(x)$ but as a new function $xsin$). And occasionally the grapher simply makes its own errors (smoothing out a discontinuous function, not extending a graph close enough to a singularity, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, overall, I try to teach using the graphing calculator as a helpful but not entirely reliable tool whose main purpose is to check answers gotten in another way. I teach these concepts by doing them in practice problems at the board. I let students guide the solution, and sometimes they get it wrong. I often just proceed, and when we discover that we have gone wrong somewhere I demonstrate how to find the error. I deliberately choose some examples that lead to grapher error, often examples that came up in previous years. This way I do not even need to be embarrassed when I make a mistake: it leads to finding the bug, and I give extra credit to the person who discovered that there was a mistake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Details: I use the TI-Nspire CX calculator, without a computer algebra system. So far this seems to have the best mix of power in graphing and in numerical calculations without allowing the student to avoid doing his own algebra. At least, it seems the best mix for high school precalculus and calculus. I use the Teacher Software to emulate the calculator on my school computer and project the results on the whiteboard. It took me years to get the funds for all this but I have a pretty decent setup now.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2015-01-18T00:44:56.730" LastActivityDate="2015-01-18T00:44:56.730" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7245" PostTypeId="1" CreationDate="2015-01-17T19:26:50.760" Score="12" ViewCount="297" Body="&lt;p&gt;I'll have a department meeting in about 10 days and I want to bring the subject of proofs up. While most teachers do proofs in the blackboard, I want to argue that we should put problems to prove in students's exercises sheets. I don't think it is enough to show them proofs, they have to do some themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this is mostly based on intuition and my own experience with learning. Is there research that backs up my statements, so that I can discuss them with other teachers, showing that proofs not only help deepen their understanding of mathematics but also helps them with other tasks in mathematics, including computations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also interested in research that might contradict my intuition, in which case I'll change my beliefs for the better.&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="47" LastEditDate="2015-01-18T15:02:43.203" LastActivityDate="2015-01-20T01:20:37.013" Title="Effectiveness of proofs in secondary education" Tags="|secondary-education|proofs|education-research|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7246" PostTypeId="2" ParentId="7243" CreationDate="2015-01-17T23:48:54.013" Score="9" Body="&lt;p&gt;When I first started asking students if they &quot;checked their answers&quot; on a test, a number of them asked me what I meant. This was specific to algebra, and I told them they should take the answer, say, the X intercepts, and put those back into the equation to see if it resulted in Y being zero. Many had never heard of this, and thought that checking simply meant to go back through every calculation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, tests are often structured in a way that the average student barely has enough time to finish the exam, let alone time to check their work. Same for homework. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, I believe the practice of checking work is a good idea and should be encouraged. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-01-17T23:48:54.013" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7247" PostTypeId="1" AcceptedAnswerId="8454" CreationDate="2015-01-18T05:58:44.950" Score="4" ViewCount="11180" Body="&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/q/1108725/85079&quot;&gt;Because I'm OK at Applied Math but failed Pure Math two times&lt;/a&gt;, I asked my instructors and a brilliant friend studying the undergraduate Math Tripos at the University of Cambridge for advice. They all said that there's no secret or shortcut, that you must know your material, practice a lot, and keep trying. But I've been doing all this and still failed two times. So I Googled to see if even  Cambridge math undergraduates fail PM. What should I learn from the quotes? Is Applied Math easier? Why or why not? I bolded the text.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://www.reddit.com/r/cambridge_uni/comments/2agxc7/switching_tripos_after_1a/civ1h0m&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stealth77 ( Emmanuel [College] ).&lt;/a&gt;&lt;br&gt;&#xA;  I'm also in Maths IA (or rather just finished). A 2:2 isn't that bad. I have friends with 3rds and know of people not even on the class list. &lt;strong&gt;Remember, you can do mostly applied courses next year.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;http://www.thestudentroom.co.uk/showthread.php?t=1314227&amp;amp;p=25988557#post25988557&quot; rel=&quot;nofollow noreferrer&quot;&gt;smilepea&lt;/a&gt;. a lot of people [,] who didn't get on very well with first year because it was too pure [,]  &lt;strong&gt;really like second year as you can focus almost totally on applied&lt;/strong&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;http://www.thestudentroom.co.uk/showthread.php?t=1314227&amp;amp;p=26007014#post26007014&quot; rel=&quot;nofollow noreferrer&quot;&gt; Xavikadavi post 20&#xA;      &lt;/a&gt;. &lt;em&gt;[ This is the person who's suffering like me and responded below. ]&lt;/em&gt;&lt;br&gt;&#xA;  What you say about the pure/applied side makes a lot of sense - for final revision I focused on my strongest topic for each paper which in every case was the applied one- I've found these so much more managable over the year (hence why I thought maybe I should be on the natsci course doing physics . I found pure impossible- esp. groups and analysis.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;http://www.thestudentroom.co.uk/showthread.php?t=1314227&amp;amp;page=2&amp;amp;p=26008064#post26008064&quot; rel=&quot;nofollow noreferrer&quot;&gt;smilepea post 22&lt;/a&gt;. &lt;strong&gt;I have a friend who only did applied this year (in IB) or as far as I can remember any way (she may have done one pure course) and got at 2.1. Doing only applied meant that she was much happier and did much better.&lt;/strong&gt; And the applied courses are much, much better in 2nd year, I know the first year applied courses aren't brilliant and certainly not what I consider applied!!&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;http://www.thestudentroom.co.uk/showthread.php?t=1314227&amp;amp;page=2&amp;amp;p=26008344#post26008344&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zhen Lin, post 23&lt;/a&gt;&lt;br&gt;&#xA;  I've found that applied questions consistently take longer to do. This isn't a disadvantage if &lt;strong&gt;you find pure questions impossible&lt;/strong&gt;, but if you can do both, then, from a mark-maximising perspective, it is a disadvantage to focus on applied.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user4655" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2019-04-10T12:45:10.610" Title="Is Applied Math easier than Pure Math?" Tags="|self-learning|" AnswerCount="5" CommentCount="10" ClosedDate="2019-04-10T17:50:31.010" ContentLicense="CC BY-SA 3.0" />
  <row Id="7248" PostTypeId="2" ParentId="7245" CreationDate="2015-01-18T17:23:07.723" Score="10" Body="&lt;p&gt;These books will be of interest to you:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/9077874224&quot;&gt;Theorems in School&lt;/a&gt; edited by P. Boero&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B001Y35IUY&quot;&gt;Teaching and Learning Proof Across the Grades&lt;/a&gt;, ed. by Despina Stylianou et. al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My subjective sense is that the majority of researchers in math education agree with your point of view: learning proofs requires hands-on practice with writing proofs, and not only improves the ability to write proofs but also supports an understanding of the interconnectedness of mathematics and supports the development of computational skills as well. For example the US elementary and secondary math teachers' trade guild, the &lt;a href=&quot;http://www.nctm.org/&quot;&gt;NCTM&lt;/a&gt;, has a statement stressing the importance of proof:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Instructional programs from prekindergarten through grade 12 should enable all students to--  &#xA;&#xA;recognize reasoning and proof as fundamental aspects of mathematics;&#xA;make and investigate mathematical conjectures;&#xA;develop and evaluate mathematical arguments and proofs;&#xA;select and use various types of reasoning and methods of proof.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://www.nctm.org/standards/content.aspx?id=26861&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2015-01-18T17:23:07.723" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7249" PostTypeId="1" AcceptedAnswerId="7253" CreationDate="2015-01-18T20:52:52.107" Score="7" ViewCount="833" Body="&lt;p&gt;What are some common mistakes students often make, which may seem logical at first?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a student myself, but I'm curious of what some of the most frequent mistakes which happens. I'm thinking of purely mathematical incorrect beliefs like $(a+b)^n = a^n+b^n$, because $(a\times b)^n = a^n\times b^n$, or misunderstanding notation like $(f\circ g)(x)$ thinking it means $g(f(x))$, because you do calculations left to right, right? Or just abuse of notation and writing $(f(g(x)))'$ instead of $(f\circ g)'(x)$, thinking $'$ is a function on $f(g(x))$ (a value).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, how do you explain these mistakes?&lt;/p&gt;&#xA;" OwnerUserId="4657" LastEditorUserId="143" LastEditDate="2015-01-18T21:40:39.413" LastActivityDate="2015-01-27T18:50:48.400" Title="What are some common fallacies students make when they learn $X$ concept?" Tags="|secondary-education|mathematical-pedagogy|primary-education|students-mistakes|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7250" PostTypeId="2" ParentId="7247" CreationDate="2015-01-18T21:22:52.773" Score="7" Body="&lt;p&gt;I do not think it would even make sense to answer your question literally. First applied and pure maths are intertwined and often difficult to tell apart. For example when we teach the spectral theorem, we often consider it as pure maths, but then there are lots of clearly applied maths where it is fundamental to understand this result intimately.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, I would say that given any two field, and any difficult concept A in one of them, it is possible to find a concept B in the second that is arguably more difficult than A. So some parts of so-called applied math could be easier than some part of so-called pure maths, but the converse is true too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, there is an interesting and common question behind yours: What may explain your feeling?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible explanation is that the expectations of teachers are different in pure and applied maths (e.g. because more pure math majors will try to do research, while applied math majors have more diverse and arguably less selective professional objectives such as engineering, or because the teachers are different).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-01-18T21:22:52.773" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7251" PostTypeId="2" ParentId="7249" CreationDate="2015-01-18T21:46:37.720" Score="13" Body="&lt;p&gt;A common error is canceling something &quot;completely&quot; instead of leaving the &quot;trivial&quot; term $1$. What I mean is the following type of &lt;em&gt;incorrect&lt;/em&gt; simplifications: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{4x + 2}{2} =2x$$ or &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$4x^3 + 3x^2 + x = x(4x^2 + 3x).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A way to counter this can be to insist that the factor is first written out at each term, like:&#xA;$$\frac{4x + 2}{2} = \frac{2 \cdot 2x + 2 \cdot 1}{2} = 2x + 1$$&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="1147" LastEditDate="2015-01-19T19:40:23.853" LastActivityDate="2015-01-19T19:40:23.853" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7252" PostTypeId="2" ParentId="7249" CreationDate="2015-01-18T22:00:53.093" Score="12" Body="&lt;p&gt;&lt;strong&gt;the freshman mistake&lt;/strong&gt;  is what a colleague of mine used to call the error that every function is linear.  Like&#xA;$$&#xA;\sin(3x) = 3 \sin x&#xA;$$&#xA;another popular version&#xA;$$&#xA;\frac{1}{x+y} = \frac{1}{x} + \frac{1}{y}&#xA;$$&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2015-01-18T23:54:38.850" LastActivityDate="2015-01-18T23:54:38.850" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7253" PostTypeId="2" ParentId="7249" CreationDate="2015-01-19T02:53:42.527" Score="9" Body="&lt;p&gt;&lt;strong&gt;Some people omit zeros from numbers.&lt;/strong&gt;  For example, they treat all of the following as being equivalent:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;015&lt;/li&gt;&#xA;&lt;li&gt;15&lt;/li&gt;&#xA;&lt;li&gt;150&lt;/li&gt;&#xA;&lt;li&gt;105&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In many computer programming languages, all of these are different.  (Many languages default to base (5+5), but will use base (5+3) if there is a leading zero, or base (5+5+5+1) if there is a leading 0x.)  Most non-programmers expect the first two examples to mean the same thing, and the last two examples to mean different things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conflating tens and hundreds.&lt;/strong&gt; Some children, when learning to count to large numbers, try to do this...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;instead of:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Confusing partial derivatives with full derivatives.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Confusing demonstrating that something is true for a special case with proving that it is true for the general case.&lt;/strong&gt;  Of course, demonstrating that something is &lt;em&gt;false&lt;/em&gt; for a special case does disprove the general case of a claim -- unless special exceptions are carved out.  But it is not enough to prove that the general case is always false.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Thinking that units don't matter.&lt;/strong&gt;  If I travel 120 miles in 2 hours, what is my average speed?  1?  60?  &lt;em&gt;Um, no.&lt;/em&gt;  Those answers should be 1 mile per minute, or 60 miles per hour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mis-squaring units.&lt;/strong&gt;  How large is a square mile?  1 mi² = 5280 ft² ?  &lt;em&gt;Um, no.&lt;/em&gt;  1 mi² = 1 * (5280 ft)² = 1 * 5280 ft * 5280 ft = 5280 * 5280 ft² = 27,878,400 ft².&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Thinking that the calculator will tell them the correct number of significant figures.&lt;/strong&gt; What is 1,000,000 - 100,000? 899,999.99999999?  &lt;em&gt;Um, no.&lt;/em&gt;  The calculator had a rounding error.  The precise answer is 900,000; rounding to the correct number of significant figures will produce either 900,000 or 1,000,000.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Rounding off intermediate values, instead of final answers.&lt;/strong&gt;  Computers are especially prone to this problem, which is why there are entire books on &lt;em&gt;Numerical Methods&lt;/em&gt;.  Intermediate values should be absurdly precise.  Care needs to be exercised in interpreting them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Throwing away the remainder.&lt;/strong&gt;  What is 10 / 9?  1? 1.1? 1.11?  &lt;em&gt;Um, no.&lt;/em&gt;  It is 1 + 1/9, or 1.11… . That … is a number, and it is not zero.  It might not be worth calculating right now, but if we multiply it by 9, we will get something useful.  (This also makes it easier to understand why 10 = 9.99…; this … equals 9 * 0.01 / 9, which equals 0.01 .)&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-01-19T03:29:39.190" LastActivityDate="2015-01-19T03:29:39.190" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7254" PostTypeId="2" ParentId="7243" CreationDate="2015-01-19T03:41:31.213" Score="8" Body="&lt;p&gt;Teach the students to perform a sanity check at the end of every problem, and a &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1834/how-do-i-teach-algebra/1861#1861&quot;&gt;check-by-substitution&lt;/a&gt; when practical at the end the problem.  If either check fails, they can use a technique for finding errors, such as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Rory's binary search for the mistaken step.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Keeping a &quot;top 10 list&quot; of most common mistakes, such as some of the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;** (not) distributing a -1&lt;br&gt;&#xA;** multiplying by 2 instead of squaring&lt;br&gt;&#xA;** adding instead of subtracting (or &lt;em&gt;vice versa&lt;/em&gt;)&lt;br&gt;&#xA;** adding instead of multiplying (or &lt;em&gt;vice versa&lt;/em&gt;)&lt;br&gt;&#xA;** slipping a decimal&lt;br&gt;&#xA;** rounding errors&lt;br&gt;&#xA;** mixing up 2s and 5s, or 2s and 7s, or 3s and 5s, or 3s and 8s, or 0s and 8s.&lt;br&gt;&#xA;** forgetting to &quot;carry the 1&quot; in addition or subtraction problems.&lt;br&gt;&#xA;** transposing digits.&lt;br&gt;&#xA;** &quot;off by one&quot; errors when setting up a problem.&lt;br&gt;&#xA;** forgetting to change the direction of an inequality when multiplying by a negative number.&lt;br&gt;&#xA;** accidentally dividing by zero, or not splitting a problem into parts&lt;br&gt;&#xA;** mistakes when cross-multiplying ratios&lt;br&gt;&#xA;** mistakes when adding/rationalizing/simplifying fractions.&lt;br&gt;&#xA;** omitting units, or mis-converting units.&lt;br&gt;&#xA;** confusing sines and cosines&lt;br&gt;&#xA;** using the wrong formula for the area or volume of a shape.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Look for spots where any of the &quot;top 10 list&quot; operations were performed, and double-check them.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;Encourage the students to keep track of some of their more common mistakes.  If they notice a pattern, provide help.  For example, if they are mixing up similar-looking digits, first try teaching them how to make numbers more clearly.  (Drafting classes teach how to make very legible numbers.) If that fails, they might have dyslexia; there are programs that can help.&lt;/li&gt;&#xA;&lt;li&gt;When working the problem, make a note when a step is hard or confusing.  If the check-by-substitution did not work, double-check those steps.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-02-08T23:22:49.837" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7255" PostTypeId="2" ParentId="6031" CreationDate="2015-01-19T08:28:32.170" Score="14" Body="&lt;h2&gt;Quadratic Formula (deterministic---no guess and check about it)&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The QF yields that $-{\frac13}$ and $5$ are roots. So $$3x^2-14x-5=c\left(x+\frac13\right)(x-5)$$ Comparing leading coefficients, $c$ must be $3$: $$\begin{align}3x^2-14x-5&amp;amp;=3\left(x+\frac13\right)(x-5)\\&amp;amp;=(3x+1)(x-5)\end{align}$$ &lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Use Parabola Vertex Form (deterministic---no guess and check about it)&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The $x$-coordinate of the vertex of the parabola $y=3x^2-14x-5$ is $-{\frac{b}{2a}}=-{\frac{-14}{2\cdot3}}={\frac73}$. The $y$-coordinate is $3\left(\frac73\right)^2-14\left(\frac73\right)-5=\frac{49}{3}-\frac{2\cdot49}{3}-5=-{\frac{49}{3}}-\frac{15}{3}=-{\frac{64}{3}}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So $y=c\left(x-\frac73\right)^2-\frac{64}{3}$. Comparing leading coefficients, $c=3$, so $$\begin{align}&#xA;y&#xA;&amp;amp;=3\left(x-\frac73\right)^2-\frac{64}{3}\\&#xA;&amp;amp;=\frac{1}{3}\left(9\left(x-\frac73\right)^2-64\right)\\&#xA;&amp;amp;=\frac{1}{3}\left(3\left(x-\frac73\right)-8\right)\left(3\left(x-\frac73\right)+8\right)\\&#xA;&amp;amp;=\frac{1}{3}\left(3x-15\right)\left(3x+1\right)\\&#xA;&amp;amp;=\left(x-5\right)\left(3x+1\right)&#xA;\end{align}$$&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Complete the Square (deterministic---no guess and check about it)&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Starting with $3x^2-14x-5$, always multiply and divide by $4a$ to avoid fractions:&#xA;$$\begin{align}&#xA;&amp;amp;3x^2-14x-5\\&#xA;&amp;amp;=\frac{4\cdot3}{4\cdot3}\left(3x^2-14x-5\right)\\&#xA;&amp;amp;=\frac{1}{12}\left(36x^2-12\cdot14x-60\right)\\&#xA;&amp;amp;=\frac{1}{12}\left(\left(6x\right)^2-2(6x)(14)-60\right)\\&#xA;&amp;amp;=\frac{1}{12}\left(\left(6x\right)^2-2(6x)(14)+14^2-14^2-60\right)\\&#xA;&amp;amp;=\frac{1}{12}\left((6x-14)^2-196-60\right)\\&#xA;&amp;amp;=\frac{1}{12}\left((6x-14)^2-256\right)\\&#xA;&amp;amp;=\frac{1}{12}(6x-14-16)(6x-14+16)\\&#xA;&amp;amp;=\frac{1}{6\cdot2}(6x-30)(6x+2)\\&#xA;&amp;amp;=(x-5)(3x+1)\\&#xA;\end{align}$$ &lt;/p&gt;&#xA;&#xA;&lt;h2&gt;AC Method (involves integer factorization and a list of things to inspect)&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;$$3x^2-14x-5$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take $3\cdot(-5)=-15$. List pairs that multiply to $-15$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(-15,1),(-5,3),(-3,5),(-1,15)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We could have stopped at the first pair, because $-15+1=-14$, the middle coefficient. Use this to replace the $-14$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3x^2-15x+x-5$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Group two terms at a time and factor out the GCF:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3x(x-5)+1(x-5)$$&#xA;$$(3x+1)(x-5)$$&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Prime Factor what you can version 1 (involves integer factorization and a list of things to inspect)&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;If $3x^2-14x-5$ factors, then prime factoring $3$, it factors as &#xA;$$(3x+?)(x+??)$$&#xA;And $(?)(??)=-5$. There are only four possibilities. $(?,??)$ is one of $$(1,-5),(-1,5),(5,-1),(-5,1)$$&#xA;Multiplying out $(3x+?)(x+??)$ for each of the four cases reveals $3x^2-14x-5=(3x+1)(x-5)$.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Rational Root Theorem (involves integer factorization and a list of things to inspect)&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;If $3x^2-14x-5$ factors, there are rational roots. They must be of the form $\pm\frac{a}{b}$ where $a\mid5$ and $b\mid3$. The only options are $\pm5,\pm{\frac53},\pm1,\pm{\frac13}$. Check these eight inputs to $3x^2-14x-5$ and find that $-{\frac13}$ and $5$ are roots. So $$3x^2-14x-5=c(x+1/3)(x-5)$$ Comparing leading coefficients, $c$ must be $3$.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Prime Factor what you can version 2 (using Rational Root Theorem to speed up version 1)&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;If $3x^2-14x-5$ factors, then prime factoring $3$, it factors as &#xA;$$(3x+?)(x+??)$$&#xA;The latter factor reveals that if the thing factors at all, one of its roots is an integer. Considering the RRT, check if any of $\pm5,\pm1$ are roots, and discover that $5$ is. Conclude $$(3x+?)(x-5)$$ and then conclude $$(3x+1)(x-5)$$&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Graphing to improve efficiency ot Rational Root Theorem method&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Using the vertex formula again, locate the vertex at $\left(\frac73,-{\frac{64}{3}}\right)$. Since $a=3$, consider the sequence $\{3\cdot1,3\cdot3,3\cdot5,3\cdot7,\ldots\}$. Extend horizontally outward from the vertex by $1$ in each direction, move up $3$ and plot a point. Extend horizontally outward again by $1$, move up $9$ and plot a point. Continue until you've plotted points that cross over the $x$-axis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/kKyC1.png&quot; alt=&quot;step plot of parabola&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you have a rough idea where the roots are. Returning to the rational root theorem approach, you can eliminate many of the potential roots now from the initial list, speeding up that approach.&lt;/p&gt;&#xA;" OwnerUserId="767" LastEditorUserId="767" LastEditDate="2015-01-19T20:35:57.473" LastActivityDate="2015-01-19T20:35:57.473" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7256" PostTypeId="1" CreationDate="2015-01-19T12:51:02.267" Score="1" ViewCount="103" Body="&lt;p&gt;There are many fields in mathematics, in which one wants to optimize a process, for example finding the shortest way in a graph etc. &#xA;However I got curious and wondered, if there are also books or papers about the topic of how to maximize the effort or steps needed to reach a certain goal, for example an algorithm that works as slowly as possible but still ends after a finite number of steps. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One has to care about loopholes, for example one can easily extend the time one who wants to walk from A to B, if he just keeps moving forward and backwards, but never reaching his goal. I am sure, there are more things one has to worry, take care about to do serious mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know one author who worked on those kinds of problems and if there is actually a  good book for beginners to that topic etc.? (if not, if there is a book especially recommendable) My question isn't very specific yet, cause first I want to gain a better overview over that field/topic. &lt;/p&gt;&#xA;" OwnerUserId="4660" LastActivityDate="2015-01-19T12:51:02.267" Title="Mathematics after Rube Goldberg (recommendation) - Question for orientation" Tags="|undergraduate-education|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7257" PostTypeId="2" ParentId="7243" CreationDate="2015-01-19T16:29:48.333" Score="2" Body="&lt;p&gt;First, let me endorse Jared's point that checking calculations and debugging computer programs have much in common. Good programmers build checks into their code &lt;em&gt;before&lt;/em&gt; it evidences a bug.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, one general technique, which only works in circumstances where at least one variable is present, is: &lt;strong&gt;&lt;em&gt;look at extreme values&lt;/em&gt;&lt;/strong&gt; of the variables. An example from a program I was just writing this morning: I knew that as $h \to \infty$, a calculated angle $\theta \to 90^\circ$. So I placed that test into the code to halt with an error message if it was not (approximately) true.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-01-19T16:29:48.333" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7259" PostTypeId="2" ParentId="7245" CreationDate="2015-01-20T01:20:37.013" Score="5" Body="&lt;p&gt;No links to research, but too long for a comment. This is an alternative approach, which doesn't have the word &quot;proof&quot; in it, but will create proofs as part of it. Proofs can sometimes (not always) be just a symbolic game, but abstraction is motivated, tested and has wide application.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;tldr;&lt;/strong&gt; they are already proving things every day, we need to teach the process of abstraction rather than just proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proofs that will be accessible to your students are just the algebra version of abstraction. Abstraction of patterns is applicable across high school curricula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First you need to recognise the proofs that your students do in every lesson. Every line of algebra deduction is a proof that the original expression or equation implies the new expression. In geometry, when they draw 2 identical circles which intersect, they have proven that those points of intersection are equidistant from the centres of those particular circles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm guessing what you are looking for are proofs of more general results. You are looking for abstraction. Once you acknowledge the proofs they are doing you can teach how to generalise them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, students can &quot;prove&quot; several special cases of a more general proof that you want to encourage, using simple algebraic deduction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(x+2)(x-2) = x^2 + 2x - 2x -4 = x^2 -4$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(x+3)(x-3) = x^2 + 3x - 3x -9 = x^2 -9$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They have proved 2 things already! But it looks like there is a pattern in their proofs. Maybe they can make a more general proof? They can introduce a variable where a constant was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(x+n)(x-n) = x^2 + nx - nx - n^2 = x^2 - n^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This equation would be the start and end for a traditional proof, but using abstraction it is just the mid-point. They have now proven several fairly concrete equations and one more general equation in a way that is motivated and easily accessible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We now need to test the applicability of the abstraction and maybe extend it. For instance, at this point they probably see this only as a proof for speeding up expanding brackets (students often see equations as implications or rewritings rather than bidirectional equivalence). They can abstract this further by flipping the equation around to prove that certain quadratics can be factorised.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x^2 - n^2 = (x+n)(x-n)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They can test it for $n^2 = 25, 49$. Irrational numbers are extra for experts, $n^2 = 10$. See how far they can take the general proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They have now proven how to factorise a monic quadratic of a certain form. As $x$ is so often used as simply an independent variable of an expression, they probably don't realise that it is generalisable further. So next they can experiment by replacing $x$ by another expression. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They now have:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(u+v)(u-v) = u^2 -v^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After all of this they have completed possibly pages of proofs, some very specific, some more general. They have further investigated the meaning of the proof to actually understand how it can be used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally don't see much point in writing down simple proofs, but the process of abstraction and application of these abstractions is key in mathematics, algorithms, and across most school subjects. The above described process will definitely deepen understanding and lead to higher order thinking. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-01-20T01:20:37.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7260" PostTypeId="2" ParentId="1002" CreationDate="2015-01-20T20:49:35.830" Score="0" Body="&lt;p&gt;A real world example of how a logic gate can be used:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) most of us have a garage light that is controlled by a switch. However, it may also automatically turn on when it senses motion during nighttime. This example has 3 inputs &#xA;    A = is the switch on?&#xA;    B = is it daytime?&#xA;    C = is there motion nearby?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We want the only output (garage light) to turn ON when:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;The switch is ON or it is nighttime and there is motion nearby&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Logic Statement = A + !BC&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody else have a similar example to this?&lt;/p&gt;&#xA;" OwnerUserId="4662" LastActivityDate="2015-01-20T20:49:35.830" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7261" PostTypeId="1" AcceptedAnswerId="7263" CreationDate="2015-01-21T01:29:36.470" Score="10" ViewCount="688" Body="&lt;p&gt;How would my Colleagues here on &lt;a href=&quot;http://matheducators.stackexchange.com&quot;&gt;Math Educators&lt;/a&gt; differentiate between mathematical skills and understanding of mathematical concepts? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a community college instructor and high school instructor looking for examples from my colleagues. I have researched several examples (one here: &lt;a href=&quot;http://aft.org//sites/default/files/periodicals/wu.pdf&quot;&gt;http://aft.org//sites/default/files/periodicals/wu.pdf&lt;/a&gt;) but I'm wondering if my colleagues here had any more.&lt;/p&gt;&#xA;" OwnerUserId="2198" LastEditorUserId="4872" LastEditDate="2015-06-14T10:35:33.853" LastActivityDate="2015-06-14T11:17:58.277" Title="How to differentiate between mathematical skills and understanding of mathematical concepts?" Tags="|secondary-education|mathematical-pedagogy|concept-motivation|transferrable-skills|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7262" PostTypeId="1" CreationDate="2015-01-21T02:54:31.073" Score="3" ViewCount="83" Body="&lt;p&gt;this is something I have been wondering about for a while now.  In my school district (urban setting in Pennsylvania) student input amounts to exactly 0% of the final assessment for observed teachers.  however, student performance, in the form of standardized test scores, amount a large percentage of a teachers final performance rating.  Do any other states/countries take student input into consideration when evaluating teachers? if so, how does this look?  I know from the peers I have spoken with that none of them have encountered student input being taken into consideration in the various rating systems at their districts.  is there any research-based reason for this?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-01-21T02:54:31.073" Title="Student input in secondary teacher evaluations" Tags="|secondary-education|reference-request|teacher-evaluations|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7263" PostTypeId="2" ParentId="7261" CreationDate="2015-01-21T04:04:06.710" Score="7" Body="&lt;p&gt;I agree with Jasper that your first paragraph is prompting for an essay, but is your second paragraph asking for examples, meaning references that discuss the difference?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are three:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Richard Skemp's article on &lt;a href=&quot;http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdf&quot;&gt;Relational Understanding and Instrumental Understanding&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The five strands of mathematical proficiency described in &lt;a href=&quot;http://www.nap.edu/openbook.php?record_id=9822&amp;amp;page=116&quot;&gt;Adding It Up&lt;/a&gt;. The first two strands are conceptual understanding and procedural fluency.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Phil Daro's 10-minute video on &lt;a href=&quot;http://vimeo.com/59262342&quot;&gt;Four Levels of Learning&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3578" LastEditorUserId="3578" LastEditDate="2015-01-21T08:14:11.957" LastActivityDate="2015-01-21T08:14:11.957" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7264" PostTypeId="2" ParentId="5884" CreationDate="2015-01-21T08:28:34.340" Score="4" Body="&lt;p&gt;I taught a class like yours when I was a graduate student over 30 years ago. I remember how I taught some of it, but not exactly how I taught ANOVA. I was happy with the class, and the students were happy, too. They brought refreshments on the last day of class, and gave me a card with kind comments on it. They were happy because most were afraid of taking a statistics class, but were comfortable in class and understood more than they thought they would. Since it was a successful class, I’ll share how I might have taught the lesson on ANOVA.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At that time, for a college class, I did mostly lecture. But I did encourage questions a lot, and if there were no questions I would sometimes say, “If you have no questions, then I’ll ask you questions,” and get someone to explain an important point. This is one way that I highlighted important points. When lecturing I might ask a question or pose a hypothetical situation for students to consider before explaining, because we process information better when it is used to answer a specific question or resolve a specific problem. If I asked a question when lecturing, we might or might not have had some whole class discussions, but less than I would have now. (The class included about 30 students.) In the spirit of collaboration, others can suggest revisions to improve this lesson and to create different versions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to get students to understand the concepts, not just to do calculations. A class in ANOVA would build on understanding of concepts taught earlier, primarily:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;em&gt;mean&lt;/em&gt; as the middle of a distribution and the best estimate of a set of scores based on the “least squares” criterion. It is computed as the sum of scores divided by the number of scores.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;em&gt;deviation score&lt;/em&gt; as the distance a score is above or below the mean. It is computed as &lt;em&gt;score&lt;/em&gt; minus &lt;em&gt;mean&lt;/em&gt; (x minus x-bar). This is a simple but key concept that shows how the mean is in the middle of interval data, since the sum of the positive deviations equals the sum of the negative deviations. It is the basis for the least squares criterion, where error is measured as the sum of squared deviations from the estimate. The error is lower when computed from the mean than from any other point. The deviation score is also the basis for computing variance.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Variance&lt;/em&gt; as a measure of dispersion, the extent that scores are spread away from the middle of the distribution. It is also a measure of error if the mean is taken as the estimate for a group. It is computed as the mean of the squared deviations (the “mean-square”): the sum of squared deviations divided by the number of scores, or by the degrees of freedom if estimating the population variance from sample data.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The logic of &lt;em&gt;hypothesis testing&lt;/em&gt; including especially the concepts of the null hypothesis, the experimental hypothesis, alternative hypotheses which are eliminated by a good research design, and alpha error. I remember using an analogy of trying to disprove someone’s claim that he was psychic. Everyone could understand that process, so mapping the terms of hypothesis testing to aspects of that process made the terms easy to understand.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Before doing any computation&lt;/em&gt;, I want to highlight several concepts: the mean of each group as the value that represents the score for that group, the variance within a group as how good an estimate its mean is as a group score, and the variance of the group means (between-group variance) as a measure of how spread apart the groups are from the grand mean and from each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I want the students to think about, if they were researchers doing a study with three groups, each getting a different treatment, and they expected each group to be significantly different from each other, would they want the within group variances to be larger or smaller, why? Would they want the between group variance to be larger or smaller, why? I would expect they would want the within groups variances to be smaller because a tighter distribution for each group results in less overlap between group scores. The larger the within variance, the less accurate is the mean as a score representing its group, and a larger separation between means is needed to have confidence in group differences. They would want the between group variance to be larger because that happens when the means of the groups are spread farther apart. (I want the students to think about it, but this is straight forward so I wouldn’t have spent much time on discussion. I would have quickly explained or summed up the discussion.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next I want the students to think about the within group variances relative to the between group variance. I ask the students to visualize the within group variances staying the same, but the between group variance starting small then getting larger and larger. I ask the students to describe what they see. I expect that they would say the groups of scores move apart, being greatly overlapped at first then separating into more distinct groups. At some point the groups become significantly different. (I would spend a little more time on discussion of the relative sizes of the variances, on this and the next questions before lecturing on the calculations, because this is conceptually important and more complex than the first set of questions.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there is time, I would ask the class to visualize the reverse situation--the within group variance staying the same this time, meaning the means of the groups stay the same, but the within group variances starting out very large then becoming smaller and smaller. What do they see? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before doing computations, I would ask the class, “If you did a study and the means of the three groups were farther apart than you expected, with two treatment groups doing better than the control group to a surprising extent, and one treatment group doing much better than the other, you would expect the statistical test to be significant. Thinking in terms of the variances, is there a situation where the statistical test is NOT significant despite the large differences in group means?” I would hope that, after the visualizations that we did of the variances changing relative to one another, the students would say that if the within group variances were extremely large, then seemingly large differences in group means may be relatively small, and there would be much overlap in group scores. For group scores to be significantly separated, the between group variance, and therefore the differences among group means, should be large relative to the within group variances, which measures the accuracy of the means as the groups’ scores. This would be a logical segue to the F-statistic, which I would explain to the class is what is used in ANOVA.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The questions and limited discussions get the students to understand the issues. I would have then just lectured on partitioning the variance and doing all the calculations, but relating it to the concepts and the discussion. I think the calculations make sense and are easier to remember when the issues are discussed first conceptually, without any numbers attached. One point in particular that would need emphasis is using degrees of freedom instead of n-size to compute a sample variance as an estimate of population variance. Otherwise, the means and sum of square calculations are similar to what they had already done in earlier parts of the class. The F-statistic calculation should make sense after discussing the importance of the relationship between the between-group and within-group variances. Calculations were shown in the textbook, and I don’t remember doing any handouts, but it would make sense to have a handout with all the calculations done for a small set of data, and a table for the F-statistic with the degrees of freedom and critical values of F highlighted for the particular data contained in the handout.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I think it is important to review effect size, n-size and the power of the test, and the difference between statistical significance and practical significance—all of which might have been covered with hypothesis testing in the introduction to inferential statistics. I don’t know that I actually did a review of these points in that class I taught (probably not), but I think it is more important for students to know these factors that affect the results of a study than it is to know how to &lt;strong&gt;compute&lt;/strong&gt; the F-statistic from raw data, since they would use a computer to do the calculations if they actually analyzed data. These factors that affect the result of a study are important to know if they actually did a study using ANOVA, or even if they only need this knowledge to evaluate studies in their field. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The class always had homework, so they would have had to do computations from raw data. I had a system where they turned in their homework at the department office and I reviewed and wrote comments on each one, and returned them by the next class. Classes were three times a week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember that I gave an optional extra credit assignment to copy the Results section of a journal article in their field and to discuss the article in terms of what we covered in class, to demonstrate their understanding of the course material. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I taught the class now, I would incorporate what technology we had available in the class. I would also try to take a few minutes for small group work to get the students to think more deeply about real world issues related to statistical testing, especially for the social science graduate students in the class. Because of time constraints, they might first discuss the question in class, but then write up their responses as homework or for optional extra credit. For example, I would refer back to the last hypothetical situation regarding large group differences that were not statistically significant. “If you did a study and your group differences were large but not statistically significant, and you were confident in your theory and thought the study should be redone, what changes would you consider to improve the study? Remember what we discussed about why large group differences might not be statistically significant.” They might research it online and that’s fine. My goal is to get them to think more deeply about the issues, not necessarily come up with solutions all on their own.&lt;/p&gt;&#xA;" OwnerUserId="3578" LastActivityDate="2015-01-21T08:28:34.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7265" PostTypeId="2" ParentId="7247" CreationDate="2015-01-21T10:49:38.267" Score="7" Body="&lt;p&gt;I have taught basic Group Theory (pure maths) to primary school kids of varied ability. It is a beautiful self-contained subject that you can play with and easily prove some interesting theorems (ie. properties of very small groups which even kids can enumerate). It is much easier than multiplying 2-digit numbers, let alone 1st year calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difficulty that you have in an area of maths depends on several things: the expectations of the lecturer, the style of presentation, and the type of leaner that you are. Given the right expectations and style of presentation, pure maths is not inherently hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found pure maths easy because there is so little to learn. Get the basic concepts and a few fundamental theorems and everything else flows. The downside is that if you don't &quot;get&quot; the key concepts, the whole thing is near impossible. Pure maths just might not have clicked for you yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it came to calculus and all the tricks to allow you to calculate the exact integral of all the special cases - I hated it! Lists of trig identities!! I thought I may as well be studying organic chemistry with everything I was supposed to memorise. You might be fine with all of that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also comes down to motivation. For a Platonist that enjoys exploring the world of pure forms, pure maths is as good as it gets, applied maths is dreary by comparison. Other people get their motivation from the way mathematics models the real world, and applied maths is awe inspiring from that perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mainstream culture in the English speaking world doesn't really care for either endeavour, but is especially antagonistic to a Platonist perspective. Even among those of us who like maths, the &quot;pure maths is hard&quot; sentiment might be an acceptable refinement of &quot;maths is hard&quot;, but in my perspective it is just as unjustified. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is better to stick with the perspective that some of us are good at some things, and not at others, but sometimes we get good at the things we previously had trouble with.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-01-21T10:49:38.267" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7266" PostTypeId="2" ParentId="5884" CreationDate="2015-01-21T11:23:00.893" Score="3" Body="&lt;p&gt;I don't have more experience than you, but hopefully you might find an alternative experience helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where I live, high schools teach this with resampling/bootstrapping methods. It allows students to get a feel for what hypothesis testing really means before using the idealised distributions. When I taught this, I was surprised at how much more immediate and comprehensible it was than using the CLT and the &quot;tedious calculations&quot; you refer to. Even fairly complex hypothesis testing of was direct and comprehensible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like all Montecarlo methods, you require a huge number of samples to get an accurate estimation of the hypothesis distribution, but that is what computers are for. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Resampling methods are easy to understand, and all the calculation is done automatically by appropriate software. It may only take a few tutorial sessions for your students to get a  good understanding of how hypothesis testing works in a variety of situations. After they understand hypothesis testing, the only hard thing is to motivate a shift to the less accurate traditional methods that are probably mandated by your syllabus!&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-01-21T11:23:00.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7268" PostTypeId="1" AcceptedAnswerId="7269" CreationDate="2015-01-21T18:30:27.493" Score="14" ViewCount="1054" Body="&lt;p&gt;I'm a private tutor working with a 7th grader who is struggling with solving equations. Given a simple equation, he is able to solve it using a formulaic procedure, but it is very obvious that he has no idea what the solution really means. Hence, if he gets a problem that's slightly different from ones he's solved before, he's completely lost. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Working with him yesterday, I realized that he doesn't understand what a variable is, or what he's doing when he's solving an equation -- he's just following the steps his teacher told him for that specific problem.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm trying to think of a way to visually demonstrate what's happening when he's solving an equation -- something visual that he can see. Kind of an algebraic equivalent to putting four coins on the table and adding one more to show 4 + 1 = 5. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any ideas? Thanks!&#xA;-Ian &lt;/p&gt;&#xA;" OwnerUserId="4664" LastActivityDate="2015-02-02T01:37:50.390" Title="How to visually demonstrate basic algebra" Tags="|algebra|tutoring|struggling-students|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7269" PostTypeId="2" ParentId="7268" CreationDate="2015-01-21T20:47:22.417" Score="9" Body="&lt;p&gt;I don't know how feasible this is or what kind of resources you have, but try finding a balance scale, some standard weights, and a collection of identical objects. You can set up the scale to represent a problem and ask the student to determine the unknown weight. The only rule is that the scale can never become unbalanced in the process (i.e. what they do to one side they must do to the other).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, to demonstrate solving $3x+6=5x+2$ start with 3 identical objects and 6 unit weights on one side and 5 of the same identical objects and 2 unit weights on the other. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the solutions is: Remove two units from each side. Remove 3 objects from each side. Split each side into two equal groups and remove a group from each side. This should leave one object on the right and 2 unit weights on the left.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, to get the scale to balance out from the beginning, you, the teacher, will have to know the weight of the objects. And you'll have to watch out for fractions and negatives.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-01-21T20:47:22.417" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7270" PostTypeId="2" ParentId="7268" CreationDate="2015-01-21T22:59:43.940" Score="4" Body="&lt;p&gt;You may find this &lt;a href=&quot;http://myplace.frontier.com/~paulgriffith2/mathmodels/math_models.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics Models I webpage&lt;/a&gt; by Paul Griffith useful, although it targets a lower educational level:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/z5eZi.gif&quot; alt=&quot;Factors&quot;&gt;&#xA;&lt;hr /&gt;&#xA;Another source is &lt;a href=&quot;http://www.soesd.k12.or.us/Page.asp?NavID=1264&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Oregon web page with many links&lt;/a&gt;, including:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/BabtG.png&quot; alt=&quot;AlgVis&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;em&gt;Caveat&lt;/em&gt;: I have not used any of these materials in the classroom myself.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-01-21T22:59:43.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7271" PostTypeId="1" AcceptedAnswerId="7284" CreationDate="2015-01-22T13:48:44.053" Score="5" ViewCount="285" Body="&lt;p&gt;&lt;strong&gt;The Question:&lt;/strong&gt; Does combining two or more topics into one question on a mathematics examination when the topics have not notably been combined in the course lecture, homework, or other assessments provide a good measure for any of the following points?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ability to combine knowledge quickly in a new situation.&lt;/li&gt;&#xA;&lt;li&gt;Demonstrate knowledge of all topics incorporated.&lt;/li&gt;&#xA;&lt;li&gt;Passing correctly demonstrates working knowledge of the topics more than independent questions on each topic.&lt;/li&gt;&#xA;&lt;li&gt;Failure correctly shows that the student might not posses the skills to succeed at later courses or does not have a working knowledge of the topics.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Any educational research included would be appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Notes:&lt;/strong&gt; The exam is for undergraduate calculus students (first or second year university) who are not all assumed to be math or science students to provide a summative assessment of their knowledge thus far in the course (see &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1899/exam-philosophy/2119#2119&quot;&gt;ME.SE Exam Philosophy&lt;/a&gt; answer by Benjamin Dickman). We may assume that this is not a multiple choice exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A Not Particularly Hard Example:&lt;/strong&gt;&#xA;Topics: Separable Differential Equations, Conic Sections&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sketch several solution curves to $\dfrac{dy}{dx}= -\dfrac{x}{2y}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Removed:&lt;/strong&gt; &lt;em&gt;My department uses multiple choice for machine grading purposes (mandatory for this course), so particular answers with this in mind would be grateful but not fully necessary.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="201" LastEditorUserId="201" LastEditDate="2015-01-23T01:30:35.220" LastActivityDate="2015-01-24T17:27:08.227" Title="Exam Writing: Combining Topics on Exams in New Ways" Tags="|undergraduate-education|mathematical-pedagogy|exams|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="7273" PostTypeId="2" ParentId="7268" CreationDate="2015-01-22T22:09:41.013" Score="4" Body="&lt;p&gt;I would backtrack and revisit substition, solving equations should fall out as a bi-product or the student hasn't grasped that topic fully. With regards to solving equations I believe choosing the numbers to provide a rich example is extremely important. Consider for example  $$\frac{x}{4}=12$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is far more likely to be answered incorrectly  than$$\frac{x}{3}=7$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why? Multiplication is the only 'sensible' option for the student in the second question while the opposite is true in the first example. Force them to confront such addictive number bonds head on.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-01-22T22:09:41.013" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7274" PostTypeId="1" AcceptedAnswerId="7276" CreationDate="2015-01-23T15:12:44.530" Score="33" ViewCount="10633" Body="&lt;p&gt;I am on a team which is writing a set of lecture notes for differential calculus.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am using a format of &quot;Break ground&quot; which poses a problem,  &quot;Dig in&quot; which develops the tools to solve the problem, and &quot;Reinforce&quot; which practices using the tools we just developed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While using an abstract mathematical problem for the &quot;Break ground&quot; section is fine, some people on my team would rather that this problem be a &quot;real application&quot; whenever possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have been trying to find such an example for l'Hôpital's rule for a while, but we have run into the following roadblocks:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Simple l'Hôpital's rule problems (which require only one differentiation) can seemingly all be solved by appealing to the definition of the derivative.  So it is only when we apply l'Hôpital's rule twice that the method seems &quot;necessary&quot;. However, such a problem seems too complicated for a &quot;first brush&quot; with l'Hôpital.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We have two &quot;real world&quot; examples of l'Hopital in action.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;a. One in deriving continuous interest as a limit of discretely compounded interest.  This is, perhaps, better done using the definition of the exponential function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b. The other involves taking the limit of the velocity function of a falling object with air resistance, as air resistance goes to $0$.  This is a great example, but there are at least two essential variables (time after the fall $t$, and the coefficient of air resistance $k$).  The presence of more than one variable could be confusing (especially when you are thinking of velocity as a function of time, but then taking the limit with respect to $k$).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Does anyone have an example of l'Hopital's rule for which it is both necessary (or at least convenient),  relatively straight forward , and is a &quot;real world application&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If no one can come up with something meeting all 3 requirements, I will probably settle for using the &quot;abstract&quot; problem&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\lim_{x \to 1} \frac{x^4-2x+1}{4x^4-15x+11}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and then motivate the idea of the rule by &quot;zooming in&quot; on the point $(1,0)$ (&lt;a href=&quot;https://www.desmos.com/calculator/klphvr1rkx&quot; rel=&quot;noreferrer&quot;&gt;play with this graph&lt;/a&gt;), and welcoming the students to try to approximate the limit by the ratio of the tangent line approximations at $x=1$.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="123" LastEditDate="2015-12-23T11:44:42.757" LastActivityDate="2018-12-11T09:00:13.853" Title="Simple &quot;real world&quot; l'Hôpital's rule problem?" Tags="|calculus|examples|" AnswerCount="10" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="7275" PostTypeId="2" ParentId="7274" CreationDate="2015-01-23T15:55:46.457" Score="9" Body="&lt;p&gt;This is a nice question, and I look forward to seeing people's answers. You suggest that if we want examples that can't be done by the definition of the derivative, we can use ones that require second derivatives. An additional possibility is a limit at infinity, although that might or might not work for your order of topics. Here's a simple real-world example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have some money, and two choices of what to invest it in. A share in Acme, Inc., costs \$7, and returns a dividend of \$1 per year. A share of Glutco costs \$30 and gives a dividend of \$2 per year. If we want to compare the long-term value of the two investments, a natural way to do it is with the limit&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \lim_{t\rightarrow\infty}\frac{-7+t}{-30+2t} \quad .$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The top represents the net return on Acme, the bottom Glutco. If this limit is greater than 1, then Acme is the better long-term investment. The constant terms are irrelevant, and the competition between the top and bottom is determined by which one &lt;em&gt;grows&lt;/em&gt; faster.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2015-01-23T18:26:16.747" LastActivityDate="2015-01-23T18:26:16.747" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7276" PostTypeId="2" ParentId="7274" CreationDate="2015-01-23T18:55:24.480" Score="21" Body="&lt;p&gt;Here's a possible problem: The Rayleigh–Jeans Law for black body radiation at a wavelength $\lambda$ was given by $$B_{RJ}(\lambda) = \frac{K}{\lambda^4}$$ where $K$ is a constant (depending on temperature, speed of light, Boltzmann's constant, but that's not important). The problem is this clearly gives divergent energy output as wavelength approaches zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Plank's Law (in light of quantum mechanics) instead gives $$B_P(\lambda) = \frac{k_1}{\lambda^5}\frac{1}{e^{k_2/\lambda}-1}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for constants $k_1$, $k_2$. What happens to Plank's Law as wavelength approaches $0$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: This is also one of those great problems you can come back to with Taylor Series and show that for large $\lambda$ (and more details about the constants above) the two agree to a first-order approximation.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-01-23T18:55:24.480" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7277" PostTypeId="2" ParentId="7274" CreationDate="2015-01-23T19:46:10.137" Score="3" Body="&lt;p&gt;Suppose a company has two new types of sweet to test out.  The company gets two jars of the sweets and a lot of children.  They allow the children to eat the sweets and frequently weigh the jars to see which sweet is most popular.  The company will sell packets of sweets according to the ratio the children eat the sweets in.  As time goes on the ratio of the weights will go towards $\frac{0}{0}$ and can not be computed.  A solution is to compare the rate of change of each weight between measurements.&lt;/p&gt;&#xA;" OwnerUserId="4672" LastEditorUserId="4672" LastEditDate="2015-01-23T19:52:06.307" LastActivityDate="2015-01-23T19:52:06.307" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7278" PostTypeId="2" ParentId="7274" CreationDate="2015-01-23T20:07:46.223" Score="5" Body="&lt;p&gt;I have a colleague whose hobby is to always figure out how to a evaluate any given limit without L'Hospital's rule. Every week in the lunch room, there is yet another example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think a very good application of the rule is to $\lim_{x\to 0}\frac{\sin(x)}{x}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I frequently teach the calculus sequence at the university level. And I am fortunate to have a daughter in high school who took AP Calculus AB as a junior and is now taking BC as a senior. On occasion I have mentioned L'Hospital to her. It has been interesting to gauge her reaction to the rule over time. For awhile a fake quotient rule took hold: $$\left(\frac{f(x)}{g(x)}\right)'=\frac{f'(x)}{g'(x)}. $$ And she would resist paying attention to the conceptual derivation that involves linearization of the numerator and denominator. But as she becomes more comfortable with all the concepts, she appreciates this viewpoint and is even becoming a clone of my senior colleague. She often has the patience to understand how to evaluate limits without the rule, but at the same time feels confident using it when appropriate. And now she even appreciates the Taylor polynomial perspective. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I think I have learned by observation is that the L'Hospital rule should be viewed as a &quot;developmental&quot; concept that threads through Calculus I and II, and the instructor (or parent!) needs to be patient. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-01-23T20:07:46.223" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7279" PostTypeId="2" ParentId="7274" CreationDate="2015-01-24T03:35:52.363" Score="12" Body="&lt;p&gt;In thermodynamics and information theory the function $-x\log x$ for $0 \le x \le  1$ occurs in the definition of entropy. Just Google &quot;entropy&quot; and see. But that formula is not really defined at $x = 0$. It is natural to declare its value at $x=0$ to be $\lim_{x\rightarrow 0^+} -x\log x$. A graph makes it very plausible that the limit is $0$. This can be justified with L'Hospital's rule once, and that one use of the rule does not look like the definition of the derivative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you let $y=1/x$ then that limit is the same as $\lim_{y\rightarrow \infty} \log(y)/y$, which is obviously $0$ without using L'Hospital's rule if you understand how slowly logarithms grow, although such understanding is normally gained by L'Hospital's rule in the first place, e.g., to show $\lim_{y\rightarrow \infty} \log(y)/y^c = 0$ (or equivalently, $\lim_{x \rightarrow 0^+} x^c\log x = 0$) for any $c&amp;gt;0$ by L'Hospital.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a different application, it's important that exponential functions (with base greater than 1) ultimately always grow faster than power functions. That is, if $a &amp;gt; 1$ and $n$ is a positive integer then $\lim_{x \rightarrow \infty} a^x/x^n = \infty$ or equivalently $\lim_{x \rightarrow \infty} x^n/a^x = 0$. This can be shown by L'Hospital's rule $n$ times (or just one time with induction). The special case $a = e$ is probably safer to focus on than the general case, and shows up in Euler's beautiful integral formula $n! = \int_0^\infty x^ne^{-x}\,dx$; it's good to know that the integrand decays to $0$ for large $x$.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2016-11-27T17:46:45.083" LastActivityDate="2016-11-27T17:46:45.083" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7280" PostTypeId="2" ParentId="7274" CreationDate="2015-01-24T05:35:33.840" Score="11" Body="&lt;p&gt;I personally don't see any reason to teach L'Hopital's rule at all until after understanding how to manipulate asymptotic expansions. Even Taylor series should come after asymptotic series as a special nice case. The reason is that asymptotic expansions can do everything that L'Hopital can do and more, including handling non-differentiable functions, and is more intuitive.&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\def\lfrac#1#2{{\large\frac{#1}{#2}}}&#xA;\def\wi{\subseteq}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;span class=&quot;math-container&quot;&gt;$\lim_{ d \to 0 } \lfrac{f(x+d)-2f(x)+f(x-d)}{d^2} = f''(x)$&lt;/span&gt; for any function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x \in \operatorname{Dom}(f)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is twice-differentiable at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. This (and its higher order generalizations) can be used to numerically estimate the derivatives of a function given its values. It may seem that this is perfect to illustrate the power of L'Hopital's rule, but in fact it illustrates how asymptotic expansions actually give more information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, let's see how L'Hopital's rule can be used:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$\lim_{ d \to 0 } \lfrac{f(x+d)-2f(x)+f(x-d)}{d^2}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$= \lim_{ d \to 0 } \lfrac{f'(x+d)-f'(x-d)}{2d}$&lt;/span&gt; [by L'Hopital because &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differentiable in a neighbourhood of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$= \frac{1}{2} \lim_{ d \to 0 } \lfrac{f'(x+d)-f'(x)}{d} + \frac{1}{2} \lim_{ d \to 0 } \lfrac{f'(x-d)-f'(x)}{-d}$&lt;/span&gt; [because these two limits exist]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$= \frac{1}{2} \lim_{ d \to 0 } \lfrac{f'(x+d)-f'(x)}{d} + \frac{1}{2} \lim_{ d \to 0 } \lfrac{f'(x+d)-f'(x)}{d}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$= f''(x)$&lt;/span&gt; [by definition of derivative of &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we get the result, but actually much more can be said about the result if we look at the asymptotic expansion of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at the given &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; If &lt;span class=&quot;math-container&quot;&gt;$f(x+d) \in f(x) + a d + b \frac{d^2}{2} + o(d^2)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$d \to 0$&lt;/span&gt; for some constants &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$\lfrac{f(x+d)-2f(x)+f(x-d)}{d^2} \in b + o(1)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$d \to 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is strictly stronger than the earlier result, because some &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; are not even once differentiable in any neighbourhood around &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; but still satisfy the asymptotic condition. For example if &lt;span class=&quot;math-container&quot;&gt;$f(t) = t+\lfrac{1}{\lfloor \frac1{t^2} \rfloor}$&lt;/span&gt; for any real &lt;span class=&quot;math-container&quot;&gt;$t \ne 0$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$f(0) = 0$&lt;/span&gt;, then as &lt;span class=&quot;math-container&quot;&gt;$t \to 0$&lt;/span&gt; we can easily obtain &lt;span class=&quot;math-container&quot;&gt;$f(t) \in t+\lfrac{1}{\frac1{t^2}+O(1)}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\wi t+\lfrac{t^2}{1+O(t^2)}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\wi t+t^2(1+O(t^2))$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\wi t+t^2+o(t^2)$&lt;/span&gt; and so &lt;span class=&quot;math-container&quot;&gt;$\lfrac{f(t)+f(-t)}{t^2} \in 1+o(1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now even if we are not interested in &quot;not nice&quot; functions, we still can say even more than L'Hopital's rule, specifically for &quot;even nicer&quot; thrice differentiable functions. It can be proven (one form of Taylor's theorem) that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; For any function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x \in \operatorname{Dom}(f)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; times differentiable at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$f(x+d) \in \sum_{k=0}^n f^{(k)}(x) \frac{d^k}{k!} + o(d^n)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$d \to 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we immediately get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; For any function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x \in \operatorname{Dom}(f)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is thrice differentiable at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$\lfrac{f(x+d)-2f(x)+f(x-d)}{d^2} \in f''(x) + o(d)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$d \to 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As well as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; For any function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x \in \operatorname{Dom}(f)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; times differentiable at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$\lfrac{f(x+d)-2f(x)+f(x-d)}{d^2} \in f''(x) + f^{(4)}(x) \frac{d^2}{12} + o(d^3)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$d \to 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice further that the approximation is exact if &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a cubic polynomial. All these precise results cannot be obtained by L'Hopital. Also don't forget that a direct use of L'Hopital will fail to simplify all but the simplest limits, and so typically some clever manipulation is necessary. The problem is how to derive those manipulations! One way is, unsurprisingly, by analyzing the asymptotic expansions...&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="1550" LastEditDate="2018-12-11T09:00:13.853" LastActivityDate="2018-12-11T09:00:13.853" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7281" PostTypeId="1" AcceptedAnswerId="7295" CreationDate="2015-01-24T05:54:16.490" Score="19" ViewCount="36597" Body="&lt;p&gt;I found it difficult to explain the difference between the fraction a / b and the ratio a : b. This subject is for pupils of grade 5. So is there a real difference between them and how to explain the difference in simple way ?!&lt;/p&gt;&#xA;" OwnerUserId="3306" LastEditorUserId="14024" LastEditDate="2020-11-17T00:41:17.103" LastActivityDate="2022-04-24T08:17:48.380" Title="How to explain the difference between the fraction a / b and the ratio a : b?" Tags="|primary-education|fractions|" AnswerCount="16" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="7282" PostTypeId="2" ParentId="7281" CreationDate="2015-01-24T13:08:39.993" Score="8" Body="&lt;p&gt;I always get students to colour in dots.  1 red and 2 blue. Here the ratio is 1:2 red to blue.  I ask, &quot;What fraction are red?&quot; Hopefully someone says $\frac{1}{2}$ and we can discuss the misconception.  In terms of how to relate $\frac{1}{2}$ to the ratio 1:2 I do it after establishing the equivalence of ratio by scaling up both sides. I want to make green from blue and yellow in the ratio 1:2. I have 2 tins of blue how many yellow? Students often say that's easy just double the tins of yellow because you have doubled the blue. Now I ask them if I had 1000 yellow or something like they say 500 but this time because they recognise there are $\frac{1}{2}$ the number of yellow. A subtle change but one that recognise the proportion of blue and yellow is the same for equivalent ratios. Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-01-24T13:08:39.993" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7283" PostTypeId="1" CreationDate="2015-01-24T15:25:34.793" Score="5" ViewCount="347" Body="&lt;p&gt;I want a way to know the time needed to solve addition, subtraction, multiplication and division problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$15 + 10$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$500 - 132$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$10 \cdot 10$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$20 \, / \, 10$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Is it possible to create a formula that take the operation type and the operands values and calculate the time needed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="4674" LastEditorUserId="47" LastEditDate="2015-01-25T00:26:12.380" LastActivityDate="2015-01-26T05:28:55.060" Title="How to estimate the time needed to solve a basic mathematics problem" Tags="|primary-education|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7284" PostTypeId="2" ParentId="7271" CreationDate="2015-01-24T17:27:08.227" Score="2" Body="&lt;p&gt;Still a comment, but expanded.  I wrote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After the first year or so of writing my own exams, I concluded that it was (for me) not a good idea to have questions combining two or more nontrivial things. Not even combinations we had done in class.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For example: in integral calculus, computing an area in the plane.  I give them some equations describing a region in the plane, then ask them to compute the area.&lt;br&gt;&#xA;&lt;strong&gt;BAD IDEA&lt;/strong&gt;&lt;br&gt;&#xA;There are two non-trivial steps: (1) find the integral to compute, then (2) evaluate the integral.  The problem for the grader (me): if there is an error in the first part, then the second part may become much easier, or much harder.  There is no sensible way to assign partial credit for that second part.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of that, I would ask two questions.  One in which they find the integral to compute, but &lt;strong&gt;do not evaluate it&lt;/strong&gt;.  Another in which they evaluate an integral.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am talking about in-class time-limited exams.  Take-home problems (which we call &quot;homework&quot;, and may or may not be graded) of course will include two- or three-part problems.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-01-24T17:27:08.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7285" PostTypeId="2" ParentId="7249" CreationDate="2015-01-24T17:56:02.053" Score="2" Body="&lt;p&gt;I think in part you answered your own question. Students make these mistakes because they seem logical. What 'seems logical' to a student will be the product of a host of factors such as age, maturity (social and  mathematical) experience and many more besides.  Whether you make these mistakes or not is not necessarily a sign of mathematical ability.  The student that writes $a^2=a2=a\times2$ has a right to be confused if they have taken on board the idea operations don't need to be written down after all small handwriting was not part of the deal and my English teacher is forever telling me to write on the line! Sir obviously has the same problem.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-01-24T17:56:02.053" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7286" PostTypeId="1" AcceptedAnswerId="7287" CreationDate="2015-01-25T14:17:47.610" Score="5" ViewCount="236" Body="&lt;p&gt;In an interesting (and, perhaps, controversial from some viewpoints) text &#xA;&lt;a href=&quot;http://arxiv.org/abs/1407.1954&quot; rel=&quot;noreferrer&quot;&gt;Calling a spade a spade: Mathematics in the new pattern of division of labour&lt;/a&gt;, Alexandre Borovik describes changes in the mathematical education forced by changes in economy and technology. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this moment, I do not want to discuss the essence of this text, but I am &#xA;puzzled by one rather tangential remark. On p.18, when discussing an apparently emerging so-called &quot;deep mathematical education&quot;, Borovik writes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... in such a system, it could be desirable to have oral examinations in place &#xA;  of written ones. The reader familiar with the British university system, for &#xA;  example, can easily imagine all the political complications that would follow.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Being unfamiliar with the &quot;British university system&quot;, I am wondering what &#xA;these &quot;political complications&quot; might be.&lt;/p&gt;&#xA;" OwnerUserId="4676" LastActivityDate="2015-01-25T15:04:33.777" Title="Political complications following from oral examinations in the British university system?" Tags="|oral-exams|cultural-differences|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7287" PostTypeId="2" ParentId="7286" CreationDate="2015-01-25T15:04:33.777" Score="6" Body="&lt;p&gt;British universities have complex systems of teaching quality assurance, mandated by the QAA (&lt;a href=&quot;http://www.qaa.ac.uk/en&quot;&gt;http://www.qaa.ac.uk/en&lt;/a&gt;).  Exams are typically set by a lecturer, checked by a colleague in the same department, and then sent for at least brief review by an External Examiner who is a senior academic at another University.  There are also policies about marking systems, procedures for reviewing the overall shape of results, and so on.  Opinions vary about how useful all this is, but in practice every University has to comply.  It would be difficult to construct a system of oral exams that could be shown to have the level of consistency that these systems aim to achieve.  &lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2015-01-25T15:04:33.777" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7288" PostTypeId="2" ParentId="1639" CreationDate="2015-01-25T20:20:30.383" Score="1" Body="&lt;p&gt;An idea my favorite psychology professor uses with great success: &lt;a href=&quot;http://en.wikipedia.org/wiki/Operant_conditioning&quot; rel=&quot;nofollow&quot;&gt;place a large bowl of candy on your desk.&lt;/a&gt; At least, this improves the return rate. &lt;/p&gt;&#xA;" OwnerUserId="9" LastActivityDate="2015-01-25T20:20:30.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7289" PostTypeId="2" ParentId="4071" CreationDate="2015-01-25T20:36:37.563" Score="4" Body="&lt;p&gt;Here are references directed at teachers more than policy-makers, that analyze teaching from the teacher’s point of view, including the decisions that teachers must make as the lesson unfolds, and how their decisions affect the students.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0300099479&quot; rel=&quot;nofollow&quot;&gt;Teaching Problems and the Problems of Teaching&lt;/a&gt; by Magdalene Lampert&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0325006709&quot; rel=&quot;nofollow&quot;&gt;Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning&lt;/a&gt; by Jo Boaler and Cathy Humphries&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=IQUezn-XAYk&amp;amp;list=PLtTnKedqG-f3kTNO9lvsT2hNRWKvn_crW&quot; rel=&quot;nofollow&quot;&gt;How to Learn Math&lt;/a&gt; videos on Youtube from Jo Boaler’s first MOOC for math teachers. Related is Boaler’s site at Stanford&lt;a href=&quot;http://youcubed.stanford.edu/category/mooc/&quot; rel=&quot;nofollow&quot;&gt;, youcubed&lt;/a&gt;, where you can sign up for  free courses.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3578" LastActivityDate="2015-01-25T20:36:37.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7290" PostTypeId="2" ParentId="7281" CreationDate="2015-01-25T21:24:47.560" Score="2" Body="&lt;p&gt;After thinking a while about this question, here comes my first answer on matheducators.SE:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, I would leave out ratios entirely, if possible. They aren't as expressive as fractions are, and fractions are more widely used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why are ratios less expressive?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With (binary) ratios, you can compare only two things: $a:b$ (scores in a game, male-to-female distribution in the class, ...), while with fractions, you can have $a/x$, $b/x$, $c/x$, the fraction of several things from a larger group $x$ (color of hair, election results, ...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course you could also do $a:b:c:...$, but I think that's too complicated?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How to explain the difference?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could emphasize the difference in operations on both: adding ratios is simple, adding fractions is not. Multiplying fractions, or a fraction with a number  makes sense, multiplying ratios doesn't.&#xA;Another key is proper use of language to make clear what is being compared: The &lt;em&gt;ratio&lt;/em&gt; of the number of one thing &lt;em&gt;to&lt;/em&gt; another number of things, versus the &lt;em&gt;fraction&lt;/em&gt; of a number of things &lt;em&gt;of&lt;/em&gt; a total number of things.&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2015-01-25T23:28:41.467" LastActivityDate="2015-01-25T23:28:41.467" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7291" PostTypeId="2" ParentId="7283" CreationDate="2015-01-25T21:25:06.250" Score="3" Body="&lt;p&gt;I will formulate my comment into a more fully fledged answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not believe that there can be a formulaic answer to how long does it the average student to do any particular problem or exam. It depends on far too many factors that are individualistic for each student such as 'how does this student study?', 'what are their motivations to study and do well?', or 'did they get enough sleep?'. It even depends on the type and difficulty of the problem itself and how to quantify that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best way to gauge how long it takes students to do particular problems is to just watch them doing it. Assign some quizzes with such problems and time how long it takes half to three-fourths or even all to finish. After doing this several times and comparing how long they take on average to how quickly you can do the same problems, you can get an estimate for how much longer you should give them compared to yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some estimates that I frequently hear are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;University Students: They usually are given 4 to 5 times how long the instructor takes in the introductory courses. It might be 2 to 3 in higher level courses.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;High School Level or Equivalent: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/148/how-can-i-estimate-the-length-of-an-exam&quot;&gt;This answer&lt;/a&gt; by benblumsmith gives an estimate of 6 to 8 times.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I've not heard any estimates given for younger students nor do I have any experience to venture a guess. &lt;a href=&quot;https://matheducators.stackexchange.com/users/1147/jasper&quot;&gt;Jasper's answer is a good reference for this.&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Addendum: I just realized that some online assignment systems (i.e., WebAssign, Connect, etc) can provide instructors averages for how long students take on assignments (I believe from first access to last access on the assignment). This could be used to gauge how long students take for homework assignments, a harder thing to measure for instructors.&lt;/p&gt;&#xA;" OwnerUserId="201" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-01-26T02:41:13.767" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7292" PostTypeId="2" ParentId="7283" CreationDate="2015-01-26T02:21:43.167" Score="1" Body="&lt;p&gt;As discussed in a question about &lt;a href=&quot;https://matheducators.stackexchange.com/questions/6089/how-to-teach-quick-multiplication-and-division-in-head/6096#6096&quot;&gt;teaching mental arithmetic&lt;/a&gt;, my fourth grade math class included a daily 3-minute exercise with 50 - 100 problems.  Each day's problems were either addition (of one- or two-digit whole numbers), subtraction (of one- or two-digit whole numbers), multiplication (of integers between 0 and 12), or division (of integers between 0 and 144 by integers between 0 and 12, yielding integers between 0 and 12).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students have been through such a program to encourage them to memorize basic math facts and perform arithmetic quickly, it is therefore reasonable to expect students age 11 or higher to correctly answer items 1, 3, and 4 in 2 - 10 seconds per question, with an error rate on the order of one percent.  Question 2 might require more steps, such as writing out the problem, and two different carries.  Thus, question 2 might take 5 - 60 seconds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would not be surprised if the time to manually perform addition or subtraction is proportional to the total number of digits involved.  Similarly, I would not be surprised if the time to manually perform multiplication is proportional to the product of the number of digits involved, multiplied by a different proportionality constant.  I expect long division to take even longer than the corresponding multiplication problem.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-01-26T05:28:55.060" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7293" PostTypeId="2" ParentId="5605" CreationDate="2015-01-26T12:53:59.570" Score="2" Body="&lt;p&gt;I used:  Axler, &quot;Linear Algebra Done Right&quot; to teach myself the material in university.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend this book because it focuses on understanding the concepts rather than memorizing the proofs. It also has great exercises. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good Luck!&lt;/p&gt;&#xA;" OwnerUserId="4683" LastActivityDate="2015-01-26T12:53:59.570" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7295" PostTypeId="2" ParentId="7281" CreationDate="2015-01-26T20:56:21.117" Score="-7" Body="&lt;p&gt;This paragraph from &lt;a href=&quot;http://www.nap.edu/openbook.php?record_id=9822&amp;amp;page=233&quot; rel=&quot;nofollow&quot;&gt;Adding It Up&lt;/a&gt; is a good overview of the context of this question. Rational numbers are complex because there are multiple interpretations (meanings) as well as multiple forms for expressing a rational number.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As we said at the beginning of the chapter, rational numbers can be expressed in various forms (e.g., common fractions, decimal fractions, percents), and each form has many common uses in daily life (e.g., a part of a region, a part of a set, a quotient, a rate, a ratio). One way of describing this complexity is to observe that, from the student’s point of view, a rational number is not a single entity but has multiple personalities. The scheme that has guided research on rational number over the past two decades identifies the following interpretations for any rational number, say 3/4: (a) a part-whole relation (3 out of 4 equal-sized shares); (b) a quotient (3 divided by 4); (c) a measure (3/4 of the way from the beginning of the unit to the end); (d) a ratio (3 red cars for every 4 green cars); and (e) an operation that enlarges or reduces the size of something (3/4 of 12). The task for students is to recognize these distinctions and, at the same time, to construct relations among them that generate a coherent concept of rational number. Clearly, this process is lengthy and multifaceted.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think it is important for students to understand the connection as well as the distinction between the fraction and the ratio. They can both be written as 3/4. &lt;em&gt;The point to make to students is that the expression 3/4 can be interpreted in different ways&lt;/em&gt;. It has different meanings that are mathematically equivalent. So the students should realize that the ratio 3:4 has the same meaning as the ratio 3/4. &lt;em&gt;A second point to make is that there are different forms for writing rational numbers&lt;/em&gt;. Using the colon and the bar are just different ways of writing the ratio. As a ratio it is a correspondence of 3 things to 4 things. The students should also realize that the expression 3/4 can be interpreted as a fraction, 3 out of 4 equal portions of some whole. The fraction is a part-to-whole ratio. There are also part-to-part ratios, like one blue dot to 2 yellow dots. Since there are three dots in total, I can also compare the blue dots to the total, giving 1 to 3 as a &lt;em&gt;ratio&lt;/em&gt;, or as a &lt;em&gt;fraction&lt;/em&gt; 1/3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Give lots of examples to make it concrete. Answer their questions to clarify the connection and the difference. Then ask them questions and let them explain their thinking. For example, “There are 12 boys and 14 girls in this class. What fractions and ratios can you make from these numbers?”&lt;/p&gt;&#xA;" OwnerUserId="3578" LastEditorUserId="3578" LastEditDate="2015-01-27T20:26:52.013" LastActivityDate="2015-01-27T20:26:52.013" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7296" PostTypeId="1" CreationDate="2015-01-26T22:45:40.377" Score="5" ViewCount="388" Body="&lt;p&gt;Apologies for the crass title.  However, there are incentives for many children in grades (comparable to the U.S.) 3-12 to learn to do arithmetic well.  (I recall Number Sense and related contests from my school days.)  I would like to mention to students I am tutoring of current programs that offer such rewards.  There are of course the science competitions of Westinghouse, Intel, Google, and such, and competitions like Putnam and (used to be called) AHSME, but I am looking for more, especially offered at a regional or local level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I choose arithmetic as an example, but I am also interested in such things for algebra, geometry, and related areas.  Further, for this question I want to disallow coding challenges and similar programs that end up requiring more than basic mathematics.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Added 2015.01.27:  I appreciate Joseph O'Rourke's take on the question.  I think that &quot;joining an elite group&quot; is differently persuasive from &quot;getting something for demonstration of ability&quot;.  Also, while I think for some tournaments and competitions can suffice for motivation, in this question I am interested in something like money, i.e. folding stuff, or scholarships, or texts, or gift cards, however that might be.  In my case, participating in number sense contests got me into other branches of science, other contests, and acquiring medals, ribbons, trophies, a chemical engineering handbook, some small scholarships, and thanks to AP tests, saving a year of tuition by placing out of many introductory college courses. It also turned into a salary bump in my first job. I would like to say to a student that there are many benefits to learning how to do arithmetic well, and that some of them turn into cash equivalents, not just things that will help them live better lives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am hoping there are merit-based scholarships, local foundations that support students based on ability, programs that give rewards and other incentives for supporting mastery of basic skills.  I can name several such programs today for reading and science for elementary school and beyond.  I am having trouble naming them for mastering skills such as algebra, geometry, and basic arithmetic that are not also general science competitions.  Thus this question: what are such math-focused programs? I hope this makes the question more clear. End Added 2015.01.27)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Scholarships Also Count As Rewards&quot; Paseman, 2015.01.26&lt;/p&gt;&#xA;" OwnerUserId="3468" LastEditorUserId="3468" LastEditDate="2015-01-27T16:59:35.393" LastActivityDate="2015-02-28T13:17:17.203" Title="Show me the money" Tags="|student-motivation|concept-motivation|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7297" PostTypeId="2" ParentId="7296" CreationDate="2015-01-27T02:38:14.530" Score="0" Body="&lt;p&gt;I know you didn't mean &quot;money&quot; literally, but nevertheless I would like to&#xA;suggest that the real incentive is to demonstrate membership in an exclusive&#xA;club based on talent &amp;amp; knowledge. At that age perhaps there is little one can&#xA;do but rank kids according to speed and accuracy, which are only indications of&#xA;possible future talent. But the payoff for the kids is joining an elite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be nice if there were some way to encourage (a) understanding, and (b) creativity, &#xA;which are rather different talents than speed and accuracy.&#xA;Perhaps someone with experience with &#xA;&lt;em&gt;&lt;a href=&quot;http://schools.nyc.gov/Academics/CommonCoreLibrary/ProfessionalLearning/DOK/default.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Norman Webb's Depth of Knowledge (DOK) model&lt;/a&gt;&lt;/em&gt;&#xA;could contrast that with the skills encouraged by math competitions.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/LHOwY.png&quot; alt=&quot;DOK&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-01-27T02:38:14.530" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7298" PostTypeId="1" CreationDate="2015-01-27T06:50:26.467" Score="6" ViewCount="179" Body="&lt;p&gt;As a TA I always try to get students to help each other during office hours, because when I was in college I never had people to work with, and I strongly believe college is a lot more fun if you have problem set buddies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if someone asks me for help with Problem 4, I'll be like &quot;That group over there is also working on Problem 4, why don't you come join us?&quot;  Or &quot;That guy is the expert on Problem 4, maybe he can help you with it.&quot;  And then they will usually start working together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But sometimes when one person helps a group, I notice they don't actually include him in the conversations, they just ask him for help.  Usually he will chip in his bit of advice, and then stand around awkwardly the rest of the time while everyone else is laughing and joking.  This happened today, and when he left the others were like &quot;maybe we should add him to our GroupMe,&quot; but I think they weren't actually going to.  And in the end they will keep working together, and he'll still have no one to work with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I integrate my students better so everyone has people to work with?&lt;/p&gt;&#xA;" OwnerUserId="3504" LastEditorUserId="4872" LastEditDate="2015-08-26T09:05:01.277" LastActivityDate="2015-08-27T12:32:25.397" Title="How to help my students form study groups during office hours?" Tags="|struggling-students|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7299" PostTypeId="2" ParentId="937" CreationDate="2015-01-27T07:37:52.437" Score="3" Body="&lt;p&gt;If you are still doing research on PD, here are references specifically related to implementation of reform recommendations. These articles are based on the fact that successful implementation of reform requires most teachers to make fundamental and perhaps even radical changes in their teaching practices.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://pdk.sagepub.com/content/92/6/81&quot; rel=&quot;nofollow&quot;&gt;Policies That Support Professional Development in an Era of Reform&lt;/a&gt; by Linda Darling-Hammond and Milbrey McLaughlin&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The vision of practice that underlies the nation’s reform agenda requires most teachers to rethink their own practice, to construct new classroom roles and expectations about student outcomes, and to teach in ways they have never taught before — and probably never experienced as students. The success of this agenda ultimately turns on teachers’ success in accomplishing the serious and difficult tasks of &lt;em&gt;learning&lt;/em&gt; the skills and perspectives assumed by new visions of practice and &lt;em&gt;unlearning&lt;/em&gt; the practices and beliefs about students and instruction that have dominated their professional lives to date. Yet few occasions and little support for such professional development exist in teachers’ environments.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://eric.ed.gov/?id=EJ620635&quot; rel=&quot;nofollow&quot;&gt;External Reform Initiatives and Teachers’ Efforts to Reconstruct Their Practice&lt;/a&gt; by James P. Spillane&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.jstor.org/discover/10.2307/1164544?sid=21105697546743&amp;amp;uid=3739632&amp;amp;uid=2&amp;amp;uid=3739256&amp;amp;uid=4&quot; rel=&quot;nofollow&quot;&gt;Reform  and Teaching: Exploring Patterns of Practice in the Context of National and State Mathematics Reforms&lt;/a&gt; by James P. Spillane and John S. Zeuli&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The latter two references describe the results of an 18-month study on teachers who claimed on a survey to be implementing the 1989 NCTM reform recommendations at a high level in their classrooms. The study found that only four of 25 teachers had actually changed the core of their practice to conform with the reform recommendations. The second article describes the elements of PD of the teachers who were successful in implementing the reforms and those who were not successful. The third article describes the three patterns of practice of the teachers, from most to least consistent with the reform recommendations, and discusses the implications for policy.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://amte.net/sites/all/themes/amte/resources/ccssm/CCSSM_SupportingImplementation.pdf&quot; rel=&quot;nofollow&quot;&gt;Supporting Implementation of CCSSM: Recommendations for Professional Development&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The recommendations that follow are intended to support large-scale, system-level implementation of professional development (PD) initiatives aligned with the CCSSM. They emerged from the work done under the auspices of a NSF-funded project, which provided the opportunity for experts from diverse fields to collaboratively address the challenge of providing high-quality mathematics PD at scale to support the implementation of the CCSSM. Over the course of the project, researchers and expert practitioners worked to integrate various perspectives on this challenge into a set of design recommendations for creating, sustaining, and assessing PD systems for practicing mathematics teachers. Generated from the coordination of research-based knowledge in different but related fields, these recommendations build on state-of-the-art research findings from mathematics education, PD, organizational theory, and policy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3578" LastEditorUserId="3578" LastEditDate="2015-01-28T20:53:45.093" LastActivityDate="2015-01-28T20:53:45.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7300" PostTypeId="2" ParentId="7187" CreationDate="2015-01-27T17:06:39.200" Score="8" Body="&lt;p&gt;Proving equivalent the below characterizations of &lt;em&gt;squarefree&lt;/em&gt; naturals is quite elementary employing cyclic inferences (assuming one knows the existence and uniqueness of prime factorizations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt; $\ $ The following are equivalent for a natural number $\,q &amp;gt; 1$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(1)\qquad\quad n^2\!\nmid  q\,\ $ for all naturals $\,n &amp;gt; 1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(2)\qquad\quad p^2\!\nmid q\ \ $ for all primes $\,p$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(3)\qquad\quad  q\,$ is a product of distinct primes&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(4)\qquad\quad q\mid n^k\Rightarrow\, q\mid n\,\ $ for all naturals $\,n,k$   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proof&lt;/strong&gt; $\ $ All $\,(n)\Rightarrow (n\!+\!1)$ are obvious. $\,(4)\,\Rightarrow\,(1)\,$ may be proved as follows&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad\ \ $ If $\ q = an^2\,$ then $\ q\mid (an)^2\,\overset{ (4)}\Rightarrow\ q\mid an\,\Rightarrow\,an^2\mid n\,\Rightarrow\,n=1$  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark&lt;/strong&gt; $\ $ For further characterizations &lt;a href=&quot;https://math.stackexchange.com/a/54105/242&quot;&gt;see here&lt;/a&gt;, which includes the little known $\, q^q\!\mid n^n\Rightarrow\, q\mid n$&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-01-27T17:06:39.200" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7301" PostTypeId="2" ParentId="7298" CreationDate="2015-01-27T18:02:31.007" Score="2" Body="&lt;p&gt;I'll quite agree with David's comment above. However, I want to make a distinction between &quot;friendship groups&quot; and &quot;working groups&quot;. You have hardly any power to add someone to a FG, but as a TA you can actually include someone in a WG. Provide some problems that actually need group-work. My favorite non-example is solving quadratic equations after one knows the general formula. There is no point to ask people to work a couple of those equations together. It is better done individually. Now, assuming that you have a number of good questions worthy of group-work, set the working groups as you wish, and in particular, include that particular student in one of the group. I am sure you can find some general advice on the web to guide you about different ways of setting working groups to get the best out of each group. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-01-27T18:02:31.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7302" PostTypeId="2" ParentId="7249" CreationDate="2015-01-27T18:50:48.400" Score="9" Body="&lt;p&gt;Indeed, it is a very very broad question with a very very long literature behind it since for a long time Math Ed research was just about recognizing such mistakes. Thus, I narrow down my answer to a general phenomenon and a favorite example of mine. The general phenomenon is &lt;strong&gt;to project (mistakenly) the properties of an old domain to a new domain&lt;/strong&gt;. An often quoted example is the idea of &quot;multiplication makes bigger&quot;. Working with vectors as scalars is another example. There are too many. A favorite of mine, happened once in one of my classes, is the one about complex numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have just seen the geometric interpretation of complex number, and the very fact that $i$ is one unit above the origin on the y-axis. Then one student used the Pythagorean Theorem for the right triangle with the &quot;legs&quot; $i$ on the y-axis, and $1$ on the x-axis, and concluded that the length of the hypotenuse is zero: $i^2+1^2=0$!!  &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-01-27T18:50:48.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7303" PostTypeId="2" ParentId="7298" CreationDate="2015-01-27T20:45:41.493" Score="2" Body="&lt;p&gt;I have to also agree DavidButlerUofA's comment, it is quite hard to make people work together with a new person.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I find is that many students do not understand that it is more beneficial to their learning to work with students other than their friends who might, perhaps, know the material a little better. The students frequently prefer to be around people they get along well with socially, perhaps to enjoy doing the required work more than if they were just independent. The issue is that this often leads to distractions and 'joking around' instead of efficient work. For this reason, it is hard to get a new student to join a study group who is not a friend or social contact of the current members. Additionally, friends frequently come to office hours together which would explain the situation that you have in that the &quot;new student&quot; in the group is awkward around them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is hard to combat if the study group meets and works nearly exclusively outside of class time. You have no control over who they choose to work with (nor should you particularly try to have any control). What you can do is to try to make them realize that they are making this choice. I would announce at the very beginning of the semester that this class is not meant for the students and their friends to hang around each other more. They need to consider how their interactions affect their grade and to consider working with people who are just as dedicated to doing well as they are. They all know of the one person in each group who slacks off and is just there to goof around. It is harder to deal with them if they are a friend than just a colleague. They should consider study groups more of a group of business colleagues than friends meaning that they are just there to do efficient work, not hang out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When in office hours, you can try to break up the group a bit more. Consider making them work in pairs instead of large clusters. This might force one of the students (many times, the poorer quality student) to work with the new student who might know the material better.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-01-27T20:45:41.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7304" PostTypeId="2" ParentId="7298" CreationDate="2015-01-27T22:01:38.800" Score="6" Body="&lt;p&gt;To repeat my comment: if you are looking for ways to help students to break into already-formed groups without awkwardness, then I think you are trying to solve a problem that is bigger than just your office hours! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, there are strategies I use to get students working together in my Maths Learning Centre and in tutorial classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find that grouping people together works best when &lt;strong&gt;everyone in the group is still working on the problem&lt;/strong&gt;. It never works nearly as well when someone has already finished it and is helping the others because the others are in &quot;receiving mode&quot;. So perhaps look out for more times when everyone in the group is struggling with the same problem, and encourage them to work together. You might need to stay to help them ask each other useful questions, but it seems to work better than the alternative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do send a more experienced student to help a group, then remember that it is most likely that the more experienced student will be acting in the teacher role, regardless of your original goal. Therefore what you need to do is help that student to &lt;strong&gt;get better at passing on what they know&lt;/strong&gt;. I find the best way to do this is to go with them. The experienced student knows they have you to fall back on if they mess it up, and you can support the other students to respond appropriately and offer more of their own thoughts, and you can help the experienced student to stay or leave when the explanation is over (thus avoiding the awkwardness). Finally, being there lets them know that you really do care about students helping each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can &lt;strong&gt;get groups working together at blackboard/whiteboards&lt;/strong&gt;. In my Maths Learning Centre, the moment anyone starts writing on a whiteboard, students will drift towards it to join in. Because working on a board is more public, people are usually more inclined to ask questions and offer opinions. It also has the advantage that they can change what's written there more easily and so construct a better solution than if they were writing on paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, you need to &lt;strong&gt;set the ground rules&lt;/strong&gt;. In my first tutorial, I would tell the students we would be working in groups at whiteboards because they needed practice at solving problems where they could get advice as they were working. I would tell them that they were expected to ask each other questions if they weren't sure, and to explain what they were doing as much as possible, and especially if anyone asks. I also told them that there were no stupid questions, but that a specific question like &quot;how did you get from this line to this line?&quot; is better than one like &quot;What's going on?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in summary:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Get people to work on problems they all haven't finished.&lt;/li&gt;&#xA;&lt;li&gt;Help more experienced students to be better teachers.&lt;/li&gt;&#xA;&lt;li&gt;Stay to support the interaction.&lt;/li&gt;&#xA;&lt;li&gt;Get people to work at whiteboards/blackboards.&lt;/li&gt;&#xA;&lt;li&gt;Set ground rules.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-01-27T22:01:38.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7305" PostTypeId="1" AcceptedAnswerId="7306" CreationDate="2015-01-27T23:44:34.433" Score="3" ViewCount="100" Body="&lt;p&gt;I am teaching Mathematics for undergraduate students at social sciences. I have two choices. I can publish all old exams with their solutions on my personal weblog or preventing students from obtaining them with tied control over students during making exam. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When old exams would be published my students may be familiar with my question style and may be succeed there. In contrast, when them be public, students may concentrate on questions only and forget learning of all mathematics. Also when exams be published old students may feel bad because of their bad chance that hadn't old exams texts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For effective teaching and motivation of students, is it good idea that old exams be published by me or not? Shall I publish exam answer keys too? may publishing answers  cause memorization instead of learning? &lt;/p&gt;&#xA;" OwnerUserId="79" LastActivityDate="2015-01-28T00:30:04.383" Title="Publishing of exams in internet" Tags="|undergraduate-education|publishing|" AnswerCount="1" CommentCount="1" ClosedDate="2015-01-28T05:54:41.893" ContentLicense="CC BY-SA 3.0" />
  <row Id="7306" PostTypeId="2" ParentId="7305" CreationDate="2015-01-28T00:30:04.383" Score="2" Body="&lt;p&gt;It is key to remember the purpose of the assessment. I presume this is an end-of-course summative assessment. You are therefore trying to find out to what extent they have learned the material you have taught which may be required for further study or employment. You don't want them to study all of mathematics, you want them to study the specific topics that you have taught, and become proficient at the achievement objectives of the course. If old exams were valid, ie. they covered the achievement objectives, studying the old exams will further that aim. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast, a formative assessment, which you could use to guide your teaching, could validly assess the general mathematical competence of your students. However this should be done near the beginning of the course, not the end. It is not the test that shows up on their report. These are two quite different purposes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to widen the breadth of topics that will be assessed in a summative assessment, it is best to specifically alert the students to such changes and possibly even give some example questions so that students can give the new topics adequate weighting in their revision. This will be more effective than hiding the previous exams and telling them to study for everything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you noted, beyond encouraging study towards the achievement objectives, it can also reduce presentation confusion and exam stress which are both confounding factors for a valid assessment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note, whether you publish or not, some students will manage to get hold of old exams, which gives them an advantage unrelated to their mathematical ability. Publishing is the only way to level the playing field. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-01-28T00:30:04.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7307" PostTypeId="1" CreationDate="2015-01-29T10:53:30.257" Score="8" ViewCount="333" Body="&lt;p&gt;I am looking for a good app that enables students to chat with one another in order to work on mathematics problems.  The idea is that, in the old days, students would work over the phone to solve math problems; these days, they are more inclined to chat using apps.&#xA;I would like to provide students with that option so that they can work collaboratively in their own voice (so to speak).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To this end:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are the best apps for students to discuss math problems?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4695" LastEditorUserId="262" LastEditDate="2015-02-14T08:11:26.980" LastActivityDate="2015-02-14T08:11:26.980" Title="What are the best apps for students to discuss math problems?" Tags="|secondary-education|mathematical-pedagogy|technology-in-education|interactive-teaching|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7308" PostTypeId="2" ParentId="7307" CreationDate="2015-01-29T12:12:13.217" Score="3" Body="&lt;p&gt;We use Google Hangouts as a part of a Google Apps for Education deployment. It allows pupils to chat in real time, and allows the school some degree of control over this space.&#xA;Apart from this, I think you can be pretty much assured that the pupils in your class will have their own strategies to solve this problem. They may not use them much for Maths, though!&lt;/p&gt;&#xA;" OwnerUserId="3571" LastActivityDate="2015-01-29T12:12:13.217" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7309" PostTypeId="1" CreationDate="2015-01-29T16:21:30.243" Score="9" ViewCount="2456" Body="&lt;p&gt;I know a talented, enthusiastic, and very very hard-wroking 7th grade student, who began working on a research project about prime numbers a month ago. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He has written numerous Pascal programs to find large prime numbers (using methods such as sieve of Eratosthenes) and twin primes, computing density of prime numbers and twin primes and their relative frequencies in successive intervals of arbitrary length, etc. His programs work quite fine and fast. He consistently improves his codes, e.g. his last code for finding prime numbers yields all primes less than 1000000 in 3 minutes (the first one took 3 hours!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, he knows/has discovered some theoretical facts about prime numbers, e.g. how to prove that there are infinitely many primes, and why the twin primes are of the form $6k-1$ and $6k+1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My Question:&lt;/strong&gt; What are the possible directions for him to continue his research project (via programming or pure mathematical practice), without facing technicalities?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2017-03-14T22:23:58.440" Title="Student Project about Prime Numbers: How to Continue?" Tags="|secondary-education|prime-numbers|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7310" PostTypeId="2" ParentId="6125" CreationDate="2015-01-29T16:33:20.663" Score="1" Body="&lt;p&gt;Thank you for the great suggestion @RoryDaulton !  Below is the project description that i decided upon for this project.  The students can pick from one of the below options for their project, or have the opportunity to come up with their own idea.  Figured I would post this for anyone who would like to get ideas for similar project ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Family Tree&lt;/strong&gt;&#xA;A Square provides very detailed descriptions of the social ladder in Flatland and how Flatlanders proceed are able to slowly climb from generation to generation (Ch. 3, Ch. 7, Ch. 11).  For this project, you will construct a family tree, starting from an isosceles triangle all the way to a circle.  Here are the requirements for this project:&#xA;Include at least 7 family members from distinct generations, including the very first isosceles triangle and the final circle&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-For each family member provide a picture, basic measurements, and a paragraph of biographical information, making sure to include what generation they are in and who their parents and children are.  Be as creative as you would like!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Provide your “rules” for your family tree.  A Square, while thorough, does not provide us with actual numbers, likely because every family is slightly different.  For your family, I would like to know the starting angles on the original isosceles triangle, how much that angle increases from generation to generation, at what generation it is considered an equilateral triangle, how the side numbers increase from generation to generation, how many sides constitute a circle, and how many total generations it took to make it to a circle (This list is not exhaustive. Make sure to provide ALL information that is needed to understand your family tree)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Worldbuilder&lt;/strong&gt;&#xA;In Chapter 11, “Concerning Our Priests”, A Square begins the chapter by apologizing to the readers that he will not be able to explain the entirety of Flatland, including such topics as “our method of propelling and stopping ourselves,...the means in which we give fixity to structures of wood, stone, or brick...our Alphabet and method of writing, ”  Your job for this project is to pick three aspects of Flatland that A Square does not provide an explanation for and come up with (they do not have to be the ones that he says in Ch. 11) a description of how you think they would work for Flatlanders.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Provide a brief background to each topic and why Flatlanders cannot perform the task or experience the phenomenon in the same way that we Spacelanders can&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-For each topic give a complete description as to how the topic is handled in Flatland.  Each topic must be at least a page in length, with a title and date&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-You can be as creative as you would like, providing your description is consistent with A Square’s own description (You can’t say “The Flatlanders just pick it up with their hands” because polygons do not have hands, at least as we know them.  If you would like to give your Flatlanders hands then you need to describe exactly what you mean by “hands” and how it fits into Abbott’s Flatland)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Creatively frame your topics by introducing yourself as a citizen of Flatland and your motivation for writing the three passages, just like Abbott did with A Square&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Phlatland: A Derivative Work&lt;/strong&gt;&#xA;For this project, you will use Flatland as an inspiration for your own creative work.  Your project does not necessarily need to be a piece of fiction, it can be a movie, a song, an epic poem or any creative mode of expressing a mathematical story.  It does not necessarily need to be structured the way Flatland is, however, if you decide to tell your story in a different world or universe you need to provide some level of explanation so that the reader/listener/watcher knows what is going on.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-A creative work of sufficient length that ties together storytelling and mathematics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-You may produce a recreation of Flatland in a different media than written fiction (comic, short film, song) provided it is consistent with the universe and story of Flatland&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-It must somehow reference Flatland or an aspect of Flatland as laid out by A Square&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Other Ideas&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-A Map of Flatland:  Create a map of an entire Flatland city including exact measurements and a brief history of the city and its context within Flatland&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Flatlander Profiles:  Detail the lives of a handful of various Flatlanders, including detailed biographical information, sketches, and different possible cross sections&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Flatland Infographic:  Create an “Everything Flatland” infographic that explains all of the details of the Flatland Universe and A Square’s journey&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-01-29T16:33:20.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7311" PostTypeId="2" ParentId="7309" CreationDate="2015-01-29T17:16:44.160" Score="4" Body="&lt;p&gt;The problems you describe are the tip of an iceberg call &quot;Computation Number Theory&quot; and a google search should turn up a variety of resources at various levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possible route is to have the student learn about polynomial time vs. non-polynomial time algorithms and then try to implement the polynomial time primality testing algorthim (called the AKS primality test after the discovers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another route is to have them look into basic number-theoretic cryptography schemes (like RSA). Implementing them requires only a bit of modular arithmetic knowledge, which would be beneficial to learn anyway and shouldn't be too hard for this student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like I was once this student (though not quite that young) so I encourage and applaud your desire to continue to feed their fire!&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-01-29T17:16:44.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7312" PostTypeId="1" AcceptedAnswerId="14259" CreationDate="2015-01-29T17:23:07.710" Score="8" ViewCount="364" Body="&lt;p&gt;I find William Dunham's book &lt;a href=&quot;http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471500305.html&quot;&gt;Journey Through Genius&lt;/a&gt; is a great text to use in a History of Mathematics course, but I wish it had more textbook-like content, e.g. reading comprehension questions, exercises, etc. for each chapter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of a companion resource to this book and/or have one they'd be willing to share?&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2018-06-26T23:05:56.277" Title="A companion on the Journey Through Genius" Tags="|textbooks|history-of-math|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7313" PostTypeId="2" ParentId="7309" CreationDate="2015-01-29T19:11:28.897" Score="5" Body="&lt;p&gt;Ask them how much they would like to discover on their own and reinvent, and how much they would like to do research and verify results as a stepping stone to becoming an independent researcher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, the student will find resources on his own, such as the Prime Pages by Chris Caldwell, Ribenboim's books on primes and Diophantine equations, Crandall and Pomerance's book on computational number theory, and many other nontechnical and technical results.  Sooner or later, this student will have to make up or adopt terminology for the concepts he will encounter, such as pseudoprimes, primality tests, prime constellations, and the like.  You can tell the student about these piecemeal to keep the fire going, or give them a wheelbarrow full of resources and see if they get overwhelmed.  I recommend that if they have a question you can't handle, that you direct them to math.stackexchange, where it may get a technical answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you visit my homepage on MathOverflow, you will find a reference to my email address there.  I have some computational projects involving coprimality, and am willing to give some of them over to youthful energy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Ask Me About Jacobsthal's Function&quot; Paseman, 2015.01.29&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-01-29T19:11:28.897" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7314" PostTypeId="2" ParentId="4295" CreationDate="2015-01-29T19:25:16.957" Score="2" Body="&lt;p&gt;You might note that some systems of measurement are mixed-base number systems, and that more complicated unit conversion is essentially changing between mixed bases.  For example, converting a quantity expressed in meters and centimeters can be expressed in a sum of yards and inches, or even yards, feet, and inches.  Currency conversion to currencies not on a decimal system is challenging until one adopts a general formula for conversion of mixed base representations.  Writing a general purpose algorithm that converts between two mixed base systems can then be applied to these and other situations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Making Simple What's Mixed Up&quot; Paseman, 2015.01.29&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-01-29T19:25:16.957" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7315" PostTypeId="2" ParentId="7309" CreationDate="2015-01-29T21:43:50.190" Score="5" Body="&lt;p&gt;&lt;em&gt;side note:&lt;/em&gt; If he is not using iPython, he might give it a go. There are free cloud based iPython notebooks and it is an excellent environment for investigations, with easy graphing etc. A couple of my students of similar age use it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Test for primality rather than finding all primes&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First I would show him how to extend his research by tweaking his current algorithm. As he can compute all primes up to $10^6$, he can evaluate primarily up to $10^{12}$ without any libraries and with basically the same algorithm. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He can therefore investigate prime conjectures much higher than he can evaluate and store a complete list of primes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mersenne Primes&lt;/strong&gt; Test these and any other related conjectures he can devise up to $2^{40}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Distribution of primes&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting case might be estimating the distribution of primes using statistical sampling. He could then try informal curve fitting. He could make predictions on the proportion of primes at various sizes of numbers and test them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Non-prime related Montecarlo investigations&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After he has got used to sampling, he can move away from primes and implement any number of Montecarlo algorithms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He can easily calculate the value of $\pi$. He should be able to handle Pythagorus to find the distance from origin of a point. A simple method could be to make a random point with x,y each in [0,1], and count the proportion that is less than distance 1 from the origin. This proportion is the area of a quarter of a circle of radius 1 ($p\times 4 = \pi$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further to the above, he could investigate how many extra trials it takes to make an improvement of a significant figure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Montecarlo methods opens up a range really interesting applications for computationally intensive mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;probability&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He can investigate the statistical distribution of rolling 1-2 dice, and possibly compare to the theoretical distributions. Lots of common statistical laws or fallacies can be investigated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or investigate basic binomial distributions with coin flips. He could find the exact distribution for 8 coin flips (maybe counting bits in integers up to $2^{8}-1$. What is the likelihood of getting 8 heads in a row? Or n heads in a row?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At some point he will need to take a statistical approach as the number of coin flips increases. What is the mean number of heads in n coin flips? He should be able to sort his list of results using a library function, and therefore easily find the interquartile range. How does this change as the number of flips increases? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you use a weighted coin how does the above change?&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-01-29T21:43:50.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7316" PostTypeId="2" ParentId="7309" CreationDate="2015-01-30T00:29:44.310" Score="9" Body="&lt;p&gt;Your student may be interested in solving the &lt;a href=&quot;https://projecteuler.net/problems&quot;&gt;Project Euler&lt;/a&gt; series of problems.  These are programming exercises to solve mathematical challenges, many of which deal with number theory.  All of the exercises are designed such that the running time on a normal personal computer should be not much more than a minute.  However, achieving reasonable running times will often require the clever use of a theorem or algorithm.  Therefore, these exercises are valuable for gaining knowledge in both pure and applied mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The solutions for Project Euler questions are discussed on many websites.  One website that I highly recommend is &lt;a href=&quot;//codereview.stackexchange.com/&quot;&gt;Code Review&lt;/a&gt;, where there are currently &lt;a href=&quot;//codereview.stackexchange.com/search?q=project+euler+is%3Aquestion&quot;&gt;over 200 questions about Project Euler problems&lt;/a&gt;.  Other programmers will give feedback, both on the coding style and mathematical principles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, Code Review questions about any programming project are welcome too — as long as the code works as intended.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note that minors under age 13 should have a &lt;a href=&quot;//meta.stackexchange.com/a/102250/148099&quot;&gt;parent&lt;/a&gt; create the account.)&lt;/p&gt;&#xA;" OwnerUserId="996" LastEditorUserId="996" LastEditDate="2015-01-30T00:35:48.477" LastActivityDate="2015-01-30T00:35:48.477" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7317" PostTypeId="2" ParentId="1639" CreationDate="2015-01-30T02:59:23.933" Score="0" Body="&lt;p&gt;In every class I've taken (or TAed), office hours were used primarily to help students with homework.  The issues may be that&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Your assignments are so easy that most students think they can figure out how to do them on their own.&lt;/li&gt;&#xA;&lt;li&gt;Your assignments are purely computational and students just copy other people's (wrong) answers instead of figuring out how to solve problems themselves.&lt;/li&gt;&#xA;&lt;li&gt;Your students are freshmen and have not yet caught on to the fact that they can get significant homework help in office hours (they may need an upperclassman to tell them this).&lt;/li&gt;&#xA;&lt;li&gt;Your office hours are held at an inconvenient time for the students, like in the morning, or five days before the problem sets are due.  If you want people to come to your office hours, the best time to hold them is the night before homework is due, or even the day of.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3504" LastActivityDate="2015-01-30T02:59:23.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7318" PostTypeId="2" ParentId="7309" CreationDate="2015-01-30T16:24:29.200" Score="1" Body="&lt;p&gt;Continuing with the prime theme there is a natural progression to &lt;a href=&quot;http://www.virtualnerd.com/algebra-2/polynomials/equations/factoring-strategies/prime-polynomial-definition&quot; rel=&quot;nofollow&quot;&gt;prime polynomials&lt;/a&gt;. Then there's &lt;a href=&quot;http://www.seanerikoconnor.freeservers.com/Mathematics/AbstractAlgebra/PrimitivePolynomials/theory.html&quot; rel=&quot;nofollow&quot;&gt;primitive polynomials&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4701" LastActivityDate="2015-01-30T16:24:29.200" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7319" PostTypeId="2" ParentId="7309" CreationDate="2015-01-31T00:37:22.143" Score="0" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Ulam_spiral&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ulam Prime Spiral&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/V7M4I.jpg&quot; alt=&quot;Prime Spiral&quot; /&gt;&#xA;&lt;br /&gt;&#xA;and its variants leads to interesting questions to explore,&#xA;e.g.,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) &lt;a href=&quot;https://mathoverflow.net/q/102075/6094&quot;&gt;Prime spiral distribution into quadrants&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) &lt;a href=&quot;https://math.stackexchange.com/q/59369/237&quot;&gt;Prime spirals on surfaces of revolution&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am sure there are many more along this vein that could be explored experimentally.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2017-03-14T22:23:58.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7320" PostTypeId="2" ParentId="7281" CreationDate="2015-01-31T12:37:04.810" Score="18" Body="&lt;p&gt;I find this diagram helpful when relating the two: &#xA;&lt;img src=&quot;https://i.stack.imgur.com/heY9L.png&quot; alt=&quot;Top-down chart that compares ratios with fractions and rates&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It comes from a model curriculum unit on Rates and Ratios for 6th graders (you can see them all &lt;a href=&quot;http://www.doe.mass.edu/candi/model/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; after registering), and I have found this particular graphic very helpful with math content professional development with 6th grade math teachers.&lt;/p&gt;&#xA;" OwnerUserId="4702" LastActivityDate="2015-01-31T12:37:04.810" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7321" PostTypeId="1" AcceptedAnswerId="7326" CreationDate="2015-01-31T14:47:21.547" Score="5" ViewCount="1481" Body="&lt;p&gt;At my institution there is an Honors program which encourages (or requires) students to petition classes for Honors. Basically, what this amounts to is the student has to write a 10 page (not set in stone, this is just ballpark) paper on some topic loosely associated to the course. I'm teaching linear algebra this semester and I have an Honors student making a request to do such a project. Now, my default response is to ask them to do some project on tensors and multilinear algebra as I know a little about all that. However, this student also has interest in &lt;em&gt;Game Theory&lt;/em&gt; and was interested in a project about Markov chains etc. The problem for me, I don't have a good sense of what would make a nice project in the direction of Game Theory. So finally, the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; what would a good book, pdf, video series etc. be to provide a framework for an expository piece on Game Theory, or related topics, for someone who is currently learning linear algebra ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm interested in book suggestions, but, also templates for study if you have some passion for this topic. At this point, the student has already studied Gaussian elimination, matrix algebra and after next week, linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^m$. The next third of the course is basic vector space theory from subspaces to a bit quotient spaces. Then the final third of the semester we study eigenvectors, orthogonality, spectral theorem and a somewhat non-rigorous treatment of the real Jordan form.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-02-09T20:12:40.160" Title="Honors project idea for linear algebra" Tags="|undergraduate-education|reference-request|examples|gifted-students|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7323" PostTypeId="2" ParentId="7208" CreationDate="2015-01-31T20:33:08.517" Score="1" Body="&lt;p&gt;I let the students discover the rules for exponentiation as a consequence of finding prime factors. Students like the shorthand notation $24=2\times2\times2\times3=2^3\times3$ and related questions arise naturally. Given the prime factors of 24 what is the prime factor decomposition of 240? I like to juxtapose $ 2+2+2+2=4\times2$ with $2\times2\times2\times2=2^4$ Why do the rules work for one set and not the other? What's the same or different between the cases? Will the rules ever fail? I let the students apply the rules with algebra after developing their own abstract ideas.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-01-31T20:33:08.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7324" PostTypeId="1" CreationDate="2015-01-31T22:00:27.070" Score="17" ViewCount="389" Body="&lt;p&gt;I am teaching an introductory course on proofs in mathematics in a mid-size American public university, and trying to develop some kind of consistent grading meta-scheme for grading proofs. I am writing to see if anyone can refer me to math ed papers about grading proofs, or if anyone has such a meta-strategy that was helpful to use in the past. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such a meta-strategy might look like: grade each proof out of 20 points, assigning 5 points to each of&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Correctness - are definitions and previous results used correctly&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Logic (micro) - does each step follow from the previous step&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Logic (macro) - has the student understood what is to be proved, and chosen a path that is appropriate (i.e. correctly set-up a proof by contradiction, or direct proof, or etc)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Readability - has the writer used sufficient language to communicate the intended arguments&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(Incidentally, this is what I came up with this morning, in an attempt to start grading the first exam, but I suspect there are educators else where who have thought longer and harder about this problem, and I am dying to hear what they have to say.)&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2015-02-01T21:02:32.157" Title="How to assign grades to proofs: what do(es) the literature/experts suggest?" Tags="|reference-request|proofs|grading|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7325" PostTypeId="2" ParentId="7321" CreationDate="2015-01-31T22:52:42.953" Score="3" Body="&lt;p&gt;I have looked at a few basic courses which use video lecture series, and you might be able to pick something useful out of them. I don't know what level you are looking at, so they may be too basic for your requirements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stanford offers a video lecture series on Game Theory (&lt;a href=&quot;https://class.coursera.org/gametheory-004/lecture&quot; rel=&quot;nofollow&quot;&gt;https://class.coursera.org/gametheory-004/lecture&lt;/a&gt; you will need to register for access) though as an intro course I don't think it covered Markov chains. They also have an advanced computational course which also uses video lectures which is not offered on Coursera and which was advertised to people who had done the first course. I didn't do that second course so I'm unsure of the syllabus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;University of Tokyo also offers Game Theory on Coursera, but I  have not looked at it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other video lecture series that I have found useful is Stanford's Social Network Analysis, also through Coursera. I can't remember if that made links to Game Theory, or whether I got those linked from independent study after the course. Either way, SNA introduces you to NetLogo which makes it easy to investigate Game Theory concepts in networks and Markov processes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking through networked game theory simulations on NetLogo has also been extremely useful. For some investigations, finding the endpoint of the network's evolution is isomorphic to finding the dominant eigenvector, though the software just as easily models non-linear game theory processes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Model Thinking, also through Coursera, gives very brief overviews of many topics, including Markov processes and Game theory, though I can't remember if they link the two concepts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of the above are video lecture series which require minimal prerequisites. There are other courses which I have not looked at which may be helpful to you, and there may be something on the EdX platform, but most of the courses that I have done have been on Coursera, so that limits the range of my recommendations. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-01-31T22:52:42.953" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7326" PostTypeId="2" ParentId="7321" CreationDate="2015-02-01T02:47:58.273" Score="4" Body="&lt;p&gt;It would make a nice project to use payoff matrices to describe the &#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/Prisoner%27s_dilemma&quot; rel=&quot;nofollow noreferrer&quot;&gt;Prisoner's Dilemma&lt;/a&gt;,&#xA;and to use that to explore &lt;a href=&quot;http://en.wikipedia.org/wiki/Nash_equilibrium&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Nash equilibrium&lt;/a&gt;.&#xA;The Kahn Academy's &#xA;&lt;a href=&quot;https://www.khanacademy.org/economics-finance-domain/microeconomics/nash-equilibrium-tutorial/nash-eq-tutorial/v/prisoners-dilemma-and-nash-equilibrium&quot; rel=&quot;nofollow noreferrer&quot;&gt;10-minute video&lt;/a&gt;&#xA;is a place to start:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Hhq5X.png&quot; alt=&quot;KahnSnap&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-02-01T02:47:58.273" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7328" PostTypeId="1" AcceptedAnswerId="7329" CreationDate="2015-02-01T09:39:49.790" Score="18" ViewCount="684" Body="&lt;p&gt;In &lt;a href=&quot;http://www.mathcounterexamples.net/counterexamples-on-real-sequences-part-1/&quot;&gt;this&lt;/a&gt; article, I give counterexamples regarding real sequences. And in &lt;a href=&quot;http://www.mathcounterexamples.net/counterexamples-on-real-sequences-part-2/&quot;&gt;that one&lt;/a&gt; some others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular counterexamples answering questions like: &quot;If for all $p \in \mathbb{N}$ $\lim\limits_{n \to +\infty} (u_{n+p} – u_n)=0$ then $(u_n)$ converge?&quot; or &quot;if $\lim (u_{2n} – u_n) = 0$ and $(u_n)$ is increasing then $(u_n)$ converges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are you favorite instructional counterexamples on sequences? That can be about convergence, limit set...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would prefer to focus on sequences of reals, complexes or finite dimensional vectors. And avoid series (I know it is a bit artificial) or sequences of maps.&lt;/p&gt;&#xA;" OwnerUserId="4705" OwnerDisplayName="Jean-Pierre Merx" LastEditorUserId="143" LastEditDate="2015-02-01T14:22:04.127" LastActivityDate="2015-02-11T07:52:32.257" Title="What are your favorite instructional counterexamples on sequences?" Tags="|calculus|examples|sequences|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7329" PostTypeId="2" ParentId="7328" CreationDate="2015-02-01T10:39:55.663" Score="12" Body="&lt;p&gt;Show that the series &#xA;$$1-\frac{1}{2}-\frac{1}{2}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}++ -- \cdots $$ converges to all of the interval $[0,1]$. This is an example  (not so much a counter-example)&#xA;of a series whose $n$-th term tends to zero, but the series diverges spectacularly: the set of limit points is the whole interval $[0,1]$.&lt;/p&gt;&#xA;" OwnerDisplayName="Venkataramana" LastActivityDate="2015-02-01T10:48:36.327" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7330" PostTypeId="2" ParentId="7328" CreationDate="2015-02-01T17:04:49.533" Score="4" Body="&lt;p&gt;My favorite counterexamples using sequences is, in fact, with regards to continuity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many starting students have this notion in their head that to be continuous is essentially being able to draw the graph of a function without being able to pick up their pencils (pens, quills?, etc). Now, this is true for many functions that they are subject to in the usual beginning calculus sequence, but I like to mention that it is possible to be continuous at only one point or even continuous on only a uncountable set based upon the definition of continuity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Consider $&#xA;   f(x) = \left\{&#xA;     \begin{array}{lr}&#xA;       x &amp;amp; : x \in \mathbb{Q}\\&#xA;       0 &amp;amp; : x \notin \mathbb{Q}&#xA;     \end{array}&#xA;   \right.&#xA;$. This function is only continuous at 0 via for all sequences going to 0, the function converges to 0. Edit: To elaborate a little more, let $\lim_{n \to \infty} x_n =0$, then $\lim_{n \to \infty} f(x_n) = 0$, no matter if the $x_n$ are rational or irrational. But if $\lim_{n \to \infty} x_n =x$ where $x \neq0$, then if $x_n \notin \mathbb{Q}$ so that $\lim_{n \to \infty} f(x_n) = 0$ while if $x_n \in \mathbb{Q}$, then $\lim_{n \to \infty} x_n =x$. Such sequences can be constructed as the rationals are dense in the reals.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Thomae's function: $&#xA;   f(x) = \left\{&#xA;     \begin{array}{lr}&#xA;       1/q &amp;amp; : p/q \in \mathbb{Q} \text{ with $p/q$ in lowest terms and $q&amp;gt;0$}\\&#xA;       0 &amp;amp; : x \notin \mathbb{Q}&#xA;     \end{array}&#xA;   \right.$.&#xA;This is continuous on the irrationals, but discontinuous on the rationals again by looking at sequences, though, there are other methods for testing continuity here.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="201" LastEditorUserId="201" LastEditDate="2015-02-02T01:48:43.503" LastActivityDate="2015-02-02T01:48:43.503" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7332" PostTypeId="2" ParentId="7324" CreationDate="2015-02-01T21:02:32.157" Score="7" Body="&lt;p&gt;This paper: &quot;&lt;a href=&quot;http://link.springer.com/article/10.1007/s10649-011-9349-7&quot;&gt;An assessment model for proof comprehension in undergraduate mathematics&lt;/a&gt;&quot; can be used as a starter. As a starter, You have to work on it to convert the ideas discussed into grades. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-02-01T21:02:32.157" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7333" PostTypeId="2" ParentId="7268" CreationDate="2015-02-02T01:37:50.390" Score="1" Body="&lt;p&gt;I would use Algebra tiles (like &lt;a href=&quot;http://illuminations.nctm.org/Activity.aspx?id=3482&quot; rel=&quot;nofollow&quot;&gt;these&lt;/a&gt;).  I've found that as students see the difference between an x tile and a unit tile, for example, they tend to make fewer mistakes with combining like terms.  When they have to physically remove the same amount from both sides of the &quot;equal sign&quot;, it's harder for them to make careless mistakes, especially with negatives.  It takes a little practice (and more so for students who have a traditional method sort of stuck in their heads), but moving to abstraction seems to take less time.&lt;/p&gt;&#xA;" OwnerUserId="4702" LastActivityDate="2015-02-02T01:37:50.390" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7336" PostTypeId="1" CreationDate="2015-02-03T05:22:01.913" Score="12" ViewCount="747" Body="&lt;p&gt;I am cross-posting from &lt;a href=&quot;https://math.stackexchange.com/questions/1131391/a-good-way-to-explain-varepsilon-delta-for-chemistry-biology-students&quot;&gt;MSE&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students in my calc classes tend to be primarily bio/chem majors, and not very much math / physics / engineering. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like there are pretty good ways to talk about and explaining the epsilon-delta definition of limits for physics, engineering students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I'm not exactly sure what could be a good one for chem / bio majors in a first year calculus class. A few examples were given in the comments in the MSE post, but I was hoping this could be expanded further.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One idea I had was something along the following lines: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f$ represent the function that represents the blood sugar level&#xA;  as a function $x$ the amount of sugar (g) consumed per day. One wishes&#xA;  to maintain a certain a sugar level $L$, which can be achieved by&#xA;  consuming $a$ grams of sugar. But since it is difficult to maintain&#xA;  such a strict diet, one would like to allow for a bit of wiggle room&#xA;  without jeopardizing the sugar level too much...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Since I know very little about biology (or chemistry), I feel like this example may be too contrived or needlessly convoluted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any feedback or suggestion would be greatly appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; One big problem I have with many of the motivating examples from out side of mathematics is that they limit and continuity seem to be mixed up. Distinguishing the idea of limit and continuity already confuses a lot of first year calculus students even without the epsilon and delta thrown in there.&lt;/p&gt;&#xA;" OwnerUserId="4714" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2023-10-10T06:49:45.497" Title="Good ways of explaining the idea of epsilon-delta limits to bio &amp; chem majors?" Tags="|calculus|concept-motivation|biology|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="7338" PostTypeId="1" AcceptedAnswerId="7340" CreationDate="2015-02-03T08:02:12.517" Score="10" ViewCount="364" Body="&lt;p&gt;Starting from&#xA;\begin{equation*}&#xA;\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots,&#xA;\end{equation*}&#xA;which holds when $|x| &amp;lt; 1$,&#xA;we can conclude that&#xA;\begin{equation*}&#xA;\frac{1}{1 + x^2} = 1 - x^2 + x^4 - x^6 + \cdots&#xA;\end{equation*}&#xA;when $|x| &amp;lt; 1$.&#xA;Now taking antiderivatives of both sides, we discover that&#xA;\begin{equation*}&#xA;\tan^{-1}(x) = C + x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots&#xA;\end{equation*}&#xA;when $|x| &amp;lt; 1$.&#xA;By plugging in $x = 0$, we see that $C = 0$, and we have found the Taylor series (centered at $0$) for $\tan^{-1}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a pretty neat calculation.  Computing the derivatives of $\tan^{-1}$ at $0$ directly would have been laborious.  What are some other neat or good examples of calculations like this (where a Taylor series is computed in an efficient way)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple other examples would be computing the Maclaurin series for $x \sin(x)$ by just multiplying the Maclaurin series for $\sin(x)$ by $x$, or computing the Maclaurin series for $\sin(x) \cos(x)$ by multiplying the series for $\sin(x)$ by the series for $\cos(x)$.  (You could also use the identity $\sin(x) \cos(x) = \frac12 \sin(2x)$.)&lt;/p&gt;&#xA;" OwnerUserId="2049" LastEditorUserId="201" LastEditDate="2015-02-03T17:01:42.780" LastActivityDate="2015-02-04T19:29:34.430" Title="What are some good or neat examples of computing a function's Taylor series?" Tags="|calculus|examples|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7339" PostTypeId="1" CreationDate="2015-02-03T13:24:37.937" Score="18" ViewCount="1321" Body="&lt;p&gt;Let me start by sharing what happened in my class today. The subject was complex number and I started with the historical problem of &quot;finding two numbers whose sum is equal to 10 and whose product is equal to 40&quot;. Okay, as you know from experience or say from this &lt;a href=&quot;https://matheducators.stackexchange.com/questions/4503/appropriate-ways-sayings-to-discourage-undergraduate-students-overreliance-on-c&quot;&gt;question&lt;/a&gt;, nowadays there is no way to avoid student's over-reliance on calculator. Thus soon after students formalized the problem and came to $x^2-10x+40=0$, the room was filled with the sound of click, click, click of inputting the equation into the calculator to find its roots. But, the calculator for the reason that you know gave students an Error message, and then naturally, they wanted to know why. Indeed, I hadn't planned for that, but suddenly it occurred to me what a good use of calculator: create a situation for misusing it purposefully. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somehow my story is a possible answer to this question: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/4503/appropriate-ways-sayings-to-discourage-undergraduate-students-overreliance-on-c&quot;&gt;Appropriate ways/sayings to discourage undergraduate students' over-reliance on calculators&lt;/a&gt; that in itself includes some other calculator's related questions. But, Here is my own &quot;question&quot; that is asking for some other such misuse of calculator that if used purposefully could create a rewarding learning situation.&lt;br&gt;&#xA;There are some , say &quot;positive&quot;, uses of calculator. For example, the problems about that broken calculator that hasn't got a numbers of keys and we want to use other keys to get a certain result. Though nice, these uses are not what the question is about. Our calculator works perfectly!  &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-02-03T20:06:51.827" Title="How to create a misuse of calculator!" Tags="|technology-in-education|calculators|" AnswerCount="3" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="7340" PostTypeId="2" ParentId="7338" CreationDate="2015-02-03T15:44:37.257" Score="4" Body="&lt;p&gt;One of my favorites also provides a useful way to do an integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the usual Taylor Series of $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$ we can plug in $x^2$ to obtain $$e^{x^2} = \sum_{n=0}^{\infty} \frac{x^{2n}}{n!}.$$ As a note, this convergence is uniform on all compact subsets of $\mathbb{R}$ as the radius of convergence $R$ is $\infty$ (using the usual analytic notation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I know in lower level calculus classes that $\int e^{x^2}dx$ is mentioned to not have a closed form solution, that is, it is not a derivative of some composition of 'standard' functions. The issue lies in that many students do have to integrate on intervals to obtain values when they have to study normal distributions. So instead of yielding to pre-computed tables, it can be analytically computed, using uniform convergence where necessary or viewing as the generic antiderivative we teach calculus students, as such:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align*}&#xA;\int e^{x^2} dx &amp;amp;= \int \sum_{n=0}^{\infty} \frac{x^{2n}}{n!} dx \\&#xA;&amp;amp;= \sum_{n=0}^{\infty} \int \frac{x^{2n}}{n!} dx \\&#xA;&amp;amp;= \sum_{n=0}^{\infty} \frac{x^{2n+1}}{n! (2n+1)}&#xA;\end{align*}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, we can do computations: &#xA;\begin{align*}&#xA;\int_a ^b e^{x^2} dx &amp;amp;\approx \sum_{n=0}^{N} \frac{b^{2n+1} - a^{2n+1}}{n! (2n+1)}&#xA;\end{align*}&#xA;with error $|\dfrac{d^{N+1} e^{x^2}}{dx^{N+1}} \dfrac{b^{N+1}-a^{N+1}}{(N+1)!}|$ for a $x \in [a,b]$.&lt;/p&gt;&#xA;" OwnerUserId="201" LastEditorUserId="201" LastEditDate="2015-02-04T14:49:53.643" LastActivityDate="2015-02-04T14:49:53.643" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7341" PostTypeId="2" ParentId="7339" CreationDate="2015-02-03T18:18:37.610" Score="6" Body="&lt;p&gt;Calculators can enrich the learning of students if properly used. A simple example; I like to get students to work $2+3\times10$ and then get them to check with a calculator. They are forced into confronting their misconception that the answer is 50 and in the process are more likely to remember the order of operation.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-02-03T18:18:37.610" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7342" PostTypeId="2" ParentId="7339" CreationDate="2015-02-03T18:30:41.777" Score="5" Body="&lt;p&gt;I'm a fan of questions like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a) Plot &lt;span class=&quot;math-container&quot;&gt;$y = x^3 - 19x^2 + 96x - 144$&lt;/span&gt; on your calculator in the standard window.&lt;/p&gt;&#xA;&lt;p&gt;b) Find an end behavior diagram for &lt;span class=&quot;math-container&quot;&gt;$y = x^3 - 19x^2 + 96x - 144$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;c) These answers conflict with each other. What went wrong?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The trick here is that in the standard window, the function looks like a parabola (it has roots at 3, 4, and 12, with 12 outside the window). But this will conflict with their end behavior diagram, and hopefully show them to be skeptical of their calculator.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-02-03T18:30:41.777" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7343" PostTypeId="2" ParentId="7339" CreationDate="2015-02-03T19:44:12.007" Score="9" Body="&lt;p&gt;Reminds me of a story a friend (who is a math teacher) told me. Student comes to him and says his calculator is broken - it shows (-5)^2 = -25.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;[C] [-] [5] [x^2] [=]                  shows -25&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Q: So, how do you know that's wrong?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A: Because my mobile says 25:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;[C] [-] [5] [*] [=]                    shows 25&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Q; And how do you know the 25 is correct, and the -25 isn't ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A: I checked with my friend's mobile, it says 25 as well ....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My friend turned this lesson into one focusing on how to use calculators correctly, and how to check whether or not the result makes sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Especially with those cheap calculator apps that come pre-installed with mobiles, and that don't handle operator precedence, you can get a lot of wrong/conflicting results.&lt;/p&gt;&#xA;" OwnerUserId="3590" LastActivityDate="2015-02-03T19:44:12.007" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7344" PostTypeId="2" ParentId="7336" CreationDate="2015-02-04T00:44:38.037" Score="11" Body="&lt;p&gt;If you must introduce epsilon/delta...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As the sciences usually deal with real measurements with errors, this might be a way in: Is the calculation increasingly stable under decreasing measurement errors?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is possible to motivate much of differentiation just from estimating errors. The percentage error (error / value) of squares is doubled, and of cubes is tripled. This is the definition polynomial differentiation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the link Feynman made use of in his famous abacus vs mental calculation anecdote. As a bonus, an interesting anecdote that shows something is not as difficult as it first appears is always good motivation!&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-02-04T00:44:38.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7345" PostTypeId="1" AcceptedAnswerId="7346" CreationDate="2015-02-04T01:14:58.220" Score="10" ViewCount="693" Body="&lt;p&gt;Today in second semester calculus, I found myself stumbling a bit to provide a natural-sounding explanation for all the letters involved in the expression&#xA;$$&#xA;\lim_{t \rightarrow \infty} \int_1^t \frac{1}{x^p} dx,&#xA;$$&#xA;where $p$ is an arbitrary real number. There are some subtle things going on.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The value of $p$ is fixed throughout the evaluation, but is arbitrarily chosen at the beginning.&lt;/li&gt;&#xA;&lt;li&gt;The value of $t$ is fixed as regards the definite integral, but is later pushed toward infinity when the limit is evaluated.&lt;/li&gt;&#xA;&lt;li&gt;The value of $x$ is not like either of $p$ or $t$, but rather is the variable of integration. It ranges over the domain $[1,t]$, which again is finite for any fixed $t$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I think these distinctions arise frequently in analysis, so I'd be interested in advice on making them more concrete to students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I think might help is to separate the limit and the integral. That is, I would first ask them to consider $\int_1^t \frac{1}{x^p} dx$ for arbitrary (but fixed) $p$ and $t &amp;gt; 1$. In this case, the only thing varying is the variable of integration. Now we have a function $A(t)$ that gives the area under $\frac{1}{x^p}$ on $[1,t]$, and we can ask how this function behaves as $t$ tends to infinity. While this viewpoint is helpful to me, I wonder if it is still too abstract for a typical Calculus student.&lt;/p&gt;&#xA;" OwnerUserId="222" LastActivityDate="2015-02-04T01:32:19.507" Title="Language to Distinguish Between Variables and Arbitrary Constants" Tags="|calculus|mathematical-pedagogy|notation|integration|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7346" PostTypeId="2" ParentId="7345" CreationDate="2015-02-04T01:32:19.507" Score="16" Body="&lt;p&gt;The logicians' words &lt;em&gt;&quot;bound&quot;&lt;/em&gt; and &lt;em&gt;&quot;free&quot;&lt;/em&gt; may help.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The variable $x$ is bound in $\int_1^t dx/x^p$, and so also bound in the expression for the limit.&lt;/li&gt;&#xA;&lt;li&gt;The variable $t$ is free in $\int_1^t dx/x^p$, but bound in the expression for the limit.&lt;/li&gt;&#xA;&lt;li&gt;The variable $p$ is free in the expressions for both the integral and the limit.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2015-02-04T01:32:19.507" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7347" PostTypeId="2" ParentId="5901" CreationDate="2015-02-04T01:44:12.710" Score="0" Body="&lt;p&gt;In a Riemann sum, we add up a bunch of skinny rectangles to get an area. For convenience, we split up the domain into equal parts of length $\Delta x$. The height of each rectangle is determined by evaluating the function $f$ at $x_i$. Rectangle $i$ therefore contributes area $f(x_i)\Delta x_i$, which gives total area&#xA;$$&#xA;\sum_{i = 1}^n f(x_i)\Delta x.&#xA;$$&#xA;The units of $f(x_i)$ and $\Delta x$ are both (say) meters, so the units of $f(x_i) \Delta x$ are square meters (which makes sense, considering it is supposed to be an area).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we want to pass from adding up finitely-many rectangles to adding up infinitely-many rectangles, we use a special notation&#xA;$$&#xA;\int_a^b f(x) dx.&#xA;$$&#xA;The summation ceases to be about a finite-number of rectangles, but rather instructs us to visit &lt;em&gt;every&lt;/em&gt; value of $x$ in the interval $[a,b]$ and ask &quot;What is the value of $f(x)$?&quot; (that is, the height of the &quot;rectangle&quot;). The units of $f(x)$ are still meters. The $dx$ is the width of this impossibly small rectangle, which has no particular numerical value, but crucially keeps its units: meters. Therefore, the units of $f(x)dx$ are square meters, so we're still talking about area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find talking about units helpful with single integrals but indispensable in motivating multiple integrals. For example, let $\rho(x,y,z)$ be the density of some solid $E$ at coordinates $(x,y,z)$. The units of $\rho(x,y,z)$ are (say) grams per cubic meter. The integral sign in $\iiint_E \rho(x,y,z) dV$ instructs us to visit &lt;em&gt;every&lt;/em&gt; point of the solid $E$ and ask &quot;What is the density here?&quot;. Since $dV = dxdydz$, its units are cubic meters, so the units of $\rho(x,y,z)dV$ are grams. Therefore, we must be computing the mass of this solid $E$.&lt;/p&gt;&#xA;" OwnerUserId="222" LastActivityDate="2015-02-04T01:44:12.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7348" PostTypeId="2" ParentId="7338" CreationDate="2015-02-04T19:29:34.430" Score="2" Body="&lt;p&gt;$f(x) = \begin{cases}\exp(-\frac1{x^2}) &amp;amp; x \neq 0 \cr 0 &amp;amp; x = 0 \end{cases}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If only to disabuse the students of the idea that smooth implies analytic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Fabius function or $F(x) := \sum_{n=0}^\infty \frac{\exp(i2^nx)}{n!}$ (or its real part) would be better examples since they're nowhere analytic, but that's a bit harder to verify. I have &lt;a href=&quot;https://math.stackexchange.com/questions/12989/a-smooth-functions-domain-of-being-non-analytic/12990#12990&quot;&gt;an answer over on MSE&lt;/a&gt; attempting to show non-analyticity of $F$, but I might have gotten the application of the Cauchy&amp;ndash;Hadamard formula wrong. &lt;/p&gt;&#xA;" OwnerUserId="1527" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-02-04T19:29:34.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7349" PostTypeId="2" ParentId="7328" CreationDate="2015-02-04T19:36:03.280" Score="8" Body="&lt;p&gt;My favorite counterexample is one about exchange of limits:&#xA;$$a_{n, m} =\frac{n}{n+m}. $$&#xA;It is indeed pretty simple to see that $\lim_{n\to\infty} \lim_{m\to\infty}a_{n, m} \neq \lim_{m\to\infty} \lim_{n\to\infty}a_{n, m} $. &lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2015-02-04T19:36:03.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7350" PostTypeId="2" ParentId="5901" CreationDate="2015-02-04T19:52:02.433" Score="1" Body="&lt;p&gt;For definite integrals I also explain that the $d$ in $dx$ does not have a meaning since it does not add any information to the expression–but the $x$ in $dx$ has one: It indicates over which variable the integration should be performed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compare the notation for summation&#xA;$$\sum_{i=n}^m a_i$$&#xA;with the notation for integrals&#xA;$$\int_a^b f(x) dx$$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Going from sum to integral, you&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;replace the sequence $a:\mathbb{N}\to\mathbb{R}$ by a function $f:\mathbb{R}\to\mathbb{R}$,&lt;/li&gt;&#xA;&lt;li&gt;the bounds $n$ and $m$ for the sum by the bounds $a$ and $b$ for the integral,&lt;/li&gt;&#xA;&lt;li&gt;the symbol $\sum$ by $\int$ and finally,&lt;/li&gt;&#xA;&lt;li&gt;the symbol $i=$ which indicates over which variable the sum is computed by $dx$ which does basically the same thing for the integral.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So, one could, in principle, say that the $d$ in the integral serves the same purpose than the $=$ below the sum. This would suggest to write the definite integral as&#xA;$$\int_{x=a}^b f(x)$$&#xA;and the indefinite integral as&#xA;$$\int_x f(x)$$&#xA;but I doubt that this would work out. However, if one would also speak about indefinite sums $\sum_i a_i$ in the same way, one could make sense of this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, it is pretty custom to write $\int f$ for the indefinite integral without mentioning the &quot;free variable&quot; at all when there is just one function involved. The $dx$ (i.e. the naming of the integration variable) is only needed for expressions like $\int_a^x (x-t)^n dt$ and such.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="3398" LastEditDate="2015-02-05T07:24:04.627" LastActivityDate="2015-02-05T07:24:04.627" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7351" PostTypeId="2" ParentId="7247" CreationDate="2015-02-04T20:22:03.850" Score="4" Body="&lt;p&gt;My two cents are: There is not a real dividing line between &quot;pure&quot; and &quot;applied&quot; mathematics. I think the real distinction is between &lt;strong&gt;mathematicians&lt;/strong&gt; (or &quot;want-to-be-mathematicians&quot;) &lt;strong&gt;who are inspired by an application&lt;/strong&gt; and &lt;strong&gt;those who are not&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this answer does not help you with the question you posed. But well, there are people who are bad at algebra (considered &quot;pure&quot;) and stochastics (considered &quot;applied&quot;) at the same time and also good in optimization (considered &quot;applied&quot;) and functions of a complex variable (considered &quot;pure&quot;)–one example is myself.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2015-02-04T20:22:03.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7352" PostTypeId="2" ParentId="5901" CreationDate="2015-02-05T13:01:45.307" Score="1" Body="&lt;p&gt;I think that a first introduction to integration should demonstrate what it can be used for in the real world, motivating it by asking how we can find the distance traveled by a train whose velocity changes over time. First we of course have to define velocity, which is the rate of change of position at the point in time, measured by taking smaller and smaller intervals of time around the point. One natural way of expressing that would end up being the derivative of position with respect to time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we employ the intuition that if a train has higher velocity than another at every point in time, it would travel at least as far in the positive direction. So we can approximate the real displacement by comparing with trains that have constant velocity over each short time interval. For each of those intervals we can see that the displacement is clearly the velocity multiplied by the time elapsed, almost by definition of velocity. Subsequently we can see that if the train's velocity is sufficiently nice we can obtain good upper and lower approximations by choosing sufficiently thin intervals. Hence we get the Riemann sum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using this intuition, all the symbols make sense, &quot;$\int$&quot; being an elongated &quot;s&quot; for sum, &quot;$dt$&quot; being the size of the thin time intervals, and the bounds of the definite integral being the endpoints of the time period under discussion. Also, it is worth pointing out that using these rectangular approximations is equivalent to finding the area under the graph.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next step is to explicitly calculate some simple examples to get a feel for what the result is like. One should then compute the displacement in terms of the time period. It is also easy to observe that like the intuition says, the displacement is additive in the time period, and so this as well as the explicitly computed examples should suggest that we should compute just the integral with a fixed lower bound, since we can obtain the general case just by a difference. This general technique of reducing the parameters needed is very useful in both mathematics and computer science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far we have built everything without mentioning anti-derivatives, and indeed we should not just yet. First note that the integral with a fixed lower bound can be used in all cases where the integrand is integrable to compute the integral with arbitrary bounds, even if the integrand is non-continuous. In mechanics, even though everything is quite continuous until down to the quantum level, in practice it makes sense to simplify the model of say a train that leaves a station with velocity being non-continuous. (And in other cases, acceleration. Or jerk. Or...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note also that the actual value of the fixed lower bound does not matter in obtaining the definite integral because the difference eliminates its effect, much as in the real world the point of reference that is used to measure displacement does not matter. Plotting the graphs obtained by using different integral lower bounds should make it clear to students that they are all the same except for a constant difference across all time, and that the constant difference corresponds to the displacement between the two chosen lower bounds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the symbols &quot;$\int f(t)\ dt$&quot; can be said to represent simply the class of functions &quot;$x \mapsto \int_c^x f(t)\ dt$&quot; for all possible $c$. I find it misleading to write &quot;$\int f(t)\ dt = \cdots + C$&quot; because it is actually a class, not one single function. In any case, this is what the symbols should be explained to mean, and &lt;strong&gt;not the anti-derivative&lt;/strong&gt;, since it may not exist. This class of functions is called the &quot;indefinite integral&quot;, precisely because it can be used to compute arbitrary definite integrals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then now it is very natural to ask if there is a way to efficiently compute the indefinite integral directly without going through the approximation process. That is of course easy for certain special integrands, namely those which are continuous derivatives of some other function. The reason is that one can notice that the indefinite integral grows at a rate that is proportional to the value of the integrand at that point quite intuitively, since the thin rectangle at that point has exactly that height. A formal analysis would give the fundamental theorem of calculus, as well as the desired result that we can use anti-derivatives &lt;strong&gt;in the case the integrand is the continuous derivative of some other function&lt;/strong&gt;. An integral may need to be split into separate regions before anti-derivatives can be used, and certain badly behaved cases should not be overlooked (the derivative of Volterra's function). These issues will be avoided if the indefinite integral is not confused with the anti-derivative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course all these can and should be made rigorous via careful proofs at some point in time, but the above should suffice for an intuition explanation of all the basic notions. The very same ideas generalize naturally to Lebesgue integration using simple functions to approximate non-negative measurable functions, or using Lebesgue measure in place of Jordan area.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2015-02-05T13:01:45.307" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7353" PostTypeId="2" ParentId="5998" CreationDate="2015-02-05T16:58:10.107" Score="0" Body="&lt;p&gt;The size does not matter as can be seen in this PRIMUS paper, &quot;&lt;a href=&quot;http://www.tandfonline.com/doi/full/10.1080/10511970.2011.639337#.VNObc9KsXTo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moore and Less&lt;/a&gt;&quot; in which I told the story of a very non-standard, absolutely student-centered multi-variable calculus course with 136 students! What matters is to create a &quot;culture&quot; for such an adventure. That is hard, indeed very hard, whether your class is small or large.&lt;br&gt;&#xA;Though, I am obviously an advocate of IBL, I'd like to draw your attention to some of its disadvantages &lt;a href=&quot;https://mathoverflow.net/a/131842/29316&quot;&gt;here&lt;/a&gt; as an answer to this question on MO: &lt;a href=&quot;https://mathoverflow.net/questions/12070/what-are-the-advantages-and-disadvantages-of-the-moore-method/131842#131842&quot;&gt;What are the advantages and disadvantages of the Moore method?&lt;/a&gt;  &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-02-05T16:58:10.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7354" PostTypeId="1" CreationDate="2015-02-06T04:35:30.673" Score="8" ViewCount="2073" Body="&lt;p&gt;some of my students refer to there being an invisible $-1$ in front of the expression $-(x + 4)$ or in the exponent of $x$.  While it is not phrased mathematically, I am ok with them saying this because it reminds them to distribute fully before simplifying etc.  It got me thinking though is there a reason to not teach students to always write in $1$ wherever there is a single variable/unknown/etc such as $1(x^1+40)-1(3x^1 - 2)$?  I know that it is not done in general because it is incredibly repetitive and annoying, but is there any reason why not to teach students to do this?  Eventually they will be comfortable enough that they can imply the $1$ but I feel like a lot of my students would benefit from this especially when simplifying exponential expressions and distributing negative signs.  Are there any downsides to this or is it just de facto mathematics education? &lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2018-11-21T22:52:47.910" Title="Teaching students to write the &quot;invisible&quot; ones" Tags="|secondary-education|mathematical-pedagogy|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7355" PostTypeId="2" ParentId="7354" CreationDate="2015-02-06T08:00:50.173" Score="4" Body="&lt;p&gt;Actually, I don't think that it is quite right that there really &quot;is an invisible $-1$&quot; in front of $-x$. I would say that one &lt;strong&gt;defines the symbol $-x$ to be the one that fulfills $x + (-x) = 0$&lt;/strong&gt; and then one shows that $(-1) x =-x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ok, you don't teach that at high-school level, but the message that $-x$ is a symbol on its own is important. If there were no such symbol $-x$ for any $x$ it would be rather strange to have it for the special case of $x=1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for teaching this way is that mathematics is precisely about the accurate and concise description of facts. Leaving out unnecessary parts it a crucial point on the way to mathematical thinking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If n is an integer greater than 1, then either n is prime or n is a finite&#xA;  product of primes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This formulation is not good in a mathematical sense, since it contains many unnecessary things. Much crisper and, I guess, even clearer, is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A positive integer is a finite product of primes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Writing concise, omitting everything unnecessary is a part of mathematical education.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2015-02-06T08:00:50.173" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7356" PostTypeId="2" ParentId="7336" CreationDate="2015-02-06T14:07:59.307" Score="5" Body="&lt;p&gt;(I should preface that I also agree with Dave Renfro (in the comments to the original question) that, given a choice, there is much more useful and relevant material to cover than the $\epsilon$-$\delta$ definition in a first calculus course for chemistry and biology majors.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the problem you suggest is a good motivational problem, but that it is not strictly speaking motivating the $\epsilon$-$\delta$ definition.  Your problem amounts to given $L$ and a specific $\epsilon &amp;gt; 0$, find a specific value of $\delta$ such that for all sugar consumption amounts $x$ that are within $\delta$ of $a$, the associated blood sugar level $f(x)$ will fall within $\epsilon$ of $L$. (FWIW, I don't know if the correlation between sugar consumption and blood sugar level is as simple as your function model assumes, though I think it sounds like a plausible first model to try.  But you might try asking biologists or medical experts whether this model is reasonably accurate, an acceptable white lie, or an outright gross oversimplification of the situation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two key respects in which your example does not fully capture the definition of a limit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) In practice, you would likely only be interested in a single $\epsilon$ value or perhaps a small finite number of $\epsilon$ values (different patients may require different levels of care for their blood sugar levels).  It is important to point out that the mathematical definition requires the existence of a $\delta$ for every $\epsilon$ (and that you are allowed to use a different $\delta$ for each $\epsilon$).  You can certainly use your example to help explain this, but it's important to recognize the quantification logic behind the definition of a limit gives students a lot of difficulty and consequently that it needs to be addressed carefully.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Your example uses a continuous function, where $L$ will be equal to $f(a)$.  The primary motivation for introducing limits is to deal with discontinuous functions, such as the average rate of change functions that arise in computing derivatives.  It is precisely because the function which computes the average rate of change of $y = f(x)$ with respect to $x$ for $x$ in the interval $a \leq x \leq a + h$ (viewed as a function of the variable $h$) is not defined (and consequently not continuous) when $h = 0$ that limits are needed.  This point does not invalidate your example, but you need to address this point.  Inevitably, one must introduce an example involving a rate of change to fully motivate introducing limits.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2016-02-19T15:24:04.853" LastActivityDate="2016-02-19T15:24:04.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7357" PostTypeId="2" ParentId="7354" CreationDate="2015-02-06T16:12:34.407" Score="4" Body="&lt;p&gt;I see this as two separate questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Are there any downsides?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Should I condone it for my students?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) As mentioned in other answers, it can make the transition into deeper areas more difficult than need be.  (Understanding for groups)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) I think the answer to the questions can only be answered by the students you teach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some students have so little confidence in themselves that building this up should be a priority for any teacher.  If leaving a $-1x$ is right for my students at that time then I will do it.  &lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-02-06T16:12:34.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7358" PostTypeId="1" CreationDate="2015-02-06T19:29:44.210" Score="7" ViewCount="598" Body="&lt;p&gt;I'm learning about groups and $a^n$ suddenly doesn't mean exponentiation anymore, but repetition of $\underbrace{a\circ a\circ \cdots\circ a}_n$. In some sense I think it would be useful to learn $a\!\overset{n}{\circ}$ first so it's clear that it's not exponentiation, but repetition of $a\!\circ$, and then learn $a^n$ later when I've become accustomed to the meaning. It's somewhat confusing to think of $4^3$ to mean $4+4+4$ in $(\mathbb{N},+)$...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm just a student, so I'm just viewing this from my perspective, but in general, is it useful to use non-standard notation to teach new concepts?&lt;/p&gt;&#xA;" OwnerUserId="4657" LastActivityDate="2015-02-07T00:00:53.407" Title="Is non-standard notation useful when teaching new concepts?" Tags="|mathematical-pedagogy|intuition|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7359" PostTypeId="2" ParentId="7354" CreationDate="2015-02-06T20:17:09.553" Score="4" Body="&lt;p&gt;It depends what you mean by &quot;teach them to do it&quot;. If you mean &quot;insist that they do it, correct them every time they don't and take off marks on a test&quot;, then no. If you mean &quot;let them know about this trick as a way to remember how to distribute correctly, but don't insist they do it if they can remember without it&quot;, then sure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You wouldn't take off marks if someone didn't write &quot;Soh Cah Toa&quot; next to all their trig problems, and you wouldn't insist someone use training wheels if they're perfectly capable of riding a bike without them. But if you have students distributing $5-5(2+x)$ as $5-10+5x$ or something rather than $5-10-5x$, then I can't really think of a &lt;em&gt;better&lt;/em&gt; way of explaining to them why this is wrong than the &quot;invisible $-1$&quot; notion.&lt;/p&gt;&#xA;" OwnerUserId="115" LastActivityDate="2015-02-06T20:17:09.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7360" PostTypeId="2" ParentId="7358" CreationDate="2015-02-06T20:21:11.973" Score="4" Body="&lt;p&gt;At the algebra level you are at, you must toss out many of your old ideas of what the symbols mean and use what they are currently defined as. Actually, I've seen many introductory algebra texts that start out with the operator $\star$ to rule out this old intuition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is common at a more mathematically mature level to abstract the idea of some concept with a new notation, symbol, or even a name. This allows for the manipulation of concepts at a level where prior knowledge might get in the way or to have students practice in thinking more abstractly. Just one brief example might be defining $\frac{a}{b} = (a,b)$ so that $(a,b) + (c,d) = (ad+bc,bd)$, highlighting a slightly different area of the concept (and allowing it to be on a ring product too e.g. $\mathbb{Z}\times\mathbb{Z}$). I have also seen professor's use uncommon notation or terminology when they want to discourage students from looking up the solutions to problems online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I don't think it would look at any lower maturity levels. Many students would grow flustered when you convert back to the more accustomed notations, thinking of it as one more thing to keep track of. They usually don't have the practice in abstraction to see that they are the same object. You can see this frustration when presenting a problem to beginning calculus students with just relabeling indices on a summation. They are the same object, but the look is just enough off than many might not realize it at first.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-02-06T20:21:11.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7361" PostTypeId="2" ParentId="7358" CreationDate="2015-02-07T00:00:53.407" Score="2" Body="&lt;p&gt;Indeed, there is nothing wrong to use personal non-standard notations to learn new concepts providing that you know how to change them to the standard ones; otherwise, as a student you don't get the mark that you deserve since your lecturer could hardly understand what you mean. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, the example you mentioned is more about teaching a general concept and then specific ones than using non-standard notations to teach. That one, I don't think has any clear-cut answer. For example, probably years ago you were taught first about some special parallelograms, say square and rectangles, before the general notion of parallelograms. Some people think the other way round is better whenever possible (for example when the general notion is not that much abstract). It depends on the teaching philosophy one advocate.       &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-02-07T00:00:53.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7362" PostTypeId="2" ParentId="41" CreationDate="2015-02-08T18:37:00.983" Score="5" Body="&lt;p&gt;I write like this:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/TCyk9.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And everybody can read what I'm writing. Especially, take look at letter &lt;code&gt;x&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here are my digits:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/XSSVi.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="4723" LastEditorUserId="4723" LastEditDate="2015-02-08T21:32:11.947" LastActivityDate="2015-02-08T21:32:11.947" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7363" PostTypeId="1" CreationDate="2015-02-08T23:38:30.983" Score="12" ViewCount="648" Body="&lt;p&gt;I am teaching a discrete math course, and doing multiply quantified statements. All the book examples are sober and forgettable:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Every real number has a reciprocal.&lt;/p&gt;&#xA;&lt;p&gt;For all triangles x [in a &amp;quot;Tarski world&amp;quot; with a bunch of shapes of different colors], there is a square y such that x and y have the same color.&lt;/p&gt;&#xA;&lt;p&gt;There is no smallest positive real number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And so on. These are good, but I would like to supplement these with examples from pop culture, song lyrics, contemporary politics, etc. -- especially anything amusing, complicated, and/or which drastically changes meaning when you change the order of the quantifiers. Something the students might remember the day after the lecture.&lt;/p&gt;&#xA;&lt;p&gt;Usually I am good at this kind of thing but for now I am drawing a blank. Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="1535" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-02-12T01:12:02.570" Title="Entertaining examples of multiply quantified statements" Tags="|logic|" AnswerCount="10" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7364" PostTypeId="2" ParentId="7363" CreationDate="2015-02-09T01:35:22.277" Score="4" Body="&lt;p&gt;You could have students draw up the quantifications for the following, taken from &lt;em&gt;Alice's Adventures in Wonderland&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;‘Then you should say what you mean,’ the March Hare went on.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;‘I do,’ Alice hastily replied; ‘at least--at least I mean what I&#xA;  say--that’s the same thing, you know.’&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;‘Not the same thing a bit!’ said the Hatter. ‘You might just as well&#xA;  say that “I see what I eat” is the same thing as “I eat what I see”!’&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;‘You might just as well say,’ added the March Hare, ‘that “I like what&#xA;  I get” is the same thing as “I get what I like”!’&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;‘You might just as well say,’ added the Dormouse, who seemed to be&#xA;  talking in his sleep, ‘that “I breathe when I sleep” is the same thing&#xA;  as “I sleep when I breathe”!’&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or the following, from &lt;em&gt;Through the Looking Glass&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;‘Who did you pass on the road?’ the King went on, holding out his hand&#xA;  to the Messenger for some more hay.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;‘Nobody,’ said the Messenger.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;‘Quite right,’ said the King: ‘this young lady saw him too. So of&#xA;  course Nobody walks slower than you.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;‘I do my best,’ the Messenger said in a sulky tone. ‘I’m sure nobody&#xA;  walks much faster than I do!’&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;‘He can’t do that,’ said the King, ‘or else he’d have been here first.&#xA;  However, now you’ve got your breath, you may tell us what’s happened&#xA;  in the town.’&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2015-02-09T09:25:37.063" LastActivityDate="2015-02-09T09:25:37.063" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7365" PostTypeId="2" ParentId="7363" CreationDate="2015-02-09T01:57:04.600" Score="9" Body="&lt;p&gt;For single quantifiers, there is the standard -- &lt;em&gt;jazz&lt;/em&gt; standard -- &lt;a href=&quot;http://quoteinvestigator.com/2013/12/11/cannot-fool/&quot;&gt;example&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Everybody loves my baby.&lt;/li&gt;&#xA;&lt;li&gt;My baby loves nobody but me.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For multiple quantifiers, a &lt;a href=&quot;http://quoteinvestigator.com/2013/12/11/cannot-fool/&quot;&gt;classic&lt;/a&gt; is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You can fool all of the people some of the time&lt;/li&gt;&#xA;&lt;li&gt;You can fool some of the people all of the time&lt;/li&gt;&#xA;&lt;li&gt;But you can't fool all the people all the time.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2015-02-09T02:10:39.473" LastActivityDate="2015-02-09T02:10:39.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7367" PostTypeId="2" ParentId="7363" CreationDate="2015-02-09T03:09:39.300" Score="1" Body="&lt;p&gt;This one might take some extra massaging to make it &amp;quot;mathy&amp;quot; (re: &amp;quot;half&amp;quot; and &amp;quot;part&amp;quot; and &amp;quot;some&amp;quot;), but I like it as a fun twist on Abe Lincoln's classic (which @MattF used in his answer):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Half of the people can be part right all of the time&lt;/p&gt;&#xA;&lt;p&gt;Some of the people can be all right part of the time&lt;/p&gt;&#xA;&lt;p&gt;But all of the people can't be all right all of the time&lt;/p&gt;&#xA;&lt;p&gt;I think Abraham Lincoln said that&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bob Dylan, in &amp;quot;Talkin' World War III Blues&amp;quot;, from &lt;em&gt;The Freewheelin' Bob Dylan&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-02-09T03:09:39.300" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7368" PostTypeId="1" CreationDate="2015-02-09T03:35:47.657" Score="20" ViewCount="6890" Body="&lt;p&gt;I'm teaching a Calc 2 class now (integration and applications) and I'm surprised that more than a handful of students seem to think the graph of $y=x^2$ on $-1\le x\le 1$ is part of a circle!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a quiz question I posed:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consider the part of the parabola $y=x^2$ that goes from the point $(-1,1)$ to $(1,1)$. Suppose I used the arc length formula to find the length of that curve and I got an answer of $2\sqrt{2}$. Explain to me how you can be sure that my answer is too &lt;em&gt;small&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here are two answers I got from students:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;The answer is too small because the graph only extends from (-1,1) to (1,1), which means that the shape is half of a circle and not a whole circle.&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;The distance between these two points is 2 units, and this portion of the parabola resembles a circle. If the diameter is 2, then the circumference should be about 2pi, or 6.28. Half of this would be 3.14, and 2 root 2 is only about 2.83.&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Where does this come from?&lt;/strong&gt; Is there evidence that students tend to confuse parabolas with circles, in general? Or is it only in particular cases, like this one where they were already asked to think geometrically?&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Is there a short presentation I can give in class to, once and for all, convince students these are two different curves?&lt;/strong&gt; Ideally, this could be just a 5-minute (max) discussion in class to say, &quot;Look at how these are different. Remember this!&quot; This could involve basic geometry or calculus or both.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Has anyone else noticed this?&lt;/strong&gt; What was your audience? How did you first notice this phenomenon? What did you do to address it?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(Note: I realize there are likely other issues at play with this particular quiz question. For instance, Student #2 might think that the only way to show some value is too small is to find the correct value and point out that it's bigger. Perhaps that made them make a mistake they wouldn't usually make: conflating parabola with circle. But the fact remains, they conflated the two, and it's that phenomenon about which I'm curious.)&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2015-03-02T21:35:18.200" Title="A parabolic arc is not semicircular. But students think so" Tags="|undergraduate-education|calculus|geometry|students-mistakes|" AnswerCount="10" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="7369" PostTypeId="2" ParentId="7368" CreationDate="2015-02-09T09:14:09.300" Score="15" Body="&lt;ol&gt;&#xA;&lt;li&gt;&#xA;This is pretty natural, I think. People understand things in terms of things that they already know, and while Calc 1/2/3 students should theoretically have a reasonably well developed 'catalog' of curves, it typically isn't the case in practice. They do all know, understand, and love circles though. Second, any infinitesimal portion of a parabola actually is a portion of a circle, ie: can be well approximated by a portion of a circle, and this is 'maximally true' (lets not be too rigorous at 5am) at and around the vertex. Third, all of the reference points which are easy to check ($(-1, 1), (0,0)$, and $(1,1)$) line up correctly.&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&#xA;Calc 2 students might, in general, be deficient in their ability to classify different curves, but I'll bet that very few of them will have trouble differentiating $x^2$ and finding that its slopes at $-1$ and $1$ are $-2$ and $2$ respectively. They also should have little trouble finding that the slopes at the 'sides' of the circle of radius $1$ centered at $(0,1)$ are infinite/undefined. Talking about this contradiction is likely to be sufficient, but be sure to be clear about drawing out 'natural language' statements about each curve, since most students at this point aren't familiar with mathematical argument.&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;...This discussion with your students might be a decent jumping-off point to start talking about curves approximating other curves and getting into Taylor series if that's part of your course.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2015-02-09T10:29:12.840" LastActivityDate="2015-02-09T10:29:12.840" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7370" PostTypeId="1" CreationDate="2015-02-09T09:58:42.287" Score="18" ViewCount="990" Body="&lt;p&gt;I'm a Math Teacher for highly talented students in High School.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's hard for me to determine how fast can I move through topics and how deeply can I dive into different math subjects for enhancing their abilities, while also keeping their interest and fondness for mathematics high. I know that it's a quite ambiguous question but I'm asking about &lt;em&gt;teaching methods and the psychology of teaching for high talents&lt;/em&gt; because I don't have much experience and I don't want to ruin their talents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anybody provide some suggestions or rules for teaching Math to those talents? Or maybe some books about teaching?&lt;/p&gt;&#xA;" OwnerUserId="5287" OwnerDisplayName="mmahdit" LastEditorUserId="64" LastEditDate="2015-02-09T14:32:15.577" LastActivityDate="2015-06-23T10:58:11.370" Title="How can I teach Mathematics to genius students?" Tags="|secondary-education|gifted-students|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7371" PostTypeId="2" ParentId="7368" CreationDate="2015-02-09T16:53:38.237" Score="12" Body="&lt;p&gt;This might be a good opportunity to inject some history into your course. Aristotle thought that cannonball trajectories were line segments, up at an angle then straight down. Look in Chapter 5 of &quot;The Day the Universe Changed&quot; by James Burke. Leonardo realized that the trajectory was curved, but he did not seem to further quantify. In about 1537 Tartaglia attempted to quantify: the cannonball angles up in a straight line, then follows the arc of a circle along the top, then follows a line segment down. There are many other references. One is &quot;Exploring the Limits of Preclassical Mechanics&quot; by Peter Damerow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then of course people used think that planets followed circular orbits, presumably because &quot;circles are perfect.&quot;  &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-02-09T16:53:38.237" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7372" PostTypeId="2" ParentId="7363" CreationDate="2015-02-09T17:14:35.500" Score="3" Body="&lt;p&gt;There's a lovely bit from &lt;em&gt;Catch-22&lt;/em&gt; that I'm quite fond of (I'll boldface the relevant part, but you really need the full passage to appreciate how wonderfully Carollian it is):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;All right,&amp;quot; said the colonel. &amp;quot;Just what the hell did you mean?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;I didn't say  you couldn't punish me, sir.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;When?&amp;quot; asked the colonel.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;When what, sir?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Now you're asking me questions again.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;I'm sorry, sir. I'm afraid I don't understand your question.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;When didn't you say we couldn't punish you? Don't you understand my question?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;No, sir. I don't understand.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;You've just told us that. Now suppose you answer my question.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;But how can I answer it?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;That's another question you're asking me.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;I'm sorry, sir. But I don't know how to answer it. &lt;strong&gt;I never said you couldn't punish me.&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Now you're telling us when you did say it. &lt;strong&gt;I'm asking you to tell us when you didn't say it.&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Clevinger took a deep breath. &lt;strong&gt;&amp;quot;I always didn't say you couldn't punish me, sir.&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;That's much better, Mr. Clevinger, even though it is a barefaced lie. Last night in the latrine. Didn't you whisper that we couldn't punish you to that other dirty son of a bitch we don't like? What's his name?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Yossarian, sir,&amp;quot; Lieutenant Scheisskopf said.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Yes, Yossarian. That's right. Yossarian. Yossarian? Is that his name? Yossarian? What the hell kind of a name is Yossarian ?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Lieutenant Scheisskopf had the facts at his finger tips. &amp;quot;It's Yossarian's name, sir,&amp;quot; he explained.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-02-09T17:14:35.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7373" PostTypeId="2" ParentId="7363" CreationDate="2015-02-09T18:50:13.203" Score="3" Body="&lt;p&gt;I remember some advertising slogans.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Everybody doesn't like something.  But nobody doesn't like Sara Lee.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There isn't a car on the road that shouldn't be using it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-02-09T18:50:13.203" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7374" PostTypeId="1" AcceptedAnswerId="7379" CreationDate="2015-02-09T19:00:43.480" Score="7" ViewCount="359" Body="&lt;p&gt;I am currently introducing function operations to my basic level Algebra students and it has been semi-disastrous.  The biggest problem I have with them is the notation that &#xA;$$(f+g)(x)=f(x)+g(x)$$&#xA;I  have been trying to explain that the reason for the notation on the left is the fact that when you add two (polynomial) functions, you get a polynomial function in return.  Hence the $(x)$ on the right is indicative of this idea of function closure under the operations of addition, subtraction, multiplication, division, and composition.  The district I teach at wants kids to understand the notation.  If I just make them do $f(x)+g(x)$ they understand, but, with products for example, if I do $(fg)(x)$ they are lost.  Any ideas how to appoach the subject with the basic learners?&lt;/p&gt;&#xA;" OwnerUserId="4728" LastEditorUserId="143" LastEditDate="2015-02-09T19:15:39.190" LastActivityDate="2015-02-09T19:48:11.780" Title="Explaining Closure to &quot;Basic&quot; Algebra 2 Students" Tags="|secondary-education|mathematical-pedagogy|algebra|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7375" PostTypeId="2" ParentId="7368" CreationDate="2015-02-09T19:06:05.643" Score="5" Body="&lt;p&gt;As for your second question, there are many ways to get a contradiction from the assumption that that curve is half a circle:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If that curve were half a circle, we should be able construct a full cirle by putting two of such curves together. However we get this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/MQ80d.png&quot; alt=&quot;not a circle&quot;&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If that curve were half a circle, the point $\left (\tfrac{1}{\sqrt{2}}, 1-\tfrac{1}{\sqrt{2}} \right)$ would have to be on it, as its distance to the center of the circle at $(0,1)$ is 1. However, $\left (\tfrac{1}{\sqrt{2}} \right )^2 ≠ 1-\tfrac{1}{\sqrt{2}}$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="75" LastActivityDate="2015-02-09T19:06:05.643" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7376" PostTypeId="2" ParentId="7368" CreationDate="2015-02-09T19:27:01.893" Score="1" Body="&lt;p&gt;This is a case where an answer that is too detailed (in a way that includes a mistake) is worse than an answer that is just detailed enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a simple proof for the original poster's question.  Notice that this proof does not describe the shape of the parabola, other than not being just two line segments.  Also, notice that this proof (with a minor tweak at step 7) is equally valid for a circle:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The graph y = x&lt;sup&gt;2&lt;/sup&gt; goes through the (x, y) points (-1, 0), (0, 0), and (1, 1).&lt;/li&gt;&#xA;&lt;li&gt;The distance between (-1, 0) and (0, 0) is √2.&lt;/li&gt;&#xA;&lt;li&gt;The distance between ( 0, 0) and (0, 1) is also √2.&lt;/li&gt;&#xA;&lt;li&gt;We have accounted for the entire alleged length of 2√2, using just two straight lines.&lt;/li&gt;&#xA;&lt;li&gt;A straight line is the shortest distance between two points.&lt;/li&gt;&#xA;&lt;li&gt;If the curve between (-1, 1) and (1, 1) goes through any points that are not on these two line segments, then the arclength must be greater than 2√2.&lt;/li&gt;&#xA;&lt;li&gt;The graph y = x^2 goes through (1/2, 1/4).&lt;/li&gt;&#xA;&lt;li&gt;The two-segment approximation goes through (1/2, 1/2).&lt;/li&gt;&#xA;&lt;li&gt;Therefore the curve between (-1, 1) and (1, 1) goes through a point that is on neither of the two line segments.&lt;/li&gt;&#xA;&lt;li&gt;The arclength must be greater than 2√2.  &lt;strong&gt;Q.E.D.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For a very informal proof, steps 6 - 8 can be summarized as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A parabola is curved; it does not have any straight-line segments.&lt;/li&gt;&#xA;&lt;li&gt;Optionally:  We know this because a straight-line has either a zero second derivative, or an infinite second derivative.  Whereas a parabola has a non-zero, non-infinite second derivative.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In terms of the difference between a parabola and a circle, that last &quot;optional&quot; explanation gets to the heart of the matter.  A circle has constant curvature -- it is everywhere bending (at the same rate) toward a single point.  A parabola has a constant second derivative -- it is everywhere bending away from a single line.  This allows the parabola to extend forever (or until it hits the ground, whichever happens first), whereas the circle wraps back around onto itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a sufficiently short distance, a parabola can approximate a circle, and &lt;em&gt;vice versa&lt;/em&gt;.  But the approximation cannot be perfect, because they are two different shapes (that are not defined piece-wise).&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-02-09T20:28:16.010" LastActivityDate="2015-02-09T20:28:16.010" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7377" PostTypeId="2" ParentId="7374" CreationDate="2015-02-09T19:35:56.667" Score="3" Body="&lt;p&gt;Are the kids familiar with vectors and coordinates?&#xA;If so, consider two vectors $\vec a,\vec b\in\mathbb R^3$, say.&#xA;If $\vec c=\vec a+\vec b$, they probably know that $\vec c_i=\vec a_i+\vec b_i$.&#xA;(It is convenient here to denote the $i$th component of $\vec c$ by $\vec c_i$ instead of $c_i$.)&#xA;But this means that $\vec c_i=(\vec a+\vec b)_i$, right?&#xA;Now it might make sense to write $\vec a_i+\vec b_i=(\vec a+\vec b)_i$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this sounds reasonable to the students, try to explain what is going on: the vectors $\vec a$ etc. are the objects that we want to study, and they are determined by their coordinates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same happens with polynomials.&#xA;The things we wish to study are polynomials, say $f$ and $g$ (like we had vectors $\vec a$ and $\vec b$ before).&#xA;If you think of polynomials like vectors (they in fact are vectors, but you might not want to mention this depending on what you want to do), then $f$ is the vector and $f(x)$ is the component.&#xA;Make sure your students understand the difference between $f$ and $f(x)$ &amp;ndash; and that of $\vec a$ and $\vec a_i$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now what might it mean to add two polynomials $f$ and $g$?&#xA;Well, it is defined by its &quot;coordinates&quot;, and the analogy from the vector side is $\vec a_i+\vec b_i=(\vec a+\vec b)_i$.&#xA;This translates to $f(x)+g(x)=(f+g)(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Alternative idea:&lt;/strong&gt;&#xA;Try to introduce vectors (if the kids are familiar with them) as functions.&#xA;Write $a(i)$ or $\vec a(i)$ instead of $a_i$ or $\vec a_i$; after that bit of new notation the jump to polynomials (and other functions) might be easier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Final remark:&lt;/strong&gt;&#xA;I think the key is making a difference between $f$ and $f(x)$.&#xA;You have little hope before they mean different things.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2015-02-09T19:35:56.667" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7378" PostTypeId="2" ParentId="7370" CreationDate="2015-02-09T19:43:36.073" Score="12" Body="&lt;p&gt;The key with students like this is to allow them to dictate the pace of teaching, while providing interesting and challenging material. Now, that is very easy to say, but here are a couple of pointers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Don't force them to go through the motions with topics they are already comfortable with. Allow them to test out of any unit, and find them extension material until their colleagues catch up.&lt;/li&gt;&#xA;&lt;li&gt;Use their natural competitiveness by giving them the same problems in the same order, while keeping track (not necessarily publicly) of progress. A useful aid to this could be the tracking features in something like Schoology.com&lt;/li&gt;&#xA;&lt;li&gt;You are going to need a lot of problems! Try any of the books on this list, especially the Mathematical Circles, the Stanford Problems, and if you are feeling insanely confident in their abilities, the IMO one:&#xA;&lt;a href=&quot;https://www.goodreads.com/list/user_vote/4298881&quot;&gt;https://www.goodreads.com/list/user_vote/4298881&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="3571" LastActivityDate="2015-02-09T19:43:36.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7379" PostTypeId="2" ParentId="7374" CreationDate="2015-02-09T19:48:11.780" Score="4" Body="&lt;p&gt;For your students, they are defined that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Particularly in the lower level algebra,&#xA;\begin{align*}&#xA;(f+g)(x) &amp;amp;= f(x) + g(x)\\&#xA; (fg)(x) &amp;amp;= f(x)g(x)&#xA;\end{align*}&#xA;are taken to be the definition of usual multiplication and addition. Simply stated, this notation can be used to talk about the names $f+g$ and $fg$ of the new functions created by adding or multiplying them. And whenever they see the above, that's what they need to write. It is similar for them with composition being $(f \circ g)(x) = f(g(x))$ Getting into the functional ring theory will only serve to confuse even though it is interesting to us.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-02-09T19:48:11.780" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7380" PostTypeId="2" ParentId="7368" CreationDate="2015-02-09T19:48:24.267" Score="4" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;depending on your students' backgrounds this could come from plain ignorance, oversimplification at a younger age, poor teaching, etc. I don't think that this necessarily matters when moving forward but it is a good thing to know and may allow you to predict future gaps your students have.  I would just simply ask them and if they don't remember or don't know then move on, but they may be able to give you some interesting insight.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think a very effective way to demonstrate that projectile motion is not semi-circular is to show them a &quot;time lapse&quot; or &quot;stop motion&quot; video of a projectile and have them trace the path on the board.  an easy way to do this is to film with an iPhone or something similar a slow-mo video from the side of you throwing a basketball through the air.  Slow the video down and project it onto the black/whiteboard in your classroom.  Have the students watch the video a few times, having them focus on the path of the ball.  Then, have a student come up and trace the path of the ball on the board as it moves through the air with a marker/chalk.  This should make it pretty clear that the path is not circular.  Well then what is it?? It is a parabola! Which can lead right into a discussion on parabolas.  Alternatively, challenge the students to throw a ball of paper or a pencil in the air so that its path is a semi-circle and they will very quickly find out it is impossible. It sounds like there is a very fundamental misunderstanding, so showing them sophisticated proofs etc. might confuse even more.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;I have definitely noticed this, as well as many other misunderstandings! the joys of teaching!&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-02-09T19:48:24.267" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7381" PostTypeId="2" ParentId="7321" CreationDate="2015-02-09T20:12:40.160" Score="2" Body="&lt;p&gt;James,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have the book, you could have your student read in David Poole's Linear Algebra, chapter 2.4, there is a section on Finite Linear Games as related to systems of equations. There is also a section on Markov Chains. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could send you a pdf of that (those?) section(s) if you are interested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Zach&lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2015-02-09T20:12:40.160" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7382" PostTypeId="2" ParentId="7312" CreationDate="2015-02-09T20:15:10.993" Score="3" Body="&lt;p&gt;John Stillwell's &quot;Mathematics and its History&quot; -- exercises and all. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have used Stillwell's book to connect with the readings in Journey Through Genius, so it is worth looking at for more context and actual problems to work. &#xA;I also have a friend who has used Dunham as a text and has created resources. I will obtain those for the OP, if Aeryk would like them.&lt;/p&gt;&#xA;" OwnerUserId="1249" LastEditorUserId="1249" LastEditDate="2015-02-10T23:50:33.493" LastActivityDate="2015-02-10T23:50:33.493" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7383" PostTypeId="2" ParentId="7368" CreationDate="2015-02-09T21:16:31.850" Score="9" Body="&lt;p&gt;What is most concerning here is that this is coming from &lt;em&gt;calculus&lt;/em&gt; students. This sort of misconception should have been removed during their algebra and calculus courses. To have this happen in a &lt;em&gt;second calculus&lt;/em&gt; course is of great concern. Calculus students seem to make most of their mistakes on their algebra anyway, so if you can, pay attention to these students and how they handle the algebra required to implement the things they're learning in your calculus class.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Now, is this common? I would expect this to be common at an algebra-level class. I would expect this to not be common once someone has gone through a trig (tangent lines should make this a giveaway, and they had to at least touch on the different conic sections, right?) or multiple algebra courses (calculating slope enough times would raise a flag that they're different).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there something you could do? Well, you could try putting a real semicircle up and showing they're two different shapes might do it!! Alternatively, you could get out a flashlight and start making shapes on the blackboard showing how the parabola turns into an ellipse that turns into a circle as you rotate the flashlight (i.e. teach them about conic sections.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="565" LastActivityDate="2015-02-09T21:16:31.850" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7384" PostTypeId="2" ParentId="7368" CreationDate="2015-02-09T23:28:31.260" Score="4" Body="&lt;p&gt;Based on the &quot;evidence&quot; given, my best guess is that the matter is not about confusing parabolas with circles. Rather, it is the result of a more general phenomenon that is indeed very common: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;If you cannot solve the problem, solve what you can solve!&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Consider that the problem explicitly state &quot;the part of the parabola&quot;, so why one should replace that with a circle. The reason is that he or she does not know the arc length for a parabola by heart, but knows that for a circle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, I strongly doubt that students confuse &quot;big&quot; part of a parabola with a circle. For years I have challenged cal 1 students by asking them to guess the shape of a hanging chain. They always come with parabola, never with circle (of course, the answer is neither of them). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third, it is likely that they don't know the formula neither for a parabola nor for a circle. Second and third together means if you still want to go with that five minutes lecture, it is better to focus on the difference in formulas not in shapes. That said, it seems that even the first student you mentioned knows the general shape of a parabola since he or she has replaced it with &quot;partly&quot; a similar looking figure, not for example, with two segments joining $(-1,1)$ to $(0,0)$ and $(0,0)$ to $(1,1)$.     &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="1217" LastEditDate="2015-02-10T00:00:17.770" LastActivityDate="2015-02-10T00:00:17.770" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7385" PostTypeId="1" AcceptedAnswerId="7422" CreationDate="2015-02-10T05:03:24.677" Score="22" ViewCount="955" Body="&lt;p&gt;Consider the following example of working &quot;directly&quot; with differentials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to approach the problem of determining the arc length of the graph of a single-variable function is to imagine the arc broken up into infinitesimal line segments. Such a line segment is the hypotenuse of a right triangle with legs of length $dx$ and $dy$ (infinitesimal changes in $x$ and $y$, respectively). By the Pythagorean Theorem, the length of the hypotenuse is &#xA;$$&#xA;\sqrt{(dx)^2 + (dy)^2},&#xA;$$&#xA;which one can rearrange to either&#xA;$$&#xA;\sqrt{1 + \left( \frac{dy}{dx} \right)^2} \ dx&#xA;$$&#xA;or&#xA;$$&#xA;\sqrt{\left( \frac{dx}{dy} \right)^2 + 1} \ dy&#xA;$$&#xA;depending on whether it is easier to integrate with respect to $x$ or $y$, respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, can one be led astray by reasoning directly with differentials in this way (as opposed to defining a Riemann sum and taking a limit)? If so, are such examples difficult to contrive or are they typical enough that arguments like the one I presented should be avoided?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: I feel my question was not clearly stated. By &quot;can one be led astray&quot;, I mean I am concerned about other applications in Calculus or Physics where a perfectly sensible-looking manipulation of differentials (like the one in my example) leads one to an incorrect conclusion.&lt;/p&gt;&#xA;" OwnerUserId="222" LastEditorUserId="1385" LastEditDate="2017-04-20T18:05:00.987" LastActivityDate="2019-09-25T14:52:38.240" Title="Non-Rigorous Use of Differentials" Tags="|calculus|intuition|integration|infinitesimals|" AnswerCount="8" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7386" PostTypeId="2" ParentId="7368" CreationDate="2015-02-10T05:49:29.220" Score="6" Body="&lt;p&gt;(this answer is fairly speculative)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;People don't always speak precisely; lacking good words to precisely express their meaning, they may be falling back on more familiar words and ideas that they believe are 'good enough' to use, and may not have expected you to read their words so literally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;e.g. the second student does even take care to talk about it being a &lt;em&gt;resemblance&lt;/em&gt; than meant literally. And his intuition is that the difference is small enough that looking at the arclength of the circle makes for a decent sanity check on the value. (and didn't realize you were asking something rigorously provable, or at maybe they did, but weren't able to come up with a better idea after getting this one in their head)&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2015-02-10T05:49:29.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7387" PostTypeId="2" ParentId="7385" CreationDate="2015-02-10T13:46:44.490" Score="4" Body="&lt;p&gt;One &lt;em&gt;can&lt;/em&gt; be misled by anything and everything.  However mathematicians have written things like&#xA;$$&#xA;ds^2 = E\, du^2 + 2F\, du\, dv + G\, dv^2&#xA;$$&#xA;for a long time without problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/First_fundamental_form&quot; rel=&quot;nofollow noreferrer&quot;&gt;LINK&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-02-10T13:46:44.490" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7388" PostTypeId="2" ParentId="7363" CreationDate="2015-02-10T14:02:19.750" Score="2" Body="&lt;p&gt;Another song: &lt;a href=&quot;https://www.youtube.com/watch?v=AWMXRV28Bzw&quot; rel=&quot;nofollow&quot;&gt;Happy Together&lt;/a&gt;, by The Turtles&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The lines: I can't see me lovin' nobody but you   For all my life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And: The only one for me is you   And you for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The fact that he says the same thing twice, instead of changing direction, always bugged me.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-02-10T14:02:19.750" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7389" PostTypeId="2" ParentId="7363" CreationDate="2015-02-10T15:39:48.170" Score="2" Body="&lt;p&gt;From &quot;Maybe This Time&quot; from the musical Cabaret: &quot;Everybody loves a winner/So nobody loved me.&quot;  (This exhibits, of course, a very common mistake in quantifier manipulation, which bothers me every time I hear the song.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-02-10T15:39:48.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7390" PostTypeId="2" ParentId="7368" CreationDate="2015-02-10T17:36:17.570" Score="2" Body="&lt;p&gt;I think your second student is closer than you think but maybe didn't couch it correctly. A parabola that passes through the two endpoints and the midpoint of a half circle is intuitively always smaller than the half circle. It's also intuitively larger than an abs function that passes through the same points. The lengths of both of those shapes are easy to find and the length of the parabolic arc should be between them. His problem is that he found the bigger one when he needed to find the smaller (the abs function has a length of 2 sqrt(2) and since the parabola is always bigger...) I would ask him to prove or disprove his implied assumption that the arc of the parabola is always smaller than the arc of the circle (ie there is no possbile transform that could make the parabola bigger than the circle) for some extra credit. &lt;/p&gt;&#xA;" OwnerUserId="4735" LastActivityDate="2015-02-10T17:36:17.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7391" PostTypeId="2" ParentId="7385" CreationDate="2015-02-10T19:51:26.623" Score="1" Body="&lt;p&gt;Before I answer I'd like to clarify that I've never taught this topic and that any teaching should be adapted to the learners' needs. That said the topic itself suggests we are talking about talented students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An important question for me is &quot;What are the students learning?&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue they are practicing manipulating variables. In particular they are realizing they can put variables in and out of root signs by either squaring or square rooting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also say they are becoming familiar with the arc length formula and calculating it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think there is anything wrong with these learning outcomes at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A different question is &quot;What could they be learning?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think there is an opportunity for them to be learning about the importance of approximations.  This to me is a deeper and more challenging skill. I struggle myself if I'm honest. I'm not necessarily suggesting they learn the full rigours of a Riemann sum but perhaps a halfway house?  I reiterate I have no experience at this level. However asking students to approximate the length of a line with fixed $\delta x=0.2$ would be a useful exercise. What happens if $\delta x =0.1$? I feel that the limiting process involved in integration is too readily forgot not least by myself. Approximating other objects or circumstances and forming other integrals might be less of a daunting prospect for some once they familiarize themselves with easier examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarize I think working with infinitesimal fine but I don't think I'd reason with them.  Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-02-10T19:51:26.623" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7392" PostTypeId="1" AcceptedAnswerId="7399" CreationDate="2015-02-11T01:54:56.670" Score="17" ViewCount="10764" Body="&lt;p&gt;Can anyone offer a concise, convincing explanation for why cutting&#xA;a Möbius strip along a line, not midway but rather one-third of the width of&#xA;the strip, and eventually joining back to itself, produces two linked, twisted&#xA;loops, one long, one short:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/DbCxU.jpg&quot; alt=&quot;MobiusThird&quot;&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;(Image from &lt;a href=&quot;http://www.wikihow.com/Explore-a-Mobius-Strip&quot; rel=&quot;noreferrer&quot;&gt;this web site&lt;/a&gt;.)&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;I am not seeking a proof, but rather an explanation that could convince&#xA;essentially anyone paying sufficient attention.&#xA;Especially the lack of symmetry in the result can be surprising.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider it a teaching challenge. :-)&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="143" LastEditDate="2016-09-28T06:48:46.210" LastActivityDate="2022-01-01T22:47:17.220" Title="Explanation for cutting a Möbius strip at one-third its width" Tags="|topology|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7393" PostTypeId="2" ParentId="7392" CreationDate="2015-02-11T02:31:46.563" Score="5" Body="&lt;p&gt;The outer two-thirds are the same piece; the inner third is a different piece from the outer two-thirds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There have to be two different pieces, because at all points along the cut line, the cut is separating a piece that &quot;contains the centerline&quot; from a piece that &quot;contains the original edge&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because both edges are the same edge, the outer two-thirds form a single piece.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because the outer piece has twice the area as the inner piece, but has the same width, it must be twice as long.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2015-02-11T02:31:46.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7394" PostTypeId="1" AcceptedAnswerId="7395" CreationDate="2015-02-11T03:09:29.293" Score="13" ViewCount="958" Body="&lt;p&gt;What the title says. I got my degree in math last year and now I'm working on a master's independent study project through my education department finding math tasks for K-12 curriculum aligned with the common core standards. These could be tasks out of textbooks, online resources, games, apps, whatever. Specifically, I'm looking for 8th grade tasks, but anything will help!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is two resources that I have (I can't post my entire list right now because I just made an account here and am restricted to 2 links until I get 10 reputation)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://illuminations.nctm.org/&quot;&gt;NCTM&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://youcubed.stanford.edu/tasks/&quot;&gt;Stanford Education&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also if you have any recommendations on textbooks, that would be great! Right now I'm just trying to gather as many resources as possible to start analyzing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;edit: now that I have more than 10 reputation, here's my complete list:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://docs.google.com/a/colorado.edu/spreadsheet/pub?key=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE&amp;amp;output=html&quot;&gt;Dan Meyer's 3-act math tasks&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://map.mathshell.org/materials/tasks.php&quot;&gt;MARS - Mathematics Assessment Resource Service&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://illuminations.nctm.org/&quot;&gt;NCTM&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://youcubed.stanford.edu/tasks/&quot;&gt;Stanford Education&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I posted this question to reddit.com/r/math earlier and got these resources from sjhatters:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.tes.co.uk/teaching-resources/&quot;&gt;Everything&lt;/a&gt; Make an account. It's a goldmine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.transum.org/Software/SW/Starter_of_the_day/index.htm&quot;&gt;Start of the day&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.coops-online.co.uk/&quot;&gt;Worksheets&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.mrbartonmaths.com/&quot;&gt;Everything&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://numberloving.co.uk/resources.php&quot;&gt;Everything&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://nrich.maths.org/frontpage&quot;&gt;Interesting problems&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://sites.google.com/site/phsalevelmathematics/complex-numbers&quot;&gt;Worksheets&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://examsolutions.net/&quot;&gt;Exam solutions&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://braingle.com/&quot;&gt;Everything&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://plus.maths.org/content/&quot;&gt;Everything&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/user/khanacademy&quot;&gt;Vids&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/user/numberphile&quot;&gt;Vids&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some are UK specific.&lt;/p&gt;&#xA;" OwnerUserId="4737" LastEditorUserId="262" LastEditDate="2015-11-09T04:27:53.583" LastActivityDate="2020-05-15T19:29:51.393" Title="Where do you find math tasks?" Tags="|secondary-education|examples|" AnswerCount="7" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7395" PostTypeId="2" ParentId="7394" CreationDate="2015-02-11T03:55:46.713" Score="5" Body="&lt;p&gt;great question, beat me to a lot of great resources! while it is broad i think that it is good to try and have a collection of sources for math tasks as this is what secondary math teachers end up spending most of their time doing.  I know that you are requesting actual tasks for your study but I will also touch upon creating tasks for fellow teachers who may read this and are trying to figure out good ways to efficiently make tasks. I tried to go more for categories of sites that you can find more of easily with a google search, rather than just posting links that may need to be updated in the future. I will post my most immediate list and update it as I find new sources or new sources are brought to my attention.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Check your or other school districts or state education departments websites for resources, anchors, benchmarks etc. &lt;a href=&quot;http://www.pdesas.org/Standard/AnchorsDownloads&quot;&gt;Here&lt;/a&gt; are links to the PA assessment anchors which surprisingly have some pretty good tasks. Also check for university and school websites, there are often class pages or archives that have all types of tasks&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The textbook that your school uses or any other text book and its supplementary material.  For Geometry we use Discovering Geometry, Algebra uses Holt, pre-calc we use Pre-Calculus Graphical, Numerical, Algebraic (only math classes we offer :(.  Many of the tasks that I use are adapted or sometimes taken directly from the supplementary material provided by these textbooks.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Math circle or other math club or organization websites generally have an archive or sampling of some sort and can give good ideas for out of the box lessons.  Here is the &lt;a href=&quot;http://www.math.ucla.edu/~radko/circles/archive.shtml?&quot;&gt;Los Angeles Math Circle archive&lt;/a&gt;, so full of great tasks and lessons!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Depending on the type of task that you are trying to create, &lt;a href=&quot;http://www.kutasoftware.com/&quot;&gt;Kuta Software&lt;/a&gt; may be helpful.  It allows you to pick the type of problem that you would like to create and offers a ton of options for each problem type and then generates questions on the fly.  It does not pull from a database so the questions are &quot;guaranteed&quot; to be unique.  I really like it for quick and dirty practice problem sheets either as a quick quiz or for homework.  It does cost money but they do offer two week trials of the full version for you to test out. Does not create word problems but it does allow you to create your own questions so you can still use it to make assessments.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-02-11T03:55:46.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7396" PostTypeId="2" ParentId="7363" CreationDate="2015-02-11T07:10:21.147" Score="1" Body="&lt;p&gt;Phew!... I got first to enter this answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://en.m.wikipedia.org/wiki/Who&amp;#39;s_on_First%3F&quot; rel=&quot;nofollow&quot;&gt;&quot;Who's on First?&quot; routine&lt;/a&gt; by Abbott &amp;amp; Costello.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2015-02-11T07:10:21.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7397" PostTypeId="2" ParentId="7363" CreationDate="2015-02-11T07:15:11.447" Score="2" Body="&lt;p&gt;Raymond Smullyan gave a nice example in one of his books (I forget now which): Two restaurants have signs on their doors. One says&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Good food is not cheap.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The other says&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Cheap food is not good.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The two statements are equivalent, but one makes you think of delicacies, while the other about disgusting greasy bites...&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2015-02-11T07:15:11.447" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7398" PostTypeId="2" ParentId="7328" CreationDate="2015-02-11T07:52:32.257" Score="1" Body="&lt;p&gt;Directly referring to the phenomenon in the title: Consider&#xA;$$x_n = \sum_{k=1}^n \frac{1}{k}.$$&#xA;Then for any $p$ and $n\to\infty$&#xA;$$|x_{n+p}-x_n| = \sum_{k=n+1}^{n+p}\frac1k \leq \frac{p}{n+1}\to 0.$$&#xA;But $(x_n)$ does not converge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, that's really about series but still is a valid example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a more applied note: If some studies optimization algorithms, then one often proves convergence of some algorithm by first noting that $\|x_{n+1}-x_n\|\to 0$ then additionally show boundedness of the $x_n$'s and then improve to $\sum_{k=1}^\infty\|x_{k}-x_{k-1}\|&amp;lt;\infty$ which shows that the sequence is Cauchy…&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2015-02-11T07:52:32.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7399" PostTypeId="2" ParentId="7392" CreationDate="2015-02-11T14:39:38.177" Score="14" Body="&lt;p&gt;The middle third is obtained by trimming the edges off the original möbius loop. It is therefore simply a thinner möbius loop (the short loop).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The outside thirds of the möbius loop are obtained by cutting the loop in half and trimming 1/3 off the edge that was not originally the outside edge. It is the same as cutting the strip in half: as the outside edges are linked by a twist, the edge is a single loop twice as long as the original.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagining that a wire runs along the centre of the original möbius loop, one can follow the path of the outside edge as you trace along the wire. The edge slowly twists around the inside loop, so that after following the wire for 2 rotations, the edge has made a complete loop around the centre of the Möbius strip, going through the middle of the wire loop. As the wire becomes the small Möbius strip, and edge becomes the long Möbius strip, the long strip loops itself once around the small strip. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-02-11T14:39:38.177" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7400" PostTypeId="2" ParentId="7363" CreationDate="2015-02-12T01:12:02.570" Score="1" Body="&lt;p&gt;Can anybody find me somebody to love?&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2015-02-12T01:12:02.570" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7401" PostTypeId="2" ParentId="7392" CreationDate="2015-02-12T02:05:12.447" Score="3" Body="&lt;p&gt;A nice way to think about the result is to take a long strip of paper, cut it into three pieces, and then twist and join them.  This results in the same configuration as cutting it after twisting and joining, but it lets you track each strand individually.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2015-02-12T15:04:33.277" LastActivityDate="2015-02-12T15:04:33.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7403" PostTypeId="2" ParentId="7392" CreationDate="2015-02-12T20:49:34.453" Score="2" Body="&lt;p&gt;To explain the linking, consider three long parallel pieces of string attached to a long rectangular piece of paper.  Joining the edges in a cylindrical fashion results in three disjoint loops, while using a Moebius twist joins two of the strings together, and does so in a linked fashion to the loop created by the middle string.  From this image it is not too hard to switch to your ribbon model and count the number of twists in each ribbon.  Perhaps someone will be kind enough to provide an image to illustrate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Twist Your Brain To See&quot; Paseman, 2015.02.12&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-02-12T20:49:34.453" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7404" PostTypeId="2" ParentId="7394" CreationDate="2015-02-12T21:01:02.493" Score="2" Body="&lt;p&gt;I don't think you had this one: &lt;a href=&quot;http://www.risps.co.uk/&quot; rel=&quot;nofollow&quot;&gt;RISPS&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll try to post more later...&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-02-12T21:01:02.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7405" PostTypeId="2" ParentId="7394" CreationDate="2015-02-13T04:41:48.213" Score="8" Body="&lt;p&gt;For teaching &lt;strong&gt;&lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot; rel=&quot;nofollow noreferrer&quot;&gt;CCSS Mathematical Practice Standards&lt;/a&gt;&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;&lt;strong&gt;&lt;a href=&quot;http://fivetriangles.blogspot.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;1. Five Triangles&lt;/a&gt;&lt;/strong&gt;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Though advertised as problems for grades 6 through 8, many of their problems are appropriate for high schoolers, too. From their About page:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Five Triangles shares a philosophy of mathematics education that&#xA;  favours concise, unambiguous, and calibrated problems that lend&#xA;  themselves to a variety of approaches, with the additional condition&#xA;  that, throughout the solution process, cognitive stress should be&#xA;  maintained.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In my secondary geometry class, we spent a good 20 minutes thinking about their &quot;&lt;strong&gt;&lt;a href=&quot;http://fivetriangles.blogspot.com/2012/04/area-problem.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;2. Odd quad&lt;/a&gt;&lt;/strong&gt;&quot; problem and ended up discussing what it means to have &quot;insight&quot; in problem solving:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;img src=&quot;https://i.stack.imgur.com/SUSSfm.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;&lt;a href=&quot;http://balancedassessment.concord.org/tasksnum.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;2. Balanced Assessment&lt;/a&gt;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Though no longer updated, this resource is a treasure trove of longer performance assessment tasks with suggested scoring guides. I am fond of their &quot;&lt;strong&gt;–Ness&lt;/strong&gt;&quot; tasks for teaching mathematical modeling. Here is an excerpt from their &quot;&lt;strong&gt;&lt;a href=&quot;http://balancedassessment.concord.org/hl002.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Square-Ness&lt;/a&gt;&lt;/strong&gt;&quot; task:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/4897T.png&quot; alt=&quot;Balanced Assessment: &amp;quot;Square-Ness&amp;quot;&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;&lt;a href=&quot;http://matharguments180.blogspot.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;3. MathArguments180.com&lt;/a&gt;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;This is a collection of quick ideas from the internet for starting discussions or &quot;arguments&quot; in the classroom. The posts vary in readiness for use, and many times, they are just a perplexing image for students to talk about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this same short-form presentation of task ideas, I recommend:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;&lt;a href=&quot;http://www.estimation180.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Estimation180&lt;/a&gt;&lt;/strong&gt;, which presents a photo, and students take a guess&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;&lt;a href=&quot;http://www.visualpatterns.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Visual Patterns&lt;/a&gt;&lt;/strong&gt;, which gives Stages 1–3, and students have to find Stage 43&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;&lt;a href=&quot;http://www.playwithyourmath.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;4. Play With Your Math&lt;/a&gt;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;This one is a poster series that a few colleagues and I started at our high school. We find problems that have &quot;play potential&quot; and try to present them in a more visual format in hopes to lower the barrier to entry. Here is an example of our design for a &lt;a href=&quot;http://matharguments180.blogspot.com/2014/02/day-17-breaking-twenty-five.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathArguments180 problem&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;img src=&quot;https://i.stack.imgur.com/pJV64m.png&quot; alt=&quot;Play With Your Math: &amp;quot;1. Split 25&amp;quot;&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="3484" LastActivityDate="2015-02-13T04:41:48.213" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7406" PostTypeId="1" CreationDate="2015-02-13T11:57:35.560" Score="9" ViewCount="1122" Body="&lt;p&gt;I am looking for a free online (Q &amp;amp; A) platform service for using in my math classes which permits the students to ask their questions, and supports Mathjax/Latex like on Mathoverflow/Math.SE/Matheducators and its platform resembles to that of stackexchange websites.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I already know the service &lt;a href=&quot;http://www.piazza.com/&quot;&gt;piazza&lt;/a&gt; but its platform is not as user-friendly as that of stackexchange websites.&#xA;If there is not such a service, suggesting any other alternative would be helpful.&lt;/p&gt;&#xA;" OwnerUserId="3353" LastEditorDisplayName="user507" LastEditDate="2020-12-02T14:09:40.817" LastActivityDate="2020-12-02T14:09:40.817" Title="More user-friendly alternatives to Piazza service" Tags="|online-instruction|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7408" PostTypeId="1" CreationDate="2015-02-14T00:38:52.840" Score="6" ViewCount="2223" Body="&lt;p&gt;I am looking in too getting a textbook for my son who I will be teaching algebra soon (homeschool). I don't like math, never did and am not very good at it. Quite frankly, my son isn't good at it either. So I am looking for a textbook that focuses on the &lt;a href=&quot;http://en.wikipedia.org/wiki/Rote_learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;rote method&lt;/a&gt; which is easier so we can both get through it together.&lt;/p&gt;&#xA;" OwnerUserId="4742" LastEditorUserId="201" LastEditDate="2015-02-14T14:48:25.987" LastActivityDate="2019-04-11T03:37:51.123" Title="Best rote high school algebra textbook?" Tags="|algebra|textbooks|homeschooling|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7409" PostTypeId="2" ParentId="7408" CreationDate="2015-02-14T05:28:32.480" Score="15" Body="&lt;p&gt;I have recently been a homeschool parent, as well as a school maths teacher so I hope I have some perspective that can help. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will suggest a textbook that does what you want, hidden somewhere in this post. But I would like you to understand a few things first. It is very likely that your son has never done any mathematics. He has done years of arithmetic. To say that he is not good at maths based on this is similar to suggesting that because he is not good at spelling, therefore he is not good at reading and writing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your child wasn't good at reading, you could find a text which included all his required vocab words and slog your way through it, or you could find something that he enjoyed reading and do it with him until he got good at reading. If he had to learn Shakespeare, you could look for good crib notes so he can learn the answers by rote, or you could find an enjoyable and relevant way of reading and actually learning Shakespeare.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a homeschooler, you have the flexibility to find out what kind of maths he really enjoys. You can also put it into contexts that he finds meaningful or enjoyable. You can make a difference in his education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my kids didn't like the arithmetic he did at school, and wasn't keen on starting algebra. So when I took him out, we didn't do any afor 2 years. We did number theory, game theory, transformational geometry, and group theory. Then we did a big project on friendship networks and he learned statistics and network(graph) theory. After that we studied Mechanics and he learned vectors, matrices, and lots of programming. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He discovered that he loved maths, and it is now his favourite subject. In the 3 months before he went back to school he quickly learned all the algebra he needed so he could study Calculus this year and continue his physics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He still doesn't like arithmetic, but it is fairly irrelevant now. He likes what he can do with it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If all you want is to get a student through SATs using lots of rote practice, then you will find Saxon is popular in the homeschool community. It is thorough and gives lots of review. I have been told some teachers use it effectively (maybe modern editions?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, when I was at school, it nearly destroyed my love of maths. It was like learning to do art with paint by numbers, or creative writing by filling in the blanks to &quot;Once ... a time&quot;. Learning by rote can teach a student what he needs to know for a simple exam, but it will probably not help him enjoy it or give understanding as deep as alternative approaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to use something like Saxon, I would highly recommend that you do it only as part of a more enriched curriculum, using creative maths resources and projects that are based around your son's interests and other abilities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want more advice on how to enrich his maths so that you both understand and enjoy it, then ask another question and you will be inundated with answers. One of my favourite resources is nrich.maths.org which has masses of interesting activities on a huge variety of topics linked to the UK curriculum.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-02-14T22:50:38.183" LastActivityDate="2015-02-14T22:50:38.183" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7411" PostTypeId="2" ParentId="7408" CreationDate="2015-02-14T05:43:49.003" Score="8" Body="&lt;p&gt;What do you mean by &quot;the rote method&quot;? If you mean lots of questions to pass a specific exam then past papers an option. These are usually available from the exam boards.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would ask you to consider other options in addition. Rote learning has a role in memory and retention but will not provide a deeper understanding.  The beauty of maths is in the rich and a suprising connections between topics. It's the thrill of spotting a new pattern or seeing things a new way. This understanding is developed from fun with maths. Play with it and don't mind getting it wrong! I always  make mistakes and it just makes the connections more rewarding when I spot them. Maths is not about 1000 identikit questions that have no sense at all. Repeating  questions will never  give you  the insight  to apply maths and it will always remain incomprehensible. I hope  it works out but please consider other teaching strategies as well.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-02-14T05:43:49.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7412" PostTypeId="2" ParentId="5898" CreationDate="2015-02-14T09:35:43.177" Score="2" Body="&lt;p&gt;Let me start with mentioning this now well-known experiment: &lt;a href=&quot;http://www.math.unm.edu/mctp/gstts/science.pdf&quot; rel=&quot;nofollow&quot;&gt;Improved Learning in a Large-Enrollment Physics Class&lt;/a&gt;. I highlight part of which that is related to your question.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;During week 12, we studied two sections whose instructors agreed to&#xA;  participate. For the 11 weeks preceding the study, both sections were&#xA;  taught in a similar manner by two instructors (A and B), both with&#xA;  above average student teaching evaluations and many years experience&#xA;  teaching this course and many others.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;More or less, both lecturers used the same way to teach. And a variety of different data showed that both groups of students were similar in many different aspects (including learning) at the end of week 11. What happened at week 12?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The control section was taught by instructor A as before. But the experiment section was taught by two instructors with a very limited teaching experience using a more active teaching method as designed. What happened? The abstract tells it all: &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;We compared the amounts of learning achieved using two different instructional approaches&#xA;under controlled conditions. We measured the learning of a specific set of topics and&#xA;objectives when taught by 3 hours of traditional lecture given by an experienced highly rated&#xA;instructor and 3 hours of instruction given by a trained but inexperienced instructor using&#xA;instruction based on research in cognitive psychology and physics education. The comparison&#xA;was made between two large sections (N = 267 and N = 271) of an introductory undergraduate&#xA;physics course. We found increased student attendance, higher engagement, and more than&#xA;twice the learning in the section taught using research-based instruction.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;But, How all these is related to your question. Let us do a thought experiment. Suppose we want to design a research to compare what you are interested in. These are the variables that we should take the same: the materials covered and the order in which they are covered, the instructional methods used, the values (instructors' views of mathematics) and so on. Having kept all these the same, what remains to compare? Basically, nothing but some social aspects.   &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-02-14T09:35:43.177" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7413" PostTypeId="2" ParentId="7385" CreationDate="2015-02-14T12:22:35.840" Score="4" Body="&lt;p&gt;Since the rearranging part of your strategy involves dividing $dy$ by $dx$ this &lt;a href=&quot;https://mathoverflow.net/questions/73492/how-misleading-is-it-to-regard-fracdydx-as-a-fraction&quot;&gt;post&lt;/a&gt; on MO is quite related to your concern. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as the arc length is concerned, I usually avoid your strategy as a teaching tool at the outset, but sometimes mention it as a way to memorize the formula. Why do I avoid this? I have no conceptual way to distinguish between $dS$ (the infinitesimal arc length) and $dx$. They are infinitesimally small so &quot;intuitively&quot; they have to have the same length; in particular $dy$ is also infinitesimally small give additional support to the intuition: In a right triangle with such an small side $dx$ is equal to the hypotenuse ($dS$). Right? Indeed, once as a student, I did something similar to this line of argument when calculating the area of a surface of revolution for the first time on my own.      &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-02-14T12:22:35.840" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7414" PostTypeId="2" ParentId="7385" CreationDate="2015-02-14T19:52:15.407" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;In general, can one be led astray by reasoning directly with differentials in this way (as opposed to defining a Riemann sum and taking a limit)? If so, are such examples difficult to contrive or are they typical enough that arguments like the one I presented should be avoided?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;People used differentials for centuries before the calculus was reformulated in terms of limits. Nothing bad happened as a result of these procedures, and they can be more formally justified in &lt;a href=&quot;http://en.wikipedia.org/wiki/Non-standard_analysis&quot; rel=&quot;noreferrer&quot;&gt;non-standard analysis&lt;/a&gt; (NSA). NSA has a rule called the transfer principle, which guarantees that any of the elementary axioms of the real number system can be applied to differentials as well. (&quot;Elemenary&quot; means anything except for the completeness property.) Since the manipulations in your question can all be justfied from the elementary axioms, they are all OK.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main thing you have to watch out for is identifying dy/dx with the derivative y', or the closely related practice of discarding squares of differentials. The more rigorous way to do this is to talk about the &lt;em&gt;standard part&lt;/em&gt; of a number, which means the unique real number that differs from it only infinitesimally. Then the derivative is defined as the standard part of dy/dx. Leibniz actually made this distinction, but other people got sloppy later on and dropped it.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2015-02-14T22:50:13.297" LastActivityDate="2015-02-14T22:50:13.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7415" PostTypeId="1" AcceptedAnswerId="7468" CreationDate="2015-02-14T20:41:25.103" Score="10" ViewCount="651" Body="&lt;p&gt;I am curious, is anyone aware of a study which determines the advantage of typed vs. handwritten material in mathematics? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, ideally, we would compare the best of both worlds. Say a beautifully $\LaTeX$ed document vs. the notes of someone whose handwriting is impeccable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my personal work, I probably create neither, but I aspire to both in various venues. At various times I hear positive things for one format over the other. I really doubt there is a quantitative case to make one way or the other. Am I wrong? Is there someone who has made the case that neat handwritten notes are inferior to cold sterile typed lecture notes? I beg the wisdom of the internet.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-02-21T23:23:12.887" Title="advantage of handwritten materials for course documents" Tags="|mathematical-pedagogy|lecture-notes|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7416" PostTypeId="1" CreationDate="2015-02-14T21:01:16.053" Score="1" ViewCount="361" Body="&lt;p&gt;So I was asked in an interview to design two questions for UK Physics A Level students studying the &lt;em&gt;suvat&lt;/em&gt; equations, that is, equations of motion with a constant acceleration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first needs to 'test understanding' of the equations, while the second needs to 'develop understanding'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of any good questions I could use? Something you might remember which is particularly innovative? Something without calculus is preferred, though an additional question involving simple integration might be helpful.&lt;/p&gt;&#xA;" OwnerUserId="4745" LastEditorUserId="2083" LastEditDate="2019-02-18T07:31:17.860" LastActivityDate="2019-02-18T09:16:58.767" Title="Designing a Good Question on Kinematics: Test and Develop" Tags="|student-motivation|examples|physical-sciences|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7417" PostTypeId="2" ParentId="7416" CreationDate="2015-02-14T23:41:31.147" Score="4" Body="&lt;p&gt;How many &lt;em&gt;suvat&lt;/em&gt; equations do you use? I have taught, in two countries in multiple schools using multiple books, four equations of constant acceleration. That seems to be standard. Each equation leaves out one variable, and the four standard equations have one question each leaving out &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; (position), &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; (final velocity), &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; (acceleration), and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; (time interval). My first question assumes that the student is taught four equations, rather than five.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A question to &lt;strong&gt;develop understanding&lt;/strong&gt; would be to derive a constant acceleration question that does not include the variable &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; (initial velocity). For true understanding, the student would need two derivations: one from the other four equations, and one from first principles. One derivation uses just mathematics, the other would include some physics. The equation, of course, is the following or an equivalent:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$s=vt-\frac 12at^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A question to &lt;strong&gt;test understanding&lt;/strong&gt; would be for the student to explain a general strategy for solving word problems of constant acceleration. For full marks the student would need to include most of these ideas:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make sure the problem truly is about constant acceleration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are five variables involved. (The student of course must list them.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some concepts, such as distance and speed, are derived from these variables but are not themselves in the equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the values of any three variables are given, the other two can be found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three values must be found from the wording of the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes a value is given implicitly rather than explicitly. Examples include starting from rest, coming to a complete stop, dropping, returning to the start position.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Care must be taken as to sign. Whether a value is positive or negative is sometimes also given implicitly rather than explicitly.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2019-02-18T09:16:58.767" LastActivityDate="2019-02-18T09:16:58.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7418" PostTypeId="2" ParentId="7394" CreationDate="2015-02-15T01:16:51.987" Score="0" Body="&lt;p&gt;I like to use some tasks from &lt;a href=&quot;http://engageny.org&quot; rel=&quot;nofollow&quot;&gt;EngageNY&lt;/a&gt;. Also, I like to use &lt;a href=&quot;http://mathematicsvisionproject.org&quot; rel=&quot;nofollow&quot;&gt;The Mathematics Vision Project&lt;/a&gt; materials on occasion.&lt;/p&gt;&#xA;" OwnerUserId="4624" LastActivityDate="2015-02-15T01:16:51.987" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7419" PostTypeId="2" ParentId="7408" CreationDate="2015-02-15T02:01:21.623" Score="5" Body="&lt;p&gt;Teach your son how to:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;understand that &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/1834#1840&quot;&gt;x is just a number you don't know yet.&lt;/a&gt;&quot;&lt;/li&gt;&#xA;&lt;li&gt;do the same thing to both sides of the equation (but don't divide by zero).&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/4448#6097&quot;&gt;solve story problems&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;use &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1834#1861&quot;&gt;Check-By-Substitution&lt;/a&gt; to check his work.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/7243#7254&quot;&gt;keep track of his common mistakes&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Consider the &lt;em&gt;Saxon&lt;/em&gt; math series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My high school math classes were taught using traditional math textbooks.  Each year, about half of the students in each class dropped the class in the first six weeks.  So of the 100 - 200 freshmen that tried Algebra I, only about 6 - 8 completed Precalculus.  One of the big problems was that the textbook would teach each topic as a single unit, with just a few days of homework.  If a student did not understand the topic &lt;em&gt;that week&lt;/em&gt;, they might never learn the topic, and would have a hard time with later topics that depended on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About a year after I took each class, the high school switched that class to use the &lt;em&gt;Saxon&lt;/em&gt; math text.  The students' success improved dramatically -- most students were able to get the hang of each of the classes.  Eventually, there were about 30 - 50 students taking Calculus each year.  &lt;em&gt;Saxon&lt;/em&gt; taught the individual topics about the same way as the traditional math textbooks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The biggest difference was in what problems were in each night's homework.  A typical &lt;em&gt;Saxon&lt;/em&gt; problem set would have 30 problems, of which only a handful were from that day's lesson.  The remaining problems were from previous days lessons.  This meant that the student would continue practicing each topic for three weeks, and have many more chances to get the hang of the topic -- and see how it related to the following topics.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-02-16T20:46:02.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7420" PostTypeId="1" CreationDate="2015-02-15T03:57:25.900" Score="3" ViewCount="193" Body="&lt;p&gt;What are some of the best resources and drills that one can use to take students from good to great scores on the SAT especially in math section. &lt;/p&gt;&#xA;" OwnerUserId="4748" LastEditorUserId="201" LastEditDate="2015-02-15T05:02:19.220" LastActivityDate="2015-05-11T21:01:23.517" Title="SAT Mathematics Section Prep" Tags="|secondary-education|exams|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7421" PostTypeId="2" ParentId="7416" CreationDate="2015-02-15T23:41:33.400" Score="2" Body="&lt;p&gt;I might suggest a question concerning what initial velocity is necessary to apex&#xA;at a given height, for either fireworks or water fountains:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/06Ojj.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/06Ojj.jpg&quot; width=&quot;400&quot; /&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;Image from &lt;a href=&quot;http://www.citizensinspace.org/2012/07/fireworks-and-suborbital-markets/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/W45xy.jpg&quot; alt=&quot;WaterFountain&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;Image from &lt;a href=&quot;http://www.mathforgrownups.com/ah-ohh-math-and-fireworks/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="6053" LastEditDate="2019-02-18T04:10:10.930" LastActivityDate="2019-02-18T04:10:10.930" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7422" PostTypeId="2" ParentId="7385" CreationDate="2015-02-16T05:26:47.257" Score="11" Body="&lt;p&gt;As @BenCrowell mentioned, the transfer principle proves that direct algebraic manipulation of infinitesimals in single variable calculus is allowed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I stumbled upon &lt;a href=&quot;http://www.theshapeofmath.com/oxford/physics/year1/calc/sepdiff&quot; rel=&quot;nofollow noreferrer&quot;&gt;this post in theshapeofmath.com&lt;/a&gt;, which shows how things can break when switching to multiple variables. It provides the following basic example of a possible error when handling partial derivatives (the same example also appears as &lt;a href=&quot;https://mathoverflow.net/a/73503/49215&quot;&gt;an answer&lt;/a&gt; in the MO question linked to by @AmirAsghari), where given an implicitly defined function&#xA;&lt;span class=&quot;math-container&quot;&gt;$F(x,y)=0$&lt;/span&gt; , we can algebraically get an expression with the wrong sign:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{dy}{dx}=\frac{\partial F}{\partial x}\frac{\partial y}{\partial  F}=\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the correct expression is:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="688" LastEditorUserId="688" LastEditDate="2019-09-24T02:52:35.537" LastActivityDate="2019-09-24T02:52:35.537" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7423" PostTypeId="2" ParentId="553" CreationDate="2015-02-16T09:40:26.737" Score="5" Body="&lt;p&gt;I think one important distinction is: Is the joke about mathematics or about mathematicians? An example of the former is the &quot;rat poison principle&quot; from  Steve Gubkin's comment or any other &quot;funny name&quot; for a mathematical principle/trick or whatever. An example of the latter is the one from the OP.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the separation is not really strict as every joke a mathematician tells about mathematics also tells something about mathematicians (&quot;these guys have weird jokes…&quot;) and also jokes about mathematicians often highlight important principles (as the one from the OP).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But my point is: Jokes that are explicitly about mathematicians may lead to the &quot;Those weird mathematicians…&quot;-effect. Jokes that help to memorize mathematical principles and do not carry to much implicit meaning about mathematical may be less harmful…&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But anyway, if a joke is harmful or not very much depends on the image the students have of their instructor. If they already think that he's a math-weirdo, jokes probably won't help.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2015-02-16T09:40:26.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7424" PostTypeId="2" ParentId="553" CreationDate="2015-02-16T14:33:46.090" Score="5" Body="&lt;p&gt;Prof. Frank Carroll at Ohio State had a routine like this in elementary undergraduate lectures.  At the beginning of class, he would write at the side of the chalkboard* a list of topics for the lecture.  Sometimes he would put &quot;Joke&quot; in the list.  Then the students were all waiting for the joke.  Supposedly they would be more likely to pay attention, in case they missed the joke.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Some old guys here may remember chalkboards.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-02-16T14:33:46.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7425" PostTypeId="2" ParentId="7385" CreationDate="2015-02-16T15:46:33.933" Score="4" Body="&lt;p&gt;It's easy to confuse yourself with differentials.  Consider the surface area of a cone given in cylindrical coordinates by $z=1-2r$.  What is the element of area?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should I visualize a cylinder tangent to the cone?&#xA;\begin{equation}&#xA;\begin{aligned}&#xA;dA&#xA;&amp;amp;= (d\text{ curved base})(d\text{ height}) \\&#xA;&amp;amp;=(r\,d\theta)\sqrt{dr^2+(2\, dr)^2} \\&#xA;&amp;amp;=\sqrt{5}\, r\, dr\, d\theta&#xA;\end{aligned}&#xA;\end{equation}&#xA;Or should I visualize a rectangle tangent to the cone?&#xA;\begin{equation}&#xA;\begin{aligned}&#xA;dA&#xA;&amp;amp;= (d\text{ linear base})(d\text{ height}) \\&#xA;&amp;amp;=r\sin(d\theta)\sqrt{dr^2+(2\, dr)^2} \\&#xA;&amp;amp;=\sqrt{5}\, r\,dr\sin(d\theta) \\&#xA;&amp;amp;&amp;lt;\sqrt{5}\, r\,dr\,d\theta\, (?)&#xA;\end{aligned}&#xA;\end{equation}&#xA;I relied on the formulas in Thomas &amp;amp; Finney for years before I could confidently use differentials in a derivation like this.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2015-02-16T15:46:33.933" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7426" PostTypeId="1" CreationDate="2015-02-16T16:26:16.117" Score="9" ViewCount="181" Body="&lt;p&gt;There are a number of skills needed in maths (I'm teaching undergraduate pure maths) that are not really topics on their own, such as interpreting a definition, taking negation, or giving counter-examples. Part of me thinks that it's useful, while covering the official content of a course, to also bring in these ideas, so that the students get some practice. On the other hand, part of me thinks this risks confusing the students, by bringing in more than one idea that they are not comfortable with at once. Is there any evidence for which, if either, is better? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally I'd like research papers, but I'm also open to personal experience. I'm also more interested in the case where there are time constraints (ie there is a quantity of material that must be covered that fills the available time).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(A connected question: what about bringing in ideas from earlier courses that students are expected to remember but don't?)&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-08-07T15:34:41.150" Title="Teaching background skills together with specific concepts" Tags="|undergraduate-education|mathematical-pedagogy|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7427" PostTypeId="2" ParentId="1794" CreationDate="2015-02-16T17:02:35.363" Score="9" Body="&lt;p&gt;I think the short answer to your question is that you are (like many people) confusing the arithmetic that you called 'mathematics' at school with the mathematics that mathematicians do. I very rarely add up numbers above 10, or write down ones with several digits. I think I possibly do own a calculator... somewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest starting with some popular-maths books, and see how you get on. Since you've already done some philosopy, maybe try some logic. (I was going to suggest a book I liked, but I've completely forgotten what it was.)&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-02-16T17:02:35.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7428" PostTypeId="1" CreationDate="2015-02-17T02:18:17.033" Score="4" ViewCount="1590" Body="&lt;p&gt;I have in mind that volume is the amount of room or space a 3-d object takes up - its &quot;outsideness&quot; and that capacity with the amount of room or space a 3-d object can hold. Then I start thinking about that capacity piece and think terms like &quot;volume by weight&quot; and start getting a bit boggled. THEN I start thinking of liquids and then I get more befuddled. Just looking for a good, quick, succinct way to explain this to my 5th graders. Perhaps the concepts ARE indeed situational specific? Perhaps I'm making too much of this?&lt;/p&gt;&#xA;" OwnerUserId="4752" LastActivityDate="2015-08-28T14:22:35.137" Title="how is volume different than capacity" Tags="|definitions|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7429" PostTypeId="2" ParentId="2322" CreationDate="2015-02-17T02:32:11.687" Score="2" Body="&lt;p&gt;Consider the notion of the &quot;tangent space&quot; to a differential manifold at a point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are various ways to make this abstract notion concrete, and points on the tangent space are often thought of as a way to make rigorous the idea of an infinitesimal displacement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in many cases we can appeal to prior notions; we learned about tangent lines and tangent planes and stuff in our elementary geometry courses, and these make a perfectly good model for the tangent space to a manifold at a point. In fact, the first time I saw tangent spaces defined in algebraic geometry, it was done in this fashion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, viewing differentials as dual to tangent vectors leads directly to, e.g. in the case of a plane curve, having $(\mathrm{d}x, \mathrm{d}y)$ coordinates for points on the tangent line.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2015-02-17T02:32:11.687" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7430" PostTypeId="1" CreationDate="2015-02-17T02:43:15.680" Score="3" ViewCount="1500" Body="&lt;p&gt;I have three students in my secondary geometry class that just destroy everything I throw at them.  I tasked them with writing the word problems for their midterms and one of the three wrote simply &quot;What are the interior angle measures of an arch made of five isosceles trapezoids?&quot; (I had to cheat to get the answer:) really astounding minds that I want to challenge with some tough geometric constructions because I think they would really get a lot out of the assignment.  They really enjoy when we do constructions in class.  What are some challenging geometric constructions that I can give to my students?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-02-17T16:04:49.937" Title="interesting/challenging geometric constructions for gifted secondary students" Tags="|secondary-education|geometry|gifted-students|constructions|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7431" PostTypeId="1" AcceptedAnswerId="7440" CreationDate="2015-02-17T09:26:52.253" Score="22" ViewCount="913" Body="&lt;p&gt;When I was a freshman in Mathematics we learned the usual integration techniques (lots of standard integrals, integration by parts, substitution, partial fractions,…). As homework we simple got a bunch of integrals we had to determine. Now I am teaching freshman Mathematics and also teach integration techniques. But given that you just type &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;integrate 'basically any function'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;into Wolfram Alpha and you do not even need any software (besides a browser) or any knowledge of syntax whatsoever, I find it worthless to give homework of the form &quot;Find antiderivatives of $\sin(x) e^x$, $1/\tan(x)$…&quot; and so forth. Of course there will be students who take the challenge and go ahead and learn something but there will also be students who just use available technology and I really can understand them. Integration and especially subtechniques like partial fractions are things that computers can do much better than humans–at least for the standard homework integrals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't want to argue about if one should still teach integration techniques or not (could be a good separate question) but &lt;strong&gt;ask under the premise that it is meaningful to learn integration techniques and that homework has to be done&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;How should you design homework today to let students learn integration techniques?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Related question with a different focus:&#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/6/how-to-assign-homework-when-answers-are-freely-available-or-attainable-online&quot;&gt;How to assign homework when answers are freely available or attainable online?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-07-16T16:52:57.003" Title="How to give homework for integration techniques?" Tags="|homework|software|integration|" AnswerCount="7" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7432" PostTypeId="2" ParentId="7431" CreationDate="2015-02-17T13:57:46.490" Score="15" Body="&lt;p&gt;First of all, I try to be honest with my students by telling them directly and explicitly about the existence of such integration machines (It is silly of me assuming that they don't know that!). Then, I add, but we don't integrate for the sake of integration. For us, it is a &lt;strong&gt;practice of problem solving&lt;/strong&gt; in which we need to choose the right techniques from the existing ones. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having this change of perspective in mind, a good homework is the one that asks students to choose integration techniques. Consider this one, $x/(x^2-9)$, from an old paper of Alan Schoenfeld (see below). How should we evaluate the integral? By substitution, partial fractions, or by a trigonometric substitution! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I learned this &quot;trick&quot; years ago from a teacher who had to teach a meaningless algorithm (at least for students) for finding square root to her students. She changed the lesson from being a lesson about finding square root to a lesson about algorithms and the way they are written for a computer. My suggestion for integration techniques was just one way to use this trick. I am sure you can come up with some other ways. If yes, please let me know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. These two papers of Schoenfeld &lt;a href=&quot;http://www.jstor.org/stable/2320787?seq=6#page_scan_tab_contents&quot;&gt;Teaching Problem-Solving Skills&lt;/a&gt; and &lt;a href=&quot;http://www.jstor.org/stable/2320344?seq=1#page_scan_tab_contents&quot;&gt;Presenting a Strategy for Indefinite Integration&lt;/a&gt; are quite related to the kind of change of perspective suggested above for integration. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-02-17T13:57:46.490" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7433" PostTypeId="2" ParentId="7430" CreationDate="2015-02-17T14:37:51.643" Score="3" Body="&lt;p&gt;There are two websites that make geometric construction into a game. NiloCK mentioned &lt;a href=&quot;http://euclidthegame.com/Tutorial/&quot; rel=&quot;nofollow&quot;&gt;euclidthegame.com&lt;/a&gt; in his comment. There is another construction game at &lt;a href=&quot;http://sciencevsmagic.net/geo/&quot; rel=&quot;nofollow&quot;&gt;sciencevsmagic.net&lt;/a&gt;, with a very different look and feel. I think some people will one enjoy one or the other much more, though I enjoy them both. I have only made it through about half the challenges on the second site.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-02-17T14:37:51.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7434" PostTypeId="2" ParentId="7430" CreationDate="2015-02-17T15:14:47.837" Score="1" Body="&lt;p&gt;Suggest posing these constructions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://fivetriangles.blogspot.com/search/label/construction&quot; rel=&quot;nofollow&quot;&gt;http://fivetriangles.blogspot.com/search/label/construction&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4753" LastActivityDate="2015-02-17T15:14:47.837" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7436" PostTypeId="2" ParentId="7430" CreationDate="2015-02-17T16:04:49.937" Score="2" Body="&lt;p&gt;It might be time to show them how algebra can support or deny constructions. I would have them start looking at Gauss's proof of the constructability of the heptadecagon. This could serve as an introduction to areas of abstract algebra and beyond.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-02-17T16:04:49.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7437" PostTypeId="2" ParentId="7408" CreationDate="2015-02-17T16:21:01.880" Score="2" Body="&lt;p&gt;'Rote math' had what effect on you?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Buy these three books, and use them to guide your instruction: &quot;What's math got to do with it?&quot; by Jo Boaler or &lt;a href=&quot;http://youcubed.stanford.edu/category/mooc/&quot; rel=&quot;nofollow&quot;&gt;take her MOOC&lt;/a&gt;  with your son, &quot;the Joy of X&quot; by Steven Strogatz (&lt;a href=&quot;http://topics.nytimes.com/top/opinion/series/steven_strogatz_on_the_elements_of_math/index.html&quot; rel=&quot;nofollow&quot;&gt;preview it&lt;/a&gt;), and &quot;Innumeracy&quot; by John Allen Paulos.  &lt;/p&gt;&#xA;" OwnerUserId="3492" LastActivityDate="2015-02-17T16:21:01.880" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7438" PostTypeId="1" AcceptedAnswerId="7947" CreationDate="2015-02-17T16:48:01.630" Score="10" ViewCount="418" Body="&lt;p&gt;Our department has decided to offer a &quot;Discrete Math&quot; course that students would take &lt;strong&gt;before&lt;/strong&gt; their &quot;Intro to Proofs&quot; course. The idea is that the Discrete Math course would focus on the computations and techniques that are used in discrete settings similar to how a typical Calculus I course focuses on the computations and techniques of continuous situations. The problem I'm having in preparing for this course is that every Discrete Math textbook I look at spends a lot of time on proof technique and contains many &quot;Prove that...&quot; exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Does anyone know a Discrete Math textbook that emphasizes computations and applications instead of proof? (Or at least, proof at a level analogous to Calculus I.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a basic run-down of the topics chosen by the department:&#xA;Boolean logic, Basic Set Theory, Counting/Combinatorics, Modular Arithmetic, Relations, Graphs/Digraphs (Eulerian Circuits, Hamiltonian Cycles, Dijkstra’s Algorithm), Trees (Minimal Spanning Tree), Sequences (Finite Difference Methods).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bonus Points:&lt;/strong&gt; Such a text that allows use of an online homework system (WebAssign, MyMathLab, WeBWorK, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; The course has only Calculus I as a prerequisite and it will be taken by freshman/sophomore majors in Math or Math Ed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt; I've found that the text &quot;Discrete Mathematics&quot; by Dossey, Otto, Spence and Vanden Eynden seems to strike the right balance. However the latest edition is over a decade old and, hence, the internet is rife with solution manuals and the Amazon.com reviews are poor. Does anyone know of a newer/better text similar to this one?&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2015-03-11T16:35:47.620" LastActivityDate="2015-04-25T18:05:46.693" Title="Discrete Math text before a proof course" Tags="|textbooks|discrete-math|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7439" PostTypeId="2" ParentId="7428" CreationDate="2015-02-17T19:51:29.850" Score="-1" Body="&lt;p&gt;Volume counts the amount of cubes in a shape and can be measured in $cm^3$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Capacity measures the amount of liquid filling the cubes in units of $ml$ for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The connection is that $1cm^3=1ml$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you had a  $cm^3$ cup it could be refilled about $330$  times from a can of coke.  &lt;/p&gt;&#xA;" OwnerUserId="4668" LastEditorUserId="4668" LastEditDate="2015-02-17T20:57:21.770" LastActivityDate="2015-02-17T20:57:21.770" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7440" PostTypeId="2" ParentId="7431" CreationDate="2015-02-17T20:37:05.153" Score="20" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I don't want to argue about if one should still teach integration techniques or not (could be a good separate question) but ask under the premise that it is meaningful to learn integration techniques and that homework has to be done: How should you design homework today to let students learn integration techniques?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One answer is trust, but verify. For example, you can assign low-stakes or zero-stakes homework on a certain technique of integration, then give a higher-stakes quiz or exam in which your students need to use the same technique in a situation where they don't have access to a computer algebra system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, presumably you are assigning work that requires both computational skills and conceptual understanding. That is, you're not just assigning a list of 50 problems that all look like $\int f(x) dx$, where $f$ is some given function. When the problems are less stylized than this, the students who lack the computational skills are the same students who will not be able to figure out all the conceptual aspects of the problem, or even how to input the problem into a CAS. For example, it's very common to see students who can't handle any variable of integration besides $x$, or who can't understand how to handle symbolic constants, e.g., $\int dz/(z-a)$. If they don't understand what they're doing, then they aren't likely to be able to even put such a question into the form required by a CAS.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of course you're doing applications, right? A CAS isn't going to be able to convert a word problem into an equation, or interpret the result of the calculation and draw a conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below is another example of a type of problem that doesn't lend itself to solution by a CAS.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In problems 1 and 2, two indefinite integrals are given&#xA;that involve functions which&#xA;look similar to one of the following:&#xA;\begin{equation*}&#xA;  e^{-x^2} \qquad x^x \qquad \frac{\sin x}{x} \qquad e^x \tan x&#xA;\end{equation*}&#xA;As discussed in section 9.3, the four functions given above can't&#xA;be integrated in closed form.&#xA;In each pair below, one can be integrated, while&#xA;the other can be made into one of the above forms by a substitution, proving that it's impossible&#xA;to integrate. Determine which is&#xA;which, integrate the one that can be done, and check your answer to that one online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 (a)&#xA; $\displaystyle \int x^{-3/4}e^{-\sqrt{x}}\: d x$&#xA; (b)&#xA; $\displaystyle \int x^{-1/2}e^{-\sqrt{x}}\: d x$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 (a)&#xA; $\displaystyle \int x^{-2}\sin\frac{1}{x}\: d x$&#xA; (b)&#xA; $\displaystyle \int x^{-1}\sin\frac{1}{x}\: d x$&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2015-02-17T20:37:05.153" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7441" PostTypeId="2" ParentId="7428" CreationDate="2015-02-18T02:18:13.467" Score="12" Body="&lt;p&gt;The situation does indeed change your interpretation of the terminology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Volume is a general concept of the amount of 3D space something takes up. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a solid object like a brick, or for a liquid there is no ambiguity and you can simply do a calculation (like $V = \pi r^2 h$ for a solid cylinder), or possibly compare its mass to the mass of a known volume (eg $1$ cm$^3$ weighs $2$ g, so $50$ g must be $25$ cm$^3$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, for an object with a hollow or hole in it, there is some ambiguity as to which part of the object you are referring to when you say &quot;the volume&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, with a coffee cup, you could interpret its volume as the amount of space it takes up including the space inside it, or as the amount of space just the porcelain takes up. In that case you should always specify which you mean (for example, &quot;the volume of porcelain&quot;, or &quot;the volume of the cup including the space inside it&quot;). Interestingly, you couldn't find the volume of the cup including the space inside it simply by weighing it, because the space wouldn't contribute anything to the weight. You'd either have to use a formula, or find a way to find the volume of porcelain and space separately and combine them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If what you want is &quot;the volume of the space inside the cup&quot;, then there is a special word for this and it is &quot;capacity&quot;. This word allows us to not have a whole sentence to specify the part we want the volume of, but it is still a volume.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in short, volume is a measure of 3D space, but if an object isn't the same throughout you have to specify which bits you want; and capacity is a kind of volume, but it refers specifically to the space inside something.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-02-18T08:58:16.170" LastActivityDate="2015-02-18T08:58:16.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7442" PostTypeId="1" AcceptedAnswerId="7443" CreationDate="2015-02-18T03:56:25.300" Score="12" ViewCount="810" Body="&lt;p&gt;I've been trying to introduce my younger sister to programming, and one of the things I said was basically that once you know how to program, you can have a computer solve equations for you as long as you know how to program it to - including math homework. She argued that this would be unacceptable cheating. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can see where a person would get this idea, not being familiar with programming and hearing only &quot;homework machine&quot;. But teaching a computer to do math beyond simple operations requires a solid grasp of mathematical concepts, and requires one to effectively know the material in and out already. All that would be lost is the monotonous labor of manually punching numbers into a calculator and pushing buttons, since a conceptual understanding is required to be capable of making such a program (assuming they don't cheat on the making of the program itself and use someone else's work).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a moral standpoint, is it acceptable to write and use a program to solve mathematical problems assigned as homework?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would/should it be against the rules of most high schools to do this?&lt;/p&gt;&#xA;" OwnerUserId="4755" LastActivityDate="2015-02-19T07:55:37.530" Title="Is it morally acceptable to complete math homework with a program oneself has written?" Tags="|secondary-education|technology-in-education|homework|" AnswerCount="4" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="7443" PostTypeId="2" ParentId="7442" CreationDate="2015-02-18T10:32:13.943" Score="8" Body="&lt;p&gt;I put my recommendation first, with my reasoning about the moral basis of school and homework underneath if you are interested.&lt;/p&gt;&#xA;&lt;h2&gt;Recommendation&lt;/h2&gt;&#xA;&lt;p&gt;It is not cheating and is probably more moral than doing the homework.&lt;/p&gt;&#xA;&lt;p&gt;I would recommend that you and your sister consider the possible benefits of practicing manually solving the maths problems for exams or more advanced study. If she decides she doesn't need more practice, she could sound out her teacher. A really good teacher may even encourage her to do the programming as a maths/science fair project. If it sounds like the teacher won't like it, just don't tell them. See Ben Crowell's comment - most teachers will give a knee-jerk reaction rather than actually thinking about it, so why upset the teacher for no reason?&lt;/p&gt;&#xA;&lt;p&gt;If she tells the teacher and the teacher disapproves, it really makes no difference. If she is sure she can do the maths well enough manually, it is unethical for the teacher to insist that she does more useless practice, and your sister has no ethical responsibility to obey an unethical requirement, especially when it is out of school time.&lt;/p&gt;&#xA;&lt;p&gt;In the end, it is up to your sister to take control of her education and do what she thinks will help her learn and get qualifications and have a good all-round life. Encourage her, but if she feels uncomfortable with it, don't push it.&lt;/p&gt;&#xA;&lt;h2&gt;General Theory of homework&lt;/h2&gt;&#xA;&lt;p&gt;The moral justification for making a student do school &amp;quot;work&amp;quot; and homework is the extent that it produces something valuable for the student, ie. skills/understanding and qualifications. School &amp;quot;work&amp;quot; has no moral value beyond this. The teacher is not an employer, and no one except the student has anything to gain by the &amp;quot;work&amp;quot;. Equally, no one is disadvantaged when the student takes a short cut on doing the &amp;quot;work&amp;quot;, except the student herself. &amp;quot;Cheating&amp;quot; implies that she will get something unfairly or that other students will be disadvantaged, but it should now be clear that there is no way to cheat on school work or homework (excluding summative assessments).&lt;/p&gt;&#xA;&lt;p&gt;Therefore education needs to balance striving for deep understanding, getting formal qualifications, and minimising the opportunity cost to the student (lost time for socialising, relaxation, and extra curricular activities). It is these factors alone that usually need to be analysed in order to determine the morality of an educational activity. What the teacher wants is not really morally relevant in the calculation.&lt;/p&gt;&#xA;&lt;p&gt;For example, in a recent meta-analysis, high homework requirements have been shown on average to have a neutral or only slight positive impact on student educational outcomes, while other studies indicate it can have a large negative effect on extra curricular activities and student health. Therefore homework itself is on shaky moral ground, and more than 2 hours of homework a day is almost certainly unethical, no matter what the teacher or school demands.&lt;/p&gt;&#xA;&lt;p&gt;From another perspective, it is questionable how much a teacher can prescribe a student's activities outside of school grounds and school hours. Consider how upset a teacher would get if parents tried to require that students do an hour of chores during class time! Unless a student is committing fraud, whether or however your sister decides to do her homework is outside the jurisdiction of her teacher to control.&lt;/p&gt;&#xA;&lt;h2&gt;Your Situation&lt;/h2&gt;&#xA;&lt;p&gt;Your sister making a &amp;quot;homework machine&amp;quot; obviously leads to a deeper understanding, and increased time for extra curricular activities (if you include programming as the extra-curricular activity!), at the possible expense of being fast at manually doing those problems when she sits the test or studies more advanced work. If she is confident in solving these problems manually, then there is no issue here. An ideal teacher would have used formative assessment to determine a more profitable use of the student's time, and so wouldn't give those drills as homework in the first place.&lt;/p&gt;&#xA;&lt;p&gt;For your sister, her moral course of action is to maximise her education while minimising the opportunity cost.&lt;/p&gt;&#xA;&lt;p&gt;The only real moral issue would be if she used the &amp;quot;homework machine&amp;quot; to fraudulently gain an advantage on summative assessments, however, I am assuming that she would use this for homework, and not smuggle her computer into the exam.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-02-18T10:32:13.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7444" PostTypeId="2" ParentId="7442" CreationDate="2015-02-18T11:56:23.810" Score="0" Body="&lt;p&gt;2 equations in 2 unknowns. There are no fewer than 3 ways to solve this. Similarly, there are many ways to solve for the roots (the x axis intercepts) of a quadratic equation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a multiple choice test, such as the SATs, the college entrance exam in the US, the question is moot. Not so much for lack of the ability to consult the computer, but for the fact that no 'work' need be shown as part of the answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of homework or a class exam, it's frequently the case that work has to be shown. The process of learning the different ways to solve these problems is as important as reaching the correct solution. Your sister's software, while perhaps brilliant, skips over this pedagogical multi-process approach. I'd question whether the same level of understanding is even achieved, as computer's programs can be written to 'brute force' solutions with no understanding by the programmer of any of the methods the teacher is trying to teach. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student is doing herself a disservice to use this for homework, morality aside.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-02-18T11:56:23.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7445" PostTypeId="2" ParentId="7394" CreationDate="2015-02-18T12:15:18.127" Score="1" Body="&lt;p&gt;Here is another resource, very tailored to the Common Core State Standards: &lt;a href=&quot;https://www.illustrativemathematics.org/&quot; rel=&quot;nofollow&quot;&gt;Illustrative Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This site is the brainchild of Bill McCallum, one of the lead authors of the Common Core State Standards for Mathematics. The idea is to be a catalogue of tasks that are aligned to the individual standards and &quot;illustrate&quot; the types of ideas and skills that the standards are aiming for. The site has a review process wherein the tasks are vetted by both educators and mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2015-02-18T12:15:18.127" CommentCount="0" CommunityOwnedDate="2015-02-18T12:15:18.127" ContentLicense="CC BY-SA 3.0" />
  <row Id="7446" PostTypeId="2" ParentId="7442" CreationDate="2015-02-18T14:14:04.797" Score="11" Body="&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, as long as&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the teacher didn't explicitly forbid it,&lt;/li&gt;&#xA;&lt;li&gt;you wrote the mathematical core of the program yourself,&lt;/li&gt;&#xA;&lt;li&gt;the technique you are using doesn't twist the intent of the problem.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher explicitly forbid such solutions:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In such case you are going against the teacher's will. In an ideal world the teacher will explain his motivation in a sensible way and convince you into doing the homework manually. In a less ideal world one have to decide for yourself whether the teacher is crazy or perhaps there is some reason to it. In such case I would suggest asking someone on case-by-case basis, someone more experienced who you can trust.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;You didn't wrote the mathematical core of the program yourself:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are libraries which allow you to do quite complex mathematical calculations without the understanding. To trivialize the matter, suppose the homework was to multiply two numbers, but you wrote a program:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;#!/usr/bin/ruby&#xA;b = gets.to_i&#xA;a = gets.to_i&#xA;print a*b, &quot;\n&quot;&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;and that program calculated the homework for you. You don't need to know how the multiplication works to code it, in fact your homework is being solved by the code implemented by someone else. In such a case it is not alright.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The technique you are using twists the intent of the problem:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you want to find the smallest positive number $n$ such that $180\cdot n$ is a perfect cube. That is easy enough to bruteforce:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;(1..180).each do |m| &#xA;    if m**3 % 180 == 0 then&#xA;        print m**3/180, &quot;\n&quot;&#xA;        break&#xA;    end&#xA;end&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;However, that would be laborious to do manually. In fact the teacher &lt;em&gt;expects&lt;/em&gt; the student to factorize $180$ into $2^2\cdot3^2\cdot5$ and find $n = 2 \cdot 3 \cdot 5^2 = 150$. If the program would involve factorization, etc. then it could be alright, but the one above sidesteps the &quot;hard&quot; part by sheer force. Mathematics is a great deal about thinking and I would not find such a computer-enhanced solution appropriate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This issue is present also in the manual solutions. For example, suppose you are given to solve some quadratic equation, but instead of using the standard formula you &quot;guess&quot; the roots using Wolfram Alpha, and then only check that these are the roots and that there are two of them, so you have all the solutions. Indeed such an answer &quot;solves&quot; the problem, but I would say that it is against the intent of the question. Please note, that it is another matter when you are given an equation like $x^2 - 20x + 99 = 0$, where you in fact &lt;em&gt;can guess&lt;/em&gt; the roots $9$ and $11$, because these can be easily seen as the factors of the constant term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a problem with knowing intentions of the teacher, but in my opinion in most cases it is clear whether the program circumvents the problem or solves it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Otherwise:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the above cases do not apply, then the ability to code such a program shows an understanding that covers at least the homework and probably much more. I would happily applaud such a method. We live in a world where mindless calculation practice stops being useful and the importance of long division algorithm comes not from it intrinsic value of calculating quotients, but from being a step to more general and abstract algorithms further on. If the student is able to come up with a general solution to a range of problems and code it down, then he/she is clearly ahead in this journey.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2015-02-18T14:14:04.797" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7448" PostTypeId="2" ParentId="2459" CreationDate="2015-02-18T16:12:59.427" Score="6" Body="&lt;p&gt;i think the answers here are all very useful and thought out, i would like to add one thing.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tablets are not and likely will never be allowed to be used on standardized tests (SAT, ACT, AP, GRE, Praxis, etc) but (most) graphing calculators are.  I would strongly suggest to at least expose students to graphing calculators and their more useful functions and capabilities because it is possible that comfort with the more advanced features of a graphing calculator could make or break a students score on said standardized test/s.&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-02-18T16:12:59.427" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7451" PostTypeId="2" ParentId="2459" CreationDate="2015-02-19T07:33:07.360" Score="1" Body="&lt;p&gt;Choose calculators. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both TI and Casio make very capable ones and it does not sound as if your school has a well-established policy or plan for deploying school-managed tablets, deploying educational software or handling the issues involved in Bring Your Own Device. Until that is worked out you need to be very wary of tablets, phablets and phones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mobile devices in the classroom can be a fabulous resource, but they are also a &lt;strong&gt;huge&lt;/strong&gt; distraction and lead to many, many places other than the lesson at hand. If you allow them in your classroom you will have the impossible task of checking whether the use is legitimate calculation or reference work or playing Temple Run. I confiscate all mobile phones unless I have given explicit permission for their use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By imposing a very short list of standard calculators (we allow two models from Casio and TI) for all your maths/science/economics classes you remove all of the &quot;but I just need my phone for this calculation&quot; excuses and you also know what the calculator is capable of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My third years (eighth grade US) have the simpler models with basic arithmetic and scientific functions, which are quite cheap. The fourth-years (ninth US) upgrade to a graphing calculator and much of our material references the calculator too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many devices (also the high-end calculators) are capable of algebra and differentiation and would allow students to answer test questions without understanding the techniques involved. Once those skills are in place and tested this becomes less of an issue.&lt;/p&gt;&#xA;" OwnerUserId="3483" LastActivityDate="2015-02-19T07:33:07.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7452" PostTypeId="2" ParentId="7442" CreationDate="2015-02-19T07:55:37.530" Score="1" Body="&lt;p&gt;I agree with Richard and dtldarek.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, back in 1988, I did use a computer to do some of my homework this way and my math teacher had no objections. It was a boring task to create diagrams for some elementary functions by hand in order to better understand their properties. I made a BASIC :-) program for each of them, then printed them out and glued into my workbook. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This way I could compute ten times more points (more smooth curves) than the homework asked for in less time, which I was proud of. I could even play with parameters and make more variants of them just for fun. Nobody doing this manually would come to the idea to do that extra drawing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conclusion: If the homework problem allows the benefit and you want to - &lt;strong&gt;program&lt;/strong&gt; the computer. If the aim is at logic, don't brute force.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2015-02-19T07:55:37.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7454" PostTypeId="2" ParentId="5986" CreationDate="2015-02-20T05:14:48.277" Score="2" Body="&lt;p&gt;Why so much emphasis on the &quot;physics&quot; aspects of vectors? Are you an engineer? Engineers are accustomed to thinking of vectors as arrows in 3D space that represent force or displacement. Teaching this way brings in the added challenge of geometry and trig.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you should start by demonstrating that vectors are just a method of representing information, and not use the Cartesian plane. Vector mathematics are true for collections of objects that might seem less &quot;physical&quot; such as ingredients in a recipe. Most kids would understand how to make their favorite dish (say pizza or cookies?). Vectors are simply a collection of objects obeying a few simple rules. For example: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;ingredients vector = {chocolate chips, eggs, flour, butter, sugar, salt, baking soda}&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;cost vector = {chocolate chips: \$4 per bag, eggs: \$2 per dozen, flour: \$1 per lb, butter: \$1.50 per stick, sugar: \$2 per lb, salt: \$1 per lb, baking soda: \$1 per lb}&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;ratios vector: {chocolate chips: 1 bag, eggs: 2/12 dozen, flour: 0.6 lb, butter: 2 sticks, sugar: 0.6 lb, salt: 0.013 lb, baking soda: 0.01 lb}&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then, you take inputs, apply an operation, and get a result. Say, a dot product:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Tell me the recipe for cookies&quot; becomes:&#xA;  recipe = {ingredients vector} dot {ratios vector}&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;How much does 3 batches of cookies cost? becomes:&#xA;  3 x cost of a batch of cookies = the sum of (costs dot ratios) x 3&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now, questions like &quot;what is the domain?&quot; or &quot;what is the vector space?&quot; are intuitive.  If they can get this, then it will be a breeze showing them how to apply to 2-component vectors in the Cartesian plane! The rules that you teach for working with vectors are the same.&lt;/p&gt;&#xA;" OwnerUserId="4762" LastEditorUserId="1147" LastEditDate="2015-02-20T11:33:06.360" LastActivityDate="2015-02-20T11:33:06.360" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7455" PostTypeId="1" AcceptedAnswerId="7461" CreationDate="2015-02-20T08:34:40.660" Score="11" ViewCount="661" Body="&lt;p&gt;I'd like to introduce Taylor polynomials by generalizing the linear approximation of a function $f(x)$ to a quadratic approximation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The linear approximation formula $L(x)=f(a)+f'(a)(x-a)$ of $f(x)$ near $a$ can be seen to be the point-slope formula of a line passing through the point $(a,f(a))$ with slope $f'(a)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I'd like to derive the formula for a quadratic approximation of $f$ near $a$.  Let this quadratic approximation be denoted by $T_2(x)$, and require that it satisfy the following $3$ conditions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$T_2(a)=f(a)$&lt;/li&gt;&#xA;&lt;li&gt;$T_2'(a)=f'(a)$&lt;/li&gt;&#xA;&lt;li&gt;$T_2''(a)=f''(a)$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What is a nice way to show that $T_2(x)$ is what we know it to be without just writing it down and showing it has the desired properties?  How can I motivate the coefficients?  In other words, we don't really have a well-known ``point-slope-concavity&quot; formula for parabolas that generalizes the point-slope formula we used to write down the linear approximation.  I know I could start with a general form of $T_2(x)=Ax^2+Bx+C$, and use the above conditions to compute the coefficients, but for $a\ne 0$ this will be messy and unintuitive.  If I could somehow motivate the switch to consider the form $T_2(x)=A(x-a)^2+B(x-a)+C$, that would be better, but I'm unsure how to convince the students that this is a natural form to consider.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best I've been able to do is start off assuming that $T_2(x)$ is of the form $$T_2(x)=f(a)+f'(a)(x-a)+c(x-a)^2$$ for some constant $c$ but I think this is too much assuming too soon.  Sure we can test that this form will satisfy $1$ and $2$ above, and that $c$ gives us the flexibility to specify $T_2''(a)$ so as to satisfy $3$, but this is still unmotivating to me.  Why did I think this would work?  It seems I've just written down what I know will work, and have only left the challenge of finding the right $c$ (which is just a computation and gives no intuition).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I motivate the form for $T_2(x)$ I've started with?  Or is there another way to do this?  I'd like to avoid viewing Taylor polynomials as truncated Taylor series, as I'm hoping to introduce Taylor polynomials before discussing infinite and power series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I'm hoping that if I can intuitively motivate the derivation of $T_2(x)$, then a generalization to $T_n(x)$ will be clear.&lt;/p&gt;&#xA;" OwnerUserId="21" LastActivityDate="2015-03-05T08:57:36.253" Title="An intuitive derivation of Taylor polynomial coefficients" Tags="|calculus|concept-motivation|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7456" PostTypeId="1" CreationDate="2015-02-20T11:45:25.970" Score="8" ViewCount="532" Body="&lt;p&gt;In some school textbooks,  before introducing the probability as a number between $0$ and $1$, the words &quot;likely&quot;, &quot;More likely&quot;, &quot;less likely&quot; etc. are used to indicate the likelihood of an event. Why are these words chosen and not the words &quot;probable&quot;, &quot;more probable&quot;, &quot;less probable&quot;? Is it intentional to avoid words having common origin with the word &quot;probability&quot;?&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2015-02-20T13:30:25.833" Title="&quot;Probable&quot; vs. &quot;Likely&quot;; Choosing the Appropriate Word" Tags="|textbooks|probability|writing|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7457" PostTypeId="2" ParentId="7456" CreationDate="2015-02-20T13:30:25.833" Score="7" Body="&lt;p&gt;Textbooks tend to try to use the language that would be familiar to readers. In natural speech the use of &amp;quot;likely&amp;quot; is much more prevalent than that of &amp;quot;probable&amp;quot;. Interestingly, though, according to Google N-Grams, that hasn't always been the case, with &amp;quot;probable&amp;quot; and &amp;quot;likely&amp;quot; starting nearly level in popularity in 1800:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/IAsZP.png&quot; alt=&quot;Language Trajectories&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;And yes, as you mentioned, as students mature, we can move them towards sentences like: &amp;quot;x has a greater probability than y&amp;quot; more easily from the use of &amp;quot;likely&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="3571" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-02-20T13:30:25.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7458" PostTypeId="1" AcceptedAnswerId="7462" CreationDate="2015-02-20T13:59:42.180" Score="2" ViewCount="2069" Body="&lt;p&gt;I apologize for the &quot;softness&quot; of this enquiry, but as mathematics educators I believe these are some of the most rewarding moments of teaching our discipline. Having said that, may I share with you a story of teaching an online high school student elementary Algebra, after which, I replied &quot;thanks for trying to learn&quot;, the student answered, &quot;I always try before I give up.&quot; :D&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am simply looking for stories, to share with other mathematics educators, that made them laugh out loud. This, imho, is part of the reward in teaching. &lt;/p&gt;&#xA;" OwnerUserId="216" LastEditorUserId="216" LastEditDate="2015-02-20T20:14:27.463" LastActivityDate="2015-02-21T00:05:01.610" Title="Funny things students say when learning mathematics" Tags="|general-pedagogy|students-mistakes|student-question|mathematical-discourse|" AnswerCount="3" CommentCount="8" ClosedDate="2015-02-21T00:22:54.037" ContentLicense="CC BY-SA 3.0" />
  <row Id="7459" PostTypeId="2" ParentId="7458" CreationDate="2015-02-20T14:26:03.750" Score="2" Body="&lt;p&gt;I once had a colleague when I was teaching in Asia who was new to teaching high school, but was a very accomplished Mathematician. In one of his first lessons he was teaching simultaneous equations, and said that he would teach them two methods to do this. He thought that offering an option would endear him to the students. A student then put his hand up, and with all seriousness said:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Thank you, but previous teacher taught &lt;em&gt;All Known Methods&lt;/em&gt;&quot;&lt;/p&gt;&#xA;" OwnerUserId="3571" LastActivityDate="2015-02-20T14:26:03.750" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7460" PostTypeId="2" ParentId="7455" CreationDate="2015-02-20T15:01:15.270" Score="2" Body="&lt;p&gt;Motivation is a slippery goal. What does your audience know? What do they appreciate? These questions are central. I advocate the view that your audience probably knows more about algebra than geometry. What is the geometry of $f^{(n)}(x)$? Will connecting to that help? Perhaps, I give a constructive approach a bit further down in this answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tend to think your approach has more motivation than you give it credit. The key idea is that the coefficient of the expansion has to do with a single power of the derivative provided we're talking about the center point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Contrast,&#xA;$$ y = mx+b \qquad \&amp;amp; \qquad y=y_o+m(x-a) $$&#xA;the point-slope formula is explicitly connected to the point whereas the slope-intercept is implicitly connected to the intercept $(0,b)$. Next, you prove a simple lemma to show $f(x)=c_o+c_1(x-a)+c_2(x-a)^2+ \cdots $ once more connects the formula of the function to data $f'(a),f''(a)$ etc.. about the center point. The advantage of the expansion in $(x-a)$ verses $x$ is that the expansion in $(x-a)$ is localized at $a$. Of course, the expansion in $x$ is localized at $x=0$. So, why the $1/n!$ factors?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here I might try a boot-strap approach. Let me explain the first step. For the linearization, a nice alternative is the theorem of Caratheodory: $f'(a)$ exists iff there is a continuous function $\phi$ for which&#xA;$$ f(x) = f(a)+\phi(x)(x-a)$$&#xA;and $\phi(a)=f'(a)$ (I assume $f$ is defined on $\mathbb{R}$ for brevity). If $f$ is twice differentiable then $\phi$ must be differentiable at $a$ hence:&#xA;\begin{align} f''(x) &amp;amp;= \frac{d}{dx}\frac{d}{dx} \left[f(a)+\phi(x)(x-a) \right]\\&#xA;&amp;amp;= \frac{d}{dx} \left[\phi'(x)(x-a)+\phi(x)\right] \\&#xA;&amp;amp;= \phi''(x)(x-a)+2\phi'(x). &#xA;\end{align}&#xA;Setting $x=a$ in the above reveals $f''(a)=2\phi'(a)$. Interesting. On the flip-side, we can apply Caratheodory's Theorem to $\phi$ at $a$ to find $\psi$ such that&#xA;$$ \phi(x)=\phi(a)+\psi(x)(x-a) = f'(a)+\psi(x)(x-a) $$&#xA;where $\phi'(a)=\psi(a)$ and so $f''(a)=2\psi(a)$. Likewise, if triply differentiable there exists $\lambda$ such that $\psi(x)= \psi(a)+\lambda(x)(x-a) = \frac{1}{2}f''(a)+\lambda(x)(x-a)$. Nesting these gives:&#xA;\begin{align} f(x) &amp;amp;= f(a)+\phi(x)(x-a) \\&#xA;&amp;amp;= f(a)+\left[f'(a)+\psi(x)(x-a)\right](x-a) \\&#xA;&amp;amp;= f(a)+ \left[f'(a)+\left(\frac{1}{2}f''(a)+\lambda(x)(x-a)\right)(x-a)\right](x-a) \\&#xA;&amp;amp;= f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2+\lambda(x)(x-a)^3.&#xA;\end{align}&#xA;If there is interest, I could attempt to show inductively how this explains the $1/n!$ in the $n$-th coefficient. That said, I think your approach is more appropriate for the second-semester calculus course. I do love these formulas for their exactness. All the nasty remainder is tied up in the function $\lambda(x)$ rather than some fuzzy $+\cdots$. I am no expert, but I think this has connection to Morse's Lemma.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-02-20T15:01:15.270" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7461" PostTypeId="2" ParentId="7455" CreationDate="2015-02-20T15:11:21.280" Score="5" Body="&lt;p&gt;It seems to me that the goal here is to exemplify that a higher degree polynomial is capable of providing a higher fidelity approximation. I think this is a good goal, but I don't think that there's a lot of value in pursuing a 'clean' process of producing the parabola. Your thoughts on that line are good, but will be difficult for most students to follow and more trouble than it's worth to instruct given that the learned technique is essentially a one-off for exploratory purposes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, this is an opportunity for a bit of cross pollination. Students are likely to be able to understand the motivation behind the three constraints $T(a) = f(a)$, $T'(a) = f'(a),$ and $T''(a) = f''(a)$, especially when compared with the constraints that produce the linear approximation at a point. They're also (very) likely to be familiar with the general $Ax^2 + Bx + C$ form of a parabola. I'd suggest working directly from here by solving the three-variable, three-equation system. This gives an understandable, workable method for producing 2nd order approximations of continuous functions at a point. An in-class worked example or two comparing the linear and quadratic approximation, paired with a homework question or two doing the same, is likely to meet your goals, eg, providing a bridge to understanding Taylor series.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2015-02-20T15:11:21.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7462" PostTypeId="2" ParentId="7458" CreationDate="2015-02-20T15:44:19.487" Score="6" Body="&lt;p&gt;From John Stembridge's web site at &lt;a href=&quot;http://www.math.lsa.umich.edu/~jrs/plans.html&quot;&gt;http://www.math.lsa.umich.edu/~jrs/plans.html&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Are we going to have to think today, or is it going to be all math?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--a student in Phil Hanlon's Math 115 class&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2015-02-20T15:44:19.487" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7463" PostTypeId="2" ParentId="7458" CreationDate="2015-02-21T00:05:01.610" Score="2" Body="&lt;p&gt;From a student learning real analysis:&lt;br&gt;&#xA;&lt;em&gt;The sequence diverges because the Cauchy criterion is dissatisfied.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-02-21T00:05:01.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7464" PostTypeId="2" ParentId="5986" CreationDate="2015-02-21T01:32:13.117" Score="2" Body="&lt;p&gt;I recognize that this is not directly answering the question, but I'll proceed regardless.&#xA;I have had some success conveying some aspects of vectors to middle-school children&#xA;(in the U.S.: 8th-9th grade), in the context of robot-arm motion in the plane.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/tbUT9.png&quot; alt=&quot;HTFIFig12.1&quot;&gt;&#xA;&lt;hr /&gt;&#xA;The figure above (from &lt;em&gt;&lt;a href=&quot;http://howtofoldit.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;How To Fold It&lt;/a&gt;&lt;/em&gt;) is an attempt to illustrate the commutativity of vector addition,&#xA;which implies that a robot arm's links can be permuted without altering&#xA;the reachable endpoints.&#xA;Once one accepts this invariance under permutations, non-obvious lemmas can be proved.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-02-21T01:32:13.117" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7465" PostTypeId="1" CreationDate="2015-02-21T17:05:03.950" Score="2" ViewCount="69" Body="&lt;p&gt;I have assembled a list of statistical and numerical illiteracy examples in the media, as they appeared in Chance News online (a statistics wiki) between 2005 and 2014.  See &quot;Collected Forsooths at &lt;a href=&quot;https://www.causeweb.org/wiki/chance/index.php/Collected_Forsooths&quot; rel=&quot;nofollow&quot;&gt;https://www.causeweb.org/wiki/chance/index.php/Collected_Forsooths&lt;/a&gt;. I would appreciate feedback about its usefulness in the classroom.&lt;/p&gt;&#xA;" OwnerUserId="4770" LastEditorUserId="1853" LastEditDate="2015-02-21T19:24:46.303" LastActivityDate="2015-02-21T19:24:46.303" Title="Feedback about statistical and numerical illiteracy-examples website?" Tags="|examples|statistics|" AnswerCount="0" CommentCount="4" ClosedDate="2015-02-24T16:40:27.167" ContentLicense="CC BY-SA 3.0" />
  <row Id="7466" PostTypeId="1" AcceptedAnswerId="7499" CreationDate="2015-02-21T20:08:13.097" Score="22" ViewCount="773" Body="&lt;p&gt;I have a student who is working in their spare time on proving or disproving a conjecture of the form&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\exists x.\forall y.\phi(x,y).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Right now their strategy is to construct an $x$ and then show $\forall y.\phi(x,y)$.  So far so good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student has several times now brought me an $x$ and a &quot;proof&quot; that it works for all $y$ only for me to quickly point out a counterexample $y$.  That's fine too; proof-writing can be tricky business.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what's more worrisome is how they react to the counterexample: the $x$ the next time we meet will usually be a slight variation on the old $x$, adjusted just enough to handle the old $y$.  Now since this problem is open, it's conceivable that such back and forth could eventually produce a valid proof.  But I'm doubtful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'd like to convey to the student is that they can't just react to the counterexamples I give them; they have to keep their eye on the bigger picture if they're going to inoculate their $x$ against any conceivable $y$.  But so far these conversations have had little effect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could fairly easily just keep generating counterexamples and hope that they'll realize what's happening.  But is there perhaps a nice exercise or analogy that would illustrate the issue faster and more clearly?  Or should I focus on the trouble they seem to be having finding counterexamples for themselves?&lt;/p&gt;&#xA;" OwnerUserId="4772" LastActivityDate="2015-02-27T16:31:07.157" Title="How can I discourage proof by patchwork?" Tags="|proofs|graduate-education|students-mistakes|computer-science|" AnswerCount="3" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="7467" PostTypeId="2" ParentId="7415" CreationDate="2015-02-21T20:11:06.153" Score="4" Body="&lt;p&gt;I'll stick my neck out:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no obvious advantage of handwritten lecture notes (written by the lecturer) over typed ones (typed by the lecturer), given infinite time, ability and willingness to create the notes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: this is not intended to be a research-based answer, and I'm not even certain I'd agree with it if I spent a long time thinking. It is essentially intended to act as a poll of general opinion, given that nothing more concrete has come forward in a week. If this is felt to be inappropriate, I'm happy to delete it.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-02-21T20:11:06.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7468" PostTypeId="2" ParentId="7415" CreationDate="2015-02-21T23:23:12.887" Score="6" Body="&lt;p&gt;I have no study to cite at this stage, but I can offer some of my experience of working with thousands of students in my Maths Learning Centre.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Organisation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, students dislike things that are difficult to follow and difficult to find information in by themselves. They like things with clear layout and organisation, with important information highlighted and summarised, and where different parts of the text (proofs, examples, worked examples) are clearly delineated. These are not just likes and dislikes either, but I have seen a change in the structure of the notes encourage more independent learning between two semesters of the same course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of being able to achieve this organisation, the good things can be achieved in both handwritten and typed materials, and badly-organised things can exist in both. Personally I have found that you have to work harder with handwritten materials to be organised, especially if you are making them up on the fly. LaTeX has a natural structure of chapter-section-subsection-etc which helps you to be more organised as you type.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Already-written maths notation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the specific case of worked examples of problem-solving, my experience is that &lt;em&gt;maths that is already written down does not make any sense&lt;/em&gt;. The eye naturally slips over equations and symbols, often not pausing to even recognise the names of the symbols. Moreover, if you are projecting maths onto a screen in your lecture, the students will never have enough time to read it and understand what is going on, no matter how slow you think you're taking it. Already-written symbols are just too difficult to parse at any speed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Leading by example&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I believe that the goal of a worked example is to show what students are supposed to do themselves, and so worked examples should look as close as possible to what you expect the students to do. It is easier to imagine doing the maths yourself if it is handwritten. Moreover, you are more able to imagine how long it takes them to do it and the difficulties they face if you are doing yourself what you have asked them to do!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could apply this argument to the information text as well. If you want them to learn the skill of reading maths (such as proofs etc), then you probably want to present those parts in the format they are likely to see in textbooks, journal articles etc -- that is, it might be better to have this part typed. [A caveat is that since maths that is already written down doesn't make sense, if I were presenting a proof in class, I would handwrite it even though it was typed in the notes. Alternatively, I would take the typed version and mark it up with highlighting and extra working just like I would if I was reading a real journal article myself.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In light of my experience, I would recommend typing most of the materials you give to students, but handwriting anything that is supposed to show them how to do something for themselves.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-02-21T23:23:12.887" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7469" PostTypeId="1" CreationDate="2015-02-22T14:08:38.397" Score="11" ViewCount="1304" Body="&lt;p&gt;&lt;strong&gt;Main question:&lt;/strong&gt; Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are there recognizable benefits that we, as teachers and/or mathematicians, can point to in the hopes of motivating students? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example:&lt;/strong&gt; Consider a simple periodic function like a square wave. This function can be easily written (e.g. defined piece-wise), but the Fourier series is an infinite series of sines and cosines, and we all know infinite series pose notorious conceptual challenges for students at all levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;A student might ask, &quot;Why bother with this format when we can quite easily deal with the given periodic function?&quot;&lt;/em&gt; In answering such a query we might want to address similar questions: Does this format tell us something useful? Does it reveal deeper behavior of this function and similar ones? etc.&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="80" LastEditDate="2015-02-24T18:50:22.227" LastActivityDate="2015-03-13T19:32:15.770" Title="How can I convince students that Fourier series are useful?" Tags="|undergraduate-education|calculus|mathematical-pedagogy|concept-motivation|series|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7470" PostTypeId="2" ParentId="7469" CreationDate="2015-02-22T14:49:24.080" Score="7" Body="&lt;p&gt;You cited the square wave, saying that it can be written as a piecewise function. True, but when given a box of resistors, capacitors, and inductors, you can't build a piecewise filter. This series is taught in engineering school and is one of the most beautiful examples of a Fourier series being used in real life. A filter creating a square wave that can get as clean as you need it depending on the number of components used to build it, i.e the number of terms of the series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$y=\sin x\space +\space \frac{\sin \left(3x\right)}{3}\space +\frac{\sin \left(5x\right)}{5}\space +\space \frac{\sin \left(7x\right)}{7}\space +\frac{\sin \left(9x\right)}{9}+\space \frac{\sin \left(11x\right)}{11}\space +\frac{\sin \left(13x\right)}{13}+\frac{\sin \left(15x\right)}{15}$..........&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/44A1w.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In sum, Fourier series are an important part of many branches of engineering, offering an answer to &quot;when is this used in real life?&quot;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-13T19:32:15.770" LastActivityDate="2015-03-13T19:32:15.770" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7471" PostTypeId="2" ParentId="4351" CreationDate="2015-02-22T15:18:12.030" Score="30" Body="&lt;p&gt;&lt;strong&gt;The &quot;Only Critical Point in Town&quot; test&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose I have a nice function $f : \mathbb R^n \to \mathbb R$.  Suppose it has only one critical point, and that is a local maximum.  Then (of course) it is the global maximum of the function.&lt;br&gt;&#xA;&lt;a href=&quot;http://mathworld.wolfram.com/OnlyCriticalPointinTownTest.html&quot;&gt;&lt;strong&gt;FALSE&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-02-22T15:18:12.030" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7472" PostTypeId="2" ParentId="7466" CreationDate="2015-02-22T16:23:55.973" Score="7" Body="&lt;p&gt;I think the situation is tricky. Without knowing more about the actual problem it may well be that this proof by patchwork will be finished after a few iterations.  This happens when every counterexample you give has another new &quot;component&quot; that the earlier ones did not have. So every new $x$ can handle a new case and there are only finitely many of these cases.  If this is the case, there would be no need to discourage this strategy.  Indeed many good proofs where discovered exactly this way (I guess).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example: Consider global of minimizers of functions on $\mathbb{R}$, so a theorem like: Every “such” function has a global minimizer. One may start with &quot;Every continuous functions&quot; but this fails since there is $f(x) = x$ on the set $\mathbb{R}$. Hence one needs to rule out functions that are unbounded from below. Then take &quot;Every continuous function that is bounded from below&quot;. But still a function like $\exp(x)$ does not  fit the bill. The problem is that the minimizer hides at infinity. There are several possibilities now e.g. like restricting the domain to a compact set or assuming a growth condition at infinity. In the end one may arrive at a nice theorem like &quot;a continuous function, bounded from below, defined on a compact set has a minimizer&quot; (and then realize that &quot;bounded from below&quot; is indeed redundant) or &quot;a continuous function $f$, bounded from below, with $f(x)\to\infty$ for $x\to\infty$ has a minimizer&quot; (and realize that one could generalize &quot;continuous&quot; to &quot;lower semi-continuous&quot;)…  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the situation may be different in that there could be an infinite sequence of counterexamples . It seems like you feel that this will be the case.  In this situation it could be best to wait until the student realizes this himself. He would learn the most this way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess it is quite an important skill to learn if &quot;proof by patchwork&quot; is going to lead somewhere or not…&lt;/p&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="3398" LastEditDate="2015-02-23T08:28:09.093" LastActivityDate="2015-02-23T08:28:09.093" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7473" PostTypeId="2" ParentId="7466" CreationDate="2015-02-22T18:59:50.163" Score="10" Body="&lt;p&gt;Given the student's computer science background, I'd draw a programming analogy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the example &lt;a href=&quot;https://softwareengineering.stackexchange.com/a/148641/143380&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You start with this:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;&amp;lt;p&amp;gt;&#xA;    &amp;lt;label for=&quot;field&quot;&amp;gt;My field&amp;lt;/label&amp;gt;&#xA;    &amp;lt;input type=&quot;text&quot; id=&quot;field&quot;&amp;gt;&#xA;&amp;lt;/p&amp;gt;&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;then you get rid of all that annoying boilerplate and put it in a function:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt; createFieldHtml( id, label )&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;this is good, I'm saving myself so many lines!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;createFieldHtml( id, label, defaultValue )&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;yea, I need a default value too, that was easy to add.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;createFieldHtml( id, label, defaultValue, type )&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;cool, I can use it for checkboxes now too&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;createFieldHtml( id, label, defaultValue, type, labelFirst )&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;UX designer said label has to be after checkbox.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;createFieldHtml( id, label, defaultValue, type, labelFirst, isDate )&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;it now renders a date picker when needed. Hm.. params are getting a bit out of hand&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;createFieldHtml( id, label, defaultValue, type, labelFirst, isDate, containerCssClasses )&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;there was this one case where I need to add CSS classes&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;createFieldHtml( id, label, defaultValue, type, labelFirst, isDate, containerCssClasses, fieldCssClasses, disabled, clearAfter, helpText, uploadPath )&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;aaaaaaaaaaaaaaaaaaaaa&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The context is different, but the lesson is the same: at some point your current approach becomes unwieldy as you keep trying to dodge special cases. A good programmer, at this point, would say it's time to &lt;em&gt;refactor&lt;/em&gt; the program, to restructure the logic of the code to achieve the same end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's easy to stretch the analogy too far, but hopefully it gets the point across that it might be time to take a different approach.&lt;/p&gt;&#xA;" OwnerUserId="1756" LastEditorUserId="-1" LastEditDate="2017-04-12T07:32:02.267" LastActivityDate="2015-02-22T18:59:50.163" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7474" PostTypeId="2" ParentId="294" CreationDate="2015-02-22T20:36:56.677" Score="2" Body="&lt;p&gt;A great book:&#xA;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691154570&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Euler's Gem: The Polyhedron Formula and the Birth of Topology&lt;/strong&gt;&#xA;by &lt;strong&gt;&lt;em&gt;David S. Richeson&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt;. It does an outstanding job at explaining serious mathematics to a general audience, it introduces Euler's Polyhedral formula as well as many concepts in topology.&lt;/p&gt;&#xA;" OwnerUserId="167" LastActivityDate="2015-02-22T20:36:56.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7475" PostTypeId="2" ParentId="7431" CreationDate="2015-02-22T22:05:41.770" Score="5" Body="&lt;p&gt;It seems to me that it is not the homework that needs any special design; it is only the way you grade it. Don't just ask for the answer, ask for a complete proof using the techniques taught in class. It is easy to check one's answer with Wolfram alpha, but I think one needs some kind of understanding to write a complete justification even when using a computer.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-02-22T22:05:41.770" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7476" PostTypeId="2" ParentId="7431" CreationDate="2015-02-23T06:14:02.720" Score="-3" Body="&lt;p&gt;You express a strong sentiment that technology gives students a reason to not need to integrate, but why do you feel like this particular topic in their mathematical careers would be different than any others prior? The same argument might be made in regards to simple arithmetic. 'Why bother with learning how to divide when a basic five function calculator can do the same thing?' I'm relatively certain that there is now a smartphone application that allows users to take a picture of a linear equation and it immediately outputs the answer. 'Why bother with learning Algebra if my phone can do it?' &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Technology and Mathematics Education have had this type of relationship for (at the very least) the last century. I would not worry about students cheating if I were in your shoes if only because if a something is seeking answers from technology for integration solutions, they have probably been doing it for earlier topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your question seems to knock on the door of 'why learn mathematics at all if technology can do it better than I ever can?' Of course, as mathematicians and mathematics educators, we understand the value in learning the content to help further the field and to apply the mathematical principles to other scenarios. To me, mathematics has always been about the journey, and not the destination. If you can instill this in your students, then you should feel comfortable giving them any problem set.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2015-02-23T06:14:02.720" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7477" PostTypeId="2" ParentId="7469" CreationDate="2015-02-23T13:09:12.597" Score="2" Body="&lt;h2&gt;on possibly infinite representations&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I once had this same question, so the following is the way I came to understand it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many ways of defining a function. A piece-wise definition (like for your square wave) or a single succinct formula that may include logarithms, polynomials, or anything else we might like to throw at it. Or we might only have some Differential Equations which can be used to generate the function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, just because we can specify functions like that doesn't mean we can do much with them. Different mathematical tools or manipulations require different representations. In particular integration is often intractable for many arbitrary functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Common formats that give us access to common &quot;Toolkits&quot; are an (often infinite) list of coordinates, an (often infinite degree) polynomial, and (often infinite) combinations of sines and cosines. That all of these representations may be infinite does not mean that they are not useful, as there are often shorter ways expressing them (Sigma notation is common for infinite Taylor series), or hopeful they &quot;converge&quot; so a finite estimate gives us enough information about the part of the function we are interested in.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;The Fourier Transform&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I'm not an expert on the Fourier Transform (I've only ever studied Discrete FT, so others feel free to change or add) but they have strengths and weaknesses just like other representations. An obvious strength is that differentiation of sine waves is easy: simply multiply the amplitude of each wave by its frequency.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your example, the infinite series of sines and cosines that represent a square wave can be simply expressed as $\frac{sin(\pi t)}{\pi t}$ (think of it like that formula gives the amplitude of the sine wave at frequency $t$). It also converges (high frequencies contribute much less to the sum). Once you have it in this form, all of the tools developed for Fourier series are at your disposal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can do signal processing on it. What will your square wave look like after it has gone through filters, or after it has been realised as an analogue waveform? If it is a sound wave, what harmonics will people hear? How will it be encoded into an MP3?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is your square wave part of a control system? How will it interact with other components in the system?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is your square wave part of a Differential Equation? You can now use Fourier methods to solve it algebraically. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want an example of how insanely flexible Fourier transforms are, look up the &quot;Ptolemy and Homer Simpson&quot; YouTube video. Now we can integrate Homer!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a page which goes into more detail about the Fourier transform with some interesting visualisations:&#xA;&lt;a href=&quot;http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face&quot; rel=&quot;nofollow&quot;&gt;http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-02-24T00:42:58.833" LastActivityDate="2015-02-24T00:42:58.833" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7478" PostTypeId="1" CreationDate="2015-02-23T19:01:40.597" Score="13" ViewCount="974" Body="&lt;p&gt;As a child, I made frequent use of a particular 'trick' in order to make short work of many different problems. The general form is to be presented a question which wants a definite (numerical) answer, where the question statement contains a free variable. The 'trick' was to recognize that the answer (which the teacher has told us exists by asking for it) doesn't depend on the free variable, so we now know that we can substitute any convenient value we like for the free variable and find the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://fivetriangles.blogspot.ca/2014/10/193-concave-quadrilateral.html&quot;&gt;This FiveTrangles problem&lt;/a&gt; is an excellent example. The height of the non-shaded triangles is unspecified, so we know straight away that it doesn't figure in the answer to the question. Since it doesn't matter, we can set it to zero and replace this question with the trivial question of finding the area of a triangle with $base = 10$ and $height = 5$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does this technique have a name? Do problems which permit it have a name? Should such questions be avoided?&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2015-05-04T21:42:07.370" Title="Name the heuristic: exploiting the legitimacy of the questioner" Tags="|mathematical-pedagogy|problem-design|problem-solving|" AnswerCount="7" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7479" PostTypeId="2" ParentId="7478" CreationDate="2015-02-23T21:39:43.413" Score="2" Body="&lt;p&gt;That some solutions are independent of a variable is not uncommon, but it does not necessarily trivialise a problem or negate its teaching value.  The following problem presents similar possibilities, but we'd argue that solving it with such a &quot;trick&quot; is actually more circuitous than finding the answer directly (and may be more enlightening, too):&#xA;&lt;a href=&quot;http://fivetriangles.blogspot.com/2012/04/area-problem.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://fivetriangles.blogspot.com/2012/04/area-problem.html&lt;/a&gt;&#xA;Here is a nice animation &lt;a href=&quot;https://i.stack.imgur.com/UHKuT.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;someone posted&lt;/a&gt; of the situation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/USwbI.gif&quot; alt=&quot;&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The shortcoming in such &quot;tricks&quot;, we agree, is that they may lead to tunnel vision, but worse, their utilisation is at the user's risk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(continued in next post)&lt;/p&gt;&#xA;" OwnerUserId="4778" LastEditorUserId="2074" LastEditDate="2015-02-24T10:42:00.940" LastActivityDate="2015-02-24T10:42:00.940" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7480" PostTypeId="2" ParentId="7478" CreationDate="2015-02-23T21:42:05.667" Score="3" Body="&lt;p&gt;(continued from previous post)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this next problem, the horizontal leg of the large right triangle and the vertical leg of the smaller right triangle are also not fixed, but setting each to zero won't lead to a solution at all:&#xA;&lt;a href=&quot;http://fivetriangles.blogspot.com/2013/04/56-obtuse-triangle-area.html&quot; rel=&quot;nofollow&quot;&gt;http://fivetriangles.blogspot.com/2013/04/56-obtuse-triangle-area.html&lt;/a&gt;&#xA;(Never mind that getting in the habit of simply setting values to zero will lead to a &quot;zero&quot; in calculus.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the following proof uses the identical triangle area concept as in the original &quot;concave quadrilateral&quot; problem, but the &quot;trick&quot; doesn't work at all:&#xA;&lt;a href=&quot;http://fivetriangles.blogspot.com/2013/10/104-equal-area-proof.html&quot; rel=&quot;nofollow&quot;&gt;http://fivetriangles.blogspot.com/2013/10/104-equal-area-proof.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4778" LastActivityDate="2015-02-23T21:42:05.667" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7481" PostTypeId="2" ParentId="7478" CreationDate="2015-02-23T21:50:16.610" Score="0" Body="&lt;p&gt;I don't know the answer to the questions about the potential names for such things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to whether such questions should be avoided: I think some types of these questions can maybe help students to think about quantifiers, or to really think about the concepts involved in the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like Chris Cunningham's example: if you tell the student that $(x+4)^2 - x^2 - 8x$ simplifies to a number, and ask them to find the number, they may just do the algebra of simplifying. But if you show them that you can get the answer by plugging in $0$, then you can teach them something about what it really means to say that &quot;$(x+4)^2 - x^2 - 8x$ simplifies to a number&quot;. Now you can talk about what it means for a statement to hold &quot;for all $x$&quot;. Things like this are useful to be familiar with. Many college students struggle with this when I ask them, for example, to show that a function is not linear!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's also nice to have this kind of logic on hand for when you need to remember things. For example, say you were taking a linear algebra exam and you knew the determinant was either the sum or the product of the eigenvalues. Well, you could just consider the $2\times2$ identity matrix and figure it out pretty quickly! This is because you knew the statement didn't depend on the matrix or even the size of the matrix!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I (and this is just opinion) don't like questions like the triangle one that you linked to. Because there, the &quot;trick&quot; from the solver's perspective is just to realize that the question asker left out some words! I don't know how useful it is to have someone apply the logic &quot;If they asked me this, it has an answer, and if they didn't say it depended on quantity $A$ then it must not depend on quantity $A$.'' It is not clear to me what the mathematical benefit is to that.   &lt;/p&gt;&#xA;" OwnerUserId="4779" LastActivityDate="2015-02-23T21:50:16.610" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7482" PostTypeId="2" ParentId="7478" CreationDate="2015-02-23T21:51:25.687" Score="0" Body="&lt;p&gt;I would be surprised if the solution strategy or problem types have a name. While it can be useful to find the correct answer, what does it accomplish from a learning perspective? Take, for instance, Chris Cunningham's problem of &quot;$(x+4)^2 - x^2 -8x$ simplifies to a number. Find that number.&quot; As an instructor, I would be looking to make sure the students accomplished the appropriate learning outcomes for the problem. For this problem, I would more than likely be looking for (1) can they multiply out $(x+4)^2$ and (2) can they simplify the addition of terms. Such a technique would not accomplish (1) or (2), so I would avoid writing problems that can (knowingly) make use of it nor would I ever teach such meta-strategies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This reminds me of a multiple choice exam question on trig identities: &quot;Which of the following is an equivalent identity to $\cot(x) + \tan(x)$?&quot; with $\csc(x)\sec(x)$ being in the list of answers. Instead of using trig identities, many students plugged in a value like $x=10$ and compared. While it works for this case, it dodges the actual learning and testing of the concept.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-02-23T21:51:25.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7483" PostTypeId="2" ParentId="7478" CreationDate="2015-02-23T21:57:21.133" Score="12" Body="&lt;p&gt;The heuristic described here is one manifestation of what Polya (1945) and others thereafter refer to as &lt;strong&gt;trying a special case&lt;/strong&gt;. I do not know of a more specific term for the context that you have put forth, but this is &lt;em&gt;often&lt;/em&gt; how one approaches a problem if you are initially unsure as to how to solve it in generality.&lt;/p&gt;&#xA;&lt;p&gt;It is a good way to get an initial foot-hold on a problem, and especially useful in the context of some standardized tests. For example, consider the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Player 1 flips a penny &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;0$&lt;/span&gt; times, and Player 2 flips a penny &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; times. What is the probability that Player 2 flips more heads than Player 1?&lt;/p&gt;&#xA;&lt;p&gt;A. 1/2&lt;/p&gt;&#xA;&lt;p&gt;B. 1/3&lt;/p&gt;&#xA;&lt;p&gt;C. 1/4&lt;/p&gt;&#xA;&lt;p&gt;D. 1/8&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since none of the choices depends on &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, you can consider &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt; and see easily that the probability is &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt;. This allows you to blaze through the question very quickly. (Indeed, this holds for all &lt;span class=&quot;math-container&quot;&gt;$n \geq 1$&lt;/span&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;Although testing well is an important life skill, your final question as to whether such problems are advisable is interesting. In my estimation, they can be good in the classroom as a way of scaffolding, as long as you press students (when appropriate) to explain &lt;em&gt;why&lt;/em&gt; their reasoning holds for the general case.&lt;/p&gt;&#xA;&lt;p&gt;As one small example, induction problems are done like this: You show the (hopefully easy) base case, and then complete the proof by setting up and using your inductive hypothesis.&lt;/p&gt;&#xA;&lt;p&gt;A possible pitfall, though, is not only the absence of reasoning by students (as in the example question above) but also a possible reinforcement around misconceptions of proof by example. (I believe this relates to the recent query about counterexamples in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7466&quot;&gt;&lt;strong&gt;MESE 7466&lt;/strong&gt;&lt;/a&gt;.) For more about some of these misconceptions, check out work by Orit Zaslavsky (&lt;a href=&quot;http://scholar.google.com/scholar?hl=en&amp;amp;q=zaslavsky+counterexamples&amp;amp;btnG=&amp;amp;as_sdt=1%2C22&amp;amp;as_sdtp=&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;google scholar&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Lastly: The issue with guessing answers based on problem statements is pervasive in school mathematics. It ranges from only seeing examples of the form &lt;span class=&quot;math-container&quot;&gt;$a + b = \square$&lt;/span&gt; and adopting a misunderstanding of the equal sign as an operator, to problems in which students just try to combine numbers in ways that might be &amp;quot;sensible&amp;quot; (without really engaging in sense-making). Cf. the classic:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; There are 125 sheep and 5 dogs in a flock. How old is the shepherd?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; (thinking: 125+5, 125-5, and 125*5 are too big; it must be 125/5) The shepherd is 25.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-02-23T21:57:21.133" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7484" PostTypeId="2" ParentId="7478" CreationDate="2015-02-23T22:03:16.983" Score="0" Body="&lt;p&gt;Discovering the irrelevance of $x$ in the expression could be achieved by getting the students (perhaps in different groups) to substitute different values in for $x$ and compare results. Was this what they expected? Which number would they choose to substitute knowing the answer will always be the same? Having aroused their curiosity the expanding brackets and simplifying can be motivated and discussed.  This could be a rich activity if the emphasis is on discovery and not just sub in $x=0$&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-02-23T22:03:16.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7485" PostTypeId="2" ParentId="6031" CreationDate="2015-02-24T05:07:35.593" Score="3" Body="&lt;p&gt;I have (just recently) become aware of a method for factoring non-monic quadratics called the &quot;Berry Method&quot;. At first glance it seems like a variation of the &lt;a href=&quot;https://matheducators.stackexchange.com/a/6035/267&quot;&gt;ac method&lt;/a&gt; discussed in a different answer, though perhaps more attainable for weaker students. I will continue to research this method and report back, but has anyone had experience using this method? Included &lt;a href=&quot;https://www.google.com/webhp?sourceid=chrome-instant&amp;amp;rlz=1C5CHFA_enUS503US506&amp;amp;ion=1&amp;amp;espv=2&amp;amp;ie=UTF-8#q=berry%20method%20factoring&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; are some early Google results.&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-02-24T05:07:35.593" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7488" PostTypeId="2" ParentId="7469" CreationDate="2015-02-25T02:11:13.173" Score="14" Body="&lt;p&gt;The Fourier coefficients contain useful information that is not at all apparent just from the shape of the periodic function itself. For example, when Fourier series are used for musical vibrations we may remove the terms in the Fourier series with coefficients having magnitude below a certain cutoff (kind of like taking out fundamental tones in a musical note at a frequency beyond the range of human hearing) and the resulting finite Fourier series may be indistinguishable from the original but require a lot less information to store it. That's the whole idea behind data compression. If your students ever listened to an MP3 file then they know why this is important without even realizing it.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2015-02-25T02:52:32.800" LastActivityDate="2015-02-25T02:52:32.800" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7489" PostTypeId="2" ParentId="7469" CreationDate="2015-02-25T12:14:20.127" Score="7" Body="&lt;p&gt;There are tons of reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Signal theory:&lt;/strong&gt; The key phrase here is &lt;em&gt;bandwidth&lt;/em&gt;.&#xA;If you want to transmit a signal over an analog channel you could modulate it onto a carrier frequency. To ensure that you do not have inference with other signals transmitted simultaneously, you should ensure that the signal you want to transmit does not have too large frequency components. In other words: For a periodic signal you need that Fourier coefficients for high frequencies are zero. Check the Wikipedia pages for &lt;a href=&quot;http://en.wikipedia.org/wiki/Bandwidth_%28signal_processing%29&quot;&gt;Bandwidth&lt;/a&gt; and &lt;a href=&quot;http://en.wikipedia.org/wiki/Frequency_response&quot;&gt;Frequency response&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Vibrations:&lt;/strong&gt; The key phrase is &lt;em&gt;natural frequency&lt;/em&gt;. If you have a physical object that is subject to excitation and start to vibrate, you could model this by finite elements and end up with certain eigenfrequencies. In a linear model the different eigenfrequencies all have their own damping coefficient i.e. oscillations at different frequencies are damped away independent of each other, each with its own speed. Coming back to the square wave: If the excitation is a square wave, then what matters for the resulting vibration is the expansion of the excitation in eigenfrequencies. A square wave has contributions from all frequencies, i.e. it excites all eigenfrequencies.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Electrical circuits:&lt;/strong&gt; Circuits consisting of resistors, inductors and capacitators behave similarly, i.e. the behavior of the circuit is described by its eigenfrequencies. Again, exciting the circuit with a square wave (that is, switching a current on and off) excites all eigenfrequencies which is good to know since this may cause undesired behaviour. Check &lt;a href=&quot;http://en.wikipedia.org/wiki/RLC_circuit&quot;&gt;RLC circuits&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Regularity of functions:&lt;/strong&gt; More on the mathematical side, the order of differentiability can be seen in the decay of the Fourier coefficients. If $f$ is $n$-times continuously differentiable, then the Fourier coefficients decay like $1/k^n$ and conversely, if the coefficients decay fast enough, the function is indeed differentiable… Check, e.g., &lt;a href=&quot;http://www-m7.ma.tum.de/foswiki/pub/M7/Analysis/Fourier13/lecture9.pdf&quot;&gt;these notes&lt;/a&gt; for details and exact statements.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2015-02-25T12:14:20.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7490" PostTypeId="1" CreationDate="2015-02-25T16:21:00.277" Score="4" ViewCount="167" Body="&lt;p&gt;Boolean logic is easily explained using truth tabels, but the underlying hardware is rarely mentioned in the same class. I would like to explain boolean logic from the point of view of actual transistor. My target audience is educated adults with no background in engineering, for a one-time public lecture. The goal is to have them understand why a &lt;a href=&quot;https://electronics.stackexchange.com/questions/36810/how-can-i-design-a-circuit-that-will-turn-on-when-a-wire-is-cut&quot;&gt;simple cut-wire alarm circuit&lt;/a&gt; is designed the way it's designed, and without going &quot;deep&quot; into solid state physics.&lt;/p&gt;&#xA;" OwnerUserId="4765" LastEditorUserId="-1" LastEditDate="2017-04-13T12:33:11.313" LastActivityDate="2015-02-26T21:17:15.980" Title="How to intuitively explain the role of transistors in boolean logic and switching?" Tags="|undergraduate-education|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7491" PostTypeId="2" ParentId="7469" CreationDate="2015-02-25T16:31:56.067" Score="3" Body="&lt;p&gt;I often use the module &quot;&lt;em&gt;How to Tune a Radio--Trigonometric integrals explain tuning a radio&lt;/em&gt;.&quot; It is in Volume 3 (Applications of Calculus) of the &lt;strong&gt;MAA Resources for Calculus Collection&lt;/strong&gt; (MAA Notes Number 29, edited by Philip Straffin).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The author of the module, Clark Benson, provides very good context to the problem of tuning a radio, connects the problem to integration, and introduces Fourier analysis. Even if you do not assign the series of problems in the module, having students read the module can help convince them of why one might want to represent a square wave as a Fourier series. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-02-25T16:31:56.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7492" PostTypeId="2" ParentId="7490" CreationDate="2015-02-25T19:08:01.763" Score="2" Body="&lt;p&gt;You can explain them as electrically controllable switches.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If a voltage is applied to gate and source, this means &lt;code&gt;switched on&lt;/code&gt;: Current may flow between drain and source. Or voltage is transferred through the transistor.&lt;/li&gt;&#xA;&lt;li&gt;If no voltage is applied, this means &lt;code&gt;switched off&lt;/code&gt;: No current may flow resp. voltage is blocked.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="439" LastActivityDate="2015-02-25T19:08:01.763" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7493" PostTypeId="2" ParentId="7490" CreationDate="2015-02-26T01:47:42.110" Score="1" Body="&lt;p&gt;This may be off the mark of your question, but I have explained&#xA;logic gates composed of (transistor) switches by using water pressure&#xA;as analogous to electricity.&#xA;Below, water will only reach the pipe-end $C$ if water pressure comes&#xA;in both along pipe $A$ AND there is water pressure&#xA;from $B$ that pushes up and closes the normally open switch, connecting $A$ to $C$.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/2jDzv.jpg&quot; alt=&quot;WaterSwitch&quot;&gt;&#xA;&lt;hr /&gt;&#xA;If either there is no water pressure at $A$, or there is no water pressure at $B$, &#xA;no water reaches $C$.&#xA;So this is an AND-gate. Etc.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-02-26T12:34:23.527" LastActivityDate="2015-02-26T12:34:23.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7494" PostTypeId="2" ParentId="1037" CreationDate="2015-02-26T11:09:01.710" Score="11" Body="&lt;p&gt;Besides mastering the material in the course one thing that the students have to learn during studies is to communicate mathematics in written form. Almost nobody comes to university and is able to write clear proofs or mathematical arguments. You need to communicate that this is part of what they have to learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may grade as harsh as you like if you are willing to engage into discussion about your grading. When I had messy sheets and graded harsh and these students came to office hours we usually had good conversation. More often than not the conversation goes roughly like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Student: What I mean here is X.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Me: I see, but that is not the right argument at that place.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Student: But at this point.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Me: Right, a similar argument would help at this point. But yours is not totally correct.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Student: But this (slightly different) argument works.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Me: Correct!&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Student: …&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Me: At this point the conclusion is not clear to me.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Student: But it is clear from Y.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Me: If it's clear from Y, write that it is clear from Y. Conclusion: If something is clear, write it down clearly. Moreover, you wanted to write Z at place B and not X at place A. So please write that down in the first place next time!&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Student: (Leaves with new insight but the same grading… In most cases happy.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The sheets got a lot less messy next time…&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2015-02-26T11:09:01.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7495" PostTypeId="2" ParentId="7490" CreationDate="2015-02-26T19:30:21.297" Score="1" Body="&lt;p&gt;You could follow the suggestions made in the other posts, but I would start with an even simpler model which you could build and demonstrate easily: levers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider building a contraption (a la Rube Goldberg, but much more boring) of remotely flipping a light switch (where down means light is off and up means light is on).  One way to do this is to attach a long rod to the switch as follows: fix one end (use a universal joint as needed) of the rod to the switch, and fix the middle of the rod to a stationary object.  You now have a logical inverter (the other end of the rod will be up when the switch is down, and down when the switch is up).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This requires a representation (0 means down and 1 means up) and a tolerance for building more complicated contraptions (concatenate two inverters to get the Boolean identity function, take two inverters and a spring or a lead weight on the end of a third inverter and make a two input AND gate by placing the three levers in a star configuration with the weight in the center; pushing on the ends on one of two of them will cause the third end to fall), but conceptually they are very simple: this only requires people having seen a lever before, and the contraptions aren't that complex.  You will need very few levers to make a model of a transistor, and many people should make the analogy between lever position and voltage level easily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Mixes Water And Electrons Mentally&quot; Paseman, 2015.02.26&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-02-26T19:30:21.297" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7496" PostTypeId="1" CreationDate="2015-02-26T19:39:35.293" Score="6" ViewCount="205" Body="&lt;p&gt;I am currently a junior at a mid-tier college and am doing reasonably well in my math major,  but not quite as well as I suspect necessary to get into well respected PhD programs as is my dream. I've been receiving a healthy mix of A's and B's among my math courses and as far as research is concerned, am only going to participate in one low caliber REU this upcoming summer. As such, I would like to explore high school teaching opportunities in case I don't get into  (and funding for) a good PhD program. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I have had my eye set on graduate school, I haven't done anything to prepare to become a teacher. I am majoring in pure mathematics and have only taken one education course. My schedule next year is tight so I wouldn't have time to take any more than one education course as an undergraduate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What can I do set up as a reasonable back up plan in teaching?&lt;/strong&gt; Please note that I would be looking to teach in public or private schools in New Jersey and that I don't have very much money so would be unable to pay for another few years of tuition to get a masters degree right off the bat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would very much appreciate any advice -- this is something I haven't ever looked into so I don't know what sort of certifications etc someone in my position would need to acquire.&lt;/p&gt;&#xA;" OwnerUserId="4785" LastEditorUserId="4785" LastEditDate="2015-02-26T20:38:03.880" LastActivityDate="2015-02-26T20:38:03.880" Title="Grad school backup: high school teacher" Tags="|teacher-preparation|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7497" PostTypeId="2" ParentId="7496" CreationDate="2015-02-26T20:28:26.823" Score="9" Body="&lt;p&gt;tl;dr do a ton of research first before making any decisions&lt;/p&gt;&#xA;&#xA;&lt;p&gt;IMHO, you shouldn't just whimsically pursue teaching as a &quot;backup plan&quot; if your current plan doesn't work out.  Teaching is A TON of work from planning, grading, teaching, tutoring, after-school activities, etc. and if you don't have a true passion for teaching and the discipline and persistence it requires, you will almost assuredly leave the profession within a few years and be right back where you started (upwards of half of all teachers leave within five years of becoming teachers).  I am not trying to turn you off from teaching by any means, I am just trying to make sure you know what you are getting yourself into.  Even if you have the required courses and know the material, there is still a certification process which takes time and money and usually requires you to apply and be accepted to a university that has been OKed by New Jersey to certify new teachers (more info &lt;a href=&quot;http://www.state.nj.us/education/educators/license/teacher/&quot;&gt;here&lt;/a&gt;).  Once you do become a teacher, the pay is almost certainly much less than you will deserve for the amount of work you will be putting in and you will likely continue to be shafted by your state's and the federal government.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would seriously suggest to do a lot of research into teaching certifications for your state and the various programs that are available before making any decisions.  If you are serious, i would suggest contacting a high school in your area, explain your situation, and see if they will let you shadow/observe for a day.  This should give you a snapshot of what teachers are expected to do each and everyday.  If you like what you see and think that teaching is for you, the most well-known program would be Teach For America, which places you in a city of their choosing, gives you an emergency cert, and drops you in a classroom with only six weeks of actual training.  However, there are many other programs, usually through universities that offer varying levels of support and instruction to get you prepared to be a teacher, most of which cost money.  The Federal Government does understand the position most teachers are in (broke, underfunded, under resourced) and as such usually offer pretty generous loans that can help pay for the various costs of certification.  There are also programs for loan forgiveness where you can essentially not have to pay your loan if you teach for X amount of years after your loan matures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I really don't want to sound negative, it is great that you are considering teaching! Its an amazing profession and so rewarding and fun. It is a lot of work, but it is a truly selfless act and I hope that after doing some research you still are interested :)&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-02-26T20:28:26.823" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7498" PostTypeId="2" ParentId="2459" CreationDate="2015-02-26T21:14:01.610" Score="2" Body="&lt;p&gt;This may not be directly answering your question, but I would suggest, if it is possible with your specific content matter, allowing students to make the choice of &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;tablet&lt;/li&gt;&#xA;&lt;li&gt;calculator&lt;/li&gt;&#xA;&lt;li&gt;smartphone&lt;/li&gt;&#xA;&lt;li&gt;computer&lt;/li&gt;&#xA;&lt;li&gt;etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://calculuscourse.maa.org/&quot; rel=&quot;nofollow&quot;&gt;The MAA's calculus text&lt;/a&gt; is a good example of one way that tablets could be used fruitfully.  I don't require ANY technology other than access to something that can do roots and logs, but even for graphing there are so many downloadable apps, as well as even Google &lt;a href=&quot;https://www.google.com/search?q=x%5E2%2By%5E2&amp;amp;ie=utf-8&amp;amp;oe=utf-8&quot; rel=&quot;nofollow&quot;&gt;plotting&lt;/a&gt; in modern browsers, that you may not need to require any one technology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that said, you may need uniformity for other reasons (such as standardized exam, internet, whatever), in which case this wouldn't apply.  But if there is no particular reason to enforce uniformity in technology, this might be a nice solution.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-02-26T21:14:01.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7499" PostTypeId="2" ParentId="7466" CreationDate="2015-02-27T16:31:07.157" Score="8" Body="&lt;p&gt;Instead of providing your student a counter-example, you could try to ask her or him about its faulty proof. Justifying every step until the faulty one is identified by the student herself might trigger some more understanding.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-02-27T16:31:07.157" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7501" PostTypeId="2" ParentId="4379" CreationDate="2015-02-28T00:07:53.073" Score="-2" Body="&lt;p&gt;You have certainly added a lot of stuff, but If I was doing this remotely, I would look at Classkick. Synchronous whiteboard sharing, but full of bugs currently.&lt;/p&gt;&#xA;" OwnerUserId="3571" LastActivityDate="2015-02-28T00:07:53.073" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7502" PostTypeId="1" CreationDate="2015-02-28T01:20:34.500" Score="11" ViewCount="412" Body="&lt;p&gt;Symbolic math apps, such as Mathcad, are extremely useful in doing exploratory/experimental math. I've frequently used it to run numerical checks on mathematical expressions I've derived; put up quick plots to understand the qualitative behavior of functions; and manipulate power series, matrices, and differential/integral operators to discover potential relations among number sequences (linear, triangular) for the OEIS and to generate new sequences. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The process of generating numerical checks often reveals gaps in my understanding, leads me to explore other perspectives, and can force me, in general, to organize my ideas more constructively, leading to a deeper understanding of a problem, much as in writing up expository math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question: Are such apps being used in teaching &quot;hands-on&quot; mathematics as opposed to or in conjunction with the traditional lecture/blackboard methods? If so, where and how?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example proplem: Have the students find approximate numerical values for a complex line integral along different easily parametrized paths (e.g., arcs, edges of a rectangle, any such paths the students can come up with) in an analytic neighborhood connecting the two end points. Using Mathcad the students will have to thoroughly understand how to parametrize the contour integral to have Mathcad do a numerical integration. Use an easily integrated function as the first example to provide a check. Then use one that can't be analytically integrated. Discuss path independence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a more general note:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the disappointing aspects of many video lessons on the Net is that they still rely on the old technique of writing out an equation (useful if you want to give the student time to copy it by hand--unnecessary for pre-recorded material) while talking though it stroke-by-stroke. Incredibly tedious and uninformative when one is already familiar with the notation. I'm interested in guided instruction rather than pure lecturing that takes the students on a journey of discovery (not the least being that of their own abilities at intuiting math). Combo: smart boards, apps, and guidance. Assumption: smart students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Software, such as Logical Zoombinis, effortlessly teach the very young to use logic in solving problems. Same for the classic board game Clue, and any number of games quickly motivate students to do simple arithmetic. Khan Academy uses innovative techniques, Dan Meyers has a great approach to interactive math learning, and any number of websites have hands-on courses for learning programming languages, such as Udacity and Code.org. Where has the math community adopted innovative techniques at the college level?&lt;/p&gt;&#xA;" OwnerUserId="4793" LastEditorUserId="4793" LastEditDate="2015-03-09T02:05:33.483" LastActivityDate="2015-03-09T02:05:33.483" Title="Use of symbolic math apps in teaching" Tags="|mathematical-pedagogy|technology-in-education|interactive-teaching|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7503" PostTypeId="1" CreationDate="2015-02-28T03:16:12.670" Score="12" ViewCount="254" Body="&lt;p&gt;I'm just starting up the academic year (yes, it starts in February here in the Southern Hemisphere) teaching a 2nd-year Introduction to Pure Mathematics class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For general background, I would like to explain the general concept of axioms and the role of axiomatisation in Pure Mathematics, and refer them to a suitable reasonably-accessible essay for more information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Googling has led me to an oldish paper by De Villiers (mzone.mweb.co.za/residents/profmd/axiom.pdf) which is quite interesting. In particular he distinguishes between &quot;constructive axiomatisation&quot; in which (essentially) the axioms come first and then logical deduction creates new mathematics and &quot;descriptive axiomatisation&quot;, where new mathematics that has arisen in an untidy, organic, uncontrolled way is reorganised and systematised with the benefit of hindsight (in other words, which ideas led somewhere, which didn't).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, he mentions criticisms of the &quot;axiomatic-deductive&quot; method in TEACHING (i.e. Lecture 1 gives the axioms for a vector space and proves that 0 is unique, etc etc) because it conveys - inaccurately - to students that mathematics is just the dry mechanical manipulation of statements starting from some &quot;god-given&quot; or &quot;magic&quot; axioms that are simply presented as a fait-accompli, rather than a creative, iterative, untidy process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I can't just give the students this link, because it assumes that the reader already KNOWS examples of descriptive and/or constructive axiomatisation, and it also uses language that I predict my students would not enjoy. (Example: Freudenthal (1973, 451) refers to this dramatic change in style of thinking, as the &quot;cutting of the ontological bonds.&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To a large degree, time constraints will force me to continue teaching essentially used the axiomatic-deductive method, but I would really like to alert the students to the fact that this is just one side of the coin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, it seems to me that this topic is something that surely has faced many many teachers of Pure Mathematics before, and that there &lt;em&gt;should&lt;/em&gt; be an essay somewhere that is aimed at beginning students, rather than their professors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is: Is there an essay, aimed at beginning students, that explains in simple terms the role of axioms and axiomatisation in mathematics?&lt;/p&gt;&#xA;" OwnerUserId="4795" LastActivityDate="2015-02-28T21:51:50.740" Title="Has someone written an essay on the role of axioms in mathematics (suitable for undergrads)?" Tags="|reference-request|axioms-foundations|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7504" PostTypeId="2" ParentId="7296" CreationDate="2015-02-28T13:17:17.203" Score="2" Body="&lt;p&gt;Here's an excerpt from an article in the &lt;em&gt;Houston Chronicle&lt;/em&gt; dated August 26, 2010.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;http://www.chron.com/news/houston-texas/article/HISD-families-could-get-up-to-1-020-if-kids-1705453.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;HISD families could get up to \$1,020 if kids master math&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Fifth-graders at select Houston ISD schools will get richer for passing math tests — and their parents will get paid, too — under a rare experiment to try to boost student performance with cash incentives totaling as much as \$1,020 per family.&lt;/p&gt;&#xA;&lt;p&gt;The Houston school board signed off Thursday on the \$1.5 million program, which is funded by the Dallas-based Liemandt Foundation. The incentives will go to students and parents at 25 elementary schools that rank among the lowest in math achievement.&lt;/p&gt;&#xA;&lt;p&gt;The pilot program — thought to be the first that offers joint incentives for parents and students — will allow fifth-graders to earn up to \$440 for passing short math tests that show they have mastered key concepts, according to the draft proposal. Parents will get slightly less money for their children doing the work, and they can earn an extra \$180 for attending nine conferences with teachers to review the youngsters' progress.&lt;/p&gt;&#xA;&lt;p&gt;Combined, the students and their parents can pocket \$1,020.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Were you looking for this kind of program?&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-02-28T13:17:17.203" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7505" PostTypeId="2" ParentId="7502" CreationDate="2015-02-28T15:09:46.550" Score="6" Body="&lt;p&gt;This is not an answer to your specific question, but rather offers one&#xA;example toward the larger question of where has software been used for&#xA;interactive explorations in educational settings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The geometry software &#xA;&lt;em&gt;&lt;a href=&quot;http://www.cinderella.de/tiki-index.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cinderella&lt;/a&gt;&lt;/em&gt; is used for hands-on&#xA;teaching at many schools around the world, especially in Germany, its source.&#xA;See this &lt;a href=&quot;http://antique.cinderella.de/en/community/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;collection of links&lt;/a&gt;.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/KvjgX.png&quot; alt=&quot;CindyExamples&quot;&gt; &#xA;&lt;hr /&gt;&#xA;I use it myself in a college-level course on computational geometry.&#xA;E.g., we use it to understand the dynamics of linkages:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Q3FlX.gif&quot; alt=&quot;3bar&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://archive.msri.org/about/computing/docs/cinderella/Texts/Mathematics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-03-01T01:59:40.977" LastActivityDate="2015-03-01T01:59:40.977" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7506" PostTypeId="2" ParentId="7503" CreationDate="2015-02-28T21:51:50.740" Score="5" Body="&lt;p&gt;Only one thing came to my mind reading your post: &lt;a href=&quot;https://books.google.co.uk/books?id=4WPDAgAAQBAJ&amp;amp;printsec=frontcover&amp;amp;dq=ian%20stewart%20concepts%20modern%20mathematics&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=iDPyVN2YCKTC7gao7oBA&amp;amp;ved=0CDQQ6AEwAA#v=onepage&amp;amp;q=ian%20stewart%20concepts%20modern%20mathematics&amp;amp;f=false&quot;&gt;Concepts of Modern Mathematics&lt;/a&gt; written by Ian Stewart. It was an eyeopener for undergraduate me, and it is now for many of undergraduate students I teach. It is about mathematics as a whole, including a section on axioms and axiomatisation (or, as it is &quot;Axiomatics&quot;). To make you curious, here is how that section starts: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Only an elephant or a whale gives birth to a creature whose weight is&#xA;  70 kilograms or more. The President's weight is 75 kilogram.&#xA;  Therefore, the President's mother was either an elephant or a whale.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-02-28T21:51:50.740" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7507" PostTypeId="1" CreationDate="2015-03-02T01:11:06.147" Score="1" ViewCount="149" Body="&lt;p&gt;I am a basic skills community college math instructor here in Los Angeles. This Spring is a very exciting time for our college. There are several campus initiatives taking place. Among them is the implementation of Assembly Bill 86, the Adult Education Consortium Program &lt;a href=&quot;http://ab86.cccco.edu&quot; rel=&quot;nofollow&quot;&gt;http://ab86.cccco.edu&lt;/a&gt; which will take effect July 1 2015. There are 5 major areas AB 86 seeks to improve:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Elementary and Basic Skills (Math &amp;amp; English)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Classes for Immigrants (ELLs)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Programs for Adults with Disabilities&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Short term CTE programs&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Programs for Apprentices &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’m meeting with the College President’s task force on Adult Education in 2 weeks and as you can imagine, at this point in the planning process there is a tremendous amount of work that must get done so that we can begin to address the goals of AB 86 by July 1st. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I plan to bring the &lt;strong&gt;pedagogy&lt;/strong&gt; aspect to the table as one of the goals. I want to make sure the president considers professor accountability - because we can’t have classes for immigrants with professors who lack strategies for learning - the program will be ineffective in my opinion - several programs such as &quot;First Year Completion&quot; go negatively criticized by their respective students because “the professor was unhelpful.” &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’m asking that you help me add professional points to improve my conversation with the committee in terms of teaching basic skill Math to English Language Learners and offering classes for immigrants by putting effective instructors, hold all instructors to mandatory professional development for best strategies for ELLs, SIOP models, cyclic instruction, etc etc. Appreciate the help in advance. &lt;/p&gt;&#xA;" OwnerUserId="2198" LastActivityDate="2015-03-03T15:34:10.027" Title="AB 86 Basic Skills Mathematics Courses at the Community College Level" Tags="|mathematical-pedagogy|community-colleges|" AnswerCount="2" CommentCount="1" ClosedDate="2015-03-10T21:08:33.183" ContentLicense="CC BY-SA 3.0" />
  <row Id="7508" PostTypeId="2" ParentId="7469" CreationDate="2015-03-02T02:21:51.363" Score="4" Body="&lt;p&gt;On one hand understanding Fourier series is very important in Engineering, Physics and Math. On the other hand, to actually calculate the coefficients by hand, instead of using software, do we really need that nowadays? We do not teach how to use an engineering sliding ruler or logarithmic tables to do operations anymore, because mastering those skills is not necessary today. I try to show my engineering students in Acoustics the meaning of the Fourier series using software (Mathematica), please see the PDF documents in the several &quot;Acoustic Spectrum&quot; sections of this link &lt;a href=&quot;http://matecmaticaacustica.weebly.com&quot; rel=&quot;nofollow&quot;&gt;http://matecmaticaacustica.weebly.com&lt;/a&gt;, perhaps those documents can be useful for you, even if you insist (or need to insist) in calculating the coefficients by hand (I mean without software)&lt;/p&gt;&#xA;" OwnerUserId="1533" LastActivityDate="2015-03-02T02:21:51.363" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7509" PostTypeId="1" CreationDate="2015-03-02T09:15:15.563" Score="20" ViewCount="3022" Body="&lt;p&gt;Are there any resources which show Fourier series approximating a given waveform? I am looking for examples which have a real  impact on students and provides motivation. I am trying to find something visual but it could be just audio based. Something to start the topic of Fourier series so that students are inspired to study this difficult topic.&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="143" LastEditDate="2015-03-02T14:28:41.263" LastActivityDate="2020-11-25T12:55:27.693" Title="Fourier Animation" Tags="|technology-in-education|mathematical-analysis|interactive-teaching|" AnswerCount="8" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7510" PostTypeId="2" ParentId="7509" CreationDate="2015-03-02T11:00:11.357" Score="9" Body="&lt;p&gt;The Wolfram demonstrations project has several, including:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://demonstrations.wolfram.com/FourierSeriesOfSimpleFunctions/&quot;&gt;http://demonstrations.wolfram.com/FourierSeriesOfSimpleFunctions/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See here for the full list:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://demonstrations.wolfram.com/search.html?query=fourier&quot;&gt;http://demonstrations.wolfram.com/search.html?query=fourier&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3571" LastActivityDate="2015-03-02T11:00:11.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7511" PostTypeId="2" ParentId="7509" CreationDate="2015-03-02T13:58:33.457" Score="9" Body="&lt;p&gt;I would suggest using a freely available program to generate and analyse signals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://audacity.sourceforge.net&quot; rel=&quot;nofollow&quot;&gt;Audacity&lt;/a&gt; is a freely available program that can generate various waveforms (sine, square, triangle, sawtooth) and can also do Fourier analysis.  It can also analyse real signals (record from microphone or import audio files).  There is also a generator that can do freqency sweeps, and effects that include simple filters for when you get to that stage.  It isn't limited to just simple filters, though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This gives you a simple, interactive way to investigate simple signals and a path towards more interesting signals.  You can look at the effects of changing sampling rate, Fourier length, and windows.  You can also play the generated signals through the PC speakers so that you can hear what you are analysing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Audacity isn't real time, but that is good when starting out - and when in a classroom.  It can get quite loud when students are recording live signals, and the recorded signals won't be clean because of all the signals and the students talking.  Using generated signals within the program is much simpler.&lt;/p&gt;&#xA;" OwnerUserId="4800" LastEditorUserId="376" LastEditDate="2016-04-29T14:08:39.927" LastActivityDate="2016-04-29T14:08:39.927" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7512" PostTypeId="2" ParentId="7507" CreationDate="2015-03-02T15:09:45.053" Score="3" Body="&lt;p&gt;Mandatory professional development is not what makes a good teacher. And if there was the same sort of criticism across multiple teachers, it may be the program and not the teacher that's the problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the most effective ways to change remedial programs is to make them shorter. Check out the information from the &lt;a href=&quot;http://cap.3csn.org/why-acceleration/&quot; rel=&quot;nofollow&quot;&gt;California Acceleration Project&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-03-02T15:09:45.053" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7513" PostTypeId="1" AcceptedAnswerId="7514" CreationDate="2015-03-02T17:45:18.850" Score="5" ViewCount="382" Body="&lt;p&gt;Let $F$ be a field. What is the best notation (in an undergraduate or graduate abstract algebra class) for a generic element of the univariate polynomial ring $F[x]$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most common notation seems to be $f(x)$. It seems strongly to me that $f$ (or any other single letter) is to be preferred: it emphasizes that elements of $F[x]$ are well-defined objects, without reference to plugging in anything for $x$, and that if you see $f(a)$, this always indicates the &lt;em&gt;evaluation homomorphism&lt;/em&gt; obtained by replacing $x$ with $a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no logical conflict here, because the tautological evaluation homomorphism $F[x] \rightarrow F[x]$ sending $x$ to $x$ is just the identity. Nevertheless, I cannot help but feel that writing $f(x)$ sends the wrong message.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, judging from a skimming of the algebra textbooks on my shelf, I seem to be in the minority. Is there reason to prefer writing $f(x)$?&lt;/p&gt;&#xA;" OwnerUserId="1535" LastActivityDate="2015-04-03T15:27:26.247" Title="Notation for an element in a polynomial ring" Tags="|notation|abstract-algebra|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7514" PostTypeId="2" ParentId="7513" CreationDate="2015-03-02T18:06:14.923" Score="6" Body="&lt;p&gt;A couple of practical points to consider: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If one does not always write the variable it can be more tricky to avoid confusion between substitution and multiplication. Consider:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f=5x^2 +7$. Further, let  $g = f(x^2 +1)$.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What is meant here,   $5(x^2+1)^2 +7$ or rather $(5x^2+7)(x^2+1)$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By contrast &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f(x)=5x^2 +7$. Further, let $g(x)= f(x^2 +1)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f(x)=5x^2 +7$. Further, let $g(x)= f(x)(x^2 +1)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;seem both clear. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another issue arises when there are parameters involved: &quot;the polynomial  $f=x^2 + ax+ a^2$&quot; is somewhat unclear. By contrast &quot;the polynomials $f(x)= x^2 + ax+ a^2$ and $f(a)=x^2 + ax+ a^2$  and $f(x,a)=x^2 + ax+ a^2$&quot; all seem clearer. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similar to the previous point it can be handy to be able to highlight in the multi-variable case when some polynomial does not depend on all variables, say like: &quot;We have a factorization $f(x,y)= g(x)h(y)$.&quot; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;That being said I do agree with the point that there is merit in pointing out that there is no actual need to write $f(x)$. And, one could avoid the issues mentioned above in other ways. Still there is also some downside, and as you asked about this I mainly talk about this.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2015-03-02T21:11:46.630" LastActivityDate="2015-03-02T21:11:46.630" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7515" PostTypeId="2" ParentId="7513" CreationDate="2015-03-02T19:49:49.030" Score="3" Body="&lt;p&gt;I would strongly present notation that is consistant with whatever book you are using. I know the most popular books use the $f(x)\in F[x]$ (Dummit and Foote, Lang), while others use just $f \in F[x]$ (Atiyah and Macdonald). Either way, they both represent the same polynomial $a_0 + a_1 x + ... + a_n x^n \in F[x]$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would strongly indicate that notation for the variable (or other object) is defined by the $F[x]$, and not the representation of the elements. You might even indicate $f = f(x) \in F[x]$ so that they can see the equivalence. Students should be warned that $f(c) \in F$ for $c \in F$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I like the notation $f(x) \in F[x]$. It highlights the evaluation map $Eval_a :F[x] \to F$ by $Eval_a (f(x))=f(a)$ instead of the map $x \mapsto a$, but for a sufficiently advanced class it shouldn't matter. The definition element-wise might be easier for beginning students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as @quid pointed out, if you mix notations, ambiguity arises. So stick with your book's notations and highlight them early.&lt;/p&gt;&#xA;" OwnerUserId="201" LastEditorUserId="201" LastEditDate="2015-03-02T20:19:43.487" LastActivityDate="2015-03-02T20:19:43.487" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7516" PostTypeId="2" ParentId="7509" CreationDate="2015-03-02T20:05:19.657" Score="24" Body="&lt;p&gt;Wikipedia always has some great animations, often by &lt;a href=&quot;https://en.wikipedia.org/wiki/User:LucasVB/Gallery&quot; rel=&quot;noreferrer&quot;&gt;User:LucasVB&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/User:Cmglee#Fourier_series_square_wave_circles_animation.gif_.E2.98.8E_4_May_2014&quot; rel=&quot;noreferrer&quot;&gt;User:Cmglee&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A few Fourier transform–related ones:&lt;/p&gt;&#xA;&lt;h1&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/File:Fourier_series_square_wave_circles_animation.gif&quot; rel=&quot;noreferrer&quot;&gt;Approximating a square wave with a Fourier transform&lt;/a&gt;&lt;/h1&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/1/1a/Fourier_series_square_wave_circles_animation.gif&quot; alt=&quot;Approximating a square wave with a Fourier transform&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;(more of these for other functions &lt;a href=&quot;https://en.wikipedia.org/wiki/User:Cmglee#Fourier_series_square_wave_circles_animation.gif_.E2.98.8E_4_May_2014&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;: &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Fourier_series_sawtooth_wave_circles_animation.gif&quot; rel=&quot;noreferrer&quot;&gt;sawtooth&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Fourier_series_triangle_wave_circles_animation.gif&quot; rel=&quot;noreferrer&quot;&gt;triangle&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Fourier_series_cubed_wave_circles_animation.gif&quot; rel=&quot;noreferrer&quot;&gt;partial cubic&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;h1&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/File:Fourier_transform_time_and_frequency_domains_(small).gif&quot; rel=&quot;noreferrer&quot;&gt;Fourier transform time and frequency domains&lt;/a&gt;&lt;/h1&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/7/72/Fourier_transform_time_and_frequency_domains_%28small%29.gif&quot; alt=&quot;Fourier transform time and frequency domains&quot; /&gt;&lt;/p&gt;&#xA;&lt;h1&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/File:Continuous_Fourier_transform_of_rect_and_sinc_functions.gif&quot; rel=&quot;noreferrer&quot;&gt;Continuous Fourier transform of rect and sinc functions&lt;/a&gt;&lt;/h1&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/a/a3/Continuous_Fourier_transform_of_rect_and_sinc_functions.gif&quot; alt=&quot;Continuous Fourier transform of rect and sinc functions&quot; /&gt;&lt;/p&gt;&#xA;" OwnerUserId="253" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-03-02T20:05:19.657" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7517" PostTypeId="2" ParentId="7368" CreationDate="2015-03-02T21:35:18.200" Score="2" Body="&lt;p&gt;A simple proof by contradiction answers your question 2. Suppose that the arc of the parabola $y=x^2$ between $A\;(1,1)$ and $B\;(-1,1)$, which touches the $x$ axis at the origin $O$, is also an arc of a circle. This arc is symmetric about the $y$ axis, and so the centre must lie on the $y$ axis: say at $C\;(0,a)$. Equating the squared radii $CO^2$ and $CA^2$ (or $CB^2$), using Pythagoras, after one line of algebraic simplification, gives $a=1$. But that would make the tangent to the parabola at $A$ or $B$ (which is perpendicular to the radius) vertical, which we know is not the case.  &lt;/p&gt;&#xA;" OwnerUserId="4804" LastActivityDate="2015-03-02T21:35:18.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7518" PostTypeId="2" ParentId="7507" CreationDate="2015-03-03T15:34:10.027" Score="2" Body="&lt;p&gt;Re. teaching basic skill Math to English Language Learners&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Teaching basic skills is a losing proposition because they can only be memorized &quot;Voodoo&quot; style. This, unless there is an immediate demand for specific skills. For instance, if your students are carpenters to be, then one skill they will need is, for instance, how to use a carpenter square. &lt;/li&gt;&#xA;&lt;li&gt;The &quot;pedagogy aspect&quot;, namely how one drills the stuff into the students'mind is not the important one because the retention is extremely limited in any case. What &quot;developmental&quot; students need is mainly to learn how to extract information from a text, maybe specifications, owner manual, building code, whatever.&lt;/li&gt;&#xA;&lt;li&gt;The best way to achieve this is to have the math course linked with a developmental read-write English course in which the text used in the math course is used as reading text in the English course.&lt;/li&gt;&#xA;&lt;li&gt;Developmental students also need to learn how and on what basis to decide that some statement is true. For instance, there is a lot of stuff out there on the web that is false, some of it to the point of being dangerous. So, students mostly need to get used to the notion of &quot;making a case&quot; somewhat like in a court of law. This means get used to &quot;demanding&quot; that a case to be made, develop an ability to decide on the merits of a case and, at the same, get used to the idea of not asserting anything for which one is not prepared to make a case.&lt;/li&gt;&#xA;&lt;li&gt;Finally, one should always keep in mind Hestenes'dictum that &quot;[That c]ourse content is taken [by many] as given [...] ignores the possibility of improving pedagogy by reconstructing course content.&quot;&lt;/li&gt;&#xA;&lt;li&gt;Thus, a VERY TALL ORDER. I wish you good luck because College Presidents are never interested in that sort of things.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-03-03T15:34:10.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7519" PostTypeId="2" ParentId="7509" CreationDate="2015-03-03T15:34:52.230" Score="6" Body="&lt;p&gt;Jack Schaedler has created a wonderful &quot;compact primer on digital signal processing&quot; - &lt;a href=&quot;http://jackschaedler.github.io/circles-sines-signals/index.html&quot;&gt;SEEING CIRCLES, SINES, AND SIGNALS&lt;/a&gt; that includes a page with visualisations of the Fourier transform:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://jackschaedler.github.io/circles-sines-signals/dft_introduction.html&quot;&gt;http://jackschaedler.github.io/circles-sines-signals/dft_introduction.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4814" LastActivityDate="2015-03-03T15:34:52.230" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7520" PostTypeId="2" ParentId="6031" CreationDate="2015-03-03T16:08:08.267" Score="3" Body="&lt;p&gt;I have never understood the infatuation of teachers with &quot;factoring quadratics&quot;. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;As a concept, &lt;em&gt;factoring on the ring of integers&lt;/em&gt; is way down on the list of what one should spend time on. Plus, most students deemed good at it stall on $x^{2}-2$ and $x^{2}+x+1$.&lt;/li&gt;&#xA;&lt;li&gt;A concept much higher on the list is that of a &lt;em&gt;change of variable&lt;/em&gt;. But then, to find the zeros of a quadratic function $q$, just &quot;localize&quot;, that is make the change of variable $x=x_{0}+u$ and kill the coefficient of $u$ to find $x_{0-\text{slope}}$. Then, $q_{x_{0-\text{slope}}}(u)$ will have no $u$ term and $q_{x_{0-\text{slope}}}(u)=0$ can be solved.&lt;/li&gt;&#xA;&lt;li&gt;The big advantage is that this takes its place in the much &quot;bigger picture&quot; of the investigation of elementary functions. E.g. the inflection of a cubic is found the same way. And it also shows the limitations of the approach: since $=$ has only two sides, we need to be able to kill enough terms so that there remain only two terms in $f(u)$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="4815" LastEditDate="2015-03-03T16:15:43.233" LastActivityDate="2015-03-03T16:15:43.233" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="7521" PostTypeId="2" ParentId="294" CreationDate="2015-03-03T16:21:34.377" Score="3" Body="&lt;p&gt;There is nothing even remotely close to T. Gowers &lt;em&gt;Mathematics, A Very Short Introduction&lt;/em&gt;. (And less than $10 so that it can be really recommended to students.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the preface: &quot;It is possible to become comfortable with [infinity, the square root of –1, the twenty sixth dimension, and curved space ] without immersing oneself in technicalities.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And it's true!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And while Gowers never even mentions &lt;em&gt;real numbers&lt;/em&gt;, he spends a whole chapter on &lt;em&gt;inﬁnite decimals&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both immense PLUS: As engineers are wont to put it, “The &lt;em&gt;real&lt;/em&gt; real numbers are the &lt;em&gt;decimal&lt;/em&gt; numbers”.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="4815" LastEditDate="2015-03-03T16:26:37.970" LastActivityDate="2015-03-03T16:26:37.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7522" PostTypeId="2" ParentId="7455" CreationDate="2015-03-03T16:51:33.210" Score="1" Body="&lt;p&gt;Why not just pose the problem of how best to approximate functions locally by polynomial functions? (In more general terms, that of asymptotic expansions.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, get $f(x_{0}+h)=\sum_{0}^{\infty}A_{i}(x_{0})h^{i}$ and just truncate at $k$ with the remainder then being &lt;em&gt;o&lt;/em&gt;[$h^{k}$]. Then, you can say that $f$ is &lt;strong&gt;$k$-differentiable&lt;/strong&gt; iff $f$ has a best polynomial approximation of degree $k$ and &lt;em&gt;define&lt;/em&gt; $f^{(k)}$ to be the function which for $x_{0}$ returns $A_{k}(x_{0})k!$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for $k!$ is to make derivatives recursively computable: $f^{(k+1)}=f^{(k)\prime}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is in fact what Lagrange did. (But much misunderstood by people who confuse &lt;em&gt;series&lt;/em&gt; and &lt;em&gt;asymptotic expansions&lt;/em&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="4815" LastEditDate="2015-03-03T20:10:16.723" LastActivityDate="2015-03-03T20:10:16.723" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="7523" PostTypeId="2" ParentId="5986" CreationDate="2015-03-03T19:54:31.487" Score="2" Body="&lt;p&gt;Re. How soon for kids?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the following &quot;basket&quot; (in a 3D space):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\left(3 \text{ apples}, 2 \text{ bananas}, \text{5 carrots}\right)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the following &quot;price list&quot; (in the dual space) where the fraction bar is read &quot;per&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\left[7 \frac{\text{ cents}}{\text{apple}},2\frac{\text{ cents}}{\text{banana}},8\frac{\text{ cents}}{\text{carrot}}\right]$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finding the total price &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\left(3 \text{ apples}, 2 \text{ bananas}, \text{5 carrots}\right)@\left[7 \frac{\text{ cents}}{\text{apple}},2\frac{\text{ cents}}{\text{banana}},8\frac{\text{ cents}}{\text{carrot}}\right]$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;does not seem to be any harder than the 1D version.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://freemathtexts.org/References/AmatycReview/MathUndergroundPDF/17-LanguageAnalysisLinAlgebra.pdf&quot; rel=&quot;nofollow&quot;&gt;http://freemathtexts.org/References/AmatycReview/MathUndergroundPDF/17-LanguageAnalysisLinAlgebra.pdf&lt;/a&gt;&#xA;for a bit more.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="4815" LastEditDate="2015-03-03T20:00:46.757" LastActivityDate="2015-03-03T20:00:46.757" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7524" PostTypeId="1" CreationDate="2015-03-04T02:30:50.100" Score="14" ViewCount="439" Body="&lt;p&gt;Has anyone had their students (high school and beyond) put expositions of math problems/topics/projects on a class blog to be critiqued by other students and revised on-the-fly to provide some interactive feedback and social incentive/synergy?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If so, what are your assessments of the pros and cons of such a project?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(For example, for students at the appropriate level of undergraduate education, an exposition of the Dirac delta function and its applications could be a potentially interesting and rewarding exercise.)&lt;/p&gt;&#xA;" OwnerUserId="4793" LastEditorUserId="4793" LastEditDate="2015-03-04T14:08:28.573" LastActivityDate="2015-03-04T18:43:07.540" Title="Students blogging on math" Tags="|undergraduate-education|mathematical-pedagogy|interactive-teaching|class-participation|classroom-interaction|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7525" PostTypeId="2" ParentId="7455" CreationDate="2015-03-04T13:50:25.443" Score="2" Body="&lt;p&gt;To motivate the writing of $T_2$ under the form $A+B(x-a)+C(x-a)^2$ (note the change of order), you can explain that the formula you seek is an approximation formula at $a$, and that $(x-a)$ is precisely the gap between the central value $a$ and the considered point $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The zeroth approximation, given that $f$ is continuous, is to simply approximate $f(x)$ by a constant $A$ when $x$ is near $a$; we want the approximation to be perfect if $x$ is equal to $a$, so that we pick $A=f(a)$ (at this point you can discuss the merits of continuity if needed). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first-order approximation naturally takes the form $A+B(x-a)$ because we want a linear approximation, and we want to compare the error made in the zeroth approximation with the gap between $x$ and $a$. (For example, $a$ can be the imperfect measurement of a physical quantity $x$: can we estimate the error $f(x)-f(a)$ in terms of $x-a$?). It also backs up nicely with the formula for the derivative. At this point it is easy to see that when $f$ is derivable, the best choice for $A$ is still $f(a)$ and the best choice for $B$ is $f'(a)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, if one has to take this process one step further, using $(x-a)$ as a gap, it makes much sense to try to approximate $f$ near $a$ by a second-order polynomial &lt;em&gt;in the gap (x-a)&lt;/em&gt;. This happens to be a second-order polynomial in $x$ also, but this is mostly a happy coincidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One benefit of this approach is that you have a good opportunity to stress that when $(x-a)$ is small (in particular less than $1$ in absolute value), $(x-a)^2$ is even smaller. One difficulty in this business is to remove this idea that squares are necessarily bigger.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-03-04T13:50:25.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7526" PostTypeId="1" CreationDate="2015-03-04T16:40:58.950" Score="10" ViewCount="330" Body="&lt;p&gt;When it comes to writing examinations, there are two options I am considering.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Write the entire exam originally (which will of course mean some well-known proofs, thereby technically not making it completely original)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use a ratio of 'original' questions and actual past paper questions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;My students have access to perhaps 50 past exams to practice from. Not necessarily exams that I have written, but indeed examinations that students have been able to keep and were provided with worked solutions sets.&lt;/p&gt;&#xA;&lt;p&gt;The theory I had behind using actual past paper questions was to encourage them to do more past papers. Perhaps not for the best of reasons (as I presume they'll be hoping that they'll get lucky and get a question in my exam that they've seen already) but motives aside, they may complete a larger volume of past papers.&lt;/p&gt;&#xA;&lt;p&gt;I'd like to hear some thoughts regarding this. Is this a good idea, is this a terrible idea? I can imagine that a few students may get lucky but in terms of the majority of the cohort (around 60), it may boost overall past paper attempts.&lt;/p&gt;&#xA;" OwnerUserId="3383" LastEditorUserId="5220" LastEditDate="2021-10-15T14:52:57.063" LastActivityDate="2021-10-21T15:57:06.077" Title="Is it a good idea to use past questions in examinations?" Tags="|exams|assessment|cheating|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7528" PostTypeId="1" CreationDate="2015-03-04T17:39:35.863" Score="48" ViewCount="21792" Body="&lt;p&gt;As part of my secondary geometry class I like to hook students by presenting real-world examples (usually images I find online or have taken myself) of different geometric shapes from real life.  For instance, a lesson on the area of a circle might start out with a picture of a pizza pie or a lesson on the midsegments of triangles might start out with a picture of the Triforce.  However, there are some geometric figures that I have had a hard time finding interesting, real-world examples of.  Those figures (and I know I am forgetting a bunch..) are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Segment of a circle&lt;/li&gt;&#xA;&lt;li&gt;Secant line&lt;/li&gt;&#xA;&lt;li&gt;Trapezoid (Isosceles or not)&lt;/li&gt;&#xA;&lt;li&gt;Inscribed angle&lt;/li&gt;&#xA;&lt;li&gt;Parallel lines cut by a transversal&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I was wondering if anyone had any ideas for these geometric figures of interesting, real world examples? Also, I think it would be great that if people are aware of really cool real world examples for the more &quot;standard&quot; geometric figures to post those as answers as well.  For instance, the Dockland Building at the Port of Hamburg is an astoundingly perfect parallelogram :)&#xA;&lt;img src=&quot;https://i.stack.imgur.com/tFbU0.jpg&quot; alt=&quot;enter image description here&quot;&gt;&#xA;Having a collection would be very helpful for teachers because I have not found a better way to get my students right into the groove by starting class off with a brief discussion about an interesting picture!&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2023-08-05T18:58:08.413" Title="Real-world examples of more &quot;obscure&quot; geometric figures" Tags="|secondary-education|geometry|" AnswerCount="39" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7529" PostTypeId="2" ParentId="7528" CreationDate="2015-03-04T17:47:52.080" Score="25" Body="&lt;p&gt;I like the Gateway Arch in St. Louis as an example of a catenary with a formula of the form $y= A \cosh(\frac{C X}{L}) -A$. More information on the wiki: &lt;a href=&quot;http://en.wikipedia.org/wiki/Gateway_Arch#Mathematical_elements&quot; rel=&quot;noreferrer&quot;&gt;Gateway Arch: Mathematical Elements.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/7pRyP.jpg&quot; alt=&quot;Gateway Arch&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-03-04T17:47:52.080" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7530" PostTypeId="2" ParentId="7526" CreationDate="2015-03-04T17:59:48.707" Score="6" Body="&lt;p&gt;It depends on your goals for the examination. As you are mentioning proof, and using some well-known ones, it seems that you are aiming for more of a &quot;show me that you've thought about all of these types of proofs&quot; style.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As they have access to a large collection of examinations, I would, perhaps, pick a few exercises from those of the most important ideas and run with those with a few 'newer' style problems. As they have a large collection, you can perhaps place emphasis more &quot;write down the core idea behind the proof of 'Big Theorem X'&quot;, or knowing how to prove a key lemma that is critical to a 'Big Theorem'. Other options are just having them prove an inductive step, if that is a tricky part, or a special case of some Big Theorem (or Large Lemma) or even a homework problem. The key point is that they can show you that they are familiar to the key proof strategies for the particular course you are teaching. While it sometimes boils down to &quot;memorize this list of 100 proofs&quot;, hopefully, along the way, they pick up the proof strategies to simplify their list.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-03-04T17:59:48.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7531" PostTypeId="2" ParentId="7524" CreationDate="2015-03-04T18:43:07.540" Score="6" Body="&lt;p&gt;In some of my graduate courses, my professors tried to get a 'post your homework solutions and other thoughts' page going. It didn't work, which is surprising considering we are graduate students massively interested in the topic. Here are my thoughts in no particular order:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pros:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Encourages students to practice mathematical typesetting (latex, etc)&lt;/li&gt;&#xA;&lt;li&gt;Helps students to formulate their thoughts coherently&lt;/li&gt;&#xA;&lt;li&gt;Hopefully encourages students who are interested to delve deeper into the subject with 'extra' work&lt;/li&gt;&#xA;&lt;li&gt;Other students critiques and critiquing help stimulate their own thoughts on the problem&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cons:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Many students need to learn to do mathematical typesetting, which can be off-putting at first&lt;/li&gt;&#xA;&lt;li&gt;Might be more time-consuming than working on homework, particularly for students lacking internet at home as they have to work elsewhere&lt;/li&gt;&#xA;&lt;li&gt;Criticism is not prompt. If the assignment is due that evening and only one student is commenting, it isn't particularly useful&lt;/li&gt;&#xA;&lt;li&gt;Need a strong, enthusiastic student base who is at their computer frequently&lt;/li&gt;&#xA;&lt;li&gt;I personally would rather talk to another student in person than via computer&lt;/li&gt;&#xA;&lt;li&gt;Lazy students can just copy other students work (if used for homework), which, in turn, decreases the good students motivation from helping (why should they carry the weight of those who are lazy?)&lt;/li&gt;&#xA;&lt;li&gt;Disinterested students will likely fall behind as they are not willing to put in that extra time&lt;/li&gt;&#xA;&lt;li&gt;The longevity of it being 'on the internet' is negligible in that it seems to me that you are looking for an alternative to a paper project or presentation&lt;/li&gt;&#xA;&lt;li&gt;Many times the webpage format itself is clunky and hard to use&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If instituted correctly with good, enthusiastic students, I think it could work. But for an average class, I would aim more at in-class presentation or turned in assignments/projects.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-03-04T18:43:07.540" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7532" PostTypeId="2" ParentId="7455" CreationDate="2015-03-04T20:08:34.007" Score="1" Body="&lt;p&gt;Given $f\colon I\to \mathbb R$ and $x\in\mathrm{int}\, I$, you could construct the unique second degree polynomial $P_\varepsilon$ that passes by the points $(x-\varepsilon, f(x-\varepsilon))$, $(x, f(x))$ and $(x+\varepsilon, f(x+\varepsilon))$ (&lt;a href=&quot;http://en.wikipedia.org/wiki/Lagrange_polynomial&quot; rel=&quot;nofollow&quot;&gt;Lagrange interpolation&lt;/a&gt;) and then let $\varepsilon \to 0$. This seems the most natural approach generalizing the idea of a tangent line and if you argue $f''(x)$ exists, it is not hard to see that $\displaystyle\lim_{\varepsilon\to 0}P_\varepsilon=T_2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another interesting thing to say is that a function may be well-approximated by a second degree polynomial in the above sense but not be twice differentiable; check $1+x+x^2+x^3\sin(1/x)$ out around zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may extend the interpolation idea to higher degrees with more work, but the computations get messy fast. The idea of constructing the second degree polynomial that passes by the points $(x+\delta, f(x+\delta))$, $(x, f(x))$ and $(x+\varepsilon, f(x+\varepsilon))$ and then letting $(\delta,\varepsilon)\to (0,0)$ is also probably too cumbersome.&lt;/p&gt;&#xA;" OwnerUserId="4826" LastEditorUserId="187" LastEditDate="2015-03-05T08:57:36.253" LastActivityDate="2015-03-05T08:57:36.253" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7533" PostTypeId="1" AcceptedAnswerId="7546" CreationDate="2015-03-04T21:37:20.590" Score="12" ViewCount="420" Body="&lt;p&gt;&lt;strong&gt;A typical student mistake&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You see this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{15}{4\sqrt{15}}=\frac{15}{4\sqrt{15}}\cdot\frac{\sqrt{15}}{\sqrt{15}}=\frac{15\sqrt{15}}{4\cdot15}=\frac{15\sqrt{15}}{120}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see that the student tried to rationalize the denominator.  They did it the hard way, and made a mistake.  You want to teach them a better way, that would have helped them with this problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Trying to get them to see the mistake(s)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You say this is not right. They look at the thing and say, oh, you're right. $4\cdot15$ is $60.$ Yes, you say, but do you think perhaps there was a way of avoiding the offending calculation altogether? Perhaps there was a way to do it in such a way that you couldn't have made a mistake because you wouldn't have had to calculate anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They look again and you see a mixture of impatience and anxiety in their eyes. They know why they made the mistake. It's because $4\cdot15$ is $60.$ Why are you bothering them and implying they're stupid? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You say, look, it's always a good idea to avoid calculations. It saves you time and spares you mistakes. Look at the two fifteens, the one at the top and the one at the bottom. What's better, cancelling them out, or multiplying the $4$ by the $15$ first? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They agree it's the former, but you can see it's not sinking in. They've done it this way for so long and it's automatic. This is the way they know. They understand your way is better, but they're unwilling to follow it. It's difficult for them to take in a new idea if they have one that works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately for them, you have one more thing to add. Look at the previous part of your solution to this problem. You showed there that $\dfrac{15}{\sqrt{15}}=\sqrt{15}$ Can you see why that is? Yes, of course they can. And they show you how to multiply the fraction by $1$ written as the quotient of two square roots of $15$. You'd like to make them appreciate the fact that $\sqrt{15}\cdot\sqrt{15}=15$ by definition and make them able to see the result without having to write the complicated algebraic manipulation. But you've tried that too many times and know what their reply will be. They'll explain to you that you're a mathematician so you see these things and they're not so they don't. You decide against even mentioning the thing this time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is there a better way to teach this?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This simply happens all the time. How do you deal with this? How can you make a student appreciate a better way of doing something they already know how to do? How do you conquer their mental inertia? &lt;/p&gt;&#xA;" OwnerUserId="4827" LastEditorUserId="1147" LastEditDate="2015-03-06T06:00:26.623" LastActivityDate="2015-03-06T06:00:26.623" Title="How do I get them to appreciate learning a new way of doing that thing?" Tags="|mathematical-pedagogy|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7535" PostTypeId="2" ParentId="7528" CreationDate="2015-03-05T00:17:19.537" Score="30" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/National_Library_of_Belarus&quot; rel=&quot;noreferrer&quot;&gt;National Library of Belarus&lt;/a&gt;,&#xA;a &lt;a href=&quot;http://en.wikipedia.org/wiki/Rhombicuboctahedron&quot; rel=&quot;noreferrer&quot;&gt;rhombicuboctahedron&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/BjqCq.jpg&quot; width=&quot;400&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/IPmNA.jpg&quot; width=&quot;400&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-03-05T00:35:43.460" LastActivityDate="2015-03-05T00:35:43.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7536" PostTypeId="2" ParentId="7533" CreationDate="2015-03-05T00:21:12.950" Score="6" Body="&lt;p&gt;The answer is: If the student isn't willing, you cannot. The problem you describe roots in the difference between breadth-first search and depth-first search. We're psychologically trained to use depth-first search if we only care for the result. And most students only care for the result of having done the task. So, if they see a task for which they know a way, the do it and have it done. We as math educators or mathematicians see the problem from a breadth-first search point of view. We try different paths in our head and find a very quick and easy was of solving. But the students don't and they know it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, don't try to make them take a breadth-first search point of view, if they are unwilling. It'll give them nothing. If they only want to have done the task, all you do to postpone this point of having it done is only creating frustration with them and you. Instead, try to make them willing. Showing off doesn't help. Make them have fun and get a positive feeling with math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they &lt;em&gt;are&lt;/em&gt; willing, first show them all the paths and make them understand the relationship between them. Then, let them try out the different paths and find the one that suits them best. Repeat with similar exercises until they know, that the best path depends on the problem.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2015-03-05T00:21:12.950" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7537" PostTypeId="2" ParentId="7509" CreationDate="2015-03-05T02:23:18.627" Score="5" Body="&lt;p&gt;Depending on the audience I like to show a video of image reconstruction, as in &#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=D9ziTuJ3OCw&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=D9ziTuJ3OCw&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to also show the alternative &quot;Euclidean&quot; reconstruction, where rather than global support you see that each component provides perfect information for only one pixel.  &lt;/p&gt;&#xA;" OwnerUserId="4828" LastActivityDate="2015-03-05T02:23:18.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7538" PostTypeId="2" ParentId="7528" CreationDate="2015-03-05T12:35:08.400" Score="18" Body="&lt;p&gt;&quot;&lt;a href=&quot;http://en.wikipedia.org/wiki/Turning_Torso&quot; rel=&quot;noreferrer&quot;&gt;Turning Torso&lt;/a&gt;,&quot; an apartment building in Malmö, Sweden designed by&#xA;architect Santiago Calatrava, following a twisting spiral.&#xA;It consists of&#xA;&quot;nine segments of five-story pentagons that twist relative to each other as it rises; the topmost segment is twisted 90 degrees clockwise with respect to the ground floor.&quot;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Pkn5J.jpg&quot; width=&quot;360&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-03-05T12:35:08.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7539" PostTypeId="2" ParentId="7533" CreationDate="2015-03-05T13:31:27.793" Score="0" Body="&lt;p&gt;Students will adopt 'better' methods and rigorous screening for simplification opportunities when it benefits them to do so. Taking the given example, how do you think your students would feel if presented a worksheet with 10 such problems? 30? 100?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somewhere there's an appropriate number and difficulty level of questions such that if it were approached naively it would be an extremely boring, arduous task, but if approached with a keener eye would become quick work, and moreover become more exciting and better educating at the same time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Give them (something approximating) this work, and prime them with the tools to cut through it like mature mathematicians instead of slogging through it like grunts. Be sure to throw a few in there which aren't susceptible to any great shortcuts as well, or else you may accidentally teach that 'problems from ymar always have 1-step solutions'.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2015-03-05T13:31:27.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7540" PostTypeId="2" ParentId="7528" CreationDate="2015-03-05T15:05:55.750" Score="13" Body="&lt;p&gt;As constrast to the catenary in Chris's answer, you could show a suspension bridge, which has a parabola...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://carolinmader.wordpress.com/2008/09/29/unser-sonntag-in-clifton-our-sunday-in-clifton/clifton-suspension-bridge/&quot; rel=&quot;noreferrer&quot;&gt;LINK&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/fHQcS.jpg&quot; alt=&quot;Image&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;added&lt;/strong&gt;&lt;br&gt;&#xA;According to &lt;a href=&quot;http://en.wikipedia.org/wiki/Parabola#Parabolas_in_the_physical_world&quot; rel=&quot;noreferrer&quot;&gt;LINK&lt;/a&gt;, the curve in a suspension bridge is generally a curve intermediate between a catenary and a parabola.&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2015-03-05T21:45:45.537" LastActivityDate="2015-03-05T21:45:45.537" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7541" PostTypeId="2" ParentId="4025" CreationDate="2015-03-05T18:41:35.023" Score="3" Body="&lt;p&gt;I think that if the Ph.D. level course were in mathematics education and focussed on using cartoons in class, then yes.  Otherwise, no.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The major reason is that most material in graduate level classes is about serious stuff.  That's not to say you can't have fun with it and make it more enjoyable, but in addition to the key concepts that can be quickly conveyed by cartoon, there is a lot of history, ancillary detail, context on when to use and not use a result, counterexamples, and other things that are not easily reduced to a cartoon.  Supplement a lecture with a cartoon or two? Sure.  Do this for every lecture? Maybe, give it a try.  Have a comic-book or graphic-novel style textbook as the main text?  I don't think so, Tim*: I think the students will miss out on some of the things they need to learn.  (Illustrated text books are different:  that is for a different question.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Reference to the American TV show &quot;Home Improvement&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Moderation In All Things Funny&quot; Paseman, 2015.03.05&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-03-05T18:41:35.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7542" PostTypeId="2" ParentId="7533" CreationDate="2015-03-05T19:45:58.643" Score="3" Body="&lt;p&gt;I'd try to create a need for simplifying first.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{15\sqrt{15}}{4\cdot15}$ doesn't motivate a student to simplify because $4\cdot15$ is easy to do anyway. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{259\sqrt{259}}{137\cdot259}$ on the other hand is a right pain if no calculators are allowed!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd have a starter (maybe a game or competition) that involved simplifying a mix of $\frac{15}{40}$ styled questions and $\frac{2\times5\times7\times11}{5\times5\times7\times11}$ styled.  I'd ask how they did each and why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully this will empower the students to make life easier for themselves later on.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately though I'd leave them to do it their own way as you only truly improve by making the connotations yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-03-05T19:45:58.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7543" PostTypeId="2" ParentId="7528" CreationDate="2015-03-05T20:26:07.437" Score="14" Body="&lt;p&gt;Really great answers! I just found this while making a lesson on &lt;a href=&quot;https://en.wikipedia.org/wiki/Annulus_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;annuli&lt;/a&gt;, an Annular Eclipse, very beautiful! and it also has interesting math behind why the sun is not entirely covered by the moon!&#xA;&lt;img src=&quot;https://i.stack.imgur.com/aVeSZ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-03-05T20:26:07.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7544" PostTypeId="2" ParentId="7528" CreationDate="2015-03-06T00:38:32.350" Score="12" Body="&lt;p&gt;(Inspired by Gerhard's comment) &lt;strong&gt;&lt;em&gt;Trapezoid&lt;/em&gt;&lt;/strong&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Zc4qJ.jpg&quot; width=&quot;360&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Image from Parth Chandran @&lt;a href=&quot;http://app.emaze.com/@AOLTILCZ/quadrilaterals#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;emaze.com&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-03-06T01:16:40.097" LastActivityDate="2015-03-06T01:16:40.097" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7545" PostTypeId="1" AcceptedAnswerId="7553" CreationDate="2015-03-06T00:51:16.437" Score="12" ViewCount="709" Body="&lt;p&gt;After years of experience in some area of expertise, you can easily forget how difficult it can be for the uninitiated to grasp some fundamental concepts, and, indeed, people often edit out of their own personal history memories of their initial confusion and fears.&lt;/p&gt;&#xA;&lt;p&gt;History can re-teach us about the inherent difficulties in assimilating new concepts, and, I think, help in tempering the novices' fears by assuring them that initial confusion and making mistakes are quite natural and in encouraging them to persevere.&lt;/p&gt;&#xA;&lt;p&gt;Using the evolution of the concept of negative numbers as an example, I'd like to present a historical note from Mathematics: The Loss of Certainty by Morris Kline (Oxford, pg. 115):&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;An interesting argument against negative numbers was given by Antoine Arnauld (1612-1694), theologian, mathematician, and close friend of Pascal. Arnauld questioned that -1 : 1 = 1 : -1 because, he said, -1 is less than +1; hence, how could a smaller be to a greater as a greater is to a smaller?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The history of mathematics is replete with similar examples. What are some of your favorites?&lt;/p&gt;&#xA;&lt;p&gt;Edit in response to comments:&lt;/p&gt;&#xA;&lt;p&gt;I'm asking for historical examples that illustrate the natural confusion present in the evolution of fundamental concepts in math and that ideally could serve both to mitigate the remedial or novice learner's reaction to his own confusion and allied fear and as a stepping stone for exploring the concept.&lt;/p&gt;&#xA;&lt;p&gt;(I know of other examples involving Leibniz and the product rule (cf. Humanizing Calculus by M. Cirillo), and Euler and log(-1), but I'll leave them for others to describe.)&lt;/p&gt;&#xA;" OwnerUserId="4793" LastEditorUserId="470" LastEditDate="2023-06-04T23:58:01.303" LastActivityDate="2023-06-04T23:58:01.303" Title="Fighting math phobia with history" Tags="|mathematical-pedagogy|history|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="7546" PostTypeId="2" ParentId="7533" CreationDate="2015-03-06T01:29:00.937" Score="5" Body="&lt;p&gt;Use a positive approach, instead of a negative approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The approach in the original post&lt;/strong&gt; is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Not only did you make a mistake,&lt;/li&gt;&#xA;&lt;li&gt;But you need to go figure out what you could have done instead (that you did not see)&lt;/li&gt;&#xA;&lt;li&gt;Oh by the way, you need to figure out something else that you could have done instead (that you did not see).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;That approach provides three pieces of negative feedback before the learner gets to solve the problem to the teacher's satisfaction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A positive approach&lt;/strong&gt; might be:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Yes, you know how to rationalize denominators.  You multiply by a fraction that equals one, where both the numerator and the denominator equal the relevant square root.&lt;/li&gt;&#xA;&lt;li&gt;But that involves a lot of math, and you might make a mistake.&lt;/li&gt;&#xA;&lt;li&gt;Sometimes you can use this shortcut.  If the numerator already includes a power of the square root, rewrite the numerator to show it in terms of the square root.  For example:  $$\frac{15}{4\sqrt{15}}=\frac{\sqrt{15}\sqrt{15}}{4\sqrt{15}}$$  Then you can cancel out the square roots right away.&lt;/li&gt;&#xA;&lt;li&gt;You won't always be able to use this shortcut.  But when you can, it makes things easier.&lt;/li&gt;&#xA;&lt;li&gt;Here are some example problems.  Circle the ones where you can use this shortcut.&lt;/li&gt;&#xA;&lt;li&gt;Practice solving the problems.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And tell them that it is OK if they don't remember the shortcut.  They can always do it the hard way -- if they are careful.  If they do it right, they will get the same answer either way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Study habits&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, encourage your students to maintain a &quot;cheat sheet&quot; of the things they learn in your class.  Tell them that if they think that this is a good trick, they should put a note about it (in their own words) in their &quot;cheat sheet&quot;.  Encourage them to read through their &quot;cheat sheet&quot; the night before an exam.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-03-06T01:46:04.307" LastActivityDate="2015-03-06T01:46:04.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7547" PostTypeId="2" ParentId="7509" CreationDate="2015-03-06T02:51:33.967" Score="3" Body="&lt;p&gt;You might be interested in a purely mechanical approach.  There are an interesting few videos by Bill Hammack on youtube showing a mechanical harmonic analyzer that combines and analyzes the first several terms of Fourier series:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=8KmVDxkia_w&quot; rel=&quot;nofollow&quot;&gt;https://www.youtube.com/watch?v=8KmVDxkia_w&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="223" LastActivityDate="2015-03-06T02:51:33.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7548" PostTypeId="2" ParentId="7528" CreationDate="2015-03-06T02:57:49.610" Score="10" Body="&lt;p&gt;For a &lt;a href=&quot;https://en.wikipedia.org/wiki/Superellipse&quot; rel=&quot;nofollow noreferrer&quot;&gt;super-ellipse&lt;/a&gt;, one example would be the fountain at Sergels torg, in Stockholm, Sweden.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/2owqT.jpg&quot; alt=&quot;Sergels torg&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a &lt;a href=&quot;https://en.wikipedia.org/wiki/Circular_segment&quot; rel=&quot;nofollow noreferrer&quot;&gt;circular segment&lt;/a&gt;, one example would be the cross-section of liquid in a horizontal-axis circular cylinder tank.  (Another picture is &lt;a href=&quot;https://i.stack.imgur.com/nWnuZ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/KwVfj.jpg&quot; alt=&quot;http://image.shutterstock.com/display_pic_with_logo/92498/172961744/stock-photo-an-open-red-wine-bottle-laying-on-the-table-172961744.jpg&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2015-03-09T13:14:31.617" LastActivityDate="2015-03-09T13:14:31.617" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7549" PostTypeId="2" ParentId="7528" CreationDate="2015-03-06T08:24:59.480" Score="19" Body="&lt;p&gt;A corkscrew (for a &lt;a href=&quot;http://en.wikipedia.org/wiki/Helix&quot; rel=&quot;noreferrer&quot;&gt;helix&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/yT04M.jpg&quot; alt=&quot;corkscrew&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A donut (for a &lt;a href=&quot;http://en.wikipedia.org/wiki/Torus&quot; rel=&quot;noreferrer&quot;&gt;torus&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/osbpc.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A football (for a &lt;a href=&quot;http://en.wikipedia.org/wiki/Spheroid&quot; rel=&quot;noreferrer&quot;&gt;spheroid&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ENTDi.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then, there's also the atomium (for which I am not sure exists a geometric name)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/B7vyF.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;cooling towers (for a &lt;a href=&quot;http://en.wikipedia.org/wiki/Hyperboloid&quot; rel=&quot;noreferrer&quot;&gt;hyperboloid&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Z5xNY.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the pentagon (well, for a &lt;a href=&quot;http://en.wikipedia.org/wiki/Pentagon&quot; rel=&quot;noreferrer&quot;&gt;pentagon&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/qQjuj.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A pyramid is, of course, a &lt;a href=&quot;http://en.wikipedia.org/wiki/Pyramid_%28geometry%29&quot; rel=&quot;noreferrer&quot;&gt;pyramid&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/sphnL.jpg&quot; alt=&quot;Pyramid&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, a soccer ball is a &lt;a href=&quot;http://en.wikipedia.org/wiki/Truncated_icosahedron&quot; rel=&quot;noreferrer&quot;&gt;truncated icosahedron&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/c0cIW.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Images by &lt;a href=&quot;http://en.wikipedia.org/wiki/Corkscrew&quot; rel=&quot;noreferrer&quot;&gt;wiki&lt;/a&gt;, &lt;a href=&quot;http://upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Glazed-Donut.jpg/250px-Glazed-Donut.jpg&quot; rel=&quot;noreferrer&quot;&gt;pedia&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="4835" LastEditorUserId="4835" LastEditDate="2015-03-10T06:08:11.553" LastActivityDate="2015-03-10T06:08:11.553" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7550" PostTypeId="1" CreationDate="2015-03-06T11:53:20.450" Score="8" ViewCount="167" Body="&lt;p&gt;Next week, a workshop about the cross sections of solids will be held in our school, for 7th grade students. These are what we've decided to include in the workshop:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Showing &lt;a href=&quot;http://www.flatlandthemovie.com/&quot; rel=&quot;noreferrer&quot;&gt;&quot;Flatlad: The Movie&quot;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Giving the students cubic pieces of polystyrene and asking them to make polygon cross sections with given number of sides.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/40kvC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Asking the students to construct given solids, using their cross sections by sheets of thin foam board.&lt;/li&gt;&#xA;&lt;li&gt;Talking about 3D-Printers and showing clips about them.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What other things can be included in our workshop? Do you have any suggestions?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2015-03-06T22:12:37.020" Title="Cross Sections Workshop" Tags="|constructions|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7551" PostTypeId="2" ParentId="7550" CreationDate="2015-03-06T12:56:34.537" Score="1" Body="&lt;p&gt;Although this has nothing to do with cross sections,&#xA;it fits the physical materials:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could have the students cut out shapes&#xA;to illustrate constructive proofs of the Pythagorean theorem:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/A8x3P.jpg&quot; alt=&quot;enter image description here&quot;&gt;&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/M8W1U.jpg&quot; alt=&quot;enter image description here&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Images from a &lt;a href=&quot;http://opinionator.blogs.nytimes.com/2010/03/14/square-dancing/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Steven Strogatz NYTimes article&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Another constructive proof is described in &lt;a href=&quot;https://matheducators.stackexchange.com/a/7203/511&quot;&gt;this MESE answer&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-03-06T13:02:10.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7552" PostTypeId="2" ParentId="7528" CreationDate="2015-03-06T14:54:27.623" Score="38" Body="&lt;p&gt;&lt;strong&gt;Trapezoid&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Machu_Picchu&quot; rel=&quot;noreferrer&quot;&gt;Native Peruvian architecture&lt;/a&gt; makes heavy use of the trapezoid for stability in earthquakes.  (The Spaniards thought they were primitive as they didn't use arches ... but most of the Spanish buildings have collapsed or had to be rebuilt).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's especially apparent in their doorways and windows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Czdmu.jpg&quot; alt=&quot;Doorways &amp;amp; windows at Machu Picchu&quot;&gt; (&lt;a href=&quot;http://en.wikipedia.org/wiki/Machu_Picchu#mediaviewer/File:124_-_Machu_Picchu_-_Juin_2009.jpg&quot; rel=&quot;noreferrer&quot;&gt;hi res&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other examples with licensing such that I don't want to embed it:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://russellbevan.photoshelter.com/image/I0000rB5g1TrtY8E&quot; rel=&quot;noreferrer&quot;&gt;http://russellbevan.photoshelter.com/image/I0000rB5g1TrtY8E&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.travel-images.com/photo/photo-peru194.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.travel-images.com/photo/photo-peru194.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Trapezoids are also found in cabinet joinery, specifically &lt;a href=&quot;http://en.wikipedia.org/wiki/Dovetail_joint&quot; rel=&quot;noreferrer&quot;&gt;dovetail joints&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Q4MGd.png&quot; alt=&quot;dovetail joinery&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Segment of a Circle&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most &lt;a href=&quot;http://leydilaverde.wikispaces.com/Oral+Presentation+(Arch)&quot; rel=&quot;noreferrer&quot;&gt;architectural arches&lt;/a&gt; are based on segments of circles, particularly those in &lt;a href=&quot;http://en.wikipedia.org/wiki/Romanesque_architecture&quot; rel=&quot;noreferrer&quot;&gt;Roman architecture&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/KuyZ2.jpg&quot; alt=&quot;Santa Maria in Cosmedin, Rome&quot;&gt; (&lt;a href=&quot;http://en.wikipedia.org/wiki/Romanesque_architecture#mediaviewer/File:Santa_Maria_in_Cosmedin-templom_belso.jpg&quot; rel=&quot;noreferrer&quot;&gt;hi res&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ie88i.jpg&quot; alt=&quot;Roman Aqueducts in southern France&quot;&gt; (&lt;a href=&quot;http://en.wikipedia.org/wiki/Roman_aqueduct#mediaviewer/File:Pont_du_Gard_Oct_2007.jpg&quot; rel=&quot;noreferrer&quot;&gt;hi res&lt;/a&gt;) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chinese architecture tends to favor segmental arches (which the &lt;a href=&quot;http://en.wikipedia.org/wiki/Alcon%C3%A9tar_Bridge&quot; rel=&quot;noreferrer&quot;&gt;Romans also used&lt;/a&gt;), rather than full rounded arches:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/3REBK.jpg&quot; alt=&quot;enter image description here&quot;&gt; (&lt;a href=&quot;http://kaleidoscope.cultural-china.com/en/10Kaleidoscope3071.html&quot; rel=&quot;noreferrer&quot;&gt;more images&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;http://en.wikipedia.org/wiki/Arch_bridge&quot; rel=&quot;noreferrer&quot;&gt;arch bridges&lt;/a&gt; and &lt;a href=&quot;http://en.wikipedia.org/wiki/Barrel_vault&quot; rel=&quot;noreferrer&quot;&gt;barrel vaults&lt;/a&gt;.  If you want sophomoric humor, also consider the &lt;a href=&quot;http://en.wikipedia.org/wiki/Groin_vault&quot; rel=&quot;noreferrer&quot;&gt;groin vault&lt;/a&gt; (when made with round barrel vaults, not pointed barrel vaults).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Parabola&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also &lt;a href=&quot;http://leydilaverde.wikispaces.com/Oral+Presentation+(Arch)&quot; rel=&quot;noreferrer&quot;&gt;Parabolic Arches&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/W7xcn.jpg&quot; alt=&quot;parabolic vault&quot;&gt; &lt;img src=&quot;https://i.stack.imgur.com/L1JJu.jpg&quot; alt=&quot;roof arch&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Parallel lines cut by a transversal&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Runways at large airports. They typically have taxiway parallel to the runway, and in windy areas have a second (or even third) pair to avoid takeoffs/landings into a cross wind.  &lt;a href=&quot;http://www.jmt.com/project-portfolio/bwi-airport-runway-safety-area-pavement-management/&quot; rel=&quot;noreferrer&quot;&gt;BWI is a good example&lt;/a&gt;, but I was having difficulty finding public domain images.  Here's one of O'Hare:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/HleAG.png&quot; alt=&quot;O&amp;#39;Hare airport from USGS&quot;&gt; (&lt;a href=&quot;http://en.wikipedia.org/wiki/O&amp;#39;Hare_International_Airport#mediaviewer/File:O%27Hare_International_Airport_(USGS).png&quot; rel=&quot;noreferrer&quot;&gt;hi res&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="4836" LastEditorUserId="77" LastEditDate="2015-03-09T01:00:30.253" LastActivityDate="2015-03-09T01:00:30.253" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7553" PostTypeId="2" ParentId="7545" CreationDate="2015-03-06T15:01:41.970" Score="11" Body="&lt;p&gt;The biggest example to me of this is the confusion brought into math when calculus was first developed by Newton and Leibniz, with Bishop Berkeley saying, &quot;And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It took mathematicians 150 years to develop a sound logical foundation (the limit) for the basic ideas used in calculus. I talk about this extensively in calc class, so my students can realize that even the people best at math can struggle with it. I also want them to realize both the importance and the difficulty of the topic of limits. I also think this history shows how important calculus was to science. People kept using it for that 150 years, even though it didn't have the logical rigor they knew was possible with geometry and algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I follow the historical path in my course, presenting the book's material on limits and continuity after we spend a few weeks exploring the idea of the derivative (using the limit notation, but without any more backup than &quot;this means infinitely close&quot;). We do a few derivative properties, graph polynomials, and then look at the limit chapter, where the squeeze theorem is introduced, before we deal with the derivatives of trig functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think calculus textbooks presenting a whole chapter on limits before students have any idea of their value shows that the texts were written by mathematicians who aren't thinking much about how most people learn. (Many of us who love math are willing to learn about the weirdest things, just because it's a big adventure, and a great puzzle. But that's not how it works for most of our students. And math lover that I am, I still would have benefited from a class like the one I teach.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-03-06T15:01:41.970" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7554" PostTypeId="2" ParentId="7528" CreationDate="2015-03-06T15:38:19.163" Score="20" Body="&lt;p&gt;There's a fair attempt at a &lt;a href=&quot;http://en.wikipedia.org/wiki/Hypercube&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hypercube&lt;/a&gt; with the &lt;a href=&quot;http://www.google.com/imgres?imgurl=http://2.bp.blogspot.com/-Aeq1oSI_Whk/UCxsq0IYc_I/AAAAAAAAAEs/twhXbedMxdg/s1600/La%252BGrande%252BArch.jpg&amp;amp;imgrefurl=http://fourdimensionalart.blogspot.com/2012/07/moving-architecture-in-4th-dimension_25.html&amp;amp;h=572&amp;amp;w=500&amp;amp;tbnid=jLIXRjHb34J0rM:&amp;amp;zoom=1&amp;amp;docid=yMJNvm0mR7iu4M&amp;amp;ei=Qsj5VIqqOcGO7QbGjIGgCg&amp;amp;tbm=isch&amp;amp;ved=0CCgQMygLMAs&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grande Arche de la Défense in Paris&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/s5UIp.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="4701" LastActivityDate="2015-03-06T15:38:19.163" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7555" PostTypeId="2" ParentId="7528" CreationDate="2015-03-06T16:56:39.817" Score="18" Body="&lt;p&gt;Sphere or hemisphere: Pantheon&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Vi2hR.png&quot; alt=&quot;Sphere or hemisphere: Pantheon&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="4837" LastActivityDate="2015-03-06T16:56:39.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7556" PostTypeId="2" ParentId="7545" CreationDate="2015-03-06T17:15:06.140" Score="4" Body="&lt;p&gt;In my experience some students have difficulty with where $m$ and $b$ come from in the slope-intercept form of a line. (In the US, we use $y=mx+b$ but this is not the same in other countries.) Apparently, the origin of $m$ is still disputed (see &lt;a href=&quot;https://www.math.duke.edu/education/webfeats/Slope/Slopederiv.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; for a more detailed explanation). The origin for $b$ is less ambiguous, and appears to follow from the case where $y=ax+b$ is used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some quick Googling appears to show that different countries use different variables, so I can't speak for those cases, but it seems to me that students who struggle with this concept are further put off by mathematics because it doesn't make sense to them. If I tell a story about where $m$ and $b$ come from, they seem to accept it a little bit more because there is a reason behind it (even if my story may not be 100% true).&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2015-03-06T17:15:06.140" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7557" PostTypeId="2" ParentId="7550" CreationDate="2015-03-06T22:12:37.020" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Conic_section#Features&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conic sections&lt;/a&gt; would seem appropriate if you're looking to motivate some of the algebra that is on the horizon for your 7th grade students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/i2zUr.png&quot; alt=&quot;Image from [Tutorvista.com](http://math.tutorvista.com/geometry/cross-section.html)&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2015-03-06T22:12:37.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7558" PostTypeId="2" ParentId="7545" CreationDate="2015-03-07T01:08:59.080" Score="10" Body="&lt;p&gt;The one I love and and I design my class around its history frame by frame is the notion of complex number. Just consider this question as a starter:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find two numbers whose sum is equal to 10 and whose product is equal&#xA;  to 40.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Students' reactions are telling, in particular, those who calculate $b^2-4ac$ for $x^2-10x +40=0$ and put the pencil down as soon as they get $-60$ for the discriminant. But, exactly when they are happy with the end, comes this crucial and for most of them shocking new start: &quot;what if we carry on with our beloved quadratic formula for the roots&quot;. At that very moment you can witness the &quot;&lt;strong&gt;mental torture&lt;/strong&gt;&quot; attributed to Cardano: &quot;No way&quot; they say.And, here is me repeating the words of Cardano saying &quot;Dismiss the torture&quot;  And a few moment later, they are adding and multiplying those &quot;objects&quot; being completely meaningless two minutes earlier:   $5 + \sqrt{-15}$   and $5 - \sqrt{-15}$ without even remembering their initial reactions!  &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="4869" LastEditDate="2015-03-13T05:51:27.417" LastActivityDate="2015-03-13T05:51:27.417" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7559" PostTypeId="2" ParentId="7528" CreationDate="2015-03-07T10:29:59.740" Score="25" Body="&lt;h1&gt;Dice&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;You get all Platonic solids, some trapecohedrons and bipyramids, and the tetrahexahedron and the rhombic triacontahedron:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/HGbRw.jpg&quot; alt=&quot;assorted dice&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2015-03-07T10:29:59.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7560" PostTypeId="1" CreationDate="2015-03-07T18:58:50.763" Score="4" ViewCount="185" Body="&lt;p&gt;If we have a standard function, like $f(x) = x$ or $g(x) = x^2$, defined between $0$ and $\pi$, then why should we be interested in extending this function to give a Fourier series that resembles this function between $0$ and $\pi$? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the whole purpose of this process? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does it have any real life application or is it just a mathematical exercise?&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="143" LastEditDate="2015-03-11T19:52:22.867" LastActivityDate="2015-03-11T19:52:22.867" Title="What is the point of using half range Fourier series for standard functions?" Tags="|undergraduate-education|student-motivation|mathematical-analysis|interactive-teaching|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7562" PostTypeId="2" ParentId="7528" CreationDate="2015-03-08T21:18:04.840" Score="17" Body="&lt;p&gt;One other poster mentioned arches; I'd like to add in the Gothic arch as an example of circular segments. These are great examples of arcs as well. I find them much more interesting, and they don't always have to have the angle shown here; the location of the circle's center can vary depending on the &quot;slope&quot; of the arch that is desired. There are also three- and four-centered arches. I can imagine that you could differentiate for your more advanced students by having them try to figure out how the more complicated arch structures were designed. Calculations related to the complicated structures could be somewhat intense but a fun challenge for a gifted student. The area underneath one of the simpler arches would be an interesting problem more at the level of the majority of the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/jujPi.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/uWoLV.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/PQv0D.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ztqUO.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2015-03-08T21:18:04.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7563" PostTypeId="1" AcceptedAnswerId="7584" CreationDate="2015-03-09T12:00:54.150" Score="29" ViewCount="1267" Body="&lt;p&gt;Like all mathematicians, I have a deep appreciation of the&#xA;beauty of mathematics. Many theorems I find amazing even after&#xA;I fully understand their proofs. (Example: Euler's formula,&#xA;$V-E+F=2-2g$. That the genus $g$ can be determined by counting&#xA;$V,E,F$ remains astounding to me.)&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/XCJYz.png&quot; alt=&quot;DavidStereo&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://www.ics.uci.edu/~eppstein/junkyard/euler/&quot; rel=&quot;noreferrer&quot;&gt;David Eppstein's webpage on Euler's theorem&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;In my teaching, I try to convey that wonder, and hope that&#xA;my students start to see mathematics in the same light.&#xA;But I wonder :-) if this approach can backfire, with students&#xA;turned off by what they see as an alien aesthetic, shared&#xA;only by math nerds.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. Have you encountered this reaction?&#xA;  If so, do you have strategies to mitigate it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Certainly I do not consider it an option to hide my enthusiasm.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-03-11T22:27:52.320" Title="Wonder as motivation" Tags="|student-motivation|concept-motivation|intuition|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7564" PostTypeId="2" ParentId="7528" CreationDate="2015-03-09T12:54:28.820" Score="14" Body="&lt;p&gt;At &lt;a href=&quot;http://en.wikipedia.org/wiki/Saddle_roof&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia: Saddle roof&lt;/a&gt; you can see images of rooftops which are a Hyperbolic Paraboloid. Other &quot;saddle-like&quot; objects may also be this shape - the primary advantage of which (like its cousin the one-sheet hyperboloid i.e. nuclear plant cooling tower) is that it can be formed from supports that are straight lines in a grid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/thumb/a/a3/W-wa_Ochota_PKP-WKD.jpg/500px-W-wa_Ochota_PKP-WKD.jpg&quot; alt=&quot;W-wa_Ochota_PKP-WKD.jpg&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At &lt;a href=&quot;http://en.wikipedia.org/wiki/Hyperboloid_structure&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hyperboloid structure&lt;/a&gt; you can see some radio towers that use the one-sheet hyperboloid as their shape.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kobe_port_tower11s3200.jpg/400px-Kobe_port_tower11s3200.jpg&quot; alt=&quot;Kobe_port_tower11s3200.jpg&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="3591" LastEditorUserId="-1" LastEditDate="2017-03-10T09:42:42.063" LastActivityDate="2015-03-09T12:54:28.820" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7565" PostTypeId="1" CreationDate="2015-03-09T13:23:45.790" Score="8" ViewCount="302" Body="&lt;p&gt;The answer to the question in the title, that then it will not easily fall down into the manhole (just try with ellipsoidal or square covers ...), is a property of circles I never learnt in school.  Yet it seems important! So the question:  Is there some good reference, book, webpages, papers, others,... for such important but forgotten properties of simple geometrical figures?  Geometry in mechanics (practical, simple, traditional mechanics, not the quantum variant ...), geometry in architecture, geometry in other applied and practical areas.  There must be something out there, but I cannot find it by googling ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar question here, but different, was &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7528/real-world-examples-of-more-obscure-geometric-figures&quot;&gt;Real-world examples of more &amp;quot;obscure&amp;quot; geometric figures&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="64" LastEditDate="2019-02-18T14:26:09.633" LastActivityDate="2019-02-18T14:26:09.633" Title="Why are manhole covers circular?" Tags="|reference-request|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7566" PostTypeId="1" CreationDate="2015-03-09T13:26:31.660" Score="7" ViewCount="2970" Body="&lt;p&gt;I am interested in examples of use of mathematics in real life situations.&#xA;To be more precise, something that could be presented to undergraduate students in order to motivate them for studying maths.&#xA;Is there some reference where one can find such examples?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A colleague of mine suggested me one of the following books:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;H. Parks, G. Musser, R. Borton, W. Silber, Mathematics in Life, society &amp;amp; the World, Prentice Hall, 1997.&lt;/li&gt;&#xA;&lt;li&gt;C . Miller, V. Heeren, J. Hornsby, Mathematical ideas, Addison Wesley Educational Publishers, 2001.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;but I cannot find a source for it online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any help with either getting a source for one of the mentioned books online, or for some other references is welcome.&lt;/p&gt;&#xA;" OwnerUserId="4847" LastActivityDate="2015-03-12T20:31:13.187" Title="Mathematics in real life" Tags="|undergraduate-education|mathematical-pedagogy|applied-mathematics|mathematics-in-daily-life|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7567" PostTypeId="2" ParentId="7566" CreationDate="2015-03-09T14:11:02.070" Score="7" Body="&lt;p&gt;First give me an example of real life then I can tell you about the math in that activity. Now, is math central to the activity? From my perspective, it is likely.&#xA;From the perspective of the person you might seek to teach math, probably not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much of math has been &lt;strong&gt;black-boxed&lt;/strong&gt; in our daily life. Cash registers are automated to the point that at least 1/4 workers cannot make change by hand (my experience). Addition of fractions and other basic arithmetic has been erased from the collective conscience thanks to misuse of calculators in early age education. We use computers, but, you most certainly do not need discrete math to use computers. This list goes on. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;We have the option to be ignorant. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The question is: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;do you want to understand the world around you?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If the answer to that is yes then finding applications of math is as simple as waking up in the morning. They're everywhere. Physics. Chemistry. Computer science. Engineering. DNA sequencing. Communications theory...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surely being able to understand the world around you is motivation enough for an educated person, or, at least a person who seeks an education. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-03-09T14:11:02.070" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7568" PostTypeId="2" ParentId="7566" CreationDate="2015-03-09T17:00:50.367" Score="1" Body="&lt;p&gt;Continuous compounding interest is a good example:&#xA;\$1.00 at an annual rate of 100% turns into \$2.00 after one year,&#xA;but compounded continuously the \$1.00 turns into $e$ dollars, \$2.72:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/PsY8a.png&quot; alt=&quot;CompoundInterest&quot;&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://en.wikipedia.org/wiki/E_(mathematical_constant)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article on $e$&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-03-09T17:00:50.367" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7569" PostTypeId="2" ParentId="7566" CreationDate="2015-03-09T19:30:26.797" Score="4" Body="&lt;p&gt;If you want to get students excited about math, why not show them some math?  Talks on applications frequently send the tacit message that &quot;you should study math because it's useful, though of course it's not really very interesting&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students who are just getting excited about math need to be reassured that it is, in fact, possible, to make a career out of math.  These &quot;applications&quot; talks, in my experience, send the message that &quot;if you study math, you'll end up doing physics/economics/engineering/biology/whatever&quot;.  I am sure that if I'd received that message in my formative years, it would have turned me away from math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To send the message that math is exciting, you need to talk about math in a way that suggests that &lt;em&gt;you&lt;/em&gt; find it exciting.  If you choose to spend your hour talking about things &lt;em&gt;other&lt;/em&gt; than math, you undercut that message.   &lt;/p&gt;&#xA;" OwnerUserId="4852" LastActivityDate="2015-03-09T19:30:26.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7571" PostTypeId="2" ParentId="7566" CreationDate="2015-03-09T22:22:36.080" Score="1" Body="&lt;p&gt;&lt;a href=&quot;http://dilbert.com/strip/1998-04-09&quot; rel=&quot;nofollow&quot;&gt;Every problem has an engineering solution&lt;/a&gt;.  I think you're better off attacking the &quot;what good is math&quot; question from the opposite direction.  You don't have to come up with examples of math being useful.  Your message can be that having mathematics in your tool belt makes you more prepared for (&lt;a href=&quot;http://xkcd.com/55/&quot; rel=&quot;nofollow&quot;&gt;almost&lt;/a&gt;) every problem you'll face in your whole life.  I suppose you'll get some resistance if your students think math is nothing more than the memorization of arcane computation rituals… but there's another teaching opportunity there.&lt;/p&gt;&#xA;" OwnerUserId="795" LastActivityDate="2015-03-09T22:22:36.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7572" PostTypeId="2" ParentId="7566" CreationDate="2015-03-09T22:51:40.840" Score="0" Body="&lt;p&gt;I sometimes feel like this and have struggled to find good talks. In my opinion learning math opens your mind in all sorts of ways that can be just appreciated when you start learning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I like this video : &lt;a href=&quot;https://www.youtube.com/watch?v=eD2hLhhYmbg&quot; rel=&quot;nofollow&quot;&gt;https://www.youtube.com/watch?v=eD2hLhhYmbg&lt;/a&gt;&#xA;In short, math touches all aspects of life.&lt;/p&gt;&#xA;" OwnerUserId="4853" LastActivityDate="2015-03-09T22:51:40.840" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7573" PostTypeId="2" ParentId="7566" CreationDate="2015-03-09T23:08:17.403" Score="1" Body="&lt;p&gt;Have you ever thought about  Realistic Mathematics Education ? That is an antique theory in Mathematics Education, which has some details how math could be applied in real life and shows some examples. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://aanhendroanto.blogspot.com/2013/09/books-of-realistic-mathematics-education.html&quot; rel=&quot;nofollow&quot;&gt;Take for example these books (online)&lt;/a&gt;. You could read them and have some idea about Mathematics in real life. &lt;/p&gt;&#xA;" OwnerUserId="1789" LastActivityDate="2015-03-09T23:08:17.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7574" PostTypeId="2" ParentId="7565" CreationDate="2015-03-10T01:00:47.593" Score="5" Body="&lt;p&gt;It is folk lore that the shape of an egg prevents it from rolling away from&#xA;the mother's nest.&#xA;For example, the article by Karen Hill,&#xA;&quot;&lt;a href=&quot;http://www.bigsiteofamazingfacts.com/why-are-eggs-egged-shaped-and-how-does-the-ovoid-shape-of-an-egg-prevent-it-from-rolling-out-of-the-nest&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why are eggs egg-shaped?&lt;/a&gt;,&quot; or,&#xA;a bit more formally,&#xA;Yutaka Nishiyama's paper&#xA;&quot;The Mathematics of Egg Shape&quot; (&lt;a href=&quot;http://www.osaka-ue.ac.jp/zemi/nishiyama/math2010/egg.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;):&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/BIXcI.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BIXcI.jpg&quot; alt=&quot;Egg Fig&quot;&gt;&lt;/a&gt;&lt;br&gt;&#xA;&lt;sub&gt;(source: &lt;a href=&quot;http://cs.smith.edu/~orourke/MathOverflow/EggInclinedPlane.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;smith.edu&lt;/a&gt;)&lt;/sub&gt;&lt;br&gt;&#xA;&lt;hr /&gt;&#xA;So this is an answer (of sorts) to the question: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why are eggs egg-shaped?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="6053" LastEditDate="2019-02-18T14:25:16.913" LastActivityDate="2019-02-18T14:25:16.913" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7576" PostTypeId="2" ParentId="7528" CreationDate="2015-03-10T12:54:36.463" Score="10" Body="&lt;p&gt;So called tensile structures in architectures are indeed &lt;strong&gt;minimal surfaces&lt;/strong&gt;. Popular examples are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the Olympiastadium in Munich: &lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/f/f0/Olympiastadion_Muenchen.jpg&quot; alt=&quot;enter image description here&quot;&gt; or&lt;/li&gt;&#xA;&lt;li&gt;the former Millenium Dome in London: &lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/6/6f/The_O2_Arena%2C_entrance.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="-1" LastEditDate="2017-03-10T09:42:42.090" LastActivityDate="2015-03-10T12:54:36.463" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7577" PostTypeId="2" ParentId="7563" CreationDate="2015-03-10T20:11:48.980" Score="7" Body="&lt;p&gt;This spoke to me, enough that I decided to answer a related question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had the privilege to watch Ole Hald in action when he taught a second year service class (linear algebra/diff e.q).  He had won at least one (university-wide, student voted) teaching award, and had a number of techniques for emphasizing and making clear much of the material.  It was clear to me that he spent a lot of energy preparing the lectures as well as encouraging teaching assistants (of which I was one) about how to help the students.  To me, he never seemed to be satisfied with his efforts.  Indeed, he surprised me when he told me something related to a comment I had made: he was worried that he could not be flexible enough to adapt his presentation to his audience; in particular he was concerned about being able to teach graduate courses adequately, being &quot;stuck in an undergraduate rut&quot; (my words).  I would have gotten satisfaction to do as good a job part of the time as he did consistently.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see in your question the drive to do better continually and reach everyone.  Don't give up the struggle, or even relax much.  Be sure to celebrate your accomplishments, know that you are reaching some, and use that as fuel to find, share, and improve upon ever-inspiring pictures.  (It isn't an alien aesthetic.  It is your aesthetic, the sharing of which will help people find and develop their own aesthetic.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;One Of Those Saluting You&quot; Paseman, 2015.03.10&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-03-10T20:11:48.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7578" PostTypeId="2" ParentId="7563" CreationDate="2015-03-10T22:28:20.403" Score="4" Body="&lt;p&gt;Even if the students are turned off by your enthusiasm, they will be even more turned off by a bored teacher. So go with your enthusiasm, realise that some kids will react badly to it, but it is probably better than the alternatives.&lt;/p&gt;&#xA;&lt;p&gt;Below I discuss some particular factors involved in the negative reaction towards enthusiasm for mathematics.&lt;/p&gt;&#xA;&lt;h3&gt;Maths curricula are too crammed to develop aesthetics&lt;/h3&gt;&#xA;&lt;p&gt;I share your enthusiasm for Euler's formula. I managed to get some kids really interested in it, and they now share our wonder of it. This is how I did it:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I taught kids who already loved maths&lt;/li&gt;&#xA;&lt;li&gt;I had unlimited time to spend on the topic, so that they could adequately explore the problem domain before we discovered the formula. Understanding the complexity of the problem is key to appreciating the simple relationship.&lt;/li&gt;&#xA;&lt;li&gt;After we had the formula we used it on a big project, finding all the possible Pythagorean and Archimedian solids and creating nets for most of them on Geogebra&lt;/li&gt;&#xA;&lt;li&gt;We spent time extending the relation in more general topological contexts (spherical/hyperbolic geometry)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I have also taught class room maths, and I have no idea how I would be able to do it in a way which communicates the aesthetic within the time constraints of a standard curriculum. Developing aesthetics takes time. Like art classes, we would need to give kids room to experiment and play around and develop their own responses to the aesthetic. We probably developed our own aesthetics in our own time rather than class time.&lt;/p&gt;&#xA;&lt;h2&gt;Maths is uncool&lt;/h2&gt;&#xA;&lt;p&gt;The bigger problem is that enthusiasm for maths often seems to alienate kids. I think there are strong cultural factors at play here. Kids have been programmed into thinking that people who enjoy the mathematics aesthetic are uncool, and therefore listening to them is uncool. By loving/valuing the beauty of maths, you are demonstrating that your values are at odds with their values, and you lose their interest.&lt;/p&gt;&#xA;&lt;p&gt;If have personally seen teachers over come this in a number of ways:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;subtly make fun of the class nerd. This shows that you have the same values as they do and you respond appropriately to those who transgress those values. You might even poke fun at other teachers who students may see as uncool, and point out how their subject is even more uncool than yours.&lt;/li&gt;&#xA;&lt;li&gt;cover the walls with pictures teens like, ie. half-naked young adults (this was a female art teacher - I don't know if a maths teacher could get away with this. And yes, it was super-creepy.)&lt;/li&gt;&#xA;&lt;li&gt;let them know you agree that it is unreasonable that they have to learn this uncool stuff, but here are some easy tricks to pass the test without really needing to understand it. This is a popular tactic for some tutors.&lt;/li&gt;&#xA;&lt;li&gt;turn a small topic into a big project that has very little maths content but which gives each student a sense of belonging which turns their lives around ... I admit, I've only ever seen this one in movies.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Those of us with a basic sense of ethics and professional responsibility don't have the flexibility to be really cool, so I guess we need to rely on our own relationships with the students. Hopefully some of them will forgive our geeky love of what we teach, and actually learn something or maybe even come to appreciate our perspective.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-03-10T22:28:20.403" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7579" PostTypeId="1" CreationDate="2015-03-11T05:02:03.217" Score="7" ViewCount="298" Body="&lt;p&gt;At the school I am student teaching at starting in 2 weeks (Algebra 1), students are strongly encouraged to rely on their calculators for nearly everything, from any operation with fractions to matching graphs with equations. I am fairly sure that even the most vehement calculator proponents would agree that this is unhealthy, for instance, in the cases where using the calculator works &quot;well enough&quot; but doing pieces by paper or in your head will vastly increase the speed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recognize some questions that are similar have been asked; however, few focus on actual pedagogical techniques that are used at the high school level. Most are aimed at convincing students at the undergraduate level that they should use their calculators less, ways to talk about it during lecture, or the amount of calculator use students should be allowed. So, my question is threefold: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What are ways that I can design assignments that discourage the over-reliance on the calculator and encourage the development of heuristics? Additionally, how would you change your answer, considering that the current class is structured on the availability of a calculator because &quot;they are allowed on the state test&quot;?&lt;/li&gt;&#xA;&lt;li&gt;What are class activities (such as group work, games) that can be done to help students develop heuristics and mental models that do not rely on calculator usage yet are still appropriate for early high school? (For instance I remember playing a sort of &lt;a href=&quot;http://everydaymath.uchicago.edu/about/understanding-em/games/beat-the-calculator/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;beat the calculator&quot; type game&lt;/a&gt; as a young elementary student that was designed for the full class.)&lt;/li&gt;&#xA;&lt;li&gt;What are other (direct?) instructional techniques that I can use that help students see the utility of avoiding their calculator if unnecessary? (possibly covered &lt;a href=&quot;https://matheducators.stackexchange.com/questions/4503/appropriate-ways-sayings-to-discourage-undergraduate-students-overreliance-on-c&quot;&gt;here&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="724" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-04-12T14:26:02.213" Title="High school level activities to decrease reliance on calculators?" Tags="|secondary-education|calculators|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7580" PostTypeId="2" ParentId="7528" CreationDate="2015-03-11T06:32:33.517" Score="12" Body="&lt;p&gt;The stone spheres (or stone balls) of Costa Rica are an assortment of over three hundred petrospheres in Costa Rica, located on the Diquís Delta and on Isla del Caño. Locally, they are known as Las Bolas (literally The Balls). The spheres are commonly attributed to the extinct Diquís culture and are sometimes referred to as the Diquís Spheres. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Palmar Sur Archeological Excavations are a series of excavations of a site located in the southern portion of Costa Rica, known as the Diquís Delta. The excavations have centered on a site known as &quot;Farm 6&quot;, dating back to the Aguas Buenas Period (300-800AD) and Chiriquí Period (800-1550 AD).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They are almost perfectly round, developed by a culture without any knowledge on geometry? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/igxMC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="4859" LastActivityDate="2015-03-11T06:32:33.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7581" PostTypeId="1" CreationDate="2015-03-11T08:35:15.120" Score="8" ViewCount="120" Body="&lt;p&gt;There is software available on the Net for 3-D printing of math expressions encoded in LaTex. What such technology is available off-the-shelf for the visually impaired to learn mathematics? And, accompanying research/anecdotes on its effectiveness?&lt;/p&gt;&#xA;" OwnerUserId="4793" LastEditorUserId="4793" LastEditDate="2015-03-11T08:48:43.293" LastActivityDate="2015-03-11T08:48:43.293" Title="3-D printing of formulas encoded in LaTex for the visually impaired?" Tags="|undergraduate-education|mathematical-pedagogy|secondary-education|specific-software|special-needs-education|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7582" PostTypeId="1" CreationDate="2015-03-11T12:34:08.797" Score="8" ViewCount="971" Body="&lt;p&gt;Is there any good reason that in educational materials, I consistently see the formula for calculating geometric series in canonical form as:&#xA;$$\sum_{k=0}^{n-1} ar^k = a \frac{1-r^n}{1-r}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While an equally correct version is given by [edit: I have now found a book that uses this form]&#xA;$$\sum_{k=0}^{n-1} ar^k =a \frac{r^n-1}{r-1}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The canonical form seems to be backwards for the understanding of most students, as initial examples usually have $r &amp;gt; 1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A case of special interest is the infinite series with $r&amp;lt;1$, with the limit of $$\sum_{k=0}^{\infty} ar^k =a\frac{1}{1-r} \text{ (when r&amp;lt;1)}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A geometric sequence is simply the discrete version of the exponential function, so a geometric series is simply the discrete version of the integral of the exponential function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I can't imagine that anyone would write [edit: an answer below shows that accountants do use it in this form - though I would think it is motivated by usefulness rather than consistency with the infinite sum]:&#xA;$$\int_{0}^{n} ar^x dx = a\frac{1-r^n}{-\ln r} + C$$&#xA;Just because a particular definite integral is of the form:&#xA;$$\int_{0}^{\infty} ar^x dx = a\frac{1}{-\ln r} \text{(when r&amp;lt;1)}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note, I've never seen integration of exponentials taught with regard to geometric series, even though they are normally taught in close proximity, and the formulae are nearly identical (x-1 is a good approximation of ln x for x close to 1). Has anyone had experience with this?&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-03-12T04:08:30.943" LastActivityDate="2015-03-12T23:32:14.003" Title="Geometric Series Formula and Calculus" Tags="|calculus|series|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7583" PostTypeId="2" ParentId="7579" CreationDate="2015-03-11T13:43:08.620" Score="1" Body="&lt;p&gt;For some ideas, look at resources relating to NZ Achievement Standard AS91027. In this external assessment, calculators are banned. It is a &quot;Level 1&quot; standard which is at a similar level to US Year 10.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A description of the standards in student language is here: &lt;a href=&quot;http://www.studyit.org.nz/subjects/maths/math1/2/&quot; rel=&quot;nofollow&quot;&gt;http://www.studyit.org.nz/subjects/maths/math1/2/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The standards documents, with previous assessments and marking schedules are here: &lt;a href=&quot;http://www.nzqa.govt.nz/ncea/assessment/view-detailed.do?standardNumber=91027&quot; rel=&quot;nofollow&quot;&gt;http://www.nzqa.govt.nz/ncea/assessment/view-detailed.do?standardNumber=91027&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A third party resource with teaching materials and example assessments is given here &lt;a href=&quot;http://www.mathscentre.co.nz/Booklets/&quot; rel=&quot;nofollow&quot;&gt;http://www.mathscentre.co.nz/Booklets/&lt;/a&gt; &#xA;There should be a download button to the bottom right of &lt;em&gt;AS91027 Apply Algebraic Procedures In Solving Problems&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for motivation to not use calculators, we have &quot;if you can't do it without calculators, you fail!&quot;. If all your state tests allow calculators, it is not so easy to motivate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know of any particularly innovative pedagogical techniques surrounding this standard. It is designed purely to ensure minimum proficiency of basic algebra necessary for further study. By contrast, the other 12 standards strongly emphasise contextualised mathematics, making them easier to motivate, but calculators are required.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for designing assessments, the big thing is that the decimals are at most 2 decimal places (for money), and most things work out to nice numbers. Everything is designed to be solved by pure mental arithmetic, not even pencil and paper algorithms should be necessary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally agree with you that mental heuristics should be emphasised, but it is not emphasised in the curriculum. Many heuristics are very contextual, making them hard to teach. Also most students don't understand the point of getting something wrong by over 50% but within &quot;Order of Magnitude&quot;, and test markers don't give credit for it.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-03-11T13:43:08.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7584" PostTypeId="2" ParentId="7563" CreationDate="2015-03-11T14:17:38.963" Score="14" Body="&lt;p&gt;I have given this issue a lot of thought over the years -- in fact a large portion of &lt;a href=&quot;http://deepblue.lib.umich.edu/handle/2027.42/64672&quot;&gt;my dissertation&lt;/a&gt; is devoted to related issues.  I think of this challenge in terms of a &quot;mathematical sensibility&quot; -- a way (more precisely, a cluster of related ways) of appreciating and participating in mathematics -- and frame the question as &quot;What is the role of school mathematics [and I would extend this to University level as well] in cultivating a mathematical sensibility?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to get a handle on this question, I identified a number of distinct dispositions or 'categories of appreciation' -- ways that mathematicians use to label or attach value to individual mathematical results.  These dispositions seem to appear in dialectical pairs; some of the ones I have described are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;surprise / confirmation&lt;/li&gt;&#xA;&lt;li&gt;simplicity / complexity&lt;/li&gt;&#xA;&lt;li&gt;utility / abstraction&lt;/li&gt;&#xA;&lt;li&gt;theory-building / problem-solving&lt;/li&gt;&#xA;&lt;li&gt;formalism / realism&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It's not at all unusual for mathematicians to refer to any and all of the above with the catch-all label &quot;beauty&quot;.  This is not really surprising; we are not trained as aestheticians, and consequently we aren't particularly skilled at teasing apart the different strands that evoke that aesthetic response.  In the case of Euler's Theorem, I see in the OP mainly the disposition of &lt;em&gt;surprise&lt;/em&gt; (&quot;amazing... That the genus g can be determined by counting $V,E,F$ remains astounding to me&quot;), with perhaps some implied appreciation also for the &lt;em&gt;simplicity&lt;/em&gt; of the formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is worth pointing out that sometimes we have much the same appreciation for mathematics that has exactly the opposite characteristics.  The construction of a faithful complex representation of the monster group, for example, is astounding precisely for its &lt;em&gt;complexity&lt;/em&gt;, and the proof of a long-standing open conjecture is appreciated for &lt;em&gt;confirming&lt;/em&gt; what we already were pretty sure was true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also worth pointing out that K-12 mathematics focuses almost exclusively on just one of these dispositions, namely, &lt;em&gt;utility&lt;/em&gt;.  In nearly all of students' mathematics education, the value of a piece of mathematics lies (if anywhere) solely in what it can be &lt;em&gt;used for&lt;/em&gt;.  This is why students ask &quot;When are we ever going to need to know this?&quot;  If the only way you have of appreciating the value of something is in terms of its instrumentality, than something that lacks an obvious application is quite literally valueless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, why is it so hard to cultivate an appreciation for these values?  Certainly part of it is personal proclivity; some people are more naturally inclined to appreciate these values than others.  But I think a larger problem is that it is extraordinarily difficult to &lt;em&gt;assess&lt;/em&gt; whether students are learning a mathematical sensibility.  Without a way of assessing something, we can't hold students accountable for it; and instruction naturally defaults toward those things for which accountability can be maintained. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If that hypothesis is true, then another way of framing the question is:  Is there a way to assess whether students are learning to appreciate mathematics?  One of the things that I found in my dissertation was that standard forms of assessment (exams, quizzes, daily or weekly homework) seem to be ill-suited for that purpose, but other forms (specifically, open-ended research projects conducted over a longer timeframe) may provide an opportunity for students to get interested in things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides my dissertation, I've written about these matters in a couple of places: &lt;a href=&quot;http://www.jstor.org/discover/10.5951/mathteacher.106.4.0269?uid=3739728&amp;amp;uid=2&amp;amp;uid=4&amp;amp;uid=3739256&amp;amp;sid=21105611002471&quot;&gt;an article for classroom teachers on 'Thinking Like a Mathematician'&lt;/a&gt; and &lt;a href=&quot;http://link.springer.com/article/10.1007/s10649-015-9599-x&quot;&gt;a new article (just out this week!) on the theory-building disposition&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-03-11T14:17:38.963" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7585" PostTypeId="2" ParentId="7582" CreationDate="2015-03-11T15:13:43.147" Score="4" Body="&lt;p&gt;The first example of this I saw was the sum of the divisors of a prime power $p^n$:&#xA;$$&#xA;1+p+p^2+\dots+p^n = \frac{p^{n+1}-1}{p-1}&#xA;$$&#xA;and in that case $p&amp;gt;1$ is certainly intended.  Then it seemed pretty strange to me when my teacher wrote it as&#xA;$$&#xA;\frac{1-p^{n+1}}{1-p}&#xA;$$&#xA;as part of his explanation of the formula.  Only later in calculus did I see that this was the natural way to write it when working on infinite series.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-03-11T15:13:43.147" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7586" PostTypeId="2" ParentId="7579" CreationDate="2015-03-11T18:31:29.920" Score="1" Body="&lt;p&gt;A general anti-calculator strategy is to give the students the answer and then adjust your  questioning accordingly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you were teaching multiplication and was trying to get them to do $23\times27$.  You could just outright give them the answer $621$ and ask them to convince you the answer couldn't be anything else.  How do you know it can't be $400$? or more difficult $641$? The calculator would be redundant or at least used creatively in that situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I appreciate that this is a simple example but I hope it gives you ideas.  &lt;/p&gt;&#xA;" OwnerUserId="4668" LastEditorUserId="1147" LastEditDate="2015-03-13T13:39:21.103" LastActivityDate="2015-03-13T13:39:21.103" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7587" PostTypeId="1" CreationDate="2015-03-11T18:51:36.183" Score="3" ViewCount="303" Body="&lt;p&gt;I came across the question below on IXL.com. (I have reproduced it in its entirety. Given were 5 possible answers to choose from.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It strikes me that there is not one single correct answer for this question. But perhaps according to the US Common Core standard there is. If so, what is the single correct answer and why? And if not, would this question simply be wrong?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/mXA0T.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user378" LastActivityDate="2015-03-13T23:45:44.763" Title="Domain of an exponential function (US Common Core, High School Math)" Tags="|common-core|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7588" PostTypeId="2" ParentId="7582" CreationDate="2015-03-11T18:54:42.273" Score="7" Body="&lt;p&gt;I agree that the form $\frac{r^n-1}{r-1}$ is in many respects more natural:  certainly it is in cases where $r&amp;gt;1$, but even when $r$ is left unspecified, I think this form arises most naturally from many &quot;customary&quot; ways of deriving the formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why is it usually taught the other way around?  I suspect the answer is precisely that, in most secondary curricula that I am familiar with, as soon as we have learned the formula for summing a (finite) geometric series we &lt;em&gt;immediately&lt;/em&gt; proceed to infinite geometric series:  we restrict attention to the case where $\vert r \vert &amp;lt;1$ and say that as $n \to \infty$ the $r^n$ term can be neglected.  If we wrote the formula out the way you want to, we would then end up with $\frac{-1}{r-1}$ as the sum, which is weird and confusing because it has an explicitly negative numerator but only an &lt;em&gt;implicitly&lt;/em&gt; negative denominator (that is, $r-1$ doesn't &lt;em&gt;look&lt;/em&gt; obviously negative, and only is seen to be so if you think about the size of $r$), and so it seems to the novice as if we are saying the infinite series has a negative sum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much better and less confusing to write the sum of the infinite series in the form $\frac{1}{1-r}$, and that is the reason, I think, for what you refer to as the &quot;canonical&quot; form: it provides cohesion between the finite and infinite cases.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-03-11T18:54:42.273" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7589" PostTypeId="2" ParentId="7563" CreationDate="2015-03-11T21:03:15.573" Score="5" Body="&lt;p&gt;A student asked me to explain how to graph a hyperbola. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{\left(x+2\right)^2}{3^2}-\frac{\left(y-6\right)^2}{2^2}=1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My first question was if she understood ellipses. And specifically the ellipse,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{\left(x+2\right)^2}{3^2}+\frac{\left(y-6\right)^2}{2^2}=1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Drawing that ellipse will give us the rectangle needed to form the two asymptotes as well as easily see the hyperbola vertices. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/tblzx.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For sake of completeness, the conjugate equation &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$-\frac{\left(x+2\right)^2}{3^2}+\frac{\left(y-6\right)^2}{2^2}=1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is also shown above. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The punchline is this student responded with, &quot;that's so cool.&quot; She went on to ask why the teacher didn't present it this way. She (the teacher) hadn't make any connection between the ellipse and the hyperbola. For me, the reward comes from aiming to have as many &quot;that's so cool&quot; moments as I can. It won't be every topic I address, nor will every student react this way. But I keep my enthusiasm and cherish those moments. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a lighter note - When asked to explain the nature of asymptotes, I explained that they think they are MC Hammer, always humming &quot;can't touch this.&quot; I'm in this to have fun and need to inject humor like that when I see the opening. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-03-11T21:03:15.573" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7590" PostTypeId="2" ParentId="7582" CreationDate="2015-03-11T21:39:28.723" Score="4" Body="&lt;p&gt;It may have something to do with the classical proof of the formula:&#xA;$$a(1+r+\dots+r^{n-1})(1-r)=a(1-r+r-r^2+\dots+r^{n-1}-r^n)=a(1-r^{n+1})$$&#xA;which makes slightly more sense this way given that we usually write the sum $\sum ar^k$ in increasing term order.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-03-11T21:39:28.723" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7591" PostTypeId="2" ParentId="7563" CreationDate="2015-03-11T22:27:52.320" Score="12" Body="&lt;p&gt;This answer is from my experience running a Maths Learning Centre. I help students learn and use maths, mostly when they are struggling, and I also hear their opinions of their lecturers and other teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am very interested in all sorts of maths, and I will take almost any opportunity to be excited about a mathematical idea. For example, I made t-shirts with mathematical things on them which I wear every day in the Centre, and only need students to show the slightest interest in what the t-shirt is about to wax lyrical about it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do the students find my sense of fun and excitement with maths alien? Yes, most of them do. A lot of them shake their heads and simply can't understand how I can love it so much. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do they find my sense of fun and excitement with maths alienating? No. Let me tell you why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, students love it when their teachers show enthusiasm. They often praise teachers who are enthusiastic about the subject matter, even if they dislike every other aspect of their teaching. They want to be confident that the person who is teaching them knows what they are doing, and cares about what they are doing, and excitement with the subject matter is a good sign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, I am not excited by only the &quot;high and mighty&quot; maths -- I am just as excited about the maths they are doing right now. I have a favourite fraction (3/8), I love that division has multiple interpretations, the classification of quadrilaterals is fascinating to me, and the fact that a set of points in the plane has an equation at all is the coolest thing ever. If I was only ever excited about things beyond what they were doing -- things they won't learn for years, if ever -- then they might get the impression that I think their problems are beneath me. Students have cited this very idea to me when they say they don't like some of their lecturers. They say they feel like the lecturer is great with their fancy cutting-edge research and is only teaching their class because they have to. Moreover, it's tricky to be inspired to wonder if you have to wait for a couple of years before you're allowed to feel it properly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thirdly, I sympathise with their experience. When they find it difficult to learn something, I tell them about how much I hated differential equations when I first learned it. When they find the lecturer's style of presentation boring, I can relate to all the times when I wished the lecturer gave more actual examples with actual numbers. And when they just want to get their assignment done, I remember what it was like with those pressures and work with them to get that bit further. Students prefer teachers who care about the students' struggles and consider what it is like to be a student. They hate it when the lecturer loves their subject but is &quot;out of touch&quot;. A particularly relevant and hated brand of &quot;out of touch&quot; is when a lecturer has such a sense of wonder that they set assignment questions based on their mathematical coolness, without regard for the difficulty students have knowing what they hell they are supposed to achieve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in short, displaying your sense of wonder is a good thing, but you need to find wonder in the things the students are doing now, and you need to balance it with a healthy dose of understanding what it is like to be a student.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-03-11T22:27:52.320" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7593" PostTypeId="2" ParentId="7582" CreationDate="2015-03-12T00:09:03.233" Score="2" Body="&lt;p&gt;Practical problems for this formula that involve 0 &amp;lt; r &amp;lt; 1 are more common than similar problems that involve r &gt; 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mortgage interest calculations (such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Amortization_calculator&quot; rel=&quot;nofollow&quot;&gt;present value interest factor of an annuity&lt;/a&gt;) involve 0 &amp;lt; r &amp;lt; 1.  These problems involve a known (or infinite) n = number of terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometric expansion problems involving r &gt; 1 do exist, but the problem formulator must arbitrarily decide on n = number of terms.  In order to have a finite answer, there must be a limit to the number of terms -- but it is often not obvious how such a limit should be chosen.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-03-12T00:20:33.587" LastActivityDate="2015-03-12T00:20:33.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7594" PostTypeId="1" AcceptedAnswerId="7607" CreationDate="2015-03-12T01:48:41.527" Score="13" ViewCount="2762" Body="&lt;p&gt;I'm teaching an Algebra II/Trigonometry class that is currently working on logarithms as the inverse function of exponential functions. Now, to keep their interest I had them struggle for a few minutes with $2^x = 10$ and trying to solve it. Trying to find the answer to that question is the driving force behind our activities with logarithms (thankfully my students &lt;em&gt;really&lt;/em&gt; want an answer to that question, so they're willing to learn whatever I have for them so long as they can get closer to answering the question posed originally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of them have noticed that there is a &quot;log&quot; function on their calculator, which I acknowledged, but I also warned them that right now if they plugged 10 into their calculator they wouldn't get the correct answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to show them tomorrow as a quick history lesson before some presentations that before we had calculators that could calculate logarithms, there were tables of logarithms that people kept in books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that they're going to ask, and as of right now my searches haven't yielded an answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How were the values in the old logarithm tables found? I've found a lot of information on how people used logarithm tables, but very little information on how those values were calculated. Can anyone help me here?&lt;/p&gt;&#xA;" OwnerUserId="4709" LastActivityDate="2015-03-13T15:48:47.807" Title="Logarithm Tables - How were the values reached?" Tags="|algebra|inverse|logarithm|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7595" PostTypeId="2" ParentId="7587" CreationDate="2015-03-12T02:20:19.943" Score="3" Body="&lt;p&gt;The correct answer is obviously all reals, since it is an exponential function. I think the big point &lt;em&gt;trying&lt;/em&gt; to be made here is that exponential functions are defined &lt;em&gt;anywhere&lt;/em&gt;, so even though the graph is (more than likely very purposefully) misleading to show that there appears to be an asymptote at x=0, The key is relying on the student's ability to reason that exponential functions all look like &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f(x)= a^{(x-c)} +d$ where a,c,d $\in \mathbb R$ and that there just is not a value that you &lt;strong&gt;&lt;em&gt;can not&lt;/em&gt;&lt;/strong&gt; plug in to the function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, the graph is horribly drawn, and I had a similar problem with my students when we started working with exponential functions because of the fact that no matter the window you use the graph just always seems to sort of stop. That's why I used &lt;a href=&quot;http://desmos.com&quot; rel=&quot;nofollow&quot;&gt;Desmos&lt;/a&gt; along with the question of &quot;do you think it stops at [whatever value is at the top of the window]? &quot; I would zoom out or pan up to show them that the graph always sort of went on, but it grew &lt;em&gt;very&lt;/em&gt; quickly, which is why it would look like it had an asymptote at some x-value.&lt;/p&gt;&#xA;" OwnerUserId="4709" LastActivityDate="2015-03-12T02:20:19.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7596" PostTypeId="2" ParentId="7582" CreationDate="2015-03-12T03:42:18.510" Score="5" Body="&lt;p&gt;As a financial practitioner where such calculations are commonplace, I would encourage you to teach both and motivate them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In finance an annuity is a stream of regularly spaced payments. An &quot;ordinary&quot; annuity, e.g. a mortgage, has the first payment incurred at the beginning of the first period but paid at the &lt;em&gt;end&lt;/em&gt; of the first period, so it includes interest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The present value of an ordinary annuity, with interest rate $i$, using the handy actuarial notation, is written&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\require{enclose}a_{\enclose{actuarial}{n}i} = (1+i)^{-1}+(1+i)^{-2}+...+(1+i)^{-n} = \frac{1-(1+i)^{-n}}{i}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it is common for actuaries to use the convention $v = (1+i)^{-1}$ and write $\require{enclose}a_{\enclose{actuarial}{n}i} = \frac{1-v^n}{i}$ instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The future value is generally written&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\require{enclose}s_{\enclose{actuarial}{n}i} = (1+i)^{n-1}+(1+i)^{n-2}+...+(1+i)^{0} = \frac{(1+i)^n-1}{i}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, alternatively $\require{enclose}s_{\enclose{actuarial}{n}i} = \frac{v^{-n}-1}{i}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These formulas have a natural symmetry and the relation $\require{enclose}a_{\enclose{actuarial}{n}i} (1+i)^n = s_{\enclose{actuarial}{n}i}$ links these two together. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The present value calculation is common for amortizing, e.g. determining level payments for a loan, while the future value calculation is common for determining level premiums or savings for a future payout, e.g. retirement goal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that present value for continuous case is analogously written&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\require{enclose}\bar{a}_{\enclose{actuarial}{n}} = \int_0^n v^t dt = \frac{1-v^n}{\delta}$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $\delta = \ln(1+i)$ is called the force of interest, i.e. the rate of accumulation is $a(t) = e^{\delta t}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you do generally find the formula expressed in a similar fashion to the &quot;I can't imagine that&quot; form you provided, albeit with a different convention for variables.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As far as derivation is concerned, the most natural seeming to me is to teach the perpetuity&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\require{enclose}a_{\enclose{actuarial}{\infty}} = v + v^2 + ...$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where the trick is to see that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\require{enclose}a_{\enclose{actuarial}{\infty}} = v + v a_{\enclose{actuarial}{\infty}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;may be used to derive &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\require{enclose}a_{\enclose{actuarial}{\infty}} = \frac{v}{1-v} = \frac{1}{i}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the relationship&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\require{enclose}a_{\enclose{actuarial}{\infty}} = a_{\enclose{actuarial}{n}} + v^n a_{\enclose{actuarial}{\infty}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;may be used to derive&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\require{enclose}a_{\enclose{actuarial}{n}} =  (1-v^n) a_{\enclose{actuarial}{\infty}} = \frac{1-v^n}{i}$$&lt;/p&gt;&#xA;" OwnerUserId="4866" LastEditorUserId="4866" LastEditDate="2015-03-12T04:33:10.383" LastActivityDate="2015-03-12T04:33:10.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7597" PostTypeId="1" AcceptedAnswerId="7602" CreationDate="2015-03-12T16:48:42.667" Score="17" ViewCount="1820" Body="&lt;p&gt;Mathematics at highschools is quite different from that in universities.&#xA;Instead of calculating numbers and finding solutions to specific problems, freshmen end up proving theorems and figuring out what happens in general.&#xA;(By in general I mean, for example, that we are not very interested in the integral function of $xe^{\sin x}\log\cosh x$, but in the fact that every continuous function can be integrated and the integral satisfies the fundamental theorem of calculus.)&#xA;I have heard statements like &quot;I like math but I always hated when we had to prove stuff&quot;, which means that the student will hate mathematics the way it is presented and studied at universities, and studying mathematics would probably be a bad idea for the student.&#xA;The way of thinking about mathematics as a whole and the collection of mathematical tools makes a big transition when one enters a university (at least in Finland), and it is easy for a teacher to make fallacious assumptions about students' proficiency in basic tools in doing and communicating mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I, as an instructor of some kind in a freshman course, help students adapt to the new situation?&#xA;Is there something that I should be very careful about?&#xA;In your experience, what is important when trying to get abstract ideas (and the need for them) across to freshmen?&#xA;I understand that it is not uncommon that a freshman class is taught by someone who has long ago forgotten how a freshman thinks, and although (because?) the teacher has made his best to make the material mathematically elegant, the students get little grasp of what is going on.&#xA;I would like to prevent this from happening in my class if I ever get assigned a freshman course.&#xA;It would be great to hear experiences about supporting students in this transition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt;&#xA;Educational systems are different in different countries, so &quot;first year university mathematics&quot; is not globally well defined.&#xA;The transition I wanted to support the students in is the one from calculation-oriented mathematics to proof-oriented mathematics.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2015-03-13T16:08:42.070" LastActivityDate="2015-03-13T18:19:11.073" Title="How to help students bridge the gap between highschool and university mathematics?" Tags="|undergraduate-education|mathematical-pedagogy|prerequisites|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7598" PostTypeId="2" ParentId="7528" CreationDate="2015-03-12T17:07:00.593" Score="5" Body="&lt;p&gt;Minimal surfaces was mentioned. Another example of minimal surfaces is soap bubbles:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Dy9NQ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Q5DCH.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2015-03-12T17:07:00.593" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7599" PostTypeId="2" ParentId="7528" CreationDate="2015-03-12T17:15:30.047" Score="7" Body="&lt;p&gt;Water towers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/OzeYC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The form comes from the need to (approximately) maintain a constant pressure.&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2015-03-12T17:15:30.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7600" PostTypeId="2" ParentId="7587" CreationDate="2015-03-12T18:33:44.477" Score="1" Body="&lt;p&gt;Because I'm a bit OCD, I had to know. @Jhecht gave enough hints for me to push for a solution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/s0Wir.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$y=10^{\left(x+1\right)}+4$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is as close at it gets to what the real equation is. And the equation makes it obvious the Y-intercept is 14. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And while it's fair enough at some level to expect an understanding that this function will have all reals as domain, I still find the graph misleading. I remain curious what the choices were for answers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(On further reflection, I have (-1,5) as a point, the OP graph is closer to (-1,5.5) so the exponent is closer to x+1.1. The point remains, the Y intercept is high)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-13T23:45:44.763" LastActivityDate="2015-03-13T23:45:44.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7601" PostTypeId="2" ParentId="7587" CreationDate="2015-03-12T18:37:36.233" Score="1" Body="&lt;p&gt;It is a poorly written question because it presumes there is some sort of universal definition of what is meant by an &quot;exponential function.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can closely replicate the graph with  $f(x)=3+e^{-1/x}$ with $x&amp;lt;0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is consensus about what we mean by polynomial and rational functions. But depending on the context, it would not be unreasonable to conceptualize an &quot;exponential&quot; function as being any function in the field of functions that we get by adjoining exp to the field of rational functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the context of the original question, one can guess that what is meant by an &quot;exponential function&quot; is something of the form $\hbox{polynomial}+Ae^{cx}$ or perhaps $\hbox{rational}+Ae^{cx}$&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-03-12T18:37:36.233" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7602" PostTypeId="2" ParentId="7597" CreationDate="2015-03-12T20:30:23.020" Score="15" Body="&lt;p&gt;First it's a good sign that you're on this website and keen to make a difference. A passionate tutor is halfway there in my opinion.  I've thought about what would have helped me as an undergraduate but everyone is different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Break the theorem-proof-theorem-proof-theorem-proof-lemma-theorem cycle.  I'm sure that many will argue that this is the bedrock of the subject but for me at least it quickly became overwhelming.  I've thought about why it became overwhelming and I can think of two reasons. Firstly I needed time to aclimatise to the new maths and wanted to just calculate with each new theorem before moving on.  Second it was just boring. There were lectures I probably could have understood if I hadn't switched off with the monotony of theorem proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Facilitate the students in forming abstractions and don't jump straight in with the abstract. I remember an abstract algebra class beginning with the axioms of arithmetic including commutativity and associativity. I remember not understanding what all the fuss was because I'd only ever dealt with real numbers and couldn't imagine anything being different.  It was only until much later when I'd studied linear algebra did it dawn on me that things could be different! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Why say it with a letter when a number will do? Never start with a general case.  If you can give examples with numbers do so. Do the matrix with number elements to calculate determinants or the differential equation with numbers where you might have put an $a$.  Later on we then it's sunken in can you generalise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Keep things in context.  I remember calculating eigenvalues from characteristic equations and had no idea what I was doing. Much later I understood the topic thinking about transforms. The point being it wasn't hard after all but I had learnt it out of any meaningful context and so ended up lost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are just a few of my thoughts, everyone is entitled to their own opinion and disagree.  If you want more then just let me know.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastEditorUserId="4668" LastEditDate="2015-03-12T20:45:03.307" LastActivityDate="2015-03-12T20:45:03.307" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7603" PostTypeId="2" ParentId="7566" CreationDate="2015-03-12T20:31:13.187" Score="3" Body="&lt;p&gt;I just ran across this book:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;W.D. Wallis, &lt;em&gt;Mathematics in the Real World&lt;/em&gt;.&#xA;  2013. Birkhäuser Basel.&#xA;  (&lt;a href=&quot;http://www.springer.com/new+%26+forthcoming+titles+%28default%29/book/978-1-4614-8528-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer link&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is a review:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;“This textbook contains a treasure trove of such topics, such as sampling, cryptography, voting, and probability. While &lt;em&gt;Mathematics in the Real World&lt;/em&gt; is intended for students with a minimal background in mathematics, the mathematics in the textbook is not dumbed down. The student will be performing calculations, using formulas, and drawing graphs. … All in all, I think Wallis has written a wonderful text. He has chosen exciting topics and explained them in a straightforward manner.”&#xA;&lt;br /&gt;&#xA;(Kara Shane Colley, &lt;em&gt;MAA Reviews&lt;/em&gt;, January, 2014)&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/6wUry.png&quot; alt=&quot;Wallis&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-03-12T20:31:13.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7604" PostTypeId="2" ParentId="7594" CreationDate="2015-03-12T22:23:21.110" Score="5" Body="&lt;p&gt;Gauss said &quot;You have no idea how much poetry there is in a table of logarithms.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first paragraph of this paper might get you pointed in the right direction&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/journals/bull/2006-43-01/S0273-0979-05-01096-7/S0273-0979-05-01096-7.pdf&quot; rel=&quot;nofollow&quot;&gt;ON THE DISTRIBUTION OF PRIMES—GAUSS’ TABLES&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="64" LastEditDate="2015-03-13T15:48:47.807" LastActivityDate="2015-03-13T15:48:47.807" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7606" PostTypeId="2" ParentId="7597" CreationDate="2015-03-13T00:31:21.937" Score="6" Body="&lt;p&gt;The more the students can reinvent theorems themselves, the better. This is the philosophy of math circles, and it's also how R.L. Moore ran his classes (&lt;a href=&quot;http://legacyrlmoore.org/reference/mahavier1.html&quot;&gt;http://legacyrlmoore.org/reference/mahavier1.html&lt;/a&gt;). He had a record number of PhD's come out of his classes I believe. (Though his legacy is problematic due to his racism.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can work somewhere in between lecture and discovery, guiding students. I tell my (lower level) students that we're going to build the quadratic formula together. And I get them to do most of the work. They feel pretty proud of that.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-03-13T00:31:21.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7607" PostTypeId="2" ParentId="7594" CreationDate="2015-03-13T05:07:38.007" Score="9" Body="&lt;p&gt;I did this with my students a while ago. First I got them to construct their own slide rules using Briggs estimation technique. Then a slightly more accurate table translating between base 10 logs, decibels, base 2 logs, and musical notes (&lt;span class=&quot;math-container&quot;&gt;$semitone^{12}=2$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;After that we studied Briggs' methods to improve the estimates and create accurate logs.&lt;/p&gt;&#xA;&lt;p&gt;My source material for Napier is &lt;a href=&quot;http://profmarino.it/Nepero/napier1619construction.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://profmarino.it/Nepero/napier1619construction.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;My source material for Briggs is a translation of Chapter 7 of his Arithmetica Logrithmica &lt;a href=&quot;http://www.17centurymaths.com/contents/albriggs.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.17centurymaths.com/contents/albriggs.html&lt;/a&gt;  I have since found a more easily understood document explaining his techniques &lt;a href=&quot;https://hal.inria.fr/inria-00543939/document&quot; rel=&quot;noreferrer&quot;&gt;https://hal.inria.fr/inria-00543939/document&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h1&gt;Napier&lt;/h1&gt;&#xA;&lt;p&gt;If I remember correctly, Napier's method was really weird. He started with a big number and basically multiplied by something like 0.999999 to get lower numbers. He did an insane number of calculations at up to 32 bit precision, so he only got parts of his table finished.&lt;/p&gt;&#xA;&lt;p&gt;Something to note, logs are about &lt;strong&gt;ratios&lt;/strong&gt;, not numbers, so the base number of a log table is arbitrary. We set the base of our ratio to be 1 to make them trivial to calculate, so &lt;span class=&quot;math-container&quot;&gt;$\log 1 =0$&lt;/span&gt; ... Napier did not :-(&lt;/p&gt;&#xA;&lt;h1&gt;Briggs method for initial estimation&lt;/h1&gt;&#xA;&lt;p&gt;Henry Briggs decided that there had to be a better way and used a number of techniques around base 10, with the base ratio set to 1. In Napier's system it was annoying to multiply or divide numbers by 10 so you could use the parts of the log tables Napier had completed, but with &lt;span class=&quot;math-container&quot;&gt;$\log 10=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\log 1=0$&lt;/span&gt; it is trivial. This not only cuts down on an extra calculation but reduces the accumulation of rounding errors.&lt;/p&gt;&#xA;&lt;p&gt;The following describes how Briggs did this hundreds of years ago &lt;em&gt;by hand&lt;/em&gt;, with much greater precision than a calculator.&lt;/p&gt;&#xA;&lt;p&gt;As a first estimate, observe that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$2^{10} \approx 10^3$$&lt;/span&gt;&#xA;Therefore we can calculate&#xA;&lt;span class=&quot;math-container&quot;&gt;$$10\log 2 \approx 3\log 10$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\log 2 \approx 0.3$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h1&gt;Briggs method for calculating correction factor&lt;/h1&gt;&#xA;&lt;p&gt;But Briggs needed to improve on that. The correction factor is &lt;span class=&quot;math-container&quot;&gt;$1.024$&lt;/span&gt;. If we find out the log of that, we can add it into the above equation. We will use 3 facts to estimate this number. First, since ancient times there has been an easy algorithm for finding square roots (for my students, it is called &amp;quot;the square root button&amp;quot;), and repeated application takes numbers close to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. Second, the log of a square root is exactly half the log of original number. Third, we have set the log of 1 to be 0. Therefore all we need to do is find a root of 10 close to 1, and then we can easily interpolate to accurately find more logs.&lt;/p&gt;&#xA;&lt;p&gt;To find a root of 10 close to 0, take repeated square roots until you get a small number, say&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \sqrt{\sqrt{\sqrt{\sqrt{{\sqrt{\sqrt{\sqrt{\sqrt{10}}}}}}}}} = 10^{(\frac 1 2 ^8)} = 1.0090$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$\log 1.0090 = (\frac 1 2 )^8$$&lt;/span&gt;&#xA;The interval between &lt;span class=&quot;math-container&quot;&gt;$\log 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\log 1.0090$&lt;/span&gt; is linear up to 4dp. Using smaller numbers rapidly improves accuracy, but this is sufficient for a class project. Briggs used 54 square roots, accurate to 32 dp.!&lt;/p&gt;&#xA;&lt;p&gt;Next we do the same to 1.024 to get a fraction close to &lt;span class=&quot;math-container&quot;&gt;$1.0090$&lt;/span&gt;, we'll use &lt;span class=&quot;math-container&quot;&gt;$\sqrt {1.024} = 1.0119$&lt;/span&gt;. We use extrapolation between&#xA;&lt;span class=&quot;math-container&quot;&gt;$\log 1 = 0 $&lt;/span&gt; and&#xA;&lt;span class=&quot;math-container&quot;&gt;$\log 1.0090 = \frac {1}{256}$&lt;/span&gt;&#xA;The interpolation formula is simplified as &lt;span class=&quot;math-container&quot;&gt;$y_0=0$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ y = y_1 \times \frac {x -x_0}{x_1-x_0}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\log 1.0119= \log 1.0090 \times \frac {0.0119}{0.0090} = 0.00516$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$\log 1.024 = 2 \log 1.0119  = 0.0103$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h1&gt;Putting it all together&lt;/h1&gt;&#xA;&lt;p&gt;We then use this correction factor to get good precision based on our initial estimate&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 2^{10} = 10^3 * 1.024$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 10 \log 2 = 3 + 0.0103$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \log 2 = 0.30103$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 10^{0.30103}=2.00000...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The technique itself magnifies the accuracy. The next to use is &lt;span class=&quot;math-container&quot;&gt;$6^9 = 10^7 * 1.0077696$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-03-13T07:23:52.157" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7608" PostTypeId="1" CreationDate="2015-03-13T13:41:13.793" Score="8" ViewCount="209" Body="&lt;p&gt;I realize it is probably a little late seeing as Pi Day is tomorrow.. but for the future, or late celebrations on Monday, what are some things that you do with your students for Pi Day (any grade level)?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2024-02-16T16:19:09.377" Title="Pi Day Celebration Ideas?" Tags="|undergraduate-education|secondary-education|primary-education|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7609" PostTypeId="2" ParentId="7608" CreationDate="2015-03-13T14:57:22.060" Score="4" Body="&lt;p&gt;Here at the university level, we had a social gathering served with quite a large quantity of pie and a speaker who gave a talk to the department colloquium and to the math club. We ended up doing this yesterday (Thursday prior) as Pi Day is Saturday and it is also spring break. We do this every year and have a great turn out amongst graduate students and faculty. &lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-03-13T14:57:22.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7610" PostTypeId="2" ParentId="7597" CreationDate="2015-03-13T18:19:11.073" Score="4" Body="&lt;p&gt;I appreciate the question and the problems it is trying to solve.  I have the feeling that it is trying to &quot;attack the symptoms rather than the cause&quot;, and that what is really needed is a more foundational approach.  However, my perspective is firmly grounded in my experience as a student in America; the problems being mentioned may actually be different in Finland and other countries, and may not benefit from considering the following ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see one issue as &quot;one course must fit all&quot;, when the intent is to provide a common foundation to be used in diverse environments.  Yes, I want the students to be able to prove some useful results and be able to communicate those results to others.  I also want them to have some approaches for tackling new problems.  I also want them to recognize situations where they can apply methods that are available to get the answers they need.  Finally, I want them to understand what directed questions to ask when they do not see how to work through a situation, be it solving a known problem, tackling an unfamiliar problem, or communicating how to think about concepts related to existing and new problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it should be made clear in the goals of the curriculum (and iterated on a small level for each course, and on a large level for each department) what are some of the expected outcomes ( e.g. you can learn where and how to ask for help, you can learn how to solve known problems, you can learn ideas on how to approach new problems, you can learn how to develop and communicate new technologies to deal with problems yet to be seen ).  In delivering a particular lecture, the priority should be set for which one of the outcomes will be reinforced, and assign as a task outside the lecture something that helps progress to the other outcomes.  If there are too many outcomes expected for a student, divide the class into those groups who wish to master that particular outcome, and have the groups perform and present to one another some results of their outcomes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The major work lies, of course, in determining and properly expressing what outcomes are desired, as well as providing opportunities for achieving that outcome even in the event that only that outcome is achieved (that is, where possible arrange things so that achieving one outcome doesn't depend on achieving another outcome first, but so that achieving any of the outcomes will make it easier to achieve the others).  To me, who am not a professional educator, this is the essence of programs in education: to find continually new ways of arranging and presenting material so that the connections within and without can be used and appreciated and properly communicated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, rather than viewing the situation as a gap between high school and university material, view it as introducing a new level of maturity where the student is taught that several outcomes result from mastery, and they should prioritize the set of outcomes in accordance with their (ever-changing) goals for their education.  Ideally, learning styles and ability can also be factored in, but I don't know enough to talk sensibly about how to accommodate such in the classroom/lecture hall.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Ask Me About System Design&quot; Paseman, 2015.03.13&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-03-13T18:19:11.073" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7611" PostTypeId="2" ParentId="7608" CreationDate="2015-03-14T01:51:58.123" Score="4" Body="&lt;p&gt;The student AMS chapter here in MN had a pizza party, and (through whatever chain of causality) asked me to give a talk... and I chose to &quot;ramp it up&quot; by talking about &quot;failing to prove the Riemann Hypothesis&quot;... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;... which, if one knows about various events, can be a substantive, non-trivial story. And, yes, people've not given up on the issue, although obviously young-ish people oughtn't pin their careers to such a thing...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, as it happens, I had/have some not-completely-trivial things to say about trying-if-only-to-fail in this case, ... and I did the dramatized-for-general-audience version.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Point: math can have broad appeal if we manage to forget about its dubious (awful) use as filter/weeder for other academic disciplines. Yes, indeed, this hatchet-man function does allow large math departments... but, srsly, folks, we should try to offer something better! Fun! Math should be fun! Not just a prank...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Don't get me started...&quot; :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-03-14T01:51:58.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7612" PostTypeId="2" ParentId="5706" CreationDate="2015-03-14T02:37:10.450" Score="5" Body="&lt;p&gt;This problem is not hard with Briggs method. It is similar to the method described by Benjamin Dickman but converges with slightly less square roots. Take repeated square roots until the number is close enough to 1 to enable linear interpolation of the exponential. Do &lt;em&gt;not&lt;/em&gt; take the same number of square roots for the other number. Instead take as many square roots as is required to get close to the first number so as to allow for more accurate interpolation. Then multiply the resulting ratio by power of 2 required to equalise the number of square roots taken.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You know how precise your calculation will be by the behaviour of each subsequent square root. If n decimal digits exactly half during the square root, then you know that it is linear over that number of digits. For instance 11 square-roots of 5 are 1.000786, while 12 square-roots is 1.000393. 393 is exactly half of 786, so I know that it is accurate for 3 sig.fig. after the 1, and 3 sig fig. should get me close enough to 326.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Get your base numbers:&#xA;$$ 5^{1/2^{12}} = 1.000393 $$&#xA;$$ 326^{1/2^{14}} = 1.000353 $$&#xA;Do the interpolation:&#xA;$$ 1.000353 = 5^{1/2^{12} * \frac{0.000353}{0.000393}}$$&#xA;Exponent to get the original number:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ 326 = 5^{(1/2)^{12} * \frac{0.000353}{0.000393} * 2^{14}}$$&#xA;Cancel out the powers of 2:&#xA;$$ 326 = 5^{2^{2} * \frac{0.000353}{0.000393}}$$&#xA;$$ 326 = 5^{4 * {353} / {393}}$$&#xA;Calculate:&#xA;$$ 326 = 5^{3.596}$$&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-03-14T02:49:51.097" LastActivityDate="2015-03-14T02:49:51.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7613" PostTypeId="5" CreationDate="2015-03-14T15:33:37.323" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-03-14T15:33:37.323" LastActivityDate="2015-03-14T15:33:37.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7614" PostTypeId="4" CreationDate="2015-03-14T15:33:37.323" Score="0" Body="For questions related to the teaching of number theory, the part of mathematics concerned with properties of the positive integers." OwnerUserId="39" LastEditorUserId="39" LastEditDate="2015-03-15T22:34:44.977" LastActivityDate="2015-03-15T22:34:44.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7616" PostTypeId="2" ParentId="369" CreationDate="2015-03-15T21:59:46.553" Score="2" Body="&lt;p&gt;I studied with SMSG materials in San Diego 1963-1968 grades 7-11 under an &quot;advanced math&quot; program. Grade 7 was an introduction to math. Grade 8 was half geometry and half algebra. Grade 9 was algebra. Grade 10 was geometry. Grade 11 was trigonometry and pre-calculus. If I had taken grade 12, it would have been calculus and computing science.&lt;/p&gt;&#xA;" OwnerUserId="4876" LastActivityDate="2015-03-15T21:59:46.553" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7617" PostTypeId="2" ParentId="369" CreationDate="2015-03-15T23:35:12.967" Score="6" Body="&lt;p&gt;As SMSG books are public domain, I have found some more PDFs. Thanks mweiss for checking it is the right document. &#xA;&lt;a href=&quot;http://babel.hathitrust.org/cgi/pt?id=mdp.39015017374235;view=1up;seq=7;skin=mobile&quot; rel=&quot;noreferrer&quot;&gt;http://babel.hathitrust.org/cgi/pt?id=mdp.39015017374235;view=1up;seq=7;skin=mobile&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an index of a number of SMSG scans:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=School%20Mathematics%20Study%20Group&quot; rel=&quot;noreferrer&quot;&gt;http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=School%20Mathematics%20Study%20Group&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original ERIC link is freely downloadable for everyone, while I've just noticed that Hathitrust only allows subscribers to do full downloads. However, the Hathitrust scans are done by Google and the quality seems to be better than the ERIC scan. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-03-16T01:00:34.837" LastActivityDate="2015-03-16T01:00:34.837" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7619" PostTypeId="1" CreationDate="2015-03-16T09:36:37.680" Score="5" ViewCount="182" Body="&lt;p&gt;This is a spin-off of the &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7563/wonder-as-motivation&quot;&gt;MESE-Q &quot;Wonder as Motivation&quot;&lt;/a&gt;, where there was some concern expressed about relating to the students. It reminded me of this rap video by a stem-cell researcher on &lt;a href=&quot;https://www.youtube.com/watch?v=pCDxarY8P3g&quot; rel=&quot;nofollow noreferrer&quot;&gt;YouTube &quot;Johnathan Garlick Stem Cell Rap&quot;&lt;/a&gt;, which led me to this thought-provoking video on motivating students &quot;&lt;a href=&quot;http://video.pbs.org/video/2362269145/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Songs for Biology - Students Write Hip Hop to Learn Science&lt;/a&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any similar attempts to get students to write rap / hip hop for math as a way to motivate interest?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Another example: &quot;&lt;a href=&quot;https://www.youtube.com/watch?v=dJHi-MhznZE&quot; rel=&quot;nofollow noreferrer&quot;&gt;Biology Rap - Learning Bio - Biology Cell Song&lt;/a&gt;&quot;)&lt;/p&gt;&#xA;" OwnerUserId="4793" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-03-16T21:04:11.397" Title="Students writing hip hop to learn biology (and math?)" Tags="|student-motivation|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7620" PostTypeId="1" AcceptedAnswerId="7621" CreationDate="2015-03-16T18:40:43.003" Score="4" ViewCount="850" Body="&lt;p&gt;Note - I am not a math teacher; I am seeking an answer from a math teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While helping my son with his 5th grade homework today we had one answer wrong. I'm not sure I agree with the book and wanted to see if there's a rule on this. The assignment is to &lt;em&gt;write the expression for the phrase&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is &lt;strong&gt;2/3 of 30 minus 11&lt;/strong&gt;. You can read that two ways:&#xA; &quot;Two thirds of [pause] thirty minus eleven&quot;&#xA; &quot;Two thirds of thirty [pause] minus eleven&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our answer was &lt;strong&gt;(2/3 * 30) - 11&lt;/strong&gt; and we are aware the parenthesis do not matter here. This also fits with the fact that 30 is divisible by 3. The book's answer is &lt;strong&gt;2/3 * (30 - 11)&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a specific reason for this? Misprint?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll add that the book lists the answer to &lt;strong&gt;1/4 times 8 increased by 11&lt;/strong&gt; as being &lt;strong&gt;1/4 * 8 + 11&lt;/strong&gt; which we have correct. Seems quite similar.&lt;/p&gt;&#xA;" OwnerUserId="4882" LastActivityDate="2015-03-16T18:46:35.423" Title="Parse out 2/3 of 30 minus 11" Tags="|arithmetic-operations|" AnswerCount="1" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7621" PostTypeId="2" ParentId="7620" CreationDate="2015-03-16T18:46:35.423" Score="9" Body="&lt;p&gt;it sounds like the textbook has some poorly written and vague questions.  there is no reason why either of the answers would be &quot;better&quot; for the first example. It is simply dependent on how you interpret the words.  You would be surprised how many textbooks actually include similar vagueness in some questions and it is concerning to say the least.... keep up the good work of teaching your son math! This might actually be a good opening to have a nice discussion with him about the ambiguities that are present and challenge him to come up with a similar question without any of the same ambiguities&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-03-16T18:46:35.423" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7622" PostTypeId="1" AcceptedAnswerId="7626" CreationDate="2015-03-16T19:41:07.720" Score="12" ViewCount="1726" Body="&lt;p&gt;My grade 8 students generally know how to use the distributive property: $a(b+c)=ab+bc$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, we're now learning exponent laws, and one of them is $(ab)^n=a^nb^n$. In order to help my students remember to raise $a$ to the $n$ and not just $b$, I often draw two  little arrows from the exponent to the two bases. Because this looks a bit like The Distributive Property, I sometimes call this action &quot;distributing.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Is it OK for me to call this action &quot;distributing&quot; as long as I try to make it clear that it is different to The Distributive Property?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Is there a better verb I could us instead of &quot;distribute&quot; in the case of the exponent law?&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="867" LastEditDate="2015-03-16T19:46:38.780" LastActivityDate="2015-03-17T18:27:36.873" Title="Can I call $(ab)^n=a^nb^n$ &quot;Distribution&quot;?" Tags="|secondary-education|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7623" PostTypeId="2" ParentId="7622" CreationDate="2015-03-16T20:33:16.197" Score="3" Body="&lt;p&gt;I would be strongly hesitant in calling $(ab)^n = a^n b^n$ distribution for students at that level. You are correct in that distribution is $a(b+c) = ab+ac$, so I would stick with that. Continuing down that road would lead to questions whether $a^{n+m} = a^n a^m$ is distribution or not (it isn't). We want to caution them that distribution is distinct due to errors such as $\sqrt{a+b} \neq \sqrt{a}+\sqrt{b}$ where a more colloquial usage of distribution leads to a more colloquial understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The more proper terms here are associativity and commutativity or just that they are associative and commutative, that is, $(ab)c = a(bc)$ for associativity and $abab = aabb$ is due to commutativity, but that might be too much for 8th graders.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would add, I recall focusing more on this material in 10th grade geometry than in 8th grade, but times may have changed.&lt;/p&gt;&#xA;" OwnerUserId="201" LastEditorUserId="201" LastEditDate="2015-03-16T20:53:32.577" LastActivityDate="2015-03-16T20:53:32.577" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7624" PostTypeId="2" ParentId="4505" CreationDate="2015-03-16T21:22:09.060" Score="2" Body="&lt;p&gt;Here is a somewhat off-the-wall approach (not really, though) that may do well for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can worry about retaining, so that you have instant (or near-instant) recall of what you have learned.  You might consider a slightly different goal of recovering what you have learned, so that you can recover what you have learned when you need it.  How exactly that goes depends on how you want to use it in the future, but I present a scenario for establishing a system for recovery, and then using that system.  Incidentally, this has the benefit of learning by teaching, primarily by teaching yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will take as an example learning about computing with permutations in group theory.  The system I describe looks a lot like annotating an existing text.  However, what you actually do is create a couple of documents (on computer, or in a paper journal, or scrawled in beet juice on the wall as you prefer), one of which is your main set of notes to remind you of particulars of the subject you are learning, and the other document is a glossary or list of references.  For example, you may want to use a.o.l as an abbreviation for &quot;acting on the left&quot;, a notion used when thinking of a permutation as a function on the group of permutations induced by left-multiplication; record the definition in the glossary document, and then use the abbreviation as needed in the main document.  Or you may want to note errata in a text that you like, but in the reference list include the edition of the volume, date of printing, and other things that focus on the text you are using.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So suppose you have your documents after a few months of study, and you print them out and put them on the shelf.  14 months later, a project pops up where computing with permutations would be useful.  You take your binder off the shelf, look at the document of references and glossary (and other high-level information you record, perhaps an index to the larger document), and locate the information you might need in the document of notes about the subject.  Or you find the pointer to the page in the textbook you need, as well as to your page of notes that contains the explanation you needed or insight required for you to understand the page.  This way you recover the important information you wish to do the computation.  You might even use this to explain to another person how to do the computation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, some variation on this is what you would do in researching a problem.  I recommend two documents to separate the kinds of info to be recorded.  If you use a computer format, you can have the machine perform certain jobs such as indexing and cross-referencing, but doing it by hand as you go is not much harder when you have a system in place for recording the information.  If you approach your study as preparing for the future, without the burden of testing your knowledge in exam format, you may find that doing recovery (with testing of the recovery system occasionally to make sure it works for you) is as good if not better a goal than recall.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Ask Me About System Design&quot; Paseman, 2015.03.16&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-03-16T21:22:09.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7625" PostTypeId="2" ParentId="4505" CreationDate="2015-03-16T21:42:56.620" Score="2" Body="&lt;p&gt;There are already several long answers here, I'll try a shot at a short one; and while your question is not strictly a good fit to this SE site, it is easy to interpret as a good fit: I'll try  to answer briefly &lt;strong&gt;&lt;em&gt;&quot;what to say to student to help them recall what maths they are supposed to learn?&quot;&lt;/em&gt;&lt;/strong&gt; (Sorry I don't have the time to edit the question in this direction).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most important thing to learn maths is to try to deepen your understanding. This goes with practicing, but more importantly with practicing in various directions (note that below I am assuming quite a commitment to the task of learning maths):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;get your hands dirty, and do the messy calculation that are needed in the standard exercises of your topic. Then drop your sheet, and the next time you'll have to do them again from the beginning,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;make connections, as was said in another answer: try to gather results by type, by similarity, by differences; connect multiplication to additions, taking powers to multiplications;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;understand why each rule you want to learn must be how it is: to understand a theorem, construct explicit examples, and seek counter-examples that show that the hypotheses are mandatory; try to construct a counter-example of a true statement you want to learn, and see why it &lt;em&gt;has&lt;/em&gt; to fail and how the particular example &lt;em&gt;does&lt;/em&gt; fail,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;to remember a formula, apply it to silly base cases and look how easy it is (e.g. for $a^{x}a^y=a^{x+y}$, expand $a^3\times a^2$ and $a^5$),&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;each time you apply a rule, ask yourself again why is that rule correct, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The bottom line is: don't do the same thing over again, but &lt;strong&gt;turn your math upside down&lt;/strong&gt; until you are familiar with all its aspects. &lt;strong&gt;You are a baker, and math is your bread dough. Bread is learned math, it needs kneading.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2015-03-17T12:24:03.077" LastActivityDate="2015-03-17T12:24:03.077" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7626" PostTypeId="2" ParentId="7622" CreationDate="2015-03-17T00:25:38.363" Score="20" Body="&lt;p&gt;Yes. It is &lt;em&gt;distribution (on one side) of exponentiation over multiplication&lt;/em&gt;, rather than the traditional &lt;em&gt;distribution (on both sides) of multiplication over addition&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For two binary operators $\oplus$ and $\otimes$, $\otimes$ is right-distributive over $\oplus$ iff&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ (a \oplus b) \otimes c = (a \otimes c) \oplus (b \otimes c) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the traditional case, $\oplus$ is addition and $\otimes$ is multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the the case you ask about, $\oplus$ is multiplication and $\otimes$ is exponentiation.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This is the right terminology, and there's no reason to call it anything else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't give the generic definition with funny-looking circles. Just write something like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ (a + b) \cdot c = (a \cdot c) + (b \cdot c) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ (a \cdot b) \hat{\text{ }} c = (a \hat{\text{ }} c) \cdot (b \hat{\text{ }} c) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the pattern is is clear. (The caret will be more familar if they use calculators that have it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may be likely to get students to &lt;em&gt;think&lt;/em&gt; about it, rather than just rearrange symbols -- and higher, littler symbols -- around according to quirky rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd much prefer explaining &lt;em&gt;multiplication distributing over addition&lt;/em&gt; and &lt;em&gt;exponentiation distributing over multiplication&lt;/em&gt;, than trying to make up a term.&lt;/p&gt;&#xA;" OwnerUserId="1005" LastEditorUserId="3468" LastEditDate="2015-03-17T18:27:36.873" LastActivityDate="2015-03-17T18:27:36.873" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7627" PostTypeId="2" ParentId="7622" CreationDate="2015-03-17T08:54:05.830" Score="5" Body="&lt;h1&gt;Operator viewpoint&lt;/h1&gt;&#xA;&lt;p&gt;Yes, you are right. See Paul Draper's answer.&lt;/p&gt;&#xA;&lt;h1&gt;Function viewpoint&lt;/h1&gt;&#xA;&lt;p&gt;Here you should use the term algebraic &lt;em&gt;linearity&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The proportional functions &lt;span class=&quot;math-container&quot;&gt;$f(x)=kx$&lt;/span&gt; are &lt;em&gt;linear&lt;/em&gt; concerning addition: &lt;span class=&quot;math-container&quot;&gt;$$f(a+b)=f(a)+f(b)$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;The power functions &lt;span class=&quot;math-container&quot;&gt;$g(x)=x^n$&lt;/span&gt; are &lt;em&gt;linear&lt;/em&gt; concerning multiplication: &lt;span class=&quot;math-container&quot;&gt;$$g(ab)=g(a)g(b)$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Remark: I used &lt;em&gt;linear&lt;/em&gt; as a translation of the German &lt;em&gt;verträglich&lt;/em&gt;. If there is a better translation, please say so in the comments or edit it straigtaway.&lt;/p&gt;&#xA;&lt;h1&gt;Ring theory viewpoint&lt;/h1&gt;&#xA;&lt;p&gt;Here it's just a generalization of commutativity and associativity. See Chris C's answer.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-03-17T08:54:05.830" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7628" PostTypeId="1" CreationDate="2015-03-17T17:59:58.620" Score="1" ViewCount="86" Body="&lt;p&gt;In which order should these topic be read if one have to understand mathematics topic well?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Differential Equations&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Game Theory&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Graph Theory&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Linear Programming&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probability&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Statistics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Vector Analysis&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Algebraic Geometry&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analytic Geometry&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Differential Geometry&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematical Analysis&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Matrices&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Combinatorics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fractals&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Set Theory&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of you would like to know why you wanna read all these topics, the answer is I wish to be a game programmer and I know it is not mandatory to know these topic but for having a deeper understanding I wish to study them so without further ado I would like to know from where should I start?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank You&lt;/p&gt;&#xA;" OwnerUserId="4885" LastActivityDate="2015-03-18T00:04:36.770" Title="In which order should I read these topics?" Tags="|reference-request|" AnswerCount="1" CommentCount="3" ClosedDate="2015-03-18T05:35:41.333" ContentLicense="CC BY-SA 3.0" />
  <row Id="7630" PostTypeId="2" ParentId="7628" CreationDate="2015-03-18T00:04:36.770" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;I wish to be a game programmer&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then your list does not (in my opinion) encompass the ideal topics.&#xA;I would include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Calculus&lt;/li&gt;&#xA;&lt;li&gt;Analytic Geometry&lt;/li&gt;&#xA;&lt;li&gt;Linear Algebra (emphasizing geometric interpretations)&lt;/li&gt;&#xA;&lt;li&gt;Physics (mechanics)&lt;/li&gt;&#xA;&lt;li&gt;Quaternions&lt;/li&gt;&#xA;&lt;li&gt;Discrete Math; Graph Theory&lt;/li&gt;&#xA;&lt;li&gt;Probability &amp;amp; Statistics&lt;/li&gt;&#xA;&lt;li&gt;Computational Geometry&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-03-18T00:04:36.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7631" PostTypeId="1" AcceptedAnswerId="7634" CreationDate="2015-03-18T09:36:07.420" Score="10" ViewCount="3066" Body="&lt;p&gt;I am looking for some software producing beautiful and exact pictures of functions, plots, etc. I really love those images in the german textbook “Höhere Mathematik 1(2)”, Meyberg, Vachenauer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/lrSHE.jpg&quot; alt=&quot;from Höhere Mathematik 1, Meyberg, Vachenauer&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/B65ur.jpg&quot; alt=&quot;from Höhere Mathematik 1, Meyberg, Vachenauer&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/V3elA.jpg&quot; alt=&quot;from Höhere Mathematik 1, Meyberg, Vachenauer&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And this one (created with POV-Ray) is truly beautiful, too:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Us2aO.gif&quot; alt=&quot;from en.wikipedia.org/wiki/Line_integral&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So—besides from using POV-Ray—what software is being used to produce high quality, clear, mathematical exact and beautiful graph plots etc?&lt;/p&gt;&#xA;" OwnerUserId="4887" LastActivityDate="2019-04-10T11:33:09.413" Title="Looking for software for plotting exact and beautiful (function-)graphs" Tags="|software|" AnswerCount="7" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7632" PostTypeId="2" ParentId="7631" CreationDate="2015-03-18T10:19:35.307" Score="6" Body="&lt;p&gt;One possibility would be to use tikz and pgfplots within $\LaTeX$.&#xA;Within the preamble say&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;\usepackage{pgfplots}&#xA;\usetikzlibrary{arrows,shapes,trees}    %some tikz extensions&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;this also loads tikz. Some more information about pgfplots:  &lt;a href=&quot;http://pgfplots.sourceforge.net/pgfplots.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://pgfplots.sourceforge.net/pgfplots.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2015-03-18T10:19:35.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7633" PostTypeId="2" ParentId="7631" CreationDate="2015-03-18T11:55:12.417" Score="11" Body="&lt;p&gt;Mathematica has impressive graphics capabilities.&#xA;Here is an example taken from&#xA;&lt;a href=&quot;https://mathematica.stackexchange.com/q/14863/194&quot;&gt;an expert's answer at Mathematica StackExchange&lt;/a&gt; (Vitaliy Kaurov):&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/pfUNP.png&quot; alt=&quot;ContourPlot&quot;&gt;&#xA;&lt;hr /&gt;&#xA;For more examples, see the many &lt;a href=&quot;http://demonstrations.wolfram.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wolfram Demonstrations&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:56:09.833" LastActivityDate="2015-03-18T11:55:12.417" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7634" PostTypeId="2" ParentId="7631" CreationDate="2015-03-18T12:05:08.837" Score="10" Body="&lt;p&gt;If you are familiar with Python coding, you can easily use &lt;a href=&quot;https://wiki.python.org/moin/NumericAndScientific/Plotting&quot;&gt;matplotlib&lt;/a&gt; to get quite good outputs. The library's website &lt;a href=&quot;http://matplotlib.org/users/screenshots.html&quot;&gt;has some demos&lt;/a&gt;. This also is a pretty good/easy way to produce animations, if you can generate the frames from an algorithm. I threw together &lt;a href=&quot;https://www.youtube.com/watch?v=iv9XNV_VU5A&quot;&gt;this animation&lt;/a&gt; with about 150 lines of actual code (not counting comments and blank lines). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally I usually use &lt;a href=&quot;http://asymptote.sourceforge.net/&quot;&gt;Asymptote&lt;/a&gt; which has a C-like syntax and readily integrates into LaTeX (for example, I can make sure that the generated images use exactly the same font sets as my main document). You can take a look at &lt;a href=&quot;http://asymptote.sourceforge.net/gallery/index.html&quot;&gt;their gallery&lt;/a&gt; for the type of things you can do. The figure on page 5 of &lt;a href=&quot;http://arxiv.org/pdf/1310.5606.pdf&quot;&gt;this paper&lt;/a&gt; is made using it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the object you want is parametrized, then many of the usual computer algebra packages (Mathematica, Maxima, MatLab) can be used to get pretty decent quality plots; to get them to look &quot;right&quot; you have to play a bit with the configuration parameters. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The software &lt;a href=&quot;http://surf.sourceforge.net/&quot;&gt;surf&lt;/a&gt; was designed for drawing algebraic surfaces; unfortunately it has not been updated for over a decade and I don't actually know for sure whether it still builds on modern machines. &lt;/p&gt;&#xA;" OwnerUserId="125" LastActivityDate="2015-03-18T12:05:08.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7635" PostTypeId="2" ParentId="7631" CreationDate="2015-03-18T12:18:18.850" Score="7" Body="&lt;p&gt;I've used iPython with matplotlib.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has a basic mode which mimics matlab, but it has facilities to make many kinds of graphs/diagrams easier. It does not have full 3D, just  2.5D, which is limiting for nicely shaded 3D graphics, but fine for most graphs and diagrams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be perfect for creating your final animation graph. As it is programatic, it is easy to create a series of graphs with a slowly varying perspective or parameter, saving each as an image, and then use ImageMagick or other program to turn it into a gif.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Its default output is not as smooth and pretty as a modern spreadsheet program, but it is very flexible, and fast to use if you know how. For an important image I would use the matplotlib for the diagrams without text, and then use a basic image editing program to add text and any effects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With numpy for calculating, it is very easy to do basic graphs. If you know some python programming you can do almost anything. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wrote a little physics graphics library for my students to use for their physics projects. The following is a student's work showing his analysis of a star's position data. It is not as pretty as it could be, but it shows what is possible. It is constructed by overlaying 2 line plots, 2 scatter plots, and 3 vector plots. Colours are set for each item of the plot to show the passing of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/igvDM.png&quot; alt=&quot;star orbiting a black hole&quot;&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not as pretty as it could be, but it gives an indication of what is possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a gallery of the type of thing that can be done with matplotlib. &lt;a href=&quot;http://matplotlib.org/gallery.html&quot; rel=&quot;noreferrer&quot;&gt;http://matplotlib.org/gallery.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-03-18T12:18:18.850" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7636" PostTypeId="1" CreationDate="2015-03-18T16:40:55.283" Score="9" ViewCount="231" Body="&lt;p&gt;My secondary geometry class is really amazing and is likely going to finish everything I have for the curriculum with a few weeks left in the school year.  I was thinking to use these last few weeks to look into some interesting Non-Euclidian geometries, such as hyperbolic, affine, elliptic etc.  Unfortunately, I do not know much about these different geometries (my training is in computer science) so...&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What are some good resources to get a good overview of non-euclidian geometries?&lt;/li&gt;&#xA;&lt;li&gt;What are some fun/interesting activities involving non-euclidian geometry?  It will be the end of the school year so i don't to get into anything too rigorous but interesting would be great!&lt;/li&gt;&#xA;&lt;li&gt;Are there any other extensions to either non-euclidian or euclidian geometry that are not in the typical secondary curriculum but which would be fun and interesting for my students?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-04-10T12:58:06.167" Title="Extensions beyond Euclidian Geometry for Secondary students" Tags="|secondary-education|geometry|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7637" PostTypeId="2" ParentId="7636" CreationDate="2015-03-18T17:11:00.403" Score="6" Body="&lt;p&gt;This might give you ideas...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From sarah-marie belcastro's &lt;a href=&quot;http://www.toroidalsnark.net/mkhyp.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Knitted Hyperbolic Surfaces&lt;/a&gt;,&#xA;a knitted &lt;a href=&quot;http://en.wikipedia.org/wiki/Pseudosphere&quot; rel=&quot;nofollow noreferrer&quot;&gt;pseudosphere&lt;/a&gt;,&#xA;a surface of constant negative curvature:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/0fb61.jpg&quot; alt=&quot;Pseudosphere&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-03-18T17:11:00.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7638" PostTypeId="2" ParentId="7631" CreationDate="2015-03-18T18:04:05.167" Score="6" Body="&lt;p&gt;Mathematica has been suggested here, but if you want a free alternative, you could try &lt;a href=&quot;http://www.sagemath.org/&quot;&gt;Sage&lt;/a&gt;. It's an Open Source program that combines lots of existing open source mathematics programs into one interface.&#xA;It is also possible to use Sage online, for example &lt;a href=&quot;http://sagecell.sagemath.org/&quot;&gt;here&lt;/a&gt;, so you don't have to install anything.&#xA;You can also easily integrate these graphs and calculations into your website and allow people to change parameters.&lt;/p&gt;&#xA;" OwnerUserId="4888" LastActivityDate="2015-03-18T18:04:05.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7639" PostTypeId="1" AcceptedAnswerId="7642" CreationDate="2015-03-18T19:04:23.517" Score="5" ViewCount="377" Body="&lt;p&gt;I am a college student. I am glad to meet math as the first subject I am ridiculously interested in. But I am unable to control myself from playing cricket and counter-strike in my leisure time. There is always a desire to study math in leisure time but I can not do so. I don't know why but I only study it when my examinations are close. When I study it, it terrifies me with its enchantment, its beauty. I want to be a mathematician but I am really confused that do I really love maths or not? I think I do but I can't conclude for myself considering above conditions, I stated. I am scared that will I be doing good with maths after my Ph.D or it's just a temporary love. If this will go on, how will I do my research work? It's so criticizing. :(&lt;/p&gt;&#xA;" OwnerUserId="4890" LastEditorUserId="4890" LastEditDate="2015-03-18T21:45:09.680" LastActivityDate="2015-03-18T23:33:45.467" Title="Do I really love Maths?" Tags="|undergraduate-education|student-motivation|" AnswerCount="1" CommentCount="11" ClosedDate="2015-03-19T01:26:17.033" ContentLicense="CC BY-SA 3.0" />
  <row Id="7640" PostTypeId="2" ParentId="7631" CreationDate="2015-03-18T19:22:14.407" Score="10" Body="&lt;p&gt;&lt;strong&gt;Gnuplot&lt;/strong&gt; is widely used in academia.&lt;br&gt;&#xA;It can plot in 2D and 3D, both analytical functions and numerical data.&#xA;While the command line interface might be a little intimidating at first, you can use it as simple as &lt;code&gt;plot sin(x)&lt;/code&gt; or &lt;code&gt;plot 'data.txt'&lt;/code&gt;. All the options to finetune the looks can be accessed through the GUI if you don't remember the name of something and of course you can save all the commands in a textfile to quickly replicate the plot later.&lt;br&gt;&#xA;It also supports LaTeX and can output to an interactive window, bitmap- or vector graphic formats.&lt;br&gt;&#xA;Another very convenient feature is being able to fit arbitrary functions, so you can plot the data and the model with fitted parameters in one go.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://gnuplot.info/screenshots/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; are some the examples of what it can do:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JPoSq.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JPoSq.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/pVYf1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pVYf1.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4889" LastEditorUserId="6053" LastEditDate="2019-04-10T11:33:09.413" LastActivityDate="2019-04-10T11:33:09.413" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7642" PostTypeId="2" ParentId="7639" CreationDate="2015-03-18T23:33:45.467" Score="6" Body="&lt;p&gt;Perhaps this question will be closed, etc., but before such a thing happens, some possibly-useful comments can be interpolated:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, and observation, young people very-reasonably have many other interests... the proverbial &quot;sex, drugs, and rock-and-roll&quot;... and we note that the human race would not still be here if not for some of those interests... so one should not &lt;em&gt;censure&lt;/em&gt; oneself for such interests.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the delicate operational point is that such external stuff can fill up the day, and one does no mathematics. Thus, one might imagine that a socially dysfunctional persona is advantageous/required... for mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be true that if, in your 30+'s and/or 40+'s you'd rather be schmoozing than doing math, then doing math would have been a bad life-choice. But at age 20-something, there's no such indicator. I suppose a risk could be that if one were sooo haute-bourgeois that one had no instinct to think of making-a-living, and didn't have any urge to do anything &quot;non-recreatinoal&quot;, then, yeah, one'd probably not be productive at much of anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summary: being young is ... to say the least ... understandable. Being a bad sort of bourgeois/aristocrat-entitled is debilitating, in terms of doing useful things in scholarly domains, it seems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, if you're &quot;just a kid&quot;, but not a trust-fund baby, interest in things other than math does not doom you at all! :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is still true that there is a tipping-point, probably occurring within a few years, at most, wherein one has to &quot;commit&quot; to something...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-03-18T23:33:45.467" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7643" PostTypeId="2" ParentId="7636" CreationDate="2015-03-19T01:58:19.803" Score="10" Body="&lt;p&gt;You are probably looking for something light and accessible. I would stick with spherical geometry as it is easier for students to conceptualise. Discrete spherical geometry makes it even easier and more hands-on.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Introduce basic graph theory and use Euler's formula to calculate the possible regular polyhedra (Platonic and Archimedean solids) which are topologically equivalent to a sphere. Spherical geometry can be studied from these polyhedra. You will also encounter 2 infinite series. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Draw graphs of some of these polyhedra and create nets to construct physical models. Some graphs are quite difficult, but can be made easier if you start at a vertex for the tetrahedron and a side for the most of the others. You want to finish drawing at a side, not a vertex. Practice drawing them yourself first. Grow the graph symmetrically around the starting figure. Look for forced lines (one line left at a vertex) and keep track of filled vertices (maybe draw a circle around them when they are full). Don't worry about curved lines, especially around the outside of the graph, as curves don't matter topologically. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Notice that the graphs of spherical surfaces become highly distorted. Draw polyhedra on a spheres (oranges/mandarins). Discuss strategies that map makers have used to address the distortion of spheres projected onto flat surfaces. You could peel and flatten an orange skin as a demonstration. Students can do it outside during recess.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Calculate the angle defect of each vertex of a polyhedra. Use Descartes theorem to link to Euler's characteristic.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Look at possible symmetries of a sphere. Discrete symmetries (of Platonic solids) correspond to discrete groups, another interesting and accessible topic. Continuous symmetries correspond to only a few Lie groups. You can make a hierarchy of symmetries based on experimentation on just Platonic solids. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Find underlying similarities between dual pairs of Platonic solids based on possible symmetries. Or transform graphs of any solid to its dual. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You can do group arithmetic on solids. Some group arithmetic is commutative, some is not! The single rotations of a prism compose with nice modular arithmetic, but see what happens when you allow a flip. Your students might even handle some simple problems using the 3 rotations of a cube. Keep in mind the full state of a cube with no reflection is the front face and orientation of the face - up/down/left/right - leading to 24 possible states, so you will not want to create a full Caley table, just do some simple arithmetic. Be very specific on your rotation directions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Decorate solids to demonstrate the discrete spherical symmetries.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Extra for experts: create graphs and maybe nets of (parts of) regular discrete hyperbolic surfaces (full hyperbolic surfaces are infinite). Graphs will collapse into something like a Poincare disc rather than exploding like a sphere. Will take some creativity to make nets that can be stuck together.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I've used Geogebra for creating nets as it is easy to draw joined polygons, but it is not necessary for simply polyhedra. Some regular polygons to trace around maybe? It was often useful to draw graphs of surfaces first before doing the net.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope a few of these ideas are at the right level of your students. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-03-19T01:58:19.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7645" PostTypeId="1" CreationDate="2015-03-19T07:00:04.903" Score="11" ViewCount="2147" Body="&lt;p&gt;I am trying to come up a method to grade a multiple choice question. I prepared a test that consists of some questions and each question has 10 options for which 3 to 5 options are correct. When I grade them, I want to give high points if the student picked correct options, and give penalty for options selected incorrectly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's an example question: &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Which of the below numbers are even? &#xA; 1.) 17 &#xA; 2.) 22&#xA; 3.) 33&#xA; 4.) 42&#xA; 5.) 57&#xA; 6.) 61&#xA; 7.) 49&#xA; 8.) 99&#xA; 9.) 13&#xA;10.) 30&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Actual correct answers are options 2, 4, 10. So, there are 3 options are correct. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's say a student picked 4 options (2, 4, 5, 10). 3 of them are correct, 1 incorrect. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a method in the literature for grading this kind of questions? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; The range of grade should be 0 to 1 (regardless of number of correct answers). &#xA;Also, can we use an exponential function where we award each correct answers and penalize each incorrect answers? &lt;/p&gt;&#xA;" OwnerUserId="4893" LastEditorUserId="201" LastEditDate="2015-03-22T16:05:49.720" LastActivityDate="2015-03-22T16:05:49.720" Title="A Method to Grade a Multiple Statement or True and False Question" Tags="|exams|grading|test-design|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7646" PostTypeId="2" ParentId="7645" CreationDate="2015-03-19T07:53:27.617" Score="11" Body="&lt;p&gt;For multiple choice questions it is much better to ask in a slightly different way, namely&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Are the following numbers even?&#xA;     yes  no&#xA; 1.)  O   O   17 &#xA; 2.)  O   O   22&#xA; 3.)  O   O   33&#xA; 4.)  O   O   42&#xA; 5.)  O   O   57&#xA; 6.)  O   O   61&#xA; 7.)  O   O   49&#xA; 8.)  O   O   99&#xA; 9.)  O   O   13&#xA;10.)  O   O   30&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The advantage is, that you explicitly open the option &quot;I don't know.&quot; (and also the possibility to say &quot;Both&quot;). Consequently, every cross set by the student has a meaning and a missing cross also says something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I read on the &lt;a href=&quot;http://de.wikipedia.org/wiki/Multiple_Choice&quot;&gt;German Wikipedia site on multiple choice questions&lt;/a&gt; that a fair grading that takes guessing into account would be:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Every correct cross gives one point and every wrong cross results in&#xA;  negative points. In this example with only two possibilities for every&#xA;  answer each wrong cross should give one negative point (hence, one&#xA;  correct and one incorrect cross cancel each other).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For questions with more possible choices, a wrong crosses should give less negative points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that there are (at least in Germany) quite some court decisions on how multiple choice test have to be graded (and not all make perfect sense…).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know any English resources on the topic…&lt;/p&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="3398" LastEditDate="2015-03-19T11:39:43.603" LastActivityDate="2015-03-19T11:39:43.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7647" PostTypeId="2" ParentId="7645" CreationDate="2015-03-19T12:45:46.577" Score="4" Body="&lt;p&gt;As Michael E2's comment already says: How you grade depends on what you want to achieve. One goal might be, that pure guessing should have the same mean value as not doing anything at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a yes-no-scheme like in Dirk's answer, that is easy. Every wrong answer gives the same points negativ, which the correct answer gives positive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If only correct answers are to be marked (and wrong answers are indistinguishable from no answer), this needs to be different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose a question has $t$ correct and $f$ wrong answer options and the question has a weight of $w$, which means giving exactly the correct answers yields $w$ points. Then each correct mark gives $\frac{w}{t}$ points and each wrong mark gives $-\frac{w}{f}$ points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: I take your example. The weight is $1$, there are $t=3$ correct answers and $f=7$ wrong answers. So each correct mark gives $\frac{1}{3}$ point, each wrong mark gives $-\frac{1}{7}$ point. The students marks all three correct options and one wrong option, so he should get $3\cdot\frac{1}{3}+1\cdot\left(-\frac{1}{7}\right)=\frac{6}{7}$ points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reasoning: If you don't discriminate the answer options, then you get the following crosstab for the points to be awarded:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;|       |mark|no mark|&#xA;----------------------&#xA;|correct| x  | 0     |&#xA;|wrong  | y  | 0     |&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The &lt;code&gt;0&lt;/code&gt; on the right side are necessary, because you don't award someone points (neither positive nor negative) for not answering. The $y$ is gonna be negative, of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's take a random guesser. Each correct options gives him an expected value of $\frac{x}{2}$. Each wrong option gives him an expected value of $\frac{y}{2}$. Suppose, that the random guesser answers each option independantly (by no way realistic if $t=1$ is not known). Than the expected value of the whole question is $E=t\cdot\frac{x}{2}+f\cdot\frac{y}{2}$. This should be $0$, so you get&#xA;$$tx+fy=0$$&#xA;Furthermore, giving exactly the correct answers earns $tx$ points, which should be the weight $w$:&#xA;$$tx=w$$&#xA;Solving these equations for $x$ and $y$ yields:&#xA;$$x=\frac{w}{t},\quad y=-\frac{w}{f}=-\frac{t}{f}x$$&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="439" LastEditDate="2015-03-20T09:29:34.750" LastActivityDate="2015-03-20T09:29:34.750" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7649" PostTypeId="2" ParentId="7645" CreationDate="2015-03-19T21:37:42.083" Score="2" Body="&lt;p&gt;Some people have given this question (and a lot of similar ones) a great deal of thought and come up with industry standards:  &lt;a href=&quot;http://en.wikipedia.org/wiki/QTI&quot; rel=&quot;nofollow&quot;&gt;QTI&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Yes, there is something in the literature about it.  &lt;/li&gt;&#xA;&lt;li&gt;You can use an exponential function (or just about anything you want).  But I would caution it against it.  Do you want to spend more time explaining the subject matter or more time explaining the grade?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3346" LastActivityDate="2015-03-19T21:37:42.083" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7650" PostTypeId="1" CreationDate="2015-03-19T23:51:48.893" Score="14" ViewCount="354" Body="&lt;p&gt;I've read some articles about it and those papers show us that there are some facts to understand the mathematical competence in problem-posing. Besides that, those investigations show that there are some indicators to assess the competence.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Brown, S. I., &amp;amp; Walter, M. I. (Eds.). (2014). &lt;em&gt;Problem posing: Reflections and applications.&lt;/em&gt; Psychology Press.&lt;/p&gt;&#xA;&lt;p&gt;Malaspina, U., Mallart, A., &amp;amp; Font, V. (2015). Development of teachers’ mathematical and didactic competencies by means of problem posing. &lt;em&gt;Proceedings of the 9th Congress of the European Society for Research in Mathematics Education&lt;/em&gt; (in press). Prague, Czech Republic.&lt;/p&gt;&#xA;&lt;p&gt;Christou, C., Mousoulides, N., Pittalis, M., Pitta-Pantazi, D., &amp;amp; Sriraman, B. (2005). An empirical taxonomy of problem posing processes. &lt;em&gt;ZDM, 37(3)&lt;/em&gt;, 149-158.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also, some researchers agree that mathematical competence in problem-posing is part of another great competence, it's problem-solving.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Is there a framework to study the mathematical competence in problem-posing for prospective teachers?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm interested in figure out how we can recognize and propose specific indicators to assess this competence, if it could be studied.&lt;/p&gt;&#xA;&lt;p&gt;I hope your help to collect some articles or thesis which could answer this question.&lt;/p&gt;&#xA;" OwnerUserId="1789" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-01-18T20:36:31.237" Title="Is there a framework to study the mathematical competence in problem-posing?" Tags="|undergraduate-education|education-research|problem-design|problem-solving|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7651" PostTypeId="2" ParentId="7650" CreationDate="2015-03-20T00:34:06.737" Score="8" Body="&lt;p&gt;Nice question! Let me add one reference to your list:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Silver, Edward A. &quot;On mathematical problem posing.&quot; &lt;em&gt;For the learning of mathematics&lt;/em&gt; (1994): &lt;strong&gt;14&lt;/strong&gt;(1) 19-28.&#xA;  (&lt;a href=&quot;http://flm-journal.org/Articles/2A5D152778141F58C1966ED8673C15.pdf&quot;&gt;PDF download link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Silver cites Hadamard's famous book, &lt;em&gt;The Psychology of Invention in the Mathematical Field,&lt;/em&gt; as recognizing that isolating key research questions&#xA;is a sign of exceptional mathematical talent, but then focuses most&#xA;of his article on fostering problem-posing as part of inquiry-based learning&#xA;in elementary education. &#xA;In his conclusion, he says,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;The process of personalizing and humanizing mathematics for students&#xA;  through the use of open-ended problem-posing tasks invites&#xA;  them to express their lived experiences, and this can have important consequences&#xA;  for teachers and for researchers.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-03-20T00:34:06.737" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7652" PostTypeId="2" ParentId="7645" CreationDate="2015-03-20T03:00:43.927" Score="2" Body="&lt;p&gt;Really great question with a great set of answers.  I would like to offer another form of multiple choice assessment that I use frequently that I am really pleased with.  I know that this will not directly answer your question as per your description, but given the title I think this answer can be useful for future MESE users who come across this question. Ultimately, I believe that you are trying to create a more rigorous multiple-choice assessment that actually digs into a students thinking and is not just a &quot;black-box of guessing&quot;, hence the penalty for incorrect answers and having multiple correct answers per question (a really great strategy that I am definitely going to use ;), but also remaining as fair as possible.  What I do is three-fold...&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Each question has five options for answers.  Four are potential solutions to the problem and the last is described in 2.  For each question there is one correct answer and one &quot;almost correct&quot; answer.  If the student chooses the correct answer they get full credit and if they choose the AC answer they get half credit.  The AC answer will typically involve making one standard error such as missing a sign, confusing operations, or a logic error you have noticed frequently happening with the class.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Provide an option E for each questions that allows a student to write in an answer that is different from those provided.  Generally, I only do this on one question per assessment and I tell the students ahead of time that there is exactly one question that requires a write-in.  If a student gets the question correct, it proves a high level of understanding with the topic of the question.  If they get it wrong, they still have a chance to prove some level of understanding, i.e. they pick the AC answer or their explanation is logical (see 3.).  Sometimes, usually with accelerated classes I will have multiple write-ins required and not inform the students of how many there are.  It is really up to you and what you on which way to go, depending on your class.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Each question has a space underneath to write one to two sentences justifying the students selected answer.  This not only shows a students real understanding but it also gives really good insight into which questions were guessed on.  It also gives students a chance to express math in words, which I believe to be a very powerful thing to learn and something shirked by many educators.  As far as grading this part, I usually will do it with a pass-fail mentality, i.e. if it is &quot;more than half logical&quot; the student will receive full credit, but anything that is illogical is circled and a note is written either explaining the fallacy if it is brief or &quot;ok but see me&quot; if it is a more serious misunderstanding.  I have had these explanations factor into the grade for the question in two ways in the past: worth some amount of bonus points (usually a fractional amount) or worth some percentage of the questions total value.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A few notes on this strategy..&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It takes a lot of time and energy to both create and grade, I will usually only do a fully multiple choice assessment like this for cumulative assessments such as midterms and finals&lt;/li&gt;&#xA;&lt;li&gt;It is very different from most multiple choice assessments your students have likely ever taken.  Because of that, ease them in putting a few questions like this on other assessments to get them used to the style of MC question&lt;/li&gt;&#xA;&lt;li&gt;Because of the nature of these questions, they should be deep and meaningful.  As a result of this, these questions will usually take students two to three times as long to complete so keep that in mind when creating these questions&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-03-20T03:00:43.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7653" PostTypeId="1" AcceptedAnswerId="7657" CreationDate="2015-03-20T15:00:13.060" Score="9" ViewCount="1707" Body="&lt;p&gt;As part of my job search, I've come into contact with universities that are beginning to offer new majors at their university such as applied mathematics or computer science. A frequent interview question is, &quot;What kind of classes would you require for such a major?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I usually tell them that I would look at other universities that have established programs and try to use them as a template, together with my own experience. However, I was wondering:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there any published research, news articles, or other reliable sources which indicate which mathematical skills are most important/most used in typical computer programmer positions in industry?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;While I appreciate colloquial advice, I am specifically looking for published information.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2015-03-20T15:35:28.243" LastActivityDate="2015-03-20T22:04:03.590" Title="Which math classes should be included in an undergraduate computer science program?" Tags="|undergraduate-education|curriculum|computer-science|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7654" PostTypeId="1" AcceptedAnswerId="7665" CreationDate="2015-03-20T16:18:02.887" Score="20" ViewCount="7238" Body="&lt;p&gt;In my experience, algebra is one of the biggest stumbling blocks to calculus students. For instance, sign errors are common, and exponent laws (and log laws!) cause a lot of headaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many courses have algebra pre-tests to let students know if they are &quot;prepared&quot; or not; however, these pre-tests often cover material that has nothing to do with the course itself.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there any published information on what algebra skills and techniques are most important for student success in calculus?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(I saw &lt;a href=&quot;https://matheducators.stackexchange.com/questions/387/&quot;&gt;this question&lt;/a&gt;, but it covered a different aspect of the issue).&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2023-01-26T22:41:01.857" Title="What basic algebra skills and techniques are most important for calculus students to know?" Tags="|undergraduate-education|calculus|algebra|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7655" PostTypeId="1" AcceptedAnswerId="7659" CreationDate="2015-03-20T16:22:58.507" Score="9" ViewCount="864" Body="&lt;p&gt;One of the best algebra-teaching games I've seen is the &quot;Four 4's&quot; game, where students have to take 4 fours and construct every number from 1-100 using only those fours and algebraic operations:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;44/44=1, 4/4+4/4=2, 4-(4/4)^4=3, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a great way to get students to explore the power of algebra. It is open ended, requires creativity, and gets students to use a variety of algebraic operations.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Q:Is there anything similar for integration techniques?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;By similar, I mean an easily-stated, open-ended task that requires students to be creative and gets them to use a variety of integration techniques.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've thought of integration bee-type contests, but those aren't open ended.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2015-03-21T06:16:03.747" Title="Open-ended tasks for teaching students about integration techniques" Tags="|calculus|examples|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7656" PostTypeId="1" AcceptedAnswerId="7658" CreationDate="2015-03-20T16:27:06.703" Score="19" ViewCount="642" Body="&lt;p&gt;It is well known that Allen Hatcher has created a free textbook for algebraic topology that is high enough quality to be used in a large number of graduate courses in the united states, saving students a large amount of money.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus is one of the most-taught courses in the united states; as such, many high-quality, expensive textbooks have been developed to teach calculus, often costing over $100.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are the major obstacles to a group of mathematics educators collaborating on an open source calculus textbook with a high enough quality to replace a commercial calculus textbook?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastActivityDate="2015-03-29T17:50:42.953" Title="What are the major obstacles to crowdsourcing a competitive, free calculus text?" Tags="|calculus|textbooks|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="7657" PostTypeId="2" ParentId="7653" CreationDate="2015-03-20T17:11:05.117" Score="4" Body="&lt;p&gt;These two references might help. Neither is aimed at industry needs specifically, but&#xA;undergraduate computer science curricula are partially driven&#xA;by industry job market needs. The first is aimed specifically at liberal arts colleges:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(1) &quot;A 2007 Model Curriculum for a Liberal Arts Degree&#xA;  in Computer Science,&quot;&#xA;  &lt;em&gt;Liberal Arts Computer Science Consortium&lt;/em&gt;, 2007.&#xA;  (&lt;a href=&quot;http://cs.wellesley.edu/~pmetaxas/LACS2007report.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is an excerpt; more detail in the report:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/imR0M.png&quot; alt=&quot;ACM07&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;The second is more recent, and not specific to liberal arts colleges&#xA;(but agrees with the above):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(2) &quot;Computer Science&#xA;  Curricula 2013:&#xA;  Curriculum Guidelines for&#xA;  Undergraduate Degree Programs&#xA;  in Computer Science.&quot; 2013. (&lt;a href=&quot;http://www.acm.org/education/CS2013-final-report.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is a snippet (p.467) of topics coverage, all of which are required&#xA;at &quot;Tier-1&quot; universities:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/rmJNa.png&quot; alt=&quot;ACMTopics&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-03-20T22:04:03.590" LastActivityDate="2015-03-20T22:04:03.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7658" PostTypeId="2" ParentId="7656" CreationDate="2015-03-20T19:22:52.163" Score="10" Body="&lt;p&gt;To directly address your question, I would venture the major obstruction to universities developing their own textbooks is the publishing industry lobbying. You can read more about this &lt;a href=&quot;http://www.educause.edu/ero/article/ten-years-later-why-open-educational-resources-have-not-noticeably-affected-higher-education-and-why-we-should-ca&quot;&gt;in Educause Review&lt;/a&gt;. They state the major roadblocks are: discoverability, quality control, bridging the last mile, and acquisition. But here is an alternative that some universities are piloting to help with those.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Developed by Michigan State University, &lt;a href=&quot;http://www.lon-capa.org/&quot;&gt;LON-CAPA or Learning Online Network with Computer-Assisted Personalized Approach&lt;/a&gt; is a course management and assessment much like Blackboard and WebAssign systems. The system is open-sourced under GPL, so has no direct licensing fees to use. At my institution, such a system is used to avoid the massive fees on the students that WebAssign currently uses as support. I just want to mention, that as a content manager, it is not meant as a complete replacement of Blackboard systems. In fact, it can work in sync with Blackboard, synchronizing grades between the systems. &lt;a href=&quot;http://www.lon-capa.org/papers/flyer_for_web.pdf&quot;&gt;Here is an flyer about the LON-CAPA system.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the content on such a system, it seems that some publishers do provide resources for the LON-CAPA system, but in more of a bulk pricing instead of price per student. A resource that I found is &lt;a href=&quot;http://www.courseweaver.com/homeworksystem/&quot;&gt;Course Weaver&lt;/a&gt;. Their &lt;a href=&quot;http://www.courseweaver.com/homeworksystem/?page=pricing&quot;&gt;pricing system&lt;/a&gt; seems much more reasonable than the large fees with ones like WebAssign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are other resources for free open source textbook as well. Check out &lt;a href=&quot;http://www.collegeopentextbooks.org/opentextbookcontent/open-textbooks-by-subject/math&quot;&gt;College Open Textbooks&lt;/a&gt;. One feature of LON-CAPA is a cross-institutional repository of content for courses, so if you pick a textbook, you can use problems from other similar courses. You can check out the &lt;a href=&quot;http://www.lon-capa.org/stepbystep.html&quot;&gt;Step-by-Step guide&lt;/a&gt; for an idea for how it works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, it seems that there is some research into the effectiveness of such a system: &lt;a href=&quot;http://www.researchgate.net/publication/251690370_Experiences_using_the_open-source_learning_content_management_and_assessment_system_LON-CAPA_in_introductory_physics_courses&quot;&gt;Experiences using the open-source learning content management and assessment system LON-CAPA in introductory physics courses - ResearchGate. [accessed Mar 20, 2015]&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I must say, this is just what I've looked into. I have not interacted with this system, so I don't have any personal perspective on it. It seems I could be using it as early as this summer or fall, I can update then with a perspective if I use it. I'm also not in a position to comment on the future course content that we use just yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another additional alternative is the &lt;a href=&quot;http://ocw.mit.edu/&quot;&gt;Open Courseware from MIT&lt;/a&gt;. I have not looked too much into what they provide, but it seems comparable.&lt;/p&gt;&#xA;" OwnerUserId="201" LastEditorUserId="201" LastEditDate="2015-03-20T19:52:04.027" LastActivityDate="2015-03-20T19:52:04.027" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7659" PostTypeId="2" ParentId="7655" CreationDate="2015-03-20T20:17:17.363" Score="11" Body="&lt;p&gt;One activity is to ask them to do one problem in multiple ways. For example, &quot;Find $\int \sin(x) \cos(x) dx$ in five ways&quot;. (Some possible ways are to use a trig identity, use integration by parts integrating cos first, use integration by parts integrating sin first, use substitution with $u=\sin(x)$, use substitution with $u=\cos(x)$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A favourite of mine is to ask to do $\int x^2 dx$ in multiple ways. My colleague calls this &quot;stupid integration&quot; (because you're not doing the most effective method).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One interesting upshot of this activity is that the results often look different, but you can show that they are actually the same, or at least only different by a constant. This is an important lesson that they often don't learn in ordinary &quot;closed&quot; integration questions.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-03-21T06:16:03.747" LastActivityDate="2015-03-21T06:16:03.747" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7660" PostTypeId="1" CreationDate="2015-03-20T20:59:10.690" Score="15" ViewCount="823" Body="&lt;p&gt;I am scheduled to teach an upper-division undergraduate class on &quot;Geometry&quot; and I get to choose more or less what that means.  Common choices seem to be non-Euclidean, hyperbolic, projective, or &lt;em&gt;Erlangen&lt;/em&gt; geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would have liked to do differential geometry, since it seems to me to be a more central part of a mathematics education, but right now I think that that would probably be too difficult.  In particular, linear algebra is not a prerequisite for the class.  Also several of the students are future high-school teachers and would probably appreciate something that feels more relevant to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I am wondering whether there is any textbook on non-Euclidean/hyperbolic/projective geometry (any or all) which at least leads up towards a &quot;modern&quot; differential-geometric point of view, e.g. the importance of a local metric and perhaps the intrinsic curvature.  All the textbooks I have looked at so far seem to emphasize either the Euclid/Hilbert axiomatic style or the Klein &lt;em&gt;Erlangen&lt;/em&gt;/transformation style of geometry, which are all well and good for the traditional (mostly homogeneous) examples; but I would like to at least expose the students to some of the ideas that lead later on to things like manifolds (and which are necessary for applications like relativity theory).&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="376" LastEditDate="2016-12-30T18:03:11.917" LastActivityDate="2022-05-04T23:40:53.390" Title="Geometry with a view towards differential geometry textbook" Tags="|undergraduate-education|course-design|textbooks|geometry|differential-geometry|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7661" PostTypeId="1" CreationDate="2015-03-21T00:41:29.940" Score="10" ViewCount="685" Body="&lt;p&gt;I have recently been engaging more and more with my creative side by drawing, writing, and (trying to) playing piano.  I have come to see mathematics as much more of a creative field of study than I have previously and it got me thinking...Is there any research into creativity or creative aspects of mathematics?  Specifically, dealing with the teaching and learning of math, as we are teachers after all :) &lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2016-02-18T05:18:10.957" Title="Creativity in mathematics" Tags="|reference-request|education-research|creativity|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7662" PostTypeId="2" ParentId="7661" CreationDate="2015-03-21T01:45:35.750" Score="7" Body="&lt;p&gt;I believe you would like Jacques Hadamard, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691029318&quot;&gt;The Mathematician's Mind&lt;/a&gt;, which has changed the name since I've read it (it was called &lt;em&gt;The Psychology of Invention in the Mathematical Field&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the synopsis:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence of unconscious mental processes in mathematical invention and other forms of creativity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2015-03-21T01:45:35.750" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7663" PostTypeId="1" AcceptedAnswerId="7664" CreationDate="2015-03-21T06:51:37.970" Score="8" ViewCount="1815" Body="&lt;p&gt;Suppose you had to teach averages to a teenager. For arguments sake let the question be; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Find the mean of $2, 3, 4, 7$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the simple answer is $\frac{2+3+4+7}{4}=4$ but in my experience the student only reinforces their addition and division skills and is left bereft of any conceptual understanding of mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Context and comparison helps;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The length of male mice tails are $2cm,3cm,4cm,7cm$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mean average female tail is $3.9cm$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compare the mice tails.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know ways in which I can instill conceptual understanding of averages. I'm always left with a feeling students just calculate and don't really understand why they are doing it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps worse is range. Practically all students will tell be it's highest take lowest but almost never attach any sensible meaning to it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do you teach averages and not the simple calculations?&lt;/p&gt;&#xA;" OwnerUserId="4668" LastEditorUserId="201" LastEditDate="2015-03-22T16:49:16.347" LastActivityDate="2015-03-23T00:05:05.320" Title="How to Teach Averages (Arithmetic Mean) to a Teenager?" Tags="|mathematical-pedagogy|secondary-education|" AnswerCount="3" CommentCount="10" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7664" PostTypeId="2" ParentId="7663" CreationDate="2015-03-21T15:04:24.617" Score="7" Body="&lt;p&gt;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/0FVq1.jpg&quot; alt=&quot;BeamMean&quot;&gt;&#xA;&lt;hr /&gt;&#xA;$\mbox{}\mbox{}\mbox{}\mbox{}\mbox{}$&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-03-21T15:04:24.617" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7665" PostTypeId="2" ParentId="7654" CreationDate="2015-03-21T16:58:33.477" Score="20" Body="&lt;p&gt;Sam Shah has this post on his blog: &lt;a href=&quot;http://samjshah.com/2012/06/01/algebra-bootcamp-in-calculus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://samjshah.com/2012/06/01/algebra-bootcamp-in-calculus/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I loved his list, but wanted to rewrite it a bit for myself. (Also, Sam finds it more effective to review the algebra ahead of time, while I think it's more effective to review once we see the need in our exploration of calculus.)&lt;/p&gt;&#xA;&lt;p&gt;I teach my calculus course in an order that I think will help students learn. I have four units:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Unit 1 includes history, graphing functions, slopes of tangent lines&#xA;by approximation, algebraically finding the derivative using the&#xA;limit (which we do not carefully define yet), seeing the similarities&#xA;between velocity, rate of change, and slope, average versus&#xA;instantaneous velocity, derivative from a graph, (estimated)&#xA;derivative from a table of values.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Unit 2 includes derivative properties needed for polynomials,&#xA;graphing, limits, product and quotient rules, and trig derivatives.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Unit 3 includes chain rule, derivatives of exponential functions,&#xA;implicit differentiation, derivatives of inverse functions (ln x and&#xA;inverse trig functions), and related rates.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Unit 4 includes integration (finding area under the curve),&#xA;anti-derivatives, fundamental theorem of calculus, and substitution&#xA;method. If there is time we include volumes of rotation (which I&#xA;think is a perfect ending for the course).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra Skills needed for Unit 1&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Determine the equation of a line given two points, or a point and a slope, or a graph of a line,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Find the average rate of change over an interval given a function or its graph,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Clearly express what is happening to an object given a position versus time graph,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Evaluate f(x+h) for any given function f(x),&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Rationalize the numerator (to find the derivative of the square root&#xA;function) ,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simplify complex fractions (to find the derivative of the 1/x&#xA;function).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra with Calculus Concepts&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Approximate, using two points close to each other, the instantaneous&#xA;rate of change at a point, given a function or its graph,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain clearly why the procedure you used gives an approximation of&#xA;the true instantaneous rate of change,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sketch a velocity versus time graph given a position versus time&#xA;graph,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Construct the formal definition of the derivative by modifying the&#xA;definition of slope,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Apply the formal definition of the derivative to simple polynomials&#xA;and to simple square root functions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra Skills needed for Unit 2&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Multiply out the expression (x+h)n (necessary to understand the proof&#xA;for the derivative of y=xn),&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Identify the holes, vertical asymptotes, x- and y-intercepts,&#xA;horizontal or slant asymptote, and domain of any rational function,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sketch the basic shape of a rational function,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Identify an equation for a rational function given a sketch of the&#xA;function,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain clearly what a hole and an asymptote are,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Construct the equation of a piecewise function given its graph,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sketch the graph of a piecewise function given its equation,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Work with inequalities,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give both triangle and circle definitions of sin x, cos x, and tan x,&#xA;and explain how they’re related,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Evaluate sin x, cos x, and tan x at all multiples of  π/6 and  π/4,&#xA;without a calculator,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Understand trigonometry identities, including and sin(x+h)=sin x cos&#xA;h + sin h cos x,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Graph y = sin x and y = cos x.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra with Calculus Concepts&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Graph a polynomial or rational function, showing its maximums,&#xA;minimums, and inflection points,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Follow complicated logic (in the definition of limit).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra Skills needed for Unit 3&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Understand composition of functions,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use logarithm properties to “break apart” a single logarithmic&#xA;expression into simple logarithms,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Understand properties of exponents,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Be able to graph exponential and logarithmic functions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra with Calculus Concepts&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Think in terms of composition of functions to determine outer and&#xA;inner functions, in order to use the chain rule.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra Skills needed for Unit 4&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Algebra&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Work with summations.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-01-25T22:41:31.210" LastActivityDate="2023-01-25T22:41:31.210" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7666" PostTypeId="1" CreationDate="2015-03-21T17:53:38.543" Score="2" ViewCount="236" Body="&lt;p&gt;What software and/or hardware is typically used for creating Khan Academy style math videos?&lt;/p&gt;&#xA;" OwnerUserId="2049" LastEditorUserId="3237" LastEditDate="2015-03-21T18:29:57.577" LastActivityDate="2015-03-21T18:29:57.577" Title="Software / hardware for creating Khan Academy style videos?" Tags="|software|" AnswerCount="0" CommentCount="4" ClosedDate="2015-03-21T18:41:15.153" ContentLicense="CC BY-SA 3.0" />
  <row Id="7667" PostTypeId="2" ParentId="7656" CreationDate="2015-03-21T18:32:44.753" Score="7" Body="&lt;p&gt;I am the author of a number of Free texts, for instance one on Linear Algebra that has some reasonable exposure.  There are a fair number of such works and more are appearing.  Check out the the &lt;a href=&quot;http://www.openmathbook.org/&quot;&gt;Open Mathbook blog&lt;/a&gt; for some start at a community for authors.  I see that the current entry references an open Calc text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such efforts would benefit from more support from the mathematics community and from administrations as scholarship, but there are some of us lucky enough to have convinced our colleagues that this is a useful contribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However your question is a bit unclear.  If you really mean crowdsourcing, that would, in my opinion, be much harder.  A classroom textbook whose style changes from section to section, or even from paragraph to paragraph, would be hard to use, it would seem to me.&lt;/p&gt;&#xA;" OwnerUserId="213" LastActivityDate="2015-03-21T18:32:44.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7668" PostTypeId="2" ParentId="7661" CreationDate="2015-03-21T21:18:48.947" Score="11" Body="&lt;p&gt;Yes, there is a growing literature at the nexus of mathematics education and creativity. The main name to know is Bharath Sriraman (&lt;strong&gt;&lt;a href=&quot;https://scholar.google.com/citations?user=YnPG9YoAAAAJ&amp;amp;hl=en&amp;amp;oi=ao&quot; rel=&quot;nofollow noreferrer&quot;&gt;google scholar&lt;/a&gt;&lt;/strong&gt;) though the classic pieces to read for &lt;em&gt;mathematical creativity&lt;/em&gt; are the book by Hadamard (see the answer of &lt;strong&gt;Lucas Virgili&lt;/strong&gt;) as well as the chapter &lt;em&gt;Mathematical Creation&lt;/em&gt; by Poincare that served to inspire the aforementioned work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may also wish to check out several earlier questions (to which I provided some sort of answer) that mention mathematical creativity; these include the following trio:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/q/4093/262&quot;&gt;&lt;strong&gt;MESE 4093: Moving From Rote Learning To Creative Thinking&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/q/2236/262&quot;&gt;&lt;strong&gt;MESE 2236: Jason Padgett's “Struck by Genius”&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/q/630/262&quot;&gt;&lt;strong&gt;MESE 630: How can mathematics educators encourage innovation and creativity?&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The above links and their many (many) references should give you a good start, and may lead you to refine your question into something more specific. I should also note that there is a literature on creativity and teaching that is not necessarily connected to mathematics, but which is in many cases (potentially) applicable! To this end, I think R. Keith Sawyer (&lt;a href=&quot;http://scholar.google.com/citations?user=JE3leMsAAAAJ&amp;amp;hl=en&amp;amp;oi=sra&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;google scholar&lt;/strong&gt;&lt;/a&gt;) is producing some interesting work; see, e.g., his book &lt;a href=&quot;http://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=ym9LrnH1PfsC&amp;amp;oi=fnd&amp;amp;pg=PR1#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Structure and Improvisation in Creative Teaching&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-02-18T05:18:10.957" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7669" PostTypeId="2" ParentId="7645" CreationDate="2015-03-22T03:44:13.500" Score="1" Body="&lt;p&gt;In &lt;a href=&quot;https://educationrealist.wordpress.com/2014/01/17/multiple-answer-math-tests/&quot; rel=&quot;nofollow&quot;&gt;https://educationrealist.wordpress.com/2014/01/17/multiple-answer-math-tests/&lt;/a&gt; a California high school math teacher found that the best way to grade multiple answer tests is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Each answer is an individual True/False question. That works really well. So if you have a six-option question, you can get 6/6, 5/6, 4/6 etc. Then you assign point totals for each option.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2015-03-22T03:44:13.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7670" PostTypeId="2" ParentId="7661" CreationDate="2015-03-22T16:19:17.460" Score="2" Body="&lt;p&gt;Decided on elaborating my comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might find the article &lt;a href=&quot;http://journal.frontiersin.org/article/10.3389/fnhum.2014.00068/full&quot; rel=&quot;nofollow&quot;&gt;The experience of mathematical beauty and its neural correlates (Front. Hum. Neurosci., 13 February 2014 | doi: 10.3389/fnhum.2014.00068)&lt;/a&gt; interesting. They found that some mathematics (such as $e^{i \pi} +1 = 0$) activates similar parts of the brain for mathematicians that viewing a work of art or other sources does. Thus there is an art to creating proofs, equations, and statements in aesthetically pleasing ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The BBC has a nice article on it: &lt;a href=&quot;http://www.bbc.com/news/science-environment-26151062&quot; rel=&quot;nofollow&quot;&gt;Mathematics: Why the brain sees maths as beauty.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-03-22T16:19:17.460" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7671" PostTypeId="2" ParentId="7663" CreationDate="2015-03-22T16:19:48.337" Score="8" Body="&lt;p&gt;Another interpretation that can help make the formula meaningful is &quot;equal redistribution&quot;.  Say a group of 4 friends have (respectively) \$2, \$3, \$4, and \$7.  How much money would they each have, if they all had the same amount of money as each other without changing the total?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or:  Tom makes hand-made wooden spoons, which he sells at a local farmer's market.  At the most recent market he sold 2 spoons in the first hour, 3 in the second hour, 4 in the third hour and 7 in the fourth (and last) hour.  If you imagine all of those spoons being sold at a steady rate over the course of the market, how many would be sold per hour?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In both cases you find the answer by summing the number of &quot;things&quot; (dollars or spoons) and dividing by the number of &quot;places&quot; (people or hours), and end up calculating the mean.  The idea that underlies both of these examples is &quot;uniformization&quot; -- you take some distribution of stuff that has an irregular pattern to it, and describe what it would be like if things were equally distributed.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-03-22T16:19:48.337" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7672" PostTypeId="2" ParentId="7660" CreationDate="2015-03-22T17:36:48.000" Score="2" Body="&lt;p&gt;I cannot answer your question. But since no one else answered, let me&#xA;reply with another question: Would it be feasible to base an undergraduate&#xA;geometry class on Thurston's great book?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;William P. Thurston. &lt;em&gt;Three-Dimensional Geometry and Topology:&#xA;  Volume 1&lt;/em&gt;.&#xA;  Edited by Silvio Levy.&#xA;  (&lt;a href=&quot;http://press.princeton.edu/titles/6086.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Princeton Univ Press link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/vZ1UU.gif&quot; alt=&quot;ThurstonCover&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-03-22T18:52:48.750" LastActivityDate="2015-03-22T18:52:48.750" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7673" PostTypeId="1" CreationDate="2015-03-22T22:16:06.120" Score="10" ViewCount="579" Body="&lt;p&gt;I am currently helping my advisees (undergraduate math majors) plan their summer.   As the obvious option, they've already applied to lots of REUs (some have gotten in and others not heard a positive response yet), but of course REUs can be very competitive to get and it would be nice to have other options I could suggest.  Any recommendations?  As background on the students, they are serious and high-achieving sophomores with an interest in pursuing graduate studies.  Ours is an applied math program at a smaller polytechnic, and the students have some research experience.  The program wouldn't necessarily have to be research-based, though - just something mathematical that they could learn and advance from and further their careers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for any and all thoughts/suggestions!&lt;/p&gt;&#xA;" OwnerUserId="4906" LastActivityDate="2015-10-20T20:17:56.860" Title="Summer opportunities for math undergrads as an alternative to REU" Tags="|undergraduate-research|advising|summer|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7674" PostTypeId="2" ParentId="7656" CreationDate="2015-03-22T22:25:40.190" Score="7" Body="&lt;p&gt;I use &lt;a href=&quot;http://scholarworks.gvsu.edu/books/10/&quot; rel=&quot;nofollow&quot;&gt;Active Calculus&lt;/a&gt; by Matt Boelkins et al, and like it a lot.  It is very good and totally open source (the author provided me the LaTeX files, which are licensed under Creative Commons, so that I could modify at will!)  I also use and like (though they are less traditionally structured calculus texts, but in a way this is one of their strong points) the &lt;a href=&quot;http://www.math.smith.edu/Local/cicintro/&quot; rel=&quot;nofollow&quot;&gt;Calculus in Context&lt;/a&gt; series.  For online homework I use &lt;a href=&quot;http://webwork.maa.org/&quot; rel=&quot;nofollow&quot;&gt;Webwork&lt;/a&gt; from MAA, which is free and open source - you can even write your own problems and contribute them to a national database.  More generally, I tend to use &lt;a href=&quot;http://aimath.org/textbooks/&quot; rel=&quot;nofollow&quot;&gt;AIM's Open Textbook Initiative&lt;/a&gt; to find open source texts.  I've had very good experiences with the texts I have found.&lt;/p&gt;&#xA;" OwnerUserId="4906" LastEditorUserId="4906" LastEditDate="2015-03-23T00:46:06.053" LastActivityDate="2015-03-23T00:46:06.053" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7675" PostTypeId="2" ParentId="7438" CreationDate="2015-03-22T23:44:22.583" Score="1" Body="&lt;p&gt;I may suggest Discrete Mathematics by Rosen, this book has a lot of computation problems, and the sections on proof are not really that necessary to continue with the advanced material on the book.&lt;/p&gt;&#xA;" OwnerUserId="3253" LastActivityDate="2015-03-22T23:44:22.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7676" PostTypeId="2" ParentId="7663" CreationDate="2015-03-23T00:05:05.320" Score="3" Body="&lt;p&gt;Once upon a time, I designed some versions of the following task to bring about the algebraic nature of 100 number grid. But, @mweiss answer gave me the idea of the possibility of using it for what you've asked for. Thus, it is clear that I've never used it to teach the &quot;average&quot;, but I thought it is worth sharing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose three children are standing on different numbers on a 100 number grid (if you are a school teacher you can indeed make such a suitable-for-standing number grid in schoolyard). The task for them is to move to and meet at the same cell of the number grid without changing the initial sum of the three numbers they were standing on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is only work when the average is a natural number. But, it is nice to see that to solve the problem both the smallest number and the biggest number should &quot;move&quot; towards the number that later on will be called mean. As a result, your students may also experience some aspects of &quot;range&quot; as well.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. Of course, you can easily use this idea with paper number gird and counters.      &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-03-23T00:05:05.320" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7677" PostTypeId="2" ParentId="7661" CreationDate="2015-03-23T02:25:05.543" Score="3" Body="&lt;p&gt;If you are looking for information about relation math and creativity, you should take in mind something about intuition. This topic is very important in the field of mathematics education. For it, you may read Fischbein's book.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fischbein, E. (1987). &lt;em&gt;Intuition in science and mathematics: An educational approach&lt;/em&gt; (Vol. 5). Springer Science &amp;amp; Business Media.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1789" LastActivityDate="2015-03-23T02:25:05.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7678" PostTypeId="1" CreationDate="2015-03-23T14:34:13.710" Score="5" ViewCount="137" Body="&lt;p&gt;Where I live, there is this competition called the &quot;Känguru Wettbewerb&quot; (german), in English, that would be &quot;Kangaroo Competition&quot;. This is a mathematics competition where the goal is to solve problems that presuppose different areas of mathematics (like geometry, algebra etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, if there are any sources for problems like that, for training. I know the that problems of recent years are hosted on their website, but I'd like to know if there are any other sources that provide similar problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.kaenguru.at/zum-herunterladen/aufgaben-der-vorjahre/2014/&quot;&gt;http://www.kaenguru.at/zum-herunterladen/aufgaben-der-vorjahre/2014/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On this site, there are the problems of 2014. Scroll down until you see the English version. The preferred level would be &quot;Junior&quot; in their system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any sources for problems like this? If yes, please post links.&lt;/p&gt;&#xA;" OwnerUserId="4908" LastActivityDate="2015-03-23T14:53:05.683" Title="Logic/Mathematics problems for training" Tags="|logic|math-puzzle|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7679" PostTypeId="2" ParentId="7678" CreationDate="2015-03-23T14:53:05.683" Score="2" Body="&lt;p&gt;I'm not sure this will help but you could try UK maths challenges&#xA;&lt;a href=&quot;http://www.ukmt.org.uk/individual-competitions/junior-challenge/&quot; rel=&quot;nofollow&quot;&gt;http://www.ukmt.org.uk/individual-competitions/junior-challenge/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-03-23T14:53:05.683" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7680" PostTypeId="1" CreationDate="2015-03-23T18:35:52.653" Score="5" ViewCount="135" Body="&lt;p&gt;This was originally part of a question on averages but it was suggested to me that it should be a question in itself. If people think the question is too similar it can be deleted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll link the questions when I've worked how to do it. &#xA;How do you teach range to students so that they get a good understanding of range as a measure of spread.  To be clear computing highest take lowest is not really the issue as almost all can do that.  I'm trying to combat statements such as  &quot;the highest range is the best&quot; in inappropriate situations.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-03-25T01:31:01.607" Title="Teaching range as a measure of spread" Tags="|mathematical-pedagogy|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7681" PostTypeId="1" CreationDate="2015-03-23T19:03:32.773" Score="8" ViewCount="237" Body="&lt;p&gt;For example, consider trying to prove that: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $A$ is a set and $F \subset P(A)$, then the relation $R := \{(a, b) \in A \times A $ such that for every&#xA;  $X \subset A - \{a, b\}$, if $X \cup \{a\} \in F$ then $X \cup \{b\} \in F\}$ is transitive.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Keep in mind that I don't want you to prove this for me. I only give it as an example, in order for you to understand the question better. This theorem is not obvious to me: I can't envision it, so to say. Am I approaching mathematics in a wrong way, by trying to envision every theorem and build an intuition on it, or is this not possible when the theorems themselves are too complex?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, if building an intuition is often not possible, how do mathematicians come up with new conjectures, ideas, etc?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; What I am saying is that I can prove some theorems by simply applying previously learned techniques, algorithmically, although I don't understand completely these theorems before I attempt to prove them. The question is: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it important to try to first understand the theorem completely, and then attempt to prove it ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4911" LastEditorUserId="262" LastEditDate="2015-03-23T19:34:03.417" LastActivityDate="2015-03-24T16:19:25.657" Title="How important is building up intuition for a theorem before trying to prove it?" Tags="|concept-motivation|proofs|intuition|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7682" PostTypeId="2" ParentId="7681" CreationDate="2015-03-23T19:50:44.827" Score="2" Body="&lt;p&gt;(Include von Neumann quote about not understanding things in mathematics so much as getting used to them.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the goal is for your understanding, then you need to break it down as you can to aid in your understanding.  I would ask why you need the relation to be transitive.  Are you going to form an equivalence relation eventually?  Do you have some idea of how you are going to use R and its presumed transitivity?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have to build up your intuition about some abstract constructs, usually by relating them to things you know. (&quot;It behaves like this other thing, see?&quot;)   Without any intuition, you have little or no chance of conveying the meaning, importance, or even correctness of a theorem to someone else, including your later self.  Yes, it would help if you understood R.  At the beginning though, you just work with it to develop your intuition and understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to try a meta-experiment, observe and record your thoughts and states of understanding along the way as you work with R.  You may find and identify certain blocks to understanding and ways to overcome them, which can be quite useful to yourself in other contexts and to the person after you who has to work with R.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Depends On Your Particular Goal&quot; Paseman, 2015.03.23&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-03-23T19:50:44.827" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7683" PostTypeId="2" ParentId="7680" CreationDate="2015-03-23T23:48:34.360" Score="1" Body="&lt;p&gt;This may be too complicated, but you could find more straightforward plots of&#xA;data with vertical ranges that make the same point.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://darkmatterdarkenergy.files.wordpress.com/2013/04/positronspectrum.jpg?w=360&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://darkmatterdarkenergy.com/2013/04/07/ams-positron-excess-due-to-dark-matter-or-not/&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMS Positron Excess&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-03-10T09:42:42.103" LastActivityDate="2015-03-23T23:48:34.360" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7684" PostTypeId="2" ParentId="7681" CreationDate="2015-03-24T09:19:23.957" Score="2" Body="&lt;p&gt;I think there are (at least) two different skills involved in what you are asking. One is being able to work with formal definitions, to be able to prove things from the theorems you know by working with the basic structure, even without understanding particularly what is going on. The second is having the understanding and intuition to picture how the proof should go. I think both these skills are useful, and different bits of maths are suited to these different approaches. There are some bits of algebra when it's best to just multiply out the terms and cancel down. I've often had student who are overwhelmed by a problem because they try to see the whole proof, when working one step at a time will lead you through. When dealing with functions of functions of functions... I find it helps to work through the formalities. On the other hand, there are many places where having no intuition will make it very hard to make any progress, especially in the sorts of proofs where you need to pluck the right object essentially out of thin air, before proving it has the properties you want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, I think doing either one has a value, but never doing the other would be more of a problem.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit after reading your comment on the other answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being able to come up with the result in the first place is a third skill, somewhat unrelated to the other two. If you are at the stage of trying to understand enough to prove the given result, you are not yet at the point where you should be worrying about coming up with the result (in my opinion). That will come later. In particular, this specific example of a transitive relation is not something you are likely to stumble upon out of nowhere. It is likely whoever came up with it had some other reason to be thinking about that particular relation, or else was actively looking for more unusual transitive relations to use for teaching.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="4746" LastEditDate="2015-03-24T15:29:36.837" LastActivityDate="2015-03-24T15:29:36.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7685" PostTypeId="2" ParentId="7681" CreationDate="2015-03-24T16:19:25.657" Score="1" Body="&lt;p&gt;It would seem to me that you're not yet expected to come up with such things yourself. You just have to prove a bunch of statements provided in a sort of &quot;black-box&quot;, that is, just accepting that there is a motivation for this that might not be clear to you at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a few ideas behind why an instructor would ask you to prove such a thing:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;To reinforce a proof strategy or concept that is critical.&lt;/em&gt; It could be that such a statement is of no particular use, but the methodology used in the proof is key to later success in the course. In fact, I know that using equivalency classes is critical in many later courses (such as Topology or Algebra).&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;To prove a lemma/theorem that will be used in a larger theorem later.&lt;/em&gt; This happens more often than not in the exercises in books. Early in the text, it will list a bunch of exercises for you to prove without much motivation other than to get a foothold in the subject. Then later, the fact will appear in a few proofs in the text or even is needed in later exercises.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thinking back to when I was first learning to prove things, there were many such statements that didn't make sense as to why they were formulated a certain way. It was not until much later that I saw that many were technical theorems/lemmas that were used in a more larger concept. It is not usually until much later, after proving and going over many statements, that you will have the intuition built to write such statements yourself. In fact, I find they're usually from asking questions about relations between the topics you are learning or a common proof strategy.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-03-24T16:19:25.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7686" PostTypeId="1" CreationDate="2015-03-24T20:48:00.240" Score="7" ViewCount="200" Body="&lt;p&gt;In &lt;a href=&quot;http://www.reddit.com/r/math/comments/2yc5am/topological_compactness_is_an_induction_principle/&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; post on Reddit, a user proposes an alternate definition of compactness, as an &quot;induction principle&quot;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A topological space $X$ is compact iff given a statement $P$ whose&#xA;  truth or falsity is defined on open subsets of $X$, we have $P(X)$ as&#xA;  soon as &lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;p&gt;For all $x \in X$ there exists open $U \ni x$ such that $P(U)$.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;$P(U_1), \dotsc, P(U_n) \implies P(U_1 \cup \dotsb \cup U_n)$.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To me this seems like such a drastically better definition of compactness: more intuitive, more mathematical, more to the point. The statement in terms of open covers is useful in proofs, but it should be secondary to this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any thoughts/opinions? Why don't we teach compactness this way, except that that's the way it's always been done?&lt;/p&gt;&#xA;" OwnerUserId="1415" LastEditorUserId="201" LastEditDate="2015-03-26T20:04:18.277" LastActivityDate="2015-03-26T20:04:18.277" Title="Teaching an alternate definition for a compactness via the induction principle" Tags="|topology|" AnswerCount="0" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="7687" PostTypeId="1" AcceptedAnswerId="7690" CreationDate="2015-03-24T22:10:49.693" Score="17" ViewCount="2322" Body="&lt;p&gt;It is a good thing for students to self-check their work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The results of some calculations can be checked easily. For example, the solutions to an equation can be substituted back into the original to see if they really work, and integrals can be differentiated to see if it comes to the original function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In having trouble coming up with effective ways to self-check derivatives. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not looking for general advice like &quot;look for minus-sign errors&quot;, but methods that will indicate that a mistake has occurred somewhere. Note also I am not talking about differentiation by first principles using limits, but using the rules of differentiation (powers, product rule etc). I would also prefer at least some things that don't require the use of a graphics calculator/computer algebra system.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-03-24T22:51:25.870" LastActivityDate="2015-03-25T16:38:50.497" Title="How can students self-check derivatives?" Tags="|calculus|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7688" PostTypeId="2" ParentId="7687" CreationDate="2015-03-24T22:22:07.843" Score="14" Body="&lt;p&gt;One way (which I strongly encourage students to use), is to plot the original function and the derivative, and see if it makes sense in terms of increasing/decreasing behavior and concavity.  This also adds a good conceptual reinforcement to what would otherwise be a mechanical problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very simple example would be $f(x)=\frac{1}{3}x^3−x$. I calculate $f′(x)=x^2$. I plot $f(x)$ and $f′(x)$. Hmm. Looks like $f(x)$ is decreasing on $(−1,1)$ while $f′(x)&amp;gt;0$ there. Must have made a mistake. Go back and check my derivative calculation. Ah, should actually have $f′(x)=x^2−1$. Plot both. Looks good now! &lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2015-03-25T16:38:50.497" LastActivityDate="2015-03-25T16:38:50.497" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7689" PostTypeId="2" ParentId="7687" CreationDate="2015-03-24T22:46:36.940" Score="4" Body="&lt;p&gt;I teach students to graph the numerical derivative of the original function as well as what they think the derivative is. If the student is correct, the two graphs should be identical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We use the TI-Nspire CX, which puts each graph in a different color and makes it pretty easy to see if the graphs overlap. I also spend time on some problem functions, where the numeric derivative is not graphed as expected or where it is easily misunderstood. Students soon learn that the graphic calculator is a great aid but is not to be completely trusted.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2015-03-24T22:46:36.940" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7690" PostTypeId="2" ParentId="7687" CreationDate="2015-03-25T00:19:18.390" Score="23" Body="&lt;p&gt;As a student currently taking calculus, something I do is select a couple of arbitrary points, and compare my derivative at those points with a small secant:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f'(a) \approx \frac{f(a+h) - f(a)}{h}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To avoid large error, I just make sure I'm not checking near any points where the derivative might be changing very quickly (eg. near 0 of $\frac{1}{x}$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually, with an $h$ of 0.01, the error is extremely small, and after checking at a couple of points, I'm confident I've got it right. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's nice is that this strategy can be extended to higher order derivatives too!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f''(a) \approx \frac{f(a+x) - 2f(a) + f(a-x)}{x^2}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, upon being asked to differentiate:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f(x)=\frac{x^2 + 2}{3x - 5}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's say I come up with the correct answer of:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f'(x)=\frac{3x^2 - 10x-6}{(3x-5)^2}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I want to check my answer.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, upon analyzing $f(x)$, I see that there's a vertical asymptote at $x=5/3$. I'll want to be staying away from that region, so I decide to check the value of the derivative at $x=0$ and $x=10$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Plugging in these values into the derivative function above I came up with:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f'(0) = \frac{-6}{25} = -0.24$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f'(10) = \frac{194}{625} = 0.3104$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I will approximate my derivative using:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f'(x) \approx \frac{f(x+0.01) - f(x)}{0.01}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And plugging in the values, I get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f'(0) \approx -0.2434607645875$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f'(10) \approx 0.3104274870155$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, the approximations are extremely close to what my derivative function is outputting, so I'm confident I differentiated correctly.&lt;/p&gt;&#xA;" OwnerUserId="4916" LastEditorUserId="4916" LastEditDate="2015-03-25T02:49:17.287" LastActivityDate="2015-03-25T02:49:17.287" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7691" PostTypeId="1" AcceptedAnswerId="7712" CreationDate="2015-03-25T00:58:07.783" Score="7" ViewCount="215" Body="&lt;p&gt;I'm writing a comparative study between mathematical teaching in my country (which sucks by the way) with others. However, I don't want to make it about today, globalization gives us a good notion of that; what I want is to understand how (mathematical) education was before WWI in England, Germany and France, which are the countries that produced the most genious mind in terms philosophy of mathematics. The problem is: I don't know any book who treats the subject. I already searched on archive.org and found nothing! Any help would be appreciated.&#xA;Have a nice day.&lt;/p&gt;&#xA;" OwnerUserId="4917" LastEditorUserId="2074" LastEditDate="2015-03-25T11:40:32.693" LastActivityDate="2016-08-17T12:23:48.483" Title="References on British, German and French educational system" Tags="|undergraduate-education|reference-request|graduate-education|education-research|reference-books|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7692" PostTypeId="2" ParentId="7680" CreationDate="2015-03-25T01:31:01.607" Score="1" Body="&lt;p&gt;In our high-school curriculum, range and especially inter-quartile range (IQR) have replaced standard deviation as the main measures of spread. It has the following benefits:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Student are able to quickly ascertain them manually, without tons of mindless calculations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is obvious what they mean, unlike the &quot;black box&quot; of standard deviation calculations, so students can intuitively discuss samples in real language based on these statistics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simple box-plots based on these range measures simultaneously display range, median, IQR, and skew. The shape of the distribution is not hidden as is the case in mean/SD summaries. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The median, range, and IQR are often more useful for quick, informal population inferences especially in the presence of outliers and non-normal distributions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Traditional confidence intervals are still easy to calculate. For example: median +/- 1.5 * IQR is equivalent to 2 standard deviations either side of mean as the basis of a 95% confidence interval on a normal distribution. On non-normal distributions mean and SD are problematic anyway. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For more advanced calculations, such as difference between medians/means, the curriculum specifies that students use bootstrapping/Montecarlo techniques to give accurate confidence intervals. Montecarlo is easy for students to understand and intuitive influences are more easily drawn. Though the CLT says the distributions &lt;em&gt;tend towards&lt;/em&gt; normality, in reality it is often far from normal and automated Montecarlo techniques can be more accurate. The resulting Montecarlo distributions can again be analysed using the median/IQR distribution summaries. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As a bonus, Montecarlo techniques can easily be extended to create confidence intervals for arbitrarily complicated hypotheses, such as regression analysis which is impossible with mean/SD. Advanced Montecarlo is not part of the high-school curriculum.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This methodology is not intended to replace traditional techniques to be learned at university, but they give a good grounding in statistical concepts of centre, range, comparing samples, and making inferences. The concepts learned are not tied to traditional mean/SD and are easily extended to modern statistical techniques. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2015-03-25T01:31:01.607" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7693" PostTypeId="2" ParentId="7687" CreationDate="2015-03-25T01:38:01.253" Score="3" Body="&lt;p&gt;Well, how about they integrate the answer to see if they get back where they started modulo a constant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More seriously though, if the derivative is correct then we can see some things in terms of even and odd functions. The derivative of an odd function is even. The derivative of an even function is odd. Beyond that, I suppose some CAS help is the inevitable, but irrelevant, answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other ideas: sometimes there is more than one way to differentiate. If they can compare product to quotient, or logarithmic to other methods that would also provide fairly disparate paths to an answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in summary, integrals are easy as differentiation to check. But, differentiation is only as safe as the student's algebra is strong. In practice, I see weakness differentiation as a common problem in the students who barely make it through the calculus sequence. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-03-25T01:38:01.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7694" PostTypeId="1" CreationDate="2015-03-25T02:42:15.783" Score="18" ViewCount="1861" Body="&lt;p&gt;A common identity in integration is $\int_0^af(x)dx=\int_0^af(a-x)dx$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The steps to prove it (algebraically, ignoring the geometric method) are as follows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $u=a-x$ so $dx=-du$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\int_0^af(a-x)dx=-\int_a^0f(u)du=\int_0^af(u)du=\int_0^af(x)dx$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My students often aren't convinced as to how $\int_0^a f(u)du$ can just become $\int_0^a f(x)dx$ because of the relationship $u=a-x$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My way of convincing them of this is to ask them to evaluate for me&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\int_0^a x^2 dx$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\int_0^a y^2 dy$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\int_0^a z^2 dz$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes this works but there have been times when they've said &quot;Oh but that's because there's no relationship between $x$, $y$ and $z$!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have you ever had this problem? Do you perhaps have any suggestions in this scenario?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I'm not too sure what tags to attach to it, some help here would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="3383" LastEditorUserId="201" LastEditDate="2015-03-25T18:14:50.947" LastActivityDate="2015-04-01T22:33:05.417" Title="How to convince students of the integral identity $\int_0^af(x)dx=\int_0^af(a-x)dx$?" Tags="|mathematical-pedagogy|calculus|integration|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7695" PostTypeId="2" ParentId="7694" CreationDate="2015-03-25T02:54:05.500" Score="12" Body="&lt;p&gt;I would ask students to consider what the graphs of $f(x)$ and $f(a-x)$ look like (and how they are related to each other) on the interval $[0,a]$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw a sketch of some arbitrary-looking function on $[0,a]$ and label it $f(x)$.  Now we want to figure out what $g(x)=f(a-x)$ looks like.  By direct computation, $g(0)=f(a)$ and $g(a)=f(0)$.  Once you have those, it is not hard to realize that the graph of $g(x)$ is just a mirror-reversal (left-to-right) of the graph of $f(x)$.  Draw that sketch and label it $f(a-x)$.  (At this point it probably wouldn't hurt to remind students of their prior experience with function transformations.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once they see that the graph of $f(a-x)$ on $[0,a]$ is simply a mirror-reversal of the graph of $f(x)$ on the same interval, it should be obvious why the areas under the two curves are equal.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-03-25T02:54:05.500" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7696" PostTypeId="2" ParentId="7687" CreationDate="2015-03-25T03:15:56.997" Score="8" Body="&lt;p&gt;They can always check that the value of their computed derivative at a few (preferably easy) points makes sense. This is exactly how I initially attempt to convince Calc I students that &#xA;$$\frac{d}{dx}e^x \neq x \, e^{x-1}.$$&#xA;Such students typically know that the graph of $y=e^x$ has positive slope at all points but the value of the purported derivative is zero at the origin.&lt;/p&gt;&#xA;" OwnerUserId="4912" LastActivityDate="2015-03-25T03:15:56.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7697" PostTypeId="1" CreationDate="2015-03-25T04:36:32.933" Score="8" ViewCount="204" Body="&lt;p&gt;Has anyone written a set of &lt;a href=&quot;https://en.wikipedia.org/wiki/Moore_method&quot; rel=&quot;nofollow&quot;&gt;Moore method&lt;/a&gt; notes for synthetic &lt;em&gt;projective&lt;/em&gt; geometry?  It seems like it would be well-suited, but I haven't been able to find any such thing on the Internet.&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="38" LastEditDate="2015-12-21T20:09:06.640" LastActivityDate="2015-12-21T20:09:06.640" Title="Moore method projective geometry" Tags="|geometry|lecture-notes|inquiry-based-learning|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7698" PostTypeId="2" ParentId="7694" CreationDate="2015-03-25T07:59:43.257" Score="5" Body="&lt;p&gt;How about this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider $\int f(a-x) dx$.&#xA;Let $F(x)$ be an antiderivative of $f(x)$.&#xA;Write $u=a-x$, which means $du = -dx$ and $-du = dx$.&#xA;Then &#xA;$$&#xA;\begin{align}&#xA;\int f(a-x) dx &amp;amp;= \int -f(u)du \\&#xA;&amp;amp;= -F(u)+C \\&#xA;&amp;amp;= -F(a-x) + C.&#xA;\end{align}&#xA;$$&#xA;Thus $-F(a-x)$ is an antiderivative of $f(a-x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &#xA;$$&#xA;\begin{align}&#xA;\int_0^a f(a-x)dx &amp;amp;= \big[-F(a-x)\big]_0^a \\&#xA;&amp;amp;= -F(a-a) - - F(a-0) \\&#xA;&amp;amp;= F(a)-F(0).&#xA;\end{align}&#xA;$$&#xA;Also &#xA;$$&#xA;\begin{align} \int_0^a f(x)dx &amp;amp;= \big[F(x)\big]_0^a \\&#xA;&amp;amp;= F(a) - F(0).&#xA;\end{align}&#xA;$$&#xA;So the two definite integrals are the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the point you're trying to make is that it doesn't matter what the variable is, even if you've mentioned it already, because it will eventually be replaced with proper numbers later. And therein lies the issue, I think: many students don't behave as if they realise that the definite integral is a &lt;em&gt;number&lt;/em&gt;. To say that $\int_a^b f(x) dx = \int_a^b f(u)du$ is to say that these two &lt;em&gt;numbers&lt;/em&gt; are equal. Actually performing the calculation in some way to have the numbers themselves may go some way to convincing them of that.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-03-25T14:23:45.173" LastActivityDate="2015-03-25T14:23:45.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7699" PostTypeId="2" ParentId="7694" CreationDate="2015-03-25T10:25:41.543" Score="13" Body="&lt;p&gt;This answer attacks not only this problem, but a lot of all others. At the expense of going against the grain, however.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A much deeper issue is this permanent grip on the concept of 'function of a variable' which &lt;a href=&quot;https://math.stackexchange.com/questions/1091873/is-a-function-changed-into-another-function-by-a-change-of-variables#comment2225539_1091873&quot;&gt;I've described in the past&lt;/a&gt; as pedagogical cancer. There's no such thing as the function $f(x)$ (unless $f$ is a functional, but that's a different matter), the function is $f$. If one wants to talk about $f$ evaluated at generic value, one takes a variable name, say $x$ (or $u$) and one writes $f(x)$ (or $f(u)$) and it doesn't matter what variable one chooses. Then some properties are proven and $\forall x(\text{something})$ is concluded, which is equivalent to $\forall u(\text{something})$, the variables are bound.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To avoid the present problem, the one linked above and many others, rid the students of this inadequate concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The symbol $\small{\displaystyle \int\limits_a^b f(x)\,\mathrm dx}$ is just another way of writing $\small{\displaystyle \int\limits_a^b f}$ and this is true for $\small{\displaystyle \int\limits_a^b f(u)\,\mathrm du, \int\limits_a^b f(z)\,\mathrm dz, \int\limits_a^b f(\text{cancer})\,\mathrm d\text{cancer}}$, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if certain requirements are met, integration by substitution says that $$\displaystyle \int\limits_a^b f=\int \limits_{u^{-1}(a)}^{u^{-1}(b)}\left(f\circ u\right) u'.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where's your $x$ now? In doing this it's not so much that the students answer the question themselves, it's more that the question will not even arise.&lt;/p&gt;&#xA;" OwnerUserId="287" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-04-01T22:33:05.417" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="7700" PostTypeId="2" ParentId="7687" CreationDate="2015-03-25T12:54:02.453" Score="10" Body="&lt;p&gt;One method, which is probably not always applicable, would be to calculate the derivative using multiple different methods. If all methods give the same answer, you've probably got it right. If not, then you've definitely got at least one of them wrong, so keep trying till all of methods give the same answer. Example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{d}{dx}\frac 1 {x^2}=\ ?$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chain rule 1:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= \frac{d}{dx}\frac 1 {(x^2)}=\left(\frac{d}{dx}x^2\right)\left(\frac{-1}{(x^2)^2}\right)=\frac{-2x}{x^4}=\frac{-2}{x^3}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chain rule 2:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= \frac{d}{dx}\left(\frac 1 x\right)^2=\left(\frac{d}{dx}\frac 1 x\right)\left(2\frac 1 x\right)=\frac{-2}{x^3}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quotient rule:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= \frac{d}{dx}\frac 1 {x^2}=\frac{0\cdot x^2-2x\cdot1}{(x^2)^2}=\frac{-2}{x^3}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Product rule:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= \frac{d}{dx} \frac 1 x \frac 1 x=\frac 1 x\frac{-1}{x^2}+\frac{-1}{x^2}\frac 1 x=\frac{-2}{x^3}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I still do this sometimes with complicated functions, if I get odd results and suspect I might have dropped a minus sign or something.&lt;/p&gt;&#xA;" OwnerUserId="115" LastActivityDate="2015-03-25T12:54:02.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7701" PostTypeId="2" ParentId="925" CreationDate="2015-03-25T15:42:06.547" Score="3" Body="&lt;p&gt;Here are a couple more suggestions to drill in students' heads that sequence means progression and series means sum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider a sequence of &quot;episodes&quot;, each of which is a set of events. Episodes can be added by taking their union. The events that have happened so far as of an episode is the sum of it and all previous episodes. We call this sum a &lt;strong&gt;TV series.&lt;/strong&gt; If your students aren't ready to accept a union as a type of sum, you could start with an &lt;a href=&quot;https://allthetropes.miraheze.org/wiki/Anyone_Can_Die&quot; rel=&quot;nofollow&quot;&gt;&quot;anyone can die&quot; series&lt;/a&gt;, define an episode's value as the number of deaths in that episode, which gives the series as the total body count.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider a sequence of &quot;ballgames&quot;, each of which has a value of 1 for a win or 0 for a loss. Then the &lt;strong&gt;World Series&lt;/strong&gt; is a sum that approaches 4 for one of the teams.&lt;/p&gt;&#xA;" OwnerUserId="4925" LastEditorUserId="4925" LastEditDate="2015-11-03T23:24:53.977" LastActivityDate="2015-11-03T23:24:53.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7702" PostTypeId="1" AcceptedAnswerId="7704" CreationDate="2015-03-25T02:29:28.947" Score="9" ViewCount="686" Body="&lt;p&gt;Weierstrass' method for handling limits with the epsilon and delta symbols is very useful for rigorous analysis of math but it is terrible in terms of any intuitive approach to limits. There are are other ways to teach limits. For instance with infinitesimals yet I don't know if that has been simplified. Why does the 'epsilon- delta' method seem to dominate many calculus courses? Are there Calculus Courses that don't use this 'epsilon-delta' method?&lt;/p&gt;&#xA;" OwnerUserId="4632" OwnerDisplayName="201044" LastEditorUserId="1385" LastEditDate="2017-04-21T05:40:38.800" LastActivityDate="2017-04-21T11:02:43.677" Title="The 'epsilon-delta' method for teaching limits" Tags="|calculus|infinitesimals|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7703" PostTypeId="2" ParentId="7694" CreationDate="2015-03-25T18:04:39.010" Score="27" Body="&lt;h2&gt;Problem of sloppy notation&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The notation is sloppy. Your students are justifiably confused. We've just gotten used to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to untangle this, we need the notion of &lt;a href=&quot;http://en.wikipedia.org/wiki/Free_variables_and_bound_variables&quot;&gt;free variables and bound variables&lt;/a&gt;. These have somewhat confusing, perhaps even counter-intuitive names. So, I will use &quot;local&quot; as synonymous with &quot;bound&quot; and &quot;non-local&quot; as synonymous with &quot;free&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we write the expression $\int_0^a f(x) dx$, the $x$ is a local (bound) variable. It does not refer to anything external to the expression. The meaning of $x$ is &quot;bound&quot; within the $\int$-operation. Local (bound) variables may be thought of as conveniences or placeholders. Their names do not matter. So, we could also write $\int_0^a f(v) dv$ and mean the exact same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the expression $\int_0^a f(x) dx$, the $f$ and $a$ are non-local (free). They refer to objects that need some external context. Another way to say this is that the expression is a function of $f$ and $a$, that is we could write $g(f,a) = \int_0^a f(x) dx$. It is not a function of $x$. We could write this function instead as $g(f,a) = \int_0^a f(v) dv$ and mean the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now consider the expression put before your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation}&#xA;\int_0^af(a-x)dx=-\int_a^0f(u)du=\int_0^af(u)du=\int_0^af(x)dx \qquad (1)&#xA;\end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In equation (1) you've asked your students to consider $u$ as non-local variable and equal to $a-x$ in the exposition for the leftmost equality.On the right hand side that context has already been dropped. In the rightmost equality, students are confused that $u$ is treated as a local variable, that is not subject to the context of $u = a - x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, if we are allowed to let context come and go, then we could continue  equation (1) as follows since $x = a - u$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation}&#xA;\int_0^af(x)dx = -\int_0^af(a-u)du \qquad (2)&#xA;\end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Combining (1) and (2) we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation}&#xA;\int_0^af(a-x)dx=-\int_0^af(a-u)du \qquad (3)&#xA;\end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And dropping context once more to consider $u$ as local to the $\int$-expression only, we'd get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation}&#xA;-\int_0^af(a-u)du=-\int_0^af(a-x)dx \qquad (4)&#xA;\end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And putting our results together we have arrived at &lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation}&#xA;\int_0^af(a-x)dx=-\int_0^af(a-x)dx \qquad (5)&#xA;\end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which of course implies that for any $f$ and $a$, $\int_0^af(a-x)dx=0$. Yikes! Your students are wise to be wary of applying and dropping context.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;Problem of missing the key transformation&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Consider instead writing $u = \phi(x) = a - x$, in other words $u$ is a function. Then, develop purely with derivatives and algebraic substitution that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation}&#xA;\int_0^af(a-x)dx=-\int_0^a f(\phi(x))\phi'(x) dx \qquad (6)&#xA;\end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we often just write $u = \phi(x)$ and $du = u'(x) = \phi'(x)dx$, so we could also write (6) as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation}&#xA;\int_0^af(a-x)dx=-\int_0^a f(u)du \qquad (7)&#xA;\end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, at this point we are yet to actually do integration by substitution. Going from (6) to (7) is just a change in notation! Continuing from (6), integration by substitution, which requires two applications of the fundamental theorem of calculus, allows the following&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align}&#xA;\int_0^af(a-x)dx &amp;amp;=-\int_0^a f(\phi(x))\phi'(x) dx  \\&#xA;&amp;amp;=-\int_{\phi(0)}^{\phi(a)} f(v) dv&#xA;\end{align}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $v$ is a local variable! Writing this same development with $u$ notation instead, that is continuing equation (7), we'd have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align}&#xA;\int_0^af(a-x)dx &amp;amp;=-\int_0^a f(u)du \\&#xA;&amp;amp;=-\int_{\phi(0)}^{\phi(a)} f(v)dv &#xA;\end{align}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, where it looks like we've simply swapped names for $u$ and $v$ and changed the limits, we actually applied the fundamental theorem of calculus twice! That is what justifies the apparent dropping of context, and it is no trivial result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No since $x$ and $v$ are local variables (bound within their own $\int$-expressions), we can now rename those as we like, such as naming $v$ as $x$ to get &lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align}&#xA;\int_0^af(a-x)dx &amp;amp;=-\int_0^a f(u)du \\&#xA;&amp;amp;=-\int_{\phi(0)}^{\phi(a)} f(v)dv \qquad \text{by integration by substitution} \\&#xA;&amp;amp;=\int_0^a f(v) dv &#xA;=\int_0^a f(x) dx&#xA;\end{align}&lt;/p&gt;&#xA;" OwnerUserId="4866" LastEditorUserId="4866" LastEditDate="2015-03-31T12:16:57.147" LastActivityDate="2015-03-31T12:16:57.147" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7704" PostTypeId="2" ParentId="7702" CreationDate="2015-03-25T18:42:41.020" Score="10" Body="&lt;p&gt;Teaching calculus using infinitesimals from nonstandard analysis has been tried, &lt;a href=&quot;http://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infinitesimal_Approach&quot; rel=&quot;noreferrer&quot;&gt;Keisler's text&lt;/a&gt; is considered a staple in this regard. But I am curious as to what is so &quot;terrible&quot; about epsilon-delta approach? Using infinitesimals invites us to envision a hidden world of numbers beyond the real numbers, into which all elementary functions miraculously extend, and their properties there are then mysteriously reflected back into the world of real numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The intuitive idea of a limit, as taught to non-majors, is &quot;$f(x)$ approaches $L$ as $x$ approaches $a$&quot;, epsilon and delta simply quantify what &quot;approaches&quot; means for $f(x)$ and $x$ respectively. Admittedly, epsilon and delta arguments are technical, but it does not mean that they are unintuitive. We know from the early history of calculus that the simplicity of intuitions about limits is misleading because those intuitions are self-contradictory. Resolving these contradictions inevitably leads to subtleties and technicalities, and refinement of intuition, under any systematic approach. At least, epsilon and delta does it without invoking extra phantom entities, that would have to be either explained or glossed over in addition to real numbers, so why is it worse than infinitesimals?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not to say that calculus with epsilon and delta can not be taught terribly, and that unfortunately it often is. But I am not sure that this is the fault of the approach rather than of the fact that there is a fundamental difficulty in the concept itself, as its 2000 year long genesis suggests. Instructors are often unable and/or unwilling to dedicate the time and the effort required to teach it properly. Especially since the algorithmic aspect of calculus is so much easier to teach, and can be taught while ignoring limits almost completely.  &lt;/p&gt;&#xA;" OwnerUserId="4621" LastEditorUserId="4621" LastEditDate="2015-03-26T00:46:41.147" LastActivityDate="2015-03-26T00:46:41.147" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7705" PostTypeId="1" AcceptedAnswerId="7708" CreationDate="2015-03-26T02:27:04.200" Score="10" ViewCount="295" Body="&lt;p&gt;In roughly two weeks, my secondary geometry class will have reached the part of the year where we discuss geometric translations: translation, rotation, and reflection.  Dilation i am going to use as a segue to similarity so I am just talking within the context of these first three transformations for the remainder of this question.  These transformations will be discussed both on and off the coordinate plane.  There will be some basic algebra to manipulate coordinates etc. but nothing too intense, some of my students have not taken the second algebra course yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to frame these transformations within the context of fractals because 1)  they are so visually appealing and intriguing! 2) the serve as a good motivator to investigate these different transformations and 3) ideally the final project for the unit will be to create your own fractal, either by hand or on the computer, using the transformations that we learned, which i think will be really fun for the kids and give me some awesome things to hang on my wall :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I do not know a ton about fractals and am not really sure how to go about tying these two subjects without getting too in depth. I have a book called Indra's Pearls, but it gets into imaginary numbers and some pretty heavy programming which my students have limited exposure to. How can I marry these two topics at a level suitable to a secondary geometry class? Any good resources that give the &quot;quick and dirty&quot; on fractals?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2015-03-26T19:23:17.150" LastActivityDate="2015-03-26T19:23:17.150" Title="Teaching geometric transformations through fractals?" Tags="|secondary-education|geometry|fractals|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7706" PostTypeId="2" ParentId="2763" CreationDate="2015-03-26T04:01:38.190" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;I would probably &lt;em&gt;introduce&lt;/em&gt; multiplication as area of a rectangle because it extends easily to mixed numbers and decimals. &lt;/li&gt;&#xA;&lt;li&gt;But multiplication is also an additive power and should be dealt with as such, perhaps initially with a different symbol. Also, eventually, one can/should introduce some kind of logarithmic idea, that is the correspondence between $3a = a + a + a$ and the multiplicative power $a^{3} = a\times a\times a$&lt;/li&gt;&#xA;&lt;li&gt;Multiplication is also an operation between a vector space (e.g. basket) and its dual (price list). Initially $3\, apples\, \times 5\, \frac{cents}{apple} = 15\, cents$ is something that can be dealt with very early. &lt;/li&gt;&#xA;&lt;li&gt;But in fact, $&amp;lt;3\, apples,\,2\, bananas,\, 5\, carrots&amp;gt;\otimes&amp;lt;5\, \frac{cents}{apple},\,7\, \frac{cents}{banana},\,2\, \frac{cents}{carrot}&amp;gt;$ is just as simple.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Regards&#xA;--schremmer&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-03-26T04:01:38.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7707" PostTypeId="2" ParentId="7694" CreationDate="2015-03-26T09:01:33.223" Score="2" Body="&lt;p&gt;In $\int_0^af(x)\,dx$, the inputs to $f$ begin at $0$ and progress to $a$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In $\int_0^af(a-x)\,dx$, the inputs to $f$ begin at $a$ and progress to $0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when considering Riemann sums with uniform-width bases and using midpoints for test points, the two collections of heights of the rectangles in the expressions are the same.  &lt;/p&gt;&#xA;" OwnerUserId="767" LastActivityDate="2015-03-26T09:01:33.223" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7708" PostTypeId="2" ParentId="7705" CreationDate="2015-03-26T10:22:25.897" Score="10" Body="&lt;p&gt;If you start directly with fractals, you'll get a hard time because dilation and similarity are an inherent feature of fractals. You should start with tesselations and friezes. They are equally visually stunning, can be used to problem-oriented and theory-oriented investigations and there are easy computer and tablet apps to generate and play with them. &lt;em&gt;And&lt;/em&gt; they only need the rigid motions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From tesselations and friezes on you can get to fractals by adding dilation.&lt;/p&gt;&#xA;" OwnerUserId="439" LastActivityDate="2015-03-26T10:22:25.897" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7709" PostTypeId="2" ParentId="7705" CreationDate="2015-03-26T10:50:08.483" Score="6" Body="&lt;p&gt;An illustration for @Toscho's nice idea, from Steven Dutch's&#xA;&lt;em&gt;&lt;a href=&quot;https://www.uwgb.edu/dutchs/symmetry/reptile1.htm&quot; rel=&quot;noreferrer&quot;&gt;Rep-Tiles&lt;/a&gt;&lt;/em&gt; webpage:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/xcm2a.gif&quot; alt=&quot;reptiles&quot;&gt;&#xA;&lt;br&gt;&#xA;&lt;sup&gt;&#xA;&quot;Four squares can be arranged around a central square to form a Greek Cross. Greek Crosses make a lovely plane tesselation, but they are not rep-tiles. We can try modifying a Greek Cross tesselation by replacing each cross with the compound of five crosses. It's closer, but still not an exact rep-tile. We can repeat the process, each time getting figures that are more and more crinkled and closer to a true rep-tiling. But at every step, we see that the perimeter of the tiling always has twice as many crinkles as each tile.&quot;&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-03-26T11:08:15.610" LastActivityDate="2015-03-26T11:08:15.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7710" PostTypeId="5" CreationDate="2015-03-26T14:30:21.770" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-03-26T14:30:21.770" LastActivityDate="2015-03-26T14:30:21.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7711" PostTypeId="4" CreationDate="2015-03-26T14:30:21.770" Score="0" Body="Courses designed to prepare students for subsequent calculus courses" OwnerUserId="39" LastEditorUserId="77" LastEditDate="2019-12-31T03:28:33.513" LastActivityDate="2019-12-31T03:28:33.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7712" PostTypeId="2" ParentId="7691" CreationDate="2015-03-26T19:21:00.520" Score="5" Body="&lt;p&gt;One reference from before WWI:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=Pl32YkKFIhsC&amp;amp;printsec=frontcover&amp;amp;dq=Rouse+ball+history+cambridge&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=elsUVcOuM4miNsrKg4gF&amp;amp;ved=0CCgQ6AEwAg#v=onepage&amp;amp;q=Rouse%20ball%20history%20cambridge&amp;amp;f=false&quot;&gt;W.W Rouse Ball, A History of the Study of Mathematics at Cambridge, 1889.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A synoptic review from the middle ages to the nineteenth century of the curricula, the mathematics studied, methods of instruction, student life, notable students in mathematics, notable professors, and reforms at Cambridge; plus a brief history of the university.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2015-03-26T23:52:08.243" LastActivityDate="2015-03-26T23:52:08.243" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7713" PostTypeId="2" ParentId="7691" CreationDate="2015-03-26T20:09:23.960" Score="3" Body="&lt;p&gt;If you want to have any idea about the mathematics education in Schools in Great Britain many years ago, you should read &lt;a href=&quot;http://www.educationengland.org.uk/documents/cockcroft/cockcroft1982.html&quot; rel=&quot;nofollow&quot;&gt;The Cockcroft Report&lt;/a&gt;. This documen is free access &lt;/p&gt;&#xA;" OwnerUserId="1789" LastActivityDate="2015-03-26T20:09:23.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7714" PostTypeId="2" ParentId="2532" CreationDate="2015-03-27T00:24:25.273" Score="5" Body="&lt;p&gt;I am an undergraduate student at Portland State University, having finished my major requirements I have decided to add a math minor. I did quite well in the lower division calculus, differential equations, and intro to linear algebra courses; yet I have had no experience with proofs. I am planning on taking a &quot;Intro to Math Reasoning&quot; course first for precisely that reason. If I had to jump into vector calculus with proofs as described above I would be ill prepared. Perhaps it is for folks like myself that there are courses like &quot;math analysis&quot; (which specifically includes: &quot;Special emphasis is placed on the ability to understand and construct rigorous proofs&quot;)&lt;/p&gt;&#xA;" OwnerUserId="4934" LastActivityDate="2015-03-27T00:24:25.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7716" PostTypeId="1" AcceptedAnswerId="7731" CreationDate="2015-03-27T16:19:19.843" Score="18" ViewCount="443" Body="&lt;p&gt;Mathematics education research is generally very different than mathematics research. I am interested in collaborating with mathematics education professors at my next institution, and possibly transitioning into mathematics education as my specialty.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What examples are available of successful transitions from mathematics research into mathematics education research? What common challenges did these individuals have to overcome in the process? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastActivityDate="2015-06-03T04:30:22.603" Title="How can a research mathematician transition into a mathematics education researcher?" Tags="|education-research|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7717" PostTypeId="1" CreationDate="2015-03-27T17:59:29.327" Score="27" ViewCount="1338" Body="&lt;p&gt;It's been said that no one explains anything so well as when they are trying to persuade you of something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my favourite textbooks is E.T. Jayne's &quot;Probability Theory: The Logic of Science&quot;. The entire book is designed to convince the reader of the superiority of Bayesian methods in probability, and the writing is particularly clear and understandable. I suspect that the quality of the writing is partially a consequence of the book's opinionated nature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example of this is &quot;Linear Algebra Done Right&quot; by Sheldon Axler. The book is written with the unusual approach of banishing determinants all the way to the final chapter. In trying to prove that this is the right way to go about things Axler has produced an excellent textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I suspect that other &quot;opinionated&quot; books might also be very well written. What other books or textbooks (at any level and in any topic) are written in a way that tries to push a particular point of view?&lt;/p&gt;&#xA;" OwnerUserId="4938" LastEditorUserId="4938" LastEditDate="2015-03-27T20:44:46.237" LastActivityDate="2016-02-20T00:36:56.280" Title="&quot;Opinionated&quot; textbooks" Tags="|textbooks|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7718" PostTypeId="1" CreationDate="2015-03-27T23:45:14.427" Score="41" ViewCount="3178" Body="&lt;p&gt;&lt;strong&gt;Context:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was recently a question on Math.SE: &lt;a href=&quot;https://math.stackexchange.com/q/1209296/26306&quot;&gt;Inferior to Other Younger and Brighter Kids&lt;/a&gt; which starts as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I'm a high school student (Junior/Grade 11) and I'm currently studying Spivak's Calculus on Manifolds. I've finished Single/Multi Variable Calculus (the basic ones without analysis/rigor), the first half of Rudin for Real Analysis and Linear Algebra (Lang) through self-study.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And then the OP writes how &lt;em&gt;inferior&lt;/em&gt; he feels compared to other younger and brighter kids. Perhaps there are some deeper issues involved, but I don't want to dwell on it, nor discuss the situation of OP publicly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The issue:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue I would like to write about is that more and more often I notice kids studying &lt;em&gt;advanced&lt;/em&gt; books. Surely gifted students will access more advanced material earlier than their peers and I would applaud the kids reading almost anything they want. Yet, I have the feeling that sometimes this is just &lt;em&gt;too early&lt;/em&gt; and reading advanced books without certain level of mathematical maturity may cause more harm than good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience isn't enough to draw any conclusions, but this is not just a feeling, I've witnessed negative effects of such behavior multiple times. Here are some example motives:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Over-reliance on the already-learned material.&lt;/em&gt; There was an introductory problem given which could be solved using some elementary method. The gifted student thought &quot;this is boring&quot; and solved it faster using an advanced method. Then there was another problem given, but the advanced method couldn't be generalized to this new problem (while the elementary way could) and the gifted student was unable to find a solution until the end of the class.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Missing fundamentals.&lt;/em&gt; Gifted student learned $X$ by himself, but the material depended on the knowledge of $Y$. Yet, the student lacked proper understanding of $Y$ and so build some bad intuitions, which had to be unlearned before progressing further.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Closed-mindness (compared to other gifted students).&lt;/em&gt; This one is hard to describe, some signs are:&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the student thinks there's always a book that explains the area in question;&lt;/li&gt;&#xA;&lt;li&gt;the student specializes in solving toy problems and (difficult) homework, but is unable to handle even simple &lt;em&gt;open&lt;/em&gt; or &lt;em&gt;research&lt;/em&gt; problems;&lt;/li&gt;&#xA;&lt;li&gt;the student is unable to form his own theories or own interpretations.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There also others, like burnout, but the causes will be more complicated, let's not diverge too much. Even these three trouble me very deeply (especially the last) and it's like some math-contest rat race rather than curious study of the beautiful area that mathematics is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, rediscovering the wheel (if not too often, human lifespan is still limited) is a necessary part of study, so that when the time comes, the kids will be able to do their own original discoveries. Getting all the best concepts laid out in a book, digested, updated and improved over the years doesn't seem like the best idea. Still, perhaps this view is invalid and unfounded?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any data or research on the effects of access to advanced material early in kid's education?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that gifted students are rare, right now even some anecdotal evidence would be great. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are your experiences in this matter?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;More concrete examples:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;@Joseph Van Name asked for examples, so here there are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Over-reliance on the already-learned material:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In a geometry class there was a guy that tried to solve every single problem using analytical approach: coordinates, trigonometry, algebra, etc. I admit that he was very good at this, but he didn't learned some techniques we practiced using simpler problems and then wasn't able to solve the more difficult ones which wouldn't yield to his standard method.&lt;/li&gt;&#xA;&lt;li&gt;During programming class there was a group who liked &lt;a href=&quot;http://en.wikipedia.org/wiki/Suffix_tree&quot; rel=&quot;noreferrer&quot;&gt;suffix trees&lt;/a&gt; too much and some students didn't learned some simpler techniques (here I mostly mean variations on the &lt;a href=&quot;http://en.wikipedia.org/wiki/Knuth%E2%80%93Morris%E2%80%93Pratt_algorithm&quot; rel=&quot;noreferrer&quot;&gt;Knuth-Morris-Pratt algorithm&lt;/a&gt;) which were necessary later.&lt;/li&gt;&#xA;&lt;li&gt;There was another guy who would try to solve all the contest inequalities using &lt;a href=&quot;http://en.wikipedia.org/wiki/Muirhead%27s_inequality&quot; rel=&quot;noreferrer&quot;&gt;Muirhead's inequality&lt;/a&gt;. He had a huge problem each time the inequality in question was not similar to any mean (e.g. all the numbers begin equal not causing equality).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Missing fundamentals:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A person who had learned more advanced probability before university disregarded a few first classes and then assumed that any distribution has a density function. There were later some problems with conditional expected values and their $\sigma$-algebras, but I don't remember it well enough.&lt;/li&gt;&#xA;&lt;li&gt;Many times students who learn asymptotic complexity by themselves don't understand some concepts properly and don't pay enough attention to details. Some common misconceptions are &#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$n \notin O(n^2)$, &lt;/li&gt;&#xA;&lt;li&gt;$n^2 + n \in \omega(n^2)$, &lt;/li&gt;&#xA;&lt;li&gt;$f_i \in O(g)$ implies $\sum_{i=1}^{n} f_i \in O(n\cdot g)$.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Frequently students that learned programming without any supervision or guidance from more experienced people have bad style (and programming contests often reinforce that bad style). It is an issue, because it's hard to unlearn and the drawbacks are not severe enough for the student to want to change it by himself. The effect is that the student penalizes himself for a much longer time (one of the reasons why summer internships or involvement in open-source projects is a good idea).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2019-01-15T00:17:58.250" Title="Effects of early study of advanced books" Tags="|secondary-education|reference-request|self-learning|gifted-students|creativity|" AnswerCount="5" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="7719" PostTypeId="2" ParentId="7716" CreationDate="2015-03-28T00:21:55.720" Score="7" Body="&lt;p&gt;One example may be &lt;a href=&quot;http://en.wikipedia.org/wiki/Roger_Evans_Howe&quot;&gt;Roger Howe&lt;/a&gt;, who is a distinguished mathematician at Yale University, most recently&#xA;at &lt;a href=&quot;http://tias.tamu.edu/facultyfellows/2013-2014/roger-howe&quot;&gt;Texas A&amp;amp;M&lt;/a&gt;.&#xA;He has written widely on education topics, including this article on&#xA;1st-grade math and the U.S. &quot;Common Core&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(1) &quot;Three Pillars of First Grade Mathematics.&quot;&#xA;  (&lt;a href=&quot;http://commoncoretools.me/wp-content/uploads/2012/02/3pillars.pdf&quot;&gt;PDF download&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and this conference presentation on teacher preparation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(2) &quot;The Mathematical Education of Teachers.&quot; CBMS National Summmit.&#xA;  (&lt;a href=&quot;http://www.cbmsweb.org/NationalSummit/Plenary_Speakers/howe.htm&quot;&gt;weblink&lt;/a&gt;.),&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;in which he says,&#xA;&quot;I want to emphasize the possibility of greater involvement of university mathematicians in K-12 education.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-03-29T00:17:32.460" LastActivityDate="2015-03-29T00:17:32.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7720" PostTypeId="2" ParentId="7717" CreationDate="2015-03-28T00:27:40.810" Score="14" Body="&lt;p&gt;Not an &quot;opinionated&quot; book &lt;em&gt;per se&lt;/em&gt;, but Tom Apostol's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471000051&quot;&gt;Calculus&lt;/a&gt; follows up the chronological order of the concepts of the calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, it starts with an example of Archimedes exhaustion method and after defines the integrals before limits as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f$ be a function defined and bounded on $[a, b]$. Let $s$ and $t $ denote&#xA;  arbitrary step functions defined on [a, b] such that&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$s(x) \leq f(x) \leq t(x)\hspace{3cm}(1)$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;for every $x$ in $[a, b]$. If there is one and only one number $I$ such that:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$\int_{a}^{b}s(x)dx \leq I \leq \int_{a}^{b}t(x)dx$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;for every pair of step functions $s$ and $t$ satisfying (1), then this number $I$ is called the integral of $f$ from $a$ to $b$, and is denoted by the symbol $\int_a^bf(x)dx$ or $\int_a^bf$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Later, after the integral is defined and proved solely based on the integration of step functions, the $\epsilon - \delta$ is introduced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is an interesting approach, but Apostol never tries to convince us that it is a superior one, so it is not really an &quot;opinionated&quot; book. However, if we consider that writing a two volume textbook is not easy, it does contain the indirect opinion that, at least, Apostol liked this approach better. Also, in his words:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In this book the subject is introduced in an informal way, and ample use is made of geometric intuition whenever it is convenient to so so.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which are now commonplace in many calculus textbooks.&lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="1890" LastEditDate="2015-03-28T12:02:22.223" LastActivityDate="2015-03-28T12:02:22.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7721" PostTypeId="2" ParentId="7717" CreationDate="2015-03-28T00:52:06.590" Score="14" Body="&lt;p&gt;Permit me to recycle a portion of &lt;a href=&quot;https://math.stackexchange.com/a/85244/237&quot;&gt;my answer to an MSE question&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Tristan Needham,&#xA;  &lt;a href=&quot;http://usf.usfca.edu/vca//&quot; rel=&quot;nofollow noreferrer&quot;&gt;Visual Complex Analysis&lt;/a&gt;, Oxford Univ. Press. 1997.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/CXmuw.jpg&quot; alt=&quot;Needham cover&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This book &quot;brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation.&quot; [quoting the jacket]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here's one figure from the book, p.135, that gives a flavor of his approach:&#xA;&lt;br /&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;img src=&quot;https://i.stack.imgur.com/HpbKO.png&quot; alt=&quot;Figure 13&quot;&gt;&lt;br /&gt;&#xA;You can almost guess the theorem from the figure:  The two spheres $S_1$ and $S_2$ are&#xA;orthgonal iff the two circles $C_1$ and $C_2$ are orthogonal.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-03-28T14:23:05.377" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7722" PostTypeId="1" AcceptedAnswerId="7728" CreationDate="2015-03-28T16:30:19.143" Score="35" ViewCount="4773" Body="&lt;p&gt;When presenting a proof, there are usually a lot of parts which look like &quot;obvious&quot;, &quot;routine&quot; manipulation to me, and between zero and two genuinely insightful steps. I want to point out the difference between these two steps and the others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I worry that calling things like this &quot;obvious&quot; or &quot;routine&quot; makes students who don't see them feel like idiots. Is there a way to discuss this distinction without making the students who are still working on these steps feel dumb?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same issue arises when helping students work out their own proofs, rather than presenting one.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2015-03-30T14:02:18.097" Title="Uninsulting way to say &quot;this will eventually be easy&quot;" Tags="|proofs|language-use|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7723" PostTypeId="2" ParentId="7722" CreationDate="2015-03-28T16:35:05.913" Score="17" Body="&lt;p&gt;Perhaps not pointing out that the obvious steps are obvious but that the insights are insights.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe students don't feel bad for not seeing the &quot;magic steps&quot; by themselves, so pointing out that those are hard is not a problem. The opposite is what you mention: they would feel bad for not seeing the obvious. Hence, only treat the difficult as difficult.&lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2015-03-28T16:35:05.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7724" PostTypeId="2" ParentId="7722" CreationDate="2015-03-28T17:48:15.120" Score="10" Body="&lt;p&gt;For presenting proofs, I would only use terminology such as &quot;obviously&quot;, &quot;clearly&quot;, &quot;routine&quot;, etc if it would be for a course just below the class had the student had 'perfect' recollection. For instance, in a introductory commutative algebra class, if the proof should be routine for the previous regular algebra class (i.e., something involving a course below algebra), then I would use such terminology. It should be used only as an indicator for how difficult the proof is relative to their expected skills. Another example, in Multivariate Calculus, I would say &quot;obvious&quot;, &quot;routine&quot;, or whatnot to Intro. Calculus concepts, but not necessarily to Calculus 2 concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it is not at this level, I would just present it without comment or just state the result without explanation. Many students at the level of proofs would question it if it wasn't obvious. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I use such terminology to indicate what I expect my students to have as prior knowledge that they should recall pretty quickly. I do this even in the calculus sequence, quickly going through prior knowledge that should be clear quicker. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the measure of &quot;material from two classes prior in a course series&quot; as &quot;obvious&quot; has worked for me so far in indicating expectations to students while not perturbing them. Sometimes it takes a semester or two for things to &quot;click.&quot; &lt;/p&gt;&#xA;" OwnerUserId="201" LastEditorUserId="201" LastEditDate="2015-03-30T14:02:18.097" LastActivityDate="2015-03-30T14:02:18.097" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7725" PostTypeId="1" AcceptedAnswerId="7726" CreationDate="2015-03-28T18:29:54.353" Score="6" ViewCount="322" Body="&lt;p&gt;Basically the textbooks in my country are awful, so I searched on the web for a precalculus book and found this one: &lt;a href=&quot;http://www.stitz-zeager.com/szprecalculus07042013.pdf&quot;&gt;http://www.stitz-zeager.com/szprecalculus07042013.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it does not cover convergence,limits etc. and those topics were briefly mentioned in my old textbooks. So what i am asking is are these topics a prerequesite for calculus or are they a part of the subject?&lt;/p&gt;&#xA;" OwnerUserId="4911" LastActivityDate="2015-04-05T08:39:49.770" Title="Are convergent and divergent sequences a prerequesite for calculus or are they a part of it?" Tags="|calculus|precalculus|sequences|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7726" PostTypeId="2" ParentId="7725" CreationDate="2015-03-28T19:01:33.317" Score="5" Body="&lt;p&gt;It depends on how Calculus is treated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At my university and others I've attended (US), the concept of the limit is usually treated early in the first Calculus class in order to talk about continuity and derivatives. The core idea of sequences is brushed over in the introductory courses (except in the advanced courses). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes (and the department is debating this here) limits and sequences are taught in precalculus, allowing them to be reinforced in calculus. But in general, they're not expected to be well known before the first calculus course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, they're good to look into algebraically before calculus, but they involve topological ideas more suited to calculus. &lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-03-28T19:01:33.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7727" PostTypeId="2" ParentId="7722" CreationDate="2015-03-28T21:00:56.373" Score="5" Body="&lt;p&gt;It may help your students if you make explicit why it is obvious/routine/etc. Have they seen it in the previous example? Does it just depend on unpacking definitions? It is a calculation/result/etc that was covered in a prerequisite class? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think there is anything wrong with pointing out which parts are routine, but saying &lt;em&gt;why&lt;/em&gt; it's routine will help them more, since this gives them some place to start when they get stuck in a proof, eg instead of asking themselves &quot;is this routine?&quot; they can ask themselves &quot;is this a matter of unpacking definitions?&quot; or &quot;have I solved a similar problem before?&quot; I think being able to ask yourselves these questions (and more!) is a big part of learning to write proofs.&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2015-03-28T21:00:56.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7728" PostTypeId="2" ParentId="7722" CreationDate="2015-03-28T23:34:59.283" Score="12" Body="&lt;p&gt;One thing that you might want to do early on in your course is think about the &lt;em&gt;classroom norms&lt;/em&gt; that you wish to establish. From your post, it seems like an example of a norm in &lt;em&gt;your class&lt;/em&gt; is that it is &lt;strong&gt;okay&lt;/strong&gt; to ask a question of the form:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What do you mean by obvious (routine, etc)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you do not try to make it clear to students from the outset that such questions are totally on task, then it is much less likely for them to ask it. (After all, &quot;What is obvious?&quot; sounds silly in most contexts.) Similarly, you may wish to establish the related norm that students are expected to know what is meant when such words (obvious, routine, etc) are used. To this end, perhaps you may want to call on students occasionally (or often, depending on how you teach...) to have them chime in to fill in those details. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that classroom norms are typically established early on in the term; if the current norm in your classes is that you will lecture unless a student asks a question, then cold-calling on someone halfway through the semester to ask &quot;what did I mean by obvious candidate?&quot; could be a bit jarring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a concrete example of the type of questions you might ask in a math class, &lt;a href=&quot;https://www.dropbox.com/s/v0x7w1110tg7v3t/topology%20text.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; is a handout I used at the start of a Topology course that used Munkres' text (mentioned earlier in &lt;a href=&quot;https://matheducators.stackexchange.com/a/2189/&quot;&gt;&lt;strong&gt;MESE 2189&lt;/strong&gt;&lt;/a&gt;). Although this handout pertains to &lt;em&gt;reading&lt;/em&gt; mathematics, I think it is relevant for &lt;em&gt;listening&lt;/em&gt; to mathematics, as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As one more concrete example, consider the following problem for a first Real Analysis course:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Prove:&lt;/strong&gt; Any infinite, closed set $A \subset [0,1]$ contains a limit point.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What do students need to know (or figure out) as they come to understand a proof, or generate a proof, or prove a more general statement (e.g., replace $[0,1]$ with any bounded set; or consider the analogous proposition in $\mathbb{R}^n$ rather than just $\mathbb{R}$)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are curious about frameworks from the mathematics education literature that could help with thinking about such questions, then Dubinsky's &lt;em&gt;APOS theory&lt;/em&gt; is one place to start; essentially, what is asked above amounts to providing a &quot;genetic decomposition&quot; (terminology originating from Piaget's work as a &quot;genetic epistemologist&quot;). Alternatively, you may wish to look at Simon et al on &lt;em&gt;hypothetical learning trajectories&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think there is about one genuine insight needed for the problem above, which is &lt;em&gt;bisecting repeatedly to pick a side, each time, with infinitely many points&lt;/em&gt;. (As mentioned in &lt;a href=&quot;https://matheducators.stackexchange.com/a/2230/262&quot;&gt;&lt;strong&gt;MESE 2226&lt;/strong&gt;&lt;/a&gt;, I think about this as specifying the limit point by finding its binary representation.) Perhaps this is really two insights that I have combined together. In any event, there are many details to be filled in: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How were the givens used? Must the set by infinite? (Yes!) Must the set be closed? (In general, yes: Consider $A$ as the set of unit fractions, for example.) Was bounding $A$ necessary? (In general, yes: Consider $A = \mathbb{N}$, for example.) Are these parenthetical counterexamples clear? Is it obvious why the point produced through repeated bisection is a limit point? (Perhaps not the first time through; can the student show as much for any given $\epsilon &amp;gt; 0$?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that unpacking questions/proofs by writing down (for yourself) what you consider to be the &lt;em&gt;key insights&lt;/em&gt; and the &lt;em&gt;routine details&lt;/em&gt; is a good way to prepare for a topic. This prepares you to emphasize the insights, and perhaps to pose follow-up problems in which they are used or adapted, and to ensure students can fill in the details, and perhaps to call on them occasionally to do as much (both for their sake, as learners, and for your sake, as you check in with your class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Establish classroom norms around questions and expectations with respect to that which could be called (by some) obvious, routine, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to partition material into &lt;em&gt;key insights&lt;/em&gt; and &lt;em&gt;routine details&lt;/em&gt; beforehand; emphasize the former, and check in to make sure the class is on board with the latter.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-03-28T23:34:59.283" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7729" PostTypeId="2" ParentId="7722" CreationDate="2015-03-28T23:43:59.363" Score="5" Body="&lt;p&gt;I find the easiest way is to just straight out say that it's not something you expect them to know offhand or immediately understand, but that as their studies advance it will become trivial. That way they understand that it's new, but nothing to be scared of. Something along the lines of:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;This may not make sense to you yet, but trust me it will soon be second nature!&quot; For now, just try to learn it/remember it/keep it in mind [as applicable].&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Don't worry if you can't do this immediately or it take a few tries, it isn't always something you &quot;just get&quot;... you'll get faster and better at it with practice/experience.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4947" LastActivityDate="2015-03-28T23:43:59.363" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7730" PostTypeId="2" ParentId="2790" CreationDate="2015-03-29T02:16:39.437" Score="4" Body="&lt;p&gt;I think one should consider two domains in parallel:  writing programs, and writing test cases.  If students became marginally proficient at both, I believe it would help crystallize their thinking mechanically.  (It would also make them much better programmers.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, it would not lend itself to the other kinds of thinking that should be present in people practicing mathematics, but it would be a good practice in developing rigor and abilities in refutation.  For producing a mathematically literate individual, one will need some history, literature, and expository skills as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Make Minds Go Both Ways&quot; Paseman, 2015.03.28&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-03-29T02:16:39.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7731" PostTypeId="2" ParentId="7716" CreationDate="2015-03-29T03:32:01.900" Score="11" Body="&lt;p&gt;As an earlier comment (&lt;a href=&quot;https://matheducators.stackexchange.com/questions/7716/how-can-a-research-mathematician-transition-into-a-mathematics-education-researc#comment15925_7716&quot;&gt;How can a research mathematician transition into a mathematics education researcher?&lt;/a&gt;) indicates, I speak from the perspective of someone who has done a smidgen of math-education research, but not in any sense as an expert.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With this caveat, I think that it is important to realise that math-education research is much closer to science research than to math research, in the sense that it is necessarily an experimental science.  As such, gathering experimental data &lt;em&gt;is&lt;/em&gt; research, and this is something that we do every day, as teachers.  (I believe that the usual term for this is 'practitioner research' (&lt;a href=&quot;https://en.wikipedia.org/wiki/Practitioner_research&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Practitioner_research&lt;/a&gt; ).)  This is not the &lt;em&gt;only&lt;/em&gt; kind of research that one can do, but it is a legitimate and valuable kind, and it is one that is accessible to you with only a little bit of extra training.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have heard a math-education teacher literally say &quot;I had an interesting experience in class today.  I wrote a paper about it.&quot;  While I am not sure that this is the kind of mentality that one should seek, it at least shows that writing a practitioner-research paper is not necessarily the huge and time-intensive process that writing a mathematics paper seems to me to be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also &lt;em&gt;strongly&lt;/em&gt; recommend working in concert with an existing math-education researcher, collaborating informally (discussing ideas together) if not formally (writing a paper together).  Such a person will be able to tell you whether questions you have are interesting; to suggest ways to elaborate on, expand, and research them; and to advise you on how to write up results or discoveries that you might find.  (A tip:  don't try to transfer your knowledge about writing mathematics papers directly to writing math-education papers.  The latter are much closer in structure (I think) to papers in the social sciences.)&lt;/p&gt;&#xA;" OwnerUserId="2070" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-06-01T02:59:04.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7732" PostTypeId="2" ParentId="4182" CreationDate="2015-03-29T06:25:49.177" Score="1" Body="&lt;p&gt;I have found the following resource online:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Fractions: pikelets and lamingtons&quot;, State of New South Wales Department of Education and Training, 2003 &lt;a href=&quot;http://www.google.com.au/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=1&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CB4QFjAA&amp;amp;url=http%3A%2F%2Fwww.schools.nsw.edu.au%2Flearning%2F7-12assessments%2Fnaplan%2Fteachstrategies%2Fyr2014%2Fimg%2Ffractions-pikelets-and-lamingtons-book.pdf&amp;amp;ei=a5QXVYScHYqi8QXuqYBo&amp;amp;usg=AFQjCNGGdE_QZEwc2AOpFNpS7twq3549ow&amp;amp;sig2=JHoheOWZ0YDodR8cSoMwPA&amp;amp;bvm=bv.89381419,d.dGc&quot; rel=&quot;nofollow&quot;&gt;PDF online, accessed 29/3/2014&lt;/a&gt; (Note that according to the acknowledgements, most of it was written by Peter Gould, though he is not listed as the author.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This document contains activities designed to support children's learning of fractions, as well as discussion of how children learn fractions and some review of literature. The activities described begin with children's natural language of &quot;half&quot; and &quot;quarter&quot;, and then move directly to sharing numbers of pikelets and lamingtons between children in such a way that they must be cut into halves and quarters. So basically, the concept of &quot;and a half&quot; is introduced before other types of fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't find further references to this idea in other literature yet, but I will add them to the answer if I do.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The following teaching sequence is described on page 61:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students’ informal notions of partitioning, sharing and measuring provide a starting point for developing the fraction concept. Drawing on the research on developing the fraction concept and the identified problem of representing fractions as a/b, the following sequence is used to develop the fraction concept.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Subdividing continuous quantities into halves and quarters using the names halves and quarters and identifying the sub-units.&lt;/li&gt;&#xA;&lt;li&gt;Sharing of numbers of continuous models. First using the halving strategy (e.g. Share 6 pikelets between 2 people, share 3 pikelets between 2 people, share 3 pikelets between 4 people) then coordinating the partitioning with the number of sharers.&lt;/li&gt;&#xA;&lt;li&gt;Recordings using sharing diagrams for the continuous model (typically circles or rectangles) showing the partitioning.&lt;/li&gt;&#xA;&lt;li&gt;Traditional recording of fractions used as “environmental print” linked to sharing diagrams. If a student choses to write one-quarter as 1/4 it is accepted as a common shorthand form.&lt;/li&gt;&#xA;&lt;li&gt;Comparison of units by re-dividing a continuous quantity, such as a paper streamer, leading to representing equivalent fractions (e.g. fraction wall).&lt;/li&gt;&#xA;&lt;li&gt;Identifying fractions as numbers and locations on a number line (comparison of location, Which is larger, 1/3 or 1/2?)&lt;/li&gt;&#xA;&lt;li&gt;Addition and subtraction of fractions using representations of equivalent fractions as subdivisions of the same unit of length ( 2/3 + 1/4). For example, addition or subtraction using the “fraction wall”. &lt;/li&gt;&#xA;&lt;li&gt;Symbolic operations with fractions based on students’ coherent and stable meanings for fractions that may be expressed symbolically. &#xA;It is important to build meaning for fractions arising from the process of division. In this&#xA;way halves, quarters, thirds and fifths arise from problems of sharing rather than being defined as abstract numerical quantities.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The teaching sequence described in the document builds fraction ideas on the foundation of equal sharing, and starts by using children's natural language of &quot;half&quot; and &quot;quarter&quot;begins with the idea of sharing things equally among people (pikelets and lamingtons actually), introducing the idea of sharing three pikelets among two people in only the second activity&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-03-29T06:25:49.177" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7733" PostTypeId="2" ParentId="7722" CreationDate="2015-03-29T14:27:07.680" Score="3" Body="&lt;p&gt;I see the question cross this and academic sites a lot about whether &quot;trivial&quot; is insulting.  Trivial, taken in the context of its literal meaning and explained as such, is exactly the word for these steps you're describing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wiktionary says the following:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;trivial (comparative more trivial, superlative most trivial)&#xA;&#xA;1. Ignorable; of little significance or value.&#xA;2. Commonplace, ordinary.&#xA;...&#xA;5. (mathematics) Of, relating to, or being the simplest possible case.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;When referring to steps of a proof, any of those three definitions apply to many of the steps.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Commonplace steps which are found in any proof of its kind, I suggest you explain thoroughly and mention that they are in fact &quot;routine&quot; and the student will see such steps frequently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also those steps which are required to provide foundation to a proof and assume familiarity with prior proofs.  They are present for the sole purpose of indicating that assumptions or entire branches of reasoning present in the proof are already well-established in earlier proofs.  I suggest those steps be pointed out as familiar once the reader is familiar with those prior proofs, and explaining what those proofs demonstrate, even if you don't go into &lt;em&gt;how&lt;/em&gt; those proofs demonstrate it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternately, they could be proving the most basic case of a proof, which to a student may seem like a red herring, but are foundational to understanding later steps. I'd suggest that you not characterize anything in this latter category as routine or obvious. Rather I'd suggest explaining them fully in the context of the proof the first few times you come across them in your teaching materials, as well as explaining the role that those steps play in proofs of that form.&lt;/p&gt;&#xA;" OwnerUserId="4952" LastActivityDate="2015-03-29T14:27:07.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7734" PostTypeId="2" ParentId="7716" CreationDate="2015-03-29T17:10:35.540" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;I agree about Howe. See for another instance his review in the September 1999 Notices of Liping Ma's book. But not everyone has Howe's stature. &lt;/li&gt;&#xA;&lt;li&gt;Math education is highly problematic. On the one hand, educando is strictly for &quot;academics&quot; and of zero use for the students. On the other hand, Hestenes'&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Course content is taken [by many] as given, so the research problem is how to teach it most effectively. This approach [...] has produced valuable insights and useful results. However, it ignores the possibility of improving pedagogy by reconstructing course content.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is an especially difficult road to take because:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a. It requires  (much) mathematical thinking of a very different nature than that necessary in research. Briefly, in reconstructing course content, one must think &lt;em&gt;globally&lt;/em&gt;, that is one must constantly try to take as many mathematical relationships into consideration as possible and then decide which to deal with. You might call this the problem of organizing the &quot;content architecture&quot; and/or that of providing a &quot;story line&quot;. (Very rewarding.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b. One then has to write the materials to be used by the students. (A lot of work.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c. One then has to cope with one's department. (Likely a lot of grief.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyone curious as to what experience I am basing this opinion on might want to check &lt;a href=&quot;http://freemathtexts.org/&quot; rel=&quot;nofollow&quot;&gt;FreeMathTexts&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best of luck.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-03-29T17:10:35.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7735" PostTypeId="2" ParentId="7656" CreationDate="2015-03-29T17:50:42.953" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;I agree very much with Hefferon but his Linear Algebra is not a very good example. Rather, it is too good an example. To write a book like this requires an immense amount of work.&lt;/li&gt;&#xA;&lt;li&gt;Re crowdsourcing. It seems to me that this would be extremely difficult because between the moment you start writing and the moment when you start to have adoptions, and you need to reach a critical mass, quite a long time will have elapsed. See for instance what is left of the NSF' &quot;Calculus Initiative&quot; of circa 1990. And it could be argued that without the millions the NSF invested, even Harvard's calculus text would not have made it. And even &lt;em&gt;it&lt;/em&gt; did not last that long.&lt;/li&gt;&#xA;&lt;li&gt;Instead of crowdsourcing, then, I would try to go for collaboration with like-minded potential authors. But that too is not easy if only because of the problem of &quot;How to write a book that is different&quot; with &quot;co-authors who might put different meaning on different&quot;.&lt;/li&gt;&#xA;&lt;li&gt;Finally, I also agree with Hefferon that a big problem is to &quot;convince our colleagues that this is a useful contribution.&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Here again, anyone interested in what experience I am basing these opinions on might want to look up &lt;a href=&quot;http://freemathtexts.org&quot; rel=&quot;nofollow&quot;&gt;freemathtexts&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-03-29T17:50:42.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7736" PostTypeId="2" ParentId="369" CreationDate="2015-03-30T07:35:36.390" Score="1" Body="&lt;p&gt;In 1963-64, I took (or was taken by) a class called Algebra II, in which we used an SMSG book.  Like SMSG Algebra I (or just plain Algebra) I found it utterly wretched and resolved to leave anything science-related, and to become chief poet in my high-school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, funny how things turn out-- I went on to do a doctorate in Biochemistry, and recently published a theoretical article (all quantum mechanics) in Journal of Chemical Physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No thanks to SMSG though.  The geometry text wasn't so bad -- how could you screw up Euclid, after all? but the the algebra was godawful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A particular hang-up of mine was irrational numbers -- the simple proofs of irrationality, say of the square root of two, seemed like sleight of hand to me (still do, in fact) but in a larger sense, I couldn't get my mind around irrationality.  When in college, a friend gave me Boyer's book on the &quot;History of the Calculus&quot; and I came to understand that the question of irrational numbers had perplexed the greatest minds of our civilization from Pythagoras to Cantor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If someone had put it to me in these terms in high school -- that these are difficult ideas that men have wrestled with for millennia-- I might have done better earlier.&lt;/p&gt;&#xA;" OwnerUserId="4955" LastActivityDate="2015-03-30T07:35:36.390" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7737" PostTypeId="2" ParentId="7722" CreationDate="2015-03-30T09:29:55.517" Score="3" Body="&lt;p&gt;You can say, &quot;This is the straightforward part; vs. this is the less obvious part....  The straightforward steps will become easier with practice.  The more experience you have with proofs, the more comfortable you'll get.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are right, that it is not helpful to say, &quot;This part is obvious/routine/trivial.&quot;&lt;/p&gt;&#xA;" OwnerUserId="4956" LastActivityDate="2015-03-30T09:29:55.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7738" PostTypeId="2" ParentId="2164" CreationDate="2015-03-30T09:56:28.490" Score="6" Body="&lt;p&gt;Xerox a couple of pages of exposition from the textbook ahead of time.  Give a five-minute-max lecture-type introduction that motivates the hand-out.  Now distribute the hand-out.  (Don't tell them at this point that it's from their textbook.) Let the students know that if they have any questions, they can raise their hands and you'll come and help.  When someone raises his hand, go to his seat and squat down next to him.  Carry on a soft, short conversation with him.  You'll probably be able to get him unstuck quickly, or be able to tell him that he asked a good question, which will become clearer a couple of paragraphs later.  If it's a large group, bring some TAs to the class to circulate with you.  If it's a large lecture hall, ask the students to leave every other row in the hall unoccupied.  You need to be able to move among the students comfortably.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After some reading time, you can re-teach the material if need be; maybe you'll be able to elicit a fair amount of the material from the students.  Don't hesitate to have students put things on the board.  Just don't put anyone on the spot, with just one student coming up to the board all by his lonesome.  Draw vertical lines to make sections of board, so that each student has his own section of board.  Let them explain their sketches or equations by turns if they want to -- this has to be optional for shy students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't do this every session, but do try it at least once every two weeks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, also take some time at least once every two weeks in the early part of the semester to have students work collaboratively in small groups during class.  This will facilitate the formation of study groups that meet outside class.  Students need to learn to talk about math with each other, and show each other math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notes:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Students get spoiled.  For many students, it's much easier to learn something from you than from the book.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Difficult things get easier with practice.  If reading the text is a bit scary (&quot;What if I don't understand anything, and it's all gobbledygook?&quot;), then have them do it the first few times in a safe environment, with you (and the TAs, perhaps) there as a safety net.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Different students learn best in different ways (auditory, visual, exploratory, etc.).  So, it is important to present the material in a variety of ways.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4956" LastActivityDate="2015-03-30T09:56:28.490" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7739" PostTypeId="1" CreationDate="2015-03-30T15:07:38.997" Score="8" ViewCount="1620" Body="&lt;p&gt;I'm looking for some animations (videos or gifs) of finding the volume of different 3d shapes. It would be super helpful if I could find something which stacks unit cubes into a rectangular prism to find how many of these unit cubes are able to fill up the entire prism. This concept leads into base area * height which leads into deriving the formula for a cylinder. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anyone has anything, I'd greatly appreciate it!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="4737" LastActivityDate="2015-04-06T00:16:53.510" Title="Gifs of finding the volume of 3d shapes?" Tags="|secondary-education|examples|geometry|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7740" PostTypeId="1" AcceptedAnswerId="7743" CreationDate="2015-03-30T17:58:12.890" Score="7" ViewCount="242" Body="&lt;p&gt;The author of an answer on math.se remarked that much of mathematical progress is a factor of the accrual of both, theorems, which other mathematicians can use in their proofs, and more efficient proofs of those theorems, which enables, students, and mathematicians, to apprehend them faster than they would have otherwise apprehended them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the paper A Beginner's Guide to Forcing, Timothy Chow introduces the phrase &lt;em&gt;open exposition problem&lt;/em&gt;, which he uses to denote all theorems for which there is no proof such that any person capable of apprehending the premises would, after s/he read the proof, disbelieve that s/he could have inferred the conclusions him/herself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering that, it seems to me that mathematicians appreciate how succinct expositions affect the progress of their discipline. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I study undergraduate math and philosophy. Next year, I'd like to submit an essay on the importance of solving open exposition problems in philosophy to an undergraduate journal. Before writing it, I'd like to read what mathematicians have written on either, the effect of the simplification of proofs on the progress of mathematics, or how to simplify proofs. Have mathematicians written on either topic? If so, could you recommend some titles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Hal&lt;/p&gt;&#xA;" OwnerUserId="961" LastActivityDate="2015-03-31T00:30:36.107" Title="Has anyone written anything notable on the relation between mathematical progress and the simplification of proofs overtime?" Tags="|proofs|writing|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7741" PostTypeId="2" ParentId="7717" CreationDate="2015-03-30T20:19:27.743" Score="8" Body="&lt;p&gt;Maybe I could title my post as &quot;formal, instead of informal&quot;, to constrast it with the suggestions presented it so far. Although, I am not as opinionated to suggest one method in exclusion to the others. I think like all good things in life, mathematical method has to be mixed and matched. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure if letters or papers will count, but &lt;a href=&quot;http://www.cs.utexas.edu/~EWD/&quot; rel=&quot;nofollow&quot;&gt;E. W. Dijkstra was a prolific writer of letters to communicate with a scientific audience&lt;/a&gt;. For those not familiar with him, he is probably most well known today for &lt;a href=&quot;http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm&quot; rel=&quot;nofollow&quot;&gt;&quot;Dijkstra's algorithm&quot;&lt;/a&gt; to find the shortest path between two nodes on a graph  (which he writes, he came up with while out for a coffee with his wife, and thus one of his most trivial achievements). More significantly, he came up with the conceptual underpinnings of modern &lt;a href=&quot;http://research.microsoft.com/en-us/um/people/lamport/pubs/garbage.pdf&quot; rel=&quot;nofollow&quot;&gt;garbage collection routines&lt;/a&gt; now common for memory management on computers and ways to reason about parallel programming algorithms (&lt;a href=&quot;http://en.wikipedia.org/wiki/Semaphore_(programming)&quot; rel=&quot;nofollow&quot;&gt;semaphores&lt;/a&gt; were invented by Dijkstra -- terminology and concept). &lt;a href=&quot;http://amturing.acm.org/award_winners/dijkstra_1053701.cfm&quot; rel=&quot;nofollow&quot;&gt;He won the Turing Award&lt;/a&gt; in computing science for:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For fundamental contributions to programming as a high, intellectual challenge; for eloquent insistence and practical demonstration that programs should be composed correctly, not just debugged into correctness; for illuminating perception of problems at the foundations of program design.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Dijkstra partially received the Turing award for the eloquence of his &lt;em&gt;opinionated&lt;/em&gt; arguments.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is worth noting that Dijkstra strongly thought of himself as a mathematician, (he was afterall, the chairman of &lt;a href=&quot;http://www.cs.utexas.edu/users/EWD/transcriptions/EWD04xx/EWD442.html&quot; rel=&quot;nofollow&quot;&gt;Mathematics Inc.&lt;/a&gt;) and that thus, he was involved in computing science as it was an application of mathematics of the highest order. However, Dijkstra was very opinionated about &lt;em&gt;discipline&lt;/em&gt; in thinking, which computing simply demanded, and that he believed that most mathematicians at the time lacked. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Here's a superbly convincing (and polemical!) paper written by him&lt;/strong&gt;: &lt;a href=&quot;https://www.cs.utexas.edu/users/EWD/transcriptions/EWD10xx/EWD1036.html&quot; rel=&quot;nofollow&quot;&gt;On the cruelty of really teaching computing science&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leslie Lamport, one of the people who collaborated with Dijkstra and was strongly influenced with him &lt;strong&gt;recently wrote a similarly opinionated piece (consider the title alone!) regarding how proofs in mathematics should be structured and presented&lt;/strong&gt;: &lt;a href=&quot;http://research.microsoft.com/en-us/um/people/lamport/pubs/proof.pdf&quot; rel=&quot;nofollow&quot;&gt;How to write a 21st century proof&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think you'll think too highly of Spivak's Calculus after you're done with Lamport and Dijkstra!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lamport in particular uses specific examples where Dijkstra did not, but the general idea of both of these men is that proof construction as taught by mathematicians of old is awfully disorganized and clumsy, and most importantly: &lt;em&gt;horrendous&lt;/em&gt; for education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;I must end with a textbook:&lt;/strong&gt; &lt;a href=&quot;http://algorithmicproblemsolving.org/aps/&quot; rel=&quot;nofollow&quot;&gt;Algorithmic Problem Solving&lt;/a&gt; by Roland Backhouse (another one of Dijkstra's disciples, if you will) presents a whole textbook (aimed for the beginner) on the type of elegant and precise proofs that Dijkstra and Lamport pushed for. He is just as opinionated as Lamport and Dijkstra, even if his language does not match the flourish of Dijkstra's. Perhaps he can be forgiven for that, given that he is writing a textbook, after all, not a piece of rhetoric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You'll find it difficult to write proofs &quot;the old way&quot;, after you are done, and you'll likely hate your high school math teachers even more than before. &lt;/p&gt;&#xA;" OwnerUserId="2013" LastEditorUserId="2013" LastEditDate="2016-02-20T00:36:56.280" LastActivityDate="2016-02-20T00:36:56.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7742" PostTypeId="2" ParentId="4351" CreationDate="2015-03-30T23:11:44.053" Score="21" Body="&lt;p&gt;Theorem: The largest positive integer is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Proof:  If &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a positive integer and &lt;span class=&quot;math-container&quot;&gt;$n \not= 1$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$n^2 &amp;gt; n$&lt;/span&gt;, so there is an integer larger than &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. Thus the largest integer has to be &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.  QED&lt;/p&gt;&#xA;&lt;p&gt;EDIT:  I learned from the answer by John B. &lt;a href=&quot;https://math.stackexchange.com/questions/2582488/will-assuming-the-existence-of-a-solution-ever-lead-to-a-contradiction&quot;&gt;here&lt;/a&gt; that this example goes back at least to Oskar Perron in 1913. See p. 532 of the reference in John's answer.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2021-01-03T19:07:14.627" LastActivityDate="2021-01-03T19:07:14.627" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7743" PostTypeId="2" ParentId="7740" CreationDate="2015-03-31T00:30:36.107" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Thurston, William P. &quot;On proof and progress in mathematics.&quot; &lt;em&gt;New Directions in the Philosophy of Mathematics&lt;/em&gt; (1998): 337-55. (&lt;a href=&quot;http://arxiv.org/abs/math/9404236&quot;&gt;arXiv abstract link&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think that Thurston's famous essay supports the notion that simpler proofs&#xA;are a mark of progress in mathematics, because simpler proofs lead to &#xA;easier communication and deeper understanding, and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;The measure of our success [in mathematics] is&#xA;  whether what we do enables people to understand and think more&#xA;  clearly and effectively about mathematics.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An example is deBranges 1984 proof of the Bieberbach Conjecture,&#xA;which was long and complicated but eventually simplified to a four-page&#xA;proof by Weinstein in 1991 (and further &quot;simplified&quot; by use of Zeilberger's&#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/Wilf%E2%80%93Zeilberger_pair&quot;&gt;computer WZ method&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-03-31T00:30:36.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7744" PostTypeId="2" ParentId="4351" CreationDate="2015-03-31T01:19:30.873" Score="8" Body="&lt;p&gt;I'm not sure if this fits, but it seems related to me:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You know the old circle cutting problem?  What is the maximum number of pieces a disc can be cut into using n straight cuts?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, let us number these regions using a binary string according to if the region is &quot;above&quot; or &quot;below&quot; the lines.  (The lines may be constructed with all the same signed slope) So &quot;1101&quot; corresponds to a region (in the n=4 case) that is above line a, above line b, below line c, and above line d.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the n=3 case illustrated: &lt;img src=&quot;https://i.stack.imgur.com/TRkKg.png&quot; alt=&quot;3 cuts of a disc&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So where is the &quot;010&quot; region?  If there is an 8th region, it must be below a, above b, below c, thus 010... but those areas do not intersect, so that 8th region does not exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is n=4, four cuts obtain 11 regions maximally: &lt;img src=&quot;https://i.stack.imgur.com/AMoti.png&quot; alt=&quot;Four cuts, 11 regions&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We &quot;miss&quot; five combinations of half-planes with four lines: those that correspond to 0010, 0100, 0101, 0110, 1010.  (I also have translated the binary strings to base 10 numerals as a easy and interesting way to keep track of what gets left out)&lt;/p&gt;&#xA;" OwnerUserId="4962" LastActivityDate="2015-03-31T01:19:30.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7745" PostTypeId="1" CreationDate="2015-03-31T01:44:07.860" Score="12" ViewCount="346" Body="&lt;p&gt;In 1992 &lt;a href=&quot;http://hplengr.engr.wisc.edu/Math_Schoenfeld.pdf&quot; rel=&quot;noreferrer&quot;&gt;Schoenfeld wrote an excellent &quot;review&quot;&lt;/a&gt; of (among other things) metacognition as it applies to mathematics: whether from the perspective of a student, or a teacher. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Metacognition, as quoted by Schoenfeld from Flavell, is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Metacognition refers to one's knowledge concerning one's own cognitive&#xA;  processes or anything related to them...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In Schoenfeld's lengthy review, the sections most relevant to metacognition are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Self-regulation, or monitoring and control (pg. 57 of &lt;a href=&quot;http://hplengr.engr.wisc.edu/Math_Schoenfeld.pdf&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Beliefs and affects (pg. 67 of &lt;a href=&quot;http://hplengr.engr.wisc.edu/Math_Schoenfeld.pdf&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Teacher beliefs (pg.71 of &lt;a href=&quot;http://hplengr.engr.wisc.edu/Math_Schoenfeld.pdf&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Societal beliefs (pg. 73 of &lt;a href=&quot;http://hplengr.engr.wisc.edu/Math_Schoenfeld.pdf&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Practices (pg. 74 of &lt;a href=&quot;http://hplengr.engr.wisc.edu/Math_Schoenfeld.pdf&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;1992 was a long while back. What are some newer resources that talk about metacognition? One book I can think of is &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/039916524X&quot; rel=&quot;noreferrer&quot;&gt;A mind for numbers, by Barbara Oakley&lt;/a&gt;. Any other recommendations?&lt;/p&gt;&#xA;" OwnerUserId="2013" LastEditorUserId="143" LastEditDate="2015-03-31T18:27:29.773" LastActivityDate="2015-03-31T18:27:29.773" Title="Books on meta-cognition that would be relevant for those involved in mathematics?" Tags="|mathematical-pedagogy|meta-cognition|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7746" PostTypeId="2" ParentId="2790" CreationDate="2015-03-31T01:53:31.850" Score="2" Body="&lt;p&gt;I don't think there is a need to begin with a programming language, because the language of mathematical proof already is like a programming language. Consider &lt;a href=&quot;http://algorithmicproblemsolving.org/aps/what-is-algorithmic-problem-solving/&quot; rel=&quot;nofollow&quot;&gt;Algorithmic Problem Solving by Roland Backhouse&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The advantage of thinking about formal mathematics &lt;em&gt;as&lt;/em&gt; a program (rather than the other way around) is that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the pedagogical approach is computer language agnostic and allows students to transfer &quot;algorithmic problem solving&quot; skills to any programming language, or formal mathematics notation;&lt;/li&gt;&#xA;&lt;li&gt;it allows students to realize that writing formal proofs is as &lt;em&gt;creative&lt;/em&gt; an endeavour as writing a program, and more fundamentally so, because it is the basis of &lt;em&gt;all&lt;/em&gt; programming!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2013" LastActivityDate="2015-03-31T01:53:31.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7747" PostTypeId="2" ParentId="4118" CreationDate="2015-03-31T01:59:22.313" Score="8" Body="&lt;p&gt;I am going to copy and paste my answer from another question on this site, because I think one would be hard pressed to beat it in terms of the number of suggestions it covers, and the general quality with which it presents these suggestions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You might be interested in the expansive answers that were generated on math.stackexchange by the questions &lt;a href=&quot;https://math.stackexchange.com/questions/828458/book-ref-request-starting-from-a-mathematically-amorphous-problem-and-comb/889732#889732&quot;&gt;Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/questions/828648/book-series-like-ams-student-mathematical-library&quot;&gt;Book series like AMS' Student Mathematical Library?&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In order to make this answer complete in its own right, and in order to reduce the number of times one has to depress a mouse button, I will summarize the suggestions from those threads here, giving credit to those who originally provided them. &lt;/p&gt;&#xA;&#xA;&lt;h1&gt;“…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/828458/book-ref-request-starting-from-a-mathematically-amorphous-problem-and-comb/889732#889732&quot;&gt;Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”&lt;/a&gt; was asked due to inspiration from from Charles Radin's &lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=STML-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Miles of Tiles&lt;/a&gt;, which has the following description:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Theme:&lt;/strong&gt; &quot;In this book, we try to display the value (and joy!) of starting from a mathematically amorphous problem and combining ideas from diverse sources to produce new and significant mathematics--mathematics unforeseen from the motivating problem ... &quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Style:&lt;/strong&gt; The common thread throughout this book is &lt;code&gt;&amp;lt;insert topic here&amp;gt;&lt;/code&gt;...the presentation uses many different areas of mathematics and physics to analyze features of &lt;code&gt;&amp;lt;insert topic here&amp;gt;&lt;/code&gt;...[as] understanding &lt;code&gt;&amp;lt;insert topic here&amp;gt;&lt;/code&gt; requires an unusual variety of specialties...this interdisciplinary approach also leads to new mathematics seemingly unrelated to &lt;code&gt;&amp;lt;insert topic here&amp;gt;&lt;/code&gt;...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Readership:&lt;/strong&gt; Advanced undergraduates, graduate students, and research mathematicians.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;mweiss further suggested Soifer's &lt;a href=&quot;http://www.ams.org/bookstore/mawrldseries&quot; rel=&quot;nofollow noreferrer&quot;&gt;How does One Cut a Triangle?&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You may enjoy Alexander Soifer's book &lt;a href=&quot;http://www.ams.org/bookstore/mawrldseries&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Does One Cut a Triangle?&lt;/a&gt;.  From my review of this on Math Reviews (MR#2548775):&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;Indeed the entire work is a sequence of problems posed and solved, with each new solution yielding, through generalization and specialization, new questions. One of the most noteworthy features of the text is its “just-in-time” approach to introducing new ideas: tools from linear algebra (linear independence and eigenvalues), Diophantine and algebraic equations, calculus (the intermediate value theorem), combinatorics (the pigeonhole principle), and affine geometry are brought in with a minimum of fuss precisely when they are most useful.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/829889/115703&quot;&gt;Conifold&lt;/a&gt; further provided an absolute wealth of suggestions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1) Books with light prerequisites&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/bookstore/mawrldseries&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stories of Maxima and Minima&lt;/a&gt; by Tikhomirov, a guided tour of extremal problems starting with Dido and the founding of Carthage all the way to convex programming with geometry, optics and mechanics visited along the way. While the author aims the book at &quot;high school students&quot; he means Russian ones perhaps. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.maa.org/book-series/classroom-resource-materials?page=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Indra's Pearls: The Vision of Felix Klein&lt;/a&gt; has Mumford (that one) for one of the authors, and a &lt;a href=&quot;http://en.wikipedia.org/wiki/Indra%27s_Pearls_%28book%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia article&lt;/a&gt; devoted to it, saves me the effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books/about/G%C3%B6del_Escher_Bach_Anniversary_Edition.html?id=aFcsnUEewLkC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gödel, Escher, Bach&lt;/a&gt; by Hofstadter is a book with almost cult following, also has a &lt;a href=&quot;http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia article&lt;/a&gt;. Very roughly, looks into how recursion and self-reference lead to expressing meaning in formal systems, music and art. Goes in depth into Gödel's incompleteness and mathematical themes of Escher and Bach, while staying a literary marvel that won a Pulitzer prize. According to Martin Gardner, &quot;a book of such depth, clarity, range, wit, beauty and originality that it is recognized at once as a major literary event&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=uVE_LiXbSpoC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fibonacci Numbers&lt;/a&gt; by Vorobiev studies the title subject by introducing modular arithmetic, recurrence relations and continued fractions, then discusses their role in approximating irrationals by fractions, Fibonacci enumeration system for integers and its application to winning a Chinese game, their appearence in geometry alongside the golden ratio, and in the theory of search.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0821838598&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Gift I-III&lt;/a&gt; by Ueno, Shiga and Morita is a well designed intuitive transition into graduate notions of geometry and topology, with highlights including Poincare-Hopf and Gauss-Bonet theorems, theories of dimension and volume (with Banach-Tarsky paradox explained), Poncelet closure theorem in projective geometry, Whitney embedding theorem, and Dehn's solution to the third Hilbert problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0817633162&quot; rel=&quot;nofollow noreferrer&quot;&gt;Felix Klein and Sophus Lie&lt;/a&gt; by Yaglom is an inspired story of how a mathematical theory is born, the theory of symmetry. The content is much broader than the title, related ideas of Galois, Poncelet, Hamilton, Grassmann, Cayley, Peirce, Clifford are thoroughly explored as well. Most insightful historical account of 19th century geometry and algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cambridge.org/ar/academic/subjects/mathematics/recreational-mathematics/series/outlooks?layout=grid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Knot Book&lt;/a&gt; by Colin Adams is a gem that takes one from knotting and braiding rope to topological invariants, Seifert surfaces, 3-manifolds by surgery and applications in biology, chemistry and physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://cms.math.ca/Publications/Books/treatises.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Excursions into Mathematics&lt;/a&gt; by Beck, Bleicher and Crowe is a collection of 6 mini-books under one cover. My favorite ones are on perfect numbers, the ancient topic that launched much of modern number theory (which still can't answer some basic questions about them), and on exotic geometries. You may like that one because it comes close to &quot;laying down the axioms and playing with them&quot; from your other question, albeit in geometry rather than algebra. From Euclid's postulates to Hilbert's axioms, what happens if some of them are modified, on to Latin squares, arithmetic of finite fields, lines and circles in finite projective spaces, and geometries they create. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=7US0cSy70hoC&amp;amp;dq=ash+gross+fearless+symmetry&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fearless symmetry&lt;/a&gt; by Ash and Gross is not a text for liberal arts majors despite the title.  It  sets out to outline a proof of the Last Fermat Theorem to non-experts with all the jazz of quadratic reciprocity, modular forms, algebraic integers, Galois group of $\mathbb{Q}$ and its representations on elliptic curves, traces of Frobenius elements, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=3RLGKcEjVIoC&amp;amp;dq=moore+zermelo%27s+axiom&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zermelo's Axiom of Choice&lt;/a&gt; by Moore. AC with its controversies and history up to Gödel and Cohen, and equivalents and consequences in algebra, topology and analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521666465&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proofs and Confirmations&lt;/a&gt; by Bressoud follows your wishes very closely. It is a thrilling story of proving a conjecture about the total number of alternating sign matrices that draws on insights about partitions, symmetric functions, hypergeometric series, lattice paths and statistical mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2) Advanced Books&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=ECnHLtiCiNsC&amp;amp;dq=hardy+ramanujan&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=zxyeU6XSK86uyASu3IHgBA&amp;amp;ved=0CB4Q6AEwAA&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ramanujan&lt;/a&gt; by Hardy is not a biography but a look at Ramanujan's enigmatic mathematical legacy by the man who knew him best. Hardy explains and ties together Ramanujan's 'magic' insights into primes, partitions, hypergeometric series, zeta function, elliptic and modular forms. Understanding the genesis of analytic number theory is a side bonus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://cms.math.ca/Publications/Books/treatises.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Exploring the Number Jungle&lt;/a&gt; by Burger. The theme is approximating irrationals by fractions with relatively small denominators, a.k.a. Diophantine approximation. But that doesn't stop Riemann surfaces, elliptic curves, Pythagorean triples, quadratic forms and $p$-adic numbers from showing up. It is unusually written: there are descriptions, questions, theorems, exercises, hints, but no proofs. On principle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.librarything.com/series/New+Mathematical+Library&quot; rel=&quot;nofollow noreferrer&quot;&gt;Radical Approach to Real Analysis&lt;/a&gt; also by Bressoud is a very unconventional exposition of the subject that starts with the crisis in mathematics posed by the discovery of Fourier series and develops ideas in a very versatile manner, highlighting perspectives lost in modern texts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=3RLGKcEjVIoC&amp;amp;dq=moore+zermelo%27s+axiom&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Coloring Book&lt;/a&gt; by Soifer, who also wrote How Does One Cut a Triangle. Coloring everything here: polygons, graphs, plane, space, integers, arithmetic progressions, but it all ties to the chromatic number of the plane. Which depends on the axiom of choice and existence of inaccessible cardinals (not kidding!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=6kCIAwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Glimpses of Soliton Theory&lt;/a&gt; by Kasman is a rare book on the subject that doesn't just throw cumbersome computations and transformations at the reader. Intuition for non-linear PDE-s is built up through examples and history, and then supplemented with ideas about elliptic curves, isospectrality, wedge products, pseudo-differential operators and the Grassmann cone. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0883850362&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tour Through Mathematical Logic&lt;/a&gt; by Wolf is a historically driven exposition of advanced modern logic including Gödel's incompleteness and constructible hierarchy, model theory, Cohen's forcing, Robinson's non-standard and Bishop's constructive analyses, large cardinals, determinacy and the Woodin program. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.librarything.com/series/Dolciani+Mathematical+Expositions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Methods of Classical Mechanics&lt;/a&gt;, Arnold's classic, is about mechanics obviously. And about differential forms, Poisson structures, symplectic manifolds, geodesic flows, Legendre transforms and singularities, to name a few. According to a MathSciNet reviewer a unique element in the intersection of &quot;the most influential books of the second half of this century, the most frequently quoted books, books that have the highest probability of surviving into the 21st century, books that are very useful in teaching, books characterized by a very strong personal style, books that provide a delightful reading experience.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Free electronic versions may be available &lt;a href=&quot;https://www.librarything.com/series/London+Mathematical+Society+Student+Texts&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; or &lt;a href=&quot;http://bib.tiera.ru/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some more suggestions were provided by &lt;a href=&quot;https://math.stackexchange.com/a/889732/115703&quot;&gt;user89&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cambridge.org/ca/academic/subjects/mathematics/logic-categories-and-sets/counting-frameworks-mathematics-aid-design-rigid-structures&quot; rel=&quot;nofollow noreferrer&quot;&gt;Counting on Frameworks&lt;/a&gt; ...fits the style of my original request. The author builds up a mathematically amorphous problem (&quot;what is a rigid structure?&quot;), progressively developing a theory (along with the participation of the reader through exercises) to describe &quot;rigidity&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.dartmouth.edu/news-resources/electronic/kpbogart/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Combinatorics through guided discovery, by Kenneth Bogart&lt;/a&gt;. It seems to be quite fun! You learn combinatorics by going through exercises, rather than by being &lt;em&gt;told&lt;/em&gt; what it is -- in other words, you make up the subject yourself, in a guided way. The book is completely free, and available online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cs.utoronto.ca/~hehner/FMSD/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Formal Methods of Software Design&lt;/a&gt;, available online for free..., is an excellent introduction to boolean logic, and general theory building -- it has been a fantastic base for learning other math. As a cool side effect (or was that the original goal?), you'll learn how to derive computer programs (yes, computer programs), from boolean specifications: very much like how you would derive any other proof! Fantastic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.coursera.org/course/learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Learning how to Learn&lt;/a&gt; ...(offered by the UC San Diego, on Coursera), which is actually, hands down, the most honest and effective course (good MOOCs are rare these days) on the subject that I have ever come across. Generally useful ideas to keep in mind while learning new mathematics!&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Book series like AMS' Student Mathematical Library&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/q/828648/115703&quot;&gt;This&lt;/a&gt; reference request was created because:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I had the joy of discovering AMS' &lt;a href=&quot;http://www.ams.org/bookstore/stmlseries&quot; rel=&quot;nofollow noreferrer&quot;&gt;Student Mathematical Library&lt;/a&gt; book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect stepping stone for learning!&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I am familar with some Springer book series (Undergraduate/Graduate Texts in Mathematics), but I think those have a much more of a textbook nature in general.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What are some great book series that fit the style of Student Mathematical Library?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/828458/book-ref-request-starting-from-a-mathematically-amorphous-problem-and-comb&quot;&gt;See this question for inspiration as to what the answers should look like.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/863799/115703&quot;&gt;Conifold&lt;/a&gt; once again delivers an excellent answer: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally what I think makes such series so good is that the format forces the authors to explain non-trivial and often non-elementary mathematics in accessible and inspiring way. The concentration of mathematically &quot;clever&quot; and &quot;cool&quot; both fascinates and challenges. They also often expose parts and perspectives of mathematics that are largely missing in standard texts and approaches. Of course, not every book in a series is equally good, so I will list some that I find particularly outstanding. But I haven't read all of them, so it doesn't mean that the rest are sub par, and my assessment of level only applies on average.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not a series per se but similar in spirit and close to the upper undergraduate level of AMS Student Mathematical Library (and cheap) are some (most are just texts) of the &lt;a href=&quot;http://store.doverpublications.com/by-subject-mathematics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dover Books on Mathematics&lt;/a&gt;: Riemann's Zeta Function; Three Pearls of Number Theory; Geometry and Light; Counterexamples in Topology; Regular polytopes; Beauty of Geometry; Asymptotic methods in Analysis; Satan, Cantor and Infinity; Hyperbolic Functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of Cambridge University Press' &lt;a href=&quot;https://www.librarything.com/series/London+Mathematical+Society+Student+Texts&quot; rel=&quot;nofollow noreferrer&quot;&gt;London Mathematical Society Student Texts&lt;/a&gt; are more than typical texts, and they are at the right level too: Prime Number Theorem, Undergraduate Algebraic Geometry, Elliptic Functions, Young Tableaux. Also good but very short are their &lt;a href=&quot;http://www.cambridge.org/ar/academic/subjects/mathematics/recreational-mathematics/series/outlooks?layout=grid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Outlooks&lt;/a&gt;, and Canadian Mathematical Society's &lt;a href=&quot;https://cms.math.ca/Publications/Books/treatises.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Treatises in Mathematics&lt;/a&gt; series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MAA and Cambridge University Press support &lt;a href=&quot;http://www.librarything.com/series/Dolciani+Mathematical+Expositions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dolciani Mathematical Expositions&lt;/a&gt;, which is freshman/sophomore level: Charming Proofs; Diophantus and Diophantine equations; Logic as Algebra. MAA's &lt;a href=&quot;http://www.maa.org/book-series/classroom-resource-materials?page=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Classroom Resource Materials&lt;/a&gt; also has some entries at this level: Paradoxes and Sophisms in Calculus; Counterexamples in Calculus; Explorations in Complex Analysis; Which Numbers are Real?; Real Infinite Series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ones below, especially Mir's, are generally less advanced, high school/freshman level. Still, I grew up reading such booklets, and learned from them more than from most formal studying, they also guided my interests later and helped select topics which I wanted to pursue in depth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;AMS's &lt;a href=&quot;http://www.ams.org/bookstore/mawrldseries&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical World&lt;/a&gt;: A Mathematical Gift, I, II, III; Mathematical Ciphers: From Caesar to RSA; Kvant Selecta (collections of best articles from Russian math journal for advanced high school kids); Stories about Maxima and Minima.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MAA's &lt;a href=&quot;http://www.librarything.com/series/New+Mathematical+Library&quot; rel=&quot;nofollow noreferrer&quot;&gt;New Mathematical Library&lt;/a&gt;: Game Theory and Strategy; Geometry of Numbers; Numbers: rational and irrational; Ingenuity in Mathematics; Geometric transformations; Uses of Infinity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mir's &lt;a href=&quot;http://mirtitles.org/category/little-mathematics-library/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Little Mathematics Library&lt;/a&gt;: Proof in Geometry; Solving Equations in Integers; Inequalities; Areas and Logarithms; Remarkable Curves.&lt;/p&gt;&#xA;" OwnerUserId="2013" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-03-31T19:32:32.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7748" PostTypeId="2" ParentId="4351" CreationDate="2015-03-31T09:54:05.100" Score="18" Body="&lt;p&gt;Solve $$0x=1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $x\in\mathbb{R}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $x$ is a solution, then $$x=x1=x(0x)=0.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have used this example myself with great effect, since at the very first instance the students realise something absurd is about to happen.&lt;/p&gt;&#xA;" OwnerUserId="4971" LastActivityDate="2015-03-31T09:54:05.100" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7749" PostTypeId="2" ParentId="7717" CreationDate="2015-03-31T10:05:18.373" Score="2" Body="&lt;p&gt;Royden's &quot;Real Analysis&quot; has a Prologue to the student which is somewhat opinated, containing the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;A third method of proof is&#xA;  proof by contradiction or reductio ad absurdum: We begin with A&#xA;  and -iB and derive a contradiction. All students are enjoined in the&#xA;  strongest possible terms to eschew proofs by contradiction! There&#xA;  are two reasons for this prohibition: First, such proofs are very often&#xA;  fallacious, the contradiction on the final page arising from an&lt;br&gt;&#xA;  erroneous deduction on an earlier page, rather than from the&lt;br&gt;&#xA;  incompatibility of A with —i B. Second, even when correct, such a proof gives&#xA;  little insight into the connection between A and B, whereas both the&#xA;  direct proof and the proof by contraposition construct a chain of&#xA;  argument connecting A with B. One reason that mistakes are so&#xA;  much more likely in proofs by contradiction than in direct proofs or&#xA;  proofs by contraposition is that in a direct proof (assuming the&#xA;  hypothesis is not always false) all deductions from the hypothesis are&#xA;  true in those cases where the hypothesis holds, and similarly for&#xA;  proofs by contraposition (if the conclusion is not always true) the&#xA;  deductions from the negation of the conclusion are true in those&#xA;  cases where the conclusion is false. Either way, one is dealing with&#xA;  true statements, and one's intuition and knowledge about what is&#xA;  true help to keep one from making erroneous statements. In proofs&#xA;  by contradiction, however, you are (assuming the theorem true) in&#xA;  the unreal world where any statement can be derived, and so the&#xA;  falsity of a statement is no indication of an erroneous deduction. &quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="122" LastEditorUserId="376" LastEditDate="2016-02-19T12:34:00.340" LastActivityDate="2016-02-19T12:34:00.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7750" PostTypeId="2" ParentId="7739" CreationDate="2015-03-31T18:36:00.860" Score="5" Body="&lt;p&gt;This is a bit beyond what you had in mind, but here are &lt;a href=&quot;http://ceemrr.com/Geometry2/Pyramid_Cone/Pyramid_Cone_print.html&quot;&gt;some GIF visualizations of the volume of cones and pyramids&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was trying to understand why cones and pyramid are one third the volume of prisms and cylinder's, this &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/elements/bookXII/propXII7.html&quot;&gt;visualization of Euclid's proof helped me&lt;/a&gt;. Unfortunately, the Java applet no longer works (at least in my browser).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I taught these volume formulae, I found that &lt;a href=&quot;http://www.hand2mind.com/item/giant-geometric-solids/1178&quot;&gt;geometric solid manipulatives that fill with water&lt;/a&gt; to be a good teaching tool.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last, your students mind find some these &lt;a href=&quot;https://www.msu.edu/~stemproj/simulations%20-%20volume.html&quot;&gt;online virtual volume manipulatives&lt;/a&gt; helpful.&lt;/p&gt;&#xA;" OwnerUserId="182" LastActivityDate="2015-03-31T18:36:00.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7751" PostTypeId="2" ParentId="4351" CreationDate="2015-03-31T19:49:30.507" Score="6" Body="&lt;p&gt;Russel's paradox &quot;Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition&quot;, akin to the halting problem from computer science seems to fit the bill&lt;/p&gt;&#xA;" OwnerUserId="4976" LastActivityDate="2015-03-31T19:49:30.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7752" PostTypeId="2" ParentId="7739" CreationDate="2015-03-31T20:07:43.213" Score="4" Body="&lt;p&gt;These are a bit slow for my tastes, but ... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Box a.k.a. Rectangular Prism&#xA;(&lt;a href=&quot;https://www.youtube.com/watch?v=kWkKGYWWGqQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;YourTube link&lt;/a&gt;):&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/83JJO.png&quot; alt=&quot;Volume&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) Cylinder&#xA;(&lt;a href=&quot;http://www.ck12.org/geometry/Surface-Area-and-Volume-of-Cylinders/lesson/Cylinders-Intermediate/&quot; rel=&quot;nofollow noreferrer&quot;&gt;ck12.org link&lt;/a&gt;):&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Zb7or.png&quot; alt=&quot;Cylinder&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-03-31T20:07:43.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7753" PostTypeId="1" CreationDate="2015-03-31T20:43:18.900" Score="12" ViewCount="353" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Monty_Hall_problem&quot;&gt;Monty Hall problem&lt;/a&gt; is a classic probability riddle and I will be gleefully explaining it to my class of discrete math students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is &lt;em&gt;apparently&lt;/em&gt; based on his classic game show &lt;a href=&quot;http://en.wikipedia.org/wiki/Let%27s_Make_a_Deal&quot;&gt;Let's Make a Deal&lt;/a&gt;. But I couldn't find any relevant footage despite &lt;a href=&quot;https://www.youtube.com/results?search_query=let%27s%20make%20a%20deal%20monty%20hall&quot;&gt;lots of old videos of the show&lt;/a&gt; being easily available online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any out there? (Bonus points if the contestant ends up with the goat.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit. Answer&lt;/strong&gt; (one of my colleagues shared a link with me): &lt;a href=&quot;https://www.youtube.com/watch?v=c1BSkquWkDo&quot;&gt;No&lt;/a&gt;, as explained by Hall himself.&lt;/p&gt;&#xA;" OwnerUserId="1535" LastEditorUserId="1535" LastEditDate="2015-03-31T21:33:02.063" LastActivityDate="2022-07-30T02:48:53.020" Title="Is there any footage of Let's Make a Deal illustrating the Monty Hall problem?" Tags="|reference-request|probability|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7754" PostTypeId="5" CreationDate="2015-04-01T03:05:05.960" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-04-01T03:05:05.960" LastActivityDate="2015-04-01T03:05:05.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7755" PostTypeId="4" CreationDate="2015-04-01T03:05:05.960" Score="0" Body="Meta-cognition is the subject of the study of cognitive processes involved in activities such as teaching, learning, or problem solving, and the study of the subsequent regulation of those cognitive processes based on knowledge regarding them." OwnerUserId="2013" LastEditorUserId="2013" LastEditDate="2015-04-01T09:10:49.693" LastActivityDate="2015-04-01T09:10:49.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7756" PostTypeId="2" ParentId="7717" CreationDate="2015-04-01T07:55:19.127" Score="8" Body="&lt;p&gt;Judea Pearl's &lt;a href=&quot;http://bayes.cs.ucla.edu/BOOK-2K/&quot; rel=&quot;noreferrer&quot;&gt;Causality&lt;/a&gt; is both the most opinionated textbook and one of the best textbooks I have ever read. The entire book is designed to convince the reader that it is causation, not correlation, that science should study, and that causation can be detected using statistical tools. It received the 2001 Lakatos Award.&lt;/p&gt;&#xA;" OwnerUserId="185" LastEditorUserId="185" LastEditDate="2015-04-01T10:40:57.757" LastActivityDate="2015-04-01T10:40:57.757" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7757" PostTypeId="1" CreationDate="2015-04-01T09:25:44.007" Score="15" ViewCount="594" Body="&lt;p&gt;An old favourite topic of mine to daydream about on pleasant afternoons is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you could completely redesign the university-level mathematics curriculum from the ground up to be as good as it could be, but without external constraints (e.g. designing a math curriculum for an isolated group of mountain monks with no interest in employment or in going to universities, but wouldn't mind winning Fields Medals), how would you do it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Constraints are as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The students do not necessarily have good pre-college background, and are not necessarily of &quot;high math aptitude&quot;, whatever that means. So the curriculum should be &lt;strong&gt;easier&lt;/strong&gt; than a regular undergraduate mathematics curriculum.&lt;/li&gt;&#xA;&lt;li&gt;There is no need to break things up into courses, and there is no need to grade or to write tests. Thus, nothing ever needs to be taught &quot;because it's easy to write easily-graded exercizes about it&quot;.&lt;/li&gt;&#xA;&lt;li&gt;There is no back-compatibility requirement. No need to conform to anyone's expectations of what student ought to learn or what will land them nice jobs.&lt;/li&gt;&#xA;&lt;li&gt;Students are assumed to be highly motivated and to always pay attention.&lt;/li&gt;&#xA;&lt;li&gt;There is no need to restrict to &quot;pure math&quot;. A physics or engineering or finance example or technique or insight are entirely admissible.&lt;/li&gt;&#xA;&lt;li&gt;The objective is to maximize student's understanding of mathematics and their abilities both to construct theories (Grothendieck) and to solve problems (Erdos).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Relaxing some of these constraints, and in particular assuming students of high mathematical aptitude, an attempt along these lines might include Bourbaki's work or Math55 in Harvard, although these are strongly biased in favour of &quot;pure math&quot; and are hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I posted on this topic on Facebook, and received a wonderful array of comments from mathematicians and math educator friends. The ideas were interesting, and collecting more would also be fantastic (but that isn't the question). I also saw &lt;a href=&quot;https://mathoverflow.net/questions/201506/epicycles-ptolemy-style-in-math-theory/201518#201518&quot;&gt;this intriguing argument by Alexandre Eremenko&lt;/a&gt; that vector calculus ought to be dropped in favour of differential forms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My actual question follows below:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;&lt;b&gt;Question&lt;/b&gt;: Has there been an attempt to design a utopian mathematics undergraduate-level curriculum, one of whose goals is to be easier for students than current curricula, with no external constraints (in particular, no need to test students and no compatibility constraints)? Has there been any research on the topic of what such a curriculum would include and how specifically it would differ from existing curricula? Could you point me towards such references? &#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="185" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-12-30T04:05:47.893" Title="Thought experiment: Utopian college-level math curriculum without external constraints" Tags="|reference-request|curriculum|" AnswerCount="4" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="7758" PostTypeId="2" ParentId="7528" CreationDate="2015-04-01T12:49:43.287" Score="9" Body="&lt;p&gt;An ellipse as a cylindric section: The top surface of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Tycho_Brahe_Planetarium&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tycho Brahe Planetariun&lt;/a&gt;, Copenhagen, Denmark.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/dZmIP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The building itself is a &lt;a href=&quot;http://mathworld.wolfram.com/CylindricalSegment.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;cylindrical segment&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="39" LastEditorUserId="39" LastEditDate="2015-04-01T15:05:06.177" LastActivityDate="2015-04-01T15:05:06.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7759" PostTypeId="2" ParentId="2164" CreationDate="2015-04-01T12:53:19.027" Score="8" Body="&lt;p&gt;I had tried a few different strategies in high school with little success.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My first attempt was having the students read the textbook and take notes on what they read in a very prescribed way. At the beginning of the class I explained that this was an attempt to teach them how to read a math textbook, and how that was different than any other kind of book. I would give them an open note quiz before we started a section. It was helpful for some students, but for the most part it was a way to reward good students and punish poor students. Overall I decided it was unhelpful at teaching the students who needed the help most of all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried something like a flipped classroom for some lessons. I would assign reading and videos for homework (after I had determined that the whole class could access the videos), and then the next day I would develop the skills beyond what they had learned on their own. This was a disaster, far too many students didn't do the reading, or watch the videos. Which left a handful of students actually prepared, but the majority of class unprepared for the more advanced topics. I still think this could be successful, but it would require more commitment from the teacher, and buy in from the parents. Also age and habits are difficult to get a handle on, older kids can better accept the responsibility, but are more ingrained with the habits of a traditional classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most recent effort I had was to increase the amount of writing. I assigned traditional homework, and provided solutions for all of the homework problems, but what I checked for credit was a card with three questions. One question would be one of the homework problems, I never revealed what this would be in advance. The last two questions would be more thought provoking: &quot;Describe the differences between ellipses and hyperbolas&quot; or &quot;When might a polynomial have no real solutions?&quot; Sometimes I would give them the writing questions the day before, and sometimes I would give the questions at the beginning of class and allow a few minutes to write. Some of the questions would reference their notes or the textbook. I would accept almost anything for credit except &quot;I don't know,&quot; if a student didn't know the answer I tried to get them to tell me what they thought might be the answer and why they weren't sure. I read all the cards and used that to help drive the lessons for the following days. This was the most effective method I tried to increase reading, but it was the most work on my part and only had an incidental effect on reading.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2015-04-01T12:53:19.027" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7760" PostTypeId="1" AcceptedAnswerId="7765" CreationDate="2015-04-01T16:00:53.377" Score="3" ViewCount="399" Body="&lt;p&gt;So, in my secondary Algebra II class we have gotten to the unit on probability and counting and as part of that we are looking at permutations.  We have looked at  permutations of $n$ objects, permutations of $n$ objects taking $r$ at a time, and permutations with identical objects.  The first two types of permutations my students understand pretty well, both intuitively and the formulas.  We talked about these in terms of &quot;slots and options&quot;, i.e. &quot;If you have three slots and ten objects, then you have 10 options for the first slot, 9 options for the second slot, and 8 options for the third slot and by the fundamental counting principle we multiply these to get the total number of permutations.&quot;  I was actually able to have the students get up and physically perform these permutations by standing in different orders in front of the class and this really seemed to cement in these first two types of permutations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem arose when we got to the idea of permutations with identical objects.  I tried to explain it as &quot;the total number of permutations for the objects ($n!$) and then divide out the number of repeats ($r_1!r_2!...$)&quot;  A few of my students were good with this but for the most part, the class was not able to see the connection between the situations we were looking at and the formula and ended up very confused.  Does anyone have a better, more intuitive way to explain permutations with identical objects? or perhaps a demonstration, similar to what I did with having my students line up in front of the class, that could make this concept clear?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-04-01T21:09:12.753" Title="Permutations with identical objects" Tags="|secondary-education|algebra|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7761" PostTypeId="1" CreationDate="2015-04-01T16:28:59.437" Score="13" ViewCount="2348" Body="&lt;p&gt;Our undergraduate university department is looking to spruce up our rooms and hallways a bit and has been thinking about finding mathematical posters to put in various spots; hoping possibly to entice students to take more math classes. We've had decent success in finding &quot;How is Math Used in the Real World&quot;-type posters (mostly through AMS), but we've been unable to find what I would call interesting/informative math posters. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I remember seeing a poster once (put out by Mathematica) that basically laid out how to solve general quadratics, cubics, and quartics. Then it had a good overview of proving that no formula existed for quintics. So not only was it pretty to look at, but if you stopped to read it you actually learned something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of a company or distributor that carries a variety of posters like this? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've tried searching online but all that comes up is a plethora of posters of math jokes. And even though the application/career-based posters are nice and serve a purpose, I don't feel like you actually gain mathematical knowledge by reading them.&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="2083" LastEditDate="2019-03-30T06:56:38.540" LastActivityDate="2019-03-30T06:56:38.540" Title="Interesting Math Posters" Tags="|student-motivation|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7762" PostTypeId="2" ParentId="7761" CreationDate="2015-04-01T17:06:10.517" Score="5" Body="&lt;p&gt;There is a series of posters made by Key Curriculum Press that each detail a different culture and their contribution to mathematics.  They are colorful, informative, and have, in my experience, been very engaging for students to look at and read.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, with the McGraw-Hill purchase of Key Curriculum a few years ago, they have been pigeonholed into only working on the GSP program, so i can't find these poster sets on their website.  However, the following two links provide contact information that may or may not still be valid...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.maa.org/publications/periodicals/convergence/multicultural-classroom-posters-sets-1-2&quot; rel=&quot;noreferrer&quot;&gt;http://www.maa.org/publications/periodicals/convergence/multicultural-classroom-posters-sets-1-2&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.maa.org/publications/periodicals/convergence/multicultural-classroom-posters-sets-3-4&quot; rel=&quot;noreferrer&quot;&gt;http://www.maa.org/publications/periodicals/convergence/multicultural-classroom-posters-sets-3-4&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not have the time to try and track these down but if you are successful in finding a place to purchase these posters, please let me know in the comments so I can update this answer!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Update:  While this is not the specific poster set that I was thinking of, these look nice and are in a similar vein to the Key Curriculum posters described above&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.enasco.com/product/TB18084T&quot; rel=&quot;noreferrer&quot;&gt;http://www.enasco.com/product/TB18084T&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2015-04-01T17:24:04.587" LastActivityDate="2015-04-01T17:24:04.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7763" PostTypeId="2" ParentId="7761" CreationDate="2015-04-01T18:32:37.137" Score="5" Body="&lt;p&gt;Various conference attendees sometimes have informative posters as part of their advertising campaigns.  I have an old poster (somewhere!) of the graph of the real and imaginary parts of the zeta function on the critical line, produced I think by Wolfram Research.  You might ask colleagues about promotional materials they have seen, and then check out the vendors and see what they have in their libraries/collections.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;And There's 'Men Of Mathematics'&quot; Paseman, 2015.04.01&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-01T18:32:37.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7764" PostTypeId="2" ParentId="7757" CreationDate="2015-04-01T18:47:11.723" Score="4" Body="&lt;p&gt;From my dim and tattered memory pops out the name of Stephen Leacock.  Assuming I got it right, he had some similar ideas placed on paper.  The economy of the situation reduced several departments and personnel to a large sitting room with comfortable chairs, a ready supply of cigars, ashtrays, and perhaps libation, and classes and lectures converted to discussions and reflections (sort of like the old Socratic school, but with better furniture).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In spite of the satirical nature of his essay (which I hope someone will recall), I think he anticipated many of your points, and that in this modern day and age, a utopian university would be electronic, hook up a student with a variety of materials as well as an appropriate set of tutors on a variety of subjects, and make the university a very large virtual sitting room, much as What's-His-Face wrote about many years ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Typed With A Straight Face&quot; Paseman, 2015.04.01&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-01T18:47:11.723" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7765" PostTypeId="2" ParentId="7760" CreationDate="2015-04-01T21:06:21.937" Score="8" Body="&lt;p&gt;Here are three ideas:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Numbered letters&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The classic problem of this type is &quot;How many ways are there to arrange the letters in MISSISSIPPI?&quot; Our previous knowledge of permutations only applies to ones where everything is different, so label the letters so we can tell them apart:$M_1I_1S_1S_2I_2S_3S_4I_3P_1P_2I_4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now there will be $11!$ arrangements, but if you took off the labels you'd find you couldn't tell them apart so you need to take that into account. Actually listing several arrangements and removing the labels on just one type of letter (say the S's) will make this clearer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Masked students&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have already had the students line up and rearrange themselves, so reuse the same idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This time give the students masks so that there are several students with each mask.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rearrange the students, and then get all the students with a particular mask to rearrange themselves. The class will see that the arrangement of masks doesn't change. So we will have counted all those permutations too many times. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Really this is just the same as &quot;numbered letters&quot; but more interactive and cute.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Choose where things go&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a completely different way to think about this situation using combinations instead of permutations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To construct an arrangement of MISSISSIPPI, you can think of the final arrangement as having 11 spots: ()()()()()()()()()()(). We have to choose which of these spots is M; then out of the 10 remaining spots, we have to choose which two of them are P; then out of the remaining 8 spots, we have to choose which four of them are I; and all the rest are S's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the number of arrangements is:&#xA;$$&#xA;{{11}\choose{1}} {{10}\choose{2}}{{8}\choose{4}}{{4}\choose{4}}.&#xA;$$&#xA;If you expand this out a whole lot of stuff will cancel and come to the formula you said. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I quite like showing students this at some point because it does highlight that there is often more than one way to do things. It also reinforces the idea that if you can have a story for how to construct the arrangement, then you can count how many ways each step of the story happens.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-04-01T21:06:21.937" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7766" PostTypeId="2" ParentId="7760" CreationDate="2015-04-01T21:09:12.753" Score="2" Body="&lt;h1&gt;Use colors!&lt;/h1&gt;&#xA;&lt;h2&gt;Variant 1&lt;/h2&gt;&#xA;&lt;p&gt;Use different objects like a red apple, a green apple and a pear. (Both apples should look the same except for their color. Let the students permute these objects. They count 6 &lt;strong&gt;different&lt;/strong&gt; permutations. Then apply a color filter, e.g. transform into grayscale. Now, both apples look the same. In your originally 6 different permutations, there are now 3 pairs of identical permutations, leaving 3 &lt;strong&gt;different permutations&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;h2&gt;Variant 2&lt;/h2&gt;&#xA;&lt;p&gt;Use identical objects like three apples and a pear, all in black-white. (The apples should look the same.) Let each of the students permute these objects. Everyone'll find 4 different permutations (the place of the peach being the differing aspect) and be pretty certain that these are all. Now let everybody color their food using red, green and yellow for the apples and a mix (or whatever) for the pear. Every one will still have 4 permutations on his sheet of paper, but after exchanging them, they'll have more, although everyone hat the same set of 4 original permutations.&lt;/p&gt;&#xA;&lt;h1&gt;Background&lt;/h1&gt;&#xA;&lt;p&gt;The real problem is not the counting or the fundamental counting principle. The real problem is the students' idea of &lt;strong&gt;identiy, equality and difference&lt;/strong&gt;. Although all apples look the same, the students think they know all apples are different. But what's the difference? What are sufficient conditions to say, two anything are equal or identical?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If all of their atoms are identical&lt;/li&gt;&#xA;&lt;li&gt;If &lt;em&gt;all&lt;/em&gt; properties are equal&lt;/li&gt;&#xA;&lt;li&gt;If all &lt;em&gt;relevant&lt;/em&gt; properties are equal&lt;/li&gt;&#xA;&lt;li&gt;if all &lt;em&gt;measurable&lt;/em&gt; properties are equal&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;After having discussed this, point them to the fact, that non of these conditions are mathematical or mathematically usable. The first one is physical. The apples on the papers aren't even real apples. They are ink on paper. What makes them apples is our imagination. The second one is impractical, as one can always invent a new property, like name (apple &amp;quot;Harry&amp;quot;, apple &amp;quot;Sandra&amp;quot; and apple &amp;quot;Nr. 5&amp;quot;.). The third one is philosophical/metaphysical. And the last one is scientific.&lt;/p&gt;&#xA;&lt;p&gt;Identity, equality and difference in mathematical terms are formal definitions. We, the mathematicians say, that these apples are equal or different. It doesn't matter, what. It does matter, what's the result of it.&lt;/p&gt;&#xA;" OwnerUserId="439" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-04-01T21:09:12.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7768" PostTypeId="2" ParentId="7761" CreationDate="2015-04-01T22:43:35.343" Score="11" Body="&lt;p&gt;One potential source is the AMS blog &lt;a href=&quot;http://blogs.ams.org/visualinsight/&quot; rel=&quot;noreferrer&quot;&gt;Visual Insights&lt;/a&gt;&#xA;run by John Baez. Each image comes with a clear mathematical story.&#xA;Most would not fit on one page, but you could make a poster by printing out&#xA;several pages and pasting them on a poster (or doing the equivalent electronically).&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/pLigc.jpg&quot; width=&quot;300&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;Schmidt Arrangement of the Eisenstein Integers – Katherine Stange.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/bxTjw.gif&quot; width=&quot;300&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;Icosahedron Illustrating Pentagon-Hexagon-Decagon Identity – Greg Egan.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-04-01T23:12:14.870" LastActivityDate="2015-04-01T23:12:14.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7770" PostTypeId="2" ParentId="7513" CreationDate="2015-04-02T00:02:54.930" Score="2" Body="&lt;p&gt;Essentially following up on Daniel Hast's comment (&lt;a href=&quot;https://matheducators.stackexchange.com/questions/7513/notation-for-an-element-in-a-polynomial-ring#comment15430_7515&quot;&gt;Notation for an element in a polynomial ring&lt;/a&gt;):  I think that the usual reason to distinguish between $f$ and $f(x)$ is to avoid a confusion between a function and its value; but, in this case, there is a further confusion to avoid, which is that between a polynomial and 'its function' (or rather its &lt;em&gt;family&lt;/em&gt; of functions, since a polynomial in $R[x]$ gives a function $A \to A$ for every $R$-algebra $A$).  I think that this argues &lt;em&gt;for&lt;/em&gt; the notation $f(x)$, because it emphasises (to those 'in the know') that $f(x)$ is &lt;em&gt;not&lt;/em&gt; a function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess the real question is:  what is your name for the obvious degree-$1$ polynomial?  It seems reasonable to call it $x$ (else what does the $x$ in the notation $R[x]$ mean?); but then, as quid points out at &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7513/notation-for-an-element-in-a-polynomial-ring#comment15434_7514&quot;&gt;Notation for an element in a polynomial ring&lt;/a&gt;, your element really and truly &lt;em&gt;is&lt;/em&gt; (say) $5x^2 + 7$, not &quot;the function whose value at $x$ is $5x^2 + 7$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, I think that, as you point out yourself, there is nothing logically wrong, by even the pickiest standards, with the notation $f(x)$, whereas there are solid reasons (well elucidated by quid at &lt;a href=&quot;https://matheducators.stackexchange.com/a/7514/2070&quot;&gt;https://matheducators.stackexchange.com/a/7514/2070&lt;/a&gt;) &lt;em&gt;not&lt;/em&gt; to use $f$; so, to the extent that the notation is yours (rather than forced on you by the need for consistency), I would advise going with the former.&lt;/p&gt;&#xA;" OwnerUserId="2070" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-04-02T00:02:54.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7771" PostTypeId="1" CreationDate="2015-04-02T00:41:33.253" Score="21" ViewCount="338" Body="&lt;p&gt;As a secondary teacher in an underfunded district, I get paid very little money for the time and energy I put in to my teaching.  I am totally cool with this, I never got into teaching for money, I enjoy a minimalist life style.  However, there are certain implements that are very well suited to instructional goals but cost a decent amount of money and therefore I always drag my feet to spend the money (whether the school's or my own) to purchase these implements.  I have been able to manage some hacks that I have found to be very useful in my classroom that cost little to no money to create.  They include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;using jelly beans and toothpicks instead of the heavy duty plastic sticks (we call them D-sticks, not sure if that is an official name or what) to create geometric solids and/or molecular structures&lt;/li&gt;&#xA;&lt;li&gt;Using webcam hardware/software on a laptop and a projector instead of having to purchase a $1000+ document camera (just have to make sure the software supports inverting the video image or else everything displays backwards)&lt;/li&gt;&#xA;&lt;li&gt;Using old folders for lots of uses including double sided straight edges of arbitrary width, rulers using printouts like &lt;a href=&quot;http://www.vendian.org/mncharity/dir3/paper_rulers/&quot;&gt;this&lt;/a&gt;, backing for using nets to build geometric solids, framing for student work, etc.&lt;/li&gt;&#xA;&lt;li&gt;Buying sidewalk chalk and a plastic chalk holder (about 12 bucks for a pack of 20 sticks and two holders) instead of the pitiful chalk my school buys.  Sidewalk chalk doesn't break, the holder keeps you hand from getting dusty, they write nice, fat characters that are readable even from the back of the room, they last forever (I have been using the same piece for over two months)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What are some other DIY/hack things that you all have done to create an implement/demo/etc that has been effective for instruction?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-08-07T01:13:48.133" Title="DIY/Hack Instructional Implements" Tags="|secondary-education|technology-in-education|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7772" PostTypeId="1" AcceptedAnswerId="7776" CreationDate="2015-04-02T04:03:38.623" Score="4" ViewCount="105" Body="&lt;p&gt;When I was taking the probability class myself, I remember my professor give us a standard &lt;em&gt;counter example&lt;/em&gt; of central limit theorem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $X_{i}\sim X$ be $i.i.d$ random variables. Let $E(X)=0, Var(X)=1$. Now by central limit theorem we have:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\sqrt{n}\overline{X}\rightarrow^{d} N(0,1)&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;counter example&lt;/em&gt; begins by assuming we can multiply $\sqrt{n}$ on both sides like a constant. Then we trivially have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\sum^{n}_{i=1}X_{i}\sim^{d} N(0,n)&#xA;$$&#xA;which is now nonsense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the problem is both left and right hand side are changing, and claiming something converge to $N(0,n)$ does not make sense. In general one needs something like &lt;a href=&quot;https://en.wikipedia.org/wiki/Slutsky&amp;#39;s_theorem&quot; rel=&quot;nofollow&quot;&gt;Slutsky's theorem&lt;/a&gt;. My question is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) From pedagogical point of view, is this still a good motivation for central limit theorem, that the shape of $\sum^{n}_{i=1}X_{i}$ is approximately a (rescaled) normal? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) For the best way to make sense of this statement, I am considering the following: Let $[A,B]\subseteq \mathbb{R}$ be any non-degenerate interval, then $$\forall \epsilon,\exists N, \forall n&amp;gt;N,|P(\sum^{n}_{i}X_{i}\in [A,B])-P(N(0,n)\in [A,B])|\le \epsilon$$&#xA;I feel this might be a good re-framing of convergence in distribution. But this is really appropriate?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask because my primary research field is not probability and I have never seen anyone approaching central limit theorem heuristically this way.&lt;/p&gt;&#xA;" OwnerUserId="3218" LastActivityDate="2015-04-02T14:37:19.043" Title="A different version of central limit theorem?" Tags="|probability|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7773" PostTypeId="2" ParentId="7757" CreationDate="2015-04-02T10:58:34.700" Score="15" Body="&lt;p&gt;Years ago, as an undergraduate student, I experienced something close to your description of a utopian mathematics undergraduate-level curriculum at Sharif University of Technology in Iran. We didn't mind winning Fields Medal. Indeed,one did: Maryam Mirzakhani. I remember, one of the courses we had was &quot;geometric analysis&quot;, quite uncommon as an undergraduate course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe the reason that  such a curriculum was designed was that the department had attracted nearly all Iranian medalists of The International Mathematical Olympiad and also a good numbers of some other highly motivated students. In a sense, as a student, you had a choice, to follow the enhanced curriculum or the standard one. Though in a sense the enhanced curriculum was very successful (apart from Maryam, I can name many other great mathematicians who experienced that programme), finally it ceased! I guess the reason was that the students of the other tribe (those who satisfied your first constraint) outnumbered and also running two parallel curriculum had many social issues.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Added. As I said in my comment below your question inspired me to document the history of those old good days. In that direction, I made a wiki in which people can write about what they experienced. It has just been started and most of the contributions at the moment has been naturally written in Persian. However, there are a few in English that I am sure give you some ideas of how strange was those days. Here is the link to one of them: &lt;a href=&quot;https://sharifmathoralhistory.wikispaces.com/-/Indoor/Class/Why+some+of+us+became+number+theorists&quot;&gt;Why some of us became number theorists&lt;/a&gt;    &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="1217" LastEditDate="2015-04-16T10:56:11.180" LastActivityDate="2015-04-16T10:56:11.180" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7774" PostTypeId="5" CreationDate="2015-04-02T14:25:14.690" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-04-02T14:25:14.690" LastActivityDate="2015-04-02T14:25:14.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7775" PostTypeId="4" CreationDate="2015-04-02T14:25:14.690" Score="0" Body="For questions on the use of mathematics in science, industry and everyday life." OwnerUserId="39" LastEditorUserId="39" LastEditDate="2015-04-02T16:33:18.953" LastActivityDate="2015-04-02T16:33:18.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7776" PostTypeId="2" ParentId="7772" CreationDate="2015-04-02T14:37:19.043" Score="3" Body="&lt;p&gt;My answer to 1) is &lt;strong&gt;yes&lt;/strong&gt;: this is what the TCL means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, your &quot;reformulation&quot; in 2) holds for far too many laws to be of any use. If you replace $N(0,n)$ by any sequence of laws such that the probability of any segment $[A,B]$ goes to $0$ (e.g. the uniform law on $[-n,n]$), then the same &quot;reformulation&quot; holds. It is in fact much, much weaker than the TCL, to the point of being useless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have to start from the formulation of the convergence of $\bar X/\sqrt{n}$, and &lt;em&gt;then&lt;/em&gt; rescale, i.e. you can compare the probability that $\bar X$ and respectively $N(0,n)$ gets into $[\sqrt{n} A,\sqrt{n} B]$; or equivalently, say that the probability that $\bar X$ gets into $[\sqrt{n} A,\sqrt{n} B]$ tends to the probability that $N(0,1)$ gets into $[A,B]$.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-04-02T14:37:19.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7777" PostTypeId="2" ParentId="7761" CreationDate="2015-04-02T14:58:29.850" Score="7" Body="&lt;p&gt;The German organization &lt;em&gt;Imaginary&lt;/em&gt; made a lovely set of posters showing algebraic surfaces. You can &lt;a href=&quot;http://www.imaginary-exhibition.com/poster.php&quot; rel=&quot;noreferrer&quot;&gt;buy a set&lt;/a&gt;, though they're in German. Rice University maths department has them on hanging in the corridors. Here's an example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/N28sv.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's an English translation available on their website, though it's not very idiomatic; I've sometimes thought of retranslating it, and I would if there was demand.&lt;/p&gt;&#xA;" OwnerUserId="369" LastActivityDate="2015-04-02T14:58:29.850" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7778" PostTypeId="1" AcceptedAnswerId="7806" CreationDate="2015-04-02T15:56:08.753" Score="19" ViewCount="1606" Body="&lt;p&gt;In France at least, there is quite an ancient tradition of teaching abstract affine spaces (e.g. as a triple $(\mathcal{E}, E, -)$ where $\mathcal{E}$ is a set, $E$ is a vector space and $-:\mathcal{E}\times\mathcal{E}\to E$ is a binary operation with the adequate properties) which somewhat continues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I liked that kind of approach as an undergrad, by I more and more feel it is artificial, and that we should restrict to study affine subspaces of vector spaces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Edit:&lt;/strong&gt; to be more precise, I am not against explaining that an affine space is like a vector space without a origin, on the contrary; but my point is that such concepts can be explained by sticking to affine subspaces of vector spaces (the vector space itself being an affine subspace, and the origin loosing its meaning in that structure).&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My first question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are some arguments in favor of teaching abstract affine spaces ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To explain more my reluctance, let me say I am turning more and more into a example-based mathematician and teacher; I am thus driven away from abstract affine spaces by the fact that I do not have a good answer to my second question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is an example of a &quot;natural&quot; affine space, which is not &quot;naturally&quot; an affine subspace of a vector space?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2015-04-03T12:51:40.013" LastActivityDate="2015-04-16T07:52:06.623" Title="Should we teach abstract affine spaces?" Tags="|undergraduate-education|geometry|" AnswerCount="11" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="7779" PostTypeId="2" ParentId="7778" CreationDate="2015-04-02T16:56:57.500" Score="12" Body="&lt;p&gt;The connections on a smooth vector bundle over a smooth manifold form an affine space with no canonical element to serve as $0$ for a vector-space structure.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2015-04-02T16:56:57.500" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7780" PostTypeId="2" ParentId="7761" CreationDate="2015-04-02T18:21:41.100" Score="6" Body="&lt;p&gt;The American Mathematical Society has a collection of beautiful posters in their Math Samplings. They are usually adding new ones every so often, and there and some new ones that are maybe a bit more what you'd call informative, that weren't there when you'd looked in the past:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/samplings/posters/posters&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.ams.org/samplings/posters/posters&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The posters are available for free to current students and instructors, and they just ask for you to provide the school name and some basic info. Don't need to make an account or anything. I've ordered posters different times and it's a seamless process of ordering them. They come quickly and in excellent shape.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few of the posters are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/8MCi4.jpg&quot; alt=&quot;enter image description here&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/4OmkS.jpg&quot; alt=&quot;enter image description here&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/n5fxZ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="4988" LastEditorUserId="6053" LastEditDate="2019-03-29T22:05:40.163" LastActivityDate="2019-03-29T22:05:40.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7781" PostTypeId="2" ParentId="7778" CreationDate="2015-04-02T20:03:28.407" Score="12" Body="&lt;p&gt;I am sure you know this, but the Euclidean plane is a prime example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Euclidean plane starts as just a collection of points together with a group of isometries.  There is no natural origin.  However, there is a natural associated vector space consisting of equivalence classes of pairs of points.  Identify $(p_1,q_2)$ with $(p_2,q_2)$ there is a translation of the plane taking $p_1$ to $p_2$ and $q_1$ to $q_2$.  These are the &quot;vectors as floating line segments&quot; we learn about in high school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This turns the Euclidean plane into an affine space via $p-q = [(p,q)]$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure that you will get too many really &quot;exciting&quot; examples, since as soon as you pick an origin you have a vector space.  There is no natural origin to Euclidean space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is time.  Assuming that there it has no beginning or end there is no natural origin.  It is an affine space for the vector space $\mathbb{R}$.  Of course, as soon as we pick an origin (like the death of a certain historical figure) it becomes a vector space.  But without human interference it is more naturally an affine space.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2015-04-02T20:03:28.407" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="7782" PostTypeId="2" ParentId="7778" CreationDate="2015-04-02T20:40:09.980" Score="4" Body="&lt;p&gt;Temperatures live in (part of) an affine space of dimension 1, and there seems to be some disagreements about what &quot;the&quot; origin should be...&lt;/p&gt;&#xA;" OwnerUserId="4990" LastActivityDate="2015-04-02T20:40:09.980" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7783" PostTypeId="2" ParentId="7778" CreationDate="2015-04-02T22:18:07.410" Score="9" Body="&lt;p&gt;The set of solutions to a nonhomogeneous linear differential equation form an affine space.  (The underlying vector space is the set of solutions to the associated homogeneous equation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every time I teach differential equations I wish the students had already learned about affine spaces.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2015-04-02T22:18:07.410" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7784" PostTypeId="1" CreationDate="2015-04-03T00:51:21.220" Score="8" ViewCount="366" Body="&lt;p&gt;There have been many innovative teaching methods tried from time to time. Twenty years ago was Calculus reform, now is flipped classrooms. We also have a resurgence of the Moore method under the name of IBL. Those are the ones I am most interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Universities have had computerized grade information for twenty years perhaps. I assume it should be relatively easy to do longitudinal studies to determine the relative efficacy of these various methods. Are there such studies? Where are they? Where is the data? What are the conclusions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I found this using Google: &#xA;&lt;a href=&quot;http://www.colorado.edu/eer/research/documents/IBLmathReportALL_050211.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.colorado.edu/eer/research/documents/IBLmathReportALL_050211.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4992" LastEditorUserId="4992" LastEditDate="2015-04-03T19:15:45.397" LastActivityDate="2015-04-03T19:15:45.397" Title="Have there been longitudinal studies comparing outcomes of various teaching methods?" Tags="|reference-request|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7785" PostTypeId="2" ParentId="7784" CreationDate="2015-04-03T11:38:06.353" Score="5" Body="&lt;p&gt;&lt;em&gt;To state a theorem and then to show examples of it is literally to teach backwards&lt;/em&gt;.&amp;mdash;E. Kim Nebeuts&#xA;&lt;hr /&gt;&#xA;This is hardly an answer to your question: I cite just one paper, emphasizing engineering rather than mathematics education,&#xA;focusing on IBL, and the work is from 2006.&#xA;Nevertheless...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Prince, M. J. and R. M. Felder (2006). &quot;Inductive teaching and learning methods:&#xA;  Definitions, comparisons, and research bases.&quot; &lt;em&gt;Journal of Engineering Education&lt;/em&gt;.&#xA;  95(2): 123-138. (&lt;a href=&quot;http://www4.ncsu.edu/unity/lockers/users/f/felder/public/Papers/InductiveTeaching.pdf&quot; rel=&quot;nofollow&quot;&gt;PDF download&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Their paper opens with the quote above.&#xA;They examine a range of IBL-like teaching techniques:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Inductive teaching and learning is an umbrella term that encompasses a range of&#xA;  instructional methods, including inquiry learning, problem-based learning, project-based&#xA;  learning, case-based teaching, discovery learning, and just-in-time teaching.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They survey the literature evaluating each teaching method. They cite some&#xA;longitudinal studies (in chemical engineering) and meta-analyses&#xA;of ~80 separate research studies. They conclude:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;While the quality of research data supporting the different inductive methods is variable,&#xA;  the collective evidence favoring the inductive approach over traditional deductive pedagogy is&#xA;  conclusive.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-04-03T12:04:01.223" LastActivityDate="2015-04-03T12:04:01.223" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7786" PostTypeId="2" ParentId="7778" CreationDate="2015-04-03T12:33:43.610" Score="5" Body="&lt;p&gt;In a vector space, you have an origin, addition of vectors, and scalar multiplication. Using a vector space structure to study something conveys the impression that the origin, addition, and scalar multiplication are actually meaningful things about the object of study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But sometimes that simply isn't true. And when it isn't, it's handy to have a way to apply linear algebra without giving the misleading impression that certain things are meaningful when they are not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Forgetting&quot; information is actually a pretty important thing in mathematics. For example, the whole field of differential geometry got started as a way to study what properties of a shape were intrinsic to a shape, and what properties were accidents of how they were drawn in Euclidean space.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2015-04-03T12:33:43.610" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7787" PostTypeId="2" ParentId="7513" CreationDate="2015-04-03T12:36:20.407" Score="0" Body="&lt;p&gt;One quirky feature of polynomial rings is that if $f \in F[x]$, $a$ is an element of an $F$-algebra, and the notation $f(a)$ means the result you get from evaluating $f$ at $a$ (i.e. applying the relevant evaluation homomorphism), then $f = f(x)$ is actually a literally true statement.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2015-04-03T12:36:20.407" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7788" PostTypeId="2" ParentId="7784" CreationDate="2015-04-03T13:03:21.900" Score="7" Body="&lt;p&gt;Like Joseph O'Rourke's answer, this one won't be complete, but, here's another possibly useful reference:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hiebert, J., &amp;amp; Grouws, D. A. (2007). &lt;em&gt;The effects of classroom mathematics teaching on students' learning.&lt;/em&gt; In F. K. Lester, Jr., (Ed.), Second handbook of research on mathematics &#xA;  teaching and learning (pp. 371 - 404). Charlotte, NC: Information Age Publishing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It makes an attempt to synthesize a large amount of research over a long period of time. It is not aimed at the collegiate level at all, so it does not specifically address IBL. (At the primary and secondary / K-12 level, the set of ideas behind IBL have been implemented in several different ways under several different names, and are not associated with the name of Moore. But the relevant ideas are evaluated by Hiebert and Grouws - specifically they attend to the question of how much math education should obligate students to struggle / grapple with the content.) It is also too old to mention the flipped classroom, which idea has really only attracted widespread interest inside of the last few years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside, I would like to say that I think the research problem you're posing here is much harder than you're giving it credit for. Variation in implementation is a serious issue because a teaching method that is effective in one instructor's class may be worse than nothing in another's. For example, anecdotally, from my years as a classroom teacher at a large public high school, both the strongest and the weakest lessons (IMHO) at the school were &quot;constructivist&quot; (which was the relevant name for an IBL-like pedagogy). I would have expected no significant difference in average outcomes of &quot;constructivist&quot; vs. &quot;traditional&quot; classrooms at that school. Measuring outcomes is another methodological hurdle.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="140" LastEditDate="2015-04-03T13:26:14.617" LastActivityDate="2015-04-03T13:26:14.617" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7789" PostTypeId="1" CreationDate="2015-04-03T13:31:05.763" Score="12" ViewCount="579" Body="&lt;p&gt;When I was a university student, I learnt that a function  is the data of three informations:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the rule that tells how to associate an object $x$ to its image $f(x)$, &lt;/li&gt;&#xA;&lt;li&gt;A domain $E$ where live the values of $x$ that are transformed by $f$,&lt;/li&gt;&#xA;&lt;li&gt;and a codomain $F$, a space where all images can lives.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So, an usual notation for an function is $ f:\begin{array}{rcl} E &amp;amp; \to &amp;amp; F \\ x &amp;amp; \mapsto &amp;amp; f(x) \\ \end{array}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, in high school textbook, as well as undergraduate textbook, a function is defined just by a formula, $y=x^2$ or $y=\sin(x)$. The domain is seldom part of the definition, but something the student has to compute, hence seen as an inherent property of the formula. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The codomain is, as far as I know, almost never mentioned. Most textbooks focus on the range. Since the range is the smallest possible codomain, I guess it may be for &quot;economical&quot; reasons, but it seems to me that it leads to me to misconceptions about functions. It also leads to shocking sentences like &quot;the function is one-to-one, so it has an inverse&quot; :(.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question :&lt;/strong&gt; what the pros and cons of not teaching the notion of codomain of a function?&lt;/p&gt;&#xA;" OwnerUserId="1538" LastActivityDate="2015-04-05T07:34:27.297" Title="Why don't we teach codomains of functions in high school?" Tags="|curriculum|functions|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7790" PostTypeId="1" AcceptedAnswerId="7800" CreationDate="2015-04-03T13:40:32.637" Score="6" ViewCount="434" Body="&lt;p&gt;I just learned the name &lt;em&gt;E. Kim Nebeuts&lt;/em&gt; from the quote at the beginning of Joseph O'Rourke's answer to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7784/have-there-been-longitudinal-studies-comparing-outcomes-of-various-teaching-meth&quot;&gt;this question&lt;/a&gt;. Curious, I google searched. All I saw on the first 2 pages of results was things like &amp;quot;inspirational E. Kim Nebeuts quotes!&amp;quot; I am left wondering &lt;em&gt;who is/was E. Kim Nebeuts&lt;/em&gt;? A math teacher, apparently, but -- where and when? And how did her / his quotes become known? Does he/she have a book? Etc.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally three quotes kept reappearing:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To state a theorem and then to show examples of it is literally to teach backwards.&lt;/p&gt;&#xA;&lt;p&gt;A good preparation takes longer than the delivery.&lt;/p&gt;&#xA;&lt;p&gt;Teach to the problem, not to the text.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="140" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-04-04T01:03:24.307" Title="Who is E. Kim Nebeuts?" Tags="|mathematical-pedagogy|history|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7791" PostTypeId="2" ParentId="7784" CreationDate="2015-04-03T14:18:44.823" Score="5" Body="&lt;p&gt;The calculus reform died without anybody noticing it and, that I know of, there were no longitudinal studies. In one case, though, if not an a longitudinal study, there was an evaluation and what followed was interesting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1988, I was the principal investigator of a small NSF calculus grant predicated on two ideas:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The very length of the standard Pecalculus 1 - Precalculus 2 - Calculus 1 sequence is of and by itself a major problem.&lt;/li&gt;&#xA;&lt;li&gt;The content of the sequence would be &lt;em&gt;integrated&lt;/em&gt; in a two-semester sequence by way of a systematic use of &lt;em&gt;polynomial approximations&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;After a few years, in the words of  the school's Office of Institutional Research, it was established that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Of those attempting the first course in each sequence, 12.5% finished&#xA;  the [conventional three semester 10 hour] sequence while 48.3%&#xA;  finished the [integrated two semester 8-hour] sequence, revealing a&#xA;  definite association between the [integrated two semester 8 hour]&#xA;  sequence and completion (chi2(1) = 82.14, p &amp;lt; .001).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Furthermore, the study mentioned that the same number of students in both sequences passed Calculus 2 but that the numbers in both sequences were too small to be meaningful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what happened next? The integrated sequence died within the next few years and no one was talking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then, a few years later, the school did a &lt;a href=&quot;http://freemathtexts.org/IntegratedA2DC/LongitudinalStudy.pdf&quot; rel=&quot;noreferrer&quot;&gt;longitudinal study&lt;/a&gt; of the entire Arithmetic - Basic Algebra - Intermediate Algebra - Precalculus 1 - Precalculus 2 - Calculus 1 sequence. And, here again, the rather dismal evidence was immediately forgotten.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-03T14:18:44.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7792" PostTypeId="2" ParentId="2164" CreationDate="2015-04-03T14:36:05.107" Score="5" Body="&lt;p&gt;Several years ago, I succeeded in having one of my classes linked with an English reading class in the hope that the reading material in the English reading class would be the text in the mathematics class (which I had writtem) so that the English instructor would be &quot;on the side of the students&quot; and could provide feedback on the text, at least from the readability viewpoint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While, curiously enough, the English instructor felt compelled to sit in on the mathematics class---which however she seemed to enjoy very much to the point that she would often ask questions in class, she did not use the mathematics text as reading material in her English reading class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though the Instructor publicly wrote later that she regretted not having used the mathematics text as reading material, I was never able to have another one of my classes linked with an English reading class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nobody knows why.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-03T14:36:05.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7793" PostTypeId="2" ParentId="7757" CreationDate="2015-04-03T15:02:06.350" Score="4" Body="&lt;p&gt;While not entirely free of &quot;external constraints&quot;, for many years now, I have been working on a three semester sequence [5-4-4] intended to parallel the Arithmetic - Basic Algebra - Intermediate Algebra - Precalculus 1 - Precalculus 2 - Calculus 1 sequence. The idea is to lead into the integrated sequence I mentioned elsewhere on this site (I don't know how to link to it) by integrating Arithmetic - Basic Algebra - Intermediate Algebra into a conceptual flow also centered on &lt;em&gt;approximations&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wanted the &quot;external constraints&quot; as a reality check. That is, I extracted, and teach every semester, two &quot;standalones&quot; from the sequence to see how students take it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the &lt;a href=&quot;http://www.ams.org/notices/201303/rnoti-p340.pdf&quot; rel=&quot;nofollow&quot;&gt;Notices&lt;/a&gt; for a relevant article and, for the materials themselves, &lt;a href=&quot;http://freemathtexts.org/&quot; rel=&quot;nofollow&quot;&gt;FreeMathTexts.org&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-03T15:02:06.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7794" PostTypeId="2" ParentId="7513" CreationDate="2015-04-03T15:27:26.247" Score="0" Body="&lt;p&gt;Why not indulge in a bit of redundancy and write for instance&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$x \xrightarrow{\hspace{5mm} f\hspace{5mm}} f(x) = -3x^2 +2x -21$&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-03T15:27:26.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7795" PostTypeId="2" ParentId="7778" CreationDate="2015-04-03T15:38:14.563" Score="0" Body="&lt;p&gt;The point of affine spaces is that they deal with &quot;free vectors&quot; as encountered in Dynamics to represent forces. But that they are worth the effort is indeed debatable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, physicists have no trouble with the Dirac and Heaviside &quot;functions&quot; and the fact that these are distributions à la Schwartz does not seem to add to their comfort. Which is not to say that the latter are not beautiful. They are. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, somehow, affine spaces seem less so to me.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-03T15:38:14.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7796" PostTypeId="2" ParentId="7725" CreationDate="2015-04-03T16:09:45.773" Score="1" Body="&lt;p&gt;Strictly speaking, the following might be slightly off topic, but &lt;em&gt;I&lt;/em&gt; think that it is directly relevant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the preface of Calculus with Analytic Geometry by John H Staib [1966],&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The development of the book is committed to the following thesis: The&#xA;  theory of limits is most easily grasped in the case of sequences.&#xA;  Therefore the theory of limits is presented first for sequences and&#xA;  then that theory is exploited in the introduction of all other limit&#xA;  concepts including integration. Thus, although most of the usual&#xA;  topics appear here, they are in a rather different order. Moreover,&#xA;  the emphasis on sequences not only provides the course with a unifying&#xA;  theme,but also give it a distinctive flavor which is reflected in many&#xA;  of the proofs and exercises. Indeed, it is the special allurement of&#xA;  sequences that I have attempted to exploit. For instance, few students&#xA;  are moved by the announcement that $\lim_{x\to 2}x^2=4$ but &quot;all&quot;&#xA;  students feel the challenge implicit in the assertion that the&#xA;  sequence $$\sqrt{2}, \sqrt{2+\sqrt{2}},  \sqrt{2+\sqrt{2+\sqrt{2}}},&#xA;&amp;gt; \dots$$ has 2 as a limit&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-03T16:09:45.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7797" PostTypeId="2" ParentId="7778" CreationDate="2015-04-03T17:48:43.827" Score="3" Body="&lt;p&gt;This is really another viewpoint for answers already given.  In statistics and data analysis, whenever you have a quantity which you can measure, and for which (arithmetic) means are meaningful, but addition is not, then those quantities form (at least a part of) some affine space. Important examples are already mentioned, temperature and time.  And this also answers the objection about absolute zero or the big bang origin of time: For most mundane purposes where we use statistics, those origins are rather besides the point, and the affineness of the spaces presents itself as the fact that averages are meaningful, but sums are not. That point have much larger significance than those far-away origins!&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2015-04-03T17:48:43.827" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7798" PostTypeId="2" ParentId="7789" CreationDate="2015-04-03T18:57:19.880" Score="6" Body="&lt;p&gt;Pros: People will potentially better understand ideas required for fairly abstract mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cons: It pretty much doesn't matter to any field outside of pure mathematics.  It would add to an already fairly gigantic list of things that we need to teach at the high school level.  It is a pretty abstract concept to teach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, range gets the concepts across well enough that if a student goes on to study mathematics in college they should be able to modify their definitions to include co-domain and properly apply it.  Most of the country will never need that in ever.  There's no reason to go beyond a basic overview of it at the high school level.&lt;/p&gt;&#xA;" OwnerUserId="4983" LastActivityDate="2015-04-03T18:57:19.880" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7800" PostTypeId="2" ParentId="7790" CreationDate="2015-04-04T01:03:24.307" Score="6" Body="&lt;p&gt;On page 131 of &lt;em&gt;Twenty Years Before the Blackboard&lt;/em&gt; by Michael Stueben, there is a quote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Noitaton Hsilop esrever diova! --  &lt;em&gt;E. Kim Nebeuts&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the book there is also reference to a Mr. Loof Lirpa, so writing things backwards seems to be a recurring theme.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it is safe to assume E. Kim Nebeuts is Michael Stueben.&lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2015-04-04T01:03:24.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7801" PostTypeId="1" AcceptedAnswerId="7808" CreationDate="2015-04-04T18:02:04.383" Score="7" ViewCount="195" Body="&lt;p&gt;Just wanted to share the link to a mathematical counterexamples site that I develop: &lt;a href=&quot;http://www.mathcounterexamples.net/&quot;&gt;Math Counterexamples&lt;/a&gt;. Any idea for improvement is more than welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the way the site looks like, on good ideas for counterexamples, on the way to split counterexamples between level on math... Or anything else!&lt;/p&gt;&#xA;" OwnerUserId="4705" LastActivityDate="2015-04-04T23:53:51.590" Title="Math counterexamples site" Tags="|mathematical-pedagogy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7802" PostTypeId="2" ParentId="7771" CreationDate="2015-04-04T20:02:50.210" Score="8" Body="&lt;p&gt;Circle generators:  disposable cups, you can trace around the rim and the base for quick circles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Patty paper is pretty cheap and totally worth it not just for constructions, but as tracing paper for all kinds of purposes.  &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B001OMEGFS&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.amazon.com/Geometry-Patty-Paper-Sheets-Single/dp/B001OMEGFS&lt;/a&gt;  Can be an equivalent measurement tool to a compass sometimes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Compass&quot;: paperclip and two pencils will work in a pinch.  You can bend it to adjust the radius &lt;em&gt;slightly&lt;/em&gt;.  Fix one pen in one loop and swing the other pen around in the other loop.  See &lt;a href=&quot;http://www.ictm.org/journal/index.php/imt/article/view/68/66&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.ictm.org/journal/index.php/imt/article/view/68/66&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;GeoGebra and Desmos (both free) can replace graphing calculators entirely (and do more).  You'll need either computers, tablets, or phones.  I found 2 students to 1 device works well, maybe even better than 1:1.  www.geogebra.org  www.desmos.com&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This one isn't cheap, but it is &lt;em&gt;cheaper&lt;/em&gt;:  You can use a Microsoft Surface and a normal projector and OneNote/Powerpoint as a &quot;Smart Board&quot;.  Note: this is much better than a iPad in two big ways: One, the pen is a separate input from your finger, so more accurate and you can interface with pen and finger.  Two, the Surface is a full laptop, in addition to a tablet, so it can run any windows program. There are other tablets that can do this as well-- I just have had good experiences with the Surface. If you're going to buy a laptop for yourself anyway, make it a Surface and you can handwrite and draw.  You can also go wireless drawing with some creative shenanigans.  a pic to illustrate: &lt;a href=&quot;https://i.stack.imgur.com/XzNtD.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://i.stack.imgur.com/XzNtD.jpg&lt;/a&gt;  Surface also can be used as a DocCam.&lt;/p&gt;&#xA;" OwnerUserId="4962" LastActivityDate="2015-04-04T20:02:50.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7803" PostTypeId="2" ParentId="7778" CreationDate="2015-04-04T20:07:45.767" Score="3" Body="&lt;p&gt;My opinion is, that no, you shouldn't teach that to undergrads in your mathematics course. The main reason for this is that it makes simple things complicated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that the &lt;em&gt;affine $d$-dimensional real space&lt;/em&gt; and the &lt;em&gt;linear $d$-dimensional real space&lt;/em&gt; are different, and also different from $\mathbb{R}^d$. However, introducing these three (or any two of them, for that sake) as a different mathematical concept is unnecessary for any calculus result you present, and also unnecessary for any exercises they're going to work on in the problem sessions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If any undergrad course should explain the differences, it should be the one that needs it, which is the Mechanics course. It's also much easier for the students to grasp the idea there, because it makes sense in mechanics that position $0$ can be anywhere, but distance $0$ or velocity $0$ is dependent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a mathematician, you can point this out at an appropriate place in the course; the students should simply realize that it corresponds to their physics experience, but shouldn't be bothered by you not distinguishing the two concepts.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As a side note: I was taught the &quot;complicated&quot; way, but when I got this question during my state exams, I made it the &quot;simple&quot; way. Two of the committee members than thanked me for making it easy and nice.&lt;/p&gt;&#xA;" OwnerUserId="818" LastActivityDate="2015-04-04T20:07:45.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7804" PostTypeId="1" CreationDate="2015-04-04T21:58:51.867" Score="12" ViewCount="1620" Body="&lt;p&gt;I'm looking for an assessment tool to ability group 4th and 5th grade students, in particular setting a &quot;top&quot; group. This group would receive more enrichment, less practice on concepts and a faster pace of instruction. Our goal is to figure out by the end of 5th grade who can successfully complete 6th, 7th and 8th grade curricula in 2 years and then take Algebra 1 in 8th grade. Preferably pencil and paper, something I can give 3 or 4 times a year to &quot;confirm&quot; the groupings and provide data to parents. I know there are loads of reading assessments that give a grade level equivalent or reading level, such as Gates-McGinity.-I'm looking for the same thing for math.  Anybody have any experience with anything like this?&lt;/p&gt;&#xA;" OwnerUserId="4997" LastEditorUserId="4997" LastEditDate="2015-04-08T13:35:21.363" LastActivityDate="2015-06-22T18:44:25.390" Title="elementary level assessment tools" Tags="|assessment|testing|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7805" PostTypeId="2" ParentId="7725" CreationDate="2015-04-04T23:02:25.367" Score="0" Body="&lt;p&gt;OK, here is a directly relevant answer: &lt;strong&gt;asymptotic expansions&lt;/strong&gt;, which are &lt;em&gt;finite&lt;/em&gt; expansions by definition, are used to approximate $f(x_{0}+h)$ and can be used with divergent series (That is what Poincaré invented them for) as well as convergent series. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second part of my answer is that, at least in calculus, but in fact in precalculus, polynomial expansions (certainly finite and in fact, rather short) should be used to describe the qualitative behavior near $x_{0}$ and $\infty$ of the usual functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a simple example, consider the function $f(x)=\frac{x^{3}-8}{x^{2}-5x+6}$.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Near $\infty$, we get by long division in &lt;em&gt;descending&lt;/em&gt; powers that $f(x) = x +5 + 19x^{-1} + o[x^{-1}]$ i.e. an oblique asymptote.&lt;/li&gt;&#xA;&lt;li&gt;Near $+2$, one of two possible poles, we get by long division in &lt;em&gt;ascending&lt;/em&gt; powers that $f(+2+h) = \frac{+12h +6h^{2} +h^{3}}{-h+h^{2}} = -12 -18h -19h^{2} + o[h^{2}]$ i.e. that $+2$ is regular.&lt;/li&gt;&#xA;&lt;li&gt;Near $+3$, the other possible pole, we get by short division that $f(+3+h) = \frac{+19 +27h +9h^{2} +h^{3}}{h+h^{2}} = \frac{+19+ o[1]}{h+ o[h]}=+19h^{-1} + o[h^{-1}]$ i.e. that $+3$ is indeed a pole.&lt;/li&gt;&#xA;&lt;li&gt;A smooth interpolation of the local graphs near $\infty$ and $+3$---the local graph near $+2$ provides only confirmation---gives the essential global graph which in turn shows that $f$ has at least one maximum and one minimum.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/fk95z.jpg&quot; alt=&quot;Essential graph&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In precalculus, rather than little ohs, I just use [...] to stand for &quot;&lt;em&gt;something too small to matter here.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See, for instance, my implementation of the above: &lt;a href=&quot;http://freemathtexts.org/Standalones/RAF/Contents.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reasonable Algebraic Functions&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-04T23:02:25.367" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7806" PostTypeId="2" ParentId="7778" CreationDate="2015-04-04T23:15:25.440" Score="5" Body="&lt;p&gt;As far as the first question is concerned I fully agree with Steven Gubkin's answer : the reason why abstract affine spaces are nice is because they match the intuition coming from elementary Euclidean geometry. The Euclidean plane is clearly not a vector space and clearly not naturally an affine subspace of a vector space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can also argue that undergrads should learn about affine spaces because they are the natural place where they should learn multi-variate calculus. In those space you can write $f(x+h) = f(x) + Df(x)(h) + o(h)$ with $x$ and $h$ having a very different nature (as they should have, and you certainly not draw them on the board as two arrows attached at the origin of a vector space).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About the second question, Andreas Blass's answer is of course the first one that comes to mind to differential geometers. But it obscures somehow its elementary nature by putting it into a differential setting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $V$ be a linear space and $E$ a subspace of $V$. Let $S(E)$ be the space of linear subspaces $F$ of $V$ such that $V = E \oplus F$. Then $S(E)$ is naturally an affine space directed by the vector space $L(V/E, E)$ of linear maps from $V/E$ to $E$. If $\pi$ denotes the projection from $V$ to $V/E$ then the action is defined by $u \cdot F = \{f + u(\pi(f)); f \in F\}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most famous example is when $E$ is a hyperplane. Then $S(E)$ is the complement in the projective space $P(V)$ of the projective hyperplane $P(E)$. That's why such complement are called &quot;affine charts&quot;: they have a canonical (functorial) affine space structure. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This example also underlies the connection example as soon as you think in terms of Ehresmann connections.&lt;/p&gt;&#xA;" OwnerUserId="4998" LastActivityDate="2015-04-04T23:15:25.440" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7808" PostTypeId="2" ParentId="7801" CreationDate="2015-04-04T23:32:43.737" Score="2" Body="&lt;p&gt;Nice! It seems you have few geometry (counter)examples. Here is one:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There are 4-dimensional convex polytopes that cannot&#xA;  be realized with rational vertex coordinates: that is, the combinatorial&#xA;  structure can only be realized with one or more irrational vertex coordinates.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In $\mathbb{R}^3$, however, every convex polyhedron's combinatorial type &#xA;can be realized with rational coordinates. (But it is unknown if&#xA;the rational coordinates can be chosen to be &quot;small.&quot;)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ziegler, Günter M. &quot;Nonrational configurations, polytopes, and surfaces.&quot; &lt;em&gt;The Mathematical Intelligencer&lt;/em&gt;. &lt;strong&gt;30&lt;/strong&gt;.3 (2008): 36-42.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/WWvpp.png&quot; alt=&quot;TobleroneTorus&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-04-04T23:53:51.590" LastActivityDate="2015-04-04T23:53:51.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7809" PostTypeId="2" ParentId="5898" CreationDate="2015-04-05T05:26:06.423" Score="0" Body="&lt;p&gt;It is often recommended to attend a different university for graduate work than the institution where one did the Bachelor's degree.  Why?  To get a different perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Extrapolating from that, I would expect variety to be helpful in the calculus sequence too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's another point.  Consider that different strokes work for different folks, but that not all math teachers know how to teach to a variety of learning styles.  A student would therefore do well to hedge his bets and try someone different each semester (unless he finds someone he's remarkably well matched with, and decides to take whatever that instructor has to offer in a particular semester).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, note that math is a bit like a foreign language.  Those learning a language will do best in the long run if they are exposed to a variety of speaking styles. This would make sense in the math context as well.&lt;/p&gt;&#xA;" OwnerUserId="4956" LastActivityDate="2015-04-05T05:26:06.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7810" PostTypeId="2" ParentId="7789" CreationDate="2015-04-05T07:34:27.297" Score="6" Body="&lt;p&gt;Codomain may be a relatively recent precision of the language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You're interpreting &quot;&lt;strong&gt;range&lt;/strong&gt;&quot; as only meaning the &lt;strong&gt;image&lt;/strong&gt;, but &quot;&lt;strong&gt;range&lt;/strong&gt;&quot; has also been used to mean the &lt;strong&gt;codomain&lt;/strong&gt;. according to a couple of cites here: &lt;a href=&quot;http://en.wikipedia.org/wiki/Range_%28mathematics%29&quot;&gt;http://en.wikipedia.org/wiki/Range_%28mathematics%29&lt;/a&gt; .  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree it can be troublesome when the books are not carefully distinguishing these ideas.  However, many students will be tasked with domain and range concepts from pre-algebra courses when their knowledge of functions is usually limited to discrete cases or linear real-valued functions.  In later courses, students will see and write things like $\frac{x^2}{x-1}$ is a function from $\mathbb{R}\to\mathbb{R}$.  This kind of serves as describing the function as &quot;real inputted&quot; with the origin set, and it provides the codomain explicitly.  (Students are usually tasked to find the domain (meaning pre-image) and range (meaning image)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A source of these confusions may be that the codomain is so often $\mathbb{R}$, that it is deemed automatic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to tease out the difference with a class is to discuss the function of $student \to birthmonth$.  Here you can say that the domain is students in the class (noting that you, the teacher, is not an element of the domain).  The codomain is the set of twelve months.  And the image is the actual set of months acquired, which may not be the entire codomain. (best if your students don't cover all 12 ;)&lt;/p&gt;&#xA;" OwnerUserId="4962" LastActivityDate="2015-04-05T07:34:27.297" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7811" PostTypeId="2" ParentId="7725" CreationDate="2015-04-05T08:39:49.770" Score="0" Body="&lt;p&gt;Just to throw out a different answer: my immediate reaction was 'neither'. My experience in the UK has been that limits and convergence get the slightest whisper when you first meet differentiation and integration at high school, and at university they sit firmly under the heading 'analysis'.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-04-05T08:39:49.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7812" PostTypeId="2" ParentId="7281" CreationDate="2015-04-05T16:11:47.010" Score="2" Body="&lt;p&gt;By itself, the word &lt;em&gt;ratio&lt;/em&gt; is indeed rather opaque and should probably be avoided. Here is what &quot;is under the hood and usually goes without saying&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Natural (&lt;em&gt;aka&lt;/em&gt; counting) numbers are used to measure a &quot;quantity&quot; (= counting number) of &lt;em&gt;discrete objects&lt;/em&gt; such as in: the &quot;quantity&quot; of the set {egg, egg, egg} is 3. What do we do when we want to measure a &quot;quantity&quot; (= real number) of &lt;em&gt;continuous stuff&lt;/em&gt; such as length? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of &lt;strong&gt;proportion&lt;/strong&gt;, the ancient Greeks would have said that the &quot;quantity&quot; (length) of a footstick  is to the &quot;quantity&quot; (length) of a yardstick the same as the &quot;quantity&quot; (count) of the set {egg}, i.e. 1, is to the &quot;quantity&quot; (count) of {egg, egg, egg}, i.e. 3. In terms of &lt;strong&gt;ratios&lt;/strong&gt;, they would have said that the ratio of footstick to yardstick is equal to the ratio of {egg} to {egg, egg, egg}. When we say that the &quot;quantity&quot; of a footstick  equals $\frac{1}{3}$ the &quot;quantity&quot; of a yardstick, we are just writing the &lt;em&gt;proportion&lt;/em&gt; another way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, the idea of &lt;strong&gt;proportion&lt;/strong&gt; was meant to reduce the measure of &quot;quantity&quot; of stuff (which they couldn't do) to the measure of &quot;quantity&quot; of sets (which they could do).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my (by now very remote) experience with 5 graders, I think that the above distinction, measuring sets versus measuring continuous stuff (which, by the way, requires the introduction of units). is quite within their reach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Warning&lt;/strong&gt;: what follows may be a bit off-topic but is tightly related to the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The trouble comes when we want to look at a fraction as indicative of a measure the same way as when we look at a natural number as indicative of a measure. For instance, when we look at 3084385 and 47975 we immediately see that the first is larger than the second. Not so immediately with fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, of course, we look at $\frac{1}{3}$ as &lt;em&gt;code&lt;/em&gt; for &quot;divide 3 into 1&quot; but the question now is &quot;where to stop the division?&quot; and that of course depends on the real-world situation. That we have rules for dealing with the &lt;em&gt;code&lt;/em&gt;, e.g. $\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}$ is nice and fortunate but not really necessary: as engineers are wont to put it, &quot;The &lt;em&gt;real&lt;/em&gt; real numbers are the decimal numbers&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See  &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0192853619&quot; rel=&quot;nofollow&quot;&gt;Gowers Mathematics, A Very Short Introduction&lt;/a&gt;. While spending a whole chapter (7) on &lt;em&gt;infinite decimals&lt;/em&gt;, Gowers never even mentions real numbers.  And, appropriately recast, the content of Gowers' Chapter 7 should be accessible to 5 grader.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="4815" LastEditDate="2015-04-05T16:19:24.283" LastActivityDate="2015-04-05T16:19:24.283" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7813" PostTypeId="2" ParentId="7188" CreationDate="2015-04-05T16:35:09.317" Score="-1" Body="&lt;p&gt;Should students be told they're wrong about an idea they have?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Absolutely not. Mathematics is not a matter of right or wrong: according to what/whom?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, they should be told, &lt;em&gt;up front&lt;/em&gt;, that, in science just like in a court of law, &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;we always have to make our case,&lt;/li&gt;&#xA;&lt;li&gt;we always have to submit our case to adversarial proceedings (peer review).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then, what I tell my students is that, when reading &lt;em&gt;their&lt;/em&gt; case, either I agree, will say so and that will be the end of that or, if I disagree, it will then be up to me to make &lt;em&gt;my&lt;/em&gt; case.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-05T16:35:09.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7814" PostTypeId="2" ParentId="7739" CreationDate="2015-04-05T23:40:52.833" Score="3" Body="&lt;p&gt;Khan academy has a pretty good exercise covering this&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-measurement-topic/cc-5th-volume-formula-intuition/e/volume-formula-intuition&quot; rel=&quot;nofollow&quot;&gt;https://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-measurement-topic/cc-5th-volume-formula-intuition/e/volume-formula-intuition&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Displays a rectangular prism which is precut into $n$ slices, by color.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Asks how many slices there are, how many cubes in each slice, and finally how many total cubes.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2015-04-06T00:16:53.510" LastActivityDate="2015-04-06T00:16:53.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7815" PostTypeId="1" CreationDate="2015-04-06T19:22:40.223" Score="9" ViewCount="511" Body="&lt;p&gt;I have an excellent undergraduate student who is suddenly being put into math classes where she does not have access to a calculator. She is fine with all the new topics, but when running into something like $\sqrt{441}$, it takes her a significant amount of time to finish the calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She wants to spend a couple weeks over the summer to learn to fend for herself without calculator assistance --- what kind of &quot;syllabus&quot; or &quot;list of topics&quot; could she use with a tutor to accomplish this goal?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2015-04-06T23:29:21.040" Title="Becoming independent of the calculator" Tags="|undergraduate-education|tutoring|calculators|" AnswerCount="2" CommentCount="12" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7816" PostTypeId="2" ParentId="7815" CreationDate="2015-04-06T21:33:31.170" Score="5" Body="&lt;p&gt;This addresses the title of the question more than the situation.  (So if, as commented, there are dyscalculia or time issues or something else, this advice won't be appropriate.  Also, if precision is wanted, the following is less than optimal.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I spent a lot of time before college doing mental calculation.  A lot of it developed from memorization, some from pattern recognition, but practice to increase speed as well as check, check, check again, and then look at it to see what might have gone wrong are the main reasons why I can still do a lot of computation with only a paper and pencil.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An important part of this is estimation: what is the size and shape of the answer?  Continuity of operations and experience speak to size, experience speaks to shape.  For the given example, $\sqrt{441}$, I would say first the answer is north of (bigger than) 20 and, if it is an integer, has to be odd.&#xA;Since I know tables of squares, memorization and experience gives me that the&#xA;answer is the smallest odd number bigger than 20, but I would try $(20+ \epsilon)^2 = (400 + 2 \cdot 20 \cdot \epsilon + \epsilon \cdot \epsilon)$ as a check for enough values of $\epsilon$ to see if my first guess was correct.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another important part is context: is an exact answer expected?  How many significant digits are wanted?  If it cannot be determined from the context, always, always, ALWAYS provide it in your answer, as whoever needs the answer also can use confirmation of the context to evaluate the answer.  Some tools work well in some contexts, not so well in others.  If you have picked the context and the tools you are using are not working well, check to see if you have the context right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For personal performance, information representation can be important, as well as noting certain patterns.  Rather than remembering half-angle formulae and other relations between trigonometric functions, I remember the trigonometric equivalent of the pythagorean theorem and a mnemonic (usually a picture) which help me remember and derive the desired relations.   Once I have the relations derived, it is easier to use mnemonics and rederive than wonder whether I remembered the relation correctly or not.  Similarly, noting cubes mod 10 (3 goes to 7 goes to 3 under cubing mod 10) helps in constructing answers and providing checks for calculations involving cubing integers, or even taking cube roots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a quick self-assessment, see if you can find some number sense tests, and perform the following experiment.  They are usually 80 questions to be done in 10 minutes, and top solvers (in my day) rarely did more than 50; use this to set your benchmark (20, 30, or even 100 minutes).  Then go through all questions once using a green pen.  Those you feel good about (having solved in ten seconds or less correctly) answer in green pen.  Any that don't, skip in five seconds or less.  Then go through the list again focusing only on the unanswered questions, using a yellow pen, and this time allow 20 seconds to compute and be sure of your answer.  Repeat with orange pen and 40 seconds, red pen and eighty seconds, black pen and 160 seconds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The time and colors don't matter, but the process above serves as a quick assessment of levels of difficulty, and hopefully shows through color-coding some patterns where you know you want to improve in a certain type of calculation.  Then you can focus on problems using those particular operations or combinations of operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If number sense tests aren't available, try some practice tests and other review tests in algebra texts and similar materials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Still Uses A Calculator Occasionally&quot; Paseman, 2015.04.06&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-06T21:33:31.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7817" PostTypeId="2" ParentId="630" CreationDate="2015-04-06T23:08:47.260" Score="2" Body="&lt;p&gt;I agree that &quot;the emphasis on mathematics education is typically on getting the right answer, and often in doing it the right way&quot;. I also agree on many of the points made in the previous answers but I am a bit surprised that one point was entirely missed: Given that, these days, mathematics exposition is by short topics without much of a &quot;story line&quot;, it is just about impossible for a student to be in a position to be creative. To be creative, it is necessary to have experienced how an idea that worked in a particular context can also work in an apparently completely different context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what Thurston had to say about something very close to it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematics is amazingly compressible: you may struggle a long time,&#xA;  step by step, to work through some process or idea from several&#xA;  approaches. But once you really understand it and have the mental&#xA;  perspective to see it as a whole, there is often a tremendous mental&#xA;  compression. You can file it away, recall it quickly and completely&#xA;  when you need it, and use it as just one step in some other mental&#xA;  process. The insight that goes with this compression is one of the&#xA;  real joys of mathematics. (Thurston, W. P. (1990). Mathematical&#xA;  Education, Notices of the American Mathematical Society, 37 7,&#xA;  844-850)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But, at least for the beginners, that joy requires that the contents in the materials available to them be themselves tightly conceptually organized to show the &quot;story line&quot;---as opposed to being a collection of topics. But such materials, sadly, are very much lacking.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-06T23:08:47.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7818" PostTypeId="2" ParentId="1" CreationDate="2015-04-06T23:25:00.707" Score="5" Body="&lt;p&gt;&lt;em&gt;Interaction&lt;/em&gt; is fundamentally incompatible with &lt;em&gt;lecture&lt;/em&gt;. So, the solution I adopted many years ago was to stop lecturing. But, even so, interaction is hard to get. So, I sort of not lecture and, bit by bit, as the semester goes on, I succeed in cajoling a number of students into interacting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More precisely: the students are to read, according to a calendar, the chapter before each class and do a homework on that chapter. In class, we discuss the questions being brought up by the students---more often than not about the homework itself, and then the students submit (or not) their homework. The main advantage, at least as far as I am concerned, is that this allows me to spend more time on the aspects which, from experience, I think are important---for whatever reason---without feeling the need to &quot;cover&quot; the whole chapter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. I should probably mention that I write the texts and the homework that go with them.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-06T23:25:00.707" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7819" PostTypeId="2" ParentId="7815" CreationDate="2015-04-06T23:29:21.040" Score="6" Body="&lt;p&gt;Perhaps reading one or more of these books on the art of approximation&#xA;(often: &lt;a href=&quot;http://en.wikipedia.org/wiki/Fermi_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Fermi estimates&lt;/em&gt;&lt;/a&gt;) would be fun and helpful:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Weinstein, Lawrence, and John A. Adam. &lt;em&gt;Guesstimation: Solving the world's problems on the back of a cocktail napkin&lt;/em&gt;. Princeton University Press, 2009.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Weinstein, Lawrence. &lt;em&gt;Guesstimation 2.0: solving today's problems on the back of a napkin&lt;/em&gt;. Princeton University Press, 2012.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Mead, Carver A., and Sanjoy Mahajan. &lt;em&gt;Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving&lt;/em&gt;. MIT Press, 2010.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/w0rKq.jpg&quot; height=&quot;256&quot; /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/3yzNs.jpg&quot; height=&quot;256&quot; /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Sagqh.jpg&quot; height=&quot;252&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-04-06T23:29:21.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7821" PostTypeId="1" CreationDate="2015-04-07T14:04:42.987" Score="7" ViewCount="227" Body="&lt;p&gt;I'm looking for some site or software which can help me create an easy, digital multiplication quiz for my students. Some things which need to be included:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The student need to be able to type in the answer him/herself, no multiple choice. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A timer which shows total time spent.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;All the questions are based on the multiplication table, so if there's a way to randomize number ranges (eg 2-10) and decide the number of questions, that would make it even easier.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="5010" LastEditorUserId="5010" LastEditDate="2015-04-27T13:49:10.513" LastActivityDate="2015-08-30T02:56:31.277" Title="Looking for a tool to create a digital multiplication quiz" Tags="|primary-education|online-instruction|multiplication-table|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7823" PostTypeId="2" ParentId="4223" CreationDate="2015-04-08T00:05:26.073" Score="7" Body="&lt;p&gt;&lt;strong&gt;EDIT 4/8/15&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I began teaching Statistics with the same difficulty that you had. I started with basic sorts of questions (e.g., age, height) but my students weren't really responding to these types of questions. I sought to find a way to have them feel more connected to the data. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I began with questions about text messaging. (1) How many different people did you text on Sunday? How many total texts did you send on Sunday? These actually led to other interesting questions that we had to have as a class. Do we include group texts? Do we include picture texts? These types of questions that lead to the the exact discussions that you want to have with your students about data collection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As this line of questioning developed, I started to have my students survey their own friends and contacts about their texting habits. This leads to other interesting data collection questions that we could discuss as a class. While we were able to get numbers we could work with, we had to be especially careful about what our population was, and how our data collection methods might not have been ideal. My last lab asked them about their flossing habits and total cavities. This didn't really provide the wide variety of data that I was hoping for, but it was suitable for my purposes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best thing about all of this what it opened up my students to start thinking about their own questions. They came up with many more questions than I could have. How many days a week do you work out? was a popular question. Shopping habits (online vs. in stores), coffee consumption (Starbucks or Dunkin), and QBR ratings for quarterbacks were all places that students pulled data from. I even had one student track her daily coffee consumption and sleeping habits for a month to use as a data source.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure what level of student that you have, but for my college students of traditional age, once I was able to ask some questions about what they actually did in their lives (i.e., texting) it opened them up to the possibility of what other aspects of their lives Statistics could answer. I think if you can do this, then you'll help them to not only learn Statistics, but to teach them how they can use Statistics in their everyday lives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I utilized a lot about their phone habits when I taught statistics recently. Number of total texts sent in one day, number of unique recipients of texts you sent, number of Instagram messages, etc. I can provide more details in an edited answer when I'm back on my computer&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="267" LastEditDate="2015-04-08T22:34:29.813" LastActivityDate="2015-04-08T22:34:29.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7824" PostTypeId="2" ParentId="7821" CreationDate="2015-04-08T01:46:36.317" Score="5" Body="&lt;p&gt;Here's a quick and dirty implementation: &lt;a href=&quot;http://nilock.github.io/MathSnips/MultQuiz/MultQuiz.html&quot; rel=&quot;nofollow&quot;&gt;http://nilock.github.io/MathSnips/MultQuiz/MultQuiz.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;edit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was a bit of a seat-of-pants stream-of-consciousness hack job, so the flexibility of the platform going forward is not great.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usage should be mostly self-explanatory, but I'll mention that the system is biased against questions involving 0 and 1 (eg, it will produce fewer such questions on average) so I suggest including them in your prescribed quizzes. The 'link' updates itself as you change the configuration values. You can configure a quiz with, say, 5 questions and multiplicands between 2 and 7, and load this quiz directly with the link &lt;a href=&quot;http://nilock.github.io/MathSnips/MultQuiz/MultQuiz.html?numQ=5&amp;amp;min=2&amp;amp;max=7&amp;amp;maxmin=10&quot; rel=&quot;nofollow&quot;&gt;http://nilock.github.io/MathSnips/MultQuiz/MultQuiz.html?numQ=5&amp;amp;min=2&amp;amp;max=7&amp;amp;maxmin=10&lt;/a&gt;. This link, in turn, can be fed into a url shortening service such as tinyurl.com in order to produce a url that can be more easily communicated / manually typed. For example, the above configuration has been fed into tinyurl which produced the link &lt;a href=&quot;http://tinyurl.com/m8esp94&quot; rel=&quot;nofollow&quot;&gt;tinyurl.com/m8esp94&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further extension of this particular page will be limited, but I'm happy to keep the current implementation in working order. Where defects are found on this page, the best option is to post them to &lt;a href=&quot;https://github.com/NiloCK/MathSnips/issues&quot; rel=&quot;nofollow&quot;&gt;https://github.com/NiloCK/MathSnips/issues&lt;/a&gt; with the New Issue button in the upper right. Posts there will land in my inbox, which is more reliably read than a Stack Exchange push notification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have ideas toward a similar but more generalized and robust system, but that's a long term project. I'll post it here should I ever get it off the ground.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2015-04-27T13:43:47.697" LastActivityDate="2015-04-27T13:43:47.697" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="7825" PostTypeId="1" AcceptedAnswerId="7835" CreationDate="2015-04-08T02:38:54.007" Score="7" ViewCount="2130" Body="&lt;p&gt;Whenever teaching or discussing parabolas, conic sections and other curves with my students, I always feel dissatisfied with the standard &quot;find vertex, pick points, connect the dots&quot; method to draw a parabola.  This also applies to higher degree curves, using local max/min, inflection points, etc.  I feel like it is equivalent to guess-and-check, which definitely has it's uses, but I would really like to give them more of a complete and continuous way to create parabolas and any other curves.  One thing I was thinking was to have students to essentially perform the mathematics behind Bezier curves by marking control points, drawing in the scaffold, subdividing, etc until they arrived at a reasonable construction.  I'm just not sure exactly how to go about doing this with regards to specific curves or if this would even be the best way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any experience, insight, or resources on constructing and sketching parabolas, conic sections, or other curves that is different from just picking points and connecting the dots?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2015-04-09T02:54:19.593" LastActivityDate="2016-12-07T11:24:36.887" Title="Constructing and sketching parabolas, conic sections and other curves" Tags="|calculus|algebra|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7826" PostTypeId="1" CreationDate="2015-04-08T07:37:36.220" Score="18" ViewCount="1569" Body="&lt;p&gt;It occurs to me that to really understand the ways that people work with fractions on paper requires a good grasp of the ideas that numbers have multiple representations and that expressions can be manipulated in various ways without changing the number they represent. These are essentially algebraic ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, adding fractions requires us to rewrite the fractions in a different form, and then essentially factorise the expression. This is the same as rearranging expressions in algebra. Dividing fractions requires us to rerepresent an operation like $\div \frac{2}{3}$ as $\times \frac{3}{2}$. This is the same as realising the connection between operations that you use to solve equations in algebra. And cancelling down before multiplying is very sophisticated rewriting relying on various associative and commutative laws.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it seems that we are really asking children to think in algebraic ways in order to understand fraction calculations well. This would seem to me to be a good reason why children and adults find it hard - they need more scaffolding in some abstract ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this a reasonable theory and has anyone written about this algebra-fractions connection before? To be clear, I am not asking if this is the &lt;em&gt;only&lt;/em&gt; reason fractions are hard, but if there is any discussion out there to draw parallels between learning algebra and learning to manipulate fractions.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-04-08T23:33:29.633" LastActivityDate="2016-03-03T17:36:36.340" Title="Are fractions hard because they are like algebra?" Tags="|algebra|fractions|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="7827" PostTypeId="2" ParentId="7826" CreationDate="2015-04-08T08:06:21.237" Score="9" Body="&lt;p&gt;Not sure about  paper references. One reason why  people don't understand fractions is because they are seemingly illogical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You score one basket out of three 1/3.&#xA;A little while later you try again and score 1/2.  Clearly you have scored 2/5 shots?  In many ways this is the correct answer.  So why shouldn't $\frac {1}{3}+\frac {1}{2}=\frac {2}{5}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;People generally don't understand equivalent fractions. It is strange for one farmer to say there are 4 sheep and another to say there are 8/2 sheep in the same field.  People assume that the  number 4 does what it says on the tin and is how we always describe 4 ness of something.  They don't understand equivalence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Partly to blame is treating fractions like conjuring tricks. If this is the question...do this,  if this is the question ...do another uncorrelated thing.  I asked my class (who seemingly could compute $\frac {2}{3}\times\frac {3}{5}$ correctly) to draw me a picture&#xA;instead of just multiplying.  No one could do it yet they all said &quot;but it's $\frac{6}{15}$ you times the top and the bottom!&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think drawing fractions is extremely  useful. Draw $\frac{2}{3}\div2$ or $2\div \frac {2}{3}$  It's not easy but I find students develop robustness eventually and begin to abstract themselves.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastEditorUserId="4668" LastEditDate="2015-04-08T08:14:15.640" LastActivityDate="2015-04-08T08:14:15.640" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="7828" PostTypeId="2" ParentId="7825" CreationDate="2015-04-08T11:00:48.960" Score="5" Body="&lt;p&gt;Drawing an ellipse with string is a fun &amp;amp; educational exercise.&#xA;Clear instructions can be found at&#xA;&lt;a href=&quot;http://www.mathopenref.com/printellipse1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathopenref.com&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/uZwzv.gif&quot; alt=&quot;EllispeString&quot;&gt;&#xA;&lt;hr /&gt;&#xA;One can also draw a parabola with string, but it is more difficult&#xA;(i.e., would take longer in the classroom).&#xA;Instructions from the Autralian Assoc. Math Teachers &lt;a href=&quot;http://cs.smith.edu/~orourke/MathOverflow/nmd11-DrawingParabola.doc&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;here (.doc download)&lt;/em&gt;&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/PmJ2J.png&quot; alt=&quot;ParabolaString&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-04-08T11:36:21.563" LastActivityDate="2015-04-08T11:36:21.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7829" PostTypeId="2" ParentId="7825" CreationDate="2015-04-08T13:04:41.630" Score="2" Body="&lt;p&gt;It could be an option to discuss the notion of an &lt;a href=&quot;http://en.wikipedia.org/wiki/Osculating_circle&quot; rel=&quot;nofollow&quot;&gt;osculating circle&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The osculating circle of a (sufficiently smooth) curve at a point is the circle that passes through this point, has the same tangent there (so its center is on the line orthogonal to the tangent) &lt;em&gt;and&lt;/em&gt; it even has the same curvature. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are geometric constructions, but espcially in a calculus context also a calculational approach could be an option.      &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2015-04-08T13:04:41.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7830" PostTypeId="2" ParentId="7825" CreationDate="2015-04-08T15:45:02.007" Score="3" Body="&lt;p&gt;I've got a construction technique for parabolas that's highly involved and requires a great deal of attention from the teacher, but it's really engaging and drives home some of the more obscure parts of the parabola, namely the directrix and focus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need rulers, wax paper, and permanent markers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Give each student a ruler, marker, and roughly square section of wax paper. Mark a point a couple inches from the bottom of the wax paper, this is your focus. Using the ruler, draw a line in between the close edge of the paper and the focus parallel to the bottom of the paper, this becomes your directrix. Put away the markers, the rulers are helpful for making good creases when we fold, but not necessary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is hard to describe so bear with me. The bulk of the paper including the focus remains on the table, pick up the part of the paper that has the directrix and fold so that the directrix passes through the focus. Make a good crease and unfold. This first fold is parallel to the directrix and makes the line that is tangent to the vertix. Then, pick up the directrix again and make another fold slightly to the side, so that the new fold is just out of parallel. Then repeat all the way up the side of the paper. Once that is done, make the fold just to the other side of parallel and repeat that process. What you're doing is making all of the lines that are tangent to your parabola, and the effect of those lines on the wax paper will reveal the parabola.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're trying to talk about the focus and directrix this is a good time to take some measurements to prove that the distance to the focus is equal to the height from the directrix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This construction will take some practice for you and you'll have to help a lot of your students. A document camera can help, although I don't have one and have done this successfully for several years. I have made a helper before if I had a student after school for another reason a couple days before this lesson, I would teach them to make it and then ask them to help instruct other students in class.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2015-04-08T15:45:02.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7831" PostTypeId="2" ParentId="7826" CreationDate="2015-04-08T22:38:22.770" Score="11" Body="&lt;p&gt;The obvious (to me) source of difficulty is that fractions are just plain complicated, more so than almost anything else in elementary education.  You have to operate with a pair of numbers, instead of a single one, and you have to keep the order straight.  Adding is quite complicated in its own right.  Things are further complicated by rules about least common denominators and least terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a little unclear about the question's emphasis on algebra.  Any sort of general rule or operation in arithmetic must have a connection to algebra, but I do not see what is intrinsically difficult about algebra that relates to numeric fractions.  Certainly some parts of algebra are hard, and some parts harder than others, algebraic fractions among them.  It seems to me that fractions are difficult because it's easy to confuse the various bits.  Even when you've got them straight, they're noticeably slower to use, take concentration, and when things have such cognitive demands, they're harder to think with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conceptually, they're a little bit odd, which is probably distracting until you get used to them.  What they represent do not seem to apply to the same things that (whole) numbers do.  Evidently fractions are not considered in this passage:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In that city, which was the oldest in the world, the cat was an object of veneration. Its worship was the religion of the country. The multiplication and addition of cats were a perpetual instruction in arithmetic. Naturally, any inattention to the wants of a cat was punished with great severity in this world and the next... -- A. Bierce, &quot;A Revolt of the Gods&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now to have one-and-a-half cats seems a very different thing than to have three halves.  In the former case, there's a good chance that the one cat you have will be alive and purring, while the same could not possibly be said about any of the halves.  No doubt such lessons are considered blasphemous in that city.  While many things may be divided into parts -- cars are a better example than cats -- not many can be divided into equivalent parts that can be used as a basis for fractions.  As we get used to fractions, as well as real numbers, we are taught to ignore this and accept statements such as &quot;the average family has 2.4 children.&quot;  Here is another example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By then, she will have shed 80 of the 240 pounds she weighed in with when she entered Peter Bent Brigham hospital obesity program.  A third of her left behind! -- &lt;em&gt;The Boston Herald American&lt;/em&gt;, 7/7/77&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The question seems to welcome references.  There are certainly several that connect fractions with algebra.  This paper,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S1364661312002653&quot;&gt;Seigler &lt;em&gt;et al.&lt;/em&gt; (2013), Fractions: the new frontier for theories of numerical development, &lt;em&gt;Trends in Cognitive Sciences&lt;/em&gt;&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is a short survey of what is known and unknown about neural bases for one's knowledge of fractions.  Whole number arithmetic knowledge has been studied, and the authors suggest that the representation of the knowledge fractions is an area ripe for investigation.  It reviews (with references) why fractions are difficult and the relation of skill at fractions to skill at algebra.  Generally -- or, rather, I only know of papers that discuss the connection in that direction, with algebra skill being dependent on fractions skill.  (OTOH, I'm not widely read in this area.)&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2015-04-08T22:38:22.770" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7832" PostTypeId="2" ParentId="7753" CreationDate="2015-04-08T23:29:45.063" Score="8" Body="&lt;p&gt;As the poster points out, &lt;a href=&quot;https://www.youtube.com/watch?v=c1BSkquWkDo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Monty Hall himself says they never did so on the show&lt;/a&gt;. (He says so at 1:45 in the linked video.)&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia has a &lt;a href=&quot;https://en.wikipedia.org/wiki/Let%27s_Make_a_Deal#Other_deal_formats&quot; rel=&quot;nofollow noreferrer&quot;&gt;long list of the sorts of deals that were offered on the show&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="182" LastEditorUserId="6053" LastEditDate="2022-07-30T02:48:53.020" LastActivityDate="2022-07-30T02:48:53.020" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7833" PostTypeId="1" CreationDate="2015-04-09T01:36:06.907" Score="2" ViewCount="337" Body="&lt;p&gt;Anyone tasked with moving Calculus I from a textbook to students sooner or later asks her/himself that question and I am curious as to what people are thinking of when they do. But to make it more explicit, I would like to make a few assumptions.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Let us assume that your students are the kind to be found in Two-Year Colleges----which, these days, include about 50% of the student population in the US, so that:&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Their &quot;conceptual maturity&quot; is practically inexistent. For example, they decide what calculation to embark on solely on the basis of template examples. While they can compute to an extent, they have no idea on how to verify their result and always want to be told whether the result of their computation is right or wrong.&lt;/li&gt;&#xA;&lt;li&gt;While they can usually read the template examples, they cannot read a text to extract conceptual information.&lt;/li&gt;&#xA;&lt;li&gt;They are totally unused to the idea that the very text of a problem will often give them an idea on how to attack it.&lt;/li&gt;&#xA;&lt;li&gt;They are totally unaware that, in mathematics---as in law, one has to make a case for whatever one asserts, be willing to present one's case to &quot;peers&quot;, and be ready to defend one's case.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Your students are not necessarily declared pre-engineering students, but can also be students who, based on what their experience in calculus 1 will have been, will decide on what to major in or, perhaps, students who have been persuaded to give it a try by a former instructor. In any case, a major issue is that many of the students will likely not continue into Calculus 2.&lt;/li&gt;&#xA;&lt;li&gt;You are not bound to use any particular textbook and in fact let us assume that you have facilities to develop the materials you need to help you reach your goal. Also, let us assume that you have the blessing of your department.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;At this point, the question of what you would want your students to learn might be, for example and certainly not to limit things, whether you primarily want:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To remedy some of the shortcomings listed above using as means some or all of the standard topics,&lt;/li&gt;&#xA;&lt;li&gt;To improve / consolidate / extend whatever computational skills they already have using as means some or all of the standard topics,&lt;/li&gt;&#xA;&lt;li&gt;To let the students acquire a &quot;coherent view&quot; of the calculus even at the cost of not covering all the &quot;standard&quot; topics,&lt;/li&gt;&#xA;&lt;li&gt;To let the students acquire the mathematical techniques they will be expected to use in following courses,&lt;/li&gt;&#xA;&lt;li&gt;etc&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2015-04-09T02:03:05.400" Title="Exactly what do I want these Calculus I students to learn?" Tags="|calculus|" AnswerCount="1" CommentCount="11" ClosedDate="2015-04-09T11:16:12.303" ContentLicense="CC BY-SA 3.0" />
  <row Id="7834" PostTypeId="2" ParentId="7833" CreationDate="2015-04-09T02:03:05.400" Score="5" Body="&lt;p&gt;I feel what needs to be said does not fit into a comment space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You seem to have a handle on the situation when you give a list of alternatives.  I have the feeling you want us to say which one of these, or two or three, are the ones to prioritize, given the scenario conditions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real answer is that it depends on the student.  All of them need to learn how to learn, whether it be to help themselves think, or find what they need, or actually do calculus computations, or understand the material well enough to explain it to their children.  Some of them will have the maturity or ability to know what specifically to get out of the course; others will be wondering and floundering at the (usually new) experience of learning something outside of high school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You, as teacher, have to deal with all of this, usually by delegation, sometimes by ignoring, and hopefully, by understanding and encouraging.  One of the universal gifts you can give the students is self-assessment, another that you can try to give is curiosity.  Go for trying to spark the imagination of some, and see how many catch fire (figuratively).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I were in your shoes, I would want them to get the high-level perspective, and learn how to use their text to solve problems they are likely to encounter, or at least need to learn as a foundation for problems they are likely to encounter.  I would not have them memorize the definition of derivative so much as have them imagine what a rate of change means to them well enough to explain to someone else and have that someone else get it.  I would have them appreciate that given a problem where finite differences are natural to use, it is ok to shift domains and phrase an analogous problem in terms of smooth functions, and have an idea how to interpret an answer in that domain back to the problem domain which involves discrete quantities.&#xA;I would be thrilled if they could solve problems using only what they remembered; I would be satisfied if, six months after the class, they could take the textbook and use it quickly to solve problems given on an exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;It's Different For Online Classes&quot; Paseman, 2015.04.08&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-09T02:03:05.400" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7835" PostTypeId="2" ParentId="7825" CreationDate="2015-04-09T02:49:34.340" Score="6" Body="&lt;p&gt;I don't know how I wasn't able to find these earlier...in no particular order.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe this is similar to @BBS great answer but also includes constructions for the other three conics by folding paper.  Has some pictures and clear instructions.&lt;br&gt;&#xA;&lt;a href=&quot;http://www.mathinscience.info/public/conic_sections/folding_conic_sections.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.mathinscience.info/public/conic_sections/folding_conic_sections.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is where things get interesting :) I found this video and was astouned! Seriously, this compass is quite marvelous.  Going to Mr. Sardeghi's website yeiled just a brief description and an email address.  While I wait for his response, I decided to see what else I could find about him and the CSC.  So the address he lists on his website is Google Los Angeles but there is no record of him working there (Linkedin and Google searches, not amazingly accurate i know...).  From what I can tell he has been a professor/inventor for the last 20 years or so and has filed multiple patents one of which was for the CSC.  However, the patent is registered in Germany.  Further searching found a US Patent 5870830 filed by a one Wu Yen-Hung which essentially details the exact tool that Mr. Sardeghi has machined out of metal in the video, albiet his has more frills.  At the end of the patent it discusses the Prior Art and mentions the ellipsiograph, another wonderful tool that has much more documentation.  Both of these tools require some effort to build/buy but I think they would be so much fun for the kids to get turns using!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Video of CSC:  &lt;a href=&quot;https://www.youtube.com/watch?v=lNSIGxImgeA&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=lNSIGxImgeA&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Video of Ellipsiograph: &lt;a href=&quot;https://www.youtube.com/watch?v=1AeAtP5Hz_Y&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=1AeAtP5Hz_Y&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cool constructions of conics using dynamic software like Geogebra or GSP&#xA;&lt;a href=&quot;http://whistleralley.com/conics/conic_construction/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://whistleralley.com/conics/conic_construction/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More dynamic geometry constructions, this time using animation and tracing&#xA;&lt;a href=&quot;http://jwilson.coe.uga.edu/emt668/EMAT6680.2002.Fall/Wright/wright6/sketchexp.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jwilson.coe.uga.edu/emt668/EMAT6680.2002.Fall/Wright/wright6/sketchexp.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interesting construction of ellipse and parabola, still some &quot;connect the dot&quot; but a much more geometric approach&#xA;&lt;a href=&quot;https://www.me.utexas.edu/~rbarr/sketching/conic.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.me.utexas.edu/~rbarr/sketching/conic.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cool construction of an oval using compass and straight edge&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=2GyWNVFQ6I4&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=2GyWNVFQ6I4&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has a great demonstration using lights shown on a wall.  Depending on the angle the light forms any of the four conic sections, even including a hyperbola if the light is shown through a lamp with a cylindrical shade.  Students could trace these on the white board as an introductory activity.&#xA;&lt;a href=&quot;http://php.math.unifi.it/archimede/archimede_NEW_inglese/curve/guida/paginaindice.php?id=2&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://php.math.unifi.it/archimede/archimede_NEW_inglese/curve/guida/paginaindice.php?id=2&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math Exchange answer for constructing ellipse&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/263307/a-construction-using-straightedge-and-compass&quot;&gt;https://math.stackexchange.com/questions/263307/a-construction-using-straightedge-and-compass&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cool constructions of conics using dynamic software like Geogebra or GSP&#xA;&lt;a href=&quot;http://whistleralley.com/conics/conic_construction/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://whistleralley.com/conics/conic_construction/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-04-09T02:49:34.340" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7836" PostTypeId="2" ParentId="7761" CreationDate="2015-04-09T20:57:43.177" Score="5" Body="&lt;p&gt;This is a slightly left of field answer but have you thought of no posters but something interactive instead?  I'd go for installing whiteboards with a hanging pen and stick a problem of the week on it. The questions could be made to be colourful, inspiring and relevant to what you're teaching. The most creative solutions to be discussed in class.  Better still have the students create the problem!  Or question for the professor? In my classroom all the walls have magnetic boards so that the students can do puzzles on them. I also have magnetic matches for sequences and flip over answers to simple questions.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-04-09T20:57:43.177" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7837" PostTypeId="1" AcceptedAnswerId="7868" CreationDate="2015-04-09T21:12:21.967" Score="46" ViewCount="12145" Body="&lt;p&gt;This question is inspired by @DavidButlerUofA's discussion of&#xA;&quot;&lt;span class=&quot;math-container&quot;&gt;$\div \frac{2}{3}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\times \frac{3}{2}$&lt;/span&gt;&quot; in&#xA;&quot;&lt;a href=&quot;https://matheducators.stackexchange.com/q/7826/511&quot;&gt;Are fractions hard because they are like algebra?&lt;/a&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. How can one best convey to beginners&amp;mdash;without algebra&amp;mdash;the flipping of denominator&#xA;  fractions, say, &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{\frac{2}{7}} = 4 \times \frac{7}{2} = 14$&lt;/span&gt;?&#xA;  In other words, what would convince a novice of the multiplication &#xA;  of the numerator by &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The image below is my attempt, but I doubt this is convincing&#xA;(the main insight being that one should partition the numerator &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;ths&amp;mdash;&lt;span class=&quot;math-container&quot;&gt;$28$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;ths&amp;mdash;and then it is clear that those &lt;span class=&quot;math-container&quot;&gt;$28$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;ths contain &lt;span class=&quot;math-container&quot;&gt;$14$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{7}$&lt;/span&gt;ths).&#xA;Those with experience teaching fractions likely have better &#xA;pedagogical routes.&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/BiH6x.jpg&quot; alt=&quot;TwoSevenths&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-01-18T01:24:03.400" LastActivityDate="2023-01-02T02:16:36.460" Title="How to explain the flipping of division by a fraction?" Tags="|primary-education|fractions|" AnswerCount="18" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="7838" PostTypeId="2" ParentId="7837" CreationDate="2015-04-09T22:23:51.950" Score="18" Body="&lt;p&gt;Here are a few comments, and then an attempt at two succinct answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, I will try to answer this using a &lt;strong&gt;measurement&lt;/strong&gt; interpretation, and then again with an &lt;strong&gt;equal sharing&lt;/strong&gt; interpretation. I prefer the former, but include the latter for completeness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Comments:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the key terms in unpacking this are &lt;strong&gt;measurement&lt;/strong&gt;, &lt;strong&gt;equal sharing&lt;/strong&gt;, and &lt;strong&gt;missing factors&lt;/strong&gt;, which are (up to some name-changing) the standard three interpretations of division of whole numbers, and which can be extended to discuss the division of rational numbers, as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have written a bit about each of them in &lt;a href=&quot;https://matheducators.stackexchange.com/a/5947/262&quot;&gt;&lt;strong&gt;MESE 5648&lt;/strong&gt;&lt;/a&gt;, and, as a comment on an answer from the question that inspired this one, I included a picture of $2 \div \frac{2}{3}$ using the former two interpretations:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/njMOa.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the measurement picture, the dividend ($2$) refers to the total number of items, the divisor ($\frac{2}{3}$) refers to the size of each single group, and the quotient ($3$) refers to the total number of groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the equal sharing picture, the meanings of the divisor and quotient switch; thus, depicted is $\frac{2}{3}$ of a group, and the quotient will be the size of a single group (drawn in with dotted lines).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Attempt at a succinct &quot;measurement&quot; answer:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two additional important terms are &lt;strong&gt;partitioning&lt;/strong&gt; and &lt;strong&gt;chunking&lt;/strong&gt;. To speak concretely, when we divide $4$ by $\frac{2}{7}$, let us use a length model (e.g., a line segment of length $4$). We wish to see how many $\frac{2}{7}$s we can measure out from this line segment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To do so, we begin with the simpler task of how many $\frac{1}{7}$s can be measured out from the length $4$ segment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is most easily done by &lt;strong&gt;partitioning&lt;/strong&gt; the $4$ unit intervals, each, into $7$ pieces of equal length. Note that the effect of this partitioning is that our number of intervals has been &lt;em&gt;multiplied&lt;/em&gt; by $7$. ($\star$)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Ensure students have ample practice dividing whole numbers by unit fractions before moving on!) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we move on to the case with $\frac{2}{7}$. Each of these pieces to be measured out can be found by taking our partitioned segment and its $4 \times 7 = 28$ intervals and &lt;strong&gt;chunking&lt;/strong&gt; them by combining them $2$ at a time. We do this because they &lt;em&gt;were&lt;/em&gt; of length $\frac{1}{7}$, and we want them to be of length $\frac{2}{7}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the effect of this chunking? Well, we are making each measured out piece $2$ times as long, which means the total number we can fit in will be only $\frac{1}{2}$ of what we had before, i.e., we divide the total number by $2$. ($\star$)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, let us look at the two starred items from above:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The partitioning led us to multiply $4$ by $7$, and the chunking led us to divide that result by $2$. In particular, we end up with $4 \times \frac{7}{2}$ as desired.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Attempt at a succinct &quot;equal sharing&quot; answer:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With equal sharing, we interpret $4 \div \frac{2}{7}$ by equal sharing the dividend ($4$) among the divisor ($\frac{2}{7}$) groups, and take the quotient to be the amount in one full group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, let us illustrate what $\frac{2}{7}$ of a group looks like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/LyKVG.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We want to equally share our $4$ within this $\frac{2}{7}$ of a group; as apparent in the shaded illustration above, this means distributing across the $2$ components, hence dividing $4$ by $2$ to alter our illustration as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/zKFWj.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, the quotient is the amount in &lt;em&gt;one full group&lt;/em&gt;; this will lead us to take our $4$ divided by $2$ (alternatively written as: $4 \times \frac{1}{2}$), and put this amount in each of the $\frac{1}{7}$ pieces for the full group:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/MHJFu.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We now see that we have multiplied the $4 \times \frac{1}{2}$ by $7$ to find the quotient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, we end up with $4 \times \frac{1}{2} \times 7 = 4 \times \frac{7}{2}$ as desired.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Post-Script:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With respect to contrasting the two interpretations, the dividend is dealt with in different ways: The measurement approach multiplies by $7$ then divides by $2$; the equal sharing approach divides by $2$ then multiplies by $7$. In each interpretation, the division by $\frac{2}{7}$ is equivalent to multiplication by $\frac{7}{2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally: Making sense of the invert-and-multiply algorithm is very difficult, and may require a fair bit of scaffolding. Certainly it cannot be done justice in this one brief response. Instead, I hope you will consider the above as a sketch of how the topic might be broached. For students really to encapsulate this idea will likely take a fair amount of time, and probably experience with standard division (and its interpretations), whole numbers divided by unit fractions, whole numbers divided by non-unit fractions, and so forth.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-04-10T20:44:30.470" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7839" PostTypeId="2" ParentId="7660" CreationDate="2015-04-10T01:50:10.610" Score="3" Body="&lt;p&gt;You might consider basing your course on the Differential Geometry lectures given by Dr. N. J. Wildberger [found on youtube, link below.] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He has some idiosyncrasies that you can clearly ignore, but his approach does not assume a large amount of linear algebra at all. He does employ matrices later on in the game, but I think with a lecture or two on the use of them, you could be fine. &#xA;His approach is super algebraic (not as in abstract algebra), which would be great for strengthening your students' skills before/concurrently with linear algebra!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His math history lectures on geometric topics might be good to flesh out the course if the latter part of his differential geometry series goes too far in depth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His course also employs a lot of projective geometry -- which I think is definitely lacking in the curriculum as a whole and would be very worthwhile!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the link to his playlist: &lt;a href=&quot;https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP&quot; rel=&quot;nofollow&quot;&gt;Diff Geo Wildberger&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2015-04-10T01:50:10.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7840" PostTypeId="2" ParentId="7636" CreationDate="2015-04-10T01:55:31.197" Score="2" Body="&lt;p&gt;Taxicab Geometry!!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Show your students how to pose problems in mathematics. Get them posing their own geometric questions. What if I change that definition/axiom, for example? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example: &lt;a href=&quot;http://www.jstor.org/stable/27970540&quot; rel=&quot;nofollow&quot;&gt;http://www.jstor.org/stable/27970540&lt;/a&gt; [Taxicab Conics!]&#xA;Or do plain old conics from a purely synthetic approach! The Dandelin Spheres are a beautiful geometric argument that they could appreciate if they got to mess with some conics first. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do some math history with the them! (Math History Wildberger on youtube)&lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2015-04-10T01:55:31.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7841" PostTypeId="2" ParentId="7636" CreationDate="2015-04-10T03:57:36.543" Score="2" Body="&lt;p&gt;The most important real-world application of noneuclidean geometry is in relativity. If you use GPS, then relativity is part of your everyday life. General relativity has a reputation for being difficult and highly mathematical, but it doesn't need to be. Here are some resources that let you get at the subject at a fun level:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486293157&quot; rel=&quot;nofollow&quot;&gt;Gardner, Relativity Simply Explained&lt;/a&gt; (fun, but dated)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521141001&quot; rel=&quot;nofollow&quot;&gt;Takeuchi, An Illustrated Guide to Relativity&lt;/a&gt; (good treatment of kinematics in special relativity, but lousy treatment of dynamics, and no discussion of general relativity)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0199236224&quot; rel=&quot;nofollow&quot;&gt;Stannard, Relativity: A Very Short Introduction&lt;/a&gt; (The treatment of general relativity is good. The treatment of special relativity is not as good as Takeuchi's.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I also have some &lt;a href=&quot;http://www.lightandmatter.com/poets/&quot; rel=&quot;nofollow&quot;&gt;lecture notes of my own&lt;/a&gt; that might be of interest.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2015-04-10T04:17:39.040" LastActivityDate="2015-04-10T04:17:39.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7842" PostTypeId="2" ParentId="7837" CreationDate="2015-04-10T04:46:21.903" Score="8" Body="&lt;p&gt;I suspect the main trick will be going from division by unit fractions to division by non-unit fractions. I would begin by considering that you have 4 groups of a certain size, and you want to know how many groups you'll have if you make a new set of groups $\frac{1}{7}$ the size of the current groups. This can be simplified further by considering each group on its own, and then it becomes obvious why division by unit fractions results in multiplication. Once the unit fraction idea is developed, split it into two steps: finding how many groups there will be at $\frac{1}{7}$ the size of the current groups, and then how many groups at double the size of the new groups. &lt;/p&gt;&#xA;" OwnerUserId="5019" LastActivityDate="2015-04-10T04:46:21.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7844" PostTypeId="2" ParentId="7837" CreationDate="2015-04-10T07:46:51.243" Score="7" Body="&lt;p&gt;If you have some apples and chop them into seven pieces, how many pieces do you have?  Well, seven times as many as you had apples, of course!  What if you pair those pieces two-by-two?  Well, okay, you now have half as many pairs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&quot;X divided by Y&quot; means &quot;how many Ys can you get out of what you started with&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Did it matter that we used seven?  Well, no, it could have been anything.  That we used two?   No, could have been any other thing.  Yay, it always works!  And we're doing algebra!)&lt;/p&gt;&#xA;" OwnerUserId="5021" LastActivityDate="2015-04-10T07:46:51.243" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7845" PostTypeId="2" ParentId="7837" CreationDate="2015-04-10T07:56:45.637" Score="15" Body="&lt;p&gt;I have no experience teaching fractions, but I think moving away from using the divide symbol makes things easier. It doesn't get used at university level (but exponents start being used, so there are still two notations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would do&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3\div\frac{2}{7} &#xA;= \frac{3}{\frac{2}{7}} &#xA;= \frac{3}{\frac{2}{7}} \times \frac{7}{7} &#xA;= \frac{3\times 7}{\frac{2}{7}\times7}&#xA;= \frac{3 \times 7}{2}&#xA;= 3 \times \frac{7}{2},$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;since you are assuming familiarity with multiplication of fractions. (Using $3$ instead of $4$ stops the point getting lost along the way.) &lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="308" LastEditDate="2015-04-10T12:15:57.243" LastActivityDate="2015-04-10T12:15:57.243" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7846" PostTypeId="2" ParentId="7837" CreationDate="2015-04-10T12:18:01.170" Score="15" Body="&lt;p&gt;When manipulating fractions, students quickly get comfortable with the idea that to combine two fractions they have to manipulate to get the denominators the same. Multiplying by 2/2 or 3/3, etc doesn't change the fraction value, obviously. I use the method below to unfraction (&amp;lt;&amp;lt; is there a word for this?) the denominator - &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{3}{\frac{2}{7}}=\frac{3\times \frac{7}{2}}{\frac{2}{7}\times \frac{7}{2}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;what do we multiply a number by to get 1?&quot; Its reciprocal, of course. So we multiply numerator and denominator by the reciprocal of that denominator. Some students will be stuck on this point and not really mature to the next step, unfortunately. The ones who quickly get comfortable with this will see the step of showing the denominator multiplied by the reciprocal can be skipped in favor of just using that number (the reciprocal) to multiply the numerator. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-04-10T12:18:01.170" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7847" PostTypeId="2" ParentId="7837" CreationDate="2015-04-10T12:55:13.457" Score="7" Body="&lt;p&gt;This might be overly practical, or it might be overly algebraic, but you could maybe teach them that that's all division ever is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, you could show that $ 4 = \frac{4}{1} $, because any number divided by 1 is the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you could say that $\frac{2}{3} = \frac{2}{1} \times \frac{1}{3} = \frac{2}{1} \div \frac{3}{1} = 2 \div 3$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this sense you don't teach them that they can flip a fraction to divide, so much as teach them that division is a convenience term we use for multiplying by the flip.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you start there, then when you go to $\frac{2}{3} \div \frac{7}{4}$ it becomes more obvious that we just do the same thing to get $\frac{2}{3} \times \frac{4}{7}$, because it's what we already did for $\frac{1}{3}$ and $3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not a technique, it's a definition.&#xA;That might not help, I'm not sure.&lt;/p&gt;&#xA;" OwnerUserId="3455" LastActivityDate="2015-04-10T12:55:13.457" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7848" PostTypeId="2" ParentId="7636" CreationDate="2015-04-10T12:58:06.167" Score="2" Body="&lt;p&gt;If you can get hold of all of it, &lt;a href=&quot;http://www.geom.uiuc.edu/video/NotKnot/&quot; rel=&quot;nofollow&quot;&gt;Not Knot&lt;/a&gt; gives an introduction to 3D hyperbolic geometry in the context of knot theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The the various programs by &lt;a href=&quot;http://www.geometrygames.org/&quot; rel=&quot;nofollow&quot;&gt;Jeff Weeks&lt;/a&gt; are also fun, if focused more towards topology. Playing pool on a hyperbolic surface does at least teach you it's different to Euclidean geometry.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-04-10T12:58:06.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7849" PostTypeId="2" ParentId="369" CreationDate="2015-04-10T14:38:21.223" Score="3" Body="&lt;p&gt;The original sequence of the high school SMSG texts was: First Course in Algebra; Geometry; Intermediate Mathematics; Elementary Functions; Matrix Algebra. The first three are year long courses, the last two were one semester courses. The sequence up to and including Elementary Functions would be what was needed for calculus when you got to college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later on, two other semester long courses were offered which could replace the Matrix Algebra course: Analytic Geometry and a course on Algorithms. Later still, a year long calculus course called Calculus of Elementary Functions was added as an option. This course mixed Elementary Functions with Calculus and was the advanced placement AB course of the time. There was also a year long Calculus course which had Intermediate Mathematics as a prerequisite. This class had most of the Elementary Functions material in appendices and was the Advanced Placement BC course. Finally, they added a Geometry with Coordinates course that was a mixture of the Geometry course with some Analytic Geometry. I got this information by studying the material on ERIC about the vision, actions, and execution of the SMSG plan. Also, a very helpful item is a price list for their materials also located on ERIC. Note also that I took the original sequence in high school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pretty much all of these course materials (student texts and teacher's commentaries) are availiable on ERIC along with the Elementary and Junior High courses and lots of supplementary materials.&lt;/p&gt;&#xA;" OwnerUserId="5024" LastEditorUserId="5024" LastEditDate="2015-04-21T14:21:51.353" LastActivityDate="2015-04-21T14:21:51.353" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7850" PostTypeId="2" ParentId="7837" CreationDate="2015-04-10T18:48:25.743" Score="3" Body="&lt;p&gt;I would start by demonstrating, with integers, that x / y = x * 1/y.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The technique for dividing by a fraction is based on that.&lt;/p&gt;&#xA;" OwnerUserId="5026" LastActivityDate="2015-04-10T18:48:25.743" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7851" PostTypeId="2" ParentId="7234" CreationDate="2015-04-10T19:38:50.473" Score="8" Body="&lt;p&gt;First, a disclaimer:  I am a mathematician, and not a math educator (at least, not beyond tutoring, and teaching algebra, statistics and some calculus as a grad student); thus, my answer is going to be colored by the experience of someone who has learned a lot more math than I have taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer depends on what you mean by &quot;higher math&quot;.  If by higher math, you mean &quot;calculus&quot;, then all I can say is that it's known that someone who isn't good at algebra is going to struggle in calculus; indeed, I have personally noticed that concepts of calculus can be pretty simple to understand, but individual problems can, algebraically speaking, get pretty scary, particularly if you make a mistake along the way.  I am not familiar enough with the differences between multiplicative reasoning, and algebraic reasoning, to comment on which is more important to understanding calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if by higher math you mean topology, abstract algebra, real and complex analysis, and so forth, then I would propose that the true gatekeeper is being able to prove things.  It is a well-known observation in mathematics that a lot of students who think they love math because they are good at calculus, algebra, differential equations, and so forth, will hit a brick wall when they hit proof-heavy classes, and many don't make the transition.  Indeed, occasionally someone who &lt;em&gt;isn't&lt;/em&gt; good at equations will somehow find themselves in a position to learn higher math, and find that they are pretty good at proofs!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, when I say &quot;proving things&quot;, I should point out that geometric reasoning, with its formal &quot;proof tables&quot;, isn't sufficient, and may even hurt interest in proofs.  Because I had an interest in learning everything I could, in my first year of college (and perhaps even before), I was trying to learn non-Euclidean geometry, graph theory, and number theory, and this gave me a foundation for linear and abstract algebra and other classes later on, and was certainly helpful in graduate school, where proofs are so central to everything.&lt;/p&gt;&#xA;" OwnerUserId="5028" LastActivityDate="2015-04-10T19:38:50.473" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7853" PostTypeId="1" CreationDate="2015-04-11T01:07:23.287" Score="10" ViewCount="976" Body="&lt;p&gt;Various branches of mathematics have mathematical beauty. Some of this are visual, such as the &lt;a href=&quot;http://en.wikipedia.org/wiki/Mandelbrot_set&quot;&gt;mandelbrot set&lt;/a&gt;, while others are logically sublime, such as the recursive simplicities of &lt;a href=&quot;http://en.wikipedia.org/wiki/Peano_axioms#Formulation&quot;&gt;peano arithmetic&lt;/a&gt; and &lt;a href=&quot;http://en.wikipedia.org/wiki/Surreal_number&quot;&gt;surreal numbers&lt;/a&gt;. What are some examples of mathematical beauty in school mathematics. I am mostly intrested in any examples for &lt;em&gt;High School&lt;/em&gt;, but any other K12 examples would be appreciated as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I am not asking for things marginally related to a school subject, but something that either helps a student understand or reinforce an existing understanding of a mathematical concept by making it &lt;a href=&quot;http://en.wikipedia.org/wiki/Mathematical_beauty&quot;&gt;aesthetically pleasing&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5030" LastEditorUserId="5030" LastEditDate="2015-04-11T02:11:59.777" LastActivityDate="2015-04-20T15:20:58.783" Title="Examples of Mathematical Beauty in School Mathematics" Tags="|mathematical-pedagogy|algebra|geometry|trigonometry|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="7854" PostTypeId="2" ParentId="7837" CreationDate="2015-04-11T01:42:14.320" Score="4" Body="&lt;p&gt;If you are looking for a concrete example, why not try &lt;em&gt;rates&lt;/em&gt;. Namely, suppose Jimmy can drink $4$ cups of soda every $2/7$s of a second. How much soda is that per second? Well, by the time $7$ of the $2/7$s of a second have passed, he would have drunk $28$ cups. But this amount of time is $2$ seconds, so in one second he drank $14$ cups. It may help to use a clock or something for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's abstract away the specific numbers first. If you drink $x$ cups every $a/b$ time units, you can drink $b*x$ cups in $a$ time units. If the students have not yet studied algebra, demonstrate with various $x$, $a$, and $b$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now abstract away rates. For any situation described by multiplication, division and fractions, $\frac x{\frac ab} = x \cdot \frac ba$. See if the students can abstract this to situations such as filling a rectangle with $4$ units of water, when the base is $2/7$, and finding the height.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, if you have enough time, abstract away from integers and rational numbers and discuss turning arbitrary integral domains into &lt;a href=&quot;http://en.wikipedia.org/wiki/Field_of_fractions&quot; rel=&quot;nofollow&quot;&gt;fields of fractions&lt;/a&gt; and beyond.&lt;/p&gt;&#xA;" OwnerUserId="5030" LastEditorUserId="688" LastEditDate="2015-10-17T14:04:20.427" LastActivityDate="2015-10-17T14:04:20.427" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7855" PostTypeId="2" ParentId="7853" CreationDate="2015-04-11T05:39:51.897" Score="3" Body="&lt;p&gt;One reason motivated me to study math, was my bad math teacher from high school. She showed me too much math, but never showed what I can do with it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you can make something as simple as the Thales theorem a relevant thing for them by showing a real application of it; instead of just focusing on shiny fractals or complex infinitesimal recurrences; that under the wrong circumstances will give the students the same feeling I had at high-school -yes, cool but useless. (my two cents).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;high school theory&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Lmbux.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Real Life Application&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ballistics, positioning and triangulation of objects in a real or simulated setting (yes, video games!). This mixed with some basic physics v = d/t can bring some nice visualizations and predictions of that torpedo will hit the other ship.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/F48eQ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="4859" LastActivityDate="2015-04-11T05:39:51.897" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7856" PostTypeId="2" ParentId="7837" CreationDate="2015-04-11T11:53:56.057" Score="5" Body="&lt;p&gt;Assuming that you've already covered&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Multiplication of fractions,&lt;/li&gt;&#xA;&lt;li&gt;Cancellation, and&lt;/li&gt;&#xA;&lt;li&gt;The idea that &lt;span class=&quot;math-container&quot;&gt;$a\div b$&lt;/span&gt; &lt;em&gt;means&lt;/em&gt; the solution to &lt;span class=&quot;math-container&quot;&gt;$bx=a$&lt;/span&gt; (though not necessarily expressed symbolically; it could also be in words such as &amp;quot;the number that we can multiply by the divisor to get the dividend&amp;quot;, which ought to be justifiable by reference to dividing integers into equal parts),&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I think a viable way would be to present flip-and-multiply not as a &lt;strong&gt;rule&lt;/strong&gt; that needs independent justification, but as a &lt;strong&gt;trick&lt;/strong&gt; that happens to work for solving &lt;span class=&quot;math-container&quot;&gt;$bx=a$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are fractions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If we want to divide &lt;span class=&quot;math-container&quot;&gt;$\frac38$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\frac25$&lt;/span&gt;, we're looking for a number that becomes &lt;span class=&quot;math-container&quot;&gt;$\frac38$&lt;/span&gt; when we multiply it by &lt;span class=&quot;math-container&quot;&gt;$\frac25$&lt;/span&gt;. We can manufacture such a number by multiplying the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and the &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, giving &lt;span class=&quot;math-container&quot;&gt;$\frac{3\times 5}{8\times 2}=\frac{15}{16}$&lt;/span&gt;. This works because if we then multiply by &lt;span class=&quot;math-container&quot;&gt;$\frac25$&lt;/span&gt; we get &lt;span class=&quot;math-container&quot;&gt;$\frac{15\times 2}{16\times 5}=\frac{30}{80}=\frac{3}{8}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Does this trick always work? Yes, because notice what happened to the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; on the way to &lt;span class=&quot;math-container&quot;&gt;$\frac{30}{80}$&lt;/span&gt;. The &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; got multiplied first by &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and then by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, and the &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; got multiplied first by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and then by &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;. But since the order of factors are immaterial, this means effectively &lt;em&gt;both&lt;/em&gt; the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; and the &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; got multiplied by &lt;span class=&quot;math-container&quot;&gt;$2\times 5$&lt;/span&gt;, which we're then allowed to cancel. So we do always get the original dividend back in this way.&lt;/p&gt;&#xA;&lt;p&gt;A convenient way to &lt;em&gt;remember&lt;/em&gt; this trick is that dividing by &lt;span class=&quot;math-container&quot;&gt;$\frac25$&lt;/span&gt; gives the same result as multiplying by &lt;span class=&quot;math-container&quot;&gt;$\frac52$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Or in other words: It's not a definition, it's a technique).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac52$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac25$&lt;/span&gt; here are useful example numbers here because the eventual cancellation is then directly visible.&lt;/p&gt;&#xA;" OwnerUserId="5032" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-09-21T09:38:51.303" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7858" PostTypeId="2" ParentId="7853" CreationDate="2015-04-11T14:47:03.450" Score="7" Body="&lt;p&gt;Proofs of the Pythagorean theorem can be beautiful. #9 at cut-the-knot is my favorite (&lt;a href=&quot;http://www.cut-the-knot.org/pythagoras/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.cut-the-knot.org/pythagoras/&lt;/a&gt;). Although this one on math stack exchange has its own beauty (&lt;a href=&quot;https://math.stackexchange.com/posts/259966/revisions&quot;&gt;https://math.stackexchange.com/posts/259966/revisions&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/B8bDZ.jpg&quot; alt=&quot;A visual proof of pythagoras theorem&quot; title=&quot;The blue, red, and black pentagons are similar. Since the red and blue triangles add to the black one, the same is true of the squares.&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-04-11T18:11:41.813" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7859" PostTypeId="2" ParentId="7837" CreationDate="2015-04-11T19:28:58.633" Score="2" Body="&lt;p&gt;I think working with a slough of examples may be effective.  I would start with patterns of whole numbers and then fractions n divided by n , then n divided by 1 and 1 divided by n.  Then try a whole number n divided by another whole number m and then contrast with m divided by n.  Note that the relation a / b = a * (1/b) holds throughout.  As a result, something divided by m/n is the same as something multiplied by n/m.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Then They Remember Through Example&quot; Paseman, 2015.04.11&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-11T19:28:58.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7860" PostTypeId="2" ParentId="7853" CreationDate="2015-04-12T01:01:30.783" Score="5" Body="&lt;p&gt;Mathematical beauty is in the eye of the beholder, so I hesitate to&#xA;respond to this question just before it gets closed.&#xA;But I'll share one example that I find aesthetically pleasing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you desire to cut out a triangle from the middle of a piece of paper,&#xA;not by punching the scissors through and cutting&#xA;the perimeter, but rather by folding the paper&#xA;and then cutting&#xA;straight through the folded paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The natural solution is to mountain-crease (red below) the angle bisectors,&#xA;and valley-crease (green dashed) a &quot;perpendicular&quot; from the incenter $x$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/h1flX.png&quot; alt=&quot;TriangleAngleBisectors&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Figure from &lt;em&gt;&lt;a href=&quot;http://howtofoldit.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra&lt;/a&gt;&lt;/em&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;What I find so pleasing is that when you perform this physically, the&#xA;angle bisectors meet at a point $x$ (the incenter), and one grasps &#xA;Proposition 4, Book IV of Euclid viscerally. Naively, it could well&#xA;be that the bisectors do not meet at a point. But careful creasing&#xA;shows experimentally that they do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found this tactile demonstration more convincing to &#xA;(U.S.) 8th-graders than a two-column Euclidean proof.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-04-12T01:01:30.783" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7861" PostTypeId="2" ParentId="7234" CreationDate="2015-04-12T12:22:09.607" Score="5" Body="&lt;p&gt;This is not a true answer to the question, rather a criticism of its premises. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If by &quot;multiplicative reasoning&quot; you mean, at least in part, the ability to work with linear dependence between variables or numbers, then there are indications that insisting on this as much as is done during the primary education may in fact &lt;em&gt;prevents&lt;/em&gt; students to succeed in math, and that this reflects way beyond the school and university walls.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I learned about this in the last issue of the &lt;a href=&quot;http://www.ems-ph.org/journals/newsletter/pdf/2015-03-95.pdf&quot; rel=&quot;noreferrer&quot;&gt;EMS Newsletter&lt;/a&gt; in the article by the Education Committee of the EMS, &quot;Solid Findings: Students’ Over-reliance on Linearity&quot; (page 51).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, students are taught into assuming linearity for the sake of giving them proficiency in &quot;multiplicative reasoning&quot; (change of term is mine, hope I did not misinterpreted it) through &quot;real-life&quot; exercises, but the side effect is that they may become unable to even think about a non-linear relation between the numbers at hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The following very clear example is given in the above article: &quot;Farmer&#xA;Carl needs approximately 8 hours to fertilise a square pasture with a side of 200 m. How many hours would he need to fertilise a square pasture with a side of 600 m?&quot; I let you read the article to see how much students can fail to solve this exercise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In conclusion, of course multiplicative reasoning is not by itself bad for math, but making it important and teaching it in some (currently common) way can do a lot of harm.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-04-12T12:22:09.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7864" PostTypeId="1" AcceptedAnswerId="7909" CreationDate="2015-04-13T16:10:07.640" Score="19" ViewCount="1174" Body="&lt;p&gt;I've noticed that some surfaces are difficult to draw in multivariable calculus. For instance, I always have trouble with hyperbolic paraboloids.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is an efficient way to draw the following surfaces:&lt;/p&gt;&#xA;&lt;p&gt;1)hyperbolic paraboloid&lt;/p&gt;&#xA;&lt;p&gt;2)hyperboloid (of one and of two sheets)&lt;/p&gt;&#xA;&lt;p&gt;3)a plane (harder than it seems! for me, at least)&lt;/p&gt;&#xA;&lt;p&gt;4)elliptic paraboloid&lt;/p&gt;&#xA;&lt;p&gt;5)ellipsoid&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Any other common surfaces you have tricks for would be helpful as well.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-11-14T04:19:55.157" Title="What is an efficient way of drawing surfaces in multivariable calculus?" Tags="|vector-calculus|board-use|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7865" PostTypeId="1" AcceptedAnswerId="7874" CreationDate="2015-04-13T16:13:21.673" Score="12" ViewCount="881" Body="&lt;p&gt;I have been teaching a ring theory course this semester, focusing on modular arithmetic and quotient rings of polynomials over fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Several students have asked me how one could graph functions from a finite field (or any $Z_n$) to itself. I've considered drawing them on a torus (since the input and output loop around onto themselves), but this would be difficult to depict on a two-dimensional chalkboard.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In your experience, is there an effective graphical representation of a function from a finite field to itself?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="78" LastEditorUserId="376" LastEditDate="2019-08-07T15:06:34.583" LastActivityDate="2019-08-07T15:06:34.583" Title="Graphing functions from a finite field to itself" Tags="|abstract-algebra|intuition|graphing|" AnswerCount="6" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7867" PostTypeId="2" ParentId="7865" CreationDate="2015-04-13T16:36:33.117" Score="3" Body="&lt;p&gt;There are dangers in using such graphical representations.  The reals are an ordered field, whereas orders are not compatible with field operations for finite fields and other fields.  It is easy to think things like &quot;this function is increasing&quot;: such thoughts are helpful if you are going to transfer something of the function to the reals, and can mislead in other cases.  Inside finite fields it is of more interest whether the function is a permutation of the nonzero elements or not.  Also zeroes of the function are important to represent.  Choose the representation to emphasize the &#xA;characteristic of importance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you put the necessary cautions in place, it should be OK to embed the graph in the upper right quadrant of the real plane.  Be sure to emphasize other views of the function as well: treating it as a polynomial, treating it as a permutation or near permutation, treating it as a fragment of a graph of a real valued function.  However, treating an object as something else, while useful, does not make that object something else.  If the toroidal nature of the ring appeals to you, use a square and identify edges of the square, and relate this to certain video displays to get them to deal with this form of representation (at, least, until the manufacturers come out with computer screens shaped like doughnuts).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Should Doughnuts Be Called Doughwheels?&quot; Paseman, 2015.04.13&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-13T16:36:33.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7868" PostTypeId="2" ParentId="7837" CreationDate="2015-04-13T17:04:37.230" Score="59" Body="&lt;p&gt;I'm going to focus on one aspect of the question that I think has not been fully appreciated:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can one best convey to &lt;strong&gt;beginners&lt;/strong&gt;—without algebra—the flipping of denominator fractions...  what would &lt;strong&gt;convince&lt;/strong&gt; a novice&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Many of the ingredients of this answer are already present in some of the other answers to this question, but are rearranged here in a fashion that (I hope) is significantly novel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What do beginners know?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplication and division of fractions is, in the United States at least, typically first encountered in either third or fourth grade; that is, at age 8 or 9.  Benjamin Dickman, in his answer, has discussed various representations of division (partitive/equal sharing, quotitive/measurement, missing factor).  I want to take a minute to talk about representations of fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the age of the students in question, I do not think there is much value in formal explanations, i.e. explanations that rely on symbolic manipulation, whether or not they are &quot;algebraic&quot; in nature.  They may learn that the algorithm is the correct one, but I don't think they will be &lt;em&gt;convinced&lt;/em&gt; by it, in the sense of believing that the algorithm &lt;em&gt;has to be&lt;/em&gt; that way and &lt;em&gt;makes sense&lt;/em&gt;.  To convince someone that &lt;span class=&quot;math-container&quot;&gt;$4 \div \frac27 = \frac{4 \times 7}2$&lt;/span&gt; you have to guide them to figure it out for themselves, in a way that makes the result seem completely inevitable and unsurprising.  And to do that you have to try to see things from the perspective of the learner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fractions students most often (perhaps exclusively?) encounter are &lt;em&gt;proper&lt;/em&gt; fractions, i.e. fractions &lt;span class=&quot;math-container&quot;&gt;$&amp;lt; 1$&lt;/span&gt;; quantities larger than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; are usually expressed as &lt;em&gt;mixed numbers&lt;/em&gt; rather than as &quot;improper&quot; fractions.  Students at that level typically understand fractions in the language of part/whole relationships; the most commonly used representations are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;a circle, divided into &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; sectors of equal area, with &lt;span class=&quot;math-container&quot;&gt;$k&amp;lt;n$&lt;/span&gt; of them shaded (corresponding to the &quot;real-world&quot; context of cutting up a pizza or other circular food);&lt;/li&gt;&#xA;&lt;li&gt;a collection of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; objects (often drawn as a group of small circles contained in a rectangular frame), with &lt;span class=&quot;math-container&quot;&gt;$k&amp;lt;n$&lt;/span&gt; of them shaded (corresponding to the &quot;real-world&quot; context of sharing a box of cookies or cupcakes);&lt;/li&gt;&#xA;&lt;li&gt;a collection of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; units of measurement (often drawn as small scoops or cups), drawn with some kind of scale or labeling that indicates that &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; of those units would comprise a single larger unit (corresponding to the &quot;real-world&quot; context of using, e.g., between three &quot;quarter-cup scoops of flour&quot; to measure &lt;span class=&quot;math-container&quot;&gt;$3/4$&lt;/span&gt; of a cup of flour).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The order in which I list the representations above is not accidental; it corresponds roughly to the order in which these representations actually occur in the classroom.  That is, the most commonly used representation is the &quot;circle cut into wedges&quot;, and the least commonly used representation is the &quot;small scoops that make up one big scoop&quot;.  Since the question is about &lt;em&gt;conviction&lt;/em&gt;, I think it is important to realize that not all representations are equally valuable for that purpose.  In particular, I think the third representation above, combined with the measurement interpretation of division, is actually the most useful for the purpose of explaining how division of fractions works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With all of that established as preamble, here is my strategy for convincing a novice that division of fractions works the way it ought to:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1. Announce the problem, but do not ask them to try to solve it yet.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Write the problem &lt;span class=&quot;math-container&quot;&gt;$6 \div \frac23 = ?$&lt;/span&gt; in large letters on a piece of paper and tell the student that we are going to figure this out, but we're going to work out way up to it by warming up with a few simpler problems first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2.  Begin by emphasizing the measurement interpretation of division.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since most students reflexively think of division in terms of equal-sharing, it is a good idea to start by explicitly activating the measurement interpretation, which is less common. Ask:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose you want to measure six cups of flour, and all you have is a two-cup scoop.  How many scoops do you need?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Most 8- to 9-year-old students will be able to answer &quot;Three&quot; immediately.  To do so, they do not need to explicitly translate the problem into &quot;Six divided by two equals what?&quot;, nor as &quot;Two times what equals six?&quot;  In fact, making such translations may seem like an unnatural complication to a simple problem of counting.  &lt;em&gt;So you have to ask them to make that translation explicit,&lt;/em&gt; with prompts like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How did you know that?&lt;/li&gt;&#xA;&lt;li&gt;What kind of arithmetic operation are you using? (Many will say &quot;addition&quot;, because they are simply thinking &quot;two plus two plus two&quot;.)&lt;/li&gt;&#xA;&lt;li&gt;What other kinds of arithmetic can you use to describe what you just did?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This conversation does not have to be a long one, but it does need to happen, and the goal is to activate (and &lt;em&gt;keep active&lt;/em&gt; in the student's mind) that &quot;&lt;span class=&quot;math-container&quot;&gt;$6 \div 2$&lt;/span&gt;&quot; can be understood as the question &quot;How many 2-cup scoops do you need to make 6 cups?&quot;.  Now you are ready to move to the next step:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3. Consider divisors that are unit fractions.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we want to vary the task, just slightly:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose you want to measure six cups of flour, and all you have is a &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; cup scoop.  How many scoops do you need?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Again most students (at this age level) will answer &quot;18&quot; almost immediately, at least if they have some degree of automaticity with the multiplication fact &lt;span class=&quot;math-container&quot;&gt;$6 \times 3 = 18$&lt;/span&gt;.  Students who do not have such automaticity may need to count by threes.  Either way they will get the answer very quickly.  Ask them how they know.  Guide the conversation to the following, very important summary:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Because the scoops only hold &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; cup, you need three scoops for each cup, so you have to multiply three times the number of cups you need.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To the extent possible, try to get the &lt;em&gt;student&lt;/em&gt; to be the one who says this, or something like it.  Don't say it &lt;em&gt;for&lt;/em&gt; them, but do &lt;em&gt;revoice&lt;/em&gt; their statement of it to make it more succinct and coherent, if necessary. Once students agree with this basic idea (and once said it usually seems completely obvious to them, so much so that they wonder why you were making such a big deal about it), you can move on to the final variation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4. Consider divisors that are not unit fractions.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One last tweak to the task:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose you need 6 cups of flour, and all you have is a &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; cup scoop.  How many scoops do you need?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some students will immediately try to do some kind of paper-and-pencil computation, whether they know an algorithm or not.  &lt;em&gt;Discourage this.&lt;/em&gt; Ask them instead to just think about the previous problem.  We already know that it will take eighteen &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; cup scoops to make six cups.  What if the scoops are &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; cups instead?  If they are still stuck, prompt:  How do the new scoops compare to the old scoops?  Usually students will eventually come up with an answer like this one:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Because the scoops are twice as large, we only need half as many of them, so we need just 9 scoops instead of 18.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Once they have said this, or something like it -- but &lt;em&gt;not before!&lt;/em&gt; -- write down &quot;&lt;span class=&quot;math-container&quot;&gt;$ 6 \div \frac23 = 9$&lt;/span&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are not done yet, though. The most important step is the next one:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5.  Summarize and generalize&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's time to look back at the expression &lt;span class=&quot;math-container&quot;&gt;$6 \div \frac23$&lt;/span&gt; and think about what happened when we tried to solve it.  Notice that there are two basic principles interacting here:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The 3 in the denominator of the divisor acted by &lt;em&gt;multiplication&lt;/em&gt;, because it takes three &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; cup scoops to fill a single cup, so if you are using scoops of that size, you need three times as many scoops as you do cups.&lt;/li&gt;&#xA;&lt;li&gt;The 2 in the numerator of the divisor acted by &lt;em&gt;division&lt;/em&gt;, because a &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; cup scoop is twice as large as a &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; cup scoop, so you need half as many of the larger scoops as you do of the smaller ones.  In other words:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To find &lt;span class=&quot;math-container&quot;&gt;$6 \div \frac23$&lt;/span&gt;, you &lt;em&gt;multiply by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; and divide by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or, transcribed symbolically,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To find &lt;span class=&quot;math-container&quot;&gt;$6 \div \frac23$&lt;/span&gt;, you can compute &lt;span class=&quot;math-container&quot;&gt;$\frac{6 \times 3}2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ask some questions to get the student to restate this result in their own words:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Where does the numerator of the &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; go?  (Into the denominator of the answer.)&lt;/li&gt;&#xA;&lt;li&gt;Where does the denominator of the &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; go?  (Into the numerator of the answer.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;At this point, the &quot;rule&quot; that &quot;To divide by a fraction, you invert the divisor and multiply&quot; should not seem like a &quot;rule&quot;, but merely a summary of something that should have been obvious all along.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Final Thoughts&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've gone into a lot of detail about this, and I think it's worth pointing out that it probably takes more time to &lt;em&gt;read&lt;/em&gt; the above description than to &lt;em&gt;implement&lt;/em&gt; it.  I've tutored dozens of kids in precisely this manner; some of them are elementary school kids learning fractions for the first time, others are high schoolers who &quot;learned&quot; how to divide fractions back when they were 8 or 9 but no longer remember what to do, or remember it imperfectly.  I've had very good success with this method.  If working with a student one-on-one, it usually takes no more than five minutes, start to finish. I think there are two reasons why it is effective:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, it begins by announcing the problem, and then immediately putting the problem on hold and instead considering simpler problems: first division by whole numbers, and then division by unit fractions. This models an important problem-solving heuristic:  &lt;em&gt;When you encounter a hard problem, consider a simpler one and see if you can get any insight from it.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, it concludes by looking back at a single example and trying to understand the general principles that make it work.  This models a second important problem-solving heuristic:  &lt;em&gt;When you have solved a hard problem, take a moment to look back at it and see if the perspective of hindsight reveals any general arguments to you.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both of these heuristics are, of course, taken directly from Polya's &lt;em&gt;How To Solve It&lt;/em&gt;. And while these heuristics are common to most problem-solving contexts, and may even be naturally-occurring for many students, it is worth recognizing that conventional school instruction (which I would caricature as &quot;Teach the rule, then do examples, then have students do many exercises, then provide an explanation of the rule&quot;) does not provide a lot of space for this kind of slow, reflective consideration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One final thought:  You may have noticed that the OP's example was &lt;span class=&quot;math-container&quot;&gt;$4 \div \frac27$&lt;/span&gt;, and I changed that for purposes of this explanation to &lt;span class=&quot;math-container&quot;&gt;$6 \div \frac23$&lt;/span&gt;.  That change was not accidental.  First of all, if you want to follow a narrative like the one I sketch above, asking about scoops that hold &lt;span class=&quot;math-container&quot;&gt;$1/7$&lt;/span&gt; of a cup is just silly.  Nobody makes or uses measuring scoops like that; the artificiality of the problem stands out and is distracting.  On the other hand, &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; cup measuring scoops are fairly commonplace (at least in the United States).  I imagine that this whole instructional sequence would have to be reconsidered from the ground up (and may be completely unworkable) if one were teaching in a context in which the metric system is consistently used.  The reason for changing the &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; in the original problem to a &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; is that the dual roles played by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; in the equation &lt;span class=&quot;math-container&quot;&gt;$4 \div 2 = 2$&lt;/span&gt; can actually be confusing.  Which &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; represents the size of groups, and which represents the number of groups?  Changing the &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; to a &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; eliminates that symmetry, and hence the ambiguity about which number represents which quantity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addendum, added September 18, 2016&lt;/strong&gt;&#xA;This afternoon I received an email from a middle school student who had found this answer online and was hoping for some further explanation.  Specifically, she wanted an explanation of how to think about division when &lt;em&gt;both&lt;/em&gt; the dividend and the divisor are fractions (in the example above, the dividend was a whole number).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First let me say that I think the existence of the email itself is proof that what seems obvious to the experienced person may not at all be obvious to the novice.  My first instinct in reading the email was to respond &quot;Well, it's exactly the same!&quot; -- but of course if it were exactly the same then there would have been no need for that email, would there?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So let's consider a new example:  Suppose we want to find &lt;span class=&quot;math-container&quot;&gt;$\frac{6}{10} \div \frac{2}{3}$&lt;/span&gt;.  Here is the essential idea that you need in order to generalize the work that was already done:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{6}{10}$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{10}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Let's see how that is useful here.  The problem &lt;span class=&quot;math-container&quot;&gt;$\frac{6}{10} \div \frac{2}{3}$&lt;/span&gt; can be interpreted as meaning:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If you want &lt;span class=&quot;math-container&quot;&gt;$\frac{6}{10}$&lt;/span&gt; of a cup of flour, and you only have a &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{3}$&lt;/span&gt; cup scoop to measure it with, how many scoops do you need?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now we have &lt;em&gt;already&lt;/em&gt; solved the problem of measuring out &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; cups of flour with a &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{3}$&lt;/span&gt; cup scoop, and found that we need &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; scoops to do it.  Our new problem asks us to measure out &lt;span class=&quot;math-container&quot;&gt;$1/10$&lt;/span&gt; as much flour total, so we need &lt;span class=&quot;math-container&quot;&gt;$1/10$&lt;/span&gt; as many scoops.  So the answer is &lt;span class=&quot;math-container&quot;&gt;$9/10$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Okay, now let's look back and think about what we did.  We took the numerator of the first fraction (&lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;) and multiplied it by the denominator of the second fraction (&lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;), then divided by the numerator of the second fraction (&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;).  This gave us &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;.  Finally we divided by the denominator of the first fraction (&lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;) to get the answer, &lt;span class=&quot;math-container&quot;&gt;$9/10$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In abstract form, the result of dividing &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\frac{c}{d}$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\frac{a \cdot d}{b \cdot c}$&lt;/span&gt;.  Another way of saying that is that we take the second fraction, &lt;span class=&quot;math-container&quot;&gt;$\frac{c}{d}$&lt;/span&gt;, and &quot;flip it&quot; to get &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{c}$&lt;/span&gt; ; then we multiply the first fraction by the upside-down second fraction.  In our example, we have:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{6}{10} \div \frac{2}{3} = \frac{6}{10} \times \frac{3}{2} = \frac{6 \times 3}{10 \times 2} = \frac{18}{20}$&lt;/span&gt;, which can be reduced to &lt;span class=&quot;math-container&quot;&gt;$\frac{9}{10}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2019-01-13T23:52:21.070" LastActivityDate="2019-01-13T23:52:21.070" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="7869" PostTypeId="2" ParentId="7865" CreationDate="2015-04-13T17:23:20.983" Score="5" Body="&lt;p&gt;&lt;strong&gt;Edited:&lt;/strong&gt; I would use a rectangular display that looks, at first, like a standard &quot;Quadrant I&quot; graph, but that can be grabbed and dragged left/right/up/down to move the viewing frame.  So, for example, if one is working over $\mathbb{Z}_7$ the horizontal and vertical scales will initially be labeled &quot;0 1 2 3 4 5 6&quot; in both direction, but the view can be dragged so that the horizontal axis reads &quot;3 4 0 1 2 3 4 5&quot; and the vertical axis reads &quot;5 6 0 1 2 3 4&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://tube.geogebra.org/student/mbA4jdzJU&quot; rel=&quot;nofollow&quot;&gt;Here is an attempt&lt;/a&gt; at creating such a tool.  It shows the function $y=x^2-x$ defined on $\mathbb{Z}_{12}$.  The graph behaves in such a way that the top and bottom edges of the plot are identified, as are the left and right edges, so the function is (essentially) graphed on a torus.  Points are labeled with $x$ and $y$ coordinates in the set $\{0, 1, \ldots 11\}$ and the &quot;axes&quot; (i.e. the lines $y=0$ and $x=0$) are drawn in as solid lines.  Were I more skilled at Geogebra, I would add tick marks to the axes with labels that run $\{0, 1, \ldots 11\}$ and repeat cyclically.  If any Geogebra gurus have suggestions on how to do that, I would appreciate the input.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2015-04-14T00:30:08.300" LastActivityDate="2015-04-14T00:30:08.300" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7870" PostTypeId="1" CreationDate="2015-04-13T17:26:43.080" Score="5" ViewCount="480" Body="&lt;p&gt;My questions are about LGBT people's situation in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there any research to show any connection between people sexual orientation and their mathematical ability? For example showing that LGBT people are better or worse in mathematics with respect to ordinary people? As far as I know there are such research about connection between being left or right handed and people's IQ, linguistic abilities, etc. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there countries that forbid LGBT people's tenure academic position or their education in mathematics?  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What are examples of prominent LGBT mathematicians?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5041" LastEditorUserId="78" LastEditDate="2015-04-14T00:00:10.827" LastActivityDate="2015-04-16T15:37:10.810" Title="Mathematics Education and LGBT Community" Tags="|cultural-differences|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7871" PostTypeId="2" ParentId="7870" CreationDate="2015-04-13T17:36:19.170" Score="2" Body="&lt;p&gt;Some scientific research shows that having a different sexual orientation &lt;em&gt;naturally&lt;/em&gt; happens in many spices including apes, cats, dogs, etc. As you mentioned it is somehow similar to being &lt;em&gt;left-handed&lt;/em&gt; in a community with &lt;em&gt;right-handed&lt;/em&gt; majority. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surely there are some physical and psychological differences between left-handed and right-handed people, but it doesn't mean that one should blame left-handed people or force them to write by their right hand just because write-handed people are majority of the community! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same is true for LGBT people. We &lt;em&gt;can't&lt;/em&gt; and &lt;em&gt;shouldn't&lt;/em&gt; deny or annoy them just because they are different from us. Conversely a civilized and developed society should provide some appropriate options and opportunities for its minorities including LGBT people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About the situation of LGBT community in different countries, as far as I know due to historical and cultural reasons there are many people with alternative sexual orientations in Israel and all Arabian countries of Middle East. However there is a big difference in their rights and degree of freedom. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Due to religious reasons in Muslim Arabian countries LGBT people are very limited and have almost no social rights but Israel is one of the most LGBT friendly countries of the world as well as being a religious state in many aspects. See the following &lt;a href=&quot;http://en.wikipedia.org/wiki/LGBT_rights_in_Israel&quot; rel=&quot;nofollow&quot;&gt;Wikipedia article&lt;/a&gt; which confirms there are too many LGBT people and too few anti-LGBT rules here and LGBT community is very active and popular in Israel society. In some sense Israel is the paradise of LGBT people! &lt;strong&gt;Tel-Aviv&lt;/strong&gt; itself has the credit of being LGBT capital of Middle East!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;LGBT people have their own rights and pride parades in almost all large cities of the country including Tel-Aviv and even Jerusalem where they recently had a LGBT parade! Note that Jerusalem is the city of Western Wall and one of the holiest cities of the world for almost all major religions! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;About mathematics education and LGBT people, I think there is no relevance and if there is any it is not negative. For example Alan Turing was a gay and a prominent mathematician. Also I personally know many LGBT mathematics teachers and students in universities of Israel who are my colleagues and have no problem in their career.   &lt;/p&gt;&#xA;" OwnerDisplayName="user5045" LastEditorDisplayName="user5045" LastEditDate="2015-04-13T23:20:27.990" LastActivityDate="2015-04-13T23:20:27.990" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7873" PostTypeId="2" ParentId="7865" CreationDate="2015-04-13T21:49:24.920" Score="13" Body="&lt;p&gt;One of the approaches taken in some areas of mathematics (e.g., in &lt;a href=&quot;http://en.wikipedia.org/wiki/Arithmetic_dynamics&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;arithmetic dynamics&lt;/strong&gt;&lt;/a&gt; and considerations of preperiodic points, etc) is to create these graphs by drawing discrete points and then using arrows to show which values map to which other ones. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Figuring out a &quot;canonical&quot; way to draw these pictures might be a bit tough (this is related in some manner to the concerns about &lt;em&gt;orders&lt;/em&gt; in &lt;strong&gt;Gerhard Paseman&lt;/strong&gt;'s answer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a sample picture of the function $x \mapsto x^2 - x$ in $\mathbb{Z}/12\mathbb{Z}$ (the same function drawn by &lt;strong&gt;mweiss&lt;/strong&gt; in his answer):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Nm9uo.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for whether this graphical representation is &lt;strong&gt;effective&lt;/strong&gt;: That will depend on how you use it and what you wish to achieve. I do think that, in the above case, some natural questions arise around why certain structures appear more than once. Can students tease out these questions? (What sort of questions do they ask?) Once curiosity has been piqued, can it be resolved by appealing to one's algebraic or graphical intuition?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;A brief excerpt related to the notable comment of &lt;strong&gt;mweiss&lt;/strong&gt; below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/BJaAP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The interested reader may wish to check the paper of Kempner, as well as a nice generalization in:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Chen, Z. (1995). On polynomial functions from Zn to Zm. &lt;em&gt;Discrete Mathematics, 137&lt;/em&gt;(1), 137-145. &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/0012365X93E0162W&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;ScienceDirect Link&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2015-05-12T14:15:47.100" LastActivityDate="2015-05-12T14:15:47.100" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="7874" PostTypeId="2" ParentId="7865" CreationDate="2015-04-14T00:18:51.997" Score="15" Body="&lt;p&gt;Just supplementing Benjamin Dickman's&#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/a/7873/511&quot;&gt;nice answer&lt;/a&gt;,&#xA;here is&#xA;$x \mapsto x^2 - x$ in $\mathbb{Z}_{18}$ in the same style:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/KNd8A.jpg&quot; alt=&quot;FieldGraphZ18&quot;&gt;&#xA;&lt;hr /&gt;&#xA;For example, the pentagon wheel reflects the fact that &#xA;$$(5+3k)^2-(5+3k) = 9k^2 + 27k + 20 = 9k(k+3) + 20 = 2\bmod 18 \;.$$&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-04-14T00:18:51.997" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7875" PostTypeId="1" CreationDate="2015-04-14T01:28:59.250" Score="6" ViewCount="63" Body="&lt;p&gt;I am taking an Economics class. We are required to plot and interpret 2D graphs. But often we have to use oddly shaped functions or have additional constraints we have to impose that make interpreting the graph tricky.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am used to working with words and theorems to prove things.  But my professor wants me to be able to articulate economic insights largely using visualization and let the graphs speak for themselves to some extent. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any resources that can explain what kinds of information usually expressed using words can be communicated visually using graphs? &lt;/p&gt;&#xA;" OwnerUserId="3471" LastEditorUserId="3471" LastEditDate="2015-05-17T04:21:32.840" LastActivityDate="2015-05-17T14:23:56.993" Title="Good learning resources on what kinds of propositional information one can communicate through graphs?" Tags="|graphing|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7876" PostTypeId="1" CreationDate="2015-04-14T06:36:25.507" Score="5" ViewCount="204" Body="&lt;p&gt;The question in the &lt;a href=&quot;https://i.stack.imgur.com/KMwos.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;picture&lt;/a&gt; is a relatively easy one to answer. But while learning Trigonometry, I had to answer countless questions asking to prove certain relations between obscure and complicated expressions. To give you another example: &#xA;If sin(y+z-x), sin(z+x-y), sin(x+y-z) are in arithmetic progression, prove that tan x, tan y and tan z are also in arithmetic progression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone explain to me what use they serve practically?&lt;/p&gt;&#xA;" OwnerUserId="5050" LastEditorUserId="521" LastEditDate="2015-04-14T10:56:08.777" LastActivityDate="2015-04-14T18:07:29.933" Title="What purpose do these kinds of question serve in mathematical training?" Tags="|trigonometry|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7877" PostTypeId="1" CreationDate="2015-04-14T08:24:57.737" Score="9" ViewCount="393" Body="&lt;p&gt;In our university there is a huge gap between two group of students. a group of them came from Math Olympiad competitions and have a very strong background from high school but others, they have just ordinary background from high school and there is a big difference. actually this gap between them causes some problems in classes. So we thought about this idea of having a class for 'Not Math Medalist students' to teach them how to think mathematically and problem solving strategies and somehow try to fill the gap (for first year students). I was searching for problems that can make them think, not in a special subject, not very hard not very easy. solvable with undergraduate information, and I guess important thing is these problems should be attractive for them to make them think. maybe mathematical modeling or problems without certain answer are good . Do you have any particular suggestion on what problems (or books) can be helpful ?&lt;/p&gt;&#xA;" OwnerUserId="5052" LastActivityDate="2015-12-27T22:18:08.500" Title="Ideas for math problem solving class for undergraduate students in university" Tags="|student-motivation|reference-books|problem-solving|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7878" PostTypeId="2" ParentId="7876" CreationDate="2015-04-14T15:55:33.770" Score="3" Body="&lt;p&gt;What do you mean by &quot;mathematical training&quot;?  Or by &quot;practically&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics is used in different ways by different people; the form of training each should receive will vary and hopefully correspond to how they intend to use the results.  However, creating patterns, recognizing patterns, and applying patterns are common themes across many of the forms that are practiced.  If you want a general answer that may make sense, that answer would be that such problems should give practice in the creation, recognition, and application of patterns.  If you want a general answer that is correct, that answer is that it depends on whether one is going to encounter items in arithmetic progression, or triples of tangents, or expressions in terms of signed sums of x,y, and z.  Mathematicians (and other practitioners of mathematics) are making connections between the past and the present all the time, and it is often unclear how today's patterns will be used tomorrow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Knowing The Future Is Boring&quot; Paseman, 2015.04.14&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-14T15:55:33.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7879" PostTypeId="2" ParentId="7876" CreationDate="2015-04-14T18:07:29.933" Score="10" Body="&lt;p&gt;I suspect the intended use of these questions was to provide practise using trigonometric identities and also the idea of arithmetic progression (which I think would both appear in the UK A-level syllabus, or maybe GCSE, I can't remember). Memorising formulae is not much fun, and having done so is not much use if you can't actually make use of them. Finding sufficiently many different questions isn't all that easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From that point of view, I'd say the question you describe is also an attempt to introduce some of the &lt;em&gt;thinking&lt;/em&gt; that is important in maths but often lacking in school maths lessons/exams. Rather than simply plugging numbers into a memorised algorithm, you need to actually conceptualise the question, to some degree at least, and apply the concepts in a different configuration. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally I'd happily take obscure over mind-numbingly-repetitive, but then I am a pure mathematician.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-04-14T18:07:29.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7880" PostTypeId="1" CreationDate="2015-04-14T18:32:15.150" Score="27" ViewCount="2717" Body="&lt;p&gt;I'm sorry if this is the wrong place to ask this but I don't know any other place to ask.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a research professor, but I enjoy teaching and put a lot of time into my classes.  I mainly teach introductory calculus and physics to mainly engineering students.  One of the things I've noticed while talking to students is their hate and animosity towards the liberal arts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked for some of my students reasoning and a lot of it was that History, English, Classics, etc. did not follow the &quot;laws of logic and reasoning.&quot;  There are other examples, but I those are the ones the most common, along with &quot;it's not applicable to life.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just want my students to learn that living in a bubble academically is not good for your knowledge, and the Colleges and Universities are there to help create educated individuals. &lt;/p&gt;&#xA;" OwnerUserId="5055" LastEditorUserId="5055" LastEditDate="2015-04-14T18:56:27.583" LastActivityDate="2016-03-02T21:05:29.223" Title="How can I teach my students that other disciplines are important too?" Tags="|undergraduate-education|" AnswerCount="9" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7881" PostTypeId="2" ParentId="7880" CreationDate="2015-04-14T20:00:50.243" Score="14" Body="&lt;p&gt;Perhaps your engineering students would listen to Steve Jobs, from a &lt;a href=&quot;http://www.nytimes.com/roomfordebate/2011/03/20/career-counselor-bill-gates-or-steve-jobs&quot;&gt;NYTimes article&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At an event unveiling new Apple products, Mr. Jobs said: &quot;It's in Apple's DNA that technology alone is not enough&amp;mdash;it's technology married with liberal arts, married with the humanities, that yields us the result that makes our heart sing and nowhere is that more true than in these post-PC devices.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;strong&gt;Added&lt;/strong&gt;. Here is a thoughtful blog post by &lt;a href=&quot;http://faculty.gvsu.edu/talbertr/Robert_Talbert,_PhD/Welcome.html&quot;&gt;Robert Talbert&lt;/a&gt;&#xA;about engineering and liberal&#xA;education, &quot;&lt;a href=&quot;https://castingoutnines.wordpress.com/2010/06/21/what-some-engineers-think-about-liberal-education/&quot;&gt;What (some) engineers think about liberal education&lt;/a&gt;&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is an entire division ... within the ASEE [American Society for Engineering Education] for Liberal Education. ... One might even say that those in charge of accrediting engineering programs want engineers to be liberally educated. ...&#xA;  What nobody seems able to explain just yet is the implicit and sometimes explicit resistance to liberal education among many engineers and engineering programs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Sounds like the OP is experiencing that &quot;resistance.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-04-16T00:55:23.483" LastActivityDate="2015-04-16T00:55:23.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7882" PostTypeId="2" ParentId="7880" CreationDate="2015-04-14T21:34:52.100" Score="7" Body="&lt;p&gt;The concept of the well rounded education is important. With a hat tip to Joseph's reference to Steve Job's remarks, there's nothing that's purely one discipline. Nothing. You are an engineer. 100% engineer. Still, you need to communicate with the outside world. You need to understand how to write in your native language to pass information along. Even in the &quot;olden days,&quot; whenever that was, there were the basic subjects that one needed to master before focusing on their chosen path. &quot; Readin' writin' arithmetic&quot; as well as history have been with us for hundreds (more, really) of years. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More than the pedagogical aspect, this is important for people to not be social outcasts. In engineering school, we've all run into the guy who was brilliant in a very narrow sense, but was unable to carry on a conversation beyond his/her narrow interest.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-04-15T16:59:03.803" LastActivityDate="2015-04-15T16:59:03.803" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7883" PostTypeId="2" ParentId="7880" CreationDate="2015-04-14T22:07:57.393" Score="11" Body="&lt;p&gt;I actually come from the almost opposite situation as you, liberal arts students who have little to no appreciation for mathematics.  Through my frustrations I have learned that mathematics has actually managed to find its way into nearly every field of study including art, history, science, philosophy and even literature.  Below are some examples of topics that I have shown my students that really got them thinking about math in the world around them&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Platonian Philosophy and the Platonic Solids&lt;/li&gt;&#xA;&lt;li&gt;Fibonacci numbers and their appearance in nature and art&lt;/li&gt;&#xA;&lt;li&gt;The golden mean, its relevance to art and the Fibonacci numbers&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Twelve-Mile_Circle&quot;&gt;The Twelve-Mile Circle&lt;/a&gt;, its history, its geography and determining its geometric center&lt;/li&gt;&#xA;&lt;li&gt;History of ancient mathematics (Egyptians, Indian, Mayan, Babylonian, Chinese, Arabic) and ancient perspectives on numbers&lt;/li&gt;&#xA;&lt;li&gt;&quot;&lt;a href=&quot;http://poetry.about.com/od/poems/l/blmillayeuclid.htm&quot;&gt;Euclid alone has looked on beauty bare&lt;/a&gt;&quot;, a poem by Edna St. Vincent Mallay&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.geom.uiuc.edu/~banchoff/Flatland/&quot;&gt;Flatland&lt;/a&gt; by Edwin Abbott Abbott and &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0805062998&quot;&gt;The Number Devil&lt;/a&gt; by Hans Magnus Enzensberger, two amazing books inspired by mathematics&lt;/li&gt;&#xA;&lt;li&gt;Origami, papercraft and the math that goes into the art&lt;/li&gt;&#xA;&lt;li&gt;The similarity between the creative process and the proof process (conjecture, experiment, assess, rinse, wash, repeat)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Math can literally be linked to just about any discipline or topic, you just have to spend the time looking and making the connections :)&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-04-14T22:07:57.393" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7884" PostTypeId="2" ParentId="7880" CreationDate="2015-04-15T01:30:29.603" Score="9" Body="&lt;p&gt;If you believe it, teach by example.  Include literate references along with your lectures and lessons.  Point to authors like Knuth who include quotations along with their technical material, as well as analyze.   Note that successful mathematicians are not only mathematicians, but writers, performers, students and enthusiasts in other subjects.  In my teaching section, I would write a quotation on the board for my students to look at during section, which would have some connection to the material being taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For those that desire application of liberal arts, note that communication, organization of presentation, efficacy of examples, and other points can rely heavily on knowledge of culture, history, and society.  Those that desire relevance of mathematics in liberal arts, note that pattern, aesthetic quality, self-referential structure, and various other formal and logical aspects are tools to help create and transmit ideas and feelings, or even evoke feelings in the recipient.  A goal of liberal arts is to interact with other minds; those minds are more capable of such interaction with a full education than with a partial education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Going To Fill My Brain&quot; Paseman, 2015.04.14&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-15T01:30:29.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7885" PostTypeId="2" ParentId="7880" CreationDate="2015-04-15T03:15:04.510" Score="16" Body="&lt;p&gt;[This is a particular problem in the UK, where specialization into STEM occurs during high school.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I fully agree that a broad education is important in and of itself. What is also true is that it is &lt;em&gt;important to an engineering career&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once gave a talk to a high school in which I talked about what I currently do in an engineering field and had a slide in which I showed various subjects I draw upon myself and look for in recruits. Turns out that every high school subject had some relevance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key is that engineering's ultimate job is to provide solutions to people. Providing those solutions requires working with people, understanding their needs and documenting in a manner both parties understand. This places a high demand on both written and spoken language skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What does that have to do with history, for example? Chances are that many of your students simply have little understanding of what historians actually do. A huge element of history today is analysis of texts. The ability to take a text and understand that various conclusions can be drawn is not the same as saying &quot;logic and reasoning do not apply&quot;, in fact the contrary. These skills apply directly to engineering specifications, standards and associated legal contracts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, the liberal arts are about the study and language and people. Engineering in the real world (as opposed due during study) requires those skills. My experience is that engineers who do not have such skills are limited in their career, particularly in regards to leadership opportunities. &lt;/p&gt;&#xA;" OwnerUserId="3587" LastActivityDate="2015-04-15T03:15:04.510" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7887" PostTypeId="1" AcceptedAnswerId="7895" CreationDate="2015-04-15T18:07:27.903" Score="15" ViewCount="255" Body="&lt;p&gt;The question in the title is immensely subjective and broad, and so I would like to narrow it to an answerable question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What measures are in common use by administrators and researchers in determining the quality of an instructor's teaching? What, if any, is the correlation between these measures?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I ask because I've had difficulty answering this question for myself. I originally thought that test scores would be the best answer, but I have seen the same teacher have the calculus section with lowest scores in a department and the calculus section with the highest scores in the department in the same semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have also seen teachers that I would identify as good teachers have bad test scores, and vice-versa.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've had theories as to why test scores may not be correlated to good teaching; for instance, poor students may prefer a good professor, and may be interested in joining such a professors class (this effect could be determined by comparing pretests and posttests, but I have not carried out this analysis).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has led me to wonder if &quot;good teachers&quot; have any positive effect on their students at all, compared to &quot;average teachers&quot;. That's why I have asked the highlighted question above; I would like to find out what others have looked at when evaluating teaching, to help me understand what a &quot;good teacher&quot; is.&lt;/p&gt;&#xA;" OwnerUserId="78" LastActivityDate="2015-04-21T04:47:36.610" Title="How can you determine the quality of your teaching, or someone else's?" Tags="|general-pedagogy|assessment|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7889" PostTypeId="2" ParentId="1" CreationDate="2015-04-15T20:44:15.630" Score="5" Body="&lt;p&gt;Here  4 of 10 suggestions by Steven G. Krantz in an appendix entitled &amp;quot;Some suggestions for encouraging class participation&amp;quot; of his book &lt;a href=&quot;http://math.bu.edu/people/changer/teasem/EncouragingParticipation.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to teach Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Get students to go to the blackboard.&lt;/li&gt;&#xA;&lt;li&gt;Have students prepare oral reports or mini-lectures.&lt;/li&gt;&#xA;&lt;li&gt;Have guest instructors.&lt;/li&gt;&#xA;&lt;li&gt;Intentionally make mistakes in order that students can fix it.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3353" LastEditorDisplayName="user7990" LastEditDate="2020-08-28T20:33:55.720" LastActivityDate="2020-08-28T20:33:55.720" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7890" PostTypeId="2" ParentId="7877" CreationDate="2015-04-15T20:45:15.147" Score="1" Body="&lt;p&gt;I feel that undergraduate mathematics is almost a calculus grind from start to finish for nearly every group of students (that aren't explicitly mathematics majors).  Introductory discrete math (sets, maps, etc.) offers so much more for students to grasp than continuous calculus from start to finish (as is the norm).&lt;/p&gt;&#xA;" OwnerUserId="5066" LastActivityDate="2015-04-15T20:45:15.147" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7891" PostTypeId="2" ParentId="7778" CreationDate="2015-04-15T21:10:06.027" Score="4" Body="&lt;p&gt;One answer is to regard this question as an instance of a more general question about whether to build embedding theorems into the foundations of a subject.  Should we define abstract manifolds, or only submanifolds of Euclidean space?  Should we define abstract groups, or only subgroups of permutation groups?  I think in all cases there is something to be gained by making the abstract definition and then &lt;em&gt;proving&lt;/em&gt; the embedding theorem, because it makes clear what aspects of a notion are &quot;intrinsic&quot; and independent of a chosen embedding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As mentioned in other answers, time is an excellent 1D example, and of course space is a higher-dimensional one.  In your comments on these answers you say that since we always measure these quantities with numbers, it makes your point about using affine subspaces of vector spaces.  But I would argue that these examples nevertheless do answer your second question, because the choice of numbers with which to measure them is not &lt;em&gt;natural&lt;/em&gt; but rather arbitrary.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2015-04-15T21:10:06.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7892" PostTypeId="2" ParentId="7880" CreationDate="2015-04-15T22:48:30.790" Score="8" Body="&lt;p&gt;You want to convince people of the importance of (a subset of) the humanities and social sciences? Bring proof. Bring scientific studies that point to the efficacy of learning these things and the results they will bring. You may be showing a failling that some of social science seems to have: trying to convice people before you know what is true.&lt;/p&gt;&#xA;&lt;p&gt;I'll give you a counter-example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.theatlantic.com/health/archive/2014/01/the-dark-side-of-emotional-intelligence/282720/&quot; rel=&quot;noreferrer&quot;&gt;http://www.theatlantic.com/health/archive/2014/01/the-dark-side-of-emotional-intelligence/282720/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;From the article&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In settings where emotions aren’t running high, emotional intelligence may have hidden costs. Recently, psychologists Dana Joseph of the University of Central Florida and Daniel Newman of the University of Illinois comprehensively analyzed every study that has ever examined the link between emotional intelligence and job performance. Across hundreds of studies of thousands of employees in 191 different jobs, emotional intelligence wasn’t consistently linked with better performance. In jobs that required extensive attention to emotions, higher emotional intelligence translated into better performance. Salespeople, real-estate agents, call-center representatives, and counselors all excelled at their jobs when they knew how to read and regulate emotions—they were able to deal more effectively with stressful situations and provide service with a smile.&lt;/p&gt;&#xA;&lt;p&gt;However, in jobs that involved fewer emotional demands, the results reversed. The more emotionally intelligent employees were, the lower their job performance. For mechanics, scientists, and accountants, emotional intelligence was a liability rather than an asset. Although more research is needed to unpack these results, one promising explanation is that these employees were paying attention to emotions when they should have been focusing on their tasks. If your job is to analyze data or repair cars, it can be quite distracting to read the facial expressions, vocal tones, and body languages of the people around you. In suggesting that emotional intelligence is critical in the workplace, perhaps we’ve put the cart before the horse.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This is narrow, sure. I am curious to see if studying history does increase ones output as an engineer. Also, if it correlates positively to some other interesting life outcomes (happiness, life expectancy ...)&lt;/p&gt;&#xA;&lt;p&gt;But I don't know it does. Do you ?&lt;/p&gt;&#xA;" OwnerUserId="5068" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-03-01T16:18:26.987" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7893" PostTypeId="2" ParentId="7880" CreationDate="2015-04-16T00:40:26.540" Score="2" Body="&lt;p&gt;As a computer science student who writes a lot, wants to publish his own fiction book soon, and who has always done well in his English/humanities classes, I completely understand the animosity that the other engineering students feel for the other side of academia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my perspective, the humanities seem useless. Who cares about old dead white men. I program things. I make things. I have a $80k/yr job lined up for me that does not strenuously test my language skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, the classes are easy. The only real English class that I have ever taken involved a 5 page paper every week and it was scaled to a B-. That is a real English class. The five other English classes I took involved the females and some males in the class talking about their feelings. There were maybe two papers and they were maybe a few pages long. I showed up for class maybe twice a semester. Didn't even read the books. Just wrote about feminism and I got my A.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Those five English classes were a waste of my time-- I did not even learn how to write from them, I just learned that I disliked liberals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is my opinion that the vast majority of English classes are a waste of time. The humanities department sets out to educate kids and teach them to read and write and think critically. They do not. The majority of humanities classes, not just English, devolve into talking about how white heterosexual males have wronged every imaginable demographic. Many engineers are are also such white heterosexual males. They would probably rather not learn about why their fathers wronged others and how they are currently wronging others through invisible slights.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the humanities classes were more rigorous and more focused on academic issues and demonstrably challenging, no engineer would ever scoff at the humanities department. They would fear it, like they fear proof-based math.&lt;/p&gt;&#xA;" OwnerUserId="5069" LastEditorUserId="5069" LastEditDate="2015-04-16T00:46:30.257" LastActivityDate="2015-04-16T00:46:30.257" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7894" PostTypeId="2" ParentId="7778" CreationDate="2015-04-16T07:52:06.623" Score="1" Body="&lt;p&gt;I would like to give an answer, which is not completely my own but that stems from other answers given here. It still seems work writing because I don't think it appears explicitly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One compelling reason to teach  in one form or another abstract affine spaces, or at least planes, is to be able to do &lt;strong&gt;basic non-analytic geometry&lt;/strong&gt; (e.g. study triangles and quadrangles in a light formalism). More precisely, in undergrad education we have to teach geometry to future high-school teachers, and we have to do so in a way that relates to what they will have to teach; and they won't be able to say to the pupils &quot;just consider $\mathbb{R}^n$&quot;! I don't think we can do that by avoiding completely abstract affine spaces. Now, there is a subsequent question that I will ask separately: how do we teach abstract affine spaces in a simple way that clearly relates to high-school geometry?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally I would like to thank all users who answered or commented. Even if I have often commented back not so positively, I was partly playing the devil's advocate, and all other answers have made me think about this issue.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-04-16T07:52:06.623" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7895" PostTypeId="2" ParentId="7887" CreationDate="2015-04-16T12:55:40.333" Score="5" Body="&lt;p&gt;I think it's important to distinguish between performance evaluation for the purpose of professional development, and for employment related decisions (hiring, firing, and wages).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rigor needed for professional development is much lower. A teacher (and their supervisor if needed) can develop goals using a variety of metrics, but the most useful tool I have experience with has been peer observations. Having another teacher from my department sit in on my classes to help me address an issue has been very valuable. Sometimes it has been something specific, so I asked a teacher who I thought had a good style with that issue, and sometimes its been more general. I've also gone to observe other teachers, if after a discussion I thought they had something I really wanted to learn from them. This requires the support of the department and administration, but is very effective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to employment decisions the feedback needs to come from a supervisor, it's not fair to expect employees to create critical evaluations of their peers. My only experience with this has been in class observations, which were not particularly effective for this purpose. However, we are changing to incorporate data from the Smarter Balance test, and a regional test that we use. What we're currently negotiating for is to use historical data from the same student, instead of the same class, for comparison. So the same student, but a new test. The idea is that this should show student growth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Meaning: Andy scored in the 75% in 8th grade and then the 75% in 11th grade, that's great Andy has learned all of the new material between 8th and 11th grade. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Barbara scored in the 75% in 8th grade and then 80% in 11th, not only did Barbara learn all of the new material, she has actually developed a greater understanding of math as the material became more complicated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chris scored in the 75% in 8th grade and then 70% in the 11th. So he has lost ground and an intervention is required to teach the newer material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally feel better about this as a method for incorporating student data into teacher evaluations. I feel comfortable teaching students with diverse learning needs, given that what I'm going to be assessed with is that those students learned something in my class. I am not comfortable being assessed by comparing this year's class with last year's class, and I think the class-to-class method disincentives inclusion of special needs students. Nothing is going to be perfect, but I do understand the need to incorporate some quantitative data.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2015-04-16T12:55:40.333" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7896" PostTypeId="1" CreationDate="2015-04-16T15:08:55.670" Score="3" ViewCount="155" Body="&lt;p&gt;There is a set of lecture notes I would like to work through the exercises very carefully. Instead of having thousands of scrap pages lying around my desk - I considered typing them up on and posting them on GitHub.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Will this make the textbook unususable for future courses based on that text?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I found this related question: &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/86/is-it-good-to-have-solutions-of-homework-published&quot;&gt;Is it good to have solutions of homework published?&lt;/a&gt;&lt;/strong&gt; which argues the solutions are beneficial check on the validity of the text.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I am not the teacher, I am the student.  Having them online is for my own pedagogical benefit and to share with specific other people.&lt;/p&gt;&#xA;" OwnerUserId="1654" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-04-16T18:21:14.930" Title="Is it okay to post my own problem set solutions?" Tags="|undergraduate-education|homework|graduate-education|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7898" PostTypeId="1" AcceptedAnswerId="7903" CreationDate="2015-04-16T15:40:11.380" Score="17" ViewCount="773" Body="&lt;p&gt;By now I'm sure everyone has run into the math puzzle where Albert and Bernard try to deduce Cheryl's birthday, which is all over social media, and even traditional media! If you don't know what I'm talking about, see e.g. &lt;a href=&quot;http://www.washingtonpost.com/news/morning-mix/wp/2015/04/14/when-is-cheryls-birthday-the-math-problem-that-stumped-the-internet/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Washington Post article&lt;/a&gt;, or the transcript below: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a nice logic puzzle, and I'm delighted to see any math fall into general public consumption, but to me it feels like hundreds of other logic puzzles. As someone who has spent lots of time unsuccessfully trying to get people into math problems, I wonder what the trick is with this one. Whatever it is, can we use it to design problems that students will be naturally puzzled by?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Here's a transcript of the puzzle: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Albert and Bernard just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates. &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;May 15, May 16, May 19 &#xA;    &lt;br&gt;&#xA;    June 17, June 18&#xA;    &lt;br&gt;&#xA;    July 14, July 16&#xA;    &lt;br&gt;&#xA;    August 14, August 15, August 17&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;Cheryl then tells Albert and Bernard separately the month and the  day of her birthday respectively. &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&lt;strong&gt;Albert&lt;/strong&gt;: I don't know when Cheryl's birthday is, but I know that Bernard does not know too.&lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;&lt;strong&gt;Bernard&lt;/strong&gt;: At first I don't know when Cheryl's birthday is, but I know now.  &lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;&lt;strong&gt;Albert&lt;/strong&gt;: Then I also know when Cheryl's birthday is. &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;So when is Cheryl's birthday?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1784" LastEditorUserId="5220" LastEditDate="2019-12-12T04:30:03.907" LastActivityDate="2019-12-12T04:30:03.907" Title="What teaching strategies can we learn from this logic puzzle going viral?" Tags="|mathematical-pedagogy|interactive-teaching|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7899" PostTypeId="2" ParentId="7898" CreationDate="2015-04-16T16:39:47.757" Score="6" Body="&lt;p&gt;One possible reason is that it is couched in &quot;everyday terms&quot;.  I am more familiar with a version that has products and sums, and it seems products and sums are less known than months, days, and birthdays.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another is that it involves a social event.  Disregarding gender preference or other possible norms, many are interested in birthdays and ages as a mode of comparison, and the choice of two (male-sounding) names considering something presented by a (female-sounding) name is a choice taken by many samples of popular and classic literature.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, it is &quot;simple&quot; enough and &quot;characteristic&quot; enough that one can explain the unique solution in a way that the audience can understand and (here comes the viral part) explain the solution to someone else.  Sudoku has a similar kind of simplicity and uniqueness about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be fruitful to discuss with people like Will Shortz what makes a good puzzle, and with people like Adrian Paenza (Leelavati prize winner) what would make it appropriately educational.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Two Cents Out Of Millions&quot; Paseman, 2015.04.16&lt;/p&gt;&#xA;" OwnerUserId="3468" LastEditorUserId="3468" LastEditDate="2015-04-16T18:29:36.213" LastActivityDate="2015-04-16T18:29:36.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7900" PostTypeId="1" AcceptedAnswerId="7904" CreationDate="2015-04-16T16:42:39.793" Score="6" ViewCount="292" Body="&lt;p&gt;This semesters I am teaching a course to 12-13 year old kids in which they are supposed to learn the basics of spreadsheet usage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am having difficult in coming up with fun / interesting exercises to teach them the basic functions, such as average, max, minimum and the like. One of the biggest problems is having only a 50 minutes long lesson every two weeks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've tried so far:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Teaching them to plot relations using a dispersion graphic: they would produce the axes and plot relations such as linear and quadratic formulas, exponential, etc.&lt;/li&gt;&#xA;&lt;li&gt;Introduced them to simple recurrences such as the Fibonacci sequence and the discrete logistic map.&lt;/li&gt;&#xA;&lt;li&gt;Using stock data to teach them some descriptive statistics.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What else would be proper, and preferably fun, to teach them? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2015-05-04T17:13:31.823" Title="What subjects can be taught using spreadsheets to 13 year old kids?" Tags="|interactive-teaching|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7901" PostTypeId="2" ParentId="7896" CreationDate="2015-04-16T18:21:14.930" Score="1" Body="&lt;p&gt;Good for whom?  A casual consideration shows that both good and harm can be done.  If the people who run the course encourage or approve your notes, then I am unlikely to cry &quot;Foul!&quot;.  Also, if you have your own notes that you wish to present, you can keep a digital copy in HTML or PDF or text format and put it on a USB flash drive and make it available to any small select group of individuals of your choosing (but be careful to avoid conflict with institutional policies on cheating).  Putting it on your personal website is another matter entirely, and at the absolute minimum should not be done without knowledge and agreement of the professor or administration offering the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A mindset to adopt is that you may wish to offer the notes as an educational product to be sold.  Then you can decide if what you have to offer is a supplement, a course critique, a wikipedia-type article, or just your own personal blog for entertainment.  Get a few others to look at it and agree with you before you decide what it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;This Is A No-Brainer, Right?&quot; Paseman, 2015.04.16&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-16T18:21:14.930" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7902" PostTypeId="2" ParentId="7898" CreationDate="2015-04-16T18:31:06.117" Score="5" Body="&lt;p&gt;In my mind there are two main reasons this became popular: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It is counter-intuitive.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is slightly ambiguous.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The same could be said about other popular math-y puzzles and paradoxes, e.g., Monty Hall problem or &quot;Tossing two coins one of them is heads what are the odds the other is heads too?&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second point is quite important for the popularity, I think. This ambiguity, however, is somewhat at odds with class-room use. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A way to put this problem to good use in a class-room setting could be to discuss different types of &quot;knowledge&quot; such as &lt;a href=&quot;http://en.wikipedia.org/wiki/Common_knowledge_%28logic%29&quot; rel=&quot;nofollow&quot;&gt;&quot;common knowledge&quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;To  elaborate on the point on ambiguity: &lt;a href=&quot;http://www.nytimes.com/2015/04/15/science/a-math-problem-from-singapore-goes-viral-when-is-cheryls-birthday.html&quot; rel=&quot;nofollow&quot;&gt;The New York Times gave the version below&lt;/a&gt;  to remove such ambiguity (I put emphasis on the main change). In my opinion, it is a lot clearer this way but I am not sure it would have had that success in this form.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Albert and Bernard just met Cheryl. “When’s your birthday?” Albert asked Cheryl.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cheryl thought a second and said, “I’m not going to tell you, but I’ll give you some clues.” She wrote down a list of 10 dates:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;May 15, May 16, May 19&#xA;&#xA;June 17, June 18&#xA;&#xA;July 14, July 16&#xA;&#xA;August 14, August 15, August 17&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;“My birthday is one of these,” she said.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Then Cheryl whispered in Albert’s ear the month — and only the month — of her birthday. To Bernard, she whispered the day, and only the day.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“Can you figure it out now?” she asked Albert.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bernard: I &lt;strong&gt;didn’t&lt;/strong&gt; know originally, but now I do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Albert: Well, now I know, too!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When is Cheryl’s birthday?&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="29" LastEditDate="2015-04-16T20:57:12.940" LastActivityDate="2015-04-16T20:57:12.940" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7903" PostTypeId="2" ParentId="7898" CreationDate="2015-04-16T21:09:15.970" Score="11" Body="&lt;p&gt;I think I would personally dispute the claim that this &quot;went viral&quot;.  I didn't see any people in my social networks talking about this problem until well &lt;em&gt;after&lt;/em&gt; the media started reporting that &quot;Everybody is talking about this problem!&quot;  Even now I don't see anybody talking about the problem; they are only talking about the coverage of the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said that, I do think there are two interesting features to this problem that contribute to the interest in it: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;There is something odd about the way in which the information is deployed &lt;em&gt;over time&lt;/em&gt;.  The fact that in Step 2 Bernard knows Cheryl's birthday tells Albert something entirely different from what he (Albert) could have inferred if Bernard had known Cheryl's birthday from the outset.  One might say that the knowledge claims interact in a noncommutative matter; each piece of information is meaningful in a precise way because of the information that came &lt;em&gt;before&lt;/em&gt; it.&lt;/li&gt;&#xA;&lt;li&gt;The second thing that I think is interesting here is the way in which meta-knowledge (knowledge about the state of others' knowledge) can be used to make inferences about the actual state of events.  In this case, the fact that Albert knows that Bernard does &lt;em&gt;not&lt;/em&gt; know is knowledge about somebody else's knowledge; it is rather surprising that Bernard can leverage that.  And once Albert knows that Bernard &lt;em&gt;does&lt;/em&gt; know, that is again a piece of meta-knowledge, and it is again surprising that Albert can leverage it.  There is a strange &quot;level-crossing&quot; phenomenon here, in which knowledge of Cheryl's birthday gets entangled with knowledge of others' knowledge of Cheryl's birthday. That kind of level-crossing makes me think of the &quot;strange loopiness&quot; in Douglas Hofstadter's book &lt;em&gt;Gödel, Escher, Bach&lt;/em&gt;, which is all about level-crossing phenomena and the ways in which they capture the imagination.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-04-16T21:09:15.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7904" PostTypeId="2" ParentId="7900" CreationDate="2015-04-17T01:47:19.140" Score="5" Body="&lt;p&gt;Based on your unfortunate lack of time my answer is going to be a list of suggestions that I believe could be completed and meaningful given the amount of time you have with your students.  Not all of them pertain directly with math (calculations, computations, formulas, etc), but I believe that 1) the real power of spreadsheets is their ability to organize, structure, and access data and 2) that the thinking behind these concepts is very mathematical.  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;You mentioned that they have already used the technology to explore fitting various functions to graphs.  I think this can open up into a great lesson on growth of functions, having them write basic formulas to model linear, log, quadratic, cubic, exponential, factorial, etc. and then graph them and see how the growth differs between the types of functions&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You mentioned some recursive sequences/series that you looked at, how about Newton's method, calculating square roots or logarithms &quot;by hand&quot;, or approximating pi by one of the many ways that are out there.  Very cool things to be able to do by hand because it takes a lot of the &quot;magic&quot; out of pressing a button on a calculator and lets the students know that there is a reason why these things work and are the way they are&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have them do more with descriptive data but let them find their own datasets for data that might be meaningful to them (possibly video game sales, sports records, music sales, etc).  It will reinforce their prior learnings and give them a sense of agency and ownership.  I would have them research during their time out of the classroom and come with a dataset ready to go.  Alternatively, you could spend one class finding datasets, discussing different biases, sample sizes, correlation, causation, etc. and then the next class analyze it&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A large part of being able to effectively use a spreadsheet is organization and performing tasks like sort and merge.  If you have not already talked about this I would suggest doing so and possibly using a classroom dataset of say, everyones height, eye color, length of name in letters, etc and either start with a poorly organized structure of the data and discuss why it is poor and how to fix it or start from scratch and discuss along the way meaningful ways to structure data&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use whichever spreadsheet software (i'm assuming google sheets or excel) as a means to talk about software, coding, and computer languages, possibly having them write a very simple macro or two after a lesson or two.  I think that the real power in these software is their extensibility, if you know how to work it.  I know you said that you are tasked with the basics, but it sounds like they are catching on so there is no reason not to push them to explore the tool&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this is helpful and I wish you the best of luck :)&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-04-17T01:47:19.140" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7905" PostTypeId="2" ParentId="7880" CreationDate="2015-04-17T09:00:45.660" Score="0" Body="&lt;p&gt;When i was doing my school/college i loved mathematics, and still love it. In my Profession as software developer i visualize things very analytically.&#xA;So, when teaching any logical subject its important to link real time examples with what is being taught. Giving exercises that can be fun to do, but to complete the exercise one has to understand and use mathematical concepts to achieve it. This way both fun at work and skills development can be achieved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you listen to the point of view of those students who are not interested in the subject, they would say what is the use of it, why should i pay attention to ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cheers,&#xA;Shambhu&lt;/p&gt;&#xA;" OwnerUserId="5075" LastActivityDate="2015-04-17T09:00:45.660" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7906" PostTypeId="1" CreationDate="2015-04-17T10:18:19.877" Score="4" ViewCount="120" Body="&lt;p&gt;&lt;strong&gt;Context:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of organizational mess, there is a high chance that I'll be preparing a ~3h class for gifted students during a spring-camp (a week-long gathering full of all kinds of nice things to do) which will be split into two 90-minutes chunks on two different days (hopefully consecutive).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have very little experience concerning such setting (previously it was either one-shot or at least semester-long courses) and I'm afraid that with all the activities available, they won't remember anything for the next day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I would like to achieve:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Although they need to declare which of the classes they want to attend (that is, either all the 3h or nothing), I &lt;strong&gt;don't&lt;/strong&gt; want to tie them up with some sort of assignments, homeworks, problems to think about, whatever. &lt;/li&gt;&#xA;&lt;li&gt;This cannot be two different one-shot classes. The topic is set and based on previous experience, for this audience I need at least 2.5h to deliver it in an understandable manner.&lt;/li&gt;&#xA;&lt;li&gt;I would like avoid repeating myself, as this audience gets bored &lt;em&gt;fast&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The quesion:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to plan the class so that it will be relatively easy to bring students up to speed again when starting the second chunk?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My baseline plan was:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;prepare a few memorable (if I can manage) problems for the three most important concepts of the first day,&lt;/li&gt;&#xA;&lt;li&gt;give some handouts that the could refer to the next day,&lt;/li&gt;&#xA;&lt;li&gt;at the end of the first chunk go over the handout and summarize the most important concepts,&lt;/li&gt;&#xA;&lt;li&gt;at the beginning of the second chunk go over the notions in the handout and give a small task for each important concept,&lt;/li&gt;&#xA;&lt;li&gt;solve each together, in the process referring to the memorable problems from the first bullet.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The problem with this plan is that it feels clumsy and will probably take more than 0.5h I can spare. Perhaps I could try interweaving the summary and the material of the second day, but that is hard and I'm afraid they might get lost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would you suggest?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Topic (edit):&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This will be a part of lectures on computer science (think algorithmic contests), and the topic is associative operators. It is about why and how to represent partial problem/solution information in a way that can be combined by associative operators. A simple example is exponentiation by squaring. That can be combined with matrices to calculate Fibonacci numbers with logarithmic number of operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact this topic is not &lt;em&gt;preset&lt;/em&gt;, but the intersection of what fits and what I can prepare on short notice is small, which means any other will be much worse.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2015-04-17T11:55:22.667" LastActivityDate="2015-04-17T15:49:59.503" Title="Continuity of classes split between different days" Tags="|secondary-education|class-participation|gifted-students|teacher-preparation|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7907" PostTypeId="1" AcceptedAnswerId="7911" CreationDate="2015-04-17T14:22:13.810" Score="10" ViewCount="901" Body="&lt;p&gt;I would like to construct an exam that has C-level, B-level, and A-level questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like a student who answers all C-level questions correctly a passing grade, a student who answers all C and B-level questions a B grade, and a student who answers all questions correctly an A-grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I would also like students to be able to combine problems; for instance, those who answer 70-80% of the C-questions and 1-2 B questions should still pass.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is a fair way of constructing such an exam? I'm primarily interested in answers drawn from experience. &lt;strong&gt;Edit:&lt;/strong&gt; By fair, I mean that a student that really doesn't understand the material won't automatically get a high grade, and that a student who knows the material well and slips up one one or two problems won't end up with a very low grade.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My ideas so far:&#xA;Have, say, 6 problems of each type. To get a C, you must get 6 questions right from any of the three categories. Similarly, to get a B you must get 6 questions right out of the A and B categories. To get an A, you would need to get all 6 A questions right.&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="78" LastEditDate="2015-04-17T16:08:17.773" LastActivityDate="2018-07-15T16:10:23.643" Title="What is a fair way of constructing exams with tiered levels of difficulty?" Tags="|exams|grading|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7908" PostTypeId="2" ParentId="7906" CreationDate="2015-04-17T15:49:59.503" Score="1" Body="&lt;p&gt;I think you should prepare it as if it were a special topic presented in a course in an undergraduate college.  Although you don't mention age in the post, your post suggests these are college-bound high school students.  I imagine one of the benefits of this camp is to give them a taste of what college classes are like, in an ideal sense.  Don't repeat; review.  Assume that they will behave like proper students, who are interested in learning because they opted to take the course.  Give them an outline of the material, mention time will be tight, make yourself available outside of class for questions, spend two minutes or less reviewing the outline at the beginning of the second class, and prepare to triage if things go not according to plan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Show Them You're Having Fun&quot; Paseman, 2015.04.17&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-17T15:49:59.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7909" PostTypeId="2" ParentId="7864" CreationDate="2015-04-17T20:12:55.420" Score="8" Body="&lt;p&gt;(This is not bounty-worthy; just consider these illustrated comments.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Smith &amp;amp; Minton's textbook &lt;a href=&quot;http://www.mhhe.com/math/calc/smithminton2e/cd/folder_structure/text/chap10/section06.htm&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Calculus&lt;/em&gt;&lt;/a&gt;&#xA;emphasizes drawing curves in each coordinate plane, and a few cross-sections&#xA;(echoing Gerhard Paseman).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To get an idea of what the graph looks like, first draw its traces in the three coordinate planes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although they still rely on software, they talk through the process of&#xA;gradually coming to understand what particular surfaces &lt;em&gt;should&lt;/em&gt; look like.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/vz1Q9.png&quot; alt=&quot;Paraboloid&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=S-6YHvMrbP8&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt; is a (long!) video of Michael Chamberlain sketching surfaces in class.&#xA;At the below snapshot, he explains how the cross-sections &#xA;of the hyperboloid are ellipses:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/8T3C3.png&quot; alt=&quot;Hyperboloid&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-04-18T01:46:30.750" LastActivityDate="2015-04-18T01:46:30.750" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7910" PostTypeId="2" ParentId="7907" CreationDate="2015-04-17T21:14:01.987" Score="3" Body="&lt;p&gt;Any way of breaking up questions and then assigning point values is going to result in the ability to break it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you go the route you mention (answer 6, minimum numbers for a grade), how will you deal with a student who answers the 6, but doesn't answer all of them correctly?  Do they now get a C for attempting the harder ones?  Is it an F because you can't tell?  If you start assigning point values and then just adding them up, there is no difference than any other method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd suggest that you make 3 questions.  One that you'll call &quot;A&quot; level, one you'll call &quot;B&quot; level, and one you'll call &quot;C&quot; level.  They can choose one to work through.  If they get it correct (that means completely correct work or with minor algebraic/labeling errors at most) then they get the grade that they attempted.  If they get it wrong (that means correct method but problem-breaking levels of misconceptions) then they get a &quot;D&quot;.  If they can't even attempt the problem correctly (that means blank or no work shown) give them an &quot;F&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What reason do you have for giving them 6 questions at each level if you just want to see what they know?&lt;/p&gt;&#xA;" OwnerUserId="4983" LastActivityDate="2015-04-17T21:14:01.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7911" PostTypeId="2" ParentId="7907" CreationDate="2015-04-17T23:09:14.143" Score="9" Body="&lt;p&gt;There is at least one way to make this question more reconciliable with reality if the implicit question is partly about how to make the students less stressed... by more-clearly delineating what their grade(s) will be at a given level of effort. (On the whole, it's a fine impulse to want to implement this, I think.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I very-strongly suspect that it will mostly not be received as such, but as merely a more-game-able system by being more laid-open. And then you yourself have the increased moral burden of figuring out what to do with the students who're less able to, or less &lt;em&gt;inclined&lt;/em&gt; to, try to game the system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An entirely analogous problem, that puts too much burden on students' larger sensibilities, is the common situation that homework does not contribute to the grade, but &quot;is recommended&quot;. Most students misjudge the situation, to their own detriment, but not intentionally!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, for your issue, more to the point would be to give an estimation of the &quot;difficulty&quot; of the various questions, if you are confident of your appraisals of it/them, to literally &lt;em&gt;give&lt;/em&gt; information to your students... rather than giving a little information, but &quot;in return&quot; &lt;em&gt;requiring&lt;/em&gt; additional exercise of judgement from them. Still, if you have a plan about scaled grading of more-difficult questions... you should be upfront about it... but, again, will students be able to gauge things properly? Especially the weaker students? Requiring additional exercise-of-judgement is a further test, after all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, in a course where any significant fraction of the population is near &quot;failure&quot;, such a strategy would likely amplify the failings of the weaker people, confuse the middle-ground, etc.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-04-17T23:09:14.143" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7912" PostTypeId="1" CreationDate="2015-04-18T02:28:16.837" Score="25" ViewCount="765" Body="&lt;p&gt;I am wondering if there is literature on the developmental age&#xA;(pre-adolescent?, adolescent?) at which the notion of a &quot;proof&quot;&#xA;can be understood? I am less interested in mathematical proofs&#xA;and more interested in &quot;water-tight arguments&quot;&amp;mdash;at&#xA;what age can students understand what constitutes a logically&#xA;correct, almost legal argument, impervious to objections?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(My own view is that this is in fact what constitutes a mathematical proof:&#xA;a &quot;water-tight argument&quot; for the appropriately educated audience.&#xA;But let's leave that aside.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly this capacity is present in U.S. 7th or 8th-grade (age: ~12-13 yrs old),&#xA;when (typically) two-column Euclidean geometry proofs are taught.&#xA;But I have sensed the capacity may be (nascently?)&#xA;present even at the U.S. 4th or 5th-grade levels (~9-10 yrs old),&#xA;in my occasional forays into that domain. &#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/epO3Q.png&quot; alt=&quot;Piaget&quot;&gt;&#xA;&lt;hr /&gt;&#xA;&lt;strong&gt;&lt;em&gt;Added&lt;/em&gt;&lt;/strong&gt;. With the help of keywords from the useful comments,&#xA;I found a 2003 MS Thesis literature review of the topic:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Kate Mansi. &quot;Reasoning and Geometric Proof in Mathematics Education:&#xA;  A Review of the Literature.&quot; &lt;em&gt;North Carolina State Univ. Masters Thesis&lt;/em&gt;. 2003.&#xA;  (&lt;a href=&quot;http://repository.lib.ncsu.edu/ir/bitstream/1840.16/2692/1/etd.pdf&quot; rel=&quot;noreferrer&quot;&gt;PDF download link&lt;/a&gt;)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Abstract&lt;/strong&gt;. Through a comparison of the theories of Piaget and van Hiele, I discuss how students&#xA;  acquire mathematical and geometric reasoning skills and how this relates to their readiness to&#xA;  produce formal proofs. I then discuss research findings, which indicate that students are not&#xA;  typically at a high enough van Hiele level to be successful with proof by the time they get to&#xA;  high school.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[&lt;strong&gt;p.16&lt;/strong&gt;:]&#xA;  Piaget (1987) claims that students progress through three levels (PL) in the&#xA;  development of their justification and proof skills.&#xA;  Unlike van Hiele’s levels of geometric&#xA;  understanding, Piaget’s levels for proof and justification coincide with the biological&#xA;  development of the student. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&#xA;Piaget, J. &lt;em&gt;Possibility and necessity. Vol 2. The role of necessity in cognitive&#xA;development&lt;/em&gt;. Minneapolis: University of Minnesota Press.&#xA;1987. &#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-04-19T23:03:05.107" LastActivityDate="2021-02-13T02:21:46.630" Title="Is there a Piagetian age at which proofs can be comprehended?" Tags="|mathematical-pedagogy|secondary-education|primary-education|proofs|" AnswerCount="1" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="7913" PostTypeId="5" CreationDate="2015-04-18T14:29:42.080" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-04-18T14:29:42.080" LastActivityDate="2015-04-18T14:29:42.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7914" PostTypeId="4" CreationDate="2015-04-18T14:29:42.080" Score="0" Body="For questions about the teaching of the function concept, function properties, and various types of functions." OwnerUserId="39" LastEditorUserId="39" LastEditDate="2015-04-18T19:34:33.917" LastActivityDate="2015-04-18T19:34:33.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7915" PostTypeId="2" ParentId="7900" CreationDate="2015-04-18T14:51:21.730" Score="3" Body="&lt;p&gt;I used Excel to solve many of the problems I encountered in &lt;a href=&quot;https://projecteuler.net/&quot; rel=&quot;nofollow&quot;&gt;Project Euler&lt;/a&gt;. I bet they'd find some of those problems interesting and accessible.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-04-18T14:51:21.730" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7916" PostTypeId="2" ParentId="7900" CreationDate="2015-04-18T21:18:24.167" Score="3" Body="&lt;p&gt;you might find this paper &quot;&lt;a href=&quot;http://link.springer.com/article/10.1007/s10758-005-8420-9&quot; rel=&quot;nofollow&quot;&gt;Designing Spreadsheet-Based Tasks for Purposeful Algebra&lt;/a&gt;&quot; (and a couple of some others written by the same authors and about the same year) useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. Originally, I meant to put this answer as a comment. But, I realized that I don't know how to hyperlink inside a comment. Could someone who knows please let me know how?&lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-04-18T21:18:24.167" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7917" PostTypeId="1" AcceptedAnswerId="7919" CreationDate="2015-04-19T21:14:26.987" Score="13" ViewCount="887" Body="&lt;p&gt;I read an article the other day about how physics professors have to teach their introductory classes in a somewhat old-fashioned manner.  This is due to pressure from the &lt;em&gt;other&lt;/em&gt; departments (other sciences, engineering, computer science, etc).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has me wondering what compromises math professors make when teaching things like calculus, linear algebra, and ODEs -- all of which are required courses for science and engineering majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This raises two questions for me:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How are introductory math courses designed so as to prepare not only math students, but also science and engineering students for their later works?&lt;br&gt;&#xA;  How might introductory courses be changed if science and engineering students were no longer part of the equation (so to speak)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Additionally, because there is no Physics Educators SE, if someone can answer the same questions for introductory physics courses, I'd be interested in that as well.&lt;/p&gt;&#xA;" OwnerUserId="5081" LastActivityDate="2015-04-20T19:41:25.483" Title="How would introductory classes change if they didn't need to be taken by non-mathematicians?" Tags="|undergraduate-education|course-design|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7918" PostTypeId="1" AcceptedAnswerId="7941" CreationDate="2015-04-19T21:46:19.983" Score="15" ViewCount="2237" Body="&lt;p&gt;Remedial students have seen quadratics before but, perhaps they don't elicit positive memories. The textbook (designed for people taking the course for the first time, not for remedial students) spends a long time on &quot;multiplication of binomials&quot; before there is any factoring. The idea  (I think) is you will see the patterns such as &quot;difference of two squares&quot; and the &quot;perfect square trinomial&quot; then, when it's time to factor, you can recognise these patterns in the new context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many remedial students lack confidence in their abilities, when they see patterns they don't seem to trust themselves to have recognised something significant.  They have no problem learning to multiply the binomials. But, this time feels wasted because when we move to do the reverse and factor I'm met with silence. In addition, multiplying binomials isn't hard, especially for students who have done it before. What's worse, I risk boring them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I start factoring examples, I always start trying to nurture their intuition. But I feel like it always ends up being an algorithm. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For $a \neq 1$, our book and syllabus recommend something called the &quot;AC Method&quot; which I had never heard of until I started planning.  It's an algorithm, my students like it. I hate it.  In the end, I'm happy to see them getting correct answers under their own power and feeling successful. That's why I teach it. That's why I skip the multiplying. But I want MORE for them. I want the math to seem a bit more tame and transparent than it was before they began my course. I didn't learn math the way that I'm feeling pressured to teach math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any &quot;methods&quot; I'm overlooking? I do think their remedial experience should shape the course and the lessons, but I would hope it could act as a springboard rather than a wall.&lt;/p&gt;&#xA;" OwnerUserId="5080" LastEditorUserId="5080" LastEditDate="2015-04-20T00:03:42.427" LastActivityDate="2018-06-28T00:02:37.290" Title="Remedial students struggle with factoring $x^2+bx+c$ and $ax^2+bx+c$" Tags="|secondary-education|algebra|remedial-courses|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="7919" PostTypeId="2" ParentId="7917" CreationDate="2015-04-19T21:52:02.140" Score="10" Body="&lt;p&gt;In a world where the audience was pure of heart:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;proof methods course is the first course&lt;/li&gt;&#xA;&lt;li&gt;matrix linear algebra is taught along-side early calculus&lt;/li&gt;&#xA;&lt;li&gt;calculus is taught with methods of analysis, never just with a primarily computational focus&lt;/li&gt;&#xA;&lt;li&gt;all courses are about proofs, at least 50 percent&lt;/li&gt;&#xA;&lt;li&gt;real analysis as an advanced calculus course ceases to exist because we already covered that in the &quot;calculus sequence&quot; (which is not the calculus you typically find, rather, the sort my Chinese colleague learned... replete with analysis from the outset)&lt;/li&gt;&#xA;&lt;li&gt;higher analysis replaces the traditional real analysis course&lt;/li&gt;&#xA;&lt;li&gt;required second course on theoretical linear algebra&lt;/li&gt;&#xA;&lt;li&gt;generally the level of courses is higher since the expectation of math from math majors ought to exceed that we have for other folks who have not declared math as their life's work. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Main Point:&lt;/strong&gt; because the major math courses are taken only by math students the audience of the major math courses would share common prerequisites. These prerequisites could be used to bring greater cohesion to the entirety of the curriculum. This cohesion would produce a student with greater sophistication at the end of the course of study (say over 4 years). Students would be challenged in ways we simply cannot hope to attain in the current mixed-audience format. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Admittedly, I have in mind more or less pure mathematics students. My focus would be a bit different if I was to consider the applied math. I don't think this program is realistic for a large major. Mostly, this sort of thing would really only be reasonable with a small group of highly motivated pure students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, to be clear, I speak merely from the viewpoint of curriculum design. Even in this specialized course of study I sketch here, the individual would still need to take the initiative to study beyond the rudiments. I do think it is time we (as in the society of mathematicians, or, at least people who care deeply about math as a professional activity) made more of modern mathematics an expectation for the undergraduate in math. In particular, I have in mind the subject GRE, tricky calculus and linear is great, but, I'd love to see more of an expectation in abstract courses. There is a wide gulf between the standard course load most places and the necessary scholarship for doctorate work. So, I'd rather see the undergraduate be a little less leisurely in the interest that the graduate study can be more meaningful. This is why I push undergraduates. But, only the ones who want a push. Of course, the battle I wish to fight is a losing one from a statistical perspective, the entry point of most students invariably declines (USA-based comment).&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2015-04-20T01:25:36.560" LastActivityDate="2015-04-20T01:25:36.560" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7920" PostTypeId="1" CreationDate="2015-04-19T22:05:03.033" Score="4" ViewCount="284" Body="&lt;p&gt;Many of my statistics students will calculate the standard deviation for a set of data using the formula when it is not required or appropriate. When faced with something like $\sqrt{56}$ they will just leave it that way, even for word problems where an approximation is ideal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They have calculators, but they do not know how to use them beyond the four basic operations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it a good strategy to simply try to overwhelm them with numbers to force them to stop calculating these things by hand and learn some of the functions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not have much class time to teach how to use the calculator although I have explained most of the basic functions. Perhaps they need this in written form?&lt;/p&gt;&#xA;" OwnerUserId="5080" LastActivityDate="2017-03-31T02:17:42.787" Title="Statistics students lack calculator skills, how to help them?" Tags="|statistics|calculators|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="7921" PostTypeId="2" ParentId="7917" CreationDate="2015-04-19T22:34:30.173" Score="6" Body="&lt;p&gt;I applaud James Cook for professing his opinion with some eloquence.&#xA;As I disagree with him on tone and some points, I (with respect for&#xA;his opinion) suggest the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Thinking is taught in the first, and every course.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Learning how to apply in some situations is taught in the&#xA;first (and every) course.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Learning how to make a proof and how to apply a result is&#xA;a skill/talent/goal that develops over time with practice;&#xA;both should be considered extensions of thinking.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Methods of analysis do not take a back seat to computations,&#xA;nor do computations take a back seat to analysis.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Real analysis is covered when an idea of how it is to be used is&#xA;presented first.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Linear algebra is taught as needed: a course heavy on applications&#xA;with some theory (and where to find more theory and proofs when&#xA;needed) for those who need to use old tools and develop new tools&#xA;for application, and a course heavy on theory and light on applications&#xA;(and where to find more problems and possible room for application)&#xA;for those who need to advance the theory for the sake of both the&#xA;theory and the applications.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Generally the level of courses is better (not higher) because we&#xA;give people the information to decide what education will be best&#xA;for their intended growth and that of the society they live in.&#xA;Even if we don't know the proper direction for society, we should&#xA;make several directions available by the courses we teach and the&#xA;actions we do.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Main point: who needs a main point? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: 2015.04.20&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than respond to the comments point by point, I&#xA;choose to explain the rationale and the nature of my&#xA;disagreement with the post of James Cook.  Understand&#xA;first that I don't think James Cook is wrong.  From his&#xA;perspective and experience, his answer may be as apt as any.&lt;br&gt;&#xA;I see his answer as producing people who are&#xA;adept at checking and generating proofs, and I take his&#xA;&quot;higher level&quot; as meaning &quot;more facility with handling&#xA;activities centered around understanding proofs&quot;.&#xA;Whether he intended this slant on it or not, in my view&#xA;this slant is wrong, and I attempt to correct it with&#xA;the list above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics is not just proofs and understanding proofs.&#xA;Mathematics is about pattern, observing, creating, &#xA;convincing ourselves, convincing others, and building&#xA;something that is both beautiful and useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want a student coming out of a mathematics course&#xA;to think &quot;I could do that.&quot;  That is not just doing&#xA;computations, or solving problems, or writing up&#xA;a proof of an assertion, or checking one's logic, or&#xA;trying to find a flaw in someone else's reasoning.&#xA;It is not a game of axioms and propositions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics is an activity which both has the purity and&#xA;potential beauty &#xA;of intellectual play admired and pursued by the Greeks&#xA;(and others) of old, and has the power of being applied&#xA;by someone who would work in the physical world and&#xA;help society through such application.  It may be&#xA;a pedagogical convenience to keep these two things&#xA;separate, but it is a mistake socially and perhaps&#xA;totally to do so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want a student coming out of a mathematics course to&#xA;think that they could imagine or create structures and&#xA;realize or apply them, check to see that they withstand&#xA;the brutal force of reason and reality, fix or modify&#xA;these ethereal visions and either apply them or give them&#xA;to someone else to use.  Most importantly, I want them&#xA;to be able to tell other people about them in a way &#xA;that helps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I prefer that education involve teaching the student&#xA;to think.  Not &quot;do I have to remember to write the&#xA;definition this way&quot; or &quot;is it pemdas or pedmas or&#xA;samdep&quot; or &quot;what chapters do I have to read the night&#xA;before the test&quot;.  I want them to do mental simulations:&#xA;&quot;I have to take a notion of continuity in one dimension&#xA;and generalize it to two dimensions.  Can I think of&#xA;counterexamples I want to avoid so that I can frame&#xA;the definition to avoid them?&quot; or &quot;This notion of a&#xA;group acting on itself is a little too self-referential.&#xA;Can I use group of symmetries to simplify how to write&#xA;up this case-by-case combinatorial proof of this other &#xA;thing that interests me?&quot; or &quot;My uncle's going to ask&#xA;me to help tomorrow in the backyard: can I use any of&#xA;this to make things better/easier/cooler for us tomorrow?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if the student will not be a theoretician, she or he&#xA;needs to explain and interact with others, and may need &#xA;a fragment of the theoreticians perspective.  Similarly,&#xA;the student may not be an engineer, or customer support&#xA;worker, or salesman, or actuary, or one of a number of&#xA;other intended destinations, but they will need to know&#xA;that others have to apply math, and they would do well&#xA;to prepare explanations for those who aren't as adept&#xA;at theory.  In short, all of the students need to be&#xA;teachers to some extent, for themselves as for others.  I do not see this as a potential result from James Cook's list.  I do see it in mine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will stop here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;End Edit: 2015.04.20&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Not A Professional Mathematics Educator&quot; Paseman, 2015.04.19&lt;/p&gt;&#xA;" OwnerUserId="3468" LastEditorUserId="3468" LastEditDate="2015-04-20T19:41:25.483" LastActivityDate="2015-04-20T19:41:25.483" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="7922" PostTypeId="2" ParentId="7918" CreationDate="2015-04-19T23:43:00.297" Score="4" Body="&lt;p&gt;My answer was just a little too long for a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd start with &quot;X puzzles&quot; (which I put in quotes because I'm not sure they have a name).  These are just big X's where you have 1 number on the top and 1 on the bottom.  Then they write 1 number on the left and 1 on the right that will add to the bottom and multiply to the top (or vice versa if you want).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that I'd go to the lattice style (which is basically algebra tiles but without the tiles themselves) and have them do the multiplication (give them the outside and work in).  Then show them a few that have the middle and have them find the outsides.  Then gradually pull back so you have just a trinomial and see what they can do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to do the difference of two squares you can do a page of just those types of problems and ask for any commonalities they find in the factors versus what the problems look like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's an example (I am not affiliated with this site, just did a quick search): &lt;a href=&quot;http://emathlab.com/Algebra/GenericRectangles/genericRectangleFactoring.php&quot; rel=&quot;nofollow&quot;&gt;http://emathlab.com/Algebra/GenericRectangles/genericRectangleFactoring.php&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4983" LastActivityDate="2015-04-19T23:43:00.297" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7923" PostTypeId="2" ParentId="7920" CreationDate="2015-04-20T02:00:04.470" Score="3" Body="&lt;p&gt;Here are two ideas.  See what you think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Find a tutorial (by googling) that gives them some specific tasks, and walks them through doing them, step by step, and assign it.  (In future, assign it early in the semester.)  You could call it a lab assignment, and ask them to write something up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask them to form small study groups and meet outside class to figure out together how to use their calculators.  Then, on a Friday, hold a special calculation race, with a couple of little prizes.  Ask for volunteers to bring popcorn to share.  Make it fun.  No one is too old for fun.&lt;/p&gt;&#xA;" OwnerUserId="4956" LastActivityDate="2015-04-20T02:00:04.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7924" PostTypeId="2" ParentId="7900" CreationDate="2015-04-20T05:27:36.243" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Write a formula that converts Centigrade to Fahrenheit or vice versa.  Plug in some common temperatures for the climate where you live.  They can print out the table of values, nice and big, and post it on the wall at home and at school.  At your next meeting, have a little Conversion Bee, with little prizes for the top three students.  Ask for volunteers to bring popcorn to share.  (Don't make it cutthroat -- keep it fun.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have them design and conduct a survey of fellow students, asking, for example, &quot;What is your favorite sport to watch?  And your favorite sport to practice?&quot;  They can work in small groups.  Each group should choose a different survey.  The survey results will be the data they will use for the descriptive stastictics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have them download nutrition facts from the standard fast food restaurants, and then analyze (e.g. there's lots of opportunity for averaging).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Same but with the various items of candy that are given out at Halloween.  This would require some sorting.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4956" LastActivityDate="2015-04-20T05:27:36.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7925" PostTypeId="2" ParentId="7917" CreationDate="2015-04-20T07:28:58.373" Score="7" Body="&lt;p&gt;You may want to take a look into the German education system, where undergraduate studies happen at universities and are mostly separate. Right now, at my (German) university, there are the following courses in the first semester:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;linear algebra (for mathematicians)&lt;/li&gt;&#xA;&lt;li&gt;calculus (for mathematicians)&lt;/li&gt;&#xA;&lt;li&gt;math for physicists&lt;/li&gt;&#xA;&lt;li&gt;math for computer scientists&lt;/li&gt;&#xA;&lt;li&gt;math for chemists&lt;/li&gt;&#xA;&lt;li&gt;…&lt;/li&gt;&#xA;&lt;li&gt;math for agronomists&lt;/li&gt;&#xA;&lt;li&gt;mechanics and thermodynamics (for physicists)&lt;/li&gt;&#xA;&lt;li&gt;physics for chemists, mathematicians and computer scientists&lt;/li&gt;&#xA;&lt;li&gt;physics for biologists, pharmacists, medics and similar&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I started studying math and physics in the last year when physicists did not have their own math course but had to attend the first two linear algebra and the first four calculus courses for mathematicians (though they had to pass the exams in two of them). These courses were already very much as described in &lt;a href=&quot;https://matheducators.stackexchange.com/a/7919/75&quot;&gt;James S. Cook’s answer&lt;/a&gt; and primarily focussed on theory and proofs. For example, I heard about modules, rings and groups in the first semester. Keep in mind though that the students had better prerequisites, e.g., I was suprised to learn that some of the other students had &lt;em&gt;not&lt;/em&gt; already done proofs in school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I later tutored the math-for-physicists series several times and the respective courses were still mainly theory- and proof-focussed¹. Mostly some aspects of linear algebra were missing (such as modules and rings) that are not that relevant to physicists. After what I heard, the first two courses from the list did not change much after physicists got their own courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I cannot really compare the contents of the courses to a system where everybody attends the same courses, as I have no experience with the latter.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&#xA;¹ which I as a physicist consider a good thing: We need people to understand the mathematics, being able to apply it is secondary. You cannot really understand quantum mechanics without linear algebra, for example.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-04-20T07:28:58.373" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7926" PostTypeId="2" ParentId="7920" CreationDate="2015-04-20T12:43:23.273" Score="1" Body="&lt;p&gt;Make a list of skills you think they lack and combine each skill with a topic you normally teach. Suggest the proper way to the use calculator when explaining the topic. In the beginning of the next class, give a short (2-3 min) assignment for the topic (1, or 2 questions). Make the task favor the use of calculator skill you wish them to learn. This way you can quickly assess their level of understanding both - the topic and the calculator. If needed - walk them through the correct solution inclusive the appropriate use of calculator. Repeat often.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2015-04-20T12:43:23.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7927" PostTypeId="1" AcceptedAnswerId="7935" CreationDate="2015-04-20T13:49:23.607" Score="12" ViewCount="1515" Body="&lt;p&gt;The following issue came up in my Intro to Proofs course and I wasn't sure how to explain my distaste of the student proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Prove that the solution for $x$ in $ax+b=c$ is unique ($a \neq 0$).&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student Proof:&lt;/strong&gt; Solving gives \begin{align*}&#xA;ax+b &amp;amp;= c\\&#xA;ax &amp;amp;=c-b\\&#xA;x &amp;amp;= \frac{c-b}{a}.&#xA;\end{align*}&#xA;Hence the solution must be $x=\frac{c-b}{a}.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The Proof I Wanted:&lt;/strong&gt; Suppose both $x$ and $y$ solve the equation, then&#xA;\begin{align*}&#xA;ax+b &amp;amp;= ay+b\\&#xA;ax&amp;amp;=ay\\&#xA;x&amp;amp;=y. &#xA;\end{align*}&#xA;Thus the solution is unique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is the student solution acceptable?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like finding an algebraic solution is not a proof of uniqueness. My gut says that there should be a good example where algebraically solving implies a unique solution when really there isn't one. (But I can only think of simple cases where you forget a $\pm$ on a square root.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way of explaining my distaste for this is that when solving choices are made to arrive at $x$. Maybe other choices would've gotten you to a different solution. Whereas in the desired proof since we assumed two solutions from the start, it's okay to make algebraic choices to show the two solutions are the same. But maybe I'm being too pedantic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Am I justified in disliking the student solution?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="262" LastEditDate="2015-04-20T20:56:18.167" LastActivityDate="2015-04-27T16:47:18.033" Title="Algebraic Solving and Uniqueness Proofs" Tags="|algebra|proofs|" AnswerCount="6" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="7928" PostTypeId="2" ParentId="7927" CreationDate="2015-04-20T14:16:27.303" Score="16" Body="&lt;p&gt;Be careful here. You're teetering close to the edge of checking whether the student has guessed your solution to the problem rather than whether the student has solved the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is easy to do, especially in cases like this where a problem is constructed with the specific purpose of exercising a particular technique (eg, your solution).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This purposeful construction inflicts the problem setter with a bit of blindness for other potential solution paths - when we solve a problem, we stop looking for different angles on it. And being presented with an alternate solution path can also be a little blow to the ego - because of the implicit suggestion that the problem doesn't serve its purpose perfectly, eg, the desired technique was not required.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My suggestion would be to congratulate the student on their straightforward approach, and explain that you had tripped yourself up with the expectation of a different line of thought.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moving forward, there's nothing wrong with prescribing a technique for problems when you want a particular technique to be used. (Excepting that one can get into a bit of a taxonomical nightmare when trying to put names on individual techniques for things. May be better to treat the technique names as disposable: eg, Tuesday's Method can be a labelled technique from the class notes).&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2015-04-20T14:16:27.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7929" PostTypeId="2" ParentId="7927" CreationDate="2015-04-20T14:25:10.157" Score="5" Body="&lt;p&gt;There are two typical ways for uniqueness proofs: showing that all answers have the same form or that any two potential answers must be the same. What your student did was if $x$ is a solution to $ax +b = c$ ($a \neq 0$), then $x= \frac{c-b}{a}$ implying that if $y$ is another solution, then $y = \frac{c-b}{a} = x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both ways should be explained as valid forms of proving uniqueness. While supposing if $x$ and $y$ are solutions, then $ax +b = c = ay+b$, the leap to setting both equations equal might be a leap for some. For instance, I personally would show that all solutions have the same form to get uniqueness. You could add that, since this is a proof class, that linear functions are one-to-one or injective to get the desired result as well. This allows for quite a bit of generalization for linear systems of equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, if the student uses a valid proof for something that is not what you were considering, congratulate them and, perhaps, point out the method you were intending.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2015-04-20T14:25:10.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7930" PostTypeId="2" ParentId="7853" CreationDate="2015-04-20T15:20:58.783" Score="2" Body="&lt;p&gt;A fundamental topic in school mathematics is the notation of area. A remarkable result that can be treated at different grade levels is the nifty theorem known as the Bolyai-Gerwien-Wallace Theorem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Intuitively, the result says that two plane polygons have the same area if and only if it is possible to cut one of the polygons up into a finite number of polygonal pieces and reassemble these pieces to form the other polygon. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This property is sometimes called equidecomposability. There are still open questions about the minimum number of pieces for getting from one shape of polygon to another shape of polygon.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is remarkable that the analog of this for 3-space is false. Hilbert's Third Problem asked whether a regular tetrahedron and cube of the same volume were &quot;equidecomposable.&quot; Max Dehn showed that the answer was &quot;no.&quot;&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2015-04-20T15:20:58.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7931" PostTypeId="2" ParentId="7927" CreationDate="2015-04-20T18:08:26.200" Score="14" Body="&lt;p&gt;I do not see any issue $\iff$  it is clear what the sequence of equations means. (Sorry for the pun.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The following is true (for $a \neq 0$, in the real numbers, say)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ax+b = c \iff &#xA;ax =c-b \iff&#xA;x = \frac{c-b}{a}$$&#xA;Hence the solution $x$ must be $x=\frac{c-b}{a}.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each transformation is an equivalence. The problem is that sometimes some do not pay enough attention and write sequences of equations like this that in fact are not equivalences. (Or perhaps they pay attention but just mean to express something different.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example the following might be written but does not give uniqueness: &#xA;\begin{align*}&#xA;x^2 +5 &amp;amp; = 9 \\&#xA;x^2  &amp;amp; = 4 \\&#xA;x &amp;amp; = 2&#xA;\end{align*}&#xA;The issue is that the last equation is &lt;em&gt;not&lt;/em&gt; an equivalence; it is false that $x^2   = 4 \iff x  = 2$ (again in the reals, that is; in the positive reals it would be true). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main issue I see, but this is the same for the students and your solution, is that it is not completely made clear (at least in what is written) what &lt;em&gt;the sequence of equation means.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, would we work over the $\mathbb{Z}/4\mathbb{Z}$ your solution would not show uniqueness  either as even for $a\neq 0$ the equation $ax=ay$ is not equivalent to $x= y$. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt; The discussion above uses equivalence ($\iff$) that is &quot;if and only if&quot; throughout. Such transformations will preserve the set of solutions &lt;em&gt;exactly.&lt;/em&gt; It can sometimes be more desirable to use implications only in one way. Then one also has a relation among the solutions, namely: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f(x)= a \implies g(x)=b$$ &#xA;will yield that every solution to $f(x)=a$ is also a solution to $g(x)=b$.  That is if the concern is only to show uniqueness it is even sufficient to have sequence of equations linked by $\implies$ and a unique solution for the final one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conversely, if one wants existence of a solution one needs a sequence of equations linked by $\Leftarrow$ and a solution for the last. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus (in the reals),&lt;br&gt;&#xA;\begin{align*}&#xA;x^2 +5 &amp;amp; = 9 \\&#xA;x^2  &amp;amp; = 4 \\&#xA;x &amp;amp; = 2&#xA;\end{align*}&#xA;can be taken to show the existence of a solution of the first equation as &#xA;$$x^2 +5  = 9 \Leftarrow&#xA;x^2   = 4 \Leftarrow&#xA;x  = 2&#xA;$$&#xA;is true, yet not uniqueness as &#xA;$$x^2 +5  = 9 \implies&#xA;x^2   = 4 \not \Rightarrow&#xA;x  = 2&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt; To me the key point is to make clear how the different equations are logically linked, and to know what this means for the respective solutions. &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2015-04-20T21:35:20.240" LastActivityDate="2015-04-20T21:35:20.240" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7932" PostTypeId="2" ParentId="7927" CreationDate="2015-04-20T18:12:07.793" Score="8" Body="&lt;p&gt;I suspect your discomfort with the student's proof is carryover from high school algebra where the logical underpinnings of &lt;em&gt;what you are actually doing&lt;/em&gt; isn't often made clear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student's work is essentially a rigorous proof of the theorem&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Theorem:&lt;/strong&gt; If $a \neq 0$ and $a x + b=c$, then $x = (c-b)/a$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and this sort of thing is the typical result of the work involved in &lt;em&gt;solving&lt;/em&gt; a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Uniqueness is an easy corollary:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Corollary:&lt;/strong&gt; If $a \neq 0$ and $ax+b = c$ and $ay+b=c$, then $x=y$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Proof:&lt;/strong&gt; $x = (c-b)/a$ and $y = (c-b)/a$, therefore $x=y$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;You should also note that any skepticism about the student's approach showing the solution for $x$ is unique should &lt;em&gt;also&lt;/em&gt; apply to your approach of showing that $x=y$ is the unique solution for $x$. e.g. after all you made choices too; maybe if you made other choices you wouldn't have arrived at $x=y$?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;All of the work demonstrated in your post can be thought of as simply applying transformations to a problem to derive &lt;em&gt;simpler&lt;/em&gt; problems that are implied by it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student opted to simplify &lt;em&gt;first&lt;/em&gt;, and produced a simpler problem for which it was trivial to prove uniqueness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You opted to invoke a proof technique for showing uniqueness &lt;em&gt;first&lt;/em&gt;, and then proceeded to simplify the ensuing system of equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe all of your concerns would be obviated if you did as I did above and explicitly tacked on the remaining, trivial argument needed to arrive at that the conclusion that the solution (if it exists) is unique.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2015-04-21T04:50:46.170" LastActivityDate="2015-04-21T04:50:46.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7933" PostTypeId="2" ParentId="7927" CreationDate="2015-04-20T19:00:51.090" Score="0" Body="&lt;p&gt;Actually, I get a bit confused about this, each step he took was reversable. But I don't think that by itself is enough. The important thing seems to be that each step he took preserved the solution set. In which case he proved that the solution set is a singleton.&lt;/p&gt;&#xA;" OwnerUserId="5085" LastEditorUserId="5085" LastEditDate="2015-04-27T16:47:18.033" LastActivityDate="2015-04-27T16:47:18.033" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7934" PostTypeId="2" ParentId="7887" CreationDate="2015-04-21T04:47:36.610" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How can you determine the quality of your teaching, or someone else's?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Give the students questionnaires to fill out (anonymously) to evaluate the teacher.  The great thing about this is that it doesn't just grade the teacher, it also gives directions for improvement.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At regular intervals during the class, observe who is talking.  If it's the teacher, put a tick mark in one column.  If it's a student, put a tick mark in the other column.  Count the tick marks in both columns.  The best teachers draw students out.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4956" LastActivityDate="2015-04-21T04:47:36.610" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7935" PostTypeId="2" ParentId="7927" CreationDate="2015-04-21T04:51:49.900" Score="9" Body="&lt;p&gt;Both are valid ways to prove the &lt;em&gt;uniqueness&lt;/em&gt; of solutions, i.e. that the set $\,S\,$ of solutions contains at most one element. Indeed, one easily proves that the following are equivalent for any set $\,S.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(1)\ \ \ |S| \le 1,\ $ i.e. $S\,$ contains at most one element.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(2)\ \ \ s\in S\,\Rightarrow\, s = t,\ $ for some fixed $\,t$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(3)\ \ \ s,t\in S\,\Rightarrow\, s = t$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student's solution uses method $(2)$ whereas your solution uses method $(3)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, such doubts about the method used in $(2)$ are not unusual. One can find many such questions in any general-level math forum such as sci.math, e.g. see &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=482659&quot; rel=&quot;nofollow noreferrer&quot;&gt;this thread&lt;/a&gt; about such a proof in Max Rosenlicht's &lt;em&gt;Introduction to Analysis&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark&lt;/strong&gt; $\ $ It would be interesting to better understand the source of this apparent bias against the method $(2).\,$ It may be related to the fact that the &lt;em&gt;general method(s)&lt;/em&gt; of proving uniqueness seem to be rarely explicitly taught. Rather, the method(s) are typically learned &lt;em&gt;en passant&lt;/em&gt; while working on specific problems. As a result, some students never master such method(s) and often miss opportunities for employing these powerful tools. To help rectify this, I often emphasize such applications, e.g. &lt;a href=&quot;https://math.stackexchange.com/search?q=user%3A242+uniqueness+powerful+&quot;&gt;over 30 answers on math.SE&lt;/a&gt;, e.g. this &lt;a href=&quot;https://math.stackexchange.com/a/1236370/242&quot;&gt;interesting simple example&lt;/a&gt; occurred only a few days ago: $\,4ab\!-\!1\mid 4a^2\!-\!1\,\Rightarrow\, a=b\,$ for integers $\,a,b &amp;gt; 0.\,$ As I explain there, it is just a special case of the uniqueness of (multiplicative) inverses (in $\,\Bbb Z/m = $ integers modulo $\,m).$&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-04-21T15:39:46.340" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7936" PostTypeId="1" CreationDate="2015-04-21T15:14:57.643" Score="15" ViewCount="561" Body="&lt;p&gt;This question is a follow-up to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7778/should-we-teach-abstract-affine-spaces/7894#7894&quot;&gt;that one&lt;/a&gt;, where I expressed doubt about the use of abstract affine geometry in undergraduate education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, future high-school teachers need to be able to relate their higher education to what they will teach, and they will have to teach affine geometry without much formalism available (e.g. they should also be able to explain how vectors relate to points without resorting to $\mathbb{R}^n$ in the first place). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A teacher should be able to use the linear part of affine transformations to quickly realize that a composition of a rotation of angle $\theta$ and a rotation of angle $-\theta$ with different center is a translation, even if he or she does not expect his or her pupils to come with that conclusion: this is an example of why teachers should know more than what they teach, so I assume an undergrad curriculum with quite some geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is a realistic and convenient framework for affine (possibly Euclidean) geometry which can be used by a high-school teacher, but relates easily to undergrad-level math curricula?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A possible answer could be a set basic objects (points, lines, etc.) and of axioms to work with, but maybe there are better, non-axiomatic ways of achieving this goal. It seems difficult to have enough axioms to cover all what one would have to teach in high-school, while not so many that it would be intractable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I seek answers that would be addressed to the teacher-to-be, not to its pupils; however it should be relatively easy to use the given framework in high-school teaching. I would already be happy with propositions that only cover plane geometry.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-10-12T16:53:25.973" Title="How to teach affine geometry to future high-school teachers?" Tags="|undergraduate-education|secondary-education|geometry|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7938" PostTypeId="2" ParentId="7864" CreationDate="2015-04-21T16:51:13.517" Score="3" Body="&lt;p&gt;Use &lt;a href=&quot;http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-a-functions-of-two-variables-tangent-approximation-and-optimization/session-25-level-curves-and-contour-plots/&quot; rel=&quot;nofollow noreferrer&quot;&gt;level curves and contour plots&lt;/a&gt; (click link for video examples of their use).  With contour plots, use color to represent third dimension.  Here's a familiar example: &lt;img src=&quot;https://i.stack.imgur.com/kBnhu.png&quot; alt=&quot;&quot;&gt;:&lt;/p&gt;&#xA;" OwnerUserId="5090" LastActivityDate="2015-04-21T16:51:13.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7939" PostTypeId="2" ParentId="4071" CreationDate="2015-04-21T17:34:33.530" Score="5" Body="&lt;p&gt;I spent two days sitting in on math classes at Phillips Exeter Academy.  It is an outstanding example of inquiry/&lt;a href=&quot;http://en.wikipedia.org/wiki/Constructionism_%28learning_theory%29#Problem-based_learning&quot; rel=&quot;nofollow&quot;&gt;problem-based&lt;/a&gt; mathematics education in practice.  Consider &lt;a href=&quot;http://www.exeter.edu/academics/72_6532.aspx&quot; rel=&quot;nofollow&quot;&gt;their curriculum.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their format is simple: students do 4-10 homework problems per night (all of these are &lt;a href=&quot;https://www.exeter.edu/academics/72_6539.aspx&quot; rel=&quot;nofollow&quot;&gt;posted online here&lt;/a&gt;), come in to class every day and each put their solution to one of the questions on the board, and explain the solution they put up to the class.  Students and the teacher ask questions and (ideally) only move on to the next problem once students understand the solution.  It appears to me that this is the only instructional format they use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Things to know about this method:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students don't know in advance which mathematical tools they will have to use to solve the homework problems.  The questions themselves provide the introduction to new ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To some extent, the pace of the material covered is determined by the students understanding of particular topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students are instructed not to spend more than 1 hour on the math homework each night.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The teachers have selected or written all the problems, and update them (and post the updates online) every year or two based on their collective experience.  They&lt;/p&gt;&#xA;&#xA;&lt;p&gt;90%+ of class time is spent with students writing up solutions to the problems on the board and explaining and discussing their solutions with the class.  The way students explain their solutions gives their teachers crucial information about what students do and do not understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students are encouraged (and do) put up solutions to homework problems they have not fully answered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem sets have no corresponding solution sets.  This seems to require teachers to solve the problems themselves before coming to class, rather than just reviewing the solution.  I expect that this exercise sharpens the teachers contribution in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The teachers rarely rise and go to the board to explain answers.  Instead, they tend ask questions to the class based on the discussion.  One teacher commented to me that sometimes they go the whole class period without having to enter the instructional conversation (in this case students are explaining the solutions to other students).  Because of this, I suspect that teachers there have a relatively low burn-out rate.  They certainly seem to expend less energy than they would in more traditional instructional formats.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I was told that) test questions often involve applications of the ideas presented in the homework questions, as opposed to being nearly identical to the homework questions.&lt;/p&gt;&#xA;" OwnerUserId="5090" LastEditorUserId="5090" LastEditDate="2015-04-28T19:24:54.883" LastActivityDate="2015-04-28T19:24:54.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7940" PostTypeId="2" ParentId="630" CreationDate="2015-04-21T18:52:13.273" Score="1" Body="&lt;p&gt;Paul Lockhart addresses these issues specifically at the high school and early university level mathematics in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0674284380&quot; rel=&quot;nofollow&quot;&gt;Measurement&lt;/a&gt;.   His work is significant in that his writing itself encourages creativity, rather than giving advice on how to encourage creativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lockhart does this to a lesser extent in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1934137170&quot; rel=&quot;nofollow&quot;&gt;A Mathematician's Lament&lt;/a&gt; (here: &lt;a href=&quot;https://www.maa.org/external_archive/devlin/LockhartsLament.pdf&quot; rel=&quot;nofollow&quot;&gt;a free pdf earlier version of this text &lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="5090" LastActivityDate="2015-04-21T18:52:13.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7941" PostTypeId="2" ParentId="7918" CreationDate="2015-04-21T23:33:33.533" Score="13" Body="&lt;p&gt;The &lt;strong&gt;AC-method&lt;/strong&gt; is often presented as an unmotivated rule-based algorithm, e.g. &lt;a href=&quot;http://www.regentsprep.org/regents/math/algtrig/atv1/Ltri3.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here.&lt;/a&gt; If that is the way that you learned it, then I can understand why you hate it. Below I explain the (little-known) algebraic ideas that lie at the heart of the algorithm, e.g. I emphasize how the non-monic factorization algorithm can be viewed as a &lt;em&gt;conjugation&lt;/em&gt; of the monic factoring algorithm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Understanding these underlying conceptual ideas may greatly assist in constructing more motivated presentations of the method. Further, it may help you to develop a more positive view of the method after learning that there is some interesting mathematics behind the method (it is intimately connected to various &lt;em&gt;refinement-based&lt;/em&gt; views of factorization and related lattice-theoretic ideas that play key roles in noncommutative generalizations - see below).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's examine the classical case, which reduces factorization of general &lt;em&gt;quadratic&lt;/em&gt; polynomials to the simpler special case when the polynomial is $\rm\color{#c00}{monic}\,$ (lead coeff $\color{#c00}{=1})$ as follows&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ {\begin{eqnarray}&#xA;\rm\: a\:f(x)\:\! \,=\,\:\! a\:(a\:x^2 + b\:x + c)\,  &amp;amp;\,=\,&amp;amp;\rm\,\ \color{#c00}{X^2}\, +\, b\:X\, +\, \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\overbrace{\color{#0a0}{ac}}^{\rm\qquad\ \ \ \ \ {\bf\color{#0a0}{AC}-method}}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! =\, g(X),\ \ \ X = a\:x \\[.5em]&#xA;\rm e.g.\qquad\qquad\ \ \ \ &#xA;f \, &amp;amp;\,=\,&amp;amp;\rm \ \,  2\ x^2-\ 3\ x\ +\  1\\&#xA;\rm \Rightarrow\ \ 2\,f\, &amp;amp;\,=\,&amp;amp;\rm\!\ (2x)^2\! -3(2x)\!+\!2\\&#xA;         &amp;amp;\,=\,&amp;amp;\rm \ \ \   \color{#c00}{X^2}-\, 3\  X\ +\ 2,\,\ \ X\, =\, 2x\\&#xA;         &amp;amp;\,=\,&amp;amp;\rm  \ \ (X-2)\ (X-\,1)\\&#xA;         &amp;amp;\,=\,&amp;amp;\rm \  (2x-2)\,(2x-1)\\&#xA;\rm \Rightarrow\ \  f\:=\: 2^{-1}\,(2\,f)\,  &amp;amp;\,=\,&amp;amp;\rm   \  \ \, (x- 1)\ (2x-1)\\&#xA;\end{eqnarray}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we denote our factoring algorithm by $\,\cal F\ $ then the above transformation is simply&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\rm {\cal F}\ f\  =\ a^{-1}{\cal F}\, a\ f\qquad\! $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus we've transformed by &lt;em&gt;conjugation&lt;/em&gt; $\rm\ {\cal F} = a^{-1} {\cal F}\ a\ \,$ the problem of factoring non-monic polynomials into the simpler problem of factoring monic polynomials. It works for any degree:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{eqnarray}\phantom{1^{1^{1^1}}}\!\!\!\!\!\!\rm a\:f(x)\:\! \,=\,\:\! a\:(a\:x^2 + b\:x + c)  \!&amp;amp;\,=\,&amp;amp;\!\!\rm\: X^2 + b\:X + \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\smash[t]{\overbrace{\color{#0a0}{ac}}^{\rm\qquad\ \ \ \ \ {\bf\color{#0a0}{AC}-method}}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! =\, g(X),\ \ \ X = a\:x \\[5px]&#xA;\rm\: a^{n-1}(a\:x^n\! + b\:x^{n-1}\!+\cdots+d\:x + c)\! &amp;amp;\, =\,\,&amp;amp;\!\!\rm\: X^n\! + b\:X^{n-1}\!+\cdots+a^{n-2}d\:X + \color{#0a0}{a^{n-1}c}&#xA;\end{eqnarray}\qquad\quad $$ &#xA;After factoring the monic $\rm\,g(X)\, =\, a^{n-1}f(x),\,$ the transformation must reverse to yield a factorization of $\rm\:f,\, $ because $\rm\ a^{n-1}$ must divide into the factors of $\rm\ g\ $ by invoking &lt;a href=&quot;https://math.stackexchange.com/a/2167166/242&quot;&gt;Gauss' Lemma,&lt;/a&gt; i.e. primes  $\rm\,p\in\rm\mathbb Z\,$ remain prime in $\rm\,\mathbb Z[X],\,$ so $\rm\ p\ |\ g_1(x)\:g_2(x)\,$ $\Rightarrow$ $\,\rm\:p\:|\:g_1(x)\:$ or $\rm\:p\:|\:g_2(x).$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The AC-method also works for &lt;em&gt;multivariate&lt;/em&gt; polynomial factorization, e.g below we combine it with $\color{#0a0}{\text{completing a product,}}$ in a &lt;a href=&quot;http://math.stackexchange.com/a/651073/242&quot;&gt;similar way to completing a square&lt;/a&gt; (from &lt;a href=&quot;https://math.stackexchange.com/q/1828554/242&quot;&gt;this MSE question&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \begin{eqnarray}  &amp;amp;&amp;amp;\rm \ \ \ a\ x\ y &amp;amp;+&amp;amp;\rm b\ x&amp;amp;+&amp;amp;\rm c\ y &amp;amp;=&amp;amp;\rm\ \  d\\&#xA;\smash{ \overset{\times\ a}\iff} &amp;amp;&amp;amp;\rm \ \ ax\,ay &amp;amp;+&amp;amp;\rm b\,ax &amp;amp;+&amp;amp;\rm c\,ay &amp;amp;\rm\,=&amp;amp;\rm ad\\&#xA;\iff  &amp;amp;&amp;amp;\rm \ \ \ {X\ Y} &amp;amp;+&amp;amp;\rm {b\,X} &amp;amp;+&amp;amp;\rm {c\,Y} &amp;amp;=&amp;amp;\rm ad,\quad X = ax,\ \ Y = ay\\&#xA;\iff &amp;amp;&amp;amp;\rm  \color{#0a0}{(X\!+\!c)}\!\!&amp;amp;&amp;amp;\rm\!\!\!\!\!\! \color{#0a0}{(Y\!+\!b)}&amp;amp;\color{#0a0}-&amp;amp;\rm \color{#0a0}{bc} &amp;amp;=\,&amp;amp;\rm ad,\quad {\rm by\ \ \color{#0a0}{completing\  the\  product}}\\&#xA;\iff &amp;amp;&amp;amp;\rm (ax\!+\!c)\!\!&amp;amp;&amp;amp;\rm\!\!\!\!\!\!(ay\!+\!b)\!\! &amp;amp;&amp;amp;\rm &amp;amp;=&amp;amp;\rm ad\!+\!bc&#xA;\end{eqnarray}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now it's easy to find all integer roots $\rm x,y$ by spliting $\rm ad\!+\!bc$ into two factors and checking if they have the form on the LHS. There are only finitely many such splittings so it is a finite process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Readers familiar with university algebra may wonder how this relates to better known factorization properties. The AC-method works not only for UFDs and GCD domains, but additionally for any integrally-closed domain satisfying the&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad\qquad$ &lt;a href=&quot;https://math.stackexchange.com/a/63330/242&quot;&gt;Primal Divisor Property&lt;/a&gt;  $\rm\ \ c\ |\ AB\ \ \Rightarrow\ \ c = ab,\ \ a\ |\: A,\ \ b\ |\ B$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Elements $\rm\,c\,$ satisfying this are called &lt;em&gt;primal.&lt;/em&gt; One easily checks that atoms are primal $\!\iff\!$ prime. Further, products of primes are primal. So &quot;primal&quot; may be viewed as a generalization of the notion &quot;prime&quot; from atoms (irreducibles) to composites.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Integrally closed domains whose elements are all primal are called  &lt;em&gt;Schreier rings&lt;/em&gt; by Paul Cohn (or &lt;em&gt;Riesz domains&lt;/em&gt;, because they satisfy a divisibility form of the Riesz interpolation property). In Cohn's &lt;a href=&quot;http://www.lohar.com/researchpdf/bezout_rings_and_their_subrings.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bezout rings and their subrings&lt;/a&gt;&#xA;he proved that if $\rm\:D\:$ is Schreier then so too is $\rm\,D[x],\:$ by using a primal analogue of Nagata's Lemma (an atomic domain $\rm\:D\:$ is a UFD if some localization $\rm\:D_S\:$ is a UFD, for some monoid $\rm\:S\:$ generated by primes). These primal and Riesz interpolation viewpoints come to the fore in a &lt;em&gt;refinement&lt;/em&gt; forms of unique factorization. They prove especially fruitful in  noncommutative rings (see Cohn's 1973 &lt;em&gt;Monthly&lt;/em&gt; survey &lt;a href=&quot;http://www.jstor.org/stable/2319253&quot; rel=&quot;nofollow noreferrer&quot;&gt;Unique factorization domains&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schreier domains can be characterized equivalently by a suitably formulated version of the above &quot;factoring by conjugation&quot; property. Curiously, this connection between this elementary AC method and Schreier domains appears to have gone unnoticed in the literature. &lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2018-06-28T00:02:37.290" LastActivityDate="2018-06-28T00:02:37.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7942" PostTypeId="2" ParentId="7936" CreationDate="2015-04-22T00:37:52.580" Score="6" Body="&lt;p&gt;On the axiomatic front, you might find the axiomatization of Desarguesian planes useful for your project. A very nice exposition is chapter 2 of Emil Artin's &quot;Geometric Algebra&quot;, which I'll now summarize.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The undefined notions are points, lines, and an incidence relation between them. The axioms for an affine plane are&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Through any two points there is a unique line.&lt;/li&gt;&#xA;&lt;li&gt;Through any point not on a line there is a unique parallel line through that point.&lt;/li&gt;&#xA;&lt;li&gt;There exist three points not all on the same line.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Using these three axioms (the second really) one can show that being parallel is an equivalence relation on lines. One can then define an &lt;em&gt;affine transformation&lt;/em&gt; $T$ to be one that preserves this relation, i.e. if $\ell_1\parallel \ell_2$, then $T\ell_1\parallel T\ell_2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More significant are the affine transformations that leave the parallelism class invariant, i.e. such that $\ell\parallel T\ell$. Artin calls such affine transformations &lt;em&gt;dilatations&lt;/em&gt;. This is because dilatations are uniquely determined by where they send two points, as then the location of the third point is uniquely determined by the parallel postulate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consequently, he defines a &lt;em&gt;translation&lt;/em&gt; to be a dilatation that has no fixed points, and a &lt;em&gt;dilation&lt;/em&gt; to be a dilatation that has one fixed point. One can show that dilations with no fixed points are uniquely determined by where the send a single point. This allows us to visualize the action of translations on the plane as arrows: an arrow for a translation indicates that the translation takes the starting point to the ending point. It is instructive to prove that any two non-collinear arrows that come from a translation can be realized as opposite sides of a parallelogram. This is the affine version of the statement that translations are specified by the data of a magnitude and direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We do not yet know that any arrow comes from a translation, however. This is precisely the content of axiom 4a of Desarguesian' geometry.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;4a. Desargues' theorem for parallel lines. Given three distinct parallel lines $\ell_1\parallel\ell_2\parallel\ell_3$, suppose we have points $A_i,B_i$ on $\ell_i$ so that $A_1A_2\parallel B_1B_2$, $A_2A_3\parallel B_2B_3$. Then $A_1A_3\parallel B_1B_3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that if the arrow $\vec{A_2B_2}$ were to come from a translation, then that translation would have to send $A_1$ to $B_1$ and $A_3$ to $B_3$. But this is actually a dilatation if and only if $A_1A_3\parallel B_1B_3$, which is exactly what this version of Desargues' theorem asserts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4b. Desargues' theorem for concurrent lines. Given three distinct concurrent lines $\ell_1,\ell_2,\ell_3$ with common point $O$, suppose we have points $A_i,B_i$ on $\ell_i$ so that $A_1A_2\parallel B_1B_2$, $A_2A_3\parallel B_2B_3$. Then $A_1A_3\parallel B_1B_3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This second version of Desargues' theorem says that giving the points $O,A_1,A_2$ does in fact specify a dilation fixing $O$ and sending $A_1$ to $A_2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An affine plane satisfying axioms 4a and 4b is called a &lt;em&gt;Desarguesian plane&lt;/em&gt;. These are significant because the translations plus the identity dilatation form an abelian group, and conjugating these by a dilation results is an endomorphism of this abelian group that preserves the trace lines of the translations. As Artin describes in detail, together with choosing the three non-collinear points $O$, $A$, $B$ from Axiom 3, this establishes a correspondence between Desarguesian planes and $2$-dimensional modules over division rings. In other words, the 5 axioms above are all you need in order to relate synthetic geometry to coordinate geometry.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Artin does further cover the fact that requiring an order relation on the points on the line of a geometry corresponds to requiring that we have an ordered division ring, which ends up being an ordered field, so something inbetween $\mathbb Q$ and $\mathbb R$. What Artin doesn't do is discuss lengths, congruences, and similarities. I worked out some of this some time ago, but I'm not full satisfied with it. In any case, the relevant axioms for lengths and angles are &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A. A translation-invariant distance function satisfying the triangle inequality (i.e. a norm on the vector space of translations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B. The axiom that across any line there is a distance-preserving reflection, where a reflection is an involutive affine map fixing that line. This establishes the necessary relation of perpendicularity (equivalent to a positive definite inner product on the vector space of translations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may seem strange, but remember that the affine plane is still the affine plane even if you look at it at an angle. Then what used to be circles now look like ellipses, but all of geometry still holds. You can get rotations as composed reflections, and this is enough to recover traditional side-angle-side congruence and angle-angle-angle similarities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. Vectors are translations.&lt;/p&gt;&#xA;" OwnerUserId="1308" LastActivityDate="2015-04-22T00:37:52.580" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7943" PostTypeId="2" ParentId="7865" CreationDate="2015-04-22T03:11:23.227" Score="4" Body="&lt;p&gt;Perhaps an approach that mirrors the standard graphing might be useful? One way that I appreciate is seen here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/T1Jij.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is used by N. J. Wildberger and others. I just snagged this off google images to demonstrate. I think this particular image is of $F_{13}$ with two &quot;lines&quot; plotted and their intersection marked at (5,2) -- but don't quote me on that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can generate some more examples if anyone is interested. It's fascinating to look at conics in these situations!&lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2015-04-22T03:11:23.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7944" PostTypeId="1" CreationDate="2015-04-22T05:53:04.527" Score="5" ViewCount="176" Body="&lt;p&gt;The textbook I'm using explains the it is &quot;impractical&quot; to have a normal distribution for all values of $\sigma$ and $\mu$, hence the standard normal distribution table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, this simply doesn't ring true. We could have a phone app, to generate a distribution for $\sigma$ and $\mu$, then highlight the desired area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://homepage.stat.uiowa.edu/~mbognar/applets/normal.html&quot;&gt;http://homepage.stat.uiowa.edu/~mbognar/applets/normal.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P(x)=\frac{1}{\sigma \sqrt{2\pi}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, is the standard normal distribution important beyond being a time-saving calculation tool? There are so many important concepts in statistics and so little time that it feels like a wasted opportunity to teach outdated calculation methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;HOWEVER: &#xA;Until there is a revision in the way that we teach elementary statistics, my students must learn these methods.  The standard normal distribution is ubiquitous in math and science education. Students may need to learn about it to understand how other people may talk about statistics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What should be the future of paper tables?&lt;/p&gt;&#xA;" OwnerUserId="5080" LastActivityDate="2015-04-23T03:13:35.097" Title="Is there a case for ditching any of the paper tables in elementary statistics courses?" Tags="|statistics|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7945" PostTypeId="2" ParentId="7944" CreationDate="2015-04-22T11:13:44.620" Score="3" Body="&lt;p&gt;As @kjetilbhalvorsen mentioned, tables are useful useful for old style pen and paper exams. They should be readily duplicated in a decent graphics calculator, but there are more important reasons that the days of the paper normal distribution table are dated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, the validity of using an exam to asses statistics has been largely jettisoned in some education systems, even where it is retained for other areas of maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, I find the conceit of using tables to get accurate values for a normal distribution, while the normal distribution itself is usually only a very rough approximation of the real life situation to be bizarre. IMHO a course specification asking that a rough approximation be applied to 3 sig.fig. is indicative of deep conceptual deficiencies in traditional statistics education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The heuristic usefulness of normal distributions can be provided by a few critical values for hypothesis testing, without the use of tables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you suggested, there are much more important things to learn.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-04-22T21:31:23.847" LastActivityDate="2015-04-22T21:31:23.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7946" PostTypeId="1" AcceptedAnswerId="7948" CreationDate="2015-04-22T13:04:52.380" Score="8" ViewCount="306" Body="&lt;p&gt;Student question to me involved &quot;standard&quot; form of radical expressions.  They already know what a radical in simplest form (no perfect square factors, no fractions, no radicals in denominator).  My thoughts after combing my literature was to tell him that standard form of a radical expression was something like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$p+\sum_{k=1}^n{q_k\sqrt{a_k}}$$&#xA;where $p,q_i$ are rational numbers and $a_1&amp;lt;a_2&amp;lt;...&amp;lt;a_n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like this would answer the question of &quot;standard&quot; while keeping the idea simple.&lt;/p&gt;&#xA;" OwnerUserId="4728" LastEditorUserId="4728" LastEditDate="2015-04-22T13:21:39.137" LastActivityDate="2015-04-22T20:17:13.853" Title="Simple question on radical expressions" Tags="|mathematical-pedagogy|secondary-education|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7947" PostTypeId="2" ParentId="7438" CreationDate="2015-04-22T13:50:43.557" Score="1" Body="&lt;p&gt;I hate answering my own question, but thought I should let others who find this know what I decided on:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://faculty.uml.edu/klevasseur/ads2/&quot; rel=&quot;nofollow&quot;&gt;&quot;Applied Discrete Structures&quot; by Alan Doerr and Kenneth Levasseur&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reasons for picking it: Light on proofs (but they're still there), good on computation problems, free online text.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-04-22T13:50:43.557" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7948" PostTypeId="2" ParentId="7946" CreationDate="2015-04-22T16:20:41.603" Score="4" Body="&lt;p&gt;The form you propose seems  reasonable to me (given your constraints). &#xA;The rational term at the start could be thought of as corresponding to $\sqrt{1}$ and then the numbers in the square-roots increase. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree with commenters that too much emphasis on &quot;standard&quot; forms might not be good, but when pressed to give an answer I would go with what you propose. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing to consider, though, is that $\sqrt{3}- \sqrt{2}$ is arguably more pleasant than $- \sqrt{2}+\sqrt{3}$, but after all we are seeking a &lt;em&gt;standard&lt;/em&gt; form not an optimal form, so this is not that much of a problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me also add that it makes sense to talk about a standard form in this case as given (distinct) square-free $a_k$ the coefficients $q_k$ (and $p$) will be uniquely determined by the sum; in other words it is not possible to have $p+\sum_{k=1}^n{q_k\sqrt{a_k}} = p'+\sum_{k=1}^n{q_k'\sqrt{a_k}}$ unless $p=p'$ and $q_k=q_k'$ for all $k$.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2015-04-22T16:31:26.717" LastActivityDate="2015-04-22T16:31:26.717" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7949" PostTypeId="2" ParentId="7946" CreationDate="2015-04-22T17:21:47.943" Score="3" Body="&lt;p&gt;There is no &lt;em&gt;standard&lt;/em&gt; form for such expressions. However, some forms may prove more &lt;em&gt;convenient&lt;/em&gt; for certain applications, e.g. transforming the radicands to be squarefree helps to ascertain multiplicative independence, hence linear independence (cf. &lt;a href=&quot;https://math.stackexchange.com/a/30707/242&quot;&gt;this theorem).&lt;/a&gt; But even that is not necessary since it can also be tested without squarefree normalization by instead verifying that no subset of the radicals has rational product.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact that there is no computationally effective normal form for such expressions is a reflection of the innate complexity of their arithmetic. The complexity of even simpler problems involving&#xA;radicals is currently unknown. For example, no polynomial time&#xA;algorithm is known for determining the sign of a sum of real&#xA;radicals $\,\sum c_i q_i^{1/r_i},\,$ where $\,c_i, q_i\,$ are rationals and&#xA;$\,r_i\,$ is a positive integer. Such sums play an important role in&#xA;various geometric problems (e.g. Euclidean shortest paths and&#xA;traveling salesman tours). Testing whether such a sum&#xA;of radicals is zero can be decided in polynomial time, but this is&#xA;of no help in determining the sign (it only shows that if sign&#xA;testing is in NP then it is already in NP $\cap $ co-NP).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally the context will determine what type of normal or standard forms prove most convenient. For example, denesting of radicals is governed by a structure theorem that employs a certain normalized form, e.g.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \sqrt{\sqrt[3]5 - \sqrt[3]4} \;\;=\; \frac{1}3 (\sqrt[3]2 + \sqrt[3]{20} - \sqrt[3]{25})$$&#xA;normalises to $$ \sqrt{18\ (\sqrt[3]10 - 2)} \;\;=\; 2 + 2\ \sqrt[3]{10} - \sqrt[3]{10}^2 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$  \sqrt{12 + 5\ \sqrt 6} \;\;=\; (\sqrt 2 + \sqrt 3)\ 6^{1/4} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;normalises to &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \sqrt{\frac{1}3 \sqrt{6}\: (12 + 5\ \sqrt 6)} \;\;=\; 2 + \sqrt{6} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/194148/242&quot;&gt;See here&lt;/a&gt; for more on this Denesting Structure Theorem.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-04-22T17:35:21.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7950" PostTypeId="2" ParentId="7946" CreationDate="2015-04-22T20:17:13.853" Score="0" Body="&lt;p&gt;For more information about nested and how to &quot;denest&quot; radicals, with references to Susan Landau's work on this topic consult:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Nested_radical&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Nested_radical&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2015-04-22T20:17:13.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7951" PostTypeId="2" ParentId="7944" CreationDate="2015-04-23T02:50:40.377" Score="4" Body="&lt;p&gt;This answer is not necessarily specific to statistical tables, but to table reading in general.  I basically learned how to read and understand all types of tables including spreadsheets, train tables, and various mathematical tables by analyzing a physical table in my high school statistics class.  The visual aspect of both the standard normal distribution, which most people will recognize without even knowing exactly what it is, and the layout of the table made it very easy for me to understand how the data was structured and organized.  We had graphing calculators that could do it, but after some practice I actually found using the tables in the back of the textbook for quick check ups.  I believe there is a lot of value in being able to read tables and the normal distribution is a great entry point into being able to really understand how to read them, and more importantly how to make them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as pedagogical practice, I would suggest to let your students choose which to use on a day-to-day basis, but I would defiantly push them to at least be comfortable with both methods for the sake of testing etc.  One thing that immediately comes to mind is to set up a weekly &quot;competition&quot; where students race using one of the two methods to solve a problem.  Having them switch between the two each week could allow them to try to get Personal Bests and would be a great way to have them make a table to keep track of their own times :)&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-04-23T02:50:40.377" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7952" PostTypeId="2" ParentId="7944" CreationDate="2015-04-23T03:13:35.097" Score="2" Body="&lt;p&gt;The standard normal distribution is very important concept that is completely separate from the standard normal table. The standard normal allows us to compare unlike variables. A typical example is comparing an athlete in one sport to another. You can understand and use the standard normal distribution without relying on the tables to do this. It is for this reason that I no longer paper tables and instead I have chosen have my students use the following normal distribution calculator:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://davidmlane.com/hyperstat/z_table.html&quot; rel=&quot;nofollow&quot;&gt;http://davidmlane.com/hyperstat/z_table.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe the table itself gets in the way of students seeing the big picture of what they are doing in using the normal distribution. The fact that the table only gives P(z&amp;lt;=Z) means extra computation steps that are basically irrelevant to understanding how the normal distribution works. It is another typical statistics example where computation obfuscates what is actually going on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To address concerns about students using apps on quizzes, I have designed my quizzes to focus on interpretation rather than computation. I will only ask students to compute values for labs and projects. This makes it so that apps are not necessary on quizzes and so I can evaluate my students ability to understand the meaning of important ideas. I have found that when computation is necessary, I can ask students to apply the empirical rules to demonstrate understanding. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A final note: This choice also depends on what your students' next courses will be. I teach an introductory high school course. I expect that my students will take an introductory college course. If that course does not use a table, great! If it does, they will have adequate conceptual preparation that will prepare them to learn it during that class. &lt;/p&gt;&#xA;" OwnerUserId="3485" LastActivityDate="2015-04-23T03:13:35.097" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7953" PostTypeId="1" CreationDate="2015-04-23T05:44:31.137" Score="13" ViewCount="194" Body="&lt;p&gt;My class is composed of international high school students coming from different countries and backgrounds. Hence different skills and needs. Our lingua franca is English and it is important to note that, in my school, no student (and actually, except for the English teachers, no teacher) is a native English speaker. They have intensive training in English, so after three years, they are fluent in English.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is about maths in English language for freshmen (around 15 years old teenagers): coming from environments with different expectations and emphasis on English education, their level vary greatly, from students that speak with ease to almost mute students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question&lt;/strong&gt;: some students asked me to use a dictionary during examination. There is no policy about this in my school; how should I consider that demand?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My current answer was: electronic dictionaries/translators are forbidden, since I cannot check whether they can use it to cheat or not. I allowed a handwritten piece of paper (precisely, A5 paper, both sides) with two columns, one with English words, and the other with their translation in the language they want. No sentence or formula is allowed. The idea is that, by actively writing their own dictionary, they will memorize vocabulary. It also help them to analyze the course material and extract the important notions from it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any smarter way to deal with this situation? I have a limited sample to test the method but it seems to work. I would be interested in other ideas to improve the efficiency of learning mathematic terminology in English. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, is teaching the use of logic and set theory symbols in a more extensive way than the usual textbooks do a good idea? My intuition is that, if they deeply assimilate logical symbols, relying this assimilation to their mother language, I could use more symbolic notations in class, and they will interpret symbols in their own language. In a sense, they will develop a “mathematics-mother language” dictionary instead of a “Mathematics English-mother language” dictionary. The risk is that, if they have troubles understanding these notations, the accumulation of it will slow down their understanding of other notions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is not exactly about Math Education per se, but I believe there are maths-specific answers.&lt;/p&gt;&#xA;" OwnerUserId="1538" LastEditorUserId="262" LastEditDate="2015-04-24T21:38:57.787" LastActivityDate="2015-04-24T21:38:57.787" Title="Mathematics dictionary use for English Language Learners" Tags="|language-use|english-language-learners|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7954" PostTypeId="1" CreationDate="2015-04-23T05:51:57.287" Score="23" ViewCount="409" Body="&lt;p&gt;I am teaching in an Eastern Asian environment (precisely, teaching Mathematics using English in Korea, with Asian students) and I figured out that my reasoning is a lot based on my language proficiency in English and latin languages. As a latin languages speaker, mistakes in the difference between definite article “the” and indefinite article “a” hurt me a lot, whereas these articles do not exist in my students' mother language (at least, in Korean). I also have a quite efficient writing style for proofs, organizing my reasoning on the multiple logical connectors that Western Europe languages provide: so, then, hence, therefore, as a consequence,...; the actual content of my sentences being as simple as possible to convey the mathematical content in the most straightforward way. More elaborated sentences are used for broad introduction of ideas, and conclusions. But other languages have a completely different structure. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be interested in references on how the structure of a language influences the logical reasoning. For example, I believe that the importance given on existence problems and unicity problems is mostly due to the emphasis on definite and indefinite articles in Western languages. When introducing a new mathematical object in class, I feel very uncomfortable having to say “a” until unicity is proved. For practical reasons, I am mostly interested in Asian languages (mainly Korean, and also Chinese and Japanese), but references for other languages may also be useful.&lt;/p&gt;&#xA;" OwnerUserId="1538" OwnerDisplayName="Taladris" LastActivityDate="2017-05-04T21:56:36.860" Title="Mathematic reasoning in nonEnglish/non Western languages" Tags="|reference-request|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="7955" PostTypeId="2" ParentId="7936" CreationDate="2015-04-23T13:08:16.880" Score="2" Body="&lt;p&gt;I would axiomatize affine geometry using the single function &quot;go $x$% of the way from $P$ to $Q$&quot;, where $x$ is a real number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can get a feel for this approach in a paper by Suppes available &lt;a href=&quot;https://suppes-corpus.stanford.edu/articles/ll/385.pdf&quot; rel=&quot;nofollow&quot;&gt;online&lt;/a&gt;.  He also refers to a book which may be your best source:  Szmielew's &quot;From Affine to Metric Geometry&quot;.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2015-04-23T13:08:16.880" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7956" PostTypeId="2" ParentId="7953" CreationDate="2015-04-23T19:10:04.870" Score="3" Body="&lt;p&gt;If possible, you should poll department members on this: whatever policy you come up with, you and the department will have to answer for it.  If they can't help you, then take the time to register your actions with the administration (cover your work-related behind).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have a somewhat free hand and understanding/permissive administration, you might try the following: crowd-source a cheat sheet.  Have several students submit the phrases and the one or two languages ahead of time, and then compile and handout for the exam the result.  If something is lacking, allow them a 3x5 notecard (or whatever is appropriate) to account for discrepancy, but ideally everyone uses the same cheat sheet.  If some of them turn in their part two weeks before the exam, that should give enough time to produce and show them the sheet before the exam.  Bonus: you can use this with few modifications for next class.  You can try a competition format, use the results to analyze student understanding, or spin this several ways for&#xA;your and the class benefit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Isn't Crowd-Sourcing A Great Idea?&quot; Paseman, 2015.04.23&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-23T19:10:04.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7959" PostTypeId="1" AcceptedAnswerId="8434" CreationDate="2015-04-25T07:28:01.800" Score="14" ViewCount="189" Body="&lt;p&gt;I'm currently involved in developing materials for a new UK tier of examination known as Core Maths.  The course is designed for 16-18 year olds to further their mathematics education but without taking the traditional A level maths (which is seen as the prerequisite for university) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be grateful for any suggestions for lesson ideas or resources that cover&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Exponentials growth decay in graph or otherwise&lt;/li&gt;&#xA;&lt;li&gt;Introduction to logarithmic scales&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The students taking this course are doing so to support their other academic studies.  It would appropriate therefore to assume only basic knowledge.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For my part (and to give a flavour of what I'm after) I am making a card sort activity involving the graphs of $y=2^x$, $y=3^x$ and $y=2^{-x}$ etc.  Students to compare and contrast formulating generalisations.  If you include the source I will make sure the originator is fully accredited. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not gaining financially (the school I work for was selected to develop good practice).&lt;/p&gt;&#xA;" OwnerUserId="4668" LastEditorUserId="1147" LastEditDate="2015-07-26T12:23:19.003" LastActivityDate="2015-07-26T12:23:19.003" Title="Resources suitable for a beginners' course with exponentials" Tags="|secondary-education|functions|logarithm|" AnswerCount="1" CommentCount="5" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7960" PostTypeId="2" ParentId="7438" CreationDate="2015-04-25T18:05:46.693" Score="2" Body="&lt;p&gt;While not, strictly speaking, a textbook, Phillips Exeter Academy publishes their &lt;a href=&quot;https://www.exeter.edu/academics/72_6539.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;full problem set&lt;/a&gt; for their non-proof based discrete mathematics course (which they offer to advanced high school students).  For more on how they use the problem set in their course, see my answer to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/4071/discovery-based-and-inquiry-based-learning/7939#7939&quot;&gt;this question&lt;/a&gt;.  Re: your &quot;Update,&quot; these problems do have the advantage of having no available solution set.&lt;/p&gt;&#xA;" OwnerUserId="5090" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-04-25T18:05:46.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7961" PostTypeId="1" CreationDate="2015-04-26T01:41:47.627" Score="5" ViewCount="355" Body="&lt;h1&gt;Background&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;I am trying to solve the following problem and explaining it to my students. However, my students (as well as I) think of my two assumptions contradictory.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&gt;&#xA;  Given 2 distinct curves $C_1: y=f(x)=e^{6x}$ and $C_2: y=g(x)=ax^2$ where $a&amp;gt;0$. The objective is to find the range of $a$ such that there exists 2 tangents, each is tangent to both given curves.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have solved it as follows.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Let $(t,e^{6t})$ be &quot;a single point&quot; on $C_1$ through which a tangent passes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The slope of the tangent is $6e^{6t}$ that is obtained from the first derivative of $f(x)=e^{6x}$ at $x=t$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The tangent is&#xA;$$&#xA;y-e^{6t} = 6e^{6t}(x-t)&#xA;$$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let $(u,au^2)$ be a point on $C_2$ through which the tangent passes. It means that the tangent and $C_2$ have only one intersection point  $(u,au^2)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have,&#xA;$$&#xA; au^2 -e^{6t} = 6e^{6t}(u-t)&#xA; $$&#xA;By arranging it, we have a quadratic equation in $u$.&#xA;$$&#xA; au^2 -6e^{6t}u + e^{6t}(6t-1) =0&#xA; $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to make the quadratic equation have twin roots, its discriminant must be equal to zero as follows.&#xA;$$&#xA;(-6e^{6t})^2-4\times a\times e^{6t}(6t-1) =0&#xA;$$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There will be 2 tangents if there are 2 roots $t_1$  and $t_2$ for the equation in $t$ below.&#xA;$$&#xA;(-6e^{6t})^2-4\times a\times e^{6t}(6t-1) =0&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can interpret finding the roots of the equation as finding the abscissa $t$ of the intersection of $\alpha(t)=a$ and $\beta(t)=\frac{9e^{6t}}{6t-1}$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using derivative, we can determine the minimum value of $\beta$ which is $9e^2$. Thus $a&amp;gt;9e^2$ is the required range.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;h1&gt;My confusion&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;In the first step I must assume there is a single $t$ but in the last (the 3rd step) step I have to assume there are 2 roots $t_1$ and $t_2$. It looks inconsistent, doesn't it? How can we explain it better without causing such a confusion?&lt;/p&gt;&#xA;" OwnerUserId="5105" LastEditorUserId="5105" LastEditDate="2015-04-26T01:57:28.397" LastActivityDate="2015-04-27T21:13:09.263" Title="My two assumptions look inconsistent. How can we explain it better?" Tags="|secondary-education|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7962" PostTypeId="2" ParentId="5676" CreationDate="2015-04-26T11:21:41.443" Score="6" Body="&lt;p&gt;A) First, let's explore the problem in more detail:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difficulty with both the presented scenarios is that they only decide to perform a test after noticing something about the data -- and then they test the hypothesis that the data led them to on the same data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When this happens, the test no longer has its nominal properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To extend the more &quot;boring&quot; example (and I do it because the probability model is less ambiguous), imagine that ten thousand students each carry our the experiment &quot;roll a six-sided die twenty times&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further imagine that if a number comes up eight times or more, the students will says &quot;that seems unfair&quot; and so test it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's ask ourselves two questions &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(i) If the dice are all fair, what proportion of students will say &quot;that seems unfair&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(ii) For the students who carry out a test, what's their type I error rate? (i.e. what proportion will reject the null?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For (i) I just did three sets of simulations of 10000 such students, and got 6.74%, 6.93% and 6.62% getting enough of one face to say &quot;that doesn't seem fair&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For (ii), we have to figure out what test they'll apply and what the nominal significance level they conduct it at might be. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I presume the students are specifically testing for too many or too few of the particular face they noticed a problem with. The sample size is too small to use a normal approximation on a proportions test; if we apply a binomial test, a two tailed test can't be symmetric (equal probability in each tail), but we can get 2.6% in the lower tail and 1.1% in the upper tail by rejecting with 0 or 8+ on the face tested (even though none of the people who test will actually have a zero, since our scenario didn't have students with 0 for a face trying to test it). So the overall nominal significance level is 3.7%.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what's the actual rejection rate for people who carry out the test (the true type I error rate)? Well, under our rejection rule they all reject ... so it's 100%!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recall working on a real-world problem like your scenario for someone once; the person's employer conducted random drug testing of drivers, and the particular person had been tested very many times -- he was certain another employee was making trouble for him, leading him to be tested much more often than randomly. If you work out the &lt;em&gt;a priori&lt;/em&gt; probability &lt;em&gt;he&lt;/em&gt; would be tested that many times, it seemed like something odd was going on. But if you worked out the probability that &lt;em&gt;some&lt;/em&gt; employee would be tested that many times, it turned out to be not at all surprising. And of course to that employee it would look strange, even though nothing need be going on. ... and of course the unluckiest of all the (&gt;100) employees would be the one asking what was up.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;B) So what can be done? The easiest way is to slightly modify the example. Instead of saying 'that doesn't seem fair, let's test that data', we instead say 'that doesn't seem fair, let's specify exactly what we want to test, and then get some new data to test it with'. Then your examples will work fine!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A store manager assigns shifts to six different employees, including you. After a month on the job, you get the impression that you're getting the &quot;bad&quot; shift way too often. The manager claims that he assigns shifts randomly, namely that each employee has an equal chance of getting assigned the &quot;bad shift.&quot; ... so you decide to collect data for the next 20 work days, and you find you're assigned the bad shift eight time. Complete a hypothesis test to assess this claim.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A similar modification can be made to the dice scenario&lt;/p&gt;&#xA;" OwnerUserId="755" LastEditorUserId="755" LastEditDate="2015-04-27T14:57:59.977" LastActivityDate="2015-04-27T14:57:59.977" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7963" PostTypeId="1" CreationDate="2015-04-26T12:18:03.110" Score="5" ViewCount="802" Body="&lt;p&gt;Is there a common and &lt;strong&gt;simple&lt;/strong&gt; way of assigning points for a test item based on the percent of students who answered the item correctly ($p$-value)? For simplicity, let us disregard for a moment the practicality (e.g., time and effort) needed and  the pedagogical consequences of grading. Let us just imagine that we are constructing the scoring system in a math competition where difficult questions are rewarded more points, where the difficulty is based on the $p$-value. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Remark: In a math competition, as opposed to a classroom setting, making the students compete for points is encouraged. So finding such a scoring system can be valuable.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, consider this hypothetical five-item test and their $p$-values. &lt;br&gt;&#xA;Item 1: 100% &lt;br&gt;&#xA;Item 2: 80% &lt;br&gt;&#xA;Item 3: 50% &lt;br&gt;&#xA;Item 4: 20% &lt;br&gt;&#xA;Item 5: 5%&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What immediately comes to mind is to assign $\frac{1}{x}$ points for an item with a $p$-value of $x$. So the scores will be &lt;br&gt;&#xA;Item 1: 1 &lt;br&gt;&#xA;Item 2: 1.25 &lt;br&gt;&#xA;Item 3: 2 &lt;br&gt;&#xA;Item 4: 5 &lt;br&gt;&#xA;Item 5: 20&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is apparent that the number of points dramatically increases as the $p$-value approaches 0. We can of course introduce immediate modifications such as getting the square root for instance, but that seems too artificial. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: The Harvard-MIT Mathematics Tournament offers such a &lt;a href=&quot;http://hmmt.mit.edu/static/scoring_technical.pdf&quot; rel=&quot;noreferrer&quot;&gt;scoring system&lt;/a&gt;. Based on the &lt;a href=&quot;http://hmmt.mit.edu/static/scoring_overview.pdf&quot; rel=&quot;noreferrer&quot;&gt;overview&lt;/a&gt;, I trust that their scoring system is indeed very effective, but I just find it too difficult to understand. &lt;/p&gt;&#xA;" OwnerUserId="4670" LastEditorUserId="4670" LastEditDate="2015-04-26T12:27:37.993" LastActivityDate="2022-02-22T22:25:16.247" Title="Way to assign points for an item based on the percent of students who answered the item correctly" Tags="|grading|" AnswerCount="4" CommentCount="8" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7964" PostTypeId="1" AcceptedAnswerId="7965" CreationDate="2015-04-27T03:12:02.390" Score="81" ViewCount="8102" Body="&lt;p&gt;An issue I see with students a lot is abuse of the equals sign. For example, one problem asked &quot;what is the degree of the polynomial: $\text{polynomial}$?&quot;, and I got answers like &quot;$\text{polynomial}=3$&quot;. I tried to explain that no, that polynomial is NOT equal to 3, the DEGREE is equal to 3. They had a difficult time understanding what was wrong. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example, same student. The problem was something like &quot;evaluate $5\times 2+5\times 7$&quot;. The student keenly asked &quot;since there's a $5$ in both, can we divide by $5$?&quot;. The question of course is, &quot;divide WHAT by $5$?&quot;. I could have said &quot;we can let $x=\text{blah}$ etc.&quot; but that would be too confusing. I just said &quot;since we don't have an equation, we can't divide, but what we can do is factor out a $5$ to get $5\times(2+7)$&quot;. They were satisfied with this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know it sounds pedantic, but conceptually equality is perhaps the most important concept in algebra to understand. Not to mention this can lead to real mistakes. For example I saw on a question like &quot;expand $x(x+1)(x-1)$&quot; the following answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x^2+x-x-1$$&#xA;$$=x^2-1$$&#xA;$$=x(x^2-1)$$&#xA;$$=x^3-x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They happened to get the right answer, but in doing things like that it's really easy to make mistakes. I tried to explain that lines 2 and 3 were clearly not equal, and again it just seemed so lost on them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is this so difficult for students, and how do I combat it? What are they hung up on?&lt;/p&gt;&#xA;" OwnerDisplayName="user5108" LastEditorUserId="3453" LastEditDate="2021-03-29T01:40:23.170" LastActivityDate="2021-03-29T01:40:23.170" Title="Issues with &quot;equals&quot;, where does this come from and how do I combat it?" Tags="|mathematical-pedagogy|algebra|notation|equal-sign|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7965" PostTypeId="2" ParentId="7964" CreationDate="2015-04-27T04:52:21.197" Score="85" Body="&lt;p&gt;A surprisingly large number of students don't know what the equals sign means. Their understanding of the symbol &quot;=&quot; is essentially &lt;em&gt;operational&lt;/em&gt;, not &lt;em&gt;relational&lt;/em&gt; — they think it means &quot;the next step&quot; or &quot;the answer&quot; or is an instruction to perform some operation. &lt;a href=&quot;http://itp.wceruw.org/Fall%2009%20seminar/Knuthetal08.pdf&quot;&gt;Knuth et al.&lt;/a&gt; (&quot;The importance of equals sign understanding in the middle grades&quot;, &lt;em&gt;Mathematics Teaching in the Middle School&lt;/em&gt;, vol. 13, no. 9, May 2008) studied middle school students' understanding of the equals sign and identified this misunderstanding as both widespread and strongly correlated with inability to correctly solve basic algebra problems. This serious conceptual error often doesn't correct itself without being directly addressed; I've seen university calculus students make the same sort of mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where does this error come from? As Knuth et al. remark:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Researchers have argued that the &#xA;  operational view of the equal sign is &#xA;  largely a by-product of students’ experiences with the symbol in elementary &#xA;  school mathematics (e.g., Baroody &#xA;  and Ginsburg 1983; Carpenter et &#xA;  al. 2003; McNeil and Alibali 2005). &#xA;  During elementary school, students &#xA;  typically encounter the equal sign in &#xA;  number sentences that have operations on the left side of the equal sign &#xA;  and an answer blank on the right &#xA;  side (e.g., 5 + 2 = __, 11 – 4 + 1 = &#xA;  __). In solving such “operations equal &#xA;  answer” equations correctly, it is not &#xA;  really necessary for students to think &#xA;  about the equal sign as a symbol of &#xA;  equivalence; rather, students need only &#xA;  perform the calculations on the left &#xA;  side of the equal sign to get an answer. &#xA;  As a result, students associate the &#xA;  equal sign with the arithmetic operations performed to get a final answer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are even examples of exercises in elementary school mathematics textbooks that make the same error, using the equals sign in a way that can &lt;em&gt;only&lt;/em&gt; be interpreted operationally. (I can't recall where I saw this; I'll add a link to an example if I find it. Edit: mweiss found &lt;a href=&quot;https://drive.google.com/file/d/0B4wpC387vwzeeE04YXhrNkhRcG8/&quot;&gt;this example&lt;/a&gt;, linked in the comments; that's not the same one I remember seeing, so there might be several textbooks doing this, or it could be from a different place in the same book.) Such textbooks have probably done quite a bit of damage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since conceptual understanding of equality is so essential to all mathematics beyond elementary arithmetic, I think it's advisable to have students solve problems that directly address the meaning of equality. I have no experience with teaching at that age, so I'll leave coming up with appropriate exercises and lessons to those who do. Instead, I'll just explicitly state the concept of equality that students must deeply internalize: &quot;S = T&quot; is the statement that S and T are &lt;em&gt;literally the same thing&lt;/em&gt;, and consequently are indistinguishable in every way and interchangeable in every context. Syntactically, this amounts to the fact that any statement involving S is true if and only if the statement formed by replacing every instance of S with T is true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note, I suspect this is also related to an overly procedural concept of what other mathematical symbols mean; I'd guess many students who don't know what the equals sign means also interpret &quot;1 + 2&quot; as an &lt;em&gt;instruction&lt;/em&gt; (&quot;add these numbers&quot;), not as a &lt;em&gt;number&lt;/em&gt;. Thus, &quot;1 + 2 = 3&quot; is being interpreted as &quot;if you perform the addition procedure on 1 + 2, you get 3&quot;, not as &quot;the number 1 + 2 is the same thing as the number 3&quot;.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastEditorUserId="3505" LastEditDate="2015-04-28T01:05:58.720" LastActivityDate="2015-04-28T01:05:58.720" CommentCount="21" ContentLicense="CC BY-SA 3.0" />
  <row Id="7966" PostTypeId="2" ParentId="7963" CreationDate="2015-04-27T13:47:54.073" Score="2" Body="&lt;p&gt;You could start by assigning the same point value to each question. Then run a multiple regression with total score earned as the dependent variable and the score earned on each question as independent variables. Perhaps then you would assign new point values to each question as a function of its R-squared value in the multiple regression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This function could be as simple as a direct proportion. In that case, any item that explained 0% of the variation in total score (such as an item that all students answered correctly) would be worth 0 points in the new set of point values.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whatever function you choose, you would then have the option to iterate the procedure, running a new multiple regression based on the new point values of the items. My only concern with that iterative approach is that the limit of the iterations might yield a scoring system in which the total score is essentially determined by a single item.&lt;/p&gt;&#xA;" OwnerUserId="5104" LastActivityDate="2015-04-27T13:47:54.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7967" PostTypeId="2" ParentId="7918" CreationDate="2015-04-27T15:02:00.213" Score="8" Body="&lt;p&gt;This is specifically in response to the question: Are there any other &quot;methods&quot; I'm overlooking?&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's how I do it.&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After I teach factoring $x^2 + bx + c$, but before I teach factoring $ax^2 + bx + c$ for $a ≠ 1$, I like to teach factoring by grouping of four-term cubic polynomials such as $2x^3 + 12x^2 + 5x + 30$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach students to start by factoring the GCF out of each pair of terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$2x^3 + 12x^2 + 5x + 30 =&#xA;2x^2 (x+6) + 5 (x+6)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then pull out the common $(x+6)$ from these two terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(x+6) (2x^2 + 5)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have the students practice this with several more examples.  Then I give them an example that doesn't work, such as:&lt;br/&gt;&#xA;$$3x^3 + 15x^2 + 4x + 8 =&#xA;3x^2 (x+5) + 4 (x+2)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students recognize that we are stuck at this point because $(x+5)$ and $(x+2)$ are not the same, so we can't pull anything out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I circle the coefficients in the original polynomial for one of the examples that worked and one of the examples that didn't work:&lt;br/&gt;&#xA;$$2x^3 + 12x^2 + 5x + 30\\&#xA;3x^3 + 15x^2 + 4x + 8$$&#xA;Then I ask how we could have known by looking at those four coefficients which one was going to work and which one was not. Most classes (sometimes with a little guidance) will arrive at something like this:&lt;br/&gt;&lt;br/&gt;&#xA;In the first example, $\frac{12}{2} = 6$ and $\frac{30}{5} = 6$ as well.&lt;br/&gt;&#xA;In the second example, $\frac{15}{5} = 3$, but $\frac{8}{4} = 2$, so they aren't the same.&#xA;&lt;br/&gt;&lt;br/&gt;&#xA;Usually then I shift to factoring $ax^2 + bx + c$ with $a ≠ 1$. I remind them that they've gotten good at factoring stuff like $x^2 + 7x + 10$, but I warn them that it's more complicated when there is a number in front of the $x^2$ (and I use my finger to point at the spot to the left of $x^2$).&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I write on the board $2x^2 + 11x + 12$. I ask them if we could use the factoring by grouping method to factor it, just like we used to factor $2x^3 + 12x^2 + 5x + 30$. When they say no, because there are only three terms, I agree that we want to have four terms, and suggest that we split up the middle term.&#xA;$$2x^2 + 11x + 12 =&#xA;2x^2 + 8x + 3x + 12$$&#xA;I make sure that they're on board with us being allowed to rewrite $11x$ as $8x + 3x$.  Usually I draw two slanted lines downward out of the $11x$ to the $8x$ and $3x$, respectively.&#xA;Then I ask them to factor this by grouping, which they are usually able to do:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$2x^2 + 11x + 12 =\\&#xA;2x^2 + 8x + 3x + 12 =\\&#xA;2x (x+4) + 3 (x+4) =\\&#xA;(x+4) (2x+3)$$&lt;br&gt;&lt;br&gt;&#xA;Then I ask whether factoring by grouping would always work no matter how we break up the $11x$; for instance, will it work if we break up the $11x$ like this:&lt;br/&gt;&lt;br/&gt;&#xA;$$2x^2 + 11x + 12 =\\&#xA;2x^2 + 2x + 9x + 12$$&#xA;I tell them to try it.&lt;br/&gt;&#xA;$$2x^2 + 11x + 12 =\\&#xA;2x^2 + 2x + 9x + 12 =\\&#xA;2x (x+1) + 3 (3x+4)$$&#xA;...and we are stuck, since $(x+1)$ and $(3x+4)$ don't match each other.&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask, how can we tell just by looking at the four coefficients that&lt;br/&gt;&#xA;$2x^2 + 8x + 3x + 12$ will work, but $2x^2 + 2x + 9x + 12$ will not work?&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Someone will give the same sort of answer as the one given for cubic polynomials.&lt;br/&gt;&#xA;The first one works because $\frac{8}{2} = 4$ and $\frac{12}{3} = 4$ as well.&lt;br/&gt;&#xA;The second one doesn't work because $\frac{2}{2} = 1$, but $\frac{12}{9} \neq 1$.&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask the class to consider $ax^2 + bx + c$ in general, where we break up the middle term to rewrite the quadratic as:&lt;br/&gt;&#xA;$$ax^2 + bx + c =\\&#xA;ax^2 + rx + sx + c$$&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask, what must be true of $a, r, s,$ and $c$? Someone usually says that&#xA;$\frac{r}{a}$ and $\frac{c}{s}$ have to be the same. So I write the equation: &lt;br/&gt;&#xA;$$\frac{r}{a} = \frac{c}{s}$$&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask them how to get rid of fractions to make the equation a little nicer. Usually someone says to cross-multiply.  If not, I just ask the class how to change the equation by cross-multiplying, and someone remembers how to do that. They arrive at:&#xA;$$r * s = a * c$$&#xA;So when we have $2x^2 + 11x + 12$, and we are trying to break it into&lt;br/&gt;&#xA;$2x^2 + rx + sx + 12$ for some numbers $r$ and $s$, &lt;br/&gt;&#xA;we need to make sure that $r * s = a * c$.&lt;br/&gt;&#xA;In this case, that means $r*s = 2*12 = 24$.&lt;br/&gt;&#xA;I ask whether we could choose $r = 4$ and $s = 6$, since those multiply to $24$.&lt;br/&gt;&#xA;$$2x^2 + 11x + 12 =\\&#xA;2x^2 + 4x + 6x + 12$$&lt;br/&gt;&#xA;I ask if that is allowed.  Someone will say no, because $4 + 6 \neq 11$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I say, oh, you're saying that $r + s$ has to equal which letter? $a$? $b$? $c$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Someone says $r + s$ has to equal $b$.&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I write on the board:&#xA;$$r + s = b\\&#xA;r * s = a * c$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I say let's see if those rules work in another example:&lt;br/&gt;&#xA;Factor $3x^2 + 8x + 4$.&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I write on the board:&lt;br/&gt;&#xA;$$r + s =$$&#xA;$$r * s=$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask them how to finish out the equations using the formulas, with no numbers yet. They tell me what to write:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$r + s = b$$&#xA;$$r * s = a * c$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I write another equals sign after each line and ask for what numbers belong there. Students tell me what to write.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$r + s = b = 8\\&#xA;r * s = a * c = 3 * 4 = 12$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By this point, they have already been factoring expressions like $x^2 + 8x + 12$, so they proceed to figure out the two &quot;mystery numbers&quot; $r$ and $s$ that add up to $8$ and multiply to $12$. They get $2$ and $6$.&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask, so, does that mean the answer is this:&lt;br/&gt;&#xA;$$3x^2 + 8x + 4 =\\&#xA;(x + 2)(x + 6)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some students may say yes. If so, I ask them to check by FOIL-ing out the $(x+2)(x+6)$. When we arrive at $x^2 + 8x + 12$, I ask if this is what we were hoping to get. Someone says, no, we wanted $3x^2 + 8x + 4$.&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I let them know that writing $(x+2)(x+6)$ at this point is a very common mistake. I remind them that the r and s in this case -- where we have a number besides 1 in front of the $x^2$ -- were intended to help us split up the middle term so that we could factor by grouping.&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3x^2 + 8x + 4 =\\&#xA;3x^2 + 2x + 6x + 4$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask students how we should continue, and usually they are able to tell me:&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3x^2 + 8x + 4 =\\&#xA;3x^2 + 2x + 6x + 4 =\\&#xA;x (3x + 2) + 2 (3x + 2) =\\&#xA;(3x + 2)(x + 2)$$&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we have time, I ask them to FOIL that out in order to check, and students are pleased when we get $3x^2 + 8x + 4$.&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: It is intentional that I don't include any negative coefficients in this part of the lesson. In general, when I'm teaching something that is complicated enough on its own, I try to remove other complications until they gain some confidence in the new topic. By the end of the lesson, or by the next day's lesson, I would start including $ax^2 + bx + c (a \neq 1)$ factoring problems with negatives. By this point, they already are comfortable handling negatives in factoring $x^2 + bx + c$, so the transition isn't too bad.&lt;/p&gt;&#xA;" OwnerUserId="5104" LastEditorUserId="5104" LastEditDate="2015-06-15T13:04:25.193" LastActivityDate="2015-06-15T13:04:25.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7968" PostTypeId="2" ParentId="7961" CreationDate="2015-04-27T15:33:44.170" Score="2" Body="&lt;p&gt;I &quot;feel&quot; there are no common tangent (in red,with a small and large; in green, the exponential).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/IR3tu.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="5112" LastActivityDate="2015-04-27T15:33:44.170" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7969" PostTypeId="2" ParentId="7961" CreationDate="2015-04-27T21:13:09.263" Score="3" Body="&lt;p&gt;Here's how I solved the problem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The tangent line to $y=e^{6x}$ at $x=x_e$ is $y=6e^{6x_e}(x-x_e)+e^{6x_e}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The tangent line to $y=ax^2$ at $x=x_q$ is $y=2ax_q(x-x_q)+ax_q^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The two lines are equal if and only if the slope and intercepts are equal. This gives the system of equations:&#xA;$$6e^{6x_e}=2ax_q$$&#xA;$$(1-6x_e)e^{6x_e} = -ax_q^2.$$&#xA;Substitute $e^{6x_e}=\frac{1}{3}ax_q$ from the first into the second to get&#xA;$$\frac{1}{3}(-1+6x_e)=x_q.$$&#xA;Then substituting this back into the first equality above gives&#xA;$$a = \frac{9e^{6x_e}}{6x_e-1}.$$&#xA;From here it's as you have above: Calculus provides the minimum is $a=9e^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When $a&amp;gt;9e^2$, there are (at least) two solutions for $x_e$ since the graph positively diverges at both $x_e=1/6$ and as $x_e \to \infty$.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-04-27T21:13:09.263" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7970" PostTypeId="1" AcceptedAnswerId="8410" CreationDate="2015-04-28T03:12:43.277" Score="6" ViewCount="261" Body="&lt;p&gt;My girlfriend is soon to become HS or secondary education public school math teacher in the USA. I was reading some HS contracts and I dont see any &quot;vacation days&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At best I only saw 2 &quot;personal&quot; days in one contact. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you not take a vacation while school is in session? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am a huge skier and was wondering if this is the end of our serious ski trip career limiting us to only when kids are off from school. &lt;/p&gt;&#xA;" OwnerUserId="5118" LastEditorUserId="143" LastEditDate="2015-04-28T12:28:53.480" LastActivityDate="2015-07-20T02:50:55.230" Title="As a Secondary public Math educator in the USA are there vacation days?" Tags="|secondary-education|usa|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7971" PostTypeId="2" ParentId="3839" CreationDate="2015-04-28T12:37:25.563" Score="9" Body="&lt;p&gt;I did this for an introductory calculus course at a US state university. My reason was that I wanted to assign more homework than my colleagues typically do. On the first day of class I warned students that there was a lot of homework, explained why I think it's important, and told the students that they could see exactly what was expected of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same semester, I also decided to promise that all the exam questions would be taken, essentially randomly, from the homework. Again, my rationale was to set a standard of being tough, but fair and highly predictable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This avoided nasty surprises on the students' part, and I was pleased with the results: the students (mostly) did the work, kept a positive attitude, and gave me good teaching evaluations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should also mention that there &lt;em&gt;were&lt;/em&gt; small modifications to the homework as the semester went on, but just deletions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This worked because I had taught calculus at the same university before, and had a near-exact plan for what I wanted to cover. As far as I could tell there were no disadvantages, and I would certainly do this again. However, I would personally not try this if I were designing a new course from scratch: as you say, it would make it difficult to switch gears if needed. &lt;/p&gt;&#xA;" OwnerUserId="1535" LastActivityDate="2015-04-28T12:37:25.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7972" PostTypeId="2" ParentId="7964" CreationDate="2015-04-28T18:10:15.363" Score="12" Body="&lt;p&gt;I think in your expansion example this is caused by wanting to solve one subpart of the problem first. In essence, they're trying to expand $(x+1)(x-1)$ first as a &quot;subroutine&quot; and then substitute it back into the original formula, but lacking a clear way to delineate this from the solution to the main problem. They chose to mark the substitution with an equals sign - what it is equal to, of course, is the original formula. Your problem with it is, reasonably, that it is &lt;em&gt;not&lt;/em&gt; equal to the immediately preceding step. Line 3 isn't equal to line 2, but line 2 is equal to line 1, and line 3 is equal to the original problem statement. They don't understand that with this kind of notation, the implied left-hand-side of the equals sign is always the immediately preceding line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They didn't &lt;em&gt;happen&lt;/em&gt; to get the right answer, and it's condescending for you to suggest this. They knew &lt;em&gt;exactly&lt;/em&gt; what they were doing in the substitution step, they just didn't know how to write it. You need to teach them the correct way to write this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I might write this as:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;  x (   (x + 1)(x - 1) )&#xA;    ( = x² + x - x - 1 )&#xA;    ( =     x² - 1     )&#xA;= x³ - x &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;I don't know how to typeset this, the outer sets of parentheses are intended to be a single large pair. And, frankly, if I were merely trying to solve it for myself, I might not write them either - the fact that the equal sign on the final line is intended to the left relative to the others would make it clear enough for my own understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe their answer was likewise arranged on the page in a way that they thought made it clear what the &quot;up-hand side&quot; of the offending $=$ sign was meant to be. Or maybe they didn't think it mattered what their intermediate steps looked like, since the question was &quot;expand ...&quot; rather than &quot;prove the expansion of ...&quot; - it was never clear to me as a student when &quot;show your work&quot; meant being graded on procedure and when it simply meant &quot;prove you didn't cheat&quot;.&lt;/p&gt;&#xA;" OwnerUserId="3591" LastEditorUserId="3591" LastEditDate="2016-09-22T20:08:24.213" LastActivityDate="2016-09-22T20:08:24.213" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7973" PostTypeId="2" ParentId="7970" CreationDate="2015-04-28T18:40:28.650" Score="10" Body="&lt;p&gt;In the United States, you get a few personal days to use as you choose, some sick days, and some &quot;professional days&quot; to improve your teaching skills. Other than that, you basically get your holidays when the children do. In fact, you lose some of those days for in-services and a few other duties. The school does need to hire substitute teachers for days you are gone, and interruptions make it more difficult for the students to learn. So plan your ski days carefully, drop skiing, or don't take a teaching job.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2015-04-28T18:40:28.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7974" PostTypeId="1" AcceptedAnswerId="7975" CreationDate="2015-04-29T17:12:06.930" Score="17" ViewCount="2058" Body="&lt;p&gt;I was asked to give extra lessons to a 5 year old boy who loves math (he says he likes 3 sports: football, swimming and math). However, I have never taught at this age and I am unfamiliar with the best theoretical way to proceed.&lt;/p&gt;&#xA;&lt;p&gt;Hence, I would like some references on teaching at this age, particularly to gifted children.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: as requested, I did take a time with him and tried to assess what he knows. He is fluent in:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;counting&lt;/li&gt;&#xA;&lt;li&gt;addition, subtraction&lt;/li&gt;&#xA;&lt;li&gt;knows fractions (in a intuitive way, he knows nothing about the notation and relation to division)&lt;/li&gt;&#xA;&lt;li&gt;he knows that negative numbers are ok in math but cannot account for a 'real' thing. In his words &amp;quot;there cannot be -2 me's&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;he has great intuition: knowing there's not a largest number and that infinity isn't a number either.&lt;/li&gt;&#xA;&lt;li&gt;he has trouble with problems which don't have a &amp;quot;precise&amp;quot; answer- such as: &amp;quot;if I toss a die many times, which number will appear the most?&amp;quot;, to which he replied &amp;quot;one&amp;quot;, but could not justify why.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;That's about it; as mentioned, I am not even sure how to properly assess what he knows, so everything helps.&lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorDisplayName="user18187" LastEditDate="2023-01-16T19:47:29.193" LastActivityDate="2023-01-16T19:47:29.193" Title="Teaching a very enthusiastic and bright 5 year old" Tags="|reference-request|primary-education|reference-books|private-lessons|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7975" PostTypeId="2" ParentId="7974" CreationDate="2015-04-29T18:33:07.300" Score="7" Body="&lt;p&gt;I would recommend &quot;&lt;em&gt;math enrichment&lt;/em&gt;&quot; instead of straightforward math acceleration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Here is one source from the US Florida school system: &#xA;&lt;a href=&quot;http://it.pinellas.k12.fl.us/schools/curlew-es/studentconnect/mathsuperstars/mathss.html&quot; rel=&quot;noreferrer&quot;&gt;Math SuperStars&lt;/a&gt; (a.k.a. &lt;em&gt;Sunshine Math&lt;/em&gt;).&#xA;It covers grades K-5, and contains weekly problem worksheets.&#xA;It is a bit too problem-based (rather than project-based) for&#xA;my tastes, but at least it gives you a consistent set of problems&#xA;at well-defined developmental levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a snippet from 1st-grade, week 5:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/pJFRF.png&quot; alt=&quot;MSS1stGradeWeek5&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) For a project, I might suggest building the Platonic solids with &#xA;&lt;a href=&quot;http://www.polydron.co.uk/&quot; rel=&quot;noreferrer&quot;&gt;polydrons&lt;/a&gt; or their functional equivalents,&#xA;and seeking &amp;amp; noticing patterns. Of course lurking beyond is&#xA;$V-E+F=2$.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/rIDxo.jpg&quot; alt=&quot;PolydronsPlatonic&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-04-29T23:27:05.747" LastActivityDate="2015-04-29T23:27:05.747" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7976" PostTypeId="1" AcceptedAnswerId="7986" CreationDate="2015-04-29T19:05:14.723" Score="16" ViewCount="1043" Body="&lt;p&gt;I'm creating a large number of practice problems for my statistics students. These problems are for an elementary stats course where students:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;measure central tendency&lt;/li&gt;&#xA;&lt;li&gt;measure dispersion&lt;/li&gt;&#xA;&lt;li&gt;use linear regression to extrapolate and interpolate &lt;/li&gt;&#xA;&lt;li&gt;calculate the mean, and standard deviation for a discrete distribution&lt;/li&gt;&#xA;&lt;li&gt;generate binomial probability distributions  &lt;/li&gt;&#xA;&lt;li&gt;use normal distributions to calculate probabilities&lt;/li&gt;&#xA;&lt;li&gt;use the central limit theorem to calculate probabilities for $\bar x$&lt;/li&gt;&#xA;&lt;li&gt;calculate confidence intervals &lt;/li&gt;&#xA;&lt;li&gt;perform simple hypothesis testing&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It is tempting to make up numbers for these problems. However, I wonder if it is better practice to use numbers based on real-world measurements? I know what when most people make up lists of values that are meant to be random, or centered about a mean, the distribution of their made-up numbers is not always the same as measured data. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, my students are not expected to deal with sets of data with more than 35 elements, and I do not want confusing examples. I must minimize student frustration. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ideal? Sets of values that seem reasonable, come from a real source and that give answers that build intuition over the core concepts in the course. Will real data help students form a more concrete mental image of the problem? Our book often has sourced data, yet I find far too many of the problems are highly technical to the point of obscuring the chapter's central concept. Complexity isn't a bad thing for the stronger students. I don't want to get hung up on converting between units, or bogged down trying hard to get my students to picture what &quot;3.7 acre-feet of water per hour per person&quot; might mean. At least not just yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found it very time-consuming to source &quot;real&quot; data. Are there any libraries online with samples that could work well in this situation? Do you have any recommendations for finding good data? I have no fanatical desire to have a real source for everything.   I seek reasonable data and with the kind of noise and randomness that arises from real-world measurements.&lt;/p&gt;&#xA;" OwnerUserId="5080" LastActivityDate="2018-05-07T15:29:14.470" Title="Where can I find realistic data for college-level elementary statistics problems?" Tags="|examples|statistics|word-problems|" AnswerCount="7" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7977" PostTypeId="2" ParentId="7976" CreationDate="2015-04-29T20:16:35.010" Score="13" Body="&lt;p&gt;&lt;a href=&quot;http://lib.stat.cmu.edu/DASL/&quot;&gt;DASL&lt;/a&gt; (pronounced &quot;dazzle&quot; and short for Data And Story Library) is an online collection of stories with matching data sets to be used for educational purposes. They are real data from real research. Searchable by statistics concept and by theme of the story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.statsci.org/data/&quot;&gt;OzDASL&lt;/a&gt; is similar, but most of the data has an Australian or New Zealandish source. Personally I find OzDASL a bit easier to navigate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The statistical package R comes with quite a few &lt;a href=&quot;https://stat.ethz.ch/R-manual/R-devel/library/datasets/html/00Index.html&quot;&gt;datasets&lt;/a&gt; built in, and so do several packages within it (for example, the MASS package). If you have R or RStudio installed you can type &lt;code&gt;data()&lt;/code&gt; to see a list of the built in data sets and &lt;code&gt;data(package = .packages(all.available = TRUE))&lt;/code&gt; to see a list of data sets inside all the available packages.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-04-29T20:16:35.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7978" PostTypeId="1" CreationDate="2015-04-29T20:44:21.833" Score="14" ViewCount="418" Body="&lt;p&gt;Are there reference books or online material for reading mathematics out loud in different languages?&#xA;For example, the expression&lt;sup&gt;1)&lt;/sup&gt;&#xA;$$&#xA;\int x^2dx=\frac13x^3&#xA;$$&#xA;is understandable to anyone who has studied mathematics, independently of their language skills.&#xA;But reading it out loud or writing it in words only is highly nontrivial for someone without mathematical education in the respective language.&#xA;I can express the equation above in English&lt;sup&gt;2)&lt;/sup&gt; or Finnish&lt;sup&gt;3)&lt;/sup&gt;, but not in any other language I speak.&#xA;I find students occasionally having trouble verbalizing formulas even in their mother tongue in which the courses are taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a complicated formula there need not exist any sensible way of reading it out loud; one has to explain it in some other way than reading through.&#xA;There rarely is a single correct verbalization of a mathematical expression.&#xA;Therefore I don't expect a &quot;complete and perfect set of rules&quot;, but just some guidelines and examples in different areas of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Main question(s)&lt;/strong&gt;:&#xA;Where can I or my students learn to pronounce mathematics in different languages?&#xA;What material for reading mathematics out loud do you suggest?&#xA;Can I find examples of mathematics spelled out in different languages?&#xA;Resources for all languages&lt;sup&gt;4)&lt;/sup&gt; are welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Side questions:&#xA;Are there sources about reading mathematics out loud (or spelling it out) in general?&#xA;Is there any research on how reading formulas aloud affects learning mathematics?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;1) I don't want to discuss if this is good notation or whether I should have a constant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &quot;The integral of x squared equals one third x to the third.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &quot;x:n neli&amp;ouml;n integraali on yksi kolmasosa x kolmanteen.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) I don't speak all languages, but I think it would be useful to collect a list of mathematics pronunciation guides in one place.&#xA;My personal interest is mainly in Latin, Finnish, English, Italian, French and Swedish.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2016-11-07T09:44:03.637" Title="Resources for reading mathematics out loud in different languages" Tags="|reference-request|reference-books|language-use|english-language-learners|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7979" PostTypeId="2" ParentId="7976" CreationDate="2015-04-29T23:45:00.370" Score="4" Body="&lt;p&gt;This is not a good answer, but I'll mention anyway that Mathematica now connects to vast &lt;a href=&quot;https://reference.wolfram.com/language/howto/UseCuratedData.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;curated&quot; data sources&lt;/a&gt;. It would take&#xA;some effort to master these sources, but they could provide endless streams of real data. Here is one example I just ran (NB: Up-to-date in that no Pluto!).&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/mnDqn.png&quot; width=&quot;400&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-04-29T23:45:00.370" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7980" PostTypeId="2" ParentId="7974" CreationDate="2015-04-30T02:01:06.550" Score="3" Body="&lt;p&gt;I was impressed by some of the material used in the first and second grades of my son's elementary school in the USA, where I suspect the material comes from the recent Common Core standards.  There were some basic enumeration problems (&quot;How many ways can you attach three squares together on their sides, so that the whole side is shared by the squares?&quot;) that were open-ended.  (I get five if the squares are rigid and I get to use three dimensions.)  The best part though was to get them to explain why, and show their work.  Ambitious for second grade even, but starting good habits sooner is usually a good thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend puzzles and other things come from discrete math.  He may not&#xA;be ready for Fibonacci numbers and binomial coefficients, but some of the ideas leading up to them it seems he can handle.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also recommend Adrian Paenza.  I imagine he has some resources available that might interest your student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Oh, To Be Five Again!&quot; Paseman, 2015.04.29&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-04-30T02:01:06.550" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7981" PostTypeId="2" ParentId="7974" CreationDate="2015-04-30T04:00:33.353" Score="6" Body="&lt;p&gt;Three books you can read with your young student:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The Cat in Numberland&lt;/em&gt;. After reading it, you get to discuss orders of infinity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The Man Who Counted&lt;/em&gt;, by Malba Tahan. (Do you and your student speak Portuguese together? This was originally in Portuguese; the author is Brazilian.) As you read each episode, try to predict what Beremiz will do to solve the dilemma.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The Number Devil&lt;/em&gt;. Lots of intriguing math concepts explored.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-04-30T04:00:33.353" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7982" PostTypeId="2" ParentId="7978" CreationDate="2015-04-30T15:03:18.407" Score="2" Body="&lt;p&gt;For English, I find a few web pages...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://howtosaymathematics.wordpress.com&quot; rel=&quot;nofollow&quot;&gt;https://howtosaymathematics.wordpress.com&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://jeromyanglim.blogspot.com/2009/05/pronunciation-guides-for-mathematical.html&quot; rel=&quot;nofollow&quot;&gt;http://jeromyanglim.blogspot.com/2009/05/pronunciation-guides-for-mathematical.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.rpi.edu/dept/math/ms_graduate/resources/SayingMath.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.rpi.edu/dept/math/ms_graduate/resources/SayingMath.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.uefap.com/speaking/symbols/symbols.htm&quot; rel=&quot;nofollow&quot;&gt;http://www.uefap.com/speaking/symbols/symbols.htm&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-04-30T15:03:18.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7983" PostTypeId="2" ParentId="7978" CreationDate="2015-04-30T15:18:06.820" Score="5" Body="&lt;p&gt;As Joseph O'Rourke pointed out in the comments, teaching videos are a partial solution.&#xA;As a practical test, I went to the Wikipedia page about complex numbers and browsed through the side panel to know the translation to some other languages.&#xA;I then searched YouTube for &quot;complex numbers&quot; in English, Italian, French, Spanish, Finnish, Swedish and Latin, and for all but Latin I relatively quickly found a video in which equations or other expressions are read out loud.&#xA;(Frankly, I would have been surprised by a Latin video about complex numbers, but I couldn't resist trying.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Videos have some drawbacks, though.&#xA;If you already know the topic and just want someone to read out loud many expressions, the videos are slow.&#xA;In some languages it is difficult to catch the correct pronunciation or the corresponding way to spell out the expression.&#xA;Videos don't make a good reference material; it's not easy to check how to pronounce some specific thing in, say, Italian.&#xA;If I want to give my students a guide to pronouncing formulas in a particular course, videos are not very useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would prefer written lists of examples, but they seem harder to come by.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2015-04-30T15:18:06.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7984" PostTypeId="2" ParentId="926" CreationDate="2015-05-01T17:24:46.550" Score="12" Body="&lt;p&gt;I've read all the existing answers long ago but still feel that none have gotten to the heart of the issue. We obtain mathematical results through a process of reasoning. &lt;strong&gt;That reasoning must be logical and enough to convince anyone that our results are correct given our initial assumptions.&lt;/strong&gt; That is the actual purpose of a proof. It does not matter what form the reasoning takes, whether using only words or only mathematical symbols or only a diagram. The requirement is simply to convince the other person. &lt;strong&gt;If we cannot do so, then our reasoning is insufficient or incorrect.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This proper attitude must start right from the basics. For example &lt;span class=&quot;math-container&quot;&gt;$\frac{1+2}{1+3} \ne \frac{\not{1}+2}{\not{1}+3}$&lt;/span&gt;. &lt;strong&gt;Explaining to the student that one cannot do that is almost useless&lt;/strong&gt;. Instead, the student should be asked: &lt;strong&gt;&amp;quot;Why do you cancel?&amp;quot;&lt;/strong&gt; and then &lt;strong&gt;&amp;quot;Why does cancelling keep the value the same?&amp;quot;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The problem is that if this is not done from the beginning of arithmetic, it simply causes students to create for themselves a deep quagmire of guesswork in order to heuristically &lt;strong&gt;write down things which they believe will get them their grades&lt;/strong&gt;. If you have seen students who try to mimic their teachers' phrasing but clearly without understanding of the meaning, or students who care only about how to get the answer and not why the method is correct, you know what I mean.&lt;/p&gt;&#xA;&lt;p&gt;As a result, very few students have a full grasp of even the fundamentals, namely the field of rationals. What I mean by this is that few are able to state all the field axioms correctly and prove results like the uniqueness of inverses (when they exist) and that &lt;span class=&quot;math-container&quot;&gt;$0 \times x = 0$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$-x \times -y = x \times y$&lt;/span&gt;. (Out of these, fewer still can give any explanation as to the rationale for the axioms, but that is another topic.)&lt;/p&gt;&#xA;&lt;p&gt;It is obvious that with a proper foundation as I briefly described above, no student would ever write &lt;span class=&quot;math-container&quot;&gt;$(a+b)^2 = a^2+b^2$&lt;/span&gt;. Why? Because they know that &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;&amp;quot; is &lt;strong&gt;defined&lt;/strong&gt; as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x \times x$&lt;/span&gt;&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$()$&lt;/span&gt;&amp;quot; are used to &lt;strong&gt;denote&lt;/strong&gt; what to do first, so &lt;span class=&quot;math-container&quot;&gt;$(a+b)^2 = (a+b) \times (a+b)$&lt;/span&gt;. Moreover, they also would know the distributivity field axiom that gives first &lt;span class=&quot;math-container&quot;&gt;$(a+b) \times (a+b) = a \times (a+b) + b \times (a+b)$&lt;/span&gt; and then after 2 more applications the full expansion, using the commutativity and associativity axioms. Likewise none of the other mistakes that you mentioned would occur.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, if students cannot handle the field axioms correctly, one might as well throw the induction axiom out of the window. The way it is taught in most textbooks and curricula is seriously lacking, precisely because it is not based on sufficiently formal reasoning. A simple example that most students who were brought up with textbook induction fail to solve is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given a function &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{Z}\to\mathbb{R}$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f(0) = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(1) = 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x+1) + 6 f(x-1) = 5 f(x)$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{Z}$&lt;/span&gt;, prove that &lt;span class=&quot;math-container&quot;&gt;$f(x) = 3^x - 2^x$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{Z}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is not hard at all, but only those who understand the logical structure of induction would be able to give a correct proof. In case anyone is wondering what I mean by textbook induction, two examples that I would consider seriously lacking are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.mathsisfun.com/algebra/mathematical-induction.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathsisfun.com/algebra/mathematical-induction.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.themathpage.com/aprecalc/mathematical-induction.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.themathpage.com/aprecalc/mathematical-induction.htm&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Finally, proper reasoning naturally requires sufficient precision, because one cannot reason logically about statements whose meaning is undefined or unclear. &lt;strong&gt;Vagueness in mathematics is one great recipe for confusion.&lt;/strong&gt; This must start with the teacher. &lt;strong&gt;A teacher who is sloppy with mathematical statements or steps in reasoning is simply telling the students that it is alright to be sloppy and by extension it is alright if they do not know what they are doing as long as they get the answer!&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;One terrible example of sloppiness in most high-school curricula is solving differential equations by &amp;quot;separating variables&amp;quot;. Try giving the following to any student:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Solve for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as a function of a real variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; given that the differential equation &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx} = 2\sqrt{y}$&lt;/span&gt; holds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You know what answer to expect, and I hope you know the correct answer. Even Wolfram Alpha gets it wrong. Now for students who give the wrong answer, tell them that it is wrong but do not tell them the correct answer, and ask if they can identify the mistake and fix it. Most will fail to identify the mistake, and fixing the mistake will require the foundation in logic that most students do not have.&lt;/p&gt;&#xA;&lt;p&gt;Here are the solution sketches for the problems I've given above. I strongly encourage one to thoroughly check one's own work to verify whether each step follows completely logically from the preceding deductions, and merely look at these solutions to confirm.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given a function &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{Z}\to\mathbb{R}$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f(0) = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(1) = 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x+1) + 6 f(x-1) = 5 f(x)$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{Z}$&lt;/span&gt;, prove that &lt;span class=&quot;math-container&quot;&gt;$f(x) = 3^x - 2^x$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{Z}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Hints&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; Induction only allows you to derive something about the natural numbers. The desired theorem is about integers. Also, if you cannot prove the implication needed for the induction, a key technique that often works is to strengthen the induction hypothesis to include enough information so that you can prove the implication step. Of course that also means that the implication you need to prove has changed!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solution sketch&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; First notice that the theorem to be proven is that &lt;span class=&quot;math-container&quot;&gt;$f(x) = 3^x - 2^x$&lt;/span&gt; for all &lt;strong&gt;integers&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and so induction in one direction is not enough! Also, notice that it is impossible to prove that &lt;span class=&quot;math-container&quot;&gt;$f(x) = 3^x - 2^x$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$f(x+1) = 3^{x+1} - 2^{x+1}$&lt;/span&gt;, and hence the induction hypothesis must contain information about at least two 'data points' for &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. The easiest one would be to let &lt;span class=&quot;math-container&quot;&gt;$P(x)$&lt;/span&gt; be &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$f(x) = 3^x - 2^x$&lt;/span&gt; &lt;strong&gt;and&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$f(x-1) = 3^{x-1} - 2^{x-1}$&lt;/span&gt;&amp;quot;. Then one must prove &lt;span class=&quot;math-container&quot;&gt;$P(x+1)$&lt;/span&gt;, which expands to &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$f(x+1) = 3^{x+1} - 2^{x+1}$&lt;/span&gt; &lt;strong&gt;and&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$f(x) = 3^x - 2^x$&lt;/span&gt;&amp;quot;. I would not accept if the student does not fully prove &lt;span class=&quot;math-container&quot;&gt;$P(x+1)$&lt;/span&gt;. This would handle the natural numbers, and a similar induction would handle the negative integers. It is of course possible to combine both inductions into one, which it should be explored, although in general it is good to keep a proof as modular as possible.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Solve for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as a function of a real variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; given that the differential equation &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx} = 2\sqrt{y}$&lt;/span&gt; holds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Hint&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; The answer is not &lt;span class=&quot;math-container&quot;&gt;$y = (x+a)^2$&lt;/span&gt;, which you would get by the method of separating variables. What went wrong? Note that the error would still be there if you used the theorem that allows change of variables in an integral. Look carefully at each deduction step. One step cannot be justified based on any axiom. Think basic arithmetic. After you get that, you need to consider cases and use the completeness axiom for reals to extend the open intervals on which the standard solution works.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solution sketch&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; The field axioms only give you a multiplicative inverse when it is not zero. Now how to solve the problem? Split into cases. Note that you need to work on intervals since having isolated points where &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is nonzero is useless. First prove that for any point where &lt;span class=&quot;math-container&quot;&gt;$y \ne 0$&lt;/span&gt;, there is an open interval around &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$y \ne 0$&lt;/span&gt;. Then we can use the completeness axiom for reals to extend the interval in both directions as far as &lt;span class=&quot;math-container&quot;&gt;$y \ne 0$&lt;/span&gt;. Now we can use any method to solve for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; on that interval. Note that the method of separating variables is formally invalid, so we should use the change of variables substitution. But the prerequisite for that is that &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; is continuous, so we need to prove that! Well, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is differentiable and hence continuous, so &lt;span class=&quot;math-container&quot;&gt;$2\sqrt{y}$&lt;/span&gt; is continuous. So we get the solution on the extended interval, and it shows that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; becomes zero in exactly one direction in this example. Hence after some checking you will get either &lt;span class=&quot;math-container&quot;&gt;$y = 0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$y = \cases{ 0 &amp;amp; if $x \le a$ \\ (x-a)^2 &amp;amp; if $x &amp;gt; a$ }$&lt;/span&gt; for some real &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Alternative subproof&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; In fact, the substitution theorem can be completely avoided as follows. On any interval &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$y \ne 0$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$y'^2 = 4y$&lt;/span&gt;, where &amp;quot;&lt;span class=&quot;math-container&quot;&gt;${}'$&lt;/span&gt;&amp;quot; denotes the derivative with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Thus &lt;span class=&quot;math-container&quot;&gt;$(y'^2)' = (4y)'$&lt;/span&gt;, which gives &lt;span class=&quot;math-container&quot;&gt;$2y'y'' = 4y'$&lt;/span&gt;, and hence &lt;span class=&quot;math-container&quot;&gt;$y'' = 2$&lt;/span&gt; since &lt;span class=&quot;math-container&quot;&gt;$y' = 2\sqrt{y} \ne 0$&lt;/span&gt;. Thus &lt;span class=&quot;math-container&quot;&gt;$y' = 2x+c$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; for some real &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, and hence &lt;span class=&quot;math-container&quot;&gt;$y = x^2+cx+d$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; for some real &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;. Note that most of the above steps are not reversible and hence we need to check all the solutions we finally obtain with the original differential equation. We would get &lt;span class=&quot;math-container&quot;&gt;$c^2 = 4d$&lt;/span&gt;. After simple manipulation we obtain the same result for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; as in the other solution. The other parts of the solution still need to be there.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="1550" LastEditDate="2021-12-24T13:45:32.397" LastActivityDate="2021-12-24T13:45:32.397" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="7985" PostTypeId="1" CreationDate="2015-05-02T05:21:58.480" Score="11" ViewCount="381" Body="&lt;p&gt;I usually teach solving linear equations by balancing both sides e.g.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{array}{cccc}&#xA;2x&amp;amp;+&amp;amp;3&amp;amp;=&amp;amp;5\\&#xA;&amp;amp;&amp;amp;\color{red}{-3}&amp;amp;&amp;amp;\color{red}{-3}\\&#xA;2x&amp;amp;&amp;amp;&amp;amp;=&amp;amp;2\\&#xA;\color{red}{\div2}&amp;amp;&amp;amp;&amp;amp;&amp;amp;\color{red}{\div2}\\x&amp;amp;&amp;amp;&amp;amp;=&amp;amp;1&#xA;\end{array}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently I have been considering teaching students to solve via factoring&#xA;$$\begin{array}{cccc}&#xA;2x&amp;amp;+&amp;amp;3&amp;amp;=&amp;amp;5\\&#xA;&amp;amp;&amp;amp;\color{red}{-5}&amp;amp;&amp;amp;\color{red}{-5}\\&#xA;2x&amp;amp;-&amp;amp;2&amp;amp;=&amp;amp;0\\&#xA;2(x&amp;amp;-&amp;amp;1)&amp;amp;=&amp;amp;0\\x&amp;amp;&amp;amp;&amp;amp;=&amp;amp;1&#xA;\end{array}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think an advantage might be that solving quadratic equations is more aligned with previous learning. However I can see students being confused by factoring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anybody taught solving &lt;strong&gt;linear&lt;/strong&gt; equations by factoring? What were the outcomes for the students and future progress?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any thoughts would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-05-22T20:37:36.127" Title="Solving linear equations by factoring" Tags="|secondary-education|solving-polynomials|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7986" PostTypeId="2" ParentId="7976" CreationDate="2015-05-02T10:43:15.240" Score="8" Body="&lt;p&gt;(I came to mention DASL, but since it's already been mentioned, I'll give some other resources.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;opendata.stackexchange.com often has mentions of useful sources of data, some of which are small (and others of which might be sampled to generate smaller data sets).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's also a good place to ask about data sets&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The datasets subreddit &lt;a href=&quot;http://www.reddit.com/r/datasets/&quot; rel=&quot;noreferrer&quot;&gt;http://www.reddit.com/r/datasets/&lt;/a&gt; often has useful data sets (and again, is a good place to ask about data sets)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The US government data portal may also have some useful data sets:&#xA;&lt;a href=&quot;http://www.data.gov/&quot; rel=&quot;noreferrer&quot;&gt;http://www.data.gov/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="755" LastActivityDate="2015-05-02T10:43:15.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7988" PostTypeId="2" ParentId="926" CreationDate="2015-05-02T12:48:42.277" Score="9" Body="&lt;p&gt;I am deeply convinced that these fallacies come from the way we teach maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is some research indicating that for the over reliance on linearity (see e.g. page 51 of the &lt;a href=&quot;http://www.ems-ph.org/journals/newsletter/pdf/2015-03-95.pdf&quot; rel=&quot;noreferrer&quot;&gt;last EMS newsletter&lt;/a&gt;, as I commented on another answer giving more references).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it seems to me that this is a particular case of a more general phenomenon: maths are just thought by many student as &lt;strong&gt;black magic&lt;/strong&gt;. It is about learning and reciting formulas, using precise methods to solve precise exercise, always in the same way. Beware if you misspell a formula, as you might summon an &lt;em&gt;efreet&lt;/em&gt; by accident (i.e. the teacher will be mad at you)! In other words, students are lacking the relation between symbols, rules, formulas, theorems in the one hand, and &lt;em&gt;the meaning&lt;/em&gt; of them in the other hand. Without such connection, there is simply no way to tell a correct formula from an incorrect one; even checking may be out of the question, as replacing letters with specific values can only be done with a certain understanding of the role of variables, as opposed to the mere ability to reproduce formal manipulations of symbols.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not know if the following hypothesis has been tested rigorously, but I also think that this is linked very closely to the way we &lt;strong&gt;evaluate&lt;/strong&gt; our students: the more we ask them to solve standardized exercises, the less sense and reasoning they would put into their maths.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2015-05-16T13:36:38.003" LastActivityDate="2015-05-16T13:36:38.003" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7989" PostTypeId="2" ParentId="7900" CreationDate="2015-05-03T01:15:23.603" Score="2" Body="&lt;p&gt;&lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0393310728&quot; rel=&quot;nofollow&quot;&gt;How to Lie with Statistics&lt;/a&gt;&lt;/em&gt;.  Many graphs in newspapers are misleading because the authors used Excel's default graph settings, instead of making sure to set the range of the y-axis to start at zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are you using a recent version of Microsoft Excel, such as Excel 2013?  Excel 2013's PowerPivot and PivotChart features are amazing.  [Note:  I currently work as a contractor for Microsoft using these tools.]  Bill Jelen (Mr. Excel) has written a book on &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0789748754&quot; rel=&quot;nofollow&quot;&gt;Excel 2013 Pivot Table Data Crunching&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have a suitable data set (like the populations of various states over time, or baseball statistics, or...) you could have the students graph interesting data changes (like the population trend of a city or state, or a batter's on-base percentage &lt;em&gt;versus&lt;/em&gt; time, or a batter's home-run rate with runners in scoring position, or...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A graph of the population of a chosen city or state over time is nice.  A PivotChart of the same thing is amazing.  You can graph how the number of teenage girls in a city has changed over time.  You can then graph how the number of 13 year-old girls and 13 year-old boys in the city has changed.  Then, you can change a single filter to get a similar graph for a different city.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my job, I try to make graphs that are very easy to understand.  That usually means:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Limiting the number of things being graphed.&lt;/li&gt;&#xA;&lt;li&gt;Avoiding having lots of lines cross each other in the plot area.&lt;/li&gt;&#xA;&lt;li&gt;A chart title that describes the point of the chart.&lt;/li&gt;&#xA;&lt;li&gt;Either an easy-to-read axis scale, or a data table&lt;/li&gt;&#xA;&lt;li&gt;Descriptive axis titles (unless the axis scale shows that the units are times)&lt;/li&gt;&#xA;&lt;li&gt;Consistently coloring matching things in different graphs.&lt;/li&gt;&#xA;&lt;li&gt;Unless I have a really good reason not to (like for date/time axes), starting axis scales at zero.&lt;/li&gt;&#xA;&lt;li&gt;Where practical, making the axis scales match in corresponding graphs.&lt;/li&gt;&#xA;&lt;li&gt;Avoiding complicated definitions for what is being graphed.  It is easier to say &quot;At-bats in May&quot; or &quot;At-bats before injury&quot; than &quot;At-bats from May 4 through June 6&quot;.&lt;/li&gt;&#xA;&lt;li&gt;Making sure that the field names (as shown on the chart) are easy to understand.&lt;/li&gt;&#xA;&lt;li&gt;Hiding redundant features (like a legend that says &quot;Total&quot; when there is only one variable being graphed).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-05-04T17:13:31.823" LastActivityDate="2015-05-04T17:13:31.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7990" PostTypeId="1" AcceptedAnswerId="7994" CreationDate="2015-05-03T02:19:56.343" Score="13" ViewCount="381" Body="&lt;p&gt;As I try to incorporate more history, science, language, computing, and art into my math class I keep finding the lessons to be very successful and my students always seem to enjoy them.  While I know a fair amount about computing and science, I do not know nearly as much about the others and as such, takes longer to make these lessons.  I am lucky if I can get one in every two weeks.  I was wondering if there are any other resources on possible curricula/lessons that take an interdisciplinary approach to math education?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="15884" LastEditDate="2021-07-17T04:45:18.247" LastActivityDate="2021-07-17T04:45:18.247" Title="Resources on interdisciplinary curricula" Tags="|secondary-education|course-design|curriculum|resource-request|interdisciplinary|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7991" PostTypeId="2" ParentId="6163" CreationDate="2015-05-03T12:08:55.563" Score="3" Body="&lt;p&gt;Maybe it has to do with congruence proofs.  Proving line segments congruent is too uncomplicated and proving quadrilaterals congruent is too complicated (at least until after one has studied proving triangles congruent).  Note that proving circles congruent is also too uncomplicated.&lt;/p&gt;&#xA;" OwnerUserId="5104" LastActivityDate="2015-05-03T12:08:55.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7992" PostTypeId="2" ParentId="7990" CreationDate="2015-05-03T16:17:43.537" Score="2" Body="&lt;p&gt;Two possible sources, perhaps(?) at too low a level for your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) The &lt;a href=&quot;https://www.teachervision.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teacher Vision&lt;/a&gt; website contains&#xA;materials, which unfortunately take a bit of effort to wade through.&#xA;They do have some suggested lessons on&#xA;&lt;a href=&quot;https://www.teachervision.com/tv/tvsearch.php?lowest_grade=109&amp;amp;highest_grade=109&amp;amp;term=170500000000&amp;amp;site=TV&amp;amp;tab=Grades&amp;amp;scn=11&amp;amp;mode=-bread&quot; rel=&quot;nofollow noreferrer&quot;&gt;history at the high-school level&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/8ZDz9.png&quot; alt=&quot;Contents&quot;&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/5I8M6.gif&quot; alt=&quot;Spiral&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) And here is another possibly useful site, &#xA;&lt;a href=&quot;http://www.yummymath.com/leslies-other-sites/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yummy Math&lt;/a&gt;.&#xA;One subset here is &#xA;&lt;a href=&quot;http://www.ldlewis.com/Teaching-Mathematics-with-Art/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching math with art&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/DvOeU.jpg&quot; alt=&quot;LeslieMathArt&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-05-03T16:17:43.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7993" PostTypeId="2" ParentId="7990" CreationDate="2015-05-03T17:29:06.760" Score="2" Body="&lt;p&gt;&lt;em&gt;&lt;a href=&quot;http://press.princeton.edu/titles/9496.html&quot; rel=&quot;nofollow&quot;&gt;Viewpoints: Mathematical Perspective and Fractal Geometry in Art&lt;/a&gt;&lt;/em&gt;, by Marc Frantz and Annalisa Crannell, looks interesting. I haven't had a chance to use it. It's college level, though.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-05-03T17:29:06.760" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7994" PostTypeId="2" ParentId="7990" CreationDate="2015-05-03T19:22:12.253" Score="8" Body="&lt;p&gt;Paul Lockhart's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0674284380&quot; rel=&quot;nofollow noreferrer&quot;&gt;Measurement&lt;/a&gt; and David Foster Wallace's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0393339289&quot; rel=&quot;nofollow noreferrer&quot;&gt;Everything and More&lt;/a&gt; incorporate history, language and art in their exploration of mathematics.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;em&gt;Measurement&lt;/em&gt; explores mathematics as art form, and covers deeply many high school mathematics ideas with an eye toward their beauty.  Excerpts could be used as required reading in various high school math classes or, for you, as sparks for lesson ideas.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;em&gt;Everything and More&lt;/em&gt; gives a scattered but entertaining history of the idea of infinity that may give you ideas for pre-calculus and calculus classes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) For a thorough (albeit dry) history of pre-calculus and calculus topics, Carl Boyer's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486605094&quot; rel=&quot;nofollow noreferrer&quot;&gt;The History of Calculus and Its Conceptual Development&lt;/a&gt; is a classic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT (more resources):&#xA;4) Phillips Exeter Academy publishes &lt;a href=&quot;http://www.exeter.edu/academics/72_6539.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;their curriculum&lt;/a&gt; online, which is a set of +1000 word problems covering high school math, including discrete math and multi-variable calculus.  You could pull some of their interdisciplinary questions, like question 4 here from their &lt;a href=&quot;http://www.exeter.edu/documents/discrete1314.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;discrete math problem set&lt;/a&gt;: &lt;img src=&quot;https://i.stack.imgur.com/F980c.png&quot; alt=&quot;discrete math question&quot;&gt;.  This is just the first in a long set of problems that explore methods of apportioning votes that were proposed as the US Federal Government developed (see pages 7-8 for the beginning of the set).  &lt;a href=&quot;https://matheducators.stackexchange.com/questions/4071/discovery-based-and-inquiry-based-learning?lq=1&quot;&gt;Click here&lt;/a&gt; and scroll down for more on how they implement this curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) For computing, check out the fascinating  &lt;a href=&quot;https://projecteuler.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Project Euler&lt;/a&gt;.  From their website:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems...The problems range in difficulty and for many the experience is inductive chain learning. That is, by solving one problem it will expose you to a new concept that allows you to undertake a previously inaccessible problem. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;6) The introduction to Ernest Nagel's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0814758371&quot; rel=&quot;nofollow noreferrer&quot;&gt;Godel's Proof&lt;/a&gt; is a lucid (and high school student accessible) explanation of the historical development and thinking behind the axiomatic method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7) Douglas Hofstadter's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0465026567&quot; rel=&quot;nofollow noreferrer&quot;&gt;Godel, Escher, Bach&lt;/a&gt; is a free-wheeling exploration of human consciousness through analogies to mathematics, computer programming, biology, visual art, and music.  It's a go-to for exploring recursive functions with your students. &lt;/p&gt;&#xA;" OwnerUserId="5090" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-05-08T14:24:17.873" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7995" PostTypeId="2" ParentId="387" CreationDate="2015-05-04T14:18:55.020" Score="10" Body="&lt;p&gt;My unofficial list of things not explicitly covered in precalculus courses that cause people trouble in calculus:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\frac{6x + 3}{3}$ is not $2x+3$, and $\frac{2x + 2}{x}$ is not 4. &lt;/li&gt;&#xA;&lt;li&gt;$\sqrt{x^2 + 9}$ is not $x + 3$.&lt;/li&gt;&#xA;&lt;li&gt;$(x - 4)^2$ is not $x^2 - 16$.&lt;/li&gt;&#xA;&lt;li&gt;$x^{-1/2}$ is not $-\sqrt{x}$.&lt;/li&gt;&#xA;&lt;li&gt;If $f(x) = x^2$, then $f(x+h)$ is not $x^2 + h$.&lt;/li&gt;&#xA;&lt;li&gt;$x^2 + x^3$ is not $x^5$, or more commonly, $e^{x+2} - e^x$ is not $e^2$.&lt;/li&gt;&#xA;&lt;li&gt;$x\sqrt{x}$ is not mysterious; it is $x^{3/2}$.&lt;/li&gt;&#xA;&lt;li&gt;$\ln x^2$ is not $(\ln x)^2$&lt;/li&gt;&#xA;&lt;li&gt;$\sin^2 x$ is not $\sin \cdot \sin \cdot x$ because that doesn't mean anything.&lt;/li&gt;&#xA;&lt;li&gt;If $\log x = 3$ then $x$ is not $\frac{3}{\log}$ because that doesn't mean anything.&lt;/li&gt;&#xA;&lt;li&gt;$\frac{x}{2/3}$ is not $\frac23 x$.&lt;/li&gt;&#xA;&lt;li&gt;$\frac{x}2$ is not mysterious; it is $\frac12 x$.&lt;/li&gt;&#xA;&lt;li&gt;If $\sin 2x = 1$, then dividing by 2 does not yield $\sin x = \frac12$.&lt;/li&gt;&#xA;&lt;li&gt;If $\sqrt{3x} = 9$, then dividing by 3 does not yield $\sqrt{x} = 3$. &lt;/li&gt;&#xA;&lt;li&gt;If either of us uses the word &quot;is&quot; or the equals sign, like &quot;$x^2$ is $2x$,&quot; we literally mean they are equal, not &quot;After applying [some operation] to $x^2$, it becomes $2x$.&quot; See also the excellent &quot;equals signs&quot; threads on this site.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2015-05-04T14:26:59.763" LastActivityDate="2015-05-04T14:26:59.763" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7996" PostTypeId="5" CreationDate="2015-05-04T15:51:12.120" Score="0" Body="&lt;p&gt;This is a special tag that is designed to be used only by the system for questions that have had all of their other tags removed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All questions in the system must have at least one tag. Normally, it is not possible to submit new or edited questions that do not have tags; the system will reject the submission until a valid tag is applied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a few unusual situations that can cause this tag to be applied. When a tag has existed for six months and only been used on one question, and no tag wiki has been created for the tag, the system deletes the tag. When the community decides that a given tag can never be appropriate &amp;#8212; for example, because it is misspelled or is off-topic &amp;#8212; moderators may remove the tag from the site &lt;em&gt;en masse&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When one of these deletions removes the last tag from a question, the system adds the &lt;a href=&quot;/questions/tagged/untagged&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;untagged&amp;#39;&quot; rel=&quot;tag&quot;&gt;untagged&lt;/a&gt; tag to that question so that the &quot;all questions must have at least one tag&quot; rule is not broken.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can help clean up some of these questions! If they're good questions, edit them to remove the &lt;a href=&quot;/questions/tagged/untagged&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;untagged&amp;#39;&quot; rel=&quot;tag&quot;&gt;untagged&lt;/a&gt; tag and add more descriptive tags. If they're not so good, use the usual tools: downvoting, flagging and voting to close/delete.&lt;/p&gt;&#xA;" OwnerUserId="4872" LastEditorUserId="4872" LastEditDate="2015-05-22T07:58:26.413" LastActivityDate="2015-05-22T07:58:26.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7997" PostTypeId="4" CreationDate="2015-05-04T15:51:12.120" Score="0" Body="This is a special tag that is designed to be used only by the system for questions that have had all of their other tags removed. Do not add this tag to existing questions, it is reserved for use by the system to identify posts with no valid tags." OwnerUserId="4872" LastEditorUserId="4872" LastEditDate="2015-05-04T16:42:42.910" LastActivityDate="2015-05-04T16:42:42.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7998" PostTypeId="1" CreationDate="2015-05-04T17:39:46.160" Score="8" ViewCount="403" Body="&lt;p&gt;It is standard to use the symbol $\Delta$ to indicate a change in a quantity between two points on a curve, two rows on a table, and so forth.  For linear functions, we write slope = $\Delta y / \Delta x$; this notation is used all over the place in Physics and Chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But when looking at exponential functions, the quantity that relates naturally to $\Delta x = x_2-x_1$ is not $\Delta y = y_2 - y_1$, but rather  $\bf{\frac{y_2}{y_1}}$.  This suggests that it would be helpful to have some kind of notation to signify &quot;multiplicative change,&quot; in analogy with using $\Delta$ to signify &quot;additive change&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there such a symbol?  Is it used in any textbooks?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="143" LastEditDate="2015-05-04T17:50:01.663" LastActivityDate="2015-05-23T05:55:30.563" Title="Is there a good notation for &quot;ratio&quot; comparable to the use of $\Delta$ for &quot;difference&quot;?" Tags="|calculus|algebra|notation|precalculus|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="7999" PostTypeId="1" CreationDate="2015-05-04T18:52:23.963" Score="6" ViewCount="1554" Body="&lt;p&gt;I've found some great advice on recording software, microphones etc. to create math videos with screen captured audio from my iPad showing my writing and audio so I can record and explanation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to have some software to edit these videos a little bit. I think they could benefit from:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Cutting out bits that have little audio or screen action.&lt;/li&gt;&#xA;&lt;li&gt;Fast forwarding the video while keeping the audio at normal speed (then removing some of the dead spaces to compensate.)&lt;/li&gt;&#xA;&lt;li&gt;Basic titles.&lt;/li&gt;&#xA;&lt;li&gt;a little public domain music.&lt;/li&gt;&#xA;&lt;li&gt;easy way to add on screen text, and still images to enrich the problems.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Most of the video editor reviews seem to center on live footage and that's not what I have. I remember from when I was much more interested in video about 10 years ago that speeding up footage could take forever-- but it must be better now? And my footage is very simple visually so it should compress better?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any programs that are much better than say iMovie or some other basic editor for this purpose? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What about programs that organize clips? Helper applications?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently discovered GeoGebra and I've been using it for all kinds of graphics for my videos. &lt;/p&gt;&#xA;" OwnerUserId="5080" LastEditorUserId="5690" LastEditDate="2015-12-18T20:27:51.337" LastActivityDate="2015-12-18T20:27:51.337" Title="Video editing software for math/science videos?" Tags="|technology-in-education|computer-programs|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8000" PostTypeId="2" ParentId="746" CreationDate="2015-05-04T19:25:03.307" Score="0" Body="&lt;p&gt;Intelligence is by no means one-dimensional. The ways that any group of people will identify intelligence are as deeply rooted in cultural norms as they are in those rare hallmarks of exceptional potential to create new mathematics. When people achieve exceptional feats with their minds, who chooses to recognize these feats and how they are recognized is again a matter of social conditioning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recognize the diversity in expressions of intelligence, I also worry that these pigeon-holes: learning styles, left/right-brained thinkers*, etc. feed in to the curiosity killing essentialism that I encounter in students who have had demoralizing early math educations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;I'm just not good with math.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found no real connection between math achievement and the likelihood that a students will utter that sentence. It could mean:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I dislike making mistkes and the precision required in maths is more exhusting than rewarding.&lt;/li&gt;&#xA;&lt;li&gt;I went to a school where students learned little in math class, there was too much noise and chaos in the room and I got bad grades.&lt;/li&gt;&#xA;&lt;li&gt;I don't think math is &quot;cool&quot; it's for people who aren't like me: analytical cold people I could never relate to. &lt;/li&gt;&#xA;&lt;li&gt;I'm quite good at math, objectively, but I have really high standards for myself so I still think I'm bad at it. &lt;/li&gt;&#xA;&lt;li&gt;I've come to view math as highly competitive: a race to be the first to be correct. I find this stressful and not fun.&lt;/li&gt;&#xA;&lt;li&gt;I've hardly ever seen anyone like me doing math, so I don't see how it could be something that I might be a part of.&lt;/li&gt;&#xA;&lt;li&gt;I'm fair to creative and artistic to be good at math. You can't be both.&lt;/li&gt;&#xA;&lt;li&gt;Everyone I know says that math is impossible and awful, they must be right. I have not found out for myself... how could all those people be wrong?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;None of these things are &quot;learning styles&quot; --none of these things are intrinsic -- some of these things might indicate that one just doesn't enjoy math which is fine. Many more show how math, the sciences and at times education in general has gate keepers and the harm that these gate keepers can do. Most of the time I do not find a students with a strange brain condition that makes numbers mysterious and dificult. I find instead students who have been walled off from picturing themselves mastering mathematics because they have been taught since birth not to imagine such a thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe we should focus on ways to open all of those closed doors. The primary focus should be achievement not &quot;genius&quot; -- &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*The left and right brain do seem to delegate tasks, but there is no evidence that anyone has a &quot;dominant hemisphere&quot; -- &lt;/p&gt;&#xA;" OwnerUserId="5080" LastActivityDate="2015-05-04T19:25:03.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8001" PostTypeId="2" ParentId="7999" CreationDate="2015-05-04T20:04:30.403" Score="3" Body="&lt;p&gt;It sounds like iMovie or a similar software would be plenty for your needs. If you want to get into other production, there is a bit of a learning curve and many resources online that can get you started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the high end video production software is expensive and does more than you could possibly need, but once you've picked a program, just jump onto YouTube and search for beginner tutorials. There are a bunch of them out there that detail different techniques and go over all the tools in the program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this said, there's a reason that A/V guys go to college to get degrees to do what you're asking for.&lt;/p&gt;&#xA;" OwnerUserId="151" LastActivityDate="2015-05-04T20:04:30.403" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8002" PostTypeId="2" ParentId="6163" CreationDate="2015-05-04T21:10:40.293" Score="3" Body="&lt;p&gt;Here is a quote from a &lt;a href=&quot;http://www.google.ca/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=2&amp;amp;ved=0CCUQFjAB&amp;amp;url=http%3A%2F%2Fwww.mathhouse.org%2Ffiles%2Ffilebox%2FFile%2FMI.pdf&amp;amp;ei=XttHVaukMsiuU7qwgcgC&amp;amp;usg=AFQjCNE6qrFhijtDk4O9AlqDfayusD52PA&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; in &lt;a href=&quot;http://link.springer.com/article/10.1007/s00283-012-9337-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Mathematical Intelligencer&lt;/a&gt; of a mathematician who is in love with triangles: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;No object has ever served mathematics better or longer. Compare the&#xA;  number of nontrivial results which are true for all topological&#xA;  spaces, rings, groups, etc, without putting extra assumptions on them&#xA;  with the number of nontrivial results which are true in any&#xA;  triangle...When it comes to deducing results in mathematics just from&#xA;  the definition of an object, nothing can hold a candle to the&#xA;  triangle. The triangle will serve mathematics forever.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It explicitly tells the value of triangles, however, it does not say anything about the way that this value could be appreciated in schools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. This answer is a part of &lt;a href=&quot;https://mathoverflow.net/a/152391/29316&quot;&gt;my wiki answer&lt;/a&gt; to a MO post. Thus, I've made this one a wiki answer as well.   &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-05-05T06:33:10.217" CommentCount="0" CommunityOwnedDate="2015-05-04T21:10:40.293" ContentLicense="CC BY-SA 3.0" />
  <row Id="8003" PostTypeId="2" ParentId="7478" CreationDate="2015-05-04T21:42:07.370" Score="3" Body="&lt;p&gt;In my SAT tutoring, I refer to such problems as &lt;em&gt;underspecified&lt;/em&gt;.  For instance, consider this problem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $\frac{a}{2} + \frac{b}{3} = 0$, then find $\frac{a}{b}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach the student that since two variables are presented but only one equation, this makes the problem &lt;em&gt;underspecified&lt;/em&gt;. In other words, a and b are not uniquely determined. This gives the student license to make up a number for &lt;em&gt;one&lt;/em&gt; out of $a$ and $b$, then use the given equation to solve for the other variable, and finally substitute both $a$ and $b$ into the expression $\frac{a}{b}$ to obtain the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also many examples of underspecified SAT geometry problems.  Often the problem will ask the student to find the value of $a+b$ or some other expression involving two variables.  Example: &lt;em&gt;The Official SAT Study Guide&lt;/em&gt; (second edition) by CollegeBoard, page 799, problem 13.&lt;/p&gt;&#xA;" OwnerUserId="5104" LastActivityDate="2015-05-04T21:42:07.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8004" PostTypeId="2" ParentId="7354" CreationDate="2015-05-04T21:53:40.517" Score="3" Body="&lt;p&gt;The invisible $1$ makes sense when students are just getting started in using certain formulas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the equation $y = x + 4$, if students are new to the idea of $y = mx + b$, then it can help to ask the class, &quot;if you had to put some number to the left of the $x$, what would it be?&quot;  Someone will probably say &quot;$1$&quot;, and then you can sneak in the &quot;$1$&quot; to the left of the $x$ in the equation that you have written on the board: $y = 1x + 4$.  Then you can ask what the slope of the line is, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same idea applies for the quadratic formula. Beginners can be encouraged to &quot;sneak in the $1$&quot; to the left of $x^2$ in $x^2 - 7x + 10 = 0$ or to the left of $x$ in $2x^2 - x - 15 = 0$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do make clear that the invisible $1$ should be removed from any final answer. I teach that there is value in formatting an answer in a standard way so that it is likely to match precisely the answer given by a fellow student or by an answer key. In many cases, that means losing the $1$.&lt;/p&gt;&#xA;" OwnerUserId="5104" LastActivityDate="2015-05-04T21:53:40.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8005" PostTypeId="1" AcceptedAnswerId="8024" CreationDate="2015-05-04T21:56:06.040" Score="17" ViewCount="459" Body="&lt;p&gt;It is well recognized that learning number names is not an easy task for native English speaking children. For example, the number name for &quot;11&quot; should be learned quite independently from the name of previous numbers, rather than as &quot;ten-one&quot; like in Chinese and a number of other languages. The latter is not only easier to remember but also goes better with the idea of place value. &#xA;Indeed as Iranian, I could explain the situation just using examples from different region of Iran. Children has to learn &quot;national&quot; language (Farsi) when they start school at the age of 7. The structure of number names in Farsi is more or less like English (even a bit worse). Thus, learning number names is not an easy task for Iranian children in general, and even harder for Iranian Turk children whom the language they grow up with (Turkish) has a different (and better) structure for number names (e.g., &quot;11&quot; is &quot;ten-one&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us move beyond early number learning. Years later, one of the algebraic problems of the national mathematics textbook is the following: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Express in words: $(a+b)^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hard for children in general, harder for Iranian Turks. In Farsi, it reads the same as for the reader of this post, outside operation first, inside operation second: &quot;square of the sum of two numbers&quot;. For Iranian Turks, it reads the other way around, inside operation first, outside operation second, something like: &quot;the sum of two numbers to the power of 2&quot;. Not wrong, but not what is expected from them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know any other examples in which the structure of native language of students is in conflict with the structure of language of mathematics that they are expected to learn?            &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-05-09T18:18:23.670" Title="Language as a barrier to learn math" Tags="|language-use|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8006" PostTypeId="2" ParentId="7237" CreationDate="2015-05-04T22:21:52.507" Score="-1" Body="&lt;p&gt;It comes up frequently in solving trigonometric equations such as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2\sin x\cos x=\sin x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Students often divide by &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;. I find myself using this line a lot:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If you're ever tempted to divide both sides by a variable expression, then what you probably need to do is use addition or subtraction to get everything on one side equal to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; on the other side&lt;/em&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Sometimes I explain why, and sometimes I don't.  If I'm teaching a class, I will certainly explain why.  If I'm tutoring a student who I see for just one hour a week, it depends on whether I feel we have time to go into it.&lt;/p&gt;&#xA;&lt;p&gt;When I do explain why, I say, like the OP says, that you can't divide by a variable expression because it might be zero, and we can't divide by zero.  If a student were to say, as in the OP's experience, &amp;quot;well obviously I didn't know that zero was an answer when I was doing the problem,&amp;quot; I might point out that this is a little like saying, &amp;quot;well obviously I didn't know the gun was loaded when I fired it.&amp;quot;  Any risk of disaster is reason enough for caution.&lt;/p&gt;&#xA;" OwnerUserId="5104" LastEditorUserId="6053" LastEditDate="2021-03-17T12:57:52.453" LastActivityDate="2021-03-17T12:57:52.453" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8007" PostTypeId="2" ParentId="8005" CreationDate="2015-05-04T23:10:56.503" Score="3" Body="&lt;p&gt;German/Deutsch: &lt;em&gt;dreihundert fünfundzwanzig&lt;/em&gt;:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/CC9et.jpg&quot; alt=&quot;German325&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-05-04T23:10:56.503" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8008" PostTypeId="2" ParentId="5794" CreationDate="2015-05-04T23:59:56.257" Score="2" Body="&lt;p&gt;I wrote a great blog post on why a negative times a negative equals a positive.  I liken it to pulling weeds.  I'll preview the post here and then provide a link.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Would you say that getting good stuff is a good thing or a bad thing?&lt;br/&gt;&#xA;(2) How about getting bad stuff?&lt;br/&gt;&#xA;(3) Losing good stuff?&lt;br/&gt;&#xA;(4) Losing bad stuff?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://lifesavertutoring.com/blog.php&quot; rel=&quot;nofollow&quot;&gt;http://lifesavertutoring.com/blog.php&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5104" LastActivityDate="2015-05-04T23:59:56.257" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8009" PostTypeId="2" ParentId="6163" CreationDate="2015-05-05T16:50:56.210" Score="5" Body="&lt;p&gt;Nearly everything in Euclidean geometry comes down to a divide-and-conquer approach:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Reduce the question to a question about triangles.&lt;/li&gt;&#xA;&lt;li&gt;Use our extensive knowledge of triangles to answer it.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Several people have mentioned coordinate geometry, but ultimately that, too, comes down to our basic ideas about triangles.  Without establishing those we haven't distinguished Euclidean geometry from, say, hyperbolic geometry, so we wouldn't be able to conclude anything nontrivial about our coordinates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there's a problem here it's not the material, but perhaps that students see this as a grab-bag of disconnected facts instead of the basis of all the math they're going to use in the future.  Every single math class from here on out relies on a strong understanding of triangles, and that becomes more and more pronounced as things progress:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;strong&gt;calc 1&lt;/strong&gt; they'll learn to show that &#xA;$$\lim_{x \to 0} \dfrac{\sin(x)}{x} = 1$$&#xA;using a geometric argument and the squeeze theorem, in order to determine that&#xA;$$\dfrac{d}{dx} \sin(x) = \cos(x).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;strong&gt;calc 2&lt;/strong&gt; they'll need a solid grasp of trig so that they can take integrals like&#xA;$$\int \dfrac{1}{1 + x^2} \, dx = \tan^{-1}(x).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;strong&gt;vector calculus&lt;/strong&gt; they'll need to be able to convert between cartesian, polar, and spherical coordinates and understand why, say,&#xA;$$dx \, dy \, dz = r^2 \sin \phi \, dr \, d\phi \, d\theta$$ &#xA;and moreover they'll need to be able to parametrize a given region in three-dimensional space in, say, spherical coordinates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;strong&gt;differential equations&lt;/strong&gt; (or possibly a physics or engineering class) they'll need to grasp the Fourier transform.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;strong&gt;linear algebra&lt;/strong&gt; they'll learn about inner products in finite dimensions.&lt;/p&gt;&#xA;" OwnerUserId="623" LastActivityDate="2015-05-05T16:50:56.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8010" PostTypeId="2" ParentId="8005" CreationDate="2015-05-05T17:12:05.733" Score="2" Body="&lt;p&gt;Given the preponderance of repetition in mathematics, reuse of symbols for various meanings in differing contexts, interesting fashions of modifying words to indicate additional properties or lack there of, I don't think the result is something that should instill pride.  When one wants to abbreviate gcd(a,b) as (a,b), as well as use (a,b) for the result of other binary operations or relations, one stretches the reader's ability to include context to interpret a particular glyph.  Is the class of near-rings inclusive of the class of rings, or a subclass, or neither?  Should I take a dot product, a cross product, a cartesian product, an ultraproduct, or a semidirect product?  Am I producing when I do this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Time For A Mathematical Esperanto&quot; Paseman, 2015.05.05&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-05T17:12:05.733" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8011" PostTypeId="1" AcceptedAnswerId="8014" CreationDate="2015-05-05T18:36:45.923" Score="30" ViewCount="1356" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/5794/how-to-explain-that-a-negative-number-multiplied-by-a-negative-number-is-a-posit&quot;&gt;This&lt;/a&gt; recent question reminded me of a question: this year several students expressed concern about the expression $\sqrt{-x}$, on the grounds that it must be undefined because $-x$ is a negative number, and negative numbers don't have square roots.  This is in the context of a calculus class for first-year college students, so these are students who ostensibly know algebra (and most of whom have taken a year of calculus already).  (In the context of this class, characterizing negative numbers as not having square roots is correct; they know about complex numbers, and also know they're off-topic.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience so far has been that these students can handle it after being reminded that it's possible that $x=-2$, so I can address it in an ad hoc way when it comes up, but I suspect it indicates a bigger conceptual gap.  So my question is:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Why do students think this?&lt;/li&gt;&#xA;&lt;li&gt;What algebra should I be reviewing to head this off before the specific case comes up?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2020-02-13T22:44:55.873" Title="When $-x$ is positive" Tags="|undergraduate-education|algebra|review|" AnswerCount="7" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="8012" PostTypeId="2" ParentId="8011" CreationDate="2015-05-05T22:57:21.837" Score="11" Body="&lt;p&gt;The following &amp;quot;definition&amp;quot; given in the Glossary of the Common Core State Standards for Mathematics (2010) give you an idea why students think this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Integer&lt;/strong&gt;. A number expressible in the form a or – a for some whole number a.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The problem is not about algebra alone, rather it is about disconnection of teaching of algebra and negative numbers in schools. It has also a long history in which some great mathematicians of the past knowingly or unknowingly avoided negative values (particularly, as input) for an algebraic letter. The following is an interesting passage from Euler (taken from an under review paper of mine that is about the exact same issue you asked).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The question which we have hitherto considered lead all to an equation&#xA;of the form &lt;span class=&quot;math-container&quot;&gt;$ax+ by=c$&lt;/span&gt;, in &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;,&lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, represent integer and&#xA;positive numbers, and in which the values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; must likewise&#xA;be integer and positive. Now, if &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is negative, and the equation has&#xA;the form &lt;span class=&quot;math-container&quot;&gt;$ax-by=c$&lt;/span&gt;, we have questions of quite a different kind,&#xA;admitting an infinite number of answers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;At the first glance, it seems Euler is being careless when he is saying &amp;quot;if &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is negative, and the equation has the form &lt;span class=&quot;math-container&quot;&gt;$ax-by=c$&lt;/span&gt;...&amp;quot;; if &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is negative in the sense that we understand it today, the equation &lt;span class=&quot;math-container&quot;&gt;$ax-by=c$&lt;/span&gt; would be like the equation &lt;span class=&quot;math-container&quot;&gt;$ax+dy=c$&lt;/span&gt;, in which &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;,&lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, represent integer and positive numbers, and Euler has already considered such an equation. But, Euler is not neglectful: when he is saying that &amp;quot;if &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is negative&amp;quot;, he uses the letter &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; as a label denoting the coefficient of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, and then the letter &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; in the equation &lt;span class=&quot;math-container&quot;&gt;$ax-by=c$&lt;/span&gt; is a parameter that represents integer and positive numbers.&lt;/p&gt;&#xA;&lt;p&gt;If there is a way to bypass a negative number as an input of a letter, people find it and use it! Thus, a short answer to your second question would be to review the algebraic nature of negative numbers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Added.&lt;/strong&gt; The following Figure taken from a paper of Warwick Sawyer, &lt;a href=&quot;http://www.marco-learningsystems.com/pages/sawyer/unbelieve.html&quot; rel=&quot;noreferrer&quot;&gt;The Importance of the Unbelievable&lt;/a&gt;, beautifully shows the importance (though often taken for granted) role of negative numbers in our most basic algebraic conceptions.I often use it in my classes to bring the unifying nature of accepting negative numbers to the fore.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/S0RoT.gif&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-05-06T09:41:35.940" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8014" PostTypeId="2" ParentId="8011" CreationDate="2015-05-05T23:42:36.417" Score="19" Body="&lt;p&gt;&lt;strong&gt;Why do they think this?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can think of a few intertwined reasons why the students think that $-x$ is negative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, psychologically the $-$ sign has a very strong pull towards thinking of subtraction and negative numbers. When they were small, your students had more than five years to get used to the idea that $-$ meant subtraction and that subtraction always made things smaller. Then on top of that they probably had a few years to connect $-$ to negative numbers before they were ever introduced to letters representing numbers. Those instincts were forged very strongly and are still stronger than anything that came after, so when they see $-x$ it is natural for them to think of negative numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even more than that, their first and most strongly-forged experience of numbers is the counting numbers, and that will always be the first thing they think of unless specifically directed to think otherwise. If you ask any person -- even a mathematician -- to name three numbers I daresay they will pick three positive whole numbers. They won't even pick 0. In fact, in my experience, 0 takes up a very small amount of mental space and is often forgotten as an option in all sorts of situations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, over in other courses like economics and chemistry, variables usually do represent positive amounts and graphs only exist in the first quadrant, so it is not unnatural for that thinking to transfer over to maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally and most importantly, I don't think that your students think of the $x$ as standing for a range of possible numbers at all most of the time. I'm pretty sure you don't either. If you really analyse your own thinking, I'm pretty sure you find that when doing algebra, you mostly think of $x$ as an object of its own kind that moves around the page, rather than a number at all. You can substitute numbers in its place, sure, and make a statement that it is equal to a number, sure, but they are not the same as actually conceiving of it as a number at all times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What can you do to prevent it?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would probably be a good idea to review the classic functions and their behaviour early in your calculus course -- you probably do this anyway. What you can do to help yourself later on is to focus explicitly on reminding students that $x$ (or y or t or whatever) represent various numbers and that one way to understand a function is to ask yourself about what happens when $x$ is various possibilities. Basically this is considering the domain and range of the function, which you probably also discuss in your calculus course anyway. You can tell them that domain and range are not just information about the function that they need to be able to find for tests, they are a fundamental way of understanding functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As the course progresses, reinforce this idea: every time you write down a function, take a moment to consider the various x-values that could go in and the various function values that come out. (I'm not saying calculate the domain and range formally, just take a moment to think about it aloud.) I recommend asking &quot;Can I put in 0? Can I put in positive numbers? Can I put in negative numbers? What about very small and very big numbers?&quot; If they are used to this happening every time, they are more likely to do it themselves and come up with the idea that $\sqrt{-x}$ is ok by subbing in $-2$ on their own.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-05-05T23:42:36.417" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8015" PostTypeId="2" ParentId="8011" CreationDate="2015-05-06T06:53:57.307" Score="7" Body="&lt;p&gt;I think it's appropriate to lead the student(s) through answering their own objections with some leading questions - perhaps questions like &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;When&lt;/em&gt; is $-x$ &lt;em&gt;negative&lt;/em&gt;?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular it helps reinforce to students that &lt;em&gt;this is the sort of question they should be asking themselves&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why do students think this?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think it's because they're applying a &quot;rule&quot; that says something vaguely like &quot;a minus in front of a number makes it negative&quot; ... but it's an internalized idea they haven't articulated so they don't really notice the flaws in it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Running into these sorts of errors is in some sense part of learning to think mathematically; it's okay to &lt;em&gt;make&lt;/em&gt; these errors, as long as they're treated as something to recognize, and try to address. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's no good drilling someone for weeks on how not to fall off a bike -- they're still going to fall off when they're learning the skill. You can try to anticipate some things, but I don't think you should try too hard to make sure they never make an error. Errors are useful!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been doing mathematics for a long time; I make new mistakes almost every time I learn something new. Being able to identify and correct misapprehensions is part of getting there. I make fewer now than I did 30 or 40 years ago, but that's because I made enough errors to finally internalize things like &quot;definitions matter&quot;.&lt;/p&gt;&#xA;" OwnerUserId="755" LastEditorUserId="755" LastEditDate="2015-05-06T08:46:06.587" LastActivityDate="2015-05-06T08:46:06.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8016" PostTypeId="2" ParentId="82" CreationDate="2015-05-06T09:09:03.140" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there any intuitive way of explaining why the means of a random variable will form a normal distribution around the true mean? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One can certainly motivate a fact that somewhat similar to this (at least under certain conditions), but that's not what the CLT says.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The actual CLT doesn't really say anything about what happens in samples of given finite size. Speaking hypothetically, it would be perfectly consistent with the CLT if sample means at no sample size ever observed - even into the trillions and higher - ever saw anything close to a normal distribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it &lt;em&gt;is&lt;/em&gt; the case that (under certain conditions), standardized sample means at finite sample sizes &lt;em&gt;do&lt;/em&gt; get closer to being standard normal - multiply the sample size by 4, say, and the distance between the CDF of the distribution of the standardized sample mean a standard normal will tend to halve. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a fact - and a very important fact - but the CLT itself does not establish that what we observe will necessarily happen. Other theorems come much nearer to explaining that we will observe it (the Berry-Esséen theorem, for example).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[The question of whether you should avoid attributing it to the CLT is a different question -- given many texts (etc) do it, it may not be worth bucking the trend over a relatively esoteric point that doesn't really alter the property you want to describe.]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I thought about using computer programming and simulations to show that it works, but I feel that sampling from a simulated data is still too abstract to be intuitive.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can get somewhere with rolling dice (means of 1,4,16 rolls, say, and using results from the whole class to get distributions of averages), but it takes a while (keeping in mind it's really cdfs that are converging; trying to demonstrate convergence of a discrete pf to a continuous distribution will get you into trouble).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Outside of that, computer simulation can be very useful, particularly if they're able to do it themselves.&lt;/p&gt;&#xA;" OwnerUserId="755" LastEditorUserId="755" LastEditDate="2015-05-10T11:35:01.863" LastActivityDate="2015-05-10T11:35:01.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8017" PostTypeId="2" ParentId="7237" CreationDate="2015-05-06T13:15:49.280" Score="3" Body="&lt;p&gt;Firstly, division must obey rules regardless of what you are dividing. If the division is performed over real numbers, then you have to respect the rules for division of real numbers, that is no division by &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. In other structures &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is not the only thing that you cannot &amp;quot;divide&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Secondly, when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a variable it should be made crystal clear what it refers to. If it can refer to any arbitrary object, then division by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is meaningless. If it can refer to a real number, whether known or unknown, whether zero or not, then division by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; must again obey real number rules.&lt;/p&gt;&#xA;&lt;p&gt;Thirdly, the rules must not be given empty and without meaning. For real numbers, multiplication corresponds to scaling or stretching something, and division is to undo that. If you multiply by zero, you are scaling something until it is gone; zero size. You can never recover the original size, so division by zero is not allowed.&lt;/p&gt;&#xA;&lt;p&gt;Fourthly, and only after the first three points are made absolutely clear, then you can give some examples to show that indeed division by zero can lead to nonsense.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Fail to solve &lt;span class=&quot;math-container&quot;&gt;$x = 2x$&lt;/span&gt; by dividing by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;!&lt;/p&gt;&#xA;&lt;p&gt;Fail to solve &lt;span class=&quot;math-container&quot;&gt;$x^2 = 0$&lt;/span&gt; by dividing by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, twice!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-05-06T13:15:49.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8018" PostTypeId="1" CreationDate="2015-05-06T15:32:03.533" Score="8" ViewCount="120" Body="&lt;p&gt;I am teaching an undergraduate course in Operations Research to business students (they are not: maths students). I want to check, if and how teaching the steady state makes any sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As in the semesters before, the challenge of this term was to make the concept of &quot;steady state&quot; plausible. Thus we wrote a very simple M/M/1 queueing simulation, repeated it, produced histograms of the queue lengths of the repeated simulation over time, analysed time-regions, where these histograms did not change very much and finally calculated the &quot;ideal steady-state histogram&quot; by solving simple systems of linear equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This made the concept of &quot;steady state&quot; clearer and brought in more joy as an exploratory task than just a calculation task or formula. However, the question came up: where in the real world would a real planner in a real application (and by &quot;real&quot; the students mean: real-real) put that much focus on what is the steady state of a queue?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some ideas from my side (counterarguments in brackets):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*the planner would try to discover, at which time his system stabilises - the supermarket opens, 20 people waiting outside, when will the cashpoints be in their &quot;normal&quot;/average state? (well, wouldn't a simple measure like variance in the simulation suffice to just explore, when a system swang in?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*the planner could use the steady state for more strategic tasks - what happens e.g. if the incoming behaviour of the queue changes e.g. during night shifts? (But still: the planner could just simulate that, calculate averages and never touch his operations-research-concept steady state again, right?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Someone's up for more realistic ideas - and how to teach these? Or should we rather defocus the concept of steady state when teaching a few weeks of basic queueing theory?&lt;/p&gt;&#xA;" OwnerUserId="5142" LastActivityDate="2015-05-06T20:35:39.300" Title="How to teach steady state in queueing (if at all?)" Tags="|statistics|probability|" AnswerCount="1" CommentCount="1" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8019" PostTypeId="2" ParentId="8018" CreationDate="2015-05-06T20:35:39.300" Score="3" Body="&lt;p&gt;This is a stretch as it is far from my area, but my understanding is that steady-state histograms play a &quot;real&quot; role in Internet packet traffic analyses.&#xA;Here is one suggestive figure from a paper cited below (the blue histogram&#xA;is steady-state buffer distribution).&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/4cfcw.png&quot; alt=&quot;SteadyStateHist&quot;&gt;&#xA;&lt;hr /&gt;&#xA;The OP would know better than I if one could make a case for packet queues&#xA;that would be more resistant to counterarguments.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hernández-Orallo, Enrique, and Joan Vila-Carbó. &quot;Network performance analysis based on histogram workload models.&quot; &lt;em&gt;Modeling, Analysis, and Simulation of Computer and Telecommunication Systems&lt;/em&gt;, 2007. MASCOTS'07. 15th International Symposium on. IEEE, 2007.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-05-06T20:35:39.300" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8020" PostTypeId="1" CreationDate="2015-05-06T21:37:23.380" Score="15" ViewCount="2408" Body="&lt;p&gt;I have long been a fan of all of the different methods for factoring quadratics, yet I hardly ever use them in my classroom. The first task they are confronted with, in factoring trinomials, is to find two numbers whos sum is this and whos product is that. I call this the sum-product problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually I teach my students to solve the sum-product problem by having them list all of the pairs of factors. The drawback with this is that students have trouble with what should be positive or negative and do they add or subtract.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I came up with this method recently and I like it because it's deterministic but I wonder about the amount of calculation involved. I'd like to know what plusses and minuses you see in it. I will give you a worst-case example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EXAMPLE: Find two numbers that add to -7 and multiply to -60.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Half of -7 is $-\frac 7 2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the two numbers must be $-\frac 7 2 - x$ and $-\frac 7 2 + x$ for some positive $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus we need to solve $(-\frac 7 2 - x) (-\frac 7 2 + x) = -60$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{49}{4} - x^2 = -60$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$x^2 = \frac{49}{4} + 60 = \frac{289}{4}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$x = \frac{17}{2}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the two numbers are&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$-\frac{7}{2} - \frac{17}{2} = -\frac{24}{2} = -12$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$-\frac{7}{2} + \frac{17}{2} = \frac{10}{2} = 5$&lt;/p&gt;&#xA;" OwnerUserId="5085" LastEditorUserId="5085" LastEditDate="2015-05-12T23:47:09.820" LastActivityDate="2015-05-13T03:05:46.413" Title="The sum - product problem" Tags="|algebra|" AnswerCount="7" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8021" PostTypeId="2" ParentId="8020" CreationDate="2015-05-07T00:23:06.470" Score="9" Body="&lt;p&gt;It is likely too early to show them this...? &#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/SyRSO.jpg&quot; alt=&quot;Hyperbola&quot;&gt;&#xA;&lt;hr /&gt;&#xA;$$x+y = -7, \;\; \text{and} \; \; x y = -60 \;.$$&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-05-08T00:40:13.570" LastActivityDate="2015-05-08T00:40:13.570" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8022" PostTypeId="2" ParentId="8020" CreationDate="2015-05-07T02:03:56.580" Score="10" Body="&lt;p&gt;The method is marvelously clever and straightforward and I can see people thinking it's very cool (which it is). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do feel it would be clearer if it began like this: &quot;Let $x$ and $y$ be the two numbers. Since they sum to $-7$, their average is $-7/2$ and so one is more than $-7/2$ and the other is less. Write $x=-7/2 + d$ and $y = -7/2 - d$.&quot; Then you're explicitly declaring that your real goal is the two numbers and this other number is just a means to an end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea that if two numbers add to $A$, then one must be a bit more than $A/2$ and the other must be a bit less than $A/2$ ... I feel this idea would be quite a conceptual leap for many students. It's certainly something that it is worth them knowing, but I can bet quite a few of them will be very confused by that particular step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's a little nuance in there that the number $x$ you have chosen is implicitly positive. Students familiar with putting a $\pm$ when they perform a square root will get two answers for $x$ and sub them both back in to get four answers (which turn out to only be two answers after all). This will waste quite a bit of time unless you explicitly point out that the $x$ you're using is positive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The calculations involved are actually exactly the same as those you do when completing the square: divide the x-coefficient by 2, square it, add this to the constant, square root, add and subtract this to minus half the x-coefficient. So in terms of calculations it is actually equivalent to completing the square. Completing the square &lt;em&gt;feels&lt;/em&gt; harder for some reason to me -- I'm not sure why. Still, completing the square is a method they probably need to know for general quadratics, so is it worth showing them something else for a particular problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The major problem to me is that the reasoning in it only applies to this specific sort of problem, and so might encourage the &quot;list of types of problems - list of solution methods&quot; approach to maths. I can't see any other context in which this type of reasoning would come up, so it seems to me that you'll be teaching them to say to themselves &quot;there's a clever trick to these ones, let's see if I can remember it&quot; rather than &quot;If I just use plain ordinary reasoning and common general methods I'm sure I can figure it out.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, there are actually thought processes that will tell you precisely whether the factors have to be positive or negative (and you always add them since it says &quot;sum&quot;). If the product is positive then it must be that both factors are positive or both are negative. Then if the sum is positive this is only possible if both factors are positive, and if the sum is negative this is only possible if both factors are negative. On the other hand if the product is negative then it must be that one factor is negative and one is positive. Then if the sum is negative it must be that the negative factor is bigger in size than the positive one, and if the sum is positive then it must be that the negative factor is smaller in size than the positive one. So in short:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;            SUM +                          SUM -&#xA;PRODUCT +   both factors +                 both factors -&#xA;PRODUCT -   big factor +, small factor -   big factor -, small factor +&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;In your example, the pairs of factors of 60 are:&lt;br&gt;&#xA;1, 60; 2, 30; 3,20; 4,15; 5,12; 6,10.&lt;br&gt;&#xA;Since the product is negative we have to have one + and one -. Since the sum is negative the big one has to be -ve. So the pairs of factors we need are actually:&lt;br&gt;&#xA;1, -60; 2, -30; 3, -20; 4, -15; 5, -12; 6, -10.&lt;br&gt;&#xA;So it must be 5 and -12.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-05-07T02:48:45.893" LastActivityDate="2015-05-07T02:48:45.893" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8023" PostTypeId="1" CreationDate="2015-05-07T07:14:54.170" Score="6" ViewCount="1128" Body="&lt;p&gt;From a Grade 3 &lt;a href=&quot;http://www.naplan.edu.au&quot; rel=&quot;nofollow&quot;&gt;NAPLAN&lt;/a&gt; test:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;strong&gt;Question 28&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Joel draws 24 rectangles.&#xA;He separates them into 2 groups so that there are 6 more in one group than the other.&#xA;How many rectangles in the larger group?&#xA;Write your answer in the box.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a multitude of questions.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Critique this question.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How does this type of question fit in the primary curriculum?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How would you teach this?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In this context are any principles of linear algebra applicable to teaching  kids without knowledge of algebra. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="998" LastEditorUserId="998" LastEditDate="2015-05-09T10:22:48.703" LastActivityDate="2015-05-09T10:24:39.247" Title="Systems of linear equations in grade 3?" Tags="|algebra|primary-education|" AnswerCount="7" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8024" PostTypeId="2" ParentId="8005" CreationDate="2015-05-07T09:34:28.123" Score="7" Body="&lt;p&gt;Some examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;&quot;One plus five squared&quot;&lt;/em&gt; could be read as $(1+5)^2$ or $1+5^2$. This is a well-known ambiguity in natural language, for example in the following sentence&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I saw a man on a hill with a telescope.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;we don't know if it was a man who had the telescope, or the speaker, or perhaps the telescope was just standing on some hill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;http://en.wikipedia.org/wiki/List_of_linguistic_example_sentences&quot; rel=&quot;nofollow&quot;&gt;this list&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Double_negative&quot; rel=&quot;nofollow&quot;&gt;Double negation&lt;/a&gt; in some languages doesn't always turn sentences into positives, while in classical logic in math we have $\neg\neg x \iff x$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;The word order may have more or less meaning and it may result in wrong interpretation. For example when you ask &lt;em&gt;&quot;Take 10 and subtract the difference between 3 and 5&quot;&lt;/em&gt; you may get both $10-(3-5) = 12$ and $10-(5-3) = 8$. This effect is even stronger, if the context suggest that negative value doesn't make sense in a non-mathematical way (e.g. number of apples).&lt;/li&gt;&#xA;&lt;li&gt;Some words have special meaning, e.g.&#xA;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;No cat has eight tails. A cat has one tail more than no cat. Therefore, a cat has nine tails.&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Mathematical expressions have highly recursive structure, while spoken natural language isn't prepared to handle more than a few levels. For example&lt;/p&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The rat the cat the dog bit chased escaped.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is an understandable sentence while dissected as follows: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The rat $\Big($the cat $\big($the dog bit$\big)$ chased$\Big)$ escaped.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The rat escaped. Which rat? The rat the cat chased. Which cat? The cat the dog bit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When the term comes from a different language, it may become confusing. A mild example would be &quot;limes&quot;, a bit more confusing could be $P$ and $V$ in &lt;a href=&quot;http://en.wikipedia.org/wiki/Semaphore_%28programming%29&quot; rel=&quot;nofollow&quot;&gt;Semaphores&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Not exactly about language, but I will include nevertheless: context matters (and different language &lt;em&gt;is&lt;/em&gt; a different context).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f(x,y) = x^2+y^2$, what would be $f(\phi, r)$?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Mathematician: $f(\phi, r) = \phi^2 + r^2$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Physicist: $f(\phi, r) = r^2$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2015-05-09T18:18:23.670" LastActivityDate="2015-05-09T18:18:23.670" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8025" PostTypeId="2" ParentId="8023" CreationDate="2015-05-07T11:25:15.697" Score="10" Body="&lt;p&gt;Some of the NAPLAN questions are supposed to be about &quot;working mathematically&quot; and so not be specifically in a particular content area, and be a little bit &quot;out there&quot; so as to test problem-solving. The minimum standard for Year 3 from the NAPLAN website is that the students can &quot;recognise and respond to routine questions addressing known facts in familiar contexts&quot;. Based on that criterion, this question is a bit beyond the minimum standard. However, it's certainly not too difficult for most children.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Putting myself into a Year 3 child's mindset, I would solve this by a sort of trial and error method. I'd write down a whole lot of ways to split 24 into two groups and stop when I found some that were 6 apart. The first one I thought of would probably be splitting it equally: 12+12. Then I'd move one from one group into the other: 12+12, 11+13, 10+14, 9+15. Ah. There it is. So there are 15 in the bigger group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of teaching, I think you'd need to spend time helping children think about solving problems they haven't seen before. If this was a problem you were doing in class, I would ask them: &quot;what does it mean to split them into two groups?&quot; &quot;What ways would there be to do that?&quot; &quot;What does it mean to have six more in one group than the other?&quot;. Perhaps they'll give examples for some of them. One of the cool things about this problem is that it requires them to think about how to add things to get a particular answer, which is helping them to think about addition backwards. This will reinforce a heap of number facts in ways you couldn't do just by telling them about it or drilling them on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more problems in a similar vein with the express purpose of helping children learn both content and problem-solving skills, you could try the &lt;a href=&quot;http://mathematicscentre.com/taskcentre/&quot;&gt;Mathematics Task Centre&lt;/a&gt; or the &lt;a href=&quot;http://www.backtofrontmaths.com.au/&quot;&gt;Back to Front Maths&lt;/a&gt; books and resources (both Australian projects).&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-05-07T18:12:17.387" LastActivityDate="2015-05-07T18:12:17.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8026" PostTypeId="2" ParentId="8023" CreationDate="2015-05-07T12:19:56.337" Score="9" Body="&lt;p&gt;First, I think the title you have chosen for your post is misleading or somehow biased. Question 28 hasn't designed to be thought of as a system of linear equations. It can be seen as a problem that fits into the so-called early algebra movement (whether the designers had it in mind or not), the point of which is to help students recognize numerical patterns and structures. It also fits into The proficiency strands advocated by &lt;a href=&quot;http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1&quot;&gt;Australian National Curriculum&lt;/a&gt;: &lt;em&gt;Understanding&lt;/em&gt;, &lt;em&gt;Fluency&lt;/em&gt;, &lt;em&gt;Problem Solving&lt;/em&gt; and &lt;em&gt;Reasoning&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Critique the question&lt;/strong&gt;. As it can be seen, I am in favor of the question. To solve it students need to decompose 24 into two addends, realize that the decomposition can be done in many different ways though it doesn't matter in which order they add the addends (commutativity of addition, saving time and effort), consider that they can make a new pair of addends from an old one (e.g., $7+17=(7+1)+(17-1)=8+16=24$), and so on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How does this type of question fit in the primary curriculum?&lt;/strong&gt; I hope you have already convinced that it fits and why. If not, I can also refer you to &lt;a href=&quot;http://achievethecore.org/page/254/progressions-documents-for-the-common-core-state-standards-for-mathematics-detail-pg&quot;&gt;Progressions for the Common Core State Standards in Mathematics&lt;/a&gt; to find out that is not just about Australian Curriculum. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How would you teach this?&lt;/strong&gt; First we are not teaching &quot;this&quot; as an isolated topic, let alone as a system of linear equations. As it can be seen in both curriculums mentioned above this kind of problem is just part of certain more general strands developing over the years.     &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-05-07T12:19:56.337" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8027" PostTypeId="2" ParentId="8020" CreationDate="2015-05-07T12:42:22.133" Score="4" Body="&lt;p&gt;Mostly I agree with @DavidButlerUofA had said in his answer. Clear up some of the language to make it more sound and it's a pretty good method for factoring quadratics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is limited by the fact that it isn't easy to apply in other contexts, but neither are many of the other techniques students use to factor quadratics. My suggestion to cope with that is to frame this in the context of the variety of techniques for factoring quadratics. When I teach this topic I try to explore a variety of strategies to factor and solve quadratics, highlighting graphing and completing the square. After some exploration we usually settle into one or two strategies for the class and that's what I'll use in class for explanations. My thought is that it helps to establish that we're choosing to use a particular strategy, and I hope it limits the harm done when students believe there is a &quot;right way&quot; to solve a problem.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2015-05-07T12:42:22.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8028" PostTypeId="2" ParentId="8023" CreationDate="2015-05-07T14:15:19.517" Score="7" Body="&lt;p&gt;The Singapore curriculum has something I want to call bar diagrams, though I might have the name wrong. The child can draw a picture, show a longer train of blocks (because she thought of the rectangles as blocks) and a shorter one, show the 6 sticking out further on the longer one, then figure out that the shorter part (of one and the whole of the other) is the rest of the blocks, and split in two equal portions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My only complaint with the problem is that using something concrete and all the same (like blocks) would have made visualizing (an important mathematical skill) easier.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-05-07T14:15:19.517" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8029" PostTypeId="2" ParentId="8020" CreationDate="2015-05-07T16:10:47.740" Score="7" Body="&lt;p&gt;This idea can actually be used to solve any quadratic equation, and in fact to obtain the quadratic formula, but for some reason it never seems to get mentioned in textbooks, even those that lay all the appropriate groundwork by spending a lot of time with &amp;quot;easily solved quadratics&amp;quot;, such as &lt;span class=&quot;math-container&quot;&gt;$(x+5)^2 - 6 = 30.$&lt;/span&gt; What follows is a &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=9650705&quot; rel=&quot;noreferrer&quot;&gt;post at Math Forum&lt;/a&gt; I made fairly recently (25 November 2014), in which the only changes I've made have been to convert the math symbols to LaTeX form.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Here's a way that can be motivated from &lt;span class=&quot;math-container&quot;&gt;$x^2 = c$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$(c \geq 0)$&lt;/span&gt; and factoring methods. One would have already observed that &lt;span class=&quot;math-container&quot;&gt;$x^2 = c$&lt;/span&gt; can be rewritten as &lt;span class=&quot;math-container&quot;&gt;$(x - \sqrt {c})(x + \sqrt {c}) = 0,$&lt;/span&gt; which gives rise to the solutions &lt;span class=&quot;math-container&quot;&gt;$x = \pm \sqrt {c}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In general, quadratic equations having no linear term are easy to solve, and students should have had plenty of practice with such equations before what follows, including equations such as &lt;span class=&quot;math-container&quot;&gt;$(x+5)^2 - 6 = 30.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In considering &lt;span class=&quot;math-container&quot;&gt;$x^2 + bx = c$&lt;/span&gt; (any quadratic equation can be rewritten this way without altering the solution set), some students want to consider &lt;span class=&quot;math-container&quot;&gt;$x(x+b) = c.$&lt;/span&gt; This is not wrong! What is wrong is to continue and try to extend the zero property to the possibly nonzero number &lt;span class=&quot;math-container&quot;&gt;$c.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In considering &lt;span class=&quot;math-container&quot;&gt;$x(x+b) = c,$&lt;/span&gt; you might point out it's a shame this isn't something like &lt;span class=&quot;math-container&quot;&gt;$(x-d)(x+d) = c,$&lt;/span&gt; because when expanding a difference of squares the linear term drops out, and then the equation would be easy to solve.&lt;/p&gt;&#xA;&lt;p&gt;Let's look closer at what's not working for a specific case.&lt;/p&gt;&#xA;&lt;p&gt;Suppose we want to solve &lt;span class=&quot;math-container&quot;&gt;$x(x+6) = 16.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now &lt;span class=&quot;math-container&quot;&gt;$x(x+6)$&lt;/span&gt; can be written as &lt;span class=&quot;math-container&quot;&gt;$(x-0)(x+6),$&lt;/span&gt; which is half-way to being a difference of squares. What messes things up is that &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; are not the same number. The lightbulb idea is to notice that if were dealing with &lt;span class=&quot;math-container&quot;&gt;$x+3$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$x,$&lt;/span&gt; then the factors &lt;span class=&quot;math-container&quot;&gt;$x-0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x+6$&lt;/span&gt; are exactly &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; left and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; right of &lt;span class=&quot;math-container&quot;&gt;$x+3.$&lt;/span&gt; In other words, instead of trying to solve for &lt;span class=&quot;math-container&quot;&gt;$x,$&lt;/span&gt; instead consider the problem of solving for what lies midway between &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x+6,$&lt;/span&gt; which is &lt;span class=&quot;math-container&quot;&gt;$x+3.$&lt;/span&gt; And note that if we can find the values for &lt;span class=&quot;math-container&quot;&gt;$x+3,$&lt;/span&gt; then we can just subtract &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; to get the values for &lt;span class=&quot;math-container&quot;&gt;$x.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x(x+6) \;\; = \;\; [(x+3) - 3][(x+3) + 3] \;\; = \;\; 16$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Expanding (not completely, but in a smart way) gives&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(x+3)^2 - 9 \; = \; 16$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(x+3)^2 \; = \; 25$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x+3 \; = \; -5, \; 5$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x \; = \; -5 - 3, \; 5 - 3$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x \; = \; -8, \; 2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Doing the same thing algebraically to &lt;span class=&quot;math-container&quot;&gt;$x^2 + bx = c$&lt;/span&gt; gives the quadratic formula for equations of the form &lt;span class=&quot;math-container&quot;&gt;$x^2 + bx - c = 0.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;(comments not included in my original post)&lt;/em&gt; Some readers will recognize that this is essentially the same technique as the standard variable substitution that gets rid of the linear term, namely &lt;span class=&quot;math-container&quot;&gt;$u = x + \frac{b}{2}.$&lt;/span&gt; Note that this is simply putting &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; equal to the average of &lt;span class=&quot;math-container&quot;&gt;$(x-r_{1})$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x-r_{2}),$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$r_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r_2$&lt;/span&gt; are the two solutions, since&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{(x - r_{1}) \; + \; (x - r_{2})}{2} \;\; = \;\; x - \frac{r_1 + r_{2}}{2} \;\; = \;\; x + \frac{b}{2},$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where the last equality follows from &lt;span class=&quot;math-container&quot;&gt;$r_1 + r_2 = -b$&lt;/span&gt; (one of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Vieta%27s_formulas&quot; rel=&quot;noreferrer&quot;&gt;Vieta formulas&lt;/a&gt;). The same idea is used to get rid of the quadratic term in a cubic equation, namely put &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; equal to the average of &lt;span class=&quot;math-container&quot;&gt;$(x-r_{1})$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x-r_{2})$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x-r_{3}),$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$r_{1},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$r_{2},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$r_{3}$&lt;/span&gt; are the three solutions, and then make use of the Vieta formula for the sum of the roots of a cubic equation in order to get the substitution explicitly in terms of the coefficients of the cubic equation (rather than explicitly in terms of the solutions).&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-05-07T16:10:47.740" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8030" PostTypeId="2" ParentId="8020" CreationDate="2015-05-07T16:38:02.100" Score="2" Body="&lt;p&gt;It seems to me that if you are going so far as to invent your own specialized completing-the-square-like method, you might as well use a widely-accepted completing-the-square-like method:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To factor &lt;span class=&quot;math-container&quot;&gt;$x^2 - 7x - 60$&lt;/span&gt;, we can use the quadratic formula to find the roots of &lt;span class=&quot;math-container&quot;&gt;$x^2 - 7x - 60 = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{7 \pm \sqrt{289}}{2} = 12 \text{ and } -5$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$x^2 - 7x - 60 = (x-12)(x+5)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-05-07T16:38:02.100" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8031" PostTypeId="2" ParentId="8023" CreationDate="2015-05-07T17:55:03.693" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Joel draws 24 rectangles. He separates them into 2 groups so that there are 6 more in one group than the other. How many rectangles in the larger group? Write your answer in the box.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Critique this question.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It's a good question, and doesn't need to involve systems of linear equations per your question title.  The question is clear and simple to understand with no complex language or strange/confusing sentence structures.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How does this type of question fit in the primary curriculum?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Grouping and counting of objects is well within this age range's comprehension and ability, and this is the grade level where formal division is approached.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How would you teach this?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Two methods:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The larger group has six more than the smaller group, so go ahead and take six and put them in one group.  Then split the rest equally between the two groups.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Alternately, a brute force approach:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Group them however you like.  Are there six more in one than the other?  If so, you're done, count the larger group and answer.  If not, move one from one group to the other, and check again.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4617" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-05-07T17:55:03.693" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8032" PostTypeId="2" ParentId="8023" CreationDate="2015-05-07T20:53:21.720" Score="5" Body="&lt;p&gt;Another (quite elementary) way to solve this problem is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;First, if we divide the $24$ rectangles into two &lt;em&gt;equal&lt;/em&gt; groups, there will be $12$ in each group.  Now, we actually want one group to have $6$ more than the other group, so we move $3$ from one group into the other.  (Why $3$ instead of $6$?  Because each &lt;em&gt;one&lt;/em&gt; that we move results in the difference between the two groups growing by &lt;em&gt;two&lt;/em&gt;.)  So we have $12-3=9$ in one group and $12+3=15$ in the other group.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems like quite a reasonable 3rd grade problem.  It only requires knowledge of two number facts ($12+12=24$ and $3+3=6$) and one important idea (that moving something from one group to another decreases the size of the first group by $1$ and at the same time increases the size of the second group by $1$, so the difference in sizes changes by $2$.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2015-05-07T21:28:32.537" LastActivityDate="2015-05-07T21:28:32.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8033" PostTypeId="2" ParentId="750" CreationDate="2015-05-07T23:08:07.217" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Have a collection of objects and ask the child to put them in order based on some attribute of them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For example, my wife Catherine (a preschool teacher) and I came up with the following activity which we did with a group of children: We got empty cardboard milk containers and filled them with different amounts of sand and then glued them shut. Then we asked the children to put them in order of how heavy they were. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We've also done it with pictures of sunflowers and asked them to order them by height, and shakers which we asked to put in order of loudness. Each of these activities were part of group sessions where a story was told or read that had something to do with what the children were ordering. For example, with the sunflowers, they read &lt;em&gt;Titch&lt;/em&gt; by Pat Hutchins.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea is that before children can learn to measure things, they have to be able to compare things, and they need to realise that there are multiple aspects of objects that can be compared.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-05-07T23:28:01.577" LastActivityDate="2015-05-07T23:28:01.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8034" PostTypeId="2" ParentId="807" CreationDate="2015-05-08T00:21:40.423" Score="2" Body="&lt;p&gt;I started a puzzle club called One Hundred Factorial at my university to give such students a place where they can have their interest in maths nurtured. Once a week, a group of staff and students come together to solve mathematical puzzles and talk maths. We used to also have an online forum, where students were encouraged to write up solutions to the puzzles and post them (though this is in remission at the moment as we search for a more appropriate platform).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It gives those &quot;bored&quot; students something else to think about and people like themselves they can relate to. Many students who have been regulars at these sessions have gone on to become postgraduate students later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An alternative to a puzzle session could be a bit like a book club, where you come together to talk about something interesting or cool that you've been reading or thinking about. If this ends up just being you and one student, then it turns into the mentoring system like Aleks Vlasev describes, which is very excellent anyway.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-05-08T00:21:40.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8035" PostTypeId="2" ParentId="8023" CreationDate="2015-05-08T00:53:53.093" Score="1" Body="&lt;p&gt;I think the wording of the question is a bit abstract. Why rectangles? How does Joel separate them after drawing them? Perhaps specifying objects or using manipulatives may be better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose I know that it doesn't require linear equations, but, surely, &#xA;that's the underlying maths involved and recognizing that can only aid the teaching of this question. As long as it doesn't go over their heads one could have a go at treating it as an algebra question. How about writing down _ + ? = 24 and _ - ? = 6 (or larger + smaller = 24 and larger - smaller = 6) and solving by, say, trial and error. It could be generalized (ie replace 24 and 6 by other numbers, or that the larger is twice as large of the smaller). In any case this type of question be used in later years too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if it's not an algebra question, is doing it on paper the best way, involving lots of erasing and mess. Otherwise, it's a part-part-whole question with part unknown, according to &lt;a href=&quot;http://www.math.niu.edu/courses/math402/packet/packet-2.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.math.niu.edu/courses/math402/packet/packet-2.pdf&lt;/a&gt; Or is a compare question. Though, this classification just confuses me.&lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="998" LastEditDate="2015-05-09T10:24:39.247" LastActivityDate="2015-05-09T10:24:39.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8036" PostTypeId="2" ParentId="8023" CreationDate="2015-05-08T11:43:03.057" Score="2" Body="&lt;p&gt;Because you can solve this problem with a system of linear equations does not mean you should. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I'm with all the respondents who have found the title of the question a little misleading. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it is interesting that this question should make you think of a system of linear equations and thereby make you question its relevance to grade 3 students (which I gather must be a young age), because, in my humble and uninformed opinion, this question should make the student think of down-to-earth and practical ways of approaching the problem, rather than mechanically reach for a rigid tool.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question, it seems to me, should teach students to 1) see that the problem is not about rectangles, but about any kind of discrete objects, like marbles or apples and 2) since 24 is a very manageable number, the student should solve it by trial and error playing with small objects and/or dots on a sheet of paper. After finding a solution, the student should think about whether there might be more than 1 solution. They may also think about numbers other than 24, think about numbers other than 6, groupings other than 2, etc..&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I doubt very much that the student is expected to compute the discriminant of a system of linear equations.&lt;/p&gt;&#xA;" OwnerUserId="553" LastActivityDate="2015-05-08T11:43:03.057" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8038" PostTypeId="1" AcceptedAnswerId="8124" CreationDate="2015-05-09T02:31:31.610" Score="9" ViewCount="831" Body="&lt;p&gt;My school district has decided to provide an iPad to every student in the district, from kindergarten to grade 3, starting next September -- although a handful of classrooms are piloting this currently.  The iPads will stay at school (i.e. they won't be sent home with the students).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I predict that some parents will want to opt out.  Let's take that as a given, i.e. a premise for my question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q1. How can a teacher accommodate the student(s) who have opted out?  This isn't a rhetorical question, and I realize it may be difficult.  I really am looking for creative ideas here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q2. If a family wishes to opt out in a more drastic way, such that their child is not exposed to the iPads at all, how can a school accommodate this?  (Same clarification as for Q1.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Context: teachers here are just starting to figure out how to incorporate iPads into their teaching at this time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I expect teachers will earn higher principal evaluation scores for using the iPads in their teaching (as opposed to stuffing them in a closet).  Note, teacher tenure decisions are now tied to principal evaluation score, and tenure can be revoked.  Thus, pressure will be high for teachers to incorporate iPads to at least some extent into the school day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit 5/18/15&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Responding to @BenCrowell:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why would parents opt out? I can imagine why they would opt out of having them brought home (don't want responsibility for an expensive device), but you're not talking about having them brought home. I can imagine why they would prefer that the whole program not exist (the program sounds like a stupid idea to me, too). But how could they imagine that their kid will benefit from being the only one in the room who doesn't have the shiny toy, who doesn't get to follow the same lesson? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A growing number of parents in our district are voicing objections to the plan; whether it's one lone student in a class who's opting out, or a more substantial number of students, I'd still like to get an idea how a teacher could differentiate instruction for the padders and the non-padders.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One parent in my district wrote the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Interaction with technology, especially with touchscreen devices, can reduce students' attention spans and their desire to sink into other more difficult, but ultimately richer experiences, such as reading for pleasure. We have noticed this effect especially strongly with our boys and other boys in general. We would like to see our elementary schools continue to provide an environment of positive values, respect for others, and intellectual curiosity, and interest in activities that will provide a lifetime of enrichment. We have found that touch screen devices are so compelling for children this age that it can take away interest from activities that do not have such quick feedback, but that provide depth in the long run. We believe that computer literacy is valuable for students, especially for students that do not have computers in their home, but we believe that iPads and other touchscreen devices are not the right tools for teaching computer literacy.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found a number of districts that have already implemented a 1:1 program, and they all have opt-out forms posted on their websites.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;I would ask contributors not to argue with the premise in this question, &quot;I predict that some parents will want to opt out.  Let's take that as a given, i.e. a premise for my question.&quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In any case, this is off-topic for the site. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I hope some SE participants will put their thinking hats on and help me try to figure out how the differentiation could be done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is going to be an issue in many, many schools, sooner than you might think.  For example, the state of New York is offering schools a sizeable subsidy for supplying personal devices for students, through the Smart Schools Bond Act.  See &lt;a href=&quot;http://programs.governor.ny.gov/smart-schools-ny&quot; rel=&quot;nofollow&quot;&gt;http://programs.governor.ny.gov/smart-schools-ny&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We also don't know what the ipads are used for, so we can't answer. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Frankly, I think the folks who participate on this site would probably be able to think up better ways to use them than your typical elementary teachers, whose forte is very rarely mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not ready to give up yet on the idea of differentiating instruction to accommodate both padders and non-padders in one classroom.  I personally haven't been able to imagine any ways to do it, though, which is why I posed the question and bounty here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And let's not forget Q2 -- I'm also trying to figure out how a teacher or a school can accommodate an individual family that's saying, I don't give permission; please keep the devices well away from my child.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit 5/24/15&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks to those who have participated so far.  Responding to some of the comments:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The parents were upset about their kids touching a computer screen,&#xA;  not seeing the computer screen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That is indeed the scenario in Q1, but in Q2, a family is attempting to opt out in a more blanket way.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The dreaded screen of the iPad&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ouch!  A bit judgmental, no?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What do you do if a parent objects to the Pearson textbook you use,&#xA;  without having read it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Speaking for myself, as a parent of a student in the state of New York, I have objected to some Pearson quasi-textbooks (EngageNY &quot;modules&quot;), but &lt;em&gt;after&lt;/em&gt; having read them.&lt;/p&gt;&#xA;" OwnerUserId="4956" LastEditorUserId="4956" LastEditDate="2015-05-24T14:50:04.060" LastActivityDate="2015-05-24T14:50:04.060" Title="1:1 iPads for K-3: how can an educator accommodate student(s) opting out?" Tags="|technology-in-education|primary-education|ipads|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="8039" PostTypeId="2" ParentId="4379" CreationDate="2015-05-09T08:09:53.773" Score="0" Body="&lt;p&gt;Consider using any screen sharing software of your flavor.  I've used RealVNC but you likely want something more feature rich.  Support for two mice or simultaneous drawing would be a bonus.  Then use it with any math drawing or presentation software you like.&lt;/p&gt;&#xA;" OwnerUserId="2071" LastActivityDate="2015-05-09T08:09:53.773" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8040" PostTypeId="2" ParentId="8038" CreationDate="2015-05-09T10:18:17.177" Score="2" Body="&lt;p&gt;(This is more of a starting point than a complete answer. Having written it, it's also mostly aimed at the wrong age group.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some tasks you might want to use the ipads for, you can replace them with a different object: a single-person whiteboard for drawing/displaying their answer, a calculator, compass and ruler for construction, tracing paper for doing translations (maybe plus a torch to do enlargement??). If you're feeling really bold, get them to do logic circuits using some construction toy - the ipad kids will want to join in then, I'd think. If you can find the time, you could perhaps use this to start a discussion on the role of technology in maths (and in society as a whole). Of course, some of this would be more of a pedagogy-redesign, rather than a coping strategy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some tasks, which are really suited to use of computers, like simulations, or exploring the effect of changing parameters, will be much harder for the students to do by hand, if at all possible. Again, this might contribute to the discussion on what part computers should play in maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the choice, I would start setting tasks where sometimes the use of a computer is better, and sometimes not (eg proof type questions), and use it to teach them to choose the right tool, and to understand the difference between experimental evidence and mathematical proof.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="4746" LastEditDate="2015-05-09T10:23:17.980" LastActivityDate="2015-05-09T10:23:17.980" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8041" PostTypeId="1" CreationDate="2015-05-09T12:25:14.197" Score="17" ViewCount="2167" Body="&lt;p&gt;I am exhausted of teaching Mathematical induction to my little brother. I have given him many examples, Domino effect, aligned shops of hot dogs etc and every time he says that he got it but when I come across the problem mathematically, he starts to gaze at me with Pin-drop silence and it breaks off my attention. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it's not my duty to teach him but he is saying that his teacher didn't tell them underlying concept. I am feeling disgraced because I couldn't let him understand it after a enormous trial. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please tell me how should I explain him?   &lt;/p&gt;&#xA;" OwnerUserId="4890" OwnerDisplayName="Sufyan Naeem" LastEditorUserId="143" LastEditDate="2015-05-09T15:37:37.293" LastActivityDate="2022-06-28T09:36:19.917" Title="How to teach Mathematical Induction mathematically?" Tags="|secondary-education|proofs|concept-motivation|" AnswerCount="10" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="8042" PostTypeId="2" ParentId="8041" CreationDate="2015-05-09T13:12:18.040" Score="2" Body="&lt;p&gt;I am not sure your analogies are good. To my taste the underlying concept of induction does not map to a process or an action at all.&lt;/p&gt;&#xA;&lt;p&gt;Induction says what is true for an arbitrary collection, if it KEEPS being true under a certain expansion procedure, then is true for any collection produced by this procedure infinitely applied to the collection.&lt;/p&gt;&#xA;&lt;p&gt;It is not an explanation, but it might give you a clue for constructing an explanation.&lt;/p&gt;&#xA;&lt;p&gt;I would begin with trivial inductive definitions of known things. For example: take a bunch of toothpicks. It is a bunch of toothpicks. Add one. Is it still a bunch of toothpicks? Could you destroy this property by ADDING one toothpick? Therefore, this property will hold under infinite addition of one toothpick. We conclude that a bunch of toothpicks will remain defined as such.&lt;/p&gt;&#xA;&lt;p&gt;Induction is the same but applied to a particular property of an initial object...&lt;/p&gt;&#xA;&lt;p&gt;You can take opposite approach (more illustrative, but less true). Take a cup of water and divide it. Water remains water under division. This is what makes deduction a legitimate way of reasoning: some property holds, if it holds once it will do so INDEFINITELY under the same circumstances. (Unfortunately the example with water invisibly change these circumstances, and it is fun to think about too)&lt;/p&gt;&#xA;" OwnerUserId="5149" OwnerDisplayName="Silly Sad" LastEditorUserId="19777" LastEditDate="2022-06-27T18:24:02.310" LastActivityDate="2022-06-27T18:24:02.310" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8043" PostTypeId="2" ParentId="8041" CreationDate="2015-05-09T14:01:46.863" Score="4" Body="&lt;p&gt;This is a version of user @Mann's comment.&#xA;I owe the ladder metaphor to Joe Malkevitch:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/yRulF.png&quot; alt=&quot;InductionBox&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&lt;sup&gt;&#xA;(From &lt;a href=&quot;http://howtofoldit.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra&lt;/em&gt;&lt;/a&gt;, p.6.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-05-09T14:01:46.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8044" PostTypeId="2" ParentId="8041" CreationDate="2015-05-09T14:30:32.313" Score="14" Body="&lt;p&gt;It seems that some people find it easier to grasp the following &quot;contrapositive&quot; equivalent of the induction principle. (I have no idea whether your brother is one of these people, but it might be worth a try.) If a statement $S(n)$ (about natural numbers $n$) is true for $n=1$ but false for some other value of $n$, say $N$, then there must be some $n$ such that $S(n)$ is true but $S(n+1)$ is false.  (Intuitive reason: Imagine counting from $1$ to $N$, and checking, for each number $n$ along the way, whether $S(n)$ is true or false. $S(n)$ starts out true ($S(1)$) but ends up false ($S(N)$), so it must have changed from true to false somewhere along the way.)&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2015-05-09T14:30:32.313" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8045" PostTypeId="2" ParentId="1550" CreationDate="2015-05-09T16:00:36.780" Score="3" Body="&lt;p&gt;How about figuring out the dimensions of a cylindrical can of a specific volume that will minimize the surface area (and therefore minimize the amount of metal material needed to make up the can)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you've taught calculus, you probably recognize this as one of the classic textbook optimization examples along with fences against barns and picture frames and the like, but the cylindrical can example seems to me the most practically relevant out of those classics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that a rectangular prism shape would make the optimization solution too boring, and any variation in which a container has an open top just seems too contrived, since stuff is never shipped in a container with an open top.&lt;/p&gt;&#xA;" OwnerUserId="5104" LastActivityDate="2015-05-09T16:00:36.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8046" PostTypeId="2" ParentId="8011" CreationDate="2015-05-09T21:36:10.447" Score="5" Body="&lt;p&gt;I tutor many students whose teachers have been telling them to pronounce -x not as &quot;negative x&quot; but instead as &quot;the opposite of x.&quot;  I assume that this semantic change is aimed at instilling in students the idea that -x need not be a negative number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can ask questions such as &quot;what is the opposite of negative 3&quot; and &quot;what is the opposite of 4&quot; to get students used to this kind of thinking before you begin teaching about opposites of variables.&lt;/p&gt;&#xA;" OwnerUserId="5104" LastActivityDate="2015-05-09T21:36:10.447" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8047" PostTypeId="2" ParentId="7420" CreationDate="2015-05-09T22:31:56.433" Score="2" Body="&lt;p&gt;I agree with @BenjaminDickman that practice tests are crucial.  It stands to reason that the most realistic practice tests will be the ones made available by the company who writes the actual SAT, CollegeBoard.  This book has ten full-length practice tests: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0874478529&quot; rel=&quot;nofollow&quot;&gt;The Official SAT Study Guide Second Edition&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Warning: this book will be most useful before CollegeBoard makes major changes to the SAT in 2016.&lt;/p&gt;&#xA;" OwnerUserId="5104" LastEditorUserId="64" LastEditDate="2015-05-11T21:01:23.517" LastActivityDate="2015-05-11T21:01:23.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8048" PostTypeId="2" ParentId="1550" CreationDate="2015-05-10T03:02:43.647" Score="17" Body="&lt;p&gt;When someone swallows a dose of a drug, it doesn't go into their bloodstream all at once. What will the drug's peak blood concentration be, and when will it be reached?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the drug is caffeine, which is absorbed and eliminated by first-order kinetics, its blood concentration $c$ rises and falls over time along the curve&#xA;$$c(t) = \tfrac{D}{1-\beta/\alpha}\left(e^{-\beta t} - e^{-\alpha t}\right),$$&#xA;where $\alpha$ is the absorption rate, $\beta$ is the elimination rate, and $D$ is the size of the dose. The expression for the time of the peak is very pretty. Once you work it out, you can check your answer on p. 108 of &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pubmed/19125908&quot;&gt;this study&lt;/a&gt;, where the authors use it to help interpret their data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For alcohol, which is eliminated by zeroth-order kinetics, the differential equation for the blood concentration curve $c(t)$ is simple enough for a first-year calculus student to solve:&#xA;$$c'(t) = D\alpha e^{-\alpha t} - \beta.$$&#xA;After checking your work against Equation 1 from &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pubmed/25869833&quot;&gt;this study&lt;/a&gt;, you can find the peak time as before.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In case the study links break, here they are for the record:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pubmed/19125908&quot;&gt;Seng et al.&lt;/a&gt; Population pharmacokinetics of caffeine in healthy male adults using mixed-effects models. Journal of Clinical Pharmacy and Therapeutics (2009).&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pubmed/25869833&quot;&gt;Uemura et al.&lt;/a&gt; Individual differences in the kinetics of alcohol absorption and elimination: A human study. Forensic Science, Medicine, and Pathology (2005).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Although finding the blood concentration curve for caffeine is probably beyond first-year calculus students, it might be a fun exercise for differential equations students. It's given by the linear system&#xA;$$\left[ \begin{array}{r} s' \\ c'\end{array} \right]&#xA;= \left[ \begin{array}{r} -\alpha &amp;amp; 0 \\ \alpha &amp;amp; -\beta \end{array} \right]&#xA;\left[ \begin{array}{r} s \\ c\end{array} \right],$$&#xA;where $s$ is the concentration of caffeine in the drinker's digestive tract.&lt;/p&gt;&#xA;" OwnerUserId="5119" LastActivityDate="2015-05-10T03:02:43.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8049" PostTypeId="2" ParentId="82" CreationDate="2015-05-10T13:28:30.473" Score="3" Body="&lt;p&gt;It should be possible to convey the idea of CLT graphically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Represent the outcome of a dice roll by a histogram, six segment of height 1/6 standing over each number from 1 to 6. Then, represent the outcome of the sum of two dices rolls the same way, and more importantly show how one goes from the first histogram to the second. For example, represent first independently the two rolls, one above the other, and consider how each outcome of the second dice affects the sum: you could draw six histograms, one for each outcome of the second dice, with segments of height 1/36 placed at 2 to 7 for the first one, 3 to 8 for the second, etc. Then sum them into the histogram representing the sum of the two rolls. At this stage, you illustrated how &lt;strong&gt;sum of random variables have a distribution obtained by convolution&lt;/strong&gt; (you don't have to say the big words).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then represent in a sequence what happens when we draw more dices, and scale down the horizontal axis and scale up the vertical axis so that the picture fits in the same box; tell the students about the scaling. At this stage, they will &lt;strong&gt;see the normal distribution emerge&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then do the same with another starting distribution, say one that looks quite random. Again, the normal distribution will appear, and this will &lt;strong&gt;show how universal the normal distribution is&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then represent two normal distributions and represent their convolution. This will &lt;strong&gt;show the normal distribution as a fixed point&lt;/strong&gt; (up to scaling and translation) of the operation of &quot;taking the sum of random variables&quot;. Then what has to be accepted is that it is an attracting fixed point, but at this stage you give a pretty decent understanding of what is going on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a nutshell, looking at sum of random variable one performs a certain operation on their distribution (hardly surprising) and rescaling to only retain the &quot;shape&quot; of the distribution, we get an operation that has an attractive fixed point, something not obvious but a plausible outcome in such a situation. The normal distribution is important because it happens to be this fixed point. The fact that the starting distribution does not matter seems less magical once one has understood this dynamical point of view; it is pretty much the same phenomenon as in simpler systems : e.g. if we start from any number, divide it by 2 and then add 1, and perform this operation over and over, whatever the starting number we end up closer and closer to 2. (You can even start with this before showing how it goes for the CLT).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last, if you want to warn them about &lt;strong&gt;how important it is to only use a theorem when its hypotheses are verified&lt;/strong&gt;, you can do represent the convolution of a Cauchy distribution with itself, to show that it gives another fixed point! This will show that the normal distribution is not universal when the second moment is not finite.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-05-10T13:28:30.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8050" PostTypeId="2" ParentId="7420" CreationDate="2015-05-10T17:42:47.033" Score="2" Body="&lt;p&gt;As far as drills go - I've made the observation that calculators are a detriment in many situations. When I am with a student who takes 30 seconds on a calculator to do a bit of simple math that I can do in my head almost instantly, I try to encourage them to learn to do mental math. Yes, calculators are allowed on the SAT, much to my chagrin, but if their use is a 30 second delay, after 30 questions, there are 15 wasted minutes. Minutes than can be used to answer more questions or go back and check work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd suggest a review of the 12x12 multiplication table. When it dropped to 10x10, I don't know, but when I went to school, it was 12x12 and we liked it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, there are techniques to look at the choices of answers and cross out 2 or even 3 of the 4 that are not candidates. For example, 2 equations, 2 unknowns. I've seen practice exams where the choices are easy to put back into the equations, and brute force elimination in seconds instead of actually solving the problem. For some problems, the answers are each an order of magnitude different. Why stop to multiply 2.879 by 3.265 when only only number is in the 9-10 range? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to learning the process, I respect &quot;show all work&quot; and will explain to students that they lost points because they didn't do just that, even though they have a correct answer. (again note, I am not the teacher, just helping after the fact). But on the SATs, it's the opposite. It's an enemy to be beaten, and the pedagogical rules don't apply. Every trick, whether to save time or eliminate incorrect answers, should be used.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Update to the &quot;no trig&quot; comment - this question is in CK-12's SAT Flex Prep Book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/3MRwV.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To solve exactly, the trig functions are needed. But. If angle A were 45$^\circ$, I'd know the base was 6, and the sides each $3\sqrt{2}$. That's about 14.4. With only one answer greater than 14.4, The answer D is obvious. And, in my opinion, avoids the not so small risk that the calculator is set to radians. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-05-10T23:31:12.607" LastActivityDate="2015-05-10T23:31:12.607" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8051" PostTypeId="2" ParentId="1550" CreationDate="2015-05-10T19:01:18.850" Score="3" Body="&lt;p&gt;The concept of impedance matching is one that is occurring in your daily life right now if you're using a wired internet connection. There are more sophisticated and less sophisticated examples of impedance matching. A very conceptually simple one is the following. Suppose that a fixed voltage is applied to resistors $x$ and $y$, which are in series. The resistance $y$ is fixed. You are free to choose any value of $x$ in order to maximize the power dissipated in $x$. The power is $V^2x/(x+y)^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example from everyday life is that you want to throw a baseball as far as possible. The range is $R=(2v^2/g)\sin\theta\cos\theta$, where $v$ is the fixed speed at which you can throw, and $\theta$ is the angle at which you throw the ball, relative to the horizontal. At what angle should you throw?&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2015-05-10T19:01:18.850" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8052" PostTypeId="1" CreationDate="2015-05-10T21:07:41.513" Score="35" ViewCount="8690" Body="&lt;p&gt;I wonder how you teachers walk the line between justifying mathematics because of&#xA;its many&amp;mdash;and sometimes surprising&amp;mdash;applications, and justifying it as the study&#xA;of one of the great intellectual and creative achievements of humankind?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have quoted to my students G.H. Hardy's famous line,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and then contrasted this with the role number theory plays in contemporary&#xA;cryptography.&#xA;But I feel slightly guilty in doing so, because I believe that even without&#xA;the applications to cryptography that Hardy could not foresee&amp;mdash;if in fact&#xA;number theory were completely &quot;useless&quot;&amp;mdash;it&#xA;would nevertheless be well-worth studying for anyone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One provocation is&#xA;Andrew Hacker's influential article in&#xA;the &lt;em&gt;NYTimes&lt;/em&gt;, &#xA;&lt;a href=&quot;http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=0&quot; rel=&quot;noreferrer&quot;&gt;Is Algebra Necessary?&lt;/a&gt;&#xA;I believe your cultural education is not complete unless you understand&#xA;something of the achievements of mathematics, even if pure and useless.&#xA;But this is a difficult argument to make when you are teaching students how&#xA;to factor quadratic polynomials, e.g., &#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/q/8020/511&quot;&gt;The sum - product problem&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to repeat, how do you walk this line?&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2021-04-30T21:11:55.230" Title="Justifications for: Why learn mathematics?" Tags="|mathematical-pedagogy|student-motivation|" AnswerCount="15" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="8053" PostTypeId="2" ParentId="1550" CreationDate="2015-05-10T23:03:58.677" Score="1" Body="&lt;p&gt;Many daily life and OR problems take the following form. One has a container of fixed size and a collection of objects each with a value, and size less than that of the container. The goal of the optimization is to decide what items to take which maximizes the value of what gets taken subject to the size constraint. Problems of this kind are known as knapsack problems, and a problem of this kind is solved every time one takes an airplane trip and also when the International Space Station has supplies sent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Knapsack_problem&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Knapsack_problem&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2015-05-10T23:03:58.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8054" PostTypeId="2" ParentId="8052" CreationDate="2015-05-10T23:49:58.970" Score="26" Body="&lt;p&gt;Most of what you learn in school isn't directly useful.  When I was in primary school I was taught the difference between warm and cold-blooded animals.  I've never used that information; should I not have been taught it?  Another example is that in any English-speaking country you still have to take English courses even after you speak the language fluently.  In day-to-day life you'll never need to know about Shakespeare or whatever literature the curriculum exposes you to.  But it's still important to develop your communication and reasoning skills, your ability to think critically and creatively, to expose you to things that are influential on our culture, to show you a field or a way of looking at life that might be interesting to you or that might influence your career choice, and so on.  All of this is applicable to mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5141" LastEditorUserId="5141" LastEditDate="2015-05-13T01:57:39.007" LastActivityDate="2015-05-13T01:57:39.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8055" PostTypeId="2" ParentId="8052" CreationDate="2015-05-11T00:10:16.693" Score="0" Body="&lt;p&gt;Maybe teachers should have some of &lt;a href=&quot;http://www.ams.org/samplings/posters/posters&quot; rel=&quot;nofollow&quot;&gt;these posters&lt;/a&gt; (available free from the AMS) hanging around the place.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-05-11T00:10:16.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8056" PostTypeId="2" ParentId="8052" CreationDate="2015-05-11T03:16:11.937" Score="5" Body="&lt;p&gt;I'm no teacher, but my position is if you're only studying what has some useful application now, how are you ever going to discover the new applications that we don't know about?  I think your Hardy example sums that up perfectly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know, that still probably somewhat misses the point of your question because it still implies that mathematics (or specific branches of it) are only worth learning for practical applications.  I certainly don't think that's the case, but there are some people you are going to have a very difficult time convincing of that (much as there are some people that don't see the value in whatever human endeavor, from art to manned spaceflight).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think at the very least you might be able to win them over somewhat with the prospect of discovery.  The whole point of studying these things is because &lt;strong&gt;we don't know&lt;/strong&gt; what they can be used for.  And the only way to find out is to get them into the heads of as many people as possible so that when each of them goes out into the world, one of them stumbles onto something.&lt;/p&gt;&#xA;" OwnerUserId="5152" LastActivityDate="2015-05-11T03:16:11.937" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8057" PostTypeId="2" ParentId="8052" CreationDate="2015-05-11T05:44:19.240" Score="4" Body="&lt;p&gt;You are not learning math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are learning the very useful, highly transferable skills of abstraction and problem solving. It's not that math can solve many problems, it's the skill you learn in order to do math are useful in so many other problems.&lt;/p&gt;&#xA;" OwnerDisplayName="user5154" LastActivityDate="2015-05-11T05:44:19.240" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8058" PostTypeId="2" ParentId="8052" CreationDate="2015-05-11T08:03:00.830" Score="12" Body="&lt;p&gt;My answer is mostly about higher education, and mostly suited for undergrads (which make the largest part of my students).&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;The main purpose of their studies is to become more intelligent.&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;This is what I say my undergrad student a lot. It applies to math as well as to any other fields; as was mentioned in other answers, many aspects what we teach, from reproducing algorithmic tasks to handle abstract reasoning, makes one mind more flexible, more powerful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of years ago, I came up with a comparison I like, but the effectiveness of which I cannot yet evaluate: I tell my students that they are like high level sport competitors, except they work their brain rather than their body. This has many interesting consequences (among which the most important to me is not related to your question: it makes it clear that the student have to work by themselves to learn, and the teacher, like a coach, can only help them, but cannot work for them), and it partially answers the &quot;why learn math&quot; question.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Knowing how to use mathematical tools is useful.&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Knowing how to use mathematical tools is indeed useful&#xA;for sciences as well as in any quantitative field. Let me digress to partially answer &quot;Lockhart's lament&quot; mentioned in a comment. that mathematicians should not conflate our research activity &quot;doing maths&quot;, and what we mainly teach (this is why I wrote about mathematical tools). We also should help students not to conflate the two, and know a little bit about what &quot;doing math&quot; really is, but for most of them this is not what we are asked to teach them. That said, it would be good to make what we teach interesting, of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to usefulness. Knowing about probabilities is useful in every day's life, just as is basic arithmetic. Let me give an example I heard about recently: a biology professor gave her students yes/no tests where each correct answer was rewarded 1 point, and each incorrect answer was penalized 0.25 point. The professor was not really aware that this grading system gives a not good, but not too bad grade on average to a monkey answering randomly. A students knowing one fourth of the answers and giving random answers to the other questions would often get a passing grade. This is a classical example of mathematical illiteracy (here, the lack of understanding of expectation and the law of large numbers), and shows that a basic understanding of a variety of basic mathematics comes right next to reading and writing in terms of usefulness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also try to give examples of the usefulness of mathematical tools as often as possible, in order to give an idea of why we chose to teach them their precise curriculum. Let me name a couple: to justify the differential equations to freshmen, I gave a dozen examples from biology (various population growth models), physics (from mechanics, radioactivity, etc.), economy (I derived the &quot;second law of economics&quot; that I recently read in Thomas Picketty's book from a differential equation, and mentioned the context to show that the example was not made up for them), chemistry (cinetics); to justify the differential geometry course on curves and surfaces, I had a session devoted to students measuring distances on a globe and a map and comparing them, before I went on to prove that no map can be faithful.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-05-11T08:03:00.830" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8059" PostTypeId="2" ParentId="8020" CreationDate="2015-05-11T11:08:16.950" Score="-1" Body="&lt;p&gt;What you done is Viete, never lieked that. I admit to like your way of putting it but... I much more prone to use this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;step 1:&#xA;get to standart form aX^2 + bX + c = 0;&lt;/li&gt;&#xA;&lt;li&gt;step 2:&#xA;d = b^2 -4ac;&lt;/li&gt;&#xA;&lt;li&gt;step 3:&#xA;X1 = (-b-sqrt(d))/2a;&#xA;X2 = (-b+sqrt(d))/2a;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Imho that is way easier&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Viete would be X1 + X2 = -b/a; X1*X2 = c/a;&lt;/p&gt;&#xA;" OwnerUserId="5156" LastActivityDate="2015-05-11T11:08:16.950" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8060" PostTypeId="2" ParentId="8052" CreationDate="2015-05-11T14:06:26.860" Score="3" Body="&lt;p&gt;Sometimes learning in general is not about the actual usefulness of the subject matter in question, but how it changes and expands your thinking.  Inspiration can also stem from many different places.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main issue is that in many High school classes, students are taught to look at a problem and work it like a machine.  For example, taking the quadratic equation and putting in all the inputs.  I believe the best way to teach is be a role model and inspire an interest in learning.  Unless there is an interest, teaching is an uphill battle against all the distractions that occur.  A great teacher can often teach students ideas and concepts that can stay with them a lifetime.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most people forget things quite quickly when they aren't put to use.  However if they can remember what the math was about, with some time and google, its pretty easy to solve any high school level problem.  For people that really don't care about math, I would say its not that hard to live in society without it -- however scary that thought might be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if you are interested in a certain subject, perhaps the first thing to consider is if you can get your students interested as well (as well as if they can actually understand it).  For the gifted teachers, this eventually becomes less of an obstacle as the level of their own interest, charisma, or teaching method is enough to spark an interest in any subject matter.&lt;/p&gt;&#xA;" OwnerUserId="5158" LastActivityDate="2015-05-11T14:06:26.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8061" PostTypeId="2" ParentId="8052" CreationDate="2015-05-11T16:09:17.183" Score="1" Body="&lt;p&gt;I wish I could find an excerpt to link you to, but the book &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1400064287&quot; rel=&quot;nofollow&quot;&gt;Made to Stick&lt;/a&gt; had a great reason as one of its examples. You learn math in order to make you better at thinking. It's like athletes lifting weights; they don't lift barbells because they are going to need to&lt;/p&gt;&#xA;" OwnerUserId="5160" LastEditorUserId="5160" LastEditDate="2015-05-11T16:41:22.160" LastActivityDate="2015-05-11T16:41:22.160" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8063" PostTypeId="2" ParentId="8052" CreationDate="2015-05-11T18:43:06.320" Score="6" Body="&lt;p&gt;&lt;strong&gt;Edit (Feb 2016):&lt;/strong&gt; Since the OP mentioned Hacker's &lt;em&gt;Algebra&lt;/em&gt; opinion piece in the NYTimes, perhaps this is a good place to point out his most recent follow-up in a similar direction (I exclude here my own assessment of either): &lt;a href=&quot;http://www.nytimes.com/2016/02/28/opinion/sunday/the-wrong-way-to-teach-math.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Wrong Way to Teach Math&lt;/strong&gt;&lt;/a&gt; by Andrew Hacker (Retrieved: 2016 Feb 28).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit (July 2015):&lt;/strong&gt; In a similar vein, here is a link to &lt;a href=&quot;http://video.pbs.org/video/2365521689/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Is Math Important?&lt;/strong&gt;&lt;/a&gt; David Leonhardt of the &lt;em&gt;New York Times&lt;/em&gt; acts as host for the panel discussion; to quote directly from &lt;a href=&quot;http://devlinsangle.blogspot.com/2015/07/is-math-important.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;this blog post&lt;/strong&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;From the mathematical world there are Steven Strogatz of Cornell University and &lt;a href=&quot;https://matheducators.stackexchange.com/a/3837&quot;&gt;&lt;strong&gt;Jordan Ellenberg&lt;/strong&gt;&lt;/a&gt; of the University of Wisconsin, and from mathematics education research there is &lt;a href=&quot;https://matheducators.stackexchange.com/a/809&quot;&gt;&lt;strong&gt;Jo Boaler&lt;/strong&gt;&lt;/a&gt; of Stanford University. They are joined by David Coleman, President of the College Board, education writer Elizabeth Green, author of the recent book &lt;em&gt;Building a Better Teacher&lt;/em&gt;, Pamela Fox, a computer scientist working with Khan Academy, and financier Steve Rattner.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(The two hyperlinks were added by me: one to an MESE answer about Ellenberg's book; the other to an MESE answer about Boaler's comments on timed tests and math anxiety.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In providing a &lt;em&gt;justification for learning mathematics&lt;/em&gt;, I would like to split this question into two pieces (even though there are certainly more) and comment briefly about one of them (even though the other may be closer to the intended question).&lt;/p&gt;&#xA;&lt;p&gt;A &lt;strong&gt;first&lt;/strong&gt; interpretation of the italicized text above: Why does a subject, mathematics, that covers so much &amp;quot;abstract&amp;quot; material, occupy such an important place in our schools?&lt;/p&gt;&#xA;&lt;p&gt;This is the question that I believe is being asked, and it is the sort of consideration that underlies the linked piece on the necessity of algebra, from which I quote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was 45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You can find other comments in this direction in an earlier opinion piece in the &lt;em&gt;New York Times&lt;/em&gt;, Garfunkel and Mumford's (2011) &lt;a href=&quot;http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;How to Fix Our Math Education&lt;/strong&gt;&lt;/a&gt;. Again, quoting directly:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.&lt;/p&gt;&#xA;&lt;p&gt;Traditionalists will object that the standard curriculum teaches valuable abstract reasoning, even if the specific skills acquired are not immediately useful in later life. A generation ago, traditionalists were also arguing that studying Latin, though it had no practical application, helped students develop unique linguistic skills. We believe that studying applied math, like learning living languages, provides both useable knowledge and abstract skills.&lt;/p&gt;&#xA;&lt;p&gt;In math, what we need is “quantitative literacy,” the ability to make quantitative connections whenever life requires (as when we are confronted with conflicting medical test results but need to decide whether to undergo a further procedure) and “mathematical modeling,” the ability to move practically between everyday problems and mathematical formulations (as when we decide whether it is better to buy or lease a new car).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A &lt;strong&gt;second&lt;/strong&gt; interpretation of providing a &lt;em&gt;justification for learning mathematics&lt;/em&gt;: Why should we encourage students to study school mathematics now?&lt;/p&gt;&#xA;&lt;p&gt;This is the question that I would like to respond to, briefly.&lt;/p&gt;&#xA;&lt;p&gt;I do not disagree with studying mathematics for its aesthetic value; I do not disagree with studying mathematics for the opportunities it provides to express ourselves and be creative; I do not disagree that pure mathematics may turn out to have important applications. But I think the strongest argument &lt;em&gt;right now&lt;/em&gt; for studying mathematics is its role as a societal &lt;strong&gt;gatekeeper&lt;/strong&gt; (&lt;a href=&quot;http://scholar.google.com/scholar?hl=en&amp;amp;q=%22mathematics%22+%22gatekeeper%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;google scholar&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;There are normative and utilitarian meta-questions about where mathematics' place &lt;em&gt;should be&lt;/em&gt; in school and academic endeavors, but the &lt;em&gt;current reality&lt;/em&gt; is that &amp;quot;learning mathematics&amp;quot; is essential to moving forward (or up) in the world; such a competence seems, to me, necessary but not sufficient for working towards a &amp;quot;successful&amp;quot; life.&lt;/p&gt;&#xA;&lt;p&gt;At present, I have been teaching mathematics to elementary school teachers. Do I try to get them excited about mathematics? Yes. Do I try to get them to think about mathematics &lt;em&gt;creatively&lt;/em&gt;? Yes. Do they sometimes latch on to applications of their own in our discussion of pure mathematical concepts? Yes: If only you could see the dawning of epiphanies (!) as many soon-to-be-married teachers in my Spring semester course began to brainstorm, collectively, about applications of LCMs and GCFs to the construction of flower arrangements and seating charts at upcoming weddings.&lt;/p&gt;&#xA;&lt;p&gt;But I also realize that many of them are teaching students at high-needs schools, and that their students' futures (in our current set-up - speaking specifically about the United States) can be derailed by problems that start with an inability to factor quadratic expressions - or, in many cases, even earlier.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-02-29T03:42:22.710" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8064" PostTypeId="1" CreationDate="2015-05-11T21:43:56.490" Score="5" ViewCount="672" Body="&lt;p&gt;I have been tutoring using my iPad and I love it. Now that I'm moving away, my students have requested that I continue tutoring them online, and I'm not sure I can do so using my iPad anymore. I would like to be able to write freely on my macbook. Any recommendations for a good, responsive, and inexpensive pen tablet that I can use to tutor online with?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, any web services for tutoring to recommend? I'm not so keen on signing up with an online tutoring site and having to pay them to use the online classroom. I've found scribblar, but I'm not sure how useful it is. &lt;/p&gt;&#xA;" OwnerUserId="5162" LastActivityDate="2020-08-31T22:36:20.633" Title="Pen tablet recommendations? (Mac OS)" Tags="|technology-in-education|tutoring|online-instruction|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8065" PostTypeId="2" ParentId="8052" CreationDate="2015-05-11T23:52:20.007" Score="8" Body="&lt;p&gt;Some reasons (random order):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Math is fun and beatiful.&lt;/em&gt; Not all marvel at the beauty of mathematics and not all enjoy working with math. In fact, daily mathematical work can be tedious, boring, uninspiring even for those in love with it. But every now and then you discover something great, and the feeling you get can be matched by very few other things (it does help if you have someone to share it with).&lt;/p&gt;&#xA;&lt;p&gt;Moreover, math can help you appreciate the world even more deeply than before. For example, during a nice clear sky night, try to imagine the size of Andromeda galaxy from how far it is (about &lt;span class=&quot;math-container&quot;&gt;$2.5\times 10^6$&lt;/span&gt; light years$) and the fact that its apparent size (angular diameter) is about six times that of full moon.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Math makes you more intelligent.&lt;/em&gt; I have no citation for this, but it seems to work that way. There are some papers linking happiness to intelligence, but to the best of my knowledge they are inconclusive, so this may be a double-edged sword.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Math rewires your brain.&lt;/em&gt; I don't know how it happens, but it is true. To give an example, usually people don't consider &lt;span class=&quot;math-container&quot;&gt;$-5$&lt;/span&gt; (negative &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;) a valid number of apples, but that would be perfectly normal to a mathematician (unless assumptions imply otherwise). In most cases such a rewiring gives a superior perspective (of course, it does not help much if it is your only one; insight is not wisdom), although it may also cause friction or trouble with communication.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Math can teach you how the world works.&lt;/em&gt; There are many non-intuitive real-life phenomena, for example, adding new roads may increase traffic jams (see &lt;a href=&quot;http://en.wikipedia.org/wiki/Braess%27s_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;Braess's paradox&lt;/a&gt;). Still, there is math that models it in an intuitive way (here it is game theory). Knowledge of such math allows you to better predict the outcome of actions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Math is useful.&lt;/em&gt; Almost everybody and everything uses math now, whether it is a small MP3 player or big spaceship. We wouldn't be where we are now without probability, eigenvalues, Fourier transform or differential equations to name a few.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Math is considered important.&lt;/em&gt; Even if math was not important by itself, people thinking it is important make it important.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Math is gatekeeper.&lt;/em&gt; As @Benjamin Dickman pointed out, math is already thought of as a gatekeeper, but this might be even more true in the future, see a nice article by Alexandre Borovik: &lt;a href=&quot;http://www.borovik.net/selecta/wp-content/uploads/2015/01/Spade_31Dec14.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calling a spade a spade: mathematics in the new pattern of division of labour&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally: &lt;img src=&quot;https://i.stack.imgur.com/Fqgi8.jpg&quot; alt=&quot;math homework&quot; /&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Also, &lt;a href=&quot;https://math.stackexchange.com/a/529525/26306&quot;&gt;this post&lt;/a&gt; is somewhat related.&lt;/p&gt;&#xA;&lt;p&gt;I hope this helps &lt;span class=&quot;math-container&quot;&gt;$\ddot\smile$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-05-12T00:16:14.017" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8066" PostTypeId="2" ParentId="8052" CreationDate="2015-05-12T04:46:28.353" Score="1" Body="&lt;p&gt;I don't know that there is such a line to walk.  Justification usually requires some specific context, and an answer that is out of context assumed by the asker usually does not convince the asker.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would not dare to convince someone that study in mathematics should have higher priority than study in physics, or biochemistry, or psychology, or architecture, or art.  Nor would I ask a funding organization that mathematics education should deserve a greater share of funds than educational efforts in other disciplines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would note that if they attempted a variety of studies without mathematics or a mathematical viewpoint, they would be missing out on an opportunity to help better understand much of their world.  I would point out that trying to understand music without ratios, to understand realism in painting without some knowledge of geometry, to understand social and economic forces without some notion of function and rate of change, to arrange and prepare parties and supplies without basic notions of recipe/algorithm and algebra, much of it would have to be invented, arranged, and taught as some class called &quot;Ratiogeomfunctionrecipalgebrology&quot;.  Fortunately, older and wiser heads already chose to group such things under &quot;Mathematics&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Head Getting Older At Least&quot; Paseman, 2015.05.11&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-12T04:46:28.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8067" PostTypeId="2" ParentId="1550" CreationDate="2015-05-12T04:57:06.773" Score="2" Body="&lt;p&gt;No one has mentioned shopping directly, although various economic optimization problems have appeared in this thread.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have $20 to spend for your breakfast budget.  See how oatmeal (or your favorite breakfast option) is packaged at different prices, whether by single serving envelope, small container at one price, large container at another.  What is your quantity to optimize, and how will it affect your purchase?  Flavor, convenience, wastage are some factors to consider, but even if the oatmeal is the same in each packaging, one of them will have the lowest ratio of price to pound.  One can take this example in many directions and still keep to the shopping domain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Come See My Shoe Collection&quot; Paseman, 2015.05.11&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-12T04:57:06.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8068" PostTypeId="2" ParentId="8041" CreationDate="2015-05-12T11:24:14.473" Score="2" Body="&lt;p&gt;I have a pet hypothesis which says that it's easier to explain induction using a non-linear order (sometimes this is called well-founded induction), rather than just natural numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, it might be plain wrong, because all the people who benefited from this approach tried to understand induction over naturals first. Still, you may want to give it a try (you didn't say how old your little brother is, the example might be too silly, I don't know).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;An escape from mysterious cave&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;You are an adventurer in a mysterious cave which looks like this (here draw a graph).&#xA;  Unfortunately, your flashlight died (it's pitch dark) and you got lost. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;However, there is a gentle breeze which indicates, for each passage, in which direction the exit is (here direct each edge so that it becomes a &lt;a href=&quot;http://en.wikipedia.org/wiki/Partially_ordered_set&quot; rel=&quot;nofollow noreferrer&quot;&gt;strict poset&lt;/a&gt; where the exit(s) is(are) the bottom element(s)) - we have to go against the wind.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Prove that you can get away from the cave.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You start with &quot;proofs&quot; that you can escape from halls near the exit, namely producing explicit paths of length 1, 2, 3, whatever necessary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then explain that you would like to produce such a path for each hall in the cave, but that would be a tedious task (the cave may have huge number of halls; in math it could even be infinite). Instead, we will show how to produce such a path if we know a path from at least one hall closer to exit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Induction can be understood as an agreement that such a way of producing paths is just as good as explicit paths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now explain why you need base: you can travel inside the cave all you want, you cannot escape it if there is no exit (draw an example).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explain also why you need &lt;em&gt;all&lt;/em&gt; the steps: if some corridor collapses, then possibly some parts of the cave might be cut off (draw an example).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Heap property&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;We say that a rooted tree (it can be binary, but it's not necessary) has &lt;a href=&quot;http://en.wikipedia.org/wiki/Heap_%28data_structure%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;heap&lt;/a&gt; property if for any vertex its value is bigger or equal the values of its children.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Prove that the root is at least as big as other values in the whole tree.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Start with drawing an example where values aren't uniformly bigger from one level to the next. Of course, we could take the value of the root and compare it with any other node and that would be enough to prove that in this particular instance it is true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next draw an example with symbols like $a, b, c, \ldots$ or $a_1, a_2, a_3, \ldots$. Now we can't compare the root value with any other (we know the inequalities between the parent and its children), but we could write down sequences like $a_{12} &amp;lt; a_5 &amp;lt; a_1$ for each node and it also would be a proof for this instance. However, if the tree is big, that would be tedious. Similarly to the cave problem, instead of providing exact paths you say how to generate them from smaller paths. (Also draw an example where a single node that violates the heap property is enough for the lemma to be untrue.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if all is going well, you can then explain how we can do here even better, that is, instead of saying how to generate chains of inequalities for this particular instance, you can describe a way to do it for &lt;em&gt;any&lt;/em&gt; tree that satisfies the heap property.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't forget to stress why both the &lt;em&gt;base&lt;/em&gt; and the &lt;em&gt;step&lt;/em&gt; are important. If your brother is eager for more, you can show him &lt;a href=&quot;https://math.stackexchange.com/a/587569/26306&quot;&gt;how the sum of the first $n$ even numbers is odd&lt;/a&gt;, you can describe why &lt;a href=&quot;http://en.wikipedia.org/wiki/All_horses_are_the_same_color&quot; rel=&quot;nofollow noreferrer&quot;&gt;all horses are the same color&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your brother wants even more and you are not afraid to confuse him a bit, you can demonstrate why relation has to be &lt;a href=&quot;http://en.wikipedia.org/wiki/Well-founded_relation&quot; rel=&quot;nofollow noreferrer&quot;&gt;well-founded&lt;/a&gt;, e.g. you could prove that zero is strictly bigger than itself.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consider the set $A = \{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, 0\}$. We prove that for all $x \in A$ we have $x &amp;gt; 0$. The base is true, because $1 &amp;gt; 0$. We have that for any $x \in A$ the next element after $x$ is strictly bigger than zero, so the step also works. Hence $0 &amp;gt; 0$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-05-12T11:24:14.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8069" PostTypeId="2" ParentId="3858" CreationDate="2015-05-12T12:05:31.527" Score="0" Body="&lt;p&gt;There is a book about statistics and the law: Morris H de Groot, Stephen E Fienberg &amp;amp; Joseph E Kadane: &quot;Statistics and the Law&quot;,  Wiley Classics Library.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471055387&quot; rel=&quot;nofollow&quot;&gt;http://www.amazon.com/Statistics-Law-Morris-H-DeGroot/dp/0471055387/ref=sr_1_1?s=books&amp;amp;ie=UTF8&amp;amp;qid=1431432099&amp;amp;sr=1-1&amp;amp;keywords=statistics+and+law&lt;/a&gt;&#xA;(amazon also has a few other similar titles)&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2015-05-12T12:05:31.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8071" PostTypeId="2" ParentId="3858" CreationDate="2015-05-12T18:00:45.283" Score="0" Body="&lt;p&gt;Writing arguments is most of the work in &lt;b&gt;some&lt;/b&gt; math courses.  In high school what is often taught is just algorithms for solving the homework problems, and that's worthless garbage except when there is some pre-identified occasion to use those algorithms outside the classroom.  In some cases there is such an occasion, e.g. in a statistics course.  But the kind of math course that emphasizes writing proofs can help lawyers learn to argue logically.&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2015-05-16T19:16:45.997" LastActivityDate="2015-05-16T19:16:45.997" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8072" PostTypeId="2" ParentId="8052" CreationDate="2015-05-12T18:11:12.610" Score="5" Body="&lt;p&gt;Quoting the question: &quot;the study of one of the great intellectual and creative achievements of humankind&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The conventional high school course except for statistics and perhaps some aspects of geometry exists only for the purpose of preparing students for calculus. 99% of the students who take calculus will not use it after the final exam. Therefore the calculus course they take should be about &quot;one of the great intellectual and creative achievements of humankind&quot;; otherwise it's worthless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, what is actually done is that students learn a bunch of algorithms for evaluating limits, derivatives, and integrals without actually understanding the subject.  The only way to fix that is to drop the requirement that students must learn all the elementary ways of finding limits, derivatives, integrals, etc., to leave room to explain why the subject is one of the great intellectual, cultural, and aesthetic achievements.&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2015-05-12T18:11:12.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8074" PostTypeId="2" ParentId="8052" CreationDate="2015-05-12T19:49:17.203" Score="8" Body="&lt;p&gt;I'm not a teacher/professor but I'd like to give my input as a person who hated math for a big part of his life, especially at high school.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like first to explain why I hated math that much and for so long. At high school we were basically receiving some input and had to spit out some output. I'm referring to the countless resolution of derivatives, integrals well design to apply some seen methods. At the end I didn't need my brain at all.   Fortunately, It didn't stop me to go in a technical field (chemical engineering) but during all my years at the university I never reconsidered my feeling for the mathematics and I simply considered it as a necessary evil, some tool only needed to do some other cool stuffs. It's only lately that I've started to &lt;strong&gt;enjoy&lt;/strong&gt; math by among other things reading about its history and doing some math puzzles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now this is what I'd have loved my teacher at high school to tell us. I'd have loved her to be bluntly honest. That there are some problems with the program, that it won't be necessary useful. That what is called math at their level is actually a tiny bit of what it is reality. Someone mentioned Lockhart's Lament, why not make them read it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, just that is not enough. I think it should go along with that other thought: humans don't do something because it's useful but because &lt;strong&gt;they enjoy it&lt;/strong&gt;. Unfortunately we also &lt;strong&gt;have to&lt;/strong&gt; do things we didn't choose to and this is they case with their curriculum. But people that are happy in their life are those who manage to find some enjoyment even in the things they have to do (are forced?). So encourage them to find a way to enjoy it (some might enjoy it because it's useful - but not much). Each of them will find something different. After all if you tell them that math is useful in chemistry you'll get the attention of only those interested in chemistry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course you should give them some pointers. For instance numerical analysis was a big revelation for me,especially the methods for solving differential equations. For years I wonder how scientists who really had to solve these kind of equations in real life were doing that. I don't say to explain it, just mention it (sometime, just knowing something exist is an eye-opener).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Others things that reconciled me with math:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.coursera.org/course/maththink&quot;&gt;Coursera's course Introduction to mathematical thinking&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;The puzzles of Martin Gardner&lt;/li&gt;&#xA;&lt;li&gt;Reading about the history of mathematics&lt;/li&gt;&#xA;&lt;li&gt;Some blogs written by fun mathematicians like this &lt;a href=&quot;http://mathwithbaddrawings.com/&quot;&gt;one&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Some resources explaining basic concept using intuition like this &lt;a href=&quot;http://betterexplained.com/&quot;&gt;one&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Some Math/Computer programming challenges like the &lt;a href=&quot;https://projecteuler.net/&quot;&gt;project Euler&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;tl;dr : I wouldn't try to show them it's useful, I'd show them it's in their best interesting to try to find &lt;strong&gt;enjoyment&lt;/strong&gt; in it while being honest about the shortcoming of the educational system.&lt;/p&gt;&#xA;" OwnerUserId="5172" LastEditorUserId="5172" LastEditDate="2015-05-13T10:18:23.797" LastActivityDate="2015-05-13T10:18:23.797" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8075" PostTypeId="2" ParentId="7865" CreationDate="2015-05-13T01:03:16.483" Score="6" Body="&lt;p&gt;I don't think it's all that bad to put your graph on ordinary everyday axes as long as the students know that the order is more or less irrelevant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are happy to break out of the page, I recommend drawing your graph on a piece of paper and rolling it up to make a cylinder. Then at least one of the axes represents the cyclical nature of the field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to have the graph on a torus on a screen, it may be better to use technology.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've used Geogebra to make &lt;a href=&quot;http://ggbtu.be/b1177009&quot; rel=&quot;noreferrer&quot;&gt;a workbook of various graphs&lt;/a&gt;, including cartesian graphs, cylindrical graphs and input-output graphs for $\mathbb{Z}_7$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The cartesian graphs looks like the ordinary graphs students would be familiar with:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/p1cpZ.png&quot; alt=&quot;cartesian graph&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The x-axis and y-axis can both be moved so that the labels cycle around, giving you the feel of the toroidal shape. UPDATE: There's now an &lt;a href=&quot;http://tube.geogebra.org/student/b1177009#material/1180134&quot; rel=&quot;noreferrer&quot;&gt;activity&lt;/a&gt; that allows you to choose the order of the ring, from $\mathbb{Z}_3$ to $\mathbb{Z}_{30}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The cylindrical graphs are like the cartesian one, only the x-axis has been rolled up to make a circle:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/WDta1.png&quot; alt=&quot;cylindrical graph&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The circular axis at the bottom can be rotated, and the vertical axis (the separate circles) can be slid up and down so the values cycle around. (It looks much better in motion than statically.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the input-output graph has a domain at the top and a codomain at the bottom with arrows to show where each point goes under the function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/TRNzU.png&quot; alt=&quot;input output graph&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can spin the top and bottom independently or together, and change the perspective.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-05-13T22:49:05.913" LastActivityDate="2015-05-13T22:49:05.913" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8076" PostTypeId="2" ParentId="8020" CreationDate="2015-05-13T03:05:46.413" Score="6" Body="&lt;p&gt;A little bit of historical context:  This is, in fact, precisely the method demonstrated by Diophantus of Alexandria in &lt;em&gt;The Arithmetica&lt;/em&gt;.  In Book I, Problem 27 states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;27&quot;&gt;&#xA;&lt;li&gt;To find two numbers such that their sum and product are given numbers.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;em&gt;Necessary condition.&lt;/em&gt; The square of half the sum must exceed the product by a square number...&lt;/p&gt;&#xA;&lt;p&gt;Given sum 20, given product 96.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; the difference of the required numbers.  Therefore the numbers are &lt;span class=&quot;math-container&quot;&gt;$10+x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$10-x$&lt;/span&gt;.  Hence &lt;span class=&quot;math-container&quot;&gt;$100-x^2 = 96$&lt;/span&gt;.  Therefore &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt;, and the desired numbers are &lt;span class=&quot;math-container&quot;&gt;$12, 8$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The above is from p. 140 of the translation of Heath (2nd edition, 1910), which may be downloaded &lt;a href=&quot;https://archive.org/details/diophantusofalex00heatiala&quot; rel=&quot;nofollow noreferrer&quot;&gt;from the Internet Archive&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;necessary condition&amp;quot; is necessary only if you are looking for whole-number solutions, of course.  If we are okay using irrational numbers, it is enough to have the weaker condition that the square of half the sum must be greater than or equal to the product.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-05-13T03:05:46.413" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8077" PostTypeId="2" ParentId="8064" CreationDate="2015-05-14T00:14:52.513" Score="4" Body="&lt;p&gt;For years Rebecca Zook has been exploring the technology available to help her do long-distance tutoring, and reporting the results &lt;a href=&quot;http://www.zooktutoring.com/blog/&quot; rel=&quot;nofollow&quot;&gt;on her blog&lt;/a&gt;. She has reviews of both hardware and software.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-05-14T00:14:52.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8078" PostTypeId="1" CreationDate="2015-05-14T06:45:28.330" Score="28" ViewCount="3187" Body="&lt;p&gt;The context of this question is an &quot;introduction to proofs and mathematics&quot; class for freshman/sophomore math majors.  Most textbooks for such a class say something about functions between arbitrary sets, which are of course central to modern mathematics and which students usually aren't that familiar with (their previous exposure to functions has usually been limited to the partial $\mathbb{R}\to \mathbb{R}$ variety).  The question is, how do you &quot;define&quot; functions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many textbooks take the route of formal set theory and define a function $f:A\to B$ to be a subset of $A\times B$ such that for all $a\in A$ there is a unique $b\in B$ such that $(a,b)\in f$.  Some such books also include denigrating remarks about how functions used to be spoken about as &quot;rules&quot;, but now we have a real definition.  Other textbooks simply embrace the notion of &quot;rule&quot;.  Which is the better approach and why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I will provide my own current opinion as an answer below.)&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="187" LastEditDate="2015-05-14T15:53:44.230" LastActivityDate="2015-05-18T09:30:34.283" Title="Should we teach functions as sets of ordered pairs?" Tags="|undergraduate-education|functions|" AnswerCount="12" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="8079" PostTypeId="2" ParentId="8078" CreationDate="2015-05-14T06:48:56.713" Score="20" Body="&lt;p&gt;In my experience, the &quot;set of ordered pairs&quot; is a difficult and confusing definition.  Moreover, I feel it's almost totally unnecessary because in practice in mathematics we &lt;em&gt;do&lt;/em&gt; always treat functions as rules.  No one ever defines a function as $f = \{ (x,y) \mid y = x^2 \}$; we always write $f(x)=x^2$.  Of course, &quot;rules&quot; in this sense are much more general than &quot;algebraic formulas&quot;, but we still think of them as rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main benefits I see to the set-of-ordered-pairs definition are that (1) it reduces the notion of &quot;function&quot; to the supposedly-previously-understood notion of &quot;set&quot;, and (2) it thereby supplies the correct criterion for equality of functions: $f=g$ means $\forall a\in A, f(a)=g(a)$.  I don't see much virtue in (1) at an introductory level, where we are not concerned with foundational questions but with understanding and practical skills; saying &quot;a function is a rule&quot; should be just as acceptable as saying &quot;a set is a collection of things&quot;.  (I might also point out, for those who care, that there do exist foundations of mathematics alternative to ZFC in which &quot;function&quot; really &lt;em&gt;is&lt;/em&gt; a primitive concept.)  Similarly for (2), the criterion for equality of functions could be presented as parallel to, rather than derived from, the criterion for equality of sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By way of analogy, consider that it's not important for these students to know that an ordered pair $(a,b)$ can be defined as $\{\{a\},\{a,b\}\}$.  That's a detail about their implementation inside ZFC set theory; the important point is that any two elements can be paired up and that they satisfy the correct criterion for equality: $(a,b) = (c,d)$ iff $a=c$ and $b=d$.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2015-05-14T06:48:56.713" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="8080" PostTypeId="2" ParentId="8078" CreationDate="2015-05-14T14:09:36.917" Score="8" Body="&lt;p&gt;I think as long as students know that:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A function $f:A \to B$ assigns one and only one element of the codomain $B$ to each element of the domain.  We write $f(a) = b$ if $a \in A$ is assigned to $b \in B$ by $f$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Two functions $f,g$ are the same if they have the same domain, codomain, and $f(a)=g(a)$ for all $a$ in their common domain.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then there is no need to further formalize this in terms of sets of ordered pairs.  I do think that this is the level where students should be starting to meet such formalizations, so it makes sense to develop the ordered pairs definition, and make sure it correctly reproduces the properties we want functions to have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may also be worth noting alternative definitions of a function, to indicate that the particular formalization is not as important as reproducing the qualities we want functions to have.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, instead of defining a function as its graph, you could define it as its cograph.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2015-05-14T14:09:36.917" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="8081" PostTypeId="1" AcceptedAnswerId="8094" CreationDate="2015-05-14T15:28:34.403" Score="18" ViewCount="307" Body="&lt;p&gt;In the role of an advisor, I am often faced with the question&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Should I retake this [undergraduate] math class?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My first-order approximation to the answer is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If this is your last math class, a C or better is fine, be done with it.&lt;/li&gt;&#xA;&lt;li&gt;If you are taking more classes after this one, a B or better is fine, but you should retake a C.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;However, there is also the possibility of independently checking the student's knowledge to help inform the decision, especially if you are one of the other faculty members teaching the course. For example, you could give the student a chain rule derivative, or a substitution integral, and see how they approach the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has some pitfalls. For a start, by doing any independent checking, you are implicitly saying that you &quot;know better&quot; than that student's instructor and that you are a better assessor of the student's knowledge than the instructor. This undermines the legitimacy of the student's grades. There are more issues than this, to be sure. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, I have been on both ends of these independent assessments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it ever a good idea for an advisor to do impromptu assessments of students? When?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-01-06T02:47:33.943" Title="When should an advisor assess a student's knowledge independently of their course grades?" Tags="|undergraduate-education|advising|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8082" PostTypeId="2" ParentId="8081" CreationDate="2015-05-14T17:03:24.800" Score="5" Body="&lt;p&gt;It helps to place such assessments in light of the goal that inspired the question.  I think this goal is most often &quot;How do I manage my time to learn topics from area X?&quot;, although it could also be from a desire to master a subject independent of time constraints.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming time management is the goal, it helps to run through some scenarios:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) the student has mastered (at a B level) enough to move to the next subjects, so he just has to review old material occasionally alongside the new material.  Think of this as taking 1 1/8 courses combined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) the student has not mastered enough to move to the next subjects.  Then he has to relearn the old material as well as learn new material as well as learn how to apply old to new.  Think of this as taking 3 courses combined instead of one.  The student will need a tutor to handle each of these three areas, as opposed to handling just the material in the new course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) the student doesn't know.  In which case, the student needs to self-assess, and find whether not retaking means a multiplicative factor of 9/8, 3, or even higher.  You can give them guidelines for where you think the number will be, but a good and quick metric is how much of the introductory material in the new course they can handle.  If they are struggling with it, a likely cause is insufficient mastery, and they need to apply some (not small) multiplicative factor or postpone the new course until the old course has been mastered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you should give them an additional assessment, not an independent one.  If you present them the model above, they may get a more realistic view of whether to retake the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Even Bad Models Are Useful&quot; Paseman, 2015.05.14&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-14T17:03:24.800" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8083" PostTypeId="2" ParentId="8078" CreationDate="2015-05-14T19:09:54.213" Score="23" Body="&lt;p&gt;I wanted to provide a different perspective on this. I just recently graduated with an undergraduate degree in mathematics, and I wanted to say a few things about what personally helped me in understanding functions, and why I think introducing them as sets of ordered pairs is important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before I took my first proofs class, I thought of functions the typical naїve way as some formula that you put a number in and get another number out of. I guess that was enough for me to muddle my way through Calculus, but I never would have thought up on my own things like the Dirichilet function because of that mindset.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Defining functions as a set of ordered pairs makes your imagination of functions more flexible, you have the freedom to define each point independently of every other point without needing to follow some rule or pattern. With that in mind, the following statements&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Almost every subset of $\mathbb{N}$ is undecidable.&lt;/li&gt;&#xA;&lt;li&gt;Almost every function mapping from $\mathbb{N}\to\mathbb{N}$ is uncomputable.&lt;/li&gt;&#xA;&lt;li&gt;Almost every real number is uncomputable.&lt;/li&gt;&#xA;&lt;li&gt;Almost all functions mapping from $\mathbb{R}\to\mathbb{R}$ are discontinuous everywhere.&lt;/li&gt;&#xA;&lt;li&gt;Almost all continuous functions from $\mathbb{R}\to\mathbb{R}$ are non-differentiable everywhere.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and others of the same ilk become much less counter-intuitive. If you have preconceived ideas that functions have to be computable using some sort of formula, or be useful, and so forth, most of these results will be counter-intuitive. Many of my classmates found them hard to believe because it takes a good deal of effort to construct an example of a continuous function that is non-differentiable everywhere, or an example of an uncomputable real number, and there is the expectation that a set of things that are easy to construct examples of will be larger than a set of things that are difficult to construct examples of. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, when functions are defined as ordered pairs mapping values arbitrarily, I feel like these results are exactly what you would expect.&lt;/p&gt;&#xA;" OwnerUserId="5175" LastEditorUserId="5175" LastEditDate="2015-05-14T19:19:09.640" LastActivityDate="2015-05-14T19:19:09.640" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="8084" PostTypeId="2" ParentId="8078" CreationDate="2015-05-14T21:02:07.380" Score="2" Body="&lt;p&gt;I suppose my view is somewhat a middle ground.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I don't see harm in defining a function by a rule...&lt;/li&gt;&#xA;&lt;li&gt;IF the rule is given an explicit domain and codomain.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The same rule $f(x)=x^2$ could easily be understood to apply to integers, rational numbers, a finite field, quaternions, complex numbers, a square matrix etc. The problem I see is not the rule, it is the assumption that $x$ is automatically a real number. In context, fine, $x$ is a real-variable. The larger conceptual hurdle to get past:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The formula $f(x)$ must define the domain of $f$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I get more bewilderment than I'm expecting when I introduce concepts of restriction and extension in junior level math courses. This should not be the case as the students have already had the proofs course. I think a big part of this is their reluctance to think of a function as a more than a formula. But, here is where I differ with some answers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The formula $f(x)$ does not need to be replaced with a cartesian product-defined function. Rather, the domain and codomain simply need to be made explicit.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So, to clarify the concept of a function, I'd rather see more questions/discussion which draws attention the necessity of giving the domain and codomain. For example, $f(x)=x^2$ is injective for which intervals $I \subseteq \mathbb{R}$? Or, $f(x)=e^x$ with $dom(f)=\mathbb{R}$ is surjective for what choice of codomain? Problems with ill-defined formulas on quotient spaces are a second order topic, I try to start with the basic issue of domain, codomain, restriction and extension.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this said, I can't take away from the fact that the function as subset viewpoint is illuminating to some. In the same way, the idea that $(a,b) = \{a,\{a,b\}\}$ was illuminating for me. Or, $\sum_{i=1}^n a_i = a_n + \sum_{i=1}^{n-1} a_i$. Other folks are content to live life just trusting finite sums work as claimed. On occasion, I find it comforting to prove they work as advertised.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2015-05-15T14:34:09.373" LastActivityDate="2015-05-15T14:34:09.373" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8085" PostTypeId="2" ParentId="8081" CreationDate="2015-05-14T21:36:41.537" Score="3" Body="&lt;p&gt;The problem with course grades is that they measure the student's knowledge of the &quot;book,&quot; or lecture notes, not the material or subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student knows things about the subject over and above the &quot;book,&quot; I would have a higher assessment of his (or her) ability than the course grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if a student knew things only by &quot;rote,&quot; and was totally outside his element outside of the class, I would lower my assessment of his ability below his course grade.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2015-05-14T21:36:41.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8086" PostTypeId="2" ParentId="8081" CreationDate="2015-05-14T22:09:16.207" Score="7" Body="&lt;p&gt;The other answers made good points... and suggest continuations and qualifications. For example, some students seemingly benefit from being informed that courses and grades are merely a stylized approximation to the genuine goals... while other students are confused or see opportunities to game the system upon hearing &quot;an advisor&quot; say any such thing. Gauging the tendencies of the given student is a very-necessary preliminary to deciding how much &quot;reality&quot; to admit/explain to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly one tipping-point is about the availability of placement exams, or testing-out exams: if they're available and seem to genuinely address the issues (as opposed to being broken), there are marginal advantages to &quot;staying within the (formal) system&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is the auxiliary issue, suggested by @TomAu's answer, about whether &quot;the advisor&quot; will seek to diagnose context-independent knowledge, or, rather, assess the very-context-dependent issues of a &quot;class&quot;. For that matter, the sequels for which a given course is prerequisite may be of wildly varying natures, so that no general prescription is possible if the chief goal is &quot;success in immediately following coursework&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Indeed, in extensive observation, it is not at all the case that orthodox success in previous courses is optimal preparation for even-the-most-orthodox success in the immediate sequel, apart from general behavioral traits and capacity.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, even in not-so-inflated grading systems, a &quot;B&quot; in a &quot;prereq&quot;, if the student (as is usual) has no outside information, is like starting a long car trip with a slow leak in a tire...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On net, try to avoid improvising assessment, ... but some students are not &quot;typical&quot;, in the sense that they &lt;em&gt;do&lt;/em&gt; know more than the grade indicates, and may be ingenuous enough so that their competence would not be accurately appraised by the staid conventional apparatus (which does often appraise compliance or conformity as much as genuine mathematical competence, in my opinion.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-05-14T22:09:16.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8087" PostTypeId="2" ParentId="8078" CreationDate="2015-05-14T23:04:16.447" Score="17" Body="&lt;p&gt;Let me give you yet another point of view which is a bit closer to &lt;em&gt;discrete&lt;/em&gt; mathematics, that is, from introduction to &lt;a href=&quot;http://en.wikipedia.org/wiki/Functional_programming&quot;&gt;functional programming&lt;/a&gt;. As this course is very function-heavy, it was worth to spend the very first class solely on the basics of functions. Please note that what I describe below builds quite different intuition than that of continuous functions (I don't know, perhaps it hampers progress during calculus and analysis courses).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shortly, I completely &lt;em&gt;detest&lt;/em&gt; the function-as-a-formula approach. It is a source of many misunderstandings, poor intuition and generally all the evil of the world ;-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordered-pairs is only slightly better, the untyped aspect of set theory wrecks havoc on gullible minds of the innocent students. As @Benoît Kloeckner already pointed out, at best, this should be deferred until some foundations of mathematics have been laid out. This is not always possible, so if somebody &lt;em&gt;presses&lt;/em&gt; for definition, I do give that one, as it is the one later introduced during other courses. However, it is crucial to stress that, until later, functions should be though of as different objects than just sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My preferred way of introducing functions is via mappings/correspondences (not necessarily one-to-one). Start with some simple examples like 4 objects assigned to some other objects. Then practice evaluation, and then more examples and more evaluation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only when students are comfortable mapping sets like $\{1, \spadesuit, \text{elephant}, \ddot\alpha\}$ to other similar silly sets, then introduce function that maps $0 \to 1$, $1 \to 2$, $2 \to 3$ and so on. Name it (whatever letter or other symbol is free), and practice evaluation. Introduce $0 \to 0$, $1 \to 2$, $2 \to 4$, and so on, assign it to another letter, and practice evaluation (usually the response is immediate). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now one can comment that if we were to describe $\mathbb{R} \to \mathbb{R}$ function, then that would be &lt;em&gt;a lot&lt;/em&gt; of writing. Even if it was possible, it would be extremely tedious, so normally we use shortcuts like $x \mapsto x+1$ or write $f(x) = x+1$. Similarly $\alpha \mapsto 2\cdot \alpha$ or $g(\bigstar) = 2\cdot\bigstar$. Nevertheless, such rules are not functions themselves, they only indicate which function we are talking about. In particular $x \mapsto 2x+2$ and $y \mapsto 2(y+1)$ describe the same mapping, the same function despite the fact &quot;the rule&quot; looks different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then, because the course is functional programming, we go on to high-order functions, that is, functions taking functions as an argument or returning functions as a result. We take a step back to the mapping approach and practice things like $$\Big\{0 \to \bigstar, \spadesuit \to \{\square \to 1, 2 \to \mathtt{x}\}\Big\}$$ (the actual notation is quite different, e.g. a big oval rather than braces) and their evaluation and after enough practice we go again into &quot;rules&quot;, but then again some examples need writing down explicit mapping, so students go back and forth a number of times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then talk about whether function can point to itself or take itself as an argument, and then about standard conventions, notations, usual letters, and similar things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It takes a whole class, but it seems worth it, the students were later much more comfortable with the functional aspect of the rest of the course (although the evidence is anecdotal, I had some comparison with the groups of other TAs). Finally, please recall this was a functional programming class, it probably wouldn't work for analysis course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2015-05-14T23:04:16.447" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8088" PostTypeId="1" AcceptedAnswerId="10258" CreationDate="2015-05-15T05:26:23.403" Score="17" ViewCount="250" Body="&lt;p&gt;The state of Illinois is currently considering a bill to mandate that all public institutions accept a score of 3 on every Advanced Placement exam. &lt;a href=&quot;http://www.ilga.gov/legislation/BillStatus.asp?DocNum=3428&amp;amp;GAID=13&amp;amp;DocTypeID=HB&amp;amp;SessionID=88&amp;amp;GA=99&quot;&gt;Bill here&lt;/a&gt;. The lobbyists in favor of the bill claim that similar things have been enacted in Ohio and Florida.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested to hear stories of how math educators have been affected by state-mandated placement policies. I don't need specifically USA answers here, although I am personally most interested in the College Board's Advanced Placement exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not interested in having a debate over whether the bill is a good idea; I don't think this site is a good forum for political disagreement. But I do think it would be possible to share experiences here.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2015-12-18T06:22:30.983" Title="What happens when the state mandates college math placement policies?" Tags="|assessment|usa|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="8090" PostTypeId="2" ParentId="8078" CreationDate="2015-05-15T06:43:45.527" Score="5" Body="&lt;p&gt;It does take some work to understand a function as a set of ordered pairs.  However, this course &lt;em&gt;is for math majors&lt;/em&gt;.  They'd better get comparatively simple stuff like this out of the way quickly if they're really going to major in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I definitely wouldn't drop the set-theory definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I have noticed texts doing a rather atrocious job of motivating the definition, when it's instead a great opportunity to teach not &lt;em&gt;just&lt;/em&gt; what a function &quot;is&quot; (i.e. how it is typically defined), but &lt;em&gt;also&lt;/em&gt; the kind of generalization of concepts that you see over and over and over in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you start with a notion of a function as a mapping from $\mathbb{R}$ or some interval therein to a polynomal of that value, and then consider splitting up the input into various parts, you can pretty quickly motivate that the most general possible way to do this is to allow an arbitrary $y$ for each $x$ (let's call it $y(x)$).  And then you quickly realize that you could leave out any arbitrary subset of the $x$s, so you have a function as a domain $X \subset \mathbb{R}$, and a labeled set ${y(x)}$ for each $x \in X$.  A couple more steps and you're at the classic definition of a function--and you've also had a preview of how incredibly weird most of these functions probably are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It neither serves the students well to just dump the definition on them &lt;em&gt;or&lt;/em&gt; to shield them from it.  They should learn the thought process behind generalizing something familiar like $f(x) = x^2$ to something less familiar like a map from $\mathbb{Q}$ to least positive numerators (and then, upon realizing that this mess is perfectly okay given the definition, they'll be motivated to think about how to add back constraints to pick out the &quot;nice&quot; functions, for some definition of nice).&lt;/p&gt;&#xA;" OwnerUserId="5021" LastActivityDate="2015-05-15T06:43:45.527" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8091" PostTypeId="2" ParentId="8078" CreationDate="2015-05-15T09:45:24.260" Score="12" Body="&lt;p&gt;While I commented otherwise, let me still give an answer which complements others, relates to a more general question, and answers some starting discussion in comments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We often consider, as mathematician, that defining everything very precisely is important, and it truly is. &lt;strong&gt;But what is often done, and in my opinion is often a mistake, is to consider that we should define objects in the context of set theory&lt;/strong&gt;, much Bourbaki style; this is exactly what defining functions by their graph is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in fact, such definitions are really the encoding in a particular foundational theory of objects that exist in our mind relatively independently of that theory. As has been stressed already by another user, we should not forget that mathematics can also be founded on other axiomatic systems than ZF, some of which take functions as basic objects instead of sets. In such a theory, sets are defined as certain type of functions (identity function,  guess). I do not think that function are sets, but that we can encode function in set theory, so that set theory can be used to describe and work with functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what else could be done? The most important thing is to determine what is relevant in the lecture to be given: pave the way to analysis? have student meet relatively abstract proofs? introduce set theory? introduce mathematical constructions, identifications, quotients and the like? Once this is determine, &lt;strong&gt;one should define functions in a degree of formalism and precision that fits this goal.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;definition by the graph&lt;/em&gt; inside set theory is relevant in some cases.  In other cases, it could be more relevant to define functions through a set of axioms (regarding e.g. composition, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In most undergraduate lecture, the goal will be to face relatively general functions and to be able to prove statements like &quot;the composition of one-to-one functions is one-to-one&quot;. For this,&#xA;&lt;strong&gt;defining a function as the data of two sets, domain and range, and of an affectation of an element of the range to every element of the domain&lt;/strong&gt; seems the best approach: close to intuition but general and precise enough that we can do proofs. Here the word &quot;affectation&quot; is not precisely defined, but let's face it: to student, &quot;set&quot; is not defined either. In fact, even in ZF, it turns out that we only define them by the $\in$ relation and axioms of how sets &quot;work&quot; together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This reasoning applies to other objects. For example, I completely stopped defining the Riemann integral to freshmen, in favor of an axiomatic introduction of integral of continuous function (by the fundamental theorem of analysis) together with an idea of why it makes sense (&lt;strong&gt;edit:&lt;/strong&gt; I first wrote &quot;definition&quot; instead of &quot;introduction&quot;, but I really do is that I claim without proof that there is a construction that from a continuous function defined on a segment gives a number called the &lt;em&gt;integral&lt;/em&gt; of the function on the segment, such that the integral satisfies the fundamental theorem; I also explain the idea of the construction and the meaning of the number in term of area, and how this relates to the fundamental theorem).&#xA;There are cases where on the contrary, the classical approach seems fine to me; e.g. defining simple quotients like $\mathbb{Z}/p\mathbb{Z}$ confronts student to hugely important ideas. They should probably see this kind of construction &lt;em&gt;before&lt;/em&gt; the abstract nonsense of functions as graph and the like.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2015-05-18T07:48:49.083" LastActivityDate="2015-05-18T07:48:49.083" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8092" PostTypeId="1" CreationDate="2015-05-15T14:02:37.090" Score="17" ViewCount="498" Body="&lt;p&gt;In my school district, and I'm sure most others, every teacher needs to have a set of &quot;emergency lesson plans&quot;, in case they are sick or need to be out for a day, so that the substitute can have something for the students to do.  They usually are not something like &quot;Go over last night's homework, then do section 10.3&quot; because&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;These emergency plans need to be in usually within the first few weeks of school so you have no idea where you will be at with students when you are sick.&lt;/li&gt;&#xA;&lt;li&gt;I don't really trust the sub to legitimately instruct my students.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What I am looking for is ideas for substitute/emergency lesson plans for stand-alone, cohesive, meaningful activities that can be completed in one day (say ~1 hour long period) and that are mostly student-driven, i.e. I don't want to rely on the sub to do the heavy lifting, they should really just be a facilitator. Any suggestions?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="6053" LastEditDate="2019-03-15T23:38:46.137" LastActivityDate="2019-03-20T00:36:20.097" Title="Good lessons/activities for one-day subs" Tags="|secondary-education|calculus|algebra|geometry|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8093" PostTypeId="2" ParentId="8092" CreationDate="2015-05-15T16:32:16.593" Score="5" Body="&lt;p&gt;Here is a suggestion:  take an interesting but related topic (or two or three),  introduce it on one day, and save the rest for sick/free days.  For example,&#xA;in studying sequences and series in calculus, one could mention the Fibonacci sequence, and Lucas-Lehmer sequences, and note their connection to factorization and other number theoretical results.  One could assign calculus problems to study rates of change and come up with estimates of how large the nth member is (or how large the difference is), as well as see the applications.  The point is to give the students a chance to explore and see what they can do with the tools: the sub just needs to ask questions and follow up questions (on a sheet only the sub sees).  You may end up visiting the class on the sick day just to see how much fun they may be having.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;The Substitute Might Never Leave&quot; Paseman, 2015.05.15 &lt;/p&gt;&#xA;" OwnerUserId="3468" LastEditorUserId="1147" LastEditDate="2019-03-20T00:36:20.097" LastActivityDate="2019-03-20T00:36:20.097" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8094" PostTypeId="2" ParentId="8081" CreationDate="2015-05-15T16:43:23.037" Score="8" Body="&lt;p&gt;I post this upon request, but I immediately must excuse myself as it doesn't give a practical resolution to what the OP asks. Having said this, this &lt;em&gt;does&lt;/em&gt; respond to  an underlying component of the question, that of using grades to &quot;assess a student's knowledge&quot;, as the title states.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My original comment,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] there's proffesional research in the way of undermining the legitimacy of grades.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;refers principally to the work of Alfie Kohn, an educator and researcher in the field. Of course, by transitivity I'm referring to &lt;em&gt;his&lt;/em&gt; sources, but I haven't read many of those directly so I will limit myself to Kohn's take.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, Kohn's work provides empirical evidence for the uselessness of almost all prevalent methods in the educational system (allowing perhaps the restriction to American schools), along with theoretical discourse, and very importantly the observation of a &lt;em&gt;lack&lt;/em&gt; of evidence in the opposite direction. Of course, &quot;uselessness&quot; has to raise some eyebrows, so I must make explicit just what &quot;use&quot; we wish to have, that is not had via grading. For indeed, students &lt;em&gt;do&lt;/em&gt; acquire information, from a bare perspective. However, it is the &lt;em&gt;interest&lt;/em&gt; in said information, &lt;em&gt;complexity&lt;/em&gt;, and &lt;em&gt;quality&lt;/em&gt; of the same that fail to be present. Taken from &lt;a href=&quot;http://www.alfiekohn.org/article/case-grades/&quot; rel=&quot;noreferrer&quot;&gt;The Case Against Grades&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;&lt;em&gt;Grades tend to diminish students’ interest in whatever they’re learning.&lt;/em&gt;  A “grading orientation” and a “learning orientation” have been shown to be inversely related and, as far as I can tell, every study that has ever investigated the impact on intrinsic motivation of receiving grades (or instructions that emphasize the importance of getting good grades) has found a negative effect.&lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;Grades create a preference for the easiest possible task.&lt;/em&gt;  Impress upon students that what they’re doing will count toward their grade, and their response will likely be to avoid taking any unnecessary intellectual risks.  They’ll choose a shorter book, or a project on a familiar topic, in order to minimize the chance of doing poorly — not because they’re “unmotivated” but because they’re rational.  They’re responding to adults who, by telling them the goal is to get a good mark, have sent the message that success matters more than learning.&lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;Grades tend to reduce the quality of students’ thinking.&lt;/em&gt;  They may skim books for what they’ll “need to know.” They’re less likely to wonder, say, “How can we be sure that’s true?” than to ask “Is this going to be on the test?”  In one experiment, students told they’d be graded on how well they learned a social studies lesson had more trouble understanding the main point of the text than did students who were told that no grades would be involved.  Even on a measure of rote recall, the graded group remembered fewer facts a week later (Grolnick and Ryan, 1987).&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I won't continue on the subject because I want to avoid a rant, but I do invite all to peruse the blog I link to!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I'll try to &quot;answer the question&quot;. A priori it's strange to do so though, because if my accusations stand, my response should be &quot;A grade B or a C means nothing, ...&quot; and that's about it because the &quot;Case Against Grades&quot; obviously advocates an entire across-the-board reform of education as we know it. And I can hardly answer with &quot;well, you should probably consider rejecting the entire collection of practices that the institution which employs you utilizes, and encourage all of your peers to do the same with the hope of forming a brand new educational system&quot;. I respect that not every teacher wants to be involved in political struggles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore let's suppose grades are here to stay. We can still take the above as a guide for how to interpret them though. Much like a psychoanalyst &quot;sees through&quot; the given &quot;face value&quot; of a patients explicit words, and constructs a picture of the subconscious desires, feelings, etc., we can go beyond grades' prescribed measure of a student's adeptness, and construct a deeper notion of how his/her mathematical maturity will cope contra the future subjects they'll face.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An easy example: you know the student semi-personally, they show profound interest and ask good questions; they're always seen with some book or some other, etc. but their grade is a flat C minus. Probably the student got lazy and didn't do enough practice exercises before the exam, but overall they have no difficulty with the concepts (most students I tutor with bad grades fall into this category). Granted, the laziness itself is a problem since at one point or other they will have to actually apply their mathematical facilities. For now though, I think the example illustrates what I mean with the psychoanalyst analogy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyhow, this has been an undergraduate-who-sometimes-tutors opinion, expanded from a comment into a longer than expected answer, I hope it can be useful. I'll end on an idea for a semi-joke that I've always wanted to expose in a mathematical community: $$\textit{Lemma: }\text{The space $\mathcal I_H$ of human intelligences is isomorphic to $([0,10],\leq)$.}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I plan on somehow incorporating the &quot;operation&quot; $a\bot b:= (a+b)/2$ into it somehow too. It's just a sarcastic remark though! (not a serious dismissal of grading, that's above).&lt;/p&gt;&#xA;" OwnerUserId="4622" LastEditorUserId="4622" LastEditDate="2015-05-15T17:52:07.763" LastActivityDate="2015-05-15T17:52:07.763" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8096" PostTypeId="2" ParentId="8092" CreationDate="2015-05-15T22:42:08.457" Score="3" Body="&lt;p&gt;You might like the &lt;a href=&quot;http://mathmamawrites.blogspot.com/2010/09/recapping-online-math-circle.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Magic Pancake problem&lt;/a&gt;, for students in Algebra II, and possibly in Algebra I. You might find many problems you'd like at &lt;a href=&quot;http://www.jrmf.org/activities/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Julia Robinson Mathematics Festival site&lt;/a&gt;, or at the &lt;a href=&quot;http://www.mathcircles.org/resources/&quot; rel=&quot;nofollow noreferrer&quot;&gt;National Association of Math Circles&lt;/a&gt; site.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2019-03-17T15:38:00.563" LastActivityDate="2019-03-17T15:38:00.563" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8097" PostTypeId="2" ParentId="8078" CreationDate="2015-05-15T23:15:09.910" Score="2" Body="&lt;p&gt;Here is a comment from a colleague: defining functions as sets of ordered pairs may not be so important on its own, but it's a convenient way to learn that functions are a special kind of &lt;em&gt;binary relation&lt;/em&gt;, and thereby to emphasize that a function involves a &lt;em&gt;relationship&lt;/em&gt; between elements of the domain and elements of the codomain.  Often students get hung up on thinking of a function solely as its output values (&quot;what you get&quot;) rather than as a relationship between the input and the output.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: now I see that this may be part of what Jared was trying to say.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2015-05-15T23:15:09.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8098" PostTypeId="2" ParentId="8078" CreationDate="2015-05-15T23:27:11.013" Score="1" Body="&lt;p&gt;I still believe that the definition of function using the set of ordered pairs is a nice way to go about it. You have to let the students know the condition imposed on the set of ordered pairs before you call it a function otherwise they may think all relations are functions. Once you let them realize that these ordered pairs must have distinct first elements, I think it drives the point home. You can easily tell when a relation defines a function. All functions are relation but the converse is not true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Warning: We are not talking about function of several variables here.&lt;/p&gt;&#xA;" OwnerUserId="5182" LastActivityDate="2015-05-15T23:27:11.013" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8099" PostTypeId="2" ParentId="8041" CreationDate="2015-05-16T06:19:54.507" Score="7" Body="&lt;p&gt;You say that your brother says he understands your explanations, but then when it comes to actually proving something mathematical with induction, he can't do it. I know two sources of this sort of non-understanding, because I see them on a regular basis in my Maths Learning Centre when students learn induction for the first time. I'll describe what these sources are and how I deal with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Philosophy&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first source is a philosophical issue with it. Some students, no matter how many analogies of dominoes or ladders you describe, will still just feel that it's somehow all a bit too unbelievable. You start by assuming it's true, and prove something with an unknown value of n, and then somehow the thing is proved for all values of n? It seems a bit too much like Baron Munchausen pulling himself up by his own hair. A common phrase the students utter is &quot;it feels like magic&quot;. They can't even consider giving it a go with an actual proof assignment question when they feel so philosophically uncomfortable with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With such students, I have had success discussing the axioms of the natural numbers. I talk about the axioms of the whole numbers, saying they are based pretty much on the foundation of &quot;There is such a thing as zero&quot;, &quot;Every whole number has a unique next number&quot; and &quot;Every whole number (except 0) has a unique previous number&quot;. Then I say that mathematicians discovered that in order for a lot of things about numbers to be provably true for all numbers, they had to have induction, and they figured out that just those three rules above weren't enough to be able to prove what they knew to be true. So they had to include induction as one of the rules in the first place. In that sense, I induction &lt;em&gt;is&lt;/em&gt; a lot like magic, because it can't be proved to work using simpler rules. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Several students have said they feel better about it after such a revelation because there is no burden on them to understand it by proving it to themselves.  It can't be proved analytically, so they don't have to try anymore. This allows them the freedom to rely on the analogy for their understanding, rather than thinking they have to also understand it analytically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Practical skills&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second source is not having any practical skills for dealing with problems of this kind. Even if a student understands the idea of how induction works, this is a wholly different thing from actually using it in an actual proof. Writing the proof itself is a different skill and often has to be learned separately.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first key skill is realising what the actual statement is that needs to be proved, and the fact that this statement appears in different forms in four different places in the proof: once with the base case, once with the induction assumption &quot;n=k&quot;, once with the induction step &quot;n=k+1&quot;, and then once in the final statement which repeats what they were supposed to prove. Without these reference points to hang the proof on, a student can feel very lost, and their proof will not be at all easy to read.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second key skill is realising how you go about proving equalities, inequalities or divisibility statements (the usual fare for induction proofs). None of them can begin with what you are trying to prove! They must start with an expression and perform manipulations on it until it is in the desired form. I find it works quite well to get students to write one part of the thing they're trying to prove with the &quot;k+1&quot; in it, and next to it a column of &quot;=&quot; signs, with  the general shape of the goal at the bottom. For example, if they were aiming to show that $7^{k+1} +2$ was a multiple of 3, then it would look like this:&#xA;$$&#xA;\begin{align}&#xA;7^{k+1} + 2 &amp;amp;= \\&#xA; &amp;amp;= \\&#xA; &amp;amp;= \\&#xA; &amp;amp;= 3( \text{something} )&#xA;\end{align}&#xA;$$&#xA;Once they have this, they can focus on filling in the bits in between, which helps them focus on just a small part at a time which stops them from being overwhelmed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The final key skill is realising that while proving the induction step, you &lt;em&gt;must&lt;/em&gt; use the induction assumption. Otherwise you'd be able to show it was true without induction. Therefore you should always be looking back to the induction assumption to see if you can figure out how to use it. This gives students a clue for how to come up with the most crucial part of the proof. &lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-05-16T06:30:47.223" LastActivityDate="2015-05-16T06:30:47.223" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8100" PostTypeId="2" ParentId="8078" CreationDate="2015-05-16T06:28:51.590" Score="1" Body="&lt;p&gt;It is my impression that the primary function (no pun intended) of thinking of functions as subsets of ordered pairs is not so much to reduce functions to sets in particular, but to reduce functions to objects about which it is psychologically easier to &lt;em&gt;reason about&lt;/em&gt;. The primary function of thinking of functions as rules, on the other hand, is to give us a way to &lt;em&gt;construct&lt;/em&gt; functions we care about in the first place. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue that at the heart of the matter is the logical distinction between open and closed terms, i.e. between terms involving variables and terms that do not. For example, $(2+3\times(7+11))\div 7$ can be thought of as a &quot;rule&quot; giving a (natural) number, but $x+3$ is not any particular natural number; rather it is a hypothetical (natural) number, and statements regarding it, and which we may try proving, are then hypothetical assertions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arguing about, or quantifying over, hypothetical natural numbers is not a terribly abstract thing to do since we do know that every natural number has a canonical form (it is one of $0$, $1$, $2$, etc.). What this means is that it is psychologically easy to imagine the possibilities whenever we argue about hypothetical natural numbers. E.g. &quot;prove that the product of three consecutive natural numbers is divisible by $6$&quot;: the full range of possibilities that this statement may apply to is (psychologically) easy to picture. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to arguing about, or quantifying over functions, if one uses the conception of functions as being given by rules, then functions do not have canonical forms, and this makes picturing the totality of the domain of all functions psychologically difficult (&quot;what are all possible rules?&quot; is too open-ended a question). The conception of functions as subsets of ordered pairs alleviates this psychological difficulty (to a certain extent) since it is actually the conception of canonical forms for the functions: any function is reduced to being a certain kind of subset of ordered pairs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, it is the access to these canonical forms, the ability to manipulate them, and construct them, that can give student the intuition for how functions behave, because these forms, unlike the rules, are psychologically concrete, much the same way that factoring and playing around with actual natural numbers can give intuition about how the natural numbers behave.&lt;/p&gt;&#xA;" OwnerUserId="1308" LastActivityDate="2015-05-16T06:28:51.590" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8101" PostTypeId="2" ParentId="8041" CreationDate="2015-05-16T15:05:22.283" Score="3" Body="&lt;p&gt;This problem may be partly because the core idea of induction is essentially meta-logical. There is a first-order induction schema that we can use to have something like induction in a first-order theory like PA, but the original intuition is not first-order at all. Instead it goes something like this:&#xA;$\def\nn{\mathbb{N}}$&#xA;$\def\imp{\rightarrow}$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given a property $P$, such that you can prove $P(0)$ and $\forall n\in\nn\ ( P(n) \imp P(n+1) )$, then you would like to be able to draw the conclusion $\forall n\in\nn\ ( P(n) )$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Why should we accept this? The meta-reasoning goes as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To get $P(1)$, you instantiate $\forall n\in\nn\ ( P(n) \imp P(n+1) )$ with $n = 0$ to get $P(0) \imp P(1)$ and then use modus ponens with $P(0)$ to get $P(1)$. Similarly, to get $P(2)$, you instantiate $\forall n\in\nn\ ( P(n) \imp P(n+1) )$ with $n = 1$ to get $P(1) \imp P(2)$ and then use modus ponens with $P(1)$ to get $P(2)$. And so on. This clearly shows that we can prove $P(n)$ for any specific natural number $n$, just by repeating these deductive steps $n$ times, because we define natural numbers to be exactly the numbers we can get by incrementing $0$ by $1$ finitely many times. In the meta-logic we are already done because we can see that every natural number satisfies $P$, but we need the axiom of induction to derive the universally quantified statement in the formal system itself.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To understand this better, you could actually perform the deductive steps in the meta-reasoning and notice that proving $P(n)$ for large $n$ requires a proof of length proportional to $n$. The induction axiom essentially says that if &lt;strong&gt;you are able to prove $P(n)$ for arbitrary natural $n$&lt;/strong&gt;, you are allowed to conclude $\forall n\in\nn\ ( P(n) )$. The part in bold is a statement in the meta-logic, which you would like to transfer to a statement in the logic.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2015-05-16T15:05:22.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8104" PostTypeId="1" CreationDate="2015-05-16T23:05:05.097" Score="14" ViewCount="377" Body="&lt;p&gt;&lt;strong&gt;The context&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In high school here in Australia, students have one mathematics class which runs all year. The students do maths class most days, and the course passes from one topic to the next, one after the other. They may study quadratic equations for a few weeks, then move on to geometry, then move on to statistics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An exception to this is in their final year. Students here in South Australia may study Mathematical Studies and concurrently also do Specialist Maths. This means they will be studying two maths topics simultaneously, one in each course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In most maths courses in first year at my university, the students study two streams simultaneously - lectures on different days alternate between calculus and linear algebra (usually with two different lecturers). The weekly assignments have half their questions on calculus and half on linear algebra. Basically the students are studying two courses concurrently which each have half as much content as a full course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently, one of these courses has been rearranged to return to the high-school style of one topic after the other. This means each topic takes half the time in terms of number of weeks (the same in terms of number of lectures of course). The assessment structure is the same with one assignment every week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With this recent change, the topics move quickly, so for some students it seems they don't have enough physical time to assimilate the information before they move on to a new topic. Over in the two-stream courses, it seems that the extra time between lectures on the same topic gives students the chance to think about them more. On the other hand, having to think about two topics simultaneously might be an added burden. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it actually an advantage to study different mathematical topics concurrently in alternate classes, rather than in blocks one after the other? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is any advantage just an artifact of the way that classes and assessment are structured? (For example, a possible advantage is having more time between classes on the same topic to think about it and ask questions, but is this just an artifact of students being able to seek help in free time between classes?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am particularly interested in answers that give clear reasons from the perspective of student experience and student learning, or which describe experience of trying this in university or high-school classes.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-05-18T19:48:51.403" LastActivityDate="2015-05-19T10:31:00.550" Title="Advantages of learning different topics simultaneously" Tags="|undergraduate-education|secondary-education|course-design|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8105" PostTypeId="1" AcceptedAnswerId="8115" CreationDate="2015-05-17T00:53:53.133" Score="7" ViewCount="1679" Body="&lt;p&gt;What are the most successful analogies that are used to teach the concept of mathematical induction? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify, I am not looking for a formal explanation of the principle of mathematical induction, but analogies that can be used to help students visualise and understand it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am particularly interested in answers that explain how successful the analogy is and/or why it is so helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Note, a &lt;a href=&quot;https://matheducators.stackexchange.com/questions/8041/how-to-teach-mathematical-induction-mathematically&quot;&gt;recent question&lt;/a&gt; asked for strategies to teach mathematical induction, but it explicitly asked not to give analogies but skills for dealing with the principle when doing a proper mathematical problem.]&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-06-02T21:15:53.583" Title="Analogies for mathematical induction" Tags="|proofs|concept-motivation|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8106" PostTypeId="1" CreationDate="2015-05-17T08:39:41.397" Score="3" ViewCount="205" Body="&lt;p&gt;I have recently started teaching. It gets totally blank in front of the big crowd. Now, I am quite confused about how to start teaching the Fourier transform and Fourier series chapter.&lt;/p&gt;&#xA;&lt;p&gt;I want students be convinced and motivated to learn this difficult concept .&lt;/p&gt;&#xA;&lt;p&gt;Can anybody give mathematical explanation about&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How Fourier Analysis is performed ? Can anybody give a simple example?&lt;/p&gt;&#xA;&lt;p&gt;How Fourier Synthesis is performed ? Can anybody give a simple example ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5190" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-05-21T16:08:21.927" Title="How to teach mathematically about Fourier analysis and synthesis?" Tags="|undergraduate-education|mathematical-pedagogy|examples|engineering-mathematics|" AnswerCount="2" CommentCount="3" ClosedDate="2015-05-21T19:53:22.843" ContentLicense="CC BY-SA 3.0" />
  <row Id="8107" PostTypeId="2" ParentId="3858" CreationDate="2015-05-17T13:41:28.803" Score="4" Body="&lt;p&gt;This is in addition to what's already been contributed, not instead of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I went to court once about a car accident.  The other driver was there too, and each insurance company had sent a lawyer.  My insurance company's lawyer had a short informal chat with me in the hall, and then when we were having the formal conversation with the judge, she helped me bring out the most salient points through her questions.  I was very impressed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She sorted through my jumble of comments in the hall, picked out the important things, and arranged them in an effective order.  You do all of that when you're outlining a proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, in both fields, it's not enough to have things clear in your mind, you also have to get good at communicating your ideas to others in a persuasive way.&lt;/p&gt;&#xA;" OwnerUserId="4956" LastActivityDate="2015-05-17T13:41:28.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8108" PostTypeId="2" ParentId="8092" CreationDate="2015-05-17T14:11:17.010" Score="4" Body="&lt;p&gt;Here are a few ideas:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Tessellations -- hands on activity, using paper, scissors and tape.  I did this as a 40-minute activity with fifth graders, and it worked.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Short Eurogames.  You can write to some of the publishers and ask for donations for your classroom.  You'll want to focus on short games, for example Fluxx.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Arithmetic in base 8, base 2, base 16.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Arithmetic with abacuses.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students work in groups to measure the classroom and make a three-dimensional paper model to scale.  Materials needed: Metal self-retracting measuring tapes -- the kind sold in hardware stores, paper, rulers, scissors, tape, glue.  Perhaps it would help to have a prep lesson where they cut out some pre-printed models and glue them, so they get the hang of shaping and folding the little flaps.  Or you could give them a flat illustration of something, with the measurements provided, and ask them to make a scale paper model.  Give each team something different.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Origami.  Origami paper and xeroxed origami instructions.  Make sure you choose models that are suitable for beginners.  You could have a couple of more challenging models in reserve for the occasional student who's done a lot of origami on his own.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give an intro to some other branch of math that they haven't done much with yet.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have some biographical sketches of famous mathematicians ready, that can be read in 20 minutes; make sure to include women and people of color.  Each student chooses one, reads it, and then shares with the class based on what they read.  It's okay if more than one student chooses the same mathematician.  Allow students to share with an alternative format -- for example, with an artistic impression rather than a summary of facts, or as a dialogue with a partner.  I've seen the dialogue work well as a journalistic-type &quot;interview&quot; with the famous person, as though s/he were still alive.  To get more women and people of color in your collection, it's okay to include scientists that had to use math in their work.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Tangrams in groups of two students.  One person creates a shape, outlines it, and gives the outline to the other student to try to reproduce.  You can stand manila folders or pocket folders up on the table or desk so the other person can't see how you're arranging the pieces.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4956" LastActivityDate="2015-05-17T14:11:17.010" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8109" PostTypeId="2" ParentId="7875" CreationDate="2015-05-17T14:23:56.993" Score="2" Body="&lt;p&gt;If we open up graphs to include infographics there's little you can't communicate, and when you say that you want to communicate insight that's what comes to mind. I don't have much insight to offer on creating complex infographics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Read some &lt;a href=&quot;http://www.edwardtufte.com/tufte/&quot; rel=&quot;nofollow&quot;&gt;Edward Tufte&lt;/a&gt; he's a leading expert in compact informative graphics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387245448&quot; rel=&quot;nofollow&quot;&gt;The Grammar of Graphics&lt;/a&gt; by Leland Wilkinson is one of leading handbooks on informative visualizations. &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0985911123&quot; rel=&quot;nofollow&quot;&gt;This book&lt;/a&gt; by Naomi Robbins is well reviewed. These are both highly technical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for graphing tools I've been using plot.ly lately and have been enjoying. Their blog is also pretty informative &lt;a href=&quot;http://www.r-bloggers.com/four-famous-laws-how-you-can-visualize-them/&quot; rel=&quot;nofollow&quot;&gt;try reading here&lt;/a&gt; for some insight on how to use it to illustrate some interesting concepts through graphs.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2015-05-17T14:23:56.993" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8110" PostTypeId="1" AcceptedAnswerId="8117" CreationDate="2015-05-17T16:13:21.720" Score="26" ViewCount="3976" Body="&lt;p&gt;I'm trying to help someone pass their final exam (analysis of functions) so they can graduate high school and move on to college. (Not a teacher, just another student, currently in high level math)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They are currently struggling with many concepts, most recently including &lt;code&gt;ln&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She is very frustrated because she feels like she needs to know what &lt;code&gt;ln&lt;/code&gt; is and how to find it by hand and how to do everything by hand. If she uses a calculator, she feels like she doesn't understand and isn't learning the math. (In our specific problem, I told her she needed a calculator to enter ln(5.4) and take that divided by 8 for her answer.) Her teacher is far less than helpful (I'm not convinced her teacher even understands it), leaving her more frustrated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I explained to her that &lt;code&gt;ln&lt;/code&gt; is a function, like an oven where you bake a cake. You put something in, and depending what you put in, you get something else out. The calculator is just the oven, and &lt;code&gt;ln&lt;/code&gt; is just a recipe. You are still the one doing the baking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She asked what people did before calculators and I told her that before calculators, you used a slide rule, and before that, you used a log table. Nobody ever did them by hand. I compared it to deriving pi by hand; possible, but pointless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She wanted to know where &lt;code&gt;ln&lt;/code&gt; comes from and what it meant, so I met her with this wonderful ms paint diagram:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/4mDY6.png&quot; alt=&quot;My fabulous ms paint skills&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seemed to help her understand finally where &lt;code&gt;ln&lt;/code&gt; comes from, after I also explained that &lt;code&gt;ln&lt;/code&gt; is just log base e.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But she was still frustrated that she couldn't do it by hand and didn't know how to find the area. I warned her that this is literally second semester college calculus but she seemed to need to know, so I briefly explained to her the infinite series used to calculate it as simply as possible, which I think she actually understood. She asked why she couldn't do it by hand, and I told her the obnoxious number of iterations she would have to do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think I finally got the concept of &lt;code&gt;ln&lt;/code&gt; through to her, but I fear she will have similar problems with the rest of her work with logs, and upcoming sigma notation (for many iterations, not wanting to do it in the n-spire), and other things. How can I convince her that the mechanics of the computations aren't what are important here?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried explaining that these questions will be answered in higher math, but seems unable (unwilling?) to grasp that understanding the chemistry of baking isn't needed to know how to make a good cake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is worse because, asking why she needs to know this, I really don't have a good answer. Most of math is practical, but I was always confused as to why we teach lns and e without the proper calculus background to really understand them.&lt;/p&gt;&#xA;" OwnerUserId="5192" LastActivityDate="2018-02-05T19:36:57.330" Title="How can I convince someone to use a calculator and not worry about the mechanics too much?" Tags="|calculators|" AnswerCount="8" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="8111" PostTypeId="2" ParentId="8078" CreationDate="2015-05-17T16:50:56.470" Score="1" Body="&lt;p&gt;I'll play devil's advocate for a bit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real trick is that, for the most part, nobody cares about &lt;em&gt;functions&lt;/em&gt;, except sometimes when working with finite things. Instead, they're interested in polynomials or rational functions or semialgebraic functions or analytic functions or linear transformations or Schwartz distributions or functions modulo negligible functions or measures or correspondences or computable functions or somesuch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;These things can be hard to describe and work with synthetically&lt;/li&gt;&#xA;&lt;li&gt;These share a lot of common features you would like to be able to study all at once&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The point of the notion of (set-theoretic) &lt;em&gt;function&lt;/em&gt;, on the other hand, is&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It's easy to define clearly and precisely, and to prove the basic facts about them&lt;/li&gt;&#xA;&lt;li&gt;All of the above ideas can either be expressed and understood as a special kind of function, in terms of functions, or by the way they generalize the notion of function&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As I understand the intent, trying to talk about a function as a rule robs it of its primary role in mathematics, and is sort of circular anyways; the only way I know to get the needed generality this way is to &lt;em&gt;define&lt;/em&gt; rule to mean function.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2015-05-17T16:50:56.470" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8112" PostTypeId="2" ParentId="8110" CreationDate="2015-05-17T17:02:57.137" Score="10" Body="&lt;p&gt;I think your title reflects your frustration. To me, it sounds like you'd like to show her the mechanics (calculator), but she wants to understand the underlying concepts. I agree with her.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the definition you used for ln x is one of multiple possible definitions. I would use a different definition. ln x means log&lt;sub&gt;e&lt;/sub&gt; x. Definition of logarithm: y = b&lt;sup&gt;x&lt;/sup&gt; is equivalent to log&lt;sub&gt;b&lt;/sub&gt; y = x. This definition is much more appropriate for her level. (For you, the question remains: How can there be multiple definitions? And, how do we prove that the two definitions are equivalent? This is not something she's ready to consider.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing things by hand is an excellent route to understanding. But it is better done with logs using simpler bases - 2, 3, 10, for example. Once she's got logs in general down, she can discuss the meaning of logs base e.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She can start to get a feel for why e is important by analyzing continuously compounded interest.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2015-05-17T19:04:12.637" LastActivityDate="2015-05-17T19:04:12.637" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="8113" PostTypeId="2" ParentId="8105" CreationDate="2015-05-17T17:24:18.987" Score="10" Body="&lt;p&gt;&quot;Is it possible to climb this whole ladder? I'm stood on the first rung.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Well, could you climb any one of the rungs?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I guess so.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;And could you climb a rung straight above that one?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Sure...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;There you go. And hey look you climbed onto the first rung. You can climb the whole ladder.&quot;&lt;/p&gt;&#xA;" OwnerUserId="5193" LastActivityDate="2015-05-17T17:24:18.987" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8114" PostTypeId="2" ParentId="8078" CreationDate="2015-05-17T17:55:36.847" Score="2" Body="&lt;p&gt;Maybe... you're asking the wrong question? Don't think of this as trying to define the notion of function, but instead as a valuable exercise in how to use &lt;em&gt;set theoretic reasoning&lt;/em&gt; to study something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have some idea of &quot;function&quot; we want to make precise. The important things are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Functions can be evaluated&lt;/li&gt;&#xA;&lt;li&gt;A function is completely determined by its evaluations&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now, the standard set-theoretic technique here is to express an object as the set of behaviors that object exhibits. Well, that's usually too big and complex; we pick out some sufficient set of behaviors that is easy to work with and is sufficient to distinguish objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here, there is an obvious candidate for the set of behaviors: we represent a function $f : A \to B$ in terms of its evaluations: as the set $G_f$ of all $(a,b)$ where $a$ is in the domain and $b$ is the value of the function at $a$. ($G$ for graph)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or maybe we believe the domain and codomain are important too, so we represent $f$ as the triple $(A, G_f, B)$. Or maybe just as the pair $(B, G_f)$, since $A$ can be recovered from $G_f$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once we've chosen a representation scheme, we then ask the question of which things actually represent functions. It's easy to see the vertical line test is necessary. And then there are two paths we might take:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Given a graph $G$ satisfying the vertical line test, it's 'obvious' there is a function whose graph is $G$.&lt;/li&gt;&#xA;&lt;li&gt;The issue of which things are actually functions is very complicated, so we'll sidestep the whole issue and first worry about what can be studied easily, and &lt;em&gt;later&lt;/em&gt; worry about the complexity&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Either way, we arrive at the conclusion that &quot;set-theoretic functions&quot; is a notion we should be studying. And if we believe the two bullet points at the top of my post, then we should believe studying set-theoretic functions should tell us &lt;em&gt;something&lt;/em&gt; about functions.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2015-05-17T18:16:08.800" LastActivityDate="2015-05-17T18:16:08.800" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8115" PostTypeId="2" ParentId="8105" CreationDate="2015-05-17T19:49:12.250" Score="10" Body="&lt;p&gt;When you set up a 'domino train' you need two things to ensure all the dominos will fall:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;You can knock the first domino.&lt;/li&gt;&#xA;&lt;li&gt;The dominoes are set up such that if one falls the next will fall.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I present both the monkey on the ladder and the domino analogies. I start with &quot;how can I prove that the monkey climbs the ladder?&quot; and &quot;how can I ensure that all the dominoes fall&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I  take suggestions from the students and usually we get close enough to the answer with some prompting. The students are far better at coming up with $P(k)\Rightarrow P(k+1)$ but I press home the importance of $P(1)$ --- be it with a scared monkey, an empty room of dominoes but more likely actual examples of induction 'proofs' where $P(k)\Rightarrow P(k+1)$ alone allows you to derive an absurd conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I could be blunt, I am very concerned if a student fails to make the connection between these analogies and an inductive proof.&lt;/p&gt;&#xA;" OwnerUserId="584" LastEditorUserId="1853" LastEditDate="2015-05-18T20:20:07.127" LastActivityDate="2015-05-18T20:20:07.127" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="8116" PostTypeId="2" ParentId="8110" CreationDate="2015-05-17T21:38:05.557" Score="8" Body="&lt;p&gt;&quot;Nobody ever did them by hand&quot;: Presumably Napier did; I think some other people did too. But now about what to tell this student: I think the first thing is to deal with some easier bases and make sure she understands  logarithms as inverses of exponential functions. For example, knowing that $2^{10}$ is just a little more than 1000, does she easily see that $\log_{10}2$ is just a little more than $0.3$ (without punching calculator buttons to get $0.301\dots$)? It might be good to do some non-trivial hand calculation of logs to base 10 or base 2, say by bisection, to give her an idea of how it can be done and also how tedious it gets.  Next, I'd suggest tackling the thorny problem of defining or explaining $e$. My immediate inclination is to draw the graphs of $x\mapsto a^x$ for several values of $a$, observe their slopes at $x=0$, and introduce $e$ as the value of $a$ that makes this slope $1$. Point out that calculuating $e$ from this definition is tedious at best, but there are alternatives ($\sum(1/n!)$) that work better and are provably equivalent.  Once the general notion of logarithm and the specific number $e$ are available, they need to be combined to get $\ln$, which leads to even messier computation.  I would not encourage her to do such really messy computations, but I would encourage her to imagine what's involved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like this student's attitude of &quot;I want to understand&quot;, and I would try to avoid suppressing it, even if the present context is not particularly friendly to this attitude.  I would probably give her the explanations she wants wherever possible.  Where it's impossible (or infeasible, given time constraints), I would at least indicate why it's difficult and apologize for not being able to give her complete answers.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2015-05-17T21:38:05.557" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8117" PostTypeId="2" ParentId="8110" CreationDate="2015-05-17T23:40:24.763" Score="21" Body="&lt;p&gt;&lt;strong&gt;A possible source of the need to do it by hand&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem might be to do with the examples she saw of logs and exponentials before she saw $\ln$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually, when logs are introduced, they are introduced as the inverse of exponential functions (which is what they are of course). However, almost all of the examples and exercises are with whole numbers. For example, the statement $\log_2 (8) = 3$ is declared to be true because $2^3 = 8$. Also, examples of figuring out logs will be with whole numbers too, and will boil down to guessing what power of a whole number produces the answer. For example, finding $\log_3 (81)$ will involve multiplying 3 by itself until you figure out that $3^4 = 81$ and so therefore $\log_3 (81) = 4$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This process of guessing what (whole number) power of a particular (whole) number in order to figure out a logarithm is probably what she is thinking when she refers to &quot;figuring it out by hand&quot;. It has served her very well up until now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, almost all calculators don't even have a function to calculate general logarithms (though they do have an option to calculate any exponential) so using a calculator to simply find the answer directly is simply not an option when logs are first learned. She just has no experience calculating logs with a calculator and probably doesn't even realise it's possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I daresay she has pretty much no experience calculating logs for anything other than whole numbers. A sudden shift to a log with base an irrational number calculated upon non-whole numbers is probably causing a considerable block.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A possible way to help her through it&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest giving her experiences with the aspects of ordinary logs she hasn't had experience with yet. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, I would talk with her about logs to non-whole number bases. Ask her to calculate things like $\log_{1.5}(3.375)$ and $\log_{0.6}(0.36)$ using her &quot;by hand&quot; method. Then she can come to see that it's ok to use non-whole numbers in the base.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, I would ask her about calculating non-whole number powers of numbers. I'd ask her how to calculate $2^{1.5}$. You can talk about how $2^{\frac12} = \sqrt{2}$ so it's also $2^1\times 2^{0.5} = 2 \times \sqrt{2}$. Then you can ask her about $2^{1.6}$. As a fraction this is $2^{1+6/10}$ so it is equal to $2^1\times 2^{6/10} = 2\times \sqrt[10]{2^6}$. So now you can calculate any power of 2. Still, you need a calculator to do the roots, of course, which will probably convince her it's ok to do fractional powers using the calculator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, I'd talk with her about how she might go about calculating $\log_2(10)$. By comparing to 8 and 16, you can figure out that the answer is somewhere between 3 and 4. Then the problem is to calculate exactly how far between 3 and 4. Referring back to your fractional powers of 2,  $2^{3.5}\approx 11.314$ and so you know that $\log_2(10)$ is below 3.5. You can make it more accurate by trying various values in the power. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this stage I would get a program that does do general logs (like Microsoft Excel's &lt;code&gt;=LOG(number,base)&lt;/code&gt;) just to show that it can be done using some computer programs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nearing the end, you can talk about how ln is log base e, and try that out with the computer programs by entering both ln and log base e of various numbers. You can also write a table of values of $e^x$ for various values of $x$, including some decimal values so you can know how big $\ln$ of some numbers will be. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally you can show how any logarithm can be calculated from $\ln$ using the change of base law -- students are often very impressed with that one and it ties it all back to their own calculator and the original meaning of log because now they can use their calculator to check answers they did &quot;by hand&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-05-18T06:47:33.943" LastActivityDate="2015-05-18T06:47:33.943" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8118" PostTypeId="2" ParentId="8106" CreationDate="2015-05-18T08:15:44.170" Score="-5" Body="&lt;p&gt;My opinion is that, as a teacher, you aren't responsible for &quot;convincing&quot; students that this is an interesting topic. You just have to teach it so that the subset of students that is actually hardworking will be able to solve the problems on the final exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can give some general motivational examples from digital signal processing, but, honestly, an introductory course doesn't remotely bring students close to modern practical application along these lines.&lt;/p&gt;&#xA;" OwnerUserId="4919" LastActivityDate="2015-05-18T08:15:44.170" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8119" PostTypeId="2" ParentId="8110" CreationDate="2015-05-18T15:23:59.003" Score="0" Body="&lt;p&gt;There's a few things here I want to comment on:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-I think your friend should know what ln means, but I don't think she should care about whether she has to use a calculator.  It's important to understand at least a basic definition of ln(x) (even as &quot;e to the power of what number equals x?&quot;) because otherwise she'll have difficulty solving logarithmic equations (or she'll only be able to solve them by repeating memorized procedures).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-The idea that decimal expressions are somehow preferable is a huge misconception (usually reinforced in mathematics teaching).  ln(5.4)/8 is a perfectly acceptable answer in some contexts, ~0.21 is an acceptable answer in others.  They both provide useful information about that number - for example, the approximation is useful for situating it on the number line (so you get an idea of how big it is; ie, you know ln(5.4)/8 is less than 1 but greater than 0).  But it can never be &quot;exact&quot; because ln(5.4)/8 has a non-periodic decimal expansion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-She wants to be able to &quot;do it herself&quot;, without the assistance of a machine.  But a calculator doesn't magically know the decimal approximation for ln(5.4)/8.  It uses an algorithm to find it, the derivation of which has to be well beyond the scope of the course she's taking.  Surely she realizes that there is other, way more complicated mathematics that she has yet to be exposed to and couldn't possibly be exposed to (either because she's unable to grasp it yet or because there's nowhere near enough time to teach that much material that early on).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-I like your definition of ln(a), and if explained properly I think it could be helpful in this situation (I'm making no judgments about whether you explained it properly, whether she just didn't &quot;get it&quot;, etc).  You define it as the definite integral of 1/x from 1 to a, which is ln(a)-ln(1)=ln(a).  So if she wanted to find an approximation for ln(5.4), she would just end up back at the same spot where she'd have to plug ln(5.4) into her calculator.  On the other hand, she could approximate it &quot;on her own&quot; by drawing a bunch of suitably thin rectangles and summing their areas (to put it informally) which anyone can do, knowing the area of a rectangle, the function y=1/x and knowing how to plug numbers into a function.  This helps demonstrate the futility of approximating it without calculator assistance, and why understanding where these values come from is unnecessary (ie, assuming she figures out how to do all this, she isn't going to feel more fulfilled or more able to solve problems involving logs).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure this could have been more consise, so sorry about that.&lt;/p&gt;&#xA;" OwnerUserId="5141" LastActivityDate="2015-05-18T15:23:59.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8120" PostTypeId="2" ParentId="8110" CreationDate="2015-05-18T19:27:15.387" Score="2" Body="&lt;p&gt;How about the following characterization of $\ln$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\ln: \mathbb{R}^+ \to \mathbb{R}$ is the unique continuous function satisfying the following properties:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$\ln(xy) = \ln(x)+\ln(y)$ for all $x,y \geq 0$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\ln(1+\epsilon) \approx \epsilon$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Property $2$ is a bit wishy washy (should really be $\frac{d}{dx} \ln(x) \big|_{x=1} = 1$), but will let her compute stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could prove that $\ln(1) = 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then prove $\ln(x^n) = n\ln(x)$ for natural numbers $n$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See if she has an idea for approximating $\ln(2)$.  My idea would be to write $\ln(2) = \ln((1+0.01)^n)$ for some very large $n$.  Then $\ln(2) = n\ln(1+0.01) \approx n(0.01)$.  You could attempt to find this $n$ with an excel spreadsheet, and then confirm with the calculator button.  You could obviously do the spreadsheet &quot;by hand&quot;, so it is not cheating much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do the same thing for $\ln(3)$.  Notice that $\ln(2)&amp;lt;1$ and $\ln(3)&amp;gt;1$, so there should be some number whose natural log is 1.  This is the definition of $e$ in this framework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She might notice that she is really making a table of values for $\ln$ when she raises $(1+\epsilon)$ to higher and higher powers.  Approximate $e$ to varying degrees of accuracy by decreasing $\epsilon$.  Compare to the calculator value of $e$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This could develop into months of mathematical exploration.  How does this definition of $e$ relate to other definitions?  How is $\ln$ related to the exponential function?  How do we know that a function satisfying these properties even exists?  How does this characterization relate to solving exponential equations? How does this tie into the area picture?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may also be interested in my math.stackexchange post here, which is at a level appropriate for a calculus student &lt;a href=&quot;https://math.stackexchange.com/questions/498339/demystify-integration-of-int-frac1x-mathrm-dx/498790#498790&quot;&gt;https://math.stackexchange.com/questions/498339/demystify-integration-of-int-frac1x-mathrm-dx/498790#498790&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-05-18T19:27:15.387" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8121" PostTypeId="2" ParentId="8110" CreationDate="2015-05-19T02:18:37.507" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;She is very frustrated because she feels like she needs to know what ln is and how to find it by hand and how to do everything by hand.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think this is a fantastic attitude, and I suspect it was shared by many great classical mathematicians! Euler, for example, routinely explored &lt;a href=&quot;https://books.google.com/books?id=x7p4tCPPuXoC&amp;amp;ots=YorcAxvLaP&amp;amp;pg=PA60#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;new functions&lt;/a&gt;, including &lt;a href=&quot;https://books.google.com/books?id=CVRZEXFVsZkC&amp;amp;ots=idywbGGwYe&amp;amp;pg=PA115#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;the logarithm&lt;/a&gt;, by doing hand calculations that seem preposterous to us digital natives. When he came up with the power series for the exponential function, one of the first things he did with it was compute $e$ to &lt;a href=&quot;https://books.google.com/books?id=x7p4tCPPuXoC&amp;amp;ots=YorcAxvLaP&amp;amp;pg=PA26#v=onepage&amp;amp;f=false&quot;&gt;twenty-three&lt;/a&gt; decimal places. Doing calculations by hand often forces you to think about familiar objects in new ways, and it can be a great way to get a feel for how mathematical objects behave.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As an example, think about how you might compute $\log_2 3$ by hand. Observing that&#xA;$$2^1 &amp;lt; 3 &amp;lt; 2^2,$$&#xA;you can sandwich $\log_2 3$ between the integers 1 and 2. For a better estimate—say, a sandwich between sixteenths of an integer—you could try to sandwich $16 \log_2 3$ between integers. Playing around with $16 \log_2 3$, using what you know about how logarithms work, you might eventually notice that it's equal to $\log_2 3^{16}$. If you're willing to use a calculator for multiplication, it only takes a few minutes to find out that&#xA;$$2^{25} &amp;lt; 3^{16} &amp;lt; 2^{26},$$&#xA;sandwiching $16 \log_2 3$ between $25$ and $26$. This shows that $\log_2 3$ is between $\frac{25}{16}$ and $\frac{26}{16}$, pinning it down to within 2%.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;One of the ways people first ran into the natural logarithm was by thinking about slow exponential growth, which can be done very computationally. Let's say you're watching an &lt;a href=&quot;http://en.wikipedia.org/wiki/Algal_bloom&quot;&gt;algae bloom&lt;/a&gt; that grows at a rate proportional to its size, increasing its mass by $\frac{1}{10}$ of its current mass every day. In symbols, the size of the bloom after $p$ days is&#xA;$$\left(1 + \frac{1}{10}\right)^p.$$&#xA;Using a calculator for multiplication, it's not hard to compute that the bloom will double in size every 7 or 8 days.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What if the algae reproduce more slowly? Let's say the bloom only grows by $\frac{1}{20}$ of its current mass each day. More calculator work reveals that the doubling time is 14 to 15 days.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the bloom grows by $\frac{1}{30}$ of its mass each day, the doubling time is 21 to 22 days. A growth rate of $\frac{1}{40}$ gives a doubling time of 28 to 29 days. A growth rate of $\frac{1}{n}$ always seems to give a doubling time close to a constant multiple of $n$! That constant multiple, which from our calculations appears to be around 0.7, is the natural logarithm of two.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;We've found a number $\ln 2$ with the property that it takes about $n \ln 2$ days for something growing at a rate of $\frac{1}{n}$ per day to grow by a factor of 2. In other words,&#xA;$$\left(1 + \frac{1}{n}\right)^{n \ln 2} \approx 2,$$&#xA;an approximation that gets better and better as $n$ grows. This phenomenon isn't limited to doubling times. For any growth factor $x$, there's a number $\ln x$ with the property that&#xA;$$\left(1 + \frac{1}{n}\right)^{n \ln x} \approx x$$&#xA;more and more accurately as $n$ grows. Looking at this expression, you might suspect that&#xA;$$\left(1 + \frac{1}{n}\right)^n$$&#xA;approaches some constant as $n$ grows. For some values of $n$, you can evaluate the expression above very quickly using clever tricks. For example, you can evaluate it for $n = 256$ by squaring $1 + \frac{1}{256}$ eight times. If you do this carefully (a calculator with a squaring button helps a lot), you can see that as $n$ grows, the expression does seem to approach a constant—something around 2.71.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;With these examples, I hope I've managed to convince you that hand computations are a fun way to get a feel for the natural logarithm. The same goes for lots of other functions you'll encounter in your classes. If you ask your classmate to keep challenging you to understand the computational fundamentals of the new functions you encounter, I think you'll learn a lot!&lt;/p&gt;&#xA;" OwnerUserId="5119" LastActivityDate="2015-05-19T02:18:37.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8122" PostTypeId="2" ParentId="8104" CreationDate="2015-05-19T10:31:00.550" Score="3" Body="&lt;p&gt;We have had the different mathematics courses concurrently at the university - analysis (calculus) and linear algebra, later numerical methods and statistics.&#xA;I have observed following advantages of this approach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) You can ask a teacher from algebra a calculus question (or vice versa) if you happen to understand his explanations better than the ones from his calculus colleague.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) If you happen to miss some school time (sick) then you get two smaller gaps to cover rather than one larger one. It was easier (at least for me) to keep the continuity of understanding that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) What you have already mentioned - (at least my) brain needs a bit of breathing air between hard topics. These seem to group in larger blocks of one course. Parallel schedule allows for interleaving - difficult stuff here - easier here and then swap. Single topic in whole week does not leave that space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have observed one disadvantage: Sometimes the prerequisites from one course were not yet available in the other one, but this happened very rarely.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2015-05-19T10:31:00.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8123" PostTypeId="2" ParentId="8110" CreationDate="2015-05-19T12:11:14.367" Score="3" Body="&lt;p&gt;What about &lt;a href=&quot;http://eulerarchive.maa.org/hedi/HEDI-2005-07.pdf&quot; rel=&quot;nofollow&quot;&gt;how Euler did it&lt;/a&gt; to find logarithms by hand. Though you may consult the paper cited personally, I'd like to add a few comments. First, notice that how Euler reduce the problem of finding logarithms to the &quot;easier&quot; problem of finding square roots using repeatedly $\log \sqrt{xy}=\frac {\log x + \log y}{2}$. Moreover, it seems funny (from a modern viewpoint) how he performs the process until he runs out letters! However, it also shows that how even such a seemingly simple thing as indexing letters has been developed during the years until it has reached the point we know it. Of course, the notion of logarithm has a more complex history, an important part of which involves calculating by hand. Respecting this part is respecting the mathematicians who helped us to understand logarithm as we know it today. Moreover, It encourages our students not to take for granted what they have (i.e., calculators). It can be also used as a bridge to understanding how calculators calculate logarithms. To be honest, I am so fond of the &quot;handy&quot; history of logarithms that whenever I teach it I ask students to do it at least once as Euler did it. I also give one minute silence to respect his work, Napier's and Brigg's. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="526" LastEditDate="2015-05-19T13:00:00.247" LastActivityDate="2015-05-19T13:00:00.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8124" PostTypeId="2" ParentId="8038" CreationDate="2015-05-19T15:53:34.360" Score="5" Body="&lt;p&gt;Short answer: With a lot of work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long answer:  There may be possibilities where students working on iPads can work next to students not working on iPads, but my experiences volunteering in elementary school suggest that it creates a division that would be more distracting than helpful.  I recommend segregation so that the people not working on the iPad are in a different room.  Ideally, an &quot;iPad lab&quot; is set up so that all iPad work is done in the lab, and equivalent work done in the regular classroom.  If you have to accommodate only a few students away from the iPad, suggest that they have special assignments that their parents encourage that they do, and make sure there are personnel to supervise their working on those assignments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To really accommodate the parents who choose to opt out, you will have to offer an iPad free environment.  The lab above is the closest I can come to what I think is a realistic accommodation, based on experiences of U.S. K-3 education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;How Smart Is Segregation Anyway?&quot; Paseman, 2015.05.19&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-19T15:53:34.360" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8125" PostTypeId="2" ParentId="8038" CreationDate="2015-05-19T17:14:42.993" Score="3" Body="&lt;p&gt;If a learning activity can be effectively carried out without iPads, then the iPad is not being used appropriately. Good educational technology does not just change how something is done, it expands what is possible. If pen and paper is just as good, maybe pen and paper should be used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an analogy, if parents seriously object to the ready availability of pencils and paper in classrooms they should not expect an innovative alternative curriculum. The range of activities available to writers simply can't be easily replicated without pencil and paper. Their kids will simply miss out. But if all you are using pen and paper for is to write down times tables, maybe you should do more of it verbally, which is a more social activity, allows for faster feedback, and eases the movement to reflection and other high level activities. Even in a modern classroom there are still many activities which are best carried out by verbal discussion, though our education system has become obsessed with writing, and now with interactive technology. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, highly interactive technology is often best used in groups that can discuss what they should do, thus combining verbal, written, and interactive tasks into a single activity. This can turn basic tasks into a higher order/reflective tasks. In this case, the non-touchscreen kid can be a navigator or advisor but not the driver (the person physically controlling the device).  No change of classroom task is needed. The parents were upset about their kids touching a computer screen, not seeing the computer screen. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For individual activities, if your students are using interactive simulations to understand multiplicative/proportional relationtionships, non-digital kids can play with manipulables or draw diagrams. If your students are using adaptive quizzes with instant feedback, your other students can have generic drill sheets which they can they swap and peer mark, or maybe flash cards. If your students are proving geometric figures with Dragon Box Elements, the other kids can... doodle in their books. It is just a qualitatively different interactive learning experience from pen and paper. The parents should be told what their kid is missing out on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your students work on a cooperative project facilitated by the iPads, the non-digital kids can bring their own technology (from a brand their parents approve of) and try to join in, but should not expect the teacher to provide tech support. Or they can write out a report by hand and a team-mate can take a photo and paste it in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Visuals can be watched on a TV connected to an iPad via AppleTV, so the children don't have to touch the dreaded screen of the iPad (as apparently touching the screen of an Apple device destroys the ability of a child to concentrate). The AppleTV can then be controlled with a traditional remote control. Or the iPad can be controlled by a friend.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope some of my ideas are usefuI, but though I accept the premise of the question that you can always find some parent to object to anything a school does (especially when it comes to the brand of technology that a school uses), I disagree with the unstated assumption that it is the teacher's job to provide alternative programmes for their kids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a parent who has previously asked for my kids to opt out of an activities, I never assumed that a teacher should provide alternative activities. I simply assumed my kids would be missing out and that they should take an alternate activity, or at least a good book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the problem is the school allowing parents to opt-out of a tool rather than an activity. It is pandering to ignorance on a par with pretending that doodling is the same as maths because they both use pen and paper. Dragon Box has little in common with YouTube surfing or Candy Crush. Parents have no right to a blanket objection to a brand or variation (touchscreen vs mouse) of safe and widely accepted education equipment, only to particular educational activities. Unless, of course, they have significant reason to believe the equipment is unsafe for their kids. Maybe they have been formally diagnosed as being ... dis-haptic?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would also be very reasonable to expect teachers to limit screen time to healthy limits, and parents could easily have input on what these limits should be. However, offering parents a blanket opt-out rather than time limit suggestions seems more like a cop-out by the school rather than something intended to produce constructive school-home dialogue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's be honest: in many cases the disagreement is probably not about educational activities at all but is a strategic argument driven by politics, ideology, or by semi-religious brand loyalty. What do you do if a parent objects to the Pearson textbook you use, without having read it? Or your 'liberal' History textbook? Or textbooks in general? They can opt-out of the class, but it is not the teachers' responsibility to change their program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where the disagreement is genuinely about child health, then a  richer dialogue about appropriate use would be much better for all involved, and I suspect all parties would usually find themselves in agreement after a more detailed explanation of how the equipment will be used has taken place. &lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-05-19T17:42:18.117" LastActivityDate="2015-05-19T17:42:18.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8126" PostTypeId="1" CreationDate="2015-05-19T19:42:38.227" Score="4" ViewCount="925" Body="&lt;p&gt;What do you think, could hypothesis testing be taught, in some way (probably informal) to that age group of kindergarten to 4th grade? Ideas, some resources, published papers, and so on? This would definitely be after they have been exposed to statistics in projects with collecting and presenting some data. Myself, I am skeptical, but what do you think?&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="122" LastEditDate="2015-05-21T10:49:45.503" LastActivityDate="2015-05-26T01:45:25.153" Title="Hypothesis testing for children (grade 1-4)" Tags="|primary-education|statistics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8128" PostTypeId="2" ParentId="8106" CreationDate="2015-05-20T05:24:33.793" Score="2" Body="&lt;p&gt;You could make the connection between Fourier synthesis and (musical) synthesizers. Maybe you could play some interesting tones from a synthesizer. How do we figure out what goes into the synthesis? Fourier analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A lot of students will (sadly) shut down while you're telling them things that &quot;won't be on the test&quot;. It would help if you could repeatedly loop back to the example as you cover topics in Fourier analysis.&lt;/p&gt;&#xA;" OwnerUserId="5205" LastActivityDate="2015-05-20T05:24:33.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8130" PostTypeId="2" ParentId="8038" CreationDate="2015-05-20T12:36:23.247" Score="2" Body="&lt;p&gt;An excellent question, and @Richard is right when he says that if the iPad (or any technology) can be easily replaced it's not being used well. So I propose that the way to use iPads with some, but not all, students is to make them one tool among many.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll try and be specific here, but K-3 is not my area of expertise. Let's say you're doing an activity with perimeter and area. You could make one activity use the iPad to explore and interact with perimeter and area, like an object where the students can change the measurements of a figure with a slider. Students would learn manipulation skills and about the relationship between side lengths and both perimeter and area. You could have another activity where students measure existing figures to determine perimeter and area. These students would learn measurement skills and the relationship between perimeter and area. Then make groups with students from each area and they must create figures with specified perimeter and area. This should require the skills learned at each station and the cooperation of the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The design principle being that iPads are a tool, the best way to use them and not in the same lesson is to develop a range of tools that students can use. Skills learned on the iPad do not need to be exactly duplicated with other tools, but would ideally compliment the skills learned using the other tools.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2015-05-20T12:36:23.247" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8131" PostTypeId="1" AcceptedAnswerId="8132" CreationDate="2015-05-21T12:49:07.727" Score="6" ViewCount="143" Body="&lt;p&gt;The title says it all, but some more details.  We need good textbooks for a mathematics course for future kindergarten teachers. The actual course is titled (my translation) &quot;Text, language and mathematics&quot; and is the only course in which these students meet mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is required that the subjects should be &quot;integrated&quot;, but still they are taught separately by teacher specialists in their own discipline with limited knowledge of the others. (In the exam all questions are integrated, so there is no specific math question, and no calculation required. This is not a course in arithmetic.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have some good textbooks in Norwegian (I think they are good, the students do not agree). They are really written with primary school, not kindergarten, in mind, but starts from the children and situations with interaction children/teacher from school (and maybe a very few from earlier age), they do not start from the mathematics. So the question is for books/teaching material in English (or Spanish, my other foreign language ...) with this sort of aim, but specifically directed at future kindergarten teachers.&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="267" LastEditDate="2015-05-22T06:12:08.363" LastActivityDate="2015-05-22T06:12:08.363" Title="Reference request: Textbooks in mathematics for future kindergarten teachers" Tags="|reference-request|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8132" PostTypeId="2" ParentId="8131" CreationDate="2015-05-21T16:41:21.817" Score="6" Body="&lt;p&gt;My book is not a textbook, but I think the future teachers would like it. &lt;em&gt;&lt;a href=&quot;http://naturalmath.com/playingwithmath/&quot;&gt;Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers&lt;/a&gt;&lt;/em&gt;, edited by Sue VanHattum. There is a story from a Montessori teacher (Trust, Montessori Style) and some of the math circles are groups of very young kids. The whole book will give future teachers a very different look at math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few more recommendations...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;http://www.betterworldbooks.com/little-kids-powerful-problem-solvers-math-stories-from-a-kindergarten-classroom-id-9780325004310.aspx&quot;&gt;Little Kids—Powerful Problem Solvers: Math Stories from a Kindergarten Classroom&lt;/a&gt;&lt;/em&gt;, by Angela Andrews and Paul Trafton. One story for each of the ten months of the school year. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;http://www.betterworldbooks.com/show-and-tell-representing-and-communicating-mathematical-ideas-in-k-2-classrooms-id-9780941355506.aspx&quot;&gt;Show and Tell&lt;/a&gt;&lt;/em&gt;, by Linda Dacy and Rebeka Eston. Another good book about communicating mathematical thinking in the classroom, this one is focused on kindergarten to second grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;http://www.betterworldbooks.com/talking-mathematics-supporting-children-s-voices-id-9780435083779.aspx&quot;&gt;Talking Mathematics: Supporting Children’s Voices&lt;/a&gt;&lt;/em&gt;, by Rebecca Corwin. It takes some real skill to get students talking about their mathematical thinking. Although this one isn't directed primarily at kindergarten, I think it will be very helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;http://naturalmath.com/TheBook&quot;&gt;Moebius Noodles&lt;/a&gt;&lt;/em&gt;, from my publisher, is another great collection of ideas for getting kids playing with mathematical ideas.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-05-21T16:41:21.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8133" PostTypeId="1" AcceptedAnswerId="8134" CreationDate="2015-05-21T17:00:58.247" Score="8" ViewCount="324" Body="&lt;p&gt;For K-3 students, perhaps it is not acceptable to introduce &lt;em&gt;multiplication by zero&lt;/em&gt; as a property or definition. Instead, the child may think about multiplication as, e.g., repeated addition.&lt;/p&gt;&#xA;&lt;p&gt;Examples of the &amp;quot;repeated addition&amp;quot; conception: &lt;span class=&quot;math-container&quot;&gt;$3 × 2 = 3 + 3 $&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ 4×3 = 4 + 4 +4$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How will students conceive of &lt;span class=&quot;math-container&quot;&gt;$ 3 × 0 $&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;How should early elementary/primary curricula deal with multiplication by zero?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3306" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-05-22T16:21:51.670" Title="Is multiplication by zero clear for and understood by K-3 students?" Tags="|primary-education|concept-motivation|curriculum|teacher-preparation|definitions|" AnswerCount="2" CommentCount="7" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8134" PostTypeId="2" ParentId="8133" CreationDate="2015-05-21T22:43:36.610" Score="11" Body="&lt;p&gt;Turning my comment into an answer as per request:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the graphical approach gets the idea across. You could represent $3\times 1$ by a row of three dots; likewise, $3\times 2$ is two rows of three dots; etc. If you asked them to guess what $3\times 0$ is, some would probably tell you you'd have &lt;em&gt;no&lt;/em&gt; rows so no dots, so the answer is zero. Humans are very good at inductive logic, even if we aren't aware of it.&lt;/p&gt;&#xA;" OwnerUserId="45" LastActivityDate="2015-05-21T22:43:36.610" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8135" PostTypeId="2" ParentId="8133" CreationDate="2015-05-22T02:02:54.917" Score="0" Body="&lt;p&gt;I'm sticking with food math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have 4 friends and need 2 candy bars per friend, so 8 bars. Similar, if say, 1 bar per friend, you have 4 bars total. If you have 0 bars per friend, it's 0. But if it's 0 per friend, no matter how many friends you have, you still need 0 candy bars. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-05-22T16:21:51.670" LastActivityDate="2015-05-22T16:21:51.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8137" PostTypeId="1" CreationDate="2015-05-22T06:21:04.197" Score="13" ViewCount="332" Body="&lt;p&gt;We will be hosting an Open Education Resources workshop at the college where I work. I am one of the two faculty that uses OERs and was asked to speak about the pros and cons of using OER. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;What textbook alternatives have you already tried? I am using College Algebra by. Carl Stitz, Ph.D. Jeff Zeager, Ph.D.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there any other open resources you are using? I am using myopenmath.com and WeBWorK. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are not using OER. What is your number one question about OER? What kinds of assessment you would like to see if you were to try using OER next term? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Anything else you want to add? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thanks,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;K&lt;/p&gt;&#xA;" OwnerUserId="5214" LastActivityDate="2016-10-29T16:06:33.570" Title="Open Education Resources" Tags="|textbooks|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8138" PostTypeId="1" CreationDate="2015-05-22T07:32:48.987" Score="22" ViewCount="886" Body="&lt;p&gt;Suppose a calculus classroom is full of students majoring in Classical Greek or music or literature or sociology or pre-medical studies or any of many subjects that do not require the course as a prerequisite, and not one of them intends to take a subject requiring as a prerequisite any of the technical skills (finding derivatives, limits, integrals, etc.) that could reasonably be taught in calculus course, and every one of them has said so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should a syllabus contain something like the following statement?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The purpose of this course is &lt;b&gt;neither&lt;/b&gt; to prepare you to &lt;b&gt;use&lt;/b&gt; your knowledge of calculus &lt;b&gt;nor&lt;/b&gt; to provide you with an opportunity to show you can work hard and overcome difficulties and thereby earn a grade to impress future employers, graduate schools, etc.  Those may be side-effects, but neither of them is the intended purpose.  The purpose is to contribute to your education by acquainting you with one of the great intellectual, scientific, cultural, and aesthetic achievements of humankind and with the kind of thinking that is used in this field.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Two concerns about this (conceivably objections to it?) are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Most students have never suspected that there are other purposes than the two that this statement says are the purpose of the course.  (In a reasonable system, they would find that out long before reaching calculus.  The main thing that is unreasonable that differs from such a system is our current practice of pressuring or even coercing students known to be unqualified to take calculus to do so, anesthetizing ourselves by telling ourselves that the systems for screening out the unqualified actually work.)&lt;/li&gt;&#xA;&lt;li&gt;The usual syllabi say one &lt;em&gt;must&lt;/em&gt; cover the following technical topics (computing certain limits, derivatives, integrals, etc.) within the allotted time: [insert list here].  The perceived imperative to cover everything in the list precludes many things that would help students understand.  When I mentioned one such thing, I was told &quot;It would be nice to include that but there's not enough time.&quot;  I said &quot;So scrap ninety-percent of the material and then there's time.&quot;  I don't know if that was taken seriously.  I don't of its being implemented anywhere.  And the usual textbooks are for the sorts of students for whom it is reasonable to insist on covering everything in a list of technical topics.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2015-05-23T14:35:24.527" LastActivityDate="2018-02-11T00:22:19.457" Title="The purpose of mathematics in a liberal education when it is not a prerequisite to other subjects" Tags="|calculus|syllabus|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8139" PostTypeId="2" ParentId="8138" CreationDate="2015-05-22T11:59:52.343" Score="3" Body="&lt;p&gt;Students majoring in Greek or music or literature are already committed to education aimed at appreciating the &quot;aesthetic achievements of humankind,&quot;&#xA;and may already be approaching the topic from that direction. Students majoring in sociology and pre-med might actually have to understand aspects of calculus in their studies. And students majoring in $X$ often end up pursuing careers in $Y$,&#xA;regardless of their intents back in college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To say that obtaining a useful &quot;knowledge of calculus&quot; may be a &quot;side effect&quot; diminishes that aspect. To me there are two purposes, roughly equally important:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) The appreciation of the &quot;aesthetic achievements of humankind&quot; you articulate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Obtaining an &lt;em&gt;understanding&lt;/em&gt; of the concepts of calculus, both as generic intellectual equipment, and for possible use in other avenues of study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning (2), I would emphasize understanding over skill at calculation.&#xA;Unfortunately, calculus courses often focus on the mechanics (integration by parts, etc.) and just hope that understanding follows. For me, understanding&#xA;only followed long after I received an $A$ in calculus.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-05-22T12:24:22.603" LastActivityDate="2015-05-22T12:24:22.603" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8140" PostTypeId="2" ParentId="8137" CreationDate="2015-05-22T14:55:34.673" Score="2" Body="&lt;p&gt;I use &lt;em&gt;&lt;a href=&quot;http://faculty.gvsu.edu/boelkinm/Home/Download.html&quot; rel=&quot;nofollow&quot;&gt;Active Calculus&lt;/a&gt;&lt;/em&gt;, by Matt Boelkins, for in-class work during our first few weeks, and &lt;em&gt;&lt;a href=&quot;http://scidiv.bellevuecollege.edu/dh/Calculus_all/Calculus_all.html&quot; rel=&quot;nofollow&quot;&gt;Contemporary Calculus&lt;/a&gt;&lt;/em&gt;, by Dale Hoffman, for homework during those weeks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that, I mainly use the textbook I'm required by my department to use (Anton). I would love to convince them to use open source textbooks, but so far have had no luck.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-05-22T14:55:34.673" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8141" PostTypeId="2" ParentId="7998" CreationDate="2015-05-22T19:50:39.623" Score="6" Body="&lt;p&gt;I likewise have needed such a thing—to converse coherently about transformation properties (actually symmetries) of power, exponential, and logarithmic functions. I did not find anything in textbooks or anything commonly used otherwise. I considered using $\rho$ for &quot;ratio&quot;, but settled on $\sigma$ (and  $\sigma_y$ as needed), because in conversing, &quot;scaling&quot; was more used than &quot;ratio.&quot; Following are examples of summary statements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An exponential function $f(x) = A b^x$ scales by the same factor over equal increments of the input. The base $b$ gives the factor over a unit increment.&#xA;The base to the power of an input increment $\Delta x$ gives the output factor: $$\sigma_y = b^{\Delta x}.$$&#xA;The graph is invariant under the family of transformations&#xA;$$(x,y) \rightarrow (x + \Delta, b^\Delta y)$$&#xA;where $\Delta$ can be any real number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We use $\Delta x$, but &quot;$\sigma y$&quot; is clumsy, so I use a subscript. I try to avoid the subscript. I would write the transformation property (&quot;equivariance&quot;) of the function as&#xA;$$f(x + \Delta) = b^\Delta f(x),$$&#xA;provided students were ready for such symbolic stuff. (Usually some are, some aren't—a chance to &quot;differentiate&quot; instruction.) In making tables of values, often a column would be labeled with a $\sigma_y$ (or $\sigma_x$ for logarithmic and power functions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With exponential functions I mostly avoid this $\sigma_y$ notation. If we have the exercise of finding the parameters for an exponential curve going through $(1,6)$ and $(3,30)$, say, I would encourage putting the data in a table and writing $\Delta x = 2$ on the $x$ side and $b^2 = 5$ somewhere, just saying 5 is the output scaling associated to an input increment of 2, rather than writing $\sigma_y = 5$. Solve for the base, and go from there.&lt;/p&gt;&#xA;" OwnerUserId="5218" LastEditorUserId="5218" LastEditDate="2015-05-22T22:24:55.873" LastActivityDate="2015-05-22T22:24:55.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8142" PostTypeId="2" ParentId="8138" CreationDate="2015-05-22T20:06:30.103" Score="9" Body="&lt;p&gt;&lt;a href=&quot;http://www.sjc.edu/&quot;&gt;St. John's College&lt;/a&gt; offers a four-year undergraduate program called the &lt;a href=&quot;https://en.wikipedia.org/wiki/St._John&amp;#39;s_College_(Annapolis/Santa_Fe)#Great_Books_program&quot;&gt;Great Books Program&lt;/a&gt; focusing on the study of western civilization's classics from primary sources. Regarding calculus, the program's curricula includes Newton's Principia Mathematica and selected writings by Leibniz, Taylor, Einstein and &lt;a href=&quot;http://www.sjc.edu/academic-programs/undergraduate/liberal-arts/&quot;&gt;others&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have only heard and read about the program and have no direct familiarity with it, so this might not be considered a full answer. But perhaps researching into this and asking someone who took that program would allow you to get better insight into such studies of calculus.&lt;/p&gt;&#xA;" OwnerUserId="688" LastActivityDate="2015-05-22T20:06:30.103" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8143" PostTypeId="2" ParentId="8138" CreationDate="2015-05-22T20:31:37.827" Score="2" Body="&lt;p&gt;I learned songs because I enjoyed singing them, and only much later did I learn songs to perform them for others.  I learned history so that I could understand some of the cultural references involving history, and also in the vain hope that I could make newer, better, and more original mistakes.  I learned some literature (but it didn't sink in until years later) to see why people are interested in some of the things they are interested in, as well as in how a story could be told well.  I learned some debate because my father told me to, but also to practice the ability to think and produce arguments in real time, as well as prepare researched arguments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I learned math because I was good at some of it, and good at explaining it.  I became good at certain modes of reasoning, which helped me with economic and engineering problems as well as more abstract problems.  I can now go back to some of the stories I've read and think &quot;If this character had known calculus, they could avoid the problem in the next chapter if they just...&quot; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your goal is to prepare a good syllabus entry, you should emphasize the difference in perspective that analytical reasoning and inductive reasoning can provide.  Even going through what seem to be mindless computations trains one in mechanical reasoning, and can be useful in trying to debug problems they find with software systems.  Not a software engineer?  What's that thing in your back pocket?  You're going to have to deal with that user interface on your phone-watch-pda somehow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Ready To Turn The Page&quot; Paseman, 2015.05.22&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-22T20:31:37.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8144" PostTypeId="2" ParentId="7985" CreationDate="2015-05-22T20:37:36.127" Score="3" Body="&lt;p&gt;I think the reason for students' confusion in the latter method depends on two parameters (assuming the students are aware of both concepts):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;In which order are the two concepts taught?&lt;/li&gt;&#xA;&lt;li&gt;How long was the gap between teaching both concepts?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It all boils down to the students' reluctance in connecting two different concepts that they were taught. Students should be trained as mathematical thinkers -- problem solvers. This attitude will help them become at ease in applying two concepts together to solve a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So your method indeed reinforces this attitude. Perhaps your students are not yet acquainted with such disposition. However, I would highly recommend the second method (and similar out of box techniques to solve seemingly mundane problems).&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2015-05-22T20:37:36.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8145" PostTypeId="2" ParentId="7998" CreationDate="2015-05-23T05:55:30.563" Score="2" Body="&lt;p&gt;I think its worth noting that the obelus symbol ($\div$) coincidentally looks like a vertical fraction and does denote division. &lt;a href=&quot;http://en.wikipedia.org/wiki/Obelus&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Obelus&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It could be used as a table header pretty easily:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$x \ \ \ y \ \ \div y$&lt;br&gt;&#xA;$2 \ \ \ 7 \ \ $&lt;br&gt;&#xA;$4 \ \  35 \ \ \ \ 5$    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But could get confusing as an inline symbol: $\div y=5$ or  $\frac{\div y}{\div x}=5/2$ don't quite look right, and try saying &quot;Ten times the y ratio&quot;: $10\div y$ We'd get confused .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe $\div$ with a triangle or circle around it?  I don't know of a premade symbol like that though.&lt;/p&gt;&#xA;" OwnerUserId="4962" LastActivityDate="2015-05-23T05:55:30.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8146" PostTypeId="2" ParentId="8105" CreationDate="2015-05-23T22:07:40.550" Score="4" Body="&lt;p&gt;The staircase metaphor (I prefer a rope ladder) is useful when associated with the concept of enumerable sets, the prototype of which is $\mathbb{N}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I usually couple my explanation with a (very simplified) reference to Peano's axiomatization of natural numbers, to shake off the conventional understanding, and to create room for the idea that we are dealing with enumerations, regardless of how exactly they are represented. I would be nice if it is possible to suggest (provocatively) that not all sets can be enumerated, even if we allow the process to be infinite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only then I´ll introduce the rope ladder, without ever losing completely the connection to the symbolic level of description:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/BxSHQ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remind the students that we have the &quot;left rope&quot; (the basic enumeration). What we need is to &quot;attach the right rope to the left, one rung at a time&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is important to emphasize that we are doing is to prove a property $P$ over a set of objects that are enumerable. We are not performing a &lt;em&gt;calculation&lt;/em&gt;, but a &lt;em&gt;demonstration&lt;/em&gt; (a &quot;logic thing&quot;, not a &quot;regular math thing&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All along, it is crucial to avoid isolating symbolic reasoning and explicative metaphor. If these are kept together, you can simplify and reiterate at will, without having to count on the students to &quot;just get it&quot;.&lt;/p&gt;&#xA;" OwnerUserId="5038" LastActivityDate="2015-05-23T22:07:40.550" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8147" PostTypeId="2" ParentId="8105" CreationDate="2015-05-23T22:19:57.740" Score="3" Body="&lt;p&gt;I tell roughly eight students to line up.  Let's say the second student in line is named Natalie. I say to the class &quot;Natalie knows a secret.  Whenever someone in this line knows a secret, they whisper it to the student behind them.  Which students will eventually know the secret?&quot;  It's up to the class then to realize that Natalie and everyone behind her will know the secret.  Sometimes a student will guess that everyone in the line knows the secret, because the first person in line tells Natalie, and Natalie tells the student behind her, and so forth.  However, other students in the class will usually argue persuasively that the first student in line does not necessarily need to know the secret.  After this icebreaker, I begin the discussion of induction a little more formally.&lt;/p&gt;&#xA;" OwnerUserId="5104" LastActivityDate="2015-05-23T22:19:57.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8148" PostTypeId="1" AcceptedAnswerId="8156" CreationDate="2015-05-24T15:52:23.860" Score="19" ViewCount="2577" Body="&lt;p&gt;Is any time being spent on the history of mathematics in high school classes today?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Few observations as a student - &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I had to discover Cantor many years after I was introduced to set theory.&lt;/li&gt;&#xA;&lt;li&gt;I had to discover Fermat's last theorem and Andrew Wiles five years after I learnt the Pythagoras theorem.&lt;/li&gt;&#xA;&lt;li&gt;I had no idea about the Konigsberg problem or Euler. That had to wait for several more years.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I am not suggesting a detailed study of Fermat's last theorem, I have no idea about Wiles' proof of the same. But I would have loved to hear these stories from my teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why do mathematics classes ignore mathematicians and exclusively focus only on the mathematics that they have handed over to us?&lt;/p&gt;&#xA;" OwnerUserId="5221" LastEditorUserId="201" LastEditDate="2015-05-25T14:11:14.150" LastActivityDate="2021-03-14T17:05:53.687" Title="Teaching the History of Mathematics in High School" Tags="|mathematical-pedagogy|secondary-education|history|history-of-math|" AnswerCount="8" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8149" PostTypeId="2" ParentId="8126" CreationDate="2015-05-24T16:37:03.820" Score="4" Body="&lt;p&gt;Basic hypothesis testing is something kids do lots every day, so it is just a matter of tapping into that and giving them a way of talking about it.&lt;/p&gt;&#xA;&lt;h2&gt;Testing with non-overlapping ranges&lt;/h2&gt;&#xA;&lt;p&gt;Basic hypothesis testing would involve comparing ranges of values from 2 groups to see if the values of one group or another group is higher. If the 2 groups have non-overlapping ranges of values, it is possible for even a Kindergartener to use that to make a valid inference.&lt;/p&gt;&#xA;&lt;p&gt;You could hypothesise that adults are taller than primary school children. You could get a sample of adults (teachers) and a sample of children (the class), get their heights and plot them on a number line (they could be marked directly on the ruler/number line that was used to measure the people). You would use 2 different colours for teachers and children. The range of heights from the adult sample will be higher than the range of heights for the children, likely with no overlap. It is fair to infer that the hypothesis has been confirmed.&lt;/p&gt;&#xA;&lt;p&gt;You could then test whether boys are taller than girls with the same method. It is likely that there will be a large overlap so that no inference can be made, and the hypothesis is not supported.&lt;/p&gt;&#xA;&lt;h2&gt;Testing with Monte Carlo matches&lt;/h2&gt;&#xA;&lt;p&gt;A more advanced method that kids might understand is to use Monte Carlo methods to test for differences between to groups. If you use boys vs girls the kids would come up with lots of hypotheses. Match up a random pair, one from each team. For each match up, give a point to the team whose representative matches the hypothesis. At the end of 10 random match-ups, has the &amp;quot;boys team&amp;quot; won all (or all except one), has the &amp;quot;girls team&amp;quot; won nearly all, or are the results mixed?&lt;/p&gt;&#xA;&lt;p&gt;Keep in mind that with a large number of tests, even 8/10 could allow several apparent positives in support of false hypotheses so only 10/10 or 9/10 should be used to support a hypothesis. They need to understand it is not about which team &amp;quot;won&amp;quot; this time, hypothesis testing is about whether that team will (almost) always win.&lt;/p&gt;&#xA;&lt;p&gt;When the score is 7/10, we could imagine that with different match ups from different classes the score will sometimes be 9/10, but it might also sometimes be only 4/10, so the other team will sometimes win. But if the score is 9/10  or 10/10 it is unlikely that the other team will ever win. This concept might be too hard for your year levels, but it might be worth a shot.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-05-24T16:37:03.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8150" PostTypeId="1" CreationDate="2015-05-24T23:43:07.273" Score="12" ViewCount="835" Body="&lt;p&gt;Suppose you need to explain &quot;&lt;em&gt;What is a logarithm?&lt;/em&gt;&quot; to an intelligent&#xA;but math-phobic adult (or a student well-prior to her introduction to logarithms).&lt;sup&gt;1&lt;/sup&gt;&#xA;I have used base-$10$, saying that, essentially, the logarithm (base-$10$) of a number&#xA;is the number of digits needed to represent it in decimal notation.&#xA;But that's not quite right at the $10^k$ transitions.&#xA;And neither floor nor ceiling makes&#xA;a clean statement about the number of digits:&#xA;&lt;hr /&gt;&#xA;$$\lfloor \log_{10}(999) \rfloor = \lfloor 2.9996 \rfloor = 2$$&#xA;$$\lfloor \log_{10}(1000) \rfloor = \lfloor 3 \rfloor = 3$$&#xA;$$\lfloor \log_{10}(1001) \rfloor = \lfloor 3.0004 \rfloor = 3$$&#xA;&lt;hr /&gt;&#xA;$$\lceil \log_{10}(999) \rceil = \lceil 2.9996 \rceil = 3$$&#xA;$$\lceil \log_{10}(1000) \rceil = \lceil 3 \rceil = 3$$&#xA;$$\lceil \log_{10}(1001) \rceil = \lceil 3.0004 \rceil = 4$$&#xA;&lt;hr /&gt;&#xA;Is there some clean way to interpret $\log_{10}$ as the number of digits,&#xA;without getting hung up on the detail&#xA;that $1000$ needs $4$ digits but $\log_{10}(1000)=3$,&#xA;or trying to explain scientific notation (which accords with the floor):&#xA;&lt;hr /&gt;&#xA;$$999=9.99\times10^2$$&#xA;$$1000=1.000\times10^3$$&#xA;$$1001=1.001\times10^3$$&#xA;&lt;hr /&gt;&#xA;The caveats I find necessary to explain the nuances destroy the clarity I seek to achieve.&#xA;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt; In response to a request to explain what I mean by &lt;em&gt;mathematically naive&lt;/em&gt;, I answered: &quot;I was thinking of a generic family gathering. So in this context, most people do not understand exponents. But everyone understands that 1,234 has 4 digits.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-05-25T17:51:35.883" LastActivityDate="2022-06-14T17:37:13.963" Title="How best to explain the logarithm to the mathematically naive?" Tags="|mathematical-pedagogy|secondary-education|logarithm|" AnswerCount="8" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="8151" PostTypeId="2" ParentId="8150" CreationDate="2015-05-25T01:41:42.250" Score="6" Body="&lt;p&gt;What's wrong with&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\log_{10}(10)=1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\log_{10}(100)=2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\log_{10}(1000)=3$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\log_{10}(10000)=4$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and explaining that log is the power to raise 10 to get a given number. I think the digit count is an unnecessary tangent to the process. The above 4 equalities lend them selves to showing how 1000/10=100 and for their logs, 3-1=2, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On re-reading your question, for integer results, the base 10 log is the number of zeroes after the 1. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been thinking about your phrase &quot;mathematically naive&quot; and who might be in this group. Are these teens who are not quite at the level they should be, or adults who are not math savvy for whatever reason asking you to explain logs? What I'm trying to parse out here is what is the starting point? Do they understand exponents at all?   &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-05-25T11:43:42.617" LastActivityDate="2015-05-25T11:43:42.617" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8153" PostTypeId="2" ParentId="8148" CreationDate="2015-05-25T17:01:44.517" Score="4" Body="&lt;p&gt;I would keep separate &quot;life of mathematicians&quot; from &quot;history of math&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the first certainly contains much narrative material, which would make the subject more &quot;attractive&quot;, its pedagogical content is, to me, doubtful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the second, on the other hand, I am inclined to say that everybody would get something from presenting math as a lively subject; and lively means also that it has a history, containing evolution, wrong turns, errors, repetitions, trends and all the like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortuntaley, at least for what concerns my country, Italy, very little emphasis is usually, at high school level, put on such a presentation of math. The reason being that it would take time away from &quot;real math&quot;. I think this is an error in perspective, since it mantains a status quo in which a large majority of students exit high school convinced that math ended somewhat in the 19th century with nothing new going on after that. As a research mathematician one of the most common questions I am asked is: &quot;is there really something new to say about math&quot;? (Guess this is common)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You mention Koningsberg problem. I devoted some time to think abouth Euler charachteristic in the context of polyhedra. If you browse the book &quot;Proofs and Refutation&quot; by Lakatos you may well find an enormous amount of informations, mathematical and historical, that could be presented in a class to show how that subject, starting from an apparantely simple remark open a whole new field of research for years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I referred to the fact that I'd rather keep separate this from &quot;life of mathematicians&quot; since I fear that encouraging history of math in high school could result in simply adding to math some spicy story on people, anedoctes, dubious interpretations (like Cantor being driven crazy by infinity) which are ok for entertaining a general audience but not for class time (imho, of course).&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2015-05-25T17:01:44.517" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8154" PostTypeId="2" ParentId="8138" CreationDate="2015-05-25T17:08:42.113" Score="6" Body="&lt;p&gt;I understand and sympathize with your point. Just, as a matter of emphasis, I think it is never so nice to present a course from the point of view of what you &lt;strong&gt;do not&lt;/strong&gt; intend to obtain. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd rather turn it into positive and explain that it is mainly through understanding how to manipulate some math you can come to appreciate &quot;&lt;em&gt;acquainting you with one of the great intellectual, scientific, cultural, and aesthetic achievements of humankind&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2015-05-25T17:08:42.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8155" PostTypeId="2" ParentId="8150" CreationDate="2015-05-25T18:43:33.390" Score="2" Body="&lt;p&gt;You are close.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\operatorname{Floor}(\log_{10}(N))+1$&lt;/span&gt; is the number of digits.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="205" LastEditDate="2022-06-14T17:37:13.963" LastActivityDate="2022-06-14T17:37:13.963" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8156" PostTypeId="2" ParentId="8148" CreationDate="2015-05-25T18:50:03.070" Score="15" Body="&lt;p&gt;I disagree wholeheartedly with Nicola. I can only speak to the US' and Canada's mathematics systems so it may not be broadly true. Mathematics is largely taught in a vacuum. In every other field of study (physics, chemistry, economics, literature, etc.), we learn about the people, the history, &lt;em&gt;and&lt;/em&gt; the ideas. In math, we might learn the names of some mathematicians and very little of the history, with most focus on the ideas only. This is a huge detriment. It makes mathematics extremely dry in my opinion. People love talking about Marie Curie because she was one of the first big shot women in the sciences; people love talking about Feynman because he was kind of crazy in a fun way; people love talking about Einstein because he had such an unlikely story; people love talking about how much of a visionary Tesla was and how completely crazy he was; people love talking about Thoreau because he was extremely misanthropic; people love talking about Rockefeller because he was a ruthless capitalist; people love talking about J. Edgar Hoover because he extremely flawed and polarizing. Having these figures attracts people to a topic. Talking about history without getting into the people behind it is extremely boring. It would be like reading the line of kings in Lord of the Rings or the family trees in the Bible - just a series of words with not much to connect to at a personal level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for &lt;em&gt;why&lt;/em&gt; this happens.. It's because of a really vicious cycle. Our educators don't know anything about the mathematicians or history behind the mathematics they teach because it was never taught to them. Thus none of the students know because they weren't taught and when a small fraction of those students become math teachers, they can't impart this knowledge because they never had it. This probably has its roots in the ivory tower aspect of mathematics. Mathematicians historically shied away from the spotlight, did math and didn't worry about anything else. Other disciplines were about understanding tangible things about the universe around us which is something we can latch on to and feel passionate about like the fate of the universe or the birth of the universe or abiogenesis or space travel. It's harder to feel passionate about abstract things like a scheme or moduli space. This is made worse by the completely abysmal state of math journalism. Other fields get fairly regular press, math gets incredibly infrequent press.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I take this topic pretty seriously and it's one that I am very passionate about. After reading Dunham's Journey Through Genius, math leapt off the page and I felt like I was watching it unfold before my very eyes. I could see the arguments and spats between mathematicians. Seeing something you love come to life like that is astounding. Unfortunately, the education system in the US is so entrenched that there's not much hope for change when it comes to math education.&lt;/p&gt;&#xA;" OwnerUserId="45" LastActivityDate="2015-05-25T18:50:03.070" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="8157" PostTypeId="2" ParentId="8150" CreationDate="2015-05-25T19:23:03.613" Score="2" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\log_3 81 = {} &amp;amp; \log_3(3\times3\times3\times3) = 4 \\&#xA;= {} &amp;amp; \text{the number of multiplications by 3 that} \\ &amp;amp; \text{amount to a multiplication by 81}. \\[10pt]&#xA;\log_3 1 = 0 = {} &amp;amp; \text{the number of multiplications by} \\ &amp;amp; \text{3 that amount to a multiplication by 1}.&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2022-06-14T17:36:38.117" LastActivityDate="2022-06-14T17:36:38.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8158" PostTypeId="2" ParentId="8150" CreationDate="2015-05-25T19:55:41.707" Score="5" Body="&lt;p&gt;If I where given some time, I would try to explain exponents and then explain logarithms, because it really explains what they are. In more details, one could say the following.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Remember how multiplication is defined? It is just iterated addition: $3\times 4$ is defined as $4+4+4$, the sum of three fours. You learned that years ago, and then you learned how to interpolate so as to define the product of non-whole numbers, e.g. $1.2 \times 3.5$. You also learned how to divide numbers, and that $12/4$ is the number by which one multiplies $4$ to get $12$, i.e. $3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may also have learned exponents; usually is not remembered that well, but it is really just the same process applied to multiplication: $4^3$ is $4\times 4\times 4$, i.e. the &lt;em&gt;product&lt;/em&gt; of three fours. It is also possible to interpolate this operation to define it for non-whole numbers, e.g. $1.03^{6.5}$. This is a very useful operation, as the last example is the number by which your savings are multiplied in six and a half years if you have a $3\%$ interest rate. So, exponentiation is to multiplication what multiplication is to addition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, a legitimate question is: what is the analogue to division for exponentiation? I.e., is there a way to find to what powers one must raise, say $10$, to get a given number, say $120$? Yes there is one, and this is, by its very definition, what a logarithm is. The base-$10$ log of $120$, often denoted by $\log(120)$ of the barbaric $\log_{10}(120)$, is precisely the number by which one has to raise $10$ to get $120$. It is even easy to have an idea of what its value is, because $100$ is $10$ to the power $2$, and $1000$ is $10$ to the power $3$, so that to get $120$ one has to raise $10$ to a power a little bigger than $2$. This operation is really almost as simple conceptually as division, but is more difficult to compute by hand.&quot;&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-05-25T19:55:41.707" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8159" PostTypeId="2" ParentId="8148" CreationDate="2015-05-25T21:40:47.197" Score="8" Body="&lt;p&gt;&lt;em&gt;Caveat&lt;/em&gt;: This is merely an anecdote, from a college teacher, not a high-school teacher.&#xA;&lt;hr /&gt;&#xA;I taught a 9&lt;sup&gt;th&lt;/sup&gt;-grade (~14 yr olds) &#xA;geometry class at the invitation of a&#xA;high-school teacher last year. Tangentially &amp;amp; interactively&amp;mdash;in the context of &quot;What is a proof?&quot;&amp;mdash;the $4$-color theorem arose,&#xA;and I started to explain the theorem and the tangled history of its proof, from Kempe, to Appel &amp;amp; Haken's proof in 1976, down to the recent developments (e.g., the Robertson, Sanders, Seymour, Thomas proof).&#xA;They were fascinated, and peppered me with questions.&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/qPefl.png&quot; alt=&quot;Stamp4Color&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;This experience led me to believe there is a hunger for this type of&#xA;personalization of mathematics, which is not currently being satisfied in the&#xA;highly constrained US curriculum.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-05-27T23:12:51.833" LastActivityDate="2015-05-27T23:12:51.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8160" PostTypeId="2" ParentId="8148" CreationDate="2015-05-25T22:08:50.540" Score="1" Body="&lt;p&gt;The question and some of the other answers make me wonder at the purpose of education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can imagine situations where you are supposed to learn &quot;something&quot; ( a body of knowledge, a set of skills, a compendium of facts;  I am currently challenged to provide an accurate description of &quot;something&quot;), that this learning takes some time, and that incidental information may get in the way.  In such situations I can see the history of mathematics getting in the way of the material;  no pedagogical purpose is satisfied, just an ancillary purpose of entertainment (which in practical settings may be needed to get a student through the semester lesson plan, but I'm looking at things a little more abstractly for the present) .  I imagine Nicola Ciccoli is making this kind of point in (gender presumption) his answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the idea of the school system is not just education, but edification and entertainment ( so that some practical and political purpose is served by  &quot;keeping kids in school and out of trouble&quot; ), then I can see where it may be&#xA;useful to inject some additional material &quot;that doesn't have to be on the exam&quot; to make the lesson plan more bearable and enticing.  Doing such in an&#xA;educationally useful way is the craft of teaching, and such craft is what I&#xA;and others hope to learn with the help of a forum like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we see high school (or whatever the non U.S. equivalent is; forgive my ignorance) as a place to continue awakening the interests and develop the&#xA;capability of the next generation, then I think a major societal reform is in order.  There are some schools and programs in the United States that help this along, but my impression is more schools are doing without it.  Technology (and more importantly, information storage and retrieval) has advanced to the point where most physical and social barriers to education are being removed, the economic barriers are being overcome, and the major barrier becomes personal: do you have the interest and personal resources to learn?  Lectures and tutoring and books will not immediately become things of the past, but videos, visual and auditory guided exercises, and the ability to bring people of like interests together to discuss and learn regardless of their physical location will only increase.  The one thing that will remain are the questions to be asked.  Many of those will be &quot;What was math like in the 21st century?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Happy Memorial Day, here and abroad.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Better Write It Down Now&quot; Paseman, 2015.05.25&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-25T22:08:50.540" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8161" PostTypeId="2" ParentId="8150" CreationDate="2015-05-25T22:25:54.357" Score="2" Body="&lt;p&gt;I would not worry about an exact correspondence, because such an exact correspondence is not needed.  There is a good but not exact correlation between the function that is logarithm to the base 10 of a positive real number and the function that counts digits of a positive integer, and the two functions are used for different purposes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For such an introductory discussion, you can talk about number of digits, because that converts things from a scale that is hard to imagine physically to one that is easy to work with.  Instead of writing something like the number of atoms in a grain of sand, the number of grains of sands on a planet, the number of planets in the universe, or the inverse ratio such as the size of a subatomic particle compared to the size of a galaxy,  where such numbers are hard to get the number of digits right, much less the exact value, logarithmic approximations are easier to use.  Even if they don't have the exact definition of logarithm at hand, the fact that they can count decimal digits, do addition and subtraction, and get close to the right answer should do for an introduction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;There's No Test Afterwards, Right?&quot; Paseman, 2015.05.25&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-25T22:25:54.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8162" PostTypeId="2" ParentId="8126" CreationDate="2015-05-26T01:45:25.153" Score="1" Body="&lt;p&gt;There was a nice example I learned the other day from someone who was affiliated with a short film some students made following the theme of &quot;discovery&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students directed and shot the film with little or no adult involvement.  They decided to &quot;discover the real Santa Claus&quot;, and went to various locations where people who looked like Santa Claus were seen.  (I imagine this was shot in early December.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They had some criteria for determining if Santa Claus was real: he had to have a beard, and he had to know that caribou were reindeer (and so of course he had caribou).  The first Santa they filmed had a fake beard, and the second did not admit to having caribou.  The third Santa had a real beard and played well into the caribou question; they must have found the real thing!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if their tests were not conclusive, the idea of preliminary validation testing struck me as quite apt, and I imagine similar examples would lend themselves to students coming up with creative tests.  Even just thinking up tests is a worthy exercise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;It Was Documented On Film&quot; Paseman, 2015.05.25&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-26T01:45:25.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8163" PostTypeId="1" AcceptedAnswerId="8166" CreationDate="2015-05-26T04:05:15.100" Score="13" ViewCount="273" Body="&lt;p&gt;When covering issues related to error estimates in a calculus course, students find the technique of making estimates (definition of limit, Newton's method, numerical integration, remainder formula for power series) quite difficult even in basic situations. I suspect they would all accept the following fake theorem of error analysis: if you apply a recursion and two successive terms have the same first n digits then the sequence converges and those common initial digits are the first n digits of the limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is of course bogus, and a slowly divergent process like the partial sums of the harmonic series provides a counterexample. What I would like to have available to show the class are examples of &lt;em&gt;convergent&lt;/em&gt; recursions from a typical (not artificially created) example of the kind met in a first-year calculus course. A parametric family of examples would be even better, but interesting individual examples would be good too. It would be especially nice if the example has terms agreeing to the first 6 or so digits but the limit is completely different than what the data suggest.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2015-06-08T15:25:24.193" Title="Counterexamples to &quot;stable digit&quot; theory of error estimates" Tags="|calculus|student-motivation|examples|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8164" PostTypeId="2" ParentId="8163" CreationDate="2015-05-26T07:59:10.297" Score="0" Body="&lt;p&gt;This one was the eye opener for me years ago: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 + 1/2 + 1/4 + 1/8 + ... The digits settle quickly but the sum is equal to 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it something you look for?&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2015-05-26T07:59:10.297" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8166" PostTypeId="2" ParentId="8163" CreationDate="2015-05-26T12:44:18.390" Score="13" Body="&lt;p&gt;How about $\frac{1}{3}x^3-\frac{1}{4}x+\frac{1}{12}+\epsilon$? When $\epsilon=0$, this cubic has two distinct roots: a single root at $-1$ and a double root at $\frac{1}{2}$. If we let $\epsilon&amp;gt;0$ be something relatively small, for example $\epsilon=0.0000001$, and start Newton's method at $1$ the procedure will get caught at the local minimum for a little while before finding the true root near $-1$. With this setup, the tenth and eleventh iterations give an approximations of 0.500529571977707 and 0.500076100908512; leading to the incorrect guess.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2015-05-26T20:27:05.777" LastActivityDate="2015-05-26T20:27:05.777" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8167" PostTypeId="1" CreationDate="2015-05-26T13:18:35.923" Score="22" ViewCount="2837" Body="&lt;p&gt;I am planning a 1-hour lesson for a group of 20 11 year olds. I would like to expose them to topology, as an area of research-level mathematics that could be accessible to them. I want to convince them that maths can be visual and beautiful!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The pupils are struggling in Maths (for example are not confident with negative numbers, decimals, algebra, graphs). So an introduction to topology that requires as little machinery as possible would be ideal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am thinking of discussing some of the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Genus&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Definition of manifold (see &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5972/teaching-manifolds-to-high-schoolers&quot;&gt;Teaching manifolds to high schoolers&lt;/a&gt; for an existing discussion on this)    &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;triangulations (&lt;a href=&quot;https://matheducators.stackexchange.com/questions/4221/how-can-i-demonstrate-triangulations-of-surfaces-with-real-hands-on-objects/4239#4239&quot;&gt;How can I demonstrate triangulations of surfaces with real hands-on objects?&lt;/a&gt;). Any tips on how to help pupils with the difficult task of accurately counting objects when exploring triangulation on difficult surfaces would be appreciated.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;Homotopy type (group letters in alphabet or various types of food up to homotopy class)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you have tips or knows of any existing resources then I look forward to hearing from you! In particular, any tips on how to build models or use everyday objects (tasty donuts) to illustrate ideas.&lt;/p&gt;&#xA;" OwnerUserId="5225" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-02-27T17:41:17.973" Title="Introduction to Topology for 11 year olds" Tags="|secondary-education|topology|" AnswerCount="8" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="8168" PostTypeId="2" ParentId="8167" CreationDate="2015-05-26T17:59:34.067" Score="3" Body="&lt;p&gt;You might try a couple of puzzles near the middle for them to think about, and see if they can come up with the answers.  One of my favorites involves a (topological deformation of) two-holed doughnut with a door handle going through both holes, and attached to a door, naturally.  The idea is to deform the doughnut so to free one of the holes.  (You may be able to make a string model to demonstrate, or draw a picture of before and after.)  Another is the trick involving a coffee cup: you hold a filled cup in your hand and twist your arm around twice while not spilling the cup's contents.  You can also talk about Euler's formula for graphs drawn on one and two-holed surfaces, and mention that it isn't a geometric result so much as a topological result.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Check Out YouTube Videos Too&quot; Paseman, 2015.05.26&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-26T17:59:34.067" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8169" PostTypeId="2" ParentId="8167" CreationDate="2015-05-26T19:29:24.017" Score="13" Body="&lt;p&gt;You could explore maps such as the London Underground where the actual distance and locations of stops are unimportant only their relationships. Perhaps this could be turned to a practical lesson somehow? Kids with playing the part of stations and holding string?  Design a similar map of their home city? Ranking maps? What makes a good map? You could take the theme in many different directions and return to a real world application.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-05-26T19:29:24.017" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8170" PostTypeId="2" ParentId="8167" CreationDate="2015-05-27T00:23:33.647" Score="12" Body="&lt;p&gt;Two ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Form a Möbius strip out of paper. And then cut it down the middle.&#xA;Restart and cut it at a third of its width. See, e.g., &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/q/7392/511&quot;&gt;Explanation for cutting a Möbius strip at one-third its width&lt;/a&gt;.&quot;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/DWdM1.jpg&quot; width=&quot;280&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://www.kidzone.ws/magic/mobius.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kidzone.ws&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) You already cited the previous question about triangulated surfaces,&#xA;presumably for verifying Euler's $V-E+F=2$.&#xA;In the absence of (expensive) construction materials (like &lt;a href=&quot;https://matheducators.stackexchange.com/a/4313/511&quot;&gt;polydrons&lt;/a&gt;), you could fold polyhedra&#xA;from nets, e.g.:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Bve2o.jpg&quot; alt=&quot;NetsPaper&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://www.minieco.co.uk/i-mathematics-platonic-solids-garland/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Minieco.co.uk&lt;/a&gt;. Note the tabs for gluing.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-05-27T00:23:33.647" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8171" PostTypeId="2" ParentId="8150" CreationDate="2015-05-27T00:29:18.580" Score="3" Body="&lt;p&gt;A simple answer is that it is calculating device to reduce multiplication to addition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What about coming at it from a historical approach.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;How about making a table of logarithms? This should be within the capabilities of a 12 year old. This can be done using Excel pretty easily. There are some good resources I've found on the net. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How about learning to  use Napier's Bones? You'll probably need to make your own.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="998" LastActivityDate="2015-05-27T00:29:18.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8172" PostTypeId="1" CreationDate="2015-05-27T16:52:16.007" Score="8" ViewCount="59" Body="&lt;p&gt;I've taken a job for the next school year, which will involve embedding secondary math (algebra, geometry and trigonometry) into technical programs such as drafting and electrician school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of any resources that would be available for me to use to learn more about this? So far my educational career has only been in a more traditional classroom.&lt;/p&gt;&#xA;" OwnerUserId="5230" LastActivityDate="2015-05-28T09:49:04.247" Title="Sources for embedding secondary math into technical classes?" Tags="|reference-request|technology-in-education|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8173" PostTypeId="1" CreationDate="2015-05-27T18:36:34.967" Score="5" ViewCount="499" Body="&lt;p&gt;(Cross-posted in &lt;a href=&quot;https://math.stackexchange.com/q/1301476/&quot;&gt;&lt;strong&gt;MSE 1301476&lt;/strong&gt;&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to start to do Olympiad type questions but have absolutely no knowledge on how to solve these apart from my school curriculum. I'm 16 but know maths up to the 18 y/o level. I think I will start learning the theory of the topics (Elementary Number Theory, Combinatorics, Euclidean Plane Geometry) then going on to trying the questions, but I need help in knowing what books to use to learn the theory. I have seen several list (such as &lt;a href=&quot;http://www.artofproblemsolving.com/wiki/index.php/Olympiad_books&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.artofproblemsolving.com/wiki/index.php/Olympiad_books&lt;/a&gt;) but does anyone know which ones are the best for my level of knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. I live in UK if that matters.&lt;/p&gt;&#xA;" OwnerUserId="5231" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2024-01-15T14:44:29.503" Title="Self Teaching Theory for Olympiad. Need advice" Tags="|secondary-education|textbooks|self-learning|number-theory|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8174" PostTypeId="2" ParentId="8172" CreationDate="2015-05-28T01:15:12.503" Score="4" Body="&lt;p&gt;There is a California (US) organization called &lt;a href=&quot;http://cteteach.cteonline.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Career Technical Education&lt;/a&gt; that offers several courses, from Agriculture&#xA;to Transportation.&#xA;Below is a snapshot from the &lt;a href=&quot;http://cteteach.cteonline.org/portal/default/Curriculum/Viewer/Curriculum?action=2&amp;amp;cmobjid=804363&amp;amp;view=viewer&amp;amp;refcmobjid=777121&quot; rel=&quot;nofollow noreferrer&quot;&gt;Engineering and Architecture course&lt;/a&gt;,&#xA;which&#xA;appears to integrate mathematics into drafting, perspective drawing, etc.&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/41voC.png&quot; alt=&quot;CTE&quot;&gt;&#xA;&lt;hr /&gt;&#xA;It appears that all the resources can be accessed for free.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-05-28T09:49:04.247" LastActivityDate="2015-05-28T09:49:04.247" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8175" PostTypeId="2" ParentId="8173" CreationDate="2015-05-28T08:28:06.603" Score="4" Body="&lt;p&gt;Since this question has been posted in Educators stack exchange, I would try to structure my response in a way that addresses different learning approaches.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Approach - 1&lt;/strong&gt; : Learn techniques first, apply learnt techniques&lt;/p&gt;&#xA;&lt;p&gt;This is the most traditional approach. Let us take the example of number theory. The chapters from a typical number theory book look like this - Introduction, Prime numbers, Congruences, ...&lt;/p&gt;&#xA;&lt;p&gt;You start with the prime numbers chapter and go through every method, technique and understand the applications,then start solving exercise problems  belonging to that particular chapter. Then move on to the next chapter.&lt;/p&gt;&#xA;&lt;p&gt;While this approach is good, I am a bit concerned about the way it affects the development of your problem solving abilities. Some of these chapters may take days to complete and the exercises at the back may focus exclusively on a few select topics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Approach - 2&lt;/strong&gt;: Learn enough theory; work on problems; go back and learn techniques&lt;/p&gt;&#xA;&lt;p&gt;Learn enough theory to understand problems. For instance, you can't solve congruence equations without having knowledge about congruences. Grappling with problems at an early stage is very advantageous. After you spend sufficient time on a problem and are unable to get any ideas, go for the solution. This is a good way to track your progress. Initially the progress may be very slow and frustrating, every problem that you try may elude a solution. But gradually you would start doing better.&lt;/p&gt;&#xA;&lt;p&gt;My suggestion would be to look at a book of problems very early on to see where you stand. If you are able to follow the solutions without too much difficulty, this would be a good indication of your aptitude for olympiad math. This is a short book by Titu Andreescu that has a good collection of problems and enough theory to back them up.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.amazon.co.uk/Mathematical-Olympiad-Treasures-Titu-Andreescu/dp/081768252X&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.amazon.co.uk/Mathematical-Olympiad-Treasures-Titu-Andreescu/dp/081768252X&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A few other resources that might help:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20180531083708/http://mathforum.org/library/drmath/view/52354.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://files.vipulnaik.com/olympiads/preparingforolympiads.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://files.vipulnaik.com/olympiads/preparingforolympiads.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Caveat&lt;/em&gt;: Though hard work is very important, a certain level of problem solving ability and intuition are indispensable when it comes to solving Olympiad mathematics. You develop them as you work. You are up against some of the most competent problem solvers. If you are doing good enough at school, I think you must certainly go for Olympiad mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5221" LastEditorUserId="6053" LastEditDate="2022-11-04T13:01:02.643" LastActivityDate="2022-11-04T13:01:02.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8177" PostTypeId="2" ParentId="8137" CreationDate="2015-05-28T20:10:12.600" Score="3" Body="&lt;p&gt;&lt;strong&gt;Some resources:&lt;/strong&gt; Have a look at the &lt;a href=&quot;https://en.wikibooks.org/wiki/Subject:Mathematics&quot; rel=&quot;nofollow&quot;&gt;textbooks of Wikibooks&lt;/a&gt;. One shall also not forget the &lt;a href=&quot;https://en.wikipedia.org/wiki/Category:Mathematics&quot; rel=&quot;nofollow&quot;&gt;math articles of Wikipedia&lt;/a&gt;, which are used by everyone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My experience:&lt;/strong&gt; I write an &lt;a href=&quot;https://de.wikibooks.org/wiki/Mathe_f%C3%BCr_Nicht-Freaks&quot; rel=&quot;nofollow&quot;&gt;open textbook&lt;/a&gt; on Wikibooks (it's in German), so I can add some pros and cons from the author's point of view:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because the textbooks is freely available (an result of being open), more people are using and reading it. The necessity to pay for a content is a big hurdle most people will not take. So having an open project is a huge advantage over a conventional textbook you need to buy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example I had other 800 thousand page views last year with my project, which is quite a success (you have to note, that the textbook isn't completed yet). With a conventional distributed textbook I could never reach such an audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, that the more people you reach, the more people you help. So if your reason for writing a textbook is helping students you shall write an open textbook, especially if you also want to help those, who do not have much money.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other people are more likely to contribute to your project when it is open. In the last 5 years there were over 300 people who participated in my project. Most of them corrected typos or little mistakes, but there were also some who wrote new chapters. To archive this, you need a platform where the reader can easily change the text (such as a wiki, for example Wikibooks).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many authors have the fear, that someone else may destroy their work, when it is open. That's not the case. I also can say: You will have the opposite problem! There are always readers correcting little mistakes, but for finding co-authors you will need to make much advertisement (e.g. write on a mailing list, ask friends...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You also can use text materials, diagrams and pictures from other open projects such as Wikipedia, &lt;a href=&quot;https://commons.wikimedia.org/wiki/Main_Page&quot; rel=&quot;nofollow&quot;&gt;Wikimedia Commons&lt;/a&gt; or posts from this forum (note, that each stackexchange forum runs under CC-BY-SA), if you project is open. This will save you a lot of time. (Note: One need to have a good knowledge of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Creative_Commons_license&quot; rel=&quot;nofollow&quot;&gt;Creative Commons licenses&lt;/a&gt; of course!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether you publish your work under a free license is always a personal question. It depends of the goals one follows. An open license may not be the best choice if one wants to make money. But then he also shall not write a textbook because you cannot make a lot of money with it. If one wants to reach a huge audience, wants to get good reputation or wants to help students he really should use open licenses because this will help him reaching his goals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Final note:&lt;/strong&gt; One should carefully chose the open license for his textbook. Creative Commons licenses are most suitable for textbooks. But not every CC license is open in the sense of &lt;a href=&quot;http://freedomdefined.org/Definition&quot; rel=&quot;nofollow&quot;&gt;freedomdefined&lt;/a&gt; (it's also the definition the Wikipedia is following). CC licenses with the modules &quot;non derivation&quot; or &quot;non commercial&quot; are not free! See &lt;a href=&quot;http://freedomdefined.org/Licenses/NC&quot; rel=&quot;nofollow&quot;&gt;The case for Free use: reasons not to use a Creative Commons -NC license&lt;/a&gt; for reasons.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2015-05-28T20:10:12.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8178" PostTypeId="2" ParentId="8163" CreationDate="2015-05-28T20:55:27.057" Score="1" Body="&lt;p&gt;Take the sequences&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$a_n=\sqrt{n+1000}-\sqrt{n}$&lt;/li&gt;&#xA;&lt;li&gt;$b_n=\sqrt{n+\sqrt{n}}-\sqrt{n}$&lt;/li&gt;&#xA;&lt;li&gt;$c_n=\sqrt{n+\frac{n}{1000}}-\sqrt{n}$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For $n&amp;lt;10^6$ we have $a_n &amp;gt; b_n &amp;gt; c_n$, but it is&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\lim_{n\to\infty} a_n = 0$&lt;/li&gt;&#xA;&lt;li&gt;$\lim_{n\to\infty} b_n = \tfrac 12$&lt;/li&gt;&#xA;&lt;li&gt;$\lim_{n\to\infty} c_n = \infty$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://math.stackexchange.com/questions/1151645/how-does-sqrtn1-sqrtn-fracn1-n-sqrtn1-sqrtn&quot;&gt;https://math.stackexchange.com/questions/1151645/how-does-sqrtn1-sqrtn-fracn1-n-sqrtn1-sqrtn&lt;/a&gt; for the technique to find the limits. Now you can take sequences like $\tfrac{1}{a_n}$ for a good counterexample to your problem.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-05-28T20:55:27.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8179" PostTypeId="1" AcceptedAnswerId="8180" CreationDate="2015-05-28T21:32:58.137" Score="5" ViewCount="235" Body="&lt;p&gt;I currently write an article where I want to introduce roots. Thus I need to motivate them. Here I said, they can be used to find solutions of equations like $x^n=a$. Now I want to make some examples, where this problem arises. So far I found:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is the side length of a square with the area of $10 m^2$?&lt;/li&gt;&#xA;&lt;li&gt;What are the roots of the polynomial $x^2+x-1$?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What examples would you take? I do not have the feeling that mine are really convincing to study roots... &lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2015-06-06T23:59:16.417" Title="Examples where roots are necessary for the solution" Tags="|calculus|student-motivation|examples|concept-motivation|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8180" PostTypeId="2" ParentId="8179" CreationDate="2015-05-29T00:00:06.573" Score="10" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Ray_tracing_(graphics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ray-tracing&lt;/a&gt;, which underlies much of high-end computer graphics (from &lt;a href=&quot;http://en.wikipedia.org/wiki/Toy_Story&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Toy Story&lt;/em&gt;&lt;/a&gt; to &lt;a href=&quot;http://en.wikipedia.org/wiki/Frozen_(2013_film)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Frozen&lt;/em&gt;&lt;/a&gt;)&#xA;relies on computing the intersection of a ray line-of-sight with&#xA;a geometric object. &#xA;For example, where does the ray $a + t v$ with parameter $t$, $a=(2,0)$, $v=(-1,\frac{1}{2})$,&#xA;intersect the unit circle $x^2 + y^2 = 1$?&#xA;Substitution of $a + t v = \left( 2-t,\frac{t}{2} \right)$ for $(x,y)$ &#xA;in the circle equation leads to&#xA;\begin{eqnarray}&#xA;(2-t)^2 + \frac{t^2}{4} &amp;amp;=&amp;amp; 1 \;, \\&#xA;\frac{5 t^2}{4}-4 t+3 &amp;amp;=&amp;amp; 0&#xA;\end{eqnarray}&#xA;whose roots are $t=2$, or $t=\frac{6}{5}$, yielding the intersection points&#xA;\begin{eqnarray}&#xA;(x,y) &amp;amp;=&amp;amp; (0,1) \;, \\&#xA;(x,y) &amp;amp;=&amp;amp; \left( \frac{4}{5},\frac{3}{5} \right) &amp;amp;=&amp;amp; (0.8, 0.6) \;.&#xA;\end{eqnarray}&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/OH5qS.jpg&quot; alt=&quot;RayCirc&quot;&gt;&#xA;&lt;hr /&gt;&#xA;Because the latter point is closer to the ray origin $a$, that &#xA;$(0.8, 0.6)$ point is the one&#xA;whose surface characteristics determine &#xA;the color/shade of the image pixel represented by the ray line-of-sight.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-06-06T18:35:09.923" LastActivityDate="2015-06-06T18:35:09.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8181" PostTypeId="2" ParentId="8137" CreationDate="2015-05-29T01:26:36.990" Score="6" Body="&lt;p&gt;The American Institute of Mathematics has an &lt;a href=&quot;http://aimath.org/textbooks/approved-textbooks/&quot; rel=&quot;noreferrer&quot;&gt;open textbook list&lt;/a&gt; with a wide variety of college-level textbooks.  Including some of the ones listed in other answers.  &lt;a href=&quot;http://danaernst.com/resources/free-and-open-source-textbooks/&quot; rel=&quot;noreferrer&quot;&gt;Here is another related list&lt;/a&gt;.  It is really a useful list.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-05-29T01:26:36.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8182" PostTypeId="1" CreationDate="2015-05-29T12:54:45.563" Score="2" ViewCount="459" Body="&lt;p&gt;In mathematics education our primary focus tends to be how to approach the concepts we teach, taking for granted the choice of subjects that enter the syllabus. Traditionally, the way the standards are determined follows a specific philosophical inclination: school mathematics is still predominantly &lt;em&gt;galilean&lt;/em&gt;. It is viewed as a language with which to &lt;em&gt;understand&lt;/em&gt; and describe objects and patterns we experience in the world. Math is still viewed as a vehicle for the expression of explanatory models of reality. That's how we define what knowledge is necessary (and thus mandatory): you can't describe movement without functions, you can't solve problems without algebra. Math is the list of things you (supposedly) &lt;em&gt;can't go without&lt;/em&gt; in order to understand how the world works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It can be argued that the developments in contemporary mathematics favor a more transformative, or &quot;creative&quot; approach to mathematical thinking. The mathematical foundations of visualization techniques, robotics, modern design, software engineering, cryptography, etc., do not deal with merely understanding, but mostly with &lt;em&gt;producing&lt;/em&gt; new ideas, and new things. Math would be the way to express &lt;em&gt;proposals&lt;/em&gt;. Still a language, but for writing &quot;fiction&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: assuming this change in perspective affects the way we teach math in school, how will it impact the way we define and organize the contents we will be working with, in the years to come? More specifically, how will we determine what is mandatory for the students of tomorrow?&lt;/p&gt;&#xA;" OwnerUserId="5038" LastEditorUserId="5038" LastEditDate="2015-05-31T13:44:33.723" LastActivityDate="2015-05-31T13:44:33.723" Title="What kind of math will be mandatory in the future?" Tags="|mathematical-pedagogy|creativity|syllabus|" AnswerCount="2" CommentCount="5" ClosedDate="2015-05-31T19:55:38.700" ContentLicense="CC BY-SA 3.0" />
  <row Id="8183" PostTypeId="2" ParentId="8182" CreationDate="2015-05-29T15:23:07.067" Score="4" Body="&lt;p&gt;The (US) National Academy of Sciences wrote a book-length report,&#xA;&lt;a href=&quot;http://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Mathematical Sciences in 2025&lt;/em&gt;&lt;/a&gt;, available free.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/ueHgc.jpg&quot; width=&quot;240&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;Chapter 4 is &quot;Important Trends in the Mathematical Sciences,&quot; whose table of contents I snapshot below.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/NsvLa.png&quot; alt=&quot;Chap4&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-05-29T15:23:07.067" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8184" PostTypeId="2" ParentId="8182" CreationDate="2015-05-29T16:19:40.777" Score="0" Body="&lt;p&gt;I think it is possible to teach innovation and change, or at least teach what it might look like.  One could trace recent developments and hope to find a pattern that might allow a more direct path to a result, or one could take an approach similar to that of James Burke (Connections, The Pinball Effect) and try for unanticipated innovation by making many connections.  In either case, a certain philosophical, historical, cultural, and mathematical foundation will be needed to appreciate or even just cope with either method.  I recommend having a series of courses such that one path focuses on direct and anticipated innovation, and another path focuses on unanticipated innovation.  You might find that you may not be able to predict the mathematics needed 50 years from now, but you can at least prepare for some of it.  That preparation is what needs to be taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Might Look Back At This&quot; Paseman, 2015.05.29&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-29T16:19:40.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8187" PostTypeId="1" CreationDate="2015-05-30T06:30:32.367" Score="11" ViewCount="349" Body="&lt;p&gt;I am sorry I am not directed with the education of math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But granted, let me ask the above question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my language (actually Japanese), the words corresponding with the &lt;strong&gt;factor&lt;/strong&gt; and &lt;strong&gt;divisor&lt;/strong&gt;, &lt;strong&gt;seem to&lt;/strong&gt; have come from English. (Not confirmed.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now when teaching (potentially for now to me now) them to kids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am sorry to say my question is just as it is at the title.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does &lt;strong&gt;factor&lt;/strong&gt; simply mean the multiplication of any functions, or numbers, &lt;strong&gt;whereas&lt;/strong&gt;, the &lt;strong&gt;divisor&lt;/strong&gt; means literally and &lt;strong&gt;only&lt;/strong&gt; &quot;divide&quot; any numbers, etc?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am afraid I am a little confused about the relationship between these two terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me kindly take number 24. The number 24 can be expressed by 24=2*2*2*3 if it is expressed by the divisor(s). Now when considering the same 24, when expressed by the factor, 24=4*6 or either 2*12.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Am I correct about understanding the &lt;strong&gt;meaning and the difference of these 2 terms??&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kindly let me thank you in advance.&lt;/p&gt;&#xA;" OwnerDisplayName="user5245" LastEditorUserId="2074" LastEditDate="2015-05-30T14:40:57.947" LastActivityDate="2015-05-30T19:01:52.437" Title="Does &quot;factor&quot; mean simply the multiplication (of any functions, numbers etc)" Tags="|terminology|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8188" PostTypeId="2" ParentId="8187" CreationDate="2015-05-30T08:37:09.863" Score="2" Body="&lt;p&gt;The terms are interchangeable as I understand. Factors are divisors.&#xA;$2\times2\times2\times3$ is a product of prime factors. The $8$ and $3$ are merely factors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said  that in terms of algebraic long division the expression you are dividing by is known as the divisor. In this sense there could possibly be a remainder.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastEditorUserId="4668" LastEditDate="2015-05-30T09:11:06.660" LastActivityDate="2015-05-30T09:11:06.660" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8189" PostTypeId="1" CreationDate="2015-05-30T18:50:31.073" Score="4" ViewCount="1847" Body="&lt;p&gt;This is something that has continued to bother me in general, and recently I had multiple occasions where saying something like &quot;2 divides 14 evenly&quot; has confused students, They expected the result to be an even number, not $7$.  I can't seem to find the concise word/wording for saying that a number divides another without a remainder?  I would like to start using this instead of &quot;evenly&quot; for my students' benefit&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-06-01T00:41:51.827" Title="Correct term to say that a number divides another number &quot;evenly&quot;?" Tags="|mathematical-pedagogy|language-use|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="8190" PostTypeId="2" ParentId="8187" CreationDate="2015-05-30T19:01:52.437" Score="7" Body="&lt;p&gt;From your examples, it seems that you generally understand the two terms and their differences/similarities. as @Karl says, they are more or less interchangeable for most audiences. Below I have elaborated a little bit more on the minute distinctions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortuneately, both of these words have two different meanings each that are subtle, yet distinct.  I fear this is where your confusion might be stemming from&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general terms, a &lt;strong&gt;divisor&lt;/strong&gt; is simply a number that divides another number, whether with remainder or without.  In the following two expressions, $7$ would be the &lt;strong&gt;divisor&lt;/strong&gt; even though the result is not an integer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{22}{7} \\ 22 \div 7 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, since $2 \cdot 3 = 6$, this also means that $2$ and $3$ are both &lt;strong&gt;divisors&lt;/strong&gt; of $6$, in that they divide it without remainders, even though there isn't actually division happening, persay.  A synonymous term for this type of &lt;strong&gt;divisor&lt;/strong&gt; is a &lt;strong&gt;factor&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Factor&lt;/strong&gt; can also used in two different ways 1) as a noun (object), which is like the $2 \cdot 3 = 6$ example above, and 2) as a verb (action).  In 2) &lt;strong&gt;factor&lt;/strong&gt; means the action of breaking a number/expression into is composite parts, such as taking the polynomial expression $x^2 + 7x + 12$ and writing it as the equivalent $(x+3)(x+4)$ where $(x+3)$ and $(x+4)$ are also &lt;strong&gt;factors&lt;/strong&gt; of the original polynomial.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see they are more or less the same however it may be worth it to show your students the difference if there are confusions that are arrising, i hope this helps!&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-05-30T19:01:52.437" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8191" PostTypeId="2" ParentId="8138" CreationDate="2015-05-31T04:38:43.310" Score="4" Body="&lt;p&gt;Some math is needed in a liberal arts degree program because of the way it helps develop thinking skills. An individual may not get a job that is directly lined up with their degree, but that background may help them land a job. In the March 17,2014 New York Times article &quot;The Walls Close In&quot;, 80,000 people applied for 2,000 job openings at a VW plant that opened in Chatanooga, TN, in 2011. How many had a degree is not given, but 78,000 people didn't get better jobs because of a lack of basic math skills. This information was verified by economics professor Matt Murray. Allen Seay&lt;/p&gt;&#xA;" OwnerUserId="5247" LastActivityDate="2015-05-31T04:38:43.310" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8192" PostTypeId="2" ParentId="8167" CreationDate="2015-05-31T15:36:43.033" Score="2" Body="&lt;p&gt;You could do something in knot theory, for example: 3-colorability of a knot. The wikipedia article &lt;a href=&quot;http://en.wikipedia.org/wiki/Tricolorability&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Tricolorability&lt;/a&gt; shows a proof of isotopy invariance which should be understandable to an 11 year old. &lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2015-05-31T15:36:43.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8193" PostTypeId="1" CreationDate="2015-05-31T21:05:02.357" Score="25" ViewCount="1524" Body="&lt;p&gt;&lt;strong&gt;NB. Some answers appear to be for a question I did not ask, namely, &amp;quot;Why is standardized testing bad?&amp;quot; Indeed, these answers tend to underscore the premise of my actual question, which can be found above.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;As a foreigner who has spent some time in the US, it seems to me that in the US, there is this knee-jerk hostility against standardized testing.&lt;/p&gt;&#xA;&lt;p&gt;(I don't have proper evidence for this assertion. And it is of course not true that everyone in the US absolutely abhors standardized testing. But nonetheless I think we can safely say that most Americans consider standardized testing a bad idea. I base my assertion only on anecdotal evidence. Take a look for example at the videos that show up in &lt;a href=&quot;https://www.youtube.com/results?search_query=standardized%20testing&quot; rel=&quot;noreferrer&quot;&gt;a search for 'standardized testing' on YouTube&lt;/a&gt;. Or some of the &amp;quot;answers&amp;quot; below.)&lt;/p&gt;&#xA;&lt;p&gt;My question is: What are some historical reasons for this?&lt;/p&gt;&#xA;&lt;p&gt;For example, in the Confucian countries in East Asia, it is possible to explain why standardized testing is widely accepted by appealing to the historical fact that there was the &lt;a href=&quot;http://en.wikipedia.org/wiki/Imperial_examination&quot; rel=&quot;noreferrer&quot;&gt;imperial examination system&lt;/a&gt;. (Now I don't know if this explanation is the best explanation or if it is even correct, but it is one I hear often.)&lt;/p&gt;&#xA;&lt;p&gt;How might one explain, through history (and perhaps otherwise), why there is such immense hostility against standardized testing in the US?&lt;/p&gt;&#xA;&lt;p&gt;Here is some more context. I was inspired to ask this question after coming across this passage in &lt;a href=&quot;https://www.richmondfed.org/publications/research/econ_focus/2014/q4/interview&quot; rel=&quot;noreferrer&quot;&gt;an interview with the economist Claudia Goldin&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The education system in the early 20th century was a decentralized system that was very open, albeit with some important exceptions, such as African Americans and certain immigrant groups. But by and large, relative to Europe, America was educating all its children. European visitors would come to the United States and be shocked by how America was wasting its resources. European countries were cherry picking which students would get a good education; they set very high standards and had national exams. We didn't. We had more of a free-for-all, grassroots, local system in which until recently there were few state exams for graduation. That served us very well by getting a large number of students to graduate from high school. By the 1950s, U.S. high school enrollment and graduation rates were relatively high, much higher than Europe.&lt;/p&gt;&#xA;&lt;p&gt;But then various European countries started looking more like the United States; they began to pull more individuals into high schools, some via technical schools but also by expanding more general education. And many of them did so without abandoning the higher standards of the more elitist period. &lt;strong&gt;The United States, on the other hand, has had a very hard time adopting uniform standards. The idea has been that the different parts of the country have different demands, so we don't need to have national standards. And it's true that we do have a far more heterogeneous population. But the enormous virtue of decentralization has more recently caused some difficulty.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user378" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-05-14T05:38:34.540" Title="What are the historical reasons for the hostility against standardized testing in the US?" Tags="|testing|" AnswerCount="8" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8194" PostTypeId="2" ParentId="8193" CreationDate="2015-05-31T22:48:58.837" Score="11" Body="&lt;p&gt;Not an answer, but perhaps a clue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Richard Feynman in one of his books (I think the title began &quot;Surely You're Joking...&quot;, or it might have began &quot;What Do You Care...&quot;; hopefully someone else will recall the correct and full title) had a couple of stories regarding foreign high-level education (Brazil? physics undergraduate or graduate?)) as well as textbooks for primary science education in the U.S.  As I best recall, he observed that the foreign students were able to recall large amounts of material, but were unable to apply them or explain possible relations that weren't covered in the material.  I won't repeat what he said about the U.S. textbooks; you should find and read that rant for yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the clue is that this country grew young and strong on innovation, inspiration, and independence, and in spite of the struggles it causes, is intending for education to be varied enough so that the skill sets produced are varied, and in particular encourage independence (and hopefully ability) of thought.  Standardized testing, while good for some evaluations, is poor enough that (in the case of multiple choice examinations) it can't tell apart a poor student from a monkey, and also has trouble distinguishing between mastery of the knowledge and mastery of test-taking skills.  This is not Feynman's opinion, but I believe the above anecdotes support such an opinion.  I also believe that such opinions are and have been harbored by many in and near the U.S. education establishment for almost as long as standardized testing has been used in the U.S.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;But I Could Be Wrong&quot; Paseman, 2015.05.31&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-05-31T22:48:58.837" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8195" PostTypeId="1" CreationDate="2015-06-01T00:26:03.443" Score="31" ViewCount="2998" Body="&lt;p&gt;When I teach a course for math majors (an analysis course out of Rudin, say), I have a more or less clear idea of what the students should take away from the course, having been in their shoes some 15 years ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But when teaching a calculus class for a mix of physics, engineering, economics, business, pre-med, and miscellaneous students, I lack anything resembling this clarity. On the one hand, I can teach them how to solve computational problems, but the majority of these students (at least at the two universities where I have taught) have had calculus in high school and already know how to do the computations, making this painfully boring for them and for me. On the other hand, I could prove theorems. Experience shows that the vast majority of students are not interested in proofs, and it's hard to make the argument that knowing how to prove the Mean Value Theorem will help them in their future non-mathematical careers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, students can potentially benefit from learning proofs because through this, they will learn how to think in a new way. However, the same could be said of learning almost any academic subject, so this does not help us with the question at hand: what material to teach, how to teach it, and what to emphasize.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;If&lt;/em&gt; I could succeed in convincing my students that proofs were interesting and/or fun, I could make the class a worthwhile experience for them. But I find that this is a losing battle for the vast majority of students, even bright ones. There will usually be one or two students in a class of 50 who become (or were already) interested in the theoretical side of things, but catering the class to the interests of such a small minority seems like a bad choice. Again, with a class for math majors, I don't have to worry as much about the students' interests (though it's always good to keep in mind) because I know what they need to know, and why. I have no such &quot;moral high ground&quot; with the proof of the Mean Value Theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The standard answer is to ride the fence between the two extremes (computations vs. proofs) by focusing on concepts while mostly avoiding rigorous proofs. I have found it very difficult to adopt this attitude in practice, falling most of the time into one extreme or the other. The one useful concept that I instill in all my students is: stop and think about a problem before trying to compute the answer. I accomplish this by showing them cases where the standard methods fail, and giving them such problems on exams. Most students absorb this lesson, and that's fine. But other than that, I feel I have nothing useful to tell these students, and judging by student feedback on evaluations, they feel the same way. (Which my colleagues all tell me not to worry about, because evaluations are typically mediocre at best for freshman calculus courses. But couldn't this be indicative of a general problem?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To try to make this question more focused: a good answer would give a big-picture idea of what benefits non-math majors (in particular, those who have already learned the computational methods in high school) can take away from calculus, and/or specific examples of concepts that are &quot;just right&quot; for emphasis in a calculus class, and good ways to teach these concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a related question here: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2428/how-to-make-calculus-ii-seem-motivated-interesting-and-useful?rq=1&quot;&gt;How to make Calculus II seem motivated, interesting, and useful?&lt;/a&gt; But I am more interested in what should actually be taught, and with what emphasis, than in motivating the material per se.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After posting, I noticed the closely related question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/8138/the-purpose-of-mathematics-in-a-liberal-education-when-it-is-not-a-prerequisite&quot;&gt;The purpose of mathematics in a liberal education when it is not a prerequisite to other subjects&lt;/a&gt;. My question is more practical and specific: what concepts, or what type of concepts, can and should non-math majors take from calculus?&lt;/p&gt;&#xA;" OwnerUserId="5249" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-07-23T20:46:00.310" Title="What are non-math majors supposed to get out of an undergraduate calculus class?" Tags="|undergraduate-education|calculus|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8196" PostTypeId="2" ParentId="8189" CreationDate="2015-06-01T00:41:51.827" Score="4" Body="&lt;p&gt;Make it clear to your students to begin with that when it comes to divisibility, we are only ever interested in whole number divisibility. Otherwise, of course any non-zero number 'divides' any other number! (I assume you're working with the reals). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it will then suffice to simply say &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;2 divides 14&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3383" LastActivityDate="2015-06-01T00:41:51.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8197" PostTypeId="2" ParentId="8195" CreationDate="2015-06-01T01:39:19.767" Score="14" Body="&lt;p&gt;I attended an interesting talk by Timothy Guasco, &#xA;who teaches physical chemistry at a US college.&#xA;He's reduced some of the high demands for mathematical&#xA;skills in PChem by displacing some math derivations with&#xA;computation and modeling. Then he needs the students to&#xA;understand the concepts of calculus, but they do not&#xA;need to be adept at calculating.&#xA;E.g., they need to be able to understand perturbation &#xA;theory&amp;mdash;partial derivatives and $\nabla$&amp;mdash;while &#xA;relying on computation after the model is detailed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It could well be that this tradeoff between mathematical&#xA;derivations and computation will be more widespread in the&#xA;future. If so, it suggests that understanding the concepts&#xA;of calculus is more important than the ability to solve&#xA;calculus problems or prove theorems.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-06-01T01:39:19.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8198" PostTypeId="1" CreationDate="2015-06-01T02:48:03.237" Score="16" ViewCount="523" Body="&lt;p&gt;I'm a math grad student, and next semester I start TAing a calculus class for the first time. We all know about the standard recitations: instructor gives short lecture on some more difficult topic from class, then spends the rest of class solving homework problems on the board. I personally find such recitations extremely boring, and I have a hard time seeing how students (who often don't even take notes!) can learn anything useful from a recitation like this. Most students I know just show up because the problems the TA solves are free marks on the homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm wondering about more interactive ways to run a recitation. Right now I'm thinking about making the students attempt all problems using think-pair-share. Students get to work for a few minutes, then pair up and talk about their solutions with someone else, and then we work through the problem together on the board. We might not get through as many problems, but it should make the students more involved in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would this be a good way to run a recitation? What are some other ways to run a recitation to encourage student participation?&lt;/p&gt;&#xA;" OwnerUserId="5250" LastEditorUserId="5256" LastEditDate="2015-06-01T20:08:09.433" LastActivityDate="2015-06-03T05:33:59.017" Title="Nontraditional calculus recitations" Tags="|undergraduate-education|mathematical-pedagogy|calculus|classroom-management|class-participation|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="8199" PostTypeId="2" ParentId="8193" CreationDate="2015-06-01T05:47:01.210" Score="8" Body="&lt;p&gt;In many cases where people are distrustful about certain forms of testing, such distrust stems from a belief that the results of such testing will be put to uses that those harboring the distrust would oppose.  In the case of standardized testing of educational performance, some people are worried that test results will be used to reward better-performing schools with increased resource allocation, thus removing resources from schools which are already performing badly, while other people are worried that test results will be used to direct resources toward schools which are performing poorly, a policy which may provide perverse incentives to those involved not to fix the problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is reasonable to oppose testing which is genuinely useless; for any kind of testing to be worthy of support, there must be a reasonable expectation that the results will be usable for some positive purpose.  If there is no agreement about how the results from academic performance testing should be used, opposition to the administration of such tests is entirely logical.&lt;/p&gt;&#xA;" OwnerUserId="1740" LastActivityDate="2015-06-01T05:47:01.210" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8200" PostTypeId="2" ParentId="1816" CreationDate="2015-06-01T11:15:28.373" Score="6" Body="&lt;p&gt;Siyavula develops excellent free online mathematics and science textbooks. These are designed to fit with the South African curriculum, grades 10 to 12 (but may be of use to those outside of South Africa). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;PDF versions can be downloaded in English or Afrikaanse from their site: &lt;a href=&quot;http://everythingmaths.co.za/&quot; rel=&quot;noreferrer&quot;&gt;http://everythingmaths.co.za/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5252" LastActivityDate="2015-06-01T11:15:28.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8201" PostTypeId="2" ParentId="8198" CreationDate="2015-06-01T15:39:36.510" Score="10" Body="&lt;p&gt;I strongly recommend consulting with the department AND Course Instructors before trying new interactive protocols.  In one of my first Teaching Assistant jobs, I tried adding a quotation or factoid at the beginning of the session for the students to look at (and think about) if they wanted.  I did some other things too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My instructor sent out a midterm poll on his performance and the TA's.  He called me in and said there very few things that could be concluded from his poll, but one of them was that the students hated me.  I don't think it was due to the quotations (there were other possible factors, including some challenging problems), but I admit I did that on my own without consulting anyone; in retrospect, I should have talked to someone before instituting anything different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In one of my last years as TA, I did something similar for a related class, and got two rewarding experiences: one student said that he preferred the hard (ungraded) practice exams since the actual exams would seem like a relief and since the practice exams would help prepare them (indeed, I anticipated an actual exam problem), and another student pushed back with a question related to one I asked, which led to some research level results in combinatorial matrix theory and in combinatorial algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't have it yet, start building your support network now (instructor, fellow TA's, therapists, administrative personnel) to help you both anticipate and cope with the downs as well as the ups.  If this is your first session, don't try anything untested by you, even if all you can do to test is practice in front of a mirror.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Take Your Enthusiasm With You&quot; Paseman, 2015.06.01&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-06-01T15:39:36.510" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8203" PostTypeId="2" ParentId="8193" CreationDate="2015-06-01T22:48:37.677" Score="10" Body="&lt;p&gt;One possible reference (mentioned here specifically for its introduction) is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;van den Heuvel-Panhuizen, M., &amp;amp; Becker, J. (2003). Towards a didactic model for assessment design in mathematics education. In &lt;em&gt;Second international handbook of mathematics education&lt;/em&gt; (pp. 689-716). Springer Netherlands. &lt;a href=&quot;http://link.springer.com/chapter/10.1007/978-94-010-0273-8_23&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Springer Link&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Side-note: The second author, &lt;a href=&quot;http://ehs.siu.edu/ci/faculty-staff/faculty-all/becker.php&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Jerry Becker&lt;/strong&gt;&lt;/a&gt;, maintains a very informative mathematics education mailing list that I would highly suggest subscribing to!) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a brief excerpt:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/xIPmq.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Yk430.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for just one of their given references, see:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Kilpatrick, J. (1993). The chain and the arrow: From the history of mathematics assessment. In &lt;em&gt;Investigations into assessment in mathematics education&lt;/em&gt; (pp. 31-46). Springer Netherlands. &lt;a href=&quot;http://link.springer.com/chapter/10.1007/978-94-017-1974-2_2&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Springer Link&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is already lengthy, but as a summary of some key reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;teaching to the test&lt;/em&gt; and the resulting impact in a variety of contexts, e.g., topics included in the mathematics curriculum and approaches to classroom teaching; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the questionable validity of standardized tests;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the use of these test scores to assess, e.g., teachers and schools; and&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;not mentioned above: the commercialization of the tests. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Lastly, an important concept that is brought up in some of the more recent arguments against standardized testing (though perhaps less relevant to the question of &quot;historical reasons&quot;) can be found in Claude Steele's work on &lt;a href=&quot;http://en.wikipedia.org/wiki/Stereotype_threat&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;stereotype threat&lt;/strong&gt;&lt;/a&gt; and subsequent studies that have confirmed its existence and articulated the implications. &lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2015-06-01T22:48:37.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8204" PostTypeId="2" ParentId="8193" CreationDate="2015-06-02T00:38:30.633" Score="7" Body="&lt;p&gt;A video clip &lt;a href=&quot;https://youtu.be/J6lyURyVz7k&quot; rel=&quot;nofollow&quot;&gt;Last Week Tonight with John Oliver: Standardized Testing&lt;/a&gt; offers an 18 minute discussion on the issues. Issues raised -&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Too many standardized tests, an average of 113 over the course of one's education through high school.&lt;/li&gt;&#xA;&lt;li&gt;Test anxiety - Test proctors are literally instructed on what to do when a child vomits on his test. &lt;/li&gt;&#xA;&lt;li&gt;Too much time &quot;teaching to the test,&quot; preparing for the test itself regardless of the course material. &lt;/li&gt;&#xA;&lt;li&gt;Theory (of &lt;em&gt;pay for performance&lt;/em&gt; for teachers) vs reality of not implementing a system that actually tracks their performance accurately. &lt;/li&gt;&#xA;&lt;li&gt;The large market share of a single testing provider, along with citations of bad questions and outright errors. As well as the cited &quot;&lt;a href=&quot;http://usny.nysed.gov/docs/the-hare-and-the-pineapple.pdf&quot; rel=&quot;nofollow&quot;&gt;The Hare and the Pineapple&lt;/a&gt;&quot; story. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Yes, the HBO video was presented in comedic fashion, not a documentary. But the facts remain, the goal of assessing our students and their progress year to year is not currently achieved via the test process in place. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In response to OP's edited note - I don't think standardized tested is perceived as bad, in and of itself, just that so far, the implementation has been a failure. Which prompts the question - &quot;how can one design  set of standardized tests to track performance and deliver accurate results?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Update - Education Week, a printed and online magazine for educators, is having a webinar titled &lt;a href=&quot;http://www.edweek.org/ew/collections/inside-opt-out/inside-opt-out-the-pushback-against-testing.html&quot; rel=&quot;nofollow&quot;&gt;Inside the Opt-Out Movement&lt;/a&gt; on June 17th 2PM EDT. Note, the webinar is recorded and available for 2 months after that date. I'll return to freshen this link after the recording is made available. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-06-10T16:11:35.617" LastActivityDate="2015-06-10T16:11:35.617" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8205" PostTypeId="2" ParentId="1816" CreationDate="2015-06-02T14:44:24.210" Score="2" Body="&lt;p&gt;Of the Common Core sources, the &lt;a href=&quot;https://ccgpsmathematics9-10.wikispaces.com/&quot; rel=&quot;nofollow&quot;&gt;Georgia materials&lt;/a&gt; are my favorite. &lt;a href=&quot;https://ccgpsmathematics9-10.wikispaces.com/&quot; rel=&quot;nofollow&quot;&gt;https://ccgpsmathematics9-10.wikispaces.com/&lt;/a&gt; They are also at &lt;a href=&quot;https://www.georgiastandards.org/Common-Core/Pages/Math-9-12.aspx&quot; rel=&quot;nofollow&quot;&gt;the state website&lt;/a&gt;, but the superintendent just came out against the CCSS so who knows what will happen to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geoff Krall, a teacher for New Tech Network schools has put together &lt;a href=&quot;http://emergentmath.com/my-problem-based-curriculum-maps/&quot; rel=&quot;nofollow&quot;&gt;a PBL map&lt;/a&gt; I find extraordinarily useful.&lt;/p&gt;&#xA;" OwnerUserId="12" LastActivityDate="2015-06-02T14:44:24.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8206" PostTypeId="2" ParentId="1302" CreationDate="2015-06-02T15:57:50.590" Score="3" Body="&lt;p&gt;Numbers and Functions (Steps into Analysis) by R. P. Burn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to learn the basic theory of real analysis, this is the best choice. If you check it out, I think you will agree.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The material is similar to that of Victor Bryant, but with full detail. Similar to Abbott, but more extensive. More basic than Bressoud. Also, it is designed for &quot;self teaching&quot;, which is what was asked for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my favorite maths books.&lt;/p&gt;&#xA;" OwnerUserId="5259" LastActivityDate="2015-06-02T15:57:50.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8207" PostTypeId="2" ParentId="8105" CreationDate="2015-06-02T21:15:53.583" Score="3" Body="&lt;p&gt;One analogy I have is for the induction step itself. I say that the induction step is like a machine that transfers the truth of the proposition from one number to the next. The machine takes as input the fact that the proposition is true for $k$ and spits out as output the fact that the proposition is true for $k+1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/aBVEa.png&quot; alt=&quot;induction step machine&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You proving the induction step is like constructing this machine. You technically haven't actually shown that anything is true, you've just made the machine that will show that something is true if you put a true statement into it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Yc8qO.png&quot; alt=&quot;proving the induction step&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea of induction is to start with a statement that you know is true, and then imagine continually feeding the output back into the input funnel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I only thought of this idea recently, but so far it has had some success with students. It seems to help them understand the part that the induction step plays in the whole proof, and the fact that it works on statements rather than on numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another picture that helps is to draw the line of numbers as open circles, and colouring them in when you know they're true. The machine can be thought of as transferring the colour from one circle to the next. This &lt;a href=&quot;http://tube.geogebra.org/student/m1202747&quot; rel=&quot;nofollow noreferrer&quot;&gt;GeoGebra activity&lt;/a&gt; illustrates the idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/GbrYd.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-06-02T21:15:53.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8208" PostTypeId="2" ParentId="8193" CreationDate="2015-06-03T00:35:12.747" Score="3" Body="&lt;p&gt;In addition to the other great answers already posted, here's another thing that should be mention: Americans strongly buy into the rags-to-riches story; the idea that anyone can become successful. There are numerous examples of individuals in U.S. history that come from background with little formal education, but nonetheless ended up as successful scientists, inventors, business men, etc. So there's an inherent distaste for a standardized test that measures success in a uniform manner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me say it another way. If there is a standardized test, what do the results mean? In some countries these exams are of utmost importance for getting into higher education. But in the &quot;land of opportunity&quot; the sentiment is that no one should be denied the chance of a successful career just because of the results of a contrived, academic exercise. Americans value the ability to do as much as (possibly more than) the ability to know. So many have an attitude that such a test means nothing in the &quot;real world&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is deeply rooted and, along with the points mentioned by others, plays a large role in the reluctance to have standardized tests.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-06-03T00:35:12.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8209" PostTypeId="1" CreationDate="2015-06-03T02:13:19.653" Score="11" ViewCount="500" Body="&lt;p&gt;Every so often I run across someone online asking what they should spend a sizable (but not enormous) amount of money on for their middle or high school math classroom. The dollar amount varies, but it's often in the 500 to 1000 dollar range. That's enough for a nice piece of technology, or a class set of something not-too-pricey, or a number of subscriptions or pieces of software, or lots of small things and supplies, or some combination thereof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many options and many good ideas. I thought that this might be a good forum to collect and upvote ideas on how to spend a chunk of change on a math classroom. Best bang for the buck. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feel free to offer either a single suggestion that you would purchase or an all-inclusive package for the full sum.&#xA;In any case, please, give a &lt;strong&gt;detailed explanation&lt;/strong&gt; why you make this specific recommendation, rather than mentioning only some specific product.&lt;/p&gt;&#xA;" OwnerUserId="67" LastEditorUserId="143" LastEditDate="2015-06-03T08:50:53.247" LastActivityDate="2015-06-03T20:26:35.933" Title="$1000 to spend on a math classroom?" Tags="|secondary-education|technology-in-education|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8210" PostTypeId="2" ParentId="8209" CreationDate="2015-06-03T02:24:19.037" Score="2" Body="&lt;p&gt;Mathalicious subscription.  Great resource for teachers to use for instruction.&lt;/p&gt;&#xA;" OwnerUserId="5261" LastActivityDate="2015-06-03T02:24:19.037" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8212" PostTypeId="2" ParentId="7716" CreationDate="2015-06-03T04:30:22.603" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Felix_Klein&quot; rel=&quot;nofollow&quot;&gt;Felix Klein&lt;/a&gt; is well-known as a mathematician, and in later years became very involved in mathematics education.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Around 1900, Klein began to take an interest in mathematical instruction in schools. In 1905, he played a decisive role in formulating a plan recommending that analytic geometry, the rudiments of differential and integral calculus, and the function concept be taught in secondary schools. This recommendation was gradually implemented in many countries around the world. In 1908, Klein was elected president of the International Commission on Mathematical Instruction at the Rome International Congress of Mathematicians. Under his guidance, the German branch of the Commission published many volumes on the teaching of mathematics at all levels in Germany.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although I am not aware of any mathematics education research that he did, his name is highly regarded in the mathematics education field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.mathunion.org/icmi/activities/awards/introduction/&quot; rel=&quot;nofollow&quot;&gt;Felix Klein Award&lt;/a&gt; is given by the &lt;a href=&quot;https://en.wikipedia.org/wiki/International_Commission_on_Mathematical_Instruction&quot; rel=&quot;nofollow&quot;&gt;International Commission on Mathematical Instruction&lt;/a&gt; to honor a lifetime achievement in mathematics education research.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://blog.kleinproject.org/&quot; rel=&quot;nofollow&quot;&gt;Klein Project&lt;/a&gt;, which &quot;aims to support mathematics teachers to connect the mathematics they teach to the field of mathematics, while taking into account the evolution of this field over the last century,&quot; is &quot;inspired by Felix Klein’s famous book, Elementary Mathematics from an Advanced Standpoint, published in 1908 and 1909.&quot;&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2015-06-03T04:30:22.603" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8214" PostTypeId="2" ParentId="8198" CreationDate="2015-06-03T05:33:59.017" Score="3" Body="&lt;p&gt;I'm assuming you are going to TA for a general level calculus class (i.e. engineers, science majors, some business people, pre-med...etc.--my guess is that there is a separate business calculus though).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you basically have five options in recitations (that last of which is &lt;em&gt;very&lt;/em&gt; broad):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;You explain homework problems&lt;/li&gt;&#xA;&lt;li&gt;You explain similar problems to the homework&lt;/li&gt;&#xA;&lt;li&gt;You have the &lt;em&gt;students&lt;/em&gt; explain homework problems&lt;/li&gt;&#xA;&lt;li&gt;You have the &lt;em&gt;students&lt;/em&gt; present solutions (similar to the homework)&lt;/li&gt;&#xA;&lt;li&gt;You try to instruct the students.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I think in an intro level class such as calculus that 3) or 4) is the best way to run a recitation...and probably 3) would be the most popular among students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be nice to be adaptive.  For instance if there is a topic that &lt;em&gt;you&lt;/em&gt; realize the students don't understand, then perhaps it would be helpful to give a lesson over that topic rather than simply working problems that day.  Often times the students will benefit from hearing the same thing said two different ways.&lt;/p&gt;&#xA;" OwnerUserId="2204" LastActivityDate="2015-06-03T05:33:59.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8215" PostTypeId="2" ParentId="8209" CreationDate="2015-06-03T14:31:45.103" Score="2" Body="&lt;p&gt;Maybe a set of calculators, one for each student (and a few extras in case of breakage).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you do a calculator exercise, pass out the calculators; but the rest of the time they are sitting on the shelf, showing that calculations should be done by paper and pencil.  If they all have the same calculator, then the teacher can explain how to do things, and the students can help each other.  Much easier than if they have many different calculators.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-06-03T14:31:45.103" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8216" PostTypeId="2" ParentId="8209" CreationDate="2015-06-03T15:11:50.900" Score="4" Body="&lt;p&gt;Games, especially ones that can be played quickly are great. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One I've become particularly fond of this year is &lt;strong&gt;&lt;a href=&quot;https://www.24game.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;24&lt;/a&gt;&lt;/strong&gt;. On each card is 4 numbers; students have to figure out how to use each number once and in whatever order to reach 24. For example, a card might have the numbers 8, 5, 8, and 7. To reach 24, you could do (7 - 5) * 8 + 8. There are more advanced versions available too (e.g., one where students might need to square/take the square root at some point).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like it because I've found that most students enjoy solving them, even those who usually do not see themselves as &quot;math people&quot;. It makes for a nice activity in the morning to help students wake up and start thinking. What I usually do is give each student a card and require they solve it before they enter the classroom.&lt;/p&gt;&#xA;" OwnerUserId="5078" LastEditorUserId="5078" LastEditDate="2015-06-03T15:38:14.813" LastActivityDate="2015-06-03T15:38:14.813" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8217" PostTypeId="2" ParentId="8209" CreationDate="2015-06-03T20:26:35.933" Score="4" Body="&lt;p&gt;Paint the walls with write on whiteboard paint so students can do more vertical non permanent working &lt;/p&gt;&#xA;" OwnerUserId="5266" LastActivityDate="2015-06-03T20:26:35.933" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8218" PostTypeId="1" AcceptedAnswerId="8219" CreationDate="2015-06-03T23:45:45.737" Score="21" ViewCount="1275" Body="&lt;p&gt;Let me explain. Suppose we want to calculate $\lim\limits_{n\to\infty} n^2-n$. Since this limit is indeterminate, one way to do it is to write it as $\lim\limits_{n\to\infty} n^2(1-1/n)$. Since $n^2$ goes to infinity and $1-1/n$ goes to $1$, the limit is $\infty$. If this was part of a bigger expression, we would leave it as it is and then at the end look at the limits of all the individual factors. This is the way I've learned it and the way I've always done it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I've noticed that some students do the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\lim\limits_{n\to\infty} n^2-n = \lim\limits_{n\to\infty} n^2(1-1/n) = \lim\limits_{n\to\infty} n^2 = \infty$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is the second equality I'm concerned with. It's not, strictly speaking, wrong: after all, all the limits here are equal to each other. And yet I've been telling them not to do it. My way is to do whatever you want with your expression, and then take all limits in a single step. Now that I think about it, however, I can't find a reason not to let the $1/n$ go to $0$ before taking the rest of the limit: all the equalities are correct, and it simplifies the expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any reason why the students should be discouraged from doing this? Or am I just enforcing a rule for no reason?&lt;/p&gt;&#xA;" OwnerUserId="5269" LastActivityDate="2018-11-02T12:24:15.007" Title="Should we tell students to never replace parts of an expression by their limits when taking a limit?" Tags="|calculus|limits|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8219" PostTypeId="2" ParentId="8218" CreationDate="2015-06-04T02:50:28.027" Score="23" Body="&lt;p&gt;The important thing is whether students' reasoning is logically valid — and in particular, that they only use the conclusion of a theorem after they've checked that all its hypotheses hold — not whether they follow any particular arbitrary rules or procedures. In this case, the relevant theorem is the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; &lt;em&gt;Let $f$ and $g$ be real-valued functions defined on a subset of the real line, and let $c$ be either a real number or $\pm \infty$. If $\lim_{t \to c} g(t)$ exists and is equal to a nonzero real number, then $\lim_{t \to c} f(t) g(t)$ exists if and only if $\lim_{t \to c} f(t)$ exists, in which case $$\lim_{t \to c} f(t) g(t) = [\lim_{t \to c} f(t)] \cdot [\lim_{t \to c} g(t)].$$&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the example you gave, the problem isn't that the students fail to adhere to the (needless) rule of taking all limits in a single step. The problem is that they omit an important part of the reasoning: since $\lim_{n \to \infty} (1 - 1/n)$ exists and is equal to $1$,&#xA;$$&#xA;\lim_{n \to \infty} n^2 (1 - 1/n) = (\lim_{n \to \infty} n^2) \cdot [\lim_{n \to \infty} (1 - 1/n)] = (\lim_{n \to \infty} n^2) \cdot 1 = \lim_{n \to \infty} n^2.&#xA;$$&#xA;In Steven Gubkin's example, neither $n$ nor $1/n$ has a limit that exists and is a nonzero real number, so the theorem doesn't apply. In both cases, the key is to use precise, logically valid reasoning.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2015-06-04T02:50:28.027" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8220" PostTypeId="2" ParentId="4036" CreationDate="2015-06-04T06:12:56.563" Score="3" Body="&lt;p&gt;Check out the site for &lt;a href=&quot;http://www.mathmidway.org&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Math Midway&lt;/strong&gt;&lt;/a&gt;, and see their &lt;a href=&quot;http://mathmidway.org/contact-math-midway.php&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;contact&lt;/strong&gt;&lt;/a&gt; page. This operation is roughly equivalent to a specialized playground designed to help increase interest on a very positive scale for late elementary and junior high or middle school age bracket children in the field of numbers. One of the questions asked in the booklet I've got is: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How do you get a 10-year-old excited about the world of numbers? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They have a portable math playground that I'm still working on trying to get into Memphis because our local student population can definitely use it! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I gave the website to my nephew who has a 1st and 3rd grader in Collierville elementary and their mother is pulling info off the website on a regular basis for them to work on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have any questions for me, I can be contacted at: jaseay@southwest.tn.edu&lt;/p&gt;&#xA;" OwnerUserId="5247" LastEditorUserId="262" LastEditDate="2015-06-04T06:27:52.783" LastActivityDate="2015-06-04T06:27:52.783" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8221" PostTypeId="2" ParentId="194" CreationDate="2015-06-04T07:01:24.917" Score="-2" Body="&lt;p&gt;Sue VanHattum, in reference to your question &quot;How can we help students who are very ANXIOUS about math?&quot; I don't know that this will answer your question, but I want to refer you to www.mathmidway.org or midway@mathmuseum.net. Phone is (631) 444-0945. One of the questions in the booklet is &quot;how do you get a 10-year-old excited about the world of numbers?&quot; As the first exhibition created by the nation's only museum centered on mathematics, the Math Midway demonstrates the power of bringing hands-on math to the public. It launches the museum's mission to enhance the museums understanding and perception of mathematics as an evolving, creative, and esthetic human endeavor.The Math Midway make math approachable and enticing, so that children and adults alike can experience the exhilarating moment of mathematical discovery. I'm trying to get the portable math midway into Memphis so it might help our local public school system. &lt;/p&gt;&#xA;" OwnerUserId="5247" LastActivityDate="2015-06-04T07:01:24.917" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8222" PostTypeId="2" ParentId="8218" CreationDate="2015-06-04T09:45:57.340" Score="6" Body="&lt;p&gt;To extend the answer by Daniel Hast: One theorem one might want to use is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ are convergent sequences then&#xA;  $$\begin{align}&#xA;\lim_{n\to\infty} (a_n \pm b_n) &amp;amp;= \lim_{n\to\infty} a_n \pm  \lim_{n\to\infty} b_n \\&#xA;\lim_{n\to\infty} (a_n \cdot b_n) &amp;amp;= \lim_{n\to\infty} a_n \cdot  \lim_{n\to\infty} b_n \\&#xA;\lim_{n\to\infty} \frac{a_n}{b_n} &amp;amp;= \frac{\lim_{n\to\infty} a_n}{\lim_{n\to\infty} b_n}&#xA;\end{align}$$&#xA;  For the last equation one also needs $\lim_{n\to\infty} b_n \neq 0$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now, one can do something like $$\lim_{n\to\infty} \tfrac 1n + \sqrt[n]{4}=\lim_{n\to\infty} \tfrac 1n + \lim_{n\to\infty} \sqrt[n]{4}=0+1=1 \qquad(1)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem here is, that one applies the above theorem before showing, that the subsequences converge. So a better way to do it would be:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\lim_{n\to\infty} \tfrac 1n = 0 \land \lim_{n\to\infty} \sqrt[n]{4} = 1 \Rightarrow \lim_{n\to\infty} \tfrac 1n + \sqrt[n]{4}=\lim_{n\to\infty} \tfrac 1n + \lim_{n\to\infty} \sqrt[n]{4}=0+1=1\qquad (2)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) is the way to actually write down the proof and (1) is the way to find the proof (This need to be taught to students because they sometimes think that a proof and the solution process are the same). (2) also prevents you from failures because $\lim_{n\to\infty} a_n=\infty$ means, that $(a_n)$ diverges. I say to students:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\infty$ is no real number. So $\lim_{n\to\infty} a_n=\infty$ means, that one cannot apply the above theorem, because $(a_n)$ diverges. But $\lim_{n\to\infty} a_n=\infty$ sometimes behave in computing with limits like it would converges. For example $\lim_{n\to\infty}(a_n+b_n)=\lim_{n\to\infty}(a_n+b_n)=\lim_{n\to\infty}a_n+\lim_{n\to\infty}b_n$ if one has limits of the form $\infty+\infty$ or $\infty+c$ [Of course one need to introduce the relevant theorems here]. But for $\infty-\infty$ one can never apply the above theorem. So if one limit is $\infty$ one needs to be careful for limits of the form $\infty-\infty$, $\infty\times 0$, $\frac{ \infty}{\infty}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="5097" LastEditDate="2015-06-05T16:06:41.027" LastActivityDate="2015-06-05T16:06:41.027" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8223" PostTypeId="2" ParentId="194" CreationDate="2015-06-04T15:55:36.370" Score="4" Body="&lt;p&gt;To my view, the most effective strategies deal with anxiety as a collective - and thus political and cultural - issue. The answers provided by adamblan and Mandy Jansen are quite to the point, in that regard, and what I'll add here is just a complement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anxiety can be defined as overreaction to falsely perceived &lt;strong&gt;risk&lt;/strong&gt;. Risk taking is socially divided, just like labor is. Risk avoidance behavior is commonly present in children. It can obviously be reinforced positively or negatively, depending on what kind of parental guidance is given, and these decisions are heavily informed by the cultural background of the parents and/or caregivers. It has been argued that biological characteristics are relevant to determining what degree of exposure to danger can a person tolerate (so there would be inclinations based on gender), but anthropological research tells us that human behavior is extremely plastic, so universal truths are scarce here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, what kind of risk is presented to a child in learning mathematics? There is, of course, the social risk of being outcast by failing to fit properly into the assigned stereotype. This fear has been brilliantly addressed by previous answers, so I'll skip it here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is possibly, though, a different kind of risk perception, more subtle, and with deeper roots than social prejudice. To put it in simple terms: the human brain consumes a lot of energy, and streams of thought that are (considered) energy efficient tend to be strongly preferred. &lt;em&gt;This also has a social signature&lt;/em&gt;: anti-intellectualism is the way societies have to indicate to the individual that to be &quot;lost in thought&quot; is dangerous to herself and to the larger group. Being considered the way of thinking that is more liberated from experiential constraints, mathematics may have been historically associated with the biggest of threats. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, if thinking &quot;complicated&quot; thoughts has been considered, &lt;em&gt;in itself&lt;/em&gt;, wasteful and possibly dangerous to the cohesion of the group - and thus risky - then thinking mathematically has to be extremely problematic. This kind of prejudice affects more directly, obviously, those that are not in a position of power. In this case, though, it won't be enough to address cultural stereotypes, because the anti-mathematical prejudice is largely unconscious, and even more pervasive. I'd say that it lies outside of what can be dealt with in a pedagogical environment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This predisposition can cause enormous suffering, inside and outside of the classroom. It is a clinical problem, and a social one. The issue is particularly urgent on our present time, when the ability to think mathematically is, maybe for the first time in human history, not an accidental feature (virtue or menace), but the most valuable resource that we have, for our continued survival on this planet.   &lt;/p&gt;&#xA;" OwnerUserId="5038" LastEditorUserId="5038" LastEditDate="2015-06-04T19:40:37.487" LastActivityDate="2015-06-04T19:40:37.487" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8224" PostTypeId="2" ParentId="194" CreationDate="2015-06-04T20:19:03.697" Score="6" Body="&lt;p&gt;When I posted this question over a year ago, I meant to post my own answer after giving others a bit of time to post. I apparently forgot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My students have had some success in decreasing their anxiety with books like &lt;em&gt;&lt;a href=&quot;http://www.betterworldbooks.com/mind-over-math-put-yourself-on-the-road-to-success-by-freeing-yourself-from-math-anxiety-id-9780070352810.aspx&quot; rel=&quot;noreferrer&quot;&gt;Mind Over Math&lt;/a&gt;&lt;/em&gt; (Kogelman, Warren), &lt;em&gt;&lt;a href=&quot;http://www.betterworldbooks.com/overcoming-math-anxiety-id-9780393313079.aspx&quot; rel=&quot;noreferrer&quot;&gt;Overcoming Math Anxiety&lt;/a&gt;&lt;/em&gt; (Tobias), and &lt;em&gt;&lt;a href=&quot;http://www.betterworldbooks.com/managing-the-mean-math-blues-id-9780130431691.aspx&quot; rel=&quot;noreferrer&quot;&gt;Managing the Mean Math Blues&lt;/a&gt;&lt;/em&gt; (Ooten).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also wanted something that directly addressed their test anxiety in math, and (after much research on math anxiety and the principles of creating guided visualizations) I developed a 14-minute guided visualization, which I titled &lt;a href=&quot;http://mathmamawrites.blogspot.com/2009/09/math-relax-guided-visualization-for.html&quot; rel=&quot;noreferrer&quot;&gt;Math Relax&lt;/a&gt;. It is available online for free. Not every student finds it helpful, but some students have felt that it changed their outlook dramatically.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-06-04T20:19:03.697" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8226" PostTypeId="2" ParentId="8167" CreationDate="2015-06-05T06:29:43.187" Score="4" Body="&lt;p&gt;I recall a cute introduction to the work of Miryam Mirzakhani that is written at the level of a younger student. Something like this could be adapted and incorporated into a 11-year-old's math curriculum. &lt;a href=&quot;https://www.matific.com/us/en-us/maryam-mirzakhani&quot; rel=&quot;nofollow&quot; title=&quot;Here&quot;&gt;Here&lt;/a&gt; is the link to the infographic on Matific. It talks about surfaces and the genus of surfaces and is reasonably well explained. Keep in mind your 11-year-olds might not have a firm grasp on the difference between two dimensions and three dimensions. Upon arriving there you will need to evaluate what the students are able to understand by possibly asking a few fun get-to-know-you questions that double as assessing their geometric ability. They might not remember the difference between a circle and a sphere without being reminded. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember that when I was 11 or so, my dad had me imagine games being played on different surfaces and how the rules would change. He also taught me about donuts and coffee cups being topologically equivalent, and had an amusing obsession with the &lt;a href=&quot;http://www.kleinbottle.com/whats_a_klein_bottle.htm&quot; rel=&quot;nofollow&quot;&gt;Klein bottle and Mobius strips&lt;/a&gt; (I think this has some decent explanations for that level, and some of the examples it describes could be used as manipulatives.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, there are many &lt;a href=&quot;http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Daps&amp;amp;field-keywords=iq%20knot%20puzzle&quot; rel=&quot;nofollow&quot;&gt;manipulative topological puzzles&lt;/a&gt; that 11-year-olds could spend a class period on trying to solve in groups. You can get these (with metal rings, ropes etc) at Barnes and Noble or many novelty stores if you don't want to order online. In fact, even the &lt;a href=&quot;http://en.wikipedia.org/wiki/Human_knot&quot; rel=&quot;nofollow&quot;&gt;Human Knot&lt;/a&gt; might be a decent introduction into knot theory. If you did it multiple times or multiple small groups, your students might realize that not all of their &quot;knots&quot; could be fully untangled into a single circle. Students could also draw Celtic knots or do the &lt;a href=&quot;http://britton.disted.camosun.bc.ca/jbhandcuff.htm&quot; rel=&quot;nofollow&quot;&gt;Handcuffs puzzle&lt;/a&gt; with each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding triangulation. Small groups of students could have a balloon and a sharpie. Instruct them to draw on about 4-10 dots then connect as many dots as possible to create a triangulation, then count dots and edges. You WILL have to give extremely clear instructions and demonstrate this along with them. You may also want to give them two colors of sharpie or give them a tool to help them avoiding miscounting so the entire time isn't spent counting edges. Doing in groups will help keep them from getting over taxed -- each student could count something different. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apologies that I don't have many examples of the advanced concepts which you have mentioned, but hopefully these are some hands-on ideas that can get them excited and interested. In general, because these 11-year-olds are struggling in math, manipulatives and hands-on activities are a good idea and will engage them much more than doing examples. One hour isn't much time with 11-year-olds, and unless you're planning on giving a lecture (which wouldn't be age-appropriate), I'd plan on covering maybe one or two of those concepts.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2015-06-05T06:29:43.187" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8227" PostTypeId="1" AcceptedAnswerId="8228" CreationDate="2015-06-05T11:53:35.987" Score="8" ViewCount="278" Body="&lt;p&gt;I am currently working with a student who is on the slowest math track at her school of the three available; however, I cannot understand why. She shows an amazing ability to reason mathematically and she can understand the underlying concepts of the material extremely quickly, even more quickly than some of the students who I have worked with on the fastest math track at her school. I am not sure what her placement in her current math class was based upon, because she seems to be extremely strong in mathematics. In the US, what is the standard procedure for placing students into their earliest math classes?&lt;/p&gt;&#xA;" OwnerUserId="5275" LastActivityDate="2015-06-05T23:22:42.837" Title="How are U.S. students placed in math classes?" Tags="|secondary-education|classroom-management|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8228" PostTypeId="2" ParentId="8227" CreationDate="2015-06-05T23:22:42.837" Score="7" Body="&lt;p&gt;At my school, in a large urban district in the United States, students are (unfortunately) placed into different math tracks based off a few different factors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;strong&gt;Performance on tests (standardized or otherwise)&lt;/strong&gt; -  In this day and age, this is likely the most indicative factor of a student's math placement. &#xA;A student's ability to take tests, especially the state mandated and facilitated ones, and receive high marks consistently will almost always be placed in the highest track.  Unfortunately, these state tests are very influential over a lot of things including teacher rating, school rating, budget appropriations, etc and as such they are deemed a large indicator of students' abilities.  If a student consistently &quot;underperforms&quot; on a set of standardized tests they will almost assuredly be placed in the remedial or middle track.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;strong&gt;Teacher recommendations&lt;/strong&gt; - I would say that the second most important factor in determining a student's track is the recommendation of past teachers, which is generally trustworthy.  However, just because a student is recommended by one teacher does not mean that they will automatically be in whatever track that teacher says.  The principal, other teachers, or parents may override a recommendation to either push a student up a level or hold them back one.  Additionally, if a class is already at the population limit or only offered once or twice a day, students may not be able to be rostered and be forced to again move levels, either up or down depending on the scheduling conflict.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3)  &lt;strong&gt;Student input&lt;/strong&gt; - Last, and at least in my school least, a student's own input is sometimes taken into consideration. Usually this will occur when a parent has unreal expectations for their child and has pushed the admins to move them up a level and the student fights back because they are not comfortable with the more accelerated pace.  I have seen this go both ways where the student wins out and also when the parent persists and has the student moved up (this almost always ends disastrously for the student) against their will.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would love to say that students' placement is determined based off of their capacity for abstraction and critical thinking skills and how well they are able to conceptualize and explain their reasoning but unfortunately, at least in my school and most other schools in my district, this is how it works.  If you feel that your friend is in a lower track than they deserve, I would highly recommend having an honest conversation with their teacher, their parents, the principal, rosterchair etc and come with proof of their abilities (written work on a problem or have them come with and demonstrate how they think about a problem).  This direct method is likely the only way to get them to where they will thrive best.  Best of luck!&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-06-05T23:22:42.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8229" PostTypeId="5" CreationDate="2015-06-06T14:29:36.240" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-06-06T14:29:36.240" LastActivityDate="2015-06-06T14:29:36.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8230" PostTypeId="4" CreationDate="2015-06-06T14:29:36.240" Score="0" Body="The necessary background to learn a certain topic or to follow a course or academic program." OwnerUserId="39" LastEditorUserId="39" LastEditDate="2015-06-06T17:50:46.410" LastActivityDate="2015-06-06T17:50:46.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8231" PostTypeId="2" ParentId="39" CreationDate="2015-06-06T19:38:52.123" Score="2" Body="&lt;p&gt;Here is a thought that might help in your development.  Write up a sample lecture/class period to describe the material and how it goes.  Get two others to help you run through a trial and videotape it.  (Or should I say &quot;Cell-phone it&quot;?)  Publish on the net the written material and representative samples from the video of how the class might go.  Include contact information for anyone who is interested in the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree with the other poster that writing down a specific set of prerequisites beyond mathematical maturity will be a challenge.  Giving an idea of what you want the course to be about should not be, and communicating the style above should be wonderful preparation for the audience, at least in determining if they want to be a part of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can flesh out the ideas for such &quot;promotional material&quot; if you are interested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Not Yet A Hollywood Producer&quot; Paseman, 2015.06.06&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-06-06T19:38:52.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8232" PostTypeId="2" ParentId="8179" CreationDate="2015-06-06T23:59:16.417" Score="2" Body="&lt;p&gt;To compute the &lt;a href=&quot;http://en.wikipedia.org/wiki/Golden_ratio&quot; rel=&quot;nofollow noreferrer&quot;&gt;golden ratio&lt;/a&gt;, we need to find the positive root of a quadratic equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two quantities $a$ and $b$ with $a&amp;gt;b&amp;gt;0$ are said to be in the &lt;em&gt;golden ratio&lt;/em&gt; if $a$ is to $b$ as $a+b$ is to $a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/A00FY.gif&quot; alt=&quot;enter image description here&quot;&gt; (Image source: &lt;a href=&quot;http://en.wikipedia.org/wiki/Golden_ratio#/media/File:Golden_ratio_line.svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Algebraically, $$\frac a b= \frac {a+b} a.$$ If $\varphi=\tfrac a b$ denotes the golden ratio, then $$\varphi = 1+\varphi^{-1}.$$ Multiplying with $\varphi$ and rearranging gives the equation $$\varphi^2-\varphi-1=0,$$ which has a positive root $$\varphi=\frac {1+\sqrt 5} 2 \approx 1.618$$&lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2015-06-06T23:59:16.417" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8234" PostTypeId="1" CreationDate="2015-06-07T05:29:41.827" Score="10" ViewCount="175" Body="&lt;ul&gt;&#xA;&lt;li&gt;How important are rewards - read marks/grades - for students in education? &lt;/li&gt;&#xA;&lt;li&gt;Can the thirst for grades/marks be replaced by a desire for learning? &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Context&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problems are at the heart of mathematics making it the most easiest subject to grade and judge students. Problems are very well defined, objective chunks that have equally well structured solutions (at least at high school level). I wonder what runs through the minds of students as they solve problems in exams. In the intense atmosphere of the examination hall, would they be able to enjoy a good problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Right from primary schools, all the way up to IMO, there is intense competition to perform at the highest level. Students and teams are constantly being judged. X scored 100%/A+ and is therefore better than Y. X derives pleasure from being at the top and this is a clear motivating factor for him/her to remain consistent. Most of the students I see try to optimize for grades and not learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit :&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The following questions need not be addressed. As rightly pointed out in the comments, I think it makes better sense to put them in a separate question. But I am still having them here for later reference.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Are there any alternatives to the grading system? &lt;/li&gt;&#xA;&lt;li&gt;Is it sensible to completely do away with the grading system (I know this doesn't make a lot of sense, but still...)?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5221" LastEditorUserId="5221" LastEditDate="2015-06-08T17:28:01.427" LastActivityDate="2015-06-08T17:28:01.427" Title="Rewards based educaton" Tags="|secondary-education|education-research|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8235" PostTypeId="1" CreationDate="2015-06-07T10:28:51.513" Score="4" ViewCount="181" Body="&lt;p&gt;This is a soft question . In the walk-in for the lectureship, I have decided to give demo lecture on the topic of &lt;strong&gt;Fourier transform&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The principal of the institution ask me to take lecture  interactively with students on the given topic for &lt;strong&gt;around 20 to 25 minutes&lt;/strong&gt;.  I should teach them interactively by which they can learn Fourier transform with interest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I could not dare to counter question him &lt;strong&gt;&quot;How to make and teach Fourier transform topic interactive one making use of given time span?&quot;&lt;/strong&gt; ;-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now,I am confused with what should be the start of the lecture? Should I take any example or any application than directly coming on equations or any other ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What  should I tell about importance of Fourier transform? The important factors of signals I.e. phase,frequency and amplitude ? So can anybody help me for it?&lt;/p&gt;&#xA;" OwnerUserId="5190" LastActivityDate="2015-08-06T20:28:24.993" Title="How to explain the topic of Fourier transform interactively?" Tags="|undergraduate-education|concept-motivation|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8238" PostTypeId="2" ParentId="8235" CreationDate="2015-06-07T19:56:04.553" Score="1" Body="&lt;p&gt;Go in with two or three ideas/themes/topics to discuss, and be prepared to abandon them to handle the classroom.  The demo (in my view) is about how you will handle less standard situations, and not so much about how to deliver a good standard lecture.  In general, there will be students who aren't interested whom you are expected (or at least hoped) to draw out and help them engage their brains.  There are some who will want to grab attention and direct it to a particular topic related to but away from the main topic: how will you help them and guide the discussion back to topic?  There are those rare few cases who will be disruptive; you will have to handle them and keep the rest of the class on track.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I imagine they will be pleased at new insights or refreshing ways to present Fourier transforms, but that would not be my priority as a member of committee who is evaluating a lecturer for hire.  If you have time and political savvy, it might help your case to send a letter to the appropriate person confirming your understanding of the time, location, and other details.  You can also slip in what you believe are the things they are looking for, and suggest that you are willing to change your presentation if there are different things they wish to see/evaluate.  (Don't send such a letter if it feels wrong or if something else suggests it is a bad idea.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhards &quot;Always More Than One Goal&quot; Paseman, 2015.06.07&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-06-07T19:56:04.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8239" PostTypeId="2" ParentId="7918" CreationDate="2015-06-07T20:47:42.930" Score="2" Body="&lt;p&gt;Manipulatives work well for exploring quadratic forms. Algetiles are the standard, but you could use base 10 blocks if algetiles aren't available. You can easily make a set of algetiles if don't have one available. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the issues with manipulatives is that students sadly don't encounter them enough, so you will probably have to run over the basics with them before you do algebra: positive and negative tiles, zero pairs, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A fast introduction to the use of algetiles for quadratics is at &lt;a href=&quot;http://mathbits.com/MathBits/AlgebraTiles/AlgebraTiles/AlgebraTiles.html&quot; rel=&quot;nofollow&quot;&gt;http://mathbits.com/MathBits/AlgebraTiles/AlgebraTiles/AlgebraTiles.html&lt;/a&gt;. Factoring amounts to taking a given set of algetiles and forming them into a rectangle, so it is analogous to factoring integers by taking a collection of unit squares and forming it into a rectangle.&lt;/p&gt;&#xA;" OwnerUserId="5205" LastActivityDate="2015-06-07T20:47:42.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8240" PostTypeId="2" ParentId="8234" CreationDate="2015-06-07T23:44:59.287" Score="4" Body="&lt;p&gt;In lieu of other responses:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;How important are rewards in education?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ok, rewards to who(m)?!? Parents, or the kids? A huge difference. And, also, significantly, what &lt;em&gt;kind&lt;/em&gt; of rewards? Macho-points? Scholar-points? Good-citizen-points? Hm, we don't have &quot;contests&quot; for scholars or good citizens...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Right from primary schools, all the way up to IMO, there is intense competition.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The human primates &quot;understand&quot; this. Especially males, etc. So, yes, on one hand, playing into this will riff on kids' impulses... on another hand, (I claim) this is almost entirely unrelated to mathematics (the thing-itself) and understanding it (which perhaps is related to education...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I do truly think that a great part of the tension/problem is the confounding of human-beings'-tendencies with the demands of understanding the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Students and teams are constantly being judged.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you'll pardon me... &quot;Whoa, ...&quot; Teams? Judged? Sure, in contest scenarios, but ... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;... X scored 100%/A+ and is therefore better than Y. X derives pleasure from being at the top and this is a clear motivating factor for him to remain consistent.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you'll pardon me, I think this is exactly an indicator of the thin-ness of the motivation toward scholarship and broader-benefit of mathematics. Doing sums faster is a very minor virtue. &quot;Beating other people&quot; accomplishes nothing at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Are there any alternatives to the grading system? Is it sensible to completely do away with the grading system (I know this doesn't make a lot of sense, but still...)?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course there are alternatives, but the question is about viability. In the U.S., and with variations in the EU, and in many other places, bureaucratic judgementalism gives huge credence to &quot;objective&quot; accomplishment (ooh, think &quot;impact factor&quot; for even grown-ups...) so we (humans...) are somewhat doomed to playing against this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Most of the students I see try to optimize for grades and not learning. Should the thirst for grades/marks be replaced by a desire for learning?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an even more over-layered presumptive question... &quot;Should&quot; desire for good grades &quot;be replaced by&quot; ... &quot;desire for learning&quot;. Ahem. If we could mandate that... um, sure. But a kid (and every organism) is fundamentally rational, given the information that it has. The awkward truth about &quot;education&quot; is that kids are going through the obvious stunningly confusing/distracting issues about puberty, so will allocate only a tiny fraction of their attention to anything else. And, lest we be too disparaging of such a viewpoint, we should not that we'd not be here if our ancestors hadn't had a suffient impulse on a regular basis over the various millenia... (whew!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, apart from all that sociological babble, ... I think that a useful answer in any particular person's context depends so hugely on that context that I have no answer...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(All this was just to explain the difficulty of an answer... lucky I don't have to make a living at this...)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-06-07T23:44:59.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8241" PostTypeId="1" CreationDate="2015-06-08T00:21:57.337" Score="23" ViewCount="1280" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Note: I myself am not a math educator, though I plan to be one someday.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://www.docdroid.net/z8ki/knuth.pdf.html&quot;&gt;this letter&lt;/a&gt;, Donald Knuth suggests an alternate way of teaching calculus, based on big-O (introduced via a related big-A notation). He says &quot;it would be a pleasure for both students and teacher if calculus were taught in this way.&quot; Is his suggestion a good idea? Would it be easier for students to understand? How much of an improvement, if any, is this over the traditional way of teaching it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Since I am new to this site, please forgive me if this question is too subjective or if it doesn't meet the site's standards.)&lt;/p&gt;&#xA;" OwnerUserId="5282" LastActivityDate="2015-06-14T17:11:44.503" Title="Is Knuth's suggestion on teaching calculus a good idea?" Tags="|mathematical-pedagogy|calculus|" AnswerCount="2" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="8243" PostTypeId="2" ParentId="8241" CreationDate="2015-06-08T07:52:04.113" Score="13" Body="&lt;p&gt;This is not a complete answer, but who can say for sure what is ``the'' best way to teach calculus, or if there is a best way?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am very tempted myself toward using big-O and small-o notations, and  I have done so recently (I had to teach Taylor series to first-year undergraduates). However, &lt;strong&gt;I tend to think that ensuring student have a firm grip on simpler notation first might be better&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main quality of these notations is also what make them difficult to understand to student : an expression like $o(x^2)$ is written in exactly the same form as $f(x^2)$ where $f$ is a function, but has a quite different interpretation. It is in fact pretty difficult to grasp that it means that there is &lt;em&gt;some&lt;/em&gt; function $\varepsilon$ with certain properties (partly implicit unless on uses the more cumbersome $o_{x\to 0}(x^2)$) such that $o(x^2) = \varepsilon(x)$, but that each time when the notation appears, the function $\varepsilon$ is different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The following confusion can then easily appear: $\sin x= x + o(x^2)$ and $\cos x= 1-x^2/2 + o(x^2)$ so &#xA;$$\sin x -\cos x = x +o(x^2) -1+x^2/2 -o(x^2) = x^2/2 + x -1,$$&#xA;right?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, this could be overcome, and it might be a good idea to teach student how to learn such a tricky notation; but one should keep in mind they tend to have difficulties with the notions of variable, unknown, function, image of a number by a numerical function, etc. and that has to be taken into account before burdening them with an additional, subtle notations.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-06-08T07:52:04.113" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="8244" PostTypeId="2" ParentId="2301" CreationDate="2015-06-09T06:35:02.863" Score="-2" Body="&lt;p&gt;In reference to the questions and comments about interaction skills, I want to refer you to this website: The Conversation .com,  (title) &quot;Feet on campus, heart at home! First generation college students struggle with divided identities.&quot;    This is written by Linda Banks-Santille&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Go to this second article (seperate from &quot;Feet on campus...&quot;) &quot; Postsecondary Access and Success for First-Generation College Students&quot;&#xA;Jennefer Engle, author.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Allen Seay at jaseay@southwest.tn.edu&lt;/p&gt;&#xA;" OwnerUserId="5247" LastEditorUserId="5247" LastEditDate="2015-06-09T07:02:01.770" LastActivityDate="2015-06-09T07:02:01.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8245" PostTypeId="1" CreationDate="2015-06-09T16:48:24.850" Score="6" ViewCount="1947" Body="&lt;p&gt;Why bother writing a given integer as the sum of two squares? Does this have any practical application? Is there an introduction on a first year number theory course which would motivate students to study the conversion of a given integer to sums of two squares?&lt;/p&gt;&#xA;" OwnerUserId="1567" LastActivityDate="2015-06-12T09:24:21.743" Title="Why convert to sums of two squares?" Tags="|mathematics-in-daily-life|teaching|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8246" PostTypeId="2" ParentId="8245" CreationDate="2015-06-09T17:50:10.590" Score="8" Body="&lt;p&gt;The following two things do not match exactly the request but are perhaps close enough: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Writing an $n$ as a &lt;em&gt;difference&lt;/em&gt; of two square is useful as it yields a typically nontrivial factor of $n$, as $n = x^2 - y^2 = (x+y)(x-y)$. &#xA;This is called &lt;a href=&quot;http://en.wikipedia.org/wiki/Fermat%27s_factorization_method&quot;&gt;Fermat's factorization method&lt;/a&gt; and is still relevant  as algorithm, in fact &lt;a href=&quot;http://en.wikipedia.org/wiki/Shanks%27_square_forms_factorization&quot;&gt;SQUFOF&lt;/a&gt; which is a development of this idea is one of the best factorization algorithms today. Note that it was Fermat who also considered the sum of two squares problem. (A sum of two squares can be used to get a factorization in the Gaussian integers but this might not be relevant in your context.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The fact that every non-negative integer can be written as &lt;a href=&quot;http://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem&quot;&gt;the sum of four squares&lt;/a&gt; is relevant as it allows for an easy intrinsic characterization of the non-negative integers among all integers; the non-negative integers are definable by the formula $x_1^2 +x_2^2 + x_3^2 + x_4^2$. This is sufficiently relevant that it appears in some  courses on mathematical logic.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The first could be used as motivation by using this as a &quot;trick&quot; to find factors of some large numbers. Then, one can discuss if this always works and so on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The latter could be presented as a challenge: find a multivariate polynomial in the integers that yields  only non-negative integers as values yet all non-negative integers. &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2015-06-09T17:50:10.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8247" PostTypeId="2" ParentId="8245" CreationDate="2015-06-09T22:59:30.660" Score="2" Body="&lt;p&gt;You are given a box of square tiles all of the same size.  As it happens, there are m different colors of tile.  The mean number of tiles of a color is n.  (Some have a couple more, some less.)  You are supposed to make a pattern with the tiles, using up all or almost all of them, and according to a particular theme.  If n were a square number, say n=k^2, no problem: you could arrange&#xA;the tiles into m squares (or almost squares) and combine the result into an arrangement that happens to fit the theme.  Unfortunately, n is far from a square.  What to do?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It so happens that n is the sum of two squares.  You can now make a polygonal shape which are two squares adjacent to one another, in such a way that the shapes of m different colors tile.  The variation is acceptable, and you have saved the day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One might argue that this is somewhat contrived of an example.  However, people have been coming up with contrivances since before they were able to make tiles and arrange them into patterns.  There are reasons within mathematics to study decompositions of integers into sums of squares, and there are applications of this work.  Two-square tiling is just one; creating a logo involving pairs of rational squares is another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Even More Creative After Coffee&quot; Paseman, 2015.06.09&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-06-09T22:59:30.660" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8248" PostTypeId="2" ParentId="8245" CreationDate="2015-06-09T23:30:39.593" Score="3" Body="&lt;p&gt;Following @celeriko's suggestion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $n$ can be written as $a^2 + b^2$ for integers $n,a,b$, then $\sqrt{n}$&#xA;can be constructed as the hypotenuse of a right triangle with base and altitude&#xA;$a$ and $b$.&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/cgVZf.jpg&quot; alt=&quot;Root13&quot;&gt;&#xA;&lt;hr /&gt;&#xA;For example, if you needed to measure out $\sqrt{13}$, you could do so with&#xA;$3^2 + 2^2=9+4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can imagine physical construction tasks that require measuring $\sqrt{n}$,&#xA;e.g., rigidifying an $a \times b$ rectangle with a diagonal brace,&#xA;or constructing a square of area $n$.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-06-09T23:55:18.650" LastActivityDate="2015-06-09T23:55:18.650" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8249" PostTypeId="2" ParentId="1689" CreationDate="2015-06-10T05:38:17.760" Score="5" Body="&lt;p&gt;Redwing, In &quot;Conceptual physics&quot;, Twelfth Edition, by Paul Hewitt, he makes the following comment on page 181, &quot;Why study this material since you&quot;ll forget most of it? My answer is that whether or not you use this knowledge, in the act of learning how to connect concepts and solve problems you're establishing connections in your brain that didn't exist before. It's the wiring in your brain that makes you an educated person. That wiring will be useful in areas you can't dream of right now&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the March 17,2014 New York Times article &quot;The Walls Close In&quot;, VW invested 1 Billion in a car assembly plant in 2011 in Chatanooga,Tn. The plant offered 2,000 job positions paying an average of 19.50/hour. 80,000 people applied for these jobs in an area where the average pay scale was/is $9.00 an hour. Many low wage workers wanted to work there but could not pass the basic math tests.&#xA;In a phone interview I had with UT Economics Professor Matt Murray, I told him I contacted the VW plant to determine what the basic level of math was and I never got a reply. He told me that the 78,000 people that didn't get the jobs  also failed their drug tests and that VW wouldn't release that math info because there is fierce competition between every company in the state of Tennessee for skilled and educated workers. Please share this information with your students. This is the way that the job market seems to be going.            Allen Seay&lt;/p&gt;&#xA;" OwnerUserId="5247" LastActivityDate="2015-06-10T05:38:17.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8252" PostTypeId="1" CreationDate="2015-06-11T01:00:48.863" Score="3" ViewCount="1159" Body="&lt;p&gt;I am working with a student who is very interested in math competitions, and I am teaching him Algebra I. I feel like doing competition problems related to a given topic is an excellent way to force him to apply the concept in a non-standard way, so I try to do it as much as possible. The main problem which I encounter with this approach is the difficulty of finding good problems related to a given topic. Skimming old competitions such as the AMC occasionally yields something interesting, but that involves sifting through a lot of problems to find just a few. I have looked at the AOPS Alcumus program to find problems related to a specific problem and that has worked fairly well, the only problem being the slow pace at which I can read through the Alcumus problems due to the format. It would be excellent if there were a place where I could simply download a list of competition problems organized by topic and select the ones which I thought were most relevant and interesting. Does a list like this exist?&#xA;Thanks&lt;/p&gt;&#xA;" OwnerUserId="5275" LastEditorUserId="238" LastEditDate="2019-09-03T23:35:56.583" LastActivityDate="2019-09-05T14:25:49.183" Title="List of math competition problems by topic" Tags="|mathematical-pedagogy|reference-request|math-contests|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8253" PostTypeId="2" ParentId="7976" CreationDate="2015-06-11T02:04:38.980" Score="1" Body="&lt;p&gt;This is one way that textbooks could (potentially) justify their high price.  If your textbook does not supply this for you, then perhaps you should switch.&lt;/p&gt;&#xA;" OwnerUserId="3346" LastActivityDate="2015-06-11T02:04:38.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8254" PostTypeId="2" ParentId="8252" CreationDate="2015-06-11T05:40:14.150" Score="3" Body="&lt;p&gt;I don't know if such a list exists, but I can recommend other math competitions - specifically the waterloo contests: &lt;a href=&quot;http://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf&quot; rel=&quot;nofollow&quot;&gt;http://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this is helpful in some way.&lt;/p&gt;&#xA;" OwnerUserId="5141" LastActivityDate="2015-06-11T05:40:14.150" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8255" PostTypeId="1" AcceptedAnswerId="8256" CreationDate="2015-06-11T09:19:07.670" Score="13" ViewCount="205" Body="&lt;p&gt;I have worked with mathematics and physics in a university and I have taught on several courses.&#xA;I recently applied to a teacher education program to get formal qualifications for teaching at highschool level to become a better teacher and to broaden employment possibilities, and I was accepted.&#xA;The program will also include a practise period during which I will follow, prepare and give lectures in a school.&#xA;There is great freedom in choosing the school, and I could in principle do it at my own mathematics department, but I think I would learn more doing it in a highshool.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some key qualities in the school and the instructing teacher that I should look for?&#xA;How can I make that period in the school as worthwhile as possible for myself?&#xA;What are some key lessons that I should make sure I learn?&#xA;I have PhD in mathematics and I feel very comfortable with the subjects to be taught; the main focus for me will be on teaching rather than the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main (possibly only) subject to be taught is mathematics and everything will take place in Finland.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2015-06-11T12:35:49.747" LastActivityDate="2015-06-11T20:23:23.887" Title="How to choose the school for practising teaching?" Tags="|secondary-education|general-pedagogy|teachers|teacher-training|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8256" PostTypeId="2" ParentId="8255" CreationDate="2015-06-11T15:14:27.417" Score="9" Body="&lt;p&gt;First off, I would like to commend your desire to teach at a high school level.  It is a very challenging and tiring profession but it is so rewarding and IMHO is one of the most useful settings that one can teach in (aside from maybe elementary school but that is a whole different ball game :)  It sounds like you already understand one of the most basic fundamentals of high school teaching when you said &quot;the main focus for me will be on teaching rather than the subject&quot;.  Unlike at university, the actual content knowledge of the teacher is much less important than their ability to be organized, engaging, equitable, and controlling of behaviors and expectations and that is why it is crucial that &lt;strong&gt;if you want to be a high school teacher that you do your residency in a high school&lt;/strong&gt;.  Pedagogical approaches taken in high school vs those taken at university are vastly different and there is often little to no overlap. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, I think that the most important thing you can do while trying to decide which school to do your residency at is &lt;strong&gt;research, research, research&lt;/strong&gt;.  As you can probably imagine, every high school has it's own culture, norms, and standards and it would be in your best interest to do research into your different options either by going to the schools' websites or calling and talking with administrators.  Better than all of these is to &lt;strong&gt;actually visit the schools in person and observe a few classes&lt;/strong&gt;.  If you can, try to observe classes that 1) are in your subject area (math), 2) taught by potential mentors and 3) in different subject areas, such as history or language.  I cannot stress how important this is as it will give you first hand experience with how the school, teachers, and students behave and interact on a regular basis, not just within the small bubble of your mentor's classroom, but from a wide array of perspectives.  While you are in the buildings, talk to the teachers about what they like and what they don't like about the school.  Talk to the students about what they like and don't like.  You might be surprised at how candid teachers and students can be about their own school and you might learn something, either positive or negative, that will sway your decision.  It is helpful to read up about different schools online, but nothing is as important as actually going in person to see what it is like and determine for yourself whether it will be a good fit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as finding a mentor, I would say the most important aspect of a good mentor is one who gives you guidance and support when needed but will also give you the freedom to experiment and go about things on your own.  It is important that they are moderately hands-off and will allow you to completely bomb lessons because that is the only way you will learn. What is important is that after you bomb a lesson, they are right there to tell you why they think it went wrong, offer advice for the future, and encourage you to keep on with it.  What you want to stay away from is someone who wants to control every aspect of your teaching because you will never learn that way.  Have these conversations with potential mentors beforehand and hopefully they will be honest with you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as things that you should be focusing on during your teacher preparation, here is my list of things that I ended up focusing on the most during my time student teaching:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Planning&lt;/strong&gt; - efficient and effective planning is one of the most important aspects of high school teaching.  With a solid plan, most lessons will go pretty well, all things considered. Without a plan, you will very quickly learn that &quot;winging it&quot; is never viable.  At first, planning is long and tedious and the only way that it will get any quicker is with practice.  Talk to your mentor, talk to other teachers, see how they go about planning a lesson, see if they have resources for you so that you don't have to plan everything from scratch.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Assessment/Grading&lt;/strong&gt; - I am not sure how the situation is in Finland, but assessment and grading is a huge part of high school teaching in the US.  Whether or not there is actually any value in number/letter grades, does not take away from the fact that at some point you do need to assess your students' knowledge so that you can know where to go in future lessons.  Assessment comes in many different forms (formative, summative, group, individual, projects, classwork) and it is important that you learn how to take all of this &quot;data&quot; (data, in teaching, does not have to be numerical, it simply refers to anything that can be learned about students from their work/discussion) and use it to influence future lessons.  Blindly planning and assessing students without any purpose or goal is useless.  Always use your students' progress and abilities to drive how you teach them.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Classroom culture&lt;/strong&gt; - Another important aspect of high school teaching is the atmosphere and culture of your classroom.  Try out different things such as changing seating charts around, moving the desks around, putting up posters, having a wall for student work, etc.  All of this helps to cultivate the physical environment of your classroom and while it might not seem like a big deal it can actually have a lot of influence on how well students learn in your room.  A quick story:  While student teaching, I had the opportunity to observe at a bunch of different schools in my district.  At one school there was a teacher without his own classroom and as such had to move throughout the day to different rooms.  I observed him in the morning in this tiny little classroom with no windows, barely enough room to walk around, and nothing on the wall except for bland grey paint.  The students were raucous, unfocused, and jittery, constantly trying to get up and walk around even though there was no room.  The lesson was not terrible but the teacher had to continually struggle to get the students on task. Later in the day, I observed the same teacher teaching the same lesson to a different class in a different room.  This was more like your typical classroom: windows, posters on the wall, ample room to move around, etc. and, compared to the morning class, it was like night and day.  The students were engaged, they were not disruptive, they were asking questions and having discussions.  Now this may have just been because of the students, but it really struck me that day how different the classrooms were and how much of an effect this had on the teacher, students, and me.  The more effort you put into making your classroom comfortable, personal, and safe, the more your students will appreciate it and the easier it will be to engage them (in general, of course)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Reflection&lt;/strong&gt; - it is super important at the end of each lesson/day to think back on what you taught, how you taught it, and how effective or not effective you think it went.  It doesn't take long, just a few minutes to jot down your thoughts in a notebook or in an electronic document.  It sounds obvious, but I have met countless teachers who I know do not think back for one second about their practice and it shows.  They continue to make the same mistakes over and over again and get frustrated with the students because they do not understand that if they just changed one little thing about their lesson things would go a lot better.  It is extremely helpful to get in to the practice of annotating lesson plans, i.e. going back and taking down little notes about what worked and what didn't.  That way, when you go to teach that same lesson in the future (which you undoubtedly will) you know what to modify. Reflection also helps to ground your perspective and is certainly something that can be done with other teachers.  Some of my most useful reflections have actually come from other teachers who have observed me or tried to teach one of my lesson plans.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Getting to know students&lt;/strong&gt; - honestly, one of the most important things you can do as a teacher is know your students.  Not only should you know their abilities in your subject, but also get to know them personally.  What sports do they play, what movies do they like, what music do they listen to, what makes them laugh, what makes them upset, etc.  Knowing all of this will directly make you a better teacher because it allows you to relate to your students and make them know that you actually care about them, and not just their performance.  I can't tell you how many times a student's entire disposition and performance in math has changed almost overnight because I started casually asking them about their day or their plans for the weekend or what type of music they were listening to.  It has probably been some time since you were in high school so it will definitely be a bit awkward interacting with your students in this way but I promise it will get much more comfortable as time goes on.  I would highly suggest that once you learn about your students more to incorporate their interests into word problems, tasks, exercises, etc.  &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Have fun!&lt;/strong&gt; - too often I see teachers get stuck in this rut of needing to make the most rigorous and challenging lessons and lose sight of the fact that these students are more or less required to be here and no matter what you do, they will not always want to listen or follow along with a lesson. I strongly encourage you to incorporate games, puzzles, challenges, videos, songs, drama, and anything else you can think of to make your lessons fun.  Obviously, you can't do these things day in and day out but making them a regular part of your lessons will pull in kids who you never thought you could pull in.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Classroom/behavior management&lt;/strong&gt; - Personally, I am under the belief (and much of my experiences show) that if you do all of the above things correctly, then behavior management is just something that will ever really be an issue; the students will be engaged because of your good planning, data-driven instruction, and classroom atmosphere and wont even have time to think of ways to be disruptive.  However, I know this is an ideal and classroom/behavior management will assuredly come up in your teacher education program.  As you will soon find out there are about a million different ways that teachers monitor student behavior and some of it works and some of it doesn't.  This will really be up to you to figure out exactly what works with you and your students.  The one thing I will tell you is that I have overwhelmingly found that &lt;strong&gt;compassion is much more effective than discipline&lt;/strong&gt;.  Other than that, it will just be trial and error for you to figure out a system that works for you.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Ok, I think that is mostly it. &quot;:)  I know it is a lot but hey, teaching high school is a huge responsibility and it is better to be aware of all of this before you jump right in and try it out.  I wish you the best of luck with your teaching and hope this information is helpful!&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="2074" LastEditDate="2015-06-11T20:23:23.887" LastActivityDate="2015-06-11T20:23:23.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8257" PostTypeId="1" AcceptedAnswerId="8260" CreationDate="2015-06-11T21:23:22.117" Score="8" ViewCount="794" Body="&lt;p&gt;I will be teaching a one-semester course on numerical methods at a liberal arts college.  The students will be primarily math, physics, and engineering majors.  Note that there is no computer science department at this institution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is my first time teaching such a course and I rarely use numerical methods in my research.  What are some of the most essential topics that should appear in the course?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to treat the course as a baby introduction to programming using Python.  There is no assumption that the students will have had previous programming experience.  The only prerequisite is Calculus II.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should probably include the following topics, as they are mentioned in the course catalog:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;round-off errors&lt;/li&gt;&#xA;&lt;li&gt;computer arithmetic with algorithm and convergence&lt;/li&gt;&#xA;&lt;li&gt;solutions of equations in one variable with polynomial approximation&lt;/li&gt;&#xA;&lt;li&gt;numerical differential equations&lt;/li&gt;&#xA;&lt;li&gt;linear systems of equations&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I’ve chosen Hamming’s &lt;em&gt;Numerical Methods for Scientists and Engineers&lt;/em&gt; as the text, as it is quite inexpensive and has excellent reviews.  It’s very thick though, and includes far more than I could cover in a single semester!&lt;/p&gt;&#xA;" OwnerUserId="5289" LastEditorUserId="3237" LastEditDate="2015-06-12T15:10:18.733" LastActivityDate="2015-06-16T14:39:17.480" Title="Important topics for first numerical methods course" Tags="|undergraduate-education|mathematical-pedagogy|course-design|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8258" PostTypeId="1" CreationDate="2015-06-11T22:30:30.130" Score="14" ViewCount="7530" Body="&lt;p&gt;I am an old man trying to learn math, starting off with prealgebra and need a good comprehensive book for it. The book should NOT contain annoying images like in most American textbooks or anything irritating, just serious math. It should be rigorous, comprehensive and above all; serious. &lt;/p&gt;&#xA;" OwnerUserId="5290" LastActivityDate="2021-07-17T08:47:04.610" Title="What is a good book to learn all of prealgebra?" Tags="|reference-request|textbooks|reference-books|prerequisites|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8259" PostTypeId="2" ParentId="8258" CreationDate="2015-06-12T00:42:45.210" Score="9" Body="&lt;p&gt;This California college has a no-frills &lt;span class=&quot;math-container&quot;&gt;$602$&lt;/span&gt;-page textbook&#xA;&lt;strike&gt;available for free PDF download&lt;/strike&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;College of the Redwoods Mathematics Prealgebra Textbook&lt;/em&gt;. 2012-13.&#xA;(&lt;a href=&quot;https://www.memoriapress.com/curriculum/math/college-of-the-redwoods-prealgebra-textbook/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;PublisherLink&lt;/em&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There &lt;em&gt;was&lt;/em&gt; a free website for download in 2015, but apparently no more.&#xA;Before its demise, I browsed through book.&lt;/p&gt;&#xA;&lt;p&gt;What &amp;quot;fluff&amp;quot; the textbook authors include is hardly fluff; rather, educational:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;         &#xA;&lt;img src=&quot;https://i.stack.imgur.com/yzP6i.png&quot; alt=&quot;Horus&quot; /&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;Note the sum &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}&#xA;= \frac{63}{64}&#xA;$$&lt;/span&gt;&#xA;is missing &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{64}$&lt;/span&gt; of Horus's eye.&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-12-30T01:28:22.347" LastActivityDate="2020-12-30T01:28:22.347" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8260" PostTypeId="2" ParentId="8257" CreationDate="2015-06-12T00:43:22.437" Score="3" Body="&lt;p&gt;Not in any order of importance (but probably as outline):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;introduction to programming languages (Python how does it work)&lt;/li&gt;&#xA;&lt;li&gt;modeling and use of data (types)&lt;/li&gt;&#xA;&lt;li&gt;Numerical evaluation (computable numbers, approximations and round off errors )&lt;/li&gt;&#xA;&lt;li&gt;Resolution of equations (Bisection, Newton Raphson, open and closed methods convergence criteria)&lt;/li&gt;&#xA;&lt;li&gt;Numerical integration ( Simpson and trapezoid rule)&lt;/li&gt;&#xA;&lt;li&gt;Numerical differentiation &lt;/li&gt;&#xA;&lt;li&gt;Matrix computations (determinant , stiffness)&lt;/li&gt;&#xA;&lt;li&gt;ODEs &lt;/li&gt;&#xA;&lt;li&gt;System of ODE&lt;/li&gt;&#xA;&lt;li&gt;PDE&#xA;&lt;ul&gt;&#xA;&lt;li&gt;individual topics and further reading (high performance computing, pandas...)&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3253" LastActivityDate="2015-06-12T00:43:22.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8261" PostTypeId="1" CreationDate="2015-06-12T03:20:27.047" Score="13" ViewCount="2038" Body="&lt;p&gt;&lt;em&gt;Note. I asked the question below on Math Stack Exchange (&lt;a href=&quot;https://math.stackexchange.com/questions/1307630/high-school-algebra-textbooks-for-gifted-students&quot;&gt;link&lt;/a&gt;), but didn't get a really satisfactory answer there, so I'm posting it here too.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for high school algebra/mathematics textbooks targeted at talented students, as preparation for fully rigorous calculus à la Spivak. I am interested in the best materials available in English, French, German or Hebrew.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, the book(s) should provide a comprehensive introduction to algebra at this level, starting from the most basic operations on polynomials. It should include necessary theory (e.g., Bezout's remainder theorem on polynomials, proof of the fundamental theorem of arithmetic, Euclid's algorithm, a more honest discussion of real numbers than usual, proofs of the properties of rational exponents, etc., and a general attitude that all statements are to be proved, with few exceptions). It should also have problems that range from exercises acquainting students with the basic algebraic manipulations on polynomials to much more difficult ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically, I am looking for something similar in spirit to a series of excellent Russian books by Vilenkin for students in so-called &quot;mathematical schools&quot; from grades 8 to 11, although I am only looking for the equivalent of the grade 8 and 9 books, which are at precalculus level. To give you an idea, here are a sample of typical problems from the grade-8 book. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Perform the indicated operations. $\frac{3p^2mq}{2a^2 b^2} \cdot \frac{3abc}{8x^2 y^2} : \frac{9a^2 b^2 c^3}{28pxy}$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove that when $a \ne 0$, the polynomial $x^{2n} + a^{2n}$ is divisible neither by $x + a$ nor by $x - a$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove that if $a + b + c = 0$, then $a^3 + b^3 + c^3 + 3(a + b)(a + c) (b + c) = 0$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove that if $a &amp;gt; 1$, then $a^4 + 4$ is a composite number.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove that if $n$ is relatively prime to $6$, then $n^2 - 1$ is divisible by 24.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simplify $\sqrt{36x^2}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simplify $\sqrt{12 + \sqrt{63}}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove that the difference of the roots of the equation $5x^2 -2(5a + 3)x + 5a^2 + 6a + 1 = 0$ does not depend on $a$. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Solve the inequality $|x - 6| \leq |x^2 - 5x + 2|$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;And here are the chapter titles for the grade 8 and 9 books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grade 8: Fractions. Polynomials. Divisibility; prime and composite numbers. Real numbers. Quadratic equations; systems of nonlinear equations; resolution of inequalities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grade 9: Elements of set theory. Functions. Powers and roots. Equations and inequalities, and systems thereof. Sequences. Elements of trigonometry. Elements of combinatorics and probability theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Broadly similar questions have been asked elsewhere, however the suggestions made there are not satisfactory for my purposes. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The English translations of Gelfand's books are good; however they are not a sufficiently broad introduction to high school algebra, and do not have enough material on computational technique. They are more in the nature of supplements to an ordinary textbook.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some 19th century books like Hall and Knight have been suggested. On conceptual material, these tend to be too old in language and outlook.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Basic Mathematics&lt;/em&gt; by Serge Lang seems more to dabble in various topics than to provide a thorough introduction to algebra.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am not inclined towards books with a very strong &quot;New Math&quot; orientation (1971-1983 France, for example). I don't think a student should need to understand the group of affine transformations of $\mathbb{R}$ to know what a line is.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Also, previous questions have perhaps focused implicitly on material in English. I have in mind a student who can also easily read French, German or Hebrew if something better can be found in those languages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit.&lt;/strong&gt; I'd like to clarify that I'm not asking for something identical to these books, just something as close as possible to their spirit. Fundamentally, this means: 1. It is a substitute for, rather than just a complement to, a regular school algebra textbook. 2. It is directed at the most able students. 3. It conveys the message that proofs and creative problem-solving are central to mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5291" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-06-20T13:14:14.623" Title="High-school level algebra textbooks for gifted students" Tags="|secondary-education|algebra|textbooks|gifted-students|problem-solving|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8262" PostTypeId="2" ParentId="8258" CreationDate="2015-06-12T03:46:07.637" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://artofproblemsolving.com/store/item/prealgebra&quot; rel=&quot;nofollow&quot;&gt;Art of Problem Solving has a pre-algebra text&lt;/a&gt; that gets glowing reviews. I would expect it to be complete and rigorous. &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-06-12T03:46:07.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8263" PostTypeId="2" ParentId="8258" CreationDate="2015-06-12T05:26:52.437" Score="4" Body="&lt;p&gt;If you don't mind something a bit old-fashioned, you can use C. V. Durell's arithmetic and algebra textbooks. These were standard in good schools in England at one time, and continued to be widely used into the fifties and sixties. They're concise and have good problems, with answers. The explanations on fractions are a bit shorter than one would like (I think because they're assumed to have been learned in elementary school), but the new subjects are explained reasonably well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find several of them online &lt;a href=&quot;http://www.knowledge-dojo.com/&quot; rel=&quot;nofollow&quot;&gt;here.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can read French, then some of the books by Lebossé and Hémery listed &lt;a href=&quot;http://manuelsanciens.blogspot.ca/p/maths-college-lycee-prepa.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; are good and downloadable online. Start with &quot;quatrième&quot; if you already know fractions well. Otherwise start with &quot;cinquième.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These books are from the fifties, but the presentation is modern. Few books I've encountered at that level are as serious about proofs (within reason for that stage of learning). The geometry sections are excellent - the French taught geometry in a much much more sensible way than the British or the Americans did (before all of them stopped teaching it anyway).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the tone of the books I've mentioned is serious, they are school books. If you prefer something more recent and specifically targeted at adults, &quot;Teach Yourself Mathematics: A Complete Introduction&quot; might fit the bill. It provides better explanations of why things are the way they are than do most texts of that variety.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A comprehensive set of grade 7/8 books from the 1970s which is mathematically sound is &lt;em&gt;School Mathematics I and II&lt;/em&gt; by Eicholz. However, they are not very concise and do have a lot of illustrations (though less than these days). The explanations are very complete for that level. It seems these books can be obtained cheaply from Abebooks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These books can be followed with &lt;em&gt;Algebra&lt;/em&gt; by Richard E Johnson, which meets your requirements perfectly except that it is for algebra rather than pre-algebra.&lt;/p&gt;&#xA;" OwnerUserId="5291" LastActivityDate="2015-06-12T05:26:52.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8264" PostTypeId="2" ParentId="8245" CreationDate="2015-06-12T08:22:34.843" Score="4" Body="&lt;p&gt;I can see several general answers to &quot;why study such concept or procedure?&quot;, let me discuss them in the case of writing an integer as a sum of two squares, and when it is possible to do so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The concept or procedure may make use of tools students ought to learn or practice.&lt;/em&gt; This is a good motivation for a teacher: set ourselves a nice goal to examplify the use of what have just been learned. This may not be enough to motivate student though, so let's consider other approaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The concept may be relevant to other domains than mathematics&lt;/em&gt; (&quot;real-life&quot; is a bit overrated in my opinion, hence the broader interpretation of applicability). As far as I know, this does not really apply here. I may be wrong, but I think it detrimental to try to force a concept into unrealistic or far-fetched applications. Doing so indeed enforces the idea that applications outside mathematics is the sole purpose of any relevant mathematical concept, and at the same time shows that the concept at hand does not really meet this expectation. Far-fetched applications are ok as long as they are presented as such, and not taken as a justification for the concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The concept may be relevant to other mathematical domains&lt;/em&gt;. Here, the geometry comes to mind (Pythagorean theorem), as was pointed out in other answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The concept my be relevant to other questions in the same mathematical domain&lt;/em&gt;, and this a key point here. I think it important to fit the question into a more general framework, here diophantine equations: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given an polynomial equation (or system of equations) with integer coefficients, when are there integer solutions? How many of them? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This theme is huge, and covers both very simple concepts that students already encountered at this stage, up to the most sophisticated arithmetic geometry. For example, the equation $ax=b$ in the unknown $x$ with integer coefficients $a,b$ has a solution precisely when $a$ divides $b$. The concept we are discussing is simply to decide for which parameter $n$ does the equation $x^2+y^2=n$ have a solution. Very similar looking equations are famous: e.g. $x^3+y^3=z^3$, or the more general Fermat equation $x^n+y^n=z^n$  which have no non-trivial integer solutions for $n&amp;gt;2$ thanks to Wiles' theorem. Finding &lt;em&gt;all&lt;/em&gt; solutions of $x^2+y^2=z^2$ is a nice problem (well-understood for a long time), elliptic curves and their group structure come next, etc. There is an endless stream of beautiful mathematics behind this, and  it would be a shame not to use it to motivate student (e.g. showing them a similar-looking problem to which no one in the world knows the answer to).&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2015-06-12T09:24:21.743" LastActivityDate="2015-06-12T09:24:21.743" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8265" PostTypeId="1" CreationDate="2015-06-12T13:46:07.197" Score="8" ViewCount="354" Body="&lt;p&gt;I read about the &lt;em&gt;Theory of Quantity&lt;/em&gt; in the booklet &lt;em&gt;Principles of Mathematics Education&lt;/em&gt; written by Kô Ginbayashi in 1984.  (A summary of it is &lt;a href=&quot;https://joelreyesnoche.wordpress.com/2015/05/27/theory-of-quantity/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.)  Instead of using pure numbers as the basic concept, it uses quantities (numeral values and units).  So, for example, instead of teaching &amp;quot;1 + 1 = 2,&amp;quot; you would teach &amp;quot;1 apple + 1 apple = 2 apples,&amp;quot; or &amp;quot;1 inch + 1 inch = 2 inches.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I think this makes a lot of sense.  Note that only quantities with the same units can be added or subtracted.  Quantities that are multiplied or divided yield new quantities.  For example, &amp;quot;1 m / 1 s = 1 m/s,&amp;quot; that is, a length divided by a unit of time is a speed.&lt;/p&gt;&#xA;&lt;p&gt;This leads to ratio and proportion as the next basic idea.  From here, it is easy to find similar relationships in&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;differential calculus (&lt;span class=&quot;math-container&quot;&gt;$f'(k)=dy/dx$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are real numbers and &lt;span class=&quot;math-container&quot;&gt;$f'(k)$&lt;/span&gt; is the differential coefficient)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;linear algebra (&lt;span class=&quot;math-container&quot;&gt;$\mathbf{y}=\mathbf{Ax}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\mathbf{x}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{y}$&lt;/span&gt; are vectors and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{A}$&lt;/span&gt; is the representative matrix)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;vector analysis (&lt;span class=&quot;math-container&quot;&gt;$\mathbf{J}=\partial\mathbf{y}/\partial\mathbf{x}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\mathbf{x}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{y}$&lt;/span&gt; are vector-valued functions and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{J}$&lt;/span&gt; is the Jacobian matrix.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Note that these are applied mathematics fields.&lt;/p&gt;&#xA;&lt;p&gt;Are you familiar with any other book that uses quantity (instead of pure number) as the basic concept?&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-10-06T02:38:48.753" Title="Quantity as the basic concept for applied mathematics" Tags="|reference-request|applied-mathematics|quantity|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8266" PostTypeId="2" ParentId="5630" CreationDate="2015-06-12T18:59:05.013" Score="5" Body="&lt;p&gt;I'm going to suggest the answer is &quot;none.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the app is cute, technology has offered increasing levels of 'cheating' available to students for some time now. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Desmos, for example, can graph an equation instantly. Yet, algebra students still need to understand the process to be able to graph a given equation when test time comes. There are even sites that will give you a derivative or integral when given an equation formatted properly.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The result in my opinion, is that teachers need to be aware of what's out there, and diligent in not letting students have their phones out during tests or even during class. Aside from that, this app is not a game changer. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-06-12T18:59:05.013" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8268" PostTypeId="1" CreationDate="2015-06-14T05:43:13.750" Score="5" ViewCount="2519" Body="&lt;p&gt;I'm going to propose some research projects to my students, addressing issues about water from a mathematical point of view .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students have just finished 7th grade and are familiar with elementary algebra, geometry and probability. They also have learned basic Pascal programming during last year, so these projects can benefit from computer programming or simulation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1. What topics do you suggest?&lt;br&gt;&#xA; 2. What are good references (books,&#xA;    articles, websites) which can be used to define the titles?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="511" LastEditDate="2015-06-16T21:36:44.440" LastActivityDate="2019-01-01T15:30:07.823" Title="Math Research Projects on Water" Tags="|secondary-education|" AnswerCount="3" CommentCount="6" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8269" PostTypeId="2" ParentId="8241" CreationDate="2015-06-14T06:06:24.850" Score="5" Body="&lt;p&gt;I'm skeptical Knuth's scheme will be much of an improvement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the problem with teaching any kind of rigorous definition of limits or derivatives to the vast majority of North American students isn't going to go away by streamlining the proofs a bit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem isn't the technical difficulty of the proofs so much as the idea of proof itself, and even the idea of working from precise definitions. Students have nothing that prepares them for this in their background, because from elementary school all through high school the idea of justifying statements is treated as unimportant. This is true even when the justification would be simple, such as for the rule $\log AB = \log A + \log B$. I've taught from textbooks where the authors ease their consciences by having a page titled &quot;Proofs in Mathematics&quot; at the end of each chapter, as if mathematics without proofs could have any meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the technical merits of having a long discussion of $O$ notation as a prelude, let me first say that Knuth misstates things slightly. $O(f(x))$ would have to be defined as a function that is $CA(f(x))$ for some constant $C$ &lt;em&gt;on some $x$-interval around the point of interest&lt;/em&gt;. Perhaps this is what he means when he later says &quot;for $\epsilon$ sufficiently small.&quot; In any case, that introduces the dependence on the variable $x$, which makes this concept not much easier than defining the statement $\lim_{h \to 0} g(h) = 0$ (which itself leads directly to the definition of finite limits in general, by the condition $\lim_{h \to 0} [g(h)- l] = 0$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And even if one were to present rigorous proofs of the laws on strong derivatives, is this really better than the traditional proofs using the limit laws? Students always find the limit laws very plausible, and once those are accepted, then the proofs of the differentiation laws become accessible, and can be particularly convincing if you write $\Delta x$ for $x - a$, $y$ for $f(x)$ and $\Delta y$ for $f(x) - f(a)$. The proofs of the differentiation laws might become more rigorous in Knuth's scheme, but I am inclined to believe that they would also be made considerably less transparent. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, presenting proofs that are meaningful to students &lt;em&gt;at their level of understanding&lt;/em&gt; will do more to foster a healthy respect for the idea of proof than will giving proofs that are technically irreproachable but too demanding technically. I also believe the few students who could handle the $O$-proofs would probably manage with the traditional rigorous approach anyway. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there is any advantage to Knuth's method, it might be that of avoiding discussion of limits early in a calculus course, when you would otherwise bore students with things that seem pointless to them, like $\lim_{x \to 3} (x^2 + 5x) = 24$. You derive these results at first using the limit laws, though every example they see at first is obtained by simply substituting a value of $x$ in the expression (perhaps after simplification).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But limits need to be dealt with eventually; statements such as $\lim_{x \to 0} \frac{\sin x}{x} = 1$ or $\lim_{x \to +\infty} \frac{\ln x}{x} = 0$ are intrinsically interesting and necessary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real problem with limits is how to teach them in such a way that students will understand from the start that there's more to them than just substituting a value in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is probably to de-emphasize limits initially by treating them very briefly, with few exercises on them, developing only enough technical proficiency to accurately compute derivatives of rational functions, computed as limits of the form $\lim_{h \to 0} \frac{1}{h}[R(x+h) - R(x)]$, and without requiring much theory to be understood. Then return to limits when it is time to talk about less trivial ones, after they have already seen the usefulness of the concept demonstrated in the theory of derivatives. The meaning of the limit laws becomes much easier to grasp when they are used in a way that is &lt;em&gt;not&lt;/em&gt; obvious. Continuity can be postponed until this point as well.&lt;/p&gt;&#xA;" OwnerUserId="5291" LastEditorUserId="5291" LastEditDate="2015-06-14T17:11:44.503" LastActivityDate="2015-06-14T17:11:44.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8270" PostTypeId="2" ParentId="7261" CreationDate="2015-06-14T10:45:16.933" Score="6" Body="&lt;p&gt;I use the term &lt;em&gt;skills&lt;/em&gt; to refer to &lt;strong&gt;procedural knowledge&lt;/strong&gt; and the term &lt;em&gt;understanding&lt;/em&gt; to refer to &lt;strong&gt;conceptual knowledge&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is &lt;em&gt;a lot&lt;/em&gt; of literature about this.  One good starting place is &lt;em&gt;Conceptual and Procedural Knowledge: The Case of Mathematics&lt;/em&gt; by James Hiebert (Hillsdale, NJ: Lawrence Erlbaum Associates, 1986).  It contains ten chapters by different authors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chapter 1 of this book (by James Hiebert and Patricia Lefevre) define &lt;em&gt;procedural knowledge&lt;/em&gt; as &quot;made up of two distinct parts.  One part is composed of the formal language, or symbol representation system, of mathematics.  The other part consists of the algorithms, or rules, for completing mathematical tasks&quot; (p. 6).  They define &lt;em&gt;conceptual knowledge&lt;/em&gt; as &quot;knowledge that is rich in relationships&quot; (p. 3).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me show how I understand this distinction.  Let's say the domain is the comparison of fractions.  When I say that $\frac{4}{7}&amp;gt;\frac{3}{7}$ because of two fractions with the same denominator, the one with the larger numerator is larger, or when I say that $\frac{3}{7}&amp;gt;\frac{3}{8}$ because of two fractions with the same numerator, the one with the smaller denominator is larger, I am using conceptual knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But when I say that $\frac{4}{7}&amp;lt;\frac{5}{8}$ because I used the standard procedure of multiplying both sides by the product of the denominators ($7\cdot 8=56$) and got $32&amp;lt;35$, I am using procedural knowledge.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2015-06-14T11:17:58.277" LastActivityDate="2015-06-14T11:17:58.277" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8271" PostTypeId="2" ParentId="5715" CreationDate="2015-06-14T14:05:26.990" Score="4" Body="&lt;p&gt;Those are all the same kind of continuity:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Given a subset $S \subseteq \mathbb{R}$ and a function $f: S \to \mathbb{R}$, we say $f$ is &lt;em&gt;continuous at a point&lt;/em&gt; $x \in S$ iff for each $\varepsilon &amp;gt; 0$, there exists $\delta &amp;gt; 0$ such that, for all $s \in S$ with $0 &amp;lt; \lvert x - s \rvert &amp;lt; \delta$, we have $\lvert f(x) - f(s) \rvert &amp;lt; \epsilon$.&lt;/li&gt;&#xA;&lt;li&gt;We say $f$ is &lt;em&gt;continuous&lt;/em&gt; iff $f$ is continuous at each point $x \in S$.&lt;/li&gt;&#xA;&lt;li&gt;More generally, given a subset $T \subseteq S$, we say $f$ is &lt;em&gt;continuous on $T$&lt;/em&gt; iff $f$ is continuous at each point $x \in T$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There's no need to give a different definition for &quot;continuous at an endpoint&quot;, because it's already encapsulated in the above definitions; any points outside the domain are irrelevant.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2015-06-14T14:05:26.990" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8272" PostTypeId="2" ParentId="8268" CreationDate="2015-06-14T17:42:18.180" Score="4" Body="&lt;p&gt;Three possible topics:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) &lt;em&gt;Tides&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not easy to understand (it awaited Newton), and&#xA;somewhat intricate. But you could present some&#xA;mathematical justification, using, e.g., the discussion &#xA;&lt;a href=&quot;http://mb-soft.com/public/tides.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&#xA;Once you understand tides, you should be able to convince them&#xA;that their own weight varies in concert with the moon's tidal pull,&#xA;and maybe calculate that variation (~&lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; mg),&#xA;from &#xA;&lt;span class=&quot;math-container&quot;&gt;$$F = G \, \frac{m_1 m_2}{r^2}$$&lt;/span&gt;&#xA;as a project.&#xA;Make that a goal: Understand tides well enough to realize our&#xA;weight changes in concert with the tides; and compute that (miniscule) change.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/Gm0gv.jpg&quot; alt=&quot;image&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://pages.uoregon.edu/jimbrau/astr121/Notes/Chapter7.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;a Jim Brau page&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) &lt;em&gt;Evaporation&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The evaporation rate (at fixed temperature, fixed pressure) is&#xA;proportional to surface area. At room temperature, with no wind,&#xA;normal humidity,&#xA;water evaporates at (very!) roughly&#xA;&lt;span class=&quot;math-container&quot;&gt;$4 \times 10^{-5} \, \mathrm{kg/s/m}^2$&lt;/span&gt;&#xA;(kilograms of water per second per meter squared of surface area).&#xA;From this one can calculate how long it would take a glass of water to&#xA;evaporate, or a swimming pool, or a lake:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://www.kidzone.ws/images-changed/water/steam.jpg&quot; alt=&quot;&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://www.kidzone.ws/water/&quot; rel=&quot;nofollow noreferrer&quot;&gt;KidZone&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Those calculations could form a project: Understand the consequences&#xA;of evaporation time-linear w.r.t. surface-area, and compute several &#xA;common instances.&#xA;(Spherical raindrop evaporation would be quite challenging for 7&lt;sup&gt;th&lt;/sup&gt;-graders.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) &lt;em&gt;Waves&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally 7&lt;sup&gt;th&lt;/sup&gt;-graders (in the U.S.) have no exposure to trigonometry,&#xA;but fortunately ocean waves are not sine waves, but instead&#xA;better approximated by&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Trochoid&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;trochoids&lt;/em&gt;&lt;/a&gt;,&#xA;for physical reasons&#xA;(which one can sense while ocean-bathing):&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/ZpoMr.gif&quot; alt=&quot;Waves&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://hyperphysics.phy-astr.gsu.edu/hbase/waves/imgwav/circonwave.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;HyperPhysics&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;I can imagine a project that explores trochoids&#xA;both geometrically and through computational simulations.&#xA;There could also be a nice experimental component:&#xA;measure the speed of water waves (&lt;a href=&quot;https://en.wikipedia.org/wiki/Phase_velocity&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;celerity&lt;/em&gt;&lt;/a&gt;),&#xA;and predict how long it would take an ocean wave to&#xA;travel from coast &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to coast &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;&amp;mdash;from Cairo to Cyprus,&#xA;or whichever geographical locations are most salient to the students.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="8571" LastEditDate="2018-12-31T03:28:35.893" LastActivityDate="2018-12-31T03:28:35.893" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8274" PostTypeId="1" CreationDate="2015-06-14T20:21:46.143" Score="12" ViewCount="322" Body="&lt;p&gt;In 2010-11, 56% of US public school teachers had a master’s or higher degree, while only 43% of US private school teachers had a master’s or higher degree (43 percent) (&lt;a href=&quot;http://nces.ed.gov/fastfacts/display.asp?id=28&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 2014, only 13% of Singapore school teachers had a master's or higher degree (4233 Master's + 104 PhD, out of 32,779 teachers) (&lt;a href=&quot;http://www.moe.gov.sg/education/education-statistics-digest/files/esd-2014.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;source: p. 15 of this document&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on these 3 data points and my possibly-erroneous presumption that Singapore schools &gt; US private schools &gt; US public schools, it would seem that there is a negative correlation between teaching effectiveness and whether teachers have master's or PhD degrees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: What does the current research actually say? Is there a positive, negative, or no correlation between a teacher's academic qualifications and his/her teaching effectiveness?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/426/why-does-high-school-teaching-in-the-usa-require-a-teaching-certificate-while-co/&quot;&gt;Why does high school teaching in the USA require a teaching certificate while college/university teaching does not?&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/130/examples-why-university-education-is-important-for-future-high-school-teachers&quot;&gt;Examples why university education is important for future high school teachers&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user378" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-06-18T15:32:29.753" Title="Is there a correlation between academic qualifications and teaching effectiveness?" Tags="|teacher-training|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="8275" PostTypeId="1" CreationDate="2015-06-14T22:10:44.990" Score="7" ViewCount="198" Body="&lt;p&gt;I came across this question (picture below, the question where the drop-down bar is highlighted) on edulastic.com. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer, apparently, was that &quot;The standard deviation is a good measure of the spread of the distribution for &lt;strong&gt;Freshmen&lt;/strong&gt;.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the rationale for this? Or does this question simply not make sense?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/BjMPr.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user378" LastActivityDate="2015-06-20T16:33:11.380" Title="Does it make sense to speak of whether the standard deviation is a &quot;good measure&quot; of some particular distribution?" Tags="|statistics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8276" PostTypeId="2" ParentId="8275" CreationDate="2015-06-14T22:43:31.683" Score="2" Body="&lt;p&gt;Take the data for the seniors. Calculate the median and standard deviation. These two number make more sense when talking about data that's scattered close to a normal distribution, a bell curve. A bi-modal distribution is valid, even interesting, but the words standard deviation don't quite apply the same way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, it's a bad question. Better would be to talk about whether data fits a normal distribution, or the degree to which it does. But using words like &quot;good&quot; in math problems seems problematic to me. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-06-14T22:43:31.683" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8277" PostTypeId="5" CreationDate="2015-06-15T03:13:39.673" Score="0" Body="&lt;p&gt;Include this tag for questions involving quantities (numerical values and units of measurement such as &quot;3 apples&quot; or &quot;3 inches&quot;) as opposed to pure numbers.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2016-03-22T19:26:49.073" LastActivityDate="2016-03-22T19:26:49.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8278" PostTypeId="4" CreationDate="2015-06-15T03:13:39.673" Score="0" Body="For questions involving numeral values and units" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2015-06-15T18:41:00.647" LastActivityDate="2015-06-15T18:41:00.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8279" PostTypeId="2" ParentId="8275" CreationDate="2015-06-15T12:19:17.263" Score="4" Body="&lt;p&gt;People do speak that way, and therefore it has some meaning, even if poorly and indirectly expressed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If someone says &quot;the standard deviation is a good measure of the spread of the distribution for Freshmen&quot;, I interpret that as roughly:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;A normal distribution gives a good summary of the spread of the distribution for Freshmen.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would prefer to talk this way instead, and I would prefer talk of medians and quartiles even more.  Meanwhile I can accept the way that people talk now.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2015-06-15T12:19:17.263" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8281" PostTypeId="2" ParentId="8274" CreationDate="2015-06-16T01:25:43.730" Score="-3" Body="&lt;p&gt;edit: My original reply was too long and didn't communicate my point clearly enough.  This topic frustrates me because many people (including colleagues of mine) think there's a tension between academic credentials and effective teaching, as if every teacher with a math degree is some snooty, uninteractive, unapproachable scholar type.  One would think that familiarity with higher level math concepts implies that you'd be more comfortable with your mathematical explanations.  It's HUGELY helpful for my ability to explain things clearly in, say, a calculus course, now that I've learned the nuances behind concepts like limits and series (concepts that are widely misunderstood even by students who have a functional knowledge of them).  Even lower-level courses teach deeper concepts like asymptotes where having less formal mathematical education will affect the quality of your teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Good teaching&quot; isn't just a dichotomy of &quot;can they explain things clearly and do they know their stuff?&quot;, since the clarity of your explanations is non-trivially affected by the depth of your mathematical knowledge.  Also, several other factors are considered when someone determines that a teacher of theirs is &quot;good&quot; or &quot;bad&quot;, like whether the teacher is approachable or whether their tests are fair.  I'd like to see a plausible reason why teachers with more academic qualifications would be worse (or fewer/better) instead of just correlations in the original post.&lt;/p&gt;&#xA;" OwnerUserId="5141" LastEditorUserId="5141" LastEditDate="2015-06-18T15:32:29.753" LastActivityDate="2015-06-18T15:32:29.753" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8282" PostTypeId="2" ParentId="8257" CreationDate="2015-06-16T14:39:17.480" Score="3" Body="&lt;p&gt;&lt;strong&gt;Introduction to Approximation:&lt;/strong&gt; Approximation, Error. Decimal rounding. Sampling. Differentiation, Integration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Numerical Algorithms&lt;/strong&gt; Bisection Method. Newton Raphson Method. Gaussian Elimination with Partial Pivoting. Trapezoidal Rule. Euler Method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Taylor Series&lt;/strong&gt; Taylor Series in several variables. Propogation of Errors. Differential Equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Curve Fitting&lt;/strong&gt; Polynomial Interpolation. Least Squares. Weierstrass Approximation Theorem. Boundary Layer Regularisation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Normal Approximation&lt;/strong&gt; Histograms. Binomial and Poisson Distributions.&lt;/p&gt;&#xA;" OwnerUserId="584" LastActivityDate="2015-06-16T14:39:17.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8283" PostTypeId="1" AcceptedAnswerId="8288" CreationDate="2015-06-16T20:49:05.657" Score="20" ViewCount="535" Body="&lt;p&gt;I would like to spend a day with my students analyzing mathematical writing. One way I might accomplish this is to offer multiple proofs (some good, some poor) of the same simple statement and ask them to critique the writing. Is there a good reference for such proofs? It would be ideal if the proofs were a part of a textbook that itself contained some commentary, just in case we need a little help getting started.&lt;/p&gt;&#xA;" OwnerUserId="222" LastEditorUserId="222" LastEditDate="2015-06-19T20:02:51.643" LastActivityDate="2015-06-19T20:02:51.643" Title="Critiquing Proof Style During Class" Tags="|reference-request|proofs|writing|critical-thinking|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8284" PostTypeId="2" ParentId="8274" CreationDate="2015-06-17T04:17:09.953" Score="11" Body="&lt;p&gt;&lt;a href=&quot;http://files.eric.ed.gov/fulltext/ED400237.pdf&quot;&gt;This heavily-cited paper&lt;/a&gt; examines data from the &lt;em&gt;National Educational Longitudinal Study of 1988&lt;/em&gt; and concludes that &quot;teachers who are certified in mathematics, and those with Bachelors or Masters degrees in math, are identified with higher test scores.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More specifically, the study finds that &quot;the results reported... show that a teacher with a BA in math or an MA in math has a statistically significant positive impact on students' achievement, while a teacher with a non-math BA or an MA has a negative impact on students' math achievement.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am also curious if the strengthening of certification and Masters degree requirements (subsequent to this study) has improved student achievement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Full cite:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Goldhaber, Dan D., and Dominic J. Brewer. &quot;Why Don't Schools and Teachers Seem to Matter? Assessing the Impact of Unobservables on Educational Productivity.&quot; (1996).&lt;/p&gt;&#xA;" OwnerUserId="182" LastEditorUserId="182" LastEditDate="2015-06-17T13:36:51.603" LastActivityDate="2015-06-17T13:36:51.603" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8287" PostTypeId="2" ParentId="8283" CreationDate="2015-06-17T18:28:30.067" Score="6" Body="&lt;p&gt;There are quite a few textbooks that have critiquing sample proofs as exercises. Here are three I know of:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Foundations of Mathematics by Thomas Q Sibley&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This past year I used the Solow text in my &quot;Intro to Proofs&quot; course because it specifically approaches the logic and proof writing from a different angle than a standard math text. About 80% of my class time was spent critiquing proofs. Where did they all come from? The students of course. Since Solow focuses on writing and proof logic more than mathematical content, I pulled a bunch of standard theorems and propositions (and false statements) from other traditional texts and students signed up for problems individually. They then submitted their proof/counterexample to be critiqued anonymously in front of (and by) the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my first piece of advice to you is that one day will not be enough. The first day was almost comical in how accepting the students were. They were so used to being spoon-fed perfect proofs that they just assumed everything was right. After about 8-10 classes, the bright students really caught on and became very good at critiquing (and, of course, at submitting better work themselves). It took the middle-level students about twice as long (and of course, some students never really got it). So it took about half of a 15-week semester doing this to get to the point where the students were appropriately skeptical of a proof's logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that we tried to work on issues like voice, phrasing, exposition, etc. That was a lot harder (for me to teach and for them to get). At some point there's an amount of personal style that comes into the writing and, of course, not every student resonated with my style (which, I had to remind myself constantly, is okay).&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-06-17T18:28:30.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8288" PostTypeId="2" ParentId="8283" CreationDate="2015-06-17T22:05:08.747" Score="12" Body="&lt;p&gt;I found this an effective teaching technique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I take a topic they know, and find a Wikipedia article discussing that topic.&#xA;If you are specifically focused on proofs, as opposed to &#xA;more generic descriptions,&#xA;you can find many proofs in Wikipedia.&#xA;E.g., of&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sperner&amp;#39;s_lemma#Proof&quot;&gt;Sperner's Lemma&lt;/a&gt;,&#xA;or &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%27s_theorem#Euclid.27s_proof&quot;&gt;Euclid's proof of $\infty$ # primes&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I project the text in class, and have the students interactively criticize a paragraph&#xA;sentence-by-sentence and suggest improvements.&#xA;And here is why they attend closely: I (or one of the students) actually edits&#xA;the Wikipedia article directly. One has to be registered to edit Wikipedia,&#xA;but this is easily accomplished.&#xA;Changing Wikipedia makes it real; and it's satisfying to &#xA;even $\epsilon$-improve the world.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-06-17T22:05:08.747" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8289" PostTypeId="2" ParentId="8283" CreationDate="2015-06-18T01:08:05.827" Score="5" Body="&lt;p&gt;Go to &lt;a href=&quot;https://math.stackexchange.com/&quot;&gt;Mathematics Stack Exchange&lt;/a&gt; or &lt;a href=&quot;http://mathoverflow.net&quot;&gt;MathOverflow&lt;/a&gt;.  There are many questions there looking for proofs and there are many different answers, some good, some bad (some are even wrong).  Ask your students to criticize the proofs (Are the proofs they consider good the ones that are highly upvoted?).  The commentary that you want is sometimes there (as comments).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aside from looking at the quality of the answers, also look at the quality of the questions.  Some posts go directly to the question without defining the terms, and you will see users commenting on how the question is unanswerable and voting to close these as &lt;em&gt;unclear what you're asking&lt;/em&gt;.  (In my opinion, good mathematical writing is clear and succinct, with all important terms defined and no unnecessary details included.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I'm assuming you have access to the internet in your class.  If not, then just print out the posts of interest beforehand.)&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-06-18T01:08:05.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8290" PostTypeId="1" CreationDate="2015-06-18T01:40:23.667" Score="12" ViewCount="669" Body="&lt;p&gt;I will be teaching a third-year introductory course on Set Theory and Logic soon and was hoping to get advice from this community.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would rate my students' proof abilities as weak and was hoping to impart on them a more concrete notion of rigour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the most important (and interesting) core-concepts that these students should learn?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are good (paid but preferably free) resources?&lt;/p&gt;&#xA;" OwnerUserId="5310" LastEditorUserId="143" LastEditDate="2015-06-21T12:54:49.643" LastActivityDate="2015-06-21T19:18:27.890" Title="What to teach in Set Theory &amp; Logic Course" Tags="|course-design|logic|set-theory|" AnswerCount="2" CommentCount="9" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8291" PostTypeId="1" AcceptedAnswerId="8294" CreationDate="2015-06-18T15:44:05.283" Score="6" ViewCount="295" Body="&lt;p&gt;Are there any existing sources for topics or projects which are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Suitable for first and second year undergraduates taking introductory math classes (possibly with some prompting or a couple pages of background to point in the right direction).&lt;/li&gt;&#xA;&lt;li&gt;Have sufficient low-hanging fruit that a group of 3-4 students will typically discover something meaningful within 3 or so hours of group work.&lt;/li&gt;&#xA;&lt;li&gt;Are open ended enough that a group won't run out of things to find.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The second condition seems to be the tricky one; with more time, students can spend a long time messing around with data collection and hunting for patterns and finding some false starts.  This is a relatively short program (45 min a week for 4 weeks), so we need things which are a bit more directed.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-06-19T23:26:34.183" Title="Topics for Discovery-based Projects" Tags="|undergraduate-education|problem-design|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8292" PostTypeId="2" ParentId="8291" CreationDate="2015-06-18T17:06:39.507" Score="3" Body="&lt;p&gt;You could pick and choose from problem books.  &quot;Elementary Number Theory, A problem-oriented approach&quot; by Joe Roberts is essentially a theme-arranged book of questions, with a little exposition and an answer section.  This has the benefit that you can predict some of the discoveries, and gauge how long they will take.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also try something that draws connections between various parts of mathematics.  I am looking at additive permutations which seems to have ties to group theory (orthomorphisms) and combinatorics (Langford sequences and transversals of certain Latin squares).  Just playing with Skolem sequences (a variant of Langford) will give some opportunities for (re-)discovery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Not Showing My Connections Yet&quot; Paseman, 2015.06.18&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-06-18T17:06:39.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8293" PostTypeId="1" CreationDate="2015-06-19T02:25:36.630" Score="8" ViewCount="395" Body="&lt;p&gt;The usual way of writing the &lt;a href=&quot;https://en.wikipedia.org/wiki/Product_rule&quot; rel=&quot;nofollow&quot;&gt;product rule&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Quotient_rule&quot; rel=&quot;nofollow&quot;&gt;quotient rule&lt;/a&gt; in differentiation is&#xA;$$(fg)'=f'g+fg'$$&#xA;$$\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}\quad\text{where}\quad g\ne 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few years ago, during a mathematics conference I attended in the Philippines, a foreign speaker presented a &quot;more symmetric&quot; way of writing these rules.  Unfortunately, I have forgotten what he wrote and an internet search did not yield it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been playing around and I've found the following &quot;nicer&quot; presentation:&#xA;$$(fg)'=fg\left(\frac{f'}{f}+\frac{g'}{g}\right)\quad\text{where}\quad f,g\ne 0$$&#xA;$$\left(\frac{f}{g}\right)'=\frac{f}{g}\left(\frac{f'}{f}-\frac{g'}{g}\right)\quad\text{where}\quad f,g\ne 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know if this is the version I saw a few years ago, but I'm pretty sure I'm not the first to think of this.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What textbooks or lecture notes do you know that use this form for the product rule and the quotient rule in differentiation?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Edit&lt;/em&gt;:  As was pointed out in the comments, the &quot;symmetric&quot; version only works when $f$ and $g$ are nonzero, so it is not exactly the same as the &quot;original&quot; version.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2015-06-19T23:53:55.693" LastActivityDate="2016-08-07T01:10:01.457" Title="Symmetric version of product and quotient differentiation rules" Tags="|calculus|reference-request|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="8294" PostTypeId="2" ParentId="8291" CreationDate="2015-06-19T03:16:47.977" Score="4" Body="&lt;p&gt;Pythagorean triples may be open-ended enough. There are certainly challenges enough, and I think there is some low-hanging fruit. Depends on how willing to persist your students are. It's very algebraic, and may need a math circle environment, where the leader can encourage them to keep trying. (Also, the typical approach, with a = m^2 - n^2, b = 2mn, c = m^2 + n^2, is not necessary. There are more natural ways to find patterns.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analyzing the game of Spot It is an interesting problem. I've &lt;a href=&quot;http://mathmamawrites.blogspot.com/2012/01/math-adventures-thinking-about-spot-it.html&quot; rel=&quot;nofollow&quot;&gt;posted about using it&lt;/a&gt; &lt;a href=&quot;http://mathmamawrites.blogspot.com/2012/07/lotsa-links-tanton-busking-cat-in.html&quot; rel=&quot;nofollow&quot;&gt;many times on my blog&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are lots of interesting problems on the &lt;a href=&quot;http://jrmf.org/problems.php&quot; rel=&quot;nofollow&quot;&gt;Julia Robinson Mathematics Festival&lt;/a&gt; site. I think many of them would work for your purposes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also enjoy &lt;a href=&quot;http://www.geometer.org/mathcircles&quot; rel=&quot;nofollow&quot;&gt;Conway's Rational Tangles&lt;/a&gt;, available on Tom Davis' site (under Miscellaneous), along with lots of other possibly great stuff. &lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2015-06-19T17:49:49.720" LastActivityDate="2015-06-19T17:49:49.720" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8295" PostTypeId="2" ParentId="8291" CreationDate="2015-06-19T09:00:48.810" Score="3" Body="&lt;p&gt;I would probably try simply constructed, yet far reaching sequences like Fibonacci, or Colatz. There are many things you can &quot;discover&quot; on your own and some really challenging conjectures if you go deeper.&#xA;Another idea might be to start here: &lt;a href=&quot;https://projecteuler.net/&quot; rel=&quot;nofollow&quot;&gt;https://projecteuler.net/&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2015-06-19T09:00:48.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8296" PostTypeId="2" ParentId="8293" CreationDate="2015-06-19T14:50:48.357" Score="6" Body="&lt;p&gt;Just to elaborate on James Cook's observation in the comments, here is how to obtain these formulas using logarithmic differentiation.  I have made the post CW since it does not really answer the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$&#xA;\begin{align*}&#xA;P(x) &amp;amp;= f(x)g(x)\\&#xA;\log(P(x)) &amp;amp;= \log(f(x))+\log(g(x))\\&#xA;\frac{d}{dx} \log(P(x)) &amp;amp;= \frac{d}{dx}\log(f(x))+\frac{d}{dx}\log(g(x))\\&#xA;\frac{P'(x)}{P(x)} &amp;amp;= \frac{f'(x)}{f(x)}+\frac{g'(x)}{g(x)}\\&#xA;P'(x) &amp;amp;= P(x)(\frac{f'(x)}{f(x)}+\frac{g'(x)}{g(x)})\\&#xA;\frac{dP}{dx} &amp;amp;= f(x)g(x)(\frac{f'(x)}{f(x)}+\frac{g'(x)}{g(x)})\\&#xA;\end{align*}&#xA;$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quotient rule is derived similarly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not thought about logarithmic differentiation in a while.  I am a little worried about domain issues.  I guess just wrapping the initial equation in absolute value signs resolve these, but this still does not cover zeros of $f$ and $g$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, zeros are one reason to avoid this form.  For example, the usual product rule applies just as well to $f(x)=0$, $g(x) = \sin(x)$, while the modified form does not.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2015-06-19T14:50:48.357" CommentCount="4" CommunityOwnedDate="2015-06-19T14:50:48.357" ContentLicense="CC BY-SA 3.0" />
  <row Id="8298" PostTypeId="2" ParentId="8291" CreationDate="2015-06-19T23:26:34.183" Score="4" Body="&lt;p&gt;&lt;em&gt;Art Gallery Theorems&lt;/em&gt; are an accessible topic with many unsolved problems.&#xA;Here is one:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How many $2$-transmitters are&#xA;  sometimes necessary, and how many always suffice, to cover a polygon of $n$ vertices?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A $2$-transmitter is a point that whose broadcasts/vision can penetrate $2$ walls.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;J. O'Rourke. &quot;Computational Geometry Column 52&quot;&#xA;  &lt;em&gt;SIGACT News&lt;/em&gt;, Vol. 43, No. 1, Mar. 2012.&#xA;  (&lt;a href=&quot;http://cs.smith.edu/~orourke/papers.php#304&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.)&#xA;  (&lt;a href=&quot;http://cs.smith.edu/~orourke/Papers/52.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/p2RBq.png&quot; alt=&quot;Fig.2&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;This problem remains wide open.&#xA;For background, see &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;T.S. Michael, &lt;em&gt;How to Guard an Art Gallery and Other Discrete Mathematical Adventures&lt;/em&gt;. Johns Hopkins Univ Press, 2009.&#xA;  (&lt;a href=&quot;https://jhupbooks.press.jhu.edu/content/how-guard-art-gallery-and-other-discrete-mathematical-adventures&quot; rel=&quot;nofollow noreferrer&quot;&gt;JHU link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/r2T3z.jpg&quot; alt=&quot;TSMichael&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-06-19T23:26:34.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8299" PostTypeId="2" ParentId="8261" CreationDate="2015-06-20T13:14:14.623" Score="1" Body="&lt;p&gt;Check out the Art of Problem Solving books. They are designed to prepare students for mathematics competitions.&lt;/p&gt;&#xA;" OwnerUserId="625" LastActivityDate="2015-06-20T13:14:14.623" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8300" PostTypeId="2" ParentId="8275" CreationDate="2015-06-20T16:33:11.380" Score="3" Body="&lt;p&gt;The standard deviation is a measure of &quot;dispersion&quot;. But &quot;dispersion&quot; needs to have a focal point, and this focal point in the case of the standard deviation is the mean. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &quot;standard deviation is a measure of dispersion around the mean&quot;, of how &quot;far on average&quot; are the actual observations from their mean value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mean itself is a useful summary measure of the distribution of the data, if it represents indeed a focal point, a &lt;em&gt;concentration&lt;/em&gt; point, a point around which the actual occurrences tend to cluster. But this implies that the mean is useful when the distribution is a) unimodal and b) has occurences &quot;to the left and to the right&quot; of its mode (maximum). &lt;em&gt;Then&lt;/em&gt; the mean gives us roughly the value around which occurences hover.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But when the data is not roughly reminiscent of a &quot;symmetric unimodal&quot; shape (as is for Freshmen),  but has the appearance of the distribution of Seniors, then the mean itself stops being really useful as a measure of concentration: evidently, Seniors occurrences do not hover around the sample mean.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if the focal point around which and in relation to which the standard deviation is defined, is in itself not that useful as a metric and it does not summarize important aspects of the distribution that the data follow, then the standard deviation itself stops being also a useful measure.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2015-06-20T16:33:11.380" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8302" PostTypeId="2" ParentId="8290" CreationDate="2015-06-17T20:30:14.747" Score="3" Body="&lt;p&gt;If your students' abilities are indeed not too strong, there might be a lot to be said for basing the course closely on a couple of the really good and accessible textbooks out there -- with the course books giving weaker students a lot of helpful back-up to class work. There's a long and detailed guide to the introductory literature on various areas of math logic at various levels at &lt;a href=&quot;http://www.logicmatters.net/tyl/&quot; rel=&quot;nofollow&quot;&gt;http://www.logicmatters.net/tyl/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For weaker students, Goldrei on logic, Goldrei or Enderton on set theory, maybe Carnielli and Epstein on conmputability/incompleteness have much to recommend them. But there are other suggestions in the Guide. &lt;/p&gt;&#xA;" OwnerUserId="5319" OwnerDisplayName="Peter Smith" LastActivityDate="2015-06-17T20:30:14.747" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8303" PostTypeId="1" CreationDate="2015-06-20T17:50:47.103" Score="3" ViewCount="185" Body="&lt;p&gt;I do not officially study mathematics, so I always rely on what's on the internet. Specifically, I follow the schedules of the Tripos &amp;ndash; the math program at Cambridge, supposedly one of the most demanding, very well-known math programs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was one thing that bothered me about the schedules:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why is combinatorics not a part of the Tripos?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As far as I can see, &lt;em&gt;graph theory&lt;/em&gt; is the only combinatorics subject in there, but it comes much later, in Part 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feel free to remove this post if it's too irrelevant.&lt;/p&gt;&#xA;" OwnerDisplayName="AsG" LastEditorUserId="143" LastEditDate="2015-06-21T12:55:41.687" LastActivityDate="2015-06-21T12:55:41.687" Title="Why is combinatorics not a part of the Tripos?" Tags="|curriculum|discrete-math|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8304" PostTypeId="2" ParentId="8290" CreationDate="2015-06-21T19:18:27.890" Score="5" Body="&lt;p&gt;If I were teaching a course in the area of Set Theory and Logic to students with weak proof abilities, I would teach &lt;strong&gt;computability&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I might start with Turing-style results on uncomputability and universal computation, and build up to P=NP.  Algorithms are concrete and good for developing rigor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By contrast, I'd stay away from set theory.  Sets are too abstract to help develop a more concrete notion of rigor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sample textbooks are Sipser, Introduction to the Theory of Computation, in hardcover, or Jones, Computability and Complexity, online.  You may find more good online resources in this area than in set theory.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2015-06-21T19:18:27.890" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8305" PostTypeId="2" ParentId="7804" CreationDate="2015-06-22T01:13:41.610" Score="4" Body="&lt;p&gt;I have taught gifted students for over 25 years.  The test that I found most useful is the TOMAGs, Test of Mathematical Abilities of Gifted Students (see below for more info)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have used this test along with many other instruments to decide who would benefit from being in a group that would go faster, receive enrichment and spend less time practicing what they've learned.  I have found that this test makes hairline distinctions between the top students and even if the children aren't gifted - it will show me who is strongest.  Occasionally it identifies students that no one else would have thought belonged and these students do very well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One disadvantage of the test is that there is only edition of the test for students in 1st-3rd and in 4th-6th.  You would have to use another test to reassess.  The test does have a scale so you can compare students in different grades and different ages. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course using one test is tricky and you should use it as part of your whole assessment including teacher recommendations, previous record, and any standardized tests given.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;TOMAGS&lt;/strong&gt; is an acronym for &lt;em&gt;Test Of Mathematical Abilities of Gifted Students&lt;/em&gt;. There is more information about it in the four page PDF excerpt &lt;a href=&quot;http://www.prufrock.com/Assets/ClientPages/pdfs/tomags/tomags-ch1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;, which is described as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/mOA65.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ryser, G., &amp;amp; Johnsen, S. K. (1998). &lt;em&gt;Test of mathematical abilities for gifted students: Examiner's manual.&lt;/em&gt; Prufrock Press.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2015-06-22T18:44:25.390" LastActivityDate="2015-06-22T18:44:25.390" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8306" PostTypeId="1" CreationDate="2015-06-22T04:19:44.487" Score="8" ViewCount="376" Body="&lt;p&gt;I would like to know if anybody has experience with the British textbook series &lt;em&gt;Extension Mathematics&lt;/em&gt; in three volumes, by Tony Gardiner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It claims to be &quot;the first structured KS3/S1-S2 programme for gifted and talented students.&quot; (That's ages 11-14.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, is that true? Are there really no comparable books, even older ones?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are these books well adapted to students who are, say, in the top 1% of ability, rather than 10-20%?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And are there any books with the same objective but designed for KS4 (ages 14-16)? Perhaps older O-level books?&lt;/p&gt;&#xA;" OwnerUserId="5291" LastActivityDate="2015-06-22T04:19:44.487" Title="&quot;Extension Mathematics&quot; by Tony Gardiner" Tags="|textbooks|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8307" PostTypeId="2" ParentId="2165" CreationDate="2015-06-22T04:59:57.043" Score="4" Body="&lt;p&gt;Comprehensive references on elementary mathematics from an advanced point of view are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the three-volume series &lt;em&gt;Fundamentals of Mathematics&lt;/em&gt; (translated from German), edited by Behnke, Bachmann, Fladt, Süss and Kunle.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the five-volume &lt;em&gt;Entsyklopediya Elementarnoy Matematiki&lt;/em&gt; (Russian, Encyclopedia of Elementary Mathematics), edited by Aleksandrov, Markushevich and Khinchin.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;A Comprehensive Textbook of Classical Mathematics&lt;/em&gt; by Griffiths and Hilton&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;More specifically for geometry, restricting myself to Western languages, I would recommend one of the following.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Géométrie euclidienne plane&lt;/em&gt; by Doneddu for discussion of a modern axiomatization along Euclidean lines, but based on isometries. The idea for this axiom system is originally due to Brisac and its most characteristic axiom is one stating that given any two &lt;em&gt;flags&lt;/em&gt; (triples consisting of a point, a ray originating from that point, and one of the two half-planes determined by the support of the ray), there is a unique isometry carrying one flag to the other. The book also contains a rigorous construction of the real numbers, starting from Peano's axioms on the natural numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Lectures on the Foundations of Geometry&lt;/em&gt;, by Pogorelov, which develops the above axiom system but also looks at Lobachevskian and projective geometry, as well as geometry in space. (This book was written to be used in the geometry course in pedagogical faculties in the USSR - something that would be impossible for probably over 95% of prospective high school teachers in North America.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Geometría elemental&lt;/em&gt; by Pogorelov (the Spanish translation of &lt;em&gt;Elementarnaya Geometriya&lt;/em&gt;) for an excellent, rigorous presentation of basic geometry which was intended to be directly usable in schools. In fact, a modified version of this system is used in Pogorelov's textbooks, which are widely used in Russian schools.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For a rigorous development of geometry using vectors, Dieudonné's &lt;em&gt;Algèbre linéaire et Géométrie élémentaire&lt;/em&gt; is excellent.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5291" LastEditorUserId="5291" LastEditDate="2015-06-22T05:15:44.470" LastActivityDate="2015-06-22T05:15:44.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8308" PostTypeId="1" CreationDate="2015-06-22T17:38:53.510" Score="5" ViewCount="409" Body="&lt;p&gt;I define a &lt;strong&gt;known&lt;/strong&gt; as a variable with a number you can plug in. I define an &lt;strong&gt;unknown&lt;/strong&gt; as a variable without a number you can plug in because none is given. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently I have taken several classes where they ask me to use formulas to compute some number. But it isn't just plug in and mindlessly compute. I have to be aware of what information I do not have and what formulas are usable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose I am considering a projectile motion problem in physics. The projectile is launched at a $45°$ angle and lands at a point $P$. Now I want to repeat the experiment but this time choose an angle $\theta$ such that the projectile bounces once with a perfectly elastic collision and lands at the same point. Visually this looks like harmonics with the first throw being $f$ and the second being $2f$, $f$ standing for frequency. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note how I am not given information about $t$ or $v_{\text{ball}}$. Now, I may be able to use formulas to determine a bit more. For instance, I found&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$t_{full}= \frac{2v_b \sin(45°)}{g}$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this formula is still in terms of a variable I have no value for. And for this particular problem, I basically went in circles because I couldn't organize what I knew and didn't know to find an arrangement of my knowns and unknowns that would a numerical result.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In such problems, many formulas look like they are applicable. But in reality, you cannot use a formula if you don't have stuff to plug into it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My Question&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an efficient way to keep track of what values you have and do not have and use this to determine what formulas are worth considering?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I read a book on algorithms once that suggested breaking a large problem down into sub-problems and solving them to solve the larger problem. I think something like that is what I am looking for, but more like a notation specifically for solving problems that makes that subdivision easier to do. &lt;/p&gt;&#xA;" OwnerUserId="3471" LastActivityDate="2015-06-23T07:57:28.637" Title="How should one manage knowns vs unknowns when solving a problem?" Tags="|problem-solving|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8309" PostTypeId="2" ParentId="7370" CreationDate="2015-06-22T18:54:41.480" Score="6" Body="&lt;p&gt;A big part of the psychology of talented students is their attitude to errors that they make.  Remember that no matter how bright the students are, they can still be overconfident and make mistakes. One of your jobs is to teach them to accept errors as part of the learning process and not an indication that they're not  as smart as they thought  It is important that you evaluate their work, and give them a chance to correct any mistakes.  Don't be surprised if they insist they're right or are unable to handle being wrong.  Make sure you model being wrong on occasion, so that they see everyone makes mistakes.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-06-22T18:54:41.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8310" PostTypeId="2" ParentId="8308" CreationDate="2015-06-22T22:01:50.120" Score="1" Body="&lt;p&gt;As others have commented, this situation is a little more general than just in mathematics.  In my view, it borders on issues on mathematical modelling and general engineering rather than just mathematics, because one has to decide how to &quot;adjust the situation&quot; and &quot;decide what constitutes a solution&quot;, rather than just executing an algorithm to compute a numerical solution to a well-understood problem with well-understood methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, one can tackle it mathematically anyway, and in a way that can be useful for those that come after.  The primary idea: DOCUMENT YOUR ASSUMPTIONS.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may have some rational for picking a particular set of tools to tackle a problem.  It is a good idea to write down your rationale for your choice of tools, and don't be afraid to say that you don't know any better methods (yet), but even if you are compelled to use a limited tool box, you will run into cases where you can't use them automatically because of the unknowns that you encounter, e.g. your tool needs four inputs and you only have three.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend organizing your data by tools and by inputs that you know or easily understand and simulate.  Inputs that you don't know you treat as variables, and if the methods you use are well understood, your initial result is a nicely-expressed function or relation in the unknown variables which you can analyze.  Depending on the goal, the problem might be finishable as you might be looking for an optimal answer: you can then conclude that the most optimal solution implies certain values for the unknowns, or that a certain relation must hold between the unknowns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that by documenting your process and assumptions, others can&#xA;replicate it or at least approximate it and check your work.  Even if your&#xA;work does not represent the last word on the matter, it might be a good initial point on which others will base their analyses.  However, they can't do it if you don't tell them.  I.E., Show Your Work!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Your First-Grade Teacher Was Right&quot; Paseman, 2015.06.22  &lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-06-22T22:01:50.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8311" PostTypeId="2" ParentId="7370" CreationDate="2015-06-22T23:44:59.397" Score="7" Body="&lt;p&gt;This is triggered by @AmyB's and @DanielHast's remarks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it would be useful in the OP's environment to understand&#xA;and foster Carole Dweck's &lt;a href=&quot;http://mindsetonline.com/&quot; rel=&quot;noreferrer&quot;&gt;growth mindset theory&lt;/a&gt;.&#xA;This theory is memorably described in Malcolm Gladwell's article, &#xA;&quot;&lt;a href=&quot;http://www.newyorker.com/magazine/2002/07/22/the-talent-myth&quot; rel=&quot;noreferrer&quot;&gt;The Talent Myth&lt;/a&gt;.&quot;&#xA;There Dweck says,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Students who hold a fixed view of their intelligence care so much about looking smart that they act dumb.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Recognizing when students have the fixed mindset, and nudging them toward the&#xA;growth mindset, can dramatically change their ultimate achievements, at the same&#xA;time as reducing their stress.&#xA;The website&#xA;&lt;a href=&quot;http://www.growthmindsetmaths.com/&quot; rel=&quot;noreferrer&quot;&gt;Growth Mindset Maths&lt;/a&gt;&#xA;offers a number of practical strategies. That website attributes&#xA;this quote to Thomas Edison (but see Benjamin's comment):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;You must learn to fail intelligently. Failing is one of the greatest arts in the world. One fails forward towards success.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-06-23T10:58:11.370" LastActivityDate="2015-06-23T10:58:11.370" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8312" PostTypeId="2" ParentId="8308" CreationDate="2015-06-23T07:48:14.340" Score="2" Body="&lt;p&gt;Since it is difficult to talk only in the abstract, let's actually write down a solution to this problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I assume that the problem intends the initial velocity $v_0$ of the ball to be held fixed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, what are you trying to achieve by changing $\theta$ from $45^{\circ}$ to something else? What you want is for the horizontal displacement $d$ of the ball when it touches down to be half as large as it was before. So it is natural to write down the relationship between $d$ and $\theta$. Of necessity, $v_0$ will intervene in this calculation. (The faster the projectile is launched, the further it will go, if $\theta$ is held constant.) There are no other variables. The angle $45^{\circ}$ is just one possible value of $\theta$, so in doing this calculation you should not assume that $\theta = 45^{\circ}$ at first, although that will be one possibility you will look at.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The result is that &#xA;$$d = \frac{v_0^2}{g} \sin 2\theta$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the question is how to change $\theta_1 = 45^{\circ}$ to an angle $\theta_2$ so that $d$ is halved. We have&#xA;$$d_1 = \frac{v_0^2}{g} \sin 2\theta_1 =  \frac{v_0^2}{g}$$&#xA;and &#xA;$$d_2 = \frac{v_0^2}{g} \sin 2\theta_2.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point, you have done the necessary calculations of $d_1$ and $d_2$ in terms of the values of $\theta$, and you must ask yourself what condition you wish to satisfy. That condition is that $d_2/d_1$ should be $1/2$. Writing down the corresponding equation, you find &#xA;$$\frac{(v_0^2/g)\sin 2\theta_2}{v_0^2/g} = \frac{1}{2},$$&#xA;which simplifies to&#xA;$$\sin 2 \theta_2 = 1/2.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This equation shows that there are two solutions, namely $\theta = 15^{\circ}$ and $\theta = 75^{\circ}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Initially, it might have seemed as though the answer about how you should change $\theta$ would depend in some way on $v_0$. From a strictly logical standpoint, there was nothing that allowed you to assume that this would not be the case. It was only a mathematical calculation that allowed you to determine this (although it might be possible to give some kind of physical argument).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's summarize the status of the different variables in this problem. $\theta$ was the variable you were required to change, so that a prescribed change would correspondingly be obtained in the variable $d$. You computed $d$ in terms of $\theta$ (since $d$ is the result of a certain physical process that can be described when $\theta$ is known.) We could say that $\theta$ was an &lt;em&gt;independent variable&lt;/em&gt; and $d$ the &lt;em&gt;dependent variable&lt;/em&gt;. The important thing is that the physical nature of the problem makes it such that $d$ can be effectively calculated provided that you temporarily pretend that $\theta$ is &lt;em&gt;known&lt;/em&gt;, even though it isn't really. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to perform this calculation of $d$, however, in truth you also required knowledge of $v_0$. We say that $v_0$ is a &lt;em&gt;parameter&lt;/em&gt;. It is a sort of background variable which you must regard as known even though it may not really be. (The number $g$ could also be regarded as a parameter, especially if you were going to perform the experiment on different planets, for example. In the problem as stated, that is beside the point.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since you were required to find $\theta$, the key step was to write down the equation entirely in terms of $\theta$ (and any parameters), rather than by using the variable $d$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Initially it seemed that the answer to the question might depend on the parameter $v_0$, but eventually the calculations showed that this was not the case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main point to consider in determining your variables was which ones were needed in order to compute $d$. This is really a physics question.&lt;/p&gt;&#xA;" OwnerUserId="5291" LastEditorUserId="5291" LastEditDate="2015-06-23T07:57:28.637" LastActivityDate="2015-06-23T07:57:28.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8313" PostTypeId="1" AcceptedAnswerId="8314" CreationDate="2015-06-23T09:45:49.573" Score="11" ViewCount="1160" Body="&lt;p&gt;When integrating and differentiating, sometimes one direction is easy and the other is harder. A nice example is $\frac{d}{dx}\tan x=\sec^2x$, where differentiating is easy but integration (without knowledge of the other direction) is difficult. So to get round the difficult direction you might integrate it thus:&#xA;$$\begin{align*}&#xA;\int\sec^2xdx&amp;amp;=?\\&#xA;u&amp;amp;=\tan x\\&#xA;du&amp;amp;=\sec^2xdx\\&#xA;\int\sec^2xdx&amp;amp;=\int du,\\&#xA;&amp;amp;=u+c\\&#xA;&amp;amp;=\tan x+c&#xA;\end{align*}$$&#xA;This is correct. However, it leaves a bitter taste in my mouth, and I would be hesitant to award any marks for this in an exam. The issue is with the &quot;oracle&quot;: the student already knows the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is twofold.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Should the above solution get full marks?&lt;/li&gt;&#xA;  &lt;li&gt;If not, how can I explain this to my students (and also to my tutors)?&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am really struggling with (2) - I am struggling to verbalise my issues with this approach*. I am asking (1) because the struggle with (2) makes me wonder if my initial assumption is incorrect (that is, I wonder if the answer to (1) is &quot;yes&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I could phrase the exam question to specifically ban this approach. However, that is beside the point - I want to better understand &lt;em&gt;why&lt;/em&gt; this approach should be banned!&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;*I chose to study maths all those years ago because I thought I wouldn't have to write much...have opinions...communicate...that sort of thing...&lt;/p&gt;&#xA;" OwnerUserId="169" LastActivityDate="2018-03-28T16:50:14.790" Title="Intergration by differentiating will get you $0$ marks - but how to explain why?" Tags="|calculus|exams|grading|" AnswerCount="8" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="8314" PostTypeId="2" ParentId="8313" CreationDate="2015-06-23T12:30:46.540" Score="43" Body="&lt;p&gt;Your student should get full marks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, I would say that even a more complicated example, like $$\int 2x\cos(x^2) dx  = \sin(x^2) + C$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;should be awarded full points as long as the student justifies this by differentiating $\sin(x^2)$.  In fact, this solution demonstrates deeper understanding of the meaning of these symbols than the variable substitution would (the student actually recognizes the chain rule in action).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, while differentiation of elementary functions is a mindless algorithm (after learning the basic ones), integration is an art, which fundamentally proceeds by exactly the kind of insight in the OP.  Namely, at some point you have to recognize something which you know to be an antiderivative.  All of the other techniques (variable sub, trig sub, partial fractions, etc) are just techniques for manipulating the integral to the point where you can clearly see it.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2015-06-23T12:30:46.540" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8315" PostTypeId="2" ParentId="8313" CreationDate="2015-06-23T13:36:32.340" Score="18" Body="&lt;p&gt;I agree with the other post that you should give full credit unless there were clear directions saying what would and wouldn't be acceptable approaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it would be very, very hard to convey to students why using the integral of tan, which they happen to know, is off limits, while using other integrals they know is allowed, other than as a detailed and arbitrary decree to make the problem hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students haven't gotten an intuition for why some methods are &quot;clever&quot;, some are &quot;good&quot;, and some are &quot;unacceptable&quot;. They're struggling just to know the difference between valid and invalid. So, in general, questions which ask them to use a particular method have to very clear, and can't depend on them sharing the professor's understanding of what approaches are preferable.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-06-23T13:36:32.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8316" PostTypeId="2" ParentId="6069" CreationDate="2015-06-23T19:22:07.360" Score="4" Body="&lt;p&gt;Math encounters are a series of lectures given at the Museum of Math in NYC, celebrating the spectacular world of mathematics. The link for these lectures is &lt;a href=&quot;http://momath.org/math-encounters/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. There is a list of lectures at the bottom of this page with links to more information.  For example Math at the Movie has the following info:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Filmmaking is undergoing a digital revolution brought on by advances&#xA;  in areas such as computer technology, computational physics, geometry,&#xA;  and approximation theory. Join Tony DeRose, Senior Scientist and lead&#xA;  of the Research Group at Pixar Animation Studios, as he uses examples&#xA;  from Pixar’s feature films to provide a behind-the-scenes glimpse of&#xA;  the role math plays in the revolution. Tony will be introduced by Ira&#xA;  Flatow, Host and Executive Producer of National Public Radio’s popular&#xA;  Science Friday radio program.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Many of these lectures can be found on YouTube or if a particular one intrigues you, you can buy it on CD.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have attended these lectures and have found that they often place math in the oddest places (juggling) and expand on the math and it's applications beyond my expectations (origami).  I suggest you take the time to look through the topics and click on the links so that you can really see what each one is about. Not all will suit your purposes, but those that do will be appreciated by your students.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-06-23T19:22:07.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8317" PostTypeId="2" ParentId="8313" CreationDate="2015-06-23T20:20:02.770" Score="8" Body="&lt;p&gt;Whether you would give this solution full marks depends on &lt;em&gt;which derivatives you expect the students to remember off by heart&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you expect the students to know off by heart that $\frac{\text{d}}{\text{d}x} \tan x = (\sec x)^2$, then you must accept any answer that directly says $\int (\sec x)^2 \text{d}x = \tan x + C$. The whole point of an indefinite integral is to write down the functions that differentiate to give the integrand, and if you have spent time making them remember derivatives, then this is therefore the same as making them remember integrals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if you &lt;em&gt;always&lt;/em&gt; expect the students to derive the derivative of $\tan x$ by first writing it as $\frac{\sin x}{\cos x}$ and using the quotient rule, and they never get full marks for any question involving the derivative of $\tan x$ unless they do this, then you are justified in feeling uncomfortable with the working you see above. However, I agree with Steven Gubkin you would still have to accept working that includes a derivation of this derivative and then does the integration directly (unless you specifically asked for substitution of course).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So analyse what derivatives you ask them to just know, and these are precisely the integrals you are implicitly expecting them to just know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: In general I would never not give marks if a student gave a clearly-described method that actually worked. For example, the integral $\int \sin x \cos x \text{d}x$ is usually done by rewriting using a trig identity, but if a student did it by substitution or parts and it worked, then I'd be fine with that.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-06-23T23:00:15.040" LastActivityDate="2015-06-23T23:00:15.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8318" PostTypeId="2" ParentId="8313" CreationDate="2015-06-23T21:50:56.917" Score="13" Body="&lt;p&gt;A bit too long for a legit comment... I think a very important question in the background here is that of whether we want to teach students that &quot;correct&quot; math involves primarily adherence to somewhat-arbitrary, even if clear, rules set down by the teacher, or whether there is an underlying reality, and best-practices, etc. Some of the &quot;rule&quot; nonsense can be dispelled by asking questions more forthrightly, rather than having them take place in a secret/tacit context, that is, say &quot;evaluate this integral by reducing to simpler cases involving only sine, cosine, and log ...&quot; ... although once the rule is spelled out, it kinda ruins the &quot;examination drama&quot; of the question. Indeed, asking questions more forthrightly also is a diagnostic for questions whose implicit rules &quot;seem silly when you say them out loud&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This happened to remind me of my bafflement as a little kid when being told that $25\times 4=100$ was not &quot;obvious&quot;. Had to show work!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true that there is a subtle, perhaps profound, difference between guessing and checking, and having a device to more-systematically determine things. The problem is, there &lt;em&gt;is&lt;/em&gt; no reasonable, human-executable algorithm to express integrals in elementary terms. Many are not so-expressible, as Liouville already found. There are a few limited machinations to rearrange an integral until it is recognizable as a derivative of a familiar thing. This is not &quot;systematic&quot;. Ever-more a mystery to me why we portray this to students as an essential intellectual enterprise... except, duh, it's easy to make textbooks that do this, and easy to teach such classes. But, seriously, let not try to make the students pretend to see something that simply isn't there.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-06-23T21:50:56.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8319" PostTypeId="2" ParentId="6069" CreationDate="2015-06-23T21:58:05.753" Score="2" Body="&lt;p&gt;Many expository articles about the applicability of mathematics can be found in SIAM News:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://sinews.siam.org/&quot; rel=&quot;nofollow&quot;&gt;http://sinews.siam.org/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and as part of the AMS Feature Column:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/samplings/feature-column/fcarc-index&quot; rel=&quot;nofollow&quot;&gt;http://www.ams.org/samplings/feature-column/fcarc-index&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2015-06-23T21:58:05.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8320" PostTypeId="2" ParentId="8313" CreationDate="2015-06-24T02:49:12.587" Score="2" Body="&lt;p&gt;I kind of had the same thought as the OP since I'm currently teaching a Cal 2 summer course.  I don't think that solving $\int \sec^{2}{x}dx$ by using prior knowledge from Cal 1 is wrong necessarily, but it's interesting to me that textbooks don't provide an alternative method of reasoning the solution.  In case anyone is interested, you could also solve it using integration by parts and letting $u=\sec^{3}{x}, dv = \cos{x}dx$.  The rest of the steps are fairly easy if you're comfortable with trigonometric integrals.  I'm not aware if there's another simpler method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know this isn't exactly what the OP asked for but I figured it was relevant and would give the other commenters an alternative solution to reference.&lt;/p&gt;&#xA;" OwnerUserId="5141" LastActivityDate="2015-06-24T02:49:12.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8321" PostTypeId="1" CreationDate="2015-06-24T03:51:22.980" Score="21" ViewCount="331" Body="&lt;p&gt;I hope this question is not deemed off-topic, but I cannot think of a more appropriate place to ask it (few tech sites care about support for MathJax and $\LaTeX$), and I think that blogging about mathematical topics is certainly part of mathematical education viewed in the broad sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, I currently have an almost-dormant blog on wordpress.com, and I would like to reinvigorate it and post more frequently. However I have realised that Wordpress itself is hampering me, largely because of its primitive support for $\LaTeX$; every expression has to be between non-standard delimiters and is then replaced by an image which is usually the wrong size and aligned incorrectly.&#xA;So this results in tediously writing not-very-reusable source-text that is rendered into ugly output - basically a lose-lose-lose situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So really I need to use something that produces MathJax, and in addition it would be better if I could write once and render to either HTML or $\LaTeX$, which seems to limit the field to various Markdown editors. The StackEdit online editor is extremely good, produces output containing MathJax tags, and has a mechanism for direct &quot;Publish to Wordpress&quot;. Almost perfect, except that it doesn't work properly for $\TeX$ - it just passes through the marked-up MathJax constructs which don't work on Wordpress. (It would work on a self-hosted Wordpress, where you can add your own Javascript, but my life is too short to get back into self-hosting.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, I had better finish with an actual question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the best way to write mathematical content, preferably using&#xA;  Markdown for formatting, and have the resulting content appear online,&#xA;  with the mathematics rendered in MathJax?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4795" LastActivityDate="2015-07-06T16:45:37.917" Title="Math Blogging - what is a decent set up?" Tags="|technology-in-education|software|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8322" PostTypeId="1" CreationDate="2015-06-25T09:26:33.853" Score="4" ViewCount="163" Body="&lt;p&gt;While developing my notes for &quot;integration by parts&quot; some times back, I used a great book that must have been titled &quot;Calculus and (with?) Analytical Geometry. I can't place the exact book anywhere since its no longer in our library.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Help me locate the true title and author of the book. Here are some of the problems I derived from the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int{{x^5\,dx}\over \sqrt{3+x^2}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int{x\sqrt{2+x}\,dx}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int{\sec^3{x}\,dx}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int{x\cos{kx}\,dx}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am absolutely grateful for any help proffered. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="5332" LastActivityDate="2015-06-26T11:36:31.867" Title="Misplaced title and author of calculus book" Tags="|textbooks|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8323" PostTypeId="2" ParentId="8313" CreationDate="2015-06-25T10:38:01.637" Score="3" Body="&lt;p&gt;To agree with most of the answers here, and offer a similar-in-spirit example from a connected field, dealing with (definite) integrals in mathematical statistics is all about re-arranging them to look like a &quot;known&quot; probability density function (PDF). In essence this is also &quot;recognizing a derivative&quot; (better, &quot;recognizing that we can arrive at a derivative&quot;) since the PDF is the derivative of the distribution function which is an integral. Sometimes we have also to transform the interval of integration in order to match it with the &quot;benchmark&quot; interval for which the distribution is known... at which point the integral &quot;magically disappears from view&quot; since it just became equal to the cumulative distribution function integrated over the support, and this always equals unity... leaving you with all the stuff you had to attach to it during the transformations -that admittedly may still include an integral, but of those that we nowadays consider &quot;closed-form&quot; since they do not have an analytical solution but are totally mapped numerically like the cumulative distribution function of the standard normal distribution.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes we arrive not at an integrand that represents a PDF (i.e. the derivative), and so at the cumulative distribution function, but at the moment generating function, which is also a known associated function with the distribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To provide the simplest example, if we are given the integral&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$I =\int_{-\infty}^{\infty} \exp\{-0.5x^2-ax\}dx$$&#xA;what we will do in mathematical statistics is  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Ah, exponential with the variable squared&quot; $\rightarrow$ normal distribution: add and subtract the normalizing factor, and separate the two terms&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$I = \sqrt{2\pi}\int_{-\infty}^{\infty}e^{-ax} \left(\frac {1}{\sqrt {2\pi}}\exp\{-0.5x^2\}\right)dx$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The term in the parenthesis is the probbaility density function of the standard normal distribution , typically denoted $\phi()$, so we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$I = \sqrt{2\pi}\int_{-\infty}^{\infty}e^{-ax} \phi(x)dx$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The integral is now the moment generating function of the standard normal (the support is the &quot;benchmark&quot; one), $MGF(s) = E(e^{sx})$ evaluated at $s=-a$ so &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$I = \sqrt{2\pi}\cdot \exp\{0.5a^2\}$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is &lt;em&gt;crucial&lt;/em&gt; to develop the skill to recognize the &quot;kernels&quot; of the probability density functions (the terms that involve the variable) as well as other exhaustively worked out entities in the field like the moment generating function, or the cumulant generating function. And so, earning marks for it is only natural. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a more complicated example, see &lt;a href=&quot;https://stats.stackexchange.com/questions/151854/a-normal-divided-by-the-sqrt-chi2s-s-gives-you-a-t-distribution-proof/151879#151879&quot;&gt;this post over at Cross Validated&lt;/a&gt;, where variable transformation is required apart from accounting for normalizing factors (especially under eq. $(3)$ there).&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="-1" LastEditDate="2017-04-13T12:44:19.443" LastActivityDate="2015-06-25T13:03:43.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8324" PostTypeId="1" CreationDate="2015-06-25T13:22:55.490" Score="19" ViewCount="1561" Body="&lt;p&gt;I am teaching an undergraduate course in linear algebra this fall. I am dissatisfied with most existing textbooks, and indeed with the way in which this subject is usually taught. I hope to find a textbook that has many or all of the following characteristics:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It emphasizes linear transformations as the central object of study. This, in my view, is what the subject is about, and I think a linear algebra course should start with them (perhaps only in R^2 or R^3) from the very beginning.&lt;/p&gt;&#xA;&lt;p&gt;Instead, most books start with solutions to systems of linear equations, which I personally believe is of secondary importance. Moreover they use &amp;quot;augmented matrices&amp;quot; which (in my mind) obscure the relationship between matrices and linear transformations. I would prefer to &lt;em&gt;never&lt;/em&gt; write down a matrix which does not represent a linear transformation in the usual way.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It takes a heavily geometric approach. Linear transformations can be visualized and drawn, and I think a linear algebra course which doesn't emphasize this is selling itself short.&lt;/p&gt;&#xA;&lt;p&gt;As an example, I think the following would make a great homework and exam question: Shown are a pair of coordinate axes and a cartoon drawn on the plane. A 2x2 matrix is given. Sketch the image of the cartoon under this linear transformation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It de-emphasizes the carrying out of algorithms by pencil and paper. I have memories of row reduced echelon forms, pivots, etc., etc. but as a working mathematician, none of this really stuck. I memorized the techniques, regurgitated them for the exam, and although I could reconstruct them if needed I don't really remember them.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It de-emphasizes axiomatics. In my opinion it is uninspiring to see a list of axioms for a vector space and to prove statements like &amp;quot;If v + 0 = 0, then v = 0&amp;quot;. This goes especially since my students won't have the mathematical background or maturity to appreciate some of the more eclectic examples of vector spaces, or why you would be interested in working over any field other than R or C. They are also somewhat unlikely to have seen examples from any area of math where intuition leads you astray.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It really tries to teach the &amp;quot;meaning&amp;quot; of a matrix. To my embarrassment, it was &lt;em&gt;after&lt;/em&gt; I finished my Ph.D. that I realized the columns of a matrix are just the images of the basis vectors. This was probably said in the book I used as an undergrad, but not really emphasized. Similarly, when we solve linear equations we write each equation as a row vector rather than a column vector because row vectors in fact represent &amp;quot;covectors&amp;quot;, i.e. elements of the dual space.&lt;/p&gt;&#xA;&lt;p&gt;I don't want to go overboard with this (i.e., to the extent of introducing category theory), but I do want to present some of the notions of duality that are at the heart of linear algebra and more advanced mathematics.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is a good book overall.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It doesn't cost $300 or whatever exorbitant price publishers are charging these days.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I should mention that my university offers &amp;quot;applied&amp;quot; (using Strang) and &amp;quot;honors&amp;quot; versions of linear algebra, and that the course I'm teaching is neither.&lt;/p&gt;&#xA;&lt;p&gt;Based on the preferences I've described, does anyone have books to recommend? Thank you very much.&lt;/p&gt;&#xA;" OwnerUserId="1535" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-06-27T18:47:19.823" Title="Linear algebra textbooks presenting an eclectic, geometric approach to the subject" Tags="|undergraduate-education|textbooks|linear-algebra|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8325" PostTypeId="2" ParentId="8322" CreationDate="2015-06-26T11:36:31.867" Score="2" Body="&lt;p&gt;what about this one?&#xA;Introduction to Calculus and Analytic Geometry by Philip Gillett&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1602299811&quot; rel=&quot;nofollow&quot;&gt;http://www.amazon.com/Introduction-Calculus-Analytic-Geometry-Gillett/dp/1602299811&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2015-06-26T11:36:31.867" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8326" PostTypeId="2" ParentId="7718" CreationDate="2015-06-26T11:42:14.753" Score="16" Body="&lt;p&gt;You asked for anecdotal evidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was a &quot;gifted student&quot;. The school told me to teach myself 11th grade math (Trigonometry and Algebra 2) in 9th grade.  I never formally learned algebra 1, but I understood it. They gave me a book for 11th grade math and I learned.  I don't think it did me any harm - but I do think I understood the concepts and didn't just learn algorithmically. It gave me a lot of self confidence to do this and kept me from being bored in math class. Everyone was delighted when I got a 97 on my 11th grade regents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 10th grade we had geometry which I didn't have a strong sense for and so I rejoined my class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 11th grade they had me teach myself calculus. When the supervising teacher asked me what I was up to, I said integrals and discovered I was pronouncing it incorrectly.  I focused on the AP test and was able to get a 5.  In hindsight, I didn't learn the concepts as deeply as I might have from class - perhaps because I was focused on a test?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I filled out college applications, I wrote about my success with independent study with great pride.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 12th grade, I was allowed to take college classes and I enjoyed linear algebra and Calculus 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fast forward to my senior year in college and I took a graduate seminar in combinatorics.  We had to read and present journal articles. I was delighted to see that my early independent study had trained me to read journal articles.  I knew that if I persevered, I would understand them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I could do it all again, I would.  I believe that independent study was a great opportunity. It taught me to persevere in my learning and gave me confidence. It  kept me from being bored. I think it helped  that in 10th grade and 12th grade I was back in a classroom that was on my level. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fast forward to the present I am a gifted math teacher. I have students at the elementary level who like to fast forward using Khan Academy.  I don't think it has the same challenge as working through a course, but it keeps them from being bored in class.  At times I give these students horizontal enrichment, which is truly what they need.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do have such students, I would encourage them, but at the same time I would design intermediate assessments for them that would highlight some of your concerns about how they are using the material.  I would also offer material that challenges them without going ahead.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-06-26T11:42:14.753" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8327" PostTypeId="2" ParentId="8324" CreationDate="2015-06-26T12:39:58.967" Score="8" Body="&lt;p&gt;I am intrigued by this book, but (a) I haven't used it myself, and (b) it lists @$200:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Shifrin, Ted, and Malcolm Adams. &lt;em&gt;Linear algebra: A geometric approach&lt;/em&gt;. Macmillan, 2ndEd, 2011.&#xA;  (&lt;a href=&quot;http://www.macmillanhighered.com/Catalog/product/linearalgebra-secondedition-shifrin&quot; rel=&quot;noreferrer&quot;&gt;Macmillan link&lt;/a&gt;.)&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/WoDVz.jpg&quot; width=&quot;150&quot; /&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Perhaps it could serve as a guide in your teaching even if you don't&#xA;ask the students to purchase it...? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/ZwiaS.png&quot; alt=&quot;Page95&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-06-27T18:47:19.823" LastActivityDate="2015-06-27T18:47:19.823" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8328" PostTypeId="1" CreationDate="2015-06-26T15:24:19.607" Score="14" ViewCount="887" Body="&lt;p&gt;I'm interested to know how the real numbers are introduced at beginner level in different countries.&#xA;In my (old) experience of teacher in Italy there was some well defined steps:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Introduce the field of algebraic numbers, showing that there are numbers (radicals) that are not rationals. The classical proof is the irrationality of $\sqrt{2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Introduce some transcendental number as $\pi$ , $e$ and other that we needs to calculate transcendental functions as exponential, logarithm, trigonometric, combined between them and with radicals. The set of this number is not defined in a really rigorous way, but it is used as a field, and we can thik that coincide with the field of computable numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this is given in the first years of high-school. The we begin with mathematical analysis and we need the Completeness Axiom for the reals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience this is introduced intuitively using the ''correspondence'' from the real numbers and the points of a straight line. Well, being an axiom, we have not to prove it, but it should be good to show that really we can do some set of ''numbers'' with such property. But the classical Dedekind or Cauchy constructions are difficult at this level so I'm interested to know how this question is solved in the real teaching experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem become more difficult when we note that a consequence of CA is that the set of real numbers is uncountable. I don't know if this fact is taught to the students a this level, but it seem to me very important. And if some student asks if this uncountable many numbers are the transcendental numbers that he knows what is the answer? We are forced to introduce the set of non computable numbers, that we can not compute in any way. Is this concept introduced at high-school level in some country ?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I've seen the similar question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/186/good-definition-for-introducing-real-numbers&quot;&gt;Good definition for introducing real numbers?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but I think that it is not a duplicate because I'm interested in the different experiences and possibly in different countries, to know if there are different approach or different ''schools of thinking''.&lt;/p&gt;&#xA;" OwnerUserId="5314" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-08-09T15:54:55.710" Title="How the real numbers are taught?" Tags="|secondary-education|real-numbers|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8329" PostTypeId="2" ParentId="8328" CreationDate="2015-06-26T15:39:22.887" Score="3" Body="&lt;p&gt;Spivack's &lt;em&gt;Calculus&lt;/em&gt; uses this approach:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the beginning are stated some properties of &quot;the number system&quot; that amount to saying it is a complete ordered field (but not using that language).  At the very end of the book (perhaps you reach it 9 months later, after doing rigorous calculus) there is a construction of  a complete ordered field, and the proof that any two are isomorphic.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-06-26T15:39:22.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8330" PostTypeId="2" ParentId="4477" CreationDate="2015-06-26T20:58:09.527" Score="14" Body="&lt;p&gt;At my university we were using the 6th edition of Stewart and the publisher informed us that they would no longer be willing to sell this book (even at exorbitant prices). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we wanted to continue with Stewart, we would have to &quot;upgrade&quot; to the 7th edition, wiping out most of the benefits of continuity: students would no longer be able to sell their used books, and instructors would not be able to reuse their lists of problem numbers. (This was especially a problem for instructors using online homework systems.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, the publisher had previously promised to hold the price of their book constant, and had then not honored that promise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, when we had to stop using Stewart 6/E, we decided to review multiple calculus books, of which Stewart's 7th edition was just one alternative considered. We ended up deciding to switch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Incidentally, at a large university, announcing to every publisher in sight that you will be changing your departmentally required calculus book is an excellent way to get an impressive quantity of unsolicited free calculus books. If your department is lacking in doorstops, or you need objects on which to place computer monitors, this might be worth considering.)&lt;/p&gt;&#xA;" OwnerUserId="1535" LastActivityDate="2015-06-26T20:58:09.527" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8331" PostTypeId="2" ParentId="4477" CreationDate="2015-06-26T23:23:53.940" Score="6" Body="&lt;p&gt;I think a possible (near?) future solution is&#xA;Rice Univ&#xA;Richard Baraniuk's&#xA;&lt;em&gt;OpenStax&lt;/em&gt;&#xA;(&lt;a href=&quot;https://en.wikipedia.org/wiki/OpenStax_College&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;;&#xA;&lt;a href=&quot;https://openstaxcollege.org/&quot; rel=&quot;noreferrer&quot;&gt;OpenStax website&lt;/a&gt;),&#xA;which I learned from a talk by Richard at &#xA;&lt;a href=&quot;http://blendedlearning.blogs.brynmawr.edu/&quot; rel=&quot;noreferrer&quot;&gt;Bryn Mawr's Blended Learning conference&lt;/a&gt;.&#xA;Despite the current availability of 25 free open-source textbooks in a variety&#xA;of fields, we have: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Calculus (forthcoming in 2016).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;https://openstaxcollege.org/textbooks/precalculus&quot; rel=&quot;noreferrer&quot;&gt;Pre-Calculus&lt;/a&gt; has been available since October 2014).&#xA;One notable aspect of this project is that publishers are joining the effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An impressive number from his talk: \$1M (\$U.S.) to develop a textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also the &lt;em&gt;NYTimes&lt;/em&gt; article, &#xA;&quot;&lt;a href=&quot;http://www.nytimes.com/2015/02/26/your-money/putting-a-dent-in-college-costs-with-open-source-textbooks.html?smprod=nytcore-ipad&amp;amp;smid=nytcore-ipad-share&amp;amp;_r=2&quot; rel=&quot;noreferrer&quot;&gt;Putting a Dent in College Costs With Open-Source Textbooks&lt;/a&gt;.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt; (&lt;em&gt;5Jun2017&lt;/em&gt;). The OpenStax Calculus books are now available:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &lt;a href=&quot;https://i.stack.imgur.com/7jHZb.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7jHZb.jpg&quot; alt=&quot;Calc123&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://cnx.org/contents/i4nRcikn@2.50:BUxRn5Hd@11/Preface&quot; rel=&quot;noreferrer&quot;&gt;Here is a link&lt;/a&gt; to the Preface of &lt;em&gt;Calculus Volume 1&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-06-06T01:01:22.213" LastActivityDate="2017-06-06T01:01:22.213" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8332" PostTypeId="2" ParentId="6069" CreationDate="2015-06-26T23:25:49.793" Score="1" Body="&lt;p&gt;I've wanted to get a copy of Satyan Devadoss's 36 Lectures on &lt;a href=&quot;http://www.thegreatcourses.com/courses/shape-of-nature.html?cid=1460&quot; rel=&quot;nofollow&quot;&gt;The Shape of Nature&lt;/a&gt; for quite some time. I've finally ordered them through an interlibrary loan, as they carry a fairly significant price tag.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems the material would certain check all the boxes you care about, the reviews are generally very positive, and I'll be able to update with a personal opinion fairly soon.&lt;/p&gt;&#xA;" OwnerUserId="5153" LastActivityDate="2015-06-26T23:25:49.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8333" PostTypeId="2" ParentId="4477" CreationDate="2015-06-27T00:01:58.237" Score="-1" Body="&lt;p&gt;Are texts even necessary anymore? With the plethora of information now available to everyone, text books seem archaic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as homework problems, I just create problem sets to post online.&lt;/p&gt;&#xA;" OwnerUserId="625" LastActivityDate="2015-06-27T00:01:58.237" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8334" PostTypeId="2" ParentId="4477" CreationDate="2015-06-27T16:58:42.707" Score="5" Body="&lt;p&gt;I myself learned from the Larson book, which I enjoyed at the time.  When I finished my degree, I realized how sparse Larson really is, particularly with proof.  Larson's visuals are very nice, but the problems themselves rarely are anything more than some algebraic manipulation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My school (a community college) was using Larson, and switched to Briggs/Cochran last year.  Again, a nice book, and the publisher was very good w/following up and providing additional resources, but most of the problems were similar and not too challenging.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have directed my students to Paul Dawkins' excellent site:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://tutorial.math.lamar.edu/&quot; rel=&quot;noreferrer&quot;&gt;http://tutorial.math.lamar.edu/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He has a freely downloadable pdf of Calc I, II and III material, plus Algebra and Diff Eq.  It's a great free resource for students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also base my courses on MIT opencourseware materials:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/&quot; rel=&quot;noreferrer&quot;&gt;http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, Prof Jason Starr's notes are incredible.  Demonstrating the limit of a Riemann Sum for f(x) = x, for example, is a perfect way to tie together several topics at once for students in this environment, and it helps &quot;scare&quot; them into appreciating the power of FTC, as opposed to just memorizing it as another calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reasons for switching would be 1) less expensive for students, and 2) designing your own problem sets as a department, from the ground up.  If the goal is for students to gain a meaningful insight into Calculus, the popular books can't do it - that's not their goal.  If the goal is just to have non-majors pass the course for a requirement, then the popular books are fine and switching doesn't really change much.&lt;/p&gt;&#xA;" OwnerUserId="5337" LastActivityDate="2015-06-27T16:58:42.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8335" PostTypeId="1" CreationDate="2015-06-28T12:17:10.403" Score="19" ViewCount="1837" Body="&lt;p&gt;When I read a theorem and read its proof and fully understand it, am I supposed to know the proof even after a long time or is it natural to forget the it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask this question as I'm a self learner who forget many many proofs of the theorems I study even after reading it and understanding it (in many cases I rewrite the proof with full details entirely in my own words but even here I forget it after some time). Forgetting proofs annoys me really and I don't know if this is the natural thing or this is a something I've to deal with . &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also wonder, What should math student do with proofs? What is she expected to do with it? read it understand it and forget it or read it understand it and keep it forever? or just keeping the idea of the proof and reconstruct the details herself when needed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be concrete, in the last few weeks, I was studying Boolean algebra, ordinals and Godel first incompleteness theorem, Is it natural that I forget most of the proofs after sometime only keeping the main ideas and the main results from those topics? Is that what math student are supposed to do or am I facing a problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm going to major in math next September and I really want to know what I'm expected to do with proofs while learning for the coming years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a wider question, What are the outcomes math students should gain from studying some mathematical topic?  &lt;/p&gt;&#xA;" OwnerUserId="4694" LastActivityDate="2019-06-13T14:25:09.793" Title="Is it natural for self-learners to forget most proofs of the theorems they learn?" Tags="|undergraduate-education|proofs|self-learning|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8336" PostTypeId="2" ParentId="8335" CreationDate="2015-06-28T17:51:09.620" Score="12" Body="&lt;p&gt;You will retain something as long as you practice it. It just so happens that for many, many theorems, it's the &lt;em&gt;statement&lt;/em&gt; of the theorem that matters more than the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think a good example is the Fundamental Theorem of Algebra. The proofs for it are obnoxiously technical compared to how easy it is to state it. (You could explain the theorem to a talented high school student). The one standard proof requires half a semester's worth of complex analysis, while the other requires a whole semester's worth of Galois Theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So for a theorem like this, it's even seen sometimes where the theorem will appear &lt;em&gt;without proof&lt;/em&gt; at the start of a book. And the result is then freely used. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is perfectly practical thing to do. The main goal of any course in mathematics should not be to labor over the technical nuances, but to give the flavor and the style of the theory. And for so much of algebra and calculus, it's not really important &lt;em&gt;why&lt;/em&gt; polynomials factor completely. It's just really useful that they &lt;em&gt;do&lt;/em&gt;. If it hurts to work with an unproven assumption, the best I can offer would be to take something like this as an axiom -- albeit a redundant one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example that comes to my mind in the study of algebraic curves is Hilbert's Nullstellensatz. (It is actually a kind of multivariable version of the fundamental theorem of algebra). The theorem, used as a black box, is the heart of the elementary theory of algebraic curves. But the proof detracts from the main content of curves and dives into commutative algebra instead. If you want to study commutative algebra, it's a great topic to cover. But if you want to study curves, take it on faith. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To backtrack a bit, I think most proofs aren't worth memorizing in full. Once you have enough experience in a particular area, you will know what techniques will generally work for what kinds of problems. You can simply forget all the details and work them out as you need them. At least, for the set of theorems which have so-called &quot;follow your nose&quot; proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some theorems have proofs with a particular trick, and it's best to remember a key phrase to remind you of the trick. In linear algebra, to prove the rank nullity theorem, all I ever keep in my head is &quot;take a basis for the nullspace, then extend it&quot;. That phrase is enough for me to reconstruct the rest of the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But all in all, don't fret over forgetting things. Focus on what you find to be important, read the rest once and forget it. You'll retain the spirit of the proof, even once all the details are gone.&lt;/p&gt;&#xA;" OwnerUserId="1018" LastActivityDate="2015-06-28T17:51:09.620" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="8337" PostTypeId="2" ParentId="7718" CreationDate="2015-06-29T20:12:58.743" Score="12" Body="&lt;p&gt;To supplement the other U.S.-centric answer: yes, the U.S. standard curriculum through high school is not now and has not been in recent years comparable to several western European or former-easter-bloc education, for a variety of reasons which are irrelevant to the question. But, either way, there are some implicit hypotheses that are (in my opinion) worth being a little skeptical about. First, I am not confident that &quot;the curriculum&quot; is a complete guide to either the structure or content of mathematics, or of the right order to look at things. E.g., neither &quot;acceleration&quot; nor &quot;breadth/depth/enrichment&quot; (in the usual very flimsy sense!) necessarily recommend themselves for very-capable kids. E.g., textbooks, especially contemporary ones in the U.S., of the sort likely to be approved by school boards, are very narrow, very doctrinaire, very authoritarian, and seldom written by anyone who really knows &quot;what will/may happen later&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even some of the most-popular undergrad texts (and their adherents) seem to accomplish more in the direction of &quot;rigor&quot; and almost peevish attention to detail than giving any sort of breathtaking vistas. Apparently one is not &lt;em&gt;allowed&lt;/em&gt; to &lt;em&gt;see&lt;/em&gt; those vistas until one has paid some dues?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, returning somewhat to the question, the dangers are not only, or not so much, the possibility of immature response to the material, but that the material itself (whether accelerated-into or widened/deepened in the stereotypical ways) is misleading in several ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Another complication in the U.S. is that the system does not require, and could not currently require, high school math teachers to be acquainted with any substantive mathematics... again, political issues, but nevermind. Nevertheless, this situation makes them almost uniformly completely unprepared to give competent advice even to even-modestly-precocious high school kids.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the most poignant issues is that (certainly in the U.S.) capable kids interested in math get a very warped picture about what it is, or, rather, might be, or, rather, would eventually become, if they had advice about where to look. In particular, mathematics need be neither a &quot;school subject&quot; (with all the authority-tainted issues) nor a &quot;contest subject&quot; (with all the entailed baggage). In particular, if one steps outside those &quot;controlled&quot; situations, one &lt;em&gt;is&lt;/em&gt; &lt;em&gt;allowed&lt;/em&gt; to read about solution of equations in radical whether or not one understands well enough to &quot;do homework&quot; or &quot;pass exams&quot;. The pop-culture mythologies that reinforce &quot;rules&quot; about legal reading of mathematics may or may not be good advice for doing well in classes, in the short term, but ... especially for kids who're not just looking at mathematics because they are told to... this inculcates a very suboptimal attitude and approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a corollary, especially an interested person can often get much insight about the sense and intent of many details by seeing what happens later, how they're used, how they interact with other ideas. Most standard textbooks are really bad at telling any of that, and, in fact, often must particularly &lt;em&gt;avoid&lt;/em&gt; any whiff of connection to other things, because then it becomes more awkward to &quot;control&quot; the curriculum, etc. Things are not so interchangeable, ... don't have clear prerequisites, heaven forbid!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(These remarks in the context of having worked at pretty decent universities for 40+ years, having had the privilege of having some very good students, both undergrad and grad, along the way, and that sort of thing... but also having been an accidentally slightly precocious kid back in pre-internet days when the public library was about as good as it got, simultaneous with U.S. high schools being much more authoritarian and rule-ish. So, for example, a person might accidentally hear about &quot;calculus&quot;, and see amazing things that it can do. Or complex numbers, DeMoivre's identities, complex analysis! ... amazing stuff. Galois theory... solving cubics in radicals... algebraic number theory... Godel's theorems... set theory... all this stuff well-known to professional mathematicians, but pushed off into the hazy distance in the standard curriculum, &lt;em&gt;and&lt;/em&gt; too often completely unknown to U.S. high school math teachers. Meanwhile, in those days, high school math &lt;em&gt;was&lt;/em&gt; memorization of various goofy formulas for things, derived by as-elementary means as possible, even in scenarios where calculus would trivialize things. Elaborate trig identities... For a long time, I was unaware that there could be any sort of profession of &quot;mathematician&quot;, since it appeared that as a school subject it was just awful, although engineers and actuaries &quot;did some math&quot;. I could not get any straight answers about &quot;where&quot; the kind of mathematics in those books in the public library might be manifest. It was only by accident some years later that it became painfully clear that the world of even &quot;enhanced&quot; high school math was/is basically unrelated to anything serious... which I think is the relevant question for truly talented kids.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conclusion: for really able kids, get outside the curriculum entirely. Don't look at the curriculum for guidance. In the U.S., this would, in my estimation, refer to high school and standard undergraduate programs, which are both highly authoritarian and very limited in what is expressed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The fact that &quot;getting outside the curriculum&quot; may short-term appear disadvantageous is counter-balanced by the anecdotally-documentable fact that less authority-driven reflection on important mathematical ideas pays off both long-term and even middle-term.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-06-29T20:12:58.743" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8339" PostTypeId="1" CreationDate="2015-07-01T09:27:32.223" Score="13" ViewCount="554" Body="&lt;p&gt;Having never been a mathematics educator, my question could be stupid and, if this is the case, please delete it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was young, from the very beginning of limits, we were teached that there are two things to look at simultaneously : what is the limit and how it is approached. This means that, even when quite young, the concept of asymptotics was in our minds and, thinking back, I have the personal feeling that this was a great idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Visiting almost daily MSE, I have the feeling that this is not the case almost anywhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question : am I totally out of date ? Should I be wrong if I tried to teach that way ?&lt;/p&gt;&#xA;" OwnerUserId="4993" LastActivityDate="2015-07-01T23:49:04.570" Title="Teaching limits and asymptotics at the same time" Tags="|undergraduate-education|community-colleges|limits|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8340" PostTypeId="2" ParentId="8339" CreationDate="2015-07-01T10:21:44.720" Score="5" Body="&lt;p&gt;If I understand you, I do often think this way when solving a problem. For example, see &lt;a href=&quot;https://math.stackexchange.com/questions/1344658/functions-of-modulus/1344846#1344846&quot;&gt;this answer of mine&lt;/a&gt; at Mathematics Stack Exchange.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, limits involving infinity (as the independent or dependent variable) seem to be losing importance in textbooks. For example, I teach from &lt;em&gt;Calculus: Graphical, Numerical, Algebraic&lt;/em&gt; by Ross L. Finney et al., and that book sticks &quot;Limits Involving Infinity&quot; into one section and largely ignores the concept otherwise. This approach has some advantages, such as not needing to distinguish between a limit of plus-or-minus infinity and an undefined limit. But you see it has some disadvantages as well.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-07-01T10:28:34.330" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8341" PostTypeId="2" ParentId="8339" CreationDate="2015-07-01T23:49:04.570" Score="4" Body="&lt;p&gt;I'd recommend not being intimidated by textbook trends! &quot;In real life&quot; (as opposed to textbook-life, for sure, and often opposed to required-curriculum &quot;school-math&quot;), _of_course_ the two things you mention, &quot;the limit&quot;, and &quot;how it is approached&quot;, matter a great deal. Do not be intimidated by silly books (written by non-mathematicians, almost entirely) to think otherwise. That is, _of_course_ we care about the asymptotics as you mention. True, &quot;sadly&quot; for immature students, those details are more complicated than the more-typical (in the U.S.) symbol-pushing, and it would be hard to convince the kids to pay attention at all if &quot;it's not on the final&quot;, ... which it cannot be, mostly, because it is not the general standard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, yes, there is an underlying reality of human understanding of these things, in which asymptotics play a large role...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-07-01T23:49:04.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8342" PostTypeId="1" CreationDate="2015-07-03T02:32:06.083" Score="12" ViewCount="1530" Body="&lt;p&gt;I suppose this is the place for my questions as much as any place is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a math student coming on my 3rd year of undergrad, and I am working as a counselor at a Summer math camp. The camp is for 12-15 year old kids. About 40 are taken from across Texas each year. The kids take 38 hours of math each week for three weeks, by the end of it completing algebra 1, geometry, algebra 2, or pre-calculus. Some kids will finish the class they had initially been placed in before the three weeks are up, and they might finish two or three different courses in those three weeks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that these kids are bright, and by bright I mean &lt;em&gt;bright&lt;/em&gt;. A handful of them will most likely become mathematicians themselves. I actually did the camp when I was 15, and it paved the way for me to enter university two years early, as well as made me want to pursue math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I want to do is give these kids a problem of the day or week, depending on what I can come up with. These should be problems the higher-performing kids will be able to do, but not easily. They should take a good amount of thought, but still be doable for very gifted 12-15 year olds. I'd rather it wasn't highly technical, i.e. that they don't have to use some calculus method or something to solve it; I also want them to seem like challenge questions, rather than just more homework. Moreover, I'd rather it were fairly tough to Google.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am, however, not an educator, so I was hoping some instructors with a history of dealing with precocious and intelligent students could offer some advice. Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="5276" LastEditorUserId="5276" LastEditDate="2015-07-03T19:03:26.747" LastActivityDate="2015-07-08T07:17:49.463" Title="Challenge questions for extremely bright kids" Tags="|problem-design|gifted-students|math-puzzle|word-problems|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8343" PostTypeId="2" ParentId="8342" CreationDate="2015-07-03T03:47:53.283" Score="11" Body="&lt;p&gt;Here are some suggestions for problem sources in English. Some of them are appropriate for very bright students studying geometry or Algebra II, but might nonetheless prove too difficult for students accelerated to this extent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Mathematical Circles&lt;/em&gt;, Fomin et al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Mathematical Problems: An Anthology&lt;/em&gt;, Dynkin et al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Problems in Elementary Mathematics&lt;/em&gt;, Lidsky et al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;A Problem Book in Algebra&lt;/em&gt;, Krechmar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Many of the books in the &lt;a href=&quot;http://www.maa.org/publications/ebooks/anneli-lax-new-mathematical-library&quot;&gt;New Mathematical Library&lt;/a&gt;. (See the link for book descriptions.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Selected Problems and Theorems of Elementary Mathematics&lt;/em&gt;, Shklyarsky et al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Challenging mathematical problems with elementary solutions&lt;/em&gt;, Yaglom and Yaglom. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Problems in Plane and Solid Geometry&lt;/em&gt;, Prasolov.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;The Stanford Mathematics Problem Book&lt;/em&gt;, Polya and Kilpatrick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Five Hundred Mathematical Challenges&lt;/em&gt;, Barbeau et al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;One Hundred Problems in Elementary Mathematics&lt;/em&gt;, Steinhaus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Mathematical Scholarship Problems&lt;/em&gt;, Burkill and Cundy. (It's fascinating to see just how far the standard geometry curriculum in England could take you at one time. &lt;em&gt;Géométrie: Classe de Mathématiques&lt;/em&gt; by Hémery and Lebossé is equally telling in this regard, with respect to France in the 1950s.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Lessons in Geometry&lt;/em&gt;, Vols. 1 and 2, Hadamard (based on the 1902 French curriculum).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Geometry&lt;/em&gt;, Vols 1, 2, Kiselev (English translation by Givental).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the bibliography &lt;a href=&quot;http://www.imomath.com/index.php?options=347&amp;amp;lmm=0&quot;&gt;here,&lt;/a&gt; as well as the one in the &lt;em&gt;Mathematical Olympiad Handbook&lt;/em&gt; by Gardiner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The books &lt;em&gt;Functions and Graphs&lt;/em&gt;, &lt;em&gt;Algebra&lt;/em&gt; and &lt;em&gt;Sequences, Combinations and Limits&lt;/em&gt; by Gelfand et al. have some interesting problems in them as well, usually moderately, but not extremely, hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some high-school level books (translated into English from Russian) which, without focusing on &quot;olympiad-level&quot; problems, do aim for much higher technical proficiency than typical American precalculus books would, and they devote appropriate attention to proofs. The first two are textbooks in their own right. Sometimes the Spanish translations are easier to locate than the English ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Algebra and Analysis of Elementary Functions&lt;/em&gt;, Potapov et al. (Despite the name, there is no calculus in it except for one chapter on limits. If I'd been in a summer camp learning pre-calculus, I'd much rather have used this book than, say, Larson or some similar American book.) Here is a &lt;a href=&quot;https://archive.org/details/AlgebraYAnalisisDeFuncionesElementales&quot;&gt;link&lt;/a&gt; to the Spanish translation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Elementary Mathematics: A Review Course&lt;/em&gt;, Skanavi et al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;em&gt;Elementary Mathematics: Selected Topics and Problem-solving&lt;/em&gt;, Dorofeev et al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an excellent textbook by Boltyanski, Sidorov and Shabunin focusing on &quot;the most difficult topics in the school curriculum of algebra and elementary functions&quot;, particularly those topics &quot;not sufficiently elucidated in textbooks.&quot; Unfortunately, this book appears never to have been translated. Neither have any of the books specifically written for mathematical schools, including the excellent algebra textbooks for grades 8-11 by Vilenkin et al., the problem book by Galitski et al. for grades 8 and 9, &lt;em&gt;Algebra and Number Theory for Mathematical Schools&lt;/em&gt; by Alfutova and Ustinov, and a number of other problem books.&lt;/p&gt;&#xA;" OwnerUserId="5291" LastEditorUserId="5291" LastEditDate="2015-07-03T10:48:24.770" LastActivityDate="2015-07-03T10:48:24.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8344" PostTypeId="2" ParentId="8342" CreationDate="2015-07-03T17:20:50.960" Score="6" Body="&lt;p&gt;I work with gifted elementary school students, but one of my favorite sites, &lt;a href=&quot;http://nrich.maths.org&quot;&gt;nrich&lt;/a&gt; has challenging problems that you could use for older gifted students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try looking at secondary problems for stages 4 and 5.&lt;br&gt;&#xA;&lt;a href=&quot;http://nrich.maths.org/9233&quot;&gt;Here&lt;/a&gt; are some suggested problems to see if you'll like the site:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also look at the Post-16 Curriculum on nrich &lt;a href=&quot;http://nrich.maths.org/9088&quot;&gt;here&lt;/a&gt;. See description below.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The links below take you to a selection of &quot;rich tasks&quot; from the NRICH&#xA;  collection, chosen because they are ideal for developing subject&#xA;  content knowledge as well as mathematical thinking and problem-solving&#xA;  (process) skills.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-07-03T17:20:50.960" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="8345" PostTypeId="2" ParentId="7718" CreationDate="2015-07-03T19:10:06.450" Score="19" Body="&lt;p&gt;I have a bit of anecdotal evidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was unfortunately not homeschooled, nor did I have a technical childhood; I spent my childhood painting and writing short stories. I was in gifted classes, but I was not seen as a particularly bright student. Due to bullying I looked for alternatives to the local high schools, and ended up applying to university early to get out of town. In 2013, I graduated university at 16. Since then, I've had the full financial support of the Thiel fellowship, which allows for completely self-directed research for those who thrive in such an environment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have no formal training in pure math (my bachelors degree is in computational physics and my research was in robotics). I've been teaching myself pure math full time since July 2014. In July, someone asked me if I knew what the Atiyah Singer Index Theorem was. I didn't! I'd never learned topology or anything past numerical analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I first started, I was quite lost in the literature. I learned from the top down, and the blog posts of John Baez were the only thing on the internet I could understand which discussed the topics I was interested in. I became obsessed, and on the way latched on to trying to understand what elliptic cohomology was -- what concepts led up to it, and what concepts flowed from it -- and how it might lead to a generalization of the Atiyah Singer Index Theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I first began, I didn't know that I had anyone to talk to. I'd print out a paper by Segal or Jacob Lurie, and underline every concept &quot;x&quot; I didn't understand (which was most of the paper). Then, I'd search &quot;john baez x&quot; and read all posts he had on each concept. I did this for the first 5 months, until I gained enough vocabulary and context to begin additionally reading Lurie's &lt;em&gt;Higher Topos Theory&lt;/em&gt;, and Stolz-Teichner's &lt;em&gt;What is an elliptic object?&lt;/em&gt;. Then I moved to Berkeley and began sitting in on seminars.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you might imagine, my knowledge base is woefully incomplete and a bit more &quot;out of order&quot; than usual (learned category theory before set theory, learned what a cyclic group was a few months ago, studied Morava E-theories before basic complex analysis, etc.) This Ravi Vakhil quote resonates with me quite deeply:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning &quot;forwards&quot;. (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think this is enough context for me to go over the main issues that I face currently. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first is that I've raced ahead, and have done so by relying on intuition. It's bittersweet, a bit of a curse, to be a young person who can pick up the &lt;em&gt;structure&lt;/em&gt; of complex things quickly by reading. If I am not strong enough, the temptation to keep doing that lures away from making the mistakes and day long fiddling sessions which seem to be useless (i.e. seemingly less productive process of digging deeper and rolling around in the dirt), but are necessary for developing intuition and &lt;em&gt;creative&lt;/em&gt; obsession. I have been only recently finding the motivation to go deeper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just started writing proofs a month ago when a professor kindly began to send me exercises and then tear my responses apart. My proofs are atrocious and sloppy as I have only recently started writing them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the moment, my understanding is not sufficiently primitive and wild for me to feel comfortable and confident in expressing my own ideas clearly. I'm starting to take seriously the disease that commonly affects people of my type:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;They race ahead, and don’t learn things precisely, and worse they get into the habit of learning things imprecisely, and even worse never learn what it means to learn something precisely. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The second issue is that I am unfamiliar with the standard undergraduate curriculum (vocabulary and grabbag of examples). This leads to awkward situations. For example, I was having an exciting conversation with a grad student about elliptic cohomology theories, and asked &quot;Pardon, what is the 'characteristic of a field'?&quot; and the previously engaged and kind person became disoriented and extremely suspicious. Similar things have happened many times, though the severity has lessened now that I have learned not to take it personally and to explain my situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The third and main issue is that I resisted learning the history and context of math for so long, because I thought that was the best way to gain an original perspective, the best way to learn the art of blundering about in the darkness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that's like a writer not reading the classics to be more original and writing the same trope over and over and over. Rather, as a writer you need to reflect on your experiences, let yourself live as your character, ask how they relate to you, ask how they would act in different situations, how they develop. It's like instead of developing an idea for a novel yourself, telling it to others and scribbling down their responses to piece together a story, which is part of it. But this obviously won't work alone, you have to reflect yourself on how to weave the story together. Even if the character is an alien living in a strange world, perhaps a key development can be taken or inspired from another novel, but you can't expect to just copy and paste a part of the novel. You have to rewrite it,adjust it to fit key elements of your character and plot. Reading just one genre won't do either. Perhaps that key snibbet, that phrase that's just right, will come from a child's comedy show, or from a note bolded in an obscure french book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have slowly begun to disband my impatience to really appreciate that mathematics really is a narrative -- most of it found in classical books and concepts. But I have met far too many young students who refuse to read the classics at all for fear of &quot;losing time,&quot; preferring the slick modern exposition to the fumbling origins of the subject.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The thing to keep in mind is that all these &quot;basics&quot; were once cutting-edge research.  And the people who got to the bottom of these matters were not trivial idiots.  They were great masters.&#xA;  The more you look at this material in this way, the more you'll find you can tolerate it in good humor, and without fearing its deadening effects on your strange, wild mind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think that's enough rambling about myself, hopefully some of it was helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit (August 2018): &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shortly after the above post was written, I was accepted into UChicago for a master's program in Maths (the Bridge Program). I struggled mightily, I didn't know the basics of proof writing, and I took many classes multiple times. I loved my reading courses, and my professors were very hard on me. I could not understand why I found some things so difficult, when they were so easy for my fellow students. This allowed me to look upon many of the techniques thought of as &quot;standard&quot; with awe and appreciation. I tried ignoring my math playtime to focus on classes but that made me crumble inside, finally I let myself play (mostly in secret) to stay alive alongside my classes. I graduated with a Masters after much sweat and blood and moved to Northwestern University, where I am now entering my 2nd year in their PhD program. I love Northwestern, it is a great fit for me, as it has other grad students interested in mathematical playtime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that I have had a few years of formal training, and a systematic working through of some key textbooks on my own, my knowledge base is now mostly indistinguishable from another my academic age who took a more &quot;straightforward&quot; path. I still occasionally find impressive and unusual holes in my knowledge base, but it becomes increasingly rare, and I know to bow my head and fill them in. &lt;/p&gt;&#xA;" OwnerUserId="5312" LastEditorUserId="5312" LastEditDate="2018-09-05T15:43:31.190" LastActivityDate="2018-09-05T15:43:31.190" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="8346" PostTypeId="1" CreationDate="2015-07-04T13:29:15.140" Score="8" ViewCount="268" Body="&lt;p&gt;At the end of the school year, I gave my students a problem set related to the law of sines where many students engaged in approximation at various points. I noted two types of precision issues in their work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Looking at the following pattern 19.1, 19.03, 18.95 and concluding that the rule is that sin(A)/a, sin(B)/b, and sin(C)/c are approximately equal. I think that in this instance students are paying excessive attention to precision, rather than trying to conclude an overall rule. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) sin(60) = .9 For the purposes of this assignment and most trigonometry problems, it seems that a one digit approximation is not sufficiently precise. (Although the student who gives a 10-digit expansion also seems to be missing something)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you think is the actual issue represented by these two examples?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What knowledge/ skills are necessary for students to be appropriately precise when giving approximate answers? How do I build this capacity in my students?&lt;/p&gt;&#xA;" OwnerUserId="3485" LastActivityDate="2015-07-04T18:12:15.950" Title="Precision in student work" Tags="|secondary-education|calculations|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8347" PostTypeId="1" CreationDate="2015-07-04T14:02:27.697" Score="20" ViewCount="921" Body="&lt;p&gt;The other day, in a conversation with colleagues, I realised that the word 'algebra' means different things to us. To me, it brings to mind the study of algebraic structures: vector spaces, groups, rings, fields, algebras, modules... On the other hand, to them it means the things they called algebra at school - mostly rearranging and solving equations. These two ideas feel very different. Is there a good way of explaining how they fit together?&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="143" LastEditDate="2015-07-06T18:45:59.853" LastActivityDate="2015-12-24T13:46:13.267" Title="Difference in meaning of 'algebra'" Tags="|algebra|terminology|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="8348" PostTypeId="2" ParentId="8347" CreationDate="2015-07-04T14:46:47.657" Score="16" Body="&lt;p&gt;Elementary algebra is the study of a few specific rings: integers, rational numbers, polynomials with rational coefficients, and rational functions. All the rules for &quot;rearranging&quot; equations are applications of the ring or field axioms. Solving equations can be seen in the framework of algebraic or analytic geometry as studying the solution sets in some affine space. So, when presented in a coherent, logically precise way, elementary algebra is the first step toward abstract algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A major conceptual shift from elementary algebra to abstract algebra is in what the basic objects of the theory are. In elementary algebra, the basic objects are individual numbers. In abstract algebra, each whole &lt;em&gt;structure&lt;/em&gt; is thought of as a basic object of study. (E.g., in ring theory, we're interested in how various &quot;number systems&quot; relate to and differ from each other, not just in elements of a single number system.)&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2015-07-04T14:46:47.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8349" PostTypeId="2" ParentId="8346" CreationDate="2015-07-04T16:27:29.350" Score="9" Body="&lt;p&gt;Students have no basis for figuring out what &quot;appropriately precise&quot; means.  How much precision is appropriate is a property of the purpose of a calculation, not the calculation itself, and students rarely think about &lt;em&gt;why&lt;/em&gt; they're doing a calculation---they're doing it because they were told to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you actually want your students to make their own choices about how much and when to approximate (and make them well), that's a serious undertaking---it's really an entire additional topic to cover.  You need to give problems which include both doing some calculations and then doing something with the conclusion (your first problem looks like the start of an example---calculate sin A/a, sin B/b, and sin C/c, and decide whether or not they're equal).  More difficult, you need to have problems where the solution isn't always &quot;more digits is better&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've seen people (more often in the sciences than math, though it's certainly appropriate in math) work with numbers with errors (1.04$\pm$0.1) and expect students to report calculations correctly---that is, is accurately as possible given the uncertainty in the original data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For your second example, I share your intuition that one digit is too little precision and ten digits is too much precision.  But how would you explain that rigorously to a student, to a level of clarity that it would be fair to mark them wrong for not getting it?&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-07-04T16:27:29.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8350" PostTypeId="2" ParentId="8347" CreationDate="2015-07-04T17:02:54.230" Score="11" Body="&lt;p&gt;The main reasons both of these are called the same thing are historical.  Group theory was more or less invented by Galois to study when one could solve polynomial equations by radicals, and ring theory was more or less invented by Hilbert to solve the main problems of invariant theory, which aimed to find ways to discover when two systems of equations were &quot;the same&quot; (meaning equivalent after an appropriate change of coordinates).  It took Noether and van der Waarden to put these on a completely abstract footing, but the original motivation was to indirectly answer specific questions about solving equations.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2015-07-04T17:02:54.230" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8351" PostTypeId="2" ParentId="8346" CreationDate="2015-07-04T18:12:15.950" Score="2" Body="&lt;p&gt;There are multiple issues here. When dealing with a series of equations, taking a power, multiplying and then entering into the next calculation, it can be pretty important to hang on to all the significant digits you can through the process. But then, it's important to know when to get back to a reasonal number of significant digits. Say you have an exponential population growth. Fractional rabbits or people make no sense, so do the math, then round to an integer. Yet, when taking the tangent of an angle, see the difference between the Tangent of 89 degrees and 89.5 degrees. &#xA;In the end, you can help them understand that significant digits need to adjust to the situation. I wouldn't discuss it as an entire lecture but as a brief note in each sets of new problems.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-07-04T18:12:15.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8352" PostTypeId="1" CreationDate="2015-07-04T20:06:30.900" Score="6" ViewCount="267" Body="&lt;p&gt;Suppose I would like to make a maths test that one can do by computer and that can be automatically checked by computer. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there some automatic theorem prover that can check the proofs students write? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think that, for example, Coq does not understand the body text. Or is the only possibility to write the question such that there is just one right answer for every question?&lt;/p&gt;&#xA;" OwnerUserId="5357" LastEditorUserId="262" LastEditDate="2015-07-07T00:39:35.933" LastActivityDate="2017-01-18T02:20:04.860" Title="Are there automatic solution checkers?" Tags="|technology-in-education|proofs|testing|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8353" PostTypeId="2" ParentId="8321" CreationDate="2015-07-04T23:49:43.610" Score="2" Body="&lt;p&gt;You could try the &lt;a href=&quot;https://wordpress.org/plugins/jetpack/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jetpack Plugin&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://tex.stackexchange.com/questions/76502/is-there-any-mathematical-blog-support-latex&quot;&gt;From this post&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;By 'supporting LaTeX', many people mean 'supporting the use of TeX-like input for mathematical symbols'. That's not the same thing: support LaTeX proper requires a TeX system, supporting just a small subset does not.&lt;/p&gt;&#xA;&lt;p&gt;Taking WordPress as an example, both TeX-like input and proper LaTeX support are available. By far the most straight-forward way here is to use a plugin to load the appropriate code from elsewhere. MathJax is the technology that is used for example on other StackExchange sites for TeX-like input, and it's available for WordPress using the JetPack plugin. This plugin makes use of the WordPress.com MathJax support.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also as a commenter pointed out you could use MathJax and while Wordpress disables JavaScript you could use the &lt;a href=&quot;https://wordpress.org/plugins/simple-mathjax/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simple MathJax &lt;/a&gt; plugin to load it.&#xA;The above post shows an example, the second answer I believe, of how to incorporate the Mathjax cdn and script into your file.&lt;/p&gt;&#xA;" OwnerUserId="5358" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-07-04T23:49:43.610" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8355" PostTypeId="2" ParentId="8167" CreationDate="2015-07-05T13:05:25.407" Score="8" Body="&lt;p&gt;Thanks everyone for your tips! I thought I would share what we decided to go for in the end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We had far more material than this (it naturally extends to the investigation into mobius strips, but in an hour this is a good amount of material. Even for top set Year 8 students, who we also did this lesson with in the end).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What surface is the game &quot;snake&quot; played on?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;As a class on the projector, we played &lt;a href=&quot;http://www.learn4good.com/games/action/snake.htm&quot; rel=&quot;noreferrer&quot;&gt;snake&lt;/a&gt;, both with the walls turned &quot;on&quot; and &quot;off&quot;.&lt;/li&gt;&#xA;&lt;li&gt;After discussion of what shape the snake might be living on when walls are off (consensus was it was a sphere), pupils each given strip of paper, told to attach one pair of opposite sides together (first predict, and then do - hard to work out it is a cylinder in advance!) What would this correspond to in the snake game?&lt;/li&gt;&#xA;&lt;li&gt;Next, pupils did &quot;cross activity&quot; &lt;a href=&quot;https://www.youtube.com/watch?v=JNtKcK27x1s&quot; rel=&quot;noreferrer&quot;&gt;see 5.13 in video&lt;/a&gt;, after good discussion on what might happen when we cut along the centre of each loop (how many bits will the paper fall into? what shapes will we see?)&lt;/li&gt;&#xA;&lt;li&gt;Played &lt;a href=&quot;http://youtu.be/eNuRurUhpRs&quot; rel=&quot;noreferrer&quot;&gt;a video&lt;/a&gt;, pausing it before giving the game away.&lt;/li&gt;&#xA;&lt;li&gt;At this stage (same point in the three lessons we did with different pupils), pupils able to work out living on doughnut (we didn't call it a torus). &lt;/li&gt;&#xA;&lt;li&gt;Lesson finishes with one pupil travelling as a snake around a square, while teacher mirrors the movement on an &lt;a href=&quot;http://www.amazon.co.uk/gp/product/B003A4YPEK?keywords=inflatable%20ring&amp;amp;qid=1436101364&amp;amp;ref_=sr_1_13&amp;amp;sr=8-13&quot; rel=&quot;noreferrer&quot;&gt;inflatable ring&lt;/a&gt; (this allows for a dramatic unveiling of the inflatable ring from its hiding place - nice)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5225" LastEditorUserId="511" LastEditDate="2015-07-06T23:12:22.970" LastActivityDate="2015-07-06T23:12:22.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8356" PostTypeId="1" AcceptedAnswerId="8357" CreationDate="2015-07-05T16:27:38.967" Score="17" ViewCount="2392" Body="&lt;p&gt;I found this question on another forum. Are there any hints how one could make pictures from geometric problems by coding? I mean, if one has a handicap in his hands and he can't use mouse very well, but he would like to learn to draw pictures like &lt;a href=&quot;https://i.stack.imgur.com/28os8.png&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;  and &lt;a href=&quot;https://problemsolversparadise.files.wordpress.com/2012/04/prop122.png?w=652&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; by coding. What would be the easiest method to do such a pictures? Python? Some LaTeX package?&lt;/p&gt;&#xA;" OwnerUserId="5362" LastEditorUserId="2074" LastEditDate="2015-07-05T18:29:16.397" LastActivityDate="2017-05-02T07:24:18.367" Title="A text-based program to draw geometric figures" Tags="|geometry|software|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8357" PostTypeId="2" ParentId="8356" CreationDate="2015-07-05T18:29:33.117" Score="9" Body="&lt;p&gt;One possibility is to use &lt;a href=&quot;https://www.geogebra.org/&quot;&gt;GeoGebra&lt;/a&gt;.&#xA;I use it for making any geometrical drawings I need, especially if I want to show them in public.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It can be used via text, too, but I've only used it via the graphic interface.&#xA;One benefit is that even in text mode this is WYSIWYG so you don't have to compile to see what you'll get.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2015-07-05T18:29:33.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8358" PostTypeId="2" ParentId="8356" CreationDate="2015-07-05T21:36:04.313" Score="7" Body="&lt;p&gt;There are indeed several LaTeX packages available for this. I have only experience with one - TikZ. The best impression of what it can do is probably the corresponding &lt;a href=&quot;http://www.texample.net/tikz/&quot; rel=&quot;noreferrer&quot;&gt;Texample page&lt;/a&gt;. If you want to learn it, at least the introductory chapters of the manual coming along with the package are rather good. While &lt;a href=&quot;https://www.ctan.org/pkg/pgf&quot; rel=&quot;noreferrer&quot;&gt;looking them up&lt;/a&gt; I noticed that there's also a shorter introduction document in there now. I haven't worked with that one, but its content at least contains something about tangency, which is probably close to what you might be interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's also (at least) PSTricks - but I've never worked with that one. It seems to have more links to postscript, so I'm a bit sceptical of its use in general circumstances...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back in school we have used the programming language Logo which comes along with a graphical toolkit (Turtle graphics). That should also work - but I haven't used it since school. Skimming over the wikipedia articles it seems to have a lot of different implementations, sometimes standalone from Logo, and sometimes with extensions for 3D graphics.&lt;/p&gt;&#xA;" OwnerUserId="5346" LastActivityDate="2015-07-05T21:36:04.313" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8359" PostTypeId="2" ParentId="8356" CreationDate="2015-07-05T22:40:37.903" Score="2" Body="&lt;p&gt;I second Geogebra. It's free, runs in a browser and quite slick. I haven't worked how to properly script with it yet, but there must be a way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, &lt;a href=&quot;https://scratch.mit.edu/&quot; rel=&quot;nofollow&quot;&gt;Scratch&lt;/a&gt; can draw in a logo/turtle graphics style.&lt;/p&gt;&#xA;" OwnerUserId="998" LastActivityDate="2015-07-05T22:40:37.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8360" PostTypeId="2" ParentId="8356" CreationDate="2015-07-06T07:51:11.227" Score="4" Body="&lt;p&gt;GeoGebra seems nice, but for that figure you posted looks as if &lt;a href=&quot;http://www.tug.org/metapost.html&quot; rel=&quot;nofollow&quot;&gt;metapost&lt;/a&gt; is the tool of choice. What I don't like about GeoGebra is, that you need to login at least when I try to install the linux version and just start it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;TikZ&lt;/code&gt; is nice, but for my taste it's better to build the figures apart from the document you want to include them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many other tools for this. See the old &lt;a href=&quot;http://www.kohala.com/start/troff/troff.html&quot; rel=&quot;nofollow&quot;&gt;troff resource&lt;/a&gt; for example. But the troff pic macros are better used for more simple diagrams and don't integrate that good with the LaTeX rendering process.&lt;/p&gt;&#xA;" OwnerUserId="5364" LastActivityDate="2015-07-06T07:51:11.227" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8361" PostTypeId="1" CreationDate="2015-07-06T12:35:33.977" Score="3" ViewCount="132" Body="&lt;p&gt;I am an undergraduate student, studying towards a Degree in Applied Mathematics and Statistics. I am studying at an online university specialised in distance learning and I am therefore developing independent study skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, after achieving my degree I would like to get a Master Degree at the University of Bonn in Germany (the teaching language is English). I am therefore independently carrying out studies in Pure Mathematics to fill the gaps in my knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please, I need detailed advice about adequately preparing myself for this goal, in particular about important topics to be covered and skills to be developed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, if anyone of you has attended undergraduate studies or a Master programme at the University of Bonn or in Germany in general, I'd like to read your advice and experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any reply will be most appreciated&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many thanks in advance&lt;/p&gt;&#xA;" OwnerUserId="5365" LastActivityDate="2015-07-06T12:35:33.977" Title="How can I adequately prepare myself for a Master in Mathematics?" Tags="|graduate-education|self-learning|" AnswerCount="0" CommentCount="4" ClosedDate="2015-07-16T14:33:05.640" ContentLicense="CC BY-SA 3.0" />
  <row Id="8362" PostTypeId="2" ParentId="8356" CreationDate="2015-07-06T13:15:43.357" Score="3" Body="&lt;p&gt;For completeness, I suggest here two other graphics languages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first one is a powerful graphics language with a C++-like syntax named &lt;a href=&quot;http://asymptote.sourceforge.net/&quot; rel=&quot;nofollow&quot;&gt;Asymptote&lt;/a&gt;. It can be easily interfaced with LaTeX and provide commands which are suitable to draw geometric figures (e.g. commands for finding intersections between curves).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it's not for the faint of heart, and the learning curve can be steep. Suggested to an experienced programmer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second is a much simpler graphics language developed by Brian Kernighan (in &lt;a href=&quot;http://dl.acm.org/citation.cfm?id=806459&quot; rel=&quot;nofollow&quot;&gt;this paper&lt;/a&gt; you can find an introduction to it) called PIC. Dwight Aplevich developed a &lt;a href=&quot;https://ece.uwaterloo.ca/~aplevich/dpic/&quot; rel=&quot;nofollow&quot;&gt;PIC interpreter&lt;/a&gt; called DPIC, which comes with a few extension to the original language. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;PIC is not as powerful as Asymptote (e.g. it lacks of commands for finding intersections), but it has a much simpler syntax.&lt;/p&gt;&#xA;" OwnerUserId="5367" LastActivityDate="2015-07-06T13:15:43.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8363" PostTypeId="1" CreationDate="2015-07-06T15:46:59.230" Score="13" ViewCount="148" Body="&lt;p&gt;I am preparing for my second year of implementing &lt;strong&gt;standards-based grading&lt;/strong&gt; (SBG) in my 5th and 6th grade math course. I was thrilled with the benefits of using SBG with my students but found that I need to do more parent education. While I present a summary of SBG at my school's Parents' Weekend, not all parents attend or remember the details. I am, therefore, trying to develop a resource that I can link to in my email signature that will be accessible year-round.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for resources, &lt;strong&gt;accessible to parents&lt;/strong&gt;, that summarizes why SBG is beneficial for students. The ideal source would be short (1-2 pages), outline the motivations for abandoning traditional grading processes, discuss the concept of growth mindset, and include details of how parents can support the SBG process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am happy to read through a big collection of articles, so please share anything you think might be pertinent.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Update: I have found some articles that are close to what I am looking for, but are not quite right. In general, most of the articles that I have found tend to spend too much time outlining a specific system for employing SBG or tend to be much too long. For example&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;a href=&quot;http://www.ncte.org/library/NCTEFiles/Resources/Journals/EJ/1031-sep2013/EJ1031Better.pdf&quot; rel=&quot;nofollow&quot;&gt;Miller - &quot;A Better Grading System&quot;&lt;/a&gt; This article is too long for my needs and spends significant time describing a SBG system&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;a href=&quot;https://www.ocps.net/lc/southwest/mso/parents/Documents/Grades%20That%20Show%20What%20Students%20Know.pdf&quot; rel=&quot;nofollow&quot;&gt;Marzano and Heflebower - &quot;Grades That Show What Students Know&quot;&lt;/a&gt; Similar issues to Miller's article above: too long and too much detailing of how a specific grading system would work.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;" OwnerUserId="5368" LastEditorUserId="5368" LastEditDate="2015-07-17T22:29:31.450" LastActivityDate="2015-07-17T22:29:31.450" Title="Seeking Concise Article on Motivations for and Benefits of Standards-Based Grading" Tags="|primary-education|grading|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8364" PostTypeId="2" ParentId="8321" CreationDate="2015-07-06T16:45:37.917" Score="2" Body="&lt;p&gt;MathJax seems to work well in Blogger, although this simple example doesn't irrefutably prove that:&#xA;&lt;a href=&quot;http://fivetriangles.blogspot.com/2014/07/171-fraction-redistribution.html&quot; rel=&quot;nofollow&quot;&gt;http://fivetriangles.blogspot.com/2014/07/171-fraction-redistribution.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5370" LastActivityDate="2015-07-06T16:45:37.917" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8365" PostTypeId="2" ParentId="6150" CreationDate="2015-07-06T16:46:19.053" Score="4" Body="&lt;p&gt;The answer to this question is likely to vary based on the way schools' departments are organized, the interests and motivations of the leaders/overseers of the math and science programs, the applicable standards for math and science courses, and many other variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have some background to comment since I have taught both math (grades 5 and 6) and science (grades 8 and 9). But my experience is only representative of one PK-9 independent school in Massachusetts. At my school, there has been little or no discussion over what mathematics is expected to be known by students when planning science instruction in my 7 years at the institution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a few occasions, I have initiated conversations with science teachers when I have become aware that they are attempting to teach concepts that required certain mathematical understanding that most students did not yet possess. For example, teaching the density formula before students had an understanding of proportion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The result of these conversations (in my experience) is often that science teachers ask the math teacher how they can explain/teach that concept concisely rather than rethinking whether it is appropriate for the students' current levels of development.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you would like to investigate this question further, I would suggest comparing the &lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot; rel=&quot;nofollow&quot;&gt;Common Core State Standards for Math&lt;/a&gt; with the &lt;a href=&quot;http://www.nextgenscience.org/&quot; rel=&quot;nofollow&quot;&gt;Next Generation Sciences Standards&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not familiar enough with the NGSS to know how familiar the authors are with the CCSSM or with math education in general, but it is likely that writers of the NGSS are more likely to have considered background mathematics knowledge than the average teacher of science. A cursory (5 minute) search of the NGSS website revealed no reference to math education.&lt;/p&gt;&#xA;" OwnerUserId="5368" LastActivityDate="2015-07-06T16:46:19.053" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8366" PostTypeId="2" ParentId="8363" CreationDate="2015-07-07T11:31:21.250" Score="8" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;From ACSD, Seven Reasons for Standards Based Grading -&lt;a href=&quot;http://www.ascd.org/publications/educational_leadership/oct08/vol66/num02/Seven_Reasons_for_Standards-Based_Grading.aspx&quot; rel=&quot;nofollow&quot;&gt;Educational Leadership&lt;/a&gt;&#xA;This article explains what SBG is and why it is beneficial.  It is written in a way that parents can relate to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the points made:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Grades and assessments become more meaningful for students, parents, and teachers.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A chart is included showing how students might perform on different assignments and how SBG will make coherent sense of the variations. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;SBG is part of the larger picture of reform.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2015-07-07T18:39:56.643" LastActivityDate="2015-07-07T18:39:56.643" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8367" PostTypeId="2" ParentId="1112" CreationDate="2015-07-07T16:56:38.080" Score="7" Body="&lt;p&gt;(based on experience, not any formal reference)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am from the UK, where the first system (absolute grades, ish) is used fairly uniformly. As I understand it, the basic idea is that the purpose of an exam is to decide whether someone has achieved the level required to gain a qualification (or is progressing at the expected rate). The question being asked is 'has this person reached the level of competency that this qualification is intended to certify?' The answer to that question does not depend on how well others in the class do. Standards should be measured by actual achievement; having everyone doing uniformly badly is not acceptable (yes there's a problem with this, in that tests focus on a small proportion of what a student is capable of).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless it is important to keep an eye on the distribution, as it's hard to consistently write assessments at exactly the same level each time, so there has to be some adjustment for that. We do also have a problem with grade inflation over time, as there is a high incentive to focus on grades alone. I think that has more to do with government targets than with the method of assigning grades.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-07-07T16:56:38.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8368" PostTypeId="2" ParentId="6" CreationDate="2015-07-07T22:33:10.880" Score="5" Body="&lt;p&gt;I actually did research on this (unpublished, but I collected hundreds of questionnaires) and students overwhelmingly claimed to use W|A to check their answers, to see the steps to a solution, etc.  In the case of websites like physicsforums or crazyproject where people may be able to copy entire proofs, the students I surveyed again claimed to be engaged in some kind of learning behaviour.  I don't see how reading a proof on the internet is any different from having your professor explain it to you in class.  In both cases there's a transfer of information from an authority to the student, and the student gains new knowledge.  It would be crazy for students to come into a calculus course expecting to develop all of the ideas and methods on their own.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students have always been able to trivialize math assignments, and it's up to them as self-interested actors to exercise discipline and do what's best for their grade in the long-term.  Copy-pasting a proof from the internet and not attempting to understand it (the absolute worst case scenario) won't help them on the final exam.&lt;/p&gt;&#xA;" OwnerUserId="5141" LastActivityDate="2015-07-07T22:33:10.880" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8369" PostTypeId="2" ParentId="8342" CreationDate="2015-07-08T06:41:48.733" Score="3" Body="&lt;p&gt;The website &lt;a href=&quot;http://mathschallenge.net/&quot; rel=&quot;nofollow&quot;&gt;mathschallenge.net&lt;/a&gt; has pdfs of tiered problems with solutions. They are nicely posed problems, often with some humour, and are based on different stages of mathematical maturity. To begin with they have no real prerequisites, but eventually they require some of the 'canonical' mathematical knowledge. I just found them to be well organised, interesting and adaptable. From the outset, the solver is asked to generalise a particular problem, which lends itself to proof techniques. &#xA;The site is from the same developers as projecteuler, to give you some idea as to its credentials. Just click on the 'problems' tab at the top of the page.&lt;/p&gt;&#xA;" OwnerUserId="901" LastEditorUserId="77" LastEditDate="2015-07-08T07:17:49.463" LastActivityDate="2015-07-08T07:17:49.463" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8370" PostTypeId="2" ParentId="8167" CreationDate="2015-07-08T14:29:35.343" Score="4" Body="&lt;p&gt;While the geometry of surfaces is a relatively quick starting topic it does not seem to me that it is the ideal way to show an 11 year old how much geometric insight can be obtained without metrical information. My favorite example for introducing topological ideas is &quot;Euler's traversability&quot; theorem which says that a graph (diagram of dots and lines) has a &quot;closed tour&quot; - one that starts and ends at the same dot (vertex) that traverses each edge of a connected (one piece) graph once and only once if and only if the graph has all its vertices even-valent (even degree). This result can be discovered by a series of examples posed to a &quot;beginner&quot; and often a beginner can see the parity argument that can be used to prove the result in one direction, and also the way to design an algorithm to find such a tour in the other direction. There is also the fact that many applied problems such as snow removal, pot-hole inspection tours etc. have this theorem as a jumping off point. For these applied problems one needs to look at what is often called the &quot;Chinese postman&quot; model, which builds on the traversability theorem. &lt;/p&gt;&#xA;" OwnerUserId="1865" LastEditorUserId="1865" LastEditDate="2015-07-10T15:22:58.253" LastActivityDate="2015-07-10T15:22:58.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8371" PostTypeId="1" CreationDate="2015-07-08T19:41:35.187" Score="19" ViewCount="3374" Body="&lt;p&gt;I've noticed that most Lagrange multiplier problems I've seen can be solved with other methods.  Often the method of Lagrange multipliers takes longer than the other available methods.  I don't like forcing my students to use Lagrange multipliers on a problem that has an easier solution via other methods, but I'm having trouble coming up with problems where the method of Lagrange multipliers is the best solution.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally I'm looking for problems that are easy enough to put on a quiz or test (some of the equation solving can be quite time consuming for these types of problems).  Does anyone have any good ideas for problems or ideas for where I could look?  &lt;/p&gt;&#xA;" OwnerUserId="1445" LastEditorUserId="262" LastEditDate="2015-07-08T20:06:38.903" LastActivityDate="2015-09-25T17:33:58.293" Title="Good examples of Lagrange multiplier problems" Tags="|undergraduate-education|calculus|problem-design|vector-calculus|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8372" PostTypeId="2" ParentId="8371" CreationDate="2015-07-08T23:05:16.997" Score="14" Body="&lt;p&gt;A place that Lagrange Multipliers comes up is in the proof of the real spectral theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Namely, let $A$ be a symmetric matrix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(v) = v^\top A v$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you maximize $f$ on the unit sphere in $\mathbb{R}^n$, the Lagrange Multiplier condition will show that the maximum is achieved at an eigenvector of $A$.  This is the hard part of the real spectral theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure how much linear algebra your students have, but this could be an appropriate exercise on a test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also important for the second derivative test later, since analyzing the definiteness of the Hessian of a function can be approached systematically by finding the eigenvalues of this Hessian.  The real spectral theorem provides a theoretical basis for the existence of these eigenvectors.  &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2015-07-08T23:05:16.997" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8373" PostTypeId="2" ParentId="8371" CreationDate="2015-07-09T01:59:06.497" Score="6" Body="&lt;p&gt;A question with two constraints might make the method seem preferable to finding a parameterization (which I assume is the &quot;easier&quot; technique you refer too in the OP).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, maximizing $f(x,y,z)=x−y$ subject to the two constraints that $x^2+y^2+z^2=1$ and $x+y+z=1$ might be difficult for students to do without Lagrange multipliers.  Is this form of the theorem treated in your course, or only one constraint?&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2015-07-09T01:59:06.497" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8375" PostTypeId="2" ParentId="8371" CreationDate="2015-07-09T14:22:09.350" Score="2" Body="&lt;p&gt;I am also vaguely aware that in economics, the actual value of $\lambda$ is often important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say you have a function $f$ which needs to be optimized subject to a constraint $g = c$.  Then it turns out that $\frac{d}{dc} \operatorname{max}(f) = \lambda$, where $\lambda$ is the lagrange multiplier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exploring this set of ideas could be interesting, and motivate the need to use Lagrange multipliers, since the other methods do not give this extra data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting way to get at this idea might be to come up with an &quot;applied&quot; problem where the constraint value $c$ could reasonably vary.  Then have them use lagrange multipliers to find the maximum of $f$ when $c=100$ and when $c=101$.  Then observe that the maximum increased by approximately $\lambda$.  Rig up your problem so that this is fairly obvious (maybe the max in one case is $1000$, $\lambda$ is $21$ and the new maximum is $1021.01352...$).&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2015-07-09T14:36:57.943" LastActivityDate="2015-07-09T14:36:57.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8376" PostTypeId="2" ParentId="8328" CreationDate="2015-07-09T14:23:10.963" Score="2" Body="&lt;p&gt;Here's my journey to understanding the real numberline:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aged 15, Year 11:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was first introduced to the idea of irrational numbers and shown that $\sqrt{2} $ and $\sqrt {3} $ are irrational. I new $\pi $ was special because it was transcendental but I knew little more about transcendence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aged 16, during lower sixth form:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My motivation for wanting to explore the question &quot;what is a real number?&quot; came from realising the difference in cardinality between the rationals and reals (through Cantor's diagonal argument). It really emphasised the difference between the set of rationals and set of reals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aged 17,  during upper sixth form:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Learning of the density of the rationals and irrationals on the reals was also fascinating. I especially liked the idea that the real  number line is full of &quot;holes' if the rationals or irrationals are omitted. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aged 18, during first year of undergraduate degree:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to the Completeness Axiom I found it fruitfull to consider and prove its equivalent statements. Personally I think &quot;every decimal expansion converges&quot; is the most intuitive statement of the axiom. However I didn't become convinced of its fundamental importance until I learnt of its equivalence to so many other propositions, especially: the principal of monotone convergence, Bolzano-Wierstrauss, Cauchy sequences, least upper bound property and nested intervals theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I advocate the book below as a highly accessible introduction to real analysis:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Numbers and Functions: Steps into Analysis, R. P. Burns&lt;/p&gt;&#xA;" OwnerUserId="3407" LastEditorUserId="3407" LastEditDate="2015-08-09T15:54:55.710" LastActivityDate="2015-08-09T15:54:55.710" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8377" PostTypeId="2" ParentId="3989" CreationDate="2015-07-09T22:15:46.733" Score="-4" Body="&lt;p&gt;I prefer 1, 4 credit course, but 2, 3 credit courses is fine, especially at the Community College level. What I &lt;strong&gt;don't like&lt;/strong&gt; is when Community Colleges, like in Connecticut, say they are teaching the 4 credit version, but in fact only teach 3 credits worth. They somehow forget about Line Integrals and Surface Integrals, even though it is clearly included in the course descriptions. Why? Put another way, &quot;Why would any College not teach the Fundamental Theorems of Multivariable Calculus in any course named Multivariable Calculus? I've been asking this question to Connecticut Department of Higher Education &quot;officials&quot; and Community College Presidents for a year now. &lt;strong&gt;No answer. Shame on them all&lt;/strong&gt;.    Professor Steven J Toce&lt;/p&gt;&#xA;" OwnerUserId="5376" LastActivityDate="2015-07-09T22:15:46.733" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8378" PostTypeId="2" ParentId="8371" CreationDate="2015-07-10T16:09:42.327" Score="2" Body="&lt;p&gt;Of course, I endorse Steve Gubkin's answers as manifestly more relevant to multivariable calculus. What follows is a second or third order motivation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One interesting bit which connects $n$-dimensional constrained analysis to $n+1$ critical point analysis is the following trick: I'll just indicate it for $n=2$: suppose we wish to find extrema of $f(x,y)$ on the bounded constraint curve $g(x,y)=c$. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Define $F(x,y,\lambda) = f(x,y)-\lambda (g(x,y)-c)$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here $\lambda$ plays the usual role of $z$. In this notation: $\nabla F = \langle \partial_x F, \ \partial_y F, \ \partial_{\lambda} F \rangle$. Then the condition $\nabla F = 0$ yields:&#xA;$$ \nabla F = \langle \partial_x f-\lambda \partial_x g, \ &#xA;\partial_y f-\lambda \partial_y g, \ g-c \rangle . $$&#xA;Which gives us both the colinearity of the two-dimensional gradients of $f$ and $g$ as well as the constraint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, one way we can understand the method of Lagrange multipliers is that it is trading a constrained problem in $n$ dimensions for an unconstrained problem in $n+1$ dimensions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the start of how this appears in Junior level mechanics see pages 275-281 of &lt;a href=&quot;https://books.google.com/books?id=P1kCtNr-pJsC&amp;amp;pg=PA275&amp;amp;lpg=PA275&amp;amp;dq=lagrange%20multipliers%20in%20classical%20mechanics&amp;amp;source=bl&amp;amp;ots=giRx08FCC-&amp;amp;sig=5iwWIcxfKOk8arOOsN4lUxJfcEQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=ZOOfVcKlCYHtgwSTm5KoDA&amp;amp;ved=0CE8Q6AEwBg#v=onepage&amp;amp;q=lagrange%20multipliers%20in%20classical%20mechanics&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Taylor's Classical Mechanics&lt;/a&gt; or the related &lt;a href=&quot;https://physics.stackexchange.com/q/89834/16779&quot;&gt;PSE question&lt;/a&gt;. Page 20 or so of &lt;a href=&quot;http://www.thphys.nuim.ie/Notes/MP350/lagrange.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jon-Ivar Skullerud's Notes from MP350 Classical Mechanics&lt;/a&gt; has a very readable presentation (quite close to what I saw in school as a physics undergrad). Some nice (mostly math) examples &lt;a href=&quot;https://en.wikipedia.org/wiki/Lagrange_multiplier#Examples&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia Example page&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, to make good on my earlier comment, the Auxiliary variable concept is behind many of the techniques used in modern theoretical physics. At the moment, I can't find a good general article on this. I recall the idea from conversations with other graduate students in physics. Roughly, the idea is when we have a global symmetry of physics then we can implement it as if it was a dynamical symmetry by introducing auxiliary fields. I think the idea is that some of these may have had interactions at an early point in the universe, but, at the moment the symmetry is frozen hence the equations governing the field are merely algebraic. Algebraic symmetries are quite important, the particles seen form representations of these symmetry groups and basically we are able to predict (at times see the $\Omega_-$ story) the existence of new particles from a pattern. All of this said, the use of the term &quot;auxiliary&quot; in physics is much more general than that particular use I sketch above. Generally, there is always a question of whether we leave a constraint unimposed, or apply the constraint. A professor I studied GR from apparently found an (unphysical) solution to quantum gravity after decades of work because he finally decided to work with the unconstrained problem and only apply the constraint at the end of the analysis. It's easy to see your wrong in related rates when you fix everything to constant and get a obviously wrong result. It's much harder to ascertain you are in error when the analysis is based on a partial fixing which leaves some dynamics. Here is a token example of such terminology: &lt;a href=&quot;https://physics.stackexchange.com/q/105669/16779&quot;&gt;BRST and auxiliary field&lt;/a&gt;. I think there must be a better article somewhere, but I leave it at this for now.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:17.873" LastActivityDate="2015-07-10T16:09:42.327" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8380" PostTypeId="1" CreationDate="2015-07-12T13:55:40.407" Score="9" ViewCount="343" Body="&lt;p&gt;I'm new math teaching assistant in faculty of engineering in Egypt. And after one term only, I found 3 big problems here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) There is no help in teaching me how to teach. It's up to my own skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) There is no obvious goal for teaching math. what is important to focus on? proofs, concepts, or applications ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Fresh students who came from school lack almost all concepts behind calculations. They don't know what is the &quot;Concept&quot; of any of the math topics from school. They can only make calculations. In addition, the majority of them lack creativity and curiosity, they just want to know what make them pass in exam!.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to know what to start with?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the most important to know about when I teach math? Any source talking about best way to teach math?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is my mission when I teach math for engineering students ? to build proof constructing mind ? to focus only on applications ? to mix between proofs and aps. ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And finally, how to deal with the problems created in school phase ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm grateful for your help. :)&lt;/p&gt;&#xA;" OwnerUserId="5383" LastEditorUserId="376" LastEditDate="2015-07-13T05:55:19.280" LastActivityDate="2019-04-10T08:10:37.680" Title="New math. teaching assistant facing big problems, what to start with?" Tags="|undergraduate-education|applied-mathematics|teacher-training|engineering-mathematics|" AnswerCount="3" CommentCount="8" ClosedDate="2015-07-18T01:28:16.413" ContentLicense="CC BY-SA 3.0" />
  <row Id="8381" PostTypeId="2" ParentId="8380" CreationDate="2015-07-12T23:12:18.833" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;What is my mission when I teach math for engineering students?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think it is possible to zig-zag between applications and appreciation of&#xA;the history and the beauty of the mathematics underlying those applications.&#xA;An example is the &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Simple_suspension_bridge&quot; rel=&quot;nofollow noreferrer&quot;&gt;simple suspension bridge&lt;/a&gt;,&quot; which follows&#xA;a &lt;a href=&quot;https://en.wikipedia.org/wiki/Catenary&quot; rel=&quot;nofollow noreferrer&quot;&gt;catenary&lt;/a&gt;.&#xA;There is interesting history here, in that Galileo realized the curve was not&#xA;a parabola, but he did not derive its analytical shape.&#xA;In fact, the two curves are &lt;em&gt;very&lt;/em&gt; close, but the catenary follows&#xA;$y = \cosh(x)$:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/ol9pE.png&quot; alt=&quot;enter image description here&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Catenary: black. Parabola: red. Image (detail) from &lt;a href=&quot;https://en.wikipedia.org/wiki/Catenary&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;One could connect to architecture via the inverted catenary arch:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/I2VPF.jpg&quot; alt=&quot;CasaMila&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(&lt;a href=&quot;http://www.house-design-coffee.com/art-nouveau-architecture-next.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gaudi Casa Mila&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;This is just one topic, with deep and rich material.&#xA;But I believe, with effort, most topics of interest to engineering students&#xA;have similar rich history and mathematics behind them.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-07-13T14:18:59.943" LastActivityDate="2015-07-13T14:18:59.943" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8382" PostTypeId="2" ParentId="8380" CreationDate="2015-07-13T07:16:38.903" Score="7" Body="&lt;p&gt;I would start with a discussion with the Engineering department about what it is they want the students to learn. Teaching different skills will need different methods, so you might as well start with learning what will help you most in the short term. Personally, my experience suggests proofs are likely to be very low priority, with a focus on calculation being what is needed for later use, and a focus on applications needed to keep the students happy. This might be very different in your context though. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of learning how to teach, there are some useful resources around, if you know where to look. One I recommend is a small &lt;a href=&quot;https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/8846/1/Alcock_Simpson_book.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;book by Alcock and Simpson&lt;/a&gt;. It isn't exactly written for your situation, but I think it is helpful in explaining how things look from a student's perspective (especially when it comes to understanding concepts).&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="376" LastEditDate="2019-04-10T08:10:37.680" LastActivityDate="2019-04-10T08:10:37.680" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8383" PostTypeId="2" ParentId="8380" CreationDate="2015-07-13T16:58:56.830" Score="5" Body="&lt;p&gt;I think you need to build a support network locally.  Even if it means going outside the university to a nearby university, you will need mentors, fellow TA's, administrators, and others to help you not just understand and solve the problems you currently see, but to anticipate problems and improve your efforts as a teaching assistant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even going outside your department, there should be SOMETHING available at your university.  Even if you are the only TA for math for the engineers, other teaching assistants may be able to answer your specific questions even if they are outside the field of the questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Point 1): there IS help teaching you to teach; you have to find it.  Asking here is a good start, as there are some online resources that you can find with a web search.  However, you will need to consult people as many answers depend on the immediate or near environment.  (To show off my ignorance, if you have to teach an example of the pigeonhole principle by having students put on or take off hats, and you have a class full of women wearing burkas or some prohibition where wearing hats is socially unacceptable, then my hats example would not be appropriate.  I don't know the cultural norms in your area, so stuff I say may turn out to be singularly inappropriate.)  You need to develop (and become) local resources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Point 2)  There are obvious goals, they just are unclear and not well formed.  An obvious goal is that the engineers need to learn the stuff to either perform certain skills or to acquire material that depends on the stuff you are teaching.  If you don't know what they need, find out.  However (and I think this is more important) you need to teach them how to recognize what is and what isn't useful, and how to acquire and apply more knowledge.  They will be addressing various engineering problems and related problems, and your job gives you an opportunity to address both kinds.  They will be better practitioners if they not only can do the calculation/process/design, but decide when and how it should be done, and where they can go to look up the stuff they need to do it.  As a brief example, in showing how to do calculations for stresses experienced on a trestle bridge built for a pipeline, they need to consider the weight of both the full and the empty pipes, so they need estimates for the volume and mass of material in the pipe, as well as the mass of the pipe itself.  (Borrowed from an incident in which poor design principles were revealed.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Point 3)  In my experience, you can't awaken curiosity on call;  it happens or it doesn't.  You can try to use some other motivations to help drive the teaching.  However, if you can recognize a motivation of your students, state that motivation, get them to agree that it is a motivation, and then show how that motivation drives the lesson you are about to give, that may be a good replacement for independent curiosity.  Here you will need  examples from the field to provide motivation for you to take this kind of approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good Luck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Local Support Is Very Important&quot; Paseman, 2015.07.13&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-07-13T16:58:56.830" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8384" PostTypeId="1" AcceptedAnswerId="8387" CreationDate="2015-07-14T06:17:19.497" Score="12" ViewCount="1384" Body="&lt;p&gt;Advanced engineering mathematics is a subject of its own, building up from simple notions of functions, series, integration techniques and brief review of linear algebra which leads to transform methods (Laplace, Fourier, Z and sometimes Hilbert), with strong emphasis of course on ODE and some elements of PDE and finally ends with complex variables and conformal mapping. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that I've just described the table of content of probably 99% of books out there on advanced engineering mathematics. Kreyszig, Zill, Stroud, Duffy et. al. falls squarely in this category. This is not to say such an approach is without value, in fact the standardization and consistency of a subject is probably a sign of very mature development of the field which is great for an introduction to engineering mathematics and provides great motivation for applied mathematics in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the lack of variation between these texts leaves me wonder if there is a book that covers material that is even more advanced, more comprehensive than what has been described above. For engineering students, it is often crucial to have motivation when it comes to studying a subject and such motivation usually comes from engineering examples (circuit theory, mechanical system, modeling) so an advanced text that is typically used and cherished by math majors just won't do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone recommend a text that is most advanced or comprehensive when it comes to engineering mathematics covering topics that goes beyond the standard curriculum that could potentially be engaging for a mathematically mature engineering student who wish to know more about techniques used in mathematics for engineering applications? Please let me know if this is a bit too much to ask.&lt;/p&gt;&#xA;" OwnerUserId="3560" LastActivityDate="2022-01-26T08:16:30.757" Title="Reference requests: Is there a text that is even more advanced than books on &quot;advanced engineering mathematics&quot;" Tags="|reference-request|engineering-mathematics|" AnswerCount="6" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8386" PostTypeId="2" ParentId="8384" CreationDate="2015-07-14T11:45:34.150" Score="9" Body="&lt;p&gt;Not certain this satisfies your set of criteria, but...&#xA;This is a challenging but wonderful book. Arnold emphasizes&#xA;the geometry of manifolds throughout:&#xA;differential forms,&#xA;Riemannian geometry, symplectic geometry, Lie groups and Lie algebras, dynamical systems, integrable systems.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Arnold, Vladimir Igorevich. &lt;em&gt;Mathematical Methods of Classical Mechanics&lt;/em&gt;. Vol. 60. Springer Science &amp;amp; Business Media, 1989.&#xA;  (&lt;a href=&quot;http://www.springer.com/us/book/9780387968902&quot; rel=&quot;noreferrer&quot;&gt;Springer link&lt;/a&gt;.)&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/jVwQA.png&quot; height=&quot;200&quot; /&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Here is an excerpt from Liviu Nicolaescu's review on Amazon:&#xA;&quot;If you are looking for an easy read, this is the wrong place. ... This book frees your imagination, and it forces you to ask yourself many questions, ... This book is as eloquent an argument on the depth and beauty of geometry as you could find anywhere.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-07-14T11:45:34.150" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8387" PostTypeId="2" ParentId="8384" CreationDate="2015-07-14T12:55:13.113" Score="9" Body="&lt;p&gt;I am unaware of &lt;em&gt;standard&lt;/em&gt; mathematical texts for engineers at a level beyond that of the usual advanced engineering mathematics texts. Perhaps the closest thing in some sense might be Strang's classic &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0961408804&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Applied Mathematics&lt;/a&gt; or his more recent &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0961408812&quot; rel=&quot;nofollow noreferrer&quot;&gt;Computational Science and Engineering&lt;/a&gt;. While the aforementioned books do overlap with typical advanced engineering mathematics, they also introduce new topics and insights, in particular revealing the ubiquity (or at least broad applicability) of linear algebra and its structures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience is that engineers tend to specialize and there is no one text at above-standard-advanced level to cover the main methods used in all domains. Springer does have a fairly recent &lt;a href=&quot;http://www.springer.com/new+%26+forthcoming+titles+%28default%29?SGWID=5-40356-404-173896003-8445&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Engineering&lt;/a&gt; series that could be of interest, but many of the titles cover fairly narrow areas, fascinating as they are. There's also Chirikjian's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/081764802X&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stochastic Models, Information Theory, and Lie Groups&lt;/a&gt; in two volumes, which could be interesting to engineers wanting to learn about stochastic processes and differential geometry, and might be a nice complement or lead-in to Arnold's &lt;em&gt;Mathematical Methods of Classical Mechanics&lt;/em&gt; mentioned by Joseph O'Rourke in his &lt;a href=&quot;https://matheducators.stackexchange.com/a/8386/376&quot;&gt;answer&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-07-14T16:08:35.227" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8388" PostTypeId="2" ParentId="8384" CreationDate="2015-07-14T16:53:22.873" Score="6" Body="&lt;p&gt;&lt;em&gt;Geometric Methods and Applications for Computer Science and Engineering&lt;/em&gt;, by J. Gallier, does the job for what concerns classical differential geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Elasticity and Geometry&lt;/em&gt; by Audoly-Pomeau addresses a classical problem with many important applications with quite advanced math techniques.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for Linear Algebra, I'd like to mention &lt;em&gt;Applied linear algebra&lt;/em&gt; by Olver-Shakiban. In this direction, however, it is quite different to draw a line between good mathematical texts with solid reference to applications and introductory books to numerical linear algebra.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2015-07-14T16:53:22.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8389" PostTypeId="2" ParentId="5554" CreationDate="2015-07-14T23:05:53.827" Score="6" Body="&lt;p&gt;You asked about online classes for Singapore math. I realize you may no longer need this info, but thought someone browsing the questions might appreciate it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend the courses at &lt;a href=&quot;http://ed2go.com&quot;&gt;ed2go&lt;/a&gt;&#xA;There are two classes one for &lt;a href=&quot;http://www.ed2go.com/online-courses/singapore-math-strategies-model-drawing-for-grades-1-6?tab=detail&quot;&gt;1st-6th&lt;/a&gt; and one for &lt;a href=&quot;http://www.ed2go.com/online-courses/singapore-math-strategies-model-drawing-for-grades-6-9?tab=detail&quot;&gt;6th-9th&lt;/a&gt;.&#xA;The courses are given repeatedly for 6 weeks cycles and are under $100 &#xA;They will break down all the steps needed to problem solve with model drawing from the simplest to fractions, ratios, percents, and more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I took this course before I started teaching Singapore math and it really helped.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-07-14T23:05:53.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8390" PostTypeId="1" CreationDate="2015-07-15T15:12:42.110" Score="6" ViewCount="363" Body="&lt;p&gt;I'm a one-time academic mathematician, used to teaching smart undergraduates who made the choice to be in my classes. You can also find me overactively answering questions on &lt;a href=&quot;http://math.stackexchange.com&quot;&gt; Mathematics SE&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently, a neighbor with a 14 year-old son (let's call him Mark) asked me to help. Mark is about to start high school in an International Baccalaureate program. Over the summer Mark has been taking an obligatory refresher course offered by his new high school: 2 hours/day, 5 days/week. His marks are mediocre, his motivation is low and his parents are at a loss. Neither of Mark's parents are able to keep up with the mathematics at this level. Mark and I get along well in general and can talk about movies and general news. Right now he's resistant to the idea that he needs or wants help in the form of talking to me or working harder at mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could anyone give me some advice on how to help?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mark is interested in consumer electronics and some basic programming. Describing the intellectual paths $$Mathematics \to Physics  \to  ElecEng  \to Consumer \ Electronics \\ \text{  and }\\ Mathematics \to Computer \ Science \to Programming$$ are possibilities. However I feel using this approach runs the risk of being too abstract for him. (That is, I can see these paths clearly as I have walked them for a couple of decades; but it's all unknown to him.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His mother tried another approach last week:  &quot;Wouldn't you like to be the smartest kid in the room?&quot; Answer: no.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have watched the lectures from Jo Boaler's Stanford course on &lt;a href=&quot;http://scpd.stanford.edu/search/publicCourseSearchDetails.do?method=load&amp;amp;courseId=30587280&quot; rel=&quot;nofollow&quot;&gt; Mathematics Education&lt;/a&gt;. It has many great ideas on how to teach well at this level. So if Mark and I were to meet once or twice a week I think I know how to teach him fairly well and hopefully provide ongoing motivation through the elegance and creativity of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But how do we get him started? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm open to suggestions!&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Update (3 days later):&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks everyone for your advice and comments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After speaking to Mark's parents we are using a 4-pronged approach:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To help him get into the right mindset about learning mathematics and overcome some of the negative messages he has heard about himself, his mother has started watching with him the excellent, short course by Jo Boaler &lt;a href=&quot;https://lagunita.stanford.edu/courses/Education/EDUC115-S/Spring2014/about&quot; rel=&quot;nofollow&quot;&gt; How to learn Math: For Students&lt;/a&gt;. I watched the lessons with the mother first to coach her through the material. &lt;/li&gt;&#xA;&lt;li&gt;I am tutoring him a couple of times a week and following Jo's advice about giving him the right kinds of messages about learning, as well as doing all the good things a teacher should do such as&lt;/li&gt;&#xA;&lt;li&gt;treating mathematics as much more than just a series of rules and procedures and helping him see connections between material in mathematics&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;helping him see connections between mathematics and the broader world, such as this week's Pluto photos and his interest in computing&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In his home life, his parents are actively managing a number of issues such as excessive screen time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;One of his parents has committed come the new school year to learn mathematics in parallel with Mark up to the end of Pre-Calculus, so they can talk to him about it and demonstrate that they believe the material is important. I plan to introduce them to Khan Academy to help with that parent learning. Khan also gives them another tool to help Mark in the future if he is stuck on a topic.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As a tactical matter, I have also given him a couple of mechanical pencils. He was writing with blunt pencils and his mathematics was messy, resulting in him often misreading his own work and making errors.&lt;/p&gt;&#xA;" OwnerUserId="3478" LastEditorUserId="3478" LastEditDate="2015-07-19T21:50:48.227" LastActivityDate="2015-07-30T06:31:31.417" Title="Generating motivation for an incoming 14 year-old IB student" Tags="|student-motivation|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="8391" PostTypeId="1" CreationDate="2015-07-15T18:08:57.717" Score="3" ViewCount="557" Body="&lt;p&gt;I am a college sophomore in the US with a major in mathematics and am an aspiring mathematician in the fields of computational complexity theory and cryptography.  I would like to seek your advice and recommendations on comprehensive, insightful, and challenging books for abstract algebra and abstract linear algebra.  I had looked through some books and pulled out some possible candidates:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Linear Algebra: Friedberg/Insel/Spence, Hoffman/Kunze, and Axler&lt;/li&gt;&#xA;&lt;li&gt;Abstract Algebra: Herstein (Topics in Algebra), Artin, Dummit/Foote, and MacLane/Birkhoff (Algebra)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This will be my first introduction to abstract algebra and linear algebra. I would like to start with challenging, detailed books as I think my mathematical maturity and knowledge will grow much better than with easy, toned-down books.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for my background, I studied single-variable calculus using Serge Lang (A First Course in Calculus) and acquired proof-writing skill using the proof-teaching book Chartrand's Mathematical Proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I apologize for any grammatical errors, and look forward to your advice and recommendations!&lt;/p&gt;&#xA;" OwnerUserId="5399" LastEditorUserId="376" LastEditDate="2016-02-21T18:57:03.567" LastActivityDate="2016-02-22T14:12:36.690" Title="Seeking your advice on books for abstract algebra and linear algebra" Tags="|textbooks|linear-algebra|abstract-algebra|reference-books|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="8392" PostTypeId="1" CreationDate="2015-07-15T19:24:24.803" Score="6" ViewCount="225" Body="&lt;p&gt;What is the best way to introduce Pell’s equation on a first elementary number theory course? Are there any practical applications of Pell’s equation? What are the really interesting questions about Pell’s equation? Are there any good resources on Pell’s equation.&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="143" LastEditDate="2015-11-05T18:37:32.177" LastActivityDate="2015-12-06T01:36:05.980" Title="Resources for Pell's equation" Tags="|reference-request|concept-motivation|number-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8393" PostTypeId="2" ParentId="8390" CreationDate="2015-07-15T20:58:36.600" Score="3" Body="&lt;p&gt;Since you're meeting in a couple of hours with the dad to figure out a strategy I thought I'd give some suggestions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Kids that age like choices and feeling in control. Perhaps you could&#xA;ask if he wants to try meeting with you twice and if he doesn't like&#xA;it, he doesn't have to continue. &lt;/li&gt;&#xA;&lt;li&gt;I think relating it to what he wants to do in the future is very&#xA;important - it may be more theoretical to him but it should still&#xA;spark some interest.&lt;/li&gt;&#xA;&lt;li&gt;Perhaps his parents could give him some reward for making an effort&lt;br&gt;&#xA;or picking up his grades. Better yet - ask him what would make him agree&#xA;to tutoring help.&lt;/li&gt;&#xA;&lt;li&gt;If there's math that's fun/interesting for him perhaps you    could&#xA;alternate between the two.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Once you are meeting with him, you can ask for ideas on how to succeed with him, but first you need to meet with him.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-07-15T20:58:36.600" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8394" PostTypeId="2" ParentId="8390" CreationDate="2015-07-15T22:11:06.120" Score="2" Body="&lt;p&gt;Does he like to be helpful?  Find something you can interest yourself in, and take it to the point where you can ask him for help on it.  This requires a bit of artifice as well as careful planning, but you may find a question that sticks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, say you were thinking of improving on DVD technology, but the manuals and specs on the current devices aren't available till next week.  How far can you get estimating the data rates, angular velocity, data density, fine control of stepper motors, and so on?  If he likes you and thinks he is helping you, that will buy you a little bit of an in.  Hopefully within three sessions (this is the tricky part) you will find something that really excites/irks/inspires him.  Alternatively, have him suggest a couple projects he is interested in, and you can show him the fun parts of working on the projects, while telling him later about the routine chores needed to push the projects through.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can't induce excitement, but you can inspire it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;In Control And In Charge&quot; Paseman, 2015.07.15&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-07-15T22:11:06.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8395" PostTypeId="2" ParentId="8390" CreationDate="2015-07-16T01:39:18.787" Score="2" Body="&lt;p&gt;I like some aspects of Gerhard's answer, but not the whole thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you should start by furthering your non-math relationship with Mark.  For example, watch a movie together that you would both enjoy, or go see a political debate together. Or make a visit to Best Buy together to see what new products there are (with the understanding that nothing will be bought).  Just something that would be fun for both of you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the moment is right (and you'll know when it is), ask him how he feels about his math refresher course.  Find out what he likes most and least about math.  Now you whip out some paper and a pencil, and you check what his level is in a topic he said he liked.  (If he said he doesn't like anything, geometry is probably a safe choice.)  Ask him some questions that give him an opportunity to think a little bit, but make sure he doesn't get stuck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea here is for the two of you to do some math together, in a low key way, and for him to get intrinsic enjoyment out of it, without it being part of any particular academic program.  It could be something that would be of no usefulness for what he's going to do in the fall.  Just something he finds intriguing and satisfying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully you'll be able to do some math together each time you get together, and a tutoring relationship will evolve naturally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also want to take a look at what they're doing in the refresher course -- he might be better off not doing it.  You won't know until you get an idea what they're doing.&lt;/p&gt;&#xA;" OwnerUserId="4956" LastActivityDate="2015-07-16T01:39:18.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8396" PostTypeId="1" CreationDate="2015-07-16T08:06:07.377" Score="3" ViewCount="138" Body="&lt;p&gt;I have a chronic visual handicap and am in need of adequate level of accessibility support while learning and teaching mathematics from and to the visually challenged as well as the sighted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm familiar with the JAWS for Windows screen reading program and the MAGic screen magnification program, both developed by the Freedom Scientific Inc. JAWS offer very little access to mathematical text (in fact none for .djvu files), whereas MAGic 9.5's magnification levels of 2x and higher are too big for my liking (as it requires too much of scrolling) and the magnification levels of below 2x tend to make the display less clearly visible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm unable to afford a newer version of MAGic and so have to content myself with the demo version.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As of now, I use the High Contrast Black (Large) display setting in Windows XP, along with the Windows Inverted (Xtra Large) mouse pointer scheme with trail and highlighting-upon-pressing-the-CTRL-key enabled.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I prefer reading books in .djvu format with the contrast enabled, or the .pdf text files with colors replaced by the Windows High Contrast color setting (but struggle to read scanned .pdf documents since the text within those pages appears black on a white background, notwithstanding the High Contrast setting).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typesetting maths is often too tedious and time consuming for me, whether it be in WinEdit (with MikeTex at the back end) or as MathJex on the Mathematics Stach Exchange website.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've also download and installed a program called WinTriangle, software developed by a Prof. Frank Gardner of the physics department at the Oregon State University. But I'm not sure how to utilise it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm aware of the BlindMaths mailing list, but over there the information is too spread out! I don't think there is any repository of audio / video tutorials, for example, on how to make the best possible of WinTriangle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During my teaching of mathematics, I often find myself to be too slow while writing on the board. Often, Drawing figures also proves to be too uphill a task. I'm not yet finished with a proof and the lecture hour goes by!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always find myself to be struggling to complete the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here in Pakistan, there is no individual or organisation working on making maths accessible to the blind and visually impaired.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this backdrop, any advice and suggestions please?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whatever help and support the SE community could offer would be most appreciated!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note: Originally posted on SU; see revisions for details.)&lt;/p&gt;&#xA;" OwnerUserId="5400" LastEditorUserId="143" LastEditDate="2016-05-26T18:02:01.060" LastActivityDate="2016-05-26T18:02:14.403" Title="What accessibility is available to the blind and visually impaired students and teachers of mathematics?" Tags="|technology-in-education|special-needs-education|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8397" PostTypeId="1" AcceptedAnswerId="8399" CreationDate="2015-07-16T19:10:39.127" Score="9" ViewCount="583" Body="&lt;p&gt;I'm wondering how many problems are needed in order to learn a subject, let's say Calculus of Several Variables. We know that the professors often assign us a list of problems to solve as homework, but I find that this is often insufficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found that solving all problems on a book or two is enough to handle (and understanding) most of the problems on a given subject and but I'm wondering if there is some more clever method to learn something because it is not always practical to do all the exercises of all subjects that we learn as math undergraduate students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any method that has proven that works? Thanks foy your help.&lt;/p&gt;&#xA;" OwnerUserId="5402" LastEditorUserId="143" LastEditDate="2015-07-18T10:55:20.767" LastActivityDate="2015-07-20T01:37:24.207" Title="How many problems do we have to do as undergraduate mathematicians in order to learn a subject?" Tags="|undergraduate-education|mathematical-pedagogy|exercises|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8398" PostTypeId="1" CreationDate="2015-07-16T22:40:46.267" Score="8" ViewCount="581" Body="&lt;p&gt;I happen to be a fan of the &quot;theme and variations&quot; approach to problem solving.  In certain cases, a certain question had drawn enough to attention to generate dozens of solutions.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra&quot;&gt;Ways to Prove the Fundamental Theorem of Algebra&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://math.cmu.edu/~bwsulliv/basel-problem.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;5 proofs that $\boxed{\sum \frac{1}{n^2} = \frac{\pi^2}{6}}$&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/1420/whats-the-best-proof-of-quadratic-reciprocity&quot;&gt;What is the Best Proof of Quadratic Reciprocity&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are some other examples of results having many different proofs but in the literature, I rarely read more than one proof of the same result. In many cases, nobody has taken the time to collect them into one place. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is &quot;compare and contrast&quot; an acceptable methodology in Mathematics?  Certainly it is in other subjects, but Mathematics is about proving new things rather than comparing what is known. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Put another way is collecting many proofs of an important result a meaningful way to spend your time?  How do I explain the advantages to others.  &lt;/p&gt;&#xA;" OwnerUserId="1654" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-07-16T22:40:46.267" Title="Can Compare and Contrast be used in Mathematics Teaching and Research?" Tags="|undergraduate-education|graduate-education|undergraduate-research|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8399" PostTypeId="2" ParentId="8397" CreationDate="2015-07-16T23:35:26.867" Score="11" Body="&lt;p&gt;I think your question is good, but a little vague.  When you say &amp;quot;how many problems,&amp;quot; would ten easy problems be counted the same as ten difficult problems?  If exactly the same problem was posed twice, would it count as one problem or two problems?  As you can imagine, the answer to your question would depend on the subject to be learned, the types of problems asked, and the background of the learner.&lt;/p&gt;&#xA;&lt;p&gt;I personally subscribe to the educational philosophy called &lt;em&gt;mastery learning&lt;/em&gt;.  From &lt;a href=&quot;https://en.wikipedia.org/wiki/Mastery_learning&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mastery Learning (or as it was initially called, “learning for mastery”) is an instructional strategy and educational philosophy, first formally proposed by Benjamin S. Bloom in 1968. Mastery Learning maintains that students must achieve a level of mastery (i.e. 90% on a knowledge test) in prerequisite knowledge before moving forward to learn subsequent information. If a student does not achieve mastery on the test, they are given additional support in learning and reviewing the information, then tested again. This cycle will continue until the learner accomplishes mastery, and may move on to the next stage.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, if your question was changed to &amp;quot;How many problems do we have to do in order to &lt;em&gt;master&lt;/em&gt; a subject?&amp;quot; then I would answer &amp;quot;as many as are necessary to &lt;em&gt;show&lt;/em&gt; mastery in the subject,&amp;quot; which often means a score of 90% or better.&lt;/p&gt;&#xA;&lt;p&gt;In practice, this number is much larger than what is usually done in typical schools.  For example, a typical school would perhaps ask a dozen questions on mixed number addition.  But to achieve mastery might involve a hundred questions or more, depending on the learner.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-07-17T00:03:31.963" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8401" PostTypeId="1" CreationDate="2015-07-16T19:00:18.517" Score="1" ViewCount="68" Body="&lt;p&gt;I'm wondering how many problems are needed in order to learn a subject, let's say Calculus of Several Variables. &#xA;We know that the professors often assign us a list of problems to solve as homework, but I find that this is often insufficient. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found that solving all problems on a book or two is enough to handle (and understanding) most of the problems on a given subject and but I'm wondering if there is some more clever method to learn something because it is not always practical to do all the exercises of all subjects that we learn as math undergraduate students.&lt;/p&gt;&#xA;" OwnerUserId="5402" OwnerDisplayName="Levent" LastEditorUserId="143" LastEditDate="2015-07-18T10:52:09.897" LastActivityDate="2015-07-18T10:52:09.897" Title="How many problems do we have to do as undergraduate mathematicians in order to learn a subject?" Tags="|undergraduate-education|mathematical-pedagogy|exercises|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2015-07-18T10:51:01.040" ContentLicense="CC BY-SA 3.0" />
  <row Id="8402" PostTypeId="2" ParentId="8397" CreationDate="2015-07-16T19:15:12.330" Score="15" Body="&lt;p&gt;This is a good question, but it is impossible to answer in general terms.  It depends upon (at least) the subject, the person, and (perhaps most importantly) what you mean by &quot;learning it&quot;.  I hope it is no secret that most undergraduate courses aim to give an &lt;em&gt;introduction&lt;/em&gt; to the subject they concern.  Things in the undergraduate curriculum which need to be learned more deeply are usually addressed as part of multiple courses.  In other words, one of the reasons that students take multi-variable calculus and/or differential equations is to &lt;em&gt;reinforce&lt;/em&gt; the differentiation and integration stuff they saw in a superficial whirlwind the year before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I said it was a good question and I meant it.  It's a good question &lt;em&gt;for you to answer for yourself&lt;/em&gt;.  In general, you should assume that homework assigned in courses is not chosen so as to be the pedagogically ideal amount but is limited by practical considerations such as time and moderate interest.  An advancing student of mathematics needs to learn how to judge for herself how well she is learning a particular topic, including how many problems to do, how quickly to move on to the next topic, and so forth.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solving all the problems in one or two books is &lt;em&gt;probably&lt;/em&gt; more than enough reinforcement for most topics beyond the calculus level.  (I have fond memories of working calculus problems as a high school student.  I loved calculus and I was good at it, but nevertheless I remember working through several hours of problems a night.  Most of them did not give me much trouble, but there were &lt;em&gt;a lot&lt;/em&gt; of them.  And I do think it was good for me...not least because I have solved so many calculus problems in a teaching capacity in my adult life.)  At a certain point, many texts stop having problems, or the problems are thin on the ground and obviously not chosen primarily for pedagogical purposes: e.g. they may be either trivial or unsolved.  &lt;em&gt;Making up your own problems&lt;/em&gt; becomes an important part of learning at an advanced level.  Spoiler alert: it makes things much more interesting.  &lt;/p&gt;&#xA;" OwnerUserId="176" OwnerDisplayName="Pete L. Clark" LastActivityDate="2015-07-16T20:53:51.757" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8403" PostTypeId="1" CreationDate="2015-07-17T19:02:14.410" Score="4" ViewCount="445" Body="&lt;p&gt;E.S.E advisers,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the computation theory and cryptography.  I apologize for this sudden interruption but I wrote this post to seek your recommendation on a problem-solving book (containing strategies and tricks) that can prepare me well for a competition mathematics, particularly the Putnam Competition (which I will take on this December).  This is my first time preparing for a mathematical competition, and I am familiar with only the basic proof skills (direct, indirect, induction, etc.).  I asked my Putnam coach and friends, and they all recommend the following books on below.  My plan is to study one of them and tackle Putnam problems at the same time.  What is your recommendation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Problem-Solving Trough Problems&quot;         Loren Larson&#xA;&quot;Problem-Solving Strategies&quot;              Arthur Engel&#xA;&quot;The Art and Craft of Problem Solving&quot;    Paul Zeitz&#xA;&quot;The Art of Problem Solving, Vol.1-2&quot;     Richard Rusczyk&#xA;&quot;Putnam and Beyond&quot;                       Titu Andreescu&#xA;&quot;Thinking Mathematically&quot;                 J. Mason&#xA;&quot;Mathematics and Plausible Reasoning:     G. Polya&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am currently reading &quot;How to Solve It&quot; by G. Polya, and I will proceed to one of them (at your recommendation) along with Putnam problems.  It is fairly natural to pick &quot;Putnam and Beyond&quot;, but I heard that it is not a good introduction for the problem-solving skills and required knowledge for the Putnam Competition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you very much for your time, and I look forward to hear back from you!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sincerely,&#xA;PK&lt;/p&gt;&#xA;" OwnerUserId="5399" LastEditorUserId="276" LastEditDate="2015-11-18T01:08:44.373" LastActivityDate="2015-11-18T01:08:44.373" Title="Seeking Your Recommendation on Problem-Solving Books (preparing for Putnam)" Tags="|undergraduate-education|textbooks|self-learning|reference-books|math-contests|" AnswerCount="0" CommentCount="5" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8404" PostTypeId="1" CreationDate="2015-07-18T19:10:18.273" Score="6" ViewCount="320" Body="&lt;p&gt;E.S.E. advisers,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am a college sophomore with a major in mathematics and an aspiring mathematician in the fields of computation theory and cryptography.  I am always curious about the importance of &quot;transition books&quot; from calculus to analysis, such as Spivak, Apostol, and Courant.  Do I need to read one of them before jumping into real analysis?  I took Calculus I-II course using Serge Lang's A First Course in Calculus, and I am currently studying Artin's Algebra and Hoffman/Kunze's linear Algebra, and preparing myself for upcoming research in the computation theory. I will be taking the theoretical linear algebra and abstract algebra for this academic year, and I will proceed to the analysis on next academic year.  However, I want to learn more about calculus since I am preparing for Putnam, and measure/approximation theory will be used on my research.  Perhaps it is naturally for me to pick up analysis books of Rudin, Apostol, or Pugh, but it seems that quite many people recommend the advanced calculus books of Apostol, Rudin, and Pugh.  Is it necessary for me to read those transition books?  Can I just jump right into the analysis books?  What are the benefits of reading the transition books first before studying the analysis?&lt;/p&gt;&#xA;" OwnerUserId="5399" LastActivityDate="2015-07-18T19:10:18.273" Title="Importance of &quot;Calculus-&gt;Analysis Transition Books&quot;?" Tags="|undergraduate-education|textbooks|reference-books|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="8407" PostTypeId="1" CreationDate="2015-07-19T20:42:32.157" Score="5" ViewCount="975" Body="&lt;p&gt;I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the computation theory and cryptography. I recently got an undergraduate research in the computation theory, which is heavily based on the algebra but also requires the extensive knowledge in the measure theory, approximation theory, and other bits of real analysis. On this upcoming academic year, I will be taking the year-long abstract algebra, linear algebra with proofs, applied linear algebra, combinatorics, and multivariable calculus (chug-and-plug level; taking on Spring 2016). I won't be taking any analysis course until next academic year. This means that I will be self-studying the real analysis by myself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My research adviser recommended me to read the following books: Rudin (RCA), Folland (Real Analysis), Royden (Real Analysis), Stein-Shakarchi (Real Analysis), Cohn (Measure Theory), Halmos (Measure Theory), and Trefethen (Approximation Theory). If my memory serves me correct, those books are considered as advanced real analysis books that even touch a functional analysis. I told my adviser that I never took a mathematical analysis course, let alone any analysis book. He then advised me to self-study the real analysis, which I am motivated and excited to do so, and he devised a following self-studying plan for real analysis:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Learn the basics of analysis (real number system, basic topology, limit, continuity, and basics of series/sequence) from the elementary analysis books (such as Rudin's PMA, Apostol's Mathematical Analysis, and Pugh's RMA), and jump directly into the real analysis books mentioned on the last paragraph, and study them in a &quot;non-linear, backward&quot; style. That means that I will study those advanced real analysis books and learn the necessary topics from basic analysis books as needed. He said that method will allow me to learn the proper real analysis, gain deeper understanding, and prepare myself for the upcoming research and even Putnam competition. He even gave me an advise that the mathematics is usually learned in a nonlinear style, and it is always a good method to learn the materials from the advanced, comprehensive books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like his plan but I fear that perhaps it is too risky. Perhaps my adviser overestimated my ability. Do you like the plan? What is your suggestion? What other advanced real analysis books do you recommend to learn with what I mentioned? Will my lack of knowledge in the multivariable calculus be a significant problem to understand those texts?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MY BACKGROUND: I am currently studying and enjoying Artin's Algebra and Hoffman/Kunze's Linear Algebra.&lt;/p&gt;&#xA;" OwnerUserId="5399" LastActivityDate="2015-07-22T11:57:22.933" Title="Inquiry about My Self-Study Plan for Real Analysis (for my research and self-enrichment)" Tags="|undergraduate-education|mathematical-analysis|self-learning|undergraduate-research|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8408" PostTypeId="2" ParentId="8397" CreationDate="2015-07-20T01:37:24.207" Score="13" Body="&lt;p&gt;As other posters have noted, the answer varies from person to person. There is, however, some knowledge from education research (much of it conducted specifically in mathematics) that you might find helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One key insight is that the number of problems isn't usually as important as the timing or spacing of your practice. Once you've &quot;worked to criterion&quot;, i.e., once you've figured out what you're supposed to figure out, working a bunch more of the same type of problems in the same session probably won't do much for your retention. The ability to remember information depends more on whether your brain has ever had to work to remember it. Your brain doesn't really have to work to remember what to do in problem 25 when 1-24 were the same idea, and so the massed practice doesn't really impact retention as much as you would think. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What will help, however, is going back and working more problems after enough time has passed that you have almost, but not quite, forgotten the ideas. When your brain has to work a bit to remember something, it strengthens the retrieval pathways in response, so your brain will be better able to dig that information out of your brain on command in the future. Thus, working five problems the week you learn something, five more the next week, and five more the week after that will probably help you more than working all fifteen or even more problems all in one block. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how many do you need? Only you will really know, but if you space them out strategically, it shouldn't be as many as you might otherwise.&lt;/p&gt;&#xA;" OwnerUserId="5417" LastActivityDate="2015-07-20T01:37:24.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8409" PostTypeId="2" ParentId="8407" CreationDate="2015-07-20T02:35:21.017" Score="2" Body="&lt;p&gt;I can not imagine a good research project for a sophomore that would use measure theory.  If you're looking for fast algorithms to square real numbers, I don't think measure theory or other analysis will help you much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The comment that &quot;it is always a good method to learn the materials from the advanced, comprehensive books&quot; seems harmful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your lack of knowledge of multivariable calculus may not be a problem in learning analysis.  However, have you ever had to manipulate $\forall x \exists \epsilon$ vs $\exists \epsilon \forall x$ in proofs?  If not, that lack of exposure to rigor in analysis may be a problem in reading the books.  I recommend learning that rigor in a classroom setting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I don't like the plan.  I suggest that you enjoy the summer without the analysis books.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2015-07-20T02:35:21.017" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="8410" PostTypeId="2" ParentId="7970" CreationDate="2015-07-20T02:50:55.230" Score="9" Body="&lt;p&gt;Schools in the US typically have summer breaks of about two months, a winter break of about two weeks, a spring break of one week, and often at least a few days for fall break. The dates for those vacations are usually known well in advance of the beginning of each school year. Therefore, schools generally expect you to plan your vacations for those times when school isn't in session, and they aren't usually sympathetic to requests for substantial time off for purely recreational activities. After all, they have to provide substitutes when you're gone, and in most districts, a substitute teacher who can do justice to secondary math instruction is a rare thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not unusual to have a small allotment of personal days that are intended for unexpected short term situations (my district gives three per year). However, regularly using the entire allotment without a pretty good reason--especially on consecutive days--would be considered less than professional by many administrators.&lt;/p&gt;&#xA;" OwnerUserId="5417" LastActivityDate="2015-07-20T02:50:55.230" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8411" PostTypeId="1" AcceptedAnswerId="8414" CreationDate="2015-07-21T03:25:30.213" Score="4" ViewCount="264" Body="&lt;p&gt;In standard usage does the word &quot;curriculum&quot; mean&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;That which ought to be taught and learned, as prescribed by authorities (i.e. teachers and textbook authors and the like); or&lt;/li&gt;&#xA;&lt;li&gt;That which actually &lt;em&gt;is&lt;/em&gt; taught and learned in conventional practice, regardless of whether it is consistent with what authorities say ought to be taught and learned; or&lt;/li&gt;&#xA;&lt;li&gt;Something else?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="205" LastEditorUserId="143" LastEditDate="2015-07-21T15:19:21.513" LastActivityDate="2015-07-21T15:19:21.513" Title="Definition of &quot;curriculum&quot;" Tags="|curriculum|terminology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8413" PostTypeId="1" CreationDate="2015-07-21T13:00:37.973" Score="14" ViewCount="354" Body="&lt;p&gt;Is there a way to measure the understandability of a proof? From a search in the internet I have only found methods for &lt;a href=&quot;http://ieeexplore.ieee.org/Xplore/defdeny.jsp?url=http%3A%2F%2Fieeexplore.ieee.org%2Fstamp%2Fstamp.jsp%3Ftp%3D%26arnumber%3D4019969%26userType%3Dinst&amp;amp;denyReason=-134&amp;amp;arnumber=4019969&amp;amp;productsMatched=null&amp;amp;userType=inst&quot;&gt;measuring the understandability of software&lt;/a&gt; or tests for &lt;a href=&quot;https://en.wikipedia.org/wiki/Readability_test&quot;&gt;measuring the readability of a text&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there similar tests or methods to measure the understandability or the complexity of a proof? My final goal is to compare to different proofs of the same theorem by their understandability.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2015-07-23T15:49:15.517" Title="How to measure the understandability of a proof?" Tags="|reference-request|proofs|testing|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8414" PostTypeId="2" ParentId="8411" CreationDate="2015-07-21T14:47:47.727" Score="10" Body="&lt;p&gt;I'm not sure what exactly &quot;standard usage&quot; means, but in educational research (and more specifically in mathematics education research) it is common to distinguish between the &lt;em&gt;intended curriculum&lt;/em&gt;, the &lt;em&gt;enacted curriculum&lt;/em&gt;.  It is also fairly common to refer the &lt;em&gt;attained curriculum&lt;/em&gt; and the &lt;em&gt;tested curriculum&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When used outside of a research context -- for example, by practitioners, school administrators, and policymakers -- &quot;curriculum&quot; (without modifiers) almost always refers to the intended curriculum.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-07-21T14:47:47.727" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8415" PostTypeId="2" ParentId="8384" CreationDate="2015-07-21T16:44:24.947" Score="5" Body="&lt;p&gt;Surprisingly not yet mentioned is&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0387989315&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Advanced Mathematical Methods for Scientists and Engineers&lt;/strong&gt;&lt;/a&gt; by Carl M. Bender and Steven A. Orszag&lt;/p&gt;&#xA;&lt;p&gt;Below are some comments about this book I made in a &lt;a href=&quot;https://groups.google.com/g/sci.math/c/rjQOTgonFT8/m/-HSzcmY48SAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;10 January 2007 sci.math post&lt;/a&gt; archived at Google Groups:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I had a course out of this book 23 years ago. If someone or some company had approached me (or I had known who to approach) to go into applied mathematics and then do the kinds of things in this book for them as a career, I might very well have followed a different career path than I did. I loved the stuff in this book (we only did about half of it in the one semester graduate course I took) and I was constantly amazed at how certain techniques were able to squeeze out so much information in situations you'd think were just hopeless. A lot of the exercises are old Putnam Exam problems also, not that it helped me any -- my Putnam days were over by then, such as it was.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[\cdots]$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One of the things I remember from this book is that it's often easier to analyze asymptotic and qualitative behavior by directly working with a differential equation than with an integral or a series expression.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2022-01-26T08:16:30.757" LastActivityDate="2022-01-26T08:16:30.757" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8416" PostTypeId="1" CreationDate="2015-07-21T18:25:36.067" Score="7" ViewCount="241" Body="&lt;p&gt;I hope this is a good place to ask about this. My institution uses WebAssign for homework in many classes. A concern has come up about being compliant with &lt;a href=&quot;http://www.dol.gov/dol/topic/disability/ada.htm&quot;&gt;ADA&lt;/a&gt;. From &lt;a href=&quot;http://www.webassign.net/manual/student_guide/c_a_accessibility.htm&quot;&gt;a page on the WebAssign site&lt;/a&gt;, it sounds like WebAssign should be ok. But I have also found a reference to &lt;a href=&quot;https://nfb.org/images/nfb/documents/pdf/miami%20teach.pdf&quot;&gt;a lawsuit&lt;/a&gt; (from 2014?) indicating that there are &quot;problems&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is basically if anyone knows whether WebAssign is ADA compliant. Before I start spending hours research this, I wanted to see if anyone has some insight.&lt;/p&gt;&#xA;" OwnerUserId="53" LastActivityDate="2015-07-21T18:25:36.067" Title="Is WebAssign ADA compliant?" Tags="|homework|online-instruction|disability|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8417" PostTypeId="2" ParentId="8413" CreationDate="2015-07-21T19:03:47.297" Score="6" Body="&lt;p&gt;I will note here a litmus test for an understandable proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A proof is understandable only when :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) you can explain it to another person, and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) that person can explain it to a third person, and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) the third person can explain it to you in a fashion that is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1) different from how you explained it, and&lt;/li&gt;&#xA;&lt;li&gt;2) convincing to you, and&lt;/li&gt;&#xA;&lt;li&gt;3) understandable by you&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are more exacting criteria, but for me, the above is the&#xA;one that matters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Science Is Meant For Sharing&quot; Paseman, 2015.07.21&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-07-21T19:03:47.297" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8418" PostTypeId="1" CreationDate="2015-07-21T21:24:51.280" Score="4" ViewCount="406" Body="&lt;p&gt;I am going to start studying the analysis texts (Rudin-PMA, Apostol-MA, Pugh-RMA) on the first week of August.  I have a good proof skills through working on Artin's Algebra and Hoffman/Kunze's Linear Algebra, but I unfortunately only took computational 1-variable calculus (Lang's A First Course in Calculus), and I did not take multivariable calculus, which I might take concurrently with Analysis I and Theoretical Linear Algebra on the upcoming Fall.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for a brief text which explains the key ideas from both 1-variable and multivariable calculus, one I can read and jump directly into the analysis texts.  Could you recommend one?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also will my lack of multivariable calculus be a problem when I tackle those analysis texts?  I seem to understand at least the beginning chapters.&lt;/p&gt;&#xA;" OwnerUserId="5399" LastEditorUserId="143" LastEditDate="2015-07-22T13:12:49.857" LastActivityDate="2015-07-27T20:17:25.890" Title="Brief book on calculus to read before studying the analysis" Tags="|calculus|textbooks|mathematical-analysis|self-learning|reference-books|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8419" PostTypeId="1" CreationDate="2015-07-21T22:20:00.787" Score="7" ViewCount="221" Body="&lt;p&gt;I'm teaching a course this fall on Discrete Structures for Computer Science. It's taught out of the math department but is a service course for the CS department with 90-95% of students being CS majors. I've taught the course before a few times, and this fall we are introducing a content unit on discrete probability that hasn't been there before. The other content units cover sets, counting, logic, and proof. Discrete probability will come last, after all the other stuff is covered. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that the book I'm using doesn't have any discrete probability in it. Which isn't really a problem as long as I can find a good, free online supplement for students to use. The learning outcomes for this unit are: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Calculate probabilities of events and expectations of random variables for elementary problems such as games of chance. &lt;/li&gt;&#xA;&lt;li&gt;Differentiate between dependent and independent events. &lt;/li&gt;&#xA;&lt;li&gt;Identify a case of the binomial distribution and compute a probability using that distribution. &lt;/li&gt;&#xA;&lt;li&gt;Apply Bayes' theorem to determine conditional probabilities in a problem. &lt;/li&gt;&#xA;&lt;li&gt;Compute the variance for a given probability distribution. &lt;/li&gt;&#xA;&lt;li&gt;Explain how events that are independent can be conditionally dependent and identify real world examples of such cases. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Those learning outcomes are taken from the ACM recommendations for discrete probability, in discrete math courses for CS majors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A brief Google search brings up a ton of free books and video resources that are way too high of a level for the course -- as in, graduate-level stuff whereas this course is mainly second-year CS majors with minimal math backgrounds. The closest match I found was &lt;a href=&quot;https://en.wikibooks.org/wiki/High_School_Mathematics_Extensions/Discrete_Probability&quot; rel=&quot;noreferrer&quot;&gt;this wikibook&lt;/a&gt; but it doesn't seem to cover all the bases. So I'm wondering if you've got anything to share. Thanks in advance. &lt;/p&gt;&#xA;" OwnerUserId="942" LastEditorUserId="77" LastEditDate="2015-08-13T03:33:09.497" LastActivityDate="2018-03-15T06:51:40.553" Title="Recommendations for free, basic resource on discrete probability for a discrete structures class?" Tags="|reference-request|probability|computer-science|discrete-math|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8420" PostTypeId="2" ParentId="8419" CreationDate="2015-07-21T22:47:00.037" Score="1" Body="&lt;p&gt;Have you looked at &lt;a href=&quot;http://www.khanacademy.org&quot; rel=&quot;nofollow&quot;&gt;Khan Academy&lt;/a&gt;?  They seem to cover just about all of this in their video lectures and follow it up with exercises to mastery. Students can have you as their &quot;coach&quot; and you can assign those exercises that you think are helpful. I've used it with elementary - high school students and they have all found it useful.  I assume you would supplement with your own teaching.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-07-21T22:47:00.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8421" PostTypeId="2" ParentId="8407" CreationDate="2015-07-22T11:57:22.933" Score="1" Body="&lt;p&gt;In my opinion measure theory does not depend much on multivariable calculus. So it is okay to do measure theory directly after studying real analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My short answer: Just do it! ;-) You will shortly note whether you can study real analysis by your own or whether you need external help in form of a course. In case you have a question you can use forums like &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;math.stackexchange.com&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you already may know: While reading a mathematics textbook you will find passages where you stuck for a long time. That's totally normal. So: Don't give up to early.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2015-07-22T11:57:22.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8422" PostTypeId="1" CreationDate="2015-07-22T18:35:49.630" Score="16" ViewCount="482" Body="&lt;p&gt;Making precise logical statements is an important part of teaching and learning mathematics.&#xA;There are many ways to write such statements, and let me divide them into two main types&lt;sup&gt;1&lt;/sup&gt;: writing in symbols ($\forall,\exists,\iff,\dots$) or writing in words (for all, there is, if and only if, &amp;hellip;).&#xA;&lt;strong&gt;Are there studies about how emphasizing one of these types affects the way students learn?&lt;/strong&gt;&#xA;What harm does it do if only one type is present in teaching?&#xA;How important is it that students can translate statements between these two languages?&#xA;What are some key skills having to do with formal logical statements that I should make sure every student learns?&#xA;Is it easier or faster for students to read statements in words than in symbols?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many style guides and older colleagues advice against using logical symbols.&#xA;I am not looking for experience or anecdotal evidence, but actual studies about teaching mathematics.&#xA;This does not exclude unpublished teaching experiments done within a department, but I would prefer more rigorous studies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my limited teaching experience I have encountered numerous students with severe problems in reading, writing and understanding logical statements.&#xA;(I also fear that some students don't see the connection between symbolic logical expressions and meaningful statements in words.)&#xA;This is clearly a grave impediment to learning mathematics, and I would like to know what is actually known to work helping students overcome it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question was inspired by &lt;a href=&quot;https://math.stackexchange.com/q/1368531/166535&quot;&gt;this question at MSE&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For concreteness, here is an example of a statement in symbols or one in words&lt;sup&gt;2&lt;/sup&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$\forall y\in\mathbb R \;\exists \delta&amp;gt;0\;\forall x\in\mathbb Q:|y-x|&amp;lt;\delta$.&lt;/li&gt;&#xA;&lt;li&gt;For any real number $y$ there is such a number $\delta&amp;gt;0$ so that for any $x\in\mathbb Q$ we have $|y-x|&amp;lt;\delta$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added clarification:&lt;/strong&gt;&#xA;The main focus here is on logical statements and logical symbols.&#xA;I am mainly interested in whether (or when) it would be good to replace logical symbols (quantifiers, implications, negations) with words.&#xA;I am quite confident that writing equations out in words would be a bad idea and university level students seem to parse equations much more reliably than logical statements.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; Some common abbreviations fall between these two classes (iff, w.r.t., s.t., &amp;hellip;) and, of course it is possible to mix symbols and words together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt; There are many conventions for punctuation and parentheses in symbolic statements and many possible wordings of the second example, but I don't want to discuss that (unless it turns out to be important for learning).&#xA;Neither do I want to discuss whether the two statements above are correct; that is left as an exercise.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-01-11T10:50:35.970" Title="Logic in symbols or words" Tags="|undergraduate-education|mathematical-pedagogy|reference-request|logic|language-use|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="8423" PostTypeId="2" ParentId="3813" CreationDate="2015-07-22T20:04:10.537" Score="6" Body="&lt;p&gt;When I taught second grade, we introduced probability without fractions.  We would talk about different events and classify them as:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;impossible&lt;/li&gt;&#xA;&lt;li&gt;unlikely&lt;/li&gt;&#xA;&lt;li&gt;equally likely and unlikely&lt;/li&gt;&#xA;&lt;li&gt;likely&lt;/li&gt;&#xA;&lt;li&gt;certain&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The students had no trouble understanding classifying events such as:&#xA; 1. the sun will rise tomorrow&#xA; 2. i will pick a blue ball from a bag with a 5 blue and 1 red balls&#xA; 3. I will pick an orange ball from a bag of blue and green balls.&#xA; 4. It will rain today (this brought many opinions)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The following year fractions were incorporated into probability. It worked very well.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-07-22T20:04:10.537" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8424" PostTypeId="2" ParentId="8413" CreationDate="2015-07-23T07:40:24.300" Score="0" Body="&lt;p&gt;One useful metric is:  &lt;em&gt;Does the proof yield a construction?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given a constructive proof and a non-constructive proof of the same length, the constructive proof is usually easier to understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This metric can be refined further, in terms of the complexity of the construction, or how obvious the construction is from the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good test case is the Fundamental Theorem of Algebra, since multiple books have multiple proofs of the theorem. &lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2015-07-23T07:40:24.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8425" PostTypeId="2" ParentId="8413" CreationDate="2015-07-23T15:49:15.517" Score="2" Body="&lt;p&gt;Lets say you give the proof as a sequence of sentences &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$S_1$$&#xA;$$S_2$$&#xA;...&#xA;$$S_n$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I consider a proof understandable, if at any point, I ask &quot;why do this?&quot; for that sentence, it should be obvious what the motivation is and where it is going to lead (IE by the time you get to sentence X, it should be obvious what sentence X+1 should be). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If that isn't the case, then usually the proof needs to be re-written, sentences rearranged, and motivation made clearer, until the entire thing jut gives off the air of &quot;this obviously the clearest way to go about it&quot;&lt;/p&gt;&#xA;" OwnerUserId="1909" LastActivityDate="2015-07-23T15:49:15.517" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8427" PostTypeId="2" ParentId="5648" CreationDate="2015-07-23T18:39:10.443" Score="23" Body="&lt;p&gt;In third grade we taught division using repeated subtraction.  To divide 6 by 2, subtract 2 until you get to 0.  6-2=4, 4-2=2, 2-2=0.  It took 3 steps so 6÷2=3.   This can also be shown on a number line, where it takes 3 steps of 2 units to go from 6 to 0.  Teaching the concept of division this way is just the inverse of what we have done for multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After we do a few simple examples and everyone has gotten the idea of division as repeated subtraction, I offer 8÷0.  We start subtracting 0 from 8: 8-0=8 and we continue 8-0=8,8-0=8,8-0=8,etc. When the whole class is giggling, I point out that this isn't working.  We can also try to go backwards from 8 on the number line by steps of 0 and somehow we never leave 8.  At this point I tell them that you can't divide by 0 and they all &quot;get it&quot; at their level.  We then try dividing by 0 on the calculator and everyone is excited to see E for error. &lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-07-23T18:39:10.443" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8428" PostTypeId="1" CreationDate="2015-07-24T04:50:36.803" Score="20" ViewCount="582" Body="&lt;p&gt;Many, perhaps most, math textbooks are very hard for most people to dive into. A few textbooks are delightful, and encourage people to learn math by their engaging tone. What textbooks have helped draw you into the subject, and what was it about them that did that? What recommendations would you give to an author based on your experience reading good and bad textbooks?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[I am asking this question because the closed question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2123/why-are-most-math-textbooks-so-uninspiring-unless-you-already-like-math&quot;&gt;&quot;Why are most math textbooks so uninspiring; unless you already like Math?&quot;&lt;/a&gt; seemed like it raised an important point - which needed reframing.]&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-11-08T01:26:31.293" LastActivityDate="2023-11-08T01:26:31.293" Title="What are some of the qualities that inviting textbooks share?" Tags="|textbooks|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="8429" PostTypeId="5" CreationDate="2015-07-24T12:51:13.153" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-07-24T12:51:13.153" LastActivityDate="2015-07-24T12:51:13.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8430" PostTypeId="4" CreationDate="2015-07-24T12:51:13.153" Score="0" Body="For questions on effective and creative use of whiteboards, blackboards, and Smart Boards, including questions on best practices for handwriting mathematics.   " OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2015-07-25T12:50:15.027" LastActivityDate="2015-07-25T12:50:15.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8431" PostTypeId="5" CreationDate="2015-07-24T13:03:35.267" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-07-24T13:03:35.267" LastActivityDate="2015-07-24T13:03:35.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8432" PostTypeId="4" CreationDate="2015-07-24T13:03:35.267" Score="0" Body="Questions related to managing time constraints for teachers both in and out of the classroom, helping students manage time and effects on students of limiting time for work and tests. " OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2015-07-24T20:09:35.160" LastActivityDate="2015-07-24T20:09:35.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8433" PostTypeId="2" ParentId="380" CreationDate="2015-07-24T13:19:05.963" Score="-1" Body="&lt;p&gt;I think the problem is partly because we often use the established notation like $\frac{dy}{dx}$ but throw away the intuition, and hence differentiation becomes a meaningless task rather than a motivated procedure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It starts with considering any two variables $x,y$ that depend on some parameter $t$ (usually position or time in the real world). Then $(x,y)$ traces a path through the state space as $t$ changes, and we are interested in the points where the path is smooth (not the mathematical sense), namely where the path is unbroken and has a direction. At any point, for any change in $t$, denoted $Δt$, there will be the corresponding changes in $x,y$, denoted $Δx,Δy$. The path being smooth at a point is then the same as saying that, at that point, $Δx,Δy \approx 0$ as $Δt \to 0$ (Of course this can be made precise, but in fact we can manipulate it axiomatically without using the usual epsilon-delta definitions!), and also the ratio $Δx:Δy$ tends to some ratio as $Δt \to 0$, where the ratio depends only on $t$ and not on $Δt$. In the specific case of $\frac{dy}{dx}$, we must have $Δx \ne 0$ and we define $\frac{dy}{dx} = r$ where $\frac{Δy}{Δx} \approx r$ as $Δt \to 0$, if such $r$ exists. (Note that we do &lt;strong&gt;not&lt;/strong&gt; need non-standard reals but just ordinary asymptotic analysis.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the above underlying framework, it is trivial to prove the basic facts of differentiation and much more. In particular, the chain rule says that given any variables $x,y,z$ that depend on a parameter $t$, whenever $\frac{dy}{dx},\frac{dz}{dy}$ are defined, then $\frac{dz}{dx} = \frac{dz}{dy} \frac{dy}{dx}$. Note that the above definition of $\frac{dz}{dy}$ does not allow the case where $y$ is constant, but this case never occurs in practice since we then do not need the chain rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example given variables $x,y$ such that $y = (x^2+3)^4$, if we let $z = x^2+3$, then $z$ is a variable and we can immediately find $\frac{dz}{dx} = 2x$ and $\frac{dy}{dz} = 4z^3$, and then conclude that $\frac{dy}{dx} = (4z^3)(2x)$. Even better still, without creating $z$ one can write $\frac{dy}{dx} = \frac{d((x^2+3)^4)}{d(x^2+3)} \frac{d(x^2+3)}{dx} = (4(x^2+3)^3)(2x)$. In both cases, it is simply a matter of a trivial instantiation of the chain rule, and the second direct method also reminds the chain rule user of the reasoning behind the rule, since the $d$s were indeed originally motivated by the $Δ$s, which cancel because they are actually nonzero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At an undergraduate level one might choose to define everything in terms of the modern epsilon-delta definitions, but in my opinion that is usually not done well. In most cases that I've personally observed, students are unable to justify the epsilon-delta definition, and many of them even produce haphazard guesswork when asked to prove a limit or derivative via the epsilon-delta definition. I would say that the latter is due to not teaching logical reasoning (basic first-order logic) before other subjects, but the former is due to the lack of motivation of defined concepts. For example, why on earth should we consider Cauchy sequences or the Bolzano Weierstrass theorem? Most students are just taught them as if they were unexplainable facts like the fact that our planet has only one moon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I do not believe in just teaching students to follow some procedure to get the answer, otherwise we are just teaching them to become a computer, and computers have already taken that kind of job. If they do not understand why they should do something (not just what they are doing), I consider that a failure in mathematics education.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2015-07-24T13:19:05.963" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8434" PostTypeId="2" ParentId="7959" CreationDate="2015-07-24T14:18:37.110" Score="8" Body="&lt;p&gt;Exponential growth or decay shows up everywhere in nature:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Temperature gradients (like in a hot water flask)&lt;/li&gt;&#xA;&lt;li&gt;Diffusion across a membrane (like in osmosis)&lt;/li&gt;&#xA;&lt;li&gt;Radioactive decay (and use in radiometric dating)&lt;/li&gt;&#xA;&lt;li&gt;Dampening (like of a vibrating string/pipe)&lt;/li&gt;&#xA;&lt;li&gt;Attenuation of a signal through a medium (like visibility in water)&lt;/li&gt;&#xA;&lt;li&gt;Uninhibited population growth (like at the start of virus epidemics)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Exponential scales include:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Loudness in decibels&lt;/li&gt;&#xA;&lt;li&gt;Octaves and other intervals in music&lt;/li&gt;&#xA;&lt;li&gt;Earthquake energy on the Richter scale&lt;/li&gt;&#xA;&lt;li&gt;Amount of information in bytes in powers of 1024&lt;/li&gt;&#xA;&lt;li&gt;Moore's 'law' (but it is going to fail soon)&lt;/li&gt;&#xA;&lt;li&gt;Acidity/basicity on the pH scale&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;They also show up in useful procedures:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Binary search (guess a number, I'll tell you whether you are high or low on each try)&lt;/li&gt;&#xA;&lt;li&gt;Decimal/binary representation (the number of symbols used compared to the value)&lt;/li&gt;&#xA;&lt;li&gt;Linearization (of suspected power relations)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I think that's more than enough significant real-world applications.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2015-07-24T14:18:37.110" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8435" PostTypeId="2" ParentId="380" CreationDate="2015-07-24T17:21:43.707" Score="2" Body="&lt;p&gt;This seems like an issue about learning objectives. What is it exactly you are wanting students to show you? It sounds like what you want is for students to show that they can approach an exercise or problem from different perspectives and using multiple methods. Is that right? For the sake of argument I'll assume it is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that assumption I'd recommend two things. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Make this expectation an explicit goal of the course in your syllabus, and use it as an organizing principle for your students' work. If a stated goal of the course is to be able to solve calculus-related (or whatever-related) problems using a variety of methods then you are allowed to restrict and direct their work to specific methods. For example if you state in the syllabus that a successful student will be able to solve calculus problems using graphical, numerical, &lt;em&gt;and&lt;/em&gt; symbolic methods -- and not just one of these -- then you can give problems that insist on a particular method, and it won't be arbitrary but rather a way of assessing how well they have mastered that particular goal. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This goes with the previous point, but you can give problems that specifically involve multiple methods and then make students do the same problem more than once, using different methods. Problems that emerge from data sets are perfect for this in calculus. For example: &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Suppose the temperature of a cup of coffee ($T$, in degrees celsius) has the following temperatures $t$ minutes after it's been brewed: &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Time 0   5   10   15   20&#xA;Temp 98  63  40   29   21&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Part 1: Estimate the instantaneous rate at which the temperature is changing at exactly 15 minutes by using the data in the table to make an estimation. [Ideal approach is a central difference estimate.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part 2: Estimate this instantaneous rate again, by plotting the data in scatterplot, fitting the function with a curve, and using the resulting graph. [Ideal approach would be to draw the tangent line to the curve they got.]  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part 3: Estimate this instantaneous rate again, by using technology to get a formula for the curve of best fit and using derivative rules. [You could specify that they must use an exponential model, for instance, if you want something other than a polynomial.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Part 4: Compare the three methods for estimating the instantaneous rate of change you used. Make sure that the results are all fairly similar. What are the pros and cons of each method? Which method seems to give the most accurate result? How can error or uncertainty propagate in each of these methods?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I bolded step 4 above because this is IMO where a &lt;em&gt;university-level&lt;/em&gt; approach to calculus needs to be reaching -- not only making computations but thinking about the relative merits and demerits of computational methods, and understanding the limitations of one's tools. By having students do problems that force them to do the same thing in different ways, you give students an opportunity to think critically about what they're doing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In some cases students have severe limitations on their short-cut methods and aren't even aware of them. For example if student are trying to find the area under a normal distribution curve and have the formula, they literally cannot answer that kind of question by hand calculations. So many students who are whizzes at taking derivatives and antiderivatives don't really get the fact that symbolic techniques are not always the best approach or even a possible approach and so they need to have a full repertoire of methods and the intellectual ability to compare them and select the best one. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2015-07-24T17:21:43.707" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8436" PostTypeId="2" ParentId="8428" CreationDate="2015-07-24T21:55:15.297" Score="6" Body="&lt;p&gt;Among many other distinctions: textbooks that are stern, hostile-skeptical, unconfiding, are obviously-reasonable construable as &quot;unfriendly&quot;. That is, if the author makes clear that you, the reader, are unworthy, and probably unable to understand the point anyway, it's off-putting. Similarly, as we know too well, it is possible to take a similar adversarial/hostile approach in &quot;teaching&quot;, and students often default into an assumption that this is the case... which combines in a bad way with such a textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The hostility can be veiled, as in text that is primarily definitions and theorem-proving with few examples, and &quot;exercises&quot; that are exclusively the working-out of the standard examples and counter-examples, but without any models in the text proper. Thus, a bait-and-switch on the reader, sometimes rationalized by the mythology that &quot;problem-solving&quot; is something that students can be coerced into, rather than shown by example, etc. Also rationalized by the unfortunate mythology that upper-division math starts to become a &quot;test of talent&quot;, offering believers in that an excuse in case they're inadequate teachers, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see the same issues of &quot;skepticism&quot;, let's call it, and &quot;(passive-aggressive) unhelpfulness&quot; appearing in texts and &quot;teaching&quot; throughout undergrad and graduate-level teaching at my R1 university. I've heard rationalizations that &quot;if there are no losers, winning is meaningless&quot;... and codeword &quot;challenging&quot; for unhelpful texts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There're also the misleadingly-thin texts whose text is definitions, with most of the major theorems left to the exercises. Ack. Really nothing attractive in such a thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sources (written or live) that show &lt;em&gt;fun&lt;/em&gt; things, &lt;em&gt;live(ly)&lt;/em&gt; examples to pique curiosity prior to &quot;definitions&quot;, rather than throwing the important examples into an ocean of exercises so they'll be lost among drill-and-kill stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many people who're &quot;successful&quot; in mathematics have developed a resistance to that toxicity, although it seems to me a sad waste of energy to have to defend oneself in such fashion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somehow the style of mathematics textbooks has mostly drifted into fairly extreme stodginess and fussiness, although some of this seems to be merely meeting societal expectations (in the U.S.). It really cannot be that &lt;em&gt;every&lt;/em&gt; detail is as important as every other, yet this seems to be a common presentation... seemingly compelling the reader to have to suspend their own judgement for the whole business to make sense, such as it may.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Worst-case: &quot;math as rules handed down by authority&quot; (The State, possibly?) No one &lt;em&gt;should&lt;/em&gt; respond positively to that. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the &quot;standard curriculum(s)&quot; I see in U.S. high schools and undergrad mostly seem to be the most inertia-laden of any part of the larger curriculum, and even many parents and students have a powerful expectation that math will &lt;em&gt;not&lt;/em&gt; be interesting or fun, and will actually react badly if it is made into something else. I've had people (both students and parents) privately tell me that the best features of &quot;math&quot; do involve it being &quot;a test of character&quot; for them or for their kids. Yikes! :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This creates a bizarre environment for successful (=adapted) text-book writing...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-07-24T21:55:15.297" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8437" PostTypeId="2" ParentId="7974" CreationDate="2015-07-24T22:02:08.660" Score="4" Body="&lt;p&gt;Have you tried KenKen puzzles? For those unfamiliar with them, these puzzles require logic and arithmetic operations to solve.  One of the best sites is &lt;a href=&quot;http://www.kenken.com&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.  You can choose puzzles of different levels of difficulty with different combinations of the four operations.  I have had a lot of success using these puzzles with first through fifth graders.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This &lt;strong&gt;edit&lt;/strong&gt;/addition is in response to a comment. In first grade I usually introduced addition only puzzles to my gifted class.  I started with 3 numbers in each row. Students who caught on quickly were given a &lt;a href=&quot;http://smile.amazon.com/Will-Shortz-Presents-Can-KenKen/dp/0312546416/ref=sr_1_5?ie=UTF8&amp;amp;qid=1437999548&amp;amp;sr=8-5&amp;amp;keywords=kenken%20books&quot; rel=&quot;nofollow&quot;&gt;book&lt;/a&gt; and many of them enjoyed working on it independently. The book's puzzle increase in difficulty and the number in each row go from 3 numbers to 6 numbers.  Some students needed guidance and weren't ready to work independently; even those children could figure out what was missing with well framed questions.  All the children enjoyed it and those who earned a book by completing several puzzles were very proud.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2015-07-27T12:24:32.393" LastActivityDate="2015-07-27T12:24:32.393" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8438" PostTypeId="2" ParentId="8428" CreationDate="2015-07-25T06:00:05.683" Score="3" Body="&lt;p&gt;I wish I could remember more about why I found these books engaging.  Although some of it is presented in a near-conversational style, I think a selection and arrangement of topics is also important.  Two examples from my undergraduate education are Martin Braun's Springer undergraduate text on differential equations and applications (I fondly recall the section on the Tacoma Narrows Bridge collapse) and Munkres's undergraduate Topology text.  A further example is Concrete Mathematics by Graham, Knuth and Patashnik.  Each of these gave me the impression they were written by humans, or at least mathematicians who were willing to show a human side.  However, I found none of these as engaging as books by Martin Gardner or Isaac Asimov.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wouldn't mind having a Gardner or Asimov-like guide to read first.  After that, I might be willing to tackle less friendly texts once I have had someone reassure me as to what was going on, and why some concepts were key to understanding the rest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Should Read More Modern Authors&quot; Paseman, 2015.07.24&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-07-25T06:00:05.683" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8439" PostTypeId="2" ParentId="8428" CreationDate="2015-07-25T23:02:39.017" Score="6" Body="&lt;p&gt;I have no doubt there are more modern textbooks,&#xA;but I found Fraleigh a fine&#xA;introduction for unsophisticated undergraduates:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fraleigh, John B. &lt;em&gt;A First Course in Abstract Algebra&lt;/em&gt;. Pearson, 2003. 7th Edition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The many short chapters, with both self-testing T/F questions, and achievable&#xA;exercises, makes it easy on both the teacher and the student. He includes historical remarks, copious examples, and distinguishes in exercises between: computation, concepts, &#xA;theory, and &quot;proof synopses.&quot;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/GAJQX.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GAJQX.jpg&quot; alt=&quot;Fraleigh&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Fraleigh, pp.66-7, Sec. 6, Groups &amp;amp; Subgroups.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-07-25T23:02:39.017" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8440" PostTypeId="2" ParentId="8428" CreationDate="2015-07-26T08:34:35.513" Score="3" Body="&lt;p&gt;Here are some examples of really accessible textbooks:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0307137643&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;DINOSAURS and Other Prehistoric Reptiles&lt;/em&gt;&lt;/a&gt;.   A Giant Golden Book.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0393310728&quot; rel=&quot;nofollow&quot; title=&quot;How to Lie with Statistics&quot;&gt;&lt;em&gt;How to Lie with Statistics&lt;/em&gt;&lt;/a&gt;.  By Darrell Huff; illustrated by Irving Geis.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B000O8UZDK&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;How to Win Friends and Influence People&lt;/em&gt;&lt;/a&gt;.  By Dale Carnegie.  1937 edition.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the best textbooks take an approach of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Limiting the scope to things that a non-expert reader is likely to find interesting.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Consciously treating the field as containing mysteries:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Asking &quot;How can we figure something like this out?&quot;&lt;/li&gt;&#xA;&lt;li&gt;Showing that even experts make mistakes, and that ordinary people can figure things out.&lt;/li&gt;&#xA;&lt;li&gt;Providing examples of the limits of what is known at the time.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Illustrate the ideas with pictures and/or examples.&lt;/li&gt;&#xA;&lt;li&gt;Show a clear, practical, step-by-step way to solve each solvable problem.&lt;/li&gt;&#xA;&lt;li&gt;Show a clear, practical way to check how accurate (or useful) each answer is.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These books tend to be well-organized, with big fonts and clear summaries.  They emphasize &quot;you can do it yourself&quot; sanity checks rather than proofs.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-08-04T18:48:51.700" LastActivityDate="2015-08-04T18:48:51.700" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8441" PostTypeId="1" CreationDate="2015-07-26T13:01:25.920" Score="0" ViewCount="327" Body="&lt;p&gt;I'm looking for an effective classroom management plan for high school teachers. Mine is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Every student should have a notebook and a pen to take notes.&lt;/li&gt;&#xA;&lt;li&gt;Food, drinks, chewing gums and electronic devices are not allowed in the classroom. If you want to drink you should bring your water bottle with you.&lt;/li&gt;&#xA;&lt;li&gt;You can't joke or act like a clown.&lt;/li&gt;&#xA;&lt;li&gt;Every student should be sitting at his table when the teacher enters the classroom.&lt;/li&gt;&#xA;&lt;li&gt;You can't leave your place without permission.&lt;/li&gt;&#xA;&lt;li&gt;I won't tolerate cheating and plagiarism.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I give it to students on the first day of school. It's a part of the syllabus together with the table of contents, the evaluation schema, the objectives, the materials needed and the methodology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm waiting to see some other classroom management plans, a more exhaustive one.&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="501" LastEditDate="2015-08-03T13:06:52.533" LastActivityDate="2015-08-03T13:06:52.533" Title="Most effective classroom management plan" Tags="|secondary-education|classroom-management|syllabus|" AnswerCount="1" CommentCount="28" ClosedDate="2015-07-27T09:44:09.387" ContentLicense="CC BY-SA 3.0" />
  <row Id="8442" PostTypeId="2" ParentId="8441" CreationDate="2015-07-26T21:00:09.973" Score="5" Body="&lt;p&gt;Suppose I invite you to dinner.  I want you to have the best experience possible.  So I will manage things to maximize what I think is this best&#xA;experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, everyone needs a place setting.  So I prepare name cards so that&#xA;you sit at your assigned place.  I think I know best who you will like to&#xA;talk to, so please stay in your seat during dinner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the food should be served in a particular order.  So everyone needs to enjoy the same course at the same time. Of course, wasting food is sinful (or socially unacceptable) so we will wait until everyone finishes their course before moving on to the next.  You may have water at anytime, but the wine/beverage only does best with certain courses.  Please don't drink the wine except during those courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, we want to observe proper table manners.  My apologies if you are left-handed (or single handed), please use the knife and spoon with the right hand and the fork with the left.  We don't want confusion or delay, and keeping diversity to the minimum should enhance everyone's experience, right?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course people will need to attend to (ahem) personal matters.  There will be restroom breaks between the fourth and fifth courses (not during the fourth course), and between the twelfth and thirteenth courses (not during the twelfth course).  A bell will ring to remind you at the appropriate time.  Please return to your seats for the beginning of the next course, so that there is no delay.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding discussion, preferably only one person speaks at a time, and every one should pay attention to that person.  We will have a microphone-shaped marker to pass around; you will be briefed on its use at the beginning of dinner.  The list of topics to talk about will be on the green sheet; the areas to avoid on the red.  Don't confuse them!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could go on, but what I have just done in my enthusiasm to do well is going to turn out to be a recipe for disaster.  People will either be too frightened or resentful to contribute, or you will spend too much time attempting to enforce conformity.  People will not remember how the food tasted or how delightful and edifying the dinner conversation was; they will remember the farce of attempted &quot;experience management&quot; that you hosted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is directly inspired by imagination and by your list.  In my (not so humble) opinion, you are concocting a plan for failure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that certain behaviour is encouraged in a classroom, that there should be rules, that to get things done in an effective fashion and not waste anyone's time, there should be a plan and guidelines.  Indeed you should have such a list.  However, if you are upfront and promoting the list before the material, I think you are not educating;  I think you are (attempting to be) controlling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are ways to achieve similar goals, but using a different approach.  Show students that this is a relaxed and inviting atmosphere, encourage certain values by actions and behaviour, recruit others to help this encouragement along, teach by modelling, and be forgiving about departures and mistakes.  This can encourage confidence as well as the desire to learn.  In my view, your list does not do that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I'm not in your shoes, and I don't know what your agenda really is.  There are situations where your list might be ideal.  Certainly I don't disagree with some of the intent behind having such rules.  Just remember how easily dinner can go bad, regardless of how first-rate the food and drink are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Class Over.  Time For Snack.&quot; Paseman, 2015.07.26&lt;/p&gt;&#xA;" OwnerUserId="3468" LastEditorUserId="3468" LastEditDate="2015-07-26T21:24:25.833" LastActivityDate="2015-07-26T21:24:25.833" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="8443" PostTypeId="2" ParentId="8418" CreationDate="2015-07-27T20:17:25.890" Score="4" Body="&lt;p&gt;I first learned calculus from the predecessor of the current college outline series Calculus, which you can get from Amazon for 59 cents, free shipping if you sign up for a trial 6 month trial of &quot;Amazon Student.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0156015560&quot; rel=&quot;nofollow&quot;&gt;http://www.amazon.com/Calculus-Harcourt-Jovanovich-College-Outline/dp/0156015560/ref=sr_1_1?ie=UTF8&amp;amp;qid=1438026539&amp;amp;sr=8-1&amp;amp;keywords=college+outline+series+calculus&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A second edition of my analysis book will be out in September.  Among other things it adds a discussion of visualizing in 4D.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best, Charles Pugh&lt;/p&gt;&#xA;" OwnerUserId="5433" LastActivityDate="2015-07-27T20:17:25.890" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8444" PostTypeId="1" CreationDate="2015-07-28T00:04:43.177" Score="10" ViewCount="535" Body="&lt;p&gt;I'm interested in gathering a list of physical objects of mathematical interest for occasional or permanent display in a classroom. Mostly I'm interested in things that are wall-mountable, but any physical objects will do.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2019-02-20T05:55:46.420" Title="Interesting Physical Implements for the Classroom" Tags="|secondary-education|primary-education|classroom-interaction|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8445" PostTypeId="2" ParentId="8444" CreationDate="2015-07-28T00:27:12.847" Score="12" Body="&lt;p&gt;Two things that I had in mind, which prompted the question in the first place, are:&lt;/p&gt;&#xA;&lt;h2&gt;A Pin Board&lt;/h2&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/dr2dN.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dr2dN.jpg&quot; alt=&quot;Pin Board&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;sub&gt;(source: &lt;a href=&quot;http://ptrow.com/articles/Galton_June_07_files/GaltonBoard2.jpg&quot; rel=&quot;noreferrer&quot;&gt;ptrow.com&lt;/a&gt;)&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;p&gt;This device demonstrates the emergence of normally distributed data from a large number of independent, random events applied in series.&lt;/p&gt;&#xA;&lt;p&gt;Image from &lt;a href=&quot;http://ptrow.com/articles/Galton_June_07.htm&quot; rel=&quot;noreferrer&quot;&gt;http://ptrow.com/articles/Galton_June_07.htm&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;An Adding Machine&lt;/h2&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://woodgears.ca/marbleadd/marble_adder.jpg&quot; alt=&quot;Adding Machine&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm currently building one of these beauties. This machine serves as an excellent starting point for binary representation of numbers, mechanized computation in general, etc. Students can gain an appreciation for the notion that computers working in binary is primarily an engineering decision (eg, it's easier to make a toggle that has 2 positions than any other number).&lt;/p&gt;&#xA;&lt;p&gt;Image and machine plans from &lt;a href=&quot;https://woodgears.ca/marbleadd/index.html&quot; rel=&quot;noreferrer&quot;&gt;https://woodgears.ca/marbleadd/index.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Both of these pieces are beautiful, loud, mathematically rich, and quite a bit of fun to watch. Their presence will stir curiosity and discussion.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-02-20T05:55:46.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8446" PostTypeId="2" ParentId="8444" CreationDate="2015-07-28T04:40:16.057" Score="8" Body="&lt;p&gt;I think &lt;a href=&quot;https://en.wikipedia.org/wiki/Polyhedron&quot; rel=&quot;nofollow noreferrer&quot;&gt;polyhedra&lt;/a&gt; and polyhedra-like objects are, although not exactly wall-mountable, quite pleasant to have around.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Polyhedra are particularly easy to build with &lt;a href=&quot;http://zometool.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;zometool&lt;/a&gt; materials (a professor of mine always had a complete set of zometool &lt;a href=&quot;https://en.wikipedia.org/wiki/Platonic_solid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Platonic Solids&lt;/a&gt; laying around in his office; they were great), as this picture (from the zometools website) shows: &lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/8irMJ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8irMJ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, you don't need fancy struts and joints, as (preferably nice cardstock) &lt;a href=&quot;http://www.korthalsaltes.com/cuadros.php?type=p&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; will suffice, and opens up whole new realms of possibility, as &lt;a href=&quot;http://georgehart.com/slide-togethers/slide-togethers.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;George W. Hart&lt;/a&gt; shows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/sRndh.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sRndh.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The latter have the advantage of being light enough to be hung from the ceiling easily and don't need a file cabinet or anything to rest on. I've built a few of his 'slide-together' structures, and they're marvelous. (Note: The above image is the object obtained from rotating each face of an icosahedron by, I believe, 60 degrees, as demonstrated in &lt;a href=&quot;http://demonstrations.wolfram.com/RotatingTheFacesOfTheIcosahedron/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; Wolfram Demonstration).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Regular_icosahedron#Cartesian_coordinates&quot; rel=&quot;nofollow noreferrer&quot;&gt;icosahedron&lt;/a&gt; can also be obtained by fitting three golden rectangles together. This is also fairly easy to implement physically, though I haven't decided how to include the edges (and it's kind of more fun to hold the three interlocking planes and convince yourself that you're looking at an icosahedron).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kUOCH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kUOCH.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, of course, who doesn't want to have a dodecahedron, &lt;a href=&quot;https://en.wikipedia.org/wiki/Ball_%28association_football%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;soccer/football&lt;/a&gt;, and golf ball around, as evidence that all &lt;a href=&quot;https://en.wikipedia.org/wiki/Goldberg_polyhedron&quot; rel=&quot;nofollow noreferrer&quot;&gt;polyhedral objects&lt;/a&gt; with only regular hexagonal and pentagonal faces, with three meeting at each vertex, must have exactly 12 pentagons?&lt;/p&gt;&#xA;" OwnerUserId="5153" LastActivityDate="2015-07-28T04:40:16.057" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8447" PostTypeId="2" ParentId="8444" CreationDate="2015-07-28T12:33:49.227" Score="4" Body="&lt;p&gt;You could hang posters of the &lt;a href=&quot;http://www.clowder.net/hop/17walppr/17walppr.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;17 wallpaper groups&lt;/a&gt;, or the &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_space_groups&quot; rel=&quot;nofollow noreferrer&quot;&gt;230 space groups&lt;/a&gt;,&#xA;hoping their presence will engender wonder and generate questions.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/c0t5b.jpg&quot; width=&quot;250&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(&lt;a href=&quot;http://www.mjsd.k12.wi.us/archive/mhs/depts/math/teachers/christensen/17wallpapergroups.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;17 Wallpaper Groups&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/waFeN.png&quot; height=&quot;500&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(&lt;a href=&quot;https://crystalsymmetry.wordpress.com/2014/08/15/the-space-group-list-project-as-a-poster/&quot; rel=&quot;nofollow noreferrer&quot;&gt;230 Space Groups&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-07-28T22:39:28.187" LastActivityDate="2015-07-28T22:39:28.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8448" PostTypeId="2" ParentId="8444" CreationDate="2015-07-29T08:27:41.100" Score="4" Body="&lt;p&gt;How about a Slide Rule:&lt;a href=&quot;https://i.stack.imgur.com/ceSgY.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ceSgY.jpg&quot; alt=&quot;slide rule&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also some wolfram posters:&lt;a href=&quot;https://i.stack.imgur.com/A7WCo.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/A7WCo.jpg&quot; alt=&quot;Riemann&amp;#39;s Zeroes&quot;&gt;&lt;/a&gt; or:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/0eDrS.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0eDrS.jpg&quot; alt=&quot;Solving the Quintic&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="998" LastActivityDate="2015-07-29T08:27:41.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8449" PostTypeId="2" ParentId="2462" CreationDate="2015-07-30T01:02:42.123" Score="5" Body="&lt;p&gt;I taught honors and enrichment math to students in 1st - 6th grades and we explored a lot of games.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My students in 4th to 6th grades have access to chrome books and can go to sites that are bookmarked to play.   Here are their favorites:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.coolmath-games.com/0-lemonade-stand&quot; rel=&quot;noreferrer&quot;&gt;Lemonade Stand at CoolMath4Kids&lt;/a&gt;.  The kids run a lemonade stand. They buy inventory, set prices and try to make a profit. Daily weather reports add to the fun. Older kids used unit pricing to find the best buys for their supplies. There are more &lt;a href=&quot;http://www.coolmath4kids.com/0-cool-math-games.html&quot; rel=&quot;noreferrer&quot;&gt;games at this site&lt;/a&gt;. &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.mathplayground.com/PartyDesigner/PartyDesigner.html&quot; rel=&quot;noreferrer&quot;&gt;Design a Party&lt;/a&gt; at Math Playground. You have to use area and perimeter to plan the layout of a party.  There are more games at this site.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://illuminations.nctm.org/Activity.aspx?id=4134&quot; rel=&quot;noreferrer&quot;&gt;The factor game&lt;/a&gt; and &lt;a href=&quot;http://illuminations.nctm.org/Activity.aspx?id=4213&quot; rel=&quot;noreferrer&quot;&gt;the product game&lt;/a&gt; These are two player games at &lt;a href=&quot;http://illuminations.nctm.org/&quot; rel=&quot;noreferrer&quot;&gt;illuminations from NCTM&lt;/a&gt; but the students can play against the computer.  The factor game require an understanding of prime and composite numbers and the ability to find factors.  The product game is a great thinking game for basic multiplication facts.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As for non-technology games: every Friday at lunch the students would come and play math games.  Here is a list of some of the most popular:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.gamewright.com/gamewright/index.php?page=game&amp;amp;section=games&amp;amp;show=203&quot; rel=&quot;noreferrer&quot;&gt;Zeus on the Loose&lt;/a&gt; - Practices addition (mental math) of one digit numbers and two digit numbers up to 100. It also uses rounding to 10, and recognizing multiples of 10.  Weaker students can play with stronger students who help with the mental math.  This was so popular I had to buy four of them! Enjoyed by gifted 1st and 2nd graders and 3rd-6th graders. You can buy it at Amazon.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B00000IRW1&quot; rel=&quot;noreferrer&quot;&gt;Presto Change-O&lt;/a&gt; - Practices recognizing money, making change, counting money.  Money amounts include nickels, dimes, quarters, ones and fives.  The object is to get $10.  Very popular among 3rd and 4th graders though it can be played with older and younger children.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B00004TDKU&quot; rel=&quot;noreferrer&quot;&gt;Dino Math Tracks Game&lt;/a&gt;  Teaches place value and very popular with 2nd graders, although good for 1st-3rd.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B005IRW0EE&quot; rel=&quot;noreferrer&quot;&gt;Bazaar&lt;/a&gt;  Students trade stones and learn algebraic concepts - good for students as young as 1st grade.  &lt;/li&gt;&#xA;&lt;li&gt;Midnight Party - Uses addition of negative numbers  An all time favorite but it is no longer being made. If you can find them on e-bay, I highly recommend it.&lt;/li&gt;&#xA;&lt;li&gt;99 - very similar to Zeus on the Loose but the rules are less complicated. Uses addition of one digit and two digit numbers and subtraction.  Excellent game for 1st and up.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.leganigames.com/parts_of_a_whole.htm&quot; rel=&quot;noreferrer&quot;&gt;Parts of a Whole&lt;/a&gt; This was used in class only. Students play&#xA;war and in the process order fractions, decimals, percents, and&#xA;parts of a whole in pictures.  Much more popular than worksheets!&#xA;This is made by &lt;a href=&quot;http://www.leganigames.com/products.htm&quot; rel=&quot;noreferrer&quot;&gt;Legani Games.&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://thinkfun.com/results?keyword=swish&quot; rel=&quot;noreferrer&quot;&gt;Swish&lt;/a&gt; This is a visual spatial game made by &lt;a href=&quot;http://thinkfun.com/&quot; rel=&quot;noreferrer&quot;&gt;Think Fun&lt;/a&gt;. It is enjoyed by my 6th graders.  Younger children find it very challenging.  &lt;a href=&quot;http://thinkfun.com/&quot; rel=&quot;noreferrer&quot;&gt;Think Fun&lt;/a&gt; has many puzzles that my students enjoy and digital links to some of their puzzles.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-07-30T01:02:42.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8452" PostTypeId="2" ParentId="8390" CreationDate="2015-07-30T06:31:31.417" Score="2" Body="&lt;p&gt;We have a 15 year old son, Grade 10, that lacks motivation to study as well. Mommy and me were the top learners at school, and have a lot of degrees and are IT professionals, so this is a total unknown state to us. His one male friend, with the same lack of motivation, has a 12 year old brother, that studies and scores 90 percent!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are 4 stakeholders that have to be in place to get a learner to be successful to an average degree: the parents, the school, the teacher and the child(and maybe his friends). We normally first blame the child for lack of performance. Move the child to another teacher or move the child to a different school, or look at the example that the parents set to the child, and then, the least controllable is how the school and the teacher treat your child!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at the modern child: they are all occupied by PC games, Internet Youtube, social media, movies, music, etc.Then, the modern boy: he is not as mature as the girls, he is struggling to find and define himself and this results in a youngster with very little self confidence. This is where your problems start! As a qualified Maths teacher I believe that kids do not master the basics of Maths during Grade 4-7 ie times tables, adding fractions, mental arithmetic, basic problem solving resulting in a kid that &quot;cannot do Maths&quot; later because of a fear of showing that they cannot do the basics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So for a 14 year old: let him enjoy life first - do not overburden him with study/extra lessons/threats. Boost his confidence by letting him do small tasks at home, fill his room with secondhand university books on Einstein, space travel, biology... Tell and go show him how wonderful it is to be a University student and that he needs to master his Maths to be able to qualify for University. Do times table drills on a Saturday. Buy old school books from his lower grades, sit down and jointly do those (sometimes trivial/boring) Maths jointly and repetitively to become masters of the notation, concepts,mechanics and problem solving thought processes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly confront the teacher (the most common problem I have found, as they are too busy to give attention to 30 x 5 children every day, or sometimes they are just rude/non-caring bullies): ask her what is she doing to foster enthusiasm and understanding of Maths in her class this week! Ask her to put your 14 year old in the front of the class, ask her to let him do some trivial task on the board to boost his confidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, once the Maths basics and self confidence improve, your boy will suddenly take off and feel that he can actually be competent in Maths! Good luck!    WDK, Pretoria, South Africa.&lt;/p&gt;&#xA;" OwnerUserId="5441" LastActivityDate="2015-07-30T06:31:31.417" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8453" PostTypeId="1" CreationDate="2015-07-30T20:08:20.447" Score="9" ViewCount="1092" Body="&lt;p&gt;I'm going to teach calculus for the first time to undergraduate students. I would like to know if there is some book about how to teach the concepts of calculus (e.g. limits, derivatives, etc.).&lt;/p&gt;&#xA;" OwnerUserId="5236" LastEditorUserId="2074" LastEditDate="2015-07-30T21:01:18.853" LastActivityDate="2021-01-25T19:49:50.853" Title="How to teach calculus (book recommendation)" Tags="|undergraduate-education|calculus|reference-books|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="8454" PostTypeId="2" ParentId="7247" CreationDate="2015-07-31T17:48:40.477" Score="4" Body="&lt;p&gt;When I was a student, I always found pure maths very difficult to study compared to applied mathematics, and as a result I gravitated mostly towards the applied courses during my degree. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possible reasons why people might find applied mathematics more straightforward than pure mathematics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The style of thinking required to succeed at pure mathematics is unfamiliar to many and can make students uncomfortable. This is particularly because it relies on logical definitions, and these definitions are very different to those that we think of in real life. Our definitions of things are pre-concieved &lt;em&gt;images&lt;/em&gt; of what those things are, rather than a definition with several properties that must be satisfied. (Worth giving &lt;strong&gt;[1]&lt;/strong&gt; a read to gain an understanding of why pure mathematics can be uncomfortable for students!)&lt;/li&gt;&#xA;&lt;li&gt;Applied mathematics courses often involve solving differential equations, and doing calculations/simulations, with considerably less emphasis on why the methods work or the conditions that have to be met in order for them work. This is much more in line with the mathematics that pre-university students are taught (in the UK at least), so the sense of familiarity can make applied maths seem much more straightforward to digest than rigorous pure mathematics. &lt;/li&gt;&#xA;&lt;li&gt;Many areas of pure mathematics are very difficult (if not impossible) to come up with a visualisation of. Many mathematics students need to rely on &quot;concept images&quot; in order to gain an intuition of what a definition means rather than blithely learning the definition (&quot;concept definition&quot;), and if they are unable to unite these, then this can make learning pure mathematics very difficult. However, there are some areas of pure mathematics that can be quite nicely visualised, which is why I really enjoyed point-set topology and aspects of analysis, but struggled much more with topics such as group theory, linear algebra or Galois theory.&lt;/li&gt;&#xA;&lt;li&gt;Pure mathematical proofs as seen in lectures are often refined and concise. However, they do not do a good job of explaining or justifying the steps and it is often left to the reader to figure that out. This can put a lot of students off, as the way proofs are presented in lectures and books can seem arbitrary; the proofs do not outline the thought process undertaken by the person who wrote them, or explain why a specific tactic of proof has been used. It's often just &quot;Proof: here it is.&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;However, one should be careful in just categorising topics as &quot;pure&quot; and &quot;applied&quot;, as the two categories can really help each other and ties between them are essential in order to delve deeper into them. For example, partial differential equations very often form the basis of applied mathematics and physics courses, even though there are many areas of pure mathematical analysis that are required in order to be able to prove their properties or make further developments. Without a pure mathematical background, the best thing you can do is run simulations or solve PDEs in very specific cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something I've felt in recent years is that I've shied away from learning pure mathematics for the reasons above, but it's a whole new beast that I still want to conquer and understand. Even after a maths degree (and a failed PhD attempt) this is something I want to overcome in the fullness of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1]: &lt;a href=&quot;https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/8846/1/Alcock_Simpson_book.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alcock, L. and Simpson, A., 2009. Ideas from mathematics education: an introduction for mathematicians. MSOR network.&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerDisplayName="user5447" LastEditorUserId="376" LastEditDate="2019-04-10T12:45:10.610" LastActivityDate="2019-04-10T12:45:10.610" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8455" PostTypeId="1" CreationDate="2015-07-31T18:01:43.310" Score="8" ViewCount="708" Body="&lt;p&gt;I recently withdrew from a maths PhD to focus on other career options, and one of them that I'm very keen on is mathematics teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I applied for a college A-level maths lecturer role (teaching students aged 16-18), in which I had to do a 20 minute micro-teach session, some exam questions and answer some interview questions about my general attitude and ethos to teaching. I wasn't offered the job but I was offered a very appealing teacher training opportunity which would allow me to train with the college and get a PGCE/CertEd at heavily reduced costs, a generous government bursary, and take part in guaranteed hours of teaching the A-levels partially due to needing to cover for a teacher who is going on maternity leave.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the education system in the UK, students are very heavily tested and so there is a large tendency for teachers to get students through exams, more so than actually teaching the subject in a more general context. I know that as a college teacher I won't be able to reinvent the wheel with the education system in this country, and I may be pressured into solely focusing on getting good marks out of the students, but I would like to take steps in order to make my maths lessons engaging, rigorous and fulfilling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone suggest some ways in which I can teach a standard A-level course in a way that allows students to succeed in their exams, but also making it unique and exciting, and without diluting on the mathematical rigour?&lt;/p&gt;&#xA;" OwnerDisplayName="user5447" LastActivityDate="2015-08-01T12:48:31.367" Title="Tips for teaching A-level Maths/Further Maths students" Tags="|mathematical-pedagogy|secondary-education|technology-in-education|community-colleges|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8456" PostTypeId="1" CreationDate="2015-07-31T19:05:46.660" Score="18" ViewCount="1265" Body="&lt;p&gt;I am a rising college junior in US with a major in mathematics.  I recently noticed a problem in my note-taking skill in the mathematics, both from the textbooks and lectures.  When I was a microbiology major, I wrote extensive amount of notes from my books and lectures since I had to memorize all of them.  I followed this trend and took extensive notes from my math books and professors' lectures.  The problem with my note-taking skill is that I basically copy down almost everything from my books.  For example, I am currently studying Apostol's Mathematical Analysis; what I am doing is that I copy down the theorems, proofs (including my footnotes), authors's remarks &amp;amp; motivation, and examples.  I tried to copy them on my own words, but the result is that I basically copy down verbatim with little change in the grammars.  I started to realized that I should not take such exhaustive notes since the textbooks contain exact materials from my notebooks, but I cannot overcome the feeling that I need to make my own version of books, by taking notes from them, and the fear of memory loss.  Also I tried to take as much notes as possible from the professors's lectures.  What I realized is that I copy down the exact words from my professors, and I often lost the concentration to focus on lectures and absorb the materials.  I also lost my concentration whenever I take the notes from my books since I need to divert my attention from books to notebooks.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am currently thinking about various alternative strategies, which are followings:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1)  Taking notes within the textbook using the blank space of books and the Post-It:  Take notes about interesting remarks, confusion about the exposition and examples, and my own interesting ideas or approach to the proofs and examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Taking notes on a separate notebook but only copying the information from 1).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3)  Do not take any notes and try to absorb the materials from the books: I noticed that I usually learn the best by reading.  Use the lecture to supplement the textbook reading and to gain different perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you help me out by commenting about my strategies and/or share your note-taking skills?  I am quite embarrassed about my note-taking problem.&lt;/p&gt;&#xA;" OwnerUserId="5399" LastEditorUserId="5399" LastEditDate="2015-07-31T20:06:56.440" LastActivityDate="2016-01-25T19:11:36.067" Title="Inquiry about my note-taking skill" Tags="|undergraduate-education|self-learning|lecture-notes|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8457" PostTypeId="2" ParentId="8456" CreationDate="2015-07-31T22:24:39.750" Score="15" Body="&lt;p&gt;As you have noticed, mathematical text is often quite concise and it can be difficult to squeeze it into tighter space or write in your own words in a nontrivial way.&#xA;Therefore it is easy to end up copying things verbatim if you want to take notes.&#xA;Taking useful notes in mathematics is quite different from many other subjects, and you are not alone with your problem.&#xA;You should not be embarrassed; you recognized a problem and want to do something about it and far too few students actively do this.&#xA;I have some suggestions for taking notes in a more useful way.&#xA;(This is, of course, based on personal experience and opinion.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reading a book and following lectures are two different ball games.&#xA;Let me start with the lectures.&#xA;It often happens that lectures move on so fast that if you want to copy everything that happens on the blackboard, it can be difficult to follow the content and &amp;mdash; most importantly &amp;mdash; the ideas.&#xA;In these days lectures should not be the sole means of delivering the content; the professor can scan his notebook and put the pdf file online for the students to download.&#xA;If he does not do this, request that he does.&#xA;Mindless copying is pointless and wastes a lot of time.&#xA;(Arguing this to some professors can be a difficult task, but it should be tried.)&#xA;If the lectures are based on slides, ask for them and bring them with you to the lectures, printed on paper.&#xA;If no such material is available, try to read the textbook or other such material in advance, so that you will have the material in mental form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found it very useful to have the lecturer's notes he is using (or very close to it) during the lectures.&#xA;This does not mean that I sit down idly.&#xA;I follow the lecture and try to read the material in advance (at least quickly).&#xA;There are always some tricky points that are not perfectly explained in the written notes.&#xA;Often the professor says helpful remarks and answers questions from the audience, explaining what is happening or why certain choices were made.&#xA;It is those gems that I try to catch and write them among the lecture notes.&#xA;The notes can be informal and do not attempt to make rigorous mathematics.&#xA;Instead, I try to capture my momentary intuition in words and explain things to myself or relate them to other things.&#xA;The kinds of notes I take depend heavily on my proficiency in the subject and the notes can be of any kind.&#xA;(Sometimes I just keep reminding myself about the nature of each object by comments: this is a function, this is a vector, this is a set of sequences&amp;hellip;)&#xA;The notes are rarely so lucid and well explained that no notes are needed, but this can happen, too.&#xA;Most of the time I just follow the professor's train of thoughts and try to understand why everything happens as it does.&#xA;Only a small portion of time is spent on making notes.&#xA;If I have to take notes to copy the content to my notebook (this is sometimes the only way to get it, unfortunately), I have much harder time understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are reading a book, the situation is different, as there is no hurry.&#xA;Making supplementary notes works here like during the lectures.&#xA;If you don't want to damage a book with your own writings, I would suggest post-it notes &amp;mdash; I at least give great value to having notes right next to the content they comment.&#xA;My notes are mostly comments.&#xA;I find copying theorems and proofs verbatim quite useless.&#xA;I would suggest reading the theorem and its proof in detail and then trying to figure out what it is about, what it means.&#xA;Make sure you understand the overall plan of the proof, not just the technical details.&#xA;If you want to take notes, put the book aside and write the theorem.&#xA;If you understood the statement and the underlying machinery well, you should be able to write the statement.&#xA;Then write a proof, at least a sketchy one.&#xA;If you get stuck, look at the book; it doesn't matter if you have to look every three lines, as long as you really try to do it yourself first.&#xA;Now you have notes about the theorem, and you have written them down in your own words and as your own thoughts.&#xA;It might be almost exactly the same proof as in the book, and that's ok.&#xA;It came from someone else, but it's your proof now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I have to choose one of your strategies, I would go with 3.&#xA;Do what feels comfortable and natural to yourself, but make sure there is a point to the notes you are taking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is one more thing that I must add.&#xA;Doing exercises is very important in learning mathematics.&#xA;(I don't know about microbiology.)&#xA;Do them, and if there are none, make some up and do them.&#xA;I think the most important notes I have taken in my years of studying mathematics and theoretical physics are solutions to exercises.&#xA;Exercises give you hand-on experience of the power of your new theorems and make you comfortable with the machinery and language of the theory you are studying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something of a summary:&#xA;Take notes only when there is your own thought in them.&#xA;If you find yourself copying without thinking, there should be another path to the bulk of content.&#xA;Use notes to explain things to yourself, not to build all of the theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addendum:&lt;/strong&gt;&#xA;Let me elaborate on making up your own exercises.&#xA;(I did this in the comments, but it deserves to be in the actual answer.)&#xA;If you are curious enough, you don't need to produce problems artificially.&#xA;Try to dig a little deeper than your course.&#xA;How are different things related?&#xA;For example, is the sum of injective functions injective, or is a connected set always measurable, or is every square root of a continuous operator continuous?&#xA;The questions can be anything, and they can be silly.&#xA;Is a smooth function a Banach space, or is a finite group measurable?&#xA;Learning what things are completely unrelated and what things don't even mean anything is useful.&#xA;I find this kind of mental gymnastics entertaining and it gives me a better touch to the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If no questions occur, another good source of questions are the main results of your material.&#xA;Can you write a list of them without looking?&#xA;Can you prove the things on your list?&#xA;Can you define all relevant objects?&#xA;Do the theorems remain true if you assume a little less?&#xA;Why not, why yes, or why is it hard to see with the tools you have?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, informal exercises are important.&#xA;The problems in exercise sets in books, courses and exams tend to be quite formal, but that doesn't mean that only formal mathematics is good mathematics.&#xA;Play with your new tools and you'll learn to work with them more efficiently.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2016-01-18T14:37:55.033" LastActivityDate="2016-01-18T14:37:55.033" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8459" PostTypeId="2" ParentId="7247" CreationDate="2015-08-01T00:47:55.573" Score="3" Body="&lt;p&gt;One guide is that &lt;em&gt;all&lt;/em&gt; areas are developed until they are near the limits of what is possible for a bright, motivated human. Be it math, playing the violin, or planting roses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More than looking for &quot;which one is harder/easier&quot;, you need to find out what you like most. Somebody said that to get to be a great piano performer, you need to put in 10,000 hours of practice [the number is made up, but large anyway]. You see biographies of virtuoso performers (or star athletes, or whatever), and see they train for untold hours a day, every day. Same for any other area, I presume. If you aren't sufficiently interested in the subject to put in such hours, you won't excel. At the very least, you'll grow miserable in the process.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-01T00:47:55.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8460" PostTypeId="2" ParentId="8455" CreationDate="2015-08-01T09:35:51.807" Score="2" Body="&lt;p&gt;Here's how I would have attempted the lesson.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First I would have given the students various cards such as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\sin(A+B)=\sin A+\sin B$&lt;/li&gt;&#xA;&lt;li&gt;$\sin(A+B)=\sin A \cos B + \sin B \cos A$&lt;/li&gt;&#xA;&lt;li&gt;$\sin(A+B)=\sin(B+A)$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As a rule of thumb 8 cards is usually good. Never do more than 12 It just gets messy. Ask the students to sort them in to piles of true and false. No calculator just discuss and provide a reason for your choice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next give them a calculator and ask them to see if they are correct.  At the level you are talking about they should be able to reason to substitute values in.  You could ask them whether or not they were initially right or what common misconceptions do they think others have? Do the rules always work or only sometimes?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To conclude I would have given them a fact or two such as $\sin30=0.5$ and$\cos45=0.707$ and challenged them to come up with as many other facts as they can.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I probably wouldn't have proved the results.  Proof, in my opinion, needs delicate care in lessons.  I'll elaborate on that if so wished later.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best looking at exam boards themselves for latest info. AQA or Edexcel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastEditorUserId="4668" LastEditDate="2015-08-01T12:48:31.367" LastActivityDate="2015-08-01T12:48:31.367" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8461" PostTypeId="2" ParentId="8384" CreationDate="2015-08-01T15:13:05.930" Score="3" Body="&lt;p&gt;I'm currently looking at Steinmann, Geometrical Foundations of Continuum Mechanics (Springer 'Lecture notes in Mathematics and Mechanics' series), as a way of making differential forms relevant and palatable to senior engineering undergraduates.&lt;/p&gt;&#xA;" OwnerUserId="5450" LastActivityDate="2015-08-01T15:13:05.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8462" PostTypeId="2" ParentId="8148" CreationDate="2015-08-02T00:03:43.720" Score="4" Body="&lt;p&gt;I believe the history of math is part of the curriculum, almost everywhere (at least it has been in the various school systems I attended as a student, and have taught in as a teacher), however it is rarely covered in depth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One reason has already been brought up by @Cameron Williams regarding the cycle of ignorance among math educators (myself included, I do not know nearly as much about the history of mathematics as I would like). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another reason is standardized testing. Unfortunately, the history of mathematics is rarely, if ever, tested. In systems where student performance is measured by a standardized test, subjects like the history of mathematics will fall to the wayside. &lt;/p&gt;&#xA;" OwnerUserId="5451" LastActivityDate="2015-08-02T00:03:43.720" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="8463" PostTypeId="2" ParentId="8456" CreationDate="2015-08-02T09:18:15.637" Score="1" Body="&lt;p&gt;When in lectures, as mentioned above, if you have access to the slides or whatever, that is great! I'd suggest bringing a tape recorder with you, and then when you take notes, either take them on the slides itself, or in your notes write&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Slide 1: lsdkfjslkfjslkfjsldkfjsljfsdlkfs&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Slide 2: sdlkfjsldfkjslfkj&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and only take what you feel at the time is relevant. If you feel you missed something later, you have the voice recording (as you get better at taking notes, and you feel more confident, you can skip the voice recording)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can do the same with your own thoughts when reading a book, and take the page numbers, or since you mentioned that you learn best by reading, read with a highlighter as it may help keep you focused. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Creating your own problems can definitely be useful - if you find it hard to do that first, then google whatever type of problems you are struggling with and find something with solutions that you can check after. There are a ton of such problems up on the internet from various courses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note:&#xA;Most universities have some sort of learning center, study skills center to help students learn various study skills. My suggestion is that you go and take a course on note taking, but also on one that could target your learning style. Although you seem to know your learning style, such courses also usually give you tips on how to best learn, and study etc, keeping your learning style in mind. &lt;/p&gt;&#xA;" OwnerUserId="5451" LastActivityDate="2015-08-02T09:18:15.637" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8464" PostTypeId="2" ParentId="630" CreationDate="2015-08-02T09:25:04.870" Score="4" Body="&lt;p&gt;Something that helps, is to create problems where it's harder to see the &quot;right way&quot; to solve it. I try to give problems to my students where it's not obvious what type of method may be best to solve them, and then we discuss whatever solutions that come up. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also ask them to solve problems in more than one way. The idea, being that if they were teaching someone else, and their student didn't understand the first method, they would have to find another. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Included the following after extremely helpful suggestions from others&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry! Here is one: &lt;a href=&quot;http://illuminations.nctm.org/Lesson.aspx?id=1037&quot; rel=&quot;nofollow&quot;&gt;http://illuminations.nctm.org/Lesson.aspx?id=1037&lt;/a&gt; Here is a lesson that I often do with my students from Grade 6 - 8. I try to have students work in groups of ~4 and try to come up with at least 2 ways to solve this. It's also interesting when they see the problem again the following year, and have to try to solve it using something new that we have learned that year. Then, we start with what we remember from the previous year, before trying to find a new method.&lt;/p&gt;&#xA;" OwnerUserId="5451" LastEditorUserId="376" LastEditDate="2015-08-04T04:16:26.210" LastActivityDate="2015-08-04T04:16:26.210" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8465" PostTypeId="2" ParentId="8148" CreationDate="2015-08-02T13:04:18.823" Score="2" Body="&lt;p&gt;Mathematicians and history of mathematics are not in the &lt;a href=&quot;http://www.corestandards.org/Math/&quot; rel=&quot;nofollow&quot;&gt;standards&lt;/a&gt; for high school math or the mathematics tests.  I think that answers why they aren't taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, the standards include things like: factoring, completing the square, directrix of a parabola, quadrilaterals inscribed in circles, medians of a triangle meeting at a point, trigonometric sums and differences.  In my view, those are all poor choices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take some topics out of the curriculum, and there'd be more time for history and biography.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2015-08-02T13:04:18.823" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8467" PostTypeId="2" ParentId="5632" CreationDate="2015-08-03T18:13:01.460" Score="1" Body="&lt;p&gt;Interpreting&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the ... the picture...?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;far too literally, an image of $1/\zeta(2)$ was included in the question,&#xA;&quot;&lt;a href=&quot;https://mathoverflow.net/q/151706/6094&quot;&gt;What fraction of the integer lattice can be seen from the origin?&lt;/a&gt;&quot;&#xA;The answer is &#xA;$\frac{6}{\pi^2} \approx 0.6079$,&#xA;i.e., about 61%:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/BIwIu.jpg&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-08-03T18:13:01.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8468" PostTypeId="2" ParentId="5632" CreationDate="2015-08-03T18:54:16.323" Score="1" Body="&lt;p&gt;There isn't any &quot;intuition&quot; behind it, it was mind-blowing when Euler came up with this (&quot;What?! The reciprocals of &lt;strong&gt;squares&lt;/strong&gt; give something belonging to the &lt;strong&gt;circle&lt;/strong&gt;??&quot;, remember in Euler's time calculus was only really starting to untangle itself from geometry). And precisely because of that is that I consider this result of a strange beauty.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-03T18:54:16.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8469" PostTypeId="1" CreationDate="2015-08-04T08:49:54.740" Score="3" ViewCount="69" Body="&lt;p&gt;First post here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've just been reading up on giving good feedback, and decided to do some feedback on an assignment that was due today (secondary school calculus). I've been trying my best on it but am not overly sure how well I've done. If I were to somehow post a pdf (if possible) of some anonymised student work samples with my feedback, as well as the original assignment questions, would anyone experienced in good feedback be willing or able to offer their feedback/opinion/advice on my feedback? Can I upload pdfs here?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="5458" LastActivityDate="2015-08-04T08:49:54.740" Title="Looking for specific feedback on my feedback" Tags="|grading|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8470" PostTypeId="2" ParentId="1589" CreationDate="2015-08-04T09:00:53.917" Score="4" Body="&lt;p&gt;Here is an elementaryproof which considers $ p(z) $ as a function of $ (x,y) $ where $ z=x+iy $.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only assumptions are that $z^{n}=r $    has a solution for all integer $n$ and real $r$ and that intermediate value theorem holds for continuous real valued functions along any continuous curve on the complex plane. The second assumption can be proved by parametrizing the curve and then using the least upper bound principle for real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Writing $ z= r e^{-i \theta} $ we can find a big enough $r=R$ such that $ Re(p(R,\theta)) $ is positive at $\theta=\frac {2\pi k} {n} + \frac {\phi} {n} $ and negative at $\theta=\frac {2\pi k} {n} + \frac {\pi} {n} + \frac {\phi} {n} $. This is because  $Re(p(R,\theta)) $ is dominated by $r^{n}cos(n\theta)$. $\phi$ is the angle of the complex coefficient of $z^{n}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now , $ p(z) $ being continous, $ Re(p(z)) $ is also continuous. So the zeros of $ Re(p(z)) $ form a continuous curves on the complex plane. The point is to show that on one such continuous curve there is a point where $ Im(p(z))&amp;gt;0 $ and another point where $ Im(p(z))&amp;lt;0 $. Then on the particular curve $ Re(p(z))=0 $ there must be a point where $ Im(p(z))=0 $. However, this is the solution of $ p(z)=0 $.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proof can be visualized in the following way-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The arc $r=R$ between $\theta=\frac {2\pi k} {n} + \frac {\phi} {n}$ and $\theta=\frac {2\pi k} {n} + \frac {\pi} {n}  + \frac {\phi} {n} $ is called a even sector for even $k$ and is called a odd sector for odd $k$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us start from the even arc with $k=0$. The values of $ Re(p(R,\theta)) $ at the ends of this arc are of opposite sign, so it much be zero somewhere on the curve. The same thing holds if we increase $r \ge R$ continuously. So we get a continuous curve for $ Re(p(z))=0 $ in the sector $\theta \in$ { $0 $, $\frac {\pi} {n}$} for $r\ge R$. Let us call this curve $f(z)$ Now, zeros of a continuous function divide the plane into two disconnected parts. One of the parts is finite in area if the curve is closed. Otherwise both areas are unbounded. Here, since we cannot have  $ Re(p(z))=0 $ for $\theta =0 $ and  $\theta=\frac {\pi} {n}$ for $r\ge R$, the curve $f(z)$ does not intersect the rays $\theta =0 $ and  $\theta=\frac {\pi} {n}$  for $r \ge R$. This means $f(z)$ cannot be closed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, $f(z)$ must cross one of the odd arcs. This can be constructed using  exhaustion (there are only finite numbers of arcs). If not, there must be another curve satisfying $ Re(p(z))=0 $ and the desired conditions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$Im(p(R,\theta)) $ is dominated by $r^{n}sin(n\theta)$. So, for big enough $r$, $Im(p(r,\theta)) $ is positive somewhere in the even sector and negative somewhere in the odd sector. So $Im(p(z)=0) $ somewhere on the continuous curve $f(z)$. This gives a solution for $p(z)=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, $f(z)$ is not asymptotic to $\theta=\frac {2\pi k} {n} + \frac {\phi} {n}$ or $\theta=\frac {2\pi k} {n} + \frac {\pi} {n} + \frac {\phi} {n}$. This can be proven by using the fact that $Re(z) &amp;lt; |z|,  \forall z \in \mathbb{C}$. If this was not true, domination of $Im(p(z))$ on $f(z)$ by $r^{n}$ would be problematic since $sin(n \theta)$ might tend towards zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, for real coefficients, if $z$ a solution, $\overline z$ is also a solution for $p(z)=0$, since $\overline p(z) = p(\overline z)=0$. So for every root found, we can do division algorithm to get another polynomial of degree $n-2$. We know that quartic polynomials are always solvable using radicals. So induction shows that all polynomials with real coefficients are solvable on the complex plane.&lt;/p&gt;&#xA;" OwnerUserId="5459" LastActivityDate="2015-08-04T09:00:53.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8471" PostTypeId="1" AcceptedAnswerId="8472" CreationDate="2015-08-05T15:59:06.313" Score="5" ViewCount="137" Body="&lt;p&gt;I run a high school outreach program at UCLA called DMAP: Diversifying Mathematics And Physics (webpage: &lt;a href=&quot;http://dmap.pbworks.com&quot; rel=&quot;noreferrer&quot;&gt;http://dmap.pbworks.com&lt;/a&gt;), which focuses on exposing groups underrepresented in advanced maths and physics to these fields. There are two groups in this program: a non-calculus group, and a calculus and beyond group. The problem that I am running into is that the students in my non-calculus group are from mathematics levels from Algebra through Pre-Calculus, and thus their skill levels vary greatly. The club meets weekly, and right now I don't have a well defined curriculum for my non-calculus group because I can't seem to solve this problem of entertaining students with a wide skill-set variation; currently I have taken a bit of a shortcut with this dilemma and I am teaching my non-calculus group introductory computational mathematics since most of them do not have considerable exposure to this. Does anyone have a suggestion for curriculum that I can use that students from Algebra through Pre-Calculus will find entertaining but not too over their heads? (i.e., specific links to coursework, and/or suggested topics?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possibility that I was thinking was to try teaching basic probability, number theory, and proof techniques because these are easy enough for anyone to learn, but are not often taught in high schools formally. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An important note perhaps is that these students come to this club voluntarily, so they are on the upper-end of the motivation level, I don't suspect that their motivation levels will be an issue; I am more concerned with suiting the variation of their skill-levels.&lt;/p&gt;&#xA;" OwnerUserId="3386" LastEditorUserId="3386" LastEditDate="2015-08-05T16:08:19.277" LastActivityDate="2015-08-05T17:51:25.470" Title="Suitable Curriculum for Algebra through Pre-Calc students in One Class" Tags="|secondary-education|algebra|geometry|precalculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8472" PostTypeId="2" ParentId="8471" CreationDate="2015-08-05T17:51:25.470" Score="5" Body="&lt;p&gt;It sounds like all or part of a standard class in Finite Mathematics may help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finite Mathematics is a grab-bag class, including multiple topics that do not require trigonometry or calculus. Some of the topics are occasionally taught in an Algebra 2 class but are usually left out of that class. Here is a list of the chapters in the Finite Math textbook that I have sometimes used in my current high school:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Linear functions&lt;/li&gt;&#xA;&lt;li&gt;Systems of linear equations and matrices&lt;/li&gt;&#xA;&lt;li&gt;Linear programming: The graphical method&lt;/li&gt;&#xA;&lt;li&gt;Linear programming: The simplex method&lt;/li&gt;&#xA;&lt;li&gt;Mathematics of finance&lt;/li&gt;&#xA;&lt;li&gt;Logic&lt;/li&gt;&#xA;&lt;li&gt;Sets and probability&lt;/li&gt;&#xA;&lt;li&gt;Counting principles: Further probability topics&lt;/li&gt;&#xA;&lt;li&gt;Statistics&lt;/li&gt;&#xA;&lt;li&gt;Markov chains&lt;/li&gt;&#xA;&lt;li&gt;Game theory&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;When I teach this class, I get a list of interests of each of my students, discuss the topics, and get votes on each chapter. That way I tailor the class to the interests of my students, and they know the teaching is geared to help them. I add some more to subjects my students really want: for example, I added reading the book &lt;em&gt;How to Lie with Statistics&lt;/em&gt; to the statistics chapter. The feedback I have gotten from former students tells me that this class is &lt;em&gt;greatly&lt;/em&gt; appreciated by my students, much more so than my calculus class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This textbook is geared to students who have taken Algebra 2, but you could skip the topics and homework questions that are too advanced for much of your class. One advantage of this class over some other options: there are many textbooks available and curricula will be easy to find.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2015-08-05T17:51:25.470" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8475" PostTypeId="2" ParentId="8456" CreationDate="2015-08-06T19:57:13.283" Score="3" Body="&lt;p&gt;What works for me may not work for you.  But here goes with what worked well for me:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I used the textbook as my notes.  It had an index and was better written (and prettier!) than my handwriting. Where the textbook was too dense or omitted steps in proofs that I needed to make explicit, I added notes in the margins. This sounds like your option #1.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What's more important than &quot;how to take notes&quot; is this: Read the text. Read everything from titles, to captions and footnotes, to the example problems. Don't let ONE WORD get past you. If there is ANYTHNG you don't understand, keep at it until you do understand it or ask for help. This sounds a little like your option #3.  And it's not necessarily mutually exclusive with your option #1.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What's more important than ABSORBING the text is working the problems.  Even if you completely understand everything in the text, you need the practice--the fluency--that working the problems gives you.  In college, I spent about 4 hrs a night 3 nights a week doing calculus homework.  But it paid off.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5465" LastActivityDate="2015-08-06T19:57:13.283" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8476" PostTypeId="2" ParentId="7964" CreationDate="2015-08-06T20:26:35.703" Score="17" Body="&lt;p&gt;I can't add much more than what has been stated already.  But  I come from a programming background, so maybe this example may add a different perspective (or explain some problems interpreting equals for other students).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In many programming languages, statements like this are perfectly acceptable:&#xA;x = x + 1&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, the programming language DOES treat the equals sign as an operator.  (You calculate the right-hand side of the &quot;equation&quot; then assign the result to the left-hand side.)  If your student is a programming hobbyist, that may help explain why they are confusing the meaning of the programming context vs. mathematical context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Daniel's example (&quot;e.g., 5 + 2 = __, 11 – 4 + 1 = __&quot;) this could sort of be interpreted as an operator: The underscore is being SET to the value on the left-hand side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some programming languages make this distinction by using == for the &quot;relational&quot; meaning, and = as the SET operator.  (Other languages require the SET or LET keyword when = is used as an operator: SET x = x + 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Early hobbyists may learn when to use one symbol vs. the other in a habitual way without fully understanding the conceptual difference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: perhaps it may be better to emphasize that the equal sign actually DOES have two meanings (at least in some contexts), and to emphasize that the mathematics you're teaching only uses the relational aspect.&lt;/p&gt;&#xA;" OwnerUserId="5465" LastActivityDate="2015-08-06T20:26:35.703" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8477" PostTypeId="2" ParentId="7771" CreationDate="2015-08-07T01:13:48.133" Score="2" Body="&lt;p&gt;I've mentioned the expensive &lt;a href=&quot;https://matheducators.stackexchange.com/a/4313/511&quot;&gt;polydrons&lt;/a&gt; before,&#xA;but in fact one can go far in exploring polyhedra&#xA;(e.g., Euler's $V-E+F=2$) via paper cutouts, which only require the&#xA;ability to print &amp;amp; copy &amp;amp; scissors &amp;amp; a bit of glue or tape.&#xA;For example, see the &lt;a href=&quot;http://www.korthalsaltes.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;website www.korthalsaltes.com&lt;/a&gt;.&#xA;Note the templates even include the gluing tabs.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/dEdkN.png&quot; height=&quot;400&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/Fcid5.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Fcid5.jpg&quot; alt=&quot;DodecaColor3D&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-08-07T01:13:48.133" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8478" PostTypeId="2" ParentId="3827" CreationDate="2015-08-07T02:10:22.487" Score="6" Body="&lt;p&gt;While I don't have a deep knowledge of peer-reviewed literature on transfer of skills from playing tournament chess, I am an experienced tournament chess player as well as a math educator. Based on my knowledge of what tournament chess players actually do, I find it very unlikely that tournament chess would provide any significant boost to mathematical skills that could not be gained by other means. Sure, the mental discipline and working memory development can't hurt, but those things are hardly specific to chess.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The simple reason is that playing a game of chess, even at the highest levels, doesn't involve thinking processes anything like those of graph theory and the Knight's tour, or even mathematics in general. A low-level tournament player will have thought processes centered around tactical calculations a few moves into the future, e.g., &quot;If I capture on e4, then he captures with the bishop, then I recapture with the queen, ...&quot;  A mid-level player will add positional thinking such as &quot;If I make him recapture on c3, his pawn structure will be weak.&quot; The strong players will be adept at choosing the right broad plan, such as &quot;I'll spend a few moves locking up the queenside, then I'll start a pawn storm on the kingside before he can get in position to defend.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't see any way that a task analysis of any of these thought processes gets even close to chess math problems such as the Knight's tour, or to any problems of higher math in general. The thought processes involved in tournament chess are very domain-specific. So-called chess math problems such as the Knight's tour may appear to be superficially related to chess, but being entirely divorced from real chess considerations such as pawn structure, capturing pieces, etc., there is little reason for even a grandmaster to have any special insight or advantage in such problems. &lt;/p&gt;&#xA;" OwnerUserId="5417" LastActivityDate="2015-08-07T02:10:22.487" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8480" PostTypeId="2" ParentId="5812" CreationDate="2015-08-07T15:20:44.243" Score="1" Body="&lt;p&gt;Perhaps not really an answer to the question, but I believe I understand where this comes from.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, such calculus courses are among the first ones, long before students have seen enough (make that &quot;anything at all&quot; if you want) of their future field to understand what &quot;realistic&quot; applications of the covered techniques are. Also, such classes are usually a mixture of majors, so a relevant example for, say, chemical engineering will be greek to civil engineers or physicists. This unfortunately leaves geometric examples, or stuff that can be understood with high-school science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the best strategy is to talk with the teachers of the relevant higher courses to suggest problems to discuss, offer their students some &quot;remedial&quot; help as appropiate, or even restructure the curricula for &quot;just in time&quot; teaching of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-07T15:20:44.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8481" PostTypeId="2" ParentId="7426" CreationDate="2015-08-07T15:34:41.150" Score="3" Body="&lt;p&gt;My experience is that teaching &quot;tools&quot; (the things you mention in this case) beforehand/separate from their applications is counterproductive. When the tools are needed, they are already forgotten, on not really assimilated because the student didn't see the point at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try to integrate discussion of the needed skills into the class, whenever appropriate. Don't separate them unless necessary (yes, that's your call), don't make a fuss of them, but show how to apply them by example whenever relevant. Perhaps you can summarize a the end what was learned, separately for &quot;technical stuff&quot; (theorem so and so, this and that solution method) and &quot;tools&quot; (dissected a definition, checked how/when it applies). The second part will probably repeat often.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-07T15:34:41.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8482" PostTypeId="1" CreationDate="2015-08-07T15:43:00.410" Score="8" ViewCount="1558" Body="&lt;p&gt;I just saw &lt;a href=&quot;https://math.stackexchange.com/q/1387703/21820&quot;&gt;this question&lt;/a&gt; and the two answers. Now I am very sure that if many teachers see these answers they would accept them. Unfortunately they are deeply flawed. The second one is clearly invalid in saying $\int -\frac{1}{x}\ dx = -\ln(x) + c$. But if we ignore that, the remainder is equivalent to the other solution. But most teachers would not have even realized that they have &lt;a href=&quot;https://matheducators.stackexchange.com/a/8017/1550&quot;&gt;not followed the rules of arithmetic and divided by zero&lt;/a&gt;! I believe this method of solving separable differential equations is widespread, having been taught it by my own teachers in the past. How do we correct this problem? As I point out in the later part of &lt;a href=&quot;https://matheducators.stackexchange.com/a/7984/1550&quot;&gt;this answer&lt;/a&gt;, actually solving differential equations rigorously is no trivial task, but difficulty does not justify using wrong methods. So my question is more of a math-educator-educator question, in that I feel educators are not sufficiently aware of the problems in their own reasoning simply because they have rote-learnt and rote-taught these methods, and I'm wondering how best to tackle this problem, since clearly students at the high-school level will not be capable of handling the correct reasoning. I'm totally not in favour of lying to the students or sweeping problems under the carpet by something like &quot;The method works for some kinds of differential equations under certain conditions, which will be the only type I will give you, so just follow the method blindly and don't ask questions!&quot;, as I firmly believe in not telling students to suppress &lt;strong&gt;correct rules of arithmetic&lt;/strong&gt; just to permit &lt;strong&gt;incorrect mathematics&lt;/strong&gt; that happen to get the right answer only sometimes (like for analytic functions).&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-08-08T13:25:42.037" Title="Commonly taught method divides by zero" Tags="|teachers|differential-equations|division|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8483" PostTypeId="2" ParentId="7406" CreationDate="2015-08-07T15:59:03.410" Score="1" Body="&lt;p&gt;I'd just give them a sort lecture on the etiquette at &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;MSE&lt;/a&gt;, caution them that the teacher and the assistants will be keeping an eye on the site and deal with &quot;Please do my homework&quot; appropriately, and let them loose. Just having the same people (class mates, assistants) answering questions gives a narrow range of answers, learning how to find relevant material by themselves is very valuable, being able to answer questions from others helps building confidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To counter &quot;copy/paste&quot; homework (which will happen regardless) my assistants select a small group &quot;at random&quot;, their grade is according to an oral interrogation on what they turned in. I'm not interested in them doing the work on their own as much as I'm interested in them understanding what they turned in.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2015-08-07T17:30:47.270" LastActivityDate="2015-08-07T17:30:47.270" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8484" PostTypeId="2" ParentId="8482" CreationDate="2015-08-07T16:03:37.797" Score="16" Body="&lt;p&gt;The method is mathematically incorrect, but whether it is wrong or not, depends on whether people know what they're doing or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The possible division by zero is a mistake, yes. But it is a mistake one can ignore if in the end you get a solution to the differential equation. To make it full proof, what I do is the following sequence of steps.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Possibly divide by zero;&lt;/li&gt;&#xA;&lt;li&gt;Find a candidate solution (after dividing by zero);&lt;/li&gt;&#xA;&lt;li&gt;Check that the candidate solution is indeed a solution, rendering the division by zero issue a non-issue. (This is step is often implied, not written and unspoken, in my mind this is the most important step);&lt;/li&gt;&#xA;&lt;li&gt;If the goal is to find all solutions to a problem, use &lt;a href=&quot;https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem&quot;&gt;Picard–Lindelöf&lt;/a&gt; or one of its variations to conclude that the solutions previously found are the only ones. (This step is also rarely mentioned and in conjunction with the third step is also extremely important);&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So to answer the question &quot;How do we correct this problem?&quot;, just make sure steps 3 and 4 are made apparent instead of latent.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;A personal comment. In my experience as a student, this was never made clear to me. It was something I had to discover on my own and I've never really seen anyone but me make the steps this clear, not even in books. I often opt to avoid this method if the differential equation can be solved by the integrating factor method, because this method is quite elementary, no need to use big theorems like the solution uniqueness or do weird things like division by zero.&lt;/p&gt;&#xA;" OwnerUserId="287" LastActivityDate="2015-08-07T16:03:37.797" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8485" PostTypeId="2" ParentId="8482" CreationDate="2015-08-07T18:37:59.587" Score="5" Body="&lt;p&gt;First off, it's not necessarily &lt;em&gt;wrong&lt;/em&gt;; there are other formalisms than pointwise manipulation of functions, and are arguably closer to actual practice too, even if not explicitly stated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That aside, even in the formalism of pointwise manipulation of functions, the methods are still correct, just missing their final steps. Both of the sample answers can be described as solving the problem on the domain of nonzero $x$ (although, the integrating factor version is a little quirky and was probably a mistake rather than taking advantage of the quirk to arrive at the simplified form of the integrating factor).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The missing final step is to see which solutions on the domain $x \neq 0$ extend to a solution on the domain of all reals. An &lt;em&gt;error&lt;/em&gt; made by both of the posts is that $c$ is meant to be &lt;em&gt;locally constant&lt;/em&gt;; on the domain $x \neq 0$ that means $c$ potentially has two different values: one on $x&amp;gt;0$ and one on $x&amp;lt;0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(although it does turns out they have to be the same if you want to extend a solution on nonzero $x$ to a differentiable function on all reals)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I imagine the best thing is not to try and tell students &quot;don't do that&quot;, but instead show them how to &quot;do that, but correctly&quot;. And give them problems where there are new solutions. e.g. a small tweak to the same problem&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ x y' - 2y = x^3 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;has total solution, if the domain of $x$ is all real numbers,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ y = \begin{cases} x^3 + c_- x^2 &amp;amp; x \leq 0&#xA;\\x^3 + c_+ x^2 &amp;amp; x \geq 0 \end{cases}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $c_-$ and $c_+$ can each be any constant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even with a boundary condition like $y = 0$ when $x=-1$, there can still be infinitely many solutions, such as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ y = \begin{cases} x^3 + x^2 &amp;amp; x \leq 0 \\ x^3 + c x^2 &amp;amp; x \geq 0 \end{cases} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe at first give them a problem that helps remind them; e.g.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Solve the equation $x y' - 2y = x^3$ such that $y = 0$ when $x = -1$ and $y = 3$ when $x = 1$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and later give them problems where you just have one boundary condition, but still expect to be given the full solution. And give them problems of the sort with &lt;em&gt;no&lt;/em&gt; solutions too, since they should be able to recognize and prove that!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A specific kind of problem of this type might be useful, since it requires a new kind of reasoning and actually comes up in real problems is something like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ x y'' + y' = 1 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where the general solution is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ x + a \log|x| + b $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(where $a,b$ are locally constant on the domain of nonzero $x$) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;but one has to deduce that getting a solution on all $x$ requires selecting $a = 0$.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2015-08-07T18:37:59.587" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8486" PostTypeId="2" ParentId="630" CreationDate="2015-08-07T19:27:07.363" Score="-1" Body="&lt;p&gt;I gave a talk a couple years ago that tried to synthesize my experience and reading. &lt;a href=&quot;http://bit.ly/goldenggb13&quot; rel=&quot;nofollow&quot;&gt;http://bit.ly/goldenggb13&lt;/a&gt; (Also on GeoGebra)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The two main pools of ideas for me in practice come via Ken Robinson and the MIT Media Lab, especially Mitch Resnick (whose work follows the great Seymour Papert).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sir Ken: creativity is ideas with value. “Teaching for creativity involves asking open-ended questions where there may be multiple solutions; working in groups on collaborative projects, using imagination to explore possibilities; making connections between different ways of seeing; and exploring the ambiguities and tensions that may lie between them.” - Sir Ken Robinson, Out of Our Minds&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Resnick: creativity in practice is a spiral: imagine -&gt; create -&gt; play -&gt; share -&gt; reflect --&gt; imagine...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students need an opportunity for choice and an opportunity to make. Teacher provides a safe place, constructive feedback, and -hopefully - connects the students with an authentic audience.&lt;/p&gt;&#xA;" OwnerUserId="12" LastActivityDate="2015-08-07T19:27:07.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8487" PostTypeId="1" CreationDate="2015-08-08T02:51:41.477" Score="5" ViewCount="986" Body="&lt;p&gt;Using LaTeX, I am attempting to write a single-variable calculus textbook that gives the reader an understanding of calculus and its applications without a lot of the fluff I have seen in many other textbooks (I should mention that I am a rising high school senior going into multivariable calculus this coming year). In my opinion, if a student is impressed by the high quality, ease of understanding, and straightforwardness of the textbook, then with good teaching, the student would successfully learn the material as opposed to memorizing facts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not just writing a textbook. I also plan to hold a class online through Google Hangouts or a similar medium that goes through the textbook as well as practice exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have already assembled a group of friends to go through the textbook as I write it. These people range from people who have not yet started AP Calculus AB but were decent in Pre-Calculus (who will read it for understandability), people between Calculus AB and BC (who will read it to make sure I haven't overcomplicated it), and people who have completed Calculus BC and above (who will read it for correctness and suggestions on additions/deletions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does the idea for this class sound viable? Also, how would you recommend I maintain the organization of the documents for each chapter/lesson?&lt;/p&gt;&#xA;" OwnerUserId="5470" LastActivityDate="2019-08-13T00:40:00.650" Title="Writing a Calculus textbook for a course I am creating" Tags="|undergraduate-education|secondary-education|calculus|textbooks|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="8488" PostTypeId="2" ParentId="8482" CreationDate="2015-08-08T13:25:42.037" Score="1" Body="&lt;p&gt;I believe this also happens when doing partial fraction decomposition.  Often students are encouraged to plug in values of x that would lead to division by zero, but this is never really mentioned or clarified.&lt;/p&gt;&#xA;" OwnerUserId="5141" LastActivityDate="2015-08-08T13:25:42.037" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8489" PostTypeId="1" AcceptedAnswerId="8492" CreationDate="2015-08-08T18:00:27.820" Score="14" ViewCount="472" Body="&lt;p&gt;I am about to start grad school and I am trying to think seriously about teaching [you know, before I get swamped with my own coursework]. I wrote a hypothetical worksheet for an introductory linear algebra class which tries to do something a little more interesting than &amp;quot;Here is a random LT I made up, compute the nullspace&amp;quot;. Here is an excerpt from the beginning:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Triangle (T).&lt;/strong&gt; We will use the following graphic to create two vector spaces and a linear transformation between them:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\qquad\qquad\qquad\qquad$&lt;/span&gt; &lt;a href=&quot;https://i.stack.imgur.com/DCKHD.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DCKHD.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The vector space &lt;span class=&quot;math-container&quot;&gt;$C_0$&lt;/span&gt; is given by &lt;em&gt;formal linear combinations&lt;/em&gt; of the labelled points. In other words, &lt;span class=&quot;math-container&quot;&gt;$(x,a ,b )$&lt;/span&gt; form a basis for &lt;span class=&quot;math-container&quot;&gt;$C_0$&lt;/span&gt;, so a typical vector in &lt;span class=&quot;math-container&quot;&gt;$C_0$&lt;/span&gt; looks like&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 5x-3a -2b  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Similarly, the vector space &lt;span class=&quot;math-container&quot;&gt;$C_1$&lt;/span&gt; is given by formal linear combinations of the line segments. By analogy to the above construction, an explicit basis might be &lt;span class=&quot;math-container&quot;&gt;$(xa , ab, bx)$&lt;/span&gt;, so that a typical vector in &lt;span class=&quot;math-container&quot;&gt;$C_1$&lt;/span&gt; looks like&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ -6xa+2ab-8bx. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Finally, consider the linear transformation &lt;span class=&quot;math-container&quot;&gt;$D:C_1\to C_0 $&lt;/span&gt; which takes an line segment &lt;span class=&quot;math-container&quot;&gt;$vw$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$v-w$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; is the vertex the arrow is pointing toward, and &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; is the vertex the arrow is pointing away from. For instance, &lt;span class=&quot;math-container&quot;&gt;$D(xa )=x-a$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;T1.&lt;/strong&gt; Compute &lt;span class=&quot;math-container&quot;&gt;$D(-6xa+2bx-8ab)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;T2.&lt;/strong&gt; Write the matrix for &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; using the two bases provided.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;T3.&lt;/strong&gt; By Gaussian elimination or otherwise, find all &lt;span class=&quot;math-container&quot;&gt;$\alpha\in C_1$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$D(\alpha)=5x-3a-2b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;T4.&lt;/strong&gt; Compute the nullspace of &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; and its dimension (nullity).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Two more examples follow, all with &amp;quot;compute the nullity of &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;&amp;quot; (&lt;span class=&quot;math-container&quot;&gt;$4\times 4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4\times 5$&lt;/span&gt;; a little large but not terrible) with the kicker at the end:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Further Thinking.&lt;/strong&gt; Can you describe a relationship between the nullity of &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; and the geometry of the graphic? Write a conjecture, using evidence (from your answers to the previous questions, or elsewhere) to justify it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ideally, this would be accompanied by a discussion of formal linear combinations in class, since the treatment above is clearly inadequate as a first introduction.&lt;/p&gt;&#xA;&lt;p&gt;I came from an undergrad program with a high emphasis on STEM, so linear algebra education was unusual, as was the math motivation of the student body. I guess what I really want to know is, from people with actual experience in a classroom, would this fly? and if not, can it be salvaged?&lt;/p&gt;&#xA;&lt;p&gt;But to make the question more amenable to the SE model:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For what style/level of linear algebra course, if any, would this worksheet be appropriate?&lt;/li&gt;&#xA;&lt;li&gt;Are there obvious local improvements to what you see here? (i.e., could one of the questions be altered slightly to probe an additional concept)&lt;/li&gt;&#xA;&lt;li&gt;Can you point me toward education research that talks about the effectiveness of incorporating high-level ideas into lower-division courses?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5472" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-08-10T03:34:54.380" Title="Worksheet: Homology in Intro Lin Al" Tags="|homework|linear-algebra|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="8491" PostTypeId="2" ParentId="8489" CreationDate="2015-08-08T21:57:37.690" Score="8" Body="&lt;p&gt;Personally, I think this seems very appropriate.  A few points:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You might reconsider using a two character name for a vector.  $xa$ looks like a multiplication.  Perhaps just label the vertices $V_1$, $V_2$, and $V_3$, and edges $E_1$, $E_2$ and $E_3$?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Since this is linear algebra, I assume you are using the reals as your field of coefficients. However, your example coefficients are all integers, which may give the wrong impression.  You could throw in a $\sqrt{2}E_1$ somewhere.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give some examples of vector operations in this space first, since it may not be completely obvious otherwise.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You might actually consider starting with the line segment, and seeing that its homology is trivial.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If your students have seen multivariable calculus already, it might actually be cool to do cohomology instead of homology, and connect this story to line integrals.  I think Hatcher might actually motivate cohomology along these lines (the de Rham analogy)?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Obvious followup would be throwing in a couple more holes, or moving up a dimension. Also do not neglect &quot;seeing&quot; how anything that looks like a circle would have a one dimensional $1^{st} $ homology group.  At least try a square as well.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would include some kind of motivation, even if it is just &quot;this is the very first step into a wide world of mathematics called Algebraic Topology.  We aim to capture some qualitative features of shapes by computing algebraic quantities associated to those shapes.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastActivityDate="2015-08-08T21:57:37.690" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8492" PostTypeId="2" ParentId="8489" CreationDate="2015-08-09T06:30:06.853" Score="9" Body="&lt;p&gt;I have a dim memory of a simple exercise I came up with as a TA.  It was something like the vector space over the reals of polynomials in two variables, including x^2, y^2, and (x+y)^2.  I then asked them to compute something like the dimension of the space S spanned by x^2 + y^2, (x+y)^2, and xy.  I also asked them if xy was in S, and then if y was in S, and to explain why or why not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The professor for the course thought it was a great problem, and that none of the students would understand it.  He put it on a final exam with my name for attribution, and had me grade the result.  He was right.  A few students came close, but none were able to demonstrate full understanding of what was going on with the problem.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think your exercise is good for an introductory graduate student or an ambitious undergraduate.  Unless you have a top rate class though, the exercise will be under-appreciated.  Or, it will take a long time to appreciate.  This is not to discourage you so much as to alert you.  If&#xA;you turned this exercise into a lengthy blog post with some more detail worked out, you might reach a suitable audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Society Needs More Good Exercises&quot; Paseman, 2015.08.08&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-08-09T06:30:06.853" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8494" PostTypeId="1" CreationDate="2015-08-09T12:07:45.263" Score="12" ViewCount="958" Body="&lt;p&gt;I am currently grading a problem in an exam on analysis for mathematics students. One subtask is to calculate a certain integral (The resulting term has to be as simple as possible.). For this part I am able to give up to $n$ points in essentially any way I see fit, provided that I do so consistently (the number $n$ is already fixed though).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course I am interested in an answer to this question in order to apply these to my specific case, but I also find the question of how to assign points fairly for questions where there are many long paths to a concrete solution interesting in general. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some facts which may be significant: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I am not allowed to give fractions of points. &lt;/li&gt;&#xA;&lt;li&gt;The exam paper does not contain any information on how the problems will be graded other than how many points are assigned to each subtask of a problem.&lt;/li&gt;&#xA;&lt;li&gt;There are significantly more questions in the exam than can reasonably be expected to be solved in the given time; in particular, one does not have to solve all exercises to obtain full marks.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have thought a great deal about how to assign points in such a situation, but none of the methods seems to be satisfactory. Here are some of these possibilities:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;All or nothing: Assign $n$ points for a perfect answer and none otherwise.&lt;br&gt;&#xA;&lt;em&gt;Assessment&lt;/em&gt;: Small mistakes are excessively penalised.  &lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The usual approach at my university to grading such questions is to record all the different routes the students have followed to carry out the calculation, then either  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;partition every such route into $n$ &quot;even&quot; chunks, and then assign one point for every successfully completed chunk; or   &lt;/li&gt;&#xA;&lt;li&gt;isolate $n$ &quot;significant steps&quot; (e.g. rewriting the integrand in such a way that it may be easily integrated), and award a point for each such step.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Assessment&lt;/em&gt;: This seems rather unfair to me for the following reason: There are infinitely many sequences of steps that one may perform, only some of which lead to a term which is sufficiently simple so that one may deem to have completed the calculation. Now, one only really knows that such a sequence leads to the solution when the sequence has been completed, so if a student simply performs the first few steps of such a sequence, he hasn't done anything qualitatively different from a student who has performed steps in any other sequence. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;In view of the assessment of the previous method, I could solely consider calculations which lead to a definite answer, and then penalise every mistake in the calculation by the subtraction of one point, so that one obtains $\max(0,n - m)$ points, where $m$ is the number of mistakes.&lt;br&gt;&#xA;&lt;em&gt;Assessment&lt;/em&gt;: If someone writes something trivial like $\int f(x) \, dx  = 0$, he will only have made one mistake, and will thus receive $n -1$ points. Also, students aren't aware during the exam that they have to complete their calculations at any cost.   &lt;/li&gt;&#xA;&lt;li&gt;Again in view of the assessment of method 2., I could again solely consider calculations which lead to a definite answer, determine to which correct route it corresponds the best, and then apply 2.2..&lt;br&gt;&#xA;&lt;em&gt;Assessment&lt;/em&gt;: Again, students aren't aware during the exam that they have to complete their calculations at any cost.&lt;/li&gt;&#xA;&lt;li&gt;Judge how much the final result differs from the correct one (by missing signs, summands etc.), and then assign points for how strongly the calculated result resembles the correct answer.&lt;br&gt;&#xA;&lt;em&gt;Assessment&lt;/em&gt;: Judging such a resemblance is highly subjective. Also, it is possible that one arrives at a solution which looks similar to the correct one by completely invalid means. &lt;/li&gt;&#xA;&lt;li&gt;Again in view of the assessment of method 2., one could award points to any sequence of steps which could be considered to constitute a promising attack on the problem.&lt;br&gt;&#xA;&lt;em&gt;Assessment&lt;/em&gt;: I would have to determine what constitutes a promising attack. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;One final observation: I have not considered this up until now, but I could create a grading scheme where there are more than $n$ things one may write which give points, and the final number of points awarded is $\min(n,m)$, where $m$ is the number of things one has written that give points. This would allow for further, perhaps more nuanced, methods. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(P.s. This is my first question on stackexchange, so I am particularly eager to read any criticism of my question.) &lt;/p&gt;&#xA;" OwnerUserId="5476" LastActivityDate="2022-11-18T08:28:21.127" Title="How do I grade long calculations in a mathematics exam in a fair way?" Tags="|exams|grading|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8495" PostTypeId="2" ParentId="8494" CreationDate="2015-08-09T14:00:40.967" Score="7" Body="&lt;p&gt;Why not assign an even integer to the problem, and then assign full credit for all steps being correct, half of the integer value for any mistake (singular), and zero credit for more than one mistake.  This is what I do after years of similar debate ...&lt;/p&gt;&#xA;&lt;p&gt;The advantage is that it reflects the expectations of a work environment.  Down the road, if students make a small error but otherwise contribute to a team with a fluid and graceful solution, it would be valuable, and accordingly they earn half the rubric points.  If each problem was assigned 6, then they earn 3.  If they made so many mistakes that one could not appreciate the grace and fluidity of the solution, then 0 is appropriate because it would not be helpful in the workplace either.&lt;/p&gt;&#xA;&lt;p&gt;Then, on top of the 0, half, full credit rewards, utilize rubrics that borrow from the NCTM, CCSS, Buck or somewhere else, etc., in order to assess the numerical value for each problem, i.e., the weight of the problem, or whether the first problem is worth 8 points but the last problem is worth 2 points, etc.&lt;/p&gt;&#xA;&lt;p&gt;This is the fairest metric to me since it does not over-emphasize correct reasoning steps at the expense of the end result, but it likewise does not fail to reward correct reasoning despite having an incorrect answer.  It also permits for different difficulty levels of problems according to known rubrics for assessing problems, and I have found the students universally consider this the fairest system because regardless of the type of mistake or mistakes they make, there is no subjective choice involved, since they are only permitted one mistake.  This makes sense because types of mistakes vary in their significance in terms of the context that they are made within, but on a test, there is generally no real-world context, only abstract practice, and hence equalizing the penalty for types of mistakes is generally well accepted by my students, since there is usually no way to evaluate the significance of one mistake versus another without workplace context.&lt;/p&gt;&#xA;" OwnerUserId="1100" LastEditorUserId="7365" LastEditDate="2022-11-17T22:16:09.087" LastActivityDate="2022-11-17T22:16:09.087" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8496" PostTypeId="1" CreationDate="2015-08-09T15:37:15.443" Score="5" ViewCount="71" Body="&lt;p&gt;How can typical undergraduate mathematics teaching resources be made accessible to visually impaired students?&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-08-09T15:58:21.610" Title="Teaching visually impaired mathematics students" Tags="|undergraduate-education|disability|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8497" PostTypeId="2" ParentId="8496" CreationDate="2015-08-09T15:58:21.610" Score="2" Body="&lt;p&gt;One person who has published on this subject is Emma Cliffe. A couple of the points she makes are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;There are difficulties here that are specific to mathematics. It is likely that no one person's job description will cover all the aspects that need to be considered, so any plans should preferably be made together by the student, disability officers and members of the mathematics department.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Not many computer file formats of mathematics can be meaningfully converted to accessible resources. As things currently stand, those that are are:&#xA;a) well-formatted word-processed documents with equations entered in the correct format;&#xA;b) well-formatted Latex using a limited range of packages;&#xA;c) html with MathJax.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-08-09T15:58:21.610" CommentCount="2" CommunityOwnedDate="2015-08-09T15:58:21.610" ContentLicense="CC BY-SA 3.0" />
  <row Id="8498" PostTypeId="2" ParentId="8453" CreationDate="2015-08-09T18:08:12.680" Score="1" Body="&lt;p&gt;Numbers and Functions: Steps into Analysis, R.P. Burn, Cambridge University Press, ISBN 0 521 78836 6.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This book takes a constructivist approach to the subject. It takes the form of a sequence of problems, introducing the key ideas step by step. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his preface Burn identifies common problems student often encounter: the apparently needlessly formal approach of topics that are familiar from school as well as the insistent use of arbitrary and contrived definitions. Accordingly the book builds on high school concepts and naturally introduces key definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally it saved me in my first year of uni.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below is a review from 'The Mathematical Gazette'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.jstor.org/stable/3621803?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;http://www.jstor.org/stable/3621803?seq=1#page_scan_tab_contents&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3407" LastEditorUserId="3407" LastEditDate="2015-08-15T18:51:27.680" LastActivityDate="2015-08-15T18:51:27.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8499" PostTypeId="2" ParentId="8148" CreationDate="2015-08-09T20:16:26.327" Score="4" Body="&lt;p&gt;In teaching a course on geometry to sophomore's in the previous year, I definitely included as much history as possible. There are numerous ways to do this.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The first thing is that -- like @Cameron Williams said -- you have to know this material yourself before you can teach it. There are many resources out there in terms of texts and online videos. Norman Wildberger's &lt;a href=&quot;https://www.youtube.com/playlist?list=PL34B589BE3014EAEB&quot; rel=&quot;nofollow&quot;&gt;History of Math&lt;/a&gt; lectures can be found on youtube at that link. They are quite good and would be great for any high school math teacher to incorporate parts of this into class. They are based on John Stillwell's Math and Its History. Journey Through Genius has already been mentioned. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I started class each day with a Historical Date in Mathematics [I might change that to 3 days a week or 2 depending on time]. I used the &lt;a href=&quot;http://www.math.wichita.edu/~richardson/timeline.html&quot; rel=&quot;nofollow&quot;&gt;Math Timeline&lt;/a&gt; as a base to help me organize myself and also provide other context for the students that they might be familiar with (e.g. pyramids were built around the same time kind of thing). They students were required to have their own timeline for the dates we talked about in class. I didn't give an assessment of their recall of the dates, but they were required to do a few research projects. In the future, I am considering having them do independent MATH research based on some historical problem. These do not have to be problems just from ancient Greece. Students can handle investigations into number theoretic problems too. Symmetry study? Sure! Methods for solving systems of linear (and non-linear, oh my) equations? Definitely. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When I introduce a topic, I try very hard to motivate it and this includes talking about why anyone considered it in the past. The reason for (more advanced example) the current state of knowledge around unique factorization domains is due to Gauss's interest in seeing systems similar to but different from the integers. This alone begins to demonstrate to students how math even happens. What is math research? -- Start with something you already know and tweak!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(really 3a) The historical motivation is sometimes the best one in my opinion. This was particularly true for logarithms. They are usually taught as inverses of exponentials and then properties are listed, some calculations and graphing are done... moving on. That's doable (and how I first learned) but often this leaves one with a feeling of... why did I just learn that... or even more so, what are logarithms REALLY? However, if one talks about their original purpose and how to arrive at them naturally, it's rather wonderful. Then suddenly their uses for things like linear regression make since -- it's their original purpose - LINEARIZE!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I have some history modules that teach concepts via historical routes that are rather good (admittedly, I change them some) if anyone is interested. Some material goes from middle school up through first year college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another great resource is: &lt;a href=&quot;https://www.math.nmsu.edu/~history/&quot; rel=&quot;nofollow&quot;&gt;Teaching Math with Original Sources&lt;/a&gt; -- This is done with an eye towards undergrads, but there are many links and resources on the page that are for high school or high school content. The site is that of authors of two sets of books that are great (again, some advanced stuff -- do some digging). I think it would be great to incorporate some original source material at least once or twice a year in a math course at any high school in the US! Talk with your history colleagues and get them in on this! (especially those AP Euro teachers, Document-Based Questions are a big essay component!)&lt;/p&gt;&#xA;" OwnerUserId="1249" LastEditorUserId="1249" LastEditDate="2015-08-10T02:18:22.483" LastActivityDate="2015-08-10T02:18:22.483" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8500" PostTypeId="2" ParentId="8148" CreationDate="2015-08-10T03:00:22.677" Score="0" Body="&lt;p&gt;Personally, the history of mathematics (and science in general) interests me quite a bit. Just biographies can be entertaining, but they don't add much to understanding the subject matter, you need to explain the development of the ideas and techniques. But there you soon run into a snag, the notation and methods used e.g. by Euler are often hairraising. To explain them, and explain why they aren't acceptable nowadays, and how to fix the &quot;proofs&quot;, requires quite a bit of explanation. Going further back, appreciating Archimedes' work on solids is hard work, as he uses methods that are quite alien to us. William Dunham has written several books that do an outstanding job at translating historical masterpieces to something a bright student can understand, but that is clearly a complex task.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the pressure to &quot;cover the required subjects&quot; I'd leave that out, with regrets. Besides, sidetracking through strange (or in retrospect downright wrong) developments presents the risk that your students, with not-too-firm grasp of the subject in the first place, get confused or end up using outdated techniques.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-10T03:00:22.677" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8501" PostTypeId="2" ParentId="7821" CreationDate="2015-08-10T03:05:29.030" Score="2" Body="&lt;p&gt;I'm working on an iOS app that does this (and a lot more); it allows kids to use their voice to provide answers and also provides a number of learning activities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see some details here: &lt;a href=&quot;http://multiplywithme.com&quot; rel=&quot;nofollow&quot;&gt;http://multiplywithme.com&lt;/a&gt; (learning activities are not on the website yet). We are beta testing the app; if you are interested in trying it out, please let me know.&lt;/p&gt;&#xA;" OwnerUserId="5480" LastActivityDate="2015-08-10T03:05:29.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8502" PostTypeId="2" ParentId="8453" CreationDate="2015-08-10T03:21:14.290" Score="7" Body="&lt;p&gt;There are so many Calc 1, 2, and Multi video lectures online now. Watch a few... take notes on what you think are some good traits of the person teaching. MIT's Calc course (super fast) might be helpful if you're teaching that style and speed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some resources&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;This text, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0821813986&quot;&gt;How to Teach Mathematics&lt;/a&gt; - S. Krantz, has many useful guides for the beginning person. I don't agree with all, but it's a great resource.&lt;/li&gt;&#xA;&lt;li&gt;Suzanne Kelton wrote a guide for the AMS. Its pdf is here: &lt;a href=&quot;http://www.ams.org/profession/career-info/grad-school/Kelton-TEACH.pdf&quot;&gt;teaching Mathematics at the college level&lt;/a&gt; - pg. 31 goes into Calc stuff!&lt;/li&gt;&#xA;&lt;li&gt;Also, do not be ashamed to search for AP Calculus teachers' resources. It may be a &quot;high school&quot; course, but some of the teachers out there are amazing and offer a lot of guidance. Specific topic choice may vary but good teachers are that no matter what. &lt;a href=&quot;http://teachingcalculus.com/&quot;&gt;Lin McCullin's website&lt;/a&gt; is an prime example. Watch those videos yourself from his site :) Ignore test-specific things. Pedagogy!&lt;/li&gt;&#xA;&lt;li&gt;If you are in the US, go to your MAA sectional meeting in the Fall! &lt;/li&gt;&#xA;&lt;li&gt;Please, please, please incorporate technology as a reasonable tool that you actually teach how to use in the right situations. This is vital and super valuable.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2015-08-10T03:21:14.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8503" PostTypeId="2" ParentId="8489" CreationDate="2015-08-10T03:34:54.380" Score="3" Body="&lt;p&gt;I am a huge fan of using tools in a current math course to show something &quot;flashy&quot; or more advanced. It's especially nice if you can really ham it up and make people think they're doing something that no one else at their level is doing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was shocked when I learned that I knew enough to do this kind of stuff and was sad that I hadn't been shown it before! So I am impressed with your idea of going down this road. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A suggestion of usage. Obviously you would want to motivate this, perhaps along with dealing with formal linear combinations you might take Steven's idea 4 and do that yourself as a lead in for everyone to see. Definitely make sure that labeling is clear -- notation is the key to mathematical clarity! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I would suggest taking the last 15-20 minutes of a class to have students begin working on this in pairs (or make it a paired homework). It is abstract enough to warrant team work, and sometimes requiring teamwork is useful. It would also be useful if you did it in class, because you could wander the room and field questions or listen in carefully to see where roadblocks are happening. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On an editing note, I would state the two vector spaces' and linear transformation's names in the first sentence. Repetition isn't bad.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(i.e.: We will use the following graphic to create two vector spaces, $C_0$ and $C_1$, and a linear transformation between them, D : $C_0 \to C_1$ )&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All hypothetical, of course ;) &lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2015-08-10T03:34:54.380" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8504" PostTypeId="1" CreationDate="2015-08-10T18:11:59.927" Score="6" ViewCount="536" Body="&lt;p&gt;I am creating an app for multiplication learning/practice. During practice students get hints if they do not answer the question within some time or if they provide a wrong answer. These hints are tied into learning strategies presented separately via a number of learning activities; the goal is to remind the student of a strategy they learned about before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering if anybody has recommendations for x8 multiplication strategies (I am looking for things anchored in research, not reciting/memorization tricks). Any pointers to papers, resources, etc. are highly appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, a strategy for multiplying by x9 could be to get to the closest known answer, in this case by multiplying by 10 and then subtracting the other factor (e.g. 6*9 = 6 * (10 - 1) = 6*10 - 6 = 60 - 6 = 54.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 1:&#xA;Focus is on single-digit multiplication.&#xA;As for the comment about the research, perhaps let me restate that I am NOT interested in memorization tricks or anything that's not based in math (e.g. reciting, etc. I've seen a number of things that may make student memorize the fact, but they may not have a clue why that's the result). (Given the audience on stackexchange maybe that comment was not needed)&lt;/p&gt;&#xA;" OwnerUserId="5480" LastEditorUserId="5480" LastEditDate="2015-08-11T04:27:55.807" LastActivityDate="2023-01-04T22:31:59.687" Title="Strategies for x8 multiplication" Tags="|technology-in-education|multiplication-table|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="8506" PostTypeId="2" ParentId="8504" CreationDate="2015-08-11T15:22:49.057" Score="2" Body="&lt;p&gt;At the risk of being cheeky, my own strategy for multiplying things by 8 is that I've internalized (memorized) the answers. $4 \times 8$ reads simultaneously to me as both the multiplication statement and as the value $32$, and I expect that's the case for most of us.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that a successful training regimen for the multiplication tables will aim to have its users &lt;strong&gt;ultimately&lt;/strong&gt; converge on the same strategy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, employing intermediate strategies such as the repeated doubling can:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Serve as a bridge techniques that both informs and motivates the eventual internalization (ie, after some number of times working out a particular product, students are likely to simply recall the answer, and students are likely to be pleased with themselves when the answer does start 'popping' into their heads, since they've saved themselves some time and demonstrated some hard-earned knowledge)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Provide high grade intellectual fodder for students to chew on: Why is it that doubling a number three times is the same as multiplying it by 8? (An especially valuable line of questioning if the students also use repeated doubling for multiplication by 4). Similar questions arise from any such technique, and I think that they're all potentially valuable in their own ways.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="308" LastActivityDate="2015-08-11T15:22:49.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8507" PostTypeId="1" CreationDate="2015-08-11T15:51:43.557" Score="4" ViewCount="163" Body="&lt;p&gt;I am trying to help some adults who have not practiced mental arithmetic for decades, but now they have a need for various reasons. I have noticed that they are considerably uneven: $2\times4$, $6\times6$, and $8\times8$ are no problem normally, but for some reason, they are often slow and/or unsure of, for instance, $7\times8$, $3\times9$, $7\times12$, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any known apps or web sites where you can define the multiplications for a certain pupil you observed being slow or unsure? I mean, where you can pick out or choose a specific set of calculations, like &quot;flash cards&quot; and the set time limits for practice of speed and accuracy?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the purpose we use IPad, and Windows 8 on PC.&lt;/p&gt;&#xA;" OwnerUserId="5485" LastEditorUserId="2074" LastEditDate="2015-08-11T20:02:20.460" LastActivityDate="2015-08-11T23:23:45.540" Title="Adult pupils with multiplication problems" Tags="|technology-in-education|software|multiplication-table|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8508" PostTypeId="1" AcceptedAnswerId="8509" CreationDate="2015-08-11T16:56:24.973" Score="9" ViewCount="1336" Body="&lt;p&gt;I'm trying to calculate $$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\sin t \cos^3 t\,dt$$ using integration by substitution $$\int_{\varphi([a;b])} f(x)dx=\int_{[a;b]} f\left(\varphi(t)\right)|\varphi'(t)|dt$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;First Method&lt;/strong&gt;&lt;br/&gt;&#xA;Let $\varphi(t)=\cos t$ which is continuously differentiable and $\displaystyle\varphi\left(\left[\frac{-\pi}{2};\frac{\pi}{2}\right]\right)=[0;1]$ so $$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\sin t \cos^3 t\,dt=\int_0^1x^3dx=\frac{1}{4}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Second Method&lt;/strong&gt;&lt;br/&gt;&#xA;Let $\varphi(t)=\cos t$ which is continuously differentiable and $\displaystyle\varphi\left(\left[\frac{-\pi}{2};\frac{\pi}{2}\right]\right)=[0;1]$ so $$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\sin t \cos^3 t\,dt=\int_{\frac{-\pi}{2}}^{0}\sin t \cos^3 t\,dt+\int_0^{\frac{\pi}{2}}\sin t \cos^3 t\,dt=\\ -\int_{\frac{-\pi}{2}}^0 f\left(\varphi(t)\right)|\varphi'(t)|dt+\int_0^{\frac{\pi}{2}} f\left(\varphi(t)\right)|\varphi'(t)|dt=-\int_0^1x^3dx+\int_0^1 x^3dx=0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is obviously $0$ as the integrand is an odd function. As mentioned in &lt;a href=&quot;https://math.stackexchange.com/questions/470411/why-should-the-substitution-be-injective-when-integrating-by-substitution&quot;&gt;another question&lt;/a&gt;, $\varphi$ isn't required to be monotone or injective. However, most high school teachers I know and most school textbooks warn students that &lt;em&gt;you should not use integration by substitution if $\varphi$ is not monotone&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you think I should go with their methods or use the second method (with absolute values) even though it may confuse many students as they usually do it like the first method.&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-08-14T07:16:57.690" Title="Should $\varphi$ be monotone in the integration by substitution?" Tags="|textbooks|students-mistakes|integration|teaching|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="8509" PostTypeId="2" ParentId="8508" CreationDate="2015-08-11T20:04:07.530" Score="13" Body="&lt;p&gt;There are two formulations for definite integrals: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_{\phi(a)}^{\phi(b)} f(x)\, dx=\int_a^b f(\phi(t))\phi'(t)\, dt$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the one you state:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_{\phi([a,b]}f(x)\,dx=\int_{[a,b]} f(\phi(t))|\phi'(t)|\, dt$$  In the second, you do need $\phi$ to be monotone. In the first formulation, you do &lt;em&gt;not&lt;/em&gt; need this assumption. Of course when you apply the first formulation to your integral, $\phi(a)=\phi(b)$  so you get $0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the first formulation is the one typically stated in high school and undergraduate calculus textbooks.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-08-11T20:04:07.530" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8510" PostTypeId="1" CreationDate="2015-08-11T20:34:35.670" Score="9" ViewCount="482" Body="&lt;p&gt;In this (&lt;a href=&quot;https://www.youtube.com/watch?v=p9sYzMG-43k&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=p9sYzMG-43k&lt;/a&gt;) session Dan Meyer uses Excel, Google and Wolfram Alpha in a class (of teachers) for calculation. He himself uses them on a laptop projected for the class to see. The skills to use these tools are fairly basic - either natural language input or limited (in excel =A2^2, autofill, in wolfram alpha SUMMATION(13*n^2,1,40)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I use Mathematica (on Raspberry Pi) as my go to. And I envisage using it some way in my classes in my future classes. Mathematica requires more know ledge of syntax, commands (often obscure, eg eqn/.x-&gt;{1,2,3}) than Google or Wolfram Alpha.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I worry that such use of higher level programming tools will bewilder and confuse my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are guidelines for the use of such more specialised tools in a high school maths class?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possibilities-&#xA;Using Mathematica (or similar) in demostrations to the class.&#xA;Having PCs in the class that students can use with Mathematica (or similar). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people have raised the cost of Mathematica and suggested free alternatives. I envisage having a few raspberry pis in class with mathematica. These could be used alongside excel, for example. So my question is really about: &lt;strong&gt;What is realistic to expect of students when it comes to using a software package?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you see in the video google and wolfram alpha accept natural language and simple maths (13*40^2+13*39^2+...+13*1^2), but what more structured commands google and wolfram alpha will accept seems unclear. (Mathematica allows connection to wolfram alpha, I haven't tried it yet.)  On the other hand Excel has it's own barriers, but it's fairly ubiquitous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it seems a balance between accessibility, cost, familiarity... &lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="998" LastEditDate="2015-08-20T01:33:04.033" LastActivityDate="2015-08-20T21:09:35.143" Title="Mathematica in the classroom?" Tags="|technology-in-education|computer-programs|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="8511" PostTypeId="2" ParentId="8494" CreationDate="2015-08-11T23:13:47.947" Score="4" Body="&lt;ul&gt;&#xA;&lt;li&gt;Break down the &quot;compute the integral&quot; into subtasks, assign points to each step&lt;/li&gt;&#xA;&lt;li&gt;Define some &quot;mistake types&quot; (e.g. wrong sign, wrong sum, ...) and subtract a given number of points for each&lt;/li&gt;&#xA;&lt;li&gt;A combination of the above&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The problem with long answers is that if a student makes a mistake, you have to pick up their thread to check if later work is correct (and the mistake might make the problem much easier or next to impossible).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I do when designing an exam is to review the subjects to be covered, and ask self-contained, limited questions that check each important point on their own (or check a few points only). Sometimes I've broken down a long development by giving intermediate results as starting points for separate questions. You can do the same by dovetailing steps in similar questions to cover a long series of steps, if you don't want to give intermediate results away. To check if they know what the steps are, ask for that (no need to carry them out explicitly).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For long developments I give them as homework, explicitly allowing the use of external tools (computer algebra system, textbook, whatever). That way mistakes shouldn't have to be carried along, as you can also encourage checking intermediate results.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-11T23:13:47.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8512" PostTypeId="2" ParentId="8487" CreationDate="2015-08-11T23:21:31.807" Score="2" Body="&lt;p&gt;First of all, this is a &lt;em&gt;huge&lt;/em&gt; task. Plan on spending a few years on this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, Newton said he was able to see farther because he stood on the shoulders of giants. Check what is available already (here are plenty of good to excellent free textbooks and lecture notes around).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the above doesn't unearth something close to what you want, by all means go ahead. But when writing a totally new/different textbook, make sure (by asking e.g. here, by reading up on pedagogy, by doing limited experiments) that your approach really is better.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-11T23:21:31.807" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8513" PostTypeId="2" ParentId="8507" CreationDate="2015-08-11T23:23:45.540" Score="1" Body="&lt;p&gt;I am working on an app (iPad/iPhone) that allows students to learn/practice multiplication facts. The app provides a number of interactive learning activities as well as a practice section where students interact with the app using their voice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are currently beta testing it so it's not in the App Store yet. Please email me if you'd like to try it out (ogi at keenresearch dot com).&lt;/p&gt;&#xA;" OwnerUserId="5480" LastActivityDate="2015-08-11T23:23:45.540" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8514" PostTypeId="2" ParentId="7964" CreationDate="2015-08-11T23:39:58.103" Score="9" Body="&lt;p&gt;I believe you are mixing up three problems here. One is the mistaken idea that &quot;mathematics is all compact, weird symbols, if I don't use them to abbreviate my text, it's wrong&quot; (your $polynomial = 3$ example), there is the &quot;next step&quot; interpretation, and finally the &quot;assign a value&quot; from programming.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="1147" LastEditDate="2017-08-11T04:42:47.560" LastActivityDate="2017-08-11T04:42:47.560" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8516" PostTypeId="2" ParentId="8504" CreationDate="2015-08-12T05:22:14.327" Score="1" Body="&lt;p&gt;For the &lt;span class=&quot;math-container&quot;&gt;$9\times$&lt;/span&gt; table, there is the well known technique of using one's hands. Drop the &lt;span class=&quot;math-container&quot;&gt;$k^{th}$&lt;/span&gt; digit (finger/thumb), and count fingers/thumbs to the left as tens, and to the right as units.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Omucx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Omucx.png&quot; alt=&quot;hands9&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this example, we have lowered the &lt;span class=&quot;math-container&quot;&gt;$7^{th}$&lt;/span&gt; finger, in order to calculate &lt;span class=&quot;math-container&quot;&gt;$9\times 7$&lt;/span&gt;. We count &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; fingers/thumbs to the left and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; to the right, giving &lt;span class=&quot;math-container&quot;&gt;$6\times10+3=63$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is extensible beyond &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; if we use slightly different methods.&lt;/p&gt;&#xA;&lt;p&gt;We can also emulate different bases, and hence multiplication by &lt;span class=&quot;math-container&quot;&gt;$b-1$&lt;/span&gt;, by pretending we have only &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; fingers.&lt;/p&gt;&#xA;&lt;p&gt;In this example, we pretend the right thumb is not there, to work in base &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; for the &lt;span class=&quot;math-container&quot;&gt;$8\times$&lt;/span&gt; table.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JRHC3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JRHC3.png&quot; alt=&quot;hands8&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, fingers to the left count as nines, and fingers to the right remain as units. We lower the &lt;span class=&quot;math-container&quot;&gt;$4^{th}$&lt;/span&gt; finger in order to calculate &lt;span class=&quot;math-container&quot;&gt;$8\times 4$&lt;/span&gt;. We have &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; fingers to the left, and &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; to the right, and so &lt;span class=&quot;math-container&quot;&gt;$8\times 4=3\times9+5=32$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Again, this technique can be extended for larger multiplications. And working in base &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is a feat in itself!&lt;/p&gt;&#xA;" OwnerUserId="5071" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-08-12T05:52:46.810" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8517" PostTypeId="2" ParentId="8487" CreationDate="2015-08-12T09:05:07.400" Score="5" Body="&lt;p&gt;I am fully aware that much of this answer does not belong in this forum. On the other hand, I believe that many mathematics educators are interested in producing materials and would benefit from this advice, which I have gained over a few years of authoring. All in all, I think mathematics educators generally would benefit from these ideas so I will post them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been writing resources for my students for a number of years and am currently experimenting with collating things online into a textbook/forum environment. For this I am using my own domain name and a WordPress website. I think that the idea of writing your textbook in $\LaTeX$ is a great idea because it will ensure a high quality product. May I suggest a couple of ideas for you:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Consider using &lt;a href=&quot;http://www.sharelatex.com&quot; rel=&quot;noreferrer&quot;&gt;ShareLaTeX&lt;/a&gt; as your editor, because it becomes easy to collaborate, to work on your document in various locations with no tex installation, and is quite nice and easy to keep organised. The two problems are: no SageTeX; and only the tex files are autosynced (the pdf can be previewed in browser but must be downloaded manually to keep, not a huge deal though)&lt;/li&gt;&#xA;&lt;li&gt;Use the &lt;a href=&quot;http://www.ctan.org/pkg/subfiles&quot; rel=&quot;noreferrer&quot;&gt;subfiles&lt;/a&gt; package for $\LaTeX$ and write each chapter as a separate .tex file - this improves preview compilation time and allows you to focus on the structure as well as the content more easily&lt;/li&gt;&#xA;&lt;li&gt;Write your preamble in a .sty file separately and call it in the preamble of the main document. Your subfiles will automatically pick up the style from the main document. This means you only need to make changes to the preamble in one place.&lt;/li&gt;&#xA;&lt;li&gt;Download geogebra, use it to draw your pictures, then export as TiKz and copy/paste into your document instead of drawing things manually. This will save a lot of time (the code often requires tweaking but it is still faster).&lt;/li&gt;&#xA;&lt;li&gt;Have you thought about how your textbook might display online? Will you use a pdf embedder or some kind of TeX2html solution? How will students refer to specific parts of your textbook? Can they copy/paste quotes? Can they right-click to see the TeX commands you have used to produce equations? &lt;a href=&quot;https://www.peterkrautzberger.org/0136/&quot; rel=&quot;noreferrer&quot;&gt;This&lt;/a&gt; is an interesting post on the topic.&lt;/li&gt;&#xA;&lt;li&gt;In your forum, can they use TeX to write mathematics online? If not, how else can they communicate mathematically using their keyboards? Check out &lt;a href=&quot;http://www.sowiso.nl/calculus/en/&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; solution - it is impressive what they have done.&lt;/li&gt;&#xA;&lt;li&gt;From a safeguarding point of view, how will you make sure that the online communication remains friendly and is not used for trolling/bullying? Does your school have guidelines for people creating online forums? Could you be held accountable for anything bad that may happen online?&lt;/li&gt;&#xA;&lt;li&gt;Writing problems for your textbook will be one of the largest tasks. Get help with this. Generally, your main concerns should be sequencing (later problems must depend upon earlier ones, not the other way round) and comprehensiveness. Be sure to account for special cases or else have a rationale for omitting them. If you can do a problem with variables from $\mathbb{N}$, what about positive fractions? then what about negative integers? Then what about negative fractions? Every changeable aspect of a problem adds a new 'dimension of variability' and it is important to account for them all. You would be amazed at how many of my students can factorise $x^{2}+x-2$ but not $-x^{2}-x+2$. How will you train readers to prepare for variation, to be adaptable?&lt;/li&gt;&#xA;&lt;li&gt;Write a manifesto for your textbook - what is it you don't like about others? Be specific: why is the existing textbook deficient for the purpose of learning mathematics? How will yours be different? Imagine that paper is a scarce resource and you are having to justify printing your book versus an existing textbook - why should they print yours and no-one else's? This will help your book remain distinct and not fall into the trap of lazy writing (it is very, very tempting to gloss over certain points so you can get to the stuff you already have good ideas for)&lt;/li&gt;&#xA;&lt;li&gt;Try and read a few academic papers about (a) the use of mathematics textbooks and (b) the problems students commonly face with calculus. There are a lot of people doing a lot of research which largely gets ignored by textbook authors (at least here in the UK, where the standard books are merely reference guides). If more textbooks were research based, the they would be far more relevant to their purpose. For example &lt;a href=&quot;http://www.jstor.org/journal/educstudmath&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; or &lt;a href=&quot;http://www.jstor.org/journal/forlearningmath&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; are two important UK based mathematics education research journals (I think you can get a free account and get a few articles per month for free)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Yours is a sprawling question which pretty much draws on everything the mathematics educator has to worry about. Don't be daunted by this, but do make sure that you start your work in the best way so that you minimise redrafting. It is better to have a clear idea about your methods and purpose before you begin or else you duplicate your workload with every revision (I have learned this the hard way). I really hope your book sees the light of day and that the online community around it becomes fruitful for all those involved.&lt;/p&gt;&#xA;" OwnerUserId="901" LastActivityDate="2015-08-12T09:05:07.400" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8518" PostTypeId="5" CreationDate="2015-08-13T03:46:15.107" Score="0" Body="&lt;p&gt;The Common Core State Standards Initiative is an educational initiative in the United States of America that details what K–12 students should know in English language arts and mathematics at the end of each grade. The initiative seeks to establish consistent educational standards across the states as well as ensure that students graduating from high school are prepared to enter credit-bearing courses at two- or four-year college programs or to enter the workforce.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2015-08-13T13:22:12.707" LastActivityDate="2015-08-13T13:22:12.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8519" PostTypeId="4" CreationDate="2015-08-13T03:46:15.107" Score="0" Body="The Common Core State Standards in the United States of America" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2019-12-31T03:31:39.180" LastActivityDate="2019-12-31T03:31:39.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8520" PostTypeId="2" ParentId="8419" CreationDate="2015-08-13T06:00:26.603" Score="3" Body="&lt;p&gt;There is Grinstead &amp;amp; Snell's &lt;a href=&quot;http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html&quot; rel=&quot;nofollow&quot;&gt;Introduction to Probability&lt;/a&gt;, Second Revised Edition. It is freely available under GNU Free Documentation License, as explained on the linked page. The level is undergraduate and the authors state in the preface:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The text can also be used in a discrete probability course. The material has been organized in such a way that the discrete and continuous probability discussions are presented in a separate, but parallel, manner.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="376" LastActivityDate="2015-08-13T06:00:26.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8524" PostTypeId="2" ParentId="4408" CreationDate="2015-08-15T04:25:40.563" Score="2" Body="&lt;p&gt;I remember this anecdotal version, I don't know whether or not it matches yours.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;Fundamental Theorem of Calculus&lt;/strong&gt; stated using &lt;strong&gt;vegetables&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_{carrot}^{potato}vegetable(turnip)d(turnip)$$&#xA;$$=stew(potato)-stew(carrot)$$&#xA;where $$stew'=vegetable$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My tutor said you end up with potato stew with no carrots. I asked what happened to the turnips?&lt;/p&gt;&#xA;" OwnerUserId="5071" LastEditorUserId="5071" LastEditDate="2015-08-15T09:25:20.680" LastActivityDate="2015-08-15T09:25:20.680" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8526" PostTypeId="1" CreationDate="2015-08-15T16:34:23.340" Score="4" ViewCount="782" Body="&lt;p&gt;I am a rising college junior with major in the mathematics.  I recently got interested to the contest mathematics through working with few Putnam problems in the real analysis and linear algebra, and I decided to prepare for Putnam Competition, which will be on this December.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I switched my major from the microbiology to mathematics on the last semester through my ongoing research in the theoretical computer science.  Since then, I have been sharpening my proof skill.  Unfortunately, Putnam Competition will be my first experience with the contest mathematics, and that means I do not have any experience with it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am currently self-studying Rudin-PMA, Apostol-Mathematical Analysis, Artin-Algebra (little bits per day), and Hoffman/Kunze-Linear Algebra.  I will be taking the Analysis I (Rudin) and Linear Algebra with Introduction to Proofs (Friedberg) on the upcoming Fall.  Since I also have to dedicate myself to the classes and undergraduate researches, I do not have enough time to read textbooks on other major topics of Putnam: number theory, combinatorics, probability theory, and mathematical statistics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My preparation plan is to read and study the proof techniques and materials covering those topics (number theory, combinatorics, etc.) in the problem-solving books like Larson, Engel, and Andreescu (Putnam and Beyond).  Then, I am going to jump directly to the past Putnam problems and pick up the additional materials on mathematical branches (such as combinatorics and number theory) embedded on Putnam problems. That way, I can sharpen my problem-solving tricks necessary for the Putnam Competition and acquire interesting knowledge in the mathematical branches like number theory in a discrete way of learning (by attempting the problems and learn the necessary concepts).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I am a rookie in the contest mathematics and my preparation plan for Putnam might be inefficient.  I would like to take this chance to seek your advice and knowledge in the Putnam Competition, and to seek your comments in my preparation plan.  Also I heard that Putnam problems generally do not require advanced mathematical branches like functional analysis and partial differential equations, and they instead require the deep understanding of basic mathematical concepts (such as limits, continuity, groups, etc.) and clever tricks.  If my guess is correct, could you give me the advice what kind of mathematical branches and clever tricks I need to study and acquire, and elaborate more on them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you very much for your time, and I look forward to hear back from you!&lt;/p&gt;&#xA;" OwnerUserId="5399" LastActivityDate="2015-08-15T16:34:23.340" Title="How to prepare for Putnam Competition?" Tags="|undergraduate-education|self-learning|undergraduate-research|math-contests|" AnswerCount="0" CommentCount="1" ClosedDate="2015-08-18T19:51:49.263" ContentLicense="CC BY-SA 3.0" />
  <row Id="8527" PostTypeId="1" CreationDate="2015-08-17T07:51:53.983" Score="8" ViewCount="118" Body="&lt;p&gt;I co-developed and co-taught an interdisciplinary course with several colleagues from several departments for junior/senior college students. This was a very interesting experience, and I am wondering if I may just report/publish how the mathematical part was organized and taught.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What journal or journals might accept publications of this nature?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5507" LastEditorUserId="262" LastEditDate="2015-08-17T10:24:12.480" LastActivityDate="2015-08-17T20:51:59.573" Title="Where can I publish on an undergraduate course that I helped develop and teach?" Tags="|undergraduate-education|education-research|undergraduate-research|publishing|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8528" PostTypeId="2" ParentId="7192" CreationDate="2015-08-17T10:58:05.250" Score="3" Body="&lt;p&gt;When Ī̲ was taught about this stuff, my teachers explained that there are two definitions of the square root function: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;arithmetic&lt;/strong&gt; (that produces non-negative results and is undefined on negative numbers)  and&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;algebraic&lt;/strong&gt; (that is defined on the whole ℂ and is 2-valued).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;From this perspective, $\sqrt{-1}$ is arithmetically undefined and algebraically $\pm i$. One may attempt to inject something in-between, namely to extend definition of the arithmetic square root in some way to obtain exactly $\sqrt{-1} = i$, but this won’t clarify the question. All the truth about $z^2 + 1 = 0$ is that the equation has exactly two roots in the extended field, and, unlike square roots of 1, both $\pm\sqrt{-1}$ are equivalent (formally speaking, switching between them generates a field automorphism). So we have to “label” one of these indistinguishable twins with $i$ to be able to write extended field’s elements explicitly.&lt;/p&gt;&#xA;" OwnerUserId="3272" LastEditorUserId="3272" LastEditDate="2015-08-17T11:16:01.870" LastActivityDate="2015-08-17T11:16:01.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8529" PostTypeId="2" ParentId="1883" CreationDate="2015-08-17T18:36:46.787" Score="3" Body="&lt;p&gt;Disadvantages of Argand Plane as the main conceptual metaphor for ℂ are the same as of the “real line” for ℝ: both are insufficiently intuitive. Why some point of the line/plane should be labelled with “0”, not another? Euclidean lines and planes are translationally invariant, whereas ℝ and ℂ are not. Why should we choose some scale (i.e. distance between 0 and 1), not another one? Why this direction, not opposite, shall be “positive”? Euclidean metaphors for numbers are indeed very helpful, even in professional mathematics, but numbers appear only as &lt;strong&gt;relations between points&lt;/strong&gt;, not points themselves. The thing ancient Pythagoreans had a good grasp of, but possibly ignored by some modern teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don’t know about English-speaking countries, but Ī̲ was taught about vectors in secondary school. On a line, if we fixed one non-zero vector (that is the difference between two distinct points), then multiplying it by any real number gives all vectors along the line. So, we actually need at least three points to represent a “real number” without applying arbitrary conventions. If we fix one point as “0”, another one as “1”, then for the variable point &lt;em&gt;X&lt;/em&gt; the vector “0 &lt;em&gt;X&lt;/em&gt;” divided by “0 1” gives an arbitrary real number. That’s how we arrive to the notorious “real line”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same about complex numbers: they are quotients of 2D Euclidean vectors just as real numbers are quotients of 1D Euclidean vectors (and quaternions are quotients of 3D Euclidean vectors, BTW). The only quirk with Argand Plane is orientation. Quotients of 2D Euclidean vectors form a field, but which order of catheti of 1:1:√2 triangle gives &lt;em&gt;i&lt;/em&gt; and which gives &lt;em&gt;−i&lt;/em&gt;? So we choose, by convention, one specific orientation (of two possible), similarly to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7192/pedagogical-quandary-with-the-definition-of-i&quot;&gt;choosing one of two roots of −1 for &lt;em&gt;i&lt;/em&gt;&lt;/a&gt;, to make the metaphor perfect. Now complex numbers can serve in planar geometry and electrical engineering (where the natural direction of time helps to distinguish &lt;em&gt;i&lt;/em&gt; from &lt;em&gt;−i&lt;/em&gt;) as &lt;em&gt;quotients&lt;/em&gt; of vectors. Or, if you like it, &lt;em&gt;scale factors&lt;/em&gt; in similarity transformations. And if you label one point on the blackboard as “0” and another as “1”, then you are able to draw “the complex plane”, but its points only &lt;em&gt;represent&lt;/em&gt; complex numbers with all aforementioned conventions. They are not numbers themselves, of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Formally, we can speak of ℂ itself as of some metric space, of which complex numbers are elements (i.e. &lt;em&gt;points&lt;/em&gt;), and prove that it’s isometric to Euclidean plane. But this requires sufficient level of mathematical abstraction, beyond secondary education IMHO.&lt;/p&gt;&#xA;" OwnerUserId="3272" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-08-17T18:36:46.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8530" PostTypeId="2" ParentId="8527" CreationDate="2015-08-17T20:51:59.573" Score="3" Body="&lt;p&gt;Assuming that one of the disciplines involved was mathematics and you have some novel insights or ideas about how to create such a course, &lt;a href=&quot;http://www.tandfonline.com/loi/upri20&quot; rel=&quot;nofollow&quot;&gt;PRIMUS&lt;/a&gt; is a good generalist option, especially if you can &quot;aim&quot; the article at other undergraduate faculty who might be interested in creating such a course but aren't sure where to start.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-08-17T20:51:59.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8531" PostTypeId="2" ParentId="7192" CreationDate="2015-08-18T14:22:58.637" Score="1" Body="&lt;p&gt;The choice of i versus -i is correlated with the geometric orientation of the the plane. One choice, i, correponds to counterclockwise orientation which matches the righthand rule for orientation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conjugation, which is reflection across the x-axis, reverses orientation and exchanges the two choices.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-08-18T14:22:58.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8532" PostTypeId="1" CreationDate="2015-08-18T14:41:51.197" Score="1" ViewCount="179" Body="&lt;p&gt;I want to create problems concerning polynomial interpolation for students. However, I need those problems to differ in difficulty for the different students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I could always do is solve some problem and classify them by difficulty. However, it would be nice if I could find a &quot;system&quot; with I could create such problems. This would also have the benefit that the problems for the students do not repeat themselves (most of the time).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students are supposed to either use the classic approach, the newton approach, or the lagrange approach. Each one of them needs to be understood.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a way to create such problems? and where in the process would I be able to increase / decrease difficulty?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problems btw only ought to consist of the data points, from which the polynomial has to be interpolated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I already tried building relatively easy polynomials and then calculating the data points for the problem, however calculation would still get complicated for the students. Nice, clean and easy numbers would be great. Numbers like 1/4, 1/2, 1/5 etc. would also be fine. I just want to avoid the students having to work with numbers such as 83.23461.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any ideas?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you in advance.&lt;/p&gt;&#xA;" OwnerUserId="5510" OwnerDisplayName="Micha" LastEditorUserId="4872" LastEditDate="2015-08-26T09:05:16.227" LastActivityDate="2015-09-03T12:10:39.857" Title="Create Problems for Polynomial Interpolation" Tags="|problem-design|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8533" PostTypeId="2" ParentId="8532" CreationDate="2015-08-19T07:56:18.523" Score="1" Body="&lt;p&gt;From my point of view as a student, you should not give many basic interpolation problems to students, since they are just, as Normal Human stated, mindless work. However, I suggest you use this plan:&lt;br&gt;&#xA;First, you ask for a polynomial interpolating 3 points, without any condition on the said polynomial. Then, you increase the number of points to interpolate to 4 or 5 and you ask a polynomial of minimal degree. These are the very basic (and slightly boring) exercises, to ensure your students understood your lecture. I recommend you give at most 4 or 5 of those to your students.&lt;br&gt;&#xA;After that, you may step up the difficulty by asking your students to find every interpolating polynomial whose degree is lesser or equal to a given constant. You may also create an exercise where you add constraints on the k$^{th}$ derivative of the interpolating polynomial at certain points, for a given k. These problems should be more challenging and interesting.&lt;/p&gt;&#xA;" OwnerUserId="5511" LastActivityDate="2015-08-19T07:56:18.523" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8534" PostTypeId="1" CreationDate="2015-08-19T08:39:03.003" Score="12" ViewCount="2096" Body="&lt;p&gt;First off I apologize if this question comes out to be off-topic. I wasn't sure whether to post it on MSE or here.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Lately I gave an exam named SSC-CGL. This exam is conducted every year in India for the selection of grade B officers. The exam is of 200 questions out of which 50 questions are asked from mathematics. The exam is conducted in four shifts. Each shift is allocated different question-set, but the syllabus is exactly same. When I compared my paper with other shift's paper I found their difficulty level was very low and questions from my paper were comparatively lengthy. As an example consider these two questions.&lt;/p&gt;&#xA;&lt;p&gt;My paper's question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$x^2+x=5$&lt;/span&gt; then find the value of &lt;span class=&quot;math-container&quot;&gt;$\dfrac{1}{(x+3)^3} + (x+3)^3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Their paper's question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If the measure of three angles of a triangle are in ratio 2:3:5 then find the angles of the triangle&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;First question requires the trick of substitution &lt;span class=&quot;math-container&quot;&gt;$x+3=t$&lt;/span&gt; but the other question can be solved by a school student because an 8th grade student knows that the sum of the angles is 180 degrees.&lt;/p&gt;&#xA;&lt;p&gt;Similarly I found every question in my paper relatively difficult than theirs, because my questions required tricks but their question required direct use of the concept. This is &lt;a href=&quot;https://attachment.fbsbx.com/file_download.php?id=1450176968644433&amp;amp;eid=ASsHyrK3OMLMj-w88yNweltqNk1SgAN7jFCCHWUSwz6D-yljGkmYA50RHMqI0IUCmJk&amp;amp;inline=1&amp;amp;ext=1439985727&amp;amp;hash=ASvD2Zc-GFjIa-xh&quot; rel=&quot;nofollow noreferrer&quot;&gt;my paper&lt;/a&gt; and here is the &lt;a href=&quot;https://attachment.fbsbx.com/file_download.php?id=935308916514631&amp;amp;eid=ASsMjzAhW_PqaIqchShmO9shq7CK5VDltZ0vo-QLTIKQnX85ho4-H8BXmI4QtaO0yno&amp;amp;inline=1&amp;amp;ext=1439985736&amp;amp;hash=ASsuOgEbMdPpLPfy&quot; rel=&quot;nofollow noreferrer&quot;&gt;other one&lt;/a&gt;, math questions start from 101st question in both the papers.&lt;/p&gt;&#xA;&lt;p&gt;The problem is that last year too there was a difference in the difficulty of different shifts but we could not give a solid argument in court to justify the fact. That is why now I'm asking here about whether there is some standard procedure in mathematics which could compare the difficulty level of questions. E.g. I think that number of steps or length of the solution can be valid parameter. Length might not signal the difficulty but in competitive exams time per question is an important parameter.&lt;/p&gt;&#xA;&lt;p&gt;Some say that difficulty is a subjective matter and hence we can't say some question is more difficult than others. But we very well know that questions asked in Mathematics Olympiad are more difficult than those asked in school exams. Whenever teachers set an exam they ask questions based upon the pre-defined difficulty of the exams. No teacher would ask an Olympiad type question in a school exam and vice-versa.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Tl;dr: Is some standard procedure in mathematics which could compare the difficulty level of questions asked from same topics(syllabus)?&lt;/p&gt;&#xA;" OwnerUserId="1642" LastEditorUserId="376" LastEditDate="2021-12-02T16:43:42.600" LastActivityDate="2021-12-02T16:43:42.600" Title="Is there some standard method to measure the difficulty level of mathematics problems?" Tags="|exams|grading|india|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8535" PostTypeId="2" ParentId="8534" CreationDate="2015-08-19T13:40:32.860" Score="9" Body="&lt;p&gt;I don't know that there are any standardized measures that work by analyzing the questions themselves (I doubt it) but there are a number of statistical approaches that work by analyzing the test-takers' performance.  For example, depending on the size of the shifts one could look to see if there are statistically significant differences in mean score or mean time to complete.  (You did not indicate whether the shifts are all taught by the same instructors; if not, then it will be hard to argue that the differences in outcomes should be ascribed to the test rather than to the teacher.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also pre-evaluate the different versions by gathering a sample of people who successfully took the exam the previous year, randomly assign each them 1 of the 4 different versions of the exam, and look to see if there are statistically significant differences in time to complete or score.  If it is possible to design the sample so that everybody in it took the same version of the exam the previous year, you could also correct based on past performance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may want to look at Wikipedia's articles on &lt;a href=&quot;https://en.wikipedia.org/wiki/Psychometrics&quot;&gt;on Psychometrics&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Item_response_theory&quot;&gt;Item Response Theory&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2015-08-19T13:46:51.890" LastActivityDate="2015-08-19T13:46:51.890" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8536" PostTypeId="1" CreationDate="2015-08-19T15:05:13.323" Score="9" ViewCount="238" Body="&lt;p&gt;I would like to know how to better demonstrate Eigenvectors. The texts that I have display the properties and methods to calculate them. There are plenty of great elementary examples to follow through before taking on larger matrices and applying them in software to large scale problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regardless of this, there are many graduates working as analysts who still say that they know how to compute them but do not have a solid ground understanding of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can they be better motivated?&lt;/p&gt;&#xA;" OwnerUserId="5514" LastEditorUserId="3272" LastEditDate="2015-08-23T15:08:41.423" LastActivityDate="2015-08-24T14:37:48.883" Title="Teaching and motivating the use of Eigenvectors" Tags="|examples|concept-motivation|linear-algebra|applied-mathematics|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="8537" PostTypeId="2" ParentId="8536" CreationDate="2015-08-19T17:11:35.020" Score="4" Body="&lt;p&gt;I personally found &lt;a href=&quot;https://en.wikipedia.org/wiki/Principal_component_analysis&quot; rel=&quot;nofollow&quot;&gt;Principal Component Analysis&lt;/a&gt; to be helpful in explaining the use of eigenvectors and eigenvalues. Jeff Jauregui at Union College wrote a great article on PCA that I've used in my Linear Algebra class. (The bird example is pure pedagogical gold.) You can find it here: &lt;a href=&quot;http://www.math.union.edu/~jaureguj/PCA.pdf&quot; rel=&quot;nofollow&quot;&gt;www.math.union.edu/~jaureguj/PCA.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: Our curriculum contains a decent amount of statistics early on, so my students breezed over the necessary stats knowledge.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-08-19T17:11:35.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8538" PostTypeId="1" CreationDate="2015-08-19T18:23:05.447" Score="10" ViewCount="266" Body="&lt;p&gt;I am intending to teach a lesson covering some topic related to &quot;Probability via Geometry&quot; and, if possible, I would appreciate references or materials (or good ideas) that can help me. The target students for this lesson are between 14 and 17 years old, so I would like to talk about basic results, presented as intuitively as possible, and for which any proofs can be understood by the audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The subject of this class is described as: &quot;strategies to teach probability concepts via geometry.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Does anyone have references, materials, or other ideas about teaching probability via geometry for students age 14-17?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5515" LastEditorUserId="262" LastEditDate="2015-08-19T19:41:24.583" LastActivityDate="2015-09-18T23:33:17.127" Title="Lesson-planning: Teaching probability concepts via geometry" Tags="|secondary-education|reference-request|concept-motivation|geometry|probability|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="8539" PostTypeId="2" ParentId="8536" CreationDate="2015-08-19T23:46:55.930" Score="2" Body="&lt;p&gt;I find that the explanations in Treil's &lt;a href=&quot;http://www.math.brown.edu/~treil/papers/LADW/LADW.html&quot; rel=&quot;nofollow&quot;&gt;&quot;Linear Algebra done Wrong&quot;&lt;/a&gt; on all this are the clearest I've seen. But that could mean restructuring quite a bit of the rest of the course.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-19T23:46:55.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8541" PostTypeId="1" AcceptedAnswerId="9676" CreationDate="2015-08-20T11:59:32.800" Score="7" ViewCount="189" Body="&lt;p&gt;I am teaching mathematics at a Danish Business College, and some of my classes belong to the international IBB-programme. However, the college has no English textbook material for my subject, so I have to find it myself. Otherwise I will have to keep the lessons in English but using a Danish textbook which I find far from ideal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This autumn, I must teach them:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Percentage, index numbers&lt;/li&gt;&#xA;&lt;li&gt;The base terminoligy about functions such as vertical line test, domains and codomains, graphing by hand and interpreting a graph visually, zeros and variation of signs&lt;/li&gt;&#xA;&lt;li&gt;Linear equations, linear functions and piecewise linear functions&lt;/li&gt;&#xA;&lt;li&gt;Descriptive statistics for numerical data, discrete and continuous. This should include mean, variance, bar charts, histograms, ogives, and box- and whiskers plots&lt;/li&gt;&#xA;&lt;li&gt;Linear and exponential models and regressions&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Can someone help me determining a strategy for finding such material? Or better yet, does someone have an online reference to some material I could use?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found a nice one on &lt;a href=&quot;http://mathrev.redwoods.edu/PreAlgText/&quot;&gt;prealgebra&lt;/a&gt;, but it only covers basic operations with numbers and letters together with simple percentages and graphing linear equations.&lt;/p&gt;&#xA;" OwnerUserId="1434" LastActivityDate="2015-09-27T04:07:43.760" Title="Which free online books cover this curriculum?" Tags="|undergraduate-education|textbooks|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8542" PostTypeId="2" ParentId="8510" CreationDate="2015-08-20T21:09:35.143" Score="2" Body="&lt;p&gt;I used Mathematica in my High School Mathematics classroom, but not primarily as a computer algebra system, more as a programming language that happens to have excellent support for mathematical functions. What I mean by this is that my primary goal was to introduce my students to the logic of algorithms and give them a tool to use in and out of the mathematics classroom, not teach specific mathematical concepts. I did, however, find that once the pupils had the basic understanding of the syntax, they were able to enrich their understanding of topics, even if I had not specifically taught how to do this within Mathematica.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My resources are therefore more focussed on the nuts and bolts of using Mathematica, with some mathematical examples thrown in. You can download them here (and soon from the British STEM Centre e-library):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.dropbox.com/sh/vpevj14bqf0opus/AADOngwTzltbIV7XSsNyuo-ma?dl=0&quot; rel=&quot;nofollow&quot;&gt;https://www.dropbox.com/sh/vpevj14bqf0opus/AADOngwTzltbIV7XSsNyuo-ma?dl=0&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, if you go down this route, I would be very interested to hear how you go about it, and what your results are!&lt;/p&gt;&#xA;" OwnerUserId="3571" LastActivityDate="2015-08-20T21:09:35.143" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8543" PostTypeId="2" ParentId="7528" CreationDate="2015-08-21T16:12:03.077" Score="21" Body="&lt;p&gt;The hexagon at the north pole of Saturn:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/f5KVH.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/f5KVH.jpg&quot; alt=&quot;SaturnHexagon&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Saturn%27s_hexagon&quot; rel=&quot;noreferrer&quot;&gt;It is known that&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;[regular shapes] form in an area of turbulent flow between ... two different rotating fluid bodies with dissimilar speeds.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and this has been proposed as an explanation for the phenomenon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, the Earth could easily fit inside the pole hexagon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added&lt;/strong&gt; (&lt;em&gt;23Sep15&lt;/em&gt;).&#xA;An &lt;a href=&quot;http://www.space.com/30608-mysterious-saturn-hexagon-explained.html&quot; rel=&quot;noreferrer&quot;&gt;article in space.com&lt;/a&gt; cites a new and apparently thorough explanation of&#xA;Saturn's polar hexagon, in &lt;a href=&quot;http://iopscience.iop.org/article/10.1088/2041-8205/806/1/L18&quot; rel=&quot;noreferrer&quot;&gt;The Astrophysical Journal Letters&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Here we present numerical simulations showing that instabilities in shallow jets can equilibrate as meanders closely resembling the observed morphology and phase speed of Saturn's northern Hexagon. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added&lt;/strong&gt; (&lt;em&gt;10Dec16&lt;/em&gt;). New images taken by &lt;a href=&quot;http://www.usatoday.com/story/tech/sciencefair/2016/12/07/saturn-storm-cassini-nasa/95089076/&quot; rel=&quot;noreferrer&quot;&gt;Cassini&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/YFtwG.jpg&quot; width=&quot;400&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-12-11T00:03:01.480" LastActivityDate="2016-12-11T00:03:01.480" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8544" PostTypeId="5" CreationDate="2015-08-22T07:02:18.273" Score="0" Body="&lt;p&gt;This is a special tag that is designed to be used only by the system for questions that have had all of their other tags removed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All questions in the system must have at least one tag. Normally, it is not possible to submit new or edited questions that do not have tags; the system will reject the submission until a valid tag is applied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a few unusual situations that can cause this tag to be applied. When a tag has existed for six months and only been used on one question, and no tag wiki has been created for the tag, the system deletes the tag. When the community decides that a given tag can never be appropriate &amp;#8212; for example, because it is misspelled or is off-topic &amp;#8212; moderators may remove the tag from the site &lt;em&gt;en masse&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When one of these deletions removes the last tag from a question, the system adds the &lt;a href=&quot;/questions/tagged/untagged&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;untagged&amp;#39;&quot; rel=&quot;tag&quot;&gt;untagged&lt;/a&gt; tag to that question so that the &quot;all questions must have at least one tag&quot; rule is not broken.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can help clean up some of these questions! If they're good questions, edit them to remove the &lt;a href=&quot;/questions/tagged/untagged&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;untagged&amp;#39;&quot; rel=&quot;tag&quot;&gt;untagged&lt;/a&gt; tag and add more descriptive tags. If they're not so good, use the usual tools: downvoting, flagging and voting to close/delete.&lt;/p&gt;&#xA;" OwnerUserId="4872" LastEditorUserId="4872" LastEditDate="2015-08-23T14:37:03.007" LastActivityDate="2015-08-23T14:37:03.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8545" PostTypeId="4" CreationDate="2015-08-22T07:02:18.273" Score="0" Body="This is a special tag that is designed to be used only by the system for questions that have had all of their other tags removed. Do not add this tag to existing questions, it is reserved for use by the system to identify posts with no valid tags." OwnerUserId="4872" LastEditorUserId="4872" LastEditDate="2015-08-23T06:04:47.777" LastActivityDate="2015-08-23T06:04:47.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8546" PostTypeId="2" ParentId="8453" CreationDate="2015-08-22T14:35:43.730" Score="2" Body="&lt;p&gt;Mathematics should essentially be fun to do and to read and I think that generally students only love to do those things which interests them. So, one basic (and I think mandatory) component of your teaching should be about making the subject interesting for the students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Presuming that by &lt;strong&gt;Calculus&lt;/strong&gt; you mean &lt;strong&gt;Real Analysis&lt;/strong&gt; I have the following suggestions for you. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;First, let the students feel themselves the importance of a rigorous study of calculus. For this, I think that the introductory chapters of Tom M. Apostol's &lt;a href=&quot;https://www.google.co.in/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=4&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CC0QFjADahUKEwjkhpLM6rzHAhXLjZQKHQxXALI&amp;amp;url=http%3A%2F%2Fwww.flipkart.com%2Fcalculus-volume-english-2nd%2Fp%2Fitmdytmvcz2qg8xf&amp;amp;ei=D37YVaTDEMub0gSMroGQCw&amp;amp;usg=AFQjCNFgEYZHNTNfUlrNAlrt7-zmZ2q8uA&quot; rel=&quot;nofollow&quot;&gt;Calculus I&lt;/a&gt; and Terence Tao's &lt;a href=&quot;https://www.google.co.in/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=5&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CDYQFjAEahUKEwixtqSK67zHAhWElZQKHcHJAew&amp;amp;url=http%3A%2F%2Fwww.amazon.in%2FAnalysis-v-Texts-Readings-Mathematics%2Fdp%2F8185931941&amp;amp;ei=kX7YVfGpJISr0gTBk4fgDg&amp;amp;usg=AFQjCNELt5xNttd8hIHOHTvUGiXSDdzodg&quot; rel=&quot;nofollow&quot;&gt;Analysis I&lt;/a&gt; are two excellent references. Also I would like to recommend reading the last three sections of Part II of the book &lt;a href=&quot;https://www.google.co.in/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=1&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CB0QFjAAahUKEwj_tuDZ7LzHAhWDjZQKHSD8AW4&amp;amp;url=http%3A%2F%2Fwww.amazon.in%2FPrinceton-Companion-Mathematics-Timothy-Gowers%2Fdp%2F0691118809&amp;amp;ei=RIDYVf-EKoOb0gSg-IfwBg&amp;amp;usg=AFQjCNHwy9gHRlUro8PuSbiq0vr2eElyTA&quot; rel=&quot;nofollow&quot;&gt;The Princeton Companion to Mathematics&lt;/a&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Second, when you will be giving assignments to your students try to be scientific in choosing the problems. To be precise, I don't recommend giving difficult (it's a relative term, though) problems to the beginners, nor do I recommend giving a set of problems which doesn't have any relation(s) among them. Though, it's difficult to express precisely what I mean by the above statements, I will nevertheless try to give an example which is intended to make the matter a bit more clear. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the following problems, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;p&gt;Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $f(x)=0$ for all $x\in\mathbb{Q}$. Show that, $f(x)=0$ for all $x\in\mathbb{R}$.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Let $T=\left\{\dfrac{m}{2^n}\mid m,n\in\mathbb{Z}\right\}$ such that we have  $f(x)=0$ for all $x\in T$. Can we say that $f(x)=0$ for all $x\in\mathbb{R}$?&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Let $S=\left\{\dfrac{a}{2^n}\mid a, n\in\mathbb{Z}\right\}$ ($a$ is a constant) such that we have  $f(x)=0$ for all $x\in S$. Can we say that $f(x)=0$ for all $x\in\mathbb{R}$?&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;These problems have a very curious property. To solve the first problem, you need to show that $\mathbb{Q}$ is dense in $\mathbb{R}$. The second problem you need to show that a certain subset of $\mathbb{Q}$ is dense in $\mathbb{R}$. Finally, the third problem asks you to prove that a subset of the subset of Problem 2 is dense in $\mathbb{R}$. So you can see that the first problem is follows from the second one, whereas the second problem follows from the third one. Can you appreciate the order in which the exercises are arranged ? Can you intuitively grasp what I meant in the above lines ? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Third, make your students familiar with some basic ideas of Mathematical Logic at the very beginning. For example, the concept of conjunction, disjunction, negation etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Finally, I think that for teaching calculus L. V. Tarasov's book &lt;a href=&quot;https://www.google.co.in/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=2&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CCQQFjABahUKEwj5vJ2Y9LzHAhVDjJQKHYBmApQ&amp;amp;url=http%3A%2F%2Ftalegari.wdfiles.com%2Flocal--files%2Fstatic%3Astart%2Fcalculus%2520-%2520basic%2520concepts%2520for%2520high%2520schools%2520-%2520tarasov.pdf&amp;amp;ei=HojYVfm1LsOY0gSAzYmgCQ&amp;amp;usg=AFQjCNFwZ3msqM5YGtmSNU4E_QL6djVC9w&quot; rel=&quot;nofollow&quot;&gt;Calculus: Basic Concepts for High Schools&lt;/a&gt; is an invaluable guide.&lt;/p&gt;&#xA;" OwnerDisplayName="user3319" LastActivityDate="2015-08-22T14:35:43.730" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8549" PostTypeId="1" CreationDate="2015-08-22T21:39:23.040" Score="3" ViewCount="555" Body="&lt;p&gt;I'm planning to get some sources which explain  beautiful problems about quadratic function. I know that there are another kind of functions, but the quadratic function has different applications in some contexts such as maximum and minimum. Also this subject is perfect for high school student. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the mathematical olympiads considered this topic as a part of the contest, that'd be great because in those events the problems are so amazing and help the students improve their skills about math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read some books, but the most of them show  us usual problems. However, there others betters. For instance, &quot;Functions and Graphs&quot;, an english translations of Gelfand's book, which contains a lot of no-routine problems. Those problems could be a model to understand what I'm looking for. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you help me, please?&lt;/p&gt;&#xA;" OwnerUserId="1789" LastEditorUserId="1789" LastEditDate="2015-08-30T22:10:52.203" LastActivityDate="2015-08-31T19:28:53.490" Title="What are sources for non-routine problems involving quadratic functions (in one variable)?" Tags="|mathematical-pedagogy|gifted-students|functions|mathematics-in-daily-life|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8550" PostTypeId="2" ParentId="7528" CreationDate="2015-08-23T09:01:21.453" Score="9" Body="&lt;p&gt;Reggio Emilia Calatrava's railway station follows some very interesting geometrical patterns, building pairs of sinusoids in phase and out of phase&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0Dxy8.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0Dxy8.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EQpLn.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EQpLn.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2015-08-23T09:01:21.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8552" PostTypeId="1" CreationDate="2015-08-23T23:11:26.937" Score="9" ViewCount="444" Body="&lt;p&gt;I am teaching English in Japan and I have a student who speaks English well, and to keep up his level, in our weekly lessons would like to learn some subjects related to my degree in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to get across reasons why mathematics can be beautiful and interesting in a way that is accessible to a non-mathematician. Ideally this would help him understand some ideas in mathematics, avoiding the rigour of undergraduate courses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any good resources, books, websites etc., which I could be pointed towards?&#xA;Or additionally does anyone have any good topics which I could look into?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One topic I have found is the development of cryptography, based on The Code Book by Simon Singh &lt;/p&gt;&#xA;" OwnerUserId="5523" LastActivityDate="2022-03-09T09:46:21.690" Title="Teaching mathematics and its charms to non-mathematicians" Tags="|undergraduate-education|mathematical-pedagogy|course-design|mathematical-discourse|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8553" PostTypeId="2" ParentId="8552" CreationDate="2015-08-23T23:40:19.503" Score="5" Body="&lt;p&gt;Perhaps (depending on background) give them one of the books by William Dunham, like &quot;Journey through Genius&quot; (Dover, 1990), &quot;The Mathematical Universe&quot; (Wiley, 1997), there are others. He won several awards for expository writing on mathematics, Aigner et al's &quot;Proofs from THE BOOK&quot; (Springer, 2003) is more demanding, but a lot of fun.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-23T23:40:19.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8554" PostTypeId="2" ParentId="8536" CreationDate="2015-08-24T10:22:51.853" Score="3" Body="&lt;p&gt;Principal motivation behind eigenvectors is, of course, eigenspaces. And eigenspaces, in turn, are an important case of invariant subspaces and one of few constructs that we can make of a linear operator using &lt;em&gt;pure linear algebra&lt;/em&gt; (without Euclidean norm, coordinates etc.). In some cases eigenspaces &lt;em&gt;classify&lt;/em&gt; invariant subspaces. Suppose we don’t know anything about an abstract vector space. What is the only way to specify a linear operator? Multiplication by a fixed scalar, i.e.&#xA; $$ v\mapsto \lambda v\,, $$&#xA;of course. And finding subspaces where some not-so-simple operator has such form shows how to describe an operator using subspaces and scalars. When linear operator is diagonalizable (i.e. hasn’t Jordan blocks, that is a general position), it provides &lt;em&gt;complete&lt;/em&gt; description of the operator, albeit over complex numbers (or another algebraically closed field).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students should know that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;An eigenvector multiplied by a non-zero scalar is an eigenvector again.&lt;/li&gt;&#xA;&lt;li&gt;A linear combination of eigenvectors with the same eigenvalue is an eigenvector again (if non-zero), with the same eigenvalue – that’s how eigenspaces arise.&lt;/li&gt;&#xA;&lt;li&gt;Linear span of some set of eigenvectors (or direct sum of eigenspaces) produces an invariant subspace, but not necessarily an eigenspace.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is an archetypical situation that reproduces, to some extent, many times in algebra: decomposition to elementary structural blocks.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Some examples requested by the original poster.&#xA;Let, over some &lt;em&gt;F&lt;/em&gt;,&#xA;$$ A = \begin{pmatrix}&#xA;0 &amp;amp;&amp;amp; 1 &amp;amp;&amp;amp; 0 \\&#xA;1 &amp;amp;&amp;amp; 0 &amp;amp;&amp;amp; 0 \\&#xA;0 &amp;amp;&amp;amp; 0 &amp;amp;&amp;amp; 1&#xA;\end{pmatrix}. $$&#xA;The simplest choice of eigenvectors of $A$ is:&#xA;$$&#xA;\begin{pmatrix}1\\1\\0\end{pmatrix},&#xA;\begin{pmatrix}0\\0\\1\end{pmatrix}\ \text{for }\lambda=1;&#xA;\quad&#xA;\begin{pmatrix}1\\-1\\0\end{pmatrix}\ \text{for }\lambda=-1.&#xA;$$&#xA;Then an arbitrary linear combination of two $\lambda=1$ eigenvectors, namely&#xA;$$\begin{pmatrix}α\\α\\β\end{pmatrix},$$&#xA;describes the $\lambda=1$ eigenspace of $A$, a two-dimensional subspace of &lt;em&gt;F&lt;/em&gt;&lt;sup&gt;3&lt;/sup&gt;. Its non-zero elements are all $\lambda=1$ eigenvectors of $A$.&lt;br&gt;&#xA;An arbitrary linear combination of the first and the third eigenvectors, namely&#xA;$$\begin{pmatrix}α+γ\\α-γ\\0\end{pmatrix},$$&#xA;describes an $A$-invariant subspace, but not an eigenspace of $A$ (assuming 2 ≠ 0) because “1” and “−1” eigenvalues are mixed.&lt;/p&gt;&#xA;" OwnerUserId="3272" LastEditorUserId="3272" LastEditDate="2015-08-24T14:37:48.883" LastActivityDate="2015-08-24T14:37:48.883" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8556" PostTypeId="2" ParentId="720" CreationDate="2015-08-24T20:36:08.400" Score="2" Body="&lt;p&gt;Indeed, it might be a problem in countries where there is rarely any native literature. Because in such countries English competes against other transnationial languages (Spanish in Hispanic America, Arabic in Islamic world, French and Swahili in parts of Africa, Russian in some post-Soviet states, …), not much against native languages of local population. Situation is not very different from the one in France, Germany, Spain etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When to encourage? Of course, only after acquiring sufficient ability to read in English through some effort lo &lt;em&gt;learn&lt;/em&gt; the language. There are some English textbooks with mathematics-related texts, possibly even &lt;a href=&quot;http://simple.wikipedia.org&quot; rel=&quot;nofollow&quot;&gt;“Simple English Wikipedia”&lt;/a&gt;… but here is not &lt;em&gt;English&lt;/em&gt; educators’ site, indeed. Starting to learn a language from reading professional-level scientific texts wouldn’t be a good idea. IMHO any person with some pretence to be educated must be able to read in more than one language (in two with advanced level, or in one with advanced and in yet two with intermediate, maybe). Given English isn’t a difficult language, students shouldn’t necessarily be exposed to it very early. Maybe there should be more prioritized foreign languages. Either due to regional circumstances (see the first paragraph), or just because learning well-structured languages (which English doesn’t belong to) may improve thinking skills. But if a student is aimed to a world-class research, then almost certainly s/he will ultimately learn English to extent of understanding scientific texts.&lt;/p&gt;&#xA;" OwnerUserId="3272" LastEditorUserId="3272" LastEditDate="2015-08-25T07:50:01.807" LastActivityDate="2015-08-25T07:50:01.807" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8557" PostTypeId="2" ParentId="8422" CreationDate="2015-08-25T00:52:09.567" Score="-3" Body="&lt;p&gt;Mathematics is a language, with it's own rules. Symbols play a role, sure. Just like technical terms play a role in any specialized area like engineering or antropology. But in the end, we want to transmit &lt;em&gt;understanding&lt;/em&gt;, and that requires explanations. Those are expressed in natural language, not just symbols.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use symbols where words are too imprecise or long (write $\frac{\mathrm{d}}{\mathrm{d} z}$ for derivatives, don't use the quaint terms e.g. Cardan used to describe his equations); use words as far as possible, as you are talking to &lt;em&gt;people&lt;/em&gt;, not some automatic proof checking system. By the way, check out some of the &quot;automatically checked proofs&quot;, and ask yourself if you understand something, or if you wanted to learn them that way.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-25T00:52:09.567" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8558" PostTypeId="1" CreationDate="2015-08-25T06:58:57.787" Score="8" ViewCount="202" Body="&lt;p&gt;I'm using the word constructivist as it is used in &lt;a href=&quot;http://dx.doi.org/10.3998/jsais.11880084.0001.104&quot;&gt;this&lt;/a&gt; paper, not in the sense used in logic. The abstract should be sufficient to understand at least roughly what the model is. The important part, more or less, is that students construct their own questions and pursue them, rather than being dictated and tested upon a standardized curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anecdote. When I was in college, I took a course on videogame culture. Many of the lectures consisted of a discussion of relevant current events, usually involving most of the audience, but pushed along by the professor when things stood still. Out of class, we wrote papers about topics of our choice, and there was a community blog that we posted on about other relevant issues. We were required to read, post, and comment in certain quotas every week, but many students became very engaged and posted more. I learned a lot in this class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been fantasizing for some time about conducting a mathematics course with a similar approach. In my fantasy, my students post questions and answers to our own private StackExchange, exploring their own ideas about pre-calculus algebra in a mathematically meaningful way. One student makes up a family of polynomials and explores its properties in a series of well-presented questions, inspiring others to write answers, as well as further questions expanding on his. One student derives a cubic formula. Gosh, while we're at it, maybe another student asks what would happen if commutativity was dropped, stumbling inadvertently into some group theory. During face-to-face meetings, we look over everyone's work as a group, highlighting the best progress and supporting those who are stuck. It's a pretty rad dream, alright. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thing is, I think this could even almost work with upperclassmen math majors, given the right tweaking and incentives. But what I would really love is to find a way to bring a constructivist learning model to lower level coursework, since many of the students are so unengaged by the conventional style. The main problem is that I suspect many of these students have insufficient skill to participate, despite the course being college level. I wonder if there is a way to adapt the model for this purpose. Could a partial integration solve this issue? Would it be impossible to achieve the literacy required to pass a traditional final exam? &lt;/p&gt;&#xA;" OwnerUserId="263" LastActivityDate="2015-09-11T03:00:15.647" Title="Can a constructivist learning model be applied to online lower division math courses?" Tags="|mathematical-pedagogy|online-instruction|precalculus|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8559" PostTypeId="2" ParentId="7298" CreationDate="2015-08-27T12:32:25.397" Score="0" Body="&lt;p&gt;I'd try to form the groups working on the problem &lt;em&gt;before&lt;/em&gt; they get to work, so nobody is in &quot;I've solved this already, you have to...&quot; mode. Office hours/class is ideal for this. Might even try to form different groups each time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is crucial to learn to work together with a random group of people, as that will be the situation they'll find themselves in as a matter of course during their future professional life. This is hardest for students in more technical oriented fields (engineering around here).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-08-27T12:32:25.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8560" PostTypeId="2" ParentId="8534" CreationDate="2015-08-27T23:54:34.243" Score="5" Body="&lt;p&gt;Good topic.&lt;/p&gt;&#xA;&lt;p&gt;Let me say something about it. This year I've been working in my thesis which involves in part an answer for your question. In my work, some approach are posed, some of them are related to recognize a good problem. Although, some scholars say this topics is complex and controversial, in Mathematics Education there are some frameworks which propose theories about how a good problem could be understood.&lt;/p&gt;&#xA;&lt;p&gt;For instance, you might read these papers:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Crespo, S., &amp;amp; Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. &lt;em&gt;Journal of Mathematics Teacher Education&lt;/em&gt;, 11(5), 395-415. (&lt;a href=&quot;https://link.springer.com/article/10.1007/s10857-008-9081-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;Kontorovich, I. (2012). What makes an interesting mathematical problem? A perception analysis of 22 adult participants of the competition&#xA;movement. In B. Roesken &amp;amp; M. Casper (Eds.), &lt;em&gt;Proceedings of the 17th MAVI (Mathematical Views) Conference&lt;/em&gt; (pp. 129-139). Bochum: Germany.&#xA;(&lt;a href=&quot;http://www.ruhr-uni-bochum.de/imperia/md/content/mathematik/Roesken/paper_kontorovich_mavi.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1789" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-07-23T11:59:05.893" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8561" PostTypeId="2" ParentId="3983" CreationDate="2015-08-28T07:05:01.303" Score="1" Body="&lt;p&gt;Many other answers mentioned ODE, but the discrete version seems simpler to introduce at an early stage. One could use system of sequences to show a magic trick and then explain it with eigenvectors, it might give both intuition and motivation. Let me be more specific.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example problem&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we have two sequences $(u_n)$ and $(v_n)$ such that for all $n\ge0$:&#xA;$$\begin{cases} u_{n+1}=3 u_n+v_n \\ v_{n+1} = 2u_n+2 v_n\end{cases}$$&#xA;( if needed insert any modelling motivation, such as evolution of a population of insects with young ones and adults, and adjust the parameters to make it somewhat realistic.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How does the sequences $(u_n)$ and $(v_n)$ evolve? Can we give exact expressions for them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Side note:&lt;/em&gt; before going on, at this point I would discuss what can easily be said without computing anything: if $u_0$ and $v_0$ are positive then the sequences are increasing and tend to $\infty$ for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Magic trick&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us define for all $n\in\mathbb{N}$: $a_n=2 u_n+v_n$ and $b_n=u_n-v_n$. Then observe that&#xA;$$a_{n+1} = 2 u_{n+1} + v_{n+1} = 8 u_n+4 v_n = 4 a_n$$&#xA;and similarly $b_{n+1}=b_n$. So, $a_n=4^n (2u_0+v_0)$ and $b_n=u_0-v_0$. The two recursions have been decoupled! Then by solving a system we get&#xA;$$\begin{cases}&#xA;  u_n = 4^n(\frac23 u_0+\frac13 v_0) + \frac13(u_0-v_0)\\&#xA;  v_n = 4^n(\frac23 u_0+\frac13 v_0) -\frac23(u_0-v_0)&#xA;\end{cases}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can now answer far more questions on the system easily (for which values of $u_0,v_0$ does $u_n$ go to infinity? Does $u_n$ grow much faster than $v_n$? etc. - one can ask this questions beforehand for more impact) But to students it should look like a magic trick: how could one think of introducing precisely $(a_n)$ and $(b_n)$ as above?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;To the matter&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then one should be in a good shape to explain eigenvectors and eigenvalues, by rewriting the system as a vectorial sequence with a recursive property of the form $U_{n+1} = A U_n$ where $A$ is a matrix. The case of diagonal matrices is easy, the whole point of eigenvector being (at this point) to diagonalize a matrix. Alternatively, one can dissect the trick, and look at how one should choose the coefficients in $(a_n)$ and $(b_n)$ to make it work. Then eigenvectors and eigenvalues show up (but one has to introduce things in matrix form at some point, of course).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-08-28T07:05:01.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8563" PostTypeId="2" ParentId="8538" CreationDate="2015-08-28T09:53:02.970" Score="7" Body="&lt;p&gt;I see at least one good topic for this: &lt;strong&gt;Monte-Carlo algorithm to compute area or volume&lt;/strong&gt;. It has a strong geometric side, is definitely probabilistic, only involves simple concepts, and is even relevant to real-life (of mathematicians).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine we want to compute the area of a region in the plane, say a oddly-shaped pool in a fancy California hotel. Problem is, we are lazy and currently staying at the 20th floor's balcony of our room.. Good news is, we have a very large number of beers available in the minibar, we know the area of the rectangular garden where the pools lies and the pool is currently closed so we do not risk harming people in what follows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let simply drop randomly beer cans in the garden and count the amount of them that fall in the pool. If we divide this by the number of cans which did not fell out of the garden altogether, assuming we dropped them somewhat uniformly, we should get a good estimate of the ratio between the areas of the pool and the garden. Then we can deduce the former by multiplying the later by the above ratio.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I phrased it in a quite unscientific context, but the point is that this kind of method turns out to be indeed useful in applied maths to compute areas, or more generally higher-dimensional volumes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this method becomes unusable in high dimensions because irregular shape tend to occupy an exponentially small fraction of the least cube they are contained in. There are other methods available then, also using probabilities (randomly bouncing billiards, hit and run) which are implemented in computer to obtain volume estimates. In general, it is a hard problem to estimate the volume of a convex set in $\mathbb{R}^n$ given by linear constraints or as convex hulls of a given set of points, but I guess we are getting aside your current issue here.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-08-28T09:53:02.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8564" PostTypeId="2" ParentId="8538" CreationDate="2015-08-28T13:34:18.520" Score="6" Body="&lt;p&gt;Here are some examples, many of them gleaned from this &lt;a href=&quot;http://www.slideshare.net/MarkRyder1/geometry-unit-108&quot;&gt;link&lt;/a&gt;. Note that this link has slides of an entire unit and would probably be very useful.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Find the probability of landing on different colors in a spinner.  To make it interesting, use different sizes for the different colors.  Probability is based on central angle size. Students can also design spinners given different probabilities.&lt;/li&gt;&#xA;&lt;li&gt;Given a target and an arrow that landed on a target, find the probability that the arrow landed in a certain section.  This probability would be based on area of concentric circles. This example is a part of a &lt;a href=&quot;https://www.youtube.com/watch?v=5pS2_e_0nY0&quot;&gt;youtube video&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Find the probability that three points will be collinear given a picture with points: some on the same line, some outside the line.&lt;/li&gt;&#xA;&lt;li&gt;Given a rectangle with different shapes inside, find the probability that a point chosen will be inside a given shape.  Probability is based on angle size.&lt;/li&gt;&#xA;&lt;li&gt;Represent a non-geometric probability problem geometrically. For example, find the probability that a traffic light will be green when you get to the corner.  Represent the times  as a line segment divided into smaller parts.  Each part has a length corresponding to the number of seconds the light will be that particular color. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-08-28T13:34:18.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8566" PostTypeId="2" ParentId="3983" CreationDate="2015-08-28T15:48:02.963" Score="3" Body="&lt;p&gt;Familiar, everyday applications of eigenpairs abound:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for young musicians:  the natural frequencies of a musical instrument string are eigenvalues (actually they're the squares, but, for teaching purposes...).   The vibration modes are the eigenvectors.   The lowest eigenvalue for the string is the fundamental frequency, the higher values are harmonics.   When plucked, the string will vibrate at a superposition of the fundamental and several harmonics - if you want the fundamental to be dominant, pluck the string at the mid-point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for destructive teenagers:  the compressive force at which something like a plastic ruler or a thin sword-blade buckles is an eigenvalue, the shape into which it buckles is an eigenvector (usually only the lowest value is seen).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for undergraduate engineers:  principal stresses are the eigenvalues of the stress state, principal directions are the eigenvectors.&lt;/p&gt;&#xA;" OwnerUserId="5541" LastActivityDate="2015-08-28T15:48:02.963" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8567" PostTypeId="1" CreationDate="2015-08-28T21:23:09.823" Score="2" ViewCount="100" Body="&lt;p&gt;How could one explain that $(-1)(-1) = 1$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to avoid arbitrary rules. I suggested to my student to think of it as a reflexion around the 0. putting emphasis on the idea that if you reflect twice you get to the point where you started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not so clear to his mind. So I ask for other approaches on the subject.&lt;/p&gt;&#xA;" OwnerUserId="5538" LastActivityDate="2015-08-28T21:23:09.823" Title="How can one explain that $(-1) (-1) = 1$" Tags="|algebra|" AnswerCount="0" CommentCount="1" ClosedDate="2015-08-29T12:52:58.387" ContentLicense="CC BY-SA 3.0" />
  <row Id="8568" PostTypeId="1" CreationDate="2015-08-28T23:13:44.933" Score="9" ViewCount="752" Body="&lt;p&gt;I'm going to be starting teaching a course called algebra COE, which is for students who didn't pass the required state algebra exam to graduate and are now seniors, to do spaced-out exam-like extended problems after extensive support. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't want to start the class out with &quot;getting down to business&quot; because I want the students to feel comfortable in the class, with me and with each other. The &quot;getting down to business&quot; will happen during the second week of class. Therefore, I'd like to start out with a class collaboration to solve a &quot;fun&quot; problem. (There are 5 students in the class)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, I don't want to start out with a problem that feels too contrived, or too much like &quot;school math&quot; problems. They have clearly been turned off from &quot;school math.&quot; I want one or some that feel more like they are doing a puzzle, yet still engage algebra-related skills and open up a discussion about problem-solving as a process and skill that can be honed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some problems I have considered, yet I believe are too &quot;math-feeling&quot;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The exponential chessboard and rice problem &lt;/li&gt;&#xA;&lt;li&gt;How many squares are there on the chessboard? (note: more than 64)&lt;/li&gt;&#xA;&lt;li&gt;The &quot;lockers&quot; problem&lt;/li&gt;&#xA;&lt;li&gt;The &lt;a href=&quot;https://itunes.apple.com/us/app/ooops/id467564672?mt=8&quot;&gt;ooops game&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Any suggestions?&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2015-09-11T03:04:54.737" Title="Problem-Solving Activities that Engage Low-Level Students?" Tags="|algebra|student-motivation|problem-solving|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8569" PostTypeId="2" ParentId="8552" CreationDate="2015-08-28T23:44:59.443" Score="3" Body="&lt;p&gt;I'm afraid that this might not be of use to you, because it's in Hebrew, but this question reminded me of the really nice &lt;a href=&quot;https://gilkalai.wordpress.com/2010/02/24/the-beauty-of-mathematics/&quot; rel=&quot;nofollow&quot;&gt;Beauty of Mathematics&lt;/a&gt; course that &lt;a href=&quot;https://en.wikipedia.org/wiki/Gil_Kalai&quot; rel=&quot;nofollow&quot;&gt;Gil Kalai&lt;/a&gt; organized at the Hebrew University in Jerusalem for non-science undergraduates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The slides from the course (in Hebrew) are available &lt;a href=&quot;https://gilkalai.wordpress.com/%D7%99%D7%95%D7%A4%D7%99%D7%94-%D7%A9%D7%9C-%D7%94%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94/&quot; rel=&quot;nofollow&quot;&gt;on his blog&lt;/a&gt; and they include various subjects such as: Euclid's proof of the infinitude of primes, a basic overview of the concept of symmetric tilings, examples of games that can be fully solved (e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Nim&quot; rel=&quot;nofollow&quot;&gt;Nim&lt;/a&gt;), n-dimensional geometry, Cantor's diagonal argument about the cardinality of infinite sets, and others.&#xA;The topics were mostly presented in a non-rigorous, but very engaging, introductory manner.&lt;/p&gt;&#xA;" OwnerUserId="688" LastActivityDate="2015-08-28T23:44:59.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8570" PostTypeId="2" ParentId="7821" CreationDate="2015-08-29T13:53:09.113" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ceBJy.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ceBJy.jpg&quot; alt=&quot;Maths Trainer&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.mathsisfun.com/flash.php?path=/numbers/images/math-trainer.swf?m=Multiply&amp;amp;w=960&amp;amp;h=750&amp;amp;col=%23FFFFFF&amp;amp;title=Math+Trainer+-+Multiplication&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.mathsisfun.com/flash.php?path=/numbers/images/math-trainer.swf?m=Multiply&amp;amp;w=960&amp;amp;h=750&amp;amp;col=%23FFFFFF&amp;amp;title=Math+Trainer+-+Multiplication&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you make a mistake, it will flash the correct answer for a second and wait for you to put the correct answer in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It will then give you a summary of how well you did and how many right and wrong. &#xA;Really helps with memorising less multiplications as it drills $5\times3=3\times5$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I will stress that it is important that the kids understand what multipying means &quot;groups of&quot; and that they can actually answer worded problems and solve unseen questions with multiplcation (something not so repetitive). It is too late for them to learn this in high school.&lt;/p&gt;&#xA;" OwnerUserId="5544" LastEditorUserId="5544" LastEditDate="2015-08-30T02:56:31.277" LastActivityDate="2015-08-30T02:56:31.277" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8572" PostTypeId="2" ParentId="5648" CreationDate="2015-08-29T14:25:30.877" Score="2" Body="&lt;p&gt;I just realised that this solution is similar to NiloCK but it is different.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Teacher: With these 6 pens if I put 2 in each bag how many bags will I have?&#xA;Students: 3 &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teacher: With these 6 pens if I put 1 in each bag how many bags will I have?&#xA;Students: 6&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teacher: With these 6 pens if I put half a pen in each bag how many bags will I have?&#xA;Students: 12&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teacher: With these 6 pens if I put 0 in each bag how many bags will I have?&#xA;Students: You can't do it!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(YAY)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;BUT some will say&#xA;   Students: infinite!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teacher: But i do this a huge number of times and keep going on and on and on, will I ever FINISH grouping the 6 pens?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... and then explain to them how infinite is a concept and not a number.&#xA;.. in senior classes I would link it to graphs and asymptotes and why some students believe that dividing by zero is infinity, and then explain why it &lt;em&gt;tends&lt;/em&gt; to infinity&lt;/p&gt;&#xA;" OwnerUserId="5544" LastActivityDate="2015-08-29T14:25:30.877" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8573" PostTypeId="2" ParentId="8568" CreationDate="2015-08-29T15:12:25.480" Score="1" Body="&lt;p&gt;What about the Monty Hall Problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=mhlc7peGlGg&quot; rel=&quot;nofollow&quot;&gt;https://www.youtube.com/watch?v=mhlc7peGlGg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not sure if you wanted an algebra related one tho. &lt;/p&gt;&#xA;" OwnerUserId="5544" LastActivityDate="2015-08-29T15:12:25.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8574" PostTypeId="2" ParentId="8568" CreationDate="2015-08-29T17:10:02.503" Score="1" Body="&lt;p&gt;Eight adults and two kids want to cross a river. There is a boat that can hold one adult or two kids, no more. Can they all cross? How? Extend to &lt;em&gt;n&lt;/em&gt; adults.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use figurines (Lego, Playmobil, ?) or coins to model the situation. &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-08-29T17:10:02.503" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8575" PostTypeId="1" AcceptedAnswerId="9621" CreationDate="2015-08-30T07:14:41.907" Score="7" ViewCount="292" Body="&lt;p&gt;No doubt that complex analysis is a tremendously useful with plenty of applications in engineering and physics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Common raw applications of complex analysis includes: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;evaluation of ordinary and partial differential equations,  &lt;/li&gt;&#xA;&lt;li&gt;fluid dynamics,  &lt;/li&gt;&#xA;&lt;li&gt;AC circuits,  &lt;/li&gt;&#xA;&lt;li&gt;quantum mechanics,  &lt;/li&gt;&#xA;&lt;li&gt;electromagnetism and acoustics via conformal mapping,  &lt;/li&gt;&#xA;&lt;li&gt;dispersion relations in photonics via Kramer Kronig relation,  &lt;/li&gt;&#xA;&lt;li&gt;Z transform and indeed all transform techniques which give rise to signal processing and control theory&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But I quickly find that the applications mentioned above is more towards an analytical approach to physical problems that uses only tiny subset of techniques and ideas in complex analysis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, one could get away in basic quantum mechanics and ODE by merely knowing the fact $\sqrt{-1} = i$. Whereas one could get away with conformal mapping with a good grasp of sets and mappings in complex plane without delving into any integration techniques. Complex analysis yields tremendous intuition for PDEs and is used to solve the Laplace equation $\triangle F = 0$, but in reality most problems are solved using meshes and numerical techniques. Actually everything mentioned above perhaps with the exception of fluid dynamics you can get away without knowing much of complex analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet every physics and engineering students must take at least one semester of complex analysis which maybe quickly forgotten due to disuse. For example, engineers seldom invoke residue theory when computing the residue for inverting the laplace transform, favoring more heuristic approach such as &quot;the covering&quot; method. Indeed, inverse fourier and laplace transform via Bromwich integral (a complex integral) is never taught in engineering. Where although complex analysis has tremendous usage in quantum field theory and string theory, the &quot;application&quot; is towards theoretical calculations and proofs such as hand computation of green's function for propagators with emphasis on &quot;hand&quot;.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Can anyone suggest some directions for using complex analysis in ways&#xA;that goes beyond doing mostly one shot, classical physics problems by&#xA;hand?  &lt;/li&gt;&#xA;&lt;li&gt;Are there any references out there that tries to apply results&#xA;in complex analysis in a more computational oriented setting?  &lt;/li&gt;&#xA;&lt;li&gt;If not, I will be happy to know if there are other areas of complex&#xA;analysis not covered in the list I have given above and please&#xA;provide some references to for me look over those applications.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3560" LastActivityDate="2016-11-24T02:34:06.950" Title="Are there more modern or computation oriented applications of complex analysis in science and engineering?" Tags="|reference-request|applied-mathematics|applications|engineering-mathematics|complex-numbers|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8576" PostTypeId="2" ParentId="466" CreationDate="2015-08-30T13:27:02.433" Score="2" Body="&lt;p&gt;For a mathematics students: &lt;em&gt;How to Read and Do Proofs&lt;/em&gt;, by D. Solow.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-08-30T13:27:02.433" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8577" PostTypeId="1" AcceptedAnswerId="8580" CreationDate="2015-08-31T10:08:05.643" Score="8" ViewCount="219" Body="&lt;p&gt;I am looking for some non-complicated second order differential equations to illustrate certain techniques for control engineering. It doesn’t matter if the differential equations are linear or non-linear; also, simplified assumptions are fine. I’d like to have a wider pool of examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware of the standard example of the spring pendulum, and also the rolling motion of a plane simplified defined by:&#xA;$$M(t) = k \ \delta_A(t)$$&#xA;and&#xA;$$M(t) = I_{xx}\ \ddot \Phi(t)$$&#xA;with $M(t)$ being the moment, $\Phi(t)$ being the roll angle acceleration and $I_{xx}$ being the inertia torque.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Can someone direct me to more realistic examples of this nature, or is there a reference or text containing a list of this sort somewhere?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5548" LastEditorUserId="262" LastEditDate="2015-08-31T22:38:58.117" LastActivityDate="2018-04-08T16:41:46.510" Title="Second Order Differential Equation Example Request" Tags="|undergraduate-education|reference-request|examples|differential-equations|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8578" PostTypeId="2" ParentId="5916" CreationDate="2015-08-31T12:36:11.187" Score="2" Body="&lt;p&gt;Kids are ready to accept the rules in a computer game&#xA;and you can use it.&lt;/p&gt;&#xA;&lt;p&gt;Check &lt;a href=&quot;http://www.math.psu.edu/petrunin/fxd/car.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;some examples&lt;/a&gt; I made using so-called &lt;em&gt;assignments&lt;/em&gt; in &lt;a href=&quot;http://car.rene-grothmann.de/doc_en/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;C.a.R.&lt;/a&gt;. (I asked the students to take a screenshot and send it to me by e-mail, but there should be better solutions.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Postscript.&lt;/strong&gt; &lt;a href=&quot;https://www.euclidea.xyz/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclidea&lt;/a&gt; is yet better.&lt;/p&gt;&#xA;" OwnerUserId="3268" LastEditorUserId="3268" LastEditDate="2022-06-08T18:28:20.460" LastActivityDate="2022-06-08T18:28:20.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8579" PostTypeId="2" ParentId="8549" CreationDate="2015-08-31T19:28:53.490" Score="4" Body="&lt;p&gt;I left a couple of problems (and their source) in an earlier &lt;a href=&quot;https://matheducators.stackexchange.com/questions/8549/what-are-sources-for-non-routine-problems-involving-quadratic-functions-in-one#comment19206_8549&quot;&gt;&lt;strong&gt;comment&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Here are a couple more:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure about the original sources for either of the following, but they should fit some interpretation of &lt;em&gt;non-routine problems involving quadratic functions&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I gave an example in an answer to &lt;em&gt;The philosophy of change of variables&lt;/em&gt; (&lt;a href=&quot;https://math.stackexchange.com/a/226389&quot;&gt;&lt;strong&gt;MSE 226343&lt;/strong&gt;&lt;/a&gt;) that we might as well update for next year:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;$$(x-1)(x-2)(x-3)(x-4) = 2016$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observe $(x-1)(x-4) = x^2 - 5x + 4$ and $(x-2)(x-3) = x^2 - 5x + 6$. If you use a $u$ substitution for the former product, then the problem becomes $u(u+2) = 2016$, which is quadratic; so you can solve for $u$ and then solve for $x$ (though you might pick a nicer number than $2016$ to set this problem up!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An alternative method of solution is to observe symmetry around $x = 3/2$; use a substitution to shift the graph $3/2$ to the left and you end up with a function symmetric about $x= 0$, i.e., even. A similar substitution reduces the problem back to a routine quadratic situation, and, in fact, the substitutions in both of the above approaches end up being equivalent.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Suppose you have $60m$ of fence and want to build a rectangular fence that maximizes the enclosed area. What are its dimensions? &lt;strong&gt;Okay&lt;/strong&gt;, you have probably seen this problem enough times for it not to be considered non-routine. But how about a slight variation: Suppose you have access to a very long wall that can serve as one side of the rectangular fence. Again, build a rectangular enclosure that maximizes the enclosed area. Guess the dimensions, and then confirm whether your intuition was correct.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The first problem is, indeed, solved by a square: Each side is $15m$. You can tackle this by labeling adjacent sides as $x$ and $30-x$, so the area is $x(30-x)$; maximizing this downward facing parabola amounts to findings its vertex, which is why this is a quadratic problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An alternate solution, equivalent to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means#Geometric_interpretation&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;AM-GM inequality&lt;/strong&gt;&lt;/a&gt;: The area of the $15m \times 15m$ rectangular enclosure is $225 m^2$; if we alter the dimensions, it makes use of additive inverses. That is to say, if we change one of the sides to $15+x$, then the adjacent side changes to $15-x$ in order to maintain a perimeter of $60$ (all measured in meters). But $(15+x)(15-x) = 225- x^2$ is clearly maximized when $x=0$ (i.e., in the aforementioned square case). In particular, a change to one of the sides by $x$ will decrease the enclosed area by $x^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the modified version, you now have access to the wall. So you can use all $60m$ to build the remaining three sides of your rectangular closure. Clearly, then, we end up with a square of dimensions $20m \times 20m$ with total area $400 m^2$. Right? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Wrong. Why? Coming up with an &quot;intuitive&quot; explanation for what &lt;em&gt;is&lt;/em&gt; the right answer perhaps qualifies as falling outside of the routine canon around quadratic function problems.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-08-31T19:28:53.490" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="8580" PostTypeId="2" ParentId="8577" CreationDate="2015-09-01T06:57:07.620" Score="8" Body="&lt;p&gt;Some examples from Erwin Kreyszig's &lt;em&gt;Advanced Engineering Mathematics&lt;/em&gt; 7th ed. (John Wiley &amp;amp; Sons, Inc., 1993):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Falling stone&lt;/strong&gt;&lt;br&gt;&#xA;$\frac{\mathrm{d}^2y}{\mathrm{d}t^2}=g$&lt;br&gt;&#xA;where $y$ is the position of the stone and $g$ is the gravitational acceleration&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hanging cable&lt;/strong&gt;&lt;br&gt;&#xA;$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=k\sqrt{1+\frac{\mathrm{d}y}{\mathrm{d}x}^2}$&lt;br&gt;&#xA;where $y$ is the curve of an inextensible flexible homogeneous cable hanging between two fixed points and $k$ is a constant depending on weight&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Vibrating mass on a spring&lt;/strong&gt;&lt;br&gt;&#xA;$m\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+ky=0$&lt;br&gt;&#xA;where $y$ is the displacement of the body from the static equilibrium position, $m$ is the mass, and $k$ is the spring modulus&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Current in an RLC circuit&lt;/strong&gt;&lt;br&gt;&#xA;$L\frac{\mathrm{d}^2i}{\mathrm{d}t^2}+R\frac{\mathrm{d}i}{\mathrm{d}t}+\frac{1}{C}i=\frac{\mathrm{d}v}{\mathrm{d}t}$&lt;br&gt;&#xA;where $i$ is the current, $R$ is the resistance, $L$ is the inductance, $C$ is the capacitance, and $v$ is the voltage of the voltage source&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pendulum&lt;/strong&gt;&lt;br&gt;&#xA;$mL\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}+mg\sin\theta=0$&lt;br&gt;&#xA;where $m$ is the mass of the bob, $g$ is the gravitational acceleration, $\theta$ is the angular displacement measured counterclockwise from the equilibrium position and $L$ is the length of the (massless) rod&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2015-09-01T06:57:07.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8581" PostTypeId="2" ParentId="8568" CreationDate="2015-09-01T07:18:13.607" Score="3" Body="&lt;p&gt;The article titled &amp;quot;Math Teachers’ Circles: Partnerships between Mathematicians and Teachers&amp;quot; by B. Donaldson, M. Nakamaye, K. Umland, and D. White in the December 2014 &lt;em&gt;Notices of the AMS&lt;/em&gt; (pp. 1335-1341) (&lt;a href=&quot;http://www.ams.org/notices/201411/rnoti-p1335.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pdf version&lt;/a&gt;) mentions a nice problem called &amp;quot;Dividing squares.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Shown below are some pictures of a square divided into smaller squares,&#xA;not necessarily of the same size.  (Image taken from the linked file.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CyFSY.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CyFSY.png&quot; alt=&quot;Squares divided into smaller squares&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A few possible questions you can ask:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;For which whole numbers &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is it possible to subdivide a square into &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; smaller squares?&lt;/li&gt;&#xA;&lt;li&gt;For which whole numbers &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is it not possible to subdivide a square into &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; smaller squares?&lt;/li&gt;&#xA;&lt;li&gt;Is it possible to perform the division so that there is a unique square of smallest size?&lt;/li&gt;&#xA;&lt;li&gt;Is it possible to perform the division in such a way that no two squares have the same size?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The problem can lead to discussions on arithmetic sequences.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-09-01T07:18:13.607" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="8582" PostTypeId="2" ParentId="8568" CreationDate="2015-09-01T23:00:33.117" Score="2" Body="&lt;p&gt;The frog jumping puzzle is nice, in that it is very simple but hides some surprising but manageable complexity. Here's a &lt;a href=&quot;http://www.smart-kit.com/s7284/frog-jumping-puzzle/&quot; rel=&quot;nofollow&quot;&gt;&quot;video game&quot; version&lt;/a&gt;. Have the students play with this, and then try to generalize to n male frogs and m female frogs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For another favorite, see the map folding puzzle of Martin Gardner as &lt;a href=&quot;https://books.google.com/books?id=fA3CAgAAQBAJ&amp;amp;pg=PA14&amp;amp;lpg=PA14&amp;amp;dq=map%20folding%20problem%20of%20martin%20gardner&amp;amp;source=bl&amp;amp;ots=QxG5dWws10&amp;amp;sig=6bAOVmSYfDmwEYBu-AbkjMwpM3Q&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CEkQ6AEwCTgKahUKEwiDsc3a99bHAhUI8IAKHZWlAo4#v=onepage&amp;amp;q=map%20folding%20problem%20of%20martin%20gardner&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;problem 29 here&lt;/a&gt;. This is a beautiful way to show the surprisingly delicate complexity arising in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've broken all sorts of posting rules here, but perhaps someone will find better links. Please feel free to modify this post accordingly!&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2015-09-01T23:00:33.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8583" PostTypeId="2" ParentId="5794" CreationDate="2015-09-02T13:24:51.190" Score="5" Body="&lt;p&gt;I'm not really understanding the aversion to using abstract analysis.  You don't need to ruin it for your students and say &quot;by the way, this is a result of Ring-theoretic axioms, which is waaaaaaay beyond the scope of anything you'll ever learn!&quot;  Just show them that (-1)*(1 - 1) = 0 (which they won't dispute), and then distribute the terms, giving you -1 + (-1)^2 = 0, meaning (-1)(-1) = 1.  Every other case of a negative multiplied by a negative reduces to this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you present it like this, it comes across as a neat trick rather than an abstract algebraic result (like when you show why 0/0 is undefined - it's enough to show that it contradicts our assumptions about real numbers, and you don't need to accompany your explanation with a real-life example).  I think trying to make it seem &quot;intuitive&quot; can be distracting, like when people try to reason why an implication being false makes a whole statement true.  Students will usually be able to poke holes in your analogies and find destabilizing examples, or they'll just get confused and blame themselves for not understanding what is likely a silly comparison.  Sometimes math concepts can't be analogized well with things in real life, and trying to force it to work is bad pedagogy.&lt;/p&gt;&#xA;" OwnerUserId="5141" LastActivityDate="2015-09-02T13:24:51.190" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8584" PostTypeId="2" ParentId="1952" CreationDate="2015-09-02T19:16:09.187" Score="3" Body="&lt;p&gt;Everything depends on which axiomatic system you build everything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say, &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Birkhoff gives SAS-similarity condition as an axiom --- no need for Thales.&lt;/li&gt;&#xA;&lt;li&gt;In the classical axiomatic approach, &#xA;Thales' theorem is proved first and &#xA;the theorem about ratios in similar triangles is its easy corollary.&#xA;(One can prove SAS directly, but likely it is better to do with Thales;&#xA;at least the experience of few generations says so.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3268" LastActivityDate="2015-09-02T19:16:09.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8586" PostTypeId="1" CreationDate="2015-09-03T05:15:59.803" Score="26" ViewCount="936" Body="&lt;p&gt;This is something that I felt like was difficult for me in some classes, especially lower division differential equations and linear algebra classes. I know professors want to motivate certain topics but I feel like offering motivation honestly confused me more than helped, and I see this with some students I tutor. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, applications are good. Applications are great. Especially for non pure mathematicians, which is the majority of people taking these classes. However applications should be taught separately from the math, otherwise it can get really confusing. Most concepts in linear algebra and differential equations are honestly really basic straightforward definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, one that really stood out to me were eigenvectors/eigenvalues. I was confused until I stopped listening to all this nonsense about stretching, compressing, rotations, pictures of the Mona Lisa (&lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Mona_Lisa_eigenvector_grid.png/320px-Mona_Lisa_eigenvector_grid.png&quot; rel=&quot;noreferrer&quot;&gt;https://upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Mona_Lisa_eigenvector_grid.png/320px-Mona_Lisa_eigenvector_grid.png&lt;/a&gt;), linear transformations, cute pictures of arrows getting stretched or turned around, differential equations, all of the answers in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/3983/what-is-the-best-way-to-intuitively-explain-what-eigenvectors-and-eigenvalues-ar&quot;&gt;What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?&lt;/a&gt;, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember SO MUCH hoopla about eigenvectors and eigenvalues. To be frank, it confused the hell out of me. Once I ignored the motivation and just accepted &quot;an eigenvalue/eigenvector of a matrix $M$ are a pair $\lambda, \vec{v}$ such that $M\vec{v}=\lambda\vec{v}$&quot;, it got so much easier and clearer. Now I understand what everyone is saying but at the time what professors were &lt;em&gt;saying&lt;/em&gt; seemed so much at odds with what I &lt;em&gt;actually had to do&lt;/em&gt;. Pictures of the Mona Lisa, compression, rotations, etc. have nothing to do with what I had to do, and that is take the determinant of a matrix and solve for the roots. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see this in students sometimes. They vaguely talk about these &quot;motivators&quot; but might not even be able to define what an eigenvalue is!, for example. It's not just linear algebra, but I feel like a lot of professors try to motivate and show connections too much, and it really is at odds with what students have to actually do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess my question is, when is there too much motivation? Is it too much motivation, or just not enough basics?&lt;/p&gt;&#xA;" OwnerDisplayName="user5108" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-06-23T02:01:39.693" Title="Too much motivation?" Tags="|mathematical-pedagogy|concept-motivation|linear-algebra|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="8587" PostTypeId="2" ParentId="8586" CreationDate="2015-09-03T10:25:42.270" Score="10" Body="&lt;p&gt;Let me propose a non-standard distinction between two terms (in the context of teaching):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;An &lt;strong&gt;&lt;em&gt;application&lt;/em&gt;&lt;/strong&gt; is a problem or a task outside the main scope of the course with a solution presented using the present tools. When a teacher gives an application, the &quot;applied problem&quot; is actually solved in front of the students.&lt;/li&gt;&#xA;&lt;li&gt;A &lt;strong&gt;&lt;em&gt;motivation&lt;/em&gt;&lt;/strong&gt; is a problem or a task outside the main scope of the course, which can be solved using the tools of the course but such a solution is not presented.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are, of course, many things between and beyond these two concepts, but let me stick to these to make a point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A motivation is vague and often plays with heuristic ideas, whereas an application actually shows how to use the course material for something else.&#xA;Motivations and applications both present topics that are related to the course, but only applications make the connection explicit and clear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A motivation is easy to give, since there is no need to go to the details and less time is required.&#xA;The teacher doesn't even have to fully understand the motivation!&#xA;A motivation is likely to be confusing, as it pours seemingly unrelated stuff into the students and helps them forget what is a definition and what is a heuristic interpretation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An application actually works (ideally) with the very tools presented in the course, so it both provides a connection to something else and gives an example of the use of the main tools and definitions.&#xA;Applications take more effort to give, but they give more to the students.&#xA;A student feels much more empowered if they can, instead of just listing related problems, actually solve a related problem with their newly found tools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion one should be very sparing with motivations (in the sense defined above).&#xA;Convert, if possible, all motivations to applications or at least give the students a chance to do it themselves.&#xA;If you wish to list several related topics to make the course seem more relevant, try to keep it separate from actual mathematics.&#xA;When preparing lecture material, make a distinction between vague motivations and well explicated applications, and think which ones are more appropriate.&#xA;Ask yourself: &quot;Can my students really see how to connect this other thing to this concept from my course? How much do I need to explain to make them see it?&quot;&#xA;Don't answer too idealistically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your question (now in the standard meaning of &quot;motivation&quot;):&#xA;I would say there is too much motivation if its connection to the course material is not made clear.&#xA;Throwing a bunch of ideas in the air is not very helpful unless you take them (or make the students take them in exercises) and show how to actually work with those ideas and the course material.&#xA;Rotations and arrows and distorted paintings can be useful in teaching eigenvalues if they are properly connected to the core content of the course.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2017-06-22T08:16:12.523" LastActivityDate="2017-06-22T08:16:12.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8588" PostTypeId="2" ParentId="8586" CreationDate="2015-09-03T12:01:32.710" Score="6" Body="&lt;p&gt;One problem with &quot;motivating the subject matter with concrete applications&quot;, specially in first courses, is that the students have no clue (yet) on the &quot;concrete applications&quot; to their main course of study. Explain e.g.somebody in the first year studying chemical engineering that differential equations model the kinetics of chemical reactions, or to a mechanical engineering student that the system of equations models the forces in a bridge, and they won't know what you are talking about. Double confusion: the subject matter of the course and the motivation. Also, when the time later comes to apply said half-understood tools, they will be forgotten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way out is some sort of &quot;just in time&quot; teaching: Teach the tools with their use, hand in hand. That is hard to organize given standard curricula. But it did work very well in some courses I took, which required mathematical tools well outside the standard fare. The rigor and completeness suffered a lot, but as I got interested I filled in much of that later on by self learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other option is to use examples at hand, using concepts they already have from high school or everyday life. Hard to come up with reasonable applications, that don't look contrived (or aiming cannons at flies). You have to be careful to get problems that are easy to understand and to model. A colleague around here had a resounding success with a course for second term students in modelling all sorts of situations mathematically, using a CAS or other tools to solve the equations and focusing on the building of models.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-09-03T12:01:32.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8589" PostTypeId="2" ParentId="8532" CreationDate="2015-09-03T12:10:39.857" Score="0" Body="&lt;p&gt;You could grab some experimental data points, and use those. That way (specially if the data have experimental errors) show that the interpolating polynomial isn't always a good option, and one should go for some sort of curve fitting instead. If you use data that are in some way interesting to the students by themselves, much better. Can even show off e.g. spline interpolation. Having tools at hand to plot the results is a requirement in any case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It isn't the case that much anymore, but reading data out of tables (and interpolating) used to be a critical skill for an engineering student like me.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-09-03T12:10:39.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8590" PostTypeId="2" ParentId="2471" CreationDate="2015-09-03T15:54:42.887" Score="2" Body="&lt;p&gt;I think that there is an important point behind this question, that seems to have been missed by other answer (however interesting they are): in mathematics we use the word &quot;and&quot; sometimes as a logical connector, sometimes (just as in regular English) to mark an enumeration, as in &quot;$\cos$ and $\sin$ are example of trigonometric functions&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is important, and often neglected, to distinguish and allow both usages of the word. Once you make that clear, a lot of trouble (including yours) should disappear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I learned this issue from reading some work by Zoe Mesnil, a French math didactic researcher (you kind find her on the internet, but she mostly writes in French I'm afraid).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-09-03T15:54:42.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8591" PostTypeId="2" ParentId="8195" CreationDate="2015-09-03T15:59:05.063" Score="6" Body="&lt;p&gt;Engineers and physical scientists need to know a lot of maths, even if just to sensibly drive the software they will use in professional practice.   However engineers (whom I teach) are driven by the need to design and make things rather than to understand things.   If I have to pray to the full moon to get something to work, I'll do so and worry about the theory later.   This is, I think, the wrong way round for mathematicians.   So, to motivate engineers you must &lt;em&gt;quickly&lt;/em&gt; relate the mathematics to a real-world application with which they are familiar.   This tends to mean going from the particular to the general, which is again (I think) the wrong way round for mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Things that will make engineering students' eyes glaze over:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;'here are some axioms/definitions - let's see what follows and then&#xA;we'll examine an application'.   This is the wrong way round for engineers and &lt;em&gt;'...and it's applications'&lt;/em&gt; textbooks often fall into this trap.&lt;/li&gt;&#xA;&lt;li&gt;'a tensor is a 'mapping from a vector space to itself' or even&#xA;'something that transforms according to...'.  I introduce tensors as&#xA;'a thing like stress', which they know from Mechanics 1, at least in the uniaxial case.&lt;/li&gt;&#xA;&lt;li&gt;eigenvalues are not 'solutions of the equation&#xA;$A\lambda=v\lambda$'.   They are (the squares of) natural vibration&#xA;frequencies.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;We could write a book about this.   Maybe later!&lt;/p&gt;&#xA;" OwnerUserId="5541" LastActivityDate="2015-09-03T15:59:05.063" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="8592" PostTypeId="2" ParentId="8195" CreationDate="2015-09-03T16:39:37.673" Score="11" Body="&lt;p&gt;I'll address one piece of your question: how to think about proofs in a class for non-majors.  I've tried to approach this by assuming that, whether or not a proof is good for the students, they haven't learned how to read and understand proofs, and can't be expected to do so on their own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So any time I include a proof, I need to not only teach them the proof, but teach them how to think about this particular proof.  Assuming the students are comfortable with calculations, that means only worrying about the proofs that are one conceptual step above the calculations, so that I can explain its value by pointing back to the calculations.  Usually these end up being proofs where either the proof is mostly a worthwhile calculation itself (&quot;this is a good example of the method we just learned...&quot;) or where the proof illustrates the use of an assumption.  If I can't explain the purpose of the proof concretely (&quot;Well, this proof helps us understand why...&quot;) then it's too abstract.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-09-03T16:39:37.673" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="8593" PostTypeId="1" CreationDate="2015-09-03T19:31:14.277" Score="9" ViewCount="263" Body="&lt;p&gt;Our university and department, like most, are required to &quot;assess&quot; our program annually.  In the past our department has done this by e.g. picking one of the learning outcomes for our program and a final exam question in one or two classes that are related to that outcome, and coming up with a numerical rubric to evaluate performance on those exam questions.  But this doesn't really tell us much that we didn't already know, and no matter how carefully chosen, one exam question can be misleading and not representative of an entire program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What successful ways have other mathematics departments found to assess their undergraduate programs and determine whether learning outcomes are being achieved?&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2015-09-05T20:50:44.473" Title="How to assess an undergraduate mathematics program?" Tags="|undergraduate-education|assessment|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="8594" PostTypeId="2" ParentId="8593" CreationDate="2015-09-04T00:16:31.397" Score="0" Body="&lt;p&gt;You declare a number of learning outcomes already (for the program and presumably for the courses too). The learning outcomes for each course should be assessed by the grading (could e.g. track them by looking at the grades in individual questions of midterms/finals, check some other outcomes by results in term papers or such).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-09-04T00:16:31.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="8595" PostTypeId="2" ParentId="8593" CreationDate="2015-09-04T00:20:03.670" Score="7" Body="&lt;p&gt;Two questions from Lynn Arthur Steen are perhaps not addressed frequently,&#xA;but seem important to me:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(1) Are students learning how to use mathematics in other&#xA;  subjects? Do students recognize similar mathematical concepts&#xA;  and methods in different contexts?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(2) Do faculty take responsibility for the quality of students’&#xA;  learning?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;These are from &quot;Asking the Right Questions,&quot; in &lt;em&gt;Supporting Assessment in Undergraduate Mathematics&lt;/em&gt;. &lt;a href=&quot;http://www.maa.org/press/ebooks/supporting-assessment-in-undergraduate-mathematics&quot;&gt;MAA Supporting link&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The MAA (Math Assoc Amer) also published &lt;em&gt;Assessment Practices in Undergraduate Mathematics&lt;/em&gt;.&#xA;&lt;a href=&quot;http://www.maa.org/press/ebooks/assessment-practices-in-undergraduate-mathematics&quot;&gt;MAA Practices link&lt;/a&gt;. The latter publication likely addresses&#xA;your &quot;What successful ways&quot; question, but I am in no position to summarize&#xA;this 292-page book.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-09-04T00:20:03.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9594" PostTypeId="1" CreationDate="2015-09-04T11:07:25.127" Score="5" ViewCount="249" Body="&lt;p&gt;Reciprocal teaching seems to have started as method of teaching how to read difficult texts (with respect to the level of the reader).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has reciprocal teaching been applied to teaching mathematics? Is there are any literature, scientific or otherwise, on this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can treat mathematics as difficult texts, of course, but I would guess that there could be differences between mathematics and other difficult texts.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2016-05-19T08:18:21.933" Title="Research on reciprocal teaching of mathematics?" Tags="|reference-request|education-research|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="9595" PostTypeId="1" CreationDate="2015-09-04T13:38:21.813" Score="2" ViewCount="56" Body="&lt;p&gt;While getting stuff ready for the soon-to-be-here school year, I came across &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Lancaster&quot; rel=&quot;nofollow&quot;&gt;Joe Lancaster&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Monitorial_System&quot; rel=&quot;nofollow&quot;&gt;Monitorial system&lt;/a&gt;, where older students who have mastered certain content areas are charged with teaching the younger students.  In one form or another, the monitorial system is at work at university level, for example TA's and Lab Instructors.  However, at the secondary level I have not seen much, if any, evidence of the monitorial system.  I know that some schools have &quot;student interns&quot; that typically just take care of administrative stuff such as grading or handing out paper.  I am more wondering about older students actually taking over and teaching younger students.  Has anyone had an encounter with this at a secondary level?  Has it been effective?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-09-04T13:38:21.813" Title="Lancasterian/Monitorial system at a secondary level?" Tags="|secondary-education|general-pedagogy|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9597" PostTypeId="2" ParentId="2528" CreationDate="2015-09-04T16:29:34.673" Score="1" Body="&lt;p&gt;Coxeter, in his book titled &lt;strong&gt;Projective Geometry&lt;/strong&gt;, describes a dictionary game called Visch (short for &quot;viscious circle&quot;):  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Point=that which has position but not magnitude&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Position =place occupied by a thing&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Place= part of space...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Space=continuous extension..&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Extension=extent&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Extent=space over which a thing extends&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Space=continuous extension&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word &quot;space&quot; is repeated so we have Vish in Seven. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Coxeter then remarks that the only way to avoid vicious circles is to regard certain primitive concepts being fundamental and left undefined. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When one understands geometry as an axiomatic system with undefined terms, one is ready to think about models of the axiomatic system. A model is an concrete interpretation of the undefined terms where the relationships postulated by the axioms hold true in the concrete setting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Tables, chairs, and beer mugs: &lt;a href=&quot;https://math.stackexchange.com/questions/56603/provenance-of-hilbert-quote-on-table-chair-beer-mug&quot;&gt;https://math.stackexchange.com/questions/56603/provenance-of-hilbert-quote-on-table-chair-beer-mug&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then one can think about properties of the geometry such as consistency, completeness, and categoricalness. In the van Hiele model of mathematics education, this would be operating at the highest level, where  &quot;students understand that definitions are arbitrary and need not actually refer to any concrete realization.&quot; &lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-09-04T16:29:34.673" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9598" PostTypeId="2" ParentId="8593" CreationDate="2015-09-05T01:40:34.230" Score="1" Body="&lt;p&gt;Honestly, if you really want to see how your program is doing you need to rely on some external testing. Perhaps the major field assessment test (MFT), or, the subject math GRE. These are a start, but, I don't think they quite uncover the higher end of undergraduate learning. For example, I took courses on manifold theory, general relativity, fiber bundles, axiomatic classical mechanics and a few other things. None of that would have even registered a blip in those tests, yet, I can tell you to a certainty they were huge in my overall education. Often these assessments are about the bulk of the students, but, what really speaks to the quality of a program is not average student's progress. In my estimation, the bigger question, what are you offering the best students? What are you offering the students whose heart is more with the study of mathematics than the acquisition of a credential? I'm not sure I know how to &quot;assess&quot; that, but, I know it matters.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-09-05T01:40:34.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9599" PostTypeId="2" ParentId="5797" CreationDate="2015-09-05T05:44:18.540" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;This leads me to wonder if it is possible to consider a terrible&#xA;  heresy: Is it possible to imagine mathematics without problems?...&#xA;  What if &quot;conversation&quot; replaced &quot;problem&quot; in the mathematics&#xA;  classroom?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm going to focus on this other question asked here. Personally, in the math courses that I teach (community college remedial algebra, college algebra, and statistics), I never refer to &quot;problems&quot;; I think that a more accurate description is &quot;exercises&quot;. Consider the traditional format of a mathematics presentation:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Definitions&lt;/li&gt;&#xA;&lt;li&gt;Theorems&lt;/li&gt;&#xA;&lt;li&gt;Proofs&lt;/li&gt;&#xA;&lt;li&gt;Applications&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Part of the political problem with the majority of secondary and undergraduate math classes is the emphasis on &quot;making it useful&quot;, i.e., &quot;real world applications&quot;, which is to say that they evolve towards standards and standard exams which only assess the 4th item, applications. The conversation about the definitions, theorems, and proofs -- arguably the real &quot;meat&quot; of mathematical thought -- becomes degraded, vestigial, and invisible. So by emphasizing the &quot;real math&quot; part you'd be running head-on against how our political and economic institutions require us all to assess proficiency in the discipline. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I don't think that computational exercises are a bad way of verifying that the concepts have been digested and understood. Generally I wouldn't call them &quot;problems&quot; unless they're of the open-ended, ambiguous, poorly-framed variety. But the truth is in our classes the answers are generally well understood and should be achievable by anyone in the room, and therefore &quot;exercises&quot; is a more accurate and more welcoming description for them. A &quot;problem&quot; only needs one person to resolve it, while an &quot;exercise&quot; is something that everyone should be expected to do. Personally I am rather puzzled why so many people want to refer to the application examples as &quot;problems&quot;, when that seems inaccurate to me. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-09-05T05:44:18.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9600" PostTypeId="2" ParentId="5768" CreationDate="2015-09-05T05:58:02.720" Score="8" Body="&lt;p&gt;For all of the community college algebra classes I teach, I certainly make proper &lt;em&gt;mathematical writing&lt;/em&gt; the number one priority; which is not to say that I have students compose everything in &lt;em&gt;English writing&lt;/em&gt;. It's already an overwhelming challenge for students to get the algebraic grammar and syntax right, such that I already feel like there's not enough time to properly exercise even that skill. So my focus is to say that our algebra class is equivalent to a &lt;em&gt;foreign language&lt;/em&gt; class; I assume they know English (or something), and now we're interested in translating those ideas into properly-formed mathematical statements (algebraic equations), with short side-justifications as appropriate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd say the fact that the mathematical transformations are more concise and convenient isn't something that should be resisted; it should be embraced as the whole &lt;em&gt;raison d'être&lt;/em&gt;. Here's one of my favorite quotes on the subject by Alfred North Whitehead, in &lt;em&gt;An Introduction to Mathematics&lt;/em&gt; (1911):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By relieving the brain of all unnecessary work, a good notation sets&#xA;  it free to concentrate on more advanced problems, and, in effect,&#xA;  increases the mental power of the race. Before the introduction of the&#xA;  Arabic notation, multiplication was difficult, and the division even&#xA;  of integers called into play the highest mathematical faculties.&#xA;  Probably nothing in the modern world would have more astonished a&#xA;  Greek mathematician than to learn that … a large proportion of the&#xA;  population of Western Europe could perform the operation of division&#xA;  for the largest numbers. This fact would have seemed to him a sheer&#xA;  impossibility … Our modern power of easy reckoning with decimal&#xA;  fractions is the almost miraculous result of the gradual discovery of&#xA;  a perfect notation. [...] By the aid of symbolism, we can make&#xA;  transitions in reasoning almost mechanically, by the eye, which&#xA;  otherwise would call into play the higher faculties of the brain.&#xA;  [...] It is a profoundly erroneous truism, repeated by all copy-books&#xA;  and by eminent people when they are making speeches, that we should&#xA;  cultivate the habit of thinking of what we are doing. The precise&#xA;  opposite is the case. Civilisation advances by extending the number of&#xA;  important operations which we can perform without thinking about them.&#xA;  Operations of thought are like cavalry charges in a battle — they are&#xA;  strictly limited in number, they require fresh horses, and must only&#xA;  be made at decisive moments.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-09-05T05:58:02.720" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9601" PostTypeId="1" AcceptedAnswerId="9630" CreationDate="2015-09-05T05:59:01.690" Score="9" ViewCount="530" Body="&lt;p&gt;In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [&lt;span class=&quot;math-container&quot;&gt;$a(b + c) = ab + ac$&lt;/span&gt;].&lt;/p&gt;&#xA;&lt;p&gt;What's bothered me for a long time is that while combining like terms [&lt;span class=&quot;math-container&quot;&gt;$ax + bx = (a + b)x$&lt;/span&gt;] is equivalent to distribution, it seems more basic and fundamental. It's used as an addition process (instead of a multiplicative one);  it seems commonsensical in terms of unit addition (3 feet + 5 feet = 8 feet), which was referenced by some as the &amp;quot;great principle of similitude&amp;quot;; and so forth.&lt;/p&gt;&#xA;&lt;p&gt;So what is the rationale for taking distribution as axiomatic, and proving combination afterward? Why is it not better pedagogy to take combining terms as fundamental, and then prove distribution from it?&lt;/p&gt;&#xA;&lt;p&gt;I've cross-posted this question on the Mathematics site:&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/1417856/why-is-distribution-prioritized-over-combining&quot;&gt;https://math.stackexchange.com/questions/1417856/why-is-distribution-prioritized-over-combining&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="2083" LastEditDate="2021-04-17T16:06:56.860" LastActivityDate="2021-04-17T16:06:56.860" Title="Why is distribution prioritized over combining?" Tags="|algebra|textbooks|definitions|community-colleges|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="9604" PostTypeId="2" ParentId="8593" CreationDate="2015-09-05T20:50:44.473" Score="3" Body="&lt;p&gt;I was put in charge of assessment for our community-college math department as of last year. One resource that was immensely useful was the UMass &quot;Program-Based Review and Assessment&quot; handbook (&lt;a href=&quot;http://www.umass.edu/oapa/oapa/publications/online_handbooks/program_based.pdf&quot; rel=&quot;nofollow&quot;&gt;link&lt;/a&gt;). There's an example of a math department assessment plan on p. 28.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, the process highlighted there is exactly as you describe: score selected questions embedded on final exams. On the one hand, we're fortunate in the math discipline that assessment is so amenable to our existing testing practices; on the other, it does look a bit silly and redundant. The silver lining for me is to know about specific skill mastery and not overall test or course averages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one thing that mystifies me is the practice of checking only one objective per year; that doesn't seem like the intent to me (although the UMass document, for example, does say &quot;if you and other members of your department can agree on only one goal, don’t let this stall your progress. Focus on that one goal – more will come later.&quot; [p. 11]). Assessing just one goal per year seems like you'd be going a half-decade between pings on any objective to know whether they'd degraded or improved or whatnot, which seems crazy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'm doing is to establish fixed questions in each of our terminal courses, and so assess every one of our objectives every year in a consistent manner. Hopefully this actually leads to less effort, because we won't have to spend mental energy creating new plans (re-inventing the wheel) every year; will also provide a &quot;dashboard&quot; of information on critical skills to compare year-to-year; and serves as a resolution to your very valid &quot;one exam question can be misleading and not representative of an entire program&quot; issue. Note also this is just the terminal courses (last ones to complete the degree), which I think meets the spirit of program assessment (what do students know when they complete our time with us), and also conveniently has a relatively smaller number of sections, and students, and instructors to wrangle for the task. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As extra steps I'm also receiving prerequisite grades and academic majors for all our students (to perform cross-correlations from the prerequisite course to identify particular strength or weakness), and sending out a survey to alumni this month for observations, how they did afterward, things they wish they were improved, and perceived strength on each of our program objectives. But that's definitely extra-credit for this task (I was expressly told by our Dean of assessment not to do that because it would be too much work), and can certainly be skipped if you just want to minimally meet accreditation demands. However, I did want to add a bit more depth/texture to our understanding of how our students are doing in later academic programs. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-09-05T20:50:44.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9605" PostTypeId="2" ParentId="8586" CreationDate="2015-09-05T21:52:42.367" Score="11" Body="&lt;p&gt;Yes, I agree that there is too much motivation... in the sense of sloppy presentations where it is very unclear to students exactly what is what. I think a lot of the time the refrain &quot;is this on the test?&quot; is a mangled sense of confusion over whether what's currently happening is a skill they're expected to master, or motivation for a skill, or justification for a skill. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally as a halfway young lecturer I've evolved to a fairly formal and ultra-old-school presentation format of definition, theorem, proof, and applications (specifically in that sequence). While there's nothing new and sexy in that with which education schools can win grants and publications, I constantly get students coming up to me and asking &quot;how did you develop this revolutionary teaching style&quot;. (!) I think that that pattern keeps people aware during any lecture cycle of exactly where we are and where we're about to go next. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it still gets moderately little traction from other instructors that I share it with. The current dogma tends to be &quot;problem-based&quot;, i.e., the opposite, and the conference presenters and so forth are in favor of pitching poorly-defined problems at students working in groups and hoping for the best. Like you, I think that the essential strength of the math discipline grows out of careful and unambiguous definitions, concise theorems, clear proofs, and then well-defined applications as a test of the subject matter at the end. I'm reminded of this quote in the preface to Stein and Barcellos, &lt;em&gt;Calculus and Analytic Geometry 5E&lt;/em&gt; (1992):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At the Tulane conference on &quot;Lean and Lively Calculus&quot; in 1986 we&#xA;  heard the engineers say, &quot;Teach the concepts. We'll take care of the&#xA;  applications.&quot; Steve Whitaker, in the engineering department at Davis,&#xA;  advised us, &quot;Emphasize proofs, because the ideas that go into the&#xA;  proofs are often the ideas that go into the applications.&quot; Oddly,&#xA;  mathematicians suggest that we emphasize applications, and the applied&#xA;  people suggest that we emphasize concepts. We have tried to strike a&#xA;  reasonable balance that gives the instructor flexibility to move in&#xA;  either direction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2015-09-08T06:01:19.090" LastActivityDate="2015-09-08T06:01:19.090" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9606" PostTypeId="2" ParentId="424" CreationDate="2015-09-05T22:33:06.497" Score="14" Body="&lt;p&gt;A year after this question was asked, the bloom is definitely off the MOOC rose. The primary finding is that the majority of people who finish one already possess a prior bachelor's degree; offering one to say, at-risk or remedial students has been a failure over and over again. Some links that you should consider:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Recent overview of the field, &quot;The MOOC revolution that wasn't&quot;: &lt;a href=&quot;http://kernelmag.dailydot.com/issue-sections/headline-story/14046/mooc-revolution-uber-for-education/&quot; rel=&quot;noreferrer&quot;&gt;http://kernelmag.dailydot.com/issue-sections/headline-story/14046/mooc-revolution-uber-for-education/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Pilot program at San Jose State to teach basic courses via UDacity MOOC gets suspended for lower-than-normal success rates: &lt;a href=&quot;http://www.latimes.com/local/la-me-0719-san-jose-online-20130719-story.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.latimes.com/local/la-me-0719-san-jose-online-20130719-story.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Remedial online courses in Philadelphia community colleges suspended: &quot;The failure rates were so high that it seemed almost unethical to offer the option&quot;: &lt;a href=&quot;http://www.insidehighered.com/news/2011/06/01/community_college_examines_high_failure_courses&quot; rel=&quot;noreferrer&quot;&gt;http://www.insidehighered.com/news/2011/06/01/community_college_examines_high_failure_courses&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;My review in 2013 of one of MOOC's vanguards, Statistics 101 as taught by UDacity founder Sebastian Thrun: &lt;a href=&quot;http://www.madmath.com/2012/09/udacity-statistics-101.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.madmath.com/2012/09/udacity-statistics-101.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2019-06-16T15:39:53.910" LastActivityDate="2019-06-16T15:39:53.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9607" PostTypeId="2" ParentId="8558" CreationDate="2015-09-05T22:52:09.023" Score="5" Body="&lt;p&gt;I would be pessimistic. An open-ended approach works well in the liberal arts and social sciences -- such as the course on video game culture --  where there are a fairly small core group of skills (reading critically, researching, analysis, writing and commentary) that can be applied broadly to a large number of subject areas, and practiced in a spiraling manner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast, mathematics and other STEM courses are &quot;dense&quot; with new skills being added on a daily basis, and in most cases being chained from prior work in a particular sequence of prerequisites. It takes a lot of work on the part of an expert instructor to scaffold this kind of material in an accessible way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example of &quot;stumbling inadvertently into some group theory&quot;, while fascinating, would be extraneous to the skills intended if we ask if someone has mastered precalculus. Moreover, if there is a list of required, distinct concepts and skills expected at the end of any mathematics course (and a relatively large list, compared with non-STEM subject areas), then it seems highly unlikely for the open-ended approach to meet that for everyone at the end of the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the kind of approach that would be great in an independent study or elective context -- and perhaps we should require everyone to experience that at least once! But for the core sequence of precalculus, calculus, etc., there is an irreducible conceptual ladder. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-09-05T22:52:09.023" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9608" PostTypeId="1" CreationDate="2015-09-06T01:50:30.877" Score="18" ViewCount="345" Body="&lt;p&gt;Is there any research out there on how an instructor's philosophical beliefs about mathematics might affect some aspect of his or her impact as a teacher?  My intended meaning of 'impact' is broad, covering not just how well students perform in test questions on a particular topic, but possibly including how likely they are to go on in math, lead successful lives, believe in ghosts, et c.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could certainly imagine different philosophies coloring how we explain mathematics.  A platonist might describe a proof as explaining and confirming why a given theorem is true, while a formalist might describe a proof as what makes the theorem true in the first place.  A realist might explain the derivative as giving the velocity of a particle from the position function, while an operationalist would explain the derivative as an invented concept that, in our observations of the world, relates these two measurements.  (Actually, the operationalist might treat velocity itself as an invented concept.) These differences may seem minor on each instance, but they can add up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking only for references to actual research on this question, if there is any.  Please keep your own speculations on these topics to yourselves.  I understand that effect sizes may simply be too small for any research to be possible.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2015-09-09T23:32:43.237" Title="Impact of philosophy of mathematics upon effectiveness of instructor" Tags="|mathematical-pedagogy|philosophy-of-maths|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9609" PostTypeId="2" ParentId="5648" CreationDate="2015-09-06T05:48:43.560" Score="0" Body="&lt;p&gt;&lt;strong&gt;Define division:&lt;/strong&gt; $a ÷ b$, also written $a/b$, means a number $x$ such that $x ∙ b = a$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few examples: (a) 6 ÷ 3 = 2 because 2 ∙ 3 = 6. (b) 20 ÷ 5 = 4 because 4 ∙ 5 = 20. (c) 72 ÷ 9 = 8 because 8 ∙ 9 = 72. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, an expression like 6 ÷ 0 is not any real number because we cannot meet the definition of x ∙ 0 = 6; the product of any real number times 0 would be 0, never 6. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-09-06T05:48:43.560" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9610" PostTypeId="2" ParentId="2471" CreationDate="2015-09-06T06:06:10.267" Score="0" Body="&lt;p&gt;Use the phrasing &quot;and/or&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, this is the language in Michael Sullivan's &lt;em&gt;Algebra &amp;amp; Trigonometry&lt;/em&gt;: &quot;Express the sum or difference as a product of sines and/or cosines.&quot; (7th Edition, Section 7.6, p. 637-638). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2015-09-06T22:15:29.560" LastActivityDate="2015-09-06T22:15:29.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9613" PostTypeId="2" ParentId="8195" CreationDate="2015-09-08T00:59:18.950" Score="8" Body="&lt;p&gt;What does an engineer need to get out of their calculus classes?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The Reynolds Transport Theorem.  All of their Physics, Chemistry, and Engineering classes have lots of hard conservation law problems.  Teach them what all of those problems have in common, and their lives will be much easier.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How to set up a story problem -- drawing a picture, labelling axes, directions, and dimensions, identifying knowns and unknowns.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Comfort recognizing field conditions and boundary conditions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Comfort working with variables.  It is much better to be able to derive an answer using variables, than to only be able to punch numbers into a calculator.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Comfort using units and dimensional analysis.  (For example, distances in meters, speeds in meters per second, accelerations in meters per second per second, jerks in …)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How to check their work.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Significant figures and error analysis.  How accurate are the inputs?  Does the algorithm mitigate or enhance errors?  How accurate are the outputs?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How to clearly explain an answer to someone who never took a calculus class.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2015-09-08T00:59:18.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9614" PostTypeId="1" AcceptedAnswerId="9615" CreationDate="2015-09-08T04:48:01.277" Score="5" ViewCount="154" Body="&lt;p&gt;My students are preparing for their SATs and have problems with a certain type of questions, i.e., questions involving a geometrical figure and $n$ (usually $n = 2,3$) straight lines passing through it. The maximum number of regions that the figure can be divided into is asked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I generally look at the options given and try a &lt;em&gt;hit and trial&lt;/em&gt; method. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there any other approach to problems like these which I can recommend?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5572" LastEditorUserId="262" LastEditDate="2015-09-16T20:54:30.707" LastActivityDate="2015-09-16T20:54:30.707" Title="Approaches to Teaching for Questions Around Partitioning Space" Tags="|secondary-education|geometry|students-mistakes|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9615" PostTypeId="2" ParentId="9614" CreationDate="2015-09-08T12:16:53.337" Score="5" Body="&lt;p&gt;See &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lazy_caterer%27s_sequence&quot; rel=&quot;nofollow noreferrer&quot;&gt;the lazy caterer's sequence&lt;/a&gt;&quot;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/HfW1k.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HfW1k.jpg&quot; alt=&quot;Pie7ths&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from Wikipedia.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;The maximum number of cells in an arrangement of $n$ lines is&#xA;$$\binom{n}{2} + \binom{n}{1} + \binom{n}{0} \;.$$&#xA;For $n=3$, this evaulates to $7$.&#xA;This sequence is &lt;a href=&quot;https://oeis.org/A000124&quot; rel=&quot;nofollow noreferrer&quot;&gt;A000124&lt;/a&gt; in OEIS.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-09-08T12:16:53.337" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9617" PostTypeId="1" CreationDate="2015-09-08T20:25:27.160" Score="3" ViewCount="52" Body="&lt;p&gt;There is a lot of certification and licensure tests for teaching (math and etc) particularly for teachers and teacher-students (PACT, Praxis &amp;amp; PPAT, edTPA, RESA, ... in US and TAT in India and etc). These tests are usually named as TPA -Teaching Performance Assessment. Where can I see a full list of these tests and where can I see these tests classified by those goals, methods or contents? Where can I see a list of especial tests for teacher-students?&lt;/p&gt;&#xA;" OwnerUserId="5574" LastActivityDate="2015-09-08T20:25:27.160" Title="clasification of teaching licensures and certifications particularly for teacher-students" Tags="|teacher-training|teaching|certification|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9619" PostTypeId="2" ParentId="9608" CreationDate="2015-09-09T23:32:43.237" Score="2" Body="&lt;p&gt;My apologies for my small sample size... but, still, 40+ years of teaching almost gets me to statistical significance? :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, early in my career-arc, I was essentially conformist, and would aver that the course objectives officially espoused were truly what they were. Students did not &quot;like&quot; this, but could easily endorse it. (Yikes...) This is not about philosophy of mathematics, though, but about the social deconstruction of the university experience...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the next stage, I still did believe that the traditional+orthodox description of &quot;mathematics&quot;, and how one learns it, and the course sequences, and so on, were essentially accurate, if belabored. It took several years for this to be beaten out of me. :)  That is, essentially none of my advanced undergrad or graduate students responded in the &quot;appropriate&quot; manner. That is, the philosophy to which I &quot;adhered&quot; did not sell, at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, try to explain the post-Wallis experimental/inductive reality of &lt;em&gt;research&lt;/em&gt; in mathematics... Well, turns out, this is repellent to many of the people who've alread self-selected for &quot;math major&quot; (based on the traditional-orthodox pop idea of what it is...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, philosophical, and socio-ideological, attitudes of the instructor have an impact on the students. Perversely, often kindness is mistaken for weakness, etc., because of popular social &quot;constructs&quot;, ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, we don't quite &lt;em&gt;get&lt;/em&gt; to any genuine philosophy of mathematics. It is hard enough to convince most students, especially undergrads, that mathematics could possibly be anything more than &quot;a school subject&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, then, the usual scenario in which lower-division math is merely a filter/weeder, rather than a genuine educational (much less scholarly/intellectual!!!) endeavor, subliminally confirms the apparent non-sensical-ness of trying to pretend that math is anything more than &quot;another brick in the wall&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Yes, I have tried to push back, against this sort of thing, but both colleagues and the students themselves have such powerful prior expectations that it's hard to accomplish much...)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-09-09T23:32:43.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9620" PostTypeId="1" CreationDate="2015-09-10T20:57:00.613" Score="7" ViewCount="131" Body="&lt;p&gt;I was wondering if there exists a book that &lt;em&gt;talks&lt;/em&gt; to highschool level audience who wish to study pure mathematics for their undergraduate years. I'm not talking about problem solving kind of books (I've found a lot, since math competitions are very famous). But more like a book that talks of what to expect in, for example, calculus or algebra.&lt;/p&gt;&#xA;" OwnerUserId="99" LastActivityDate="2015-10-02T15:43:21.600" Title="Looking for a reference for highschool level readers" Tags="|undergraduate-education|secondary-education|reference-request|reference-books|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9621" PostTypeId="2" ParentId="8575" CreationDate="2015-09-10T22:29:24.760" Score="4" Body="&lt;p&gt;In computer science, specifically in combinatorics (much used in algorithm analysis) one important task is to derive asymptotic behaviour of sequences, which in turn are easiest to get in form of a generating function. The techniques used are heavily based on complex analysis. See for example Flajolet and Sedgewick's &lt;a href=&quot;http://algo.inria.fr/flajolet/Publications/book.pdf&quot; rel=&quot;nofollow&quot;&gt;&quot;Analytic Combinatorics&quot;&lt;/a&gt; (Cambridge University Press, 2009).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-09-10T22:29:24.760" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9622" PostTypeId="2" ParentId="8558" CreationDate="2015-09-11T03:00:15.647" Score="3" Body="&lt;p&gt;I don't know about the online part, but &lt;a href=&quot;https://www.artofmathematics.org/&quot; rel=&quot;nofollow&quot;&gt;Discovering the Art of Mathematics&lt;/a&gt; definitely has some ideas that are low-level (non-major, even) but yet stumble upon very interesting things indeed.  I'm sure they'd be interested in discussing an appropriate online pedagogy.  &lt;a href=&quot;http://www.inquirybasedlearning.org/&quot; rel=&quot;nofollow&quot;&gt;The AIBL&lt;/a&gt; would be a good resource.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be sure, one doesn't have to be a strict 'constructivist' or 'constructionist' or any of that to use these, but there are aspects of them that really would resonate with your question - including e.g. precalc or calc.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-09-11T03:00:15.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9623" PostTypeId="2" ParentId="8568" CreationDate="2015-09-11T03:04:54.737" Score="1" Body="&lt;p&gt;There are many, many examples of this in the resources at &lt;a href=&quot;https://www.artofmathematics.org/&quot; rel=&quot;nofollow&quot;&gt;The Art of Discovering Mathematics&lt;/a&gt;, including things like the game of Hex.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-09-11T03:04:54.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9624" PostTypeId="2" ParentId="5794" CreationDate="2015-09-11T03:09:40.077" Score="3" Body="&lt;p&gt;Forgive me if this is already above.  I only see one reference to it in an answer&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Instead of teaching it as flipping with respect to the vertical line at zero, I recommend teaching it as the rotation of 180∘ about the point zero.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would argue that (the well-known analogy of) flipping/reflecting about the vertical line at zero is a very good way, actually, and I do this often.  Here are three reasons.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Kinesthetic - simply take your arms and (say) flip your elbow to your other elbow.  That's $-1$; do it again for $-(-1)=1$.  Do it with the ends of your arms for $-(-2)=2$.&lt;/li&gt;&#xA;&lt;li&gt;Doesn't require wondering about why we are &lt;em&gt;rotating&lt;/em&gt; when there is no &quot;meaning&quot; (in the eyes of students likely to ask this) of things &quot;above&quot; the number line.&lt;/li&gt;&#xA;&lt;li&gt;It's easy.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I agree that if one is going to be thinking of negation in a broader context, with students already very comfortable with $\mathbb{R}^2$ or even $\mathbb{C}$ as such (rather than as the place where graphs of functions live) then rotation is the way to go, since $-(i)=-i$ would just be wrong under this analogy.  But I find it very useful for a quick rationale.  (And if they bring up $-(1,1)=(-1,-1)$ doesn't work, you have a live one!)&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-09-11T03:09:40.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9625" PostTypeId="2" ParentId="8195" CreationDate="2015-09-11T03:20:27.380" Score="5" Body="&lt;p&gt;This depends hugely on what calculus class you are teaching.  I'll speak to a non-major-focused calculus class (as opposed to non-majors in some other calc class).&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Idea that we can get a huge amount of information about functions' behavior from calculus ideas.&lt;/li&gt;&#xA;&lt;li&gt;The concept of marginality - not necessarily using that word, though it's great if you have econ/business majors.  This is far more fundamental than we really understand, in my opinion.&lt;/li&gt;&#xA;&lt;li&gt;Totals and rates are inverse operations, in some sense.&lt;/li&gt;&#xA;&lt;li&gt;Assuming you have people with the right interests, that even simple models using rates (derivatives) can be useful (either symbolic or numerical integration, which doesn't have to be as intense as Runge-Kutta but could be essentially Euler's method in a spreadsheet).  Models from fisheries to ice sheets to adoption of technology are all appropriate.&lt;/li&gt;&#xA;&lt;li&gt;The notion that how to talk about infinity might matter.  (Cue their intro philosophy class and Aristotle or Hobbes, a theology class and Aquinas, or a history class and Galileo.)  Everyone seems to have something to say about Zeno, even if &lt;a href=&quot;http://existentialcomics.com/comic/82&quot; rel=&quot;noreferrer&quot;&gt;not everyone thinks it's solved by calculus&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(Which my colleagues all tell me not to worry about, because evaluations are typically mediocre at best for freshman calculus courses. But couldn't this be indicative of a general problem?)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes.  Calculus is a course looking for a reason to exist, in many contexts - it's more of a gatekeeper than anything else.  If your students don't know why they should care about functions once they are done memorizing them for an exam, it's trouble.  That's why the modeling idea is key, though unfortunately not very easy to implement.  See &lt;a href=&quot;https://www.nostarch.com/mg_calculus.htm&quot; rel=&quot;noreferrer&quot;&gt;the Manga guide to calculus&lt;/a&gt; for some very interesting ones that unfortunately probably won't make sense to your students until after they've actually had calculus.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-09-11T03:20:27.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9626" PostTypeId="2" ParentId="9601" CreationDate="2015-09-12T10:33:15.810" Score="3" Body="&lt;p&gt;As the above comments explain, that it is called &quot;distributive law&quot; is just a name. Distributing and combining (factoring) are logically equivalent. The identity can be applied in both directions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A practical difference is that distributing (multiplying out) can be done mechanically, factoring (combining) often has several potential common factors among different sets of terms, selecting &quot;the right one&quot; isn't always obvious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, it is certainly worthwhile to emphasize that identities work both ways, and factoring complex expressions is obviously an important skill that has to be trained.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-09-12T10:33:15.810" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9627" PostTypeId="2" ParentId="8538" CreationDate="2015-09-12T14:31:25.347" Score="3" Body="&lt;p&gt;The classic: what is the probability that two points chosen at random from the interval $[0,1]$ will sum to at least $1$?&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-09-12T14:31:25.347" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9628" PostTypeId="2" ParentId="9620" CreationDate="2015-09-12T21:20:53.383" Score="4" Body="&lt;p&gt;The book &quot;How to think like a mathematician&quot; by Kevin Houston. It was recommended reading during the summer before I enrolled in my undergraduate math program, and I found it incredibly helpful. It discusses how math at university level differs from math in high school. You get a nice introduction to what mathematicians mean by definitions, lemmas, theorems, and a number of different proof techniques. It's written specifically to help bridge the gap between high school and university, but it can also help when you want to see what math at university is like.&lt;/p&gt;&#xA;" OwnerUserId="5250" LastActivityDate="2015-09-12T21:20:53.383" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9629" PostTypeId="2" ParentId="8538" CreationDate="2015-09-13T18:32:29.443" Score="2" Body="&lt;p&gt;It can be powerfully impressive to approximate $\pi$ by dropping matchsticks&#xA;on a ruled surface:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/8r2iG.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8r2iG.jpg&quot; alt=&quot;Buffon&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Buffon%27s_needle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia image&lt;/a&gt; by McZusatz.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Here, $17$ sticks were tossed, the separation between the lines is the stick length, and $\pi \approx (2 \cdot 17) / 11 = 34/11 \approx 3.1$.&#xA;See, e.g., &lt;a href=&quot;http://www.sciencefriday.com/blogs/03/12/2014/estimate-pi-by-dropping-sticks.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this description&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-09-13T18:32:29.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9630" PostTypeId="2" ParentId="9601" CreationDate="2015-09-13T21:51:07.983" Score="9" Body="&lt;p&gt;This grew a bit long for a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(My first note is similar to Alexander Woo's remark about factoring polynomials; perhaps he intended &quot;polynomials&quot; to subsume the case here, in which we add constant functions...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given $413 + 91$, it may not be clear that this can be re-written as $7 \times 59 + 7 \times 13 = 7(59+13)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Plenty of people seem to believe that $91$ is prime; an earlier &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2718/examples-for-reasoning-by-analogy-going-wrong#comment7203_2729&quot;&gt;&lt;strong&gt;comment&lt;/strong&gt;&lt;/a&gt; cites J. Conway as observing as much, and I would guess it is related to memorizing the $10\times10$ or even $12\times12$ times tables.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Meanwhile, $7(59+13)$ can have the $7$ distributed mechanically. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, the act of viewing an equation from both perspectives is certainly important, and I think what you observe here is not altogether different from the tendency to write $1 + 1 = 2$ significantly more often than $2 = 1 + 1$, which is known to cause problems with viewing the equal sign, $=$, as an operator, i.e., &lt;em&gt;operationally rather than relationally&lt;/em&gt;, cf. &lt;a href=&quot;https://matheducators.stackexchange.com/a/7965&quot;&gt;&lt;strong&gt;MESE 7964&lt;/strong&gt;&lt;/a&gt; and the nice response of D. Hast.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You did ask for some rationale; perhaps we can look historically to Euclid's &lt;em&gt;Elements&lt;/em&gt;. If we are to believe the translation provided &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/elements/bookII/propII1.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;, then we have Euclid remarking in the same order. A similar translation is found, e.g., in the following source:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Drucker, T. (Ed.). (2009). &lt;em&gt;Perspectives on the history of mathematical logic&lt;/em&gt;. Springer Science &amp;amp; Business Media.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A note from the aforecited:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NRXgB.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NRXgB.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, a separate but related comment: I observe a &lt;em&gt;prioritizing&lt;/em&gt; in the presentation of the difference of squares and their factorization, i.e., more often than not I see: $a^2 - b^2 = (a+b)(a-b)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This phenomenon seems to prioritize &quot;combining&quot; over &quot;distribution,&quot; and is often accompanied by the same for the sum and difference of cubes. Anyhow, there are consequences in this case, as well. Few students consider such a property in computing, e.g., $47 \times 53$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Ask students to compute that product in a few ways and see if it even comes up!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel confident that most important will be for students to see that the same &lt;em&gt;mathematical&lt;/em&gt; information is presented by an equality regardless of which expression is on which side, and to give students the opportunity to consider both presentations and their various ramifications.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-09-14T03:37:16.103" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9631" PostTypeId="2" ParentId="1472" CreationDate="2015-09-14T14:23:35.020" Score="0" Body="&lt;p&gt;Different notations show their strengths in different contexts.   In equations of motion, for example, Newton's (dot) notation is compact, Leibniz' is less so but has advantages elsewhere, for example when using a chain rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In vector mechanics, for example, I hop between Gibbs', matrix and subscript/index notations depending on what aspect of the vector(s) I'm trying to emphasis.   Gibbs' notation is good, for example, when considering the relationship between vectors, matrix notation is good when doing computations with components, subscript notation (given a few rules) shows explicitly how to do the computation.   I expect students to be(come) familiar with each notation and so be able to read different textbooks, each of which might show a different preference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same is true with technical words.   I can't count the number of times I've come across an unfamiliar technical term (&quot;the Trefftz stress&quot;) and only realised later on that it was something I knew by another name (&quot;the 2nd Piola-Kirchhoff stress&quot;).&lt;/p&gt;&#xA;" OwnerUserId="5541" LastEditorUserId="5541" LastEditDate="2015-09-14T14:33:20.313" LastActivityDate="2015-09-14T14:33:20.313" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9632" PostTypeId="2" ParentId="1472" CreationDate="2015-09-14T17:01:00.820" Score="6" Body="&lt;p&gt;My perspective is teaching at an open-admissions community college, and I have to say that it's an almost impossible challenge to get students to properly read and write even &lt;em&gt;one&lt;/em&gt; elementary notation. In the rare cases that we want them to know a secondary notation (e.g., radicals and rational exponents), then we have to devote an hour-long lecture &amp;amp; practice period on the particular topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I must say that even as a student I would not want arbitrarily mixed notation in a given course or lecture series. Given a particular context I would prefer to not be distracted by that switching cost. A quick comment, dictionary, or footnote on what you may see in other texts is fine and helpful. If the student is advanced and expects to be reading multiple texts in the future, then they will have the maturity to look it up or reference it as needed (i.e., they will already be past the point of needing the productive struggle of discovering that different languages exist in the world). Don't create notational Spanglish. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quote from Steven Krantz in &lt;em&gt;How to Teach Mathematics&lt;/em&gt; (Sec. 2.12):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You really are obliged to follow the notation and definitions and&#xA;  other paradigms in the text you have chosen. Otherwise all but the&#xA;  gifted students will be lost.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2019-07-16T03:51:11.850" LastActivityDate="2019-07-16T03:51:11.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9633" PostTypeId="2" ParentId="9601" CreationDate="2015-09-14T20:41:38.530" Score="2" Body="&lt;p&gt;Having an inquisitive, but impatient 3rd grader can help you to get the answer pretty quickly: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Distribution can be explained and applied directly out of the box, because &lt;strong&gt;a&lt;/strong&gt;, &lt;strong&gt;b&lt;/strong&gt; and &lt;strong&gt;c&lt;/strong&gt; are all known when starting on the left. In order to work from the right hand side, the combination requires finding (isolating) the common part (i.e. denominator), which might not be obvious on the spot, especially when starting with numbers as commented by Alexander Woo.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is unneccessary distraction while trying to explain this equivalence on its own, so more or less knowingly, the explanations prefer distribution over combination.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2015-09-14T20:41:38.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9634" PostTypeId="2" ParentId="5794" CreationDate="2015-09-15T07:32:28.687" Score="4" Body="&lt;p&gt;This might be contained in some other answer in one form or another, but I think it might be useful to simply appeal to the definition of $-x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Definition:&lt;/strong&gt; whatever the number $x$ is, we denote by $-x$ the number which yields $0$ when added to $x$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is both mathematically sound in a very general setting, simple enough for pupils (I guess), and easy to relate to all interpretations that where given (North/South, debt/credit, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, this definition makes it easy to deduce $-(-x)=x$ ; I'll make it on an example since this is how I would proceed myself.&#xA;By its very definition, $-(-3)$ is the number which yields $0$ when added to $-3$. We know that $-3+3=0$ (this is the definition of $-3$) so that the number we seek is $3$. I.e. $-(-3)=3$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a bit formal in some way, but I feel that getting back to the basic definition and relate it to the meaning of a mathematical object of notation should never be avoided unless necessary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there is something hidden behind this: the definition only makes sense if we know for sure that there exist a unique number with the given property. But this can be kept behind the scene for quite a few years I guess.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will not get at multiplying negative numbers, as I have little to add to other answers.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-09-15T07:32:28.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9635" PostTypeId="1" CreationDate="2015-09-15T11:58:56.427" Score="18" ViewCount="1104" Body="&lt;p&gt;I teach mechanics (including large deformation and flow of continua) to mechanical engineering students and have a continuing mission to drag the teaching of mechanics into the 20th century (I'll worry about the 21st later) by introducing 'modern' ideas (physicists may laugh!) on differential forms and the like (co- versus contra-variance in oldspeak).   I have most of the standard texts and well-known papers on the subject but the applications they consider (if any) tend to be on electromagnetics, general relativity or some other topic of little interest to my students.   My questions are then:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;is anyone else doing anything similar?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;does anyone know of any specifically mechanics-oriented material?   &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;has anyone any experience of teaching diff forms and geometric algebra to engineers or physicists.   If so, what is more useful in practice?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit 20 Sep 15&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To expand on the OP, I don't find any of the existing texts suitable for mech engineers, whether students or professionals:  they all start with some formalism about diff forms and then (just maybe) do applications that are irrelevant to mech eng (Frankel's &lt;em&gt;Geometry of Physics&lt;/em&gt; has a section on stress but it's a bit of a dangling appendage).   I've argued elsewhere that, for engineers, &lt;strong&gt;the motivation is the application&lt;/strong&gt;.   My approach is then to start with the need to calculate the work done by a particle moving in a force field.   In 'traditional' vector calculus, to compute the dot product in the the line integral, we change the column matrix that usually denotes the force 'vector' to a row matrix.   This is usually glossed over but I emphasise that this isn't a trivial change - it's effectively saying that force is a 1-form.   I want to then do fluid flow in $R^2$ then in $R^3$, then strain and stress, introducing 2-forms, Hodge dual, Lie derivative, etc only when needed.   I've trawled the web, looked at every textbook on diff forms I can find but nothing seems to develop along these lines (though Metzler's YouTube clips are instructive).   So either:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm barking up the wrong tree (or possibly just barking).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm breaking new ground in teaching diff forms to mech eng undergraduates.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I might be talking myself into writing &lt;em&gt;Diff Geom for Mech Engineers&lt;/em&gt; but, if someone has already done something similar, I'd really like to see it - and put it on the 'compulsory purchase' list for Mechanics 5.&lt;/p&gt;&#xA;" OwnerUserId="5541" LastEditorUserId="5582" LastEditDate="2015-09-20T19:50:24.850" LastActivityDate="2015-12-04T03:35:11.543" Title="Differential forms in mechanics?" Tags="|undergraduate-education|vector-calculus|engineering-mathematics|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9636" PostTypeId="1" CreationDate="2015-09-15T14:39:05.060" Score="19" ViewCount="401" Body="&lt;p&gt;Ideally, I'd like my students' grades to be based on more than just exams, because I think that certain tasks, such as writing proofs, aren't best performed in a test environment.  Using homework for assessment seems like an obvious way to accomplish this.  But my idealized view of homework, formed when I was a student, seems to have little to do with modern reality.  Educators on this site suggest that instructors solve homework problems during office hours, and my students seem to expect that I will do this, but the ability of students to transcribe solutions that I provide them with does not demonstrate understanding.  Working on homework in groups seems to be nearly universally recommended and very widely practiced, but how do I differentiate the signal of Student A's understanding from the noise of his/her study partners' understanding?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any studies that compare students' perception of how homework problems are to be solved (e.g, by the student him/herself or by others) today vs., say, 30 years ago?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there &quot;best practices&quot; for making valid assessments of student understanding outside the environment of an exam? &lt;/p&gt;&#xA;" OwnerUserId="5587" LastActivityDate="2017-12-30T04:18:23.777" Title="Can homework play a substantial role in the assessment of understanding in university math?" Tags="|homework|assessment|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9637" PostTypeId="2" ParentId="9636" CreationDate="2015-09-15T15:53:56.150" Score="5" Body="&lt;p&gt;This is not a full answer to your question, but these comments do not fit in the comment box.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Homework and exams measure different things.&#xA;If you want to make sure that it is the student's own thoughts that you are reading on a paper, you should have a controlled exam.&#xA;I think such control is often justified, so an exam is a good way to assess students, but it need not be the sole method.&#xA;Taking homework as a part of your grading scheme gives value to the students' activity with the problems.&#xA;Even if a student is copying the solution from someone else, he is putting at least some amount of thought into the subject and going through a solution.&#xA;This is good; the student could do better, but this is far better than nothing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of course if you want to grade the homework, you should refuse to give much help with at least some (if not all) of it.&#xA;Having two sets of exercises, ones that the instructors help with and others that have to be handed in for grading, seems to work well here.&#xA;After solving some problems with help, the students can be expected to solve more problems without help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think activity with homework is a sound criterion for assessment.&#xA;This for (at least) two reasons: (1) activity correlates with the competence you want to create in your students, and (2) giving points for activity motivates activity and causes learning as a side effect.&#xA;Student assessment is not only for seeing who knows their trade and who doesn't, but also for pushing students in the right direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is often some social pressure against copying solutions from others.&#xA;And in group homework most students will master at least some part of the topic, and that is often enough.&#xA;If you want to make sure that everyone masters the same thing, an exam is in order.&#xA;It seems that my answer to your first question is this: it is not always that important to differentiate the signal from the noise (the student's understanding from his friend's understanding).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you use both an exam and graded homework, the role of the exam can be very different.&#xA;You can just test the understanding of the basic concepts and ideas in the course and the homework project; leaving the goriest details to homework changes the tone of the exam quite a bit.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2015-09-15T15:53:56.150" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9638" PostTypeId="2" ParentId="9614" CreationDate="2015-09-15T21:21:54.607" Score="5" Body="&lt;p&gt;Since Joseph O'Rourke, in &lt;a href=&quot;https://matheducators.stackexchange.com/a/9615&quot;&gt;&lt;strong&gt;his response&lt;/strong&gt;&lt;/a&gt;, has linked to the lazy caterer's sequence, let me also mention its three-dimensional analogue, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cake_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;cake number&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, we are dividing three-dimensional space with planes. The original question asks:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I generally look at the options given and try a hit and trial method. Is there any other approach to problems like these which I can recommend?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;main reason&lt;/strong&gt; that I post this is to point to a video from the Mathematical Association of America (MAA, 1966) entitled, &lt;em&gt;George Polya in &quot;Teaching Us A Lesson&quot;&lt;/em&gt; or &lt;em&gt;Let Us Teaching Guessing: A Demonstration with George Polya&lt;/em&gt; (&lt;a href=&quot;https://vimeo.com/48768091&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;vimeo link&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(If you open the video, the former title includes a misspelling of Polya's first name, and the latter title lists the year as 1965, though &lt;a href=&quot;http://www.imdb.com/title/tt0410217/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;IMDB&lt;/strong&gt;&lt;/a&gt; has it as 1966. But let us not get bogged down in the details!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since you ask about approaches to this problem, perhaps there is something to be gleaned from Polya's approach. His advocating for &quot;guessing&quot; may seem similar to your approach of &quot;hit and trial,&quot; but Polya distinguishes between &#xA;&lt;em&gt;wild guessing&lt;/em&gt; and &lt;em&gt;careful guessing&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Polya is an important figure in mathematics education (and mentioned with some frequency on MESE; see, e.g., &lt;a href=&quot;https://www.google.com/?gws_rd=ssl#q=site:matheducators.stackexchange.com+polya&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;) and so I recommend this video, one, because it is directly related to the original query, but two, because it provides the opportunity to see (what I — and at least &lt;a href=&quot;https://www.maa.org/external_archive/columns/launchings/launchings_12_07.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;David Bressoud&lt;/strong&gt;&lt;/a&gt;) consider to be a favorite student activity!&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-09-15T21:21:54.607" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9639" PostTypeId="1" CreationDate="2015-09-16T00:19:05.460" Score="3" ViewCount="109" Body="&lt;p&gt;I am looking for any (and all) books (on math, physics and Computer Science) that discuss how to teach and selects methods based on empirical research and solid evidence. My biggest interest right now is on the second half of primary education and secondary education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd prefer books that constrast different approaches and ponder their effectiveness empirically, rather than books that propose one particular approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I teach at an NGO, and have interest in teaching, but my formal education is in Computer Science. I am trying to approach the field with as much a evidence-based approach as possible)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I know that (math and physics and CS) is broad. My reason for asking is that I'd like to have as many examples as possible, so that I can better understand the what empirical approaches to STEM education exist, and choose which approaches to further study. &lt;/p&gt;&#xA;" OwnerUserId="1662" LastEditorUserId="1662" LastEditDate="2015-09-16T17:37:28.520" LastActivityDate="2015-09-16T17:37:28.520" Title="Textbooks for mathematical/computing/physics teaching that are based on empirical research" Tags="|mathematical-pedagogy|" AnswerCount="1" CommentCount="4" ClosedDate="2015-09-23T14:46:19.230" ContentLicense="CC BY-SA 3.0" />
  <row Id="9640" PostTypeId="2" ParentId="8586" CreationDate="2015-09-16T10:16:39.093" Score="5" Body="&lt;p&gt;In the preface to &lt;em&gt;Intro to Applied Maths&lt;/em&gt;, Gil Strang says 'the equations of applied mathematics are better understood when the algorithms that solve them are presented at the same time'.   In the preface to my lecture notes, I've extended this to 'in engineering, both equations and algorithms are better understood when they are driven by the need to model engineering systems'.   One of the commonest complaints I hear from student engineers - of whom I've taught over 10,000 - is 'why are we doing this?'   For them, the application &lt;strong&gt;is&lt;/strong&gt; the motivation and comes first.   Of course, pedagogically (if not in real life), the application must be appropriate to the level of knowledge of the student; that's why we have structured curricula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, student engineers need an application to motivate the maths.   In early years, this must be something quite simple;   in later years, a more complex application might motivate deeper study of the same mathematical topic (eigenvalues are a good example).   This is not, I think, how mathematicians approach a topic.&lt;/p&gt;&#xA;" OwnerUserId="5541" LastActivityDate="2015-09-16T10:16:39.093" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9641" PostTypeId="2" ParentId="9639" CreationDate="2015-09-16T14:40:41.667" Score="2" Body="&lt;p&gt;For physics, I've found the following helpful:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mazur, Peer Instruction: A User's Manual&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arons, Teaching Introductory Physics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Mazur book is really the evidence-based book you're looking for. Unfortunately it's out of date. You can absorb the general pedagogical research results without necessarily using exactly Mazur's method of teaching. Basically the research supports active engagement rather than lecturing.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2015-09-16T14:40:41.667" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9642" PostTypeId="2" ParentId="330" CreationDate="2015-09-18T02:17:01.603" Score="5" Body="&lt;p&gt;For a change, a research-based answer!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.edweek.org/ew/articles/2015/09/09/researchers-test-out-primary-sources-for-math.html?cmp=eml-enl-eu-news2-RM&quot;&gt;This article from &lt;em&gt;Education Week&lt;/em&gt;&lt;/a&gt; describes an NSF-funded pilot program in which students will study primary sources in mathematics, from Euclid to Pascal.  From the article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Previous NSF-funded work found some evidence that college math&#xA;  students were motivated by the use of primary sources and showed some&#xA;  performance gains in subsequent math classes—but the data were mostly&#xA;  anecdotal.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[Jerry] Lodder, who also worked on a prior NSF grant on which this study is based, said he believes students will gain &quot;more enduring knowledge&quot; through the use of primary sources. &quot;When you read the formula or algorithm in a textbook, sure, you memorize it for the exam, but a few days later you forget it because there was no context,&quot; he said. &quot;When students see the original problem or solution it sticks with them longer.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There doesn't seem to be any firm evidence (yet) that this approach yields any benefits in terms of student understanding or retention, but the article makes it clear that part of the grant is devoted to measuring the efficacy of this approach:  &quot;Researchers will track students' learning growth and compare results with those for students in nonparticipating math classes.&quot;&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-09-18T02:17:01.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9643" PostTypeId="2" ParentId="8586" CreationDate="2015-09-18T05:31:10.103" Score="4" Body="&lt;p&gt;I am a retired educator. I have found that we do not really teach subjects; we teach STUDENTS.  We actually present some of the content of our subjects and we study those subjects ourselves, and how best to present their content.  I find it true that we REALLY learn what we prepare to teach.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Attempting to motivate students, is not the same as successfully (two letters &quot;c&quot; IS correct! ...contrary to the spell-checker) motivating them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As teachers, we may have subjectively idealistic goals and measurable objectives.  Students may have very different goals-and-objectives from one another as well as ourselves.  What stimulates and motivates one student may confuse and demotivate another.  That is just the way it is.  Yet, there is more, both hope and complexity…&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also the issue of &quot;personality conflict&quot; between students as well as between student and instructor.  Teachers usually address their own styles of learning when presenting subject material.  Some of us are visually oriented others are subjectively abstract or objective literalists.  The real problem is providing for students who perceive and think differently than we do.  This is now being better provided for in public schools and many undergraduate programs.  It still requires getting to know our students ( &gt; sigh &amp;lt; ) which is time consuming.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I did better in Geometry and Trigonometry than in beginning algebra.  Algebra made much more sense to me AFTER studying Geometry and Trigonometry and doing real navigation with an E6B.  For others it was quite different; algebra prepared them for Geometry and Trigonometry.  We all enjoyed studying together anyway.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My solution was “over-preparing” or being ready to meet a multiplicity of needs when they surfaced without presenting everything “I-had-up-my-sleeve” each and every time I presented.  Presenting too much tends to confuse many students while others need more as stimulus and clarification, which I offered individually.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some times, if your faculty is large enough, you can swap students with a professor, instructor or T.A. who is very different from yourself and all of you may get really wonderful results!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope some of this is helpful....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doc&lt;/p&gt;&#xA;" OwnerUserId="5598" LastActivityDate="2015-09-18T05:31:10.103" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9644" PostTypeId="2" ParentId="424" CreationDate="2015-09-18T05:59:33.217" Score="5" Body="&lt;p&gt;Honestly I feel like most MOOC fall short. Not necessarily that they &lt;em&gt;must&lt;/em&gt; be bad, but the ones I've seen are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like the problem is that they're in a weird inbetween; too time consuming to be something fun, but not rigorous enough to get something from it. If you want to learn rigorous mathematics, you have to spend a lot of time and effort and it's something MOOCs just haven't been providing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also the average person I know interested in MOOCs are those who already are educated and interested. I think MOOCs are a great idea but there really isn't too much of a market for it.&lt;/p&gt;&#xA;" OwnerDisplayName="user5108" LastActivityDate="2015-09-18T05:59:33.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9645" PostTypeId="2" ParentId="8552" CreationDate="2015-09-18T22:17:22.270" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=zO4ApMb3B8QC&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;&quot;Mathematics, a Human Endeavor: A Textbook for Those Who Think They Don't Like the Subject&quot;&lt;/strong&gt;&lt;/a&gt; is a great book. Lots of fun/interesting problems that are accessible to a wide range of math levels. The description says it's targeted at liberal arts math classes that focus on problem-solving:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;For instructors of liberal arts mathematics classes who focus on problem-solving, Harold Jacobs's remarkable textbook has long been the answer, helping teachers connect with of math-anxious students. Drawing on over thirty years of classroom experience, Jacobs shows students how to make observations, discover relationships, and solve problems in the context of ordinary experience.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5599" LastEditorUserId="262" LastEditDate="2015-09-18T23:23:34.897" LastActivityDate="2015-09-18T23:23:34.897" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9646" PostTypeId="2" ParentId="8538" CreationDate="2015-09-18T23:33:17.127" Score="3" Body="&lt;p&gt;You might look at this &quot;module&quot; published by COMAP, The Consortium for Mathematics and Its Applications:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.comap.com/product/samples/6659.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.comap.com/product/samples/6659.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2015-09-18T23:33:17.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9647" PostTypeId="2" ParentId="9635" CreationDate="2015-09-19T19:17:12.797" Score="2" Body="&lt;p&gt;It is an error to begin courses on differential forms, thinking that the multilinear algebra language is easily grasped. A good sequence of knowledge is to take on vector duality and rank two tensors. Here one can&#xA;develop the basis on mechanics. Try Alessandra Iozzi's notes called &lt;em&gt;Multilinear algebra and applications&lt;/em&gt; that can be found on &lt;a href=&quot;https://www2.math.ethz.ch/education/bachelor/lectures/fs2014/other/mla/ma.pdf&quot; rel=&quot;nofollow&quot;&gt;Notes&lt;/a&gt;, to see a version.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Already with those experiences you could advance on electromagnetism and both relativity's theories. Hand by hand with differential geometry of curves and surfaces à la Cartan of moving frames.&lt;/p&gt;&#xA;" OwnerUserId="5582" LastEditorUserId="5582" LastEditDate="2015-12-04T02:16:58.877" LastActivityDate="2015-12-04T02:16:58.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9649" PostTypeId="2" ParentId="8552" CreationDate="2015-09-20T12:54:05.990" Score="1" Body="&lt;p&gt;People are often keen to develop their knowledge of their current interests so, if you can find out what interests your student (sport, cars, travel, cookery, whatever), you can hang some elegant mathematics on that.&lt;/p&gt;&#xA;" OwnerUserId="5541" LastActivityDate="2015-09-20T12:54:05.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9650" PostTypeId="2" ParentId="9635" CreationDate="2015-09-20T18:56:46.480" Score="4" Body="&lt;p&gt;I'd suggest having a look at &lt;a href=&quot;http://wwwf.imperial.ac.uk/~dholm/books/index.html&quot; rel=&quot;nofollow&quot;&gt;Geometric Mechanics&lt;/a&gt; by Darryl Holm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The theme of developing mechanics in the framework of differential geometry has a considerable record at the graduate level, as indicated in Dave Renfro's comment.  At a level suitable for undergraduates, though,  the pickings may be slim. Somewhat more accessible than Abraham and Marsden might be &lt;a href=&quot;http://www.springer.com/us/book/9780387986432&quot; rel=&quot;nofollow&quot;&gt;Introduction to Mechanics and Symmetry&lt;/a&gt; by Marsden and Ratiu.&lt;/p&gt;&#xA;" OwnerUserId="717" LastActivityDate="2015-09-20T18:56:46.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9651" PostTypeId="2" ParentId="248" CreationDate="2015-09-20T22:18:15.487" Score="7" Body="&lt;p&gt;I like Gowers' &lt;a href=&quot;https://gowers.wordpress.com/2011/11/20/normal-subgroups-and-quotient-groups/&quot; rel=&quot;nofollow noreferrer&quot;&gt;fake history&lt;/a&gt; of normal subgroups.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/1014791&quot;&gt;This&lt;/a&gt; is also good. Especially if you can relate it to change-of-basis, and Weyl's famous quotation  &amp;quot;The introduction of a coördinate system is an &lt;a href=&quot;https://web.archive.org/web/20170708175940/http://isomorphism.es:80/post/95450363264/a-sudoku-of-linear-functionals&quot; rel=&quot;nofollow noreferrer&quot;&gt;act of violence&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://33.media.tumblr.com/0f5677189fc35ca88c3bafabe303e552/tumblr_inline_napx6dPrL21qbdydv.png&quot; alt=&quot;the introduction of a coördinate system is an act of violence&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some resources I like for the basis-free approach:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://linear.axler.net&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://linear.axler.net&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://unapologetic.wordpress.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://unapologetic.wordpress.com&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://sites.google.com/site/winitzki/linalg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://sites.google.com/site/winitzki/linalg&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://faculty.luther.edu/%7Emacdonal/laga/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://faculty.luther.edu/~macdonal/laga/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For an application to finance, note that &amp;quot;change of basis&amp;quot; = &amp;quot;change of numeraire&amp;quot;. Aaron Brown &lt;a href=&quot;https://books.google.com/books/about/Red_Blooded_Risk.html?id=X3sGahpmkQ0C&quot; rel=&quot;nofollow noreferrer&quot;&gt;calls&lt;/a&gt; change-of-numeraire &amp;quot;the most important tool&amp;quot; or something like that. (Also note in Emanuel Derman's advice to quants that saying complete markets exist because of Girsanov's theorem is considered bad form (putting the model above reality).&lt;/p&gt;&#xA;&lt;p&gt;Every currency, or perhaps any bond or share class, can be considered as a &amp;quot;numeraire&amp;quot; (every publicly-traded corporation could be viewed as issuing its own currency. The ability to buy eg other firms partly by giving stock and not only having to use cash, is one of the principal reasons cited to go public.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Change-of-basis also shows up famously in &lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-31-change-of-basis-image-compression&quot; rel=&quot;nofollow noreferrer&quot;&gt;image compression&lt;/a&gt; so again there's a practical reason to look at change-of-basis. Quotients of kernels can then be sold as &amp;quot;deeper, simpler thinking&amp;quot; about change-of-basis, once it's been accepted as important.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So this is perhaps a sketch for one long detour &lt;em&gt;off&lt;/em&gt; the path to normal subgroups &lt;em&gt;per se&lt;/em&gt;, onto the topic of change-of-basis. But if Gowers treats the small (&lt;span class=&quot;math-container&quot;&gt;$S_3$&lt;/span&gt;) case extremely well, and my analogy between change-of-basis and commutative subgroups is correct, then you can wave your hands in that direction with some pointers to further reading.&lt;/p&gt;&#xA;" OwnerUserId="5129" LastEditorDisplayName="user7990" LastEditDate="2020-09-17T12:16:29.017" LastActivityDate="2020-09-17T12:16:29.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9652" PostTypeId="2" ParentId="1348" CreationDate="2015-09-21T05:57:46.737" Score="0" Body="&lt;p&gt;Arguably, fractions provide a good first-pass at thinking abstractly, &lt;em&gt;without&lt;/em&gt; having to develop complete intuition by manipulatives. (Priming students for the algebra that comes next.) As a community college lecturer teaching this in our remedial classes, I give a single visual example of each operation, and then we need to focus on practicing the mechanics of those operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But one thing I do hammer on throughout the semester is estimations for double-check and self-correction purposes (which most of my students find overwhelming, baffling, and frequently never heard of before). So a lecture later there's a beautiful opportunity when multiplying/dividing mixed numbers to practice more estimation double-checks (round and operate on whole numbers), which hopefully serves to bolster confidence that the process is giving reasonable results.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-09-21T05:57:46.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9653" PostTypeId="2" ParentId="9636" CreationDate="2015-09-21T15:59:34.357" Score="3" Body="&lt;p&gt;Making your students to learn something is more important than grading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grading homework is good for one thing: &lt;em&gt;it makes your students to do the homework&lt;/em&gt;. No matter how much they cheat, it helps them to learn something. Therefore you have to grade it. &lt;/p&gt;&#xA;" OwnerUserId="3268" LastActivityDate="2015-09-21T15:59:34.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9654" PostTypeId="2" ParentId="9636" CreationDate="2015-09-21T16:37:18.103" Score="3" Body="&lt;p&gt;I like @Joonas explanation of why homework is valuable, and how it can philosophically fit in with exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that there are few students who go to the effort of copying an assignment, but not put in enough effort in understanding the assignment that they deserve no credit. Most students will at least attempt an assignment, or will make zero effort and not produce anything. So I am comfortable with giving a completion credit to any assignment that is turned in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I generally do not like grading homework for more than completion because I feel that places too much of an incentive to copy answers. I have wanted to make homework more effective and worth a greater portion of students' grades. In an effort to reduce the incentive on copying assignments and make homework more valuable I tried a couple strategies: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a tried to make grading work by checking homework at the end of sections, allowing time for students to complete work and fix mistakes if they didn't get the homework assignment at first. This worked reasonably well but was a great burden on my time. Grading tests and detailed notebooks at the same time was difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I required students to choose a problem from a subset of their homework to copy and submit. I liked this and have continue to use it for certain classes. Works well with other entrance/exit card strategies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One idea I have had, but never tried, would be to create quizzes that are similar to homework problems and allow the students to complete the quizzes with open notes. I have not tried this because of the class time required. I don't like the idea of how long it can take some students when presented with the option of using their notes, and I don't like the idea of making it timed and cutting them off.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2015-09-21T16:37:18.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9658" PostTypeId="2" ParentId="1245" CreationDate="2015-09-22T20:32:11.027" Score="-2" Body="&lt;p&gt;-1. Ask what &lt;span class=&quot;math-container&quot;&gt;$D^{50} \left\{ \ x^{100} \ \right\}$&lt;/span&gt; is. Stop them from actually multiplying the coefficients; just leave them unmultiplied the way you would for a prime decomposition. Define factorial and show how &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \prod_{50}^{100} \bullet = {100! \over 50!}$&lt;/span&gt; shows up naturally in this context of repeated derivatives. This proves that arbitrary number of &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;'s of a polynomial is easy.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;By fundamental theorem of algebra, &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Polynomials}$&lt;/span&gt; cover all functions (so, school algebra can work in the real world)&lt;/li&gt;&#xA;&lt;li&gt;By &lt;a href=&quot;http://qr.ae/RoMkR3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Taylor&lt;/a&gt;'s approximation, &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Finite\ Polynomials}$&lt;/span&gt; can get you close (so, it &lt;em&gt;actually&lt;/em&gt; could work, in the real-real world)&lt;/li&gt;&#xA;&lt;li&gt;Each polynomial &lt;a href=&quot;http://isomorphism.es/post/16482880064/polynomials-sequences-of-numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;is &quot;just a&quot; sequence&lt;/a&gt;. (The covector of coefficients to the one basis-vector &lt;span class=&quot;math-container&quot;&gt;$(\ldots, x^{-1}, x^0, x^1, x^2, \ldots)$&lt;/span&gt; (with &lt;span class=&quot;math-container&quot;&gt;$x^0$&lt;/span&gt; &lt;a href=&quot;https://medium.com/@isomorphisms/what-is-an-algebraic-variety-c8be09c662c9&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;like a dowsing rod…&quot;&gt;moving the root&lt;/a&gt; to wherever you want&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://medium.com/@isomorphisms/what-is-an-algebraic-variety-c8be09c662c9&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;pictures of moving the «root-finding stick» around&quot;&gt;&lt;img src=&quot;https://cdn-images-2.medium.com/max/995/1*36fhbW9FRk4IYOs88jM5UQ.jpeg&quot; alt=&quot;roots here&quot;&gt; &lt;img src=&quot;https://cdn-images-2.medium.com/max/999/1*BcBlUxj7srtZt9P-ujIxBQ.jpeg&quot; alt=&quot;roots there&quot;&gt; &lt;img src=&quot;https://cdn-images-2.medium.com/max/995/1*hrrodgUNWUGwM8yn98P2rg.jpeg&quot; alt=&quot;roots elsewhere&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;We can understand sequences. (There is an analogy to digits, possibly negative, in base &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, but that could be ignored.) We have a shift / lag operator (&lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;) on these sequences. (−1) So symbolic differentiation &amp;amp; integration of these letters / strings / symbols, gives us an &quot;Easy way to do calculus&quot; (No hard integrals! Only polynomials). If most functions can be written this way (1), and we can even know &lt;em&gt;beforehand&lt;/em&gt; (in the planning phase) how good of a job this approach will do (2), &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Now play around with them — do &lt;a href=&quot;http://tmblr.co/ZdCxIyy2FhcN&quot; rel=&quot;nofollow noreferrer&quot;&gt;ℂ→ℂ Wegert-plots&lt;/a&gt; and Cartesian ℝ→ℝ plots, &lt;em&gt;and&lt;/em&gt; &lt;a href=&quot;http://isomorphism.es/post/38251878380/differential-pushforward&quot; rel=&quot;nofollow noreferrer&quot;&gt;input/output tables &lt;img src=&quot;https://33.media.tumblr.com/7b578ae7121a9eedd86c7f1fd92385b9/tumblr_inline_mf8rimE0T81qbdydv.png&quot; alt=&quot;seq,id&quot;&gt; &lt;img src=&quot;https://33.media.tumblr.com/10630c27257412bf5404e5388bb42847/tumblr_inline_mf8td3dGsT1qbdydv.png&quot; alt=&quot;sine, circle&quot;&gt; &lt;/a&gt; to see how varying the &quot;sliders&quot; (entries in the covector of coefficients) changes the shape of the output in the various views (Cartesian, Wegert, tabular). A computer/interactive setup could be the best here. Just mess around and try to get the output to do cool stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/DHgET.png&quot; alt=&quot;exp(ζ)/(ζζζ−ζζ−ζ+i)&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;exp(ζ)/(ζζζ−ζζ−ζ+i)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://40.media.tumblr.com/ef47436d03b4283701fa5498f81b0717/tumblr_muwotcAaJE1qc38e9o4_1280.png&quot; alt=&quot;{ ζζζ−ζζ−ζ+i } / { log(ζ)+i }&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;{ ζζζ−ζζ−ζ+i } / { log(ζ)+i }&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(For the ℂ→arg ℂ plots you want to tell them that arguments swirl around the roots (&lt;span class=&quot;math-container&quot;&gt;$\arg \zeta$&lt;/span&gt; takes on every argument at 0) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://qph.is.quoracdn.net/main-qimg-4441055b1ebb2f5667997ff531b6b75a?convert_to_webp=true&quot; alt=&quot;some ℝ polynomial&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For applications, you can point to some interesting things to do with ℂ maps (dynamical systems, root finders, some of &lt;a href=&quot;http://arminstraub.com/math/what-is-column&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://arminstraub.com/math/what-is-column&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further applications you can discuss net present value, the geometric series (and &lt;a href=&quot;http://isomorphism.es/post/820377329/geometric-series&quot; rel=&quot;nofollow noreferrer&quot;&gt;the dumb geometric-series trick&lt;/a&gt;), the second dumb trick of subtracting two &lt;a href=&quot;http://isomorphism.es/post/78869644070/learn-calculus-first-time-no-background&quot; rel=&quot;nofollow noreferrer&quot;&gt;shifted&lt;/a&gt; geometric series to get a finite one (since no-one lives forever), and how although rates-of-return might vary in reality, we've now carved out some &quot;easy cases&quot; that we can do back-of-the-envelope, as quickly as &lt;span class=&quot;math-container&quot;&gt;$1 \over 1-r$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$- \mathrm{ shift} \left\{ 1 \over 1-r \right\}$&lt;/span&gt; to put some easy bounds on real-world cases.   (If your students don't already know how to compute a NPV in Excel, then here is your chance to teach them the only mathematics they will actually need to know in 90% of office jobs.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also tell them they now understand (finally) what &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is. All the little kids are just made to use it, they can now explain it. (Identify it with the covector &lt;span class=&quot;math-container&quot;&gt;$(1,1,1,1,1,\ldots)$&lt;/span&gt; once you adjust by &lt;span class=&quot;math-container&quot;&gt;$/ \sim k!$&lt;/span&gt;.) And show how &lt;span class=&quot;math-container&quot;&gt;$D(e^{\bullet}) = e^{\bullet}$&lt;/span&gt; via &lt;span class=&quot;math-container&quot;&gt;$(1,1,1,1,\ldots) \overset{D}{\mapsto} (1,1,1,\ldots)$&lt;/span&gt;. Reinforce the shift-arithmetic on symbols p.o.v., mention why this makes &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt;'s a good basis for ODE's, and mention the &quot;slick&quot; version of instantiating &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; (simply assume it) --- versus seeing it in series. (So which is better? Chance to make a value judgement / form an opinion on mathematical beauty.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;It's not a bad time either to talk about the history of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;: it was only possible to discover this &quot;algebraic&quot; / &quot;axiomatic&quot; / &quot;slick&quot; thing (with so many implications) by using stuff from another part of mathematics (analysis, but let's call it series / string-ops). By staring at and doodling around (mention Ramanujan) with the &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt; series, &lt;em&gt;only if&lt;/em&gt; you've &lt;em&gt;also&lt;/em&gt; tried to approximate the circle height function (&lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt;) and its cousin by infinite-approximation (mention Aristotle)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; comes from the intersection of different areas of mathematics, i.e. from different ways of thinking being tied together by writing down on paper. This is a thesis about the way mathematics works as well as about creativity, exploration, proof, dealing with your own ideas, how paper can help you organise your thoughts. (in other domains perhaps, as well)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now the discussion has gone from meaningless symbol-scratching to high-level.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The usual approach (Stewart) is, in my opinion, insane. I would show the Stewart chapters as enrichment. This is heavy symbol scratching with no motivation why it matters at all. Yet another exercise in making people do mechanical calculations, instead of teaching them anything about mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Series are here to make our lives easier. They either make the impossible possible (formal power series, Ramanujan stuff) , make hard integrals turn into shift-operators, or they give you some peace-of-mind that (pretty much) any ℝ→ℝ mapping the real world might throw at you, can be reduced to some stuff you could do on paper, in a symbolic differentiator, etc. Reduced to essentially a string / list.&lt;/p&gt;&#xA;" OwnerUserId="5129" LastEditorUserId="6053" LastEditDate="2019-01-01T14:36:40.920" LastActivityDate="2019-01-01T14:36:40.920" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9659" PostTypeId="1" CreationDate="2015-09-23T02:58:06.037" Score="5" ViewCount="212" Body="&lt;p&gt;At the behest of a &lt;a href=&quot;https://mathoverflow.net/questions/218983/teaching-an-abstract-algebra-class-involving-modules-best-way-to-introduce-oper#comment539746_218983&quot;&gt;comment&lt;/a&gt; on Mathoverflow, I am posting this here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am about to teach a few classes on modules and their operations, namely the following: direct product, direct sum, and finitely generated modules. The following is the &quot;skeleton&quot; version of how I plan to talk about them.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The direct sum of modules $M_1, M_2$ is defined to be$$M_1 \oplus M_2 = \{(m_1, m_2) : m_1 \in M_1, m_2 \in M_2\}.$$The action of $A$ is given by $a(m_1, M_2) : = (am_1, am_2)$. More generally, for any set $I$ and any collection of $A$-modules $\{M_i\}_{i \in I}$, one has a &lt;strong&gt;direct product&lt;/strong&gt; $A$-module$$\prod_{i \in I} M_i = \{(m_i \in M_i)_{i \in I}\}.$$We will often write an element of $\prod_{i \in I} M_i$ in the form $\sum_{i \in I} m_i$ instead of $(m_i \in M_i)_{i \in I}$.&lt;/li&gt;&#xA;&lt;li&gt;The $A$-module $\prod_{i \in I} M_i$ contains a submodule$$\bigoplus_{i \in I} M_i = \{(m_i \in M_i)_{i \in I} : m_i = 0 \text{ for all but finitely many }i\},$$called &lt;strong&gt;direct sum&lt;/strong&gt;. In particular, for any $n \ge 1$, we write$$M^n := \underbrace{M \oplus \dots \oplus M}_{n \text{ times}}.$$&lt;/li&gt;&#xA;&lt;li&gt;The sum of a collection of submodules $M_i \subseteq M$, $i \in I$, is defined to be$$\sum_{i \in I} M_i = \{m_{i_1} + \dots + m_{i_j} : m_{i_j} \in M_{i_j}\}.$$&lt;/li&gt;&#xA;&lt;li&gt;The submodule generated by a fixed element $m \in M$ is defined to be$$Am = \{am : a \in A\}.$$More generally, for any collection $\{m_i\}_{i \in I} \subset M$, of elements of $M$, one has$$\sum_{i \in I} Am_i \subset M,$$the submodule generated by the $m_i$. We say that $M$ is &lt;strong&gt;finitely generated&lt;/strong&gt; if there exist $m_1, \dots, m_k \in M$ such that $M = Am_1 + \dots + Am_k$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What are some good things to add about these things which other people have found helpful when teaching/learning about these things?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best,&#xA;C&lt;/p&gt;&#xA;" OwnerUserId="5613" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-09-23T02:58:06.037" Title="Teaching an abstract algebra class involving modules, best way to introduce operations on modules?" Tags="|intuition|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9660" PostTypeId="2" ParentId="1245" CreationDate="2015-09-23T12:58:25.547" Score="1" Body="&lt;p&gt;I motivate series for their primary application in the course, representing functions as power series.  And I motivate power series in a couple of ways.  First, they are a vast generalization of polynomials whose differentiation and integration rules are basically as simple as the power rule for polynomials (plus the implicit linear structure of differentiation and integration).  Never underestimate the value of something that can be easily computed!  Second, Taylor series are refinements of the key idea of calculus -- linear approximation.  They are used to approximate transcendental quantities using only the four basic arithmetic operations.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2015-09-23T12:58:25.547" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9661" PostTypeId="2" ParentId="530" CreationDate="2015-09-23T12:58:42.290" Score="6" Body="&lt;p&gt;After coming back to this question, it strikes me that in fact the best thing might be to explicitly teach both. Not to accommodate the advantages of $\tau$ with the entrenchment of $\pi$, but simply because it makes clearer that pi is a convention, has a definition, and is not just some magical formula given to us. It is a choice. The real thing that is given as a fact is that a circle twice as large is twice as long, after that we name one ratio  or the other (perimeter over diameter or over radius), but it does not matter that much which one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once had to teach some trigonometry using complex numbers, and asked myself for one quarter of an hour how damn to prove that $e^{i\theta}$ equals $1$ precisely at $\theta=2\pi$, until I realized that in that setup it was nothing else than the definition of $\pi$. I guess many students also need to be reminded that once in a while, and teaching $\tau$ is probably a good opportunity.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-09-23T12:58:42.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9664" PostTypeId="2" ParentId="224" CreationDate="2015-09-25T15:15:25.443" Score="0" Body="&lt;p&gt;There's a paper available that might be relevant:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Henz, M., &amp;amp; Hobor, A. (2011). Teaching Experience: Logic and Formal Methods with Coq. In J.-P. Jouannaud &amp;amp; Z. Shao (Eds.), Certified Programs and Proofs (Vol. 7086, pp. 199–215). Berlin, Heidelberg: Springer Berlin Heidelberg. &lt;a href=&quot;http://doi.org/10.1007/978-3-642-25379-9_16&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1007/978-3-642-25379-9_16&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5625" LastActivityDate="2015-09-25T15:15:25.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9665" PostTypeId="2" ParentId="8371" CreationDate="2015-09-25T17:33:58.293" Score="0" Body="&lt;p&gt;You may be able to tweak an ordinary problem into one that fits your needs.  Here are a some suggestions, the first two involving optimizing with linear functionals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Optimize on an interesting parameterized convex surface, such as an ellipsoid or an appropriate portion of a hyperboloid.  You might even compare that with solving on a sphere.  The idea is to &quot;follow the normal until you fall off the constraint&quot;, but for a sphere that point is easy to compute in ones head, while not so much for an ellipsoid or section of a hyperboloid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Draw a picture of a surface (say two hills), and put a table of 14 points marked on the surface, with a table of the labels and the coordinates of the points.  Give a linear functional F and ask which points of the surface are likely to be values that yield near-optimal values for F on the surface.  Although they can solve it by computing F on each of the 14 points, the exercise is to imagine the hyperplane moving through the surface and finding points of tangency.  This tests a geometric intuition for what is going on, and gives them a chance to verify it by computing F at the one or two points that are near optimal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Try a squared &quot;distance&quot; from a point, where squared &quot;distance&quot; is determined by some quadratic form ((x -x')^2 + 2(y-y')^2 is a simple variation).  Even if they change coordinates to solve the problem more simply, the have Lagrange multipliers to fall back on to confirm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Also Check Points On Boundary&quot; Paseman, 2015.09.25&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-09-25T17:33:58.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9666" PostTypeId="2" ParentId="8504" CreationDate="2015-09-25T18:06:13.550" Score="1" Body="&lt;p&gt;Similar to the double it, double it again, double it again, but a little more advanced. If students know their 4x tables, they can use those as well. Example: I don't remember 8×7, but I know 4x7=28. So, 8×7=2(4×7)=2×28=40+16=56. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, like the 9 strategy, students can use their 10's. Example: I don't remember 8x7, but I know 10x7=70, and I know 2x7=14. So, 8×7=(10×7)-(2×7)=70-14=60-4=56. &lt;/p&gt;&#xA;" OwnerUserId="5618" LastActivityDate="2015-09-25T18:06:13.550" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9667" PostTypeId="1" AcceptedAnswerId="9674" CreationDate="2015-09-26T10:10:42.160" Score="15" ViewCount="40591" Body="&lt;p&gt;To contextualize: I know someone who is ten years old, and needed to use repeated addition to compute $4 \times 8$, i.e., needed to calculate it as $4 \times 8 = 8+8+8+8$.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; By what age should a kid have memorized the multiplication table?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5628" LastEditorUserId="262" LastEditDate="2015-09-27T00:48:01.073" LastActivityDate="2020-01-22T02:42:32.187" Title="When should a kid have memorized the multiplication table?" Tags="|primary-education|multiplication-table|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="9668" PostTypeId="1" AcceptedAnswerId="9670" CreationDate="2015-09-26T10:57:47.240" Score="7" ViewCount="276" Body="&lt;p&gt;I am writing a &lt;a href=&quot;http://sklivvz.com/posts/two-scary-signs-that-a-site-is-unsafe&quot; rel=&quot;nofollow noreferrer&quot;&gt;blog post about cryptography&lt;/a&gt; and I need to show some examples that involve sums modulo N.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like (if possible) to use a different &quot;plus&quot; symbol for these sums. The kind of examples I am writing are tabular look like this&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CvzAt.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CvzAt.png&quot; alt=&quot;example sum&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which is a base 36 sum modulo 36. If it was in line I could just write $( A + 4 )_{mod\ 36} = E$, but that's not a possibility in the context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it appropriate the circled plus ($\oplus$) in this case? &lt;/p&gt;&#xA;" OwnerUserId="24" LastEditorUserId="24" LastEditDate="2015-09-27T09:52:17.613" LastActivityDate="2015-09-27T09:52:17.613" Title="What symbol is appropriate to represent a sum modulo N?" Tags="|notation|" AnswerCount="3" CommentCount="7" ClosedDate="2015-09-27T18:51:52.250" ContentLicense="CC BY-SA 3.0" />
  <row Id="9669" PostTypeId="1" CreationDate="2015-09-26T12:24:10.080" Score="14" ViewCount="365" Body="&lt;p&gt;Has there been any attempts at developing a curriculum for teaching analysis (here let us be narrow and say real analysis in the sense of rigorous integral and differential calculus) in a multipass, spiral approach?&lt;/p&gt;&#xA;&lt;p&gt;If there is, has there been any studies on its efficacy? Or any guidelines on best practices? Has any textbook been written from this point of view?&lt;/p&gt;&#xA;&lt;p&gt;To a certain extent that is what we do a bit already: in many institutions the students are expected to first learn calculus and multivariable calculus from the algorithmic, computational point of view, and then re-learn it in an analysis class with more rigorous foundations. But frequently this is a large jump from 0 to 1 in terms of rigor. I am wondering whether someone has developed a textbook or curriculum for a 3 or 4 pass spiral approach to the increase in &amp;quot;rigor&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;An example for what I am thinking about is, on the topic of integration, maybe doing a multipass approach would lead to something like&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The first pass: integration by taking antiderivatives; fundamental theorems of calculus used axiomatically and not justified, ditto mean value theorems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The second pass: Riemann integration of (uniformly) continuous functions, with tagged partitions and so on. The first mean value theorem and the first fundamental theorem proven in detail. Prove that the evaluation of the Riemann integral is linear. Some basic approximation/quadrature theorems and error estimates.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The third pass: Darboux version, start with approximations by piecewise linear or piecewise constant functions, take the sup/inf appropriately. Define the class of (Riemann) integrable functions, prove that it forms a real algebra. Reprove in this generality the first mean value theorem and the second (the monotonic one) mean value theorem, as well as the first and second fundamental theorems, and the Lebesgue criterion (in a suitable formulation).&lt;/p&gt;&#xA;&lt;p&gt;Possibly also talk about directed sets and nets (useful for taking the directed limit over partitions for defining the Darboux integral, and also for topology later) and various other techniques.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Fourth pass (optional): Riemann-Stieltjes integration, redo the relevant theorems, integration by parts for RS integration, and expose the second mean value theorem as essentially one integration by parts plus the first mean value theorem.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="125" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-10-19T17:52:44.030" Title="Spiral learning in real analysis" Tags="|reference-request|curriculum|general-pedagogy|mathematical-analysis|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9670" PostTypeId="2" ParentId="9668" CreationDate="2015-09-26T14:25:28.713" Score="5" Body="&lt;p&gt;To use a special plus-symbol like $\oplus$ can be appropriate in this context, depending on what you want to high-light. It is quite common at least at the start of presenting some new structures to use &quot;special&quot; symbols for the operations and often they are derived from the usual symbols for addition and multiplication. This serves well as a reminder to the reader that this is not good-old addition but something else, yet still some kind of addition. Later on, one will often fall-back to the standard symbols on the grounds that &quot;it is clear from context&quot; what is meant.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The specific symbol you propose can be a good choice, as it is a standard symbol visually close to $+$, provided you do not want to discuss &lt;a href=&quot;https://en.wikipedia.org/wiki/Direct_sum&quot;&gt;direct sums&lt;/a&gt; or related. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A common use of $\oplus$ is to denote the direct sum of (commutative) groups. For example, it is common to write things like $\mathbb{Z}/\mathbb{7}Z \oplus \mathbb{Z}/49\mathbb{Z}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There should be no actual risk of confusion though. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternative ideas include: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\hat{+}$&lt;/li&gt;&#xA;&lt;li&gt;$+_{n}$ with $n$ the modulus. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="143" LastActivityDate="2015-09-26T14:25:28.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9671" PostTypeId="2" ParentId="9668" CreationDate="2015-09-26T20:19:02.000" Score="4" Body="&lt;p&gt;Depending on your approach, you could also use $(A+4) \% 36 = E$, with $\%$ the binary operator $A\%B$ being the &lt;em&gt;reduction&lt;/em&gt; of $A$ modulo $B$. At least this is a very common programming-language symbol.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast, although some introductory sources use $\oplus$ or $+$ with funny marks on it, these are absolutely not standard, &lt;em&gt;and&lt;/em&gt; I have witnessed people believe implicitly that all mathematical notation is universal and &quot;global scope&quot;, so they have trouble recovering from thinking that $\oplus$ is funny kind of operation on funny numbers (as opposed to direct sum of suitable objects...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For that matter, my experience teaching basic crypto to computer science and engineering people suggests to me that it is almost pointlessly difficult to get across the idea of integers-mod-$N$ as equivalence classes, and the operation of addition therein as something in its own right. Instead, for most basic purposes I found that the $n\% N$ operation was more palatable, and could be explained quickly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then if you're addressing more sophisticated people, very possibly you can use plain-old $+$ and just say &quot;in $\mathbb Z/36$&quot;, rather than have non-standard symbols.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-09-26T20:19:02.000" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9672" PostTypeId="2" ParentId="8268" CreationDate="2015-09-27T01:16:31.890" Score="3" Body="&lt;p&gt;Students could investigate some simple hydraulics such as the pressure per square inch relationships used to power hydraulic jacks. The calculation of these is simple, staying within the use of proportions and that should be within the abilities of seventh graders. Just learning the calculation of lifting power could be very dry, (pun intended), but the ability to manipulate a very heavy object would get kids' attention. Many theme parks, (notably Disney World) have a sculpture of a large ball, usually made of granite or other HEAVY material. This ball is located in a socket and water is pumped in to the reservoir under the ball effectively lifting the ball on a thin layer of water. Since the friction between the ball and the water has a very small value, it is easy to spin the heavy ball.&#xA;There could be investigation into the friction coefficients of wet and dry surfaces leading into car safety discussions.&#xA;Students could also construct a working physical version of this.&lt;/p&gt;&#xA;" OwnerUserId="5629" LastActivityDate="2015-09-27T01:16:31.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9673" PostTypeId="2" ParentId="9668" CreationDate="2015-09-27T02:29:27.277" Score="1" Body="&lt;p&gt;In brief: it's your article, you can do what you want.  You also reap the reward.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the idea is to just mention arithmetic in base 36, without any other context, you can use your own algebraic symbols.  That there is a relation to base 10 addition might suggest your using a symbol with a plus shape, but by itself it may not matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the idea is to do arithmetic in base 36 and in other bases, it might make sense to use different universes and different symbols.  So your example might be instead box{A} box{+} box{4} box{=} box{E} .  I use box{} to indicate that for each symbol that normally represents an A or a 4, I am using a different symbol that is related to the symbol I normally use, so that I have some familiar handle to use to grapple with these unfamiliar objects and strange new operations and relations.  For each different base, use a different box.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the idea is to do arithmetic in many bases, AND SHOW HOW THEY ARE RELATED TO WHAT YOU ARE DOING, then I recommend a different approach.  You should use a + for something familiar, another symbol for an operation in an unfamilar base, and a procedure or mapping or way of translating one problem to another.  You should say that normally you might do a base 10 computation like 10+4 to get 14, but because it is base 36, you have to &quot;box&quot; everything and so the computation, while on some level is the same, has a different representation: box{10+4} gives box{10} box{+} box{4} which is A_ +_ 4_ , which is E_ in our new representation.  Hey, box{14} is also E_!  Maybe this new world isn't so strange after all!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It really matters what connections you are trying to establish.  I think people should know about the mapping as well, so that they can make the translation themselves.  Hide that mapping, and the reader is left with wondering why bother with a different base.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Notation Is Important; Intent Importanter&quot; Paseman, 2015.09.27&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-09-27T02:29:27.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9674" PostTypeId="2" ParentId="9667" CreationDate="2015-09-27T02:34:39.883" Score="26" Body="&lt;p&gt;As an example of one curriculum, the Common Core standards say that students, &quot;By the end of Grade 3, know from memory all products of two one-digit numbers.&quot; (&lt;a href=&quot;http://www.corestandards.org/Math/Content/3/OA/C/7/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.corestandards.org/Math/Content/3/OA/C/7/&lt;/a&gt;) This matches my personal experience, too, for what it's worth, in the 1970's. In the U.S., this would be around age 8 or 9 years old. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, as someone who teaches community college remedial math, I'll note that in practice there are large cohorts of any age who have not actually memorized the times tables (age 18, 20, 30, etc.). This skill is taken as the fundamental prerequisite for our lowest-level arithmetic courses. As an informal observation, people who think that times table memorization is optional likely aren't involved in teaching algebra or higher-level subjects (including work with fractions, factoring, etc.). Students who haven't memorized the times tables are close to helpless when they get to one of our algebra classes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To support my students in college remediation I established a website with timed quizzes, having times tables as the first and most fundamental skill. For some students it's the first occasion that they've been made aware that automaticity in &quot;knowing times tables&quot; is &lt;i&gt;not&lt;/i&gt; the same as &quot;being able to repeatedly add&quot;. See: &lt;a href=&quot;http://www.automatic-algebra.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.automatic-algebra.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-01-22T02:42:32.187" LastActivityDate="2020-01-22T02:42:32.187" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="9675" PostTypeId="2" ParentId="352" CreationDate="2015-09-27T03:08:47.080" Score="22" Body="&lt;p&gt;Reasons for having automaticity with single-digit times tables (from the perspective of a community college lecturer with many remedial courses):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Long multiplication algorithm &lt;/li&gt;&#xA;&lt;li&gt;Long division algorithm, and thus:&lt;/li&gt;&#xA;&lt;li&gt;Convert fractions to decimals&lt;/li&gt;&#xA;&lt;li&gt;Understand why rational numbers have repeating decimal expansions&lt;/li&gt;&#xA;&lt;li&gt;Understand the proof why $\sqrt{2}$ is irrational&lt;/li&gt;&#xA;&lt;li&gt;Factor integers, and thus:&lt;/li&gt;&#xA;&lt;li&gt;Understand the Fundamental Theorem of Arithmetic&lt;/li&gt;&#xA;&lt;li&gt;Reduce fractions by finding the LCD, and thus:&lt;/li&gt;&#xA;&lt;li&gt;Add, subtract, multiply, divide, and compare fractions&lt;/li&gt;&#xA;&lt;li&gt;Factor polynomials, and thus:&lt;/li&gt;&#xA;&lt;li&gt;Reduce rational expressions&lt;/li&gt;&#xA;&lt;li&gt;Solve higher-degree equations&lt;/li&gt;&#xA;&lt;li&gt;Understand the Fundamental Theorem of Algebra&lt;/li&gt;&#xA;&lt;li&gt;Estimations to double-check technology output&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Perhaps on a deeper level I'd say that the base-10 place value writing system was architected specifically to easily support these operations, assuming that single-digit elementary operations were memorized (like phonemes) -- so if someone hasn't done that, they're really not using the language correctly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students should certainly know how to multiply any arbitrary number by 10. For products of two numbers above 10, I would agree that it's not really critical -- although the 11-table is trivial and knowing the 12-table may be handy when dealing with clocks, inches, and units in dozens (and $12^2$, one gross, does pop up a lot). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2015-09-27T03:50:20.987" LastActivityDate="2015-09-27T03:50:20.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9676" PostTypeId="2" ParentId="8541" CreationDate="2015-09-27T04:07:43.760" Score="5" Body="&lt;p&gt;I'm growing fond of the OpenStax open-source books. I haven't used them in class yet, but they've recently gotten good enough where I'd be happy to do so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One option is to use bits from two of their books. College Algebra Ch. 2, 3, 4, 6 (equations, functions, linear functions, exponential functions), and Introductory Statistics Ch. 2, 12 (descriptive statistics, linear regression) would seem to cover most of your needs. The initial percentage discussion you'd have to fill in yourself (or from your Prealgebra book).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://openstaxcollege.org/books&quot; rel=&quot;noreferrer&quot;&gt;https://openstaxcollege.org/books&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-09-27T04:07:43.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9677" PostTypeId="2" ParentId="6089" CreationDate="2015-09-27T04:16:30.053" Score="8" Body="&lt;p&gt;&quot;She knows the 10x10 multiplication table in principle, but it takes a huge amount of time till she recalls the result. Sometimes she even needs some seconds to calculate 2x3=6.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is absolutely &lt;em&gt;not&lt;/em&gt; knowing the multiplication table. As a community college lecturer with lots of remedial courses, I see this a lot; a student will &lt;em&gt;say&lt;/em&gt; they know the times tables, but they're really sequentially adding mentally. Frankly, they don't understand that &quot;knowing times tables&quot; means instant recall, total automaticity. Later on they'll be near-helpless when it comes to factoring, fractions, estimating, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the first thing in this case is that she really needs to sit down and memorize the times tables for real. I tell my students to practice it in a timed environment every day until they never make any mistakes. Here's the website I set up for that; note that the first and fundamental skill is times tables: &lt;a href=&quot;http://www.automatic-algebra.org/&quot; rel=&quot;noreferrer&quot;&gt;http://www.automatic-algebra.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-09-27T04:16:30.053" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9678" PostTypeId="2" ParentId="9667" CreationDate="2015-09-27T10:02:37.953" Score="7" Body="&lt;p&gt;Age is not a good marker of mathematical ability as abilities can differ so much.  In addition I'd add that having a good memory obviously an advantage in the mathematical  development of a child but it is not a prerequisite nor a guarantee of success.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On to 'knowing' your times tables.  It is tempting to say that someone that recalls $8\times 4 = 32$ knows that times table but the reality is more complex.  A child that knows their times tables should be able to answer questions like, &quot; Which times table is $32$ , $56$ and $96$ in? &quot;  Could there be more than one? Why is $8\times 4=2\times 16$ ? There are other examples but the point is that recalling the facts isn't enough to 'know' them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be clear recalling multiplication  facts is an integral part of future development but caution is required or frustration can set in.  &lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2015-09-27T10:02:37.953" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9679" PostTypeId="1" AcceptedAnswerId="9681" CreationDate="2015-09-28T14:50:34.600" Score="3" ViewCount="245" Body="&lt;p&gt;Since I will be creating a calculus course, I am hoping to find calculus material used in the best colleges across the globe as a reference. Lecture notes and exercise sheets are highly appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know any college that showcases their course?&lt;/p&gt;&#xA;" OwnerUserId="4645" LastEditorDisplayName="user507" LastEditDate="2016-11-24T21:45:15.323" LastActivityDate="2016-11-24T21:45:15.323" Title="Colleges that showcase their calculus material online" Tags="|calculus|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9680" PostTypeId="1" CreationDate="2015-09-28T16:14:43.850" Score="4" ViewCount="114" Body="&lt;p&gt;I am presently teaching eleventh grade (XI standard) students an introductory course in co-ordinate geometry with a focus on preparations for competitive exams. I have seen books like S.L.Loney's co-ordinate geometry which is like an encyclopedia of co-ordinate geometric results. I feel that an encyclopedia does not actually give a good taste of a field. So I wanted interesting results and techniques that arise from co-ordinate geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My present focus is points and straight lines. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have observed that co-ordinate geometry has very mysteriously simple formulas for many problems. For example, if $L(x,y) = ax+by-c$ where $a^2+b^2 =1$, if $P$ is any point on the plane, $L(P)$ represents the perpendicular distance of the point from the plane (even the sign has meaning). It follows that if two such lines $L_1$ and $L_2$ are given, then $L_1 \pm L_2 = 0$ represent the two angle bisectors through the intersection point of $L_1$ and $L_2$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I thought that this behavior is due to vectors and started teaching the course by viewing a line $L$ as $(a,b)\cdot(x,y)^T = c \implies \overline{w}\cdot \overline{r} = c.$ Now associated with every line $L: \overline{w}\cdot \overline{r} = c$ not passing through origin, we have a unique vector $\frac{\overline{w}}{c}$. From now on we will assume $c=1$. A cool theorem here is: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A family of lines $L(\overline{w})$ is concurrent at a point iff the&#xA;  associated vectors $\overline{w}$ are collinear.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I was enthralled by this observation. However this quickly started to fade when I realized I could not get a meaningful translation to all the formulae of the triangle. I am looking at other approaches like Mobius's barycentric co-ordinates and projective geometric notions (by going to three dimensions and then doing two dimensional geometry on a sliced plane).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I have the following precise questions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1) Are there simple formulas for the medians, altitudes and other cevians&#xA;  in terms of $\overline{w}$ if one is given three lines&#xA;  $L(\overline{w_1}),L(\overline{w_2}),L(\overline{w_3})$ which form the&#xA;  sides of the triangle?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;2) Are there alternative approaches to points and straight lines which&#xA;  makes certain proofs (and concepts) simpler?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; &#xA;I am aware of the duality between lines and points used in modern expositions of vector spaces (functional analysis) and I know that is related to my exposition. I am also aware of Poncelot duality and the projective geometric idea of duality in theorems about collineations (Pascal's theorem, Desargues theorem and so on). I am aware of Hilbert spaces which are complete vector spaces endowed with a notion of angle and Banach spaces are complete spaces endowed with a distance. However, I want to connect these modern viewpoints to concrete non-trivial elementary examples. So I would be happy if a knowledgeable reader shares examples of theorems in co-rdinate geometry which have deeper analogs in advanced mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S: I asked this question on math stackexchange today morning and I was directed to this site. See the question here: &lt;a href=&quot;https://math.stackexchange.com/questions/1454723/on-teaching-elementary-co-ordinate-geometry&quot;&gt;https://math.stackexchange.com/questions/1454723/on-teaching-elementary-co-ordinate-geometry&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="368" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-09-28T16:14:43.850" Title="Making co-ordinate geometry interesting for XI grade students" Tags="|mathematical-pedagogy|analytic-geometry|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9681" PostTypeId="2" ParentId="9679" CreationDate="2015-09-28T19:16:52.543" Score="5" Body="&lt;p&gt;MIT offers a lot of their class materials online through their &lt;a href=&quot;http://ocw.mit.edu/index.htm&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;OpenCourseWare initiative&lt;/strong&gt;&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find a list of their calculus classes &lt;a href=&quot;http://ocw.mit.edu/courses/find-by-topic/#cat=mathematics&amp;amp;subcat=calculus&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastEditorUserId="143" LastEditDate="2015-09-30T10:14:33.357" LastActivityDate="2015-09-30T10:14:33.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9682" PostTypeId="2" ParentId="9679" CreationDate="2015-09-29T14:14:13.340" Score="1" Body="&lt;p&gt;There are several &quot;free&quot; (or very cheap) &quot;textbook&quot; initiatives, like &lt;a href=&quot;http://trillia.com&quot; rel=&quot;nofollow&quot;&gt;Trillia&lt;/a&gt;, William Chen has a set of &lt;a href=&quot;https://rutherglen.science.mq.edu.au/~maths/Chen-notes/ln.html&quot; rel=&quot;nofollow&quot;&gt;lecture notes&lt;/a&gt; that I like a lot, asking Google for &quot;free mathematics textbooks&quot; gives many hits. Searching for specific topics will net you lots of more or less formal notes, slides, webpages for courses (often with homework and exams and solutions).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-09-29T14:14:13.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9683" PostTypeId="2" ParentId="9620" CreationDate="2015-09-29T16:04:35.190" Score="1" Body="&lt;p&gt;I find the books by William Dunham (&quot;Euler: Master of Us All&quot;, &quot;Journey Throug Genius&quot;, &quot;The Mathematical Universe&quot;, there are others) very inspiring. Mostly at high-school level, but they require quite a bit of effort to really grasp the details. Aigner and Ziegler's &quot;Proofs from THE BOOK&quot; is a bit harder, but shows off many dazzling proofs. None are really about &quot;what is math at college like&quot;, but they do show a bit of the history and some gems.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-09-29T16:04:35.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9684" PostTypeId="2" ParentId="9667" CreationDate="2015-09-29T23:23:27.090" Score="11" Body="&lt;p&gt;I hope I can contribute an answer with a story.  I failed one grading period in 3rd grade math because I stubbornly refused to memorize the multiplication tables.  At that age I understood that multiplication was explained as repetitive addition. My reasoning was &quot;why memorize that which I can derive?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I failed because quizzes were timed, and I couldn't complete the repetitive addition fast enough.  My teacher could see what I was doing, since the I did the addition in the margins.  It still amazes me that she never pulled me aside and explained WHY memorization was important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is memorization important?  I can still only think of two related reasons: (1) It's needed for the fluency required for higher math; and (2) division (and other kinds of factoring) requires the immediate recognition afforded by memorization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when should students memorize multiplication tables?&#xA; 1. After they have mastered addition. &#xA; 2. Before they are introduced to division.&lt;/p&gt;&#xA;" OwnerUserId="5465" LastActivityDate="2015-09-29T23:23:27.090" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="9685" PostTypeId="2" ParentId="1520" CreationDate="2015-09-30T03:45:35.153" Score="2" Body="&lt;p&gt;I would call this an issue of measurement granularity. Consider this blurb from the Six Sigma process site:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Establishing the adequacy of your measurement system using a&#xA;  measurement system analysis process is fundamental to measuring your&#xA;  own business process capability and meeting the needs of your customer&#xA;  (specifications). Take, for instance, cycle time measurements: It can&#xA;  be measured in seconds, minutes, hours, days, months, years and so on.&#xA;  There is an appropriate measurement scale for every customer&#xA;  need/specification, and it is the job of the quality professional to&#xA;  select the scale that is most appropriate.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Consider the case where a practitioner in the field is measuring angles with a protractor or similar instrument, based on radians, i.e., approximately just 6 units per complete circle. Obviously that would be too imprecise, so you'd need subdivisions: perhaps the protractor is marked in hundredths of a radian? So then almost every measurement one makes would have to document some decimal places (e.g., 1.57 radians). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Difficulties of this approach: It's a bit inefficient to track a &quot;point&quot; for every single measurement. Important angles cannot be exactly expressed in this system -- 1.57 radians is only approximately a right angle, but you can't be any more precise with this level of granularity. As a result, it's also a bit difficult to calibrate or manufacture the protractor (granted that a full circle is actually an irrational number, not constructable by compass-and-ruler geometry, etc.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So for convenience sake, it's reasonable to rescale the measurement to a finer granularity, such that practitioners are always dealing with whole numbers -- hopefully neither too big nor too small (see the example above for different options used for time measurements). One option would be to simply scale by a factor of 100 for the protractor in hundredths of a radian -- in fact, that's exactly analogous to the utility of using units of &quot;percent&quot; to express ratios that are usually smaller than one. Call this &quot;centiradians&quot; if you like: approximately 628 per circle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the remaining problem is that this level of granularity on the protractor is still only approximate for a full circle, half circle, right angle, etc. So better options would be to arbitrarily define the circle as a convenient whole number on the order of a few hundred. One option is 400 (gradians); as others point out, 360 (degrees) is nice for having lots of divisors (including 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, etc.) -- that is, many important angles will be a whole-number of degrees. Thus, no fractions or decimals are needed in most cases, important angles can be exactly expressed, and it's about the most efficient measurement scale to document and communicate the angles that most people will be dealing with. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Links on Measurement System Analysis:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.isixsigma.com/tools-templates/measurement-systems-analysis-msa-gage-rr/measurement-system-analysis-resolution-granularity/&quot; rel=&quot;nofollow&quot;&gt;http://www.isixsigma.com/tools-templates/measurement-systems-analysis-msa-gage-rr/measurement-system-analysis-resolution-granularity/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.isixsigma.com/tools-templates/measurement-systems-analysis-msa-gage-rr/proper-data-granularity-allows-stronger-analysis/&quot; rel=&quot;nofollow&quot;&gt;http://www.isixsigma.com/tools-templates/measurement-systems-analysis-msa-gage-rr/proper-data-granularity-allows-stronger-analysis/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.isixsigma.com/tools-templates/measurement-systems-analysis-msa-gage-rr/measurement-systems-analysis-process-industries/&quot; rel=&quot;nofollow&quot;&gt;http://www.isixsigma.com/tools-templates/measurement-systems-analysis-msa-gage-rr/measurement-systems-analysis-process-industries/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2015-09-30T06:28:04.023" LastActivityDate="2015-09-30T06:28:04.023" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9686" PostTypeId="1" CreationDate="2015-09-30T05:20:37.257" Score="4" ViewCount="173" Body="&lt;p&gt;I´m looking for some good and clear example/s about using graph theory in scheduling. (I´m interested in good books/web/PDF in this way)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My target is to show the utility of graph theory to some people in a little college/academy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="1760" LastActivityDate="2015-09-30T19:53:45.813" Title="looking for good and clear examples about graph theory apllied to scheduling" Tags="|community-colleges|graphing|scheduling|" AnswerCount="1" CommentCount="3" ClosedDate="2015-10-01T17:02:50.023" ContentLicense="CC BY-SA 3.0" />
  <row Id="9687" PostTypeId="2" ParentId="7528" CreationDate="2015-09-30T17:08:25.277" Score="6" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Gate_of_Europe&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Puerta de Europa&lt;/em&gt; (Gate of Europe)&lt;/a&gt; in Madrid&#xA;consists of two &lt;span class=&quot;math-container&quot;&gt;$26$&lt;/span&gt;-floor prisms inclined &lt;span class=&quot;math-container&quot;&gt;$15^\circ$&lt;/span&gt;,&#xA;a nice example of a &lt;em&gt;spatial shear transformation&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TvE5C.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TvE5C.jpg&quot; alt=&quot;PuertaEuropa&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;(Image from [archiseek.com](http://archiseek.com/2009/1996-puerta-de-europa-madrid/).)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Designed by architects Senteri Cardillo, Philip Johnson and John Burgee.&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2023-07-07T23:57:13.570" LastActivityDate="2023-07-07T23:57:13.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9688" PostTypeId="2" ParentId="9686" CreationDate="2015-09-30T19:53:45.813" Score="4" Body="&lt;p&gt;You might try getting the students to understand &lt;a href=&quot;https://en.wikipedia.org/wiki/Braess%27_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;Braess's Paradox&lt;/a&gt;, which can &#xA;be phrased as a paradox about traffic flow, modeled by a weighted graph.&#xA;The paradox is that the addition of a &quot;short cut&quot; road leads, under&#xA;individual rational behavior by each driver, to everyone taking longer&#xA;to reach their destination.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Y47xH.png&quot; alt=&quot;BraessParadox&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;The above image is from: Frank Kelly. &quot;The mathematics of traffic in networks.&quot; &lt;em&gt;The Princeton Companion to Mathematics&lt;/em&gt; (2008). (&lt;a href=&quot;http://www.statslab.cam.ac.uk/~frank/PAPERS/PRINCETON/pcm0052.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download link&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Braess%27_paradox#Example&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia example&lt;/a&gt; might be even clearer for your purposes, because it is phrased without mentioning &quot;Wardrop equilibrium.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-09-30T19:53:45.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9689" PostTypeId="1" AcceptedAnswerId="9718" CreationDate="2015-09-30T21:41:03.503" Score="15" ViewCount="2719" Body="&lt;p&gt;For my purposes, I am interested mostly in a medium-sized liberal art college setting. My students have mostly seen this before, but it is not something they understand. When discussing parallel lines, I have them try to find a point of intersection, fail to find such a point, and conclude that the lines never meet, which I think is convincing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to find ways that they can convince themselves that two lines with negative reciprocated slopes are perpendicular. Do you have a method that you like?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(also: not sure how to tag this question)&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2020-11-16T18:26:01.487" Title="How do you explain why perpendicular lines have negative reciprocated slopes?" Tags="|undergraduate-education|secondary-education|analytic-geometry|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9690" PostTypeId="2" ParentId="9689" CreationDate="2015-09-30T23:16:47.400" Score="23" Body="&lt;p&gt;If we have two lines $l$ and $l'$ with slopes $m&amp;gt;0$ and $m'= - \tfrac 1 m$ respectively, then we can always make the following diagram&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CCjGx.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CCjGx.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where the blue line is parallel to the $x$-axis and the green lines are parallel to the $y$-axis. That the lines $l$ and $l'$ are perpendicular now follows from that the two triangles are congruent and that the sum of angles in a triangle is $180^{\circ}$. The Pythagorean Theorem is not needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the converse, we can use almost the same diagram. Assume the lines $l$ and $l'$ are perpendicular and $l$ has slope $m&amp;gt;0$. We can then make the diagram&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/H4y42.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/H4y42.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where the blue line is still parallel to the $x$-axis and the green lines are parallel to the $y$-axis. This time an Angle-Side-Angle comparison shows that the two triangles are congruent. Therefore $BD=1$, and the slopes are negative reciprocals of each other. &lt;/p&gt;&#xA;" OwnerUserId="39" LastEditorUserId="39" LastEditDate="2015-10-02T12:24:39.243" LastActivityDate="2015-10-02T12:24:39.243" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9691" PostTypeId="2" ParentId="9689" CreationDate="2015-09-30T23:20:03.570" Score="3" Body="&lt;p&gt;I am interpreting the question as:&#xA;&lt;em&gt;negative reciprocal slopes&lt;/em&gt; $\implies$ &lt;em&gt;perpendicular&lt;/em&gt;&#xA;(rather than the converse).&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(1) If they understand the dot product already,&#xA;as the projection of one vector on another,&#xA;then the following might convince.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you have two lines with negative reciprocal slopes.&#xA;They are not parallel, so they cross.&#xA;Translate the lines so the crossing point is the origin.&#xA;Now take two points on the lines,&#xA;$a=(x,y)$ and $b=(-y,x)$, so their slopes are $y/x$ and $-x/y$ respectively.&#xA;Viewing $a$ and $b$ as vectors from the origin, we have&#xA;$a \cdot b = -xy + xy = 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) If they don't understand the dot product, &#xA;they might be persuaded by the calculation&#xA;$$| a-b |^2 = (x+y)^2 + (y-x)^2 = (x^2+2xy+y^2) + (y^2 -2xy + x^2) = 2 x^2 + 2 y^2 \;.$$&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/z3kce.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/z3kce.jpg&quot; alt=&quot;Pythag&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-10-01T20:44:37.223" LastActivityDate="2015-10-01T20:44:37.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9692" PostTypeId="2" ParentId="9689" CreationDate="2015-10-01T01:26:51.090" Score="10" Body="&lt;p&gt;One approach is to start with the fact that a 90-degree rotation results from a reflection across the $x$-axis followed by a reflection across the line $y=x$. (In general, reflecting across two intersecting lines results in a rotation through an angle $2\theta$ where $\theta$ is the angle between the two lines.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can get students to this place, then the rest is easy: reflecting across the $x$-axis takes $y$ to $-y$, and reflecting across the diagonal line exchanges $x$ and $y$. So $\frac{y}{x}$ is transformed into $\frac{x}{-y}$. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-10-01T01:26:51.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9693" PostTypeId="2" ParentId="9689" CreationDate="2015-10-01T03:13:02.207" Score="6" Body="&lt;p&gt;I find the following approach straightforward geometrically &amp;mdash; it's more a demonstration than a formal proof, but it explains the intuition well enough. It's similar to Dag Oskar Madsen's answer but isn't based on a congruence argument, which I think makes it easier for students to follow. (Besides, on some syllabuses the perpendicular gradient rule is taught before congruence!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simply take a right-angled triangle $ABC$ such that $B$ is to the right of $A$ and $C$ is vertically above $B$. Now rotate by $90^{\circ}$ counter-clockwise about $A$ to produce $A'B'C'$. Since $AB$ was horizontal with $B$ right of $A$, then $A'B'$ will be vertical with $B'$ above $A'$. Similarly, since $BC$ was vertical with $C$ above $B$, then $B'C'$ will be horizontal with $C'$ left of $B'$. And since $A'C'$ was produced by rotating $AC$ by $90^{\circ}$, then $AC$ and $A'C'$ will be perpendicular. Also note that $A'B'$ is the same length as $AB$, and so on. This is a lot of text for something which is obvious from the diagram. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xxOL6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xxOL6.png&quot; alt=&quot;Rotating right-angled triangles to prove perpendicular gradient rule&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{Gradient of } AC = \frac{BC}{AB} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{Gradient of } A'C' = -\frac{A'B'}{B'C'} = -\frac{AB}{BC} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the gradients are negative reciprocals of each other, as required.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make this even clearer, write onto your original diagram that $AB=1$ and $BC=m$ so that the gradient of $AC$ is $m$. Then when you draw the rotated triangle, label on $A'B'$ as $1$ and $B'C'$ as $m$, and it is immediately clear that the gradient of $A'C'$ is $-\frac{1}{m}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you've got a projector in your classroom you can animate the rotation of your gradient triangle. If not, why not use a large cardboard cut-out? I've drawn this on a blank background, but if you're working through a problem in the classroom you might want to superimpose them on the lines you're actually finding the gradients of (similar to Dag Oskar Madsen, but obviously my alignment is different).&lt;/p&gt;&#xA;" OwnerUserId="5640" LastEditorUserId="5640" LastEditDate="2015-10-01T03:21:27.253" LastActivityDate="2015-10-01T03:21:27.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9694" PostTypeId="2" ParentId="9689" CreationDate="2015-10-01T10:44:11.833" Score="10" Body="&lt;p&gt;One slightly different perspective, but with most of the same mechanics, is to imagine any set of perpendicular lines as having been translated and/or rotated from the x- and y-axes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UzAtC.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UzAtC.png&quot; alt=&quot;perp lines with slopes&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you pick any point on one line and form a triangle with the nearest axis, then because they've both rotated the same angle, the other line must form the same triangle with the other axis. This triangle, though has its rise and run swapped and a left-to-right run has become a top-to-bottom drop. So the slope must be the negative reciprocal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar rotational reasoning can be applied to two parallel lines. Any two parallel lines can be thought of as being rotated from two horizontal lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/HJ2u8.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HJ2u8.png&quot; alt=&quot;parallel lines with slopes&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since they're rotated the same way, they'll have the same slopes. (Less compelling than the perpendicular lines, but I thought I'd put it in for completeness.)&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-10-01T10:44:11.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9695" PostTypeId="1" CreationDate="2015-10-01T12:33:21.387" Score="3" ViewCount="304" Body="&lt;p&gt;Do you know of any Universities offering an undergraduate degree in Maths by distance learning other than the Open University.It is important that the content is in English language and any tutors will be able to deliver any online tutorial in English. &lt;/p&gt;&#xA;" OwnerUserId="5637" LastActivityDate="2016-05-21T02:42:06.297" Title="Looking for undergraduate degree in Maths through distance learning" Tags="|undergraduate-education|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9696" PostTypeId="1" CreationDate="2015-10-01T16:48:48.783" Score="17" ViewCount="277" Body="&lt;p&gt;I am tutoring a Grade 2 girl in arithmetic. She has demonstrated an ability to add two-digit numbers with carrying. For example:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\;\;14\\&#xA;+27\\&#xA;=41$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked her to write this out horizontally, and this is what she produced.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$12+47=41$$  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;She evidently is failing to  see the numbers and is confounding the vertical addition of the digits with the horizontal reading of the numbers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With practice, and prompting, she is able to get this right, but it seems as though she sees the sum as a matrix of four digits, and is missing the &lt;em&gt;numbers&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any insights on how to help her?&lt;/p&gt;&#xA;" OwnerUserId="5642" LastActivityDate="2015-10-10T15:04:25.097" Title="How do I help a 6-year-old understand the meaning of her sums" Tags="|tutoring|arithmetic-operations|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9697" PostTypeId="2" ParentId="7660" CreationDate="2015-10-01T18:18:04.517" Score="1" Body="&lt;p&gt;You might try to do intrinsic geometry of convex/saddle surfaces.&#xA;The subject is very visual and it makes perfect nontechnical introduction to differential geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with Alexandrov's &quot;Convex Polyhedra&quot;, and continue as to convex surfaces and Shefel's theorem on saddle surfaces if you time allows.&lt;/p&gt;&#xA;" OwnerUserId="3268" LastActivityDate="2015-10-01T18:18:04.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9698" PostTypeId="2" ParentId="9696" CreationDate="2015-10-02T01:33:54.017" Score="6" Body="&lt;p&gt;Does she have difficulty with reading or other taking in other visual information?  Her problem might have nothing to do with  understanding numbers and everything to do with a learning difference in how she perceives visual information.  I've had students who could answer questions but couldn't handle a worksheet (with the same questions) if it was organized in an unusual way.  You might check with the child's teacher to see if this is an issue and/or if she has an IEP (Individualised Education Plan). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the reason that I suspect this, is that she seems to understand the numbers when you question her. I suggest you rule out learning issues and continue to reinforce the different formats using one and tens as Gerhard suggested in his comment. &lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="688" LastEditDate="2015-10-04T14:58:29.253" LastActivityDate="2015-10-04T14:58:29.253" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9699" PostTypeId="1" AcceptedAnswerId="9703" CreationDate="2015-10-02T15:07:47.013" Score="9" ViewCount="178" Body="&lt;p&gt;While going over how to &lt;em&gt;simplify expressions with radicals&lt;/em&gt; with my precalculus class, I ran into the problem of how to explain that $\sqrt[4]{9}$ can be simplified to $\sqrt{3}$. The best I could come up with was along the lines of you &quot;&lt;em&gt;just have to see that the fourth-root can be broken up into two square roots&lt;/em&gt;&quot;. So something like:&#xA;$$&#xA;  \sqrt[4]{9} = \sqrt[2]{\sqrt[2]{9}} = \sqrt[2]{3}&#xA;  \quad\quad\text{or}\quad\quad&#xA;  \sqrt[4]{9} = 9^{\frac{1}{4}} = (9^{\frac{1}{2}})^{\frac{1}{2}} = &#xA;  (3)^{\frac{1}{2}} = \sqrt{3}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The step of breaking up the one-fourth seems kinda &lt;em&gt;tricky&lt;/em&gt;, and like something students wouldn't notice if they are in the mindset to &lt;em&gt;simplify&lt;/em&gt; an expression. Is there a better way to explain this so that students will more immediately see that $\sqrt[4]{9}$ can be simplified? I am looking for an explanation that is more intuitive than just stating the rule, &quot;&lt;em&gt;If the power of any divisors of the radicand divides the index of the radical, then ...&lt;/em&gt;&quot;, because it seems like there should be some simpler explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be better to say&#xA;$$&#xA;  \sqrt[4]{9} = 9^{\frac{1}{4}} = (3^{2})^{\frac{1}{4}} = &#xA;  (3)^{\frac{2}{4}} = (3)^{\frac{1}{2}} = \sqrt{3}&#xA;$$&#xA;but turning the $9$ into $3^2$ will still run counter to the students' mindset of &lt;em&gt;simplify&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2015-10-02T18:15:21.523" LastActivityDate="2015-10-02T18:15:21.523" Title="How to show that a radical can be partially simplified" Tags="|undergraduate-education|algebra|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9700" PostTypeId="2" ParentId="9620" CreationDate="2015-10-02T15:43:21.600" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0195105192&quot; rel=&quot;nofollow&quot;&gt;What is Mathematics&lt;/a&gt; by Courant and Robbins is what I usually recommend to my highschoolers, closely followed by &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0395929687&quot; rel=&quot;nofollow&quot;&gt;The Mathematical Experience&lt;/a&gt; by Davis and Hersh.  Both of these would seem to fit the bill quite neatly - a good overview of topics like topology, abstract algebra, real analysis etc with some nice problems to work through (although not strictly necessary).&lt;/p&gt;&#xA;" OwnerUserId="5646" LastActivityDate="2015-10-02T15:43:21.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9701" PostTypeId="1" AcceptedAnswerId="9714" CreationDate="2015-10-02T15:45:30.027" Score="14" ViewCount="1688" Body="&lt;p&gt;I'm tutoring a Grade 11 Math student in BC, Canada, and we're going over parabolas. He's having difficulty with finding the vertex of a parabola - not how to find it, but WHY it works. And I'm having a hard time finding a good way to explain it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With Calculus I can explain this in a heartbeat - we're just finding where the derivative is zero. But this student definitely has no knowledge of calculus, and teaching him derivatives is probably out of the equation (math pun, har har).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a non-calculus way to derive the solution for the x coordinate of the vertex of a parabola (y is easy to explain graphically). A hand-wavy or purely graphical depiction is fine, so long as it seems to 'make sense' without any knowledge of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cheers!&lt;/p&gt;&#xA;" OwnerUserId="5645" LastEditorUserId="143" LastEditDate="2015-10-03T10:42:03.970" LastActivityDate="2016-12-02T14:26:13.683" Title="Simpler explanation for finding the vertex of a parabola" Tags="|secondary-education|geometry|functions|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="9702" PostTypeId="1" CreationDate="2015-10-02T16:27:18.880" Score="23" ViewCount="2449" Body="&lt;p&gt;This is the question that was not asked &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7431/how-to-give-homework-for-integration-techniques/7432&quot;&gt;here&lt;/a&gt;.  Also related is &lt;a href=&quot;https://matheducators.stackexchange.com/questions/4326/direct-applications-and-motivation-of-trig-substitution-for-beginning-calculus-s/4332&quot;&gt;this question&lt;/a&gt;, but both presuppose that it will be taught and ask about how best to do it.  My question here is, &lt;em&gt;suppose we are designing a calculus curriculum; should we include integration by trigonometric substitution?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some arguments in favor:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It teaches abstract pattern-recognition skills&lt;/li&gt;&#xA;&lt;li&gt;With multiple integration techniques in a &quot;toolbox&quot; the students can learn the experience of being given a problem and having to choose the right tool&lt;/li&gt;&#xA;&lt;li&gt;It's fun to see how an integral that doesn't seem to involve trig at all can be integrated using it&lt;/li&gt;&#xA;&lt;li&gt;It's necessary if you want to actually evaluate by hand any but the most trivial integrals in multivariable calculus (possible rebuttal: why would you want to do that?)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some arguments against:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;No one is ever going to use it; in practice nowadays we just reach for a CAS&lt;/li&gt;&#xA;&lt;li&gt;It's hard and difficult to motivate, and only math majors will enjoy or appreciate it; to everyone else it's just one more useless thing to memorize, lowering their opinion of math still further&lt;/li&gt;&#xA;&lt;li&gt;It takes time away from other topics that are more likely to be useful.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="38" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-10-12T17:03:16.040" Title="Should we teach trigonometric substitution?" Tags="|undergraduate-education|calculus|curriculum|integration|" AnswerCount="8" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="9703" PostTypeId="2" ParentId="9699" CreationDate="2015-10-02T17:21:32.130" Score="6" Body="&lt;p&gt;Which is simpler, $\sqrt[4]{243}$ or $3^{5/4}$ ?  (Or even $3\sqrt[4]{3}$?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for teaching such rules at all is in the context of larger terms, where $\sqrt[4]{9}$ might cancel or combine with $\sqrt{3}$ in another place in the same or adjoining term.  Further, it is useful to know some of the &quot;notational patterns&quot;, because $9$ might be replaced by $(n^2 + 2n + 1)$ at some point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if they are not going to be applying the rule often, there is a situation when they may need it.  In this case noting that $\sqrt[stuff]{junk^{blah}}=junk^{blah/stuff}$ (with appropriate contextual restrictions) is really the relation to know, and your example of applying this relation to rewrite a term is one they may need to use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Variable Names Need More Letters&quot; Paseman, 2015.10.02&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-10-02T17:21:32.130" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9704" PostTypeId="2" ParentId="9702" CreationDate="2015-10-03T03:05:56.767" Score="1" Body="&lt;p&gt;&lt;strong&gt;No, we should not teach trigonometric substitution.&lt;/strong&gt;  Your arguments against are overwhelming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can teach a toolbox of techniques for numerical integration instead.  E.g.:  Find three ways to answer to an accuracy of .01, what is the area of the region with $\sin(x)&amp;gt;y^2$ and $|\,x\,|&amp;lt;\pi$?&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2015-10-03T03:05:56.767" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="9705" PostTypeId="2" ParentId="9702" CreationDate="2015-10-03T08:33:32.257" Score="9" Body="&lt;p&gt;My first feeling on this is that, although in any &lt;em&gt;given&lt;/em&gt; calculus course there might be a reason not to introduce trig substitution, in practice it is still a valuable technique that takes a whole huge class of integrals and &quot;make sense&quot; of them.  For instance, to tell whether you have just fed your CAS junk - misplace an $x^2$ by and $x^3$ and it's all over!  So rather than being reliant on a black box for all these things with square roots of $x^2$ you at least know why you should expect logs or other such stuff as an answer.  I find it hard to believe it's hard to motivate - surely there are many real-life physics integrals which need precisely this technique?  Naturally it might not be quite as fun for econ or bio majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Myself what I like to do is give some 'greatest hits' of it.  But perhaps it's not so useful to spend two very long days of class doing it any more, that is true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also have a suspicion (untested) that even the most powerful CASes might have some situations which their pattern-matching does not recognize as a trig substitution where a human might.   It would be interesting to see some such examples.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-10-03T08:33:32.257" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9706" PostTypeId="2" ParentId="9701" CreationDate="2015-10-03T16:51:46.117" Score="1" Body="&lt;p&gt;Consider $f(x)=x^2$ then $f(x-a)+b=(x-a)^2+b$ and geometrically corresponds to translating the graph of $y=f(x)$ by a vector $[a,b]^T$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular the vertex $(0,0) \mapsto (a,b)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cases such as $kf(x)$ were omitted for simplicity but follow similar reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="4668" LastEditorUserId="4668" LastEditDate="2015-10-03T16:58:32.030" LastActivityDate="2015-10-03T16:58:32.030" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9707" PostTypeId="1" CreationDate="2015-10-03T17:01:11.247" Score="8" ViewCount="123" Body="&lt;p&gt;I'm a third year philosophy student and considering pursuing an academic career. I'm comfortable with formal logic, set theory, and the calculations taught in high-school math. I've noticed that my best philosophical insights often come from the bit of math I know. I'd like to know more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a consequence of the classes that my school offers, it seems that linear algebra is my gateway to more advanced math. (The alternative is to enroll in the honors-stream algebra class. However, the grading in that class is notoriously tough: D- in honors algebra and a B- in linear algebra satisfy the same prerequisites. The viability of my plans depends on keeping my grades in the A range, and so linear algebra seems like the more prudent option) Unfortunately, the linear algebra course mostly, if not, only, teaches calculations. I don't mind studying the calculations, but I'd really prefer to know the ideas that underpin them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been looking for a linear algebra book that focuses more on theory. However, the books that do focus on theory use only proofs and examples from other areas of math to make their points. I'm comfortable with the proofs, but given that I'm trying to learn linear algebra in order to learn more math, those examples often don't help a great deal. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a book that uses proofs and real-world analogies to teach the underlying theory of linear algebra?&lt;/p&gt;&#xA;" OwnerUserId="961" LastActivityDate="2015-10-03T18:39:21.600" Title="Is there an introductory book on linear algebra that makes use of simple concrete examples to teach the underlying theory?" Tags="|reference-request|textbooks|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9708" PostTypeId="2" ParentId="9707" CreationDate="2015-10-03T18:39:21.600" Score="1" Body="&lt;p&gt;I happen to like Treil's &lt;a href=&quot;http://www.math.brown.edu/~treil/papers/LADW/LADW.html&quot; rel=&quot;nofollow&quot;&gt;&quot;Linear Algebra Done Wrong&quot;&lt;/a&gt;. Best thing, it's free.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-10-03T18:39:21.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9709" PostTypeId="2" ParentId="9701" CreationDate="2015-10-03T18:57:01.107" Score="1" Body="&lt;p&gt;This response is similar to some of what has already been written, but I thought it might be helpful to connect finding the vertex's $x$-coordinate with figuring out the roots given by the quadratic equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finding the $x$-coordinate of the vertex for $x \mapsto wx^2$ shouldn't be too tough: it is at $x = 0$. Observe also that the $x$-coordinate is not affected by shifting the curve up by $z$, i.e., for $x \mapsto wx^2 + z$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difficulty with a curve like $x \mapsto ax^2 + bx + c$ is the presence of that pesky $b$ term; so let us shift $x$ by some distance so as to get a new curve where the coefficient of $x^1$ is $0$. Call the distance $d$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we have (after expanding and combining like terms):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a(x+d)^2 + b(x+d) + c = ax^2 + (2ad + b)x + ad^2 + bd + c$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To achieve our goal means ensuring $2ad + b = 0$, i.e., $d = -b/2a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we end up with a curve${^1}$ whose vertex's $x$-coordinate is $0$; returning to our original curve means shifting back by $-b/2a$ and the vertex's $x$-coordinate shifts along with it &lt;em&gt;to&lt;/em&gt; the desired $x = -b/2a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And so the original question is answered: the $x$-coordinate of the vertex is at $x = -b/2a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1$. The curve we end up with after using $d = -b/2a$ is: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x \mapsto ax^2 + 0 + \frac{b^2}{4a} - \frac{b^2}{2a} + c = ax^2 - \frac{b^2}{4a} + c$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what $x$ values does this return $0$? Setting this equal to $0$ and multiplying through by $4a$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$4a^2 x^2 - b^2 + 4ac = 0 \Longrightarrow x^2 = \frac{b^2 - 4ac}{4a^2} \Longrightarrow x = \pm\frac{\sqrt{b^2 - 4ac}}{2a}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shifting back by the $-b/2a$ now gives the roots found in the quadratic equation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = \frac{-b}{2a} \pm\frac{\sqrt{b^2 - 4ac}}{2a}$$&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2015-10-03T18:57:01.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9710" PostTypeId="1" AcceptedAnswerId="9712" CreationDate="2015-10-03T20:07:45.487" Score="15" ViewCount="585" Body="&lt;p&gt;I was taught in a school that one has to use different brackets in expressions like $\{[(3+4)\cdot 4]^4\}^{1/2}$ to denote the order which subexpression is evaluated first. But can this be recommended in current mathematics? I guess no as one can create arbitrary complex expressions such that one would require arbitrary many different bracket notations. Also, I think that is good to teach that $()$ is for evaluating order, $[]$ is for floor function and matrices and $\{\}$ is for the set notation. So is it wrong if I suggest students to use the notation $(((3+4)\cdot 4)^4)^{1/2}$&lt;/p&gt;&#xA;" OwnerUserId="5654" LastActivityDate="2023-01-17T13:51:37.757" Title="How to teach brackets?" Tags="|notation|" AnswerCount="7" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="9711" PostTypeId="1" CreationDate="2015-10-03T22:13:40.693" Score="8" ViewCount="183" Body="&lt;p&gt;I'm teaching a quantitative reasoning course at a two-year community college and I'm endeavoring to be as inquiry-based as possible in my teaching. We have access to class sets of iPads, and I'd like to use them in a few class sessions, but I need to find the right apps. I'm looking for apps that do something like the &lt;a href=&quot;https://teacher.desmos.com/tilepile&quot;&gt;Tile Pile&lt;/a&gt; exercise on proportional reasoning over at Desmos.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've seen a few apps that are basically textbooks with interactive problems, and a few other apps that are just exercises disguised as games. What I really want is an app that allows students to explore the concepts on their own. The Tile Pile example is really perfect in this regard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a list of different topics I'll be covering to help you suggest different apps.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Proportional reasoning (ratios, percentages, rates)&lt;/li&gt;&#xA;&lt;li&gt;Number Sense (place value, scientific notation, rounding, understanding big numbers)&lt;/li&gt;&#xA;&lt;li&gt;Graphs and charts (creating, interpreting, and analyzing)&lt;/li&gt;&#xA;&lt;li&gt;Inequalities&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="21" LastActivityDate="2016-01-17T17:45:15.600" Title="Good apps on the iPad for inquiry-based exploration in a quantitative reasoning course" Tags="|technology-in-education|inquiry-based-learning|ipads|quantitative-reasoning|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9712" PostTypeId="2" ParentId="9710" CreationDate="2015-10-04T09:07:15.097" Score="9" Body="&lt;p&gt;It is definitely not wrong to teach your students to just use (). I would possibly argue the converse. Your experience at school flys in the face of convention. Teaching your students that way will not prepare them well for meeting the rest of the world.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-10-04T09:07:15.097" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9713" PostTypeId="2" ParentId="9710" CreationDate="2015-10-04T13:14:28.430" Score="11" Body="&lt;p&gt;I teach the students to use parentheses only. They are widely acceptable and it is what I prefer to use.  One argument against brackets is that some students find it hard to remember how to draw different kinds of brackets and spend inordinate amounts of time shaping their brackets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I find that it is important to show students that different brackets may be used as parentheses and are not only used for matrices or the floor function.  When students find an expression with different brackets they need to be able to understand it and know how to evaluate correctly.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-10-04T13:14:28.430" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9714" PostTypeId="2" ParentId="9701" CreationDate="2015-10-04T17:27:17.583" Score="8" Body="&lt;p&gt;Take your equation and complete the square to get it into the form $y=a(x - h)^2 + k$.  Assume first that $a$ is positive.  That means that $a(x - h)^2$ is positive so your $y$ values are always $k$ plus some positive number.  The original equation has its minimum when the &quot;something&quot;, $a(x - h)^2$, is equal to 0 which occurs when $x = h$.  That makes the minimum value $y=a(h - h)^2 + k = k$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the case where $a$ is negative, the argument is the same only you're looking for the maximum value.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastEditorUserId="39" LastEditDate="2016-12-02T14:26:13.683" LastActivityDate="2016-12-02T14:26:13.683" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9715" PostTypeId="2" ParentId="9701" CreationDate="2015-10-04T18:07:54.383" Score="9" Body="&lt;p&gt;Curiously enough, I wrote a short article about this that has just been published in &lt;em&gt;The Mathematics Teacher&lt;/em&gt; (&lt;a href=&quot;http://www.nctm.org/Publications/Mathematics-Teacher/2016/Vol110/Issue5/Solving-and-Graphing-Quadratics-with-Symmetry-and-Transformations/&quot; rel=&quot;nofollow noreferrer&quot;&gt;available here&lt;/a&gt;, although you may need subscriber access, or access to a library with such access, in order to get the full text).  The advantage of this approach is that it requires &lt;em&gt;very&lt;/em&gt; little algebra -- and in particular does not require students to complete the square or use the quadratic formula.  Instead, the approach relies on three basic ideas:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Parabolas are symmetric&lt;/li&gt;&#xA;&lt;li&gt;Adding a constant term to a quadratic function shifts its graph vertically&lt;/li&gt;&#xA;&lt;li&gt;Quadratic equations with no constant term are very easy to solve by factoring&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Here is the approach, in detail:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you want to find the vertex of $y=5x^2 - 20x + 12$.  Ask yourself:  Could I solve an easier, related problem?  One such easier, related problem comes by dropping the constant term entirely.  So let's ask ourselves:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Where is the vertex of $y = 5x^2 - 20x$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Well, we know that parabolas are symmetric.  A consequence of that is that if a parabola has two $x$-intercepts, those will be located equally spaced on opposite sides of the line of symmetry.  Or, to put it another way, if you can find the two $x$-intercepts, the line of symmetry will be halfway between them (i.e. at their average).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fortunately, the $x$-intercepts of $y = 5x^2 - 20x$ are easy to find:  We just solve&#xA;$$5x^2-20x = 0$$&#xA;$$5x(x-4) = 0$$&#xA;$$x = 0,4$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and that means that the line of symmetry is at $x = 2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, how does that help us solve the original problem?  Here is the crucial idea:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Restoring the $ + 12 $ term to the original equation &lt;em&gt;shifts the parabola vertically&lt;/em&gt; but &lt;em&gt;does not change its line of symmetry&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So for the problem we are interested in, $y = 5x^2 - 20x + 12$, the line of symmetry is &lt;em&gt;also&lt;/em&gt; at $x = 2$.  Since the vertex is &lt;em&gt;on&lt;/em&gt; the line of symmetry, its $x$-coordinate is $2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, we can find the $y$-coordinate by simply plugging $x=2$ into the formula for the parabola:  $ y= 5 \cdot 2^2 - 20(2) + 12 = -8$.  So the vertex is at $(2, -8)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having gone through this for a single example, now let's consider the general problem.  If we want to find the vertex of $ y = ax^2 + bx + c $, let's ask ourselves first the simpler problem obtained by dropping the constant term:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Where is the vertex of $y=ax^2 + bx$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The answer is:  it will have an $x$-coordinate halfway between the $x$-intercepts of the parabola.  We can find those easily:&#xA;$$ax^2 + bx = 0$$&#xA;$$x(ax + b) = 0$$&#xA;$$x = 0,  -b/a $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore the line of symmetry is at $x = -\frac{b}{2a}$.  Because adding or subtracting a constant term just shifts a parabola vertically, this is &lt;em&gt;also&lt;/em&gt; the line of symmetry for the parabola we are actually interested in, and therefore is also the $x$-coordinate of its vertex, too.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2016-12-02T02:38:20.120" LastActivityDate="2016-12-02T02:38:20.120" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9716" PostTypeId="2" ParentId="9696" CreationDate="2015-10-04T18:28:46.540" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;She evidently is failing to see the numbers and is confounding the vertical addition of the digits with the horizontal reading of the numbers.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;With practice, and prompting, she is able to get this right, but it seems as though she sees the sum as a matrix of four digits, and is missing the numbers. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That may indeed be what she is doing, but there is an alternative hypothesis to consider, which is that she sees and interprets the arrangement of the digits perfectly well, but does not understand what &lt;em&gt;you&lt;/em&gt; mean when you ask her to write the problem horizontally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good way to probe what is going on is to ask her to &lt;em&gt;read aloud&lt;/em&gt; the original problem.  If she reads it as &quot;fourteen plus twenty-seven equals forty-one&quot; then that means she is correctly parsing the vertically-written sum, and suggests that she just doesn't understand what you are asking her to do.  In that case, try revoicing her answer back to her, and ask something like &quot;Could you write fourteen plus twenty-seven equals forty-one on one line?&quot;  &lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-10-04T18:28:46.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9717" PostTypeId="1" AcceptedAnswerId="9791" CreationDate="2015-10-04T18:36:38.437" Score="7" ViewCount="283" Body="&lt;p&gt;I've decided to create a concept map of a chapter I covered in a textbook, it's about basic set notation. What I want is suggestions on how to improve the presentation of the map. It seems quite convoluted at the moment, and I feel to anyone apart from me it may be unreadable. I think it may be improved if I addressed the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Organisation of concepts — conceptually should you go from general to specific? Should the structure be hierarchical?&lt;/li&gt;&#xA;&lt;li&gt;Granularity — What is the level of detail one should aim for?&lt;/li&gt;&#xA;&lt;li&gt;Linkages — as you can see in my map there are concepts&#xA;lower in the hierarchy that are linked with concepts higher in the hierarchy, with large swooping arrows, is this preferable? or should linkages only be made with local concepts?&lt;/li&gt;&#xA;&lt;li&gt;Are there other useful tools that may help me visualise knowledge structures?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Any other suggestions are welcome, thank you.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/KUGwt.jpg&quot; alt=&quot;Concept map for basic set notation&quot; /&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;EDIT: You need to zoom in with your browser to see the concept map properly, as StackExchange automatically scales it down. If you zoom in the writing will be legible without a problem.&lt;/p&gt;&#xA;&lt;p&gt;Alternatively, you can view the image directly &lt;a href=&quot;https://i.stack.imgur.com/KUGwt.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1677" LastEditorUserId="17657" LastEditDate="2022-04-01T14:36:44.040" LastActivityDate="2022-04-01T14:36:44.040" Title="How can I improve my concept map?" Tags="|mathematical-pedagogy|self-learning|intuition|aids|concept-maps|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9718" PostTypeId="2" ParentId="9689" CreationDate="2015-10-04T19:35:23.420" Score="11" Body="&lt;p&gt;A proof following from the Pythagorean Theorem:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Claim&lt;/strong&gt;: If two lines are perpendicular, then their slopes are negative reciprocals of one another.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof&lt;/strong&gt;: Assume WLOG that two nonvertical lines with slopes &lt;span class=&quot;math-container&quot;&gt;$m_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m_2$&lt;/span&gt; intersect at the origin and form a right angle. Construct the line &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt;. Then we have a triangle with coordinates &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(1,m_1)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(1,m_2)$&lt;/span&gt;. Moreover, the triangle is a right triangle with legs &lt;span class=&quot;math-container&quot;&gt;$\sqrt{1^2+m_1^2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{1^2+m_2^2}$&lt;/span&gt;, and hypotenuse &lt;span class=&quot;math-container&quot;&gt;$(m_1-m_2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;By the Pythagorean theorem:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{1^2+m_1^2}^2+\sqrt{1^2+m_2^2}^2=(m_1-m_2)^2$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$2+m_1^2+m_2^2=m_1^2-2m_1m_2+m_2^2$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$2=-2m_1m_2$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{-1}{m_1}=m_2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/QBwxj.png&quot; alt=&quot;Pythagorean Theorem and Slope&quot; /&gt;&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="867" LastEditDate="2020-11-16T18:26:01.487" LastActivityDate="2020-11-16T18:26:01.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9720" PostTypeId="2" ParentId="7218" CreationDate="2015-10-05T16:08:07.393" Score="2" Body="&lt;p&gt;With the benefit of hindsight, the one thing I wish I knew was the following:  &quot;Bits of this are hard.  Ask for help with them.  Share what you know (don't know).&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The last part (about sharing what you know) was easy for me, especially since I ran into fewer hard bits than many of my students.  However in the latter part of my undergraduate and graduate studies, I ran into hard bits and did not ask for help (and did not share with others that I was lacking).  Floundering, addressing, and overcoming challenges single-handedly can be quite satisfying and emotionally appealing (and even character building), but is not always a healthy mindset.  Thinking that you have to do it by yourself is unnecessarily overwhelming.  If you are trying to learn mathematics to achieve a reasonable goal, I think you will enjoy it more if you learn to ask for help along the way.  Form study groups, talk to graduate students, participate in the online forums.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Sharing Isn't Just One Way&quot; Paseman, 2015.10.05&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-10-05T16:08:07.393" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9721" PostTypeId="2" ParentId="8265" CreationDate="2015-10-06T02:38:48.753" Score="3" Body="&lt;p&gt;The Elkonin-Davydov mathematics curriculum from Russia &amp;quot;is characterized&#xA;by the early development of quantitative and algebraic reasoning through measurement of continuous quantities&amp;quot; (&lt;a href=&quot;http://www.csuchico.edu/sced/documents/Seminar%203-13%20Math%20Ed.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I was lead to it by a recent article (&lt;a href=&quot;http://www.nctm.org/Publications/Teaching-Children-Mathematics/2015/Vol22/Issue3/Units-Matter/&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;, access to which might need a subscription) in the October 2015 (Vol. 22, Issue 3) issue of &lt;em&gt;Teaching Children Mathematics&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The article (&amp;quot;Units Matter&amp;quot; by Ji Yeong I, Barbara J. Dougherty, and Zaur Berkaliev) has the subtitle &amp;quot;The meat and potatoes of fraction multiplication is the change of units.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Here are some relevant quotes from it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The distinguishable characteristic between additive and multiplicative reasoning is &lt;em&gt;the change of unit of measure&lt;/em&gt; (Berkaliev 2008).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] A unit cannot be separated from a number, and a number is dependent on its unit. This explicit use of unit is usually observed in scientific measurement activities, but units are also used implicitly in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;In the measurement perspective, all numbers involve the concept of unit. Berkaliev (2008) indicated that “a number itself is not just an absolute and final entity, but is only an expression of a relationship between two different quantities” (p. 9). When multiplying two numbers, one number is the unit of measure and the other is a quantity to be measured [...].  [...] It also exemplifies the difference between addition and multiplication: Addends in addition simply use one common unit, and multiplication involves more than one unit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-10-06T02:38:48.753" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9722" PostTypeId="1" AcceptedAnswerId="9733" CreationDate="2015-10-06T11:30:03.090" Score="13" ViewCount="695" Body="&lt;p&gt;Later this year, high school teachers from the region will visit our university for a day and be given special lectures on different topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been asked to give a talk with the title &quot;How to think mathematically?&quot;. In addition to mathematical examples, they also want to hear about the newest Math Ed research regarding students who struggle to think mathematically. The talk is supposed to be light and inspiring rather than heavy and theoretical. Other than the above, I haven't been given much to go on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you give me any ideas I could use? Do you have inspiring examples of mathematical thinking and/or references to Math Ed research about the topic?&lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2015-10-10T14:30:03.493" Title="How to think mathematically?" Tags="|secondary-education|education-research|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9723" PostTypeId="1" CreationDate="2015-10-06T14:58:19.020" Score="5" ViewCount="196" Body="&lt;p&gt;I understand that the question in the title, and of course, the longer version of the question (below) is too broad and subject to being closed.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Will the low-achieving children always be there, is it their fault or&#xA;  the education systems fault?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Fortunately, neither of them is the question that I am directly interested in! &#xA;Here is the story. I am working with a group of first year mathematics students in a course that is somehow about mathematics education. As part of their assessment, they have to write a short essay on a problem that they are personally interested in finding an answer for. One of the students came up with the question above. I realize that it is too hard for a short essay over a short time. But, I am so intrigued by the question that I've decided to find a way to help her to delimit the question and does something about it. That is why I am here, to get help to make the question approachable. To keep answers &quot;closed to discussion&quot; please just give the following: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The angle(s) that the question can be approached from +  some good resources (papers, books, etc) related to the angle(s) you suggest &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please feel free to add tag. If I knew which tag to use for this question, I would have advised her already!&lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-10-14T00:10:24.100" Title="Will the low-achieving children always be there?" Tags="|advising|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="9724" PostTypeId="2" ParentId="9722" CreationDate="2015-10-06T15:01:44.010" Score="3" Body="&lt;p&gt;This seems appropriate, but I haven't read it yet:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Byers, William. &lt;em&gt;How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics&lt;/em&gt;. Princeton University Press, 2010.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/FKFRk.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FKFRk.gif&quot; alt=&quot;BookCover&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Book cover image from &lt;a href=&quot;http://press.princeton.edu/titles/8386.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Princeton Press&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Donal O'Shea: &quot;Ambitious, accessible and provocative...[In] &lt;em&gt;How Mathematicians Think&lt;/em&gt;, William Byers argues that the core ingredients of mathematics are not numbers, structure, patterns or proofs, but ideas...&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-10-06T15:01:44.010" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9726" PostTypeId="2" ParentId="9723" CreationDate="2015-10-06T17:25:39.757" Score="6" Body="&lt;p&gt;I think the question is one of perspective, and that really the answer should be about changing perspective and not about trying to answer the question from the perspective suggested by the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To illustrate, supposed that some system is put in place to study and answer the question as asked.  They might decide that students who routinely score in the bottom 10% of a number of trials are &quot;low-achieving&quot;, and then decide to address that by remediation or by something else that is meant to improve performance in such situations.  I see this as socially harmful, not just for the individuals so labeled, but for a healthy society in general.  While the idea that &quot;everyone should have a minimal competency in every subject&quot; is worth considering, making such distinctions early and impressing them on the distinguished strikes me as a bad strategy.  Instead, use it as an evaluating tool and suggest that that subject be covered later or not at all in the student's education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the question is a springboard for a number of discussions practical as well as philosophical.  I would like to steer it from &quot;recognizing that this person is not good in this subject and fixing it&quot; to &quot;seeing certain lack of ability and channelling the student to address that by switching subjects or learning styles for that subject&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, that is what teachers should be and probably are trying anyway, but in the U.S. there seems to be an emphasis on fixing the specific lack rather than working around it.  We should have enough data now for knowing how to work around such things in a socially responsible and beneficial fashion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Is Math Really That Important?&quot; Paseman, 2015.10.06&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-10-06T17:25:39.757" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9727" PostTypeId="2" ParentId="9722" CreationDate="2015-10-06T17:40:19.843" Score="1" Body="&lt;p&gt;I like to think that a subject in its own right is &quot;how to find an easy way&quot;, where the goal can be not just to solve but to explain.  This leads to several possibilities for both, which is why mathematicians cover the same problems over and over again, looking for an easier way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have loads of time, you could guide students through a couple of ways in the problem I present below.  Take ten trees and plant them in five rows with four trees in each row.  A really careful and interactive discussion of this would take twenty minutes.  If you have only five minutes or less to devote (and are willing to have the problem spoiled), I would recommend you cover the following points in this order.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What happens when you try planting trees?  Can you find a solution easily?&lt;/li&gt;&#xA;&lt;li&gt;Try planting rows instead. What should a row look like?&lt;/li&gt;&#xA;&lt;li&gt;Can you have two rows or more that do not share a tree?&lt;/li&gt;&#xA;&lt;li&gt;Try drawing three rows, each of which share a tree.&lt;/li&gt;&#xA;&lt;li&gt;See if you can extend this to four and then five rows.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The upshot is that taking the perspective of placing lines rather than points (rows rather than individual trees) leads to an easy and general solution (five lines of which no two are parallel and no three concurrent) to the problem, and that one part of mathematical thinking (in both solving and education) is to shift perspectives, always looking for &quot;the easy way&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Sometimes Finds The Easy Way&quot; Paseman, 2015.10.06&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-10-06T17:40:19.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9728" PostTypeId="1" AcceptedAnswerId="9731" CreationDate="2015-10-06T17:54:21.290" Score="25" ViewCount="4189" Body="&lt;p&gt;I've seen in a lot of questions about &quot;which textbook to use for intro analysis&quot;, and inevitably Rudin's &lt;em&gt;Principles of Mathematical Analysis&lt;/em&gt; comes up, with the (almost cliche) rejoinder that &quot;&lt;em&gt;today's&lt;/em&gt; incoming US freshmen cannot handle Rudin.&quot; [emphasis mine].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a product of the crumbling edifice of pre-college US mathematics education, I can attest that I was not well versed in methods of proof, and I found geometry, which was proof based, quite awful when I took it (I'm referring to the &lt;em&gt;experience&lt;/em&gt;, not the subject matter). My experience certainly did not endear me to proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is...&lt;strong&gt;was there a generation of high school students in the US who &lt;em&gt;could&lt;/em&gt; jump right into Rudin&lt;/strong&gt; without &quot;a bloodbath&quot; (to quote an &lt;a href=&quot;http://www.maa.org/press/maa-reviews/principles-of-mathematical-analysis&quot;&gt;MAA review of Rudin's PMA&lt;/a&gt;)? If so, was it the &quot;new math&quot; era in the 1960's that put the first cracks in our math system? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;OR&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it the case that the more widespread use of math in our society led to a need to &quot;democratize&quot; it, pulling it out of its rigorous, ivory tower to a level that is more focused on applications, of which there were becoming many. Was this &quot;industrialization&quot; of math perhaps the real issue?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conjectures on my part, but I keep hearing about how &quot;un-prepared&quot; my generation  is, so I'd like to know the genesis of this statement (besides the usual &quot;..in my days we walked to school with newspaper on our feet..&quot; sentiments).&lt;/p&gt;&#xA;" OwnerDisplayName="user5661" LastActivityDate="2017-07-09T18:22:53.637" Title="When did US mathematics programs start failing to prepare incoming students for books like &quot;Baby&quot; Rudin?" Tags="|mathematical-analysis|history|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="9729" PostTypeId="2" ParentId="9702" CreationDate="2015-10-06T18:02:09.750" Score="6" Body="&lt;p&gt;I think teaching trigonometric substitutions is a good idea - that is, I presume you mean working out integrals such as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\displaystyle\int\frac{1}{x^2+a^2}\mathrm{d}x$ and $\displaystyle\int\frac{1}{\sqrt{a^2-x^2}}\mathrm{d}x$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think learning to work out integrals such as these is useful from a mathematical point of view, because it exposes students to the idea that an integral can be evaluated by using a substitution of the form $x=x(t)$ rather than using $u=u(x)$. Then students can calculate a greater class of integrals, and they need to more carefully consider the limits and what $\mathrm{d}x$ is in terms of $\mathrm{d}t$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These integrals also emphasise the importance of calculating the limits, which involves needing to take &lt;em&gt;inverses&lt;/em&gt; of the function used for the substitution, rather than just the function itself.&lt;/p&gt;&#xA;" OwnerDisplayName="user5447" LastActivityDate="2015-10-06T18:02:09.750" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9730" PostTypeId="1" CreationDate="2015-10-06T18:13:14.140" Score="0" ViewCount="95" Body="&lt;p&gt;I am currently doing MS in Mathematics. As far as I know there are two approaches which anyone can take to do Mathematics- Analysis and Algebra. Choosing which area to work on seems difficult to me. Currently I like both but I have to decide which area to pursue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wanted to know if we can switch our fields later in our career?&#xA;For example, if I do my MS thesis in Algebra how feasible will it be for me to switch to Analysis in my PhD?&lt;/p&gt;&#xA;" OwnerUserId="5663" LastActivityDate="2015-10-06T19:19:45.470" Title="Algebra or Analysis" Tags="|graduate-education|" AnswerCount="1" CommentCount="5" FavoriteCount="0" ClosedDate="2015-10-06T19:00:02.113" ContentLicense="CC BY-SA 3.0" />
  <row Id="9731" PostTypeId="2" ParentId="9728" CreationDate="2015-10-06T18:17:59.460" Score="22" Body="&lt;p&gt;Opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There never was a generation of high school students in the US who could jump right into Rudin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There would be (and still is) a small portion of the top high school graduates who could.  And maybe a larger portion of the graduates from a few elite high schools.  But that's it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Baby Rudin would be used (if at all) for advanced undergraduates or even beginning graduate students.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-10-06T18:17:59.460" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9732" PostTypeId="2" ParentId="9730" CreationDate="2015-10-06T18:40:12.753" Score="1" Body="&lt;p&gt;First, there are many more fields in mathematics than just Abstract Algebra and Analysis. For example, Number Theory is also very suitable for an MS thesis or PhD. There are several other fields, but they are somewhat less common. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your PhD thesis is not really meant to be a follow up on the MS thesis. In both theses you are using the things you've learned in Algebra and Analysis class to go into the depth and investigate something. Since should have had plenty of courses in both fields, so you know the basics of each fields, you can switch form fields. However, since the Master thesis doesn't have to be original research, but the PhD has to be, you can use the MS thesis to review and expand your knowledge on one of the subjects so that if you write the PhD thesis about the same subject, you have a better ground to stand on. This might save time in you PhD thesis. In short, you can switch, but it might cost you some time and effort when you're writing your PhD thesis.&lt;/p&gt;&#xA;" OwnerUserId="5664" LastEditorUserId="5664" LastEditDate="2015-10-06T19:19:45.470" LastActivityDate="2015-10-06T19:19:45.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9733" PostTypeId="2" ParentId="9722" CreationDate="2015-10-06T21:34:55.983" Score="12" Body="&lt;p&gt;I believe the classic reference from the mathematics education literature is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Cuoco, A., Goldenberg, E. P., &amp;amp; Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. &lt;em&gt;The Journal of Mathematical Behavior, 15&lt;/em&gt;(4), 375-402. &lt;a href=&quot;http://nrich.maths.org/content/id/9968/Cuoco_etal-1996.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt; (no pay-wall).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The authors are all out of &lt;a href=&quot;http://ltd.edc.org/people?tid_1=2044&amp;amp;field_dir_person_last_name_value=&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;EDC&lt;/strong&gt;&lt;/a&gt; (Cuoco and Goldenberg on the linked page; Mark on the next one).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before listing the habits of mind, they write:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/a0Zrb.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/a0Zrb.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the paper for more information (including examples) but the habits of mind, in order, are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Students Should Be Pattern Sniffers&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students Should Be Experimenters&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students Should Be Describers&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students Should Be Tinkerers&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students Should Be Inventors&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students Should Be Visualizers&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students Should Be Conjecturers&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students Should Be Guessers&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The next section is entitled, &lt;strong&gt;Mathematical Approaches to Things&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, I suggest reading the paper for examples, but here are the headers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematicians Talk Big and Think Small&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematicians Talk Small and Think Big&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematicians Use Functions&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematicians Use Multiple Points of View&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematicians Mix Deduction and Experiment&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematicians Push the Language&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematicians Use Intellectual Chants&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Next, the article provides approaches that are specific to &lt;em&gt;geometric thinking&lt;/em&gt;, and then they do the same for &lt;em&gt;algebraic thinking&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, you can find plenty more by checking the papers that have cited this piece (&lt;a href=&quot;https://scholar.google.com/scholar?cites=8576760172544403926&amp;amp;as_sdt=40000005&amp;amp;sciodt=0,22&amp;amp;hl=&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;google scholar&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2015-10-06T21:34:55.983" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9734" PostTypeId="2" ParentId="1572" CreationDate="2015-10-06T22:04:29.497" Score="4" Body="&lt;p&gt;I would check out some of the work of Edward Tufte (&lt;a href=&quot;http://www.edwardtufte.com&quot; rel=&quot;nofollow&quot;&gt;http://www.edwardtufte.com&lt;/a&gt;).  His book &lt;em&gt;The Visual Display of Quantitative Information&lt;/em&gt; is replete with examples of deliberate and accidental mis-use of graphs which mislead readers.  I know this is going to sound like a commercial.  I have no connection to the author in any way.  Unfortunately, I'm only aware of his hardbound books.  It's not easy to find online examples for you to look through.  You may be able to find copies at a local library. In the book mentioned above, chapter 2 &quot;Graphical Integrity,&quot; shows many practical examples of misleading graphs taken from pictures, newspapers, articles, etc.  &lt;/p&gt;&#xA;" OwnerUserId="5465" LastActivityDate="2015-10-06T22:04:29.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9735" PostTypeId="2" ParentId="1572" CreationDate="2015-10-07T02:53:17.990" Score="9" Body="&lt;p&gt;Recent meme that spending 360 million dollars to give 317 million people health care means that you're spending over 1 million dollars person:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.dailymail.co.uk/femail/article-3180161/Maddening-online-debate-cost-Obamacare-turns-simple-math-question-complex-equation-able-solve.html&quot;&gt;http://www.dailymail.co.uk/femail/article-3180161/Maddening-online-debate-cost-Obamacare-turns-simple-math-question-complex-equation-able-solve.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-10-07T02:53:17.990" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9736" PostTypeId="2" ParentId="9728" CreationDate="2015-10-07T04:00:27.847" Score="7" Body="&lt;p&gt;My recollection of the Math program at Harvey Mudd College (I was in CS, but lots of math major friends), is as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The core Calc I included a taste of analysis-like material, in building up the epsilon-delta definition of a limit and proving it for various functions&lt;/li&gt;&#xA;&lt;li&gt;Linear algebra was part of the common core taken by all students, and the first portion of it was taken in the first semester (first or second half, depending on placing out of Calc I). It was very heavy on proofs.&lt;/li&gt;&#xA;&lt;li&gt;Discrete math had LinAl as a pre-req, and was even heavier on proofs. It was required for CS and Math majors.&lt;/li&gt;&#xA;&lt;li&gt;Intro analysis, using Rudin, had Discrete as a pre-req. Math majors were required to take it, and it was typically taken relatively early.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So, sophomore-level math majors who've already dipped their feet in the water.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not quite the 'advanced' undergraduates of Gerald's answer, but certainly not just-enrolled freshman, either.&lt;/p&gt;&#xA;" OwnerUserId="5665" LastActivityDate="2015-10-07T04:00:27.847" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9737" PostTypeId="2" ParentId="9722" CreationDate="2015-10-07T16:11:28.137" Score="5" Body="&lt;p&gt;I've found &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691156662&quot; rel=&quot;noreferrer&quot;&gt;&quot;The 5 Elements of Effective Thinking&quot;&lt;/a&gt; to be a good start. It was originally written to address exactly what you ask about: to explain how mathematicians think. However (according to the author whom I heard speak once) the editors quickly recognized that the ideas and methods were good ways of thinking for more than just mathematicians and so the text was re-written with a broader slant. Knowing that, however, you can still read it to learn about how mathematicians think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a similar vain, I've heard good things about &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0143127535&quot; rel=&quot;noreferrer&quot;&gt;&quot;How Not to Be Wrong: The Power of Mathematical Thinking&quot;&lt;/a&gt; but haven't taken the chance to read it myself yet.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-10-07T16:11:28.137" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9738" PostTypeId="1" CreationDate="2015-10-07T21:14:09.937" Score="12" ViewCount="726" Body="&lt;p&gt;Since this does not appear to have been asked here before, I would like to solicit suggestions and recommendations, ideally with rationales, for a textbook in a first course in linear algebra.  In my university, students will not necessarily have taken any proof-based course (we do not have an introduction to proofs course).  Our course is expected to cover basic computational issues including Gaussian elimination and eigenvectors/eigenvalues.  We also cover theoretical material including an introduction to vector spaces over $\mathbb{R}$ (as well as over $\mathbb{C}$ to deal with diagonalization) culminating, ideally, with a statement and proof of the Spectral Theorem.  We expect students to learn how to carry out fairly straightforward proofs that primarily follow from the definitions or directly from one of the theorems covered in class.  Because we have a large number of engineers and other science majors in the course, I do plan to spend some time on practical issues as well.  I like to discuss rotations and reflections in $\mathbb{R}^3$ using an eigenvector/eigenvalue analysis.  I will definitely cover the singular value decomposition and would like to spend some time discussing modified Gaussian elimination to handle numerical problems due to rounding error, though the latter is likely to be omitted because of time constraints.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've tried several different books before (Strang, Anton) and yet to have found one that had a presentation close to what I have in mind for the course.  Any suggestions for books whose aims are aligned with the goals above would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2015-10-08T11:53:25.530" LastActivityDate="2015-10-16T16:32:41.363" Title="Textbook for first course in linear algebra" Tags="|textbooks|linear-algebra|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9739" PostTypeId="2" ParentId="9702" CreationDate="2015-10-07T22:09:42.213" Score="4" Body="&lt;p&gt;I taught a calc II course at my university using Mathematica for evaluating integrals.  It allowed us time to consider a more vast array of applications of integrals and the students learned more about how to model a problem using an integral, since we saved so much time that would have been wasted evaluating integrals.  Besides mentioning a few numerical integration techniques and their error estimates, I didn't teach how numerical integration really works.  Student feedback was overwhelmingly positive and I was very happy with the final exam performance.  I will definitely do it again.  In short, to answer the OP, I say, don't teach trig substitution.&lt;/p&gt;&#xA;" OwnerUserId="5672" LastActivityDate="2015-10-07T22:09:42.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9740" PostTypeId="2" ParentId="9738" CreationDate="2015-10-07T22:28:54.857" Score="10" Body="&lt;p&gt;Hefferon is a wonderful book, and it's also free: &lt;a href=&quot;http://joshua.smcvt.edu/linearalgebra/&quot;&gt;http://joshua.smcvt.edu/linearalgebra/&lt;/a&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I think it's a bad idea to write down a list of topics you want and then look for a book that covers them all. If the topics are standard, this is a waste of time, because they'll be covered in every book. If the topics are not standard, then you're shooting yourself in the foot by ruling out almost all possible books. If you have a few nonstandard topics you want to do, just write up lecture notes on them and put them online.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2015-10-07T22:28:54.857" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9741" PostTypeId="2" ParentId="9738" CreationDate="2015-10-07T22:40:52.393" Score="3" Body="&lt;p&gt;Look at Treil's &lt;a href=&quot;http://www.math.brown.edu/~treil/papers/LADW/LADW.html&quot; rel=&quot;nofollow&quot;&gt;&quot;Linear Algebra done Wrong&quot;&lt;/a&gt;, it's free.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-10-07T22:40:52.393" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9742" PostTypeId="2" ParentId="9738" CreationDate="2015-10-08T02:19:13.033" Score="4" Body="&lt;p&gt;I think Lay's and Poole's books are easily the best of the common ones on the market.  However, if you have stronger students than we do, Strang or Bretscher might be more appropriate.  (Not having a introduction to proofs class is frequently an indication of stronger students.)&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2015-10-08T02:19:13.033" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9743" PostTypeId="2" ParentId="9702" CreationDate="2015-10-08T03:07:48.577" Score="14" Body="&lt;p&gt;In reality, I think this is not the most important topic, and if I was designing a curriculum from scratch, I would probably omit it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We rarely have that luxury however.  In my state, for example, all state colleges must meet certain common standards to be able to transfer credit:  one of these standards is teaching trig sub.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I had to include it, I think that teaching trig sub could be valuable in the context of a larger story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could tell them how not every function is integrable in terms of elementary functions.  Then a natural question is &quot;Can we characterize all those functions which are?&quot;.  This is too hard for this beginning course.  However, we can at least find some large families of functions which we can integrate. Certainly all polynomials.  With some work, all polynomials in $\sin$ and $\cos$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another big class to try and tackle is rational functions. At the start of the day, there is no reason to think these should all be integrable in elementary functions. But the fundamental theorem of algebra tells you that you can factor denominator as a product of linear terms and irreducible quadratic terms.  Partial fraction decomposition further simplifies.  Finally, we need trig sub to knock out the terms with quadratics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So trig sub becomes valuable as part of a story about how to integrate any rational function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you know how to integrate any rational function, it turns out you can also integrate any rational function of $\sin$ and $\cos$ using the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tangent_half-angle_substitution&quot;&gt;tangent half-angle substitution&lt;/a&gt;, which has connections to some basic algebraic geometry (Rational parameterization of the unit circle).&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2015-10-08T03:07:48.577" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9744" PostTypeId="2" ParentId="9702" CreationDate="2015-10-08T03:18:12.920" Score="9" Body="&lt;p&gt;&lt;strong&gt;Yes.&lt;/strong&gt; We should teach trigonometric substitution. But, I take it a step further, I think we should also teach hyperbolic substitution. With this additional technique the idea of the substitution is much clearer. Also, teach it with confidence. Teach it as if they can all understand it... because they can.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why do students have trouble with this topic? I submit, the reason is often &lt;strong&gt;ignorance&lt;/strong&gt; of trigonometry. Especially an inability to manipulate basic algebraic identities. Should we face this inequity of their past education in second semester calculus? I think yes. I could throw up my hands in despair and just say they're too far gone, but, I refuse. It's not too late to learn elementary algebra and trigonometry. That is &lt;strong&gt;all&lt;/strong&gt; trig-substitution entails. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Almost any topic we cover in calculus, or algebra for that matter, can be done with a CAS. However, to think that the point of calculus is to solve calculus problems misses the mark. The real reason is that these students (I am not talking about the life-science calculus crowd here, rather those students destined for science-heavy technical majors or math) &lt;strong&gt;need&lt;/strong&gt; to refine their algebra skills. When they rely on a CAS to work around it, they make the course a waste of time. Much like a calculus III student I talked with today who is using a CAS to do his homework, fine, but, I wonder how will the test go? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basic knowledge of graphs of sine, cosine, tangent etc as well as their values is in short supply with many of this generation. I believe this is in no small way tied to overuse of calculators in the precalculus mathematics. On the other hand, it also tied to the fact that their education has been &lt;strong&gt;eviscerated&lt;/strong&gt; by the removal of more and more topics which we deemed too difficult by &quot;educators&quot;. Surely, by now it has been clearly demonstrated that the removal of hard topics from elementary school does &lt;strong&gt;not&lt;/strong&gt; bring us a society of like-educated people. In fact, with lower standards we all get left behind because none of us really know the math anymore. I've seen the texts my colleague used as a child. It's sobering to see how far we've allowed the standards to drop. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hold the line.&lt;/strong&gt; Teach trigonometric substitution. If nothing else, as an opportunity to teach some much needed trigonometry in calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should qualify, I do not view university &lt;strong&gt;education&lt;/strong&gt; as a job-training program. I want my students to &lt;strong&gt;think&lt;/strong&gt;. If they want to solve problems without thinking later, that's their business. The reason we solve problems in calculus is not for their applicability. Rather, we solve problems in the interest that the student learn to think. If the problem happens to be really really real-world applicable then great, but, that is &lt;strong&gt;not&lt;/strong&gt; the primary purpose. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-10-08T03:18:12.920" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="9745" PostTypeId="2" ParentId="9738" CreationDate="2015-10-08T05:12:57.630" Score="4" Body="&lt;p&gt;&quot;Introduction to Linear Algebra&quot; by Professor Gilbert Strang &lt;/p&gt;&#xA;&#xA;&lt;p&gt;along with the accompanying video lectures &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5673" LastActivityDate="2015-10-08T05:12:57.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9747" PostTypeId="2" ParentId="9636" CreationDate="2015-10-08T17:34:51.133" Score="0" Body="&lt;p&gt;No. In high school, you do homework. In college, you study. This is precept #11 in the list of differences between high school and college. For the full list: &lt;a href=&quot;http://www.public-domain-materials.com/folder-the-difference-between-high-school-and-college.html&quot; rel=&quot;nofollow&quot;&gt;http://www.public-domain-materials.com/folder-the-difference-between-high-school-and-college.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2039" LastActivityDate="2015-10-08T17:34:51.133" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9748" PostTypeId="2" ParentId="5692" CreationDate="2015-10-08T18:42:51.580" Score="1" Body="&lt;p&gt;I have created a website of public domain mathematics teaching materials. Most of the material is in the K-12 range. (I have been developing these materials since the late 1970’s.) The address is: &lt;a href=&quot;http://www.public-domain-materials.com/&quot; rel=&quot;nofollow&quot;&gt;http://www.public-domain-materials.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the items in my website is autocorrected exercises in proving theorems:&#xA;&lt;a href=&quot;http://www.public-domain-materials.com/folder-student-exercise-tasks-for-mathematics-language-arts-etc---autocorrected.html&quot; rel=&quot;nofollow&quot;&gt;http://www.public-domain-materials.com/folder-student-exercise-tasks-for-mathematics-language-arts-etc---autocorrected.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The materials are more “open” than “open source”, since they are in the public domain. You can use the materials, in whole or in part, for any purpose whatsoever, including commercial purposes, without notifying me or even acknowledging source. It is just as if you yourself had written the materials, except that you cannot restrict anyone else’s use of the materials.&lt;/p&gt;&#xA;" OwnerUserId="2039" LastActivityDate="2015-10-08T18:42:51.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9749" PostTypeId="2" ParentId="9728" CreationDate="2015-10-08T19:22:40.840" Score="2" Body="&lt;p&gt;As a group &quot;high school students&quot; (at least in the United States) are not ready to jump into &quot;Baby Rudin&quot; (basically junior level college and up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a high schooler, I was admitted into a summer math program, where the two &quot;introductory&quot; courses were Number Theory and Abstract Algebra. The better students in that group were ready for &quot;Baby Rudin.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that is an exceptional group of people. One of my roommates went to Harvard, and went on to help Bill Gates found Microsoft.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2015-10-08T19:22:40.840" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="9750" PostTypeId="2" ParentId="9722" CreationDate="2015-10-08T19:24:51.187" Score="1" Body="&lt;p&gt;To some, &quot;mathematics&quot; is foreign &quot;language.&quot; One way to teach mathematics is through an &quot;immersion&quot; program, whereby people express themselves &quot;mathematically,&quot; 24/7, even when they are having dinner, or going about their &quot;normal&quot; activities.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2015-10-08T19:24:51.187" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9751" PostTypeId="1" CreationDate="2015-10-08T22:51:30.013" Score="17" ViewCount="332" Body="&lt;p&gt;Today I realized that many of my students in an upper-division undergraduate (projective) geometry class don't really understand how to use pictures in proofs.  Some draw no pictures at all, making their proofs nearly impossible to read; some oppositely draw a picture but don't explain it at all; and others try to explain their picture but get confused about things like which elements of the picture are givens and which are constructed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that I've noticed this problem, it seems obvious and that I should have expected it.  Probably no one has ever &lt;em&gt;taught&lt;/em&gt; them how to use pictures in proofs!  I can't remember ever seeing this topic discussed in a logic textbook or an introduction-to-proofs class.  And the correct use of pictures in proofs is actually a very subtle thing, e.g. how a picture should be &quot;generic&quot; so as to not give false intuitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone point to any good resources (books, papers, web pages, videos, whatever) that teach students how to use pictures in proofs?&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2015-10-09T14:46:52.530" Title="Teaching the use of pictures in proofs" Tags="|undergraduate-education|proofs|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9752" PostTypeId="1" CreationDate="2015-10-09T06:43:41.100" Score="3" ViewCount="148" Body="&lt;p&gt;I would like to give some private math classes online and for this purpose I currently use this pen: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B002OOWC3I&quot; rel=&quot;nofollow&quot;&gt;Wacom Bamboo Pen Tablet&lt;/a&gt;. The problem is I cannot see directly where I am writing. I'm thinking about buying a &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B007M506WS&quot; rel=&quot;nofollow&quot;&gt;Samsung Galaxy Note 10.1&quot;&lt;/a&gt; and using its pen to give my classes. I would like to know if there is another tablet available which would better suit my purposes, or if there is good software to use with Galaxy note in order to give classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'm looking for is software similar to this &lt;a href=&quot;https://www.youtube.com/watch?v=IjJxjcbVM0w&quot; rel=&quot;nofollow&quot;&gt;site&lt;/a&gt; [the video is in portuguese].&lt;/p&gt;&#xA;" OwnerUserId="5236" LastEditorUserId="3505" LastEditDate="2015-10-10T15:53:42.470" LastActivityDate="2015-10-24T14:06:48.013" Title="What is the best hardware/software available to give online classes" Tags="|technology-in-education|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9753" PostTypeId="2" ParentId="9751" CreationDate="2015-10-09T12:13:06.780" Score="6" Body="&lt;p&gt;The first thing comes to (my) mind is &lt;a href=&quot;http://www.maa.org/publications/books/proofs-without-words&quot;&gt;Proofs without Words I&lt;/a&gt; (Exercises in Visual Thinking) and &lt;a href=&quot;http://www.maa.org/publications/books/proofs-without-words-ii&quot;&gt;II&lt;/a&gt; (More Exercises in Visual Thinking).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also you (not your students) might find interesting the section &quot;Visualization and Diagrammatic Reasoning in Mathematics&quot; of the chapter &quot;Proof: Its Nature and Significant&quot; of the book &quot;&lt;a href=&quot;https://books.google.co.uk/books?id=wPhwJdjI-dIC&amp;amp;printsec=frontcover&amp;amp;dq=proof%20and%20other%20dilemmas&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CCAQ6AEwAGoVChMI3Jviia61yAIViMcUCh0jHwbQ#v=onepage&amp;amp;q=proof%20and%20other%20dilemmas&amp;amp;f=false&quot;&gt;Proofs and other Dilemmas: Mathematics and Philosophy&lt;/a&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll try to add to this list later on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again you might find this one worthy of reading: &quot;&lt;a href=&quot;http://ukcatalogue.oup.com/product/9780199285945.do&quot;&gt;Visual Thinking in Mathematics&#xA;An Epistemological Study&quot;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The videos used in &lt;a href=&quot;http://cosmolearning.org/courses/visual-introduction-projective-geometry/video-lectures/&quot;&gt;this personal website&lt;/a&gt; about projective geometry might be handy to show your students some pictures hoping that they use them in proof. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="1217" LastEditDate="2015-10-09T13:06:04.397" LastActivityDate="2015-10-09T13:06:04.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9754" PostTypeId="2" ParentId="9751" CreationDate="2015-10-09T14:46:52.530" Score="1" Body="&lt;p&gt;This is not an answer to your question (sorry!), but rather two example&#xA;books that to me illustrate(!) the power of pictures in proofs:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Tristan Needham,&#xA;&lt;a href=&quot;http://usf.usfca.edu/vca//&quot; rel=&quot;nofollow noreferrer&quot;&gt;Visual Complex Analysis&lt;/a&gt;, Oxford Univ. Press.&#xA;&lt;hr /&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/CXmuw.jpg&quot; alt=&quot;Needham cover&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) Nathan Carter, &lt;a href=&quot;http://www.maa.org/publications/ebooks/visual-group-theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Visual Group Theory&lt;/em&gt;&lt;/a&gt;, MAA.&lt;br&gt;&#xA;&lt;hr /&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/kDwQ3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kDwQ3.png&quot; alt=&quot;CarterImage&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-10-09T14:46:52.530" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9755" PostTypeId="2" ParentId="9702" CreationDate="2015-10-09T18:18:31.993" Score="5" Body="&lt;p&gt;Just to show what it could be like, Math 116 at Michigan is a Calc II course which does not cover trig integrals, and which I consider one of many possible good courses. You can read the syllabus &lt;a href=&quot;http://www.math.lsa.umich.edu/courses/116/1Syllabus-F15.html&quot;&gt;here&lt;/a&gt; which, combined with the table of contents for the &lt;a href=&quot;https://www.wileyplus.com/WileyCDA/Section/Calculus-Single-and-Multivariable-6th-Edition.id-816385.html?descCd=toc&quot;&gt;textbook&lt;/a&gt;, should give you an idea of what is covered. Chapters 7.3 and 7.4 are what I think of as the standard &quot;techniques of integration&quot; material, and they are not included.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the most recent &lt;a href=&quot;http://www.math.lsa.umich.edu/courses/116/Exams/u2015winter/FinalExam/final_W15.pdf&quot;&gt;final exam&lt;/a&gt;, to give an idea of what students learn to do by the end of the course.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2015-10-09T18:18:31.993" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9756" PostTypeId="2" ParentId="9722" CreationDate="2015-10-10T14:30:03.493" Score="3" Body="&lt;p&gt;I recently bought &lt;em&gt;Thinking Mathematically&lt;/em&gt; by John Mason with Leone Burto and Kaye Stacey. I have the revised edition, but I believe there is a newer edition. I am an educator who loves math, but I am not a math educator. I've found this book invaluable. There are many simple concepts that I was never taught as a high school student despite being someone who did well in math and took calculus in high school. I wish I had a formal course that taught these courses when I was younger. I am finding it much easier to learn higher level math now as an adult because I started with bigger picture issues like how to think mathematically first.&lt;/p&gt;&#xA;" OwnerUserId="5679" LastActivityDate="2015-10-10T14:30:03.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9757" PostTypeId="2" ParentId="9696" CreationDate="2015-10-10T15:04:25.097" Score="2" Body="&lt;p&gt;I agree that you should ask her to read the orginal problem aloud. Does she understand that the problem reads &quot;fourteen plus twenty-seven?&quot; If you write a problem both &quot;horizontally&quot; and &quot;vertically&quot; and then ask her to read both versions aloud, does she read them the same? Can she demonstrate that problem with unifex cubes or on a number line? Can she explain in some other way why 14 + 27 = 41?&lt;/p&gt;&#xA;" OwnerUserId="5679" LastActivityDate="2015-10-10T15:04:25.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9759" PostTypeId="2" ParentId="7431" CreationDate="2015-10-11T10:36:54.477" Score="4" Body="&lt;p&gt;Give problems that a CAS won't be able to evaluate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;a href=&quot;http://www.wolframalpha.com/input/?i=integrate+x%5E%7B10%5E%7B10000%7D%7D&quot; rel=&quot;nofollow&quot;&gt;look&lt;/a&gt; how Wolfram Alpha fails to give &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\int x^{10^{10000}} \mathrm{d}x = \frac{x^{10^{10000}+1}}{10^{10000}+1}+C$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Several other integral calculators I tried also fail to give it. &#xA;Of course this is rather a simple integral, but one may also apply it to other integrals, like $\sin(10^{10000}x)$.&lt;/p&gt;&#xA;" OwnerUserId="5664" LastActivityDate="2015-10-11T10:36:54.477" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9760" PostTypeId="1" CreationDate="2015-10-07T23:44:51.203" Score="8" ViewCount="427" Body="&lt;p&gt;This question is related to &lt;a href=&quot;https://math.stackexchange.com/questions/476471/teaching-a-4-year-old-maths&quot;&gt;https://math.stackexchange.com/questions/476471/teaching-a-4-year-old-maths&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My 3 years old seems interested in math (remembering numbers, counting stuff and more). I want to encourage this. However, it seems that it would be best if the kid can have some fun game with his small children's tab (not an Apple one). I wanted to know if there are any video games/apps that can get kids interested in math (and also science)? Though I am looking for some free ones, but I would also like to know if there are some paid stuff out there.&lt;/p&gt;&#xA;" OwnerUserId="5686" OwnerDisplayName="dbm" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2020-04-04T12:53:16.360" Title="Are there any video games/app to teach children at age 3-4-5?" Tags="|technology-in-education|software|preschool-education|" AnswerCount="9" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="9761" PostTypeId="2" ParentId="9760" CreationDate="2015-10-11T13:10:38.727" Score="9" Body="&lt;p&gt;Not formally an answer (&lt;strong&gt;so if you have one, do not let this prevent you to post it&lt;/strong&gt;), but I think challenging the implicit assumption that it would be good to have your kid use any such software deserves more than a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, I remember that several studies failed to prove any improvement on teaching math using computer technologies. This lets me suspect that tablets and educative apps might be less of a great teaching tool than they are told to be by those selling them, but is not completely to the point, as the comparison in those studies was with other form of teaching, not with nothing at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More importantly, quite some researchers strongly advise against exposing young children to screens, even for educational purposes (my reference is unfortunately in French: &lt;a href=&quot;http://www.lemonde.fr/idees/article/2013/02/08/laisser-les-enfants-devant-les-ecrans-est-prejudiciable_1829208_3232.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.lemonde.fr/idees/article/2013/02/08/laisser-les-enfants-devant-les-ecrans-est-prejudiciable_1829208_3232.html&lt;/a&gt;, written by Michel Desmurget (neuroscientist at Inserm), Laurent Bègue (Professor in social psychology), Bruno Harlé (pedopsychiatrist)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To prevent or severely limit the use of a tab before a few more years might be the best thing you can do to ensure your child's mind is able to thrive latter on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Clarification added in edit:&lt;/strong&gt; my reason to write this is that I think that this information is relevant to the question even if it does not answer it, and that it should be brought to the attention of anyone interested in the question in the first place. Feel free to flag  (and to delete) if you think this is against the best practice on this site to use answer in that way.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2015-10-13T20:54:26.120" LastActivityDate="2015-10-13T20:54:26.120" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9762" PostTypeId="2" ParentId="7431" CreationDate="2015-10-11T19:33:56.960" Score="2" Body="&lt;p&gt;I think there's an assumption here that as the instructor you are: (1) collecting homework, (2) correcting the homework, and (3) giving grades for the homework. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Honestly, I wrestled with doing that for about a decade and I decided that I simply had to give up on it. The overall process was so time-consuming, so distasteful, and so difficult for me to parse poorly-written student work that I was simply tearing my hair out. I think there was a Thanksgiving break where I had a quasi-breakdown over it and said to myself that it just had to stop. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think that a perfectly legitimate strategy at the college level is this: Don't collect, correct, or give points for homework. The homework exists as an exercise for the student to prove to themselves that they've mastered the skill. My students don't wait for me for confirmation: immediately check the answer at the back or, yes, type it into Wolfram Alpha (for, say, even-numbered problems). If they don't get it right immediately, then fix  it. If they can't fix it on their own, have a study group or go to the math workshop. If they can't fix it then, ask about it in the next class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I start every class with a call for any questions about the homework, and as long as a single student asks about something, I try to lavish them with praise and attention as we work through their problem. Hopefully this demonstrates the expected interaction and primes the pump for other future questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But grading is purely on the basis of quizzes and in-class tests (where of course, well-written justifications must be made in each case). As one example, my college algebra classes almost uniformly fail the first test, no matter how much I emphasize every day what the grading criteria will be. But generally after this &quot;shock treatment&quot; they do bounce back, mostly start doing their own homework, and asking questions in class about the best way to write or present results. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's no way to verify that students are doing their own homework, so I think it's best to be honest about that fact and just say that the homework is for their own benefit to exercise and prove their knowledge to themselves. The graded tests are how we later assess that the skills and knowledge are really present.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; To be clear, my point is that in discarding protocols 1-3 (the expectation that homework will be collected, corrected, and awarded grades), then the existence of automated math engines is non-problematic, and traditional math homework is seen to be as meaningful as ever. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2015-10-12T06:12:08.283" LastActivityDate="2015-10-12T06:12:08.283" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9763" PostTypeId="2" ParentId="7936" CreationDate="2015-10-12T16:02:19.397" Score="2" Body="&lt;p&gt;There is a long list of definitions of &quot;affine space&quot; &lt;a href=&quot;http://ncatlab.org/nlab/show/affine+space&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;; I don't know whether any of them fits your needs.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2015-10-12T16:02:19.397" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9764" PostTypeId="2" ParentId="7936" CreationDate="2015-10-12T16:53:25.973" Score="2" Body="&lt;p&gt;In &quot;that one&quot; question you reference, you mention becoming an examples based mathematician and teacher.  Examples and how they are used are part of what needs to be taught, and (I believe) are more easily grasped by the bulk of the classroom than (a presentation that heavily favors) an axiomatic framework and derivation.  The framework you seek is a theory which allows a form of presentation of the examples so that the students can derive them as well as learn them.  (The one thing I remember from a projective geometry class that I graded was an application for railroad tunnel making: how to extend a line past a bounded obstacle using geometric construction.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The teachers need to know a list of important examples and the accompanying derivations.  You can use an axiomatic framework as a basis, but for the teacher to internalize and communicate it, you need (could really use to great effect) mottos and mnemonics.  For the translation by two rotations, you can use &quot;Ballerina twirls across the room&quot; to convey the idea.  You can devise similar mnemonics to communicate key ideas of the result derivation from the axiom system you care to use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have a list of such results, mottos, mnemonics, and derivations, you can prioritize and see which ones give you the greatest pedagogical mileage.  Present those to the teachers, and assign the rest as exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Mentally Dances Through The Answer&quot; Paseman, 2015.10.12&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-10-12T16:53:25.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9765" PostTypeId="2" ParentId="9702" CreationDate="2015-10-12T17:03:16.040" Score="3" Body="&lt;p&gt;Not necessarily an answer, but some points:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If further courses require students to be able to do trigonometric substitution, then it does need to be taught; the calculus curriculum may or may not be the place to do it. (but one should probably ensure that it will happen elsewhere before one excises it from the calculus courses!)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If one expects students to rely on CAS to solve problems that require trigonometric substitution, then there must be a place in the curriculum where students learn how to use CAS effectively.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In order to get the most use out of CAS, it helps to have an idea of what sorts of things they can solve; both so that they know what to aim for when analyzing a problem, and where they can stop doing analysis and just solve.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2015-10-12T17:03:16.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9766" PostTypeId="1" AcceptedAnswerId="9770" CreationDate="2015-10-13T17:43:51.940" Score="15" ViewCount="700" Body="&lt;p&gt;Before teaching the chapter on determinants in a linear-algebra course for beginning undergraduate students (mathematics and computer science, more specifically) I would like to give a small introduction and convince the students that this is an interesting notion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What could one tell beginning undergraduate students in 15 to 30 minutes to get them more interested in determinants? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What then would follow are  the &quot;usual things&quot; (special cases for small dimension, multi-linearity, computation via row and column transform, computation by developing along a line/column, Leibniz formula, Cramer's rule and formula for the inverse matrix). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I might modify this slightly if it helps to make the motivation make sense in retrospect, but to the extent possible I would like to keep the question focused on an introductory presentation (rather than the actual content), as&#xA;I am aware of the question &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/1149/is-there-a-good-way-to-explain-determinants-in-an-elementary-linear-algebra-clas&quot;&gt;Is there a good way to explain determinants in an elementary linear algebra class?&lt;/a&gt;&quot; which covers the aspect how to teach determinants. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To high-light and to reiterate, the intended difference between the two questions is that while the referenced one asks about how to actually teach the subject I am mainly interested in &lt;strong&gt;advice for a short (non-technical) introduction&lt;/strong&gt; to be given before the subject of determinants is actually discussed. Thus, I would be willing to do some hand-waving and over-simplifications to convey an application of determinants and the related ideas also in areas currently beyond the students reach. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something I would like to say is: Knowing the determinant is important to doing ThisInterestingThing? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am mainly looking forward to answers in this sense, but more general considerations are welcome, too. &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-10-20T19:50:28.903" Title="How to get students in a under-graduate linear algebra course interested in determinants?" Tags="|undergraduate-education|concept-motivation|linear-algebra|" AnswerCount="6" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="9767" PostTypeId="1" CreationDate="2015-10-13T18:45:45.783" Score="10" ViewCount="394" Body="&lt;p&gt;So far I have found an offline version of Khan Academy, but that´s about it. Does anyone know of similar offline programs that one can use to teach mathematics? How about mathematics assesment? I have not been able to find any yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I´m looking for exclusively offline programs that can be used in remote areas where there is no internet available.&lt;/p&gt;&#xA;" OwnerUserId="5692" LastActivityDate="2020-04-03T19:11:13.213" Title="Are there free, offline mathematics assesment and learning programs for grades 1 to 12?" Tags="|software|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9768" PostTypeId="2" ParentId="9760" CreationDate="2015-10-13T21:46:14.840" Score="6" Body="&lt;p&gt;(Migrated from the comments.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are three specific suggestions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.teachley.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Teachley&lt;/strong&gt;&lt;/a&gt; (company page)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.tiggly.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Tiggly&lt;/strong&gt;&lt;/a&gt; (company page)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bedtime_Math&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Bedtime Math&lt;/strong&gt;&lt;/a&gt; (wikipage)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I mentioned the first two in an earlier answer to &lt;a href=&quot;https://matheducators.stackexchange.com/a/843&quot;&gt;&lt;strong&gt;MESE 750&lt;/strong&gt;&lt;/a&gt;; I became aware of these two companies because the founders (all three at #1; one at #2) worked with my graduate advisor (&lt;a href=&quot;https://en.wikipedia.org/wiki/Herbert_Ginsburg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Herb Ginsburg&lt;/strong&gt;&lt;/a&gt;, whose expertise is on age 3 to grade 3 mathematical thinking, the work of Piaget and the clinical interview technique, and teaching mathematics with software) which means, in particular, these are examples of mathematics teaching apps based on &lt;em&gt;actual&lt;/em&gt; research from educators and teachers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest the first two, to start with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot speak directly to #3, but I have heard it spoken of in positive terms.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-10-13T21:46:14.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9769" PostTypeId="2" ParentId="9723" CreationDate="2015-10-14T00:10:24.100" Score="1" Body="&lt;p&gt;Perhaps a more interesting (and specific) question would be, does schooling cause (or increase) gaps in achievement? And one way to approach that would be to look at the range of achievement (leaving out the top and bottom x%) in different countries.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-10-14T00:10:24.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9770" PostTypeId="2" ParentId="9766" CreationDate="2015-10-14T00:22:19.460" Score="18" Body="&lt;p&gt;I have found it motivates to explain the determinant as computing a volume.&#xA;One can work through and convince for $2 \times 2$ and $3 \times 3$ matrices,&#xA;and perhaps only hint at the &#xA;$n \times n$ generalization, when $|\det(M)|$ is the volume of the $n$-dimensional&#xA;parallelepiped spanned by the column vectors of $M$.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/ZOzBj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZOzBj.png&quot; alt=&quot;Parallelepiped&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Determinant&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia's Determinant article&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;It is an added benefit that shuffling the column vectors&#xA;clearly does not change the parallelepiped's absolute&#xA;volume. One can then talk about signed area and signed volume.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These volume determinants are heavily used in computer graphics,&#xA;providing more motivation. &#xA;For example, signed area leads to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Shoelace_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;shoelace formula&lt;/a&gt;&#xA;for computing the area of a simple polygon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be a challenge to find a 3D computer game that does &lt;em&gt;not&lt;/em&gt;&#xA;use the signed volume determinant, e.g., to detect whether&#xA;an object/point $p$ is inside or outside another object. Objects&#xA;are triangulated, so one is computing the signed volume of a tetrahedron&#xA;determined by a surface triangle $T$ and point $p$, which is a corner &#xA;($\frac{1}{6}$-th) of the parallelepiped.&#xA;In other words, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Knowing the determinant is important to doing...&quot;&#xA;  &lt;em&gt;collision detection in 3D games&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-10-14T11:55:03.370" LastActivityDate="2015-10-14T11:55:03.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9771" PostTypeId="2" ParentId="9766" CreationDate="2015-10-14T02:28:46.530" Score="2" Body="&lt;p&gt;Knowing the determinant of a matrix lets you:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Solve the corresponding set of linear equations.&lt;/li&gt;&#xA;&lt;li&gt;Find the eigenvalues of the matrix.  The eigenvalues correspond to the growth rates of the vector solutions as the matrix is repeatedly multiplied by itself.&lt;/li&gt;&#xA;&lt;li&gt;Invert the matrix.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2015-10-14T02:28:46.530" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9772" PostTypeId="2" ParentId="9766" CreationDate="2015-10-14T16:24:41.497" Score="6" Body="&lt;p&gt;Maybe begin by saying the determinant is a test of whether a matrix is singular.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;anti Loop Space:&lt;br&gt;&#xA;Determinants must be retained, since they are routinely used in branches of learning other than abstract algebra.  Physics, engineering, even ODE and PDE.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree we should not calculate large determinants by hand.  But after completing your course, a student should have no problem understanding if his future lecturer says: &quot;by Cramer's Rule, we see that the solution is a rational function of the variables&quot;.  [an example from my personal experience]&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2015-10-14T16:24:41.497" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9773" PostTypeId="2" ParentId="9766" CreationDate="2015-10-14T18:31:05.250" Score="8" Body="&lt;p&gt;To give a brief list of interesting applications:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Volume&lt;/strong&gt; obviously the lead application. It is not unreasonable to say determinants are volumes. Of course, they're more than that, their &lt;strong&gt;signed-volumes&lt;/strong&gt;. For example, if $v,w$ are three dimensional vectors tangent to a plane and we want to decide which side of the plane a third vector $u$ is on then the &lt;strong&gt;sign&lt;/strong&gt; of $det(u|v|w)$ tells us if $u$ is &lt;strong&gt;up&lt;/strong&gt; or &lt;strong&gt;down&lt;/strong&gt;. This idea persists in $n=2$ (it's CW verses CCW there) and $n&amp;gt;3$.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Cramer's Rule&lt;/strong&gt; can be used in both numerical and symbolic systems of equations. For example, it lies at the heart of the mysterious variation of parameter formulas you see in the differential equations course.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Exterior Algebra&lt;/strong&gt; with the determinant in hand you can define wedge products as particular multilinear maps with formulas given by determinants. This is strongly connected to the larger program of detecting linear dependence by vanishing determinants or, vanishing wedge products.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Super slick formulas&lt;/strong&gt; there are too many to list. Slater determinants, the two-point formula for a line, the three point formula for a plane, a differential equation with a given solution set, the inverse of a matrix, triple product identity, Lagrange's identity,...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Anyway, so much of the mystique stems from defining the determinant via Laplace's expansion by minors. Instead, we should introduce it in its raw multilinear glory.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-10-14T18:31:05.250" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9774" PostTypeId="2" ParentId="9766" CreationDate="2015-10-15T09:17:57.387" Score="4" Body="&lt;p&gt;One very practical area (ok, mathematically practical...) where determinants are used quite often is in multivariable statistics -- and I'm not talking about advanced mathematical statistics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, lets say you have a random &lt;strong&gt;vector&lt;/strong&gt; (i.e., its components are random variables), the determinant allows you to calculate a single number that is the multivariate equivalent of a &quot;variance&quot; for this vector. That this can be gainfully done is not something that is immediately obvious for random vectors. Specifically, if $\Sigma$ is the covariance matrix, then $|\Sigma|$ is analogous to a univariate variance. This is especially apparent if you compare the univariate and multivariate Gaussian densities:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f(x) = \frac{1}{\sqrt{2\pi \sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f_{\mathbf x}(x_1,\ldots,x_k) = \frac{1}{\sqrt{(2\pi)^k|\boldsymbol\Sigma|}} \exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu}) \right)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another important use of determinants is when you want to &lt;a href=&quot;http://ece-research.unm.edu/bsanthan/ece340/note1.pdf&quot; rel=&quot;nofollow&quot;&gt;compute functions of random variables&lt;/a&gt;. For example, if $\mathbf{x}$ is a continuous random vector, where $\mathbf{x}$ has pdf $f(\mathbf{x})$ and we want to calculate the density function of $A\mathbf{x}+\mathbf{b}$, where $A$ is a non-singluar matrix, it turns out that it is not as bad as you'd initially think:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$A\mathbf{x} + \mathbf{b} \sim \frac{1}{|A|}f(A^{-1}[\mathbf{x}-\mathbf{b}])$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quantity $\frac{1}{|A|}$ is called the &lt;em&gt;Jacobian&lt;/em&gt; and ensures that when we integrate the transformed density function, it still integrates to 1 (see the connection to volume again?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, to the extent that students see statistics as &quot;practical&quot; or &quot;interesting&quot;, the above may be motivating examples of the everyday use of determinants.  &lt;/p&gt;&#xA;" OwnerDisplayName="user5661" LastActivityDate="2015-10-15T09:17:57.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9775" PostTypeId="2" ParentId="5825" CreationDate="2015-10-16T02:52:57.920" Score="2" Body="&lt;p&gt;There are many flaws in this system, but there aren't many other ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As @celeriko pointed out, testing on material that has not been taught yet is dangerous. It can result in not promoting a fast learner who is truly &quot;gifted&quot;, but rather promoting a kid who worked for hours the week before the test on the next year's textbook, on the way attaining many holes in their math and hurting them in the long run. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the flip side, if a student is truly talented and learns fast, they should have taken initiative, possibly through their parents, to explore higher math by now and should be comfortable with the next year's concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your concerns are valid, and they should definitely be also integrating teacher input and grade on the current class to ensure social maturity, capability of learning, and consistent performance. In summary, the test itself is fairly useless without external validation, at which time it becomes useful enough to make the decision.&lt;/p&gt;&#xA;" OwnerUserId="5708" LastActivityDate="2015-10-16T02:52:57.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9776" PostTypeId="1" CreationDate="2015-10-16T11:44:27.120" Score="11" ViewCount="203" Body="&lt;p&gt;Consider the following piece of reasoning:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x+3 = 5$$&#xA;$$x=5-3$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on my own experience back in school, and helping people with their math homework, as well as in talking to other people about math education, it seems that in the minds of most students, when they do the above, they are &quot;moving the three to the other side of the equation&quot;. I've seen students act genuinely surprised when you point out that actually, you're not &quot;moving&quot; anything, you're simply saying that it is a true fact that if $x$ is a number such that $x+3=5$, then it must also be the case that $x=5-3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that students perceive a lot of math as just being &quot;rules&quot;, like the rules of chess, rather than &quot;facts&quot;, things that are actually true. When they solve a problem, they think of themselves as playing a game using certain allowed moves, rather than proving that something is true based on what they already know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do students know - do they feel in their gut - that the Pythagorean theorem is &lt;em&gt;true&lt;/em&gt; in the sense that it would check out if you measured the actual side lengths of an actual right angle triangle? Or do they just think of it as a rule for finding $c$ in an exercise?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What I'm looking for here are scientific studies&lt;/strong&gt;, or any research at all into this distinction between &quot;facts&quot; and &quot;rules&quot; - not anecdotal evidence. I want to know if this distinction has been discussed before in the literature, and what research has been done into finding out if my suspicions are correct.&lt;/p&gt;&#xA;" OwnerUserId="115" LastActivityDate="2019-04-10T08:08:52.543" Title="Has any research been done on the topic of &quot;fact-based&quot; versus &quot;rule-based&quot; understanding?" Tags="|reference-request|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9777" PostTypeId="2" ParentId="9776" CreationDate="2015-10-16T15:45:32.077" Score="2" Body="&lt;p&gt;I think what you are thinking of is found in maths research described as 'processes' and 'objects'. See, for example, Chapter 2 of &lt;a href=&quot;https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/8846/1/Alcock_Simpson_book.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alcock--Simpson&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The big idea is that many (most?) students will start from think of &lt;span class=&quot;math-container&quot;&gt;$5+3$&lt;/span&gt; as the process of starting with &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and then adding &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; to it. That is different to seeing &lt;span class=&quot;math-container&quot;&gt;$5+3$&lt;/span&gt; as a number (that we more conveniently call &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;). The change in thinking from one to the other (or better, to being able to switch between the two ideas at will) comes with familiarity, but can take a very long time if nothing happens to show the limitations of the first approach.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="376" LastEditDate="2019-04-10T08:08:52.543" LastActivityDate="2019-04-10T08:08:52.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9778" PostTypeId="2" ParentId="9738" CreationDate="2015-10-16T16:32:41.363" Score="1" Body="&lt;p&gt;I would back up @Alexander-Woo for Poole's text as well as offer the suggestion for &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shifrin's &quot;Linear Algebra - A Geometric Approach&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;None of these books are really affordable. There was a suggestion earlier of a free text, and I say... supplement that, frankly. More work, but it's worth considering your students and their finances at this point. &lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2015-10-16T16:32:41.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9779" PostTypeId="2" ParentId="280" CreationDate="2015-10-16T18:51:03.540" Score="5" Body="&lt;p&gt;Solve for a required grade at the end of a course. A common question that many of us get from students is, &quot;What do I need to score on the final exam to get an A in the course?&quot; or somesuch. My response is now always, &quot;You've just asked me an algebra question, you should be able to solve that yourself.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I set up my grading formula specifically to support this exercise: $W = 15\%Q + 50\%T + 35\%F$, where W = weighted total for the course, Q = quiz average, T = test average, F = final exam score. A course &quot;A&quot; grade requires at least W = 90, &quot;B&quot; at least W = 80, etc. In the last week, if the student has a particular target grade then that dictates W, and Q and T are known, so the only unknown is F.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my elementary algebra class, I spend an hour on this as its own exercise around mid-semester. In higher-level courses, if a student asks the question in the last week, then I'll help remind/set up the grading formula, and let them solve it themselves. Sometimes this gets a pretty intense &quot;wow!&quot; reaction, like it's the first time when a problem they personally instigated is being solved by algebra; sometimes phones come out and they take photos of it, etc.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-10-16T18:51:03.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9780" PostTypeId="1" AcceptedAnswerId="9781" CreationDate="2015-10-17T21:32:23.847" Score="11" ViewCount="302" Body="&lt;p&gt;I am teaching &lt;em&gt;Theory of Computation&lt;/em&gt; for the first time soon, and need a short&#xA;introduction and justification to motivate students who have only &lt;em&gt;Introductory Programming&lt;/em&gt; behind them.&#xA;I am torn between:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1) Emphasizing the &lt;strong&gt;limits&lt;/strong&gt; of computation, i.e., there is no universal debugging program (Turing, Halting.)&lt;/p&gt;&#xA;&lt;p&gt;(2) Emphasizing the &lt;strong&gt;practical&lt;/strong&gt; value of searching webpages to gather data using regular expressions, leading to automata, coding, etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The choice is between emphasizing the &lt;strong&gt;negative&lt;/strong&gt;—the ultimate limits on computation—or&#xA;emphasizing the &lt;strong&gt;positive&lt;/strong&gt;—what's achievable with proper understanding of the theory.&#xA;Suggestions welcomed!&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-10-24T20:27:45.223" Title="How to motivate 'Theory of Computation'?" Tags="|undergraduate-education|student-motivation|concept-motivation|computer-science|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9781" PostTypeId="2" ParentId="9780" CreationDate="2015-10-17T23:56:01.613" Score="4" Body="&lt;p&gt;They need at least a passing acquaintance with both aspects. There are many &lt;em&gt;practical&lt;/em&gt; uses (regular expressions for patterns, context free grammars revolutionized programming languages in ways that are hard to appreciate unless you learned classic BASIC or early FORTRAN and later saw Algol or Pascal for the first time) to highlight. The &lt;em&gt;limits of computation&lt;/em&gt; aspects (and problems like P = NP, complexity theory in general) are in my opinion central in any deeper understanding of computing, and (particularly NP completeness) are required knowledge for any professional programmer.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-10-17T23:56:01.613" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9782" PostTypeId="2" ParentId="280" CreationDate="2015-10-18T10:20:36.863" Score="-1" Body="&lt;p&gt;&quot;Alright class: what do you get when you divide 1 by 0?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Almost inevitably you will get the response: &quot;Infinity!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this is wrong, and you can use algebra to show it:&#xA;The question itself is algebra in disguise. It is asking to solve for x:&#xA;$$0*x = 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point, you might write on the board the most basic of algebra equations:&#xA;$$Solve\quad for\quad x:\quad\quad\quad ax = b$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;This, class, is an algebra equation. It can also be written like this...&quot;&#xA;$$x=b/a$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;In our problem, then, b=1 and a=0.&quot;&#xA;$$0*x = 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Now if I plug in x = 1 million, will this solve the equation?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course not: $0*(10^{6})=0\ne 1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;How about if I plug in x = 1 billion?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still &quot;no&quot;: $0*(10^{9})=0\ne 1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;That's right, even if we plug in 1 trillion billion zillion +1, multiplying it by 0 gives us 0, which is not equal to 1.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proper answer in the purely algebraic context is that x is indeterminate: there is no real number that will solve this equation. In fact, the answer &quot;infinity&quot; can only arise in the pre-calc/calc context of limits:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = \lim_{a\downarrow 0}\frac{1}{a}=\infty$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this is the answer to &quot;What is the limit of 1/a as a goes to zero,&quot; Which is not the same question as what we originally asked, which is &quot;What number x do we get when we divide 1 by 0.&quot; &lt;/p&gt;&#xA;" OwnerUserId="5715" LastActivityDate="2015-10-18T10:20:36.863" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9783" PostTypeId="1" CreationDate="2015-10-18T12:35:56.067" Score="7" ViewCount="191" Body="&lt;p&gt;I have to teach someone whose English isn't her native language (and her English skills aren't advanced) the concept of negative numbers + exponents, and how to read scientific / engineering numbers. Do you know of any good website for beginners / children that uses simple English to achieve this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I googled and couldn't find any website that is simple enough and uncluttered by ads (she might not be able to differentiate the ads from teaching text).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should mention that I am not a teacher. I'm an expat computational engineer who, in addition to my contractual engineering tasks, has to perform some kind of &quot;technology transfer&quot; to local engineering trainees who don't use English as a first language. Their main skills are CAD / engineering design. They help me with engineering analysis while they learn on the job, but I am tired of having to explain simple math concepts to them. I hope this community might be able to lead me to the resources I need.&lt;/p&gt;&#xA;" OwnerUserId="5717" LastEditorUserId="122" LastEditDate="2017-07-08T13:22:27.630" LastActivityDate="2017-07-08T13:22:27.630" Title="Online Resources to Explain Negative Numbers and Scientific Notation" Tags="|secondary-education|primary-education|online-instruction|negative-numbers|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9784" PostTypeId="1" AcceptedAnswerId="9799" CreationDate="2015-10-18T16:03:04.557" Score="21" ViewCount="1182" Body="&lt;p&gt;When I taught courses on algebra giving a first exposition to Galois theory I usually included some discussion of classical results showing the impossibility of constructing certain points with ruler and compass, such a the cube root of $2$ (Delian problem) and the trisection of a (generic) angle.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, sometimes I think maybe this is not that good an allocation of time. It took me quite some time to recall and/or to introduce in precise terms what it means to construct something, which feels a bit like an isolated subject in such a course.  Likewise, the results, while historically important and interesting, do not seem of much use later on.&#xA;Since as said I did include the subject, needless to say, I can also see some merit in it. Yet, I never quite managed to make up my mind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, I would like further input on this subject:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are reasons for or against teaching the impossibility of certain geometric constructions in a course on algebra that covers the basics of  Galois theory?&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2015-10-20T15:05:50.037" LastActivityDate="2021-08-18T09:42:07.287" Title="Impossibility of trisecting the angle, doubling the cube and alike, what are reasons for or against discussing them in a course on algebra?" Tags="|undergraduate-education|course-design|geometry|abstract-algebra|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="9785" PostTypeId="1" AcceptedAnswerId="9787" CreationDate="2015-10-18T19:58:35.477" Score="10" ViewCount="294" Body="&lt;p&gt;One of my peers is studying for the GRE and ran into the following problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When is $|x-4|$ equal to $4-x$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I tried to attempt teaching this by developing the intuition for why&#xA;$$|x| = \begin{cases}&#xA;x, &amp;amp; x &amp;gt; 0 \\&#xA;-x, &amp;amp; x &amp;lt; 0 \\&#xA;0, &amp;amp; x = 0 \end{cases}$$&#xA;by taking positive numbers, negative numbers, and $0$ and putting them into the absolute value function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I told her, replace the $x$ with $x-4$. At this point, I lost her.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any suggestions for teaching this to someone with a high-school-math background?&lt;/p&gt;&#xA;" OwnerUserId="1907" LastActivityDate="2015-10-21T17:06:31.360" Title="Teaching about absolute values not centered at $0$" Tags="|algebra|precalculus|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9786" PostTypeId="2" ParentId="9784" CreationDate="2015-10-18T20:30:47.487" Score="10" Body="&lt;p&gt;Perhaps rather than spend time establishing that trisecting an angle is impossible via Euclidean (ruler-compass) constructions,&#xA;you could instead (a)&amp;nbsp;Make that claim without proof,&#xA;and (b)&amp;nbsp;Mention that different axioms &lt;em&gt;do&lt;/em&gt; permit angle trisection: &#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/paqju.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/paqju.png&quot; alt=&quot;Fig.19.1&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Figure from &lt;a href=&quot;http://www.gfalop.org/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Geometric Folding Algorithms: Linkages, Origami, Polyhedra&lt;/em&gt;&lt;/a&gt;, p.286.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;This would (temporarily) turn the topic away from Galois theory and toward the history and philosophy of mathematics, which could be an illuminating interlude.&#xA;You could even have the students trisect an angle in-class via Abe's construction! :-) ~15 minutes.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-10-18T20:30:47.487" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="9787" PostTypeId="2" ParentId="9785" CreationDate="2015-10-18T20:36:42.627" Score="8" Body="&lt;p&gt;First, concerning your tentative explanation: the status of '$x$' is something difficult to convey and inherently subtle. I guess you ran into this issue at least as much as into the absolute value issue. One thing that needs to be told is that as a variable, $x$ is a placeholder just meant to say that its various apparitions in a formula like the one you wrote, all denote the very same number. Try explaining that by replacing it by another letter which you replace by $x-4$, or use sentences like &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given any number, the absolute value of this number is another number; if this number is non-negative then that number is equal to this number by definition, and if this number is negative then we define that number to be minus this number. In particular that number is always non-negative and has the same magnitude than this number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then observe that giving a name to &quot;this number&quot; (say, $y$ or $\square$ or whatever) and &quot;that number&quot; (usually $|y|$ or $|\square|$ or $|\mathrm{whatever}|$) makes things easier to understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning the absolute value issue, you can add other ways to see it (like above: a non-negative number having the same magnitude as the given number ; or the distance to the origin) and explain why they all give the same result. Once this meaning is understood, you can go with formal resolution of the problem. In my experience, trying to get to the formal manipulation of mathematical symbols before they make sense usually causes trouble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added in edit:&lt;/strong&gt; I must say that reading other answers makes me realize that, while I still endorse mine, I kind of fall in the French pitfall of being quite abstract and formal. One should definitely read also the answers below, and notably consider visual explanations and tackle particular cases.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2015-10-21T08:17:45.763" LastActivityDate="2015-10-21T08:17:45.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9788" PostTypeId="2" ParentId="9785" CreationDate="2015-10-18T21:44:51.323" Score="8" Body="&lt;p&gt;I think it is very important to stress that the absolute value of a difference of two numbers computes the distance between these two numbers on the number line.  For one, it gives a concrete geometric meaning to an otherwise abstract concept.  As well, the distance interpretation generalizes well to higher dimensional situations encountered later on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your particular case, $|x - 4|$ is equal to the distance between $x$ and $4$ on the number line.  It will be equal to $4 - x$ precisely when $4$ is at least as big as $x$ (or equivalently, $4$ is to the right of or equal to $x$ on the number line).  That occurs precisely when $x \leq 4$.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2015-10-18T21:44:51.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9789" PostTypeId="1" AcceptedAnswerId="9790" CreationDate="2015-10-18T23:25:13.127" Score="13" ViewCount="958" Body="&lt;p&gt;So, I'm a mathematics tutor, and I always run into this mental block that students have: they don't seem to understand what &quot;in terms of&quot; means. Also, the phrase &quot;in terms of&quot; is so useful, and it kind of frustrates me when student don't understand it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Let me explain why I think it is so useful. Sometimes, when I am working with a word problem, we will define a variable to be something like &quot;x=number of apples&quot; or &quot;y=cost of toaster oven&quot;. Then to help set up an equation to work through the word problem, I will ask them to tell me what the price of 5 toaster ovens &lt;em&gt;in terms of &lt;code&gt;y&lt;/code&gt;&lt;/em&gt;. Sometimes I can work around this phrase, but sometimes I just really wish I could use it. (I guess I could just say &quot;the answer depends on &lt;code&gt;x&lt;/code&gt;&quot;, but that kind of has the same set of problems.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's two examples:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bobby buys 20 apples for \$15 and eats &lt;code&gt;x&lt;/code&gt; apples. He sells the rest for \$2 each. (1) How much money does he make total? Write your answer in terms of &lt;code&gt;x&lt;/code&gt;. (2) How many apples can Bobby eat but still make a profit?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solve the following equation for &lt;code&gt;y&lt;/code&gt; in terms of &lt;code&gt;x&lt;/code&gt;: &lt;code&gt;2y+3x=1&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now while it is fairly easy to get a student to solve such problems, so many students just don't seem to understand the concept of &quot;in terms of&quot;. Since it is such an important concept (it is essentially precursor to functions) I really want to convey some kind of understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any good ideas and advice for explaining this or working around the use of the phrase?&lt;/p&gt;&#xA;" OwnerUserId="5720" LastEditorUserId="5720" LastEditDate="2015-10-19T03:29:47.577" LastActivityDate="2015-10-20T23:04:52.713" Title="Explaining the phrase &quot;in terms of&quot;" Tags="|undergraduate-education|secondary-education|word-problems|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9790" PostTypeId="2" ParentId="9789" CreationDate="2015-10-18T23:40:47.340" Score="7" Body="&lt;p&gt;I do avoid using this phrase in all of my math classes. Not that I've ever thought of it as a particular goal, but I would want to reserve the word &quot;term&quot; to specify an addend. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your first example, I would specify, &quot;Write a formula using the variable $x$&quot;. In your second example, I specify, &quot;Solve the following equation for $y$&quot; (end of direction), having previously defined &quot;solving&quot; as isolating the indicated variable on one side.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-10-18T23:40:47.340" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="9791" PostTypeId="2" ParentId="9717" CreationDate="2015-10-19T00:26:45.247" Score="3" Body="&lt;p&gt;I found this to be a highly readable concept map. The major problem I see is that it clearly has two main concepts: Sets and The set of Real Numbers. Each one of the concepts has an extensive tree of sub-concepts built from it. Additionally, there are only three connections between these two trees and these are all weaker connections: &quot;can be used...&quot;, &quot;is an example of&quot;. You could improve this concept map by breaking it into two maps, one for each core concept. (I suspect you combined these because they were combined in the textbook. Realizing that these concepts which are co-taught are only marginally connected is one valuable thing that can come out of making a concept map.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To address your questions about concept maps in general:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Going from general to specific concepts is a good idea. However, given the highly connected nature of mathematical ideas, you will have lateral connections. Being able to represent lateral connections is advantage that concept maps have over outlines. When possible, arranging your concept map so that the lateral connections are short and don't cross is desirable. Putting the main concept in the center of the page (Radial hierarchy) can help with this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For reflecting on a concept map, seeing the entire map at once is helpful. Include as much detail as you can while keeping the map to one page or screen.&lt;/p&gt;&#xA;" OwnerUserId="5603" LastActivityDate="2015-10-19T00:26:45.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9792" PostTypeId="2" ParentId="9789" CreationDate="2015-10-19T02:59:44.240" Score="-1" Body="&lt;p&gt;What's important is what came before. Early on, algebra students should be comfortable with slope-intercept form, y=mx+b. They may also have seen the notation f(x)=something. In both cases, all x's, powers of x with their coefficients are on the right of the equal sign. These equations are present &quot;in terms of x&quot; or &quot;as a function of x.&quot;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-10-19T02:59:44.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9793" PostTypeId="2" ParentId="1068" CreationDate="2015-10-19T10:03:47.823" Score="6" Body="&lt;p&gt;As others have suggested, it's good to begin with special cases that the students already know, at least implicitly, like $\mathbb Z/12$ (clocks) and $\mathbb Z/2$ (odd and even).  Another implicitly known example is $\mathbb Z/10$.  When you add two integers (by the usual procedure from elementary school), the last digit of the sum depends only on the last digits of the summands.  So there is a notion of &quot;last-digit addition&quot;.  Similarly for multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd recommend discussing several such examples before bringing up the general notion of equivalence classes.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2015-10-19T10:03:47.823" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9794" PostTypeId="2" ParentId="7673" CreationDate="2015-10-19T17:51:36.043" Score="4" Body="&lt;p&gt;Some local (or not) businesses may have fairly mathematical internships.  For instance, a biotech company may have nontrivial graph theory or other discrete computational problems to solve that will require something analogous to math research, though without proofs.  (I'm not making this example up.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certain computer science internships would be heavily mathematical as well.  For instance, the Google Summer of Code often partners with open-source mathematical software, such as &lt;a href=&quot;http://wiki.lmona.de/get_involved/gsoc&quot; rel=&quot;nofollow&quot;&gt;lmona.de&lt;/a&gt; and &lt;a href=&quot;http://wiki.sagemath.org/GSoC&quot; rel=&quot;nofollow&quot;&gt;SageMath&lt;/a&gt; - I think Sympy and others have also gotten these.  Some of these projects can be &lt;em&gt;quite&lt;/em&gt; mathematically technical.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-10-19T17:51:36.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9795" PostTypeId="2" ParentId="9669" CreationDate="2015-10-19T17:52:44.030" Score="2" Body="&lt;p&gt;I don't know if there is a specific curriculum that adheres to your exact ideas, but there are some historically-oriented ones that could work.  In particular, I think David Bressoud's &lt;a href=&quot;http://www.macalester.edu/~bressoud/books.html#aratra&quot; rel=&quot;nofollow&quot;&gt;Radical Approach to Real Analysis&lt;/a&gt; and &lt;a href=&quot;http://www.macalester.edu/~bressoud/books.html#integratio&quot; rel=&quot;nofollow&quot;&gt;Radical ... Lebesgue Integration&lt;/a&gt; could conceivably help fill that kind of gap.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I would imagine there are also combinations of inquiry-based notes that could work, but that wouldn't be a formal curriculum.)&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-10-19T17:52:44.030" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9796" PostTypeId="2" ParentId="82" CreationDate="2015-10-19T19:55:27.483" Score="3" Body="&lt;p&gt;In my community-college statistics course I motivate this by taking a minimal discrete population {1, 2, 3} and listing the sampling distributions with replacement for sample sizes 1, 2, and 3 (which are respectively: uniform, triangular, and bell-shaped). Specifically: I do n = 1, students do n = 2, then I present a slide for n = 3 and have students name the shape. I think this concrete example is about the best thing we can do to lend intuition about what we're discussing. I got the idea from this Wikipedia article: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Illustration_of_the_central_limit_theorem#Illustration_of_the_discrete_case&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Illustration_of_the_central_limit_theorem#Illustration_of_the_discrete_case&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-10-19T19:55:27.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9797" PostTypeId="2" ParentId="9785" CreationDate="2015-10-20T19:15:02.483" Score="3" Body="&lt;p&gt;I am a big believer in visual learning. Showing instead of telling. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/48vMM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/48vMM.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is $y=|x|$. (I sometimes start with $y=x$ and then discuss what the absolute function means). This function translates, shifts, like any other. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iWWCc.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iWWCc.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is $y=|x-4|$ and hopefully, the student's algebra level have them understand why the shift was right. If not, you have a bit of explaining. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/i71eR.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/i71eR.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last, is $y=4-x$ and by comparing the two graphs, you can see where and why graph 2 and graph 3 are identical for X&amp;lt;4 (or equal) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have many students walk in to my room and ask about a topic that leads me to whip up fast graphs. More often than not, they ask why the teacher didn't do this from the start. (And I know that statistically, I'm only getting students who are not understanding, so it's likely that 90% are getting it, but for those who don't, this offers the visual way to explain)&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-10-20T19:15:02.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9798" PostTypeId="2" ParentId="9766" CreationDate="2015-10-20T19:50:28.903" Score="2" Body="&lt;p&gt;Warning: I am not a design specialist or professional: use the fuzzy description below to inspire research, not replace it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In manufacturing and operations research (and likely some branches of statistics) one is interested in economical designs for testing.  This is sort of like testing in batches 100 glasses of wine in 7 tests to find one bad glass; in industry often several different characteristics are desired to be learned by sampling a large population.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A design is an array, usually object or characteristic for each column, and each row is a grouping: each entry indicates a value that is related to testing the group including that object.  For the wine example, 0 means don't test and 1 means do test. In some situations, a D-optimal design is a square array of values which maximizes information gathered from a sequence (or even partial sequence) of tests: numerically this is by maximizing the determinant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even outside the context of designs, there are some open problems that are computationally feasible, but still take many cycles to run.  Determinants of 0-1 matrices are related to the Hadamard maximal determinant problem and the determinant spectrum problem, both of which are open and for which even for matrices of order less than 20 a lot is left to known.  Combinatorial matrix theory could benefit from some well-organized use of computer instruction cycles on small examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Determinant Means Volume Of Information&quot; Paseman, 2015.10.20&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-10-20T19:50:28.903" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9799" PostTypeId="2" ParentId="9784" CreationDate="2015-10-20T20:07:46.010" Score="15" Body="&lt;p&gt;Let me first say how I have taught this, and then why it was worth doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the stripped down version I speak of. The first block of bullet points is one day.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Forget about straight edge and compass, no one uses them. Fortunately, I think we still have another few years of students who have still seen the standard five function calculator: &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{ \ }$&lt;/span&gt;. So the question was: using these keys, and the ability to type in integers or recopy numbers that you have computed before, what can you compute? Can you compute &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{2}$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$\cos (20^{\circ})$&lt;/span&gt;? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We'll show that the answer is no! Suppose that we could compute &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{2}$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$\theta_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\theta_2$&lt;/span&gt;, ..., &lt;span class=&quot;math-container&quot;&gt;$\theta_N$&lt;/span&gt; be the sequence of numbers displayed on our calculator. So each &lt;span class=&quot;math-container&quot;&gt;$\theta_i$&lt;/span&gt; is made up of one or two previous &lt;span class=&quot;math-container&quot;&gt;$\theta_i$&lt;/span&gt;, using the five operations above. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$K_i$&lt;/span&gt; be the field &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}(\theta_1, \theta_2, \ldots, \theta_i)$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$[K_{i+1}:K_i]=1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; for each &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$[K_N:\mathbb{Q}]=2^r$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{2} \in K_N$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}(\sqrt[3]{2}) \subseteq K_N$&lt;/span&gt;. But then &lt;span class=&quot;math-container&quot;&gt;$3 = [\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$2^r$&lt;/span&gt;, a contradiction.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Tell students that the historical version of this problem is using straight-edge and compass, not calculators. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Here ends that day, but later follow up:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;After we have introduced Galois groups, have them prove the converse: If &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Gal}(K/\mathbb{Q})$&lt;/span&gt; is a &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-group, then elements of &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; can be constructed using the &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;-function calculator.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Work out the construction of the &lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt;-gon using the above.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When we get to solvability, point out that &quot;&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-group&quot; means &quot;all Jordan-Holder constituents are &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}/2\mathbb{Z}$&lt;/span&gt;&quot;, and that we are now looking for an analogous description of fields that involve any radical extension.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The most recent time I taught this course, I did it as an IBL class. Here is the main &lt;a href=&quot;http://www.math.lsa.umich.edu/~speyer/594/19_594_W_20_worksheets.pdf&quot; rel=&quot;noreferrer&quot;&gt;worksheet corresponding to this material&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So, why do I think this was worth it? &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I think it is a moderately interesting question, and it can be attacked just using basic field theory tools like degree of an extension and minimal polynomials. It's nice to reach an application before we bring in the Galois groups. If you've never seen it done, the idea of proving that no formula exists can seem like a miracle, and the quintic case is hard enough that I don't think most students internalize the full argument.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The story about sequentially computing a sequence of numbers on your calculator makes the tower of field extensions natural.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The followup of proving that &quot;Galois group is a &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-group&quot; implies &quot;tower of square root extensions&quot; is a good preparation for Kummer theory. It also makes the question &quot;what criterion describes a tower of radical extensions&quot; natural.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2020-05-08T12:16:38.520" LastActivityDate="2020-05-08T12:16:38.520" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9800" PostTypeId="2" ParentId="7673" CreationDate="2015-10-20T20:17:56.860" Score="1" Body="&lt;p&gt;The tags say &quot;undergraduate-research&quot;, but the question seems broader than that. I'd like to point out that summer teaching is an excellent alternative to research, which can give the students a much better understanding of the material they think they know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the advanced end, summer programs like &lt;a href=&quot;http://www.promys.org/counselors&quot; rel=&quot;nofollow&quot;&gt;PROMYS&lt;/a&gt;, &lt;a href=&quot;http://u.osu.edu/rossmath/counselors/&quot; rel=&quot;nofollow&quot;&gt;Ross&lt;/a&gt;, &lt;a href=&quot;https://www.mathcamp.org/employmentopportunities/&quot; rel=&quot;nofollow&quot;&gt;Mathcamp&lt;/a&gt; and &lt;a href=&quot;http://www.hcssim.org/job-applications.html&quot; rel=&quot;nofollow&quot;&gt;Hampshire&lt;/a&gt; hire undergraduate counselors, provide a great environment for learning mathematics, and are recognized as prestigious in graduate admissions. (Disclosure/endorsement I worked for PROMYS for four summers, and had a great experience.) Down a step, many local colleges run summer programs (my own university has &lt;a href=&quot;http://www.math.lsa.umich.edu/mmss/&quot; rel=&quot;nofollow&quot;&gt;MMSS&lt;/a&gt;) and hire undergraduate assistants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of course, more basically, tutoring jobs exist and usually the tutor winds up learning more than the students.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2015-10-20T20:17:56.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9801" PostTypeId="2" ParentId="9785" CreationDate="2015-10-20T20:49:08.333" Score="1" Body="&lt;p&gt;Going beyond the good answers here that have been given, and which work for teaching in a classroom, one could instead focus on pliable strategies that work well on GRE exams. When I tutor students for GRE math prep I always emphasize tables of values as a tool to remember. In this case a quick production of a table like this&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{array}{c|c|c}&#xA;x&amp;amp;|x-4|&amp;amp;4-x\\&#xA;\hline&#xA;2&amp;amp;2&amp;amp;2\\&#xA;3&amp;amp;1&amp;amp;1\\&#xA;4&amp;amp;0&amp;amp;0\\&#xA;5&amp;amp;1&amp;amp;-1&#xA;\end{array}$$&#xA;should help the exam taker to organize her thoughts enough to tackle the GRE question.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-10-20T20:49:08.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9802" PostTypeId="2" ParentId="9780" CreationDate="2015-10-20T21:32:42.267" Score="2" Body="&lt;p&gt;(I should write this when I have more than a few minutes.  Hopefully updates will follow.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are too many aspects to cover: big data, human computer interaction, computing using conventional architectures, computing using not-yet-realized architectures (nanocomputing, swarm- and cloud- and quantum-computing), computing with limited resources, complexity classes, privacy concerns, the list goes on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to have a handle on any of these, one should have some acquaintance with the foundations of algorithms, philosophy of computing, mathematical logic, and basic theory on simple architectures.  Even if your course is required to focus on one section, you can tie in to several others.  A. K. Dewdney's &quot;The New Turing Omnibus&quot; is a must read for any mathematician/educator interested in classical computer science.  I hope a new one comes out for the present generation so that I can understand some of current computer science before I miss out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Not Ready To Be Old&quot; Paseman, 2015.10.20&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-10-20T21:32:42.267" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9803" PostTypeId="2" ParentId="9789" CreationDate="2015-10-20T22:02:28.637" Score="1" Body="&lt;p&gt;I had similar problems with the phrase &quot;such that&quot; as taught in set-builder notation.  Still--40 years later--I don't like the expression &quot;such that.&quot;  It's just not a phrase I would ever use naturally.  I learned to read (in my head) the vertical bar in set builder notation as &quot;where.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't experienced the problem you described with any students I've taught.  But I might apply the same solution: give your students a number of equivalent expression they can substitute in their heads.  For example, &quot;in terms of x&quot; = &quot;using units of x&quot; = &quot;where the variable x appears in the answer.&quot; For the students that do seem to grasp concept, ask them to suggest equivalent phrases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three other observations:&#xA;1) The word &quot;term&quot; has at least two meanings.  One meaning is a part of a polynomial that is expressed without addition or subtraction operators.  But a second meaning is &quot;a specific word or expression,&quot; as in &quot;vocabulary terms.&quot;  The phrase &quot;in terms of&quot; uses the 2nd meaning.  If you have emphasized the first meaning with your students, it may be helpful to state explicitly that you're using the 2nd meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) I understand this may be an artifact of typing your question, but I found your example &quot;x apples&quot; a little confusing myself.  In this case, x is a unitless number, and the &quot;units&quot; are apples.  So asking for the answer &quot;in terms of x&quot; seems like you're asking for an answer &quot;in terms of a unitless number.&quot;  I don't think you mean &quot;in terms of x.&quot; In that particular example, I think you mean &quot;in terms of apples&quot; = &quot;using units of 'apples'.&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Have your students been exposed to dimensional analysis (a.k.a. unit analysis) in your course or previous ones?  The concept of a conversion factor is closely related to this question.  Your students are probably used to converting between inches and centimeters or other units of measure.  Perhaps you can give them a serious of word problems involving unit conversion, and the word problems can use the phrase &quot;in terms of.&quot;  This may help get them accustomed to the meaning of the phrase.  For example, &quot;Express the capacity of a 10 gallon gas tank in terms of liters.&quot;&lt;/p&gt;&#xA;" OwnerUserId="5465" LastActivityDate="2015-10-20T22:02:28.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9804" PostTypeId="1" CreationDate="2015-10-20T22:44:06.530" Score="11" ViewCount="166" Body="&lt;p&gt;I'm tutoring a 6th grader in math at the moment and because she never has a ton of homework I like to give her some interesting extra problems to do.  It seems she really enjoyed a problem I showed her the other day where we counted up how many ways there were of getting between two points in a city without actually counting them all up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the thing she really liked about it was that we had a picture in front of us (a roadmap for a madeup town with ridiculously few roads) and could use some simple math to conclude things about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any other visual problems like this that you guys recommend that only involve arithmetic skills (preferably involving fractions or decimals so that she gets some more practice with those)?  Or is there a website or book that has several of these types of problems?  Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="5726" LastActivityDate="2015-10-25T13:49:40.583" Title="Visual questions for 6th graders" Tags="|secondary-education|primary-education|arithmetic-operations|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9805" PostTypeId="2" ParentId="9789" CreationDate="2015-10-20T23:04:52.713" Score="1" Body="&lt;p&gt;Your students may be wondering &quot;What is the &lt;em&gt;value added&lt;/em&gt; of the expression 'in terms of'.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the example you gave: Solve the following equation for y in terms of x: 2y+3x=1, you could just as easily say &quot;Solve for y.&quot;  Since there are only two variables in the expression &quot;solve for y&quot; and &quot;solve for y in terms of x&quot; are the same (in this particular example).  Adding the expression &quot;in terms of y&quot; is redundant.  If the students are used to seeing just &quot;solve for x,&quot; they may be trying to figure out how adding the phrase &quot;in terms of y&quot; is changing what they were doing before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get over that, you may want to try systems of equations...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;x + 2 = y&lt;/p&gt;&#xA;&#xA;&lt;p&gt;y + 5 = z&lt;/p&gt;&#xA;&#xA;&lt;p&gt;...and then you can demonstrate the difference between &quot;solve for x in terms of y&quot; vs. &quot;solve for x in terms of z.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to word problems: Word problems are supposed to reflect real-world situations for students.  In a word problem, that deals with real objects and real things, I would never say, &quot;express your answer in terms of ex.&quot;  I would use the actual objects, &quot;express your answer in terms of apples&quot; or &quot;what is the cost of 5 toasters in terms of dollars?&quot; Mixing variable names with real-world objects in word problems sort of defeats the purpose of word problems and could be confusing to students.&lt;/p&gt;&#xA;" OwnerUserId="5465" LastActivityDate="2015-10-20T23:04:52.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9806" PostTypeId="1" CreationDate="2015-10-21T03:35:08.040" Score="18" ViewCount="924" Body="&lt;p&gt;I want to introduce calculus students to computer algebra systems (CAS) like Sage, Geogebra, and Wolfram Alpha in college Calculus 1 and 2.  While I believe in the value of learning to do calculus by hand to a certain extent, I also don't want to pretend that we don't live in the 21st century.  However, most textbooks I have seen that discuss CAS at any length mainly just list some examples where a CAS gets a wrong or unhelpful answer unless you know what you're doing.  While this is undoubtedly an important point, I think it comes across as a sort of desultory and apologetic afterthought.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, I feel there are important skills that could be learned here, that could potentially transfer and outlast any content knowledge about calculus: how to use a computer as a tool, translating a problem into a form that a computer can understand and interpreting the answer, what a computer can and can't do, experimenting with a computer, using a computer to visualize an answer or a problem, and so on; in short, how to use a computer as an extension of your brain (rather than a replacement for it).  Are there any textbooks or other resources out there that utilize CAS in such a way throughout a calculus 1 or 2 curriculum?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I'm not talking so much about the &lt;em&gt;teacher&lt;/em&gt; using a CAS to produce nice pictures or animations to illustrate a concept (although that's certainly a valid use), but about teaching the &lt;em&gt;student&lt;/em&gt; to use a CAS in productive and creative ways.  (This is related to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7502/use-of-symbolic-math-apps-in-teaching&quot;&gt;this question&lt;/a&gt;, but it's both more general (being about all uses of CAS rather than just exploratory/experimental) and more specific (being particularly about calculus 1 and 2).)&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="38" LastEditDate="2017-07-23T19:05:13.513" LastActivityDate="2017-07-26T15:49:17.537" Title="How to use a CAS in teaching calculus" Tags="|undergraduate-education|calculus|technology-in-education|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="9807" PostTypeId="1" AcceptedAnswerId="9808" CreationDate="2015-10-21T13:17:10.927" Score="13" ViewCount="625" Body="&lt;p&gt;Is it mathematically correct to write&#xA;$$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx},$$&#xA;rather than&#xA;$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}?$$&#xA;If not, what is the difference?  If so, why isn't this notation used from the beginning?  My feeling for the latter is that it would align the derivative more with the inverse of the indefinite integral.&lt;/p&gt;&#xA;" OwnerUserId="5727" LastEditorUserId="1853" LastEditDate="2015-10-26T18:53:29.643" LastActivityDate="2016-02-11T13:37:37.660" Title="Using $dx$ for $h$ in the definition of derivative" Tags="|calculus|notation|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9808" PostTypeId="2" ParentId="9807" CreationDate="2015-10-21T14:09:05.377" Score="27" Body="&lt;p&gt;To use $$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx}$$ is mathematically correct &lt;strong&gt;if&lt;/strong&gt; $dx$ is the name for a real variable. (If it should be something else it needs to be made clear what it should be.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would also be correct to say &#xA;$$f'(x)=\lim_{\text{small}\to 0}\frac{f(x+\text{small})-f(x)}{\text{small}}$$ with the understanding that $\text{small}$ is the name of a real variable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An issue I see with what you propose is though that $dx$ is &lt;em&gt;not&lt;/em&gt; a common notation for a real variable, but rather something else. What exactly $dx$ means, depends on the context, but typically it is not used to denote a real number. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very related notation that is more common is &#xA;$$f'(x)=\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea is that the $\Delta x$ is a finite difference, as opposed to an infinitesimal difference that might be denoted by $dx$ (where the latter notion might or might not be made precise). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally I prefer to use just an $h$ or something similar, to emphasis that it is just some real parameter there nothing special or mysterious. To name it $dx$ or $\Delta x $ goes counter this so I would not do it. &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2015-10-21T14:09:05.377" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9809" PostTypeId="1" CreationDate="2015-10-21T15:13:44.563" Score="0" ViewCount="226" Body="&lt;p&gt;I have made a big mistake by signing up to an advanced functional analysis class, when I do not have the background. This is my first analysis course ever, and aside from that my math background is very light (one semester calculus, two semesters stats, one semester game theory). Alas, the die has been cast and now I am trying to keep my head above water and survive(pass) the course. I have a midterm in about 20 days from the date of this publication. I would like some opinion on how I could muster a semi-decent-not-really grasp of what is going on to get half the marks(just enough to pass). We have basically gone through Metric, Banach, and Hilbert spaces(their topological properties and key theorems to analyse them) and some applications. I have been told that the exam will be to apply the properties of these to economic problems (I am an economics major).&#xA;Any advice on how I could approach my studying to be efficient would be greatly appreciated. Thank you! &lt;/p&gt;&#xA;" OwnerUserId="5728" LastActivityDate="2015-10-21T15:13:44.563" Title="Minimum preparation to pass advanced functional analysis?" Tags="|functional-analysis|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9810" PostTypeId="2" ParentId="9785" CreationDate="2015-10-21T17:06:31.360" Score="0" Body="&lt;p&gt;One observation that should be made: Absolute value does not produce negative numbers.  If 4-x is negative, it can't equal the output of the absolute value of any expression.  So 4-x must not be negative (if it is equal to the absolute value of something).  So 4-x &gt;= 0.  So the first consideration shows that 4 &gt;= x is necessary.  If the student does not understand this, there are bigger issues that need resolving first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once necessity is made apparent, you can move on to sufficiency using material from the other answers. For example, knowing that absolute value (in the context of x-4 being a real number) is either x-4 or -(x-4)=4-x, then you check that (4-x) &gt;= 0 is sufficient: if 4-x &gt;= 0, then x-4 is negative or zero, and you can check those cases mechanically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;Simple Necessity Is Necessarily Simple&quot; Paseman, 2015.10.21&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-10-21T17:06:31.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9811" PostTypeId="2" ParentId="9806" CreationDate="2015-10-21T20:33:05.810" Score="4" Body="&lt;p&gt;I suspect a good answer to this question means rewriting the curriculum and standards to incorporate CAS.  Otherwise, you just use the tool to implement the current curriculum, which to my mind means experimenting/numerical verification, or producing pretty pictures (which can be useful, but one learns by doing more than by looking).  Are you prepared to do that?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say that one needs to cover Taylor's theorem with Lagrange form of the remainder (and some other form for purpose of this discussion).  One can set up step-by-step exercises illustrating some examples, hammering home some essential points and covering what might be important.  However, who is ever going to do this outside of class?  Even if you are in engineering, numerical analysis, or mathematical modeling, you are likely to use a package someone else developed to determine the remainder without thinking much about it.  What you should REALLY teach is the idea of approximation with error control, and that two specific methods in calculus involve these two forms of remainder.  That bit of knowledge (in my biased opinion) is vastly more important than the specific remainders.  In teaching this bit of knowledge, one has the opportunity to teach CAS skills along with this bit.  To my mind, the class will have intellectual integrity if this &quot;bit of knowledge&quot; which is not calculus specific is spelled out in the curriculum.  This is an example of what I mean by rewriting the curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you should consider foundational issues in answering this question, which shouldn't be &quot;what foundations and tools should I use and how do I use to teach calculus&quot;, but &quot;what calculus should I use and how do I use to teach foundations and tools&quot; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerhard &quot;That's How I See It&quot; Paseman, 2015.10.21&lt;/p&gt;&#xA;" OwnerUserId="3468" LastActivityDate="2015-10-21T20:33:05.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9812" PostTypeId="2" ParentId="7528" CreationDate="2015-10-21T22:48:23.717" Score="9" Body="&lt;p&gt;The Mito Art Tower consists of&#xA;&lt;span class=&quot;math-container&quot;&gt;$28$&lt;/span&gt; congruent, stacked regular tetrahedra, each with edge length about &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;m.&#xA;It is in Mito, Ibaraki, Japan.&#xA;Architect: Arata Isozaki.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/KhAMN.jpg&quot; /&gt; &#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/cA4TB.jpg&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;Left image from [www.panoramio.com](http://www.panoramio.com/).&#xA;&lt;br /&gt;&#xA;Right figure from Elgersma &amp; Wagon. &quot;The Quadrahelix:&#xA;A Nearly Perfect Loop of Tetrahedra.&quot; 2016. [arXiv abstract](https://arxiv.org/abs/1610.00280).&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Known as the&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Boerdijk%E2%80%93Coxeter_helix&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boerdjik-Coxeter helix&lt;/a&gt;.&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-11-09T11:12:27.813" LastActivityDate="2020-11-09T11:12:27.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9813" PostTypeId="2" ParentId="2338" CreationDate="2015-10-22T08:59:54.707" Score="11" Body="&lt;p&gt;Here's one problem with trying to replace indefinite integrals with definite ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the indefinite integral, we can say&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \int \frac{\mathrm{d}x}{1+x^2} = \arctan x + C $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $C$ can be any constant. However, the only antiderivatives we can express as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \int_a^x \frac{\mathrm{d}x}{1+x^2} = \arctan x - \arctan a $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;are those with $C \in [-\pi/2, \pi/2]$. (or $C \in (-\pi/2, \pi/2)$ if you don't want to allow $a = \pm \infty$)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does anyone even imagine writing something like $$\sum_n n^3= \frac{n^2(n+1)^2}4+k?$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes, actually; it is a very natural thing to do when what you're looking for is an anti-difference. Even if your problem actually called for a definite summation, keeping track of the boundary can be more complicated than just leaving things up to a constant which you solve for later by plugging in values.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2015-10-22T08:59:54.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9814" PostTypeId="1" CreationDate="2015-10-22T12:45:09.703" Score="23" ViewCount="851" Body="&lt;p&gt;We have the good fortune of having &quot;lab sections&quot; here at my college. I'm interested in conducting some activities in the spirit of &lt;a href=&quot;https://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en&quot;&gt;this talk&lt;/a&gt;. However, even in my stash of inquiry-based learning resources I can't easily find a large number of &lt;em&gt;natural&lt;/em&gt; questions outside the usual suspects (falling bodies and other physics examples) that aren't somehow artificial-looking. I'm mostly interested in questions that lead into topics before related rates and optimization...since those topics have been around so long in calculus courses that the relevant ideas can be found &quot;canned&quot; easily.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are some very good inquiry-based activities for calculus that lead students to the discovery of basic definitions in the subject via natural examples?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One more point of clarification: Ideally, I'd like to see problems that lead to calculus topics through a &quot;Martin Gardner&quot; feel...if this helps...rather than some contrived artifice meant to jam the definition of derivative down the throats of unwilling students...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark: Perhaps this will be helpful: &lt;a href=&quot;http://www.iblcalculus.com/&quot;&gt;http://www.iblcalculus.com/&lt;/a&gt;. The bottle graph activities are nice (things like this can be found in Hughes-Hallett) and the traffic camera activity is pretty good, too. You can imagine a student wondering, outside of a mathematics class, how traffic cameras can measure a car's speed. That is another litmus test for the quality of activity...would a student come up with it on his or her own under minimally invasive circumstances?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a &quot;lowbrow&quot; example that seemed to work well today: I drew a circle on the board, then the tangent line to the circle at a point and a radius of the circle connecting the origin to that point. Then I asked the students to use the definition of derivative to verify that the angle formed by any radius and the tangent line to the circle at the terminus of that radius is a right angle. This question was very simple to state, as well as straightforward and natural, but incorporates and reinforces several concepts from the course. &lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2015-10-23T00:36:17.833" LastActivityDate="2015-10-31T01:27:13.873" Title="Natural, rich, calculus questions" Tags="|calculus|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="9815" PostTypeId="2" ParentId="9814" CreationDate="2015-10-23T00:19:57.333" Score="3" Body="&lt;p&gt;I recall that Mount Holyoke College built a fascinating unit around&#xA;witchcraft, but I cannot find curricular details at the moment. Perhaps these&#xA;references might indicate the potential.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(1) Schwartz, Robert M. &lt;em&gt;History and Statistics: The Case of Witchcraft in Early Modern Europe and New England&lt;/em&gt;. Research Foundation of State University of New York, 1992. (&lt;a href=&quot;https://www.mtholyoke.edu/courses/etownsle/qr/witchcraft%20chi-square%20chapter.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mount Holyoke web excerpt&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(2) There is a detailed list in Wikipedia of &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_people_executed_for_witchcraft&quot; rel=&quot;nofollow noreferrer&quot;&gt;those executed for witchcraft&lt;/a&gt;&#xA;by date and nationality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) Here is a chart specific to Salem:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/qRjwv.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qRjwv.gif&quot; alt=&quot;Accusations&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://www.tulane.edu/~salem/AccExec%20Discussion.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Salem Witchcraft Site&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-10-23T00:19:57.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9816" PostTypeId="2" ParentId="9814" CreationDate="2015-10-23T02:11:57.423" Score="6" Body="&lt;p&gt;There are quite a few good resources that could be adapted, from various &quot;projects&quot;.  I haven't read all of them but some of these could work.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.maa.org/press/ebooks/calculus-an-active-approach-with-projects&quot;&gt;book on calc with projects&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.maa.org/press/ebooks/calculus-mysteries-and-thrillers&quot;&gt;book of 'mysteries'&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.math.smith.edu/Local/cicintro/&quot;&gt;Calculus in Context&lt;/a&gt; has some projects as part of its exercises&lt;/li&gt;&#xA;&lt;li&gt;The Hughes-Hallett applied text actually has some interesting projects along these lines&lt;/li&gt;&#xA;&lt;li&gt;A couple different books called &quot;Calculus: The Language of Change&quot; - &lt;a href=&quot;http://homepage.math.uiowa.edu/~stroyan/CTLC3rdEd/ctlc.htm&quot;&gt;one from Iowa&lt;/a&gt; CC licensed, and &lt;a href=&quot;http://math.jbpub.com/calculus/realworld_calc.cfm&quot;&gt;one that is not&lt;/a&gt; but perhaps more edited.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As to topics, here are a few econ/business ones that could be interesting for you - though finding complete stuff ready-made may be tough thus far, there are many texts nowadays that have fairly realistic versions of this that could be discussed.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Revenue optimization derived solely from a demand function and theory&lt;/li&gt;&#xA;&lt;li&gt;Deriving elasticity - I find this quite well-suited to IBL&lt;/li&gt;&#xA;&lt;li&gt;Storage costs for inventory - could start with VERY simple cost functions and go to more and more realistic ones to motivate integration&lt;/li&gt;&#xA;&lt;li&gt;Degree-days!  But you have to live in a cold-weather state for this to gain traction.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-10-23T02:11:57.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9817" PostTypeId="2" ParentId="1268" CreationDate="2015-10-23T02:14:41.383" Score="2" Body="&lt;p&gt;This isn't exactly a full answer for you, but &lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=acalc&quot; rel=&quot;nofollow&quot;&gt;Approximately Calculus&lt;/a&gt; has a large bit on the prime number theorem as a motivation, which was motivated by Gauss' thinking of prime distribution probabilistically and where using integrals doesn't make sense unless you are thinking of integrating a pdf.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-10-23T02:14:41.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9818" PostTypeId="2" ParentId="9807" CreationDate="2015-10-23T09:54:38.530" Score="7" Body="&lt;p&gt;In the other examples, one can make sense of $\mathrm{d}x$ in a consistent way: e.g. if we model it as differential forms, then $\frac{\mathrm{d}y}{\mathrm{d}x}$ is a ratio between two differential forms,  $\int \sin x \, \mathrm{d} x$ is the anti-derivative of a differential form, and $\int_a^b \sin x \, \mathrm{d} x$ is the integral of a differential form along a (directed) path. One can even reasonably extend differentiable functions to the exterior algebra, and make literally true statements like $f(x + \mathrm{d}x) - f(x) = f'(x) \mathrm{d}x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if one were to write the calc-1 definition of derivative as&#xA;$$ \lim_{\mathrm{d}x \to 0} \frac{f(x + \mathrm{d}x) - f(x)}{\mathrm{d} x} $$&#xA;the $\mathrm{d}x$ here doesn't represent anything resembling a differential form at all. The superficial similarity to my last remark in the previous paragraph makes things &lt;em&gt;worse&lt;/em&gt;, not &lt;em&gt;better&lt;/em&gt;, since it would conflate the distinct ideas, and thus make it harder to intuitively arrive at what $\mathrm{d} x$ means.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2015-10-23T09:54:38.530" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9819" PostTypeId="2" ParentId="4333" CreationDate="2015-10-23T12:19:52.193" Score="5" Body="&lt;p&gt;If you see this in calculus, there is a rather natural approach: to use the methods of calculus to study it!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Significant features of $f(x) = \frac{ax + b}{(x - c)(x - d)}$ happen around $x=c$ and $x=d$. Suppose we study the $x \approx c$ case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One basic thing is to observe that $x-c$ is the only part of the expression that behaves interestingly; consequently, we should expect&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ x \approx c \implies f(x) \approx \frac{ac+b}{(x-c)(c-d)} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Following the pattern of subtracting off features, you write&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ f(x) = \frac{ac+b}{(x-c)(c-d)} + g(x) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and have some expectation that $g(x)$ doesn't have a 'feature' at $c$, or at least, has a simpler feature. Then, either you do a few examples to see that this really does eliminate the pole at $c$, or maybe you do something with Taylor/Laurent series, or you estimate it with asymptotic analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more generic lemma is probably simpler to use, though:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \frac{r(x)}{x-c} = \frac{r(c)}{x-c} + \frac{r(x)-r(c)}{x-c} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and we recognize the second term as a difference quotient, which should remain well-behaved at $x=c$, as it converges to $r'(c)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The remaining clever idea to convey (and it &lt;em&gt;is&lt;/em&gt; a clever idea until you're used to such things) is that once you know the end form of the manipulation, you can skip straight there and solve for the unknowns by hook or by crook. (or even by the formula indicated by the lemma above)&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2015-10-23T12:19:52.193" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9820" PostTypeId="2" ParentId="9804" CreationDate="2015-10-23T23:24:48.577" Score="3" Body="&lt;p&gt;Highly recommend the Anne Joshua books - Enrich-e-matics.&#xA;She has things like balance scales to solve equations (using shapes), with fractions she draws out the bars and labels them very well. &#xA;They start from easy to hard and Book two is an extension of book one. &lt;/p&gt;&#xA;" OwnerUserId="5544" LastActivityDate="2015-10-23T23:24:48.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9822" PostTypeId="2" ParentId="4351" CreationDate="2015-10-24T06:58:32.623" Score="4" Body="&lt;p&gt;The derivative of $x \mapsto |x|$ at zero, or, say, the mean (not in a principal-value sense) of a Cauchy-distributed variable or the limit (in the finite real line) of $x_n = 2x_{n-1}; x_0 \neq 0$.  Symmetry considerations show the result must be 0 if it were to exist, but to casually assume existence/convergence gets one burnt in places such as these.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can also be used to show that while $f''(x) = \lim_{h \to 0; h &amp;gt; 0} \frac{f(x-h) - 2f(x) + f(x+h)}{h^2}$ and similar formulae can be used to calculate higher-order derivatives, they don't make good definitions -- applied to, e.g., the function that is -1 for negative arguments, +1 for positive arguments, and 0 at zero, you find the pathologies cancel and reach an absurd conclusion.  (Or more simply $f'(x) = \lim_{h \to 0} \frac{f(x + h/2) - f(x - h/2)}{h}$ on the absolute value, as OP notes below.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can interpret the fundamental theorem of the calculus in various forms as another instance of this phenomenon.  The statement typically requires that the integrand be integrable&lt;sup&gt;0&lt;/sup&gt;, so one has, example,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $-\infty &amp;lt; a \leq b &amp;lt; +\infty$, and let $f,F :  [a,b] \rightarrow \mathbb{R}$ be functions.  If&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(I)  &lt;code&gt;&lt;/code&gt; $F$ is continuous (everywhere), if &lt;br&gt;&#xA;  (II)  &lt;code&gt;&lt;/code&gt; $F'(x)$ exists and equals $f(x)$, except at a finite number (possibly none) of points $x \in [a,b]$, and if &lt;br&gt;&#xA;  (III)  &lt;code&gt;&lt;/code&gt; $f$ is Riemann integrable,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;then $\int_a^b f(x) \, dx = F(b) - F(a)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Viewing this as $\big( \text{(I) and (II)} \big) \Rightarrow \big( \text{(III)} \Rightarrow \text{formula} \big)$, the theorem rules out all but one number as a possible value of the integral, but kids tend to neglect or forget the subtle difference and claim, e.g., that $f :  [0, 1/\pi] \rightarrow \mathbb{R}$ given by $f(0) := 0 , f(x) := \sin (1/x) - (1/x) \cos (1/x)$ has Riemann integral $\lim_{t \to 0; t &amp;gt; 0} \left( \left( x \sin (1/x) \right) \bigg|_t^{1/\pi} \right) = 0$.  (No it doesn't!  &lt;em&gt;If&lt;/em&gt; the integral exists then it is zero as we have so lovingly described.  Except it doesn't!  (At least not properly, i.e., according to the definition they wrote down yesterday.)  It's not even Lebesgue integrable, we just went over the $x \sin (1/x)$ function in an example on total variation.  Aarg!)  Similarly, if a real function has an (relative) interior extremum, then either it fails to be differentiable at that point, or the derivative there is zero.  (I tend by nature to put failure and 'strings-attached' clauses and hypotheses first.  It helps very little if any, but at least it seems not to impede.  Oh well.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;0&lt;/sup&gt;  even assuming (global) differentiability of the antiderivative, unless your formulation uses the &lt;a href=&quot;https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral&quot; rel=&quot;nofollow&quot;&gt;gauge integral&lt;/a&gt;, in which case integrability very nicely becomes a conclusion as opposed to an assumption, but this integral is unpopular or passe or something (granted, it doesn't generalise as well as Lebesgue's owing to the structure it requires on the domain space, but (&lt;strong&gt;a&lt;/strong&gt;) so long as one already is interested only in sticking to the real line, why not?  and (&lt;strong&gt;b&lt;/strong&gt;) I love it!!  Are those alone not reasons enough??).&lt;/p&gt;&#xA;" OwnerUserId="4679" LastEditorUserId="4679" LastEditDate="2016-05-01T02:57:05.550" LastActivityDate="2016-05-01T02:57:05.550" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9823" PostTypeId="2" ParentId="9767" CreationDate="2015-10-24T13:58:22.613" Score="1" Body="&lt;p&gt;You may use any &lt;a href=&quot;https://en.wikipedia.org/wiki/Open_educational_resources&quot; rel=&quot;nofollow noreferrer&quot;&gt;OER&lt;/a&gt; content. See the recommendations of the following questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/8137/open-education-resources&quot;&gt;Open Education Resources&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/1479/open-source-math-textbooks%E2%80%8C%E2%80%8B&quot;&gt;Open-Source Math Textbooks&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/4482/how-can-i-choose-a-free-calculus-textbook&quot;&gt;How can I choose a free calculus textbook?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Normally you can download an offline version of the OER-textbooks / OER-contents. If not, you can do it by yourself:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;With &lt;a href=&quot;https://rg3.github.io/youtube-dl/&quot; rel=&quot;nofollow noreferrer&quot;&gt;youtube-dl&lt;/a&gt; you can download Youtube videos and playlists (also from other sites as Vimeo)&lt;/li&gt;&#xA;&lt;li&gt;With &lt;a href=&quot;https://en.wikipedia.org/wiki/Wget&quot; rel=&quot;nofollow noreferrer&quot;&gt;wget&lt;/a&gt; or similar programs you can make an offline mirror of a website, see &lt;a href=&quot;http://fosswire.com/post/2008/04/create-a-mirror-of-a-website-with-wget/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://fosswire.com/post/2008/04/create-a-mirror-of-a-website-with-wget/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Because of the free license of OER websites, you can download and redistribute their content. Mostly &lt;a href=&quot;https://en.wikipedia.org/wiki/Creative_Commons&quot; rel=&quot;nofollow noreferrer&quot;&gt;Creative Commons&lt;/a&gt; licenses are used. Have a look at the project's license page to see, what is under which circumstances allowed...&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-10-24T13:58:22.613" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9824" PostTypeId="2" ParentId="9752" CreationDate="2015-10-24T14:06:48.013" Score="1" Body="&lt;p&gt;The youtube video you linked to is a &lt;a href=&quot;https://en.wikipedia.org/wiki/Screencast&quot; rel=&quot;nofollow&quot;&gt;screencast&lt;/a&gt;. You can create such videos with &lt;a href=&quot;https://en.wikipedia.org/wiki/Camtasia_Studio&quot; rel=&quot;nofollow&quot;&gt;Camtasia&lt;/a&gt; but also with the &lt;a href=&quot;https://en.wikipedia.org/wiki/VLC_media_player&quot; rel=&quot;nofollow&quot;&gt;VLC player&lt;/a&gt; you can record a screencast (Camtasia offers features to edit an recorded screencast which you might find useful). Google &quot;screencast&quot; or &quot;record + screencast + [your operating system]&quot; and you will find a lot of resources about this topic. See also &lt;a href=&quot;https://en.wikipedia.org/wiki/Comparison_of_screencasting_software&quot; rel=&quot;nofollow&quot;&gt;this wikipedia list&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For making drawings, you need a software like &lt;a href=&quot;https://en.wikipedia.org/wiki/Xournal&quot; rel=&quot;nofollow&quot;&gt;xournal&lt;/a&gt;. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Comparison_of_notetaking_software&quot; rel=&quot;nofollow&quot;&gt;this list&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2015-10-24T14:06:48.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9825" PostTypeId="2" ParentId="9807" CreationDate="2015-10-24T14:16:55.313" Score="4" Body="&lt;p&gt;A sitenote: When you leave classical analysis and you take &lt;a href=&quot;https://en.wikipedia.org/wiki/Non-standard_analysis&quot; rel=&quot;nofollow&quot;&gt;non-standard analysis&lt;/a&gt;, then you have infinitesimals in your underlying theory (they are part of the so called &lt;a href=&quot;https://en.wikipedia.org/wiki/Hyperreal_number&quot; rel=&quot;nofollow&quot;&gt;hyperreals&lt;/a&gt;). In this theory you can write $$\frac{f(x+dx)-f(x)}{dx}$$ with $dx$ being an infinitesimal. This gives a number which difference to $f'(x)$ is only an infinitesimal (see the comments to this question). No &lt;a href=&quot;http://www.tienchiu.com/wp-content/uploads/2015/05/then-a-miracle-occurs-logic.jpg&quot; rel=&quot;nofollow&quot;&gt;hocus pocus&lt;/a&gt; necessary ;-)&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="5097" LastEditDate="2015-10-25T23:11:50.840" LastActivityDate="2015-10-25T23:11:50.840" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="9826" PostTypeId="2" ParentId="9814" CreationDate="2015-10-24T19:58:18.873" Score="6" Body="&lt;p&gt;I was just now working on my presentation for the &lt;a href=&quot;http://jointmathematicsmeetings.org/jmm&quot; rel=&quot;nofollow&quot;&gt;Joint Mathematics Meetings in January&lt;/a&gt;, and put together links to some blogs that have great resources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Check out:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://bowmandickson.com/category/teaching/calculus/&quot; rel=&quot;nofollow&quot;&gt;Bowman in Arabia&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://shawncornally.com/wordpress/?page_id=306&quot; rel=&quot;nofollow&quot;&gt;ThinkThankThunk&lt;/a&gt; (I'd love to try the &lt;a href=&quot;http://shawncornally.com/wordpress/?p=110&quot; rel=&quot;nofollow&quot;&gt;rolling chair races&lt;/a&gt;.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://samjshah.com/tag/calculus/&quot; rel=&quot;nofollow&quot;&gt;Sam Shah's Blog&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://mathteachermambo.blogspot.com/2015/06/calculus-final-project.html&quot; rel=&quot;nofollow&quot;&gt;Math Teacher Mambo&lt;/a&gt;  (Link is to one project. Search on calculus for more.) &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://mathteachermambo.blogspot.com/2015/06/calculus-final-project.html&quot; rel=&quot;nofollow&quot;&gt;Math Mama Writes&lt;/a&gt; (This is my blog. Link is to optimization project. &lt;a href=&quot;http://mathmamawrites.blogspot.com/2014/01/calculus-solids-of-revolution.html&quot; rel=&quot;nofollow&quot;&gt;I also recommend this&lt;/a&gt;.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2015-10-30T23:26:47.580" LastActivityDate="2015-10-30T23:26:47.580" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9827" PostTypeId="2" ParentId="9804" CreationDate="2015-10-25T13:49:40.583" Score="2" Body="&lt;p&gt;Check out the &lt;a href=&quot;http://mathmunch.org/&quot; rel=&quot;nofollow&quot;&gt;Math Munch blog&lt;/a&gt;. Their subtitle is A Weekly Digest of the Mathematical Internet, but I think they originally focused on middle school. You might try looking under their &lt;a href=&quot;http://mathmunch.org/tag/puzzles/&quot; rel=&quot;nofollow&quot;&gt;puzzles tag&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-10-25T13:49:40.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9828" PostTypeId="1" CreationDate="2015-10-25T14:07:03.833" Score="13" ViewCount="3381" Body="&lt;p&gt;I am tutoring a high school student in Algebra 2.  Her class has just covered systems of linear equations, mainly via the &quot;substitution&quot; and &quot;elimination&quot; methods.  Her textbook concludes that unit with Cramer's rule (which is only used for 2 x 2 and 3 x 3 systems).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My student likes using Cramer's rule, and asks me if there is any easy way to explain why it works.  I'm having a hard time coming up with an explanation that doesn't require undergraduate-level linear algebra.  Is there a &lt;em&gt;simple&lt;/em&gt; way to explain it?&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-10-31T04:06:44.617" Title="A simple explanation or derivation of Cramer's rule, suitable for secondary Algebra 2?" Tags="|algebra|linear-algebra|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="9829" PostTypeId="1" AcceptedAnswerId="9864" CreationDate="2015-10-25T19:43:11.407" Score="18" ViewCount="4073" Body="&lt;p&gt;My kids (12, 14) attend a school in the US in which the mathematics department has a policy of &lt;strong&gt;not&lt;/strong&gt; returning tests to the kids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have very strong feelings about this (I think it is an obviously bad policy),&#xA;but would like to gather any research/evidence that supports either case. By&#xA;support I mean leads to a positive outcome for the children.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any research or hard evidence would be most welcome. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: I should mention that the kids get a look at their tests in the classroom after the tests are graded. However, I think this puts the onus on the kids to quickly spot errors and improvements. I know that I often have to stare at something for a long time before I see the problem.&lt;/p&gt;&#xA;" OwnerUserId="1836" LastEditorUserId="1836" LastEditDate="2015-10-25T21:34:25.563" LastActivityDate="2015-10-29T02:32:52.723" Title="Evidence showing that returning tests helps students" Tags="|testing|" AnswerCount="5" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="9830" PostTypeId="2" ParentId="41" CreationDate="2015-10-25T20:42:43.783" Score="4" Body="&lt;p&gt;The key is neatness. I am a student, and I take for granted that my lecturers will use neat handwriting. I don't like it when a lecturer's handwriting is scrawled hastily and messily because it makes it difficult to read. I also don't like it when a lecturer neglects to write down something they have said, or something that is helpful with understanding what is being communicated (e.g. writing &quot;&lt;em&gt;Proof&lt;/em&gt;&quot; before a proof to let the class know that the theorem has ended). Below I have given some pictures of the best handwriting that I have encountered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/bXcoUl.jpg&quot; alt=&quot;Picture 1&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Picture 1: this lecturer gives a tail on his v's and w's. The only problem with this handwriting is that the difference between a u and a v can be confusing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The following two pictures show writing done by a lecturer who is exceptionally neat when he writes on the board. His style of teaching is to have a lot of discussion, and to only write down the mathematical symbols which he is talking about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9ENqUl.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9ENqUl.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/6VMm2l.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6VMm2l.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5074" LastActivityDate="2015-10-25T20:42:43.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9831" PostTypeId="2" ParentId="9829" CreationDate="2015-10-26T13:28:08.337" Score="3" Body="&lt;p&gt;Sorry I don't have links to specific research that you would like (mweiss' comment above seems a likely reason). My kids had the exact same issue, but it was easily resolved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reason not returned - need to keep for record keeping. Solution - photocopy&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A rationale as to why exams should always be returned if asked for has two cases. Exams are either formative or summative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Formative: exams must be returned so students can see what they did wrong and improve. I can't understand the point of a formative assessment that is not returned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summative: Students have the right to ask for an exam to be remarked in case of mistakes (maybe not in US?). Every exam leading to a qualification in high school or university was returned to me. All previous years' exams were available for photocopying, published in affordable revision books, and are now available for free download.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Availability of exams is also considered an incentive for students to do more study of exam style questions. This levels the playing field as otherwise students who (by whatever means) gain access to banks of exam questions have an exam advantage. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used to think that &quot;teaching to the exam&quot; undermined pure education, but I am now firmly of the opinion that having some time to devoted to understanding likely exam questions reduces the impact of variables unrelated to maths knowledge and therefore makes exams much more reliable indicators of domain learning. I specifically remember one of my university lecturers writing some previous exam questions on the board - and strongly hinting that there would be identical questions on this year's exam. He wanted students who could learn how to do those proofs and problems to pass - not to fail because they couldn't guess what they were supposed to study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problems for teachers: how to get new exam questions for summative assessment? In national high school exams and universities, there should be the budget to pay people to create new exam questions each year. For internal exams, most teachers here belong to a Maths teachers' association, which provides exam questions to participating high schools each year for internal exams. A few hundred teachers can easily create an exam each year to be shared by all their schools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't have a national exam and the teachers can't cooperate to create an exam - I can see that there could be problems for the teachers. But even by yourself - there are massive banks of questions available on the internet that you can copy and paste into an exam. Daniel R. Collins mentioned Pearson Testgen. Set aside a few teacher only days each year with teachers at the same school to copy paste the exams. If you have too many summative assessments to make that a viable option - consider whether you are over assessing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There may be other factors that I haven't thought of, but I really can't think of good reasons not to hand back exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Looking a BBS's answer, some schools have trouble synchronising the use of summative assessments, to the extent that even an embargo does not work. In this case it would indeed be impossible to give the test back, though this would also seem to cause problems in allowing students to ever see their exams, even during class, unless strict exam conditions were maintained in those revision classes.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-10-26T16:28:42.700" LastActivityDate="2015-10-26T16:28:42.700" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9832" PostTypeId="2" ParentId="9829" CreationDate="2015-10-26T13:30:23.837" Score="3" Body="&lt;p&gt;Any teacher will tell you if a student doesn't understand his mistakes, all he has practiced is making that mistake.  Clearly that is not the goal of the classroom.  However changing the policy of not sending home tests is a losing battle in my opinion.  What follows is my own experiences as a teacher and some suggestions of what you can fight for in the classroom to correct the situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I taught we gave &quot;book tests&quot; and were not allowed to send them home, because the tests were reused every year. This was in elementary school and many teachers were not math specialists and it wasn't realistic to expect them to make up tests.  Since I was the &quot;math person&quot; if an alternate test was needed, I was called up to make it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my classroom, the test is returned with only x's and checks. Students were expected to figure out and correct their errors. If they needed support then hints were given. In that case students often had additional work to do to practice what they got wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to suggest using this model as an alternative to challenging the system of not sending tests home. Sending tests home is likely a department decision that allows all teachers to use the book's test.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask the school for the following&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Can your children correct their test mistakes before being told what&#xA;they did wrong?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Can the students/you have a list of the kinds of mistakes they made? (e.g. arithmetic, didn't follow instructions, didn't know law of sines etc.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;Can the students be given additional practice work in the areas that they had the most trouble with.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;By approaching things this way, you are not challenging the system of using the book tests, but instead you are focused on having your children learn from their mistakes.  Good luck!&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-10-26T13:30:23.837" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="9833" PostTypeId="2" ParentId="9829" CreationDate="2015-10-26T14:26:44.000" Score="10" Body="&lt;p&gt;Without defending this decision too much, I must say that I come from a school with the same policy, and I had a hand in implementing it as a policy. Let me explain why we did it, and provide some of our suggested solutions for parents and students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We wanted to reduce variability between teachers and improve our assessments overall. So we started to implement common assessments for all teachers. We wrote and revised the tests in teams, allowing input from all teachers. In order to get buy-in from all of the teachers we had to make a couple compromises. Namely, we agreed that teachers would not send tests home while other teachers had not given the test yet. In practice this meant we didn't send tests home as it was difficult to tell if every teacher had finished with a test or not. Overall this did meet the goals we had, it improved the quality of assessments and reduced variability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first suggestion I would give is very situational. At our school we encouraged teachers to give assessments back before the midterm and final if they hadn't yet. Find out if there is some kind of embargo period, and maybe you can get the test back later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next suggestion is to get involved personally. We frequently gave tests back to parents after explaining our concerns about sharing the test with other students. We felt that once parents were familiar with the idea that their kid would be found cheating if another student got the test from them, they would be sufficiently careful with the test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another suggestion would be to see if the teacher has some time to work with the student on the test. We would frequently have students after school to detail the issues with their test and go over problems with them.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2015-10-26T14:26:44.000" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9834" PostTypeId="2" ParentId="1649" CreationDate="2015-10-26T18:11:47.540" Score="2" Body="&lt;p&gt;&lt;strong&gt;Squaring natural numbers around 50&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let n be an integer &quot;around 50&quot; (which can be anywhere from 30 to 70).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we write $n$ as $n = 50+m$ then $n^2= 2500+100m+m^2$. So just multiply $m$ by 100, square $m$ (most people know the squares of the natural numbers up to 20), add the two and then add the result to 2500.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To illustrate, let $n = 62$, then $m=12$ and $100m=1200$ and $m^2=12^2=144$, so $100m+m^2=1344$ and now just add 2500 to get your final answer 3844!&lt;/p&gt;&#xA;" OwnerUserId="5749" LastEditorUserId="2074" LastEditDate="2015-10-26T20:44:56.790" LastActivityDate="2015-10-26T20:44:56.790" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9835" PostTypeId="1" CreationDate="2015-10-26T18:54:51.540" Score="17" ViewCount="1399" Body="&lt;p&gt;Teaching my precalculus class today, I noticed something very simple that I hadn't taken into account previously. The definition in our textbook read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;A linear function is a function defined by an equation of the form &lt;span class=&quot;math-container&quot;&gt;$$y = Ax + B.&quot;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This strikes me as a common definition, and while this is certainly an alright algebraic definition for the equation of a &lt;em&gt;line&lt;/em&gt;, it is certainly not consistent with the definition of a &lt;em&gt;linear function&lt;/em&gt; in general, which I give as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;A linear function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a rule of correspondence from a set &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to a set &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; such that&lt;/p&gt;&#xA;&lt;p&gt;1.&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; relates to each element of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; a unique element of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;,&lt;br /&gt;&#xA;2. &lt;span class=&quot;math-container&quot;&gt;$f(a+b) = f(a) + f(b)$&lt;/span&gt;&lt;br /&gt;&#xA;3. &lt;span class=&quot;math-container&quot;&gt;$f(a \cdot x) = a \cdot f(x)$&lt;/span&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We take for granted that &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cdot$&lt;/span&gt; are carried out with respect to their respective sets and need not coincide.&lt;/p&gt;&#xA;&lt;p&gt;In this more general definition the elements need not be real numbers. Indeed, one can show some linear functions that take as their arguments complex numbers, matrices, vectors, other functions, and even more exotic objects.&lt;/p&gt;&#xA;&lt;p&gt;It's easy to show that a line with equation &lt;span class=&quot;math-container&quot;&gt;$y = Ax$&lt;/span&gt; satisfies both definitions, but &lt;span class=&quot;math-container&quot;&gt;$y = Ax +B$&lt;/span&gt; fails to satisfy the general definition.&lt;/p&gt;&#xA;&lt;p&gt;Today, I decided to tell my students that &amp;quot;every line is a &lt;em&gt;translation&lt;/em&gt; of the graph of a linear function over the real numbers.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-08-30T15:32:07.100" Title="Why do we teach that every line is a linear function?" Tags="|calculus|algebra|linear-algebra|precalculus|" AnswerCount="7" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="9836" PostTypeId="2" ParentId="9835" CreationDate="2015-10-26T19:07:44.773" Score="25" Body="&lt;p&gt;The usage you object to is, in fact, the &lt;em&gt;original&lt;/em&gt; meaning of &quot;linear&quot;.  &quot;Linear&quot; means &quot;having to do with lines&quot;.  The notion of &quot;linear&quot; in the sense of &quot;linear transformation&quot; is a more modern, restricted notion.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-10-26T19:07:44.773" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9837" PostTypeId="1" CreationDate="2015-10-26T19:35:26.300" Score="12" ViewCount="1071" Body="&lt;p&gt;In the last 30 years more and more countries introduced graphing calculators and then CAS systems to their high-school students. But are there already any examples of a trend in the opposite direction? That is, examples where a state or country decided to row back and deemphasize or discontinue the use of CAS systems or graphical calculators in teaching mathematics (at the secondary level). If so, what were the (official) reasons for this decision? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please note that I don't want to discuss advantages or disadvantages of CAS systems in high school math, but just the very specific question above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of this phenomenon I am aware of is in  a state of Germany, more specifically Baden Württemberg: from 2004 on at every Gymnasium [a type of high-school] graphical calculators were allowed (e.g. TI-83/84) in the final examination (Abitur) in one part of the exam. Yet, it has been decided that from the Abitur 2019 on only a simple scientific calculator will be allowed. Since this tool will not be allowed in the Abitur it changes the style of teaching. CAS tools or software like geogebra will be used from time to time on a PC/tablet/smartphone but most of the time, calculations will be done manually as well as in tests. &lt;/p&gt;&#xA;" OwnerUserId="1203" LastEditorUserId="1203" LastEditDate="2015-10-27T13:27:58.750" LastActivityDate="2024-01-16T06:37:13.750" Title="Are there examples of countries where the use of CAS systems or graphing calculators was deemphasized or discontinued?" Tags="|secondary-education|technology-in-education|exams|calculators|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="9838" PostTypeId="1" AcceptedAnswerId="9839" CreationDate="2015-10-26T20:59:05.927" Score="11" ViewCount="1609" Body="&lt;p&gt;A while back I stumbled on teacher's website that advocated a different way to teach factorization. Rather than jumping straight to factorization practice, the teacher first had their student's practice finding pairs of numbers with a given product and sum. That is, they would give them tons of problems of the form:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $a \times b = 2$ and $a+b=3$, then what are $a$ and $b$? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Rather than teaching them a method for finding $a$ and $b$, the goal was to get students to 'see' how a given sum and product can be formed. The idea was that when the students were introduced to factorization, they would have extensive experience with one of the basic skills required, and they could focus on understanding the conceptual structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thing is, I can't find the website again. Is there a standard name or reference for this kind of method? &lt;/p&gt;&#xA;" OwnerUserId="5750" LastEditorUserId="143" LastEditDate="2015-10-26T23:07:31.200" LastActivityDate="2015-10-26T23:07:31.200" Title="Method for teaching factorization" Tags="|mathematical-pedagogy|reference-request|arithmetic-operations|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9839" PostTypeId="2" ParentId="9838" CreationDate="2015-10-26T22:28:23.397" Score="8" Body="&lt;p&gt;One way this is sometimes done is with &lt;a href=&quot;http://www.pleasanton.k12.ca.us/avhsweb/kiyoi/algebra_powerpoints/Factor_Diamonds&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;factor diamonds&lt;/strong&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Google or google image: &lt;strong&gt;factor diamond method&lt;/strong&gt;. Here is a &lt;a href=&quot;http://mrsgalgebra.pbworks.com/w/page/12019048/Diamond%20Method%20for%20Factoring%20Polynomials&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;sample result&lt;/strong&gt;&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your question, you gave the example of $a \times b = 2$ and $a + b = 3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could present this as the following puzzle:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gSVIQ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gSVIQ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I made this image with the goal of including relevant vocab (&lt;em&gt;product&lt;/em&gt; and &lt;em&gt;sum&lt;/em&gt;) and I purposefully avoided writing $a = \square$ and $b = \square$ because I thought it might be a nice opportunity to vary the presentation of the equal sign (to help avoid some of the problems mentioned in &lt;a href=&quot;https://matheducators.stackexchange.com/a/7965&quot;&gt;&lt;strong&gt;MESE 7964&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;strong&gt;Note:&lt;/strong&gt; These puzzles are sometimes used to scaffold towards factoring quadratic expressions.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-10-26T22:28:23.397" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9840" PostTypeId="2" ParentId="9835" CreationDate="2015-10-27T05:09:30.387" Score="4" Body="&lt;p&gt;The function $y(x)=ax+b$ is linear under &lt;a href=&quot;https://en.wikipedia.org/wiki/Homogeneous_coordinates#Homogeneity&quot; rel=&quot;nofollow&quot;&gt;homogeneous coordinates&lt;/a&gt;; more generally, $y = g(x_1,x_2,...,x_k)=b_0+b_1x_1+b_2x_2+...+b_kx_k$ is linear in that sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically, rendering $g$ in homogeneous coordinates, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$g(x_1,x_2,...,x_k)=f(x_1,x_2,...,x_k,1)$,  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;and $y=f(x_1,x_2,...,x_k,1)$ is linear in the usual sense.&lt;/p&gt;&#xA;" OwnerUserId="755" LastActivityDate="2015-10-27T05:09:30.387" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9841" PostTypeId="2" ParentId="9814" CreationDate="2015-10-27T05:46:41.657" Score="3" Body="&lt;p&gt;An example that I once started a calculus class with was how to build a reflecting telescope.  Imagine the light from a distant object arrives as parallel rays, and you want to build a mirror that will focus all of them at a single point.  What shape should the mirror be?  Obviously it will have to be curved; but then how do we figure out at what angle light will be reflected from a curved mirror?  We know how light bounces off of a &lt;em&gt;straight&lt;/em&gt; mirror &amp;mdash; incidence equals refraction &amp;mdash; so maybe we can pretend that a curved mirror is approximately straight.  Etc.  Once you can find the derivative of $x^2$, checking that a parabolic mirror really does focus parallel rays at a point is not too hard of a trigonometry problem (having the additional advantage of forcing them to remember trigonometry and its geometric meaning).&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2015-10-27T05:46:41.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9842" PostTypeId="1" AcceptedAnswerId="9949" CreationDate="2015-10-27T08:39:34.133" Score="24" ViewCount="4715" Body="&lt;p&gt;My daughter showed me the picture below, which came from &lt;a href=&quot;http://9gag.com/gag/aDmN0LK?ref=tp&quot; rel=&quot;noreferrer&quot;&gt;9GAG&lt;/a&gt;.  It shows a question on an exam asking the student to &quot;use the repeated addition strategy to solve: 5 x 3.&quot;  The student answered &quot;5+5+5&quot; and the teacher gave the answer a one-point deduction, stating that the correct answer is &quot;3+3+3+3+3.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1wSYp.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1wSYp.jpg&quot; alt=&quot;repeated addition&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not seen any textbook that defines the notation $a\times b$ (where $a$ and $b$ are positive integers) as $b$ added $a$ times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I prefer to use $a\times b$ to represent $a$ added $b$ times.  This is to be consistent with using the notation $a^b$ (where $a$ and $b$ are positive integers) to mean $a$ multiplied $b$ times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What official document defines $a\times b$ (where $a$ and $b$ are positive integers) as $b$ added $a$ times?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastActivityDate="2017-05-13T12:06:14.953" Title="Repeated addition: standard notation?" Tags="|notation|arithmetic-operations|" AnswerCount="7" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="9843" PostTypeId="2" ParentId="9842" CreationDate="2015-10-27T10:58:37.753" Score="13" Body="&lt;p&gt;A reason why this form might be preferred is the way one says it: &#xA;$5 \times 3$ is read out  &quot;five times three&quot; so it says take $3$ five times, hence it &quot;is&quot; $3+ 3+ 3 + 3 + 3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I doubt there is any real standard. For what it's worth Wikipedia disagrees with itself. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;On the page on &lt;a href=&quot;https://en.wikipedia.org/wiki/Multiplication&quot;&gt;Multiplication&lt;/a&gt; it has $a \times b$ as $b + \dots + b$.&lt;/li&gt;&#xA;&lt;li&gt;On the page on &lt;a href=&quot;https://en.wikipedia.org/wiki/Peano_axioms&quot;&gt;Peano axioms&lt;/a&gt; when it defines multiplication recursively it states  $a \times S(b)= a + a\times b$, leading to $a \times b$ when unrolled being $a+ \dots + a$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Personally, I am very used to thinking of $a \times b$ as $b+ \dots + b$ as it lines up with scalar multiplication in vector-spaces (or more generally the fact usually modules over commutative rings are written as left-modules). &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2015-10-27T10:58:37.753" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="9844" PostTypeId="2" ParentId="9835" CreationDate="2015-10-27T13:13:00.323" Score="4" Body="&lt;p&gt;I believe the distinction that's bothering you is now called &quot;direct variation.&quot; This term is used to describe a line that goes through the origin, i.e. Y=mX, and is considered a subset of linear functions Y=mX+b. I graduated HS in 1980, and don't recall direct variation being mentioned. Now, our local HS math curriculum includes it, but the HS a few towns away where I tutor doesn't seem to mention it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You mention precalc which can either be in high school or college. In my opinion, math students struggle enough with the material, and adding a layer of these distinctions might not be appropriate at this stage. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-10-27T13:29:56.233" LastActivityDate="2015-10-27T13:29:56.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9847" PostTypeId="1" CreationDate="2015-10-27T19:20:55.793" Score="27" ViewCount="18041" Body="&lt;p&gt;Without knowing the Pythagorean theorem, and in presenting reasons why the theorem might be true (without giving a full proof), is there any way to give examples of triangles that are intuitively understandable to be right triangles?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, one can easily show (proof by &lt;em&gt;elementary&lt;/em&gt; picture with elementary geometry) that two area 1 squares and one area 2 square (side &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; but no need to mention that) form an isosceles right triangle, by dissecting the unit squares into four smaller congruent isosceles right triangles and the area 2 squares into eight of the same triangles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LnnK6.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LnnK6.gif&quot; alt=&quot;tiling showing one instance of PT: http://www.geom.uiuc.edu/~demo5337/Group3/tileproof.GIF&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is 'obvious that the abc triangle is right, and that the two smaller squares sum to the larger (with units the triangle tiles). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I don't know of a similar tiling for a 345 triangle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/IlAGf.gif&quot; width=&quot;375&quot; height=&quot;400&quot; alt=&quot;3 4 5 right triangle: jwilson.coe.uga.edu/emt668/EMAT6680.2001/Meyers/Emat%206690/Instructional%20Unit/image6.gif&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The square on the hypotenuse is skew and impossible to visually validate. There's nothing to sum. It is simple calculation to note that &lt;span class=&quot;math-container&quot;&gt;$3^2 + 4^2 = 5^2$&lt;/span&gt;, but its picture doesn't 'say' that the angle between the 3 and 4 sided square must be 90 degrees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how do you know that a 3/4/5 triangle is right without proving the full force of the Pythagorean theorem?&lt;/p&gt;&#xA;" OwnerUserId="2134" LastEditorUserId="6053" LastEditDate="2019-01-02T18:52:11.347" LastActivityDate="2022-01-16T21:15:09.977" Title="Given a 3 4 5 triangle, how do you know that it is a right triangle?" Tags="|geometry|visual-proof|" AnswerCount="18" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="9848" PostTypeId="2" ParentId="9847" CreationDate="2015-10-27T19:31:12.980" Score="-4" Body="&lt;p&gt;I would think you'd approach it the reverse direction:  Draw a side straight up of length 4, draw another straight to the right of length 3, and then calculate the distance between the ends and show that it's 5--assuming you have the distance formula or permit yourself to measure the line lengths physically.&lt;/p&gt;&#xA;" OwnerUserId="5680" LastActivityDate="2015-10-27T19:31:12.980" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9849" PostTypeId="2" ParentId="9842" CreationDate="2015-10-27T19:38:12.777" Score="0" Body="&lt;p&gt;If you want an official document from the Math universe your best bet is Bourbaki.  However, that probably isn't useful, since almost no educator works from Bourbaki.  You may want an official document from the Math education world, but I don't know anything about that and it probably differs from place to place anyway.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And in reality, this only matters at all for a very brief time until they realize that multiplication is commutative, so I kind of feel like you can pick whichever convention you want and if it conflicts with anyone else's convention you can just argue that they're effectively equivalent.&lt;/p&gt;&#xA;" OwnerUserId="5680" LastActivityDate="2015-10-27T19:38:12.777" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9850" PostTypeId="2" ParentId="9842" CreationDate="2015-10-27T19:51:59.843" Score="15" Body="&lt;p&gt;While there may be legitimate reasons behind the convention&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In $a \times b $ the $a$ denotes the number of terms and the $b$ denotes the individual terms&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;the larger issue is the mismatch between the teacher's enforcement of that convention and the &lt;em&gt;expressly stated purpose of the formative assessment, which is written at the top of the very same page&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;I can use multiplication strategies to help me multiply&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is clear that the student &lt;em&gt;is&lt;/em&gt; using multiplication strategies to help her multiply; she is even using the exact multiplication strategy called for in the item.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know this is not directly an answer to your question, but this teacher clearly does not understand what &quot;Formative Assessment&quot; means, nor what this particular formative assessment is meant to assess.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-10-27T19:51:59.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9851" PostTypeId="1" CreationDate="2015-10-27T23:06:38.747" Score="21" ViewCount="972" Body="&lt;p&gt;This semester I'm teaching introductory linear algebra for engineering students, and I don't think I'm doing a good job explaining why these topics are important; specifically, everything having to do with linear transformations. I'm a physicist (physics student, really), and so when thinking about linear algebra the first thing that comes to mind is eigenvectors, which are not covered in this course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This week I have to teach some of the most abstract stuff yet: Kernels of linear transformations, images and preimages (and showing that they are subspaces of $\mathbb{R}^n$), defining a linear transformation by its action on a basis, the rank-nullity theorem, composing linear transformations, etc. Personally I love this subject, because I think that once you really understand it, everything fits together in the most beautiful way and all the connections between the theorems seem obvious; sadly, this isn't very useful for my students who are still getting used to multiplying matrices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some examples I can give to show why this stuff is useful in engineering? I'm particularly interested in the subjects I mentioned above. Please note that not teaching them is not an option: It's not up to me to decide what is covered and what isn't, and I don't write the exams.&lt;/p&gt;&#xA;" OwnerUserId="5269" LastActivityDate="2016-03-14T09:22:02.183" Title="Applications and motivation of abstract linear algebra topics for engineers" Tags="|linear-algebra|engineering-mathematics|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9852" PostTypeId="2" ParentId="9851" CreationDate="2015-10-27T23:41:36.450" Score="9" Body="&lt;p&gt;&lt;strike&gt;Two&lt;/strike&gt; Four ideas:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) &quot;&lt;strong&gt;composing linear transformations&lt;/strong&gt;&quot;: Use rotation, scaling, and shearing. If you extend to homogenous coordinates, you can include translations. Fundamental to all computer graphics. Explore which combinations of these transformations always commute, and which sometimes do not commute. In $\mathbb{R}^2$ and in $\mathbb{R}^3$. E.g., rotations in $\mathbb{R}^3$ in general do not commute.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) &quot;&lt;strong&gt;the rank-nullity theorem&lt;/strong&gt;&quot;: Derive the consequence that there is no injective (1-to-1) linear transformation from $\mathbb{R}^3 \to \mathbb{R}^2$. And the reverse, that there is no surjective (onto) linear transformation&#xA;from $\mathbb{R}^2 \to \mathbb{R}^3$. These are quite natural conclusions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) &quot;&lt;strong&gt;Kernels of linear transformations&lt;/strong&gt;&quot;: They play a role in error-correcting codes, but that would not be easy to explain. One intuitive example is: the kernel of projection onto a plane is the set of vectors orthogonal to the plane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) &quot;&lt;strong&gt;its action on a basis&lt;/strong&gt;&quot;: To understand reflection of a ray in a mirror,&#xA;it is easier to use basis vectors tangent to and perpendicular to the plane of the mirror, rather than a standard basis.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-10-28T23:07:16.967" LastActivityDate="2015-10-28T23:07:16.967" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9853" PostTypeId="2" ParentId="9847" CreationDate="2015-10-28T00:27:39.430" Score="4" Body="&lt;p&gt;Not an answer; rather an observation.&#xA;&lt;hr /&gt;&#xA;If you can convince that the blue circle of radius $3$, and the black circle of&#xA;radius $4$,&#xA;meet the red circle of radius $\frac{5}{2}$ at the same point, then the right&#xA;angle follows from Thales' Theorem.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/wOPCN.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wOPCN.jpg&quot; alt=&quot;Pythag345&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;But I have to admit, I don't see how to intersect the circles without&#xA;using the Pythagorean theorem in some hidden form.&#xA;If the blue circle is centered at $(0,0)$ and the black at $(5,0)$,&#xA;they intersect at $(\frac{9}{5},\frac{12}{5})$, which is of course&#xA;distance $\frac{5}{2}$ from the center of the red circle.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-10-28T10:22:09.673" LastActivityDate="2015-10-28T10:22:09.673" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="9854" PostTypeId="2" ParentId="9847" CreationDate="2015-10-28T01:19:44.533" Score="-1" Body="&lt;p&gt;At this level students benefit from exploring physical models.  There is an inexpensive product called &lt;a href=&quot;http://www.geoleg.com/&quot; rel=&quot;nofollow&quot;&gt;geoleg&lt;/a&gt; that allows the students to build PHYSICAL triangles with specific side and angle measurements. You specify what the students build and they can see if there is more than one possible triangle (eg SSA) or only one possible triangle. Specify a triangle that is right and has two legs that are 3 and 4, and the students will see that the hypotenuse must be 5.  Specify a triangle that is 3, 4, 5 and the students will see that it must be a right triangle (angle measures are built in). &lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2015-10-28T17:47:08.023" LastActivityDate="2015-10-28T17:47:08.023" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="9855" PostTypeId="2" ParentId="9847" CreationDate="2015-10-28T01:44:25.837" Score="4" Body="&lt;p&gt;According to the Wikipedia entry for &lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagorean_theorem#cite_ref-29&quot; rel=&quot;nofollow&quot;&gt;Pythagorean theorem&lt;/a&gt;, a proof of the converse of the Pythagorean theorem without assuming the Pythagorean theorem can be found in Stephen Casey, &quot;&lt;a href=&quot;http://www.jstor.org/stable/27821791?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;The converse of the theorem of Pythagoras&lt;/a&gt;,&quot; &lt;em&gt;The Mathematical Gazette&lt;/em&gt;, Vol. 92, No. 524 (July 2008), pp. 309-313.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2015-10-28T01:44:25.837" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9856" PostTypeId="2" ParentId="9829" CreationDate="2015-10-28T02:28:04.213" Score="7" Body="&lt;p&gt;As Richard pointed out, it may be helpful to understand the difference between &lt;em&gt;formative&lt;/em&gt; and &lt;em&gt;summative&lt;/em&gt; assessment. Here's a good &lt;a href=&quot;http://www.cmu.edu/teaching/assessment/howto/basics/formative-summative.html&quot;&gt;link&lt;/a&gt; if you're not familiar with the terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your question suggests the following proposition: &lt;em&gt;Students should receive their summative assessments (i.e. tests) back &lt;strong&gt;so that&lt;/strong&gt; they can learn from them in the same way they learn from their formative assessments (i.e. quizzes, homework, teacher observations, etc).&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the learning model suggested by &lt;em&gt;formative&lt;/em&gt; and &lt;em&gt;summative&lt;/em&gt; assessment, this statement can be analyzed in two cases:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;1) &lt;em&gt;The students &lt;strong&gt;do not&lt;/strong&gt; receive formative assessment on a regular basis.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Possible problem&lt;/strong&gt;: If this is the case, then students are likely to become frustrated, since they may not understand where and how their mistakes happen and as a result may not do well in summative assessments. However, &lt;strong&gt;the fundamental problem may not be that students don't get to keep their &lt;em&gt;summative&lt;/em&gt; assessments, but rather that their &lt;em&gt;formative&lt;/em&gt; assessments are lousy.&lt;/strong&gt; Perhaps the teacher doesn't return homework promptly, or she gives sparse feedback in class, or he never provides quizzes or practice questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Possible solution&lt;/strong&gt;: By encouraging the teacher to &lt;strong&gt;improve his or her formative assessment&lt;/strong&gt; (or providing it yourself since you seem like an awesome dad), your children may have more confidence going into summative assessments, and if they do get seriously tripped up on questions, then hopefully the teacher is reasonable enough to let them look at the assessment outside of class time and help them understand their mistakes a little better. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;2) &lt;em&gt;The students &lt;strong&gt;do&lt;/strong&gt; receive formative assessment on a regular basis.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Comment&lt;/strong&gt;: If this is the case, there (theoretically) isn't a problem with the teacher keeping the summative assessments. As others have pointed out, there are a lot of reasons why teachers might want to keep formative assessments after allowing students to peek at them. As a teacher required to collect and keep departmental-wide tests, I've found that there have been times when it has been helpful to look back and analyze old tests as a way to inform my instruction. And students are always welcome to visit me outside of class, see any of their old tests, and ask questions. I probably wouldn't keep the tests if the decision was up to me, but I understand the department's decision.&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2015-10-28T02:28:04.213" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9857" PostTypeId="2" ParentId="9847" CreationDate="2015-10-28T04:00:19.493" Score="2" Body="&lt;p&gt;Cut out four identical 345 triangles and put the biggest angle from each triangle together at a point. Then observe that all 4 identical angles fit to make a revolution, which implies each of the angles is a right angle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A picture is provided below. More work would be required to prove this construction, but hopefully this is the level of proof you were looking for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XE3da.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XE3da.png&quot; alt=&quot;345 Triagles&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, it is interesting to note that &lt;a href=&quot;https://en.wikipedia.org/wiki/Rope_stretcher#The_Egyptian_rope_trick&quot; rel=&quot;nofollow noreferrer&quot;&gt;ancient Egyptians apparently used 345 triangles&lt;/a&gt; for laying foundations, possibly even in religious ceremonies. I'm not sure if they realized a circular rope with 12 equally-spaced knots and be stretched into a 345 triangle made a right angle, but I imagine that they had some idea of the above line of reasoning.&lt;/p&gt;&#xA;" OwnerUserId="867" LastEditorUserId="867" LastEditDate="2015-10-28T05:55:56.313" LastActivityDate="2015-10-28T05:55:56.313" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9858" PostTypeId="2" ParentId="9847" CreationDate="2015-10-28T04:29:00.987" Score="22" Body="&lt;p&gt;Here is a simple construction. Adjust to taste.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1.&lt;/strong&gt; Draw a line $l$ passing through a point O.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2.&lt;/strong&gt; Construct circles of radius 3 and 4 with centre O. Call them $C_3$ and $C_4$. Let the intersection of $C_4$ and $l$ be A. (Note that OA is 4 units.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SJCLI.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SJCLI.png&quot; alt=&quot;Figure 1&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3.&lt;/strong&gt; Construct an interval of length 5 from A to $C_3$. Call it B. (Note that OB is 3 units, and AB is 5 units.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RZKi9.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RZKi9.png&quot; alt=&quot;Figure 2&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4.&lt;/strong&gt; Construct an interval of length 5 from B to $C_4$. Call it C. &lt;strong&gt;Do not assume C lies on $l$.&lt;/strong&gt; (Note that OC is 4 units, and BC is 5 units.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/heEub.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/heEub.png&quot; alt=&quot;Figure 3&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;OAB and OBC are both 3-4-5 triangles. By symmetry, $\angle\text{AOB}=\angle\text{BOC}$. So if C lies on $l$, then $\angle\text{AOB}$ must be a right angle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you draw everything accurately, you will of course find that C lies on $l$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If 3-4-5 were not a right triangle, you would get a result were C did not lie on $l$, like this (a 4-4-5 triangle):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tifiL.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tifiL.png&quot; alt=&quot;Figure 4&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5 (optional).&lt;/strong&gt; For further assurance, go all the way round the circle and see if you get back to A.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qbwHp.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qbwHp.png&quot; alt=&quot;Figure 5&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The intuitive argument here is that anything &lt;em&gt;but&lt;/em&gt; a right triangle would either go &quot;too far&quot; or &quot;not far enough&quot; if you put 2 or 4 of them together. Of course, there will be a lot of &quot;not quites&quot; due to inaccuracies (those confounded cheap school compasses always seem to open up slightly during use!), which is a good opportunity to discuss the limitations of proof-by-picture. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Missing_square_puzzle&quot; rel=&quot;noreferrer&quot;&gt;missing square puzzle&lt;/a&gt; is a fantastic example.&lt;/p&gt;&#xA;" OwnerUserId="5756" LastEditorUserId="5756" LastEditDate="2015-10-28T08:32:53.013" LastActivityDate="2015-10-28T08:32:53.013" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9859" PostTypeId="2" ParentId="9847" CreationDate="2015-10-28T06:43:03.457" Score="-3" Body="&lt;p&gt;Can you use congruence of triangles?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since any triangle with sides of length 3, 4, 5 have equal length side pairs to the right angled triangle with sides of length 3, 4, 5, this triangle must be congruent to the right angled triangle.&lt;/p&gt;&#xA;" OwnerUserId="5757" LastActivityDate="2015-10-28T06:43:03.457" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9860" PostTypeId="2" ParentId="9847" CreationDate="2015-10-28T07:41:57.340" Score="23" Body="&lt;p&gt;The Chinese came up with the following a long time ago. Probably something better, but this is the gist of it.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Let's start with a right triangle with height b=4 and base a=3. We know it has some hypotenuse, c, but we don't know it's length because that's what we're trying to prove.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SudWZ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SudWZ.png&quot; alt=&quot;Right triangle, b=4, a=3.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Now we're going to make 3 copies, and rotate them each 90°.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WyeJY.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WyeJY.png&quot; alt=&quot;Original plus 3 copies rotated 90° each.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Next we're going to shove them all together so their sides touch. Because we defined them as right triangles, and rotated them exactly 90°, we know all the sides that are touching are parallel.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/x1R5J.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/x1R5J.png&quot; alt=&quot;Four right triangles making a square.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Finally, we construct a square around the entire construct, ensuring the sides are parallel/perpendicular to the a/b sides of the right triangles.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ff2WI.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ff2WI.png&quot; alt=&quot;Big square surrounding the angled squares.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here I've shaded it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BYTn9.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BYTn9.png&quot; alt=&quot;Shapes are filled in with color.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;We can see that the sides of the large square must be a+b = 7 in length. Necessarily, the area of the square (green+orange+yellow) is 49.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One of the orange triangles forms a rectangle with a green triangle, with sides 3 by 4, so the area of either an orange triangle or a green triangle is (3*4)/2 = 6.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The four green triangles have a total area of 6*4 = 24. The area of the middle square (orange+yellow) must be 49-24 = 25. Alternately, you could just add the obviously 1*1 = 1 area of the central square (yellow) to the area of the orange triangles which is also 6*4 = 24, for a total of 25.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Because of symmetry, the sides of the middle square must be length c. We know the area of the square is c^2 and we also know it's 25, so c^2 = 25. Because 5^2 = 25, c=5. We've also managed to show at least one case of Pythagorean's theorem is true in the process.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5758" LastActivityDate="2015-10-28T07:41:57.340" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9861" PostTypeId="2" ParentId="9847" CreationDate="2015-10-28T13:08:39.747" Score="23" Body="&lt;p&gt;The fact that there is a 3-4-5 triangle that is a right triangle is unique to the Euclidean plane. There is no such triangle in the spherical or hyperbolic planes. Since the Pythagorean theorem is equivalent to the parallel postulate, any proof that a 3-4-5 triangle is a right triangle will somehow depend on the Pythagorean theorem/parallel postulate.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-10-28T13:08:39.747" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9862" PostTypeId="2" ParentId="9851" CreationDate="2015-10-28T17:57:47.083" Score="3" Body="&lt;p&gt;Engineers at our school are not required to take the full Linear Algebra course. However, they take a 1-credit course called &quot;Linear Algebra for Differential Equations.&quot; So I think the answer might be to look into some Differential Equations material where the entries of the matrices, the basis elements, and the eigenvectors are all functions (usually with understandable conceptual meanings).&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-10-28T17:57:47.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9863" PostTypeId="2" ParentId="9847" CreationDate="2015-10-28T19:21:23.783" Score="20" Body="&lt;p&gt;Graph paper (or square floor tiling) to the rescue! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof by picture for a 3 4 5 triangle:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BrNPx.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BrNPx.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because the drawing is on the grid and not the skew tiling of the square on the hypotenuse, determining the area is not inscrutable. The blue square (it is a square by adding angles of the triangle) is area 25 by adding the four blue triangles (obviously 6 each) and the single unit square in the middle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of course this can be immediately generalized to any right triangle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I asked for the determination of the largest angle of the 3 4 5 triangle (and this visual proof shows the other direction that the hypotenuse is a square on 5), I think the visual intuition is enough to go both directions, that showing a 3 4 rt triangle has hypotenuse 5 is enough (intuitively) to show the 3 4 angle of a 3 4 5 triangle is right.&lt;/p&gt;&#xA;" OwnerUserId="2134" LastActivityDate="2015-10-28T19:21:23.783" CommentCount="2" CommunityOwnedDate="2015-10-28T19:21:23.783" ContentLicense="CC BY-SA 3.0" />
  <row Id="9864" PostTypeId="2" ParentId="9829" CreationDate="2015-10-29T02:32:52.723" Score="4" Body="&lt;p&gt;Here is my best attempt at collecting the evidence requested.&lt;/p&gt;&#xA;&lt;p&gt;OP has stated under David Ebert's answer that they would be happy with formative assessments being returned. However, many of the objections to returning assessments are with summative assessments (reduced cheating, &amp;quot;reduced variability&amp;quot;...). The point of formative assessment is not reliable measurement, but increased achievement.&lt;/p&gt;&#xA;&lt;p&gt;There is empirical data on the benefits of well managed formative assessment in partnership with students (&amp;quot;achievement gains of one-half to two standard deviations&amp;quot; see below). Part of this is &amp;quot;continuous access&amp;quot; to the information, which I take to be the intent of the OP.&lt;/p&gt;&#xA;&lt;p&gt;An easy to understand explanation of formative assessment:&#xA;&lt;a href=&quot;http://education.jhu.edu/PD/newhorizons/Better/articles/Spring2011.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://education.jhu.edu/PD/newhorizons/Better/articles/Spring2011.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Margaret Heritage explains how formative assessment is carried out by teachers (tests and informal activities) and students (peer and self evaluation), and used by teachers (to improve teaching) and students (to guide their own learning).&lt;/p&gt;&#xA;&lt;p&gt;Some principals of formative assessment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;First, whatever method teachers use to elicit evidence of learning, it should yield information that is actionable by them and their students.&amp;quot; &amp;quot;...students can take active steps to advance their own learning.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From all the above we can see that formative assessment is not solely created by or for the sole use of the teacher.&lt;/p&gt;&#xA;&lt;p&gt;Rick Stiggins (below) explains that formative assessment is created in partnership with students, and with students having &amp;quot;continuous access&amp;quot; to the information.&lt;/p&gt;&#xA;&lt;p&gt;A widely referenced paper is:&lt;/p&gt;&#xA;&lt;p&gt;Rick Stiggins, From Formative Assessment to Assessment FOR Learning: A Path to Success in Standards-Based Schools, Phi Delta Kappan, Vol. 87, No. 04, December 2005, pp. 324-328.&lt;/p&gt;&#xA;&lt;p&gt;sourced from:&#xA;&lt;a href=&quot;http://bibliotecadigital.academia.cl/bitstream/handle/123456789/586/Rick%20Stiggins.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://bibliotecadigital.academia.cl/bitstream/handle/123456789/586/Rick%20Stiggins.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some quotes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Assessment FOR learning rests on the understanding that students are data-based instructional decision makers too.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Students &amp;quot;must have &lt;strong&gt;continuous access&lt;/strong&gt; to evidence of what they believe to be credible academic success, however small.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Students partner with their teacher to continuously monitor their current level of attainment in relation to agreed-upon expectations so they can set goals for what to learn next and thus play a role in &lt;strong&gt;managing their own progress&lt;/strong&gt;. Students play a special role in &lt;strong&gt;communicating evidence... to their families&lt;/strong&gt;... &amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;When consistently carried out as a matter of routine within and across classrooms, this &lt;strong&gt;[Assessment for Learning] has been linked to achievement gains of one-half to two standard deviations&lt;/strong&gt; on high-stakes tests, and the largest gains made are by low achievers.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The thrust of the article is that as long as teachers see formative assessment as something that is owned by the teachers and primarily for use of the teachers, they are really missing the point of formative assessment.&lt;/p&gt;&#xA;&lt;h2&gt;Property Rights&lt;/h2&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.copyrightlaws.com/us/copyright-ownership-in-students-work/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.copyrightlaws.com/us/copyright-ownership-in-students-work/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;From further reading, it is clear that it is the board - the institution - that is seeking ownership rights. Many teachers have apparently stated that they disagree with the boards position. Legal opinion also disagrees, and most commentators make it clear that society as a whole disagrees.&lt;/p&gt;&#xA;&lt;p&gt;So, from a property perspective it is acknowledged that students are the owners of the material that they create while they are learning, unless it is produced under &amp;quot;work-for-hire&amp;quot;, or the ownership has been explicitly signed over to the institution. The refusal to hand over material that has been created by students and is requested by those students, in absence of substantial overriding concerns (a necessary embargo period on the return of summative assessments) is a breach of property rights.&lt;/p&gt;&#xA;&lt;p&gt;One university gets around this by including &lt;a href=&quot;http://www.gadsdenstate.edu/current-students/legal/intprop.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this clause&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When a student submits work as a course requirement, the student retains ownership of the work, but ownership of the physical or electronic document shall be vested in the College. The College is granted a perpetual, royalty-free license by the submitting student to make copies of the work for administrative and educational purposes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the university can hold on to coursework, as the physical document belongs to them. However it is clear that the university has no intention of permanently disallowing students from having access to material that they have produced. Allowing students to have a copy for themselves is necessary for them to practice their ownership of copyright.&lt;/p&gt;&#xA;&lt;h2&gt;Educational Philosophy&lt;/h2&gt;&#xA;&lt;p&gt;Several answers indicate that their personal preference is for students to have full access to tests, while departmental policies prohibit this. Like any institution, schools seek to keep as much information as possible internal to the organisation. This increases the power of the institution - which is not necessarily a bad thing, as empowered teachers can help students.&lt;/p&gt;&#xA;&lt;p&gt;The fundamental error comes from the institutional (not personal) perspective that students are primarily an object that is taught, so an empowered teacher is more important to success than an empowered learner. It is this institutional attitude that is probably more damaging for student achievement than any particular policy on whether or not students have access to formative assessments.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-10-29T02:32:52.723" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="9865" PostTypeId="2" ParentId="9837" CreationDate="2015-10-29T02:39:39.163" Score="4" Body="&lt;p&gt;New Zealand used to have a CAS version of calculus exams. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;old exam example:&#xA;&lt;a href=&quot;http://www.nzqa.govt.nz/nqfdocs/ncea-resource/exams/2010/90834-exm-2010.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.nzqa.govt.nz/nqfdocs/ncea-resource/exams/2010/90834-exm-2010.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was actually much harder than the non-CAS version. I believe is was discontinued because it was much harder than the standard exam - how many kids want a harder exam? The regular exam assumes that students have access to a graphing calculator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I forgot to mention that there has also been recently created a pre-calculus algebra exam in which they have banned all calculators entirely! This is amazingly popular with schools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;first exam sample: &lt;a href=&quot;http://www.nzqa.govt.nz/nqfdocs/ncea-resource/exams/2011/91027-cat-A-2011.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.nzqa.govt.nz/nqfdocs/ncea-resource/exams/2011/91027-cat-A-2011.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect it is simply coincidence that the calculator-fee exam was introduced the same year that the CAS exam with withdrawn.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2015-10-29T14:38:56.973" LastActivityDate="2015-10-29T14:38:56.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9867" PostTypeId="2" ParentId="9847" CreationDate="2015-10-29T04:10:02.183" Score="13" Body="&lt;p&gt;A complete different approach reasoning with the area $A$ of the triangle 3-4-5.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Use Heron's Formula to show that $A =6$.&lt;/li&gt;&#xA;&lt;li&gt;Conclude that the height to the side with length 4 must be 3 since $A = ah/2$.&lt;/li&gt;&#xA;&lt;li&gt;Thus, the length of the side (3) equals the length of the height, which finishes the proof, since the height is perpendicular by definition.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The crucial step of this proof is the use if Heron's formula, which can be shown without using Pythagorian Theorem, see &lt;a href=&quot;http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Umberger/MATH7200/HeronFormulaProject/GeometricProof/geoproof.html&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5765" LastActivityDate="2015-10-29T04:10:02.183" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9869" PostTypeId="1" AcceptedAnswerId="9872" CreationDate="2015-10-29T08:25:11.400" Score="14" ViewCount="1089" Body="&lt;p&gt;I'm not a course instructor, just a TA of the first quarter calculus course who lead discussion sections and grade exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When grading the midterm, I found large number of students showed some understanding of the concepts/problem solving strategies, but writing them using poor notations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Question: Find the horizontal asymptotes for the function $h(x)=\frac{x-1}{x^2-6x+8}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some students' answers are similar to&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Answer: $$\lim_{x\to \infty}\frac{x-1}{x^2-6x+8}=\lim_{x\to \infty}\frac{x-1}{(x-2)(x-4)}=\frac{\infty}{\infty\cdot\infty}=\frac{1}{\infty}=0$$ so H.A. at 0.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Answers like this usually&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a). take lots of time to grade and hard to decide how many points (out of 10, for example) should be given. (Imagine yourself trying to grade consistently and seeing 10 different variations of answers like that in a 300 people class)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b). result in students submitting regrade requests asking why 6 points instead of 7 points are given, therefore extra workload for TA.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is, in the discussion section I'm leading, what teaching strategy can I use to reduce the amount of answers like that in the future exams I'm grading?&lt;/p&gt;&#xA;" OwnerDisplayName="user2139" LastActivityDate="2017-03-16T04:22:41.743" Title="As a TA, how to reduce imprecise notations/statements in students' exams" Tags="|calculus|exams|grading|notation|teaching|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="9870" PostTypeId="2" ParentId="9847" CreationDate="2015-10-29T08:46:40.617" Score="13" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UQXfu.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UQXfu.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Diagram shows that there exists a 345 triangle that is right-angled.&lt;/p&gt;&#xA;&lt;p&gt;It is clear by inspection that an angle greater than 90 between 3 and 4 leads to a hypotenuse longer than 5. Similarly an angle less than 90 leads to a hypotenuse of less than 5&lt;/p&gt;&#xA;" OwnerUserId="5768" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-10-29T08:46:40.617" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9871" PostTypeId="2" ParentId="9851" CreationDate="2015-10-29T09:14:06.340" Score="10" Body="&lt;p&gt;Of course it depends on how much time you're willing to spend on this. If the answer is &quot;very little&quot; then no chance that you can say something more than &quot;in the future this will be useful for you&quot;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I start from the idea that you can devote at least one hour to this. And it better be a well prepared one, otherwise time will not be enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Codes&lt;/strong&gt;: I do not think that it is difficult to explain as Joseph's says, at least if you do not really expect to arrive to the point of error-correcting ones. The idea of check digit in a linear code is exactly equivalent to the kernel of a linear transformation. If they have a book with them they have an ISBN code written on it and you can have then check that the last digit satisfies a linear equation in the vector space $\mathbb Z_{11}^{11}$: &lt;a href=&quot;https://en.wikipedia.org/wiki/International_Standard_Book_Number&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/International_Standard_Book_Number&lt;/a&gt;&#xA;Similarly for bar-codes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Linear optimization&lt;/strong&gt;: you can show how to use matrices and linear equations to solve an easy optimization problem, e.g. the &quot;diet problem&quot; with restrictions. There is no need to develop a full theory. Just build up an example with resources and consumption of resources, and formalize it with a system of linear equations. Look at some examples here. You will have your way to decide whether you want to prove inequlities or only equalities and can easily adapt them... &lt;a href=&quot;http://www.shelovesmath.com/algebra/advanced-algebra/linear-programming/&quot; rel=&quot;nofollow&quot;&gt;http://www.shelovesmath.com/algebra/advanced-algebra/linear-programming/&lt;/a&gt;&#xA;(BTW if you students are addicted to some MORPG you can use a word problem with resources appearing in that game)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Least squares method&lt;/strong&gt;. It is an approximation problem turned into a linear algebra problem. Not too difficult to explain (you better have had some discussions about norms, though) and easy to see that it can lead to linear systems with a huge number of equations: good place to remind them why Gauss reduction is much more effective than algorithms based on determinants...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Markov chains&lt;/strong&gt; in disguise: though you do not have eigenvalues and eigenvectors you can show that matrix powers have a role in understanding the behaviour of some linear dynamical system. The examples I do prefer are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;dynamic of a population: divide a population in young, adult, old. Define a fertility number and a dying rate to each of this and see how the population evolves starting from a fixed initial vector distribution. This is computing powers of a 3x3 matrix.&lt;/li&gt;&#xA;&lt;li&gt;reallocation of resources: say you're car renting in NY-Chicago-LA and have a monthly rate of cars being returned at the same rental or to another in the country. Starting from any distribution of cars at time 0 determine the distribution of cars after a fixed amount of time.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="465" LastEditorUserId="376" LastEditDate="2016-03-14T09:22:02.183" LastActivityDate="2016-03-14T09:22:02.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9872" PostTypeId="2" ParentId="9869" CreationDate="2015-10-29T13:13:16.330" Score="12" Body="&lt;p&gt;Let me echo Benjamin's comment that any proactive step that you take should be done with the instructor's permission.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a practical level, I think there are ways to address issues (a) and (b).  For (a), make a rubric (either in advance or a running one as you go) which lays out the criteria for awarding points.  This allows you to be consistent with how you distribute points and, since the assignment of points itself is subjective in nature, a reasonable rubric enforced as consistently as possible usually results in good grading.  Your instructor can give you a sense of whether your overall grading standards are too harsh or too lenient.  For (b), if you have followed (a) and have the support of your instructor, I think it would be a reasonable policy not to consider changes to homework grades unless a substantial error was made (e.g. you misread a student's response and marked a correct method as being incorrect).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But regardless of grading issues, we would certainly like calculus students to produce better responses than the one you used as an example.  In my view, one needs to take a perspective similar to triage in medicine.  Often, students have many deficiencies relative to the level of performance we would like to see from them, so prioritizing the &lt;em&gt;most important&lt;/em&gt; issues to address is essential to good instruction.  Since examples are often easier to discuss than general principles, I'll stick with your specific example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see one glaring issue that is very concerning, and that is the use of the step $\frac{\infty}{\infty \cdot \infty} = \frac{1}{\infty}$.  Not only is it an invalid thought process, it is likely to produce incorrect conclusions if the student attempts to extend it to other contexts.  Would $\lim\limits_{x \rightarrow \infty} \frac{(x - 1)(x + 4)}{x^3}$ equal $\frac{\infty \cdot \infty}{\infty}$ or $\frac{\infty \cdot \infty}{\infty^3}$?  What would happen if the denominator is $\sqrt[4]{x}^3$ instead of $x^3$?  It's not clear that the student would use the right &quot;power&quot; of $\infty$ to be able to compute the limit.  (If they are able to, that's a positive sign -- at least they have some sense of what is going on, even if their articulation of the reasoning is very flawed.)  In comparison, if a student writes $\lim\limits_{x \rightarrow \infty} \frac{1}{x} = \frac{1}{\infty} = 0$, I'm not thrilled with the inappropriate use of $\infty$ (according to the standard definitions), but I consider it mostly harmless.  There is reasonable sense that can be made to $\frac{1}{\infty} = 0$ that is much more palatable than nonsense such as $\frac{\infty}{\infty \cdot \infty} = 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other issue that I would want to address is that &quot;H.A. at 0&quot; is not very precise and I would prefer a student write that &quot;the line $y = 0$ is a horizontal asymptote&quot; (&quot;H.A.&quot; in place of &quot;horizontal asymptote&quot; is fine), but some instructors may prefer the less formal means of expression.  After all, someone could be pedantic with my approach and say that $y = 0$ is not actually a line, so we should write &quot;the line which is the locus of all points where $y = 0$&quot; or some such and now we've simply confused the students entirely.  In practice, I would accept an answer such as &quot;H.A. at 0&quot; from students, but expect myself and my TAs to be more precise and refer to the line $y = 0$ as the horizontal asymptote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you've identified the issues you want to address, here are some concrete ways you can obtain improvements:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Be sure you are very careful in your own explanations of solutions.  I've had TAs who wanted to mark very harshly for sloppy writing on homework ... and then I observed them engage in the same sloppiness on the board during their recitation periods.  When I brought this issue up, they argued, &quot;Well of course I have to be sloppy in class because I am under time pressure, but students doing homework have the time to write better.&quot;  The flaws in this reasoning are many, but in brief (1) it is generally better to write explanations clearly on the board and cover less problems than to sloppily solve many problems; (2) students cannot possibly learn to write well if they do not see a consistent high standard [and certainly not if they see sloppiness is acceptable for an instructor or TA]; (3) students do not have the time or do not choose to use their time to write polished solutions to every calculus problem they are assigned -- and it's not realistic to expect that [so if good writing has not become a learned habit, don't expect it to happen].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you start by providing a good example yourself.  Then you perform triage and identify the worst offenses that you will mark off on homework.  The cancellation of factors of $\infty$ in your example would certainly qualify.  You should take time to address common mistakes.  I recommend a brief amount of time addressing such issues in class (a list of things not to do and 1-2 sentence explanations of why for each item on the list), with an offer to explain more thoroughly the rationale in office hours.  Students will respond to the incentive of losing points, and they are generally accepting if you can show that there is a reasonable rationale behind it (even if they don't fully understand or accept that rationale).  Just don't appeal to authority or superiority.  If you cannot give an explanation that a reasonably strong student can understand, you are probably setting an unrealistic expectation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a TA, it should be added, that you should generally rely on your instructor's opinion to perform your triage assessment.  Certainly it is inappropriate to simply decide your own priorities without any consultation of the instructor.  The instructor often cannot micromanage every single prioritization decision you make, but a good conversation with your instructor can leave you with a good understanding of how to prioritize (and a good instructor will understand that at the level of small details, you two may not come to the exact same conclusions and that is okay as long as you are in harmony overall).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, by the time exams come around, students have had good instruction to see what an appropriate solution looks like for the types of problems they encounter, and the homework has identified the most egregious errors in writing up solutions.  Then the markings for exams should generally follow the same standards.  (In practice, I prefer to be have a somewhat more harsh grading standard for homework than exams.  Harsh feedback on homework is more likely to get their attention and lead to improvement before the next exam.  And that improvement will hopefully lead to increased retention of conceptual understanding for use later in the course and beyond.) &lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2015-10-29T21:48:30.213" LastActivityDate="2015-10-29T21:48:30.213" CommentCount="19" ContentLicense="CC BY-SA 3.0" />
  <row Id="9873" PostTypeId="2" ParentId="9847" CreationDate="2015-10-29T14:43:16.613" Score="4" Body="&lt;p&gt;If you have  &lt;a href=&quot;https://en.wikipedia.org/wiki/Pick&amp;#39;s_theorem&quot; rel=&quot;nofollow&quot;&gt;Pick's formula&lt;/a&gt; at your disposal, you can draw your favourite right triangle on grid paper and count. Actually, you can do the counting for &lt;em&gt;any&lt;/em&gt; triangle with grid point vertices, but of course (by what &lt;em&gt;we&lt;/em&gt; know) we get equality iff the triangle is a right one.&lt;/p&gt;&#xA;" OwnerUserId="4642" LastEditorUserId="262" LastEditDate="2015-10-29T20:26:17.347" LastActivityDate="2015-10-29T20:26:17.347" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9874" PostTypeId="1" AcceptedAnswerId="9896" CreationDate="2015-10-29T21:02:57.187" Score="18" ViewCount="3304" Body="&lt;p&gt;I realise that teachers tend to focus on right-angled triangles when introducing trig functions, and for those I can see that the most intuitive approach seems to be starting with the opposite and adjacent sides of a triangle matching the right and bottom of a rectangle. But otherwise, it's always struck me as odd that we go anti-clockwise from the right, rather than clockwise from the top - which gives the same results, but with sine corresponding to the horizontal direction, and cosine on the vertical. This is how bearings are conventionally given, after all, and even in the digital age the analogue clock-face is something almost everyone's familiar with. And it's not &lt;em&gt;so&lt;/em&gt; hard to draw a triangle with its point facing downwards, right? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm just wondering if I'm missing something here. Would I be doing my students a terrible disservice if I introduced them to trig functions treating them as going clockwise from 12 o'clock?&lt;/p&gt;&#xA;" OwnerUserId="5772" LastActivityDate="2015-11-01T16:02:28.383" Title="Why do we conventionally treat trig functions as going anti-clockwise from the right?" Tags="|trigonometry|" AnswerCount="12" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="9875" PostTypeId="2" ParentId="9874" CreationDate="2015-10-29T21:33:11.657" Score="3" Body="&lt;p&gt;It's starting on the positive branch of the real number line; and passing through the positive quadrant first. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-10-29T21:33:11.657" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9876" PostTypeId="2" ParentId="9874" CreationDate="2015-10-29T21:47:29.230" Score="12" Body="&lt;p&gt;My guess is that it comes from drawing complex numbers. Putting real numbers on the $x$-axis from left to right and imaginary numbers on the $y$-axis from bottom to top matches with the way we tend to think (in cultures based on Latin at least). Once you establish that, then the function $x\mapsto e^{2\pi ix}$ takes you anti-clockwise from the positive $x$-axis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if I'm not missing something, a clock face is based on living in the northern hemisphere.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-10-29T21:47:29.230" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="9877" PostTypeId="2" ParentId="9874" CreationDate="2015-10-30T02:25:42.163" Score="26" Body="&lt;p&gt;Regarding the second part of your question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Would I be doing my students a terrible disservice if I introduced them to trig functions treating them as going clockwise from 12 o'clock?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think it's important to stress that the convention is just a convention, and there is no intrinsic reason why one convention is better than another.  But at the same time, this particular convention is a &lt;em&gt;nearly universal&lt;/em&gt; one, and teaching your students a nonstandard convention is roughly equivalent to a language teacher teaching his or her students a dialect that nobody else speaks.  As soon as they leave your classroom they will find themselves in a mathematical world that does things differently, and you &lt;em&gt;are&lt;/em&gt; doing them a disservice if they are not thoroughly used to the conventions that everybody uses. &lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-10-30T02:25:42.163" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="9878" PostTypeId="2" ParentId="9874" CreationDate="2015-10-30T02:32:38.223" Score="2" Body="&lt;p&gt;Perhaps the convention is rooted in how the big dipper rotates around the north star--in a counterclockwise direction. Also think about how the earth rotates around its axis--in a counterclockwise direction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the origins of trigonometry are inextricably tied to navigation, you would think our modern convention might descend from this. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-10-30T02:32:38.223" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9879" PostTypeId="2" ParentId="9874" CreationDate="2015-10-30T03:56:01.170" Score="1" Body="&lt;p&gt;There are many possible reasons why it should be defined that way, but I'm not sure what were the initial motivating reasons.&#xA;$\def\nn{\mathbb{N}}$&#xA;$\def\rr{\mathbb{R}}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, $(\cos,\sin)$ is a basis for all the real-valued functions satisfying $f''=-f$. It is then natural to choose the basis elements so that $\cos(0) = 1$ and $\cos'(0) = 0$ and $\sin(0) = 0$ and $\sin'(0) = 1$. Another viewpoint is that this differential equation immediately implies that if they have a Taylor series $x \mapsto \sum_{k\in\nn} a_k \frac{x^k}{k!}$, then the coefficients $(a_n)_{n\in\nn}$ would have to satisfy $a_{n+2}=a_n$ for any $n\in\nn$. If we do not want to invoke the Taylor theorem we could also have defined them by the infinite series, prove that it converges, and then prove directly that it satisfies the differential equation. Either way, we obviously want the basis elements to be $(1,0,-1,0,\cdots)$ and $(0,1,0,-1,\cdots)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note, harmonic motion is often started off with nonzero displacement but zero velocity, corresponding to a multiple of $\cos$, whereas $\sin$ corresponds to zero initial displacement but nonzero initial velocity. In some sense, rest is simpler than motion, just as first-order terms in a Taylor series are 'more' important than second-order terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, $\exp(it)$ goes round the unit circle for $t\in\rr$. We can see this from the properties of $\exp$ that follows from the infinite series definition, which is motivated by the differential equation $f'=f$. The other definitions do not reveal the underlying structure. We can then choose $\cos,\sin$ as the $x,y$ coordinates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In both reasons above, they do not fix a starting point or direction for the path traced by $(\cos,\sin)$ in the plane. What fixes it is the choice of our coordinate system where the $y$-axis is $90^\circ$ anti-clockwise from the $x$-axis, and we plot a pair with the first element as $x$-coordinate and the second element as $y$-coordinate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, a lot of mathematical objects that seem tied to the anti-clockwise direction are nowhere near arbitrary but have to do with the way we have chosen to position the coordinate axes on paper. For example signed area is positive for anti-clockwise traversal of a non-self-intersecting polygon in the plane. The relation between the anti-clockwise contour integral around a pole and the residue there as given by the Laurent series is another. Not coincidental at all. (I decided to put this paragraph in because amazingly people don't seem to get that my answer explains all these seemingly arbitrary conventions at one go.)&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="1550" LastEditDate="2015-11-01T14:00:43.177" LastActivityDate="2015-11-01T14:00:43.177" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="9880" PostTypeId="2" ParentId="9869" CreationDate="2015-10-30T04:18:09.260" Score="4" Body="&lt;p&gt;Michael Joyce's answer above is excellent. Let me sketch out a few extra thoughts. Assume in principle that you're the instructor, or else that you can discuss and come to agreement with the instructor on the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The points assessed on each item should be scaled in advance to the length of the item, such that grading rubric is either self-evident or easy to specify, and the grader wastes no time in debate on what points to take off where. In the example problem, I wouldn't assign it more than 5 points, since it only takes about 5 steps of work to generate the answer; since the middle 3 steps are all clearly fallacious, I'd probably take off 3 points in this example. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If required to assess it at 10 points, then I'd grade in 5 2-point increments each, and take off 6 points for the example work; no need to sweat over finer granularity than that. (Personally, I would probably set up this problem at only 3 points on my own tests.) Furthermore, distribute a complete answer sheet with grading rubric (just note how many points at each step), so that (a) students can see the complete, correct answer, and (b) ascertain that the grading was done in a reasonably objective manner -- hopefully heading off complaints or requests for re-grading. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Michael indicates, definitely model the proper writing in class, and question/ask for directions on proper writing at each step from the class. I do think that grading the details is the only way to get students to attend to proper care in their writing and reasoning. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-10-30T04:18:09.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9881" PostTypeId="2" ParentId="9828" CreationDate="2015-10-30T05:59:12.980" Score="7" Body="&lt;p&gt;Just solve a general $2$ by $2$ linear system using substitution, and out pops Cramer's rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with:&#xA;\begin{align}&#xA;ax + by = u ,\\&#xA;cx + dy = v.&#xA;\end{align}&#xA;Multiplying both sides of the first equation by $c$, and both sides of the second equation by $a$, then subtracting, we find that&#xA;\begin{equation}&#xA;(ad - bc)y = av - uc.&#xA;\end{equation}&#xA;Assuming that the determinant $ad - bc$ is not $0$, we find that&#xA;\begin{equation}&#xA;y = \frac{av - uc}{ad - bc} = \frac{ \begin{vmatrix} a &amp;amp; u \\ c &amp;amp; v \end{vmatrix}}{\begin{vmatrix} a &amp;amp; b \\ c &amp;amp; d \end{vmatrix}}.&#xA;\end{equation}&#xA;The formula for $x$ can be derived similarly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is the easiest way to discover the determinant in the first place, and to recognize its relevance to solving linear systems.&lt;/p&gt;&#xA;" OwnerUserId="2049" LastEditorUserId="2049" LastEditDate="2015-10-30T10:17:36.043" LastActivityDate="2015-10-30T10:17:36.043" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9882" PostTypeId="2" ParentId="8011" CreationDate="2015-10-30T11:36:08.077" Score="6" Body="&lt;p&gt;Instead of thinking about $-x$, consider the expression $(0-x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having (hopefully) being exposed to expressions of the form $(a-b)$ for a long time, where $a$ and $b$ can be arbitrary reals, there's no reason to assume that the value of the expression is positive (or negative) without further information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, when they see $\sqrt{-x}$, they can now think of $\sqrt{-x}$ $\equiv$ $\sqrt{(0-x)}$, and realise that there is a real root where $x$ has a negative value.&lt;/p&gt;&#xA;" OwnerUserId="5781" LastActivityDate="2015-10-30T11:36:08.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9883" PostTypeId="2" ParentId="9874" CreationDate="2015-10-30T16:34:16.873" Score="1" Body="&lt;p&gt;I did a google search for &quot;right triangle&quot;. The first image I got is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DMdLT.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DMdLT.png&quot; alt=&quot;a right triangle &quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suitable for viewing angles counterclockwise from East or clockwise from South.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;*: I use the compass points for direction, since I find phrases like &quot;clockwise from down&quot; awkward, and did not want to use &quot;counterclockwise from the positive $x$ axis&quot;, since sometimes people do have the $y$ axis pointing East and the $x$ axis pointing either North or South&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second image I got was&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cLMoz.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cLMoz.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;suitable for viewing angles clockwise from West, or counterclockwise from South.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;&lt;em&gt;twenty-fourth&lt;/em&gt;&lt;/strong&gt; image was&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oYGgjm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oYGgjm.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and is the first triangle suitable for having an angle measured clockwise from North.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is by no means conclusive, but if we were to believe the idea of measuring angles from coordinate axes and extending the trig functions to the whole circle came about as an extension of measuring right triangles, it's plausible that the way we tend to write triangles had some influence.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2015-10-31T09:51:18.683" LastActivityDate="2015-10-31T09:51:18.683" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9884" PostTypeId="2" ParentId="8011" CreationDate="2015-10-30T18:10:00.390" Score="7" Body="&lt;p&gt;I'd conjecture that classes have not given sufficient time to the specific translation skill between the English and Algebra languages, especially the meaning of relation symbols like equals (=). Two days ago I had to spend a few minutes clarifying this in my College Algebra class.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Q: &quot;Algebraically, how would one say that $c$ is a negative number?&quot;&lt;/li&gt;&#xA;&lt;li&gt;A: &quot;$-c$&quot; (this from one of my best students)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This brings up a few problems. Primarily, this response isn't any kind of assertive statement; they fail to intuit relational symbols (=, &amp;lt;, &gt;, etc.) as the necessary verbs in a sentence no matter how much I say, drill, or grade on it (obviously, &quot;$-c$&quot; is merely an expression, equivalent to a sentence fragment). Generally when I first meet students they place or omit equals signs pretty randomly on the page, as if it's mostly a decorative element, and are frequently bewildered at my grading for this on tests. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I write that the English &quot;c is a positive number&quot; is equivalent to $c &amp;gt; 0$, then most of my students can correctly answer that &quot;c is a negative number&quot; is equivalent to $c &amp;lt; 0$. For at least one weaker student I had to go over this again after class, but she did seem to have an &quot;aha&quot; moment after I wrote down an explicit table (English vs. Algebra columns).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary: Practically none of my students can intuit that &quot;c is a negative number&quot; would be written $c &amp;lt; 0$ without some direct instruction on the issue. Without that, their first guess is indeed that $-c$ has that meaning.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-10-30T18:10:00.390" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9885" PostTypeId="1" AcceptedAnswerId="10960" CreationDate="2015-10-30T19:16:46.870" Score="3" ViewCount="923" Body="&lt;p&gt;I am an undergraduate student and since this is a site of math educators, I thought of putting this question (more accurately a query) here. Can you list  some good books on analytical solid geometry?&lt;/p&gt;&#xA;" OwnerUserId="5786" LastEditorUserId="376" LastEditDate="2015-12-24T14:13:49.210" LastActivityDate="2020-05-07T06:29:56.343" Title="Which book should I refer to for analytical solid geometry?" Tags="|undergraduate-education|textbooks|geometry|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="9886" PostTypeId="2" ParentId="9874" CreationDate="2015-10-30T19:44:28.163" Score="5" Body="&lt;p&gt;I would avoid teaching it &lt;em&gt;only&lt;/em&gt; as clockwise.  If you wanted to teach it that way, I would recommend showing both throughout the process, and pointing out that while clockwise might make more intuitive sense at first, the world will expect you to think of it anti-clockwise later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I point it out is not a mathematics one.  I am an engineer, who deals with things like coordinate conventions every day of my life.  Even after doing it for &lt;em&gt;years&lt;/em&gt;, coordinate conventions still confuse engineers enough to make machines break and planes crash every year (hopefully simulated machines and simulated planes!  We do like to fix these things before they get manufactured!)  The topics we have to deal with are so exacting that we oft do not have the spare brain-power to consider what might happen if our coordinate system is wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How bad is it?  I work on a simulation where we support multiple frames / coordinate systems because the work done in the simulation is too complicated to do them in the wrong frame.  We support, at any time, at least 5 frames and another dozen or so coordinate systems associated to those frames.  We handle Euler angles and Tait-Bryan angles, all 1000+ combinations thereof (did you know there were that many?  Most people I work with are aware of 2 or 4!).  I admit I get a sadistic gleam in my eye whenever anyone asks me to read in some Euler angle data, and I ask &quot;which convention are you using?&quot;  It's especially satisfying when I make them go back to the vendors to make sure both vendors supplying data are using the same definitions!  (usually they do.  Most of our vendors follow the DIS &lt;em&gt;convention&lt;/em&gt;, thank goodness)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can call it a very powerful group-think, which it is, but its important for young engineers to learn the conventions we use, because we will use them.  Circular, but true.  Initially teach it however your instincts recommend, but please make sure your students are comfortable with the conventional way of doing it before they leave your class.  In the best of worlds, a new engineer who is totally comfortable switching conventions is the best of all.  However, a new engineer who has an alien convention is a liability until they can be trained to group-think like the rest of us.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And that's where I leave it to you.  I'm no educator.  I'm just a guy in the field.  If you think you can train students to switch conventions in their mind without sweating it, please do so.  Minds that can switch on a whim are tremendously valuable.  If you think some of the students might have great trouble switching later on, I'd think twice (especially if you think any of those students might enter STEM fields).&lt;/p&gt;&#xA;" OwnerUserId="5787" LastEditorUserId="5787" LastEditDate="2015-10-30T23:06:19.887" LastActivityDate="2015-10-30T23:06:19.887" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9887" PostTypeId="2" ParentId="9814" CreationDate="2015-10-31T01:27:13.873" Score="2" Body="&lt;p&gt;I find the calculus based explanation for why we can't go into space just by bundling together a bunch of low power rockets (as it turns out ISP matters a lot!).  I found it especially neat because it was the first bit of calculus based physics that really clicked as &quot;oh, this is why we need calculus,&quot; because the chain rule shows up when calculating accelerations as the mass changes.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2015-10-31T01:27:13.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9888" PostTypeId="2" ParentId="9828" CreationDate="2015-10-31T04:06:44.617" Score="0" Body="&lt;p&gt;You can look at &lt;a href=&quot;http://www-math.mit.edu/~djk/calculus_beginners/chapter17/proof01.html&quot; rel=&quot;nofollow&quot;&gt;this short proof&lt;/a&gt;. Basically, all you need to know is that $\det$ is unchanged by adding any multiple of any row to any other row, and then use that to manipulate the original system of equations to get row $k$ being all zero except for the $k$-th entry, which is the trivial 1-dimensional case. But the big question is how students at that level be possibly expected to know and understand the determinant!&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2015-10-31T04:06:44.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9889" PostTypeId="2" ParentId="9874" CreationDate="2015-10-31T10:27:35.393" Score="-1" Body="&lt;p&gt;I am so interested in finding an answer for the first part of the question that I copy (as community wiki) my &quot;answer&quot; to this somehow the same &lt;a href=&quot;https://math.stackexchange.com/q/1504483/83875&quot;&gt;MSE question&lt;/a&gt;, hoping that it helps us to understand the roots of this particular &quot;convention&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is by no means answering the question as it is. However, It is just to give a historical piece (taken from &quot;&lt;a href=&quot;http://www.ingelec.uns.edu.ar/asnl/Materiales/Cap03Extras/Stokes-Katz.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The history of Stokes' Theorem&lt;/a&gt;&quot; written by Katz) that might come handy when thinking of the first part of the question (CW vs. CCW or ACW). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KTDH8.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KTDH8.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As it can be seen, Green's theorem as introduced by Cauchy (1846) is undecided about the orientation of the curve. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I repeat that I didn't attempt to answer the question as asked. I just tried to extend the question a bit hoping that it helps us to find the answer (if any) to the original question. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-10-31T10:27:35.393" CommentCount="0" CommunityOwnedDate="2015-10-31T10:27:35.393" ContentLicense="CC BY-SA 3.0" />
  <row Id="9891" PostTypeId="2" ParentId="9847" CreationDate="2015-10-31T12:43:37.793" Score="-1" Body="&lt;p&gt;To prove the converse of the Pythagorean theorem, you could use the cosine rule. Let's consider the 3-4-5 triangle. We have $\cos C =\frac{3^2+4^2-5^2}{2\times 3\times 4}=0$. Since $C$ is an interior angle of a planar Euclidean triangle, $0\leq C\leq \pi$. Thus $C=\frac{\pi}{2}$ as required.&lt;/p&gt;&#xA;" OwnerUserId="5791" LastActivityDate="2015-10-31T12:43:37.793" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9892" PostTypeId="2" ParentId="9874" CreationDate="2015-10-31T13:22:45.333" Score="2" Body="&lt;p&gt;Question 1 - Why: it has to do with the right-handed coordinate system (Ampere's invention). X and Y are not the only axes, there is a third one- Z. In a right handed CS, if X points to the right and Y points up, then Z must point out of the wall (towards observer). &#xA;Almost everything in physics, engineering and math relies on the right handed CS. The Cross-product X(cross)Y = +Z, Y(cross)Z = +X, Z(cross)X = +Y, whereas Y(cross)X = -Z. &#xA;Positive moments are defined following the cross product notation: positive moment (or rotation) about Z axis goes from X towards Y axis. Take a screw for example - rotate it counterclockwise and it goes out of the wall towards the observer (positive Z direction), rotate it clockwise - it goes into the wall (negative Z direction).&#xA;Question 2 - Since this convention is ubiquitous across all the scientific fields, teaching in a left handed CS will do a disservice to students, since re-learning is more difficult than learning from scratch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS&#xA;I know at least one public clock that goes the other way around.&lt;/p&gt;&#xA;" OwnerUserId="5792" LastActivityDate="2015-10-31T13:22:45.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9893" PostTypeId="2" ParentId="9874" CreationDate="2015-10-31T13:30:07.970" Score="-1" Body="&lt;p&gt;I'm looking at the unit circle -&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BxZpn.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BxZpn.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and it seems to me that if we deemed Y to be cosine, and X, sine, i.e. flip the graph about the Y=X line, we would achieve your goal. Right? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, as Hurkyl indicates, the orientation of a 45 degree right triangle results in &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LnXq6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LnXq6.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and that would appear to be a bit disorienting to students. In the end, it's the combination of Cartesian coordinates being established as well as the 'comfortable' orientation for the 0-90 degree right angle triangles that dictated the placement. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-10-31T20:19:23.433" LastActivityDate="2015-10-31T20:19:23.433" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9894" PostTypeId="2" ParentId="9874" CreationDate="2015-10-31T16:13:21.140" Score="1" Body="&lt;p&gt;A large source of problems that trigonometry solved was astronomy. Our solar system is &quot;right handed&quot;. If you point your thumb &quot;north&quot;, your fingers will curl in the direction that the planets revolve around the sun. The orbits of most (not all) of the planets in our solar system are also right handed. It makes sense then to use the right hand rule to describe the direction in which angles increase.&lt;/p&gt;&#xA;" OwnerUserId="5144" LastActivityDate="2015-10-31T16:13:21.140" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9895" PostTypeId="2" ParentId="9847" CreationDate="2015-10-31T19:35:32.497" Score="6" Body="&lt;p&gt;I don't think I can do better than Giles answer, but here is an answer which gets the converse of PT without proving PT first:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $AB=3$, $BC=4$ and $AC=5$. Draw a point $D$ on the line segment $AC$ with $AD=9/5$ and $DC=16/5$. Then triangles $ABC$ and $ADB$ are similar, since they have the same angle at $A$ and proportional sides. So $\angle ABC = \angle ADB$. Similarly, $\triangle ABC \sim \triangle BDC$ so $\angle ABC = \angle BDC$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have $\angle ADB+\angle BDC = 180^{\circ}$ (a straight line), so $\angle ABC = 90^{\circ}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this can be adapted to any triangle with $a^2+b^2=c^2$. (Mark a point dividing the hypotenuse into $a^2/c$ and $b^2/c$.)&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2015-10-31T23:32:39.707" LastActivityDate="2015-10-31T23:32:39.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9896" PostTypeId="2" ParentId="9874" CreationDate="2015-11-01T05:05:25.643" Score="11" Body="&lt;p&gt;There is no such thing as a natural sign/direction convention until long after the fact.  Consider another answerer's comment that anyone using the left hand rule to construct the cross product of two vectors would be mistreated.  Of course, the left hand rule is exactly the correct rule to use for the path of electrons in a magnetic field?  Why?  Because the convention of positive and negative charge was set before anyone knew that the mobile charge carriers had been called &quot;negative&quot;.  Thanks, Ben Franklin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consequently, conventions are made and the ones that achieve critical mass succeed and overwhelm the alternatives.  Frequently, it is the ones that &lt;em&gt;can&lt;/em&gt; be described as somehow natural.  In this context:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I teach my trigonometry students that all three pairs of (cos, sin), (adjacent, opposite), and ($x$,$y$) are in alphabetical order.  &lt;/li&gt;&#xA;&lt;li&gt;I teach them that angles are measured from the positive half of the first ($x$-) axis in the direction of the positive half of the second ($y$-) axis.&lt;/li&gt;&#xA;&lt;li&gt;Reference triangles have two points on the $x$-axis, in fact, their first two points.  That is, start at the origin with the first point, go along the first axis to the second point, then turn at right angles and go parallel to the second axis to the third point.  For some students, it even helps to see this as a sequence: &quot;$(0,0), \xrightarrow{x}, 1, \xrightarrow{y} 2$&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Later, I explain to them that bearings are different and for a very practical reason:  bearings are measured from the direction compasses point (North).  I also warn them that there are many, many mismatched conventions.  (Nearly every &quot;reasonable&quot; reference vector has been used in a navigation system at some point in the past 600 years).  Although North is a common reference vector now, for astronomy and stellar navigation, it was common to use South as the reference vector, as it was then zero at the same time that the hour angle of a star was zero.  Then, once one is in this mode of comparing bearings to times and star positions, it is only sane to measure angles clockwise so that positive offset times correspond to positive offset angles.  That is, the conventions of navigation in the past were based on the need to have an easily interpretable table of stars and times, but the conventions of navigation in the present are based on the underlying convention that North is the basis of bearing measurements (which is largely a consequence of gradually narrowing cartographic convention in the last half of the second millenium).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, none of that explains why angles are measured up from right.  Early applications of trigonometry were to astronomy and navigation (see prior paragraph) and architecture.  In architecture, the problem is to pile up building materials (either in preparation to construction or placed in the constructing itself).  Consequently, one is measuring &lt;a href=&quot;https://en.wikipedia.org/wiki/Grade_%28slope%29&quot;&gt;grades&lt;/a&gt; of piled up or placed materials.  Then one measures angles in quadrants I or II up from the $x$-axis.  One &lt;em&gt;always&lt;/em&gt; puts the angles at the origin so that the sine and cosine actually correspond to the coordinates of the third point (projected onto the unit circle) because the practical historical problem is to measure an angle up from the horizon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can see this in Euclid's Elements, Prop. 1.  The construction of an equilateral triangle from a given line segment AB.  The point $A$ is drawn at the lower left, the segment AB proceeds to the right, and the third point is constructed above the segment AB (even though constructing it below is just as easy).  That is, the author finds it natural to place the starting segment horizontally with the starting point at the left.  (The next few triangle propositions are about isosceles triangles, which are drawn with the first point on a vertical symmetry axis -- i.e., modeling a pile of stuff.  I recall reading that this was intended, but I have no hope of dredging up a reference.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The final reason to do this is so that derivatives work in the expected way in Calculus:  a slope of zero does not correspond to a vertical tangent line.  Having to unlearn this will be a significant disservice to your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit:  (More on the Calculus reason)  Pick a point on a function at which the function is continuous and translate that point to the origin (just to make the rest of this clearer).  Positive derivatives at that point correspond to tangent lines lying in quadrants I and III.  Negative derivatives correspond to tangent lines lying in quadrants II and IV.  Further, a derivative of zero corresponds to a horizontal tangent line.  If we arrange angles to be measured in the right-handed orientation from the horizontal axis, then a zero angle coincides with a zero derivative, a positive angle with a positive derivative, and a negative angle with a negative derivative.  We can actually finish this correspondence by observing that the tangent of the angle (measured in the standard way) of a tangent line &lt;em&gt;is&lt;/em&gt; the slope of that tangent line. That is, the slope is the $\frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \tan \theta$.  And I've seen &quot;the lightbulb&quot; for students when they realize the tangent line's angle's tangent is the slope.  (I don't know that this is a valid &quot;Aha!&quot;.  There's linguistic repetition, but the ideas are really unrelated.  But some students seem to make progress with this association.)&lt;/p&gt;&#xA;" OwnerUserId="5800" LastEditorUserId="5800" LastEditDate="2015-11-01T16:02:28.383" LastActivityDate="2015-11-01T16:02:28.383" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="9898" PostTypeId="1" AcceptedAnswerId="9904" CreationDate="2015-11-01T19:53:49.313" Score="21" ViewCount="1072" Body="&lt;p&gt;&lt;strong&gt;Background information&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm currently teaching common core geometry, which assumes that a student has algebraic knowledge coming in. Clearly, we shouldn't expect students to retain everything from their algebra class before they take geometry. I've found myself reteaching the entirety of the class what slope is and how to calculate it. Ok, I can understand if about 80% of the students forgot entirely how to create a triangle on a graph and count how long its sides are. That's fine. But there are other concerning observations about some students in my classes, some of whom apparently failed algebra 1 but are being passed along into my class anyway... &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Forgetting which axis is x and which is y, and had difficulty plotting (x,y) coordinates at the beginning of the year. &lt;/li&gt;&#xA;&lt;li&gt;Not being able to determine the length of an unknown line segment -- (2) and (?) next to each other, with a total length of (11). She guessed 5.5. Even after I told her the smaller segments add up to 11. &lt;/li&gt;&#xA;&lt;li&gt;Not being able to solve the equation x * 1.2 = 24 with any amount of prompting until I literally rewrote the equation as 1.2 x = 24 (I had suggested that she do so herself, but she didn't and we had been talking about this equation together for several minutes).&lt;/li&gt;&#xA;&lt;li&gt;Not being able to determine whether a shape looks like it has been rotated or reflected, even though they demonstrate knowledge of what the words &quot;rotate&quot; and &quot;reflect&quot; mean (spatial skills). &lt;/li&gt;&#xA;&lt;li&gt;Could not graph an equation in the form y=mx+b at the beginning of the year, and still require prompting and guidance to do so.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I guarantee that some of the students mentioned above will have extreme difficulty in geometry class. Not only do we use algebra in most sections now that it is &quot;common core,&quot; but some of them are missing the &lt;em&gt;arithmetic&lt;/em&gt; reasoning that they need for geometry. Several failed algebra 1, but passed the &quot;EOC&quot; exam, so they got placed in my classes anyway. They will struggle to pass my class as much or more than they struggled to pass algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Main question(s)&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;At what point is it a disservice to pass a student into a class they are not ready for? &lt;/li&gt;&#xA;&lt;li&gt;How does someone determine the prerequisite knowledge for a math class while recognizing that the knowledge that would be best for students to have is not necessarily the knowledge they will enter with?&lt;/li&gt;&#xA;&lt;li&gt;Is having students repeat low-level math classes like algebra &lt;em&gt;before moving on to geometry&lt;/em&gt; beneficial, or should students be able to succeed at common core geometry without algebraic background?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2015-11-02T00:25:07.900" LastActivityDate="2015-11-02T17:14:10.977" Title="At what point is it a disservice to pass someone on to the next math class?" Tags="|geometry|struggling-students|prerequisites|common-core|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="9901" PostTypeId="2" ParentId="122" CreationDate="2015-11-02T07:16:08.030" Score="5" Body="&lt;p&gt;The introduction of calculators has turned calculus, algebra, and trigonometry into &quot;which buttons do I push&quot; and questions like &quot;Why am I getting 0.8509 for the sine of 45 degrees?&quot; It is, probably unintentinally, de-emphasizing the importance of theoretical mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sine of 45 degrees is 0.7071 to four significant digits. This is not the same thing as saying that the sine of 45 degrees is 0.7071. There are times and places where that distinction is important. Students should understand that.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Addendum&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Recently I was tutoring a student in geometry. She had been given the graph of a line (with two &quot;nice&quot; points indicated) and was asked to find the equation of the line. She came up with $y=2x+5$. I asked her to look at the line and tell me what the slope was. Using the two points, she came up, correctly, with $-2$. I then asked her what was wrong with her answer. It took more than a few heartbeats for her to figure out what was wrong. Would it be right to argue that she could have plugged the two coordinates into Wolfram Alpha and gotten the equation of the line, so why bother teaching her analytical skills? There is a mathematical difference between $1.414$ and $\sqrt 2$. It's the job of a mathematics student to understand that at more than a superficial level.&lt;/p&gt;&#xA;" OwnerUserId="5144" LastEditorUserId="5144" LastEditDate="2017-01-28T16:20:13.167" LastActivityDate="2017-01-28T16:20:13.167" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9902" PostTypeId="1" CreationDate="2015-11-02T11:06:02.893" Score="8" ViewCount="1140" Body="&lt;p&gt;This is to some extent a cross-posting from English Language &amp;amp; Usage. &lt;a href=&quot;https://english.stackexchange.com/questions/284303/how-do-you-pronounce-xth&quot;&gt;How do you pronounce “xth”?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am asking a slightly different question -- but only slightly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was attempting to offer ways of dealing with the difficulty of pronouncing 'xth' as in &quot;Please take note of the xth element in the series.&quot; This can be tricky, not only for learners of English but even many native speakers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It occurred to me that this must be bread and butter to maths educators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What is the correct/customary pronunciation of 'xth'?  (Is it eksth)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In general are all such terms pronounced as the letter name followed by a 'th' sound. For example is 'wth' pronounced 'double-youth'?  Would 'Δth' be pronounced 'deltuth' etc.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What advice do you offer to someone who simply cannot pronounce 'xth' because they find it an impossible tongue-twister. How can they make themselves understood?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5798" LastEditorUserId="-1" LastEditDate="2017-04-13T12:38:45.993" LastActivityDate="2015-11-02T21:42:42.830" Title="Correct pronunciation of 'xth' (and workarounds for those who find it a tongue-twister)" Tags="|terminology|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9903" PostTypeId="2" ParentId="9902" CreationDate="2015-11-02T12:10:34.450" Score="11" Body="&lt;p&gt;For things in a list, in my experience it is very uncommon to have things indexed by letters other than m, n, i, j, k and sometimes r. Letters whose names end in a vowel sound (like i, j, k) and letters whose names end in a glide (like m, n, r) offer most people no particular problems with pronouncing the &quot;-th&quot; construction. This even goes for unusual ones like &quot;wth&quot; which I would pronounce as &quot;double youth&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I do come across the &quot;-th&quot; construction with letters whose names end in consonants (f, h, s, x, z), I often pronounce them with a small gap between the letter and its &quot;th&quot;. So &quot;xth&quot; is &quot;eks..th&quot; and &quot;zth&quot; is &quot;zed..th&quot;. Note that it's an actual moment of silence, not a vowel sound, so the &quot;th&quot; is not &quot;eth&quot; or &quot;uth&quot; or &quot;ith&quot;, but just the &quot;th&quot; sound all by itself. Think saying &quot;x thin&quot; but just stopping at the &quot;th&quot; without saying the &quot;in&quot;. It sounds very strange to my ear to say something like &quot;eksuth&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In many situations, I avoid the &quot;-th&quot; construction altogether and just say &quot;element x&quot;, or &quot;interval number i&quot;, or &quot;the number at position j&quot;. It takes more to say and write, but is unambiguous and perfectly pronounceable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is especially true for other situations where you might use a &quot;-th&quot; construction such as powers or fractions - most people I know would avoid the &quot;-th&quot; and use an alternative. For example, instead of &quot;The xth power of 2&quot;, say &quot;2 to the power of x&quot;; instead of &quot;two mths&quot;, say &quot;two over m&quot;. Roots are a different thing and the only natural way I can think to pronounce &quot;$\sqrt[x]{2}$ is &quot;the xth root of 2&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I actually think this is the perfect fuel to talk to students about an often underappreciated aspect of mathematical writing -- reading it aloud. In other disciplines you are encouraged to read your own writing aloud, and I think this is still useful advice for maths. The existence of easy-to-write-but-difficult-to-pronounce constructions might just help students consider the needs of their reader a little.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-11-02T13:30:00.060" LastActivityDate="2015-11-02T13:30:00.060" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9904" PostTypeId="2" ParentId="9898" CreationDate="2015-11-02T13:53:47.723" Score="7" Body="&lt;p&gt;Social promotion is an interesting topic, and I know my personal views are unpopular in my school. So they probably will be here too, but I'll go ahead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Due to developmental issues with children I think that social promotion is acceptable before high school. It should be coupled with remediation, but I think there are too many issues that can arise from having a contingent of older students in lower grades. I think that it is part of our responsibility in high school to end social promotion. It is our responsibility to hold students to a higher standard and should start holding them back is they do not meet expectations. Making them repeat courses or take alternative tracks. I have taught a lot of first year classes, and I have come to expect a wide range of abilities in the students sent to us and that's OK.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, what we've done within our school to deal with the ranges is a little different. We've found that meeting the minimum requirements of a class is not always a good predictor of success in a future class. So we've added that in order to take the next class in a sequence a student must have received a C or better. Students who end up below this threshold may repeat the class, and we're trying to create more avenues for students to receive remediation without repeating the whole class. Some students who do not meet the grade requirement are passed on by teacher recommendation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We've determined the standards necessary for the classes that follow and we are trying to improve the ways we can use these as objectives instead of a raw grade requirement. I think these learning objectives will eventually replace the raw grade requirement, but for now using the grade is a lot easier for guidance and parents to understand. Also it could allow us to better deal with students who generally understand the material, but for one reason or another have critical gaps in their knowledge.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2015-11-02T13:53:47.723" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9905" PostTypeId="2" ParentId="9898" CreationDate="2015-11-02T16:26:55.037" Score="5" Body="&lt;p&gt;Indeed it is futile to expect students to successfully catch up with the prerequisite mathematical understanding for a course if they are too far behind. This is because many students think that they would rather focus on getting their grades for the current course despite not having any proper foundation in prerequisite courses. (This is real and happens even in top universities.) So it is a disservice to allow students to take a class they are not ready for, because it is a waste of everybody's time (especially the poor teacher's).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certain fundamental concepts cannot be neglected. Basic arithmetic and algebra need to be there, otherwise you get the kind of problems you described in your question. I would say that if students magically pass a geometry course without being able to solve quadratic equations, then I can safely conclude that they have nearly zero understanding of geometry as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for determining what concepts are really necessary prerequisites, that is a very vague question that has to be dealt with on a case-by-case basis. I would think that even very basic Euclidean geometry requires propositional logic, arithmetic and algebra. Anything more, including dealing with degenerate cases properly (such as in Pascal's theorem and Desargue's theorem and their converses), would require a firm grasp of full first-order logic. In general, all areas of mathematics require rigorous logical reasoning, which is somehow not taught in most high-school curricula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One solution is not to force everyone to proceed through education at a fixed pace, but rather take as many or as few classes as they want to each term, and require that students are allowed to take a class only if they understand the core concepts in each prerequisite class (this may be assessed by grades as in BBS's answer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, the classes must ensure that students are not simply taught about mathematical objects and techniques, but rather taught &lt;strong&gt;why&lt;/strong&gt; one should want to know about the objects in the first place, and &lt;strong&gt;how&lt;/strong&gt; to devise the techniques &lt;strong&gt;without&lt;/strong&gt; knowing them, otherwise most students will form a justified opinion of mathematics classes that they are useless (because they don't know when to use the stuff) and meaningless (because no one ever explains to them the &lt;strong&gt;reasons&lt;/strong&gt; for what is done).&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2015-11-02T16:26:55.037" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9906" PostTypeId="2" ParentId="9898" CreationDate="2015-11-02T17:14:10.977" Score="7" Body="&lt;p&gt;There is a point at which material is so far beyond students' skills that they learn less. Imagine a freshman calculus student plopped into a graduate topology seminar: they will learn almost nothing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider &lt;a href=&quot;https://en.wikipedia.org/wiki/Mihaly_Csikszentmihalyi&quot; rel=&quot;noreferrer&quot;&gt;Csikszentmihalyi's&lt;/a&gt; flow model:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NQiUs.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NQiUs.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We'd like students to be in that upper right corner, where they are highly skilled (relative to the task) and highly challenged.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When students are so unprepared that promoting them to the next class will cause them to learn less than keeping them back, they should be held back.&lt;/p&gt;&#xA;" OwnerUserId="182" LastActivityDate="2015-11-02T17:14:10.977" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9907" PostTypeId="1" CreationDate="2015-11-02T19:54:32.067" Score="34" ViewCount="10521" Body="&lt;p&gt;I'm student teaching this semester, and so far I'm loving it! Our next section in the book teaches point-slope formula, and my cooperating teacher (a 24-year veteran teacher) is convinced that point-slope formula serves no purpose. He said he has been at a constant &quot;battle&quot; with the other teachers at the high school because there is no purpose. At first, I challenged him, and asked, &quot;what if you have a point far off from the origin?&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, the question would provide you with a point and a slope. He said you can just plug those into slope-intercept form (which he loves) and solve for b (the y-intercept). I did some googling and found that the point-slope form reappears in Calc 2 and Calc 3 with polynomials and linear approximation functions. When I showed that to him, he said, &quot;well by the time they get there, they'll just learn it new&quot;. So he's STILL not convinced that we should be teaching point-slope formula to these students! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I even challenged him by asking, &quot;isn't it important for these students to see multiple ways to write equations for lines?&quot;. No, slope-intercept is all they need to get through the class. &quot;Well, if you're given a point and a slope (say x1,y1 and m), you could plug them into the equation to get y-y1=m(x-x1) and type that straight into desmos and get a line without having to do any work&quot;. Well, my technology argument was a no-go as well. So I'm at a loss.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: After reading a few replies, I showed him how simple it is to derive the point-slope formula from the slope form m=(y2-y1)/(x2-x1), and then from there you could derive the slope-intercept form, but his response was, &quot;or they could just know slope-intercept form and use it every time&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Am I crazy in thinking that point-slope formula is important for these students in 9th grade honors algebra? How can I convince my cooperating teacher?&lt;/p&gt;&#xA;" OwnerUserId="4737" LastEditorUserId="262" LastEditDate="2015-11-10T07:01:39.243" LastActivityDate="2021-07-25T23:38:39.350" Title="Should I be teaching point-slope formula to high school algebra students?" Tags="|secondary-education|algebra|concept-motivation|" AnswerCount="13" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="9908" PostTypeId="2" ParentId="9907" CreationDate="2015-11-02T20:41:40.283" Score="36" Body="&lt;p&gt;As someone who teaches calculus to college students, I expect my students to have seen point-slope form. We just start using it (because it's the right way to talk about tangent lines and linearization) without teaching it, because we consider it part of the standard algebra curriculum, so students who haven't seen it are at a disadvantage. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, students who haven't seen the idea that there are multiple ways to write equations for lines are at a bigger disadvantage, because they have to grasp the idea that there are multiple representations, and a particular new representation, and the actual new course material, all at once.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-11-02T20:41:40.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9909" PostTypeId="2" ParentId="9902" CreationDate="2015-11-02T20:50:44.400" Score="2" Body="&lt;p&gt;I'm not a native English speaker, but I typically teach technical topics in English to international students. In this context, in addition to the problem of (my lousy) pronunciation, there is another issue. Many students have no idea that, e.g., the &lt;em&gt;n&lt;/em&gt;th element is written in that way: they might not have yet read a technical book in English or that piece of information might not have sunk in. Because of this, hearing &quot;enth&quot;, however pronounced, might not trigger any familiar word or symbol.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If I can support what I'm saying by writing on the blackboard, I say &quot;en-th&quot;, clearly separating the two parts and emphasizing the &quot;th&quot;. Meanwhile, I write on the blackboard which element I'm referring to.&lt;/li&gt;&#xA;&lt;li&gt;If I cannot write while speaking, I avoid &lt;em&gt;n&lt;/em&gt;th altogether by saying &quot;the element number n&quot; or &quot;the element n&quot;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5367" LastActivityDate="2015-11-02T20:50:44.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9910" PostTypeId="2" ParentId="9907" CreationDate="2015-11-02T20:55:13.207" Score="19" Body="&lt;p&gt;As someone else teaching calculus and higher math to college students, I use point slope form repeatedly:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In the/a definition of derivative, I use point-slope form. I have students think about &lt;span class=&quot;math-container&quot;&gt;$y_1-y_2=m(x_1-x_2)$&lt;/span&gt; rewritten as &lt;span class=&quot;math-container&quot;&gt;$f(x_1)-f(x_2)=m(x_1-x_2)$&lt;/span&gt; rewritten as &lt;span class=&quot;math-container&quot;&gt;$f(x+\Delta x)-f(x)=m(x+\Delta x-x)$&lt;/span&gt; rewritten with limits to describe the slope &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; as the derivative.&lt;/li&gt;&#xA;&lt;li&gt;In linear approximation, I likewise use point-slope form.&lt;/li&gt;&#xA;&lt;li&gt;I make my students apply these ideas using specific functions, like &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^p$&lt;/span&gt;. Yes, it's just applying the definition of derivative but I make them draw pictures to see that it's the point-slope form of a line coming up again.&lt;/li&gt;&#xA;&lt;li&gt;In discussing the fundamental theorem of calculus, I use point-slope form.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I tell my first-semester calculus students that if they have point-slope form down cold they can probably pass the whole class. I also tell them that it's the beautiful and lazy way to write down the equation of a line, and I don't want to see slope-intercept for the first five weeks. If they still love slope-intercept by the end of five weeks they can go back... but most don't :)&lt;/p&gt;&#xA;" OwnerUserId="5810" LastEditorUserId="8571" LastEditDate="2018-12-27T14:54:28.997" LastActivityDate="2018-12-27T14:54:28.997" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9911" PostTypeId="2" ParentId="9907" CreationDate="2015-11-02T21:09:56.117" Score="11" Body="&lt;p&gt;Personally, I went many years not teaching the point-slope formula (college algebra and remedial algebra), and I just flipped on the issue this past weekend. Here's why: In making sure that I could justify all the basic graphing principles, I found that this formula was the easiest way to establish that lines can indeed be written as linear equations (restricting to defined slope for simplicity):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Theorem: Assume line L has defined slope $m$ and some point $(x_1, y_1)$. There is a linear equation such that any point on the line is a solution to the equation, specifically: $y - y_1 = m(x - x_1)$. Proof: Given any other point $(x_2, y_2)$, we know $m = \frac{y_2 - y_1}{x_2 - x_1}$ because slope is constant on a line. Check that the points are solutions to this equation: For the first point we get $y_1 - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x_1 - x_1) \Rightarrow 0 = 0$, while for the second point we get $y_2 - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x_2 - x_1) \Rightarrow y_2 - y_1 = y_2 - y_1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Likewise, I discovered that the point-slope formula was about the easiest way to get started on several of the other proofs, working through the basic graphing principles from an axiomatic basis. It was just yesterday that I rewrote all my lecture notes to include it going forward -- and this quick proof that lines are indeed linear equations, whereas for the prior decade I was just hand-waving the issue intuitively. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-11-02T21:09:56.117" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9912" PostTypeId="2" ParentId="9902" CreationDate="2015-11-02T21:23:17.257" Score="0" Body="&lt;p&gt;When using letters as variable ordinary numbers, say the letter and add th. (Example: x=ecksth, b=beeth)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't want to say that, say &quot;take note of the element in the x position in the series&quot;&lt;/p&gt;&#xA;" OwnerUserId="5811" LastActivityDate="2015-11-02T21:23:17.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9913" PostTypeId="2" ParentId="9902" CreationDate="2015-11-02T21:42:42.830" Score="-3" Body="&lt;p&gt;X would be the vowel in this word. When x is used as a vowel, it is pronounced like the letter z. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the correct pronunciation of xth would sound like: zith.&lt;/p&gt;&#xA;" OwnerUserId="5812" LastActivityDate="2015-11-02T21:42:42.830" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9914" PostTypeId="2" ParentId="9847" CreationDate="2015-11-03T00:05:39.440" Score="0" Body="&lt;p&gt;Building off of the circle construction, but physical like you request:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only possible triangle that has sides of 3, 4, and 5 must be &quot;right&quot;.  There are blocks out there (you could replicate blocks with strips of paper if unattainable) and place those down with the corners touching. (&lt;a href=&quot;https://en.wikipedia.org/wiki/Cuisenaire_rods&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Cuisenaire_rods&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take a 3 a 4 and a 5, and arrange it into a triangle.  We can't &quot;see&quot; that it's right.  But, if you continue around the circle, and create a triangle on it's opposite side, then on the opposite of the new triangle, etc... all the way around, you should end up with 4 triangles, completing a &quot;360 degree&quot; turn, and splitting it into 4 pieces... which by definition is 360/4, or a 90 degree triangle.  Proof by visuo-physical means. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(this is nothing more than the circle construction proposed by Artelius, or the Triangle solution proposed by Ebert.  The only difference is, instead of starting with a triangle, or compass and straightedge, you have the blocks which serve as lengths in a non-triangle format, which I believe is what you want.)&lt;/p&gt;&#xA;" OwnerUserId="5814" LastActivityDate="2015-11-03T00:05:39.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9915" PostTypeId="2" ParentId="9835" CreationDate="2015-11-03T00:07:30.890" Score="5" Body="&lt;p&gt;In math and physics people tend to name things similarly in different field's and sub-fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be best to stop using the term &lt;strong&gt;linear function&lt;/strong&gt; and instead call them &lt;strong&gt;&lt;em&gt;linear equations&lt;/em&gt;&lt;/strong&gt;, for polynomials of order zero or one, and &lt;strong&gt;&lt;em&gt;linear operators&lt;/em&gt;&lt;/strong&gt; [or some other term depending on what branch of science you come from], for operators where f(a+b) = f(a) + f(b).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I would explain to your students that many terms have multiple definitions (e.g. Euler's numbers) or multiple terms can refer to a single definition (Normal/Gaussian distribution). They will see this often through out their academic career. It only gets worse as you delve deeper into scholarly pursuits. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The number of things named after Euler is a good example of this pattern, where in, there are 4+  Euler numbers, a multitude of theorems, and various synonyms (Euler's formula =/= Euler's equation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler#Euler.27s_formulas&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler#Euler.27s_formulas&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5813" LastEditorUserId="5813" LastEditDate="2015-11-03T00:15:03.623" LastActivityDate="2015-11-03T00:15:03.623" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9917" PostTypeId="2" ParentId="9869" CreationDate="2015-11-03T06:47:25.453" Score="-4" Body="&lt;p&gt;As far as I see, most of you disagree, but I do not find $\frac{\infty}{\infty \cdot \infty}$ wrong. I could write it myself. It leads to the correct answer, and not by accident. If the student can not use this method in other cases, he will lose points when the exam questions include them. Here, in the worst case, I would understand noting that this notation is not very precise and requires quotation marks (or some talking about nonstandard analysis and probably using other sign than $\infty$), but myself I would just write &quot;a bit risky trick&quot; without subtracting points. Would you give the student full points if he did not write this two steps with infinity? Why punish him for showing his reasoning?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, as the final answer I would expect at least something like &quot;two-sided horizontal asymptotics y=0&quot; or &quot;2-s. H.A. y=0&quot;. The answer given does not show that the student knows that there can be two horizontal asimptotes (even $\rightarrow \pm \infty$ in the first limit could be enough) and that he knows what horizontal asymptotics is - he could just write the number that he gets from the formula. &quot;At&quot; is a good sign, so I would give 8.5/10: -1 for no signs of seeing the possibility two asymptotes and -0.5 for giving an answer that looks &lt;em&gt;rather&lt;/em&gt; like giving random number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find understending (which he does not show clear enough) and correct answers (which he gives) more important than not writing something that is against standard rules, but makes sense.&lt;/p&gt;&#xA;" OwnerUserId="4681" LastActivityDate="2015-11-03T06:47:25.453" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="9918" PostTypeId="1" AcceptedAnswerId="9919" CreationDate="2015-11-03T14:40:12.783" Score="9" ViewCount="1080" Body="&lt;p&gt;Which topics - what skillset in mathematics need the students to possess to be able to proceed with learning quantum mechanics without hitches like need for explaining notation or understanding the underlying calculations? I.e. what should a Mathematics teacher include in the curriculum to make the Physics teacher able to smoothly teach quantum mechanics?&lt;/p&gt;&#xA;" OwnerUserId="5820" LastActivityDate="2015-11-05T01:49:40.117" Title="What are the mathematical prerequisites to quantum mechanics?" Tags="|curriculum|physical-sciences|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="9919" PostTypeId="2" ParentId="9918" CreationDate="2015-11-03T16:21:14.757" Score="10" Body="&lt;p&gt;There are no clearly defined mathematical prerequisites that are needed in order to learn quantum mechanics. QM can be taught at a variety of levels. Here are some examples of levels at which it can be taught:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a very nice book by Hewitt, Conceptual Physics, that&#xA;presents a complete survey of physics for a gen ed course with no&#xA;more mathematics than very basic algebra. Hewitt does a presentation&#xA;of quantum mechanics that works nicely at that level.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In a typical freshman college calculus-based course, students learn quantum mechanics at a level that is slightly higher than what's in Hewitt, but that still omits a lot of mathematical detail. Students at this level do not know differential equations or linear algebra, so they aren't ready to deal with either the complete Schrodingger picture or the Heisenberg picture.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In an upper-division course for physics majors, one would expect differential equations and linear algebra as prerequisites.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;At the lower mathematical levels, the approach I take is that we have two basic quantum mechanical relationships, $E=h\nu$ and $p=h/\lambda$, and everything follows from that. At the lower levels, one has to work around certain difficulties. For example, at the gen ed level, it's not realistic to discuss things in terms of complex wavefunctions. Real wavefunctions work fine for standing waves, but you can't develop a completely accurate description of traveling waves without complex wavefunctions. This means that topics like the WKB approximation have to be omitted, or treated less rigorously.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As happens with many topics in modern physics, the big issue is not mathematical background but physics background. Many mathematicians do not seem to appreciate this, and imagine that they can follow a royal road to understanding advanced topics in physics, simply because they know a lot of math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A difficulty even at the upper-division undergrad level is that photons are inherently relativistic, and therefore there are foundational issues in trying to present the photon within the same framework as material particles. For example, there is no wavefunction for a photon, if you insist on all the usual axioms of QM. The only really rigorous and detailed way to deal with this is by teaching quantum field theory, but that's unrealistic for undergrads, not because they don't know enough math but because they haven't been exposed to enough physics. As a workaround, one approach is to use the heuristic of treating the electromagnetic wave as the wavefunction of the photon, but this is not rigorously correct.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I'd prefer an answer that contains a more comprehensive list - where first lessons in QM (not just &quot;descriptive&quot;/&quot;popular science&quot; but with actual mathematics behind the phenomena) could start, and what other problems would be solvable with extra additions. Nevertheless, the (stretch) goal is to reach quantum computing and ability to understand some easiest quantum algorithms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't think there's any clear distinction of the type you're imagining between a &quot;descriptive&quot; treatment and one with &quot;actual mathematics.&quot; But for the topics you have in mind, you probably don't need any QFT at all, just nonrelativistic quantum mechanics. The mathematical prerequisites are linear algebra and differential equations, and you need the ability to handle those topics over the complex numbers, not just the reals.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2015-11-03T16:33:18.737" LastActivityDate="2015-11-03T16:33:18.737" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9920" PostTypeId="2" ParentId="9907" CreationDate="2015-11-03T19:06:07.083" Score="11" Body="&lt;p&gt;I am someone whose mind seems to work similar to your cooperating teacher.  Why learn two forms of something when you can learn one and manipulate it?  I made it through all of algebra 2 never learning the quadratic formula.  Instead, I learned how to derive it by completing the square, and I started every test by re-deriving it!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think its essential that a student who learns point slope learns that it's nothing magical.  It's just solving for a variable, as you noticed.  However it has two major features:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There's less algebra, more arithmetic.&lt;/li&gt;&#xA;&lt;li&gt;Its in a form which mirrors the way a problem often reveals itself.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;From a teaching perspective, the former might be a disadvantage.  This might be what your cooperating teacher is picking up on.  The more you practice algebra, the better you are at it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, now that I am no longer a student, I find problems often present themselves in the form of a point and a slope.  Knowing that there is a quick arithmetic way to solve these sorts of equations draws my attention to them more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's also very helpful as a computer programmer.  My software can't always solve an equation for a variable, but it can do arithmetic.  I rely heavily on the point-slope formula for these situations.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2015-11-03T19:06:07.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9921" PostTypeId="2" ParentId="9907" CreationDate="2015-11-03T21:12:00.773" Score="29" Body="&lt;p&gt;Point slope form emphasizes the actual meaning of slope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Literally,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;y - b = m(x -a)&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Says&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The change in the outputs ($y-b$) is equal to the slope ($m$) times the change in the inputs ($x-a$)&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Translating between a verbal statement like this and an equation is essential.  Understanding slope is essential. Point slope form of a line is essential.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It links to transformations of functions, which should be &lt;strong&gt;the&lt;/strong&gt; main story in precalculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is, ridiculously, the first order taylor expansion of the function about $x=a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More than any technical details about this particular problem, however, this focus on one form betrays a whole way of thinking.  Namely, that we shouldn't confuse students, and we should give them dependable algorithms that always work.  From this perspective, your collogue may be correct, and only using one form may be &quot;easier for the students&quot;.  But if the goal is to get the students to think, understand, explain, explore, then they are very incorrect.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2015-11-03T21:12:00.773" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9922" PostTypeId="2" ParentId="9783" CreationDate="2015-11-03T21:27:45.100" Score="4" Body="&lt;p&gt;Consider Khan Academy: they offer high-quality ad-free math videos and exercises in English, Spanish, Portuguese, French, Turkish, and Norwegian. (The content is more limited in non-english languages.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For negative numbers, start here: &lt;a href=&quot;https://www.khanacademy.org/math/arithmetic/absolute-value&quot; rel=&quot;nofollow&quot;&gt;Negative Numbers and Absolute Value&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For scientific notation, start here: &lt;a href=&quot;https://www.khanacademy.org/math/pre-algebra/exponents-radicals/scientific-notation/v/scientific-notation-old&quot; rel=&quot;nofollow&quot;&gt;Scientific Notation&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may interest the teachers here to know that Khan Academy offers support for teachers to monitor their students work via the &quot;coaching&quot; feature, which may be also helpful for the OP as he helps these folks learn.&lt;/p&gt;&#xA;" OwnerUserId="182" LastActivityDate="2015-11-03T21:27:45.100" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9923" PostTypeId="2" ParentId="9918" CreationDate="2015-11-03T22:07:56.837" Score="6" Body="&lt;p&gt;As stated above QM can be taught and understood at a variety of mathematical levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a minimum the required concepts are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Linear Algebra &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Probability Theory&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Calculus (basic derivatives and integrals, multivariate would be even better)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Newtonian physics and vector mathematics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now all of these topics can be explored at a variety of levels also, but I would say that 1 semester at a college level for each of these is sufficient. &lt;/p&gt;&#xA;" OwnerUserId="5824" LastActivityDate="2015-11-03T22:07:56.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9924" PostTypeId="2" ParentId="9907" CreationDate="2015-11-04T04:04:29.887" Score="9" Body="&lt;p&gt;I remember being slightly giddy the first time I was told that the equation of the line with x-intercept a and y-intercept b is $\frac xa + \frac yb = 1.$ I love this equation and the generalizations it implies. My guess is that less than one percent of the algebra students in a school would be impressed with this equation. So, occassionally, I teach to the one percent. Not really for them, but because I get a kick when that one student gets that look on his/her face.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I got to thinking about this. If you can find two &quot;nice&quot; points $(x_1, y_1)$ and $(x_2, y_2)$, then the equation of the line passing through those two points is&#xA;$$\frac{x-x_1}{x_2-x_1} + \frac{y-y_2}{y_1-y_2} = 1$$ &lt;/p&gt;&#xA;" OwnerUserId="5144" LastEditorUserId="5144" LastEditDate="2015-11-05T15:06:28.767" LastActivityDate="2015-11-05T15:06:28.767" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9925" PostTypeId="2" ParentId="9907" CreationDate="2015-11-04T07:15:17.207" Score="6" Body="&lt;p&gt;It can't be that important because I have never heard of it before (and teach at university level). I believe textbooks will have some things because they provide an easy form of question rather than because they actually matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: The more I think about this, the more I agree with your teacher, in that I think it is unhelpful to teach both given that only one is needed. What I mean is, the 'point-slope' equation is the same equation as the 'slope-intercept' equation. As such, I think it is unhelpful to teach one concept as if it is two different ones. I believe that telling students incorrectly that these are different will encourage their natural tendency to think of the equation of a line as a string of characters, rather than a mathematical statement that is true for points on the line and false otherwise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, emphasising the difference makes out that there is an important difference, when all the difference is is basic algebra. I would think this sends the message that the algebra is difficult and should be expected to be difficult (already far too wide-spread an opinion). Instead we should be teaching that the algebra is not difficult and is not the important feature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My suggestion, then, would be to teach both ways of writing the equation, showing that one or other may be more convenient depending on the information you have available, and can help in different ways to aid intuition. Emphasise that the two only differ by rearranging the formula, and that they can easily move between the two to get the viewpoint they find helpful. Also make it clear that mathematical convention is to write $y=mx+c$, and so they should give their answer in this form, whichever way they arrive at it. (Before I get objections to this last point: if a student wrote something else I would consider it laziness, and if I saw anything else in a paper without reason I would be very surprised.)&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="4746" LastEditDate="2015-11-04T09:33:21.933" LastActivityDate="2015-11-04T09:33:21.933" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="9926" PostTypeId="2" ParentId="9907" CreationDate="2015-11-04T13:41:29.227" Score="3" Body="&lt;p&gt;I have always argued for teaching it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Like others have mentioned it has a concrete connection to the slope formula. This can be a nice teaching moment on deriving formulas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) It relates to the standard form of many other types of curves. Students who have a good grasp of the different forms of lines tend to do better when we introduce the conic sections, and I think that they're recognizing some of those similarities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) It is a good stepping off point for talking about transformations. It's similar to other common forms because those forms show the transformation form the origin for that family of curves. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) It is helpful for some students who struggle with abstract concepts. When teaching this in high school, it is helpful for some students to be encouraged to &quot;get something on paper.&quot; This form requires the least specific information to get some kind of equation down on paper.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2015-11-04T13:41:29.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9927" PostTypeId="2" ParentId="5989" CreationDate="2015-11-04T18:17:21.723" Score="5" Body="&lt;p&gt;I believe St. Johns college does. They are a liberal arts school that focuses on historical texts. From their sylabus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Junior Mathematics: Calculus and its Foundations Junior Mathematics&#xA;  concerns itself with questions about the continuity of motion, the&#xA;  infinite, and the infinitesimal, which lead to a new form of&#xA;  mathematics, the calculus. The initial sequence of readings&#xA;  (Aristotle, Galileo, and Leibniz) leads to the primary text, Newton’s&#xA;  Principia, which offers a sweeping vision of the mechanical motions of&#xA;  the universe. The year concludes with Dedekind’s Essays on the Theory&#xA;  of Numbers, which attempts to establish the continuity of number and&#xA;  prompts students to revisit questions about the nature of the infinite&#xA;  and the infinitesimal. Some classes continue this inquiry with a brief&#xA;  study of Cantor at the close of the second semester.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also you should probably come up with a name other than &quot;true infinitesimal calculus.&quot; It has the stench of &quot;No true Scotsman&quot;. Perhaps reference it after the scholar that popularized it.&lt;/p&gt;&#xA;" OwnerUserId="5813" LastActivityDate="2015-11-04T18:17:21.723" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9928" PostTypeId="2" ParentId="5989" CreationDate="2015-11-04T19:40:22.273" Score="4" Body="&lt;p&gt;This does not directly address the question about where nonstandard-analysis-based Calculus is taught, nor the request for educational studies comparing the two approaches, but it &lt;em&gt;is&lt;/em&gt; one of the few (maybe only?) studies that take an epistemological perspective on nonstandard analysis:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ely, Robert. (2010)  Nonstandard student conceptions about infinitesimals.  &lt;em&gt;Journal for Research in Mathematics Education&lt;/em&gt; 41 (2), 117-146.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From the abstract:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This is a case study of an undergraduate calculus student’s&#xA;  nonstandard conceptions of the real number line. Interviews with the&#xA;  student reveal robust conceptions of the real number line that include&#xA;  infinitesimal and infinite quantities and distances. Similarities&#xA;  between these conceptions and those of G. W. Leibniz are discussed and&#xA;  illuminated by the formalization of infinitesimals in A. Robinson’s&#xA;  nonstandard analysis. These similarities suggest that these student&#xA;  conceptions are not mere misconceptions, but are nonstandard&#xA;  conceptions, pieces of knowledge that could be built into a system of&#xA;  real numbers proven to be as mathematically consistent and powerful as&#xA;  the standard system. This provides a new perspective on students’&#xA;  “struggles” with the real numbers, and adds to the discussion about&#xA;  the relationship between student conceptions and historical&#xA;  conceptions by focusing on mechanisms for maintaining cognitive and&#xA;  mathematical consistency.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Article available at &lt;a href=&quot;http://www.jstor.org/stable/20720128&quot; rel=&quot;nofollow&quot;&gt;http://www.jstor.org/stable/20720128&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-11-04T19:40:22.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9929" PostTypeId="1" CreationDate="2015-11-05T00:56:48.897" Score="6" ViewCount="194" Body="&lt;p&gt;I have a section of 8th grade engineering/mathematics. The class is meant to be a support/enrichment environment rather than instructional, they have a separate math teacher for pre-algebra/algebra.  We have been doing a lot of fun engineering projects, but i would like to work more mathematics in.  Rather than going through engineering by discipline (chemical, electrical, mechanical, etc) the class is project-based where we learn about a specific topic and then complete a project with the engineering design cycle in mind.  For example, we did a unit on water and the water crisis where students built water desalinators and DIY water filters. Then we learned about hydropower and dams and did some calculations of energy potential given the size of a waterfall and plant and compared it to how many houses it could power.  I am open to any type of math that can be worked into an engineering curriculum because I have been given freedom to write my own curriculum, which is still in progress.  Also, any one-off ideas where a little bit of engineering knowledge and a little bit of math can come together in an hour or so can be fit in between units.&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2015-11-06T01:44:59.453" LastActivityDate="2015-11-07T03:34:37.520" Title="Mathematics that can be worked into 8th grade engineering course" Tags="|secondary-education|curriculum|engineering-mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9930" PostTypeId="2" ParentId="9918" CreationDate="2015-11-05T01:49:40.117" Score="5" Body="&lt;p&gt;As for almost any course, the best person to ask is whoever is &lt;em&gt;teaching&lt;/em&gt; the following course. They are the only ones who really are in position to know what is needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, it will happen that they forget to mention some critical topic. And very, very often what they tell you is that &quot;they should know about ...&quot; when they really require the students to be intimately familiar with the application of the technique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to get a sample of exams/homework (with solutions) to work through with your informant(s).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, the smattering of introductory quantum mechanics I got as part of the undergraduate studies required at least three or four full semesters of math, in the lines of what @rushimg states. For an in-depth course it will be quite a bit more (and much physics besides).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-11-05T01:49:40.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9931" PostTypeId="2" ParentId="9929" CreationDate="2015-11-05T02:24:26.957" Score="6" Body="&lt;p&gt;Here are two activities I have used. The second does not fit in an hour, however.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Give the students maybe two different sizes and thickness of paper.&#xA;Ideally, the two are quite different.&#xA;Their goal is to design a paper airplane that flies the longest distance&#xA;when launched from a standard height.&#xA;Depending how ambitious you are, you could even brush&#xA;up against the &lt;a href=&quot;https://en.wikipedia.org/wiki/Reynolds_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reynolds number&lt;/a&gt;.&#xA;At a simpler level, if the paper sizes/weights are quite different, the&#xA;winning designs will be different. Scaling up the best lightweight design&#xA;does not yield the best heavyweight design.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/jyaYJ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jyaYJ.jpg&quot; alt=&quot;PaperAirplane&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://thekidshouldseethis.com/post/84516623332&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) This is hardly original, but I gave highschool students standard&#xA;cardboard rectangles. Their task was to cut and crease one rectangle to fold into&#xA;a single&#xA;structure&amp;mdash;no glue or tape, all one piece, gripped with slots etc.&amp;mdash;that folded over and protected a single egg. Then we dropped them from a height of about&#xA;5m to see which protected the egg from cracking.&#xA;The requirement that it be all one piece, without attachments, requires&#xA;considerable geometric ingenuity (and lots of trial and error!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't have an appropriate illustrative image. This will have to suffice:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/Z4gZT.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Z4gZT.jpg&quot; alt=&quot;Eggs&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://makezine.com/2012/09/12/new-egg-carton-design-uses-less-cardboard/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;This activity took all morning, approximately 3 hours.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-11-05T02:24:26.957" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9935" PostTypeId="1" CreationDate="2015-11-05T22:27:15.937" Score="11" ViewCount="232" Body="&lt;p&gt;I'm creating a list of &quot;gotcha words&quot; that are often used in writing proofs (particularly quantifier proofs), but frequently in more than one possible way, and that beginners frequently misuse or misunderstand.  But it seems likely that there are some I will forget to include, so I'm wondering whether anyone else has already created such a list and perhaps spent more time ensuring its completeness?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far, here are the words I've got:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;suppose&quot; is used when proving a $\forall$ statement (suppose $x$ is an arbitrary real number) or using a known $\exists$ statement (we may suppose $x$ is a real number such that $x^2=2$).&lt;/li&gt;&#xA;&lt;li&gt;&quot;assume&quot; is used like &quot;suppose&quot;, but more commonly restricted to statements rather than variables.&lt;/li&gt;&#xA;&lt;li&gt;&quot;choose&quot; is used when proving an $\exists$ statement (to show that there exists a $y$ with $xy=y+1$, choose $y = 1/(x-1)$) and using a known $\exists$ statement (we may choose an algebraic closure of the field $K$)&lt;/li&gt;&#xA;&lt;li&gt;&quot;define&quot; is used when proving an $\exists$ (define $y=1/(x-1)$)&lt;/li&gt;&#xA;&lt;li&gt;&quot;arbitrary&quot; is used when proving a $\forall$ (let $x$ be an arbitrary real number)&lt;/li&gt;&#xA;&lt;li&gt;&quot;let&quot; is used when proving a $\forall$ (let $x$ be an arbitrary real number), proving an $\exists$ (let $y=1/(x-1)$), or using an $\exists$ (let $L$ be an algebraic closure of $K$).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; The current draft of my list is &lt;a href=&quot;http://home.sandiego.edu/~shulman/quantifier-words.pdf&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="38" LastEditDate="2015-11-06T05:55:05.007" LastActivityDate="2015-11-06T12:29:45.310" Title="Words used in quantifier proofs" Tags="|undergraduate-education|proofs|logic|language-use|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9937" PostTypeId="2" ParentId="8392" CreationDate="2015-11-06T00:39:02.690" Score="3" Body="&lt;p&gt;Keith Conrad gave a presentation in 2008 that addresses your question,&#xA;e.g.:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Pell solutions lead to good rational approximations to $\sqrt{d}$:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/fhneD.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fhneD.png&quot; alt=&quot;KConrad&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(PDF download presentation from &lt;a href=&quot;http://www.math.uconn.edu/~kconrad/ross2008/pelltalkOSU.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.uconn.edu/~kconrad/&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-11-06T00:39:02.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9938" PostTypeId="2" ParentId="9935" CreationDate="2015-11-06T01:59:25.143" Score="4" Body="&lt;p&gt;(This is not exactly an answer to your question, but I hope it is useful.)&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Fix, Set&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;(These are how I understand these terms.  I'm not sure if my usage is popularly used.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Fix&lt;/em&gt; means &quot;choose an arbitrary value for this constant.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fix an $m\in\mathbb{N}$. Note that $\frac{\mathsf{d}}{\mathsf{d}x}m=\frac{\mathsf{d}}{\mathsf{d}x}(\underbrace{1+1+\ldots+1}_{m\text{ times}})=0$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Set&lt;/em&gt; means &quot;choose this specific value for this variable.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Set $r=1$ for the case of a unit circle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h3&gt;Because (since), Therefore (thus)&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Because&lt;/em&gt; ($\because$) or &lt;em&gt;since&lt;/em&gt; means &quot;given the premise.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Therefore&lt;/em&gt; ($\therefore$) or &lt;em&gt;thus&lt;/em&gt; means &quot;the logical consequence is.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Therefore_sign&quot; rel=&quot;nofollow noreferrer&quot;&gt;For example&lt;/a&gt;, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\because$ All gods are immortals. $\because$ Zeus is a god. $\therefore$ Zeus is an immortal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Therefore_sign#Related_signs&quot; rel=&quot;nofollow noreferrer&quot;&gt;Also&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To denote logical implication or entailment, various signs are used in mathematical logic: $\rightarrow$, $\implies$, $\supset$, $\vdash$, $\models$. These symbols are then part of a mathematical formula, and are not considered to be punctuation. In contrast, the therefore sign is traditionally used as a punctuation mark, and does not form part of a formula.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;More information can be found &lt;a href=&quot;https://math.stackexchange.com/q/238872/18398&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-11-06T06:15:25.870" CommentCount="7" CommunityOwnedDate="2015-11-06T06:15:25.870" ContentLicense="CC BY-SA 3.0" />
  <row Id="9939" PostTypeId="2" ParentId="9935" CreationDate="2015-11-06T05:12:35.677" Score="3" Body="&lt;p&gt;Have you looked at Charles Wells's &quot;Handbook of Mathematical Discourse&quot;?  It has quite a bit on this (and more generally on the use of language in mathematics teaching.)  &lt;/p&gt;&#xA;" OwnerUserId="5841" LastEditorUserId="77" LastEditDate="2015-11-06T12:29:45.310" LastActivityDate="2015-11-06T12:29:45.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9940" PostTypeId="1" CreationDate="2015-11-06T14:07:00.517" Score="23" ViewCount="712" Body="&lt;p&gt;Even though I studied maths, I became familiar with programming in my junior year at university. I'm indebted to the professor who encouraged me in that pursuit, as it provided me with an avenue not just for problem solving but for exploration that many of my classmates neglected. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In many K-8 math and science classrooms, coding is beginning to creep into the curriculum in the form of fun and kid-friendly applications like scratch and initiatives like Hour of Code.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any scholarly research or professional education publications that report that 9-12th grade maths classrooms are beginning to incorporate coding as a standard tool for problem solving, as the graphing calculator was in the 1990's, or that recommend that maths teachers move in this direction? &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2015-12-22T00:45:16.233" LastActivityDate="2018-07-04T17:15:22.003" Title="&quot;Calculators are so twentieth century.&quot;" Tags="|technology-in-education|education-research|computer-programs|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9941" PostTypeId="5" CreationDate="2015-11-06T20:22:54.910" Score="0" Body="&lt;p&gt;This is a special tag that is designed to be used only by the system for questions that have had all of their other tags removed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All questions in the system must have at least one tag. Normally, it is not possible to submit new or edited questions that do not have tags; the system will reject the submission until a valid tag is applied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a few unusual situations that can cause this tag to be applied. When a tag has existed for six months and only been used on one question, and no tag wiki has been created for the tag, the system deletes the tag. When the community decides that a given tag can never be appropriate &amp;#8212; for example, because it is misspelled or is off-topic &amp;#8212; moderators may remove the tag from the site &lt;em&gt;en masse&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When one of these deletions removes the last tag from a question, the system adds the &lt;a href=&quot;/questions/tagged/untagged&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;untagged&amp;#39;&quot; rel=&quot;tag&quot;&gt;untagged&lt;/a&gt; tag to that question so that the &quot;all questions must have at least one tag&quot; rule is not broken.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can help clean up some of these questions! If they're good questions, edit them to remove the &lt;a href=&quot;/questions/tagged/untagged&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;untagged&amp;#39;&quot; rel=&quot;tag&quot;&gt;untagged&lt;/a&gt; tag and add more descriptive tags. If they're not so good, use the usual tools: downvoting, flagging and voting to close/delete.&lt;/p&gt;&#xA;" OwnerUserId="4872" LastEditorUserId="4872" LastEditDate="2015-11-07T00:00:39.293" LastActivityDate="2015-11-07T00:00:39.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9942" PostTypeId="4" CreationDate="2015-11-06T20:22:54.910" Score="0" Body="This is a special tag that is designed to be used only by the system for questions that have had all of their other tags removed. Do not add this tag to existing questions, it is reserved for use by the system to identify posts with no valid tags." OwnerUserId="4872" LastEditorUserId="4872" LastEditDate="2015-11-07T00:00:45.707" LastActivityDate="2015-11-07T00:00:45.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9943" PostTypeId="2" ParentId="9940" CreationDate="2015-11-07T02:11:56.727" Score="3" Body="&lt;p&gt;You have perhaps heard of &lt;a href=&quot;https://code.org/&quot; rel=&quot;nofollow&quot;&gt;code.org&lt;/a&gt;.&#xA;They have developed&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a curriculum which teaches algebraic and geometric concepts through computer programming.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is described here:&#xA;&lt;a href=&quot;https://code.org/curriculum/algebra&quot; rel=&quot;nofollow&quot;&gt;Computer Science in Algebra&lt;/a&gt;.&#xA;Here is a bit more detail:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Algebra goes beyond just solving for &lt;em&gt;x&lt;/em&gt;, and &lt;code&gt;Code.org&lt;/code&gt; &lt;em&gt;CS in Algebra&lt;/em&gt; goes beyond this writing code. Through learning to program, students will also practive problem decomposition, clear communication through documentation, testing their own functions against example cases and input/output tables, and much more. Many of the practices employed by &lt;em&gt;CS in Algebra&lt;/em&gt; directly correlate to key concepts in the &lt;strong&gt;Common Core&lt;/strong&gt; math standards: data types teach domain and range, test cases introduce students to input/output tables, and our block based programming language allows students to write functions with manipulatives.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have no direct experience with this curricular approach myself,&#xA;but it may be an indication of a nascent beginning of&#xA;&quot;incorporat[ing] coding as a standard tool for problem solving&quot;&#xA;(to quote the OP's question).&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-11-08T01:08:13.887" LastActivityDate="2015-11-08T01:08:13.887" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9944" PostTypeId="2" ParentId="9929" CreationDate="2015-11-07T03:34:37.520" Score="2" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Make a horizontal cantilevered beam.  Place it against a horizontal vs. vertical grid.  Put progressively heavier weights on the end of it.  Trace the shape of the cantilever for each weight.  Fit a polynomial to the shape of the curve.  Notice whether the deflection is proportional to the weight.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use plastic Slinkies to illustrate S-waves and P-waves.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Confirm that the period of a pendulum really is nearly constant.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Demonstrate leverage using a balance-scale.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Demonstrate leverage using a hydraulic jack.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Drop something just dense enough that you can measure its terminal velocity.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use a Scotch Yoke, a piston, and a chart recorder to show how circular motion can be converted into a sine wave.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2015-11-07T03:34:37.520" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9947" PostTypeId="2" ParentId="9842" CreationDate="2015-11-07T12:27:34.987" Score="11" Body="&lt;p&gt;One place in math where this issue actually does come up is in defining &lt;a href=&quot;https://en.wikipedia.org/wiki/Ordinal_arithmetic#Multiplication&quot;&gt;ordinal multiplication&lt;/a&gt;. From an ordinal perspective, the ordinal $5$ is the order type $a&amp;lt;b&amp;lt;c&amp;lt;d&amp;lt;e$, the ordinal $3$ is the order type $x&amp;lt;y&amp;lt;z$, and $5 \times 3$ is &#xA;$$a_x &amp;lt; b_x &amp;lt; c_x &amp;lt; d_x &amp;lt; e_x &amp;lt; a_y &amp;lt; b_y &amp;lt; c_y &amp;lt; d_y &amp;lt; e_y &amp;lt; a_z &amp;lt; b_z &amp;lt; c_z &amp;lt; d_z &amp;lt; e_z.$$&#xA;I would definitely describe this as more like $5+5+5$ (the student's answer), than $3+3+3+3+3$ (the teacher's), although they are isomorphic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, ordinal multiplication is distributive from the left: $a \times (b+c) = a \times b + a \times c$, but not from the right: $(a+b) \times c$ need not equal $a \times c + b \times c$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see that this issue can matter, consider $2 \times \omega$ and $\omega \times 2$, where $2$ is the order type $p&amp;lt;q$ and $\omega$ is $0 &amp;lt; 1 &amp;lt; 2&amp;lt; 3 &amp;lt; \cdots$. Indeed, $\omega \times 2 = \omega + \omega$ and $2 \times \omega \neq \omega + \omega$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course (1) this is a pretty specialized topic which is very distant from elementary school math and (2) the decision about which way $\alpha \times \beta$&#xA;should be defined was arbitrary in the first place. But it is a nice example of how issues that seem like philosphical hairsplitting on a lower level can actually matter if you get far enough into math.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2015-11-07T12:27:34.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9948" PostTypeId="2" ParentId="925" CreationDate="2015-11-07T18:26:12.777" Score="5" Body="&lt;p&gt;Well, the first step to recovery is admitting that you have a problem.  Just by acknowledging that students have this misconception, you're already well on your way to resolving it.  Students are a lot less likely to make this mistake if you draw attention to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd offer 2 main explanations as to why this occurs in the first place:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The words &quot;sequence&quot; and &quot;series&quot;, in common parlance, can mean the same thing.  Nobody would distinguish between the phrases &quot;sequence of events&quot; and &quot;series of events&quot;.  Because of this, students who are either struggling or half paying attention might assume you're using two words for the same concept.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As others have pointed out, you can have fun with this and point out the absurdity of saying things like &quot;TV sequence&quot;, &quot;World Sequence of Poker&quot;, etc.  Noting that this is a misconception students have to overcome forces them to be more vigilant on exams whenever dealing with sequences or series.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Series involve sequences in a number of confusing ways:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(i) A series is, colloquially, formed by adding the terms of a sequence together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(ii) Formally, the definition still involves a sequence of partial sums.  Even though students almost never fully understand this (they rely on their informal understanding of a series as a sum of infinitely many terms, however erroneous that may be), most instructors feel obligated to at least mention the formal definition, which can cause confusion if you aren't careful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(iii) The convergence or divergence of a series $\sum{a_{n}}$ can be determined by properties of the &lt;em&gt;sequence&lt;/em&gt; $a_n$ - for example, if $\displaystyle\lim_{n\to\infty}a_{n} \neq 0$, $\sum{a_{n}}$ diverges.  This gets complicated by the fact that students confuse the limit (sum) of the series with the limit of the sequence on which the series is based.  So you have to reassure students that a series can converge if its sum is nonzero, but not if the limit of $a_{n}$ is nonzero.  Having to constantly refer to related but distinct mathematical objects and how they affect each other makes it all quite garbled.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Series are easily my favorite topic, and I taught a group with a 90% pass rate and 100% positive course evaluations - and I still struggled to get this distinction across to some students.  So, good luck with that.&lt;/p&gt;&#xA;" OwnerUserId="5141" LastEditorUserId="5141" LastEditDate="2015-11-07T18:31:52.957" LastActivityDate="2015-11-07T18:31:52.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9949" PostTypeId="2" ParentId="9842" CreationDate="2015-11-07T19:56:54.743" Score="6" Body="&lt;p&gt;You asked for an official document, and I can't give that. But I will try to speak for the teacher here. I don't agree with deducting points, but want to point out that the strategy of identifying axb with b+b+...+b (a of them) may be useful for students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics generalizes, but it is helpful if it starts with something concrete. If some of the kids think 5x3 means 5+5+5 and others think it means 3+3+3+3+3, then it may be hard for them to discuss the ideas with each other. If they all start from the same place, and are led to notice that these two different sorts of problems always end up with the same answer, then they get to discover commutativity, instead of having it forced on them. (What I remember of the 'new math' of the 60's is having to write &quot;this is true because multiplication is commutative&quot;. Gag.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Some kids will already have thought about this, and will already feel that both are the same. It will be hard for those kids to be put back in a box. With my bad memory, I would never remember which way was &quot;right&quot;. That's why they shouldn't have points deducted.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My friend wrote a great couple of blog posts referencing this. &lt;a href=&quot;http://jd2718.org/2013/12/03/a-childs-non-commutative-model-of-multiplication/&quot; rel=&quot;noreferrer&quot;&gt;His niece thought of multiplication this way&lt;/a&gt;. &lt;a href=&quot;http://jd2718.org/2014/05/24/where-my-niece-got-her-non-commutative-model-of-multiplication/&quot; rel=&quot;noreferrer&quot;&gt;He later saw the classwork behind that.&lt;/a&gt; In the comments on that post, he and I discuss the merits of this approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although my friend is in New York, the school he references at his blog turns out to be in Connecticut. For New York, on &lt;a href=&quot;http://engageny.org/resource/grade-3-mathematics&quot; rel=&quot;noreferrer&quot;&gt;this site&lt;/a&gt; you can find: Common Core Learning Standard 3.OA.1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For Connecticut, I'm not sure.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2015-11-10T02:38:26.217" LastActivityDate="2015-11-10T02:38:26.217" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9950" PostTypeId="1" AcceptedAnswerId="9953" CreationDate="2015-11-07T22:38:55.117" Score="8" ViewCount="3570" Body="&lt;p&gt;In teaching Middle School students (often around year 8 or 9), the topic of surds comes up here (I have to teach this topic) - and is often met with derision on commencement of the topic and during the topic. When asked, the students' derision is usually related to the fact this is a 'dry' subject with a perception of it being not practical and not of any use (usually with the 'when are we ever going to use this in our life?' being asked).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found that the assessment (test) results are not all that great compared to other topics - with questions about these either unanswered or not completed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a bounty of websites that have the basic tutorials as to what surds are - for example, the &lt;a href=&quot;https://www.mathsisfun.com/surds.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math is Fun page on surds&lt;/a&gt; provides a colorful, yet still dry representation of the definition&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When we can't simplify a number to remove a square root (or cube root etc) then it is a surd.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The surds have a decimal which goes on forever without repeating, and are Irrational Numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;With examples, such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{4} = 2$&lt;/span&gt; is not a surd&lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{2} = 1.4142135...$&lt;/span&gt; is a surd&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is there a practical means to teach surds?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user5799" LastEditorUserId="64" LastEditorDisplayName="user5799" LastEditDate="2019-03-17T18:17:41.477" LastActivityDate="2019-03-17T18:17:41.477" Title="How to practically teach surds?" Tags="|secondary-education|real-numbers|" AnswerCount="3" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="9951" PostTypeId="2" ParentId="9950" CreationDate="2015-11-07T23:50:36.570" Score="5" Body="&lt;p&gt;In my experience (remedial and college algebra), these topics are always resisted and show poor performance -- very analogously to the topic of fractions (irreducible divisions). One thing I do is show the proof that $\sqrt{2}$ is irrational and tell the story of Hippasus; the lesson, as Lowell put it, &quot;Thoughts that great hearts once broke for, we/ Breathe cheaply in the common air&quot;. That's my one stab at making the subject visceral and memorable. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-11-07T23:50:36.570" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9953" PostTypeId="2" ParentId="9950" CreationDate="2015-11-08T05:05:52.130" Score="5" Body="&lt;p&gt;Chapter 2 of &lt;em&gt;Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory&lt;/em&gt; (Edwards, H. M.) is dedicated to surds and their application to Euler's treatment of the problem of whether there are integers $x,y,z$ such that $x^3+y^3=z^3.$ I would look into whether this book, or the chapter in question, can be read or captured in an accessible form in the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the problem is student engagement, a lot of people are willing to go along with what you're doing if it's relevant to a story, accepting the motivations of historical characters as valid. Especially if they see the struggle from their point of view - the simplicity of the statement, accessibility to reasoning, and the centuries of perspective shifts it took to prove the thing. The chapter also deals with proof and mistakes in reasoning, which are often overlooked aspects of the nature of mathematics at that level of education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It deals with square roots of negative numbers. It doesn't assume the background of an education in complex numbers, but rather introduces the way they were manipulated in terms of &quot;this is what they did, and they did it by defining the operations this way, and it worked.&quot;&lt;/p&gt;&#xA;" OwnerUserId="727" LastActivityDate="2015-11-08T05:05:52.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9954" PostTypeId="1" AcceptedAnswerId="9959" CreationDate="2015-11-09T04:53:15.600" Score="8" ViewCount="759" Body="&lt;p&gt;I would like to convince a high school student that $i$ is a number, broadly put.  I'm not going to define what I mean by &quot;number&quot; unless he asks, but I just want to convince him that it's somehow meaningful and useful.  Part of this I've already done by showing the geometric interpretations of complex numbers as points on a complex plane with geometrically sensible and interesting interpretations of addition and multiplication.  That's probably the most important part.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I'd like to take it a step further if that's not asking too much, and I know that complex numbers started getting studied by studying cubic equations, particularly because we know that the curve $y=x^3$ must intersect any line $y=mx+b$ somewhere.  Simple suggestive graphs convince us of this.  So can someone give me an example of such a cubic equation that is most easily solved using complex numbers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not exactly sure what I have in mind for that (Will I ask him to use the complex root with polynomial division to find the other roots?  Or is there some other assignment that is perhaps easier for a high schooler to grasp?).  Any recommendations would be appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Icing on the cake is if the equation had some kind of meaning in Physics or Geometry, like modeling the volume of a certain box.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Edit: Perhaps I should elaborate with how I came to be focused on getting some kind of an example with a cubic.  I've read in some history of Math sources that the study of complex numbers arose from the study of certain cubics and the search for a general solution.  I figure, if the study of cubics is what compelled people in history to take complex numbers at least a little seriously (and then later on, take them even more seriously) then it could serve as a helpful example.  However, when I looked at the cubic and the method of solution Bombelli used to find the solutions of $x^3=15x+4$, it was clearly too complicated to give most of the high school students I work with.  So I was wondering if there was some similar type of magic that was simple enough for an introductory lesson in complex numbers.  But I'm starting to guess than the answer is no.]&lt;/p&gt;&#xA;" OwnerUserId="5680" LastEditorUserId="5680" LastEditDate="2015-11-09T17:26:27.073" LastActivityDate="2015-11-09T17:33:40.027" Title="Convincing a high schooler that $i$ is a number" Tags="|examples|complex-numbers|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="9955" PostTypeId="2" ParentId="9954" CreationDate="2015-11-09T06:39:14.460" Score="6" Body="&lt;p&gt;You probably cannot get away with having a different definition of &quot;what a number is&quot; than your student has.  Your student almost certainly has a very clear understanding of what a number is.  By this point in his education, &lt;strong&gt;your student probably defines a &quot;number&quot; to be &quot;a real number&quot;&lt;/strong&gt;.  (But he might define it as &quot;a positive integer&quot;, &quot;a non-negative integer&quot;, &quot;an integer&quot;, &quot;a non-negative rational number&quot;, or &quot;a rational number&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;You need to work with your student's understanding of what a number is.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;First,&lt;/strong&gt; you need to admit that &lt;em&gt;i&lt;/em&gt; is &lt;strong&gt;not&lt;/strong&gt; a real number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Second,&lt;/strong&gt; you can explain that &lt;em&gt;i&lt;/em&gt; is &lt;strong&gt;almost&lt;/strong&gt; a number -- it obeys &lt;strong&gt;most&lt;/strong&gt; of the rules that real numbers obey.  (Be able to show the student a list of the axioms that &lt;em&gt;i&lt;/em&gt; obeys, and the axioms that &lt;em&gt;i&lt;/em&gt; does not obey.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From an engineer's perspective, &lt;strong&gt;&lt;em&gt;i&lt;/em&gt; is useful for taking derivatives and integrals of sinusoidal functions&lt;/strong&gt;.  Specifically, we can pretend that sinusoids are exponential functions (with complex powers), and use the (relatively easy) method of taking derivatives (or integrals) of an exponential instead of the (relatively hard) method of taking the derivative of a sinusoid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, the key concept you should show them is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The slope of e^x = e^x.&lt;/li&gt;&#xA;&lt;li&gt;The slope of e^ax = a·e^ax.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;The slope of the slope of e^ax = a·a·e^ax.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;The slope of sin ax = a cos ax.&lt;/li&gt;&#xA;&lt;li&gt;The slope of cos ax = -a·sin ax.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;The slope of the slope of sin ax = -a·a·sin ax.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;Engineers need to do lots of calculations using exponentials and sinusoids.  For example, alternating current has sinusoids, repeating waves can be thought of as sums of sinusoids, &lt;em&gt;et cetera&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Engineers need to do lots of calculations using slopes of functions, and slopes of slopes of functions.  (distance -&gt; velocity -&gt; acceleration, momentum -&gt; force, &lt;em&gt;et cetera&lt;/em&gt;)&lt;/li&gt;&#xA;&lt;li&gt;That formula for calculating the slope of an exponential is awfully nice.  Wouldn't it be nice if we could use a similar formula for the sines and cosines?&lt;/li&gt;&#xA;&lt;li&gt;We can, &lt;strong&gt;if&lt;/strong&gt; we can &lt;strong&gt;imagine&lt;/strong&gt; something like a number that is the square root of minus one.&lt;/li&gt;&#xA;&lt;li&gt;Thus, &lt;em&gt;i&lt;/em&gt;.  We define &lt;em&gt;i&lt;/em&gt; to be that missing square root of minus one.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some of the cool features of complex numbers (such as &lt;strong&gt;Euler's Rule, &lt;em&gt;e^πi&lt;/em&gt; = -1,&lt;/strong&gt; and &lt;strong&gt;rotation tricks&lt;/strong&gt;) can be derived from the idea that a sinusoid is an exponential with an imaginary power.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rotation tricks imply that the &lt;em&gt;i&lt;/em&gt; axis is orthogonal to the real number line.  Thus, the &lt;em&gt;a + bi&lt;/em&gt; notation was an early way of expressing two-dimensional vectors.  This was further extended to the &lt;em&gt;a + bi + cj + dk&lt;/em&gt; notation for expressing three- and four-dimensional vectors.  Treating a = 0 results in the i-axis being the x-axis, the j-axis being the y-axis, and the k-axis being the z-axis.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-11-09T06:56:07.230" LastActivityDate="2015-11-09T06:56:07.230" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="9956" PostTypeId="2" ParentId="9954" CreationDate="2015-11-09T15:01:50.603" Score="9" Body="&lt;p&gt;If I were faced with this situation, I'd have a discussion with the student about what a number is. We believe that 1,2,3,4,… are numbers. It is a stretch to say that 0 is a number…which is why it took people so long to think about using it as a number. Think about it…0 behaves VERY DIFFERENTLY from any of the other familiar numbers. Admitting it &quot;numberhood&quot; is very strange. After this, why do we say -2 is a number? That's also bizarre. One may say that subtraction is the root of both of these problems…the point is that certain abstract properties of numbers are still seen if one generalizes the notion of addition and subtraction to include these new entities. The extended notion is a generalization of familiar properties to a larger class of objects. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, to me the problem with i is not that it is so terribly exotic. The problem is that we are too familiar and comfortable with 0 and negative numbers to realize how bizarre they are in comparison to the positive integers. If you can clarify the precise reasons why the non-positive integers are strange, i ought to be a bit more acceptable.&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2015-11-09T15:01:50.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9957" PostTypeId="2" ParentId="9954" CreationDate="2015-11-09T15:18:09.577" Score="4" Body="&lt;p&gt;The formula (in terms of radicals) for roots of cubics generically can involve square roots of negative numbers, even when the coefficients are real and the three roots are real. This is the &quot;casus irreducibilus&quot; or something similar. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, in contrast, one could simply declare $x^2+1=0$ to have no solutions at all, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A little more complicated: to understand the periodicity of the &quot;sine&quot; function defined indirectly by $x=\int_0^{{\rm arcsin}\, x} {dt\over \sqrt{1-t^2}}$, it is simplest to understand the ambiguity of this integral, up to path integrals enclosing &lt;em&gt;both&lt;/em&gt; the $\pm 1$. These are integer multiples of $2\pi$, by complex analysis. Thus, the sine function itself, the inverse function, is $2\pi$-periodic. (This is in analogy with treatment of the doubly-periodic elliptic integrals...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more sophisticated example is Riemann's Explicit Formula, simplified by von Mangoldt, and generalized by Guinand, Weil, and others, giving asymptotics for various expressions involving &lt;em&gt;primes&lt;/em&gt; in terms of the (non-real!) zeros of zeta functions and L-functions. If those zeros were &quot;not real&quot;, or were somehow &quot;optional&quot;, then these formulas would degenerate enormously.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-11-09T15:18:09.577" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9958" PostTypeId="2" ParentId="9954" CreationDate="2015-11-09T16:15:49.250" Score="11" Body="&lt;p&gt;One of the historically compelling arguments for complex numbers was that they can be used to find &lt;em&gt;real valued&lt;/em&gt; solutions to polynomials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's a nice discussion on &lt;a href=&quot;http://www.clarku.edu/~djoyce/complex/cubic.html&quot;&gt;this site&lt;/a&gt;.  For example, the cubic $x^3=cx+d$ has, in general, a solution&#xA;$$x=\sqrt[3]{d/2+\sqrt{(d/2)^2-(c/3)^3}}+\sqrt[3]{(d/2)-\sqrt{(d/2)^2-(c/3)^3}}.$$&#xA;(This formula can be verified by plugging in and doing a whole bunch of algebra, while completely ignoring whether or not $e=(d/2)^2-(c/3)^3$ is negative or not.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you try this with $x^3=15x+4$, you end up with $(d/2)^2-(c/3)^3=-121$, but&#xA;$$x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}=4,$$&#xA;and you can easily check that this is a root.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a calculation which goes through the imaginary numbers, but ends up giving you a real number which actually works.  This is like how one detects things in physics that can't be directly observed: the formula &lt;em&gt;acts like&lt;/em&gt; the complex numbers really do exist.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-11-09T16:15:49.250" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9959" PostTypeId="2" ParentId="9954" CreationDate="2015-11-09T16:55:22.430" Score="9" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I just want to convince him that it's somehow meaningful and useful.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is probably going to require coming up with an example that is applied to something in the real world, not just an example that is applied to some abstract topic in mathematics (roots of polynomials, or some of the more advanced examples referred to in Paul Garrett's answer). You only need one example, but it has to be a convincing one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example I would pick is addition of sinusoidal functions. It is very easy to come up with real-world examples of this. If you want something very concrete (these are high school students, after all), you could make something out of mechanical linkages. Here is an animation of a slider-crank linkage: &lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/0/01/Slider_Crank_animation.svg&quot;&gt;https://upload.wikimedia.org/wikipedia/commons/0/01/Slider_Crank_animation.svg&lt;/a&gt; . If you tie three of these together in series, you have a very concrete example of why you would want to take an expression such as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\cos(t)+3\cos(t+\pi/2)+\sqrt{2}\cos(t+\pi/4)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and simplify it into the form&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A\cos(t+\delta).$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course it's not totally obvious why you'd want to link together three such linkages in this way, but it's not implausible. For a more compelling motivation, you could use diffraction of a wave through three slits. However, this may be not as concrete and easy for high schoolers to get, since it lies outside their experience. Probably just showing a video of three-slit diffraction in a ripple tank would be enough motivation for a problem of this form. (I've been looking around for such a video for a while, and would be interested if anyone knows of one.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So assuming you have the motivation, I think you could go ahead and solve the problem simply by using trig identities, and make the point that it's horribly nasty and grotty. Then you show them that the whole problem can be solved in a single line by writing down the sum of the amplitudes expressed as complex numbers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(1)+(3i)+(1+i).$&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2015-11-09T16:55:22.430" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9960" PostTypeId="1" CreationDate="2015-11-09T16:58:19.947" Score="16" ViewCount="1299" Body="&lt;p&gt;I was introduced to the SMSG math curriculum at Topeka High School between 1965 and 1966. my recollection (somewhat defective for medical reasons) was that the Topeka (KS) school system rolled the curriculum out across all grade levels the same year. Our teacher went to a summer class to prepare for the new material. He was also an athletics coach and taught math as his academic job. The students had not come up through the curriculum and the teacher had no clue. Some of the brightest students in the class were confused, the rest were lost. &#xA;The next summer, I tried to get my hands on the other textbooks to attempt to catch up. No joy!&#xA;The curriculum was abandoned. I have read some stories on the Internet that suggest that some students who experienced more of the curriculum and its introduction and exploration of math concepts went on to enjoy and flourish in the study of math and astrophysics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is the one in the title: Did any school districts actual teach the curriculum as planned and what were the results for the teachers and students?&lt;/p&gt;&#xA;" OwnerUserId="5565" LastEditorUserId="29" LastEditDate="2015-11-13T14:33:53.770" LastActivityDate="2023-03-25T21:44:24.183" Title="SMSG: Did any school districts actual teach the curriculum as planned and what were the results for the teachers and students?" Tags="|secondary-education|history|new-math|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="9961" PostTypeId="2" ParentId="9954" CreationDate="2015-11-09T17:33:40.027" Score="5" Body="&lt;p&gt;Ignoring the specifics of the question to concentrate on the title, I would explain that negative numbers, rational numbers, real numbers and complex numbers all answer the same need with respect to the precious number set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have non-negative integers, and you know how to add things. Then sometimes you need to substract, which exactly amounts to solve a certain equation: $a-b$ is the number x which solves $b+x=a$. Except sometimes such a number does not exist (in the nonnegative integers). At some point, one just introduces new numbers as solutions to such equations, and beautifully it all makes sense: with little adaptation, all previously known rules (distributivity, etc.) still holds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once one is used to work with possibly negative integer, the need for rational comes as soon as one wants to divide any number by (almost) any other number. We just introduce new numbers to ensure all equations $xb=a$ have a solution (when $b\neq 0$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then one needs to enlarge the set of numbers when one realizes that $\sqrt{2}$ is not a rational, i.e. there is no number (in the rationals) which solves $x^2=2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Introducing $i$ is not worse: we just want to have a set of numbers where $x^2=-1$ also have a solution. Very beautifully, just adding solutions to this equations gives solutions to &lt;em&gt;all&lt;/em&gt; polynomial equations! This is an amazing theorem, when you think of it that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not exactly how things occurred in the history of maths, but it makes a lot of sense.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-11-09T17:33:40.027" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9962" PostTypeId="1" AcceptedAnswerId="9965" CreationDate="2015-11-09T19:24:42.340" Score="33" ViewCount="5647" Body="&lt;p&gt;This week I'm going to give a talk on fiber bundles, and I found myself with an unexpected problem. Since I'm not using slides, I'll need to draw a Möbius band on the blackboard. Usually what I do is simply draw a rectangle with some arrows to indicate the identification, but I think I could do better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a method that can help me to draw a good Möbius band on the blackboard? I'm looking for something that looks good and is easy to draw.&lt;/p&gt;&#xA;" OwnerUserId="5855" LastEditorUserId="6166" LastEditDate="2019-02-23T05:42:15.663" LastActivityDate="2019-02-23T05:42:15.663" Title="What is a good method for drawing a Möbius band on the blackboard?" Tags="|undergraduate-education|topology|board-use|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="9964" PostTypeId="2" ParentId="9962" CreationDate="2015-11-09T20:57:21.730" Score="12" Body="&lt;p&gt;I searched google images and found many nice renditions.&#xA;Here one that you may prefer from this &lt;a href=&quot;http://www.math.lsa.umich.edu/~rlg/images/mathforkidsimages/moebiusstrip.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Nb6HU.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Nb6HU.jpg&quot; alt=&quot;Valid XHTML&quot;&gt;&lt;/a&gt;&lt;br&gt;&#xA;&lt;sub&gt;(source: &lt;a href=&quot;http://www.math.lsa.umich.edu/~rlg/images/mathforkidsimages/moebiusstrip.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;umich.edu&lt;/a&gt;)&lt;/sub&gt;  .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I especially like that it shows the width of the paper and doesn't draw the strip as a line or piece of string.&#xA;Having the students make them is always a hit.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="6053" LastEditDate="2019-02-22T14:40:41.013" LastActivityDate="2019-02-22T14:40:41.013" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9965" PostTypeId="2" ParentId="9962" CreationDate="2015-11-09T22:23:02.830" Score="60" Body="&lt;p&gt;Draw the bottom three-quarters of an oval:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/9mZHk.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9mZHk.png&quot; alt=&quot;mobius step 1&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Flesh that out to make the bottom half of the strip:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/9EKXF.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9EKXF.png&quot; alt=&quot;mobius step 2&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Connect one of the open ends at the top to the bottom on the other side:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/EJLpQ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EJLpQ.png&quot; alt=&quot;mobius step 3&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now draw a straight line across the top:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/CvYKa.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CvYKa.png&quot; alt=&quot;mobius step 4&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally fill in the last edge at the back:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/WHOpE.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WHOpE.png&quot; alt=&quot;mobius step 5&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An advantage to this approach is that it highlights the fact that the mobius strip starts out as an ordinary strip, but has a twist in one spot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note I figured out these steps by making various physical mobius strips with very thin width relative to their length, and manipulating them so that they were as loop-like as possible. One of the key features is the straight line at the top. That is due to how a curve in paper looks from the side -- same as at the sides of the picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I echo everyone else's sentiments: you should bring strips of paper so that everyone can make a real physical mobius strip of their own. There's nothing like seeing the physical real object to support the imagination!&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-11-10T01:45:15.977" LastActivityDate="2015-11-10T01:45:15.977" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="9966" PostTypeId="2" ParentId="9962" CreationDate="2015-11-09T22:48:11.497" Score="8" Body="&lt;p&gt;Since I use Seifert's algorithm to construct surfaces a lot, I tend to go for the flat approach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8WBAQ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8WBAQ.png&quot; alt=&quot;flat Mobius band&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-11-09T22:48:11.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9968" PostTypeId="2" ParentId="9962" CreationDate="2015-11-10T05:24:04.407" Score="5" Body="&lt;p&gt;Instead of drawing on the blackboard, use a 3D model and pass it around when you're finished using it for demonstration.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the class is small enough, you can bring in a strip of paper for each student to make his/her own. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The demonstration will be much more powerful this way and the lesson more likely to stick.&lt;/p&gt;&#xA;" OwnerUserId="5861" LastActivityDate="2015-11-10T05:24:04.407" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9969" PostTypeId="1" AcceptedAnswerId="9970" CreationDate="2015-11-10T11:41:08.710" Score="1" ViewCount="5718" Body="&lt;p&gt;[&lt;em&gt;This may be the wrong SE for this question. But as a non-mathematician I feel it may be too simple for MathOverflow, and that I might benefit from a school-level explanation. Also, it might make a fun seasonal class assignment.&lt;/em&gt;]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If everyone in a classroom is involved in a Secret Santa style present exchange then, intuitively, it seems to make sense that it requires an even number in order to be fair. Someone gives a present, another receives: that's a pair. If there's an odd number in the class, someone will lose out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Except they won't. Let there be three people in the class: pupils A, B and C. A gives to B, B gives to C, C gives to A.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of &quot;common sense&quot; thinking this appears to be a paradox. Obviously it isn't. Can someone explain why not, mathematical terms?&lt;/p&gt;&#xA;" OwnerUserId="833" LastActivityDate="2015-11-21T06:06:49.383" Title="The secret Santa present-swapping paradox" Tags="|secondary-education|" AnswerCount="3" CommentCount="4" ClosedDate="2015-11-23T22:07:31.497" ContentLicense="CC BY-SA 3.0" />
  <row Id="9970" PostTypeId="2" ParentId="9969" CreationDate="2015-11-10T14:24:20.493" Score="6" Body="&lt;p&gt;If someone gives a present and another receives it, they are indeed a pair.  If there are n pairs, it would seem there are 2n students.  The problem with that is the pairs overlap and every student is in two pairs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is shown in your counterexample.  A gives to B, B gives to C, and C gives to A.  There are 3 pairs and 3 students.  The can be generalized to n pairs and n students. For example if there are 5 students: A gives to B, B gives to C, and C gives to D, D gives to E and E gives to A.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;IF n is odd, the number of pairs will be odd. Thus there can be an odd number of students and an odd number of pairs.  &lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-11-10T14:24:20.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9971" PostTypeId="1" CreationDate="2015-11-10T20:08:38.137" Score="12" ViewCount="466" Body="&lt;p&gt;Warning: a lot of this post borrows heavily from education theory. I'm in my student teaching semester right now, so a lot of what I explain is taken from research papers and things like that. So how realistic it all is, I don't know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been thinking a lot recently about different ways of running a secondary mathematics classroom. I've heard some teachers have great success with a flipped model, and some teachers hate it. Here is my current understanding how it all works:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students are expected to go home and watch a video and do some problems as homework a couple times a week. Then when they get to class, the time is reserved for reviewing the videos and homework, and then spending the rest of the time doing projects and applying the knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In theory, it sounds great. Less lecturing in the classroom, and more project-based instruction. But I see a few drawbacks as well:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can see students not wanting to do homework that is &quot;new&quot; to them. Students are used to getting problems in class, and very similar problems for homework. This is a problem in mathematics education, but it is probably the most prevalent model used. Come to class, review homework, see new problems, go home, do problems similar to class for homework, rinse and repeat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another issue I see with a flipped classroom model is that it doesn't really allow students to discover the math on their own. Sure, they're learning new material on their own at home via some video and some problems, but it's really not that different from an in-class lecture. In my mind, an ideal mathematics education would have students seeing real problems, and create an atmosphere where the students are asking the questions, and &quot;reinventing&quot; the mathematics required to solve the problem. (This concept is known as RME - Realistic Mathematics Education and/or IBL - Inquiry Based Learning).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It almost seems as if the ideal flipped classroom would be backwards. What I mean is that students begin with projects in class, and then their homework is the lecture which &quot;formalizes&quot; what they discovered in class. This borrows from the idea of something called &quot;The Iceberg Model&quot; or &quot;Progressive Formalization&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would love to hear what you guys think about this backwards flipped classroom idea, and what you think the ideal classroom would be like. How is class time spent? What is for homework? What does the week look like for a student?&lt;/p&gt;&#xA;" OwnerUserId="4737" LastEditorUserId="688" LastEditDate="2015-11-13T21:19:47.370" LastActivityDate="2015-11-16T02:48:47.033" Title="What are your opinions of a flipped classroom at the secondary level?" Tags="|mathematical-pedagogy|secondary-education|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="9972" PostTypeId="2" ParentId="9950" CreationDate="2015-11-10T20:16:30.900" Score="1" Body="&lt;p&gt;IF you have time (which you may not), I recommend doing something with continued fractions and surds.  There is a &lt;a href=&quot;https://en.wikipedia.org/wiki/Periodic_continued_fraction#Reduced_surds&quot; rel=&quot;nofollow&quot;&gt;nice theorem&lt;/a&gt; (which you won't have to prove or even state) that quadratic surds and repeating continued fractions are nicely related.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your context, of course, you wouldn't say that or mention Galois.  But creating an experimental worksheet where they use their calculators to create some continued fractions (this is actually nowhere near as hard as it sounds) out of e.g. $\sqrt{5}$, $\sqrt{10}$, $\sqrt{17}$, $\sqrt{26}$ will lead to nice patterns.  So at least they see something pretty and maybe that helps with motivation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Going the other direction, from a repeated continued fraction to a quadratic surd, would be a lot harder because to have that make sense you need things like $$1+1/(1+1/(1+1/(1+\cdots)))=1+1/(itself)$$ which invokes all kinds of nasty infinitude.)&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-11-10T20:16:30.900" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9973" PostTypeId="2" ParentId="9971" CreationDate="2015-11-11T03:19:07.707" Score="11" Body="&lt;p&gt;From this statement&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I've been thinking a lot recently about different ways of running a&#xA;  secondary mathematics classroom.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and that you said you are student-teaching, it sounds like you want to try out a flipped classroom with your students but are hesitant because of your lack of experience.  In that sense, I read your question as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Should I flip my classroom?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now, I do not know anything about your school, area, student demographics, etc. but the first thing that you need to ask yourself is &lt;strong&gt;&quot;Why do I want to flip the classroom?&quot;&lt;/strong&gt; Maybe your students are completing their work but not performing well on tests.  Maybe they are doing well with mechanical tasks but they are struggling with conceptual understandings. Or maybe nothing is working at all.  Either way, you should have a justifiable reason for making a major change to your classroom structure and environment. This is both to give motivation to continue with flipping even if it is rocky at first and to cover your butt for administration and parents.  Flipping of a classroom will almost surely raise a few eyebrows (unless this is already a part of the culture of the school) and it is in your best interest to be able to &quot;prove&quot; why you decided to flip. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have a reason the next question you need to ask yourself is &lt;strong&gt;&quot;Who are my students?&quot;&lt;/strong&gt;.  This is crucial because if you don't have a real understanding of the capabilities and resources available to your students, the flipping will almost surely be a fail.  Things like having a lack of a computer/internet access, taking care of family members, or having jobs are all things that can complicate flipping, mainly because it means students will probably not do the homework consistently and without that, flipping really doesn't work.  However,  if you gauge your situation and judge that flipping, i.e. doing substantial work at home each night, is a reasonable expectation then I think that you should absolutely try it out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as my experience goes, the best flipped classrooms are ones where:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Students consistently do the homework&lt;/li&gt;&#xA;&lt;li&gt;The homework is used to introduce topics and give students a base exposure to working with the concept&lt;/li&gt;&#xA;&lt;li&gt;Class time is used to review the homework, assess understanding and dig deeper into the more conceptual aspects, usually through projects, activities, and collaboration&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As far as running a flipped classroom, here are my tips:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Start out slow.&lt;/strong&gt;  Try out 1 or 2 days a week where the class is flipped and see how it goes.  Explain to your students exactly what this &quot;new thing&quot; is, why they are doing it, what you hope to achieve from it, and what the expectations are.  If it goes well then step it up to 3, 4, 5 days a week as you see fit. However, the more likely outcome is that it will not go well at first, and in that case you should&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Stick with it.&lt;/strong&gt;  At least for a half-dozen cycles or so.  At that point you should definitely either feel it still dragging and not really working or you will start to see things slowly improve. To help things improve, i suggest you regularly&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Talk with your students.&lt;/strong&gt;  Ask them what they like about it?  What they don't like about it?  Their ideas to make it more useful/engaging?  You don't necessarily have to follow through with everything they say but it is essential to have that dialogue and to know what they are thinking.  All that being said, the last tip is to&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Know when to throw in the towel&lt;/strong&gt;.  If it is the second month of flipping your classroom and you still are having the same problems you were two months ago, it is probably time to stop.  Again, you are the ultimate judge of when enough is enough and it will depend on each individual situation.  At some point it becomes damaging to your students' educations to continue with consistently ineffective pedagogy.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Without knowing your specific situation, that is about as much as I can say.  If you are this interested in flipping, I think you should try it. Especially with teaching, the best way to learn how effective something is is to just try it out and see how it goes.  I hope this helps, I hope you decide to try out flipping, and I hope you learn from the experience.  Best of luck with your student teaching!&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-11-11T03:19:07.707" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9975" PostTypeId="1" CreationDate="2015-11-11T16:44:07.643" Score="6" ViewCount="257" Body="&lt;p&gt;These five imagined persons have accompanied me for some time. We've had a bunch of laughs and a few tears. I love them dearly. That said, I'd like to retire them in favor of a more culturally diverse crowd before I start teaching myself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far I'm thinking that Ahmed should be included. After that I'm in trouble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This is not a technical question about pedagogy, but I think it has some relevance to practitioners. I don't know of any other place where I'm likely to  encounter someone who has already given this some consideration.)&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2017-07-01T16:32:09.720" Title="What is a less anglo-centric collection of persons than Andy, Beth, Carl, Debby and Earl?" Tags="|examples|terminology|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="9976" PostTypeId="1" CreationDate="2015-11-11T22:08:33.087" Score="12" ViewCount="402" Body="&lt;p&gt;I have a chronic visual handicap and am in need of adequate level of accessibility support while learning and teaching mathematics from and to the visually challenged as well as the sighted.&lt;/p&gt;&#xA;&lt;p&gt;I'm familiar with the JAWS for Windows screen reading program and the MAGic screen magnification program, both developed by the Freedom Scientific Inc. JAWS offers very little access to mathematical text (in fact none for .djvu files), whereas MAGic 9.5's magnification levels of 2x and higher are too big for my liking (as it requires too much of scrolling) and the magnification levels of below 2x tend to make the display less clearly visible.&lt;/p&gt;&#xA;&lt;p&gt;I'm unable to afford a newer version of MAGic and so have to content myself with the demo version.&lt;/p&gt;&#xA;&lt;p&gt;As of now, I use the High Contrast Black (Large) display setting in Windows XP, along with the Windows Inverted (Xtra Large) mouse pointer scheme with trail and highlighting-upon-pressing-the-CTRL-key enabled.&lt;/p&gt;&#xA;&lt;p&gt;I prefer reading books in .djvu format with the contrast enabled, or the .pdf text files with colors replaced by the Windows High Contrast color setting (but struggle to read scanned .pdf documents since the text within those pages appears black on a white background, notwithstanding the High Contrast setting).&lt;/p&gt;&#xA;&lt;p&gt;Typesetting maths is often too tedious and time consuming for me, whether it be in WinEdit (with MikeTex at the back end) or as MathJax on the Mathematics Stack Exchange website.&lt;/p&gt;&#xA;&lt;p&gt;I've also downloaded and installed a program called WinTriangle, software developed by a Prof. Frank Gardner of the physics department at the Oregon State University. But I'm not sure how to utilise it.&lt;/p&gt;&#xA;&lt;p&gt;I'm aware of the BlindMaths mailing list, but over there the information is too spread out! I don't think there is any repository of audio / video tutorials, for example, on how to make the best possible use of WinTriangle.&lt;/p&gt;&#xA;&lt;p&gt;During my teaching of mathematics, I often find myself to be too slow while writing on the board. Often, drawing figures also proves to be too uphill a task. I'm not yet finished with a proof and the lecture hour goes by!&lt;/p&gt;&#xA;&lt;p&gt;I always find myself to be struggling to complete the course.&lt;/p&gt;&#xA;&lt;p&gt;Here in Pakistan, there is no individual or organisation working on making maths accessible to the blind and visually impaired.&lt;/p&gt;&#xA;&lt;p&gt;With this backdrop, any advice and suggestions, please?&lt;/p&gt;&#xA;" OwnerUserId="5400" LastEditorUserId="19777" LastEditDate="2022-09-19T18:48:36.177" LastActivityDate="2022-09-19T18:48:36.177" Title="How can I be more prolific as a maths teacher with my visual handicap?" Tags="|teaching|special-needs-education|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9977" PostTypeId="2" ParentId="9976" CreationDate="2015-11-12T01:23:34.587" Score="4" Body="&lt;p&gt;I can only address one point - writing on the board slowly.  I am also slow writing on the board, not because of any handicap. A few years ago we got a smart board (interactive white board) and I was able to prepare parts of the lecture in advance on smart notebook software. You can put things on different pages and save it to use again for your next class.  They also have a screen that you can use to slowly unveil the parts that you want to show so it is almost like you the students are watching you write it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't have access to a Smart Board, perhaps you can do what I did when the Smart Board needed repair and I couldn't use it.  I borrowed an easel from the preschool and used am exta-large pad that fit over the easel.  I wrote on different pages in advance and found this helpful.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point I don't know how I managed to conduct a class without writing in advance.  Good luck.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-11-12T01:23:34.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9978" PostTypeId="2" ParentId="9962" CreationDate="2015-11-12T10:24:55.770" Score="3" Body="&lt;p&gt;A Möbius band can be constructed by cutting a cylinder and then rejoining its ends after a twist. So draw it the same way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw a cylinder. This should be easy enough to do. Cut it by rubbing out parts of it. And then link it back with a twist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is in the same spirit as David's answer, with the added benefit that your audience can follow its construction with their imagination.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the use of blackboard may present a problem, as erasing tends to make an area less readable. Perhaps a different colour for the twist would help.&lt;/p&gt;&#xA;" OwnerDisplayName="user5871" LastActivityDate="2015-11-12T10:24:55.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9979" PostTypeId="1" CreationDate="2015-11-12T16:52:41.520" Score="5" ViewCount="179" Body="&lt;p&gt;I taught a student last year who has since decided to home school.  She was very bright and at 16 had already finished all of the math credits required by our district.  This leaves two years for her to finish up her other requirements (english, social studies, science, etc), and as such, will basically not be doing any math for the next two years.  She has come to me looking for opportunities for math practice/enrichment so that she will not get rusty over the next two years.  Ideally these would be things that would be accessible to her on her own, as her parents are by no means mathematicians.  My only idea was to participate in a local math circle.  What other ways are there for home-schooled students to get enrichment in math?&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-11-17T17:35:51.487" Title="Resources for math enrichment for home-schooled student" Tags="|secondary-education|homeschooling|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9980" PostTypeId="2" ParentId="9979" CreationDate="2015-11-12T18:05:59.343" Score="3" Body="&lt;p&gt;The University of Waterloo (Canada's MIT, roughly speaking) keeps a large catalog of high quality math problems online at &lt;a href=&quot;http://www.cemc.uwaterloo.ca/contests/past_contests.html&quot; rel=&quot;nofollow&quot;&gt;http://www.cemc.uwaterloo.ca/contests/past_contests.html&lt;/a&gt;. A strategy for a student in the situation you've described could be to download and attempt one contest every week. These aren't as curriculum focused as some other materials might be, but they'll definitely prevent any atrophy of mathematical thinking skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, they also publish a set of challenge problems on a weekly basis, which are available via email subscription or download at &lt;a href=&quot;http://www.cemc.uwaterloo.ca/resources/potw.php&quot; rel=&quot;nofollow&quot;&gt;http://www.cemc.uwaterloo.ca/resources/potw.php&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A word of warning - these materials are quite challenging and can lead to frustration or burnout, especially in the case of a young student unaccustomed to failure and lacking immediate mathematical mentorship. The contests, though, have the nice feature that the difficulty ramps up in each individual one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A bright high school student might do well to start by writing the grade nine contests, where she could treat the introductory questions as exercises, the intermediate ones as more substantive work, and the final sections as serious problems for deeper consideration. If it turns out that these contests are too easy or too difficult, then she can move up or down a grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;edit: (Solutions are also hosted on the site, which I've got mixed feelings about. Sometimes it's nice to see people tortured with mathematics.)&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2015-11-12T18:26:56.187" LastActivityDate="2015-11-12T18:26:56.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9981" PostTypeId="2" ParentId="9960" CreationDate="2015-11-12T23:42:31.063" Score="13" Body="&lt;p&gt;Your question is twofold:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1) Did any school districts actual teach the curriculum as planned and (2) what were the results for the teachers and students?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;At the moment, I have an answer for (1): Yes.&lt;/p&gt;&#xA;&lt;p&gt;For (2), the results for teachers and students &lt;em&gt;exist&lt;/em&gt; in various forms, but I have not combed through them carefully. And so your second question is certainly tractable, though I do not know the answer.&lt;/p&gt;&#xA;&lt;p&gt;To find out about your query, I passed it on to &lt;a href=&quot;http://www.maa.org/about-maa/governance/maa-presidents/henry-otto-pollak-1975-1976-maa-president&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Henry Pollak&lt;/strong&gt;&lt;/a&gt;  since (quoting from the aforelinked):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pollak was heavily involved with the &lt;a href=&quot;http://www.maa.org/archives-spotlight-the-school-mathematics-study-group-records&quot; rel=&quot;noreferrer&quot;&gt;School Mathematics Study Group&lt;/a&gt; (SMSG), an attempt to improve primary and secondary American mathematics education. Invited by Edward Begle and Albert Tucker, he participated in the initial four-week writing session of the SMSG, creating new curriculum (1958-60). He later was a member of the SMSG advisory board (1961-64, 1967-72) and its chairman (1963-64).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The take-away seems to be that, yes, the curriculum was taught in some places, and assessments of its creation and implementation were carried out by ETS, the Minnesota National Laboratory, the NSF, and SMSG itself. (&lt;strong&gt;Note&lt;/strong&gt;: You may be interested in consulting Phillips' &lt;em&gt;The New Math: A Political History&lt;/em&gt;; google books begins its preview with &lt;a href=&quot;https://books.google.com/books?id=IQswBQAAQBAJ&amp;amp;pg=PA96&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Chapter 5&lt;/strong&gt;&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;You can find an example of an SMSG report (funded by the NSF) from 1959 &lt;a href=&quot;http://files.eric.ed.gov/fulltext/ED088706.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;; please note that there are many such documents (cf. the archive at U of Texas Austin &lt;a href=&quot;https://www.lib.utexas.edu/taro/utcah/00284/cah-00284.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;) and some are quite long. (More precisely: I have not read through them in granular detail, so I am confident that a more informative response than mine can be posted to answer your question.)&lt;/p&gt;&#xA;&lt;p&gt;I paste below Pollak's response, since it draws from his first-hand experience, and I am sure its availability on MESE is more valuable than confining it to my inbox!&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The work on SMSG started in the summer of 1958.  I was a member of the team writing the first course on algebra and did not become a member of the overall advisory board for the project until about 1962.  But I have both written materials and recollections relevant to your questions, both of them incomplete and full of gaps.  In summary, there was an enormous amount of teaching (1) to provide formative evaluation for revisions, (2) to carry out major evaluations, by both Educational Testing Service  and the Minnesota  National Laboratory, and (3) to allow schools and teachers to use the materials.  I have recollection that experimental teaching began immediately, and proof of careful testing for both the National Science Foundation and SMSG itself first carried out no later than 1959.  The preparation by school districts for adopting the materials was highly variable, and [the emailed question] itself provides an example.&lt;/p&gt;&#xA;&lt;p&gt;Let me note parenthetically that UICSM, the University of Illinois Committee(?) on School Mathematics, which began around 1956, so at least two years earlier, initially required a full year of training at Urbana of any teacher  wishing to teach its materials.&lt;/p&gt;&#xA;&lt;p&gt;I have summaries of both the ETS and the Minnesota evaluations published in 1961, but I do not own the full reports themselves.  I have further evaluation summaries from 1963.  I have summaries of state-by-state programs for in-service teachers from 1964 from 23 states.  Kansas is not one of them.&lt;/p&gt;&#xA;&lt;p&gt;People still active who may have more information and copies of evaluations from the 1960's are Jeremy Kilpatrick and Jim Wilson,  both at the University of Georgia (but possibly retired).&lt;/p&gt;&#xA;&lt;p&gt;Henry&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(I am not sure how one would obtain the full reports alluded to above; it is possible that they already exist in a digitized form, or are somewhere in the Briscoe Center archives. Hopefully another MESE user can fill in more details.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-11-12T23:42:31.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9982" PostTypeId="2" ParentId="9907" CreationDate="2015-11-13T03:19:03.657" Score="2" Body="&lt;p&gt;Tangent at a point.  The (currently) most-voted answer mentions this, but not in the &lt;em&gt;multivariable&lt;/em&gt; context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The easy generalization of $$L(x)=m(x-a)+K$$ to the multivariate context is $$L(x,y)=m_x(x-a)+m_y(y-b)+K$$ (and so forth for more variables, eventually you get into matrix/vector version...) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Naturally, you could also solve some $ax+by+c=z$ here, but it gets less and less connected to reality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I will grant that for most high-school teachers in the US this will not seem the most compelling argument, since the small percentage of students who go on to this will probably survive no matter what version you use in 9th grade.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2015-11-13T03:19:03.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9983" PostTypeId="1" CreationDate="2015-11-13T03:22:41.983" Score="8" ViewCount="163" Body="&lt;p&gt;I see this a lot with students, &quot;solve this derivative&quot;, &quot;solve this integral&quot;, &quot;solve the following expression&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can compute a derivative, evaluate an integral, simplify an expression, solve an equation. How do you encourage students to use more verbs other than just &quot;solve&quot; for everything? &lt;/p&gt;&#xA;" OwnerDisplayName="user5108" LastActivityDate="2015-11-13T07:34:50.653" Title="How do you encourage students to use verbs other than &quot;solve&quot;?" Tags="|vocabulary|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="9984" PostTypeId="2" ParentId="7864" CreationDate="2015-11-13T04:38:33.377" Score="15" Body="&lt;p&gt;Below are instructions for drawing various quadric surfaces. They're not sketches but drawings designed to look (sort of) pretty and be easy to draw. You can see all of them in one YouTube playlist if you want: &lt;a href=&quot;https://www.youtube.com/playlist?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/playlist?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hyperbolic paraboloid&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Oi3dw.png&quot; alt=&quot;hyperbolic paraboloid&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/ZofoRuDJXhM?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/ZofoRuDJXhM?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you cut a hyperbolic paraboloid with a circular cutter, the outside edge is two cycles of a cos/sin curve. Slices parallel to the x axis and y axis will be parabolas, facing up in one direction and down in the other.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Draw a shallow parabolic arc.&lt;/li&gt;&#xA;&lt;li&gt;The left-end is the peak of the sin/cos curve. Draw the down-curve at the back.&lt;/li&gt;&#xA;&lt;li&gt;Draw the down-curve at the front, looping around to reach a peak at the other end of the original parabolic arc.&lt;/li&gt;&#xA;&lt;li&gt;Draw the down-curve at the back, flattening out a little as you reach the middle.&lt;/li&gt;&#xA;&lt;li&gt;At the bottom of the original parabolic arc, draw one side of a down-facing parabolic arc.&lt;/li&gt;&#xA;&lt;li&gt;Fill in the down-facing parabolic sections.&lt;/li&gt;&#xA;&lt;li&gt;Fill in the up-facing parabolic sections.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hyperboloid of one sheet - emphasis on rotation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KScyT.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KScyT.png&quot; alt=&quot;hyperboloid of one sheet, emphasis on rotation&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/zQ127ElV6Ns?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/zQ127ElV6Ns?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you view a hyperboloid of one sheet from the side, you can see a hyperbola. Horizontal cross sections are ellipses and sections through rotating vertical planes are also hyperbolas.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Draw a hyperbola opening sideways - remember hyperbolas are almost straight at the edge and turn quite sharply in the middle.&lt;/li&gt;&#xA;&lt;li&gt;Draw an ellipse at the top, and an ellipse arc at the bottom.&lt;/li&gt;&#xA;&lt;li&gt;Fill in horizontal sections - it works well to start in the middle.&lt;/li&gt;&#xA;&lt;li&gt;Fill in rotating vertical sections - a vertical line in the middle and a hyperbolic arc on each side.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hyperboloid of one sheet - emphasis on vertical sections&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6dqSZ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6dqSZ.png&quot; alt=&quot;Hyperboloid of one sheet - emphasis on vertical sections&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/2XNxBGmtItE?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/2XNxBGmtItE?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you view a hyperboloid of one sheet from the side, you can see a hyperbola. Horizontal cross sections are ellipses and sections in vertical planes are hyperbolas in different directions or just two lines.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Draw a hyperbola opening sideways - remember hyperbolas are almost straight at the edge and turn quite sharply in the middle.&lt;/li&gt;&#xA;&lt;li&gt;Draw an ellipse at the top, and an ellipse arc at the bottom.&lt;/li&gt;&#xA;&lt;li&gt;Fill in horizontal sections - it works well to start in the middle.&lt;/li&gt;&#xA;&lt;li&gt;Draw two lines crossing on the centre horizontal section.&lt;/li&gt;&#xA;&lt;li&gt;Draw upper and lower branches of a hyperbola inside the two lines.&lt;/li&gt;&#xA;&lt;li&gt;Draw left and right branches of a hyperbola outside the two lines.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hyperboloid of two sheets - emphasis on rotation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nTHCV.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nTHCV.png&quot; alt=&quot;Hyperboloid of two sheets - emphasis on rotation&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/XpsgENVJuGA?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/XpsgENVJuGA?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you view a hyperboloid of two sheets from the side, you can see a hyperbola. Horizontal sections are ellipses and rotating vertical sections are hyperbolas, with one branch on each nappe of the hyperboloid.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Draw a hyperbola opening upward - remember hyperbolas are almost straight at the edge and turn quite sharply in the middle.&lt;/li&gt;&#xA;&lt;li&gt;Draw an ellipse at the top, and an ellipse arc at the bottom.&lt;/li&gt;&#xA;&lt;li&gt;Draw in ellipse horizontal sections on the two nappes.&lt;/li&gt;&#xA;&lt;li&gt;Fill in rotating vertical sections on the bottom nappe - they all cross at a point, and are only really curved at the very top.&lt;/li&gt;&#xA;&lt;li&gt;Fill in rotating vertical sections on the top nappe - you can't see where they cross.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hyperboloid of two sheets - emphasis on vertical sections&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zyv3b.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zyv3b.png&quot; alt=&quot;Hyperboloid of two sheets - emphasis on vertical sections&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/kjuJ1pOTiZI?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/kjuJ1pOTiZI?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you view a hyperboloid of two sheets from the side, you can see a hyperbola. Horizontal sections are ellipses and vertical sections are hyperbolas, with one branch on each nappe of the hyperboloid.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Draw a hyperbola opening upward - remember hyperbolas are almost straight at the edge and turn quite sharply in the middle.&lt;/li&gt;&#xA;&lt;li&gt;Draw an ellipse at the top, and an ellipse arc at the bottom.&lt;/li&gt;&#xA;&lt;li&gt;Draw in ellipse horizontal sections on the two nappes.&lt;/li&gt;&#xA;&lt;li&gt;Fill in sections on the bottom nappe - resist the urge to make them curved except at the very top.&lt;/li&gt;&#xA;&lt;li&gt;Fill in sections on the top nappe - try to line them up with the sections on the bottom nappe.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Elliptic paraboloid&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bDe3w.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bDe3w.png&quot; alt=&quot;Elliptic paraboloid&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/QIlz6A4sqy0?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/QIlz6A4sqy0?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you view an elliptic paraboloid from the side, you see a parabola. Horizontal sections are ellipses and vertical sections are parabolas.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Draw a parabola - make it quite pointy.&lt;/li&gt;&#xA;&lt;li&gt;Draw an ellipse at the top, then fill in the horizontal sections.&lt;/li&gt;&#xA;&lt;li&gt;Draw the vertical sections as not-quite-half-parabolas starting with a very tiny one in the corner.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Ellipsoid&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LjSxr.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LjSxr.png&quot; alt=&quot;Ellipsoid&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/0_KWp-2PaP4?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/0_KWp-2PaP4?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you view an ellipsoid from the side, you see an ellipse. All the sections are ellipses.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Draw an ellipse.&lt;/li&gt;&#xA;&lt;li&gt;Starting roughly in the middle, draw the bottom of an ellipse. Fill in a curved arc below this.&lt;/li&gt;&#xA;&lt;li&gt;Draw horizontal sections above the middle line. Include more of the end-curve each time.&lt;/li&gt;&#xA;&lt;li&gt;Draw the vertical sections. Include less and less of the end-curve each time.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Elliptic cone -- emphasis on rotation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AQ49w.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AQ49w.png&quot; alt=&quot;elliptic cone with rotating vertical sections&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/TLLGpa4yAdk?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/TLLGpa4yAdk?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you view a cone from the side, you can see two intersecting lines. The horizontal sections are ellipses, and rotating vertical sections through the vertex always give two lines.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Draw two intersecting lines.&lt;/li&gt;&#xA;&lt;li&gt;Draw an ellipse at the top and an elliptical arc at the bottom.&lt;/li&gt;&#xA;&lt;li&gt;Fill in the ellipse horizontal sections. It is easier to start near the top on the top nappe and near the bottom on the bottom nappe, so you can follow the curve of the existing lines.&lt;/li&gt;&#xA;&lt;li&gt;Draw straight lines to indicate rotating vertical sections.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Elliptic cone -- emphasis on vertical sections&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2fX06.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2fX06.png&quot; alt=&quot;elliptic cone with parallel vertical sections&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/7wHxjqB-H1M?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/7wHxjqB-H1M?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you view a cone fromt the side, you can see two intersecting lines. The horizontal sections are ellipses, and vertical sections are two lines if through the vertex or hyperbolas if not.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Draw two intersecting lines.&lt;/li&gt;&#xA;&lt;li&gt;Draw an ellipse at the top and an elliptical arc at the bottom.&lt;/li&gt;&#xA;&lt;li&gt;Fill in the ellipse horizontal sections. It is easier to start near the top on the top nappe and near the bottom on the bottom nappe, so you can follow the curve of the existing lines.&lt;/li&gt;&#xA;&lt;li&gt;Draw a line through the vertex off-centre on the top nappe, and another line on the same side but on the bottom nappe.&lt;/li&gt;&#xA;&lt;li&gt;Draw hyperbolic sections on both nappes. Make them mostly straight except where they meet the edge of the cone.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-11-14T04:19:55.157" LastActivityDate="2015-11-14T04:19:55.157" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9985" PostTypeId="2" ParentId="9979" CreationDate="2015-11-13T04:40:33.670" Score="2" Body="&lt;p&gt;Many homeschoolers attend community college while of high school age. She could take whatever math class she wants at a community college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The local math circle sounds like a good idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.girlsangle.org/&quot; rel=&quot;nofollow&quot;&gt;Girls' Angle&lt;/a&gt; posts interesting math problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;She might like my book, &lt;em&gt;&lt;a href=&quot;http://naturalmath.com/playingwithmath/&quot; rel=&quot;nofollow&quot;&gt;Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-11-13T04:40:33.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9986" PostTypeId="2" ParentId="9983" CreationDate="2015-11-13T07:25:43.157" Score="6" Body="&lt;p&gt;Every time a student uses it incorrectly, I quickly correct them. &quot;You mean simplify this expression&quot;, or the like, and then go on to answer whatever question they meant to ask. Furthermore I verbally check in on the exact definition of &quot;solution&quot; multiple times. Some book exercises that I hand out occasionally have warm-up questions on this topic (Bittinger Intermediate Algebra, Martin-Gay Introductory Algebra). And every time I present an exercise on the board I always include a formal written direction, e.g., &quot;Divide the polynomials and check&quot;, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This encourages, but does not entirely solve, the problem. I suppose the next level of escalation would be to include it in some multiple-choice quiz questions, but to date I don't think its priority is high enough to bump another item out of my quizzes. The truth is students at this level rarely put effort into an issue unless points are on the line. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2015-11-13T07:34:50.653" LastActivityDate="2015-11-13T07:34:50.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9987" PostTypeId="2" ParentId="9979" CreationDate="2015-11-13T12:27:32.540" Score="3" Body="&lt;p&gt;Here are some resources:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://nrich.maths.org&quot; rel=&quot;nofollow&quot;&gt;nrich&lt;/a&gt; They have a wide range of rich problem solving explorations for students in primary through secondary and some post secondary.  They have student pages and teacher pages.  &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://illuminations.nctm.org&quot; rel=&quot;nofollow&quot;&gt;NCTM illuminations&lt;/a&gt;  Explore the interactives for grades 9-12. They have tools to explore fractals, a game to explore vector addition as a boat travels through a current, etc.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.khanacademy.org/&quot; rel=&quot;nofollow&quot;&gt;Khan Academy&lt;/a&gt;  This is a wonderful place to practice different skills so they don't become rusty.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.ohiorc.org/for/math/stella/&quot; rel=&quot;nofollow&quot;&gt;Stella's Problems&lt;/a&gt; Wonderful non-routine problems for all levels.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-11-13T12:27:32.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9989" PostTypeId="2" ParentId="9979" CreationDate="2015-11-13T20:30:14.877" Score="2" Body="&lt;p&gt;In addition to the above recommendations, I'd add resources from &lt;a href=&quot;http://www.artofproblemsolving.com&quot; rel=&quot;nofollow&quot;&gt;the Art of Problem Solving&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(BTW, the answer to this question depends heavily on the homeschooler. I was homeschooled and found my way to mathematics through physics. I didn't like pure mathematics &lt;em&gt;at all&lt;/em&gt; until I needed it for a better understanding of physics. Only much later did I develop a taste for pure mathematics. Somehow I ended up a pure mathematician so my perspective has changed. I figured I should share my memory of this…since I'm not sure how the old me would have responded to the material at AOPS above…)&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2015-11-13T20:37:06.023" LastActivityDate="2015-11-13T20:37:06.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9990" PostTypeId="1" AcceptedAnswerId="9992" CreationDate="2015-11-13T20:50:33.650" Score="21" ViewCount="772" Body="&lt;p&gt;Consider the following problem:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Maria always buys ice-cream when she goes to the beach. She bought ice-cream today. So, she must have gone to the beach.  &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Obviously this statement is wrong. Maria could have gone to other place and bought an ice-cream. You don't need any math tool to arrive at this conclusion, all you need is reasoning.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, several adults (18~50 years old) with difficulty in math, also have a really hard time to solve/understand such kind of problems. For them, learning math is the same as memorizing rules and formulas. Anything different than that (ex: reasoning) is extremely painful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, is it possible to make such students correctly answer &lt;a href=&quot;https://en.wikipedia.org/wiki/Graduate_Management_Admission_Test&quot;&gt;GMAT&lt;/a&gt; style questions?&lt;/p&gt;&#xA;" OwnerUserId="5878" LastEditorUserId="2083" LastEditDate="2015-11-16T16:02:37.280" LastActivityDate="2021-12-17T14:58:00.310" Title="Students problems with reasoning, not exactly math" Tags="|mathematical-pedagogy|students-mistakes|quantitative-reasoning|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9991" PostTypeId="2" ParentId="59" CreationDate="2015-11-13T21:07:11.923" Score="3" Body="&lt;p&gt;Perspective: I'm a lecturer at a large urban community college. My opinion is that open-book tests are fundamentally a trap for students. It reduces complaints and makes it seem like you're a &quot;nice guy&quot;, but it gives the impression that students won't have to study in advance, and can just look the answers up in the book during the test. I think what generally happens is that they get lost shuffling pages during the test, unable to get started because they really don't know how.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like we need to communicate the fact that there is some body of conceptual knowledge that practitioners carry around in their head, and can converse and read about with &quot;automaticity&quot;, without looking it up. On the other hand, I'm a huge proponent of providing a 2-4 page formula card (in common to all the students) as a tool which is short and clarifying and not likely for someone to get &quot;lost&quot; in. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-11-13T21:07:11.923" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9992" PostTypeId="2" ParentId="9990" CreationDate="2015-11-13T21:17:51.620" Score="19" Body="&lt;p&gt;You can say that this is &amp;quot;just reasoning&amp;quot;, but the truth is that this is a specific application of basic logic, in particular the implication (if/then) relation. I have a colleague with a PhD in logic who says, &amp;quot;Implication is tricky!&amp;quot; when I bring this up. And I do think that it's a major problem that schools don't teach basic logic as a high-school (or earlier) requirement; it really puts all their later coursework on a foundation of shifting sand without that.&lt;/p&gt;&#xA;&lt;p&gt;If I had complete dominion at the community college where I teach, then personally I would mandate a 1-credit seminar in basic logic (at least drills with and-or-not-if/then statements) for incoming students. At times I've tried to find an hour in my basic math classes to work on this, but unfortunately at the moment other priorities take precedence for that time.&lt;/p&gt;&#xA;&lt;p&gt;Some blog articles I've written on this subject:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.madmath.com/2012/07/teach-logic.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.madmath.com/2012/07/teach-logic.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.madmath.com/2013/11/branching-decisions-in-algebra.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.madmath.com/2013/11/branching-decisions-in-algebra.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.madmath.com/2014/05/basic-logic-errors.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.madmath.com/2014/05/basic-logic-errors.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="14091" LastEditDate="2021-12-15T21:01:59.693" LastActivityDate="2021-12-15T21:01:59.693" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9993" PostTypeId="1" CreationDate="2015-11-14T11:36:33.240" Score="7" ViewCount="149" Body="&lt;p&gt;I am currently helping a friend's child with his schoolwork. He is currently in primary school and being taught the topic of speed. I would like to give him the following problem as practice but I am not sure how to explain the solution to him at his level:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A and B competed in a race. Every time A ran $7$ m, B ran $3$ m. If B took $14$ minutes to complete the race, how long did A take?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My proposed solution: the ratio of A's speed to B's speed is $7:3$. So A will take a shorter time than B. Since distance $=$ speed $\times$ time, we have distance covered by A $=$ A's speed $\times$ A's time $=$ distance covered by B $=$ B's speed $\times$ B's time, i.e. $7$ $\times$ A's time is equal to $3$ $\times$ B's time. Hence the ratio of A's time to B's time is $3:7$. So, $7u=14,\ 1u=2$ and hence $3u=6$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do I rephrase my solution to make it more student-friendly? I am trying not to use any formula. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think many students have difficulty grappling with the fact that speed is inversely proportional to time. If A's speed is $x$ times B's speed, where $x$ is whole number, then most students can easily see that A's time is $1/x$ times of B's time. But if $x$ is not a whole number, most will get stuck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I appreciate all advice. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="5879" LastEditorUserId="77" LastEditDate="2015-11-17T04:02:11.397" LastActivityDate="2015-11-17T04:43:34.227" Title="Advice needed to teach a problem about speed" Tags="|primary-education|proportional-reasoning|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="9994" PostTypeId="1" CreationDate="2015-11-15T17:59:20.660" Score="8" ViewCount="613" Body="&lt;p&gt;This question arose from discussion of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/9869/as-a-ta-how-to-reduce-imprecise-notations-statements-in-students-exams&quot;&gt;this question&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How long would it take you to teach typical undergratuate (calculus) students the difference between the following two calculations?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$$\lim_{x\to \infty}\frac{x+2}{x(x-1)}=\frac{\infty}{\infty^2}=\frac{1}{\infty}=0$$&#xA;versus&#xA;$$\lim_{x\to\infty}\frac{x+2}{x(x-1)}=\lim_{x\to\infty}\frac{x^{-1}+2x^{-2}}{1-x^{-1}}=\frac{0-0}{1-0}=0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm happy to accept different interpretations of 'long' and 'typical' (please give an indication of what they mean to you).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By 'teach' I mean such that the students properly grasp why one solution is better than the other, rather than simply that they remember that 'the teacher told us to do it this way' and can do the calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Justification from research if possible would be nice, but I'd be interested to hear personal experience too.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-11-25T06:07:10.213" Title="How long would it take to teach proper limit calculations?" Tags="|undergraduate-education|limits|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="9995" PostTypeId="2" ParentId="9994" CreationDate="2015-11-16T00:44:02.700" Score="6" Body="&lt;p&gt;It might be enough to show an example or two of what can go wrong with the logic of the first solution:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\lim_{x\to\infty} \frac{e^x}{x^2}=\frac{\infty}{\infty^2}=\frac{1}{\infty}=0$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course this is incorrect!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's one more, in case they don't know the exponential-polynomial comparison:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\lim_{x\to\infty}\frac{x^2}{\sqrt{x+1}\sqrt{x+2}\sqrt{x+3}}=\frac{\infty^2}{\infty^3}=\frac{1}{\infty}=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wrong again!&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2015-11-16T00:44:02.700" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="9997" PostTypeId="2" ParentId="9971" CreationDate="2015-11-16T02:48:47.033" Score="2" Body="&lt;p&gt;While it may be true that sometimes 'backwards' flipping might be effective, sometimes it might rather be preferable to provide some of the low level content in a video watched beforehand.  This would allow students to engage with the material at a higher level in class.  You could try pre- and post- class videos.  It might depend on the level of difficulty of the topic which method works best.  Even within a single course, you wouldn't have to do the same thing with every topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally I'd usually prefer to have the students arriving at class time with some idea of the topic.  (I flip some lower level college math classes.)&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2015-11-16T02:48:47.033" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="9998" PostTypeId="2" ParentId="9994" CreationDate="2015-11-16T14:24:20.110" Score="9" Body="&lt;p&gt;I think it is too hasty to dismiss manipulation of &quot;$\infty$&quot; out-of-hand, although, yes, there is a widespread tendency among students to &lt;em&gt;over&lt;/em&gt;-simplify, thus crossing various lines into trouble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first-presented version is not so bad as a heuristic, and might have been written by Euler or Lagrange. For that matter, it can be made more focused by $\lim{x-2\over x(x-1)}=\lim{x\over x^2}=\lim {1\over x}=0$, with or without the flourish of $\lim {1\over x}={1\over \infty}=0$ at the end. The latter is certainly the way to &lt;em&gt;remember&lt;/em&gt; the conclusion, as opposed to &quot;for all $\epsilon&amp;gt;0$ there exists $N$ such that for all $N'\ge N$...&quot; which is too quantifier-rich.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd claim that there are two distinct issues here, apart from whether or not $\infty$ is a number, which should be separated. First, seeing that $x(x-1)\sim x^2$, or that $x-2\sim x$ (as $x$ becomes large), is an important skill, despite &quot;asymptotic expansions&quot; falling somewhat outside the contemporary version of &quot;tradition&quot;. But this is very important in genuine mathematical, scientific, and engineering practice: seeing through the noise of details. Second, the comparison of a range of &lt;em&gt;not&lt;/em&gt;-noisy functions as $x\to+\infty$: powers of $x$ versus exponentials versus logs, &lt;em&gt;without&lt;/em&gt; the &quot;$x-2$&quot; noise to weary and confuse the novice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If suitable use of &quot;$\infty$&quot; is short-hand for the latter, what's the problem? Ok, yes, students seem to be eager to corrupt things, but, ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: and the second choice is really not the most intuitive, and all the inverses suddenly create lots more symbols. Simplifying the asymptotics of both numerator and denominator &lt;em&gt;first&lt;/em&gt; eliminates the need for this, and reduces the chances for student errors simply in the relevant elementary algebra, which I've found to be alarmingly high, perhaps the main bottleneck in doing the second sort of argument.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2015-11-16T15:09:44.877" LastActivityDate="2015-11-16T15:09:44.877" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10000" PostTypeId="1" CreationDate="2015-11-16T16:36:01.487" Score="5" ViewCount="116" Body="&lt;p&gt;&lt;strong&gt;Definition&lt;/strong&gt; (Finite series). Let $m,n$ be integers, and let $(a_i)_{i=m}^{n}$ be a finite sequence of real numbers, assigning a real nmber $a_i$ to each integer $i$ between $m$ and $n$ inclusive. Then we define the finite sum $\sum_{i=m}^{n} a_i$ by the recursive formula $\sum_{i=m}^{n} a_i = 0 $ whenever $n &amp;lt; m$ and $\sum_{i=m}^{n+1} a_i = \sum_{i=m}^{n} a_i + a_{n+1}$ whenever $ n \geq m-1$ . &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lemma&lt;/strong&gt; Let $m \leq n &amp;lt; p$ be integers, and let $a_i$ be a real number assigned to each integer $m \leq i \leq p$. Then we have $\sum_{i=m}^{n} a_i + \sum_{i=n+1}^{p} a_i =\sum_{i=m}^{p} a_i$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;MY QUESTIONS&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What are the typical questions one would ask of himself in trying to understand such a typical sequence of definition and lemma?&lt;/li&gt;&#xA;&lt;li&gt;Why did the author write $m \leq i \leq p$ and not $(n+1) \leq i \leq p$? Is the author trying to generalize $i$? If yes what is the benefit of this generalization?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5882" LastActivityDate="2015-11-17T00:33:24.637" Title="What are the questions one should ask of oneself in trying to understand definitions and lemmas?" Tags="|self-learning|definitions|" AnswerCount="2" CommentCount="2" ClosedDate="2015-11-20T06:12:18.883" ContentLicense="CC BY-SA 3.0" />
  <row Id="10001" PostTypeId="2" ParentId="10000" CreationDate="2015-11-16T18:12:07.793" Score="5" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Questions to ask yourself: &quot;Can I create a concrete example myself? Why are the hypothesis sufficient? Are they necessary? Can I find a counter-example if I change them?&quot; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The index of summation ($i$) must be between the bounds of summation ($m$ and $p$). The integer $n$ is just where the summation is being split, so it will be an upper-bound for $i$ in the first summation, and a lower-bound for $i$ in the second summation. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="98" LastActivityDate="2015-11-16T18:12:07.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10002" PostTypeId="2" ParentId="10000" CreationDate="2015-11-17T00:33:24.637" Score="4" Body="&lt;p&gt;Mathematics, in contrast to the wasy it is presented in many textbooks, is not a collection of sacred rules of conduct, nor a sacred dictionary of meaning-of-words, but a certain form of an account of what mathematics is about, and what mathematicians do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true that there has been &quot;optimization&quot; of these accounts over the last 200 years, ... but that in itself is a problem, in some cases! That is, the back-story is tricky enough so that without knowing it (which would be daunting already) the &quot;calm&quot; sequel is incomprehensible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then there's the stylistic conceit about definitions/lemmas/proofs. This can easily degenerate into a sort of acolyte-seeking-to-propitiate-whimsical-gods scenario. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While we're here, although &quot;the Moore method&quot; has its fans, I am not among them. In fact, I would claim that the implicit hypothesis in this is that mathematics is (conceptually) &quot;shallow&quot;, in the specific technical sense (not necessarily insulting, though it sounds like it!) that not much prior experience or prior knowledge (the big two &quot;priors&quot;) are relevant. (Another discussion: necessity versus relevance.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most-most-immediate answer to such questions, I'd claim, is to ask what &quot;real question&quot; this &quot;stylized/formal&quot; question is hiding. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually, the answer is that there's no real issue. More like whether you're sitting down... much less having fastened the seat-belt... before driving.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is certainly true that other philosophical approaches to mathematics dictate or suggest &lt;em&gt;other&lt;/em&gt; views, but, I'd recommend not thinking in terms of those things, but in terms of the &quot;what are we trying to do?&quot; criterion. I know, yes, these are not the usual things in &quot;school&quot;, ...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-11-17T00:33:24.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10003" PostTypeId="2" ParentId="9993" CreationDate="2015-11-17T04:43:34.227" Score="5" Body="&lt;p&gt;My teaching philosophy is to start at a level below the learner's assumed mathematical maturity.  I will assume that the student has not mastered the concepts of ratio or basic algebra.  But I will assume that the student is familiar with the concepts of speed and arithmetic equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Review the definition of speed as the quantity equal to the distance traveled divided by the time it took to travel.  Then, noting that most problems like these involve equations (statements that two quantities are equal), ask the student what quantities are equal in this situation.  The student should be led to discover that the distances traveled by A and B should be equal.  Then note that the distance traveled is equal to the speed multiplied by the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are not given the speeds of A and of B.  But note that the speed of A is $7$ meters for some unknown but fixed unit of time, and the speed of B is $3$ meters for the same unit of time.  Thus, the distance traveled by A is $\frac{7\mathsf{~meters}}{\mathsf{unit~of~time}}\times\mathsf{?~minutes}$; and the distance traveled by B is $\frac{3\mathsf{~meters}}{\mathsf{unit~of~time}}\times 14\mathsf{~minutes}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Noting that $3\times 14=3\times 2\times 7=6\times 7=7\times 6$, it can be seen that A would take $6$ minutes.  (For brevity, I skipped a lot of steps here.  You should supply the missing steps.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that we only dealt with a specific case, that is, we did not use a generalization.  Have the student solve, say, ten more specific cases (exactly the same as this problem, but with different numbers).  (You should carefully choose the numbers so that the answers are whole numbers.)  By the end of these &quot;drills,&quot; the student should be able to construct the generalization by himself/herself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also note that we did not use division or the concept of &lt;em&gt;inversely proportional&lt;/em&gt;.  Actually, we did not even use the terms &lt;em&gt;ratio&lt;/em&gt; or &lt;em&gt;proportion&lt;/em&gt;.  We just used multiplication (and factoring) and the concepts of &lt;em&gt;speed&lt;/em&gt;, &lt;em&gt;distance&lt;/em&gt;, and &lt;em&gt;time&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2015-11-17T04:43:34.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10004" PostTypeId="1" CreationDate="2015-11-17T16:23:01.300" Score="20" ViewCount="2371" Body="&lt;p&gt;What could be good non-mathematical analogy/analogies to explain the difference among the words - theorem, proposition, lemma and corollaries to high school students? &#xA;I am looking for analogies that would explain their meaning and their usage by mathematicians.  &lt;/p&gt;&#xA;" OwnerUserId="5882" LastActivityDate="2018-08-11T15:59:44.157" Title="What could be good non-mathematical analogies to explain the difference between the words theorem, proposition, lemma and corollaries?" Tags="|mathematical-pedagogy|terminology|" AnswerCount="12" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10005" PostTypeId="2" ParentId="10004" CreationDate="2015-11-17T16:57:49.623" Score="7" Body="&lt;p&gt;From Computer Science:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lemma:&lt;/strong&gt; Built-In Function (highly technical, but needed)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proposition:&lt;/strong&gt; Sub-routine (a good tool to have)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem:&lt;/strong&gt; Main Program (the main project you're building)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Corollary:&lt;/strong&gt; A smaller program that calls the Main Program as a sub-routine (the cute things you can use your final project to do)&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="1147" LastEditDate="2015-11-17T20:15:33.347" LastActivityDate="2015-11-17T20:15:33.347" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10006" PostTypeId="2" ParentId="10004" CreationDate="2015-11-17T17:24:49.973" Score="2" Body="&lt;p&gt;&lt;strong&gt;Theorem:&lt;/strong&gt; A business deal that has been signed, sealed, and delivered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lemma:&lt;/strong&gt; A product listing in a catalog, with a known price and availability.  A businessman can include this product as part of a larger deal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proposition:&lt;/strong&gt; A proposal.  It might or might not work out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Corollary:&lt;/strong&gt;  An easy follow-up deal, based on a big deal.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2015-11-17T17:24:49.973" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10007" PostTypeId="2" ParentId="9979" CreationDate="2015-11-17T17:35:51.487" Score="2" Body="&lt;p&gt;Unfortunately, all too often what is done with bright students (especially middle school and beyond) is that their mathematical clocks are pushed - that is if they are in &quot;grade x&quot; they are shown the mathematics at grade (x+1) because they are already proficient with the mathematics at grade x. There are similar issues for the home schooled. My advice is to show students who are outpacing the mathematics at their grade level the books of Martin Gardner. These are available at many public libraries. Here is a partial list:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.york.cuny.edu/~malk/biblio/martin-gardner-biblio.html&quot; rel=&quot;nofollow&quot;&gt;http://www.york.cuny.edu/~malk/biblio/martin-gardner-biblio.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gardner's books often show people a different vision of what mathematics is and can be. My only reservation about his work is that he does not do justice to using applications of mathematics as an entry point to creating excitement and interest in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2015-11-17T17:35:51.487" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10008" PostTypeId="2" ParentId="10004" CreationDate="2015-11-17T18:48:18.113" Score="3" Body="&lt;p&gt;The distinction is (much more than) somewhat arbitrary. Some people call everything &quot;proposition&quot; (or &quot;result&quot;, or even something else). The &lt;em&gt;idea&lt;/em&gt; is that in &quot;orderly development&quot; your goal is to prove a &lt;strong&gt;theorem&lt;/strong&gt;. Most of the time this can't be done from cold start, you'd need some auxiliary results, not important in themselves, to get to the goal, &lt;strong&gt;lemmata&lt;/strong&gt;. It often happens that once you have your goal (theorem), you can use it in a simple step/application to derive another important result (or use some intermediate step in the proof as easy starting point), giving rise to a &lt;strong&gt;corollary&lt;/strong&gt;. Then you have &lt;strong&gt;conjectures&lt;/strong&gt;, which are essentially educated guesses. And there are important theorems that are called something else, like the &quot;division algorithm&quot; in elementary number theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the &lt;em&gt;intent&lt;/em&gt; of the author, what happens afterwards is anybody's guess. It is said that the mathematician's wet dream is to become known by a lemma, not a theorem (often lemmas turned out to be much more important than the long forgotten theorems they were stepping stones for, like the various Gauss' lemmata, or Burnside's lemma). And some famous conjectures have their own names, like the Riemann Hypothesis, or the all-time favorite &quot;Last (or great) theorem of Fermat&quot; (which was proved recently).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trying to find &quot;everyday parallels&quot; is futile, this are technical terms with precise meaning in mathematics (or mathematics writing). You might as well ask for &quot;everyday equivalents&quot; for the distinctions between, say, carving tools.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-11-17T18:48:18.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10009" PostTypeId="2" ParentId="10004" CreationDate="2015-11-17T19:01:17.373" Score="10" Body="&lt;p&gt;From cooking:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt;: recipe, something that produces a desired result&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lemma&lt;/strong&gt;: a technique (e.g. creating a &lt;em&gt;roux&lt;/em&gt;, proofing dough, preparing a &lt;em&gt;mirepoix&lt;/em&gt;, etc.), a procedure that one uses in many different recipes, but might not produce something interesting/tasty by itself&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Corollary&lt;/strong&gt;: (really stretching here) extra garnish/styling to slightly modify the dish&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not going to try for proposition.&lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="1147" LastEditDate="2015-11-18T14:32:49.793" LastActivityDate="2015-11-18T14:32:49.793" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10010" PostTypeId="2" ParentId="10004" CreationDate="2015-11-17T19:56:27.883" Score="13" Body="&lt;p&gt;Origami has things like lemmas, theorems and corollaries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Definition/axiom&lt;/strong&gt;: The basic folds such as book, mountain, valley etc&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lemma&lt;/strong&gt;: a folding procedure that is used as part of another, such as the bird base or the water bomb base.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt;: a folding procedure that produces something you wanted, such as the traditional crane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Corollary&lt;/strong&gt;: a folding procedure that follows on from something you made, such as modifying the traditional crane by expanding the tail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proposition&lt;/strong&gt;: To me this is just the name of the statement you are trying to prove before you've proven it (regardless if whether it's a lemma, a theorem or a corollary), so in that sense it's like the photo of the finished product at the top before the instructions start.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some use it to mean a result that isn't necessarily in the sequence of things you're trying to prove. So perhaps this would be in the category of &quot;other things you can make from the bird base&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The advantage of origami is that you can bring physical models to illustrate your point!&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-11-19T05:53:54.817" LastActivityDate="2015-11-19T05:53:54.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10011" PostTypeId="2" ParentId="10004" CreationDate="2015-11-17T23:27:51.877" Score="10" Body="&lt;p&gt;&lt;strong&gt;Lemma:&lt;/strong&gt; A wheel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proposition:&lt;/strong&gt; A gear-box.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem:&lt;/strong&gt; A car. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Corollaries:&lt;/strong&gt; Photos of the car at interesting destinations. &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2015-11-17T23:27:51.877" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10013" PostTypeId="2" ParentId="9990" CreationDate="2015-11-17T23:36:55.017" Score="3" Body="&lt;p&gt;There is a paradox here.  Implication IS tricky.  Part of the reason is it's REALLY HARD to avoid implicit assumptions in human scenarios. The only absolute way to avoid assumptions is to translate the English into meaningless symbols (p, q) and rigorously apply logic rules.  The rules themselves are tricky.  You pretty much need to prove them once to know they are correct, then apply them by rote from then on.  Your example is on the cusp what most people can handle in their head. For anything much more complex, most of us will need to manipulate symbols using logical rules to &quot;reason correctly.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your example, some of your students will intuitively see the correct premise and conclusion.  Others will need some help.  All of us will need some help if the scenario gets complicated enough.  (Or if it is sneaky enough to trick us into making unwarranted assumptions.  It happens.)  I'll probably lose points for politicizing this, but consider the recent debate on gay marriage.  There were some not irrational arguments along the line of &quot;If you want to have children, then you should be married.&quot;  But there was an implicit assumption of bi-implication.  Not once did I hear anyone--on either side of the debate--point out that this is not the same thing as saying &quot;If you want to get married, then you should have children.&quot;  So this sort of logical error is very easy to make, every by well-trained people. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree with Daniel and his blogs.  The only solution is formal training in logic.  If only to make people aware of how EASY it is to make mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're looking for the most bang for your teaching buck, what I have personally found the most useful is this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What implication is: An if-then statement.  Show how to identify the premise and the conclusion.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Show the differences between the INVERSE, CONVERSE, and CONTRAPOSITIVE.  My first exposure to these three conditionals was a real eye-opener.  It did more than anything else in my life to avoid the kind of error shown in your example.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5465" LastActivityDate="2015-11-17T23:36:55.017" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10014" PostTypeId="2" ParentId="5989" CreationDate="2015-11-17T23:39:39.940" Score="2" Body="&lt;p&gt;Calculus and Differential Equations for Biology (MATH 1251 and sequels) at Northeastern University in Boston, coordinated by Dr. Samuel Blank, are taught using infinitesimals and infinite numbers rather than limits. The textbook in use, however, is a standard limits-based textbook, and is only loosely followed. The material on infinitesimals is provided in online notes.&lt;/p&gt;&#xA;" OwnerUserId="748" LastActivityDate="2015-11-17T23:39:39.940" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10015" PostTypeId="2" ParentId="10004" CreationDate="2015-11-18T00:40:39.787" Score="2" Body="&lt;p&gt;The existence of a Lost City of Atlantis is a &lt;strong&gt;proposition&lt;/strong&gt;. It has not been proven or dis-proven to exist. But if one day, someone discovers a route to get there, we would know it exists. When this happens, its existence becomes a &lt;strong&gt;theorem&lt;/strong&gt;. The route that we take to get there is the &lt;strong&gt;proof&lt;/strong&gt;. Along the way we also discover Davy Jones' Locker. So the existence of said locker is a &lt;strong&gt;lemma&lt;/strong&gt;. When we explore the lost city, we discover it was built by aliens. So the fact that it was built by aliens is a &lt;strong&gt;corollary&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5890" LastEditorUserId="5890" LastEditDate="2015-11-18T01:27:54.197" LastActivityDate="2015-11-18T01:27:54.197" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10016" PostTypeId="2" ParentId="5852" CreationDate="2015-11-18T01:46:22.093" Score="18" Body="&lt;p&gt;It has been nearly a year now since I've made this question, and I think I've discovered a 'magical cure'. This one simple trick has worked for &lt;strong&gt;all&lt;/strong&gt; of my classes with great success (I am feeling more like a scam-advertisement as I am typing this), though I admit I don't fully understand why it works so well, though I do have theories.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Simply kneel down when answering or checking their work.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ever since I've been doing this, my students have been &lt;strong&gt;significantly&lt;/strong&gt; more responsive and almost eager for me to have a look.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How I discovered this actually (although a little off-topic, though I thought it may be interesting) is that I was one day very tired and my feet were very sore, but still had to teach a class. One of my more shy students asked a question and I knelt down (to save my feet) to answer her question. I was blown away by how receptive she was because usually she is one of those students who has a habit of saying &quot;I &lt;em&gt;think&lt;/em&gt; I get it...&quot; when it is very clear that they don't know what's happening. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried this for the rest of the class and it worked amazingly! I then tried it for all of my classes and it yielded the same result. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My theory as to why this works is that by kneeling, you are totally committing your focus to that one student whilst also 'being on their level' rather than standing up and pointing at whatever mistake they've made. &lt;/p&gt;&#xA;" OwnerUserId="3383" LastEditorUserId="3383" LastEditDate="2015-11-18T03:30:52.383" LastActivityDate="2015-11-18T03:30:52.383" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10017" PostTypeId="1" CreationDate="2015-11-18T08:31:46.290" Score="6" ViewCount="409" Body="&lt;p&gt;&lt;strong&gt;The Plan&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the study of any special issue in mathematics, two things may be of importance, namely, &lt;strong&gt;subjects&lt;/strong&gt; and &lt;strong&gt;order of them&lt;/strong&gt;. I just wrote down a plan to study the theory of partial differential equations in a &lt;strong&gt;treatise&lt;/strong&gt; manner. My program is as follows, considering the subjects and their order based on the prerequisites and logical relationships between them:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Logic&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Topology&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Linear Algebra&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematical Analysis&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Real Analysis&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Functional Analysis&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Calculus of Variations&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Classical Theory of Ordinary Differential Equations (ODEs)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Classical Theory of Partial Differential Equations (PDEs)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I am not sure that I have chosen the proper order for &lt;strong&gt;self-studying&lt;/strong&gt; and also there may be things that I even don't know about to cover. So please tell me subjects that I should cover based on what you think and the ideal order of learning them.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It will be really nice if you suggest just &lt;strong&gt;one&lt;/strong&gt; reading in each subject that you prefer is &lt;strong&gt;the best&lt;/strong&gt; for a &lt;strong&gt;theoretical&lt;/strong&gt; study.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How much time does it take to do the whole thing? 1, 2, 3 or 4 years?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Notification&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any other plan or thought you have is welcome. Please write &lt;strong&gt;whatever&lt;/strong&gt; you think &lt;strong&gt;as answers not comments&lt;/strong&gt;! :)&lt;/p&gt;&#xA;" OwnerUserId="5892" LastEditorUserId="376" LastEditDate="2018-07-16T15:05:23.633" LastActivityDate="2018-07-18T11:43:52.547" Title="A Plan for a Treatise Study of the Classical Theory of PDEs" Tags="|mathematical-analysis|self-learning|differential-equations|" AnswerCount="2" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="10018" PostTypeId="1" CreationDate="2015-11-18T13:50:54.960" Score="4" ViewCount="4943" Body="&lt;p&gt;I am currently teaching basic mathematics like numbers and tables to a $5$ year old girl. She is in nursery. But I am facing troubles while teaching her what comes after or before of a given number, say $71$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Questions like &quot;What comes after $71$?&quot; or &quot;__ is before $78$&quot; she couldn't answer properly. She knows what &quot;after&quot; or &quot;before&quot; means. Once I taught her, she remembers it for few days then forget it again. Is there any easier way to teach her so that she remembers it properly, like some graphic way that a child can remember? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many thanks!&lt;/p&gt;&#xA;" OwnerUserId="5894" LastActivityDate="2015-11-18T15:35:19.140" Title="How to teach &quot;after-before&quot; of numbers to a $5$ year old?" Tags="|primary-education|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10019" PostTypeId="2" ParentId="10004" CreationDate="2015-11-18T13:59:10.747" Score="1" Body="&lt;p&gt;The first thing that came to mind was food, let's take Cheerios for example (dry breakfast cereal for those not aware).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt; - regular old Cheerios&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Corollary&lt;/strong&gt; - Honey Nut Cheerios, Multi-grain Cheerios, Banana Nut Cheerios (any offshoot flavor of cheerios)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lemma&lt;/strong&gt; - Oat flour, salt, sugar, flavors (the ingredients in Cheerios)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proposition&lt;/strong&gt; - An unreleased, being-tested-by-General-Mills flavor of Cheerios&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that this isn't perfect but i think it is very accessible for high school students and is appropriate for just being introduced to the concepts.  Later on, the definitions can be refined.&lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="1147" LastEditDate="2018-08-11T09:10:17.437" LastActivityDate="2018-08-11T09:10:17.437" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="10020" PostTypeId="2" ParentId="10018" CreationDate="2015-11-18T15:35:19.140" Score="9" Body="&lt;p&gt;It would seem you are being overly ambitious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to this link at &lt;a href=&quot;http://www.pbs.org/parents/childdevelopmenttracker/five/mathematics.html&quot;&gt;PBS&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Some children at the beginning of this year are still learning how to&#xA;  verbally count by ones up to &quot;ten.&quot; The average five-year-old,&#xA;  however, will be able to use the &quot;teen&quot; pattern to accurately count to&#xA;  &quot;20.&quot; (Some children may not be able to count up to &quot;20&quot; until age&#xA;  six.) Other children will be able to use repeating patterns to&#xA;  accurately count up to &quot;42,&quot; with the average child able to do this by&#xA;  the second half of this year. (Some children may not be able to count&#xA;  up to &quot;42&quot; until age six.) In the second half of this year, a few&#xA;  five-year-olds will be able to use repeating patterns to accurately&#xA;  count up to &quot;200.&quot; (The average child can count to &quot;200&quot; at age six.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This refers to counting.  Actually understanding and using the numbers at this age:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At age five, some children may still be gaining an understanding of&#xA;  the number words up to &quot;four&quot; (e.g., distinguishes one-four items from&#xA;  &quot;many&quot;; can identify collections of up to four items with a&#xA;  corresponding number; asks for up to &quot;four&quot; of something; knows age;&#xA;  can put out &quot;one,&quot; &quot;two,&quot; &quot;three,&quot; or &quot;four&quot; items upon request).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although there are gifted children - it is not clear what value there is teaching the child something s/he is not ready for if it is not expected for children of this age. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is lot more information at the link. There is additional information for children of other ages and for other subjects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used this link because it is geared for parents and the teachers at my school find it useful and accurate.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-11-18T15:35:19.140" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10021" PostTypeId="1" AcceptedAnswerId="10042" CreationDate="2015-11-18T16:32:07.347" Score="87" ViewCount="25891" Body="&lt;p&gt;This forum already has many&#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/q/220/511&quot;&gt;good, simple examples of induction&lt;/a&gt; proofs, a great resource.&#xA;As I am soon to teach induction for the $n^\textrm{th}$ time&amp;mdash;this time to some perhaps under-prepared college students (in the US)&amp;mdash;I would like to&#xA;understand &lt;em&gt;why&lt;/em&gt; induction proofs are often challenging for students to&#xA;grasp. Perhaps I have been at it so long that I am losing sight of&#xA;what makes it difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also be interested to hear if you think in fact induction proofs are&#xA;not challenging to your students.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2022-12-02T17:30:39.447" Title="Why are induction proofs so challenging for students?" Tags="|undergraduate-education|proofs|induction|" AnswerCount="21" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="10022" PostTypeId="2" ParentId="10004" CreationDate="2015-11-18T16:56:08.617" Score="-2" Body="&lt;p&gt;A love comparative is on the next TED video: &lt;a href=&quot;https://www.youtube.com/watch?v=jej8qlzlAGw&quot; rel=&quot;nofollow&quot;&gt;https://www.youtube.com/watch?v=jej8qlzlAGw&lt;/a&gt; (It's with English subtitles)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The video is a must be seen and the speaker is mathematician who mainly works doing outreach, so it's well explained. I include the explanations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It's forever&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proposition&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If never becomes a theorem, you might divorce, but I think will be forever.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of my own creation:&#xA;&lt;strong&gt;Lemma&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One thing the couple do together&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Corollary&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the documents you may decide to sign for your wedding in order to organize your marriage (that may be a theorem or a proposition)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5895" LastEditorUserId="143" LastEditDate="2015-11-18T17:07:56.760" LastActivityDate="2015-11-18T17:07:56.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10023" PostTypeId="2" ParentId="10021" CreationDate="2015-11-18T17:28:29.640" Score="39" Body="&lt;p&gt;In my experience, the biggest issue is that students don't have a clear grasp of quantifiers, so they don't see the distinction between &quot;for all n P(n)&quot; and &quot;consider an n such that P(n)&quot;.  This leads to common errors like using P(n) to prove P(n), or to thinking the method is circular because we assume P(n) to prove P(n).&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-11-18T17:28:29.640" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10024" PostTypeId="2" ParentId="10021" CreationDate="2015-11-18T19:56:52.203" Score="10" Body="&lt;p&gt;Seems to me that there are (at least) two types of induction problems: 1) Show something defined recursively follows the given explicit formula (e.g. formulas for sums or products), and 2) induction problems where the relation between steps is not obvious (e.g. Divisibility statements, Fund. Thm. of Arithmetic, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For 1) the problem typically seems to be that students don't understand the recursion, e.g. that $\sum_{i=1}^{n+1}f(i) = \sum_{i=1}^nf(i)+f(n+1)$. If they do understand this (but still can't do it), then they are typically just weak in the algebra skills required to manipulate after the inductive substitution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For 2) the problem is just that: the relation between steps is not obvious and some intuition/experience is required to find the $n$th statement within the $(n+1)$th case. (Heck, even I struggle with this at times when going over homework problems.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A note on base cases: There are a few instances I've noticed where the base case is so simple that students struggle to know what to write down to prove it. For example, they'll write something trivial like $0=0$ with a check mark instead of $1^2-1=0=\frac{1}{3}(1-1)(1)(1+1)$. I think this just comes from years of schooling that emphasized the end result (and omitting mental calculations in work shown) instead of the process that brought it about.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-11-18T19:56:52.203" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10025" PostTypeId="2" ParentId="10021" CreationDate="2015-11-18T20:24:32.090" Score="31" Body="&lt;p&gt;There is a fair amount of research on students' understanding of (and difficulties with) proof by induction.  Some good places to start:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Palla, M., Potari, D., and Spyrou, Panagiotis. (2012) Secondary school&#xA;  students' understanding of mathematical induction: Structural&#xA;  characteristics and the process of proof construction. &lt;em&gt;International&#xA;  Journal of Science and Mathematics Education&lt;/em&gt; 10(5), 1023-1045.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://link.springer.com/article/10.1007%2Fs10763-011-9311-2&quot;&gt;The abstract of this paper&lt;/a&gt; reads:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In this study, we investigate the meaning students attribute to the&#xA;  structure of mathematical induction (MI) and the process of proof&#xA;  construction using mathematical induction in the context of a&#xA;  geometric recursion problem. Two hundred and thirteen 17-year-old&#xA;  students of an upper secondary school in Greece participated in the&#xA;  study. Students’ responses in 3 written tasks and the interviews with&#xA;  18 of them are analyzed. Though MI is treated operationally in school,&#xA;  the students, when challenged, started to recognize the structural&#xA;  characteristics of MI. In the case of proof construction, we&#xA;  identified 2 types of transition from argumentation to proof,&#xA;  interwoven in the structure of the geometrical pattern. In the first&#xA;  type, MI was applied to the algebraic statement that derived from the&#xA;  direct translation of the geometrical situation. In the second type,&#xA;  MI was embedded functionally in the geometrical structure of the&#xA;  pattern.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;http://link.springer.com/article/10.1007/s10857-007-9034-z&quot;&gt;second good paper&lt;/a&gt; is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Stylianides, G., Stylianides, A. and Philippou, G. (2007)  Preservice&#xA;  teachers’ knowledge of proof by mathematical induction.  &lt;em&gt;Journal of&#xA;  Mathematics Teacher Education&lt;/em&gt; 10(3), 145-166.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The abstract reads:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is a growing effort to make proof central to all students’&#xA;  mathematical experiences across all grades. Success in this goal&#xA;  depends highly on teachers’ knowledge of proof, but limited research&#xA;  has examined this knowledge. This paper contributes to this domain of&#xA;  research by investigating preservice elementary and secondary school&#xA;  mathematics teachers’ knowledge of proof by mathematical induction.&#xA;  This research can inform the knowledge about preservice teachers that&#xA;  mathematics teacher educators need in order to effectively teach proof&#xA;  to preservice teachers. Our analysis is based on written responses of&#xA;  95 participants to specially developed tasks and on semi-structured&#xA;  interviews with 11 of them. The findings show that preservice teachers&#xA;  from both groups have difficulties that center around: (1) the essence&#xA;  of the base step of the induction method; (2) the meaning associated&#xA;  with the inductive step in proving the implication P(k) ⇒ P(k + 1) for&#xA;  an arbitrary k in the domain of discourse of P(n); and (3) the&#xA;  possibility of the truth set of a sentence in a statement proved by&#xA;  mathematical induction to include values outside its domain of&#xA;  discourse. The difficulties about the base and inductive steps are&#xA;  more salient among preservice elementary than secondary school&#xA;  teachers, but the difficulties about whether proofs by induction&#xA;  should be as encompassing as they could be are equally important for&#xA;  both groups. Implications for mathematics teacher education and future&#xA;  research are discussed in light of these findings.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You may also want to look at &lt;a href=&quot;http://eric.ed.gov/?q=proof%20induction&amp;amp;id=ED489630&quot;&gt;this link to a PME-NA paper by Tommy Dreyfus and Gila Ron&lt;/a&gt; which describes &quot;explanations, models and examples that distort the underlying mathematical ideas and show teachers’ conceptual difficulties&quot; with mathematical induction.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2015-11-21T14:26:28.277" LastActivityDate="2015-11-21T14:26:28.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10026" PostTypeId="2" ParentId="10021" CreationDate="2015-11-18T23:18:40.007" Score="16" Body="&lt;p&gt;As someone who took math courses but does not teach, I would claim that inductive techniques are taught with two rather separate approaches:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A step-by-step recipe to take a problem that tells you to use induction and write down the series of symbols such as &quot;n&quot; and &quot;n+1&quot; in the right order to get full credit for the problem.&lt;/li&gt;&#xA;&lt;li&gt;An understanding of how induction actually works, and why we use it.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Induction is a very axiomatic process.  If you try to dig down to the &quot;rule&quot; that shows that it works, it ends up being the Axiom of Induction from the Peano Axioms.  Its actually the only second order axiom in the entire set of Peano Axioms!  Induction is so fundamental that we don't even have an explanation for why it works; we accept it as an axiom for use in proofs involving natural numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student has not developed a strong inductive reasoning skillset, this axiom will appear foreign to them.  they may be able to see that f(0) f(1) f(2) and f(3) are all true, leading onwards, but the step to f(N) is true for all N is actually a leap of logic requiring students to develop a way to use their brains that they didn't use before.  (On the other hand, if a student has strong inductive reasoning skills, they may pick it up quite easily, frustrating all of their friends.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do know that many approaches for teaching mathematics decry intuitive approaches such as induction until the day they show inductive proofs.  In many situations (especially proofs), its deductive logic or bust!  If a student is not aware that it's okay to think inductively in mathematics, they may have put up barriers to prevent themselves from doing something that gets them a bad grade.  Inductive proofs are deemed an acceptable way to put inductive reasoning into a field that is otherwise taught as deduction-dominated, so it takes a while for them to click.  You may have to nurture them to encourage it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just to brainstorm a potential solution (I haven't tried it yet), try to give students a simple induction word problem, and ask them to &quot;convince themselves&quot; that the statement in the problem is true.  They don't have to &quot;prove&quot; it mathematically, they need to prove it to themselves enough to feel comfortable that the statement is true, and write down their logic on the paper.  Then you can look at what they have done, and try to identify the spark of induction in their work, and try to kindle it into a form that you can give traditional mathematical syntax to.  We all do inductive logic (its one of the major branches of thought as defined by the Greek philosophers), so its just a matter of convincing them its safe to use it.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2015-11-18T23:18:40.007" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10028" PostTypeId="2" ParentId="10021" CreationDate="2015-11-19T04:57:42.400" Score="10" Body="&lt;p&gt;The reason why learning mathematical induction is difficult is developmental. Your audience of under-prepared college students probably accept &quot;proof by verifying the first few cases&quot; as more convincing than proof using the formalism of mathematical induction. Sure, they grasp that &quot;proof by verifying the first few cases&quot; is somehow inadequate, even though they find it convincing. But proof by mathematical induction to them is too abstract and formal, and hence not emotionally convincing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It just takes time for mathematical maturity to develop. Gradually mathematical induction becomes less abstract and eventually it seems intuitive and obvious. Analogy: it was really hard to learn how to tie my shoe laces when I was young, but now I am bewildered when I see a young person having difficulty learning how to tie their own shoelaces.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-11-19T04:57:42.400" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10029" PostTypeId="2" ParentId="10004" CreationDate="2015-11-19T05:15:45.817" Score="1" Body="&lt;ul&gt;&#xA;&lt;li&gt;A proof is a path of stepping stones over a creek.&lt;/li&gt;&#xA;&lt;li&gt;Every stone is a proposition. &lt;/li&gt;&#xA;&lt;li&gt;Whenever you get to a milestone, you get to a lemma.&lt;/li&gt;&#xA;&lt;li&gt;The stone right after a milestone is a corollary. &lt;/li&gt;&#xA;&lt;li&gt;When you get to the other side, you get to the theorem. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="961" LastActivityDate="2015-11-19T05:15:45.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10030" PostTypeId="2" ParentId="5768" CreationDate="2015-11-19T13:20:37.600" Score="5" Body="&lt;p&gt;Another thought occurred to me regarding this, after having read &lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=mbk-83&quot; rel=&quot;nofollow&quot;&gt;Reuben Hersh's collection of essays&lt;/a&gt;. There is a quote of Bill Thurston, which I paraphrase as &quot;thinking is the same as seeing&quot;. In a sense, having the logic of one's proof &quot;at the tip of the tongue&quot; produces a sense of unity in the proof that is analogous to an object one can mentally manipulate. This sense is like the &lt;em&gt;reification&lt;/em&gt; phenomenon observed by mathematics educators: the proof process is &quot;reified&quot; into an object (the proof) that can be thought about as a coherent whole. The process of &quot;reifying&quot; a proof can be greatly aided by mathematical writing that shines light on potentially obscure details that threaten the coherence or consistency of the object...how it fits together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From this viewpoint, mathematical writing and calculation stand on an equal footing. Both serve as &lt;em&gt;processes to be reified into mental objects&lt;/em&gt;. Oddly, our mathematical activity wants to change verbs into nouns. We've all experienced this phenomenon when we sit to calculate something in order to gain intuition or feelings for a mathematical object or theory. The calculation is incorporated in our memory as an &lt;em&gt;aspect&lt;/em&gt; or &lt;em&gt;vantage point&lt;/em&gt; of the theory. We are looking at a new part of the theory. The attitude here is that whenever we DO something in order to understand a mathematical object, we are LOOKING at another facet of the object. These facets fit together as our mental picture of the object as a whole. We experience this feeling of the object as a whole as mathematical intuition, and get a sense of &quot;what's going on&quot;.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2015-11-19T14:43:06.107" LastActivityDate="2015-11-19T14:43:06.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10031" PostTypeId="2" ParentId="1455" CreationDate="2015-11-19T13:36:44.493" Score="2" Body="&lt;p&gt;If the error is a little one then I pass it off as intentional, &lt;strong&gt;for their learning&lt;/strong&gt;. However, if the error is more substantial then I go stand in the corner for a little while. The students then gain enjoyment from my suffering and all is well. When you teach as much as many of us do, making some errors is an inevitability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I am most annoyed by is when I make an error and &lt;strong&gt;nobody&lt;/strong&gt; says anything. This is one of the few things which makes me angry. Usually this doesn't happen, but, if it does happen it seems that the end of class is the prime time. Naturally, many students would rather let my mistake fester than correct it and go overtime by a minute.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-11-19T13:36:44.493" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10032" PostTypeId="2" ParentId="10004" CreationDate="2015-11-19T14:09:27.893" Score="2" Body="&lt;p&gt;The first thing to bear in mind is that a lemma &lt;em&gt;is&lt;/em&gt; a theorem, and usually so is a proposition (although in certain uses in English it might make sense to talk about an &quot;incorrect proposition&quot;, which then naturally isn't a theorem, rather it's a failed conjecture). Everything proved is a theorem. But we only give a theorem a name like &quot;Pythagoras' Theorem&quot; if we think it's really important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming every thing is proved or will be proved by the end of the exercise, though, then:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A theorem is something you are constructing.&lt;/li&gt;&#xA;&lt;li&gt;Propositions and lemmas are components that themselves require construction before incorporating them into the main thing. The difference between the two is usually not important, but the more impressive your proposition is the more likely you are to call it a lemma.&lt;/li&gt;&#xA;&lt;li&gt;Corollaries are things that your theorem can be incorporated into the construction of.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;quid's answer gives a specific example of this pattern with &quot;car&quot; as the theorem, but you should be able to apply the pattern to any construction process that's complex enough to have recognisable component parts. In particular, one might wish to reflect the fact that some theorems use their lemmas more than once in their proof, by using duplicated parts (wheels for cars, arches for cathedrals, seats for movie theatres, whatever) for the lemma in the analogy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the interesting things about lemmas is that you don't always write your proof such that the lemma is proved before it is used (hence the name, I believe). You can leave the proof until the end if it make the presentation clearer. It's probably useful to draw attention to this fact in the analogy: one might design and even part-construct a car knowing that it's going to &lt;em&gt;have&lt;/em&gt; wheels with a certain specification, but actually leave off designing and constructing the wheels until long after the point where we've staked our reputation on them appearing before the end. One cannot complete the car without putting actual wheels in: what we have before that is an incomplete car with missing parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often a conjecture is something that we hoped to use as a lemma, but find we cannot prove even though we've done a lot of work proving that &lt;em&gt;if&lt;/em&gt; the conjecture is true, &lt;em&gt;then&lt;/em&gt; some useful theorem follows. By (facetious) analogy there might be a fantastic new underwater car that we'll build just as soon as we can figure out how to make waterproof windows. Then these waterproof windows, that nobody knows how to make or if it's even possible, are a conjecture and the underwater car is a consequence (or corollary) of the conjecture being true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We often use the word &quot;corollary&quot; to indicate a theorem that's very easily proved from our main theorem. &quot;Corollary 1 of Theorem 12&quot;, &quot;Corollary 2 of Theorem 12&quot;, etc. We're using the term to help organise our thoughts. We also sometimes use it for something more involved: &quot;Theorem 93 is a corollary of Theorem 12, the proof follows ... [12 pages later] QED&quot;. What we really mean then, is that Theorem 12 can be used to prove Theorem 93. So in the analogy it would probably be useful to offer more than one corollary, at different &quot;distance&quot; from the theorem.&lt;/p&gt;&#xA;" OwnerUserId="321" LastEditorUserId="321" LastEditDate="2015-11-19T14:33:45.137" LastActivityDate="2015-11-19T14:33:45.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10033" PostTypeId="2" ParentId="10021" CreationDate="2015-11-19T17:03:03.227" Score="15" Body="&lt;p&gt;&lt;strong&gt;Logic foundation&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, the only way for anyone to really understand induction is to really understand the logical structure behind it. So a prerequisite is a complete grasp of working in first-order logic, for which I recommend both boolean algebra and natural deduction (&lt;a href=&quot;https://math.stackexchange.com/a/1684204/21820&quot;&gt;Fitch-style&lt;/a&gt;) in conjunction. It is unfortunate that many people, teachers and students alike, don't actually appreciate how important this is, and hence they try to teach induction to students who are so weak in logic that they cannot possibly know what they understand or do not understand about induction.&lt;/p&gt;&#xA;&lt;p&gt;In contrast, for someone who is capable of working in first-order logic, the axiom of induction would be trivial to state and easy to use with confidence. Not only that, when students are at that point they will understand the subtle difference between reasoning in first-order logic and reasoning outside, which is crucial to a full understanding of induction. The reason is that induction has a meta-logical basis, not a logical one. All the other deduction rules are trivial to check via truth-tables, but the underlying intuition for induction is not at all the same.&#xA;&lt;span class=&quot;math-container&quot;&gt;$\def\nn{\mathbb{N}}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\def\imp{\rightarrow}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Intuition for Induction&lt;/strong&gt; (after the logic foundation is firm)&lt;/p&gt;&#xA;&lt;p&gt;Basically we could &lt;strong&gt;define&lt;/strong&gt; natural numbers as exactly those that we can get by counting upwards from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. What does counting upwards mean? Well, the usual one, which is incrementing the previous number by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. Using the decimal system it would go &lt;span class=&quot;math-container&quot;&gt;$0,1,2,3,4,5,6,7,8,9,10,\cdots$&lt;/span&gt;. For simplicity let's just assume that for any natural number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; is the next natural number after &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now let's say we have some 1-input predicate &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; on the natural numbers and we have proven:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(0)$&lt;/span&gt;. (1)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall n \in \nn\ ( P(n) \imp P(n+1) )$&lt;/span&gt;. (2)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Without induction&lt;/strong&gt;, what can we do? Certainly, one can prove:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(0) \imp P(1)$&lt;/span&gt; [from instantiating (2)]. (3)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(1)$&lt;/span&gt; [from (1) and (3)]. (4)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(1) \imp P(2)$&lt;/span&gt; [from instantiating (2)]. (5)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(2)$&lt;/span&gt; [from (4) and (5)]. (6)&lt;/p&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is now clear that given any natural number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; written in decimal notation, one can prove &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; by continuing the above proof up to a length of &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; lines. Of course, this fact about &lt;strong&gt;what we can prove&lt;/strong&gt; is not something proven within the formal system itself, but is an &lt;strong&gt;external fact observed from the outside&lt;/strong&gt;. All this fact requires is that we have the same common understanding about strings from a fixed alphabet (with which we form proofs) and how we can manipulate strings (joining them together, reading each symbol in order, ...). But it is &lt;strong&gt;not a fact provable within the formal system&lt;/strong&gt;. In other words, this fact is a meta-logical fact. &amp;quot;Meta-logic&amp;quot; means &amp;quot;about logic&amp;quot;, since we're now thinking not within the logical framework we set up but thinking &lt;strong&gt;about&lt;/strong&gt; the logical framework from the outside.&lt;/p&gt;&#xA;&lt;p&gt;So there are two questions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(A) Is it true that for any &lt;span class=&quot;math-container&quot;&gt;$n \in \nn$&lt;/span&gt;, there some proof that &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; is true? Yes, and we've shown above how to construct one.&lt;/p&gt;&#xA;&lt;p&gt;(B) Is it true that there some proof that &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; is true for any &lt;span class=&quot;math-container&quot;&gt;$n \in \nn$&lt;/span&gt;? No!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And this is &lt;strong&gt;why&lt;/strong&gt; we want to have the axiom of induction, because if our deductive rules are sound, then according to our definition of natural numbers the answer to (A) shows that in fact &lt;span class=&quot;math-container&quot;&gt;$\forall n \in \nn\ ( P(n) )$&lt;/span&gt; is a true assertion, and so we are &lt;strong&gt;justified in adding induction as an axiom&lt;/strong&gt; so that the answer to (B) becomes &amp;quot;Yes&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Together with the earlier comment about the meta-logical intuition behind induction, it should now be even clearer why the induction axiom is really a meta-logical axiom intrinsically. The quantification in (A) is &lt;strong&gt;outside&lt;/strong&gt; the formal system with the usual deductive rules, while the quantification in (B) is &lt;strong&gt;inside&lt;/strong&gt; the formal system. It essentially stipulates that whenever we can systematically generate a proof of &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$n \in \nn$&lt;/span&gt; in the above manner, we can &lt;strong&gt;transfer this meta-logical universal quantifier inside the formal system itself&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Notes&lt;/strong&gt; (only for the teacher, not the student learning induction!)&lt;/p&gt;&#xA;&lt;p&gt;I used &amp;quot;decimal system&amp;quot; in my above explanation so that it is understandable to those who are not familiar with formal systems, but of course one has to be careful with the difference between the natural number and an encoding of it. In formal logic, usually the language of arithmetic only has a single constant &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, the successor function symbol &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;, and addition and multiplication function symbols, and of course equality. In that case there may be no such thing as decimal notation in the formal system itself, but it's no harm to add them in, for example we use &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;&amp;quot; instead of &lt;span class=&quot;math-container&quot;&gt;$S(S(S(0)))$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Also, induction does not imply that (A) always implies (B). If (A) always implies (B), then the formal system is ω-complete. However, PA is ω-incomplete if it is consistent, because it cannot prove &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Con}(PA)$&lt;/span&gt;, which is a sentence of the form &lt;span class=&quot;math-container&quot;&gt;$\forall n\ ( \neg \mathrm{CodesAProofOfFalse}(n) )$&lt;/span&gt;, even though it can prove &lt;span class=&quot;math-container&quot;&gt;$\neg \mathrm{CodesAProofOfFalse}(n)$&lt;/span&gt; for each natural number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; encoded as &lt;span class=&quot;math-container&quot;&gt;$\underbrace{1+1+\cdots+1}_{\text{$n$ times}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Finally, note that throughout this answer, in the meta-logic &amp;quot;natural number&amp;quot; refers to &lt;strong&gt;the true natural numbers&lt;/strong&gt; (which is ultimately undefinable), while in the logic &amp;quot;natural numbers&amp;quot; are simply defined by the properties they obey, such as the axiom of induction. The axioms including induction serve as part of how we characterize the natural numbers, but no recursive axiomatization can ever fully characterize them. So although the true natural numbers are exactly those that we can get by counting upwards from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, there will never be a systematic non-circular definition of them. (In particular, &amp;quot;we can get by counting upwards&amp;quot; presupposes natural steps.)&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="1550" LastEditDate="2022-12-02T17:30:39.447" LastActivityDate="2022-12-02T17:30:39.447" CommentCount="20" ContentLicense="CC BY-SA 4.0" />
  <row Id="10034" PostTypeId="2" ParentId="10021" CreationDate="2015-11-19T17:30:26.523" Score="25" Body="&lt;p&gt;For CS students specifically, there is another approach that would work better than the usual way induction is taught, namely by teaching structural induction, which goes like this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you want to prove that a collection &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; of finite structures (such as binary trees) satisfy a property &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, then all you have to do is to show that for &lt;strong&gt;any arbitrary&lt;/strong&gt; structure &lt;span class=&quot;math-container&quot;&gt;$x \in S$&lt;/span&gt; that &lt;strong&gt;I give you&lt;/strong&gt;, you can do either of the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Show me that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; satisfies &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; &lt;strong&gt;without any assumptions&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Break &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; down into strictly smaller structures such that, if &lt;strong&gt;I show you&lt;/strong&gt; that all of them satisfy &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, then &lt;strong&gt;you can show me&lt;/strong&gt; that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; also does.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now this may seem silly, as it is even stronger than induction, and involves the same essential steps. But it is very different, because students &lt;strong&gt;always&lt;/strong&gt; (&lt;strong&gt;wrongly&lt;/strong&gt;) take a structure of size &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; satisfying &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and build a structure of size &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; from it and shows that it satisfies &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, and then claim that by induction all the structures satisfy &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;! Using this formulation of strong induction, that mistake is prevented from happening, because they have to start from an arbitrary structure and &lt;strong&gt;reduce it&lt;/strong&gt; to smaller structures.&lt;/p&gt;&#xA;&lt;p&gt;This doesn't solve the problem of not understanding the logical underpinnings of induction, but at least it solves the problem of using induction wrongly. The other important point is that it presents the quantification involved as a game, which some students grasp much faster than quantifier chains.&lt;/p&gt;&#xA;&lt;p&gt;There is another advantage that this has for CS students over the usual presentation of induction, which is that following the proof it is easy to construct the recursive algorithm corresponding to it. For example the proof that every finite binary tree has a depth translates straightforwardly and cleanly into the recursive algorithm. Also, the recursive structure lends itself immediately to the idea of storing the relevant data in each substructure, in this case the depth of each subtree.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-11-19T17:40:17.957" CommentCount="25" ContentLicense="CC BY-SA 3.0" />
  <row Id="10035" PostTypeId="2" ParentId="10021" CreationDate="2015-11-20T00:36:58.420" Score="5" Body="&lt;p&gt;I'll not touch the problem of convincing your students that induction actually proves anything, but just some practical problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually you start: A (0) is true, A (n) implies A (n+1) for all n ≥ 0, =&gt; A (n) is true for all n ≥ 0. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often the induction step doesn't work for n = 0 or n = 2 or other small n. Student must learn that instead they can prove A (n) is true for n = 0, n = 1, ..., n = k. A (n) implies A (n+1) for all n ≥ k. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often the induction step doesn't work at all. Student must learn that often you can find a stronger statement B (n) which implies A (n) and which can be proved by induction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often a step n-&gt;n+1 doesn't work. Student must learn that they can instead prove: If A (k) is true for all k ≤ n then A (n+1) is true. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also good to learn is another principle: If A (n) is &lt;em&gt;not&lt;/em&gt; true for all n ≥ 0 then there is a smallest such n. Assume that there is a smallest n where A (n) is false and prove that there is an even smaller one. Which is of course complete equivalent to induction. &lt;/p&gt;&#xA;" OwnerUserId="1723" LastEditorUserId="1723" LastEditDate="2015-11-20T00:42:10.123" LastActivityDate="2015-11-20T00:42:10.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10036" PostTypeId="2" ParentId="9969" CreationDate="2015-11-20T00:51:14.300" Score="0" Body="&lt;p&gt;Notice the word &quot;Secret&quot; in &quot;Secret Santa&quot;: The recipient is not supposed to know who his or her Santa is. If presents were &lt;em&gt;swapped&lt;/em&gt; then the receiver of my present would be presenting the present for me, so I would know who my Secret Santa is and so would they. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's why the receivers are a permutation of the givers, with no fixed point (because nobody should buy their own present), and nobody being told that they are part of a cycle of length 2. &lt;/p&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2015-11-20T00:51:14.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10037" PostTypeId="1" AcceptedAnswerId="10043" CreationDate="2015-11-20T02:37:51.883" Score="9" ViewCount="454" Body="&lt;p&gt;The title says it all, I suppose. I'm interested to know when/where the term/technique &lt;em&gt;cross multiply&lt;/em&gt; came into use. Sources would be nice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case it's unfamiliar to anyone, or in case the usage of the term varies, I'm referring to the compound arithmetic operation where an equation of two fractions is multiplied on both sides by its two denominators. It's named because it can be illustrated with a big cross:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{a}{b} = \frac{c}{d}$$&#xA;&lt;a href=&quot;https://i.stack.imgur.com/6VSk8.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6VSk8.png&quot; alt=&quot;The cross!&quot;&gt;&lt;/a&gt;&#xA;$$ad = cb$$&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2015-11-20T02:43:40.007" LastActivityDate="2015-11-21T00:56:02.097" Title="When did the term and taught technique 'cross multiplication' enter into common use?" Tags="|mathematical-pedagogy|terminology|vocabulary|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10039" PostTypeId="1" CreationDate="2015-11-20T05:50:57.373" Score="9" ViewCount="3474" Body="&lt;p&gt;I am working on some research on implementation of oral assessment in mathematics classroom, and I was wondering are there any questions/problems/concepts in mathematics that can be only assessed through Oral Examination (or types of questions where oral assessment would be more sufficient than written)?&lt;/p&gt;&#xA;" OwnerUserId="5913" LastEditorUserId="5913" LastEditDate="2015-11-20T06:15:22.453" LastActivityDate="2015-11-20T21:15:47.320" Title="Questions for Oral Examination in Mathematics" Tags="|examples|exams|assessment|oral-exams|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10040" PostTypeId="1" CreationDate="2015-11-20T12:18:42.270" Score="21" ViewCount="1713" Body="&lt;p&gt;One of my main research areas is algebraic thinking at different levels. Yet, from time to time, I observe something that I cannot relate to anything else that I know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the story of one of these things that happened today in my calculus class. As part of using the ratio test, students needed to work with factorials. One of them interpreted $(2k-1)!$ as $1\cdot3\cdot5\cdots(2k-1)$. It took quite a while for me to help him to see the correct interpretation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, it does not look like it was just a mistake. If not a mistake, what was it? Can you see it as an example of a more general algebraic (or something else) misconception? Or alternatively, can you point to a &quot;similar&quot; phenomenon (mistake, error, misconception, or whatever you want to call it)? &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="4746" LastEditDate="2015-11-20T19:53:45.867" LastActivityDate="2015-11-20T19:53:45.867" Title="Is this just a mistake or a more serious misconception?" Tags="|algebra|students-mistakes|functions|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10041" PostTypeId="2" ParentId="10039" CreationDate="2015-11-20T13:13:59.123" Score="5" Body="&lt;p&gt;I worked with elementary school and preferred oral assessment with the following types of questions.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Students had to explain their thinking.  Very often students had an easier time explaining what they thought when they could say it, then when they had to write it.&lt;/li&gt;&#xA;&lt;li&gt;Doing mental math - It doesn't make sense to give them a pencil and paper and tell them to do mental math.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I also found oral assessment easier for student who have test anxiety and for those who had trouble writing (perhaps because of a learning disability or because they hurt their hand).&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-11-20T13:13:59.123" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10042" PostTypeId="2" ParentId="10021" CreationDate="2015-11-20T13:40:23.590" Score="86" Body="&lt;p&gt;The following list comes from a combination of reading various research articles and my own experience helping students in my Maths Learning Centre for the last seven years.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Some reasons why students find induction difficult:&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Many students don't know what proof is.&lt;/li&gt;&#xA;  &lt;li&gt;Many students don't realise it's actually about statements.&lt;/li&gt;&#xA;  &lt;li&gt;Many students don't have experience in manipulating inequalities and divisibilities.&lt;/li&gt;&#xA;  &lt;li&gt;Many students are philosophically uncomfortable with it.&lt;/li&gt;&#xA;  &lt;li&gt;Many students find analogies unhelpful.&lt;/li&gt;&#xA;  &lt;li&gt;Many students don't know what to focus on to come up with something to prove.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And here's some further explanation of each:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1. Many students don't know what proof is&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For many students, the problem with induction proofs is wrapped up in their general problem with proofs: they just don't know what a proof is or why you need one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most students starting out in formal maths understand that a proof convinces someone that something is true, but they use the same reasoning that convinces them that everyday things are true: empirical reasoning. That is, in everyday life they can use several pertinent examples to convince themselves something is true, so they do the same with maths. If you ask them to prove that, say, all prime numbers beyond 3 are one more or one less than a multiple of 6, they'll say &quot;7 = 6+1, 11 = 12-1, 13 = 12+1, 17 = 18-1, so yes it's true&quot;. They transfer this approach to mathematical induction too. [2]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After some experience with maths teachers, students become aware that this is unacceptable. For many it is literally unacceptable in the sense that &quot;my maths teacher doesn't accept this sort of thing&quot;. So, they learn to apply other methodologies, such as manipulating symbols [2], and think that now this constitutes proof, because it is what is convincing to their teachers. They will attempt to apply symbol manipulations to an induction problem, which often won't work, because the logical structure of induction is different to their previous experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's probably a good idea to get the students to discuss what constitutes a proof, and bring out ideas of why it's necessary to use certain ways of arguing in maths.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2. Many students don't realise it's actually about statements.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can define mathematical induction as being sure the statement &quot;true for n=1&quot; is the truth, being able to transform the statement of &quot;true for n=k&quot; into the statement &quot;true for n=k+1&quot;. As such, it's actually something you do to statements, rather than objects or numbers per se. That means it's much closer to propositional calculus than, say, a geometry or algebra proof. If your students have learned any propositional calculus, this may help them. [1]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, knowing it's about statements helps them to structure their proof appropriately because they know it hangs upon certain statements in certain places. This is something many students don't realise about proofs in general, and I find it really helps them to focus their attention when they prove almost anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3. Many students don't have experience in manipulating inequalities and divisibilities&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether students are attacking induction proofs operationally or with an understanding of how the idea actually works, they often don't have the operational skill with the types of algebraic manipulations required. [3] Standard fare for induction proofs are inequalities and divisibilities, and most students I meet simply don't know how to reason with these, even if they've done the highest level of maths at school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, consider a sequence of working like this:&#xA;$$&#xA;\begin{align*}&#xA;(n+1)^2 &amp;amp;= n^2 + 2n+1\\&#xA;&amp;amp;&amp;gt; 2n + 2n + 1\\&#xA;&amp;amp;\geq 2n + 6 + 1 \quad \text{Since } n \geq 3\\&#xA;&amp;amp;&amp;gt; 2n + 2\\&#xA;&amp;amp;= 2(n+1)&#xA;\end{align*}&#xA;$$&#xA;Most students don't understand that each equality/inequality symbol only refers to the step immediately before it, so that the final statement is actually $(n+1)^2 &amp;gt; 2(n+1)$. They also don't realise you can replace the &quot;6+1&quot; with &quot;2&quot; because you want to make a statement that one number is bigger than another. Moreover, they don't have well-developed instincts for what they should choose to do next to advance towards the goal. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Divisibility proofs generally involve rewriting things so that they have the divisor out the front, both for the k and the k+1 case. Many students don't realise this is what divisibility means, and also have trouble seeing how to split up the expression to sub in the induction hypothesis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you show any examples of doing a proof by induction in these situations, you're going to have to be extremely explicit about your ordinary algebraic reasoning at every step, and how you made the decision of what to do on every line, so that you can help them develop these skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4. Many students are philosophically uncomfortable with it&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some students, no matter how many analogies of dominoes or ladders you describe, will still just feel that it's somehow all a bit too unbelievable. You start by assuming it's true, and prove something with an unknown value of n, and then somehow the thing is proved for all values of n? It seems a bit too much like Baron Munchausen pulling himself up by his own hair. A common phrase the students utter is &quot;it feels like magic&quot;. They can't even consider giving it a go with an actual proof assignment question when they feel so philosophically uncomfortable with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Partly this is due to them not understanding that you actually aren't assuming it's true for any specific value of $n$ at all. You're doing a thought experiment of what would happen if it &lt;em&gt;was&lt;/em&gt; true for $n=k$.  [3] I use the phrase &quot;suppose true when n=k&quot; rather than &quot;assume true when n=k&quot; to emphasise this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if they do realise this, some students still want to somehow prove to themselves that it's ok to just say &quot;therefore it's true for all n&quot;. A mathematically mature student will often understand things by proving to themselves that they work (such as deriving the quadratic formula). However, you &lt;em&gt;can't&lt;/em&gt; derive mathematical induction. It's an axiom. Telling them this removes the burden of proof and they can feel slightly better about using it. Alternatively, you could prove it from the well-ordering of the natural numbers, and that might make them feel better if they think the well-ordering is a more obvious sort of axiom to use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5. Many students find analogies unhelpful&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everyone always advocates using the domino analogy for teaching induction, but there are problems with this. One is that some students simply will have never seen dominoes set up in a line to knock down! If you're going to use this analogy, bring actual dominoes! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A second problem is that if you are not very specific about exactly what in the domino analogy corresponds to what in the induction proof, students will develop unhelpful ideas about induction. This of course goes for any analogy you might use, such as climbing a ladder or a game of whispers. [4]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Worse than this is the problem that no amount of very salient analogies will actually help them to attack an actual problem when faced with it. [5] Analogies are really only good for helping them understand how induction works in a general way. You're going to have to give them examples and experiences of the act of coming up with a proof because that is a separate skill from describing how induction works in a general way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;6. Many students don't know what to focus on to come up with something to prove&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students can actually become quite successful in solving your standard identity, inequality and divisibility induction proofs. But anything other than this leaves them completely stumped. Mostly this is because they have learned some standard classes of problems and how to deal with them. (Many students actually think this is what mathematics &lt;em&gt;is&lt;/em&gt; -- a list of problems and how to solve each one!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When faced with something entirely different (eg the Tower of Hanoi, or a visual pattern), they don't know what to focus on. First, you need to help them find where the natural number is that describes the possible situations. [3] Induction only works on natural numbers, and is the gold standard of proving things for all natural numbers, so you really need to look for a natural number! But it has to describe the situation well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After this, some authors suggest getting them to focus on a &lt;em&gt;process pattern generalisation&lt;/em&gt;. [2,3] That is, figure out how you would move from size 1 to size 2, from size 2 to size 3 etc, and see if you can figure out if the process of moving from one size to the next follows a pattern. This pattern is the shape of your proof of &quot;true for n=k implies true for n=k+1&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An &lt;em&gt;unhelpful&lt;/em&gt; thing to focus on is verifying that the formula works for each number in turn. This is just reinforcing their empirical reasoning, which won't help them figure out how to prove! [2]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;References:&lt;/em&gt;&lt;br&gt;&#xA;[1] Dubinsky, E. (1986) Teaching Mathematical Induction I, Journal of Mathematical Behavior, 5, 305-317&lt;br&gt;&#xA;[2] Harel, G. (2002) The Development of Mathematical Induction as a Proof Scheme: A DNR-Based Instruction, in Campbell, S. &amp;amp; Zazkis, R. (Eds.) Learning and Teaching Number Theory: Research in Cognition and Instruction, Greenwood Publishing Group&lt;br&gt;&#xA;[3] Palla, M., Potari, D. &amp;amp; Spyrou, P. (2012) Secondary school students' understanding of mathematical induction: structural characteristics and the process of proof construction, International Journal of Science and Mathematics Education, 10, 1023-1045&lt;br&gt;&#xA;[4] Ron, G. &amp;amp; Dreyfus, T. (2004) The use of models in teaching proof by mathematical induction, in Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education,  113-120&lt;br&gt;&#xA;[5] Segal, J. (1998) Learners' difficulties with induction proofs, Journal of Mathematical Education in Science and technology, 29, 159-177&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2015-11-27T18:42:14.697" LastActivityDate="2015-11-27T18:42:14.697" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="10043" PostTypeId="2" ParentId="10037" CreationDate="2015-11-20T14:08:06.420" Score="8" Body="&lt;p&gt;The earliest example of the OP's use I can find on Google Books is 1908. It was also used to describe adding fractions about the same time (although there is a use in a professional mathematics article in 1833).  It starts to be used often in the late 1930s and shoots up again in mid 1970s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some early references to school mathematics from Google Books:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com/books?id=z-hAAQAAMAAJ&amp;amp;pg=PA126&amp;amp;dq=%22cross+multiply%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;adding fractions (1903)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com/books?id=wjpRAAAAYAAJ&amp;amp;pg=PA222&amp;amp;dq=%22cross+multiply%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;adding fractions (1905)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com/books?id=Zt1HAAAAIAAJ&amp;amp;pg=PA149&amp;amp;dq=%22cross+multiply%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;solving fractional equations (1908)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Earlier uses of the term in professional articles.  They seem merely descriptive, although the usage to describe multiplying number systems with place values seems common (ones-tens-hundreds, feet-inches-parts).  The 1833 paper uses the term to describe an operation in simplifying an fairly complicated algebraic fraction.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com/books?id=0v5bAAAAQAAJ&amp;amp;pg=PA267&amp;amp;dq=%22cross+multiply%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;multiplying lengths in the English system (1803)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com/books?id=IkEwAAAAIAAJ&amp;amp;pg=PA182&amp;amp;dq=%22cross+multiply%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;adding fractions (1833)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com/books?id=E4qVSfNEX3YC&amp;amp;pg=PA282&amp;amp;dq=%22cross+multiply%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;systems of equations (1843)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com/books?id=Ov0YAQAAIAAJ&amp;amp;pg=PA427&amp;amp;dq=%22cross+multiply%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;multiplying digits (1851)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/ngrams/graph?content=cross+multiply&amp;amp;year_start=1800&amp;amp;year_end=2000&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t1%3B%2Ccross%20multiply%3B%2Cc0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ngram viewer results&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PqZ7R.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PqZ7R.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="3237" LastEditDate="2015-11-21T00:56:02.097" LastActivityDate="2015-11-21T00:56:02.097" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10044" PostTypeId="2" ParentId="10040" CreationDate="2015-11-20T15:52:03.873" Score="16" Body="&lt;p&gt;It seems clear that there is a certain conceptual gap in the student's understanding.  My suspicion is that the student is essentially running the following program in his mind:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Initialize factorial = 1&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For $j = k$ to $1$&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;  factorial = factorial * (2j - 1)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Output factorial&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The correct program should read: for $j = 2k - 1$ to $1$, factorial = factorial * $j$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, the student is decreasing $k$ by 1 in $(2k-1)!$ to form each successive factor, rather than decreasing $n = 2k - 1$ by $1$ at each step of the recursive algorithm.  (This is not quite the right way to phrase it, but I'm at a loss for better words at the moment.)  Perhaps explaining the difference between the two algorithms will solve the issue.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2015-11-20T16:12:01.113" LastActivityDate="2015-11-20T16:12:01.113" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10045" PostTypeId="2" ParentId="10039" CreationDate="2015-11-20T16:07:33.477" Score="4" Body="&lt;p&gt;I teach calculus at a community college. I've decided to give my students at least two chances on each test. It's hard to make up multiple ways of asking them to explain the definition of the derivative. Because I think that's a vital thing for them to truly understand, I allow them to do an oral exam if they haven't quite gotten that on the two written test attempts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tell them they must pretend I'm an algebra student, and they must explain how the definition of derivative is connected to the definition of slope, and draw diagrams to help me understand. (They cannot use any notes.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of them do well. Many don't. If they don't, I explain it all to them, and tell them they can come back next week to try again.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-11-20T16:07:33.477" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10046" PostTypeId="2" ParentId="10040" CreationDate="2015-11-20T16:57:40.757" Score="13" Body="&lt;p&gt;As per my comment above, it looks to me like the student saw $2k−1$ which is a pretty standard mathematical way to say &quot;odd numbers&quot; and then saw the factorial and thought to themselves &quot;looks like i need to factorial only the odd numbers&quot;.  While I have never encountered this particular misconception, I categorize it into a general category of an &quot;illogical leap&quot; (not an official term, just one that I use when thinking about these classroom issues).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, the illogical leap occurs when a student recognizes individual pieces of an equation/expression/relation but they don't understand the whole or how to correctly connect the pieces.  Because of this, they take what they know about the individual pieces and try to connect them in an incorrect and illogical fashion.  In this case, they recognized the factorial sign and the common expression $2k-1$ and made the illogical connection that this should mean to factorial the odd numbers.  I don't think that this is an incredibly unreasonable jump, but it should obviously be corrected.  I have found the best way to correct a student who makes an illogical leap is to first recognize the information they understood.  This is big because it encourages them that they did get something right and makes them more responsive to your corrections.  Make sure that you have a legitimate explanation for exactly why they were wrong and how to correct their thinking.  Simply saying &quot;This is not correct, here is the correct way&quot; is not very helpful and will probably not actually help their thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some other examples I have seen of this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Thinking that there is no real number solution to $ - \sqrt{4} $.  They recognize that the square root of a negative number is imaginary but they fail to recognize the negative needs to be under the radical for this to be true. There are plenty of others involving radicals (ex. $\sqrt[3]{8} = \pm 2$)&lt;/li&gt;&#xA;&lt;li&gt;Take the equation: $$8x - 9 = 3x$$  I have seen many students rewrite this as $$-9 = 3x + 8x$$  They recognize that the variables should be consolidated but they think &quot;I'll just move it over here&quot; without recognizing the actual operation that needs to occur to make this &quot;move&quot; possible, i.e. subtracting $8x$&lt;/li&gt;&#xA;&lt;li&gt;Thinking that $2^{-3} = -8$.  They recognize that there is a cubic in there but they mistake the negative sign to mean the sign of the answer not that the cube should be performed in the denominator.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are probably hundreds of other examples out there but these were the first three that came to mind.  Unfortunately, there is no one-size-fits-all answer  that will correct students thinking every time.  The best that I can suggest is to: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Look at their mistake and try to approach the problem from their point of view.  Ask yourself what they got correct and how that manifests in the answer provided.  Use that to determine what &quot;leaps&quot; they would have to take to get from their correct knowledge to the incorrect answer.  Then make sure to correct those leaps.  Lastly, if you have no idea what they were thinking, then ask them!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2015-11-20T17:13:56.423" LastActivityDate="2015-11-20T17:13:56.423" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10047" PostTypeId="1" CreationDate="2015-11-20T17:36:04.807" Score="5" ViewCount="113" Body="&lt;p&gt;Knowing that pedagogy for each age group is different, I will say right off the bat I am talking about working adults.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am noticing more and more, that despite people's phobias about math, they are often quite good about dealing with those special cases specific to their subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today I will talk about &lt;strong&gt;units&lt;/strong&gt; and &lt;strong&gt;measurement&lt;/strong&gt;.  The typical discussion of measurement uses Area and Length and feet and inches.  These are considered &quot;universal&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/j3KFC.jpg&quot; width=&quot;300&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice, these units of measurement are not the most useful.  Here is an example from a page on website, related to a course on Web design I am reading:&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Example: Mozilla CSS docs on &lt;a href=&quot;https://developer.mozilla.org/en/docs/Web/CSS/length&quot; rel=&quot;noreferrer&quot;&gt;length&lt;/a&gt; (Relative and Absolute Units)&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/r8sXb.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/r8sXb.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9bfOe.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9bfOe.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;These are not your typical units of measurement.  We have the &lt;strong&gt;inch&lt;/strong&gt; and &lt;strong&gt;centimeter&lt;/strong&gt; but also &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the &lt;strong&gt;pixel&lt;/strong&gt; which is measure relative to the device you are using and &lt;/li&gt;&#xA;&lt;li&gt;the &lt;strong&gt;em&lt;/strong&gt; which is relative to the font being typed.  &lt;/li&gt;&#xA;&lt;li&gt;There are also &lt;strong&gt;percentage&lt;/strong&gt; units which are measured relative to the width and height of your window, or even whether you are holding your phone straight or sideways.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These units of measure are &lt;strong&gt;dynamic&lt;/strong&gt; since these lengths or proportions are based on information that is changing with time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A web designer's job is to take an aesthetic or conceptual description of a web site and turn it into a precise and more quantitative version for the computer to understand.  Moreover, the same set of proportions have to work under a variety of circumstances of which they do not have any control.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How can we discuss the geometry of websites when there is so much variation?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems are textbooks are do not reach far enough in addressing real-life practical measurement.  &lt;strong&gt;Why are web designers able to understand delicate geometric concepts in the context of the theory of Cascading Style Sheets but not within the traditional mathematics framework?&lt;/strong&gt; &lt;/p&gt;&#xA;" OwnerUserId="1654" LastActivityDate="2015-11-20T17:36:04.807" Title="the role of context in mathematical discussions of units and measurement in web design" Tags="|undergraduate-education|secondary-education|geometry|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10048" PostTypeId="2" ParentId="10039" CreationDate="2015-11-20T21:15:47.320" Score="1" Body="&lt;p&gt;In some circumstances, oral exams have the advantage that you can adjust the later questions on the basis of the answers to earlier ones.  If a student has trouble with a question, you can follow up with questions that either serve as hints for the student or enable you to isolate what exactly is causing the trouble.  On the other hand, if the answers to the first few questions make it obvious that the student has completely mastered the material, you might just omit later questions on the same material (or you might replace them with more difficult questions, to see just how much the student actually knows).&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2015-11-20T21:15:47.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10049" PostTypeId="2" ParentId="621" CreationDate="2015-11-21T05:50:56.300" Score="0" Body="&lt;p&gt;There are many existence proofs that are formally not difficult but I think could be useful examples  .E.g., Euclid's proof that there is no largest prime.Rolle's theorem (calculus) and its main corollary,the Mean Value theorem. There are also some puzzles that should only be solved analytically rather than constructively, such as &quot;Can a chessboard, with 2 diagonally opposite corner squares removed, be tiled by non-overlapping 1x2 dominoes?&quot; (Black/white square parity problem.)&lt;/p&gt;&#xA;" OwnerUserId="5926" LastActivityDate="2015-11-21T05:50:56.300" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10050" PostTypeId="2" ParentId="9969" CreationDate="2015-11-21T06:06:49.383" Score="0" Body="&lt;p&gt;Mathematically you are looking at a bijection G (for Giving) with no fixed point, on a finite set S with at least 2 members, and its inverse bijection R (for Receiving). When S has at least 3 members it is not inevitable that a &quot;gift-pair&quot;  exists. A gift-pair is a pair (x,y) of members of S for which G(x)=y and G(y)=x.(Or equivalently R(x)=y and R(y)=x.) And if S has an odd number of members, at least  3 them do not belong to any gift-pairs.&lt;/p&gt;&#xA;" OwnerUserId="5926" LastActivityDate="2015-11-21T06:06:49.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10051" PostTypeId="2" ParentId="621" CreationDate="2015-11-21T07:21:56.387" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Edit: I think of first-year undergraduate students which are&#xA;  confronted with formal definition and proof for their first time and&#xA;  ask &quot;why?&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Responding to your edited side-question: I'm always happy to have this question asked, because it cuts to the heart of the mathematical discipline. It's worth spending a few minutes on anytime it comes up. My current answer ticks off several bullet points:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To guarantee that our theorem/shortcut is actually correct.&lt;/li&gt;&#xA;&lt;li&gt;To explain exactly why the theorem works in simpler steps.&lt;/li&gt;&#xA;&lt;li&gt;To come to a communal agreement before moving on.&lt;/li&gt;&#xA;&lt;li&gt;To practice old skills and methods that will be used later.&lt;/li&gt;&#xA;&lt;li&gt;As an aid for remembering the structure of the theorem or formula. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In many ways the mathematics discipline is wonderful because it's the only field where we can accomplish all of this, essentially for free, by just talking and pencil and paper. Only math can actually prove things for certain (other sciences at best develop theories from inductive evidence). Only math can establish the principles by pure logical thought, without resorting to expensive or complicated experimental apparatus. Only math allows everyone to comment and participate, without trusting some authority that outside evidence was acquired correctly. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-11-21T07:21:56.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10052" PostTypeId="1" CreationDate="2015-11-21T08:32:40.433" Score="14" ViewCount="1644" Body="&lt;p&gt;Everybody who is in graduate mathematics had a moment where they realized that mathematics was &quot;their thing&quot;, and they decided to dedicate their academic career to it. I don't know of many people who decided this before their fourth semester or so, but I do know a few who have had a life dream of being a mathematician since high school (they went to some charter school or something). For me personally, I entered undergrad as a physics major, and when I took my first abstract algebra course (as a just-for-fun elective) in my fourth semester, I loved it, I felt this feeling of &quot;How have I not seen this [abstract mathematics] before?&quot; I had had 13 years of mathematics education - from first grade up elementary school through three semesters of calculus and differential equations in my first two years of college - and never once had I even heard of these abstract mathematical structures - groups (-&gt; universal algebra, free term algebras, commutative algebra), topological spaces (-&gt; algebraic topology -&gt; category theory), analysis (-&gt; hardcore set theory), combinatorics (-&gt; finite set theory)... etc. etc. etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why? Why had I never seen these before? Was it because I wouldn't have understood them before? I certainly don't think this is true. I think as early as middle school, but definitely by 9th grade, if I had been introduced (in a suitable &quot;fun&quot; way, perhaps) to basic set theory, groups (set with an operation), topology (set with some sub-structure), combinatorics (playing with finite sets)... I would have understood these, I think. The standard two-semester undergraduate series in these topics, maybe with a slight slow-down to accommodate the lack of &quot;mathematical maturity&quot;, could be understood by an (advanced) high schooler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have heard of examples where a student's prodigous interest in math in middle school will get recognized by a teacher and they will get put on some magic track to math success - they will design a cross-curriculum with the local university which allows them to spend the mornings doing their rote high school graduation requirements - biology, history, etc. - but in the afternoons they will go to a class or two at college - usually number theory or something basic like that. In the extreme (e.g. &lt;a href=&quot;https://www.wku.edu/academy/&quot;&gt;Gatton Academy&lt;/a&gt;) the students are effectively attending college for the last two years of their high school career.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this system is obviously completely ad-hoc cannot reach everyone. Back to me personally, I never met a single &quot;real mathematician&quot; until I came to college. I was a &quot;smart kid&quot; - I got a 5/5 on the AP Calc test, 35/36 on my math ACT, 800/800 math SAT, etc. I did all the things &quot;smart kids&quot; are supposed to do - in particular I was a member of the math club at my high school. But, what happens in a math club? Who runs the math club? Who selects the topics? I'll spare you the sob story: it was essentially an engineering club. All of the fun activities were fundamentally engineering - launching rockets, designing bridges, etc. etc. I did not learn any math in the math club.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I entered college and decided to be a physics major. Here for my first two years I took calculus 1-3, diff eq, matrix algebra. Even in these classes, it felt very rote and &quot;computational&quot; (possibly because this university's engineering department is so big that it dictates the curriculum of the calculus courses since 75% of the students taking them are in the college of engineering...? just a crackpot theory). The professors would drop subversive hints sometimes - &quot;They don't really want me to teach you guys this, but I'm going to spend today talking about the epsilon-delta definition continuity. This will not be on the exam.&quot; - but even then, I only got the feeling that there was some secret world behind &lt;em&gt;calculus in particular&lt;/em&gt; - still no inkling that there existed a grand structure of abstract mathematics underlying everything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, all this is a vague rant to suggest that somehow there should be opportunity for smart, interested students to access &quot;deeper depths&quot; of mathematics - to know that mathematics is a field &lt;em&gt;where people still do research today&lt;/em&gt;, and that, bluntly, this shit goes mad deep. Ideally at a very early level - the earlier you start, the further head-start you have on your finite lifespan. I am already 21 years old! (Galois already made substantial advancements in mathematics by my age, and even had enough free time remaining outside of mathematics to &lt;em&gt;die&lt;/em&gt;!) If I had learned about this stuff in high school, I could have had a 2-4 year head start on where I am now! It makes me feel stupid by comparison to &quot;what I could have been&quot;. I feel like a large part of my life was wasted (not the point of depression, of course, but it is a nagging thought I've been having recently).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Of course, it could be argued that such a structure would be impossible to implement without forcing the &quot;uninterested&quot; students through so much math that &quot;they will never need in everyday life&quot;. After all, only a very small percentage of the world's children grow up to be abstract mathematicians.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Hypothetically the argument could be made, &quot;Seventh graders just aren't ready for abstract math - they still have to learn about adding and multiplying and fractions and slope of a line in the plane.&quot; I don't believe this for a second. All it takes is a sufficiently strong teacher-student connection (which can often be obtained by simply a good enough or passionate enough teacher).]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the education-theoretic position on this phenomenon? Are there any avenues in place to combat this sad occurrence?&lt;/p&gt;&#xA;" OwnerUserId="5927" LastEditorUserId="5927" LastEditDate="2015-11-21T08:46:37.890" LastActivityDate="2017-06-11T13:28:36.433" Title="How early to start &quot;abstract&quot; math education, or, How to prevent smart kids from never getting exposed to math?" Tags="|mathematical-pedagogy|secondary-education|student-motivation|curriculum|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10053" PostTypeId="2" ParentId="621" CreationDate="2015-11-21T10:31:51.887" Score="4" Body="&lt;p&gt;Here's one that just occurred to me: differentiation. First-year undergraduates are likely to have been told that the derivative of $x^n$ is $nx^{n-1}$, and that this is the gradient of the graph $y=x^n$, but I can't think how you would explain &lt;em&gt;why&lt;/em&gt; that is true without writing out the definition and seeing what happens.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-11-21T10:31:51.887" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10054" PostTypeId="2" ParentId="10052" CreationDate="2015-11-21T15:38:19.000" Score="13" Body="&lt;p&gt;&lt;a href=&quot;https://www.mathcircles.org/&quot;&gt;Math circles&lt;/a&gt; are an attempt to bring the beauty of math to young people. My book, &lt;em&gt;&lt;a href=&quot;http://naturalmath.com/playingwithmath/&quot;&gt;Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers&lt;/a&gt;&lt;/em&gt; is an attempt to bring the joy of math to those who don't yet see it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of us work at this problem. We will not reach everyone. There will be people who miss out for too many years. Teaching basic math in ways that relate it more to the beauties of abstract mathematics is possible. But it takes math teachers who enjoy math. I see many of them online, but they are not a representative sample.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I saw the beauty of math when I was young through logic puzzles and a book titled &lt;em&gt;Mathematics: A Human Endeavor&lt;/em&gt;. But my introduction to higher-level math (at the University of Michigan) was unpleasant (too hard, proofs too long and unconnected to anything I deeply understood), and made me think I didn't like math. I am very lucky that I fell back in love with math at a less elite school (where I got my masters' degree). &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-11-21T15:38:19.000" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10055" PostTypeId="1" CreationDate="2015-11-21T17:21:22.400" Score="17" ViewCount="901" Body="&lt;p&gt;I would like students to be critical and not believe that every proof they see is correct.&#xA;Lecturers make mistakes and students should not think: &quot;That must be a valid argument/proof/syntax because it was given in the lectures/lecture notes/book.&quot;&#xA;I hope that this kind of critical thinking would make them better at deciding whether the proofs they write themselves are correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To this end, I would like to present invalid proofs and results to the students.&#xA;But how should I go about this?&#xA;What kinds of problems have you had using false proofs?&#xA;I have never really tried this or seen this used, so I would like to hear experiences and recommendations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Context: University students that have received or are close to receiving a bachelor's degree.&#xA;Possible courses might cover, for example, point set topology, measure theory or differential geometry.&#xA;A typical course will consist of lectures and exercise sessions (for which the students have received the problems a week or so in advance).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some scattered ideas I have:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;One option is to give a false theorem in (almost) every lecture and ask the students to find it.&#xA;This may have some benefits, but the big drawback is that then there might be no lecture material that the students can completely rely on.&#xA;It is often desirable that students be able to use the main theorems of the course even if the proofs are too complicated for them to grasp.&#xA;I think this approach would cause too much confusion, especially when results are built on earlier results.&lt;/li&gt;&#xA;&lt;li&gt;I could give false theorems and &quot;proofs&quot; for them as exercise problems and ask them to figure out what went wrong.&#xA;They could start by finding a counter-example to the claim and then proceed to finding a &quot;corresponding&quot; error in the proof.&#xA;It might be too much of a burden to give the students five proofs or so and make the judge which are correct; this takes time away from other exercises.&lt;/li&gt;&#xA;&lt;li&gt;I want to point out to students that there are false proofs of correct statements.&#xA;Such problems should be included.&lt;/li&gt;&#xA;&lt;li&gt;Maybe I could make the students read proofs from other students and evaluate them.&#xA;This would require some rearranging of the work flow of the course.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There is already a question about &lt;a href=&quot;https://matheducators.stackexchange.com/q/2206/2074&quot;&gt;examples of bad proofs&lt;/a&gt;.&#xA;I'm not asking for examples but for good ways to make &quot;critical proof reading assignments&quot; for relatively advanced courses.&#xA;Assume I already have a bunch of bad proofs; I want to focus this question away from finding good bad proofs.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-11-24T14:35:15.457" Title="How to use false theorems or proofs?" Tags="|mathematical-pedagogy|proofs|problem-design|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10056" PostTypeId="2" ParentId="10021" CreationDate="2015-11-21T18:13:06.477" Score="9" Body="&lt;p&gt;I have another explanation for why it is so difficult to get right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know how it is in other countries, but usually students (here in Germany) learn it the wrong way, either at school already or in Bachelor studies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is how that happens:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some teacher or even assistant of a professor takes an example and then says something like: &quot;Whatever we add to this, we always end up with another thing that satisfies what we need it to satisfy. Then we can do the same with the resulting thing and so on. So it's true for all of them.&quot; This initially sounds very logical and since you're learning something new, you might not spot the mistake they all make at that point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's only been in one lecture in my Master studies (which was mostly about graph theory), that a professor (a mathematician and physicist) explained in detail what's wrong with such a proof. It's lacking something. It's lacking the additional proof you'd have to bring for the example you start out with. You'd have to prove, that this example can stand for all others you could possibly pick and that proof can be very had to do. He recommended us to always use a &quot;destructive&quot; way of doing it, which is to say: &quot;I can any example, take it apart in any way I want, and then all the parts satisfy the conditions and whatever we add to those, it still satisfies what needs to be satisfied.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since it's been a while, I don't fully remember the procedure anymore, but I think it's exactly what user21820 described as reducing. To go the other way around always requires more proving and is way more difficult to get correct than this way of reducing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because this is so often taught in a wrong way, with examples, where going the wrong way around works, students think they know how to do it, but then it is simply wrong. Then they'll get confused, when you tell them how it's actually done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also there are simply so many steps with several names for each of them (at least in German), that it is difficult to keep their order and and names and what to do in them in memory for a long time. In Computer Science, you'll learn this method of proving something at some point, but never really use it on a day to day basis as you use programming languages and frameworks. This is why everytime you repeat learning it in some lecture, it is like something new, that needs to be understood first. I once understood it, when that professor explained it in detail and also explained why the other way, which everyone learned so far was wrong, but now I also don't know how to do this kind of proof anymore. It's been two semesters since I understood it ... To be honest, if I had the choice of asking a mathematician, I'd always go for that, instead of trying to do that myself and making mistakes. That's what mathematicians are for. Let them do what they're good at, which should be proving stuff and let computer scientists do the computer stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see from my probably quite imprecise way of describing things, I am not a mathematician myself ;)&lt;/p&gt;&#xA;" OwnerUserId="5931" LastEditorUserId="376" LastEditDate="2016-01-12T18:00:58.967" LastActivityDate="2016-01-12T18:00:58.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10057" PostTypeId="2" ParentId="10021" CreationDate="2015-11-21T20:27:09.223" Score="22" Body="&lt;p&gt;Lots of good answers here (I've upvoted many). I'm won't try to add to the discussion about why induction is hard, but I can suggest some approaches that have helped some of my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many have seen induction as an algebra exercise - for example, summing the first $n$ squares. Those examples are tedious and work badly. Students have trouble with the hypothetical &quot;Suppose it's true for $n$ ...&quot; and you find yourself trying to answer their &quot;How do you know it's true for $n$?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to start with well ordering, without even calling it induction. Students have little trouble accepting the fact that a nonempty set of positive integers must have a least element. Then you can prove lots of things by showing that a minimal counterexample leads to a contradiction. (I think that's worth doing in spite of my preference for direct rather than indirect proofs.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example you can start the fundamental theorem of arithmetic (FTA)- every integer is a product of primes in at least one way - by looking at the smallest integer for which it fails. (One pedagogical problem here is that they don't think the FTA needs to be proved. Examples of rings in which it fails usually take more class time than one can afford.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The uniqueness in the FTA follows from the same kind of argument if you grant the lemma that a prime dividing a product must divide one of the factors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That lemma follows from the extended form of the Euclidean algorithm, asserting that the $\text{gcd}(m,n)$ is an integral combination of $m$ and $n$. You can prove that by induction - a minimal counterexample leads to a contradiction with one application of division with remainder. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The actual computation of the coefficients for the linear combination giving the gcd is a classic recursive program - well worth doing in a course with both math and cs students. Just reading the code in an easy language like python is instructive - no need for them to write it. As has been noted in other answers. learning recursion and learning induction reinforce each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With well ordering in hand you can tell the classic joke: every positive number is interesting. If not, then the least boring integer is an interesting number. Be sure to distinguish between least (boring integer) and (least boring) integer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well ordering helps students understand the traditional formulation of induction and implies it: consider proposition $P(n)$ about positive integral $n$. If $P(1)$ is true and the truth of $P(n)$ implies that of $P(n+1)$ for all $n$ then $P(n)$ is true for all $n$. Just look for a minimal counterexample. To prove that the traditional formulation implies well ordering, consider a set $S$ with no least element and let $P(n)$ be the proposition &quot;$S$ has no elements less than or equal to $n$&quot;. Then $P(1)$ is true, since if $1$ were in $S$ it would be the least element.  If $P(n)$ is true then $P(n+1)$ is too, lest $n+1$ be the least element of $S$. The truth of all the statements $P(n)$ clearly implies $S$ is empty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have time, teach Fermat's method of descent by going through Euler's proof that $x^4 + y^4 = z^2$ has only trivial solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well ordering also helps focus on the &quot;inductive step&quot; rather than the base case (as another answer recommends). That inductive step is central in lots of combinatorial arguments. You can show that the cardinality of an $n+1$ element set is twice that of an $n$ element set without knowing a formula for the latter. The recursion for counting combinations works the same way. Then you have a nice inductive proof of the formula by showing it satisfies the same recursion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The pattern I'm recommending might be called &quot;induction in disguise.&quot; The formal structure is there, but appears as simple common sense rather than formidable logic with quantifiers and hypotheses and algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="4984" LastEditorUserId="4984" LastEditDate="2015-11-22T15:08:33.760" LastActivityDate="2015-11-22T15:08:33.760" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10058" PostTypeId="2" ParentId="10021" CreationDate="2015-11-21T21:22:57.540" Score="21" Body="&lt;p&gt;I think the main problem students have with induction proofs is that the ordinary direct proof works by reducing a statement with unknown truth value to one that is known as true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bulk of an induction proof however is reducing a statement with unknown truth value to a statement with unknown truth value.  And not just that, it's even the same statement, just with a trivial variable substitution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So essentially the bulk of the induction proof appears like a big tautology only very superficially masked with a rather transparent shell game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now confronted with all this major bulk of self-referential weirdness it is easy to forget that all of it hinges, after all, on the truth of the proposition for $n=0$.  So suddenly some specificness creeps into all that circular logic (which actually is not circular as much as spiralling).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bulk of the work seems mostly of the kind that you have been painstakingly taught to recognize as meaningless because of being tautological.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's like opening window blinds by drilling a tiny hole in them.  Particularly if you are a skilled handiworker, it appears quite absurd as an interface to seeing the whole light.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fifth Peano axiom and the abstract induction proof structure are easy to pin down and have some intuition.  But the bulk of the actual hard work in an induction proof looks like something else entirely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Distinguishing an actual tautology from a valid induction step requires a deep understanding separate from the skills needed for doing the bulk of the work.  Being secure in the mapping from the proof structure to the required work requires practice, confidence, and understanding.  With most other math work, you can learn the steps first and get the understanding some time later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's a bit more danger-fraught with induction.&lt;/p&gt;&#xA;" OwnerUserId="5935" LastActivityDate="2015-11-21T21:22:57.540" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10059" PostTypeId="2" ParentId="10021" CreationDate="2015-11-22T00:07:22.920" Score="13" Body="&lt;p&gt;A short, true story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two years ago, working with a colleague, we found what we&#xA;thought was a beautifully clean induction proof of an old theorem that had,&#xA;'til then,  only a&#xA;complex, intricate proof, standing unimproved for ~25 yrs. We started to write it up for publication, only to realize that &#xA;the base case&amp;mdash;which we thought was obvious&amp;mdash;could sometimes fail.&#xA;We were unsuccessful in repairing the new proof, and have since pretty much abandoned the pursuit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is my only experience with an induction proof floundering on the shoals of the base case.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-11-22T00:07:22.920" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10060" PostTypeId="2" ParentId="10055" CreationDate="2015-11-22T00:22:00.953" Score="11" Body="&lt;p&gt;Wondering &lt;a href=&quot;https://matheducators.stackexchange.com/q/10021/511&quot;&gt;Why are induction proofs so challenging for students?&lt;/a&gt;,&#xA;I thought of trying this as a possible assignment after introducing induction:&#xA;&lt;hr /&gt;&#xA;Find the flaw in this induction proof.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Claim&lt;/em&gt;: $3n = 0$ for all $n \ge 0$. &lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Base&lt;/em&gt;: for $n=0$, $3n = 3(0) = 0$.&lt;/li&gt;&#xA;&lt;li&gt;Assume&#xA;&lt;em&gt;Induction Hypothesis&lt;/em&gt; (&lt;em&gt;IH&lt;/em&gt;): $3k = 0$ for all $0 \le k \le n$. &lt;/li&gt;&#xA;&lt;li&gt;Write $n+1 = a + b$ where $a &amp;gt; 0$ and $b &amp;gt; 0$ are natural numbers each less than $n+1$. &lt;/li&gt;&#xA;&lt;li&gt;Then&#xA;$3(n+1) = 3(a + b) = 3a + 3b$. &lt;/li&gt;&#xA;&lt;li&gt;Apply &lt;em&gt;IH&lt;/em&gt; to $3a$ and $3b$, showing that $3a=0$&#xA;and $3b=0$. Therefore, $3(n+1)=0$. ∎&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;The advantage here is that the &lt;em&gt;Claim&lt;/em&gt; is obviously false, but the flaw in the&#xA;proof is not (I think) obvious.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-11-22T02:23:45.693" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10061" PostTypeId="2" ParentId="10055" CreationDate="2015-11-22T00:36:51.113" Score="16" Body="&lt;p&gt;I would not recommend putting false proofs onto the board unless you immediately (within the same class period) point out their falsity, and make an assignment to find it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For smaller mistakes that I make unintentionally, I give the students 'donut points'. 30 donut points (over the semester) and I bring in donuts for the class. I tell them that my main purpose is to get them to question what I say.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a proof-oriented class, like linear algebra, I like giving them lots of practice with &quot;prove or disprove&quot;. At each test, I give them two smaller tests, one on the content, and one that is a 'prove or disprove'. The three proof tests count as one test. Before each test we do what is often called '&lt;a href=&quot;http://function-of-time.blogspot.com/2009/10/speed-dating.html&quot;&gt;speed dating&lt;/a&gt;' among math educators. The end of each chapter (in Lay) has a whole bunch of true false questions, which make great 'prove or disprove' problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have the students put the desks in a U-shaped configuration, so the outside students can stay put and the inside students rotate. (Our room is tiny, and I was delighted to find a way to make this work.) Everyone discusses the first problem with a partner for two minutes, then moves. With their second partner they try to write a proof, again in two minutes. Then we discuss as a class. On to their next partner for the next problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students work much harder and more actively than they would if they stayed in their own seats. When I did this last semester, there were students who thought it was juvenile and they kind of wrecked the atmosphere of it. This semester they love it, and are getting a lot out of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have made a handout of about 6 student proofs after a test, and asked them to evaluate. They didn't do very well, but I think it was helpful. I should probably do it again. If you can get them to seriously evaluate each others' proofs, that could be a powerful learning experience.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-11-22T00:36:51.113" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10063" PostTypeId="1" AcceptedAnswerId="10064" CreationDate="2015-11-22T16:49:15.990" Score="5" ViewCount="275" Body="&lt;p&gt;I'm currently teaching basic probability and after that I would like to do some simple hypothesis testing. Frankly, probability theory and statistics have never really interested me but I feel like my students like how it can very often be applied to everyday life, so I want to teach them as much as possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've only learned about hypothesis testing at university level, and there it was just at the end of a very theoretical course on probability. I know some parametric and non-parametric tests from my studies (and from talking with friends), for example $t$-tests, sign-tests, $\chi^2$-tests. I usually have to look up what they are about, when I can apply them and how they are applied. This is a huge disadvantage when trying to answer the question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Which statistical tests should I thoroughly teach?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, it is very important to me that my students do not only blindly follow some algorithm and have no idea why (as seems to be the rule for high schoolers doing statistics). I want to teach them why exactly a certain test works the way it does. This obviously requires myself to understand why certain tests work the way they do. The problem is, most sources I've found have rather little detail on &quot;why&quot; and much more on &quot;how&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even worse, some reasonings behind certain tests seem to be out of scope for most high schoolers (within a normal curriculum). I looked up what the idea for the $t$-test is and I think it would be rather confusing to point out that by $z$-transforming the sample mean, the random variable is no longer normally distributed and then to introduce the Student's $t$-distribution. So maybe, a more accurate question would be:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Which statistical tests can I thoroughly teach while not skipping the idea behind the test?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is high school level, 12th grade. They already know single-variable calculus, so at least integrals can be understood properly. &lt;/p&gt;&#xA;" OwnerUserId="493" LastEditorUserId="5321" LastEditDate="2015-11-22T18:00:42.477" LastActivityDate="2015-11-22T18:00:42.477" Title="What should I cover when teaching hypothesis testing in high school?" Tags="|secondary-education|statistics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10064" PostTypeId="2" ParentId="10063" CreationDate="2015-11-22T17:15:59.553" Score="3" Body="&lt;p&gt;Consider developing the $z$-test first. This takes as a simplifying assumption that the population standard deviation $\sigma$ is known; and supports sketching the sampling distribution of the sample mean as a standard normal curve for visual intuition and double-check purposes. Afterward we can bootstrap to the more practical case of using the sample standard deviation $s$ to estimate $\sigma$, which then gives rise to Student's $t$-distribution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is the pedagogical standard in any statistics textbook that I've looked at. It only requires very minimal probability content, and some kind of rude understanding of density curves and the normal distribution. In my community college statistics courses, I cut out most of the optional probability content (it's not a probability course, after all), so we can get directly to inferential statistics including hypothesis tests. I develop the $z$-test pretty carefully, and in fact only test on that, with lots of drills about when it's appropriate and when it isn't. Then on the last day I pitch the $t$-test as an example the myriad other statistical tests out there, and note that the human user will always be responsible for checking assumptions, requirements, and interpretations. Of course, your mileage may vary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a pretty good resource that basically follows the standard progression, consider &lt;a href=&quot;https://openstaxcollege.org/textbooks/introductory-statistics&quot; rel=&quot;nofollow&quot;&gt;OpenStax Introductory Statistics&lt;/a&gt; (Ch. 7-9). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-11-22T17:15:59.553" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10065" PostTypeId="1" CreationDate="2015-11-22T18:34:37.163" Score="9" ViewCount="196" Body="&lt;p&gt;I am looking for pedagogically  motivated examples (like elementary proofs,or simple problems) of &quot;abstraction in action&quot; ? I am looking for good specific examples (pre-university level or first year undergraduate level ) that could very clearly address the following questions&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When did the abstraction occur?&lt;/li&gt;&#xA;&lt;li&gt;What prompted the particular abstraction?&lt;/li&gt;&#xA;&lt;li&gt;What or which property was abstracted?&lt;/li&gt;&#xA;&lt;li&gt;Why was the abstraction necessary? I mean how the abstraction was necessary to continue further?&lt;/li&gt;&#xA;&lt;li&gt;What could have been other possible abstractions but they were rejected because they couldn't have been very useful to continue further.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Essentially,my question is an attempt to &quot;show few examples (like elementary proofs,or simple problems) which appear rich in abstractions when observed through the lens of abstraction?&quot;&lt;/p&gt;&#xA;" OwnerUserId="5882" LastEditorDisplayName="user507" LastEditDate="2015-11-22T21:38:01.673" LastActivityDate="2015-12-08T00:34:14.113" Title="Specific examples (like elementary proofs,or simple problems) which appear rich in abstractions when observed through the lens of abstraction" Tags="|mathematical-pedagogy|examples|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10066" PostTypeId="2" ParentId="10065" CreationDate="2015-11-22T18:43:07.317" Score="5" Body="&lt;p&gt;I don't know if this is what you're looking for, but way back in the 80's (before the internet...), I  attended a marvelous talk, by John H. Hodges, where he used the Konigsberg Bridge problem to talk about mathematical creation. He gave out a ditto, which I transcribed a few years ago. It looks like I never finished. Here's what's in my file:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5 Major Steps in Mathematical Creation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Notes given by John H. Hodges for a talk at a math conference, probably in the 1980’s.)&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Original Stimulation: A problem or question that comes from the real world or from mathematics itself&lt;/li&gt;&#xA;&lt;li&gt;Abstraction: A description of the problem that represents essence, and eliminates extraneous details&lt;/li&gt;&#xA;&lt;li&gt;Generalization: Consideration broadened to a whole class of problems (with the original problem as a particular case)&lt;/li&gt;&#xA;&lt;li&gt;Inductive &amp;amp; Deductive Reasoning: Inductive reasoning used to make conjectures, deductive reasoning used to prove theorems&lt;/li&gt;&#xA;&lt;li&gt;Application of general results: Does the mathematical work done solve the original problem? Does it raise new problems? Formulate those new problems, and begin anew…&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The Konigsberg bridges provide a great example of these steps in the process of mathematical creation.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-11-22T18:43:07.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10067" PostTypeId="1" AcceptedAnswerId="10068" CreationDate="2015-11-22T21:51:43.723" Score="33" ViewCount="1969" Body="&lt;p&gt;On a recent first-semester calculus exam, I gave a bunch of limits. The student was supposed to use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other method. One of the problems was this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\lim_{x\rightarrow\infty}\frac{e^{x+1}}{e^x}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roughly 90% of them asserted that the limit was equal to 1, usually with some variation on the argument that the $+1$ would become negligible compared to $x$ as $x$ got big. Only one student correctly found the limit to be $e$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was surprised by the near-universal repetition of the same mistake. I would be interested for others' insights on why this would happen. I suppose a contributing factor might be lack of facility with manipulating exponents. It would have seemed obvious to me (once I'd realized that L'Hospital wouldn't help) to simplify the expression inside the limit, and it also seems obvious to me that exponentials turn addition into multiplication. Probably they are also forming some kind of false analogy with limits of expressions such as $(x+1)/x$. One or two seemed to be using bogus symbolic reasoning with the $\infty$ symbol, simplifying the expression to $e^\infty/e^\infty$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also surprised me that so many students made mistakes on these that they could have easily caught simply by plugging in some sample numbers on a calculator, such as $e^9/e^8$ in this example. Maybe it would be helpful to assign homework in the future on which they were required to do this kind of check.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/9994/how-long-would-it-take-to-teach-proper-limit-calculations&quot;&gt;How long would it take to teach proper limit calculations?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-11-24T18:14:45.957" Title="Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$" Tags="|calculus|students-mistakes|limits|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10068" PostTypeId="2" ParentId="10067" CreationDate="2015-11-22T22:47:15.797" Score="29" Body="&lt;p&gt;While all of your students at this point will have done (extensive) units on manipulating and simplifying expressions with exponents, &lt;em&gt;this is the limits unit&lt;/em&gt;. When doing &lt;em&gt;limits&lt;/em&gt; questions, most students are only searching their brains for &lt;em&gt;limits&lt;/em&gt; techniques - they're extremely unlikely to come up with the $\frac{x^a}{x^b} = x^{a-b}$ rule that had been driven into them by rote some 3 years ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(see my question: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/3938/&quot;&gt;Compartmentalization in K-12 curriculum&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the correct bit of reasoning isn't readily available, students are likely to substitute a question that they can answer for this one that they aren't sure about. The (rather beautiful) fact that $\frac{x+1}{x}$ goes to $1$ as $x$ grows is very fresh in their minds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd say that many of your students had some idea that this fact didn't jive exactly with this question, but in an exam situation you've sometimes got to just go with what's immediately available to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: see &lt;a href=&quot;http://uweb.cas.usf.edu/%7Edrohrer/pdfs/Rohrer2009JRME.pdf&quot; rel=&quot;noreferrer&quot;&gt;this paper&lt;/a&gt; by Doug Rohrer for some extended commentary on the 'induced blindness' that your students seem to have been suffering from on this question.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-11-22T23:21:20.283" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10069" PostTypeId="2" ParentId="10067" CreationDate="2015-11-22T23:38:02.563" Score="16" Body="&lt;p&gt;First of all, I think that the problem statement can be confusing.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other method&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It can be misleading for students because it is easy to assume that if a fairly complicated method does not work (L'Hospital's rule), what you should do is use an even more complicated method (otherwise, why should you use the complicated method in the first place ?). So students tend to fall into the following reasoning :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Try L'Hospital's rule&lt;/li&gt;&#xA;&lt;li&gt;L'Hospital fails &lt;/li&gt;&#xA;&lt;li&gt;Search for more complicated method&lt;/li&gt;&#xA;&lt;li&gt;Do not find it&lt;/li&gt;&#xA;&lt;li&gt;Panic&lt;/li&gt;&#xA;&lt;li&gt;Get stuck into the washing machine&lt;/li&gt;&#xA;&lt;li&gt;Answer by analogy&lt;/li&gt;&#xA;&lt;li&gt;Answer 1&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Anyway, it can not explain everything, that is why I introduced the concept of &quot;answering by analogy&quot;. Some (most ?) students tend to survive in maths class without understanding the basic rules of mathematics or why it is important to follow them. How is it possible that a student, with false reasoning, gives a correct answer ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just use follow the following reasoning&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Look at the problem&lt;/li&gt;&#xA;&lt;li&gt;Search a formula, which looks like the problem, but for which you have a solution&lt;/li&gt;&#xA;&lt;li&gt;Adapt the formula to make it fit your problem&lt;/li&gt;&#xA;&lt;li&gt;Use the adapted solution as a solution to your problem&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For many problems it will work just fine. It works very well for factorization for example. But sometimes it fails.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your example, when they are searching for something looking like $\frac{e^{x+1}}{e^x}$, they naturally find $\frac{x+1}{x}$ in their mind (as mentioned by NiloCK). And if it looks like the same, it must behave the same, &lt;em&gt;that is how mathematics work, isn't it ?&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience it is a real issue, since this kind of practice is hard to spot, and generally when you spot it, it is hard to explain why a method that just worked fine for years suddenly break down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edited : conclusion rephrased and extended&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To conclude on a more general remark, it feels very natural for me to teach as an absolute rule to always simplify everything, and only when it is done to start using more advanced methods. With this in mind, the statement of the problem seems really strange, since it asks to violated what I consider as an absolute rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do understand the relevance of such a problem, for example to show that l'Hospital's rule does not always help, or to verify explicitly if students have understood a given method, but it can help answer why so many students fails at that particularly point.&lt;/p&gt;&#xA;" OwnerUserId="5941" LastEditorUserId="5941" LastEditDate="2015-11-24T18:14:45.957" LastActivityDate="2015-11-24T18:14:45.957" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="10072" PostTypeId="2" ParentId="10065" CreationDate="2015-11-24T00:14:37.947" Score="2" Body="&lt;p&gt;(1) Euclid's &lt;em&gt;Elements&lt;/em&gt; represented a giant step in abstraction, starting from the five axioms for plane geometry.&#xA;And then much later, through the efforts of Gauss, Lobachevsky, and Bolyai,&#xA;attempts to prove that 5th axiom led to non-Euclidean geometries, another step in abstraction,&#xA;an abstraction perhaps completed with Klein.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) And then there is &lt;em&gt;abstract algebra&lt;/em&gt;, developed in struggle to understand&#xA;the solvability of algebraic equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) Perhaps these examples are too general for your purposes. A more specific&#xA;example is the struggle to understand and prove $V-E+F=2$ for polyhedra,&#xA;so memorably described in Imre Lakatos's &lt;em&gt;Proofs and Refutations&lt;/em&gt;.&#xA;In a crude sense, it was necessary to define what constitutes a &lt;em&gt;vertex&lt;/em&gt;, an&#xA;&lt;em&gt;edge&lt;/em&gt;, and a &lt;em&gt;face&lt;/em&gt;, before a convincing proof could be achieved.&#xA;And this definitional process required careful abstraction from many examples&#xA;and counterexamples.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lakatos, Imre. &lt;em&gt;Proofs and refutations: The logic of mathematical discovery&lt;/em&gt;. Cambridge University Press, 1976.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-11-24T00:14:37.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10074" PostTypeId="1" CreationDate="2015-11-24T10:34:42.307" Score="11" ViewCount="324" Body="&lt;p&gt;I've been making some mathematical presentation in prezi, for example: &lt;a href=&quot;https://prezi.com/aexypzd25tmd/spectral-theorem-from-proofs-of-the-book/&quot;&gt;Spectral Theorem (from proofs of the book)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a prezi that I presented today, but it took me a hell lot of work, to get LaTeX working in prezi. I was wondering if there are any prezi alternatives that are more LaTeX friendly? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a programmer myself, so if they don't exist, I may try to make one myself some day, but I want to make sure that I don't miss out on some great already existing software.&lt;/p&gt;&#xA;" OwnerUserId="1712" LastEditorUserId="2074" LastEditDate="2015-11-24T12:03:25.417" LastActivityDate="2015-11-30T15:48:04.963" Title="Searching for prezi like software that supports LaTeX natively" Tags="|software|presentation|specific-software|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10075" PostTypeId="2" ParentId="10065" CreationDate="2015-11-24T11:41:47.100" Score="2" Body="&lt;p&gt;Most abstraction has to do with concepts rather than specific problems. Some random examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Brackets and infix notation. (Would you prefer &quot;The product of five and two more than three&quot;?)&lt;/li&gt;&#xA;&lt;li&gt;Variables (Who would be able to do mathematics efficiently without them?)&lt;/li&gt;&#xA;&lt;li&gt;Functions (A function is itself an object and not just its inputs and output.)&lt;/li&gt;&#xA;&lt;li&gt;Operations (E.g. we handle not permutations of specific objects but permutations themselves.)&lt;/li&gt;&#xA;&lt;li&gt;Ring action (E.g. natural numbers on objects, matrices on vectors, ...)&lt;/li&gt;&#xA;&lt;li&gt;Graph theory (Abstraction of the concept of objects and their connections.)&lt;/li&gt;&#xA;&lt;li&gt;Formal systems (As a logical basis for mathematical work, previously based a lot on intuition.)&lt;/li&gt;&#xA;&lt;li&gt;Type theories (As foundations for our intuitive notions of typing.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;On the other hand, there are specific mathematical entities whose abstractions reveal more, such as:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Algorithms (Abstracted as Turing machines or lambda calculus expressions.)&lt;/li&gt;&#xA;&lt;li&gt;Factorization (It abstractly generalizes to irreducible factorizations and prime factorizations.)&lt;/li&gt;&#xA;&lt;li&gt;Fermat's Little or Euler's phi theorem (They generalize to Lagrange's theorem for groups.)&lt;/li&gt;&#xA;&lt;li&gt;Strong induction (It generalizes to structural induction or transfinite induction.)&lt;/li&gt;&#xA;&lt;li&gt;[Weighted] Max-QM-AM-GM-HM-Min (All specific instances of the [weighted] power mean!)&lt;/li&gt;&#xA;&lt;li&gt;Boolean algebra (It generalizes to boolean lattices and then partial-orders with join/meet.)&lt;/li&gt;&#xA;&lt;li&gt;Differentiation (The abstract formal differentiation is often used in algebra and combinatorics!)&lt;/li&gt;&#xA;&lt;li&gt;Geometric transformations (The common ones are affine transformations.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I don't know enough about mathematical history to present historical details of the above abstractions, but presumably it could be found out or vaguely guessed. In my opinion the most important thing is that students can grasp and appreciate the motivation for, utility of and beauty in the abstractions.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2015-11-24T11:41:47.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10076" PostTypeId="2" ParentId="10067" CreationDate="2015-11-24T12:12:45.877" Score="1" Body="&lt;p&gt;I believe the problem is as usual, namely that your students have no proper logic foundation, so they just followed what they believe are rules for symbolic manipulation but without any real understanding of what those manipulations may mean, not to say when they are valid or invalid. In finding limits, there are two commonly taught but erroneous ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Error 1&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One error is to use the &quot;$\infty$&quot; symbol in the midst of a computation. There is never any reason to use it. For example, always use:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\dfrac{x^2+3}{4x^2-5} = \dfrac{1+\frac{3}{x^2}}{4-\frac{5}{x^2}} \to \dfrac{1}{4}$ as $x \to \infty$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And never:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\color{red}{\dfrac{x^2+3}{4x^2-5} \overset{???}{\to} \dfrac{\infty^2+3}{4\infty^2-5} \overset{???}{=} \dfrac{\infty^2}{4\infty^2} \overset{???}{=} \dfrac{1}{4}}$ as $x \to \infty$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Because the latter is &lt;strong&gt;so very wrong in so many ways&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best way is always to teach students to identify the significant terms and keep them in front in each subexpression, and manipulate the expression without changing the value in order to align the significant terms or reduce them. All this should be done without attempting to take limits, until the very end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So for your example students should have done:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\dfrac{e^{x+1}}{e^x} = \dfrac{e^x e}{e^x}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Simply because $e^x$ is the significant part of $e^{x+1}$ and it also aligns with the $e^x$ in the denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Error 2&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other 'error' is to teach the limit laws as usually presented to students who are unable to understand the proofs completely. Instead of manipulating limits by reducing subexpressions to their limit values, students should be made to manipulate the expressions themselves without reducing anything but keeping enough information about everything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\color{red}{\lim_{x\to 0} (1+x^5)(2-3x^4) = \lim_{x\to 0} (1+x^5) \times \lim_{x\to 0} (2-3x^4) = 1 \times 2}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which conveys almost zero meaning whatsoever, students should be taught to think:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$(1+x^5)(2-3x^4) \in (1+o(1))(2-o(1)) = 1 \times 2 + o(1)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Where you read each &quot;$o(1)$&quot; as &quot;some little bit&quot; and emphasize that the little bits may be different and may not be positive. And one should clearly explain why it is true:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$(1+o(1))(2-o(1)) = 1 \times 2 - 1 \times o(1) + o(1) \times 2 - o(1) \times o(1) = 2 + 4 o(1)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Because you have literally $4$ little bits there, whose sign you don't know, so they don't cancel one another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end, when the limit is wanted, only then do we look at what is the significant term and ignore the little bits. The reason for doing this is so that the concepts and techniques extend naturally and easily to asymptotic expansions, where we want finer distinction between the little bits. (Some little bits are littler than others.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With both of these in place, it is simply impossible for students to have any incorrect conceptual notions about limits, and the kind of error that you observed will never occur (unless they're not interested in learning).&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2015-11-24T12:12:45.877" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10077" PostTypeId="2" ParentId="9994" CreationDate="2015-11-24T12:36:48.587" Score="2" Body="&lt;p&gt;In the spirit of &lt;em&gt;paw88789&lt;/em&gt;'s answer, first show:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\color{red}{ 1 = \lim_{x\to\infty} 1 = \lim_{x\to\infty} \dfrac{x}{\sqrt{x}\sqrt{x}} \overset{???}{=} \dfrac{\infty}{\infty\infty} \overset{???}{=} \dfrac{1}{\infty} = 0 }$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And &lt;strong&gt;ask&lt;/strong&gt; the students what is wrong with it, and &lt;strong&gt;attempt&lt;/strong&gt; to argue with them that this reasoning is correct. By forcing them to pinpoint the problem &lt;strong&gt;on their own&lt;/strong&gt;, you automatically make them much better at avoiding the same erroneous thought process themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Almost surely students will be able to 'point out' that the &quot;$\infty$&quot; on top is bigger than each of the individual &quot;$\infty$&quot;s below. Ignoring that $\infty$ isn't well-defined, &lt;strong&gt;pursue&lt;/strong&gt; the matter and ask them, &lt;strong&gt;why&lt;/strong&gt; so, and &lt;strong&gt;why&lt;/strong&gt; should it make any difference? They should say that it was &quot;$x$&quot; on top but &quot;$\sqrt{x}$&quot; below, and that if you don't distinguish the 'infinities' then you get the incorrect conclusion that the top is 'smaller' than the bottom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then it is easy to now tell them that using a single symbol &quot;$\infty$&quot; has resulted in &lt;strong&gt;loss of information&lt;/strong&gt; of how big $x$ is getting, so we simply cannot use &quot;$\infty$&quot;. This should be enough to convince students of the futility in using the first 'method' in your example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the correct solution, I do not recommend the one you propose but rather:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\dfrac{x+2}{x(x-1)} = \dfrac{x(1+\frac{2}{x})}{x^2(1-\frac{1}{x})} = \dfrac{1}{x} \dfrac{1+\frac{2}{x}}{1-\frac{1}{x}} \to 0$ as $x \to \infty$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;With the following thought process:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Identify the significant terms in each subexpression. On the top, for large $x$ the $2$ is &lt;strong&gt;comparatively&lt;/strong&gt; insignificant to the $x$, and at the bottom likewise. We want to see how fast it grows, so we isolate that &lt;strong&gt;main growth rate&lt;/strong&gt;. On top it is $x$, and at the bottom it is $x^2$. As we can see, at the bottom the leftover is $1$ plus/minus something small, so indeed $x^2$ has the same &lt;strong&gt;significant part&lt;/strong&gt; (digits) as the original denominator. After simplification, we are left with $\frac{1}{x}$ multiplied by something close to $1$ then divided by something close to $1$. Intuitively it is clear that the overall growth is significantly the same as $\frac{1}{x}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Why is this better? Because students get a &lt;strong&gt;real feel&lt;/strong&gt; for the actual growth rate rather than attempting to blindly churn out the answer. Not only that, it &lt;strong&gt;generalizes readily&lt;/strong&gt; without difficulty to asymptotic expansions, which are widely used in engineering, physics and chemistry but that students are hardly taught to be able to manipulate!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After practicing with many examples, it would be a good idea to show students the following error:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\displaystyle \color{red}{ 1 = \lim_{n\to\infty} 1 = \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n} \overset{???}{=} \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{\infty} = \lim_{n\to\infty} \sum_{k=1}^n 0 = \lim_{n\to\infty} 0 = 0 }$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Again as before, it is best to draw the students into a discussion and try to 'convince' them that it is correct. It should become clear that not only is $\infty$ dangerous to try using, it is still &lt;strong&gt;very dangerous&lt;/strong&gt; even if only the limiting variable is substituted by $\infty$! This idea is a &lt;strong&gt;very common but fatal one&lt;/strong&gt;, as the following examples will show any students that remain stubbornly unconvinced that $\infty$ is not to be trifled with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\displaystyle \color{red}{ \lim_{n\to\infty} \sum_{k=1}^n \frac{k}{n^2} \overset{???}{=} \sum_{k=1}^\infty \frac{k}{\infty^2} \overset{???}{=} {???} }$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And if they argue that $k$ is on average $\frac{1}{2} \infty$ in the above summation then show them:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\displaystyle \color{red}{ \lim_{n\to\infty} \sum_{k=1}^n \frac{k^2}{n^3} \overset{???}{=} \sum_{k=1}^\infty \frac{(\frac{1}{2}\infty)^2}{\infty^2} \overset{???}{=} \infty \times \frac{\infty^2}{4\infty^3} \overset{???}{=} \frac{1}{4} }$. [FALSE!!!]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="1550" LastEditDate="2015-11-24T16:22:44.040" LastActivityDate="2015-11-24T16:22:44.040" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10078" PostTypeId="2" ParentId="10021" CreationDate="2015-11-24T14:26:09.823" Score="15" Body="&lt;p&gt;I have no evidence for this, but I speculate that one reason students are so bad at induction is that it is so unlike real world reasoning. In ordinary English, we say things like &quot;If I miss the bus, I'll be late to work.&quot; Well, maybe not! Maybe a friend will happen to drive by and offer me a ride! Maybe there will be a fire alarm at work, and everyone will start an hour late today. What one really means is &quot;If I miss the bus, there is a high probability (say 95%) that I'll be late to work.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, start stacking those implications: &quot;If I miss the bus, I'll be late to work.&quot; &quot;If I'm late to work, the students who are waiting for my office hours will be annoyed with me.&quot; &quot;If my students are annoyed with me, they'll write poor evaluations of my course.&quot; ... and eventually you end with &quot;If I miss the bus, I'll be living homeless in People's Park.&quot; Most of us understand that stringing together long chains of causation like this is invalid, roughly because $(0.95) \times (0.95) \times (0.95) \times  \cdots  \times (0.95)$ gets small pretty fast.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that this innate revulsion to long logical chains is a problem for students getting used to any kind of mathematical proof. But it is particularly bad for induction, because the chain gets longer at every step. Outside of mathematics, is there any chain of argument with a dozen if/then links which you would accept?&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2015-11-24T14:26:09.823" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10079" PostTypeId="2" ParentId="10055" CreationDate="2015-11-24T14:35:15.457" Score="6" Body="&lt;p&gt;There is a common type of exercise in French's mathematical secondary educators curriculum: give the student a (true or alleged) student's answer to a typical problem in your class, and ask them to analyze it. Is the answer correct? Near correct? What is good or wrong about it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The nice thing is that they have to think critically even more than they would when asked to debug a faulty proof: here the proposed answer can a priori be perfect, or ok with a harmless error, or completely wrong even if the result is right, or the result might be wrong. Make sure all these possibilities turn out, and even the smallest set of basic exercises turns into a very complete training (just write down many genuine student's answers for later use if you don't want to forge them yourself).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-11-24T14:35:15.457" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10081" PostTypeId="1" CreationDate="2015-11-24T16:22:48.603" Score="27" ViewCount="1253" Body="&lt;p&gt;The word &amp;quot;if&amp;quot; is used in two meanings in mathematics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Definition.&lt;/strong&gt; A topological space is compact &lt;em&gt;&lt;strong&gt;if&lt;/strong&gt;&lt;/em&gt; every open cover has a finite subcover.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; A topological space is compact &lt;em&gt;&lt;strong&gt;if&lt;/strong&gt;&lt;/em&gt; it is finite.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Remark.&lt;/strong&gt; There are infinite compact topological spaces.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In a definition &amp;quot;if&amp;quot; usually means &amp;quot;if and only if&amp;quot; (or more properly something like &amp;quot;this is what we mean by this word&amp;quot;).&#xA;In a theorem &amp;quot;if&amp;quot; means implication; it is the usual logical if.&#xA;Some people prefer to use &amp;quot;if and only if&amp;quot; in definitions, but this is not a universal practice so I cannot completely avoid the phenomenon.&lt;/p&gt;&#xA;&lt;p&gt;Questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Can students tell the difference between the two ifs? Research is preferred but experience is accepted.&lt;/li&gt;&#xA;&lt;li&gt;In your experience, has this double meaning of the word &amp;quot;if&amp;quot; actually led to significant confusion? That is, is there a reason to worry that this might confuse someone?&lt;/li&gt;&#xA;&lt;li&gt;Has it been studied whether this issue slows down (or prohibits) learning what definitions are and how to use them?&lt;/li&gt;&#xA;&lt;li&gt;Would it be enough to remind students about this issue (once in every course, say) or should I do more? This will probably become unnecessary after a couple of years at the university, but it may be necessary at first.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Context: Consider bachelor level courses in universities.&#xA;There are no separate &amp;quot;introduction to proofs&amp;quot; courses here, just courses about areas of mathematics.&#xA;(The closest equivalent is &amp;quot;introduction to mathematics&amp;quot; which is the very first course and covers some basic proof techniques, but I cannot rely on students remembering all details from that short course.)&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="60" LastEditDate="2021-02-23T22:02:42.327" LastActivityDate="2021-02-23T22:02:42.327" Title="Can students tell the difference between the &quot;definition if&quot; and the &quot;theorem if&quot;?" Tags="|undergraduate-education|reference-request|logic|definitions|language-use|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="10082" PostTypeId="2" ParentId="9994" CreationDate="2015-11-24T16:28:16.280" Score="2" Body="&lt;p&gt;Elaborating on some of my earlier comments:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, the difficulty is not so much with exposing what is wrong with method (1), but the challenge of having students genuinely understand method (2) beyond merely manipulating symbols.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To genuinely understand method (2), one should have (a) have a precise definition of limit, (b) understand how the limits laws follow from this definition, and (c) understand how to apply the limit laws in this problem along the lines of method (2).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typically, when I teach calculus, I only attempt part (c) of this process, and students either don't appreciate why method (2) is reliable in a way that method (1) is not or (worse, in my opinion) accept the limit laws as some magic and completely unmotivated dogma that is introduced completely ad hoc.  I do try to motivate the limit laws and explain that they can be proven using a precise definition of limit, even though we do not cover the details of this in the course.  Still, I worry that this falls on deaf ears far too often.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice, the students who best understand the material typically get that method (2) works in all the types of problems that they see, whereas method (1) can lead to wrong answers and so is not reliable.  Thus, students have an empirical basis for trust in method (2), but almost never a truly mathematical basis for belief in the reliability of method (2).  Arguably, this is all that can be accomplished in a first introduction to calculus (at least in American universities similar to mine), given the practical needs to cover other material in the course.  At the end of the day, calculus is fundamentally about rates of changes, accumulation, and the deep connection between these two ideas.  Limits are necessary to fully understand these concepts and so need to be substantially developed, but not at the expense of preventing the students from spending an adequate amount of time studying the primary themes of the course.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2015-11-24T16:28:16.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10083" PostTypeId="2" ParentId="10081" CreationDate="2015-11-24T22:40:50.150" Score="16" Body="&lt;p&gt;Not formal research, but some decades of experience teaching both undergrad and graduate level courses, and &quot;editing&quot; PhD theses and such:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It appears that even many serious professional mathematicians do not understand the difference between a &quot;definitional&quot; iff and an &quot;assertive&quot; iff. This is entirely parallel to an &lt;em&gt;assignment&lt;/em&gt; equality versus an &lt;em&gt;assertive&lt;/em&gt; equality, for which mathematics has surprisingly mediocre accommodations, in contrast to computer science. (I'm not a fan of colon-equality or variants, but prefer clearer, less fragile, context-setting.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of definitional-iff, nowadays I prefer (as @DavidSteinberg suggests) entirely different language, often &quot;Say that X is a Y &lt;em&gt;when&lt;/em&gt; Z-property holds.&quot;&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-11-24T22:40:50.150" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10084" PostTypeId="2" ParentId="10081" CreationDate="2015-11-25T05:38:28.903" Score="6" Body="&lt;p&gt;&lt;strong&gt;iff and if&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In my experience, students who have a solid grasp of first-order logic have absolutely no problem with the inconsistent use of &amp;quot;if&amp;quot; in definitions. The problem is that most students don't, and as I've personally observed, often confuse between &amp;quot;if&amp;quot; and &amp;quot;iff&amp;quot; precisely because of such notational issues. Therefore the answer to your first two questions is &amp;quot;No&amp;quot; and &amp;quot;Yes&amp;quot; for many students.&lt;/p&gt;&#xA;&lt;p&gt;The reason it is extremely inconsistent to use &amp;quot;if&amp;quot; when &amp;quot;iff&amp;quot; is meant is because we commonly have the following two types of definitions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We say that an X is A iff it satisfies P.&lt;/p&gt;&#xA;&lt;p&gt;If an X is A, then define f(X) to be Y. [Here &amp;quot;f(X)&amp;quot; may be meaningless for an X that is not A!]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you want concrete examples:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We say that a real number is rational iff it is the ratio of two integers.&lt;/p&gt;&#xA;&lt;p&gt;If a set of real numbers &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is finite, define &lt;span class=&quot;math-container&quot;&gt;$\max(S)$&lt;/span&gt; to be the maximum element of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Clearly, in the second type of definition, &amp;quot;if&amp;quot; is certainly not &amp;quot;iff&amp;quot;. In a technical sense it is not even the material implication, but to satisfactorily deal with that we would have to use some form of type theory, which is beyond the scope of this answer.&lt;/p&gt;&#xA;&lt;p&gt;As for your third question, I don't know any formal study conducted on this issue, but I have encountered some very ambiguous definitions before, where usage of &amp;quot;if&amp;quot; made it unclear whether &amp;quot;iff&amp;quot; was meant. At the moment I can't recall them, but I think it was an &amp;quot;if&amp;quot; that was buried in some complicated definition. Also, when definitions are understood as &amp;quot;iff&amp;quot; statements, then many proofs become trivial at least at the lower levels since often simply unfolding definitions yields the proof after one or two other small logical steps. In my experience, many students do not understand how to unfold the definitions both forward and backward to meet in the middle, and hence find many logically trivial theorems very hard to prove!&lt;/p&gt;&#xA;&lt;p&gt;For your last question, you should simply use &amp;quot;iff&amp;quot; whenever you mean &amp;quot;iff&amp;quot;, and tell students that some people don't and so they have to be careful when they read other mathematical writings and always check that they are certain which is meant.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Let and take&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;On an unrelated note, there is a similar notational inconsistency between the two common uses of &amp;quot;let&amp;quot;. One is an instantiation of an existential assertion and the other is the creation of a new context of universal quantification:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[Where &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is a non-empty set:] Let &lt;span class=&quot;math-container&quot;&gt;$a \in S$&lt;/span&gt;. [This instantiates the non-emptiness of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;.]&lt;/p&gt;&#xA;&lt;p&gt;Take any &lt;span class=&quot;math-container&quot;&gt;$r \in \mathbb{Q}$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$r^2 = 6$&lt;/span&gt;. [Many people use &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$r \in \mathbb{Q}$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$r^2 = 6$&lt;/span&gt;.&amp;quot;.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I personally strongly disapprove of using the same notation in both cases, since in the second example it is totally false to assert that there is such an &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, and empirically students often do not know the logical structure of proofs and have great difficulty distinguishing between the two types if &amp;quot;let&amp;quot; is used in both cases. Some authors are careful to distinguish them, and one way is to use &amp;quot;take&amp;quot; or &amp;quot;given&amp;quot; for universal quantification.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Side remarks&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A number of logicians I've spoken to agree with me on the above two notation issues, but they also say that they don't know what can be done about it, and their main reason is that &amp;quot;most of the textbooks and papers use it&amp;quot;. I don't think that's too important in considering whether or not to continue using it, and indeed some of them thereafter try to unambiguously distinguish these in their own teaching.&lt;/p&gt;&#xA;&lt;p&gt;The main problem is that teachers &lt;strong&gt;already know&lt;/strong&gt; what they mean when they write &amp;quot;if&amp;quot; or &amp;quot;let&amp;quot;, but they don't realize that most students don't know. The few that eventually grasp it fully are usually the only ones that become teachers of the next generation... And unlike a compiler that tells the programmer in his face that his program is meaningless (syntax error), mathematics teachers try hard to guess the students' meaning, usually giving benefit of doubt, and hence many students get away with random guesswork, simply because the teachers know how to modify their proofs to make it work.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, you will never find the same kind of error in the programming world, because instantiation is equivalent to an assignment of a variable with the result of a function call, whereas quantification corresponds to a for-each loop. Similarly, programmers were the first to realize the great importance and utility of scoping, indicated by either braces or indentation, usually both. In the same way it would be excellent for mathematics students to learn this, and one way would be through teaching natural deduction in Fitch-style. Sadly, this is almost never done.&lt;/p&gt;&#xA;&lt;p&gt;Lest you disbelieve me, I've regularly overheard students discussing among themselves how to prove a theorem, and invariably it goes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To find limits we always write down &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$ε &amp;gt; 0$&lt;/span&gt;.&amp;quot; first...&lt;/p&gt;&#xA;&lt;p&gt;The question is about injection, so write down &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$f(a) = f(b)$&lt;/span&gt;&amp;quot;...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And worse still:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let's try induction. If &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is true, ..., therefore &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; is true.&lt;/p&gt;&#xA;&lt;p&gt;Proof by induction: If &lt;span class=&quot;math-container&quot;&gt;$n = n+1$&lt;/span&gt;, ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Imagine if people who have learn programming for 3 months still wrote:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;int n; input(n);&#xA;if( n==true ) { ... }&#xA;if( n==n+1 ) { ... }&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;They don't, but why? It's because their compilers force them to learn something that mathematics teachers don't ensure their students learn, which is actual logical reasoning including conscious type-checking.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-11-25T05:38:28.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10085" PostTypeId="2" ParentId="9994" CreationDate="2015-11-25T06:07:10.213" Score="2" Body="&lt;p&gt;In my opinion, the moment you write $\frac{1}{\infty}$ on the blackboard and treat it as a first-class mathematical object, you have lost. Students are going to misuse it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teach them that &quot;$\lim_{x\to a} f(x) = \infty$&quot; is a notation for an epsilon-delta statement meaning that $f(x)$ is unbounded, and that a theorem says that if $a(x)\to A \in\mathbb{R}$ and $b(x)\to\infty$, then $\frac{a(x)}{b(x)}\to 0$. You may then point out that $\frac{1}{\infty}$ is a shorthand to recall the theorem, or a slogan, but using it in actual calculations is just humbug.&lt;/p&gt;&#xA;" OwnerUserId="4930" LastActivityDate="2015-11-25T06:07:10.213" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10086" PostTypeId="1" CreationDate="2015-11-21T10:43:26.170" Score="7" ViewCount="393" Body="&lt;p&gt;What is the best way to teach the standard arithmetic operations ($+ - \times \div$) to a 3 year old child? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Also:&lt;/strong&gt; How we do know if the child has really understood it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;can a kid be 3 years old prodigy?&lt;/p&gt;&#xA;" OwnerUserId="5928" OwnerDisplayName="vishal" LastEditorUserId="5928" LastEditDate="2015-11-27T16:22:22.553" LastActivityDate="2015-11-29T12:40:57.123" Title="Teaching arithmetic operations ($+ - \times \div$) to a 3 year old" Tags="|assessment|gifted-students|arithmetic-operations|preschool-education|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10087" PostTypeId="2" ParentId="10086" CreationDate="2015-11-21T11:07:09.343" Score="4" Body="&lt;p&gt;Maria Montessori pioneered the study of childhood education, and the materials she developed for mathematics remain state-of-the-art.  In &lt;a href=&quot;http://mariamontessori.com/mm/?p=1416&quot; rel=&quot;nofollow&quot;&gt;this article&lt;/a&gt; the author encourages taking an actual lesson from a Montessori-trained teacher, which I would also encourage.  I participated in an event called &quot;The Journey&quot; at a local Montessori school where parents were invited to witness how children are taught the materials from ages 3-14.  Some parents literally wept as they (1) understood certain mathematical concepts, such as the &lt;a href=&quot;http://www.wikisori.org/index.php/Binomial_cube&quot; rel=&quot;nofollow&quot;&gt;binomial cube&lt;/a&gt;, for the first time, and (2) understood that their children were developing an understanding of math that they never had.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original text of the whole method provides a context for understanding the specifics of &lt;a href=&quot;https://www.gutenberg.org/files/39863/39863-h/39863-h.htm#CHAPTER_XIX&quot; rel=&quot;nofollow&quot;&gt;the teaching of numeration&lt;/a&gt;, which consists of 11 pages.  I realize that the rules of a place like math.stackexchange.com frown on citing by reference.  I'm not sure that math educators are greatly served by too much cutting and pasting of original literature, nor by the casual attempt to rephrase such in the interest of brevity.  But here goes...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maria Montessori observed that ”all children pass through 'sensitive periods' that stimulate them to carry out certain activities&#xA;and to acquire certain experiences.”  Crucially, she observed and understood that when children followed these &lt;em&gt;inner&lt;/em&gt; directives, which characterize sensitive periods, the child is able to structure their development in relation their environment, which is the essence of learning.  She understood that efforts of a teacher to either pour knowledge into &quot;an absorbent mind&quot; (or worse, to force it) works not only against nature, but also against the child's own sense of self.  Montessori's genius was in reporting that it is through the child’s own activity that development transpires. In the literature, sensitive periods have four basic characteristics:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;They provide the intrinsic motivation for a specific aspect of human development. These human acquisitions relate to order, sensory perception, language, and movement--particularly of the hand and walking upright. The result is human intelligence.&lt;/li&gt;&#xA;&lt;li&gt;Internal mental growth precedes external manifestation of activity. Organizational brain activity takes place prior any visible expression of the sensitive period.&lt;/li&gt;&#xA;&lt;li&gt;They are unconscious; the child is completely unaware of them. Not only is the child unaware of why he wants to do a particular activity; adults cannot directly influence the onset of sensitive periods.&lt;/li&gt;&#xA;&lt;li&gt;They are temporary windows for optimal development. After the sensitive period has passed, the particular acquisition is much more difficult, if it is possible at all.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The construction of a Montessori environment is all about optimizing development according to sensitive periods.  To wit: the environment must contain all the necessary elements that pertain to a given sensitive period, and the child must have access to the elements as well as the agency to engage the elements in the environment.  Montessori wrote, “If the child lacks suitable external means, he will never be able to make use of the great energies with which nature has endowed him. He experiences the instinctive urge toward activities worthy of engaging all his energy, because this is the way nature incites him to perfect the acquisitions of his faculties. But if there is nothing there to satisfy the impulse, what can the child do but dissipate his energies aimlessly, with disorderly unruliness?”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proper construction of a Montessori environment is the subject of many books, but suffice it to say that the materials for teaching mathematics are specifically designed to engage the sensitive periods related to numeracy, not as &quot;this is first grade, this is second grade, this is third grade&quot; but rather as holistic elements which can be understood with increasing levels of sophistication, starting at age 3 and continuing into middle school.  Moreover, the environment is child-centered, meaning that it affords the child a choice in the materials they engage, and a liberally allotted work cycle in which to explore.  The child is not constantly interrupted by a classroom bell as is so common in conventional classrooms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first step is to introduce the concepts of numbers as tangible things that can be counted (including zero), typically done with &quot;golden beads&quot;.  The second step is to engage the child in simple activities that have mathematically observable consequences, such as making change.  This gets you to + and -.  The third step is to use didactic materials, such as sets of rods with lengths 1 to 10 (and zero if you use your imagination), one set being red and one set being blue.  These rods can confirm the understanding of addition and subtraction as the child plays with all possible ways to add numbers 1 to 9 to get 10.  Then the child can try to construct a series of numbers by using two rods to do so.  The child may discover that 1+1 = 2, 2+2 = 4, 3+3 = 6, 4+4 = 8, 5+5 = 10, and may thus learn the concept of &quot;two things of one kind equal one thing of another kind&quot;.  This unlocks * and /.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most importantly, it is the child who pushes forward and asks the questions, which can happen when they enjoy an environment that is tuned to their sensitive periods.&lt;/p&gt;&#xA;" OwnerUserId="5952" OwnerDisplayName="Michael Tiemann" LastEditorUserId="5952" LastEditDate="2015-11-29T11:38:36.903" LastActivityDate="2015-11-29T11:38:36.903" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10088" PostTypeId="2" ParentId="10086" CreationDate="2015-11-25T08:08:57.533" Score="12" Body="&lt;p&gt;My immediate response is 'wait a few years'. I've spent a fair amount of time with 3 year olds, and most of them are busy learning how to be a person in their own right, how to have a conversation, what the difference is between real and make-believe, and (often) how to tell when they need the toilet. I've read that they can't understand metaphors by that age, and my guess is that abstraction is a fairly similar ability.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-11-25T08:08:57.533" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10089" PostTypeId="2" ParentId="3938" CreationDate="2015-11-25T22:50:44.397" Score="1" Body="&lt;p&gt;I don't agree with it, but &quot;flow chart mathematics&quot; and &quot;flow chart learning&quot; generally, seems to be the order of the day, with the rampant computerization of society. It started about 40 years ago, and is coming to a culmination as we speak. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The advantage of it is that it is practical for solving linear, or straight line, problems using logical reasoning. Its disadvantage is that it inhibits creative thinking using analogies, or involving hidden relationships, or even connections. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In their book, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0688119123&quot; rel=&quot;nofollow&quot;&gt;&quot;Generations,&quot;&lt;/a&gt; regarding the Anglo-American world, probably including Canada (the late) William Strauss and Neil Howe postulated that linear (flow chart) and critical thinking alternated in roughly 40-year cycles (that is, over the course of a total 80 year cycle). The development of critical thinking in schools has been at a low ebb, and may be about to make a comeback.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2015-11-25T22:50:44.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10090" PostTypeId="2" ParentId="9851" CreationDate="2015-11-25T23:02:19.353" Score="3" Body="&lt;p&gt;My father, a retired engineering professor, told me that engineers think largely in terms of systems of equations. The theory is important, but only to the extent that it serves that end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, students will want to know how and why a system of equations &quot;span&quot; a particular space. As for solution methods, the foundation method is nowadays called &quot;lower echelon row reduction&quot; (my father used a different term 50 years ago), using linear transformations. The ideas of kernels and images really have to do with which equations (and co-efficients) are left in the system after the reduction process. That's what engineers are most interested in.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2015-11-25T23:02:19.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10091" PostTypeId="2" ParentId="10086" CreationDate="2015-11-26T03:24:17.463" Score="1" Body="&lt;p&gt;Assuming the kid can count, and also understands that the sequence of utterances 'one', 'two', ... corresponds to amounts of stuff, rather than being rote, then I'd say there are lots of ways to talk about these things age appropriately.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I take three of these apples, how many will be left?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The cow has four legs! How about two cows? Three cows?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You and Mom and I are going to share these cookies. How many do we each get? ... But that's not fair! Mom has more than me! Make it fair!&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2015-11-26T03:24:17.463" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10092" PostTypeId="1" CreationDate="2015-11-27T02:21:35.570" Score="6" ViewCount="538" Body="&lt;p&gt;I plan to study for 6-12 months to take a high school mathematics teaching license exam. This one to be exact.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20151123152128/http://www.fl.nesinc.com/pdfs/math6-12_tig_6thedition_051413_doe.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Florida Teacher Certification Examinations Test Information Guide for Mathematics 6–12&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Been out of school for roughly 8 years. Never took calculus before. Was never really the studious type in school, but managed out alright.&lt;/p&gt;&#xA;&lt;p&gt;Nonetheless, I am wanting to put a lot of time and effort into re-learning these subjects from the ground up not only to pass the test but also to help me be an effective teacher.&lt;/p&gt;&#xA;&lt;p&gt;Thus I am looking for the best textbooks to have to help aid me in this process.&lt;/p&gt;&#xA;&lt;p&gt;Thanks for the recommendations and any advice really. Cheers&lt;/p&gt;&#xA;" OwnerUserId="5957" LastEditorUserId="745" LastEditDate="2022-01-13T09:48:06.230" LastActivityDate="2022-01-15T17:42:32.867" Title="Aspiring HS Math Teacher: Textbooks for learning Algebra, Geometry, Trigonometry, and Calculus?" Tags="|secondary-education|textbooks|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="10093" PostTypeId="2" ParentId="10092" CreationDate="2015-11-27T03:21:03.410" Score="2" Body="&lt;p&gt;If you need it, consider Martin-Gay's &lt;em&gt;Prealgebra &amp;amp; Introductory Algebra&lt;/em&gt;; it's about the most complete and clearest presentation I've seen, and moreover provides a template for teaching those subjects when needed (it's my starting point for lectures in those subjects). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the next step, consider Michael Sullivan's &lt;em&gt;Algebra &amp;amp; Trigonometry&lt;/em&gt;; it's a comprehensive and pretty widely-used text. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If budget and time is a consideration, consider the analogous free texts at &lt;a href=&quot;https://openstaxcollege.org/books&quot; rel=&quot;nofollow&quot;&gt;OpenStax&lt;/a&gt;: &lt;em&gt;College Algebra&lt;/em&gt; and &lt;em&gt;Algebra and Trigonometry&lt;/em&gt;. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-11-27T03:21:03.410" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10094" PostTypeId="1" AcceptedAnswerId="10097" CreationDate="2015-11-27T10:46:33.783" Score="17" ViewCount="1986" Body="&lt;p&gt;I taught three basic trigonometric identities. One of my 9th grade students asked: How can we say $$\sec^2x-\tan^2x=1$$ is an identity since when we plug in $x=\frac{\pi}{2}$ the identity fails which creates an ambiguity for the definition of an identity in mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How should I try to explain him without entering into calculus part?&lt;/p&gt;&#xA;" OwnerUserId="5666" LastEditorUserId="2074" LastEditDate="2015-11-27T20:31:02.403" LastActivityDate="2017-07-06T01:11:03.870" Title="How to convince my student that this is an Identity : $\sec^2x-\tan^2x=1$?" Tags="|secondary-education|trigonometry|" AnswerCount="9" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10095" PostTypeId="2" ParentId="10094" CreationDate="2015-11-27T10:59:19.380" Score="0" Body="&lt;p&gt;For a right triangle with angle $x$, hypotenuse $H$, adjacent side $A$ and opposite side $O$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$sec(x) = H/A$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$tan(x) = O/A$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$sec^2(x) - tan^2(x) = 1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$H^2/A^2 - O^2/A^2 = 1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$H^2 = A^2 + O^2$, which is just the Pythagorean Theorem&lt;/p&gt;&#xA;" OwnerUserId="5959" LastActivityDate="2015-11-27T10:59:19.380" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10096" PostTypeId="2" ParentId="10094" CreationDate="2015-11-27T13:00:25.507" Score="3" Body="&lt;p&gt;$π/2$   and $−π/2$ are not in the domain. I'd take a step back and offer examples from algebra where, when observing an equation, the first discussion is &quot;what is the domain?&quot; &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-11-27T13:00:25.507" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10097" PostTypeId="2" ParentId="10094" CreationDate="2015-11-27T13:24:37.503" Score="30" Body="&lt;p&gt;An identity always holds for some values of the free variables. Sometimes the allowed values are all real numbers, sometimes something else. In this case the identity holds whenever $\sec x$ and $\tan x$ are defined (they are defined in the same set). In a certain limiting sense the identity is also true at $x=\pi/2$, but you probably don't want to go into that with your student. The point is: &lt;strong&gt;An identity should always come with information about when it is supposed to hold. It's not just an equation.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even with the more common identity $\sin^2x+\cos^2x=1$ you can start asking questions:&#xA;This holds for geometrical reasons for $x\in(0,\pi/2)$, but does it hold for all $x\in\mathbb R$?&#xA;Ok, does it hold for all $x\in\mathbb C$ as well?&#xA;Well, what if $x$ is a matrix (you can still define the functions via the matrix exponential)?&#xA;The answers are all yes, but certainly only the real case is meant when it is first encountered.&#xA;It is not clear how far an identity should extend unless you explicitly tell what you mean by it.&#xA;You cannot always trust that your students will intuitively see what values the variables should take.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For comparison, the Pythagorean theorem does not state just $a^2+b^2=c^2$.&#xA;It states this equation and tells what the three variables should be for this to hold.&#xA;The equation only becomes meaningful once there is a triangle with a right angle and the lengths of the sides are named appropriately &amp;mdash; it is all this information together that makes the theorem.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2017-07-05T07:27:25.103" LastActivityDate="2017-07-05T07:27:25.103" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="10098" PostTypeId="2" ParentId="10094" CreationDate="2015-11-27T13:41:20.647" Score="7" Body="&lt;p&gt;One answer is to say that, properly, the identity is $\sec^2x=1+\tan^2x$, where the sides fail to be defined at the same values.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="77" LastEditDate="2015-11-27T22:46:09.107" LastActivityDate="2015-11-27T22:46:09.107" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10099" PostTypeId="2" ParentId="10094" CreationDate="2015-11-27T15:30:55.363" Score="10" Body="&lt;p&gt;If $x$ is meant to be a real variable, then your student is right; $\sec^2 x - \tan^2 x = 1$ is &lt;strong&gt;&lt;em&gt;not&lt;/em&gt;&lt;/strong&gt;&lt;sup&gt;*&lt;/sup&gt; an identity; it suffers from the same sort of problems that $\frac{x}{x} = 1$ does.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It fails to be an identity in two different ways, depending on fine details of what one actually means: either the left hand side is undefined because $x$ is not restricted to the domain of $\sec$ and $\tan$, or the identity fails because the domain of the left hand side differs from the domain of the right hand side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(the former is syntax/semantics based on total functions, the latter on partial functions)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make a correct statement, limit $x$ appropriately: e.g. $\sec^2 x - \tan^2 x = 1$ whenever $x \neq (2n+1) \frac{\pi}{2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As &lt;a href=&quot;https://matheducators.stackexchange.com/a/10098/797&quot;&gt;another answer suggested&lt;/a&gt;, the version $1 + \tan^2 x = \sec^2 x$ has the advantage of actually being an identity in the partial function syntax/semantics when $x$ is a real variable, since both sides have the same domain. Think of it as saying &quot;if one side is defined, then so is the other and they're both equal&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;*: Technically, there are other settings where this would be an identity; e.g. if the two sides don't refer to functions, but instead to equivalence classes of functions where two functions are equivalent if they differ on a set of measure zero, or if every operation is to be followed by &quot;... then continuously extend the result&quot;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-11-27T15:30:55.363" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10100" PostTypeId="2" ParentId="10094" CreationDate="2015-11-27T18:19:49.230" Score="0" Body="&lt;p&gt;Look at the &quot;identity&quot;.  On the right-hand-side, you apparently have the number 1 and on the left hand side, you have the difference of two functions (which is also a function).  A number is not a function, so the stated identity would seem to be wrong.  Undoubtedly the student is trying to make sense of this by thinking of &quot;=&quot; as some sort of &lt;em&gt;process&lt;/em&gt; where it really means (or so s/he thinks) &quot;plug in numbers on the left-hand-side and the result should be the same as the numerical value on the right&quot;.  This is &lt;em&gt;wrong&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The identity is in fact asserting equality of the two sides.  You just have to be careful about defining what the sides are.  The right hand side, which is represented by the symbol &quot;1&quot;, is really the constant function giving 1 on a specified domain.  It would be better to represent it as something like $1_D$ where $D$ is defined to be the domain that matches the domain of the left-hand-side.&lt;/p&gt;&#xA;" OwnerUserId="5961" LastEditorUserId="5961" LastEditDate="2015-11-27T21:03:06.400" LastActivityDate="2015-11-27T21:03:06.400" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10101" PostTypeId="2" ParentId="10021" CreationDate="2015-11-27T19:04:49.100" Score="10" Body="&lt;p&gt;I've noticed that the main problem is in how people teach induction.  Often teachers use examples with number theoretical identities or other examples which by their nature require understanding something else in addition to induction.  In my opinion, induction should be taught by itself first in isolation with hypothetical &quot;Hilbert hotel&quot; type scenarios.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, consider an infinite lunch line of students.  A student in line will laugh if the student in front of him/her laughs.  If the first student in line laughs, is there a student that will not eventually laugh?  Have your students think this out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now consider the same example, but now a student in line will laugh if two or more students ahead of him/her laughs.  The first student in line laughs, what happens now?  Can you be sure every student end up laughing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a final example.  Suppose the rule is now that a student will laugh for certain if there is a student two spots ahead in line and that student laughs.  The first student in line laughs.  Which students will laugh?  Suppose instead of the first student, the second student laughs.  What happens?&lt;/p&gt;&#xA;" OwnerUserId="5961" LastActivityDate="2015-11-27T19:04:49.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10102" PostTypeId="2" ParentId="10094" CreationDate="2015-11-27T22:07:06.950" Score="4" Body="&lt;p&gt;For &lt;em&gt;typical&lt;/em&gt; inputs the equality holds. Thus, whatever further qualifications or discussion may be interesting or necessary, we're &quot;most of the way there&quot;. To &lt;em&gt;deny&lt;/em&gt; that it's &quot;an identity&quot; is to pretend to disqualify the fact on some technical grounds that do not address &lt;em&gt;significant&lt;/em&gt; issue.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-11-27T22:07:06.950" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10103" PostTypeId="2" ParentId="10086" CreationDate="2015-11-28T08:14:01.800" Score="2" Body="&lt;p&gt;For my own child, who was counting past 20 at the age of 2, I have found that visual learning and using games has worked well.  The ideas of addition and subtraction are simple to introduce by using familiar objects that can be counted (e.g. marbles, toys, fruit, etc).  At the age of three, we would practice questions like &quot;how many will you have left if I take away this many?&quot;  or &quot;how many will you have if we put these two piles together?&quot;.  When first learning these concepts, children will need to re-count the piles of objects to determine the new amount.  This is not really adding/subtracting in a typical sense, but it helps a new learner understand the concepts and verify the answers.  Eventually, with practice, a 3 year old can visualize the answer in their head without having to move pieces around.  Though, we stick with numbers below 20.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplication and division are not so simple.  In order to develop this skill, we have been counting pairs/matches when we do memory games.  The concept of division makes sense to our (now) 4 year old (e.g. she knows that if she has 10 cards, when she splits them into pairs, she has 5 piles).  However, multiplication has not interested her, and she does not yet understand multiplication even in basic form.  We have been working with laying out the cards to play the memory matching game in different grids (e.g. 4 x 6, 5 x 3), but she sees no benefit to thinking about it any other way than just &quot;counting&quot;.   Me feeling is that in order for multiplication to make much sense to her, she will need to be able to count by 2s, 3s, etc., which she can do a little.  But, in general, I feel the real complication is finding an activity that is both interesting and educational.  In general, if it helps her win the game (or determine if she has won), she will try to learn the concept.  So, I would like to find a children's game that could put multiplication to use.  I've read here: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/750/mathematical-problems-for-preschoolers?rq=1&quot;&gt;Mathematical problems for preschoolers&lt;/a&gt; about the use of legos (which she loves) so we will explore that idea.&lt;/p&gt;&#xA;" OwnerUserId="1595" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-11-29T12:40:57.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10104" PostTypeId="2" ParentId="10092" CreationDate="2015-11-28T08:22:49.577" Score="1" Body="&lt;p&gt;I highly suggest &quot;Hurricane Calculus&quot; by John B. Hahn (1995).  It was introduced to me in Calc II, but I wish it had been available for my introductory Calculus class.  The writing style is not &quot;textbook&quot;.  Instead it is written more like a person talking to you.  There are also many examples at varying difficulty level, which is extremely helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7GuGN.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7GuGN.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hurricane Calculus covers the full range of first year college calculus, from the concept of limits to the often complex partial derivatives and multiple integrals. It tells you in plain, simple terms what the basic calculus concepts are, provides you with step-by-step problem solving guidelines, and shows you plent of completely worked out examples.&#xA;  Hurricane Calculus may be the most readable book ever written on the subject. There's very little of the mathematical jargon that often causes confusion. The book is full of examples drawn from actual real world applications. Its easy-going style is aimed directly at the reader who may be new to calculus or needs clarification.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1595" LastActivityDate="2015-11-28T08:22:49.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10105" PostTypeId="2" ParentId="10092" CreationDate="2015-11-28T17:28:58.317" Score="1" Body="&lt;p&gt;Do you really need to learn all of this material or just brush up for the standardized test you are planning to take? The answer would probably change the advice you get here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a full learning effort, the &lt;a href=&quot;http://www.ck12.org/&quot; rel=&quot;nofollow&quot;&gt;CK-12 series of books&lt;/a&gt; is a great start, and will help you re-learn what you feel you forgot. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For your state exam, I recommend going to Amazon and searching on &lt;code&gt;FTCE mathematics 6-12&lt;/code&gt; which will give you the study guides targeted to the exam. When I took my state's 6-12 exams (2 different levels) the study guides gave an overview at a level that made more sense than to go through a text similar to what a student would use. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-11-28T17:28:58.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10106" PostTypeId="1" CreationDate="2015-11-29T17:01:18.440" Score="8" ViewCount="698" Body="&lt;p&gt;Is it important to convert subtractions to addition, while simplyfing algebraic expressions (For example converting $3x-2y+7x-7y$ to $3x+(-2y)+7x+(-7y)$)? If yes, why and when? &lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2022-06-11T19:52:59.570" Title="Subtraction to Addition Conversion; how important is it?" Tags="|algebra|negative-numbers|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10107" PostTypeId="2" ParentId="10106" CreationDate="2015-11-29T19:16:38.210" Score="6" Body="&lt;p&gt;My opinion is that it should be exercised and thereafter done mentally (not written down) as part of reading an expression. This then allows students and practitioners to directly follow the order of operations, simplifying at each step, and then in the last step recall and apply only the sign rules for addition. Note that the issue of reading/writing additions and subtractions is identical whether we discuss algebraic expressions or purely arithmetic ones.&lt;/p&gt;&#xA;&lt;p&gt;For example, say we need to simplify &lt;span class=&quot;math-container&quot;&gt;$-10-(-12)+(-7)-4$&lt;/span&gt;. First we treat all the juxtaposed signs as multiplying and simplify to get &lt;span class=&quot;math-container&quot;&gt;$-10+12-7-4$&lt;/span&gt;. Then we can read this as, by definition, additions of positive and negative terms to get &lt;span class=&quot;math-container&quot;&gt;$-9$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Note: (1) this follows the standard order of operations (multiply, then add), (2) the entire process requires only remembering the sign rules for two operations (multiply and addition), (3) at each step the expression gets shorter (notably, removing parentheses in the multiply step), and (4) in practice we don't write down the separate added terms, but should be reading it that way mentally.&lt;/p&gt;&#xA;&lt;p&gt;When I introduce and need to &lt;em&gt;explain&lt;/em&gt; the subtraction, then I'll write something like &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$2 - 6 = -4$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$2 +(-6) = -4$&lt;/span&gt;&amp;quot;, so as to highlight that the latter is not something we'll customarily write down, but is rather part of the mental reading that we do.&lt;/p&gt;&#xA;&lt;p&gt;A blog article I wrote about this issue: &lt;a href=&quot;http://www.madmath.com/2014/07/automatic-negatives.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MadMath: Automatic Negatives&lt;/a&gt;, which motivates the structure of the quiz on &amp;quot;Negative Numbers&amp;quot; at &lt;a href=&quot;http://www.automatic-algebra.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Automatic Algebra&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2022-06-11T15:43:39.237" LastActivityDate="2022-06-11T15:43:39.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10108" PostTypeId="2" ParentId="1040" CreationDate="2015-11-29T19:30:43.930" Score="17" Body="&lt;p&gt;Late answer, but since it comes up every semester I have a stock response that I'd like to share. The most common mistake in this vein is for a student to write $\sqrt{16} = \sqrt{4} = 2$ or $\sqrt{81} = \sqrt{9} = 3$. Frequently if I try to correct this they'll ask, &quot;Where did the radical sign go?&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When this happens, I go to the board and write something like $5 - 3 = 2$, and get them to agree that's true. Then I ask &quot;Explain to me, where did the minus sign go? Why didn't we need to rewrite the minus sign?&quot; And thinking about explaining that does seem to help most students. When they're done I try to frame it as, &quot;When we perform a particular operation, then the symbol goes away because we're done with it.&quot; &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-11-29T19:30:43.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10109" PostTypeId="1" CreationDate="2015-11-29T19:36:51.100" Score="6" ViewCount="196" Body="&lt;p&gt;I'm convinced that, at any given time, there are hundreds (perhaps thousands) of English-speaking students looking for real-time chat-based math help for free.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question: how do I find them and help them for free?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More info: every so often, I get bored and feel like spending an hour or so helping people with math online. Sites like openstudy.com sort of allow me to do this, but don't have a good real-time chat interface, so communication is ugly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;openstudy.com also discourages people from giving students external links (eg, I can help you at my scribblar.com whiteboard at ... ).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a good solution here?&lt;/p&gt;&#xA;" OwnerDisplayName="user3282" LastActivityDate="2016-01-19T20:01:05.287" Title="Where can I help students for free in real-time? (not openstudy.com)" Tags="|tutoring|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10111" PostTypeId="1" CreationDate="2015-11-30T00:17:05.040" Score="8" ViewCount="246" Body="&lt;p&gt;I was helping a high school student with some fundamental concepts of planes and lines, when I realized I am rusty on some definitions myself. I found some minimal coverage in his high school math and precalculus textbook, and I also found a section on my vector calculus text by Marsden. However, none of these coverage seems to be comprehensive - in particular, I could not even find a definition of what it means for two planes to be perpendicular. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometry books like Hilbert and Hartshorne are not at the right level for a senior high student or a freshman. I would imagine there are some books out there that lays out the fundamental definitions, theorems, and proofs involving the perpendicularity of lines and planes, and their parametric and symmetric equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What kind of books cover these things now? Or are students expected to learn this from the Internet?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;: I'm looking for something that can be a reference for me and the student. I think these linear algebra notes are perfectly reasonable level-wise. I was not able to find a definition for &quot;perpendicular planes&quot; in my linear algebra text or these notes, however. In the linear algebra course I took a while back, &quot;orthogonal complements&quot; and &quot;orthogonal subspaces&quot; are covered, but these two are distinct from the concept of &quot;perpendicular planes&quot;.&lt;/p&gt;&#xA;" OwnerUserId="5966" OwnerDisplayName="FreshAir" LastEditorUserId="4872" LastEditDate="2016-01-22T14:45:31.067" LastActivityDate="2016-01-22T14:45:31.067" Title="Coverage of Fundamentals of Lines and Planes" Tags="|geometry|reference-request|" AnswerCount="2" CommentCount="7" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10112" PostTypeId="1" CreationDate="2015-11-30T14:41:17.953" Score="11" ViewCount="423" Body="&lt;p&gt;If this is the wrong forum for this post I apologize but I'm not sure of another well-suited medium for this question (and any reference to one is appreciated).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering if any research in mathematics education has been done about the value of seeing proofs in a course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the one hand, seeing proofs done by mathematicians seems very valuable. However this takes up time and perhaps is better discovered on one's own than seen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, suppose that you just discussed statements instead of their proofs and actually used the statements to prove new results. This also has merit and is more likely what we want students to do within the scope of the course and to some extent as a mathematician.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Has research been done in say a proofs class where one class sees proofs of certain statements and (possibly some/fewer) applications and another class only sees the statements and their applications?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have actual experience with this question?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ideal answer here would be a paper reference to this topic but my searches of this have yielded no fruit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Ideally a first proof course would be the topic of this question (so say first or second year of undergraduate mathematics) but any experiences would be interesting. Maybe even a second course (but something close to elementary definitions like an Elementary Number Theory course let's say where a student would have already taken a proofs course but has no other knowledge of the subject)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 2: Thanks for all the posts thus far! The ideal situation would be a study where say a class does the proof of Fermat's Little Theorem (for example; maybe more than one example) in class and maybe one instance of its use and another class does the statement of Fermat's Little Theorem and does some number of instances of its use. Then on say a final exam both classes get tested on:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Questions using Fermat's Little Theorem&lt;/li&gt;&#xA;&lt;li&gt;Questions that require some sort of proof technique similar to that of Fermat's Little Theorem (maybe like a similar binomial theorem + induction type question).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;and see how students differ in these respects. I suspect there's more factors involved than just seeing the proofs and that the likely situation is that this factor alone is negligible but I'm still curious. Thanks for the current references and I'll be sure to check them out!&lt;/p&gt;&#xA;" OwnerDisplayName="user82634" LastEditorUserId="-1" LastEditDate="2015-12-01T22:32:40.957" LastActivityDate="2015-12-02T23:52:28.393" Title="Effectiveness of students seeing proofs - reference request" Tags="|undergraduate-education|proofs|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10113" PostTypeId="2" ParentId="10074" CreationDate="2015-11-30T15:48:04.963" Score="4" Body="&lt;p&gt;Not aware of any software but here's a simple hack to achieve what you want:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Go to &lt;a href=&quot;http://www.codecogs.com/latex/eqneditor.php&quot; rel=&quot;nofollow&quot;&gt;codecogs equation editor&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Write your LaTeX equation.&lt;/li&gt;&#xA;&lt;li&gt;Copy the url generated (it will be exported as swf).&lt;/li&gt;&#xA;&lt;li&gt;In import dialog box in Prezi, enter the url&lt;/li&gt;&#xA;&lt;li&gt;Voila!&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Someone has written a detailed blog post about it &lt;a href=&quot;https://dawes.wordpress.com/2011/11/06/latex-in-prezi/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2015-11-30T15:48:04.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10114" PostTypeId="2" ParentId="10112" CreationDate="2015-12-01T08:15:09.577" Score="11" Body="&lt;p&gt;Perhaps not a perfect answer, but &#xA;&lt;a href=&quot;http://scimath.net/articles/32/323.pdf&quot;&gt;http://scimath.net/articles/32/323.pdf&lt;/a&gt; is a study of whether the number of maths modules students take affects how good they are at proofs. The result is roughly that students start off bad at proofs and end up slightly less bad at proofs, but those who do more maths modules (being on a maths degree rather than a teaching degree) do improve more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.nctm.org/Publications/journal-for-research-in-mathematics-education/2012/Vol43/Issue4/Expert-and-Novice-Approaches-to-Reading-Mathematical-Proofs/&quot;&gt;This&lt;/a&gt; looks at how students go about reading a proof; the result is they tend to focus on the mathematical expressions and ignore the logical structure, so they cannot easily identify whether a proof is correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't think of other references right now. But the main message I've picked up is that students don't by default know what to do when you show them a proof, so unless you actually teach them how to benefit, they will not gain much from reading/watching a proof.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-12-01T08:15:09.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10115" PostTypeId="2" ParentId="10106" CreationDate="2015-12-01T08:19:20.813" Score="1" Body="&lt;p&gt;Understanding that $3-2=3+(-2)$ becomes relevant when learning group theory, where you need to have additive inverse elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Off the top of my head I can't remember really needing it before that, although I can imagine it might be important in some ways of explaining operations.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-12-01T08:19:20.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10116" PostTypeId="2" ParentId="10106" CreationDate="2015-12-01T12:39:33.177" Score="3" Body="&lt;p&gt;&lt;strong&gt;Why&lt;/strong&gt;? It much depends on the background. Here are a couple of scenarios.&lt;/p&gt;&#xA;&lt;p&gt;You have the philosophy of &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/New_Math&quot; rel=&quot;nofollow noreferrer&quot;&gt;new math era&lt;/a&gt;&amp;quot; in mind in which &lt;span class=&quot;math-container&quot;&gt;$x-y$&lt;/span&gt; should be taught as &lt;span class=&quot;math-container&quot;&gt;$x+(-y)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$-y$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$(-1).y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You have used the so-called &lt;a href=&quot;https://nrich.maths.org/5947&quot; rel=&quot;nofollow noreferrer&quot;&gt;counters model&lt;/a&gt; to teach the addition and subtraction of positive and negative numbers. In this case, you might want your students to see &lt;span class=&quot;math-container&quot;&gt;$3x-2y+7x-7y$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$3x+(-2y)+7x+(-7y)$&lt;/span&gt; since then they can think of &lt;span class=&quot;math-container&quot;&gt;$3x$&lt;/span&gt; plus &lt;span class=&quot;math-container&quot;&gt;$7x$&lt;/span&gt; as counting the number of blue counters (representing positive), and &lt;span class=&quot;math-container&quot;&gt;$(-2y)$&lt;/span&gt; plus &lt;span class=&quot;math-container&quot;&gt;$(-7y)$&lt;/span&gt; as counting the number of red counters (representing negatives).&lt;/p&gt;&#xA;&lt;p&gt;Please consider the above scenarios is just scenarios not what I am suggesting as a teaching method.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;When&lt;/strong&gt;? If the point of the question is &lt;strong&gt;to simplify&lt;/strong&gt; an algebraic expression like the one you have given, I guess there would be no must unless you have a pure algebraic algebra lesson in mind. However, if the point of the question is to &lt;strong&gt;see a certain structure&lt;/strong&gt;, then it might be useful to convert subtraction to addition. Here are a couple of scenarios.&lt;/p&gt;&#xA;&lt;p&gt;consider writing &lt;span class=&quot;math-container&quot;&gt;$1-x+x^2-x^3+x^4- ...$&lt;/span&gt; in sigma notation. We need to see it as &lt;span class=&quot;math-container&quot;&gt;$1+(-x)+x^2+(-x^3)+x^4+ ...$&lt;/span&gt; and then as &lt;span class=&quot;math-container&quot;&gt;$1+(-1) x+ x^2+(-1) x^3+ x^4+ ...$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Suppose you want to discuss with your students that an expression like &lt;span class=&quot;math-container&quot;&gt;$a-b+c$&lt;/span&gt; (whether numeric or algebraic) can be structured in two, though equivalent, ways: &lt;span class=&quot;math-container&quot;&gt;$(a-b)+c$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$a + (-b+c)$&lt;/span&gt;. In this case, rewriting the expression as &lt;span class=&quot;math-container&quot;&gt;$a+(-b)+c$&lt;/span&gt; would relate the situation to the associative property. And you know,  associativity comes into play in one way or the other even when you just want to simplify an expression.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Added&lt;/strong&gt;. Or you want to show them &lt;span class=&quot;math-container&quot;&gt;$x .(y-z)$&lt;/span&gt; is just &lt;span class=&quot;math-container&quot;&gt;$x.(y+(-z))$&lt;/span&gt;, thus we use the distribution property for subtraction. Or &lt;span class=&quot;math-container&quot;&gt;$(x-y)^2$&lt;/span&gt; can be seen as &lt;span class=&quot;math-container&quot;&gt;$(x+(-y))^2$&lt;/span&gt;, thus we can get the identity for the former from well-known identity for the latter.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The bottom line&lt;/strong&gt; is your students need to see that they can and sometimes they must convert subtraction to addition though they might rarely need it explicitly for the sake of calculations&lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-12-04T08:46:41.920" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10117" PostTypeId="1" CreationDate="2015-12-01T15:49:16.643" Score="10" ViewCount="188" Body="&lt;p&gt;Let's say we study a reccurence relation such as&#xA;$$a_{n+1}=a_{n}+n, n \geq 1$$&#xA;I find many students are having difficulties when you ask them to find $$a_{n}$$ in terms of $$a_{n-1}$$ where they just have to&#xA; replace $n$ by $n-1$ in the above relation, and get $$a_{n}=a_{n-1}+n-1, n \geq 2$$. They don't get that $n$ is a just a variable, some placeholder for a value, and could be substituted with whatever we want, even with something also depending on $n$.&#xA;They get it if we replace $n$ by some specific number such as 2 or 3 but they don't see replacing $n$ with another expression which also contains $n$ such as $n-1$ legit.&#xA;What could be done to make them understand this better?&#xA;Maybe use another letter, say replace $n$ by $k-1$? And then work only with $k$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested in another ways of making them understand this substitution better.&lt;/p&gt;&#xA;" OwnerUserId="5972" LastActivityDate="2015-12-02T03:23:28.393" Title="Substitution in recurrence relations" Tags="|sequences|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10118" PostTypeId="2" ParentId="10117" CreationDate="2015-12-01T16:58:35.413" Score="8" Body="&lt;p&gt;One idea: Try using function notation instead $$a(n+1)=a(n)+n.$$ It's more familiar than subscripts. Then ask for $a(m)$ possibly with the hint that $m=n+1$.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-12-01T16:58:35.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10119" PostTypeId="1" CreationDate="2015-12-02T01:32:39.110" Score="4" ViewCount="65" Body="&lt;p&gt;I am working as a volunteer, teaching children programming once a week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The attendence is somewhat irregular (and the children are a bit restless :P) so, instead of more traditional lectures, we try to give the children self-paced activities at the computer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This class is goes on for 1hour 20mins, in which we usually have the students trying to complete a series of &quot;programming challenges&quot; aimed at teaching a particular concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often, however, children get tired in mid activity.&#xA;What can we do to re-energize the children?&#xA;How long is it resonable to expect an activity to hold a child's attention?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(ages: 11 up to 15. But mostly 11 and 12)&lt;/p&gt;&#xA;" OwnerUserId="1662" LastEditorUserId="1662" LastEditDate="2015-12-02T02:05:25.067" LastActivityDate="2015-12-02T04:01:37.537" Title="How to better pace activities" Tags="|class-participation|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10120" PostTypeId="2" ParentId="10117" CreationDate="2015-12-02T03:23:28.393" Score="4" Body="&lt;p&gt;The students may need some support to imagine what the recurrence relation is telling them. Many may not realise that the sequence of a's is in some sense already there, and the recurrence relation tells them how to calculate one term from others no matter what part of the sequence they are looking at.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience is that pictures often help students feel that mathematical things are real and so allows them to do new things with them. It also gives them something they can &quot;hang their understanding on&quot; - something that they can go back to until they understand in a symbolic way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a suggestion based on that idea that may help. (I'd appreciate if you try it telling me if it does help!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw a picture like this to represent the sequence itself:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/IVwWw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IVwWw.png&quot; alt=&quot;sequence as boxes&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also draw a general part of the sequence centred at a$_n$:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/5XYgL.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5XYgL.png&quot; alt=&quot;sequence as boxes at general n&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The recurrence relation tells you how to calculate a$_{n+1}$ from a$_n$, and you can represent this as an arrow. I put a curl in the arrow to indicate that something possibly complex happens in between:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/89IVY.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/89IVY.png&quot; alt=&quot;recurrence relation as arrow&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key point students need to realise is that this applies no matter what $n$ is. To start off with, you can calculate a$_2$ using a$_1$, you can calculate a$_3$ using a$_2$, and so on. Slowly add the arrows from left to right to indicate how this happens: &#xA;&lt;a href=&quot;https://i.stack.imgur.com/gAbUE.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gAbUE.png&quot; alt=&quot;recurrence relation as arrow beginning&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can say that it works anywhere in the sequence, even if you don't know what n is. You calculate a$_{n+1}$ using a$_n$, you can calculate a$_n$ using a$_{n-1}$ and so on. So now you can add extra arrows to the left and right of the picture with general n:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/10GMz.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/10GMz.png&quot; alt=&quot;recurrence relation as arrow anywhere&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you can circle the bit you want, which is the bit that shows how to go from a$_{n-1}$ to a$_n$, to highlight that this is what you are looking for.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-12-02T03:23:28.393" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10121" PostTypeId="2" ParentId="10119" CreationDate="2015-12-02T04:01:37.537" Score="5" Body="&lt;p&gt;I've seen in a couple places that children tend to have attention spans of under 10 minutes, while teenagers and adults may have attention spans of up to 20 minutes (&lt;a href=&quot;http://www.wisegeek.org/what-is-considered-a-normal-attention-span.htm&quot;&gt;example&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At least at the community college level, I try to frame all by presentations in 30 minute blocks, with about 20 minutes of me lecturing/presenting, and about 10 minutes of students doing exercises on their own (with me circulating with checks and advice). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So that may not be exactly what you want in your situation. But if you can find some way of switching between different contexts every 20 minutes or so (maybe lecture vs. exercise, or individual vs. group work, or something), that might help.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-12-02T04:01:37.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10122" PostTypeId="1" CreationDate="2015-12-02T15:40:54.113" Score="14" ViewCount="552" Body="&lt;p&gt;I am tasked with teaching an introductory differential equations course.  In my experience, this course tends to be a laundry list of methods that apply to specific forms of differential equations where each form needs to be memorized.  I think such a course lacks continuity and appeal, so my question is: what are possible themes for a first differential equations course?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some ideas I have come up with are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Modeling&lt;/em&gt;.  The course should be focused on taking real-life situations and producing differential equation-based models.  I suspect most of these models would be unsolvable, so maybe this course would also have to focus on numerical methods.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Reversing Calculus&lt;/em&gt;. Cast each method as undoing a method from regular calculus.  This would work for several of the methods, but maybe not for things like Laplace transforms...&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;We can solve very little&lt;/em&gt;.  The focus of a course could be on the fact that most equations we cannot solve, but there a couple of types that we can, so we should work to approximate the equations we cannot solve with ones that we can.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Change of basis&lt;/em&gt;. Make the course about the Laplace transform and the Fourier transform and converting back and forth between calculus problems and algebra problems.  I don't know whether this subsumes all of the traditional techniques or not...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have very limited experience with differential equations, so I'd appreciate comments from people with a broader view of the subject&amp;mdash;I could be missing the point entirely!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;: I am teaching the only section of first-year honors diffeq, so I have some flexibility. The students are studying sciences, so mostly physics, chemistry, and biology. A few of them are math majors. &lt;/p&gt;&#xA;" OwnerUserId="5978" LastEditorUserId="143" LastEditDate="2016-01-22T19:39:15.197" LastActivityDate="2018-12-03T22:02:11.977" Title="Theme of a Differential Equations Course" Tags="|differential-equations|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10123" PostTypeId="2" ParentId="10111" CreationDate="2015-12-02T16:58:16.250" Score="3" Body="&lt;p&gt;In Finland this is covered (very briefly) in gymnasium, in the third course of extended mathematics, so most such course books should cover the subject. These are not rigorous treatments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one I'm looking at right now (Pyramidi 3) does the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Fail to define the angle between two lines. One exercise does concern this, but it is left up to the teacher to mention. This is not emphasized at all in the text.&lt;/li&gt;&#xA;&lt;li&gt;Define the normal (line) to a plane. The angle between the normal and the plane is 90 degrees.&lt;/li&gt;&#xA;&lt;li&gt;Define the angle between a line and a plane (using a projection, which I doubt has been defined anywhere, so it is left as a heuristic throw-away).&lt;/li&gt;&#xA;&lt;li&gt;Through any point outside the plane there is exactly one parallel plane.&lt;/li&gt;&#xA;&lt;li&gt;The angle between parallel planes is 0 degrees.&lt;/li&gt;&#xA;&lt;li&gt;The angle between two intersecting planes is the angle between their normals.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There is one example and a total of 3 exercises, after which the subject changes. Everything takes two spreads (4 pages, page size somewhat smaller than A4): one spread for the definitions and one for the example and the exercises and one huge picture of a book that relates to one exercise but is not essential.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't remember explicit treatment of this in the university either, but might be that I have forgot. Linear algebra courses focused on other matters.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2015-12-18T17:02:50.020" LastActivityDate="2015-12-18T17:02:50.020" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10124" PostTypeId="1" AcceptedAnswerId="10127" CreationDate="2015-12-02T20:07:15.830" Score="22" ViewCount="1878" Body="&lt;p&gt;I am writing some Calculus content, and I would like a &quot;big list&quot; of useful functions which are defined by definite integrals, but are not elementary functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two examples of such functions are&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\mathrm{Erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\mathrm{d} t&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which is fundamentally important to statistics, and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\mathrm{Si}(x) = \int_0^x \frac{\sin(t)}{t} \mathrm{d} t&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which comes up all the time in signal processing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to be able to sketch such functions, express some definite integrals (like $\int_0^1 e^{-4t^2} dt$) in terms of such functions, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what other functions are important enough to have their own name, and are given as integrals of elementary functions?&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2015-12-02T20:51:00.523" LastActivityDate="2016-03-27T19:18:53.730" Title="Good examples of functions defined as definite integrals of elementary functions?" Tags="|calculus|reference-request|problem-design|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10125" PostTypeId="2" ParentId="10124" CreationDate="2015-12-02T20:29:15.750" Score="11" Body="&lt;p&gt;The function $\displaystyle \text{Li}(x) = \int_2^x \frac{1}{\log t} \,\, dt$ comes up in the study of the distribution of primes.  Specifically, the number of prime numbers less than $x$ is asymptotic to $\text{Li}(x)$ and a major consequence of the Riemann Hypothesis would be the sharpest possible bound for the difference between these two functions as $x \rightarrow \infty$.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2015-12-02T20:29:15.750" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10126" PostTypeId="2" ParentId="4070" CreationDate="2015-12-02T23:03:34.987" Score="0" Body="&lt;p&gt;First, we find an integer &lt;em&gt;universally&lt;/em&gt; coprime to any prime $\,p.\,$ A moments thought yields the solution $\color{#c00}1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, we generalize to larger solutions. Since the coprimality of $\,n\,$ to $\,p\,$ depends only its remainder modulo $\,p,\,$ the above solution generalizes to $\, \color{#c00}1+mp,\,$ for any integer $\,m.$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To generalize that to multiple primes $\,p_1,\ldots,p_k\,$ we replace $\,mp\,$ by $\,mp_1\cdots p_k\,$ to obtain an integer that leaves remainder $\color{#c00}1$ when divided by every $\,p_i,\,$ yielding Euclid's classical result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With appropriate hints, each of the above steps can be motivated quite naturally. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark&lt;/strong&gt; $\ $ In fact one could just as easily motivate a more general solution.&#xA;Euclid employed $\,1 + p_1\cdots p_n$ is coprime to $\,c = p_1\cdots p_n.\,$ Stieltjes noted the generalization that, furthermore,  $\ \color{#c00}{p_1\cdots p_k} +\, \color{#0a0}{p_{k+1}\cdots p_n}\,$ is coprime to $\,c\,$ too, which motivates the following &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Key Idea&lt;/strong&gt; $\, $ Coprimes to $\,c\,$ arise by partitioning  into $\rm\color{#c00}{two}\ \color{#0a0}{summands}$ all prime factors of $\,c,\,$ i.e.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt; $\,\ \ \color{#c00}a+\color{#0a0}b\ $ is coprime to $\ c\:$ if every prime factor of $\,c\,$ divides $\,a\,$ or $\,b,\,$ but not both.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proof&lt;/strong&gt; $\ $ If not then  $\,a+b\,$ and $\,c\,$ have a common prime factor $\,p.\,$ By hypothesis $\,p\mid a\,$ or $\,p\mid b.\,$ Wlog, say $\,p\mid b.\,$ Then $\,p\mid (a+b)-b = a,\,$ so $\,p\,$ divides both $\,a,b,\,$ contra hypothesis. $ $ &lt;strong&gt;QED&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that Euclid's idea is just the special case $\,k=0.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This extra generality proves quite useful in many applications, e.g. see &lt;a href=&quot;https://math.stackexchange.com/a/855462/242&quot;&gt;this answer.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can generalize further, e.g. from $\,p_i$ distinct primes to pairwise coprimes.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-12-02T23:27:44.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10127" PostTypeId="2" ParentId="10124" CreationDate="2015-12-02T23:03:50.140" Score="17" Body="&lt;p&gt;It seems that the key term here may be the somewhat non-specific-sounding &lt;em&gt;special functions&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By googling for a few examples (Erf, Si, Li) I came across a &lt;a href=&quot;http://math2.org/math/integrals/specialfuns.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Table of Special Functions&lt;/strong&gt;&lt;/a&gt; and, on the &lt;em&gt;Lists of integrals&lt;/em&gt; wikipage, there is a sub-section on &lt;a href=&quot;https://en.wikipedia.org/wiki/Lists_of_integrals#Special_functions&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Special Functions&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a related remark, one reason that functions may be presented and/or defined in terms of integrals is that it may not be possible to express them using elementary functions. Actually &lt;em&gt;proving&lt;/em&gt; that an antiderivative cannot be written using elementary functions is generally done using Liouville's criterion; in the spirit of the OP's mention of $\text{Si}(x)$, I include here a pointer to &lt;a href=&quot;https://math.stackexchange.com/a/832601&quot;&gt;&lt;strong&gt;MSE 694915&lt;/strong&gt;&lt;/a&gt; in which I provided a reference with a proof that the antiderivative of $\sin(x)/x$ cannot be expressed using elementary functions alone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For one other example, see Peter Mueller's proof that the antiderivative of $x \tan x$ cannot be written using elementary functions in &lt;a href=&quot;https://mathoverflow.net/a/108616&quot;&gt;&lt;strong&gt;MO 108598&lt;/strong&gt;&lt;/a&gt; (for an additional reference proving this fact, see the answer that I provided at the same question). Although I am not aware of an important function defined directly in terms of an integral of $x \tan x$, one can use the relation between its antiderivative and the dilogarithm to show that the latter, too, cannot be expressed in terms of elementary functions.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-12-02T23:03:50.140" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10129" PostTypeId="2" ParentId="10111" CreationDate="2015-12-03T00:06:10.693" Score="7" Body="&lt;p&gt;If the student attends a school that has a distinct Geometry course (typically taken in 9th or 10th grade) then it will probably use a textbook similar to &lt;a href=&quot;http://www.mheducation.com/prek-12/program/MKTSP-GAR20M0.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Glencoe Geometry&lt;/a&gt; or &lt;a href=&quot;http://www.hmhco.com/shop/k12/HMH-Geometry/9780544385818&quot; rel=&quot;nofollow noreferrer&quot;&gt;HMH Geometry&lt;/a&gt;.  I have one of those open in front of me now; looking at it, I see that the sequence of topics runs like this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;First, &quot;perpendicular lines&quot; are defined as lines that intersect to form 90° angles.&lt;/li&gt;&#xA;&lt;li&gt;Second, &quot;a line perpendicular to a plane&quot; is defined to mean a line that is perpendicular to every line in the indicated plane.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Parallel planes&quot; are defined, and one of the postulates in the book is that if two planes are non-parallel then they intersect in a single line.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;But (somewhat to my surprise) &quot;two perpendicular planes&quot; does not appear to be defined or ever used anywhere in the context of the book.  I don't know that it's possible to say &lt;em&gt;why&lt;/em&gt; that concept is never invoked, but I have a conjecture:  The typical (?) definition of &quot;perpendicular planes&quot; goes like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A planes $M$ is perpendicular to a plane $N$ if there exists some line $m$ in $M$ that it perpendicular to every line $n$ in $N$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That, at least, is the definition I found in the first few Google search results for &quot;perpendicular planes definition&quot; (examples: &lt;a href=&quot;https://www.mathsisfun.com/geometry/parallel-perpendicular-lines-planes.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://www.cliffsnotes.com/study-guides/geometry/fundamental-ideas/parallel-and-perpendicular-planes&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). Notice, first, that this definition involves two quantifiers: An existential quantifier on $m$ (we only need to find &lt;em&gt;one&lt;/em&gt; line $m$ in $M$ that meets the required condition), and a universal quantifier on $n$ (once we have chosen $m$, &lt;em&gt;every&lt;/em&gt; line $n$ in $N$ needs to be perpendicular to it). That's conceptually quite subtle, arguably on a par with the well-known conceptual difficulties around the quantification in epsilon-delta proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, notice that although the property being defined is symmetric (i.e. &quot;$M$ is perpendicular to $N$&quot; means exactly the same thing as &quot;$N$ is perpendicular to $M$&quot;), the &lt;em&gt;definition is not&lt;/em&gt;.  In fact it is not at all obvious that the existence of an $m$ in $M$ that is perpendicular to every $n$ in $N$ implies the existence of a specific $n$ in $N$ that is perpendicular to every $m$ in $M$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that the conventional wisdom among curriculum designers, to the extent that it has given any thought to the matter at all, is that the potential benefit of &lt;em&gt;defining&lt;/em&gt; perpendicular planes is outweighed by the difficulties entailed by the definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that another possible definition would be to say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose planes $M$ and $N$ intersect in line $l$, and choose any point $P$ on $l$.  Let $m$ be the line in $M$ that is perpendicular to $l$ at $P$, and let $n$ be the line in $N$ that is perpendicular to $l$ at $P$.  Then planes $M$ and $N$ are called &lt;strong&gt;perpendicular&lt;/strong&gt; if the lines $m$ and $n$ are perpendicular.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This definition has the advantage of being obviously symmetric in $M$ and $N$. It also generalizes, with very little work, to define a dihedral angle:  The angle between $M$ and $N$ is defined to be the angle between $m$ and $n$.  With this definition, $M$ is perpendicular to $N$ if the dihedral angle between them is 90°.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But whatever advantages this other definition may have, they are also probably outweighted by the semantic complexity of the definition, which requires &lt;em&gt;six geometric objects&lt;/em&gt; (not only the two planes $M$ and $N$ but also the three lines $l, m$ and $n$ and the point $P$) to define a single relationship.  For 9th or 10th graders, it's a disaster waiting to happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it does not surprise me that Geometry textbooks avoid the problem of &lt;em&gt;defining&lt;/em&gt; what it means for two planes to be perpendicular.  On the other hand it &lt;em&gt;does&lt;/em&gt; surprise me that they avoid the concept altogether.  High school geometry typically treats rigor as a kind of aspirational value, not an absolute commitment.  I think it's pretty surprising that I don't find any exercises that show, for example, a three-dimensional solid (say a frustrum) and ask the student to identify which planes are parallel, which are perpendicular, and which are neither.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, all of the above assumes you are interested in the geometry of lines and planes in space from a &lt;em&gt;synthetic&lt;/em&gt; or &lt;em&gt;Euclidean&lt;/em&gt; perspective.  But if you want &quot;parametric and symmetric equations&quot; for planes (which I see was part of your question), then a Geometry course is the wrong place to look -- you want to look at a &lt;em&gt;Precalc&lt;/em&gt; book.  I have one in front of me:  It's the Glencoe &lt;em&gt;Advanced Mathematical Concepts: Precalculus with Applications&lt;/em&gt;.  In this textbook we find that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;An equation of the form $ax+bc+cz=d$ defines a plane in 3-dimensional space&lt;/li&gt;&#xA;&lt;li&gt;So a system consisting of two such equations defines two planes, which either are parallel (in which case there is no solution), or intersect in a single line (in which case there are infinitely many solutions that can all be expressed in terms of a single parameter), or coincide (in which case there are infinitely many solutions that can all be expressed in terms of two parameters)&lt;/li&gt;&#xA;&lt;li&gt;And a system consisting of three such equations defines three planes, which either do not intersect, or intersect in a line, or intersect in a single point, or coincide.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A typical treatment looks something like this:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/tUcWO.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tUcWO.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But notice that in this context -- solving systems of equations in three variables -- it &lt;em&gt;does not matter whether the planes are perpendicular or not&lt;/em&gt;.  All that matters is whether they &lt;em&gt;intersect&lt;/em&gt;; the &lt;em&gt;angle&lt;/em&gt; of intersection is irrelevant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much later in a Precalc book, we encounter information about vectors, dot products, and cross products.  In this context students learn that two vectors are perpendicular if and only if their dot product is zero.  They also learn that the cross product of two vectors is perpendicular to the plane in which those vectors lie:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/exz38.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/exz38.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such properties make it possible for students to answer questions like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Given two vectors, determine if they are perpendicular&lt;/li&gt;&#xA;&lt;li&gt;Given three points in space, determine if any of the vectors joining those points are perpendicular to each other&lt;/li&gt;&#xA;&lt;li&gt;Given two vectors, find a third vector perpendicular to both&lt;/li&gt;&#xA;&lt;li&gt;Given three points in space, find a vector perpendicular to the plane containing the three points&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Students also learn to:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Given a point $P$ and a vector $\vec{v}$, write a pair of equations for the line through $P$ in the direction of $\vec{v}$ in parametric form&lt;/li&gt;&#xA;&lt;li&gt;and vice versa.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This all gets pretty close to what you are asking for.  But I still don't see in this textbook any place where students encounter the definition of a normal vector to a plane, or study the relationship between the equation of a plane and its orientation in space, or talk about what it means for two planes to be perpendicular.  I think that material is simply beyond the scope of high school mathematics in the United States.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One more thought:  You asked about &quot;Fundamental definitions, theorems, and proofs&quot;.  Quite apart from content, you should probably be aware that very few mathematics textbooks at the secondary level (with the exception of Geometry!) are organized along &quot;definitions, theorems, and proofs&quot;.  Students learn techniques for solving problems, but those techniques are not organized into a logico-deductive structure, in which some properties are taken as primitive and others derived.  That is not to say that such a structure does not exist at all, but it is not made explicit; the word &quot;Theorem&quot; is almost never used, and when it &lt;em&gt;is&lt;/em&gt; used, normally a Theorem is merely &lt;em&gt;illustrated&lt;/em&gt;, not &lt;em&gt;proved&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-12-03T00:06:10.693" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10130" PostTypeId="2" ParentId="10124" CreationDate="2015-12-03T02:34:32.020" Score="17" Body="&lt;p&gt;The gamma function is very useful in counting problems (among others) and is seen as an extension of the factorial function into the reals.  It is defined as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,.&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Incidentally, this is the example of how to use MathJax in the help section.)&lt;/p&gt;&#xA;" OwnerUserId="5981" LastActivityDate="2015-12-03T02:34:32.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10131" PostTypeId="1" CreationDate="2015-12-03T06:01:11.437" Score="1" ViewCount="104" Body="&lt;p&gt;I am self-teaching myself precalculus.My biggest sticking point is to move lines within a diagram, especially, functions in the coordinated plane. For instance, doing horizontal and vertical shifts of functions. Could there be a base line skill that I am lacking? Now I have two theories for this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) As the diagram becomes more abstract we are no longer, &quot;naming the parts of the diagram&quot; but the parts are implied, like the book calls it &lt;strong&gt;graph of a function&lt;/strong&gt; as opposed to the &lt;strong&gt;line representing the function.&lt;/strong&gt; Therefore, I might get lost. &#xA;(2) A lack of concentration to enable myself to move the line around, thereby, having difficulty manipulating the line around the coordinate plane. Maybe a specific spatial skill? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am passionate about finding conceptual issues in learning and I really hope you guys can help me! Or point me out to literature that touches on this topic. &lt;/p&gt;&#xA;" OwnerUserId="5982" LastActivityDate="2015-12-08T23:47:21.000" Title="Moving lines in schematic diagrams" Tags="|self-learning|students-mistakes|notation|precalculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10132" PostTypeId="2" ParentId="10124" CreationDate="2015-12-03T14:11:25.320" Score="12" Body="&lt;p&gt;My first take is&#xA;$$&#xA;\ln(x) = \int_1^x\frac1t dt.&#xA;$$&#xA;Granted, some texts introduce the natural log of the inverse of $\exp$ but other texts define $\ln$ as above and the $\exp$ as the inverse. If I remember correctly, the definition of the logarithm by the integral was historically first.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2015-12-03T14:11:25.320" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10133" PostTypeId="2" ParentId="10122" CreationDate="2015-12-03T18:40:59.507" Score="4" Body="&lt;p&gt;These are some of the questions I would use to guide the course:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is a differential equation ? In particular, what language should we use to communicate the structure of a differential equation.&lt;/li&gt;&#xA;&lt;li&gt;What is a solution to a differential equation ? In particular, what object is best suited to capture our intuitive concept of a solution?&lt;/li&gt;&#xA;&lt;li&gt;Can we free ourselves of coordinates in both of the items above?&lt;/li&gt;&#xA;&lt;li&gt;For each example we study, what is the relation between different solutions? What is the structure of the general solution?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have not fully embraced this program in my teaching, it would be difficult with the general audience of the course. But, I think there is room for creative implementation of symmetry methods in the elementary DEs course. I want someone to write the book which does this, while not sacrificing standard techniques.Of course in addition, it seems obligatory to add:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;where do differential equations appear in other math and the &lt;em&gt;real world&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This question is actually the easiest I've listed.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-12-03T18:40:59.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10134" PostTypeId="2" ParentId="9635" CreationDate="2015-12-04T03:35:11.543" Score="1" Body="&lt;p&gt;You might find &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0817637079&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edwards' book on advanced calculus&lt;/a&gt; using differential forms to have useful stuff.  The very first section is about work in a constant vector field - I haven't used it as a text and I'm not an engineer so I can't say whether it will fit your needs as a supplement, but at the very least it could give ideas for your proposed writing of a book...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Borrowed from my answer at &lt;a href=&quot;https://math.stackexchange.com/questions/1525309/multivariable-calc-second-course-that-does-differential-forms/1525326#1525326&quot;&gt;math.SX&lt;/a&gt;, which might be another place to ask about such things.  Good luck.)&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-12-04T03:35:11.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10135" PostTypeId="2" ParentId="10124" CreationDate="2015-12-04T09:42:50.520" Score="8" Body="&lt;p&gt;No one has still mentioned Fresnel functions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$S(x)=\int_0^x \sin(t^2)dt$ and $C(x)= \int_0^x \cos(t^2)dt$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They are (of course) very relevant in signal analysis and in studying diffraction. What is less often mentioned is that the parametric function $x\mapsto (S(x),C(x))$ gives you a beautiful curve, the Cornu spiral, which is used by engineers in roads and railorads design and which is a very nice curve from the differential geometric point of view since its curvature is a linear function.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2015-12-04T09:42:50.520" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10136" PostTypeId="1" AcceptedAnswerId="10140" CreationDate="2015-12-04T23:07:26.407" Score="6" ViewCount="1505" Body="&lt;p&gt;My supervisors want to see coding integrated into the ninth grade Geometry class. This class is mostly concerned with proofs--not too much algebra. These students know a decent amount of the visually oriented language: &lt;a href=&quot;https://processing.org/&quot; rel=&quot;noreferrer&quot;&gt;Processing&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I integrate their coding ability into the geometry class as a project?&lt;/p&gt;&#xA;" OwnerUserId="5995" LastEditorUserId="62" LastEditDate="2015-12-09T17:12:10.633" LastActivityDate="2017-04-18T17:20:19.537" Title="Integrate Coding into the Geometry Curriculum" Tags="|secondary-education|curriculum|geometry|computer-programs|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10137" PostTypeId="2" ParentId="10136" CreationDate="2015-12-05T01:48:18.577" Score="2" Body="&lt;p&gt;I misread the question, and this doesn't answer it (because Dopapp specificially&#xA;wants to use the &lt;a href=&quot;https://processing.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;Processing&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt; language). But I will leave it here for the links.&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(1) The Khan Academy has 10 drawing and animation projects based on &lt;a href=&quot;http://www.w3schools.com/js/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;JavaScript&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt;.&#xA;See their curriculum &lt;a href=&quot;https://www.khanacademy.org/computing/computer-programming/programming&quot; rel=&quot;nofollow noreferrer&quot;&gt;at this link&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) &lt;a href=&quot;https://scratch.mit.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;Scratch&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt; is even easier to learn than JavaScript. Some intro projects can be&#xA;found at &lt;a href=&quot;http://www.edutopia.org/blog/scratch-programming-drawing-2d-shapes-dylan-ryder&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eutopia's link&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) Then there is &lt;a href=&quot;https://www.geogebra.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;Geogebra&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt;, with a broad community and many resources available. Maybe start &lt;a href=&quot;http://wiki.geogebra.org/en/Main_Page&quot; rel=&quot;nofollow noreferrer&quot;&gt;at this link&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) &lt;a href=&quot;http://www.dynamicgeometry.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;Geometer's Sketchpad&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt;, dynamic geometry software. (Thanks to user &lt;em&gt;celeriko&lt;/em&gt;).&#xA;They have &lt;a href=&quot;http://www.keycurriculum.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;curricula and considerable supporting resources&lt;/a&gt; provided by&#xA;McGraw-Hill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(5) &lt;a href=&quot;http://cinderella.de/tiki-index.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;Cinderella&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt;: Powerful dynamic geometry&#xA;software, used widely in Germany and Japan, perhaps too sophisticated for your purposes.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/ZTprc.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZTprc.jpg&quot; alt=&quot;Cinderalla&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-12-06T20:45:56.920" LastActivityDate="2015-12-06T20:45:56.920" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10138" PostTypeId="1" AcceptedAnswerId="10144" CreationDate="2015-12-05T08:58:09.967" Score="7" ViewCount="209" Body="&lt;p&gt;When I talk to other research mathematicians, there is pretty uniform agreement that we want our (university) students to understand the maths we teach them rather than just memorise processes. As far as I can tell, we have a reasonably consistent idea of what that means (although much less agreement of what to expect of students in a given context). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in the context of curriculum discussions, I've found that non-mathematicians are unwilling to accept 'understanding' as a suitable word to describe the desired outcome. I've also read a couple of times that students don't necessarily share our view of the term, although I haven't had time to check out the references.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there alternative phrases that can be used to describe what we mean by mathematical 'understanding' (at undergraduate level) that are better at communicating the meaning to non-mathematicians?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have come across the phrases 'relational understanding' and 'instrumental understanding', which I think are helpful, but would only be meaningful to the minority who have come across them before.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="376" LastEditDate="2015-12-05T11:22:42.367" LastActivityDate="2015-12-06T22:47:04.833" Title="Alternative terms for 'mathematical understanding'" Tags="|undergraduate-education|terminology|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10139" PostTypeId="2" ParentId="3865" CreationDate="2015-12-05T15:06:41.573" Score="3" Body="&lt;p&gt;&lt;em&gt;Mathematics for secondary School Teachers&lt;/em&gt;, put out by MAA, is a good text used at our universities here.  Also, &lt;a href=&quot;https://math.berkeley.edu/~wu/&quot; rel=&quot;nofollow&quot;&gt;Hung-Hsi Wu&lt;/a&gt; at Berkeley has some good stuff for the elementary level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Currently the &lt;a href=&quot;http://www.aplu.org/projects-and-initiatives/stem-education/mathematics-teacher-education-partnership/&quot; rel=&quot;nofollow&quot;&gt;Mathematics Teacher Education Partnership&lt;/a&gt; has a team that is currently working to develop materials that are more aligned to teaching pre-service teachers that special MKT.&lt;/p&gt;&#xA;" OwnerUserId="6001" LastActivityDate="2015-12-05T15:06:41.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10140" PostTypeId="2" ParentId="10136" CreationDate="2015-12-05T19:54:12.390" Score="3" Body="&lt;p&gt;Here is one idea for a project. Have them implement &lt;a href=&quot;https://en.wikipedia.org/wiki/Descartes%27_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Descartes' Theorem on four tangent circles&lt;/a&gt;:&#xA;Given three tangent circles, the theorem can be used to construct a fourth:&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/HqDop.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HqDop.png&quot; alt=&quot;4TangentCircs&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Descartes%27_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;You could start with the special case when one of the first three circles has&#xA;an infinite radius and so is a straight line. Then only two tangent circles touching that line are needed to get started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So a separate problem is: Given $r_1$ and $r_2$, construct two circles with&#xA;those radii tangent to each other and resting on the $x$-axis. &#xA;That seems accessible to $9^\textrm{th}$ graders.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This allows the project to start with a relatively easy part, move on&#xA;to the more challenging general case, and you might entice them beyond with&#xA;the &lt;a href=&quot;https://en.wikipedia.org/wiki/Apollonian_gasket&quot; rel=&quot;nofollow noreferrer&quot;&gt;Apollonian gasket&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/ekAeX.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ekAeX.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Apollonian_gasket&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;The project would require using Processing's &lt;code&gt;line( )&lt;/code&gt; and &lt;code&gt;ellipse( )&lt;/code&gt; functions,&#xA;as well as some algebraic calculations.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-12-06T12:38:56.787" LastActivityDate="2015-12-06T12:38:56.787" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10141" PostTypeId="1" AcceptedAnswerId="10142" CreationDate="2015-12-06T21:15:42.483" Score="41" ViewCount="4493" Body="&lt;p&gt;It seems like this is exclusively how (most) people teach graduate/upper div math classes. They go through the proof of some big theorem, sometimes it might take two lectures. It's obviously important. But I honestly have no idea what I am supposed to be getting from this. It honestly seems useless to me. I know it isn't. I know it's a rite of passage to prove xyz theorem. But is this really worth sometimes two class periods? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I honestly don't know what I'm supposed to be learning. These proofs require unique logic a lot of the time, and this logic is difficult to transfer to other problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In lower div classes, it would be informal explanations of things followed by a lot of examples. I'm not saying that that's the way to teach functional analysis for example. But at least it made sense to me. I know what I was supposed to be learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can go through all the details myself, I go to lecture for motivation/intuition/things you can't normally get from a textbook. The proof of Riesz representation theorem for example can be found in any text. Why can't I just read it on my own? It's much easier to read proofs like this on your own rather than in class.&lt;/p&gt;&#xA;" OwnerDisplayName="user5108" LastActivityDate="2015-12-08T17:29:51.937" Title="What am I supposed to be learning with long proofs of the main theorems in class?" Tags="|proofs|graduate-education|" AnswerCount="6" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10142" PostTypeId="2" ParentId="10141" CreationDate="2015-12-06T21:57:04.013" Score="24" Body="&lt;p&gt;I agree with the sentiment in this question. I too often feel that lecturers go through a detailed proof because they think that everything must be proven pedantically to be able to use it. Sometimes unnecessarily complicated proofs are skipped, but not often enough to my taste.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there is a point to proving these &quot;big theorems&quot;. The proofs contain new ideas and those ideas will often become useful later, however unlikely that feels when you first encounter the ideas. The new clever ideas should be explained in lecture time, but the entirety of a technical proof may be too much. Reading a book or lecture notes with supporting exercise problems seems like a better way to go than a full lecture proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope lecturers would feel less of an urge to prove everything. What to skip and how to give students access to the proofs if they want them is another issue. I want to skip this question because the OP has the point of view of a student. It can be hard to persuade the teacher to change their ways; I will let it for you to judge whether or not that is reasonable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a student, if you feel that a proof given in the lectures is too much, try to look at it from a distance. Try to see the big picture instead of trying to understand all technical details. If the proof can also be found in a textbook or other source, there is little need to take notes. Ask the lecturer for intuition and explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you feel that the proof is way too hard for you and you don't get the big picture either, you can ignore it entirely. It's perfectly fine to occasionally take a theorem as a black box and just learn to use it (as opposed to learning how to prove it and understanding why it is true). If the situation keeps bugging you, you can go back to the result later and fill the gap. A year or two more of studying mathematics will make the proof more accessible to you. I see no reason to hurry to learn everything; it can deter you from making the progress you can make.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2015-12-06T21:57:04.013" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10143" PostTypeId="2" ParentId="10138" CreationDate="2015-12-06T22:15:31.920" Score="3" Body="&lt;p&gt;You mentioned: I have come across the phrases 'relational understanding' and 'instrumental understanding'...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdf&quot; rel=&quot;nofollow&quot;&gt;The original article, by Richard Skemp&lt;/a&gt;, is well worth reading. I think you'll find your own favorite way of explaining this to non-mathematicians once you've read Skemp's piece, and perhaps a few of the many blog posts about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://mathmamawrites.blogspot.com/2012/06/teaching-for-understanding.html&quot; rel=&quot;nofollow&quot;&gt;Math Mama Writes&lt;/a&gt; (my blog) &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.blog.republicofmath.com/richard-skemps-relational-understanding-and-instrumental-understanding/&quot; rel=&quot;nofollow&quot;&gt;Republic of Math&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://mathhombre.blogspot.com/2011/02/instrumental-vs-relational.html&quot; rel=&quot;nofollow&quot;&gt;Math    Hombre&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://christopherdanielson.wordpress.com/2011/05/11/dear-christina-open-letter-to-a-student/&quot; rel=&quot;nofollow&quot;&gt;Overthinking My Teaching&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-12-06T22:15:31.920" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10144" PostTypeId="2" ParentId="10138" CreationDate="2015-12-06T22:16:03.020" Score="5" Body="&lt;p&gt;When desired learning outcomes are listed in a curriculum, who are they listed for? It depends on the situation, but often the target audience includes not only those who teach the course, but also administrators and prospective students. While your colleagues who are ordered to teach a course might understand what 'understanding' means, the other groups of people might not, and they are not to be ignored. I think understanding is a good goal but you need to explain it to others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to make 'understanding' more concrete by asking yourself and your colleagues how it manifests itself (how it can be measured). The word as I understand it means several things and cannot &amp;mdash; and should not &amp;mdash; be captured in a single term. And even mathematicians may disagree on what the term actually means. Maybe something along the following lines would communicate your goal better than &quot;The students should learn mathematical understanding&quot;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The students should be able to explain ideas behind proofs instead of being only able to repeat the proofs when prompted.&lt;/li&gt;&#xA;&lt;li&gt;The students should be able to use ideas from proofs they have seen to prove results they have never encountered before.&lt;/li&gt;&#xA;&lt;li&gt;The students should have a collection of examples readily available and be able to use them to quickly check if a given mathematical statement is plausible.&lt;/li&gt;&#xA;&lt;li&gt;The students should recognize false and meaningless mathematical statements. They should be able to judge whether a proof is valid or not.&lt;/li&gt;&#xA;&lt;li&gt;The students should communicate their reasoning clearly and following the conventions of the field.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I believe this list would be more understandable to non-mathematicians than just the word 'understanding'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was just my list. You can make yours. (I can append my list if there are good suggestions.)&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2015-12-06T22:47:04.833" LastActivityDate="2015-12-06T22:47:04.833" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10145" PostTypeId="2" ParentId="10141" CreationDate="2015-12-06T22:52:49.730" Score="7" Body="&lt;p&gt;I view &#xA;&lt;em&gt;avid19&lt;/em&gt;'s frustration as an argument for presenting proofs within some historical&#xA;context.&#xA;Few major theorems have been achieved without a struggle, often involving&#xA;several mathematicians over an extended period of time.&#xA;Of course there is rarely the freedom to sketch&#xA;out these struggles within the time-constraints&#xA;of a specific course. But even a &lt;em&gt;nod&lt;/em&gt; toward&#xA;that historical context and struggle would help motivate &quot;long proofs&#xA;of ... main theorems.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are two books, separated by 50 years,&#xA;that illustrate the struggle, the first for&#xA;&lt;a href=&quot;https://plus.maths.org/content/eulers-polyhedron-formula&quot; rel=&quot;noreferrer&quot;&gt;Euler's $V-E+F=2$&lt;/a&gt;, the second Villani's Fields Medal&#xA;nonlinear Landau-damping theorem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(1) Imre Lakatos. &lt;em&gt;Proofs and Refutations: The Logic of Mathematical Discovery&lt;/em&gt;. Cambridge University Press, 1976.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;a href=&quot;https://i.stack.imgur.com/y5dqH.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/y5dqH.png&quot; alt=&quot;VEF2&quot;&gt;&lt;/a&gt;&#xA;  &lt;br /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;sup&gt;&#xA;  (Image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler_characteristic&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.)&#xA;  &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(2) Cédric Villani, &lt;em&gt;Birth of a Theorem: A Mathematical Adventure&lt;/em&gt;.&#xA;  Farrar, Straus and Giroux, 2015.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/S8Hux.jpg&quot; width=&quot;200&quot; /&gt;&#xA;  &lt;br /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;sup&gt;&#xA;  (Image from &lt;a href=&quot;http://www.learningpersonalized.com/category/educators/page/2/&quot; rel=&quot;noreferrer&quot;&gt;www.learningpersonalized.com&lt;/a&gt;.)&#xA;  &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-12-07T18:47:08.173" LastActivityDate="2015-12-07T18:47:08.173" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10146" PostTypeId="2" ParentId="10141" CreationDate="2015-12-07T08:24:48.697" Score="28" Body="&lt;p&gt;Hmm apparently I will be the dissenter here. I think that long proofs taught in lectures are very much a good thing. This is particularly true for hard proofs. I will try and split the reasons why I think so into a couple of points.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Hard proofs are no only hard to create but also hard to learn on your own.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Have you tried learning a complex proof on your own from the book? This is usually quite hard. It gets much harder when the proof is just in an article.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When learning things on your own you often miss many details.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Quite often when you are reading a proof in a book you will miss places where the author performs a non-trivial step which might seem intuitive because they don't point it out. You can also miss steps which are actually in the book but may not be emphasized.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Very few theorems you learn in upper div/lower grad level classes have proofs which are not eminently reproducible.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Pretty much every proof you will do in a class at this level will have methods and ideas which are the stock in trade of the field. Whether these are bounding methods for analysis, diagram chasing for algebra or c.c.c. proofs in set theory these methods come up time and again in the field. You can never see too many of them.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When the professor explains a proof you can ask questions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This in my mind is the biggest boon of these proofs. You can ask about the proof as it's being built. Any nagging doubt, any confusion or missed step can be clarified on the spot. Often you can ask questions which highlight misunderstanding or extend your knowledge of why and how a given method is used. This is never the same if you read the proof on your own and then ask questions later.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Good professors will highlight the essential points of the proof presented.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is another important thing. When reading a proof it is much harder to tell which part is just easy formula pushing and where the main idea of the proof is. A good professor can also explain where you might be able to relax or strengthen conditions and what results that can give.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All in all I think these reasons explain why teaching proofs in class is a good idea.&lt;/p&gt;&#xA;" OwnerUserId="5896" LastActivityDate="2015-12-07T08:24:48.697" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10147" PostTypeId="2" ParentId="10141" CreationDate="2015-12-07T13:47:58.297" Score="8" Body="&lt;p&gt;This is a great question. Here are some thoughts on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A theorem statement is a sign of an idea that tends to be useful in the pattern of mathematical inquiry in a given subdomain. A good theorem adds clarity to a subject in the sense that it can often be used to answer questions that arise. This is really the main reason point in demarcating theorems…they point to ideas that explain things &lt;em&gt;often&lt;/em&gt; where other ideas fail to explain or resolve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One might take the extreme view (as Halmos once wrote…perhaps facetiously) that one should never read the proof of a theorem. One should try to prove it for oneself and then, after checking that it is true, just use it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another perspective is that a useful theorem is a signpost for a cluster of useful ideas. The proof of a major theorem by its very nature is a small world of ideas that were essential to the resolution of a problem. One might regard learning about the theorem as living in this world for a while. One can internalize the ideas and then look around (in the exercises perhaps) for further implications of these ideas. Since graduate courses should ideally be about teaching students how to think like a research mathematician, this process of revisiting and internalizing the world of a proof with the intent on exploring its implications and structure can be a valuable experience. From this perspective, the proof of a major theorem is a &quot;mini course&quot; in itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the energy and time is not being taken by the professor to include the context for the result, though, you may be victim of someone trying to easily fill their teaching time so they can get back to their research. There is no easier way to fill a class period than stepping through a proof that could be routinely read in a book...&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2015-12-07T13:47:58.297" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10148" PostTypeId="2" ParentId="8052" CreationDate="2015-12-07T23:25:34.120" Score="4" Body="&lt;p&gt;I am not a formal teacher or a mathematician, but a mechanical engineer who loves to learn and derives great satisfaction from mentoring other up-and-coming “S.T.E.M.” professionals/students. Due to my lack of educator credentials or particular knowledge on the subject, my response reflects personal views and experiences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was terribly bored by math during my primary and secondary education. I was much more prone to want to tinker with things to see how they worked. I used the knowledge that I gained to build and create things that I enjoyed or found useful. When the time came to select a major while filling out my university application I chose mechanical engineering because it would extend my understanding of the mechanical objects that had captured my fancy. I made this choice in spite of my mathematical insecurities because of my passion for the mechanical and my desire to understand it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My first math and math based courses were extremely difficult for me, but I persevered and received good grades. However, my grades were more a result of my ability to regurgitate information effectively than a result of a sound understanding. Here and there a light would come on and things would make sense, but by far and large I was being led on a blindfolded path, copying the recipes that I was instructed to follow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Through it all, I did feel very empowered as I learned that I could make, design, or mathematically model nearly anything that I wanted to if I could just find an equation in some textbook for the problem. This spurred me to develop a can-do attitude toward anything that I wanted to do. This new mental paradigm and tenacity was soon to be called to good use in awakening my understanding of what math really could be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While studying heat transfer I was introduced to PDE’s. I had actually previously had a formal introduction in a class on differential equations (focusing of course on ODE’s), but the regurgitation education model had left me with virtually no recollection of what a PDE even was so it was like discovering it for the first time. I was intrigued, so I enrolled in a PDE’s math class during the last semester of my undergrad. My professor had a doctorate of art in mathematics, which contrasted greatly with my purely applied engineering background.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At first I scoffed at the professor’s insistence that math was a creative form, but as the class went on I was forced to dig up and actually learn a great deal of the topics that had been presented in earlier math classes. As I did, I began to really understand a few concepts and realize that they actually made sense in and of themselves. Math began to take on a meaning of its own beyond just its applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the biggest revelations was the ideas of vector spaces and that of variable transformations to map from one vector space to another. After asking my professor for help with just such a problem, I asked how he had known that that particular variable transformation would work. His answer changed the way that I have looked at math ever since. He said “I defined it that way because it was convenient”. The idea was so foreign to me that I had to think about it for some time before it really made sense. I had always thought that there was only one correct variable transformation, one correct proof of every mathematical fact, one best way to solve every problem. The idea that I could pick something, define it, and work out the implications on my own was amazing to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My interest in PDE’s led me to begin a study of them on my own. This study has led to a multitude of other topics that are interrelated with PDE’s including higher dimensional vector spaces and other wondrous ideas that lead the imagination to ask a lot of “what if” type questions. One of my most recent reads was a book on the history of mathematics. It was such an eye opener. Math and science progressed hand in hand in most eras (to me, those seemed to be the most fruitful). Many mathematicians were also classified as scientists or experts in other disciplines. Their processes for discovery varied, some liked rigorous proofs, others relied on inspiration/intuition and worked out the details later, some even published erroneous solutions to problems, notations changed and evolved, and creativity flourished. No longer was math a cold, exact, and deterministic subject. It had come to life for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I apologize for taking so long to get to my point, but I felt the background was necessary to justify my position since I do not know any educational theories. My point is that I think the best way to balance the applications and the art (the art part is a newly discovered part to me, but I am so glad that I have come to be able to view math in such a way) is to follow the historical development itself. Don’t ask students to solve totally stupid and uninteresting problems about Sally and Rob’s ages, travel distances, etc. Why not use the real questions of the giants that paved the way for us? There are a great deal of artistic and applied problems that spurred the development (and may I add understanding) of mankind. Why rob students of the richness of that history? I think their minds will tend to evolve in understanding over time much the way the ancients did. They can then see our wealth of knowledge as accessible to them. They can ponder on things and question about how the ancients determined the mass of the earth, the percent of gold in a crown, the value of pi or even the fact the ratio of the circumference of a circle to its diameter is constant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, I think the best way to discover math is to relive the discovery of it with a little help from a skilled mentor so as to avoid the intellectual pitfalls of the ancients and of course so it will not take thousands of years to get an education. My opinion is that this would turn math from a dry subject into an interesting narration. It would also give a balance of art and application since neither deserves full credit for the present state of our knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What if a student will never use it again? My answer to that is that from an art and creativity perspective it will open their mind to consider all the possibilities to any given scenario they may meet and from an applied perspective that it will enhance their appreciation for the world around them just like a study of art an poetry turns a mundane artwork into something with meaning.&lt;/p&gt;&#xA;" OwnerUserId="6016" LastActivityDate="2015-12-07T23:25:34.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10149" PostTypeId="2" ParentId="10065" CreationDate="2015-12-08T00:34:14.113" Score="0" Body="&lt;p&gt;I'm going to interpret your question a little differently than the answers already posted and understand you to be asking about abstraction as a problem-solving strategy (i.e., &quot;simplify the problem by applying a homomorphism&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proofs about &lt;strong&gt;grid puzzles and grid games&lt;/strong&gt; are useful for motivating this kind of abstraction because there are lots of possible homomorphisms from marked grids and/or positions on a grid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some examples that students could probably solve on their own without introduction:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;On an infinite &lt;a href=&quot;https://en.wikipedia.org/wiki/Chess&quot; rel=&quot;nofollow&quot;&gt;chessboard&lt;/a&gt;, white has only a knight, and black has only a pawn.  From what positions can white eventually capture the pawn?&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;On an infinite &lt;a href=&quot;https://en.wikipedia.org/wiki/Checkers&quot; rel=&quot;nofollow&quot;&gt;checkers&lt;/a&gt; board, each side has one king and is trying to end the game in as few moves as possible.  From what positions will black win?&quot;  (This is more or less the previous one rotated 45°.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Show that there is a non-losing &lt;a href=&quot;https://en.wikipedia.org/wiki/Tic-tac-toe&quot; rel=&quot;nofollow&quot;&gt;tic-tac-toe&lt;/a&gt; strategy for X that has them play in the center for their first move and across from O's move for their second.&quot;  The case analysis on this one is a lot easier if we only consider the board up to rotation and reflection.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Eight_queens_puzzle#Related_problems&quot; rel=&quot;nofollow&quot;&gt;The $n$-rooks puzzle&lt;/a&gt;: &quot;How many ways can $n$ rooks be arranged on an $n\times n$ chessboard such that no rook threatens another?&quot;  The first step in almost any treatment is an abstraction of candidate positions to permutations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And some examples that such students could probably handle if given the appropriate nudges:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mutilated_chessboard_problem&quot; rel=&quot;nofollow&quot;&gt;The mutilated chessboard problem&lt;/a&gt;: &quot;Is there a domino covering for a chessboard with opposite corners removed?&quot;  In the standard solution, we abstract the board to two numbers: the number of black squares and the number of white squares.  It's usually helpful to not stop there, but compare the solution to a related, but not entirely analogous problem, like &quot;Eight $3\times 1$ polyminos are packed into a $5\times 5$ box; why must the empty space be in the center?&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;For what values of $n$ is a $n\times 1$ &lt;a href=&quot;https://en.wikipedia.org/wiki/Lights_Out_%28game%29&quot; rel=&quot;nofollow&quot;&gt;lights-out&lt;/a&gt; board solvable from every position?&quot;  Here, we abstract a position to three booleans: whether there are an odd number of lit spaces at locations $\{0, 3, 6, \ldots\}$, at locations $\{1, 4, 7, \ldots\}$, and at locations $\{2, 5, 8, \ldots\}$.  This and the next one make for good discussion on your point #2: &quot;Where did the number three come from?&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;For what values of $n$ is a $3n\times 2$ block of pegs a solvable position on an infinite &lt;a href=&quot;https://en.wikipedia.org/wiki/Peg_solitaire&quot; rel=&quot;nofollow&quot;&gt;peg solitaire&lt;/a&gt; board?&quot;  This demonstrates well with two abstraction steps: a first like the one above, but talking about pegs instead of lights, a second that takes a triple of booleans to a boolean indicating whether they are all the same or not.  (I chose $3n\times 2$ instead of $n\times 2$ since I assume you're not interested in the induction side of the argument.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Besides exploiting the grid structure, sometimes you can also get additional homomorphisms from the rules of the puzzle or game:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;How many arrangements of nine $1$s can be extended to a &lt;a href=&quot;https://en.wikipedia.org/wiki/Sudoku&quot; rel=&quot;nofollow&quot;&gt;Sudoku&lt;/a&gt; solution?&quot;  A nice strategy here is to abstract away the position of the $1$ in the upper-left box (nine possibilities), then whether the $1$s ascend or descend in the top row of boxes (two possibilities), in the middle row (two possibilities), etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;An [insert your favorite &lt;a href=&quot;https://en.wikipedia.org/wiki/Fairy_chess_piece&quot; rel=&quot;nofollow&quot;&gt;fairy chess piece&lt;/a&gt; here] is placed by itself on a chessboard of arbitrary shape and dimensions.  Every time it moves, the color of the square it left changes from black to white or vice versa, as does the color of the square it lands on.  After an unknown number of moves, is it possible to determine from the board and the piece's position where it was originally placed?&quot;  This is virtually the bridges of Königsberg, just in disguise.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4772" LastActivityDate="2015-12-08T00:34:14.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10150" PostTypeId="1" CreationDate="2015-12-08T04:25:05.787" Score="16" ViewCount="589" Body="&lt;p&gt;After teaching induction and then strong induction (i.e. the version where you assume $\forall k&amp;lt;n, P(k)$ and prove $P(n)$), one of my students asked why we ever use ordinary induction, since strong induction is stronger.  I said that some people just find it simpler to think about, and often it's good enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it made me wonder: would it be ridiculous to teach strong induction right off the bat, and only then specialize it to ordinary induction? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible advantage of this that I can see is that it would separate the &quot;inductiveness&quot; from the &quot;division into cases&quot; aspect of ordinary induction.  There's something &quot;cleaner&quot; about strong induction: we only have to give one method, and it works for all natural numbers.  Then we can just say well, it happens often that when doing a (strong) induction, we have to divide into cases based on whether $n=0$ or $n&amp;gt;0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible disadvantage I can see is that it might be hard to find very many examples of strong inductions that &lt;em&gt;don't&lt;/em&gt; require such a case split.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone ever taught induction this way?  If so, was it a success or a failure?&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2023-05-18T00:31:23.430" Title="Teaching strong induction instead of induction" Tags="|proofs|induction|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10152" PostTypeId="2" ParentId="10141" CreationDate="2015-12-08T09:12:16.497" Score="5" Body="&lt;p&gt;In honesty, two class periods does not seem much to me. But on the other hand, I have a four year mathematics degree that I've rarely used in the last 25 or so years. That's my perspective. You'd think that I regret spending a month on wavelets, a year on iterative function systems, etc, when I'm not actively using it, but I don't. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Studying mathematics is like studying Zen. Don't resist it. Open your mind.&#xA;You will grow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if you can not cope with the current situation, free your self from it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once dropped out of a class in mathematical statistics because I thought it was a waste of time. Took a job, worked for half a year. Then sat down with my textbooks and finished the course on my own, because I needed it for my exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of all the courses I took, guess which has been most useful to me?&lt;/p&gt;&#xA;" OwnerUserId="6020" LastActivityDate="2015-12-08T09:12:16.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10153" PostTypeId="1" CreationDate="2015-12-08T15:11:47.893" Score="5" ViewCount="136" Body="&lt;p&gt;I apologize if this is a silly question. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I look for free online web page for a &lt;strong&gt;&lt;em&gt;differential calculus&lt;/em&gt;&lt;/strong&gt; and theirs &lt;strong&gt;&lt;em&gt;rules&lt;/em&gt;&lt;/strong&gt; for higher school. By google shearch, I don't have an intersting results. For instance:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.sporcle.com/games/14111184/calculus&quot; rel=&quot;nofollow&quot;&gt;http://www.sporcle.com/games/14111184/calculus&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The french pages gives more famous things.&#xA;Thank's for any help!&lt;/p&gt;&#xA;" OwnerUserId="4995" LastEditorUserId="39" LastEditDate="2016-03-04T19:32:37.160" LastActivityDate="2016-04-03T19:54:47.057" Title="interactive web page for differential calculus" Tags="|calculus|technology-in-education|interactive-teaching|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10154" PostTypeId="2" ParentId="10141" CreationDate="2015-12-08T17:12:46.150" Score="-1" Body="&lt;p&gt;No idea how long these class periods are in your country (the US, supposedly), but I guess of the order of magnitude of one hour. Then multiple class periods are an excessive amount of time. During my time at the university as an engineering student, I rarely experienced proofs taking longer than one hour, as far as I remember.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, the usefulness would be the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;getting the mathematical way of thinking, and being able to apply this knowledge for other proofs or problems&lt;/li&gt;&#xA;&lt;li&gt;understanding that things can be long and tedious. This is important. One of the mistakes I made initially, was assuming a proof or other type of answering strategy to a question was incorrect because it did not take me to the result fast enough. And while fast and elegant solutions are the preferred ones, sometimes things are just hard end tedious (maybe until some clever person finds a faster way). For me it was useful to understand this.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6030" LastEditorUserId="2074" LastEditDate="2015-12-08T17:29:51.937" LastActivityDate="2015-12-08T17:29:51.937" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10155" PostTypeId="2" ParentId="10124" CreationDate="2015-12-08T21:09:35.523" Score="2" Body="&lt;p&gt;Mathematica has many special functions built into its language. There's a whole list here: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://reference.wolfram.com/language/guide/SpecialFunctions.html&quot; rel=&quot;nofollow&quot;&gt;https://reference.wolfram.com/language/guide/SpecialFunctions.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of these are defined from definite integrals. You can find the definitions in the details section of a specific function.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2015-12-08T21:09:35.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10156" PostTypeId="2" ParentId="10131" CreationDate="2015-12-08T23:47:21.000" Score="1" Body="&lt;p&gt;I'm not 100% sure I understand the question, but here's my hunch.  The base skill to be learned in moving lines within a diagram is two-fold:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Learning to the see the graph of a function in your head, based just on the equation.  This is done be recognizing variables within an equation as points on a graph.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For example, in the slope-intercept form of a line y = mx + b, the m variable gives you a sense of how steep the line is, which direction it leans, and the y-intercept tells you where the line crosses the y-axis.  You can get a good visualization of a line just from the equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a parabola y = a(x-h)&lt;sup&gt;2&lt;/sup&gt; + k, the sign of the a variable tells you if the parabola opens up or down, and the vertex is at point (h, k).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a circle, (x-h)&lt;sup&gt;2&lt;/sup&gt; + (y-k)&lt;sup&gt;2&lt;/sup&gt; = r&lt;sup&gt;2&lt;/sup&gt;, you get a sense of the size of circle from the r variable, and the center is at point (h, k).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can look at an equation, and say (for example) &quot;that's an ellipse,&quot; that's a good thing.  If you can look at the equation and say &quot;that's an ellipse with the center in the 4th quadrant and the major axis at about a 45 degree angle from the x-axis,&quot; then you will do well in class!  You will have a feel for the properties of an equation, know how it behaves if one of the variables is perturbed.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;A 2nd reason to become efficient at moving lines is the idea of &quot;translating coordinate systems.&quot;  In physics and in programming, you may get an equation based on one coordinate system, but it's helpful to visualize a graph of the same shape, but in a coordinate system with a different origin.  This is done more often than you realize.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In both of the above cases, if you can achieve fluency through studying theory alone, you're a better man than me.  For me (and maybe most people) it's takes drilling.  That's why your textbooks have you go through these exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Congratulations on teaching yourself precalculus!  I am awed and impressed!&lt;/p&gt;&#xA;" OwnerUserId="5465" LastActivityDate="2015-12-08T23:47:21.000" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10157" PostTypeId="1" AcceptedAnswerId="10160" CreationDate="2015-12-09T02:28:53.983" Score="11" ViewCount="2341" Body="&lt;p&gt;I had a bit of a disagreement with some of my colleagues over the correct way to represent the solution to a system of two linear equations, e.g.:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align}&#xA;x + y = 1\\&#xA;3x + 2y = -1&#xA;\end{align}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We should find that the solution is $x = -3$ and $y = 4$.  I had been telling my students that they should give the solution as:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;x = -3&#xA;y = 4&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;(or really anything that put $x = -3$ and $y = 4$ close together to show the solution compactly--as opposed to just finding the $x$ and $y$ in random parts of their work and never summarizing their findings)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my students informed me that another teacher told them that they should give the solution as an ordered pair, i.e. $(-3, 4)$.  I explained that yes, assuming that we understand that an ordered pair is $(x, y)$, that that could be a way of representing the solution but that I didn't think it was correct and that giving $x = $ and $y = $ is a more correct way of giving the solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I further explained that the ordered pair $(x, y)$ represents a point on a coordinate plane.  Thus the ordered pair would be a correct answer if the question had stated something to the affect of &quot;If $(x, y)$ represents points on a coordinate plane, then find the point that represents the solution to the given system of equations.&quot;  I further elaborated that an ordered pair represents a point, so we can get the solution from the ordered pair (e.g. if we were to solve by graphing), but that the ordered pair does not represent the solution--similar to the idea that the x-intercept is a point and we can get the zeros of a function from its x-intercepts but that an x-intercept does not directly represent a zero of the function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is whether or not it's proper to represent the solution as an ordered pair when asked to solve a system of two equations or are both ways correct since it's pretty well accepted that ordered pairs are of the form $(x, y)$ (or is it even &lt;em&gt;more&lt;/em&gt; correct to give the ordered pair rather than $x = $ and $y = $).&lt;/p&gt;&#xA;" OwnerUserId="2204" LastActivityDate="2015-12-23T05:01:25.970" Title="What is the proper way to represent the solution to a system of two equations" Tags="|algebra|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10159" PostTypeId="2" ParentId="10157" CreationDate="2015-12-09T03:55:30.263" Score="2" Body="&lt;p&gt;I've seen it both ways. Students calculate the value of x and write &quot;x=3&quot; and then calculate y. Some teachers will accept this, others will subtract a point out of the 5 the question might be worth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The better teachers will be explicit in how they require an answer. Similar to the instruction &quot;rationalize the denominator&quot; &quot;leave in radical form&quot; etc. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-12-09T03:55:30.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10160" PostTypeId="2" ParentId="10157" CreationDate="2015-12-09T05:24:59.980" Score="16" Body="&lt;p&gt;A good question to ask. The definitions in play, and the exact wording of the question, are very important. In most algebra contexts that I've seen, the ordered-pair is actually the &lt;em&gt;definition&lt;/em&gt; of a solution to a system (see below). My opinion is that this makes sense, because it is the most concise  expression of the solution in $x$ and $y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider carefully different possible English phrasings for a question in this area (which I'm pretty sure need to be included for any well-defined math problem). For example: If the question is &quot;What is the solution set of $x+3=7$?&quot;, then a response of &quot;$\{4\}$&quot; precisely answers that question, and is commonly used (similar to answering, &quot;What color is the sky?&quot; with the one-word response, &quot;Blue&quot;). Note that this differs from a direction of: &quot;Solve $x+3=7$&quot;, to which the correct response is indeed the assertive statement &quot;$x = 4$&quot;. Likewise, if the question asks for the solution to an independent system of equations, then the ordered pair is perfectly acceptable, and preferable by virtue of being more concise. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly a naked $4$ should not be any line of the &lt;em&gt;mathematical&lt;/em&gt; work (even if it is a correct response to the first English-language question above). The algebraic writing should always be fully-formed statements involving some relation symbol (most commonly equals), or proper English sentences (or whatever is the local language).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Be aware that the use of ordered pairs for system solutions is in fact the standard &lt;strong&gt;definition&lt;/strong&gt; in most algebra books and standardized tests that I see, for example: Martin-Gay &lt;em&gt;Introductory Algebra&lt;/em&gt;, Bittinger &lt;em&gt;Intermediate Algebra&lt;/em&gt;, Sullivan &lt;em&gt;College Algebra&lt;/em&gt;, Ratti &amp;amp; McWaters &lt;em&gt;Precalculus&lt;/em&gt;, etc. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;A &lt;strong&gt;solution&lt;/strong&gt; of a system of two equations in two variables is an&#xA;  ordered pair of numbers that is a solution of both equations in the&#xA;  system.&quot; -- Martin-Gay, Sec. 7.1.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;A &lt;strong&gt;solution&lt;/strong&gt; of a system of two equations in two variables is an&#xA;  ordered pair that makes &lt;em&gt;both&lt;/em&gt; equations true.&quot; -- Bittinger, Sec.&#xA;  3.1.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;A &lt;strong&gt;solution of a system of equations in two variables&lt;/strong&gt; $x$ and $y$&#xA;  is an ordered pair of numbers $(a, b)$ such that when $x$ is replaced&#xA;  with $a$ and $y$ is replaced with $b$, the resulting equations are&#xA;  true.&quot; -- Ratti &amp;amp; McWaters, Sec. 8.1&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In each of these texts, all exercises in systems of equations have their solutions given as ordered pairs (or triplets). Assuming that you're working in conjunction with some textbook, then I would encourage you to read carefully and follow the practice of the book being used in the course.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="1853" LastEditDate="2015-12-09T07:24:20.457" LastActivityDate="2015-12-09T07:24:20.457" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10161" PostTypeId="1" AcceptedAnswerId="10166" CreationDate="2015-12-09T08:57:05.853" Score="14" ViewCount="481" Body="&lt;p&gt;Background: I am a fourth year math graduate student in the US.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps you will think this question silly, but I intend for it to be a serious question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In teaching courses such as calculus and analysis, I frequently find myself wanting to use colored chalk.  Unfortunately, all the colored chalk I can get seems to be of exceptionally poor quality.  It tends to be very light and brittle, and quite difficult to erase.  Too be honest, it's nearly unusable.  (I suspect it's been designed for children, rather than teachers.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of the topologists at my university have a much better type of colored chalk, but none of them seem to know where to buy more of it.  I'm aware that &lt;a href=&quot;http://gizmodo.com/why-mathematicians-are-hoarding-this-special-type-of-ja-1711008881&quot;&gt;Hagoromo&lt;/a&gt; chalk exists, but I'm looking for something that's more easily obtainable.  I don't need it to be the best chalk in the world, just something useable would be great.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, can anyone recommend a brand that is both &lt;em&gt;easily obtainable&lt;/em&gt; and &lt;em&gt;reasonably useable&lt;/em&gt;?  Links to online retailers are wonderful, but any guidance whatsoever is much appreciated.&lt;/p&gt;&#xA;" OwnerUserId="5988" LastActivityDate="2015-12-09T14:46:02.030" Title="Colored chalk recommendations" Tags="|topology|board-use|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10162" PostTypeId="1" CreationDate="2015-12-09T11:48:07.187" Score="2" ViewCount="292" Body="&lt;p&gt;I want to know what is the best way to teach simple trigonometric equations, such as $$\sin x=0$$.&#xA;Should I use the trigonometric circle or the sine graph? Which is better?&lt;/p&gt;&#xA;" OwnerUserId="5972" LastEditorUserId="262" LastEditDate="2015-12-09T17:57:25.530" LastActivityDate="2016-07-15T22:06:35.597" Title="Explaining trigonometric equations" Tags="|concept-motivation|trigonometry|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10163" PostTypeId="1" CreationDate="2015-12-09T12:05:43.037" Score="36" ViewCount="5925" Body="&lt;p&gt;My six year old started school a few months back and he's loving it. This first year is more about social skills than anything academic and I like that approach. But we're spending some time at home with letters and numbers. In the numbers department there's a sense of excitement on his part and I want to kindle it. Here are some things he finds amazing:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The numbers never end. Not even after a 1000! You can go on forever.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You can get to a number in many different ways. For example you can get to 6 by adding 4 and 2, but you could also do it by adding 3 and 3.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The number zero.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Numbers feel different. They're like personalities.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The other day I told him about odd and even numbers. We talked about the fact that there's an infinite number of even numbers but that if you add in the odd numbers you'll get twice as many numbers. That's crazy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I'm just looking for more stuff to keep him going. I'm not looking for puzzles to solve as much as things to marvel at.&lt;/p&gt;&#xA;" OwnerUserId="1925" LastEditorUserId="262" LastEditDate="2015-12-09T18:00:50.423" LastActivityDate="2023-01-24T17:25:26.280" Title="Imbuing a six year old with a sense of mathematical wonder" Tags="|mathematical-pedagogy|primary-education|examples|creativity|recreational-learning|" AnswerCount="24" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="10164" PostTypeId="2" ParentId="10163" CreationDate="2015-12-09T13:11:31.010" Score="6" Body="&lt;p&gt;The earlier question, &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/q/7974/511&quot;&gt;Teaching a very enthusiastic and bright 5 year old&lt;/a&gt;&quot; could help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Building and manipulating shapes enhances geometric imagination. Consider&#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/a/4313/511&quot;&gt;polydrons&lt;/a&gt;, or&#xA;snap-cubes:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/l3xtt.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/l3xtt.jpg&quot; alt=&quot;SnapCube&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;sup&gt;&lt;a href=&quot;https://www.learningresources.com/home.do&quot; rel=&quot;nofollow noreferrer&quot;&gt;Learning Resources Snap Cubes&lt;/a&gt;&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-12-09T13:11:31.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10165" PostTypeId="2" ParentId="10163" CreationDate="2015-12-09T14:35:06.277" Score="6" Body="&lt;p&gt;Below are some great and inspiring books by an excellent mathematician. (In the Really Big Numbers book, on the page where counting by tens is discussed there is an inspiring error (?)…Big Bird is right, everyone makes mistakes!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/reallybignumbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;Really Big Numbers&lt;/a&gt; and &lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=MBK-90&quot; rel=&quot;nofollow noreferrer&quot;&gt;You Can Count on Monsters&lt;/a&gt; by Richard Evan Schwartz.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me also add this wonderful activity JDH did with his child's class:&#xA;&lt;a href=&quot;http://jdh.hamkins.org/tag/graph-coloring/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jdh.hamkins.org/tag/graph-coloring/&lt;/a&gt;. Graph coloring is an excellent way to engage small children with the feel of mathematical thought. The idea of a coloring book incorporating this was just fantastic! For 4 year olds it is a bit early, but the activity is just wonderful.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2017-01-21T14:53:00.460" LastActivityDate="2017-01-21T14:53:00.460" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10166" PostTypeId="2" ParentId="10161" CreationDate="2015-12-09T14:38:19.230" Score="8" Body="&lt;p&gt;What you want is &lt;em&gt;Omega Colored Chalk&lt;/em&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/fPWqZ.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fPWqZ.jpg&quot; alt=&quot;Colored Chalk&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;sup&gt;&#xA;&quot;This non-toxic, low-dust chalk delivers extra-smooth and easily erasable writing in eight vibrant colors.&quot;&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;I won't include a link, but if you search, you will find many office-supply&#xA;outlets that carry it in the US, for about $4.&#xA;Good stocking stuffer! &lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-12-09T14:38:19.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10167" PostTypeId="2" ParentId="10161" CreationDate="2015-12-09T14:46:02.030" Score="4" Body="&lt;p&gt;For a long time, until I transferred to a school with white boards and dry erase markers, I actually used sidewalk chalk like these&#xA;&lt;a href=&quot;https://i.stack.imgur.com/MVcKr.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MVcKr.jpg&quot; alt=&quot;Colored Sidewalk Chalk&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is cheap, lasts a long time, incredibly durable, and has nice bright colors. Sidewalk chalk also draws really thick lines making your writing more visible from further away. Additionally, many sidewalk chalk packs come with plastic chalk holders that help to keep your hands from getting dusty and to give the chalk pieces a little extra support.  I mostly used Crayola brand and I found it no more difficult to erase from the board than any other brand/type of chalk.&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-12-09T14:46:02.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10168" PostTypeId="2" ParentId="10163" CreationDate="2015-12-09T16:28:48.423" Score="15" Body="&lt;p&gt;How about:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Numbers go the other way, too (negative)&lt;/li&gt;&#xA;&lt;li&gt;You can cut numbers in half, forever&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What if you cut a number into three pieces?&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;1 million is a thousand thousands (100 is ten tens)&lt;/li&gt;&#xA;&lt;li&gt;If you don't know the number, call it a letter (or name it :) )&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You can add letters together too&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;What if you cut triangles in half?&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Rectangles?&lt;/li&gt;&#xA;&lt;li&gt;Can you make a box from squares?&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Start at 1. Every day, double it. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6039" LastEditorUserId="6039" LastEditDate="2015-12-11T04:39:20.553" LastActivityDate="2015-12-11T04:39:20.553" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10169" PostTypeId="1" CreationDate="2015-12-09T16:44:49.680" Score="5" ViewCount="109" Body="&lt;p&gt;I can't find this ZDM article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Vinner, S., &amp;amp; Hershkowitz, R. (1983). On concept formation in geometry. &lt;em&gt;Zentralblatt für Didaktik der mathematik&lt;/em&gt;, 83(1), 20-25.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Could you help me?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was searching in some sites, but no success. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in this one, it isn't available: &lt;a href=&quot;http://www.emis.de/journals/ZDM/zdmcont.html&quot; rel=&quot;nofollow&quot;&gt;http://www.emis.de/journals/ZDM/zdmcont.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1760" LastEditorUserId="262" LastEditDate="2015-12-09T17:38:14.083" LastActivityDate="2015-12-10T09:55:22.063" Title="Looking for a classic paper &quot;Vinner and Hershkowitz&quot;" Tags="|reference-request|geometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10170" PostTypeId="2" ParentId="10163" CreationDate="2015-12-09T20:32:40.597" Score="2" Body="&lt;p&gt;While this is going to above the &quot;age: 6&quot; mark, I ran across this Youtube video just the other day that really fits the &lt;em&gt;topic&lt;/em&gt; of this question, although for a different age group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For those of you old enough to understand infinite sums, this is &lt;a href=&quot;http://youtube.com/watch?v=XFDM1ip5HdU&quot; rel=&quot;nofollow&quot;&gt;What it Feels Like to Invent Math&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6044" LastActivityDate="2015-12-09T20:32:40.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10171" PostTypeId="2" ParentId="10163" CreationDate="2015-12-09T23:36:59.107" Score="3" Body="&lt;p&gt;Try folding a piece of paper in half, then in half again and see how many times you can do it. start with A4 then find larger pieces-newspaper then wallpaper perhaps.&lt;/p&gt;&#xA;" OwnerUserId="6045" LastActivityDate="2015-12-09T23:36:59.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10172" PostTypeId="2" ParentId="10163" CreationDate="2015-12-10T04:35:54.407" Score="3" Body="&lt;p&gt;Base 2 counting on fingers - gets you up to 1023.&lt;/p&gt;&#xA;&lt;p&gt;Base 12 counting - need two more numbers flip &amp;amp; flap. 1, 2, 3, 4, 5, 6, 7, 8, 9, flip, flap, flap-one, flap-2, flap-3, flap-4, flap-5, flap-6, flap-7, flap-8, flap-9, flap-flip, and flap-flap !!!!&lt;/p&gt;&#xA;&lt;p&gt;Review of &amp;quot;Mathsemantics&amp;quot; by Edward MacNeal:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The language of mathematics seeks precision--only one interpretation. Everyday language such as English, colorful, emotive, and artistic, allows many interpretations.&lt;/p&gt;&#xA;&lt;p&gt;Edward MacNeal's Mathsemantics presents a whole new way of looking at math that liberates mathphobes from math anxiety, enables businesspeople to do their jobsmore effectively, entertains and informs math buffs, and provides educators with the tools to teach match without instilling fear.&lt;/p&gt;&#xA;&lt;p&gt;This divorce between numbers and what they mean begins in childhood and pervades every aspect of life from then on.  One result is innumeracy.  LIke John Allen Paulos's bestseller of that name, this book describes the symptoms of the problem, but unlike Innumeracy, Mathsemantics offers a solution: a revolutionary way of looking at math as a language, something we've all heard but which never made sense until now.&lt;/p&gt;&#xA;&lt;p&gt;Mathsemantics takes off from a quiz that was given to job applicants for the author's consulting firm who described themselves as &amp;quot;good at numbers.&amp;quot;  Most of them, it turned out, weren't in fact good at numbers, because they couldn't draw conclusions about what the numbers meant.  The good news is that many people who think they're terrible at numbers will find after reading this book that they aren't so bad after all.  They'll learn how to one-up the number crunchers.&lt;/p&gt;&#xA;&lt;p&gt;Mathsemantics is that rare book that will change the way readers look at the world.  It provides the most promising and entertaining answer yet to the problem of American innumeracy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Might be little advanced, but with some parental guidance it should do the trick. (Flip-flap example drawn from the book)&lt;/p&gt;&#xA;" OwnerUserId="6049" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-12-11T06:29:00.953" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10173" PostTypeId="2" ParentId="10163" CreationDate="2015-12-10T09:53:42.920" Score="5" Body="&lt;p&gt;Another book, to add to @Jon Bannon's list, that my 4 year old daughter and I can recommend is &#xA;&lt;a href=&quot;http://www.amazon.co.uk/Introductory-Calculus-For-Infants-Inouye/dp/0987823914&quot;&gt;Introductory Calculus For Infants&lt;/a&gt;. We also seem to discuss &lt;a href=&quot;https://www.youtube.com/watch?v=XTeJ64KD5cg&quot;&gt;Graham's number&lt;/a&gt; a bit after watching the Numberphile videos&lt;/p&gt;&#xA;" OwnerUserId="6050" LastActivityDate="2015-12-10T09:53:42.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10174" PostTypeId="2" ParentId="10169" CreationDate="2015-12-10T09:55:22.063" Score="4" Body="&lt;p&gt;Finally, the author posted it in www.researchgate.net &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.researchgate.net/publication/284382026_On_concept_formation_in_geometry&quot; rel=&quot;nofollow&quot;&gt;https://www.researchgate.net/publication/284382026_On_concept_formation_in_geometry&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you to all.&lt;/p&gt;&#xA;" OwnerUserId="1760" LastActivityDate="2015-12-10T09:55:22.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10175" PostTypeId="2" ParentId="10163" CreationDate="2015-12-10T10:16:19.300" Score="2" Body="&lt;p&gt;I have been teaching my daughter mathematics now for nearly a decade. She is 13.  I have been telling her that mathematics is a language- with symbols for imagining big numbers and patterns.   Very much like spoken languages in which we can spell her name.  The one that I remember vividly was that I showed her that fractions are very special types of numbers, different from the other numbers she had used (integers).  I showed her that 1/2, 2/4, 3/6,...are all &#xA;symbols to represent the same concept - half.  It was based on a story of she and her friend sharing a soup on a day when she had money only to buy a soup.  I asked her how she would share it equally and she had a method - get another bowl and pour out some into it so that the level are the same.  I told her she could write the experience as 1/2, read out as One-by-Two.  Then, the next day she had money to buy two soups, but 4 friends (obviously, the word her spread that she was buying soups).  How would she split the two equally among the four- sure, she had a clear answer.  But, did she have more on the second day or on the first day?  Same on both days.  How did she write it? Two-by-Four. So, one-by-two is same as two-by-four.  But English is a funny language- we have agreed on this often.  Mathematics is precise so we write 1/2 = 2/4.  Then, we discussed the value of this crazy possibility.  I believe I made rational number arithmetic very intuitive. Today we are reading &quot;Challenge and Thrill of Pre-College Mathematics&quot; by Pranesachar et al, &quot;Mathematical Circles&quot; by Fomin, Iternberg, Genkin. &#xA;Most of my teaching has been when I was jogging and she was cycling with me, and these days she reads mathematics herself. I only need to help her read carefully once in a while - especially when it comes to a new concept.  &lt;/p&gt;&#xA;" OwnerUserId="6051" LastActivityDate="2015-12-10T10:16:19.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10176" PostTypeId="2" ParentId="10162" CreationDate="2015-12-10T10:35:12.453" Score="0" Body="&lt;p&gt;Start with &lt;strong&gt;&lt;em&gt;triangular&lt;/strong&gt; approach&lt;/em&gt; first , and then go to &lt;strong&gt;&lt;em&gt;unit&lt;/strong&gt; circle approach&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="6052" LastActivityDate="2015-12-10T10:35:12.453" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10177" PostTypeId="2" ParentId="10163" CreationDate="2015-12-10T13:13:34.030" Score="5" Body="&lt;p&gt;Arranging counters into groups (multiplication and division), so arranging 12 counters into 6x2, 3x4 etc, and realising that there are some numbers that cant' be arranged, no matter how hard you try (prime numbers). Then, how many prime numbers are there, can you work out which number is going to be prime?&lt;/p&gt;&#xA;" OwnerUserId="6054" LastActivityDate="2015-12-10T13:13:34.030" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10178" PostTypeId="2" ParentId="10150" CreationDate="2015-12-10T13:14:44.090" Score="7" Body="&lt;p&gt;The main problem with teaching strong induction as you define it is its logical complexity.  What you seem to be doing is replacing quantification over a single integer by quantification over a set of integers (namely all those less than $n$). This is of course only appearances but appearance of difficulty can be daunting to a beginning student also.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To respond to Mike's question, in my experience as a teacher I only taught the ordinary induction, never &quot;strong induction.&quot; In my experience as a highschool student, I recall first encountering &quot;strong induction&quot; and finding it rather confusing.  In retrospect, I feel this was because &quot;all integers smaller than $n$&quot; is harder to encompass in one mind's eye than &quot;$n-1$&quot;. This only resembles the distinction between first and second order quantification, but psychologically may involve the same kind of barrier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just a quick additional comment: &quot;strong induction&quot; not only seems to involve added complexity because of additional quantification, but actually does involve additional quantification, though of course only of first order.  Namely, you do have to say &quot;for all&quot; integers less than $n$. This could be compared to the difference between Cauchy's infinitesimal definition of continuity and the epsilon-delta paraphrase introduced by Weierstass, with its hair-raising (for students) quantifier alternations.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2015-12-10T17:20:45.920" LastActivityDate="2015-12-10T17:20:45.920" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10179" PostTypeId="2" ParentId="10163" CreationDate="2015-12-10T13:38:38.317" Score="7" Body="&lt;p&gt;There is a great old Disney short called Donald Duck in Mathmagic Land. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As well as being delightfully drawn in the traditional Disney style, it contains lots of useful and occasionally surprising information about where math can be found in everyday life. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally I found it informative and inspiring, I imagine there would be plenty of conversations to had from watching it. &lt;a href=&quot;https://www.youtube.com/watch?v=AJgkaU08VvY&quot;&gt;YouTube&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6055" LastActivityDate="2015-12-10T13:38:38.317" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10180" PostTypeId="2" ParentId="10163" CreationDate="2015-12-10T17:27:50.943" Score="4" Body="&lt;p&gt;Just to complete the list of books. Some time ago I read &lt;a href=&quot;https://books.google.es/books/about/The_Number_Devil.html?id=Vtcmvzl7s1oC&amp;amp;source=kp_cover&amp;amp;hl=en&quot; rel=&quot;noreferrer&quot;&gt;The Number Devil&lt;/a&gt; and liked very much. I think it is the proper book for your needs as the argument shows how a &quot;math devil&quot; shows math concepts to a child in a way that makes the child more interested in new concepts.&lt;/p&gt;&#xA;" OwnerUserId="6058" LastEditorUserId="6058" LastEditDate="2015-12-10T17:37:01.683" LastActivityDate="2015-12-10T17:37:01.683" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10181" PostTypeId="1" AcceptedAnswerId="10186" CreationDate="2015-12-10T18:04:10.593" Score="9" ViewCount="358" Body="&lt;p&gt;&lt;strong&gt;I am looking for a function of on an interval with several local optima that appears in some mathematical model and which you can at least imagine that you want to optimize.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am teaching a calculus course right now, and as usual there is a short chapter on optimization. I wanted to have give a list of examples of functions on an interval $[a,b]$ that you might want to optimize and which display different behavior. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I find it easy to give examples of monotone functions from applications and it is also not difficult to find good examples of applied problems where the functions you optimize is U-shaped. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I actually got stuck trying to find good examples of more complex behavior with several local optima with an applied motivation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally the function I am looking for should come from a real application in science or otherwise and you should at least be able to imagine that you want to optimize it. It has to be a single-variable function and it would be better the less knowledge one needs to understand what the function is modelling. &lt;/p&gt;&#xA;" OwnerUserId="6057" LastEditorUserId="77" LastEditDate="2017-12-26T16:01:59.487" LastActivityDate="2017-12-26T21:15:43.353" Title="Good examples of non-convex optimization" Tags="|calculus|examples|optimization|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10182" PostTypeId="2" ParentId="10163" CreationDate="2015-12-10T18:25:28.743" Score="16" Body="&lt;p&gt;I remember being excited about the following at a young age.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you add consecutive numbers you get triangle numbers.&#xA;Triangle numbers are fun.&lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/eJlzS.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eJlzS.jpg&quot; alt=&quot;triangle numbers&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you put two consecutive triangle numbers together you get a square number.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Suzhn.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Suzhn.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also make a square number by adding the next odd number.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/hVPrD.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hVPrD.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-12-10T18:25:28.743" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10183" PostTypeId="2" ParentId="10162" CreationDate="2015-12-10T18:30:22.650" Score="5" Body="&lt;p&gt;Since the students have already been exposed to the unit circle and the graph, I would give them the equation and ask for discussion of how to figure it out.  If the students come up with the answers themselves they are most likely to remember it far better than if you tell them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they only come up with one way, present the other way as well.  If they have no idea, present both ways as tools and ask which would help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe self-discovery is best.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-12-10T18:30:22.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10184" PostTypeId="2" ParentId="10163" CreationDate="2015-12-10T21:52:29.187" Score="2" Body="&lt;p&gt;Do an exercise a couple times per day with a hidden number. For example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I just wrote a secret number down on this little piece of paper. I can't tell you what it is, but I'll tell you a secret: If you add 1 to this number, then you get your own age! Can you guess what it is?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It may take a couple examples, but he will eventually understand this concept. And this will give him an advantage later on in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Count by 2's to 100. Now count by 5's to 100. Now 10's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you get into any multiplication (even very simple numbers), teach him how to do 11. The trick is to split your number then add into the middle. For example take 21. Split it and get 2_1. Now stick 2 + 1 into the middle and get 231. Never give him two numbers that add to 10 or higher and he'll be able to impress family and have fun doing it.&lt;/p&gt;&#xA;" OwnerUserId="4882" LastActivityDate="2015-12-10T21:52:29.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10185" PostTypeId="2" ParentId="10163" CreationDate="2015-12-10T22:15:15.957" Score="1" Body="&lt;p&gt;At some point you need to remember why we should even educate at all -- because there are useful things to know so we can do things in life.  The more practical and relevant things you can demonstrate and integrate into one's life, the more one is going appreciate it and hang on to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, there is a lot of art in all this, but the kiddo's going to either appreciate it or not on the level of the mechanics and novelty of various facets... that'll be the child's preference.  What I'm saying is focus on &quot;why&quot;, on how useful mathematics is, because it truly is amazing what you can do with it, especially the deeper you go when you have a problem to solve.  You can at least foster some long-lived appreciation and life skills.  That goes for the future engineer, scientist, and everyday Joe. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that all said, here's a fun starter: How to use algebra to derive a way to calculate post-sales tax prices in one step... maybe a little early for the 6 years old, but hey. :)&lt;/p&gt;&#xA;" OwnerUserId="6061" LastEditorUserId="6061" LastEditDate="2015-12-10T22:24:03.400" LastActivityDate="2015-12-10T22:24:03.400" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10186" PostTypeId="2" ParentId="10181" CreationDate="2015-12-11T01:44:58.303" Score="10" Body="&lt;p&gt;The graph below shows the distance from the origin (black dot) to&#xA;the (blue) convex polygonal curve, which alternately has vertices (red) on the unit-radius&#xA;semicircle (brown), and slightly inside that semicircle.&#xA;The distance $d$ is plotted with respect to $\theta$, the angle w.r.t.&#xA;the $x$-axis. You can see it is multimodal.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/w7W39.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/w7W39.jpg&quot; alt=&quot;MultiModalDists&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;Slight perturbations of the vertices (red) would break the&#xA;symmetries, if you prefer more variation in the peaks and valleys.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This multimodal distance function $d(\theta)$ plays a role in algorithms to find the&#xA;diameter of a convex polygon. &lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-12-11T01:44:58.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10188" PostTypeId="2" ParentId="10163" CreationDate="2015-12-11T09:09:20.507" Score="2" Body="&lt;p&gt;I would try Pascal's triangle. It's easy to set up (requires only addition), and it has a &lt;a href=&quot;https://en.wikipedia.org/wiki/Pascal%27s_triangle#Overall_patterns_and_properties&quot; rel=&quot;nofollow&quot;&gt;bunch of cool patterns&lt;/a&gt; that your son can find.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of those patterns is the Fibonacci sequence, which may be worth diving into in its own right. Like Pascal's triangle, it only requires addition to set up. It's fun, and it'll come up again and again in his education -- starting with the fact that it shows up in Pascal's triangle! I always liked &quot;themes&quot; like that in math, those things that just seem to crop up again and again unexpectedly. And it's always neat to come across a &quot;new&quot; concept in class that you already know from home -- it'll make your son feel smart (I'm not implying he's not, of course! He sounds like he is).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And while we're on recursion, you can venture out a bit from the world of numbers and take a look at the Tower of Hanoi. You may be able to help him come up with the recursive solution to it, and you can tie it back to the world of numbers by seeing if there's a pattern in how quickly the number of steps grows as the number of levels grows. This also shows the connection between numbers math, puzzles, and the physical world -- which is a neat connection at any age!&lt;/p&gt;&#xA;" OwnerUserId="6065" LastActivityDate="2015-12-11T09:09:20.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10189" PostTypeId="1" CreationDate="2015-12-11T09:35:10.357" Score="7" ViewCount="460" Body="&lt;p&gt;This question was inspired by a different question on this site which asked by what grade times tables had to be learned. The consensus seemed to be that it is essential that you learn basic times tables for single digit numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm asking mainly because I'm in my mid thirties and still don't really feel like I remember my times tables and often spend some amount of time even deciding what 6*4 or 8*4 is, never mind things like 7*8 or 7*6. In many cases I end up either performing some arithmetic trick (6*4=2*12=24, 8*4=10*4-2*4=32) or just add/subtract from a known value (7*6=6*6+6=42, 8*7=8*8-8).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said that I hold two Masters in mathematics (one in abstract algebra and one in set theory) from different universities and I have never had any issues with dealing with fractions, algebraic manipulation or similar. My arithmetic tends to be slow, but that seems to rarely be an issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To further elaborate on the question and make it more distinct from the possible duplicate. Is there any actual research that causally connects not learning multiplication tables by rote with having problems with further abstract mathematics? Any such research would obviously have to eliminate the possibility that the students that haven't learned multiplication tables by rote are also in general less suited for further mathematical education (that is there isn't some underlying cause of both and instead there is a causal link).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: I'm not sure that the question suggested as duplicate isn't a duplicate but the answers to that question are in my opinion unsubstantiated and incomplete. The only answer substantiated by anything other than &quot;this is what I think/from my experience&quot; just cites a paper which doesn't even answer the question at all. Though that paper cites many papers that might answer the question that seems a bit of a long stretch.&lt;/p&gt;&#xA;" OwnerUserId="5896" LastEditorUserId="5896" LastEditDate="2015-12-11T15:58:01.687" LastActivityDate="2015-12-14T10:08:21.720" Title="Why are times tables thought to be important?" Tags="|multiplication-table|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="10190" PostTypeId="2" ParentId="10163" CreationDate="2015-12-11T14:59:42.800" Score="1" Body="&lt;p&gt;I find this video by VSauce on logarithmic thinking really interesting. I dont know if its appropriate for a 6 year old, but in the video he mentions this is the way we naturally understand the world, and that children before 4 years old think in this way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also the story of Gauss figuring out how to efficiently find the sum of the first 100 whole numbers when he was 6(?) is really interesting. This link is one I found with a quick google search: &lt;a href=&quot;http://mathcentral.uregina.ca/QQ/database/QQ.02.06/jo1.html&quot; rel=&quot;nofollow&quot;&gt;http://mathcentral.uregina.ca/QQ/database/QQ.02.06/jo1.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6068" LastActivityDate="2015-12-11T14:59:42.800" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10191" PostTypeId="2" ParentId="10163" CreationDate="2015-12-11T16:36:53.483" Score="1" Body="&lt;p&gt;Inspiring question..&lt;/p&gt;&#xA;&lt;p&gt;Try explaining bisecting: Get a number between 1 and 10(00) and let him guess by cutting the range in half.&lt;/p&gt;&#xA;&lt;p&gt;Summing a sequence of numbers (1 + 2 + 3 + 4 + 5 + 6) = 1+6 + 2+5 + 3+4&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20171225201205/http://mathforum.org/dr.math/faq/faq.monty.hall.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Monty Hall&lt;/a&gt; (and increase the number of doors to 10 and leave one closed).&lt;/p&gt;&#xA;&lt;p&gt;induction puzzles: Like &lt;a href=&quot;https://en.wikipedia.org/wiki/Induction_puzzles&quot; rel=&quot;nofollow noreferrer&quot;&gt;The king's wise man&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Oh yeah I loved it when my dad let me discover PI by cutting out cardboard wheels and rolling and measuring distance. But i think I was 10 at that age.&lt;/p&gt;&#xA;" OwnerUserId="6069" LastEditorDisplayName="user18187" LastEditDate="2023-01-24T17:25:26.280" LastActivityDate="2023-01-24T17:25:26.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10192" PostTypeId="2" ParentId="10163" CreationDate="2015-12-11T19:01:01.867" Score="1" Body="&lt;p&gt;As a parenting question (rather than a formal teacher question), we found it good to put the number in context, or rather, extract the numbers from our local context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, when driving home from a long journey you can ask what are the distances to places when the sign boards come by. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can as how much will be left to get to the further away place when you get to the next place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You say how far 'home' is from some place listed and ask how far is it to home (really good for subtraction if home is before the listed place). You can ask how many times you have to do the little bit you did before you get home (5 miles from last village, 20 miles to go, -. we'll have to do that same distance four times'...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can show them tricks you know and use, such as rounding to get a handle on things. Here we have petrol/gas/deisel in litres but everyone talks in miles per gallon, so try some approximations - we got 300 miles on that fuel fill, we filled with 22.5 litres, how many miles can we go for every gallon? If the tank has 60 litres can we drive all the way to grannies - she's 800 miles away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same can be done when shopping (and it's good budgetting).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Estimate the beans in the jar...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key here is to use the context you have, so it's normal and natural.&lt;/p&gt;&#xA;" OwnerUserId="6072" LastActivityDate="2015-12-11T19:01:01.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10194" PostTypeId="1" CreationDate="2015-12-11T20:00:43.223" Score="6" ViewCount="1609" Body="&lt;p&gt;I  find that the notion of trigonometric angles of rotation is a bit confusing for the students.&#xA;In my curriculum, students learn about angles first time at geometry in middle-school. The angles are &quot;static&quot;(two rays with a common vertex), and have a measure between 0 and 180 degrees. Then in 7th grade the trigonometric functions(Ratios) are introduced in right-triangles and students solve a lot of problems applying them in triangles, and they seem to even like these problems and trig ratios.&#xA;But then in 9th grade the study of trig functions is continued using the unit circle approach. First of all, the notion of angle is &quot;Extended&quot;, we talk about angles greater than 180 degrees and negative angles regarded as &quot;angles of rotation&quot; of a ray(with initial side, terminal side, etc). &#xA;&lt;em&gt;But i feel these &quot;angles of rotation&quot; are a bit confusing for the students.&lt;/em&gt; First of all, because in practical math applications we work with angles greater than 180 very little. With negative angles we don't work at all.(that's the curriculum in my country, it doesn't insist on practical applications, but rather on theoretical ones, it is supposed that students make this practical math in pshysics classes).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, &lt;em&gt;there appears some confusion in student's mind between the two types of angles: geometric angle and angle of rotation.For example,  when we do the reduction to the first quadrant, we deal with the reference angle, which is somehow no more a rotation angle but rather an ordinary one, because we apply to it the trig ratios in a right triangle&lt;/em&gt; &#xA;For example in this picture:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MRKk8.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MRKk8.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;the 150 angle is a &quot;rotation angle&quot;, but the...30 is somehow an ordinary angle. Which may produce confusion from a rigorous point of view. How should we deal with this? Make the 30 also rotation angle by putting an arrow?&#xA;Also, in this picture:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/fxLIV.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fxLIV.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;somehow (-x) is a rotation angle, but at the same time there's an &quot;underlying&quot; ordinary geometrical angle of x which we use when we assert that the two points are symmetrical with respect to the x-axis. So there's again some confusion between two types of angles.&#xA;How should we  eliminate this confusion?&lt;/p&gt;&#xA;" OwnerUserId="5972" LastEditorUserId="64" LastEditDate="2015-12-12T21:11:03.217" LastActivityDate="2015-12-21T16:05:36.777" Title="Trigonometric angles of rotation" Tags="|trigonometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10195" PostTypeId="1" AcceptedAnswerId="10197" CreationDate="2015-12-11T20:23:40.923" Score="9" ViewCount="717" Body="&lt;p&gt;The usual assumption in education is that grades follow some sort of normal curve.&#xA;My question is about situations where the grade distribution curve is a rising one. More A's than B's, and more B's than C's, etc. In educational setting: Is there a name for this situation? Are there studies or methodologies related to it? Any references are welcome. (My question is not about the mathematical detail of a probability distribution function, it is about the occurrence of such distributions of grades in education, or about teaching styles that are related to them etc.)&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2015-12-11T20:45:04.023" LastActivityDate="2015-12-11T21:34:26.803" Title="When higher grades are more common than lower grades" Tags="|undergraduate-education|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10196" PostTypeId="2" ParentId="10163" CreationDate="2015-12-11T20:25:31.743" Score="1" Body="&lt;p&gt;There are many different ways to write numbers - base-10, binary, hexadecimal. You can have any number of digits 2 or greater, as long as one of those digits is 0.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Numbers that have an exact decimal representation in one system may not in another. For example 1/3 in base-10 repeats forever, but in base-3 it is just 0.1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, Roman numerals, tally marks, other ancient ways of writing numbers.&lt;/p&gt;&#xA;" OwnerUserId="6075" LastActivityDate="2015-12-11T20:25:31.743" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10197" PostTypeId="2" ParentId="10195" CreationDate="2015-12-11T20:30:51.603" Score="11" Body="&lt;p&gt;The student grades could be following an &lt;strong&gt;exponential distribution&lt;/strong&gt; or a &lt;strong&gt;Poisson distribution&lt;/strong&gt;.  Both of these distributions have been extensively analyzed statistically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also possible that the distribution is &lt;strong&gt;one tail of a normal distribution&lt;/strong&gt;, where the mean is greater than 100%.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are three scenarios that can result in such a grade distribution.  The three scenarios can be combined:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;a. The algorithm for solving problems is fault-tolerant.  There are implicit or explicit checks built into each problem.  If a student makes a small number of mistake(s), they will catch them in time, and not lose any points.&lt;BR&gt;&#xA;b. Multiple independent mistakes are required to mess up problem(s) badly enough that points are taken off.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The student population has been pre-filtered, so that only students in the bottom half of a bell curve are in the class.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The grading standards emphasize &quot;allowing students to succeed&quot; or &quot;minimizing an achievement gap&quot;.  In other words, the &quot;bar is set low&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Standardized tests are designed to have bell-curve grade distributions.  Some ways they do this are to:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Have lots of chances to score small amounts of points.&lt;/li&gt;&#xA;&lt;li&gt;Have an average score that is in the middle of the possible range.&lt;/li&gt;&#xA;&lt;li&gt;Create time pressure.  Slower students will score worse because they run out of time, or do not have time to check their work.&lt;/li&gt;&#xA;&lt;li&gt;Have lots of trick questions (especially questions with double-negatives).  These questions tend to test &quot;test-taking ability&quot; or &quot;general intelligence (IQ)&quot; instead of the specific subject matter.&lt;/li&gt;&#xA;&lt;li&gt;Have lots of questions with an &quot;obviously correct&quot; answer that is actually wrong.&lt;/li&gt;&#xA;&lt;li&gt;Have a variety of difficulty levels of questions, so there are some questions that most students answer correctly, and some that most students answer incorrectly.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-12-11T21:34:26.803" LastActivityDate="2015-12-11T21:34:26.803" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10198" PostTypeId="2" ParentId="10195" CreationDate="2015-12-11T21:17:54.863" Score="6" Body="&lt;p&gt;Although I have serious issues with &quot;outcomes based&quot; education, its philosophy does combat the pernicious notion that grades should have a normal distribution centered on &quot;average.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not unreasonable to expect that most students will meet the learning objectives of a class. If most do not, then there might be something flawed with the preparation of the audience, or with the instructor, or with the purpose of the course. In this perspective, one might expect most grades to be A or B.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-12-11T21:17:54.863" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10200" PostTypeId="2" ParentId="10163" CreationDate="2015-12-12T02:06:39.543" Score="2" Body="&lt;p&gt;You could introduce your child to CircleMath, invented by Dr Stephen Taylor. It is a very simple process for using 2 bases at once, eg 9 &amp;amp; 10, 10 &amp;amp; 11, 9 &amp;amp; 11 is best because it has many more patterns and involves 'Greek 9s'. The method relies on estimation, which I suggest is the starting point for all calculations. This works in any base... and, until metrics and decimals and calculators took over, everyone once used multiple bases every day... pints/gallons, inches/feet, days/weeks, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I first met Dr Taylor in 1985  he was tutoring a half dozen 5-year olds in his lounge using a whiteboard. The children could do 'sums' like 37x58 in their heads... in under 10 seconds... and tell me, a complete stranger - how they did it. As I was at Teachers' College at the time, preparing to teach high school math, I realised I knew very little... So, Dr Taylor and I became friends, I have most of his books, and still teach CircleMath when I get a chance. I recently wrote a PowerPoint intro - for adults - if anyone would like to see it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I frequently get amazed looks as people realise how intuitively obvious it is, but it has been overgrown by dull modern mono-base thinking... and there is always a real buzz about this 'new thing'. It's the buzz all teachers and parents love...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an HoD Math I put this to the test at one of our regional math assoc meetings, and talked about 15 capable, qualified math leaders through the basics, building up to this one: 114622/514. Following the very simple rules they got the answer correct in about 20 seconds... then looked at me and each other in silence... then said 'how the hell does that work?' Tada! CircleMath :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It had taken only 45 mins, and they all got it correct, but few really got it at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sad to say, for years afterwards some of them joked about that session as the workings of a crazed mind... not at all open for people whose job it was to open students' minds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit for clarification: there was a link but has lapsed since Dr Taylor died in the last couple of years. I'll contact his son who was also brilliant at this stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit2 for clarification: Dr Taylor wrote a book Theory of Mind, in which he identified and promoted the human brain's 'intrinsic base', meaning if things are couched in the right way then they resonate with the brain's OS, as it were, thus obviating the need for us to try to engender or pander to the child's interest. Things put in the right way will be interesting to the child. It is a provocative book about what is intrinsic and extrinsic to mind, learning and understanding.&lt;/p&gt;&#xA;" OwnerUserId="6081" LastEditorUserId="6082" LastEditDate="2015-12-14T09:42:08.400" LastActivityDate="2015-12-14T09:42:08.400" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10202" PostTypeId="2" ParentId="10163" CreationDate="2015-12-12T16:47:09.543" Score="-3" Body="&lt;p&gt;Here are a list of books and articles he might enjoy reading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.springer.com/gp/book/9789462391536&quot; rel=&quot;nofollow&quot;&gt;Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.springer.com/gp/book/9783319181318&quot; rel=&quot;nofollow&quot;&gt;Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hamiltonian Partial Differential Equations and Applications&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Approximation of Stochastic Invariant Manifolds&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd suggest that you replace bedtime stories with these articles as they are guaranteed to instill a sense of mathematical wonder in your child, and put him right to sleep...&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2015-12-12T16:47:09.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10204" PostTypeId="1" CreationDate="2015-12-12T19:01:59.773" Score="4" ViewCount="80" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Van_Hiele_model&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Van Hiele model&lt;/strong&gt;&lt;/a&gt; (wikipage) &quot;is a theory that describes how students learn geometry.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would appreciate further insights into the original model, later models that expanded or re-worked it, and the influence of the model on geometric thinking. To this end, my question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are the best expository pieces related to the Van Hiele models?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1760" LastEditorUserId="262" LastEditDate="2015-12-13T20:06:47.120" LastActivityDate="2015-12-13T20:06:47.120" Title="What are the best expository pieces related to the Van Hiele models?" Tags="|reference-request|geometry|history|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10205" PostTypeId="2" ParentId="10163" CreationDate="2015-12-13T09:05:42.463" Score="1" Body="&lt;p&gt;Pull out a couple of &lt;a href=&quot;https://en.wikipedia.org/wiki/M%C3%B6bius_strip&quot; rel=&quot;nofollow&quot;&gt;Möbius strips&lt;/a&gt; at the dinner table. There is nothing like a &quot;magic trick&quot; to inspire wonder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Know all of the fun stuff you can do cutting them, double looping them etc. If you need a refresher, look &lt;a href=&quot;http://h2g2.com/approved_entry/A337592&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://www.google.com/?gws_rd=ssl#q=mobius+strip+video&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6091" LastEditorUserId="6091" LastEditDate="2015-12-13T11:07:14.387" LastActivityDate="2015-12-13T11:07:14.387" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10206" PostTypeId="1" CreationDate="2015-12-13T11:54:16.777" Score="27" ViewCount="5746" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;From a pedagogical point of view, when is the &quot;right&quot; moment to introduce letters and variables to school children? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Let's say we taught arithmetic, the four operations, did computation exercises, or even &quot;disguised&quot; equations, when in place of a variable we use a box, and ask what numbers should we put in that box. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How should we introduce the practice of using letters, to make the transition from arithmetic to algebra?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The context:&lt;/strong&gt; I'm a high school teacher, I do my job pretty well and have good results with my high school students, but recently I've been tutoring a fifth grader, and for the future it is possible that I would have to teach also in middle school (grades 5-8). I find it is kind of hard to work with this student I tutor; he is having a hard time doing/understanding even simple stuff (simple for me, of course), and I find that variables/letters &quot;intimidate&quot; him. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How should I introduce and motivate using letters?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I feel that even high schoolers are sometimes &quot;intimidated&quot; by letters. I once had a student that said, &quot;teacher, I really loved mathematics in the past, but only when we worked with numbers. Since we were introduced to letters in math, I don't like it anymore.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What should we do to avoid this issue? i.e., How should we introduce letters/variables in order for children to understand them better and not hate algebra? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5972" LastEditorUserId="262" LastEditDate="2016-04-21T21:37:02.487" LastActivityDate="2023-02-02T18:16:43.090" Title="When should we first teach variables in school math? And how?" Tags="|mathematical-pedagogy|reference-request|algebra|" AnswerCount="8" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="10207" PostTypeId="2" ParentId="10194" CreationDate="2015-12-13T13:57:53.557" Score="2" Body="&lt;p&gt;This is the unit circle I use to teach this topic. And specifically, to explain your issue with reference angles. Reference angle is simply the acute angle closest to the X axis. Therefore, 30, 150, 210, and 330 degrees all have 30 as their reference angle. Now, if you look at the (X,Y) values, comparable to the second graph you offered, only filled in for the common angles, you'll see that the absolute values of these 4 pairs of values are the same. More important, you can see that the sine value is the same for reference angle in quadrant I and II, and again for III and IV. Cosine is the same for quadrant I and IV, and then for II and III. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not until specific problems are offered that the importance of this concept comes to light. The student is given some complex problem which simplifies to $sin(x) = .5$  He looks at his calculator, which offers 30 degrees. It's by understanding the reference angle that he'll also get the second answer, 150 degrees. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other way I describe all this - &quot;Calculate the value of $cos(x)$ at 150 degrees. By hand.&quot; The points are still on the circle, but the values are removed. The student drops a line from the point on the circle, and forms a triangle with the base angle of 30. But I asked for the cosine of 150. Exactly. To get the values needed, the triangle itself has an angle of 30, and a base with a negative length. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iu5cl.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iu5cl.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the classroom, the introduction and explanation of reference angles should have a flow from the discussion of all first quadrant observations. For the students who are attentive, there's no leap in understanding, just a continuation of learning. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-12-13T13:57:53.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10208" PostTypeId="2" ParentId="10206" CreationDate="2015-12-13T18:19:51.823" Score="14" Body="&lt;p&gt;You don't need 1-letter variable names to do algebra.  Basically, as soon as you start giving story problems to children, you need to start teaching algebra techniques.  You can teach them as &quot;easier ways&quot; to solve the problems, that help kids &quot;keep track of things&quot; and &quot;avoid mistakes&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most computer programming languages (including spreadsheets with named cells) allow using long variable names.  Languages like Logo have been successfully taught to primary school children.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key is to clearly distinguish the long variable names from other text that might represent operations.  Parentheses can suffice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After not very long, children will get tired of writing out complete variable names.  At this point, you can show them how to:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Draw a picture if it helps to understand the variables.&lt;/li&gt;&#xA;&lt;li&gt;Explicitly abbreviate their variable names.&lt;/li&gt;&#xA;&lt;li&gt;At the end of the problem, write a short summary of the answer, and circle it in a cloud.  For example:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At the beginning of the problem:&lt;BR&gt;&#xA;  (# apples) = (number of apples)&lt;BR&gt;&#xA;  …&lt;BR&gt;&#xA;  At the end of the problem:&lt;BR&gt;&#xA;  &lt;em&gt;There are 7 apples.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And later on:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At the beginning of the problem:&lt;BR&gt;&#xA;  (a) = (number of apples)&lt;BR&gt;&#xA;  …&lt;BR&gt;&#xA;  At the end of the problem:&lt;BR&gt;&#xA;  &lt;em&gt;There are 7 apples.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And later on:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At the beginning of the problem:&lt;BR&gt;&#xA;  a = number of apples&lt;BR&gt;&#xA;  …&lt;BR&gt;&#xA;  At the end of the problem:&lt;BR&gt;&#xA;  &lt;em&gt;There are 7 apples.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2015-12-13T18:19:51.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10209" PostTypeId="1" CreationDate="2015-12-13T18:27:59.670" Score="19" ViewCount="2262" Body="&lt;p&gt;I've taught integral calculus a few times, and in every course the students are confused about the distinction between Taylor series and power series.  It's something I remember being confused by as a student.  Now that I have a little more background, it seems to me that there's no difference at all, and that one could and should replace the phrase &quot;Taylor series of&quot; with &quot;power series representation of.&quot; Of course non-analytic functions won't be equal to their Taylor series in any neighborhood of the center, so calling it a &quot;representation&quot; of the function might be a bit misleading.  But most or all functions we meet in a basic integral calculus course &lt;em&gt;are&lt;/em&gt; analytic (I'd wager all $C^\infty$ functions you show your class are analytic, unless you cook up one of the standard counter-examples for that purpose), and it would be easy to simply state this caveat for non-analytic function.  In fact, you could dispense with the word &quot;representation&quot; and just talk about the &quot;power series of f(x) with center a&quot;.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So are there good reasons to keep the confusing terminology which seems to imply a distinction without a difference? &lt;/p&gt;&#xA;" OwnerUserId="5969" LastEditorUserId="5969" LastEditDate="2015-12-13T18:56:34.000" LastActivityDate="2019-06-25T20:26:29.107" Title="What good is the phrase &quot;Taylor series&quot;?" Tags="|calculus|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10210" PostTypeId="2" ParentId="10209" CreationDate="2015-12-13T19:22:51.427" Score="5" Body="&lt;p&gt;In French we have three phrases which are clear and relevant to different situations: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;développement limité&quot; which is translated as &quot;Taylor series&quot; and means a relation of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x) = \sum_{k=0}^n a_k(x-x_0)^k +o(x^n),$$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Théorème de Taylor-Young&quot; (easy to translate) which is the statement that a function which is derivable at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; has a &quot;développement limité&quot; up to its derivability order at the considered point, with the coefficient given by the normalized derivatives at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;séries entière&quot; which is, I think, translated by &quot;power series&quot; and means an expression of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{k=0}^\infty a_k z^k$$&lt;/span&gt;&#xA;in a formal sense, i.e. independently from the convergence radius of the series and with &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; a placeholder.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I think this gives a precise and distinct role to each part of the theory. In particular, a Taylor series has &lt;em&gt;finite order&lt;/em&gt; and is a mean to approximate a function, while power series representation is an exact equality with an infinite sum.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="8571" LastEditDate="2019-06-25T20:26:29.107" LastActivityDate="2019-06-25T20:26:29.107" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="10212" PostTypeId="2" ParentId="10209" CreationDate="2015-12-14T00:24:52.067" Score="17" Body="&lt;p&gt;Taylor series are an extension of the concept of a Taylor polynomial. A Taylor polynomial is a polynomial which approximates a function, and is constructed from the derivatives of the function at a specific point. It is reasonable to think that there would be many different polynomials that might approximate a particular function, and the Taylor polynomials are the ones constructed via a certain process using derivatives. When you take your polynomial to infinite degree, you get a Taylor series. Calling it a Taylor series preserves this connection to Taylor polynomials and derivatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the statement &quot;every power series that converges in an interval is a power series representation of some function&quot; -- well duh! If it converges then it has a function value. Compare to the statement &quot;every power series that converges in an interval is a Taylor series for some function&quot; -- this brings to mind the idea of the derivatives and the $n!$, which is more meaningful. Moreover, it somehow creates more of an expectation that you might sometimes be able to guess the name of the function. Something like &quot;Oh! It's the Taylor series for $\ln(x)$ centred at $x=1$!&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, regardless of whether you use the terminology in your local classroom, you do still need to tell students about the other terminology, so that they can read and use other people's resources. (It's an interesting exercise to google &quot;Taylor series&quot; and &quot;Power series representation&quot; separately.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal preference is to use &quot;power series representation&quot; most of the time, but sometimes say Taylor series too, mostly when I want to make the point about the derivatives, but sometimes just to mix it up.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-12-14T00:24:52.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10213" PostTypeId="2" ParentId="10209" CreationDate="2015-12-14T00:32:05.817" Score="1" Body="&lt;p&gt;Indeed we could get by without referring to Taylor series, and instead just always refer to a &quot;power series.&quot; But then maybe we should also stop referring to Fourier series and simply refer to &quot;trigonometric series.&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that &quot;Fourier series&quot; draws attention to the particular formula that relates the coefficients to a function that is being represented or approximated. The same is true for &quot;Taylor series.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to the redundancy of &quot;Maclaurin series&quot; since they are a special case of Taylor series, we could take this a step further and forever replace &quot;Taylor&quot; with &quot;Laurent&quot; since Laurent series are more general. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think mathematics is culturally enriched by references to constructs of Taylor series, Maclaurin series, Laurent series, Fourier series, Puiseux series, etc. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-12-14T00:32:05.817" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10214" PostTypeId="2" ParentId="10209" CreationDate="2015-12-14T06:31:18.533" Score="12" Body="&lt;p&gt;&quot;Power series&quot; is a very general term. I'm not a mathematician, but as I understand the terminology, the term &quot;power series&quot; refers to any polynomial, not just one derived as a Taylor series. Yes, all Taylor series are power series, but not all power series are Taylor series. And the process of generating a Taylor series is very specific and important. It makes sense to give it a name.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a way, your question is like asking, &quot;Why should we have the term 'Labrador retriever'? It's just a dog, isn't it? Why don't we just call it a dog?&quot; Yes, we could, and in some contexts that would be fine. But sometimes we want to be more specific.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if the student doesn't remember all the names, I think it is helpful when learning to have specific names for specific things. Giving something a name makes it easier to talk about it, and I think aids comprehension.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like, in math we often talk about the difference between integers, rationals, reals, and so on. Why don't we just say &quot;numbers&quot; and leave it at that? They're all numbers. But we want to make the point that specific kinds of numbers have specific properties. By giving them names we make it easier to talk about them.&lt;/p&gt;&#xA;" OwnerUserId="6096" LastActivityDate="2015-12-14T06:31:18.533" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10215" PostTypeId="2" ParentId="10189" CreationDate="2015-12-14T10:08:21.720" Score="2" Body="&lt;p&gt;I taught gifted students in 4th-6th grades and it was important for them to know their times tables.  We simplified fractions, performed long division, identified prime numbers and performed calculations with multi-digit numbers.  Students who didn't know the facts practiced on &lt;a href=&quot;http://www.flashmaster.com/&quot; rel=&quot;nofollow&quot;&gt;Flashmasters&lt;/a&gt;, an excellent electronic flashcard system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said that, my standard for knowing the facts is lower than many of my colleagues.  Most of my colleagues insist on the ability to instantly recall facts and their standard is 100 facts in 3 minutes.  My standard was a test of all the facts up to 10 by 10 (45 facts in all) in 5 minutes (although most could pass in 3 minutes).  Given what you said about using mathematical tricks to find your facts, I believe you would have passed the standard for knowing the facts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There will always be students who could recall facts in under 2 seconds (as confirmed by the &lt;a href=&quot;http://www.flashmaster.com/&quot; rel=&quot;nofollow&quot;&gt;flashmaster&lt;/a&gt;), but those who took a little longer whether because they figured them out or they just weren't as fast did fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My conclusion it is necessary to know the facts, but there are different levels of knowing them and your level is fine.  Many of my colleagues would disagree, but perhaps you are a good example of the fact that my standards work.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2015-12-14T10:08:21.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10217" PostTypeId="2" ParentId="352" CreationDate="2015-12-14T13:58:12.563" Score="6" Body="&lt;p&gt;As someone who moved from computer science and programming into teaching, I always think of times tables as &lt;a href=&quot;https://en.wikipedia.org/wiki/Lookup_table&quot; rel=&quot;noreferrer&quot;&gt;lookup tables&lt;/a&gt;. There's nothing there that you couldn't work out using repeated addition, but it's &lt;em&gt;faster&lt;/em&gt; if you can simply recall that &quot;6 times 7 is 42&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering how fundamental single-digit multiplication is to other algorithms (multi-digit multiplication, factorisation, etc.), the speed gains add up.&lt;/p&gt;&#xA;" OwnerUserId="1946" LastActivityDate="2015-12-14T13:58:12.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10218" PostTypeId="1" CreationDate="2015-12-14T19:35:36.030" Score="1" ViewCount="491" Body="&lt;p&gt;Im looking for a sign that is the reciprocal of '$\ll$'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where $a\ll b$ means that $a$ is very much smaller than $b$ (and perhaps may be neglected), im looking for a sign that expresses that $a$ is only a bit less than $b$.&#xA;This symbol should not be $\sim$, as this would include 'a bit greater than' as well&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any suggestions?&lt;/p&gt;&#xA;" OwnerUserId="6099" LastEditorUserId="205" LastEditDate="2015-12-25T05:07:36.177" LastActivityDate="2015-12-25T05:07:36.177" Title="Is there a symbol for 'a bit smaller than'?" Tags="|algebra|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10220" PostTypeId="2" ParentId="10157" CreationDate="2015-12-14T19:43:29.963" Score="1" Body="&lt;p&gt;A good way to avoid confusion could be to say &lt;em&gt;&quot;The solution $(x,y)$ is ...&quot;&lt;/em&gt; and &lt;em&gt;&quot;The solutions $(x,y)$ are ...&quot;&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5586" LastActivityDate="2015-12-14T19:43:29.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10221" PostTypeId="2" ParentId="10218" CreationDate="2015-12-14T22:19:48.010" Score="3" Body="&lt;p&gt;I suggest that you don't introduce new symbols; it adds too much to cognitive load. Instead you could say $b-\epsilon&amp;lt; a &amp;lt; b$ for some $\epsilon &amp;gt;0$.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2015-12-14T22:19:48.010" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10222" PostTypeId="2" ParentId="10218" CreationDate="2015-12-15T03:04:46.557" Score="7" Body="&lt;p&gt;I doubt there is, largely because the concept is ill-defined:  Is 5 &quot;a bit less than&quot; 6?  Is 995 &quot;a bit less than&quot; 996?  Is the important point here the absolute difference $|x-y|$, or the relative difference $\frac{|x-y|}{y}$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, having said that, it has to be acknowledged that the exact same criticism could be laid against the notation $x&amp;lt;&amp;lt;y$, which I have always been uncomfortable with for the same reason.  Just how much smaller than $y$ does $x$ need to be?  But despite this, people seem to be generally comfortable with writing $x&amp;lt;&amp;lt;y$, and letting context clarify its meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here are two proposals:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;You could, as Benjamin Dickman suggests in comments, use the symbol $\lesssim$.  It does not have a &quot;standard&quot; meaning, but you could establish a local convention that it means &quot;less than but approximately equal to&quot;.  As long as you establish (in whatever context you are using the symbol) what it means &lt;em&gt;in that context&lt;/em&gt; it would seem to serve your purpose well.&lt;/li&gt;&#xA;&lt;li&gt;A second option is to appropriate the existing usage of $&amp;lt;&amp;lt;$ as follows:  To say that &quot;$x$ is a little bit less than $y$&quot; you could write either $y-x&amp;lt;&amp;lt;1$ (if what you are trying to say is that $y$ is a bit less than $x$ in &lt;em&gt;absolute terms&lt;/em&gt;), or $y-x&amp;lt;&amp;lt;y$ (if what you are trying to say is that $y$ is a bit less than $x$ &lt;em&gt;relative to the scale of $y$&lt;/em&gt;).  Either of those seem to capture well the sense you are looking for, provided you have already established a contextual convention that the notation $a&amp;lt;&amp;lt;b$ is only used when both $a$ and $b$ are non-negative.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2015-12-15T03:28:39.837" LastActivityDate="2015-12-15T03:28:39.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10223" PostTypeId="2" ParentId="352" CreationDate="2015-12-15T03:52:22.947" Score="19" Body="&lt;p&gt;Someone already mentioned this but I don't think people understand the importance: The purpose to know things off by heart in Maths is to recognize the inverse applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I explain to my students that recognizing 121 as the square of 11 is like spotting a friend in a crowd. If you did not know his face he will just be another number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is true for multiplication tables (even up to 12 in my opinion) and the powers $n^m \text{ for } \left \{ n,m\in Z| n^m&amp;lt;1000\right \}$ and especially for formulas and theorems. I especially see it with factorization in Algebra, all Euclidean Geometry and the compound angle formulas in Trigonometry. So if the person asking this is a primary school teacher, please understand that by teaching them to know these things off by heart is introducing them to the friends that will help them with Maths later on in life.&lt;/p&gt;&#xA;" OwnerUserId="6101" LastActivityDate="2015-12-15T03:52:22.947" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10224" PostTypeId="2" ParentId="10218" CreationDate="2015-12-15T07:53:33.987" Score="0" Body="&lt;p&gt;As &lt;em&gt;mweiss&lt;/em&gt; said, it's ill-defined, but if you make it precise then you would probably find that you want to say &quot;$a \nearrow b$ as $x \to c$&quot;, where $x$ and $c$ are some expressions that you have in mind but didn't specify. This parallels &quot;$\frac{a}{b} \to 0$ as $x \to c$&quot; or &quot;$a \in o(b)$ as $x \to c$&quot; to precisely express &quot;$a \ll b$&quot; with all the necessary context.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2015-12-15T07:53:33.987" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10225" PostTypeId="2" ParentId="10150" CreationDate="2015-12-15T15:02:13.077" Score="5" Body="&lt;p&gt;I'm not sure how much this experience is worth, because it is such a different environment than the standard college proof environment, but the high school math program &lt;a href=&quot;http://promys.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;PROMYS&lt;/a&gt; does this. I believe &lt;a href=&quot;http://u.osu.edu/rossmath/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ross&lt;/a&gt;, which an older program that PROMYS is strongly influenced by, uses the same model.&lt;/p&gt;&#xA;&lt;p&gt;More precisely, in the first two weeks of PROMYS, students are challenged to prove statements about integers and allowed to keep a list of what axioms they choose to use. In the second week, we start discussions about which axioms might be redundant or better in one way or another. (The statement &lt;span class=&quot;math-container&quot;&gt;$x \times 0 = 0$&lt;/span&gt; is on a lot of axiom lists at first, but is eventually realized to follow from the other ring properties.) By the end of the second week, we converge to the standard PROMYS axioms of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt; which are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt; is a commutative ordered ring, and every nonempty set of positive integers has a least element.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here are some problems which occur during the first two weeks, and are substantially easier with strong than weak induction:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are positive integers and &lt;span class=&quot;math-container&quot;&gt;$x|y$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x \leq y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is an integer and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; a positive integer, there exist integers &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$0 \leq r &amp;lt; b$&lt;/span&gt;, such that &lt;span class=&quot;math-container&quot;&gt;$a = qb+r$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Every integer &lt;span class=&quot;math-container&quot;&gt;$\geq 2$&lt;/span&gt; is divisible by a prime.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Every integer &lt;span class=&quot;math-container&quot;&gt;$\geq 2$&lt;/span&gt; can be written as a product of primes.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are integers, then there are integers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$ax+by$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and divides &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;We always get a number of students who have already seen standard induction in high school, so I got used to (a) showing how to deduce strong induction from standard induction and (b) showing how strong induction proofs of the above statements were nicer to write than standard ones. I also think that PROMYS made a good choice in choosing the phrasing:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Every nonempty subset of positive integers has a least element.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;rather than&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you can prove the implication &lt;span class=&quot;math-container&quot;&gt;$\forall_{m &amp;lt;n} S(m) \implies S(n)$&lt;/span&gt; then you can prove &lt;span class=&quot;math-container&quot;&gt;$\forall_n S(n)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strike&gt;They are logically the same, but I think that thinking about manipulating statements &lt;span class=&quot;math-container&quot;&gt;$S( \ )$&lt;/span&gt; like that is confusing.&lt;/strike&gt; In the comments below, katz points out that there are two differences between the statements: I have switched from first order to second order logic, as well as taking the contrapositive. See the discussion below.&lt;/p&gt;&#xA;&lt;p&gt;However, I would caution that these are really smart and dedicated kids, so they probably aren't a good model for a college class.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2023-05-16T17:05:37.477" LastActivityDate="2023-05-16T17:05:37.477" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="10226" PostTypeId="1" CreationDate="2015-12-15T16:31:51.063" Score="7" ViewCount="8220" Body="&lt;p&gt;&quot;Why $x^2+3xy^2+4xy+7x^2y$ can not be simplified? Why can these terms not be simplified?&quot;&#xA;I would like an explanation that is understandable by 8th-grade students. The only proof I know (based on linear Independence of powers of any variable) is not elementary at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. I don't use &quot;simplification&quot; as a mechanical and by-the-law procedure. The question is deeper. For example &quot;why  $x^2+3xy^2+4xy+7x^2y \neq x^3-3xy+9x^2y^3$?&quot;. Why can't we reduce the number of mononomials?&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="1229" LastEditDate="2015-12-15T18:55:30.600" LastActivityDate="2023-10-29T19:28:05.450" Title="Why unlike terms cannot be simplified?" Tags="|algebra|" AnswerCount="12" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10227" PostTypeId="1" CreationDate="2015-12-15T18:15:23.453" Score="10" ViewCount="177" Body="&lt;p&gt;I would like to stress the kind of reference I am looking for: In statistics there are lots of motivating (and sometimes unexpected) examples that are interesting for everyone such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Birthday_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Birthday Problem&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Simpson&amp;#39;s_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simpson Paradox&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Secretary_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;secretary problem&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/St._Petersburg_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;St Petersburg Paradox&lt;/a&gt; that easily motivate people from different fields. I love the examples that come from physics or economics, but usually they are usually interested only for students of engineering (or physics) and economics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suggestions of papers from American Mathematical Monthly, American Scientist or periodicals that specialize in teaching are very welcome!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This question was previously &lt;a href=&quot;https://math.stackexchange.com/q/1574483&quot;&gt;posted&lt;/a&gt; in Math.StackExchange and someone suggested to post it here. Since I have received no answer there, I am following this suggestion.)&lt;/p&gt;&#xA;" OwnerUserId="5034" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-01-06T16:29:07.767" Title="Could you suggest books, papers or problems that could be used as good &quot;general&quot; motivating examples of calculus application?" Tags="|calculus|reference-request|concept-motivation|recreational-learning|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10228" PostTypeId="2" ParentId="10226" CreationDate="2015-12-15T18:17:35.613" Score="0" Body="&lt;p&gt;Thank you for clarifying that by &quot;simplify&quot;, you mean &quot;reduce the number of mononomials&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that you are defining &quot;like terms&quot; to have enough in common that you could use the distributive property to combine them into a single mononomial.  Thus, the answer is obvious:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Unlike terms&quot; do not have enough in common to use the distributive property to factor out all the variables, and combine the constants into a single constant expression.  Thus, you cannot combine two unlike terms into a single mononomial.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-12-15T19:17:06.390" LastActivityDate="2015-12-15T19:17:06.390" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10229" PostTypeId="2" ParentId="10226" CreationDate="2015-12-15T18:22:03.387" Score="1" Body="&lt;p&gt;We can &quot;simplify&quot; the sum of two or more terms, combining them into single term, if they are each numerical multiples of the &lt;em&gt;same algebraic expression&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 1&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can simplify $2x^2y + 5x^2y$ because $2x^2y$ and $5x^2y$  are numerical multiples of the same algebraic expression, namely $x^2y$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For any given combination of values for $x$ and $y$, you will always get the same result for $x^2y$. If $x=2$ and $y=3$, you will always get $x^2y= 12$. If $x=3$ and $y=7$, we will always get $x^2y=63$. So, for whatever values $x$ and $y$ may have, $x^2y$ will be a common factor in the expression $2x^2y + 5x^2y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, we can write:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$2x^2y + 5x^2y=x^2y(2+5)=(2+5)x^2y= 7x^2y$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 2&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We cannot simplify $2x^2y + 5x^3y$ because $2x^2y$ and $5x^3y$  are &lt;em&gt;not&lt;/em&gt; numerical multiples of the same algebraic expression. Notice that the exponents are different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $x=2$ and $y=1$, we have different values for $x^2y$ and $x^3y$, namely $x^2y=4$ and $x^3y=8$. So, there is no common factor in the expression $2x^2y + 5x^3y$.&lt;/p&gt;&#xA;" OwnerUserId="6104" LastEditorUserId="6104" LastEditDate="2015-12-15T21:01:03.243" LastActivityDate="2015-12-15T21:01:03.243" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10230" PostTypeId="2" ParentId="10226" CreationDate="2015-12-15T20:16:55.093" Score="0" Body="&lt;p&gt;To the &quot;Why is $...\neq ...$&quot;: just find a counter example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The simplification you are referring to seems to be the distibutive law &quot; backwards&quot;. Do your students know this already? Without this law, you probably have to go with the mechanical approach.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2015-12-15T20:16:55.093" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10231" PostTypeId="2" ParentId="10226" CreationDate="2015-12-15T21:07:26.363" Score="13" Body="&lt;p&gt;Since you want to negate an implicit universal quantifier, what you have to do is nothing else than showing that there is a counter example to any simplification equality one could come up with. Instead of a proof, at this stage in one's curriculum giving a strong conviction based on a meaningful insight is probably the best possible expectation. The best idea I can come up with is to use one of the rigorous proofs to cook up a counter-example which shows the idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, take the proof by using asymptotic behavior of monomials&#xA;as $x$ and $y$ go to infinity. Why  $x^2+3xy^2+4xy+7x^2y \neq x^3−3xy+9x^2y^3$? Let's take $x=1$ and $y=1000$. Then clearly $x^2+3xy^2+4xy+7x^2y$ is about $3$ million, while $x^3−3xy+9x^2y^3$ is roughly $9$ billion. After a few cases, the idea appears and a smart kid may be able to construct the counter-examples him or herself, and see why such a simplification cannot happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that the proof of the independence of monomial functions is really simple; it is only quite cumbersome to write out precisely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side point, note that this proof relies on a somewhat subtle property of real numbers (their order), for a good reason: the statement fails in finite fields, even for one-variable polynomials. If you don't know about finite fields, let's say they are sets of &quot;numbers&quot; with rules for addition and multiplication that have the same basic properties as for usual numbers, but with only finitely many numbers. The point of this side note is to stress that your desire to prove the non-simplification statement is really sane: this is by no way obvious, and cannot be proved just by applying the usual rules of computation (factorization, distributivity, etc.) which also holds in finite fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side side note, I think this side note is a good example why it is nice for a teacher to have a much higher education than the level at which one teaches: it helps distinguishing subtle things from trivial ones and gives perspective.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-12-15T21:07:26.363" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10232" PostTypeId="2" ParentId="10226" CreationDate="2015-12-15T21:18:40.133" Score="2" Body="&lt;p&gt;The &quot;high level&quot; answer is that a polynomial algebra is a free commutative algebra generated by the indeterminates. But this would probably not be a very satisfying answer to 8th graders.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible discussion starts with the cliche that &quot;you cannot add apples and oranges.&quot; And you cannot &quot;add lengths and weights.&quot; But you can multiply them! After all, we do use &quot;foot$*$pound&quot; as a unit in physics (at least in the US), and so we could just as well talk about &quot;apple$*$orange&quot; if we wanted to. So we can combine $x$ and $y$ into a single unit $x*y=xy$. But $x+y$ is just kind of stuck out there with no place to go.  &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2015-12-15T21:18:40.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10233" PostTypeId="2" ParentId="10226" CreationDate="2015-12-15T21:26:48.897" Score="1" Body="&lt;p&gt;The main reason why you can or can't do things to algebraic expressions is because you can or can't do those things with numerical expressions. The rules for manipulating algebraic expressions are exactly the same rules for manipulating numeric expressions, with some extra restrictions introduced by the fact that you don't know what some of the numbers are (or that they represent all possible numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest talking through the ways that you can rearrange expressions made entirely of numbers without changing their values. A good start is to talk through different strategies to calculate expressions where &quot;like terms&quot; will make it easier to do the calculation, such as $3 + 2\times3 + 3\times7$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can give students an expression and ask them to rearrange it so that the &quot;answer&quot; is the same. Get many different students to talk through their rearrangement and why it is the same. After that, give them several different expressions which are similar and ask them to decide if they have the same answer without actually calculating the final result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you could take the same activities above and replace copies of the same number with a letter and ask the same question. This might help to explain what you can and can't do with algebraic expressions, including the inability to add unlike terms.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2015-12-15T21:26:48.897" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10234" PostTypeId="1" CreationDate="2015-12-15T23:27:05.173" Score="24" ViewCount="563" Body="&lt;p&gt;I have found that when I give problems that require multiple steps or ideas to solve, students often give up quickly and come to office hours begging for hints.  Sometimes I break up such problems into steps, with hints for each step; this enables the students to solve them, but (as a colleague pointed out) has the unintended consequence of teaching them (or, rather, reinforcing the lesson they learned in high school) that all solvable problems can be solved in 5 minutes.  What are some ways to teach students to keep plugging away at a problem even if they don't seem to be making progress?&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2016-02-07T20:26:29.620" Title="How to teach perseverance?" Tags="|undergraduate-education|student-motivation|general-pedagogy|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10235" PostTypeId="1" CreationDate="2015-12-15T23:32:07.583" Score="18" ViewCount="1734" Body="&lt;p&gt;It's very common in learning mathematical induction to prove statements like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ 0^2+1^2+2^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found that very frequently, on this sort of problem, when students get to the induction step they will write the induction hypothesis as &quot;$n^2= \frac{n(n+1)(2n+1)}{6}$&quot; and the goal to prove as &quot;$(n+1)^2= \frac{(n+1)(n+2)(2n+3)}{6}$&quot;, totally ignoring the ellispis &quot;$\cdots$&quot; and everything before it.  I have never been able to figure out where this mistake comes from.  More importantly, what can I say to help them overcome this mistake?&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="2187" LastEditDate="2015-12-16T05:44:33.073" LastActivityDate="2015-12-28T22:54:07.093" Title="Why do students only see the last term of a sum abbreviated with an ellipsis?" Tags="|mathematical-pedagogy|students-mistakes|induction|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10236" PostTypeId="2" ParentId="10235" CreationDate="2015-12-16T00:17:11.650" Score="13" Body="&lt;p&gt;Encourage your students to actually read the problem to themselves, similar to how they would if they were reading a book.  Starting from left to right, read the problem: &quot;Zero squared plus one squared plus two squared plus dot dot dot plus n squared equals...&quot; etc. I find that this not only reinforces that there is that &quot;...&quot; there and it needs to be addressed in further procedures, but it also helps students from making &quot;stupid mistakes&quot; like dropped signs etc.  You might be surprised how many students cannot actually read mathematical expressions/equations out loud, at least at a high school level where I teach.  Adding an auditory element to math can also help students who learn better by hearing (gardner's theory of multiple intelligences and such).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would not only advise them to do this themselves but also model it for them by doing it every time a problem is discussed in class.  If you use chalk/whiteboards it works very efficiently to say it out loud as you write it on the board.  You can even have students say it out loud and write exactly what they say, which can lead to some interesting things on the board.  Sometimes all I need to do is ask a confused student to read a problem out loud to themselves and they will instantly see the answer and go &quot;Ahhh :)&quot;&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2015-12-16T00:17:11.650" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10237" PostTypeId="2" ParentId="10235" CreationDate="2015-12-16T00:29:49.383" Score="7" Body="&lt;p&gt;Perhaps they are misinterpreting the ellipses in&#xA;$$ 0^2+1^2+2^2+\color{red}{\cdots}+n^2 = \frac{n(n+1)(2n+1)}{6}.$$&#xA;as meaning&#xA;\begin{eqnarray}&#xA;0^2 &amp;amp; = &amp;amp; \frac{n(n+1)(2n+1)}{6} = \frac{0(1)(1)}{6} = 0\\&#xA;1^2 &amp;amp; = &amp;amp; \frac{1(2)(3)}{6} = 1\\&#xA;&amp;amp; \color{red}{\cdots} &amp;amp;\\&#xA;n^2 &amp;amp; = &amp;amp; \frac{n(n+1)(2n+1)}{6}&#xA;\end{eqnarray}&#xA;So I hypothesize this is more a syntactic issue than it is a semantic issue.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-12-16T00:29:49.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10239" PostTypeId="2" ParentId="10226" CreationDate="2015-12-16T02:18:05.337" Score="2" Body="&lt;p&gt;Let's say we have this expression: $$2 + 3 + x + 2x + 2x^2 + 2x^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, we know that we can combine $2 + 3$ as $5$, but let's take a closer look. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We're going to break the $2$ and the $3$ up into smaller pieces. To make things easy, let's do this so that all of the pieces that we wind up with are exactly the same size: $1$ is an easy number to work with, so that's the size that our pieces should be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The $2$ breaks up in to&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$1 + 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the $3$ breaks up in to&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$1 + 1 + 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we put these groups of pieces together and get ready to add them, we have this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$1 + 1 + 1 + 1 + 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I count five $1$'s, which means that our&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$2 + 3$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which became&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$1 + 1 + 1 + 1 + 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is equal to $5$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you take things down to the most basic level, this is how addition works. You break things up into &lt;strong&gt;units&lt;/strong&gt; and then you count up the number of those units that you have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's do the same thing with $x + 2x$, but this time our smallest piece will be $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x + 2x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;becomes&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x + x + x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I count three $x$'s, and we write that result as $3x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next up is our $2x^2 + 2x^2$. Our pieces are going to be $x^2$ for this part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$2x^2 + 2x^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;becomes&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x^2 + x^2 + x^2 + x^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I count four $x^2$'s, and we write that result at $4x^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's put our parts together...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$5 + 3x + 4x^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can we combine, or add together, $3x$ and $4x^2$? In order for it to work, we'd need to be able to break each of them up into some identical parts, or units, and then count how many units that we have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We've seen that we can break the second thing up into $x$'s. We've seen that we can break the third thing up into $x^2$'s. Unfortunately, these units don't match. Maybe we can keep breaking things up until we get units that match.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can break $x$ up into &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac 12 x + \frac 12 x$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and I can break $x^2$ up into &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac 12 x^2 + \frac 12 x^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but we're still not going to find a match.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We could break these things up in to any size parts that we want, meaning that we could choose any unit size that we want to break up these terms in to, but we'll never wind up with a unit from $3x$ that matches a unit from $4x^2$. If we can't do that, we can't add them together.&lt;/p&gt;&#xA;" OwnerUserId="6106" LastActivityDate="2015-12-16T02:18:05.337" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10241" PostTypeId="2" ParentId="10234" CreationDate="2015-12-16T08:55:08.647" Score="15" Body="&lt;p&gt;I believe that my students can learn best when they display perseverance (sometimes called grit). I'll discuss how I teach, model, and encourage mathematical perseverance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Tell a story about perseverance&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I speak to students who are struggling, especially if they failed the class before, I tell them the story of Ella, one of my algebra students. When Ella took my algebra class for the first time, she struggled with math and eventually gave up on passing the class. Later, she took the class with me a second time. I must confess that I had my doubts--but she came to me and said &quot;this time is going to be different.&quot; I told her that I was glad to hear it, but she must have heard the doubt in my voice, because she felt the need to explain: &quot;I have a notebook,&quot; she said. &quot;In it I'm going to write down every single problem, carefully and step-by-step.&quot; Sure enough, whenever I saw her doing her homework (which was on the computer, by the way--no notebook needed), she carefully wrote down every problem step-by-step. As the weeks went by, she worked through all the sections that she struggled with before, eventually acing material that had caused her to quit. I'm delighted to report that Ella passed the algebra class the second time with an A. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can we can get more students to do display such perseverance? How can we get students to learn from their mistakes and struggles and do better next time?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Communicate the value of hard work&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes in mathematics, we worship genius and intellect. With good reason: the math we study and love is the result of mathematical insights from great minds like Newton and Gauss. We too easily forget that mathematical insight is also the fruit of hard work--Newton didn't invent calculus in a single moment of brilliant insight, rather he spent years of study developing calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's &lt;a href=&quot;http://www.scientificamerican.com/article/the-secret-to-raising-smart-kids1/&quot;&gt;strong evidence that telling students they are smart is actually very detrimental to their education&lt;/a&gt;. If we give students the idea that success in math depends on their intelligence, we're telling them that their success comes down to how lucky they got in the genetic lottery. Although intelligence doubtless has a strong effect on students' success in math, we should encourage students to focus on what they can change--how they work--and compliment them on their hard work when they succeed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Model perseverance&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I sometimes insert errors into my lectures--sometimes even intentionally--to make sure they are paying attention, but also to demonstrate how to deal with mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an algebra class, you might write $x^2-4 = (x-2)^2$. Keep going as if nothing is wrong--they should catch you soon. Ask them: why this is false? How should we have proceeded?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a calculus class, begin &quot;solving&quot; $\lim_{x \to 0} \frac{x^2}{e^x}$ using repeated applications of L'Hopital's rule. You students probably won't realize that L'Hospital's rule doesn't apply, so you may get all the way to &quot;solution&quot;. Then tell them there is a mistake somewhere in your work. Ask them to find it and then discuss the conditions for applying L'Hopital's rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I ask students questions, I occasionally write what they say on the board even if it's wrong. I'll play along a little while until another student catches the mistake. Then I'll ask: who's right? How do we know?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;This TED talk conclusively proves it&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you'd like a very nice introduction to the research on grit and its important role in fostering student, check out &lt;a href=&quot;http://www.ted.com/talks/angela_lee_duckworth_the_key_to_success_grit&quot;&gt;Angela Lee Duckworth's TED Talk&lt;/a&gt; on the subject.&lt;/p&gt;&#xA;" OwnerUserId="182" LastActivityDate="2015-12-16T08:55:08.647" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10242" PostTypeId="2" ParentId="7274" CreationDate="2015-12-16T09:17:49.473" Score="8" Body="&lt;p&gt;This suggestion uses integrals and the fundamental theorem of calculus. If that is not yet known to the students, my example will not fit so well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Prime_number_theorem#Prime-counting_function_in_terms_of_the_logarithmic_integral&quot; rel=&quot;noreferrer&quot;&gt;The story says&lt;/a&gt; that Dirichlet (in a note written on a paper) around 1838 suggested to Gauss that the number of primes less than or equal to $x$, usually denoted $\pi(x)$, is for large $x$ approximately given by the integral&#xA;$$&#xA;\int_2^x\frac{1}{\ln t}\,dt.&#xA;$$&#xA;On the other hand, the prime number theorem states that&#xA;$$&#xA;\lim_{x\to+\infty}\frac{\pi(x)}{x/\ln x}=1.&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Exercise:&lt;/strong&gt; Show that the integral suggested by Dirichlet is indeed a good approximation, in the sense that&#xA;  $$&#xA;\lim_{x\to+\infty}\frac{\ln x}{x}\int_2^x\frac{1}{\ln t}\,dt=1.&#xA;$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Comment:&lt;/strong&gt; This limit appeared on &lt;a href=&quot;https://math.stackexchange.com/q/1543955/97236&quot;&gt;math.SE&lt;/a&gt;, and it is indeed also solved easily using integration by parts. Nevertheless, I think it is a nice application of the l'Hospital rule.&lt;/p&gt;&#xA;" OwnerUserId="6113" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-12-16T09:17:49.473" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10243" PostTypeId="2" ParentId="10157" CreationDate="2015-12-16T13:58:10.477" Score="2" Body="&lt;p&gt;I think it doesn't matter for linear systems. However, for non-linear systems it's a different matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the equations $x^2+y=3$ and $y^2-x^2=-1$. It would be ambiguous to write $x=\pm\sqrt{2},\pm\sqrt{5}, y=1, -2$ because the values $x=\sqrt{2}, y=-2$ do not satisfy the given system. Using ordered pairs $(x,y) = (\pm\sqrt{2},1), (\pm\sqrt{5},-2)$ is clearer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps using ordered pairs for linear systems instills &quot;good practice&quot; for the future.&lt;/p&gt;&#xA;" OwnerUserId="6114" LastActivityDate="2015-12-16T13:58:10.477" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10244" PostTypeId="1" CreationDate="2015-12-16T14:46:25.213" Score="21" ViewCount="1137" Body="&lt;p&gt;I have been teaching introductory group theory to undergraduates. We reached cosets several weeks ago, but the combination of the textbook, my explanations and various practice questions has left the students still very uncertain about cosets. I've pretty much run out of ideas for how to try and help them.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is it about cosets that students find hard to understand? What ways of thinking about cosets make sense to them?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-05-16T09:14:44.677" Title="What makes cosets hard to understand?" Tags="|undergraduate-education|abstract-algebra|" AnswerCount="10" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10245" PostTypeId="2" ParentId="10244" CreationDate="2015-12-16T17:27:35.590" Score="5" Body="&lt;p&gt;This is not an answer to your question of what makes cosets difficult,&#xA;but whatever it is, visualization may help.&#xA;For example, &#xA;Nathan Carter's &#xA;&lt;em&gt;Visualizing Group Theory&lt;/em&gt; presentation &#xA;(&lt;a href=&quot;http://groupexplorer.sourceforge.net/TalkWithoutWords.ppt&quot; rel=&quot;noreferrer&quot;&gt;Download .ppt&lt;/a&gt;)&#xA;illustrates cosets in $A_4$ over several slides. Here is one slide:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ID6WW.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ID6WW.jpg&quot; alt=&quot;Cosets&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2015-12-16T17:27:35.590" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10246" PostTypeId="2" ParentId="10235" CreationDate="2015-12-16T17:35:45.470" Score="19" Body="&lt;p&gt;I suspect that the issue is not so much the ellipsis &lt;em&gt;per se&lt;/em&gt; but a problem with notation in general, and in particular with the correct use of the equals sign.  At the risk of repeating &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1040/how-to-cure-students-from-the-idea-that-root-and-squaring-are-identity-operators/1058#1058&quot;&gt;what I wrote in this answer&lt;/a&gt;, students often regard the equals sign not as a symbol meaning &amp;quot;these two expressions are the same&amp;quot; but rather as a symbol separating a question (on the left) from an answer (on the right).&lt;/p&gt;&#xA;&lt;p&gt;To illustrate what I mean, let me change contexts for a moment to something that may seem at first to be unrelated.  Think of how students often write down their work when solving a story problem:  If the problem is &lt;code&gt;&amp;quot;Pens cost 75 cents each and pencils cost 20 cents each.  If Andy spends a total of $5.15 on a combination of 12 pens and pencils, how many of each did he buy?&amp;quot;&lt;/code&gt;, it is not at all uncommon to see student work like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pens = x&lt;/p&gt;&#xA;&lt;p&gt;Pencils = 12-x&lt;/p&gt;&#xA;&lt;p&gt;Pens = 75x&lt;/p&gt;&#xA;&lt;p&gt;Pencils = 20(12-x)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now all of us cringe at the fact that the student does not distinguish between &amp;quot;number of pencils&amp;quot; and &amp;quot;cost of pencils.&amp;quot;  We say things like &amp;quot;'Pencils' is not a number!&amp;quot; and &amp;quot;How can Pens be equal to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and also equal to &lt;span class=&quot;math-container&quot;&gt;$75x$&lt;/span&gt;?&amp;quot;  But students do it anyway.  What are they thinking?  What are they &lt;em&gt;doing&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;p&gt;What they are doing is using &amp;quot;Pens = &amp;quot;  as a &lt;em&gt;label&lt;/em&gt; for a formula.  When they write &amp;quot;Pens = &amp;quot; they internally verbalize it as &amp;quot;The formula for pens is...&amp;quot;  They are essentially using the equals sign like a colon -- what they mean is something like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Quantities&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Pens:  x&lt;/p&gt;&#xA;&lt;p&gt;Pencils:  12-x&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Cost&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Pens:  75x&lt;/p&gt;&#xA;&lt;p&gt;Pencils:  20(12-x)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;but they omit the headers (more precisely, the headers are &amp;quot;mentally present&amp;quot; but not written down, because you don't 'do' anything with them) and use an equals sign instead of a colon.&lt;/p&gt;&#xA;&lt;p&gt;Notice that &lt;em&gt;if a student wrote it down this way&lt;/em&gt; (with the headers and the colons) &lt;em&gt;we would probably be inclined to regard their work as correct&lt;/em&gt;.  So the issue is really &lt;em&gt;not&lt;/em&gt; that they do not understand the problem, but that they do not know an appropriate way to write down their understanding.&lt;/p&gt;&#xA;&lt;p&gt;Okay, let me bring this back to the actual OP's question.  I think when students write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$n^2 = \frac{n(n+1)(2n+1)}{6}$$&lt;/span&gt;&#xA;what they are thinking is something like this:  There is a list of formulas.  Each formula on the list is named or labeled by the last term in a sum.  So, there is the &lt;span class=&quot;math-container&quot;&gt;$1^2$&lt;/span&gt; formula, the &lt;span class=&quot;math-container&quot;&gt;$2^2$&lt;/span&gt; formula, the &lt;span class=&quot;math-container&quot;&gt;$3^2$&lt;/span&gt; formula, and so on.  If that list of formulas were written out, it might look something like this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1^2:  \frac{1 \cdot 2 \cdot 3}{6}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2^2:  \frac{2 \cdot 3 \cdot 5}{6}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3^2:  \frac{3 \cdot 4 \cdot 7}{6}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and the &lt;span class=&quot;math-container&quot;&gt;$n^{th}$&lt;/span&gt; formula in the list would be:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$n^2:  \frac{n(n+1)(2n+1)}{6}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But instead of using a colon (:), they use an equals sign (=).  The fact that &lt;span class=&quot;math-container&quot;&gt;$2^2$&lt;/span&gt; does not equal &lt;span class=&quot;math-container&quot;&gt;$\frac{2 \cdot 3 \cdot 5}{6}$&lt;/span&gt; does not bother them at all, because (a) when they write down &lt;span class=&quot;math-container&quot;&gt;$2^2$&lt;/span&gt; they do not literally mean &lt;span class=&quot;math-container&quot;&gt;$2^2$&lt;/span&gt; but just &amp;quot;the &lt;span class=&quot;math-container&quot;&gt;$2^2$&lt;/span&gt; formula&amp;quot;, i.e. the formula that ends with / is labelled by &lt;span class=&quot;math-container&quot;&gt;$2^2$&lt;/span&gt;; and (b) they are not thinking of &amp;quot;equals&amp;quot; as meaning &amp;quot;they are the same thing&amp;quot;, but simply as indicating a correspondence of some sort.&lt;/p&gt;&#xA;&lt;p&gt;I think a solution to this problem might lie not in fighting the tendency but in recognizing that students are looking for some way to label the &lt;span class=&quot;math-container&quot;&gt;$n^{th}$&lt;/span&gt; formula in a list, and &lt;em&gt;that impulse is not a bad one&lt;/em&gt;.  It's just that they don't know a correct way to do it, so they are grasping at whatever notation is available and misusing it.  So let's teach them a &lt;em&gt;proper&lt;/em&gt; way to write what they want to say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(1):  1^2 = \frac{1 \cdot 2 \cdot 3}{6}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(2):  1^2 + 2^2 = \frac{2 \cdot 3 \cdot 5}{6}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(3):  1^2 + 2^2 + 3^2 = \frac{3 \cdot 4 \cdot 7}{6}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course students will not know how to read such statements without being taught to do so.  Notation does not speak for itself!  Some of the key things to point out about this notation are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; means &amp;quot;the &lt;span class=&quot;math-container&quot;&gt;$n^{th}$&lt;/span&gt; property in the list&amp;quot; -- it is the label for each formula&lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;label&lt;/em&gt; for each formula is separated from the formula itself by a colon&lt;/li&gt;&#xA;&lt;li&gt;The formula is an &lt;em&gt;equation&lt;/em&gt;, so it contains an equal sign separating two expressions&lt;/li&gt;&#xA;&lt;li&gt;The expression on the left side of each equation is not a single term but a sum.&lt;/li&gt;&#xA;&lt;li&gt;The expressions on the opposite sides of the equals sign are equal to each other; they should be encouraged to actually compute them and verify that this is so (e.g. to compute both &lt;span class=&quot;math-container&quot;&gt;$1^2 + 2^2 + 3^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{3 \cdot 4 \cdot 7}{6}$&lt;/span&gt; and confirm that they are both equal to &lt;span class=&quot;math-container&quot;&gt;$14$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Give them plenty of practice writing out statements like this by giving them formulas like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(n):  n! - (n-1)! = (n-1)(n-1)!$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(n):  1 + 2 + 4 + \dots + 2^n = 2^{n+1}-1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and asking them to write out the first several instances of the formula, replacing &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; with natural numbers, and verify that they are true.  (You might also want to mix in some formulas that are &lt;em&gt;false&lt;/em&gt;, or that are true only for the first few values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, and have them determine whether for a given value of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;,  &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; is true or false.)  This will help them build their comfort with this kind of notation, and with the distinction between an &lt;em&gt;expression&lt;/em&gt;, an &lt;em&gt;equation&lt;/em&gt;, and a &lt;em&gt;labeled formula&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;TL;DR version of the above:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Students are using the expression &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$n^2 =$&lt;/span&gt;&amp;quot; as a shorthand for labelling &amp;quot;the &lt;span class=&quot;math-container&quot;&gt;$n^{th}$&lt;/span&gt; formula in the list is: &amp;quot;.  If you teach them an &lt;em&gt;appropriate&lt;/em&gt; alternative way to write that label, they may stop misusing notation the way that they are.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-12-16T17:35:45.470" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10247" PostTypeId="1" AcceptedAnswerId="10248" CreationDate="2015-12-16T19:04:40.203" Score="24" ViewCount="2769" Body="&lt;p&gt;I've recently taken on the task of helping out in my school's Math Center. The courses I assist in range from Algebra to Calculus. While I'm younger (in my 20's), most of the students at the school are 40+.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first students I worked with were taking a Calculus Prep class. Teaching them was fairly easy - most of what I did was show them that the steps required to solve a problem were steps they already knew how to do and were familiar with - the problem was just being presented in a unique manner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I recently got tasked with helping out students taking the lowest level math we offered - and I found it surprisingly difficult. Teaching a younger child math, you can tell them that a rule is a rule, and that they must accept it in math and go forward. From there, they might start to test that rule, and see that it applies in all cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, when teaching adults, I've found that I can't just tell them &quot;this is the way it's done, get used to it.&quot; I have students that are struggling with topics like adding and subtracting negatives, or determining if a function is odd or even - essentially why $-(x^2)$ is different from $(-x)^2$. I can try to explain negatives using a number line, or walk them through PEMDAS again, but it just doesn't seem to stick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems like children just need to know &lt;em&gt;how&lt;/em&gt;, but adults need to know &lt;em&gt;why&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any advice for teaching simple concepts and rules to adults, specifically working with negatives or proper application of PEMDAS?&lt;/p&gt;&#xA;" OwnerUserId="6110" LastEditorUserId="6110" LastEditDate="2015-12-16T19:46:15.483" LastActivityDate="2015-12-19T17:41:27.270" Title="How to Teach Adults Elementary Concepts" Tags="|undergraduate-education|algebra|tutoring|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10248" PostTypeId="2" ParentId="10247" CreationDate="2015-12-16T20:52:55.033" Score="32" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;But, when teaching adults, I've found that I can't just tell them&#xA;  &quot;this is the way it's done, get used to it.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Good!  Students (at &lt;em&gt;any&lt;/em&gt; age) should &lt;em&gt;never&lt;/em&gt; be satisfied with &quot;This is the way it's done, get used to it&quot;, and teachers should never give that as an explanation.  More than any other academic subject, &lt;em&gt;everything in math should make sense.&lt;/em&gt;  Scientific truth is subject to experimental verification or falsification, but in mathematics &quot;being reasonable&quot; is &lt;em&gt;literally&lt;/em&gt; the only criterion for distinguishing between true and false.  If something doesn't make sense, it's being taught wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All mathematics teaching, regardless of its level or the specific course content, ought to be largely about the art of sense-making.  The teacher’s job is to ask questions, and in so doing to model for students how to ask their own questions:  not only how to solve a particular kind of problem, but also why the problem is worth solving, why a given method works, whether there are other methods that might work as well or better, and so on.  Without that, mathematics is little more than a bag of rules and procedures to be memorized and (probably) forgotten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, you asked:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Any advice for teaching simple concepts and rules to adults, specifically working with negatives or proper application of PEMDAS?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As I've said above, I don't think adults are any different from younger students in this regard: they all deserve coherent explanations that explain &lt;em&gt;why&lt;/em&gt;, not just &lt;em&gt;how&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For  specific issues having to do with negative numbers, see (for example) &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5794/how-to-explain-that-a-negative-number-multiplied-by-a-negative-number-is-a-posit&quot;&gt;How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2497/how-to-teach-someone-that-3-4/2523#2523&quot;&gt;How to teach someone that $-3&amp;gt;-4$?&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/8011/when-x-is-positive/9882#9882&quot;&gt;When $-x$ is positive&lt;/a&gt;, all of which deal with conceptions &amp;amp; misconceptions about signed integers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For PEMDAS, see &lt;a href=&quot;https://math.stackexchange.com/questions/1385549/what-is-the-reason-behind-the-current-order-of-operations-pemdas&quot;&gt;https://math.stackexchange.com/questions/1385549/what-is-the-reason-behind-the-current-order-of-operations-pemdas&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You didn't ask about fractions, but see &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7837/how-to-explain-the-flipping-of-division-by-a-fraction&quot;&gt;How to explain the flipping of division by a fraction?&lt;/a&gt; for examples of that.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2015-12-16T20:52:55.033" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10249" PostTypeId="2" ParentId="10247" CreationDate="2015-12-16T21:34:29.953" Score="11" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Any advice for teaching simple concepts and rules to adults,&#xA;  specifically working with negatives or proper application of PEMDAS?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This will be an adjunct to mweiss' answer, which I fully agree with (please go upvote that), but this will be a little too long for a comment, I think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When some people go to teach these pre-algebra arithmetic courses for the first time, they frequently think &quot;this will be easy&quot;, but it's not; it's likely to be the hardest teaching experience, because you really need all the details right; the students won't be able to meet you halfway or fill in gaps, so you really need near-perfect explanations. There are details that you might perform on sight without thinking; perhaps you simply learned them as rules to follow without reason. But this is a terrific opportunity to realize that that's generally &lt;em&gt;incorrect&lt;/em&gt;, and should take the time to investigate, reflect, and really think through first principles and come up with &lt;em&gt;good explanations&lt;/em&gt; for all of them. (Again, mweiss is on the right track by giving you links to explanations for particular topics; you're just going to have to dig in on a case-by-case basis and thoroughly think them through.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With a small number of (axiomatic) exceptions: The order-of-operations is indeed an arbitrary grammatical convention that we do have to accept. Generally we also accept the meaning/properties of the equals sign (=), and also a small number of properties about real numbers (commutativity, associativity, and distribution). We really should be able to explain anything else in the course based on those original principles (given proper definitions of other new objects or symbols). Look for opportunities to explicate this, and connect later work back to these founding principles. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other thing that I will hypothesize, based on my experience, is that some proportion of the population you're dealing with is likely to have some learning disabilities like dyslexia, dyscalculia, vision problems, ADD, etc. This is likely explanatory to them being in this class situation in the first place. It can be frustrating that it seems like they just flat-out can't remember material from day to day. In this regard you'll just have to be patient and review whatever material they ask about whenever it comes up subject to your available time for the job. There's no silver bullet, and like a doctor you have to realize that you can't save everyone, no matter how much you may want to. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While these classes are in some sense the most challenging due to needing really high-quality, fully-detailed explanations of things you may not normally think about, I also find them among the most rewarding, and the students among the most appreciative when you respond to their questions respectfully and thoughtfully, and treat them like colleagues. I feel like I've really made some discoveries and vastly improved my teaching and writing because of teaching these classes.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-12-16T21:34:29.953" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10250" PostTypeId="2" ParentId="10247" CreationDate="2015-12-16T22:14:44.923" Score="5" Body="&lt;p&gt;Break it down into it's simplest level, go over it and over it. Start with small numbers, make them do it until they can see the patterns. Gradually increase the difficulty, and when they get flustered(they will!), revert to a comfortable level and work back up again. As for the specifics:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PEMDAS - If you do things in a different order, you will get a different answer. It is important that we can tell what order to do them in, so we get the same answer. Like if you put the lid on a bottle BEFORE you put the water in.&#xA;Additonal: Written math is like any language - when me(you, the teacher) or anyone else writes equations, we mean a very specific thing, so everyone can get the same (correct) answer. If I meant a different order than PEMDAS, I would have use a parenthesis, so that my meaning was clear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-(x)&lt;sup&gt;2&lt;/sup&gt; vs (-x)&lt;sup&gt;2&lt;/sup&gt; Make this an extension of PEMDAS. The order is important, and show them an example, with it being different. They are saying &quot;why&quot; because they don't understand PEMDAS, exponents, or negative numbers (as you mentioned).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Negative numbers: Multiplying a positive by a negative is easy. Say a candy bar costs 1 dollar. I buy three of them. On my balance sheet, I have three negative values. 3 * -1 is negative three. Start from there and do bigger numbers. Bigger number cause irrational fear! You could immediately ask some students, what 3 * -1000 is, and many would throw up there hands in frustration! Work them through it till they see the pattern and are comfortable with it. Expect them to forget and need to be reminded, over and over.&#xA;-3 * -3 is more abstract, you could try something like buying three fewer candy bars, but it's probably easier to just say at this point they &quot;undo&quot; each other, and work them through it until they just get used to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The people you are teaching have poor math skills as adults because of attitudes and fears, not a fundamental lack of intelligence. Those attitudes are going to be more challenging than simply teaching material would ever be on its own. Listen to what they ask, but take everything with a grain of salt, they just don't want to look/feel dumb.&lt;/p&gt;&#xA;" OwnerUserId="6118" LastActivityDate="2015-12-16T22:14:44.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10251" PostTypeId="2" ParentId="10244" CreationDate="2015-12-16T23:09:59.350" Score="8" Body="&lt;p&gt;One of the most famous mathematics education theoretical constructs, namely APOS, offers an answer for this question, in fact, the exact same question. A quick search leads you to some papers and even one or two books. Here is a very short summary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;APOS stands for Action, Process, Object, Schema. In simple terms, the difficulty relies on coming from forming (as an action/process) a coset to conceptualizing it as an object (O in the APOS). You can observe, more or less the same difficulty in early number learning where children need to move from counting say 5 items to conceptualize 5 as an object. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2015-12-16T23:09:59.350" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10252" PostTypeId="2" ParentId="10226" CreationDate="2015-12-17T13:36:33.767" Score="4" Body="&lt;p&gt;I start by asking them how they would order wall paper or floor tile. &quot;Square Feet&quot;. &#xA;Then we talk about the third dimension, height. This gives us volume as opposed to just area. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I take a step back and ask what the distance is to school. Is its units 'square feet'? Of course not. It's measured in feet or miles. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I then ask how they'd add 12 feet to 9 square feet. At that point they start to see the question is absurd. My 2000 sq ft house and the 2000 ft I walk to the end of my street can't be combined in any way. But if your house is 2500 sq ft, I can add them, and say that combined, we'd have 4500 sq ft of living space. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's then a small move to explain how X, x^2, and x^3 represent length, area and volume. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-12-17T13:36:33.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10253" PostTypeId="2" ParentId="10244" CreationDate="2015-12-17T16:42:10.733" Score="17" Body="&lt;p&gt;I strongly suspect the difficulty is not with cosets specifically, but with working with equivalence relations generally, especially when combined with objects that they have only recently become acquainted with.  So apologies in advance that this answer deals more generally with equivalence relations and my opinion that they often need more coverage than they get before cosets, rather than with cosets themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask a student the following question:  Is $\frac{534389}{832323} = \frac{3740723}{5826262}$?  Give them a calculator that displays 6 decimal places (or tweak the example to make the fractions agree to a larger number of places), and how many would look at their calculators and conclude they are equal?  How many would realize that they could check their answer by cross-multiplying, but then give up because their calculator could not store enough digits for the multiplications?  How many would realize that you have to cross-multiply and that it is immediate that the two products do not agree in their ones digit, so they cannot be the same?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this example illustrates the varying levels of understanding a student may have for identifying equivalence classes.  In the first case, the student knows how to distinguish equivalence classes when they are &quot;obviously&quot; different or when a certain test applies, but they do not have a general algorithm to distinguish equivalence classes.  In the second case, they have a sound theoretical understanding of what differentiates equivalence classes, but lack the tools to test identity of classes as quickly as possible.  In the third case, the student actually understands the theory and has the ability to apply other theoretical concepts for more effective calculations and deeper understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, for most students, it is quite hard to get to that final stage and often takes a long time to get there, if they make it there at all.  If they cannot solve this simple problem above at least at the second level (and ideally at the third level), then asking them to be able to manipulate cosets, which involve algebraic objects that are much less familiar to them than fractions, often becomes an exercise in futility.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2015-12-17T16:42:10.733" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10254" PostTypeId="2" ParentId="10226" CreationDate="2015-12-17T16:59:35.493" Score="1" Body="&lt;p&gt;I am going to disagree (slightly) with Benoit Kloeckner's observation that the OP's implicit claim is false in finite fields.  If we are going to allow ourselves the sophistication to look at fields other than $\mathbb{R}$, we should also oblige ourselves to distinguish between a &lt;em&gt;polynomial&lt;/em&gt; and a &lt;em&gt;polynomial function&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Give any ring $R$, a &lt;em&gt;polynomial&lt;/em&gt; in $R$ is an element of the ring $R[x]$.  A polynomial can be written uniquely as a finite sum of terms of the form $a_n x^n$, where $a_n \in R$ and $x$ is a formal variable.  A &lt;em&gt;polynomial function&lt;/em&gt; is what you get when you interpret a polynomial as a function on $R$ in the obvious way.  A polynomial is an expression, but a polynomial function is (at the formal level) a set of ordered pairs.  Every polynomial naturally induces a polynomial function, but they remain different kinds of objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we talk about &quot;simplifying polynomials&quot; we are referring to operations in $R[x]$.  For example, the fact that $(x-1)(x+1)(x+2)$ can be simplified to $x^3+2x^2-x-2$ is a consequence of the way the ring operations are defined in $R[x]$ (assuming $R$ is $\mathbb{Z}$, $\mathbb{R}$, or some other ring in which the symbols $1$ and $2$ have a natural interpretation).  The fact that $x^3+2x^2-x-2$ cannot be further reduced to an expression with fewer terms is also true in $R[x]$ for any such $R$.  Now, it turns out (surprisingly!) that if $R=\mathbb{Z}/(3)$ this polynomial induces the exact same function as does $2x^2-2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But (and here is where my disagreement with Benoit lies) despite the fact that $x^3+2x^2-x-2$ and $2x^2-2$ are identical &lt;em&gt;when considered as functions over&lt;/em&gt; $\mathbb{Z}/(3)$, they are still &lt;em&gt;different polynomials&lt;/em&gt;.  They have different degrees, different number of terms, and different leading coefficients, and they generate different ideals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So while it is true that $t^3+2t^2-t-2 = t^2-2$ may be true for all $t \in \mathbb{Z}/(3)$, I would not describe the replacement of the left-hand side by the right-hand side as &quot;simplification&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having gone through all of these preliminaries, let's go back to the OP's original question, which we can now tease apart into two different questions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;p&gt;Why is it that &quot;unlike terms&quot; in a polynomial can't be combined into a single term?&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Why is it that when working over the real numbers, two different polynomials never induce the same function?&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To answer the first question, I would turn it back onto the asker (or his students):  Why do you think they &lt;em&gt;should&lt;/em&gt; be combined?  What do you think they should be combined into?  The fact that like terms (e.g. $5x^2y + 3 x^2y$) can be combined (continuing the example, into $8x^2y$) depends on the distributive property, but the OP does not want an explanation that has to do with formal properties.  So that means we first need an informal explanation for why you &lt;em&gt;can&lt;/em&gt; combine like terms; then we can try to see why the same reasoning does not work for unlike terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Informally, &quot;combining like terms&quot; (as in our example $5x^2y = 3x^2y = 8 x^2y$) can be explained by noting that five &lt;em&gt;things&lt;/em&gt; plus three &lt;em&gt;things&lt;/em&gt; equals eight &lt;em&gt;things&lt;/em&gt;, regardless of what the 'things' are.  Five bags of 100 marbles, plus three bags of 100 marbles, add up to 8 bags of 100 marbles. Five 12-packs of bottled water plus three 12-packs of bottled water add up to eight 12-packs of bottled water.  In this case, the &lt;em&gt;things&lt;/em&gt; are expressions of the form $x^2y$.  If you have five of them and three of them, then you have eight of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you have &lt;em&gt;unlike&lt;/em&gt; things -- say, five bags of 100 marbles and three 12-packs of bottled water -- what can you combine them into?  There are eight &quot;somethings&quot;, but the &quot;somethings&quot; cannot be given a simple name.  Likewise, if you want to combine $5x^2y+3xy^2$, there may be a strong temptation to want to say that there are 8 &quot;somethings&quot;, but how can you say what the &quot;somethings&quot; are?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That, at least, is how I would &lt;em&gt;informally&lt;/em&gt; address the &lt;em&gt;formal&lt;/em&gt; question of why unlike terms in a polynomial cannot be combined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which brings us to the second question:  Why is it that two different polynomials over the reals never induce the same function?  As has been noted, this is a rather special property of the reals.  It is false when working over any finite ring; it is true when working over any infinite ring that is also an integral domain.  The most general conditions under which this property holds are given in the answers to &lt;a href=&quot;https://mathoverflow.net/questions/160986/rings-for-which-no-polynomial-induces-the-zero-function&quot;&gt;this question on MathOverflow&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-12-17T16:59:35.493" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10255" PostTypeId="1" AcceptedAnswerId="10256" CreationDate="2015-12-17T19:49:42.990" Score="25" ViewCount="701" Body="&lt;p&gt;I am a volunteer math tutor.  My student is in fifth grade.  He was evaluated in August and found to be at the first grade level in math.  (His reading is very close to grade level.)&lt;/p&gt;&#xA;&lt;p&gt;He has epilepsy, memory problems and ADD, and has made slow but good progress.&lt;/p&gt;&#xA;&lt;p&gt;This week I discovered he has a right-left problem that I wasn't aware of before.  We were doing some color coding and there was a need to write the word &amp;quot;Red&amp;quot; on the board to label something.  He wrote a backwards R, crossed it out, and said he couldn't do that.  I asked if he'd like to write it lower case, and then he did, more happily.  That was the first time I observed a left-right reversal.&lt;/p&gt;&#xA;&lt;p&gt;The next day we were working on a multiplication activity, laying out chips in an array to model multiplication. We rolled the dice and got 7 * 3.  He arranged the chips on the table in a 7 by 3 grid.  Then I said, let's imagine that this shows us the apple trees a farmer planted.  Let's say the farmer planted three rows of seven apple trees.  He said, &amp;quot;There are 21 trees.&amp;quot; (He's very proud of having learned how to multiply -- which we accomplished by skip-counting.)&lt;/p&gt;&#xA;&lt;p&gt;I wrote the number 21 on a piece of paper, and asked him to multiply by 100.  I reminded him of the shortcut we had done the day before, drawing two zeros after the number.  He did that fine.  I asked if he wanted to put the comma in, and he placed it wrong initially, so I helped him with that.  Here's what the final result looked like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;2,100&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then I asked him to read the number out loud.  He made two attempts:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;One thousand, two hundred&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;One hundred, two thousand&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I checked to make sure he wasn't trying to be funny -- he wasn't.&lt;/p&gt;&#xA;&lt;p&gt;Then I wrote a number in words, and asked him to write the number.  The text number was&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thirty-two thousand, six hundred forty-seven&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here's what he wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;000,032,647&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When I suggested that the leading zeros aren't needed, and that erasing them wouldn't change how big the number is, he said, &amp;quot;No, don't erase them, I need them or I'll feel very confused.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Now, we did similar exercises about a month ago without any trouble.  He was okay with numbers below a million.  I noticed that past a million, all the digits started to swim around in his visual field.&lt;/p&gt;&#xA;&lt;p&gt;It seems the right-left reversal is a symptom that comes and goes.  But I need to find a way he can remind himself that numbers get bigger to the left.  But without using the word &amp;quot;left,&amp;quot; and remembering that anything involving right and left is liable to get confused.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I think I need a simple graphic mnemonic he can use to remind himself which side the ones column goes on, and which direction we go in order to get bigger numbers.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Suggestions?  Please, nothing too fancy.  This is a student who needs simple procedures.  When his working memory was measured, it was found to be at the 6th percentile for boys his age.&lt;/p&gt;&#xA;&lt;p&gt;What he can reliably do now: he can add one, two or three digit numbers together vertically with pencil and paper.  He can carry reliably.  He can add two digit numbers mentally in the context of the game, as long as there's no carrying.  If there's carrying, he usually prefers to use pencil and paper.&lt;/p&gt;&#xA;&lt;p&gt;He can also plot a number up to one thousand fairly reliably on a number line, and also go the other direction.  It seems that once we get over three digits, things start to go haywire for him.&lt;/p&gt;&#xA;" OwnerUserId="4956" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-12-17T21:27:02.773" Title="Tutoring elementary student who reverses left and right" Tags="|tutoring|disability|place-value|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="10256" PostTypeId="2" ParentId="10255" CreationDate="2015-12-17T21:27:02.773" Score="15" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I think I need a simple graphic mnemonic he can use to remind himself&#xA;  which side the ones column goes on, and which direction we go in order&#xA;  to get bigger numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Suggestion: Start making routine use of the &lt;em&gt;decimal point&lt;/em&gt;, and perhaps instruct the student to always write that as a guidepost; note that it always goes at the end of an integer (to the right), in the same place as a period in an English sentence (thereby leveraging his intuition in that realm). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has the advantage that it's not too distracting to other readers if he does it all the time (e.g., many calculators do this; as opposed to the lead zeroes thing). He can use this as a flag about where the &quot;ones&quot; place is (immediately next to it). If commas are desired, he can be instructed to always count from this anchor point. And then hopefully later on he'll have a head start in learning about decimal fractions that come after the point, if need be.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-12-17T21:27:02.773" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10257" PostTypeId="2" ParentId="10226" CreationDate="2015-12-18T03:17:59.477" Score="0" Body="&lt;p&gt;Suppose $x$ and $y$ represent lengths, then terms can represent geometric shapes and explain why unlike terms cannot be added together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you get when you add a square and two more identical squares? Three squares. So $x^2+2x^2=3x^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you get when you add a square and a rectangle? A Shape that cannot necessarily be defined in terms of either the original shapes. And so $x^2+xy$ cannot be simplified into one term. Although in this case it could make a new rectangle with length $x$ and width $x+y$, so $x(x+y)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do  you get when you add a square ($x^2$) plus three rectangular prisms ($xy^2$) plus four rectangles ($xy$) plus 7 different rectangular  prisms ($x^2y$)? &lt;/p&gt;&#xA;" OwnerUserId="6101" LastActivityDate="2015-12-18T03:17:59.477" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10258" PostTypeId="2" ParentId="8088" CreationDate="2015-12-18T05:23:40.570" Score="4" Body="&lt;p&gt;In 2007, Ohio passed &lt;a href=&quot;http://codes.ohio.gov/orc/3333.163&quot; rel=&quot;nofollow&quot;&gt;Revised Code 8333.163&lt;/a&gt;, which required the Ohio higher education system to grant students award credit for passing advanced placement (AP) exams. In 2008, the Ohio university system issued &lt;a href=&quot;https://www.ohiohighered.org/sites/ohiohighered.org/files/Directive_2008-10.pdf&quot; rel=&quot;nofollow&quot;&gt;Directive 2008-010&lt;/a&gt;, implementing the law. The directive specified that a score of 3 or higher would give students college credit, but that students might be advised to to retake courses if e.g. they got a 3 in AP Calculus and intended to major in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ohiohighered.org/sites/ohiohighered.org/files/uploads/transfer/documents/AP_Alignment_Recommendations_04-01-14.pdf&quot; rel=&quot;nofollow&quot;&gt;Here are the current (April 2014) recommendations for students at Ohio's public institutions&lt;/a&gt;. They recommend that students getting a score of 3 on AP Calc AB or higher receive credit for Calc 1; a score of 3 or higher on AP Calc BC receive credit for Calc 1 and 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been unable to find data on success rates in subsequent classes for students awarded AP credit (e.g. How do students passing AP Calc AB do in university Calc 2?). &lt;a href=&quot;http://regents.ohio.gov/transfer/ap/documents/AP_CollegeBoard.pdf&quot; rel=&quot;nofollow&quot;&gt;This appears to be a presentation from the College Board to the Ohio Board of Regents&lt;/a&gt;--coming as it does from the College Board (who run the AP tests), it has nothing but good things to say about AP students, but the data seems cherry-picked. Sure, AP students outperform other students, but typically high schools put their best students in AP classes. How do AP students earning credit in high school compare to students earning those same credits at university?&lt;/p&gt;&#xA;" OwnerUserId="182" LastActivityDate="2015-12-18T05:23:40.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10259" PostTypeId="2" ParentId="8088" CreationDate="2015-12-18T06:17:07.063" Score="4" Body="&lt;p&gt;This story is not technically a state-mandated policy, but the CUNY system of colleges in New York is centrally managed and closely tied to city and state government (most of the Board of Trustees is appointed by either the mayor or governor, with one seat each set aside for faculty &amp;amp; student representation). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the past few years, more of the basic curriculum decisions have been removed from faculty and mandated by central administration. For example, one initiative called Pathways was put in place over faculty objections to set maximum credit hours for all programs and core classes, and also mandate acceptance of all of these credits in transfers from junior to senior colleges. To meet this requirement, programs have been forced to cut science lab courses, foreign language requirements, etc. In mid-2013 the CUNY-wide faculty held a vote on the issue, &lt;a href=&quot;http://www.psc-cuny.org/latest-news/92-vote-no-confidence-pathways-cunys-new-curriculum&quot; rel=&quot;nofollow&quot;&gt;with 92% voting &quot;no confidence&quot; in Pathways&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the particular area of math requirements, CUNY is open-admissions and required to accept all students from the New York system with a high school diploma or equivalent. A majority cannot pass basic entry skills tests in reading, writing, algebra, or arithmetic. These students thus enter remedial classes for these skills, the success rate is very low, and therefore the graduation rate is very low (around 20% over 6 years, in line with other community colleges around the nation). Many faculty will assert that most students are effectively unable to read college-level texts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this causes political pressure on the trustees to raise graduation rates, which is seen as a failure on the part of the institution and faculty. A great deal of &quot;churn&quot; has occurred in the last few years with &lt;a href=&quot;http://cuny.edu/academics/testing.html&quot; rel=&quot;nofollow&quot;&gt;changes to math remediation&lt;/a&gt;; the entry tests have changed, the acceptance scores have been modified up and down, the exit exam to remediation has been changed by central fiat several times, etc.; in each case hoping that a majority of students would begin passing, but success rates stay intransigently the same. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in my experience the more the political elements begin mandating certain acceptance criteria, the likelihood is great that they will inspect and complain about why the &lt;em&gt;next&lt;/em&gt; level then has low success rates (in the aftermath of lowered entry standards). This pressure rolls in the direction of mandating higher passing rates, which is ultimately accomplished by lowering standards for that next level, and so on. For example, the newest CUNY community college (Guttman) has no faculty governance, no disciplinary departments, no remediation courses, and no basic skills requirements for graduation -- and it is likely that meetings are currently being held to expand on that model and remove basic writing and algebra/arithmetic requirements throughout the system, which would hopefully raise the desired graduation rates. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short: The more the political class tightens its grip on acceptance and graduation criteria, the more academic systems will slip through its fingers. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2015-12-18T06:22:30.983" LastActivityDate="2015-12-18T06:22:30.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10260" PostTypeId="2" ParentId="5645" CreationDate="2015-12-18T10:31:39.807" Score="-2" Body="&lt;p&gt;Learning anything needs patience and continuous effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we talk about Mathematics, It is having Abstract things. We accept one as symbol &quot;1&quot; in it. To teach mathematics most important thing is you have to mix yourself with nature. There are patterns and shapes in nature which will help you to understand and taught others.&#xA;For example, shape of archimedian spiral originally it is derived from the shell of a nautilus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Check this &lt;a href=&quot;https://www.youtube.com/watch?v=cy-8lPVKLIo&quot; rel=&quot;nofollow&quot;&gt;history of mathematics on youtube&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6133" LastEditorUserId="6133" LastEditDate="2015-12-18T12:08:10.347" LastActivityDate="2015-12-18T12:08:10.347" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10261" PostTypeId="2" ParentId="5645" CreationDate="2015-12-18T14:17:46.127" Score="3" Body="&lt;p&gt;Malcolm Gladwell addresses this issue briefly in&#xA;&lt;a href=&quot;http://gladwell.com/outliers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Outliers&lt;/em&gt;&lt;/a&gt;.&#xA;Here I take some quotes from a&#xA;&lt;a href=&quot;http://www.zooktutoring.com/malcom-gladwell-on-math-and-persistence-1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zook Tutoring&lt;/a&gt; posting, quoting from &lt;em&gt;Outliers&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;a href=&quot;http://gsi.berkeley.edu/programs-services/hsl-project/hsl-speakers/schoenfeld/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alan Schoenfeld&lt;/a&gt;, a math professor at Berkeley, made a videotape of a woman named Renee as she was trying to solve a math problem. [...] Twenty-two minutes pass from the moment Renee begins playing with the computer program to the moment she says, “Ahhhh. That means something now.” [...]&lt;/p&gt;&#xA;&lt;p&gt;We sometimes think of being good at mathematics as an innate ability. You either have “it” or you don’t. But to Schoenfeld, it’s not so much ability as attitude. You master mathematics if you’re willing to try. That’s what Schoenfeld attempts to teach his students. Success is a function of persistence and doggedness and the willingness to work hard for twenty-two minutes to make sense of something that most people would give up on after thirty seconds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-12-18T15:13:36.700" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10262" PostTypeId="2" ParentId="10157" CreationDate="2015-12-18T14:49:13.720" Score="2" Body="&lt;p&gt;This began as a comment, but became long-winded enough that I decided to post an answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A wonderful thing about mathematics (and being a mathematics teacher) is that the &lt;em&gt;proper&lt;/em&gt; form for students to give any symbolic solution is that in which they are prompted, and they should be conditioned both to identify a form that is appropriate for a given problem and give any answer in a variety of ways if called upon to do so.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that I try to communicate to my students is that &lt;em&gt;symbols&lt;/em&gt; and notations &lt;em&gt;are arbitrary&lt;/em&gt;. In fact, this is really a &quot;eureka&quot; moment in mathematics education: when students realize that a variety of systems which bear wildly different mathematical descriptions can be solved using a single algorithm. In addition, many measurable physical models which vary wildly in derivation and application bear identical mathematical representation. Really, for students to understand these and apply them in general is the end of our instruction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this problem specifically, to give an answer as a point $(x_o,y_o)$, the intersection of vertical and horizontal lines $x =x_o , y=y_o$, a vector $\langle x_o, y_o\rangle$ or in any other form should depend on how the question was posed and any geometric considerations that have been given. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider this a teachable moment- ask &lt;em&gt;them&lt;/em&gt; why they gave their answer in a certain form and if they think there is a better way in subsequent questions. This is especially useful in geometry courses that &quot;skirt around&quot; definitions of objects like points. You might be surprised at some of the insightful answers you get- I usually am. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2015-12-23T05:01:25.970" LastActivityDate="2015-12-23T05:01:25.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10263" PostTypeId="2" ParentId="10136" CreationDate="2015-12-18T16:36:42.480" Score="1" Body="&lt;p&gt;Here's a Processing sketch that allows users to draw circles and line segments for geometric constructions.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt; boolean lineMode;&#xA; boolean circleMode;&#xA; boolean dragMode;&#xA; boolean last_shape_line;&#xA; boolean last_shape_circle;&#xA; boolean mouse_Is_Pressed;&#xA;&#xA; int xSize = 1200;&#xA; int ySize = 640;&#xA; int clicknumber;&#xA; int numberoflines=0;&#xA; int numberofcircles =0;&#xA; Circle[] circles;&#xA; Line[] lines;&#xA;&#xA; float startX=0;&#xA; float startY=0;&#xA; float releaseX=0;&#xA; float releaseY=0;&#xA;&#xA; class Circle {&#xA;   float x, y, radius;&#xA;   Circle(float xpos, float ypos, float r) {&#xA;   x = xpos;&#xA;   y = ypos;&#xA;   radius = r;&#xA;   }&#xA;   void display() {&#xA;     ellipse(x, y, radius*2, radius*2);&#xA;     point(x,y);&#xA;   }&#xA; }&#xA;&#xA; class Line {&#xA;   float x1, y1, x2, y2;&#xA;   Line(float xStart, float yStart, float xEnd, float yEnd) {&#xA;     x1 = xStart;&#xA;     y1 = yStart;&#xA;     x2 = xEnd;&#xA;     y2 = yEnd;&#xA;   }&#xA;   void display() {&#xA;     line(x1, y1, x2, y2);&#xA;   }&#xA; }&#xA;&#xA; void mouseClicked() {&#xA;   if (circleMode &amp;amp;&amp;amp; clicknumber == 1){&#xA;     circleMode = false;&#xA;     releaseX = mouseX;&#xA;     releaseY = mouseY;&#xA;     circles[numberofcircles] = new Circle(startX,startY, sqrt(pow(mouseX-startX,2)+pow(mouseY-startY,2)));&#xA;     numberofcircles +=1;&#xA;     clicknumber = 0;&#xA;     last_shape_circle = true;&#xA;     last_shape_line = false;&#xA;   }&#xA;   if (lineMode &amp;amp;&amp;amp; clicknumber == 1){&#xA;     lineMode = false;&#xA;     releaseX = mouseX;&#xA;     releaseY = mouseY;&#xA;     lines[numberoflines] = new Line(startX,startY,releaseX,releaseY);&#xA;     numberoflines +=1;&#xA;     clicknumber = 0;&#xA;     last_shape_line = true;&#xA;     last_shape_circle = false;&#xA;   }&#xA;   if ((circleMode || lineMode) &amp;amp;&amp;amp; clicknumber == 0){&#xA;     startX = mouseX;&#xA;     startY = mouseY;&#xA;     clicknumber = 1;&#xA;   }&#xA; }&#xA;&#xA; void setup() {&#xA;   size(1200, 640, P3D);&#xA;   circles = new Circle[1000];&#xA;   lines = new Line[1000];&#xA; }&#xA;&#xA; void keyPressed() {&#xA;   if (key == 'd') {&#xA;     if (!dragMode &amp;amp;&amp;amp; !circleMode &amp;amp;&amp;amp; !lineMode) {&#xA;       dragMode = true;&#xA;     }&#xA;   }&#xA;   if (key == 'l') {&#xA;     if (!circleMode) {&#xA;       lineMode = true;&#xA;       clicknumber = 0;&#xA;     }&#xA;   }&#xA;   if (key == 'c') {&#xA;     if (!lineMode) {&#xA;       circleMode = true;&#xA;       clicknumber = 0;&#xA;     }&#xA;   }&#xA;   if (key =='e') {&#xA;     circleMode = false;&#xA;     lineMode = false;&#xA;   }&#xA;   if (key =='u') {&#xA;     if (last_shape_line &amp;amp;&amp;amp; numberoflines&amp;gt;0) {&#xA;       lines[numberoflines-1] = null;&#xA;       numberoflines += -1;&#xA;       last_shape_line = false;&#xA;     }&#xA;     if (last_shape_circle &amp;amp;&amp;amp; numberofcircles &amp;gt; 0) {&#xA;       circles[numberofcircles-1] = null;&#xA;       numberofcircles+=-1;&#xA;       last_shape_circle = false;&#xA;     }&#xA;   }&#xA; }&#xA;&#xA; void draw() {&#xA;   if (dragMode) {&#xA;     fill(255);&#xA;     text(&quot;drag&quot;, mouseX+10, mouseY+10, 300,300);&#xA;     noFill();&#xA;   }&#xA;   background(0);&#xA;   if (lineMode) {&#xA;     text(&quot;line&quot;, mouseX+10, mouseY+10, 300,300);&#xA;     if (clicknumber&amp;gt;0){&#xA;       strokeWeight(1);&#xA;       line(startX,startY,mouseX,mouseY);&#xA;     }&#xA;   }&#xA;   if (circleMode) {&#xA;     text(&quot;circle&quot;, mouseX+10, mouseY+10, 300,300);&#xA;     if (clicknumber&amp;gt;0){&#xA;       line(startX,startY,mouseX,mouseY);&#xA;       strokeWeight(3);&#xA;       ellipse(startX,startY,2*sqrt(pow(mouseX-startX,2)+pow(mouseY-startY,2)),2*sqrt(pow(mouseX-startX,2)+pow(mouseY-startY,2)));&#xA;       strokeWeight(1);&#xA;     }&#xA;   }&#xA;   noFill();&#xA;   stroke(255);&#xA;   strokeWeight(3);&#xA;   for (int i=0; i&amp;lt;numberofcircles; i=i+1){&#xA;     circles[i].display();&#xA;   }&#xA;   for (int i=0; i&amp;lt;numberoflines; i=i+1){&#xA;     lines[i].display();&#xA;   }&#xA;   text(&quot;press 'c' to draw a circle&quot;, 10, 10, 300, 20 );&#xA;   text(&quot;press 'l' to draw a line&quot;, 10, 30, 300, 20 );&#xA;   text(&quot;press 'u' to undo the last shape&quot;, 10, 50, 300, 20 );&#xA; }&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;If you want to focus on programming in the curriculum, you might consider exposing your students to methods of proof that are altogether different than what you may have been using in the past by teaching them to do computational coordinate proofs using symbolic variables. Processing can compile Python, which is excellent for symbolic and numerical computing. You can work with them on expressing theorems and &quot;given&quot; statements in terms of symbolic quantities, then use the equivalence operator to determine equivalence with the &quot;result&quot; statements. Here's an example for the theorem &quot;&lt;em&gt;In an isosceles triangle, the base angles are congruent&lt;/em&gt;.&quot; You'll have to coach the students to (1) generate the appropriate symbolic coordinates, and (2) generate the appropriate symbolic statement of equivalence, then (3) implement these into a computational proof. Ideally, you'll be developing programs as you go along so, so most of the work represented by the sketch below would already be done and just copied in from a class workflow file. In order to run this sketch, you'll have to copy the numpy and sympy modules into the sketch directory and install the Python compiler in your Processing IDE. Here is the proof: &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;  import numpy&#xA;  import sympy&#xA;&#xA;  def add(u,v):&#xA;      #computes the sum of vectors u,v&#xA;      sum = [0]*len(u)&#xA;      for i in range(len(u)):&#xA;          sum[i] = u[i] + v[i]&#xA;      return sum&#xA;&#xA;  def subtract(u,v):&#xA;      #computes the difference of vectors u,v&#xA;      return [u[0]-v[0],u[1]-v[1]]&#xA;&#xA;  def dot(u,v):&#xA;      # computes the dot product of u anv v.&#xA;      return u[0]*v[0] + u[1]*v[1]&#xA;&#xA;  def length(u):&#xA;      # computes the length of vector u&#xA;      return dot(u,u)**(0.5)&#xA;&#xA;  def angle(p,q,r):&#xA;      # returns the measure of angle pqr for points p = [p[0],p[1]], q = [q[0],q[1]], r = [r[0],r[1]]&#xA;      u = subtract(p,q)&#xA;      v = subtract(r,q)&#xA;      return sympy.acos(dot(u,v)/ (length(u)*length(v)) )&#xA;&#xA;  # In order to prove this statement, construct an arbitrary isosceles triangle PQR and then&#xA;  # prove that it has the desired property. Triangle PQR will have a base of length b and a &#xA;  # height of length h. &#xA;&#xA;  #(1) Define any symbolic variables you will use in your proof:&#xA;  b = sympy.symbols('b')&#xA;  h = sympy.symbols('h')&#xA;&#xA;  #(2) Assign coordinates to determine your isosceles triangle:&#xA;  P = [-b/2, 0]&#xA;  Q = [0,h]&#xA;  R = [b/2, 0]&#xA;&#xA;  #(2.5, optional) Demonstrate that your triangle is indeed isosceles:&#xA;  length(subtract(P,Q)) == length(subtract(R,Q))&#xA;&#xA;  #(3) The base angles for this triangle are angle QPR and angle QRP, so we want to show &#xA;  # that both angles have the same measure. We will use the function we've already &#xA;  # written to give the angles. If the statement is true, the compiler will print &quot;true&quot;.  &#xA;  if angle(Q,P,R) == angle(Q,R,P):&#xA;      print(&quot;true&quot;)&#xA;  else:&#xA;      print(&quot;false&quot;)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;When students run the sketch, Processing will return &quot;True&quot; in the console. Quod erat demonstrandum. This scheme should in principle work for any geometric proof that your students encounter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students will have difficulties with the notion of an &quot;arbitrary&quot; representative object, but I find that once they become familiar with assigning symbolic coordinates (which is already a part of most elementary geometry curricula), and you tell them that's what constitutes a proof, as opposed to giving a concrete example, (and that you won't give any points for the latter...) they do okay. If they've worked with objects in Java then they should be better primed for abstraction in mathematics, I think.&lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-06-25T00:08:35.440" LastActivityDate="2016-06-25T00:08:35.440" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10264" PostTypeId="2" ParentId="7999" CreationDate="2015-12-18T17:15:47.810" Score="0" Body="&lt;p&gt;Try &lt;a href=&quot;https://www.algoriddim.com/vjay-ipad&quot; rel=&quot;nofollow&quot;&gt;Vjay by algoriddim&lt;/a&gt;. It costs $9.99 on the iPad app market. The only requested feature that I'm not sure it has is on-screen text. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastActivityDate="2015-12-18T17:15:47.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10265" PostTypeId="2" ParentId="9711" CreationDate="2015-12-18T17:21:46.320" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://itunes.apple.com/us/app/mandelpad-fractal-imaging/id372066030?mt=8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mandelpad&lt;/a&gt; plots the Mandelbrot set and associated Julia sets in the complex plane, and could spark a few interesting discussions (it has in my class), or even a &quot;fractal art&quot; project. In coordination with some other visual processing apps I've used it to create some very beautiful and intriguing images to use as cover photos, backgrounds, and the like on my class websites. I actually used it this year to make a &quot;festive fractal&quot; holiday greeting card.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VNnsC.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VNnsC.jpg&quot; alt=&quot;festive fractal&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://itunes.apple.com/us/app/geometry-pad/id517461177?mt=8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometry pad&lt;/a&gt; is a fantastic iPad geometry platform. Students can construct geometric objects, and purchase additional features that make constructions easier. I've relied on this tool heavily for teaching and demonstrations in my geometry classes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://itunes.apple.com/us/app/good-grapher-scientific-graphing/id416996329?mt=8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Good Grapher&lt;/a&gt; is hands down the best and most versatile graphing tool I've come across. It permits graphing in two or three dimensions, parametric and implicitly defined graphs, inequalities, and includes a scientific calculator, polynomial solver, and linear system solver. There is a free version, but I use the paid version with more features. It has been an indispensable for demonstrations in my multivariate calculus class.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll try to update this answer if I come across anything else I really like. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2015-12-18T17:29:47.050" LastActivityDate="2015-12-18T17:29:47.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10266" PostTypeId="1" CreationDate="2015-12-18T19:39:51.243" Score="2" ViewCount="124" Body="&lt;p&gt;I was reading this &lt;a href=&quot;http://www.cs.ubc.ca/%7Emurphyk/Bayes/economist.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt;, and I decided to use the “canonical example” of a “precocious newborn observing his first sunset” as a problem for my students (undergrads in an intro to inductive logic course). As noted &lt;a href=&quot;http://allendowney.blogspot.com/2012/03/sun-will-probably-come-out-tomorrow.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; though: it doesn’t seem to lend itself too well to a basic analysis using Bayes’ rule.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, I came up with what’s below. Am I missing anything?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Deep in the forest of northern Mississippi, it was a dark and stormy night. A bolt of lightning struck the middle of a swamp; the water bubbled and sparks flew. Suddenly, an arm reached out of the muddy water and grabbed onto a log! The electrical current from the lightning must have rearranged the molecules in the swamp! Swampman---a creature looking just like an adult human male, but created only moments before by the fortuitous rearrangement of molecules in the swamp---was pulling himself onto the shore!&lt;/p&gt;&#xA;&lt;p&gt;By the next evening, Swampman has cleaned himself up and gotten some clothes. All day he has observed his surroundings. He noticed that, when he crawled out of the swamp, it was dark, but then the sun rose, moved across the sky, and disappeared below the horizon. He wonders if this will happen again—will the sun rise again the next morning? He forms this hypothesis, which he refers to as H, &lt;em&gt;the sun will rise tomorrow&lt;/em&gt; or, equivalently, since each day has a tomorrow, &lt;em&gt;the sun rises every day&lt;/em&gt;. Since he has no other basis for setting the probability that H is true except to give each possibly---the sun will rise tomorrow or it won’t---equal weight, he makes P(H) = .50. He quickly sees that, given H---that is, if the sun rises everyday---then the probability of the sun rising on any particular morning is 1.0.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, if H is false, the sun still might rise on any given morning, but the chance of that happening is less than 100 percent. He has already seen the sun rise once, though, so P(R|not H) has to be greater than 0. (R, which is the evidence, is &lt;em&gt;the sun rose that morning&lt;/em&gt;.) Swampman mulls it over and decides to put the probability of the sun rising on any particular morning, given that H is false (i.e., the sun does not rise every day), at 50 percent also.&lt;/p&gt;&#xA;&lt;p&gt;What are P(A), P(R|H), and P(R|not H)?  What is the probability that H is correct, given that the sun rose Swampman’s first morning? That is, what is P(H|R)? The next morning, Swampman observes the sun rise again. Now, what is P(H|R)? How many times does Swampman have to observe the sun rising for P(H|R) to be higher than 95 percent?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Answer&lt;/strong&gt;&#xA;P(H) = .50, P(R|H) = 1.0, P(R|not H) = .50&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$P(H\mid R) = \frac{P(H) \times P(R\mid H)}{(P(H) \times P(R\mid H)) + ((P(\text{not } H) \times P(R\mid\text{not } H))} = \frac{.50 \times 1.0}{(.50 \times 1.0) + (.50 \times .50)} = .67$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Subsequent iterations produce &lt;span class=&quot;math-container&quot;&gt;$P(H\mid R) = .80$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$.89$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$.94$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$.97$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6139" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2015-12-24T08:51:50.980" Title="Basic Bayes rule problem" Tags="|undergraduate-education|secondary-education|probability|" AnswerCount="0" CommentCount="1" ClosedDate="2015-12-29T16:47:39.620" ContentLicense="CC BY-SA 3.0" />
  <row Id="10267" PostTypeId="2" ParentId="9940" CreationDate="2015-12-19T01:53:01.067" Score="3" Body="&lt;p&gt;At an advanced undergrad level, I have had an encounter with &quot;programming-versus-math&quot; in the crypto course and equally-&quot;applied&quot;-but-different error-correcting codes course that I developed in the late 1990s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, let me say that I myself find contemporary computers substantially helpful &quot;even&quot; for purposes that existed prior to them. Communication! Typesetting!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Computational issues in number theory (my general field of interest) are low-hanging fruit in terms of marketability to students, or to amateurs, and have considerable interest to any sensible person. But, of course, there's a limit to what experiment can suggest, all the while thinking that (if we were practical physicists) a think untestable by experiment is nearly worthless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somewhat surprisingly to me, I found a bifurcation in the attitudes of students, between computational/experimental verification (or experiment!) and &quot;proof&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, much of the push-back against experiment from undergrads who claimed to exclusively endorse &quot;proof&quot; could be understood, under closer examination, to be a simple-minded resistance to &quot;confusing inputs&quot;. In particular, a significant portion of the population was the sort of &quot;math major&quot; who &quot;likes math&quot; because &quot;there are rules&quot; and no general sensibility is relevant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a pathetic counterpoint, the people who showed up with some capacity to program (but mostly inefficient graphical interfaces, silly things, nothing high-performance, by a mile...) could not imagine that there would ever be any need to &quot;prove&quot; anything. In fact, the most bizarre conception I'd encountered up to those dates was that ... in computer science students' belief system... computers are so fast/good/whatever that no task is impossible... The reason I had trouble catching-on to this conceit was that I'd have thought they'd have known that it's not clear that P is or is not NP, not to mention that the security of various security systems depends on tasks being difficulty for &lt;em&gt;anyone&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sad summary: the majority of kids attracted to math as an undergrad major were attracted for reasons violently opposite to computer-science reasons, while the computer-science kids had belief systems that made them unable to understand why/how mathematics was necessary or useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, apparently, there's a bifurcation in the general population between these ways of thinking. The time I've spent trying to cajole people into seeing the opposite possibility .. has not been repaid in any way, sadly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, yes, in the ideal new world, people would learn a few computer languages, especially some scripting languages, and learn some mathematics, and be able to do things... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, yet again, I find that the implicit aspects of the question are the real trouble... Yes, I've tried to have an impact on curriculum, and on &quot;attitude&quot;, but &quot;computing&quot; and &quot;math&quot; each do seem to have angry, uncooperative constituencies already established... Whah?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, I honestly think a person would be a fool to not learn about computing (not to mention communication) if they could encompass it, whether or not they were interested in mathematics. Srsly, this is a sort of tail obviously wagging the dog thing, for most purposes. (And, with luck, I will become rich by using Fourier analysis to break the stock market, of course.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-12-19T01:53:01.067" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10268" PostTypeId="2" ParentId="10247" CreationDate="2015-12-19T17:41:27.270" Score="3" Body="&lt;p&gt;The way I explain order of operations is that the strongest things come first:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Addition and subtraction are weakest, so they come last &lt;/li&gt;&#xA;&lt;li&gt;Multiplication (sometimes like repeated addition) and division are stronger &lt;/li&gt;&#xA;&lt;li&gt;Exponentiation is stronger yet&lt;/li&gt;&#xA;&lt;li&gt;And grouping symbols are the most powerful, so deal with those first&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="60" LastActivityDate="2015-12-19T17:41:27.270" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10269" PostTypeId="2" ParentId="4367" CreationDate="2015-12-20T15:42:44.320" Score="4" Body="&lt;p&gt;For completeness I'll add a reference to Rick Garlikov's use of the Socratic method to teach binary arithmetic to a third grade class.  It took 75 questions in the instance he describes.  A complete transcript and a summary of his thoughts on the process are at &lt;a href=&quot;http://www.garlikov.com/Soc_Meth.html&quot; rel=&quot;nofollow&quot;&gt;The Socratic Method: Teaching by Asking Instead of by Telling&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6071" LastActivityDate="2015-12-20T15:42:44.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10271" PostTypeId="2" ParentId="4367" CreationDate="2015-12-20T21:10:19.047" Score="2" Body="&lt;p&gt;I haven't tried this in a class but it worked in one-on-one conversation with kids at basically all ages (well, ability to count  is needed but that's basically it). It also has the benefit, that you do not need to speak about positional notation or the 10 based system at all for the beginning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;State:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I can count to 1023 with my fingers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To show how, hold your closed fists, showing the backs of your hands. Show your right thumb and say &quot;one&quot;. Fold the thumb in and the index out and say &quot;two&quot;. Let the index out and also show the thumb and say &quot;three&quot;, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At some point some kids get where this is going and figure out how the 1023 comes into play. Some other kids need some more time but in my experience, basically anybody can figure out what the system is by doing it himself. Of course, many kids will find it funny to show the number 4 or (even better) 132. You can also ask how far two kids can count if they use all four hands…&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From that point on you can take different. You could go into the direction that the fingers really do not matter here and use another binary notation. You could also investigate numbers that the single fingers represent. No matter how you do, you'll end up explaining the binary system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am also in Germany and confirm that this is a topic in 5th grade and also that the books I've seen make no reference to &quot;counting with your fingers to 1023&quot; at all.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2015-12-20T21:10:19.047" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10272" PostTypeId="2" ParentId="1587" CreationDate="2015-12-21T05:51:15.380" Score="3" Body="&lt;p&gt;Here's a late response, but this question keeps taunting me for a direct and difficult-to-craft answer.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a significant difference in the rigor of these courses as&#xA;  taught in college and high school (i.e., is it not just my&#xA;  imagination)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Likely yes; although this will vary by location, institution, and instructor. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If there is a difference, is there any published research supporting&#xA;  such a difference, or statements by those in charge of curriculum at&#xA;  an institution giving an explanation?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My best reply is this: The published research supporting this difference is &lt;em&gt;the entire body of literature for the discipline of mathematics since Ancient Greece&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To expand: College professors are likely to be &lt;em&gt;active mathematicians&lt;/em&gt; and have a perspective on the structure and development of mathematics that high school instructors are less likely to have. Moreover, college professors are principally incentivized and promoted on the basis of &lt;em&gt;doing math&lt;/em&gt;, whereas modern high school teachers are judged on the basis of &lt;em&gt;passing students&lt;/em&gt;. At the high school level, the algebra course is likely terminal for most students; whereas at the college level, it will be viewed as only the beginning of a much larger body of knowledge and exploration. The college faculty will want this foundation to be firmly set, with correct explanations and formulations, so as to support later work throughout the discipline. At high school this is unlikely to be a goal or even an observation; if the system can get the student past a given standardized exam, it is considered &quot;job done&quot;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;But perhaps the greatest accomplishment during the first three&#xA;  hundred years of Greek mathematics was the Greek notion of a logical&#xA;  discourse as a sequence of statements obtained by deductive reasoning&#xA;  from an accepted set of initial statements assumed at the onset of the&#xA;  discourse... Certainly the most outstanding contribution of the early&#xA;  Greeks to mathematics was the formulation of the pattern of material&#xA;  axiomatics and the insistence that mathematics be systematized&#xA;  according to this pattern. The concept of axiomatic development in&#xA;  mathematics must be ranked as one of the very greatest of the GREAT&#xA;  MOMENTS IN MATHEMATICS.&quot; -- Howard Eves, &lt;em&gt;Great Moments in&#xA;  Mathematics&lt;/em&gt; (Before 1650), Lecture 7.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To the college professor, math without a logical axiomatic development &lt;em&gt;doesn't even count as mathematics at all&lt;/em&gt;. Personally, I refer to the memorize-this-formula-without-proof approach as &quot;faith-based mathematics&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if you look for some modern literature on teaching algebra at the college level, it will be very easy to find publications arguing &lt;em&gt;exactly the opposite&lt;/em&gt;. That is: Most students won't &quot;use&quot; algebra, practical applications are the only thing we need to consider, deep understanding is extraneous, taken-for-faith procedures are acceptable, college professors should be removed from remediation efforts, etc. But this overlooks the whole point of mathematics -- &lt;em&gt;the search and sharing of proper justifications of our knowledge&lt;/em&gt;. This oversight is &lt;em&gt;exactly how rigorous math has been removed from high schools in the last few decades&lt;/em&gt;, resulting in the jarring wake-up call when students jump the chasm to college-level work. There is definitely a movement to &lt;em&gt;remove real mathematics&lt;/em&gt; from college academics for the general student in the future, because of this ongoing difficulty; but the idea that most students will &lt;em&gt;never even be introduced to real mathematics&lt;/em&gt; is almost overwhelmingly difficult for the dedicated math practitioner to swallow. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short: Math without logical proof is not really mathematics at all. The fact that today's high school programs leave the majority of students completely unaware of that fact is indeed an ongoing tragedy, one that we can only hope won't be forced as a contamination into our higher education programs.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-12-21T05:51:15.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10273" PostTypeId="2" ParentId="10227" CreationDate="2015-12-21T12:57:37.813" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;With calculus, we have the ability to find the effects of changing   conditions on a system. By studying these, you can learn how to control a system to make it do what you want it to do. Because of the ability to model and control systems, calculus gives us extraordinary power over the material world.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(adapted from &lt;a href=&quot;http://www-math.mit.edu/~djk/calculus_beginners/chapter01/section02.html&quot; rel=&quot;nofollow&quot;&gt;http://www-math.mit.edu/~djk/calculus_beginners/chapter01/section02.html&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Calculus is the language of engineers, scientists, and economists. The work of these professionals has a huge impact on our daily life - from your microwaves, cell phones, TV, and car to medicine, economy, and national defense.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(from &lt;a href=&quot;https://books.google.nl/books?id=pE_nAwAAQBAJ&amp;amp;lpg=PP5&amp;amp;dq=Calculus%20is%20the%20language%20of%20engineers%2C%20scientists&amp;amp;pg=PP5#v=onepage&amp;amp;q=Calculus%20is%20the%20language%20of%20engineers,%20scientists&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Calculus Equations and Answers&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This link will be very helpful for you: &lt;a href=&quot;http://tutorial.math.lamar.edu/Classes/CalcI/RateOfChange.aspx&quot; rel=&quot;nofollow&quot;&gt;http://tutorial.math.lamar.edu/Classes/CalcI/RateOfChange.aspx&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6148" LastEditorUserId="376" LastEditDate="2016-01-06T16:29:07.767" LastActivityDate="2016-01-06T16:29:07.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10275" PostTypeId="1" AcceptedAnswerId="10281" CreationDate="2015-12-21T18:42:51.120" Score="8" ViewCount="220" Body="&lt;p&gt;I am preparing to give an abstract algebra course, which should be mainly focused on Group Theory. However, I want to cover other algebraic structures (magmas, quasigroups, etc.) as well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering when I should cover them. If I do it in the beginning, it will connect nicely to the other basic properties of a group other algebraic structures may or may not have. However, I am afraid that an overload of algebraic structures will confuse the students. But I have to cover monoids somewhere in the beginning anyway to get to the free monoids, which are usually covered somewhere in the beginning. &#xA;On the other hand, if I do it in the end, then there will be some repetition of the basic properties, which is nice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How should I approach this?&lt;/p&gt;&#xA;" OwnerUserId="5664" LastActivityDate="2015-12-22T19:00:21.853" Title="When to cover other algebraic structures in an abstract algebra course?" Tags="|undergraduate-education|abstract-algebra|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10277" PostTypeId="2" ParentId="8347" CreationDate="2015-12-21T20:24:44.490" Score="1" Body="&lt;p&gt;A nice time to discuss this with students is when you talk about matrix operations. Matrices, of course, form a non-commutative ring. You can write out the list of ring axioms: $A+B=B+A$, $A+(B+C)=(A+B)+C$, $A(BC) = (AB)C$, etcetera... and have them all be true statements about matrices, which pointing out that, in general, we don't have $AB=BA$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can point out that the true axioms imply many of the identities they are used to. For example, $(A+\mathrm{Id})^2 = A (A + \mathrm{Id}) + \mathrm{Id} (A+\mathrm{Id}) = A^2 + A + A + \mathrm{Id} = A^2+2A + \mathrm{Id}$ just like they are used to. But not always! $(A+B)(A-B) = A^2 +BA - AB - B^2$, which is usually not equal to $A^2 - B^2$. All of this just feels like a slightly funky version of high school algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After a bit of time practicing this sort of thing, you can say &quot;if you like thinking about what sort of different laws mathematical operations might obey in different contexts, and what the consequences of those laws should be, this is the subject of what modern mathematicians call Algebra. The introductory course at this university is Math XXX.&quot;&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2015-12-21T20:24:44.490" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10281" PostTypeId="2" ParentId="10275" CreationDate="2015-12-21T23:12:16.953" Score="12" Body="&lt;p&gt;I think you are correct to worry about an overload of algebraic structures, especially if they are not well-motivated.  I would strongly encourage you to keep the naming of various algebraic structures to a minimum in a first course in abstract algebra.  In my view, the goal of the course is for them to see the power of abstraction in a fairly concrete setting, illustrating how it illuminates certain structures (e.g. the study of symmetry via group theory, the similarity in structure between integers and polynomials in one variable with coefficients in a field via ring theory).  I think it is much more useful to get to something non-trivial and engaging for the students such as the wallpaper groups, Platonic solids, or Rubik's Cube rather than trying to convince of the austere beauty of abstract algebraic structure for its own sake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you insist on introducing a bunch of different structures, my guideline for when it is appropriate to move on from one structure to another is when students, at a minimum, can do the following:&#xA;(1) distinguish between objects which have the structure and which do not. (e.g. is such-and-such set with such-and-such operation a quasigroup?  is such-and-such subset of a group a subgroup?)&#xA;(2) be able to work with simple ideas related to the given structure such as substructures, quotient structures (though this is challenging in a first course) and homomorphisms. (e.g. is the following set-theoretic map a homomorphism of groups?  is the following quasigroup a Moufang loop?  what is the kernel of the following group homomorphism?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If students do not become proficient at both (1) and (2) for a given algebraic structure, then it is not clear to me what is the point of introducing said algebraic structure in the course.  As I suggested in my first paragraph, I am very skeptical that students can actually acquire significant mastery of multiple algebraic structures during the course of a semester.  I think it is much more valuable to students to understand group theory at a deep level than to have a broad, but shallow, acquaintance with a wide variety of algebraic structures.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2015-12-22T19:00:21.853" LastActivityDate="2015-12-22T19:00:21.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10282" PostTypeId="2" ParentId="5700" CreationDate="2015-12-21T23:22:29.843" Score="1" Body="&lt;p&gt;I like the answer by @DavidButlerUofA above, but here's how I'd be prone to do this (in synch with similar issues like solving by factoring and completing the square). I'd throw up the initial theorem: &quot;For all real $a &amp;gt; 0$, $|u| = a$ means $u = a$ or $u = -a$.&quot; (Or using the plus-or-minus symbol for shorthand.) I'd point out a procedure: to solve such an equation we first need to isolate the absolute-value expression. And then I'd run a few warm-up recognition exercises of the ilk: &quot;Is the absolute-value expression isolated? Can we use this transformation right now?&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I have recently evolved to require checking solutions all the time, in an attempt to make sure students have that as a tool to try to catch these errors. In the specific case of your supported students, I'd totally let them go through with the faulty transformation, then check at the end, and seeing that it doesn't check, then thinking through where they broke the process. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-12-21T23:22:29.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10283" PostTypeId="2" ParentId="4453" CreationDate="2015-12-22T07:05:44.320" Score="-1" Body="&lt;p&gt;For basics: A Course in Ordinary Differential Equations - &#xA;Bindhyachal Rai, ‎D. P. Choudhury, ‎Herbert I. Freedman&lt;/p&gt;&#xA;" OwnerUserId="6133" LastActivityDate="2015-12-22T07:05:44.320" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10284" PostTypeId="2" ParentId="10275" CreationDate="2015-12-22T12:45:22.340" Score="3" Body="&lt;p&gt;When I took a first course in abstract algebra, we were taught the words &quot;semigroup&quot; and &quot;monoid&quot; along the way to learning what a group is. This was mainly a matter of convenience of language. It allowed us to say things like, &quot;A ring is an abelian group under addition and a semigroup under multiplication&quot;. (Our rings didn't need an identity.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The actual &lt;em&gt;theory&lt;/em&gt; of semigroups is of course a giant topic, almost all of which I think is not appropriate for a first group theory course.&lt;/p&gt;&#xA;" OwnerUserId="6153" LastActivityDate="2015-12-22T12:45:22.340" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10285" PostTypeId="2" ParentId="4453" CreationDate="2015-12-23T06:37:43.033" Score="1" Body="&lt;p&gt;You can try this pdf book for differential equations .&#xA; &lt;a href=&quot;https://www.math.ust.hk/~machas/differential-equations.pdf&quot; rel=&quot;nofollow&quot;&gt;https://www.math.ust.hk/~machas/differential-equations.pdf&lt;/a&gt;.&#xA;It has also the important formulas and questions with examples.&lt;/p&gt;&#xA;" OwnerUserId="6148" LastActivityDate="2015-12-23T06:37:43.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10286" PostTypeId="2" ParentId="10226" CreationDate="2015-12-23T06:59:04.423" Score="0" Body="&lt;p&gt;James Brennan explains &lt;a href=&quot;http://www.jamesbrennan.org/algebra/intro%20to%20algebra/simplifying_algebraic_expression.htm&quot; rel=&quot;nofollow&quot;&gt;Simplifying Algebraic Expressions&lt;/a&gt; in the following way:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By “simplifying” an algebraic expression, we mean writing it in the most compact or efficient manner, without changing the value of the expression. This mainly involves collecting like terms, which means that we add together anything that can be added together. The rule here is that only like terms can be added together.&#xA;  Like terms are those terms which contain the same powers of same variables. They can have different coefficients, but that is the only difference.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Examples: $3x$, $x$, and $–2x$ are like terms.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$2x^2$, $–5x^2$, and are like terms.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$xy^2$, $3y^2 x$, and $3xy^2$ are like terms.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$xy^2$ and $x^2 y$ are NOT like terms, because the same variable is not raised to the same power.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Combining Like terms&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Combining like terms is permitted because of the distributive law. For example,&#xA;  $3x^2 + 5x^2 = (3 + 5)x^2 = 8x^2$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What happened here is that the distributive law was used in reverse—we “undistributed” a common factor of x2 from each term. The way to think about this operation is that if you have three x-squareds, and then you get five more x-squareds, you will then have eight x-squareds.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Example:  $x^2 + 2x + 3x^2 + 2 + 4x + 7$&lt;br&gt;&#xA;    Starting with the highest power of $x$, we see that there are four x-squareds in all $(1x^2 + 3x^2)$. Then we collect the first powers of $x$, and see that there are six of them $(2x + 4x)$. The only thing left is the constants $2 + 7 = 9$. Putting this all together we get&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$ x^2 + 2x + 3x^2 + 2 + 4x + 7  = 4x^2 + 6x + 9$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6148" LastEditorUserId="143" LastEditDate="2015-12-24T13:54:41.430" LastActivityDate="2015-12-24T13:54:41.430" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10288" PostTypeId="2" ParentId="10234" CreationDate="2015-12-23T11:53:22.207" Score="1" Body="&lt;p&gt;Take a look at Pólya's classic &quot;How to Solve It&quot; (Princeton University Press, 1945). It gives hands-on advice on strategies on how to solve complex problems, mostly in the setting of problems understandable/solvable by high school students.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-12-23T11:53:22.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10289" PostTypeId="1" CreationDate="2015-12-24T01:30:33.583" Score="10" ViewCount="594" Body="&lt;p&gt;I posted &lt;a href=&quot;https://math.stackexchange.com/questions/1515013/what-topics-should-be-included-in-a-course-matching-these-specifications&quot;&gt;this question&lt;/a&gt; on &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;m.s.e.&lt;/a&gt;, where I upvoted the two answers, both of which said rather little by comparison to what the question asks.  Hence this present posting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say you have a calculus classroom full of liberal-arts majors who are not particularly mathematically inclined.  Your goal is &lt;b&gt;NOT&lt;/b&gt; to teach them everything on a list of topics that will be needed in other courses (either later calculus courses or courses in physics or engineering or statistics or biology or other subjects to which calculus is applied).  (As anyone with any common sense would do) you will include only ten percent or less of the topics in the usual lists and perhaps examine each included topic in more depth than it might normally get, but perhaps also proceed more slowly than you would if you needed to complete all the topics in the usual list.  Rather, your goal is to impress them with &lt;b&gt;(1)&lt;/b&gt; the ways in which calculus has played a role and continues to play a role in the world --- in the sciences and engineering and philosophy, etc. --- and with &lt;b&gt;(2)&lt;/b&gt; the fact that calculus is a considerable intellectual and aesthetic achievement.  This might necessitate presenting a few applications in the sciences, not usually found in the first-year calculus text, rather than concentrating on techniques.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: What topics would you include in such an (in the present day) unusual course?&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="1147" LastEditDate="2017-12-02T10:26:17.687" LastActivityDate="2017-12-02T16:35:01.763" Title="What topics should be included in a course matching these specifications?" Tags="|undergraduate-education|calculus|curriculum|syllabus|liberal-arts|" AnswerCount="8" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10290" PostTypeId="2" ParentId="10289" CreationDate="2015-12-24T02:49:01.790" Score="3" Body="&lt;p&gt;This will also say little in comparison to the breadth of your question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would include some of the history of how the notion of limit of a function,&#xA;$\lim_{x \to a} f(x)$, emerged over time.&#xA;I was surprised to learn this concept was not present&#xA;(or at least not clear) in the work of either Newton or Liebnitz. It took&#xA;another hundred$+$ years for this to reach its modern $\epsilon$-$\delta$ form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Including the major historical struggles &#xA;fits in well with your 2nd goal, to convey the&#xA;&quot;intellectual and aesthetic achievement&quot; of calculus.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2015-12-24T12:50:26.563" LastActivityDate="2015-12-24T12:50:26.563" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10291" PostTypeId="2" ParentId="10289" CreationDate="2015-12-24T03:44:31.110" Score="3" Body="&lt;p&gt;One really long answer to your question that I am fond of is the book &lt;a href=&quot;http://www.math.smith.edu/Local/cicintro/&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Calculus in Context&lt;/em&gt;&lt;/a&gt;.  Selected topics from the first four chapters might make for a nice one-semester course, though perhaps it has too much of an emphasis on technique for what you have in mind.  But I like the explanation of why calculus is relevant and why successive approximation is important through the concrete SIR model.  And I think that some understanding of technique is necessary to truly have a meaningful appreciation of why calculus is necessary and useful in applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of all, I like their philosophy that with the appropriate use of a calculator or computer, one can get a sense for how computations are carried out and how complex systems are analyzed.  If students come away with a sense for how differential equations naturally model the world around us and how basic techniques such as Euler's method can be used to approximate solutions and why computers are naturally suited to carrying out such computations if they are programmed properly, then I think the course has served a significant purpose.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2015-12-25T04:16:02.903" LastActivityDate="2015-12-25T04:16:02.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10292" PostTypeId="2" ParentId="10289" CreationDate="2015-12-24T09:07:27.343" Score="5" Body="&lt;p&gt;My first idea would be to aim to include applications at different levels (ones they might be able to calculate, ones they can get maths software to do, high-level maths research) and also across different subjects. Some ideas:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;relationship between area/volume/perimeter/surface-area of circles and spheres, maybe cones&lt;/li&gt;&#xA;&lt;li&gt;some fluid dynamics, maybe in relation to aeroplanes&lt;/li&gt;&#xA;&lt;li&gt;some modelling, such as populations in ecosystems&lt;/li&gt;&#xA;&lt;li&gt;the &lt;a href=&quot;http://www.numberphile.com/videos/embedding_torus_nash.html&quot;&gt;Numberphile discussion of John Nash's embedding of a torus&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;designing a suspension bridge, or similar&lt;/li&gt;&#xA;&lt;li&gt;finding an equilibrium point of some physical system, and testing whether it is stable (and therefore safe to build)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Other initial ideas, which might or might not be sensible, would be something in economics/related to financial markets, and designing a route for a spaceship (ie variable acceleration, change of mass... generally a situation where the formulae they have previously met don't work).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also want to show them that in some cases approximation does not work, so the full theory does have value. I might also discuss the Fundamental Theorem of Calculus - that it's not obvious why rates-of-change should be related to areas-under-graphs, although I'm not sure whether the students would appreciate that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would probably not include all the various calculation rules (chain rule, integration by parts etc) beyond what is directly needed for any calculations you are doing.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-12-24T09:07:27.343" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10293" PostTypeId="2" ParentId="10289" CreationDate="2015-12-24T11:44:02.933" Score="3" Body="&lt;p&gt;For a look at the intellectual development of calculus (&lt;em&gt;not&lt;/em&gt; a textbook nor a skeleton for a course, but something to peek at for the instructor), I'd recommend Dunham's &quot;The Calculus Gallery&quot; (2005). His books on the history of mathematics are outstanding expositions, accessible with a high-school background (and interest enough to work through the material).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-12-24T11:44:02.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10294" PostTypeId="2" ParentId="10289" CreationDate="2015-12-24T13:52:03.240" Score="3" Body="&lt;p&gt;Have you checked out the &lt;a href=&quot;https://www.artofmathematics.org/books/calculus&quot; rel=&quot;nofollow&quot;&gt;Discovering the Art of Mathematics project&lt;/a&gt;? Their 11 books, one of which is Calculus, are free for educational use, and they provide lots of supportive community. If we taught a liberal arts calculus course, this is what I'd want to use, I think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see that your question was about topics, not textbooks. They have lots of experience teaching courses like this, and would be good people to ask that question of. They encourage building your own textbook by using chapters from various books in their series, so will have people in their community who have created all sorts of hybrid courses.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="376" LastEditDate="2015-12-24T14:01:19.013" LastActivityDate="2015-12-24T14:01:19.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10295" PostTypeId="2" ParentId="10289" CreationDate="2015-12-24T22:44:27.617" Score="4" Body="&lt;p&gt;Key points for goal #1:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Professor Krugman's intuitive summary of how a derivative is the limit of a sequence of differentials.  Start with the graph of a function you want to differentiate.  Find the slope over a modest delta X.  Repeat for the next delta X. Optionally repeat again.  The shape of the derivative should now be obvious, but also obviously has kinks or steps (because it was computed using finite length steps).  &lt;strong&gt;&quot;Now we take off our glasses…&quot;&lt;/strong&gt; and the derivative looks like the true curve.  (For this level of class, the example curve is probably smooth.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teach students that they can &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7243&quot;&gt;check their work&lt;/a&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Always do a sanity check.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If possible, &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1834#1861&quot;&gt;check by substitution&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Integrals can be used to check derivatives, and &lt;em&gt;vice versa&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How to set up problems&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;How to identify boundary conditions and field conditions.  If you cannot find enough independent conditions, your solution will give you a range of possibilities instead of a single answer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teach students to keep track of their units.  (This also means that students usually should not use calculators until the end of a problem.)  Derivatives usually involve dividing by length or time.  Integrals usually involve multiplying by length or time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Common applications&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Position -&gt; Velocity -&gt; Acceleration -&gt; Jerk.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Energy -&gt; Power&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The importance of conservation laws.  Simple examples of the Reynolds Transport Theorem.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Exponentials, and common applications&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The derivative of an exponential curve.  The integral of an exponential curve.  If profits are proportional to capital invested, and a fixed portion of the profits are reinvested, you get exponential growth.  Demonstrate the power of exponential growth.  Ask the students to consider real-world limits to exponential growth.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The integral (from time = now to time = forever in the future) of an exponential decay is inversely proportional to the decay rate.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The integral (from time = 30 years in the future to time = forever in the future) of the same exponential decay is the previous integral, times the amount of decay in 30 years.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Applications of the exponential decay integrals to mortgages, radiation exposure, ….  Dare the students to come up with other examples where exponential decay might be a good model.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Sine waves, and common applications&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Show how having the second derivative be proportional to the original value, but opposite in sign, results in sine and cosine waves.  Show the forms of what became known as Maxwell's Equations, and that they have this property.  Point out that this allowed Maxwell to describe light as a wave, and understand its speed.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Point out that there had to be a negative feedback in there somewhere, or it would have exponentially grown out of control.  (And that would violate a conservation law.)  Dare the students to come up with other examples of wave phenomena.  Dare the students to guess where the negative feedback comes from.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Optionally mention some cool stuff&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Any repeating pattern can be modeled using a sum of sine and cosine waves.  &lt;strong&gt;Fourier transforms&lt;/strong&gt; are used to digitize songs, to compress images, to transmit data via radio waves, &lt;em&gt;et cetera&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2015-12-25T05:05:16.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10297" PostTypeId="1" AcceptedAnswerId="10304" CreationDate="2015-12-25T14:19:52.103" Score="8" ViewCount="705" Body="&lt;p&gt;Are there any mathematical departments which present the course &quot;elementary physics&quot; for pure math undergraduate students, separately? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a way to present this course with the most pure mathematical manner?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember finding a PDF file, many years ago, titled &quot;Lectures on physics&quot;. It was a graduate level lecture, written by a mathematician. In that lecture, one could not find any technical physical words. Instead, it contained only mathematical descriptions of physical systems, for example: Hamiltonian system, symplectic geometry, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately I lost that PDF file and cannot find it again. It was a lecture from Michigan university. I do not even remember the name of that author but if I am not mistaken his name was 'Idaglu' (or some thing like this) from Michigan. Can anyone help me find that lecture again?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an undergraduate version of the above graduate level lecture?&lt;/p&gt;&#xA;" OwnerUserId="6154" LastEditorUserId="4746" LastEditDate="2015-12-30T10:46:13.137" LastActivityDate="2019-04-13T07:34:55.333" Title="Elementary physics course for pure math student" Tags="|undergraduate-education|mathematical-pedagogy|reference-request|physical-sciences|teaching|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10298" PostTypeId="2" ParentId="10289" CreationDate="2015-12-25T18:10:18.757" Score="1" Body="&lt;p&gt;Do hand-waving-in-the-air lectures building up to the calculus of variations.  Then get them to &lt;em&gt;build&lt;/em&gt; two slopes - one with a straight path and one with a cycloid path that they can actually roll two balls down side-by-side to &lt;em&gt;see&lt;/em&gt; the fastest one for real.&lt;/p&gt;&#xA;" OwnerUserId="5783" LastActivityDate="2015-12-25T18:10:18.757" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10300" PostTypeId="2" ParentId="10094" CreationDate="2015-12-26T08:51:36.333" Score="3" Body="&lt;p&gt;Start with $\sin^2(x) + \cos^2(x) = 1$, which is defined for $0\leq x\leq 2\pi$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then divide through by $\cos^2(x)$ to get $\tan^2(x) + 1 = \sec^2(x)$, which is not defined when either of the denominators $= 0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rearrange to get $\sec^2(x) - \tan^2(x) = 1$, as per the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason it is an identity is because it derives from an identity but is undefined when trying to divide by zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I always taught an identity with an = sign with 3 bars ($\equiv$). This may not be common in other parts of the world, as suggested by some of the answers above, which are a bit stuck on the meaning and purpose of the 'ordinary' = sign: I suggest don't use it.&lt;/p&gt;&#xA;" OwnerUserId="6082" LastEditorUserId="2074" LastEditDate="2015-12-26T10:52:30.427" LastActivityDate="2015-12-26T10:52:30.427" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10302" PostTypeId="2" ParentId="7877" CreationDate="2015-12-26T18:12:30.160" Score="4" Body="&lt;p&gt;'Thinking Mathematically' by Mason, Burton and Stacey sounds like a good match. It has a large collection of problems/investigations using high-school level maths, and discusses how to go about the thinking process of doing maths. It isn't about contest maths, but my understanding is that contest mathematics is not the same thing as university maths (some people are good at one and not the other, so some of your first group might also benefit from your proposed class).&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2015-12-26T18:12:30.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10303" PostTypeId="2" ParentId="7877" CreationDate="2015-12-27T00:09:57.013" Score="1" Body="&lt;p&gt;Perhaps the &lt;a href=&quot;https://openstaxcollege.org&quot; rel=&quot;nofollow&quot;&gt;OpenStax&lt;/a&gt; textbooks (discovery-based, available for free, made up of independent chapters to mix and match) serve as a starting point.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2015-12-27T00:09:57.013" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10304" PostTypeId="2" ParentId="10297" CreationDate="2015-12-27T14:02:31.953" Score="6" Body="&lt;p&gt;While I did not attend any &lt;em&gt;elementary&lt;/em&gt; physics courses by mathematicians for mathematicians, I did attend a curriculum of courses called &quot;Mathematical Physics&quot; held by mathematicians for mathematicians and physicists. In terms of physics and mathematics, they were some kind of advanced level course - it helped if you knew linear algebra, calculus and the basic concepts of theoretical physics. The courses were:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Classical Mechanics (including geometrical aspects, dynamical systems and Hamiltonian and Lagrangian mechanics). The lecture notes (in German) can be found &lt;a href=&quot;http://www.physikmethoden.de/uploads/1/1/0/9/11093021/mechanik_knauf_seiler.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Statistical Mechanics (including topics like entropy and phase transitions). The lecture notes (in German) can be found &lt;a href=&quot;https://www.studium.math.fau.de/fileadmin/min/users/knauf/Skripte/mp2_script.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Quantum Theory/Operator Theory for Hilbert spaces: Based on the books by Reed/Simon, this was a 1.5-year course providing all the important tools for quantum mechanics and dipping into some interesting aspects of quantum mechanics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Quantum Information Theory. There are preliminary lecture notes by A. Knauf in German, which I'm not going to link here.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="108" LastEditorUserId="376" LastEditDate="2019-04-13T07:21:23.990" LastActivityDate="2019-04-13T07:21:23.990" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="10306" PostTypeId="2" ParentId="7877" CreationDate="2015-12-27T22:18:08.500" Score="1" Body="&lt;p&gt;One of my favorite question for this kind of situation is: &quot;what is the triangle with least perimeter among triangles of given area?&quot;. There are several ways to attack it, problem reduction is a must, solutions may use geometry, analysis and basic topology, and it has so many follow-ups (&quot;what about quadrilaterals?&quot;) that you can feed student's curiosity for years, and more (&quot;what about domains in a 5-dimensional simply connected manifold of non-positive curvature?&quot; is an open research question).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2015-12-27T22:18:08.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10307" PostTypeId="1" AcceptedAnswerId="10311" CreationDate="2015-12-28T22:00:25.973" Score="10" ViewCount="4613" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://philosophy.stackexchange.com/a/540/8572&quot;&gt;[Source:]&lt;/a&gt; Having studied maths gives you a particular way of thinking through problems that Intro to Logic just doesn't.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Will someone please explain and explicate the quote above? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please pardon me if I ought to have posted this in Philosophy SE; I posted here because I am conjecturing that you have more scholars of maths than Philosophy SE?&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="-1" LastEditDate="2017-04-13T12:42:23.083" LastActivityDate="2015-12-29T03:22:29.950" Title="What does maths teach you that logic does not?" Tags="|logic|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10308" PostTypeId="2" ParentId="10235" CreationDate="2015-12-28T22:54:07.093" Score="5" Body="&lt;p&gt;The other answers may indeed be right, but another thing just occurred to me, namely that when they prove the &lt;em&gt;base&lt;/em&gt; case of the induction, the sum on the left-hand side &lt;em&gt;does&lt;/em&gt; generally reduce to a single term (in my example, $0^2$).  Perhaps they then approach the inductive step by looking at the base case and saying &quot;okay, wherever there was a $0$, now I put an $n$&quot; rather than correctly looking all the way back to the original statement to be proven.  (Of course, different students might be making the same mistake for different reasons.)&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2015-12-28T22:54:07.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10309" PostTypeId="2" ParentId="10307" CreationDate="2015-12-29T00:53:50.537" Score="7" Body="&lt;p&gt;I think the only sense in which the quote is accurate is if you interpret &quot;maths&quot; broadly and &quot;Intro to Logic&quot; narrowly. Intro to Logic would only introduce limited proof techniques tailored to elementary logic. Mathematics is far richer in both content and techniques than would be encountered in Intro to Logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you replace &quot;Intro to Logic&quot; with &quot;mathematical logic,&quot; then the claim is&#xA;less supportable. Certainly mathematical logic has a content more narrow than all of mathematics, but this is true for any field of mathematics. The proof techniques used throughout mathematical logic are quite broad, although certainly not encompassing all proof techniques encountered throughout mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get a sense of the breadth of mathematic logic, look at the answers to&#xA;the MO question, &lt;a href=&quot;https://mathoverflow.net/q/227083/6094&quot;&gt;&quot;What are some important but still unsolved problems in mathematical logic?&quot;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2015-12-29T00:53:50.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10310" PostTypeId="2" ParentId="10307" CreationDate="2015-12-29T01:14:17.110" Score="3" Body="&lt;p&gt;To supplement @JosephO'Rourke's good answer, I would note that the &quot;logicist&quot; philosophy of mathematics is by no means inclusive of mathematical practice... by a long shot... despite various enthusiasts' tendency to implicitly or explicitly claim that first-order logic is the underpinning for mathematics, that formal set theory (axiomatized in first-order logic) is the next layer, ... and, supposedly, everything else is superstructure. An afterthought.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, srsly, apart from the haute-bourgeois pretenses (however clever) of the (slave-holding, not-working...) classical Greeks, it is easily arguable that the true appeal of mathematics for human civilization is its explanatory power about actual phenomena in the world... even phenomena human-abstracted from chaotic physical events.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would claim that &quot;mathematical logic&quot;, and, to a much lesser degree, &quot;logic&quot; in a more general sense, is a program of backfilling upon realization of weaknesses in prior understanding. Not with the expectation that very much is truly in error, but that the same truths need deeper foundations from some viewpoint...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, in particular, to only look at &quot;(mathematical) logic&quot; is to look at a sort of symptom rather than cause or most substantial effect. Thus, analysis of this symptom, no matter how energetic, can scarcely manage to divine the real events, causes, and causalities.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2015-12-29T01:14:17.110" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10311" PostTypeId="2" ParentId="10307" CreationDate="2015-12-29T02:53:42.137" Score="6" Body="&lt;p&gt;In the United States, some universities offer &lt;strong&gt;&quot;Introduction to Logic&quot;&lt;/strong&gt; courses.  These courses are often offered to undergraduates who are not majoring in mathematics, as a way that the undergraduates can satisfy requirements that they take a certain number of &quot;math&quot; and/or &quot;philosophy&quot; classes.  Some of these courses &lt;a href=&quot;https://faculty.washington.edu/smcohen/120/syllabus.htm&quot;&gt;teach&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A notation for representing logical statements.  This notation is very compact, with single characters representing things like &quot;is a member of&quot;, &quot;is not true&quot;, &lt;em&gt;et cetera&lt;/em&gt;.  Unfortunately, this notation is not the same as the notation used by computer programmers, nor is much of this notation used by many non-mathematicians.&lt;/li&gt;&#xA;&lt;li&gt;How to set up simple logic puzzles using the notation.&lt;/li&gt;&#xA;&lt;li&gt;Some proofs of basic concepts.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Similar length &lt;strong&gt;Math and Computer Science courses cover a lot more.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice, a good introductory university-level course in relational databases covers about four times as much material, and may cover (in an incidental manner) all of the logic puzzles and basic concepts taught in the &quot;Introduction to Logic&quot; course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, many secondary schools teach &quot;Geometry&quot; as a &quot;how to write a mathematical proof&quot; class.  The proofs are at a level that non-mathematicians consider thorough; mathematicians often have much more detailed proofs for the same concepts.  In the process of doing these proofs, the students learn most of the logic concepts taught in the &quot;Introduction to Logic&quot; course discussed above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Statistics&lt;/strong&gt; courses teach approaches that allow drawing inferences with less than 100% confidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A good math course teaches &quot;ways of thinking through problems&quot;&lt;/strong&gt; that are not taught in the &quot;Introduction to Logic&quot; course discussed above.  For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Given a story problem (written in words that talk about a real-life problem), how to identify aspects of the problem that can be measured and compared.&lt;/li&gt;&#xA;&lt;li&gt;How to identify what you want to know about a problem.&lt;/li&gt;&#xA;&lt;li&gt;How to identify things you already know about a problem.&lt;/li&gt;&#xA;&lt;li&gt;Drawing a picture of the problem.&lt;/li&gt;&#xA;&lt;li&gt;Labelling variables.&lt;/li&gt;&#xA;&lt;li&gt;Identifying directions in which variables increase.&lt;/li&gt;&#xA;&lt;li&gt;Identifying relationships between variables.&lt;/li&gt;&#xA;&lt;li&gt;Identifying the precision to which variables can be accurately measured.&lt;/li&gt;&#xA;&lt;li&gt;Identifying &quot;field conditions&quot; and &quot;boundary conditions&quot; that restrict the possible answers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The process of using the relationships (found above) to solve for what you want to know follows a logical process.  This process often includes naming intermediate concepts, and looking at things from multiple points of view.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Explaining what the answer means.&lt;/li&gt;&#xA;&lt;li&gt;Confirming that the answer is consistent with the original information.  (A.k.a. &quot;Check by substitution&quot; and &quot;sanity checks&quot;.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&quot;Checks by substitution&quot; follow a logical process.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2015-12-29T03:22:29.950" LastActivityDate="2015-12-29T03:22:29.950" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10312" PostTypeId="1" AcceptedAnswerId="10321" CreationDate="2015-12-29T04:19:29.557" Score="4" ViewCount="225" Body="&lt;p&gt;The following is from &lt;a href=&quot;http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty&quot; rel=&quot;nofollow noreferrer&quot;&gt;an article in &lt;em&gt;The New Yorker&lt;/em&gt; on Y. Zhang and his proof on gaps between primes&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://www.math.rutgers.edu/people/?id=175&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rutgers University Professor [Henryk] Iwaniec&lt;/a&gt; and his friend, John Friedlander, a professor at the University of Toronto, read with increasing attention. &lt;strong&gt;“In these cases, you don’t read A to Z,”&lt;/strong&gt; Iwaniec said. $\color{green} { \text {  “You look first at where is the idea.&quot;  }   }$ There had been nothing written on the subject since 2005. The problem was too difficult to solve. As we read more and more, the chance that the work was correct was becoming really great. Maybe two days later, we started looking for completeness, for connections. A few days passed, we’re checking line by line. The job is no longer to say the work is fine. We are looking to see if the paper is truly correct.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am confused: If &lt;strong&gt;you don’t read A to Z&lt;/strong&gt;, then how is it possible to $\color{green} { \text {  “look first at where is the idea&quot;  }   }$? &lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user155" LastEditDate="2017-01-27T06:11:36.000" LastActivityDate="2017-01-27T06:11:36.000" Title="If you do not 'read A to Z', then how can you discover the idea?" Tags="|proofs|self-learning|" AnswerCount="2" CommentCount="5" ClosedDate="2015-12-31T04:14:52.420" ContentLicense="CC BY-SA 3.0" />
  <row Id="10315" PostTypeId="1" AcceptedAnswerId="10316" CreationDate="2015-12-29T13:48:35.360" Score="6" ViewCount="461" Body="&lt;p&gt;Given the rational function $f(x)=\frac{x^2-1}{x+1}$. The expression can be simplified to $g(x)=x-1$ and thus the singularity at $x=-1$ is removed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would personally claim that $f$ and $g$ are the &lt;strong&gt;same&lt;/strong&gt; function and that $f$ is defined on all $\mathbb R$ with no discontinuities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in some text book I read, the author stated that by doing so, one obtained the continuous extension of $f$ and would &lt;strong&gt;remove&lt;/strong&gt; the discontinuity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware that this question is somehow opinion based, but: Where does the creation of a new (extended) function start and where are we just refactoring terms while maintaining a given function?&lt;/p&gt;&#xA;" OwnerUserId="6184" LastEditorUserId="143" LastEditDate="2015-12-29T17:43:55.400" LastActivityDate="2015-12-30T03:58:33.973" Title="Is simplifying a rational function considered as a continuous extension?" Tags="|secondary-education|functions|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10316" PostTypeId="2" ParentId="10315" CreationDate="2015-12-29T16:06:49.543" Score="8" Body="&lt;p&gt;I would say they are definitely not the same function.&#xA;Let's take an even simpler example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$h(x)=\frac{x}{x}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This function is defined for all $x \ne 0$, and at such points it is equal to $1$.  But it is &lt;em&gt;not the same function&lt;/em&gt; as the constant function&#xA;$$k(x)=1$$&#xA;because the two functions have different domains.  See also my answer at &lt;a href=&quot;https://math.stackexchange.com/questions/1525054/why-are-removable-discontinuities-even-discontinuities-at-all/1525156#1525156&quot;&gt;https://math.stackexchange.com/questions/1525054/why-are-removable-discontinuities-even-discontinuities-at-all/1525156#1525156&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2015-12-29T16:06:49.543" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10317" PostTypeId="2" ParentId="10315" CreationDate="2015-12-29T16:55:23.320" Score="0" Body="&lt;p&gt;This is really a matter of convention. But a usual convention is that for polynomials  $g,h$ one considers the rational mapping&lt;br&gt;&#xA;$$\frac{g(x)}{h(x)}$$ &#xA;to have domain $\mathbb{R}\setminus \{x \colon h(x)\neq 0\}$ and likewise in the case one is working over the complex numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could also set things up so that one reduces &#xA; $$\frac{g(x)}{h(x)}= \frac{\overline{g}(x)}{\overline{h}(x)}$$ &#xA;where $\overline{g}, \overline{h}$ are co-prime and then  have domain $\mathbb{R}\setminus \{x \colon \overline{h}(x)\neq 0\}$ but this is just not what is common. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not only but also as generalizing from polynomials $g,h$ to more general functions there is no clear-cut analog for the co-prime condition. &#xA;Say already with &#xA;$$\frac{x}{\sin(x)}$$&#xA;it feels more natural to consider this as defined on $\mathbb{R} \setminus \pi \mathbb{Z}$ rather than to special case $0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is however true that as elements of the algebraic structure field of rational functions $\mathbb{R}(X)$ the two expressions &#xA;$X-1$ and and $\frac{X^2-1}{X+1}$ denote the very same element.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;To sum it up:&lt;/strong&gt; the convention is to read such expressions as quotient of the &quot;visible&quot; (polynomial) maps and not as representing an elements of the field of rational functions to which then a map is assigned to.  &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2015-12-29T16:55:23.320" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10318" PostTypeId="1" CreationDate="2015-12-29T16:55:45.420" Score="9" ViewCount="415" Body="&lt;p&gt;I have a handful of high school students that are all prospective math/physics majors and have pooled their resources to hire me to teach them a proof based math course because it has become apparent to them in my physics class that proof and derivations are important. Basically I meet with them 2 hours a week and run it like a socratic method or students have to prove the theorems with my limited guidance and so far I have covered the following topics:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;GCD&lt;/li&gt;&#xA;&lt;li&gt;Euclid's Lemma&lt;/li&gt;&#xA;&lt;li&gt;Well Ordered Principle and Mathematical Induction&lt;/li&gt;&#xA;&lt;li&gt;Fundamental Theorem of Arithmetic&lt;/li&gt;&#xA;&lt;li&gt;Theorems of Elementary Arithmetic a*0=0, (-a)(-b)=ab etc.&lt;/li&gt;&#xA;&lt;li&gt;Arithmetic mean - Geometric Mean Inequality&lt;/li&gt;&#xA;&lt;li&gt;Pythagorean Theorem&lt;/li&gt;&#xA;&lt;li&gt;Cauchy Schwartz Inequality&lt;/li&gt;&#xA;&lt;li&gt;Irrationality of sqrt(p) where p is prime&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I will have only a limited amount of time with these students and I need to decide on what topics would be most valuable for them to be exposed to, here is the list of topics, which do you think would be most valuable:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Conic Sections - going from the geometric definitions to the algebraic representations of conics&lt;/li&gt;&#xA;&lt;li&gt;Proofs of Archimedes: Areas of Circle, Quadrature of the Parabola, On the Sphere and Cylinder&lt;/li&gt;&#xA;&lt;li&gt;Exploring the completeness property of reals&lt;/li&gt;&#xA;&lt;li&gt;Sets, Nested Intervals and the Uncountability of the Reals&lt;/li&gt;&#xA;&lt;li&gt;Exploring sequences and series - in particular using telescoping series to derive Σi, Σi^2 , etc&lt;/li&gt;&#xA;&lt;li&gt;Area under curves Riemann Sums&lt;/li&gt;&#xA;&lt;li&gt;Counting and Binomial Theorem&lt;/li&gt;&#xA;&lt;li&gt;Sets and the Axioms of Probability&lt;/li&gt;&#xA;&lt;li&gt;Limits, Continuity (delta-epsilon proofs)&lt;/li&gt;&#xA;&lt;li&gt;Differentiability&lt;/li&gt;&#xA;&lt;li&gt;Properties of Exponential and Logarithmic Function using power series definition of the exponential function&lt;/li&gt;&#xA;&lt;li&gt;Vectors, Vector Spaces, Linear Operators&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6185" LastEditorUserId="6185" LastEditDate="2022-01-10T04:18:36.173" LastActivityDate="2022-01-10T04:18:36.173" Title="Advice on Proof-based Math Topics for High Schoolers" Tags="|proofs|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="10319" PostTypeId="2" ParentId="10315" CreationDate="2015-12-29T16:56:52.120" Score="3" Body="&lt;p&gt;The expression $f(x) = \frac{x^2 - 1}{x + 1}$ is a formula (with two free variables, $f$ and $x$), not a function. It is not the same formula as $f(x) = x - 1$, but neither is $1 + 1 = 2$ the same formula as $2 = 1 + 1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By convention, such formulas are sometimes taken to represent functions in contexts where the domain and codomain can be inferred from context. In this case, one could say more precisely that $f: \mathbb{R} \setminus \{-1\} \to \mathbb{R}$ is the function defined for each $x \in \mathbb{R} \setminus \{-1\}$ by $f(x) = \frac{x^2 - 1}{x + 1}$. This is a continuous function. (The &quot;missing point&quot; in the domain, $-1$, is a (removable) &lt;em&gt;singularity&lt;/em&gt;, not a discontinuity; it makes no sense to talk about continuity of a function outside its domain.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we instead tried to say that $f: \mathbb{R} \to \mathbb{R}$ is the function defined for each $x \in \mathbb{R}$ by $f(x) = \frac{x^2 - 1}{x + 1}$, we have spoken nonsense, because the expression $\frac{x^2 - 1}{x + 1}$ has no meaning if $x = -1$. However, we could instead take the domain to be literally any subset of $\mathbb{R} \setminus \{-1\}$, or even a subset excluding $-1$ of some other field (who said we're working in the real numbers?), provided we change the codomain to match.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, given any $D \subseteq \mathbb{R}$, one can say $g: D \to \mathbb{R}$ is the function defined for each $x \in D$ by $g(x) = x - 1$. Of course, this is a different function for each choice of $D$; from a structural standpoint, it doesn't even make sense to compare functions with different domains or codomains (and if one works in a framework where doing so does make sense, they're never equal anyway).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, if we choose $D = \mathbb{R}$, then $f$ and $g$ are not the same function. But they are related, in that $f$ is the restriction of $g$ to the domain of $f$; in other words, $g$ is an extension of $f$ (and, as it happens, a continuous extension).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if we choose $D = \mathbb{R} \setminus \{-1\}$, then $f$ and $g$ are literally the same function, because they have the same domain and codomain, and $x - 1$ and $\frac{x^2 - 1}{x + 1}$ are the same number for every $x$ in that domain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if we interpret $f$ and $g$ not as &lt;em&gt;functions&lt;/em&gt;, but as &quot;rational functions in one indeterminate $x$&quot;, i.e., elements of the field $\mathbb{R}(x)$, then $f = g$. Note, however, that &quot;rational functions&quot; in this sense are not functions, but equivalence classes of formal fractions of polynomials.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2015-12-29T16:56:52.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10320" PostTypeId="2" ParentId="10312" CreationDate="2015-12-29T17:45:34.690" Score="4" Body="&lt;p&gt;As Jon Bannon says in his answer, to the expert there should be a lot of &quot;standard&quot; techniques that are very familiar and can be at least recognized at a glance. Then there should be new techniques that are unfamiliar and give new leverage. You can sort of imagine this if you were reading a text in English, and then suddenly there's a part in italicized Latin -- it would stand out to you as important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best-written papers should make the &quot;new ideas&quot; plain up front, possibly in the abstract, introduction, or header statement(s). Richard Lipton writes about this frequently and has called this the &quot;Elevator Pitch&quot;. Some articles on the subject that I found interesting to read:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://rjlipton.wordpress.com/2009/02/16/proofs-and-elevator-rides/&quot;&gt;&quot;Proofs and Elevator Rides&quot;&lt;/a&gt; (Richard Lipton)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://rjlipton.wordpress.com/2010/07/11/proving-a-proof-is-a-proof/&quot;&gt;&quot;Proving a Proof is a Proof&quot;&lt;/a&gt; (Richard Lipton)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.scottaaronson.com/blog/?p=458&quot;&gt;&quot;Eight Signs a Claimed P≠NP Proof Is Wrong&quot;&lt;/a&gt; (Scott Aaronson)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2015-12-29T20:16:21.927" LastActivityDate="2015-12-29T20:16:21.927" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10321" PostTypeId="2" ParentId="10312" CreationDate="2015-12-29T19:10:54.950" Score="6" Body="&lt;p&gt;I think this question, with editing, has a lot to do with mathematics education, in that it points to the problem of learning to read and interpret mathematical writing. I would instead ask something like: &quot;How does one glean the ideas in a mathematical paper without reading it line by line?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now for a somewhat rambling answer to this question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that mathematical work is steeped in context, and that without familiarity with the standard ideas and tricks surrounding the paper in question, it is nearly impossible to find the idea in the paper. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reading line by line can allow the reader to perhaps, with a LOT of work, reconstruct a consistent mental model that reproduces the results of the paper (perhaps this is what mathematics is, making such models). What is meant by &quot;first finding the idea&quot; is that much of what is written in a paper is standard in the community of persons working on the problems in question. The new idea in the paper stands out against the standard &quot;background noise&quot;. In order to find it, you have to first know what is standard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stated another way: If you are using only standard techniques tried by many experts to solve a reticent problem, then you are very likely to be wrong. So your proposed solution is first evaluated by whether it is novel against the &quot;background noise&quot; and whether it hangs together logically as a strategy. A mathematical proof has a basic logical outline that contains &quot;the idea&quot;, and much of the detail needed to complete the idea is necessary tinkering with standard techniques to make it all work out. I hope this helps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for mathematics education, this is very important insofar as it encodes &quot;big picture thinking&quot;, as opposed to &quot;algorithmic nose-following&quot; that many think is mathematics. See Paul Zeitz's book &quot;The Art and Craft of Problem Solving&quot; for a discussion of strategy, tactic and tool. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without a sense of the living ideas encrypted in the writing of a mathematical paper, which sadly are not found in traditional mathematical writing, one is flying blind. The belief that reading papers line by line will yield an understanding of a subject is pernicious, and this question (if properly asked) has the educational benefit of dealing with that pernicious belief.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Sorry for the behemoth comments…which now appear as the present answer.)&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2015-12-29T19:10:54.950" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10322" PostTypeId="2" ParentId="10315" CreationDate="2015-12-30T03:58:33.973" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Given the rational function $f(x)=\frac{x^2-1}{x+1}$. The expression&#xA;  can be simplified to $g(x)=x-1$ and thus the singularity at $x=-1$ is&#xA;  removed.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I would personally claim that $f$ and $g$ are the &lt;strong&gt;same&lt;/strong&gt; function&#xA;  and that $f$ is defined on all $\mathbb R$ with no discontinuities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is false (and not a matter of opinion). When $x = -1$, then $f(x)$ is &lt;em&gt;undefined&lt;/em&gt; (not any number), but $g(x) = -2$. They are clearly not the same function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can only make the simplification $\frac{x^2-1}{x+1} = x-1$ so long as &lt;em&gt;$x$ is not $-1$&lt;/em&gt;. You sort of admit as much when you use the language &quot;the singularity at $x=-1$ is removed&quot;, which recognizes that some change has been made. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may the most common error at the level of precalculus? &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Whenever one learns a new mathematical operation, it is imperative&#xA;  also to learn the limitations under which the operation may be&#xA;  performed. Lack of this additional knowledge can lead to the&#xA;  employment of the new operation in a blindly formal manner in&#xA;  situations where the operation is not properly applicable, perhaps&#xA;  resulting in absurd and paradoxical conclusions. Instructors of&#xA;  mathematics see mistakes of this sort made by their students almost&#xA;  every day... (Howard Eves, &lt;em&gt;Great Moments in Mathematics&lt;/em&gt;, Lecture 32)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2015-12-30T03:58:33.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10323" PostTypeId="2" ParentId="7757" CreationDate="2015-12-30T04:05:47.893" Score="3" Body="&lt;p&gt;I can't speak to existing research. But, I can't help but think of Frenkel's account of the community of mathematicians in the time of the cold war. The idea of meeting at night with a large group of practicing mathematicians to engage in mathematical conversation, guided by the masters, it's fascinating. Perhaps this still exists at some elite universities, but, in my limited experience the closest thing is a department colloquium. I suppose in graduate school, there is a time before the thesis needs to be written and jobs need to be found, when not studying for quals, where we just brainstorm with our peers. But, it seems to me, the time and social aspect of the Russian school is much more serious. It seems math was their priority.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, from a math perspective, that is utopia. In other words, the MSE and MO are a bit of utopia for us to enjoy when the tasks of life allow us leave to play.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other aspect of Frenkel's account which merits our attention is the mentor he acquired early in his highschool (I forget the precise details, and one of my students currently has my copy of &lt;em&gt;Love and Math&lt;/em&gt; which you should read if you haven't already). This mentor skipped past all the usual topics and immediately challenged him with interesting, cutting edge, math. Now, obviously the mentor had to know that Frenkel had unusual ability. It's interesting to read about how the mentor enticed Frenkel deeper and deeper into the world of beautiful mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me collect some guidelines which are useful. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Find some community of mathematics where you can sharpen your math by the constructive criticism of others. Math is a social enterprise at a certain level, it is nice to have some sense of what other like-minded individuals find worthwhile. Here I mostly mean for the student to develop a mature style, their ideas must ultimately be their own. A good REU experience, or a cluster of excellent students can go a way towards the goal I outline here. It's equally important to learn how not to be a jerk, and, yet, to still say something.&lt;/li&gt;&#xA;&lt;li&gt;Find a master. Learn everything they know. Repeat.&lt;/li&gt;&#xA;&lt;li&gt;Free your mind of this nonsense like &quot;I'm an applied mathematician&quot; or &quot;I like algebra, but not analysis&quot; or (my vice early on) &quot;I'm just interested in calculations, tell me how to calculate&quot;. Embrace a liberal arts education in &lt;strong&gt;mathematics&lt;/strong&gt;. Learn everything.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;You might criticize my answer in that it is not about a community, it's really just advice for an individual. But, that is my last and most important point: one size does not fit all. Education, especially utopian, must be individualized. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is why homeschooling, when done correctly, will dominate conventional schooling. Especially now that we have shackled teachers from enforcing difficult standards...but, I probably digress. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2015-12-30T04:05:47.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10324" PostTypeId="1" AcceptedAnswerId="10332" CreationDate="2016-01-01T14:13:57.587" Score="7" ViewCount="261" Body="&lt;p&gt;This semester I am teaching a basic geometry course for design students (interior and industrial), which I am making strongly project-based since I believe they need to learn how to use geometry &quot;in the field&quot; rather than do too many constructions and proofs on paper.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a result, it doesn't seem appropriate to feature traditional exams heavily, but at the same time I do want to ensure that people become proficient at basic algebra and geometry, so I need something to fill the role of exams.  I was thinking about using Khan Academy badges, but when I go in there to take a look, it appears impossibly broad and complicated to use, and I can't figure out whether there is some streamlined way to have someone earn a badge in, say, lines and slopes without wading through an infinity of questions about subtracting two decimal numbers, scrolling through knowledge maps, etc etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a bit lost!  I still like the concept of badges.  I'm not sure if I am missing something, or if Khan Academy just is (what seems to me in light of my intended use) cumbersome by nature and perhaps not best suited to my purpose.  Has anyone used Khan or similar for the purpose I have in mind?  What was your experience?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you, and happy new year! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:  I realized I should have clarified that these are undergraduate interior and industrial design majors at a polytechnic institute.&lt;/p&gt;&#xA;" OwnerUserId="4906" LastEditorUserId="4906" LastEditDate="2016-01-02T23:58:55.277" LastActivityDate="2016-01-03T12:04:12.993" Title="Experience using Khan Academy badges for basic math courses" Tags="|technology-in-education|geometry|exams|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10325" PostTypeId="1" CreationDate="2016-01-01T14:22:12.373" Score="7" ViewCount="216" Body="&lt;p&gt;In John Dewey's &lt;em&gt;&lt;a href=&quot;http://ruby.fgcu.edu/courses/ndemers/colloquium/experienceducationdewey.pdf&quot; rel=&quot;nofollow&quot;&gt;Experience and Education&lt;/a&gt;&lt;/em&gt;, he proposes a qualitative measure for an experience (please simply understand this in common usage for the purposes of this question) to be &lt;em&gt;educative&lt;/em&gt;. Namely, an experience is educative to the extent that it opens the possibility for further experiences. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the following interesting quotation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The belief that a genuine education comes about through experience does not mean that all experiences are genuinely or equally educative. Experience and education cannot be directly equated to each other. For some experiences are miseducative. Any experience is miseducative that has the effect of arresting or distorting the growth of further experience. An experience may be such as to engender callousness; it may produce lack of sensitivity and of responsiveness. Then the possibilities of having richer experience in the future are restricted. Again, a given experience may increase a person's automatic skill in a particular direction and yet tend to land him in a groove or rut; the effect again is to narrow the field of further experience. An experience may be immediately enjoyable and yet promote the formation of a slack and careless attitude; this attitude then operates to modify the quality of subsequent experiences so as to prevent a person from getting out of them what they have to give. Again, experiences may be so disconnected from one another that, while each is agreeable or even exciting in itself, they are not linked cumulatively to one another. Energy is then dissipated and a person becomes scatter- brained. Each experience may be lively, vivid, and &quot;interesting,&quot; and yet&#xA;  their disconnectedness may artificially generate dispersive, disintegrated, centrifugal habits. The consequence of formation of such habits is inability to control future experiences. They are then taken, either by way of enjoyment or of discontent and revolt, just as they come. Under such circumstances, it is idle to talk of self-control.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Reuben Hersh's position that &lt;a href=&quot;http://bookstore.ams.org/MBK-83&quot; rel=&quot;nofollow&quot;&gt;Pragmatism provides a decent philosophy of Mathematical Practice&lt;/a&gt; leads one to wonder whether the overarching goal of mathematics education ought to aim to train students to find highly educative mathematical experiences (i.e. good problems and questions?) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Despite the heavy preamble above, I'd like some references to the math education literature that develops connections with Dewey's above point of view. I am aware of, and very much in favor of, &lt;a href=&quot;https://en.wikipedia.org/wiki/Inquiry-based_learning&quot; rel=&quot;nofollow&quot;&gt;IBL&lt;/a&gt; methods in mathematics education. I'd be interested in papers related to problem-posing. I am particularly interested in studies of activities aimed at &lt;strong&gt;teaching students to evaluate a problem or direction of inquiry for fecundity.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify this last point a bit, I wonder if anyone has studied the ability of students to work on problems with a view toward seeing how many distinct interesting ideas they generate, rather than with an eye toward solving the problems. This sort of evaluation is somewhat essential to the practice of (pure) mathematics. The ability to &lt;em&gt;find&lt;/em&gt; problems that generate interesting mathematics and abandon those problems that don't should be a central emphasis of an experience (inquiry?)-based approach to mathematics education. Of course this depends on individual student experience and maturity…but it seems that this is something that should be actively assessed as a primary goal of instruction. In fact, the ability of a student to perform this selection in a way closer to a professional mathematician may be deemed a major criterion for measuring &quot;mathematical maturity&quot;. Dewey's above quote and the related discussion may reinforce this point.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="2083" LastEditDate="2016-01-05T10:47:24.050" LastActivityDate="2016-01-05T10:47:24.050" Title="John Dewey and Educative Mathematical Experience" Tags="|reference-request|inquiry-based-learning|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10326" PostTypeId="1" CreationDate="2016-01-01T14:39:06.337" Score="4" ViewCount="258" Body="&lt;p&gt;&lt;em&gt;This is not a question; so I perfectly understand that it could be deleted almost immediately.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yesterday, I was attending a New Year eve party and I was introduced to a young and brilliant lawyer. Hearing that I am (&lt;em&gt;supposed to be&lt;/em&gt;) a mathematician, he seriously told me : &quot;So, could you, by the end, explain to me why is $ax^2+bx+c$ always equal to $0$?&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I took it as a very good joke. The problem is that this was not a joke at all : the guy was serious !!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not an educator but I think that there are a lot of questions coming out from it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would really appreciate your opinions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since questions are required : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;what can we do to make better understanding of mathematics starting with young people&lt;/strong&gt; ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;how can we make mathematics more attractive ?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="4993" LastEditorUserId="4993" LastEditDate="2016-01-01T17:07:20.730" LastActivityDate="2016-01-01T17:07:20.730" Title="And the joke was NOT a joke" Tags="|general-pedagogy|students-mistakes|" AnswerCount="0" CommentCount="18" ClosedDate="2016-01-01T16:54:12.040" ContentLicense="CC BY-SA 3.0" />
  <row Id="10327" PostTypeId="2" ParentId="10325" CreationDate="2016-01-01T18:02:03.633" Score="2" Body="&lt;p&gt;Not exactly a &quot;study&quot;, but you may be interested in &lt;a href=&quot;http://eric.ed.gov/?id=EJ1000869&quot; rel=&quot;nofollow&quot;&gt;a short article&lt;/a&gt; I wrote with Deborah Moore-Russo a few years back that touches on issues relating to problem-posing, in particular the last section (&quot;Students as researchers&quot;) seems to be pertinent.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-01-01T18:02:03.633" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10328" PostTypeId="2" ParentId="4424" CreationDate="2016-01-01T18:16:13.683" Score="5" Body="&lt;p&gt;While the first point is fairly straight-forward to address, I suspect the other two will not be easy to change. The whole set-up of schooling (in the UK at least) generally favours getting the answer right, at least in the short term, so the incentive for the students is to use the book to obtain their answers rather than to benefit their learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it's likely that there will always be some students who are lazy, and no amount of effort on your part will make them want to use the answers correctly for their own sake. However, there are probably some other students who will rely on the answers too much for other reasons. Talking these through with the students might be helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One reason might be a lack of confidence. If you are afraid of always getting things wrong, you are likely to keep checking how you are doing, to avoid making too big an error. The aim-of-the-game for you is to minimise risk, rather than to maximise potential gain. One thing such students might need telling is that textbooks can contain errors, because if they encounter a wrong answer they cannot match, they may well assume they themselves are wrong and lose hope. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another reason might be a misunderstanding of what the aim of mathematics is. Many students will have picked up from their schooling that mathematics is about performing meaningless calculations rapidly and accurately, and that the rewards come for doing so. Even if they personally would prefer to take the time to build understanding, they may feel that they are not allowed to do so. It could help to tell the students that you value their understanding of the material, not just a line of correct answers. I'd be tempted to try and show that in action too, perhaps by giving them questions to answer of the form 'what have you learned in the last lesson/week/month', or by encouraging and allowing students to discuss their intuition with each other.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-01-01T18:16:13.683" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10329" PostTypeId="2" ParentId="4424" CreationDate="2016-01-01T22:38:09.463" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What guidelines can I give my students so that they can use the answer&#xA;  key in the back of their books correctly?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here's what I do: Document it on the syllabus. Tell them explicitly on the first day of class. Occasionally remind them at the start of a class session (e.g., &quot;Were there any homework problems you had trouble with, or were surprised by the answer at the back of the book?&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that won't solve the problems that you list in the lead-up to your question; other classroom procedures also have to support these learning objectives. In particular: Not just grading on the correct final answer (which is what the possibility of &quot;Plagiarising from the back of the book&quot; sounds like). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my community college classes (mostly remedial, i.e., 6th-8th grade level), I'm dedicated to grading the justification (&quot;work&quot;) for any piece of mathematical writing line-by-line; the final answer is just one guidepost by which we can check whether the procedures are correct or broken. In order to accomplish this, I've had to reduce the amount of homework I grade, currently at &quot;none&quot;, but commit to giving complete scored feedback on any tests. This sends the signal that the &lt;em&gt;logical connections&lt;/em&gt; are the important thing (both to the work itself and communicating it to others), and copying someone else's answers and presenting them as if by magic won't be rewarded. This has to be a daily point of emphasis, and not everyone gets it, but it at least gets some traction with a good proportion of students. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-01-01T22:38:09.463" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10330" PostTypeId="2" ParentId="10324" CreationDate="2016-01-02T20:48:43.410" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://ixl.com&quot; rel=&quot;nofollow&quot;&gt;ixl.com&lt;/a&gt; would be a good try since it is very skills specific.  Students can earn badge for certain skills.  It is not free though.  For individual plan it is around \$80 per student. Add \$40 it would include language. I don't about plan for school.   &lt;/p&gt;&#xA;" OwnerUserId="6199" LastEditorUserId="39" LastEditDate="2016-01-02T21:10:15.507" LastActivityDate="2016-01-02T21:10:15.507" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10331" PostTypeId="2" ParentId="10206" CreationDate="2016-01-03T09:01:52.780" Score="8" Body="&lt;p&gt;I'm not an expert, but I understand the curriculum in Russia has gone back and forth on this point, with a lot of debate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally, grades 1 to 4 are spent on the basic arithmetic operations, while grades 5 and 6 are mainly focused on: familiarizing students with the algebraic properties of &lt;em&gt;numbers&lt;/em&gt; in a concrete way, systematizing and extending knowledge of fractions and decimal numbers, and studying signed numbers. Algebra starts in earnest in grade 7.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the earlier grades, letters are used freely to state the general properties of numbers, to state formulas, and to write down simple equations. But they are not used systematically to manipulate expressions and equations and isolate an unknown as is done in algebra. In the 1970s, there was an effort to introduce algebra in grades 5 and 6 (called grades 4 and 5 at that time, since school then started at age 7). But it was found that by turning the process of solving arithmetic problems into a kind of algorithm, it took away a lot of the opportunity for kids to experiment and develop intuition for numbers. So there was a kind of &quot;back to basics&quot; reaction in the 1980s on this point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me give an example to show what I'm talking about. Your sister had twice as many books as you do until a friend gave her 6 books today, and now she has 24 books. How many books do you have? Before she was given the six books, she had 24 - 6 = 18 books. So you have 18/2 = 9 books. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you compare this solution with the mathematically equivalent steps in solving $2x + 6 = 24$, the difference is that the steps have a concrete meaning in the first solution, but possibly not in the second. So the first solution helps develop intuition for numbers in a way that the second may not. In fact, the second kind of solution is more likely to make sense to you if you are already proficient in the first kind and can therefore make the connection between the two. There needs to be a phase in which kids are encouraged to think concretely about problems rather than turning them into equations and going on auto-pilot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there is a role for equations where they clarify the meaning of an operation. For example, it's useful to say that subtracting $7$ from $12$ means finding the number $x$ for which $7 + x = 12$. Or that dividing $\frac{4}{7}$ by $\frac{3}{5}$ means finding the number $x$ for which $\frac{3}{5} \times x = \frac{4}{7}$. Using formulas like $d = vt$ or $A = \frac{1}{2}bh$ also accustoms students to the use of variables without turning everything into algebra.&lt;/p&gt;&#xA;" OwnerUserId="6201" LastActivityDate="2016-01-03T09:01:52.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10332" PostTypeId="2" ParentId="10324" CreationDate="2016-01-03T12:04:12.993" Score="4" Body="&lt;p&gt;I have used Khan Academy and believe it can work for you.  If you do use Khan Academy here are some tips.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You don't want the students to earn badges, you want the students to master specific skills. Mastering skills, means practicing the skill several times (and getting 5 correct) and then doing a mastery challenge to &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Become the students' coach.  That way you can assign specific skills to the students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should master the skills you want the students to work on, so that you can see what Khan Academy expects of the skills and you can anticipate and understand their questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student needs to go back and work on an old skill, it will become clear; it doesn't have to be complicated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a coach you can monitor your students - I have told students that they have to work on Khan Academy for 15 minutes and then checked that they did so.  It will show you the mistakes your students made etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I encourage you to try it.  If you do have questions you can of course ask them here, but there are also places to ask them on Khan Academy.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-01-03T12:04:12.993" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10333" PostTypeId="1" CreationDate="2016-01-03T14:20:36.683" Score="11" ViewCount="1932" Body="&lt;p&gt;Arriving at a rigorous definition of the concept of angle for high school students is not as easy as expected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Google search provided me with many definition that are &lt;a href=&quot;https://math.stackexchange.com/a/583119/103861&quot;&gt;too technical&lt;/a&gt; or &lt;a href=&quot;http://www.mathopenref.com/angle.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;too vague&lt;/a&gt; IMO.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is my current version: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Measurement of the counterclockwise rotation of one ray from another&#xA;  ray with the same endpoint (vertex). Usually measured in degrees from&#xA;  $0$ to $360$ or radians from $0$ to $2\pi$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I ask for your feedback. Does my definition fall (break) down in certain (relevant) settings? Does it obfuscate? Is there a better way to provide intution for the concepts.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One problem I see is that depending on the ray you decide &lt;em&gt;has rotated&lt;/em&gt; you get different angles. For example how do I know if the angle is $90°$ or  $270°$.&lt;/p&gt;&#xA;" OwnerUserId="5731" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-01-12T20:42:15.003" Title="Rigorously defining the concept of an angle for high school students" Tags="|geometry|trigonometry|definitions|" AnswerCount="5" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10334" PostTypeId="2" ParentId="10333" CreationDate="2016-01-03T16:23:28.853" Score="15" Body="&lt;p&gt;Many high school geometry textbooks define an angle as simply&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the union of two rays with a common endpoint&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The advantage of this definition is its simplicity.  Among its disadvantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It does not serve well for capturing the idea of a &quot;direction&quot;:  That is, there is no way to distinguish between a clockwise and a counterclockwise rotation.&lt;/li&gt;&#xA;&lt;li&gt;It more or less forces you to define &quot;angle measure&quot; in such a way that it is bounded between $0°$ and $180°$ (or between $0$ and $\pi$ radians).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;However, the second objection can be remedied somewhat by introducing the notion of an &lt;em&gt;angular region&lt;/em&gt;, defined as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a subset of the plane bounded by angle&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One then has that every angle divides the plane into two angular regions. If the angle is not a straight angle, then those regions can be distinguished as the &quot;interior&quot; and &quot;exterior&quot; of the angle (by using betweenness as a criterion). You can then define the measure of an angular region according to the two cases: an interior angular region has the same measure as the angle that bounds it, and an exterior angular region has measure equal to $360°$ minus the measure of the angle that bounds it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With this machinery in place, you can distinguish between a $270°$ and a $90°$ angular region. But you still can't capture the idea of direction.  Nor can you discuss angles with negative measure, or angles with measure larger than $360°$.  &quot;Angular regions&quot; are essentially &lt;em&gt;static&lt;/em&gt; objects; if you want to capture the idea of &quot;rotation&quot; they will not work well for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;strong&gt;Edited to add&lt;/strong&gt;: The approach taken above is more or less identical to the definition that followed in &lt;em&gt;Geometry&lt;/em&gt; by Lang &amp;amp; Murrow, &lt;a href=&quot;https://books.google.com/books?id=ntA5AlD3p4AC&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.com/books?id=ntA5AlD3p4AC&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&lt;/a&gt;.  Thanks to user @whatever for providing a link to it in the comments.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said all of that, though, I think it is worth rethinking the premise of this question.  I do not know your students, but I strongly doubt that &quot;rigor&quot; is their main criterion for deciding how to trust you and how motivated to be.  Remember:  &lt;em&gt;They do not know, yet, what rigor is&lt;/em&gt;, and they are &lt;em&gt;not expecting it&lt;/em&gt;.  They probably want a teacher who is clear and considerate, not someone who is rigorous and precise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that what I have just written is not an argument &lt;em&gt;against&lt;/em&gt; rigor.  I think there are good reasons to be rigorous in a high school mathematics classroom -- but I am not sure you have named one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You also might want to consider the proposition that &quot;rigor&quot; is not an absolute criterion but a relative one.  Some definitions are more rigorous than others; typically the more rigorous, the less clear a definition is.  The question you need to ask yourself is not &quot;What is a rigorous definition  for high school students?&quot; but &quot;How rigorous can I be without sacrificing clarity and confusing my audience?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Supplement:&lt;/strong&gt;  As Ben Crowell points out, the answer above define &quot;angle&quot;, but does not really address the question of &lt;em&gt;angle measure&lt;/em&gt;.  In most contemporary high school geometry curricula (at least in the United States) this is handled by virtue of something called a &quot;Protractor Postulate&quot;, which asserts (in slightly more formal terminology than would be used in high school) the existence of a mapping that assigns to each angle $\angle ABC$ a real number, denoted $m\angle ABC$, with $0 \leq m\angle ABC \leq 180$ (if using degrees). Depending on the definition of &quot;angle&quot;, the upper bound might be $360$; also, again depending on how &quot;angle&quot; is defined, the upper and lower bounds may or may not be strict.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The properties of the mapping $\angle ABC \mapsto m\angle ABC$ are further specified by additional postulates and/or definitions.  One important property that we need is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Two angles are congruent if and only if they have the same measure&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Whether the above is a postulate, a definition, or a consequence depends a lot  on how the rest of the theory is structured -- in particular whether &quot;congruent angle&quot; is taken as an undefined relation, or defined in a measurement-free way, or whether the above is taken &lt;em&gt;as&lt;/em&gt; the definition of congruent angles.  For the sake of this discussion, let's call it the &quot;Angle Congruence Postulate&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another property we need is an &quot;Angle Addition Postulate&quot;, which can be stated as saying that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let three distinct rays have a common vertex. Then any two of the rays define two angular regions, with the third ray lying in exactly one of those regions and dividing it into two subregions.  In this situation, the sum of the measures of the two subregions is equal to the measure of the full angular region.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that the Protractor Postulate, Angle Congruence Postulate and Angle Addition Postulate don't actually &lt;em&gt;define&lt;/em&gt; the measure function, but they do &lt;em&gt;characterize&lt;/em&gt; it, at least enough to make possible everything you need to do high school geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A historical note:  As far as I know the approach above was first introduced by Birkhoff &amp;amp; Beatley in &lt;em&gt;Basic Geometry&lt;/em&gt; (1941), but was largely ignored for twenty years or so, until the School Mathematics Study Group (SMSG) adopted the &quot;Ruler &amp;amp; Protractor Postulates&quot; as part of the axiomatic framework in the New Math era Geometry textbooks, and it has remained the standard approach ever since.  In the introduction to the first volume of the SMSG &lt;em&gt;Geometry&lt;/em&gt; text, the editors wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The basic scheme in the postulates is that of G. D. Blrkhoff. In this scheme, it is assumed that the real numbers are known, and they are used freely for measuring both distances and angles…  It has been correctly pointed out that Euclid’s postulates are not logically sufficient for geometry, and that the treatments based on them do not meet modern standards of rigor. They were improved and sharpened by Hilbert.  But the foundations of geometry, in the sense of Hilbert, are not a part of elementary mathematics, and do not belong in the tenth-grade curriculum.  If we assume the real numbers, as in the Birkhoff treatment, then the handling of our postulates becomes a much easier task, and we need not face a cruel choice between mathematical accuracy and intelligibility.   (Allen et. al., 1965, p. 10).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The last two sentences are perhaps most relevant for any one considering the question of how to bring mathematical rigor to the secondary classroom without sacrificing clarity.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2016-01-04T19:25:01.757" LastActivityDate="2016-01-04T19:25:01.757" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="10335" PostTypeId="2" ParentId="10333" CreationDate="2016-01-03T22:41:55.937" Score="4" Body="&lt;p&gt;My answer will probably go deeper than a high-school student will be exposed to, but I think it is still relevant to a high-school teacher. The most conclusive part is far below in bold, if the rest is too long. Bottom line is, angles are tricky and easy to underestimate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, it may be preferable to distinguish the geometric object from the measurement. I will therefore use &lt;em&gt;angle&lt;/em&gt; for the geometric object (see below) and &lt;em&gt;angle measure&lt;/em&gt; for the measurement (in radians or degrees or whatever).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, it is important to realize that there are many different kind of angles, defined by e.g. pairs of incident (i.e. sharing their starting point) half-lines, pairs of vectors, pairs of intersecting lines, and the ordered version of each of those. You may want to go with angular sectors (aka component of the complement of a pair of incident half-lines), as mentioned in a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, one can ask when to consider two angles equal (e.g., when defining an isosceles triangle). Usually, one defines an angle measure and defines equality of angles by equality of measures, but this makes very difficult to understand the difference between the different situations above: in fact the measures do not live in the set of real numbers but in a quotient set: as everyone knows, angle measures of $2\pi$ and $0$ are the same really (except in cases they are not, see below). But when it comes to ordered pairs of lines, in fact $\pi/3$ and $-2\pi/3$ are also the same (more on that later): the various situations lead to various quotient sets where the measurement has its meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the universal (in the sense: for all the cases above, other approaches to the whole topic are possible) definition of equality of angles is when one can be mapped to the other by a displacement (aka direct isometry of the plane). For example for angles as unordered pairs of vectors, two are equal when there is a displacement mapping each of the vectors of the first pair to a vector of the second pair (disregarding which is mapped to which). I fact, I used the word &quot;equality&quot; because from a deeper, more formal point of view we define angles as of the geometric objects under the action of the group of displacements of the plane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, one tries to capture this equivalence relation between angles by an angle measure. For an ordered pair of vectors, the usual definition is to consider the &quot;angle&quot; of the rotation mapping the first one to the second one. This seems a recursive definition, but is not when one has enough background: a rotation can be defined as a displacement with a fixed point, one proves that it is a affine map whose linear part has a matrix of the form $$\begin{pmatrix}\cos \theta &amp;amp; -\sin\theta \\ \sin \theta &amp;amp; \cos\theta\end{pmatrix}$$&#xA;for some number $\theta$ (in any direct orthonormal basis), and then calls $\theta$ the angle of the rotation. Of course, for this one needs to define $\cos$ and $\sin$, usually via analytic tools, and that's a moral problem to have everything patch up together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When one considers angles defined by an ordered pair of intersecting lines, then there are two possible ways to get an ordered pair of vectors: when one takes one vector in the direction of each line, then changing exactly one into its opposite is just as valid a choice, and adds (or substracts) the angle measure by $\pi$. So one ends up working in $\mathbb{R}/\pi\mathbb{Z}$. Unordered pairs are even messier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The easiest case (in which one have no &lt;em&gt;oriented&lt;/em&gt; angles) is to consider angular sectors and use the Greek definition of angle measure: take a circle centered at the vertex of the sector, and divide the length of its intersection with the sector by its radius. The number obtained does not depend on the chosen circle, and is the most geometric possible definition of an angle measure.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is basically the point of view proposed by mweiss, but with an actual definition of angle measure (which is a non-obvious concern, as I hope I made clear).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that with this definition the angle measure belongs to $[0,2\pi]$, with the two extreme points corresponding to different angles (when the two half-lines are the same, and one looks either in between them or outside them) and being difficult to get into the definition (and this way of measuring angles does not exactly fit into the above framework with quotients).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main reason why I wrote the long answer to end to this relatively easy solution, is that angles are much, much more tricky than any one can realize before trying to write a comprehensive and consistent lecture about this. I had to teach this subject to future high-school teachers, started thinking it would be a piece of cake, then wondered how I could make it fit in my lecture time and student's background without lying to them, and ended lying to them (a bit).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-01-03T22:41:55.937" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10336" PostTypeId="2" ParentId="10206" CreationDate="2016-01-03T23:41:54.017" Score="7" Body="&lt;p&gt;The letters intimidate him because he doesn't recognise them or their meaning. Only introduce them when he demonstrates that he is ready.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask him what his initials are. When he tells you, ask him if he would recognise he was being addressed if you used only his initials. When he (hopefully) says yes, then ask him if distance might have an 'initial' he could suggest so that when you used that letter, he would know you were referring to distance. Hopefully he will suggest 'd'. And so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could even introduce the lesson by having every student only address other students using their initials... good fun, but it breaks down the resistance to the dreaded 'letters' :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then only introduce variables which have &lt;em&gt;immediately obvious&lt;/em&gt; 'initials', such as p for price or pizza, h for height, l for length, a for adults, h for hours, etc, depending on the context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only after he has begun to associate letters with practical measurements and quantities should you mention that 'algebra is just the same... it just uses abbreviations to simplify things, just like his initials. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any talk of x and y and z will re-introduce the fear of symbols. It's better to wait until he gets frustrated with 'all these words' and the idea of an abbreviation is his idea...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I often tell students that they have been using algebra 'for ages', only they didn't realise it. Then I tell them a word story like this:&#xA;Jimmy goes to the bank and gets out enough money for three pizzas. On his way to the pizza shop, he finds 8 dollars on the footpath (sidewalk). When he gets to the shop he has 41 dollars. How much was each pizza going to cost him?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Almost all students who can add and subtract will get it correct very quickly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then say &quot;I'm going to write the algebra abbreviation for that story, and write 3p + 8 = 41 on the board.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say &quot;if he hadn't found the 8 dollars, would he have more or less?' They all will say 'less'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then write &#xA;3p = 41 - 8&lt;/p&gt;&#xA;&#xA;&lt;p&gt;NB! don't introduce any rules about sides and signs and what you do to one side, etc, same for words like solve and expression and equation. These should only come after the process is understood using things they already know... or you'll turn him off all over again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and ask how much money did he get from the bank? They will say '33'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3p = 33&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The say 'So, how much was each pizza going to cost? They will say '11 dollars'. Trust them to refer to what they already do at the shops, not what you have to tell them about dividing, or whatever. Trust that their experience already contains algebraic processes, only they didn't call it that... but they were perfectly good at getting it right. Hint! money and food questions are very familiar to almost every student everywhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then say 'That's all algebra is: an abbreviation for words, using a consistent set of rules.' They will get it... and then wait for them to ask about or discover or intuit or teach each other the rules. Teaching is as much about asking and listening and enabling as it is about telling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then make up similar stories that you read out and they write the abbreviations for. Include things like 'on the way he lost 12 dollars', etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have done this for 25+ years and it it has 'worked' almost every time with almost every student, so long as they can add, subtract, multiply and divide competently. If not, then address that first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only introduce abstract concepts and 'letters' when students are more than comfortable... and you have exhausted most of the other letters and they are ready for some meat in the sandwich. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps&lt;/p&gt;&#xA;" OwnerUserId="6082" LastActivityDate="2016-01-03T23:41:54.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10338" PostTypeId="2" ParentId="10333" CreationDate="2016-01-04T15:19:12.857" Score="0" Body="&lt;p&gt;None of the proposed definitions that I've seen so far (by snoram, mweiss, and Benoît Kloeckner) actually gives a rigorous definition of the measure of an angle. The define a geometrical figure, but don't define its measure. A clear verbal, conceptual definition is of course helpful to students and needs to be given, but it doesn't constitute a rigorous definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One good rigorous approach is to do what Euclid does, but throw out all the bogus pseudo-definitions given in &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#defs&quot; rel=&quot;nofollow&quot;&gt;Book I, definitions 1-23&lt;/a&gt;. That is, we let notions such as &quot;point,&quot; &quot;line,&quot; and &quot;angle&quot; be undefined primitive terms. This is the approach used in modern synthetic treatments such as &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012&quot; rel=&quot;nofollow&quot;&gt;Tarski's&lt;/a&gt;, although the number of primitive notions is reduced and usually does not include the measure of an angle. Note that Euclid doesn't verbally distinguish the measure of a figure from the figure itself, but the distinction is always implicitly clear. For example, &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI2.html&quot; rel=&quot;nofollow&quot;&gt;Book I, Proposition 2&lt;/a&gt; is &quot;To place a straight line equal to a given straight line with one end at a given point.&quot; Here he proves that line AL equals line BC, which in more typical modern terminology would be that the measure of line segment AL equals the measure of line segment BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For perspective, it may be helpful to consider the question of what is meant by the measure of a line segment. This may seem more basic than the measure of an angle, but logically they have exactly the same status within the axiomatic framework of Euclid. If you want to make one of them a fundamental notion and define the other in terms of it, you can do that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is actually a case where I don't think rigor conflicts with education. If you haven't done it, a really interesting experience is to present to a class of students one of Euclid's bogus definitions, and ask them whether they think it's valid and rigorous. When I've done this, the reaction has been that yes, it's rigorous. This shows that, as mweiss has pointed out, &quot;They do not know, yet, what rigor is.&quot; After eliciting that reaction, one can explain why modern mathematicians do not accept this kind of definition, and maybe follow up with the famous quote by Hilbert, &quot;One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, one thing that seemed really ugly to me when I looked at the treatment of geometry in my kids' school books was that they assumed two independent mathematical systems, the real numbers and the Euclidean plane. This is completely redundant, and approaching it this way requires lots of &quot;glue&quot; in terms of notation and definitions. The theory of the reals &lt;em&gt;is&lt;/em&gt; the theory of Euclidean geometry. They're exactly logically equivalent, and that's why Euclid never had to bother setting up a separate structure for his number system.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2016-01-04T16:09:40.967" LastActivityDate="2016-01-04T16:09:40.967" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10339" PostTypeId="1" AcceptedAnswerId="10340" CreationDate="2016-01-04T17:26:43.643" Score="13" ViewCount="428" Body="&lt;p&gt;There is considerable evidence that &lt;a href=&quot;https://en.wikipedia.org/wiki/Pair_programming&quot; rel=&quot;noreferrer&quot;&gt;pair programming&lt;/a&gt;, when&#xA;executed properly, both increases the&#xA;accuracy of the code produced and enhances the learning of both participants.&#xA;I wonder if anyone has explored similar pairing for math homework assignments, say,&#xA;in a college Discrete Math course?&#xA;The two domains are rather different, and it is quite possible the advantages&#xA;disappear in a course where proof assignments are not uncommon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[&lt;strong&gt;Updated&lt;/strong&gt; to specify &lt;em&gt;graded&lt;/em&gt; assignments.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use group work in the classroom, but have not permitted it for &#xA;graded homework&#xA;assignments. If anyone has experience, positive or negative, with permitting collaboration on graded assignments, or can point me to relevant literature, I would&#xA;appreciate it.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-01-05T22:21:27.343" LastActivityDate="2019-07-21T11:51:22.410" Title="Collaboration on math homework assignments?" Tags="|undergraduate-education|proofs|homework|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="10340" PostTypeId="2" ParentId="10339" CreationDate="2016-01-04T18:46:58.950" Score="6" Body="&lt;p&gt;As I see it, the key issue with allowing collaboration on homework assignments is how to suitably assign grades, and in particular how to be seen to be 'fair' in the process of doing so. The approach I've been introduced to is to use peer assessment to modify the marks from the mark assigned to the group piece of work to an individual mark for each student. The idea is that the students report on how well each person contributed to the group, by whatever criteria of 'well' you set up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://webpa.ac.uk/&quot; rel=&quot;noreferrer&quot;&gt;WebPA&lt;/a&gt; is a system designed to implement this process. The documentation contains some discussion of the relevant research.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-01-04T18:46:58.950" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10341" PostTypeId="1" CreationDate="2016-01-04T21:19:55.103" Score="5" ViewCount="390" Body="&lt;p&gt;Our undergraduate mathematics program has recently seen a large drop-off in majors (suspected reason: our growing (but separate) undergraduate statistics program is seen as being a more employable degree). In an attempt to appeal to students looking for a more employable math degree, we've discussed adding more modeling to our program. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I have to admit that (as a pure theory person) I'm not clear what's modeling versus application versus problems in context. Consider the following standard problems from Calculus:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Farmer Brown has some fence and wants to put it in a rectangle that encloses the largest area. Determine the dimensions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Find the work required to build an Egyptian pyramid given the dimensions and the density of stone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Pure water is flowing into a salt-water tank at a given rate. At the same time, salty water is leaving the tank at a given rate. Determine the salinity at time $t$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) What's the velocity of a car that's traveling down a straight highway with given position at time $t$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problems 2 &amp;amp; 3 seem to me like applications. Problems 1 &amp;amp; 4 seem to me like problems with context. Are any of them modeling? If not (or if so), then what makes modeling modeling?&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="2083" LastEditDate="2023-09-18T05:02:53.050" LastActivityDate="2023-09-18T05:02:53.050" Title="Modeling vs. Application vs. Context" Tags="|curriculum|terminology|applied-mathematics|applications|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10342" PostTypeId="2" ParentId="352" CreationDate="2016-01-04T22:11:57.953" Score="7" Body="&lt;p&gt;From &lt;em&gt;A Brief History of American K-12 Mathematics Education in the 20th Century&lt;/em&gt; by David Klein:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Some proponents of the Activity Movement [of the early twentieth century] did not even acknowledge that reading and learning the multiplication tables were legitimate activities. As in the 1990s, there was public resistance to the [traditional] education doctrines of this era. Among the critics were Walter Lippman, one of the nation's most widely respected commentators on public affairs, and literary critic, Howard Mumford Jones.&lt;/p&gt;&#xA;&lt;p&gt;In the 1940s it became something of a public scandal that army recruits knew so little math that the army itself had to provide training in the arithmetic needed for basic bookkeeping and gunnery.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Klein Cites &lt;em&gt;Judging Standards for K-12 Mathematics, In: What's at Stake in the K-12 Standards Wars: A Primer for Educational Policy Makers&lt;/em&gt; in the last sentence. It appears that the progressive ideals emerging from the late twentieth century that include teaching only &amp;quot;useful&amp;quot; or &amp;quot;necessary&amp;quot; mathematics and omitting the rote memorization of multiplication tables, like those similar initiatives that came nearly a century earlier, will certainly result in a public that lack basic skills necessary for homeland defense.&lt;/p&gt;&#xA;&lt;p&gt;If the army found that these recruits were unable to carry out basic bookkeeping and gunnery, their classmates with the same deficiencies certainly had difficulties with the following basic but important household tasks:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;keeping and updating a home inventory&lt;/li&gt;&#xA;&lt;li&gt;calculating the cost of a grocery bill with many items&lt;/li&gt;&#xA;&lt;li&gt;checking the accuracy of a bill or financial statement&lt;/li&gt;&#xA;&lt;li&gt;calculating long term costs of services billed at regular intervals&lt;/li&gt;&#xA;&lt;li&gt;planning to provide food or beverage items for an event with many attendees&lt;/li&gt;&#xA;&lt;li&gt;preparing short- or long-term emergency provisions for a household&lt;/li&gt;&#xA;&lt;li&gt;estimating the rate of speed one must travel to arrive at a destination in a given time&lt;/li&gt;&#xA;&lt;li&gt;estimating the time required to travel between two distant locations&lt;/li&gt;&#xA;&lt;li&gt;estimating the long term consequences of making changes to a budget&lt;/li&gt;&#xA;&lt;li&gt;modifying a recipe to accommodate more or less people&lt;/li&gt;&#xA;&lt;li&gt;estimating the cost of home improvements&lt;/li&gt;&#xA;&lt;li&gt;estimating the cost of a series of medical appointments&lt;/li&gt;&#xA;&lt;li&gt;planning or estimating the cost of travel for a household&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;While it may not be &lt;em&gt;absolutely necessary&lt;/em&gt; for all students to memorize multiplication tables because of technology or reference tables, or because the internet culture allows households to &amp;quot;outsource&amp;quot; these tasks, the &lt;em&gt;prudent&lt;/em&gt; choice is to teach multiplication tables and require students to memorize them, if only because we've already seen that omission of such basic skills results in  less educated and less able graduates. Teachers and administrators should evaluate their goals for education: if the goal is independence, individuation, and empowerment of the student, and freedom from necessary assistance for life tasks, then multiplication tables should be taught and memorized, or students at least should be comfortable quickly calculating the products of single digit numbers mentally.&lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-01-05T12:42:16.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10344" PostTypeId="2" ParentId="10339" CreationDate="2016-01-05T07:54:52.370" Score="6" Body="&lt;p&gt;What level of mathematics are you looking at? When you say discrete math is this an undergrad course for general students, math majors or a grad course?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience has been that for undergrad courses for general students it was enough to make sure they &lt;em&gt;DID&lt;/em&gt; their homework whether or not they collaborated seemed to make little difference as an indicator of test performance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For grad courses the problems were usually hard enough that we would collaborate and just each write up our own solution. That again would usually be enough to distinguish between those that actually came up with the solutions vs. those that just listened to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the trickiest part is math majors. There the problems might be easy enough to be fairly easy to copy while still being hard enough that you want to the students to do more then just write them down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt; In response to clarified/amended OP. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I TA'd/taught I would grade in a very minimalist way (choose 5 problems and grade them 0/1 add that up and add up to 2 points for neatness and up to 3 points for completeness). In this case I felt that whether or not they collaborated was not particularly important with regards to the homework grade (which would usually comprise at most 20% of their course grade). This is usually appropriate if each problem is fairly easy (derivatives, integrals, combinations, permutations etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the problems are hard enough to necessitate a decent write up (a page+ in mathematical notation) you can probably get away with letting the students collaborate and insist they write the problems up separately. This works better then you would expect since usually they come up with the solution together, but actually  do write them up separately (copy and paste would be obvious and since they've got their notes it's easy enough to write up on their own right? wrong) and that surprisingly gives you more then enough to grade since if they didn't understand the solution properly they will make mistakes in the write up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I still think there is a middle ground here where the problems aren't quite hard enough for write ups to make a difference yet they are hard enough that just checking they copied it down correctly isn't enough. I admit I don't know what to do then.&lt;/p&gt;&#xA;" OwnerUserId="5896" LastEditorUserId="5896" LastEditDate="2016-01-06T22:18:55.360" LastActivityDate="2016-01-06T22:18:55.360" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10345" PostTypeId="2" ParentId="10341" CreationDate="2016-01-05T08:00:19.630" Score="2" Body="&lt;p&gt;These questions could technically be called modelling, in the same sort of way that questions like 'simplify $2x-4+3(x-1)$' can be called algebra. They are not modelling in the sense of being useful or interesting to students as examples of modelling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematical modelling is the process of turning a (physical) problem into a mathematical question you can analyse. You first work out what sort of objects are involved, and attempt to establish relationships between them. You then see what outcomes you get, compare with the situation you started with, and make adjustments accordingly, in a cycle until your model matches reality to an accuracy you're happy with. Writing this, it occurs to me that modelling might be said to be the mathematical version of the scientific method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typically modelling questions in maths classes lead to differential equations, and so often it is paired with that, but it would be possible to model other types of systems, although I'd have to think about how.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="376" LastEditDate="2016-01-05T09:46:47.703" LastActivityDate="2016-01-05T09:46:47.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10346" PostTypeId="2" ParentId="10227" CreationDate="2016-01-06T14:05:30.463" Score="1" Body="&lt;p&gt;Depending on your audience, perhaps Dunham's &quot;The Calculus Gallery:&#xA;Masterpieces from Newton to Lebesgue&quot; (Princeton University Press, 2008) might fill the slot. It is more geared towards someone who has seen a bit of calculus, and is curious about it's history and some more exotic developments that aren't part of the &quot;normal&quot; curriculum.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-01-06T14:05:30.463" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10347" PostTypeId="2" ParentId="10339" CreationDate="2016-01-06T16:24:33.903" Score="4" Body="&lt;p&gt;In order to take full advantage of group homework, I think an instructor needs to think carefully about the purpose of such an assignment and how it fits in with the general goals of the course.  The folks who teach calculus at the University of Michigan, for example, have thought a lot about this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, when I teach Calculus I assign students individual homework using WebWork (these are the more straightforward computational problems and the like) and assign team problems in the style of University of Michigan Calculus Problems (these are messy real-world data-driven problems).  The team problems, in addition to being messy, also have more structure: groups consist of four people with one of four rotating roles, the write-ups have to be written in carefully and rigorously and clear expectations are laid out about how much time these should take.  So the main goals of these team problems are: (1) they are different kinds of problems than just ones that deliver content, (2) they require students to write careful explanations, (3) they encourage students to think about team work in the abstract and (4) they give students the opportunity for peer instruction.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2016-01-06T16:24:33.903" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10348" PostTypeId="1" CreationDate="2016-01-07T13:28:20.613" Score="24" ViewCount="3648" Body="&lt;p&gt;Unless you have taught highschool algebra in Iran, you could not make sense of the phrase: &lt;strong&gt;Elephant and Teacup Identity&lt;/strong&gt;! This is what teachers use to refer to the following identities: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ (a+b)(a^2-ab+b^2)=a^3+b^3$  and $ (a-b)(a^2+ab+b^2)=a^3-b^3$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such reference is so common that today a colleague of mine (in a discussion about students' algebraic difficulty) referred to it assuming that I know what she is referring to. Whether or not such references would be of any help to students is an important question, but not my question now. For this post, the question is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know any of these linguistic references for communicating mathematics? It could be something that you use in your own class, or you have heard that someone else uses. Thus, it doesn't matter whether its usage is limited to just one class, or is as popular as the one I gave.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;. The first attempt to clarify the question. The question is looking for &quot;non-mathematical&quot; terms or phrases that are used to refer to mathematical objects (of any kind) mainly for educational purposes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;. The second attempt to clarify the question. Admittedly the question is a bit vague. Do examples like &quot;continuity&quot;, &quot;saddle point&quot;, &quot;horseshoe map&quot;, or &quot;hairy ball theorem&quot; count? I guess not. They are now formal terms belonging  to Mathematics culture here, there and everywhere. What if we call what this question is looking for &quot;mathematical slang&quot;. Here is a dictionary definition of slang: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A type of language consisting of words and phrases that are regarded&#xA;  as very informal, are more common in speech than writing, and are&#xA;  typically restricted to a particular context or group of people.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Interesting, after coming with the term, I found this paper &quot;&lt;a href=&quot;http://www.jstor.org/stable/27953399?seq=1#page_scan_tab_contents&quot;&gt;The blight of mathematical slang&lt;/a&gt;&quot;, that gives the expression &quot;cross-multiply&quot; as an example. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;. Following a number of suggestions for using a more informative title (see comments below), I changed it in a way that also better reflects the final version of the question (previous edit).   &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="77" LastEditDate="2016-02-07T05:21:47.320" LastActivityDate="2022-11-10T15:22:50.630" Title="Examples of Mathematical Slang" Tags="|examples|terminology|language-use|" AnswerCount="17" CommentCount="21" ContentLicense="CC BY-SA 3.0" />
  <row Id="10349" PostTypeId="1" CreationDate="2016-01-07T15:46:58.827" Score="29" ViewCount="1964" Body="&lt;p&gt;Every-so-often I come across proofs of the form&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Assume $X$ is false.&lt;/li&gt;&#xA;&lt;li&gt;Prove $X$ is true (without using that it is false).&lt;/li&gt;&#xA;&lt;li&gt;This contradicts that $X$ is false.&lt;/li&gt;&#xA;&lt;li&gt;Hence $X$ is true.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I've seen students write such proofs, and I've seen them read this structure into a proof even when it isn't there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why do students seem to prefer this type of proof by contradiction to direct proof?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of ideas I thought might be behind this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;'proof by contradiction' is a thing they can give a name to, whereas direct proof doesn't have its own 'identity' in the same way;&lt;/li&gt;&#xA;&lt;li&gt;in a proof by contradiction you see at the start what you are aiming for, ie the opposite of your assumption (this won't always work, but is probably true of most cases they see);&lt;/li&gt;&#xA;&lt;li&gt;proof by contradiction changes the logic to something they find easier to think about (I came across this with a standard uniqueness proof - I think the idea that I had two objects that in the end turn out to actually be the same object was too confusing as an idea that was true, but it made more sense as a contradiction).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2022-01-03T19:10:33.923" Title="Why do students like proof by contradiction?" Tags="|undergraduate-education|proofs|" AnswerCount="12" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10351" PostTypeId="2" ParentId="10348" CreationDate="2016-01-07T21:00:12.203" Score="7" Body="&lt;p&gt;My elementary students always wanted to know the name of the symbol shown here:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/jPNnF.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jPNnF.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;We called it the &lt;strong&gt;division house&lt;/strong&gt; as did many of my colleagues, but my students wanted a mathematical name.  We therefore wrote to Dr. Math at Drexel.  We were told there is no name and were referred to this &lt;a href=&quot;https://jeff560.tripod.com/operation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I subsequently held a contest (on election day) and the winner was a sixth grade girl, who named it the &amp;quot;&lt;strong&gt;parenticulum&lt;/strong&gt;&amp;quot;  because it is a contraction of parentheses and vinculm which is what the symbol is name for.&#xA;After the contest, we all called it the &amp;quot;&lt;strong&gt;parenticulum&lt;/strong&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;For more about the origin of our name, see the following from &lt;a href=&quot;https://web.archive.org/web/20180119061901/http://mathforum.org/library/drmath/view/58409.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dr. Math&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;You might be able to call the horizontal line in the division symbol a&#xA;vinculum, but I don't think there is a name for the whole thing. In&#xA;fact, in the following page about the history of symbols, it is not&#xA;named, but drawn,  and the alternate text in the HTML calls it &amp;quot;a&#xA;close parenthesis attached to a vinculum&amp;quot;See Jeff Miller's &lt;a href=&quot;https://jeff560.tripod.com/operation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Earliest&#xA;Uses of Symbols of Operation&amp;quot;&lt;/a&gt;&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="6053" LastEditDate="2022-11-10T11:24:37.557" LastActivityDate="2022-11-10T11:24:37.557" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="10352" PostTypeId="1" AcceptedAnswerId="10358" CreationDate="2016-01-07T22:35:32.650" Score="11" ViewCount="3661" Body="&lt;p&gt;I teach high school geometry and see many of my students fall in to the trap of &quot;it looks like it, so it's true&quot; -- a &lt;a href=&quot;https://en.wikipedia.org/wiki/Van_Hiele_model&quot; rel=&quot;noreferrer&quot;&gt;Van Hiele Level 0 to 1&lt;/a&gt; thought process. For instance, when talking about parallel line angle relationships, once they realize that all pairs of angles (when you have 2 parallel lines and a transversal) are either congruent or supplementary, asking them to justify why the angles are congruent is &quot;They're parallel and both look acute.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I continually remind them that explaining their ideas based off of earlier ideas is BETTER than using visual information, and provide not-to-scale diagrams to emphasize that fact, but I'd like to create a poster to point to to say something like &quot;It's good that you noticed they look the same, but we're looking for at least a level 2 explanation here.&quot; I'd like to make explicit to my students my goal of moving them &quot;up the scale&quot; to a higher level of thinking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For ease, the Van Hiele levels as presented to educators can be phrased as follows: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wqSB7.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wqSB7.jpg&quot; alt=&quot;Van Hiele Levels&quot;&gt;&lt;/a&gt;&#xA;(&lt;a href=&quot;http://www.nctm.org/Publications/teaching-children-mathematics/2014/Vol21/Issue5/Linking-the-Van-Hiele-Theory-to-Instruction/&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can these be rephrased concisely and accurately in student-friendly language? For instance, I'm not sure how I feel about the rephrasingseen in even the table I provided. I have some ideas of my own, but would prefer to collaborate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is communicating this goal with students a good idea, in your opinion?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At what level do higher-level educators expect their students to be by the time they reach their college level courses, so I could possibly add this to the poster?&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2016-01-08T16:08:38.737" Title="Phrasing the Van Hiele levels in student-friendly language" Tags="|secondary-education|geometry|language-use|" AnswerCount="4" CommentCount="4" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10353" PostTypeId="2" ParentId="10352" CreationDate="2016-01-08T00:26:46.960" Score="2" Body="&lt;p&gt;Unable to resist continuing my comment-rant: it is perverse to disbelieve a thing that appears to be true. Sure, appearances can be deceiving, but... mostly not so much. It is also perverse for mathematics teachers to create situations in which appearances are deceiving... since, in my opinion, the goal is not to teach paranoia, but to teach the amazing things that mathematical methodologies can uncover. Often (notoriously in physics) an idea is &quot;only&quot; a heuristic, but a very powerful one (as Dirac's quantum theory...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also perverse to claim that obviously-true things &quot;must be proven before we can move on&quot;. Yes, it is incidentally a test of our &quot;proving-power&quot; to prove &quot;obvious things&quot;, ... but, here is the point, this is certainly not the first or primary point of mathematics. That is, mathematics really took off after Newton's epiphanies that it explained nature. Prior to that, the bourgeois games of the ancient Greeks (preserved for 2,000 years) were mostly a game played by those with sufficient leisure. Didn't explain anything about the world, and that didn't matter. Newton's and Leibniz' calculus &quot;had issues&quot;, that were not fully resolved until mid-20th century, when A. Robinson and E. Nelson showed that infinitesimals are legitimate. The epsilon-delta debauche of the late 19th century was a slightly awkward first attempt...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, I worry that this purported summary of students' competence is based on a significantly misguided notion of the nature and history of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-01-08T00:26:46.960" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10354" PostTypeId="1" CreationDate="2016-01-08T00:35:52.383" Score="6" ViewCount="8306" Body="&lt;p&gt;I am looking for realistic applications of the average AND instantaneous rate of change, that can serve as an entry point to calculus for students.&#xA;The main-idea is to show them a (simplified) problem of the real world that needs(!) to be solved by:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Modeling the situation upfront from measurements (Turning measurement into a function and a graph)&lt;/li&gt;&#xA;&lt;li&gt;Calculating the average rate of change&lt;/li&gt;&#xA;&lt;li&gt;Calculating the instantaneous rate of change&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So I'm looking for a big question to pursue/problem to solve while learning about those three aspects. Every time we move to the next aspect we learn something new to solve this big problem.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;What do I mean by &quot;realistic&quot;?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;For example, I could create a hypothetical situation including a car and a mountain. I could go on and tell the students that we could figure out the average steepness of the mountain's silhouette and even the precise steepness of a single point of that curve by using derivatives in order to find out whether or not the car will make it up the hill. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now here's the big question: Is this really a realistic application? The main issues I'm having with this is, that there are too many open questions:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is it really done this way? Why would anyone model a function for the mountain and calculate the rate of change instead of just measuring the angle of the steepest looking part using a spirit level? This hypothetic situation doesn't answer the question why it &lt;strong&gt;needs&lt;/strong&gt; to be solved using calculus.&lt;/li&gt;&#xA;&lt;li&gt;Why would we want to calculate the precise steepness of a single point? Just for the sake of being more precise? Why don't we just stick with an almost as precise approximation by using the average rate of change with a really small interval? Are there or have there ever been any real consequences because someone used such an &quot;approximation&quot;?&lt;/li&gt;&#xA;&lt;li&gt;If the values on road gradient signs are really calculated this way, who is responsible for doing this calculations?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So what I mean by realistic is, that I'm looking for an example of an application, where real persons in the real world &lt;strong&gt;really need&lt;/strong&gt; to mathematically model a situation before they &lt;strong&gt;need&lt;/strong&gt; to calculate the rate of change.&lt;/p&gt;&#xA;" OwnerUserId="6215" LastEditorUserId="6215" LastEditDate="2016-01-08T13:06:59.467" LastActivityDate="2016-01-09T05:46:40.550" Title="Looking for realistic applications of the average and instantaneous rate of change" Tags="|calculus|course-design|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10355" PostTypeId="2" ParentId="10352" CreationDate="2016-01-08T02:24:44.967" Score="5" Body="&lt;p&gt;I think one problem with this whole approach is that one of the fundamental tenets of the van Hiele model is what the van Hieles called &quot;the principle of &lt;em&gt;separation&lt;/em&gt;&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a teacher who is reasoning at one level speaks a different &quot;language&quot;&#xA;  from a student at a lower level, preventing understanding. When a&#xA;  teacher speaks of a &quot;square&quot; she or he means a special type of&#xA;  rectangle. A student at Level 0 or 1 will not have the same&#xA;  understanding of this term. The student does not understand the&#xA;  teacher, and the teacher does not understand how the student is&#xA;  reasoning, frequently concluding that the student's answers are simply&#xA;  &quot;wrong&quot;. The van Hieles believed this property was one of the main&#xA;  reasons for failure in geometry. Teachers believe they are expressing&#xA;  themselves clearly and logically, but their Level 3 or 4 reasoning is&#xA;  not understandable to students at lower levels, nor do the teachers&#xA;  understand their students’ thought processes. Ideally, the teacher and&#xA;  students need shared experiences behind their language.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;https://en.wikipedia.org/wiki/Van_Hiele_model&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming this is correct, it would seem to follow that a student currently reasoning at level $n$ cannot &lt;em&gt;possibly&lt;/em&gt; understand what you are asking for if you tell them you want them to be at a level higher than $n+1$.  They would be literally incapable of understanding what you mean, because individuals must pass through the levels in sequence.  The levels only make sense retrospectively, to a person who has already passed through them.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-01-08T02:24:44.967" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10356" PostTypeId="2" ParentId="10352" CreationDate="2016-01-08T03:18:22.597" Score="3" Body="&lt;p&gt;This model is really describing three different concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1-2:  What does it look like?  This is &quot;basic&quot; shape recognition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2-3:  Practical geometry.  How many sides does a shape have?  What kinds of angles does the shape have?  Can shapes be put next to each other without any gaps?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4-5:  Proofs, for the sake of proofs.  The examples might have to do with geometric shapes.  But the exercise of doing the proofs could be done just as well with algebra, or with matrices, or with sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The model is missing a step between 3 and 4:  Constructive geometry, like the methods used by &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486281388&quot; rel=&quot;nofollow&quot;&gt;origami masters&lt;/a&gt;, old-fashioned &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0022321500&quot; rel=&quot;nofollow&quot;&gt;hand draftsmen&lt;/a&gt;, and ancient Greek compass-and-straightedge geometers.  These are practical exercises, which someone at level 2 can see the use of, and someone at level 4 can check the accuracy of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A translation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Visual.  Name that shape.  What does it kinda look like?&lt;/li&gt;&#xA;&lt;li&gt;Descriptive.  Tell me about that shape.  How many sides does it have?  Are the sides straight?  What kinds of angles does it have?&lt;/li&gt;&#xA;&lt;li&gt;Categorizing.  What do these shapes have in common?  Can they be joined together?  What gaps would they have?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;3.5  Construction.  Can you modify a shape to have particular properties?  Can you make drawings of a 3-D object carefully enough that someone can make another object based on your drawings?  What are the tolerances?&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Proofs.  Can you prove things about the shape?  If you say something is true about kind(s) of shapes, can you be sure that nobody can find an exception?&lt;/li&gt;&#xA;&lt;li&gt;Creative proofs.  What rules can you change about the proofs?  How does this change what you can prove?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Real-world applications&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Most human beings need to be at least at level 1.  There is a reason that babies can recognize shapes before they can talk.&lt;/li&gt;&#xA;&lt;li&gt;Most hands-on workers need to be at least at level 3.&lt;/li&gt;&#xA;&lt;li&gt;Designers of physical things (including tailors, seamstresses, drafters, engineers, and architects) need to be at least at level 3.5.  (The missing &quot;constructive geometry&quot; level.)&lt;/li&gt;&#xA;&lt;li&gt;Students studying mathematics, science, computer programming, or engineering in a college-preparatory or college program need to be at least at level 4.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2016-01-08T16:08:38.737" LastActivityDate="2016-01-08T16:08:38.737" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10357" PostTypeId="2" ParentId="10348" CreationDate="2016-01-08T04:39:54.363" Score="12" Body="&lt;p&gt;How about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Shoelace_formula&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;shoelace formula&lt;/em&gt;&lt;/a&gt; for the area of an arbitrary&#xA;simple polygon?&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/UJjT4.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UJjT4.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Shoelace_formula&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;The formula computes the area from the coordinates of the vertices,&#xA;essentially by a cross product to compute (signed) areas of triangles.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-01-08T04:39:54.363" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10358" PostTypeId="2" ParentId="10352" CreationDate="2016-01-08T11:41:39.647" Score="7" Body="&lt;p&gt;The argument has been made that this is sort of a misappropriation of the terms, because the levels are meant to define levels of understanding rather than levels of detail. I'm assuming what you're aiming for is that you can say to your student(s) &amp;quot;I'm looking for a level 2 explanation here&amp;quot;, and they would provide you with an answer that would &lt;em&gt;prove&lt;/em&gt; that they have achieved a level 2 understanding of that concept.&lt;/p&gt;&#xA;&lt;p&gt;Thus, your table is going to look slightly different because it describes the &lt;strong&gt;type of answer you must give in order to prove a level of comprehension&lt;/strong&gt;, rather than describing the level of comprehension itself. Note I may have misunderstood the levels above to some extent as I am familiar with the concept of levels of comprehension but haven't seen it formally listed in this form before (irony!). I think, even if you disagree slightly with my interpretation, the overall exercise will still be useful (I hope!):&lt;/p&gt;&#xA;&lt;h1&gt;Level 1 - It looks like it's true (&lt;a href=&quot;https://en.wikipedia.org/wiki/Duck_test&quot; rel=&quot;noreferrer&quot;&gt;The Duck Test&lt;/a&gt;)&lt;/h1&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;If it looks like a duck, and quacks like a duck, it's probably a duck.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's a good phrase because it's memorable and it well-defines the concept. It also reinforces the lack of concrete data associated with this level of comprehension: &amp;quot;I can't prove it's true, but my understanding is that it's unlikely to not be true&amp;quot;. This proves level 1 comprehension because... well, they have made a correct basic assumption based on the appearance of the problem.&lt;/p&gt;&#xA;&lt;h1&gt;Level 2 - I can deduce that it's true&lt;/h1&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;I have gathered some data on the objects in question, and can deduce that this is the case&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note: Deduce, not prove.&lt;/p&gt;&#xA;&lt;p&gt;To show Level 2 comprehension, the student must be able to provide detail to support their Duck Test. In the original example given, it would be to say &amp;quot;I have calculated the angles, I know they're both acute so my duck test holds&amp;quot;. They don't have to then prove that the corollary (angles are congruent) holds to show this level of comprehension. This proves level 2 because they have used actual data to make a correct assumption (well, technically an assumption).&lt;/p&gt;&#xA;&lt;h1&gt;Level 3 - I understand the Importance of these properties&lt;/h1&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;Not only can I deduce that this is true, I can extend my deduction to related abstract concepts&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;To show Level 3 comprehension (IMO, there may be disagreement here), the student must show that they understand the relevance of what they are seeing. What are the consequences of the these two angles being congruent? Can the student make the jump of comprehension to the other properties that must therefore be true (related pairs of supplementary angles, the types of geometrical shapes that would be formed from sets of congruent angles, which lines must be parallel if these two angles are congruent). Rather than just seeing it's true, how can I &lt;em&gt;USE&lt;/em&gt; the fact that it's true?&lt;/p&gt;&#xA;&lt;p&gt;Again, note that the student isn't required to prove any of this yet, just understand the significance of the data. I might be able to deduce from the sound on the roof that it's raining pretty hard, but will it occur to me to take an umbrella when I go outside?&lt;/p&gt;&#xA;&lt;p&gt;This proves level 3 because they can volunteer consequences of the properties they are seeing, showing an understanding of the abstraction - they understand what it &lt;em&gt;means&lt;/em&gt;, not just what it says.&lt;/p&gt;&#xA;&lt;h1&gt;Level 4 - I can Prove that it's true&lt;/h1&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;I know that this is true, and I can prove it&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The student shows comprehension of both the consequences of the properties &lt;strong&gt;and&lt;/strong&gt; the base principles behind the properties. They can use these base principles to prove their duck test rather than just deducing from &amp;quot;stuff I know is true&amp;quot;. F.Ex., I can prove that these angles are congruent using my knowledge of angles and parallel lines, rather than simply deducing it's true because I've been told that two parallel lines bisected by another line at an angle forms a set of congruent angles.&lt;/p&gt;&#xA;&lt;p&gt;This shows level 4 comprehension because they can prove their deductions rather than relying on &amp;quot;facts&amp;quot; that they have been told (Ability to prove the theorem from axioms, definitions, and previously proven theorems, as opposed to merely knowing how to use the theorem).&lt;/p&gt;&#xA;&lt;h1&gt;Level 5 - I don't know that this is true (yet), but I can prove it&lt;/h1&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;I know that this is true (and I can prove it), but can I prove whether this is true?&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Student shows absolute comprehension of the subject, to the point that they can apply what they have been taught to prove things that they haven't been taught. Using their knowledge, they can make good assumptions about other things that might be true, and go on to prove and disprove them.&lt;/p&gt;&#xA;&lt;p&gt;This shows level 5 comprehension because the student understands enough about the system to try and extend their knowledge beyond its current level.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;I know that this holds in this case, but can I be sure that it &lt;em&gt;always&lt;/em&gt; holds?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;This configuration looks interesting, can I use my knowledge of geometry to prove anything about it?&amp;quot;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;It may not be exactly what you're looking for, but it's what I've understood your question to be asking for, and I hope it helps regardless! The unfortunate truth is that (in my experience) most students won't ever reach level 5, particularly in a compulsory subject, because it requires a significant level of enthusiasm to reach it. Maybe by showing the decent but relatively uninterested students that there is something to aim for beyond level 4 (which is sufficient to get you through pretty much any exam), it will encourage more of them to take that leap. If that's the case, then this will all have been worthwhile!&lt;/p&gt;&#xA;" OwnerUserId="6023" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-01-08T11:41:39.647" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10359" PostTypeId="1" CreationDate="2016-01-08T13:37:51.203" Score="13" ViewCount="1059" Body="&lt;p&gt;Amir Asghari recently asked &lt;a href=&quot;https://matheducators.stackexchange.com/q/10348/77&quot;&gt;a question about &lt;em&gt;mathematical slang&lt;/em&gt;&lt;/a&gt;.  He was &quot;looking for &quot;non-mathematical&quot; terms or phrases that are used to refer to mathematical objects (of any kind) mainly for educational purposes.&quot;  He defined slang as &quot;a type of language consisting of words and phrases that are regarded as very informal, are more common in speech than writing, and are typically restricted to a particular context or group of people.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I am looking for examples of &lt;em&gt;mathematical education slang&lt;/em&gt;, where the terms refer not to mathematical objects but to mathematical education objects.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An example of what I am interested in is &quot;&lt;strong&gt;drill-and-kill&lt;/strong&gt;.&quot;  A &lt;a href=&quot;https://matheducators.stackexchange.com/q/10348/77&quot;&gt;Wikipedia article&lt;/a&gt; defines this as &quot;an unhealthy focus on excessive repetition of simple, isolated skills.&quot;  I first heard this term from a teacher who did not like teaching that focuses on procedures.  (I later heard of the more positive sounding term &quot;drill and skill,&quot; but I don't know which of the two terms came first.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another term that I often hear is &quot;&lt;strong&gt;chalk-and-talk&lt;/strong&gt;,&quot; used to describe the traditional way of teaching through lectures.  I first heard the term from the teacher mentioned above who used it in a derogatory manner.  (However, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Chalk_talk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; about it currently seems to be in favor of it.)&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2020-09-05T17:38:59.503" Title="Mathematical education slang" Tags="|mathematical-pedagogy|examples|general-pedagogy|terminology|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10360" PostTypeId="2" ParentId="10349" CreationDate="2016-01-08T13:40:17.870" Score="6" Body="&lt;p&gt;This is unavoidably going to be a subjective answer, but I always liked Proof by Contradiction because it's often easier to prove that &lt;strong&gt;one&lt;/strong&gt; case &lt;em&gt;isn't&lt;/em&gt; true than it is to prove &lt;strong&gt;all&lt;/strong&gt; cases &lt;em&gt;are&lt;/em&gt; true. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof by contradiction allows the prover to focus on one specific case and show that the identity doesn't hold, as opposed to having to show that the converse holds for all, which can be more difficult to grasp due to the greater generality.&lt;/p&gt;&#xA;" OwnerUserId="6023" LastActivityDate="2016-01-08T13:40:17.870" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10361" PostTypeId="2" ParentId="10354" CreationDate="2016-01-08T13:52:28.730" Score="6" Body="&lt;p&gt;I've always found mechanics to be my friend when it came to explaining the necessity of calculus. What about something like a rocket taking off into outer space? &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The process needs to be modelled up front because of the cost and risk to life involved in a failed mission.&lt;/li&gt;&#xA;&lt;li&gt;The average rate of change needs to be calculated in order to ensure that the rocket gains enough speed to reach escape velocity, otherwise the mission will fail.&lt;/li&gt;&#xA;&lt;li&gt;The instantaneous rate(s) of change need to be calculated in order to ensure that the rocket materials and crew can cope with the stress of acceleration.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Pretty much everyone knows what a rocket is and most people find them at least vaguely exciting. Everyone has at least a passing experience with gravity and acceleration, so the real world necessity is relatively obvious.&lt;/p&gt;&#xA;" OwnerUserId="6023" LastActivityDate="2016-01-08T13:52:28.730" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10362" PostTypeId="2" ParentId="10348" CreationDate="2016-01-08T14:16:18.200" Score="3" Body="&lt;p&gt;The identities $(a + b) (a - b) = a^2 - b^2$ and $(a + b)^2 = a^2 + 2 a b + b^2$ (and sometimes $(x - a) (x - b) = x^2 - (a + b) x + a b$) are called &quot;productos notables&quot; (notable products) in Spanish. This is sometimes extended to the products mentioned in the question, and even higher order ones.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-01-08T14:16:18.200" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10363" PostTypeId="2" ParentId="10359" CreationDate="2016-01-08T14:29:19.297" Score="11" Body="&lt;p&gt;Similar to 'drill-and-kill', a common one is 'plug-and-chug'. I guess this refers not so much to the method of teaching (drilling students) as to the method of completing the exercises (running the prescribed algorithm without thought to what it does or why).&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-01-08T14:29:19.297" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10364" PostTypeId="2" ParentId="10348" CreationDate="2016-01-08T15:07:55.553" Score="3" Body="&lt;p&gt;My school talked about &lt;strong&gt;sausages and cocktail-sticks&lt;/strong&gt;, meaning questions where you have to find the formula for the $n$th term in the sequence. I never worked out what the point was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7EAsl.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7EAsl.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-01-08T15:07:55.553" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10365" PostTypeId="2" ParentId="10359" CreationDate="2016-01-08T16:48:01.673" Score="11" Body="&lt;p&gt;There is the constructivist &quot;guide on the side&quot;, contrasted with the traditional &quot;sage on the stage&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These phrases have been popular in secondary and primary education in recent years, but they originated over 20 years ago, and were originally aimed at post-secondary education. Alison King published an &lt;a href=&quot;http://www.jstor.org/stable/27558571&quot;&gt;article introducing these terms&lt;/a&gt; in the Winter 1993 issue of &lt;em&gt;College Teaching&lt;/em&gt; journal.&lt;/p&gt;&#xA;" OwnerUserId="3593" LastActivityDate="2016-01-08T16:48:01.673" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10366" PostTypeId="5" CreationDate="2016-01-08T17:39:55.297" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2016-01-08T17:39:55.297" LastActivityDate="2016-01-08T17:39:55.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10367" PostTypeId="4" CreationDate="2016-01-08T17:39:55.297" Score="0" Body="For questions about discrete mathematical structures and applying to courses with the title &quot;Discrete Mathematics&quot; or equivalent. Since this tag is fairly broad, consider using a more specific tag if appropriate." OwnerUserId="376" LastEditorUserId="376" LastEditDate="2016-01-08T19:10:06.607" LastActivityDate="2016-01-08T19:10:06.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10368" PostTypeId="2" ParentId="10354" CreationDate="2016-01-08T20:02:48.640" Score="1" Body="&lt;p&gt;This follows up on I Stanley's mention of mechanics, and particularly kinematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure if this works well for you, but, if you're willing to discuss two-dimensional motion (where the direction changes with time), the difference between the average and the instantaneous rates of change becomes obvious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basic examples could include circular motion (e.g. a planet's orbit) or a ballistic trajectory. The latter perhaps being a particularly good example, since in the x-axis you have a constant velocity, such that the average explains everything, while in the y-axis, the average velocity is zero, so you must examine the instantaneous change to make sense of the motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This also reminded me of Steven Strogatz's wonderful discussion of chase problems in the Pursuit chapter of &quot;The Calculus of Friendship&quot;. It is very easy to formulate and explain such examples that arouse curiosity, but require a deep dive into calculus to solve.&lt;/p&gt;&#xA;" OwnerUserId="688" LastActivityDate="2016-01-08T20:02:48.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10369" PostTypeId="2" ParentId="10348" CreationDate="2016-01-08T22:33:55.960" Score="9" Body="&lt;p&gt;If you simplify a term by adding and subtracting something you call this a &quot;&lt;strong&gt;nahrhafte Null&lt;/strong&gt;&quot; in German (probably translates to &quot;&lt;strong&gt;nutritious null&lt;/strong&gt;&quot;?). &lt;/p&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="3398" LastEditDate="2016-01-11T08:30:18.430" LastActivityDate="2016-01-11T08:30:18.430" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10370" PostTypeId="2" ParentId="10348" CreationDate="2016-01-08T22:57:49.067" Score="6" Body="&lt;p&gt;I first heard the term &quot;&lt;strong&gt;stars and bars&lt;/strong&gt;&quot; a few years ago in Mathematics Stack Exchange.  From &lt;a href=&quot;https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the context of combinatorial mathematics, stars and bars is a graphical aid for deriving certain combinatorial theorems. It was popularized by William Feller in his classic book on probability. It can be used to solve many simple counting problems, such as how many ways there are to put $n$ indistinguishable balls into $k$ distinguishable bins.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-01-08T22:57:49.067" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10371" PostTypeId="2" ParentId="10348" CreationDate="2016-01-09T01:05:14.443" Score="12" Body="&lt;p&gt;In Central Mexico, the expression&#xA;\begin{equation}&#xA;x_{\pm} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}&#xA;\end{equation}&#xA;that solves quadratic equations of the form $ax^2 + bx + c = 0$ is called &quot;fórmula del chicharronero&quot; (formula of the &lt;em&gt;chicharronero&lt;/em&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;chicharronero&lt;/em&gt; is the guy who sells salty snacks made of wheat (called &lt;em&gt;chicharrones&lt;/em&gt;). Outside most schools there is always a &lt;em&gt;chicharronero&lt;/em&gt; selling snacks to kids (here is a photo of a &lt;em&gt;chicharronero&lt;/em&gt;; to the right of the picture there are chips; to the left the famous &lt;em&gt;chicharrones&lt;/em&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mxOUw.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mxOUw.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is said that the formula is so famous that even the &lt;em&gt;chicharronero&lt;/em&gt; knows about it; thus the name. &lt;/p&gt;&#xA;" OwnerUserId="5922" LastEditorUserId="5922" LastEditDate="2016-01-09T05:38:54.143" LastActivityDate="2016-01-09T05:38:54.143" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10372" PostTypeId="2" ParentId="10354" CreationDate="2016-01-09T05:46:40.550" Score="5" Body="&lt;p&gt;I would appreciate a really good answer to this.  Somehow my brain is not up to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, as far as I can tell, all &lt;strong&gt;real&lt;/strong&gt; rates are average rates.  Every observed change takes place in a time and the magnitude of the change divided by the magnitude of the time gives us the average rate.  In an instant, there is no change and the magnitude of the time is zero (or possibly infinitesimal, if you're of that bent).  Instantaneity is the result of our imaginations inventing models that are logically consistent.  They are fictional, in a sense, but more like historical novels, they sometimes approximate reality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason instantaneous rates are important is that, because of the Mean Value Theorem, derivatives are good approximations to average rates over small intervals of time.  And derivatives are easier to think with than finite differences.  I don't say they're easier to compute with.  That may be true, sometimes.  They usefulness is in how easy they are to think about, as well as in their effectiveness at approximating the changes we observe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;real&lt;/strong&gt; application to me is how we use them to think about problems.  This has led to advances in mathematics.  Try to think of all the times the MVT is used.  I'm preparing to teach numerical analysis, and it is amazing to see how much of what we understand about effective numerical approximation is based on the MVT.  This is real work, done by real people, to develop methods that help others solve problems within specified tolerances.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, the actual real applications are theoretical.  And  to understand the world and set up its problems in a form to be solved, you need a workable mathematical model.  When I was a young person, this was what attracted me into science and engineering, that the mathematics I was learning was giving me a grasp of the world that would allow me to solve some of its problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many applications of derivatives.  In some, an equation with derivatives is set up to be integrated.  In others, something, such as energy, is to be minimized.  Sometimes, the derivative of the potential function indicates the force on the system.  As alluded to before, the derivative may be used to help bound the error from discretization or truncation of continuous processes.  In applications outside mathematics, the calculus problem has usually been solved in the form of an algebraic formula to be applied.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Persons who merely carry out the work of calculating the solution, filling out the reports and so forth, they just need good training in their specific tasks.  Sometimes you get the training on the job; sometimes you enroll in a certificate program.  The ones who need to solve problems need to understand the principles and to be able to set the problem up.  One concern is that when you do anything, whether you have the first or second type of job, you're likely to run into problems now and then.  If you've worked anywhere for a length of time, you've probably run into people who had trouble solving problems outside their ordinary tasks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To circle back again to my point.  The real application is to use derivatives to think about things.  Many things are subject to mathematical formulations.  They may be in mathematics.  They may be in the sciences.  Derivatives may be used to solve problems and advance our understanding in mathematics and the sciences.  Or they be used to understand mathematical principles in the sciences, which are just problems that other people have solved.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;An example, taken from the real world, but without formulas and so not quite like what the OP seeks: It should be clear to one who knows Newton's second law, force equals the mass times the acceleration $x''(t)$, that turning off the water is potentially dangerous.  It never seems to be, but that's because of building codes; so one might be forgiven for never thinking so.  A whole column of water, stretching back to who know where accelerates when the taps are opened.  That momentum has to be stopped when the tap is turned off.  Turn it off quickly and the water has to stop quickly.  So the acceleration $x''(t)$ has be large, and thus the force has to be large.  So the pipes have to be strong (enough).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was talking with a civil engineer who had built dams in Iran in the 1950s.  He said he never had used calculus on the job.  But I was thinking that turning off a valve on a dam must be quite a dangerous thing.  He told me about one young engineer who he had told to go shut off a valve.  He soon saw the pressure gauge at the station shoot up and get pegged at max.  Then it shot down and was pegged at zero.  Then up. Finally it went down and stayed down.  Then he knew.  He went to the window and saw a geyser shooting up from the busted pipe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example: When you're in a tight turn in a car and the driver suddenly straightens out the steering wheel, your body tends to be thrown toward the other side of the car.  Why, when the acceleration in the car has changed to zero?  It's because of the tension in the muscles in your torso that are exerting a force to supply the centripetal acceleration to your body to keep it in the same trajectory as the car.  When the car straightens out, the tension in your body continues the centripetal acceleration, until you relax. So if the third derivative is too great, the acceleration changes faster than your body can react, and your torso tends to follow the extension of the curve that car used to be tracking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's much easier to think in these terms than in terms of finite differences (average rates).&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2016-01-09T05:46:40.550" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10373" PostTypeId="1" CreationDate="2016-01-09T09:18:59.253" Score="6" ViewCount="312" Body="&lt;p&gt;In &lt;a href=&quot;https://calculus123.com/wiki/Fantasy_math&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fantasy math&lt;/a&gt;, Peter Saveliev remarks:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Removing structures from consideration is hard; adding is much easier. Compare how you start with point-set topology -- by removing the geometry from the Euclidean space -- and how you start with inner product spaces -- by adding this structure to vector spaces.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt; Has research been carried out on the difference in difficulty, if any, between removing and adding structures when it comes to learning a mathematical concept or topic? If so, what were the conclusions? Is it inherently easier to think about additional structure on a familiar object, and inherently harder to let go of already familiar structures? Are any aspects of this special to mathematics or is it merely an example of a more general phenomenon of difficulty in unlearning what you have already learned?&lt;/p&gt;&#xA;&lt;p&gt;(Mathematical examples and counterexamples are welcome.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Note: while I am using the term &amp;quot;structure&amp;quot; somewhat loosely, it may be helpful to see the Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_structure&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; on mathematical structure(s). I quote the opening sentences below:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In mathematics, a &lt;strong&gt;structure&lt;/strong&gt; on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.&lt;/p&gt;&#xA;&lt;p&gt;A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="376" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-11-21T14:55:29.840" Title="Difference in difficulty between removing and adding structures?" Tags="|mathematical-pedagogy|reference-request|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="10374" PostTypeId="2" ParentId="10348" CreationDate="2016-01-09T09:37:42.590" Score="17" Body="&lt;p&gt;One of the most colorful names I have heard is the &lt;strong&gt;Chicken Mc Nugget theorem&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;for any two relatively prime positive integers &lt;span class=&quot;math-container&quot;&gt;$m,n$&lt;/span&gt;, the greatest integer that cannot be written in the form &lt;span class=&quot;math-container&quot;&gt;$am + bn$&lt;/span&gt; for nonnegative integers &lt;span class=&quot;math-container&quot;&gt;$a, b$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$mn-m-n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Coin_problem#Examples&quot; rel=&quot;noreferrer&quot;&gt;link1&lt;/a&gt;, &lt;a href=&quot;http://www.artofproblemsolving.com/wiki/index.php/Chicken_McNugget_Theorem&quot; rel=&quot;noreferrer&quot;&gt;link2&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From the links:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The story goes that the Chicken McNugget Theorem got its name because in McDonalds, people bought Chicken McNuggets in 9 and 20 piece packages. Somebody wondered what the largest amount you could never buy was, assuming that you did not eat or take away any McNuggets. They found the answer to be 151 McNuggets, thus creating the Chicken McNugget Theorem.&lt;/p&gt;&#xA;&lt;p&gt;The McNuggets version of the coin problem was introduced by Henri Picciotto, who included it in his algebra textbook co-authored with Anita Wah. Picciotto thought of the application in the 1980s while dining with his son at McDonald's, working the problem out on a napkin.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4930" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-01-09T09:37:42.590" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10375" PostTypeId="2" ParentId="10348" CreationDate="2016-01-09T12:44:04.107" Score="6" Body="&lt;p&gt;I've just remembered that &quot;Donkey Theorem&quot; is used to refer to triangle inequality in geometry textbooks in Iran. The name implies that even a donkey which is on one corner of a triangle chooses the straight  path (rather than the broken one) to get to the other corner where there is some hay to eat. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I checked to see if it  is used elsewhere and I learned from this &lt;a href=&quot;https://math.stackexchange.com/questions/66181/what-is-the-donkey-theorem&quot;&gt;MSE&lt;/a&gt; post that it is also used in Turkey.     &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-01-09T13:05:55.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10376" PostTypeId="2" ParentId="10359" CreationDate="2016-01-09T14:47:00.590" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://www.brown.edu/sheridan/teaching-learning-resources/teaching-resources/classroom-practices/active-learning/interactive&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Think, Pair, Share&lt;/em&gt; and &lt;em&gt;Exit Tickets&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2020-09-05T17:38:59.503" LastActivityDate="2020-09-05T17:38:59.503" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="10377" PostTypeId="2" ParentId="10244" CreationDate="2016-01-10T00:32:51.523" Score="2" Body="&lt;p&gt;Jessica,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Norman Wildberger has a few interesting approaches to thinking about quotient groups, leading up to him talking about homology in his intro Algebraic Topology course. I cannot remember which video he begins quotients, but the first of the (I think 4) videos is here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=Kcyitn0LHSM&amp;amp;list=PL6763F57A61FE6FE8&amp;amp;index=14&quot; rel=&quot;nofollow&quot;&gt;https://www.youtube.com/watch?v=Kcyitn0LHSM&amp;amp;list=PL6763F57A61FE6FE8&amp;amp;index=14&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2016-01-10T00:32:51.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10379" PostTypeId="2" ParentId="10348" CreationDate="2016-01-10T03:53:56.127" Score="10" Body="&lt;p&gt;I often refer to the identities $(AB)^{-1} = B^{-1}A^{-1}$ or $(AB)^T = B^TA^T$ as the &lt;strong&gt;socks-shoes&lt;/strong&gt; identity. I'm not sure how wide-spread this is, I certainly did not invent it and I'm pretty sure I've read at least one of these in at least one text.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2016-01-10T03:53:56.127" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10380" PostTypeId="2" ParentId="2105" CreationDate="2016-01-10T16:14:11.923" Score="2" Body="&lt;p&gt;Discrete mathematics (some combinatorics, at least) is a requisite for probability and statistics. So it should be covered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For computer science, combinatorics, techniques to evaluate sums, recurrences, and some graph theory are a must, and should be covered early in some detail.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-01-10T16:14:11.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10381" PostTypeId="2" ParentId="10333" CreationDate="2016-01-10T16:23:25.467" Score="3" Body="&lt;p&gt;Going all &quot;rigorous definition&quot; in front of students who are being exposed to a concept for the first time in a more formal setting is a serious mistake. All you'll get is glassed over eyes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You want them to develop an &lt;em&gt;intuition&lt;/em&gt; of what angle (or variable, or equation, or polynomial, whatever) is all about, and why the concept is useful. Once that is solid, you can go for rigorous definitions, and show that they agree with the intuitive idea. Once &lt;em&gt;that&lt;/em&gt; is internalized, you go for a rigorous axiomatization, and make clear that &quot;angle&quot; is just something unspecified that satisfies the axioms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That process takes time. Calendar time, not class/study time. Across subjects. Mostly called &quot;developing mathematical maturity&quot;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-01-10T16:23:25.467" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10382" PostTypeId="2" ParentId="10349" CreationDate="2016-01-10T22:29:28.623" Score="12" Body="&lt;p&gt;&quot;For all&quot; statements are hard to grasp because they don't really exist outside of mathematics (or philosophy or other abstract domains).  In any vaguely empiricist epistemology, you can have &quot;there exist&quot; statements, and you can have &quot;for some&quot; or &quot;for most&quot; or &quot;for every case we have seen&quot; statements, but even the last is not &quot;for all&quot; statement in the mathematical sense, and you can't verify empirically a &quot;for all&quot; statement in the mathematical sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence for someone who is not familiar with mathematics, the easiest way to conceive of a &quot;for all&quot; statement is as an absolute negation of a &quot;there exist&quot; statement.  In other words, once we break a student out of thinking of a &quot;for all&quot; statement as an empirical &quot;for every case we have seen&quot; statement, the next step is for the student to think of the &quot;for all&quot; statement as a &quot;there cannot logically exist a counterexample&quot; statement.  If you think of the statement you are trying to prove as a &quot;there cannot logically exist a counterexample&quot; statement, then you are naturally led to proof by contradiction.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2016-01-10T22:29:28.623" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10383" PostTypeId="2" ParentId="10349" CreationDate="2016-01-11T05:03:19.387" Score="3" Body="&lt;p&gt;When students prove results by contradiction, it might just be following what they see in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the first two theorems they see are &quot;there are infinitely many primes&quot; and &quot;$\sqrt{2}$ is irrational&quot;, with both proved by contradiction.  Then they naturally use that technique as a default.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an alternative, one can prove the infinitude of primes in the Euclidean way, as &quot;the number of primes is more than any assigned magnitude&quot;.  And instead of proving irrationality, one can start with an example like &quot;there is a sequence of 100 consecutive composite numbers&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This shows students a key ingredient in direct proof, and it has a name that they can learn too:  &lt;strong&gt;proof by construction&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastEditorDisplayName="user173" LastEditDate="2016-01-11T05:10:26.433" LastActivityDate="2016-01-11T05:10:26.433" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10384" PostTypeId="2" ParentId="10348" CreationDate="2016-01-11T06:16:38.030" Score="1" Body="&lt;p&gt;In Uruguay (at least) famous formulas are often referred to using the name of a person who has been associated to their initial discovery.  For instance (see my comment above), the quadratic formula is called Bhaskara and the corollary to the first part of the Fundamental Theorem of Calculus $\int_a^b f(t) dt = F(b)-F(a)$ is called Barrow.  Mathematicians there say &quot;by Barrow&quot; or &quot;by Bhaskara&quot; which I find much more interesting than &quot;by the Corollary to the first part of the Fundamental Theorem of Calculus&quot; or &quot;by the quadratic formula&quot;.  It feels more human to me.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2016-01-11T06:16:38.030" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10385" PostTypeId="2" ParentId="10349" CreationDate="2016-01-11T07:39:51.323" Score="19" Body="&lt;p&gt;Suppose you're in an unfamiliar city without a map.  You're trying to get to a particular address, which you know is within five blocks of you, but you have no idea how the streets are laid out, whether they're all orthogonal to one another, in which direction the addresses increase, or anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to start out you'll probably walk in one direction until you can find at least two buildings on the same street with their addresses written on them.  (This is surprisingly hard, if you haven't tried recently; often there may only be a single building on a block that has its address displayed anywhere.) Then you at least know which direction the numbers increase in.  Next, you might figure this out in a perpendicular direction, and so forth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eventually, you'll reach your destination, but you likely won't have a solid mental picture of how you got there.  And the path you took would look convoluted and absurd to someone familiar with the city, who knows that all the numbered streets in this city all run east-west and are all on the south side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Someone setting out to write a proof for the first time cannot hold an entire proof in his head, even of something completely trivial.  A student may be able to start out from the hypotheses and get to the conclusion while having no idea how what happened along the way.  Since contradiction is often helpful, it makes sense to start out by trying this.  By the time the proof is done, the student doesn't know whether this got used or not, because the student doesn't have the mathematical maturity to see the proof as anything more than a long sequence of steps, only one or two of which can be held in the mind at once.&lt;/p&gt;&#xA;" OwnerUserId="623" LastActivityDate="2016-01-11T07:39:51.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10386" PostTypeId="2" ParentId="3858" CreationDate="2016-01-11T08:00:01.623" Score="3" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Coase_theorem&quot; rel=&quot;nofollow&quot;&gt;Coase theorem&lt;/a&gt;, in some sense, underlies nearly all of the law, and an explicit understanding of this sort of thing is becoming increasingly important for legal scholars.  Judge Richard Posner is a leading thinker on the economic foundations of the legal system and has written a large number of books on the subject.  I took a class using his book &lt;em&gt;Economic Analysis of Law&lt;/em&gt; in college, but I can't remember now how much economic jargon it relied on.&lt;/p&gt;&#xA;" OwnerUserId="623" LastActivityDate="2016-01-11T08:00:01.623" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10387" PostTypeId="2" ParentId="10244" CreationDate="2016-01-11T08:11:07.027" Score="3" Body="&lt;p&gt;The problem I generally see with cosets is that people generally are shown the group axioms, then they're shown a couple simple examples of groups (cyclic groups, dihedral groups, symmetric/alternating groups, and maybe the quasidihedral group or something), and then they're asked to think about cosets and quotients.  In particular, they don't have any idea what a group &quot;is&quot; or have any intuition for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not surprising to me that students don't understand, because it seems like we've made a completely arbitrary definition and asked them to compute with it.  In particular, groups are often presented in such a way that students don't understand that group actions are fundamental and a group is just something with exactly the right amount of structure to act on something.&lt;/p&gt;&#xA;" OwnerUserId="623" LastActivityDate="2016-01-11T08:11:07.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10388" PostTypeId="2" ParentId="10348" CreationDate="2016-01-11T08:27:21.623" Score="4" Body="&lt;p&gt;What is called &quot;fórmula del chicharronero&quot; in Central Mexico (see the answer by Rodrigo Zepeda) is called &quot;&lt;strong&gt;Mitternachtsformel&lt;/strong&gt;&quot; (&quot;&lt;strong&gt;midnight formula&lt;/strong&gt;&quot;) in middle school in some parts of Germany. This is because, if someone wakes you up at midnight and asks what are the roots of a parabola you &lt;strong&gt;have&lt;/strong&gt; to know this in a second.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After second thought, I think that the Mitternachtformel is&#xA;$$&#xA;x_{1/2} = -\tfrac{p}2\pm\sqrt{\tfrac{p^2}{4}-q}&#xA;$$&#xA;for the roots of&#xA;$$&#xA;x^2+px+q=0.&#xA;$$&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2016-01-11T08:27:21.623" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10389" PostTypeId="2" ParentId="8422" CreationDate="2016-01-11T10:50:35.970" Score="5" Body="&lt;p&gt;&lt;em&gt;Note: This came up in my &quot;hot questions&quot; feed - I didn't notice how old the question was until after I answered it. Sorry if this isn't required anymore or digs up a dead and buried debate!&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Studies (and my own experiences) would suggest that using more words than symbols (to start with) would help everyone.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's a page &lt;a href=&quot;http://www.learningideas.me.uk/clearmaths/&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt; that deals specifically with guidelines on how to write mathematics (and consequentially, logic). There's an awful lot there, with examples and preferences over many different cases, and there is a list of sources to papers and studies at the bottom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The general gist of the page is that mathematical language has a tendency to be too succinct when it comes to beginners - professional level communication (which assumes knowledge of the subject) is too often used with beginners. Words are left out, context isn't given and a glut of symbols is used where words could be more clear, if not more descriptive to someone with professional knowledge of the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;If you're only interested in literature to back up a point of view, stop here&lt;/strong&gt;. If you want some more input and specific answers to your questions within your question, feel free to read on, but be aware that the rest is more opinion based on personal experience.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This backs up my own personal experiences on the subject - I studied Maths in university and had a module on Mathematics Education. Our lecturer said she has started the course because there weren't enough good maths educators in the world; either you understood the language - and then found it hard to understand when someone else didn't - or you didn't, and while you could then empathise and help someone else who didn't, you never truly felt comfortable with it. People who both &quot;Got it&quot; AND could see how to explain it to someone who didn't are incredibly rare, at least in the western world - the amount of people out there convinced they &quot;can't do maths&quot; is proof of that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this being said, &lt;strong&gt;it is still extremely important to learn the symbols properly if you are going to converse in the mathematical world&lt;/strong&gt;. While it might be best to start with words to encourage understanding, at some point that understanding needs to be extended to the symbolic representations of the same concepts - learning the language of maths is in many ways just as important as learning the concepts themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To address the questions in your question directly (and succinctly - if you want me to expand on any let me know, I'm just way of the length of this answer):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Are there studies about how emphasizing one of these types affects the way students learn?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Yes, at least partially - see the link at the start of this answer (&lt;a href=&quot;http://www.learningideas.me.uk/clearmaths/&quot; rel=&quot;noreferrer&quot;&gt;This one!&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;What harm does it do if only one type is present in teaching?&lt;/em&gt; &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In most (those who don't &quot;find maths easy&quot;), only teaching the symbols will prevent true understanding and in many cases any understanding of what the student is learning. Only teaching the words prevents students from learning the language they need to converse in the mathematical community and will also ostracise those students who are gifted at maths but find language hard (quite common).&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;How important is it that students can translate statements between these two languages?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I would say extremely important, but that's a personal opinion. For one, it shows sufficient comprehension. If the student can convert a symbolic statement into language, it shows they understand the concept correctly, and if they can convert from language to symbols, it shows they understand the symbols correctly.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;What are some key skills having to do with formal logical statements that I should make sure every student learns?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I don't feel fully qualified to answer this, as I'm not a trained mathematics educator (I've just studied the stuff), and I cannot claim any list I make to be remotely definitive and may even be vastly misleading due to glaring omissions. All I can say is that you're on the right track with this question in the first place - communication and clarity are key in logic and maths. If a student learns to phrase their statements in a way that makes the content as clear and obvious as possible, they've done the hard part.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Is it easier or faster for students to read statements in words than in symbols?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;for an absolute beginner, words will always be easier as they know what the words mean. As they do more, learn more of the symbols, the ratio of words:symbols that they find &quot;fastest&quot; and &quot;easiest&quot; to read will tend more towards symbols than words, because words take longer to read, but the actual ratio will likely depend on the person. For me, the below is easier to read than either of the examples in your question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$∀y∈R, ∃δ&amp;gt;0$ such that $∀x∈Q : |y−x| &amp;lt; δ$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's because I understand all the symbols and the connotations of the symbols that would take a lot of words to properly write out, but I still feel it benefits from a little English in the middle, just to give the &quot;sentence&quot; some direction. Of course, the advantage of pure mathematical/logical symbolism is that it transcends language; if I can write the whole thing without using any English, it will be absolutely clear to a French mathematician, which has its own benefits (and is another reason to learn the symbols!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In conclusion, I would say that studies support the concept that words are (for the general population) far easier to understand (at least at first) than symbols, and that if your aim is to help spread the love of maths to more than the disappointingly small portion of the population that currently understand it, then advocating a &quot;words, then symbols&quot; approach would probably be vastly beneficial, even to those who would understand it without the wordy approach, because it may help them get their ideas across when communication with symbols breaks down.&lt;/p&gt;&#xA;" OwnerUserId="6023" LastActivityDate="2016-01-11T10:50:35.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10391" PostTypeId="2" ParentId="10373" CreationDate="2016-01-11T16:26:52.747" Score="0" Body="&lt;p&gt;It is hard to remove the structures, usually knee-jerk manipulations, $x^n \leadsto nx^{n-1}$ type of thing, when teaching calculus at the college level. Another example is un-teaching them to use l-Hopital without checking the hypotheses first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example perhaps more along the lines of what the OP had in mind is the tendency to attribute certain characteristics to the word &quot;number&quot;, for example assuming that the term can only refer to a finite quantity.  Paradoxically, this is the opposite of what the greats of the past considered intuitive; thus, Leibniz and Euler naturally assumed that a number can be infinite, and moreover used such numbers in numerous articles.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2016-01-11T19:05:36.700" LastActivityDate="2016-01-11T19:05:36.700" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10392" PostTypeId="2" ParentId="10333" CreationDate="2016-01-11T18:57:20.080" Score="0" Body="&lt;p&gt;Just a suggestion for the purposes of discussion...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;em&gt;A, B, C&lt;/em&gt; be points in the Euclidean plane. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then &lt;em&gt;(A,B,C)&lt;/em&gt; is said to be an &lt;em&gt;angle&lt;/em&gt; with &lt;em&gt;vertex&lt;/em&gt; &lt;em&gt;B&lt;/em&gt; if and only if  &lt;em&gt;B=/=A&lt;/em&gt; and &lt;em&gt;B=/=C&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Let &lt;em&gt;A, B, C, D&lt;/em&gt; be points in the Euclidean plane such that: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;em&gt;A,B,C&lt;/em&gt;) is an angle with vertex &lt;em&gt;B&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;D&lt;/em&gt; is not on &lt;em&gt;Ray(B,A)&lt;/em&gt; or on &lt;em&gt;Ray (B,C)&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then &lt;em&gt;D&lt;/em&gt; is said to be an &lt;strong&gt;&lt;em&gt;interior&lt;/strong&gt; point&lt;/em&gt; of &lt;em&gt;angle (A,B,C)&lt;/em&gt; if and only if there exist points &lt;em&gt;E, F&lt;/em&gt; such that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;B=/=E&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;B=/=F&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;E&lt;/em&gt; is on &lt;em&gt;Ray(B,A)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;F&lt;/em&gt; is on &lt;em&gt;Ray(B,C)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;D&lt;/em&gt; is on &lt;em&gt;LineSegment(E,F)&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2016-01-12T20:42:15.003" LastActivityDate="2016-01-12T20:42:15.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10393" PostTypeId="2" ParentId="10348" CreationDate="2016-01-11T23:28:42.097" Score="4" Body="&lt;p&gt;The phrase &lt;strong&gt;chain rule&lt;/strong&gt; in Calculus is a mathematical slang.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rule goes like this: If $f(x) = g(h(x))$, then $f'(x) = g'(h(x))h'(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I learned the rule at school, my teachers just called it &quot;the function of a function rule&quot;, which describes precisely the situation in which you use it. When I got to university in a different state, they kept mentioning this &quot;chain rule&quot; and I had not the slightest clue what they were talking about.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2016-01-11T23:28:42.097" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10394" PostTypeId="2" ParentId="10021" CreationDate="2016-01-12T06:11:50.210" Score="9" Body="&lt;p&gt;Because for us students it seems that there is no structure for it. We get the impression that there is a lot of assumption that one might do in order to prove by induction and there is no formula for it. Every question is different and brings in a new factor or a new way of creating the base case, or splitting the equation to be proved, or inserting a random number at the end that makes the equation still valid, etc. There is always something that we haven't seen in previous proofs that makes it really challenging for us to just come up with this new &quot;trick&quot; for every new thing we try to prove. And it is frustrating. There seems to be no method to learn it. We see other people doing it and think it is a natural talent that we don't have and no matter how much we practice we will never learn. When I see someone doing a proof by induction, during the the process I am often like: &quot;Where did you get that from?!?!&quot;. And the answer is always like &quot;Oh, that is simple, you just do this, this, and that... tadah!!&quot; And what the person does is always something I have never seen before. Always. How did he come up with that?!! So, yeah, that is why I struggle with induction.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2016-01-13T19:37:14.390" LastActivityDate="2016-01-13T19:37:14.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10395" PostTypeId="1" CreationDate="2016-01-12T12:03:47.023" Score="5" ViewCount="166" Body="&lt;p&gt;$\newcommand{\RR}{\mathbb{R}}&#xA;\newcommand{\dd}{\mathrm{d}}$&#xA;In teaching multivariable integration on sub-manifolds in $\RR^n$, i.e. integrals over $k$-dimensional surfaces $M\subset \RR^n$ you define the integral of $f$ over $M$ as follows: If $\phi:T\to M$ is a paremetrization (i.e. $T\subset\RR^k$ is open, $D\phi$ has full rank), then the integral is&#xA;$$&#xA;\int_M f(x)\dd S(x) = \int_T f(\phi(t))\sqrt{\det G(t)}\dd t&#xA;$$&#xA;where $G$ is the matrix with entries&#xA;$$&#xA;G_{ij}(t) = \langle\partial_i\phi(t),\partial_j\phi(t)\rangle.&#xA;$$&#xA;Once it is clear that $\sqrt{\det G(t)}$ is the $k$-dimensional volume of parallelotope spanned by the tangential vectors $\partial_j\phi(t)$ it seem clear why this should be the surface element.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But my question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there an intuitive way to see that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gramian_matrix#Gram_determinant&quot;&gt;Gramian&lt;/a&gt; $\det G(t)$ is the squared volume of said parallelotope?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2016-01-21T20:33:37.620" Title="How to motivate the surface element" Tags="|vector-calculus|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10397" PostTypeId="1" CreationDate="2016-01-12T18:27:25.290" Score="6" ViewCount="374" Body="&lt;p&gt;There are many questions about how best to teach functions, for example &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7789/why-dont-we-teach-codomains-of-functions-in-high-school&quot;&gt;why don't we teach codomains in HS&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/8078/should-we-teach-functions-as-sets-of-ordered-pairs&quot;&gt;should we teach them at all&lt;/a&gt;.  On &lt;a href=&quot;https://math.stackexchange.com/&quot;&gt;Math.SX&lt;/a&gt; there are questions about the &quot;right&quot; name for this concept, such as &lt;a href=&quot;https://math.stackexchange.com/questions/59432/domain-co-domain-range-of-a-function&quot;&gt;this one&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/questions/440235/range-of-function-vs-target-of-function?lq=1&quot;&gt;that one&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is different.  Assuming that one &lt;em&gt;is&lt;/em&gt; going to introduce functions as a special type of relation between two sets $A$ and $B$, with $A$ the domain (there aren't other words for &lt;em&gt;that&lt;/em&gt;, are there?), what should we call $B$, &lt;em&gt;from a pedagogical point of view&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no right answer here, but I do feel like people have &lt;a href=&quot;https://math.stackexchange.com/a/1412590&quot;&gt;strong opinions&lt;/a&gt; about this issue that are rooted in good pedagogical practice, which is after all why we're lurking on this site, right?  So perhaps this question can serve as a good place to gather the 'range' of good pedagogical reasons for choice of terminology in this case.  Discussions about how this is resolved (or if it's unnecessary) in non-Anglophone contexts are also welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth, because of the ambiguity in the meaning of the term my main feeling is, as a professor of mine once said, to leave the range to the cattle.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-01-16T10:29:06.207" Title="Name to use for codomain/range/target" Tags="|terminology|functions|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10398" PostTypeId="2" ParentId="10348" CreationDate="2016-01-12T18:49:48.387" Score="4" Body="&lt;p&gt;I was about to write a comment to @Amy B division house, saying that in Iran we use &lt;a href=&quot;https://i.stack.imgur.com/EjnEz.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EjnEz.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;to denote $a$ divided by $b$, that a colleague entered the room asking me what I am doing. I explained and she told me that in their primary school (somewhere in  Leicestershire in England) they called it (the symbol drawn by Amy B) &quot;&lt;strong&gt;bus stop&lt;/strong&gt;&quot; where the bigger number needs to be covered and the smaller number remains outside! I thought it is worth mentioning as a separate answer.    &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2016-01-12T18:49:48.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10399" PostTypeId="1" CreationDate="2016-01-12T19:10:15.310" Score="4" ViewCount="177" Body="&lt;p&gt;These are my instructions for a problem set that I am giving for a Math 60 (Intro Algebra) class at a community college.  What do you think?  Are they clear?  What would you add or subtract?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;These are hard problems.  I ask that you exert a substantial amount of effort on these problems before seeking help.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are allowed to look in your text or on the Internet for help.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are not allowed to copy another person's work, whether that person is a classmate, a tutor, or someone on the Internet.  Your write-up should be your own work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Classmates (and tutors) are allowed to discuss these problems with each other but one person should not be telling another person what specifically to write down.  If a student wants more specific help, then the person helping should find a similar problem to solve, and not solve the one given below. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grading criteria:&#xA;Each question is worth 4 points.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;4 points = accurately explained and solved&lt;/li&gt;&#xA;&lt;li&gt;3 points = a mathematically sound, substantial attempt, but has a small error&lt;/li&gt;&#xA;&lt;li&gt;2 points = a mathematically sound, substantial attempt, but has substantial errors &lt;/li&gt;&#xA;&lt;li&gt;1 points = not mathematically sound, not a substantial attempt&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3265" LastEditorUserId="123" LastEditDate="2016-01-12T20:28:43.393" LastActivityDate="2016-01-13T18:06:56.747" Title="Instructions for a problem set" Tags="|assessment|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10400" PostTypeId="2" ParentId="10341" CreationDate="2016-01-12T20:13:43.357" Score="1" Body="&lt;p&gt;In my opinion, what differentiates modeling from applications or contexts is data. To be a modeling problem, you need data. Either the students should be collecting data, or the data can be given to them. One possible &quot;flow&quot; for a modling problem is: &#xA;data -&gt; patterns/relationships -&gt; creating mathematical model -&gt; using the model to predict -&gt; going back to the &quot;physical world&quot; to determine accuracy/results&lt;/p&gt;&#xA;" OwnerUserId="6230" LastActivityDate="2016-01-12T20:13:43.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10402" PostTypeId="2" ParentId="10399" CreationDate="2016-01-12T20:31:20.553" Score="2" Body="&lt;p&gt;What is &quot;copying&quot;? Is looking at Wikipedia and paraphrasing what is said there &quot;copying&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should perhaps ask for citations of the materials referenced/used, be it classmate discussion, tutor, textbook, lecture note elsewhere, whatever.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-01-12T20:31:20.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10403" PostTypeId="2" ParentId="10397" CreationDate="2016-01-12T21:11:11.180" Score="6" Body="&lt;p&gt;I think &quot;codomain&quot; is great.  If $f: A \to B$, then I have only ever heard &quot;the range of $f$&quot; defined as the subset of the codomain $\{b \in B : \exists a \in A  \textrm{ with } f(a) = b \}$.  It is helpful to have names for these things, and I think these are pretty much standard in higher level mathematics.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-01-12T21:11:11.180" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10404" PostTypeId="2" ParentId="10399" CreationDate="2016-01-13T13:38:14.203" Score="1" Body="&lt;p&gt;You say students should &quot;exert a substantial amount of effort on these problems before seeking help.&quot;  That is open to interpretation and might mean a half hour of staring at the problems without any ideas to one student and trying different methods to another students.  You might clarify this in class so that you can have a discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also you say students can discuss this problem, but it isn't clear if that is considered seeking help or if that is something they can do before exerting that effort.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-01-13T13:38:14.203" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10405" PostTypeId="2" ParentId="10399" CreationDate="2016-01-13T18:06:56.747" Score="1" Body="&lt;p&gt;I have updated my instructions.  I'll see how it goes.  Thanks for your feedback.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;These are hard problems.  Please make an attempt on paper before seeking help from another person.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are not allowed to look at or copy another person's work, whether that person is a classmate, a tutor, or someone on the Internet.  Your write-up should be your own work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Classmates (and tutors or other outside help) are allowed to discuss these problems with you.  The helping person may say something like “The first thing I did was add 5 to both sides” or “I think the answer is 5.”  The helping person may not tell the student what to write down line by line, as that is the same as copying.  In other words, the helping person may discuss the outline of a solution, but the student has to fill in the details.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you received outside help, please cite the source, whether the source is a classmate, a tutor, or someone on the Internet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grading criteria:&#xA;Each question is worth 4 points.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;4 points = accurately explained and solved&lt;/li&gt;&#xA;&lt;li&gt;3 points = a substantial attempt using a valid method with a small error&lt;/li&gt;&#xA;&lt;li&gt;2 points = a substantial attempt using a valid method with substantial errors&lt;/li&gt;&#xA;&lt;li&gt;1 points = not a valid method, or not a substantial attempt&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3265" LastActivityDate="2016-01-13T18:06:56.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10406" PostTypeId="2" ParentId="10373" CreationDate="2016-01-14T15:36:57.710" Score="4" Body="&lt;p&gt;There is a sense in which it is not fair to say that one &quot;removes structure&quot; when one considers a more general class of mathematical objects. Usually one tries to do analogous things with the more general class that one was able to do with the more specific class, and this requires &lt;em&gt;new&lt;/em&gt; definitions and techniques.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's take the example of &quot;removing the geometry from Euclidean space&quot;. Just forgetting about distance and angles etc. is easy to do. But if I want to actually do anything in this new geometry-less world, that's another story. Let's say, as in the question, that I've moved into point-set &lt;em&gt;topology&lt;/em&gt;. First need to know what a topology is, and that takes a certain level of mathematical maturity to grasp. Once I understand what a topology is, I will probably learn that there are many different possible topologies on a set. I will also learn that the notion of &quot;distance&quot; I used to have in Euclidean geometry gives one such topology. Then I can check my understanding by convincing myself that some of these new notions I've learned about really do specialize to the things I used to do when I was doing geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a common pattern in mathematics: generalize with an eye towards recovering old results or analogs of old results in the generalized setting. This almost invariably leads to a net gain in &lt;em&gt;number of possible structures&lt;/em&gt;, and these possible structures are the things that wind up being studied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a more concrete level, Jessica B's comment also addresses a difficulty that can occur when removing structure: we can't use the rules we used to use. Consider a linear algebra student encountering matrices and matrix algebra for the first time. The student would like to do algebra with matrices just like with numbers. But now there are things that are not invertible, and even when a matrix is invertible the student can't &quot;divide&quot; both sides of an equation by a matrix because they don't have commutativity. So the student needs to adjust the way that they've always done algebra.&lt;/p&gt;&#xA;" OwnerUserId="6235" LastActivityDate="2016-01-14T15:36:57.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10409" PostTypeId="1" CreationDate="2016-01-15T15:10:04.367" Score="9" ViewCount="192" Body="&lt;p&gt;Suppose that a secondary school intends to hire a new math teacher, math department head, or math and science director. Do any professional mathematics or educational organizations such as the MAA, NCTM, or NAIS publish a list or summary of competencies, skills, and qualifications that schools should seek in such new hires? &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-01-15T23:01:30.230" LastActivityDate="2016-01-15T23:01:30.230" Title="Do mathematics teaching professional organizations publish hiring recommendations?" Tags="|secondary-education|teachers|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10410" PostTypeId="2" ParentId="10409" CreationDate="2016-01-15T18:00:12.653" Score="8" Body="&lt;p&gt;The MAA has a document (for undergraduate institutions) titled &quot;Guidelines for Programs and Departments in Undergraduate Mathematical Sciences&quot; and addresses some of this in Section C: &quot;Program Faculty and Staffing&quot;. See &lt;a href=&quot;http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/guidelines-for-undergrad-programs#c&quot;&gt;http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/guidelines-for-undergrad-programs#c&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2016-01-15T18:00:12.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10411" PostTypeId="2" ParentId="10341" CreationDate="2016-01-15T20:18:04.473" Score="7" Body="&lt;p&gt;&lt;strong&gt;What is Mathematical Modeling?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might think this to be a simple, straightforward answer, but unfortunately we have no such luck. The definition of &lt;em&gt;mathematical modeling&lt;/em&gt; varies depending on the author unlike other more clearly defined mathematical terms like &lt;em&gt;prime&lt;/em&gt; or &lt;em&gt;group.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pollak Definition&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arguably, one leading mathematician in the field of modeling is &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_O._Pollak&quot; rel=&quot;noreferrer&quot;&gt;Pollak&lt;/a&gt;, former Bell Labs director who has also been an educator for the past 30 years or so. In his chapter &lt;em&gt;A History of the Teaching of Modeling&lt;/em&gt; in Stanic and Kilpatrick's &lt;a href=&quot;https://books.google.com/books/about/A_History_of_School_Mathematics.html?id=59PaAAAAMAAJ&quot; rel=&quot;noreferrer&quot;&gt;A History of School Mathematics&lt;/a&gt;, he opens with providing a clean definition for mathematical modeling. To quote from here:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What distinguishes modeling from other forms of applications of mathematics are (1) &lt;em&gt;explicit&lt;/em&gt; attention at the beginning to the &lt;em&gt;process&lt;/em&gt; of getting from the problem outside of mathematics to its mathematical formulation and (2) an explicit reconciliation between the mathematics and the real-world situation at the end. Throughout the modeling process, consideration is given to both the external world and the mathematics, and the results have to be both mathematically correct and reasonable in the real-world context. (p. 649)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Pollak goes on to describe an 8-step process that form mathematical modeling. These 8 steps can be split apart into different groupings to form other types of mathematics [e.g., &quot;'Applied mathematics'&quot; is traditionally a name for a collection of fields of mathematics that arise frequently in step 5.&quot; (p. 650)].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An abridged version of these 8 steps are below. [For the full version see H. Pollak, Solving problems in the real world, in &lt;em&gt;Why Numbers Count: Quantitative Literacy for Tomorrow's America&lt;/em&gt;, (ed.) L. Steen, College Board, New York, 1997, pp. 91-105.]:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Identify the real-world problem;&lt;/li&gt;&#xA;&lt;li&gt;Identify important factors on which to focus;&lt;/li&gt;&#xA;&lt;li&gt;Select which factors and which interrelations to keep and which to toss away;&lt;/li&gt;&#xA;&lt;li&gt;Translate the real-world problem into a mathematical one;&lt;/li&gt;&#xA;&lt;li&gt;Identify the mathematics required;&lt;/li&gt;&#xA;&lt;li&gt;Use mathematics to determine results;&lt;/li&gt;&#xA;&lt;li&gt;Translate results back into the real world; and&lt;/li&gt;&#xA;&lt;li&gt;Determine if your answer is reasonable. If it is, report on this. If not, repeat the process.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Common Core Definition&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More recently, the authors of the Common Core have brought mathematical modeling to the forefront by placing it prominently within their standards document. It appears in two distinct locations: First, as one of the 8 &lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot; rel=&quot;noreferrer&quot;&gt;Standards for Mathematical Practice&lt;/a&gt; and then again as a &lt;a href=&quot;http://www.corestandards.org/Math/Content/HSM/&quot; rel=&quot;noreferrer&quot;&gt;High School Standard&lt;/a&gt; unto itself. What is unique about this second appearance is that it does not provide bite-sized content pieces (e.g., CCSSM.HSG.C.A.4) as do all of the other sections (e.g., Geometry or Grade 4). The authors do define mathematical modeling in the High School section as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;the process of choosing and using appropriate mathematics and&#xA;  statistics to analyze empirical situations, to understand them better,&#xA;  and to improve decisions.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that while there are similarities here to Pollak's definition above, the &lt;em&gt;emphasis&lt;/em&gt; on the movement from the real world to the mathematical one, and back again, does not appear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Common Core authors do choose to include what I call &quot;visual maps&quot; of mathematical modeling. (Note, it was originally included in the PDF versions of the Standards but has become difficult to find on the online version.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EJXVB.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EJXVB.png&quot; alt=&quot;http://2.bp.blogspot.com/-yrpvvVk3uPg/T7gUWqbLeVI/AAAAAAAAAaw/djAmW9xor7I/s1600/MathModelingModel.png&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This image does call to Pollak's definition of re-cycling through the steps if needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Are any of your Calculus questions modeling?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on Pollak's definition, I would say as stand alone problems, that &lt;strong&gt;no&lt;/strong&gt; these are not modeling problems. Being given specific problems like these takes away from the earlier steps in the modeling process that Pollak indicates. It is certainly possible to use these problems as parts of the modeling process, and even to adjust them to include many or all of Pollak's requirements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on the Common Core definition, I might claim that &lt;strong&gt;yes&lt;/strong&gt; all of these could be considered modeling problems, as long as these are based on empirical and not theoretical situations (which could be argued in and of itself).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for my personal opinion, I think that the Common Core definition is (understandably) lacking in some of the areas that Pollak's definition seems to detail. On the other hand, Pollak's definition might make it difficult to ever deem any textbook or teacher provided problem as modeling. The definition of mathematical modeling is very much in the eye of the beholder.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2016-01-15T20:18:04.473" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10412" PostTypeId="2" ParentId="1572" CreationDate="2016-01-15T21:20:11.770" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/aiAG8.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aiAG8.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was making the rounds on Twitter this week. There's so much wrong with this, I don't know where to start. It certainly goes back to the manipulation of large numbers, which people in general tend to struggle with. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note - in the US, our lottery hit a record $1.5 billion dollar prize. Our billion has 9 zeros.)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-01-16T03:30:41.397" LastActivityDate="2016-01-16T03:30:41.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10416" PostTypeId="2" ParentId="10397" CreationDate="2016-01-16T10:29:06.207" Score="3" Body="&lt;p&gt;In French, the most common terminology in my experience is &quot;ensemble de départ&quot; (&quot;starting set&quot; or maybe &quot;departure set&quot;) for the domain and &quot;ensemble d'arrivée&quot; (&quot;ending set&quot; or maybe &quot;arrival set&quot;) for the codomain. Sometimes, especially in high school where it is a common exercize to determine for which values of the variable a given expression makes sense, a function might only be defined on a subset of its &quot;ensemble de départ&quot; which is then called &quot;ensemble de définition&quot;. This is a little confusing when in university we tend to mean that a function is defined on its whole &quot;ensemble de départ&quot; (especially if physicist use the other definition).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I know, &quot;domain&quot; is almost only used for the &quot;ensemble de définition&quot; of an unbounded operator on a linear space.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-01-16T10:29:06.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10417" PostTypeId="2" ParentId="10244" CreationDate="2016-01-16T10:56:18.843" Score="3" Body="&lt;p&gt;I usually find it useful to give concrete examples of the philosophy that cosets are what's left when one wants to neglect the elements of the considered subgroup.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most basic example is the following: turn the light of the room on and off repeatedly and fast a number of time, and then ask the students: &quot;how many time did I hit the button?&quot; You want to have done it in a way that they won't be able to answer. Then ask: &quot;did I hit the button an even number of time or an odd number of time?&quot;, and then the answer only depends on the state of the light before and after the operation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that the light does not care exactly how many time you hit the button, it only cares if this number is even or odd, or in other words the light does exactly as if 2 where equal to 0. The state of the light is described by a quotient, whose elements are cosets (all the numbers of it inducing a given state of the light).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lights you tap to cycle through dim light/bright light/off give a similar example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possibility for orbits with a non-abelian group is to find several standard and a non-standard dices (most dices have the numbers arranged in the same patterns, but not all of them). Ask your student if two dices are the same : they will try to rotate them so that they match. The point is that not any position will do, and if they succeded it means they applied an element of the group of displacements of the cube (aka $S_4$, but that is another story of course) to a labeling of the faces by $1,\dots, 6$, to obtain another labeling. If they (provably) can't, then it means that the two dices belong to different orbits. The point again is that how to try answer the question should be obvious to the students. Then one can frame their methods in the mathematical framework and hope that they make the connection between what they did and the object of the course, and then the later may start making sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem with this example is that the set of labelings has no natural group structure, so it does not fit your question well. I guess there are other possible examples which may do.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-01-16T10:56:18.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10418" PostTypeId="2" ParentId="1572" CreationDate="2016-01-16T13:37:04.233" Score="9" Body="&lt;p&gt;There was &lt;a href=&quot;http://www.capecodtimes.com/article/20090430/NEWS/904300313&quot;&gt;this story&lt;/a&gt; from 2009, where a town mistakenly believed it had passed a measure which required support from two-thirds of the voters, when it only had support from 66%.  (The town accountant (!) calculated two-thirds by multiplying by .66.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems to be a common problem.  One of my colleagues was consulted by a representative of the Pennsylvania judicial system to sort out a standing dispute about how to calculate fees which were capped by law at one-third: some judges multiplied by .33, others by .333, and they wanted to know what was right.  (They were allegedly impressed by the suggestion that they could simply divide by three.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-01-16T13:37:04.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10419" PostTypeId="2" ParentId="9842" CreationDate="2016-01-16T20:00:34.573" Score="1" Body="&lt;p&gt;I am not a math educator; however I wonder if at times it might make sense to use the commutative law first before expanding the terms.  Please solve&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1,000,000,000 x 3&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Might make sense to use the commutative law&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3 x 1,000,000,000&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1,000,000,000 + 1,000,000,000 + 1,000,000,000  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However the Teacher may prefer the student use their method for this operation. &lt;/p&gt;&#xA;" OwnerUserId="6247" LastEditorUserId="6247" LastEditDate="2016-01-16T20:59:37.730" LastActivityDate="2016-01-16T20:59:37.730" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10420" PostTypeId="2" ParentId="10244" CreationDate="2016-01-17T02:47:17.790" Score="5" Body="&lt;p&gt;I think that one unfortunate aspect about the presentation of cosets in many elementary books is that they don't show any &quot;geometric&quot; examples. Thus the students try to treat the cosets entirely formally, which is much more challenging for them, particularly because most undergraduates have not mastered formal reasoning about sets before they reach group theory. Without intuition or a visual picture to guide them, they find it hard to understand what anything means. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To combat this, I refer to cosets early on as &quot;shifted subgroups&quot;, and work through examples such as the cosets of lines through the origin viewed as subgroups of the group $\mathbb{R}^2$ under addition. These cosets are just lines parallel to the original subgroup, easy to see. This gives the students something to visualize geometrically, which is harder to see with cosets inside finite groups. They can also see the fact that &quot;$aH = bH$ if and only if $a-b \in H$&quot; in this special case first, where I can draw a diagram with vectors $a$, $b$, and $a-b$. &lt;/p&gt;&#xA;" OwnerUserId="5042" LastEditorUserId="5042" LastEditDate="2016-01-17T02:54:24.407" LastActivityDate="2016-01-17T02:54:24.407" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10421" PostTypeId="1" CreationDate="2016-01-17T15:35:49.737" Score="4" ViewCount="973" Body="&lt;p&gt;I am math major junior considering math grad school. I need to decide whether to take complex analysis or cryptology this semester. Complex analysis seems to be a recommended course for people considering graduate school, but cryptology is strongly recommended by my advisor because 1. I will be exposed to many different fields of math in crypto so it is better for my overall mathematical maturity and 2. my current research interest is in probability and I may want to study probability in grad school. Also, I will see lots of applications of probability in cryptology and possibly find a new research idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides these two classes, by the time I graduate I will have taken calc, linear, modeling, diffeq, probability, math stats, sets and proofs, topology, graph theory, abstract algebra, real analysis, stochastic process, abstract algebra II, combinatorics, and numerical analysis/dynamical systems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any suggestions on whether I should take complex analysis or cryptology?&lt;/p&gt;&#xA;" OwnerUserId="6250" OwnerDisplayName="angie123" LastEditorUserId="262" LastEditDate="2016-01-17T20:15:50.770" LastActivityDate="2017-06-29T23:58:45.880" Title="Soft question: take complex analysis or cryptology?" Tags="|course-sequence|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10422" PostTypeId="2" ParentId="10421" CreationDate="2016-01-17T15:49:28.470" Score="4" Body="&lt;p&gt;As you seem to have already studied essentially (expected to be) applied math, I suggest you keep going that way and maybe learn complex analysis aside by yourself for your own culture. I would have recommended complex analysis if you had been into physics or (analytic) number theory, where complex analysis is an essential tool.&lt;/p&gt;&#xA;" OwnerDisplayName="Sylvain JULIEN" LastActivityDate="2016-01-17T15:49:28.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10423" PostTypeId="2" ParentId="10421" CreationDate="2016-01-17T19:20:56.290" Score="10" Body="&lt;p&gt;I will vote for complex analysis.  The main theme of complex analysis is that differentiable complex functions are incredibly rigid, which is completely different from the behaviour of differentiable real functions.  The main results of complex analysis are truly astonishing, when you think about them properly.  In my opinion, complex analysis is the first subject in the usual undergraduate sequence where the main results have this character.  &lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2016-01-17T19:20:56.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10424" PostTypeId="2" ParentId="10421" CreationDate="2016-01-17T20:51:59.013" Score="8" Body="&lt;p&gt;An argument can be made in favor of Cryptography, not because it is more fundamental than Complex Analysis, or more beautiful, but because:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) You have not yet taken Number Theory, and you will learn quite a bit of Number Theory in a course on Cryptography. Here's a typical course:&#xA;&lt;a href=&quot;https://cs.brown.edu/courses/csci1510/syllabus.html&quot; rel=&quot;noreferrer&quot;&gt;&quot;Introduction to Cryptography and Computer Security&quot;&lt;/a&gt;, Brown Univ., 2014.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) It looks like you have learned a lot of math but not seen it applied directly to practical problems in your courses.&#xA;The latest research moves from fundamental&#xA;number theory to practical questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, there is no wrong choice here: Either course would be both interesting and beneficial.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-01-18T13:28:01.260" LastActivityDate="2016-01-18T13:28:01.260" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10425" PostTypeId="1" AcceptedAnswerId="10513" CreationDate="2016-01-17T23:05:44.610" Score="14" ViewCount="609" Body="&lt;p&gt;When teaching calculus, I often present the indefinite integral&#xA;$$ \int \sec x \, dx = \ln | \sec x + \tan x | + C. $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for ways to motivate this anti-derivative rather than just tell the students the answer and show them how to verify it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the best I have come up with:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are trying to find a function whose derivative is $\sec x$, so let's think of all the functions we know whose derivative involves $\sec x$.  Well, $\frac{d}{dx} \sec x = \sec x \tan x$ and $\frac{d}{dx} \tan x = \sec^2 x$.  If we play around a bit, we might stumble upon noticing that $\frac{d}{dx} (\sec x + \tan x) = \sec x (\sec x + \tan x)$ and if we are clever enough, we might realize how that can help us, since if&#xA;$$ \sec x = \frac{\frac{d}{dx} (\sec x + \tan x)}{\sec x + \tan x},$$&#xA;then since we know that $\frac{d}{dx} \ln |f(x)| = \frac{f'(x)}{f(x)}$, we can realize that&#xA;$$ \sec x = \frac{d}{dx} \ln |\sec x + \tan x|.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a simpler way to motivate the result?  Is there a better explanation or reason for taking the linear combination $\sec x + \tan x$ than just trying simple combinations in the hope that something good will happen?&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2016-01-18T03:46:55.427" LastActivityDate="2016-02-05T03:35:02.023" Title="What's the most effective way to introduce/motivate the anti-derivative of $\sec x$?" Tags="|undergraduate-education|calculus|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10426" PostTypeId="2" ParentId="10425" CreationDate="2016-01-18T00:49:09.870" Score="9" Body="&lt;p&gt;In my opinion, a good story to tell about techniques of integration in general goes something like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We know that some functions (like $x^2$) have elementary antiderivatives, and some  (like $e^{x^2}$) do not.  This raises an obvious question:  which functions do have elementary antiderivatives?  We cannot answer the question in general in this course, but something we can do is tackle some large families of functions.  For instance, we can anti differentiate all polynomials.  Can we find elementary antiderivatives for all rational functions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First learn partial fraction decomposition to reduce rational functions into simpler terms.   Those involving linear functions in the denominator can be solved instantly using variable substitution.  Those involving quadratic functions in the denominator require a new technique, trig sub.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conclude that all rational functions can be integrated via this algorithm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This also means that, if we can make a clever change of variables to yield a rational function, we can be assured of our ability to integrate it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For any rational function of sine and cosine, using the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tangent_half-angle_substitution&quot;&gt;tangent half angle substitution&lt;/a&gt; yields a rational function, and so we can integrate all of these functions as well.  One can ominously foreshadow connections with algebraic geometry here, to lend the subject a suitable air of excitement and mystery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The integral of secant is a corollary of this kind of work, and an illustration of the power of the method.  We would have never been able to guess such a bizarre form for the antiderivative, but this grand structure yields the intuition needed to derive the result.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-01-18T00:49:09.870" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10427" PostTypeId="1" CreationDate="2016-01-18T04:59:13.860" Score="4" ViewCount="385" Body="&lt;p&gt;I have given a problem in limits to my students:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\lim_{x \to \infty} \frac{e^x+e^{-x}}{e^x-e^{-x}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the students used direct substitution and identified that it is an indeterminate form  $\frac{\infty}{\infty}$ and tried to apply l'Hôpital's Rule. They observed that it will fail, since the numerator and denominator get swapped, which again leads to the same indeterminate form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I gave the solution as $$\lim_{x \to \infty}\frac{e^x+e^{-x}}{e^x-e^{-x}}=\lim_{x \to \infty}\frac{1+e^{-2x}}{1-e^{-2x}}=1.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But students' query was: is there any way to solve this limit using l'Hôpital's rule. Can any one give some comments on this?&lt;/p&gt;&#xA;" OwnerUserId="5666" LastEditorUserId="123" LastEditDate="2016-01-18T11:35:14.897" LastActivityDate="2016-01-18T14:53:38.097" Title="How can I explain $\lim_{x \to \infty} \frac{e^x+e^{-x}}{e^x-e^{-x}}$ using L'Hôpital's Rule?" Tags="|calculus|limits|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10428" PostTypeId="2" ParentId="10395" CreationDate="2016-01-18T10:26:20.190" Score="2" Body="&lt;p&gt;$\newcommand{\RR}{\mathbb{R}}\renewcommand{\span}{\mathrm{span}}&#xA;\newcommand{\vol}{\mathrm{vol}}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is my own answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's assume that you know that the volume of an $n$-dimensional parallelotope $A$ spanned by $n$ vectors $a_1,\dots,a_n\in\RR^n$ is $\vol_n(A) = \det(a_1,\dots,a_n)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get the $k$-dimensional volume $\vol_k(A)$ of the parallelotope $A$ spanned by $a_1,\dots,a_k\in\RR^n$ (still in $\RR^n$!) do the following: Do $QR$-factorization of $A=[a_1,\dots,a_k]$ to get&#xA;$$&#xA;A = QR&#xA;$$&#xA;with $Q\in\RR^{n\times n}$ and $R\in\RR^{n\times k}$ upper triangular. This is nothing else than turning (and mirroring) the parallelotope around by $Q^T$ such that&#xA;$$&#xA;\begin{array}{cl}&#xA;r_1 = Q^T a_1&amp;amp;\in\span(e_1)\\&#xA;r_2 = Q^Ta_2&amp;amp;\in\span(e_1,e_2)\\&#xA;\vdots &amp;amp; \vdots\\&#xA;r_k=Q^Ta_k&amp;amp;\in \span(e_1,\dots,e_k).&#xA;\end{array}&#xA;$$&#xA;Since $Q^T$ is an isometry, we have that the $k$-dimensional volume of the parallelotope spanned by $a_1,\dots,a_k$ is the same as the one spanned by the columns of $R$. These lie in $\RR^k$ anyway, and hence, this volume is the determinant of the upper $k\times k$ part of $R$, i.e. the determinant of the diagonal entries of $R$. In other words: We have&#xA;$$&#xA;R = \left[\begin{matrix}R_1\\ 0\end{matrix}\right].&#xA;$$&#xA;and the volume is $\vol_k(A) = \det(R_1)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, we see that&#xA;$$&#xA;\det(A^TA) = \det(R^TQ^TQR) = \det(R^T I R) = \det(R_1^T R_1) = \det(R_1)^2 = \vol_k(A)^2.&#xA;$$&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2016-01-18T10:26:20.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10429" PostTypeId="1" CreationDate="2016-01-18T12:12:12.703" Score="5" ViewCount="286" Body="&lt;p&gt;In the context of an undergraduate class focused on proof-writing, a small change in what is written donw can make a big difference to the meaning. Are there any guidelines in use on how to mark such work for students with dyslexia (or similar), who are prone to such errors?&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-01-23T19:37:58.020" Title="Marking dyslexic mathematical writing" Tags="|undergraduate-education|grading|disability|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10430" PostTypeId="2" ParentId="10427" CreationDate="2016-01-18T14:53:38.097" Score="6" Body="&lt;p&gt;As noted, l'Hopital directly fails.  So you have to transform somehow first.   For example,&#xA;$$&#xA;\frac{e^x+e^{-x}}{e^x-e^{-x}} = \frac{e^{2x}+1}{e^{2x}-1}&#xA;$$&#xA;then l'Hopital works.  But of course the same method&#xA;$$&#xA;\frac{e^x+e^{-x}}{e^x-e^{-x}} = \frac{1+e^{-2x}}{1-e^{-2x}} \to \frac{1+0}{1-0}&#xA;$$&#xA;solves it without l'Hopital at all.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2016-01-18T14:53:38.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10431" PostTypeId="2" ParentId="10429" CreationDate="2016-01-18T15:19:19.723" Score="2" Body="&lt;p&gt;This is certainly a question for the Disability Services Office at your college or University. They will have any existing policy for your school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience (at 2 different schools), I recieve a letter from Disability Services detailing the accommodations a student is entitled to. I can't say that I've ever knowingly taught a student with dyslexia, because it is none of my business what the disability is and the accomodation letter doesn't (and shouldn't) say. I can say that I've never recieved a letter asking me to &lt;em&gt;grade&lt;/em&gt; something differently. The most common accomodations I see are extended time on assignments and notetakers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have had a few conversations with Disability services offices before about students, and in those instances the Disability Services people told me specifically &lt;em&gt;not&lt;/em&gt; to go beyond the scope of the official accomodation letters (i.e. don't grade any differently).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student has told you (the instructor) that they have a disability, but they haven't gone to Disability Services, I would direct them to that office as soon as possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This answer is based on experiences at American Universities.)&lt;/p&gt;&#xA;" OwnerUserId="6251" LastActivityDate="2016-01-18T15:19:19.723" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10432" PostTypeId="2" ParentId="10395" CreationDate="2016-01-18T17:00:40.167" Score="2" Body="&lt;p&gt;My proof is basically the same as yours, but perhaps this will be a bit more intuitive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We want to find the volume of a $k$-parallelepiped spanned by $v_1,v_2,...,v_k$ in $\mathbb{R}^k$.  We are armed only with the determinant, which lets us find the (signed) volume of $m$-parallelepipeds in $\mathbb{R}^m$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The crucial observation is that, for any good definition of volumes, we should have that the &quot;$m$ -volume&quot; of $v_1,v_2,...,v_m \in \mathbb{R}^n$ should be the same as the &quot;$m+1$-volume&quot; of $v_1,v_2,...,v_m,w \in \mathbb{R}^n$ if $w$ is perpendicular to all the $v_j$, and is of length $1$. In other words &quot;Volume equals base times height&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the volume of $v_1,v_2,...,v_k \in \mathbb{R}^n$ is the same as the volume of $v_1,v_2,...,v_k,w_1,w_2,...,w_{n-k}$, where each $w_i$ is orthogonal to all the other vectors in the list, and is of norm $1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we could just say that the volume is $Det(v_1,v_2,...,v_k,w_1,w_2,...,w_{n-k})$.  This is a recipe for how to find the volume, but it suffers some flaws as a definition and as a computational tool of $k$-volumes.  As a definition, it is difficult because we would need show that the result is independent of the choice of $w_i$, and because the sign is meaningless (it depends on the $w_k$).  As a computational tool, one actually needs to carry out the laborious process of calculating $w_i$ which work.  It seems we should be able to find a formula which does not involve actually calculating these auxiliary vectors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The insight is to realize that the determinant of the transpose is the same as the determinant of the original matrix.  So letting $M = (v_1 v_2 ... v_k w_1 ... w_{n-k})$ and $A = (v_1 v_2 ... v_k)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\textrm{Vol}^2 = Det(M^\top M)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But when we actually compute the matrix $M^\top M$, we see that it consists of $A^\top A$ and then ones down rest of the diagonal.  So its determinant is just $Det(A^\top A)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus $\textrm{Vol}^2 = Det(A^\top A)$.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-01-18T17:00:40.167" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10433" PostTypeId="1" CreationDate="2016-01-18T18:59:46.677" Score="2" ViewCount="129" Body="&lt;p&gt;I am a junior math major considering math grad school, possibly in probability theory. Would the admission commitee of math grad schools view the absence of complex analysis as a significant disadvantage?&lt;/p&gt;&#xA;" OwnerUserId="6250" LastActivityDate="2016-01-18T18:59:46.677" Title="Is the absence of complex analysis a significant disadvantage in math grad school application?" Tags="|undergraduate-education|probability|applications|" AnswerCount="0" CommentCount="4" ClosedDate="2016-01-19T23:35:21.210" ContentLicense="CC BY-SA 3.0" />
  <row Id="10434" PostTypeId="1" CreationDate="2016-01-19T13:58:19.500" Score="11" ViewCount="11304" Body="&lt;p&gt;There are 4 division symbols that I have learned/taught.&#xA;Below is 18 divided by 3, shown with 4 different symbols.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TG9f8.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TG9f8.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question was sparked by the comments on my answer to the question on &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10348/examples-of-mathematical-slang/10351#10351&quot;&gt;examples of mathematical slang&lt;/a&gt;. I wrote about the second symbol and it  seemed that there were mathematical professionals who hadn't seen it before.  I know that it was useful for teaching long division.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering when teachers introduce the 4 symbols in the US and other countries. I am also wondering the value of teaching each symbol.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: To clarify, I was hoping for anecdotal evidence of when teachers introduce different symbols.  &lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-01-20T14:51:47.837" Title="When and Why are different division symbols taught?" Tags="|secondary-education|primary-education|notation|division|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10435" PostTypeId="2" ParentId="10434" CreationDate="2016-01-19T14:51:44.177" Score="5" Body="&lt;p&gt;1$\;$ When students begin learning arithmetic, they first learn how to do $18+3$, then $18-3$, and $18*3$. Rather than using fraction form to teach division, it's useful to have a symbol that can be used like $+, -,$ and $*$ to teach division. This is where $÷$ comes into play. It's easier for them to see $18÷3$ than $\frac{18}{3}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 $\;$The second method isn't used so much as to represent division as it is to perform division. This is used to teach students how to divide larger numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3/4$\;$These are the same symbol, really, but often the fourth is used because it's difficult to type the third without MathJax. The third is obviously more useful as students progress beyond arithmetic to topics like algebra and calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as the origin of these symbols, most came about simply because different individuals across the planet had to generate notation on their own to work mathematics, and naturally they didn't all choose the same notation.&lt;/p&gt;&#xA;" OwnerUserId="6257" LastEditorUserId="6257" LastEditDate="2016-01-20T03:02:43.373" LastActivityDate="2016-01-20T03:02:43.373" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10436" PostTypeId="1" AcceptedAnswerId="10441" CreationDate="2016-01-19T17:43:51.863" Score="4" ViewCount="636" Body="&lt;p&gt;This is my first year teaching calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a recent quiz, I asked my students to give an argument that $\int^0_1(1/x)dx$ does not exist. I was looking for arguments that appealed to Riemann sum definition of the integral. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;since $1/x$ is unbounded as $x \to \infty$, the area under the curve given by  $y = 1/x$ must also be unbounded. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which makes sense to the intuition, but is actually incorrect. Using the integral evaluation formula for example, we know that $\lim_{x \to 0}(1/\sqrt{x})$ does not exist, but $\int^0_1(1/\sqrt{x})dx = 2.$ While constructing this counterexample, I considered the fact that $\int_a^\infty (1/x^2)dx$ exists, and $(1/\sqrt{x})$ is the inverse of $(1/x^2)$ on $(0,\infty),$ so $\int^0_1(1/\sqrt{x})dx$ must be a finite area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After introducing the counterexample, I then began to show the class that $\int^0_1(1/x)dx$ does not exist using the Riemann sum definition. So, we construct a Riemann sum for the function on $[0,1]$, $$S = \sum_{k=1}^{n}(1/c_k)(1/n) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the left endpoint of the interval of integration is $0$, the first term of any left-handed Riemann sum is unbounded, and since the rest are positive since $1/x$ is positive on $(0,1]$, the value of the left-handed sums must be unbounded for every value of $n$, and so $\lim_{n \to \infty} S$ does not exist. Since the limit of any Riemann sum of a function on an interval converges to the value of the integral of the function over the interval whenever the integral exists, either all Riemann sums of $1/x$ on $[0,1]$ must have the same limit as $n \to \infty$, namely they are all unbounded, or they converge to different values and the integral does not exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However on closer inspection, this argument doesn't seem much better than the one the students gave above, since the first term of the left hand Reimann sums of $1/\sqrt{x}$ on $[0,1]$ are also unbounded. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Is there an error in my argument above that $\int^0_1(1/x)dx$ does not exist?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Appealing to the Riemann sum definition of the integral, what is a good way to demonstrate that $\int^0_1(1/x)dx$ does not exist, but $\int^0_1(1/\sqrt{x})dx$ does? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My next inclination is to consider limits of Riemann sums for these functions over $[a,1]$ as the left endpoint approaches $0$, but I want to keep the discussion as intuitive as possible to support the definition of the integral and to avoid moving forward into a discussion of improper integrals just yet. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-01-19T18:34:00.147" LastActivityDate="2016-01-29T02:46:29.637" Title="Demonstrating that integrals of some unbounded functions exist, and others do not" Tags="|calculus|integration|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10437" PostTypeId="1" CreationDate="2016-01-19T19:09:17.740" Score="4" ViewCount="230" Body="&lt;p&gt;Is there a meeting place where all fraction education stakeholders (students, teachers, parents, researchers, fraction software developers) meet and collaborate? If not, do you have an advice on setting up such a place? (And, if you are interested in starting such an endeavor, please, indicate so.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Details: &#xA;I would like for such a site to capture general principles of and scientific recommendations for teaching fractions off- and online (constructivist framework), as well as best practices; to draw from the past, depict the present conditions, and form the future of fraction education; to serve for people of different backgrounds (cultures, ages, levels incl. remedial and special ed); to arch over platforms for different forms of collaboration; possibly some Intelligent tutoring systems; and, finally, to provide materials for learning and teaching, including working with error, lesson plans, and assessments, and context problems open books, which could be used by all involved parties. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From what I researched to my thesis, there is no such a collaborative place and, I believe, it would be nice if there was one. Examples somewhat close in some feature of the idea in quasi-random order: &lt;a href=&quot;https://en.wikiversity.org/wiki/Primary_mathematics/Fractions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikiversity&lt;/a&gt; and &lt;a href=&quot;https://en.wikibooks.org/wiki/Primary_Mathematics/Fractions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikibooks&lt;/a&gt; on Primary Mathematics for open editing, &lt;a href=&quot;https://www.khanacademy.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Khan Academy&lt;/a&gt; for tracking student progress and coaching, &lt;a href=&quot;http://mathagogy.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathagogy&lt;/a&gt; and &lt;a href=&quot;https://www.mooc-ed.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friday Institute MOOCs&lt;/a&gt; for teacher collaboration, &lt;a href=&quot;https://www.edmodo.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edmono&lt;/a&gt; for social network for teachers and students and for teachers group, &lt;a href=&quot;http://www.artofproblemsolving.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Art of Problem Solving&lt;/a&gt; for the student collaboration, &lt;a href=&quot;https://www.youtube.com/user/1veritasium&quot; rel=&quot;nofollow noreferrer&quot;&gt;Veritasium&lt;/a&gt; for working with misconceptions, &lt;a href=&quot;http://polymathprojects.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Polymath&lt;/a&gt; for collaborative exploration, &lt;a href=&quot;https://projecteuler.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Project Euler&lt;/a&gt; for automatically graded competitive problem solving, &lt;a href=&quot;https://matheducators.stackexchange.com/&quot;&gt;stack overflow&lt;/a&gt; for answering questions by community of volunteers, &lt;a href=&quot;https://scratch.mit.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scratch website&lt;/a&gt; for collaborative project based learning, South African &lt;a href=&quot;http://researchspace.csir.co.za/dspace/bitstream/10204/5045/1/Butgereit1_2011.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dr. Math&lt;/a&gt; for free chat with mentors, &lt;a href=&quot;http://www.oise.utoronto.ca/aphd/report_writer/Recommendations_Glossary/MATHEMATICS/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Debuggy&lt;/a&gt; and &lt;a href=&quot;https://www.researchgate.net/profile/Paula_Durlach/publication/263202753_Reinforcing_Math_Knowledge_by_Immersing_Students_in_a_Simulated_Learning-By-Teaching_Experience/links/546774750cf2397f782be890.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bella&lt;/a&gt; for working with error and learning by teaching, &lt;a href=&quot;http://www.public.asu.edu/~mtchi/papers/Craig_Chi_Vanlehn2009.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Craig, Chi, and VanLehn's paper&lt;/a&gt; for collaborative observing a dialogue, &lt;a href=&quot;https://www.researchgate.net/publication/247502551_Two_Wrongs_May_Make_a_Right_If_They_Argue_Together&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schwarz's paper&lt;/a&gt; for justifying that two people with wrong opinions can arrive at a right solution, &lt;a href=&quot;https://scribblar.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scribblar&lt;/a&gt; for collaborative blackboard, &lt;a href=&quot;https://education.microsoft.com/skypeintheclassroom&quot; rel=&quot;nofollow noreferrer&quot;&gt;Skype in the classroom&lt;/a&gt; for safe classrooms' cooperation, &lt;a href=&quot;https://www.cs.cmu.edu/~bmclaren/pubs/AdamsEtAl-UsingErrExToImproveMathLearningWithWebTutoringSys-CHB2014.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;ErrEx&lt;/a&gt; for erroneous examples, &lt;a href=&quot;https://www.radixendeavor.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Radix&lt;/a&gt; for immersive setting, &lt;a href=&quot;https://www.theschoolinthecloud.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;SOLE&lt;/a&gt; for inquiry learning, &lt;a href=&quot;http://www.wolframalpha.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wolfram&lt;/a&gt; for seeing the demonstrations and automatic answers, &lt;a href=&quot;https://github.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;github&lt;/a&gt; for cooperation on development, &lt;a href=&quot;http://www.slatescience.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Slate&lt;/a&gt; math and &lt;a href=&quot;http://web.stmath.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math ST&lt;/a&gt; for visual game learning, &lt;a href=&quot;http://fractionhub.weebly.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fraction Hub&lt;/a&gt; for the teacher side of the idea, &lt;a href=&quot;http://trojsten.sk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trojsten&lt;/a&gt; for Slovak site on correspondence competitions (many times with proofs; corrected by volunteer university students), &lt;a href=&quot;http://www.worldscientific.com/doi/suppl/10.1142/6339/suppl_file/6339_chap01.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Japanese lesson study&lt;/a&gt; for improving teacher education, &lt;a href=&quot;https://www.teachertube.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teacher's tube&lt;/a&gt; and &lt;a href=&quot;http://ed.ted.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;TED-Ed&lt;/a&gt; for educational videos, &lt;a href=&quot;https://www.desmos.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dan Mayer&lt;/a&gt; and &lt;a href=&quot;http://www.slideshare.net/jdewaard/what-is-realistic-mathematics-education-national-mathematics-conference-swakopmund&quot; rel=&quot;nofollow noreferrer&quot;&gt;Realistic Math Education&lt;/a&gt; for contextual problems, &lt;a href=&quot;https://www.teachingchannel.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching Channel&lt;/a&gt; for examples of teaching, &lt;a href=&quot;http://timss.bc.edu/timss2011/international-results-mathematics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;TIMSS&lt;/a&gt; for international comparisons, &lt;a href=&quot;http://www.cehd.umn.edu/ci/rationalnumberproject/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rational Number Project&lt;/a&gt; for in depth exploration in fractions (along with the learning paths, research in &lt;a href=&quot;https://books.google.sk/books?id=7v2gRGuxw4sC&amp;amp;pg=PA347&amp;amp;dq=kieren+fractions&amp;amp;hl=sk&amp;amp;sa=X&amp;amp;ved=0ahUKEwiKncf0zbbKAhUHpg4KHSg1B38Q6AEITzAH#v=onepage&amp;amp;q=kieren%20fractions&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;How People Learn&lt;/a&gt;, &lt;a href=&quot;http://math.coe.uga.edu/Olive/Preface_CFK.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Steffe and Olive&lt;/a&gt;, &lt;a href=&quot;https://books.google.sk/books?hl=sk&amp;amp;lr=&amp;amp;id=xvHESHH7m5IC&amp;amp;oi=fnd&amp;amp;pg=PA1&amp;amp;dq=cognitive%20science%20education%20mathematics&amp;amp;ots=oBQVk72PXl&amp;amp;sig=fmxzEfdqJH6JoUKgDq6WvU1MYGE&amp;amp;redir_esc=y#v=onepage&amp;amp;q=cognitive%20science%20education%20mathematics&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schoenfeld&lt;/a&gt;, &lt;a href=&quot;http://www.psy.cmu.edu/~siegler/NCTM.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Siegler&lt;/a&gt; and &lt;a href=&quot;http://files.eric.ed.gov/fulltext/ED496933.pdf#page=297&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hejny&lt;/a&gt; and many others), Socratic Questioning (e.g. &lt;a href=&quot;http://www.garlikov.com/Soc_Meth.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Garlikov&lt;/a&gt;), &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Montessori_education&quot; rel=&quot;nofollow noreferrer&quot;&gt;Montessori&lt;/a&gt; for aids and creation within boundaries, Philosophy and History of Fraction and Fraction Education for creating a context, blogs for metacognition, Facebook for messaging and collaboration, twitter for news, google hangout for collaboration &quot;face to face,&quot; &lt;a href=&quot;https://unhangout.media.mit.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;unhangout&lt;/a&gt; for group problem solving, wiki for summarizing the gist of the knowledge and multilingual versions, &lt;a href=&quot;https://www.teacherspayteachers.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;teachers pay teachers&lt;/a&gt; for shared materials,  &lt;a href=&quot;http://www.innovationsforlearning.org/tutormate&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tutormate&lt;/a&gt; for virtual volunteer tutoring (in reading), &lt;a href=&quot;https://www.facebook.com/groups/freemathtutoring/?fref=ts&quot; rel=&quot;nofollow noreferrer&quot;&gt;free math tutoring groups on fb&lt;/a&gt; and math groups and different sites on tutoring e.g. &lt;a href=&quot;https://helphub.me/&quot; rel=&quot;nofollow noreferrer&quot;&gt;helphub&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware to create such a central for learning and teaching would be very demanding, but I think it is important and that it can be slowly done if enough people took part in development. Any ideas on what else there is important to think about or how to start?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;&#xA;&lt;p&gt;stanya &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. This is my first post and it does not allow me to post links, so I will post them below. Thank you for understanding, and if I made any mistakes, I will be glad if you point them out.&lt;/p&gt;&#xA;" OwnerUserId="6258" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2018-08-24T00:15:30.757" Title="Is there a meeting place for fraction education stakeholders?" Tags="|technology-in-education|students-mistakes|fractions|teachers|struggling-students|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10438" PostTypeId="2" ParentId="10318" CreationDate="2016-01-19T19:43:09.460" Score="2" Body="&lt;p&gt;I have experience from my high school that the most helpful preparation for majoring in math in college was not at all learning the college math beforehand. It was the reasoning skills I learned while solving hard high school problems (somewhat under IMO level, but hours and hours of time spent on them). Some problems that you could give your students from are from Art of Problem Solving Website. &lt;/p&gt;&#xA;" OwnerUserId="6258" LastActivityDate="2016-01-19T19:43:09.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10439" PostTypeId="2" ParentId="10109" CreationDate="2016-01-19T20:01:05.287" Score="2" Body="&lt;p&gt;I was also searching for a place to teach virtually for quite a long time. I have finally found Cheery Education Center in Kibera, Nairobi, Kenya, that appreciated a little help in education (and still does). I found it through Skype in the classroom, there I searched for teachers who mystery Skype and looked at their profile.&lt;/p&gt;&#xA;" OwnerUserId="6258" LastActivityDate="2016-01-19T20:01:05.287" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10441" PostTypeId="2" ParentId="10436" CreationDate="2016-01-20T06:16:14.753" Score="12" Body="&lt;p&gt;When we write $\int_0^1 \frac{1}{\sqrt{x}} d x = 2$, we do not mean that the Riemann integral of $\frac{1}{\sqrt{x}}$ on $[0,1]$ exists and is equal to $2$, because, as you note, this is false.  One can construct sequences of riemann sums for this function which converge to any number $&amp;gt;2$ than you wish, or sequences which diverge to infinity, or sequences which alternate between $2$ and $3$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is unfortunate, but we actually use the same symbol for a different mathematical object.  In this case the integral sign is an &quot;improper integral&quot; which has a different definition.  The definition of this improper integral is &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\lim_{a \to 0} \int_a^1 \frac{1}{\sqrt{x}} d x&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which you can prove converges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\int_0^1 \frac{1}{x} dx$ is also an improper integral, but this one diverges because the corresponding limit diverges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So your argument via riemann sums is indeed flawed:  these integrals are not riemann integrals, so the riemann sum argument you gave is irrelevant.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-01-20T06:16:14.753" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10442" PostTypeId="1" AcceptedAnswerId="10447" CreationDate="2016-01-20T17:22:45.440" Score="8" ViewCount="1930" Body="&lt;p&gt;I want to write an article which explains the &lt;a href=&quot;https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior&quot; rel=&quot;noreferrer&quot;&gt;limit superior&lt;/a&gt;. I also want to present the intuition behind this concept. Currently I would describe the limit superior as the &lt;em&gt;&quot;least upper bound of a sequence at infinity&quot;&lt;/em&gt; which therefore can be used to estimate the sequence &lt;em&gt;&quot;at infinity&quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;To explain my idea:&lt;/strong&gt; Take a sequence $(a_n)$. The supremum $\sup_{n\in\mathbb N} a_n$ is the least upper bound of all sequence elements and thus can be used to estimate the sequence elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly we can define an &lt;em&gt;&quot;upper bound of $(a_n)$ at infinity&quot;&lt;/em&gt; as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A real number $c$ is an upper bound at infinity of the sequence $(a_n)$, iff for all $\epsilon &amp;gt; 0$ almost all sequence elements $a_n$ are less then or equal $c+\epsilon$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Given this definition, $\limsup_{n\to\infty} a_n$ is the smallest of all upper bounds at infinity (analogously as the supremum is the least upper bound of a sequence).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The limit superior can also be used to estimate the sequence at infinity: Having $\limsup_{n\to\infty} a_n = c \in \mathbb R$ we know that almost all sequence elements are less than or nearly equal $c$ and that $c$ is the best of all those upper estimates. Here we have to use &quot;nearly equal&quot; instead of &quot;equal&quot; because $c$ must not be an upper bound for $(a_n)$. Take $a_n=1+\tfrac 1n$ as an counterexample.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question:&lt;/strong&gt; How would you describe the intuition behind the limit superior? What do you think about the above description of the limit superior as the &lt;em&gt;&quot;least upper bound at infinity&quot;&lt;/em&gt;? Are there characteristics of the limit superior I have missed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that the presented intuition still needs to be elaborated. But that's also the reason I am asking the above questions here... ;-)&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="5097" LastEditDate="2016-01-20T17:42:55.557" LastActivityDate="2022-10-06T14:59:58.827" Title="What is the intuition behind the limit superior?" Tags="|concept-motivation|mathematical-analysis|definitions|sequences|limits|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10443" PostTypeId="2" ParentId="10442" CreationDate="2016-01-21T00:11:14.077" Score="2" Body="&lt;p&gt;I would probably avoid the phrase &quot;at infinity&quot;, since it's kind of vague and I'm not sure if it'd help someone who doesn't already understand the concept. Instead, I would talk more explicitly about numbers that are an upper bound &lt;em&gt;outside a bounded segment&lt;/em&gt; — in other words, an upper bound if we throw out some finite initial piece. This works for both sequences and functions, as long as we take &quot;finite&quot; in the sense of &quot;finite length&quot; for functions. (&quot;Bounded&quot; makes sense for both as is.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word &quot;least&quot; is also a bit problematic, since the limsup itself might &lt;em&gt;not&lt;/em&gt; be an &quot;upper bound at infinity&quot; itself — it's just the greatest lower bound of the set of &quot;upper bounds at infinity&quot;. This is genuinely different than for the least upper bound: by completeness of the reals, if a set has an upper bound, then the set of its upper bounds really does contain a least element. (That is, the least upper bound &lt;em&gt;is&lt;/em&gt; an upper bound itself.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This latter issue seems a bit harder to intuitively describe for the limsup. If your audience is already comfortable with the notion of supremum and infimum, you can just say it's the infimum of all the numbers that are upper bounds outside a bounded interval. If not, I'm not sure what's the best way to explain it... however, I suspect someone who has trouble understanding supremum and infimum will inevitably have a hard time with limsup and liminf no matter how it's presented, because it's a strictly harder concept.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-01-21T00:11:14.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10444" PostTypeId="1" AcceptedAnswerId="10454" CreationDate="2016-01-21T00:27:20.430" Score="14" ViewCount="356" Body="&lt;p&gt;When I was a student in college (very recently), my college had implemented a program called WebWork to give immediate feedback on homework problems. Other systems like MyMathLab exist as well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Distinguishing features of such a system tend to include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Online&lt;/li&gt;&#xA;&lt;li&gt;Provides immediate feedback on whether or not an answer is right or wrong&lt;/li&gt;&#xA;&lt;li&gt;Is able to create a similar problem with different numbers for different students (preventing copying answers)&lt;/li&gt;&#xA;&lt;li&gt;Can provide multiple attempts to try the same problem or variants of the same problem&lt;/li&gt;&#xA;&lt;li&gt;Can create clear cut deadlines for assignments&lt;/li&gt;&#xA;&lt;li&gt;Does not generally give a lot of &quot;process&quot; feedback so much as &quot;answer&quot; feedback. &lt;/li&gt;&#xA;&lt;li&gt;Can require a number, drop down, multiple choice answer (large variety)&lt;/li&gt;&#xA;&lt;li&gt;Hopefully customizable (WebWork is, don't know about MyMathLab). (e.g. in Webwork you can use flash applets and have students draw a graph on the screen)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am contemplating trying to implement WebWork at my high school once we go one-to-one with devices. However, I want to foresee any possible pitfalls AND have research to back up my proposal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are you currently aware of any educational research supporting systems such as this in the math learning environment? Do you have experience with using these systems as an educator? If so, how well do you see a system such as this working in a high school environment, where it is also difficult to provide timely feedback to students? What pitfalls have you found with this approach, and what benefits? &lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2016-01-30T19:11:47.653" Title="Research/Anecdotes supporting WebWork, MyMathLab and other computerized, immediate feedback HW" Tags="|secondary-education|technology-in-education|homework|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10445" PostTypeId="1" CreationDate="2016-01-21T00:32:47.507" Score="9" ViewCount="600" Body="&lt;p&gt;In looking at the definition of vertical tangent lines in some popular calculus texts, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I noticed that there are a few different definitions for this term, including the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A function $f$ has a vertical tangent line at $a$ if &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$\;f$ is continuous at $a$ and $\displaystyle\lim_{x\to a}\;\lvert f^{\prime}(x)\rvert=\infty$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\;f$ is continuous at $a$ and $\displaystyle\lim_{x\to a} f^{\prime}(x)=\infty$ or $\displaystyle\lim_{x\to a} f^{\prime}(x)=-\infty$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\;\displaystyle\lim_{h\to0}\frac{f(a+h)-f(a)}{h}=\pm\infty$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I would like to ask if there is a standard definition of this term, and whether or not the definition should include continuity at $a$ and should not include the situation where the graph has a vertical cusp at $a$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Here are some examples where these definitions lead to different conclusions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) $\;f(x)=x^{2/3}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) $\;f(x)=\begin{cases}1&amp;amp;\mbox{, if }x&amp;gt;0\\0&amp;amp;\mbox{, if }x=0\\-1&amp;amp;\mbox{, if }x&amp;lt;0\end{cases}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This &lt;a href=&quot;https://math.stackexchange.com/q/1618929&quot;&gt;question&lt;/a&gt; has also been posted on Math Stack Exchange.)&lt;/p&gt;&#xA;" OwnerUserId="6265" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-02-14T20:32:05.660" Title="Defining vertical tangent lines" Tags="|calculus|definitions|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10446" PostTypeId="2" ParentId="10445" CreationDate="2016-01-21T00:55:20.013" Score="4" Body="&lt;p&gt;Personally, I like option $2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you point out, $1$ is satisfied by functions with cusps as in your example $a$, which does not match up with my intuition about tangent lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$3$ is the wrongest, because of your example $b$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not think the definition is standardized however.  As is often the case in mathematics, it is probably best just to clarify what you mean if one of these situations arise.  Definitions change from person to person.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-01-21T00:55:20.013" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10447" PostTypeId="2" ParentId="10442" CreationDate="2016-01-21T07:18:40.980" Score="11" Body="&lt;p&gt;I have two intuitions to offer:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A sequence $(a_n)$ may have cluster points (these are points such that every neighborhood contains infinitely many elements of the sequence, or, more precisely, for every neighborhood and every index $N$, there is an element $a_n$ with an index $n&amp;gt;N$ which is in this neighborhood). The $\limsup$ is the &lt;strong&gt;largest of these cluster points&lt;/strong&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Consider the tail of the sequence $(a_n)_{n\geq N}$. This has a supremum $\sup_{n\geq N} a_n$. This supremum gets smaller and smaller as $N$ grows since the set over which we take the supremum gets smaller. The $\limsup$ is the infimum of these suprema (equivalently, it's the &lt;strong&gt;limit of the suprema of the tails&lt;/strong&gt;). You may visualize this by a kind of decaying straicase which lies on top of the sequence and the step number $N$ of the stair is as low as possible while still above all elements with index $\geq N$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;While both intuitions are really valid definitions of the $\limsup$ I find both of them intuitive (at least as intuitive as I find any definition of the limit).&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2016-01-21T07:18:40.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10448" PostTypeId="2" ParentId="10442" CreationDate="2016-01-21T13:49:55.370" Score="3" Body="&lt;p&gt;Every sequence $\langle u_n: n\in\mathbb{N}\rangle$ has a natural extension $\langle u_n: n\in{}^\ast\mathbb{N}\rangle$ where ${}^\ast\mathbb{N}$ are the hypernatural numbers (positive hyperintegers). The limsup of the sequence (when it is finite) is then exactly the sup of the shadows of $u_H$ for infinite indices $H$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would suggest explaning this as follows: look at terms in the sequence for indices that are so huge that you can't express them even using a computer the size of the universe, in the total time allotted to our civilisation, so that for all practical purposes they are infinite.  Evaluate the sequence at those indices, and take the maximal value among the ones you get.  That's the limsup. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can point out that a way of writing this precisely depends on which background formalisation of analysis you use.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2016-01-21T17:32:59.677" LastActivityDate="2016-01-21T17:32:59.677" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10449" PostTypeId="2" ParentId="10444" CreationDate="2016-01-21T16:27:30.047" Score="2" Body="&lt;p&gt;I only have anecdotal experience, but perhaps it will be useful.  I've used MyMathLab a number of times in introductory college calculus classes, and generally been happy with the results, but there are some definite caveats.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thing that I find works really well, and that keeps me coming back to it, is the ability to give students quick feedback and then let them try again, and repeat until they get it right.  MyMathLab also (for many problems) will help them by breaking the problem into steps, walking them through the steps, and then generating a similar problem for them to try again with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, some issues to watch out for:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;These programs work best for very algorithmic problems.  Even when the software nominally contains harder problems, the benefits go away (it can't generate multiple versions as well or at all, it can't walk them through it, the feedback is less meaningful).  The standard solution is to have &lt;em&gt;some&lt;/em&gt; problems be done online and harder ones done by hand.  This always sounded great to me: it's easier for the students to drill the basic stuff, &lt;em&gt;and&lt;/em&gt; it emphasizes that conceptual problems should be approached differently because they're literally being done in a different format.  I've never quite made that work in practice: my students often seem to be surprised that the &quot;written homework&quot; is harder than the &quot;online homework&quot;, and if anything it amplifies their frustration that the written homework is actually hard.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;These programs always seem to be a bit glitchy (my students probably uncover 1-2 actual errors in MML problems every semester, which suggests that there must be a lot there), and students have &lt;em&gt;very low&lt;/em&gt; frustration tolerance for computer errors grading their problems.  There are some things you can do to mitigate this: at least with MML, it's pretty easy to go in and check what a student wrote and change their grade, so I always tell them repeatedly that I'll fix any computer errors for them.  Also, you can turn up error tolerance (nothing seems to piss students off more than being told that the answer is 12.213, not 12.21, when the problem didn't even say how many significant figures to include).  Finally, I've found that MML isn't very responsive to students, but is really, really responsive to faculty.  (They know who their real clients are...)  So putting in some time being an intermediary to the company smooths that over (at the cost of time).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you let students try problems repeatedly, the problems generated have patterns, and they'll sometimes hack the patterns rather than actually solving the problem.  I got better results by splitting the problems into a homework and something I called a &quot;quiz&quot;, where the quiz allowed only a few retries and most of the help was turned off (but was the same basic problems they'd just practiced on the homework).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-01-21T16:27:30.047" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10450" PostTypeId="2" ParentId="10395" CreationDate="2016-01-21T20:27:40.470" Score="3" Body="&lt;p&gt;Here is an argument that has convinced (or at least silenced :-)) students in office hours. Making it into a rigorous proof would require more time talking about what volume MEANS than I do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, suppose I have $k$ vectors $\vec{v}_1$, $\vec{v}_2$, ..., $\vec{v}_k$ in $\mathbb{R}^k$. Let $A$ be the $k \times k$ matrix with columns $\vec{v}_1$, ..., $\vec{v}_k$. Then the volume of the parallelotope is&#xA;$$|\det(\vec{v}_1, \ldots, \vec{v}_k)| = |\det A| = \sqrt{\det(A^T A)} = \sqrt{\det(\vec{v}_i \cdot \vec{v}_j)}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The nifty thing is that each entry $\vec{v}_i \cdot \vec{v}_j$ of the $k \times k$ matrix is invariant under rotation of coordinates. So we have expressed the rotationally invariant volume in terms of much simpler rotationally invariant quantities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the vector $\vec{v}_1$, ..., $\vec{v}_k$ are now in $\mathbb{R}^n$ for some $n&amp;gt;k$, it is still true that the volume is $\sqrt{\det(\vec{v}_i \cdot \vec{v}_j)}$, since we can just choose orthogonal coordinates in the $k$-plane spanned by the $\vec{v}$'s and ignore the larger ambient space. (On one occasion, I drew two vectors on a notepad, shaded the parallogram and tilted the notepad at various angles, to point out that the area and the dot products don't care how I hold the notepad in space.) And we can still write this as $\sqrt{\det(A^T A)}$.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2016-01-21T20:33:37.620" LastActivityDate="2016-01-21T20:33:37.620" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10451" PostTypeId="2" ParentId="9695" CreationDate="2016-01-21T22:48:23.233" Score="1" Body="&lt;p&gt;Western Governors University. One of my coworkers has a masters from them, and our school is also currently hosting a student teacher who is getting his undergrad through them.&lt;/p&gt;&#xA;" OwnerUserId="6269" LastActivityDate="2016-01-21T22:48:23.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10453" PostTypeId="2" ParentId="10442" CreationDate="2016-01-22T02:15:17.580" Score="5" Body="&lt;p&gt;The idea behind the limsup that you write is not simple and will not convey a concise intuition. The shortest description of the limsup of a sequence needs two steps:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) The audience has to know what a limit point (also called accumulation point or cluster point) of a sequence is, and that a sequence can have many limit points, by examples. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Granting the technical term introduced in (1), the limsup of a sequence is its largest limit point. (Similarly, the liminf is the smallest limit point.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's it! This is the same idea as Dirk describes in his answer. He also offers another intuition, but I do not find anything other than (2) to be so immediately intuitive, cutting right to the heart of what the limsup of a sequence means in such a concise way. The other possibilities offered here illustrate this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had known real analysis for a couple of years in college years before I read (2), and until then I never felt fully comfortable with limsups and liminfs, or solving problems involving them. After reading (2) I immediately understood this notion and never found it hard to use again. &#xA;Ironically, the place where I found (2) was in a book on $p$-&lt;em&gt;adic analysis&lt;/em&gt;! (See the first page of Chapter IV in Koblitz's Springer GTM on $p$-adic numbers, where he explains Hadamard's limsup formula for the radius of convergence $r$ of a power series in terms of $1/r$ being the greatest accumulation point of a sequence.)&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2016-01-22T02:15:17.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10454" PostTypeId="2" ParentId="10444" CreationDate="2016-01-22T04:21:24.353" Score="5" Body="&lt;p&gt;Some of these links are old but I don't think WebWork has changed that much:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://math.rutgers.edu/~weibel/ww.html&quot;&gt;research&lt;/a&gt; at Rutgers about their calculus courses&lt;/li&gt;&#xA;&lt;li&gt;this &lt;a href=&quot;https://www.maa.org/external_archive/columns/launchings/launchings_04_09.html&quot;&gt;post&lt;/a&gt; has useful summaries and description of various aspects of WebWork&lt;/li&gt;&#xA;&lt;li&gt;the bibliography at the bottom of &lt;a href=&quot;http://www.csis.pace.edu/~scharff/webwork/WWtutorial.htm&quot;&gt;this&lt;/a&gt; page has a number of references for its effectiveness.  &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2016-01-22T04:21:24.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10455" PostTypeId="2" ParentId="4326" CreationDate="2016-01-22T05:58:58.123" Score="2" Body="&lt;p&gt;Trigonometric substitution is related to how Newton first showed the derivative of $\sin x$ is $\cos x$. See &lt;a href=&quot;https://hsm.stackexchange.com/questions/3174/how-were-derivatives-of-trigonometric-functions-first-discovered?rq=1&quot;&gt;https://hsm.stackexchange.com/questions/3174/how-were-derivatives-of-trigonometric-functions-first-discovered?rq=1&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.147" LastActivityDate="2016-01-22T05:58:58.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10456" PostTypeId="1" CreationDate="2016-01-22T14:11:45.580" Score="3" ViewCount="78" Body="&lt;p&gt;I'm planning some interdisciplinary options for future students at a small independent school to be implemented in the next few years. The courses would be offered as independent studies and students would consult with faculty in both departments. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately I've come up short in finding calculus and statistics texts that contain enough of the cross-disciplinary material but are appropriate for advanced high-schoolers. I came across a text called &quot;Calculus for Biology and Medicine&quot;, which is an example (at least in theory- I haven't gotten my hands on a copy) of the type of text I'm looking for. Other disciplines I would like to be available to students are economics &amp;amp; calculus, engineering &amp;amp; calculus, social sciences &amp;amp; statistics, ecological science &amp;amp; statistics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ideal text would not simply teach calculus or statistics with exercises in the cross discipline, but utilize principles in both disciplines to teach topics in both. Introductory undergraduate texts in these disciplines are alright as long as they contain enough material approachable to advanced high school students to fill a year-long course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) What sorts of programs like this have been successfully implemented in the past?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) What books should serve as texts four courses like these? &lt;/p&gt;&#xA;" OwnerUserId="5690" LastActivityDate="2016-01-22T14:11:45.580" Title="Interdisciplinary Calculus and Statistics Texts for Advanced Secondary Students" Tags="|secondary-education|textbooks|interdisciplinary|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10457" PostTypeId="5" CreationDate="2016-01-22T14:14:32.427" Score="0" Body="&lt;p&gt;For questions about utilizing core principles from other disciplines such as physical sciences, laboratory sciences, economics, computer science, and social sciences to teach and demonstrate mathematical topics. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-01-22T15:09:16.137" LastActivityDate="2016-01-22T15:09:16.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10458" PostTypeId="4" CreationDate="2016-01-22T14:14:32.427" Score="0" Body="For questions about utilizing core principles from other disciplines such as physical sciences, laboratory sciences, economics, computer science, and social sciences to teach and demonstrate mathematical topics. " OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-01-22T15:09:09.153" LastActivityDate="2016-01-22T15:09:09.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10459" PostTypeId="1" CreationDate="2016-01-22T16:31:03.130" Score="5" ViewCount="56" Body="&lt;p&gt;I was wondering if anyone knew of a source for pdfs of graph paper with both the triangular lattice and its dual hexgonal lattice superimposed over each other?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like this to hand out to my students and have them draw subgraphs (and their duals), but am having trouble finding or generating this.&lt;/p&gt;&#xA;" OwnerUserId="6273" LastActivityDate="2016-01-22T16:31:03.130" Title="Resource request: superposition of triangular lattice and its hexagonal dual lattice" Tags="|reference-request|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10461" PostTypeId="2" ParentId="10429" CreationDate="2016-01-23T03:04:01.680" Score="3" Body="&lt;p&gt;I work in disability services as an advisor. Your campus disability services office can certainly help you in terms of talking through how to assess and assign a grade. But, as user6251 said, this needs to be in the letter of accommodation, so the student should register with the disability services office. If the office on your campus has never worked with a case like this, there are ample resources available and they can find out how other universities have dealt with the same issue. AHEAD (Association on Higher Education and Disability) may have some resources on their website as well. Recommendations for specific accommodations should come from the student's clinician (usually the psychologist who diagnosed the dyscalculia).&lt;/p&gt;&#xA;" OwnerUserId="5679" LastActivityDate="2016-01-23T03:04:01.680" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10462" PostTypeId="2" ParentId="10429" CreationDate="2016-01-23T16:09:55.827" Score="3" Body="&lt;p&gt;I doubt that any such marking guidelines specialized for dyslexic students at the college level are likely to exist. (I'm not on expert on learning disability issues, but due to the dearth of answers here, I'm going to take a stab at it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the secondary school level in the U.S., it is required by law that students have an Individualized Education Program (IEP), and this can in fact result in different grading criteria for different students based on their ability level. See the ASCD paper, &lt;a href=&quot;http://www.ascd.org/publications/educational-leadership/oct03/vol61/num02/Grading-Students-with-Disabilities.aspx&quot; rel=&quot;nofollow&quot;&gt;&quot;Grading Students with Disabilities&quot;&lt;/a&gt;, for suggestions. Warning: As an anecdote, this may result in the experience of an acquaintance of mine: her middle-school son has an autistic classmate who is effectively nonfunctional, disruptive on a daily basis, and simultaneously carrying the highest grade in the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the college level in the U.S., IEP's do not exist (to my knowledge; definitely not as a matter of law). The emphasis here is generally on a uniform assessment and reporting rubric, with accommodations for doing the work from the accessibility office. In the last few years, the recommendation has come in the form of Universal Design for Learning (UD), possibly including flexible assignments options for different students. (For example, this is now highly recommended by our school's Accessibility Office.) See a primer on UD from the DO-IT foundation at the University of Washington, &lt;a href=&quot;http://www.washington.edu/doit/academic-accommodations-students-learning-disabilities&quot; rel=&quot;nofollow&quot;&gt;&quot;Academic Accommodations for Students with Learning Disabilities&quot;&lt;/a&gt;. Warning: When I've studied UD in the past, I've found the recommendations for math classes to be effectively incoherent, and clearly disconnected from anything a math educator would say (e.g., let students talk to each other during tests; have another instructor come in and write formulas on the board during testing; eliminate word problems because that doesn't really count as math). Also, part of the attraction of UD for accessibility specialists is that it reduces their workload by having instructors do it instead. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary: For the U.S., individuated grading procedures on the same assignment are sometimes seen at the secondary level and below, but they are generally not a consideration at the college level. However, allowing students to choose from among several possible assignment options may replicate that effect.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-01-23T19:37:58.020" LastActivityDate="2016-01-23T19:37:58.020" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10463" PostTypeId="2" ParentId="10445" CreationDate="2016-01-24T16:33:03.060" Score="5" Body="&lt;p&gt;I suppose if one had a definition of a tangent line, defined for all curves, then one would only need to know which way is vertical.  [Thurston, &lt;a href=&quot;http://doi.org/10.2307/2311405&quot; rel=&quot;nofollow&quot;&gt;On the Definition of a Tangent-Line&lt;/a&gt; (1964)] explores several definitions.  A geometric one is the following, where an arc is defined to be the homeomorphic image of a closed, bounded interval.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The line $L$ through a point $C$ of an arc is a geometrical tangent to the arc at $C$ if, given any positive number, there is a circle with centre $C$ such that the angle between the line $PC$ and the line $L$ is less than the given number for every point $P$ (other than $C$) which is both on the arc and in the circle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This would mean there is a tangent line at a cusp, and, in the case of $y = f(x)$, it is equivalent to def. 1.  Thurston points out, more or less, that in the parametric approach one requires that &quot;the ratio $X'(c) \colon Y'(c)$ exists,&quot; which it won't at a cusp.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The OP asks about tangent line explicitly in terms of a &lt;em&gt;function&lt;/em&gt; and not of &lt;em&gt;curve.&lt;/em&gt;  It is easy to talk about a nonvertical tangent in terms of the explicit linear function $f(c) + f'(c)\,(x-c)$, provided $f'(c)$ is defined.  Thurston calls the corresponding line an &quot;explicit tangent.&quot;  But it seems to hard me to talk about vertical tangents without considering the graph of the function as a curve in the plane, except strictly in terms a definition like one those given in the OP.  Note, too, that the definition of the tangent line as the limit of the secant line leads to def. 1, assuming that the notion of the limiting position of a variable line can be made sufficiently clear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess I have tended to use def. 1 myself, probably because of the textbooks I've used.  It's easier for me to adapt to the textbook than for the students&#xA;to adapt to my idiosyncrasies, especially first-year students.&#xA;But I could live with def. 2. &#xA;I don't believe there is consensus&#xA;about a standard definition.  It's not clear to me how much the parametric situation should influence the single-variable function case.  It's fairly clear to me that the distinction between def. 1 and 2 in first-year calculus is not that important (except in each particular class, terminology and instructions have to be clear).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting aside, not particularly relevant to the OP's question, is that while any explicit tangent line $y = f(c) + f'(c)\,(x-c)$ is equivalent to a tangent line according to the parametric definition, there is a parametrization of a curve that has a tangent line at a certain point $C$ but which is not locally equivalent to a graph $y = f(x)$ with an explicit tangent at $C$.  The parametrization involves one of my favorite functions and I leave the reader to ponder it (or read Thurston's paper):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\cases{X(t) = t^2 \sin t^{-1} + {1\over2}\,t &amp;amp; if $0&amp;lt;|t|\le1$, and $X(0)=0$ \cr&#xA;Y(t) = t\,X(t)\cr}&#xA;$$&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2016-01-30T03:53:45.383" LastActivityDate="2016-01-30T03:53:45.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10464" PostTypeId="2" ParentId="8456" CreationDate="2016-01-24T23:59:50.763" Score="3" Body="&lt;p&gt;For many courses, note taking is concerned with the organization of data for the purposes of recall. Let's call such organized data &quot;information&quot;. In mathematics, the purpose of writing is more, borrowing the language of John Dewey, to marshal resources toward a purpose more specific than recall. Let's call information aimed at a specific purpose &quot;knowledge&quot;. Of course, one person's knowledge is another's information. Knowledge is subjective to the kinds of problems or actions an individual is motivated to solve or do (again taking from Dewey, a purpose forms when someone plans their way around an obstacle to satisfying an initial impulse). You can see this in the many mathematical blog posts and textbook treatments of the same basic proofs in mathematics. Writing about these things allows the author to arrange certain facts for ready use for the problems the author is interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To risk sounding overly male or violent, this perspective of knowledge as &quot;weaponized information&quot; seems to be a good motivation for mathematical writing. You should have a purpose for your note taking…some problems you (or perhaps the community) want to see solved, or ideas that you want to have a use for, and that the information collected marshals facts aiming to the solution to these problems, or the use of these ideas. The necessary selectivity for good note taking comes from having such purposes, and a sure sign that you don't have them is that you feel like you are copying everything down. (I speak from personal experience!) The thing to do under these circumstances is to ask your own questions in response to the text, and to the instructor, to stimulate purpose. A question is such a seed of an impulse that, since you cannot immediately see an answer, yields a purpose as understood above. Just about any question you genuinely ask will help with this. A good many of the concrete suggestions in the above answer of Joonas will help, as well.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2016-01-25T19:11:36.067" LastActivityDate="2016-01-25T19:11:36.067" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10465" PostTypeId="1" CreationDate="2016-01-25T05:18:56.207" Score="13" ViewCount="508" Body="&lt;p&gt;After some research on the recent history of math education in the U.S., from the new math movement to the beginning of the 21st century, I understood that the historic flow of the math education paradigm was something like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;during the cold war, the new math movement brought abstract and formal ideas to early years seeking to improve scientific education to compete with the eastern block.(60s-mid 70s)&lt;/li&gt;&#xA;&lt;li&gt;Hard criticism on the new math movement brought back a new and more radical version of constructivism (mid 70s-80s)&lt;/li&gt;&#xA;&lt;li&gt;Opposition from mathematicians and parents brought back a discussion on the effectiveness of constructivism, in face of a decreasing performance on basic math skills. No consensus exists on whether to trust on constructivism or traditional (pre-new math and pre-old constructivism) methods of math teaching.(90s-early 00's)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm from Latin America and making contact with local math education history experts to confirm a similar trend on my country. Thinking of the cold war as a pivot event for this flow of the math teaching paradigm on the U.S., I ask: was this movement (new math, constructivism, revival of traditional ideas) a trend on the western block of U.S. aligned countries during that period?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here in Brazil we have lots of material on new math from the 60's and the 70's, and from the early 90's to today the federal government supports a more constructivist approach when setting the standards for national education.&lt;/p&gt;&#xA;" OwnerUserId="6279" LastEditorUserId="39" LastEditDate="2016-01-25T17:25:16.977" LastActivityDate="2023-06-04T12:14:28.960" Title="Was math education following a western trend?" Tags="|mathematical-pedagogy|primary-education|history|new-math|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10466" PostTypeId="2" ParentId="1572" CreationDate="2016-01-26T12:04:09.140" Score="2" Body="&lt;p&gt;This was told in an interview I read with a successful businessman (a dane). He told that he learned business from his grandfather. They sold roses on the street, his grandfather told him to offer &quot;a rose for 75 øre (that is, cent of a krone), three for only 2.50 kroner&quot;.&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="5321" LastEditDate="2016-01-26T14:01:51.953" LastActivityDate="2016-01-26T14:01:51.953" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10467" PostTypeId="1" CreationDate="2016-01-26T13:43:52.313" Score="10" ViewCount="292" Body="&lt;p&gt;I am currently designing a proof-based Math course for my University. I already designed and ordered all of the theoretical content in the course and included some ad hoc exercises for practicing each of the particular topics in the course. However, I have also designed a long final assignment that introduces a problem covering roughly all of the topics in the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is on how helpful is it for the student to work on these kind of general assignments. Assuming he already understood each of the topics individually, will it be beneficial spending some weeks analyzing an application involving almost all of the topics seen? Is there any research on the educational benefits of these type of exercises?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be grateful if someone could cite a research paper that talks about the benefits of showing these kind of general applications involving many topics in undergraduate courses.&lt;/p&gt;&#xA;" OwnerUserId="6280" LastActivityDate="2019-08-12T04:42:26.200" Title="How important is it to show students an application of the topics seen in an undergraduate course?" Tags="|undergraduate-education|reference-request|course-design|teaching|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10468" PostTypeId="2" ParentId="5981" CreationDate="2016-01-26T15:16:14.347" Score="8" Body="&lt;p&gt;To see the reason why Lebesgue integral is preferred in probability theory one must go beyond the setting of real functions &lt;span class=&quot;math-container&quot;&gt;$f \colon \mathbb{R} \mapsto \mathbb{R}$&lt;/span&gt;.  In this setting both the Riemann and the Lebesgue integral can be defined, and reasons for choosing one of them are quite subtle.&lt;/p&gt;&#xA;&lt;p&gt;But in probability the interest is in random variables, which are functions &lt;span class=&quot;math-container&quot;&gt;$X(\omega) \colon \Omega \mapsto \mathbb{R}$&lt;/span&gt;, say. Here &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is a probability space, which is an abstract space which can be pretty much anything (with a &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra included). Generally we cannot assume that this space carries a structure necessary to define a Riemann integral! To define the Riemann integral we would need to be able to define intervals and partitions of them in  &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;, and we cannot.&lt;/p&gt;&#xA;&lt;p&gt;But the Lebesgue integral do not need such things, as we use the structure of the value space (here &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;) to define it. That way it can be used in the great generality we need in probability theory.&lt;/p&gt;&#xA;&lt;p&gt;One can in part get away with defining expectation in a bits-and-pieces manner, like one integral when the random variable &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; has a density, a sum when &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; has a probability mass function, some weird beast when it is a mixture of those two cases, ...  But using the Lebesgue integral we can get one, unifying  definition.  I think that is the real reason for preferring the Lebesgue integral, not the subtleties with the convergence theorems (which is an added premium).&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="122" LastEditDate="2022-02-28T23:33:43.603" LastActivityDate="2022-02-28T23:33:43.603" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="10469" PostTypeId="2" ParentId="1700" CreationDate="2016-01-27T01:52:40.027" Score="1" Body="&lt;p&gt;Students should be graded on the quality of their work (their ability to meet the desired learning targets) rather than how punctual the assignment is. Late Penalties lead to inaccuracy, which leads to deflated grades, which distorts the students’ achievement; their true ability to meet the intended learning outcomes.  Some students predictably struggle with deadlines. Quality work should trump timeliness. &#xA;By the way: The flood of assignments at the end of the year that you think you are going to get; it won’t happen, at least that wasn’t my experience.  In fact, in every school I’ve worked in where teachers eliminated their late penalties they did not experience the flood. &lt;/p&gt;&#xA;" OwnerUserId="6283" LastActivityDate="2016-01-27T01:52:40.027" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10470" PostTypeId="2" ParentId="1700" CreationDate="2016-01-27T12:19:52.410" Score="0" Body="&lt;p&gt;Regarding the question in the title on how to deal with &lt;strong&gt;late submissions&lt;/strong&gt;:&#xA;First some background. I collect hand written homework and no electronic submissions. Usually the the submission is before class (in the lecture room) or via some mailbox. I do not accept late submissions after class and empty the mailbox at the time I announced (give or take some minutes). So my policy with late submissions is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Late submissions without warning are not accepted.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Most students have phones and can call, text or write emails from anywhere. If I get an email saying &quot;I stuck in a traffic jam due to some accident and my submission will be late.&quot; I am going to accept the submission (unless there is evidence that the student lies, e.g. seeing him on campus at the same time). One important thing is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This is not an official rule.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If it were, people could easily abuse it. So I handle all exceptional late submissions individually. By not making this a rule, the students need be proactive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there is another guideline that I find helpful here: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Once does not count but twice is a general phenomenon.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is not meant per individual but in the following sense. If some reason for a late submission pops up once, I treat it individually. But if the same reason pops up twice (with a different student), I think about a general procedure how to handle these cases and make a rule with an announcement. This has the advantage that I do not have to make too many rules since many exception just occur once. (Finally: There are exceptions to this rule - sometimes even twice or thrice does not count…)&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2016-01-27T12:19:52.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10471" PostTypeId="1" AcceptedAnswerId="10516" CreationDate="2016-01-27T18:23:42.580" Score="11" ViewCount="8397" Body="&lt;p&gt;I am currently teaching a very difficult class, with young people (16 years old) who are deeply unmotivated and restless. I should work with them with polynomials, but I'm having a troublesome time; maybe some game-based activity would be an idea. Do you have some ideas to share?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am currently teaching in a professional school in Italy. On the average, there are around 20-25 pupils per class, with a 20% (i.e. at least 4 per class) of them having learning disabilities and at least one per class with moderate-to-severe cognitive problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am substituting their teacher for a month, and we are working with simple manipulations of polynomials i.e. adding similar terms and multiplying them toghether. I have the idea that a practical activity would help them find more interest in the process, however I myself am out of imagination on how to apply polynomials to real-life scenarios (apart from simple areas and perimeters of shapes). I'd be glad if you could point me to some game or playful setting that involves simple polynomial manipulation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT N.2:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the two interesting answers below (up to today Feb 3 2016), I feel I should clarify further the situation I'm working in:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) We don't have access to computers, so only &quot;old school&quot; methods are possible;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Student level is extremely basic: they have a lot of difficulties in calculating, for instance, $(x+1)^2$. Some of the students don't yet apply correctly the basics of algebra such as distributive property; others still make errors such as assuming that $x^2=2x$.&lt;/p&gt;&#xA;" OwnerUserId="6286" LastEditorUserId="6286" LastEditDate="2016-02-03T17:47:35.233" LastActivityDate="2019-07-15T00:53:14.590" Title="How to work with polynomials in difficult classes?" Tags="|student-motivation|class-participation|games|solving-polynomials|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10472" PostTypeId="2" ParentId="1700" CreationDate="2016-01-27T23:12:03.043" Score="2" Body="&lt;p&gt;To a degree I think it depends on the age of the students. I would not have the same policy for high school and college. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For high school I had a longstanding policy of on time for full credit, 1 day late half credit and none after that. The only exceptions were excused absences, and for students that missed significant time I would usually work out a plan for catching up that contained modified homework assignments and due dates for groups of assignments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wouldn't say that this helped with student motivation. I chose this because I couldn't manage the homework load when it was being turned in at odd times. I needed to reduce variability in order to effectively check the assignments quickly. Also, students can become overwhelmed if they're trying to complete too many assignments. I wanted to limit the number of items that could be on a students plate at any time. When a student missed assignments I tried to talk to them about &quot;what are we going to do differently?&quot; and move them away from mistakes that they may have made previously. I don't have any proof that this helped, but I think it did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more advanced classes this was more problematic, students were less likely to be able to catch up on their own. I still kept the same policy, but it felt like I was less successful at getting students to change their behavior and had to make exceptions more often to offer credit for late assignments if a student was willing to make the effort to catch up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last, homework does not count for much in my classes. My homework assignments were not busywork, but they tended to be short reviews of work we did together. When I have tried more significant home assignments I have used different criteria for lateness. I have given a few application problems as take home quizzes, and most often those have had a generous but firm deadline for full credit and no late assignments accepted.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastEditorUserId="4980" LastEditDate="2016-01-27T23:19:17.240" LastActivityDate="2016-01-27T23:19:17.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10473" PostTypeId="1" CreationDate="2016-01-28T15:27:24.927" Score="5" ViewCount="106" Body="&lt;p&gt;I am just marking a couple of linear algebra papers in which one of the questions is to determine whether $(A+B)^2=A^2+2AB+B^2$ is true for 2-by-2 matrices or not. One of the students has chosen two numeric matrices one of which is double the other and concluded that the statement is true. I want to write something for her and I wonder which one of the following statements would describe the situation better: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have chosen two &lt;strong&gt;specific&lt;/strong&gt; matrices.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have chosen two &lt;strong&gt;particular&lt;/strong&gt; matrices. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Please&lt;/strong&gt; consider that the question is not actually about what I should write for her or how I can help her and so on. The question is whether or not the words &quot;specific&quot; and &quot;particular&quot; are used interchangeably in mathematical (con)texts.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can ignore this comment for now. I have an old interest in this question as you can see in &lt;a href=&quot;http://amirasghari.com/wp-content/uploads/2015/07/FLM-2012.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this FLM paper&lt;/a&gt; and &lt;a href=&quot;https://english.stackexchange.com/q/292505/78203&quot;&gt;this ELU question&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:38:45.993" LastActivityDate="2016-01-28T22:13:14.127" Title="Can we use &quot;specific&quot; and &quot;particular&quot; interchangeably all the time?" Tags="|algebra|language-use|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10474" PostTypeId="2" ParentId="10122" CreationDate="2016-01-28T16:55:45.153" Score="2" Body="&lt;p&gt;I'll just give a rough description of my first ODE's course, take from it what you will!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Luckily we had lessons split up into theory and problem solving, so during the entire first month the problem solving classes were solely dedicated to liquidating the laundry list you mention (basically 4-5 known quadratures for 1D equations). During this time, the theory lessons were doing something completely different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At first the theory was just an introduction to the &quot;ODE problem&quot;: order, dimension, normal form, IVP, etc. Then we studied fundamental theorems. These were Peano's theorem, the Picard-Lindelöf theorem, and regularity of flow. We finished this at the same time that the problem lessons finished with the laundry list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From then on problem lessons followed what theory was doing, just with specific instances of ODE's. So I'll just describe the theory now. We took up the topic of first order linear IVP's (in any dimension): $\exists !$ for all time, fundamental matrices, vector space structure of solutions, Gronwall's inequality (in general, then applied to linear), Liouville's theorem, the exponential of a matrix (and how to take advantage of Jordan classification), and finally classification of critical points (I say it as one phrase but of course it took weeks). The next topic we did was sort of an indulgence of our teacher who specialized in it. It was on non-linear periodic ODE's, so things like the monodromy matrix, stroboscopic map, etc. were looked at. Lastly we looked at (what little can be said of) nonlinear ODE's. I think a good summary would be that the culmination of this topic was for us the Hartman-Grobmann theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's about it, I think. As for what to take from this, I'd say the most important tool for a general audience is a good understanding of linear ODE's, even moreso than the laundry list for 1D order $n$ ODE's. I've continually appreciated in later courses the knowledge (practical and theoretical) of linear ODE's. And it's not hard to motivate: the only damn thing we know how to do is linearize problems, right?&lt;/p&gt;&#xA;" OwnerUserId="4622" LastActivityDate="2016-01-28T16:55:45.153" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10476" PostTypeId="2" ParentId="10473" CreationDate="2016-01-28T21:43:29.973" Score="7" Body="&lt;p&gt;As a non-native speaker I'd tend towards &lt;em&gt;particular&lt;/em&gt; in this context. Any 2x2 matrix would be a specific matrix, but the one used by the student was a particularly bad example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Citing &lt;a href=&quot;http://www.merriam-webster.com/dictionary/particular&quot;&gt;Merriam Webster&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[particular]: distinctive among other examples or cases of the same general category:  notably unusual&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To me, &lt;em&gt;specific&lt;/em&gt; is &quot;any fixed number/matrix/...&quot;, not neccessarily a special case, whereas &lt;em&gt;particular&lt;/em&gt; always conveys some sort of &quot;out-of-the-ordinary-ness&quot;. Example: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For any specific $\varepsilon$, one can find n such that...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;vs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the particular $\varepsilon = 0.123$, it turns out to be n=123&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Any particular ...&quot; doesn't sound right to me, but &quot;any specific ...&quot; does. So they are not always interchangeable.&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2016-01-28T22:13:14.127" LastActivityDate="2016-01-28T22:13:14.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10477" PostTypeId="2" ParentId="10436" CreationDate="2016-01-29T02:46:29.637" Score="1" Body="&lt;p&gt;If there is a singularity in the integration range, the integral can only be defined by the appropriate limit(s). And limits do or do not exist, for various reasons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is by far the cleanest way to explain this, reduce the new phenomenon to already known pieces, and use what is known about them.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-01-29T02:46:29.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10478" PostTypeId="1" CreationDate="2016-01-29T14:10:43.607" Score="41" ViewCount="5091" Body="&lt;p&gt;The default approach to teaching abstract algebra seems to be groups first, then rings. However, occasionally a textbook pops up (e.g. Childs' &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0387745270&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Concrete Introduction to Higher Algebra&lt;/a&gt;, Hodge et al's &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1466567066&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abstract Algebra: An Inquiry Based Approach&lt;/a&gt; or Aluffi's recent &lt;a href=&quot;https://www.cambridge.org/nl/academic/subjects/mathematics/algebra/algebra-notes-underground?format=PB&quot; rel=&quot;noreferrer&quot;&gt;Algebra: Notes from the Underground&lt;/a&gt;) that starts with rings, based on the familiar examples of integers and polynomials.&lt;/p&gt;&#xA;&lt;p&gt;What are the pros and cons of taking a rings-first approach?&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2022-08-05T15:34:43.353" LastActivityDate="2022-08-05T15:34:43.353" Title="Rings before groups in abstract algebra?" Tags="|undergraduate-education|course-design|abstract-algebra|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="10479" PostTypeId="2" ParentId="10478" CreationDate="2016-01-29T15:25:59.257" Score="27" Body="&lt;p&gt;My favorite textbook for an undergraduate course in Abstract Algebra, Ted Shifrin's &lt;em&gt;Abstract Algebra: A Geometric Approach&lt;/em&gt;, uses a rings-first approach.  The primary pro is that students are much more familiar with examples of rings (integers, polynomials) than they are with the standard examples of groups (symmetries of simple shapes, permutations).  Indeed, the groups that students are familiar with tend to be the additive groups of known rings and the multiplicative groups of non-zero elements of known fields.  Another pro is that the extra structure in a ring allows students to have more tools to work with when constructing proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The primary con of this approach is that groups have a simpler list of defining axioms, so proofs are easier in the sense that there are less wrong directions to veer off into.  Another con of the rings-first approach is that it limits how much time you can spend on groups in the first-semester course.  To get to interesting group theory (in-depth studies of symmetry / Sylow Theorems / classification of finite groups of small order), one generally needs an entire semester.  Doing groups first allows an instructor to do that in the first semester, and then the second semester can be devoted to rings and fields and an introduction to Galois theory (or, of course, one can go in other directions after introducing rings).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should add that in Shifrin's book, rings-first really means commutative rings with unity first.  Imo, treating only commutative rings with unity is quite justified in an introductory course in abstract algebra.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2016-01-29T15:25:59.257" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10480" PostTypeId="1" CreationDate="2016-01-29T18:28:55.820" Score="6" ViewCount="83" Body="&lt;p&gt;Many scholarly articles have been written about the relationship between low working memory and math learning difficulties.  There's not much written about what to do about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to write a scholarly article that is a case study of a child who tested with single-digit working memory (due to epilepsy), and who was four years behind grade level in math.  The article would describe specific activities and approaches that worked well for him, that other teachers and tutors can use to guide their work in similarly challenging situations.  What are some possible journals for a case study?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related information: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10255/tutoring-elementary-student-who-reverses-left-and-right&quot;&gt;Tutoring elementary student who reverses left and right&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4956" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-12-03T19:39:45.127" Title="Appropriate journals for case study article about teaching math to learning-disabled student" Tags="|education-research|publishing|disability|journals|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10481" PostTypeId="2" ParentId="10021" CreationDate="2016-01-29T22:06:59.250" Score="7" Body="&lt;p&gt;In my experience, the most difficult part of mathematical induction is understanding where to focus my attention when creating proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the two courses I've taken that teach mathematical induction, the professor has begun with an analogy, and then moved on to perform a series of examples. It was fairly easy to understand the math involved with the examples, and analogies such as dominoes made enough intuitive sense that I believed in the principle. But when i was left to create my own proofs on homework, I found it very difficult. It just never seemed clear what the next step was. I became doubtful that I understood the concept at all, and wasn't sure how to get better. After working through many problems with examples, however, I started to see patterns and developed a mental catalog of common strategies. I think I would have benefited more from explanations on the processes used for developing inductive proofs.&lt;/p&gt;&#xA;" OwnerUserId="6294" LastActivityDate="2016-01-29T22:06:59.250" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10482" PostTypeId="1" AcceptedAnswerId="10523" CreationDate="2016-01-29T23:05:28.200" Score="12" ViewCount="805" Body="&lt;p&gt;I'm teaching a first-year calculus course, that is mid-way between a first intro to university-level calculus, and intro to real analysis (I'm based in Australia, for reference). We assume the intermediate value and extreme value theorems, but do proofs for Rolle's, MVT, l'Hôpital and other elementary theorems. (A slightly less than rigorous definition of limit is given, but I explained the idea of the epsilon-delta definition in words.) I was wondering if anyone has seen examples of a function or functions written down that displays a combination of all the features one might consider illustrating in such a course? A vain hope is that one could have a small collection of functions (no more than two or three) that combine, among them all the features one might want to illustrate over the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure this would be helpful, but I feel it would be interesting to students to see a more complicated function than familiar elementary functions, but which they gain all the tools to analyse over various intervals in the domain over the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; perhaps I wasn't clear: I would be interested in hearing if people have seen a short list--&lt;em&gt;no more than three or so&lt;/em&gt;--of explicit functions displaying as many features that get taught in first-year calculus as possible. I don't mean families or classes of functions. If the answer is no, comments to that effect from people who can say 'I've been teaching for 40 years in country X (and Y and...) and never seen such a thing' would be useful. Or perhaps someone can offer sound grounded and sound practice-driven advice on why such a thing would be useless or even detrimental to students appreciation, even if offered alongside a plethora of small and simpler bite-sized examples.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Thank you for your consideration&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1426" LastEditorUserId="1426" LastEditDate="2016-01-30T23:16:35.443" LastActivityDate="2018-05-11T21:00:53.190" Title="Example of function with *all* the features of differential calculus at first-year level" Tags="|undergraduate-education|calculus|examples|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10483" PostTypeId="2" ParentId="10444" CreationDate="2016-01-30T19:11:47.653" Score="2" Body="&lt;p&gt;I likewise have mainly anecdotal experience to offer, having used 3 different online homework systems in the last 3 years, including Khan Academy, WebAssign, and WileyPlus. Here are the top 3 issues I think about relating to online homework:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hidden Costs&lt;/strong&gt;. Online homework can be costly. WileyPlus costs $80 per student per semester. Since I'm at a university, we externalize the cost to students, which is difficult for some students and some cases causes students to dislike the program from the start. A few years ago I tried using (free) Khan Academy with my high school students, but that project was sunk as soon as I realized (a) many of my students did not have internet at home and (b) the school internet wasn't fast enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student Perseverance&lt;/strong&gt;. Nobody likes change, especially when it involves moving to an online homework system that is at best picky and at worst glitchy. But in my experience, students adapt to the quirks of an online homework system. In some cases my students have developed a war mindset, working to &lt;em&gt;defeat&lt;/em&gt; their tricky online homework. To reward perseverance we usually allow 15 to infinitely many guesses per problem. I suppose this leads to random guessing sometimes, but I doubt such problems are much worse than the problems that arise when doing homework another way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher Perseverance&lt;/strong&gt;. &lt;a href=&quot;https://www.sri.com/sites/default/files/publications/2014-03-07_implementation_briefing.pdf&quot; rel=&quot;nofollow&quot;&gt;This research on Khan Academy&lt;/a&gt; describes teachers' responses to online homework. While noting problems such as curriculum alignment, teachers like Khan Academy for its ability to engage students and provide instant feedback. In my experience, online homework has helped me to efficiently assess my students on a day-to-day basis so that I can put more effort into the most important things like planning engaging lessons, promoting problem solving, and preparing students for exams.&lt;/p&gt;&#xA;" OwnerUserId="867" LastActivityDate="2016-01-30T19:11:47.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10484" PostTypeId="2" ParentId="10482" CreationDate="2016-01-31T01:31:36.587" Score="8" Body="&lt;p&gt;I'm a fan of hidden case-wise formulas. Principle building block in this world is the absolute value function $|x| = \sqrt{x^2}$. Already, we have a function which is continuous, but, not differentiable at $0$. This gives a counter-example for a misuse of the Mean Value Theorem: if $f(x) = \sqrt{x^2}$ then $f(-1)=f(1)=1$ yet, nowhere in $[-1,1]$ do we find zero derivative. Oops, it seems I just gave an example of Rolle's Theorem failing. Not really. Of course $f$ is not differentiable everywhere in $(-1,1)$. The formula seems innocent enough, it's just the square and root function. Differentiation yields another misbehaving function:&#xA;$$ \frac{d}{dx}\sqrt{x^2} = \frac{2x}{2\sqrt{x^2}} = \frac{x}{\sqrt{x^2}}$$&#xA;Let $g(x) = \frac{x}{\sqrt{x^2}}$ then this function is discontinuous at $x=0$ but elsewhere constant. Note, $g'(x)=0$ for $x \neq 0$, yet, $g$ is not globally constant. This weirdness is possible since the domain of $g$ is disconnected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Penrose's &lt;em&gt;Road to Reality&lt;/em&gt; he studies the function $h(x) = x|x| = x \sqrt{x^2}$. This gives us a differentible function which is not twice differentiable at $x=0$. Indeed, the function $f_n(x) = x^n|x|$ is an easy example of the function which has $n$-derivatives at $x=0$, but, the $(n+1)$-th derivative does not exist at $x=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not quite sure what you're after, but, I think I've made the case for the absolute value function. We have a simple formula, but, simple twists on it give rather varied behaviour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beyond this, I also think it is interesting to study the interplay between vertical and horizontal behaviour of the function and its inverse function. For example, horizontal tangents of sine translate to vertical tangents of inverse sine. Or, just the simple idea of graphing the same shape horizontally verses vertically. The connection between the slopes, increasing paired with increasing, or decreasing with decreasing. Much to explore here there is.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2016-02-06T18:17:03.380" LastActivityDate="2016-02-06T18:17:03.380" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10485" PostTypeId="2" ParentId="10471" CreationDate="2016-01-31T01:56:57.650" Score="4" Body="&lt;p&gt;Have you looked into &lt;a href=&quot;https://teacher.desmos.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Desmos&lt;/strong&gt;&lt;/a&gt; for possible resources?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular: How about the &lt;a href=&quot;https://teacher.desmos.com/search?q=marbleslides&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Marbleslides&lt;/strong&gt;&lt;/a&gt; activities?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure whether this satisfies the criteria in your case (specifically: I do not know if you have access to the requisite technology) but the activity (it seems to me) can be modified RE:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I'd be glad if you could point me to some game or playful setting that involves simple polynomial manipulation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It came up in a reddit post (&lt;a href=&quot;https://www.reddit.com/r/math/comments/3wz65d/try_out_this_game_marbleslides_from_desmos_i/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;OP&lt;/strong&gt;&lt;/a&gt;) that linked to &lt;a href=&quot;https://student.desmos.com/?prepopulateCode=gdak&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;this site&lt;/strong&gt;&lt;/a&gt; about a month ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find more on Dan Meyer's blog &lt;strong&gt;dy/dan&lt;/strong&gt; in the post: &lt;a href=&quot;https://blog.mrmeyer.com/2015/marbleslides-is-here/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Marbleslides is here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a gif directly from that blog post:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://blog.mrmeyer.com/2015/marbleslides-is-here/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;a href=&quot;https://i.stack.imgur.com/6MGdI.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6MGdI.gif&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;br&gt;&#xA;&lt;sub&gt;(source: [mrmeyer.com](http://blog.mrmeyer.com/wp-content/uploads/151213_1.gif))&lt;/sub&gt;&lt;br&gt;&#xA;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="6053" LastEditDate="2019-07-15T00:53:14.590" LastActivityDate="2019-07-15T00:53:14.590" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="10486" PostTypeId="2" ParentId="10478" CreationDate="2016-01-31T18:40:14.700" Score="7" Body="&lt;p&gt;I think groups first is the right approach. My reasoning is that, groups are very &quot;strange&quot; structures in that they don't feel natural to work with at first. Rings are much more intuitive because as some people have already pointed out, there are much more familiar examples to work with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I actually see this as the reason to work with groups first. Getting students familiar with the type of abstractions they will see in algebra and then further along in algebraic topology is a good thing. Some of the structures we create are incredibly unintuitive and it took me a long time before I could really feel like I was okay with the abstraction taking place in algebra. Starting with groups forced me to learn to become more comfortable with these abstract structures. Moving on to rings can then be a step where things actually become easier (instead of more difficult if you're going rings -&gt; groups) because the structures become more &quot;natural&quot;.&lt;/p&gt;&#xA;" OwnerUserId="5958" LastActivityDate="2016-01-31T18:40:14.700" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10487" PostTypeId="2" ParentId="10478" CreationDate="2016-01-31T21:10:20.630" Score="18" Body="&lt;p&gt;I have taught both groups first and a rings first course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was a post-doc at Rutgers University, I taught their standard introduction to modern algebra course using Hungerford's undergraduate algebra text. I was kind of annoyed at the time that I would (unless I wanted to fight the textbook) have to teach using a rings first approach. By the end of the semester I was convinced that rings first is the way to go.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Full disclosure: That was the only time I've taught rings first. My department (at Appalachian State University) has adopted a groups first approach (we use Gallian's Contemporary Abstract Algebra) and I respect those choices. But I still believe that a ring first approach is generally better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, &quot;better&quot; is subjective and highly depends on the type of course you're teaching and who your target audience is. Generally, intro to modern algebra courses (in my experience) contain a mixed bag of math and math education majors. Math education majors tend to have difficulties with the groups first approach. They see no value in the course content -- with good reason. If you start with rings, the course sells itself. Applications are immediate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pro: For rings first is you have immediate access to a large class of examples (number rings, polynomial rings, matrix rings, etc). With groups first you must spend a large amount of time building examples before you can really do anything. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pro: Immediate applications. There are many easily accessible applications that are appealing to students. For groups your applications tend to be more obscure geometry/combinatorics or cryptography (where arguably it's not applying group theory as much as number theory). On the other hand, for example, with rings you can immediately get into the theory of factoring polynomials and related things that will seem quite appealing to secondary education majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Con: A ring's axiom system is longer and more complicated. Honestly, I don't really accept this as a serious con. We teach vector spaces in linear algebra before groups -- even more complicated! The fact that a ring is a group plus ... is typically not exploited in introductory courses. As an example, do you see undergraduate texts mention that the distributive laws are merely statements that multiplication operators are group homomorphisms (under addition) and thus since the identity maps to the identity we have $r0=0$ for all $r$? I think not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pro: Rings first, familiar examples are less weird. Every time I teach (groups first) I spend more time that I'd like reminding confused students that $\mathbb{R}$ (and other number rings) is a group under addition not multiplication (&quot;rule of thumb -- does it contain zero?&quot;). When you tell a student that $\mathbb{Z}$ is a group under addition, the immediate question that comes to mind is &quot;What about multiplication?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I rather enjoyed teaching out of undergraduate Hungerford. He presents the integers and basic number theory first. Then modular arithmetic. Next, he presents polynomials -- you get the division and Euclidean algorithms again. Then quotients of polynomial rings. Now with two very concrete classes of examples in hand (with a lot of concrete calculations behind you) you do abstract ring theory and hit quotients again. By the time you cover abstract quotient rings, you've seen quotients in the context of integers and polynomials so that the abstraction seems like a natural idea. Finally, group theory is unrolled as the next step of abstraction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I felt that this approach not only benefitted the math majors in the audience (there was plenty of time to deal with abstract systems and write abstract proofs) but it also carried along a large group of students that normally get turned off before you get to define a kernel and homomorphism. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now if you're just teaching talented math majors -- groups first is more efficient and if they can't handle it -- they should change majors (cough cough). I prefer rings first for a general audience. Groups first for honors/pure math people.&lt;/p&gt;&#xA;" OwnerUserId="453" LastActivityDate="2016-01-31T21:10:20.630" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10488" PostTypeId="1" AcceptedAnswerId="10491" CreationDate="2016-02-01T00:34:05.433" Score="15" ViewCount="707" Body="&lt;p&gt;I asked a student I tutor what $\sqrt{x^2}$ was so I could show him why the solution is $|x|$ instead of just $x$. He ended up changing the problem to $(x^2)^{1/2}$ and then multiplied the exponents to get $x^{2/2}=x^1=x$. Now I'm well aware why the answer to the problem I gave him is $|x|$ and explained it, and the answer to $(x^2)^{1/2}$ is likewise $|x|$ but I wasn't able to give a real good explanation why he couldn't use the rule $\sqrt[n]{a^m}=a^{m/n}$.&lt;/p&gt;&#xA;" OwnerUserId="6287" LastEditorUserId="77" LastEditDate="2016-02-01T00:48:54.040" LastActivityDate="2018-04-17T16:16:47.900" Title="Student converted $\sqrt{x^2}$ and ended up with just $x$ instead of $|x|$" Tags="|algebra|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="10489" PostTypeId="2" ParentId="10488" CreationDate="2016-02-01T01:12:36.683" Score="7" Body="&lt;p&gt;The identity &lt;a href=&quot;https://en.wikipedia.org/wiki/Exponentiation#Identities_and_properties&quot;&gt;$(x^m)^n=x^{mn}$&lt;/a&gt; is true for real numbers $m,n$ and $x\ge 0$.  It is not necessarily true otherwise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An &lt;a href=&quot;https://en.wikipedia.org/wiki/Nth_root&quot;&gt;$n$th root&lt;/a&gt; of $b$ (where $n$ is a positive integer and $b$ is a real number) is a value of $x$ such that $x^n=b$.  Note that if $b\ne 0$, then there are $n$ distinct values of $x$, some of which may be complex.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the expression $x^n=b$ implies many possible values for $x$, the notation $x=\sqrt[n]{b}$ implies &lt;em&gt;only one&lt;/em&gt; value of $x$, the principal $n$th root of $b$.  This is the unique positive real value of $x$ (if it exists), or the unique negative real value of $x$ otherwise (if it exists).  (In my experience, the use of the notation $x=\sqrt[n]{b}$ to denote a unique value of $x$ is often done in primary and secondary education.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if $x^2=4$, then this means that either $x=2$ or $x=-2$.  But $x=\sqrt{4}$ means that $x=2$ (and not $x=-2$).  If $x^3=-8$, then there are three solutions (only one of which is real), but the notation $x=\sqrt[3]{-8}$ means $x=-2$ (and no other values).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The equation $x^2=b$ (where $b&amp;gt;0$) means that either $x=\sqrt{b}$ or $x=-\sqrt{b}$.  If it is the former, then $\sqrt{x^2}=x$ but if it is the latter then $\sqrt{x^2}=-x$.  So one cannot claim that, in general, $\sqrt{x^2}=x$.  However, $\sqrt{x^2}=|x|$ holds for either case, so it is, in general, true.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2016-02-01T03:15:15.667" LastActivityDate="2016-02-01T03:15:15.667" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10490" PostTypeId="2" ParentId="10488" CreationDate="2016-02-01T03:20:23.170" Score="7" Body="&lt;p&gt;The identity $(x^m)^n = x^{mn}$ is only true if $x&amp;gt;0$.  In fact pretty much all identities involving exponents are only true for positive bases -- another example that often trips students up is $x^r \cdot y^r = (xy)^r$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are several ways to explain why these formulas are only true for positive bases, but at a fundamental level it has to do with the question of how we even &lt;em&gt;define&lt;/em&gt; $x^r$ when $r$ is a real number.  The conventional definition is that we define&#xA;$$x^r = \exp \left(r \ln x \right) $$&#xA;(although in order to make this definition non-circular we have to have some way of defining what $\exp$ and $\ln$ are without invoking exponents.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on the above definition it can be shown (for positive $x$) that $x^{r+s}=x^r x^s$.  From this it can further be deduced that if the exponent $r$ is some integer $n$ then the definition above is consistent the more elementary notions based on &quot;repeated multiplication.&quot;  It further follows that $x^{1/n}$ is the unique positive $n^{th}$ root of $x$, and we can likewise prove that $(x^r)^s = x^{rs}$ for any real $r$ and $s$.  But &lt;strong&gt;all of these provable properties hinge on how exactly we define $x^r$&lt;/strong&gt;, and note that the definition given above &lt;em&gt;only makes sense&lt;/em&gt; for $x&amp;gt;0$, because $\ln x$ is not defined (or, more precisely, is not single-valued) for negative values of $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt;  Actually, the situation is more complicated than what I have written above; see &lt;a href=&quot;https://math.stackexchange.com/a/2085269/124095&quot;&gt;https://math.stackexchange.com/a/2085269/124095&lt;/a&gt;, and scroll down to the second update, for more information regarding the question of in which contexts the various exponent rules apply.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2018-04-04T19:26:15.490" LastActivityDate="2018-04-04T19:26:15.490" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10491" PostTypeId="2" ParentId="10488" CreationDate="2016-02-01T03:21:35.003" Score="11" Body="&lt;p&gt;This is a surprisingly tricky issue, and the exact answer &quot;why&quot; depends on the context (real or complex numbers?) and definitions in play (varies by textbook). I'm looking for a good general answer to effectively the same issue on &lt;a href=&quot;https://math.stackexchange.com/questions/1628759/what-are-the-laws-of-rational-exponents&quot;&gt;this question here&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Restricting the discussion purely to real numbers, the most popular and straightforward response seems to be that the identity $(a^r)^s = a^{rs}$ requires some additional restriction on it, for example, that both $a^r$ and $a^s$ be defined in real numbers. Observing that restriction, the student's simplification is invalid for negative $x$, because then $x^\frac{1}{2}$ is undefined in real numbers. It's possible that may only be implied (not explicitly stated) in a given textbook that you might use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some exposition of the complex-valued case, consider the &lt;a href=&quot;https://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identities&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article here&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Note that as defined in complex numbers, $(z^m)^\frac{1}{n} = (z^\frac{1}{n})^m$ will be true if and only if $m$ and $n$ are relatively prime (assumes $n$, $z$ nonzero;  Silverman, 1975). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-02-03T20:08:54.727" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10492" PostTypeId="1" AcceptedAnswerId="10493" CreationDate="2016-02-02T05:02:43.317" Score="10" ViewCount="1162" Body="&lt;p&gt;Related but not duplicate: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1680/what-courses-require-multivariable-analysis&quot;&gt;What courses require multivariable analysis?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;By multivariable analysis I mean the rigorous version of multivariable calculus (something equivalent to Ch.9-10 in baby Rudin or topics cover in Analysis on Manifolds by Munkres).&lt;/p&gt;&#xA;&lt;p&gt;My question is,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why do advance analysis courses(at least those I took) emphasize so little in topics related to multivariable analysis, compare to topics like measure theory and Lebesgue integral? Is there any reason undergraduate/beginning graduate math education is designed this way?&lt;/p&gt;&#xA;&lt;p&gt;Aren't they equally important prerequisite for different advanced topics(e.g. Measure Theory to Probability/Stochastic Process, Multivariable Analysis to Differential Geometry, etc.)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My experience and observation,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In my undergraduate institution, 1st semester real analysis is first few chapters of baby Rudin, 2nd semester is introduction to measure theory. 1st semester graduate level is point set topology/measure theory, 2nd semester graduate level(which I didn't finish) is closer to functional analysis.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In the graduate program I'm currently in, 1 year sequence of graduate real analysis and qualifying exam focus almost exclusively on Measure theory and Lebesgue Integral. Topics in multivariable analysis are included in 3rd quarter of honor undergraduate/intro. graduate course where graduate students are only require to take it if not passing (preliminary) assessment exam.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The focus of real analysis qualifying exam in many places seem to be more on Measure theory and Lebesgue integral, too.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user2139" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2022-11-27T16:52:45.547" Title="Why is multivariable analysis often omitted?" Tags="|undergraduate-education|mathematical-analysis|graduate-education|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10493" PostTypeId="2" ParentId="10492" CreationDate="2016-02-02T07:11:02.863" Score="6" Body="&lt;p&gt;$\newcommand{\RR}{\mathbb{R}}$I am not really sure if I understood your question correctly, especially if we have the same understanding of what &quot;multivariable calculus&quot; is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If by multivariable calculus you mean that topic where you do differential calculus for maps $f:\RR^n\to\RR^m$, i.e. introducing total differentiability, the Jacobian, the Hessian, Taylor's formula for such maps, paths and tangents, the implicit function theorem and such, then I don't understand your question as these are frequently taught in analysis courses (as far as I see, but this may not be correct).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if by multivariable calculus you mean the multivariable theory of integration then things get more complicated. There are different choices to order the content: Do integration of smooth functions over smooth structures with an ad hoc approach using Riemann integrals and Fubini type arguments without appealing to measure theory at all. However, when moving to integrals over surfaces one needs some theory to back up what the surface element is and I don't know any proper motivation without the change of variables formula. This suggests to introduce Lebesgue integration theory first. Personally, I find the approach with ad hoc integration not very satisfying. What I like about integration is that it can deal with very rough functions. Moreover, using Lebesgue's approach one can somehow &quot;rectify&quot; several things that seem a bit obscure in basic analysis (e.g. the fundamental theorem is much nicer for absolutely continuous functions, the notion of Lebesgue point shows how well defined measurable function really are and also there are Lusin and Egoroff who show what's behind continuity and uniform convergence). But, in essence, I agree that the choice &quot;Lebesgue vs. differential forms&quot; is somehow a matter of taste (or, agreement in the department).&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2016-02-02T07:11:02.863" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10494" PostTypeId="1" CreationDate="2016-02-02T17:24:32.130" Score="6" ViewCount="377" Body="&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/q/1637562&quot;&gt;This&lt;/a&gt; question (and answers) on MSE got me thinking on &lt;em&gt;simple&lt;/em&gt; examples of different ways of proving the same (hopefully somewhat interesting) result, as examples to be discussed on difference in clarity or whatever. This for use in a first course in discrete mathematics, for students in the middle of the typical college calculus sequence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Particularly interesting would be proofs that use different techniques (contradiction, contrapositive) but that one would consider be &quot;the same proof, written up differently&quot;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-07-28T11:45:56.977" Title="Different approaches to proofs that &quot;are the same&quot;?" Tags="|proofs|discrete-math|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10495" PostTypeId="2" ParentId="10494" CreationDate="2016-02-02T18:04:34.530" Score="0" Body="&lt;p&gt;A comment on the question you link to makes a valid point: it's important to define what you mean by &quot;different ways of proving&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the logical standpoint, the difference depends on the axioms or first principles from which the theorem is deduced. Two proofs that argue from the same axioms are logically equivalent. If one wants to prove that $p \Rightarrow q,$ it may be possible to demonstrate this directly, or one may show that $!p \Rightarrow !q$, which is a proof by contraposition. Logically, these are equivalent proofs, so they may not necessarily be thought of as &quot;different.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you were to deduce a statement from a set of axioms, then from an independent set of axioms, that would certainly constitute a different proof by any definition. This is certainly possible in many cases, for example showing that a certain object does not satisfy a set of axioms (group, ring, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, any statement of set theory that can be deduced from the well-ordered principle can also be deduced from the principle of induction, because it can be shown that those two axioms are logically equivalent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this reason, I tend to think of axioms as sort of a &quot;logical basis,&quot; where &lt;em&gt;basis&lt;/em&gt; here is meant in the algebraic sense of a spanning set, in this case the theorems are the span of the axioms, if you will. As you search for examples, try to identify when the proofs follow from the same &quot;basis&quot; set, and when they do not. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastActivityDate="2016-02-02T18:04:34.530" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10496" PostTypeId="2" ParentId="10494" CreationDate="2016-02-02T21:52:15.093" Score="2" Body="&lt;p&gt;A source of simple example is in basic set theory. Let $A,B,C$ be sets and $f:A\to B$, $g:B\to C$ maps. Consider assertions of the kind &quot;if $f$ and $g$ are one-to-one, then $g\circ f$ is one-to-one&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can prove it directly using this classical definition &quot;$f$ is one-to-one when for all $x,y\in A, f(x)=f(y) \implies x=y$&quot;. Assume $f$ and $g$ are one-to-one and let $x,y\in A$ be such that $g(f(x))=g(f(y))$. Since $g$ is one-to-one, we have $f(x)=f(y)$. Since $f$ is one-to-one, we thus have $x=y$. We have proved that $g\circ f$ is one-to-one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can reformulate using the contrapositive definition &quot;$f$ is one-to-one when for all $x,y\in A, x\neq y\implies f(x)\neq f(y)$&quot;. Assume $f$ and $g$ are one-to-one and let $x\neq y\in A$. Since $f$ is one-to-one, $f(x)\neq f(y)$. Then since $g$ is one-to-one, $g(f(x))\neq g(f(y))$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can prove by contradiction: assume $f$ and $g$ are one-to-one and that there are $x\neq y\in A$ such that $g(f(x))=g(f(y))$. Either $f(x)= f(y)$, contradicting the assumption that $f$ is one-to-one, or $f(x)\neq f(y)$&#xA;and $g(f(x))=g(f(y))$ contradicts the assumption that $g$ is one-to-one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can use a contrapositive: assume $g\circ f$ is not one-to-one, and prove that either $f$ or $g$ is not one-to-one (as above).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are probably others... and I would say that all these proofs are the same under different guises. I believe that there exist no proofs of this fact that I would not qualify as essentially the same as the above, but I would be delighted to be proven wrong in a comment!&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-02-02T21:52:15.093" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10497" PostTypeId="1" CreationDate="2016-02-03T06:21:45.770" Score="5" ViewCount="226" Body="&lt;p&gt;For an undergraduate mathematics textbook, what are the pitfalls of inserting all of the exercises in the text? (As opposed to grouping every exercise at the end of the section). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;IMO I feel it is better to insert exercises in the text because it breaks up the monotony of reading, and the problems enable the student to check their understanding before moving on, but since this fad has not caught on I was wondering what stops most professors from doing it.&lt;/p&gt;&#xA;" OwnerUserId="5746" LastEditorUserId="143" LastEditDate="2016-02-14T19:19:24.387" LastActivityDate="2016-02-14T19:19:24.387" Title="In Text Exercises" Tags="|mathematical-pedagogy|textbooks|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10498" PostTypeId="2" ParentId="10497" CreationDate="2016-02-03T06:28:06.503" Score="5" Body="&lt;p&gt;As a starting answer, my understanding is that students generally read maths texts very poorly, often looking only at the worked examples and the sets of exercises (plus answers). Also, some lecturers like to set homework directly from sets of exercises.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-02-03T06:28:06.503" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10499" PostTypeId="2" ParentId="10494" CreationDate="2016-02-03T09:49:27.530" Score="1" Body="&lt;p&gt;One important example is the equivalence of standard bottom-up induction and top-down induction. It is interesting that bottom-up induction does not extend immediately to ordinals (since successors and limits must be handled differently), while top-down induction extends without any change. On the other hand, top-down induction is usually easier to learn without error, especially in the form of structural induction (see &lt;a href=&quot;https://matheducators.stackexchange.com/a/10034/1550&quot;&gt;this explanation&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Closely related is the equivalence of induction and well-ordering. Here there is an interesting disparity, because in some cases considering the smallest counter-example with sizes from some well-order yields much easier proofs, partly because it is essentially induction wrapped in a non-constructive proof by contradiction, which is in some sense strictly stronger than constructive induction, namely when in a constructive logic. This may seem irrelevant, but in many cases it is far easier to see a proof by well-ordering (also called extremal principle) than to see the equivalent proof by induction (see &lt;a href=&quot;http://www.artofproblemsolving.com/wiki/index.php/Extreme_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;this short list on AOPS&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For analysis in particular, it's interesting to look at the proofs of basic results such as that continuous functions on a compact set are bounded and attains its extrema, and are uniformly continuous. Because of the equivalent notions of compactness for metric spaces, it is interesting to look at the resulting differences in proofs. One would notice though that the core of each proof is pretty much the same for dimension at least 2. However, in 1 dimension there are rather different proofs that I discovered long time ago that do not extend (at least I don't see how), which I give as an example below.&lt;/p&gt;&#xA;&lt;p&gt;There are many ways to precisely quantify sameness of proofs. One way that is good for students to learn is to see exactly where each part of the theorem and which axioms are used in the proof, and also count the number of times each of them is used. Two proofs are essentially the same when they are minimal in that respect and use each part the same number of times. This can be done at any level of detail, but usually we do not include any of the logical axioms except for double negation elimination (equivalently proof by contradiction in the form &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\neg P \rightarrow \bot \vdash P$&lt;/span&gt;&amp;quot;), and often we work over a specific axiomatization such as that of the real numbers (second-order completeness). Here is an example:&#xA;&lt;span class=&quot;math-container&quot;&gt;$\def\rr{\mathbb{R}}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\def\less{\smallsetminus}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Any continuous function &lt;span class=&quot;math-container&quot;&gt;$f : [0,1] \to \rr$&lt;/span&gt; is bounded on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Proof&lt;/em&gt; [with reason on the right of each line]:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Take any continuous function &lt;span class=&quot;math-container&quot;&gt;$f : [0,1] \to \rr$&lt;/span&gt;. [logically necessary]&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$S = \Big\{ p : p \in [0,1] \land \text{$f$ is bounded on $[0,p]$} \Big\}$&lt;/span&gt;. [by comprehension]&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$0 \in S$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$\forall x \in [0,0]\ ( |f(x)| \le |f(0)| )$&lt;/span&gt;. [this uses the left endpoint of &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;]&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is non-empty. [by definition of non-empty]&lt;/p&gt;&#xA;&lt;p&gt;Also, for any &lt;span class=&quot;math-container&quot;&gt;$p \in S$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$p \le 1$&lt;/span&gt; by construction of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;. [this uses the right endpoint of &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;]&lt;/p&gt;&#xA;&lt;p&gt;Thus &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; has a least upper bound &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\rr$&lt;/span&gt;. [by completeness of real numbers]&lt;/p&gt;&#xA;&lt;p&gt;Firstly &lt;span class=&quot;math-container&quot;&gt;$c \ge 0$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$0 \in S$&lt;/span&gt;. [this uses that &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is an upper bound]&lt;/p&gt;&#xA;&lt;p&gt;Secondly &lt;span class=&quot;math-container&quot;&gt;$c \le 1$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is an upper bound for &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\rr$&lt;/span&gt;. [this uses that &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the least one]&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$δ &amp;gt; 0$&lt;/span&gt; such that, for any &lt;span class=&quot;math-container&quot;&gt;$x \in [0,1]$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$|x-c| \le δ$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$|f(x)-f(c)| \le 1$&lt;/span&gt;. [this uses continuity but is much weaker (*)]&lt;/p&gt;&#xA;&lt;p&gt;Thus &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is bounded on &lt;span class=&quot;math-container&quot;&gt;$[0,1] \cap [c-δ,c+δ]$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$|f(c)|+1$&lt;/span&gt;. [by triangle inequality]&lt;/p&gt;&#xA;&lt;p&gt;Also &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is bounded on &lt;span class=&quot;math-container&quot;&gt;$[0,c-δ]$&lt;/span&gt; by definition of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;. [this is why we defined &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; like that]&lt;/p&gt;&#xA;&lt;p&gt;Thus &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is bounded on &lt;span class=&quot;math-container&quot;&gt;$[0,1] \cap [0,c+δ]$&lt;/span&gt;. [by basic properties of reals]&lt;/p&gt;&#xA;&lt;p&gt;Thus &lt;span class=&quot;math-container&quot;&gt;$\min(1,c+δ) \in S$&lt;/span&gt; by definition of S. [in the reverse direction]&lt;/p&gt;&#xA;&lt;p&gt;Thus &lt;span class=&quot;math-container&quot;&gt;$\min(1,c+δ) \le c$&lt;/span&gt; and hence &lt;span class=&quot;math-container&quot;&gt;$1 \le c$&lt;/span&gt;. [by basic properties of reals]&lt;/p&gt;&#xA;&lt;p&gt;Therefore &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is bounded on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; since &lt;span class=&quot;math-container&quot;&gt;$[0,1] \subseteq [0,c+δ]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(*) This is a clear example of an important point that students should learn to identify and understand. Continuity of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is essentially an assertion of the form &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\forall ε &amp;gt; 0\ ( \text{$f$ is $ε$-continuous} )$&lt;/span&gt;&amp;quot;, and here all we used was &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-continuity of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; was arbitrary except that it was more than &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. Students with a proper foundation in logic would not only be able to prove the theorem but also realize via this observation that we can strengthen it to:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given any &lt;span class=&quot;math-container&quot;&gt;$ε &amp;gt; 0$&lt;/span&gt;, any &lt;span class=&quot;math-container&quot;&gt;$ε$&lt;/span&gt;-continuous function &lt;span class=&quot;math-container&quot;&gt;$f : [0,1] \to \rr$&lt;/span&gt; is bounded on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Notice that in the above proof every part of the theorem statement has been used exactly once. It is instructive to check that dropping any single part, such as continuity of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; being closed on each end, results in a false assertion. With a suitable notion of shortness, this implies that the proof is more or less minimal over &lt;a href=&quot;https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach&quot; rel=&quot;nofollow noreferrer&quot;&gt;the standard axiomatization of the reals&lt;/a&gt;, which anyone reading the proof would expect.&lt;/p&gt;&#xA;&lt;p&gt;Note however that theorems and proofs do not live in isolation, and so one can consider any particular body &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; of mathematics as a whole, and ask what is the shortest proof of all the theorems in &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;. This minimization tends to have all the key theorems factored out, much like in &lt;a href=&quot;https://en.wikipedia.org/wiki/LZ77_and_LZ78&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lempel-Ziv compression&lt;/a&gt;. What we are minimizing is essentially the &lt;a href=&quot;https://en.wikipedia.org/wiki/Kolmogorov_complexity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kolmogorov complexity&lt;/a&gt; of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; with respect to the underlying formal system (such as ZFC).&lt;/p&gt;&#xA;&lt;p&gt;For any collection &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; of theorems over an existing body &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; has a minimal proof, one can look at the proof of all the theorems in &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;. Two proofs are then reasonably considered identical if they use the same axioms and the same proven sentences in &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;. To make sure this notion coincides with our intuition, we can require that a sentence must be labelled before it can be used as a theorem, so that only major results will be labelled in a minimal proof of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If the two proofs &lt;span class=&quot;math-container&quot;&gt;$P,Q$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are not considered identical according to the above, we can measure the difference via some reasonable proof distance, such as one that I just thought of that takes into account differences between theorems used. First let &lt;span class=&quot;math-container&quot;&gt;$S,T$&lt;/span&gt; be the collection of all theorems in &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; that &lt;span class=&quot;math-container&quot;&gt;$P,Q$&lt;/span&gt; respectively depend on directly. Then let &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; be a minimal proof of &lt;span class=&quot;math-container&quot;&gt;$S \cup T$&lt;/span&gt;. (Give each equal weight if there is more than one.) Then the proof distance between &lt;span class=&quot;math-container&quot;&gt;$P,Q$&lt;/span&gt; is the edit distance between theorems and axioms used in &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; ignoring order, plus the total length of the segments of &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; that are used only for one of &lt;span class=&quot;math-container&quot;&gt;$S,T$&lt;/span&gt;. This definition ensures that two proofs that have the same essential ingredients have small proof distance, and proof distance is only slightly affected if the same theorem is proven and used under different names, since minimizing the proof of &lt;span class=&quot;math-container&quot;&gt;$S \cup T$&lt;/span&gt; automatically finds the common ingredients.&lt;/p&gt;&#xA;&lt;p&gt;Likewise, and in fact more easily, we can define similarity in mathematical content of theorems. Given any two collections &lt;span class=&quot;math-container&quot;&gt;$A,B$&lt;/span&gt; of theorems, let &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; be a minimal proof of &lt;span class=&quot;math-container&quot;&gt;$A \cup B$&lt;/span&gt;. Then the content distance between &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is just the total length of the segments of &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; that are used only for one of &lt;span class=&quot;math-container&quot;&gt;$A,B$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is quite technical, but I believe it quite accurately captures what we mean when we say things like &amp;quot;Compactness and completeness are essentially equivalent theorems about first-order logic.&amp;quot;. Of course all true assertions are equivalent, including &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$0 = 0$&lt;/span&gt;.&amp;quot;, but we all know that the statement is trying to say that the easiest way to prove both compactness and completeness is to prove one and then derive the other as a corollary. Indeed if we let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; be the singleton containing &amp;quot;compactness&amp;quot; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; be the singleton containing &amp;quot;completeness&amp;quot;, then since there is a short proof of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, the shortest proof of &lt;span class=&quot;math-container&quot;&gt;$A \cup B$&lt;/span&gt; is thus only a few lines longer than the common ingredients of the individual proofs, and thus there is negligible content difference between compactness and completeness, as expected.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-02-03T09:49:27.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10500" PostTypeId="2" ParentId="10421" CreationDate="2016-02-03T10:44:42.153" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Your advisor may be recommending one instructor over another; if so, trust that advice.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Complex analysis will prepare you better than cryptology for classes in math grad school.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For research topics in probability, an undergraduate math class is a poor source of ideas.  You'd get more ideas from a class in econometrics or public health.  If you want research ideas about probability in cryptology specifically, read up on Bitcoin debates.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I usually recommend applications, but the cryptography course doesn't seem to match your goals.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2016-02-03T10:44:42.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10501" PostTypeId="1" AcceptedAnswerId="10502" CreationDate="2016-02-03T13:18:48.187" Score="8" ViewCount="578" Body="&lt;p&gt;I find it extremely time consuming to grade a homework in an undergraduate real analysis course without a rubric. Several instructors I worked with did not have a clear rubric in their mind at all. Instead, they usually just want &quot;as much feedback (for the students) as possible&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are my questions: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What would be a reasonable amount of time (in terms of hours per week) for grading homework for a TA in graduate school? (How much time would be a sign of &quot;too much&quot;?)&lt;/li&gt;&#xA;&lt;li&gt;For grading a homework in undergraduate level real analysis (or complex analysis), what would be a good rubric?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;[P.S. I'm talking about universities in the US.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The size of the class is around 25 people. A particular homework problem set is like the following&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;6.2.1, 6.2.2, 6.2.3, 6.3.3, 6.3.5, 6.4.2, 6.4.3 in Abbott's &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1493927116&quot; rel=&quot;nofollow&quot;&gt;Understanding Analysis&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user5897" LastEditorDisplayName="user5897" LastEditDate="2016-10-03T16:15:11.887" LastActivityDate="2016-10-13T03:41:45.297" Title="Rubric for grading an undergraduate real analysis course" Tags="|undergraduate-education|mathematical-analysis|grading|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10502" PostTypeId="2" ParentId="10501" CreationDate="2016-02-03T14:16:06.147" Score="5" Body="&lt;p&gt;Many departments have a set of expectations for the amount of time a graduate student TA should be spending grading per week.  It varies from place to place, but you should find out if your department has a clear set of guidelines and, if so, respectfully ask the instructor that he agree to follow them.  I personally think 3 hours per week per course TAed is not unreasonable (and I've heard of higher), though I personally try to lessen my TAs grading responsibilities when possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are arguments for and against a rubric, largely dependent on how much experience you have with grading.  I personally do not think a predesigned rubric is optimal in such a course, but rather that you should make a rough plan of how you expect to grade each problem, and then make specific notes for how you handled specific errors as you go along (so you can deduct the same amount if the same or a very similar error appears in someone else's work).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One solution that your instructor might agree to is to come to a predetermined amount of time that you should spend each week grading problems, and have you grade as many different problems as you can within that time constraint, perhaps with some feedback as to which problems absolutely must be graded and/or which problems are relatively easily sacrificed.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2016-02-03T14:16:06.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10503" PostTypeId="2" ParentId="10471" CreationDate="2016-02-03T15:12:36.653" Score="5" Body="&lt;p&gt;For multiplying polynomials and combining like terms, &lt;a href=&quot;http://www.mathguide.com/lessons/Polynomials.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this website&lt;/a&gt; presents a simple level-appropriate tabular method. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, to multiply the polynomials $x - 2$ and $2x^2 -3x + 1$ construct the following table:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/dFwI6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dFwI6.png&quot; alt=&quot;empty table&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then fill the table with the products, like the multiplication tables they learned in primary school:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MSztf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MSztf.png&quot; alt=&quot;full table&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like terms will be located along diagonals (not including the terms of the original polynomials), so students can add them easily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To gamify this technique, you might distribute a number of versions of worksheets with fourth or fifth degree polynomials to multiply, with the empty tables already constructed and a few lines under each table to add the like terms and write the product. Complete all of the worksheets ahead of time, then cut each table box into an individual piece and deposit all of the pieces into a container. You can pick pieces out one at a time and have students stamp or color the corresponding box on their worksheet to make a BINGO type game. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-02-03T22:45:00.220" LastActivityDate="2016-02-03T22:45:00.220" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10504" PostTypeId="1" CreationDate="2016-02-03T23:07:44.707" Score="11" ViewCount="345" Body="&lt;p&gt;I'm thinking of general rules on how to write mathematics books to people that teach themselves, and are quite alone with only the single aid of the printed book (no YouTube, no software, and so on). Any clue?&lt;/p&gt;&#xA;" OwnerUserId="2044" LastEditorUserId="1147" LastEditDate="2017-06-13T03:51:28.650" LastActivityDate="2017-06-13T03:51:28.650" Title="How to write a book for someone who learns math alone?" Tags="|self-learning|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10505" PostTypeId="2" ParentId="10471" CreationDate="2016-02-03T23:14:24.660" Score="4" Body="&lt;p&gt;Well, you could start by explaining what $x^2$ means geometrically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I had a square with side length $x$, then the &lt;strong&gt;area&lt;/strong&gt; would be $x^2$.  Explain to them how this is completely different from $2x$.  Picture often help students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to explain to them that $(x+1)^2=x^2+2x+1$?  Once again, I would go with a visual:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/B0vba.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/B0vba.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pictures are often better than numbers, as I have found.  Also, using words instead of numbers helps:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$x+2x=3x\to$One apple plus 2 apples equals 3 apples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this is how you could apply polynomials into geometry.  Furthermore, if you feel confident, volume becomes a $3$rd degree polynomial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not quite sure how you could gamify this, and I'm not that great with gamifying math in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the use of colors helps students understand why which numbers go where, as opposed to strict math tables like the other answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an idea for a game, you could make a rectangle with some numbers (where $x$ should've been) and have each area shaded a different color.  Then ask your students what each color represents.  Ask them which colors have similar $x$'s, like reducing $x$ and $2x$ into a single area of $3x$, or something along those lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I noticed you could &lt;em&gt;add&lt;/em&gt; on different shapes together to get different polynomials.  This should also help out with their algebra I would think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would be cool is if you had a lot of squares and rectangles, so you could take away pieces or add on pieces as you wished.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="6220" LastEditDate="2016-02-04T21:44:21.063" LastActivityDate="2016-02-04T21:44:21.063" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10506" PostTypeId="2" ParentId="3933" CreationDate="2016-02-03T23:29:35.157" Score="1" Body="&lt;p&gt;I say you are not being too harsh on the pre-algebra students because showing the work is the key to success.  After all, for most students, algebra can be very confusing.  Those who don't show their work and get the right answers should probably be in the Algebra class, a class more advanced where it appears they already understand basic manipulations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Be careful on assuming that just because a student in Algebra gets the correct answer that they know &lt;em&gt;how&lt;/em&gt; to get it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience as a student, I knew many students actually cheated their way through much of Algebra.  If asked how they got it, they would most likely be unable to answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, Algebra is where cheating in math usually starts.  Those who don't cheat in Algebra will either struggle, succeed, or lie somewhere in the middle.  Those students, however, probably won't cheat in mathematics in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just some food for thought.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2016-02-03T23:29:35.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10507" PostTypeId="2" ParentId="10504" CreationDate="2016-02-04T00:03:52.637" Score="5" Body="&lt;p&gt;Well, start by introducing your topic (what is your topic, as in a description).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then go on about things your reader needs to know before they can study the topic about to be explained.  Be sure to include enough information so that the targeted audience will be able to understand you.  If you are writing a book about Ramanujan summations, you don't need to explain how the basic algebra works because your audience is probably experienced with summation methods.  But do explain what a Ramanujan summation method &lt;em&gt;is&lt;/em&gt; and things the reader should understand before trying to understand the Ramanujan Summation method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, go on explaining &lt;strong&gt;why&lt;/strong&gt; a Ramanujan Summation method works the way it does.  Go in depth as to the fundamental properties it has, such and such.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may want to sprinkle the writing with interesting things where relevant, such as $1+2+3+\dots=-1/12$ (a Ramanujan summation).  State this before and/or after introducing a topic.  For example, state the cool math thing (above) before explaining to get your reader interested in the summation of divergent series.  Then, revisit the question after the reader has a thorough understanding of the methods.  Or, use this to guide the reader through the process of summating a divergent series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Be sure to use common symbols and similars, if you are unclear as to write $a^b$ or $exp_a(b)$, ask &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;here,&lt;/a&gt; another stack exchange site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember to explain what $a^b$ means if you feel your reader doesn't understand it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;CHOOSE ONE TARGET AUDIENCE AND STICK WITH IT!&lt;/strong&gt;  It does &lt;strong&gt;not&lt;/strong&gt; please a well educated mathematician to skip the first $50$ pages to get to the content they want to read.  On the other hand, it does not please a reader to read the first page and understand nothing.  Use the table of contents or simply write multiple books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;It's probably better to explain more and be long-winded than to be short and unexplanatory.&lt;/em&gt;  The whole purpose of the book is to teach, so remember to teach your reader through every &quot;step.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's about all I can think of for now.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2016-02-04T00:03:52.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10508" PostTypeId="2" ParentId="2382" CreationDate="2016-02-04T00:53:20.273" Score="1" Body="&lt;p&gt;Well, many students simply won't &lt;em&gt;want&lt;/em&gt; to learn more math during the summer, the chance that students will be more distracted is higher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have to consider &lt;strong&gt;who&lt;/strong&gt; it will be beneficial to before considering &lt;strong&gt;if&lt;/strong&gt; it will be beneficial.  A student who needs the extra practice for memory may want to use the books.  A student who is failing or struggling may also want to use the books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just throw the books at them and expect them to learn?  Of course not!  I personally recommend doing math-related activities, possibly show more applications of math.  You may find many &lt;a href=&quot;https://math.stackexchange.com/questions/tagged/applications&quot;&gt;applications of math here.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, as a student, books?  +Math?  During summer?  You can see what I'm thinking, can't you?  A student doesn't like to study through books, most students don't &lt;strong&gt;want&lt;/strong&gt; to study math, and studying during the summer is generally a child's nightmare!  Make it &lt;strong&gt;fun&lt;/strong&gt;, that's the key.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-02-04T00:53:20.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10509" PostTypeId="2" ParentId="10504" CreationDate="2016-02-04T01:01:37.080" Score="4" Body="&lt;p&gt;A tangential half-answer: if the goal is to teach mathematics, rather than to prepare people to cope with school-mathematics... at the very least, &lt;em&gt;avoid&lt;/em&gt; all the things that are mandatory in mathematics as a school subject. The meta-requirements are especially invidious to actual mathematics. On the other hand, if the goal is exactly to prepare students to cope with hurdles, filters, obstacles in their lives posed in the form of mathematics, that's a very different thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me a sad irony that the attitudes of mind that optimize understand of mathematics, or of any serious enterprise, are often opposite to the traits rewarded by routine school situations, where compliance, obedience, are &quot;important&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is importantly true that exceptional preparation in genuine mathematics makes possible &quot;high testing&quot; even in conventional school-bound situations, at least for people who can cope with the cognitive dissonance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among other things, one should &lt;em&gt;not&lt;/em&gt; teach people to distrust their own (well-earned) physical intuition, which enables them to walk and chew gum at the same time, but only to recommend &lt;em&gt;refinements&lt;/em&gt;. It is certainly not the case that mathematics is anti-intuitive, but, at &quot;worst/most-provocative&quot;, subtler than anticipated. So we &lt;em&gt;refine&lt;/em&gt; our intuition, and now we can walk, chew gum, and integrate discontinuous functions at the same time.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-02-04T01:01:37.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10510" PostTypeId="1" AcceptedAnswerId="10511" CreationDate="2016-02-04T01:05:20.967" Score="14" ViewCount="6465" Body="&lt;p&gt;At first glance, I thought Khan academy was a good site for teaching people math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After trying it out, I quickly found that the topics were very limited.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So is Khan academy actually as good as I hear some claim to me?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you decide to take one or the other side of the argument, please explain why you hold your beliefs.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2021-03-05T15:27:50.423" Title="Is Khan academy really good?" Tags="|software|online-instruction|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="10511" PostTypeId="2" ParentId="10510" CreationDate="2016-02-04T03:28:55.393" Score="23" Body="&lt;p&gt;I taught gifted elementary students for over 25 years and feel my 4th-6th grade students benefited greatly from using Khan Academy with some direction from me. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first time I used Khan Academy was with my sixth graders who were having trouble understanding why the distributive property was true.  I showed  them the Khan Academy's video on the topic so they could get a different perspective.  The video really clicked with them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that in general the videos explain concepts, but too often students just do the exercises and look at hints, resulting in much less conceptual understanding.  I didn't have this problem with my students because Khan Academy supplemented what I taught and my students were often interested in the videos as a way to supplement understanding.  In other classrooms, I witnessed students ignoring the videos going straight to the exercises and trying to learn from exercises and hints. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My gifted students were often perfectionists who couldn't also admit that they made mistakes. Careless mistakes often were dismissed by students who said, &quot;I understand it, but I was just careless.&quot;  Having to do 5 exercises in a row that were correct, with the computer grading them, was a way to insure that they really mastered accuracy and facts which are so important at the elementary and middle school level. Having to come back the next day to get mastery badges and points, motivated them to keep plugging away at skills they might not have practiced otherwise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gifted students who were ahead of the class were given chromebooks to explore new topics and practice old ones. This made running the class so much easier.  I was also able to give everyone a chromebook and differentiate among the students. Everyone was doing the topics I recommended for them on Khan Academy.  One advantage of Khan Academy for the teacher is that you can recommend topics to your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students were also happy to get Khan Academy for homework.  This is homework that I didn't have to grade, because the computer did it for me.  I could check on the students and see how much time they spent at home on the topics that I assigned and how much trouble they had with the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, Khan Academy has wonderful tools for the teacher including recommending topics, tracking student time and progress, and built in grading.  Elementary students can develop accuracy, become motivated, gain understanding, and build confidence.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also enjoyed Khan Academy and spent countless hours exploring different topics.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-02-04T03:28:55.393" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10512" PostTypeId="2" ParentId="3933" CreationDate="2016-02-04T03:38:21.023" Score="7" Body="&lt;p&gt;Short answer: &lt;strong&gt;No&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I'm of the opinion that the common phrase &quot;show your work&quot; is really a misstatement of what we're aiming for. The point is neither to just evidence sweat, toil, and tears, nor purely as proof of non-cheating. Rather, the point of written mathematics is to &lt;em&gt;prove, explain and convince other human beings of something&lt;/em&gt;. As Benjamin Pierce put it, a keystone idea that we should be making clear to our students:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematics is the science that draws necessary conclusions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, we should not be developing students to be in the habit of pronouncing some answer by fiat, that might as well be delivered by ESP or divine providence, and expect that it will be accepted or used by colleagues or fellow citizens. (Nor expect that they can deal with skeptics by means of mere rhetoric, intimidation, or call to authority.) Rather, we should be sharing the scientific method with our students, that they can convince others around them by means of &lt;em&gt;clear written evidence and logic&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To my mind, that is the real point of writing more than the final answer; the student should be writing a series of logical statements, in the grammar of mathematical statements, that will &lt;em&gt;convince&lt;/em&gt; a reader that their answer &lt;em&gt;must&lt;/em&gt; be correct. And if wrong: this process of writing allows the interlocutor an opportunity to participate, clarify, refine, and possibly improve our thinking. Many or most students fail to see that mathematical equations are actually assertive statements, directly equivalent to natural-language sentences (but more concise). And yet, to my mind, this is &lt;em&gt;far more important than the answer to any particular question&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After lengthy consideration, the direction on all of my tests doesn't say &quot;show your work&quot;, rather it says, &lt;strong&gt;&quot;justify your work with well-written math&quot;&lt;/strong&gt;, and of course I grade on that (uniformly regardless of level: from remedial arithmetic to college algebra and statistics). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Never mind pre-algebra, it's even more important to assess writing at the level of algebra and above. Frankly, I'm of the opinion that the only reason this is being eliminated from classrooms is some combination of the instructors being lazy, unaware, or the institutions engaging in fraud to &quot;fake&quot; students through math courses. The end result is a disastrous misunderstanding of math, and schools effectively giving up on students being able to even hope that they can master it. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-02-04T03:50:00.767" LastActivityDate="2016-02-04T03:50:00.767" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10513" PostTypeId="2" ParentId="10425" CreationDate="2016-02-04T07:48:27.753" Score="10" Body="&lt;p&gt;To expand on Gerald Edgar's comments above:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume that you have already covered partial fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Develop a general strategy for integrals of powers of trig functions:  If a power is odd, you can use pythagorean identity and a substitution.  If both powers are even, use half angle reduction formulas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\begin{align*}&#xA;\int\sin^{2k+1}\cos^m(x) dx &amp;amp;= \int (1-\cos^2(x))^k \cos^m(x) \sin(x) dx\\&#xA;&amp;amp;=\int -(1-u^2)^k u^m du&#xA;\end{align*}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where we made the substitution $u = \cos(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with some positive integer examples.  Here, you just get a polynomial, so you can integrate by expanding and integrating termwise. Then observe the same strategy works for all integers, you just get rational functions of $u$ if an exponent is negative, which might have to be handled using partial fraction decomposition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of this general strategy is integrating secant!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\begin{align*}&#xA;\int \sec(x) dx &amp;amp;= \int \cos(x)^{-1} dx \textrm{ This is an odd power of cosine}\\ &#xA;&amp;amp;= \int \cos(x)^{-2}\cos(x) dx\\&#xA;&amp;amp;=\int (1-\sin^2(x))^{-1}\cos(x) dx\\&#xA;&amp;amp;=\int \frac{1}{1-u^2} du\\&#xA;&amp;amp;=\frac{1}{2}\int \frac{1}{1-u}+\frac{1}{1+u} du\\&#xA;&amp;amp;=\frac{1}{2}\left( \ln(1+u) - \ln(1-u)\right)+C\\&#xA;&amp;amp;=\frac{1}{2}\ln(\frac{1+\sin(x)}{1-\sin(x)})+C\\&#xA;&amp;amp;=\frac{1}{2}\ln(\frac{(1+\sin(x))^2}{\cos^2(x)})+C\\&#xA;&amp;amp;=\ln\left| \frac{1+\sin(x)}{\cos(x)} \right|+C\\&#xA;&amp;amp;=\ln\left| \sec(x) +\tan(x)\right|+C&#xA;\end{align*}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So now the integral of secant fits into a more general story:  it is an example of the general strategy we have for integrating any function of the form $\sin^n(x)\cos^m(x)$&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2016-02-04T07:57:17.093" LastActivityDate="2016-02-04T07:57:17.093" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10514" PostTypeId="2" ParentId="3933" CreationDate="2016-02-04T12:52:31.673" Score="1" Body="&lt;p&gt;I think it's up to the teacher to decide which students are actually capable of doing the math they're doing by their own work, if they're leaving out steps. I say this because when I was in high school I hated the idea of &quot;showing all my work&quot; for tedious steps that I could do in my head. Most of my teachers understood early on that I was capable of actually doing this, and not just copying my answers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I really appreciated that but looking back I can see why some teachers want every student to show work. They just want to be sure students aren't cheating, and that their students are learning to go step by step in their thought processes. But it's part of the teacher's job to get to know their students, and figure out who's actually capable of what, because sometimes the extra writing just feels like a waste to a student who's not challenged.&lt;/p&gt;&#xA;" OwnerUserId="5958" LastActivityDate="2016-02-04T12:52:31.673" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10515" PostTypeId="2" ParentId="10425" CreationDate="2016-02-04T17:34:55.453" Score="5" Body="&lt;p&gt;&lt;strong&gt;I&lt;/strong&gt;. The history of the integral of the secant and the loxodrome is an excellent approach to this difficult, and historically difficult &amp;amp; important, integral.  See Rickey and Tuchinsky, An Application of Geography to Mathematics: History of the Integral of the Secant, &lt;em&gt;Mathematics Magazine&lt;/em&gt;&#xA;Vol. 53, No. 3 (May, 1980), pp. 162-166, &lt;a href=&quot;http://www.jstor.org/stable/2690106&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jstor.org/stable/2690106&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One lesson for students is that when a difficult problem is solved, the method of solution can be shared with others, making things easier for future generations, &lt;em&gt;including them.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;II&lt;/strong&gt;.  Once I accidentally stumbled upon the following. I was looking for an example of an integral to differentiate (with respect to the upper limit).  I wanted something not too complicated but one that a Calculus I student couldn't antidifferentiate.  And then I wanted to put a function in the upper limit and apply the chain rule.  I thought it would be nice if it simplified, too.  So I hit upon&#xA;$$\int_0^x {1 \over 1-t^2} \; dt \quad \hbox{and} \quad \int_0^{\sin x} {1 \over 1 - t^2} \; dt \,.$$&#xA;And of course the last one simplifies to $\sec x$ (and is rather similar to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10425/whats-the-most-effective-way-to-introduce-motivate-the-anti-derivative-of-sec#comment22781_10425&quot;&gt;Michael Joyce's comment&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/a/10513&quot;&gt;one of Steven Gubkin's answers&lt;/a&gt;).&#xA;This occurs shortly before we get to substitution and therefore before the students know the antiderivative of $\sec x$.  So I could point to the integral, and say, &quot;Hey, this is an antiderivative of secant? Do we know a function that is antiderivative of secant? That's interesting.  I wonder what that integral is.  I guess we'll have to wait and see.&quot;  It is motivating in that it piques their curiosity.  Even if we don't get to partial fractions in Calc. I or prepare them for the Calc. I method, it gets the students to want to know the answer.&#xA;It's more effective if most of your students haven't already had AP calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;III&lt;/strong&gt;.  Interestingly, according to Rickey and Tuchinsky, Isaac Barrow came up with essentially the same method as Joyce/Gubkin and was the first use of partial fractions in integration:&#xA;$$\int {dx \over \cos x} = &#xA; \int {\cos x \; dx \over \cos^2 x} = \cdots = &#xA; {1\over 2} \int {\cos x \over 1 - \sin x} + {\cos x \over 1 + \sin x} \; dx = &#xA; \text{etc.}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;IV&lt;/strong&gt;.  A geometric approach.  Let $s = \sec\theta$, $t = \tan\theta$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/st5mx.png&quot; alt=&quot;Mathematica graphics&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then from the differential triangle and the similar $1$-$s$-$t$ triangle, we have&#xA;$$&#xA;{s \; d\theta \over 1} = {dt \over s} = {ds \over t} \,.&#xA;$$&#xA;Okay, maybe it is easier to deduce the proportion from the derivative rules,&#xA;given the current conditions in math. ed.&#xA;Nevertheless, the symmetry of the last two terms of the proportion suggests combining them. Hence we get&#xA;$$&#xA;{s \; d\theta \over 1} = {dt + ds \over s + t} = {d(s+t) \over s+t} = d(\ln |u|)\,,&#xA;$$&#xA;where $u = \sec\theta + \tan\theta$.  Integrating gives the result.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-02-05T03:35:02.023" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10516" PostTypeId="2" ParentId="10471" CreationDate="2016-02-04T18:30:43.257" Score="5" Body="&lt;p&gt;There is a great &lt;a href=&quot;https://www.youtube.com/watch?v=Mv21qXyDZMc&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;game about polynomials&lt;/strong&gt;&lt;/a&gt; presented by Rachel Kaplove on the eHow YouTube channel. The rules are very simple:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Each student gets a card with either expanded polynomial, for example $x^2-81$, or one of its factors, for example $x-9$ and $x+9$.&lt;/li&gt;&#xA;&lt;li&gt;If a student has a polynomial, he/she needs to find classmates with the right factors.&lt;/li&gt;&#xA;&lt;li&gt;If a student has a factor, he/she needs to find a classmate with the right polynomial.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You don't need computers for this game and students should work in groups to achieve the final goal. But, of course, students need to know the factorization of polynomials for this activity.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Regarding the &lt;strong&gt;lack of motivation&lt;/strong&gt;, I would consider three reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Students don't find algebra useful because the tasks are not related to real-world problems.&lt;/li&gt;&#xA;&lt;li&gt;Students aren't excited about algebra because the tasks lack the gaming aspect.&lt;/li&gt;&#xA;&lt;li&gt;Students aren't good with algebra because they don't understand its basic principles.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These issues can be solved with the following &lt;strong&gt;approaches&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Polynomials in economics (coming a nice example from agriculture!)&lt;/li&gt;&#xA;&lt;li&gt;Playing with maths using computers (there is no excuse not to use them)&lt;/li&gt;&#xA;&lt;li&gt;Didactic vs Project-based learning (explaining the concept of powers)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Now let's cover each point in detail:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Polynomials are extremely important in economics to model economical behaviors. Since the author of the question teaches in the agriculture-oriented school, I found the book &lt;a href=&quot;http://www.uky.edu/~deberti/agrod5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Agricultural economics&quot;&lt;/a&gt; and the word &quot;polynomial&quot; was mentioned there 21 times! For example, a short-run total cost function of modern production technologies can be realistically described by a third-degree polynomial like this: $TC = 2q^3 - 15q^2 + 50q +50$ (taken from &lt;a href=&quot;https://books.google.be/books/about/Maths_for_Economics.html?id=XXmcAQAAQBAJ&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Maths for Economics (3rd edition)&quot;&lt;/a&gt; by Geoff Renshaw, p. 141). The graph of this function shows the most efficient output range (green) where the total cost $TC$ doesn't vary much with the output $q$ as well as the least efficient output range (red) where output increase would cause significant production cost increase.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iXQSE.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iXQSE.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Instead of doing boring calculations by hand, let the students play with maths using computers and programming. Check this inspiring &lt;a href=&quot;http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers?language=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;TED talk&lt;/a&gt; by Conrad Wolfram about Computer-Based Maths! There are no excuses not to use computers for two reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Software for mathematical programming is cloud-based and free (for example, &lt;a href=&quot;https://www.wolfram.com/programming-lab/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wolfram Programming Lab&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;li&gt;Computers are dirt cheap nowadays (for example, &lt;a href=&quot;https://www.google.com/chromebook/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chromebook&lt;/a&gt; for \$150 or &lt;a href=&quot;https://www.raspberrypi.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Raspberry Pi&lt;/a&gt; for \$25).&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maths is not fun when you don't know how to use it. Explain basic concepts to students (didactic learning) and then let them explore the topic themselves so they could really understand it (project-based learning). Let's take, for example, the concept of powers. There is some simple code for the Wolfram Programming Lab that can be executed to explain the basics:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;3*3&#xA;3*3*3&#xA;(* Now, how to calculate 3 multiplied by itself 100 times? *)&#xA;3^100&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;After students understand the definition of power, let them explore the following questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How much is &lt;code&gt;3^0&lt;/code&gt;? What about &lt;code&gt;2^0&lt;/code&gt;? Why are both expressions equal to &lt;code&gt;1&lt;/code&gt;? Is it valid for any number &lt;code&gt;x^0&lt;/code&gt;? If you want them to tackle really an open question, ask about &lt;code&gt;0^0&lt;/code&gt;.&lt;/li&gt;&#xA;&lt;li&gt;How much is the number &quot;googol&quot;? Where can you find it in daily life?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Ideally, students should work in groups of 3-4 people using one computer per group. After their research on the above questions, each group should present their results to the whole class. This method is called &lt;a href=&quot;https://www.theschoolinthecloud.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Self-Organized Learning Environment (SOLE)&lt;/a&gt; invented by Sugata Mitra.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;That's a lot of stuff for one answer but I hope that some of the ideas will help you. And don't forget to accept my post as an answer if you liked it ;) &lt;/p&gt;&#xA;" OwnerUserId="6306" LastEditorUserId="6306" LastEditDate="2016-02-06T21:08:39.390" LastActivityDate="2016-02-06T21:08:39.390" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10517" PostTypeId="1" CreationDate="2016-02-04T18:53:08.817" Score="4" ViewCount="191" Body="&lt;p&gt;I'm going to be teaching measurement to 4th grade students and I want something a little more hands on. I believe that they would enjoy that more. So any helpful ideas or things you think would work will be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="6317" LastEditorUserId="262" LastEditDate="2016-02-10T22:16:33.240" LastActivityDate="2016-02-10T22:16:33.240" Title="Teaching measurement to 4th grade students by projects?" Tags="|technology-in-education|primary-education|interactive-teaching|teacher-preparation|teachers|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10518" PostTypeId="2" ParentId="10501" CreationDate="2016-02-04T19:50:48.380" Score="4" Body="&lt;p&gt;In my department, homework graders are hired for a specific number of hours.  I usually assign more homework than the grader can grade in the allotted time, so I give the grader a prioritized list of problems, from most important to least important.  I tell the grader to begin grading with the most important problems and work down the list as far as his/her time allows.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In many (though not all) of my classes, I give relatively little weight to homework grades (compared to exams).  For these classes, I ask the graders to grade on the basis of two points per problem (rather than the ten points that many people are used to).  That way, the grader doesn't need to worry about detailed assignment of partial credit.  &lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2016-02-04T19:50:48.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10519" PostTypeId="2" ParentId="3933" CreationDate="2016-02-05T18:36:54.997" Score="0" Body="&lt;p&gt;A student should follow the directions given. On a standardized test, there are multiple choices (usually) and a student has the ability to beat the test while not necessarily knowing how to solve the problem the &quot;correct&quot; way. If the instructions for the exam or homework say &quot;show all work,&quot; the unspoken implication is that it's one step shy of a two column proof. That may seem harsh from the student's perspective, but with the advent of the TI84 type of calculators, there's a lot that be plugged in, and the answer appears. Solutions to a quadratic equation? The calculator will offer the roots. Should the teacher simply accept &quot;$x=4, x=8$&quot; with no work? During this part of the algebra curriculum, for example, there are multiple ways to solve, and teachers can expect students to learn each way and solve by the method requested. Spell out the quadratic equation, work it out, and give the solutions, but if the test section said &quot;solve by factoring,&quot; any or all points are lost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yesterday, I got back an exam I proctored, and one question was&#xA;$\log_3 5 = x$. The answer was there, $1.46$, to $2$ digits as requested, but no work. I asked &quot;did you do this on scrap paper?&quot; hoping to just staple it to the test. He said he guessed and checked. Knowing that $\sqrt{3} \approx 1.73$, and so $3^{1.5} \approx 5.20$, he just hacked down to $1.46$. Clever, but he forgot how to manipulate logs. If I were to grade this, I'd have to ask how he showed any knowledge of logs, even though his gut was correct for exponents.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-02-05T21:26:25.660" LastActivityDate="2016-02-05T21:26:25.660" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10520" PostTypeId="1" CreationDate="2016-02-05T20:39:24.743" Score="19" ViewCount="248" Body="&lt;p&gt;Has there been any research into comparing the effectiveness of using &quot;structured proofs&quot; or &quot;outlined proofs&quot; in higher level mathematics education, compared to traditional &quot;prose&quot; proofs?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the purpose of this question:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A &quot;structured proof&quot; is something akin to what Lamport described in &lt;a href=&quot;http://research.microsoft.com/en-us/um/people/lamport/pubs/proof.pdf&quot;&gt;How to write a 21st century proof&lt;/a&gt;. I don't necessarily want something strictly as what he defines, but you should keep in mind a proof format where a proof is broken down to smaller steps, each step individually justified, with the &lt;em&gt;whole proof presented in a way that the hierarchical nature of the statements is made clear&lt;/em&gt;. (Traditional proofs often do invoke lemmata and propositions, but the presentation is not as organized.)&lt;/li&gt;&#xA;&lt;li&gt;An &quot;outlined proof&quot; is a proof given in several iterations. At the first iteration a sketch of the main proof is given. At the second the sketch of the proofs of the various steps using in the main proof is given. And so on. So if you think of a proof sort of as a tree-like data structure, a structured proof presents it &quot;depth first&quot; while a outlined proof presents it &quot;breadth first&quot;. &lt;/li&gt;&#xA;&lt;li&gt;A &quot;prose proof&quot; is what we see in most textbooks or journal articles. &lt;/li&gt;&#xA;&lt;li&gt;Higher level mathematics education I refer to &quot;Intro to proofs&quot; and beyond in a US system. This would be proof-based courses that often leads to the completion of an undergraduate degree in mathematics. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Motivation: anecdotally I have observed that students often struggle when doing proof writing because they have no grasp of the &quot;big picture&quot;. This either manifests in a written proof that gets derailed (the student losing track of what he or she is trying to prove) or is circular (the student losing track of where they are in the proof). The students often cannot concisely explain what they have learned from reading a proof. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of this I think can be safely blamed on the non-transparent nature of traditional proof writing and reading. If you give me a random sample of three consecutive sentences from a published mathematical proof in a journal, out of context, most likely I will not be able to tell you whether there are any logical relation between the three of them (is one justifying another? Are they three independent conditions?) One may imagine making the proofs available to students in a more structured format, and making the students practice writing some proofs in the same format, can make them better at the mathematical craft. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have people seen anecdotal/empirical justification for this? Are there published research concerning this aspect of mathematical teaching?&lt;/p&gt;&#xA;" OwnerUserId="125" LastActivityDate="2016-02-05T20:39:24.743" Title="Research on the use of outlined / structured proofs in instruction" Tags="|mathematical-pedagogy|reference-request|proofs|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10521" PostTypeId="1" CreationDate="2016-02-06T12:20:11.657" Score="13" ViewCount="4055" Body="&lt;p&gt;In most high schools (in America), I think it is safe to say that the highest math subject offered is calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But why is it calculus rather than number theory or some other branch of mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While calculus has applications, I'm sure other branches of math have applications just as well.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorDisplayName="user507" LastEditDate="2016-02-06T16:26:08.697" LastActivityDate="2017-11-29T22:53:32.870" Title="Why do we teach calculus in high school rather than a different math course?" Tags="|secondary-education|calculus|curriculum|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10522" PostTypeId="2" ParentId="10521" CreationDate="2016-02-06T18:12:12.283" Score="4" Body="&lt;p&gt;I think calculus is just seen as the most relevant, and not for bad reason. The majority of students who are going to take calculus are engineering students. These students need ideas from calculus more than they need to know about something like axiomatic set theory (although linear algebra would be a good class to offer in addition to calculus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, engineers need to calculate the occasional integral. Although they can do this with a computer nowadays, they should still know what an integral actually is. They're never going to need to know how to solve a Diophantine equation on the spot&lt;/p&gt;&#xA;" OwnerUserId="5958" LastActivityDate="2016-02-06T18:12:12.283" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10523" PostTypeId="2" ParentId="10482" CreationDate="2016-02-06T21:52:28.790" Score="7" Body="&lt;p&gt;I guess I'm not going to get get quite a small list as I'd hoped for, but I'm falling in love with the family of functions $f_{p,n}(x) = p(x)e^{-1/x}/x^{2n}$, for $p$ a polynomial and $x&amp;gt;0$, and $f(x) = 0$ otherwise. The case $n=0$ and $p(x)=1$ is the classic example of a smooth but nonanalytic function, and all its derivatives fall in this family. One can ask all sorts of questions about a function from this family (and, I guess, one could replace $p$ by any function that has all derivatives at $0$). Considering even just $f_{1,0}$, it's bounded but has no maximum; has an inflection point, even though $e^x$ doesn't; one can talk about continuity at 0, differentiability at 0 and continuity of the differential; one can consider its Taylor expansions around positive $a$, and at zero, and discuss radii of convergence, and so on. This function is important, looking at higher levels of analysis and geometry, for constructing bump functions, limits of which are indicator functions of intervals, for constructing partitions of unity etc.&lt;/p&gt;&#xA;" OwnerUserId="1426" LastActivityDate="2016-02-06T21:52:28.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10524" PostTypeId="1" AcceptedAnswerId="10525" CreationDate="2016-02-07T00:20:38.057" Score="7" ViewCount="248" Body="&lt;p&gt;Is there any data available on the majors of students taking freshman calculus, including information on changes over time? In the past I think engineering students were probably a hefty majority, but I suspect that that has changed quite a bit within the last 20 years, at least here in California. Circa 1996, the University of California system started requiring its biology majors to take calculus and a new flavor of calculus-based physics (a 1-year survey with calculus as a corequisite). If the population has changed a lot, then one question that could be raised is whether the traditional structure of a one-year calculus course is appropriate for the new population.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2016-02-07T01:26:31.643" Title="Population of students taking freshman calculus" Tags="|calculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10525" PostTypeId="2" ParentId="10524" CreationDate="2016-02-07T00:42:09.003" Score="7" Body="&lt;p&gt;&lt;a href=&quot;http://www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf&quot;&gt;This&lt;/a&gt; should have all the information you need (demographic data as well as possible interventions, etc.)  The linked-to file is the summary report of a multi-year multi-institutional study to understand the way calculus is taught in colleges and universities in the US. On page 11 (and coincidentally in Table 11) you will find that the 31% folks taking calculus want to be engineers, 30% want to be in the biological/life sciences and the other (ten or so) disciplines come in at the single digits (2% are math majors, 8% are business majors, etc).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answers the demographic question, I hope.  Your general question about the traditional one year calculus structure is addressed repeatedly throughout the linked-to file but maybe not with particular majors in mind.  They make the argument that the challenges Calculus-teaching in the US faces are complicated and are best answered locally (perhaps) through site-dependent methods of placement, student support, curriculum and pedagogy. &lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="1537" LastEditDate="2016-02-07T01:26:31.643" LastActivityDate="2016-02-07T01:26:31.643" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10527" PostTypeId="2" ParentId="10234" CreationDate="2016-02-07T20:26:29.620" Score="4" Body="&lt;p&gt;For teaching perseverance, I would recommend works of Barbara Oakley.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She doesn't advocate working on a problem 24/7 but rather using smart perseverance techniques when you switch between &lt;em&gt;two modes of your brain&lt;/em&gt;: focused and diffuse. The first mode allows you to stick to your methodology while the second one - lets you explore other methodologies when the current one doesn't work or not efficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She also explains the science behind the worst enemy of perseverance - &lt;em&gt;procrastination&lt;/em&gt;, and gives tips and tricks for preventing it, for example the Pomodoro technique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The last but not least in her list of efficient learning techniques is &lt;em&gt;sleep&lt;/em&gt; - something that we often neglect in our struggles to finish our tasks no matter what.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below you can find online resources about her methods and much more:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For a quick introduction, check &lt;a href=&quot;https://www.youtube.com/watch?v=O96fE1E-rf8&quot; rel=&quot;nofollow&quot;&gt;Barbara's TED talk&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;For additional details, watch her &lt;a href=&quot;https://www.youtube.com/watch?v=vd2dtkMINIw&quot; rel=&quot;nofollow&quot;&gt;Talk at Google&lt;/a&gt; or read her bestselling book &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B00G3L19ZU&quot; rel=&quot;nofollow&quot;&gt;&quot;A Mind For Numbers&quot;&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Finally, her famous MOOC &lt;a href=&quot;https://www.coursera.org/learn/learning-how-to-learn&quot; rel=&quot;nofollow&quot;&gt;&quot;Learning How to Learn&quot;&lt;/a&gt; is available for free on Coursera.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6306" LastActivityDate="2016-02-07T20:26:29.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10528" PostTypeId="2" ParentId="10482" CreationDate="2016-02-08T10:15:24.007" Score="5" Body="&lt;p&gt;A classical example: the functions $f_k$ defined by $f_k(0)=0$ and $f_k(x)=x^k\sin(1/x)$ for non-zero $x$, with $k=1,2,3$. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Its continuity is easy but not of the &quot;as sum and product and composition of continuous functions&quot; kind, and it is unbounded but has no (even infinite) limit in $\pm\infty$,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;with $k=2$ we get a function that is derivable everywhere but not $C^1$; one can check that the derivative satisfies the conclusion of the intermediate value theorem (as any derivative must!) even if it is not continuous (note that I vaguely remember being told that at some point in history the IVT was taken as the &lt;em&gt;definition&lt;/em&gt; of continuity),&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;with $k=3$ it is the classical example to illustrate that a continuous function which is $C^1$ outside a point, with derivative converging at that point must be $C^1$; it also yields an example of function having a second-order Taylor expansion without being twice differentiable.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I guess some other stuff can be further illustrated with those, but it feels already good for three functions (four if you include $x\mapsto x\cos(1/x)$ which shows up in the derivative of $f_3$).&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2018-05-11T21:00:53.190" LastActivityDate="2018-05-11T21:00:53.190" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10529" PostTypeId="1" CreationDate="2016-02-08T15:27:39.597" Score="7" ViewCount="227" Body="&lt;p&gt;[I am not an mathematics educator; but because the process of learning is educating yourself, I'm posting it here]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Visual Complex Analysis's preface, the author gives an analogy with pseudo-deaf musicians and follows the same to mathematics. Mathematics today, he argues, is mostly build on abstract symbolic manipulation rather than much more stronger visual (intuition) method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it a good notion to rely on visualization and intuition solely on graduate and postgraduate mathematics on topics like Analysis, Abstract Algebra and other fields (Topology and differential geometry is pretty obvious - it is probably visual per se) ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If yes, then how to visualize and develop intuition in graduate and postgraduate mathematics in fields mostly relying on (stupid; ?) symbolic manipulation ?&lt;/p&gt;&#xA;" OwnerDisplayName="user6330" LastEditorUserId="376" LastEditDate="2016-02-08T15:57:53.547" LastActivityDate="2016-02-09T01:27:54.373" Title="The role of visualization and intuition in graduate and postgraduate math and developing it" Tags="|graduate-education|intuition|visual-proof|" AnswerCount="1" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10531" PostTypeId="1" AcceptedAnswerId="10533" CreationDate="2016-02-08T21:57:49.700" Score="15" ViewCount="1259" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Euclidean_algorithm&quot; rel=&quot;noreferrer&quot;&gt;Euclid's algorithm&lt;/a&gt; is a way to find the greatest common divisor of two natural numbers $a$ and $b$.&#xA;In the usual version of the algorithm one tries to find $p,q\in\mathbb N$ so that $a=pb+q$ and $0\leq q&amp;lt;b$ (assuming $a&amp;lt;b$).&#xA;Iterating this &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclidean_division&quot; rel=&quot;noreferrer&quot;&gt;Euclidean division&lt;/a&gt; until $q=0$ produces the GCD.&#xA;(I will not describe the algorithm here in more detail unless requested.)&#xA;I call this classical version a rigid version of the algorithm.&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a slight preference for choosing $p$ and $q$ freely without any definite rule.&#xA;As long as the equation $a=pb+q$ is satisfied and $|q|&amp;lt;|a|$, the algorithm works perfectly fine.&#xA;If the choice doesn't have to be optimal, one can make rough approximations and avoid doing any exact division.&#xA;Especially in the absence of calculators it is faster to divide roughly and take a few extra steps.&#xA;This version is somewhat non-deterministic, and I call it the flexible version.&#xA;I have given an example below&lt;sup&gt;2&lt;/sup&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;When teaching Euclid's algorithm, should the flexible version be included as well? Has anyone actually tried teaching the flexible version?&lt;/strong&gt;&#xA;Here are some thoughts on the subject:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The algorithm is (at least in Finland) introduced in highschool and then again in the university. The different school levels may call for different approaches.&lt;/li&gt;&#xA;&lt;li&gt;The flexible method can be confusing for some students, so the rigid version should probably be included anyway.&lt;/li&gt;&#xA;&lt;li&gt;The flexible method makes it (hopefully!) more transparent that the algorithm is all about using the identities $\gcd(a,b)=\gcd(a-pb,b)$ and $\gcd(a,b)=\gcd(|a|,|b|)$ to simplify the problem recursively. I don't really want to present the algorithm as a black box unless I need to. (Proving that the algorithm works doesn't mean that the students won't use it as a black box with no idea of what's going on.)&lt;/li&gt;&#xA;&lt;li&gt;The flexible method works with relatively simple mental arithmetic, as there is no need to get the integer part of the division exactly right.&lt;/li&gt;&#xA;&lt;li&gt;I recently followed a highschool class being introduced to the algorithm. The students became quite quickly fluent with the method but I doubt they had any idea what they were doing. Especially in highschools there is a temptation (and need due to tight schedules) to give the method without any justification for why it works. Giving the most advanced students the flexible version might open their eyes and help them &lt;em&gt;understand&lt;/em&gt; the algorithm.&lt;/li&gt;&#xA;&lt;li&gt;Flexible use of the algorithm (which I had to come up with myself) made me see what it's all about.&lt;/li&gt;&#xA;&lt;li&gt;The flexible version can be hard to use because you can make arbitrary choices. Some like more rigid instructions.&lt;/li&gt;&#xA;&lt;li&gt;I hate it when students resort to a calculator when they need to decide how many (integer) times 97 goes into 324. I fear that a too rigid algorithm turns their brain off, but the more flexible approach forces them to think.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; There are variations of the algorithm.&#xA;In one of them one choose $p$ and $q$ so that $|q|$ is minimal.&#xA;This is not what I'm after, and I would classify these variations as rigid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt; Here is a flexible way to find that $\gcd(7893,897)=3$:&#xA;\begin{align}&#xA;7893&amp;amp;=10\cdot897-1077\\&#xA;1077&amp;amp;=1\cdot897+180\\&#xA;897&amp;amp;=5\cdot180-3\\&#xA;180&amp;amp;=60\cdot3.&#xA;\end{align}&#xA;All calculations can be done mentally.&#xA;The standard rigid version is actually a bit longer:&#xA;\begin{align}&#xA;7893&amp;amp;=8\cdot897+717\\&#xA;897&amp;amp;=1\cdot717+180\\&#xA;717&amp;amp;=3\cdot180+177\\&#xA;180&amp;amp;=1\cdot177+3\\&#xA;177&amp;amp;=59\cdot3&#xA;.&#xA;\end{align}&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2016-02-09T14:36:39.593" LastActivityDate="2016-02-09T22:08:17.467" Title="Should Euclid's algorithm be taught as rigid or flexible?" Tags="|undergraduate-education|secondary-education|number-theory|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10532" PostTypeId="2" ParentId="10531" CreationDate="2016-02-08T22:50:33.117" Score="7" Body="&lt;p&gt;(&lt;strong&gt;Summary:&lt;/strong&gt; I would suggest exploring it flexibly, but ensuring students also know the &quot;rigid&quot; version.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than directly addressing Euclid's algorithm for the $\gcd$ of two whole numbers, I believe one witnesses similar phenomena when covering the standard algorithm for division. For example, I observe analogs with your remarks of:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The flexible method can be confusing for some students, so the rigid version should probably be included anyway.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The same could be said of the &lt;strong&gt;scaffolding approach&lt;/strong&gt; (sometimes called &lt;strong&gt;partial quotients&lt;/strong&gt; since there are &lt;strong&gt;partial sums, differences&lt;/strong&gt; and &lt;strong&gt;products&lt;/strong&gt; explored en route to the other operations' standard algorithms) on the way to covering the standard algorithm for whole number division.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The flexible method works with relatively simple mental arithmetic, as there is no need to get the integer part of the division exactly right.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is &lt;em&gt;also&lt;/em&gt; true of the &lt;strong&gt;scaffolding approach&lt;/strong&gt;: You don't need to &quot;guess&quot; exactly correctly.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Flexible use of the algorithm (which I had to come up with myself) made me see what it's all about.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'd argue that this sort of sense-making occurs through the &lt;strong&gt;scaffolding approach&lt;/strong&gt; as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case others are unfamiliar with what is meant throughout by the approach to long division that I have described here, I excerpt the following from:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Otto, A., Caldwell, J., Hancock, S. W., &amp;amp; Zbiek, R. M. (2011). &lt;em&gt;Developing Essential Understanding of Multiplication and Division for Teaching Mathematics in Grades 3-5&lt;/em&gt;. National Council of Teachers of Mathematics. Reston, VA. &lt;a href=&quot;http://eric.ed.gov/?id=ED521512&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;ERIC&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Strategy (a) below (from page 46) illustrates the &lt;strong&gt;scaffolding procedure&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wnXWp.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wnXWp.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal in this approach to whole number division is to explore strategy (a) in a way that &lt;a href=&quot;https://en.wikipedia.org/wiki/Instructional_scaffolding#Theory_of_scaffolding&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;scaffolds&lt;/strong&gt;&lt;/a&gt; towards the more formal approach in strategy (b). I maintain that this line of pedagogical reasoning from the elementary grades can be applied to your question about undergraduate mathematics, too!&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-02-08T22:50:33.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10533" PostTypeId="2" ParentId="10531" CreationDate="2016-02-08T23:16:04.743" Score="9" Body="&lt;p&gt;There is some value beyond the algorithm to insist on the fact that  quotient and remainder in an Euclidean division are uniquely determined as soon as  one settles on some convention on the remainder. What they are exactly depends on ones convention (non-negative remained, smallest absolute value, or still something else) but if one fixes a convention then they are unique. &#xA;I feel this is especially relevant in an algorithmic context, since to be aware of this is an issue when writing code or even just using a CAS. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, I would not give up on the &quot;rigid&quot; version completely, and as you imply for some students it is easier to use a completely deterministic algorithm rather than to have some flexibility. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;However,&lt;/em&gt; in the literature there are various versions of the Euclidean algorithm. One of them is the called the &quot;subtractive Euclidean algorithm&quot;; this version does not use division but just subtraction/addition, based on the identity you recalled that $\gcd(a,b)= \gcd (a-b,b)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it could make sense to discuss this version in addition, maybe even to start there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have this, you then can discuss your version as carrying out several steps of the subtractive version at once. Like, if we want to calculate $\gcd(234,21)$ we anticipate we will do subtraction by $21$ at least $10$ times so we do not write all these steps but go directly to $\gcd(24, 21 )$.&#xA;(And, if somebody  does not anticipate it, no harm is done either.) &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2016-02-08T23:16:04.743" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10534" PostTypeId="2" ParentId="10517" CreationDate="2016-02-08T23:32:43.753" Score="1" Body="&lt;p&gt;&quot;What type of things are measured? In What units? and &quot;Where do these units come from?&quot;   These are good jump off points for this age group. After that decide as a group (or small groups) the development of your own measurement system. The discussion of a standard will lead to 'problems'. As an example if a decision is to make a linear measurement system you will need a standard 'unit of measure' one that the teacher will hold as the 'gold standard' and accurate copies to be used in the 'field'.  Example: Brand x paperclip. Keep the standard and hand out a copy to each child/group to use. Decide on a name for the standard.  (say clip') Ask for dimensions of the desk/table in terms of clips. Ask for other measurements, small or large .. the desk...the room, the hallway, the length of your fingernail, hair.  etc.  The need will come apparent for multiples and divisions of units, sticks/rulers and decisions as to how many divisions/multiples and their names. Create tables of individual units  ie. x clips = 1 dip,  etc. These choices will affect the 'ease' of conversion between other systems later on in the project. Carry this to an appropriate endpoint.  Extend to liquid measurements (later to weights etc. and the obvious need need to make a unit that is not 'long' but can contain a liquid...think thimble etc.  Take it to appropriate end point but be sure to include conversions to and from your system to inches, cm. etc&lt;/p&gt;&#xA;" OwnerUserId="6333" LastActivityDate="2016-02-08T23:32:43.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10535" PostTypeId="2" ParentId="10529" CreationDate="2016-02-09T01:27:54.373" Score="4" Body="&lt;p&gt;First, I am a great fan of &lt;em&gt;Visual Complex Analysis&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, I disagree with the statement that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematics today [...] is mostly built on abstract symbolic manipulation rather than on [...] visual (intuition) &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Even the areas of mathematics that do not easily lend themselves&#xA;to visualization&amp;mdash;say, abstract algebra, or mathematical logic&amp;mdash;still rely on intuition, just not visual intuition. Someone adept at group theory&#xA;develops strong intuitions about group structures, strong enough to sense&#xA;whether a claim or conjecture might be true, whether a proof is sound, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am convinced by Needham that visual thinking can greatly enhance understanding of complex analysis. I am not convinced the same is true when &quot;complex analysis&quot; is replaced by other areas of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-02-09T01:27:54.373" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10537" PostTypeId="1" CreationDate="2016-02-09T06:37:01.800" Score="15" ViewCount="2713" Body="&lt;p&gt;In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also geometric constructions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;As a teacher&lt;/strong&gt;, to what extent is it important and useful for me to make &lt;strong&gt;my drawings exact&lt;/strong&gt; and precise? By this I mean: Draw circles with a compass, make right-angled triangles actually right-angled by using a suitable device when drawing them, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, suppose I am teaching how to calculate the area of a triangle. I want to demonstrate that I can choose any side to be the base and then determine the associated height of the triangle. The product of these (halved) is always the same.&#xA;Should I take the time to draw the triangle exactly, or is a less exactly drawn conceptual image (which clearly is a triangle, even though the sides are drawn with free hand) equally effective as a teaching tool?&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2016-04-10T04:51:48.153" LastActivityDate="2016-04-10T10:08:43.417" Title="Are precise drawings important in geometry?" Tags="|mathematical-pedagogy|secondary-education|geometry|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10538" PostTypeId="2" ParentId="10531" CreationDate="2016-02-09T22:08:17.467" Score="3" Body="&lt;p&gt;To guide understanding of the &lt;em&gt;why&lt;/em&gt; the Euclidean algorithm works on integers, I first discuss a &quot;subtractive&quot; version, based on the identity $\gcd(a, b) = \gcd(a  - b, b)$. The question of how many times to subtract $b$ to speed up the process suggests itself, leading more or less directly to what you call the &quot;rigid&quot; algorithm (even selecting the remainder with least absolute value). Your &quot;flexible&quot; approach isn't far off, but I've never considered that in class.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-02-09T22:08:17.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10539" PostTypeId="2" ParentId="10501" CreationDate="2016-02-09T22:49:16.717" Score="2" Body="&lt;p&gt;I write up detailed solutions to homework and exam questions, and (at least rough) breakdown of grade assignment. Solving the question helps convincing yourself that the problem is solvable within the allotted time, and helps keeping consistency when grading. Publishing solutions allows students to compare their work with a correct solution, and in theory learn from their mistakes (in practice, they only look if their grades are too low for passing, but there isn't much to be done about that).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-02-09T22:49:16.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10540" PostTypeId="1" CreationDate="2016-02-10T07:01:31.363" Score="6" ViewCount="352" Body="&lt;p&gt;Consider two possible lesson outlines:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Check homework.&lt;/li&gt;&#xA;&lt;li&gt;Show a visual demonstration for the area of a circle, e.g. &lt;a href=&quot;https://tube.geogebra.org/student/m279&quot; rel=&quot;nofollow&quot;&gt;https://tube.geogebra.org/student/m279&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Calculate the area of a circle as an example.&lt;/li&gt;&#xA;&lt;li&gt;Students do exercises.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;-&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Check homework.&lt;/li&gt;&#xA;&lt;li&gt;Calculate the area of a circle as an example.&lt;/li&gt;&#xA;&lt;li&gt;Students do exercises.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Also, at some point, write the/a formula for the area somewhere where the students can see it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose that due to various constraints these are the only two possibilities - we can't, for example, have the students do the demonstration, or have an interactive demonstration. (Possible constraints: planning time, lesson duration, coming up with interesting exercises for several courses is overwhelming for a new teacher, students don't have the equipment to use an online version, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Particular question&lt;/strong&gt;: Is step number 2 in the upper lesson plan useful enough to justify the reduced time that students have to complete exercises, during which the teacher gives personal feedback and helps at least some of them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;General question&lt;/strong&gt;: How useful is it to have the teacher demonstrate (at middle school level and above) or even prove (at, say, high school level and above) things, when the demonstrations are unlikely to be grasped by (all or most of) the students, given that the demonstrations have an opportunity cost: there's less time for exercises and personal guidance?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit: Thus far, the responses say that of course one should justify formulae, since doing that is exactly the essence of mathematics, and keeping unjustified formulae in mind is much harder (in the long run).&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Even if the area of a circle, let us say $A = \pi \cdot r \cdot r =\pi r^2$ (once powers are available), is taken as a black box, the exercises will include reasoning and inference. The simplest example is the case where only the diameter of a circle is known.&lt;/li&gt;&#xA;&lt;li&gt;The answers thus far do not consider the trade-off at all. Certainly there is a benefit to doing more exercises (especially the more difficult ones). A good answer should, in addition to arguing why spending more time on justifying the formulae is good, also argue why it is better then time spent on exercises.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There was also a suggestion to use less time on homework and more on the justification, but since that can be applied to both scenarios above, it does not really answer the question.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2016-02-12T07:04:00.210" LastActivityDate="2016-03-03T22:01:38.220" Title="Should one justify formulae in middle school?" Tags="|mathematical-pedagogy|secondary-education|proofs|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10541" PostTypeId="2" ParentId="10537" CreationDate="2016-02-10T08:28:08.637" Score="9" Body="&lt;p&gt;A purely personal opinion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your title asks about the importance of exact pictures in geometry. I would say &lt;em&gt;exact&lt;/em&gt; pictures are really not needed, because the point of geometry is to learn to reason from geometrical principles without needing to rely on precise measurements. Indeed, I would say that obviously spending time on accurate drawings would suggest that is an important part of what they are learning, which might be sending completely the wrong message.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I would say it is important to make the diagrams look plausible. The diagram does help to provide the intuition for the problem, and your students will need a lot of practice to relate the diagrams to intuition and the intuition to reasoning/calculation. So a right angle should be close enough to a right angle that when you look at the picture it is easy enough to remember that it is meant to represent a right angle. An acute angle should look acute, an obtuse one obtuse. The centre of a circle should be the right side of the axes, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Over time students should get better at relating the picture to the abstract mathematical setting, so accuracy will become less vital, at least for stronger students. For example, I would expect most students initially to need to treat acute and obtuse angles separately. Some will eventually become comfortable enough that they can see why the same argument would hold in both cases (or otherwise) despite the corresponding pictures being different.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-02-10T08:28:08.637" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10542" PostTypeId="2" ParentId="10537" CreationDate="2016-02-10T13:01:46.367" Score="5" Body="&lt;p&gt;I've gotten used to &quot;drawing not to scale&quot; and perhaps a bit too complacent in accepting this. Unless we are out to play tricks on students, a square should have its sides &quot;close enough&quot; so when it's stated that we have a square, there's no doubt. Don't offer a square image and declare beneath it that the rectangle is 8 x 4. The same for any other image. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll offer a painful recent example I saw.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hIP3z.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hIP3z.jpg&quot; alt=&quot;bad drawing&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For members who have forgotten their high school geometry, a tangent line (x) forms a right angle with a circle's radius at the point of intersection. i.e. points A and B should occur closer to 2 o'clock on these circles. And angles A and B are both right angles. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure there's a happy medium, where a criticism is uncalled for. But in this age of free drawing software, the effort to be precise should be practiced. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-02-10T13:01:46.367" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10543" PostTypeId="2" ParentId="10540" CreationDate="2016-02-10T18:38:46.077" Score="9" Body="&lt;p&gt;Yes, of course. As Benjamin Pierce famously put it: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematics is the science that draws necessary conclusions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you don't justify your reasoning, then &lt;em&gt;you're not really doing math at all&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or as I put: then you're doing &quot;faith-based mathematics&quot;, which is a &lt;em&gt;much harder&lt;/em&gt; task. Most people can't remember a bunch of raw abstract formulas, and those who attempt to proceed in that way inevitably crash and burn at some later point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If some prioritization must be made, then I would more highly recommend discarding item #1, checking homework in class. Either offload this to out-of-class, or prompt if students had specific inquiries on any items. Require justifications from students on all work to get them in the habit of thinking mathematically.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-02-10T18:45:31.647" LastActivityDate="2016-02-10T18:45:31.647" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10544" PostTypeId="2" ParentId="10540" CreationDate="2016-02-10T21:04:22.300" Score="6" Body="&lt;p&gt;I'd like to answer this question by quoting from something I wrote a couple of years ago as part of the &quot;Teaching Philosophy&quot; that I submitted with my job search:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I look back on my time teaching Nira, and many students like her, with mixed emotions.  On the one hand I am proud to have played a role in teaching her that mathematics is, at its core, the science of pattern, reason, and logical argument.  On the other hand I am saddened that Nira did not make this discovery until her final year of high school.  All of the mathematics she had learned prior to that point had failed to impress her as anything more than a collection of techniques to be memorized, practiced, and deployed when necessary.  Even the proofs she wrote in her 9th grade Geometry class had been disconnected from the making of reasoned arguments:  a proof, for her, was a particular kind of exercise, requiring a solution produced according to an idiosyncratic and not particularly sensible set of rules.  Proofs were never about anything — other than “doing proof” itself.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;All mathematics teaching, regardless of its level or the specific&#xA;  course content, ought to be largely about the art of sense-making.&lt;/strong&gt; &#xA;  The teacher’s job is to ask questions, and in so doing to model for&#xA;  students how to ask their own questions:  not only how to solve a&#xA;  particular kind of problem, but also why the problem is worth solving,&#xA;  why a given method works, whether there are other methods that might&#xA;  work as well or better, and so on.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If all you are doing is showing students &quot;a collection of techniques to be memorized, practiced, and deployed when necessary&quot;, without any investigation into why those techniques work, then you are teaching your students that, fundamentally, mathematics does not need to make sense.  In the long run that does much more harm than any gain that might come from having your students get more practice in the use of those techniques.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-02-10T21:04:22.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10545" PostTypeId="2" ParentId="10537" CreationDate="2016-02-11T08:45:18.907" Score="3" Body="&lt;p&gt;My point of view on the issue is that it depends on the purpose of the drawing. In general I follow the rule of drawing &quot;almost-everywhere&quot; correct drawings. I see essentially two scenarios:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;1) Getting an intuition of a geometry problem&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When trying to solve a geometry problem (i.e. finding collinear points, intersecting lines etc) it is useful to draw a sketch of the situation even if not with compass and straight edge. It is fundamental, in my view, to focus on the relationship between elements of the drawing than on the drawing itself. In these cases I usually don't le t students use straight edges or similar auxilia since a pretty drawing is not essential and sometimes misleading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;2) Studying geometrical constructions&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this setting it's (obviously) essential to get things done the most precise way possible, since it's the goal of the activity to get lines to intersect exactly where you want them or sides to be exactly the required length.&lt;/p&gt;&#xA;" OwnerUserId="6286" LastActivityDate="2016-02-11T08:45:18.907" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10546" PostTypeId="2" ParentId="9807" CreationDate="2016-02-11T13:37:37.660" Score="1" Body="&lt;p&gt;When Leibniz (co)invented the calculus, he also introduced the symbol $dx$ for an infinitesimal increment of the variable $x$. Today when the calculus is presented there is an immediate asymmetry between the independent variable $x$ and the dependent variable $y$ but to Leibniz $x$ and $y$ had equal rights; he was mostly working with curves in the $x,y$ plane. When you are dealing with more than one variable, it is important to indicate which variable is undergoing the increment.  Therefore the symbols $dx,dy$ are indispensable.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2016-02-11T13:37:37.660" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10547" PostTypeId="2" ParentId="4409" CreationDate="2016-02-11T13:40:48.353" Score="2" Body="&lt;p&gt;An interesting historical tidbit is Cantor's attitude to infinitesimals, which he variously described as &quot;cholera bacillus of mathematics&quot;, &quot;paper numbers&quot;, and even &quot;abomination&quot; (see P. Ehrlich's article from 2006).  Cantor claimed that he proved infinitesimals to be inconsistent.  He even published a paper to the effect.  The reportedly great philosopher Russell was convinced by Cantor's claim and even reported in his book &lt;em&gt;Principles of Mathematics&lt;/em&gt; that infinitesimals are inconsistent. Maybe they were right...&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2016-02-15T07:40:38.870" LastActivityDate="2016-02-15T07:40:38.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10548" PostTypeId="2" ParentId="2428" CreationDate="2016-02-11T13:43:11.130" Score="0" Body="&lt;p&gt;The OP mentioned the issue of $0.999\ldots=1$ in his question.  With regard to the problem of making calculus intereting, in my experience, students react with much more interest when one argues that zero followed by an infinite tail of 9s can indeed be a number that falls a little bit short of 1 (by an infinitesimal amount). A relevant paper is that by Ely &lt;a href=&quot;http://www.nctm.org/Publications/journal-for-research-in-mathematics-education/2010/Vol41/Issue2/Nonstandard-Student-Conceptions-About-Infinitesimals&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Other relevant papers are &lt;a href=&quot;http://link.springer.com/article/10.1007/s10649-010-9239-4/fulltext.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://www.math.umt.edu/tmme/vol7no1/TMME_vol7no1_2010_article1_pp.3_30.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-08-14T16:42:21.473" LastActivityDate="2017-08-14T16:42:21.473" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10550" PostTypeId="2" ParentId="7385" CreationDate="2016-02-11T13:50:55.610" Score="4" Body="&lt;p&gt;It has long been a puzzle to historians how Leibniz could not have been misled by using differentials, given that to him &quot;differential&quot; meant &quot;infinitesimal&quot; and infinitesimals were thought to be inconsistent. About 50 years ago Abraham Robinson clarified the picture by providing a justification for infinitesimals that's rigorous by current standards. There is little room today to think that  a justification by means of differentials or infinitesimals should be more prone to error than any other kind of justification.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2016-02-11T13:50:55.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10552" PostTypeId="1" CreationDate="2016-02-12T01:32:25.403" Score="2" ViewCount="203" Body="&lt;p&gt;I was wondering if there are any special math scholarships in the United States of America for High School students with high capacity in mathematics.  At the age of merely 16, I see great potential in my future mathematics courses as you may be able to tell from my activity on &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;math.stackexchange.com&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because I am able to perform math that most students my age aren't able to and I am already learning math topics at a fast rate, I already fear that I won't be able to have math courses left by next year if I so chose, which would only be my junior year of high school!  I'm quite unsure if I am able to take more math courses as a dual-enrolled student at my local high school simply because such courses don't lend to my high school diploma.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get an idea of how fast I learn, I taught myself Calculus One in about a month... without neglecting my other subjects.  Oddly, I haven't taken Calculus yet, so I think it will be a breeze.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I keep my skills sharp of course, using this site to help me learn and teach at the same time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also looking towards becoming a math major and teaching a math course as a professor or High School teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Preferably, I'd like a scholarship or grant that is primarily math based but if science tags along I'm ok.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you, Simple Art&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="6220" LastEditDate="2016-02-12T21:48:41.327" LastActivityDate="2016-02-12T21:48:41.327" Title="Math Scholarships for the highly advanced" Tags="|gifted-students|student-question|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10553" PostTypeId="2" ParentId="2428" CreationDate="2016-02-12T07:52:55.717" Score="5" Body="&lt;p&gt;If the mathematical content is not interesting, then focus more on useful meta-level skills:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Clearly state what is known and what one wants to figure out or proof. In case of a proof this means explicitly writing down the assumptions and the claim. In case of word problems this means writing down the known information (and naming variables) and giving a name to what one wants to find out. When done, explicitly write down the solution and check that it is plausible.&lt;/li&gt;&#xA;&lt;li&gt;Justify each and every step of an argument.&lt;/li&gt;&#xA;&lt;li&gt;Translate a word problem or a real world problem into the language of mathematics. Calculus has plenty of applications; use them and make the students write problems from their own fields that use calculus, and then solve the problems posed by the others.&lt;/li&gt;&#xA;&lt;li&gt;Have students work in groups of mixed majors. Teach them how to communicate a problem from their own field to a mathematician, solve the problem together, translate the solution back to the language of applications, and then check it for plausibility.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;You can even make this explicit to the students: &quot;We will learn these sorts of meta-skills during this course. But, since practicing them with only the mathematics we know already is not very meaningful, we will also learn some integration techniques along the way.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When moving on to sequences and series, you can make the following analogy: Translating an applied problem to the language of mathematics is pretty similar to turning a theorem, such as every monotone bounded sequence converging, into something one can prove (with epsilonistics).&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2016-02-12T07:52:55.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10554" PostTypeId="2" ParentId="10552" CreationDate="2016-02-12T09:13:21.513" Score="3" Body="&lt;p&gt;I don't know about scholarships in the USA, but here's an alternative: Consult a maths department at some university (maybe college is more appropriate in USA?), preferably one that is close enough for you to visit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two ways to get in touch:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The mathematics teachers at your high school might have connections. Ask them.&lt;/li&gt;&#xA;&lt;li&gt;Go there, try to find some friendly people or staff members (PhD/doctoral students are a good candidate, since they know the faculty and remember what it is like to be a student, so they also know the courses available there). Discuss your situation with them.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;You can also try sending emails without visiting, but that is less likely to be effective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These two methods are also good ways of learning about any possible grants.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;My general suggestion is to study other subjects and learn as much as you can about everything. Physics can give you good intuitions and problem solving skills in mathematics also, and advanced mathematics requires good reading comprehension and some writing skills. Foreign languages are always useful - I've used works in French and German (and Finnish, but I'm Finnish) and a friend has used Russian references in research mathematics. Travelling is also a big part of research mathematics. Knowing foreign languages makes that easier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If mathematics lessons are too easy, then ask for more advanced material from your teacher (or ask who is the most knowledgeable teacher and ask more material from them) and focus on teaching your classmates. Teaching others will sharpen your own math skills, since explaining to others requires you to understand the subject quite well, and will teach you how to communicate mathematics, which is a crucial skill if you want to teach mathematics or do research.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2016-02-12T09:13:21.513" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10555" PostTypeId="1" CreationDate="2016-02-12T18:33:10.587" Score="17" ViewCount="814" Body="&lt;p&gt;I recently observed a group of students being introduced to logarithms for the first time.&#xA;Some of them had trouble writing $\log_ab$ properly, and it looked more like $\log a^b$.&#xA;All logarithms have a base, so $\log a^b$ doesn't even mean anything (the natural logarithm in that context was denoted by $\ln$ and base 10 by $\lg$), but that might not be obvious to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This made me wonder:&#xA;&lt;strong&gt;Do students think that there is an $a^b$ in the notation $\log_ab$?&#xA;If yes, how does this manifest itself?&lt;/strong&gt;&#xA;I'm mainly looking for actual teaching experiences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My concern is that logarithm is related to exponentiation and there is something that looks like exponentiation in the notation, and students could make false connections.&#xA;In particular the students might turn the rule $\log_ab^r=r\log_ab$ into $\log_ab=b\log a$ or something of the kind.&#xA;It would be better to use parentheses to separate $\log_a(b)$ from $\log(a^b)$ (or even $\log_{a^b}(\cdot)$), but the textbooks do not use parentheses with logarithms and trigonometric functions unless the input is long and students have to cope with that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another concern is that the logarithm is defined so that&#xA;$$&#xA;\log_ab=x\iff a^x=b,&#xA;$$&#xA;but the &quot;$a^b$ in the logarithm&quot; might make students think that there ought to be an $a^b$ in the exponential equation.&#xA;This could turn the definition into&#xA;$$&#xA;\log_ab=x\iff a^b=x&#xA;$$&#xA;in the students' heads.&#xA;(I know this second definition makes no sense, but that rarely keeps students from doing things.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Are these concerns unnecessary? If you have found that the logarithm notation indeed confuses students, how have you been able to help them?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2016-07-12T20:56:06.420" Title="Do students confuse $\log_ab$ and $\log a^b$?" Tags="|secondary-education|students-mistakes|notation|logarithm|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10556" PostTypeId="1" AcceptedAnswerId="10557" CreationDate="2016-02-12T23:14:18.393" Score="12" ViewCount="522" Body="&lt;p&gt;I may one day be interested in teaching students who are preparing for, say, the IMO or other math competitions.  However, I haven't found a particularly clear statement about exactly what material is on it.  I know they contain algebra, geometry, combinatorics, and number theory, but is there any other well-defined subject that is tested in these competitions?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know there are a number of resources for preparing students and teachers for these competitions, but reading a document that is exclusively about the competition feels too narrow--I'd rather learn the subjects that are tested instead, and then apply that knowledge to teaching the competition material.&lt;/p&gt;&#xA;" OwnerUserId="5680" LastEditorUserId="143" LastEditDate="2016-02-13T01:42:15.410" LastActivityDate="2016-02-14T20:43:52.110" Title="Which math subjects should I know well to tutor competitive mathematics?" Tags="|secondary-education|math-contests|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10557" PostTypeId="2" ParentId="10556" CreationDate="2016-02-13T02:38:29.643" Score="6" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I know they contain algebra, geometry, combinatorics, and number theory, but is there any other well-defined subject that is tested in these competitions? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Consulting Gelca and Andreescu's (2007) &lt;em&gt;Putnam and Beyond&lt;/em&gt;, there are also:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;methods of proof&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;real analysis&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;trigonometry&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;probability&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and, within each of these &quot;well-defined subjects,&quot; many sub-categories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although non-exhaustive - and intended for a particular mathematics competition, i.e., the &lt;em&gt;Putnam&lt;/em&gt; - I believe this provides a reasonable skeleton. Moreover, drawing again from this same reference, here are the chapter-by-chapter breakdowns of content.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Click the first image for a higher-resolution version.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EsitM.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EsitM.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/g6NPA.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/g6NPA.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-02-13T02:38:29.643" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10558" PostTypeId="2" ParentId="10555" CreationDate="2016-02-13T08:42:03.460" Score="3" Body="&lt;p&gt;The more general issue is that your students may simply be unfamiliar with reading and writing subscripts (irrespective of their particular use for logarithms). For example, I know that in my statistics classes the notation $z_{\alpha / 2}$ will frequently be transcribed incorrectly as $z \alpha /2$... or related manglings, like if we want $z_{\alpha / 2}$, and $\alpha = 0.10$, then $z = 0.05$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose if you had all the time in the world, then you could have a writing exercise where everyone in the class writes down three or four copies of a given (subscript) expression, and checks with a student next to them to see if they've written it correctly. Personally, I find that I never have time for that level of in-class inspection, even though I've attempted it/written it in my lecture notes in the past. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my recommendation is that you've got to commit to getting into these details when grading and take points off if this syntax is incorrect (even if the final answer is otherwise correct). That's the only way that you'll draw a student's attention to fix the issue; and otherwise they'll be permanently confused by what they're reading and writing. I'm pretty sure that I could write large and talk until I'm blue in the face to be careful of this, and some students would still be blind to the subtlety of the notation until they see points coming off the test from it. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-02-13T08:42:03.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10559" PostTypeId="2" ParentId="2428" CreationDate="2016-02-14T03:25:00.493" Score="3" Body="&lt;p&gt;I have heard of people teaching Calculus II with the theme of &lt;strong&gt;approximation&lt;/strong&gt;.  Integrals and Riemann sums (as well as other numerical integration techniques) are approximations of each other, and functions and their Taylor polynomials are also approximations of each other.  Integration techniques don't fit so well here, but my opinion is that integration by parts can be taught as approximation with &lt;a href=&quot;https://math.stackexchange.com/questions/254062/explain-this-proof-without-words-of-integration-by-parts-to-me&quot;&gt;the right picture&lt;/a&gt;, and trig substitutions and partial fractions should be left to complex analysis where they both become applications of the Residue Theorem.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-02-14T03:25:00.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10560" PostTypeId="2" ParentId="4409" CreationDate="2016-02-14T18:23:32.220" Score="2" Body="&lt;p&gt;Complex numbers and quaternions are very different beasts, don't go this way. Perhaps explain how complex numbers came to be recognized by solution to &lt;em&gt;cubics&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-02-14T18:23:32.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10561" PostTypeId="2" ParentId="10556" CreationDate="2016-02-14T19:53:38.790" Score="4" Body="&lt;p&gt;First of all, it might be worth pointing out that algebra can be split in inequalities, polynomials and functional equations. Not all problems fall in one of these three categories, but I think that approximately four-fifths does. The others are mainly about sequences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some introduction to graph theory (as a part of combinatorics) would also be nice. There aren't many problems about it, but you don't want to get an graph question on the IMO and barely knowing what a graph is. Those graph questions are usually not too hard if you have some experience with them. &lt;/p&gt;&#xA;" OwnerUserId="5664" LastActivityDate="2016-02-14T19:53:38.790" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10562" PostTypeId="2" ParentId="10445" CreationDate="2016-02-14T20:09:57.793" Score="0" Body="&lt;p&gt;Suppose function $f$ is continuous at $a$. Then $f$ has a vertical tangent line at $a$ iff $$\displaystyle\lim_{h\to0}\frac{h}{f(a+h)-f(a)}=0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;i.e. the inverse of the slope of the tangent there will be $0$.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2016-02-14T20:32:05.660" LastActivityDate="2016-02-14T20:32:05.660" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10563" PostTypeId="2" ParentId="10556" CreationDate="2016-02-14T20:35:39.940" Score="2" Body="&lt;p&gt;Knowing your math subjects is one thing, but experience with creative problem solving itself is key.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Polya's 'How to Solve It', while a bit dated in its language, is still a good reference for anyone interested in improving as a problem solver or helping others to improve. This book is light on example problems or mathematical work, but talks generally about the processes which serve us when confronted with difficult problems - reduction to smaller case, finding analogy with simpler problems which we're already able to solve, etc. I'd say that it's required reading in this area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's been a while since I read it, but I really enjoyed and benefitted from Paul Zeitz's 'Art and Craft of Problem Solving'. This book contains a fair number of problems which exemplify different specific strategies for solving problems - finding and exploiting a symmetry, induction, etc. This one would be useful for preparing lectures / lessons on specific mathematical topics and techniques.&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2016-02-14T20:43:52.110" LastActivityDate="2016-02-14T20:43:52.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10564" PostTypeId="1" AcceptedAnswerId="10566" CreationDate="2016-02-15T09:10:13.730" Score="15" ViewCount="3290" Body="&lt;p&gt;It's just an observation, but I'll be highly appreciated if anyone with experience in teaching (or TA-ing) lower-division calculus can explain this phenomenon in detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm one of a TAs who's responsible to write up a section quiz last week. According to the course instructor's email one of the main topics in lectures was partial fraction. He went over examples that involve repeated root in the denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I composed the problem $\int\frac{2x^2+7x+6}{x^3+x^2}\mathrm{d}x$ then realized it might be too challenged for a short quiz. To make it easier, I broke the problem into two parts, and partially gave the answer.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1(a) $$\frac{2x^2+7x+6}{x^3+x^2}=\frac{1}{x}+\frac{c}{x^2}+\frac{1}{x+1}$$ Compute $c$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and the next part was evaluating the integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The interesting thing I observed when I graded was, about a quarter of students wrote down something similar to,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\frac{2x^2+7x+6}{x^3+x^2}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some of them messed up the calculation after that and some of them forgot the &quot;$B$&quot; they wrote was the &quot;$c$&quot; they had to compute.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thing I don't understand is why people want to make an easy problem harder, so I come up with the explanation I write in the title of this post. Because most partial fraction problems in textbook/lecture set up that way, they want to do all the related problems using identical notations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any insight to this phenomenon is welcome.&lt;/p&gt;&#xA;" OwnerDisplayName="user2139" LastEditorUserId="5664" LastEditDate="2016-02-15T19:19:11.440" LastActivityDate="2016-02-16T22:15:00.953" Title="Students strictly follow the steps and notations in sample problems without understanding them" Tags="|calculus|teaching|" AnswerCount="7" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10565" PostTypeId="2" ParentId="10564" CreationDate="2016-02-15T09:23:29.003" Score="2" Body="&lt;p&gt;That's the reduce-to-the-previous-case syndrome: &lt;em&gt;A mathematician walks into the break room to make a cup of tea. He takes the kettle off the table, boils water, grabs a teabag and leaves, forgetting the kettle on the stove. A second mathematician wants tea too, he goes to the break room, moves the kettle back to the table and declares the problem trivial.&lt;/em&gt; See comment by rudistroyer666 at &lt;a href=&quot;https://www.reddit.com/r/math/comments/1ose5f/favorite_math_jokes/&quot; rel=&quot;nofollow&quot;&gt;https://www.reddit.com/r/math/comments/1ose5f/favorite_math_jokes/&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2016-02-15T09:23:29.003" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10566" PostTypeId="2" ParentId="10564" CreationDate="2016-02-15T14:22:50.230" Score="22" Body="&lt;p&gt;The 25% of students did not think of the problem as easier.  It was harder because it did not exactly follow the form of the example problems.  As such, it required some (little tiny bit) of creative thought and understanding.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was an accident.  I recommend learning from this, and asking more such questions. The students who are actually thinking will have an easier time on these questions, and the students who are not will have a harder time.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to actually test understanding of the material, this is ideal.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-02-15T14:22:50.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10567" PostTypeId="2" ParentId="10564" CreationDate="2016-02-15T17:49:58.680" Score="10" Body="&lt;p&gt;I have a colleague who is fond of asking people this question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/r56u3.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/r56u3.png&quot; alt=&quot;Right triangle with hypotenuse 4, legs 3 and c; solve for c?&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you might guess, a lot of people make the same notational mistake with the Pythagorean theorem, in that the $c$ in the customary formula does not map to the $c$ in this problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue of &lt;a href=&quot;https://en.wikipedia.org/wiki/Free_variables_and_bound_variables&quot; rel=&quot;noreferrer&quot;&gt;free and bound variables&lt;/a&gt; is legitimately tricky. I might prefer to discuss this complication separately, and not have this kind of collision come up by accident when I'm otherwise trying to quiz on some new skill. In fact, I would want to support students being in the habit of writing down a general and shared formula for the kind of problem they're working on. So: If asking this again in the future I'd be prone to change the $c$ to $r$ or something&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's a separate issue from whether some students failed to recognize that half of the work in the partial fraction decomposition process had already been done for them. If so, then your quiz question is a legitimate instrument for diagnosing those who simply failed to understand why the process works, and providing them with appropriate feedback for fixing that.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-02-16T02:16:24.577" LastActivityDate="2016-02-16T02:16:24.577" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10568" PostTypeId="2" ParentId="10564" CreationDate="2016-02-15T17:54:49.120" Score="2" Body="&lt;p&gt;When I was a grad student, I asked my advisor if there was such a thing as Zen mathematics to explore, and I honestly didn't appreciate that I was very bright, that that entailed thinking very differently from how students thought and various things that &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1885705026&quot; rel=&quot;nofollow&quot;&gt;Please Understand Me II&lt;/a&gt; might, if I was lucky, have sensitized me to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I functioned as something of a mathematical Zen master, intentionally breaking not exactly the mind of reason but the mind of mindless symbol manipulation, as an attempt to share with students what was the beauty of mathematics, what poetry most low level practical math classes utterly fail to convey of what draws mathematicians to mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one problem I remember was where we had covered matrix multiplication, and I had a 1x12 and 12x1 matrix multiplied together, and besides stating &quot;Credit will not be awarded for taking long time to do unnecessary work,&quot; specifically said of the matrices, &quot;Without writing out the result, what do you get when you multiply these two of the above matrices?&quot; and in some cases got an answer of &quot;The times table&quot; and in other cases got the times table listed out. One student acted shocked and betrayed when I docked points for writing out the multiplication table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I got reamed in the end-of-class reviews, and I believe students' words about &quot;Now it's payback time&quot; were justified.&lt;/p&gt;&#xA;" OwnerUserId="1767" LastActivityDate="2016-02-15T17:54:49.120" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10570" PostTypeId="2" ParentId="10564" CreationDate="2016-02-16T01:14:43.017" Score="1" Body="&lt;p&gt;I'd suggest you vary the problems asked. You could teach them to use a computer algebra system (they will do it anyway, if they are the kind of smart students you want to have) to cut down on routine tasks, freeing up time to teach (and review, and train) &lt;em&gt;understanding&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-02-16T01:14:43.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10571" PostTypeId="1" AcceptedAnswerId="10572" CreationDate="2016-02-16T03:07:11.553" Score="7" ViewCount="842" Body="&lt;p&gt;I recently learned how to use a slide rule and I have since been wondering why they are not commonly used to teach properties of logarithms. It seems that they are very clear examples of the $\log(a) + \log(b) = \log(ab)$ rule and several other similar ones. This seems like a concept that many students struggle with, and the slide rule seems to provide both a visual and possibly even kinetic representation of it. I have not yet seen an application of slide rules in the classroom despite the obvious usability for logarithms. Is there a reason that they are used so uncommonly? (or am I simply oblivious to the slideruling masses?)&lt;/p&gt;&#xA;" OwnerUserId="5275" LastActivityDate="2019-04-17T15:30:04.257" Title="Sliderule for teaching logarithms" Tags="|technology-in-education|logarithm|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10572" PostTypeId="2" ParentId="10571" CreationDate="2016-02-16T03:24:57.997" Score="10" Body="&lt;p&gt;Anecdote: For some years I've been trying to get my hands on a slide rule and I can't find any vendor who sells them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This highlights the fact that for general purposes the electronic calculator has made the slide rule defunct, to the extent that I don't think anyone even manufactures them in scale any more. So making them available in the classroom for a fairly limited exercise would be difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or to turn the issue around, the biggest reason historically that logarithms were taught in the first place was to support calculations using the slide rule (or equivalent tables). Granted that calculators have superseded slide rules, logarithms aren't even taught that widely anymore (I got them in high school in the late 1980's, but that was presented as an &quot;optional advanced&quot; topic then.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there are web/smartphone applications that simulate functioning slide rules, so you might consider looking for those instead. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-02-16T03:24:57.997" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10573" PostTypeId="2" ParentId="10564" CreationDate="2016-02-16T08:39:57.660" Score="6" Body="&lt;p&gt;You're probably familiar with muscle memory, where a certain type of motion becomes instinctive after enough repetition, right? The same thing can happen with problem-solving procedures. If you run through a particular procedure enough times in practice, your brain &quot;optimizes&quot; that procedure, and you can do it almost without thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in all the repetition, you learn that some things vary from case to case and some things are always the same, and if you run into a situation where one of the things that was always the same in practice is now different, the neural pathways don't work anymore. It's no longer instinctive; now you actually have to think, and that's harder and more error-prone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can imagine, students taking a quiz want to take the quickest, easiest, and most reliable route they can find to the solution. If their brain has &quot;optimized&quot; a procedure that involves writing fractions with $A$, $B$, $C$, etc., then they're going to look for a way to cast any partial fraction problem in terms of that procedure, because it &lt;em&gt;is&lt;/em&gt; the quickest, easiest, and most reliable route to the solution &lt;em&gt;for them&lt;/em&gt;. The extra effort involved in translating the problem is more than offset by the ease of using the procedure they know well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, one would hope that eventually the students are able to generalize these optimized procedures into a flexible mental model that allows them to do all sorts of algebraic manipulations. As a TA, you probably did this a long time ago, but most of your students likely have not studied this material long enough to do the same. So it's not surprising that they will fall back on stored procedures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this suggests that the students have probably been using the $A$, $B$, $C$ decomposition method a &lt;em&gt;lot&lt;/em&gt; in their practice exercises, and have internalized it. If you want to break them of that habit, try encouraging them to look at partial fraction problems in a different way, e.g. as plain old algebra.&lt;/p&gt;&#xA;" OwnerUserId="463" LastActivityDate="2016-02-16T08:39:57.660" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10575" PostTypeId="2" ParentId="4409" CreationDate="2016-02-16T10:39:01.053" Score="5" Body="&lt;p&gt;There is a fabulous story that is bound to captivate the students about Cauchy's alleged &quot;error&quot; in claiming that a convergent sequence of continuous functions necessarily converges to a continuous function. This story is recycled in many history and calculus textbooks. Detlef Laugwitz was one of the first to show that this anecdote about Cauchy is anachronistic and involves imposing modern notions on 19th century work. Laugwitz points out that Cauchy clarified the correct hypothesis in a 1853 paper.  A recent study of the issue on Cauchy's own terms can be found &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-015-9473-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another great tale is about the Greek philosopher Hippasus who is usually credited with the discovery that $\sqrt{2}$ is irrational. For this, according to legend, he was thrown overboard by enraged Pythagorians. Many students today are cautious to avoid a similar fate at the hand of enraged Weierstrassians, as argued in &lt;a href=&quot;http://scholarship.claremont.edu/jhm/vol7/iss1/7&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-03-13T17:20:04.543" LastActivityDate="2017-03-13T17:20:04.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10576" PostTypeId="2" ParentId="10564" CreationDate="2016-02-16T20:28:20.287" Score="2" Body="&lt;p&gt;In many respects this problem reminds me of the sort of exercises that came out of the calculus reform movement of the 1990s. The problem is &quot;conceptual&quot; in the sense that it cleverly does not rely on what David Z describes as &quot;muscle memory.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this partial fraction problem is better suited for short group quiz with formative assessment in mind, rather than in the form of an individual summative assessment.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2016-02-16T22:15:00.953" LastActivityDate="2016-02-16T22:15:00.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10578" PostTypeId="2" ParentId="7207" CreationDate="2016-02-17T19:00:14.713" Score="1" Body="&lt;p&gt;In addition to Needham's famous book mentioned in Gerald Edgar's &lt;a href=&quot;https://matheducators.stackexchange.com/a/7212/376&quot;&gt;answer&lt;/a&gt;, there is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/3034801793&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Visual Complex Functions: An Introduction with Phase Portraits&lt;/em&gt;&lt;/a&gt; by Elias Wegert.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book is richly illustrated with colour diagrams of complex functions, especially the &quot;phase portraits&quot; mentioned in the title, as can be seen on the author's &lt;a href=&quot;http://www.visual.wegert.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt; for the book.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-02-17T19:25:41.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10579" PostTypeId="2" ParentId="7207" CreationDate="2016-02-17T19:40:43.777" Score="0" Body="&lt;p&gt;For topology, Peter Saveliev has written:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1495188752&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Topology Illustrated&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A partial draft is available on the author's &lt;a href=&quot;https://calculus123.com/wiki/Topology_Illustrated&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt;, in case you would like a preview of some sections.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;True to the title, it is heavily illustrated. While there is certainly plenty of explanatory text, the large number of pictures and diagrams could be seen as a form of infographics.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2019-11-21T14:08:33.237" LastActivityDate="2019-11-21T14:08:33.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10580" PostTypeId="1" CreationDate="2016-02-17T22:04:10.787" Score="16" ViewCount="1446" Body="&lt;p&gt;Many students I have spoken with who are drawn to becoming mathematics teachers chose their mathematics major because they enjoyed doing routine exercises in high school. The comfort of a definite and systematic general procedure that eliminates uncertainty is somehow a major motivation for these students to pursue mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above motivation stands in stark contrast to the attitudes of math olympiad enthusiasts and Eastern European colleagues raised on a diet of competition-type problems. It seems to me that the major motivation for these sorts of mathematics students is to collect an enormous store of problems that each stand out because of a distinguishing solution twist. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being overly simple minded, let me assume that for the first group the comfort in mathematics comes from the general coherence of a system, and from the second it comes from the charm of particular phenomena. (Forgive the binary nature of this, but I would like to use it as a way to tease out what I want to isolate.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q: Are there papers in the literature that study the motivation of students for studying mathematics in terms of their placement on the &quot;general-particular&quot; spectrum suggested by the above extremes? If so, can someone please point me to some of these papers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask this because I'm interested in &quot;converting&quot; some students from the first to second motivating viewpoint, and the first point of view seems rather persistent. (I am trying to do this not because I believe that the problem solving culture is necessarily &quot;best&quot;, but that it seems a better alternative to the first point of view.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit: On a somewhat related note that I will post here because it should not take up space elsewhere, at the root of this question is the more fundamental question of &quot;sustainable mathematical motivation&quot;. Making the (certainly misguided) assumption that the &quot;telos&quot; of a mathematics student should be a productive research mathematician, the &quot;general&quot; or perhaps &quot;scientific&quot; viewpoint may be more sustainable (think about a Bourbaki approach) since one can follow naturality and pretty much continuously record observations about a mathematical question until something comes out of it…and this can probably be done steadily for an entire career, assuming one does not become disenchanted facing the tremendous stamina needed for such an approach. On the other hand, the ability to find delight in shorter bursts throughout the problemist literature can lead to the continual building of technical strength that can be brought to bear on many different problems, and such an approach is perhaps more sustainable due to the little bursts of drama found in each problem. In reality, mathematicians lie on a spectrum in their tastes and time-management inclinations…so the present question is slanted by the assumption that the latter sort of taste is the more human, and thus more sustainable. (A bad Erdos is better than a bad Grothendieck...)&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2016-02-18T02:17:59.100" LastActivityDate="2016-08-15T15:46:17.923" Title="The general and particular in the psychology of mathematics education" Tags="|reference-request|student-motivation|teacher-preparation|teacher-training|teacher-motivation|" AnswerCount="1" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="10581" PostTypeId="2" ParentId="7207" CreationDate="2016-02-18T02:28:59.950" Score="0" Body="&lt;p&gt;Traditionally (say up to the last 40, 50 years?) images were extremely expensive to produce and distribute (line graphs had to be drafted by hand, other images had to be created by expensive methods, colour meant special pages in the book), and thus used sparingly. Creating, distributing and viewing videos was much worse. As a result, people streamlined methods to make their point &lt;em&gt;without&lt;/em&gt; images. That is hard to unlearn.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-02-18T02:28:59.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10582" PostTypeId="2" ParentId="7207" CreationDate="2016-02-18T02:35:14.700" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://www.math.toronto.edu/drorbn/Talks/&quot; rel=&quot;nofollow&quot;&gt;The slides of Dror Bar-Natan&lt;/a&gt; are excellent examples of such mathematical &quot;infographics&quot;. &lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2016-02-18T02:35:14.700" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10583" PostTypeId="1" CreationDate="2016-02-18T05:57:31.027" Score="9" ViewCount="471" Body="&lt;p&gt;My students are often getting confused while using chain rule for complicated functions. For example&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f(x)=\tan^3\left(\sqrt{x^2+x+1}\right)$$ Most of the students wrote $f'(x)$ wrongly as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f'(x)=3 \tan^2\left(\sqrt{x^2+x+1}\right) \times \frac{1}{2\sqrt{x^2+x+1}} \times (2x+1)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can I have any comments on how better I can teach chain rule?&lt;/p&gt;&#xA;" OwnerUserId="5666" LastEditorUserId="39" LastEditDate="2016-02-20T20:10:08.603" LastActivityDate="2016-02-20T20:10:08.603" Title="How can we neatly explain chain rule of differentiation" Tags="|calculus|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10584" PostTypeId="2" ParentId="10583" CreationDate="2016-02-18T06:40:37.287" Score="9" Body="&lt;p&gt;Here is one way to write the solution which cuts down on errors:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\frac{d}{dx} \tan^3(\sqrt{x^2+x+1}) = \frac{d(\tan^3(\sqrt{x^2+x+1}))}{d(\tan(\sqrt{x^2+x+1}))} \frac{d(\tan(\sqrt{x^2+x+1}))}{ d(\sqrt{x^2+x+1})} \frac{d\sqrt{x^2+x+1}}{d (x^2+x+1)} \frac{d(x^2+x+1)}{dx}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Forcing the student to set up all of the computations first, before performing them, seems to cut down on the number of errors.  I think this helps to solve the problem of remembering which step you are on while you compute a given derivative. &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-02-18T06:40:37.287" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10585" PostTypeId="1" CreationDate="2016-02-18T11:24:26.957" Score="13" ViewCount="650" Body="&lt;p&gt;From what I see in the curriculum we use for my children if we stay on track with the current trajectory they'll finish by grade 8 what is usually called &lt;em&gt;Precalculus&lt;/em&gt; (USA terminology, includes solutions of algebraic equations, factoring techniques, graphing rational functions, basic matrix arithmetic, solutions of inequalities, basic linear programming, trigonometry, logs, exponentials, basic theory of functions). Hypothetically, this leaves 4 years of high school to learn mathematics beyond this foundation.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; which course sequence would you suggest for a high school student who is ready to take calculus in their first year of high school ? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Assume the children are home-taught with a parent who has a graduate education in mathematics for the convenience of your answer if you wish. More to the point, don't worry much about the plausibility of your answer to &quot;real-world&quot; school situations. In particular, set-aside worries about keeping all students on the same track and/or a lack of expertise in the instructors in your ideal school. The goal here is not to create mathematicians, but, to give mathematics a proper scholarly introduction. To borrow &lt;a href=&quot;http://www.edwardfrenkel.com/lovemath/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frenkel's&lt;/a&gt; terminology, to show them artistry, not just fence painting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://matheducators.stackexchange.com/q/10521/128&quot;&gt;Why do we teach calculus in high school rather than a different math course?&lt;/a&gt; I made a comment which prompted the creation of this question. &lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-06-30T00:18:25.143" Title="Beyond Calculus, an Invitation to Dream Higher for High School" Tags="|secondary-education|curriculum|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10586" PostTypeId="1" CreationDate="2016-02-18T18:12:47.900" Score="8" ViewCount="257" Body="&lt;p&gt;Can someone please point to research papers that analyze different ways of expressing informal proofs from an educational point of view? I am particularly interested in proofs by induction but I would like to read about different ways of expressing logical inferences in a way that students understand them better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, a commonly used one is the calculational style. Are there any others? How do they compare to this one?&lt;/p&gt;&#xA;" OwnerUserId="6280" LastActivityDate="2016-05-19T02:57:20.937" Title="Educational styles for writing proofs" Tags="|undergraduate-education|reference-request|proofs|logic|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10587" PostTypeId="1" CreationDate="2016-02-18T18:32:21.943" Score="7" ViewCount="161" Body="&lt;p&gt;I'm working with a middle school student (grade 8) who recently displayed a misunderstanding of place value in decimal numbers. The student believes, for example, that $0.125$ is bigger than $0.12$ in the same proportion that $125$ is bigger than $12$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some lines of thinking / questioning that might undo this confusion? I have an idea or two of my own, but I'm not totally satisfied with them so I'd like some input.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student suffers from no specific cognitive impairments, but has a history of poor performance in math and would be a poster-child for math anxiety.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2016-03-02T21:43:50.390" Title="Specific Intervention(s) for Middle School 'Place Value' confusion" Tags="|real-numbers|proportional-reasoning|place-value|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10588" PostTypeId="2" ParentId="10583" CreationDate="2016-02-18T22:33:41.577" Score="9" Body="&lt;p&gt;This is a slight variation on Steven Gubkin's suggestion, incorporating David Z's suggestion in his comment on that answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suggest to the students that whenever they encounter a complicated nexted function, they explicitly write it  as a chain of several simpler functions, e.g.:&#xA;$$y=u^3$$&#xA;$$u=\tan(v)$$&#xA;$$v=\sqrt{w}$$&#xA;$$w=x^2+x+1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the chain rule says simply that the derivative of the first variable $y$ with respect to the last variable $x$ is the product of the derivaives of all of the individual links in the chain:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dv}\cdot \frac{dv}{dw}\cdot \frac{dw}{dx}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So&#xA;$$\frac{dy}{dx} = 3u^2 \cdot \sec^2(v) \cdot \frac{1}{2\sqrt{w}} \cdot (2x+1)$$&#xA;and the problem can be completed by back-substituting in the expressions for each of the variables in terms of the simpler ones.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-02-18T22:33:41.577" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10589" PostTypeId="2" ParentId="10585" CreationDate="2016-02-18T23:44:03.280" Score="6" Body="&lt;p&gt;You can explore combinatorics, like through Bogart's &lt;a href=&quot;http://www.math.dartmouth.edu/news-resources/electronic/kpbogart/ComboNoteswHints11-06-04.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;Combinatorics through Guided Discovery&quot;&lt;/a&gt;. Or you could take a peek at Chen's &lt;a href=&quot;https://rutherglen.science.mq.edu.au/~maths/Chen-notes/ln.html&quot; rel=&quot;noreferrer&quot;&gt;lecture notes&lt;/a&gt;, they cover a hefty part of college math curriculum in detail, and are well written.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-02-18T23:44:03.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10590" PostTypeId="2" ParentId="10586" CreationDate="2016-02-18T23:59:16.860" Score="3" Body="&lt;p&gt;All &quot;real&quot; proof are (more or less) informal. You should prove things more or less the same way as a professional would, but for the sake of newbies (and not constrained by procrustean page limits, as in formal papers) you should spell more steps out, and perhaps add some further comments and motivation for the steps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You don't just want to &lt;em&gt;show&lt;/em&gt; the proof, you want to show the scaffolding used to construct it, teach techniques to find proofs: work from both ends, hoping they'll meet in the middle; work through some examples; try to find counterexamples (not being able to find them might hint at why it is true). &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-02-18T23:59:16.860" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10591" PostTypeId="2" ParentId="10587" CreationDate="2016-02-19T00:44:26.513" Score="5" Body="&lt;p&gt;I always emphasize comparing decimals with the same number of places to my students.  Therefore before comparing 0.12 and 0.125, I would first teach them that 0.12 = 0.120.  Students can than grasp that 0.120&gt;0.012.  This is further brought home when students read it as 12 hundredths is greater than 12 thousandths.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest you work on converting decimals to different numbers of places with this student to make it easier for him to think about decimals.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-02-19T00:44:26.513" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10592" PostTypeId="2" ParentId="10587" CreationDate="2016-02-19T05:26:41.870" Score="1" Body="&lt;p&gt;I think that the overriding principle would be to model &lt;em&gt;comparing like terms&lt;/em&gt;, one term at a time. Fundamentally that's the key to any comparisons, additions or subtractions -- not just in decimals, but more generally for fractions, mixed units, etc., etc. So I'd go through and model looking at those digits one at a time, and make sure that the student was doing the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;@AmyB's answer is also useful to give additional context. Chiefly the student needs to be disabused of the idea that just looking at string-length can tell which decimal is larger. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a common enough problem that there's a speed-quiz for this skill for my incoming community-college statistics students on my site Automatic-Algebra. Consider having your student drill there for a few days to prove they've mastered it:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.automatic-algebra.org/decimalrelations.htm&quot; rel=&quot;nofollow&quot;&gt;http://www.automatic-algebra.org/decimalrelations.htm&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-02-19T05:26:41.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10593" PostTypeId="2" ParentId="10585" CreationDate="2016-02-19T06:03:38.950" Score="2" Body="&lt;p&gt;Where I live, many fences are works of art.  If someone around here bothers to paint a fence, it will be a pretty fence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some practical skills for scientists and engineers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Tolerances and estimation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Significant figures, and the difference between accuracy and precision.&lt;/li&gt;&#xA;&lt;li&gt;How to estimate the number of bits in a tolerance.&lt;/li&gt;&#xA;&lt;li&gt;Linear interpolation along non-linear functions (such as Production Possibility Frontiers in Economics)&lt;/li&gt;&#xA;&lt;li&gt;How to estimate the roots of functions&lt;/li&gt;&#xA;&lt;li&gt;Error analysis, and how to choose algorithms that mitigate errors&lt;/li&gt;&#xA;&lt;li&gt;Big-O notation for the amount of time or space required to perform an algorithm&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Calculus through the Reynolds Transport Theorem&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Integration and differentiation of polynomials in one-dimension&lt;/li&gt;&#xA;&lt;li&gt;Integration and differentiation of exponentials and sinusoids in one-dimension&lt;/li&gt;&#xA;&lt;li&gt;Area integrals&lt;/li&gt;&#xA;&lt;li&gt;Volume integrals&lt;/li&gt;&#xA;&lt;li&gt;Flux (Surface integrals, with time derivatives)&lt;/li&gt;&#xA;&lt;li&gt;The Reynolds' Transport Theorem&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Convolutions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Macaulay's notation for integrals and derivatives of singlets and doublets (e.g., Dirac delta functions, step functions, and ramp functions)&lt;/li&gt;&#xA;&lt;li&gt;A few especially useful sequences and series:  Taylor series, polynomial decay, exponential decay&lt;/li&gt;&#xA;&lt;li&gt;Laplace transforms (and inverse Laplace transforms) or Fourier transforms (and inverse Fourier transforms)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Statistics and Data Analysis&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;How to Lie with Statistics&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Basic statistics (R², variance, slope, meaning of standard deviation, normal distribution, binomial distribution, central limit theorem, Poisson distribution, Weibull distribution, cumulative density function, probability density function, control charts as used in statistical process control, how to use a &quot;significance test&quot; in a misleading way)&lt;/li&gt;&#xA;&lt;li&gt;Relational Algebra or Relational Calculus or SQL coding, through being able to write SQL code to perform a left anti semi join.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2016-02-19T06:30:10.680" LastActivityDate="2016-02-19T06:30:10.680" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10594" PostTypeId="2" ParentId="7336" CreationDate="2016-02-19T12:48:38.980" Score="1" Body="&lt;p&gt;An example: For an experiment you need $N$ cells of a certain bacterial species. You cultivate them under ideal conditions in a growth medium so that their number increases exponentially. Let $P(t) = c \cdot a^t$ be the number of bacteria at time $t$. First you answer the question how long you have to wait until you have $N$ bacteria cells. The $\epsilon$-$\delta$-definition can be motivated by asking the question: &lt;em&gt;How accurate do you have to be so that the error in the number of cells is less than $\epsilon$? What happens when due to increased requirements to the experiment you need do decrease the maximal error $\epsilon$?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the course I would use concrete values for $N$, $\epsilon$ etc.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2016-02-19T12:48:38.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10595" PostTypeId="2" ParentId="3952" CreationDate="2016-02-19T13:30:53.527" Score="2" Body="&lt;p&gt;When you look at the formal definitions of several concepts in real analysis you see that those concepts are rooted in the limit of sequences (or better to say, these concepts can be defined in a way that they all base on the concept of the limit for sequences):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The limit of a series $\sum_{k=0}^\infty a_k$ is defined to be the limit of the sequence of the partial sums: $\sum_{k=0}^\infty a_k := \lim_{n\to\infty} \sum_{k=0}^n a_k$&lt;/li&gt;&#xA;&lt;li&gt;We have $\lim_{x\to a} f(x)=c$, iff for each sequence $(x_n)$ with $\lim_{n\to\infty} x_n = a$ we find $\lim_{n\to\infty} f(x_n) = c$.&lt;/li&gt;&#xA;&lt;li&gt;After we have defined the expression $\lim_{x\to a} f(x)=c$ with the limit of sequences we can define the derivative via $\lim_{h\to0} \tfrac 1h [f(x+h)-f(x)]$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thus, when you want to teach calculus hierarchically in a way that you only define new concepts with already defined terms, you need to start with the limit of sequences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, when we already know important concepts like the derivate or the integral, the need of talking about the limit of sequences can be easily motivated: We need this concept to define the other concepts rigorously. This motivation can be hardly given when we begin calculus with the limit of sequences. Note also that in mathematics a concept was often first used intuitively before the formal definition of this concept was established.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2016-02-19T13:30:53.527" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10596" PostTypeId="2" ParentId="2625" CreationDate="2016-02-19T13:45:02.907" Score="1" Body="&lt;p&gt;I guess you will be interested in the book &lt;a href=&quot;http://www.springer.com/us/book/9780387945514&quot; rel=&quot;nofollow&quot;&gt;Analysis by Its History&lt;/a&gt; by Ernst Hairer and Gerhard Wanner which does exactly what you are looking for.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2016-02-19T13:45:02.907" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10597" PostTypeId="1" AcceptedAnswerId="10615" CreationDate="2016-02-19T22:30:19.257" Score="20" ViewCount="1663" Body="&lt;p&gt;In order to teach continuity of real valued functions $f:D\to\mathbb R$ one may start with the &lt;a href=&quot;https://matheducators.stackexchange.com/a/1686/5097&quot;&gt;(in some sense wrong)&lt;/a&gt; intuition&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$f$ is continuous when its graph can be drawn without lifting the pen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I guess this is the intuition many students start with when they first encounter continuity. However, this isn't the formal definition of continuity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, my question is: &lt;em&gt;How can we motivate the formal definition of continuity starting from the above intuition?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify my question: Imagine yourself in a time when there was no formal definition of continuity (when for example people thought that &lt;a href=&quot;https://hsm.stackexchange.com/questions/59/are-there-written-19th-century-sources-expressing-the-belief-that-the-intermed&quot;&gt;functions fulfilling the intermediate value property are continuous&lt;/a&gt;). How would you argue that the formal definition of continuity must be of the form we have it today?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thereby, with &quot;formal definition&quot; I mean either the &lt;a href=&quot;https://en.wikipedia.org/wiki/Continuous_function#Definition_in_terms_of_limits_of_sequences&quot; rel=&quot;noreferrer&quot;&gt;definition in terms of limits of sequences&lt;/a&gt; or the &lt;a href=&quot;https://en.wikipedia.org/wiki/Continuous_function#Weierstrass_definition_.28epsilon.E2.80.93delta.29_of_continuous_functions&quot; rel=&quot;noreferrer&quot;&gt;epsilon-delta definition&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My attempt for answering the question:&lt;/strong&gt; From the above intuition one may deduce several properties, continuous function should have:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Continuous functions should fulfill &lt;a href=&quot;https://en.wikipedia.org/wiki/Darboux&amp;#39;s_theorem_%28analysis%29#Darboux_function&quot; rel=&quot;noreferrer&quot;&gt;the intermediate value property&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The graph of a continuous function should be a connected subset of $\mathbb R^2$. (Although this is not true for continuous functions in general, this property makes sense when one starts from the above intuition)&lt;/li&gt;&#xA;&lt;li&gt;Small changes in the input result in small changes in the output.&lt;/li&gt;&#xA;&lt;li&gt;...&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Now we take this property as the definition of continuity which is (in our opinion) the most characteristic property of continuity. Here we might chose property (3) from which the $\epsilon$-$\delta$-definition can be deduced (note that (3) is similar to the  &lt;a href=&quot;https://hsm.stackexchange.com/a/3140/1772&quot;&gt;definition of continuity by Cauchy&lt;/a&gt; which is one of the first formal definitions of continuity).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you think about this kind of explanation? When it is in your point of view a good explanation: Why should one use property (3) as the most characteristic property for continuity and not (1) or (2) or another property?&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="5097" LastEditDate="2017-04-24T12:58:00.397" LastActivityDate="2018-09-08T17:47:47.540" Title="How can I motivate the formal definition of continuity?" Tags="|undergraduate-education|concept-motivation|mathematical-analysis|intuition|definitions|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10598" PostTypeId="1" AcceptedAnswerId="10599" CreationDate="2016-02-20T12:37:53.703" Score="4" ViewCount="1123" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Continuous_function#Definition_in_terms_of_limits_of_sequences&quot; rel=&quot;nofollow noreferrer&quot;&gt;formal definition of continuity&lt;/a&gt; is a &lt;a href=&quot;https://math.stackexchange.com/questions/1657346/what-is-the-definition-of-a-local-property-for-a-function&quot;&gt;local property&lt;/a&gt; (the definition of continuity at a point is a property of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Germ_%28mathematics%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;germ&lt;/a&gt; of the function at this point). Why is it a good decision to make the definition local?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To motivate my question: The starting intuition behind continuity for a real valued function $f:D\to\mathbb R$ is often:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$f$ is continuous when its (whole) graph can be drawn without lifting the pen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This intuition suggests that the definition should be global in a sense that it takes the whole function into account. This intuition does also not explain, why we should define continuity at a point. So:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;What is the benefit of having a local definition of continuity?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; This is a follow-up question of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10597/how-can-i-motivate-the-formal-definition-of-continuity&quot;&gt;How can I motivate the formal definition of continuity?&lt;/a&gt; Although both questions are highly connected, I think the answers to both questions will be a different. That's why I made a new question. I hope that's okay...&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-02-20T14:49:22.160" Title="Why is continuity defined as a local property?" Tags="|undergraduate-education|concept-motivation|mathematical-analysis|intuition|definitions|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10599" PostTypeId="2" ParentId="10598" CreationDate="2016-02-20T12:58:09.737" Score="6" Body="&lt;p&gt;Firstly, I don't think it entirely makes sense to ask why a property is defined to be local rather than global. Being local or global emerges from the definition itself. Continuity asks about whether a function is well-behaved in a particular way, which is determined locally. We use the word 'continuous' to mean 'well-behaved everywhere', but we also have ways of expressing finer concepts, like 'it's well-behaved here but it isn't there'. So I'd say the main point is that we've chosen the simplest term to refer to the case of good behaviour everywhere, because that is one we want to say a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, it might make more sense if you switch language. I'm told that some languages (I've forgotten which) have the primary term meaning 'discontinuous' and then make continuity the negation of that. To me that actually seems more natural. The key point of the definition of continuity is that, although we have an intuition for it, it is not straight-forward to define. On the other hand, we have a more concrete idea of what a jump in a function is.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-02-20T12:58:09.737" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10600" PostTypeId="1" AcceptedAnswerId="10601" CreationDate="2016-02-20T15:06:10.007" Score="4" ViewCount="1380" Body="&lt;p&gt;As mentioned in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1538/counterintuitive-consequences-of-standard-definitions&quot;&gt;this question&lt;/a&gt; students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \to \mathbb R: x \mapsto \tfrac 1x$ is continuous although it has a &quot;jump&quot; at $x=0$ (&lt;a href=&quot;https://matheducators.stackexchange.com/a/1686/5097&quot;&gt;cf. this answer with more details&lt;/a&gt;). So:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Why is continuity only defined on the function's domain? What's the benefit?&lt;/em&gt; How should a lecturer answer to such a question of a student?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My attempt to answer the question:&lt;/strong&gt; I would give two arguments:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When we take the &lt;a href=&quot;https://en.wikipedia.org/wiki/Continuous_function#Definition_in_terms_of_limits_of_sequences&quot; rel=&quot;nofollow noreferrer&quot;&gt;sequence limit definition of continuity&lt;/a&gt; $\lim_{n\to\infty} f(x_n) = f\left(\lim_{n\to\infty} x_n\right) = f(x_0)$, then this definition makes only sense when $x_0 = \lim_{n\to\infty} x_n$ is in the domain of $f$.&lt;/li&gt;&#xA;&lt;li&gt;The concept students have in mind is &quot;continuous continuation&quot; and not &quot;continuity&quot;. Thus, one have to distinguish between both concepts.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What do think about my answer? Have I missed something or are there other good arguments?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; This is another follow up question of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10597/how-can-i-motivate-the-formal-definition-of-continuity&quot;&gt;How can I motivate the formal definition of continuity?&lt;/a&gt; I hope that's okay since I ask here for another aspect of continuity. I want to write an introductory article for continuity. That's the reason why I ask all these questions here...&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-02-21T10:53:11.450" Title="Why is continuity only defined on its domain?" Tags="|undergraduate-education|concept-motivation|mathematical-analysis|intuition|definitions|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10601" PostTypeId="2" ParentId="10600" CreationDate="2016-02-20T16:06:22.697" Score="8" Body="&lt;p&gt;&quot;Is the sine function continuous at Mount Everest?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This or some modification thereof could be a playful start for a discussion about the fact, which you addressed more technically, that there must be some meaning to the function where it is considered. Put differently where else should continuity be considered? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One reason for the confusion may be that the students mainly think about functions defined on the real numbers with some exceptional points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An actual example to discuss, beyond the playful start, could be the (real) logarithm function: &quot;Is $\log$ continuous or discontinuous at $-27$?&quot;   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Discussing such examples could help to get the concept across that the function must have some agreed upon meaning at the point where continuity is considered. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The specific example of  $1/x$ is a bit tricky as the question &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/4165/is-the-reciprocal-function-continuous&quot;&gt;is the reciprocal function continuous&lt;/a&gt;&quot; does not have that clear-cut an answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this specific function, and related ones, it may make sense to explain that the type of phenomenon one has for $1/x$ around $0$ has a particular name: pole (or more generally isolated singularity).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may help them to categorize functions. Once $1/x$ becomes continuous on $\mathbb{R} \setminus \{0\}$, with a pole at $0$, it may be less confusing that it should be a continuous function &quot;just like&quot; $x^2 + 3$, since the one has a pole while the other does not. &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-02-20T16:06:22.697" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10602" PostTypeId="2" ParentId="41" CreationDate="2016-02-20T22:42:05.733" Score="4" Body="&lt;p&gt;One thing I have run into is, when learning or tutoring probability, being able to distinguish the random variables with their density functions' support.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To that end, I typically put serifs on my capital letters only, and that helps me tell the difference. One big issue is telling the difference between $W$, $w$, and $\omega$. I typically put serifs on $W$, keep the angles crisp on the $w$, and exaggerate the round parts for the $\omega$.&lt;/p&gt;&#xA;" OwnerUserId="6371" LastActivityDate="2016-02-20T22:42:05.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10603" PostTypeId="2" ParentId="10597" CreationDate="2016-02-20T23:54:39.883" Score="7" Body="&lt;p&gt;The prototypical way for a function to not be continuous is that of a jump discontinuity. Imagine a jump discontinuity on the order of a few micrometers, like the width of a hair. If you are tracing the graph of the function with an everyday pencil, you would slide right across the discontinuity without even noticing its presence. However, if you shrunk yourself and your pencil to the micrometer scale, you would suddenly notice the discontinuity. So the width of the pencil corresponds to $\epsilon$. The parameter $\delta$ is a localizing parameter that allows one to focus on the region near the discontinuity. There might be another location in the domain where there is a jump discontinuity on the order of a few angstroms, so the localizing parameter $\delta$ might need to be smaller. In order to rule out a jump discontinuity at $x_0$ you have to look at all small scales to make sure you are not overlooking anything, hence &lt;em&gt;for all&lt;/em&gt; $\epsilon&amp;gt;0\ldots$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think a natural way to develop the informal definition of continuity to the formal definition is to focus on the intuitive &lt;em&gt;negation&lt;/em&gt; of continuity in the form of a jump discontinuity. But as you probably know, the formal definition of continuity permits some continuous functions that only baffle the mind when we try to visualize. For example continuous functions that are nowhere differentiable, or a function that is continuous only at one point.  &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-02-20T23:54:39.883" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10604" PostTypeId="2" ParentId="10597" CreationDate="2016-02-21T10:34:19.663" Score="5" Body="&lt;p&gt;To give a partial answer to my question: In &quot;A radical approach to real analysis&quot; David Bressoud gives a good explanation why the intermediate value property (IVP) is no good choice for continuity (pp. 91 ff):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The image of a closed interval might not be bounded.&lt;/li&gt;&#xA;&lt;li&gt;The sum of two functions with the IVP might not be a function with the IVP.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For (1) take $f:[0,1]\to\mathbb R: x\mapsto \begin{cases} \tfrac 1x \sin\left(\tfrac 1x\right) &amp;amp; ;x\neq 0 \\ 0 &amp;amp; ;x=0 \end{cases}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For (2) take $f:\mathbb R\to\mathbb R: x\mapsto \begin{cases} \sin^2\left(\tfrac 1x\right) &amp;amp; ;x\neq 0 \\ 0 &amp;amp; ;x=0 \end{cases}$ and $g:\mathbb R\to\mathbb R: x\mapsto \begin{cases} \cos^2\left(\tfrac 1x\right) &amp;amp; ;x\neq 0 \\ 0 &amp;amp; ;x=0 \end{cases}$ so that $f(x)+g(x)=\begin{cases} 1 &amp;amp; ;x\neq 0 \\ 0 &amp;amp; ;x=0 \end{cases}$&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="5097" LastEditDate="2017-08-30T20:25:32.743" LastActivityDate="2017-08-30T20:25:32.743" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10605" PostTypeId="2" ParentId="10600" CreationDate="2016-02-21T10:53:11.450" Score="8" Body="&lt;p&gt;A somewhat analogous point of view is the one of &quot;continuous extensions&quot;. What looks weird in saying &lt;em&gt;$\frac 1 x$ is a continuous function&lt;/em&gt; is, of course, what is happening around $0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is happening around $0$ can be summarized by saying:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;there exists no continuous function on the whole $\mathbb R$ extending $\frac 1 x$&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The whole idea of &lt;em&gt;removable discontinuity&lt;/em&gt; is centered on this idea of extending a continuous function to another continuous function on a larger domain. This point of view has its strength in reconciling with the idea of something non continuous going on here...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, in general, a continuous extension may not exist, may exist and not be unique and whatever else. From this point of view $\log$ can be made continuous at $-27$ (e.g. extending it via $\log|x|$), but not at $0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can of course use this point of view to play around a little bit with subtle ideas like &quot;maximal extensions&quot; (where maximal is referred to the domain), gluing of continuous functions and the like. &lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2016-02-21T10:53:11.450" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10607" PostTypeId="2" ParentId="10597" CreationDate="2016-02-21T14:24:00.377" Score="4" Body="&lt;p&gt;There is something to be learned from the history of the concept.  The modern concept of continuity was introduced by Cauchy in 1821. This concept is probably best explained in two stages:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Cauchy's definition of continuity as infinitesimal $x$-increment always producing an infinitesimal change in $y$;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) once students understand the basic intuition, the epsilon-delta &lt;em&gt;paraphrase&lt;/em&gt; can be explained in terms of &lt;em&gt;trough&lt;/em&gt; and &lt;em&gt;trap&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In more detail, in order for the value of the function to end up in a &lt;em&gt;trap&lt;/em&gt; of a certain size, we must &quot;feed&quot; it values from a trough that's sufficiently small.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the absolute value function $|\;|$ is continuous at the origin because for every infinitesimal $\Delta x$, the corresponding $\Delta y=|\Delta x|$ is similarly infinitesimal.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-04-24T12:59:49.457" LastActivityDate="2017-04-24T12:59:49.457" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10608" PostTypeId="2" ParentId="8391" CreationDate="2016-02-21T20:39:13.370" Score="2" Body="&lt;p&gt;I'm only familiar with the abstract algebra books you listed, not so much the ones on linear algebra. All of those abstract algebra books are awesome, but they are quite advanced and they might not be suitable for someone just setting foot into algebra for the first time. You might try one that's a little more aimed at the average undergraduate first, then keep those other ones around when you are ready for meatier fare. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to recommend &lt;a href=&quot;http://amzn.com/1466567066&quot; rel=&quot;nofollow&quot;&gt;Abstract Algebra: An Inquiry-Based Approach&lt;/a&gt; which was written by my colleagues Jon Hodge, Steve Schlicker, and Ted Sundstrom at Grand Valley State University. It's really written so that you are supposed to interact with the book through activities and exercises, and I think you'll get a good initial grounding in the subject this way. Also there are some sections on cryptography in it, so that will satisfy some of your end goals. I used this book to teach both first- and second-semester abstract algebra here at GVSU and both the students and I really liked it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you are ready for the deeper stuff don't forget &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387905189&quot; rel=&quot;nofollow&quot;&gt;Algebra&lt;/a&gt; by Thomas Hungerford. This was my graduate-school text in algebra and I still use it! It has a pretty decent treatment of abstract linear algebra, and I think most graduate level textbooks will as well. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2016-02-21T20:39:13.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10609" PostTypeId="2" ParentId="10478" CreationDate="2016-02-21T23:30:15.960" Score="15" Body="&lt;p&gt;I've done it both ways, although I do rings-first now and for the foreseeable future. I think the pros and cons have a lot to do with the audience, especially if there are a lot of pre-service mathematics teachers taking the course (like there are at my place). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main &quot;pro&quot; is pedagogical: Rings are more familiar objects to students and, for pre service teachers, much closer to the math that they will be teaching in middle and high schools than groups are. The &quot;everyday&quot; objects we use in math -- integers, polynomials, matrices -- are all instances of rings. That also makes them groups of course, if you strip away multiplication, but it's more natural to work with these objects on their own terms without stripping away operations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea is that students should work with what is familiar to them and work outward toward abstraction, rather than face abstraction first and then see familiar things as instances of those abstract classes. I would say that for most students I've seen, this is definitely the case and especially for future teachers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main &quot;con&quot; is that once you learn some abstract algebra with familiar objects it can be hard to make the jump to abstraction -- students tend to want to map integer-like operations and properties onto everything because we started with objects that behave like the integers. Nowhere is this more apparent than when moving from rings to groups, where students have a hard time in my experience grasping the idea of having only one operation to work with. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For audiences that are mostly &lt;em&gt;not&lt;/em&gt; future teachers but future professional mathematicians, or maybe computer scientists, this is something to keep in mind -- those students need to learn to handle abstraction up front and doing groups first (or even semigroups? Or categories?) might be better. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer: I work in the same department as Hodge, Schlicker, and Sundstrom and I am pretty partial to their book. &lt;/p&gt;&#xA;" OwnerUserId="942" LastEditorUserId="376" LastEditDate="2016-10-18T10:18:24.070" LastActivityDate="2016-10-18T10:18:24.070" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10610" PostTypeId="1" AcceptedAnswerId="10612" CreationDate="2016-02-22T13:00:26.477" Score="9" ViewCount="601" Body="&lt;p&gt;In a comment on a recent post, Steven Gubkin pointed out that in doing mathematics he likes to develop &lt;em&gt;stories&lt;/em&gt;. This motivation for mathematics is perhaps familiar to many practicing mathematicians. Calculations can even become stories. I do not mean, necessarily, that we tell the story about the mathematician who developed a bit of mathematics…which may be part of the story…but that the drama of the mathematics itself is a focus for continuing to investigate. The development of the story, with unexpected mathematical plot twists and seamless interface with humans engaging with the mathematics, as well as the cultural setting of the mathematics, seems to be front and center for mathematicians. This is perhaps because we have a developed mathematical cultural identity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please note that I am not asking about the role of &lt;em&gt;writing&lt;/em&gt; in learning mathematics. It is possible for traditional mathematical writing, even when such writing is of decent quality, to completely avoid the aspect of mathematics as story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I confess that this question, like many of my questions, arises because I am rather ignorant of sources in mathematics education. (Sorry!) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q: What are some good studies and references that explore the role of developing mathematical stories as a central approach in mathematics education?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in all levels, here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S.: The recent passing of &lt;a href=&quot;https://en.wikipedia.org/wiki/Umberto_Eco&quot; rel=&quot;nofollow noreferrer&quot;&gt;Umberto Eco&lt;/a&gt; brought on this question, in part. An observation in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0151008124&quot; rel=&quot;nofollow noreferrer&quot;&gt;his book on literature&lt;/a&gt;, together with Steven Gubkin's observation mentioned above, points directly at this question in our domain. Particularly, Eco observes that storytelling and narrative writing is not illogical or unstructured, but that there is a strong need to maintain the reality and consistency with what has been written. In some sense, I think this is the kind of consistency and logic that rules (and delights) the lives  of practicing mathematicians. We have an idea and follow it up in our calculation and writing, and find out where our idea is wrong or inconsistent with what has been established. Thus unfolds a rich reality to which we respond.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Here are a few things that might be helpful. &lt;a href=&quot;https://books.google.com/books?id=BXYucAAhFCgC&amp;amp;pg=PR11&amp;amp;lpg=PR11&amp;amp;dq=Mathematics%20is%20commonly%20thought%20to%20be%20the%20pursuit%20of%20universal%20truths&amp;amp;source=bl&amp;amp;ots=KVIhnrcf-i&amp;amp;sig=ZJNKCtZAUo7gOcbVb-QszsFqaNY&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiH_OjQ-JnQAhUl2IMKHSMrB6AQ6AEIGzAA#v=onepage&amp;amp;q=Mathematics%20is%20commonly%20thought%20to%20be%20the%20pursuit%20of%20universal%20truths&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;One&lt;/a&gt; is a book introduction writtend by Bill Thurston, and &lt;a href=&quot;https://www.sensepublishers.com/media/1019-teaching-mathematics-as-storytelling.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;the other&lt;/a&gt; is a book on teaching mathematics as story.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2016-11-08T20:04:53.307" LastActivityDate="2016-11-08T20:04:53.307" Title="Developing mathematical stories" Tags="|mathematical-pedagogy|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10611" PostTypeId="2" ParentId="8391" CreationDate="2016-02-22T14:12:36.690" Score="2" Body="&lt;p&gt;You might want to take a look at Sergei Treil's &lt;a href=&quot;https://www.math.brown.edu/~treil/papers/LADW/LADW.html&quot; rel=&quot;nofollow&quot;&gt;&quot;Linear algebra done wrong&quot;&lt;/a&gt; for linear algebra. It is sort of an answer to Sheldon Axler's &lt;a href=&quot;http://www.springer.com/us/book/9783319110790&quot; rel=&quot;nofollow&quot;&gt;&quot;Linear Algebra done Right&quot;&lt;/a&gt; (Springer, 3rd edition 2015).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-02-22T14:12:36.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10612" PostTypeId="2" ParentId="10610" CreationDate="2016-02-22T15:33:26.377" Score="7" Body="&lt;p&gt;With regard to &quot;mathematics as story&quot; in the context of mathematics education, one place you might look to is the work of Leslie Dietiker (&lt;a href=&quot;https://scholar.google.com/citations?user=qN5-AgwAAAAJ&amp;amp;hl=en&amp;amp;oi=ao&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;google scholar&lt;/strong&gt;&lt;/a&gt;) on &lt;em&gt;mathematics curriculum as story&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, here is the abstract from Dietiker's (2015) &lt;a href=&quot;https://www.researchgate.net/profile/Leslie_Dietiker/publication/281477033_Mathematical_story_a_metaphor_for_mathematics_curriculum/links/560c29e708aea68653d357f6.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Mathematical story: a metaphor for mathematics curriculum&lt;/strong&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/u2RMV.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/u2RMV.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also check the more recent piece, Dietiker's (2016) &lt;a href=&quot;http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1309&amp;amp;context=jhm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Role of Sequence in the Experience of Mathematical Beauty&lt;/strong&gt;&lt;/a&gt;, which investigates a first grade lesson and its aesthetics by framing the lesson as a &lt;em&gt;mathematical story&lt;/em&gt; in keeping with the previous article. For example, consider the following excerpt, for which the citations may be of interest, as well (pp. 153-154):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3s8iu.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3s8iu.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The latter article ends with a nice quotation that I interpret as being in keeping with your general query. Specifically, Dietiker writes (p. 171):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In other words, does there exist a mathematical story that would similarly captivate students with a non-special number, such as $68$? Or with a function, such as $y = -\frac{1}{2}x + 6$? In mathematics education, when a mathematical lesson is neither stimulating nor interesting, is it a result of a poorly designed mathematical story or is it the result of poor delivery?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Rather separately, I also point back to my response in &lt;a href=&quot;https://matheducators.stackexchange.com/a/2189&quot;&gt;&lt;strong&gt;MESE 2164&lt;/strong&gt;&lt;/a&gt; on &lt;em&gt;reading&lt;/em&gt; mathematics textbooks, and, in particular, how reading a &quot;math book&quot; is or is not similar to reading a novel.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-02-22T15:33:26.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10613" PostTypeId="2" ParentId="9760" CreationDate="2016-02-22T19:25:53.603" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://ipkl.gu.se/english/Research/research_projects/codac/fingu&quot; rel=&quot;nofollow&quot;&gt;Fingu&lt;/a&gt; is an iPad game developed by a group of researchers in Kristianstad and Gothenburg, Sweden.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The game aims to improve 4-8 year old children’s basic arithmetic&#xA;  competencies but can even be played and enjoyed by younger and older&#xA;  children as well as adults. The competencies that are trained are:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;The ability to ‘see’ numbers of objects without counting them, so called ‘subitizing’&lt;/li&gt;&#xA;  &lt;li&gt;The ability to ‘feel’ a representation of the numbers up to 10 in your hands&lt;/li&gt;&#xA;  &lt;li&gt;Coordinating seeing the number of objects with showing it with your fingers&lt;/li&gt;&#xA;  &lt;li&gt;The ability to add two numbers with sums up to 10&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="39" LastActivityDate="2016-02-22T19:25:53.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10615" PostTypeId="2" ParentId="10597" CreationDate="2016-02-23T18:13:06.683" Score="11" Body="&lt;p&gt;Have a look at the paper written by Nunez et all: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://link.springer.com/article/10.1023%2FA%3A1003759711966&quot;&gt;EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In essence, they argue that it is better to be causious if you want to &quot;motivate the formal definition of continuity starting from the intuition&quot; you have suggested in your question. In the following passage, &quot;natural continuity&quot; refers to drawing without lifting the pen.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For the purposes of this article, the pedagogical problem can be&#xA;  summarized as follows: students are introduced to natural continuity&#xA;  using concepts, ideas, and examples which draw on inferential patterns&#xA;  sustained by the natural human conceptual system. Then, they are&#xA;  introduced to another concept – Cauchy-Weierstrass continuity – that&#xA;  rests upon radically different cognitive contents (although not&#xA;  necessarily more complex). These contents draw on different&#xA;  inferential structures and different entailments that conflict with&#xA;  those from the previous idea. The problem is that students are never&#xA;  told that the new definition is actually a completely different&#xA;  human-embodied idea. &lt;strong&gt;Worse, they are told that the new definition&#xA;  captures the essence of the old idea&lt;/strong&gt;, which, by virtue of being&#xA;  ‘intuitive’ and vague, is to be avoided. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If I had more time, I would summarize their argument. For the time being, I make this a community wiki to invite other people to complete this answer. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2016-02-23T18:13:06.683" CommentCount="0" CommunityOwnedDate="2016-02-23T18:13:06.683" ContentLicense="CC BY-SA 3.0" />
  <row Id="10616" PostTypeId="2" ParentId="9760" CreationDate="2016-02-23T23:53:17.120" Score="1" Body="&lt;p&gt;Suggest Monkey Math School Sunshine available in the  Google Play.&#xA;Although it is recommended for ages 5 and up, I believe your child would be ready for it.  I think your child would enjoy it and could grow with it,&#xA;According to the review on Common Sense Media:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Parents need to know that Monkey Math School Sunshine is an early math&#xA;  education game where kids do a series of math problems, covering&#xA;  topics such as shapes, number recognition, counting, patterns,&#xA;  addition, subtraction, comparisons, writing numbers, and recognizing&#xA;  sets. When kids have trouble, the game guides them with gentle hints.&#xA;  The questions are of varying difficulty. Some of the questions are&#xA;  presented in a straightforward manner, while others are more like&#xA;  games. After kids answer enough questions, they can choose a prize for&#xA;  their aquarium. The aquarium starts out empty, and kids get to fill it&#xA;  with fish, plants, and decorations. Kids can revisit their aquarium at&#xA;  any time. The aquarium can be reset to allow another child to have a&#xA;  go. Seeing their aquarium filled with the prizes they chose helps to&#xA;  encourage kids to solve more problems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-02-23T23:53:17.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10620" PostTypeId="1" CreationDate="2016-02-25T17:03:30.787" Score="24" ViewCount="7275" Body="&lt;p&gt;Let's assume our students are actual aspiring mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why would we introduce our students to Calculus rather than Real Analysis?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After all, &quot;Calculus is a subset of Real Analysis&quot;. He will have to learn everything that he had already learned in Calculus once again. In addition to this, the student learns about mathematical concepts like axioms or proofs and mathematical working in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's like saying: &quot;We assume you're stupid, so learn Calculus first.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I think this will make math harder instead of easier. For instance, If we just assume the real numbers as they are but do not show where they come from and how they can be constructed, the student will have a harder time understanding the key concepts of calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's take the extreme case, where we just tell the student the differentiation rules. This might seem to be easier. But the student will ask himself what he's actually doing and why. Because of the missing theory, he can't grasp it. Because of that he will maybe see himself as a failure or even start hating math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, historically we did not have any formalism. But our students aren't people from 2000 B.C.&lt;/p&gt;&#xA;" OwnerUserId="6385" LastEditorUserId="376" LastEditDate="2016-09-19T06:25:22.943" LastActivityDate="2017-08-16T06:54:09.530" Title="Why would you teach Calculus before teaching Real Analysis?" Tags="|calculus|mathematical-analysis|" AnswerCount="10" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10621" PostTypeId="2" ParentId="10620" CreationDate="2016-02-25T17:31:00.173" Score="61" Body="&lt;p&gt;You may as well ask:  Why teach elementary school children how to perform whole-number arithmetic without teaching them the Peano axioms first?  Why teach high-school Algebra without starting with the basics of groups, rings and fields?  Why, for that matter, teach children to read first, instead of starting with the fundamentals of grammar and linguistics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everything we know about how learning takes place tells us that &lt;em&gt;complicated ideas&lt;/em&gt; are formed and absorbed by building them on a foundation of &lt;em&gt;simpler ones&lt;/em&gt;.  It is important to realize that &quot;simpler&quot; does not mean here &quot;more fundamental&quot;, i.e. logically more primary; rather &quot;simpler&quot; is here used in the sense of &lt;strong&gt;&quot;easier&quot;&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A human being is not a formal system, and our minds do not develop according to deductive principles.  Almost nobody understands &lt;em&gt;anything&lt;/em&gt; complicated the first time they encounter it; we master things by revisiting them, over and over again, gaining deeper insight into it each time.  We teach people Calculus first because if we taught them Analysis first they would have no experiential or conceptual substrate on which to build.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edited to add:&lt;/strong&gt;  In addition to these points, I'd like to echo (and somewhat restate) an observation made (by Jon Bannon and by Austin) in the comments below the OP's post.  The opening sentence:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let's assume our students are actual aspiring mathematicians.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;describes a situation that I think needs closer scrutiny.  In my experience, anybody who is an aspiring mathematician has &lt;em&gt;already taken Calculus&lt;/em&gt;.  It is extremely rare, I think, for a person to formulate such an aspiration on the basis of nothing more than high school Algebra and Geometry.  I am not sure there is data on this, but I expect most people who eventually become mathematicians did not reach that decision until after at least one semester of Calculus, possibly much more.  So the question essentially addresses a hypothetical audience that does not exist.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2016-02-26T03:55:40.973" LastActivityDate="2016-02-26T03:55:40.973" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10622" PostTypeId="2" ParentId="7826" CreationDate="2016-02-25T17:34:33.570" Score="-2" Body="&lt;p&gt;Fractions are easier to relate to life and hence, should be easier to understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with slices of pizza, which even a first grader can understand, that if he/she ate one slice out of a total of eight, 1/8 of pizza was eaten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next is the concept of equivalency. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any subtraction or additional expression can be evaluated once you understand the concept of LCM(least common multiple of two numbers) and use to it turn the denominator to a common multiple of the both/many denominators of various fractions. Same concept is applied while comparing the fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Visit &lt;a href=&quot;https://www.biglearners.com/worksheets/grade-5/math/fractions&quot; rel=&quot;nofollow&quot;&gt;https://www.biglearners.com/worksheets/grade-5/math/fractions&lt;/a&gt; to practice the same basic concepts.&lt;/p&gt;&#xA;" OwnerUserId="6386" LastEditorUserId="6386" LastEditDate="2016-02-25T17:40:06.817" LastActivityDate="2016-02-25T17:40:06.817" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10623" PostTypeId="2" ParentId="10620" CreationDate="2016-02-25T19:10:05.587" Score="9" Body="&lt;p&gt;In many respects this is what is done in European universities, where the degree is often a three year program. First year students reading maths take analysis. Of course they probably learned calculus as part of their A-levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Years ago, as a US citizen I did a first degree as a math major in the United States, and then did a second undergraduate degree in Oxford, reading maths. I remember being amused that many of the maths students, who were very bright and talented, seemed to know little about the Riemann integral since the British university curriculum started with the Lebesgue integral. On the other hand, I was very comfortable with the Riemann integral but had never heard of the Lebesgue integral.   &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-02-25T19:10:05.587" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10624" PostTypeId="2" ParentId="10620" CreationDate="2016-02-25T19:23:22.417" Score="18" Body="&lt;p&gt;In addition to other good answers, I do think that what &quot;calculus&quot; usually refers to is what people &lt;em&gt;did&lt;/em&gt; with calculus prior to about 1830, as opposed to the &lt;em&gt;foundations&lt;/em&gt; of it. Namely, they solved a lot of physically meaningful/intuitive problems, as well as geometrical problems, problems in mechanics, celestial mechanics, fluid flow, ... and beginning of applications to number theory, as well as the beginnings of complex analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, for me, and I suspect for many, that notion of &quot;calculus&quot; is very persuasive that it is &lt;em&gt;useful&lt;/em&gt;, whatever its foundational interest or challenges may be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, we could ask why we should care about the axioms for something if some sort of utility... whether to physics or to geometry or to number theory, let's say! :) ... hasn't been demonstrated. Sure, study of axiomatic systems has some intrinsic interest, but perhaps more so after one sees why we'd care to be sure our machinery works as specified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For that matter, some of those &quot;applications&quot; help us detect &quot;stress points&quot; in the theoretical development.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-02-25T19:23:22.417" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10625" PostTypeId="2" ParentId="10620" CreationDate="2016-02-26T00:27:33.860" Score="6" Body="&lt;p&gt;Learning is easiest and the most enjoyable when it answers questions one already had. A student who has played around with functions before, and wondered how he might find their maxima, or wondered about how to express the amount by which something changes when some other thing changes, will be thrilled to learn of the powerful techniques of calculus which solve just that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you try to teach someone the rigor of formal real analysis, he will find it boring, pointless and difficult. Unless, of course, he has already grown a healthy appreciation for calculus, so that he can appreciate analysis for being the foundation of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since calculus is the foundation of the physical world, you definitely want to teach it as early as humanly possible. In my humble opinion, students - and especially those who are curious, enthusiastic, and mathematically oriented - should be taught calculus quite early. Failure to teach calculus early enough is, IMO, a problem with the curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So instead of teaching analysis when we're now teaching calculus, we should just teach calculus earlier - when students are still too young and inexperienced to follow rigorous analysis. It's really &quot;We assume you're a kid, so learn Calculus first.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;He will have to learn everything that he had already learned in Calculus once again&quot; part also makes no sense - if indeed those are the same things, then the second time will be that much easier for having learned this before, so you can spend less time on it. Besides, learning isn't a one-off thing, to truly understand and retain knowledge you have to approach it several times, from different angles.&lt;/p&gt;&#xA;" OwnerUserId="6392" LastEditorUserId="6392" LastEditDate="2016-02-26T12:33:17.563" LastActivityDate="2016-02-26T12:33:17.563" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10626" PostTypeId="1" CreationDate="2016-02-26T03:53:02.597" Score="8" ViewCount="197" Body="&lt;h2&gt;If rigor doesn't mean more challenging problems, then what does it mean?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;There is a big push for rigor in common core mathematics, but I'm not sure exactly what rigor means (I'm pretty sure it has to do with precision, which would be ironic). &lt;/p&gt;&#xA;" OwnerUserId="6394" LastEditorUserId="39" LastEditDate="2016-02-26T07:25:59.693" LastActivityDate="2016-03-02T20:53:02.617" Title="Rigor in secondary mathematics" Tags="|mathematical-pedagogy|secondary-education|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10627" PostTypeId="2" ParentId="10626" CreationDate="2016-02-26T06:13:42.687" Score="9" Body="&lt;p&gt;Rigor refers to use of the axiomatic method -- being aware of what initial definitions are taken, and being explicit about the chain of logical reasoning that one uses to achieve results. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Rigour&quot;&gt;Wikipedia&lt;/a&gt; says this about intellectual rigor in general:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Intellectual rigour is a process of thought which is consistent, does&#xA;  not contain self-contradiction, and takes into account the entire&#xA;  scope of available knowledge on the topic. It actively avoids logical&#xA;  fallacy. Furthermore, it requires a skeptical assessment of the&#xA;  available knowledge. If a topic or case is dealt with in a rigorous&#xA;  way, it means that it is dealt with in a comprehensive, thorough and&#xA;  complete way, leaving no room for inconsistencies.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Scholarly method describes the different approaches or methods which&#xA;  may be taken to apply intellectual rigour on an institutional level to&#xA;  ensure the quality of information published. An example of&#xA;  intellectual rigour assisted by a methodical approach is the&#xA;  scientific method, in which a person will produce a hypothesis based&#xA;  on what they believe to be true, then construct experiments in order&#xA;  to prove that hypothesis wrong. This method, when followed correctly,&#xA;  helps to prevent against circular reasoning and other fallacies which&#xA;  frequently plague conclusions within academia. Other disciplines, such&#xA;  as philosophy and mathematics, employ their own structures to ensure&#xA;  intellectual rigour. Each method requires close attention to criteria&#xA;  for logical consistency, as well as to all relevant evidence and&#xA;  possible differences of interpretation. At an institutional level,&#xA;  Peer review is used to validate intellectual rigour.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And this about mathematical rigor in specific:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematical rigour can be defined as amenability to algorithmic proof&#xA;  checking. Indeed, with the aid of computers, it is possible to check&#xA;  some proofs mechanically.[4] Formal rigour is the introduction of high&#xA;  degrees of completeness by means of a formal language where such&#xA;  proofs can be codified using set theories such as ZFC (see automated&#xA;  theorem proving).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Most mathematical arguments are presented as prototypes of formally&#xA;  rigorous proofs. The reason often cited for this is that completely&#xA;  rigorous proofs, which tend to be longer and more unwieldy, may&#xA;  obscure what is being demonstrated. Steps which are obvious to a human&#xA;  mind may have fairly long formal derivations from the axioms. Under&#xA;  this argument, there is a trade-off between rigour and comprehension.&#xA;  Some argue that the use of formal languages to institute complete&#xA;  mathematical rigour might make theories which are commonly disputed or&#xA;  misinterpreted completely unambiguous by revealing flaws in reasoning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the context of Common Core objectives, rigor has been defined as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Rigor:&lt;/strong&gt; In major topics, pursue &lt;strong&gt;conceptual understanding&lt;/strong&gt;, procedural skill and &lt;strong&gt;fluency&lt;/strong&gt;, and &lt;strong&gt;application&lt;/strong&gt; with equal&#xA;  intensity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See the document below for several examples in mathematics education, each of which emphasizes understanding/explaining the reasoning (in addition to computational fluency):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.louisianabelieves.com/docs/common-core-state-standards-resources/guide---math-ccss-self-learning-module-3.pdf?sfvrsn=4&quot;&gt;Louisiana Teacher Self-Learning Series&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-02-26T06:22:41.517" LastActivityDate="2016-02-26T06:22:41.517" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10628" PostTypeId="2" ParentId="10620" CreationDate="2016-02-26T06:27:52.187" Score="5" Body="&lt;p&gt;I think calculus is actually more important for a working analyst than 'real' real analysis. You can (and should) always learn more about why you are allowed to do things and how they can be constructed from simple principles but to get into the flow of figuring things out yourself, you cannot have yourself looking up the divergence theorem, substitution rule, or Cauchy-Schwarz. You have to be at absolute ease (and intuition) with all kinds inequalities, ODE, series, integrals. No matter how 'easy' those are conceptually, and no matter if you do stochastic processes, dynamical systems, modelling, statistics, or whatever other applied mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer: This of course is coming from a child of the German curriculum that never learned Cauchy-Schwarz in an analysis class  'because you did it in linear algebra' (and how we did it. 'Take this miraculous expression, insert it into this sesquilinear form, and there it is'. Duh), that learned the divergence theorem without having the slightest clue what a divergence is, that took two classes in PDE and knew topological vector spaces, negative order Sobolev spaces, real interpolation before learning about, say, the transport equation (what? there are PDE that cannot be solved with Lax-Milgram?? I actually once took a class abroad and in a mid-term exam solved a hyperbolic equation ending with 'all that is left to be done is to show cooercivity') )&lt;/p&gt;&#xA;" OwnerUserId="6395" LastActivityDate="2016-02-26T06:27:52.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10629" PostTypeId="2" ParentId="10620" CreationDate="2016-02-26T13:21:28.570" Score="19" Body="&lt;p&gt;&lt;strong&gt;Because your core assumption is bad and most students are not aspiring mathematicians&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your opening statement  - &quot;Let's assume our students are actual aspiring mathematicians&quot; - is almost certainly a very bad assumption to make about students at the age where calculus is taught. In fact, very few of the students who are taught calculus will go to study mathematics at degree level and fewer still will go on to become mathematicians. So Mathematics (like other subjects) needs to be taught in a way that is beneficial to students who do not go on to specialise in it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus is used as a tool in a wide range of subjects such as Biology, Physics and Engineering and many more students will go on to study these subjects than will go on to study Mathematics. Calculus, as a skill, is thus much more useful to a larger group of your students than Real Analysis is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the pedagogical arguments made in other answers also hold true but I think the error in your core assumption is more critical.&lt;/p&gt;&#xA;" OwnerUserId="6399" LastActivityDate="2016-02-26T13:21:28.570" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10630" PostTypeId="2" ParentId="176" CreationDate="2016-02-26T14:32:12.960" Score="6" Body="&lt;p&gt;As for examples: I taught gifted students for many years. Some were ahead of others and I would sometimes ask them to work together and teach each other. Some students were natural teachers but others struggled. The ones who struggled would sometimes give up and say they can't explain anything and at other times just give step by step instructions of what to do.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have heard that if you can't explain math - you don't really understand it; if you can explain it you have a deeper understanding.  This idea has been .repeated at workshops I've attended, by my administration, and throughout my elementary curriculum.  From experience- I have seen gifted math students who can't explain what they are doing (and therefore can't teach) However through repeated questioning  their understanding and ability to explain will come through.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-02-26T14:32:12.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10631" PostTypeId="2" ParentId="993" CreationDate="2016-02-26T15:05:57.413" Score="2" Body="&lt;p&gt;&quot;Infinitesimal Calculus&quot; by James Henle&#xA;A short but complete introduction to Calculus which starts by defining a hyper-real number system. Very enjoyable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had the pleasure of taking a course using this as a textbook, taught by the author's brother, Michael Henle, at Oberlin College. It was not my first calculus course though. I found the hyper-real number system stuff fascinating. The use of it as a basis for Calculus was boring to me at that point.&lt;/p&gt;&#xA;" OwnerUserId="1854" LastActivityDate="2016-02-26T15:05:57.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10632" PostTypeId="1" AcceptedAnswerId="10633" CreationDate="2016-02-27T01:45:18.397" Score="11" ViewCount="236" Body="&lt;p&gt;Let's assume that a group of students need to learn to solve a certain type of mathematical problem for which there is two general methods of solving it, $X$ and $Y$. We also assume that $Y$ is more conceptually difficult to learn than $X$, but once understood, $Y$ makes the problem far easier to solve. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in teaching a group of students who are capable of learning $Y$, is there any educational benefits of teaching them how to solve problems using $X$ first?&lt;/p&gt;&#xA;" OwnerUserId="5746" LastActivityDate="2016-03-03T19:25:39.590" Title="Pedagogical Purpose in Making Students Do Problems in A Less Efficient Way First" Tags="|undergraduate-education|mathematical-pedagogy|curriculum|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10633" PostTypeId="2" ParentId="10632" CreationDate="2016-02-27T03:48:54.153" Score="13" Body="&lt;p&gt;&lt;strong&gt;Yes.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The primary reason in my mind is that you want to have multiple procedures with which you can double-check work later. Knowing only a single procedure makes for very fragile understanding; on the day that you make an error, you have no way of recognizing or fixing that error. So I would claim that you want &lt;em&gt;at least two or three methods&lt;/em&gt; of sanity-checking an answer. (E.g., a rough estimation, an algebraic solution, and a graphing solution.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, in many places you would use the initial, tedious procedure as motivation and appreciation for how much more powerful and elegant the later procedure is. Likewise if students have difficulty understanding the sophisticated procedure, then $X$ can be used to confirm that $Y$ does give correct results. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thirdly, if you have a disagreement with someone over a result, you should have a more basic shared understanding that you can fall back on to similarly check that the answer is correct another way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, in my community-college classes I find that students frequently do resist this, and either zone out on the secondary procedure, or occasionally become actively hostile. Actually, just this week I updated my statistics final (which I've been using for a few years) to require graphing checks of all the problems. Previously that was only recommended, not required -- the weaker students skipped it, basically just hoping they were right, and had no way of knowing how far off they were in most cases. I'm hoping if they know that the secondary check will be required, they will then be forced to develop that skill, have better intuition, and produce better results. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-02-27T05:30:54.587" LastActivityDate="2016-02-27T05:30:54.587" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10634" PostTypeId="2" ParentId="570" CreationDate="2016-02-27T15:42:18.847" Score="0" Body="&lt;p&gt;I am very curious to learn the flaw in a Beth El &quot;positive real numbers as pairs of reciprocals&quot; diagram which seems to imply there are right triangles to which the Pythagorean Theorem may not always apply. More specifically, the diagram at HTTP//Mobius-ity.us/BENums.png uses an unorthodox but seemingly mathematically valid way of portraying the positive real number line. It does so by using a horizontal base part of the positive real number line that goes from 0 and comes to 1 and then it coninues on up by using a vertical elevation part of the positive real number line that vertically goes from 1 and comes towards infinity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If, e.g., per the diagram, you use the value x=2, you get a 1/2, 1, 2 right triangle, one for which the length of its hypotenuse seems to be the taxicab metric 2 - 1/2, not the Pythagorean length. That seems to be so because each such right triangle's three vertices are by definition on a single one-dimensional line, even though that line looks two-dimensional IF and when you choose to disregard the important fact that the right side of each such triangle begins at 1 and not 0.&lt;/p&gt;&#xA;" OwnerUserId="6408" LastActivityDate="2016-02-27T15:42:18.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10635" PostTypeId="2" ParentId="570" CreationDate="2016-02-27T16:15:27.933" Score="5" Body="&lt;p&gt;If this is already posted, my apologies, but, I didn't see it. I'm fond of this example as a warning against following intuition through limits:&lt;a href=&quot;https://i.stack.imgur.com/jKx6W.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jKx6W.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; The idea is that if you follow the purely horizontal or purely vertical paths in a manner which goes like a staircase from bottom to top then the total path length is always $3+4$. Imagine this picture continuing to make the stairs close to the hypotenuse smaller and smaller then as this process continues we obtain stair-steps arbitrarily close to hypotenuse. But, the path length of this sequence of paths is the constant $7$. The limit of a constant sequence is constant hence the length of limiting path of the sequence of paths must also be $7$. Furthermore, &lt;strong&gt;intuitively it is geometrically clear from the picture&lt;/strong&gt; the path limit is identical to the hypotenuse. The length of the hypotenuse is given by $\sqrt{3^2+4^2} = 5$. Thus we have proved, $5=7$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course this is not my idea, I'm not sure where this originates. Another nice picture for this is found at &lt;a href=&quot;http://www.futilitycloset.com/2009/07/21/pythagoras-disproved/&quot; rel=&quot;noreferrer&quot;&gt;futility closet, pthagoras disproved&lt;/a&gt;. I like setting $a=3$, $b=4$ to make it easier for students to follow.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2016-02-27T16:15:27.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10636" PostTypeId="1" AcceptedAnswerId="10641" CreationDate="2016-02-27T16:31:35.953" Score="20" ViewCount="3626" Body="&lt;p&gt;I'd like to give my Calculus 1 class periodic multiple choice questions that really test conceptual understanding. Ideally, I'd like these questions to require very little computation. I know that a lot of textbooks have true false questions, which I like, but I'm hoping to find a source of questions with more than two possible answers. Something along the following lines:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; are such that &lt;span class=&quot;math-container&quot;&gt;$f'(x)&amp;gt;g'(x)$&lt;/span&gt; for all real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Which of the following statements must be true?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The graphs of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; intersect exactly once.&lt;/li&gt;&#xA;&lt;li&gt;The graphs of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; intersect no more than once.&lt;/li&gt;&#xA;&lt;li&gt;The graphs of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; do not intersect.&lt;/li&gt;&#xA;&lt;li&gt;The graphs of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; may intersect any number of&#xA;times.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For a student who understands the derivative well, this question could be answered in under one minute with absolutely no computations.&lt;/p&gt;&#xA;&lt;p&gt;Anyone know of a good source to find a bank of such questions? It would be ideal if the TeX code was available too, or if the problems were already encoded into WeBWorK or some other online homework system.&lt;/p&gt;&#xA;" OwnerUserId="21" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2023-09-03T15:21:58.030" Title="Source of conceptual, multiple choice calculus questions" Tags="|calculus|test-design|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="10637" PostTypeId="2" ParentId="7837" CreationDate="2016-02-27T17:35:34.193" Score="6" Body="&lt;p&gt;So far no-one has mentioned the &lt;em&gt;double number line&lt;/em&gt; representation of division. I think this works best in an &quot;equal sharing&quot; context, so I'll add such an example.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Question: If I travel (with my bike) $14$ kilometers in $7/12$ of an hour, what is my average speed?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Answer: We can use proportional reasoning to figure out how many kilometers in one hour this corresponds to. The situation is represented on the double number line below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/h62Pb.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/h62Pb.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not easy to see the answer directly. As an intermediate step we divide by $7$ and find the distance traveled in $1/12$ of an hour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jPMO4.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jPMO4.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that we can multiply by $12$,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PsroS.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PsroS.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and get the answer $\boxed {24}$ kilometers per hour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comment: What we have done is of course the division $14 \div \tfrac 7 {12}$ and found it to be equivalent to &quot;$14$ divided by $7$ and then multiplied by $12$&quot;. &lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2016-02-27T17:35:34.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10638" PostTypeId="2" ParentId="10636" CreationDate="2016-02-27T17:38:20.420" Score="5" Body="&lt;p&gt;I tend to download testbanks for Pearson's TestGen application (requires &lt;a href=&quot;http://www.pearsonhighered.com/educator/main_content/instructor_resources_download_info.page&quot; rel=&quot;noreferrer&quot;&gt;instructor account at Pearson&lt;/a&gt;, available for almost any text they produce) and hunt through them for conceptual questions for this purpose. Sometimes it's a bit slim pickings in this regard, but it at least gets me started for a first semester, and then more ideas occur to me as I teach the course, and I personalize the quizzes more over time. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-02-27T19:00:39.280" LastActivityDate="2016-02-27T19:00:39.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10639" PostTypeId="1" CreationDate="2016-02-28T13:02:31.030" Score="3" ViewCount="212" Body="&lt;p&gt;I'm teaching at high school level, and the next topic will be differential equations. I would like my students to be able to solve some simple linear and non-linear ODEs using the ansatz &lt;span class=&quot;math-container&quot;&gt;$x(t) = e^{\lambda t}$&lt;/span&gt;, separating variables, varying the constant, etc. Also, I would like to treat both homogeneous and inhomogeneous linear differential equations. Lastly, I'd like to finish with some simple &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; systems of differential equations.&lt;/p&gt;&#xA;&lt;p&gt;Many of these sections are strongly connected or can be applied to other sections, so I am having troubles choosing a good order in which I should teach the topics. The first order I've come up with is as follows:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Definitions and many examples&lt;/li&gt;&#xA;&lt;li&gt;Direct integration&lt;/li&gt;&#xA;&lt;li&gt;Separating variables&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x(t) = e^{\lambda t}$&lt;/span&gt; ansatz&lt;/li&gt;&#xA;&lt;li&gt;Principle of superposition&lt;/li&gt;&#xA;&lt;li&gt;Varying the constant&lt;/li&gt;&#xA;&lt;li&gt;Inhomogeneous linear ODEs&lt;/li&gt;&#xA;&lt;li&gt;Systems in matrix notation and how to solve them&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; Is there anything important that is missing?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; Is this order fine, or would you go for a (completely) different one? If so, why?&lt;/p&gt;&#xA;" OwnerUserId="493" LastEditorUserId="7365" LastEditDate="2022-11-10T15:47:57.420" LastActivityDate="2022-11-10T15:47:57.420" Title="In what order should I teach methods for solving (linear) ODEs?" Tags="|secondary-education|differential-equations|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="10640" PostTypeId="2" ParentId="10639" CreationDate="2016-02-28T13:24:00.977" Score="4" Body="&lt;p&gt;You left out linear first order ODE's, with their integrating factors. I use this often, compared to the other techniques. This probably should be done just after separation of variables, though I have seen it done before that topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see why you would want to delay this topic until after you study inhomogeneous linear ODE's, for consistency. But the importance of this topic argues for an earlier look, even at the expense of this consistency. After all, you also put separation of variables early.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2016-02-28T13:24:00.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10641" PostTypeId="2" ParentId="10636" CreationDate="2016-02-28T19:04:23.663" Score="19" Body="&lt;p&gt;Cornell's Good Questions Project has a great question bank for &lt;a href=&quot;https://pi.math.cornell.edu/%7EGoodQuestions/materials.html#GQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;conceptual questions&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-09-03T15:21:58.030" LastActivityDate="2023-09-03T15:21:58.030" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="10642" PostTypeId="2" ParentId="10620" CreationDate="2016-02-28T19:09:12.123" Score="5" Body="&lt;p&gt;I took an honors calculus course in my first year at the University of Michigan (and my high school didn't have calculus). It was truly an analysis course. It was so far from how I was used to thinking that it felt over my head.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The simple beauty of calculus was lost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I love teaching calculus, and I often point out to the students where we are leaving out details that they'll encounter in analysis if they go on in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I emphasize velocity, slope, rate of change, graphing, and optimization.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2016-02-28T19:09:12.123" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10643" PostTypeId="2" ParentId="10639" CreationDate="2016-02-28T20:19:30.980" Score="3" Body="&lt;p&gt;i think it would be nice for them to see the geometric idea of the integral curve through plotting the slope field. it works well for autonomous equations like the exponential growth and logistic model. i think it is important to do the first order equation really well and spend good amount of time in it; it pays dividend later when you are dealing with matrix equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;here is an example. i first deal with $\frac{dx}{dt} = ax + f$ using undetermined coefficients and variation of parameters in finding particular solution. in fact the variation of parameter formula $x = \int_0^t e^{a(t-s)}f(s) \, ds$ readily extends to $\frac{dx}{dt} + Ax + f.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;the other system to spend time is the conservative system $\frac{d^2x}{dt^2} = -\frac{dV}{dx}.$ here the pendulum equation comes in handy.&lt;/p&gt;&#xA;" OwnerUserId="3441" LastEditorUserId="3441" LastEditDate="2016-02-29T20:54:22.580" LastActivityDate="2016-02-29T20:54:22.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10644" PostTypeId="2" ParentId="10636" CreationDate="2016-02-29T02:39:36.450" Score="5" Body="&lt;p&gt;Though not nearly as good as some of the other suggestions here, the &lt;a href=&quot;https://apstudent.collegeboard.org/apcourse/ap-calculus-ab/exam-practice&quot; rel=&quot;noreferrer&quot;&gt;practice exams for AP Calculus&lt;/a&gt; contain decent problems. The sample multiple choice ones are buried inside the &quot;AP Calculus Course Description&quot; pdf.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2016-02-29T02:39:36.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10645" PostTypeId="1" CreationDate="2016-02-29T16:29:16.927" Score="6" ViewCount="220" Body="&lt;p&gt;A search in Google returns lots of studies and thinkings about teaching in undergraduate schools. Could anybody come up with any research/references about &lt;strong&gt;teaching in graduate schools&lt;/strong&gt; (graduate courses taught by professors)? (When I search &quot;teaching in graduate schools&quot; in Google, what returns is mostly about teaching in undergraduate school by graduate students, which is &lt;em&gt;not&lt;/em&gt; the topic I'm talking about. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A related book I have found so far is &lt;a href=&quot;http://www.hup.harvard.edu/catalog.php?isbn=9780674728981&quot; rel=&quot;nofollow&quot;&gt;The Graduate School Mess: What Caused It and How We Can Fix It&lt;/a&gt;. But it is not directly about graduate education in mathematics.)&lt;/p&gt;&#xA;" OwnerDisplayName="user5897" LastEditorDisplayName="user5897" LastEditDate="2016-10-03T23:28:47.270" LastActivityDate="2016-10-03T23:28:47.270" Title="References for graduate education" Tags="|reference-request|graduate-education|teaching|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10646" PostTypeId="2" ParentId="10645" CreationDate="2016-02-29T17:02:34.050" Score="2" Body="&lt;p&gt;You might want to look at the book &lt;em&gt;One Field, Many Paths: U. S. Doctoral Programs in Mathematics Education&lt;/em&gt; (Reys &amp;amp; Kilpatrick, eds.).  (See &lt;a href=&quot;http://www.ams.org/books/cbmath/009/&quot; rel=&quot;nofollow&quot;&gt;http://www.ams.org/books/cbmath/009/&lt;/a&gt;).  I don't have access to the book right now, but the table of contents suggests that it may contain relevant research on your question.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-02-29T17:02:34.050" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10647" PostTypeId="2" ParentId="10645" CreationDate="2016-02-29T19:19:30.513" Score="4" Body="&lt;p&gt;As far as I remember Krantz's book &lt;a href=&quot;http://bookstore.ams.org/mbk-89&quot; rel=&quot;nofollow&quot;&gt;How to Teach Mathematics&lt;/a&gt; is not restricted to undergraduate courses and contains a lot of great advice. His book &lt;a href=&quot;http://www.maa.org/press/books/a-mathematician-comes-of-age&quot; rel=&quot;nofollow&quot; title=&quot;A mathematician comes of age&quot;&gt;A Mathematician Comes of Age&lt;/a&gt; is not focused on teaching but on &quot;mathematical maturity&quot; and how to develop it. Consequently, it contains many thoughts about teaching and mentoring on the graduate level.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="3398" LastEditDate="2016-02-29T19:54:35.190" LastActivityDate="2016-02-29T19:54:35.190" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10649" PostTypeId="2" ParentId="10636" CreationDate="2016-03-01T15:38:32.097" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://www.wileyplus.com/WileyCDA/Section/Calculus-Single-and-Multivariable-6th-Edition.id-816385.html?descCd=description&quot; rel=&quot;noreferrer&quot;&gt;The Calculus textbook of Hughes-Hallett, Gleason and McCallum et. al.&lt;/a&gt; includes ConcepTests that are rather good for this purpose. Sorry this is not a free online resource, but I hope it helps.&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2016-03-01T15:38:32.097" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10650" PostTypeId="2" ParentId="570" CreationDate="2016-03-01T15:59:21.217" Score="4" Body="&lt;p&gt;I believe the following &quot;&lt;a href=&quot;https://mathoverflow.net/a/27265/29316&quot;&gt;common false belief&lt;/a&gt;&quot; suggested by Tao as an answer to this now famous &lt;a href=&quot;https://mathoverflow.net/q/23478/29316&quot;&gt;Gowers' MO question&lt;/a&gt; can be also mentioned here.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The closure of the open ball of radius r in a metric space, is the&#xA;  closed ball of radius r in that metric space.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I guess if you check other answers of the Gowers' question, there would be some more of these.  &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2016-03-01T15:59:21.217" CommentCount="0" CommunityOwnedDate="2016-03-01T15:59:21.217" ContentLicense="CC BY-SA 3.0" />
  <row Id="10652" PostTypeId="1" CreationDate="2016-03-01T22:03:52.507" Score="13" ViewCount="306" Body="&lt;p&gt;I am teaching a course for sophomore-level math majors this quarter, and next quarter, I'll be teaching the same thing. I haven't been in this situation before, and I'm wondering to what degree I can reuse exercises from this quarter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the one hand, I provide detailed solutions to all the homework I assign, and it seems likely that these solutions will find their way to at least some of next quarter's students, one way or another. And using the same (or almost the same) exercises feels lazy on my part. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, a lot of these exercises were chosen for a good reason, and I consider them important. Sometimes I can change some numbers to get a different problem with the same flavor, but this is often not possible in a proof-based course like this. And most of the exercises were taken from the (well-known) course textbook, for which the solutions can surely be found online anyway. Even for the exercises I wrote myself, the students can ask for help on math.stackexchange or the like, or copy their friend's homework from the same class. For these reasons, I already only have homework counting 20% toward the final grade. But each exam contains one question quoted verbatim from their homework (and I warn them about this), so it's in their interest to understand the solution whether they came up with one honestly or not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that to say, I understand not all students will be 100% ethical with their homework, and I try to work around this fact rather than fight it. But does giving homework assignments that are say 75% the same as the assignments from the previous quarter make it too easy to be unethical?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would be &quot;due diligence&quot; in this situation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(When it comes to exams, I already know what I'll do: distribute last quarter's exam as a practice exam a week ahead of time, and distribute solutions three days ahead of time, so that no one has an unfair advantage, and to force myself to make a totally new exam.)&lt;/p&gt;&#xA;" OwnerUserId="6429" LastActivityDate="2016-03-02T21:21:23.843" Title="Reusing homework exercises from previous terms" Tags="|undergraduate-education|homework|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10653" PostTypeId="2" ParentId="10652" CreationDate="2016-03-02T04:12:36.370" Score="4" Body="&lt;p&gt;I guess it depends on how you grade the homework. I don't change mine at all, unless I have thought of a way to improve it. But I don't give it much weight. I stamp it each day, and count the stamps at test time. They are supposed to check answers in the back of the book. Homework is 10% to 15% of the grade.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2016-03-02T04:12:36.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10655" PostTypeId="2" ParentId="10652" CreationDate="2016-03-02T06:11:22.243" Score="7" Body="&lt;p&gt;For me, the answer is simply that homework is not collected, graded, or scored. I assign odd-numbered problems from standard textbooks, expect that students check the answers at the back, and I ask for discussion of any problems or surprises in the next class. This basically wipes the entire logistical/ethical difficulty off the table for me; and it highlights to students that the homework is a communal learning opportunity, as it should be. I think you want to be efficient with your time in this regard. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-03-02T21:21:23.843" LastActivityDate="2016-03-02T21:21:23.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10658" PostTypeId="2" ParentId="8167" CreationDate="2016-03-02T18:16:46.640" Score="3" Body="&lt;p&gt;How about giving them the &lt;a href=&quot;http://www.westpoint.edu/math/AY%202014%20Problems/POTW17%20Solution.pdf&quot; rel=&quot;nofollow&quot;&gt;five room puzzle&lt;/a&gt; on the plane and then on the torus? This is a rather visceral way to appreciate why &quot;holes matter&quot;, and may segue nicely into topics in topology.&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2016-03-02T18:16:46.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10659" PostTypeId="2" ParentId="10652" CreationDate="2016-03-02T18:25:42.817" Score="0" Body="&lt;p&gt;I would avoid giving detailed written solutions.  (I realize it's a late for that now but it's something you can do going forward.)  Instead, I would suggest discussing the solutions in the next class.  Students will have the solutions in their notes but it will be their interpretation of the solutions.  Not only will this prevent your technically complete solutions from going into the wild but the students will understand it better if they filter it through their understanding of the material.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2016-03-02T18:25:42.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10660" PostTypeId="2" ParentId="8052" CreationDate="2016-03-02T20:42:56.460" Score="3" Body="&lt;p&gt;My simplest answer to any student asking &quot;Why study math?&quot; is this&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Math gives you power to get what you want.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is not the most &quot;politically correct&quot; way of &quot;selling math&quot;, but it's both true and often persuasive to students and adults who want more power (which ends up being a majority of students and adults).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real question being asked here is this: &quot;How do we sell math?&quot; We are not just interested in selling math to students, but also in selling it to people in general. Some parts of math are easier to sell because the sales pitch is simple: you need to know how money works so that you can buy and sell things without feeling like you've made a &quot;bad deal&quot; or feeling like you've been &quot;cheated&quot;. Most of the economic reasons for learning math are based on our inherent fear of loosing economic security or our inherent want for economic power.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Outside of this economic sphere there are deep social reasons for studying mathematics. Perhaps the most valuable is that of forethought. Mathematics, unlike history, demonstrates that by systematically applying a basic  problem solving process like Polya's you can put together facts which show clearly and exactly the future consequences of present actions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far I've discussed the economic motivation, and the general problem solving motivation, now I move into the motivation from love. When someone asks a question like &quot;Why study math?&quot; we might hear, in the back of our curious mind, another question &quot;Why do anything?&quot; and this is sometimes what a pesky philosopher is apt to ask. I do not know how to answer the question &quot;Why study x?&quot; or &quot;Why do x instead of y?&quot;, but I do know of at least one statement which provides a popularly accepted way of approaching these general problems:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The good life is one inspired by love and guided by knowledge. - Bertrand Russell&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From this simple statement I say that we study math because some of us are inspired to love it: some people study math &lt;em&gt;because&lt;/em&gt; they love it. Ask them why they love it and I'm sure they can tell you all sorts of silly facts that really get their blood pumping, but the truth is that they do just love it. Why do we love anything? Why do we do anything for love? It seems that there is often no use in questioning the love that inspires us, but only in using the knowledge that we have to guide that inspiration towards the good life.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-02T20:42:56.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10661" PostTypeId="2" ParentId="10626" CreationDate="2016-03-02T20:53:02.617" Score="1" Body="&lt;p&gt;Rigor is the standard of mathematical practice, just like experiment is the standard of scientific practice. Rigor is popularly described as that which Euclid did in his Elements. In practice, rigor means being explicit about what you have assumed, what you've demonstrated, and why you believe your demonstration follows only and solely from what you've already assumed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rigor, in the mathematical sense, is a distillation of the practice of human arguments. When people argue with each other they have all sorts of tools at their disposal, the two largest collections of which are Logic and Rhetoric. Mathematical arguments are different from arguments in other parts of human activity: they are arguments about whether or not a certain convention which we've agreed upon (the assumptions) is being properly applied in a given situation (the proof).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The standards of these conventions are given by formal systems with axioms and inference rules. These technical terms are often much more frightening than they ever need to be, as they are really just another way of saying &quot;If we all agree to do things in this way by following these rules, then have we followed those rules in this case?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are plenty of reasons for why mathematical rigor has developed into the &quot;standard&quot; of axioms and inference rules that we use today, but they can all be traced back to a single human desire: that of certainty, or that of eliminating possible or future doubts. We want to have a feeling that once we make an argument in mathematics we need not ever question its use ever again, and this means eliminating as much as possible any doubts that we've done as we've said we done. This need to be absolutely clear about each and every thing that we've assumed and demonstrated is what has developed into modern formalisms of mathematics using symbolic notation.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-02T20:53:02.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10662" PostTypeId="2" ParentId="7880" CreationDate="2016-03-02T21:05:29.223" Score="2" Body="&lt;p&gt;The best way to teach a student that other disciplines are important to is to live your life in a way that demonstrates that other disciplines are important too. This might seem like an obvious or stupid answer, but it's one of the few methods of &quot;convincing&quot; or &quot;teaching&quot; others to follow one course of action rather than another that just works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, a student who questions the &quot;relevance&quot; of anything other than a STEM topic is one whose teachers have clearly failed them. I say clearly, because all it takes for a student to &quot;value&quot; or see non-STEM topics as &quot;important&quot; is for them to have had a single personal connection to any piece of homework, classwork, discussion, or event related to a non-STEM topic. For example, if there is ONE, JUST ONE, book that a student reads in their English class and which they REALLY connect with in an emotional way, then they no longer need any convincing that there is an intrinsic value to non-STEM subjects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way I like to explain this to students is as follows &quot;A life without love is not a life worth living. Seek out all that you love, and your life will be filled with the love that you seek.&quot; &lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-02T21:05:29.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10663" PostTypeId="2" ParentId="10632" CreationDate="2016-03-02T21:12:47.557" Score="0" Body="&lt;p&gt;This is a problem of design. Specifically, the fact that there is any difference between the method that is &quot;more conceptually difficult&quot; and the method which is &quot;easier&quot; is a problem of design. By claiming that there is a necessary distinction between what is &quot;more conceptually difficult&quot; and &quot;easy&quot; you have already committed yourself to a bad design constraint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Think of it this way: before the iPhone came out there were plenty of ways to do things like call people, surf the internet, and listen to music. People accepted that these were things that happened on different devices, but Apple wasn't convinced and showed that you can do all of these things in a way that is actually simpler than any of the individual ways we used to do them. Now, my summary is a bit loose on the use of the term &quot;simpler&quot; just as your question is loose with the term &quot;easier&quot; or &quot;conceptually difficult&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Outside the design of the mathematics itself, there is also ample opportunity to design a killer presentation of the methods and topics which eliminates the entire problem you're worried about. Question your assumptions and see if you can't find a way to actually reach for the ideal.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-02T21:12:47.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10664" PostTypeId="2" ParentId="10620" CreationDate="2016-03-02T21:25:33.087" Score="2" Body="&lt;p&gt;This is one of the most fundamental questions in modern mathematics pedagogy today, and I think that it is high time we seek out new alternatives to old methods. But, surprisingly or not, this question has really been addressed in large part by Goodstein in his HISTORIC text &lt;em&gt;Mathematical Analysis&lt;/em&gt;. It is not well known, and that makes me sad, but I do what I can to help people seek it out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic problem in modern Calculus/Analysis is that at the popular conception of Calculus is one without rigor: it's a mix of algebra and geometric intuition. Specifically, the notion of limits, derivatives, and integrals are, in modern treatments of Calculus, tied to the graphs and pictures that we draw. Though this makes it easier (in a sense) to communicate some of the results of Calculus/Analysis to students, it does little to reinforce the most basic principle of mathematics: rigor. The common belief is that the rigor of Real Analysis is inappropriate at the &quot;Calculus Level&quot;. I am one such individual: real analysis is so far from the math leading up to what we now conceive of as calculus so as to render it absolutely inaccessible to even some aspiring mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What Goodstein does was to take a revolutionary and new perspective on the whole Calculus/Real Analysis problem. I believe that, in time, most calculus/real analysis will be taught using a version of his methods in &lt;em&gt;Mathematical Analysis&lt;/em&gt; whether by direct reference or through some circumlocutions route.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Goodstein's approach is as follows: rather than thinking of continuity as the corner stone of mathematical analysis, we can construct a surprisingly familiar theory based entirely on uniform continuity. The result is this, we get a &lt;em&gt;Mathematical Analysis&lt;/em&gt; which is not only simpler than standard treatments of Real Analysis, but which makes it possible to present Calculus rigorously without leaping too far from the math commonly learned before taking a calculus class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a further solution to this general problem: the introduction of rigor at an earlier stage in student's education. Rather than presenting algebra and number in a hacked mash of geometric intuition and half-baked formalisms, we must seek out a design of a formal system (specifically one which is not set-theoretic) that is both concrete to students while being fully rigorous. The biggest problem/barrier to such a design is the belief that no such design is possible, or that no such design exists.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-02T21:25:33.087" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10665" PostTypeId="2" ParentId="10587" CreationDate="2016-03-02T21:43:50.390" Score="0" Body="&lt;p&gt;Decimal notation is an ingeniously efficient tool for working with rational quantities, but too often students and teachers forget just why/where it came from. There are a bunch of ways to say what a decimal like &quot;0.125&quot; means, but, assuming this student has a &quot;classical&quot; background in fractions the only way to bring concrete meaning to a notation like &quot;0.125&quot; is to relate it to dividing a whole into parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case &quot;0.125&quot; is a way of writing the fraction 125/1000 or the ratio 125:1000 and &quot;0.12&quot; is a way of writing the fraction 12/100 or the ratio 12:100. If the student is unable to identify which of these fractions is greater than the other, then you must review those concepts first as they are more fundamental than the mechanics of decimal notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, the computational method is to produce a common denominator and use the numerators of the fractions with common denominator to decide whether one is greater than the other. The appeal here is to the physical origin of these notations: 125/1000 means 125 parts out of the 1000 equal parts we've broken a whole into. Thus 12/100 is equal in quantity to 120/1000 and from here it is easily seen that 125 of 1000 parts is greater than 120 of 1000 parts of the same whole.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regards to the place value: terminating decimals are sums of fractions. So 0.123 is an abbreviation of the following expression&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\frac{1}{10} + \frac{2}{100} + \frac{3}{1000}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and it is clear that here, to calculate or &quot;know&quot; the sum is to find a common denominator and add them all together, BUT THIS IS EXACTLY WHY WE INVENTED POSITIONAL NUMERAL NOTATION, so that we can read off from 0.123 the corresponding fraction 123/1000.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope that helps.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-02T21:43:50.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10666" PostTypeId="2" ParentId="10610" CreationDate="2016-03-02T21:52:49.137" Score="1" Body="&lt;p&gt;This youtube video created by the now popular &quot;Primitive Technology&quot; account showcases EXACTLY how a story about completing a task can be made both interesting, exciting, and informative: &lt;a href=&quot;https://youtu.be/GzLvqCTvOQY&quot; rel=&quot;nofollow&quot;&gt;https://youtu.be/GzLvqCTvOQY&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal in presenting and teaching mathematics is to achieve this level of clarity and engagement while &quot;doing math&quot;. The path to this goal is exactly the same as used in any of the Primitive Technology youtube videos: show the concrete actions which take place and which solve the given problem. There is nothing simpler, and yet there is nothing harder to keep in mind while you're flipping madly through textbooks for &quot;the method&quot; or &quot;the solution&quot; or any other such nonsense. You must engage the senses that students have IN THE PRESENT and to do so requires that you are equally engaged in the physical actions which lead towards a solution to a mathematical problem. Anyone who believes mathematics is &quot;abstract&quot; is greatly confused about what mathematics is: mathematics is nothing more than the living symbolisms we use on a daily basis, it is only and solely the stories we tell about the symbols we use.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-02T21:52:49.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10667" PostTypeId="2" ParentId="10585" CreationDate="2016-03-02T22:05:08.930" Score="4" Body="&lt;p&gt;My current opinion is that the ideal curriculum is one which returns to the material learned in the past with an eye towards establishing the relevance of formal methods in addressing such troubling questions as &quot;What are numbers?&quot; or &quot;What is a function?&quot; or, gasp, &quot;What is proof?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B.F. Skinner demonstrated that a pigeon is capable of performing ever more complex sequences of behaviors, and that, in this way, a completely &quot;mindless&quot; bird could behave as if it &quot;knew&quot; what it was doing, or perhaps even &quot;understood&quot; what it was doing. But, I would find it hard for anyone to admit into science the statement &quot;That pigeon knows it's playing ping pong like a boss&quot; without some trepidation or clarification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point is that it is one thing to rehearse the mathematical methods of the past, but it is something completely different to see, within yourself, why humans perform the mathematical behaviors that they do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An alternate way of saying this is that what we must strive to develop is all that would be lost to the world if everyone over 50 suddenly disappeared.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-02T22:05:08.930" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10668" PostTypeId="2" ParentId="10521" CreationDate="2016-03-02T22:15:11.743" Score="1" Body="&lt;p&gt;Calculus is the greatest piece of human technology in history, and it is taught because it continues to be not only the most efficient technology but also the most broadly applicable. A highly &quot;applications&quot; based perspective on calculus would say that as a technology, a symbolic technology, more problems have been solved with the tools of Calculus throughout history than any other part of mathematics. It is important to note that the problems solved by calculus that I hint at are not at all limited to mathematical problems: calculus has solved the largest number of human problems in science and society than any other part of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It happens to be the tool of our times, and perhaps in the future there will be some other such symbolism which can earn even higher respect for is use as a technology, but so far we are lucky that Calculus continues to be the most efficient tool in human history.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-02T22:15:11.743" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10669" PostTypeId="2" ParentId="10521" CreationDate="2016-03-02T23:50:43.433" Score="8" Body="&lt;p&gt;David Bressoud wrote about this last year on his blog in a multipart series called &lt;em&gt;Calculus in Crisis&lt;/em&gt;. &lt;a href=&quot;http://launchings.blogspot.com/2015/06/calculus-at-crisis-ii-rush-to-calculus.html&quot; rel=&quot;noreferrer&quot;&gt;This post&lt;/a&gt; deals specifically with &quot;the rush to Calculus&quot;. If you trace his arguments across all the posts in that series, the line of reasoning that emerges is this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Economic pressures are driving more and more college students to pursue STEM careers (whether or not they are ready for them).&lt;/li&gt;&#xA;&lt;li&gt;The rapid growth in the number of STEM majors has caused a boom in enrollments in calculus (again, whether or not students are ready).&lt;/li&gt;&#xA;&lt;li&gt;High school students end up taking calculus in high school because they feel the pressure to major in STEM disciplines, and taking calculus in high school lessens the load when they get to college. &lt;/li&gt;&#xA;&lt;li&gt;And finally, since there have been so many students in the past taking calculus in high school (Bressoud estimates 40-45% of traditional first-year college students are students who took calculus in high school), there is now an &lt;em&gt;expectation&lt;/em&gt; that high school students will take calculus, and that expectation is reinforced by parents, administrators, and guidance counselors. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are no such forces pushing for the adoption in high school of number theory or abstract algebra or whatever, so that's mainly why you don't see them -- where's the demand? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think that statistics is making serious inroads into high school mathematics -- I don't know the numbers of how many AP Statistics courses are being offered right now or what their enrollments are, but anecdotally it seems like a lot more of my students are coming in with AP Stats credit than they used to. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And personally I would like to see high school math look a little more like computation + data science + discrete math and linear algebra, some combination that is accessible to a larger group of people and more broadly useful. &lt;/p&gt;&#xA;" OwnerUserId="942" LastEditorUserId="942" LastEditDate="2016-03-03T19:39:00.317" LastActivityDate="2016-03-03T19:39:00.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10670" PostTypeId="2" ParentId="7826" CreationDate="2016-03-03T03:41:04.657" Score="0" Body="&lt;p&gt;As far as I can tell, your intuition about fractions being &quot;algebraic&quot; is born from the fact that, like integer arithmetic, fraction arithmetic is built atop the arithmetic of the natural numbers. Specifically, every statement which we make about fractions can be transformed into a statement about natural numbers. The fundamental tool being that &#xA;$$&#xA;\frac{a}{b} = \frac{c}{d}&#xA;$$&#xA;can be transformed into &#xA;$$a \times d = b \times c$$ &#xA;(a statement about multiplication and equality of natural numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The conceptual origin of fractions is both simple and complex. Most kids have experience with cutting a cake or cookie into equal halves so that everything is &quot;fair&quot;. What rarely, if ever, happens is this: two kids have one cookie and decide to split it into four equal parts so that each kid can take two. This situation would only occur to a child if there was one cookie and four kids i.e. a cookie shared amongst four kids is broken into four equal parts and each kid gets one. The &quot;confusion&quot; and &quot;complexity&quot; comes from the fact that, in our experience, we don't often break things into any more equal pieces than we absolutely need so that we rarely deal with alternate &quot;representations&quot; of the same quantities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is for this reason that I've proposed that we focus on &quot;ratios&quot; rather than fractions because it is easier to talk of &quot;equivalent&quot; ratios as looking at the same &quot;situation&quot; from a different perspective. For example, a bowl with six oranges and two apples has a apple to orange ratio of &quot;2:6&quot; but notice that for each apple there are three oranges giving an apple to orange ratio of &quot;1:3&quot;. We haven't broken anything into pieces, or talked of parts of a whole: rather we've just identified that the arithmetic of ratios (which is the same as that of fractions) is built atop the arithmetic of natural numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some motivations for the additive arithmetic of ratios:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I give you one dollar every two hours and one dollar every three hours then then you have five dollars every six hours i.e. 1:2 + 1:3 = 5:6&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If Anne puts one apple in the bucket every time Bob puts two oranges in the bucket and Charles puts one apple in the bucket every time Bob puts three oranges in the bucket, then five apples are put in the bucket every time Bob puts six oranges in the bucket: 1:2 + 1:3 = 5:6.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastEditorUserId="6433" LastEditDate="2016-03-03T17:36:36.340" LastActivityDate="2016-03-03T17:36:36.340" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10671" PostTypeId="2" ParentId="7912" CreationDate="2016-03-03T18:31:32.393" Score="3" Body="&lt;p&gt;The simplest answer to this question is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Proofs can be comprehended at the same age rules can be comprehended.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, I must say why this is the case.&lt;/p&gt;&#xA;&lt;p&gt;Although it may seem that the difficulty in answering this question is identifying what is meant by &amp;quot;comprehending&amp;quot; with regards to &amp;quot;proofs&amp;quot;, the actual difficulty is in the word &amp;quot;proof&amp;quot; itself. As a human behavior &amp;quot;proof&amp;quot; is not well defined. To be more precise, there are plenty of human behaviors that can be clearly defined based on an an observable sequence of events e.g. the behavior &amp;quot;push a button&amp;quot; can be recorded and identified without the intervention of any human being. The problem with the behavior &amp;quot;give a proof&amp;quot; is that, except in the most technical circumstances, we are unable to say what constitutes the behavior of &amp;quot;give a proof&amp;quot; much less to say how we might go about observing or measuring this behavior.&lt;/p&gt;&#xA;&lt;p&gt;(Note, it has been the goal of mathematicians since the dawn of mathematics to say clearly and exactly just what constitutes the behavior &amp;quot;give a proof&amp;quot; and so far our best efforts are found in Hilbert's formalism, but there is a huge difference between proof as argument and proof as formalism.)&lt;/p&gt;&#xA;&lt;p&gt;Certainly, there is some agreement among humans as to what arguments are, but when we look at them closely, we are hard pressed to say why one argument &amp;quot;is better&amp;quot; or &amp;quot;more water-tight&amp;quot; than another, except to say that &amp;quot;we know it when we see/hear/feel/sense it.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Suppose we ignore all forms of classical Rhetoric with regards to a definition of argument. Historically, with the elimination of Rhetoric from an argument, we are left only with its &amp;quot;logical structure&amp;quot; and, it has been believed, that from this logical structure alone we can judge whether an argument is or is not &amp;quot;water-tight&amp;quot;. Furthermore, we can use this judgement to say whether an argument is &amp;quot;good&amp;quot; or &amp;quot;bad&amp;quot; based on their respective use of Logic. The problem now becomes: by what standards are we to judge one logical argument as being &amp;quot;good&amp;quot; or &amp;quot;bad&amp;quot;? In other words, what rules are there to making a logical argument and how do we know when we've followed those rules and whether those rules are the &amp;quot;right&amp;quot; rules or not?&lt;/p&gt;&#xA;&lt;p&gt;It is a common, and dangerous, misconception that mathematical statements are statements of fact; that the statement &amp;quot;Two plus two equals four.&amp;quot; is like the statement &amp;quot;I have two pennies in my pocket.&amp;quot; That this IS NOT THE CASE is just one of the many barriers to addressing your very important question.&lt;/p&gt;&#xA;&lt;p&gt;The rules we use in mathematics are conventions. All that mathematics &amp;quot;is&amp;quot; is part of the statements of convention in a language. The conventions are specified by giving rules for the use of some part of the language. In arithmetic, the rules we give are the conventions for a part of a language playing the role of a number, of addition of numbers, of multiplication of numbers etc. For example, what makes a sign of a language a natural number is not its shape, color, or sound but rather the rules we follow when using that sign e.g. a sign of a language is a natural number variable if it follows the following rules:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;x    x&#xA;-  ---&#xA;0   Sx&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;In fact, these rules characterize what it means to play the role of zero, the role of successor, and the role of variable all at the same time. Outside of the question &amp;quot;does this sign of this language follow these rules&amp;quot; there is no way to say whether a sign like &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; is on its own a &amp;quot;numeral variable&amp;quot;. (As a side note, a natural number is a sign which results from the elimination of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; via the two rules given.)&lt;/p&gt;&#xA;&lt;p&gt;I've said all of this so that I can conclude with the simplest answer to your question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Proofs can be comprehended at the same age rules can be comprehended.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, thus far I've been limiting myself to math because there is a distinct difference between a mathematical proof, which is only and solely tested based on whether we have or have not followed a certain collection of conventional rules, and proof in the broader &amp;quot;legal&amp;quot; or &amp;quot;unobjectionable&amp;quot; or &amp;quot;logically correct&amp;quot; sense that you hinted at in your question.&lt;/p&gt;&#xA;&lt;p&gt;The toughest thing to accept is this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There is no experiment which can test whether a rule is &amp;quot;right&amp;quot; or &amp;quot;wrong&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This sounds counter intuitive at first, but the only thing we can do with a rule is to test whether or not it has been followed in a specific situation. So you might make a rule in your classroom like &amp;quot;Students must raise their hands before they speak.&amp;quot; and sure enough you will observe that some students raise their hands before they speak, others speak without raising their hands, and some raise their hands and wait to be called on before they speak, but none of these observations tests whether the rule &amp;quot;Students must raise their hands before they speak.&amp;quot; is &amp;quot;good&amp;quot; or &amp;quot;bad&amp;quot; or &amp;quot;right&amp;quot; or &amp;quot;wrong&amp;quot;. All of these features which we ascribe to rules, specifically rules from science, are not based on any external observation, rather we choose, as individuals and a community, to say &amp;quot;This rule is good&amp;quot; or &amp;quot;This rule is bad&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In addressing this question we say that there is nothing which makes an argument &amp;quot;logical&amp;quot; rather we decide as individuals and as a community that &amp;quot;An argument which follows these rules is 'logical', i.e. 'good', and an argument that follows these rules is 'illogical' i.e. 'bad'.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So even in the more general case, when considering arguments of a less mathematical nature, the only thing required for a student to judge whether an argument is logical or not (&amp;quot;good&amp;quot; or &amp;quot;bad&amp;quot;) is whether they can say whether or not a certain set of rules has or has not been followed i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Arguments can be comprehended at the same age that rules can be comprehended.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The difficulty then becomes &amp;quot;What is the most efficient way to familiarize students with the conventional rules of our society?&amp;quot; and then there is the more difficult question of &amp;quot;Which rules should we teach as 'the conventional rules' and why?&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="6433" LastEditorUserId="8571" LastEditDate="2021-02-13T02:21:46.630" LastActivityDate="2021-02-13T02:21:46.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10672" PostTypeId="2" ParentId="10632" CreationDate="2016-03-03T19:25:39.590" Score="1" Body="&lt;p&gt;&lt;strong&gt;No&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be useful to go through the hoops of the &quot;hard&quot; method (perhaps it helps in understanding the problem better, maybe it is a stepping stone in the path to the more sophisticated one, or it is in fact easier in some cases), but as a general rule try to teach the &quot;best&quot; method. Be careful to teach what is actually &lt;em&gt;used&lt;/em&gt;, even if the other method is elegant/instructive. Your task is to teach useful skills first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your student's time (in class, self study, homework) is &lt;em&gt;limited&lt;/em&gt;, you have to make the best of it. If by skipping the longwinded method you gain time to have them learn the shortcut better (or even at all), I'd consider it an overall win. You might give the traditional method as a homework, or publish a few examples for contrast to cater to interested students.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-03-03T19:25:39.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10673" PostTypeId="1" CreationDate="2016-03-03T19:56:59.493" Score="18" ViewCount="1340" Body="&lt;p&gt;There are two kinds of answers I'm looking for:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What do students think a variable is?&lt;/li&gt;&#xA;&lt;li&gt;What do YOU, the teacher, think a variable is?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I'm also interested in why you think a variable is what you think it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More Context: Historically, there has been a great deal of confusion surrounding the ill defined notion of &quot;variable&quot;. The word has been used so loosely even in modern treatments of algebra and logic, that it is not only difficult to identify &quot;which definition is 'right'&quot; but also how one is to develop a student's understanding of variable towards &quot;the 'right' definition&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps it is impossible to give a final definition of the term variable, but I do believe there is no better place to start than by asking educators and students just what they think variables are.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastEditorUserId="262" LastEditDate="2016-03-03T23:49:33.403" LastActivityDate="2016-03-08T13:15:08.470" Title="What is a variable?" Tags="|mathematical-pedagogy|algebra|logic|definitions|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10674" PostTypeId="1" CreationDate="2016-03-03T20:07:22.423" Score="4" ViewCount="169" Body="&lt;p&gt;When teaching geometry it is common to use pictures/figures to &quot;show&quot; the problem and its solution. It's also common to say things like &quot;more than one figure can be shown which demonstrates the same problem&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a formal connection between geometric figures and geometric proofs?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has been suggested that geometric figures are just another notation for the geometric relations we write in a geometric proof e.g. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ptdbr.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ptdbr.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is just a different notation for $A \perp B$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To what extent has this notational equivalence been exploited formally and in an educational setting?&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2018-04-19T06:59:14.177" Title="How are geometric proofs related to geometric pictures?" Tags="|mathematical-pedagogy|geometry|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10675" PostTypeId="2" ParentId="10673" CreationDate="2016-03-03T20:16:06.733" Score="15" Body="&lt;p&gt;This is a very difficult question to answer; I recommend as a first place to look:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. &lt;em&gt;The ideas of algebra, K-12, 8&lt;/em&gt;, 19. &lt;a href=&quot;http://qrc.depaul.edu/algebrainitiative/Articles/Usiskin_Conceptions_of_School_Algebra.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt; (no paywall).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As Usiskin writes (emphasis in original):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My thesis is that the purposes we have for teaching algebra, the conceptions we have of the subject, and the uses of variables are inextricably linked. &lt;em&gt;Purposes for algebra are determined by, or are related to, different conceptions of algebra, which correlate with the different relative importance given to various uses of variables&lt;/em&gt; (p. 9).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The paper explores your question further, and contains a few nice examples. One more excerpt:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hqPqO.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hqPqO.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Really, the question at the heart of all of this is how one conceives of &lt;em&gt;algebra&lt;/em&gt; writ large. To this latter end, there is an interesting trio of sources tackling a similar conception as follows:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Usiskin&lt;/strong&gt;, from the aforelinked:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GSKE7.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GSKE7.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Chazan&lt;/strong&gt;, using similar language in a later publication:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OX0Sl.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OX0Sl.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And even in a &lt;a href=&quot;http://mathwithbaddrawings.com/2015/05/27/one-word-math-classes/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;blog post&lt;/strong&gt;&lt;/a&gt; (underline added) describing math classes with a single word:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/urX1r.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/urX1r.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;What is &lt;strong&gt;my&lt;/strong&gt; answer to the question? First, here is an excerpt from my personal communication with D.J. Velleman (author of the popular &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0521675995&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;How to Prove It&lt;/strong&gt;&lt;/a&gt; textbook):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On variables:  In proofs, I think of variables as standing for fixed but unspecified objects.  For example, suppose you're proving that A is a subset of the union of A and B.  You think of A and B as being fixed but unspecified sets.  The first sentence of the proof is &amp;quot;Let x be an arbitrary element of A.&amp;quot;  You're now thinking of x as being a fixed but unspecified element of A.  You proceed to prove that x is in A U B.  Then you say that since x was arbitrary, this is true for all x in A, so A is a subset of A U B.&lt;/p&gt;&#xA;&lt;p&gt;I find that students don't always understand variables this way.  I've had students tell me that in this proof x stands for all elements of A simultaneously, or something like that.  And when proofs get complicated, with multiple variables that are introduced in a particular order, with some depending on others (for example, delta = epsilon/3), I think this way of thinking about things is hopeless.&lt;/p&gt;&#xA;&lt;p&gt;I've been wondering if students have trouble with the meaning of variables in proofs because the way we use variables in earlier courses is different.  For example, in calculus we might say something like &amp;quot;Let y = x^2.  Then dy/dx = 2x.&amp;quot;  Here x and y &lt;em&gt;don't&lt;/em&gt; stand for fixed numbers.  In fact, the whole point is to think of x and y as changing.  So if variables don't mean the same thing in calculus as they do in proofs, then what do they mean in calculus?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My &lt;strong&gt;own&lt;/strong&gt; answer, perhaps somewhat appropriately, is that the meaning of &lt;em&gt;variable&lt;/em&gt; does, in fact, &lt;em&gt;vary&lt;/em&gt; according to its context (e.g., across different classes such as Velleman's example of proofs versus calculus, and within a single class such as Usiskin's approaches to algebra). Rather than trying to pin down a single definition, my own feeling is that we should explore these different meanings and challenge students to articulate the ramifications of adopting different conceptions at different times.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-03-03T20:27:53.827" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10676" PostTypeId="2" ParentId="6017" CreationDate="2016-03-03T20:16:56.267" Score="2" Body="&lt;p&gt;I strongly recommend looking at &lt;a href=&quot;http://www.geogebra.org&quot; rel=&quot;nofollow&quot;&gt;http://www.geogebra.org&lt;/a&gt; it is free, available on almost every OS, and can be used to PLAY with geometric constructions. Playing with geometric constructions is probably the simplest way to &quot;motivate&quot; the general methods of geometric construction.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-03T20:16:56.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10677" PostTypeId="2" ParentId="10597" CreationDate="2016-03-03T21:41:42.470" Score="4" Body="&lt;p&gt;There is an entirely different perspective on this entire problem which is revealed by asking: Is continuity what we should teach students? That is, before we think about motivating a formal definition of continuity, we might wish to question whether continuity is the concept that we really need students to know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An alternate perspective is that the concept which we want to motivate is actually uniform continuity, and that we can approach this concept using much more concrete methods than with the current conception of continuity at a point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The motivation for uniform continuity comes from calculating values of functions (like the square root function $\sqrt{x}$) to a certain number of decimal places and finding that if we have an approximation to say 2 decimal places then this approximation is constant over an interval of arguments. Another way of saying this is that a function is uniformly continuous if we expect that its decimal representation &quot;settles down&quot; over a certain interval of arguments. Said yet another way: we do not give a function arguments which are not rational i.e. we do not give endless decimals to a function when we're calculating, rather we suppose that a function giving an endless decimal is constant up to a certain number of decimal places over some finite interval. There can be some complexity in the fact that an infinite decimal number can have more than one representation e.g. 1.999... is a synonym for 2.0000... ) but these can be addressed just as concretely as the motivation for uniform continuity.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-03T21:41:42.470" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10678" PostTypeId="2" ParentId="10540" CreationDate="2016-03-03T22:01:38.220" Score="1" Body="&lt;p&gt;The standard of mathematics is rigor, and the standard of science is experiment. Between these two there is one fundamental commonality: clarity and exactness. Without a demand or a demonstration of clarity and exactness there is nothing to gain from an education in mathematics or science. The elimination of vagueness together with doubt is the cornerstone of any quality education in the basic methods of math. Outside of mathematics, education at any level is more a matter of character than content. We teach to give knowledge to students that allows them to live a &quot;good life&quot; as one &quot;inspired by love and guided by knowledge.&quot; To teach is to reveal the benefit of guiding one's love inspired actions by knowledge. If your goal as a math teacher is to empower students to develop that personal character needed to live a good life, then clarity and exactness are the standards to which you should hold yourself and others. &lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-03T22:01:38.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10679" PostTypeId="1" CreationDate="2016-03-03T22:35:46.283" Score="27" ViewCount="1298" Body="&lt;p&gt;What methods are effective in identifying and eliminating severe math anxiety, this most terrible and unfortunate part of modern mathematics education? This question is not about ordinary math anxiety but about a severe version of the condition that manifests itself in such symptoms as panic attacks, physical symptoms, or inability to function.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastEditorDisplayName="user507" LastEditDate="2016-03-05T21:47:35.857" LastActivityDate="2016-08-12T20:00:38.180" Title="What methods successfully identify and eliminate severe math anxiety?" Tags="|mathematical-pedagogy|psychology|anxiety|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="10680" PostTypeId="1" CreationDate="2016-03-03T22:43:39.597" Score="27" ViewCount="3483" Body="&lt;p&gt;I'm looking to find more examples of books which bridge the gap between &quot;story&quot; and &quot;mathematics&quot; using narrative and all those other wonderful features we might find in Harry Potter or some other well known children's book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far, Knuth's Surreal Numbers is one of the few attempts I've seen from a professional mathematician to produce a work of fiction which is also a work of serious mathematics: &lt;a href=&quot;http://www-cs-faculty.stanford.edu/~uno/sn.html&quot;&gt;http://www-cs-faculty.stanford.edu/~uno/sn.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any other similar books would be a good place to look when thinking about &quot;how do I sell this mathematical idea to kids&quot;?&lt;/p&gt;&#xA;" OwnerUserId="6433" LastEditorUserId="77" LastEditDate="2017-12-13T14:47:41.917" LastActivityDate="2021-05-03T11:44:47.837" Title="What books are like Knuth's Surreal Numbers?" Tags="|mathematical-pedagogy|student-motivation|reference-books|" AnswerCount="15" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10681" PostTypeId="2" ParentId="10680" CreationDate="2016-03-03T23:06:59.683" Score="19" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Flatland&quot;&gt;Flatland&lt;/a&gt; by Edwin Abbott Abbott&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a story about a 2-dimensional being's encounter with a three-dimensional being. (Well, there's a class(?) allegory at the beginning, but you can skip that.)&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2016-03-03T23:06:59.683" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10682" PostTypeId="2" ParentId="10680" CreationDate="2016-03-03T23:12:18.037" Score="7" Body="&lt;p&gt;&quot;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0805062998&quot;&gt;The Number Devil&lt;/a&gt;&quot; by Hans Magnus Enzensberger is right up your alley.  It is written for and is 100% accessible for children and introduces many math concepts such as recursion, infinity, and symmetry.&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2016-03-03T23:12:18.037" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10683" PostTypeId="2" ParentId="10673" CreationDate="2016-03-04T00:25:44.283" Score="8" Body="&lt;p&gt;I teach people (informally) how to make iOS apps. A lot of the people I teach are not people who were good math students. Of course in programming variables are important and anyone with a basic understanding of algebra picks up the concept pretty quickly. The term &quot;variable&quot; is pretty scary for even for people who did ok in algebra because it's a math term and that frightens a lot of people. I lose a lot of people when I bring up variables and constants. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way I usually explain it in terms of computers helps a little bit. Basically, in CS to make the computer do something you need to write code. The code is usually long. I show them that they can store all that code in a simple variable so they have to write less code, and use that variable when I want to refer to that long piece of code. This helps them understand a little better the benefit of a variable. I explain it as a little basket that can hold any &quot;object&quot;. So visually, I draw some baskets on a piece of paper, say the basket has a name, draw an animal in basket (dog, cat) and tell that they can change the animal for that basket. Then I contrast it with the concept of constants. You can use the concept of a bus as the variable and the number of passengers as a bus, and that in a lot of cases, you won't know the number of passengers or that it can change. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In CS, a variable is literally a place in the computer's memory that holds information (not exactly but conceptually this works as an explanation). Rathe than treat is an an unknown as it is usually presented, present it as a container for things. You can address visual and kinesthetic learners this way. &lt;/p&gt;&#xA;" OwnerUserId="6441" LastActivityDate="2016-03-04T00:25:44.283" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10684" PostTypeId="2" ParentId="10679" CreationDate="2016-03-04T00:29:40.007" Score="27" Body="&lt;p&gt;In regards to &quot;math anxiety&quot;, the 1990 paper by Ray Hembree helped me out a lot. It's a large meta-study of about 150 papers and a total of 25,000 students. Summary of the results, as I wrote on my blog previously:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Whole-group interventions are not effective&lt;/strong&gt; (curricular changes, classroom pedagogy structure, in-class psychological treatments). The only thing that was effective is &lt;strong&gt;out-of-classroom, one-on-one treatments&lt;/strong&gt; (behavioral systematic desensitization; cognitive restructuring); these have a marked effect at both lowering anxiety and boosting actual math-test performance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The lesson that I personally take from this is that &lt;strong&gt;addressing math anxiety is largely out of the hands of the classroom teacher&lt;/strong&gt;. Unless the student has access, or the institution provides access, to one-on-one behavioral desensitization therapy, no group-level interventions are found to be effective. When discussing the subject with students one-on-one outside of class, my number one priority is just to get them to physically relax at the moment, and I've become more prone to recommend that they arrange a meeting to see a college counselor. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hembree, Ray. &quot;The nature, effects, and relief of mathematics anxiety.&quot; &lt;em&gt;Journal for research in mathematics education&lt;/em&gt; (1990): 33-46. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-03-04T00:29:40.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10685" PostTypeId="2" ParentId="10680" CreationDate="2016-03-04T00:50:48.640" Score="7" Body="&lt;p&gt;To follow up on David's recommendation of &lt;em&gt;Flatland&lt;/em&gt;, how about&#xA;Ian Stewart's &lt;a href=&quot;https://en.wikipedia.org/wiki/Flatterland&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Flatterland&lt;/em&gt;&lt;/a&gt;.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/w6Lc3.jpg&quot; width=&quot;260&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;An amazon.com review says,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Along his fictional path, Stewart touches on Feynman diagrams, superstring theory, time travel, quantum mechanics, and black holes, among many other topics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Another review mentions fractal geometry and cosmic strings.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-03-04T00:50:48.640" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10686" PostTypeId="2" ParentId="10680" CreationDate="2016-03-04T04:07:49.617" Score="9" Body="&lt;p&gt;&lt;a href=&quot;http://bentobooks.com/mathgirls/&quot;&gt;Math Girls&lt;/a&gt;! It gave me a small taste of generating functions, and I want more! Very challenging math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surreal Numbers might be the best way to learn about Conway's system. Math Girls does not have math that is as original. But it is high-level math, and it's fun. (The storyline is more engaging than that of Surreal Numbers. I love both books, in different ways. The Number Devil is great for inspiring younger students with the beauty of math, but the math is not as high level. Flatland never moved me, personally.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2016-03-04T04:07:49.617" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10687" PostTypeId="2" ParentId="10679" CreationDate="2016-03-04T04:10:28.023" Score="6" Body="&lt;p&gt;A while back I posted the related question &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/8224/60&quot;&gt;How can we help students who are very anxious about math?&lt;/a&gt;&lt;/strong&gt;, so that I could offer up a few answers of my own. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My suggestions include a few good books:&#xA;My students have had some success in decreasing their anxiety with books like Mind Over Math (Kogelman, Warren), Overcoming Math Anxiety (Tobias), and Managing the Mean Math Blues (Ooten).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And a soundtrack I created:&#xA;I also wanted something that directly addressed their test anxiety in math, and (after much research on math anxiety and the principles of creating guided visualizations) I developed a 14-minute guided visualization, which I titled &lt;a href=&quot;http://mathmamawrites.blogspot.com/2009/09/math-relax-guided-visualization-for.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Relax&lt;/a&gt;. It is available online for free. Not every student finds it helpful, but some students have felt that it changed their outlook dramatically. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether math anxiety is moderate or extreme, it comes from past experiences that we replay in our subconscious (or conscious). Guided visualizations help us to rewrite that script.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-08-12T20:00:38.180" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10688" PostTypeId="2" ParentId="10680" CreationDate="2016-03-04T06:20:40.547" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Diamond_Age&quot;&gt;&lt;em&gt;The Diamond Age&lt;/em&gt;&lt;/a&gt;, a science fiction novel by Neal Stephenson, includes a subplot that involves mechanical Turing machines that are described in great detail and that are used in solving key puzzles that move the subplot along. The subplot involving Turing machines is only one part of a complex and multilayered story, but I thought I'd bring it up in case you might find it useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another Stephenson novel, &lt;a href=&quot;https://en.wikipedia.org/wiki/Anathem&quot;&gt;&lt;em&gt;Anathem&lt;/em&gt;&lt;/a&gt; includes a great deal of philosophical discussions about mathematics, as well as discussing directed acyclic graphs, configuration space, and other mathematical topics. It is a long and complex novel, but unusually detailed (even for a sci fi novel) discussions of mathematical topics play a significant role.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't read &lt;a href=&quot;https://en.wikipedia.org/wiki/Cryptonomicon&quot;&gt;&lt;em&gt;Cryptonomicon&lt;/em&gt;&lt;/a&gt; (also by Stephenson), but &lt;a href=&quot;http://www.ams.org/notices/199911/rev-kasman.pdf&quot;&gt;this review in Notices of the AMS&lt;/a&gt; notes that it includes quite a bit of mathematics, particularly cryptography.&lt;/p&gt;&#xA;" OwnerUserId="1603" LastEditorUserId="1603" LastEditDate="2016-03-04T06:45:09.913" LastActivityDate="2016-03-04T06:45:09.913" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10689" PostTypeId="2" ParentId="10680" CreationDate="2016-03-04T07:57:52.793" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Uncle_Petros_and_Goldbach%27s_Conjecture&quot; rel=&quot;nofollow&quot;&gt;Uncle Petros and Goldbach's Conjecture&lt;/a&gt; is a (very) light introduction to Number theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0312303025&quot; rel=&quot;nofollow&quot;&gt;The Parrot's Theorem&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="729" LastEditorUserId="729" LastEditDate="2016-03-04T08:03:33.463" LastActivityDate="2016-03-04T08:03:33.463" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10690" PostTypeId="2" ParentId="10680" CreationDate="2016-03-04T08:48:12.353" Score="20" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach&quot;&gt;Gödel, Escher, Bach: an Eternal Golden Braid&lt;/a&gt; by Douglas Hofstadter is a book about Mathematics, Art, Music, Computer Science and Artificial Intelligence.  It alternates story chapters and descriptive chapters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion the most valuable parts of the book are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a very understandable presentation of Gödel's incompleteness theorem.&lt;/li&gt;&#xA;&lt;li&gt;a good discussion of the arguments around Artificial Intelligence.&lt;/li&gt;&#xA;&lt;li&gt;a large number of reproduced art works, many from Escher, but also many others.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Others will undoubtedly find other things to like about it.&lt;/p&gt;&#xA;" OwnerUserId="5818" LastEditorUserId="5818" LastEditDate="2016-03-04T09:01:50.637" LastActivityDate="2016-03-04T09:01:50.637" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10691" PostTypeId="2" ParentId="10680" CreationDate="2016-03-04T09:39:04.880" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Incandescence_%28novel%29&quot; rel=&quot;nofollow&quot;&gt;Incandescence&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Greg_Egan&quot; rel=&quot;nofollow&quot;&gt;Greg Egan&lt;/a&gt; describes how a technologically unsophisticated alien civilisation, under the right circumstances (in this case, closely orbiting a neutron star) might discover general relativity based on simple observations and experiments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On Earth, where relativistic effects are too small to easily observe, an analogy might be drawn with Newton's discoveries (where the effects were &quot;easily&quot; observable to a civilisation that lacked advanced technology).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has &lt;a href=&quot;http://www.amazon.co.uk/Incandescence-Greg-Egan/dp/0575081643&quot; rel=&quot;nofollow&quot;&gt;mixed reviews&lt;/a&gt; and is one of Egan's least popular books, but I think most of the negativity comes from people who wanted more of a work of fiction, and less hard science. Egan is known for his hard science, and this book is heavier on it than most of his. I greatly enjoyed reading it.&lt;/p&gt;&#xA;" OwnerUserId="6451" LastEditorUserId="6451" LastEditDate="2016-03-04T09:45:07.130" LastActivityDate="2016-03-04T09:45:07.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10692" PostTypeId="1" CreationDate="2016-03-04T14:12:54.350" Score="8" ViewCount="6501" Body="&lt;p&gt;In the graph theory section of my Discrete Math course I'll be covering Prim's Algorithm for finding the minimal spanning tree. I'd like to impress upon the students just how special it is that the greedy approach actually works. To do this, it'd be nice to have a variety of easy to understand problems where the greedy algorithm fails to produce the optimal solution.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are some good examples of problems where the greedy algorithm fails?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The only example I can think of is Egyptian fraction decomposition where taking the next largest unit fraction does not always give the shortest decomposition. But the students don't have much number theory background and I feel like this problem already requires too much introduction. Ideally it'd be nice to have 2 or 3 problems that could be demonstrated quickly in one class period.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(More details: The students in the course have had Calculus and we will have covered Logic, Sets, Recursion, and the basics of Graph Theory by then.)&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2016-03-05T20:54:22.717" Title="Counterexamples to the Greedy Algorithm" Tags="|examples|discrete-math|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10693" PostTypeId="2" ParentId="10692" CreationDate="2016-03-04T14:21:23.567" Score="9" Body="&lt;p&gt;The greedy algorithm fails quite spectacularly for the &lt;a href=&quot;https://en.wikipedia.org/wiki/Travelling_salesman_problem&quot; rel=&quot;noreferrer&quot;&gt;Traveling Salesman Problem&lt;/a&gt; (TSP):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bang-Jensen, Jørgen, Gregory Gutin, and Anders Yeo. &quot;When the greedy algorithm fails.&quot; &lt;em&gt;Discrete Optimization&lt;/em&gt; 1.2 (2004): 121-127.&#xA;(&lt;a href=&quot;http://www.cs.rhul.ac.uk/~gutin/paperstsp/anti8.pdf&quot; rel=&quot;noreferrer&quot;&gt;PDF download&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Abstract&lt;/strong&gt;. We provide a characterization of the cases when the greedy algorithm&#xA;  may produce the unique worst possible solution for the problem&#xA;  of finding a minimum weight base in an independence system when the&#xA;  weights are taken from a finite range. We apply this theorem to TSP&#xA;  and the minimum bisection problem. The practical message of this paper&#xA;  is that the greedy algorithm should be used with great care, since&#xA;  for many optimization problems its usage seems impractical even for&#xA;  generating a starting solution (that will be improved by a local search&#xA;  or another heuristic).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is a little example from S.K.Basu's book,&#xA;&lt;em&gt;Design Methods and Analysis of Algorithms&lt;/em&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/yHQuy.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yHQuy.jpg&quot; alt=&quot;BasuFig&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;From $A$, the greedy cycle is $ABDCA$ of length $9$,&#xA;while $ACBDA$ has length $8$.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-03-05T20:54:22.717" LastActivityDate="2016-03-05T20:54:22.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10694" PostTypeId="2" ParentId="10680" CreationDate="2016-03-04T15:26:52.327" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Phantom_Tollbooth&quot; rel=&quot;nofollow&quot;&gt;The Phantom Tollbooth&lt;/a&gt; (1961) by Norton Juster is a children's fantasy story that mixes wordplay and mathplay somewhat in the style of Lewis Carroll. It's suitable for anyone, but resonates well with the 8-12 age group, according to the &lt;a href=&quot;https://www.commonsensemedia.org/book-reviews/the-phantom-tollbooth/user-reviews/child&quot; rel=&quot;nofollow&quot;&gt;GoodReads reviews&lt;/a&gt; by children.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Milo travels to the fantastic land called Kingdom of Wisdom, where he is joined by the &quot;watch dog&quot; Tock, who has a giant alarm clock on his side. They meet the feuding wizard brothers Azaz, the letters wizard who is King of Dictionopolis, and the Mathemagician, the numbers wizard who rules Digitopolis. Milo solves math riddles, eats Subtraction Stew, and has other adventures in his quest to settle the brothers' feud.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's not a particular math concept that is illustrated, but if you want to get the 8-12 age group interested in learning more about math (and words), this might be a good place.&lt;/p&gt;&#xA;" OwnerUserId="3593" LastActivityDate="2016-03-04T15:26:52.327" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10695" PostTypeId="2" ParentId="10680" CreationDate="2016-03-04T16:10:30.750" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://kasmana.people.cofc.edu/MATHFICT/default.html&quot; rel=&quot;nofollow&quot;&gt;The Mathematical Fiction&lt;/a&gt; web site, maintained by Alex Kasman of the College of Charleston, has summaries of hundreds of stories and movies that in some way include or discuss mathematics. It is searchable by keyword and indexed by mathematical topic(s) discussed. &lt;/p&gt;&#xA;" OwnerUserId="1603" LastEditorUserId="1603" LastEditDate="2016-03-08T18:03:05.497" LastActivityDate="2016-03-08T18:03:05.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10696" PostTypeId="1" CreationDate="2016-03-04T17:26:28.787" Score="6" ViewCount="296" Body="&lt;p&gt;A teacher has recently asked me to provide some suggestions for math movies to screen for a set of high school seniors. My first inclination was &lt;a href=&quot;http://www.flatlandthemovie.com/&quot;&gt;Flatland&lt;/a&gt; before I decided to do some basic searching. I came across &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_films_about_mathematicians&quot;&gt;this&lt;/a&gt; list, as well as the movies &lt;a href=&quot;http://www.imdb.com/title/tt0241317/&quot;&gt;The Colours of Infinity&lt;/a&gt; and &lt;a href=&quot;http://www.imdb.com/title/tt0125425/&quot;&gt;N is a Number&lt;/a&gt;--both seem interesting--but I can't help but think I might be missing something. Does anybody else have any other suggestions?&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2016-03-06T20:34:50.393" Title="Motivating Mathematics Movies" Tags="|student-motivation|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10697" PostTypeId="1" CreationDate="2016-03-04T18:18:41.400" Score="4" ViewCount="1723" Body="&lt;p&gt;The fundamental trigonometric functions $\sin(x)$ and $\cos(x)$ are used throughout the sciences, but I believe students are often introduced to a very limited initial understanding where it is thought of as a purely mechanical process (e.g. &quot;make sure your calculator is in degrees and put this in your calculator when you want to find out the length of this side of a triangle from this other side of a right triangle&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My first question to you is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How do you introduce the basic trigonometric functions to students?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As a corollary to this question, I ask:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is your definition of the basic trigonometric functions?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Outside of this usage to calculate (1) the &lt;em&gt;side lengths of triangles&lt;/em&gt;, students also learn early on to define these functions as (2) &lt;em&gt;models for periodic motion&lt;/em&gt;; and later on as (3) &lt;em&gt;coordinates on the unit circle&lt;/em&gt; and (4) the &lt;em&gt;analytic sum of some power series&lt;/em&gt; (or some other limiting approximation of the classical power series definition of sine and cosine).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With these four definitions in mind, I ask these additional questions.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What methods are used to discuss the differences and similarities between these four (and possibly more) definitions of trigonometric functions?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What methods are there for motivating the connection between a geometric definition of the trigonometric functions and an analytic definition of the trigonometric function?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How are these definitions (1) and (4) connected in a (late high school or early college) student's calculus course?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is it useful for students to see the connection between definitions (1) and (4), or is it enough to provide them with the tools and knowhow to use them and just let them use them as they are commanded to in their science classes?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6433" LastEditorDisplayName="user507" LastEditDate="2018-03-12T02:09:32.377" LastActivityDate="2020-11-03T17:10:17.217" Title="How are the basic trigonometric functions introduced to students?" Tags="|mathematical-pedagogy|calculus|trigonometry|precalculus|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10700" PostTypeId="2" ParentId="10697" CreationDate="2016-03-04T18:55:35.733" Score="6" Body="&lt;p&gt;&lt;strong&gt;How do you introduce the basic Trigonometric functions to students?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my geometry courses, as in most American secondary school geometry courses of which I am aware, the sine and cosine functions are introduced as ratios of lengths of sides of right triangles. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The tangent function can be introduced simply as a ratio of those two ratios, however the tangent function is realizable as a geometric length if one considers a similar right triangle to that from which the angle comes such that the hypotenuse is unity. The value of tangent of the angle is the signed length of the vertical line segment that is tangent to the unit circle and whose endpoints lie on the x-axis and the terminal side of the angle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given a right triangle, label measures of the acute angles $\alpha$ and $\beta$, the length of the side opposite $\beta$ label $b$, and length of the side opposite $\alpha$ label $a$. Label the length of the hypotenuse $h$. Then $$ \sin \alpha = \frac{a}{h} $$ $$ \cos \alpha = \frac{b}{h}$$ $$ \sin \beta = \frac{b}{h} $$ $$ \cos \beta = \frac{a}{h}$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that cosine of an acute angle gives the &quot;complementary sine,&quot; or the sine of the complementary angle. Because every right triangle is similar to a triangle with hypotenuse one (multiply the length of each side by $\dfrac{1}{h}$ ), and because corresponding angles of similar triangles are congruent, we can discuss the sine or cosine of an angle without reference to any particular right triangle.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First in geometry, students learn to compute the sine or cosine of an acute angle in a right triangle by writing a ratio of the triangles sides. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In precalculus algebra, after students have been familiarized with functions, the definition of these ratios is extended to include right and obtuse angles, by describing angles in the unit circle. By relating any angle to acute or right angles in the first quadrant via reference angles, the student can then find the vertical or horizontal displacement of an infinite number of arbitrarily close points on the unit circle without the use of a computational device after establishing identities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is your definition of the basic Trigonometric functions?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Formally, trigonometric functions take equivalence classes of real numbers as their argument, with the equivalence class representative given by the real number modulo $2\pi$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider a ray initiating at the origin. Since the ray is continuous and has length greater than 1, the ray must intersect the unit circle at some point, say $(x,y)$. Define the &lt;em&gt;measure of the angle&lt;/em&gt; to be the length of the arc of the circle that initiates at $(1,0)$ and terminates at $(x,y)$. A degenerate angle has a measure of 0, and every other angle has a measure in the interval $(0,2 \pi)$, since the circumference of the unit circle is $2 \pi$. Using these definitions, $\sin(\theta)$ maps the interval $[0,2\pi)$ to the value of $y$ associated with the intersection of the ray and the unit circle. $\cos(\theta)$ maps the interval $[0,2\pi)$ to the value of $x$ associated with the intersection of the ray and the unit circle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once these functions have been defined as functions of an angle, one can consider arguments less than $0$ or greater than $2 \pi$. For real numbers $\alpha$ and $\beta$, let $\alpha$ relate to $\beta$ if $\alpha \equiv \beta \mod 2 \pi$. This is an equivalence relation and thus gives equivalence classes over the real numbers, each with a unique representative in $[0,2 \pi)$. Then for $\theta &amp;lt; 0$ or $\theta \geq 2 \pi$, let $\sin\theta = \sin (\theta \mod 2 \pi)$. A similar definition is given for cosine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After the functions have been extended over the real numbers, it is useful to consider the case that the angle parameter $\theta$ is a linear function of time. Such functions are immediately applied to describing the dynamic of a simple harmonic oscillator. By considering alternative parameterizations of the angle parameter, one can obtain useful function models of a number of oscillating systems that yield important results in many areas of science. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, once analytic definitions are demonstrated, as detailed below, there is no difficulty associated with considering a complex parameter, whose value is given as a complex number in terms of the hyperbolic sine and hyperbolic cosine functions. So, it is even reasonable to discuss &lt;em&gt;complex angles&lt;/em&gt;, which is convenient, for example, when considering elements of a vector space over a field of complex numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What methods are their for motivating the connection between the geometric definition of the trig functions and the analytic definition of the trigonometric function?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With my students, I relate these functions to power series through Euler's identity $$e^{i \theta} = \cos \theta + i \sin \theta.$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I introduce this identity in precalculus without explicit proof. By that time, my students have discovered complex numbers as zeros of polynomials. Initially in precalculus, we discuss the roots of unity in rectangular form $a + ib$ without any mention of the polar form $e^{iθ}$ or the trig functions. When we introduce the trig functions and unit circle later in the course, I include a discussion of the polar form $e^{iθ}$ and show that the polar numbers also solve those polynomials, and we see that the two forms are equal. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euler's identity, when considered in the context of the &lt;em&gt;complex&lt;/em&gt; unit circle, has incredible pedagogical value as it constitutes a unifying theme of algebra and geometric trigonometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How are these two definitions connected in a student's calculus course?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The series definition for the exponential function is demonstrated early in my calculus course. Returning to the long established relationship between the complex exponential function and the trigonometric functions, one can obtain the series representations of $\sin \theta$ and $\cos \theta$ by begining with the series representation for $e^x,$ with which most students are comfortable, then substituting $x = i \theta$ in that series. After some elementary algebraic simplification, one obtains a sum of two series, one a real series of even power functions, and the other an imaginary series of odd power functions. Returning to the original statement of the identity, the students then discover that the even series must be the power series representation of the cosine function, and the odd series must be the representation of the sine function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is it useful for students to see a connection between these two different definitions, or is it enough to give them the tools and let them use them as they are commanded to in their science classes?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For students who wish to study science, mathematics, and engineering, an understanding of the geometric and analytic properties of the trigonometric functions and how they emerge from study of the complex unit circle are indeed crucial. Students will encounter many methods for modeling and problem solving in later courses and in their industries that presume such an understanding. Some examples are the Laplace Transform integral solution of a differential equation, Fourier Analysis, electric currents and fields, vectors and vector analysis, representations of cyclic groups, matrix representations of linear transformations, and the list goes on. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-03-11T13:20:23.323" LastActivityDate="2016-03-11T13:20:23.323" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10701" PostTypeId="2" ParentId="10153" CreationDate="2016-03-04T19:12:45.577" Score="2" Body="&lt;p&gt;Jim Fowler, Bart Snapp, and I collaborated on a project called &amp;quot;Mooculus&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The main page is here:  &lt;a href=&quot;https://mooculus.osu.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mooculus.osu.edu/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is a free open source textbook (mostly Bart), youtube &amp;quot;lecture&amp;quot; videos (mostly Jim), and Khan Academy style practice exercises (mostly me).&lt;/p&gt;&#xA;&lt;p&gt;Here are some highlights from the interactive stuff.  Note that you have to log in with a google account for any of this to work unfortunately.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://mooculus.osu.edu/explorations/derivative&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mooculus.osu.edu/explorations/derivative&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This let's a student enter any elementary function, and we algorithmically compute the derivative step by step, while citing the appropriate derivative rule.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://mooculus.osu.edu/exercises/tangentLineIdea&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mooculus.osu.edu/exercises/tangentLineIdea&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Forces the student to graph a tangent line to a function&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mooculus.osu.edu/exercises/derivativeOracle&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mooculus.osu.edu/exercises/derivativeOracle&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Gives student access to a &amp;quot;black box&amp;quot; function, and asks them to approximate the derivative&lt;/p&gt;&#xA;&lt;p&gt;There are many more for you to explore on the webpage. For instance, chain rule problems with tables, problems where they have to differentiate an arbitrary elementary function (we have an expression input), problems that force them to think about derivatives of inverse functions geometrically, etc. We have not done much maintenance, so there may be some bugs (I think, sadly, a mathjax update killed some of our multiple choice questions, for instance).&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-03-04T19:12:45.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10702" PostTypeId="2" ParentId="10597" CreationDate="2016-03-04T19:33:48.887" Score="4" Body="&lt;p&gt;I believe it can be easier, at first, to look at the formal definition of continuity in terms of function application commuting with limits, which says that $\lim_{x \to A} f(x)$ is the same as $f(A)$. The explanation is more clear if also you write $f(A)$ as $f(\lim_{x \to A} x)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$f(A) = f(\lim_{x \to A})$ is &quot;where you are&quot; at at time $A$&lt;/li&gt;&#xA;&lt;li&gt;$\lim_{x \to A} f(x)$ is &quot;where you should be&quot; at time $A$, based on where you were at nearby moments of time.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If someone else naturally followed the function, then at time $A$ they ought to arrive at $\lim_{x \to A} f(A)$. But this means that if $f(A)$ is not the same as $\lim_{x \to A} f(A)$, then the actual function would require &quot;lifting a pen&quot;, in some informal sense.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, as with any explanation, this requires the students to apply some amount of intuition. And it makes more sense if we already know that the function is continuous everywhere else, so that it is possible to trace &quot;the rest&quot; of the function.  But, if students are already able to visualize limits, this helps them visualize continuity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, because the limit form of the definition of continuity is vital for applying continuity to compute limits in the context of calculus, emphasizing the link between continuity and limits can have other benefits, compared to the $\epsilon-\delta$ definition. &lt;/p&gt;&#xA;" OwnerUserId="5042" LastEditorUserId="376" LastEditDate="2016-03-05T07:24:17.790" LastActivityDate="2016-03-05T07:24:17.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10704" PostTypeId="2" ParentId="10696" CreationDate="2016-03-04T20:51:07.670" Score="6" Body="&lt;p&gt;There is a list at &lt;a href=&quot;http://www.qedcat.com/moviemath/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Mathematical Movie Database&lt;/strong&gt;&lt;/a&gt; that includes (for some of the entries) the relevant time stamps and dialogue. I found the site by looking for one particular movie:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://www.qedcat.com/moviemath/its_my_turn.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;It's My Turn&lt;/strong&gt;&lt;/a&gt; (1980) &lt;a href=&quot;http://imdb.com/title/tt0080936/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;IMDB&lt;/strong&gt;&lt;/a&gt; ***** Light romantic movie about a female mathematician. Famous among mathematicians because the full proof of the Snake Lemma is given in the first scene.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(I learned of this reference several years ago in &lt;a href=&quot;https://mathoverflow.net/q/106943&quot;&gt;&lt;strong&gt;MO 106943&lt;/strong&gt;&lt;/a&gt;: &lt;em&gt;Who named it the Snake Lemma?&lt;/em&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's another list called &lt;a href=&quot;http://www.math.harvard.edu/~knill/mathmovies/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Mathematics in Movies&lt;/strong&gt;&lt;/a&gt; that was included among others at &lt;a href=&quot;https://mathoverflow.net/questions/77279&quot;&gt;&lt;strong&gt;MO 77279&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2016-03-04T20:51:07.670" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10706" PostTypeId="2" ParentId="10680" CreationDate="2016-03-04T22:41:55.360" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wgAit.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wgAit.jpg&quot; alt=&quot;Cover of To Mock a Mockingbird&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might want to consider the books of Raymond Smullyan. They often take the form of a sequence of puzzles tied together loosely by a narrative. The puzzles are typically intended to work towards deeply understanding some topic. For example, &lt;a href=&quot;https://en.wikipedia.org/wiki/To_Mock_a_Mockingbird&quot; rel=&quot;noreferrer&quot;&gt;To Mock A Mockingbird&lt;/a&gt; disguises a course on &lt;a href=&quot;https://en.wikipedia.org/wiki/Combinatory_logic&quot; rel=&quot;noreferrer&quot;&gt;Combinatory Logic&lt;/a&gt; (closely related to lambda calculus) as a tale about birds and their calls.&lt;/p&gt;&#xA;" OwnerUserId="6359" LastActivityDate="2016-03-04T22:41:55.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10707" PostTypeId="2" ParentId="10692" CreationDate="2016-03-04T23:18:11.483" Score="7" Body="&lt;p&gt;There are tons of tasks where greedy algorithms fail, but the best in my opinion is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Change-making_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;change-making problem&lt;/a&gt;. It is great, because whether the obvious greedy algorithm works depends on the input (i.e. the denominations). For example, if you have coins $1,6,8$, then $12=6+6$ is better than $12=8+1+1+1+1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some other tasks:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Shortest paths where the edge labels do not have to be positive.&lt;/li&gt;&#xA;&lt;li&gt;Maximum independent set (strategy: consider sequentially vertices ordered by their degree).&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/273/42&quot;&gt;This problem&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-03-04T23:18:11.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10708" PostTypeId="2" ParentId="10680" CreationDate="2016-03-05T03:16:34.103" Score="1" Body="&lt;p&gt;A K Dewdney's &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Planiverse&quot; rel=&quot;nofollow&quot;&gt;Planiverse&lt;/a&gt; and Rudy Rucker's &lt;a href=&quot;https://en.wikipedia.org/wiki/White_Light_(novel)&quot; rel=&quot;nofollow&quot;&gt;White Light&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6464" LastActivityDate="2016-03-05T03:16:34.103" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10710" PostTypeId="2" ParentId="10680" CreationDate="2016-03-05T07:53:32.633" Score="2" Body="&lt;p&gt;Some scholars think that &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1619490226&quot; rel=&quot;nofollow&quot;&gt;Alice in Wonderland&lt;/a&gt; is entirely an allegory on mathematical topics by the Oxford logician and mathematician Charles Lutwidge Dodgson (Lewis Carroll). &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.npr.org/templates/story/story.php?storyId=124632317&quot; rel=&quot;nofollow&quot;&gt;NPR: The Mad Hatter's Secret Ingredient: Math&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.maa.org/external_archive/devlin/devlin_03_10.html&quot; rel=&quot;nofollow&quot;&gt;MAA: The Hidden Math Behind Alice in Wonderland&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-03-05T07:53:32.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10711" PostTypeId="2" ParentId="10349" CreationDate="2016-03-05T20:36:45.960" Score="8" Body="&lt;p&gt;My impression is that proofs by contradiction are easier to find – within a direct proof you need to reach some particular conclusion, while it is enough for a proof by contradiction to reach just any contradiction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, rarely the reasoning in some direct proof is non-constructive. Frequently we use non-constructive theorems, but the &quot;glue&quot; is usually constructive. In my experience very few of proofs actually directly invoke the law of excluded middle or something similar. Thus, a direct proof can be indeed considered stronger/harder.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2016-07-13T19:48:02.853" LastActivityDate="2016-07-13T19:48:02.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10713" PostTypeId="2" ParentId="10696" CreationDate="2016-03-06T01:57:46.240" Score="5" Body="&lt;p&gt;I highly recommend &lt;a href=&quot;http://www.greenfusefilms.com/&quot; rel=&quot;nofollow&quot;&gt;Between the Folds&lt;/a&gt;, on origami, math, and science.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2016-03-06T20:34:50.393" LastActivityDate="2016-03-06T20:34:50.393" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10714" PostTypeId="2" ParentId="10673" CreationDate="2016-03-06T03:45:00.713" Score="3" Body="&lt;p&gt;Suggestion: Instead of using single letters for variables, you might consider using meaningful words or phrases. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of &lt;em&gt;s=d/t&lt;/em&gt;, for example, write &lt;em&gt;speed = distance / time&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Computer programmers always use this technique. They are required to consistently assign meaningful names to variables so that another programmer reading their code can more easily understand what it is supposed to do. It also helps in debugging your own code.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe teaching some rudimentary programming or spreadsheet skills would help. Of course, you have to somehow initially avoid things in programming languages like x=x+1. Maybe some language that uses a different symbol for this kind of assignment of values. I don't think spreadsheets do this -- not sure. It's been a while.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can start abbreviating the words as students get used to the idea of a variable. &lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2016-03-06T15:50:56.550" LastActivityDate="2016-03-06T15:50:56.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10715" PostTypeId="2" ParentId="10673" CreationDate="2016-03-06T17:10:48.747" Score="2" Body="&lt;p&gt;I am both a teacher and a student so I will answer 1. and 2. at once:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;an $S$-variable $x$ is a possible quantity taken from set $S$. For example, suppose $x$ is a real variable. If $x^2+1=0$ then no such $x$ exists, whereas if $x^2-1=0$ then $x=\pm 1$. In contrast, if $x$ is a complex variable and $x^2+1=0$ then $x = \pm i$.&lt;/li&gt;&#xA;&lt;li&gt;an indeterminant $x$ is a symbol which is used as a place-holder for expressions. For example, $a+bx+cx^2 = (a,b,c,0,..)$ where $1 = (1,0,0,..)$, $x=(0,1,0,..)$ and $x^2=(0,0,1,0,..)$. We can capture the algebra of polynomials either via expressions with $x$ or with sequences with only finitely many nontrivial terms. In this context, $x$ is not an arbitrary element of some set $S$, it is the sequence $(0,1,0,...)$. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My example of an indeterminant is by no means general! You could think of indeterminants $x,y$ for which $xy \neq yx$ etc... In any event, an indeterminant can be made into a variable by limiting $x$ to assume the value inside some given set (through the appropriate valuation homomorphism).&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2016-03-06T22:29:39.757" LastActivityDate="2016-03-06T22:29:39.757" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10716" PostTypeId="1" CreationDate="2016-03-07T12:05:37.950" Score="12" ViewCount="1156" Body="&lt;p&gt;I am currently writing an independent project investigating the teaching and learning of algebra for students with a visual impairment. I am struggling to find literature specifically about teaching mathematics to VI. Are there any relevant journals or authors that I'm missing?&lt;/p&gt;&#xA;" OwnerUserId="6472" LastEditorUserId="2083" LastEditDate="2016-03-07T18:35:47.633" LastActivityDate="2020-03-31T21:28:22.883" Title="Teaching algebra to visually impaired or blind students" Tags="|reference-request|algebra|education-research|teaching|special-needs-education|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10718" PostTypeId="1" CreationDate="2016-03-07T20:43:45.080" Score="1" ViewCount="31" Body="&lt;p&gt;I am interested in creating an applet to illustrate how polar graphs get formed.  I was able to make some videos of what I envision using &lt;i&gt;Interactive Physics.&lt;/I&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But they are not really interactive once the video is exported.  Still, you can see what I am aiming for here:  &lt;a href=&quot;http://wp.me/p4uvY7-9V&quot; rel=&quot;nofollow&quot;&gt;http://wp.me/p4uvY7-9V&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you were starting from scratch and you wanted to make this into a web-based applet where students could choose the function and change the parameters on the fly, how would you begin?  Is this something that I could do in Geogebra?  Do I need to learn to write Java code?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you. &lt;/p&gt;&#xA;" OwnerUserId="6236" LastActivityDate="2016-03-07T20:43:45.080" Title="Environment for Applet development" Tags="|graphing|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10719" PostTypeId="2" ParentId="10716" CreationDate="2016-03-07T22:14:21.827" Score="2" Body="&lt;p&gt;I do not have any sources, but I do have some critical arguments on this topic that can produce insight into this and related problems in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than limiting ourselves to the blind, let us assume that we wish to teach math to someone who is unable to see or hear. The question &quot;How do we teach math to someone who is unable to see or hear?&quot; is surprisingly similar to the question &quot;How do we communicate with people who are deaf blind?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason these two questions are similar is because, as the old adage goes, math is a language. Math, as a human activity, is a way of using language according to certain rules. The first rules we introduce a student to, be they blind, deaf, deaf blind, or otherwise, are those of arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though there are arguments, and you can find many of them, which claim that geometry plays some fundamental role in elementary mathematics, there is nothing in our intuitive notions of geometry which reflect the formal rigor which is found in a mathematical study of geometry. Furthermore, the rigor which is the cornerstone of mathematics is found immediately in the most elementary arithmetic operation: addition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The simplest arithmetical act of addition is addition by one, which is given the technical name &quot;succession&quot;. The principle problem in teaching any student about math is to introduce them to the rules that must be followed in order to add one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A tool which is surprisingly useful at introducing this rule to a student is the abacus (it is also one of the most efficient tools to use when introducing a positional numeral systems). Though, for the deaf blind some minor modifications significantly improve the learning experience. Rather than having an abacus whose beads are lose and easy moved by bumps or brushings of fingers, it is best to put a piece of felt underneath the frame of beads so that the beads are held firm by the felt, but can still be moved so as to operate the abacus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The essential part of learning the rules of arithmetic is to recognize that a process each of whose steps might be something simple like &quot;add one&quot; can transform one configuration of beads into a surprisingly different configuration, and that all of these states can be understood as abbreviations for successive actions of &quot;add one&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While we use geometric terms to describe the shapes of physical objects, these terms are rarely, if ever, used in a mathematical sense. Rather, they are introduced just as words like &quot;fluffy&quot; or &quot;soft&quot; are introduced: to communicate an informal similarity between observations of events.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The remaining complexities of mathematics are introduced alongside the introduction of ever more sophisticated communication techniques. Ultimately, there are often a large number of unique and nonuniform methods of communication that combine to give the deafblind the power to connect with others.&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-07T22:14:21.827" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10720" PostTypeId="5" CreationDate="2016-03-07T23:18:43.210" Score="0" Body="&lt;p&gt;&lt;a href=&quot;http://www.afhe.org/about_afhe_definitions.html&quot; rel=&quot;nofollow&quot;&gt;According to the Arizona Families for home education:&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Homeschoolers are parents or legal guardians who choose to educate&#xA;  their own children at home in at least the required subjects of&#xA;  reading, grammar, math, science, and social studies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;https://www.google.com/search?q=homeschooling%20math&amp;amp;ie=utf-8&amp;amp;oe=utf-8&quot; rel=&quot;nofollow&quot;&gt;google search of homeschooling math&lt;/a&gt; shows lots of resources, but not all resources are created equal. Questions on the value of different resources can be asked here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other questions that might be asked include:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Questions on Pedagogy - the method and practice of teaching math to&#xA;homeschooled children  &lt;/li&gt;&#xA;&lt;li&gt;Questions about teaching math to gifted homeschooled children as well as homeschooled children with learning differences. &lt;/li&gt;&#xA;&lt;li&gt;Questions about math related trips a that homeschooled parent might arrange.  &lt;/li&gt;&#xA;&lt;li&gt;Questions on enrichment topics in math for homeschooled children.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Math Education Stack Exchange is wonderful forum for finding guidance from Math Educators on everything related to a homeschooled student's education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that parents who are interested in supplementing their children's math education may have questions that are appropriate for math educators stack exchange but these &lt;strong&gt;are not&lt;/strong&gt; questions that should be tagged with homeschooling.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2016-03-08T21:46:31.117" LastActivityDate="2016-03-08T21:46:31.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10721" PostTypeId="4" CreationDate="2016-03-07T23:18:43.210" Score="0" Body="For questions about math education in the context of homeschooling." OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2016-03-08T04:58:14.833" LastActivityDate="2016-03-08T04:58:14.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10722" PostTypeId="1" CreationDate="2016-03-07T23:56:21.103" Score="13" ViewCount="848" Body="&lt;p&gt;I notice that geometry students frequently have difficulty with representations of 3-dimensional objects in 2 dimensions. Today, we worked with physical manipulatives in order to help visualize where right triangles can occur in 3 dimensions in both pyramids and rectangular prisms (the focus is on fluency with the Pythagorean Theorem and noting its application in many contexts.) I chose to create physical manipulatives instead of finding online 3d manipulatives because &lt;em&gt;I felt as if the physical manipulatives would provide more insight than simply seeing a draggable, yet still 2d, projection.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For clarity: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Physical manipulative in conjunction with 2d drawing:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/PMWeo.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PMWeo.jpg&quot; alt=&quot;Photo of maipulative&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some online examples of &quot;virtual manipulatives&quot;: &lt;a href=&quot;https://www.learner.org/interactives/geometry/3d_pyramids.html&quot; rel=&quot;noreferrer&quot;&gt;1&lt;/a&gt; &lt;a href=&quot;http://www.learnalberta.ca/content/mejhm/index.html?l=0&amp;amp;ID1=AB.MATH.JR.SHAP&amp;amp;ID2=AB.MATH.JR.SHAP.SURF&amp;amp;lesson=html/object_interactives/surfaceArea/explore_it.html&quot; rel=&quot;noreferrer&quot;&gt;2&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: &lt;em&gt;Is there research supporting my intuition?&lt;/em&gt; Are students who have difficulty translating between 2D and 3D more benefited by a physical model than a virtual manipulative, or the other way around?&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2016-03-08T00:04:35.493" LastActivityDate="2016-08-18T11:53:30.863" Title="Physical vs. Virtual manipulatives in the high school classroom" Tags="|secondary-education|reference-request|geometry|manipulatives|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10723" PostTypeId="2" ParentId="10555" CreationDate="2016-03-08T00:26:27.967" Score="9" Body="&lt;p&gt;I don't believe students subconsciously misread the notation as an exponent with a word in front. I think they are not sure at all how to read the notation!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When using logarithms (whether in a basic, introductory algebra course, or an advanced vector calculus course) I always use parentheses to distinguish the argument of the operation, e.g. $\log_a(b)$. But still, there are always plenty of students who (i) don't know how to read that expression verbally, (ii) don't know how to interpret it conceptually and, consequently, from both, (iii) have no idea how to evaluate or manipulate such an expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try to stress the exponential aspect, introducing examples like $\log_2(8)$ and reading it out loud as, &lt;strong&gt;&quot;To what exponent would we need to raise 2 to have 8 as an outcome?&quot;&lt;/strong&gt; Sometimes this helps. But I even worry that the &lt;em&gt;grammar&lt;/em&gt; of that verbal reading of the expression confuses some students just as much! I mean, is this any better: &lt;strong&gt;&quot;What exponent do we need to raise 2 to, to get 8 out?&quot;&lt;/strong&gt; I think not. (And the fact that I picked 2 as the base here only makes it worse; the confusion is just as bad with other bases that don't sound like &quot;to&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's even more shocking to me is that, every semester, I find students &quot;solving a problem&quot; and getting an answer that's something like $x=\frac{4}{\log}$. &lt;strong&gt;I wish I were joking!&lt;/strong&gt; But no, their confusion is so extensive that they don't see anything wrong with such an expression, with an &quot;empty&quot; log like that. I typically respond by saying, &quot;This is like writing $x=3+\sqrt{}$ with nothing underneath; it has no meaning.&quot; But I so rarely get to follow up with those particular students, so I can't be sure whether that comment helps them better understand the concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I wouldn't say your concerns are &lt;em&gt;unnecessary&lt;/em&gt;; rather, I think the root problem(s) precede confusion about super/subscripts. Below, I list a few things that I believe may be related to this confusion, and try to make some suggestions about how to address them.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Problem: Poor arithmetic skills, especially with basic exponentiation.&lt;/strong&gt; If a student doesn't recognize immediately that, say, $81=3^4$, then they're not going to see any sense in the expression $\log_3(81)=4$. It will look like black magic to them. &lt;em&gt;Addressing this: Take the time to show that $3^4=3\cdot 3\cdot 3\cdot 3=81$ when you introduce this example. It will take more time in class, but it will help prevent some from getting lost right off the bat.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Problem: Difficulty understanding the notion of an &lt;em&gt;inverse operation&lt;/em&gt;, or being able to ask/answer a question that probes this concept.&lt;/strong&gt; Look at the sentences I used above: &quot;To what exponent should we raise ... to get ...?&quot; This seems like a simple question to teachers like us who can move &quot;forward and backward&quot; with operations. But to a student, they're used to evaluating expressions &lt;em&gt;forward&lt;/em&gt;, plugging and chugging. This may be the first time (except for square roots) that they face an operation that asks them to &quot;undo&quot; something. &lt;em&gt;Addressing this: Perhaps it's best to introduce the fundamental idea of an inverse operation before tackling exponentials and logarithms. Most College Algebra textbooks I've worked with tend to have Inverses as part of their chapter about exponents/logs, but perhaps we can do a better job of synthesizing all those ideas and presenting them as one.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Problem: Perhaps $\log_a(b)$ is just genuinely bad notation.&lt;/strong&gt; I hesitate to say this because I don't have a good proposal for a replacement. But after several years witnessing the same conceptual and procedural errors from students, I truly wonder whether the notation itself is part of the problem. Our temptation is to say something like, &quot;Well, it's just the way we write down the concept; if they understood what it meant underneath, they would understand the notation.&quot; Or, we might say, &quot;Well, that's how it's written everywhere; we can't change it now.&quot; But I just don't know. Hopefully, someone else can link to some actual research about this idea. &lt;em&gt;Addressing this problem: I don't know. Does anyone have a good proposal for a new notation and reasonable evidence for why it's better for students learning the concept?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="80" LastActivityDate="2016-03-08T00:26:27.967" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10724" PostTypeId="2" ParentId="10722" CreationDate="2016-03-08T00:32:58.943" Score="6" Body="&lt;p&gt;This is not what you seek, because it compares two different&#xA;physical manipulatives, rather than physical vs. virtual.&#xA;But I find it interesting partly because &#xA;my own research involves studying nets of polyhedra.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Scott, Jacqui, Anton Selvaratnam, and Lynden Rogers. &quot;Using Bendable and Rigid Manipulatives in Primary Mathematics: Is One More Effective Than the Other in Conceptualising 3D Objects from Their 2D Nets?.&quot; &lt;em&gt;TEACH Journal of Christian Education&lt;/em&gt; 6.1 (2012): 10. (&lt;a href=&quot;http://research.avondale.edu.au/cgi/viewcontent.cgi?article=1028&amp;amp;context=teach&quot; rel=&quot;noreferrer&quot;&gt;Article link&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/PRory.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PRory.jpg&quot; alt=&quot;Manipulatives&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Abstract excerpts&lt;/em&gt;&lt;/strong&gt;. The purpose of this study was to compare&#xA;  the effectiveness of two different types of&#xA;  manipulatives, bendable and rigid, as aids for&#xA;  the conceptualisation of 3D solids from 2D nets...Contrary to initial expectations, the bendable&#xA;  nets, although more attractive to pupils, did not&#xA;  prove superior to the rigid variety. In fact, the&#xA;  most noticeable advances in conceptualisation&#xA;  followed teaching experiences using the rigid&#xA;  nets.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They cite an article I couldn't find:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Shaw, J. M. &quot;Manipulatives enhance the learning of mathematics.&quot; Houghton Mifflin Mathematics. (2002).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-03-08T00:32:58.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10725" PostTypeId="2" ParentId="10673" CreationDate="2016-03-08T13:15:08.470" Score="5" Body="&lt;p&gt;Ok, let me take my comment seriously and have a try at a personal answer to the second question (I really don't know what students think a variable is, and I feel it is a question that needs difficult research to be answered more than anecdotally).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h3&gt;Short version&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Constants&lt;/em&gt; and &lt;em&gt;variables&lt;/em&gt; are names (often a single letter) given to mathematical object for various reasons. &lt;em&gt;Constants&lt;/em&gt; are mostly a convenience, while different kind of &lt;em&gt;variables&lt;/em&gt; are used to denote object whose identity is not completely specified. I distinguish (at least) three kind of variables: &lt;em&gt;placeholders&lt;/em&gt;, &lt;em&gt;arguments&lt;/em&gt; and &lt;em&gt;unknowns&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Placeholders&lt;/em&gt; are variables used to structure a general reasoning around a fixed but unspecified object. &lt;em&gt;Arguments&lt;/em&gt; are used in functions, as a way to express what part of the expression of the image is the object taken as entry by the function. &lt;em&gt;unknowns&lt;/em&gt; denote putative objects, that may or may not exist, in order to express an equation (or more generally a set of constraints) we want to study (e.g. to determine whether said equation has solutions).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Here is the long version, hopefully useful by its examples (which are addressed to students, for use by teachers). We should distinguish at least the following terms: &lt;em&gt;variable&lt;/em&gt;, which will be the most generic term, and three specific kind of variables: &lt;em&gt;placeholder&lt;/em&gt;, &lt;em&gt;unknown&lt;/em&gt;, &lt;em&gt;argument&lt;/em&gt; (the terminology might not be completely standard, but that is about what fulfills my needs). I won't speak about the indeterminants, which are still another beast and somewhat more confusing (at least to me), and that I would not consider as variables.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;First, the common between these objects.&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;They all are about &lt;em&gt;giving a name to mathematical objects&lt;/em&gt;, but with slightly different goals and convention. That name is usually a Latin or Greek letter, but it may in principle be any symbol or word (e.g. $\square$). They thus relate to &lt;em&gt;constants&lt;/em&gt;, which are names given to well-defined, fixed mathematical objects. For example the golden ratio is defined as $\frac{1+\sqrt{5}}{2}$, and writing $\phi$ for it is just a shortcut. The number $\pi$ has conveniently been given a one-letter name to avoid having to constantly use periphrases such as &quot;the ratio of any circle's perimeter to its diameter&quot; as well as recalling that said ratio does not depend on the chosen circle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What distinguish &lt;em&gt;variables&lt;/em&gt; from constants is that their value is &lt;em&gt;not fixed, or unclear, or unspecified.&lt;/em&gt; The reason why we use a letter is not mere convenience, it is a way to speak about mathematical object before we know much about them. Before going further, let us take an example: $\sqrt{2}$. This looks like a bad example, as it is a constant, whose name has a quite different form from a $x$ or a $y$. But thinking about it, we name it as soon as we are able to define it (i.e. as soon as we can prove or accept that there is a unique positive number whose square is $2$), &lt;em&gt;and then&lt;/em&gt; we can start studying it. For example, if we want to give the first decimal digits of $\sqrt{2}$, we will use its definition and thus will need its existence, and we will use the name we have created before starting the computation. That is an example why we may need to give names to quantities or objects we don't yet know perfectly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difference between the three words to be defined lies in the type of unspecification we deal with. What is confusing is that these different situations are something mathematician are so used to that they don't need to really think about them: they have build an intuitive understanding of how to use them, to the point they don't feel the need to use different words. And that is how we tend to call &lt;em&gt;variables&lt;/em&gt; very different things that &lt;em&gt;look&lt;/em&gt; the same, but which are better distinguished at first.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;The main point: differences between different kind of variables&lt;/h2&gt;&#xA;&#xA;&lt;h3&gt;placeholders&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;This is probably what most mathematically educated people think of when told about variables. A placeholder is a name given to an object of some specified type (we ask that it belongs to some set, for example &quot;let $x$ be a real number&quot;), possibly with some specified property (e.g. &quot;assume $x$ is positive&quot;), but whose identity is not specified either to ensure that the ensuing reasoning can be applicable generally, or because its precise value cannot be determined without specifying completely previously introduced placeholders. Placeholder thus enable us to do one reasoning, and apply it at once to all possible values in the specified set with the specified properties. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, assume we want to prove that the square of an even integer is even. We would let $n$ be an even integer, then apply the definition of $n$ to state that there exists an integer $k$ such that $n=2k$, then compute $n^2=4k^2=2\times(2k^2)$. Observing that $2k^2$ is an integer, we deduce from the definition that $n^2$ is even. In this proof, $n$ is a placeholder that ensures generality, while $k$ is a placeholder whose value depends on $n$ (and is unique given $n$, in this case).&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Arguments&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;When we define a function, we specify (a domain, a target set, and) an assignation to each element of the domain of an element of the target set. We can for example define a function $f$ by stipulating that it maps each given real number to the sum of the square of the given number, minus three times the given number, and two. We see that this is not a very comfortable way of phrasing things. What we do is we use an &lt;em&gt;argument&lt;/em&gt;: we choose any symbol (that does not provoke collision with previously used symbols), say $x$, and use it instead of &quot;given number&quot; above. The example above can then be written $f : x\mapsto x^2-3x+2$; or we say that $f$ is the function defined by $f(x)=x^2-3x+2$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An important point is that the chosen symbol must appear both on the domain side and on the target side, so that we know what it means; for example, assume that there is a &lt;em&gt;parameter&lt;/em&gt; $t$ in our context, which is known to be an unspecified real number. Then we could want to study the function $g$ defined by $g(x)=x^2-tx+2$, and then we have to distinguish clearly between the roles of $x$ and $t$.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Unknowns&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;An &lt;em&gt;unknown&lt;/em&gt; appears notably when one wants to study an equation: it is a name for a putative object of a specified nature (i.e. belonging to a specified set) which satisfies a given constraint or set of constraints (often equations). It is important to stress that the object is putative, in that it may or may not exist, depending whether the equation (or set of constraints) admits a solution or not. But we need to name the thing just to phrase the equation, thus before determining whether it exists or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, consider the question: &quot;which real numbers have their square equal to themselves plus one?&quot; It is rephrased as &quot;find the solutions of the equation $x^2=x+1$ in the real unknown $x$&quot;. (We should always specify the set of allowed values of the unknown, but this is often neglected).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are situations where the different kind of variables are difficult to separate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take for example the following problem: &quot;determine which pairs of real numbers are the sum and the product of two real numbers&quot;. To solve the problem, we can first reformulate it with variables: &quot;let $a$ and $b$ be real numbers, and consider the equation system&#xA;\begin{cases} a =x+y \\ b = xy \end{cases}&#xA;in the real unknowns $x,y$.&quot; Here $a,b$ are placeholders and $x,y$ are unknowns. But then we work a little, and arrive at the conclusion that &quot;the above system has a solution if and only if $a^2-4b\ge0$.&quot; So the answer to the original question is best phrased as an equation on the placeholders. This is ok in the phrasing above, because $a^2-4b\ge0$ is an assertion rather than an equation to be solved. But the difference is thin and can be overlooked by the reader, or the writer may phrase things more loosely. This is probably the kind of situation where extreme confusion can happen, with teachers struggling to understand what is unclear to the students, because teachers have a much better intuitive understanding of these matters, but an understanding that they only rarely phrase explicitly.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-03-08T13:15:08.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10726" PostTypeId="2" ParentId="220" CreationDate="2016-03-08T15:33:19.360" Score="4" Body="&lt;p&gt;Although building up to it might be beyond the time available in many courses introducing induction, an interesting elementary example that seems unbelievable but is very easy to verify in any given case is &lt;a href=&quot;https://web.archive.org/web/20171111004722/http://mathforum.org/library/drmath/view/65670.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pick's Theorem&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="6053" LastEditDate="2022-11-12T21:01:49.510" LastActivityDate="2022-11-12T21:01:49.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10730" PostTypeId="1" CreationDate="2016-03-09T18:58:37.850" Score="6" ViewCount="263" Body="&lt;p&gt;While the question is stated with reference to the iPhone, my actual question is about phones in general. Just as there was much talk about the use of Computers in the classroom over the past fifty years (or so), how much talk is there about the use of cellphones in the classroom?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What benefits are there to students being able to access the internet from their fingertips in a math class?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What methods are there for teachers controlling the use of phones for the expressed purpose of academic interests rather than social interests?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do phones help to forge a mathematical community within the classroom?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can cellphones be used to dissuade the disruptors from damaging the learning experience of other students?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it possible to use phones ethically in the math classroom?&lt;/p&gt;&#xA;" OwnerUserId="6433" LastActivityDate="2016-03-14T18:49:11.970" Title="Do iPhones help students in their math class?" Tags="|mathematical-pedagogy|secondary-education|technology-in-education|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10733" PostTypeId="2" ParentId="2079" CreationDate="2016-03-10T14:45:21.217" Score="1" Body="&lt;p&gt;Note that some programs in pure math allow one to receive a Math PhD with concentration in Mathematics Education (or maybe even a Math Ed PhD?).  Being the Northeast USA, I happen to know of &lt;a href=&quot;http://www.math.uconn.edu/research/research-areas/mathematics-education/&quot; rel=&quot;nofollow&quot;&gt;the University of Connecticut&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that &lt;a href=&quot;http://ceps.unh.edu/mathematics-statistics/mathematics-education-phd&quot; rel=&quot;nofollow&quot;&gt;UNH&lt;/a&gt; also allows one to &quot;slide over&quot; from a Math PhD program to Math Ed, though I'm not sure how formalized that is, and so &quot;ranking&quot; might be a relative concept here.  I suspect there are other programs that would fall under one of these categories which may not make it into lists in the same way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This isn't a direct answer to your questions, but nonetheless I hope it can help answer the underlying question of where to look for such programs.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-03-10T14:45:21.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10734" PostTypeId="2" ParentId="10730" CreationDate="2016-03-10T15:50:48.140" Score="7" Body="&lt;p&gt;Some anecdotes.  I have volunteered with 5th grade math students for a few years.  The general rule is, they do not use electronics unless instructed to do so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We came to a problem where it says to use calculators.  But they had not brought their calculators.  So one guy pulls out his iPhone, with a calculator app.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was a lesson on the period of a pendulum; different lengths, different weights, different amplitudes.  We needed to measure time.  So one guy had an iPhone with a stopwatch app.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was a lesson on estimation.  To begin, we have this drawing with lots of flowers in it, and the students are supposed to think of many possible ways to estimate how many flowers are in the picture.  So one guy takes out his iPhone and talks to it: &quot;Siri, look at this picture...&quot;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2016-03-10T15:50:48.140" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10735" PostTypeId="2" ParentId="10730" CreationDate="2016-03-10T17:33:38.447" Score="6" Body="&lt;p&gt;I teach undergraduate students (ages ~17-22), with class sizes ranging from 10 to 30. I have come to realize that students will have their phones with them no matter what, so it would be unnecessarily punitive of me to try to limit their use. Instead, I have accepted the fact that they're in the classroom and have sought to put them to use whenever I can.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Meanwhile, if a student is going to distract themself with their phone, that's their prerogative; they're nascent adults, so it's really their choice whether they want to be a responsible student. Plus, if I merely see a phone out, how am I to know whether they're texting a friend, checking Facebook, or looking up a definition on Wikipedia? Or checking the arithmetic of a problem we just did? I suppose these reactions might not apply to a classroom with younger students, but with college students, I feel comfortable saying, &quot;It's on you now.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, here are some ways that I think phones/laptops/tablets can be helpful in the classroom, from my experiences:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Following along with slides:&lt;/strong&gt; For all my courses, I post lecture slides well before class time so students can peruse them. I also tell them it would be beneficial to have them open during class so they can follow along and only focus on jotting down important notes in their notebook. They don't have to transcribe everything on the slides because they have them right in front of them. Furthermore, I put a slide counter in the corner, so their notes can say, &quot;March 10, Slide #5: ....&quot;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;&lt;a href=&quot;https://www.polleverywhere.com/&quot;&gt;polleverywhere.com&lt;/a&gt;:&lt;/strong&gt; This is a free site/app for &quot;Clicker&quot; questions. (A paid account lets you use this for &gt;25 people, but with my classes, the free version suffices!) I don't use this in all courses, but when I do, the students really enjoy it. They respond by texting their answer to a given phone #.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Software demonstrations:&lt;/strong&gt; In some classes, I demonstrate techniques on Excel spreadsheets, or on other software, or websites. Allowing the students to open these sites/programs on their laptops at the same time surely makes sense.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Taking pictures instead of transcribing:&lt;/strong&gt; I see students taking pictures of the whiteboard all the time! Rather than trying to copy down everything as I write it, and subsequently missing out on what I'm saying out loud, I notice them paying careful attention to my words, and then when I've stepped aside, they snap a photo of the full board.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="80" LastActivityDate="2016-03-10T17:33:38.447" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10736" PostTypeId="1" CreationDate="2016-03-11T14:10:08.533" Score="7" ViewCount="497" Body="&lt;p&gt;I've often wondered about the &quot;devise a plan&quot; part of Polya's &quot;How to solve it&quot; outline. What we call &quot;problem solving&quot; can be thought of as &lt;em&gt;what to do when you have no idea what to do&lt;/em&gt;. From this standpoint, the plan making activity, particularly the &lt;em&gt;&quot;Do you know a related problem?&quot;&lt;/em&gt; heuristic, seems to beg the question a little. (If we &lt;em&gt;really&lt;/em&gt; didn't have any idea what to do, we would not know a related problem...since this would be an idea of what to do.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This observation is not new. If we have never seen anything like a given problem before, then we perhaps have no real chance of a solution to that problem. The belief that we &lt;em&gt;do&lt;/em&gt; have a chance despite knowing no related problem is, probably, some kind of belief in genius. Most mathematicians would probably not believe in such discontinuities and would say that the solution of a new problem comes from a connection with a dense neighborhood of nearby solved problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder about the above heuristic in connection with Inquiry Based Learning. In an IBL context, the dense neighborhood of previously solved problems are arranged to be discovered by the learner. Ideally, the &quot;first intuition&quot; building block problems are very close to common sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Contrasting IBL to standard methods of instruction, the main difference may be efficiency of recall: an IBL student may have greater perceived ownership of solutions to nearby problems, allowing easier recall when they are needed to solve new problems, whereas a student of more standard methods &lt;em&gt;may&lt;/em&gt; not have the emotional investment in solutions to previously encountered problems to be able to recall them when needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Said another way, IBL methods give students a smaller store of more &quot;deeply experienced&quot; problems to draw from when using the &quot;nearby problem&quot; heuristic, whereas a more content-driven approach exposes students to a larger store of &quot;nearby problem&quot; candidates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that the ability to plan a solution to any problem is related to the ability to recall relevant related problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Has there been any study of the correlation of the ability of students to apply Polya's &quot;related problem&quot; heuristic with the intensity of the students' emotional engagement with the &quot;related problems&quot; recalled? (Here, emotional intensity is admittedly ill-defined.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question aims at helping sort out whether it is best to have students work on many, many problems or to focus on solving a small number of problems with the hope of developing better recall of the techniques learned. This is, of course, a matter of personal taste in problems...but I wonder about what general capacities are encouraged by either approach.&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2016-04-29T07:42:45.300" Title="Polya's &quot;Nearby Problem&quot; Heuristic and Inquiry Based Learning" Tags="|mathematical-pedagogy|psychology|problem-solving|inquiry-based-learning|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10737" PostTypeId="1" AcceptedAnswerId="10748" CreationDate="2016-03-13T00:27:56.480" Score="14" ViewCount="944" Body="&lt;p&gt;Teaching &lt;em&gt;Theory of Computation&lt;/em&gt; for the first time, I encountered&#xA;a phenomenon which perhaps is familiar to others in different contexts.&#xA;I realize most MESE participants are not conversant with &lt;em&gt;Th.Comp.&lt;/em&gt;,&#xA;so I will just sketch the issue at a high level, without explaining the technical terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two important theorems in &lt;em&gt;Th.Comp.&lt;/em&gt; are (1) Every NFA is equivalent to a DFA,&#xA;and (2) Every NFA is equivalent to a regular expression.&#xA;In both cases, a well-chosen example illustrates the equivalence&#xA;and is entirely convincing. &#xA;Then launching into an inductive proof requires introducing&#xA;formidable notation, and in my recent experience, rendered the theorem &lt;em&gt;less&lt;/em&gt;&#xA;clear than it was before the proof presentation.&#xA;This even though the inductive proof was ultimately straightforward.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q.&lt;/em&gt;&lt;/strong&gt; Are there other important theorems that are more convincingly presented&#xA;  via a well-chosen example, than they are via a formal proof?&#xA;  If so, is the burden of formal abstract notation the barrier&#xA;  (as I sense it is in my examples)?&#xA;  Could one justifiably argue that the well-chosen example&#xA;  supplants the need for a formal proof?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-08-02T12:58:50.513" Title="Proofs that make theorems less clear" Tags="|undergraduate-education|proofs|discrete-math|induction|" AnswerCount="4" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="10739" PostTypeId="1" CreationDate="2016-03-14T03:45:22.667" Score="7" ViewCount="244" Body="&lt;p&gt;I was thinking today applications of $\textbf{Ax}=\textbf{b}$ where $\textbf{A}\in\mathbb{R}^{m \times n}$. Specifically, I am interested to know what applications one might give to students, who don't know anything about the topic, when the dimension is large. For instance, I would typically introduce all the necessary apparatus, e.g. multiplication or linear combination, row-reduction, to solve the problem and present the geometric implications for $\textbf{A} \in \mathbb{R}^{3 \times 3}$. However, if you wanted to introduce numerical techniques, motivated by large coefficient data, then what tangible and accessible problem might you lay out for a young student? Something analogous to the intersection of planes in space, which even a young student could meaningfully fiddle with outside of an instructional setting. &lt;/p&gt;&#xA;" OwnerUserId="6496" LastEditorUserId="64" LastEditDate="2016-03-17T22:37:47.813" LastActivityDate="2016-03-29T13:51:04.640" Title="Solving linear systems - Applications" Tags="|undergraduate-education|secondary-education|linear-algebra|applications|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10740" PostTypeId="2" ParentId="10739" CreationDate="2016-03-14T14:02:03.557" Score="6" Body="&lt;p&gt;One possible class of examples of high dimensional systems is discretizations  of infinite dimensional ones.&#xA;Differential equations, as suggested in the comments, are one option, but let me propose another one that hopefully seems interesting and useful to students.&#xA;This is a linear problem that is solved several times every day in hospitals around the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/CT_scan&quot;&gt;X-ray tomography&lt;/a&gt; one attempts to construct a three dimensional image of an object from measuring the attenuation of X-rays passing through it.&#xA;The problem can &amp;mdash; under ideal circumstances &amp;mdash; be easily reduced to recovering a function in $\mathbb R^n$ (with $n=2$ or $n=3$) from its integrals over lines.&#xA;The function is physically the (non-constant) attenuation coefficient.&#xA;The key object here is the &lt;a href=&quot;https://en.wikipedia.org/wiki/X-ray_transform&quot;&gt;X-ray transform&lt;/a&gt;, an integral transform that takes a function on the Euclidean space to a function over the set of all lines in that space.&#xA;The value of the X-ray transform of a function at a line is the integral of that function over that line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is then whether the X-ray transform is injective.&#xA;If it is, ideal measurements indeed uniquely determine what we wanted to know.&#xA;The X-ray transform is linear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It takes a while to set up a theory of the X-ray transform with proper function spaces and such.&#xA;But that is not necessary.&#xA;You can start building a model starting from the simpler end.&#xA;For example, take a $3\times3$ grid with unknown numbers in each square (compare to a sudoku) and take some lines through that square.&#xA;Those numbers are the values of the unknown functions in the squares, and outside the bigger square the function vanishes.&#xA;Given your set of lines (pick horizontal, vertical and diagonal ones), can you determine the nine numbers from the integrals?&#xA;You need to have at least nine lines, but more can make the problem easier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we want to see inside a human patient, nine pixels is not quite enough.&#xA;What resolution should we have?&#xA;How many lines would we need?&#xA;Will the X-ray transforms still be injective?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students can't tackle the problem by hand in realistic scales, but they can play with it with small resolution and learn to appreciate the problem.&#xA;If the course (or your students) is numerically oriented, you can have them play with a &lt;a href=&quot;http://fips.fi/dataset.php&quot;&gt;real X-ray tomography dataset&lt;/a&gt;.&#xA;The walnut looks a bit like the human brain inside a skull, doesn't it?&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="2074" LastEditDate="2016-03-14T14:15:22.857" LastActivityDate="2016-03-14T14:15:22.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10741" PostTypeId="2" ParentId="10730" CreationDate="2016-03-14T18:49:11.970" Score="2" Body="&lt;p&gt;I have had very good results with banning both iphones and computers in the classroom.  Usually I announce that university regulations prohibit the use of iphones, computers, guns, pistols, and invite the students to close them all.  Both the attention span and the grades go up.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2016-03-14T18:49:11.970" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10742" PostTypeId="1" AcceptedAnswerId="10756" CreationDate="2016-03-14T21:26:36.690" Score="1" ViewCount="560" Body="&lt;p&gt;I am having trouble with taking notes in class. I believe this is because we are using slides for the course. Slides bombard students with all this information, and at the same time the teacher is speaking. Moreover, the teacher may be writing down stuff on the board as well. So I usually don't read the slides and try to listen but the notes I take from the board are just mathematical symbols which make no sense to me because there are no headers, just many calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know if you have any tips on note taking when teacher does slides and blackboard and speech simultaneously. Imo this approach on slides gives me no time to absorb the material enough for me to follow it. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="5728" LastActivityDate="2016-03-17T16:58:35.147" Title="How to take notes when teacher uses slides?" Tags="|lecture-notes|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10743" PostTypeId="2" ParentId="10737" CreationDate="2016-03-14T22:28:08.343" Score="9" Body="&lt;p&gt;There are many situations in which we have a clear collective understanding of &lt;em&gt;intention&lt;/em&gt;, or &lt;em&gt;goals&lt;/em&gt;, and examples which persuade us that these goals are plausible, as well as illustrating apparent causal mechanisms identifiable as &quot;reasons&quot; for things being the way they are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For many reasons, the contemporary style of proof-writing, especially in intensely didactic situations, is wildly different from any attempt at &quot;persuasion&quot;, or invocation of &quot;intuition&quot;. Indeed, on one hand, one certainly wants whatever &quot;physical&quot; intuition is available, but, on the other, one eventually wants a proof that independent of literal physical phenomena or analogies. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fundamental theorem of calculus, the intermediate value theorem, and many other of the basics of calculus, were visibly more-or-less true long before they were formalizable at all... and their utility gave some interest to eventual formal proofs, even while those formalized arguments, needing also formalized contexts, do not (to my perception) clarify or enhance the physical intuitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At least one particular pattern of events produces unhelpful, unpersuasive proofs: situations in which some interesting, important examples are fairly immediate, but it is less clear what &quot;the general case&quot; might be, and how to describe it. For example, the topic of &quot;topological vector spaces&quot; is arguably very practical, insofar as many natural spaces of functions should appear here, but there has been sufficient abstraction and eponymous-adjectification so that statements of theorems, and their proofs, &lt;em&gt;can&lt;/em&gt; be couched in terms that do not obviously refer to any example of interest... and it often requires some scholarly chops to figure out the relevance of the iconic theorems to any particular tangible example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a different vein, the Cayley-Hamilton theorem? Historically/originally, just proven for the two-by-two case and three-by-three, and in the two-by-two the most naive possible computation succeeds. The three-by-three case can also succeed by most-naive means, but it is ghastly. The general case needs at least one of several more-abstract devices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not really undergrad, but certainly discussion of Sobolev spaces and regularity properties on Riemannian manifolds is best introduced by $\mathbb R^n$ and $n$-fold products of circles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generalities about automorphic forms and $L$-functions, and &quot;Whittaker models&quot; are arguably best introduced via discussion of the holomorphic case, with explicit exponentials for &quot;Whittaker functions&quot;, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Riemann surfaces are probably best introduced via the square root function, logarithm, and so on, prior to &quot;covering by an atlas of charts...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And from the other side, it is invariably wise to check an idea by considering examples... and if one cannot find any tangible example, or cannot determine the truth or falsity of the immediate question, that in itself is telling.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-03-14T22:28:08.343" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10744" PostTypeId="2" ParentId="9847" CreationDate="2016-03-15T04:54:53.823" Score="-1" Body="&lt;ol&gt;&#xA;&lt;li&gt;Perpendicular bisector of the 3 sides taken 2 at a time circumscribe circle about the triangle.&lt;/li&gt;&#xA;&lt;li&gt;Note that longest side = 2r = d&lt;/li&gt;&#xA;&lt;li&gt;Big angle is inscribed and =1/2 180&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6333" LastActivityDate="2016-03-15T04:54:53.823" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10745" PostTypeId="2" ParentId="10739" CreationDate="2016-03-15T11:32:50.373" Score="3" Body="&lt;p&gt;A relatively low dimensional example is the problem of oil mixing: when preparing a certain quality of gas to be sold, an oil company mixes several oils of various composition which have been extracted and refined, and aims to get a certain composition. The question is in which ratio to mix the oils to get the wanted gas (I hope I got the vocabulary right, I am not a native English speaker).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-03-15T11:32:50.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10746" PostTypeId="2" ParentId="10739" CreationDate="2016-03-15T16:50:08.147" Score="6" Body="&lt;p&gt;&lt;strong&gt;Balancing Chemical Equations&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Balancing chemical equations is an effective and useful application, especially for students who are also taking chemistry. This &lt;a href=&quot;http://www.iiste.org/Journals/index.php/JNSR/article/viewFile/20721/21315&quot; rel=&quot;nofollow&quot;&gt;article from the Journal of Natural Sciences Research&lt;/a&gt; gives a detailed explanation and provides multiple examples. From the conclusion of that article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This allows average, and even low achieving students, a real chance at success. It can remove what is often a&#xA;  source of frustration and failure that turns students away from chemistry. Also, it allows the high achieving to&#xA;  become very fast and very accurate even with relatively difficult equations. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The practical superiority of the matrix procedure as the most general tool for balancing chemical&#xA;  equations is demonstrable. In other words, the mathematical method given here is applicable for all possible&#xA;  cases in balancing chemical equations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems like a cumbersome method, but students get the knack of writing the matrix directly from the chemical equation, and using a calculator to put the augmented matrix into rref allows for students to balance any equation in about a minute, an incredible improvement over the guess-and-check method that most chemistry teachers employ. It's an invaluable method that I teach all my students that are also in AP Chemistry, since they are allowed to use their calculator on the exam. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.chembuddy.com/?left=balancing-stoichiometry-questions&amp;amp;right=balancing-questions&quot; rel=&quot;nofollow&quot;&gt;This site&lt;/a&gt; lists about 260 unbalanced chemical reactions from which you can pick exercises.  &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-03-29T13:51:04.640" LastActivityDate="2016-03-29T13:51:04.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10748" PostTypeId="2" ParentId="10737" CreationDate="2016-03-15T23:25:21.067" Score="15" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;A well-chosen example illustrates ... and is entirely convincing.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For me, all of what is usually called &quot;generic proof&quot; satisfy your criterion. Consider the Euclidean algorithm for finding the greatest common divisor of two numbers. A well-chosen example tells your students all they need including why the algorithm gives the greatest common divisor, how to use the algorithm, what is the structure of &quot;proof&quot;, etc. The symbolic proof does not tell them anything more in terms of the content of the algorithm. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me give another example in the other direction where it is better to use symbols from start (of proof). Suppose you want to prove that the product of two sums of two squares is also a sum of two square. Of course, you can use numerical examples to show that is true. For example, &lt;span class=&quot;math-container&quot;&gt;$2=1^2+1^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$5=1^2+2^2$&lt;/span&gt;, and for these, we have &lt;span class=&quot;math-container&quot;&gt;$2 \cdot 5=10=1^2+3^2$&lt;/span&gt;. I bet if your students are like mine, that example would be &quot;entirely convincing&quot; for them! But, it is not what you want since, first of all, they shouldn't be convinced just by a couple of examples, and in this case, because the &lt;strong&gt;structure&lt;/strong&gt; of the proof is in fact can be used to write the product as a sum of two squares. I believe, to &lt;strong&gt;illustrate the proof&lt;/strong&gt; of this one, it is better not to use an example since concrete examples would hide the &lt;strong&gt;structure of the proof&lt;/strong&gt;; using numbers make it hard for students to see how the product should be reorganized since they are naturally tempted to calculate all the intermediate steps rather that to keep the numbers as they are.    &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="12399" LastEditDate="2019-08-02T12:58:50.513" LastActivityDate="2019-08-02T12:58:50.513" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="10749" PostTypeId="1" CreationDate="2016-03-16T10:05:26.813" Score="15" ViewCount="1815" Body="&lt;p&gt;Terrence Tao commented of internalizing [here: &lt;a href=&quot;https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/&quot;&gt;https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/&lt;/a&gt; ] &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;It is true that some mathematicians can be vastly more efficient than&#xA;  others at learning material, but I feel this is more due to experience&#xA;  and an efficient means of study than to any innate genius ability,&#xA;  though of course innate talent is still a contributing factor. For&#xA;  instance, among the graduate students I have advised, the first paper&#xA;  they read in a subject often takes a month or so to read (and they&#xA;  have a question on almost every page on the paper); but after a few&#xA;  years, they can get the gist of a new paper in the subject within a&#xA;  day, skimming past all the “standard” (or at least “plausible”)&#xA;  portions of the argument and focusing on the key new ideas. &lt;strong&gt;The key,&#xA;  I think, is to find one or more efficient ways to internalise the&#xA;  subject – either by using formalism, or geometric intuition, or&#xA;  physical intuition, or some other analogy or heuristic.&lt;/strong&gt; Each&#xA;  mathematician has their own different way of doing this. Ramanujan,&#xA;  for instance, apparently performed a tremendous number of numerical&#xA;  computations, and derived much of his intuition from the patterns he&#xA;  observed from those computations. The intuition wasn’t always correct&#xA;  (for instance, he famously gave an incorrect formula for the n^th&#xA;  prime), but he did discover a number of amazing results this way, some&#xA;  of which took a long time to prove rigorously.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How one effectively internalizes portions of math to make it as intuitive as breathing, as suggested in the quotation ?&lt;/p&gt;&#xA;" OwnerDisplayName="user6330" LastEditorUserId="1397" LastEditDate="2016-03-18T15:39:18.223" LastActivityDate="2016-03-21T19:35:00.130" Title="How to effectively internalize math?" Tags="|self-learning|intuition|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10751" PostTypeId="1" CreationDate="2016-03-16T13:23:20.027" Score="19" ViewCount="2328" Body="&lt;p&gt;I've been teaching the Pythagorean Theorem in my 8th grade class, and I noticed something odd. In the book I'm using, the sequence goes something like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Motivate the idea of distances on a grid&lt;/li&gt;&#xA;&lt;li&gt;Talk about areas on a lattice grid&lt;/li&gt;&#xA;&lt;li&gt;Work on finding the areas of irregular shapes -- including tilted squares&lt;/li&gt;&#xA;&lt;li&gt;Talk about the relationship between squares and side lengths, introduce irrationals and things like that.&lt;/li&gt;&#xA;&lt;li&gt;Look at squares coming off of a triangle. By now, students can find the area of all 3 squares coming off the right triangle, and derive the equation a^2+b^2=c^2 just by observation from a few example problems.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;While I think this is very cool, it's not really a proof. They see a few example cases, and then the sum of the squares coming off of the legs of the triangle are exactly the same as the area of the square off the hypotenuse! And if we introduce labels a, b, and c instead of numbers, they can derive the Pythagorean Theorem this way. I noticed that a lot of teachers teach the Pythagorean Theorem this way. There are a lot of visual gifs and videos showing that the areas of the squares coming off the triangle add up, but is that technically a proof? At this point, the students are actually very close to seeing a more algebraic proof by the time they get to that last bullet point! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All they would need to do is take the tilted square (c^2) and find the area with respect to labels a and b. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Qxkme.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Qxkme.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's two ways to do this. The &quot;inside&quot; way and the &quot;outside way&quot;. The inside way is adding up all the inside areas using labels a and b. The outside way is taking the outer box and subtracting off the corner triangles. Using real numbers, this isn't very difficult, but using a, b, and c as the numbers is a huge &quot;cognitive leap&quot; for these students. So at the end of the day, my question is: Should I be leading students through an actual proof of the Pythagorean Theorem? Or is an exploration of deriving the formula good enough?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this all made sense! If not, I'll do my best to clear things up.&lt;/p&gt;&#xA;" OwnerUserId="4737" LastEditorUserId="143" LastEditDate="2016-03-16T14:04:47.963" LastActivityDate="2021-12-24T06:43:01.840" Title="Should my 8th graders see a proof of the Pythagorean Theorem?" Tags="|secondary-education|proofs|geometry|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10752" PostTypeId="2" ParentId="10737" CreationDate="2016-03-16T14:41:24.467" Score="3" Body="&lt;p&gt;I think that whether or not a proof makes a theorem more convincing depends on who exactly you are trying to convince. While teaching an introductory math course to non-mathematicians (like say, calculus I in the U.S.), the emphasis is often on &lt;em&gt;using&lt;/em&gt; the results. I think it is rare that a proof helps a student use a result, so proofs might not help a student go, &quot;Oh, I see that result is true!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the example of the product of sums of squares in the answer of Amir Ashgari is not quite fair. Imagine if instead you told students that the theorem was $$(b^2+c^2)(e^2+f^2) = (cf+be)^2+(ce-bf)^2.$$ &#xA;Then there is no need to find anything useful in the proof. Moreover, with this statement, examples do illustrate the proof well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In calculus, we don't (or I don't) say to students that the derivative of the product of two functions can be expressed using the derivatives of the the two individual functions and the functions themselves, and then leave the actual formula to be discovered in the proof. Instead, I give them a formula and illustrate it with examples that can be checked &quot;by hand&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, if the main result of a theorem is not a formula, then this discussion doesn't apply as readily. I am fine with the statement that every metric space is Hausdorff. I think the proof of this statement does a better job than any specific example. (Because any specific example would likely generalize to the proof in general. But there are lots of metric spaces, so maybe the student would not be convinced by the specific example.) Now to criticize my own example, I could have stated this result as: Any two distinct points in a metric space are separated by open balls of small enough radius. This way, the proof is basically in the statement, just like the theorems whose punchline is a formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I guess my (very weak) answer to all of the questions in the post is that it just depends on the audience. The audience determines how one should state theorems as well as how one should illustrate the truth of any given statement.&lt;/p&gt;&#xA;" OwnerUserId="6511" LastActivityDate="2016-03-16T14:41:24.467" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10753" PostTypeId="2" ParentId="10751" CreationDate="2016-03-16T14:51:26.437" Score="16" Body="&lt;p&gt;If you are in the United States, at a public school, then you should explain a proof because this is one of the common core state standards:  &lt;a href=&quot;http://www.corestandards.org/Math/Content/8/G/B/6/&quot;&gt;http://www.corestandards.org/Math/Content/8/G/B/6/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest using the following proof:  use a line perpendicular to the hypotenuse and passing through the vertex of the right angle.  This divides the triangle into two similar right triangles.  Using this similarity, one can prove the theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This proof is somewhat preferable to others because it connects to the larger story of similarity, instead of being a one off proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if common core were not an issue, I think that mathematics is about making sense of patterns.  Surely you should investigate, and notice the pattern first.  Just noticing the pattern is not enough:  math is not an experimental science.  We should seek to understand why the pattern holds.  So yes, we should try and find an explanation (a proof).&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-03-16T14:51:26.437" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10754" PostTypeId="2" ParentId="10751" CreationDate="2016-03-16T16:55:12.373" Score="7" Body="&lt;p&gt;I think the usual visual proof of PT uses something like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ojdtE.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ojdtE.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(From &lt;a href=&quot;http://www.mathsisfun.com/geometry/pythagorean-theorem-proof.html&quot; rel=&quot;noreferrer&quot;&gt;Math Is Fun&lt;/a&gt; website)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make cardboard cutouts of the outer and inner squares and the 4 identical triangles. Fit them together as shown. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comparing areas, you can obtain:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a+b)^2 = c^2 + 4(ab/2)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Getting rid of the brackets and cancelling 2ab from bothsides, you obtain as required:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a^2 + b^2 = c^2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe one student in the class will &quot;get it?&quot; Might be worth it, though, to light a fire under that one student. Make it quick though. I would think that most students at that level won't get it no matter how long you take, but it may help them remember the formula if your presentation is sufficiently animated. &lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2016-03-16T17:24:04.040" LastActivityDate="2016-03-16T17:24:04.040" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10755" PostTypeId="2" ParentId="10749" CreationDate="2016-03-17T12:16:10.187" Score="9" Body="&lt;p&gt;This is an excellent question. Some good advice on this can be found in the writing of Bill Thurston, some of which I have posted in an answer to &lt;a href=&quot;https://mathoverflow.net/questions/220052/essays-and-thoughts-on-mathematics/220066#220066&quot;&gt;this question on Math Overflow&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The opening of the quotation I posted there is particularly telling: &quot;Mathematics is a paradoxical, elusive subject, with the habit of appearing clear and straightforward, then zooming away and leaving us stranded in a blank haze.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This feeling of being abandoned by intuition, or feeling that mathematics is a wall keeping you outside, is common even to older mathematicians. I, personally, struggle with this all the time. When in the throes of a problem about which I have developed an intuitive way to think, life is never better. Other times, though, I feel somewhat lost. The literature looks like a lot of noise that does not promise the sense of meaning one gets from having that intuitive idea that is making sharp predictions that prove theorems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point, though, is to keep trying to make your own intuitive way into the subject via SOME channel. Just keep trying. I think this is what mathematics is, at its heart: trying to develop these human-friendly ways to think about mathematical problems. This trying is connected to a certain &lt;em&gt;feeling of freedom&lt;/em&gt; to play with one's surroundings using one's natural capacities. &lt;a href=&quot;http://www.thebookshelf.auckland.ac.nz/docs/Maths/PDF2/mathschron014-001.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;This essay&lt;/a&gt; may help with some first steps to try.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Extension of Answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are other things to read that may be helpful. One that comes to mind is Hadamard's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486201074&quot; rel=&quot;nofollow noreferrer&quot;&gt;Psychology of Invention in the Mathematical Field.&lt;/a&gt;. In this essay, descriptions of several mathematicians' thought processes are described. Again, read as much Thurston as you can, as well as some essays by Gowers and Tao's blog, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a starting assumption, we can take that mathematics resides in the minds of living mathematicians. The lifeblood of mathematics is how mathematicians think about the problems and propositions that are recorded in proofs and calculations. To differing degrees, this can appear as residing inside these proofs and calculations. (Why else would someone as great as Hilbert even dream of suggesting his formalist programme?) Mathematicians have the capability of making associations and analogies that lead to new mathematical connections. Most such ideas occur in the minds of mathematicians, minds which have evolved to have various modules for interacting with the &quot;external&quot; world via sensory data. The spatial sense, perhaps with the sense of touch yields geometric intuition, the sense of time, language and hearing (think music) gives rise to an algebraic intuition. The beauty of these is that they are not evenly subdivided: an algorist's sense of formal beauty sometimes can come from a &quot;geometric&quot; appreciation of formulas, and there is certainly a temporal aspect of drawing a diagram that might communicate more than seeing a static picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some mathematicians have cast mathematics as a dance between the continuous and discrete, or perhaps &quot;potential&quot; versus &quot;actual&quot; infinities. This is certainly related to the interplay of this sense of time and space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Hadamard's essay above, he reports that most mathematicians don't think verbally, but instead somehow fashion &quot;vague images&quot; to manipulate in the mind. One mathematician desribed these as &quot;muscular contractions&quot;. Interestingly, George Polya (who wrote 'How to Solve It') was the only mathematician surveyed by Hadamard who reported working largely verbally. Others reported working with some sort of semi-linguistic thinking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most mathematicians would advocate for computing &quot;to get feelings&quot; for some bit of mathematics. Why does this work? Why does computing give us feelings? (See Whitney's excellent essay linked to above for a bit on this.) Some mathematicians can't think without a pencil and paper nearby on which to scribble. Other mathematicians have to go for a long walk to try to imagine what a computation is likely to &quot;look like&quot; before sitting down to actually compute (otherwise they would immediately feel overwhelmed by the complexity). I think it was Gowers (please, someone, correct my if I'm wrong) who wrote that a mathematician tries to &quot;imagine the outline of a proof, no matter how vague&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We all want a sense of &quot;gestalt&quot; to the mathematics we do. We want to feel the entirety of the proof we are working on as a whole in our mind. That said, we have to work on the pieces, to compute and to piece together a chain of steps. (This feels somewhat, again, like the reconciliation of &quot;space&quot; and &quot;time&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal position is that to feel like we are &quot;living in the mathematics&quot; and not &quot;held outside the walls&quot;, mathematics must be done with the whole person (or at least the whole brain), in the sense that the fullness of experience we undergo while struggling with a problem makes impressions on our mind that give the problem a &quot;face&quot;. We can move around in our minds to see the different parts of this face and see how calculations are to work out before we even begin them. We build this &quot;face&quot; by trying to do mathematics in deeply engaging ways over a very long period of time. We know the face of our most beloved problems like we know the face of a loved one. We know its every contour and yet still long to know more. Each &quot;feeling&quot; we gain from a calculation or more bizarre attempt to understand (think of Terry Tao rolling around on the floor of his office to understand a PDE…he reported doing this somewhere on MO) is stored to be accessed later. All the people we talk to, pictures we draw, things we smell while working, places we worked, pain we endured while working etc… all of these things are rolled up and embedded in OUR mathematics. To a mathematician, genuine mathematics is an expression of the whole self and cannot be anything less.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to develop human-friendly ways of thinking about things? Struggle mightily to be able to explain and use them in as many ways as you can. As you attempt to understand a proof, the total of the attempts you make to get used to the reasoning, the myriad angles you try in order to &quot;understand&quot; it, all the angles you try to see it from, all the failed attempts, will give it a face that will become part of you. I don't think this process ever ends. You just have to fight to keep trying. I think there is a lot to say about Whitney's &quot;freedom&quot; in his essay, in this regard. As the cares of life multiply, the ability to fight for mathematics is threatened. If you can keep your life simple enough to protect this freedom, this is probably the best way protect the ability to continue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I apologize if this is all too &quot;subjective and argumentative&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Second edit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In light of the bounty comments, I can add a bit more to this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We may interpret &lt;em&gt;internalization&lt;/em&gt; as a feeling that the mathematics is part of us, and not something we are superficially tinkering with external to ourselves without any feeling of understanding. I will share how I activate each of Tao's modes in my own work, phrasing them as specific things to try:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Formalism: Given a problem you are interested in, take the simplest nontrivial example you can think of, and just compute with it on a sheet of scrap paper. Writing down these details formally will give you a sense of the problem that can be felt as a sort of muscle-memory. To amplify this effect, write details on a large blackboard. This will amplify your spatial sense to aid with memory of the written details. Also, speak the formal logical arguments out at different scales, to activate your auditory sense. I find it good to speak out loud details while you are taking a long drive or walk. Try to picture the vague shape of the calculations and rapidly predict how they should look when complete. Roughly, push symbols around on paper and in your mind. Focus on how the symbols interact and the balance of the symbols and their analogy with other computations you have seen before in the literature. Choose different notations. Just calculate. Make formal substitutions. Doing this will contribute to your &quot;feelings&quot; for your problem. Insight here will rely on &quot;formal analogy&quot;, in that formulas and calculations that remind you of other formulas and calculations will lead to insights (Read the Poincare essay. His remarks about &quot;Theta-Fuschian functions&quot; discuss such a formal analogy.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Geometric intuition:&lt;/em&gt; Simply put, draw pictures. Try to get away from the symbolic and instead focus on the spatial. Scribble pictures that express the relative relationships of things on paper. If you are working in higher dimensions, try to capture some aspect of the problem in a 2 or 3-dimensional picture. (E.g. if you are working on a closed convex cone in an infinite-dimensional Hilbert space, just draw a picture of a cone and try to think about this low-dimensional analogy.) Gesticulate with your hands while trying to visualize the relations in three dimensions. Try to animate these pictures. Make sound effects (don't laugh, this can help sometimes) for how certain parts move. Even when doing analysis, draw sequences and limit points etc…and not just inequalities. Try to engage your visual sense. This often requires &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/087477523X&quot; rel=&quot;nofollow noreferrer&quot;&gt;shutting off your verbal sense.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Physical intuition:&lt;/em&gt; This is a lot like geometric intuition, but incorporates our sense of &lt;em&gt;force&lt;/em&gt; and &lt;em&gt;motion&lt;/em&gt; more actively. Try to interpret the problem in a way where physical sense, or the ability to predict what should happen as a result of a physical experiment, can suggest a solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Other heuristics&lt;/em&gt;: A heuristic is a rule of thumb. Analogy is a powerful tool for mathematics, and some thinkers believe that analogy is the basis for nearly all thought. &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691025096&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical analogy&lt;/a&gt; is a comparison of two mathematical structures such that the component parts making up the structures are compared, as well as how they fit together. Any way we can bring abstract mathematics into analogy with sensory experience in such a way that we can more intuitively think about the mathematics can be useful. For example, when we learn to count as children, we are not taught about equivalence classes or von Neumann ordinals but instead witness a manifestation of the abstract in counting stones with our hands. The feeling of understanding that comes from feeling and seeing and hearing and smelling and speaking the numbers yields a sense of understanding. Seeking other analogies to make abstract things concrete is one of the principle tasks of practicing mathematicians. Connecting back to more concrete intuitions is key. This is the role of working with the most concrete examples possible. This said, the ability to rapidly read papers in an area is tied to the internalization of certain key fruitful analogies developed by the leading experts in the field. Talking with these people, you can quickly gain their intuitions and use these intuitions to help make sense of the literature. For example, Sorin Popa once described his &quot;deformation/rigidity strategy&quot; as follows: Imagine a bucket of dark water in which you know there is a bowling ball. If you put your hands in the water and swish them around but never feel the bowling ball, you know the bowling ball has to be where your hands never went. This simple picture (of very crude physical intuition type) maps pretty faithfully to one of the most successful and productive ideas in my subject's history. Of course, a lot of hard work goes into applying the idea. Without the idea, though, the literature can look very opaque to an outsider. Often you can get a lot of mileage from getting good organizing analogies from &quot;the big guys&quot;. &lt;a href=&quot;https://mathoverflow.net/questions/234125/expert-intuitive-organizing-analogies?noredirect=1#comment579253_234125&quot;&gt;I've asked an MO question that should try to gather some more such analogies from different areas&lt;/a&gt; of expertise in the hope that these will help you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Language and writing:&lt;/em&gt; In addition to the above, I think that writing mathematics down (and writing it up) can help make it intuitive, as it improves memory and harnesses our considerably developed linguistic sense. A turn of phrase or slogan or other written &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Intuition_pump&quot; rel=&quot;nofollow noreferrer&quot;&gt;intuition pump&quot;&lt;/a&gt; can help us feel more connected with our problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To test whether we are internalizing mathematics is somewhat impossible, as it is a matter of our subjective experience of the world. You know that you are internalizing mathematics when you feel it. It is for this reason that I say it is intensely personal. Your experience and mine are very different, and this is the great benefit of having many mathematicians at work. Our individual experiences provide uncountably many perspectives that may lead to new interpretations and discoveries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Amir Asghari &lt;a href=&quot;https://mathoverflow.net/questions/38639/thinking-and-explaining?lq=1&quot;&gt;reminded me of this fabulous MO question&lt;/a&gt;, which is probably a better answer to the present question than all of what I've written above! &lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2016-03-21T19:35:00.130" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10756" PostTypeId="2" ParentId="10742" CreationDate="2016-03-17T16:58:35.147" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8IWhn.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8IWhn.png&quot; alt=&quot;binder&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Print the slides on ordinary paper in portrait orientation with two slides per page. Print only on one side of the paper.&lt;/li&gt;&#xA;&lt;li&gt;Use a three hole-punch to punch the paper on the opposite side you normally would, so that the slides are on the &quot;back&quot; of each sheet. &lt;/li&gt;&#xA;&lt;li&gt;Bind the printed slides in three-ring binder, and include additional paper. After turning the first page, the first two slides will be visible on the left side. Take notes on the blank sheet (the back of the next page of slides) on the right side in the space opposite each slide. If you need room for additional notes, bind a sheet of additional paper before the next page of slides. &lt;/li&gt;&#xA;&lt;li&gt;When reviewing, both slides and notes are simultaneously visible and the system is neat and organized.&lt;/li&gt;&#xA;&lt;li&gt;Use your awesome notes to diligently study, and get an A+.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5690" LastActivityDate="2016-03-17T16:58:35.147" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10757" PostTypeId="2" ParentId="10749" CreationDate="2016-03-18T01:16:46.953" Score="5" Body="&lt;p&gt;Ok, speaking from personal experience at digesting math related topics quickly, I'd say that the key is to think about it often.  I perform this by being an active user on the &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;math SE site.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Use it or lose it.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a well known quote, and it translates as &quot;Use your knowledge, or lose that knowledge.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another quote that should exist would be &quot;A sword used often does not rust.&quot;  You will find that if you don't practice mathematical topics often, at least weekly, for example, you will find it hard to learn mathematics on your own at an accelerated pace.  You must learn and practice your skills often to hone them to the best of your ability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Persistence is key&quot;  Remember that if you don't persist, you will surely fall short.  Of course, don't forget to do other important things, but practicing and learning must be done continuously and without too much time between each learning session.  I personally aim for about 1-2 hours per day dedicated to my study of mathematics.  More or less depending on if I feel I've hit something I want to learn or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't try to confuse yourself.  Aim for topics you can somewhat understand, then ask for assistance if needed.  I ask for help concerning different topics all the time on the math SE site.  Go for something slightly above comfort level and use online resources to assist you where your knowledge falls short.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I taught myself Calculus, the key was to simply keep trying and to think about what I could use this new knowledge to do.  For example, I was able to take what I learned from Calculus to show that the trigonometric identities were true by differentiating both sides.  Simple things like that made Calculus more than a study of the shape of a graph, it became a tool I could use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also note that you can learn a lot by simply &lt;em&gt;being&lt;/em&gt; on the math SE site.  If you see a question you can answer, attempt to answer it, and later, compare your answer to other answers.  Alas, some questions are beyond my level, but often times I can learn a small bit of something from reading those answers.  Start by going through tags that contain topics you are already familiar with.  Then learn what you can from it and slowly that knowledge will build.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I started out by being an active user under the tags algebra-precalculus, sequences-and-series, trigonometric-functions, and exponential-function, all topics I was familiar with.  Eventually, I learned about an amazing number of ways to evaluate summations, I even learned how to evaluate some divergent summations.  I learned how to include complex numbers into my trig and exponential functions, and I learned some miscellaneous topics from the algebra-precalculus tag.  Eventually, I would branch off by finding tags in questions I understood and following them to see where they could take me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, you don't have to follow this path, I merely do it as a recreational activity, but it has benefitted my very much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By reading your post, I assume you are someone who doesn't want to memorize random formulas, but your main goal is to be able to swallow down the information in a short amount of time.  There are two obvious things that pop into my head.  First, use a lot of your time.  Secondly, you should try to 'understand' the math, which I can imagine you are already doing well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Understanding the topic generally requires that you think about two things.  You should try to understand 'why' something, binomial expansion for example, works the way it works.  Secondly, you should try to use it or apply it, as I have shown you could do by thinking outside the box with something like Calculus or by being a user on the math SE site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's my personally path to my status quo, but your own path my very well be different.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2016-03-18T01:16:46.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10758" PostTypeId="1" CreationDate="2016-03-18T14:55:47.597" Score="25" ViewCount="1120" Body="&lt;p&gt;All of us know, when teaching, the &amp;quot;right&amp;quot; choice of examples is important. Though, making the &amp;quot;right&amp;quot; choice is one of those things that are easier said than done. Here is the story of a series of unfortunate examples I used in my class yesterday. My class was for undergraduates students, but what I observed is quite general. Thus please consider an answer for this question even if you are just involved in teaching school mathematics. The subject of the class was double integrals. Here are the three &amp;quot;simple&amp;quot; examples that I used one after the other to exemplify changing the order of integration in double integrals.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^2 {\int\limits_0^1 f(x,y) dxdy }$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^1 {\int\limits_0^{y-1} f(x,y) dxdy }$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^1 {\int\limits_x^{2x} f(x,y) dydx }$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;They look three innocent examples. But let us see what happened. We happily changed the order of integration in the first integral, accompanying it with all the usual explanations. Thus:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^2 {\int\limits_0^1 f(x,y) dxdy }$&lt;/span&gt; changed to &lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^1 {\int\limits_0^2 f(x,y) dydx }$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As you might guess, what the students got from the whole process was only how the final answer looks. As a result:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^1 {\int\limits_0^{y-1} f(x,y) dxdy }$&lt;/span&gt; changed to &lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^{y-1} {\int\limits_0^1 f(x,y) dydx }$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We draw the region of integration, discussed the problem again and came to the following correct answer:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^1 {\int\limits_0^{y-1} f(x,y) dxdy }$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^1 {\int\limits_0^{x-1} f(x,y) dydx }$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As you might guess again, what the students got from the whole process was only how the final answer looks. As a result, they got the following answer for the next problem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^1 {\int\limits_x^{2x} f(x,y) dydx }$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^1 {\int\limits_y^{2y} f(x,y) dxdy }$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Unlucky me, the function that I had used, for the sake of simplification,was &lt;span class=&quot;math-container&quot;&gt;$f(x,y)=1$&lt;/span&gt; which gives the same numerical answer whether you calculate&#xA;&lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^1 {\int\limits_x^{2x} 1 dydx }$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\int\limits_0^1 {\int\limits_y^{2y} 1 dxdy }$&lt;/span&gt;!!&lt;/p&gt;&#xA;&lt;p&gt;It wasn't the first time that I observed such a phenomenon, but perhaps it was the most bizarre one so far. In fact, this is a famous phenomenon with some famous examples. All of us have heard of those students who think all the triangles should have a base parallel to the edge of the paper since all the triangles they had seen were as such. I am also sure, every teaching day, we observe one or two of these &lt;strong&gt;mis-generalizations based on examples&lt;/strong&gt;. The purpose of this post is to share these observations.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is your series of unfortunate examples?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="2083" LastEditDate="2021-03-27T15:43:19.667" LastActivityDate="2021-03-28T00:45:57.513" Title="A Series of Unfortunate Examples!" Tags="|undergraduate-education|secondary-education|examples|primary-education|students-mistakes|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10759" PostTypeId="1" AcceptedAnswerId="10760" CreationDate="2016-03-18T18:59:38.490" Score="10" ViewCount="344" Body="&lt;p&gt;The &lt;em&gt;adjugate matrix&lt;/em&gt; of an $n \times n$ matrix $A$ is defined by $(\mathrm{adj}\ A)_{k\ell} = (-1)^{k+\ell}\,\det M(\ell,k)$, where $M(\ell,k)$ is the minor matrix obtained from $A$ by deleting row $\ell$ and column $k$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The obvious application of the adjugate is the identity $A (\mathrm{adj}\ A) = (\mathrm{adj}\ A)A = (\det A)I$. Arguably this gives ample motivation, but only after a 'rabbit-from-hat' definition.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a good way to motivate the adjugate matrix in a first course on matrix algebra?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="640" LastEditorUserId="475" LastEditDate="2017-11-14T12:18:26.473" LastActivityDate="2017-11-14T12:18:26.473" Title="How can one motivate the adjugate matrix?" Tags="|concept-motivation|linear-algebra|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10760" PostTypeId="2" ParentId="10759" CreationDate="2016-03-18T19:44:16.750" Score="10" Body="&lt;p&gt;Here is one way to put the rabbit in the hat before pulling it out:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Derive the general formula for the inverse of an invertible matrix.  It ends up having the form $A^{-1} = \frac{1}{\textrm{Det}(A)} \textrm{Adj}(A)$, where $\textrm{Adj}(A)$ is something we just now discovered.  Hey, the formula for  $\textrm{Adj}(A)$ makes sense whether $A$ is invertible or not, so it might be worth studying...&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2016-03-18T20:26:12.367" LastActivityDate="2016-03-18T20:26:12.367" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10761" PostTypeId="2" ParentId="10758" CreationDate="2016-03-18T20:42:32.530" Score="22" Body="&lt;p&gt;Personally, I refer to this phenomenon as students &quot;submarining&quot; a broken understanding on a particular kind of problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example #1:&lt;/strong&gt; Our in-house elementary algebra textbook, in its first edition, had this problem: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $1.05x = 22.05$, then $x = ?$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that the result is the same whether the student correctly divides both sides by 1.05, or incorrectly thinks that they can subtract 1.05 ($x = 21$). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example #2:&lt;/strong&gt; Many textbooks will have an example of reducing rational expressions like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Reduce $\frac{x^2 - 9}{x+3}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Observe that the numerator is a difference of squares, and the denominator is one of its factors. I went several years before one group of students pointed out an &quot;easier&quot; (incorrect) way of accomplishing the reduction: just divide the numerator and denominator term-wise. That is: $x^2/x = x$, and $-9/3 = -3$, so the answer is $x-3$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that more disturbingly than just this one example, this broken process does actually generate the correct answer for &lt;em&gt;any&lt;/em&gt; problem of this form. &lt;em&gt;Any&lt;/em&gt; difference of squares on top (any numbers), and &lt;em&gt;any&lt;/em&gt; of its factors on the bottom (regardless of which sign appears), will behave the same way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion:&lt;/strong&gt; It really is critically important for instructors to work out the details of any exercise in advance, and pay close attention to any mechanics that may be confusing or obfuscate the initial mechanism. Students being able to generate the correct answer by some broken process will allow the broken understanding to be &quot;submarined&quot; and not visible. As an instructor, I feel like a major part of the job is to be able to function like a forensic specialist and &quot;dredge up where the bodies are buried&quot; with a high-quality series of in-class exercises. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-03-18T20:42:32.530" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10762" PostTypeId="2" ParentId="10758" CreationDate="2016-03-19T01:08:07.797" Score="11" Body="&lt;p&gt;One mathematical example that has been explored is the somewhat &lt;em&gt;pathological&lt;/em&gt; nature of &quot;&lt;a href=&quot;http://mathworld.wolfram.com/AnomalousCancellation.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;anomalous fractions&lt;/strong&gt;&lt;/a&gt;&quot; where digit cancellation produces correct simplification. For instance:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{16}{64} = \frac{1}{4}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why?&lt;/strong&gt; Well... just cancel the $6$s!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Wolfram link above to &lt;strong&gt;anomalous cancellation&lt;/strong&gt; is missing a few important citations. I dug these up while co-writing a paper with Bharath Sriraman (&lt;a href=&quot;https://scholar.google.com/citations?user=YnPG9YoAAAAJ&amp;amp;hl=en&amp;amp;oi=ao&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;google scholar&lt;/strong&gt;&lt;/a&gt;) which has been sent off:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Sriraman, B. &amp;amp; Dickman, B. (in press). “Mathematical Pathologies as Pathways into Creativity.” &lt;em&gt;ZDM, 49&lt;/em&gt;(2).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Sriraman had known of such an example via the somewhat recent paper of Osler:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Osler, T. (2007). Lucky fractions: Where bad arithmetic gives correct results. &lt;em&gt;Journal of Mathematics and Computer Education, 41&lt;/em&gt;(2), 162-167 &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I had been aware of an earlier book, which perhaps fits the OP's question more generally:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Borasi, R. (1996). &lt;em&gt;Reconceiving mathematics instruction: A focus on errors&lt;/em&gt;. Greenwood Publishing Group. &lt;a href=&quot;https://books.google.com/books?id=jKCA3leU3hMC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Google Books&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.genealogy.math.ndsu.nodak.edu/id.php?id=95762&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;author&lt;/strong&gt;&lt;/a&gt; was a doctoral student of &lt;strong&gt;Stephen Brown&lt;/strong&gt;, whose name may be familiar from a wonderful reference: &lt;strong&gt;Brown&lt;/strong&gt; and Walter's &lt;em&gt;The Art of Problem Posing&lt;/em&gt; (and its various reprintings and follow-up compendium, &lt;em&gt;Problem Posing: Reflections and Applications&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the excerpt from &lt;em&gt;Reconceiving&lt;/em&gt; as follows: (p. 3):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gUep6.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gUep6.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book looks at a variety of &quot;error case studies&quot; (p. 12) for which I have mentioned F/O (click through for larger resolution):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/R6fi5.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/R6fi5.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About the fraction cancellation, Borasi footnotes: &quot;Further inquiry about possible generalizations of the 'error' discussed here was also pursued by a student in one of my teacher education courses and later published (see Johnson, 1985)&quot; (p. 71). That reference goes to:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Johnson, E. (1985). Algebraic and Numerical Explorations Inspired by the Simplification: 16/64 = 1/4. &lt;em&gt;Focus on Learning Problems in Mathematics, 7&lt;/em&gt;, 15-28.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So:&lt;/strong&gt; I recommend this family of citations on anomalous cancellation, in particular, and suggest looking through Borasi's book more generally to see if there are other &quot;unfortunate examples&quot; of interest!&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-03-19T01:08:07.797" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10763" PostTypeId="2" ParentId="10758" CreationDate="2016-03-19T11:02:21.297" Score="9" Body="&lt;p&gt;&lt;strong&gt;mis-generalizations based on examples&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;From calculus course (I'm TA)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There is a limit problem on midterm $$\lim_{x\to \infty} \sqrt{x+1}-\sqrt{x}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is 0 and the problem is expected to be solved by multiplying $\frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}}$ but this question makes many people believe they can do $\infty-\infty=0$.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;From abstract algebra course. (I'm student)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Example given in lecture: $\mathbb{Z}[x]$ is not a PID because $(2,x)$ cannot be generated by an element.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Midterm: Prove or disprove every ideal in $\mathbb{Z}[x]$ can be generated by 2 elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer: Consider $(4,2x,x^2)$......not hard to do the rest. Many people include myself believed the statement is true because we only remembered the &quot;standard example&quot; given in lectures.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;From the same abstract algebra course.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Example given in lecture: Let $p$ be prime, $f(x)=x^{p-1}+...+x+1$ is irreducible in $\mathbb{Z}[x]$. To prove it, use Eisenstein Criterion on $f(x+1)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Midterm: True or False, let $p$ be prime, $f(x)=x^{p-1}+...+x+1$ is irreducible in $(\mathbb{Z}/p\mathbb{Z})[x]$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer: False, $f(1)=0$ in $\mathbb{Z}/p\mathbb{Z}$ therefore...... and before you ask, yes, I got tricked again. :(&lt;/p&gt;&#xA;" OwnerDisplayName="user2139" LastActivityDate="2016-03-19T11:02:21.297" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10764" PostTypeId="1" CreationDate="2016-03-19T12:32:14.837" Score="2" ViewCount="126" Body="&lt;p&gt;I have to teach sketching paraboloids on paper by looking at it's equation. Last year when I taught this topic no one was interested in learning this particular thing. They felt the topic difficult and utter waste of time with so much hard work to do. I know that at first it's difficult to visualize this topic and it becomes a pain to sketch them on paper. But I think it really develops an analytic approach towards coordinate geometry and mathematics in general. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do I motivate students to take it seriously?&#xA;Which online source can I refer to supplement my knowledge about this particular topic?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't want to repeat what happened last year.&lt;/p&gt;&#xA;" OwnerUserId="6481" LastActivityDate="2016-03-19T12:32:14.837" Title="Sketching paraboloids on paper" Tags="|reference-request|concept-motivation|geometry|graduate-education|teacher-preparation|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10765" PostTypeId="1" CreationDate="2016-03-20T13:45:41.400" Score="12" ViewCount="217" Body="&lt;p&gt;I write a math textbook for which I want to make an overview chapter for each part of the textbook. In those overview chapters I want to motivate and introduce the new mathematical concepts. I also want to answer questions like: Why do we need those mathematical concepts? How did the concept evolve in history? What will be the content of the next chapters?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this context, will the overview sections facilitate learning? How important are overview chapters in textbooks for the success of learning? Is there any research about it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note:&lt;/em&gt; I asked the &lt;a href=&quot;https://cogsci.stackexchange.com/questions/13342/do-overview-sections-increase-learning-outcome&quot;&gt;same question&lt;/a&gt; on &lt;a href=&quot;http://cogsci.stackexchange.com&quot;&gt;cogsci.stackexchange.com&lt;/a&gt;. Since I haven't got an answer there I want to reask this question here. I hope that's okay.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="-1" LastEditDate="2017-04-13T12:56:18.120" LastActivityDate="2016-06-09T02:59:13.287" Title="Do &quot;overview&quot; sections increase learning outcome?" Tags="|reference-request|textbooks|concept-motivation|general-pedagogy|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10766" PostTypeId="1" AcceptedAnswerId="10767" CreationDate="2016-03-20T14:06:53.983" Score="13" ViewCount="5278" Body="&lt;p&gt;This question is related to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10597/how-can-i-motivate-the-formal-definition-of-continuity&quot;&gt;How can I motivate the formal definition of continuity?&lt;/a&gt; Imagine a student asks the question why it is worth it to study continuity. What is a good response to this question?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;My attempt to answer the question:&lt;/em&gt; I would argue that there are two main reasons to study continuity:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Continuous functions have &quot;beautiful&quot; properties like the intermediate value property (for connected domains), the limit can be put inside the function (i.e. $\lim_{n\to\infty} f(x_n) = f(\lim_{n\to\infty} x_n)$), compositions like sums and products of continuous functions are again continuous etc.&lt;/li&gt;&#xA;&lt;li&gt;In topology continuous functions are exactly those functions which preserve topological structures.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;However, the second reason can only be fully understand when the student already has studied topology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did I miss something? How would you argue why continuity is important for mathematics?&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-03-23T21:50:43.653" Title="Why should we study continuity?" Tags="|undergraduate-education|concept-motivation|mathematical-analysis|intuition|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10767" PostTypeId="2" ParentId="10766" CreationDate="2016-03-20T18:04:01.523" Score="11" Body="&lt;p&gt;Most functions that are studied by physicists and other scientists are continuous. However, more and more discontinuous functions are appearing in the various sciences. This is due to:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Computers and their digitization of data. Many computer routines produce discontinuous output, even if the data is near-continuous.&lt;/li&gt;&#xA;&lt;li&gt;Quantum theory is a mixture of the continuous and the discontinuous. Schrodinger's equation assumes continuity and differentiability, but in the Copenhagen interpretation of quantum theory any measurement causes a collapse of the waveform and things become discontinuous: the spin is either up or down, etc.&lt;/li&gt;&#xA;&lt;li&gt;The study of Chaos, resulting from weather and other such topics, gives results that are practically discontinuous, even if continuous in theory.&lt;/li&gt;&#xA;&lt;li&gt;Catastrophe theory, developed in the 1970's, shows there are a variety of ways to get discontinuity, with a variety of applications.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Any student who does not directly think about continuous and discontinuous functions early in his/her math training may assume that all functions are continuous. This student will get into trouble when this is not the case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example story: when I begin to teach the Intermediate Value Theorem for continuous functions, I give the example of me driving to school from my home. Students assume that the distance from my home is a continuous function, so I must at some point be exactly ten miles from my home (school is 22 miles away). This assumption is correct, but when I ask them they cannot even imagine any discontinuous possibility such as a Star Trek beam transporter. This silly example shows me every year that my students do not understand how some functions can act differently from others. They have to be shown the differences between continuity and the lack of it.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2016-03-21T09:19:38.570" LastActivityDate="2016-03-21T09:19:38.570" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10768" PostTypeId="2" ParentId="10766" CreationDate="2016-03-20T19:40:19.793" Score="4" Body="&lt;p&gt;Continuity is a property that many real functions or relations have on practically relevant scales, i.e., if there are discontinuities, they are so small that they have no impact on most applications. For example, we treat space and time as continuous, though they may be discretised on levels that we cannot yet access with measurements, and we can treat matter as continuous on many scales, if even though it is not continuos on the atomic level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, it is only natural to have a word for this property and to properly define it to ensure that everybody is talking about the same thing. Moreover, it is worth studying whether any useful properties follow from continuity – which is indeed the case – as those properties would have a broad application. Without having a concept of continuity, we would have to appeal to intuition, make assumptions or explicitly show the fact, whenever we would normally apply the intermediate-value theorem or swap limits and function evaluation. Moreover, only by knowing what we usually take as given due to the ubiquity of continuity, we know what we cannot expect anymore in the rare case that continuity is absent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above motivation translates to almost all of abstract mathematics for most people and most non-mathematicians: We detect and abstract some properties of natural objects and see whether we can derive some new abstract statements from those properties, which we can in turn translate back to application, where they may be useful. In this sense the motivation to study continuity is not much different than the motivation to study, e.g., integers: Not every number occuring in real life is an integer but many are and properties of integers may thus be of general interest.&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2016-03-20T19:40:19.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10769" PostTypeId="2" ParentId="10766" CreationDate="2016-03-20T22:47:47.803" Score="10" Body="&lt;p&gt;&lt;strong&gt;Imagine a student asks the question why it is worth it to study continuity. What is a good response to this question?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It matches human intuition. We use continuity very frequently without realizing we're using it. It's the foundation of most approximations we make in physical science. Whenever we think $f(x)$ can be approximated by $f(y)$ because $x$ can be approximated by $y$ we inherently assume $f$ is continuous.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It provides a natural way of understanding irrational exponents (or more generally, extend a function defined on rational number to real number). How do you define $3^{\sqrt2}$ or $7^{\pi}$? When we're first introduced exponential, we're taught $a^b$ means &quot;$a$ multiply $b$ times&quot; when $b$ is an integer. When b is a rational number, like $5^{1/2}$, it's also not too hard to think it's a positive number that &quot;get 5 by multiplying itself&quot;. However it's very difficult to explain what $7^{\pi}$ means naturally. To me, the easiest way to understand it is &quot;that is a number where this sequence $7^3, 7^{3.1}, 7^{3.14}, 7^{3.141}, 7^{3.1415}......$ eventually get closed to.&quot; More generally and rigorously, we use the following corollary from continuous function,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Continuous functions preserve limit of sequence. Any real number is a limit of a sequence of rational numbers. Therefore, if $f$ is a real value continuous function and we know it's value on each rational number, we know $f$. In other words, continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ can be uniquely determined by it's value on $\mathbb{Q}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;IMO it's a much more natural way of understanding exponential than the &quot;power series way&quot; or &quot;inverse function of antiderivative of $\frac{1}{x}$ way&quot; many calculus textbooks use. That kinds of attempts to avoid &quot;completeness of $\mathbb{R}$&quot; are unintuitive and often create confusion to high school/freshman college students.&lt;/p&gt;&#xA;" OwnerDisplayName="user2139" LastEditorDisplayName="user2139" LastEditDate="2016-03-21T00:02:05.113" LastActivityDate="2016-03-21T00:02:05.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10770" PostTypeId="2" ParentId="10766" CreationDate="2016-03-21T01:54:53.140" Score="6" Body="&lt;p&gt;In our intuition, everything is continuous.  But if we think long enough about real life, we discover that actually lots of things aren't.  Computers generally replace continuous things with discontinuous approximations.  When a driver hits the car's brake, acceleration is discontinuous.  The stock market has lots of discontinuous jumps (ignore that at your peril).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much of the mathematical study of continuity was developed in trying to understand why something unexpected happened when in fact things weren't continuous.  When a function is continuous, a large set of tools is available to us to use.  When it is not continuous, these same tools will lead us astray.  It is important for us to understand when we are and aren't allowed to use these tools.  To understand it, we need to understand what continuity means.&lt;/p&gt;&#xA;" OwnerUserId="6522" LastActivityDate="2016-03-21T01:54:53.140" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10771" PostTypeId="1" CreationDate="2016-03-21T14:13:12.923" Score="4" ViewCount="346" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Geometric_algebra&quot; rel=&quot;nofollow&quot;&gt;Geometric Algebra&lt;/a&gt; brings together algebra, geometry, vectors, complex numbers, and linear algebra. It provides a single unification of all elementary math and serves as an excellent basis for physics, robotics and computer graphics. Are there any Geometric Algebra resources available for teaching high school students?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;------ 3/24/16 addition ------&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any Vector Algebra resources that include the outer product (wedge product) before the cross product?&lt;/p&gt;&#xA;" OwnerUserId="6401" LastEditorUserId="6401" LastEditDate="2016-03-24T19:57:15.040" LastActivityDate="2016-03-25T01:24:19.477" Title="Geometric Algebra Resources" Tags="|secondary-education|geometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10773" PostTypeId="2" ParentId="10771" CreationDate="2016-03-22T01:07:48.480" Score="4" Body="&lt;p&gt;I have taught basic Geometric Algebra to high school students. I have not been able to find any appropriate resources. I would only do this with very able students, as otherwise you are risking some serious confusion which could interfere with understanding more basic concepts.  As prerequisites, I would suggest that complex numbers and 2D vectors (addition/subtraction) should be understood. I did not need dot product at the most basic level, but I'll need to introduce it when doing trig with GA.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most difficult concept for students to accept was non-commutative operations. Don't underestimate how big this change is for students. You may be able to demonstrate this with a non-commutative group (the symmetries of a 2D shape including rotations and flips), but this is a rather big detour. You can show how rotating a rubiks cube is non-commutative. I emphasised that they are not allowed to take any shortcuts in rearranging the order of multiplied vectors - just follow the rules:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\hat x  \hat x=1$$&#xA;$$\hat y  \hat y=1$$&#xA;$$\hat x  \hat y= -\hat y \hat x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I revised how vectors can be written in terms of basis vectors: $(3,4) = 3\hat x +4\hat y$, and how addition/subtraction works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did some simple vector multiplication to demonstrate that distributive and associative laws still work fine, and to practice the anti-commutative property of basis vectors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I then gave them 3 example vectors: $(1,0),(3,4),(0,1)$ and discussed where they would rotate to. They then had to experiment to find what multiplication could rotate any vector $90^\circ$ anti-clockwise, ie.$(1,0)\cdot \Box = (0,1)$. This was a too hard given the small amount of practice that they had, so I gave them some hints.  A demonstration of how $\hat x \hat y$ works can be given (ie. &lt;em&gt;the first $\hat x$ cancels the $\hat x$ of the vector, and then multiply by $\hat y$ to get the desired result&lt;/em&gt;), then show it automatically works for the others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The final step is to ask &lt;em&gt;&quot;what is $(\hat x \hat y)^2$?&quot;&lt;/em&gt;. After 30 seconds a triumphant student exclaimed &quot;It's $i$!!&quot; (referring to $\hat x \hat y$). Which was of course the whole point of the exercise, to demonstrate why $i$ works - because of the underlying unity of geometric algebra. I did not discuss how complex numbers are confusingly used to encode both rotations and vectors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not introduced dot (symmetric) products or areas and wedge (anti-symmetric) products, though I will touch on this when we get to Trig II. You can easily calculate the cos and sine of the angle between vectors. It is quite beautiful when studying sine and cosine rules - especially the sine rule becomes obvious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not a curriculum I'm afraid, but I hope it helps.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2016-03-25T01:24:19.477" LastActivityDate="2016-03-25T01:24:19.477" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10774" PostTypeId="2" ParentId="9695" CreationDate="2016-03-22T01:32:50.423" Score="2" Body="&lt;p&gt;You could look at &lt;a href=&quot;http://www.engine.ac.nz/aperfectexample/locations/distance-learning/&quot; rel=&quot;nofollow&quot;&gt;Massey University&lt;/a&gt;.&#xA;I have known many people to use this university's distance program, undergrad and postgrad, and they have had good experiences. It was even recommended by my local university as the best option for a non-contact course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below are the available Maths/Stats distance papers this year - enough for a full degree.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;160.101 Calculus I&#xA;160.102 Linear Mathematics&#xA;160.103 Introductory University Mathematics&#xA;160.111 Mathematics 1A&#xA;160.112 Mathematics 1B&#xA;160.203 Calculus&#xA;160.204 Differential Equations I&#xA;160.211 Linear Algebra&#xA;160.212 Discrete Mathematics&#xA;160.301 Analysis&#xA;160.314 Combinatorics&#xA;160.318 Differential Equations II&#xA;160.320 Mathematics in Education&lt;/p&gt;&#xA;&#xA;&lt;p&gt;161.120 Introductory Statistics&#xA;161.130 Introductory Biostatistics&#xA;161.200 Statistical Models&#xA;161.220 Data Analysis&#xA;161.221 Applied Linear Models&#xA;161.223 Introduction to Data Mining&#xA;161.304 Advanced Statistical Modelling&#xA;161.322 Design and Analysis of Surveys and Experiments&#xA;161.327 Generalised Linear Models&#xA;161.342 Forecasting and Time Series&#xA;161.380 Statistical Analysis Project&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2016-03-22T01:32:50.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10775" PostTypeId="2" ParentId="10766" CreationDate="2016-03-22T13:06:41.533" Score="4" Body="&lt;p&gt;I agree with user @BenCrowell that continuity is less essential in freshman calculus than other topics such as differentiability.  Perhaps it is for this reason that Keisler treats differentiability first in &lt;a href=&quot;https://www.math.wisc.edu/~keisler/calc.html&quot; rel=&quot;nofollow&quot;&gt;his online textbook&lt;/a&gt; which we have used for the past three years to teach freshman calculus (to a total of almost 300 students by now).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To address the OP's question specifically, I would therefore suggest motivating differentiability first, which is an easier task since it suffices to mention applications in physics (velocity, etc) and other fields. Then one could point out that a useful larger class includes continuous functions which are useful for technical reasons, such as for example wanting to work with the absolute value function or the cube root function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible objection that continuity is more general and therefore should be treated first is not convincing; after all we don't start freshman calculus with Lebesgue integrable functions.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2016-03-22T13:06:41.533" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10776" PostTypeId="2" ParentId="10751" CreationDate="2016-03-22T15:19:00.813" Score="2" Body="&lt;p&gt;A broader answer regarding the teaching of the Pythagorean Theorem (including the proof):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.magicalmaths.org/could-this-be-the-best-pythagoras-theorem-lesson-ever/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.magicalmaths.org/could-this-be-the-best-pythagoras-theorem-lesson-ever/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And an image from the proof -&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hgm8l.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hgm8l.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1249" LastEditorUserId="64" LastEditDate="2019-05-10T14:03:54.567" LastActivityDate="2019-05-10T14:03:54.567" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="10777" PostTypeId="1" CreationDate="2016-03-23T02:33:24.900" Score="19" ViewCount="785" Body="&lt;p&gt;Where I live and work, there is a widely-accepted and often-repeated claim that there are two kinds of students:  &quot;algebra people&quot; and &quot;geometry people&quot;.  This claim sometimes gets expressed in different ways; sometimes it is articulated as a difference in &lt;em&gt;ability&lt;/em&gt; (&quot;The students who are good at algebra tend not to be good at geometry and vice versa&quot;) and other times as a difference in &lt;em&gt;interest&lt;/em&gt; (&quot;The students who like algebra tend not to like geometry and vice versa&quot;).  This belief is often conflated with claims about some students being &quot;visual learners&quot; (despite the fact that &lt;a href=&quot;http://www.changemag.org/archives/back%20issues/september-october%202010/the-myth-of-learning-full.html&quot;&gt;so-called &quot;learning styles&quot; do not exist&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On its surface, this belief seems to me to be almost certainly a myth -- at least when expressed in terms of ability.  But it also seems like it could be empirically tested.  If people who are good at algebra were bad at geometry and vice versa, that would surely show up through an analysis of the grades they get in school, or how students perform on standardized tests that include both kinds of content.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as &lt;em&gt;interest&lt;/em&gt; goes, the claim has slightly more surface plausibility.  I know plenty of adults who will testify that the only math class they ever enjoyed was Geometry, and others who will say with equal conviction that Geometry was the only math class they &lt;em&gt;hated&lt;/em&gt;.  But a few anecdotes is hardly  compelling evidence.  This, too, seems like it could be easily studied empirically with some kind of attitudinal survey.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Have there been such studies?  Is there any evidence for or against the claim that people tend to be either &quot;geometry people&quot; or &quot;algebra people&quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2016-03-23T13:23:35.810" LastActivityDate="2016-03-29T22:47:43.703" Title="Evidence for or against the claim that some students are &quot;algebra people&quot; and others are &quot;geometry people&quot;" Tags="|secondary-education|reference-request|algebra|geometry|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10778" PostTypeId="2" ParentId="10777" CreationDate="2016-03-23T06:15:05.730" Score="2" Body="&lt;p&gt;I don't know if this has been studied. Here's how to do it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the data required for this might very well be public in Finland. At least the final grades of pupils are public knowledge. I don't know about the situation elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could go to a school and ask for grades of mathematics courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two forms of the hypothesis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The strong one is that geometry and algebra grades are negatively correlated. This is very unlikely, since practically all grades across subjects are positively correlated (e.g. history and mathematics), at least in the compulsory school levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The weak one is: After accounting for the average mathematics grade, the geometry and algebra grades are negatively correlated. This might even be true, but I would be surprised if the effect was large.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2016-03-23T06:15:05.730" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10779" PostTypeId="2" ParentId="10777" CreationDate="2016-03-23T07:58:41.917" Score="2" Body="&lt;p&gt;Not quite the same question, but along the same lines is &lt;a href=&quot;http://link.springer.com/article/10.1007%2Fs10699-008-9149-4&quot; rel=&quot;nofollow&quot;&gt;On the Persuasiveness of Visual Arguments in Mathematics&lt;/a&gt;. It looks at how convincing different mathematicians find visual proofs compared to written ones.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-03-23T07:58:41.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10780" PostTypeId="1" CreationDate="2016-03-23T21:01:00.657" Score="1" ViewCount="176" Body="&lt;p&gt;Recent developments in Geometric Algebra have extended vector algebra to include the outer product (wedge product) and bivectors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a Vector Algebra text (preferably at the advanced high school level)where the more recently developed material (outer product and bi-vectors) is taught before more &quot;traditional&quot; material such as the cross product? &lt;/p&gt;&#xA;" OwnerUserId="6401" LastEditorUserId="1333" LastEditDate="2016-05-09T03:51:20.913" LastActivityDate="2016-05-09T03:51:20.913" Title="Vector Algebra Text" Tags="|textbooks|vector-calculus|" AnswerCount="1" CommentCount="3" ClosedDate="2016-03-24T12:03:35.587" ContentLicense="CC BY-SA 3.0" />
  <row Id="10781" PostTypeId="1" AcceptedAnswerId="10790" CreationDate="2016-03-23T21:26:41.850" Score="7" ViewCount="373" Body="&lt;p&gt;I am a new teacher teaching mathematics (College Algebra, Calculus, etc.) in a college. My wife has made an interesting suggestion that I would like to have a discussion here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is it appropriate to assign an extra credit homework question on the &quot;next&quot; session?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say that I am assigning homework questions for Section 3.1; is it appropriate to assign one routine question from Section 3.2, which will be taught in the next class period, to encourage the students to read the materials before the class?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2016-04-05T13:30:10.737" Title="Is it appropriate to assign an extra credit homework question on the &quot;next&quot; session?" Tags="|student-motivation|homework|community-colleges|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10782" PostTypeId="2" ParentId="10766" CreationDate="2016-03-23T21:50:43.653" Score="3" Body="&lt;p&gt;As others, I think it is not necessarily relevant to actually &lt;em&gt;study&lt;/em&gt; continuity. I do mention it in order not to lie when I have to (e.g. when stating the fundamental theorem of calculus), but often barely more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, depending on the context, there are reasons one can chose to study it in details. The most important I can see is that &lt;strong&gt;proving things around continuity is a very good model of what mathematics, especially analysis, looks like.&lt;/strong&gt; For example, proving that the product of two continuous functions is continuous gives already gives a rather sophisticated proof (for freschmen). The main question I would ask myself if I had to teach a course where I could, or could not treat continuity in some depth, is whether I want to show my students such proofs (I could answer positively even for non-math majors), and whether I want to ask them to be able to perform them (this would probably be restricted to math majors).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-03-23T21:50:43.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10783" PostTypeId="2" ParentId="10781" CreationDate="2016-03-23T21:53:40.843" Score="3" Body="&lt;p&gt;No good can come of this. The students who are already at the top of the class will easily grab the points, and those who actually wait for you to lecture on the topic to learn it from you, will resent the steepening of the curve. &#xA;What do you really hope to gain from this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit - part of my knee-jerk reaction is from my own experience. A freshman class where the teacher tried a similar approach, multiple chances to earn &quot;extra points&quot;. As a freshman, I understood the material, felt no need to do extra, and tried to balance my time toward the classes that were a struggle. In this class, a 96 (combined HW grades and exams) resulted in a B. The teacher graded on a curve, and in hindsight, the &lt;em&gt;extra credit&lt;/em&gt; was anything but. 35 years later and that experience stuck with me. If you decide to do anything with extra credit, I urge you to spell out how you intend to grade, to make clear the impact of doing that work, vs skipping it. If you are not curving, keep in mind, the result will be a positive shift toward higher grades, and your supervisor may question your grade distribution. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-03-25T00:30:12.930" LastActivityDate="2016-03-25T00:30:12.930" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="10784" PostTypeId="2" ParentId="10780" CreationDate="2016-03-24T10:32:42.527" Score="2" Body="&lt;p&gt;Though I would encourage you to stick with your previous question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10771/geometric-algebra-resources&quot;&gt;Geometric Algebra Resources&lt;/a&gt; ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Short answer - there will not be a Vector Algebra text that addresses Geometric Algebra (GA) at High School level. Vector Algebra texts address 3D vectors. In my experience most High School students are barely equipped for basic 3D vector algebra - let alone 3D GA. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am also not aware of any high school maths curriculum with enough &quot;free time&quot; to spend the months required to address 3D GA at even a basic level. Therefore I doubt someone would create a text like this designed for high school students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have only taught this as an extension for some advanced high-school students, and then only in 2D. For most students, attempting to teach this could be &lt;em&gt;extremely&lt;/em&gt; counter productive as it would interfere with their later education when it is someone else's turn to teach them a traditional curriculum. I have shown you part of my methodology in my answer to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10771/geometric-algebra-resources&quot;&gt;Geometric Algebra Resources&lt;/a&gt; which attempts to augment a traditional curriculum rather than replace traditional vector algebra.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-03-24T10:32:42.527" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10785" PostTypeId="2" ParentId="10759" CreationDate="2016-03-24T17:35:26.157" Score="5" Body="&lt;p&gt;Slight variant on Steven's answer: You don't actually have to &lt;b&gt;derive&lt;/b&gt; the general formula. Make them work out the inverse of $\left( \begin{smallmatrix} x_{11} &amp;amp; x_{12} \\ x_{21} &amp;amp; x_{22} \end{smallmatrix} \right)$. (if they don't have it memorized already). Point out that they could clearly do $3 \times 3$ and $4 \times 4$ if they worked harder. Pass out paper with the $3 \times 3$ and $4 \times 4$ results printed on it (&quot;I'll save you the time&quot;). Ask them what patterns they see.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2016-03-24T18:23:16.570" LastActivityDate="2016-03-24T18:23:16.570" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10786" PostTypeId="1" CreationDate="2016-03-24T20:19:57.697" Score="11" ViewCount="2072" Body="&lt;p&gt;There is a smart way of learning, and it consists in first finding out what are the most valuable pieces of knowledge to acquire. The ones that will give you the highest value for your investment in attention and time. For instance, when you are learning a new language, it is typical to find that 2% of the words  make 95% of conversations.  You'd do far better if you knew what those words are when acquiring vocabulary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is whether there are mathematical concepts, results, techniques, tricks, etc., that give such high payoffs.  It could be because they are used everywhere, or because they are really easy to learn considering how much they worth, etc. As was pointed out in the comments, the meaning of &quot;payoff&quot; depends on the goals of your mathematical tools. Techniques like frequency analysis might bring the bread to the table of an engineer, but be irrelevant to a number theorist. I, in particular, plan to become a sort of applied mathematician in neuroscience and perhaps other &quot;soft&quot; sciences. I don't know beforehand exactly what I will be doing. That's why I want to gather a set of skills that are useful in a broad, applied, sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might express the efficiency of a bit of knowledge in the following way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{\text{how much is it worth mathematically}}{\text{how much do I need to invest to learn it}}$$  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;&quot;how much is it worth&quot;&lt;/strong&gt; would be the weighted sum of&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The extension of mathematics in which is required to know it&lt;/li&gt;&#xA;&lt;li&gt;The number of applied problems whose solutions require it&lt;/li&gt;&#xA;&lt;li&gt;The work time you could save by using it, instead of acting in a naive way&lt;/li&gt;&#xA;&lt;li&gt;The average amount of errors you could stop making if you use it&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;&quot;how much do I need to invest&quot;&lt;/strong&gt; is the time it would take me to master this skill, trick, knowledge, etc. provided I already have all the prerequisites assumed in a standard introduction to the topic.  Avoid considerations about how much money books or papers would cost, that is not the issue here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Current List (to be filled with the answers)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Visual evaluation&lt;/strong&gt; of 2D and 3D graphs (see Jasper's answer)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Dimensional analyis&lt;/strong&gt;: you will have an edge in problems related to physical magnitudes, with little extra effort. You could also find the functional form of some solutions without calculations&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Logic&lt;/strong&gt;: most common and dangerous fallacies&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Algebra&lt;/strong&gt;: Elementary Algebra (how to manipulate expressions, and doing it without errors) Linear Algebra (Matrices, Vector Spaces, etc.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Calculus&lt;/strong&gt;: The very vital concepts of limit, derivative and integral. Numerical procedures&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Statistics&lt;/strong&gt;:  &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0393310728&quot; rel=&quot;nofollow&quot;&gt;How to lie&lt;/a&gt;, &lt;a href=&quot;http://www.dummies.com/how-to/content/the-bootstrap-method-for-standard-errors-and-confi.html&quot; rel=&quot;nofollow&quot;&gt;Resampling techniques(from Richard's answer)&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Optimization&lt;/strong&gt;: &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Approximation methods&lt;/strong&gt;:  like in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/026251429X&quot; rel=&quot;nofollow&quot;&gt;Mahajan. Street fighting mathematics&lt;/a&gt;, they are easy to use, but can give you a decent answer to a hard problem&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Fourier Transforms&lt;/strong&gt;:  are used in a lot of areas and not hard to grasp&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Elementary Programming&lt;/strong&gt;: It's not properly maths, but needed everywhere. You could learn high level open source languages like Python and R in around a semester, and use them for a lot of things. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6537" LastEditorUserId="6537" LastEditDate="2016-03-31T17:45:51.180" LastActivityDate="2016-03-31T17:45:51.180" Title="Mathematical concepts and techniques that **pay off the most**?" Tags="|mathematical-pedagogy|self-learning|" AnswerCount="4" CommentCount="7" ClosedDate="2016-03-31T02:20:57.870" ContentLicense="CC BY-SA 3.0" />
  <row Id="10787" PostTypeId="2" ParentId="10786" CreationDate="2016-03-24T21:48:09.310" Score="8" Body="&lt;p&gt;I guess that (by the question's metric) the most valuable things to learn are visual and conceptual, instead of the ability to perform detailed computations.  Some concepts are very easy to learn, and have substantial value, so the ratio is very high.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Some of the most common logical fallacies.&lt;/li&gt;&#xA;&lt;li&gt;Some of the most dangerous logical fallacies.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0393310728&quot;&gt;How to Lie with Statistics&lt;/a&gt;&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given a 2-D graph:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;** Is it a function?&lt;br&gt;&#xA;** Where is the slope zero? infinite?&lt;br&gt;&#xA;** Where are the inflection points?&lt;br&gt;&#xA;** Does it have asymptotes?&lt;br&gt;&#xA;** Is it symmetrical? even? odd?&lt;br&gt;&#xA;** Is it periodic? What is the period? frequency? amplitude? phase shift?  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given a 3-D shape:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;** What portion contributes the most to its volume? its stiffness? its strength?&lt;br&gt;&#xA;** Removing which material would weaken the shape the most?&lt;br&gt;&#xA;** How does doubling any dimension change the load the shape can carry?&lt;br&gt;&#xA;** How does doubling any dimension change how much the shape will sag under a load?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Always keep track of units when doing word problems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;You can always multiply by one.  (This lets you do most unit conversions.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2016-03-25T20:05:01.050" LastActivityDate="2016-03-25T20:05:01.050" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10788" PostTypeId="1" CreationDate="2016-03-25T03:47:59.950" Score="9" ViewCount="459" Body="&lt;p&gt;I have recently worked with some students motivating the development of $e^t$ and $e^{t i}$ as summing change over time, basically informally solving differential equations. My motivation for this is that the students are already used to programming physics simulations which is basically numerically solving differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The definition that students developed was:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When calculating 100% interest continuously compounded for one year, your money is calculated by $m_1 = m_0 \cdot \lim_{\Delta t \to 0} \left(1+ \Delta t\right)^{\frac 1 {\Delta t}}$ &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is simply the usual definition of $e$ replacing $n$ with $\frac1{\Delta t}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on this the definition, the exponential that they developed was: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$m_t = m_0 \cdot \lim_{\Delta t \to 0} \left(1+ \Delta t\right)^{\frac t {\Delta t}}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which calculates $e^t$. However this is different from the way I usual derive $e^x$ in which the $x$ is &lt;em&gt;inside&lt;/em&gt; the exponential rather being the exponent $$\lim_{n \to \infty} \left(1+ \frac x n\right)^n$$&#xA;This is really only substituting $n=\frac t {\Delta t}$ and $x=t$, but massively changes what is looks like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It occurs to me that this could have potential advantages:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It is immediately obvious that $t$ in $\lim_{\Delta t \to 0} (1+ \Delta t)^{\frac t {\Delta t}}$ is an exponent. The only task is to substitute $ u = \lim_{\Delta t \to 0} \left(1+ \Delta t\right)^{\frac 1 {\Delta t}}$ and the above can be rewritten as $u^t$. This is much easier than proving that the $x$ in $\lim_{n \to \infty} \left(1+ \frac x n\right)^n$ acts like an exponent.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The limit looks like a normal calculus-style limit of $\Delta t \to 0$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;potential disadvantages:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The next task was to calculate position on the complex plane where $\frac {\Delta d}{ \Delta t}$ is always &lt;em&gt;perpendicular&lt;/em&gt; to the displacement. They already know that a way to create a perpendicular vector is to multiply by $i$ so this quickly led to the formula $\lim_{\Delta t \to 0} \left(1+ i\Delta t\right)^{\frac t {\Delta t}}$. Unfortunately in this formulation as the $t$ is not inside the base of the exponent, it does not immediately follow that the analytic formula is $e^{it}$ as it does in the formulation $\lim_{n \to \infty} \left(1+ \frac {ix} n\right)^n$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It seems like there are 4 equivalent ways of writing this formula by mixing and matching the following options.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$x$ or $t$ inside the base or as an exponent&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the limiting variable: ${n \to \infty}$ or ${\Delta t \to 0}$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;h1&gt;Questions&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;What do people think is the best combination of the above (or other) options?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are additional advantages/disadvantages or various options?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are other good ways to motivate the definition of $e^x$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there better direct ways of demonstrating $e^{\theta i}$ (not Taylor series!)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is some existing discussion at &lt;a href=&quot;https://matheducators.stackexchange.com/questions/3886/the-definition-of-natural-log-and-e&quot;&gt;The definition of natural log and e&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="8571" LastEditDate="2017-12-05T04:46:49.510" LastActivityDate="2017-12-05T07:36:18.123" Title="Alternative limit for e" Tags="|undergraduate-education|secondary-education|calculus|definitions|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10789" PostTypeId="2" ParentId="10786" CreationDate="2016-03-25T11:08:13.027" Score="8" Body="&lt;p&gt;Estimation: +1 for previous mentions of estimation / Street Fighting Maths / Dimensional Analysis. To the list I would add numeric techniques for Calculus and resampling for Statistics. When you combine the basic estimation techniques with these numeric techniques the range of what you can intelligently estimate greatly expands.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basic numeric techniques can be used with only HS pre-calc maths combined with a few weeks of programming knowledge (how a basic loop operates and how variables work). This is a very low time investment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for what you get - numeric calculus moves static and linear estimation techniques into the dynamic world. High school science usually tops out at idealised parabolic and simple harmonic motion - with numeric techniques you can overlay as many forces as you want and get a picture of real world motion. You have problems with 3 bodies?...  how about examining the motion of 300. Struggling to find the right U substitution to integrate a shape - set $\Delta x$ to a micro-metre and let the computer do the hard work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something else I really value is the sensitivity analysis provided by numeric techniques. Using basic estimation techniques is fine, but by augmenting them with numeric techniques you can find out how much more information you actually need before you get a good enough estimate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The complementary techniques of numeric statistics allows you to work with real rather than idealised distributions. You can calculate probabilities by directly (re)sampling the problem space rather than working with p-values of some idealised problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When students learn more advanced maths, the numeric techniques continue to magnify the power of their knowledge. The cost is very small, the pay off is huge.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Added reference and a few other concepts I've found useful:&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Resampling a basic introduction: &lt;a href=&quot;http://www.dummies.com/how-to/content/the-bootstrap-method-for-standard-errors-and-confi.html&quot; rel=&quot;nofollow&quot;&gt;http://www.dummies.com/how-to/content/the-bootstrap-method-for-standard-errors-and-confi.html&lt;/a&gt;&#xA;This page also links to a computer program and text about resampling techniques. I am not a specialist in statistics, so if anyone can recommend a text, I'll add it in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wider than just resampling, there are other amazing &lt;strong&gt;Monte Carlo&lt;/strong&gt; techniques built on the same premise - that by doing something simple a huge number of times we can get a good estimate of complex behaviour. They can also apply to calculus for estimating areas etc. and their application in statistical mechanics really opened my eyes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have especially found an understanding of &lt;strong&gt;Markov Chains&lt;/strong&gt; and random walks to been very useful. It is fairly simple to understand, but as with other techniques, all you need to do is estimate the model, and an algorithm does all the hard lifting to estimate your distribution.&lt;/p&gt;&#xA;" OwnerUserId="3493" LastEditorUserId="3493" LastEditDate="2016-03-29T02:33:00.293" LastActivityDate="2016-03-29T02:33:00.293" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10790" PostTypeId="2" ParentId="10781" CreationDate="2016-03-26T00:36:11.427" Score="6" Body="&lt;p&gt;In general, I'd think that anything that helps students to break out of a tendency to passivity is good. Initiative should be rewarded, looking ahead is good, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sure, if there is a pervasive assumption that everything is &quot;curved&quot; in an invidious way, there are even larger problems... but the &lt;em&gt;possibility&lt;/em&gt; of &quot;curving&quot; unfairly (don't do it) is not a sufficient reason to try to keep students in lock-step for worry (?) about weaker students being disserved. Simply don't punish the weaker students for the opportunities given to stronger students to entertain themselves, at least.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;oh, there'll be chances later for the good students to go into things in depth, it's just a first-year course&quot; I think miss the point, that first-year courses are powerful PR about what the course-of-study &lt;em&gt;is&lt;/em&gt;. If it's rules and no-looking-ahead, who'd be interested? One does not have to punitively grade weaker students while making important points to better students about the larger/forward picture of the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, surely one can keep focused on the goal of education, rather than grading... especially in the degenerate sense of the latter as &quot;filtering&quot; or &quot;gate-keeping&quot;. There is no compulsion to make a richer picture of a subject entail more grueling gate-keeping.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also worth making the point to students that, in contrast to some remarks, it is rarely wise to &quot;wait to have the material explained&quot;, rather than reading to see what will be discussed. A terrible habit to even passively endorse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, sure, give people incentive to look ahead a little. At the same time, there's no compulsion to actually &lt;em&gt;punish&lt;/em&gt; those who don't take the advice to do so, by distorted &quot;curving&quot;. That's a separate issue.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-03-26T00:36:11.427" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10791" PostTypeId="2" ParentId="10124" CreationDate="2016-03-27T13:22:47.697" Score="6" Body="&lt;p&gt;A &lt;a href=&quot;https://en.wikipedia.org/wiki/Caustic_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;(cata)caustic&lt;/a&gt; is formed by the reflection of light, such as &lt;a href=&quot;https://math.stackexchange.com/questions/1550259/cardioid-in-coffee-mug&quot;&gt;the cardioid in this coffee cup&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Abxv9.jpg&quot; width=&quot;300&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;G.B. Airy showed in [Airy,&#xA;&quot;On the intensity of light in the neighbourhood of a caustic,&quot;&#xA;&lt;em&gt;Transactions of the Cambridge Philosophical Society&lt;/em&gt; &lt;strong&gt;6&lt;/strong&gt; (1838) 379--402]&#xA;that the intensity of the light along a normal line to the caustic is proportional to the square of the integral&#xA;$$&#xA;\int_0^{\infty } \cos \frac{\pi}{2}  &#xA;   \left(w^3-m w\right) \; dw&#xA;$$&#xA;where $m$ is proportional to the displacement from the caustic toward the convex side.  This led to the &lt;a href=&quot;http://dlmf.nist.gov/9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Airy functions&lt;/a&gt; and the Airy differential equation.&#xA;The modern definition is&#xA;$$&#xA;\mathop{\mathrm{Ai}}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}%&#xA;\mathop{\cos}\nolimits\!\left(\tfrac{1}{3}t^{3}+xt\right)dt&#xA;$$&#xA;and the differential equation is $y''=x\,y$.  A second independent solution to the differential equation $Bi(x)$ also has &lt;a href=&quot;http://dlmf.nist.gov/9.5#E3&quot; rel=&quot;nofollow noreferrer&quot;&gt;an integral representation&lt;/a&gt;, but it is not so nice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note, &#xA;if your calculus or differential equations courses teach students how&#xA;to think with factorial-like products that multiply every other integer (e.g. the power series of Bessel functions, which also have &lt;a href=&quot;http://dlmf.nist.gov/10.9#E1&quot; rel=&quot;nofollow noreferrer&quot;&gt;an integral representation&lt;/a&gt;) or as in this case, multiply every third, Airy's example shows a natural application of such formulas.&#xA;The Airy functions are linear combinations of the series&#xA;$$&#xA;y_1 = 1+\frac{1}{3!}x^{3}+\frac{1\cdot 4}{6!}x^{6}+%&#xA;\frac{1\cdot 4\cdot 7}{9!}x^{9}+\cdots&#xA;$$&#xA;$$&#xA;y_2=x+\frac{2}{4!}x^{4}+\frac{2\cdot 5}{7!}x^{7}+\frac{2%&#xA;\cdot 5\cdot 8}{10!}x^{10}+\cdots&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Remark:&lt;/em&gt; &#xA;While the Airy (and Bessel) functions are (or can be) defined by definite integrals, they are &quot;complete&quot; integrals (i.e., with fixed limits of integration) that are functions of a parameter in the integrand, whereas the OP's examples have one limit of integration that varies.  The gamma function is also of the first class (i.e. &quot;complete&quot;).&#xA;The significance is that the OP's class of examples can be used in single-variable calculus. The other class requires differentiating under the integral sign, which is usually not taught until multivariable calculus or later.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-03-27T19:18:53.730" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10793" PostTypeId="2" ParentId="10786" CreationDate="2016-03-27T18:22:08.977" Score="3" Body="&lt;p&gt;So, you want a skill-set which allows you to solve a wide variety of problems. You want to develop mental toughness. The few, the proud, the physics major I think. That's my recommendation, the study of physics exposes you to mathematical techniques and concepts of modeling which are on target with your goals. My evidence? I would point you to the many jobs outside physics which physics PhD's tend to thrive in. I have a friend who pursued a PhD in physics and now is successful in a job which is essentially actuarial in nature. You could say similar things about a PhD in math, but, it seems to me Physics as a discipline is more on target with your goals. In short, I would add:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;develop physical intuition&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As far as I know, besides intrinsic capability, the way to do this is to take courses and listen to physicists. Beyond books and homework problems there is a culture of physics. This is part of what forms the intuition. For me, being part of the physics major was a big part of how I developed what physical intuition I now possess. Short of pursuing a degree, at least take the major courses in physics.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2016-03-27T18:22:08.977" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10794" PostTypeId="1" CreationDate="2016-03-27T21:49:25.590" Score="3" ViewCount="198" Body="&lt;p&gt;Which trigonometric identities are necessary for the k-12 student?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to make a list of trigonometruc identities for k-12 student. My list is below, please help me to improve the list by proposing&lt;br&gt;&#xA;&quot;useful formulas and necessary formulas&quot;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$\sin^2x + \cos^2x = 1$ &lt;/li&gt;&#xA;&lt;li&gt;$\tan^2x+1=\sec^2x=\frac{1}{\cos^2x}$&lt;/li&gt;&#xA;&lt;li&gt;$\cot^2x+1=\operatorname{cosec}^2x=\frac{1}{\sin^2x}$&lt;/li&gt;&#xA;&lt;li&gt;$\sin(2x)=2\sin x . \cos x $&lt;/li&gt;&#xA;&lt;li&gt;$\cos(2x)=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$ &lt;/li&gt;&#xA;&lt;li&gt;$\tan(2x)=\frac{2\tan x}{1-\tan^2x}$&lt;/li&gt;&#xA;&lt;li&gt;$\sin(x \pm y)=\sin x . \cos y \pm \cos x . \sin y$&lt;/li&gt;&#xA;&lt;li&gt;$\cos(x \pm y)= \cos x . \cos y - (\pm)\sin x. \sin y$&lt;/li&gt;&#xA;&lt;li&gt;$\tan(x +y)=\frac{\tan x + \tan y}{1-\tan x . \tan y}$&lt;/li&gt;&#xA;&lt;li&gt;$\tan(x -y)=\frac{\tan x - \tan y}{1+\tan x. \tan y}$&lt;/li&gt;&#xA;&lt;li&gt;$\tan x+ \cot x=\frac{1}{\sin x . \cos x}=\frac{2}{\sin 2x}$&lt;/li&gt;&#xA;&lt;li&gt;$\sin x +\sin y=2 \sin(\frac{x+y}{2}) . \cos(\frac{x-y}{2})$&lt;/li&gt;&#xA;&lt;li&gt;$\sin x-\sin y= 2\cos(\frac{x+y}{2}).\sin(\frac{x-y}{2})$&lt;/li&gt;&#xA;&lt;li&gt;$\cos x + \cos y=2\cos(\frac{x+y}{2}) . \cos(\frac{x-y}{2})$&lt;/li&gt;&#xA;&lt;li&gt;$\cos x - \cos y=-2\sin(\frac{x+y}{2}). \sin(\frac{x-y}{2})$&lt;/li&gt;&#xA;&lt;li&gt;$\sin x .\cos y=\frac{1}{2}(\sin(x+y) + \sin(x-y))$&lt;/li&gt;&#xA;&lt;li&gt;$\cos x . \cos y=\frac{1}{2}(\cos(x+y)+ \cos(x-y))$&lt;/li&gt;&#xA;&lt;li&gt;$\sin x . \sin y =-\frac{1}{2}(\cos(x+y) - \cos(x-y))$&lt;/li&gt;&#xA;&lt;li&gt;$a \sin x +b \cos x= \frac{|a|}{a}\sqrt{a^2+b^2}\sin(x+\tan^{-1}(\frac{b}{a}))$&lt;/li&gt;&#xA;&lt;li&gt;$\cos^2x=\frac{1+\cos(2x)}{2} \sin^2x=\frac{1-\cos(2x)}{2}$&lt;/li&gt;&#xA;&lt;li&gt;$\tan^2x=\frac{1-\cos(2x)}{1+\cos(2x)}$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Law of sine $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2R} \space\space\space\space A+B+C=\pi$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Law of cosine &#xA;$$\cos A=\frac{b^2+c^2-a^2}{2b.c}$$ &#xA;$$\cos B=\frac{a^2+c^2-b^2}{2a.c}$$&#xA;$$\cos C=\frac{b^2+a^2-c^2}{2b.a}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\cos(\frac{\pi}{2}-x)=\sin x$$ $$\sin(\frac{\pi}{2}-x)=\cos x$$ by dividing 2 last formula $$\tan(\frac{\pi}{2}-x)=\cot(x)$$&lt;/p&gt;&#xA;" OwnerUserId="6544" LastEditorUserId="64" LastEditDate="2016-04-02T19:24:38.690" LastActivityDate="2016-04-02T19:24:38.690" Title="Necessary trigonometric identities for k-12 students" Tags="|secondary-education|trigonometry|" AnswerCount="2" CommentCount="9" ClosedDate="2016-04-04T16:18:57.017" ContentLicense="CC BY-SA 3.0" />
  <row Id="10795" PostTypeId="2" ParentId="10794" CreationDate="2016-03-28T00:23:05.813" Score="2" Body="&lt;p&gt;I recommend you add these, though they are not equally &quot;necessary&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Reciprocal Identities:&lt;/strong&gt;&#xA;$$\sin\theta=\frac 1{\csc\theta}, \quad \csc\theta=\frac 1{\sin\theta}$$&#xA;$$\cos\theta=\frac 1{\sec\theta}, \quad \sec\theta=\frac 1{\cos\theta}$$&#xA;$$\tan\theta=\frac 1{\cot\theta}, \quad \cot\theta=\frac 1{\tan\theta}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Quotient Identities:&lt;/strong&gt;&#xA;$$\tan\theta=\frac{\sin\theta}{\cos\theta}, \quad \cot\theta=\frac{\cos\theta}{\sin\theta}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Odd-Even Identities:&lt;/strong&gt;&#xA;$$\sin(-\theta)=-\sin\theta, \quad \csc(-\theta)=-\csc\theta$$&#xA;$$\cos(-\theta)=\cos\theta, \quad \sec(-\theta)=\sec\theta$$&#xA;$$\tan(-\theta)=-\tan\theta, \quad \cot(-\theta)=-\cot\theta$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Period Identities:&lt;/strong&gt; (where $k\in\Bbb Z$)&#xA;$$\sin(\theta+2k\pi)=\sin\theta, \quad \csc(\theta+2k\pi)=\csc\theta$$&#xA;$$\cos(\theta+2k\pi)=\cos\theta, \quad \sec(\theta+2k\pi)=\sec\theta$$&#xA;$$\tan(\theta+k\pi)=\tan\theta, \quad \cot(\theta+k\pi)=\cot\theta$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Half-Period Identities:&lt;/strong&gt;&#xA;$$\sin(\theta+\pi)=-\sin\theta, \quad \csc(\theta+\pi)=-\csc\theta$$&#xA;$$\cos(\theta+\pi)=-\cos\theta, \quad \sec(\theta+\pi)=-\sec\theta$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Half-Angle Identities:&lt;/strong&gt;&#xA;$$\cos\frac{\theta}2=\pm\sqrt{\frac 12(1+\cos\theta)}$$&#xA;$$\sin\frac{\theta}2=\pm\sqrt{\frac 12(1-\cos\theta)}$$&#xA;$$\tan\frac{\theta}2=\frac{\sin\theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin\theta}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most important reciprocal and quotient identities are those that define tangent, cotangent, secant, and cosecant in terms of sine and cosine. Some identities can be memorized in words, such as &quot;Cosine and secant are even functions; the others are odd functions&quot; and &quot;Tangent and cotangent have periods of pi; the others have periods of two pi.&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Note that your Law of Sines is missing an equals sign, and the two different parts are not separated clearly. I recommend that you use the Laws of Cosines in the non-fraction form, since they are then true even for degenerate triangles with zero-length sides.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2016-03-28T13:58:40.670" LastActivityDate="2016-03-28T13:58:40.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10796" PostTypeId="2" ParentId="10794" CreationDate="2016-03-28T12:22:51.230" Score="3" Body="&lt;p&gt;Keep in mind that the Law of Sines and Law of Cosines are not identities in the same sense that your items 1-21 are identities. Rather these two laws are geometrical facts that are particular to Euclidean geometry. Items 1-21 are analytical facts. If you define sine and cosine in terms of the exponential function, then these identities follow formally without any appeal to the geometric axioms.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-03-28T12:22:51.230" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10797" PostTypeId="2" ParentId="10786" CreationDate="2016-03-28T18:50:18.100" Score="3" Body="&lt;p&gt;Some points not already in the other excellent answers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basic algebra!  Most students in elementary university courses are still struggling with the elementary algebra, if you really can do this without problems you will have an edge/learn faster in all other courses.  Then, even if not really a mathematics topic, &lt;strong&gt;learn programming&lt;/strong&gt;. Whatever you will do within applied mathematics,  programming will be an important part. Language is not so important, but learn it in connection with some mathematics course, such as for example numerical analysis.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, linear algebra. This is used in aboslutely all parts of applied ( and pure) mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basic statistics. especially visualization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probability.&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2016-03-28T18:50:18.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10798" PostTypeId="2" ParentId="925" CreationDate="2016-03-28T22:24:57.210" Score="5" Body="&lt;p&gt;Sequence is just a function of the type $f:\mathbb{N} \to \mathbb{R}$. It is common to list the elements of this sequence as $$(a_1,a_2,a_3,\ldots,a_n)\,.$$ One example is the sequence of all even numbers: $(0,2,4,6,8,10,\ldots)$. However some sequences may be defined in a different form when there is no easy formula for expressing the terms, like the sequence of prime numbers $(2,3,5,7,11,13,\ldots)$, defined verbally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Series means summation of terms, this is clear if we use sigma notation, in which the terms are defined by a law that resembles a sequence. For example: $$\sum_{i=1}^{n} i^2$$ is the sum of the sequence of squares $(1,4,9,16,25,\ldots,n^2)$. We can expand the RHS for the sake of clearness: $$\sum_{i=1}^{n} i^2=1^2+2^2+3^2+4^2+\cdots+n^2.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often the series has a formula that depends only on the upper limit, so we can easily find the result without adding the terms, for the example above we know that $$\sum_{i=1}^{n} i^2=1^2+2^2+3^2+4^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also the term infinite series, that is simply the limit $$\lim_{n\to\infty}\sum_{i=1}^{n} a_i\,,$$ from this many concepts arrises, but it is another discussion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the sake of didactic you can visualize a sequence as a stone path, where each stone has a number, when talking about sequences what matters is in what stone you are, like in the number 3 or more generally, $a_n$. On the same example, series is the path you travel to reach a determined stone, i.e., if you are supposed to go to the stone numbered 9 (from the origin), series is the sum of the stones you stepped, in this case: $(1+2+ 3+\ldots+8+9)$, or $(a_1+a_2+a_3+\ldots+a_8+a_9)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope I could help you more. I could, so I edited.&lt;/p&gt;&#xA;" OwnerUserId="6547" LastEditorUserId="6547" LastEditDate="2016-03-31T19:42:45.653" LastActivityDate="2016-03-31T19:42:45.653" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10799" PostTypeId="2" ParentId="7528" CreationDate="2016-03-29T03:10:31.197" Score="13" Body="&lt;p&gt;A (cata)caustic is the envelope of lines reflected in a curve.&#xA;The caustic formed by parallels lines reflected in a semicircle is a cardioid,&#xA;such as can be seen in the bottom of this &lt;a href=&quot;https://math.stackexchange.com/questions/1550259/cardioid-in-coffee-mug&quot;&gt;MSE coffee mug&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Abxv9.jpg&quot; width=&quot;300&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other envelopes include evolutes.  An evolute is the envelope of the normal lines to a given curve; the given curve is the involute of the evolute.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/sEzKL.png&quot; height=&quot;200&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Zj9ak.png&quot; height=&quot;200&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A famous involute is the cycloid, which the involute of itself (and therefore the evolute of itself, too). Because the cycloid is a &lt;a href=&quot;https://en.wikipedia.org/wiki/Tautochrone_curve&quot; rel=&quot;nofollow noreferrer&quot;&gt;tautochrone&lt;/a&gt;, Huygens used it to design a clock (left, Fig. II), which Coster made (right):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/xOjOx.jpg&quot; height=&quot;350&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/l3Gxa.jpg&quot; height=&quot;350&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The involute of a circle (the smaller ones) can be used to design gear teeth that roll off each other without slipping (thus minimizing heating due to friction):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ER6Qk.gif&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-03-29T03:10:31.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10800" PostTypeId="1" AcceptedAnswerId="10801" CreationDate="2016-03-29T09:59:58.590" Score="11" ViewCount="249" Body="&lt;p&gt;I'm teaching my 6 year old how to tell the time. She can read analogue clocks comfortably.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We want her to be able to tell the time so when she wakes, if the time is past 6:15 then she is allowed to get up. So I'm trying to help her understand what time is in relation to &lt;em&gt;this&lt;/em&gt; time (is it before or after). If I ask her the following questions&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is ten past five before or after quarter past six&lt;/li&gt;&#xA;&lt;li&gt;Is seven thirty before or after quarter past six&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then she gets them right because (I think) she works the answer out by the hour (in the first example it's 5 which is before 6, in the second example the hour is 7 which is after 6).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue I have is if I ask her the following&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is five minutes to six before or after quarter past six&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This type of question, where the hour remains at 6, tricks her and she guesses the answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A clock with moveable hands does not help as she'll happily spin the clock round a dozen times to show it was after as she's moved the clock hands forward!!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a common way (ideally visual) to teach children this concept?&lt;/p&gt;&#xA;" OwnerUserId="6551" LastEditorUserId="6551" LastEditDate="2016-04-02T06:04:22.077" LastActivityDate="2017-04-20T06:09:37.713" Title="How can I teach the concept of before and after a given time" Tags="|primary-education|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10801" PostTypeId="2" ParentId="10800" CreationDate="2016-03-29T15:38:56.003" Score="12" Body="&lt;p&gt;Don't use the constructions of &quot;five minutes to six&quot; and &quot;quarter past six&quot;, which are tricky and somewhat slang-y. Use the proper terminology of &quot;five fifty-five&quot; and &quot;six fifteen&quot;, etc., and the problem should resolve itself. Perhaps later she can add the slang comparisons to her vocabulary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, there's a reason we got away from naming numbers as in Latin like &quot;one from twenty&quot; (undeviginti, 19) and &quot;two from twenty&quot; (duodeviginti, 18). The comparisons are indeed easier with proper and consistent use of place values; and this also explains decimalization in our own age. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And likewise: The simple linguistic difference in naming numbers in Chinese (standard places: &quot;ten-one, ten-two, ten-three...&quot;) versus English (nonstandard places: &quot;eleven, twelve, thirteen...&quot;) has been demonstrated to delay counting skills for English-speakers, on average, at young ages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/usgyx.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/usgyx.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Miller, Kevin F., et al. &quot;Preschool origins of cross-national differences in mathematical competence: The role of number-naming systems.&quot; &lt;em&gt;Psychological Science&lt;/em&gt; 6.1 (1995): 56-60. (&lt;a href=&quot;http://pss.sagepub.com/content/6/1/56.short&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-03-29T17:42:53.223" LastActivityDate="2016-03-29T17:42:53.223" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10802" PostTypeId="2" ParentId="10777" CreationDate="2016-03-29T22:47:43.703" Score="4" Body="&lt;p&gt;This is the story of an attempt to find some evidence for something &quot;similar&quot;. Let me start with an anecdote. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Personal anecdote&lt;/strong&gt;. Yeeeears ago, as a math student I was good at continues mathematics (e.g., analysis) and bad at discrete mathematics (e.g., algebra, graph theory). To be honest, I wasn't even able to accept the given proofs in graph theory as &quot;real&quot; proofs, let alone to attempt for one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why this is related to your question&lt;/strong&gt;. I don't know if there is an established difference between continuous and discrete mathematics, but I feel the distinction between geometry and algebra as you mentioned is somehow related to it. If not, you can see the rest of my answer as an attempt to address a similar concern. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Research.&lt;/strong&gt; Yeeears after my student time, I was a mathematics educator working in a math department where I had a chance to tell the anecdote above to my mathematician colleagues. Interestingly, many of them knew someone else with the same experience. Thus I decided to find some hard evidence to show there is something called &quot;continuous thinking&quot; and something called &quot;discrete thinking&quot; where some people are good at one but not necessarily at the other. A master student of mine &quot;chose&quot; this as the topic of her dissertation. We focused on understanding of proof in analysis and algebra. We saw a proving activity as a problem solving activity and accordingly we use some of the variables in research on problem solving (as introduced &lt;a href=&quot;http://scholar.google.co.uk/scholar?hl=en&amp;amp;q=%22Variables%20and%20Methodologies%20in%20Research%20on%20Problem%20Solving%20%22&amp;amp;btnG=&amp;amp;as_sdt=1%2C5&amp;amp;as_sdtp=&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; by Kilpatrick) to study whether there is any difference in students' thinking of proofs in analysis and algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Design of the experiment.&lt;/strong&gt; We gave students several pairs of proofs. In each pair, one proof was from algebra and the other from analysis. Both proofs had the &quot;same structure&quot;. Students didn't know that the pairs are purposefully related. They were reading the proofs and then answering some questions designed in the light of some of the variables (given by Kilpatrick).The variables were too many, we just tried some of them. Unfortunately, I couldn't remember now which variables we used, perhaps because of the result of the research. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Result&lt;/strong&gt;. We didn't find any difference noticeable! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion.&lt;/strong&gt; I still believe there is some differences there that we have failed to find!!&lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2016-03-29T22:47:43.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10803" PostTypeId="1" CreationDate="2016-03-30T02:01:20.150" Score="5" ViewCount="369" Body="&lt;p&gt;I'm explaining to a student that all functions can be divided into odd and even (symmetric and anti-symmetric) components. It is easy to prove (basic algebra or Taylor series) but is not referenced in the text. I'm using it in relation to the definition of trigonometric functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does this theorem have a name? Are there any good online resources for this  that cover this at a High School/Calc I level (i.e. no Fourier series). Or is this considered simply too obvious to cover formally?&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2016-04-04T12:19:01.500" Title="Functions can be divided into odd and even components - name of theorem?" Tags="|undergraduate-education|secondary-education|calculus|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10804" PostTypeId="2" ParentId="10803" CreationDate="2016-03-30T02:34:53.480" Score="11" Body="&lt;p&gt;Take the function $f(x)$ and write the functions $\frac{1}{2}(f(x)-f(-x))$ and $\frac{1}{2}(f(x)+f(-x))$. They are odd and even, respectively, and their sum is $f(x)$. Nothing fancy required so I doubt it is named.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: If you wanted to generalize this to the action of $n$th roots of unity as @KCd suggests, you could decompose the function $f(z)$ into functions $f_k(z)$ of the form&#xA;$$ f_k(z) = \frac{1}{n} \sum_{j=0}^n \omega^{kj} f(\omega^j z) $$&#xA;where $\omega$ is a primitive $n$th root of unity. It is easy to check that $f= \sum_k f_k$ and that $f_k(\omega z) = \omega^k f(z)$. The odd/even case then corresponds to $n=2$, $\omega=-1$. &#xA;You might even be able to explain that to advanced high school students if you can get them to understand why $\sum_{j=0}^n \omega^j = 0$. However, at that stage, the potential for confusion is probably not worth the reward.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2016-04-04T12:19:01.500" LastActivityDate="2016-04-04T12:19:01.500" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10805" PostTypeId="1" CreationDate="2016-03-30T12:12:20.207" Score="3" ViewCount="190" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;A vector is a collection of elements whose order matters. If the elements represent comparable measurements it can be said that they together represent a direction and a magnitude (length). The Pythagorean theorem is used to find the length of a vector.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is to be represented to high school students. I want to ask for opinions so the definition can be as useful as possible while not misleading them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: forgot to mention that the students are familiar with what a set is.&lt;/p&gt;&#xA;" OwnerUserId="5731" LastEditorUserId="5731" LastEditDate="2016-03-30T12:48:11.460" LastActivityDate="2016-03-31T07:52:29.020" Title="Defining/introducing vectors informally" Tags="|linear-algebra|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10806" PostTypeId="2" ParentId="10805" CreationDate="2016-03-30T12:47:35.520" Score="4" Body="&lt;p&gt;The first part of your definition is strange and unclear. What is the meaning of &quot;whose order matters?&quot; The second sentence is much better for a high-school understanding: a quantity with magnitude and direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I teach vectors in my 12th grade Physics class, I do this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;State the &quot;quantity with magnitude and direction&quot; definition.&lt;/li&gt;&#xA;&lt;li&gt;Show a &lt;a href=&quot;https://www.youtube.com/watch?v=A05n32Bl0aY&quot; rel=&quot;nofollow&quot;&gt;YouTube clip&lt;/a&gt; that gives the same definition. (Most of my students are already familiar with this clip.)&lt;/li&gt;&#xA;&lt;li&gt;State that the usual way to represent a vector is an arrow whose length and direction matter but whose starting point does not matter. I use actual arrows (which may not be allowed in your school) and slide them around the white board to show equivalent and non-equivalent vectors.&lt;/li&gt;&#xA;&lt;li&gt;Discuss real-world examples of vectors such as displacement, velocity, and force.&lt;/li&gt;&#xA;&lt;li&gt;Explain how to show a vector variable: with bold-face if in print ($\mathbf v$), with an over-arrow if hand-written ($\vec v$).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;All that can be done pretty quickly. I then take more time to discuss vector operations, such as addition, negation, subtraction, scalar multiplication, decomposition into components, and re-composition from components. I briefly state that other operations exist (e.g. dot and cross product) but we will not discuss them further in my class. I also state that other, more advanced and complicated definitions of vectors exist but that my definition suffices for the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works pretty well, with no resulting confusion about vectors. The difficulty comes in their application, a little later in the chapter. None of my students who do further math and science in college has ever told me that they were confused or mislead by my treatment of vectors, and I do ask about such things.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2016-03-30T12:52:40.093" LastActivityDate="2016-03-30T12:52:40.093" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10807" PostTypeId="2" ParentId="10805" CreationDate="2016-03-30T17:13:44.203" Score="6" Body="&lt;p&gt;I think the &quot;ordered $n$-tuple&quot; notion of &quot;vector&quot; is not only unnecessarily abstract, it actually distorts the meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I informally define a vector as directed line segment (or &quot;arrow&quot;) that &quot;floats&quot; in space.  A vector has magnitude and direction, but no specific position.  Crucially, the &lt;em&gt;components&lt;/em&gt; of a vector are only defined relative to a coordinate system, and sometimes it is helpful to choose different coordinate systems for different problems.  For an objects sliding down a ramp, the &quot;best&quot; coordinates are parallel to and perpendicular to the ramp; for an object flying through the air, the &quot;best&quot; coordinates are horizontal and perpendicular.  (Sometimes you may even change coordinates in the middle of a problem!)  So a single vector may have more than one representation as an ordered list.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-03-30T17:13:44.203" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="10808" PostTypeId="1" CreationDate="2016-03-30T22:36:39.490" Score="1" ViewCount="82" Body="&lt;p&gt;I'm interested in discovering how many hours of contact math educators have with their students each week. I've done some research and gotten some averages by country, but I can't find enough data (distributions) to determine if my classroom time is significantly below average.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realize than many countries (and states in the US) have determined a minimum number of annual hours of &quot;instruction&quot;; however, this often includes lunch breaks, study halls, and other time that is not specifically used for instruction.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is an overview of general guidelines as to how many contact hours I should have with my students? How many hours do you have and how does it work?&lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-03-31T12:00:28.817" LastActivityDate="2016-03-31T12:00:28.817" Title="How many contact-hours should I have with students? (Secondary Ed.)" Tags="|statistics|scheduling|time-management|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10810" PostTypeId="1" CreationDate="2016-03-30T22:43:26.447" Score="3" ViewCount="133" Body="&lt;p&gt;If you segment each class period regularly, what is your method? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, do you always do a 10 min HW check, a 15 minute review, a 25 minute lesson, then a 10 min learning assessment? Do you switch the order of the class segments? Do you add certain events regularly, like a notebook check, quiz, or group-work? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do you schedule your daily class time?&lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-03-31T01:08:16.373" LastActivityDate="2016-03-31T01:58:17.783" Title="How do you segment your classroom time? (Secondary Ed.)" Tags="|secondary-education|time-management|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10811" PostTypeId="2" ParentId="10810" CreationDate="2016-03-31T01:47:28.160" Score="3" Body="&lt;p&gt;I have taught both standard 45-50 minute periods and 90-100 minute block periods and this answer will cover my general structuring for both.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I will always start off with a &quot;warm-up&quot; (review problem usually, but sometimes a video, image, etc), which lasts 5 minutes for standard periods.  I then review any homework questions which can take anywhere from 5 minutes to nearly an entire period.  It is important to continually think about the skill level of your students and the difficulty of the homework when planning for the next day's homework review time.  If you assign 1-100 odds, be prepared for a lot of questions taking a lot of time, whereas if you assign 3, 5, and 6, then five minutes is probability a safe bet. In general, I usually allocate about 10 minutes to homework review, leaving 30ish minutes left for my core lesson.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I structure lessons in what I call &quot;chunks&quot; which last roughly 10 minutes each for a standard period.  A chunk is an isolated cycle of &lt;strong&gt;introduce-example-practice-review&lt;/strong&gt;.  So for instance, when teaching a new theorem in Geometry, the first two minutes would be for introducing the theorem, possibly walking through a proof if it was basic enough.  Then I would work an example problem for about three minutes on the board.  Students are then presented with two example problems to work on for three minutes independently or in groups and then we review the problems in the last two minutes of the chunk.  Then in the next chunk we repeat the same process but for either a more advanced application of the concept, a related concept, or even the same concept if it is more essential and deserves extra time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can usually fit in two ten-minute chunks for a standard period (45-50 minutes). The last 10-15 minutes of a standard period are spent on mixed practice of the material covered in the two chunks.  Sometimes I will do examples on the board, have students present their solutions, discuss specific solutions in-depth, or any other number of things but this last section of class is always spent reinforcing the topics introduced in that lesson.  I will usually end the standard period class with a 1 or 2 minute &quot;exit ticket&quot; which is usually a question such as &quot;What is one thing you have learned today?&quot; or &quot;The sum of the interior angles of a triangle is __________________&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I structure block periods (90-45 minutes) slightly differently.  In the district I teach and I'm sure many others, block periods are either done on a semester basis or on an every-other-day basis, so in either sense you only see students on half of the amount of days.  While it is theoretically the exact same amount of time as standard periods meeting everyday, anyone who has taught blocks knows that the time is much different than in a standard period. Because of this I tend to follow the same general structure of a standard period, except that I go more in-depth during each phase of the lesson and thus spend more time in each phase.   The warm-up will typically last 10 minutes, and will have either more problems than a standard period warm up or more in-depth problems for the students to work on.  Homework goes the same as it did for standard periods, however I typically assume 20 minutes for review as I almost always assign more homework to block classes.  I still do two chunks, however each chunk lasts roughly seventeen minutes.  Four minutes for introduction of topic, 5 minutes for example, 5 minutes for independent work, and 3 minutes for review.  This leaves roughly 30-40 minutes left in the period.  I will usually give a 5 minute break in the middle of the period and when students come back we will essentially do the same thing as for the standard period, use the last portion of class reinforcing the topics that were introduced earlier.  Spend roughly 25 minutes on a mixture of independent, group, and class work. One benefit of blocked periods is that you do have more time for activities and explorations because this last portion of the period is twice the length as a standard period.  Because of the length of the period, I end block periods with a more formal exit-ticket, usually two problems that test a basic understanding of the material in the lesson and this takes roughly 5 minutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As with everything in teaching, this structure needs to be flexible, which is fine, provided that you have already anticipated many of the &quot;pitfalls&quot;, &quot;time sinks&quot; and &quot;rabbit holes&quot;.  If your students were having a tough time with the previous day's material, you will probably want to allocate more time to review in the front half of the class, maybe making one chunk a review chunk.  If there is an activity that you would like to do make sure you allocate enough time to complete it and discuss, possibly eliminating one of the two chunks.  If you are introducing a lot of smaller but related concepts, doing three or four chunks in a period is not unheard of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say that I stick to this general structure roughly 85% of the time.  The other 15% of the time, will usually be spent on an activity or exploration that takes thirty or more minutes or on summative assessments.  Prior to a test, I will usually spend an entire period reviewing and answering any questions that students have.  Tests will almost always take an entire period.  For block periods, I will use the first 45 minutes of class for review and the last 45 to take the test.  For quizzes, it depends.  Roughly half the time, I will have them in the beginning of class, when I am not anticipating too much struggling and there is content that I would like to cover afterwards. The other half of the time I will do quizzes at the end of the period, either because they are a quiz on what was learned in the first half of the period or because I feel that I need to review certain things before the assessment is administered.  &lt;/p&gt;&#xA;" OwnerUserId="3237" LastEditorUserId="3237" LastEditDate="2016-03-31T01:58:17.783" LastActivityDate="2016-03-31T01:58:17.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10812" PostTypeId="1" AcceptedAnswerId="10816" CreationDate="2016-03-31T02:24:09.743" Score="4" ViewCount="154" Body="&lt;p&gt;I am of a mature aged learner and want to learn probability and statistics at a senior high school (k12) and undergraduate level for my own general understanding. I have discovered in the past that I learn best by having large well graduated problem sets where the variations between problems are small and obvious, rather than having relatively few problems with significant differences between them. Based on previous testing I can conclude that I am of moderately above average general intelligence but I am a slow learner. I can be patient. Most texts do not have enough problems and the steps from problem to problem are to great for me to learn efficiently. Can anyone suggest a good text book. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="6555" LastActivityDate="2016-04-01T14:13:50.710" Title="suggestion for probability and stats text to suit my learning style." Tags="|textbooks|general-pedagogy|self-learning|statistics|probability|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10813" PostTypeId="2" ParentId="10812" CreationDate="2016-03-31T04:22:14.130" Score="1" Body="&lt;p&gt;I use Neil Weiss, &lt;em&gt;Introductory Statistics&lt;/em&gt; in my classes. It's my favorite textbook that I use. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0321393619&quot; rel=&quot;nofollow&quot;&gt;Introductory Statistics, 8th Edition at Amazon&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-03-31T04:22:14.130" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10814" PostTypeId="2" ParentId="10800" CreationDate="2016-03-31T07:33:04.127" Score="1" Body="&lt;p&gt;Since she was getting confused about what was before (quater past 6 or quater to 6) I felt she may not have understood what time is or why we show it on a clock. The approach I took (which definatley helped) was to draw time out on a piece of paper. Something like&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;5, 5 past 5, 10 past 5, quarter past 5, 20 past 5 etc&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;This went on for as long as I could (until I ran out of paper). I then explained to my daughter that we couldn't fit any more time on the page so that was a bit rubbish and that since time doesn't stop we would need more paper. It didn't take long for her to realize that that no piece of paper would ever be big enough. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We tried a few other things but came to the same conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally we tried with a piece of string. We put 12 colors on the string, evenly spaced, and numbered them 1 - 12, but we ran out of string so couldn't repeat. I could then show her we could bend (loop)  the string to a circle to show it can be represented in a different way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I then stuck the string on her clock face to show how it matched up with the hours and she then realized why the clock was a circle (this meant preparing the string etc).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This helped to reduce her errors and I'm sure with practice she would have understood but the accepted answer was obviously the best solution.&lt;/p&gt;&#xA;" OwnerUserId="6551" LastEditorUserId="6551" LastEditDate="2017-04-20T06:09:37.713" LastActivityDate="2017-04-20T06:09:37.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10815" PostTypeId="2" ParentId="10805" CreationDate="2016-03-31T07:52:29.020" Score="1" Body="&lt;p&gt;I think the elements of your suggested definition are ok, but I would want to split them up and put a lot more explanation in between.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would perhaps start by saying that there are different ways of thinking about vectors, and that while they are all the same in some sense (and so it's reasonable to use the same word), it might not be obvious to everyone until they get used to switching the way they think about them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, according to time/syllabus, I'd introduce the different ways of looking at vectors in the contexts that each is used: position vectors in coordinate geometry, free-floating arrows in coordinate-free geometry, direction/magnitude in mechanics, $n$-tuple of objects in sequences or programming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For each one I'd get them doing questions in that context. After that, I'd move on to questions that involve changing between two of the descriptions, eg Cartesian and polar coordinates. Doing such questions should allow students to mentally connect the definitions, although some won't manage that level of understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I think your definition is potentially confusing is that it mixes the different viewpoints, at a point when the students don't have the understanding to see the connections. If you're just working with $n$-tuples, direction and length don't have any meaning, particularly if the entries might not be real numbers, and making the shift in thinking is likely to take longer than reading one sentence. Using Pythagoras only makes sense when you have some kind of coordinates, and seeing why it has anything to do with something other than triangles will take time too. The students need to get their heads around the different forms before they can put the different sentences together, if they are to be happy with a definition presented that way.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-03-31T07:52:29.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10816" PostTypeId="2" ParentId="10812" CreationDate="2016-03-31T22:38:25.367" Score="2" Body="&lt;p&gt;My AP Statistics class uses &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1464108730&quot; rel=&quot;nofollow&quot;&gt;The Practice of Statistics (Josh Tabor)&lt;/a&gt;. It has a large variety of exercises at the end of each chapter and they are definitely in the order of difficulty. They are also presented in the same order as the ideas are presented in the chapter itself. There are definitely both &quot;practice&quot; type questions and &quot;deep thinking&quot; type questions, and in my opinion the problems are &quot;graduated.&quot; There are generally two very similar problems back-to-back, one even one odd, with only the odd ones having the answer in the back of the book.&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2016-04-01T14:13:50.710" LastActivityDate="2016-04-01T14:13:50.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10817" PostTypeId="1" CreationDate="2016-04-01T08:54:51.353" Score="7" ViewCount="230" Body="&lt;p&gt;I've been thinking if it would be possible to teach something about the basic mathematics behind Neural Networks / Deep learning to a class of 6th graders (12-13ys old). A 90-minute (max) lecture must be assumed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that it maybe sounds like too difficult/impossible -- but I'd love to hear if there are any good ideas. We can assume that the students can each have their personal tablet in-class, and also there is a projector in-class, if that helps in any way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal, assuming that a valid solution does exist, would be showing students one more cool application of maths; especially one that is everywhere those days.&lt;/p&gt;&#xA;" OwnerUserId="6556" LastActivityDate="2016-04-02T22:08:00.950" Title="Teaching Neural Networks to 6th grade students" Tags="|technology-in-education|primary-education|computer-science|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10819" PostTypeId="2" ParentId="10800" CreationDate="2016-04-02T05:41:44.117" Score="1" Body="&lt;p&gt;If she knows how to compare numbers, as you said, then your task is easy. If the hour is the same, then tell her to compare the minutes and if those are also the same then compare the seconds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With this strategy she will better understand&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-how do we compare ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Which is the largest amount ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Which two numbers are equal ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and this will also help her for the next standard.&lt;/p&gt;&#xA;" OwnerUserId="6561" LastEditorUserId="5321" LastEditDate="2016-04-02T17:21:35.690" LastActivityDate="2016-04-02T17:21:35.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10821" PostTypeId="2" ParentId="10817" CreationDate="2016-04-02T22:08:00.950" Score="1" Body="&lt;p&gt;Neural networks are largely about doing a large number of regression problems --- sometimes linear regression --- so you may be able to appeal to some intuitive sense about what a best fit line/plane and go from there. It feels like it would be tough to get students to understand how the network learns (back propagation) in a meaningful way without some notion of gradient descent. (Aren't parabolas introduced in later grades?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if what you want to do is show them a modern data analysis tool, I would suggest k-means clustering instead. &lt;a href=&quot;https://en.wikipedia.org/wiki/K-means_clustering&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/K-means_clustering&lt;/a&gt; The most advanced concepts used there are distance and averaging (the mean). &lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2016-04-02T22:08:00.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10822" PostTypeId="1" CreationDate="2016-04-03T02:22:29.390" Score="4" ViewCount="146" Body="&lt;p&gt;I am dealing with some inner city kids who  are all over the place. I want to use a concept map to slowly develop their math skills in an organized manner. If anybody can point me to Math Concept maps something that is visual , that will be very helpful. Thanks much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good question, @ Tommi Brander ,@ Joe&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students are between the ages of 9-12 years. Ramsey county, Minnesota.&lt;/p&gt;&#xA;" OwnerUserId="6567" LastEditorUserId="6567" LastEditDate="2016-04-04T00:21:13.203" LastActivityDate="2016-04-04T00:21:13.203" Title="Math concept map" Tags="|concept-maps|" AnswerCount="0" CommentCount="6" ClosedDate="2016-04-06T19:47:38.610" ContentLicense="CC BY-SA 3.0" />
  <row Id="10826" PostTypeId="1" CreationDate="2016-04-04T18:31:32.413" Score="-1" ViewCount="117" Body="&lt;p&gt;a)I am trying to get a book that would give describe all the coordinate systems and their transformations(e.g. cartesian,polar,spherical,homogeneous,curvilinear,generalized,etc).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) And I need to know about some good books on multi-variable calculus which exclude vector calculus and operates only on scalar field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All suggestions are welcome although books on an undergraduate level are preferred.&lt;/p&gt;&#xA;" OwnerUserId="6560" LastActivityDate="2016-04-11T17:23:01.997" Title="books of mathematics" Tags="|undergraduate-education|calculus|" AnswerCount="1" CommentCount="3" ClosedDate="2016-04-06T19:46:53.373" ContentLicense="CC BY-SA 3.0" />
  <row Id="10827" PostTypeId="2" ParentId="10826" CreationDate="2016-04-04T20:52:53.503" Score="1" Body="&lt;p&gt;Related is the math stackexchange question &lt;a href=&quot;https://math.stackexchange.com/questions/1501301/book-on-coordinate-transformations&quot;&gt;Book on coordinate transformations&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In two comments there I suggested looking at some of the many lengthy treatises on mathematical methods for physicists for beginning graduate students in physics, in particular Morse/Feshbach's 2-volume treatise &lt;strong&gt;Methods of Theoretical Physics&lt;/strong&gt;. You can find reviews at amazon.com for &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/007043316X&quot; rel=&quot;nofollow noreferrer&quot;&gt;Part I&lt;/a&gt; and &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0070433178&quot; rel=&quot;nofollow noreferrer&quot;&gt;Part II&lt;/a&gt; and &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0976202107&quot; rel=&quot;nofollow noreferrer&quot;&gt;Parts I &amp;amp; II&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kris writes (in part) in a review at the &quot;Parts I &amp;amp; II&quot; page:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;These books go into great detail on what are now considered esoteric coordinate systems (elliptical, parabolic, etc); this may be the best place to find material on this topic.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;(ADDED A WEEK LATER)&lt;/em&gt; I happened to be at a university library this weekend and while there I looked at Morse/Feshbach. Pages 656-666 of Part I has summaries of the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Rectangular&lt;/strong&gt; Coordinates, &lt;strong&gt;Circular Cylinder&lt;/strong&gt; Coordinates, &lt;strong&gt;Elliptic Cylinder&lt;/strong&gt; Coordinates, &lt;strong&gt;Parabolic Cylinder&lt;/strong&gt; Coordinates, &lt;strong&gt;Spherical&lt;/strong&gt; Coordinates, &lt;strong&gt;Conical&lt;/strong&gt; Coordinates, &lt;strong&gt;Parabolic&lt;/strong&gt; Coordinates, &lt;strong&gt;Prolate Spheroidal&lt;/strong&gt; Coordinates, &lt;strong&gt;Oblate Spheroidal&lt;/strong&gt; Coordinates, &lt;strong&gt;Ellipsoidal&lt;/strong&gt; Coordinates, &lt;strong&gt;Paraboloidal&lt;/strong&gt; Coordinates, &lt;strong&gt;Bispherical&lt;/strong&gt; Coordinates, &lt;strong&gt;Toroidal&lt;/strong&gt; Coordinates.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-04-11T17:23:01.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10828" PostTypeId="1" CreationDate="2016-04-04T23:41:42.973" Score="9" ViewCount="2735" Body="&lt;p&gt;Hello fellow educators,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been hearing a lot recently that many students struggle with Algebra 1 in the US. The dropout rate is supposedly very high and some educators argue that Algebra 1 and higher math should be pulled from the school system because it's just &quot;too hard&quot; for students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone here know the actual failure rates of 9th graders in Algebra 1? I'm looking for anything recent within the past few years. Furthermore, how many students drop out of high school because they struggle in math? Any help is appreciated!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="4737" LastActivityDate="2018-12-07T22:28:15.023" Title="What is the failure rate of students in Algebra 1?" Tags="|secondary-education|algebra|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10829" PostTypeId="2" ParentId="10781" CreationDate="2016-04-05T13:30:10.737" Score="0" Body="&lt;p&gt;I would alter the presumption of the question that &quot;extra credit&quot; is the only way to do this.  There are several techniques people use to get students to &quot;read the book&quot; ahead of time which involve &quot;regular&quot; credit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tommy Ratliff and Matt Boelkins have &lt;a href=&quot;http://www.maa.org/how-we-get-our-students-to-read-the-text-before-class&quot; rel=&quot;nofollow&quot;&gt;a nice article about a good method&lt;/a&gt; - nowadays your LMS can handle this for you.  The short version is that you have a required writing assignment (long or short) where they would do something related to the routine problem you are asking about.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have found that a binary grading scheme works well for the assignments: a student earns a 1 for sincerely attempting to answer the questions (independent of whether the answers are correct), or receives a 0 if no such attempt is made. In addition, the assignments count for 5% of a student's final grade in the course.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So this isn't &quot;extra credit&quot; in the same sense.  In fact, it's very appropriate at the college level to simply expect students to do this.  It's also not realistic to expect all students to do it.  But the penalty is relatively minimal and proportional to expectations, and it seems realistic to use this as part of a 'participation' grade, if you use that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(That said, I do think that &lt;em&gt;very&lt;/em&gt; straightforward questions, for regular - not extra - credit are also appropriate no matter what the instruction level, assuming that they are completely procedural and you can give instructions as to where to find models.  As students mature, they could be less procedural.  I rarely get complaints about these as long as I am sure to cover them in class, or you could grade them on completion, not correctness.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I won't bother to link to the zillions of people discussing &quot;flipped classrooms&quot; in math, but this is another method that is much more intense for the instructor, but which also gets at &quot;reading ahead&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-04-05T13:30:10.737" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10830" PostTypeId="1" CreationDate="2016-04-05T18:13:59.730" Score="0" ViewCount="245" Body="&lt;p&gt;I am working on some introductory notes for group theory. Comments on my initial approach here and any errors so far would be appreciated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I begin with the group axioms:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall a,b\in G:[ a+b\in G]$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall a,b,c\in G:[ (a+b)+c=a+(b+c)]$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$0\in G$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall a\in G: [a+0=a \land 0+a=a]$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall a\in G: \exists b\in G: [a+b=0 \land b+a=0]$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then I prove:&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall a,b \in G:[a+b=a \implies b=0]$ (All right identities are equal to $0$)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall a,b \in G:[a+b=b \implies a=0]$ (All left identities are equal to $0$)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall a,b,c\in G:[a+b=c+b \implies a=c]$ (+ is right-cancelable)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall a,b,c\in G:[a+b=a+c \implies b=c]$ (+ is left-cancelable)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall a,b,c\in G:[a+b=0 \land a+c=0 \implies b=c]$ (right inverses wrt $0$ are unique)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\forall a,b,c\in G:[a+b=0 \land c+b=0 \implies a=c]$ (left inverses wrt $0$ are unique)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;12.$\exists !inv: \forall a\in G: [inv(a)\in G\land a+inv(a)=0 \land inv(a)+a=0]$ (unique inverse operator exists)&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2016-04-05T20:28:49.813" LastActivityDate="2016-04-09T06:37:58.517" Title="Comments on my approach to Group Theory notes?" Tags="|undergraduate-education|abstract-algebra|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="10832" PostTypeId="1" AcceptedAnswerId="10846" CreationDate="2016-04-06T06:02:52.297" Score="13" ViewCount="22863" Body="&lt;p&gt;I am asking this question here to obtain a nice reference list. I am asking for examples of words which have different connotations  or meanings in Mathematics compared to English as taught in literature class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even: can be written as 2n, where n is an integer&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even (if): make flat or smooth, or to show extreme degree&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Series: the sum of the terms of an infinite sequence&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Series: a set of related television or radio programs, especially of a specified kind&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; The purpose of this question is to help my teaching to low level ESL students; nothing to do with mathematics is a language debate. I am asking for examples to help me be more aware of my language when teaching AP Calculus and AP Statistics to low functional second language students. My students are at grade or above mathematical ability in their native language but most function between third to sixth grade (American grade levels) English ability. &lt;/p&gt;&#xA;" OwnerUserId="6586" LastEditorUserId="262" LastEditDate="2016-04-16T10:38:04.873" LastActivityDate="2020-11-07T19:53:44.153" Title="Examples of vocabulary that have different meanings in Mathematics compared to &quot;everyday&quot; English" Tags="|reference-request|vocabulary|english-language-learners|" AnswerCount="23" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10833" PostTypeId="2" ParentId="10832" CreationDate="2016-04-06T08:27:37.573" Score="10" Body="&lt;p&gt;This can get quite some list…&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a few:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;function&lt;/strong&gt;: a mapping in mathematics vs. a &quot;feature&quot; a &quot;purpose&quot; and &quot;functionality&quot; in everyday language &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;root&lt;/strong&gt;: e.g. a point where a function is zero vs. the vegetable or a source of something&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;character&lt;/strong&gt;: see &lt;a href=&quot;https://en.wikipedia.org/wiki/Character_%28mathematics%29&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; vs. &quot;person&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;matrix&lt;/strong&gt;: while this is probably not used very much in &quot;everyday&quot; English, this can have quite some different meanings (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Matrix&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;), especially the movie with the same name may not refer to the mathematical meaning but probably more to meaning in biology or geology.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;or&lt;/strong&gt;: In everyday language this often means the exclusive or while is mathematics this is always non-exclusive.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;real&lt;/strong&gt;: You know what a real number is, but real numbers are in no sense any more or less real that other numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;complex, rational, natural&lt;/strong&gt;: Same as above for these numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="3398" LastEditDate="2016-04-06T13:41:15.783" LastActivityDate="2016-04-06T13:41:15.783" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10834" PostTypeId="2" ParentId="10832" CreationDate="2016-04-06T09:59:46.707" Score="9" Body="&lt;p&gt;This could be a huge list but Il add one :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Volume. How loud something is or the number of cubes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think an interesting connected question is which maths words are used by the public in a mathematical sense but incorrectly?   &lt;/p&gt;&#xA;" OwnerUserId="4668" LastActivityDate="2016-04-06T09:59:46.707" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10835" PostTypeId="2" ParentId="10832" CreationDate="2016-04-06T11:00:19.460" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://www.cut-the-knot.org/language/hersh.shtml&quot; rel=&quot;nofollow&quot;&gt;This&lt;/a&gt; is a rather good collection. It is an essay by Reuben Hersh entitled Math Lingo vs. Plain English: Double Entendre.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2016-04-08T19:45:15.143" LastActivityDate="2016-04-08T19:45:15.143" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10836" PostTypeId="2" ParentId="10832" CreationDate="2016-04-06T14:53:54.743" Score="7" Body="&lt;ol&gt;&#xA;&lt;li&gt;Angle - Formed by 2 rays joined at a vertex, vs. a viewpoint, vs. tilt&lt;/li&gt;&#xA;&lt;li&gt;Yard -A unit of measure equal to 3 feet vs the grassy area around a house &lt;/li&gt;&#xA;&lt;li&gt;Constant - A number that doesn't change (not a variable) vs occurring continuously over a period of time vs. a situation that doesn't change&lt;/li&gt;&#xA;&lt;li&gt;Digit - Any of the numerals from 0 through 9 vs. a finger or toe.&lt;/li&gt;&#xA;&lt;li&gt;Gross - 12 dozen vs. yuck&lt;/li&gt;&#xA;&lt;li&gt;Key or Legend - List of labels on a graph or items in a map vs. what you use to open the door or a great story.&lt;/li&gt;&#xA;&lt;li&gt;Orientation - the angle of an object relative to the axes vs. training for new students vs. feelings and beliefs.&lt;/li&gt;&#xA;&lt;li&gt;Power - the number of times a base is multiplied by itself vs. the capacity to direct others.&lt;/li&gt;&#xA;&lt;li&gt;Similar - having the same shape but not the same size vs. resembling without being exactly the same.&lt;/li&gt;&#xA;&lt;li&gt;Table - Mathematical information organized in columns and rows vs furniture with a flat top and legs.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-04-06T14:53:54.743" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10838" PostTypeId="1" CreationDate="2016-04-07T01:49:50.687" Score="3" ViewCount="1032" Body="&lt;p&gt;I'm teaching AP Calculus BC for the first time this year. The AP curriculum is an attempt to cover most of the material in a two-semester university freshman calculus series, and so I am reminded of my own Calculus II course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As we finish up the curriculum, we are discussing the convergence of series. For the most part, the curriculum has a natural flow from functions to limits, then to the derivative, derivative rules and applications, integrals, integration techniques, integration and solving differential equations, improper integrals, then... infinite series representation of functions. It seems to me to be a disruption in what was otherwise a smooth progression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that for students in a math program it is &lt;em&gt;very&lt;/em&gt; important to learn the theory of infinite series. I recall that I did not understand infinite series and convergence until taking analysis as a first semester junior. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also observed that while the method of finding series solutions to differential equations is included in many textbooks it was not included in the undergraduate differential equations course I took, nor the few that I TA'd as a graduate student. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is so much attention given to infinite series in calculus II? &lt;/p&gt;&#xA;" OwnerUserId="5690" LastActivityDate="2016-04-07T05:43:15.193" Title="Why is the convergence of infinite series covered in Calculus II?" Tags="|calculus|curriculum|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10839" PostTypeId="2" ParentId="10838" CreationDate="2016-04-07T05:30:35.460" Score="5" Body="&lt;p&gt;One could make a lot of arguments either way. However, with respect to the AP curriculum as it stands, it makes sense to include infinite series, because Taylor Series are there. Why do I say this? Why include Taylor Series at this venture?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, the main viewpoint of the AP calculus curriculum should be local linearity. It should be how the major rules for derivates, L'Hopital's Rule, and even the FTC are justified -- not absurd limit arguments (have them see the &quot;clever zero&quot; somewhere else). Please see Dan Teague's resources from the North Carolina School of Mathematics or comment if you want me to post some resources for those things. Taylor polynomials naturally flow from this discussion, because you might be curious as to which functions accept a &quot;locally quadratic&quot; approximation. Why stop there?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that you've justified and played with getting ever better Taylor Polynomials and approximations for all sorts of practical purposes (including discussion of error!), one begins to conjecture what happens as the number of terms goes to infinity and bam! Convergence is necessary. Then you step back and talk about series convergence so that way you can attack all of those lovely Taylor Series with justification for their convergence to functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I 100000% disagree with just randomly starting with infinite series. That is not logical and is completely disconnected. It's a tool for justifying things you really want. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could also find other reasons to talk about infinite series (not on the AP test) such as the Basel problem of Euler which gives us the lovely &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I HIGHLY recommend looking at the pdf here: &lt;a href=&quot;http://venus.unive.it/pasinigi/teaching/Baselprob.pdf&quot; rel=&quot;noreferrer&quot;&gt;Basel Problem&lt;/a&gt; and having your class go through the document as part of the review for the AP exam. Maybe split them up into groups and have each group present on a section or something? If there isn't enough time, go over it together as a comprehensive, single problem (going through its historical development in sections 1-4 of the pdf) that requires many of the tools from the latter part of the BC curriculum.  The exercises are great review problems!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Section 5 is short and after the firepower of section 4, one looks at &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^3}$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;thinking awesome! I have the tools to attack this. Wait. No I don't! Mathematics isn't dead --- no one knows how to do this, and it's been centuries since Euler solved the rather similar problem! A good way to end a BC calc course in my mind, emphasizing there is a lot of math to be learned, because it's a vibrant, living subject where a simple change in familiar, solved problems results in something unknown.&lt;/p&gt;&#xA;" OwnerUserId="1249" LastEditorUserId="1249" LastEditDate="2016-04-07T05:43:15.193" LastActivityDate="2016-04-07T05:43:15.193" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10841" PostTypeId="1" AcceptedAnswerId="10867" CreationDate="2016-04-08T00:12:30.770" Score="9" ViewCount="397" Body="&lt;p&gt;This question is very similar to one I posed two years ago: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7324/how-to-assign-grades-to-proofs-what-does-the-literature-experts-suggest&quot;&gt; How to assign grades to proofs: what do(es) theliterature/experts suggest?&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to ask the more general question of: what do you actually do? Specifically, what do you look for from your math majors in an undergraduate intro-to-proof course? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Closely related: Do you have (graduate or undergraduate) homework graders for that course? What do you tell them? &lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-04-13T08:35:05.617" Title="How do you assign a grade to a proof?" Tags="|undergraduate-education|proofs|grading|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10842" PostTypeId="1" AcceptedAnswerId="10844" CreationDate="2016-04-08T00:20:25.607" Score="14" ViewCount="281" Body="&lt;p&gt;I am teaching an intro to proof course for undergraduate math majors at a medium-sized american research university. I would like to provide my students with some incorrect proofs for the purpose of having them critique or correct the work of others. Can anyone recommend references or resources? &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2017-03-16T02:28:02.683" Title="Resource request: incorrect &quot;proofs&quot; for undergrads to correct/critique" Tags="|undergraduate-education|reference-request|proofs|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10843" PostTypeId="2" ParentId="10832" CreationDate="2016-04-08T04:26:56.207" Score="4" Body="&lt;p&gt;&quot;For an arbitrary x&quot;: in mathematics, usually &quot;for all x&quot;;in English, usually &quot;for a given x&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1573" LastActivityDate="2016-04-08T04:26:56.207" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10844" PostTypeId="2" ParentId="10842" CreationDate="2016-04-08T13:04:23.277" Score="5" Body="&lt;p&gt;&lt;em&gt;A Transition to Advanced Mathematics&lt;/em&gt; by Smith, Eggen, and Andre contains numerous (usually between five and fifteen) such exercises at the end of each exercise set. Since it is a regular feature of these exercise sets, topics in these proof exercises span properties of numbers, set theory, functions, groups, rings &amp;amp; fields, and even the completeness of the real numbers. The book contains hundreds.&lt;/p&gt;&#xA;" OwnerUserId="5690" LastActivityDate="2016-04-08T13:04:23.277" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10845" PostTypeId="2" ParentId="10832" CreationDate="2016-04-08T14:08:35.970" Score="5" Body="&lt;p&gt;I'm going with this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/afsrx.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/afsrx.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When dealing with teens, someone is bound to make a remark regarding asymptotes since its pronunciation starts with ass. This one, on the other hand, inevitably causes a bit of uproar. Clearly a different meaning than expected. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-04-08T14:08:35.970" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10846" PostTypeId="2" ParentId="10832" CreationDate="2016-04-09T00:17:45.950" Score="19" Body="&lt;p&gt;This is a problem for some English language learners:  The triangle on the left is also a right triangle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/L0I9x.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/L0I9x.png&quot; alt=&quot;left triangle, right triangle&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-04-09T00:17:45.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10847" PostTypeId="2" ParentId="10830" CreationDate="2016-04-09T06:37:58.517" Score="4" Body="&lt;p&gt;My comment is: what are you aiming to achieve with these notes? My instinctive reaction is that you are doing too much and not getting the student to do enough. Writing out a full exposition is fine if the notes are for reference or you only care about memorisation, but won't help that much if the aim is to learn to do mathematics.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-04-09T06:37:58.517" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10848" PostTypeId="2" ParentId="10537" CreationDate="2016-04-09T07:46:43.073" Score="3" Body="&lt;p&gt;Reasons why drawing precise pictures is important:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students need to learn how to draw with ruler and compass, as it is a necessary (or at least useful) skill in other subjects and in the world outside school. As mentioned by marco trevi, precise drawings are also important for geometrical constructions. The teacher should be an example.&lt;/li&gt;&#xA;&lt;li&gt;Students need to learn how to measure distances and angles. A good skill to learn is to check one's answers, and often a given angle can both be measured and calculated through other means. Precise drawings and pictures make this possible.&lt;/li&gt;&#xA;&lt;li&gt;Deciphering unclear pictures adds more cognitive load to students, who might be struggling with the notation, the concepts and the calculations, which are very important.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2016-04-09T07:46:43.073" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10849" PostTypeId="2" ParentId="10537" CreationDate="2016-04-09T16:33:46.407" Score="3" Body="&lt;p&gt;I think Jessica B's answer is spot-on (and it relates with me strongly with respect to my research in geometry), but I would like to add something in that direction, which I get from Anne Parreau who was teaching math to future primary schools teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;There is no such thing as an exact drawing.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are precise drawing, there are enlightening drawings, there are beautiful drawings, but all of them cheat. They cheat a lot more than one expects before doing the following experiment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask your student to draw a triangle with sides 3, 5 and 8 and observe. There should be three categories: those who say that it is not possible because two circles won't meet; those who will draw a very flat, but non-degenerate triangle, and those who will draw a flat triangle by cheating (i.e. not using fairly the rule and compass, but using their geometric &lt;em&gt;knowledge&lt;/em&gt; to know what they should see, and then adapting  there drawing to fit that knowledge). If you get even a single student who draws a fair (inexact) picture and says that it is not what should appear and ask for help, then you should congratulate him a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another one (again mentioned to me by Parreau): have your student draw a triangle with sides 3, 4 and 6, say, and then have them measure the angles and compute the sum. You should observe that many of them will change their measurement a posteriori to fit what they &lt;em&gt;know&lt;/em&gt;, that the sum should be 180°. If you get some that discuss imprecisions or even just wonder why this is happening, you are blessed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me finally add a personal touch. The previous example could be done with a triangle formed on the surface of earth by three cities, on a globe, realized by strings. You would need as large a globe as you can get to choose a not too large triangle, so that the sum of angles is definitely not 180° but not too far. Then you have some ground to discuss the assumptions of geometry and mathematical theorems in general, the questions of precisions, and why we want proofs rather than only measurement.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-04-09T16:33:46.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10850" PostTypeId="2" ParentId="10537" CreationDate="2016-04-09T21:19:26.760" Score="3" Body="&lt;p&gt;&lt;strong&gt;First&lt;/strong&gt;. Since you have naturally come to the question you asked, this paper of Fischbein, &lt;a href=&quot;http://www.google.co.uk/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=2&amp;amp;ved=0ahUKEwil6vfptoLMAhWGFiwKHcQUAHkQFggkMAE&amp;amp;url=http%3A%2F%2Fweb.math.unifi.it%2Fusers%2Fdolcetti%2FFischbein.pdf&amp;amp;usg=AFQjCNH26dUfgNZgMLHrtDQFs9-wUZWyHw&amp;amp;sig2=7DXZ771SQeEi8cmsQDMcZA&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Theory of Figural Concepts&lt;/a&gt; is a must read. Here is the abstract of the paper.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;ABSTRACT&lt;/strong&gt;. The main thesis of the present paper is that geometry deals with mental entities (the so-called geometrical figures) which&#xA;possess simultaneously conceptual and figural characters. A&#xA;geometrical sphere, for instance, is an abstract ideal, formally&#xA;determinable entity, like every genuine concept. At the same time, it&#xA;possesses figural properties, first of all a certain shape. The&#xA;ideality, the absolute perfection of a geometrical sphere cannot be&#xA;found in reality. In this symbiosis between concept and figure, as it&#xA;is revealed in geometrical entities, it is the image component which&#xA;stimulates new directions of thought, but there are the logical,&#xA;conceptual constraints which control the formal rigour of the process.&#xA;We have called the geometrical figures figural concepts because of&#xA;their double nature. The paper analyzes the internal tensions which&#xA;may appear in figural concepts because of this double nature,&#xA;development aspects and didactical implications.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Second&lt;/strong&gt;. I guess the reason that you couldn't find what you have suggested in your answer in the answers of others is that your question is not clear enough. In particular, the second paragraph and the third paragraph lead the reader in two quite different directions. In a way, most answers chose to react to your third paragraph. But, in fact, if we just think of your second paragraph, the answer would be &amp;quot;&lt;strong&gt;Yes,  it important and useful to make drawings exact&lt;/strong&gt;&amp;quot; If by drawing you mean: &amp;quot; Draw circles with a compass, make right-angled triangles actually right-angled by using a suitable device when drawing them, and so on.&amp;quot; I believe the most important things that my daughter ,who is a student of fashion now, learned from her geometry lessons were exactly the drawings you exemplified. And of course, my daughter is just one example among many similar ones.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Third&lt;/strong&gt;. There is a very important difference between drawing, say a circle, by hand or by a compass. Drawing by free hand is more or less based on the gestalt understanding of a circle (i.e., circle as a round object), while by a compass is based on an understanding of a circle as a locus. Thus, the answer to your question relies on the level of geometric thinking your students are. If they are at a gestalt level (i.e., seeing geometric objects as a whole without noticing the details), drawing by tools could make them aware of details. If they have already passed that stage, moving into a stage in which they need to prove something, exact drawing by tools would have limited applicability (as it is already pointed in other answers).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;ps&lt;/strong&gt;. The third point is based on my understanding of &amp;quot;The van Hiele Levels of Geometric Thought&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-04-10T10:08:43.417" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10854" PostTypeId="1" AcceptedAnswerId="10859" CreationDate="2016-04-10T17:18:25.473" Score="48" ViewCount="12914" Body="&lt;p&gt;It's the time of year for prospective college freshman in the US to make campus visits, and I'm once again confronted with my lamentable ignorance when the students and their parents ask, &quot;So what do math majors do after graduation?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A web search easily turns up dozens of pages talking about what you &lt;strong&gt;can&lt;/strong&gt; do with a math major, but here I'm more interested in practice than theory.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does anyone know of any statistics about what jobs/careers undergraduate math majors actually &lt;strong&gt;do&lt;/strong&gt; go into?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Before anyone else suggests it, it would certainly also be useful for me to know what the math majors at my university specifically do after graduation; I don't know whether anyone here collects data on such things, but I plan to look into it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt; I don't see how I could have been any clearer, but many of the answers below suggest that some people haven't understood my question.  I am &lt;strong&gt;not&lt;/strong&gt; asking for information about what people &lt;strong&gt;can&lt;/strong&gt; do with a math major, nor for anecdotes about what any specific individuals have done.  What I want is &lt;strong&gt;statistics&lt;/strong&gt; about which jobs math majors actually &lt;strong&gt;do&lt;/strong&gt; go into, and at what rates.  Jessica B's answer is exactly the kind of thing I'm looking for, but since I'm in the US, something comparable for US math majors would be even better.&lt;/p&gt;&#xA;" OwnerUserId="129" LastEditorUserId="129" LastEditDate="2016-04-12T12:22:43.863" LastActivityDate="2016-04-14T16:40:25.537" Title="What do math majors (actually) do after graduation?" Tags="|undergraduate-education|" AnswerCount="8" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="10855" PostTypeId="2" ParentId="10854" CreationDate="2016-04-10T17:41:24.493" Score="8" Body="&lt;p&gt;High school students may well ask the same thing...   One recommendation is to have available (in your classroom, or in the school library) books like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.maa.org/press/books/101-careers-in-mathematics&quot;&gt;101 Careers in Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2016-04-10T17:41:24.493" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10856" PostTypeId="2" ParentId="10854" CreationDate="2016-04-10T19:46:48.003" Score="10" Body="&lt;p&gt;Some resources, which also have further links.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://stats.bls.gov/ooh/Math/Mathematicians.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bureau of Labor Statistics Occupational Outlook Handbook (Mathematicians)&lt;/a&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.bls.gov/ooh/math/mathematicians.htm#tab-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;&lt;a href=&quot;http://www.bls.gov/ooh/math/mathematicians.htm#tab-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Work Environment&lt;/a&gt;&quot;&lt;/a&gt; for professional mathematicians  (&lt;a href=&quot;https://matheducators.stackexchange.com/questions/10854/what-do-math-majors-actually-do-after-graduation/10856?noredirect=1#comment23930_10856&quot;&gt;J W's&lt;/a&gt; link)&lt;/li&gt;&#xA;&lt;li&gt;Careers in Math/CS: &lt;a href=&quot;http://www.bls.gov/careeroutlook/2013/summer/art03.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;College to career:&#xA;Projected job openings in occupations that typically require a bachelor’s degree&#xA;&quot;&lt;/a&gt; (2013), p. 38.&lt;/li&gt;&#xA;&lt;li&gt;Stats on Math/phys. sci. majors: &lt;a href=&quot;http://www.bls.gov/careeroutlook/2008/summer/art02.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;The class of 1993: Earnings and occupations&#xA;by college major,&#xA;1 and 10 years after graduation&quot;&lt;/a&gt; (2008), p. 22.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/yA1QL.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yA1QL.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.maa.org/careers&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA Careers&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.ams.org/profession/career-info/career-index&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMS Career Information&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.ams.org/programs/students/students/&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMS for HS/College students&lt;/a&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;has a dead link to &lt;a href=&quot;http://www.ams.org/profession/data/other-sources/what-mathdegree.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;What can I do with a math degree?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.siam.org/careers/thinking.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;SIAM Career in Applied Mathematics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.amstat.org/careers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;ASA Career Center&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.beanactuary.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Be an Actuary&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A book: &lt;a href=&quot;https://books.google.com/books?id=N4nmGjq5dWsC&quot; rel=&quot;nofollow noreferrer&quot;&gt;She Does Math!&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="-1" LastEditorDisplayName="user173" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-04-13T12:52:18.727" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10858" PostTypeId="2" ParentId="10832" CreationDate="2016-04-11T05:12:53.867" Score="10" Body="&lt;p&gt;The term &quot;statistically significant&quot; is significant in this regard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the previous sentence, the first example of &quot;significant&quot; means &quot;unlikely to have occurred by chance&quot;, and the second denotes importance.&lt;/p&gt;&#xA;" OwnerUserId="6477" LastEditorUserId="1147" LastEditDate="2019-01-09T02:03:35.833" LastActivityDate="2019-01-09T02:03:35.833" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10859" PostTypeId="2" ParentId="10854" CreationDate="2016-04-11T10:48:33.490" Score="31" Body="&lt;p&gt;From the UK:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.prospects.ac.uk/careers-advice/what-can-i-do-with-my-degree/mathematics&quot; rel=&quot;noreferrer&quot;&gt;Prospects&lt;/a&gt; offer some very basic stats.&#xA;This &lt;a href=&quot;http://www.cms.ac.uk/files/News/article_5630c69e789971.96989222.pdf&quot; rel=&quot;noreferrer&quot;&gt;CMS&lt;/a&gt; report has some graphs broken down a bit further, from p36 onwards.  For instance, the graph below shows that of majors in math, statistics or operations research, about half went into education and financial services.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/djJti.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/djJti.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="4746" LastEditDate="2016-04-11T17:40:53.100" LastActivityDate="2016-04-11T17:40:53.100" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10860" PostTypeId="2" ParentId="10832" CreationDate="2016-04-11T19:56:12.690" Score="3" Body="&lt;p&gt;&lt;strong&gt;factor&lt;/strong&gt; -- &quot;this is a factor of the polynomial&quot; versus &quot;this is a factor that contributes to poverty&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next few are more about how these are used by mathematicians (rather than having a technical meaning): &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;obvious&lt;/strong&gt; -- &quot;you can figure it out ten minutes and pencil&quot; versus &quot;everyone would immediately agree this is the case&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;trivial&lt;/strong&gt; -- see above&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;intuitive&lt;/strong&gt; -- &quot;if you've taken the standard math sequence, this should make sense&quot; versus actual intuition&lt;/p&gt;&#xA;" OwnerUserId="6605" LastActivityDate="2016-04-11T19:56:12.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10861" PostTypeId="2" ParentId="10854" CreationDate="2016-04-11T22:14:20.900" Score="3" Body="&lt;p&gt;Earned a BS in Mathematics with CS minor from a Liberal Arts University.  I was interested in a career in software development.  I found in my interviews that my degree in Mathematics was a competitive edge over CS Majors due to the difficulty and rigor.  Any deficiency in my CS background would be made up with on-the-job experience and training.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have to say that there hasn't been a need for my experience in Real or Complex Analysis in my 16 years of professional experience, but my degree gave me a competitive edge, a boost at the beginning of my career, that has paid off very well over the years. &lt;/p&gt;&#xA;" OwnerUserId="6607" LastActivityDate="2016-04-11T22:14:20.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10862" PostTypeId="2" ParentId="10854" CreationDate="2016-04-12T05:41:31.700" Score="4" Body="&lt;h2&gt;&lt;strong&gt;Web developers!&lt;/strong&gt;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Having a PhD in applied mathematics and having struggled quite a lot just last year in order to find my first job outside of the academia (guess what), I felt somehow compelled to post this answer. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me add some context and background to back it up. First of all, what follows is purely based on personal experience. However, I believe that having spent 10 years in the &quot;math environment&quot;, hunted for a job in the industry, and eventually found one, should give enough credit to what I'll say.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Notes:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I have been doing research/working in Japan for the past 4 years, but I think this should not be a big discriminant.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I will focus on the private sector, that is, I will not consider research centers etc. that are also an obvious career possibility. I also won't consider the education sector (public or private) in general.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Anyway, what I realized during my job-hunting quest as a mathematician is that the majority of jobs for what you will be actually considered (in my case meaning that I had at least one interview) comes from the following type of industries: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;IT;&lt;/li&gt;&#xA;&lt;li&gt;Insurance;&lt;/li&gt;&#xA;&lt;li&gt;Finance;&lt;/li&gt;&#xA;&lt;li&gt;Pharmaceutical;&lt;/li&gt;&#xA;&lt;li&gt;Consulting.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I would say that more or less that's it. In most cases though, whatever the type of industry, most of the employers will be interested in two things when looking at a mathematician: programming and problem solving (mindset) skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only area in which I honestly have the feeling that some serious mathematics is actually employed on a regular basis is finance (you can google for &quot;quant jobs&quot; and find tons of information). I had a few quant interviews and I was actually asked quite high-level purely mathematical questions. Those times only. Even in the quant case though, often programming skills are essential... At this point you should be able to start seeing where my original answer is coming from.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, I have started working in a Fintech company, where eventually I might join a quantitative team and at least have the feeling of doing a job that is  related to my background. For now, I am working as a web developer doing web applications. The level of mathematics I use is zero, and math comes into play only in the sense that it helps when it comes to logical thinking (that is obviously at the core of programming). So although there is an obvious pinch of irony in my first answer, there is at the same time a little bit of disappointment. The truth is that the business world and mathematics are still quite far from each other, despite what some will say. At least in the majority of cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even among friends or former colleagues, if I think about people that now work in the industry they all belong to one of the above categories (many of them not doing or using any math at all). So I guess it is quite true in the end, you indeed find math majors in a variety of fields.  It is how you get there to be tricky.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Off topic:&lt;/strong&gt; The above was just my personal experience to give a small insight of what math majors do after graduation.&#xA;However, I would like to add that what I have learned, in the end, is that the most helpful thing a math major could do to himself/herself is to understand in advance what he/she wants to do next and work his/her math in that direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I never really given too much thought to that and when I decided to drop research I realized the hard way that finding a job was much more difficult than I expected. Definitely having a PhD and research experience was a double edged sword for many reasons, but this is another story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, I would boil it all down to this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If you love math as a subject and that's what you want to do, I&#xA;hardly see any career other than the standard academic one.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you want to find a regular job in a company, it definitely helps to&#xA;   improve coding skills focusing on maybe numerical analysis and/or&#xA;   applied mathematics. And be prepared not to see any math at all&#xA;   again.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When it comes to money, maybe the most financially rewarding career&#xA;   for a mathematician is still in the financial field working as a&#xA;   quant. They are notoriously tough jobs to get, but knowing in&#xA;   advance that that's your goal might be an advantage so that you can&#xA;   for example drop that algebraic topology class and attend stochastic&#xA;   calculus instead.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6612" LastEditorUserId="6612" LastEditDate="2016-04-12T23:15:08.173" LastActivityDate="2016-04-12T23:15:08.173" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10863" PostTypeId="2" ParentId="10854" CreationDate="2016-04-12T12:04:40.240" Score="2" Body="&lt;p&gt;My brother is a Math major, from the Old Dominion University, Virginia and now he is in the strategy management department in the US navy.&#xA;Math can get you anywhere in any field because as long as you have a good logical mind, you will be welcomed anywhere.&#xA;The organizations look for specialisations in a particular field so that the one who they are hiring doesn't need a lot of time to come in pace but in the long shot, everyone kinda know that the people with Math majors can pay off better that the others. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This &lt;a href=&quot;https://www.math.uci.edu/math-majors/math-career-resources#sec%201.1&quot; rel=&quot;nofollow&quot;&gt;list&lt;/a&gt; provided by UCI shows various areas you can think to work in.&lt;/p&gt;&#xA;" OwnerUserId="6613" LastEditorUserId="2074" LastEditDate="2016-04-12T13:35:14.400" LastActivityDate="2016-04-12T13:35:14.400" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10864" PostTypeId="2" ParentId="10854" CreationDate="2016-04-12T16:53:44.137" Score="4" Body="&lt;p&gt;No one mentioned NSA, which reportedly employs more mathematicians than anyone else in the US.  They're also Baltimore Gas and Electric's biggest customer.&lt;/p&gt;&#xA;" OwnerUserId="6617" LastActivityDate="2016-04-12T16:53:44.137" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10865" PostTypeId="2" ParentId="10842" CreationDate="2016-04-12T23:50:16.963" Score="4" Body="&lt;p&gt;There are some &quot;evaluation of proof&quot; questions in Sundstrom's book, &lt;a href=&quot;http://scholarworks.gvsu.edu/books/7/&quot; rel=&quot;nofollow&quot;&gt;Mathematical Reasoning: Writing and Proof&lt;/a&gt;, typically at the end of each list of exercises (starting at 3.1, when proofs take centre stage). &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2016-04-12T23:50:16.963" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10866" PostTypeId="1" CreationDate="2016-04-13T07:12:17.060" Score="12" ViewCount="234" Body="&lt;p&gt;There are several styles of delivering a lecture:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The lecturer uses the blackboard and the students write their own lecture notes.&lt;/li&gt;&#xA;&lt;li&gt;The lecturer uses the blackboard and also provide a script so that the students are not forced to create their own script (but they can when they wish).&lt;/li&gt;&#xA;&lt;li&gt;The lecturer uses a slide show and uploads it to his homepage. In those lectures the student normally doesn't create his own script.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Each lecture style differs in the question, whether the students have the possibility to write their own script or whether they are forced to do so. How do those ways to lecture influence the learning outcome?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make my question concrete: Is there any research investigating how important the act of writing his own script during the lecture is for the learning outcome? Is it advisable to give the students the possibility to write their own lecture notes or should they forced to do so?&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2017-11-01T21:00:19.643" Title="How important is the act of writing his own script for the learning outcome?" Tags="|undergraduate-education|reference-request|course-design|writing|lecture-notes|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10867" PostTypeId="2" ParentId="10841" CreationDate="2016-04-13T08:35:05.617" Score="11" Body="&lt;p&gt;In France we barely have intro-to-proof courses, but we ask for proofs in other courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually, each proof has little granularity in the grading, and I tend to avoid giving half the points which sends a mixed signal, so most of the time small proofs get either zero, one-third, two-third or full credit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically I try to give one-third credit when the core argument is not present in a reasonable form, but some good idea is, without major issues; and two-third credit when the core argument is well written but there are minor, significant issues. Full credit is for clean, proper and correct reasoning with a level of details suited to the level of the course, and zero credit for everything else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My most constant guidelines are :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;zero credit when there is a clear bluff (e.g. student cites a dozen properties and conclude one can apply some theorem: even if all hypotheses of the theorem belong to the cited properties, but not more prominently than irrelevant ones, I call a bluff),&lt;/li&gt;&#xA;&lt;li&gt;no more than one-third credit, and in some cases zero credit when a huge misconception or mistake, central to the course or previous courses, is present (e.g. &quot;$A$ is not closed, therefore it is an open set&quot; or &quot;the function $f$ is not non-decreasing, thus it is non-increasing&quot; -- here the English wording for monotonicity shows all its badness, and the same sentence looks far more egregious in French),&lt;/li&gt;&#xA;&lt;li&gt;no more than two-third credit if the writing is insufficient (e.g. computations not introduced by any word or assertions not clearly indicated whether they are being claimed, denied, or assumed),&lt;/li&gt;&#xA;&lt;li&gt;no more than two-third credit if there are insufficient justifications, that could be present in the student's mind but is not recalled in the paper (the flexibility depend on the course's level, e.g. failing to mention why a quantity that is used as denominator does not vanish); no more than one-third credit if they are clearly absent from the student's mind (e.g. divides by something that can vanish, without distinguishing cases).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-04-13T08:35:05.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10869" PostTypeId="2" ParentId="10854" CreationDate="2016-04-14T16:40:25.537" Score="4" Body="&lt;p&gt;At  &lt;a href=&quot;http://ncsesdata.nsf.gov/sestat/sestat.html&quot; rel=&quot;nofollow&quot;&gt;http://ncsesdata.nsf.gov/sestat/sestat.html&lt;/a&gt; you can query the National Science Foundation's database for the National Survey of College Graduates.  For example, it indicates that only about 1% of those graduating with undergraduate degrees in mathematics end up with &quot;mathematician&quot; as their job title.&lt;/p&gt;&#xA;" OwnerUserId="6625" LastActivityDate="2016-04-14T16:40:25.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10870" PostTypeId="2" ParentId="10832" CreationDate="2016-04-15T11:06:10.430" Score="13" Body="&lt;p&gt;&quot;In general&quot; or &quot;generally&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematics, if we say a specific result holds in general, we mean there are no exceptions to the result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In every-day non mathematical discussions, if someone makes a claim and says it is true in general, they mean it is true most of the time but with possibly a few exceptional cases. Exactly the opposite of the mathematical meaning!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematicians often use &quot;generically&quot; to mean essentially what nonmathematicians mean by &quot;generally&quot;.&lt;/p&gt;&#xA;" OwnerUserId="6626" LastActivityDate="2016-04-15T11:06:10.430" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10871" PostTypeId="2" ParentId="1498" CreationDate="2016-04-15T12:00:47.267" Score="3" Body="&lt;p&gt;I would answer it with a story:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My aunt whom I stay with is a Math lecturer for last 21 years and she tells me things happening everyday. From all that, what I could collect is this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The starting respect: She says &quot;Compared to any other, the students have a respect for you from the starting day as helloooo- you are eligible enough to teach a subject which most people are afraid of.&quot;&lt;/li&gt;&#xA;&lt;li&gt;Keeping it interactive: As the subject is a bit dry, she always cracks jokes, smiles a lot and not lazy to answer something more than once. She thinks that the students need a bit of an extra care to keep them active during the class hours.&lt;/li&gt;&#xA;&lt;li&gt;Tough moments: As we are not god to know everything, if a questions stays unattended and the students give a strange look of &quot;who made this person a teacher&quot; she always came back the next day to give a fantastic answer in such a way that all the trust you lost regains.&lt;/li&gt;&#xA;&lt;li&gt;Emotions: She says &quot;Dont expect too much that you will get response from your students after they pass out. As they will choose a particular stream, they will remember the faculties of that stream more than you as yours was a base subject. But there will be few who who will understand that without you they could not be anything in life and when you hear from them, it'll make your day.&#xA;The response from these people will be genuine than the others as only few understand what an important part you have played in your life. Math is the base for all and we always need to know it.&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;From this, I hope you would get a vibe of the answer you were looking for.&lt;/p&gt;&#xA;" OwnerUserId="6613" LastEditorUserId="2074" LastEditDate="2016-04-15T14:02:44.953" LastActivityDate="2016-04-15T14:02:44.953" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10873" PostTypeId="2" ParentId="1498" CreationDate="2016-04-15T23:53:54.863" Score="11" Body="&lt;p&gt;Since one point was not made very forcefully in the other answers: I like teaching mathematics because, even on the worst days, I get to talk/think/engage about &lt;em&gt;mathematics&lt;/em&gt;... which I somehow find endlessly entertaining. &lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-04-15T23:53:54.863" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10875" PostTypeId="1" CreationDate="2016-04-16T08:48:23.750" Score="4" ViewCount="143" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;We know that $\dfrac{(x^2+1)x^3}{x^2+1}=x^3$ and $(\sqrt{|x|})^4=x^2$ for every $x\in \mathbb{R}$. Can $\dfrac{(x^2+1)x^3}{x^2+1}$ and $(\sqrt{|x|})^4$ be called polynomials?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there a general name referring to a function which are identical to a polynomial on a proper subset of $\mathbb R$  (e.g. $\dfrac{x^2-1}{x+1}$ which is identical to $x-1$ if $x \in \mathbb{R}-\{0,-1\}$)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2016-04-16T19:20:20.493" Title="In Polynomial Form, After Simplification (But Not Before!)" Tags="|definitions|functions|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10876" PostTypeId="1" CreationDate="2016-04-16T12:06:50.517" Score="20" ViewCount="467" Body="&lt;p&gt;In teaching undergraduate mathematics, I implemented some strategies to encourage the students to look at errors they made or at &quot;typical errors&quot; in the current topic. One attempt was to compile a commented list of errors which appeared in the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the moment I am looking for literature on learning by analysing errors in undergraduate mathematics education. I found a lot of literature focused on mathematics education at school but very few on undergraduates. I would be especially interested in evaluations of teaching strategies.&lt;/p&gt;&#xA;" OwnerUserId="6002" LastEditorUserId="77" LastEditDate="2016-08-21T02:39:42.513" LastActivityDate="2016-08-22T03:33:11.990" Title="Literature on learning from errors in mathematics" Tags="|reference-request|students-mistakes|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10877" PostTypeId="1" AcceptedAnswerId="10878" CreationDate="2016-04-16T12:23:31.020" Score="10" ViewCount="305" Body="&lt;p&gt;Is there a literature database like &lt;a href=&quot;http://www.ams.org/MathSciNet/index.html&quot;&gt;MathSciNet&lt;/a&gt; for mathematics education?&lt;/p&gt;&#xA;" OwnerUserId="6002" LastEditorUserId="6002" LastEditDate="2016-04-19T08:24:14.273" LastActivityDate="2020-07-09T10:46:08.937" Title="Is there a literature database like MathSciNet for Mathematics Education?" Tags="|education-research|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10878" PostTypeId="2" ParentId="10877" CreationDate="2016-04-16T12:36:23.337" Score="12" Body="&lt;p&gt;Yes, there is one called &lt;a href=&quot;https://www.zentralblatt-math.org/matheduc/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathEduc&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Quote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;MathEduc (formerly MATHDI) is the only international reference database offering a world-wide overview of literature on research, theory and practice in mathematics education. MathEduc also covers education in computer science on the elementary level. The scope includes literature for all school levels up to university education, teacher education, general and vocational education. About 500 journals from all over the world are evaluated as well as books and teaching material.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Update 09 July 2020:&lt;/p&gt;&#xA;&lt;p&gt;There is now the following message at the linked page.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;MathEduc no longer available&lt;/strong&gt;.&#xA;At this address, the MathEduc database, published by FIZ Karlsruhe and the German Society for Didactics of Mathematics (GDM), has been available until December 31, 2019.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="39" LastEditorUserId="39" LastEditDate="2020-07-09T10:46:08.937" LastActivityDate="2020-07-09T10:46:08.937" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="10879" PostTypeId="2" ParentId="10876" CreationDate="2016-04-16T13:55:58.430" Score="10" Body="&lt;p&gt;You might be interested in &lt;em&gt;systematic errors&lt;/em&gt;.  From page 14 of Kurt VanLehn's &quot;Mind Bugs: The Origins of Procedural Misconceptions&quot; (MIT Press, 1990):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Systematic errors appear to stem from consistent application of a faulty method, algorithm, or rule.  Slips are unsystematic &quot;careless&quot; errors (for example, fact errors, such as 7-3=5).  Because slips occur in expert performance as well as student behavior, the common view is that they are due to inherent &quot;noise&quot; in the human information processor.  Systematic errors on the other hand are taken as stemming from mistaken or missing knowledge, the product of incomplete or misguided learning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(An example of a systematic error is writing the remainder of a division of two integers as a decimal fraction, such as saying that $17\div 5=3.2$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many researchers have studied these.  A few are John Seely Brown, Richard R. Burton, and Kurt VanLehn.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-04-16T13:55:58.430" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10880" PostTypeId="2" ParentId="10875" CreationDate="2016-04-16T18:09:53.327" Score="11" Body="&lt;p&gt;I think this is a great question because it is really about the ambiguous/confusing way we use the word polynomial.  So I want to use the terms &quot;polynomial&quot; and &quot;polynomial function&quot; separately here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &lt;em&gt;polynomial&lt;/em&gt; is an algebraic expression.  It consists of variables and coefficients and only uses the operations of addition, subtraction, multiplication, and non-negative integer exponents.  That is &lt;a href=&quot;https://en.wikipedia.org/wiki/Polynomial&quot;&gt;Wikipedia's&lt;/a&gt; definition and is in the plainest language that I found, but it matches definitions found in undergraduate algebra textbooks like &lt;a href=&quot;http://www.springer.com/us/book/9780387220253&quot;&gt;Lang&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &lt;em&gt;polynomial function&lt;/em&gt; is a function that can be defined by evaluating a polynomial.  It &lt;strong&gt;can&lt;/strong&gt; be defined that way; it doesn't have to.  However, you pick up any precalculus or calculus textbook and this is not what you'll find.  For example, in Stewart's Calculus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A function $P$ is called a &lt;em&gt;polynomial&lt;/em&gt; if $P(x) = a_nx^n + ... + a_0$ where $n$ is a nonnegative integer and the numbers $a_i$ are called the &lt;em&gt;coefficients&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So he uses the term differently.  However, this definition is effectively the same as Wikipedia's definition of polynomial function if we think about what it means to be equal as functions.  Two functions are equal if $f(x) = g(x)$ for every $x$ in the domain of $f$ and $g$.  The domains have to match, and every value has to match.  Those functions are the same.  They are one function with two different expressions.  In your example, $f(x)=\frac{(x^2+1)x^3}{x^2+1}$ is equal to $g(x)=x^3$ for every real number, so they are both polynomial functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we're talking about algebraic expressions, on the other hand, the quality of being a polynomial doesn't have to be preserved when you alter the expression.  $\frac{(x^2+1)x^3}{x^2+1}$ is equivalent to $x^3$, but they are not the same expression.  One is a polynomial and the other is not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, a fancier example from Lang:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In some cases, it may be that two polynomials can be distinct, but give rise to the same polynomial function on a given ring.  For example, let $\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$ be the field with $p$ elements.  Then for every element $x\in\mathbb{F}_p$ we have $x^p=x$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So you have different polynomials, but the same polynomial function.&lt;/p&gt;&#xA;" OwnerUserId="4703" LastEditorUserId="4703" LastEditDate="2016-04-16T19:20:20.493" LastActivityDate="2016-04-16T19:20:20.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10881" PostTypeId="2" ParentId="1498" CreationDate="2016-04-17T13:23:56.420" Score="9" Body="&lt;p&gt;With respect to Benjamin's answer, the top reason in his graphic &quot;to make a difference.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IBG3h.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IBG3h.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I received the note above at the end of the last school year. The first few reasons listed are slight variations of this 'difference' idea. For me, it's a part time job, just 2 days a week, and so far, in 3 years, I've never had a bad day. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've met with students who are learning disabled, and when I thought I let them down, they visit again, and the week after, the teacher stops by and asks how I got the student to pass his last exam. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To the 'full potential' (no. 3 on that slide), I know that most students are not going to gain a love for math, (the slide no.2) so while I stick with that enthusiasm, to the student who comes in with the &quot;I hate math&quot; attitude, I explain that my role is to help her understand, pass, and get the best grade she's capable of. Last week, such a student finished a session with me, and said &quot;I still hate this.&quot; I asked &quot;But do you understand it better now than when you walked in, 90 minutes ago?&quot; And she answered yes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one thing missing from the graphic - to show students respect and caring. One of my jobs is to proctor missed exams. Some of the students are allowed extra time and often, if they come during the last period, I might need to stay late. Not one such student has ever taken it for granted. When the bell rings to end the day at 3:20, and at 4:00 I tell them to take their time, and be sure to check the work. &quot;My family time starts at 6 tonight, I can get caught up on my emails here with you or at my desk at home. No difference to me.&quot; I don't always know what their life is like with friends, family, etc. I just know that in my room, I can have them feel that they are important, my top priority.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Update - I was in the office area, and an English teacher (whom I hadn't yet met) introduced himself, and told me he gave his class an assignment, to write an essay on who has made a difference in your life. A student made me the topic of half her paper. I had to keep from tearing up as he described how this student felt I had impacted her life. This is why I do what I do. To make a difference. To let these students know they are important to me. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-06-02T23:30:46.750" LastActivityDate="2017-06-02T23:30:46.750" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10883" PostTypeId="2" ParentId="10832" CreationDate="2016-04-19T02:53:56.090" Score="8" Body="&lt;p&gt;For many calculus theorems, you'll need to refer to open or closed sets.  Those terms have nothing to do with the English meanings.  I've found that even native English speakers can be confused because the math terms (unlike the English terms) aren't mutually exclusive, i.e. a set can be open, closed, open and closed or neither open nor closed where a door has to be one or the other.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2016-04-19T02:53:56.090" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10885" PostTypeId="1" CreationDate="2016-04-19T07:32:00.547" Score="3" ViewCount="185" Body="&lt;p&gt;Following the premise that mathematics is an art as well as a science, I want to encourage students to produce not only correct proofs but also to try to find a particular beautiful/elegant proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apart from pointing out particularly beautiful proofs, which strategies for promoting elegant or beautiful proofs do exists? I would be particularly interested in strategies which have been evaluated.&lt;/p&gt;&#xA;" OwnerUserId="6002" LastActivityDate="2016-04-19T07:32:00.547" Title="How to promote more elegant and beautiful proofs by students?" Tags="|undergraduate-education|proofs|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10889" PostTypeId="2" ParentId="10832" CreationDate="2016-04-20T10:01:34.997" Score="5" Body="&lt;p&gt;By &lt;strong&gt;sphere&lt;/strong&gt; a mathematician means a two-dimensional surface. In everyday speech, a solid three-dimensional ball is often called a &lt;strong&gt;sphere&lt;/strong&gt;. This difference sometimes causes confusion in introductory calculus classes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By a &lt;strong&gt;manifold&lt;/strong&gt; a mathematician means an abstract notion generalizing a two-dimensional surface. In everyday speech, a &lt;strong&gt;manifold&lt;/strong&gt; means part of the intake or exhaust system on a car.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2016-05-13T08:54:52.780" LastActivityDate="2016-05-13T08:54:52.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10890" PostTypeId="1" CreationDate="2016-04-20T19:05:23.217" Score="7" ViewCount="373" Body="&lt;p&gt;It is easy to find/construct cases that can be proven by nested induction, i.e., some variation of the theme to prove the statement $P(m, n)$ you prove $P(1, n)$ by induction as a base case for $m$, and given $P(m,n)$ you prove $P(m + 1, n)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested in cases in which it is natural/simpler to define some order between pairs $(m, n)$ and the induction proof is along this order, not two nested inductions as outlined above.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2016-04-22T15:36:11.367" Title="Examples of multiple induction" Tags="|examples|induction|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10891" PostTypeId="2" ParentId="7528" CreationDate="2016-04-21T00:53:53.590" Score="4" Body="&lt;p&gt;A just-released image of a crudely&amp;mdash;but recognizably &lt;em&gt;hexagonal&lt;/em&gt;&amp;mdash;crater&#xA;(the &quot;Haulani Crater&quot;) on the dwarf planet Ceres (between Mars &amp;amp; Jupiter),&#xA;taken by the Dawn spacecraft.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/YzkS0.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YzkS0.jpg&quot; alt=&quot;HexagonalCrater&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;http://www.space.com/32621-ceres-stop-sign-crater-photos-dawn.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;One article&lt;/a&gt; says it &quot;look[s] oddly like a stop sign,&quot;&#xA;but we know stop signs (in the U.S.) are octagons.&#xA;How a physical process (asteroid collision) could result in &#xA;an approximate hexagon is (I think?) not yet understood.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cf. &lt;a href=&quot;https://matheducators.stackexchange.com/a/8543/511&quot;&gt;Saturn's north-pole hexagon&lt;/a&gt;,&#xA;which is better understood (at least conjecturally).&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-04-22T21:03:03.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10892" PostTypeId="2" ParentId="10206" CreationDate="2016-04-21T06:27:17.953" Score="3" Body="&lt;p&gt;Many elementary students have a problems with the equals sign.  They may not be able to solve for the missing number 3+2=__+1.  They see = and think &quot;the answer is&quot;. Therefore they will fill in the blank with 5, which is of course incorrect.&#xA;If you add to the difficulty of this problem with a variable 3+2=x+1, many students and teachers will think the problem is that there is an x and not realize the student just doesn't understand how equality works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;BTW this is why students often in a multi-step problem may write:&#xA;   3+2=5-2=3+17 =20.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest that you introduce variables first where they are the answer to the problem, so that you have 3(77) = x   and 3(42-4.1)=x  etc.  If the student is overwhelmed by a letter, replace with a box and then show that they are the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once the student is comfortable with x as the missing answer, go on to problems where one side of the equality is a single number and the other side is an expression containing a variable. Start with simple problems that the students can do mentally and then proceed to more complex examples. For example 3+x=5 and   x- 7 =2.   Proceed to more complex problems such as 3x+2 =8.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally make sure the students understands how equality works, by giving problems such as 3+11 = __*7.  If the student understands that, then you can proceed to replacing these examples with variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Above all be patient.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2016-04-21T06:36:53.143" LastActivityDate="2016-04-21T06:36:53.143" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10893" PostTypeId="2" ParentId="10206" CreationDate="2016-04-21T11:09:53.047" Score="3" Body="&lt;p&gt;I would start early with &quot;boxes&quot; -- just empty squares and playing &quot;what goes in the box?&quot; for simple equations. Works well for kids as young as second grade and definitely by 4th. It makes it much easier to introduce letters as variables later, with the understanding that they act like containers. And, as someone else said, you can bridge the gap by modeling story problems with known quantities (at least that's what my son gets in 4th grade). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should note, I've used boxes for adults in GED class, equally effective. They really liked &quot;what did I start with?&quot; where I'd write [BOX]+7-2+4-9+6=12 or something and have them try to work out a general method.&lt;/p&gt;&#xA;" OwnerUserId="6651" LastActivityDate="2016-04-21T11:09:53.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10895" PostTypeId="1" CreationDate="2016-04-22T08:45:10.650" Score="23" ViewCount="6485" Body="&lt;p&gt;I teach for a company along with a large group of teachers, almost all of which are people who have graduated with the standard Bachelor level education in Education and Science/Mathematics. I am currently teaching senior high school and undergraduate mathematics whilst studying towards becoming a professional mathematician.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, a very large number of my coworkers, especially the junior high school teachers, refer to themselves as 'mathematicians' to both other staff members and to students. These students go into senior high school and enter my classes, where I hope to get them interested in studying mathematics beyond the high school level. However, when talking to them about mathematics beyond high school, I often hear something along the lines of&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Being a mathematician seems a bit boring and easy. My Grade 9 teacher&#xA;  is a mathematician and it really doesn't seem like an interesting or difficult occupation. There's only so much you can learn about fractions and trigonometry.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This bothers me quite a bit, as I feel that it trivialises what mathematicians really do. It also turns students away from the field even before they get a chance to do it properly with me. Sometimes I have to resort to saying &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Teacher $X$ is not a mathematician, despite what he/she claims&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which does not feel like a very nice thing to say. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you know, a professional mathematician actually &lt;strong&gt;works&lt;/strong&gt; with mathematics for a living and at the very least have doctorates in mathematics. Many of these so-called 'mathematicians' are unable to teach senior high school, let alone award themselves such a label. You could not call a Biology teacher a 'Biologist'. You would not call an Economics teacher an &quot;Economist&quot;. Similarly, you would not call a mathematics teacher a &quot;Mathematician&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know precisely which teachers are making these claims, but I am not sure how to approach them in a nice way about this. I am getting a bit tired of having to justify and explain my career choices to these misinformed students. Should I just bite the bullet and let it go?&lt;/p&gt;&#xA;" OwnerUserId="3383" LastActivityDate="2020-03-21T14:10:01.047" Title="How do I nicely tell my coworkers that they are NOT mathematicians?" Tags="|secondary-education|" AnswerCount="8" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="10896" PostTypeId="2" ParentId="10895" CreationDate="2016-04-22T09:37:58.417" Score="39" Body="&lt;p&gt;I do not know if there is an accepted definition of what a mathematician is. There are teachers of mathematics and professors of mathematics, for example, and most people agree that people of the latter category are mathematicians while I am not sure if there is any consensus about the former category.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my suggestion would be to &lt;em&gt;not&lt;/em&gt; got into the trouble of arguing who is a mathematician and who is not but, instead answering to&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Being a mathematician seems a bit boring and easy. My Grade 9 teacher is a mathematician and it really doesn't seem like an interesting or difficult occupation. There's only so much you can learn about fractions and trigonometry.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;something like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Teaching mathematics is just one thing a mathematician can do. There are many many other things mathematicians do. If you finish a university degree in mathematics, you may enter research mathematics which is an incredibly large field. You may also work as a mathematician in companies doing very different things like data analysis, technical development, code development but also management, strategy development and other things.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="3398" LastEditDate="2018-12-23T14:49:21.323" LastActivityDate="2018-12-23T14:49:21.323" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="10898" PostTypeId="2" ParentId="10832" CreationDate="2016-04-22T17:10:44.057" Score="4" Body="&lt;p&gt;&lt;strong&gt;Strategy&lt;/strong&gt; -- in everyday usage it's a general plan of action to deal with uncertainty, whereas in game theory it's an integer sequence or an algorithm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Neighborhood&lt;/strong&gt; -- open ball vs. well, neighborhoods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To say nothing of &lt;strong&gt;Group, Ring, Field&lt;/strong&gt;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're interested in how we connect everyday language to formal mathematical terms, I highly recommend &lt;em&gt;Where Mathematics Comes From&lt;/em&gt; by Rafael Nunez and George Lakoff (2000) or even Nunez' &quot;Mathematical Idea Analysis&quot; plenary address from 1999. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, “Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.”― Johann Wolfgang von Goethe&lt;/p&gt;&#xA;" OwnerUserId="6651" LastActivityDate="2016-04-22T17:10:44.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10899" PostTypeId="2" ParentId="10895" CreationDate="2016-04-22T20:57:03.387" Score="3" Body="&lt;p&gt;You could say &quot;Well, there are high school level 'biologists,' 'mathematicians,' 'physicists,' etc. (just entertain their loose definitions).  The people doing exciting work (and maybe describe what PhD graduates are doing) are PhDs, and studied years beyond what your teachers did.&quot;&lt;/p&gt;&#xA;" OwnerUserId="6659" LastActivityDate="2016-04-22T20:57:03.387" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10900" PostTypeId="2" ParentId="10895" CreationDate="2016-04-23T07:57:13.450" Score="3" Body="&lt;p&gt;I would personally distinguish between 'professional mathematician', meeting the definition you seem to be using, and 'mathematician', as someone who thinks that way, but may or not do that as their profession. Someone might be a musician without being primarily employed as such.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think I would focus on talking about what real mathematics is with the students (and maybe also the teachers), rather than convincing the teachers to stop identifying as mathematicians. They could be mathematicians who spend most of their time teaching arithmetic instead. Or its possible they've never really encountered nice maths for themselves. Generally (in my view) too much of the population believes that mathematics is what you do at school (and there's nothing better!). So with students I'd take the line of 'you have only seen one example of things you can do with a maths degree, there are many many others'.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-04-23T07:57:13.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10903" PostTypeId="1" CreationDate="2016-04-23T15:24:21.773" Score="2" ViewCount="199" Body="&lt;p&gt;Are there any schools out there successfully using CGI with a 5th grade curriculum? Our students have poor number sense, and we'd like to remediate using &lt;a href=&quot;https://en.wikipedia.org/wiki/Cognitively_Guided_Instruction&quot; rel=&quot;nofollow&quot;&gt;CGI&lt;/a&gt;. I'd love to compare sequences of lessons with anyone involved in the same endeavor! There are some resources out there, notably, &lt;a href=&quot;http://www.heinemann.com/ChildrensMath/&quot; rel=&quot;nofollow&quot;&gt;Children' Mathematics&lt;/a&gt; (Carpenter et al), but I am more interested in sharing our first hand experiences.  &lt;/p&gt;&#xA;" OwnerUserId="6665" LastEditorUserId="1217" LastEditDate="2016-04-24T20:04:25.037" LastActivityDate="2016-04-24T20:04:25.037" Title="Cognitively Guided Instruction (CGI) for 5th Grade" Tags="|course-design|primary-education|course-sequence|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10904" PostTypeId="2" ParentId="10895" CreationDate="2016-04-23T17:52:37.477" Score="7" Body="&lt;p&gt;You probably can't change the behavior of your colleagues, for a number of reasons (one vs. many; people get defensive/entrenched; you're likely only part-time; etc.). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if this is an ongoing irritation for you, then you should be sensitive to that, and think about how you can ju-jitsu this into a learning opportunity for your students. Perhaps make this a point of discussion/assignment, and remediate it head-on early in the semester with your new students. &quot;I find that many people have incorrect impressions of what it means to be a mathematician. What do you think? How can we investigate that?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a resource: &lt;a href=&quot;http://www.ams.org/profession/career-info/math-work/math-work&quot;&gt;&quot;What Do Mathematicians Do?&quot;&lt;/a&gt;, by the AMS. The page is short; consider assigning that as a reading to your students. They point out that &quot;mathematician&quot; was identified as the #1 job in America a number of times (on &lt;a href=&quot;http://www.careercast.com/jobs-rated/jobs-rated-report-2016-ranking-200-jobs&quot;&gt;Careercast.com&lt;/a&gt;, where it is defined as &quot;Conducts research to develop and understand mathematical principles&quot;; currently #6, with the top #1-2 jobs now being &quot;data scientist&quot; and &quot;statistician&quot;). The AMS page also says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Many people are familiar with mathematicians in academia, but&#xA;  mathematicians also work in many other fields, including:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Astronomy and space exploration&lt;/li&gt;&#xA;  &lt;li&gt;Climate study&lt;/li&gt;&#xA;  &lt;li&gt;Medicine&lt;/li&gt;&#xA;  &lt;li&gt;National security&lt;/li&gt;&#xA;  &lt;li&gt;Robotics&lt;/li&gt;&#xA;  &lt;li&gt;Animated films&lt;/li&gt;&#xA;  &lt;li&gt;and in a wide range of businesses...&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The AMS page emphasizes members who have advanced graduate degrees like the doctorate (likely not your junior-high school colleagues):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How Many Mathematicians Are There in the U.S.?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;There are over 35,800 individual members of the four leading&#xA;  professional mathematical sciences societies in the U.S.---the AMS,&#xA;  the Association for Women in Mathematics, the Mathematical Association&#xA;  of America, and the Society for Industrial and Applied Mathematics.&#xA;  Most would call themselves mathematicians; &lt;strong&gt;many received their&#xA;  doctoral degrees outside the U.S.&lt;/strong&gt; There are at least 10,000 more&#xA;  members of the societies who are graduate students or in other&#xA;  categories, and there are also mathematicians who are not members of&#xA;  any of these societies.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Although they have advanced degrees in mathematics, many of those&#xA;  employed in academia might call themselves professors instead of&#xA;  mathematicians&lt;/strong&gt;, and similarly, those in industry and government may&#xA;  not have &quot;mathematician&quot; in their job title...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that the title of the AMS page is taken by an &lt;a href=&quot;http://www.math.nus.edu.sg/~matberic/ResNewsletter95.htm&quot;&gt;essay from A.J. Berrick&lt;/a&gt; (one of several links on the AMS page), part of whose thesis is that we can't simply answer what mathematicians &lt;em&gt;are&lt;/em&gt; (like many other disciplines, according to target of study), but rather what they &lt;em&gt;do&lt;/em&gt; (according to specific strategies applied). He writes in his conclusion, emphasizing the goal of new discoveries:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The discovery of a unifying pattern can be like lightning flashing&#xA;  from one discipline to another. The difference is that it can&#xA;  illuminate both subjects forever. So there is a simple message for the&#xA;  nonmathematical researcher reading this article. When all seems&#xA;  cloudy, contact a mathematician (as broadminded as possible). Then&#xA;  stand by for flashes of lightning!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Surely, that doesn't sound boring; and it also doesn't really sound like what your colleagues do. More generally: As a college math teacher, I think you'll find that a large part of your career will be spent remediating/fixing hideous misinformation that K-12 teachers have foisted upon their students. Best to use this for practice in that regard now. Good luck!&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-04-23T17:52:37.477" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10905" PostTypeId="2" ParentId="10895" CreationDate="2016-04-24T07:45:49.147" Score="6" Body="&lt;p&gt;A quarterback and a center have completely different jobs, but they are both football players. &quot;A mathematician&quot; covers a lot of territory. I think they are mathematicians and they deserve some respect for the level of competence that they have achieved - however small compared to, say, Euler or Hofstadter or Angel Muleshkov.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are upset about the wrong thing. No one is going to become a mathematician because he read a good definition of what a mathematician is. And there is no polite way to tell someone who thinks of himself as a mathematician that you don't think he is. If you want to get your students interested in mathematics, explain to them what mathematics is about.&lt;/p&gt;&#xA;" OwnerUserId="5144" LastEditorUserId="5144" LastEditDate="2016-04-25T07:33:58.127" LastActivityDate="2016-04-25T07:33:58.127" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10906" PostTypeId="2" ParentId="8041" CreationDate="2016-04-24T08:24:28.333" Score="1" Body="&lt;p&gt;I do it by presenting and discussing this &quot;Theorem&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $X \subseteq \mathbb W$ where $\mathbb W = \{0, 1, 2, \dots\}$ and suppose $X$ has the following two properties&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$0 \in X$&lt;/li&gt;&#xA;&lt;li&gt;If $n \in X$ then $n+1 \in X$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then $X = \mathbb W$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We don't really prove it, but we try to convince ourselves that it is true. Then we consider similar theorems for sets other than $\mathbb W$. Once I am pretty sure that they understand the paradigm, then I introduce mathematical induction.&lt;/p&gt;&#xA;" OwnerUserId="5144" LastActivityDate="2016-04-24T08:24:28.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10907" PostTypeId="2" ParentId="8041" CreationDate="2016-04-24T14:18:42.513" Score="1" Body="&lt;p&gt;For some reason, I always found it easier to understand if I started with the inductive step, &lt;em&gt;then&lt;/em&gt; did the base case. I think it's because I had trouble with the idea of &quot;&lt;em&gt;if&lt;/em&gt; it's true for a given step, &lt;em&gt;then&lt;/em&gt; it's true for the next step.&quot; This may be because of the use of the word &quot;assume&quot; in mathematics. I interpreted the usual base case --&gt; inductive step as &quot;Show it's true for the first case. Now just &lt;em&gt;assume&lt;/em&gt; it's (miraculously) true for some case further down the line.&quot; By reversing it, it becomes &quot;if it happens to be true for some case, it also happens to be true for the next case. Oh, but it's true for the first case, therefore...&quot;&lt;/p&gt;&#xA;" OwnerUserId="6651" LastActivityDate="2016-04-24T14:18:42.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10908" PostTypeId="2" ParentId="10206" CreationDate="2016-04-24T17:29:15.447" Score="7" Body="&lt;p&gt;Sorry for adding another answer, I'm new here. This paper: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Carraher, D. W., Schliemann, A. D., Brizuela, B. M., &amp;amp; Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics education, 87-115.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is a good start. Lots of references to existing research. I can add a link but not sure if it's behind a paywall.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quick glance tells me that the gist is that you can introduce variables as early as 9-10 years of age but that it's important to build algebraic thinking by looking for patterns in number properties first. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Adding some detail as per Ben's comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is something of a rabbit hole. It seems that very young children can solve equations with boxes, question marks, or even letters to represent unknowns, however, they may not be truly engaging with variables in and algebraic way. The question becomes whether they are actually operating on unknown values or using &quot;pure-arithmetic&quot; strategies like counting and working backwards. The authors present several sources claiming to show that young children (at least as young as third grade) can work with symbols for unknowns in an algebraic way, and that algebra and arithmetic are not entirely separate modes of thinking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For anyone wanting answers to the posted question, I'd suggest reading the brief literature review in the above article. The article itself is really focused on using functional relationships to teach arithmetic, rather than variables in particular. The authors seem to feel that there is strong evidence that students can engage with variables in a meaningful way from about age 8 or 9, but that we need to be careful about what we define as &quot;meaningful,&quot; since researchers who argue that younger children aren't &quot;developmentally ready&quot; may have somewhat rigid views about meaningful use of variables and the nature of the relationship between arithmetic and algebra.&lt;/p&gt;&#xA;" OwnerUserId="6651" LastEditorUserId="6651" LastEditDate="2016-04-26T19:01:14.377" LastActivityDate="2016-04-26T19:01:14.377" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10910" PostTypeId="1" CreationDate="2016-04-24T23:12:11.097" Score="12" ViewCount="996" Body="&lt;p&gt;The concept of &quot;reification&quot; in mathematics education is interesting. Roughly, if I understand this correctly, one &quot;reifies&quot; processes into mathematical objects. Very recently, it occurred to me that although this reification seems like a clear thing mathematicians do…perhaps without thinking about it…I don't have a very clear picture of the impetus for reification. What leads us to &quot;reify&quot;? Is it that we perform a process often enough that it becomes a recognizable pattern and thus is given a name and becomes more of a noun? This would make sense, but I'm not sure it is the only reason for reification…there may be deeper reasons connected to the compressive function of mathematical thought, similar to the use of highly leveraged notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry for the blathering. I am writing to ask mathematics educators for a summary of the current understanding of the mechanism for reification, understood as the impetus for reification and the process by which it takes place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware of &lt;a href=&quot;http://www.pat-thompson.net/PDFversions/1994Reification.pdf&quot;&gt;some controversy surrounding this&lt;/a&gt;, but a summary statement may help me understand my students' resistance to thinking about objects rather than processes. I imagine what is needed may be similar to the transfer from the mindset of classical procedure-based programming to object oriented programming. I apologize that this question is somewhat vague, but I'd appreciate any answer to it that may help me understand the various mechanisms...&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2016-04-24T23:55:38.170" LastActivityDate="2017-02-07T08:24:53.707" Title="Summary of the mechanism of reification" Tags="|mathematical-pedagogy|psychology|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10911" PostTypeId="2" ParentId="10832" CreationDate="2016-04-25T05:59:07.203" Score="1" Body="&lt;p&gt;There are many examples from statistics, one already mentioned in other answer (significant versus statistically significant).  Words from English in daily use, hijacked for a highly technical meaning:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;sufficient statistic&lt;/strong&gt;, &lt;strong&gt;efficient estimator&lt;/strong&gt;,  &lt;strong&gt;unbiased estimator&lt;/strong&gt; (more troublesome since unbiased in daily use have value  connotations), &lt;strong&gt;regression&lt;/strong&gt; another case with a big divergence in meaning!, &lt;strong&gt;likelihood&lt;/strong&gt; is a technical concept very distinct from probability while in daily use more or less synonymous with chance or probability, ...&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="122" LastEditDate="2020-11-07T19:53:44.153" LastActivityDate="2020-11-07T19:53:44.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10912" PostTypeId="1" CreationDate="2016-04-25T16:42:52.240" Score="27" ViewCount="790" Body="&lt;p&gt;One of the most frustrating things about my experiences teaching math (at the university level, if that matters) is that students seem very reluctant to actually learn the material. Instead, they seem to want to be presented with a series of examples, then generalize from those examples, rather than the other way around. This works well up till calculus and then fails completely afterward, when math becomes more than simply a set of computational algorithms. I've never had any success getting students to think mathematically; any suggestions? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, like anyone who's taught mathematics, or even answered questions on stackexchange, I've been frustrated (and irrationally annoyed) by students who come to me with no ideas whatsoever about a problem, saying that they &quot;don't know where to start.&quot; A bit of prodding often reveals that they don't understand one of the terms in the problem. I'm not sure how they were planning on solving it without that, but my pet theory is that they tried to compare that problem to their list of examples, found nothing, and gave up. (If they had found such an example, that might have allowed them to solve the problem without the missing definition, effectively treating the example as a black box.) Is there any credence to that idea, or is it something else entirely?&lt;/p&gt;&#xA;" OwnerUserId="6674" LastEditorUserId="6674" LastEditDate="2016-04-25T20:43:41.787" LastActivityDate="2016-04-28T08:22:12.390" Title="Breaking students from the habit of relying on examples" Tags="|undergraduate-education|examples|" AnswerCount="3" CommentCount="19" ContentLicense="CC BY-SA 3.0" />
  <row Id="10913" PostTypeId="1" CreationDate="2016-04-25T20:58:19.137" Score="13" ViewCount="645" Body="&lt;p&gt;&lt;em&gt;This question is an attempt at implementing my suggestion that the second paragraph of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10912/breaking-students-from-the-habit-of-relying-on-examples&quot;&gt;that question&lt;/a&gt; could be turned into an important question. The discussion starting in comments there would be more interesting under the form of developed answers, in my opinion.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think most higher educators in maths have felt the despair of seeing so many students&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;not being able to check a definition on the most basic examples, &lt;/li&gt;&#xA;&lt;li&gt;being clueless on how to solve an exercise that turns out to contain terms they do not understand but that they did not thought of looking up,&lt;/li&gt;&#xA;&lt;li&gt;being unable to apply correctly a theorem except by copy-pasting the exact sentence that was used in a previous exercise (and which of course is ill-suited to the case at hand), etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If we assume that these problems are entirely theirs to deal with, then we only have to bear it and leave it be. More constructively, let us assume that it is part of our job to help them solve this issues (I do believe it &lt;em&gt;is&lt;/em&gt; part of our jobs, and probably the main part at that).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What can we do about it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A good answer should start with a diagnosis as to where this issues are coming from, and suggest a course of action or at least hints we can use with students. Suggestions that can be carried on individually are especially welcome, but suggestion that involve changes in curriculum or collective action from a whole pedagogical team can be interesting too (they just probably won't happen).&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-04-28T21:48:46.287" Title="How to help student get to grips with exercises, definitions, theorems?" Tags="|undergraduate-education|" AnswerCount="4" CommentCount="1" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10914" PostTypeId="2" ParentId="10913" CreationDate="2016-04-25T21:14:39.453" Score="7" Body="&lt;p&gt;One hypothesis I've seen (unfortunately, I can't find the paper at the moment) is that students sometimes don't understand that when a mathematician uses a word, it means just what they define it to mean--- neither more nor less. The definitions in mathematics are performative acts, not descriptions as even in other areas of science. When one defines, for example, continuity of real functions using the $\delta$-$\epsilon$ formulation, that &lt;i&gt;is&lt;/i&gt; continuity; it's not some fuzzy, general idea of &quot;not having any gaps in its graph&quot; that the definition is merely groping for. In my own experience, some students attempt to learn or even define concepts by reading through some examples or worked problems, then coming up with a working definition themselves by looking for the similarities among them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've never found anything particularly effective. The best proposal I've seen is to aggressively show students counterintuitive examples. In the example of continuity above, we could consider (expanding the definition to more general metric spaces or topological spaces) maps to and from discrete spaces, the $p$-adic topology, the Zariski topology, etc. I don't like the idea, though, of getting away from inferring concepts from examples by showing more examples. Maybe the idea is to get further away from concrete, specific examples and remain in the abstract. It's not hard, for example, to use naive ideas or intuition to guess whether functions like $\sin x, 1/x$, $\operatorname{sign}(x)$, etc. are continuous. When you want to prove, for example, that the image of a connected set is connected, you're pretty much forced to actively use the definitions abstractly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus maybe part of the hang-up is the discrepancy between insisting on formal definitions and abstractions, yet invoking them in situations where students don't feel they're necessary or desirable. For that matter, up to a point students don't necessarily distinguish (or have the tools to distinguish) between an argument, computation, etc. that's inherently invalid, versus one that's just arbitrarily disallowed by the teacher. I'm sure most people here have some memory in high school of using a clever, justifiable trick that the teacher disallowed because it wasn't what was then being tested, was outside the curriculum, or something similar.&lt;/p&gt;&#xA;" OwnerUserId="6674" LastEditorUserId="6674" LastEditDate="2016-04-25T22:55:21.453" LastActivityDate="2016-04-25T22:55:21.453" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10915" PostTypeId="2" ParentId="10206" CreationDate="2016-04-25T22:15:19.063" Score="5" Body="&lt;p&gt;I'm a teacher at a small Christian school and so I fill a lot of gaps. I've taught 4th, 5th, 6th, 7th, Alg. 1, and Geometry, as well as tutoring everything in between up to Pre-Calc with great success. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My 4th graders just recently got introduced to algebra. I've found that at that age, a lesson dedicated to the subject matter is more than sufficient for mastery of the simple idea that a letter represents some unknown or variable amount. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our curriculum is very good at building from the &quot;boxes&quot; mentioned in other's posts, and lends opportunity for a teacher to get students excited earlier in the year if the class seems ready, by telling them, &quot;What you're really doing is Algebra! The only difference, is that in Algebra, a letter is used instead of a box!&quot; This takes the intimidation factor out of algebra, makes the students feel smart, and excited about furthering their math knowledge. They beg for the lesson on algebra before you get there, and then as a teacher, you have an easy time keeping their attention and making sure correct steps are used in finding the correct answer that will build a foundation in 4th grade that they'll use all the way through Algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've also found while teaching higher grades, that students without this foundation have a much harder time of developing it after they've developed instead, an aversion to algebra in general, but a similar approach can definitely be done in a tutoring scenario.&lt;/p&gt;&#xA;" OwnerUserId="6675" LastActivityDate="2016-04-25T22:15:19.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10917" PostTypeId="2" ParentId="10913" CreationDate="2016-04-25T23:05:43.957" Score="12" Body="&lt;p&gt;One hypothesis is that many students are functionally illiterate, and in fact cannot parse or understand the carefully-constructed sentences that we present to them. They might be succeeding in many other courses using an examples-induction approach, and only in the math discipline (and perhaps computer science) does the lacuna become unavoidable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A second hypothesis that I've been working on is that many in the math discipline generally don't spend enough time emphasizing the importance of those definitions. I've been in multi-disciplinary teachers conferences where someone will present &lt;a href=&quot;https://cft.vanderbilt.edu/guides-sub-pages/blooms-taxonomy/&quot; rel=&quot;nofollow&quot;&gt;Bloom's Taxonomy&lt;/a&gt; and say, &quot;We spend most of our time at the lowest level of definitions; the struggle is to get our students into the higher tiers of applications, et. al.&quot;, at which everyone at the table nods. To this I think: In the math discipline, our students are routinely jumping directly into applications without really understanding the foundational terms of discussion well enough. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should instructors be spending more time testing on the definitions directly? Do most instructors lack such time due to other heavy curricular requirements? Should questions like that be on our final exams (almost uniquely lacking, compared to other disciplines)? Are math instructors so skilled at instantly remembering definitions that this passes under our radar much of the time?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The remediation that I've attempted is to commit to presenting a definition, and then immediately running a gauntlet of yes-or-no questions to the class about whether a series of examples meets the definition or not. (Using presentation software here helps for time purposes; when previously doing it by hand I frequently ran out of time.) For example: Define polynomials, then immediately ask: Is this a polynomial? (a) $5x^2 + 7x$, (b) $\frac{2}{x}$, (c) $\frac{1}{2}x$, (d) $8$. Then proceed to any theorems/proofs/applications after that point. Also: I have weekly online quizzes that test most conceptual issues arising directly from the definitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I have not found that to be making a great difference in end-results. Full disclosure: This is the first semester where I've been able to fully present things that way, and the class grades so far are the lowest that they've been in over 4+ years since I've been keeping current master records. That's likely due to the course population this spring semester, though. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-04-28T21:48:46.287" LastActivityDate="2016-04-28T21:48:46.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10918" PostTypeId="2" ParentId="10913" CreationDate="2016-04-26T00:03:08.417" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;Emphasize definitions in homework (eg, penalizing people who don't invoke the definition)&lt;/li&gt;&#xA;&lt;li&gt;Repeat definitions in class (eg, whenever I work out an example, I pause and ask the class what the definition is, and then write it down) &lt;/li&gt;&#xA;&lt;li&gt;Ask for definitions on exams, and let them know that you will ask for definitions on exams.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I hadn't thought too hard about it before, but if you wanted to further emphasize this, you could keep a running list of definitions that you expect them to know, and even encourage them to make a list of such things &lt;em&gt;along with the definitions themselves&lt;/em&gt; as a homework/study aid. &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2016-04-26T00:03:08.417" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10919" PostTypeId="1" CreationDate="2016-04-26T03:16:03.090" Score="9" ViewCount="303" Body="&lt;p&gt;I'm a homeschool mom, my daughter is 7. She will be in second grade next year. I appreciate the different ways of doing math, rather than just stacking the numbers and adding/subtracting. I'm not sure how to go about teaching it. Any good resources out there for a homeschool parent? Thanks!&lt;/p&gt;&#xA;" OwnerUserId="6677" LastEditorUserId="262" LastEditDate="2016-05-02T01:27:14.013" LastActivityDate="2017-07-13T19:54:24.693" Title="Primary/Elementary Math Resources for Homeschool" Tags="|primary-education|common-core|homeschooling|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10920" PostTypeId="2" ParentId="10919" CreationDate="2016-04-26T06:28:59.010" Score="1" Body="&lt;p&gt;Stacking the numbers, and carrying the tens, &lt;strong&gt;works&lt;/strong&gt; and is &lt;strong&gt;understandable&lt;/strong&gt; to generations of people.  You can explain each step as being correct by itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend teaching your daughter how to &lt;strong&gt;count on her fingers using roman numerals&lt;/strong&gt;.  This will let her see how she can &quot;carry the one&quot;.  Her four fingers on her right hand are Is; her right thumb is a V; her four fingers on her left hand are Xes; and her left thumb is an L.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whatever method you teach, you should make sure that every step has a clear meaning, and that it is clear why the expression at that step has the same value as the expression at the previous step.  You should &lt;strong&gt;make sure that your daughter can explain each step&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addition example&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;            1        1      1 1      1 1&#xA;  537      537      537      537      537&#xA;+ 547 =&amp;gt; + 547 =&amp;gt; + 547 =&amp;gt; + 547 =&amp;gt; + 547&#xA; ====     ====     ====     ====     ====&#xA;             4       84      084     1084&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;In the first step, 7 + 7 = 14.  She can check this by doing the math on her fingers (using roman numerals).  The 4 goes down below, and the 10 is &quot;carried&quot;, by putting a 1 above the tens place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the second step, 1 + 3 + 4 = 8.  (or 10 + 30 + 40 = 80, because all three numbers are in the tens place.)  The 8 goes down below.  Because 8 is less than 10, there is no need to carry anything from this step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the third step, 5 + 5 = 10.  (or 500 + 500 = 1,000, because both 5s are in the hundreds place.)  The 0 goes down below, and the 10 is &quot;carried&quot;, by putting the 1 above the thousands place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the fourth step, 1 + 0 + 0 = 1.  (or 1,000 + 0 + 0 = 1,000, because this step is for the thousands place.)  The 1 goes down below, and nothing needs to be carried further.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teach your daughter to &lt;strong&gt;check her own work&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Casting out nines is an old-fashioned way.&lt;/li&gt;&#xA;&lt;li&gt;Performing the inverse calculation is a complete way -- but might allow making the same mistake in reverse.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Teach your daughter to &lt;strong&gt;sanity check&lt;/strong&gt; her own work.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Estimating checks the rough size of the answer.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Have your daughter keep a log of her most common mistakes&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2016-04-26T06:51:18.027" LastActivityDate="2016-04-26T06:51:18.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10921" PostTypeId="2" ParentId="10913" CreationDate="2016-04-26T09:02:44.670" Score="5" Body="&lt;p&gt;One thing that I more and more realize is necessary but often left untold, is to explain some basic principles about how to approach an exercise they don't know at first how to solve. My experience makes me think they often don't guess a few things we assume are obvious, or we assume where told them in high school or previous classes. A few students get it by themselves, many don't and are thus driven toward mere reproducing of standard exercises, and consider it unimaginable to solve anything remotely different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a tentative list of hints they need to be told explicitly (at various stages of their studies):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;look up or recall or write down the definition of every term appearing in the statement which is not crystal clear to you,&lt;/li&gt;&#xA;&lt;li&gt;a mathematical object can be called the name defined in a mathematical definition if, and only if, it exactly and precisely matches the requirements written in the definition,&lt;/li&gt;&#xA;&lt;li&gt;to apply a theorem (or proposition), find which objects in the case at hand which you want to play the role of each object in the theorem to be applied, then check the assumption of the theorem on these, then you get the conclusion of the theorem,&lt;/li&gt;&#xA;&lt;li&gt;search your notes for results that uses  the assumptions of the statement to be proven,&lt;/li&gt;&#xA;&lt;li&gt;search your notes for results that produce the sort of conclusion claimed by the statement to be proven,&lt;/li&gt;&#xA;&lt;li&gt;if the above ends don't meet, sum up the properties you have and the ones you can get from applying theorems, and the properties you would need to apply theorems yielding the desired conclusion, and iterate,&lt;/li&gt;&#xA;&lt;li&gt;if things get messy, one option is to try the statement on simple examples and work them out, then proceed to more complex examples until you get an idea,&lt;/li&gt;&#xA;&lt;li&gt;if you have to decide whether all objects of a kind satisfy a given property, answering no only needs to produce one counter-example; failing repeatedly to produce one might indicate the answer is yes, and understanding what fails in the tried examples might help you find out a proof.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-04-26T09:02:44.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10922" PostTypeId="2" ParentId="10919" CreationDate="2016-04-26T14:12:16.727" Score="2" Body="&lt;p&gt;This is coming from an educator that isn't fond of common core for various reasons. What I believe is at the heart of common core math tho, is that the &quot;Why?&quot; is important. Math isn't just a series of steps to get to an arbitrary answer that someone says works, but it's a logical method of using order built into numbers to arrive at the correct answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal philosophy is that students first and foremost need the building blocks to USE to think. Ensure that they know their multiplication tables like their own name! But... once they have that foundation on which to think, give them all the &quot;why's&quot; you can; not just &quot;these are the steps.&quot; My 2 cents anyway. &lt;/p&gt;&#xA;" OwnerUserId="6675" LastEditorUserId="6675" LastEditDate="2016-04-26T17:58:36.977" LastActivityDate="2016-04-26T17:58:36.977" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10923" PostTypeId="1" CreationDate="2016-04-26T16:51:29.390" Score="10" ViewCount="330" Body="&lt;p&gt;I'm planning on showing my students a &quot;deep zoom&quot; video of the Mandelbrot set.  The video is about 15 minutes long and, at the end, shows an image that is zoomed by a factor of $10^{220}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to convey somehow just how staggeringly huge that scale is.  The best I have do far is that if the image shown at the beginning of the video were the size of the Planck scale (many orders of magnitude smaller than a proton), then after scaling it up by a factor of $10^{62}$ it would be about $100$ billion light years across, which is the size of the observable universe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose I could now iterate:  take a speck the size of the Planck scale in the &lt;em&gt;enlarged&lt;/em&gt; video, and scale &lt;em&gt;it&lt;/em&gt; up until it's the size of the observable universe.  Now we are at a scale of about $10^{124}$.  Do this two more times; now you have some idea of how large $10^{220}$ is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering if anybody has any other ideas for how to convey the enormity of this scale.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="6053" LastEditDate="2018-05-15T17:40:07.890" LastActivityDate="2018-05-17T18:36:55.990" Title="Communicating to students the meaning of extremely large numbers" Tags="|intuition|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10924" PostTypeId="2" ParentId="10923" CreationDate="2016-04-26T17:13:30.560" Score="10" Body="&lt;p&gt;My favorite video for this is &lt;a href=&quot;https://m.youtube.com/watch?v=0fKBhvDjuy0&quot;&gt;powers of ten&lt;/a&gt; from 1977. Though we can get a little smaller today, I think it still does an excellent job with getting the scale of things starting from what we know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They should pick up on that these numbers are far far far bigger than the largest scale in the video. &lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2016-04-26T17:13:30.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10925" PostTypeId="2" ParentId="10923" CreationDate="2016-04-26T17:35:17.590" Score="6" Body="&lt;p&gt;I was looking for a link for the book &quot;&lt;a href=&quot;http://www.maa.org/press/ebooks/the-lore-of-large-numbers&quot; rel=&quot;noreferrer&quot;&gt;The Lore of Large Numbers&lt;/a&gt;&quot; that I read years ago and I thought might give you some ideas that I came to this new book &quot;&lt;a href=&quot;http://www.maa.org/press/maa-reviews/really-big-numbers&quot; rel=&quot;noreferrer&quot;&gt;Really Big Numbers&lt;/a&gt;&quot; that seems fascinating too. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also like to mention a story from the years that I was a middle school teacher. To give an idea of how the powers of a number get bigger and bigger so quickly we worked on the idea of folding a paper (that gives us powers of two) and supposed that we can do that physically as many as we like. It was amazing for my students to see that just with &quot;a few&quot; folds we could climb to the moon! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also find &lt;a href=&quot;https://www.facebook.com/TheScienceWorld/videos/1173654972667774/&quot; rel=&quot;noreferrer&quot;&gt;this fascinating video&lt;/a&gt; (viewed by about 42 million) useful. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="1217" LastEditDate="2016-04-27T08:16:38.340" LastActivityDate="2016-04-27T08:16:38.340" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10926" PostTypeId="1" AcceptedAnswerId="10932" CreationDate="2016-04-26T17:57:35.727" Score="4" ViewCount="258" Body="&lt;p&gt;I am teaching matrices, determinants and systems to a class of 16-year-olds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can expect, they do not know calculus or linear algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What they do know is a little bit of matrices (how to multiply and invert them, for instance), some plane geometry, some spacial geometry, some analytical geometry, and most elementary functions: linear, quadratic, exponential, modular, trigonometric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had to explain what subjects they have studied because it must vary wildly from country to country.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this context, I am having great trouble in finding real-world applications and examples of determinants being applied to do anything at all. The only examples I could show are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Area of a triangle given the coordinates A,B,C;&#xA;Cramer's rule for solving systems of linear equations.&#xA;That being said, if anyone can think of any example I could use in my class, please tell me! Thanks in advance for your help.&lt;/p&gt;&#xA;" OwnerUserId="6681" LastActivityDate="2016-05-03T17:45:25.630" Title="Determinant applications for 16 year olds" Tags="|secondary-education|linear-algebra|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10927" PostTypeId="2" ParentId="1498" CreationDate="2016-04-26T22:40:51.513" Score="5" Body="&lt;p&gt;Since the other posts here are positive, I'll go ahead and say that it's not, or at least not necessarily, gratifying. You're talking about undergrad-level education, presumably something like calculus, complex analysis, etc. Someone commented above that you can learn more about the subject by teaching it to students; but you presumably have a PhD in mathematics, and you're honestly not going to learn anything from teaching introductory calculus. Someone commented above that it's a chance to do mathematics; but teaching a class on century-old mathematics is not the same as actually doing mathematics itself, whether it's solving new problems or completely original research. Someone commented above that it's great to teach students who can surpass you and help them up the ladder; but those students are going to be quite rare; besides, there's not much you need to do to teach them, since they're going to be able to pick up the material themselves, independently. Someone commented above that you can make a difference in students' lives; but you're also going to be teaching dozens of students who are only there because it's a requirement, and really just want you to spoonfeed them answers for an upcoming exam. Someone commented above that it's great to be around intellectually engaged people, and that's certainly true; but teaching an undergrad math class is not like crashing a math conference. And so on.&lt;/p&gt;&#xA;&lt;p&gt;You may disagree, and that's totally fine. If so, great: You should seriously look into mathematics education, whether at the university level or below. My point is simply that it isn't all great, or even partially great, for everyone. There are rewards to it, whether concrete or in warm fuzzy feelings. If those rewards are things you want, go for it; if not, try something else. Just don't expect to have those rewards guaranteed or continual.&lt;/p&gt;&#xA;" OwnerUserId="6674" LastEditorUserId="6674" LastEditDate="2023-02-02T15:33:06.370" LastActivityDate="2023-02-02T15:33:06.370" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10928" PostTypeId="2" ParentId="10912" CreationDate="2016-04-26T22:56:03.153" Score="11" Body="&lt;p&gt;This question is very usefully provocative, as evidenced by the comments, and the pro-example versus [sic] pro-abstraction notions... and the apt comment(s) suggesting that, in particular, the genuine issue is not that students have not &quot;learned the definition&quot;, but that their intellectual methodologies are inadequate to the task... or maybe some really are a bit lazy or work-avoidant (sounds better!) while not being actively opposed to the mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.g., I was a bit baffled by the comment that implied that it is perhaps dangerously deficient to characterize things as: continuity is that the graph has no jumps, and a differentiable functions' graph has no corners... considering that this is exactly my own intuition, although/and I can conform to the ambient standards-of-proof as need be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would claim that the underlying problem is that students are typically rewarded more for compliance and conformity than for other forms of astuteness. Thus, they have often learned to ignore their own intuition, since the instructor/exam/homework is actively trying to prank them in that regard. &quot;Definitions&quot; are therefore not at all codifications of extensive prior experience of serious people, but just set-ups for pratfalls of the victims, ... EDIT: for example, prank questions might have very little mathematical content, but, rather, play upon delicacies of wording, or the tension/conflict/ambiguity between mathematical use and colloquial use. No way to &lt;em&gt;reason&lt;/em&gt; these out. Rather, one must know what the examiner is thinking. Similarly, textbooks can create conventions/rules/definitions with little genuine mathematical content, but, nevertheless, with very precise boundaries, the latter lending themselves to questioning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, in all my experience with standard courses/textbooks/exams, it is &lt;em&gt;not&lt;/em&gt; the case that definitions [sic] are &lt;em&gt;helpful&lt;/em&gt;, much less clarify pre-existing ambiguities or troubles or confusions. Instead, for exam and such purposes, they are far too often (and, then, this &quot;taints the well&quot;) used to prank students in a quasi-legalistic sense. An example of the worst sort of crap is any question about &quot;how many axioms does a group satisfy?&quot;. EDIT: again, such a question has essentially no mathematical content, because any finite list can be put together into a single axiom by conjunction, of course. But students may be simply required to remember what a particular book says on a particular page (as opposed to intrinsic mathematical assertions). Similarly, notational conventions can be &quot;tested&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other issue, which is maybe not literally &quot;laziness&quot;, but &quot;passivity&quot;, is (I claim) partly a result of years of being beaten down by petty-despot &quot;teachers&quot; of mathematics, who portray the subject as consisting of ineffable, uncontestable rule-sets... It is surely fairly futile, but I think any approach that does not tell kids to &quot;trust, but maybe refine/improve, their intuition&quot;, is doomed to disengagement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same appears to be true of grad students in mathematics at good universities, etc., btw.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, I'm resisting answering the question as asked... but eminently willing to edit, considering that I do think this is an important sort of question.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2016-04-27T13:16:05.713" LastActivityDate="2016-04-27T13:16:05.713" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10929" PostTypeId="2" ParentId="10912" CreationDate="2016-04-27T19:29:33.450" Score="4" Body="&lt;p&gt;$c^2=a^2+b^2-2ab\cos C\\$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The law of cosines. When I lay out the problem so that a, b, and c don't line up with the equation, half the students are suddenly lost. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, I offer a right triangle, but play an awful trick, I label the hypotenuse with the letter A, and the 2 legs are B and C. I watch as $a^2+b^2=c^2\\$ is followed regardless, with nonsensical results. It gets marginally better if I start with say, x, y, z, but not much. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples are great, until the student is stumped with the similar triangle question, confused, as they solved 4 problems each with a man and flagpole with shadows for each, but now you've presented a building with its shadow. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My best approach is to vary the examples so much that they don't fall into the traps I tried to illustrate. Find ways for the example to be just that, but help them open their mind to apply the example to other concrete examples. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-04-27T19:29:33.450" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10930" PostTypeId="2" ParentId="10912" CreationDate="2016-04-28T08:22:12.390" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Instead, they seem to want to be presented with a series of examples, then generalize from those examples, rather than the other way around.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To be honest: This does not seem &lt;em&gt;that&lt;/em&gt; wrong in general. Most of the generalisations offered by mathematics are motivated by examples, e.g., the concept of continuity is motivated by the fact that the vast majority of real-life relations is continuous (at least in very good approximation). Moreover, examples are a good way to discover proofs. That being said, the power and importance of mathematics comes from abstraction and generalisation, but if you cannot illustrate this power by connecting the example level with the abstract one, your students will not appreciate the latter. This holds in particular, if your students are not actually aspiring to become mathematicians, but study physics, computer science, or similar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain to them the way mathematics works:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;examples motivate definitions;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;statements based on suitable definitions have wide, easy, and robust applicability (in a sense, this is why we bother with abstract mathematics in the first place), while appealing to example-based intuition is prone to errors;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;examples are a good way to discover proofs, but they never &lt;em&gt;are&lt;/em&gt; proofs themselves.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For a practical example: Once we show that some structure complies with the vector-space axioms, the whole apparatus of linear algebra becomes available to us and due to the rigidity of mathematics, we do not have to spend any thought on whether we can actually apply it – example-based intuition cannot do this.¹&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Guide them by example (yes, yes, I know): Do not only present them polished proofs, but show them how to come up with a proof. Write down all the relevant definitions and known properties, and continue from there. Also try to give an example where you can translate some example-based intuition to a proper proof.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For students of other disciplines like physics or computer science, debunk the myth that they only need mathematics as a computational tool. For example, explain to physics students that vector spaces are the basic language of quantum mechanics and they cannot properly understand it without understanding vector spaces.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;¹ Blatant self-advertising: For an even more practical example for this example, see &lt;a href=&quot;http://dx.doi.org/10.1088/0143-0807/36/3/035008&quot; rel=&quot;nofollow&quot;&gt;this didactics paper of mine&lt;/a&gt; (&lt;a href=&quot;http://arxiv.org/abs/1504.02332&quot; rel=&quot;nofollow&quot;&gt;preprint&lt;/a&gt;).&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2016-04-28T08:22:12.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10931" PostTypeId="2" ParentId="7281" CreationDate="2016-04-28T09:04:53.363" Score="0" Body="&lt;p&gt;Ratio is just a single fraction.&#xA;Whereas proportion is a ratio between 2 fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example - 5 boys and 3 girls in a class.&#xA;Ratio of boys is 5/8&#xA;Ratio of girls is 3/8&#xA;But proportion of boys to girls is 5:3&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic difference is Ratio is always with respect to some large quantities or superior quantities, whereas proportion is between same kind of quantities.&lt;/p&gt;&#xA;" OwnerUserId="6693" LastActivityDate="2016-04-28T09:04:53.363" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10932" PostTypeId="2" ParentId="10926" CreationDate="2016-04-28T12:38:58.777" Score="5" Body="&lt;p&gt;I think you can go deeper than area of a triangle. This leads to computing the area of a polygon, and the volume of a polyhedron, two very common&#xA;and important calculations.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/OxXbx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OxXbx.png&quot; alt=&quot;TetraVol&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;sup&gt;&lt;a href=&quot;http://demonstrations.wolfram.com/TetrahedronVolume/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wolfram Demo by Ed Pegg&lt;/a&gt;&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;See also &lt;a href=&quot;https://matheducators.stackexchange.com/a/9770/511&quot;&gt;this discussion re determinants&lt;/a&gt; here on MESE.&#xA;Nearly every video game uses determinants somewhere deep in the code:&#xA;The signed tetrahedron volume indicates whether the apex is to the&#xA;positive or negative side of the plane containing the base.&#xA;And deciding sidedness is a key calculation for collision detection.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-05-03T17:45:25.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10933" PostTypeId="1" CreationDate="2016-04-28T15:31:12.460" Score="3" ViewCount="465" Body="&lt;p&gt;as I can see there are mainly two ways to introduce fractional terms. Two examples to demonstrate the two variants:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$\frac{a^2+3}{a};  \frac{3}{2c}$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$T(a) = \frac{a^2+3}{a}; T(c) = \frac{3}{2c}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In fact, the word &quot;function&quot; or &quot;functional term&quot; or &quot;equation with variables&quot; has not been learned at this point, but I think that variant two is able to demonstrate better that any term can be seen as a &quot;number machine&quot;, in which you &quot;throw&quot; a number and get a certain result. I could imagine that the first variant leads to young learners being unsure what exactly to do with this experssion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I made a quick research in some school books here (in Germany) and found those two variants. So even though this is really just a small difference, do you have any opinion or experience?&lt;/p&gt;&#xA;" OwnerUserId="6695" LastEditorUserId="5220" LastEditDate="2016-11-02T01:16:41.217" LastActivityDate="2017-01-01T02:27:48.717" Title="How to explain &quot;fractional terms&quot;?" Tags="|algebra|notation|terminology|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10934" PostTypeId="2" ParentId="10926" CreationDate="2016-04-28T18:33:59.137" Score="2" Body="&lt;p&gt;I'm not sure if you would count it as a 'real world' example, but you have enough to define the Alexander polynomial and so prove (modulo invariance) the non-triviality of small knots (or bigger ones with a computer).&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-04-28T18:33:59.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10935" PostTypeId="1" CreationDate="2016-04-28T20:05:46.137" Score="9" ViewCount="282" Body="&lt;p&gt;I'm about to give a first-semester calculus lecture covering the mean value theorem for integrals:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $f$ is continuous on $[a,b]$, then there is some $c\in(a,b)$ such that $(b-a)f(c)=\int_a^b f(x)\,dx$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In past semesters, I've shown examples in which I confirm that this theorem holds for some specific $f(x)$ and $[a,b]$, by solving for $c\in(a,b)$.  But this is just checking the theorem -- not actually applying it.  A &quot;real&quot; application occurs, for example, in the proof of Taylor's remainder formula, but my students aren't ready for that example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is a good &quot;real&quot; application of this theorem, suitable for students in first-semester calculus?&lt;/p&gt;&#xA;" OwnerUserId="6696" LastActivityDate="2016-05-19T08:31:08.140" Title="Applications of MVT for Integrals, suitable for calculus 1" Tags="|undergraduate-education|calculus|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10936" PostTypeId="2" ParentId="7650" CreationDate="2016-04-29T04:21:44.213" Score="5" Body="&lt;p&gt;The answer to your question is &lt;strong&gt;yes&lt;/strong&gt;. Check out the recent textbook:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Singer, F. M., Ellerton, N. F., &amp;amp; Cai, J. (Eds.). (2015). &lt;em&gt;Mathematical problem posing: From research to effective practice&lt;/em&gt;. Springer. &lt;a href=&quot;https://books.google.com/books?id=NU7gCQAAQBAJ&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Google Books&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In particular, &lt;strong&gt;Part III&lt;/strong&gt;, which contains 10 chapters, is described as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8C3KT.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8C3KT.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the beginning of the first chapter in that section:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YV7Iz.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YV7Iz.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the actual article for their &lt;em&gt;theoretical&lt;/em&gt; framework, as well as the rubrics developed for problem-solving (Appendix B) and problem-posing (Appendix C). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are various theoretical frameworks and methodologies described in the several chapters in this section of the textbook, though many connect back to the work of Silver and colleagues. You may also appreciate the review chapter included later on in this section:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tWuv7.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tWuv7.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-04-29T04:21:44.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10937" PostTypeId="2" ParentId="10736" CreationDate="2016-04-29T06:32:48.240" Score="4" Body="&lt;p&gt;(This answer has &lt;strong&gt;two&lt;/strong&gt; parts: The &lt;strong&gt;first&lt;/strong&gt; one is about existing research, and &lt;em&gt;probably&lt;/em&gt; relevant, but succinct; the &lt;strong&gt;second&lt;/strong&gt; one is about a problem solved in practice, and &lt;em&gt;possibly&lt;/em&gt; relevant, but definitely rambling. I will leave the determination of what constitutes a &quot;related&quot; answer to the reader!)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Part I&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I perceive Polya's (1945) &lt;em&gt;How to Solve It&lt;/em&gt;, the chief weakness is precisely that which you point to: how does one &quot;devise a plan&quot; in the first place, and how helpful is the advice to use a heuristic such as &quot;think of a related problem&quot; when one is stuck from the outset?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I know, the (an) answer is that even with the explicit suggestion to think of a related problem, students will struggle if they have an ill-conceived notion of relatedness, which, in fact, is the case for many students: cf. the comment I provided above to my earlier answer (&lt;a href=&quot;https://matheducators.stackexchange.com/a/1256&quot;&gt;&lt;strong&gt;MESE 380&lt;/strong&gt;&lt;/a&gt;) and the sources cited there. Specifically, work by Silver and Schoenfeld on looking at problem relatedness. The latter went on to write a book, &lt;em&gt;Mathematical Problem Solving&lt;/em&gt;, a few years later (1985) and has a more recent book, &lt;em&gt;How We Think&lt;/em&gt; (2010), on similar information from the perspective of decision-making.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than phrasing this answer using your term (&quot;emotional engagement&quot;) I point out instead that Schoenfeld's contribution to the problem solving literature included the argument that our knowledge (factual, procedural, etc: called &quot;resources&quot;) and strategies (called &quot;heuristics,&quot; although grouped under &quot;resources&quot; in the 2010 book) are insufficient with respect to determining how well students will problem solve. More precisely, his assertion is that there are two more categories that are both necessary and sufficient for identifying when students will be successful problem solvers: those categories are &quot;beliefs and belief systems&quot; and &quot;control decisions and metacognition&quot; (to the latter end, see also: &lt;a href=&quot;https://matheducators.stackexchange.com/a/2399&quot;&gt;&lt;strong&gt;MESE 2397&lt;/strong&gt;&lt;/a&gt; and my pointer -- sorry for all the links! -- to Schoenfeld's &lt;em&gt;What's All the Fuss about Metacognition?&lt;/em&gt;). I suspect that work on beliefs (especially) and metacognition (perhaps) can essentially subsume what you refer to as &lt;em&gt;intensity of emotional engagement&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above contains references to four substantial papers and two books; below, I recount a brief experience from my course on &lt;em&gt;problem solving in mathematics&lt;/em&gt; (taught to graduate students in mathematics education) in the hope that it will be illustrative of ... something relevant! Still, I separate these two parts in case only the top half is of interest.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Part II&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked students for their observations around the product of four consecutive natural numbers; this is a list that begins with $1\cdot 2\cdot 3\cdot 4 = 24$ and $2\cdot 3\cdot 4\cdot 5 = 120$. One student observed that each is one short of a square, i.e., $24+1 = 5^2$ and $120+1 = 11^2$. We checked a few more examples to see if this continued to be the case (yes) and then decided we had a conjecture worth pursuing. Since the OP contains a reference to IBL, I believe it is fair to say that &quot;IBL&quot; is a roughly accurate description of the context and educational environment at hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So:&lt;/strong&gt; How do you solve this problem? (i.e., prove or refute the conjecture that the product of any four consecutive natural numbers, plus one, is a perfect square)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The classroom setup is something like 4 tables of 5 students each, so they went about their attack on this problem in different ways. I sometimes write down the hashtag &lt;strong&gt;#WWGPD&lt;/strong&gt; (What Would George Polya Do?) on the whiteboard, and did so for this problem. What seems like a &lt;em&gt;related&lt;/em&gt; problem here?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All students began to use notation (recall Hadamard: &quot;[Polya] finds that a proper notation – that is, a properly chosen letter to denote a mathematical quantity – can give him similar help&quot;) although their approaches even in this respect differed, slightly. All but one table denoted the product as $n(n+1)(n+2)(n+3)$. I happen to believe that even this step is quite non-trivial; it indicates a number theoretical or algebraic approach from a mathematical perspective, but it also indicates an &lt;em&gt;I can get something out of this problem&lt;/em&gt; approach from an &quot;emotional&quot; (to use your term) perspective. (I still think of this as &quot;beliefs&quot; and &quot;metacognition&quot; to use Schoenfeld's terms, though a search for &quot;mathematical mindset&quot; could yield some useful papers, as well.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One student multiplied $n(n+3)$ and $(n+1)(n+2)$ separately, in an effort to find some sort of balance among terms; indeed, this yields the equivalent expression $(n^2 + 3n)(n^2 + 3n + 2)$ whence a $u$ substitution for the first parenthetical term gives $u(u+2) = u^2 + 2u$, which is factored as a square when $1$ is added, &quot;&lt;strong&gt;QED&lt;/strong&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But most students did &lt;em&gt;not&lt;/em&gt; have such an approach. One table tried to prove it by mathematical induction. I think this is a reasonable idea; the conjecture is about natural numbers, and we had proved other propositions about squares using induction, e.g., the sum of the first $n$ odds is $n^2$. This turned out to be a bit of a mess, though, so that table abandoned the strategy. My own wonder was whether there was a belief that the related problems solved with induction were neat enough to indicate that it was not a viable strategy here; of course, induction &lt;em&gt;could&lt;/em&gt; be carried out for this problem, but it might not be so quick. (For the aforementioned example around summing odds, one summarizes that $1 = 1^2$ and an inductive hypothesis gives that the first $k+1$ odds sum to $k^2 + (2k+1)$, which is $(k+1)^2$ — &quot;&lt;strong&gt;QED&lt;/strong&gt;&quot; rather succinctly!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The subsequent approach of that table was to multiply everything out, and then use Mathematica to factor it. This worked. A student at another table &lt;em&gt;also&lt;/em&gt; multiplied everything out, but factored it by hand. This can be done by observation/inspection, by matching up coefficients in a procedural way, or by noticing that the factored quartic expression is of the form $(n^2 + an + 1)^2$ and using the initial observation that $n=1$ yields $24$ to find $a$ rather quickly. That is, $(1^2 + a + 1)^2 = 25$, so that $2+a=5$ gives $a=3$ (should the conjecture hold). To speak again of my own wonder: Did the table that resorted to factoring by Mathematica (&lt;a href=&quot;http://www.wolframalpha.com/input/?i=sqrt%28n%28n%2B1%29%28n%2B2%29%28n%2B3%29+%2B+1%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;e.g.&lt;/strong&gt;&lt;/a&gt;) feel &lt;em&gt;emotionally&lt;/em&gt; worn out by the involved nature of the arithmetic in their (reasonable) attempt at mathematical induction, thereby priming them for a calculator-style solution? Or are they tech-savvy and able to recognize the problem (or &quot;similar&quot; i.e. &quot;related&quot; ones) as quickly reduced to a computation from that point on? I do not know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I spoke of one table that wrote the expression differently. There was a problem that might be construed as relevant that asked the class to prove that for a prime $p \geq 5$, the expression $p^2 - 1$ is divisible by $24$. In fact, some number of problems around divisibility were solved using a difference of squares, and so the last table re-wrote the given expression as $(n-1)n(n+1)(n+2)$. From here one multiplies the first and third expression, and the second and fourth expression, to find: $(n^2 - 1)(n^2 + 2n)$. The table managed to finagle a solution (proof) using this approach, although they articulated their initial thinking as trying to balance the terms by shifting the natural number denoted as $n$; note that this draws from a related problem, uses a reasonable heuristic (the first student mentioned also tried to balance terms), but ultimately leads to a matching of two products that is both non-routine and (in terms of brevity) non-ideal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one believes in drawing from related problems to balance terms, then perhaps a &lt;em&gt;different&lt;/em&gt; shift would be better: Substituting $n=m-1.5$ yields $(m-1.5)(m-0.5)(m+0.5)(m+1.5)$. This latter factorization, to my eye, makes the seemingly opaque nature of why it is beneficial to multiply the first and last term transparent (or, at least, translucent); here, we end up with $(m^2-2.25)(m^2-0.25)$ from which another substitution of $w$ for the first term gives $w(w+2)=w^2 + 2w$, so that adding $1$ gives a square as desired.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No student came up with that last substitution; indeed, the other approaches can be found in Zeitz's (2006) &lt;em&gt;The Art and Craft of Problem Solving&lt;/em&gt; as &lt;strong&gt;Example 1.2.1&lt;/strong&gt;. The book has a few approaches, my class had a few approaches, but there was no shift by $1.5$. Why not?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hypothesize that the phrasing around natural numbers and the use of $n$ (rather than $x$) are to blame. If you think of the problem using $x$ instead, then it looks a bit like a function; in that context, one may consider zeroes -- perhaps even a graph! -- and then the idea of shifting the (induced) function so that it is even (i.e., the substitution suggested) fits in more reasonably. In the context of natural numbers, I do not think that shifting by a non-whole number is a &quot;nearby&quot; heuristic; indeed, one may believe some justification is needed for why the $(w+1)^2$ factorization is really the square of an integer. (One way to do this, of course, is to unravel the substitutions and write everything in terms of $n$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not have a grand point with all of this; I record the above in part as an anecdotal record. But in thinking about the role of &quot;emotional intensity&quot; and relatedness, I wonder whether there was a way to scaffold instruction in non-routine problem solving so that the last strategy described would arise organically. How can the $1.5$ shift be viewed as a &quot;nearby&quot; heuristic? The problem was within (at least almost) everyone's zone of proximal development, and the notions around balancing terms and factorizations using the differences of squares were all floating in the ether. Is it possible, or probable, that the &quot;emotional intensity&quot; attached to words like &lt;em&gt;natural numbers&lt;/em&gt; or notations like '$n$' is enough to sway solvers so that their ultimate choice of heuristic is doomed only to whiff the heuristic described? (Relegating &quot;nearby&quot; to something akin to &quot;so close yet so far&quot;...) I do not know!&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-04-29T07:42:45.300" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10938" PostTypeId="2" ParentId="7509" CreationDate="2016-04-29T11:44:39.477" Score="2" Body="&lt;p&gt;A bit late, but here's another one:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://weitz.de/dft/&quot; rel=&quot;nofollow&quot;&gt;http://weitz.de/dft/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have a choice of different input and window functions and you can interactively manipulate parameters like the frequency of your input or the width of the window.  The output, i.e. the result of the (discrete) Fourier transform, is shown in real time.  This is mainly intended for an audience of engineers, but I think it can help anyone who has to deal with this subject for the first time.&lt;/p&gt;&#xA;" OwnerUserId="2089" LastEditorUserId="2089" LastEditDate="2016-04-30T10:57:14.120" LastActivityDate="2016-04-30T10:57:14.120" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10939" PostTypeId="1" CreationDate="2016-04-29T19:12:25.613" Score="18" ViewCount="1565" Body="&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=1oP2aqI9LBsC&amp;amp;lpg=PA189&amp;amp;dq=what&amp;#39;s%20all%20the%20fuss%20about%20metacognition&amp;amp;pg=PA189&amp;amp;hl=en#v=onepage&amp;amp;q=what&amp;#39;s%20all%20the%20fuss%20about%20metacognition&amp;amp;f=false&quot;&gt;Metacognition&lt;/a&gt; seems pretty universally positive. I'm wary of viewing it as such. Aside from the obvious criticisms like &quot;you can't learn to ride a bicycle by thinking about and writing a 200 page treatise on riding bicycles&quot; &lt;strong&gt;I'd like to know some concrete places where metacognition should be discouraged in mathematics education.&lt;/strong&gt; Wittgenstein, for example, may have argued that we acquire the ability to do mathematics by implicitly learning what &lt;em&gt;not&lt;/em&gt; to think about or reach for in a very unconscious way akin to natural language use. Even if one views Wittgenstein's views on mathematical practice as fringe, one might note that the typical (or perhaps I should say &quot;ideal&quot; in the sense of Reuben and Hersh) mathematician's aversion to philosophizing about mathematics reflects a general attitude against metacognition in mathematical practice…perhaps fearing one's never getting down to business because of being lured by the sirens of philosophy. Here, though I would like more concrete criticisms in the everyday practice of teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being an odd mathematician in terms of my attitude toward philosophy, I would love it if metacognition were universally good for students, as then we could just have them write in order to build in some continuity of concepts. I pause, though, when presented with a formalist critique akin to a Wittgenstein-like position above.  &lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2016-04-29T21:52:09.090" LastActivityDate="2021-12-31T12:31:59.420" Title="Is metacognition ever bad?" Tags="|undergraduate-education|mathematical-pedagogy|philosophy-of-maths|meta-cognition|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10940" PostTypeId="1" CreationDate="2016-04-30T12:57:39.760" Score="2" ViewCount="73" Body="&lt;p&gt;One of the major benefits of a good &lt;a href=&quot;http://math.uchicago.edu/~boller/IBL/&quot; rel=&quot;nofollow&quot;&gt;IBL script&lt;/a&gt; is that it effectively organizes the experience of students so that their effort does not need to be used on such organization. A good script leads from very humble beginnings straight through the heart of a subject, hitting all the high points organically…hiding these high points in very harmless looking and natural problems. Students need only worry about one problem at a time, and the entire subject opens before them via their efforts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although there are &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471789011&quot; rel=&quot;nofollow&quot;&gt;good textbooks&lt;/a&gt; for teaching students about IMO/Putnam type exam problems, I don't know of one with the property that if a student worked serially through it he or she would efficiently pick up the necessary techniques and experience to become proficient. I'd like such a source that has the above IBL flavor, in that it is comprised of a series of individual problems to be tackled in order, with each problem innocuous-looking but revealing, in an appropriately graded way, the main tactics and techniques. At the end of the sequence, if completed, a student should have vastly improved ability in solving such competition problems. Is there any such source?&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2016-04-30T13:06:13.120" LastActivityDate="2016-04-30T13:06:13.120" Title="Request for IBL Script for Putnam-type Problem Solving Course" Tags="|secondary-education|reference-request|exams|problem-solving|inquiry-based-learning|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10941" PostTypeId="1" AcceptedAnswerId="10942" CreationDate="2016-04-30T13:14:40.987" Score="7" ViewCount="840" Body="&lt;p&gt;This is a High School Trig problem asking for the solution to an otherwise simple equation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{\left(1+\cos x\right)}{\sin x}$=-1 (per comment - the domain was specified as greater than -180 degrees, less than or equal to 180 degrees.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to simplify to a single function, I manipulate and square both sides  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\left(1+\cos x\right)=-\sin x$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\cos ^2x+2\cos x+1=\sin ^2x$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then by substituting $1-\cos ^2x$ for $\sin ^2x$  I have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$2\cos ^2x+2\cos x=0$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and using zero product rule&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\left(2\cos x\right)\left(\cos x+1\right)=0$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first term giving $-90^{\circ}$,$90^{\circ}$ and the second $180^{\circ}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In hindsight, I know that when we multiply to remove the denominator (cross-multiplying) that a potential division by zero is lost. We need to review the results and invalidate the 180 degrees for this reason. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The $-90^{\circ}$ is fine, the correct answer, as cos is 0 sin is -1 and the equation fits. But, the second result, $90^{\circ}$ is also wrong. My explanation is that by using squaring as a method of seeking the roots of this equation we run into the dreaded 1 = -1 paradox, and I quickly show them how the squares of these 2 numbers are equal, but they are not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My explanation for the 2 erroneous solutions was fine for 3 of the 7 students who asked me to explain this. The other 4 wanted a better understanding of when &lt;em&gt;we can just accept the results&lt;/em&gt;, note that this was #20 of 30 problems, and the first one that resulted in this issue. It was also the first that offered a fraction like this, prior question were solved by factoring. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question now asks - for this level of math are the two issues I described the only two that result in such erroneous solutions? And so the two rules -&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Check if the results cause a 'division by zero' in original equation&lt;/li&gt;&#xA;&lt;li&gt;Check if you've created a result from the &quot;1 = -1 paradox&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(I think I've articulated the question fairly well. Comments or edits are welcome. Unless they are &quot;why are they solving such problems?&quot; I don't have a say in the material. The choice of degrees vs radians wasn't mine either.)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-04-30T16:48:37.683" LastActivityDate="2016-05-27T14:42:40.903" Title="Rules to eliminate erroneous solutions in Trig equations?" Tags="|secondary-education|trigonometry|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="10942" PostTypeId="2" ParentId="10941" CreationDate="2016-04-30T16:13:54.010" Score="8" Body="&lt;p&gt;This isn't specific to trigonometry at all; it's really about understanding what it means to &quot;solve an equation&quot;. Given functions $f, g: \mathbb{R} \to \mathbb{R}$, to &lt;em&gt;solve the equation&lt;/em&gt; $f(x) = g(x)$ means to determine exactly which numbers $x$ make the equation true, i.e., to compute the set $\{x \in \mathbb{R}: f(x) = g(x)\}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two steps to solving an equation:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Assume&lt;/strong&gt; $x$ is a number such that $f(x) = g(x)$. Under this assumption, deduce that $x$ must be an element of some set $S$ (which we want to be as small as possible to make step 2 easier).&lt;/li&gt;&#xA;&lt;li&gt;For each element $x \in S$, check whether $f(x)$ is actually equal to $g(x)$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;After step 1, we have a set $S$ that contains all the solutions, but it could also contain some things that aren't solutions. Step 2 is &lt;em&gt;always&lt;/em&gt; necessary, because everything in step 1 is done under the assumption that we're &lt;em&gt;already&lt;/em&gt; working with a solution to the equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, it's possible to do both steps simultaneously using biconditionals — for example, if $h: \mathbb{R}^2 \to \mathbb{R}$ is a function such that $y \mapsto h(x, y)$ is injective for each $x \in \mathbb{R}$, then $f(x) = g(x)$ if and only if $h(x, f(x)) = h(x, g(x))$. (This makes precise the idea of &quot;doing the same invertible operation to both sides&quot;. Of course, one would present this in less concise language to a high school class, but the idea is the same.) Many common student errors amount to failing to check injectivity, e.g., squaring isn't injective (although it is if we restrict to &lt;em&gt;non-negative&lt;/em&gt; reals).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, there are two main options: either stick to using only invertible operations (and check that all operations used really are invertible), or do step 2 explicitly by checking each possible solution. In either case, the key is to &lt;strong&gt;use reasoning&lt;/strong&gt;, which requires understanding what it means to &quot;solve an equation&quot;. A list of rules will just give the students another arbitrary thing to memorize and won't address the real issue, which is a lack of reasoning in what they're doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, to address their question directly: When can we just accept the results? Never! One way or another, we must always show the results are correct.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-04-30T16:13:54.010" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10943" PostTypeId="2" ParentId="10941" CreationDate="2016-05-01T00:22:44.960" Score="5" Body="&lt;p&gt;The basic issue here is that there are two kinds of manipulations on equations: reversible ones and non-reversible ones.  Really, when students write out a sequence of equations, there's an implicit logical argument connecting the equations - &quot;if this equation holds then the next equation does&quot;, and the reversible steps are bi-implications while the non-reversible ones one one directional implications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your students need to know the difference: squaring both sides is a non-reversible step, so it can create extra solutions.  Multiplying by a number-which-might-be-0 is a non-reversible step.  Most other steps they're using are reversible.  If, at any point, they use a non-reversible step, they need to check the answer at the end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This isn't the most intellectually rigorous approach, but the entire framework of manipulating equations isn't rigorous either.  The underlying mathematics is that they should be making arguments, not manipulating equations, but getting them to do that conflicts with a curriculum that also wants them to solve a lot of problems in a short amount of time.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2016-05-01T18:42:13.483" LastActivityDate="2016-05-01T18:42:13.483" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10944" PostTypeId="1" AcceptedAnswerId="10945" CreationDate="2016-05-01T08:10:49.990" Score="11" ViewCount="1607" Body="&lt;p&gt;I'd like to able to follow discussions/arguments about maths education, but many of them revolve around the transition to new math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was taught in the UK in the early 90s, and none of the examples or explanations look substantially different to how I do it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was 'old math' and how was it taught? &#xA;When an older person says they don't understand all of this 'places' and 'borrowing' nonsense,  what are they expecting to see instead?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is the difference primarily in notation or in the actual algorithm used?&lt;/p&gt;&#xA;" OwnerUserId="4638" LastEditorUserId="39" LastEditDate="2016-05-01T16:27:36.240" LastActivityDate="2016-05-06T01:46:02.983" Title="What on earth was Old Math?" Tags="|history|new-math|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10945" PostTypeId="2" ParentId="10944" CreationDate="2016-05-01T11:11:38.447" Score="11" Body="&lt;p&gt;New math was introduced in the US in the 1960s.  Before it was introduced, math was taught very algorithmically with little emphasis on understanding. Student were taught a method to do an arithmetic problem and drilled until they remembered the process with no understanding of what was being taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;New math introduced many new topics including boolean logic, other number bases, modular arithmetic, and matrices.  Many elementary school teachers were ill prepared for these new topics.  Many of these are no longer taught in the standard elementary curriculum.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-05-01T11:11:38.447" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10946" PostTypeId="2" ParentId="10919" CreationDate="2016-05-01T22:55:44.490" Score="1" Body="&lt;p&gt;There are so many mathematical directions you can go. I have found &lt;a href=&quot;https://groups.yahoo.com/group/LivingMathForum/&quot; rel=&quot;nofollow&quot;&gt;Living Math Forum&lt;/a&gt; (a yahoo group) to be incredibly eye-opening. There are about 5,000 members, and questions usually get a variety of answers from other homeschoolers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do you want to approach math? There are various curricula available, there are &quot;math readers&quot;, there are activities that help kids make sense of the math, and activities that make it feel useful to them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And how comfortable with math are you?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My answers to you would depend very much on details of your situation and interests.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You also &lt;em&gt;might&lt;/em&gt; be interested in my book, &lt;a href=&quot;http://naturalmath.com/playingwithmath/&quot; rel=&quot;nofollow&quot;&gt;Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2016-05-01T22:55:44.490" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10948" PostTypeId="2" ParentId="10919" CreationDate="2016-05-02T02:03:46.047" Score="2" Body="&lt;p&gt;I rather like The Art of Problem Solving's &lt;a href=&quot;https://www.artofproblemsolving.com/store/list/beast-academy&quot; rel=&quot;nofollow&quot;&gt;Beast Academy&lt;/a&gt; series. It is a complete curriculum, but in a &quot;graphic novel&quot; format, with problem solving emphasis. This can lead into other resources from AOPS for middle and high school. (I was myself homeschooled, homeschool both of my children ages 5 and 8, and am a mathematics professor. In my experience, the perspective of these books will begin to develop the sort of attitude toward mathematics that yields playful and productive discovery, as well as the development of core skills.)&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2016-05-02T02:10:53.020" LastActivityDate="2016-05-02T02:10:53.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10949" PostTypeId="1" AcceptedAnswerId="10957" CreationDate="2016-05-02T05:08:21.197" Score="9" ViewCount="5216" Body="&lt;p&gt;Mathematicians does something when they mean by doing &quot;abstraction&quot; or &quot;generalization&quot;. I can't exactly decipher what they mean, but I suppose&lt;/p&gt;&#xA;&#xA;&lt;p&gt;i. &lt;strong&gt;Abstraction&lt;/strong&gt; is of finding patterns from some objects(and conctretizing something is probably adding structure to the pattern to produce some examples)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Ex:&lt;/strong&gt; &lt;em&gt;From Sun, Blood, Traffic light stopping color etc the pattern red could be found (which is an abstraction). Reversly, adding structure to the red will produce Blood/Sun (Concrete)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ii. &lt;strong&gt;Generalization&lt;/strong&gt; is probably making some constant variable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Ex:&lt;/strong&gt; &lt;em&gt;The binomial theorem can be thought of a good generalization of $(1+x)^3, (x+p)^{13}, (a-b)^{50}$ (Random)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is my notions of abstraction and  generalization right ? If not, then please explicitly state what it is with a lot of example. (An example) What can be treated as a generalization and (separately) abstraction of $1^2 + 2^2 + 3^2 + 4^4 + ... + 66^2 $ (The last term is quite arbitrary)?&lt;/p&gt;&#xA;" OwnerDisplayName="user6330" LastActivityDate="2017-03-06T04:44:18.540" Title="What is abstraction and generalization ?" Tags="|self-learning|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10951" PostTypeId="2" ParentId="10949" CreationDate="2016-05-02T17:29:27.113" Score="-2" Body="&lt;p&gt;I'm not sure what you mean by abstraction, but you can make a universal generalizations from a specific case in one of two ways:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Direct proof:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start by assuming for that sake of argument that $P(x)$ is true, where $P$ is a logical predicate (or truth function) and $x$ is a newly introduced free variable (not previously used in your proof). Suppose that, by using whatever rules and axioms you may have at your disposal, you infer $Q(x)$ is also true, where $Q$ is another logical predicate and $x$ is the same object introduced the initial premise above, and $Q(x)$ has no free variables introduced &lt;em&gt;after&lt;/em&gt; that initial premises. Then you can conclude that, for &lt;em&gt;all&lt;/em&gt; $x$, $P(x)$ implies $Q(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof by contradiction:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start by assuming for that sake of argument that $P(x)$ is true, where $P$ is a logical predicate (or truth function) and $x$ is newly introduced free variable (not previously used in your proof). Suppose that, by using whatever rules and axioms you may have at your disposal, you infer a contradiction of the form $Q$ and not $Q$, or $Q$ if and only if $\neg Q$, where $Q$ is any logical proposition. Then you can conclude that, there does not exist an $x$ such $P(x)$ is true.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6314" LastEditDate="2016-05-04T13:21:14.003" LastActivityDate="2016-05-04T13:21:14.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10952" PostTypeId="2" ParentId="10935" CreationDate="2016-05-02T21:11:55.517" Score="8" Body="&lt;p&gt;I think the best &quot;real&quot; application of the mean value theorem for integrals is to make a rigorous proof of the fundamental theorem of calculus. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-05-02T21:11:55.517" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10953" PostTypeId="1" CreationDate="2016-05-02T21:31:48.487" Score="15" ViewCount="259" Body="&lt;p&gt;I am looking for a study or reference on gender in math problems given in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In math texts or even on math exams, if there is a word problem involving people, these people or &quot;characters&quot; in the problem are given names and pronouns.  For example, if a professor is in the problem, the professor might be male.  If there is a high school teacher or a nurse, this person is given a female pronoun or name.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am particularly interested in studies about how this affects women or minorities in the study of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="228" LastEditorUserId="228" LastEditDate="2016-05-02T21:57:01.420" LastActivityDate="2016-05-25T22:35:37.087" Title="Reference request for studies on gender in math examples, homework problems, or math exams" Tags="|mathematical-pedagogy|reference-request|word-problems|women|gender|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="10955" PostTypeId="2" ParentId="10935" CreationDate="2016-05-03T00:10:47.200" Score="0" Body="&lt;p&gt;You can show that a function $f(x)$ that is continuous and differentiable everywhere with two roots has at least one value such that $f'(c)=0$.  Follows from the mean value theorem since $(b-a)f'(c)=f(a)-f(b)$, or $(b-a)f'(c)=0$ since $x=a,b$ are roots.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2016-05-03T00:10:47.200" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10956" PostTypeId="2" ParentId="2015" CreationDate="2016-05-03T13:10:45.540" Score="4" Body="&lt;p&gt;I teach (see my profile) at the high school level in the US. This is the sequence we use, and I think works.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Triangles - students have already taken geometry, and are familiar with the 45/45/90 and 30/60/90 (degrees, of course) triangles. With the rules of similarity, we introduce the 3 basic trig functions and the initialism SOHCAHTOA which helps students remember the relationship for each. In my opinion, there's something concrete about the examples we offer here. The flagpole, tree, etc in sample problems. &lt;/li&gt;&#xA;&lt;li&gt;Unit circle - this seems a logical progression, as a way to get beyond the 30/45/60/90 angles, and introduce the cyclic nature of the trig functions. I've shown students how a rough sketch can easily remind them of the 30/45/60 sin/cos numbers with little to no memorizing. &lt;/li&gt;&#xA;&lt;li&gt;Periodic sinusoid graph - This seems the logical next step, graphing the sine or cosine value on the Y axis, vs degrees or radians on the X. It segues nicely to more complex equations where the period, amplitude, phase shift, etc, are introduced, and the use of the unit circle wouldn't be sufficient. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-05-03T13:36:58.110" LastActivityDate="2016-05-03T13:36:58.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10957" PostTypeId="2" ParentId="10949" CreationDate="2016-05-04T04:09:40.467" Score="13" Body="&lt;p&gt;&lt;strong&gt;By abstraction, we mean that we step back from concrete objects to consider a number of objects with identical properties simultaneously.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, consider the following three objects:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The set of functions $A,B,C$ defined on the set $\{1,2,3\}$ by $A(1)=1,\;A(2)=2,\;A(3)=3$,&lt;br&gt;&#xA;$B(1)=2,\;B(2)=3,\;B(3)=1$,&lt;br&gt;&#xA;$C(1)=3,\;C(2)=1,\;C(3)=2$,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The set of complex numbers $a = 1, b = e^{i2\pi/3},c = e^{i4\pi/3}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The set of matrices $ \alpha =\left( \begin{array}{c c}1 &amp;amp; 0\\ 0&amp;amp; 1 \end{array}\right),  \beta = \left( \begin{array}{c c}0 &amp;amp; -1\\ -1&amp;amp; -1 \end{array}\right),  \gamma = \left( \begin{array}{c c}-1 &amp;amp; -1\\ -1&amp;amp; 0 \end{array}\right)$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Consider these notations: AB means A(B(x)), $ab$ is ordinary multiplication of complex numbers, and $\alpha \beta$ means ordinary matrix multiplication. Verify the following &quot;multiplication&quot; tables:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{array}{c | c c c}&amp;amp; A &amp;amp; B &amp;amp; C\\ \hline A &amp;amp; A &amp;amp; B &amp;amp; C \\ B &amp;amp; B &amp;amp; C &amp;amp; A\\C &amp;amp; C &amp;amp; A &amp;amp; B \end{array}\;\;\;\;\;\begin{array}{c | c c c}&amp;amp; a &amp;amp; b &amp;amp; c\\ \hline a &amp;amp; a &amp;amp;b &amp;amp; c \\ b &amp;amp; b &amp;amp; c &amp;amp; a\\c &amp;amp; c &amp;amp; a &amp;amp; b \end{array}\;\;\;\;\;\begin{array}{c | c c c}&amp;amp; \alpha &amp;amp; \beta &amp;amp; \gamma \\ \hline \alpha &amp;amp; \alpha &amp;amp; \beta &amp;amp; \gamma \\ \beta &amp;amp; \beta &amp;amp; \gamma &amp;amp; \alpha\\\gamma &amp;amp; \gamma &amp;amp; \alpha &amp;amp; \beta \end{array}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that these tables are identical. Then let us by &lt;em&gt;abstraction&lt;/em&gt; define an abstract object which is the set of three elements $\{e,g,g^{-1}\}$ paired with a binary operator $\cdot$ such that set acts on itself in the following manner with respect to the operator:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{array}{c | c c c} \cdot &amp;amp; e &amp;amp; g &amp;amp; g^{-1}\\ \hline e &amp;amp; e &amp;amp; g &amp;amp; g^{-1} \\ g &amp;amp; g &amp;amp; g^{-1} &amp;amp; e\\g^{-1} &amp;amp; g^{-1} &amp;amp; e &amp;amp; g \end{array}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Group Theory an object with such structure is called the cyclic group of order three. Then the examples above are representations of this abstract object. It is an abstract object because while we have now given it a definition, notice that it is itself a stand-in for a variety of objects that have the properties that it demonstrates. You might even consider the abstract object to be more of a set of properties or a structure than an object itself, since its elements can refer to many (indeed, in this case an infinite number of) actual sets. This abstraction is powerful, since proving any property of this abstract object (like the fact that $x^3$ = e for any element $x$) demonstrates this property for each and every realization, in the same way that observing the properties of an artist's posable figurine allows the artist to determine how to sketch all sorts of different people in many different configurations because it is an abstraction of a human figure. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Generalization, on the other hand, is carried out when we consider an object under inspection in the context of a larger domain for the parameters involved.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, we might define the sine function as the ratio of the side opposite to the acute base angle to the side adjacent to said angle in a right triangle. With this definition, we can only consider the sine of an acute angle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By considering reference angles, we &lt;em&gt;generalize&lt;/em&gt; our definition of the sine of an angle to &quot;the height of a point on the unit circle intersected by a ray at the terminal side of the angle in standard position&quot;. With this more general definition, all of our previous results still hold and we can consider more values of the sine function- any real number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generalizing further, we find that defining sine as a power series admits us to evaluate the sine of a complex number. We see that our original definition was very limited in scope, and by generalizing we obtained new and broader domains which agree with our more narrow previous domains, but in which we find richer or more widely applicable discussion of our query.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a few more examples of generalization:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The improper Riemann integral generalizes the definite Riemann integral, since the definite Riemann integral is only defined for a continuous function on a closed interval. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Lebesgue integral is a generalization of the Riemann integral. For a function defined over a Lebesgue measurable set, if the function is Riemann integrable then the values of the two integrals agree. But the Lebesgue integral provides meaningful results in some cases that the Riemann integral is meaningless. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The norm is a generalization of distance. The $l^p$ norms are generalizations of Euclidean distance, and there are some interesting norms that allow us to define a &quot;distance&quot; between objects that are not geometrical, like the Hamming distance. It turns out that all norms are equivalent up to a scalar and so one can consider a single &lt;em&gt;abstract&lt;/em&gt; norm for many purposes. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The inner product allows one to generalize the notion of an angle. Using the inner product, one can, for example, find the angle between two vectors or two matrices with complex entries or describe which polynomials are &quot;orthogonal&quot;. Orthogonal functions are important in the solution of partial differential equations. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Algorithms are generalizations of solutions to particular problems. The Euclidean algorithm, for instance, allows one to find the greatest common divisor of any two integers, and it is &lt;em&gt;always&lt;/em&gt; a solution. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2017-03-05T23:34:27.127" LastActivityDate="2017-03-05T23:34:27.127" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10958" PostTypeId="1" CreationDate="2016-05-04T08:13:59.233" Score="5" ViewCount="469" Body="&lt;p&gt;In order to introduce the fundamental theorem of Calculus to a 18 year-old class I was thinking about starting with simple telescoping sums (finite sums). It would be important to have examples that are really basic and don't need much algebra and notation to be developed. The most simple example I can think of is the Mengoli series $\sum \frac{1}{n(n+1)}$ (in some finite version) but maybe something more basic could be found that doesn't involve decomposition of algebraic fractions. Any idea?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Connections between telescoping sums and FTC: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The telescoping sum formula is a discrete equivalent of the FTC where integral are replaced by sums and derivative by increments $a_{n+1}-a_n$: the formula $\sum_{k=0}^N(a_{n+1}-a_{n})=a_N-a_0$ has the same structure of $\int_a^bf'(t)dt=f(b)-f(a)$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Telescoping sums can be also used to prove the FTC by evaluating the integral $\int_a^bf'(t)dt$ with a Riemann sum and replacing &#xA;$$&#xA;\sum_k f'(t_k) \Delta t \approx \sum_k [f(t_{k+1})-f(t_k)] =f(b) - f(a) &#xA;$$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5601" LastEditorUserId="5601" LastEditDate="2016-05-07T07:49:50.703" LastActivityDate="2016-05-07T07:49:50.703" Title="Telescoping sums to introduce integral and derivative relationship" Tags="|calculus|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10959" PostTypeId="2" ParentId="10958" CreationDate="2016-05-04T08:27:13.313" Score="4" Body="&lt;p&gt;$\sum_{k=1}^N 2^{-k}=1-2^{-N}$&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-05-04T08:27:13.313" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10960" PostTypeId="2" ParentId="9885" CreationDate="2016-05-04T16:41:24.910" Score="2" Body="&lt;p&gt;There are large numbers of books available in the market for learning mathematics or the geometry. But to be good at this, you should start from basic level such as following CBSE curriculum in India. websites like link &lt;a href=&quot;http://www.khanacademy.com&quot; rel=&quot;nofollow&quot;&gt;http://www.khanacademy.com&lt;/a&gt; or &lt;a href=&quot;http://www.studyrankers.com/&quot; rel=&quot;nofollow&quot;&gt;http://www.studyrankers.com/&lt;/a&gt;  can help you in this. These websites have solutions to NCERT, tips and tricks for Maths including solution to RD Sharma, RS Aggarwal and cover many topics. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="6704" LastActivityDate="2016-05-04T16:41:24.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10961" PostTypeId="2" ParentId="10944" CreationDate="2016-05-04T16:44:22.833" Score="-3" Body="&lt;p&gt;Vedic maths techniques which had been used by Aryabhatta can also be described as old Maths. Though, new Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s.&lt;/p&gt;&#xA;" OwnerUserId="6704" LastEditorUserId="64" LastEditDate="2016-05-06T01:46:02.983" LastActivityDate="2016-05-06T01:46:02.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10962" PostTypeId="1" AcceptedAnswerId="12208" CreationDate="2016-05-04T17:14:18.773" Score="14" ViewCount="733" Body="&lt;p&gt;Copying from &lt;em&gt;Calculus Made Easy&lt;/em&gt; by Silvanus Thompson (2nd ed., 1914):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;CHAPTER I:TO DELIVER YOU FROM THE PRELIMINARY TERRORS&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The preliminary terror, which chokes off most fifth-form boys from even attempting to learn how to calculate, can be abolished once for&#xA;  all by simply stating what is the meaning -in common-sense terms-of&#xA;  the two principal symbols that are used in calculating. These dreadful&#xA;  symbols are:&#xA;  &lt;strong&gt;(1)&lt;/strong&gt; $d$ which merely means &quot;a little bit of.&quot; Thus $dx$ means a little bit of $x$; or $du$ means a little bit of $u$. Ordinary&#xA;  mathematicians think it more polite to say &quot;an element of,&quot; instead of&#xA;  &quot;a little bit of.&quot; Just as you please. But you will find that these&#xA;  little bits (or elements) may be considered to be indefinitely small.&#xA;  &lt;strong&gt;(2)&lt;/strong&gt;, $\int$ which is merely a long $S$, and may be called (if you like) &quot;the sum of.&quot; Thus $\int dx$ means the sum of all the little&#xA;  bits of $x$; or $\int dt$ means the sum of all the little bits of t.&#xA;  Ordinary mathematicians call this symbol &quot;the integral of.&quot; Now any&#xA;  fool can see that if $x$ is considered as made up of a lot of little&#xA;  bits, each of which is called $dx$, if you add them all up together&#xA;  you get the sum of all the $dx$'s, (which is the same thing as the&#xA;  whole of $x$). The word &quot;integral&quot; simply means &quot;the whole.&quot; If you&#xA;  think of the duration of time for one hour, you may (if you like)&#xA;  think of it as cut up into $3600$ little bits called seconds. The&#xA;  whole of the $3600$ little bits added up together make one hour. When&#xA;  you see an expression that begins with this terrifying symbol, you&#xA;  will henceforth know that it is put there merely to give you&#xA;  instructions that you are now to perform the operation (if you can) of&#xA;  totalling up all the little bits that are indicated by the symbols&#xA;  that follow.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;That's all.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I was wondering whether there has been (specific) research as regards the educational merits (or demerits) of such an approach to calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering that basic computational and calculation skills are necessary for the reproduction of society, I can understand why one could approach arithmetic like that (in an attempt to make all citizens literate on arithmetic). But once we are into differential and integral calculus, it is not about basic computational skills anymore.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So does such an approach make for perhaps a fabulous ride in the beginning, only to be regretted later when symbols become more abstract in content and their manipulation must become more rigorous? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or does it instill in the students an attitude like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What? The integral should not necessarily be thought of as a sum? Great, because we were getting bored – let’s see what else we can do with it&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Again, I am not asking for opinions (I find opinions interesting and useful, since they usually come with arguments – at least on this site –, but they are not allowed here as far as I know), but for any research results on the matter.&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="1385" LastEditDate="2017-04-20T13:45:56.813" LastActivityDate="2017-04-20T13:45:56.813" Title="Is there research for or against such an approach in teaching calculus?" Tags="|mathematical-pedagogy|calculus|infinitesimals|" AnswerCount="4" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="10963" PostTypeId="2" ParentId="10958" CreationDate="2016-05-04T20:26:08.837" Score="2" Body="&lt;p&gt;This is not more basic than your example,&#xA;but you can generalize it by replacing $n+1$ by $n+k$.&#xA;Using&#xA;$$&#xA;\frac{1}{n (n+k)} = \frac{1}{k} \left( \frac{1}{n} - \frac{1}{n+k} \right)&#xA;$$&#xA;one can derive &#xA;$$&#xA;\sum_{n=1}^\infty \frac{1}{n (n+k)} = \frac{1}{k} \sum_{n=1}^k \frac{1}{n} = \frac{1}{k} H_k&#xA;$$&#xA;which is certainly not obvious (to me).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For $k=1$, the infinite sum is $H_1/1=1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For $k=2$: &#xA;$$&#xA;\frac{1}{2} \left[&#xA;\left( \frac{1}{1}-\frac{1}{3} \right) +&#xA;\left( \frac{1}{2}-\frac{1}{4} \right) +&#xA;\left( \frac{1}{3}-\frac{1}{5} \right) + &#xA;\left( \frac{1}{4}-\frac{1}{6} \right) + \cdots&#xA;\right]&#xA;= \frac{1}{2} \left( 1 + \frac{1}{2}&#xA;\right)&#xA;$$&#xA;which is&#xA;$H_2/2=(\frac{3}{2})/2=\frac{3}{4}$. &#xA;In other words, everything cancels in the infinite sum &#xA;except $\sum_1^k \frac{1}{n}$.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-05-05T00:17:38.187" LastActivityDate="2016-05-05T00:17:38.187" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10964" PostTypeId="1" CreationDate="2016-05-04T20:56:55.167" Score="12" ViewCount="207" Body="&lt;p&gt;I was wondering if there is any benefit to having undergraduate students swap homework and quizzes and grading them, with solutions that I provide.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking it might be helpful for them to compare another student's, or their own, solutions to my own, so that they can better learn how to write a solution, or maybe see different ways of coming to the same solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach undergraduate math classes that have the students learning how to do proofs, and I was thinking it might be helpful for them to do this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't tried this yet, which is why I was wondering if this is beneficial or detrimental.&lt;/p&gt;&#xA;" OwnerUserId="228" LastEditorUserId="228" LastEditDate="2016-05-04T21:17:26.707" LastActivityDate="2016-05-09T12:05:01.407" Title="Benefits of students grading each other's (or their own) homework or quizzes?" Tags="|homework|grading|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10965" PostTypeId="2" ParentId="10962" CreationDate="2016-05-04T22:41:42.830" Score="5" Body="&lt;p&gt;From what I know of Nonstandard Analysis this seems to be similar. Your notion of &quot;a little bit of&quot; seems very close to infinitesimals. There were attempts to ground Calculus with NSA, and teach it that way, but this way of teaching has not caught on, for whatever reason.  Below is some research on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;H. Jerome Keisler, Elementary Calculus: An Infinitesimal Approach. First edition 1976; 2nd edition 1986.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edward Nelson: Radically Elementary Probability Theory, Princeton University Press, 1987.&lt;/p&gt;&#xA;" OwnerUserId="1816" LastActivityDate="2016-05-04T22:41:42.830" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10966" PostTypeId="2" ParentId="10716" CreationDate="2016-05-05T00:46:48.543" Score="2" Body="&lt;p&gt;Here's an article/video that I found helpful as I teach math to a visually impaired student:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.perkinselearning.org/videos/webcast/teaching-math-students-who-are-blind-or-visually-impaired&quot; rel=&quot;nofollow&quot;&gt;http://www.perkinselearning.org/videos/webcast/teaching-math-students-who-are-blind-or-visually-impaired&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best of luck to you and your students!&lt;/p&gt;&#xA;" OwnerUserId="6707" LastActivityDate="2016-05-05T00:46:48.543" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10967" PostTypeId="2" ParentId="10962" CreationDate="2016-05-05T11:08:31.267" Score="3" Body="&lt;p&gt;I am posting this as an answer to not overburden my question, but also  to add content here that I consider useful. I found the following paper&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.jstor.org/stable/3615117?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;Schwarzenberger, R. L. E. (1980). Why calculus cannot be made easy. The Mathematical Gazette, 64(429), 158-166.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(some info and links on the author can be found &lt;a href=&quot;https://en.wikipedia.org/wiki/Rolph_Ludwig_Edward_Schwarzenberger&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Despite the title, the paper is not a polemic against the book. Rather it starts by noting that two years earlier &quot;A Course of Pure Mathematics&quot; by G.H. Hardy had been published, and the two books represented the end-points of the spectrum as regards the approach to mathematics as a teaching subject, and (quote) &lt;em&gt;&quot;both books shared a tone of evangelical enthusiasm&quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The author summarizes his argument as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;&quot;&lt;strong&gt;(i)&lt;/strong&gt; The set $\mathbb R$ of real numbers possesses simultaneously many different structures.&lt;br&gt;&#xA;  &lt;strong&gt;(ii)&lt;/strong&gt; Any attempt to explain calculus&#xA;  &quot;easily&quot; will use some of these structures but not others, and so will&#xA;  make certain aspects of calculus more easy but other aspects more&#xA;  difficult.&lt;br&gt;&#xA;   &lt;strong&gt;(iii)&lt;/strong&gt; This applies particularily to those aspects of&#xA;  calculus which depend on the completeness of $\mathbb R$, and any&#xA;  attempt to avoid explicit mention of completeness puts a heavy burden&#xA;  on subjective and intuitive ideas about $\mathbb R$.&lt;br&gt;&#xA;   &lt;strong&gt;(iv)&lt;/strong&gt; Small&#xA;  differences in the way different students think about sets and&#xA;  functions may grow into large -and even irreconcilable- differences in intuitive&#xA;  ideas about calculus.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He then discusses the concepts of &quot;Instrumental understanding&quot; vs &quot;relational understanding&quot;, (see &lt;a href=&quot;http://www.skemp.org.uk/&quot; rel=&quot;nofollow&quot;&gt;Skemp, R. R.&lt;/a&gt; (1976). &quot;Relational understanding and instrumental understanding&quot;, &lt;em&gt;Mathematics Teaching 77, 20-26&lt;/em&gt;), cautions about their use, and ends by arguing for&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;&quot;(...)the freedom to teach a particular group of students, with particular intuitive ideas and particular applications of calculus in&#xA;  mind, in a manner appropriate to their particular needs.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Well, certainly such a &quot;flexibility ideal&quot; is indeed only an ideal in any human activity that must be done &lt;em&gt;en masse&lt;/em&gt;, structured as a system, &lt;em&gt;standardized&lt;/em&gt; (some would even raise issues of &quot;equitable treatment&quot; of students if &quot;freedom to adapt&quot; was granted to teachers). Still, &lt;em&gt;some&lt;/em&gt; degree of flexibility must exist (and I suspect it indeed does, at least in practice and unofficially, if not officially sanctioned).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But to an outsider like me, points $(i)$ and $(ii)$ above strike an abstract structural chord as to why &quot;things like calculus&quot; can not be made &quot;universally easy&quot;.&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2016-05-06T15:54:52.737" LastActivityDate="2016-05-06T15:54:52.737" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="10968" PostTypeId="2" ParentId="7281" CreationDate="2016-05-05T13:52:33.807" Score="5" Body="&lt;p&gt;I want to build a factory and I go to a bank for a loan, to finance part of the investment cost.&lt;/p&gt;&#xA;&lt;p&gt;Bankers tend to think in &lt;span class=&quot;math-container&quot;&gt;$a:b$&lt;/span&gt;: &amp;quot;How many euros we will lend for every euro the company will invest&amp;quot;. And they tend to have rules of thumb on the matter, say a &lt;span class=&quot;math-container&quot;&gt;$3:1$&lt;/span&gt; rule. From the point of view of the company, this could be written &lt;span class=&quot;math-container&quot;&gt;$1:3$&lt;/span&gt; and here, confusion may arise more easily, because &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt;&amp;quot; should be interpreted as &amp;quot;the company will invest one third of what the bank will lend&amp;quot; and &lt;em&gt;not&lt;/em&gt; as &amp;quot;one third of the total cost of the factory&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;To arrive at this last magnitude the relation is always&#xA;&lt;span class=&quot;math-container&quot;&gt;$a:b \rightarrow \frac {a}{a+b}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;More generally, I think a fraction &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt; (which then can be written also as a number, a decimal, etc), is meaningful only when &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; measure same entities in nature (in my example, money in the same currency).  But the concept represented usually by &lt;span class=&quot;math-container&quot;&gt;$a:b$&lt;/span&gt; can bring together items that are not alike (say, &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; car-accident deaths per &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; kilometers of highways), in which case there is no meaningful interpretation of &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="376" LastEditDate="2022-04-22T09:11:51.140" LastActivityDate="2022-04-22T09:11:51.140" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10969" PostTypeId="2" ParentId="10962" CreationDate="2016-05-05T22:24:51.757" Score="7" Body="&lt;p&gt;Ideas, devices, methods, etc., under the name &quot;method of exhaustion&quot; were &lt;em&gt;the&lt;/em&gt; effective form of &quot;calculus&quot; for 1500+ years, successfully answering many questions both within mathematics and in continuum models of the world. I myself am very fond of &quot;finding out what is true&quot;, even if I recognize that there are &quot;gaps&quot; or ambiguities. Then one looks at the latter, to resolve them, if there's any doubt, or if the conclusion is sufficiently scandalous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I have often said, and will say again, if the informal (pre-Robinson, pre-Skolem, et-al) idea of infinitesimals had not been so wildly successful, the epsilon-delta episodes in the 19th century would have been of interest to specialists only. (As it is, despite the stories we often tell students, being super-careful with calculus was by far &lt;em&gt;not&lt;/em&gt; the main point of either Cauchy's or Weierstrass' work.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, when I have taught calculus intermittently over the last 40+ years, my first goal is to encourage people to use their physical intuition (and middle-school algebra, and basic geometry) to first-of-all get to an answer, or else discover serious obstacles/dangers. After all, which is better, having an understanding/answer of which one may be only 65% sure, or having no answer at all? Refining a flawed understanding/answer &lt;em&gt;iteratively&lt;/em&gt; is what happens in real life, and in real mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very poignant, and distressing, anecdote about &quot;rigor&quot; in calculus occurred in the mid-1980's when I was teaching a &quot;rigorous math experience for future grade-school teachers&quot;. One could imagine that there was virtue in this. At the outset, a diagnostic &quot;exam&quot; showed that everyone in the room could carry out the usual operations of basic calculus, especially with polynomials and such. Long-story-short, after a quarter's discussion of epsilons and deltas, about half the population came to doubt themselves... to the amazing extent that they balked at doing things they could do a few months earlier. (Sure, maybe I did a bad job talking about things, but I don't think that's the dominant feature, since I was as much a pragmatist then as now...) This stunned me, and not in a good way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, after all, the epsilon-delta biz is just one possible rule-set, with nothing about it more sacred than its current quasi-popularity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet further, one could ask why lower-level mathematics (e.g., through undergrad and basic grad) is so rule-based, so prohibition-based, with quite a few of the rules nearly unguessable to non-initiates. I really do not like the &quot;secret club&quot; aspects... since the genuine difficulties are great enough.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-05-05T22:24:51.757" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10970" PostTypeId="1" CreationDate="2016-05-06T07:16:19.160" Score="9" ViewCount="1014" Body="&lt;p&gt;I think that an international certification of math level would be really important also for non-math educators. Here in Italy primary school teachers actually teach math without being proficient at it. I was wondering, does there exist an international math certification program, analogous to the Common European Framework of Reference for Languages (CEFR)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I'd be a politician, I'd require teachers of every discipline to have at least a B2 in maths :)&lt;/p&gt;&#xA;" OwnerUserId="6286" LastEditorUserId="6286" LastEditDate="2016-05-06T12:36:15.190" LastActivityDate="2022-08-21T18:01:25.153" Title="Does there exist an international certification of mathematical skill?" Tags="|teacher-preparation|certification|teacher-evaluations|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10972" PostTypeId="1" CreationDate="2016-05-06T09:48:36.270" Score="3" ViewCount="348" Body="&lt;p&gt;Recently, I read a paper about student's understanding of fraction. Most  students mentioned in the paper had given $5/8$ as their answer to the following question: &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;  What fraction is shaded? &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/n3Sb9.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/n3Sb9.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For reasons that only an &quot;educated eye&quot; might see, the author of the paper was expecting $5/4$ as &lt;strong&gt;the&lt;/strong&gt; answer. Surprised of the conviction of our educated eyes, I put a voting question (on social media) where &quot;normal&quot; people could vote for $5/8$, $5/4$, or $other$. The result, &lt;a href=&quot;http://amirasghari.com/number-conception/12-2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://amirasghari.com/number-conception/%DA%86%D9%87-%DA%A9%D8%B3%D8%B1%DB%8C-%D8%B1%D9%86%DA%AF%DB%8C-%D8%A7%D8%B3%D8%AA%D8%9F/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, is telling. To make the voters more curious, I used different quantifications of the shaded area to &quot;prove&quot; &lt;a href=&quot;http://amirasghari.com/number-conception/12-explained/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; that $1=2$. For a similar argument, see also &lt;a href=&quot;https://matheducators.stackexchange.com/a/10976/1217&quot;&gt;Benjamin's answer&lt;/a&gt; below. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am now in the position of explaining the problem to more than 400 voters and the aim of this question is to get help to do so. Obviously, the question is about &lt;strong&gt;what&lt;/strong&gt; &lt;strong&gt;fraction of what&lt;/strong&gt; is shaded. More generally, the question is about unit conversion in disguise. So far so good. The problem started when I was going to write about the meaning of the equal sign in unit conversion in comparison with its meaning in arithmetic. Then I realized that I don't know (&lt;strong&gt;and this is my question&lt;/strong&gt;) how these meanings are similar to and/or different from each other. Going into more details, I even don't know, I should think of $5/8$ &lt;strong&gt;and&lt;/strong&gt; $5/4$, or of $5/8$ &lt;strong&gt;or&lt;/strong&gt; $5/4$ as possible answers. Similarly, in writing $1m=100cm$ for a length, is the length quantified by $1$ &lt;strong&gt;and&lt;/strong&gt; $100$, or, $1$ &lt;strong&gt;or&lt;/strong&gt; $100$? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just in case that you are wondering why I have asked this question here, I shall add that because we are the only community that rightly cares about different meanings of equal signs and the difficulties that they cause for our students, and of course, sometimes for us :)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added&lt;/strong&gt;. It's worth mentioning why I perceive this post related (at least partly) to unit conversion. I hope the following equalities tell it all: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1 REC(tangle) = 2 SQU(are)$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, $5/8 REC = 5/8 * 2 SQU = 5/4 SQU$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, you might rightly question what is measured by these fictional units.  For this, we might create some fictional story as well, say measuring pizza in Pizzaland where asking for 2 RECs pizza or equivalently 4 SQUs pizza is quite meaningful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;PS.&lt;/strong&gt; I hope it can be seen now why $1=2$ in the original title, and in the body of the post, is not a contradiction if understood in the context of unit conversion. &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-05-10T23:46:43.600" Title="The meaning of equal sign in unit conversion" Tags="|primary-education|fractions|equal-sign|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="10973" PostTypeId="2" ParentId="10958" CreationDate="2016-05-06T14:02:15.020" Score="3" Body="&lt;p&gt;Exponentials seem the simplest way to go, let me start from Jessica B's answer and make it clearly about a telescoping sum (which it is as soon as one recall how to compute the sum of a geometric series!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $a_n=\frac{1}{2}3^n$, for $n\in \mathbb{N}$. The discrete derivative of this sequence is given by $a'_n=a_{n+1}-a_n=3a_n-a_n= 3^n$. Now, the discrete integral of this new sequence is classically&#xA;$$\sum_{n=0}^{N-1} a'_n = \sum_{n=0}^{N-1} 3^n = \frac{3^N-1}{3-1} = \frac{3^N-1}{2}$$&#xA;but this can be retrieved using the discrete analogue of FTC, aka telescoping sums:&#xA;$$\sum_{n=0}^{N-1} a'_n= \sum_{n=0}^{N-1} \big(a_{n+1}-a_n \big)= a_N-a_0 = \frac{1}{2}3^N-\frac{1}{2}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason to use $3$ rather than $2$ as base is that with $a_n=2^n$, you get $a'_n=2^n$ too and this is confusing. This is also the occasion to explain in a different way the denominator in the sum of geometric series, and to recall that exponentials of base different from $e$ have a constant dropping of at differentiation. In some sense, $2$ is the natural discrete base while $e$ is the natural continuous base.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-05-06T14:02:15.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10974" PostTypeId="2" ParentId="10244" CreationDate="2016-05-06T14:59:40.813" Score="0" Body="&lt;p&gt;I wonder if some geometric examples might be useful. Then the cosets would be something that students could draw.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most basic would be to take the Lie group $U(1)$, thought of as the unit circle in the complex plane. For any $n$, there is an isomorphic copy of $\mathbb{Z}/n\mathbb{Z}$ inside, given by the multiplicative group generated by $e^{2\pi i / n}$. What are the cosets of these  $\mathbb{Z}/n\mathbb{Z}$s?: Each coset is the vertices of a regular $n$-gon. You can even talk about what does the &lt;em&gt;space&lt;/em&gt; of cosets looks like. (Topologically an $S^1$ and isomorphic to $U(1)$, but we should probably think it as having $1/n$ times the linear measure of the $U(1)$ we started with.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another geometric example would be to look at a copy of $U(1)$ inside of $SU(2)$. (It's geometric since $SU(2)$ is diffeomorphic to the 3-sphere.) The copy of $U(1)$ in $SU(2)$ I am thinking about is the one which contains matrices of the form&#xA;$$ \left[\begin{matrix} e^{i\theta} &amp;amp; 0 \\ 0 &amp;amp; e^{-i\theta} \end{matrix}\right]$$&#xA;This is a quotient (the coset space is a 2-sphere) which shows up naturally in physics: &lt;a href=&quot;https://en.wikipedia.org/wiki/Bloch_sphere&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Bloch_sphere&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since each of the cosets is an unknotted curve in $S^3$ and cosets link each other once, it also shows up naturally in knot theory: &lt;a href=&quot;https://en.wikipedia.org/wiki/Hopf_fibration&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Hopf_fibration&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2016-05-06T14:59:40.813" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10975" PostTypeId="2" ParentId="10972" CreationDate="2016-05-06T20:20:38.620" Score="5" Body="&lt;p&gt;Your findings, as far as I'm concerned, have nothing particular to say about fractions and nothing at all to say about unit conversion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You started with a question that was not well-posed. Since we are generally expected to answer questions even if they are not well-posed (mathematicians not doing so annoys people), people voting chose what they interpreted as the most likely intended question, and answered accordingly. What you have actually done is a psychology experiment into the metric people have on a particular type of maths problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My guess would be that the key determining factor is what people have been taught at school. It is not uncommon for practice questions to also be somewhat ambiguous, and typically we have drummed into us how to interpret certain wording to remove the ambiguity. I would also expect the details of how they were presented with the two questions to have some bearing on the results - the two different answers become more obvious when the two diagrams are compared. Also, presenting only two possible answers (it is implicit that 'other' is almost certainly not the 'right' answer) is probably not the best way of testing what people see.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no need to explain to anyone why they are wrong, because no-one is wrong. The question was ambiguous, so there is no unique answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. There is a standard proof that $0=1$, from which $1=2$ is easily deduced. In the book &lt;a href=&quot;https://en.wikipedia.org/wiki/Zero:_The_Biography_of_a_Dangerous_Idea&quot; rel=&quot;noreferrer&quot;&gt;Zero: the biography of a dangerous idea&lt;/a&gt; this proof is used to show that Winston Churchill is a carrot.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-05-06T20:20:38.620" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10976" PostTypeId="2" ParentId="10972" CreationDate="2016-05-06T21:03:09.480" Score="6" Body="&lt;p&gt;You write the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Obviously, the question is about &lt;strong&gt;what fraction of what&lt;/strong&gt; is shaded. More generally, the question is about unit conversion in disguise. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I believe what you have recorded here is the importance of specifying the &lt;em&gt;unit&lt;/em&gt; when broaching fractions. This is one of the principal sources of difficulties for students (and some teachers) in dealing with fractions. One of the places in which this is specified is in the NSF-funded &lt;strong&gt;Elementary Pre-service Teachers Mathematics Project&lt;/strong&gt; (&lt;a href=&quot;http://www.elementarymathproject.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;EMP&lt;/strong&gt;&lt;/a&gt;) where I find the following sample item:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IqaZ0.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IqaZ0.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my perspective, a description of your finding should be as simple as specifying what constitutes a &lt;em&gt;unit&lt;/em&gt; (or &lt;em&gt;whole&lt;/em&gt; or &lt;em&gt;one&lt;/em&gt; -- this last term sharing an etymology with &lt;em&gt;unit&lt;/em&gt;). Thematically, this arises repeatedly as students (and teachers) work their way through not only the equivalence and comparison of fractions, but also in carrying out fraction arithmetic. It occurs in the example you give and in the &lt;strong&gt;EMP&lt;/strong&gt; example I have given with respect to parts of a whole when dealing with discrete items, but it can also rear its head, e.g., during the process of partitioning a number line and using that to add or subtract fractions, or partitioning an area diagram in the process of multiplying fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an example with the number line, one sometimes sees the assertion that fractions placed on two number lines are &lt;em&gt;equivalent&lt;/em&gt; if (and only if) they are directly above (or below) one another. The basic idea of this is pretty accurate, but a bit more needs to be said about alignment; consider the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JsOhK.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JsOhK.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see from this example, literal adherence to the earlier description of &lt;em&gt;equivalence&lt;/em&gt; would yield $1 = 0$. Students are generally able to argue back that the number lines need to be &quot;lined up&quot; somehow; the key in ensuring you have accomplished this, though, involves attending to the notion of the unit or whole that is partitioned:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XK6TH.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XK6TH.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this picture, the latter &lt;strong&gt;?&lt;/strong&gt; can be seen to correspond to $1/2$, but there is not enough information to identify the former &lt;strong&gt;?&lt;/strong&gt;: Even though it &quot;lines up&quot; with the bottom number line, the top number line has not established what a &lt;em&gt;unit&lt;/em&gt; is; so we might (as teachers) assign it the value of $1$ and ask whether the two number lines now assert that $1 = 1/2$. Clearly this is not the case, but the true question is &lt;em&gt;what is still missing&lt;/em&gt; from our notion of equivalence. As I perceive matters, the issue is once again analogous to the one that you have pointed to: The top number line and bottom number line are using a different representation of one &lt;em&gt;unit&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-05-06T21:03:09.480" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10977" PostTypeId="1" CreationDate="2016-05-06T23:51:34.690" Score="15" ViewCount="2657" Body="&lt;p&gt;When I taught calculus, I posted my notes after the lecture.  Then I had the students fill out a mid-quarter evaluation, and a lot of them wanted me to post my notes before class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I started doing was printing and handing out the notes to them, leaving the examples blank so they can fill those in.  Many of them expressed to me that they liked it, so that they can concentrate on the problem instead of trying to write everything down. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just read my course evaluations and I got mixed reviews about the notes.  Some students said they liked them and found them helpful, and an equal amount said that they didn't like it and preferred to take their own notes, and learned better that way. I did tell them that they don't have to use me notes if they don't want them, and to let me know if they don't want to use them so I can save paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;In your experience, what has worked well, to provide notes, let the students make their own notes, or give them the option by posting it before class to they can print out notes if they want to use it?&lt;/strong&gt;   I usually print out the notes, a copy for each student, and pass it out to all of them individually, but I am wondering if I should change this.&lt;/p&gt;&#xA;" OwnerUserId="228" LastEditorUserId="376" LastEditDate="2016-07-23T05:43:11.877" LastActivityDate="2020-09-16T20:09:13.123" Title="Is it better to provide students with guided notes or to have them write their own notes, or both?" Tags="|teaching|lecture-notes|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="10978" PostTypeId="2" ParentId="10964" CreationDate="2016-05-07T12:31:18.673" Score="1" Body="&lt;p&gt;Teaching good peer review skills is a very noble goal. I'm all for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students need to step back from their own point of view to be able to follow the path their peer took. You could start the students with simple problems of type &quot;calculate ...&quot; where about the only kind of error is &quot;wrong number&quot; or &quot;dropped a minus&quot;. Repeated exposure to these tasks will help them later when they work on more complex problems, or when they need to check their own work to find out why that solution is so impossibly far off the target that there must be an error in the calculation leading to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wish that would be taught much earlier than &quot;undergraduate&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1479" LastActivityDate="2016-05-07T12:31:18.673" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10980" PostTypeId="2" ParentId="7281" CreationDate="2016-05-07T16:50:35.587" Score="2" Body="&lt;p&gt;Cooking examples work great here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rice is made with 1 part rice 2 parts water. The ratio is 1:2 rice:water. Or we can say 1/2 the water is the volume of rice to use. Or, as is often the case, you have an odd amount of rice, use twice that amount of water. I once saw my sister fill a 1 cup measure with rice, throw away the rest, and then add 2 cups of water. I asked why she threw out what looked like another 1/2 cup of rice. It wasn't enough for a &quot;recipe&quot;. I don't know if she was a bad cook or bad at math, but she could have just measured 1-1/2 cups rice and double to 3 cups water to use. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We use 2:1 for Margarita's as well. 2 parts Mix to 1 part Tequila. Mix to Tequila is 2:1, and the final drink has 1/3 Tequila, 2/3 Margarita Mix. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's key to understand that (ratio) 2:1 results in 3 parts of stuff, made up of (fraction) 1/3 one ingredient, 2/3 the other. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-05-08T14:41:52.040" LastActivityDate="2016-05-08T14:41:52.040" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10981" PostTypeId="1" CreationDate="2016-05-08T17:45:21.087" Score="11" ViewCount="13108" Body="&lt;p&gt;In my first full-time teaching post, it is very likely that I'll need to be teaching a small amount of GCSE Mathematics to students retaking it. One thing that has been bugging me is that I can't seem to find any sort of &quot;proof&quot; or explanation of why the BODMAS (PEMDAS, for Americans) rule for doing calculations works. I learned this rule in school, and applying it is second nature. It's obvious to me that multiplication precedes addition, and brackets proceeds powers, via elementary properties of real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would be a good way of explaining this rule to students who hadn't seen it before, and does anyone have any understanding-focused ways of explaining why calculations are ordered this way? &lt;/p&gt;&#xA;" OwnerDisplayName="user5447" LastEditorUserId="6725" LastEditDate="2016-05-09T07:14:02.553" LastActivityDate="2019-03-06T08:15:50.973" Title="Proof of why BODMAS (or BIDMAS) works?" Tags="|notation|arithmetic-operations|calculations|" AnswerCount="8" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="10982" PostTypeId="2" ParentId="10981" CreationDate="2016-05-08T20:24:50.477" Score="3" Body="&lt;p&gt;It works by avoiding the ambiguity that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 + 3 x 6 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;would otherwise have. If we simply said we calculate left to right, we'd have a result of 30. With the priority, multiplication higher, we have agreement the above resolves to 20. There's no more complicated origin than this. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-05-08T20:24:50.477" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10983" PostTypeId="2" ParentId="10981" CreationDate="2016-05-08T20:36:15.907" Score="52" Body="&lt;p&gt;It's purely a matter of how we choose to define the notation. The main reason for it is that it lets us write polynomial expressions (which are extremely common) without parentheses, e.g., $x^3 + 3x^2 y - 41x + 2z$ rather than $(x^3) + (3(x^2)y) - (41x) + (2z)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, what really matters is that the notation is clear and unambiguous, so expressions like $a/bc$ should be avoided (and replaced with something like $\frac{a}{b} c$ or $\frac{a}{bc}$ depending on what's intended) rather than trying to rigidly adhere to one convention or the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trying to &quot;prove&quot; a notational convention — or treating it as anything more than an agreed-upon way to efficiently communicate meaning — would represent a fundamental misunderstanding of the way mathematical notation works.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-05-08T20:36:15.907" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="10984" PostTypeId="2" ParentId="10981" CreationDate="2016-05-08T22:02:16.857" Score="11" Body="&lt;p&gt;Perhaps it is worth pointing out that every programming language defines&#xA;an &lt;em&gt;operator precedence&lt;/em&gt; structure to avoid ambiguities.&#xA;An example table for C and C++ can be found &lt;a href=&quot;https://en.wikipedia.org/wiki/Operators_in_C_and_C%2B%2B#Operator_precedence&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&#xA;Ambiguities must be avoided in order for the language parser to create the&#xA;correct compiled (or interpreted) machine code to implement the expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the expression $4+3*2$ could be interpreted as either&#xA;$(4+3)*2=14$ or $4+(3*2)=10$. Operator precedence in most languages follows&#xA;the mathematics convention of ranking multiplication higher than addition, leading to the second interpretation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The operator precedence structure goes well beyond arithmetic, dealing with&#xA;complex expressions such as (in C):&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;a = b &amp;lt; c ? * p + b * c : 1 &amp;lt;&amp;lt; d ()&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Here the first &lt;code&gt;*&lt;/code&gt; is a unary operator, whereas the second &lt;code&gt;*&lt;/code&gt; is a binary operator. (Example from &lt;a href=&quot;https://en.wikibooks.org/wiki/Introduction_to_Programming_Languages/Precedence_and_Associativity&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-05-09T21:00:42.177" LastActivityDate="2016-05-09T21:00:42.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10985" PostTypeId="1" CreationDate="2016-05-08T23:20:39.853" Score="3" ViewCount="103" Body="&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt; I'm helping engineers use software to create 3D geometry in a programmatic way (similar to &lt;a href=&quot;http://www.openscad.org/documentation.html&quot; rel=&quot;nofollow&quot;&gt;OpenSCAD&lt;/a&gt;). The functions they need to call have inputs which are low-level geometry concepts: 3D vectors, coordinate systems, and rigid transforms. This comes very naturally to some, but others struggle; they may have learned these concepts back in high school or college, but they've &#xA;since forgotten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, most of the literature I see introduces these concepts too abstractly for someone to recognize what it means in a 3D context. Naturally, transforms are far more general than three dimensions allow, but seeing simple examples of rotations, translations, and scalings would immediately be helpful. Similarly, vectors are not always 3D, and coordinate systems are not always cartesian, but these cases lend themselves really well to visual, intuitive examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Are there resources (ideally free and online) that define and explain the concepts of 3D geometry primarily inside that context?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="1373" LastActivityDate="2016-07-08T12:31:04.243" Title="Resources on 3D transforms, vectors, coordinate systems" Tags="|technology-in-education|geometry|linear-algebra|intuition|trigonometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10986" PostTypeId="1" AcceptedAnswerId="10987" CreationDate="2016-05-09T01:49:42.427" Score="8" ViewCount="556" Body="&lt;p&gt;I am teaching myself mathematics, and I don't really know anyone who is interested in mathematics. I find that the hardest part about teaching myself is that I am lacking the social aspect of being in a classroom. How can I compensate for the lack of a study group?&lt;/p&gt;&#xA;" OwnerUserId="6708" LastActivityDate="2016-05-15T17:20:26.507" Title="I'm a self learner, what are some ways that I can compensate for the lack of a study group?" Tags="|student-motivation|self-learning|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10987" PostTypeId="2" ParentId="10986" CreationDate="2016-05-09T02:28:58.140" Score="5" Body="&lt;p&gt;For self-learners who have connections to the internet, but otherwise do not have access to other people who share the same interest in mathematics, I suggest participating in online communities such as Stack Exchange.  This will provide you answers to your questions, feedback to your work, and insights from those who are more experienced.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-05-09T02:28:58.140" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10988" PostTypeId="2" ParentId="10986" CreationDate="2016-05-09T03:42:46.607" Score="2" Body="&lt;p&gt;You could try to form a study group with some people online who want to learn about the same thing as you. I've seen people form groups through reddit.com/r/math then use applications like slack.com to center their study groups around. It's a bit hard to keep these types of groups active, but it still could be worth thinking about&lt;/p&gt;&#xA;" OwnerUserId="5958" LastActivityDate="2016-05-09T03:42:46.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10989" PostTypeId="2" ParentId="10986" CreationDate="2016-05-09T06:59:55.540" Score="4" Body="&lt;p&gt;Use stack exchange - it gets brutal sometimes, but at least people don't hold the positive/negative back.&lt;/p&gt;&#xA;" OwnerUserId="6728" LastActivityDate="2016-05-09T06:59:55.540" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10990" PostTypeId="2" ParentId="10981" CreationDate="2016-05-09T10:01:44.227" Score="0" Body="&lt;p&gt;You can prove certain properties of the operations, such as commutativity and associativity. From that we can identify that there are ambiguities in standard notation that need to be resolved. However, the resolution of those ambiguities could be entirely arbitrary, there is no proof for any particular resolution as it is merely a selection, a convention. It may be worth discussing the reasoning behind the convention, though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Brackets take highest priority by virtue of enabling overriding of the convention. If they did not take first precedence they would be ineffective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exponentiation (ordinals) is next due to being the strongest operator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplication and division are of the same relative strength, however, multiplication is commutative whereas division is not, so we give multiplication priority.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Addition and subtraction are last in strength, but again we prioritize addition over subtraction due to its commutativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With these heuristics in place, we can guess at where other operators might fall in order of precedence, for example roots are the same strength as exponents, whereas a factorial we would prioritize above exponents.&lt;/p&gt;&#xA;" OwnerUserId="6732" LastActivityDate="2016-05-09T10:01:44.227" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="10991" PostTypeId="2" ParentId="10985" CreationDate="2016-05-09T11:46:25.140" Score="1" Body="&lt;p&gt;Every computer graphics textbook, course, or tutorial includes such material,&#xA;usually explained for beginners with little mathematical preparation.&#xA;You can find a light tutorial at &lt;a href=&quot;http://www.tutorialspoint.com/computer_graphics/3d_transformation.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;TutorialsPoint&lt;/a&gt;,&#xA;and courses notes all over the web by searching for, e.g.,&#xA;&quot;computer graphics transformations.&quot;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7fX4s.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7fX4s.png&quot; alt=&quot;3DRots&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;sup&gt;&#xA;Snapshot from Berkeley course notes (James O'Brian): &#xA;&lt;a href=&quot;http://www-inst.cs.berkeley.edu/~cs184/fa05/lectures/lecture-04.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-05-09T11:46:25.140" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10992" PostTypeId="2" ParentId="10964" CreationDate="2016-05-09T12:05:01.407" Score="7" Body="&lt;p&gt;A benefit that can be expected is to help student realize how important it is to write mathematics in a language (e.g. English; French in my case). I have often been confronted by students who considered I was being unduly demanding about sentences, correct use of « therefore » or $\implies$, etc. in my grading. I think that having to grade their peers (who will write about as poorly as they do at first) will help them realize how essential it is for the writer to be precise, if he or she wants the reader to be able to get what he or she means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This feeling is backed up somewhat indirectly by the reports written by students who took a one-month internship in a high school, where they observed a teacher, then assisted her or him, including by grading an exam. They were amazed to realize that the poorly written exams made them want to scream to the pupils that they should write things properly if they want to get those damn points!&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-05-09T12:05:01.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10993" PostTypeId="1" CreationDate="2016-05-09T22:04:24.533" Score="11" ViewCount="807" Body="&lt;p&gt;Imagine a child aged 11. They have just finished primary education and now moving into secondary education. This child has shown a great mathematical talent/disposition since a very young age. By the end of primary school they have completed studies that a top tier 13 year old would be doing (in my experience (UK) primary schools are better than secondary schools for giving stretching pieces of work like this).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How should the school/parents deal with this situation? The problem with learning with children their own age is that they are unlikely to learn anything new in class for two years, may get bored with the subject, and lose the enthusiasm they had for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One option is that the parents (or a tutor) could teach them extra maths outside of school. However the child may resent having to do extra learning, and will still be bored in class. The other factor is, what is the end-game in all of this? To do undergraduate maths at an earlier age? I'm not sure that's a good idea, nor that you should be pointing an 11-year-old down such a narrow path. &lt;/p&gt;&#xA;" OwnerUserId="6740" LastActivityDate="2020-07-08T17:25:27.847" Title="How to deal with a talented 11-year-old pupil?" Tags="|secondary-education|gifted-students|" AnswerCount="8" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="10994" PostTypeId="2" ParentId="10993" CreationDate="2016-05-10T03:53:06.180" Score="12" Body="&lt;p&gt;Here in the U.S. there's been a rise in the last decade of &quot;math circles&quot;; extracurricular math clubs with students of the same age, with some amount of play/competitiveness to hone their interest. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disadvantages: They may not solve the problem of being bored in regular class; they may be expensive (not available to the economically disadvantaged); and in some sense they take pressure off public schools to provide high-quality education (increasing inequity between rich and poor students). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.theatlantic.com/magazine/archive/2016/03/the-math-revolution/426855/&quot;&gt;The Atlantic: The Math Revolution&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-05-10T03:53:06.180" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="10995" PostTypeId="1" CreationDate="2016-05-10T06:51:49.043" Score="3" ViewCount="91" Body="&lt;p&gt;I am developing three lectures on &lt;a href=&quot;https://en.wikipedia.org/wiki/Dynamic_programming&quot; rel=&quot;nofollow noreferrer&quot;&gt;dynamic programming&lt;/a&gt;. What is a good place to let peers and/or colleagues and/or students review the material online? Material includes: work sheets, exercises, quizzes, programming code (in R), and last but not least: the script and timeline for applying material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any platforms similar to code review stack exchange - see &lt;a href=&quot;https://codereview.stackexchange.com/&quot;&gt;https://codereview.stackexchange.com/&lt;/a&gt;? Moreover: is mathematics educators &lt;em&gt;the&lt;/em&gt; potential platform for such a review?&lt;/p&gt;&#xA;" OwnerUserId="5142" LastEditorUserId="-1" LastEditorDisplayName="user173" LastEditDate="2017-04-13T12:41:08.550" LastActivityDate="2016-05-11T02:03:42.377" Title="A good (open) place to let colleagues/peers review the material of one lecture?" Tags="|meta|" AnswerCount="0" CommentCount="7" ClosedDate="2019-12-02T06:56:11.367" ContentLicense="CC BY-SA 3.0" />
  <row Id="10996" PostTypeId="2" ParentId="10981" CreationDate="2016-05-10T08:03:48.647" Score="9" Body="&lt;p&gt;To build on other answers, you might show how other conventions exist. Use an H.P. calculator for example (postfix), the LISP family of languages (prefix), and the APL language (all right-associative), all of which &lt;em&gt;do not&lt;/em&gt; have differing precedence of operators at all, and write expressions in different ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given 4 parallel translations of the same expression, the students may better appreciate that the &lt;em&gt;notation&lt;/em&gt; is a communications convention, different from the underlying meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you think &quot;BODMAS&quot; is a chore to remember, take a look at &lt;a href=&quot;https://en.wikipedia.org/wiki/Operators_in_C_and_C%2B%2B#Operator_precedence&quot; rel=&quot;noreferrer&quot;&gt;this chart&lt;/a&gt;!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best way to &lt;em&gt;learn&lt;/em&gt; it is not to use a silly nmenonic,  but to grasp that the order is a convention adoped because people found it handy in their work. Other answers have pointed out that this comes from polynomials. Just knowing of polynomials you &lt;em&gt;know&lt;/em&gt; the convention from that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also look how Einstein came up with &lt;a href=&quot;https://en.wikipedia.org/wiki/Einstein_notation&quot; rel=&quot;noreferrer&quot;&gt;his own convention&lt;/a&gt; which again entails leaving out explicit symbols and groupings and just writing stuff next to each other: in that kind of problem domain, that's a common thing, so the notation can be simplified to that end.  Consider this the same idea as the polynomial: in $2x^3$ where are the symbols and grouping notes?  And then Direc invented a notation that covers the kind of work he was doing.  People will, and continue to, streamline the notation to match the kind of work being done.&lt;/p&gt;&#xA;" OwnerUserId="6744" LastEditorUserId="6744" LastEditDate="2016-05-10T08:14:06.307" LastActivityDate="2016-05-10T08:14:06.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="10997" PostTypeId="2" ParentId="10981" CreationDate="2016-05-10T11:34:24.590" Score="2" Body="&lt;p&gt;It is kind of arbitrary which operation goes first, but my guess is that our current system is just a little more concise in most problems. As an example imagine two systems: a system A that is like our current system, multiplication goes before addition and the multiplication sign can be left out and a system B where addition goes first and the addition sign can be left out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First imagine you want to multiply a bunch of numbers and then add $x$ to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In system A it would look like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a\times b\times c\times d+x=abcd+x$$&#xA;and in system B it would look like this:&#xA;$$(a\times b\times c\times d)+x=(a\times b\times c\times d)x$$&#xA;The parentheses need to be added, because otherwise $x$ would be added to $d$ before being multiplied. Clearly system A wins in this example, but look what happens when the numbers are added first and then multiplied by $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;System A:&#xA;$$(a+b+c+d)x$$&#xA;System B:&#xA;$$abcd\times x$$&#xA;This time system B wins, so it appears it just depends on the situation. Also note that system B looks quite ugly, but only because we are so used to system A. System A still has a couple of tricks up its sleeve though, due to distributivity the second example can be expanded whereas in system B there is no such thing possible.&#xA;$$(a+b+c+d)x=ax+bx+cx+dx$$&#xA;And lastly polynomials (as mentioned in other answers) just don't work as well in system B.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;System A:$$f(x)=ax^2+bx+c$$&#xA;System B:$$f(x)=(a\times x^2)(b\times x)c$$&lt;/p&gt;&#xA;" OwnerUserId="6748" LastActivityDate="2016-05-10T11:34:24.590" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="10998" PostTypeId="1" CreationDate="2016-05-10T16:44:49.217" Score="7" ViewCount="274" Body="&lt;p&gt;The last time I taught the first semester of calculus, I decided to go the route of teaching them limits of functions via characterization by sequences.  I found that many students were able to grasp the concepts much better.  Since we teach a lower level Real Analysis course, they get the $\varepsilon$-$\delta$ definition there.  My issue was that I couldn't find a textbook that used this approach.  I had to resort to TeXing some notes up to supplement the class.  This became too time consuming.  Are there any Calculus texts that define limits using sequences?&lt;/p&gt;&#xA;" OwnerUserId="494" LastEditorUserId="205" LastEditDate="2016-07-22T13:53:20.010" LastActivityDate="2016-07-22T13:53:20.010" Title="Calculus Text that Uses Sequences to Define Limits" Tags="|calculus|textbooks|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="10999" PostTypeId="2" ParentId="10993" CreationDate="2016-05-10T17:52:48.250" Score="11" Body="&lt;p&gt;I might disagree with several implicit hypotheses: that mathematics is only a school subject; that the there is a single linear course through it; that the main option is just the &lt;em&gt;speed&lt;/em&gt; with which one goes through the standard curriculum; that &lt;em&gt;contests&lt;/em&gt; (&quot;competition&quot;) (invariably problem-solving with time constraints) are the main alternative; that some sort of traditional teacher-student relationship (with its challenge-response aspects, if not actually adversarial) is the way that adults can help kids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How about the &quot;adult&quot; role being to find not-necessarily-textbook math books written by real mathematicians about real mathematics, to recommend to the student? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although it might be nice to have a social aspect to a kid's mathematical life, all my observations indicate that this is difficult in many ways. For one thing, kids have been taught (implicitly or explicitly) that school is about competition in the first place, and mathematics perhaps especially so. But not everyone enjoys math-as-competition, and contest-math tends to be heavily caricatured math. Further, although it's nice to be quick, mathematics is not only about speed, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I'm not a fan of most textbooks, either, since they most often make their topics dreary and authority-bound, and more complicated than they really are. Or exaggerated worries and insistence about &quot;rigor&quot; in relatively trivial situations. (E.g., a rigorous proof of a &lt;em&gt;boring&lt;/em&gt; thing is not nearly as interesting as a compelling heuristic for an &lt;em&gt;interesting&lt;/em&gt; thing.) &lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-05-10T17:52:48.250" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11000" PostTypeId="2" ParentId="10993" CreationDate="2016-05-10T23:32:22.147" Score="4" Body="&lt;p&gt;Permit me to support &lt;a href=&quot;https://matheducators.stackexchange.com/a/10999/511&quot;&gt;&lt;em&gt;paul garrett&lt;/em&gt;'s response&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;not-necessarily-textbook math books written by real mathematicians about real mathematics&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;with an unusual suggestion:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Skiena, Steven S. &lt;em&gt;Calculated bets: Computers, gambling, and mathematical modeling to win&lt;/em&gt;. Cambridge University Press, 2001.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I read this book in one day, I found it so engaging. It uses quite a bit of&#xA;mathematical modeling, but always in the pursuit of a practical goal (which he achieves):&#xA;winning Jai-Alai bets against the odds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/xcTK8.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xcTK8.jpg&quot; alt=&quot;Carom&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Calculated Bets, p.12)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;A nice interaction between mathematics, statistics, gambling&amp;mdash;and algorithms and practical programming.&lt;sup&gt;1&lt;/sup&gt;&#xA;It could form a nice math-adult-supervised reading.&#xA;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;&#xA;Skiena is the author of: &lt;em&gt;The Algorithm Design Manual&lt;/em&gt;. Springer Science &amp;amp; Business Media, 1998.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-05-10T23:32:22.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11001" PostTypeId="2" ParentId="10972" CreationDate="2016-05-10T23:46:43.600" Score="0" Body="&lt;p&gt;You might get better answers by:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;drawing some crust around the edges and asking &lt;em&gt;&quot;how many &lt;a href=&quot;http://en.wikipedia.org/wiki/Detroit-style_pizza&quot; rel=&quot;nofollow&quot;&gt;Detroit pizzas&lt;/a&gt;?&quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;drawing some crust around new circular borders and asking &lt;em&gt;&quot;how many pies?&quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;drawing 100m for comparison and asking &lt;em&gt;&quot;how many hectares?&quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Turning this into a word problem develops a more useful skill with an easier question.&lt;/p&gt;&#xA;" OwnerDisplayName="user173" LastActivityDate="2016-05-10T23:46:43.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11002" PostTypeId="2" ParentId="10993" CreationDate="2016-05-11T10:16:14.693" Score="2" Body="&lt;p&gt;This happened to me at an earlier stage - in an attempt to keep me occupied before I was old enough to go to school,  my mum did everything up to long division with me by the time I was 3. She stopped after that as I had other things to keep me busy, but I can still remember the crisis point I hit when we got to long division at school, because I wasn't used to there being something mathematical that I didn't already know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are lots of great specific answers here on things you can do, but the important thing is yes, you need to keep them learning and experiencing things they don't know, otherwise they will struggle when they are next out of their comfort zone. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally speaking, there are two areas you can explore - Applications and Theory. I know from my own education that school maths tends to be lacking in both of these areas, so looking at an area of practical maths (physics and programming are the obvious ones, but maths is everywhere!), or delving into the theory behind what they have already learned (axioms of arithmetic, real analysis) can provide an extremely useful grounding that makes learning more complex ideas a lot easier. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's an awful lot of resources for both of these online - even good old Wikipedia tends to be incredibly accurate when it comes to maths - so it needn't be a task that takes up a lot of the mentor's time. Set research projects and ask them to give you a presentation/short essay on what they found - writing about maths and reporting is another area that isn't necessarily taught very well and is a very useful skill!&lt;/p&gt;&#xA;" OwnerUserId="6023" LastActivityDate="2016-05-11T10:16:14.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11004" PostTypeId="2" ParentId="4223" CreationDate="2016-05-12T13:36:08.620" Score="2" Body="&lt;p&gt;How many hours a day do you read&lt;/p&gt;&#xA;" OwnerUserId="6756" LastActivityDate="2016-05-12T13:36:08.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11006" PostTypeId="2" ParentId="10998" CreationDate="2016-05-12T22:11:49.957" Score="2" Body="&lt;p&gt;Here is first year calculus text which uses sequences to define continuity of functions: &lt;a href=&quot;http://www.synechism.org/wp/difference-equations-to-differential-equations/&quot; rel=&quot;nofollow&quot;&gt;http://www.synechism.org/wp/difference-equations-to-differential-equations/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found the exposition in this book to be very clear and every step to the concept of the limit is made explicit.&lt;/p&gt;&#xA;" OwnerUserId="6759" LastActivityDate="2016-05-12T22:11:49.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11007" PostTypeId="2" ParentId="10244" CreationDate="2016-05-13T04:19:25.127" Score="2" Body="&lt;p&gt;I propose that it's not cosets themselves that are hard to understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, the problem is that they often arise not because they are a thing of interest but instead as a technical device &amp;mdash; a trick for systematically converting problems about congruence relations into problems about equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the memories are hazy, I recall that I personally didn't really gain any facility in working with cosets until I did the reverse &amp;mdash; ignore the coset structure and instead treat them as funny notation for working with a congruence relation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And honestly, I think that better reflects actual real-world practice for doing computations in many cases, such as modular arithmetic. Which version of Fermat's little theorem would you expect to see and use?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$a^p \equiv a \pmod p$&lt;/li&gt;&#xA;&lt;li&gt;$[a^p]_p = [a]_p$&lt;/li&gt;&#xA;&lt;li&gt;$a^p + p \mathbb{Z} = a + p\mathbb{Z}$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(in fact, I've since become enamored with the idea of working with &lt;a href=&quot;https://en.wikipedia.org/wiki/Setoid&quot; rel=&quot;nofollow&quot;&gt;setoids&lt;/a&gt; instead of sets)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I opine that using cosets for the purpose of this technical trick would be easier to swallow if students first had experience actually working with a congruence relation, and then emphasis was placed on the mechanics of how working with congruence classes can be used as a substitute for working with the congruence relation, and that the congruence classes are cosets of the class of zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, cosets as being objects of interest their own object of interest needs to be introduced somewhere along the line too.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2016-05-13T06:55:42.807" LastActivityDate="2016-05-13T06:55:42.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11008" PostTypeId="2" ParentId="10832" CreationDate="2016-05-13T22:51:13.770" Score="7" Body="&lt;p&gt;The adjective &lt;strong&gt;simple&lt;/strong&gt; is heavily overloaded (to use programming terminology)&#xA;in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &lt;em&gt;simple polygon&lt;/em&gt; is non-self-intersecting. It can be arbitrarily complicated, but still called &quot;simple.&quot;&#xA;Analogously, a &lt;em&gt;simple geodesic&lt;/em&gt; is non-self-intersecting.&#xA;So a closed geodesic is a geodesic that forms a &lt;em&gt;simple&lt;/em&gt;, closed curve.&#xA;A Jordan curve is a plane, &lt;em&gt;simple&lt;/em&gt;, closed curve.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;em&gt;simple polytope&lt;/em&gt; in $\mathbb{R}^d$ has each vertex adjacent to $d$&#xA;faces. So in $d=3$, every vertex is trivalent.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;em&gt;simplicial polytope&lt;/em&gt; has every facet a simplex. So in $d=3$, this&#xA;is a triangulated polyhedron.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;em&gt;simple group&lt;/em&gt; has no normal subgroups (except the trivial subgroup&#xA;and the group itself).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;em&gt;simple graph&lt;/em&gt; has no loops or multiple edges.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;em&gt;simple root&lt;/em&gt; of a polynomial is one with multiplicity $1$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;em&gt;simple pole&lt;/em&gt; is a pole of order $1$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A &lt;em&gt;simple algebra&lt;/em&gt; is one with no nontrivial ideals.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One could easily extend this list...&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-05-13T23:40:21.177" LastActivityDate="2016-05-13T23:40:21.177" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11009" PostTypeId="2" ParentId="10981" CreationDate="2016-05-14T06:03:24.123" Score="8" Body="&lt;p&gt;I understand this is not a realistic suggestion, but can you avoid &quot;teaching&quot; &quot;PEMDAS&quot; or &quot;BOMDAS&quot; altogether, and teach your students just the math instead? As pretty much everybody already said, this is not actually a rule -- this is a mnemonic device that's supposed to help students remember the actual rules of the order of operations (in the traditional math sense; programming languages and software packages are a whole different story). And then the order of operations is again not as much rules as conventions, intended to simplify notation.&#xA;The problem I have with &quot;PEMDAS&quot; is that in my experience it does more harm than good. Way too many students follow it literally and actually evaluate 8-2+1 to 5 because &quot;A&quot; in the &quot;PEMDAS&quot; &quot;rule&quot; is before &quot;S&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this is merely a single example, and not even the worst one, of a much deeper problem when students learn math superficially by rote memorization of rules instead of understanding and internalizing the content.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer: as a college math instructor, I'm on the receiving side of what comes out of school education, but I'm not involved in school education.&lt;/p&gt;&#xA;" OwnerUserId="6762" LastActivityDate="2016-05-14T06:03:24.123" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11011" PostTypeId="1" CreationDate="2016-05-15T04:59:41.227" Score="4" ViewCount="242" Body="&lt;p&gt;Consider the case of a college remedial/developmental course; traditionally arithmetic and elementary algebra. If we had the opportunity to pick a new program for the course, and wanted to include some basic logic exposition in it, is there a textbook like that?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In sum, the course would cover some combination like: Logic, basic operations, solving equations by inverses, graphing, estimations and double-checks, etc. Anything like that? Open educational resource (OER) options are also open to consideration.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-05-15T05:11:16.500" LastActivityDate="2016-05-16T17:07:25.990" Title="Is there a remedial college textbook that includes basic logic?" Tags="|undergraduate-education|textbooks|community-colleges|remedial-courses|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11012" PostTypeId="2" ParentId="771" CreationDate="2016-05-15T05:44:43.843" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;My Question: Have other CC teachers noticed a similar trend? For those&#xA;  of you who have--and maybe have more experience dealing with this&#xA;  problem--what have you done when confronted with this issue?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have definitely seen exactly this trend, in exactly those skills, in my two decades off-and-on of teaching at this same level. My response in the last year or so has been to develop a website designed to drill students on these basic skills with &lt;em&gt;timed&lt;/em&gt; quizzes. (esp.: Times tables, negative numbers, and order of operations.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For many it's the first time that an instructor has clarified that these skills are meant to be &lt;em&gt;automatic&lt;/em&gt;; that is, not done by counting on their fingers (which is quite common). Giving exercises and personal feedback every half-hour in class gives me a chance to diagnose and direct students to the website for practice when I see these problems pop up. It also helps if I introduce the site in class on an overhead projector and run through a quiz collectively with the whole class calling out answers. See:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.automatic-algebra.org/&quot; rel=&quot;nofollow&quot;&gt;Automatic-Algebra.org&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-05-15T05:44:43.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11013" PostTypeId="2" ParentId="10986" CreationDate="2016-05-15T16:33:53.560" Score="1" Body="&lt;p&gt;Since you asked this question, I am assuming you have the internet in your pocket (or wherever) and that you can use &lt;a href=&quot;http://math.stackexchange.com&quot;&gt;math.stackexchange.com&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simply participating on this site has made me much more educated in math, and more importantly, I think more like a real mathematician.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to get some goals down on the table to:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to answer a few questions per week.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to ask an intelligent question each week.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you just do those two goals each week on the math SE site, you'll start learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another interesting thing you could do is to try and get a notebook (or 5...) and fill it with everything you know about math.  Just dump it, from summations to pi to rational functions etc.  And every time you have a problem, try to work it out in that notebook.  Also write down interesting things you see on the math SE site that you can understand.  (You'll find you probably don't know so much about pi, so you could look that up)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if you have school, take a notebook with you too.  Always have a notebook with empty pages so you can write down whatever pops to your mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And use pen.  This way, you can see your mistakes.  If you mess up, don't scratch it out, just put one big line through it so you know its wrong but you can still see it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't be afraid write trivial statements, like $m=\frac{y_2-y_1}{x_2-x_1}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There will be a point when certain things you just know, like the quadratic formula.  Use your notebook to show why that is the case, why the quadratic formula works, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And some silly ideas might actually seem profound to you, like if you haven't taken Calculus and I had told you &quot;use the slope formula from algebra to find the slope of $y=x^2$ at $x=2$&quot; you should actually try it.  Feel free to try it before viewing the answer.&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; $m=\frac{y_2-y_1}{x_2-x_1}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; $m=\frac{(x_2)^2-(x_1)^2}{x_2-x_1}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; $m=x_2+x_1$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt;$=2+2=4$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Slope at $x=2$ means we plug that in for both $x_2$ and $x_1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of course, use colored pens as needed, because sometimes the page gets messy, and colored ink is more fun to use anyways.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="6220" LastEditDate="2016-05-15T17:20:26.507" LastActivityDate="2016-05-15T17:20:26.507" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11014" PostTypeId="1" CreationDate="2016-05-15T17:14:50.187" Score="3" ViewCount="150" Body="&lt;p&gt;how to create videos for teaching maths and what are the Hardware and software required. I know to prepare slides but not free flow.I want to preapre videos just like teaching in a class room but without the teacher in the video.How to insert connecting arrows or how to highlight in the presentation&lt;/p&gt;&#xA;" OwnerUserId="6767" LastActivityDate="2016-05-15T17:14:50.187" Title="videos for teaching and publishing in youtube" Tags="|interactive-teaching|" AnswerCount="0" CommentCount="3" ClosedDate="2016-05-18T00:44:40.910" ContentLicense="CC BY-SA 3.0" />
  <row Id="11015" PostTypeId="2" ParentId="10244" CreationDate="2016-05-16T09:14:44.677" Score="2" Body="&lt;p&gt;In fact, seeing this question again, I think that there is one aspect which may explain a big chunk of why student struggle with cosets and seems not to have been mentioned: &lt;strong&gt;coset might be hard to understand because with them, we try to change the very notion of &quot;mathematical object&quot; in the students' mind.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar problems arise in many occasions when a similar cause can be blamed:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;students struggle with functions, which turn into object the operations that where applied &lt;em&gt;to&lt;/em&gt; their only objects thus far, numbers; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;they struggle with propositional and basic logic, which turns assertions into objects that can be manipulated just like numbers, although with their own set of operations;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;they struggle with basic set theory, which turns sets into objects to be manipulated while they where mere containers, or symbols only meant to give the type of a number;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;they struggle with cosets, which are subsets, meant to precise the type of an element of a set, and (after a previous struggle) could be subject to some operations such as intersection and union, but are now meant to be subject to algebraic operations, as if they where numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-05-16T09:14:44.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11016" PostTypeId="2" ParentId="11011" CreationDate="2016-05-16T17:07:25.990" Score="3" Body="&lt;p&gt;You might be able to find a &quot;math appreciation&quot; or &quot;math for elementary education majors&quot; type of text that works for you. Around 1993-1996 I taught perhaps 12 to 15 semester courses from an earlier edition of &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0321693817&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Mathematical Ideas&lt;/strong&gt;&lt;/a&gt; by Miller/Heeren/Hornsby for a 2-semester course sequence for students (in nursing, education, and some other related majors) that was their first math class &lt;em&gt;after&lt;/em&gt; finishing the remedial/developmental courses. Students found this material &lt;em&gt;easier&lt;/em&gt; than their remedial/developmental courses (and in many ways it was actually easier in an absolute sense, and not just in a relative sense due to the students having more math background when they took the courses), so I can easily see this text working for some remedial/developmental programs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, instead of trying to find a single text for everything, perhaps look for suitable logic material independently from looking for a text for the other topics, such as something from the internet, or one or more book chapters copied with permission and bound by your bookstore (or library), or something written by your department and bound by your bookstore (or library) and sold cheaply to students, etc.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2016-05-16T17:07:25.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11017" PostTypeId="1" AcceptedAnswerId="11019" CreationDate="2016-05-16T17:54:03.610" Score="4" ViewCount="449" Body="&lt;p&gt;If a maths lecturer is lecturing a higher-level undergraduate course outside their area (i.e. has not taken a course in themselves or learned themselves over the course of their research), how could you justify to the students that they are suitably qualified to teach and assess them?&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="4746" LastEditDate="2016-05-17T06:13:01.337" LastActivityDate="2019-07-01T06:54:15.183" Title="Justifying non-expert lecturer" Tags="|undergraduate-education|" AnswerCount="5" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="11018" PostTypeId="2" ParentId="11017" CreationDate="2016-05-17T11:15:18.507" Score="9" Body="&lt;p&gt;Usually undergraduate level courses are not very difficult for experienced researchers/lecturers. Even without previous research experience in the specific topic, lecturers can learn the material very fast and be able to teach the topic to students well enough afterwards. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it may be the case that the lecturer &lt;em&gt;appears&lt;/em&gt; to be unqualified or unfit to teach the material, but based on my personal experience I think this often reflects on the lecturer's ability to &lt;em&gt;teach&lt;/em&gt; rather than their knowledge on the subject.&lt;/p&gt;&#xA;" OwnerUserId="6769" LastActivityDate="2016-05-17T11:15:18.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11019" PostTypeId="2" ParentId="11017" CreationDate="2016-05-17T14:52:57.610" Score="13" Body="&lt;p&gt;The comments are very interesting, but you are talking about how to justify to students.  That may not always be necessary, but here is a stab at it if someone really asked (which I find somewhat unlikely).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The instructor has a PhD in mathematics, which means that s/he has learned not just how to learn independently, but how to become an expert in mathematical subject matter.  Given the interconnectedness of mathematics, I am confident that Dr. X will do an excellent job not only with the subject at hand, but how it fits into your studies in general, both before and after this class.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some may disagree with this characterization, but anyway I think it's more true than not.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously the first time won't be the best, but that is something that can be said for any course when taught the first time, including freshman calculus/college algebra/finite mathematics or whatever your &quot;first&quot; college course happens to be.  How many people really &quot;get&quot; the Mean Value Theorem until they have to teach it?  Proving random stuff with it in some undergraduate real analysis course also likely lies years behind instructor Y or Z, not just instructor X.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-05-17T14:52:57.610" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11020" PostTypeId="2" ParentId="11017" CreationDate="2016-05-17T16:26:49.703" Score="2" Body="&lt;p&gt;Assessment is the easy part, anyone who is well versed in the logic of mathematics should be able to vet the student's proofs and computations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teaching is less clear. You want the lecturer to know what the important parts are. If they aren't familiar with the subject, how are they going to know where the most fertile ideas can be found? (This can be remedied if there is a good textbook available.) Still, you can always run the class --- not as a lecture --- but as a seminar where everyone is upfront about the fact that everyone, including the professor, is learning the subject together. I've certainly had classes like that in graduate school, for emerging topics. They were fun!&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2016-05-17T16:26:49.703" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11021" PostTypeId="1" CreationDate="2016-05-17T22:31:37.340" Score="7" ViewCount="286" Body="&lt;p&gt;I am an professional that has rarely used math over the last 20 years. Well, anything outside a spreadsheet and basic formulas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My son, who is quite bright, is now getting to an age where I am finding it difficult to help him with this studies. So, I want to relearn what I lost. From what I remember, the progression is Pre-Algebra, Algebra, Geo., Trig., Pre-Calc, Calc... That right? Any new and fun ways to learn? Just pick up books and go for it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am doing this for me and him, so either way, I going to try. Just want to make it as efficient as possible.&lt;/p&gt;&#xA;" OwnerUserId="6770" LastActivityDate="2017-06-11T07:23:56.217" Title="Looking for a progression of learning to get back my math skills" Tags="|self-learning|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11022" PostTypeId="1" CreationDate="2016-05-17T23:04:54.530" Score="13" ViewCount="937" Body="&lt;p&gt;Like it is just as primitive as makes intuitive sense in a way similar to division. I feel like teaching 4th graders the modulo function would complement learning division extremely well, as it helps make the idea of remainders and such concrete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But first time I learned about modulo was comp sci in 11th grade. Calculators have five big buttons– +, -, x, ÷, = –but no modulo. PEMDAS (even excusing its flaws) includes no mention of it, and I've never, in any math class, plotted the graph of X%3. I feel like learning it a younger age makes using it more second-nature and gives a greater familiarity with the way numbers work. It's a good conceptual tool. In fact, why don't we teach the floor function too? It makes sense especially when considering things like time and its weird base 24-60-60-10 system&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why is it so obscure? Is it just not that applicable (I mean, I can think of lots of examples where it is…) Is it just one of those things that hasn't occurred to anyone?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Really, my question is not even about the practicality or the benefits. It just seems like a very basic and straightforward function that has simply been ignored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note: Some responses have noted that my use of the word modulo is questionable. Sorry about that. If it's unclear, by &quot;modulo&quot;, I mean &quot;remainder&quot;, which I had previously thought was a synonym of the aforementioned word.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="6771" LastEditorUserId="6771" LastEditDate="2016-05-27T12:28:45.393" LastActivityDate="2016-06-09T00:53:58.113" Title="Why is modulo not an elementary operation?" Tags="|arithmetic-operations|philosophy-of-maths|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="11023" PostTypeId="2" ParentId="11022" CreationDate="2016-05-18T01:47:40.127" Score="5" Body="&lt;p&gt;Here's a few things you might consider that distinguish the modulo operation (&lt;a href=&quot;https://en.wikipedia.org/wiki/Modulo_operation&quot; rel=&quot;nofollow&quot;&gt;as referred to in computing&lt;/a&gt;, that is, remainder after division) from the elementary operations (add, subtract, multiply, divide):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The graph of modulo is not continuous.&lt;/li&gt;&#xA;&lt;li&gt;Modulo does not have a derivative everywhere (not analytic).&lt;/li&gt;&#xA;&lt;li&gt;The function x ◦ 3 is one-to-one for the elementary operators, but not for modulo.&lt;/li&gt;&#xA;&lt;li&gt;Because of the preceding, modulo does not have an inverse (it is not paired with another operator the same way elementary operations are).&lt;/li&gt;&#xA;&lt;li&gt;You can't double-check it the way you can the elementary operations.&lt;/li&gt;&#xA;&lt;li&gt;Modulo is infrequently seen in formulas for common applications (physics, business, biology, statistics, geometry, etc.)&lt;/li&gt;&#xA;&lt;li&gt;It is not an operator seen in polynomials, upon which much of the theory for algebra and number systems is built.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The same can be said for the floor function.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-05-31T16:52:17.797" LastActivityDate="2016-05-31T16:52:17.797" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="11024" PostTypeId="2" ParentId="130" CreationDate="2016-05-18T09:47:49.977" Score="14" Body="&lt;p&gt;While teaching polynomials to future high-school teachers, I realized one example of pretty advanced mathematics that is needed to understand why a certain important (but often kept implicit) result in high school mathematics is a result, and not something so obvious that it need not be even stated: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Two real polynomials taking the same value at each point are equal coefficient-wise.&lt;/strong&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It has proved hugely difficult to make my students realize there where something to state here, because they did not question the notion of equality, and did not realize that the definition of equality for functions and for polynomials might conflict each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, once one has seen &lt;strong&gt;finite fields&lt;/strong&gt;, one can be given an example of a non-zero polynomial whose polynomial function vanishes identically. I don't see any better way to show that there is something to prove in the real case.&#xA;There are some tricks, such as using trigonometric polynomials for comparison ($\cos^2+\sin^2$ is coefficient-wise different from $1$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the finite-field example made it also possible for me to convince my student that they would never, ever be able to prove that coefficient-wise equality and functional equality agree just by using algebraic manipulations, because they would be able to apply their proof to the finite field case. This helped them accepting that it is worth giving a proof, but also that this proof &lt;em&gt;has&lt;/em&gt; to use something more than the field operations (I gave two flavor of proofs, one using polynomial algebra and the fact that there are infinitely many real numbers, and one using topological arguments).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, as high-school teachers they will not teach finite fields, and will probably not insist on the distinction between equality as polynomial and functions. But they have learned that there is something they will be pushing under the rug, making them certainly better at presenting the material, understanding some possible misconceptions by students, and answering some questions if they turn out. These look like invaluable teaching skills.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-05-18T09:47:49.977" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11025" PostTypeId="2" ParentId="11022" CreationDate="2016-05-18T12:19:06.987" Score="7" Body="&lt;p&gt;Modulo is not a well-behaved operation when thought of as being solely defined in terms of integers, in the context of how it interacts with the other basic operations.  Taking as the definition that $mod(m,n)$ is the unique number $r$ with $0 \leq r &amp;lt; n$ and $m - r$ is divisible by $n$, then you run into unpleasantries such as $$mod(m_1 + m_2, n) \neq mod(m_1,n) + mod(m_2,n).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is much better conceptually to fix a modulo $n$ and define an arithmetic structure (typically called the ring of integers modulo $n$) with a map from integers to this ring which essentially associates an integer $m$ with its remainder $\overline{m}$ upon dividing by $n$, but the remainder should be thought of as an element of this new arithmetic structure.  Then, inside this new arithmetic structure, you have nice arithmetic properties such as $$\overline{m_1 + m_2} = \overline{m_1} + \overline{m_2}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, you can certainly introduce modular arithmetic without getting into this issue in technical detail. But still, it is important to realize that some adjustment has to be made because of it. For example, 5 hours after 10:00 is not 15:00 (the primary exception in the U.S. being if you're in the military!), but rather 3:00.  Thus, hours on a clock do not add in the &quot;obvious&quot; way, but instead one has to account for passing 12:00 when adding.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2016-05-18T22:07:59.607" LastActivityDate="2016-05-18T22:07:59.607" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11026" PostTypeId="1" AcceptedAnswerId="11031" CreationDate="2016-05-18T15:46:27.977" Score="32" ViewCount="2365" Body="&lt;p&gt;I teach to predominantly non-majors in college algebra, precalculus, and calculus.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can one possibly incentivize or rationalize assigning practice problems outside of class when this software is readily available?  Math/engineering folk need to understand the algebraic manipulation behind differentiation and integration, but not the social science or bio-tech majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student asks me &quot;Why do I need to know how to do this if Wolfram Alpha can just do it for me?&quot;.  I honestly don't know how to answer anymore.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I take great issue with answers like &quot;You won't have wolfram alpha on the test&quot; or &quot;Maybe the internet will break or the power will go out&quot;.  I think we need to embrace it to allow us to go deeper, like the Ti8X's did.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;WAlpha has also rendered any meaningful outside-of-class assessment useless, as I will just get regurgitated output, or &quot;trained&quot; results from the tutors in the math center.  Any knowledge demonstrated on these assignments is fleeting, such is seen on the exams and in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How have you solved this problem?&lt;/p&gt;&#xA;" OwnerUserId="6590" LastActivityDate="2022-03-16T03:51:14.340" Title="How to solve the problem of Wolfram Alpha?" Tags="|technology-in-education|student-motivation|assessment|" AnswerCount="8" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="11027" PostTypeId="2" ParentId="9594" CreationDate="2016-05-19T08:18:21.933" Score="1" Body="&lt;p&gt;I am by no means an expert, so the references listed below are likely to be extremely incomplete. Hopefully they can help in hunting down further references. (These are based on a short essay written with Eero Hakavuori.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reciprocal teaching/learning/education was introduced in Palincsar's thesis [1] and in an article of Brown and Palincsar [2]. Their original motivation was helping poor readers to perform better by using reading strategies (I think tactics is more accurate word here) that more skilled readers use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The method of teaching the tactics: First teacher demonstrates them, and then gradually, step by step, gives more independence to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that the reciprocal teaching process and the tactics taught through it are called reciprocal teaching; at least in the articles I read the concepts were not explicitly separated and named.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There exists a general review of reciprocal teaching [3].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is some research on word problems in specific [4,5].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We did not find anything related to understanding or composing proofs, or more general mathematical problem-solving.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;[1] Annemarie Sullivan Palincsar. Improving the reading comprehension of junior high students through the reciprocal teaching of comprehension-monitoring strategies. PhD thesis, University of Illinois, 1982.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[2] Annemarie Sullivan Palincsar ja Ann L. Brown. Reciprocal teaching of comprehension-fostering and comprehension-monitoring activities. Cognition and instruction, 1(2):117–175, 1984.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[3] Barak Rosenshine ja Carla Meister. Reciprocal teaching: A review of the research. Review of educational research, 64(4):479–530, 1994.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[4] Delinda van Garderen. Focus on inclusion: reciprocal teaching as a comprehension strategy for understanding mathematical word problems. Reading &amp;amp; Writing Quarterly: Overcoming learning difficulties, 20(2):225–229, 2004.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[5] Allan Collins, John Seely Brown ja Susan E. Newman. Cognitive apprenticeship: Teaching the crafts of reading, writing, and mathematics. In a book edited by Lauren B. Resnick: Knowing, learning, and instruction: Essays in honor of Robert Glaser, pages 453–494. Lawrence Erlbaum associates, publishers, Hillsdale, New Jersey, USA, 1989.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2016-05-19T08:18:21.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11028" PostTypeId="2" ParentId="10935" CreationDate="2016-05-19T08:31:08.140" Score="0" Body="&lt;p&gt;The construction of the delta distribution as a limit of box functions. Consider&#xA;$$h_{c}(t) = \begin{cases} 0 &amp;amp; t &amp;gt; |c|,\\ \frac{1}{2c} &amp;amp; t \leq |c|.\end{cases}$$&#xA;One wants to motivate defining the delta distribuion as the limit (of the linear functionals given by integrating against) $h_{c}(t)$ when $c \to 0$. For any function $f$ continuous in a neighborhood of the origin, by the mean value theorem for integrals and the continuity of $f$, the limit of &#xA;$$\int_{-\infty}^{\infty}f(t)h_{c}(t)\,dt = \frac{1}{2c}\int_{-c}^{c}f(t)\,dt$$&#xA;when $c \to 0$ exists and equals $f(0)$. This shows how to define $\delta(t)$ as the nonexistent limit $\lim_{c\to 0}h_{c}(t)$, by interpreting functions as functionals via integration. It indicates the possibilities of extending the function concept in a useful way. Students that study physics or engineering will later see the delta function handled formally in those contexts, and it might help them to understand its formal properties to have been explained its relation to the usual function concept.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2016-05-19T08:31:08.140" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11029" PostTypeId="2" ParentId="11026" CreationDate="2016-05-19T08:45:24.153" Score="12" Body="&lt;p&gt;It doesn't exactly solve the problem, but one way to make positive use of Wolfram would be to use some &lt;a href=&quot;http://demonstrations.wolfram.com/index.html&quot;&gt;Wolfram Demonstration Projects&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are &lt;a href=&quot;http://demonstrations.wolfram.com/CalculusGame/&quot;&gt;some&lt;/a&gt; that attempt to gamify calculus calculation practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://demonstrations.wolfram.com/SecantAndTangentLines/&quot;&gt;Some&lt;/a&gt; help with intuition. If the students can see how the ideas could be of use to society as a whole (say through higher-dimensional calculus), even if not directly to themselves, it might make them resent the course a bit less.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://demonstrations.wolfram.com/PredatorPreyModel/&quot;&gt;Others&lt;/a&gt; could possibly help to link the ideas to contexts they might be more interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I read somewhere (I've completely forgotten where) that 'why do we have to do this?' often is a symptom of 'we're struggling to make any sense of this'.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-05-19T08:45:24.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11030" PostTypeId="2" ParentId="11026" CreationDate="2016-05-19T13:07:44.193" Score="8" Body="&lt;p&gt;You can also use Wolfram Alpha to look up definitions of words. So by reductive reasoning, there is no need for infants to learn vocabulary because when they are old enough to use a smartphone, people can always look up the meaning of words in Wolfram Alpha.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being less reductive and less cynical, there is no need to learn a foreign language because you can use google.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hey life is good! There is no need to learn anything anymore.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students in college algebra can appreciate the fallacy in these arguments, and appreciate how vocabulary and translation parallel algebraic manipulation.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-05-19T13:07:44.193" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11031" PostTypeId="2" ParentId="11026" CreationDate="2016-05-19T23:12:01.190" Score="38" Body="&lt;p&gt;First of all, I believe this question is quite similar to the question &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/q/7431/1217&quot;&gt;How to give homework for integration technique?&quot;&lt;/a&gt;. I avoid the temptation of repeating my &lt;a href=&quot;https://matheducators.stackexchange.com/a/7432/1217&quot;&gt;answer&lt;/a&gt; for that question. Instead, I try to give an answer from a different angle based on a recent experience I had in a numeracy class with adult students. One of the questions I asked was inspired with the way my son calculated $142 \times 39$ (that was one of his school procedural homework). Basically, he started from $140 \times 40$ and found his way through to the answer of the initial question. Following that, the question for the class was &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Knowing that $140 \times 40=5600$ suggest a way to calculate $142 \times 39$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course, they all had smart phones and on their phones they had a calculator. But, no smart phone was of any help to solve the problem at hand, they needed smart minds. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics is about solving problems not repeating exercises. For the latter you cannot beat Wolfram and the likes; for the former, it would be beneficial to know how to use them (if needed), but you need something more. Thus, it is one of our responsibilities (as lecturer/teacher) to give our students problems which need that something more.   &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-11-29T08:35:48.987" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11032" PostTypeId="2" ParentId="11026" CreationDate="2016-05-20T10:46:44.950" Score="7" Body="&lt;p&gt;Show your students some simple questions that Wolfram Alpha cannot answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wolfram Alpha currently cannot satisfactorily answer some problems stated in words.  For example, the query &quot;&lt;a href=&quot;http://www.wolframalpha.com/input/?i=find%20the%20maximum%20area%20of%20a%20rectangle%20given%20its%20perimeter&quot; rel=&quot;noreferrer&quot;&gt;find the maximum area of a rectangle given its perimeter&lt;/a&gt;&quot; is interpreted as &quot;rectangle | area&quot; and has the result &quot;A = a b.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The query &quot;&lt;a href=&quot;http://www.wolframalpha.com/input/?i=find%20the%20maximum%20area%20of%20a%20rectangle%20given%20a%20perimeter&quot; rel=&quot;noreferrer&quot;&gt;find the maximum area of a rectangle given a perimeter&lt;/a&gt;&quot; is interpreted as &quot;rectangle | perimeter x | maximal area&quot; and has the result &quot;1/2 h (x-2 h) (assuming height h).&quot;  While this is a more informative answer than that for the previous case, it still does not explicitly state that the maximum area is when the rectangle is a square.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, perhaps a future version of Wolfram Alpha will be able to answer the question above.  But I doubt that it will be able to answer a question such as &quot;Given a fence of fixed length, find the largest rectangular plot of land that it can surround.&quot;  To answer a question like this, Wolfram Alpha needs some common sense, for example, it needs to know that a fence is like a perimeter, and that a large plot of land is one that has a large area.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-05-20T10:46:44.950" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11033" PostTypeId="2" ParentId="11026" CreationDate="2016-05-20T10:54:41.310" Score="27" Body="&lt;p&gt;Imagine you had to look up every word you wanted to use, because you had a poor vocabulary. This would get old, very quickly.&lt;/p&gt;&#xA;&lt;p&gt;The trouble is, many people don't have a genuine need to internalize computational technique as a way to &lt;em&gt;see&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As a researcher, I find one of the main values of manual calculation is as error correction for reasoning, and a source of intuition. When I am in love with a certain idea, carrying out calculations related to this idea is a way to slow down and become objective about the idea and to test it from many angles. I am constantly guessing what should happen and checking this against the results of the calculation, which I attempt to do accurately and honestly to bolster or debunk my belief.&lt;/p&gt;&#xA;&lt;p&gt;At some stage, when trying things by hand to develop intuition and generate ideas that will bypass computation, it may become appropriate to use technology to continue this process. Usually, though, moving to technology without having developing feelings by calculating many pages by hand feels very icky.&lt;/p&gt;&#xA;&lt;p&gt;I'd like, as an educator, to try to motivate students to calculate for similar reasons to why researchers do. Unfortunately, I often don't do a terribly good job of tying &amp;quot;getting the right answer&amp;quot; to &amp;quot;debugging reasoning&amp;quot;. There ends up being very little student metacognition or conjecture to be tested...only whether the number matches the answer key.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, this is similar to why we ought to care about grammar and even spelling...for metacognitive reasons. Grammar, despite its conventional aspects, provides logical checksums that can rudder thought. Subtle differences in meaning can be detected by one's critical consideration of turns of phrase. Mindlessly relying on spell check or denying the need for grammar has similar detrimental effects to the blind use of Wolfram Alpha and calculators. Of course, these things are helpful, but can easily erode one's mental acuity if one is not very careful.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps the way to go is (1) Focus on reasoning about mathematics, and thinking about what it means; (2) Give problems that require thought and ideas, and that would benefit from accurate exploratory calculation.&lt;/p&gt;&#xA;&lt;p&gt;Standard Calculus often fails this miserably, I'm afraid.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2022-03-01T12:07:32.707" LastActivityDate="2022-03-01T12:07:32.707" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11034" PostTypeId="2" ParentId="11021" CreationDate="2016-05-20T17:22:38.627" Score="5" Body="&lt;p&gt;I'd recommend the Khan Academy exercises, and in particular their &lt;a href=&quot;https://www.khanacademy.org/exercisedashboard&quot; rel=&quot;noreferrer&quot;&gt;Knowledge Map&lt;/a&gt;. You can start by trying the exercises at any point on the map, and based on whether you do well or not, either move further down, or back up towards any rusty prerequisites.&lt;/p&gt;&#xA;" OwnerUserId="688" LastActivityDate="2016-05-20T17:22:38.627" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11035" PostTypeId="2" ParentId="11017" CreationDate="2016-05-23T08:50:49.377" Score="8" Body="&lt;p&gt;In France, my answer would be that &lt;strong&gt;we only hire faculty who can teach every mathematical subject at every undergraduate level&lt;/strong&gt; (possibly with quite some work, of course). This is most usually included in the job description, and is pretty accurate in practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that makes it relatively easy to ensure that a good majority of faculty have this broadness is that a good majority of us have taken &lt;em&gt;Agrégation&lt;/em&gt;, a national competition intended to hire high school teachers of higher mathematical level, where one has to demonstrate a good mastery of a big chunk of undergraduate mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another point is that the higher education in &lt;em&gt;Écoles Normales Supérieures&lt;/em&gt; is very broad: one does not specialize too much too early; and a good majority of faculty in France went to one of these &lt;em&gt;ENS&lt;/em&gt;. Of course, we also hire people whose higher education was at a university, and foreign mathematicians, but many of them are equally broad, and all in all I think the vast majority of faculty is able to teach any undergraduate course, if needed.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-05-23T08:50:49.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11036" PostTypeId="1" AcceptedAnswerId="11041" CreationDate="2016-05-24T09:34:07.603" Score="13" ViewCount="3035" Body="&lt;p&gt;During the internship I recently finished, I came to realise how important it is to have a good and structured use of the blackboard when teaching mathematics to 12-14 year pupils. Circumstances forced me to only use the blackboard, a pretty classic chalkboard. My writing on that blackboard can definitely use some improvement. I was hoping for some tips, besides exercise, on how to improve the writing itself and make the total scheme more structured, neat and orderly.&lt;/p&gt;&#xA;" OwnerUserId="6785" LastEditorUserId="7365" LastEditDate="2022-11-17T22:16:05.980" LastActivityDate="2022-11-17T22:16:05.980" Title="Tips to improve blackboard writing" Tags="|mathematical-pedagogy|writing|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11037" PostTypeId="2" ParentId="2279" CreationDate="2016-05-24T09:59:12.870" Score="2" Body="&lt;p&gt;I haven't made any video's in wich my face appears on the screen (on purpose). &#xA;I also haven't done many video's in wich i explain things and not even that many in general. &#xA;However i have made a few short video's to motivate my students, aged roughly 12-16. When making these motivational video's i simply create images by placing different pictures and texts over eachother using simply Paint, Windows moviemaker , Word or PIXLR (free 'photoshop' online software). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's very time consuming at first but you get faster at it.&#xA;Sometimes it really gets the pupils exited and therefor it's usually been worth the effort. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(For an example of the video's i made : &lt;a href=&quot;https://www.youtube.com/channel/UCio66FwDtvZnm7puStBV7ow&quot; rel=&quot;nofollow&quot;&gt;https://www.youtube.com/channel/UCio66FwDtvZnm7puStBV7ow&lt;/a&gt; ) &lt;/p&gt;&#xA;" OwnerUserId="6785" LastActivityDate="2016-05-24T09:59:12.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11038" PostTypeId="2" ParentId="2730" CreationDate="2016-05-24T10:37:28.217" Score="1" Body="&lt;p&gt;Also: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Poker hands (high card is far less valuable than a full house for example) &lt;/li&gt;&#xA;&lt;li&gt;Rankings in sports or games (ex: the winner of the worldcup vs someone who ended in the semi finals , prestigelvl in online gaming, ...)&lt;/li&gt;&#xA;&lt;li&gt;Meals during the day (first breakfast, .... ) &lt;/li&gt;&#xA;&lt;li&gt;Ranks in the militairy or in youth movements &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6785" LastActivityDate="2016-05-24T10:37:28.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11039" PostTypeId="2" ParentId="9760" CreationDate="2016-05-24T10:43:50.690" Score="2" Body="&lt;p&gt;The Monster Math Squad is an English tv-show. &#xA;Episod's are roughly 13 minutes long and during a 'quest' children aged 2 to 5 can learn about diffrent subjects such as weight (diffrence between heavy and light), size ( small, medium, large), sorting etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most episodes are also available on youtube or other channels.&lt;/p&gt;&#xA;" OwnerUserId="6785" LastActivityDate="2016-05-24T10:43:50.690" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11040" PostTypeId="2" ParentId="11036" CreationDate="2016-05-24T11:30:01.820" Score="10" Body="&lt;p&gt;Blackboard writing 101: &lt;strong&gt;always break the chalk&lt;/strong&gt; (to reduce the likelihood of squeaking)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other points will depend on how much space you have available. Working consistently from left to right helps, although an alternative is working right to left (assuming you are right-handed and have a fair amount of space) to keep your body away from what you've most recently written. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a whiteboard I use different colours for different blocks of ideas, to provide more of a visual divide. That might not work so well on a blackboard, depending on the chalk available.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often at undergraduate level there are places where you need to do rough working separately from your main solution. I like to do that on the other side of the board, in a different colour. One of my lecturers would draw the back of an envelope on the board. I guess this is less applicable for 12-14 year olds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A personal pet-peeve is the smudge marks on a blackboard once it's been erased. I try to be sure to wipe the board using horizontal strokes, so at least the backdrop to the next set of writing is not too chaotic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You probably need to write bigger than you think you do (although I had one lecturer who wrote way too large). I also took to a completely different style of handwriting to what I use for myself - I think of it as 'baby writing'. Don't be afraid to change your letters if you need to. I've recently started crossing my &lt;strong&gt;z&lt;/strong&gt;s on the board, even though I've never liked that, after confusing one of my own for a 2.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-05-24T11:30:01.820" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11041" PostTypeId="2" ParentId="11036" CreationDate="2016-05-24T14:18:07.390" Score="12" Body="&lt;p&gt;There are probably many tips on this; I remember some discussions of font choice happening here. Let me give only one: &lt;strong&gt;take your time&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One easily feels that writing on the board is too slow, and is thus inclined to deform letter, take shortcuts in sentences, and use abbreviations. Most of the time, these are mistakes because it makes the meaning intrinsically more obscure, and it also gives less time for students to understand what is on the board. In most circumstances, teachers should find the writing on board &lt;em&gt;painfuly&lt;/em&gt; slow for it to be acceptable for student. Writing full sentences (except at your envelop-back/draft zone as mentioned in Jessica B's answer), taking time to form and space letters will make it easier to have a clean board.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-05-24T14:18:07.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11042" PostTypeId="2" ParentId="11036" CreationDate="2016-05-24T18:26:26.580" Score="8" Body="&lt;p&gt;Let students see what you're writing as you're writing.  Otherwise, they're bored while you write and must hurry to read when you finish.  Instead of facing the board, stand with your side to the board and your chalk hand in plain view, sort of like a &quot;Wheel of Fortune&quot; host.  This posture also makes it much easier to talk to your students and not to the chalkboard, which is good policy as well.&lt;/p&gt;&#xA;" OwnerUserId="6787" LastActivityDate="2016-05-24T18:26:26.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11043" PostTypeId="2" ParentId="6062" CreationDate="2016-05-25T13:09:47.297" Score="8" Body="&lt;p&gt;My favorite little tidbit on this topic is due to Jean Pierre Serre: One should aim to be &quot;precise, yet informal&quot;. Here, &quot;informal&quot; means not using symbols and notation, i.e. not using &quot;formalism&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the steps suppressed in mathematical writing, this is very sensitive to the audience one is addressing. Routine arguments that can be checked by a reader of appropriate background do not need to be elaborated on (provided you are absolutely sure they are correct). It is generally a good practice keep a draft with all the messy details, and abridge them in a final draft, in a way sensitive to one's intended audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should elaborate a bit more on which details should be hidden. Generally, people read papers on topics they are interested in, either because they have nearby expertise or the topic has some bearing on their own work. Most of us immediately go looking for what is new in a paper, and appreciate finding it as soon as we can. If you don't know your audience and just write like an automaton, giving all details, it looks like you are a novice and have no idea what is routine and what is new. Worse yet, this sort of approach that avoids nuance makes your paper look more error-prone...because it looks like you are treating everything with the same amount of care. As a rule, one should omit or abridge those things that are sufficiently routine (after checking them to death), include those things that are delicate (i.e. potential pitfalls that should be treated with care...most experts are aware of these things in their areas...knowing these gives your paper credence with these experts) and last, but not least, highlight the new ideas so they are clearly visible. A paper that does these things looks more like a mathematician wrote it...indeed it would be very difficult for an amateur to effectively write such a paper convincingly!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, regarding Serre, his talk &quot;&lt;a href=&quot;https://www.youtube.com/watch?v=tJZpdXWm4Gg&quot; rel=&quot;noreferrer&quot;&gt;How to Write Mathematics Badly&lt;/a&gt;&quot; is fun.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2016-05-25T13:17:31.790" LastActivityDate="2016-05-25T13:17:31.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11044" PostTypeId="2" ParentId="10977" CreationDate="2016-05-25T19:43:23.013" Score="3" Body="&lt;p&gt;Well, what I've got during my classes, like GeraldEdgar already said, is that you don't have one solution to solve it all. What you could do is posting your notes the night (or maybe two nights) before the lecture. That way the students who want to use your notes, can print it and take it with them. They're will always be some other students who wish to make their own notes, because they know by themselves that they will learn better from that. I had experienced that some students scored a little worse then usual after giving everyone the educator's notes. Furthermore I think it's not a bad idea to let sometimes all students make their own notes. If there's a more complex proof, you could give the student's one way to proof it by giving them your notes, but I'm sure they would understand the proof better if you could build up the proof on a different way during your lecture.&lt;/p&gt;&#xA;" OwnerUserId="6516" LastActivityDate="2016-05-25T19:43:23.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11045" PostTypeId="2" ParentId="10953" CreationDate="2016-05-25T22:35:37.087" Score="2" Body="&lt;p&gt;To directly answer your question: here's an oft-cited work from 1999 that showed that the gender in the word problems can influence performance through stereotype threat (although it did not account for the entire achievement gap in gender):  Walsh, M., Hickey, C., &amp;amp; Duffy, J. (1999). Influence of Item Content and Stereotype Situation on Gender Differences in Mathematical Problem Solving. Sex Roles, 41(3–4), 219–240. &lt;a href=&quot;http://doi.org/10.1023/A:1018854212358&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1023/A:1018854212358&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to complexify the question and consider a less binary concept of gender: Indigo Esmonde's work would be a great starting point.  Here's a paper to get you started:&#xA;&lt;a href=&quot;http://lib-ocs.lib.sfu.ca:8087/fedcan/index.php/csse2010/csse2010/paper/download/2645/673&quot; rel=&quot;nofollow&quot;&gt;http://lib-ocs.lib.sfu.ca:8087/fedcan/index.php/csse2010/csse2010/paper/download/2645/673&lt;/a&gt;&#xA;which I think eventually became:&#xA;Esmonde, I. (2011). SNIPS AND SNAILS AND PUPPY DOGS’TAILS: GENDERISM AND MATHEMATICS EDUCATION. For the Learning of Mathematics, 31(2), 27–31.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but the version I linked to above is longer and more comprehensive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a similar question: one might wonder if you should include queer culture and queer families in math problems, and here's one author who discusses what that might entail (as well as how to move beyond that approach):&#xA;Rands, K. (2009). Mathematical Inqu[ee]ry: beyond “Add-Queers-and-Stir” elementary mathematics education. Sex Education, 9(2), 181–191. &lt;a href=&quot;http://doi.org/10.1080/14681810902829646&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1080/14681810902829646&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my own experience: I know for myself, when I write math problems for middle school students, when putting names and/or genders in the problems, I try to make a point of once in a while having boys do &quot;girl things&quot; and girls doing &quot;boy things&quot;, not necessarily designing problems around that, but just being aware of it as I write the problems and occasionally switching it up a bit.  I also occasionally when teaching math bring up social justice issues that are currently in the media and ask students to consider them mathematically, so that's another thing to think about - might, say, using math to tackle issues of gender equality have more of an impact in the classroom than just putting women in math problems...?  &lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2016-05-25T22:35:37.087" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11048" PostTypeId="1" AcceptedAnswerId="11051" CreationDate="2016-05-26T03:57:54.850" Score="22" ViewCount="2823" Body="&lt;p&gt;Today I happen to have an interesting discussion with a primary school kid. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked him &quot;Which is the smallest one - digit integer?&quot; He instantly replied $-1$. I told him that he's wrong and the answer is $-9$. He then replied then &quot;why&quot;? He constantly kept saying that $9$ is greater than $1$ and therefore, the smallest integer should also be -$1$.  To him number with greater value is greater. He doesn't care about $+$ or $-$ sign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I then explained things to him by drawing number line and then explained him that all the numbers to the right hand side of a particular number on the number line are greater. He then again asked &quot;why?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end I was speechless and couldn't satisfy his curiosity because I had no proper trick to put this into his brain. Kindly suggest me some methods to explain him everything he wants to know. I just want him to feel satisfied.&lt;/p&gt;&#xA;" OwnerUserId="6481" LastEditorUserId="2083" LastEditDate="2016-05-26T12:08:12.163" LastActivityDate="2018-08-30T05:07:16.753" Title="Explaining the order of negative integers" Tags="|primary-education|negative-numbers|" AnswerCount="12" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11049" PostTypeId="2" ParentId="11048" CreationDate="2016-05-26T04:48:38.520" Score="6" Body="&lt;p&gt;If he's ignoring the + or - signs, then that needs to get remediated. My approach is to:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Define negative numbers as running to the left on the number line. (In other words, the &quot;-&quot; means &quot;in the reverse direction&quot;). Exercise this thoroughly first, finding numbers like +5 or -3 on a marked number line.&lt;/li&gt;&#xA;&lt;li&gt;Define the relation lesser-than as meaning further left on the number line. Exercise this, comparing various integers drawn on the number line. &lt;/li&gt;&#xA;&lt;li&gt;If one wants to consider just the number part, then introduce the absolute-value operator for that (that is, magnitude: distance from the origin). Exercise writing various expressions using absolute-value notation.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Applications of integers may help: Temperature, elevations, football, money. Which is colder: -1 or -9 degrees (a thermometer may help)? Which is lower: 50 feet below sea level or 100 feet below sea level? Which football team has done better: One who lost 2 yards on the drive, or one who lost 5 yards? Whatever is familiar to your student. The reason we need signed numbers is to distinguish different directions in cases like these. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-05-26T04:48:38.520" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11050" PostTypeId="2" ParentId="6062" CreationDate="2016-05-26T05:03:45.850" Score="5" Body="&lt;p&gt;Mathematical writing is very different from other kinds of writing, including other kinds of technical writing. Aside from low-level textbooks, the audience is generally assumed to be other competent mathematicians, with their expertise in the particular subfield depending on how technical the work is. It's awful form, and it will infuriate your readers, if you try to hold their hand and assume that they need helpful analogies and metaphors (which invariably just cloud the issue more) to deal with the concepts. Just present the data straightforwardly and without narration, giving enough details to be precise but not too much to be pedantic. (Note in particular that a math paper or book intended for other mathematicians is not a problem set to be graded; it's actually preferable to omit trivial details that a competent reader should be able to fill in themselves.) The line between the two is hard to define but easy to spot in practice. Think of it as something like a legal document: The audience is other lawyers, who are assumed to be competent enough to handle the material if it's presented directly, and who expect and want the terse style and precise vocabulary that presents the argument without beating them over the head with it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best-written math textbook I've read is Bott and Tu's &lt;em&gt;Differential Forms in Algebraic Topology&lt;/em&gt;. Try copying the style there. &lt;/p&gt;&#xA;" OwnerUserId="6674" LastActivityDate="2016-05-26T05:03:45.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11051" PostTypeId="2" ParentId="11048" CreationDate="2016-05-26T07:04:35.163" Score="9" Body="&lt;p&gt;He wants to know the why, not per se the logic behind it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So give him a reason he can understand. Explain to him that many hundreds of years ago the concept of zero didn't exist. There was no number for having nothing of something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, someone thought of the zero, a simple value, that meant you had nothing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, things can have less than nothing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw a simple number line:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;There was a worker, who wanted to buy a new tool. He has 5 coins. The tool costs 10 coins.   &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10  &#xA;-----------------------------------------^----------| &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;How much extra coins does the worker need to be able to buy the tool?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The person selling the tool can lend him the remaining 5 coins. This means he owes the person selling still 5 coins.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The worker hands the salesman his five coins. His money in his pocket is now zero&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10  &#xA;-------------------------------^---------------------&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The worker gets his tool. But now owes the salesman 5 coins, which he will have to pay out of his future salary. Until this is paid to the sales man he will not be able to have money in his pocket. He has -5 coins.  &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10  &#xA;-----------------^-----------------------------------&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;If he gets a coin for his work, he will have to pay the sales man first, making his coins -4. He can only get coins in his pocket again when his count reaches zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what is less, 0 coins or -5 coins?&lt;/p&gt;&#xA;" OwnerUserId="6796" LastEditorUserId="5117" LastEditDate="2016-05-27T06:18:38.063" LastActivityDate="2016-05-27T06:18:38.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11052" PostTypeId="2" ParentId="11048" CreationDate="2016-05-26T07:57:46.923" Score="14" Body="&lt;p&gt;It seems to me that what you asked wasn't really right.  -9 is the &quot;&lt;em&gt;lowest&lt;/em&gt; one-digit integer&quot; but (at least it can reasonably argued) 0 is the &lt;em&gt;smallest&lt;/em&gt;. Maybe making this difference explicit would clear confusion: &quot;big&quot; negative numbers are &lt;strong&gt;a long way&lt;/strong&gt; less than zero and therefore the lowest. &lt;/p&gt;&#xA;" OwnerUserId="6797" LastActivityDate="2016-05-26T07:57:46.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11053" PostTypeId="2" ParentId="11048" CreationDate="2016-05-26T08:07:05.360" Score="44" Body="&lt;p&gt;To my mind, the problem is the word &lt;em&gt;smallest&lt;/em&gt;. If you asked me which is smaller, $-1$ or $-9$, I'd ask you to clarify in what sense. Colloquial use of &lt;em&gt;small&lt;/em&gt; refers to magnitude rather than ordering. It is not true that $-9$ is smaller than $-1$ in magnitude, although it makes sense to say that $-9$ is less than $-1$ although it is bigger in magnitude. The kid's perspective is captured by dropping the &quot;in magnitude&quot; and speaking with less precision. When comparing electric charges $1$ and $-2$, which would you say is smaller? One says that $sin(t)$ has smaller amplitude than $-2cos(t)$ although there is no order relation between the two. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explain the difference between order and size - one refers to relative position, the other to the distance from the origin - that by &quot;smaller&quot; you mean &quot;less than&quot; in reference to ordering, and not magnitude. The kid is likely to reject this (to his mind) misuse of &quot;small&quot;, so an alternative is to not use the problematic word &quot;smaller&quot; except to refer to magnitude. In mathematics words have meanings that are precise and limited, and often different from their meanings in ordinary speech. &quot;Less than&quot; also has a mathematical meaning that doesn't conform well with ordinary speech, where &quot;less&quot; also often refers to size, but its use in relation to ordering is palatable. Communicate that in mathematical speech, &quot;less than&quot; usually refers to order. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2016-05-26T08:07:05.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11054" PostTypeId="1" AcceptedAnswerId="11061" CreationDate="2016-05-26T08:56:22.963" Score="4" ViewCount="659" Body="&lt;p&gt;Let me use this example,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Solve $x^3-4x&amp;gt;0$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After factorization, we have $$(x+2)x(x-2)&amp;gt;0$$, in order to have product of several numbers positive, even(0,2,4,...) of them have to be negative numbers. In this case, none of them is negative or two of them are negative, we can conclude $\displaystyle x\in(-2,0)\cup (2,\infty)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I don't find the above approach too popular, more often than not, the following approach is what I see people doing,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start by solving equality, $x^3-4x=0$, same factorization gives us $x=-2,0,2$, therefore we make partition $\mathbb{R} \backslash \{-2,0,2\}=(-\infty,-2)\cup(-2,0)\cup (0,2)\cup (2,\infty)$. In order to see whether $f(x)=x^3-4x$ is positive or negative on each interval, plug in a number in each interval. For example $f(-10)=-960&amp;lt;0$ so $f$ is negative on $(-\infty,-2)$, $f(-1)=3&amp;gt;0$ so $f$ is positive on $(-2,0)$,etc. In the end you get the same conclusion $\displaystyle x\in(-2,0)\cup (2,\infty)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although perfectly valid, there are two main reasons I don't like the second approach,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In order to solve equality, we still need to factorize the left hand side. Once we have the factorization, it's much easier to check positive/negative than to compute values of function.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In order to conclude $f(x)&amp;gt;0,\forall x\in I$(where $f(x)\neq 0$ on interval $I$) from $\exists c\in I, f(c)&amp;gt;0$, $f$ needs to satisfy intermediate value property(all continuous functions have it by IVT). I don't see why we need any result from calculus to solve any precalculus problem.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Therefore my question is, despite the reasons I list, why is the second approach more popular, not only among students but also instructors and TAs (of calculus/precalculus class)? There should be some advantages of this method that I don't know about. Any insight is welcome.&lt;/p&gt;&#xA;" OwnerDisplayName="user2139" LastActivityDate="2019-05-17T13:30:05.607" Title="Why &quot;plug in numbers&quot; when solving inequality?" Tags="|calculus|precalculus|teaching|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11055" PostTypeId="2" ParentId="11048" CreationDate="2016-05-26T10:21:17.277" Score="2" Body="&lt;p&gt;There is a great board game that I use with my third graders which teaches negative numbers.  It's called Midnight Party or Ghost Party.  In the game the students can get captured and get sent to a place with negative points of different amounts.  The later they get captured, the less negative points they get.  So the first piece captured lands on the -10 space and the second piece would get -9 and so on.  The kids have to add up their negative points at the end of the round.  &lt;strong&gt;Everyone understands negative points are bad and would rather have fewer negative points.&lt;/strong&gt; In addition they can land on one or two spaces that have positive points and they understand that more positive points are good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can try to get the game or just create your own board game with a path on oak tag.  There should be a path with a few positive spaces and spaces for capture.  There should be a place where they are sent and get negative points. If it is just the two of you, you should each have 5 pieces so that there is something to add up.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am confident that after playing this game a few times, the nine year old start to understand that negative points are bad and therefore it's better to have -1 than -8, just as it's better to have 8 than 1.  This should help him make the leap that you are looking for.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-05-26T10:21:17.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11056" PostTypeId="2" ParentId="11048" CreationDate="2016-05-26T13:25:37.173" Score="1" Body="&lt;p&gt;The student is correct: -9 is larger in magnitude that -1 (in conventional notation |-1|&amp;lt;|-9|). It's possible that the student isn't yet making the distinction between larger in magnitude and larger in the usual ordering, or just that the student finds the absolute value comparison more natural.  But the student is expressing a legitimate view of ordering on the integers, it just happens not to be the one the teacher had in mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think we turn students off to math when we tell them their (valid but underdeveloped) intuitions are &quot;wrong&quot; when we really mean that they aren't using the official language for expressing them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd prefer to emphasize that these are both meaningful orderings, and so they have different names because they make sense in different situations.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-05-26T13:25:37.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11057" PostTypeId="2" ParentId="11048" CreationDate="2016-05-26T14:14:57.660" Score="5" Body="&lt;p&gt;Help him first to understand that &lt;em&gt;subtracting 1 make numbers lower&lt;/em&gt;, lower and lower. Each time you subtract 1, you have less than before. Just like &lt;em&gt;adding 1 will make them higher&lt;/em&gt;, higher and higher. (In my opinion the terms &lt;em&gt;low&lt;/em&gt; and &lt;em&gt;high&lt;/em&gt; will be less confusing as kids already have the association &lt;code&gt;bigger figure (absolute) =&amp;gt; bigger number&lt;/code&gt; in their heads). More you add, the more you have. More you subtract, the less you have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you go subtracting 1, even though the figure increases (by figure I mean the absolute number), the number gets lower, lower and lower. It means, you have less, less and less.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;owing&lt;/em&gt; concept might help him to understand that you can take -1 from quantities even when you go below zero, and that means &quot;you have even less&quot; than before. So if they have 5 apples and need to give you 5, they will then have 0. But they can also have 5 apples and owe you 8. You will subtract 8 from them. So after they give you all (5), the next 3 apples they get must be given to you as well, because they are due. It means that having -3 apples is worst than having 0, because it means less apples (even though the absolute number is bigger).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After they understand that even having 5 apples, you can try to subtract 8 (so they own 3 apples to you), ask if they would prefer that you subtract 8 or 20 (and how much that would represent). They should pick 8. Because if they pick 20, they will have so, so much less apples because of owing you... And that may help to understand that having -8 somehow means to have more than -20.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just an insight, a slap dash explanation for you to work on. But hope it helps.&lt;/p&gt;&#xA;" OwnerUserId="6801" LastActivityDate="2016-05-26T14:14:57.660" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11058" PostTypeId="2" ParentId="5794" CreationDate="2016-05-26T15:18:31.380" Score="2" Body="&lt;p&gt;None of the other answers get at the basic psychological problem: People begin with an understanding of numbers as &lt;strong&gt;magnitudes&lt;/strong&gt; (i.e amounts of things).  When introduced to zero and negative numbers, they can usually fit zero in as the magnitude of 'nothing left', but they try to grasp negatives as somehow pushing magnitude beyond nothing, which they find utterly baffling.  What you have to do is get them to understand that the sign of a number &lt;em&gt;has nothing to do with magnitude&lt;/em&gt;, it is a completely new and different feature of a number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since this is about time the number line gets introduced, the most natural approach is to say that numbers can have direction as well as magnitude, and sign indicates direction.  So, for example: The difference between 1, +1, and -1 is exactly analogous to the difference between 'a mile', 'a mile east', and 'a mile west'.  &lt;em&gt;[if they ask 'what about a mile north?', tell them they'll get to that a few years down the road &lt;strong&gt;:-)&lt;/strong&gt; ]&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can then talk about numbers using the number line via the metaphor of walking: The number 3, for example, means starting at the origin and walking three paces; if you walk east, that's +3, if west, that's -3 (zero means don't walk).  Addition means you start at one number and walk the other, using sign to know in which direction to walk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you get to '-(-x) == x', you can explain that applying 'minus' in multiplication means &lt;em&gt;flipping the entire number line over about the origin&lt;/em&gt; - east becomes west, west becomes east.  So if 'x' is 'right of origin, facing east', '-x' flips it over so now it is 'left of origin, facing west'.  So naturally, '-(-x)' flips it over twice, leaving you back with what you started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;[note: if you can get kids to use this model early, they'll probably have a much better time if and when they come to complex numbers]&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="6802" LastActivityDate="2016-05-26T15:18:31.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11060" PostTypeId="1" CreationDate="2016-05-26T18:06:09.867" Score="4" ViewCount="84" Body="&lt;p&gt;I recently read the book &quot;Peak: Secrets from the New Science of Expertise&quot; by Karl Anders Ericsson and Robert Pool. Ericsson is a respected researcher in the field of expertise formation, this is why I take his words as valuable. In a paragraph of the book it reads:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Some activities, such as playing music pop in music groups, solving crossword puzzles, and folk dancing, have no standard training approaches. Whatever methods there are seem slapdash and produce unpredictable results. Other activities, like classical music performance, &lt;strong&gt;mathematics&lt;/strong&gt; and ballet, are blessed with highly developed, broadly accepted training methods. If one follows these methods carefully and diligently, one will almost sure become an expert.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I´ve written him an email, but, in the mean while, do you have any idea what  &quot;highly developed, broadly accepted methods&quot; is he talking about? A simple reference to a book, article or website to start with might suffice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of my question is that I feel I'm learning maths without any &quot;highly developed&quot; approach, just by following my intuition. If such approach existed I would really like to know about it, to improve my learning, and perhaps teaching, in the future.&lt;/p&gt;&#xA;" OwnerUserId="6537" LastActivityDate="2016-05-26T18:06:09.867" Title="Accepted mathematical training methods?" Tags="|self-learning|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11061" PostTypeId="2" ParentId="11054" CreationDate="2016-05-27T04:53:47.580" Score="11" Body="&lt;p&gt;I, personally, wouldn't solve this specific inequality in &lt;em&gt;either&lt;/em&gt; fashion; I would instead decide the sign on one of the four intervals &amp;mdash; e.g. by plugging in $x = +\infty$ &amp;mdash; and recognize that each of the roots is a simple crossing so that the sign alternates between adjacent intervals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the precalculus courses at my university, yet another method was taught: rather than having to reason out what possible combinations of signs there are and fill in the detail you glossed over with &quot;we can conclude&quot;, they construct a table&#xA;$$\begin{matrix}&#xA;&amp;amp; (-\infty, -2) &amp;amp; (-2, 0) &amp;amp; (0, 2) &amp;amp; (2, \infty)&#xA;\\ x-2 &amp;amp; - &amp;amp; - &amp;amp; - &amp;amp; +&#xA;\\ x &amp;amp; - &amp;amp; - &amp;amp; + &amp;amp; +&#xA;\\ x+2 &amp;amp; - &amp;amp; + &amp;amp; + &amp;amp; +&#xA;\end{matrix} $$&#xA;so they can just tally up the signs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, the important thing is that students have a method that &lt;em&gt;works&lt;/em&gt;, and furthermore can be executed in a systematic and straightforward fashion on problems complicated enough that they can't just see the answer directly.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Despite your assertion, you do &lt;em&gt;not&lt;/em&gt; need to factor to do sign analysis. Obtaining roots can be faster, and factorization could well be rather complicated. For example, solving $x^8 - 1 &amp;gt; 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your second point looks like a non sequitur; I'm going to assume your objection is more about using methods whose underpinnings use results you haven't proven... but when the point is to have a method that students can use and understand, the theoretical underpinnings aren't really relevant unless they get in the way of students' ability to use and understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're really interested in full rigor, there are worse things to worry about, such as the existence of square roots, or the definitions and properties of transcendental functions.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;TL;DR:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;there are numerous approaches&lt;/li&gt;&#xA;&lt;li&gt;none seem dramatically superior to the others&lt;/li&gt;&#xA;&lt;li&gt;an advantage of the second versus first approaches of your post is that the algorithm is explicit: it tells students &lt;em&gt;how&lt;/em&gt; to conclude&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2016-05-27T04:53:47.580" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11062" PostTypeId="1" CreationDate="2016-05-27T09:31:30.357" Score="8" ViewCount="113" Body="&lt;p&gt;Recently I've moved to another university and next semester I will have my first class there. The problem is that I don't know the level of the students that will be attending my class. Of course, I've asked other faculty members, but I'm afraid that their previous performance might not be indicative of how will they do in my class. In particular, because it will be their first really theoretical subject (except some basic math course) in a rather applied-oriented curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, I'm looking for a way to asses their level and their knowledge of things I will depend on (probably I will need to make a recap if they forgot some material along the way). Of course, I don't expect such an assessment to be precise, however, I would like to at least to have a ballpark of what I could do with them (esp. useful when planning future lectures).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could make a non-graded exam during their first class, but I doubt it will be representative, and it might feel like a bad start (I would like them to like my class). Do you have any other suggestions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My class will be directed to masters-students, but general-approach answers will also be useful to me.&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2016-05-27T11:20:10.967" Title="Assessing the level of students in a class" Tags="|graduate-education|assessment|class-participation|classroom-interaction|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11063" PostTypeId="2" ParentId="11062" CreationDate="2016-05-27T11:20:10.967" Score="2" Body="&lt;p&gt;My approach currently is to give a diagnostic test in the first week &lt;em&gt;online&lt;/em&gt; via my school's learning-management system (LMS, i.e., Blackboard). I make it count as a quiz grade so there's some amount of motivation for students to try their best (also: it's timed unlike my other online quizzes). Depending on the class, I may use either a research-based diagnostic or a simulacrum of the prerequisite course's final exam, or portion thereof. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pros:&lt;/strong&gt; This gets the diagnostic information I want in the first week without sucking up the first class meeting for that purpose. The first class meeting can still actually set the tone for the major topics and how the course will normally be run (i.e., seriously and efficiently). I can scan the results for any outliers who need counseling or skipped the prerequisite course. Having the questions in digital form allow me to easily use it for in-house research and correlations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cons:&lt;/strong&gt; Being done out-of-class, there's no way to prevent students potentially working together, sharing answers or materials, or using a calculator if that's otherwise unwanted (but I think the time limit makes most of that less feasible). The first week of class students may be unfamiliar with the LMS or not have access accounts in some cases; I almost always have to reset a few accidental blank submissions, and be on the lookout for technical outliers that need to be dropped from summary statistics. Occasionally a student takes offense or unusual anxiety at the exercise, complaining that they're unfairly rusty or were never taught certain items (I drop a few quiz scores, including possibly the diagnostic, partly to alleviate this). In your case you may or may not have time right now to learn your new school's LMS and spend time constructing/uploading a diagnostic quiz to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-05-27T11:20:10.967" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11065" PostTypeId="2" ParentId="11048" CreationDate="2016-05-27T13:18:14.950" Score="2" Body="&lt;p&gt;We want inequalities to be a property of the &lt;em&gt;relative&lt;/em&gt; position of numbers — it shouldn't change if we translate the whole picture by the same amount. More precisely, we want inequality to have the property that, for any numbers $r, s, t$, if $r + t &amp;lt; s + t$, then $r &amp;lt; s$. This is clearly true for positive integers, and once we've decided to keep this property, there's only one way to extend the definition to zero and negative integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, we want inequalities to be preserved by scaling by a positive number, i.e., if $rt &amp;lt; st$ and $0 &amp;lt; t$, then $r &amp;lt; s$. This also leaves us with only one choice for how to extend the definition of inequality to rational numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The general pattern here is: to extend a definition to a new setting, identify some key properties you want to preserve. (A related example is exponentiation: we define $r^s$ for positive real $r$ and arbitrary integers $s$ by saying we want $r^{s+t} = r^s r^t$, and similarly for rational $s$ by saying $(r^s)^t = r^{st}$.) It's important to note that there is a new &lt;em&gt;definition&lt;/em&gt; being made here, and not just exploring consequences of an old definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student is using a different definition, then it should be pointed out that this is due to different definitions, not an error in reasoning — which is a good opening to discuss the advantages of one definition over the other. Usually, when there are two conflicting definitions, they're trying to capture different notions.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-05-27T13:18:14.950" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11066" PostTypeId="2" ParentId="10941" CreationDate="2016-05-27T14:42:40.903" Score="2" Body="&lt;p&gt;An alternative approach would be to solve the equation $1 + \cos{x} = -\sin{x}$ without squaring. This avoids completely the introduction of a spurious solution. One solution, that works directly with the geometric meaning of the sine and cosine, goes as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If $\cos{x}$ is positive, then $1 + \cos{x}$ is bigger than $1$, so cannot equal the sine of any number. Thus $\cos{x}$ must be nonpositive, so $x$ must lie in the closed left half plane.&lt;/li&gt;&#xA;&lt;li&gt;Since $1 + \cos{x}$ must be nonnegative, $\sin{x}$ must be nonpositive, so $x$ must lie in the closed lower half plane.&lt;/li&gt;&#xA;&lt;li&gt;The preceeding remarks imply that $x$ lies in the closure of the lower left quadrant. That is $x \in [\pi, 3\pi/2]$.&lt;/li&gt;&#xA;&lt;li&gt;For any angle $t$, the triangle inequality implies that $|\cos{t}| + |\sin{t}| \geq 1$ with strict inequality except when $|\cos{t}|$ and $|\sin{t}|$ are the side lengths of a degenerate triangle, that is when one of them equals $1$ and the other equals $0$. (This all can be explained geometrically, referring to a right triangle with hypotenuse $1$.) Since $1 = -\cos{x} - \sin{x} \leq |\cos{x}| + |\sin{x}|$, this implies $x$ is an integer multiple of $\pi/2$. &lt;/li&gt;&#xA;&lt;li&gt;The only possible $x$ satisfying 3. and 4. and in $[-\pi, \pi]$ are  $\pi$ and $-\pi/2$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;After having deduced the solution this way, one could then show the trick of squaring and observe that it can be used, provided one is careful to note that it possibly introduces false solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The explanation above should work in a classroom setting. On the other hand, students probably will not be able to reproduce it on their own without substantial practice on similar problems, and so students wanting step by step recipes or thoughtless algorithms might find it frustrating.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2016-05-27T14:42:40.903" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11067" PostTypeId="1" CreationDate="2016-05-27T17:56:59.970" Score="11" ViewCount="256" Body="&lt;p&gt;After teaching mathematics for a year, where in each class I had opportunities for my students to earn extra credit, I am reflecting on whether this has any value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason why I am questioning this is that most of the students that do the extra credit are students that have good study skills, work hard, and are excelling in the class.  I think extra credit is a good opportunity for students to try harder problems, but I also gave extra credit with the hope that the weaker students have other opportunities to show their understanding of the material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe I have to reevaluate my motivation for providing extra credit, but &lt;strong&gt;I wanted to know if there was evidence or reasearch on whether or not providing extra credit is valuable in an undergraduate mathematics class.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To differentiate my question from the post, &lt;a href=&quot;https://matheducators.stackexchange.com/questions/508/is-extra-credit-appropriate&quot;&gt;Is extra credit appropriate?&lt;/a&gt;, I want to know if there is any research on this.&lt;/p&gt;&#xA;" OwnerUserId="228" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-06-17T21:31:22.357" Title="Is there any research on the value of extra credit in the college mathematics classroom?" Tags="|undergraduate-education|mathematical-pedagogy|reference-request|grading|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11068" PostTypeId="2" ParentId="5794" CreationDate="2016-05-27T18:32:20.523" Score="3" Body="&lt;p&gt;I won't try to give a specific explanation here because I think that depends highly on the level of knowledge of the student. Rather, I'd like to emphasize some general points that I think are helpful for teachers to keep in mind when explaining and motivating these and related matters about the extension of &quot;number systems&quot; (here the extension from positive to negative integers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, it is often helpful to look at the history of how these concepts evolved. For example, we need to have available negative integers, rationals, and (imaginary) quadratic surds in order to give a single (uniform) formula for the solution of a quadratic equation (prior ancient solutions of the quadratic bifurcated into motley cases because they had to avoid working with not yet understood negative or &quot;imaginary&quot; or &quot;irrational&quot; numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet, once we have such numbers available, the solution of the quadratic is childs-play, using simple algebra by completing the square. Deserving explicit emphasis about such algebra is that the arithmetical laws used (commutative, associative and distributive laws) hold true for &lt;em&gt;all&lt;/em&gt; the types of numbers involved (i.e. they are all &lt;em&gt;commutative rings&lt;/em&gt;). Because of the &lt;em&gt;persistence&lt;/em&gt; of these laws, even though we may make detours through fractions, &quot;irrational&quot; and &quot;imaginary&quot; numbers, any results we deduce about &quot;natural&quot; integers still hold true. This is one of the primary reasons that we desire to preserve these laws when we make extensions to our number systems. It allows us to give &quot;universal&quot; proofs that work in any of these rings. Further, problems may simplify when passing to extension rings, e.g. &lt;a href=&quot;https://math.stackexchange.com/a/199771/242&quot;&gt;nonlinear problems may reduce to linear problems&lt;/a&gt; in algebraic extension rings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the old days (pre-axiomatic method) the constraint to preserve the laws of arithmetic when we extend our number systems was sometimes called the &lt;a href=&quot;https://math.stackexchange.com/q/751624/242&quot;&gt;Permanence Principle&lt;/a&gt;, attributed to Hankel or Peacock. Nowadays, with the axiomatic method available, we simply express it in that language, e.g we desire rational, quadratic and complex numbers to be &lt;em&gt;commutative ring&lt;/em&gt; extensions of the ring of integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This notion of the great utility of the preservation of arithmetical laws can be explained at an elementary level. Doing so yields motivated explanations that are a bit  more faithful to the underlying mathematics (and its history) than do alternative &quot;real world&quot; answers to questions like the OP.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, below I copy &lt;a href=&quot;https://math.stackexchange.com/a/878844/242&quot;&gt;one way I explain&lt;/a&gt; the Law of Signs, emphasizing the crucial role played by the distributive law. With a little effort one could simplify the exposition to make it much more elementary, yet still be faithful to the &quot;law preservation&quot; viewpoint.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Law of Signs&lt;/strong&gt; proof: $\rm\,\ (-x)(-y) = (-x)(-y) + x(\overbrace{-y + y}^{\large =\,0}) = (\overbrace{-x+x}^{\large =\,0})(-y) + xy = xy$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Equivalently, evaluate $\rm\:\overline{(-x)(-y) +} \overline{ \underline {x(-y)}} \underline{ +\,xy}\, $ in 2 ways, noting each over/under term $ = 0$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Said more &lt;em&gt;conceptually&lt;/em&gt;, $\rm\:(-x)(-y)\ $ and $\rm\:xy\:$ are both inverses of $\rm\ x(-y)\ $ so they are equal by &lt;strong&gt;uniqueness of inverses&lt;/strong&gt;: if $\,a\,$ has two additive inverses $\,\color{#c00}{-a}\,$ and $\,\color{#0a0}{-a},\,$  then&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\color{#c00}{-a}\, =\, \color{#c00}{-a}+\overbrace{(a+\color{#0a0}{-a})}^{\large =\,0}\, =\, \overbrace{(\color{#c00}{-a}+a)}^{\large =\,0}+\color{#0a0}{-a}\, =\, \color{#0a0}{-a} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This proof of the Law of Signs uses well-known laws of positive integers (esp. the &lt;em&gt;distributive law&lt;/em&gt;), so if we require that these laws &lt;em&gt;persist&lt;/em&gt; in the larger system of positive &lt;em&gt;and&lt;/em&gt; negative integers, then the Law of Signs is a logical consequence of these basic laws of positive integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These fundamental laws of &quot;numbers&quot; are axiomatized by the algebraic structure known as a &lt;a href=&quot;http://en.wikipedia.org/wiki/Ring_%28mathematics%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;ring&lt;/a&gt;, and various specializations thereof. Since the above proof uses only &lt;em&gt;ring laws&lt;/em&gt; (most notably the &lt;em&gt;distributive law&lt;/em&gt;), the Law of Signs holds true in every ring, e.g. rings of polynomials, power series, matrices, differential operators, etc. In fact every nontrivial ring theorem (i.e. one that does not degenerate to a theorem about the underlying additive group or multiplicative monoid), must employ the distributive law, since that is the only law that connects the additive and multiplicative structures that combine to form the ring structure. Without the distributive law a ring degenerates to a set with two completely unrelated additive and multiplicative structures. So, in a sense, the distributive law is a &lt;em&gt;keystone&lt;/em&gt; of the ring structure.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-05-27T20:21:14.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11070" PostTypeId="1" CreationDate="2016-05-28T04:04:37.123" Score="5" ViewCount="1595" Body="&lt;p&gt;I am a game developer currently making a game based on basic arithmetic problems for kids and teens. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any research papers on the average time to solve basic arithmetic problems or on how to evaluate success and improvement rate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That would help us a lot in creation of this game! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: What I mean by basic arithmetic is simple question like 2+3+0+4, 3*3, 9-4-2, 9/3 etc, you get the idea.&lt;/p&gt;&#xA;" OwnerUserId="6810" LastEditorUserId="6810" LastEditDate="2016-06-02T03:36:24.210" LastActivityDate="2016-06-08T17:54:02.823" Title="Average time to solve basic arithmetic problem" Tags="|secondary-education|primary-education|education-research|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11071" PostTypeId="1" CreationDate="2016-05-28T04:35:27.003" Score="3" ViewCount="1769" Body="&lt;p&gt;I want to find and know about teaching models in mathematics. There's such a model for teaching science: the 5 E's model:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/VRlCb.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VRlCb.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any suggestion (books, references, etc.) would be greatly appreciated.&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2016-07-21T16:29:42.243" Title="Teaching Models for Mathematics (like 5 E's in Science)" Tags="|teaching|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11073" PostTypeId="2" ParentId="5794" CreationDate="2016-05-28T05:35:05.740" Score="1" Body="&lt;p&gt;Here's what my math teacher once told me: if a good thing happens to a good person it's a good thing. If a bad thing happens to a bad person it's a good thing. If a good thing happens to a bad person or a bad thing happens to a good person it's a bad thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This doesn't help in any way shape or form of you're trying to explain why these things happen; however, if the student is not naturally mathematically inclined then this is the easiest analogy to help them remember the rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student in question does not struggle in math then this analogy will just mess them up.&lt;/p&gt;&#xA;" OwnerUserId="6811" LastEditorUserId="64" LastEditDate="2016-06-21T16:42:58.367" LastActivityDate="2016-06-21T16:42:58.367" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11074" PostTypeId="2" ParentId="11026" CreationDate="2016-05-28T05:42:28.830" Score="4" Body="&lt;p&gt;The answer to this question is identical to the answer to the question &quot;why should I learn trigonometry if I'm not going to be an engineer&quot;. We do not learn math for the sake of being able to solving equations. We learn math to train our minds and develop mathematical reasoning. Wolfram alpha will solve an equation but it will not train you to comprehend complex situations in real life.&lt;/p&gt;&#xA;" OwnerUserId="6811" LastActivityDate="2016-05-28T05:42:28.830" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11075" PostTypeId="2" ParentId="11048" CreationDate="2016-05-28T06:56:00.570" Score="11" Body="&lt;p&gt;All 8 answers so far seem to have missed the following issue:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ask the student why he picked $-1$ rather than $1$ or $0$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If he changes his mind and says that $0$ is the smallest, then he is using a &lt;strong&gt;valid&lt;/strong&gt; notion of &quot;smallest&quot;, namely &lt;strong&gt;smallest in magnitude&lt;/strong&gt;. Then it is interesting to discuss when this is useful (distance from origin) and when it is not useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If however, he says that $-1$ is strictly smaller than $1$, ask him &lt;strong&gt;why&lt;/strong&gt;? Invariably the only answer he can give is that $-1$ is negative. Then the next question is, &lt;strong&gt;why&lt;/strong&gt; are negative numbers smaller than positive numbers? It is entirely possible that the student was &lt;strong&gt;rote-taught&lt;/strong&gt; this, but his teacher never gave an explanation as to &lt;strong&gt;why&lt;/strong&gt;, and therefore it was messily mixed with the intuitive notion of magnitude and produced a weird ordering:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$-1 &amp;lt; -2 &amp;lt; -3 &amp;lt; \cdots &amp;lt; 0 &amp;lt; 1 &amp;lt; 2 &amp;lt; 3 &amp;lt; \cdots$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Along the way, the student should be asked to order the above seven numbers, which should reveal clearly his perspective. Ultimately, exploring the students' perspectives is crucial to a successful mathematical education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, for the original question of what is the smallest 1-digit number, the official answer is &lt;strong&gt;wrong&lt;/strong&gt; because the correct answer is $0$. $-1$ is not a 1-digit number; it does not only consist of one digit! If you want to argue with me, simply do a Google search and see that almost everyone says &quot;$k$-digit number&quot; to exclude negative integers (there may be some who include zero, and some who do not, but nobody includes negative integers).&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2016-05-28T06:56:00.570" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11076" PostTypeId="2" ParentId="5794" CreationDate="2016-05-28T10:37:57.810" Score="1" Body="&lt;p&gt;Here is another way to show/prove this by treating multiplication as repeated addition:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$3 \cdot 5 = 5+5+5 = 15$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;as &quot;adding $5$ three times&quot;. Now&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$3 \cdot (-5) = (-5) + (-5) + (-5) = (-15)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is clear because &quot;adding $-5$ three times is $-15$&quot;. Then we want the commutative law to hold for negative numbers as well, so&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(-5) \cdot 3 = (-15)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but we have to leave the &quot;repeated addition&quot; model of multiplication here, because adding something $-5$ times does not make sense. If we now flip the sign of the second factor again, we want the product to flip its sign as well, just as it happened in the second step. $(-5) \cdot 3$ shouldn't be the same as $(-5) \cdot (-3)$. We get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(-5) \cdot (-3) = 15 \qquad \text{&quot;qed&quot;}$&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2016-05-28T10:37:57.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11077" PostTypeId="1" CreationDate="2016-05-28T11:58:01.487" Score="2" ViewCount="640" Body="&lt;p&gt;I am teaching English in a Japanese primary school and they have a Nepalese child who doesn't understand written Japanese especially so they got me to translate some maths problems for them. I had a lot of those sentence type maths growing in Australia and I wondered what the point of writing sentences that need to parsed by students to extract the mathematics is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I read somewhere that they think it's real although there would be situations where someone would write a sentence on a situation just to be mathematically analyzed. I can't think of an engineering (my first degree) mathematics situation like that. For instance, in calculus there is no way to have a true sentence problem---it would take many paragraphs to express the many complicated expressions and that would be very difficult for a university student to understand; they don't have that enough time in exams. It's not the same as having an engineering problem to solve---not that we did many.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many ways to include context into a maths question (without using sentences with lots of extraneous detail)---diagrams, tables, short phrases (e.g., 240V +-15, 60Hz according a standard)---so that it isn't that hard to linguistically interpret. An applied math problem is not the same as a pure math problem with the same nominal mathematics.  For example, $2+2=4$ but the applied maths example is you need $2$ lots of $2$ m of steel and the only available lengths of steel don't include $4$ metre lengths rather only $5$ metre lengths then it's effectively $2+2=5$ (+extra $1$ metre) in metres.     &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You also have to differentiate between reading and writing. It's a bit easier to write than read---you understand your own writing more and writing summary statements is not what the question is about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sure there are some documents in some fields where there are numbers but it's unclear whether pushing primary school children with the simplistic sentence type problems that are normal helps them do such problems.  Please explain the value of sentence type problems as against other contextual forms for preparing young children for possible real problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone spot what's wrong with the Paint problem below?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FttXz.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FttXz.jpg&quot; alt=&quot;Paint problem&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\frac56$ dL ($1$ dL $=0.1$ L) of paint covers $\frac34$ m$^2$.  How far does $1$ dL of paint cover?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It should have shown a paint can and the painting done from it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's wrong with the Steel problem below?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/yaDtp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yaDtp.png&quot; alt=&quot;Steel problem&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\frac27$ m of steel bar weighs $\frac45$ kg. How much does $1$ m of steel bar weigh?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It should have had a metre length of steel and its weight with an arrow. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is wrong with the answer key (teacher's copy is shown) is that it doesn't include any details of the context except units.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is wrong with the context is that primary school students wouldn't have any idea about paint economics or steel usage. These are totally foreign to them and so students know to skim out the numbers. But this is what happens when we insist on having sentence problems---they have to be artificial and superficial otherwise the effort of teaching fractions is very diluted with teaching context and sentence structure.  &lt;/p&gt;&#xA;" OwnerUserId="6813" LastEditorUserId="77" LastEditDate="2016-06-05T04:59:48.313" LastActivityDate="2016-06-05T10:18:31.463" Title="What is the value of sentence type maths problems for primary school students?" Tags="|word-problems|" AnswerCount="8" CommentCount="38" ClosedDate="2016-06-04T17:49:55.157" ContentLicense="CC BY-SA 3.0" />
  <row Id="11078" PostTypeId="2" ParentId="11077" CreationDate="2016-05-28T22:15:45.647" Score="4" Body="&lt;p&gt;There are three direct benefits, as far as I can tell.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Word problems answer questions like &quot;why do I need to know this&quot;. If you have a student who thinks that learning math is pointless then you can use word problems to help them understand why math is important&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For some students math is very difficult when's it's just abstract numbers. For more linguistically oriented students the word problems may help them visualize the problems and gain an understanding of the underlying mathematical concepts. For more mathematically oriented students word problems may be distracting and more difficult than simple equation - based questions, meaning that they will be forced to develop their linguistic reasoning skills.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As others have pointed out, the ability to study through excess information to find the actual salient details is a valuable skill.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Just a note: for points 1 &amp;amp; 3 it may be best to tell the student directly that these are part of the reasons for word problems. If you just give students the questions without clarification they may just think that word problems are pointless and irritating.&lt;/p&gt;&#xA;" OwnerUserId="6811" LastActivityDate="2016-05-28T22:15:45.647" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11079" PostTypeId="1" CreationDate="2016-05-28T22:32:01.837" Score="7" ViewCount="262" Body="&lt;p&gt;Throughout my mathematical education I came across a very common problem. Essentially what happens is that a student will fail to properly learn a certain concept and will fall behind. Each year that the student is in math classes he or she will fall further and further behind the rest of the class until, eventually, the gap between students is so wide that the student has too stop taking classes altogether. My question is: what can be done to stop this problem? How can a student be brought up to the correct grade level without making the rest of the class go through remedial course work as well? Assume that the school in question has very limited resources. (i.e. no special remedial classes or in- class assistants)&lt;/p&gt;&#xA;" OwnerUserId="6811" LastEditorDisplayName="user2139" LastEditDate="2016-05-29T11:42:47.393" LastActivityDate="2016-05-29T11:42:47.393" Title="Students who fall behind" Tags="|struggling-students|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11080" PostTypeId="2" ParentId="11079" CreationDate="2016-05-29T09:09:49.657" Score="4" Body="&lt;p&gt;something you can do is make fun and motivating papers of the theory. This could be, for example on one A4 paper . Good clear explanations with lots of color and some clear examples develop step by step. It takes a lot of your time, but it's something that you can use every year. &#xA;What you can also do is scan your empty exercise papers and  write tips. The students can then use your tips and this can help him solve the exercises in the class. For example, at an exercise on number theory where there is an ‘ – ‘ before the fracture &quot;watch out for the minus before the fracture and add ( ) !!!!&quot; are small but helpfull things that can make a big difference . When studying the student can read the tips again and again the theory is refreshed each time.&lt;/p&gt;&#xA;" OwnerUserId="6799" LastActivityDate="2016-05-29T09:09:49.657" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11081" PostTypeId="2" ParentId="11048" CreationDate="2016-05-29T15:12:23.920" Score="8" Body="&lt;p&gt;So $-9$ is &quot;smaller&quot; than $-1$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By that terminology, a bank account overdrawn by $9$ dollars (balance $-9$) would be said to carry a &quot;smaller debt&quot; than a bank account overdrawn by $1$ dollar (balance $-1$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept of &quot;smaller&quot; and &quot;bigger&quot; corresponds naturally to magnitude.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the complex plane, it corresponds to modulus. A complex number $z_0$ is smaller than $z_1$ if $|z_0| &amp;lt; |z_1|$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We should avoid calling $-9$ &quot;smaller&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Between the integer $-9$ and $-1$, $-9$ is &lt;em&gt;the more negative&lt;/em&gt; of the two, and has the &lt;em&gt;greater magnitude&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can also say that $-9$ is &lt;em&gt;less than&lt;/em&gt; $-1$ (the pronunciation of $-9\lt -1$).  If you carry a bigger debt, you have less money (are less rich) than if you carry a smaller debt.   A good way to explain why we call this &quot;less&quot; when the magnitude is getting bigger is that in measurements of measured or counted quantities like money, voltage or temperature, zero is an &lt;em&gt;arbitrary point&lt;/em&gt;.  For instance, just because you have zero dollars in a bank account, that doesn't mean you are broke: you hold money in other places, such as other bank accounts, investments or cash.  If all your other assets are held equal, then a $-9$ account makes you poorer than a $-1$ account, even if you have a billion dollars.  The zero is just &lt;em&gt;relative to that account&lt;/em&gt;.  In temperature, $-9^o C$ is less warm than $-9^o C$ because zero is also arbitrarily set; both temperatures are greater than absolute zero. If these measurements are expressed as absolute temperature, such as $^oK$, then the lower one indeed has the lower magnitude.&lt;/p&gt;&#xA;" OwnerUserId="6816" LastActivityDate="2016-05-29T15:12:23.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11082" PostTypeId="2" ParentId="11077" CreationDate="2016-05-29T15:37:31.813" Score="2" Body="&lt;p&gt;The point to word problems is scalable testability. You need to state some problem once and can test thousands of students, compare their answers and produce impressive statistics. Whether this has something to do with real-life situations, where &quot;you have to science the shit out of this&quot; (quoting The Martian) is debatable.&lt;/p&gt;&#xA;" OwnerUserId="1479" LastActivityDate="2016-05-29T15:37:31.813" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11085" PostTypeId="2" ParentId="11077" CreationDate="2016-05-30T06:43:56.623" Score="3" Body="&lt;p&gt;Perhaps an example would clarify what you mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question 1:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What number is obtained after reducing 40 by 5%?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Question 2:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Farmer John has lately been having problems selling his corn.  He needs to start selling more of them soon, or they will start to go bad.  It seems that the retail price of 40 cents is too high, so he decides to start offering a 5% discount.  What would the new price of the corn be?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If I understand you correctly, you feel that these two questions are mathematically equivalent (that is, they test for the same mathematical skills and understanding).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that you believe that the second one is inferior compared to the first because it has a lot of irrelevant (extraneous) text that serve only to distract and confuse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But another way to look at it is to say that the second question provides context and relevance by presenting the math in a (supposedly) real-world situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue that sometimes these &quot;irrelevant&quot; texts can sometimes actually aid the student with the math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, compare:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question 3:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$812\div 4=$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Question 4:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Alice, Bob, Charlie, and Dan are a close group of friends.  Alice recently won \$ 812 in a lottery and would like to share the amount equally among the members of the group.  How much should each friend get?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It so happens that many students answer question 3 incorrectly.  (They answer &quot;23,&quot; a split-dividend error.)  But if you were, say, Dan, and your friend Alice wanted to divide \$ 812 equally among the four of you, would you be satisfied with getting \$ 23?&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-05-30T06:43:56.623" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11086" PostTypeId="1" CreationDate="2016-05-30T13:51:51.973" Score="7" ViewCount="183" Body="&lt;p&gt;I recall reading a paper a few years ago (though I think it was in physics) on a class where they decided to repeat a midterm test question on the final exam and students performed noticeably worse on the final exam than on the midterm (in fact only one person's grade noticeably increased I think).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone help jog my memory with which paper this might have been? Alternatively, if you know of another paper that replicated this effect in mathematics and could let me know that would work as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="6822" LastEditorUserId="6822" LastEditDate="2016-05-30T15:24:32.107" LastActivityDate="2016-06-27T12:35:42.223" Title="Paper on Replicating Test Questions" Tags="|reference-request|testing|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11087" PostTypeId="2" ParentId="11077" CreationDate="2016-05-30T14:00:17.800" Score="19" Body="&lt;p&gt;Consider these 5 problems:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What is $ 35 \div 10 $ ?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We are in the store and want to buy packages of party plates for a birthday party.   The plates come in packages of 10.  There will be 35 children at the party.  How many packages should we buy so that everyone will have a plate?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;10 friends go out to get ice cream. The bill comes to $35.  If the friends want to divide the bill evenly, how much should each friend pay?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Leah has 35 oz. of juice.  She has glasses that hold 10 oz.  How many glasses can she fill?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;David has 35 large cookies.  He can put 10 on a platter.  After he has filled all the platters he can, he leaves the remaining cookies on the counter for his family to enjoy.  How many cookies are left on the counter?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Note that all of these problems can be solved by solving the first expression&#xA;$$ 35 \div 10  $$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet the solution to problems 2, 3, 4, and 5 are all different.&#xA;The answers are:&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;4 packages&lt;/li&gt;&#xA;&lt;li&gt;$3.50&lt;/li&gt;&#xA;&lt;li&gt;3 full glasses&lt;/li&gt;&#xA;&lt;li&gt;5 cookies&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The word problems give us a context for how to handle the remainder in problem number 1.  Without that context, teaching these different division methods would harder to grasp. It also gives the student a chance to decide in what context they should use each method.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2016-06-01T04:34:51.593" LastActivityDate="2016-06-01T04:34:51.593" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11088" PostTypeId="2" ParentId="11070" CreationDate="2016-05-30T16:03:42.297" Score="3" Body="&lt;p&gt;I'd expect that such data would be quite useless for your game. &quot;Reliable&quot; numbers would originate from large-scale tests which are usually quite far away from the situation of a kid competing in a game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To appeal to the broadest population, you should keep individual timings for each player and only use them as comparison.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another point to consider: If you're doing &quot;basic arithmetic&quot; the game is probably colorful and flashy, possibly mousy-pointer-clicky. This captivating gameplay will distract in the beginning and become more and more ignored with more practice. So part of the &quot;improving timings&quot; will be due to the audience learning to deal with the game mechanics and not only with improve numeric skills.&lt;/p&gt;&#xA;" OwnerUserId="1479" LastActivityDate="2016-05-30T16:03:42.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11089" PostTypeId="2" ParentId="11070" CreationDate="2016-05-30T18:00:41.750" Score="6" Body="&lt;p&gt;This question is almost totally unanswerable unless you clarify what is meant by &quot;basic arithmetic problems for kids and teens&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But let's consider a lower-bound case where you're talking about basic one-digit addition and multiplying facts. In instances like this, the knowledge should be &lt;em&gt;automatic&lt;/em&gt;, that is, instantly recalled without effort (at least by the 3rd grade). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few years ago, when developing a similar gamey practice website (&lt;a href=&quot;http://www.automatic-algebra.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Automatic-Algebra.org&lt;/a&gt;), I made a survey of several educators' and practice materials' suggestions for times-tables drills. That's summarized in the chart below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/v3yWG.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/v3yWG.png&quot; alt=&quot;Multiplying drill rate statistics&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short: Simple times-tables facts should be answerable in 2 or 3 seconds on average. Personally, I set up my drill to give twice that much time as a safety buffer (30 seconds for 5 questions, that is, one every 6 seconds). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best practice would be to test and refine your game with actual users. In the most general sense, questions might take anywhere between zero and infinity seconds, depending on what you're asking. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-05-30T22:36:31.000" LastActivityDate="2016-05-30T22:36:31.000" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11090" PostTypeId="2" ParentId="11077" CreationDate="2016-05-30T18:45:48.660" Score="11" Body="&lt;p&gt;Professional work in general, and scientific and mathematical work in particular, is done principally &lt;em&gt;in writing&lt;/em&gt;. Requirements, specifications, orders, field reports, case studies, journals, grants, etc., are all disseminated and documented &lt;em&gt;in writing&lt;/em&gt;. Symbolic mathematical notation is inherently a specialized system of &lt;em&gt;concise writing&lt;/em&gt; (arguably, it is nigh-impossible to speak it with proper precision). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Treating mathematical writing as a specialized language, there can be no more important skill than being able to translate from natural language to mathematical language and back again. Without that skill, mathematical notation is entirely useless, an isolated island without meaning. Word problems are precisely the exercise one needs to develop this critical skill. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider these words by William Kingdon Clifford (&lt;em&gt;The Common Sense of the Exact Sciences&lt;/em&gt;, 1823), which appear at the top of my course syllabi in the next semester:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We may always depend on it that algebra, which cannot be translated&#xA;  into good English and sound common sense, is bad algebra. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Consider, too, George Polya's famous book &lt;em&gt;How to Solve It&lt;/em&gt; (1945), which essentially organizes problem-solving into four stages (also reminiscent of the software development life cycle; quote &lt;a href=&quot;https://en.wikipedia.org/wiki/How_to_Solve_It&quot;&gt;here&lt;/a&gt;): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;First, you have to understand the problem.&lt;/li&gt;&#xA;  &lt;li&gt;After understanding, then make a plan. &lt;/li&gt;&#xA;  &lt;li&gt;Carry out the plan. &lt;/li&gt;&#xA;  &lt;li&gt;Look back on your work. How could it be better?&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that every step other than #3 is a reading/writing task. If you dig into Polya's details more, Step #1 includes &quot;Do you understand all the words used in stating the problem?&quot;; Step #2 is essentially translation to an equation; and Step #4 is restating the answer in natural language so as to check for reasonability. The translation between natural language and back is inherent throughout the process. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the OP says, &quot;I can't think of an engineering situation like that - my first degree.&quot;, that's a statement that frankly beggars the imagination, and leaves this writer entirely incredulous and aghast. Surely any real engineering work is initially a problem posed in natural language. Even if we just open a random physics, calculus, or statistics book, we will find that &lt;em&gt;most or all of the problems&lt;/em&gt; after a certain point are phrased as natural-language word problems. Let's take a survey from a few, using OpenStax open-source texts as a resource (available for &lt;a href=&quot;https://openstax.org/subjects/&quot;&gt;free download here&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;OpenStax College Physics, Chapter 2 (Kinematics), Example 2.1:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A racehorse coming out of the gate accelerates from rest to a velocity&#xA;  of 15.0 m/s due west in 1.80 s. What is its average acceleration?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;OpenStax Calculus Volume 1, Chapter 4 (Applications of Derivatives), Example 4.1:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A spherical balloon is being filled with air at the constant rate of 2&#xA;  cm^3/sec (Figure 4.2). How fast is the radius increasing when the&#xA;  radius is 3 cm?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;OpenStax Introductory Statistics, Chapter 9 (Hypothesis Testing with One Sample), Example 9.21:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In a study of 420,019 cell phone users, 172 of the subjects developed&#xA;  brain cancer. Test the claim that cell phone users developed brain&#xA;  cancer at a greater rate than that for non-cell phone users (the rate&#xA;  of brain cancer for non-cell phone users is 0.0340%). Since this is a&#xA;  critical issue, use a 0.005 significance level. Explain why the&#xA;  significance level should be so low in terms of a Type I error.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Anywhere we look at core prerequisites for any standard engineering curriculum, we find that the work comes to us in written, natural-language form first, and developing the capacity to parse and translate that language to mathematical language is essential.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-05-30T18:45:48.660" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11091" PostTypeId="2" ParentId="11048" CreationDate="2016-05-31T01:32:18.523" Score="6" Body="&lt;p&gt;I will only add to the other excellent answers that even the words &quot;&lt;em&gt;less than&lt;/em&gt;&quot; (the conventional name of the $&amp;lt;$ sign) can reinforce the (incorrect) notion that $&amp;lt;$ is used to compare the &lt;em&gt;magnitude&lt;/em&gt; of two numbers.  For non-negative numbers this is perfectly correct and reasonable, but as soon as negative numbers enter the conversation it becomes problematic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this reason, it may sometimes be helpful to avoid pronouncing &quot;$a&amp;lt;b$&quot; as &quot;$a$ is less than $b$&quot; and instead pronounce it as &quot;$a$ &lt;strong&gt;comes before&lt;/strong&gt; $b$&quot;.  (Or, if you want to be fancier, as &quot;$a$ &lt;strong&gt;precedes&lt;/strong&gt; $b$.&quot;  This focuses the attention on $&amp;lt;$ as a symbol denoting &lt;em&gt;order&lt;/em&gt;, not &lt;em&gt;size&lt;/em&gt;.  So we would say that $-9$ comes before $-1$, which comes before $3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Truth in advertising:  I have never used this way of speaking with elementary school students, but I have used it with graduate students in a real analysis class, when discussing ordered fields other than $\mathbb{R}$.  In that context the order may really be based on something other than magnitude, so the phrase &quot;less than&quot; / &quot;greater than&quot; is not just confusing, it is outright inaccurate.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2018-08-30T05:07:16.753" LastActivityDate="2018-08-30T05:07:16.753" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11093" PostTypeId="1" CreationDate="2016-05-31T19:07:45.680" Score="7" ViewCount="199" Body="&lt;p&gt;I have studied long ago different method for subtracting numbers, for instance the borrow method or the Austrian method (I hope I am using the right names; I am not an English native speaker and these do not correspond to the terminology used in french) and others whose names I don't know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be interested in knowing how countries have decided what is the &quot;main&quot; method to teach in school, and if there is some kind of international study trying to determine whether some method provides better results than another.&lt;/p&gt;&#xA;" OwnerUserId="6825" LastEditorUserId="262" LastEditDate="2016-05-31T19:31:38.430" LastActivityDate="2016-05-31T21:06:59.583" Title="Subtraction Methods in School Mathematics" Tags="|reference-request|primary-education|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11094" PostTypeId="2" ParentId="11093" CreationDate="2016-05-31T19:30:10.640" Score="6" Body="&lt;p&gt;For the United States, I suggest as a resource:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Karp, K., Caldwell, J., Zbiek, R. M., &amp;amp; Bay-Williams, J. (2011). &lt;em&gt;Developing Essential Understanding of Addition and Subtraction for Teaching Mathematics in Pre-K-Grade 2&lt;/em&gt;. National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is a screenshot of page 45:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wMNHe.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wMNHe.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you may notice from the image above, the book &lt;em&gt;does&lt;/em&gt; cover the &quot;Austrian method&quot; about which you ask. You can also see in &lt;strong&gt;Reflect 1.14&lt;/strong&gt; at the top of the page that there is another method discussed called &quot;partial differences.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The section on &lt;strong&gt;Properties and Algorithms [of Subtraction]&lt;/strong&gt; (p. 35) includes a number of approaches and representations with respect to subtracting. Besides the Austrian Method and partial differences, there is also the so-called Standard Algorithm for Subtraction, as well as the use of additive inverses in what is called Compensation or Friendly Numbers (in the book, this is called the &quot;same change algorithm&quot;). In terms of presentations, these include number lines, hundreds charts, place-value blocks, descriptions in words, and symbolic representations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For completeness, here is a picture of the book's table of contents:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NgZig.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NgZig.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-05-31T19:30:10.640" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11096" PostTypeId="1" CreationDate="2016-06-01T05:14:24.420" Score="-2" ViewCount="133" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Consider a square : four points in a plane constructed with classical means (compass and straightedge). Since no point is different from others (no coordinates, no labels...) it seems that we can not define any (some) symmetry without additional mathematical structure.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;With distance involved, the two nearest points to a given point are equivalent and so this new object has two lines of symmetry only.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In case all four points are taken as different we have our four lines of symmetry...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The question is : what is the best way to introduce the minimum additional mathematical structure needed, to go from 1 to 2 and from 2 to 3 without reaching to physics and other extra-mathematical illustrations (paper folding, physical rotation etc.) ? Maybe there are more than one good road to achieve this goal ? What are the key / most persistent / misconceptions in introducing symmetry of a square to elementary graders as well as the silent assumptions underlaying such introductions ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any reference would be very much appreciated.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Let me try to clarify myself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal is to explain, in simple but hopefully, mathematically correct way, why and under which conditions one may say that a square has four lines of symmetry. So, let start with those four points that geometrically &lt;em&gt;define&lt;/em&gt; square. They are all indistinguishable as lay on a plane. We may label them (and so make them physically distinguishable) but lets try not to step outside mathematics.  We can not rotate this &quot;square&quot;, can not translate it - it is quite a primitive object. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Introduce, at a step No. 2, a simple metric (distance) in this set X of our four points. (map d: X x X -&gt; R) Question: what is the most simple metric here ? The one in which the two adjacent point of any given point of our square are still indistinguishable (equivalent) and the fourth one, is the furthest one from our vantage point. Question: with this most simple metric, how many lines of symmetry does this (new) object (four points plus simple metric) has ? I say &lt;em&gt;two&lt;/em&gt;, one along the diagonal and the other cutting the opposite sides at half. We can not discuss any rotation at this level as we did not have coordinates for &lt;em&gt;all&lt;/em&gt; points of our plane. Now we may select another point as reference point of a coordinate system, make axes, units along these axes and label the points with coordinates in respect of this coordinate system. This is a lot of additional mathematical structure. Now we have square with distinguishable points, rotations and - voila : four lines of symmetry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, that is. From geometry, to metric spaces, real numbers, continuous symmetry etc. Quite a lot, i would say, to be introduced at one step to. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Key question : how to make these steps, from a most basic square (four points only) to our useful, ordinary square (with circumference and area) as simple and as precise as possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regards.&lt;/p&gt;&#xA;" OwnerUserId="6830" LastEditorUserId="6830" LastEditDate="2016-06-01T15:05:50.263" LastActivityDate="2016-06-01T15:15:26.313" Title="symmetry of a square - is it possible pure geometric approach in didactics?" Tags="|geometry|" AnswerCount="1" CommentCount="7" ClosedDate="2016-06-01T20:33:08.277" ContentLicense="CC BY-SA 3.0" />
  <row Id="11097" PostTypeId="2" ParentId="11086" CreationDate="2016-06-01T06:48:40.387" Score="1" Body="&lt;p&gt;What you describe sounds like a standard &lt;strong&gt;pre-test&lt;/strong&gt; and &lt;strong&gt;post-test&lt;/strong&gt; analysis. Typically these are done to check the impact of a specific &lt;strong&gt;intervention&lt;/strong&gt;. You first give a pre-test to the target group. Then you make the intervention. (You might have a control group too which may not undergo intervention). After the intervention you give the same test again which is called post-test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the performance in post-test is better than pre-test you could say the intervention worked. If it's not the case then probably the intervention was ineffective (&lt;em&gt;as seems to be the case in the paper you're looking for&lt;/em&gt;). There are lots papers which do pre-test and post-test analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a couple to get you started:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.2167/le678.0&quot; rel=&quot;nofollow&quot;&gt;http://www.tandfonline.com/doi/abs/10.2167/le678.0&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://search.proquest.com/openview/91bc411ad432f947bf80358805ec2d1f/1&quot; rel=&quot;nofollow&quot;&gt;http://search.proquest.com/openview/91bc411ad432f947bf80358805ec2d1f/1&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a couple more where post-test performance was lower than pre-test:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://dl.acm.org/citation.cfm?id=971328&quot; rel=&quot;nofollow&quot;&gt;http://dl.acm.org/citation.cfm?id=971328&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://goo.gl/DIsw8W&quot; rel=&quot;nofollow&quot;&gt;https://goo.gl/DIsw8W&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2191" LastEditorUserId="2191" LastEditDate="2016-06-01T06:56:31.477" LastActivityDate="2016-06-01T06:56:31.477" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11098" PostTypeId="1" AcceptedAnswerId="11099" CreationDate="2016-06-01T09:11:11.597" Score="14" ViewCount="1196" Body="&lt;p&gt;I am concluding my second year in mathematics at the university of Milan.&#xA;I also happen to be an educator for 8 - 12 year old children (as a &lt;a href=&quot;http://www.agesci.it/&quot;&gt;Scout&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently I have tried to fill some dead time by asking my kids some soft math/logic questions and having them argue about them among themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Little did I know that my tiny project would have so much success that they now demand more questions, especially at lunch, and I have exhausted my store of puzzles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also found that while I sometimes manage to come up with a new one, it is often too easy or too hard for their age.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know of some resource covering this kind of... thing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Straight-up question suggestions are also welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a sample of the questions I gave them (brutally stripped of the playful, narrative parts):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A certain frog doubles in size every day. When thrown into a well, it fills the well in 10 days. How many days does it take to fill a well, if I throw two frogs in it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How high up can you count using only the 10 fingers of your hands? (Really &quot;how many numbers can you represent&quot;, made them discover the binary counting).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How many fingers are there in one hand? And in two? And in ten? (evil trick question: almost everybody answers &quot;a hundred&quot; to the last question).&lt;/p&gt;&#xA;" OwnerUserId="6703" LastActivityDate="2016-07-05T18:28:08.107" Title="Soft questions for 8 - 12 year olds" Tags="|reference-request|primary-education|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11099" PostTypeId="2" ParentId="11098" CreationDate="2016-06-01T12:32:19.237" Score="7" Body="&lt;p&gt;An excellent ressource is Vladimir Arnold's list, which is (at least in part) available in a recent EMS Newsletter:&#xA;&lt;a href=&quot;http://www.ems-ph.org/journals/newsletter/pdf/2015-12-98.pdf&quot;&gt;http://www.ems-ph.org/journals/newsletter/pdf/2015-12-98.pdf&lt;/a&gt;&#xA;(page 14).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A classical variant of your frog question is there, as well as many others. Some are said by Arnold to be easier for kids than for academics, and he might be right!&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-06-01T12:32:19.237" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11100" PostTypeId="2" ParentId="11096" CreationDate="2016-06-01T15:15:26.313" Score="3" Body="&lt;p&gt;The indistinguishability of the points is precisely the reason &lt;em&gt;why&lt;/em&gt; the square has symmetries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's labels one of the four points of the square $A$.  To this you object (I think, if I have understood you correctly):  &quot;But there is no way to tell the points apart, so how do we know which one is $A$?&quot;  And to this I respond:  &quot;I agree that we don't know which one is $A$.  It is an arbitrary choice.  &lt;em&gt;Remember that for later.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, let's label one of the remaining three points $B$ -- but let's be sure to choose one of the points with the property that the segment joining $A$ to $B$ is one of the sides of the square, not its diagonal.  (Notice that having made a choice of $A$, we are now able to &lt;em&gt;partly distinguish&lt;/em&gt; among the remaining three points, because two of them are adjacent to $A$ and the third one is diagonally across from $A$.)  We still don't have any basis for distinguishing between the two candidates for $B$, so just pick one of them arbitrarily, and &lt;em&gt;remember for later that we made this choice&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having chosen $A$ and $B$, let $C$ be the point diagonally across from $A$ and let $D$ be the point diagonally across from $B$.  Notice that this identification is now &lt;em&gt;unambiguous&lt;/em&gt;; having chosen $A$ (arbitrarily) and $B$ (partly arbitrarily), the identification of $C$ and $D$ is now completely determined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, what if two different people, looking at the same (unlabeled, &quot;naked&quot; square in a bare plane with no coordinates) had followed the same set of instructions above, but made different initial choices about which point to call $A$ and which point to call $B$?  Then they would have labeled the squares differently, of course.  All together there are $8$ different possible labelings they could have chosen.  Those $8$ labelings are precisely the $8$ symmetries of the square.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-06-01T15:15:26.313" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="11101" PostTypeId="2" ParentId="11098" CreationDate="2016-06-01T17:20:11.213" Score="5" Body="&lt;p&gt;Here are some of my favorites:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A man is given 3 pills and told to take them every half hour starting at 9 AM.  When will he finish?   (He'll finish at 10 AM &lt;strong&gt;and not at 10:30&lt;/strong&gt;)&lt;/li&gt;&#xA;&lt;li&gt;What number comes after ninety-nine? four hundred ninety-nine? four thousand ninety-nine?? (The answers are 100, 500, and 4,100 &lt;strong&gt;not 5000 for the last one&lt;/strong&gt;)&lt;/li&gt;&#xA;&lt;li&gt;A man has 17 sheep.  All but 8 die of sickness.  How many are left? (8 &lt;strong&gt;and not 9&lt;/strong&gt;)&lt;/li&gt;&#xA;&lt;li&gt;I have 2 coins totaling 35 cents.  One of them is not a quarter.  What are the coins?   (A dime and a quarter - &lt;strong&gt;one is not a quarter but the other is&lt;/strong&gt;).&lt;/li&gt;&#xA;&lt;li&gt;I have 9 coins. One is counterfeit and is lighter than the others.  I have a balance scale.  What is the least number of weighings that I can use to find the counterfeit coin?  (It can be done in 2 weighings)&lt;/li&gt;&#xA;&lt;li&gt;A pencil and an eraser cost $1.10. The pencil costs one dollar more than the eraser. How much does the eraser cost?  (5 cents and &lt;strong&gt;not 10 cents&lt;/strong&gt;)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Have fun. I will try to find a reference for these and others like it for you.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-06-01T17:20:11.213" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11102" PostTypeId="2" ParentId="11077" CreationDate="2016-06-01T23:03:26.220" Score="7" Body="&lt;p&gt;I think you have a very different understanding of what &quot;mathematics&quot; is than I do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the following questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What kinds of geometric properties are held by the figure you get when you join the midpoints of adjacent sides of a polygon?&lt;/li&gt;&#xA;&lt;li&gt;Under what conditions on the parameters $a,b$ does an equation of the form $a^x = x^b$ have a rational solution?&lt;/li&gt;&#xA;&lt;li&gt;How many different ways are there to make \$3.45 using only three different kinds of coins?&lt;/li&gt;&#xA;&lt;li&gt;If you know a 5th-degree polynomial has real solutions only at $x=3, x=5$ and $x=-2$, what else can you say about the polynomial?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am not sure how &lt;em&gt;any&lt;/em&gt; of these questions could be expressed without using &quot;sentences&quot;.  Do you consider these valuable mathematical questions?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2016-06-02T01:52:05.873" LastActivityDate="2016-06-02T01:52:05.873" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11103" PostTypeId="2" ParentId="11070" CreationDate="2016-06-02T13:01:25.760" Score="3" Body="&lt;p&gt;At the risk of posting an answer that doesn't directly answer the question, I'm going to suggest that as a game or computer based software, you consider levels. Start with, say, 10 seconds per problem, and at each level, retire the time a bit, .5 seconds or so, to produce the number of levels you wish. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Better yet, have the reward points count down. At the moment the problem appears, the screen shows they can earn 10 points, and it drops as time passes, to zero when time runs out and the next question pops up. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-06-02T13:01:25.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11104" PostTypeId="1" CreationDate="2016-06-02T13:59:02.100" Score="10" ViewCount="1257" Body="&lt;p&gt;I have been reading a lot of books about deliberate practise recently like Angela Duckworth's &quot;Grit&quot;, &quot;Talent is Overrated&quot; - Colvin and &quot;Peak&quot; - Ericsson and Pool. I want to apply these principles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you suggest some concrete steps to implement the principles  deliberate practise in my mathematical studies ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I understand, deliberate practise has two main principles :&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It is focussed - For example, if you're a tennis player, it isn't just to play the most games or to hit the most number of balls as fast as you can. Rather, it would be to specifically practise the motion of the forehand and focus on the wrists, elbows and shoulder to ensure that the form is as correct as it can be. If you're a cricketer, it isn't to just hit the most number of balls, but to ensure that the bat lift is straight, the weight transfer is right, head is steady  and that the elbows, wrists and feet move as accurately as possible for the best possible shot. It is to put deliberate focus into those things and correct the form, rather than just focus on hitting the ball the most number of times.&#xA;While doing mathematics, what can I choose as specific goals to improve ? In sports, it's easy to choose a small part of the form to improve. But, it's not so obvious what this goal should be in Mathematics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It has constant feedback - If you're a tennis player, you become conscious of which part of your swing is wrong exactly and decide if the ball doesn't have too much top spin. Having gotten that feedback, you specifically start tailoring that part of your swing consciously to increase the amount of top spin. If you're a writer, you don't just write more. You analyse your writing, and realise that your character psychology is fine but your description of events and settings isn't. So, you find some writings where those parts are excellent and specifically focus on improving that aspect of your writing.&#xA;How do I analyze what mathematics I'm doing as good or bad ? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I want a set of concrete steps I can follow to improve my mathematical skills. I should note, that by mathematical skill, I mean ability to solve problems (Like the Olympiad type, which will later help me in research). I'm not asking for advice to increase my grade or marks. I'm interested in improving my mathematical ability.&lt;/p&gt;&#xA;" OwnerUserId="6833" LastEditorUserId="6833" LastEditDate="2016-06-05T01:39:46.983" LastActivityDate="2017-06-11T13:32:33.367" Title="How can I implement the principles of deliberate practise in my mathematical studies?" Tags="|undergraduate-education|student-motivation|self-learning|" AnswerCount="4" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="11105" PostTypeId="2" ParentId="11098" CreationDate="2016-06-02T14:32:54.637" Score="2" Body="&lt;p&gt;I think that you might find some good puzzles (some of which are quite active) in some Math Circles resources.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://bookstore.ams.org/mcl-11/&quot; rel=&quot;nofollow&quot;&gt;Appropriate for this age group&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://bookstore.ams.org/mcl-5/&quot; rel=&quot;nofollow&quot;&gt;Surprisingly, much in here could work well with your age group (e.g. the robot stuff)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://bookstore.ams.org/mcl-2/&quot; rel=&quot;nofollow&quot;&gt;Circle in a Box&lt;/a&gt; - I think the &quot;Criss-Cross&quot; and &quot;Double Time&quot; activities are fun&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now, some of these (most?) will be more time than you would have available.  But if you are reasonably creative you should be able to turn a lot of them into something that fits the dead time available.  Good luck!&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-06-02T14:32:54.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11106" PostTypeId="1" AcceptedAnswerId="11109" CreationDate="2016-06-03T11:29:01.480" Score="9" ViewCount="211021" Body="&lt;p&gt;In this part of the course I am just trying to show that we actually see a lot of sequences and series every day in our regular life.&#xA;I already found some examples such as the house numbers when you drive down a street, the number of people you reach in those 'chain mails', the value of your block in the game &lt;a href=&quot;https://gabrielecirulli.github.io/2048/&quot; rel=&quot;nofollow noreferrer&quot;&gt;2048&lt;/a&gt;, ...&lt;/p&gt;&#xA;&lt;p&gt;I am still looking for some fun, everyday sequences or series that without much context give an idea of what those sequences are.&lt;/p&gt;&#xA;" OwnerUserId="6785" LastEditorUserId="7365" LastEditDate="2022-11-10T15:47:42.147" LastActivityDate="2022-11-10T15:47:42.147" Title="Examples of arithmetic and geometric sequences and series in daily life" Tags="|examples|mathematics-in-daily-life|series|" AnswerCount="7" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11107" PostTypeId="2" ParentId="11077" CreationDate="2016-06-03T12:22:37.670" Score="4" Body="&lt;p&gt;I think there's a very basic answer to this. It's the very origin of math and numbers. Try to teach a child that 1+1=2 without the visual (or tactile) example of using fingers to show the concept of 1 and 2, and how the oneness of a single finger has something in common with the oneness of a single block. It's only when this sinks in, do we get comfortable with the multiplication table, although we still like to use physical aids to show how say, 3 rows of 4 eggs and 4 rows of 3 eggs are the same dozen eggs. Can I explain the commutative property of multiplication without eggs? I suppose, but why would I want to? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm specifically addressing &quot;what is the point of writing sentences that need to parsed by students to extract the mathematics?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because in grade school and high school, the math is meant to apply to real life, not to be abstract. In my experience, it's just the opposite (of your objection), students will be more engaged when a problem is offered in a way that occurs to solve a real situation. &quot;shirts cost \$X, pants cost \$Y, how many of each did Jane and Jack buy given $Z?&quot; This type of problem is preferable to only having lists of 2 equations in 2 unknowns. More than engaging the student, it helps them apply math to situations they are likely to encounter on their own. In high school, we have the issue of &quot;when will I need the law of cosines in real life?&quot; and I struggle at times to honestly say that more than a select few careers with call for its use. (That's another issue/question). But - in primary school, nearly every last bit of math turns into the math we actually should master to understand our own lives. How to balance my checkbook, go shopping and compare unit cost, handle my finances, order floor tiles for my bathroom project. Strip out the words from the math, and you find you've produced an adult who, as a child, could add 8% of a number to itself, but never made the connection this is how you calculate sales tax on a purchase. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To come full circle, when I proctor exams (in a high school), I'm sensitive to the students for whom English isn't a first language, and I tell them that for word problems, they can come ask for clarification on questions where their math is fine, but the vocabulary is an issue. In your case, this is what you need to do, to help translate the words into the student's native language. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-06-05T10:18:31.463" LastActivityDate="2016-06-05T10:18:31.463" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11108" PostTypeId="2" ParentId="11077" CreationDate="2016-06-04T15:30:19.363" Score="-3" Body="&lt;p&gt;I work in a highly technical field, and (realising this is an unpopular view) regard this type of question as an attempt to create bias in favour of (a) native English speakers, and (b) less mathematically able students. The 2nd point may be slightly less intuative, but frequently this question style is ambiguous when parsed in a litteral/formal way, and also the padding can be very distracting to anyone on the autistic/aspergers spectrum.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If someone asks me to solve this sort of problem in real life, I will &lt;em&gt;insist&lt;/em&gt; they ask a well defined problem rather than give me their life story. If they can't focus, I will not try and work out what they want.&lt;/p&gt;&#xA;" OwnerUserId="6520" LastActivityDate="2016-06-04T15:30:19.363" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11109" PostTypeId="2" ParentId="11106" CreationDate="2016-06-04T16:52:59.397" Score="10" Body="&lt;p&gt;Here are a few more examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the amount on your savings account ;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the amount of money in your piggy bank if you deposit the same amount each week (a bank account with regular deposits leads you to arithmetico-geometric sequences) ;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the size of a population in exponential growth, e.g. bacteria in a Petri dish (or in your leftovers if you find Petri dishes not &quot;every day life&quot; enough) ;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the intensity of radioactivity after $n$ years of a given radioactive material (with application to determining the age of mommies!).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-06-04T16:52:59.397" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11110" PostTypeId="2" ParentId="11106" CreationDate="2016-06-05T18:12:45.273" Score="17" Body="&lt;p&gt;I tutored a student who came with a kind of problem I had never seen before and found quite refreshing. It was something like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A child is being pushed on a swing by their father, reaching a maximum height of 4 feet. The father stops pushing, and the maximum height of the swing decreases by 15% on each successive swing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't remember the question itself, but the main idea was that the sequence of maximum heights of the child was a geometric sequence! (Perhaps the goal was to find the total vertical distance traveled by the child.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You get a related example considering a bouncing ball:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/R277u.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/R277u.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;br&gt;Image taken from &lt;a href=&quot;https://commons.wikimedia.org/wiki/File%3ABouncing_ball_strobe_edit.jpg&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It turns out that a bouncing object loses an approximately constant fraction of its remaining energy with each bounce, and in turn the sequence of maximum heights is (approximately) geometric! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More info on the latter &lt;a href=&quot;http://feucht.us/writings/bouncing_ball.php&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5153" LastActivityDate="2016-06-05T18:12:45.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11111" PostTypeId="2" ParentId="11104" CreationDate="2016-06-05T19:42:40.630" Score="2" Body="&lt;p&gt;I haven't heard of deliberate practice before but what you describe sounds pretty much like what we do in math courses at the university where I am (and also at most others in Germany):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We give &lt;strong&gt;homework that is focused&lt;/strong&gt; on specific topics, notions and techniques. There are problems to train proof by induction, there are exercises to grasp the notion of continuity, there are problem sets on linear maps,...  Basically any homework I give is focused on a special thing and I usually note the topic of the specific exercise in the title. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We provide &lt;strong&gt;weekly feedback&lt;/strong&gt; for the exercise. The feedback is written down on the homework sheet and also communicated during the exercise classes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There may be something that is different from the technique you described, namely an issue with time. Performing a formal proof by induction takes some minutes even when you really know what to do and can take much longer if you have to figure out how it works. Most students don't have to time to do many exercises on the same topic. &lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2016-06-05T19:42:40.630" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11113" PostTypeId="2" ParentId="10765" CreationDate="2016-06-07T16:00:12.920" Score="5" Body="&lt;p&gt;Speaking from personal experience, I found such introductions extremely valuable, exactly for the reasons you give. You might even be able to set up larger problems which are then solved piece by piece during the following chapters; this would make a whole out of the parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also consider wrapping things up at the end of each part with a compact blurb as to what the results mean for the problems in the introduction. &lt;/p&gt;&#xA;" OwnerUserId="1479" LastEditorUserId="1479" LastEditDate="2016-06-07T21:07:41.640" LastActivityDate="2016-06-07T21:07:41.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11114" PostTypeId="2" ParentId="11067" CreationDate="2016-06-07T16:45:24.160" Score="2" Body="&lt;p&gt;Your question opens up the field of &lt;strong&gt;motivation&lt;/strong&gt; in education, on which much research has been performed. In my opinion, &quot;extra credit&quot; is an &lt;em&gt;extrinsic motivation&lt;/em&gt; tool and I refer you to the work of Jere Brophy to find out how to appropriately use it. Contrast this with intrinsic motivation in your search. Unfortunately, you may find that having one policy for &quot;improving the grade&quot; of students will have a limited effect: you should be ready to give students several options depending on your assessment of their skills. Full disclosure: I am a high school teacher. However, the research done on theory of motivation applies to adult learners as well.&lt;/p&gt;&#xA;" OwnerUserId="6845" LastActivityDate="2016-06-07T16:45:24.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11115" PostTypeId="2" ParentId="11104" CreationDate="2016-06-07T16:54:24.273" Score="-1" Body="&lt;p&gt;You also need a &lt;strong&gt;goal&lt;/strong&gt; to work towards (what is the &quot;end game&quot;?). You can be (1) focused and (2) receive quality feedback, but you need to create a clear objective regarding your studies. For example, what is the research that you would like to be involved in (as you mentioned in the OP)? It should be a specific thing, like &quot;research in AI focusing on assistance of people with disabilities&quot;. That is, it should be measurable and clearly stated. Then, work backwards from your goal and apply deliberate practice to achieve it.&lt;/p&gt;&#xA;" OwnerUserId="6845" LastActivityDate="2016-06-07T16:54:24.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11116" PostTypeId="2" ParentId="11026" CreationDate="2016-06-07T17:48:18.093" Score="4" Body="&lt;p&gt;&lt;strong&gt;Different needs for different students&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your question in another way, consider what is most important for students to be able to do or understand. Do you have students that will ultimately be &lt;strong&gt;users&lt;/strong&gt; &lt;em&gt;or&lt;/em&gt; &lt;strong&gt;creators&lt;/strong&gt; of mathematical knowledge (using &lt;em&gt;or&lt;/em&gt; in the mathematical sense, of course)? Non-majors will need, at worst, access and skills to technology that can solve repetitive/rudimentary problems for them (e.g. calculating alpha levels for hypothesis testing). In addition, they will need abilities to collaborate and co-create solutions to problems with their peers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Guiding questions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are you creating opportunities for them to work together? Are you creating opportunities for them to use technology to solve menial tasks and focus on the big ideas/problems? Are there groups of students in similar areas of study that could explore big issues / big questions in their field? In this way, you can emphasize what role Wolfram Alpha plays in their learning: a tool that can solve &quot;simple&quot; questions, so that the complex, &quot;big&quot; questions can be focused on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;An empathetic analogy&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this forum existed years ago, and an instructor had asked the same question, had the same concerns, but replacing all instances of Wolfram Alpha with &quot;calculators&quot;, what would your answer have been for him? Calculators were a game-changer for how mathematics was done. It changed &lt;em&gt;how&lt;/em&gt; we teach mathematics: we could finally focus on certain types of complex, difficult questions more often. The same is true for Wolfram Alpha. So, with the understanding that technology-assisted learning is here to stay, what skills and abilities are most important for students to learn that TA learning cannot give them? The answer this question is the answer to yours.&lt;/p&gt;&#xA;" OwnerUserId="6845" LastActivityDate="2016-06-07T17:48:18.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11117" PostTypeId="2" ParentId="11104" CreationDate="2016-06-07T19:32:18.417" Score="5" Body="&lt;p&gt;Following up on (and repeating part of) a discussion-in-comments just below the original question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I can tell, contest-problem math is significantly different from genuine scholarly-study-and-research math. The goals, constraints, and sources are significantly different, although there is some overlap at the low end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Practicing contest-math problem-solving is maybe not so good as a means to understand genuine mathematics. Even as generalized problem-solving, it is solving problems deliberately contrived by other clever people, designed to have specific difficulties overcome-able by things youngish people might know... with time being a significant, yet also artificial, constraint. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, some people who are good at math contests are also good at other things, including genuine mathematics. But this misappraises causality to some degree, and confuses quickness with insight, potentially. It is true that being quick &lt;em&gt;can&lt;/em&gt; be an advantage, but only if one persists proportionately longer. Quick and impatient, or quick and disinterested, or quick but lazy, and so on, are not productive combinations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original question in the title does not really seem to be the question. That is, the questioner expressed some disinterest in repetition. Yet the mathematical analogue of &quot;muscle memory&quot; is exactly what can afford speed-ups and &quot;correct reflexes&quot; in a variety of situations. Extensive rehearsals of &quot;variations on a theme&quot; seems to me the best way to assimilate new ideas... if there really are ideas rather than mere stunts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, &quot;deliberate practice&quot; is plausibly at-least-partly synonymous with &quot;rehearsal of variations on a theme&quot;... until one can react reflexively in an appropriate fashion without much conscious effort. It does seem, in my experience, that one's subliminal capacity is much greater than one's explicit conscious capacity, but that it takes much &quot;rehearsal&quot; to assimilate things to a subliminal level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, the exercises given in many of the standard upper-division and beginning graduate-level texts present an ocean of mix-and-match details, as though every possible logical combinations of hypotheses deserved equal consideration. This is bad training for sensibility, I think, in part because it does hark back to the contest-math side-channel information that there really is a human adversary who has contrived the problems to &quot;challenge&quot; us... not necessarily to edify us by bringing to our attention important issues that will help us subsequently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the contemporary style of mathematics seems to needlessly disparage heuristics, and exaggerate the importance of &quot;proof&quot;, often elevating the supposed importance of proof far beyond any alleged significance or utility of the things being proven. To be able to allow oneself to optimistically skip over innocent-seeming intermediate steps to try to get an interesting conclusion is good practice! Also, after all, what's the point of worrying about a proof if we don't care about the conclusion? There are plenty of seemingly-trivial &lt;em&gt;important&lt;/em&gt; things whose proofs are surprisingly subtle. (The intermediate value theorem from calculus, for example.) Given that people have finite capacity, it may be a bad allocation of resources to &quot;practice&quot; proving pointless or boring or artificial things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in particular, I'd not advocate &quot;drills&quot; nor &quot;doing all the exercises&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather, it seems to me that &quot;understanding&quot; mostly involves assimilation to a subliminal level, which usually occurs only after considerable rehearsal of the ideas. Not adversarial-style &quot;exercises&quot;, but rehearsal. Something like the way one might rehearse lines for a play, or an explanation of a technical idea, or a piece of music for performance. As we know, mere &quot;memorization&quot; is a flimsy and inadequate version of assimilation of such things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the other end, exercising one's own critical judgement is another thing not really promoted in typical math coursework and textbooks, since there's typically a pretense of absolutism. Learning to look at arguments or computations &quot;with eyes open&quot;, even if one wrote it oneself, is necessary. At the same time, one shouldn't sabotage a good heuristic by squelching it at the outset due to minor gaps (as opposed to literal errors). Nothing constructive about that.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-06-07T19:32:18.417" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11118" PostTypeId="2" ParentId="11022" CreationDate="2016-06-08T00:33:11.380" Score="4" Body="&lt;p&gt;The other (correct) answers explain why the modulus operation doesn't fit with the other ordinary operations. Nonetheless there are very good reasons for introducing it at many places in the K-8 curriculum, with an appropriate degree of rigor at each place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've had a lot of fun with &quot;clock arithmetic&quot; on various sizes of clock. The first hurdle (and kids like it) is starting at 0 and numbering through 11 rather than starting at 12. The two hour clock has just a 0 and a 1. You can write out the addition and multiplication tables, see lots of patterns and learn a lot. Kids soon see that serious computation mod $n$ is a lot faster if you reduce all along the way, particularly for powers. They think it's cool to calculate, say $2^{10} \pmod 7$  without ever having to double a number greater than $6$. They can take advantage of the fact that you have a ring homomorphism, without having to deal explicitly with the abstraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explaining why 1/2 on a seven hour clock is 4 is a very good way to move from fractions as cutting up pies to fractions as solutions to equations: 1/2 is the thing you multiply by to get to 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Making change for \$1 in a country that has bills with denominations \$m and \$n motivates a discussion of greatest common factor (often neglected in favor of least common multiple, since the latter is what's needed to add fractions). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the exercises at &lt;a href=&quot;https://www.youcubed.org/task/counting-cogs/&quot; rel=&quot;nofollow&quot;&gt;https://www.youcubed.org/task/counting-cogs/&lt;/a&gt; can be a good place to begin a discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With a small group of good fifth graders, Fermat's little theorem is within reach. Last year I was able to do RSA cryptography with that group. Using Wolfram Alpha they could break the code and solve this problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Eratosthenes' RSA public key is $$ n = 10967535067, \ \  e = 1051. $$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Archimedes encrypts his message with this key and sends Eratosthenes&#xA;  $$ C = 1963501580 .$$ &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What did he say?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4984" LastEditorUserId="4984" LastEditDate="2016-06-09T00:53:58.113" LastActivityDate="2016-06-09T00:53:58.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11119" PostTypeId="2" ParentId="11070" CreationDate="2016-06-08T17:54:02.823" Score="1" Body="&lt;p&gt;If you would like to create a game that adapts to its user's skill level, then the best way to go is to research &lt;strong&gt;Computerized Adaptive Assessment&lt;/strong&gt; and &lt;strong&gt;Item Response Theory&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a starting point for this, you would need to create a database of questions that are graded (i.e. in order of difficulty) and that are cross-referenced (so that game can make a decision on the selection of follow-up questions). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, a way to give appropriate &lt;em&gt;feedback&lt;/em&gt; to the user with respect to their progress would be ideal.&lt;/p&gt;&#xA;" OwnerUserId="6845" LastActivityDate="2016-06-08T17:54:02.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11120" PostTypeId="2" ParentId="10555" CreationDate="2016-06-08T23:34:47.220" Score="2" Body="&lt;p&gt;I frequently repeat the line &quot;it's all about the base,&quot; which gets a chuckle, but then I say the goal of logs is to solve for the power. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;A log is a power.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The power that gets us from that base to the result, so the second, false, equation would obviously be wrong. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-06-08T23:34:47.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11121" PostTypeId="2" ParentId="10765" CreationDate="2016-06-09T02:59:13.287" Score="5" Body="&lt;p&gt;Are you familiar with the term &quot;Dunning-Kruger effect&quot;? To answer your question: yes there have been studies that show that many topics including mathematics can benefit from the Rule of the 3 T's: Tell 'em what your gonna tell them, Tell them, and the Tell them what you told them. Writing a good textbook actually is very difficult, and not necessarily related to having expert knowledge of the subject of the textbook. You may know this. Here is a question for you to ponder: which is more difficult mathematics or the teaching of mathematics? (for extra credit, prove your conclusion) It takes innate talent to be able to write a textbook without an enormous amount of trial, feedback, and error correction. Few do a great job without a lot of student input. There's no formula.&lt;/p&gt;&#xA;" OwnerUserId="6848" LastActivityDate="2016-06-09T02:59:13.287" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11122" PostTypeId="1" CreationDate="2016-06-09T19:49:10.767" Score="9" ViewCount="707" Body="&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: Is there a way we can teach polynomials, avoiding the &quot;watch me do it &amp;amp; now you do it&quot; training method, that will allow students to anticipate and predict the existence or formulation of polynomials? (as it applies to, say, middle school mathematics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are some properties of polynomials that may not be fully realized or proved at the grade 7 - 9 level. This is fine. I am more concerned about the &lt;em&gt;introduction&lt;/em&gt; of polynomials to students. What I want to avoid is the particularly robotic and soulless routine of (some) math teaching which goes something like this: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;here is a thing that's unfamiliar or you've never seen before&lt;/li&gt;&#xA;&lt;li&gt;here is what we call some of the parts of this thing&lt;/li&gt;&#xA;&lt;li&gt;here are some things you can do with them&lt;/li&gt;&#xA;&lt;li&gt;now watch me do it and you do it until you get it right&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6845" LastActivityDate="2016-06-24T03:24:43.713" Title="In what ways can educators introduce polynomials in grades 7 to 9?" Tags="|mathematical-pedagogy|secondary-education|" AnswerCount="3" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="11123" PostTypeId="2" ParentId="11122" CreationDate="2016-06-10T07:59:21.247" Score="12" Body="&lt;p&gt;One of the things I discovered with polynomials when working with them as a high-school student is that &lt;em&gt;they generalize the idea of a place value system&lt;/em&gt;. This is a very cogent way to introduce the subject to middle-school students, who have probably been exposed to working in multiple bases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, $11$ times $11$ is $121$ in every base except binary where the 2 in the middle gets carried out. This isn't a coincidence — it's because $(x + 1)(x + 1) = x^2 + 2x + 1$. Same with $12 \times 13 = 156$ for bases 7 and higher — it's just saying that $(x + 2)(x + 3) = x^2 + 5x + 6$, and evaluating it for $x = 10$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what you could do is to do a series of classes about working in different bases, and then ask — what if we work in a base with an unknown number? What happens then? Can we make arithmetic statements that work in &lt;em&gt;all&lt;/em&gt; bases?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;If you use this place value paradigm, teaching polynomial arithmetic becomes easy as well — it's simply the algorithms they're used to, but you never carry anything like you would in a positional number system with a fixed base.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To multiply, say, $2x + 3$ by $x + 6$, you simply treat it as multiplying $23$ by $16$ in &quot;base $x$&quot;, in which case you get $2\ 15\ 18$ as the &quot;digits&quot;, or $2x^2 + 15x + 18$ in standard notation. They can verify this for themselves by plugging $x = 10$ and seeing that $23 \times 16 = 368$, which you would expect if you were to carry everything out. They can also plug in $x = 100$ to see that $203 \times 106 = 21518$, and that the carrying only happens when $x = 10$ because the digits are all scrunched together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It will also help explain why certain mental tricks with multiplication work, such as the trick where you multiply a 2-digit number by $99$ by subtracting 1 from it, then appending the new number's difference from $99$ to the end (e.g. $43 \times 99 = 4257$). It works because multiplying a monomial $ax + b$ by $x^2 - 1$ gives you $ax^3 + bx^2 - ax - b$, which you can rewrite as $(ax + b - 1)x^2 + x^2 - ax - b$, and then $(ax + b - 1)x^2 + (x^2 - 1) - (ax + b - 1)$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Afterwards, you can discuss further differences between polynomials and regular place values, such as what happens when you use fractional coefficients. Why can't you represent, say, $\frac12 x^2$ as a polynomial with lower degree? What happens when you divide 100 by 2 in several different bases? Do they ever return the same thing, or even something similar?&lt;/p&gt;&#xA;" OwnerUserId="856" LastEditorUserId="856" LastEditDate="2016-06-10T16:57:34.130" LastActivityDate="2016-06-10T16:57:34.130" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11124" PostTypeId="1" AcceptedAnswerId="11179" CreationDate="2016-06-12T21:57:50.557" Score="10" ViewCount="303" Body="&lt;p&gt;I am interested in ideas what to teach when the task is to teach a bit of (linear) optimization to third year undergraduate mathematics students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More specifically: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Assume 'a bit' means I'd have about eight hours of lecturing time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The students know standard linear algebra results and some multivariable real analysis (including basic optimality criteria via derivatives, and also Lagrange multiplier method). They do not know much (or any) numerical mathematics in a narrow sense, but at least in part have some knowledge of programming and algorithms.    &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;An immediate idea is discussing the simplex algorithm. But what else?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am tangentially familiar with various mathematically interesting subjects around this, but I do not know what could be feasible to discuss. &#xA;I am also not strictly committed on the idea that it is all about &lt;em&gt;linear&lt;/em&gt; optimization. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answers could include a plan for these lectures or also recommendations for individual subjects that could be feasible to cover in the given context. Pointers to lecture notes in that direction or other relevant literature are welcome as well. &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="1147" LastEditDate="2016-06-25T01:19:39.673" LastActivityDate="2016-10-11T09:25:50.367" Title="Ideas for teaching a bit of linear optimization to advanced undergraduates?" Tags="|course-design|optimization|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11125" PostTypeId="2" ParentId="11122" CreationDate="2016-06-13T01:05:08.293" Score="3" Body="&lt;p&gt;What the OP left aside was: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Who cares? Why would we ever need to manipulate polynomials?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Permit me to answer this unasked question. Every&lt;sup&gt;*&lt;/sup&gt; piece of text you see rendered in PDF on your laptop screen, or printed out on a printer, draws each letter either with Adobe &lt;em&gt;Postscript&lt;/em&gt; cubic &lt;a href=&quot;https://en.wikipedia.org/wiki/B%C3%A9zier_curve&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bézier polynomials&lt;/a&gt;, or Apple/Microsoft's &lt;em&gt;TrueType&lt;/em&gt;, which uses quadratic Bézier polynomials.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/ll2MA.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ll2MA.jpg&quot; alt=&quot;Fonts&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Images from &lt;a href=&quot;http://www.slideshare.net/Mark_Kilgard/23typography-12534604&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mark Kilgard's presentaion&lt;/a&gt;.&#xA;The dots are control points for the Bézier curves.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;We can hardly get through a day without seeing polynomials in action! :-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an advanced tanget,&#xA;one could discuss the advantages/disadvantages of cubic vs. quadratic&#xA;Bézier curves: continuity, control points, evaluation complexity, etc.&#xA;&lt;hr /&gt;&#xA;&lt;sup&gt;*&lt;/sup&gt;&lt;em&gt;Nearly&lt;/em&gt; every.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-06-13T01:05:08.293" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11126" PostTypeId="1" CreationDate="2016-06-13T18:22:24.680" Score="19" ViewCount="1784" Body="&lt;p&gt;My question is in the title.  Let me elaborate and give some context:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm teaching a first differential equations course, essentially for engineers, at the university.  I'm developing the syllabus and content for the course. I'm at liberty to pretty much teach what I want; there's no departmental exam to teach to.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will experiment with using computers - Mathematica by Wolfram, in particular - in order to bypass tedious computations and get to the meat of the subject, and my goal for the course is for every topic covered to have interesting, practical applications.  This brings me to the Laplace Transform.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After studying mechanical vibration and resonance caused by a sinusoidal forcing function, it would be nice to also teach the students how to work with other periodic forcing functions - e.g. square waves &amp;amp; sawtooth waves - and Laplace Transforms are, to my knowledge, the best way to deal with these.  Not to mention the dirac delta &quot;impulse&quot; functions, which are immensely practical.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, after having tutored a student in a differential equations course earlier, and after having attempted to write notes on Laplace Transforms, I am face-to-face with the reality that this is a computationally-intense topic.  Even worse, it is somewhat resilient to Mathematica.  Mathematica can compute Laplace and Inverse Laplace Transforms, but the results are ugly and hard to interpret.  I'm thinking that, perhaps, they are so hard to interpret that we might as well not even come up with the exact solution at all and just use a numerical ODE solver (Mathematica's NDSolve[] function).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been told the Laplace Transform is useful, but does its use justify the class time (out of a six week course, no less!) that would be necessary explaining the difficult-to-understand shifts between t-domain and s-domain and the staggering computations that accompany these shifts?  Thoughts, experiences, references... all will be appreciated.  &lt;/p&gt;&#xA;" OwnerUserId="5672" LastEditorUserId="122" LastEditDate="2017-06-25T07:09:27.667" LastActivityDate="2017-06-25T07:09:27.667" Title="Should I teach Laplace Transforms? How much?" Tags="|mathematical-pedagogy|applications|differential-equations|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11127" PostTypeId="2" ParentId="11126" CreationDate="2016-06-13T20:56:49.720" Score="14" Body="&lt;h1&gt;Consider something besides an &quot;all or nothing&quot; approach.&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Here's what I did a couple of times when the topic was optional and I didn't have much time, but I still wanted to give students an introduction to the method. Simply restrict yourself to introducing the method by defining the Laplace transform, computing it for some simple examples, explaining what the inverse Laplace transform is, and then showing students how to use the method to solve simple differential equations such as $y' – y = e^{-t}$ with $y(0) = 1,$ or $y'' + y = e^{-t}$ with $y(0) = y'(0) = 0.$ Incidentally, even when I covered the topic briefly in this manner, I still had time to discuss how the Laplace transform was a linear operator (all my students had either completed linear algebra or were currently taking linear algebra) and to emphasize how linearity allowed the method to work (e.g. take Laplace transform of both sides of the ODE, then apply linearity). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you provide appropriate &lt;em&gt;short&lt;/em&gt; handouts to your students that give them all the necessary basics, you should be able to cover this in 2 lectures. Incidentally, the two equations I gave above came directly off the last major test (from May 1999) in one of the classes that I covered Laplace transforms briefly in this way. I realize you are going to feel compelled to try and justify Laplace transforms by getting into sawtooth graphs and impulse functions and the like, but your students will likely see Laplace transforms later in an advanced undergraduate level mathematical methods for engineers course (and perhaps also in a later mechanics course). Even a brief introduction to Laplace transforms now will help them a lot when they see it later.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2016-06-13T20:56:49.720" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11128" PostTypeId="2" ParentId="11124" CreationDate="2016-06-14T07:12:35.223" Score="3" Body="&lt;p&gt;I have not tried anything similar myself, but if I had to, I would start by looking at Gilbert Strang's book &quot;Introduction to applied mathematics&quot; in  chapter 8 &quot;Optimization&quot; (8.1 is &quot;Introduction to linear programming&quot;). My version is probably not the latest one and misses some recent developments (it seems to be written when interior point methods were just invented) but it really is a pleasant reading and has the right tone for students.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="1147" LastEditDate="2016-06-25T01:20:31.177" LastActivityDate="2016-06-25T01:20:31.177" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11129" PostTypeId="1" CreationDate="2016-06-14T17:53:57.863" Score="5" ViewCount="218" Body="&lt;p&gt;Did anyone out there have experience with SMSG's Calculus text? Our school system  (Amherst, MA) used SMSG texts from my 6th grade class onward. But for HS Calculus (1969) we didn't use the two-part SMSG Calculus text (copyright 1966). Was there some issue with it? Too hard, too &quot;abstract&quot; etc.?&lt;/p&gt;&#xA;" OwnerUserId="6863" LastEditorUserId="77" LastEditDate="2018-04-05T12:26:02.257" LastActivityDate="2018-04-05T12:26:02.257" Title="SMSG Calculus Usage?" Tags="|secondary-education|calculus|textbooks|history|new-math|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11131" PostTypeId="2" ParentId="5794" CreationDate="2016-06-15T15:33:21.963" Score="1" Body="&lt;p&gt;Proof that $-(-x)=x$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By definition, for all $a\in R$, we have $-a\in R$ and $a+(-a)=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apply this definition for $a=x$ where $x\in R$ to obtain $x+(-x)=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, apply this definition for $a=-x$ to obtain $-x + (-(-x)) =0 $.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By substitution, $-x + (-(-x))=x+(-x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cancelling $-x$ from both sides, we obtain $-(-x) = x$.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastActivityDate="2016-06-15T15:33:21.963" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11133" PostTypeId="2" ParentId="9960" CreationDate="2016-06-15T21:37:45.997" Score="5" Body="&lt;p&gt;Just a short addition. Ed Begle published a valuable study through MAA/NCTM entitled &quot;Critical Variables in Mthematics Education&quot;, which was an attempt (in part) to sort out some of the lessons of the SMSG.&lt;/p&gt;&#xA;" OwnerUserId="6863" LastActivityDate="2016-06-15T21:37:45.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11134" PostTypeId="1" CreationDate="2016-06-16T15:09:50.713" Score="4" ViewCount="180" Body="&lt;p&gt;I was wondering, how would you explain to young high school students or grade school kids the relationship between math and English?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, both are systems that follow certain rules. Math is often referred to as a language and you learn it by using it, just as you would any other language. Furthermore, one can help in decoding the other. For example, if you have an equation, it doesn't really mean anything until we give it some sort of context for your problem. Without a story behind the equation, it's just math for math's sake and it seems pointless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm looking for something along those lines. I will be teaching a summer school class soon for the first time and I have to come up with something on the fly, since last year's materials were lost.&lt;/p&gt;&#xA;" OwnerUserId="6867" LastActivityDate="2016-06-23T23:53:45.217" Title="How to get kids thinking about the relationship between basic math (algebra) and English (or language in general)" Tags="|language-use|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11135" PostTypeId="2" ParentId="10977" CreationDate="2016-06-16T17:01:33.710" Score="2" Body="&lt;p&gt;&lt;strong&gt;My answer&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, I have used guided notes with success, as long as students straddled the &quot;sweet spot&quot; of giving them ample opportunities to interact with the topic and avoiding the creation of mindless, note-taking robots. With guided notes, my guess is that your goal is to accelerate students by saving them time from writing. I think that this is a valuable tool for long paragraphs of text, and you should continue using it as long as it serves a purpose in your plan. I personally prepare the notes before hand, but this may lead to wastage in some cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Extended Answer&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it's helpful to understand the stages of learning for a particular concept and to &lt;strong&gt;understand what role your lecture (and guided notes) plays in this development&lt;/strong&gt;. There is an anxiety that wells up in us to concentrate and distill our lessons into the shortest time possible. You only have students for approximately 45 to 90 minutes, but successful learning of a concept could take somewhere between 120 to 300 minutes (my own estimation).  In short, students will probably go through the following stages (read: Bloom's taxonomy):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Basic Knowledge, including definitions and new symbolic representation&lt;/li&gt;&#xA;&lt;li&gt;Understanding and Application&lt;/li&gt;&#xA;&lt;li&gt;Connections to other fields of study (synthesizing knowledge)&lt;/li&gt;&#xA;&lt;li&gt;Analyzing and critiquing&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Consider: &lt;strong&gt;in designing your lecture or lesson&lt;/strong&gt;, to what part of these stages will you apply the time that you have with them? How much preparation will (can) be done before the lecture to prepare students? Maybe one of your lectures will focus only on one of the stages, having already given students an assignment to understand the definitions and foundations of your lecture (i.e. the whole hour can be devoted to application and analysis).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Learning can be empirically improved&lt;/strong&gt; (search &quot;effect of feedback on learning&quot;) when timely and appropriate feedback is given to students. An idea: stop your lecture, and give students several multiple choice questions to answer on the screen; connect it to an online polling system so that they can see their own (anonymous) results. Then, you can make the best use of your time as a lecturer in the remaining amount of time, either focusing on clarification or continuing on with the lesson in confidence that everyone has the prerequisite understanding. Use this with your guided notes and you'll have a lecture to be envied.&lt;/p&gt;&#xA;" OwnerUserId="6845" LastActivityDate="2016-06-16T17:01:33.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11136" PostTypeId="2" ParentId="11134" CreationDate="2016-06-17T02:59:12.957" Score="1" Body="&lt;p&gt;Assign exercises explicitly translating natural-language expressions/sentences into algebraic expressions/equations. Assign exercises like number puzzles requiring that the student &quot;translate to an equation&quot; and then solve and check/interpret. These are fairly standard, useful, but perhaps not sufficiently emphasized. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-06-17T02:59:12.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11137" PostTypeId="2" ParentId="10977" CreationDate="2016-06-17T06:06:38.817" Score="3" Body="&lt;p&gt;I think what works depends on the local culture to a fair extent, ie what the expectations of the students are. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a student myself, I almost always had to copy down everything in lectures, and no-one seemed to have a problem with that. One lecturer gave us gappy notes, which we hated, because it made the easy bits (which were printed) way too slow, and the hard bits (examples to fill in) way too fast. I think that has more to do with the pacing of the lecture though, so could be corrected by the lecturer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, last year I released 'outline notes', so students had some structure but still needed to pay attention in lectures. That was not at all popular. I got complaints that they wanted 'proper lecture notes', because all they had was 'what we write down in lectures'(!). I guess the difference was a combination of how much the students were willing to work and what they were accustomed to from my colleagues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This year I decided to change the emphasis of the 'lectures' to contain less writing and more thinking. I've given the students textbooks (essentially) that I follow generally but not to the letter. In lectures I use slides that alternate (roughly) between bits of theory and practice questions, using the board to discuss ideas and solutions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've used this with two groups so far. The first group didn't like certain details, but the overall style seemed to be ok. One student would just make notes to copy onto the slides (I haven't yet convinced myself to post slides in advance, although I can see it would benefit some students, because I'm afraid of it being unhelpful to others, but I release them at the time of the class). I haven't yet had the comments from the second group. I think some have chosen to read the full text to avoid lectures, and some have come to lectures to avoid the text, and a few have made use of both. This is in line with a paper I read at some point - where multiple resources are available, students don't tend to make use of more than one (although presumably at least some choose the one that is best for them).&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-06-17T06:06:38.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11138" PostTypeId="2" ParentId="11134" CreationDate="2016-06-17T07:49:54.537" Score="0" Body="&lt;p&gt;I am not sure it answers your question, but I strongly advocate to learn as soon as possible that &lt;em&gt;mathematic is expressed &lt;strong&gt;in&lt;/strong&gt; a language&lt;/em&gt;. You can phrase mathematics in English, in French, but there is no more way to do mathematics without a language to express it than there is, say, to do politics. Too often, people have the idea that mathematics should look like Russel and Whitehead's Principia: they don't, and pupils should know that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To take an example, compare&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$n$ odd $n=2k+1$. $n^2=(2k+1)^2=4k^2+4k+1=2(2k^2+2k)+1=2k'+1$. $n^2$ odd.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $n$ is an odd integer, then there is an integer $k$ such that $n=2k+1$. Then it holds: $$n^2=(2k+1)^2=4k^2+4k+1=2(2k^2+2k)+1=2k'+1$$&#xA;  where $k'=2k^2+2k$ is an integer. It follows that $n^2$ is odd. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The first writing is commonly found in students' papers, and would be inadmissible from a teacher. I think it should be inadmissible from students too, because it won't enable them to articulate their reasonings clearly. This may be a slightly too complicated example, but it should be clear enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, in the context of mathematics, there are some additional conventions to be followed, in particular some specific vocabulary and a more rigid use of some terms. But even this last point is overrated. Let me take the example of the word &quot;and&quot;. Many would say that in mathematics, &quot;and&quot; is used only in the very precisely defined sense of logical conjunction. &lt;strong&gt;This is not true&lt;/strong&gt;, as witnessed by the following example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The solutions of the equation $x^2-1=0$ in the unknown $x$ are $1$ &lt;strong&gt;and&lt;/strong&gt; $-1$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here, it is important to point out that &quot;and&quot; is not a logical conjunction, but a mere enumeration. So we use that word in a common way, not specific to maths.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-06-17T07:49:54.537" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11139" PostTypeId="1" CreationDate="2016-06-17T08:56:27.127" Score="3" ViewCount="230" Body="&lt;p&gt;I will go to an American High School (Ohio) this summer for a year and I will probably be a junior. I got a list of all classes, but there are really many classes to choose especially math classes. I have to take something which would be similar to my German math class&#xA;(Analysis &amp;amp; Function:&#xA;sine, cosine, exponential, powerfunction,differential calculus, polynomial function&#xA;Analytic Geometry &amp;amp; linear Algebra:&#xA;Euclidean vector&#xA;These are some topics I would do in my German school.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My high school in America (Ohio) offers: &#xA;CP Algebra,&#xA;CP Geometry,&#xA;CP Algebra II,&#xA;CP Statistics,&#xA;CP Honors Geometry,&#xA;CP Honors Algebra II,&#xA;CP Pre Calculus,&#xA;Concepts of Algebra,&#xA;College Algebra,&#xA;Transition to College Algebra,&#xA;AP Calculus&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am really overwhelmed with all these classes, could you give me an advise?&lt;/p&gt;&#xA;" OwnerUserId="6871" LastActivityDate="2016-06-21T18:41:10.187" Title="Which math class should I take as an exchange student in the USA (OH)?" Tags="|algebra|geometry|statistics|functions|" AnswerCount="3" CommentCount="3" ClosedDate="2016-06-27T16:16:55.527" ContentLicense="CC BY-SA 3.0" />
  <row Id="11140" PostTypeId="2" ParentId="11139" CreationDate="2016-06-17T11:25:43.917" Score="2" Body="&lt;p&gt;Most of the German course sounds like a College Algebra class (with trigonometry), with smatterings of Precalculus and one bit of (AP) Calculus. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-06-17T11:25:43.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11142" PostTypeId="2" ParentId="11067" CreationDate="2016-06-17T20:43:50.513" Score="4" Body="&lt;p&gt;Marian Minar kind of beat me to it -- I don't know how much research is done specifically on extra credit but there is a huge body of literature on motivation, and I think that's at the heart of the question (Which I think of as, &quot;Is extra credit valuable in the sense of getting students to engage with their work?&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Self-determination_theory&quot; rel=&quot;nofollow&quot;&gt;self-determination theory&lt;/a&gt; motivation can be thought of in terms of &lt;strong&gt;intrinsic&lt;/strong&gt; versus &lt;strong&gt;extrinsic&lt;/strong&gt; motivation, that is, motivation based on a self-determined value inherent in the task itself versus motivation based on a value outside the task; and in terms of &lt;strong&gt;autonomous&lt;/strong&gt; versus &lt;strong&gt;controlled&lt;/strong&gt; motivation, that is, motivation that is determined by the person versus that which is determined by another. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Autonomous motivation contains all of what we classify as intrinsic motivation, as well as extrinsic motivation that has &quot;buy-in&quot; from the person. For example, a student who studies hard for a calculus test because she finds calculus really fascinating or because she simply enjoys the process of practicing math problems is intrinsically (therefore autonomously) motivated. A student who studies hard for a calculus test, not because he enjoys calculus but because he has a dream of being an engineer and realizes that doing well in calculus is important for becoming a successful engineer, is &lt;em&gt;extrinsically&lt;/em&gt; but &lt;em&gt;autonomously&lt;/em&gt; motivated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By contrast, a student who studies for a calculus test only for a grade (in which he places no intrinsic value) or because failing a test would cause feelings of shame or uselessness, is experiencing controlled motivation -- any motivation that occurs is there on someone else's terms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as extra credit goes -- does extra credit motivate students? If so is this autonomous motivation or controlled? This, I don't know, and I don't have any research references handy -- but it would be interesting to find out, and this is what you'd search for. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking of which, a &lt;a href=&quot;https://eric.ed.gov/?q=title%3A%22extra%20credit%22%20or%20abstract%3A%22extra%20credit%22&amp;amp;pr=on&quot; rel=&quot;nofollow&quot;&gt;quick ERIC search&lt;/a&gt; for peer-reviewed articles with &quot;extra credit&quot; in either the title or abstract brings up 95 results. Just skimming these, the theme for many of them seems to be that students were given some interesting activity, often involving active learning, to do as extra credit and it seemed to improve learning. If those results hold up, I think the learning gains would be more attributable to the active learning than they would be to the fact that they are extra credit. In other words, maybe what people are doing for extra credit ought to be the normal assignments and activities in the class! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;References: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ryan, R., &amp;amp; Deci, E. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. &lt;em&gt;The American Psychologist&lt;/em&gt;, 55(1), 68–78. &lt;a href=&quot;http://doi.org/10.1037/0003-066X.55.1.68&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1037/0003-066X.55.1.68&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="942" LastEditorUserId="942" LastEditDate="2016-06-17T21:31:22.357" LastActivityDate="2016-06-17T21:31:22.357" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11143" PostTypeId="1" CreationDate="2016-06-17T21:26:58.803" Score="12" ViewCount="262" Body="&lt;p&gt;&lt;strong&gt;How do you incorporate the results from a final exam into evidence of learning in a valid and justifiable way (i.e. grade determination)?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;This question is open to all education levels&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is your weighting and the justification for this weight? How do you incorporate past evidence of learning and the results from the final exam into the overall grade? If past assessments showed that a student was proficient, but the final exam showed otherwise, what would you do about the discrepancy? How do you assess a students' ability of working within a group of people on a final exam? (isn't this an important skill?)&lt;/p&gt;&#xA;" OwnerUserId="6845" LastEditorUserId="2083" LastEditDate="2017-04-13T17:59:22.193" LastActivityDate="2017-04-14T11:15:31.057" Title="How do you incorporate the results from a final exam into evidence of learning in a valid and justifiable way (i.e. grade determination)?" Tags="|exams|grading|assessment|testing|" AnswerCount="2" CommentCount="4" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11144" PostTypeId="2" ParentId="11139" CreationDate="2016-06-18T18:31:16.620" Score="0" Body="&lt;p&gt;I think pre-calculus sounds like it'd be a good fit&lt;/p&gt;&#xA;" OwnerUserId="5958" LastActivityDate="2016-06-18T18:31:16.620" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11145" PostTypeId="2" ParentId="11098" CreationDate="2016-06-20T12:46:35.313" Score="2" Body="&lt;p&gt;Whenever I want to fill time with kids, simple reverse-problems come to my mind. Imagine a number, say 5. Which computation might lead to &quot;5&quot;?&#xA;One might start with 2+3. That's right, let's see if they know another way. 1+4. Right again. :-)&#xA;Maybe they come up with 3+2 and you may discuss whether this is the same as 2+3.&#xA;In just searching for new ways of computing the result 5, you may find patterns (just as 2+3=3+2). To most of the kids I played this game with, the game was rather adaptive towards their level. The kids managed to search for possible answer on exactly their highest level (since more is not possible for them and less is boring to them). What about the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1+1+1+1+1&lt;/li&gt;&#xA;&lt;li&gt;6-1&lt;/li&gt;&#xA;&lt;li&gt;x+(5-x)&lt;/li&gt;&#xA;&lt;li&gt;5/2 + 5/2&lt;/li&gt;&#xA;&lt;li&gt;$\sqrt{5}^2$&lt;/li&gt;&#xA;&lt;li&gt;...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can easily go on finding a way expressing 5 as an expected value in stochastics or as an integral or whatever. &#xA;If you have many kids, this might create an interesting pool of creative and individual statements which need to be evaluated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I think there is no clear learning goal, this may deepen kids understanding of numbers or whatever is involved.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2016-06-20T12:46:35.313" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11146" PostTypeId="1" AcceptedAnswerId="11151" CreationDate="2016-06-20T14:04:02.953" Score="3" ViewCount="75" Body="&lt;p&gt;Are there any expository books or articles on computability that are accessible to the general public on the order of Simon Singh's math popularizations (The Code Book)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm looking for one with lots of pictures/diagrams (like Scientific American graphics) and low on mathematical notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It doesn't have to be a dedicated book. A book chapter, short article, or a (mostly self contained) series of blog posts is fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It should get across the concepts of Turing Machines and calculations, universal computation, and undecidability.&lt;/p&gt;&#xA;" OwnerUserId="2134" LastEditorUserId="2134" LastEditDate="2016-06-21T12:44:00.703" LastActivityDate="2016-06-21T12:44:00.703" Title="Expository/accessible source on computability/undecidability" Tags="|reference-books|computer-science|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11147" PostTypeId="2" ParentId="11126" CreationDate="2016-06-20T18:18:45.950" Score="8" Body="&lt;p&gt;Also having a short period of time to introduce my DE students to Laplace Transforms I began with two 'Axioms': &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) $\mathcal{L}\{c_1y_1(t)+c_2y_2(t)\}=c_1Y_1(s)+c_2Y_2(s)$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) $\mathcal{L}\{y^\prime\}=sY(s)-y(0)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From these two we derived the Laplace transforms for polynomial, exponential and sinusoidal functions and proceeded to use those results and their corresponding inverse transforms to solve a select assortment of differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only after this did I introduce the 'definition' via the integral &lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation}&#xA;\mathcal{L}\{y(t)\}=\int_0^{\infty}e^{-st}y(t)\,dt&#xA;\end{equation}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The DE course was a prerequisite for 'Engineering Mathematics I, II, III and IV' in which the students would get a more complete treatment of Laplace transforms.&lt;/p&gt;&#xA;" OwnerUserId="6736" LastActivityDate="2016-06-20T18:18:45.950" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11148" PostTypeId="1" AcceptedAnswerId="11152" CreationDate="2016-06-20T20:13:10.367" Score="8" ViewCount="1444" Body="&lt;p&gt;I am looking for a good antonym for reducing/simplifying equivalent fractions: 'reduce' and 'simplify' both make sense to me when dividing, but I'm struggling to name what it is we do when we multiply a fraction by 2/2, or 7/7, or any of their Unity Sisters!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideas that came to mind: increasing?  complexifying?&lt;/p&gt;&#xA;" OwnerUserId="6885" LastEditorUserId="143" LastEditDate="2016-06-21T10:56:23.540" LastActivityDate="2016-06-22T12:04:32.707" Title="A good antonym for reducing/simplifying equivalent fractions" Tags="|mathematical-pedagogy|terminology|fractions|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11149" PostTypeId="2" ParentId="11146" CreationDate="2016-06-20T23:30:26.530" Score="2" Body="&lt;p&gt;Here are two books for &quot;the intelligent layman&quot; that explain those topics. They would be too difficult for many people but still explain the concepts in an interesting and basically non-technical way. As a bonus, these are both excellent, very famous books.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Gödel, Escher, Bach&lt;/em&gt; by Douglas R. Hofstadter. This book combines your ideas with those of art (Escher) and music (Bach) to analyze artificial intelligence.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;The Emperor's New Mind&lt;/em&gt; by Roger Penrose. This book combines your ideas with those of quantum mechanics to argue against some claims about artificial intelligence.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2016-06-21T12:08:06.283" LastActivityDate="2016-06-21T12:08:06.283" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11150" PostTypeId="2" ParentId="11148" CreationDate="2016-06-21T00:53:40.637" Score="4" Body="&lt;p&gt;How about saying &quot;find equivalent fractions to&quot;?  For example, to find equivalent fractions to $\frac13$ means giving examples of $\frac{n}{3n}$ for different values of integers $n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if we are asked to find equivalent fractions to, say, $\frac69$, this would include $\frac23$ (which is in simplest form) and, say, $\frac{10}{15}$ (where the numerator $10$ is not an integer multiple of $6$) so perhaps more clarifications are needed if this is what you intend to mean.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-06-21T00:53:40.637" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11151" PostTypeId="2" ParentId="11146" CreationDate="2016-06-21T01:48:24.990" Score="2" Body="&lt;p&gt;Not a direct answer, but still interesting: An exposition of the&#xA;undecidability of the halting problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;The Freeze App Does Not Exist&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;by Pål Grønås Drange and Jan Arne Telle from Bergen Univ., translated and&#xA;rephrased by Thore Husfeldt, posted &lt;a href=&quot;https://thorehusfeldt.net/2012/06/25/the-freeze-app-does-not-exist/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/9Iwzi.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9Iwzi.png&quot; alt=&quot;Freeze App&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;I also like the (now aging)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;The New Turing Omnibus: Sixty-Six Excursions in Computer Science&lt;/em&gt;, by A.K.Dewdney. &lt;a href=&quot;http://us.macmillan.com/thenewturingomnibus/akdewdney&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacMillan link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-06-21T01:48:24.990" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11152" PostTypeId="2" ParentId="11148" CreationDate="2016-06-21T10:36:24.823" Score="6" Body="&lt;p&gt;I thought the standard terminology was &lt;strong&gt;&lt;em&gt;expanding&lt;/em&gt;&lt;/strong&gt; fractions, at least it is in my language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some support for this term found by googling:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.wyzant.com/resources/lessons/math/elementary_math/fractions/expanding_and_reducing_fractions&quot;&gt;Expanding and Reducing Fractions&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.personal-math-online-help.com/expanding-fractions.html&quot;&gt;Expanding Fractions&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2016-06-21T10:36:24.823" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11153" PostTypeId="2" ParentId="11071" CreationDate="2016-06-21T11:50:11.280" Score="-1" Body="&lt;p&gt;Do you mean something like this:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/jmPAW.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jmPAW.jpg&quot; alt=&quot;Modelling Cycle&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The source is provided in the picture and there is a vast body of literature on modelling.&#xA;You might want to specifiy your question:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Which institution / age?&lt;/li&gt;&#xA;&lt;li&gt;Is your interest on how to model modelling or on how to teach modelling or on specific examples of modelling sessions in school?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Let me know, maybe I can add some ressources.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2016-06-21T11:50:11.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11154" PostTypeId="2" ParentId="1899" CreationDate="2016-06-21T12:01:17.917" Score="3" Body="&lt;p&gt;There will be definitely some material on assessment in school. Since my research focus is on tertiary education, I present some resources from calculus and post-calculus:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Abramovitz, B., Berezina, M., &amp;amp; Berman *, A. (2005). How not to formulate multiple choice problems. International Journal of Mathematical Education in Science and Technology, 36(4), 428–437. &lt;a href=&quot;http://doi.org/10.1080/0020739042000322655&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1080/0020739042000322655&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Bergsten, C. (2002). Critical factors and prognostic validity in mathematics assessment. ICTM2, at Crete. Retrieved from &lt;a href=&quot;https://www.researchgate.net/profile/Christer_Bergsten2/publication/228528166_Critical_factors_and_prognostic_validity_in_mathematics_assessment/links/0046352a4fa67a25e4000000.pdf&quot; rel=&quot;nofollow&quot;&gt;https://www.researchgate.net/profile/Christer_Bergsten2/publication/228528166_Critical_factors_and_prognostic_validity_in_mathematics_assessment/links/0046352a4fa67a25e4000000.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Brown, G. A., Bull, J., &amp;amp; Pendlebury, M. (2013). Assessing Student Learning in Higher Education. Routledge.&lt;/li&gt;&#xA;&lt;li&gt;Challis, N., Houston, K., &amp;amp; Stirling, D. (2003). Supporting Good Practice in Assessment in Mathematics, Statistics and Operational Research. University of Birmingham. Retrieved from &lt;a href=&quot;http://cebe.cf.ac.uk/sites/default/files/supportinggoodpractice.pdf&quot; rel=&quot;nofollow&quot;&gt;http://cebe.cf.ac.uk/sites/default/files/supportinggoodpractice.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Cretchley, P. (1999). An Argument for More Diversity in Early Undergraduate Mathematics Assessment. In Proceedings of the 2nd Australian Symposium on Modern Undergraduate Mathematics. Queensland, Australia. Available online at: &lt;a href=&quot;http://www.sci.usq.edu.au/staff/spunde/delta99/Papers/cretchl.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.sci.usq.edu.au/staff/spunde/delta99/Papers/cretchl.pdf&lt;/a&gt;. Citeseer. Retrieved from &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.159.4387&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow&quot;&gt;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.159.4387&amp;amp;rep=rep1&amp;amp;type=pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Griffiths, H. B., &amp;amp; McLone, R. R. (1984). A critical analysis of university examinations in mathematics. Educational Studies in Mathematics, 15(4), 423–442. &lt;a href=&quot;http://doi.org/10.1007/BF00311116&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1007/BF00311116&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Iannone, P., &amp;amp; Simpson, A. (2013). Students’ perceptions of assessment in undergraduate mathematics. Research in Mathematics Education, 15(1), 17–33. &lt;a href=&quot;http://doi.org/10.1080/14794802.2012.756634&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1080/14794802.2012.756634&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Iannone, P., &amp;amp; Simpson, A. (2015). Mathematics lecturers’ views of examinations: tensions and possible resolutions. Teaching Mathematics and Its Applications, 34(2), 71–82. &lt;a href=&quot;http://doi.org/10.1093/teamat/hru024&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1093/teamat/hru024&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Shorter, N. A., &amp;amp; Young, C. Y. (2011). Comparing assessment methods as predictors of student learning in an undergraduate mathematics course. International Journal of Mathematical Education in Science and Technology, 42(8), 1061–1067. &lt;a href=&quot;http://doi.org/10.1080/0020739X.2010.550946&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1080/0020739X.2010.550946&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Smith, G., &amp;amp; Wood, L. (2000). Assessment of learning in university mathematics. International Journal of Mathematical Education in Science and Technology, 31(1), 125–132. &lt;a href=&quot;http://doi.org/10.1080/002073900287444&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1080/002073900287444&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Staff, N. R. C. (1993). Measuring What Counts: A Conceptual Guide for Mathematics Assessment. Washington: National Academies Press.&lt;/li&gt;&#xA;&lt;li&gt;Tallman, M. A., Carlson, M. P., Bressoud, D. M., &amp;amp; Pearson, M. (2016). A Characterization of Calculus I Final Exams in U.S. Colleges and Universities. International Journal of Research in Undergraduate Mathematics Education, 2(1), 105–133. &lt;a href=&quot;http://doi.org/10.1007/s40753-015-0023-9&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1007/s40753-015-0023-9&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Thomas, M. O. J., &amp;amp; Klymchuk, S. (2012). The school–tertiary interface in mathematics: teaching style and assessment practice. Mathematics Education Research Journal, 24(3), 283–300. &lt;a href=&quot;http://doi.org/10.1007/s13394-012-0051-6&quot; rel=&quot;nofollow&quot;&gt;http://doi.org/10.1007/s13394-012-0051-6&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These papers maybe supplement the post of Benjamin Dickman as they are mostly specific to mathematics. I guess not all, but maybe some may help you.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2016-06-21T12:01:17.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11155" PostTypeId="2" ParentId="11071" CreationDate="2016-06-21T15:54:03.677" Score="2" Body="&lt;p&gt;I have seen 5E's used for math and have used it myself when I wished to take a more constructivist approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of which models I've also seen used for math, I'd say that the &lt;a href=&quot;http://thesecondprinciple.com/teaching-essentials/models-teaching/madeline-hunter-lesson-plan-model/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Madeline Hunter&lt;/a&gt; model was most suggested by my education professors and other math teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Madeline Hunter model has 7 basic steps that essentially follow the order of &quot;getting students ready to learn, instruction and checking for understanding, and independent practice.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eghfw.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eghfw.jpg&quot; alt=&quot;Madeline Hunter Model&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This model can be effective if you want students to develop a specific skill, but is less effective for developing deep understanding of the topic. I tend to use it when I wish for my students to have a rote ability to do something automatically. (Such as, for instance, simplifying square root expressions. The understanding is helpful and I include it, but unless they become fluent in the skill it won't help them). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This model is more effective with struggling students, and (by personal experience / distaste) very ineffective with gifted students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.onetohio.org/library/Documents/Dr%20Madeline%20Hunter%20Article1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here is a little more reading about the Madeline Hunter model.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a second response, I have found decent success using Dan Meyer's &lt;a href=&quot;http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Three Acts&lt;/a&gt; lesson plan. I don't know if this falls quite into the realm of a teaching model, particularly since I don't believe it has any supporting research, but I have managed to have several very successful lessons following this lesson plan format. I've summarized the Acts how I personally have used the lesson plan. It fits well into an 80-minute block, with time left over for independent practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Act one: Set the stage. A strong visual piques interest and students pose mathematical questions about the scenario. The scenario can be carefully selected so that some of the questions that arrive naturally are those that require the mathematics you plan on teaching. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Act two: Help students develop the mathematical tools they need to answer the question. Ask them what information they would look for, and teach them the new mathematics they need to finish the job. Direct instruction is fine here. You could even fit a mini-Madeline Hunter plan in here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Act three: &quot;Resolve the conflict and set up the sequel/extension.&quot; Once the concept has been taught, ask the students to solve the problem posed in act 1, guiding them through it and encouraging them to work together. Encourage them to ask and answer more advanced questions if there is time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope that these fit some of what you are looking for. For the record, I have only just finished my first year of teaching high school math, so who knows -- in 3 years I may look back at this post and cringe. &lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2016-06-21T15:59:48.337" LastActivityDate="2016-06-21T15:59:48.337" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11156" PostTypeId="2" ParentId="11139" CreationDate="2016-06-21T18:41:10.187" Score="1" Body="&lt;p&gt;It sounds like Precalculus is the course you should take. A typical Precalculus course covers the topics you listed (without the differential calculus). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However my best advice is to talk with  (or email) the school you're planning on attending. Ask the person(s) who teach these courses and see what they think. You might find that you're ready for AP Calculus or that Precalculus is right or that both are a bad idea.&lt;/p&gt;&#xA;" OwnerUserId="453" LastActivityDate="2016-06-21T18:41:10.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11158" PostTypeId="2" ParentId="11148" CreationDate="2016-06-22T12:04:32.707" Score="3" Body="&lt;p&gt;How about : rescaling the fraction. This could refer to going in either direction.&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2016-06-22T12:04:32.707" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11159" PostTypeId="1" AcceptedAnswerId="11162" CreationDate="2016-06-22T20:03:37.953" Score="10" ViewCount="9441" Body="&lt;p&gt;I'm from Italy and I'm 13 years old. I'm good in Math and I'm good in languages (I know Italian, English and Russian and I think I'm good at them). I'm a programmer and I know HTML, CSS, JS and Python. I've created some program (like webapp), but unfortunately my problem solving and logical thinking isn't good. I've read on the web that when you're younger, it is possible to improve problem solving abilities and logical thinking. What are some steps that I can take to improve problem solving and logical thinking ?&lt;/p&gt;&#xA;" OwnerUserId="6892" LastEditorUserId="6833" LastEditDate="2016-06-24T08:38:42.477" LastActivityDate="2017-04-20T12:00:13.267" Title="Is it possible to improve logical thinking and problem solving abilities?" Tags="|self-learning|logic|problem-solving|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11160" PostTypeId="2" ParentId="4497" CreationDate="2016-06-22T21:21:00.513" Score="7" Body="&lt;p&gt;First: there seems to be a traditional belief that &quot;pure&quot; math fusses over tiny uninteresting details that &quot;applied&quot; math takes for granted, etc. Sure, we &lt;em&gt;can&lt;/em&gt; operate this way, and make &quot;pure math&quot; as irrelevant as we want, or caricaturize it as such. Oppositely, if we try, we can caricaturize &quot;applied math&quot; as slip-shod fuzzy-thinking. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I've ranted elsewhere around here, I think that the reality is not nearly as silly or binary. In particular, complex analysis is almost singular among ideas arising in 19th century mathematics, in that everything seems to work out in the best possible way. In particular, for various technical reasons that are &lt;em&gt;not&lt;/em&gt; well-delineated by a low-tech picture of &quot;rigor&quot;, essentially all the operations one would want to perform, ... interchange of limits, passing differentiations inside integrals, and so on, ... are legal. The problem is that the legality is not so trivial to demonstrate, and, again, low-tech demonstrations are very often semi-incomprehensible/unnatural.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, the contemporary spirit of &quot;(textbook) real analysis&quot; runs off the road very often, apparently teaching us that we should be afraid to do anything, even the obvious. This is not constructive. The potential hazards that &quot;real analysis&quot; brings to our attention mostly do &lt;em&gt;not&lt;/em&gt; occur in complex analysis... but, again, it's not so easy to &lt;em&gt;prove&lt;/em&gt; that they don't occur... and low-tech (=undergrad, early grad-level) proofs are often extremely opaque.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would encourage &quot;pure&quot; and &quot;applied&quot; math people alike to first approach complex analysis in a worry-free, happy-heuristic way, to see all the amazing things that it makes possible. To the extent that one worries about edge cases or technicalities, one will (but it's not so easy to see this in advance, nor by low-tech means) find that the things that might have gone wrong (in a different universe) do not do so here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, don't worry about &quot;pure/rigorous&quot; complex analysis (and certainly not about undergrad-level &quot;real analysis&quot;, which may inhibit you more than it helps you).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(My own notes on complex analysis &lt;a href=&quot;http://www.math.umn.edu/~garrett/m/complex/&quot; rel=&quot;noreferrer&quot;&gt;http://www.math.umn.edu/~garrett/m/complex/&lt;/a&gt; are aimed at math grad students (without pure/applied distinction) who've not taken any course in complex analysis previously. Those notes do not really touch on &quot;engineering&quot; applications, but &lt;em&gt;do&lt;/em&gt; try to illustrate how complex analysis is &quot;the correct&quot; fulfillment of what &quot;calculus&quot; should have been. Not worrying toooo much about &quot;analysis/estimates&quot;, because it turns out there's rarely any serious issue. The notes are freely available, anyway.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-06-22T21:21:00.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11161" PostTypeId="2" ParentId="11159" CreationDate="2016-06-23T08:16:19.533" Score="4" Body="&lt;p&gt;I've just happened to come across this &lt;a href=&quot;https://www.futurelearn.com/courses/logical-and-critical-thinking&quot; rel=&quot;nofollow&quot;&gt;free online course on logical and critical thinking&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Excerpt from the site's &quot;About the course&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We are constantly being given reasons to do and believe things: to believe that we should buy a product, support a cause, accept a job, judge someone innocent or guilty, that fairness requires us to do some household chore, and so on. Assessing the reasons we are given to do or believe these things calls upon us to think critically and logically.&#xA;  Improve your logical and critical thinking skills&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Even though we’re called upon to use our critical and logical thinking skills all the time, most of us are not that good at it. This free online course aims to help you develop and improve these skills.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;You’ll learn how to:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;identify and avoid common thinking mistakes that lead to the formation of bad beliefs;&lt;/li&gt;&#xA;  &lt;li&gt;recognise, reconstruct and evaluate arguments;&lt;/li&gt;&#xA;  &lt;li&gt;use basic logical tools to analyse arguments;&lt;/li&gt;&#xA;  &lt;li&gt;and apply those tools in areas including science, moral theories and law.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;Associate Professor Tim Dare and Dr Patrick Girard from the University of Auckland take us on an informative and engaging eight week journey through the worlds of logical and critical thinking helping us to avoid these common obstacles and fallacies and improve our logical and critical thinking skills.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Throughout the course, Tim and Patrick provide videos, articles, and assignments to lead us through the thickets of logical and critical thinking.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="143" LastEditDate="2016-06-25T12:29:01.380" LastActivityDate="2016-06-25T12:29:01.380" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11162" PostTypeId="2" ParentId="11159" CreationDate="2016-06-23T12:13:27.533" Score="13" Body="&lt;p&gt;First of all I want to laud you on your knowledge of programming. You know a lot more than I did when I was your age. I tried to learn Italian after watching The Godfather but lost interest after a while because there's no one to talk in it with. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two types of mentalities about intelligence and success in life. Studies have shown that children who's parents praised them for being hard working do better in the long run than children who were praised for being intelligent. This is because the second group develop the mentality that they understand things because they are smart and when they have difficulty feel it's beyond their capabilities and give up, while the former group persevere. Fortunately, it's possible to change this mentality. From now on, I want you to think about problem solving skills and logical thinking as skills that can be improved with hard work, much like playing an instrument. Then, you will no longer say, &quot;I am not good at problem solving.&quot; You'll say, &quot;I haven't spent enough time working on improving my problem solving abilities.&quot; It's actually a massive difference in thinking and produces dramatically different results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that I've gotten the mental aspect out of the way, let me tell you some of the things you can do. I too, liked programming but I soon became so impressed with problem solving that I ended up liking Mathematics much more. My advice to you will be simple and uncomplicated - &lt;strong&gt;Spend time solving puzzles and problems.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, mathematics is a wonderful platform to develop these skills. You need to work on problems which require elementary knowledge but deep thinking. Problems that appear in mathematical contest are perfect for this ! Here are some recommendations to get you started off : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Puzzles :&#xA;&lt;strong&gt;Aha! Insight&lt;/strong&gt; - &lt;em&gt;Martin Gardner&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lady or the Tiger&lt;/strong&gt; - &lt;em&gt;Raymond Smullyan&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both of these authors have a wide body of work in recreational mathematics. If you like these books, explore their other books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Puzzling Adventures of Dr. Ecco&lt;/strong&gt; - &lt;em&gt;Dennis Shasha&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Puzzle Math&lt;/strong&gt; - &lt;em&gt;George Gamow&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problem Solving :&#xA;&lt;strong&gt;Mathematical Circles&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's also a journal called &lt;strong&gt;Parabola&lt;/strong&gt; (by UNSW) which has a problem section every issue. Try working through it. Their issues are available for free on the Internet. Bookmark that website on your Browser.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's an interesting YouTube channel called Mind Your Decisions, which deals with tricky problems and game theory. That will be useful for you too. Subscribe to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you'd like, I could recommend more books, journals, YouTube channels and podcasts about mathematics but I don't want you to get overwhelmed. Often, it's better to be concise to know exactly where to start. I have the problem of not finishing books I start because I simply have too many of them. I'm working on choosing one at a time so I can finish some of them off !&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also try learning some strategic game like chess and Go. Try playing Sudoku. However, don't get addicted. After you've played Sudoku for about two months, it can be tempting to continue solving them but they will not continue to increase your problem solving abilities as dramatically. It's important to move on after that period. I got addicted to playing Sudoku on the bus ride and got pretty good at it. After a while, it is no longer a challenge and you hit a plateau. This doesn't happen with games like chess and Go though.&lt;/p&gt;&#xA;" OwnerUserId="6833" LastEditorUserId="6833" LastEditDate="2016-07-06T06:03:55.803" LastActivityDate="2016-07-06T06:03:55.803" CommentCount="19" ContentLicense="CC BY-SA 3.0" />
  <row Id="11163" PostTypeId="2" ParentId="4497" CreationDate="2016-06-23T15:08:08.123" Score="5" Body="&lt;p&gt;Since &quot;epsilon-delta&quot; has been mentioned a couple of times by the question asker, I just thought I'd add my opinion that the epsilon-delta business should not be seen as a divider between pure and applied math. Understanding how sensitive the outputs of a function are to the inputs must certainly be important in real-world applications. Also, one cannot do analysis of any kind (pure or applied, real or complex) without being able to give bounds on various things (such as errors in approximations), which often requires the same types of skills as one encounters while learning to do epsilon-delta proofs in calculus. If you ever do a calculation like &quot;how much wiggle room can I allow in quantity 1 in order to ensure that quantity 2 stays within an acceptable range&quot;, then you are doing an epsilon-delta type calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This sort of addresses your question 2. I would consider epsilon-delta a good topic for everyone, including &quot;applied&quot; people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should add that I'm not advocating for a highly technical nitpicky approach to learning complex analysis. I agree with the points in paul garrett's answer (which I will avoid paraphrasing so I don't misrepresent his views.) I just don't think one should look for textbooks by filtering to avoid epsilon-delta altogether.&lt;/p&gt;&#xA;" OwnerUserId="6894" LastActivityDate="2016-06-23T15:08:08.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11167" PostTypeId="2" ParentId="3916" CreationDate="2016-06-23T22:20:01.287" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Which other skills at the middle school level, particular dealing with such matters as &quot;number sense&quot;, can be taught using the Stock Market Game (or its generic equivalent)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would say that &quot;measures of equivalence&quot; can be taught using the Stock Market Game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the amateur investing world, it's easy to believe that a \$5 stock is cheap and a \$100 stock is expensive (relatively). I think it's useful to have students develop an understanding of &quot;appropriate measure&quot;. Which is stock is cheaper? The \$5 stock or the \$100 stock? How can we compare one to another?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A non-stock market related example would be the length of a line at a checkout counter. One line is 20 feet long and the other line is 40 feet long. But is that how we determine which line is longer? Shouldn't we perhaps consider number of people in the line? Or the total number of items that will be scanned across all people in each line?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leads itself into a larger general notion of &quot;value&quot;. What is good value? How do we measure it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I needed one gallon of milk, should I buy the one gallon container for \$5 from brand MOO or two half-gallon containers for \$2.55 each from brand MOO? For most adults, the one gallon container is the better choice. However, when I've asked my (college) students this question, at least one student feels that the \$2.55 is better because &quot;they are getting two things for cheap&quot;. We can find an equivalent age appropriate question for middle schoolers.&lt;/p&gt;&#xA;" OwnerUserId="6898" LastActivityDate="2016-06-23T22:20:01.287" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11169" PostTypeId="2" ParentId="11134" CreationDate="2016-06-23T23:53:45.217" Score="1" Body="&lt;p&gt;I'll break this down into three parts with respect to &amp;quot;relationship between basic math and English&amp;quot;.&lt;/p&gt;&#xA;&lt;h1&gt;Translation and Transliteration&lt;/h1&gt;&#xA;&lt;p&gt;I've found that it is a useful exercise to have students simply take a mathematical expression and say / write it out with as little mathematical symbolism as possible. In my introductory Set Theory class, we will take a statement like &lt;span class=&quot;math-container&quot;&gt;$\{x : x &amp;gt; 0, x \in \mathbb{N}\}$&lt;/span&gt; and read it as &amp;quot;The set of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s such that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is greater than zero and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is in the set of natural numbers&amp;quot; (or other equivalent readings).&lt;/p&gt;&#xA;&lt;p&gt;I further make the point that while the mathematical statement written in mathematical symbolism is short, it is not a short statement. In fact, it is a lot of information, compactly written. As an aside, I often tell my students that while the word &amp;quot;cat&amp;quot; is a three character word which we can read at a glance, it is usually not the case with mathematical statements. For example, &lt;span class=&quot;math-container&quot;&gt;$x = 5$&lt;/span&gt; is also three non-whitespace characters, but it reads as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and five are equal to each other&amp;quot; (or the more common &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; equals five&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Similarly it is helpful to go the other way. Start with a statement with as little mathematical symbolism as possible and try to write it in the most condensed way --- just as an exercise. Somewhere in between is something that is readable.&lt;/p&gt;&#xA;&lt;h1&gt;Abuse of Vocabulary and Symbolism&lt;/h1&gt;&#xA;&lt;p&gt;I also discuss multiple definitions of words with my students. In a Calculus class, I will first ask for my students' understanding of the word &amp;quot;continuous&amp;quot; and show that there is a difference between colloquial use and the mathematical use.&lt;/p&gt;&#xA;&lt;p&gt;In an Algebra class, you may want to try this with words like &amp;quot;infinity&amp;quot;, &amp;quot;and&amp;quot; vs &amp;quot;or&amp;quot;, etc.&lt;/p&gt;&#xA;&lt;p&gt;Then, there is the abuse of vocabulary within mathematics itself. There is the word &amp;quot;graph&amp;quot;, for example which has a formal and general usage within Graph Theory as an object and the slightly more colloquial and different usage in an Algebra course: &amp;quot;Graph the function ...&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Similarly, there is the abuse of symbolism and the biased use of symbolism. Break your students out of the habit that &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the independent variable and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is the dependent variable&amp;quot;. Use different letters and change the context. Make &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; the dependent variable; let the vertical axis be associated with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, as for instances.&lt;/p&gt;&#xA;&lt;h1&gt;Left to Right Reading&lt;/h1&gt;&#xA;&lt;p&gt;English is read from left to right and then from top to bottom, like this response. Other written languages are read from right to left and from to bottom. And still others are read top to bottom and then left to right. And there may be other cases.&lt;/p&gt;&#xA;&lt;p&gt;Mathematical statements aren't necessarily read in any of these ways. Consider &lt;span class=&quot;math-container&quot;&gt;$\Big(\sum_{k=1}^{n}k^{2}\Big)^{\frac{1}{2}}$&lt;/span&gt;. Ack! This is &amp;quot;the square root of the sum of the squares of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$k = 1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&amp;quot;. &amp;quot;Diagramming&amp;quot; this would be not be so &amp;quot;linear&amp;quot; [purposeful use of mathematical term in a technically, non-mathematical way].&lt;/p&gt;&#xA;&lt;p&gt;It is helpful to get students to understand this notion and to see and read the mathematical phrase differently than they would English. This is why I say &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and five are equal to each other&amp;quot; rather than &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; equals 5&amp;quot;. The former talks about the two objects as equal, the latter has a subtle bias of causation [case in point is that when one write &amp;quot;5 = &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; there is typically an open revolt from students since they reject the idea that &amp;quot;five can equal &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;].&lt;/p&gt;&#xA;" OwnerUserId="6898" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-06-23T23:53:45.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11170" PostTypeId="2" ParentId="4438" CreationDate="2016-06-24T00:48:18.163" Score="2" Body="&lt;p&gt;An elementary introduction to Graph Theory might not be a bad idea. It has at least the following pros:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;unlike the other math classes&lt;/li&gt;&#xA;&lt;li&gt;will likely never see this in undergraduate coursework unless if they are going into Computer Science or Mathematics (maybe Electrical Engineering)&lt;/li&gt;&#xA;&lt;li&gt;ample topics and applications associated with graphs&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the internet&lt;/li&gt;&#xA;&lt;li&gt;social networks&lt;/li&gt;&#xA;&lt;li&gt;map software that gives &quot;optimal&quot; routes&lt;/li&gt;&#xA;&lt;li&gt;map coloring&lt;/li&gt;&#xA;&lt;li&gt;knights tour&lt;/li&gt;&#xA;&lt;li&gt;traveling salesman&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;nice puzzles&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Konigsberg bridges&lt;/li&gt;&#xA;&lt;li&gt;you can also make up your puzzles!&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;word games like &quot;get from paper to sketch in as few steps as possible by changing one letter at a time to construct a new valid word&quot; (can this be done?)&lt;/li&gt;&#xA;&lt;li&gt;exploration of board games&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ticket To Ride&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;is accessible to almost all math skill levels (I've taught Graph Theory units at the college level and my experience has been that no matter how &quot;weak&quot; the student may be with the traditional math curriculum, a basic introduction to Graph Theory is still within their reach)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6898" LastActivityDate="2016-06-24T00:48:18.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11171" PostTypeId="2" ParentId="11122" CreationDate="2016-06-24T03:24:43.713" Score="1" Body="&lt;p&gt;I like Joe Z's answer. It's how I introduce polynomials as well. In the interest of other approaches and since the idea is to introduce students to polynomials, here is another thought:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider a geometric approach and start with squares and cubes. If we have a square with side length $3$ units, then the area is $3^{2} = 9 \mbox{ units}^{2}$. A cube with side length $3$ would have volume $3^{3} = 27 \mbox{ units}^{3}$. A next step would be to use $x$ in place of the side length to obtain $x^{2}$ and $x^{3}$ for the area and volume, respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, we can start to tease out some rules about polynomials if we keep the units hanging around. For example, we could ask what in the world it would mean to have $x^{3} + x^{2}$. How can we add volume with area? We can't in any sensible way, and hence we must maintain $x^{3} + x^{2}$, or analagously, $27 \mbox{ units}^{3} + 9 \mbox{ units}^{2}$ --- these units don't mix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if I had two squares, I could join them and have their areas add to $2x^{2}$. Or have two cubes and have their volumes add to $2x^{3}$. All the while, we would carry our units along. Then we could really do some mind bending with some computation. Ask the question, &quot;If I had three squares, each with side length three, what would my total area be?&quot; The answer would be $27 \mbox{ units}^{2}$. Then we compare this result with one cube of side length three. The only difference is that, while in value, both quantities are equal in dimensionality they are not --- apples and oranges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplying polynomials can be brought back to finding the area of a square or the volume of a sphere. $x$ is a polynomial of degree 1. Thus, $x\times x = x^{2}$ and $x\times x\times x = x^{3}$, but $x\times x\times x = x^{2} \times x = x^{3}$ or $x\times x\times x = x\times x^{2} = x^{3}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't really had much occasion to offer this type of explanation, but I keep it in reserve in case the approach as outlined by Joe Z just doesn't resonate. One distinct &quot;advantage&quot;, is that it gets students to think about dimension and units.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This allows us to bring the conversation back to simple numerical substitution. What if we had $x^{3} + x^{2}$ and were asked to substitute $x = 3$? Why is the answer $27 + 9 = 36$ a valid answer, but in the case of our square and cube, we were forced to leave it as the nonsensical $27\mbox { units}^{3} + 9\mbox{ units}^{2}$? The answer is that $x = 3$ is dimensionless, whilst $x = 3 \mbox{ units}$ is not.&lt;/p&gt;&#xA;" OwnerUserId="6898" LastActivityDate="2016-06-24T03:24:43.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11172" PostTypeId="1" AcceptedAnswerId="11177" CreationDate="2016-06-24T08:26:50.027" Score="17" ViewCount="986" Body="&lt;p&gt;As a TA who led calculus* 1 and 2 discussion section and holds office hour** in the previous year, I heard the following (wrong) arguments several times.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to \infty} \sqrt{x+1}-\sqrt{x}=0$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$\infty-\infty=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to \infty} x^{1/x}=1$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$\infty^0=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int_1^{\infty}f(x)dx$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\int_1^{\infty}g(x)dx$&lt;/span&gt; both diverge so &lt;span class=&quot;math-container&quot;&gt;$\int_1^{\infty}f(x)+g(x)dx$&lt;/span&gt; diverge.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I usually explain the arguments are not true in general by providing a (very trivial) counter example, for example,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to \infty} f(x)=\infty$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to \infty} g(x)=\infty$&lt;/span&gt; does not guarantee &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to \infty} f(x)-g(x)=0$&lt;/span&gt;, for example, &lt;span class=&quot;math-container&quot;&gt;$f(x)=x+1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)=x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to \infty} f(x)=\infty$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to \infty} g(x)=0$&lt;/span&gt; does not guarantee &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to \infty} f(x)^{g(x)}=1$&lt;/span&gt;, for example, &lt;span class=&quot;math-container&quot;&gt;$f(x)=2^x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)=1/x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;False in general, for example &lt;span class=&quot;math-container&quot;&gt;$f(x)=-g(x)=1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After giving explanations like that I sometime heard &amp;quot;But in your examples you can cancel the expression/formula...&amp;quot; and I was not sure how to continue. I tried the following methods, non of them seem to work very well.&lt;/p&gt;&#xA;&lt;p&gt;a. Provide a much more complicated counter example which requires a few minutes of calculation to get the answer. This often leads to further confusion.&lt;/p&gt;&#xA;&lt;p&gt;b. Just say that is the wrong way to do it. It sounds like &amp;quot;I'm the teacher so believe me.&amp;quot; and doesn't do too much.&lt;/p&gt;&#xA;&lt;p&gt;c. Show them the correct way to do their problems. This is almost like b (Why is your way the right way and mine is the wrong way?).&lt;/p&gt;&#xA;&lt;p&gt;I'm looking for a better way to deal with questions like these.&lt;/p&gt;&#xA;&lt;p&gt;*&lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; definition is not introduced.&#xA;** Office hour is in tutoring center where I'm also responsible for students take the class from the professors I'm not TA'ing for.&lt;/p&gt;&#xA;" OwnerDisplayName="user2139" LastEditorUserId="-1" LastEditorDisplayName="user2139" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-07-04T08:53:27.167" Title="What is a better way to explain these claims about limit are not true in general?" Tags="|undergraduate-education|calculus|teaching|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11174" PostTypeId="2" ParentId="11172" CreationDate="2016-06-24T13:22:58.063" Score="4" Body="&lt;p&gt;Wanted to write this as a comment, but I'm not strong enough [reputation]. This may be a bit heavy handed or you're already doing it, but what about challenging the very notion of arithmetic when dealing with infinities before going towards counterexamples? That's usually what the fuss tends to be about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside, when teaching probability theory, the Cauchy distribution tends to flummox students because we run into problems when trying to compute the mean even though the distribution is symmetric. Why does there have to be a central tendency? It challenges a notion that they've held to be implicitly true. When they can't answer this general question of &quot;why&quot;, it often helps to break the cognitive dissonance and they are more receptive to the idea that, yes, perhaps the mean doesn't have to exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, with limits involving infinities, I would ask students what rule says that $\infty - \infty = 0$ every time? Then, trivial examples as you give can help to seal the deal and disavow them of such a notion. Perhaps? Basically and I'm not sure if you already do this, but I would first challenge, philosophically, what students have considered to be axiomatically / dogmatically true before showing counterexamples.&lt;/p&gt;&#xA;" OwnerUserId="6898" LastActivityDate="2016-06-24T13:22:58.063" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11175" PostTypeId="2" ParentId="11172" CreationDate="2016-06-24T16:46:31.627" Score="11" Body="&lt;p&gt;Often in courses like this, the problem is that students are almost exclusively asked to do problems that involve rote computation using procedures they've been taught, not problems that require reasoning about the concepts involved or even constructing their own examples. So, how about assigning problems that require students to come up with examples refuting various false notions like the ones mentioned? (Or at least discuss problems like these in class.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm thinking of problems like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;Give an example of functions $f$ and $g$ such that $\lim_{x \to \infty} f(x) = \infty$, $\lim_{x \to \infty} g(x) = 0$, and $\lim_{x \to \infty} f(x)^{g(x)} = 5$.&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;Give an example of functions $f$ and $g$ such that $\lim_{x \to \infty} f(x) = \infty$, $\lim_{x \to \infty} g(x) = \infty$, and $\lim_{x \to \infty} (f(x) - g(x))$ does not exist.&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-06-24T16:46:31.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11176" PostTypeId="2" ParentId="11172" CreationDate="2016-06-24T17:34:31.197" Score="2" Body="&lt;p&gt;The first thing I would say is that principally you want to stick with strategy (a), and not veer into (b) or (c). You definitely want to dig into, and spend time on, these trouble spots with proper mathematical reasoning; certainly not &quot;Because I said to do it this way&quot;, as that doesn't actually count as math at all. Counterexamples are logically the fundamental way one knocks down conjectures like those, so students should exercise and be trained in that method of dialogue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I do in a situation like this is try to jot down those trouble spots and turn them into quizzes for the whole class. Usually if I can turn it into (a) a binary &quot;true or false&quot; question, and (b) get the whole class in on the discussion, then we very quickly come to a consensus on the correct answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're solely working in a tutorial/one-on-one situation, then this becomes trickier to implement. Perhaps write down the statement as a &quot;true or false&quot; question and try to pin the student down to the one correct response (no &quot;but this case was different&quot; allowed). Possibly leverage one or two other nearby students to get their input on the discussion. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-06-24T17:34:31.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11177" PostTypeId="2" ParentId="11172" CreationDate="2016-06-24T17:44:43.520" Score="6" Body="&lt;p&gt;This is a difficult problem, because the students you're dealing with aren't really prepared to think about logical reasoning and the role of counterexamples.  (For example, I find students at that level often don't distinguish between examples and arguments.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm of the view that you can't actually expect students to understand material at a level much deeper than it's being taught.  If an algorithm works on all problems they'll see then, in a real sense, the algorithm is correct for purposes of the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As it happens, the examples you're describing are probably &lt;em&gt;not&lt;/em&gt; correct, even &quot;locally&quot;, because they're going to be expected to deal (if not immediately then later in the course) with problems for which that algorithm doesn't work, so you do need to correct these misconceptions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My suggestion would be:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Focus on giving counterexamples.  It's important to avoid trivial problems.  Students treat algorithms as heuristics, and have a hierarchy from easier to harder heuristics; you have to choose problems which trigger the faulty heuristic, and on which it fails.  There's a natural supply of these: the problems which students will later have to solve using more sophisticated methods (L'Hospital's rule problems for a and b, other integrals for c).  You don't need to work out the solution; you can say &quot;Okay then what's the solution to this problem?&quot; and then when they get the wrong answer, say &quot;Actually, we're going to learn later that it's really ...&quot;.  Instead of doing the calculation, you can draw (or even better, use a computer to generate) a graph of the function which supports your answer (don't present it as a proof, of course, just emphasize that it's suggestive, and the rigorous method will come later).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another meaningful kind of counterexample, if you can use it, is problems where the shortcut doesn't apply.  In other words, don't give a counterexample to its accuracy, give a counterexample to its usefulness.  You don't want to say &quot;this is right because I'm the teacher&quot;, but it's reasonable to say &quot;we need this method to solve hard problems, so we're going to practice it on easy problems, even if there's a shortcut&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-06-24T17:44:43.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11178" PostTypeId="2" ParentId="11172" CreationDate="2016-06-25T08:47:49.730" Score="2" Body="&lt;p&gt;I find it in general a bit backwards if a teacher has to explain &lt;em&gt;why such and such claim isn't true&lt;/em&gt;. For any given statement that that can't be proven, the default assumption should be that it's not true&lt;sup&gt;&amp;dagger;&lt;/sup&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The entire confusion could be avoided by never talking about &amp;ldquo;infinite results&amp;rdquo; in the first place. If you replace $\lim f(x) = +\infty$ with simply &lt;em&gt;the limit doesn't exist&lt;/em&gt;, then it's not possible to forge that into an unsound expression like $\infty^0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, of course what we don't want to do is make it more difficult to arrive at &lt;em&gt;correct&lt;/em&gt; results, such as l'Hospital. But IMO, it's actually &lt;em&gt;helpful&lt;/em&gt; if one doesn't start with infinite limits of some $f$ and $g$ for such theorems: the theorem is all the more impressive if it doesn't just give you a way to &lt;em&gt;calculate&lt;/em&gt; some particular result, but actually &amp;ldquo;generates&amp;rdquo; that result, which you previously had no reason to assume existed!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This kind of proof-oriented thinking was never really explained to me at school, and I only started really enjoying maths when I understood it at university.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&amp;dagger;&lt;/sup&gt;&lt;sub&gt;Of course, one should not assume that it's &lt;em&gt;wrong&lt;/em&gt; either. It's often good &lt;a href=&quot;https://en.wikipedia.org/wiki/Constructivism_(mathematics)&quot; rel=&quot;nofollow&quot;&gt;not to think in a true/false dichotomy&lt;/a&gt;...&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="6902" LastActivityDate="2016-06-25T08:47:49.730" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11179" PostTypeId="2" ParentId="11124" CreationDate="2016-06-25T13:12:41.400" Score="5" Body="&lt;p&gt;Four things come to mind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) You could focus on how the linearity of the objective function and the convexity of the feasible region yields that local optima are global, and how the linear nature of the constraints allow you to restrict your attention to corner points. This would accent the fundamental differences between linear and nonlinear optimization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) If you want to show how mathematics is interrelated -- you can show how optimal mixed strategies for two person zero-sum games can be found by linear programming. A famous anecdote which you could share with the students is how, when Dantzig was explaining linear programming to Von Neumann, the latter intuited the link with game theory and conjectured that there must be some underlying duality in linear programming (which Dantzig proved about a year later). Von Neumann's Minimax theorem and the fundamental duality theorem of linear programming are essentially the same thing. There are a couple of texts such as &quot;An Introduction to Linear Programming and Game Theory&quot; by Thie and Keough which explore this connection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) The &lt;a href=&quot;https://en.wikipedia.org/wiki/Klee%E2%80%93Minty_cube&quot;&gt;Klee-Minty&lt;/a&gt; example of linear programs for which the simplex algorithm takes exponentially many steps is really quite interesting. The simplex algorithm is the most quotable case of a practical algorithm for which worst-case and average-case complexity are radically different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) The &lt;a href=&quot;https://en.wikipedia.org/wiki/Assignment_problem&quot;&gt;assignment problem&lt;/a&gt; is fairly accessible. There is some interesting linear algebra in explaining how you get integer solutions in this case even though the simplex algorithm will often give fractional answers for other problems. This problem is a good lead-in to combinatorial optimization and integer programming. &lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2016-06-25T13:12:41.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11180" PostTypeId="2" ParentId="2262" CreationDate="2016-06-26T13:08:56.067" Score="5" Body="&lt;p&gt;I know that this is an old question, but it should be pointed out that in the United States at least, &quot;early transcendentals&quot; vs. &quot;late transcendentals&quot; refers to the placement of exponential, logarithmic, and inverse trigonometric functions. I've never seen a textbook which didn't introduce the derivatives of trigonometric functions in the first semester, but I have seen a number of texts which delay exponential and logarithmic functions until the second semester. As a simple example -- compare the table of contents of James Stewart's calculus texts. The one with &quot;Early Transcendentals&quot; in its title covers logarithms and exponential functions before integration and the one without that in the title covers them just after integration is introduced. Both cover trigonometric functions just after polynomials. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where I teach we use the early transcendentals approach. I was resistant at first since I had both originally learned and previously taught using the other approach, but I now see the advantages of the earlier approach. The chief advantage is that it is better for the science and engineering students who (at least where I teach) often make up the majority of a first semester calculus class. In particular, our biology majors are only required to take a single semester of calculus and it would be a shame if they ended their calculus without seeing the link to exponential growth and decay. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see two disadvantages to early transcendentals. One is that it pushes back integration to the very end of the first semester. I am chronically pressed for time towards the end of the first semester. I am able to get through the Fundamental Theorem of Calculus, but often at the cost of being very hand-wavy about things like Riemann sums. The early transcendentals tends to force me to cover integration more superficially than I would like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A second disadvantage is that it is almost impossible to give a rigorous approach to exponential and logarithms without knowing about the Fundamental Theorem of Calculus. I don't know of any adequate definition of &lt;code&gt;a^x&lt;/code&gt; for &lt;code&gt;x&lt;/code&gt; irrational that can be shown to make sense using only the tools of early calculus. At best you can say something like &lt;em&gt;if&lt;/em&gt; &lt;code&gt;a^x&lt;/code&gt; exists and &lt;em&gt;if&lt;/em&gt; certain limits exist, &lt;em&gt;then&lt;/em&gt; it is a differentiable function. The entire discussion is an almost hopeless hand-wave. In contrast, with the late transcendentals approach you can &lt;em&gt;define&lt;/em&gt; &lt;code&gt;ln(x)&lt;/code&gt; as the definite integral of &lt;code&gt;1/t&lt;/code&gt; from 1 to &lt;code&gt;x&lt;/code&gt; and work out the properties of both logarithms and exponential functions from there. This is clearly a better approach for pure math majors. In addition to being more rigorous, it leaves them with a better understanding of what the Fundamental Theorem actually means.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2016-06-26T13:08:56.067" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11181" PostTypeId="2" ParentId="11086" CreationDate="2016-06-27T12:35:42.223" Score="1" Body="&lt;p&gt;I have not read this paper.&#xA;My assumption is that this is an issue of retention and exam mpreparation among the students.&#xA;At the time of the midterm, the topic would be fresh.&#xA;At exam time, the newer topics were fresher and the old stuff had not been retained.&#xA;This effect can be mitigated by teaching strategies that help students to retain.&#xA;- rereading notes (5 times per page in a 2 week period)&#xA;- practicing questions from earlier in the course&#xA;-etc.&lt;/p&gt;&#xA;" OwnerUserId="6896" LastActivityDate="2016-06-27T12:35:42.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11182" PostTypeId="1" AcceptedAnswerId="11184" CreationDate="2016-06-28T13:41:45.877" Score="24" ViewCount="1972" Body="&lt;p&gt;A function is not &lt;em&gt;really&lt;/em&gt; a function unless it's defined everywhere on its domain. So consider these three questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the square root function $f(x) = \sqrt{x}$. What is its domain?&lt;/li&gt;&#xA;&lt;li&gt;Let $f: [0, \infty) \rightarrow \mathbb{R}$ be the square root function $f(x) = \sqrt{x}$. What is its domain?&lt;/li&gt;&#xA;&lt;li&gt;Let $f$ be the square root function $f(x) = \sqrt{x}$. What is its domain?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Question 1 is ill-posed, because f is not a function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question 2 is trivial; the question gives you the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question 3 is ambiguous. The domain could be $[0,\infty)$ or $\mathbb{C}$ or something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what is the best way to ask this question? I've just recently found out about &lt;a href=&quot;https://en.wikipedia.org/wiki/Partial_function&quot;&gt;partial functions&lt;/a&gt; but I'm not sure it would be a good idea to introduce this concept to my students. &lt;/p&gt;&#xA;" OwnerUserId="6889" LastActivityDate="2022-07-06T11:12:03.910" Title="What is the proper way to ask a &quot;find the domain&quot; question?" Tags="|undergraduate-education|functions|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11183" PostTypeId="2" ParentId="11182" CreationDate="2016-06-28T15:28:30.163" Score="19" Body="&lt;p&gt;user52817, in the comments, has this exactly right:  In the context of Precalculus, the &lt;em&gt;implied domain&lt;/em&gt; of a function is the largest subset of $\mathbb{R}$ on which the function is defined.  So introduce that notion, and practice it with questions like &quot;What is the implied domain of $f(x)=\sqrt{x}$?&quot;  (You can also let your students know that sometimes people are lazy and omit the word &quot;implied&quot;, but there are good reasons not to do that.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-06-28T15:28:30.163" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11184" PostTypeId="2" ParentId="11182" CreationDate="2016-06-28T16:30:22.373" Score="33" Body="&lt;p&gt;It's not really a question about functions and domains, but about &lt;em&gt;valid expressions&lt;/em&gt;. The question is, &quot;For which real numbers $x$ is the following expression valid/well-defined/well-formed?&quot; So, for example, $\frac{x + 1}{x^2 - 1}$ is valid exactly for $x \notin \{-1, 1\}$, and $\sqrt{x}$ (assuming this notation is established to mean the real-valued square root) is valid for all $x \geq 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bringing &quot;functions&quot; (which aren't actually functions unless the domain is already specified) into it is a distraction. Sure, you could try to formalize it as something like &quot;the largest subset of the real numbers to which the formula for the function can be continuously extended&quot;, but this isn't what's really being asked: $\frac{x + 1}{x^2 - 1}$ isn't valid if $x = -1$, but can be continuously extended to all $x \neq 1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, if you want to ask questions like this, recognize it as the syntactic question it is: the point of these questions isn't to work with functions themselves, but to become familiar with the notation used in mathematical expressions — and in particular, when it is and isn't valid/well-defined.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-06-28T16:30:22.373" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11185" PostTypeId="2" ParentId="11182" CreationDate="2016-06-28T17:05:47.967" Score="3" Body="&lt;p&gt;Really, the right way to go about things is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find the set $X$ of points $x$ for which there exists a $y$ satisfying $y^2 = x$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and then proceed to define $f(x) = \sqrt{x}$ for $x \in X$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, for the example of $\frac{x+1}{x^2-1}$, the &lt;em&gt;right&lt;/em&gt; question is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find the set $X$ of points $x$ for which $x^2 - 1 \neq 0$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and only ever utter the expression $\frac{x+1}{x^2-1}$ when $x$ is restricted to $X$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I honestly believe that introductory materials that would ask questions like &quot;what is the domain of this expression&quot; like this are &lt;em&gt;trying&lt;/em&gt; to teach partial functions, but are handicapped by trying to do so in the language of (total) functions.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2016-06-28T17:05:47.967" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11186" PostTypeId="2" ParentId="11182" CreationDate="2016-06-29T13:17:50.613" Score="15" Body="&lt;p&gt;Although there are good answers already, I think the ambiguity of the vocabulary is being underestimated. This might be because French definitions have more ambiguity than others, I can't tell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In France, at least up to some years ago, the words &lt;em&gt;function&lt;/em&gt; (&quot;fonction&quot;)  had various meaning depending on the context. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In high school (and probably some colleges), one would write let $f:\mathbb{R}\to\mathbb{R}$ be the function defined by $f(x)=\sqrt{x}$, because the first $\mathbb{R}$ would not mean the domain, but would be a mere notation for specifying the &lt;em&gt;type&lt;/em&gt; of argument $f$ could be fed with. Then one can ask for the &lt;em&gt;domain&lt;/em&gt; (&quot;domaine de définition&quot;) of $f$, which is the largest subset of $\mathbb{R}$ for which the given expression makes sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At university, one would use the current mathematical convention for functions and domain, thus making more intricate to ask domain questions (but they are assumed to be well understood at that point -- not a very wise assumption unfortunately).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The vocabulary &lt;em&gt;should&lt;/em&gt; be chosen so as to express easily what one wants to express, so while I have been uncomfortable with the above state of affairs, it has quite some ground. Take for example the case of unbounded operators in functional analysis: here the vocabulary matches quite closely the French high school vocabulary, and this is very useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think it is either wise nor plausible to try have a completely formalized mathematical language: one will always abuse notation at some point (If I had more time I would dare anyone disagreeing to point to any mathematical text of at least a few pages that would be a counterexample, but I don't). If we could make students understand that $f$ has the type of a function and $f(x)$ has the type of a number, it would be a good thing. But we should allow ourselves and students &lt;em&gt;some&lt;/em&gt; slack.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To finally answer the primary question, a reasonable phrasing could be: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Determine the largest real domain where a function $f$ can be defined by $f(x)=\sqrt{x}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2016-06-30T09:35:24.787" LastActivityDate="2016-06-30T09:35:24.787" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11187" PostTypeId="2" ParentId="11182" CreationDate="2016-06-29T19:16:22.610" Score="6" Body="&lt;p&gt;If you don't want to call $f$ a function, because it's not, call it what it is, a partial function:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given the partial function $f : \mathbb R \to \mathbb R$ where $f(x) = \sqrt{x}$, what is the domain of $f$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(This depends of course on what definition of domain you've selected for partial functions in your course.  You might have to ask 'for what $x$ is $f(x)$ defined' instead.  But you can avoid calling $f$ a function when it's not.)&lt;/p&gt;&#xA;" OwnerUserId="4736" LastActivityDate="2016-06-29T19:16:22.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11188" PostTypeId="2" ParentId="1589" CreationDate="2016-06-29T19:57:27.167" Score="3" Body="&lt;p&gt;I admit I'm a bit late to the party, but for completeness let me add a link to an interactive version of one of the visual proofs:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://weitz.de/fund/&quot; rel=&quot;nofollow&quot;&gt;The fundamental theorem of algebra - a visual proof&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Disclaimer: I'm the author of that page.)&lt;/p&gt;&#xA;" OwnerUserId="2089" LastActivityDate="2016-06-29T19:57:27.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11189" PostTypeId="2" ParentId="11172" CreationDate="2016-07-04T08:53:27.167" Score="3" Body="&lt;p&gt;I think your first strategy, giving counter-example, is a good start; you may have more success by &lt;strong&gt;giving counter-examples that carry meaning&lt;/strong&gt; and &lt;strong&gt;relate to things the students already know&lt;/strong&gt;. In the case at hand, you can also try to make the point that &lt;strong&gt;limits are not about arithmetic, but about asymptotic behavior.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will consider the first fallacy you mention for illustration.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;To give counter-examples that carry meaning&lt;/strong&gt;, you can draw two graphs of functions, and have the student guess what their limits are (which you have taken to be $+\infty$). Then point (ask him or her to show you) on the graph the difference between the two functions: it is represented by the vertical space between the two graphs. Now you can draw examples where both functions go to $+\infty$, but the gap between them goes to zero, or $+\infty$, or any given number, or does not have a limit. Then you can ask (or explain how) to turn these drawings into precise counter examples. Hopefully, symbols will start to carry a meaning in the students mind rather than being purely abstract. One of the thing we teachers too often forget to tell to our students is the process that made us propose a counter example. We are like magicians pulling a rabbit out of our hat, while our true goal is to attract their attention to the trick. By the way, I prefer by far to use hand drawn graphs, because a computer generated graph needs you to think of the formula first, feed it to the computer, and then observe. What we really do is to think of the general form of the graph, and then come up with a formula. Also, students are far more likely to feel they can use this methods of reasoning themselves with hand drawn graphs than with a computer, hand you want them to be able to debunk their next fallacies themselves.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;To relate the fallacies with things they already know,&lt;/strong&gt; you can ask them if it is true that when two (positive) functions go to $+\infty$, their quotient goes to $1$. They may know that the answer is negative, and you can start get them to explain it to you (to themselves, really). Then the refreshment can be used in the case at hand, which is in fact awfully close.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;To make the point that &lt;strong&gt;limits are not about arithmetic, but about asymptotic behavior&lt;/strong&gt;, you can try to separate two things: the meaning of the limit (what is the behavior of the function in a certain point or direction?) and the tools to determine limits. I guess that student often confuse the two (in a wide array of topics, not only limits; as an example, many student end up considering eigenvalues as being &lt;em&gt;defined&lt;/em&gt; as the roots of the characteristic polynomial, leaving them unable to prove that the vector with all entries $1$ is an eigenvector of the matrix with all entries $1$: I have had almost all of my students failing this question because they wanted to compute the characteristic polynomial first).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can thus try to rephrase the first fallacy: &quot;if two functions go to $+\infty$ as the argument goes to $+\infty$, then the gap between the two &lt;em&gt;must&lt;/em&gt; go to zero&quot;. They may realize they have no reason to believe that (but the drawings above may be necessary to do that). Relate to the true fact that when the common limit is not $+\infty$ but a real number, then the difference must go to zero. Have them understand that the use of arithmetic is made possible then because in this situation, the conclusion is inevitable. Even if they don't have the tools to prove it, they can see it. There you are at the root of the problem (they applied a rule they learned in a situation it does not apply to, without seeing that there is a missing hypothesis; such reasoning by similarity is very common). Once they realize the difference between the two situations (finite and infinite common limit), they may accept to at least doubt what they first said. The confusion might have been entrenched in their mind because they have seen the cases when one of the two functions goes to $\infty$ and the other has a finite limit, so make them look at this case two (without formulas at first, just graphs).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-07-04T08:53:27.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11190" PostTypeId="1" AcceptedAnswerId="11193" CreationDate="2016-07-04T17:41:01.710" Score="4" ViewCount="122" Body="&lt;p&gt;Sometimes students from different programmes take the same course together. They may have a different background and/or different interests in the course. One example could be a basic mathematics course which is taken by physicists, mathematicians and teacher training students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a reasonable way of providing all these groups with (e.g. homework) problems without boring the other students. Would it be acceptable if in addition to a common body of problems each group gets additional &quot;special&quot; questions? Is there any research on the question of how such a strategy might influence the students' motivation and their learning outcome? How to deal with the problem that some students might think that the &quot;special&quot; exercises of (one or more) of the other groups are too easy in comparison?&lt;/p&gt;&#xA;" OwnerUserId="6002" LastEditorUserId="6002" LastEditDate="2016-07-04T17:57:05.947" LastActivityDate="2016-07-05T01:45:05.993" Title="Providing interesting problems for all students in heterogeneous groups" Tags="|undergraduate-education|student-motivation|heterogeneous-groups|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11191" PostTypeId="5" CreationDate="2016-07-04T17:56:26.473" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2016-07-04T17:56:26.473" LastActivityDate="2016-07-04T17:56:26.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11192" PostTypeId="4" CreationDate="2016-07-04T17:56:26.473" Score="0" Body="For questions about problems which arise from teaching a heterogeneous group of students." OwnerUserId="6002" LastEditorUserId="6002" LastEditDate="2016-07-05T17:48:48.217" LastActivityDate="2016-07-05T17:48:48.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11193" PostTypeId="2" ParentId="11190" CreationDate="2016-07-05T01:45:05.993" Score="2" Body="&lt;p&gt;It's a broad question and as such is tricky to answer specifically. Thus, broadly speaking, I would do the following, speaking from experience not from published research on the matter&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I wouldn't advertise the problems as &quot;here are problems for Group A, here are for Group B, etc.&quot;. I've never liked to broadcast to students how they are being classified.&lt;/li&gt;&#xA;&lt;li&gt;Instead, I would ask &quot;hard&quot; problems and separately give a &quot;hints&quot; document. The purpose of the hints document is to guide students through the steps. For example, let's say that deep into a Calculus II class, after seeing the bulk of the integration techniques, I would ask $$\int \sqrt{x^{2} + 1}\ dx$$ and then on a hints page, I would give a series of hints, starting with a trig substitution, perhaps. The general instruction / philosophy would be, &quot;If you get stuck, take a look at the hint and try to continue from there. If you are still stuck, look at the next hint.&quot; Thus, by going through the hints, they (ideally) automatically fall into the right bucket of ability.&lt;/li&gt;&#xA;&lt;li&gt;I generally don't collect homework for grading, so I tend to assign pretty much the book for homework with general instructions about what I would consider to be easy, medium, hard problems [this lets them understand their own ability without explicitly calling to attention the ability of other students --- I do my best to avoid non-self comparisons]. I tell students to answer as many problems as they feel they can with the objective that they be able to get to the hard problems.&lt;/li&gt;&#xA;&lt;li&gt;I don't find anything wrong with giving reach problems for the advanced students. I just don't advertise that &quot;these problems are for advanced students&quot; [I avoid classifying students to the students.].&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If the concern is exams and grading, then I would look into if there is a possibility to have a split course. I have seen this done where if students are on, say, a BS in Math track then their course objectives are necessarily different from those on, say, a BA in Math Ed track. If you're able to do this, then I'd say it's pretty straightforward to give assignments based on which version of the course the student is enrolled. [For example, while universities may have Honors courses vs Standard courses, some universities combine these courses into one [probably for economical reasons] and those in the &quot;Honors&quot; version of the course have extra lab hour requirements or something of the sort.] In other words, students can take the same course for 3 credits or 4 credits depending on their major. The transferability of the course grade only goes down [4 credit hour version can count towards the 3 credit hour, but not the other way around].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My two cents. Hope this helps in part at least!&lt;/p&gt;&#xA;" OwnerUserId="6898" LastActivityDate="2016-07-05T01:45:05.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11194" PostTypeId="2" ParentId="10136" CreationDate="2016-07-05T02:15:22.893" Score="3" Body="&lt;p&gt;I hope I'm not too late to offer some resources. I wrote a book, Hacking Math Class with Python, to promote the use of programming in math education in high school and beyond, and I'm also a big fan of Processing and p5.js for learning math using art, games and interactive, dynamic graphics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've posted a bunch of sketches on openprocessing.org which I did with students at my Coder School and on my own. Lots of fractals and stuff borrowed from books. Here's the portfolio:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.openprocessing.org/user/49425&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.openprocessing.org/user/49425&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of dozen tutorials on my YouTube channel, but most are in Python:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/user/1houraweekmathclass/videos&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/user/1houraweekmathclass/videos&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6920" LastEditorUserId="122" LastEditDate="2017-04-18T17:20:19.537" LastActivityDate="2017-04-18T17:20:19.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11195" PostTypeId="1" AcceptedAnswerId="11197" CreationDate="2016-07-05T04:16:02.880" Score="12" ViewCount="688" Body="&lt;p&gt;Dealing with &lt;a href=&quot;https://math.stackexchange.com/questions/1849323/how-do-we-prove-that-int-01-lnx-left1-over-lnx1-over-1-x-right&quot;&gt;a recent question&lt;/a&gt; I spotted a very nice exercise for Calc-2 students, i.e. to find the mistake in the following lines.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Lemma 1.&lt;/strong&gt; For any $n\in\mathbb{N}$, we have:  $$ \int_{0}^{1} x^n\left(1+(n+1)\log x\right)\,dx = 0. $$&#xA;   &lt;strong&gt;Lemma 2.&lt;/strong&gt; For any $x\in(0,1)$ we have: $$ \frac{1}{1-x}=\sum_{n\geq 0}x^n,\qquad \frac{\log x}{(1-x)^2}=\sum_{n\geq 0}(n+1) x^n\log(x). $$&#xA;   By Lemmas &lt;strong&gt;1&lt;/strong&gt; and &lt;strong&gt;2&lt;/strong&gt; it follows that:&#xA;   $$\begin{eqnarray*}(\text{Lemma 1})\quad\;\;\color{red}{0}&amp;amp;=&amp;amp;\int_0^1 \sum_{n\geq0} x^n\left(1+(n+1)\log x\right)\,dx\\[0.2cm](\text{Lemma 2})\qquad&amp;amp;=&amp;amp;\int_0^1 \left(\frac{1}{1-x} + \frac{\log x}{(1-x)^2}\right)\,dx\\[0.2cm](x\mapsto 1-x)\qquad&amp;amp;=&amp;amp;\int_0^1 \left(\frac{1}{x}+\frac{\log(1-x)}{x^2}\right)\,dx\\[0.2cm](\text{Taylor series of }x+\log(1-x))\qquad&amp;amp;=&amp;amp;-\int_0^1 \frac{1}{x^2} \sum_{k\geq2}\frac{x^k}k \,dx\\[0.2cm](\text{termwise integration})\qquad&amp;amp;=&amp;amp;-\sum_{k\geq 2} \frac{1}{k(k-1)}\\[0.2cm](\text{telescopic series})\qquad&amp;amp;=&amp;amp;-\sum_{m\geq 1} \left(\frac{1}{m}-\frac{1}{m+1}\right)=\color{red}{-1}.&#xA; \end{eqnarray*}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now the actual questions: &lt;em&gt;were you able to locate the fatal flaw at first sight&lt;/em&gt;?&lt;br&gt;&#xA;&lt;em&gt;Do you think it is a well-suited exercise for Calculus-2 (or Calculus-X) students&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerUserId="6921" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-07-05T16:26:13.883" Title="A very tricky pseudo-proof of $0=-1$ through series and integrals" Tags="|calculus|integration|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11196" PostTypeId="1" AcceptedAnswerId="11204" CreationDate="2016-07-05T14:48:14.287" Score="2" ViewCount="949" Body="&lt;p&gt;I hope I've picked the right group...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My daughter is signed up for Integrated Math II next year in a California school.  She's coming from a french school where the curriculum is different and is worried about being prepared.  I suspect she won't have any problems but am happy to provide her with a review of Integrated Math I if I can find something. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Does anyone have any recommendations for a review book on the topics covered in Integrated Math I?&lt;/strong&gt;  Free is best, but I could order something on Amazon if necessary.  Books or videos would work.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm afraid I don't know the topics as the curriculum was different when I was in California schools.  An advanced student entering 9th grade (like my daughter) would have taken Geometry in my day.  My understanding is the Integrated Math curriculum mixes geometry and algebra much more so than the old system.  If so, it's more like the french curriculum!  If I had to guess, I'd say that because of the french curriculum, she's comparatively advanced in geometry, and comparatively weaker in exponents, logs, and long division.  But of course, I really need to see the curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for all suggestions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dave&lt;/p&gt;&#xA;" OwnerUserId="3405" LastActivityDate="2016-07-08T18:51:58.777" Title="Seeking a review of Integrated Math I" Tags="|algebra|textbooks|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11197" PostTypeId="2" ParentId="11195" CreationDate="2016-07-05T16:26:13.883" Score="10" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;[...]&lt;em&gt;were you able to locate the fatal flaw at first sight&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Given the context I was a priori quite certain it would be an issue in interchanging limits. What else could it be? You would not make an index error or something like this. Yet,  this is a bit 'cheating' though. It is a nice problem.   &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Do you think it is a well-suited exercise for Calculus-2 (or Calculus-X) students&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would likely not give this as a regular exercise, and certainly not as an exam question, as the 'variance' is too large (it can be quick and easy, or one can spend enormous amounts of time based in part on getting lucky to start in the right way). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, as some bonus or optional exercise. Sure, as long as the students have a grasp of the individual bits and pieces, that is Taylor series etc. Maybe I would even consider it &lt;em&gt;before&lt;/em&gt; discussing interchanging integrals and limits. Then it can serve as an example that motivates the need. &#xA;In this form one might even present it in a lecture, not as exercise.   &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2016-07-05T16:26:13.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11198" PostTypeId="2" ParentId="11098" CreationDate="2016-07-05T18:28:08.107" Score="1" Body="&lt;p&gt;When you look in a mirror, you notice that you are reflected right to left, that is, the writing on your shirt is backwards in your reflection.  But the mirror doesn't reverse you from top to bottom.  How can this be?  Why does the mirror seem to care about right-left but not up-down?&lt;/p&gt;&#xA;" OwnerUserId="4906" LastActivityDate="2016-07-05T18:28:08.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11199" PostTypeId="1" CreationDate="2016-07-05T22:12:30.610" Score="9" ViewCount="446" Body="&lt;p&gt;I am teaching an introduction to number theory for high schoolers right now, and there seems to be quite a bit of confusion on what the difference between the natural numbers, the integers, the rational numbers, the real numbers, the complex numbers, and Gaussian integers right now. Is there a good and quick way to explain the difference to high schoolers who might be muddled amongst these differently defined number systems, as so far as working with them in the context of elementary number theory?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="6924" LastActivityDate="2017-05-30T15:25:13.073" Title="Explaining difference between natural numbers, integers, rationals, reals, complex numbers, Gaussian integers" Tags="|mathematical-pedagogy|secondary-education|number-theory|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11200" PostTypeId="2" ParentId="5725" CreationDate="2016-07-05T23:56:54.340" Score="6" Body="&lt;p&gt;The correct answer is that it's a &lt;em&gt;definition&lt;/em&gt;. This is clearly labelled as such in any textbook I've ever seen (e.g., Martin-Gay &lt;em&gt;Prealgebra &amp;amp; Introductory Algebra&lt;/em&gt;, Bittinger &lt;em&gt;Intermediate Algebra&lt;/em&gt;, Sullivan &lt;em&gt;Algebra &amp;amp; Trigonometry&lt;/em&gt;, Ratti &amp;amp; Mcwatters &lt;em&gt;Precalculus&lt;/em&gt;, Stein &amp;amp; Barcellos &lt;em&gt;Calculus and Analytic Geometry&lt;/em&gt;, etc., etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, consider the following article by Esther Levenson (2012). In it, Levenson uses specifically this question in interviews with experienced junior-high teachers (in this case, Israeli) to assess whether they understand the status of different kinds of mathematical statements.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Abstract&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;This paper focuses on three junior high school mathematics teachers&#xA;  and their knowledge of the nature of definitions. The mathematical&#xA;  context of exponentiation is used as a springboard for discussing two&#xA;  aspects of definitions: their corresponding domains and the&#xA;  distinction and relationships between definitions, proofs, and&#xA;  theorems. Through interviews it was shown that some teachers are not&#xA;  aware that definitions and domains are intrinsically connected and&#xA;  some teachers believe that definitions may be proved. Findings also&#xA;  indicate that knowledge of the nature of definitions may be dependent&#xA;  on the context.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Highlights&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Some junior high school teachers are not aware that $a^0 = 1$ is a definition. Instead, they believe that they may prove this equality. &lt;/li&gt;&#xA;  &lt;li&gt;Some junior high school teachers are not sensitive to the issue of the domain of a definition. &lt;/li&gt;&#xA;  &lt;li&gt;Some junior high school teachers believe that some definitions may be proved. &lt;/li&gt;&#xA;  &lt;li&gt;Junior high school teachers’ knowledge of definitions and proofs is more stable within the context of geometry than it is within the&#xA;  context of exponentiation.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Levenson, Esther. &quot;Teachers’ knowledge of the nature of definitions: The case of the zero exponent.&quot; &lt;em&gt;The Journal of Mathematical Behavior&lt;/em&gt; 31.2 (2012): 209-219. (&lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0732312312000028&quot;&gt;Link&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-07-05T23:56:54.340" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11201" PostTypeId="2" ParentId="11199" CreationDate="2016-07-06T00:11:33.907" Score="1" Body="&lt;p&gt;This is a good opportunity to emphasize the critical status of starting with unambiguous definitions as the first step of any proper mathematical presentation (*). Be sure to provide them with the definitions as stated in the textbook that you're using. Then, granted that students may not be accustomed to reading with such laser-intense care to each word in a definition, it's likely important to run a rather large series of exercises to emphasize that each word of the definitions is important, and to internalize the meaning of each type of number. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This needs to be an &quot;automatic&quot; skill for students, such that they can instantaneously recognize each type of number later on. I would recommend the timed quiz site that I developed so as to exercise exactly this topic (and others like it):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.automatic-algebra.org/setsofnumbers.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Automatic Algebra: Sets of Numbers&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, look for opportunities throughout the course later on to use these definitions and reflect on them again. For example: Perhaps we allow exponents $a^n$ where the base $a$ is real and the exponent $n$ is an integer. Review: What does that mean? Define radicals $\sqrt[n]{a}$ where the index $n$ is natural and the radicand $a$ is real. What does that mean? Define polynomials as a sum of terms $ax^n$ where the coefficient $a$ is real and the power $n$ is natural or zero. What does that mean? Run a series of introductory exercises in each case where students must identify whether expressions qualify for each definition or not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, this subject is among your best chances to highlight the importance of careful definitions, and how they will be used to define and illuminate later concepts throughout the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(*) Fine print: It should go without saying that any presentation starts with some undefined intuitive terms; in this case it's the natural numbers. But integers and rationals can then be rigorously defined, and this is a prime opportunity to practice the axiomatic thought process in a basic algebra class. Real numbers will be somewhat hand-waved at this level. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-05-30T15:25:13.073" LastActivityDate="2017-05-30T15:25:13.073" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11202" PostTypeId="2" ParentId="11199" CreationDate="2016-07-06T20:01:33.110" Score="5" Body="&lt;p&gt;I am not sure whether this really answers your question, but I could think of the following strategy of introducing these sets of numbers. It is not based on any kind of research and just based on personal experience with first year university students where I sometimes give something similar as a &quot;naive idea&quot; in addition to the proper definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The natural numbers can be introduced as the numbers which appear &quot;naturally&quot; during counting and then the &quot;definition&quot;&#xA;$$&#xA;\mathbb{N} := \{0,1,2,\ldots\}&#xA;$$&#xA;could be given. I do not think that introducing the Peano axioms or something similar would be really appropriate for high school students but it might be of interest to specially interested or gifted students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I would use the theme of &quot;extending the set of numbers so that more equations have solutions&quot;: Working with the natural numbers there are many equations which can be solved, but unfortunately the equation&#xA;$$&#xA;x+1 = 0&#xA;$$&#xA;does not have a solution. This problem can be solved by adding negative numbers which yields the integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the same steps can be made from the integers to the rational number and (slightly cheating) from the rational numbers to the real numbers and from the real numbers to the complex numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on whether the students already know the concept of limits you can avoid the cheating in the extension from the rational numbers to the real numbers, but talking about the completion is much more difficult. Of course I think that it should be mentioned that the step from the rational numbers to the real numbers is a bit different and that just &quot;adding square roots&quot; is not the right construction.&lt;/p&gt;&#xA;" OwnerUserId="6002" LastEditorUserId="6002" LastEditDate="2016-07-06T21:11:02.893" LastActivityDate="2016-07-06T21:11:02.893" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11203" PostTypeId="1" AcceptedAnswerId="11206" CreationDate="2016-07-07T17:53:43.270" Score="9" ViewCount="921" Body="&lt;p&gt;What is the definition of an equation (as a mathematical terminology)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been using this term, equation, for a long time.&#xA;I don't even remember when and where I have learned this term (Possibly, I have learned about it in my grade 4 in a different language.) &#xA;Now I am preparing for teaching Introductory Algebra to undergrads and found that the text book has a different definition for this term, equation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you share how you have learned about it and what is the definition of it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are two definitions of an equation that I found:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) From Wikipedia:  a statement of an equality containing one or more variables (this is the one I have learned.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) From the text book that I am using: a statement that two quantities are equal. (this seems to be  the one taught in elementary schools in Canada)&lt;/p&gt;&#xA;" OwnerUserId="6370" LastEditorUserId="262" LastEditDate="2016-07-07T18:38:12.123" LastActivityDate="2021-09-26T21:41:37.517" Title="Definition of the term, equation" Tags="|reference-request|concept-motivation|definitions|equal-sign|" AnswerCount="8" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11204" PostTypeId="2" ParentId="11196" CreationDate="2016-07-07T17:56:35.230" Score="2" Body="&lt;p&gt;I think &lt;strong&gt;Gerald Edgar&lt;/strong&gt; has the best idea: Contact the school (or even check their website, if they are particularly organized) to see what the textbooks are. Perhaps you can even get a hold of the 9th grade Integrated Math &lt;em&gt;syllabus&lt;/em&gt; used by the school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To guess at your question: Probably the books are published by Houghton Mifflin Harcourt; see &lt;a href=&quot;https://www.hmhco.com/country/us/california/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. You could get a textbook in advance, or look for previews online, or check to see if there are videos of math teachers going through these HMH texts on YouTube; but I would not do any of this without first contacting the school simply to ask, again, &lt;em&gt;what text do students in [whatever class] use&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Side-note: This is not an endorsement of the HMH texts, which I have never used.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You write in a comment that it is: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;All part of the &quot;common core&quot; curriculum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But Common Core is a set of Standards, and different places in the United States have created different curricula to tackle them. For example, there are quite a number of helpful resources (in my opinion) found through the $\text{engage}^\text{ny}$ site on &lt;em&gt;their&lt;/em&gt; Common Core Curriculum; see &lt;a href=&quot;https://www.engageny.org/common-core-curriculum&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; for links to both the English Language Arts (ELA) and Mathematics materials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have ever read the actual Standards (if not, &lt;a href=&quot;http://www.corestandards.org/Math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; they are for Mathematics) then you will see that they are broken down by grade before high school, and topic (or strand) thereafter. And so the New York link has Pre-K, K, and Grades 1-8; but, subsequently: Algebra, Geometry, Algebra II, and Precalculus/Advanced Topics. So, nothing that is specifically designated as &lt;em&gt;integrated&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may also be interested in the NYC course overviews; see &lt;a href=&quot;https://www.engageny.org/resource/grades-9-12-mathematics-curriculum-map-and-course-overviews&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. Again, this is all New York based although I recognize your question is about California; still, maybe it is of general interest if you have not looked through Common Core before (and maybe contrasting this with what you find in CA could be of interest). As an example of what is covered in NY, below is an excerpt of their &lt;strong&gt;curriculum map&lt;/strong&gt; from &lt;em&gt;A Story of Functions Curriculum Map and Overview 9-12&lt;/em&gt; (p. 4). Click through for a larger image.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7WoYP.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7WoYP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt; You could review the topics by directly purchasing an HMH Integrated Math 1 textbook; you could review the topics by looking through the Common Core State Standards for Mathematics to see what is being covered; or you could review the topics by going through the free materials available through the $\text{engage}^\text{ny}$ website. &lt;strong&gt;If&lt;/strong&gt; you ask the prospective school about the materials that they use &lt;strong&gt;and&lt;/strong&gt; edit your question with this additional information, &lt;strong&gt;then&lt;/strong&gt; perhaps I, or others, can give a more substantive answer about recommendations for review textbooks and resources.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-07-07T17:56:35.230" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11205" PostTypeId="2" ParentId="11203" CreationDate="2016-07-07T18:34:08.670" Score="4" Body="&lt;p&gt;Having taught math in elementary school, I can assure you the second definition is correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://www.mathopenref.com/equation.html&quot; rel=&quot;noreferrer&quot;&gt;Math Open Reference&lt;/a&gt; (and many other math sites)&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;&quot;An equation is a mathematical statement that two things are equal&quot;&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Of course this definition includes equations with variables.  It also includes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1+3 = 4 &lt;/li&gt;&#xA;&lt;li&gt;1 + 3 = 11 - 7&lt;/li&gt;&#xA;&lt;li&gt;4 = 19 - 15&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This question is a great example of why you shouldn't rely on Wikipedia.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-07-07T18:34:08.670" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11206" PostTypeId="2" ParentId="11203" CreationDate="2016-07-07T18:34:56.417" Score="6" Body="&lt;p&gt;Your question is, in some sense, purely mathematical: What &lt;em&gt;is&lt;/em&gt; &amp;quot;the&amp;quot; definition of an equation? You identify two different definitions, so I do not know if any single one of them can be claimed as &lt;em&gt;canonical&lt;/em&gt;. Instead, here are some remarks about defining 'equation' from the perspective of the teaching and learning of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;The first name that comes to my mind around a question such as this (i.e., related to &lt;em&gt;equations&lt;/em&gt; or &lt;em&gt;equality&lt;/em&gt;) is &lt;strong&gt;Carolyn Kieran&lt;/strong&gt;. For example, see:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Herscovics, N., &amp;amp; Kieran, C. (1980). Constructing meaning for the concept of equation. &lt;em&gt;The Mathematics Teacher, 73&lt;/em&gt;(8), 572-580. &lt;a href=&quot;http://www.jstor.org/stable/27962179&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;JSTOR&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is the first page:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/j4FLh.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/j4FLh.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And here is the definition suggested, though you would need to read the full article to see what is meant by, e.g., &lt;em&gt;arithmetic identity&lt;/em&gt; (although I think you could well guess it):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wYUzD.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wYUzD.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is pretty much in keeping with, e.g., the Common Core State Standard &lt;a href=&quot;http://www.corestandards.org/Math/Content/6/EE/B/5/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;6.EE.B.5&lt;/strong&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Grade 6 » Expressions &amp;amp; Equations » Reason about and solve one-variable equations and inequalities. » 5&lt;/p&gt;&#xA;&lt;p&gt;Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Probably it does not jibe so well with dictionary definitions; e.g., Merriam-Webster on &lt;strong&gt;equation&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;mathematics: a statement that two expressions are equal (such as &lt;span class=&quot;math-container&quot;&gt;$8 + 3 = 11$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$2x – 3 = 7$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think that this is a pretty common definition (asserting that two expressions are equal) although the reference above distinguishes the two Merriam-Webster examples by referring to the former as an &lt;em&gt;arithmetic identity&lt;/em&gt; and the latter as an &lt;em&gt;equation&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is also worth pointing out, as Herscovics and Kieran do later in the article, that the definition provided around hiding a number in an arithmetic identity leads only to &lt;em&gt;equations that have solutions&lt;/em&gt;. Here are the related remarks, which concludes with their proposed &amp;quot;general definition of equation&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/HQlwQ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HQlwQ.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As a closing note, again from the mathematics education literature, here is one more related Kieran reference from around that time, although this one is specifically about &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; as a symbol:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kieran, C. (1981). Concepts associated with the equality symbol. &lt;em&gt;Educational studies in Mathematics, 12&lt;/em&gt;(3), 317-326. &lt;a href=&quot;http://link.springer.com/article/10.1007/BF00311062&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Springer Link&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-07-07T18:34:56.417" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11207" PostTypeId="2" ParentId="11203" CreationDate="2016-07-07T21:37:18.493" Score="8" Body="&lt;p&gt;I'd say that an equation is a statement that two things are equal. This is slightly more general than Amy B's answer, since I don't insist that the statement be mathematical, and it's more general then the second definition in the question, since I don't insist that the two things be quantities. So for example, I consider &#xA;$$\text{Andreas Blass}=\text{the author of this answer}$$&#xA;to be an equation. (It's not important that this particular equation is true; replace &quot;Andreas Blass&quot; with &quot;Napoleon&quot; and you still have an equation --- a false equation, but an equation nevertheless.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the special case where the equation contains one or more variables (as in the first definition in the question), the context, and in particular the meaning of the variables, can affect how the equation is to be understood.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One common situation is that the variables are understood as unknowns, and the intended meaning of the equation is the problem: &quot;Find all values for the variables that make the equation true.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another common situation is that the variables are understood as universally quantified, so the intended meaning of the equation is that it's true for all values of the variables. For example, some people state the commutative law of multiplication as $xy=yx$, meaning $(\forall x)(\forall y)\,xy=yx$.  Equations with this interpretation are what I'd call identities.&lt;/p&gt;&#xA;" OwnerUserId="326" LastEditorUserId="326" LastEditDate="2016-07-08T02:18:44.673" LastActivityDate="2016-07-08T02:18:44.673" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11208" PostTypeId="2" ParentId="11203" CreationDate="2016-07-08T06:11:16.207" Score="1" Body="&lt;p&gt;I believe there are different things we use the term 'equation' to mean over the course of education, and students can get confused because it's usually never explained where the distinctions come in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a simple level, an equation is any statement that two things are equal. It is a logical statement with a true/false value that uses 'equals' as the relation, rather than, say, less-than. It's worth remembering that 'equals' means different things in different contexts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So then different things can happen. One option is that the logical statement is false, e.g. $2+1=5$. We don't use these that much in education, so students may well get confused unless it is made clear that they are being taught to evaluate rather than being taught facts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another option is that the statement is true, e.g. $2+1=3$ or $2x+x=3x$. We start teaching simpler forms of these at primary school, and since this is pretty much the only form around, 'equation' does the job, if needed at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The third option is that we have decided to assume that the equation is true by hypothesis, and wish to determine the consequences of that hypothesis, e.g. $2x+1=3$. We introduce these later as 'algebra'. Here students will see both cases: $2x+x=3x=5$. These are generally simple enough that they can distinguish between 'true' and 'true by hypothesis' without realising that there's a distinction. In this context we tend to use 'solve' to mean the action of determining the value(s) of the variable. But we also use 'solve' for questions that aren't obviously of this form, so students may not pick up the technical meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After a bit, we start to introduce other statements like $\sin^2(x)+\cos^2(x)=1$, which are of the always-true type, but are not obviously so. We call these 'identities'. These are, in a sense, equations that 'cannot be solved' (more accurately, solving them would not yield much useful information about the value of the variable). Some students will not pick up this distinction, so 'identity' becomes a word they don't understand but think they should (and therefore possibly scary).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Along the way, we also introduce 'equations' with other relations, e.g. $2x\leq 3$. Since we typically use only a limited range of relations, and may not supply an alternative term, these might also get categorised as 'equations'. After all, basically the same methods are used to 'solve' them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, we bring in questions where the solution requires a sequence of logical statements rather than simply a sequence of equal expressions. Again, students may not notice the change, and come to think of '$=$' as meaning 'the next bit of the solution'.  So 'equation' is no longer 'a statement with $=$'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So then they reach university, and start getting picked up on these distinctions. Maybe they then start to catch on. Meanwhile, they meet definitions. And after a bit one might ask 'what is the definition of 'equation'?'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps, then, the answer is partly that there isn't a single definition, because 'equation' is both a mathematical term but also a meta-mathematical term. That is, we use it to talk about doing maths, as well as about the content of maths. And education terms aren't that well suited to rigorous definitions.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-07-08T06:11:16.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11209" PostTypeId="2" ParentId="11203" CreationDate="2016-07-08T17:39:07.587" Score="1" Body="&lt;p&gt;The Wikipedia definition seems very nonstandard in my experience (and also unnecessarily complicated). The definition with which I'm familiar, and which seems much simpler, is as follows:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Equation:&lt;/strong&gt; A statement that two expressions are equal.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;We may then observe three subcategories of equations:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Identity:&lt;/strong&gt; An equation true for &lt;strong&gt;all&lt;/strong&gt; possible values of any variables.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Conditional:&lt;/strong&gt; An equation true for &lt;strong&gt;some&lt;/strong&gt; possible values of any&#xA;variables.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Inconsistent:&lt;/strong&gt; An equation true for &lt;strong&gt;no&lt;/strong&gt; possible values of&#xA;any variables.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This is common usage throughout any college-level book for algebra or precalculus that I've seen. Some examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ratti &amp;amp; McWatters, &lt;em&gt;Precalculus&lt;/em&gt;, Section 1.1.&lt;/li&gt;&#xA;&lt;li&gt;Martin-Gay, &lt;em&gt;Introductory Algebra&lt;/em&gt;, Section 2.3.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;OpenStax College Algebra&lt;/em&gt;, Section 2.2.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Moreover, it's common throughout those textbooks to take certain equations &lt;em&gt;with&lt;/em&gt; variables, and manipulate it into a form &lt;em&gt;without&lt;/em&gt; variables -- saying that the equations are &lt;em&gt;equivalent&lt;/em&gt;, and thereby concluding that both the initial and final form are &lt;em&gt;identities&lt;/em&gt;. A few examples of this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Ratti &amp;amp; McWatters, Section 1.1, Example 4 (Solution): &amp;quot;We have shown that &lt;span class=&quot;math-container&quot;&gt;$0 = 0$&lt;/span&gt; is equivalent to the original equation [with variables]... Thus, the original equation is an identity.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Martin-Gay, Introductory Algebra 3E (Sec 2.3, Example 7): Having solved an equation and produced the statement &lt;span class=&quot;math-container&quot;&gt;$−12 = −12$&lt;/span&gt;, she writes: &amp;quot;Again, the final equation contains no variables... thus [the initial equation with variables] is an &lt;strong&gt;identity&lt;/strong&gt;...&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In summary, the distinction between equations with and without variables is not followed by any college-level algebra textbook I've seen, and doing so would actually interrupt the reasoning by which one recognizes identities there.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-09-26T21:41:37.517" LastActivityDate="2021-09-26T21:41:37.517" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11210" PostTypeId="2" ParentId="11196" CreationDate="2016-07-08T18:51:58.777" Score="0" Body="&lt;p&gt;For those looking at this post in the future, I include one link to online book I found myself:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://grade8im1.blogspot.com/2014/09/textbook-online.html&quot; rel=&quot;nofollow&quot;&gt;http://grade8im1.blogspot.com/2014/09/textbook-online.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3405" LastActivityDate="2016-07-08T18:51:58.777" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11211" PostTypeId="1" CreationDate="2016-07-08T20:15:55.250" Score="12" ViewCount="1562" Body="&lt;p&gt;Among the mathematically ignorant one often finds a mistaken proposed definition of &quot;irrational number&quot;, which says that it is a number whose decimal expansion does not terminate or repeat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The definition found in all textbooks I've seen and used by mathematicians is that an irrational number is one that cannot be expressed as a ratio of two integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, one can &lt;em&gt;prove&lt;/em&gt; that a number is irrational if and only if its decimal expansion neither terminates nor repeats.  But that's lousy as &lt;em&gt;definition&lt;/em&gt;, because it doesn't say motivate the concept as well and it makes it appear that we're singling out base $10$ rather than some other base, and that takes the whole account of the matter down a path that doesn't bear a lot of fruit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The wrong definition has also in some cases led students to wonder if a number can be irrational in base $10$ and rational in some other base.  That is confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;b&gt;So my question is:&lt;/b&gt; Why does this mistaken idea persist for some many decades without ever being taught?  (Or might I be mistaken in thinking that it is never taught? (And if so, is there solid evidence that such teaching is widespread?))&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="143" LastEditDate="2016-07-08T20:33:57.000" LastActivityDate="2016-09-04T20:49:21.777" Title="Why does the widespread erroneous definition of &quot;irrational number&quot; persist without being taught?" Tags="|students-mistakes|rational-numbers|arithmetic|" AnswerCount="6" CommentCount="21" ContentLicense="CC BY-SA 3.0" />
  <row Id="11212" PostTypeId="2" ParentId="11211" CreationDate="2016-07-08T20:49:55.657" Score="18" Body="&lt;p&gt;I can think of two related reasons: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The characterization via the decimal expansion might be perceived more strongly like a property of the number: &quot;This number is irrational, because this number's decimal expansion does not terminate.&quot;  The other one is rather a non-property and thus not perceived as a definition &lt;em&gt;of some thing.&lt;/em&gt;  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The characterization via the decimal expansion seems more readily useable. The idea &quot;I look at the decimal expansion and then I 'see' if it is periodic or not.&quot; seems natural and appealing. Instead the other one might lead to a:  &quot;How in the world should I check there are no integers that give this number as a fraction?&quot;  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Let me add that I agree with OP that the definition of irrational should be a (real) number that is &lt;em&gt;not&lt;/em&gt; rational. The above answers the question why this might not stick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How rigorous a notion of real number one has in a given context seems orthogonal to me. In any case, there ought to be one available if one can talk about non-periodic decimal expansion.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2016-07-13T14:39:24.327" LastActivityDate="2016-07-13T14:39:24.327" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11213" PostTypeId="2" ParentId="10349" CreationDate="2016-07-08T21:27:51.930" Score="6" Body="&lt;p&gt;This mistake is frequently made by mathematicians in the following way.  First consider this proof:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Let $S$ be any finite set of prime numbers.  (For example, one could have $S=\{2,7,19\}$.)&lt;/li&gt;&#xA;&lt;li&gt;Prove that the prime factors of $1+\prod S$ are not members of $S$. [Details omitted.]&lt;/li&gt;&#xA;&lt;li&gt;Conclude that every finite set $S$ of prime numbers is a subset of some larger finite set of prime numbers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Euclid wrote that proof. It is not by contradiction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dirichlet in the 19th century and many eminent mathematicians since then have rearranged Euclid's proof to read as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Assume only finitely many prime numbers exist.&lt;/li&gt;&#xA;&lt;li&gt;Then insert the argument described above here, proving there are infinitely many prime numbers.&lt;/li&gt;&#xA;&lt;li&gt;This is a contradiction; hence the assumption was false.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Dirichlet and many later authors wrote that Euclid's proof was by contradiction.  That is a historical error.  Rearranging it into a proof by contradiction just makes it more complicated and serves no purpose.  It also leads to errors, in the following way.  An author writes something like &quot;The number $1+\prod S$, having no prime factors, must be prime itself.  But that contradicts the assumption [etc.].&quot; Then students think that it has been proved that if you multiply the first $n$ prime numbers and then add $1$, the result is always prime.  But that is false: $1+(2\times3\times5\times7\times11\times13) = 59\times 509$ and there are many other counterexamples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without the assumption that $S$ contains &lt;b&gt;all&lt;/b&gt; primes numbers, one would not conclude that a number not divisible by any member of $S$ is not divisible by any prime, and without that, one would not conclude that it must itself be prime.  Hence rearranging the proof into a proof by contradiction has introduced a substantial error.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One student proposed to prove infinitely many twin primes exist by saying that $\pm1+\prod S$ are both prime whenever $S$ is the set of the first $n$ primes.  (But that of course is false.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another student went to Wolfram and found numerous counterexamples and claimed that therefore Euclid's proof was wrong.  But in fact, Euclid's proof was right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Catherine Woodgold and I published a paper examining the history of this error committed by otherwise respectable mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why do &lt;b&gt;mathematicians&lt;/b&gt; &quot;read this structure into a proof even when it isn't there&quot;?&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2016-07-08T21:27:51.930" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11215" PostTypeId="2" ParentId="11211" CreationDate="2016-07-09T01:22:59.787" Score="11" Body="&lt;p&gt;Note that this definition of rational and irrational numbers is most commonly presented to high school students, who tend to have a strong and natural intuition of numbers in base $10$. At this stage, it's much easier to understand what it means for a number to be irrational in terms of its decimal expansion rather than the statement &quot;a number which cannot be expressed as $\frac mn$ where $m,n \in \mathbb{Z}$ &quot;. This more 'classical' definition raises the deeper question of what precisely it means for a number to be irrational, and, indeed, whether or not such numbers exist isn't apparent. On the other hand, it's easy to &quot;imagine&quot; numbers which have non-terminating non-repeating decimal expansions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A deeper analysis of the real number system, in which one considers the notion of a number in a more general and abstract sense than a mere decimal expression,  is generally beyond the scope of high school study, and consequently the more &quot;intuitive&quot; decimal explanation is preferred. &lt;/p&gt;&#xA;" OwnerUserId="6155" LastActivityDate="2016-07-09T01:22:59.787" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11216" PostTypeId="1" AcceptedAnswerId="11219" CreationDate="2016-07-09T03:04:27.807" Score="12" ViewCount="371" Body="&lt;p&gt;A recent publication:&lt;br&gt;&#xA;&lt;em&gt;Anita Bright.&#xA;Education for Whom? Word Problems as Carriers of Cultural Values.&#xA;Taboo: The Journal of Culture and Education.  (Spring 2016).  pp. 6-22.&lt;/em&gt; &#xA;&lt;a href=&quot;http://www.freireproject.org/download/journals/taboo/05bright.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt; No paywall.&lt;br&gt;&#xA;observes that many textbook math word problems are from &quot;middle or upper class perspectives&quot;.&#xA;The first example word problem is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;Two art students are touring Paris. They each buy a one-day museum pass for \$14.&#xA;Each student also buys a ticket to the Eiffel Tower for \$11 and a boat ticket for \$3.&#xA;How much do the two students spend altogether? Explain.&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From my reading of this paper, there is an &lt;em&gt;implicit&lt;/em&gt; claim that these word problems (with middle or upper class perspectives)&#xA;have a negative impact on the learning of some less-privileged students.&lt;br&gt;&#xA;Is there evidence for this claim ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The hard evidence could be a study similar to those in the answers to a similar &lt;a href=&quot;https://matheducators.stackexchange.com/q/10953&quot;&gt;question about gender&lt;/a&gt;.&#xA;There could be comparisons of student performance on&#xA;different variations of the same word problem, &#xA;with the same math calculations but with different characters and situations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The soft evidence could be anectodal, perhaps something like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a student is stuck on the term &quot;one-day museum pass&quot;&lt;/li&gt;&#xA;&lt;li&gt;the teacher suggests the student replace it by &quot;movie ticket&quot;&lt;/li&gt;&#xA;&lt;li&gt;the student now solves the problem&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thank you for your replies.&lt;/p&gt;&#xA;" OwnerUserId="6932" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2018-01-12T21:22:16.130" Title="request for evidence about class perspectives in math word problems" Tags="|mathematical-pedagogy|reference-request|word-problems|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11217" PostTypeId="2" ParentId="10349" CreationDate="2016-07-09T13:52:53.493" Score="2" Body="&lt;p&gt;The most obvious reason of &lt;em&gt;attraction&lt;/em&gt; towards the 'proof by contradiction' may be:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1$.A Student knows the first line of proof which is just the &lt;strong&gt;inverse of the statement to be proved&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$2$. At least he/she knows what is the last line of proof, which is just the &lt;strong&gt;inverse of point $1$ or contradiction to a known mathematical fact&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5999" LastActivityDate="2016-07-09T13:52:53.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11218" PostTypeId="2" ParentId="10349" CreationDate="2016-07-09T23:07:39.897" Score="1" Body="&lt;p&gt;I can understand the impulse to do so, because of the novelty and scandalousness of &quot;proof by contradiction&quot;. Quite exotic and unlike any sort of rhetorical devices used in ordinary life. Also, as other answers and comments have observed, given typical prior human context, an assertion about an individual entity may be easier to contemplate than a universal assertion whether positive or negative.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-07-09T23:07:39.897" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11219" PostTypeId="2" ParentId="11216" CreationDate="2016-07-10T08:07:28.083" Score="10" Body="&lt;p&gt;Yes, there is evidence for the claim; for example, consult the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Abedi, J., &amp;amp; Lord, C. (2001). &lt;em&gt;The language factor in mathematics tests. Applied Measurement in Education, 14&lt;/em&gt;(3), 219-234. &lt;a href=&quot;http://www.wou.edu/~jherold08/ED633/DO%20Abedi%20language%20factor.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt; (no paywall).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You will find in this piece that the authors, as you considered in your &quot;soft evidence&quot; proposal, modified mathematical questions (from the NAEP) and checked to see the impact on student understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are two images; I will mark some relevant components in red. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first image is of the title and abstract:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pCVti.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pCVti.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second image is of the beginning of the &lt;em&gt;Discussion&lt;/em&gt; section (PDF 13/17):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oFFvR.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oFFvR.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are additional subtleties to answering your question; for example, consider the intersection of socioeconomic status and the language spoken by students at home, and you may see that ELLs (cf. bullet-point 1 above) are disproportionately low-SES. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The suggested article may have other references of interest; as far as subsequent articles that cited this piece, &lt;a href=&quot;https://scholar.google.com/scholar?q=math&amp;amp;btnG=&amp;amp;hl=en&amp;amp;as_sdt=40000005&amp;amp;sciodt=0%2C22&amp;amp;cites=5613802947736896360&amp;amp;scipsc=1&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; is a link to google scholar's &lt;em&gt;Cited by&lt;/em&gt; (with the results cut down by searching for the term &quot;math&quot;). I cannot think of a &quot;seminal article&quot; offhand that addresses your question (although there may well be one!) but I am quite confident (for scholarly and anecdotal reasons) that the answer is, yes, privilege (as measured e.g. by some SES metric) will be significant when looking at student performance on certain types of word problems.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-07-10T08:07:28.083" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11220" PostTypeId="1" CreationDate="2016-07-11T18:30:12.920" Score="2" ViewCount="84" Body="&lt;p&gt;I would like to teach some young and talented pupils basics of competition mathematics. What kind of methods, topics and theorems should I teach to those who are good at school mathematics but have no experience on competitions? How accurately I should teach?&lt;/p&gt;&#xA;" OwnerUserId="6940" LastActivityDate="2016-07-11T18:30:12.920" Title="What should I teach 13 to 15 years old pupils to learn competition mathematics?" Tags="|teaching|" AnswerCount="0" CommentCount="4" ClosedDate="2016-07-22T10:24:47.200" ContentLicense="CC BY-SA 3.0" />
  <row Id="11221" PostTypeId="2" ParentId="10349" CreationDate="2016-07-11T20:25:48.850" Score="3" Body="&lt;p&gt;My other answer isn't quite at the heart of the topic, so I'll add this: Often (usually) students in math courses are uncertain how to proceed, so they just try whatever they've recently been told about in class.  If proof by contradiction was introduced recently, they'll use it.&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2016-07-11T20:25:48.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11222" PostTypeId="2" ParentId="11211" CreationDate="2016-07-11T20:28:45.437" Score="10" Body="&lt;p&gt;The definition of an irrational number as a &quot;number which is not rational&quot; is not without its own difficulties. It presumes that we have a clear definition of a real number.  The audience you refer to probably does not know anything about Cauchy sequences or Dedekind cuts. So the complement of &quot;rational&quot; is at best only intuitively defined for them.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-07-11T20:28:45.437" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="11223" PostTypeId="2" ParentId="11216" CreationDate="2016-07-11T21:16:25.003" Score="-2" Body="&lt;p&gt;I began teaching math when SMSG just came in. I mention this for background only (and as a clue to my age!). Before this and to this time, I am sure, word problems were always the nemesis for students. I always suggested (and practiced) this philosophy.&lt;br&gt;&#xA;&quot;Word problems are easier to solve after you turn them into math.  The problem is that you need to strip away the &quot;garbage&quot;.  As teachers you (and your students) should practice this and then you and they will not have to deal with PC (politically correct) and in the process will eliminate what may be perceived PC factors.  With:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Two art students are touring Paris. They each buy a one-day museum pass for \$14. Each student also buys a ticket to the Eiffel Tower for \$11 and a boat ticket for $3. How much do the two students spend altogether? Explain.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After reading this You might say, What is important (to the solution) about: &quot;Two art students are touring Paris.&quot;  Paris? Touring? students? 2?  etc..  continue this and perhaps you/they get to the short version:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2(14 + 11 + 3) or something like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you might ask them to write the word problem that corresponds to this:&#xA;2(6 + 17) - 3   and have them read it.&lt;/p&gt;&#xA;" OwnerUserId="6333" LastEditorUserId="262" LastEditDate="2016-07-11T22:19:43.977" LastActivityDate="2016-07-11T22:19:43.977" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11224" PostTypeId="1" CreationDate="2016-07-12T01:51:27.130" Score="20" ViewCount="1826" Body="&lt;p&gt;So I am going to be teaching the basics of group theory to high schoolers in a few weeks, and I want to hear what the Stack Exchange network has to say on the matter.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What are the applications and insights I should show them in order to excite them about the subject?&lt;/li&gt;&#xA;&lt;li&gt;What would be a good possible outline of the syllabus, including the rote aspects of group theory, and what are some possible sample problems for the non-rote aspects, i.e. this is why group theory is nifty, these are the deeper connections, these are intriguing applications, etc.?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="6943" LastActivityDate="2016-08-16T05:50:36.657" Title="Group theory for high schoolers, want the opinion of other educators" Tags="|mathematical-pedagogy|secondary-education|course-design|algebra|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11225" PostTypeId="2" ParentId="11224" CreationDate="2016-07-12T07:03:00.757" Score="4" Body="&lt;p&gt;Here are some applications that may interest students :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The three great geometrical constructions - Group theory provided impossibility proofs to the three great problems the Greeks couldn't conquer - squaring the circle, trisecting the angle and doubling the cube (all with a compass and unmarked ruler). I would recommend you give your students the task of performing these constructions, and when they can't ask them to come up with impossibility proofs. And, then tell them how a proof for these questions took many centuries and it came with the apparatus of Group Theory. Tell them about the &lt;strong&gt;lune of Hippocrates&lt;/strong&gt; and how he came close to solving the problem of squaring the circle, but didn't quite do it.&lt;/li&gt;&#xA;&lt;li&gt;Rubik's cube. That would surely impress a lot of students. Toys like this are very popular with students, and you could explain some facts about how it's a group of permutations. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6833" LastEditorUserId="6833" LastEditDate="2016-07-12T18:14:16.070" LastActivityDate="2016-07-12T18:14:16.070" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11226" PostTypeId="1" CreationDate="2016-07-12T08:42:48.767" Score="6" ViewCount="8488" Body="&lt;p&gt;I have sometimes seen a notation where $AB$ could mean either the line segment or its length. Why do the same notation can be mean both? Should one teach pupils to use for example notation $d(A,B)$ or $|AB|$ to mean the length of the line segment $AB$? I saw once in a text where $AB$ had two different meanings inside one solution.&lt;/p&gt;&#xA;" OwnerUserId="6946" LastActivityDate="2022-11-21T16:11:48.537" Title="Notation of line segment and its length" Tags="|geometry|notation|teaching|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11227" PostTypeId="2" ParentId="10555" CreationDate="2016-07-12T20:56:06.420" Score="4" Body="&lt;p&gt;When I learned about logarithm, my math teacher used a rather unconventional notation to prevent us from doing such mistakes: instead of writing down $\log_ab$ or $\log_a(b)$, he insisted on writing it as $\log(b\ base\ a)$. From my experience, this notation has both advantages and drawbacks. When it comes to advantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;There is no possible confusion or misreading.&lt;/strong&gt; It is especially not possible to confuse $\log(b\ base\ a)$ with $\log(a^b\ base\ ?)$, whereas it is indeed rather easy to misread $\log_a b$ as $\log a^b$. Generally speaking, I can't think of any possible misunderstanding of the notation itself, which was one of your biggest concerns. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;It is much easier to read out loud.&lt;/strong&gt; As a direct consequence of the &quot;no misunderstanding&quot;, there is no possibility for an out loud pronunciation of $\log(b\ base\ a)$ apart from &quot;the logarithm of b to base a&quot;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;It is impossible to mistake log as an independent mathematical object.&lt;/strong&gt; You do not raise this particular issue in your question; however it does appear in brendansullivan's answer, as well as in Benjamin Dickman's &lt;a href=&quot;https://repository.nie.edu.sg/bitstream/10497/69/1/ME-8-2-53.pdf&quot; rel=&quot;nofollow&quot;&gt;link&lt;/a&gt; (student's misconceptions about the logarithm). For example, it is impossible to cancel the $\log$ in $\frac{\log(100\ base\ 2)}{\log(10\ base\ 2)}$, since $\frac{(100\ base\ 5)}{(10\ base\ 5)}$ &lt;em&gt;cannot&lt;/em&gt; make sense.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;When it comes to drawbacks:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;It is an unofficial notation.&lt;/strong&gt; Therefore, it is highly likely that the students will have to learn the correct notation when the time comes. However, this should stand no real difficulty, given that when you have &lt;em&gt;understood&lt;/em&gt; what you're talking about, the notation isn't that much of a big deal. They may even find the correct notation shorter and more pragmatic.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;It is rather long and not pragmatic.&lt;/strong&gt; The usual notation has been designed to meet the needs of higher levels practice. It is therefore difficult to find a more efficient one. However, the usual exercises you practice on when you're learning about logarithm don't require a page-long proof or discussion, and a bit longer notation doesn't hurt that much.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To put it in a nutshell, considering the drawbacks and advantages, my advice would be to &lt;strong&gt;force&lt;/strong&gt; yourself and your students into using the $\log(b\ base\ a)$ notation at first, to slowly introduce the topic and the use of logarithm. Once you've made sure that all of your students do understand what is going on and how the logarithm is used, including the formula you've presented, you can tell them about the usual notation. My opinion is that they will adapt very quickly to the real notation, and that they'll have overall less trouble in learning about the logarithm's properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although it might not &lt;em&gt;directly&lt;/em&gt; help you, given that the textbook you're using is already written in a different way, I hope this will be useful when it comes to your student's understanding of the logarithm.&lt;/p&gt;&#xA;" OwnerDisplayName="user6938" LastActivityDate="2016-07-12T20:56:06.420" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11228" PostTypeId="2" ParentId="11224" CreationDate="2016-07-12T23:30:38.257" Score="3" Body="&lt;p&gt;Addressing your 1st question (&quot;excite them about the subject?&quot;),&#xA;I hope you can make use of Brian Hayes's delightful &#xA;&lt;em&gt;American Scientist&lt;/em&gt; article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Brian Hayes, &lt;a href=&quot;http://www.americanscientist.org/issues/pub/group-theory-in-the-bedroom&quot; rel=&quot;nofollow noreferrer&quot;&gt;Group Theory in the Bedroom&lt;/a&gt;.&#xA;  &lt;em&gt;American Scientist&lt;/em&gt;, Volume 93, Number 5, 2005, p.395ff.&#xA;  Reprinted in: &lt;a href=&quot;http://grouptheoryinthebedroom.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://grouptheoryinthebedroom.com/&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/DLnRN.jpg&quot; width=&quot;400&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;Figure 1. The mattress-flipping method recommended by a number of manufacturers and retailers promises to &quot;turn it over and end-to-end,&quot; suggesting that the maneuver rotates the mattress around two axes at the same time. In fact the algorithm has the same effect as a single end-over-end flip.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-07-12T23:30:38.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11229" PostTypeId="2" ParentId="10349" CreationDate="2016-07-14T00:55:19.160" Score="6" Body="&lt;p&gt;When I prove a theorem by contradiction, I have more information available to use in my deductions --- not only the hypotheses of the theorem but also the negation of the conclusion. Having that extra information can't hurt (except aesthetically) and might help, so I might as well assume it.  If, after I complete the proof, I see that I never used that extra information, I have (or at least I like to think that I have) enough aesthetic sense and enough respect for constructive ideas to rewrite the proof as a direct proof. But when a beginning student completes such a proof, the sense of relief at having succeeded may outweigh any aesthetic sense, and so the useless detour via contradiction will survive.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2016-07-14T00:55:19.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11230" PostTypeId="2" ParentId="10349" CreationDate="2016-07-14T14:43:40.073" Score="5" Body="&lt;p&gt;One issue is that students typically don't &lt;em&gt;polish&lt;/em&gt; successful proofs. They might start proving a proposition of the form &lt;code&gt;P =&amp;gt; Q&lt;/code&gt; by assuming that &lt;code&gt;P&lt;/code&gt; is true and &lt;code&gt;Q&lt;/code&gt; is false (perhaps because they have had success with such arguments) and then be satisfied if they reach a contradiction of the form &lt;code&gt;P and not P&lt;/code&gt; or &lt;code&gt;Q and not Q&lt;/code&gt;. Unless explicitly enjoined to do so, few students will make the effort to streamline the proof, hence it wouldn't even occur to them to revisit their argument and notice if their proof failed to use the assumption P (making it essentially a contrapositive proof) or failed to use the assumption Q (making it essentially a direct proof). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not really sure what could be done about it. Perhaps a grading policy in which correct but suboptimal proofs are not given full credit could help. Personally, I'm so overjoyed when I see a correct proof that I give it full credit and content myself with writing comments on the proof. &lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2016-07-14T14:43:40.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11231" PostTypeId="2" ParentId="11226" CreationDate="2016-07-14T15:07:24.090" Score="1" Body="&lt;p&gt;The situation you describe is common for mathematics. Take the notation $\sum_{k=1}^\infty a_k$ which has two different meanings:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the sequence of partial sums, i.e. $\sum_{k=1}^\infty a_k= \left(\sum_{k=1}^n a_k\right)_{n\in\mathbb N}$&lt;/li&gt;&#xA;&lt;li&gt;the limit of the series, i.e. $\sum_{k=1}^\infty a_k= \lim_{n\to\infty} \sum_{k=1}^n a_k$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Another example is the symbol $\subset$. Some authors use it for the subset relation and some for the proper subset relation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot answer your question, why this happens. Since there is no overall style guide in the mathematics community for notations, it happens that different authors use the same notation for different mathematical concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest the following: As an author or lecturer I would avoid notations with different meanings or notations which are used for different concepts in the literature (if this is possible). So instead of $\subset$ I would use $\subseteq$ or $\subsetneq$ since there is is no ambiguity how to interpret these symbols...&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2016-07-14T15:07:24.090" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11232" PostTypeId="2" ParentId="10162" CreationDate="2016-07-15T22:06:35.597" Score="1" Body="&lt;p&gt;&lt;strong&gt;Start with the unit circle&lt;/strong&gt; to give them an intuition for what it means (&amp;amp; see &lt;a href=&quot;https://math.stackexchange.com/a/1859834/32946&quot;&gt;this answer&lt;/a&gt; for some diagrams that might help).  Then teach students how to plot a function from input-output pairs (if they don't know that already), and put that skill to work so that they are &lt;strong&gt;deriving the graph&lt;/strong&gt; from the intuition.  This will permit a deeper understanding than presenting the graph first, and help students be able to figure out key points (like, &quot;does this one start at 0 or 1?&quot;) when they've forgotten memorized graph specifics but retained the original intuition of where the curves comes from.&lt;/p&gt;&#xA;" OwnerUserId="5861" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-07-15T22:06:35.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11233" PostTypeId="2" ParentId="11224" CreationDate="2016-07-16T02:14:30.763" Score="4" Body="&lt;p&gt;Counting problems as application of Burnside theorem usually excites high school students. Even Polya's pattern theorem can be stated without proof and applied to coloring of geometrical shapes.&lt;/p&gt;&#xA;" OwnerUserId="6956" LastActivityDate="2016-07-16T02:14:30.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11234" PostTypeId="2" ParentId="11211" CreationDate="2016-07-16T03:11:49.477" Score="8" Body="&lt;p&gt;I Googled &quot;high school algebra textbook PDF&quot; and immediately found &lt;a href=&quot;http://hs-math-honors2.ism-online.org/files/2013/08/Student-Textbook-Holt.pdf&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ZBUip.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZBUip.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here they confusingly define &quot;rational&quot; as the ratio of two integers and &quot;irrational&quot; as the decimal expansion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also there is &lt;a href=&quot;http://www.jamesbrennan.org/algebra/numbers/real_number_system.htm&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KeKus.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KeKus.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somehow defines &quot;irrational&quot; as both definitions, with no mention that they're equivalent or which one is the &quot;real&quot; definition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is &lt;a href=&quot;http://mathrev.redwoods.edu/ElemAlgText/ElementaryAlgebra.pdf&quot; rel=&quot;noreferrer&quot;&gt;another example&lt;/a&gt; on the first page of Google&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9kn2s.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9kn2s.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here they with no proof or explanation say &quot;doesn't repeat or terminate thus is irrational&quot;. This one is a little better as they formally define the &quot;correct&quot; definition. But there is no explanation of the two. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think the issue isn't that some books teach the &quot;erroneous&quot; definition it's that they don't teach any single definition at all! They never even mention that the two definitions are different, or why one might expect the two to mean the same thing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the question isn't &quot;why are they presenting definition A instead of definition B&quot;, it is &quot;&lt;strong&gt;why aren't they presenting ANY coherent definition at all??&lt;/strong&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be very interested to find &lt;strong&gt;ANY&lt;/strong&gt; high school algebra textbook that presents a single coherent definition of irrational number.&lt;/p&gt;&#xA;" OwnerDisplayName="user5108" LastEditorDisplayName="user5108" LastEditDate="2016-07-16T03:17:09.693" LastActivityDate="2016-07-16T03:17:09.693" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11236" PostTypeId="2" ParentId="11226" CreationDate="2016-07-16T15:47:25.413" Score="1" Body="&lt;p&gt;If you want to use a notation which clearly distinguishes between a line segment and its line, you can use the notation&#xA;$$&#xA;[A,B]&#xA;$$&#xA;for the segment and&#xA;$$&#xA;d(A,B)\quad\text{or}\quad \|A-B\|&#xA;$$&#xA;for its length. In my experience this notation is used in many textbooks on Linear Algebra.&lt;/p&gt;&#xA;" OwnerUserId="6002" LastActivityDate="2016-07-16T15:47:25.413" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11238" PostTypeId="1" CreationDate="2016-07-18T04:25:15.303" Score="1" ViewCount="101" Body="&lt;p&gt;$p \to q$ that means (among others)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$p$ is a sufficient condition for $q$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To show the sufficiency, I teach my study by determining the set for $p$, the set for $q$ first and comparing their cardinal numbers. If the former has lower cardinal number then $p \to q$ is a correct proportion rather than $q \to p$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;p: I am in Tokyo, q: I am in Japan. The set for $p$ just contains a single city Tokyo but the set  for $q$ contains many cities such as Tokyo, Osaka, Sapporo, etc. As the former set has small cardinal number then &quot;I am in Tokyo.&quot; is a sufficient condition for &quot;I am in Japan.&quot; or $p\to q$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;p: $x=2$, q: $x^2=4$. The set for $p$ just contains a single element 2 and the set for $q$ contains 2 elements (2 and -2). Therefore, &quot;$x=2$&quot; is a sufficient condition for &quot;$x^2=4$&quot; or $p\to q$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;h1&gt;Questions&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Now consider the following&lt;/p&gt;&#xA;&#xA;&lt;p&gt;p: I am a vegetarian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;q: I don't eat pork.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students are asked to determine the correct implication whether &quot;$p \to q$&quot; or &quot;$q \to p$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;My attempt&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the set for p is {vegetarian}&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the set for q is the set of people not eating pork = {vegetarian, Moslem, people who are allergic to pork, etc}&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As the cardinal number of p is lower than q then $p\to q$ is the correct implication.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;My student attempt&lt;/h1&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the set of p is the set meats the vegetarian don't eat = {pork, beef, fish, etc}&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the set of q is {pork}&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As the cardinal number of q is lower than p then &quot;$q \to p$&quot; is the correct implication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realize that my attempt is correct and the student's attempt is wrong. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a teacher, how should I explain their fallacy in determining the set?&lt;/p&gt;&#xA;" OwnerUserId="5696" LastActivityDate="2016-07-18T14:00:14.667" Title="Determining sets to show sufficiency of a condition?" Tags="|logic|teaching|set-theory|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11239" PostTypeId="2" ParentId="11238" CreationDate="2016-07-18T07:29:08.127" Score="3" Body="&lt;p&gt;The elements of the set describing a statement are things that &quot;make the statement true.&quot;  The statement &quot;I am in Japan&quot; is described by the set {Tokyo, Osaka, Sapporo, ...} because &quot;I am in Osaka&quot; makes &quot;I am in Japan&quot; true.  The statement &quot;$x^2=4$&quot; is described by the set {-2,2} because &quot;$x=-2$&quot; makes &quot;$x^2=4$&quot; true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The statement &quot;I don't eat pork&quot; is described by the set {vegetarian, Moslem, a person who is allergic to pork,..} because &quot;I am a Moslem&quot; makes &quot;I don't eat pork&quot; true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student claims that the statement &quot;I am a vegetarian&quot; is described by the set {pork, beef, fish, ...}.  This is incorrect:  &quot;I don't eat beef&quot; does not (necessarily) make &quot;I am a vegetarian&quot; true, because, for example, I could be allergic to beef but still not be a vegetarian.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-07-18T07:29:08.127" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11240" PostTypeId="2" ParentId="11238" CreationDate="2016-07-18T13:57:08.887" Score="6" Body="&lt;p&gt;First, although you talk a bunch about cardinality, I don't see how that makes sense, so I'm going to assume you mean that you have them determine if the set corresponding to p is a &lt;em&gt;subset&lt;/em&gt; of the set corresponding to q.  (Otherwise, in your second example, you'd also have $x=2$ implies $x^2=9$, for instance.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In formal terms, your method requires translating the sentences into predicates with a free variable and then comparing the sets defined by those predicates.  With the math example, this is easy because there is a free variable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the English examples, though, it's less obvious.  In the implication &quot;I am in Tokyo&quot;/&quot;I am in Japan&quot; there are really two potential variables: the underlying form is &quot;X is in Y&quot;.  The actual comparison you want is the one with a single free variable and two predicates: you want to compare &quot;X am in Tokyo&quot; to &quot;X am in Japan&quot;, that is, your sets are &quot;the set of people in Tokyo&quot; versus &quot;the set of people in Japan&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead you've chosen to work with the single predicate &quot;I am in Y&quot; for two different values of Y.  There's no way to make that work in pure logic: the reason it works in your example is that &quot;X am in Y&quot; is &lt;em&gt;monotone&lt;/em&gt; in the predicate Y.  Which means that it happens to work in that example, but for a completely different reason than the method you're trying to teach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your students are doing exactly the same thing with &quot;I am a vegetarian&quot; and &quot;I don't eat pork&quot;.  They follow you in parsing this as &quot;I don't eat {pork, chicken, ...}&quot; and &quot;I don't eat pork&quot;.  Again, this is a two place predicate &quot;X don't eat Y&quot;.  If you think your first example is right, you have to think their parsing, of &quot;I don't eat Y&quot; is correct.  The problem is that &quot;X don't eat Y&quot; is &lt;em&gt;antimonotone&lt;/em&gt; in Y.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you aren't going to insist that you compare on the common free variable, there's no way to make this work without getting into questions of monotonicity of predicates, which is actually (while quite interesting, and potentially accessible to children - I recently saw a Dr. Seuss themed talk on the subject by Larry Moss) rather complicated.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-07-18T13:57:08.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11241" PostTypeId="2" ParentId="11238" CreationDate="2016-07-18T14:00:14.667" Score="6" Body="&lt;p&gt;I would not recommend to teach this method since there are some downsides. Take&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$A(x) \iff x \text{ is divisible by } 2$&lt;/li&gt;&#xA;&lt;li&gt;$B(x) \iff x \text{ is divisible by } 42$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Is $A(x) \implies B(x)$ or $B(x) \implies A(x)$? Since there are infinitely many $x\in\mathbb Z$ fullfiling $A(x)$ and $B(x)$ this cannot be answered unless your students already know about the cardinality of infinite sets. Actually the cardinality of $\{x\in \mathbb Z: A(x)\}$ and $\{x\in \mathbb Z: B(x)\}$ is the same. Does this mean, that $A(x) \iff B(x)$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is another example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$C(x) \iff x = 23$&lt;/li&gt;&#xA;&lt;li&gt;$D(x) \iff x = 42 \text{ or } x = 102$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are two objects fulfilling $D(x)$ and one object for which $C(x)$ is true. So we have $C(x) \implies D(x)$, right?!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The set-theoretic counterpart of the implication is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Subset&quot;&gt;inclusion&lt;/a&gt;. You have $A(x)\implies B(x)$ whenever $\{x:A(x)\} \subseteq \{x:B(x)\}$. However, the fact that the cardinality of $\{x:A(x)\}$ is less or equal to the cardinality of $\{x:B(x)\}$ does not imply $\{x:A(x)\} \subseteq \{x:B(x)\}$. That's the reason why your method does not work in the above examples.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2016-07-18T14:00:14.667" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11242" PostTypeId="2" ParentId="11211" CreationDate="2016-07-19T07:48:30.397" Score="3" Body="&lt;p&gt;Many good arguments have been presented. I would like to add that working with the decimal expansion does require much less understanding of what a real number is. The decimal expansion gives you rather touchable things to work with so that even without being able to see a &quot;sequence of digits&quot; as a number (which would allow arithmetic operations), pupils can participate in evaluating whether this sequence should represent an irrational number. In Zachary's post, you see that shortly after introducing irrationality, you can discuss the irrationality of 0.424224222422224...&#xA;Proofs of irrationality based on the a/b-definition are much more complex, I think. Since being operational is always a good criterion for a definition, this definition may not be so bad in the context of school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, although from a more conceptually point of view, the a/b-definition is much more appropriate, for teaching in schools with a mix of gifted and not-so-gifted students, the alternative definition might better fit the purpose of enabling the whole class to participate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess, some might find it pointless to present a concept that allows students to participate withouth thorugh understanding of the foundations. This, however, seems to be standard in schools around the globe and is a discussion in its own right...&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2016-07-19T07:48:30.397" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="11244" PostTypeId="1" AcceptedAnswerId="11245" CreationDate="2016-07-20T09:52:09.837" Score="12" ViewCount="203" Body="&lt;p&gt;In order to evaluate new educational material the contentment of students with this material is often measured. However, just because a student is contented doesn't mean that he/she has actually learned something. Is there any research investigating the correlation between students contentment and the educational quality of the presented material?&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="5097" LastEditDate="2016-07-21T11:40:12.007" LastActivityDate="2016-07-22T04:37:47.677" Title="What is the correlation between students' contentment and educational quality?" Tags="|reference-request|general-pedagogy|education-research|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11245" PostTypeId="2" ParentId="11244" CreationDate="2016-07-20T16:40:00.237" Score="13" Body="&lt;p&gt;From Clark, Richard, Paul A. Kirschner, and John Sweller. &quot;Putting students on the path to learning: The case for fully guided instruction.&quot; (2012):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Even more disturbing is evidence that when learners are asked to&#xA;  select between a more-guided or less-guided version of the same&#xA;  course, less-skilled learners who choose the less-guided approach tend&#xA;  to like it even though they learn less from it... Similarly,&#xA;  more-skilled learners who choose the more-guided version of a course&#xA;  tend to like it even though they too have selected the environment in&#xA;  which they learn less.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Reference cites Clark, Richard E. &quot;Antagonism between achievement and enjoyment in ATI studies.&quot; &lt;em&gt;Educational Psychologist&lt;/em&gt; 17.2 (1982): 92-101.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-07-20T16:40:00.237" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11246" PostTypeId="2" ParentId="11244" CreationDate="2016-07-20T16:50:11.907" Score="4" Body="&lt;p&gt;I would recommend you to specify the term &quot;educational quality&quot;. I think there is a study indicating that in German schools, the two educational goal variables of students' motivation and students' mathematical knowledge are negatively correlated across clases, so you have a trade-off there. I guess, students' contentment would be positively correlated with their motivation. Correlations are not transitive, however, so I cannot say anything about contentment and mathematical learning. You might also benefit from specifying your target group. The correlation you asked for my differ across groups of students (school, university, vocational education, pensioners).&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2016-07-20T16:50:11.907" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11247" PostTypeId="2" ParentId="11244" CreationDate="2016-07-22T04:37:47.677" Score="1" Body="&lt;p&gt;Vast numbers of students are trained from early childhood to engage in a ruthless competition for grades.  When such students are required to take math courses, if they're not interested in understanding math, the course degenerates into one in which getting good grades depends only on believing and obeying and working hard, rather than on understanding.  Such students learn that mathematics consists of meaningless algorithms.  That is a lie.  They are &lt;em&gt;content&lt;/em&gt; when they excel in the competition for grades.  That differs from understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what is the correlation?  Has a study been done?  I wonder if there is anyone with expertise either in education or in mathematics who is competent to conduct such a study.&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2016-07-22T04:37:47.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11248" PostTypeId="2" ParentId="11203" CreationDate="2016-07-22T04:49:34.637" Score="1" Body="&lt;p&gt;&quot;Equation&quot; and &quot;solve&quot; are among the words most frequently thrown about promiscuously by people without understanding.  They write things like &quot;How can I solve this equation: $\sum_{n=0}^\infty r^n$?&quot; (which is not an equation at all, and which admits &lt;em&gt;evaluation&lt;/em&gt; rather than &lt;em&gt;solution&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An &lt;b&gt;equation&lt;/b&gt; is an equality that has some solutions but of which not every number or function or other appropriate object is a solution.  Thus $x^2 - 10x + 24 = 0$ is an equation of which $4$ and $6$ are solutions.  And &lt;b&gt;identity&lt;/b&gt; is an equality that is true of everything (every number or every function or every other appropriate object).  For example, $\log(ab) = \log a + \log b$ or $\cos(a+b)=\cos a\sin b + \sin a \cos b$ or $(x+1)^2 = x^2+2x+1$ are identities.  Both equations and identities are examples of &lt;b&gt;equalities&lt;/b&gt;, which are statements that one thing equals another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(In the field of typography, the word &lt;em&gt;equation&lt;/em&gt; often means what illiterate laypersons think it means: anything written in mathematical notation.)&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2016-07-22T04:49:34.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11249" PostTypeId="2" ParentId="11216" CreationDate="2016-07-24T14:41:21.460" Score="6" Body="&lt;p&gt;I only have anecdotal evidence but since you said it was welcome, I am adding my 2 cents from both the perspective of a teacher and an item writer, hired to write questions for standardized tests.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I taught math to gifted elementary students. A small fraction of them were gifted in math and had reading levels below grade level. Students with poor reading levels often have poor vocabularies and have a harder time with unfamiliar words.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These students felt safe asking me about the meaning and context of all sorts of words that made no sense to them even on tests.  Before I instituted this policy, these students did very poorly in word problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An older student or adult might be able to solve a problem without knowing all the words, but younger students get stuck on word meanings and can't seem to move on the problem until they can read it through and know the meaning of all the words.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I noticed in class that students were often stuck on multi-cultural names and words, such as unfamiliar foods. Names were a particular problem if there was a girl or a boy, because the student wouldn't know which was he or she.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I wrote questions for standardized tests, I was given instructions about wording from different companies. I was told to be sensitive to gender and that no one in my problems could have a gender; telephone repairman became telephone technicians and instead of Emma and Jake, there were just two students. I was told to be sensitive to students who were poor; brunch became lunch because poor students didn't have brunch. I was told to use very simple sentence structure and language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I wrote questions for my own students, I used their names and hobbies. I found this delighted them and piqued their interest.  If a student needed a makeup assignment, I wrote 6 or 7 problems with just that student's name, which was always a hit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my experience language and context in word problems can motivate and delight students or frustrate them and put obstacles in their path.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-07-24T14:41:21.460" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11250" PostTypeId="1" CreationDate="2016-07-25T00:23:46.070" Score="14" ViewCount="1990" Body="&lt;p&gt;I can see arguments both for and against classifying constant functions as linear functions.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Against&lt;/strong&gt;:  &quot;Linear function&quot; means &quot;first-degree polynomial function&quot;, and constant functions are not first-degree.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;For&lt;/strong&gt;:  &quot;Linear function&quot; means &quot;polynomial function of degree &lt;em&gt;at most 1&lt;/em&gt;, and constant functions &lt;em&gt;do&lt;/em&gt; match that definition.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Against&lt;/strong&gt;:  But we don't define &quot;quadratic function&quot; to mean &quot;polynomial function of degree at most 2&quot;; if we did, then linear functions (under whatever definition) would also be quadratic, and nobody uses the word &quot;quadratic&quot; that way.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;For&lt;/strong&gt;:  Nevertheless, the graph of a constant function is a line, so it's a linear function.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Against&lt;/strong&gt;:  But the graph of $x=c$ is also a line, and that's not a function at all, let alone a linear one!&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;For&lt;/strong&gt;:  The notion of &quot;slope&quot; is well-defined for constant functions, and the slope just happens to be $0$.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Against&lt;/strong&gt;:  See above re: quadratic functions.  Writing a constant function in the form $y=mx + b$ with $m=0$ is analogous to writing a linear function in the form $y=ax^2 + bx + c$ with $a=0$.  Yes, you &lt;em&gt;can&lt;/em&gt; do that, but we generally &lt;em&gt;don't&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To some extent I can see this as just a matter of taste or convention.  Does anybody see any compelling arguments one way or another that I have overlooked?  Is there a &quot;best&quot; approach?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Background:  I recently put the following True/False question on an exam:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If a function is linear, then it is invertible.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Whether this is true or false depends on whether you regard constant functions as &quot;linear&quot; or not.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-07-25T23:42:52.460" Title="For purposes of teaching, should constant functions be considered &quot;linear functions&quot;?" Tags="|definitions|functions|precalculus|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="11251" PostTypeId="2" ParentId="11250" CreationDate="2016-07-25T01:03:56.530" Score="12" Body="&lt;p&gt;While I haven't done a systematic survey, my impression is that the overwhelming majority of pre-calculus and calculus texts define a linear function to be one of the form $f(x) = mx + b$ with no stipulation that $m \neq 0$. Thus, if you define linear to be a polynomial of degree 1 you are likely to be contradicting whatever textbook you are using.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a more abstract point of view, if you define linear functions as polynomials of degree 1 then you lose the nice correspondence between linear functions and affine self-maps of $R$. You also lose closure under addition and scalar multiplication, and thus a nice example of a vector space over $R$. A little less abstractly, you also lose the fact that the antiderivative of a constant function is always linear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For these reasons (as well as the other reasons you enumerated) I would go with defining linear functions in such a way that it allows for constant functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said all that, I think that your True/False question is a bit too subtle for a lower-level class (unless you had really emphasized the point).&lt;/p&gt;&#xA;" OwnerUserId="6891" LastEditorUserId="6891" LastEditDate="2016-07-25T01:15:38.473" LastActivityDate="2016-07-25T01:15:38.473" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="11252" PostTypeId="2" ParentId="11250" CreationDate="2016-07-25T08:04:21.080" Score="15" Body="&lt;p&gt;A linear function is not necessarily a first degree polynomial function: zero function is also linear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In France the terminology is more appropriate than the traditional English one: a &lt;em&gt;linear function&lt;/em&gt; is a function of the form $f(x) = ax$, while a function of the form $f(x) = ax + b$ is called an &lt;em&gt;affine function&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, strictly speaking, constant functions are affine functions, but in the traditional English terminology they are also linear functions (though they do not preserve linear combinations).&lt;/p&gt;&#xA;" OwnerUserId="6987" LastActivityDate="2016-07-25T08:04:21.080" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11253" PostTypeId="2" ParentId="11250" CreationDate="2016-07-25T09:54:21.923" Score="4" Body="&lt;p&gt;A &quot;linear &lt;strong&gt;function&lt;/strong&gt;&quot; is a function satisfying both&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f(x + y) = f(x) + f(y)$&#xA;and&#xA;$f(a x) = a f(x)$ for all $a$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a useful name, because it is a &quot;function&quot; that satisfies all the requirements of being &quot;linear&quot; (namely the ones given above). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your example of the form&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f(x) = a x + b$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is usually called &quot;linear &lt;strong&gt;equation&lt;/strong&gt;&quot; instead. Notice that for non-zero values of $b$, a linear equation does &lt;em&gt;not&lt;/em&gt; represent a linear function, but an &quot;affine function&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personaly, I think that mathematics requires careful and strict use of labeling and naming for mathematical objects. Arguments of the form&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Linear function&quot; means &quot;first-degree polynomial function&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;are not helpful because you do not gain anything from it. There already are clear and precise names for the objects you are talking about (&quot;first-degree polynomial function&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could think of &quot;Linear function means first-degree polynomial function&quot; as a mathematical insight like:&#xA;&quot;Any linear function can be expressed as a first-degree polyonomial function&quot;, but that would not help you in deciding whether &quot;A constant function is a linear function&quot; is true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact that you list &quot;pros&quot; and &quot;cons&quot; for deciding whether a mathematical object has a certain mathematical property indicates that you are missing strict definitions. Whatever you decide &quot;linear function&quot; means should give you a clear way of deciding whether or not an object satisfies that definition.&lt;/p&gt;&#xA;" OwnerUserId="6989" LastActivityDate="2016-07-25T09:54:21.923" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11254" PostTypeId="2" ParentId="11250" CreationDate="2016-07-25T18:16:39.867" Score="2" Body="&lt;p&gt;The &quot;problem&quot; here is that there are two similar-but-distinct uses. It's more than the uses distinguished by the French usage noted by @Alexey. I think (at least) the English contemporary use of &quot;linear&quot; (or &quot;quadratic&quot;, and others) is genuinely ambiguous, as it may refer to both polynomials, or to functions. Yes, polynomials also give functions, and we teach beginners to effectively identify the two, also.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to polynomials, it would be unfortunate if the sum of two linear polynomials (or any linear combination, ahem) were not guaranteed to be linear, again. Or any linear combination of degree $n$ (-or-less, ahem) polynomials again considered to be of that degree. That is, shouldn't these form vector spaces?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, &quot;linear functions&quot; should be ... um... linear maps between vector spaces, probably. (Here we might also have &lt;em&gt;affine&lt;/em&gt; as a possible legitimate modifier.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Except that, yes, with the identification of polynomials with functions, $x\to ax+b$ is &quot;linear&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;In real life&quot;, one might feel the need to set the context by adding modifiers like &quot;homogeneous&quot; (of polynomials) or &quot;non-zero homogeneous&quot; (of polynomials). Linear maps (between two fixed vector spaces) can be added, but also &lt;em&gt;composed&lt;/em&gt; ($X\to Y\to Z$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, yes, also, as Benoit Kloekner commented, the question of whether &quot;a linear (map/function? polynomial?) is invertible&quot; probably has no well-defined answer without clarifying the context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: that is, to be clear, in English especially, there is genuine ambiguity in usage. Thus, to base a question (to absolute novices) on the pretense of somehow dismissing this ambiguity is not so good, not useful, not constructive. That is, the quiz-question in the question above is not reasonably answerable as &quot;yes&quot; or &quot;no&quot;... I claim.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-07-25T18:16:39.867" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11255" PostTypeId="2" ParentId="11250" CreationDate="2016-07-25T23:42:52.460" Score="0" Body="&lt;p&gt;For a lateral answer... the term &quot;linear &lt;em&gt;polynomial&lt;/em&gt;&quot; unambiguously excludes the constant polynomials.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2016-07-25T23:42:52.460" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11256" PostTypeId="2" ParentId="11124" CreationDate="2016-07-26T05:28:18.930" Score="2" Body="&lt;p&gt;Just an idea: If your students know some statistics, including least squares regression, you could introduce optimization via some machine learnings methods, such as the lasso.  The lasso can be solved as a quadratic programming problem, but simpler methods like coordinate descent work perfectly well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One book where this is detailed is &quot;Statistical Learning with Sparsity&#xA;The Lasso and Generalizations&quot;  by Trevor Hastie, Robert Tibshirani and Martin Wainwright (CRC Press).&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="122" LastEditDate="2016-07-27T00:08:31.570" LastActivityDate="2016-07-27T00:08:31.570" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11258" PostTypeId="1" AcceptedAnswerId="11271" CreationDate="2016-07-27T23:28:01.237" Score="4" ViewCount="144" Body="&lt;p&gt;My 13 year old son is scheduled to take Algebra I next year, and this amounts to skipping one year of math. To prepare for that he has been leaning the material that he is skipping over the summer. I would like you to settle a debate my son is having with me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you recommend that people sit at a desk or table and sit on a chair designed for a desk or table when using a laptop computer to study or take a test? Would it be just as good if they have the laptop on their lap while sitting in a rocking chair?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, would listening to music while studying or taking tests cause them to not learn the subject as well? Please let me know if you are a teacher, because I am looking for replies from people with experience in education.&lt;/p&gt;&#xA;" OwnerUserId="7001" LastActivityDate="2016-07-29T01:56:44.237" Title="Recommended setting for studying" Tags="|testing|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11259" PostTypeId="2" ParentId="4409" CreationDate="2016-07-28T04:38:26.050" Score="1" Body="&lt;p&gt;Some of the previous answers are of the nice-to-us-teachers-but-irrelevant-to-students type. I favor more concrete tidbits:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In calculus it makes sense to use exponent notation for radicals and reciprocals, because derivative rules are much more straightforward. Indeed, $(x^{1/2})'$ and $(x^{-1})'$ are concrete cases of the power rule. Compare this to learning yet one more formula for $\sqrt{x}'$. Students readily assimilate this &quot;trick&quot; when I explain that the radical sign is (truly is!) a stylized version of the letter &lt;strong&gt;r&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2016-07-28T04:38:26.050" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11260" PostTypeId="2" ParentId="10494" CreationDate="2016-07-28T04:48:44.493" Score="3" Body="&lt;p&gt;The Vandermonde binomial identity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Wikipedia page gives three proofs; an algebraic one (symbol pushing), a combinatorial one (counting in two different ways), and a geometric one (where a certain family of paths is decomposed recursively). All three proofs are short, simple, and insightful, and they complement each other.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2016-07-28T04:48:44.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11261" PostTypeId="2" ParentId="4409" CreationDate="2016-07-28T07:08:19.680" Score="3" Body="&lt;p&gt;I think a more intriguing idea is to understand why more popular textbooks like the canonical Stewart textbook present materials paying no attention to historical development.  In doing so, students are exposed to mathematics as a completely abstract discipline that exists outside of human beings.  While some work has been done to correct this overtly White European reconstruction of history, much more needs to be done.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historical tidbits is perhaps not the only way to accomplish this, but instead maybe we can communicate the messy development of the discipline that has taken place over thousands of years all over the world.  Toeplitzs' genetic approach is a starting point, but so are Hairer and Wanner's more recent text on analysis, Edwards text &lt;em&gt;The Historical Development of the Calculus&lt;/em&gt;, and in particular David Perkins &lt;em&gt;Calculus and Its Origins&lt;/em&gt; from the MAA is an inspirational textbook that I continue to use with my undergraduate Calculus classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With many mathematics educators discussing issues of race, gender, and equity, it is important to understand that people other than white europeans have accomplished enormous feats relevant to the calculus curriculum.  Two examples might be ibn Al Haytham and his computation of solids of revolution and Jyesthadeva and his approximation of pi through the use of an alternating series.  From a more philosophical standpoint, see Serres &lt;em&gt;The Birth of Physics&lt;/em&gt;, though this is pretty white as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More than anything, we should continue to see the curriculum as a work in progress rather than a static set truth as presented in its best image through the work of those like Stewart and Larson whose texts cost hundreds of dollars and are inaccessible to many.  Reorganize these ideas from what you believe is a more impassioned presentation of mathematics and share with all! &lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2016-07-28T07:08:19.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11262" PostTypeId="1" CreationDate="2016-07-28T07:17:23.347" Score="5" ViewCount="482" Body="&lt;p&gt;Constructing triangles (or other shapes) seems to be quite an obsolete topic, and yet, they feature in almost every high school math competition, to disappear completely in college. In recent years in the IMO, there have also been combinatorial geometric problems, like &lt;a href=&quot;https://static01.nyt.com/packages/other/crossword/puzzleblogimages/IMO_2016_Questions.pdf&quot;&gt;problem 2016/6&lt;/a&gt;. In my opinion these require deeper proofs that can be useful later as well, so why not teach combinatorial geometry in high school instead of constructions?&lt;/p&gt;&#xA;" OwnerUserId="7002" LastEditorUserId="1905" LastEditDate="2016-07-28T10:47:13.747" LastActivityDate="2016-08-09T17:51:11.257" Title="Should the construction of triangles be taught?" Tags="|secondary-education|geometry|curriculum|constructions|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11263" PostTypeId="1" CreationDate="2016-07-28T09:32:28.353" Score="2" ViewCount="142" Body="&lt;p&gt;This is a follow up of my &lt;a href=&quot;https://math.stackexchange.com/questions/1865440/proving-convergence-or-divergence-of-series-tips-and-tricks&quot;&gt;question on MSE&lt;/a&gt;. Which tips and recommendations would you give students who want to investigate series about convergence or divergence? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far we have collected:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It is good to &lt;a href=&quot;https://math.stackexchange.com/a/1865452/32951&quot;&gt;start with the term test&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;For Series of the form $\sum_{k=1}^\infty a_k$ &lt;a href=&quot;https://math.stackexchange.com/a/1865463/32951&quot;&gt;the root test is often useful&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/1867233/32951&quot;&gt;$\sum_{k=1}^\infty \frac{1}{k^p}$ converges for $p&amp;gt;1$ and diverges for $p\le1$&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/1865566/32951&quot;&gt;Krummer's test is also useful&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Which recommondations would you add to the above list?&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-08-18T10:08:16.427" Title="Proving convergence or divergence of series: tips and recommendations" Tags="|undergraduate-education|calculus|examples|mathematical-analysis|big-list|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11264" PostTypeId="2" ParentId="10494" CreationDate="2016-07-28T11:45:56.977" Score="0" Body="&lt;p&gt;Euclid's proof that there are infinitely many primes is typically presented as a proof by contradiction but sometimes as a positive proof which provides a recipe for constructing primes not in a given set of primes (which yields the infinitude as an immediate corollary). The argument is essentially the same in both cases, but the two ways of presenting the proof have very different flavors.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2016-07-28T11:45:56.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11265" PostTypeId="1" CreationDate="2016-07-28T12:21:05.840" Score="12" ViewCount="11530" Body="&lt;p&gt;What is a good answer to the question: Why should one study &lt;a href=&quot;https://en.wikipedia.org/wiki/Ordinary_differential_equation&quot;&gt;ordinary differential equations&lt;/a&gt;?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I would give the answer: ODEs are used in many models to determine how the state of this model is changing (regarding time or another variable). Thus, ODEs are important for many scientific fields because they arise whenever a relation is given for the change of a model/system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is one application of ODEs (though a very important one). Am I missing another application or are there other reasons for studying ODEs? &lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="376" LastEditDate="2016-08-12T15:16:08.300" LastActivityDate="2019-04-10T12:46:44.480" Title="Why do we study ordinary differential equations?" Tags="|undergraduate-education|calculus|concept-motivation|mathematical-analysis|differential-equations|" AnswerCount="9" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11266" PostTypeId="2" ParentId="11265" CreationDate="2016-07-28T13:57:49.730" Score="10" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;ODEs are used in many models to determine how the state of this model is changing (regarding time or another variable).&lt;/p&gt;&#xA;&lt;p&gt;[…]&lt;/p&gt;&#xA;&lt;p&gt;Am I missing another application […]?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This may be somewhat pedantic, but I think that your answer is so generic that it covers every possible application &lt;strong&gt;per construction&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The fundamental formulation of an ODE has the change of the state as the unknown component (except if you flip it and want to seek an ODE that has a given solution) – so every application has to be about this.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Every application of mathematics is inevitably based on models (i.e., mathematical theories of reality). Of course, some of these models may be so universally accepted or have become their own fields that we are not used to treating them as such anymore.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, apart from this, there can only be inner-mathematical applications, such as being a background for more complex differential equations (PDEs, DDEs, …). As those are probably not the kind of motivation that appeals most to your students, I suggest to address the topic of ODE models in more details:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Give some specific examples from different fields such as equations of motion (physics), predator–prey models (ecology), neuron models (biology/medicine), and so on.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mention the field of dynamical systems, large parts of which are about understanding and classifying the solutions to ODEs.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mention that physicists have developed an entire arsenal of formalisms aimed at obtaining (not solving) ODEs.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="75" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-07-28T13:57:49.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11267" PostTypeId="2" ParentId="11265" CreationDate="2016-07-28T14:33:20.140" Score="6" Body="&lt;p&gt;Mention Newton's Second Law. $F = ma$ &lt;em&gt;is&lt;/em&gt; an ODE (a trivial one if $F$ is constant, but nontrivial when $F$ or $m$ depend on position or velocity). Since one of the fundamental laws of nature is a differential equation, it makes sense that you should study differential equations if you want to understand nature. I don't know who said it first but I've always liked the quote &quot;The laws of nature are expressed as differential equations&quot;&lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2016-07-28T14:33:20.140" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11268" PostTypeId="2" ParentId="11258" CreationDate="2016-07-28T15:21:29.390" Score="2" Body="&lt;p&gt;You'll find plenty of research mathematicians curled up with their laptop in their lap (although not so many in rocking chairs, I think). &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Erd%C5%91s&quot; rel=&quot;nofollow&quot;&gt;Paul Erdos&lt;/a&gt; famously asserted that drugs helped his maths (and he was pretty good at it). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing maths while listening to music might not be as productive as doing maths not listening to music, but it's more productive than not doing maths at all.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-07-28T15:21:29.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11269" PostTypeId="2" ParentId="11265" CreationDate="2016-07-28T16:47:58.083" Score="5" Body="&lt;p&gt;We should study Ordinary Differential Equations because it is beautiful mathematics which clearly illustrates the wondrous connection between analysis and algebra. Linear algebra, or perhaps matrix theory, when combined with calculus provides abstractions of ordinary functions which behave in ways similar yet fantastically different than ordinary functions. The matrix exponential in particular, we find simply from combining natural questions in calculus with those which arise from structure theory in linear algebra. Working out the modification for the laws of exponentiation leads straight to the Baker-Campbell-Hausdorff relation and hence the rudiments of Lie Algebras and Lie Groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, there are applications. But, ODEs is a natural topic which is interesting on its own.  &lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2016-07-28T19:37:27.590" LastActivityDate="2016-07-28T19:37:27.590" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11270" PostTypeId="2" ParentId="11258" CreationDate="2016-07-28T20:35:09.793" Score="4" Body="&lt;p&gt;Speaking as someone who didn't start off on the right foot with math in high school or even college, but who went on to successfully complete a PhD in mathematics and then taught at university for while, I would ask you to consider what's at stake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's easy to get caught up in short term goals and a doom and gloom story-line about how not getting into algebra can mean missed opportunities with AP courses in high school, perhaps that might mean not getting into the best college, etc.  But that's all speculation.  And, I'm not sure that that's the best lesson for your son to be learning now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If skipping a year of math and studying over the summer is his idea, then I would suggest a bit of a hands-off approach.  Perhaps sit down and work out what he wants to achieve.  Work out goals.  A goal of skipping a year of math is not the best goal.  Learning to be self motivated, how to manage time, how to approach learning a subject outside of the structure of the classroom,... these are goals that translate into skills that will help him move forward---not just in mathematics, but in other areas of study too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the scenario where he studies as you believe he should.  And let's say he does a pretty good job of it in what remains of the summer.  He's still likely to struggle in Algebra I.  Is that because he lacks background and is ill-prepared?  Or, is that because algebra is new and novel and challenging?  Who knows!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say let him study as he thinks works best for him (even if it makes you cringe or question whether he's getting anything out of it).  Have him work out goals, and (very importantly) articulate those goals he wants reach by the end of the summer.  Just saying, &quot;I want to spend an hour every day working on mathematics&quot; is a big deal.  And if he can come close to achieving, I believe it will pay off in the long run.  How much math he actually learned is irrelevant.  He will struggle one of these days in mathematics:  it might be in Algebra I, it might not be until he takes a 400 level college course in real analysis.  But the investment of starting young and learning to work toward a goal will be invaluable.  He will know that, when he puts his mind to it, he can work &lt;strong&gt;independently&lt;/strong&gt; and achieve something.  And the key here is the &lt;em&gt;independence&lt;/em&gt;.  Let him discover what he can do, and encourage him as you see him succeed (even the success seems to have little to do with gaining mathematical knowledge).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's say he only manages to put in a half hour every week until school begins, this can still be viewed as a success.  When he returns to school and he struggles in algebra I, &lt;strong&gt;don't&lt;/strong&gt; say, &quot;now if only you had put in a little more effort this summer then this wouldn't be so bad.&quot;  You don't know that.  &lt;strong&gt;Instead&lt;/strong&gt; say, &quot;hey remember how you set those goals this past summer and you put your mind to it and studied on your own?  You can do that now too.&quot;  Let him be the one who says, &quot;but I didn't do much, I only put in a half hour each week.&quot;  Just respond, &quot;yes, that's true.  But still, you still managed to put something in.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Learning that he can motivate himself enough to study on his own, even if not to the full extent hoped for, will help him as he moves forward and must face far greater challenges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember, he's only 13 years old.  He still discovering who he is and what he is capable of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I may not have settled your debate (or maybe I have), but I would say your debate isn't all that important.  Let him learn how best he learns.&lt;/p&gt;&#xA;" OwnerUserId="3491" LastActivityDate="2016-07-28T20:35:09.793" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11271" PostTypeId="2" ParentId="11258" CreationDate="2016-07-29T01:56:44.237" Score="3" Body="&lt;p&gt;(I'd like to preface this answer by saying that I'm not sure this is an argument that is either winnable against a 13-year-old or even worthwhile having. As others have already stated in comments and answers, the fact that he's motivated enough to work on mathematics on his own during the summer, is probably such a commendable venture for a student of this age, that you let him do so in whatever position he feels most comfortable.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I digress:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Little research concerning physical ergonomics of notebook computer use has been conducted, so recommendations for use are currently limited and not strongly supported by objective evidence. (Found in the Abstract of “Effects of notebook computer configuration and task on user biomechanics, productivity, and comfort,” International Journal of Industrial Ergonomics 30 (2002) pp 7-31)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that this article is already over a decade old, but a cursory search has yet to surface more relevant research. I'd be at least a little interested to know if there has been more research on the topic in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As it pertains to music listening:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Some studies supporting this technique have shown that background music promotes cognitive performance while other studies have shown that listening to music while engaged in complex cognitive tasks can  impair performance. (Found in the Abstract of &lt;a href=&quot;http://www.inquiriesjournal.com/articles/762/the-impact-of-listening-to-music-on-cognitive-performance&quot; rel=&quot;nofollow&quot;&gt;&quot;The Impact of Listening to Music on Cognitive Performance&quot;&lt;/a&gt; Inquiries Journal, 2013, VOL. 5 NO. 09)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The jury seems to still be out on this as well. Neither of these refers specifically to mathematics study, but I have a hunch that few, if any, such specific studies exist. (Though I will gladly eat crow if someone can prove me wrong.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own opinion as an educator and former (reasonably successful) student, is to find whatever comfortable position works best for the individual. I used to do my homework in High School while sitting on the floor and spreading out on the coffee table. Others might prefer laying down, working at a desk, or in a rocking chair. I don't think I'd force anyone to alter a style they were comfortable with if they were already showing success. Offering up suggestions to add to or alter his style might be the best you can hope for.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2016-07-29T01:56:44.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11272" PostTypeId="1" CreationDate="2016-07-29T11:28:14.017" Score="10" ViewCount="1105" Body="&lt;p&gt;I want to know if there is any research on whether students' learn better by answering selected response (multiple choice) questions than constructed response (written) questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My gut feeling is no. Well-structured written questions bring out much more reflection and learning than multiple choice. There are written questions that cannot be converted to decent multiple choice questions, such as proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I may be wrong. Is there any evidence on this? &lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="5563" LastEditDate="2016-07-31T00:16:28.100" LastActivityDate="2019-05-12T16:27:49.743" Title="Effectiveness of multiple-choice versus short answer questions" Tags="|reference-request|assessment|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="11274" PostTypeId="2" ParentId="11262" CreationDate="2016-07-30T02:47:17.193" Score="4" Body="&lt;p&gt;I am a strong advocate for teaching constructions.  At least when it comes to high school geometry.  My experience in teaching geometry to freshmen and sophomore high school students has shown me how much can be gained from constructions.  Of course, it also depends on how you teach it.  You can micromanage it and take all the fun out of it.  Or you can be more liberal in your approach and let the students learn to use the tools and explore what they can create.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometry has the potential to open students up to a very different view of what mathematics is or can be.  And it's a view that can be easily lost as they move on to other courses which tend to hammer in a more algebraic approach to mathematics.  In the high school where I taught, calculus was really the only next opportunity for them to discover a novel way of thinking about mathematics, but, depending on who's teaching it, that perspective is easily lost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The world outside of high school seems to fall into four basic camps: those who hate math, those who become mathematicians, those whose favorite high school math class was algebra, and those whose favorite high school math class was geometry.  (Obviously these are not mutually exclusive camps.)  There's a certain structure to algebra, a formality, a love of rules applied to manipulating symbols that seems to appeal to many.  But there are others for whom the quest to reduce things to fundamental first principles has a strong appeal, the lovers of words and who enjoy weaving them together into an ordered fashion that can &lt;em&gt;seemingly&lt;/em&gt; lead to new ideas and discoveries (&quot;seemingly&quot; since philosophically what's happening is hard to pin down).  Then there are the students who have a more artistic bent: they like to draw, to make the abstract concrete, to view things in how they relate spacially.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students who may have hated algebra can fall in love with geometry (I had a student once draw a picture of the week: each week she drew a new &lt;strong&gt;geomeTREE&lt;/strong&gt; that somehow embodied the ideas covered.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Constructions create a kinesthetic approach mathematics.  Many young minds have a hard time understanding the diagrams in the textbook.  Taking time to draw the diagrams helps them dissect what's going on and make connections.  Constructions show the students how to reproduce those diagrams &lt;em&gt;exactly&lt;/em&gt; as they appear in the textbook.  The constructions help to remind the students of the various relations they should be looking for: congruent segments, congruent angles, parallel lines, bisectors,....  But also, if they are encouraged to experiment and play, they can create all sorts of art on their own.  They then can discover tessellations or fractal mathematics  and from their art new lessons in mathematics can be explored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A regular emphasis on constructions creates an atmosphere in the classroom where it's a place to draw and doodle.  And I would encourage that.  Keeping a keen eye out for the sorts of doodles the students are doing, you can direct the students subtly in different directions.  And then, if enough students doodle and the work adorns the walls, the students begin to create abstract art with mathematics embedded within.  New relations arise.  The students are, to a degree, the ones discovering mathematics.  How easy is it to encourage such exploration and experimentation in an algebra class?  (Not in my experience.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The constructions also break students out of the bubbles of what they might consider themselves capable of.  At the beginning of the year when I would throw them into constructions, students would apologize for their poor constructions by saying things like, &quot;I'm not an artist&quot;.  But by encouraging them to keep trying and not spending just two weeks out of the school year on constructions, but revisiting the topic week after week throughout the school year, by the end of the year all the students have improved greatly and a significant portion of the class indulges in doodling to a degree I don't usually see in my math classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, you can get students for whom &lt;a href=&quot;https://www.youtube.com/watch?v=ahXIMUkSXX0&quot; rel=&quot;nofollow&quot;&gt;Vi Hart&lt;/a&gt; can become a hero of sorts and encourage them to see the world in an entirely new light.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are constructions a life-long skill the students need?  Probably not.  But by introducing the students to this approach to geometry, vista open up for them particularly in a phase of their maturity when abstract reasoning is unfamiliar territory.  The connection between abstraction and understanding is made manifestly concrete in their very hands.&lt;/p&gt;&#xA;" OwnerUserId="3491" LastEditorUserId="3491" LastEditDate="2016-07-30T02:52:39.153" LastActivityDate="2016-07-30T02:52:39.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11275" PostTypeId="2" ParentId="11262" CreationDate="2016-07-30T06:38:59.313" Score="3" Body="&lt;p&gt;I've recently committed, in my college algebra classes, to presenting proofs for as many or all topics that I possibly can. This has made me much more aware of how often basic geometry constructions (transversals, etc.) are required. This includes: Pythagorean theorem, distance formula, midpoint formula, that any line is a linear equation, slope is constant on a line, parallel relationship, perpendicular relationship, circle formula, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In conjunction with that realization, and in trying to introduce students to the idea of general proofs in college mathematics, I find that I'm building more explicit bridges between college algebra and the high school geometry course -- reminding students of that course, building upon it, and highlighting what should have been the primary message (importance and accessibility of proofs). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short: Yes, the basic geometry constructions and proofs are fundamental to later work in more than one sense (both the results themselves, as well as the introductory idea of mathematical proof). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-07-30T13:21:41.653" LastActivityDate="2016-07-30T13:21:41.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11276" PostTypeId="2" ParentId="11124" CreationDate="2016-07-30T16:50:35.940" Score="2" Body="&lt;p&gt;One of my classes actually has linear optimization on their corriculum (Switzerland, Berufsmatur). I teach according to the book &quot;Mathematik in der Berufsschule&quot;. It's focused on students who want to go into economics.&#xA;The way it's taught is very visual and seems well suited for your class if you have but 2 variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every linear condition is drawn on a graph with the corresponding side that possible colored in or marked with an arrow. Then you color the space of feasable solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you draw the equation of what you want optimized, but add a variable &quot;c&quot;. Thus you can draw any parallel line. Now take a ruler and slide it in parallel until you find the last point that's still in the space of feasable solutions. For Maxima, you slide it up, for Minima you slide it down.&lt;/p&gt;&#xA;" OwnerUserId="7013" LastActivityDate="2016-07-30T16:50:35.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11277" PostTypeId="2" ParentId="11265" CreationDate="2016-07-30T18:25:28.910" Score="7" Body="&lt;p&gt;Models, as you mention, are a huge source of applications. One can mention some in physics (free fall, radioactivity, pendulum, ...), in biology (cell growth), in chemistry (kinetics) among other. The beauty of the subject, tying together several branches of maths that student tend to consider separate, is of course also a strong motivation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me stress three further points: a particular example in economics, the educational virtues of ODE, and its applications in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;The second law of capitalism&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;An example I like a lot (and used in my calculus lectures) is from Economics; the reason I like it is that it is a simple, static law which is better understood thanks to an ODE model. In Picketty's &lt;em&gt;Capital in the XXIth century&lt;/em&gt; (where I read it) is stated as the second law of capitalism the relation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \beta =\frac{s}{g}$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; is the ratio of the stock of capital &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; to the annual income &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; produced in the considered economy, &lt;span class=&quot;math-container&quot;&gt;$g=R'/R$&lt;/span&gt; is the economic growth and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; is the proportion of income which is saved, defined by &lt;span class=&quot;math-container&quot;&gt;$C'=sR$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, this law is a limiting law, which may only hold true at equilibrium. But what if &lt;span class=&quot;math-container&quot;&gt;$g=0$&lt;/span&gt;? What if &lt;span class=&quot;math-container&quot;&gt;$g&amp;lt;0$&lt;/span&gt;? It then makes no sense whatsoever!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, one can consider where this law comes from. Assuming &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; are constant, since &lt;span class=&quot;math-container&quot;&gt;$\beta := C/R$&lt;/span&gt;, we get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\beta' = \frac{C'}{R}-\frac{CR'}{R^2} = s-\frac{gC}{R} = s-g\beta$$&lt;/span&gt;&#xA;Now, studying the ODE &lt;span class=&quot;math-container&quot;&gt;$\beta'=s-g\beta$&lt;/span&gt;, one sees that the only stationary solution is &lt;span class=&quot;math-container&quot;&gt;$\beta=s/g$&lt;/span&gt;, the second law of capitalism. But studying it in detail, one sees how the above absurd cases now make sense: if &lt;span class=&quot;math-container&quot;&gt;$g=0$&lt;/span&gt;, then there is no stationary solution, and &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; increases linearly. If &lt;span class=&quot;math-container&quot;&gt;$g&amp;lt;0$&lt;/span&gt;, then there is no positive stationary solution and &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; increases exponentially fast. Then, one can also study the speed at which the stationary solution is approached, which economically is important to decide whether the stationarity of &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; an &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; can be assumed (their characteristic variation time should be much larger than the characteristic time of &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Functions as mathematical objects&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;The notion of function is a difficult one to grasp for student, and a central one. They have mainly learned to use them as processes, but at university we insist that they consider them as mathematical objects on their own. They have already applied processes to functions (e.g. differentiation), but we go much further (defining and using properties of function, operations on functions, relations on functions such as asymptotic comparison, etc.). ODEs are an important step in this process: &lt;em&gt;we now ask them to consider functions as unknowns&lt;/em&gt;. Of course, other equations can do that (functional equations) but only few of them can be dealt with. (The &lt;a href=&quot;https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/8846/1/Alcock_Simpson_book.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;very nice and short book by Alcock and Simpson&lt;/a&gt; was important for me to figure that out, I warmly recommend it). This is an important step on the path to further mathematical abstraction.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Applications in matheamtics&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;One should not forget that ODE are important not only on their own, but also as a crucial framework for much other mathematics. I am a differential geometer, so I need them:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;to construct diffeomorphisms from vector fields,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;to construct the exponential map of a Riemannian manifold,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;to relate curvature and volume, ...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="187" LastEditorUserId="376" LastEditDate="2019-04-10T12:46:44.480" LastActivityDate="2019-04-10T12:46:44.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11278" PostTypeId="2" ParentId="11262" CreationDate="2016-08-01T00:59:40.880" Score="3" Body="&lt;p&gt;I suggest that emphasizing plane geometry problems and their solutions&#xA;could be more effective than a systematic introduction to constructive geometry.&#xA;For example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Alexander Shen. &#xA;  &lt;em&gt;Geometry in Problems&lt;/em&gt;. &#xA;  MSRI Mathematical Circles Library.&#xA;  Volume: 18; 2016.&#xA;  &lt;a href=&quot;http://bookstore.ams.org/mcl-18&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMS link&lt;/a&gt;.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;  &lt;a href=&quot;https://i.stack.imgur.com/a0PmU.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/a0PmU.jpg&quot; alt=&quot;Cover&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Here us a typical problem (out of $750$ problems):&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/s29Ch.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/s29Ch.jpg&quot; alt=&quot;#369&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-08-01T00:59:40.880" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11279" PostTypeId="2" ParentId="11262" CreationDate="2016-08-01T09:01:09.743" Score="1" Body="&lt;p&gt;I strongly believe in teaching construction problems. They can be used to stimulate lively discussions in the class. For example, see the several construction problems in Geometric Transformations, Vol I by Yaglom (MAA publication) (&lt;a href=&quot;http://maa.org/press/ebooks/geometric-transformations-i&quot; rel=&quot;nofollow&quot;&gt;http://maa.org/press/ebooks/geometric-transformations-i&lt;/a&gt;). Also, given orthocenter, circumcenter and one of the vertices, is it possible to construct the triangle? Is there one or many solutions? These kind of questions help students understand Euclidean geometry much better and at the same time teach &quot;problem solving&quot;.&lt;/p&gt;&#xA;" OwnerUserId="6957" LastActivityDate="2016-08-01T09:01:09.743" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11280" PostTypeId="2" ParentId="11265" CreationDate="2016-08-01T16:53:04.040" Score="10" Body="&lt;p&gt;Of course I agree that one motivation for studying ODEs is that they have applications. But it might be useful to also point out another fact that students do not always think about: ODEs are often the most succinct way of defining functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could use the exponential function as an example, because your students will have seen it already. Ask them to give you a precise description of the exponential function. They might start by telling you about the shape of its graph, but under further questioning they will quickly realise that they would need to give you an infinite amount of information to specify the graph exactly, until they hit upon the idea of saying that at every point its slope equals its value, so they formulate their first ODE. At that point students often get excited and wonder &quot;what if the slope is equal to the square of the value?&quot;, i.e., they start looking at other ODEs by themselves. Unfortunately it is then sometimes a bit difficult to tell them to take it slow and stick with linear ODEs for a while.&lt;/p&gt;&#xA;" OwnerUserId="7018" LastActivityDate="2016-08-01T16:53:04.040" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11281" PostTypeId="2" ParentId="11265" CreationDate="2016-08-02T23:14:37.407" Score="8" Body="&lt;p&gt;Echoing many of the points made in the other excellent answers... but just to focus on one (to me very significant) aspect: differential equations &lt;em&gt;characterize&lt;/em&gt; functions in (physically?) operationally (as it turns out) meaningful ways. Other answers have noted the &quot;dynamical&quot; aspect, and @gustav noted in particular that some things are best described by being solutions to a differential equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That characterization is certainly true in an immediate sense, based on physics and mechanics. But, to-me-surprisingly, the same definitiveness of characterization is important in &quot;pure mathematics&quot; (for me, simply &quot;mathematics&quot;, in the same way that &quot;applied language&quot; is just &quot;language&quot;, whether it is poetic or ugly or not-to-the-point). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd tend to claim that the next level of &quot;description&quot; of functions after thinking (after Lagrange and many other luminaries) of functions as inevitably lying in some completion (not in their terms) of polynomials, was to tell the ODE they satisfied. Manifestly, this is the case for many of the classical special functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, due to the subtly structure situations in which those special functions arose, they did also have significant features related to their various origins. Among other very, very weighty manifestations, the Casselman-Wallach subrepresentation theorem is a (fairly stunning) generalization of the early 20th century (but highly non-trivial) asymptotics results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, as a fairly fancy example, much of the contemporary theory of automorphic/modular forms relies upon such things. Which is to say that much modern number theory relies... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, &quot;description (of an important thing)&quot; is quite often &quot;by a differential equation&quot;.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-08-02T23:14:37.407" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11282" PostTypeId="1" CreationDate="2016-08-04T14:51:53.307" Score="2" ViewCount="184" Body="&lt;p&gt;I'm looking for rich educational resources similar to core-plus mathematics, for grades 7, 8, and 9. Is there any such material?&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2018-04-24T19:02:14.090" Title="Core-Plus Mathematics: Similar Resources for grades 7-9" Tags="|textbooks|teacher-preparation|teaching|common-core|teacher-development|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11283" PostTypeId="2" ParentId="8293" CreationDate="2016-08-07T01:04:23.290" Score="5" Body="&lt;p&gt;I've been going through my papers and have found the source of the idea.  Roger Eggleton presented it in a talk at the &lt;em&gt;Severino V. Gervacio Conference on Graph Theory and Combinatorics 2009&lt;/em&gt; held in Manila, Philippines from April 24, 2009 to April 25, 2009.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aside from the two equations in the original post, he also presented the two equations below:&#xA;$$\frac{(fg)''}{fg}=\frac{f''}{f}+\frac{g''}{g}+2\frac{f'g'}{fg}$$&#xA;$$\frac{(f/g)''}{f/g}=\frac{f''}{f}-\frac{g''}{g}-2\left(\frac{f'}{f}-\frac{g'}{g}\right)\frac{g'}{g}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems the original reference for this is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Roger Eggleton and Vladimir Kustov, &quot;&lt;a href=&quot;http://www.maa.org/sites/default/files/Eggleton-CMJ-2011.pdf&quot;&gt;The Product and Quotient Rules Revisited&lt;/a&gt;,&quot;  &lt;em&gt;The College Mathematics Journal&lt;/em&gt;, Volume 42, Number 4, September 2011, pp. 323-326.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2016-08-07T01:10:01.457" LastActivityDate="2016-08-07T01:10:01.457" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11284" PostTypeId="1" CreationDate="2016-08-08T13:08:25.970" Score="15" ViewCount="647" Body="&lt;p&gt;Is every mathematical concept, even the complex ones, explainable?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As someone who will be needing to explain my line of work for a position to a committee who is very, very, educated, just not in math (I would say over a decade or so years removed from taking a calculus I course, I'm having a hard time optimizing efficiency of my words and impact of my statement. I have a &lt;strong&gt;lot&lt;/strong&gt; of applications knowledge that is relevant to them and might get them interested, but I also have to lay the foundation for what a PDE is.&lt;/p&gt;&#xA;" OwnerUserId="7027" LastEditorUserId="376" LastEditDate="2016-08-12T15:24:51.570" LastActivityDate="2017-07-16T16:08:12.987" Title="How would you explain what a PDE is to a very educated layman with no math background?" Tags="|examples|mathematical-analysis|differential-equations|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11285" PostTypeId="2" ParentId="11284" CreationDate="2016-08-08T15:26:11.810" Score="22" Body="&lt;p&gt;I would say something like this:  &quot;Often in complicated systems one needs to study multiple quantities, each of which varies at rates that depend on the other quantities and on how fast they are varying.  For example the rate at which a drug metabolizes may depend not only on how much of the drug is in the body but also on how much blood sugar is in the body, how much of certain other hormones are present, and so forth, and each of those quantities is also changing at a rate that depends on all of the others.  Because everything is interdependent, to understand how one quantity changes over time you need to study the interactions of the whole system.  The study of partial differential equations provides a set of tools for studying such systems.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the example above is for whatever reason inappropriate for the audience, replace it with another one involving multiple interconnected systems: for example, complicated financial markets, or weather patterns, etc.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-08-08T15:26:11.810" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11286" PostTypeId="2" ParentId="11284" CreationDate="2016-08-09T00:29:25.840" Score="4" Body="&lt;p&gt;Imagine a surface, for example, a horse saddle.  Note that the &quot;curvature&quot; of the surface differs not only when you measure it at different points, but also when you measure it along different directions.  For example, at the &quot;center&quot; of the saddle, the surface is curving upwards if measured along the &quot;horse's spine&quot; and the surface is curving downwards if measured perpendicular to that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tell the very educated layman with no math background that many systems can be modeled as surfaces and that partial differential equations answer questions about the &quot;curvature&quot; of such surfaces.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-08-09T00:29:25.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11287" PostTypeId="2" ParentId="11284" CreationDate="2016-08-09T06:44:38.100" Score="3" Body="&lt;p&gt;Here's my attempt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's about building a big picture of a situation when all you know is small-scale relationships. For a simple example, imagine a Mars rover has crashed, so most of it is broken. All it can do is move about, and it can tell how steeply it's going up or down. NASA want to make the best use of the equipment they can, so they send it up and down, back and forth, taking lots of transects recording the up/down information as it goes. Although the rover only knows what's going on immediately around it, NASA can build up a relief map of the whole area.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-08-09T06:44:38.100" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11288" PostTypeId="2" ParentId="11265" CreationDate="2016-08-09T13:52:17.720" Score="4" Body="&lt;p&gt;I'm going to take a more pragmatic view:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We study ordinary differential equations, because we &lt;em&gt;can&lt;/em&gt; study ordinary differential equations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That is, it is a subject that is well amenable to study, a fairly accessible theory, and a wealth of methods of exact calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then, once we understand ordinary differential equations, we can then start &lt;em&gt;look for&lt;/em&gt; ordinary differential equations &amp;mdash; like the old adage goes, if you have a hammer, everything looks like a nail. While some problems might &lt;em&gt;naturally&lt;/em&gt; take the form of an ordinary differential equation, we solve other problems by finding ordinary differential equations that give us information about them.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2016-08-09T13:52:17.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11289" PostTypeId="2" ParentId="11262" CreationDate="2016-08-09T17:51:11.257" Score="0" Body="&lt;p&gt;I'd like to note that construction geometry can be very helpful for IMO level geometry problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always make a large diagram (picture) and construct all lines and points given in the problem as precise as I can, so that I can develop conjectures on what points be on one line, what quadrilaterals might be cyclic, which two lines might be parallel, etcetera. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing this precisely is only possible if you know the constructions well. Also, I like how constructions apply knowledge of geometry, for example, when proving that the construction is correct or when even thinking of efficient constructions yourself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true that combinatorial geometry is gaining a lot of popularity. Note that IMO 2015/1 was also combinatorial geometry. On the other hand, I think that combinatorial geometry is more combinatorics than geometry, so I would be careful when replacing geometry with combinatorial geometry since you are then actually replacing geometry with combinatorics. &lt;/p&gt;&#xA;" OwnerUserId="5664" LastActivityDate="2016-08-09T17:51:11.257" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11290" PostTypeId="2" ParentId="11265" CreationDate="2016-08-10T04:48:19.107" Score="3" Body="&lt;p&gt;After going through several integration techniques and standard forms of integrals, students wonder &quot;Why are these needed?&quot;. Once answer to this question may be provided by showing real applications using ODE. The book &quot;Chases and Escapes&quot; by Paul Nahin contains many applications of ODE that would convince the student of the utility of studying ODE.&lt;/p&gt;&#xA;" OwnerUserId="6957" LastActivityDate="2016-08-10T04:48:19.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11291" PostTypeId="2" ParentId="5645" CreationDate="2016-08-10T15:05:20.247" Score="1" Body="&lt;p&gt;Since posting my question, I have come across some material on contemplative mathematics that I will share as an admittedly-incomplete answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Section 3 of &lt;a href=&quot;http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1111&amp;amp;context=jhm&quot; rel=&quot;nofollow&quot;&gt;On Contemplation in Mathematics&lt;/a&gt; (Wolcott, 2013), Luke Wolcott discusses meditation in education and describes an informal experiment he conducted in an introductory differential equations class, as well as the feedback collected on a weekly basis. See also &lt;a href=&quot;https://contemplativemathematicspedagogy.wikispaces.com/Mindfulness+at+Lawrence+with+Wolcott&quot; rel=&quot;nofollow&quot;&gt;Mindfulness at Lawrence with Wolcott&lt;/a&gt; and Justin Brody's &lt;a href=&quot;https://contemplativemathematicspedagogy.wikispaces.com/Contemplating+Infinity&quot; rel=&quot;nofollow&quot;&gt;Contemplating Infinity&lt;/a&gt;. On the latter topic, I was intrigued to find out about &lt;a href=&quot;http://www.gardensofinfinity.com/infinities.html&quot; rel=&quot;nofollow&quot;&gt;Gardens of Infinity&lt;/a&gt;, a collaboration between Wolcott and interaction designer Luke Dressel.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2016-08-10T15:05:20.247" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11299" PostTypeId="1" CreationDate="2016-08-13T21:08:05.553" Score="4" ViewCount="96" Body="&lt;p&gt;I'm creating a course on differential equations, and I'm painfully aware that there is a LOT to teach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As of right now, I'm considering splitting the course into two, in the following manner:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;theory with visuals; growth models, equilibrium points (and their classifications), bifurcations, how computers solve equations (numerically)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;strategies for solving equations by hand; separable equations, integrating factor, parameter variation, undetermined coefficients&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I realize that both of these have untold depths in their own respects, but that is reasonable because since this will be a video course, I can add videos indefinitely. I might have to further split each part, but that's a problem for another day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this a logical way of structuring a course on differential equations into two parts?&lt;/p&gt;&#xA;" OwnerUserId="796" LastActivityDate="2016-08-13T21:08:05.553" Title="Split differential equations into two video courses; theory and calculation?" Tags="|course-design|differential-equations|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11300" PostTypeId="1" CreationDate="2016-08-15T12:38:53.417" Score="6" ViewCount="434" Body="&lt;p&gt;I'm teaching geometry at high school for the first time soon and am struggling to find a reasonable notation for lines, line segments and rays defined by two points $A$, $B$ (and a direction). At the high school I'm teaching at, everyone just uses the notation they're familiar with already, and most of geometry is taught without a textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've come across rays starting at $A$ being denoted by $\overrightarrow{AB}$, which I'm not a fan of, because many people also use this for vector notation. Some people use $\overline{AB}$ for lines, some for line segments. Those people then use $AB$ for line segments and lines, respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What should I do about this?&lt;/p&gt;&#xA;" OwnerUserId="493" LastEditorUserId="376" LastEditDate="2016-08-15T17:33:20.163" LastActivityDate="2016-08-21T20:08:26.870" Title="How to reasonably denote lines, line segments and rays?" Tags="|secondary-education|geometry|notation|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="11301" PostTypeId="2" ParentId="10580" CreationDate="2016-08-15T15:46:17.923" Score="2" Body="&lt;p&gt;To the question about whether or not there is existant literature around some of these ideas, I believe there is a great deal.  It seems you are getting at conversations around proof and how this plays out in teachers beliefs about mathematics. Several researchers have engaged in such pursuits.  Immediately two things come to mind.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The ICMI study on proof in mathematics education: &lt;a href=&quot;https://books.google.com.sa/books?id=qA_lLat89BMC&amp;amp;printsec=frontcover&amp;amp;hl=ar&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;https://books.google.com.sa/books?id=qA_lLat89BMC&amp;amp;printsec=frontcover&amp;amp;hl=ar&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Patricio Herbst and Daniel Chazan have both done research in these areas, you'll find a good deal of references in their work as well.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2016-08-15T15:46:17.923" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11302" PostTypeId="1" CreationDate="2016-08-15T17:17:55.570" Score="8" ViewCount="262" Body="&lt;p&gt;Some background: I recall becoming much more adept at the concepts of area and measurement during high school geometry. However, as I scour the Common Core standards, &quot;area&quot; only shows up in high school standards when talking about the sectors of a circle. I could also argue that concepts of area and surface area must be understood in order to perform the majority of modeling tasks, but it is not explicitly mentioned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Area is supposed to be covered in the pre-high school standards; however, experience shows that students entering the high school geometry classroom have forgotten the majority of formulas they've learned and, though they relearn the formulas reasonably quickly, they appear to have never or rarely applied these formulas to compound shapes. Only one or two had ever applied a decomposition argument to justify the area of a triangle, much less a parallelogram or trapezoid (which are 6th-8th grade standards)!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a few related questions that should be taken in this context, in addition with the knowledge that learning from the middle grades is often forgotten: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Presuming a student has actually experienced their math education in a CC-aligned manner, what role should the idea of area play in his or her geometry education at the high school level?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Presuming a student's math education was hastily shoehorned into CC alignment sometime around the 6th or 7th grade, what role should the idea of area play into his or her geometry education at the high school level?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This question is in the context of common core alignment, but can also be answered in the context of college preparedness, developmental need, etc.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2016-09-01T17:44:49.777" Title="The role of &quot;area&quot; in a Common-Core aligned high school classroom" Tags="|secondary-education|geometry|common-core|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11303" PostTypeId="2" ParentId="11300" CreationDate="2016-08-15T17:44:36.360" Score="5" Body="&lt;p&gt;The most important thing, in my opinion, is to adopt a notation that is consistent with whatever is most common in the broader culture.  Geometry notation is highly context-specific; notation used at the secondary level tends to be different from that used at the undergraduate level, and there are country-to-country differences as well.  If there is any kind of national or regional school-leavers exam (comparable to A-levels in the UK or to the SAT in the United States) you should use whatever notation is standard on that exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know much about the Swiss educational system, but I find it frankly hard to believe that there is no standard notation in use in your context.  The Swiss geometry curriculum was one of nine national curricula compared in:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hoyles, C., Foxman, D. and Küchemann, D. (2002) &lt;em&gt;A comparative study of geometry curricula&lt;/em&gt;. Qualifications and Curriculum Authority, London. ISBN 1858385091 &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;...and while I don't have access to that book at the moment the mere fact that it exists suggests to me that there &lt;em&gt;is&lt;/em&gt; a national geometry curriculum in Switzerland.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you genuinely do teach in a vacuum, free of external encumbrances and with no cultural norms to align yourself with, then I guess you are free to choose whatever makes sense to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the United States, standard notation at the &lt;em&gt;secondary&lt;/em&gt; level is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\overline{AB}$ denotes the line segment from $A$ to $B$&lt;/li&gt;&#xA;&lt;li&gt;$\overrightarrow{AB}$ denotes the ray with initial point $A$ and passing through $B$ (but note that a conventional Geometry class in the United States typically does not include vectors, so there is no conflict with that notation)&lt;/li&gt;&#xA;&lt;li&gt;$AB$ denotes the length of the segment $\overline{AB}$&lt;/li&gt;&#xA;&lt;li&gt;$\overleftrightarrow{AB}$ denotes the line through $A$ and $B$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;At the advanced undergraduate or graduate level, different notations tend to be used; for example, I believe Greenberg's text uses $AB$ to denote a line, rather than the length of a segment, but otherwise uses the conventions above.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2016-08-15T18:41:07.873" LastActivityDate="2016-08-15T18:41:07.873" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11305" PostTypeId="1" AcceptedAnswerId="11320" CreationDate="2016-08-16T15:54:29.827" Score="4" ViewCount="114" Body="&lt;p&gt;I am in need to find references for my thesis proposal. &#xA;My topic is &quot;Virtual manipulative versus physical manipulative: which way do students learn faster?&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know any authors that worked on this topic or books about it?&lt;/p&gt;&#xA;" OwnerUserId="7050" LastEditorUserId="376" LastEditDate="2016-08-18T12:34:10.983" LastActivityDate="2016-08-18T12:34:10.983" Title="References on the effectiveness of manipulatives (virtual vs. physical)" Tags="|reference-request|education-research|manipulatives|" AnswerCount="1" CommentCount="10" ClosedDate="2016-09-03T20:45:12.457" ContentLicense="CC BY-SA 3.0" />
  <row Id="11306" PostTypeId="1" CreationDate="2016-08-16T16:23:42.120" Score="7" ViewCount="1402" Body="&lt;p&gt;I am considering using Basic Mathematics by Serge Lang as the primary text for my High School Pre-Calculus course. My students have all spent a year working through the first six books of Euclid and have passed Algebra II. Is there anyone who has used this book in a class? What recommendations would you have based on your experience?&lt;/p&gt;&#xA;" OwnerUserId="7051" LastEditorUserId="7051" LastEditDate="2016-08-26T23:12:19.843" LastActivityDate="2016-08-26T23:12:19.843" Title="Teaching a Pre-Calculus Course using Basic Mathematics by Serge Lang" Tags="|textbooks|precalculus|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11307" PostTypeId="1" CreationDate="2016-08-17T00:23:54.787" Score="9" ViewCount="510" Body="&lt;p&gt;This question is meant to explore the intuition that mathematical thought does not most naturally proceed from writing in one's native language. The hackneyed and not entirely satisfying slogan that mathematics is a language is also relevant here, in that I am trying to tease out how &amp;quot;linguistic thought strategies&amp;quot; are central in the mathematical practice of successful mathematicians. Things like &lt;a href=&quot;http://www.mathunion.org/ICM/ICM1990.2/Main/icm1990.2.1665.1672.ocr.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Manin's essay here&lt;/a&gt; are interesting if you have wondered along similar lines.&lt;/p&gt;&#xA;&lt;p&gt;The basic observation I am interested in appears already in the learning of &amp;quot;math facts&amp;quot;, which obey a substitution grammar that is very different from the one appearing in everyday language use. One might, following a Wittgenstein language game approach, regard a response of &amp;quot;5&amp;quot; to the question &amp;quot; What is 2+3?&amp;quot; as somehow analogous to a response of &amp;quot;Fine&amp;quot; to &amp;quot;How are you?&amp;quot; in English. In the sense that there is a linguistic command or stimulus that has a known, standard, response, the above two are similar.&lt;/p&gt;&#xA;&lt;p&gt;In the above situation, there is an aspect of the learning of a foreign language, in that there is no &amp;quot;translation stage&amp;quot; in either case. This feels similar to a native English speaker learning Mandarin Chinese, where it is most efficient to directly learn the new language with very little translation. (Of course, this ignores the fact that in both cases, Chinese and English may point to the same physical objects or analogous human situations and mathematics may not so obviously do this.)&lt;/p&gt;&#xA;&lt;p&gt;As a professor in a liberal arts college, I deeply believe that teaching students to write will inevitably teach them to think. The trouble is, I have a nagging feeling that having students write in their native language about mathematics in order to think about mathematics is very inefficient, and if Poincaré and Hadamard are correct, most mathematicians avoid thinking explicitly in language, perhaps because it may somehow emphasize serial reasoning over a holistic intuition. (The suppression of linguistic tendencies by mathematicians is reminiscent of Betty Edwards' &lt;a href=&quot;http://drawright.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Drawing on the Right Side of the Brain&lt;/a&gt;.) If emphasizing writing too much can impede the mathematical investigation in similar ways to how language may impede drawing ability, then we should be very wary of such emphasis.&lt;/p&gt;&#xA;&lt;p&gt;It may be, though, that mathematicians are using their linguistic faculty in a way similar to learning foreign languages that share Wittgensteinian family resemblances. By learning solutions to problems in a subdomain, a mathematician stores, in a &amp;quot;language game&amp;quot;-like linguistic fashion, a set of promising responses to collections of problems in the subdomain. The responses are like various acceptable responses to &amp;quot;How are you?&amp;quot; in English. Some of these work out to lead to solutions to problems (the mathematician's language includes in its kernel the grammar of basic logic, so the proof is possible). It may even be that mathematicians are always including the use of their linguistic faculty to combine various symbols (here understood as formal entities that can be communicated, in principle...which may include pictures and various sounds) in potentially promising ways in response to problems. Any such mental manipulation of a mathematical object is somehow a &amp;quot;linguistic move&amp;quot; in that it involves a distinct transformation of something that can be communicated. It seems that solutions to problems provide patchwork conversations to learn the &amp;quot;foreign language moves&amp;quot; needed to learn mathematics.&lt;/p&gt;&#xA;&lt;p&gt;As a person stockpiles enough of these responses at the level of mini foreign languages with family resemblances, he or she eventually becomes a what is called a mathematician.&lt;/p&gt;&#xA;&lt;p&gt;I apologize for the above preamble but have not been able to distill it after thinking about it over some time. The point is, writing in one's native language, describing mathematics in order to try to learn it, seems similar to writing about playing the piano without actually training to learn to play...which is also learning something that resembles language. It may directly impede the learning of mathematics!!! It seems that one must learn the language of mathematics with as little translation as possible, and then try to write in that language, to learn mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Here is the question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;: In the education literature on writing to learn mathematics, what are some solid references that compare the above &amp;quot;foreign language acquisition&amp;quot; aspect of writing to learn with an approach of writing about mathematics primarily in one's native language?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I realize that this question is (as I am always guilty of) very dualistic and therefore kind of ridiculous. I cannot seem to settle the issue in my teaching, though, because attempts to be (quoting J.P. Serre) &lt;em&gt;precise yet informal&lt;/em&gt; and describe mathematics using one's native language is a translation activity that may fortify human understanding, which is the ultimate goal of developing mathematics in the first place. Well-written mathematics very effectively bootstraps native language intuition without losing precision...and the thought of blind algorithmic manipulation is repugnant to any red-blooded mathematician if such manipulation is not being carried out to develop intuition (which may actually be related to fluency in the foreign language being acquired).&lt;/p&gt;&#xA;&lt;p&gt;Another reason the question may be ridiculous is that it is dangerously close to being something we all have wondered about (&amp;quot;drill versus thought&amp;quot; approaches to math education). I ask, though, because it feels to me like there is something subtle here about which I'd like to read more...but I haven't been able to find a place to do so after searching the web several times.&lt;/p&gt;&#xA;&lt;p&gt;This question is also related to nearly all of my others along these lines, particularly my &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5768/calculation-versus-writing-in-mathematics&quot;&gt;&amp;quot;Calculation versus Writing&amp;quot;&lt;/a&gt; question and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10939/is-metacognition-ever-bad&quot;&gt;&amp;quot;Is Metacognition Ever Bad?&amp;quot;&lt;/a&gt;. The distinction of the present question from these two is that here I am mostly interested in the intuitive similarity of feel of learning a foreign language without first translating and ALL mathematical activity, including activities that may not be initially interpreted as calculation. Drawing diagrams and manipulating vague mental entities have this quality of trying to almost combinatorially obtain fluency with something foreign with the desire to eventually succeed in a form of communication that is solving the problem. (This is analogous to the feeling of being able to program a computer. A solution of a mathematical problem is somehow having adequate fluency to be able to successfully communicate in a foreign tongue.) This question can be interpreted as unpacking the description of &lt;a href=&quot;http://www.math.harvard.edu/conferences/unityofmath_2003/talks/gelfand-royal-talk.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Israel Gelfand of mathematics as &amp;quot;adequate language&amp;quot;&lt;/a&gt; (although, not just for science). (EDIT: &lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=AKKOFOcBb2sC&amp;amp;oi=fnd&amp;amp;pg=PA131&amp;amp;dq=mathematics%20as%20an%20efficient%20language%20gelfand&amp;amp;ots=uG7xWdNGo3&amp;amp;sig=-VYBwyB1pp-tKgefvr3YrZEaedw#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;This essay of Ed Effros&lt;/a&gt; seems to be close to what I want...so more things like that would help, I think...)&lt;/p&gt;&#xA;&lt;p&gt;EDIT: I think the intuition I am chasing here, and the one I hope the literature can help clarify, is related to the reference to mathematics as &lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/mathematics-as-the-science-of-patterns-mathematics-as-the-science-of-patterns&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;the science of patterns&amp;quot;&lt;/a&gt;. If a pattern is understood as Poincaré describes it as an &amp;quot;atom of perception&amp;quot;, something that we attribute stability to by naming it and delineating it in either space or time (in the field of our senses or thought), then it is very much connected to language by this delineation. Native language plays with the perception of new patterns by the appearance of analogy (like seeing familiar shapes in clouds) and this may be something of the role of native language in mathematics, where patterns are identified and named. The foreign language aspect of this comes about by trying to develop an adequate and efficient language for the patterns of thought that emerge. Most importantly in these observations, the pattern requires delineation, a stable entity arising by a linguistically defined boundary that can be manipulated as a meaningful idea. This naming or encapsulating process, much like the philosophical process of assigning order via language to vague intuitions, yields mathematical ideas. The sense of mathematical beauty is reflected in the ability to effectively perform this delineation...and the &amp;quot;choices&amp;quot; made in such delineation are much like those of a poet. In mathematics, the desire for recombination and something of the foreign language aspect (necessitated by the desired leverage that the adequate language of mathematics, its body of organizing principles, provides) is more important than in philosophical investigation, which proceeds more closely to the native language.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="14466" LastEditDate="2020-08-27T11:45:12.533" LastActivityDate="2023-04-01T04:59:35.513" Title="Native language, writing, and mathematical problem solving" Tags="|mathematical-pedagogy|reference-request|writing|problem-solving|philosophy-of-maths|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11308" PostTypeId="2" ParentId="7691" CreationDate="2016-08-17T12:23:48.483" Score="3" Body="&lt;p&gt;There is an article about the educational system for studying mathematics in Germany in the 19. century:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lorey, W. (1916). Das Studium der Mathematik an den deutschen Universitäten seit Anfang des 19. Jahrhunderts (pp. 85-86). &lt;a href=&quot;http://dx.doi.org/10.1090/S0002-9904-1924-03856-7&quot; rel=&quot;nofollow&quot;&gt;http://dx.doi.org/10.1090/S0002-9904-1924-03856-7&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Don't worry about the German title, the article is in English :-) You can also find a &lt;a href=&quot;http://www.ams.org/journals/bull/1924-30-03/S0002-9904-1924-03856-7/S0002-9904-1924-03856-7.pdf&quot; rel=&quot;nofollow&quot;&gt;PDF online&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastActivityDate="2016-08-17T12:23:48.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11309" PostTypeId="1" CreationDate="2016-08-17T12:25:49.590" Score="10" ViewCount="1120" Body="&lt;p&gt;For me it seem to be obvious that exercises and exercise courses are important for undergraduates to learn mathematics and skills like finding a proof or writing it down. Was there any research showing that exercises are necessary for math education (undergraduate education, math majors) or that they are very effective? &lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="143" LastEditDate="2016-08-17T12:30:08.723" LastActivityDate="2016-09-01T15:08:57.617" Title="Importance of exercises for learning mathematics" Tags="|undergraduate-education|reference-request|education-research|exercises|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11310" PostTypeId="1" CreationDate="2016-08-17T13:33:48.050" Score="10" ViewCount="3035" Body="&lt;p&gt;I am going to teach some grade 9 students about solving linear and quadratic equations. I am looking for a question from every day life (of a teenager) or a puzzle which is hard to solve without using algebra. There are of course loads out there in textbooks and the internet, but I haven't yet found one which is really intriguing and which could arouse the interest even of a student who has other things on his/her mind and who has a general dislike for school-mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example questions concerning the respective speeds, distances and time-periods of two vehicles with respect to each other are classic examples motivating linear equations, but they don't really seem to occur in real life nor are they particularly fascinating (at least for someone who isn't interested in mathematics anyway).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to begin the subject with such a question and let the students work on it together for maybe half an hour or so (i.e. the question shouldn't be too easy to solve); hopefully this way they will see themselves how useful it can be to introduce variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any ideas? &lt;/p&gt;&#xA;" OwnerUserId="7053" LastEditorUserId="262" LastEditDate="2016-08-24T03:15:23.003" LastActivityDate="2020-02-18T14:06:42.117" Title="Everyday Example Problems for Solving Linear and Quadratic Equations" Tags="|secondary-education|algebra|student-motivation|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11311" PostTypeId="2" ParentId="11310" CreationDate="2016-08-17T15:02:01.597" Score="10" Body="&lt;p&gt;The best example I've found is the question, &quot;What do I need on the final exam to get an [A, B, C, pass] in this class?&quot; at the end of a term -- which really does get asked by one or more students every semester. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I now respond to that with, &quot;Well, you're asking an algebra question, so you should be able to solve that yourself&quot;, and I've set up my grading formula specifically to make that tractable for the student. Say the weighted formula is W = 15%Q + 50%T + 35%F, where W is the weighted total, Q the quiz average, T the test average, and F the final exam score. Going into the last week, Q and T are fixed and W is known by the student's goal for overall grade (say, 90% for an A, etc.). Then one can solve for the necessary F score algebraically in a straightforward fashion, but it's really hard to guess otherwise (likely involves decimals). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I get some surprising responses when this comes up and I respond that the asking student should be able to do it on their own. In almost all cases I have to remind them of our grading formula, write it down, and possibly start the substitution step. Some students then rip off the answer in a few seconds. Some can't accomplish it after a day of effort (even at the end of an elementary algebra course). In a college algebra class, I once set up the formula on the board and people were amazed and gasped and took out phones and snapped photos. Other students have complained, &quot;Are you seriously going to be that hard on me?&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: Be aware that even with this very practical exercise, there's an enormous difference in reaction between doing it as a theoretical exercise in class (mostly the usual &quot;meh&quot; reaction) and doing it in response to an immediate needy student's request (which can be &quot;wow&quot;, etc.). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-08-17T22:11:29.533" LastActivityDate="2016-08-17T22:11:29.533" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11312" PostTypeId="1" CreationDate="2016-08-17T18:17:32.493" Score="4" ViewCount="134" Body="&lt;p&gt;&lt;del&gt; I would like to be able to illustrate why additivity is a natural property to assume for probability measures. While it is relatively simple to give a short hand waving intuitive analogy to volume or mass I feel this is somewhat unsatisfying, not least since it does not really illustrate how you would use it. &lt;/del&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;del&gt; What seems most reasonable to me is to show what amounts to an application of the law of total probability. The problem is that all examples I can think of rely on conditioning or independence, and in the standard measure theoretic (Kolmogorov axiom) formalism the topic of additivity is, by necessity, raised before one can get to independence or conditioning. I would prefer to be able to give at least one motivating example of additivity before going on to the next topic. &lt;/del&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;del&gt; The best substitute I have been able to think of is to take a proof of some basic proposition for probability measures that invokes additivity (such as monotonicity with respect to ordering of events) and specialise it to some concrete events. I worry, though, that students with little mathematical background (the ones I'm teaching this semester have not even seen basic calculus) would find such arguments overly contrived or circuitous. &lt;/del&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;del&gt; Note that I realise that one could approach the subject completely without considering assignment of probabilities as something reifed as a measure. That being said, since the course is also to cover basic statistics I feel like this is more or less necessary considering how statistics requires one to deal of multiple alternative (sets of) measures corresponding to different hypotheses. &lt;/del&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: In hindsight I think I made the mistake of trying to anticipate responses, thus making the question much more convoluted than necessary. Let me therefore try to state it more concretely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it possible to give an example of how additivity/disjointness can be used to analyse a complex event by partitioning it into simpler parts, &lt;em&gt;before&lt;/em&gt; conditioning and/or independence has been introduced. Since the idea is to illustrate how additivity/disjointness encode something important/natural I want an example with some mathematical content (i.e. I would like to avoid a pure appeal to intuition or prototypical examples) but not something purely mathematical (i.e. not only derive additional nice properties measures will have).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example I ended up using was, if I remember correctly, one that was also hinted at here. Namely that it allows one to motivate the 1/6 probability for a die roll from symmetry of the die. I don't think this worked particularly well though, since I think many people did not even realise this probability assignment was something that had to be justified. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Failing this, can one give some other example (not in terms of partitioning a complex event) achieving the same goal?&lt;/p&gt;&#xA;" OwnerUserId="7055" LastEditorUserId="7055" LastEditDate="2016-12-14T11:15:48.970" LastActivityDate="2016-12-14T11:51:48.257" Title="Example illustrating use of additivity of (probability) measures before introducing independence and conditioning" Tags="|undergraduate-education|examples|probability|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11313" PostTypeId="2" ParentId="11272" CreationDate="2016-08-17T21:47:52.277" Score="2" Body="&lt;p&gt;In most fields, I think the research indicates the opposite, though I don't know of any research that is &lt;em&gt;directly&lt;/em&gt; on point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the weaknesses of multiple choice tests is that they allow the student to write the answer when they recognize it among the choices. This is easier than retrieving it without answer cues, and so the well-established benefits of retrieval practice (a.k.a. the testing effect) for retaining information aren't gained. However, in math, even multiple choice questions typically require some calculation rather than mere recognition, so this is likely less of an issue in math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I don't know of research directly on point here either, I also suspect that multiple choice tests give students an excuse to guess on a problem and stop thinking earlier than they might if they had to come up with something rather than leave a free response question blank. Since even unsuccessful initial struggles with a problem promote retention (assuming appropriate feedback is soon provided), I would imagine that this would be a problem even in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, while very skilled test writers can find ways to write test questions to test a wide range of thinking processes in math, it isn't easy. Even some government or commercial sources often produce multiple choice questions in math whose intent can be evaded by a clever student. (A blatant example: figuring out which choice is the solution to an equation by plugging each choice in turn into the equation rather than solving the equation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for anything that indicates superiority of multiple choice questions in some respect, I don't know of any. They're really used more for convenience than because they're better at promoting learning.&lt;/p&gt;&#xA;" OwnerUserId="5417" LastEditorUserId="5417" LastEditDate="2016-08-18T10:09:33.793" LastActivityDate="2016-08-18T10:09:33.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11314" PostTypeId="2" ParentId="11310" CreationDate="2016-08-17T21:54:39.057" Score="0" Body="&lt;p&gt;How about &lt;a href=&quot;http://www.ccs.neu.edu/home/fell/CSU540/programs/RayTracingFormulas.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;ray-tracing&lt;/a&gt;&#xA;quadratic equations,&#xA;e.g., for a ray-sphere reflection?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/Jv3YY.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Jv3YY.gif&quot; alt=&quot;RayTracing&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://www.scratchapixel.com/lessons/3d-basic-rendering/introduction-to-ray-tracing/implementing-the-raytracing-algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;scratchapixel.com&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Wikipedia has an article detailing the equations &lt;a href=&quot;https://en.wikipedia.org/wiki/Line%E2%80%93sphere_intersection&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-08-24T12:22:36.397" LastActivityDate="2016-08-24T12:22:36.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11315" PostTypeId="2" ParentId="11263" CreationDate="2016-08-17T22:52:12.680" Score="1" Body="&lt;p&gt;The heuristics I emphasize in my AP Calculus class are as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Check the nth term test before you do anything else. (As noted in your list)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Check to make sure you don't have a straightforward p-series, alternating series, or geometric series before proceeding with the other alternatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) A limit comparison test to an appropriate p-series almost always works for quotients involving polynomials and similar expressions (by similar, I mean expressions which have fractional powers instead of just non-negative integers, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) The ratio test almost always works for series involving exponential expressions and factorials. (Yes, the root test is also excellent for exponentials, but I tend to emphasize the ratio test since it's more versatile. I would not advise having students jump to the root test in most other situations.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a handful of oddballs for which these heuristics don't work, but these cover a vast majority of the commonly encountered problems.&lt;/p&gt;&#xA;" OwnerUserId="5417" LastEditorUserId="5417" LastEditDate="2016-08-18T10:08:16.427" LastActivityDate="2016-08-18T10:08:16.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11316" PostTypeId="2" ParentId="11310" CreationDate="2016-08-18T00:00:58.053" Score="1" Body="&lt;p&gt;Very difficult algebra (or maybe just longer thinking time) could prove the following is true:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\underbrace{1+3+5+7+\dots}_n=n^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where we simply add the first $n$ odd numbers.  Hopefully they will understand what $n^2$ means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof I'd probably follow:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\underbrace{1+3+5+7+\dots}_{n-1}=(n-1)^2=n^2-2n+1=\underbrace{1+3+5+7+\dots}_n-(2n-1)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And simply see that $2n-1$ would be the last odd number, and when subtracted out, we get what we started with.  Then, explain how induction works, so that since we know $1=1^2$, then $1+3=2^2$, since $1+3=2^2$, then $1+3+5=3^2$, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possibly you might want to give them a hint after 10 minutes, and wink wink point towards the sign on the wall that says &quot;FOIL = First Outside Inside Last&quot; on the wall (a must have for algebra teachers).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;If your students know what $i=\sqrt{-1}$ is, challenge them to find $\sqrt{i}$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Challenge your students to uncover a secret message where the letters are all systems of equations of the other letters and solving for just a few letters doesn't necessarily give you the solution to all of the letters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a=3b+2c$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$b=7e+g+h$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$c=\dots$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And the code to break could be&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$5,20,31,13,7,17|4,3,20,4|\dots$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $|$ means a space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not particularly impossible with substitution, but just time consuming and possibly fun.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="6220" LastEditDate="2016-08-18T20:31:40.303" LastActivityDate="2016-08-18T20:31:40.303" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11318" PostTypeId="2" ParentId="7243" CreationDate="2016-08-18T07:05:59.140" Score="6" Body="&lt;p&gt;One possibility to encourage sanity checks is to practise sanity-check-type questions and include them in tests. An integration-related example could be:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A student has calculated the area bounded by $y=x^2$, the $x$-axis, $x = 0$ and $x = 4$ to be 128 square units. Without using an integral, explain why 128 square units is too large to be a correct answer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Sketching the graph and calculating the area of the rectangle with base 4 and height 16 reveals the area to be at most 64 square units. (In fact, an improved upper bound thanks to convexity is 32, found by calculating the area of the triangle of base 4 and height 16.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2016-08-19T08:40:02.067" LastActivityDate="2016-08-19T08:40:02.067" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11319" PostTypeId="2" ParentId="10722" CreationDate="2016-08-18T11:53:30.863" Score="4" Body="&lt;p&gt;Here's a piece comparing virtual manipulatives to traditional teaching without manipulatives, in the context of community college remedial courses:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Violeta Menil and Eric Fuchs, &quot;Teaching Pre-Algebra and Algebra&#xA;  Concepts to Community College Students through the Use of Virtual&#xA;  Manipulatives&quot;, Improving Undergraduate Mathematics Learning, CUNY&#xA;  Office of Academic Affairs, 2012. (&lt;a href=&quot;http://www.cuny.edu/news/publications/imp.pdf&quot; rel=&quot;nofollow&quot;&gt;Link&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Statistically significant improvements were found regarding exam performance and student attitude for prealgebra/arithmetic classes. None are found for either in elementary algebra classes. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-08-18T11:53:30.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11320" PostTypeId="2" ParentId="11305" CreationDate="2016-08-18T11:55:12.307" Score="4" Body="&lt;p&gt;Here's a piece comparing virtual manipulatives to traditional teaching without manipulatives, in the context of community college remedial courses:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Violeta Menil and Eric Fuchs, &quot;Teaching Pre-Algebra and Algebra&#xA;  Concepts to Community College Students through the Use of Virtual&#xA;  Manipulatives&quot;, Improving Undergraduate Mathematics Learning, CUNY&#xA;  Office of Academic Affairs, 2012. (&lt;a href=&quot;http://www.cuny.edu/news/publications/imp.pdf&quot; rel=&quot;nofollow&quot;&gt;Link&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Statistically significant improvements were found regarding exam performance and student attitude for prealgebra/arithmetic classes. None are found for either in elementary algebra classes. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-08-18T11:55:12.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11321" PostTypeId="1" AcceptedAnswerId="11322" CreationDate="2016-08-18T15:29:31.037" Score="23" ViewCount="2406" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What does current evidence suggest: doing group work in mixed/balanced gender groups or doing group work in single gender groups? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Setting&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;College level mathematics/science course&lt;/li&gt;&#xA;&lt;li&gt;Group size approximately 3 or 4 (so for mixed gender groups there will often be a solo-male or solo-female; if there's research on which of these are better/more damaging I would also welcome that). &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Motivation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at &lt;a href=&quot;http://digitalcommons.ilr.cornell.edu/cgi/viewcontent.cgi?article=1429&amp;amp;context=articles&quot;&gt;this review article&lt;/a&gt; from 1999 for example, there are competing theories of how gender balance affect performance in group work. There are some that advocate single-sex groups to minimize discrimination and conflict, or to maximize satisfaction and supportive behavior; there are some that suggest that majority-minority groups will lead to the minority members being ostracized; there are some that suggest precisely the opposite. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When planning group works for groups of sizes 3 or 4 college students in an introductory mathematics course, is it better to have &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;all male, all female groups versus mixed groups;&lt;/li&gt;&#xA;&lt;li&gt;1 male 2 female versus 2 male 1 female versus 2 male 2 female groups?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Please only answer with evidence-based research, and not theoretical speculation. &lt;/p&gt;&#xA;" OwnerUserId="125" LastActivityDate="2016-08-19T15:03:38.620" Title="Gender and groupwork" Tags="|reference-request|gender|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11322" PostTypeId="2" ParentId="11321" CreationDate="2016-08-18T17:27:26.153" Score="16" Body="&lt;p&gt;Here is one article in &lt;em&gt;PNAS&lt;/em&gt;. The final sentence quoted below is a summary: &quot;creating small groups with high proportions of women [...] is one way to keep women engaged [...]&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dasgupta, Nilanjana, Melissa McManus Scircle, and Matthew Hunsinger. &quot;Female peers in small work groups enhance women's motivation, verbal participation, and career aspirations in engineering.&quot; &lt;em&gt;Proceedings of the National Academy of Sciences&lt;/em&gt; 112.16 (2015): 4988-4993.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Some quotes from the Abstract:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We provide experimental evidence showing that gender composition of small groups in engineering has a substantial impact on undergraduate women’s persistence. Women participate more actively in engineering groups when members are mostly female vs. mostly male or in equal gender proportions. Women feel less anxious in female-majority groups vs. minority groups, especially as first-year students. Gender-parity groups are less effective than female-majority groups in promoting verbal participation. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;These data suggest that creating small groups with high proportions of women in otherwise male-dominated fields is one way to keep women engaged and aspiring toward engineering careers. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-08-18T17:27:26.153" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11323" PostTypeId="2" ParentId="11307" CreationDate="2016-08-19T03:01:01.587" Score="5" Body="&lt;p&gt;Based on a quick Google Scholar search, it seems like there was a flurry of activity in this area (&quot;education literature on writing to learn mathematics&quot;) circa 1990, with at least 3 book-sized collections on the subject:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Connolly, Paul, and Teresa Vilardi. &lt;em&gt;Writing To Learn Mathematics and&#xA;  Science&lt;/em&gt;. Teachers College Press, 1234 Amsterdam Ave., New York, NY&#xA;  10027, 1989.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Sterrett, Andrew. &lt;em&gt;Using Writing To Teach Mathematics&lt;/em&gt;. MAA Notes,&#xA;  Number 16. Mathematical Association of America, 1529 18th Street NW,&#xA;  Washington, DC 20036., 1992.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Countryman, Joan. &lt;em&gt;Writing To Learn Mathematics: Strategies That&#xA;  Work&lt;/em&gt;. Heinemann, 361 Hanover St., Portsmouth, NH 03801-3912., 1992.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I haven't read them, so I don't what or how solid the findings are. More can be found here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://scholar.google.com/scholar?hl=en&amp;amp;q=writing+in+mathematics+education&quot; rel=&quot;noreferrer&quot;&gt;https://scholar.google.com/scholar?hl=en&amp;amp;q=writing+in+mathematics+education&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-08-19T03:01:01.587" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11324" PostTypeId="2" ParentId="11321" CreationDate="2016-08-19T03:13:24.297" Score="8" Body="&lt;p&gt;Peter Liljedahl has done &lt;a href=&quot;http://www.peterliljedahl.com/wp-content/uploads/Visibly-Random-Groups.pdf&quot; rel=&quot;nofollow&quot;&gt;research&lt;/a&gt; showing that visibly random grouping increases student participation. I have been doing this in some math classes, and I like how it is working out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Citation: Liljedahl, P. (in press). The affordances of using visually random groups in a mathematics classroom. In Y. Li, E. Silver, &amp;amp; S. Li (eds.) Transforming Mathematics Instruction: Multiple Approaches and Practices. New York, NY: Springer.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2016-08-19T15:03:38.620" LastActivityDate="2016-08-19T15:03:38.620" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11326" PostTypeId="1" CreationDate="2016-08-19T12:44:42.867" Score="12" ViewCount="259" Body="&lt;p&gt;&quot;Course-based undergraduate research experiences&quot; (CUREs, or CBEs) are being explored&#xA;in various STEM fields, especially &lt;a href=&quot;https://curenet.cns.utexas.edu/content/about-curenet&quot; rel=&quot;noreferrer&quot;&gt;biology&lt;/a&gt;,&#xA; chemistry, geology. &#xA;Here is one geology link that gives a flavor: &quot;&lt;a href=&quot;http://blogs.agu.org/geoedtrek/2015/07/15/are-you-looking-for-a-cure-course-based-undergraduate-research-experience/&quot; rel=&quot;noreferrer&quot;&gt;Are you looking for a CURE?&lt;/a&gt;&quot; In the US, some of this activity&#xA;is funded by the National Science Foundation (NSF); e.g., see this &lt;a href=&quot;http://www.lifescied.org/content/13/1/29.full&quot; rel=&quot;noreferrer&quot;&gt;Meeting Report&lt;/a&gt;. One characteristic of CUREs is that the research is &quot;authentic&quot;: this is not just problem-based learning,&#xA;but rather explorations whose outcome is unknown both to the students and to the instructor, and ideally unknown to anyone.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Has anyone attempted course-based research in undergraduate math courses?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am particularly interested in lower-level math courses, e.g., Discrete Math,&#xA;but would appreciate hearing of (or links to) any explorations along these lines.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-08-19T16:06:40.003" LastActivityDate="2017-09-12T04:40:55.240" Title="Course-based undergraduate research experiences in math" Tags="|mathematical-pedagogy|course-design|undergraduate-research|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11327" PostTypeId="2" ParentId="7243" CreationDate="2016-08-19T19:43:01.857" Score="3" Body="&lt;p&gt;This is something that I've been specifically grappling with in my college remedial algebra classes for the last few years. JoeTaxpayer's observations in his answer very much match my own (that many students have never heard of checking solutions to equations until I make a topic out of it -- in fact, I'm somewhat embarrassed by how many years I spent thinking it was &quot;obvious&quot; before I discovered this and made it a priority). The thing I want to add is: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;One needs to get away from multiple-choice testing.&lt;/strong&gt; If you already do this, that's great, but the institutional culture is such that many students see nothing but multiple-choice tests for the first 14+ years of math classes. The problem here is that it short-circuits the need for checking in the first place: if one's answer is in the multiple-choice options, then that itself serves as a kind of check. If it's &lt;em&gt;close&lt;/em&gt; to an option then you're usually correct in picking that one. Otherwise, just picking an option at random is far less work than a proper check and justification. And yes, this is under a significant time constraint. So students never experience having something truly high-stakes depending on a single solution, with no outside safety net, and the value of &lt;em&gt;really knowing that you're right&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Amazingly, I find that even if I do make this a priority in my elementary algebra classes, and promise (and follow through) that it's &lt;em&gt;required on every single test all semester&lt;/em&gt;, and those tests are indeed not multiple-choice, many of my students simply give up on checking solutions, and leave that part on all the tests blank over and over again. &quot;It's just too hard; I don't get it&quot;, they'll say; even after practicing the skill effectively every day for an entire semester. It's incredibly discouraging, and I wish there were some solid research on why that is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I should say that there's an equivalent &quot;block&quot; that happens in our remedial college arithmetic classes when I emphasize checking answers by rounding/estimation on a daily basis. Students are agog, claim to have never heard of it before, cannot grok it all semester long, leave those sections blank on every test, etc. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-05-26T13:45:33.603" LastActivityDate="2017-05-26T13:45:33.603" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11328" PostTypeId="2" ParentId="7188" CreationDate="2016-08-21T04:46:37.137" Score="5" Body="&lt;p&gt;My philosophy is that, yes, students should be told when they're wrong. On the one hand, we surely all have self-discovery experiences like the OP's (with the alleged &quot;perpetual motion machine&quot;) that were transformative, highly memorable, and likely very dear to each of us. But: We simply don't have the time to self-construct &lt;em&gt;all&lt;/em&gt; knowledge that way. Indeed, the whole power of language and writing is to be able to share in others' experiences without re-inventing the wheel individually every time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the way that this is explained or revealed to the student can of course take many approaches. Requiring a detailed justification of each step is one way. Asking pointed questions around things they've overlooked is another way. But in general, the risk of a student proceeding forward with mistaken conceptions, and the avalanche of problems that can cause down the road, is so great that I wouldn't want someone leaving my classroom thinking that something wrong is right. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note: With my remedial college students, I find it rather stunning the degree to which they will &lt;em&gt;completely tune me out&lt;/em&gt; until I specifically say the magic words, &quot;Your answer is wrong&quot;. For example: In many cases during in-class exercises, if I start pointing out mistakes in student writing (say: mis-transcriptions, substitutions, etc.), then they'll nod blankly and go &quot;Yeah, I get it&quot; or something, as though I'm being vacantly pedagogical. Not until I say &quot;Your answer is wrong&quot; do they sit up and go &quot;Really?&quot; and then start looking to make corrections. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-08-21T06:47:59.693" LastActivityDate="2016-08-21T06:47:59.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11330" PostTypeId="1" CreationDate="2016-08-21T19:59:01.457" Score="7" ViewCount="7698" Body="&lt;p&gt;I want to introduce Monte Carlo methods for a group of 16-18-years-old high school students.&#xA;Besides classic examples (coin flips and count of heads/tails, rolls of a pair of dice) which other examples could I do?&#xA;I thought to present the solution of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Birthday_problem&quot;&gt;birthday problem&lt;/a&gt; by simulation, but I would &#xA;like some other problem that mimics a real life situation. Any suggestion?&lt;/p&gt;&#xA;" OwnerUserId="7067" LastActivityDate="2016-10-22T13:17:31.467" Title="Monte Carlo real life examples" Tags="|examples|" AnswerCount="7" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11331" PostTypeId="2" ParentId="11300" CreationDate="2016-08-21T20:08:26.870" Score="-2" Body="&lt;p&gt;I suggest not using a notation at all. &quot;line $AB$&quot;, &quot;segment $AB$&quot;, &quot;ray $AB$&quot; seems OK to me, and it forces you to write your proofs in text rather than using shorthand and over-formalization.&lt;/p&gt;&#xA;" OwnerUserId="4930" LastActivityDate="2016-08-21T20:08:26.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11332" PostTypeId="2" ParentId="10876" CreationDate="2016-08-22T03:02:09.397" Score="8" Body="&lt;p&gt;A colleague of mine did his dissertation on such matters. His book, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1534803173&quot;&gt;Mathematical Misconceptions of College-Age Algebra Students&lt;/a&gt; might be of interest. &lt;/p&gt;&#xA;" OwnerUserId="6949" LastEditorUserId="6949" LastEditDate="2016-08-22T03:33:11.990" LastActivityDate="2016-08-22T03:33:11.990" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11333" PostTypeId="1" CreationDate="2016-08-22T11:33:23.023" Score="2" ViewCount="61" Body="&lt;p&gt;I am looking to calculate an OP Score (Overall Position). It is an outdated Queensland system of post-secondary-entrance rankings due to be phased out in 2018. None the less I am looking for &lt;strong&gt;specific&lt;/strong&gt; information in regards to calculation an OP. The &lt;a href=&quot;http://www.qcaa.qld.edu.au/senior/tertiary-entrance/op&quot; rel=&quot;nofollow&quot;&gt;basics from the Queensland Curriculum and Assessment Authority (QCAA)&lt;/a&gt; is of little help in actually calculating a score.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anyone is knowledgeable in this area it would be of great help if you could outline the basic of such calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The information I have:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Subject Rankings (SA#, HA#, VHA#...)&lt;/li&gt;&#xA;&lt;li&gt;School QCS Average (~173)&lt;/li&gt;&#xA;&lt;li&gt;Position in each Subject (1st, 2nd, 3rd, 4th...)&lt;/li&gt;&#xA;&lt;li&gt;OAI cutoffs (124 = OP1, &amp;lt;120 = OP2....)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;From my interpretation of the resources at hand (QCAA):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A SAI is deductible from subject ranking and positioning&lt;/li&gt;&#xA;&lt;li&gt;A scaled SAI is deductible from School QCS 173&lt;/li&gt;&#xA;&lt;li&gt;An OAI is deductible from interschool scaling (Unknown process)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Any specifics on calculating an OP score will be of great help.&lt;/p&gt;&#xA;" OwnerUserId="7069" LastEditorUserId="143" LastEditDate="2016-08-22T13:09:44.973" LastActivityDate="2016-08-22T13:09:44.973" Title="OP (Overlall Position) Calculations for Queensland's post-secondary-entrance rankings" Tags="|secondary-education|" AnswerCount="0" CommentCount="4" ClosedDate="2016-08-24T19:36:29.577" ContentLicense="CC BY-SA 3.0" />
  <row Id="11334" PostTypeId="2" ParentId="11310" CreationDate="2016-08-22T16:50:14.067" Score="5" Body="&lt;p&gt;Here is a real-life example that I used in a class with some success. The enrollees were middle school and high school teachers, who were in their first year of teaching (not having arrived with necessarily strong coursework in mathematics, but let us not diverge for the moment). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on the interests of the (Boston-based) class, I phrased it in terms of scalping (Red Sox) baseball tickets. Students seemed engaged, and one of them emailed me the next day to say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For what it's worth: I felt like Will Hunting when I got that Red Sox ticket markup problem yesterday. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(So, some anecdotal evidence that it was received okay!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I phrase it below in terms of expensive jeans at a vintage clothing store, but I'm sure you can identify the underlying mathematics and modify the context as you see fit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Real Life A:&lt;/strong&gt; Your boss at a vintage clothing store tells you a particular brand of jeans have become very popular, and to mark them up by 10%. A few weeks pass, and she asks you to mark them back down by 10%. Is the final price more than, less than, or the same as the original price?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Real Life B:&lt;/strong&gt; Same as &lt;strong&gt;A&lt;/strong&gt; but the scenario is marking down by 10% first, and back up by 10% later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Real Life C:&lt;/strong&gt; Suppose the jean price in &lt;strong&gt;A&lt;/strong&gt; is initially $200; leaving aside whether this is an &lt;em&gt;exorbitant&lt;/em&gt; amount to pay for a pair of pants, what will be the final price?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a pretty common misconception that the before- and after-prices will be the same, so you could weave this into a tale of your boss (or maybe a nefarious manager, or whomever else) chastising you for messing up the prices. (&quot;I told you to mark it back up by 10%; they should all say $200!&quot; etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part &lt;strong&gt;C&lt;/strong&gt; is probably the easiest among the three, since it deals with concrete numbers. But I have a habit of wondering about &quot;inverting&quot; problems; in the $200 case of &lt;strong&gt;C&lt;/strong&gt;: &lt;em&gt;Can you use the percentage increase and decrease to determine the final price?&lt;/em&gt; With an eye towards inversion, this led me to ask whether you could use the final price to determine the percentage increase and decrease:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Real Life D:&lt;/strong&gt; A friend of yours is explaining that she was asked to mark a pair of $200 jeans up by some percentage, then back down by the same percentage, and her boss is convinced something went wrong because the final price is now less than the original. The boss is even talking about docking her pay, which seems unfair since your friend has done &lt;em&gt;nothing&lt;/em&gt; wrong; she followed directions exactly, and now the jeans are priced 98 cents less than they were at the outset.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In scenario &lt;strong&gt;D&lt;/strong&gt;, this is where you'd tune out and wonder: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Just what &lt;em&gt;was&lt;/em&gt; the percentage by which the jeans were marked up and then marked down?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the kind of problem that (I think) might fascinate a mathematically inclined individual (especially if there was scrap paper -- or a napkin -- nearby). I think it is a nice place to point out where one might become curious and ask a mathematical question; even if your students do not share in this mathematical curiosity initially, it is a habit of mind that I think is worth fostering. Plus, you can try to work it out and then check correctness by asking your friend for the percentage at the end!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Omitting units, and an explanation around the setup, here is one algebraic approach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$200(1+p)(1-p) = 199.02 \iff 200(1-p^2) = 199.02 \iff 200 - 200p^2 = 199.02$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\iff 0.98 = 200p^2 \iff 0.0049 = p^2 \iff 0.07 = p$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the percentage change is 7%.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The initial equation is a true quadratic that can be solved with the quadratic equation or a calculator or whatever else; the solving method above proceeds by drawing from the knowledge around the product $(1+p)(1-p)=(1-p^2)$, and there is a lot more than be unpacked from the given approach: figuring out how to write the initial equation, figuring out how to solve it by hand (how do you hand compute $0.98/200$? well, you could divide by $2$ then divide by $100$), possibly thinking of $0.0049 = 49/1000$ for which finding the sqrt to be $7/100 = 0.07$ might be easier, investigating the other possible &quot;solution&quot; of -7%, and, more generally, this problem has all the components that I think one hopes a student can do by the end of an algebra unit on solving quadratics: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Solve a problem in more than one way (e.g., the method above as well as with the quadratic equation, though a thoughtful version of guess-and-check by recalling &lt;strong&gt;C&lt;/strong&gt; and trying another value like 5% can work, too); represent a real life problem using mathematical notation; and sense-making fluently with decimals, fractions, and percentages in carrying out the algebra.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-08-22T16:50:14.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11335" PostTypeId="2" ParentId="11330" CreationDate="2016-08-22T20:24:38.710" Score="3" Body="&lt;p&gt;Here are 2 computer examples using random numbers and a unit square.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;!. Find the area of a circle in the unit square. &#xA;2. Two people decide to meet between noon (0) and 1 o'clock at a certain location with the understanding that the longest either should wait for the other is 15 minutes (.25 of a unit.) What is probability of a meeting taking place?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In both cases 2 random numbers are generated (x and y)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case 1 the distance from (0.5, 0.5) to x,y is computed and if &amp;lt;= .5^2 then plot &amp;amp; color the point green else red.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case 2 the x and y represent the different times that each arrives.&lt;br&gt;&#xA;if |x-y| &amp;lt;= 0.25 then plot &amp;amp; color the point green else red.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In both case count the number of red/green points.&lt;/p&gt;&#xA;" OwnerUserId="6333" LastEditorUserId="6333" LastEditDate="2016-08-22T20:50:48.370" LastActivityDate="2016-08-22T20:50:48.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11336" PostTypeId="2" ParentId="11330" CreationDate="2016-08-23T07:36:10.963" Score="5" Body="&lt;p&gt;Anything with probabilistic estimates should work. As a demonstration, an idea mentioned already by @Joseph O'Rourke, that is, estimating $\pi$ using &lt;a href=&quot;https://en.wikipedia.org/wiki/Buffon%27s_needle&quot; rel=&quot;noreferrer&quot;&gt;Buffon's needle&lt;/a&gt; is excellent. Estimating area of a shape by calculating the number of random points that fall into it, could also work, but it is not as illuminating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some examples that are closer to real life, you could estimate the proportion of red cards in a stack by picking a few random samples (explain connection to pools and voting). You could also estimate average height in the class by picking a few random samples and explain connection to various statistical estimates on population and how it differs from census data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Game-related algorithms use Monte Carlo a lot. You could make them play multiple &lt;em&gt;random&lt;/em&gt; games of tic-tac-toe and for each put $+1$ on every square of color that won and $-1$ on every square of color that lost (zero otherwise). Sum all these numbers and see how it corresponds to good/bad moves (a few hundred playouts should be ok, so it would work for a big class; for a smaller class use symmetries and average the four corners and the four middles). See also a small script &lt;a href=&quot;https://repl.it/CqZL/0&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have too few $\geq 9$ sided dice, teach them first how to make $12$-sided die from a $6$-sided die and a coin (or two $6$-sided dice). Then also teach them how to make $5$-sided die from a $6$-sided die simply by repeating the throw if the result &quot;falls outside the range&quot; (or in general $k$-sided die from $n$-sided die for any $2\leq k &amp;lt; n$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also try them various algorithms that use Monte Carlo approach like quicksort:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;You are given a sequence of numbers.&lt;/li&gt;&#xA;&lt;li&gt;Sort it naively if the sequence is of length $\leq 4$ (to make it less tedious).&lt;/li&gt;&#xA;&lt;li&gt;Pick a random number $x$ from it and split the sequence into two subsequences $\leq x$ and $&amp;gt; x$.&lt;/li&gt;&#xA;&lt;li&gt;Repeat recursively for each subsequence.&lt;/li&gt;&#xA;&lt;li&gt;Write down $\mathrm{result}_1, x ,\mathrm{result}_2$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="42" LastEditorUserId="42" LastEditDate="2016-08-23T08:47:43.097" LastActivityDate="2016-08-23T08:47:43.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11339" PostTypeId="2" ParentId="11330" CreationDate="2016-08-24T15:59:04.073" Score="3" Body="&lt;p&gt;Disease transmission might work well. Assume a certain percentage of unvaccinated students (some localities have high single-digit percentages) in the school when one person contracts measles, for example. About 90% of unvaccinated people who have close contact with a measles patient in the first four days or so will catch the disease. If students keep track of who they contact daily in classes, sports, etc., you could run a few cycles to see how fast an outbreak spreads given various levels of unvaccinated populations. Not only is this highly relevant to real-life vaccination decisions and policy, but it could be extended to more sophisticated models if you consider how soon patients are quarantined, etc.&lt;/p&gt;&#xA;" OwnerUserId="5417" LastActivityDate="2016-08-24T15:59:04.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11340" PostTypeId="1" CreationDate="2016-08-24T18:17:15.133" Score="5" ViewCount="664" Body="&lt;p&gt;Throughout elementary, middle and high school mathematics is quite merely about memorizing concepts and formulas, understanding the theorems (without their proofs) and applying acquired knowledge in examinations. It is not perfect, I concede, but it is almost as good as is needed. I am a graduate student who is devoted to pure mathematics, and I have about 4 years of teaching experience with a great variety of students. I hope that this is gives me enough credit to discuss a matter of this sort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is (to some extent) true that mathematics is “art”; it is about creating patterns and connecting them with each other, etc. This is what most professional pure mathematicians like to think and say to the public. Now, there is the harsh truth that a great portion of pure mathematics undergraduates drop out or, after graduating, either run away to some applied area of mathematics or decide to teach mathematics in middle or high schools; “real” mathematics is very difficult even for people who were very good/excellent in high school math. Moreover, many of the students who safely land in graduate school find it difficult to get accepted in the graduate programs they wanted to get in because of their incompetency, and so they end up doing something they did not really want to do. &lt;a href=&quot;https://www.youtube.com/watch?v=n12bfWTw9Hk&quot; rel=&quot;nofollow&quot;&gt;I am not trying to suggest that advanced pure mathematics is a nightmare&lt;/a&gt;, but my point is that if the education in school was made so that students would learn to “appreciate” the beauty of mathematics and get motivated to pursue pure mathematics, most of them will fail later on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One asks: why, then, are most professional pure mathematicians discontent with the current style of math education? In my opinion, the answer is: most of them are geniuses and have no idea what mathematics looks like to inferior beings, and the rest simply forgot what kind hard work is required for learning and digesting advanced concepts and are unaware of how it might be much more difficult for less capable people. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My conclusion is that mathematics education is in a fine state in today’s world. It is not “fundamentally wrong” and “delusive”; it is just realistic. If you are not extremely comfortable dealing with numbers, elementary functions and geometric figures, you do not stand a chance in advanced mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make this a question, allow me to pose following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do you disagree with this? What important points am I missing in my defense of the current school education system? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7077" LastEditorUserId="7077" LastEditDate="2016-08-24T19:25:50.033" LastActivityDate="2016-08-24T19:41:20.850" Title="Is the current education system as bad as most critics and famous pure mathematicians try to convey?" Tags="|mathematical-pedagogy|secondary-education|teaching|" AnswerCount="1" CommentCount="7" ClosedDate="2016-08-25T04:53:19.143" ContentLicense="CC BY-SA 3.0" />
  <row Id="11342" PostTypeId="1" CreationDate="2016-08-24T19:16:47.830" Score="10" ViewCount="2244" Body="&lt;p&gt;Why don't we discard the traditional pencil and paper method of graph plotting in high schools and for freshers at colleges since there are many electronic devices doing the graphing? And please where do I get citation on studies that show students graphing ability&lt;/p&gt;&#xA;" OwnerUserId="7079" LastEditorUserId="376" LastEditDate="2016-08-24T20:58:49.187" LastActivityDate="2016-08-26T06:37:30.227" Title="Is there any advantage of plotting graphs using traditional paper and pencil method?" Tags="|undergraduate-education|secondary-education|reference-request|graphing|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11343" PostTypeId="2" ParentId="11340" CreationDate="2016-08-24T19:41:20.850" Score="15" Body="&lt;p&gt;I think this question is, probably accidentally, responding to a strawman argument.  It presumes that the criticism of math education coming from pure mathematicians is some combination of &quot;students aren't learning sufficiently advanced math&quot;, &quot;students aren't sufficiently prepared to become mathematicians&quot;, and &quot;students don't appreciate the beauty of math&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people certainly make these claims, but they're not the main ones being made by mathematicians who criticize math education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main criticism of math education is that that students end up learning rote procedures without any understanding or ability to use them outside of the classroom.  Students don't need to learn more advanced math for these problems to become evident: consider the many students who won't notice if a calculator spits out an answer which is off by orders of magnitude, or who have great difficulty solving word problems even when they have no difficulty with the corresponding equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, the complaint isn't &quot;why do we spend so much time getting our students comfortable with numbers when we want them to do advanced mathematics?&quot;, it's &quot;why aren't our students comfortable with numbers?&quot;&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-08-24T19:41:20.850" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11344" PostTypeId="2" ParentId="11342" CreationDate="2016-08-25T00:36:59.400" Score="3" Body="&lt;p&gt;Students are expected to graph a quadratic equation, a parabola, and indicate on the graph the five points of interest - &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;vertex&lt;/li&gt;&#xA;&lt;li&gt;2 intercepts&lt;/li&gt;&#xA;&lt;li&gt;y-intercept&lt;/li&gt;&#xA;&lt;li&gt;symmetric point&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you graph this on a computer or smart phone, you are given the answers, same as plugging in the equation and asking a calculator to spit out the two solutions. I can't cite a study, but it seems to me the jumping to an answer with no work in between isn't as learning as it should be. (And I'm aware I just a assigned a different part of speech to the word &lt;em&gt;learning&lt;/em&gt; than you'll see in the dictionary.)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-08-25T03:16:19.320" LastActivityDate="2016-08-25T03:16:19.320" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11345" PostTypeId="2" ParentId="11342" CreationDate="2016-08-25T03:25:34.237" Score="9" Body="&lt;p&gt;Knowing basic graphs is a simple way to geometrically understand a wealth of data about functions. If a student understands what a point on a graph means as it relates to a function then there is so much that can be remembered by a simple picture. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, $\sin \theta = \frac{1}{2}$. For the careless calculator user, they'll answer $\theta = \pi/6$ or $30$ degrees. But, if you think graphically, then you understand $\sin \theta = \frac{1}{2}$ is the intersection of two graphs. The sine function and the horizontal line $y=\frac{1}{2}$ intersect infinitely many times. The calculator produced zero percent of the total solution set. If the student trusts calculators and doesn't have a lot of mathematical common sense then they'll have a hard time telling when they have been tricked by the calculator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, calculators will improve. However, there will always be a need for students of math to understand math. Graphing by hand is in fact hands-on experience of math. When you create a graph it gives you a chance to get to know a function in some sense. Enough experience in this regard eventually brings the concept of a function to life. If all you do is punch formulas into a blackbox and like a robot sloppily copy a window for your &quot;graph&quot; it hardly allows you to internalize much. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there is a right way to use a graphing calculator and there is also a wrong way to graph by hand. I graphed by hand AND used a graphing calculator to explore possibilities. I don't see many students really doing this sort of exploration. I often see students in my classes who are graphically clueless. I assume this is in large part because they used graphing calculators too much in their youth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, the interplay between algebra and graphing is fascinating and I hate to see students deprived of their intuition for the sake of some supposed efficiency. Technology should be reserved for ugly graphs. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2016-08-25T03:25:34.237" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11346" PostTypeId="2" ParentId="11330" CreationDate="2016-08-25T12:40:59.660" Score="5" Body="&lt;p&gt;How about the famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Monty_Hall_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Monty Hall Problem&lt;/em&gt;&lt;/a&gt;&amp;mdash;stick, or switch?&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Vu6x5.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Vu6x5.jpg&quot; alt=&quot;MontyGoat&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://www.youtube.com/watch?v=4Lb-6rxZxx0&quot; rel=&quot;nofollow noreferrer&quot;&gt;YouTube&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;There is a nice Python simulation described &#xA;&lt;a href=&quot;https://fiftyexamples.readthedocs.io/en/latest/monty-hall.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;at this link&lt;/a&gt;.&#xA;A sample run verifies the counterintuitive $\frac{2}{3}$ vs. $\frac{1}{3}$ probabilities:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/2NLqY.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2NLqY.png&quot; alt=&quot;MontyHall&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-08-25T13:37:02.253" LastActivityDate="2016-08-25T13:37:02.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11347" PostTypeId="1" CreationDate="2016-08-25T13:59:46.587" Score="6" ViewCount="181" Body="&lt;p&gt;Selden &amp;amp; Selden (2011) listed 41 difficulties their students had in an experimental proving course into 9 categories. Unfortunately I haven't found similar work. Thus, my question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a (more detailed) catalog of the misconceptions, typical errors or problems undergraduates have / make while proving?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Selden, Annie, and John Selden. &quot;Mathematical and non-mathematical university students’ proving difficulties.&quot; &lt;em&gt;Proceedings of the 33rd annual conference of the North American chapter of the international group for the psychology of mathematics education. Reno, NV.&lt;/em&gt; 2011. Retrieved from &lt;a href=&quot;http://ejournal.narotama.ac.id/files/Mathematical%20and%20non-mathematical%20university%20students&amp;#39;%20proving%20difficulties.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://ejournal.narotama.ac.id/files/Mathematical%20and%20non-mathematical%20university%20students'%20proving%20difficulties.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="5097" LastEditDate="2016-08-25T14:36:36.587" LastActivityDate="2016-08-25T14:36:36.587" Title="Catalog of undergraduate's misconceptions / problems while proving" Tags="|undergraduate-education|reference-request|proofs|students-mistakes|undergraduate-research|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11348" PostTypeId="1" CreationDate="2016-08-25T17:14:20.573" Score="5" ViewCount="272" Body="&lt;p&gt;Do you know anyone or any institution interested in accepting a donation of a relatively complete set of SMSG Materials?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I post offering to donate a small collection of School Mathematics Study Group materials which my wife inherited from her father, James C. McCaig. Jim's second career was as a 7/8th Grade Math Teacher in the Cupertino Union School District in Northern California. During his tenure he participated as a co-author of a “Special Edition” of the SMSG Secondary School Mathematics curriculum developed “…explicitly for low-acheiving mathematics students at the 7th and 8th grade levels”.  This edition was sponsored by Stanford University and copyrighted in 1970 though not formally published for distribution to schools. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our collection includes the the complete Special Edition including both Student Text Books and Teacher’s Commentaries for all 18 Chapters of the Special Edition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition to the Special Edition the collection includes all or part of:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1959 - 61   Junior High School Curricula Texts and matching Teacher’s commentaries&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1962 - 64         Secondary Curriculum Texts and Teacher’s Commentaries for Algebra, Matrix Algebra, and Geometry&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1965            Revision to Secondary Mathematics Curriculum&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1970            Revision to Secondary Mathematics Curriculum&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1964 - 70         Total of 36 &quot;Supplementary and Enrichment” or “Reprint” Booklets to extend the Secondary Curriculum&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1961 - 69   Total of 14 Reports and Studies on development of    Mathematics Curriculums&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The complete collection fits in (2) 1.5 cu.ft. book boxes. We can provide more detail in a spreadsheet for interested parties that reply to this post.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are interested or know someone who is please reply to this post.&lt;/p&gt;&#xA;" OwnerUserId="7082" LastEditorUserId="7082" LastEditDate="2016-08-26T15:41:31.690" LastActivityDate="2016-08-29T00:49:45.923" Title="SMSG Materials Available" Tags="|reference-books|struggling-students|new-math|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11349" PostTypeId="2" ParentId="11342" CreationDate="2016-08-26T01:53:26.160" Score="6" Body="&lt;p&gt;I once worked in a factory that used &quot;six sigma&quot; to improve its production processes and yields.  &quot;Six sigma&quot; involves a lot of graphs.  In half of the factory, the manufacturing engineers used computers to prepare blank charts, and the line workers drew the graphs in the charts themselves.  In the other half of the factory, the manufacturing engineers used computers to both format and prepare the graphs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the half of the factory where the workers drew the graphs:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The workers figured out which data collection was useful, and convinced the manufacturing engineers to simplify the data collection and graphing accordingly.&lt;/li&gt;&#xA;&lt;li&gt;The workers noticed problems sooner.&lt;/li&gt;&#xA;&lt;li&gt;More suggestions for process improvements were made.&lt;/li&gt;&#xA;&lt;li&gt;More suggestions for process improvements were implemented.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The line workers' foremen were the ones who noticed that the teams that graphed their own data made more progress.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2016-08-26T01:53:26.160" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11350" PostTypeId="2" ParentId="11342" CreationDate="2016-08-26T06:37:30.227" Score="8" Body="&lt;p&gt;By replacing hand-graphing with computer graphing, we place a black box between equations and their graphs in the experience of our students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Black boxes are bad for understanding. In fact, this is the entire point of a black box - to create a level of abstraction that frees a user from the gritty details of some particular task. Obviously there are times where this is appropriate. Lots of times. Who needs to know, while driving, how the fuel injection of their carburetor is working? Seriously considering this complicated mechanical process, while driving, would probably be dangerous (distracted driving, etc). Luckily all of our cars' complicated moving parts have been tucked away under the hood - a black box.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;School mathematics is maybe the best possible example of a place where black boxes - the removal of complexity from the user's view - are not appropriate. (Or at least not universally appropriate). Here are some points to that effect, pertaining to the computer graphing vs manual graphing question.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;An equation and a graph are different, but equivalent, representations of some relationship between quantities. Students who have had regular practice at sketching graphs will know and understand this, &lt;em&gt;with or without being told&lt;/em&gt;. Students whose graph production has been dominated by computers &lt;em&gt;may&lt;/em&gt; know this, but I promise that many will not. 'Graphs' will be one more abstract thing which must, for some reason, be produced for sacrifice at the math alter. And in any case, I believe that the students who do still know and understand this will have this understanding in a diminished way, as follows:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Manual graphing of unfamiliar functions or combinations of known functions reveals the particular 'forces' which inform the shape of the graph. When $ y = -5x + x^2 - 2^\frac{x}{5} $ is graphed manually, you can feel each term taking control of the action as you move along.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The above example will seem unreasonable to some - how should a student be prepared to graph a curve whose interesting bits occur near the origin and ~(50, 1300)? The answer is investigation, estimation, etc. Take some derivatives, find (even estimates for) maxima/minima/inflection points, do thought experiments on where different terms must overpower one another, etc. These 'equation debugging' skills are the product of cumulative experience, and, crucially, they are all highly transferable skills, and part of the general 'rigorous investigatory ethic' which in my mind is the entire point of mathematics education.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To put a question back to your question: why are your students producing graphs at all? If the purpose is to crank out graphs, then of course the computer is the way to go. But if the purpose is to produce something that enriches the mental model of the relationship, then the computer is lacking in some ways - the production itself lends a lot to the mental model of the relationship.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2016-08-26T06:37:30.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11351" PostTypeId="2" ParentId="11330" CreationDate="2016-08-26T13:04:10.837" Score="3" Body="&lt;p&gt;Here is an Excel example. It simulates the winnings of a person (or persons) who buys a \$1 &quot;Pick 3&quot; lottery ticket (which wins \$500 with probability 1/1000, hence nets \$499 with probability 1/1000). For 1 player it looks like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TRN3g.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TRN3g.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In A2 I have the formula &lt;code&gt;=IF(RAND()&amp;lt;0.001,499,-1)&lt;/code&gt;. In A3 I have the formula &lt;code&gt;=A2+IF(RAND()&amp;lt;0.001,499,-1)&lt;/code&gt;, which I copied down 363 rows and then plotted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also did it for 10 players, with the same formulas plus a sum of the final results (with conditional formatting to display red if a net collective loss). It looks something like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bfb9m.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bfb9m.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point of the simulation was to better understand the following &quot;paradox&quot;. I worked in a factory with several hundred employees for two summers when I was an undergraduate (this was in the mid 80s before the rust belt became rusty). Many times during those summers I heard of workers there winning the lottery. This created the strong psychological impression that winning this lottery was a fairly easy thing to do. I don't have the numbers, but I got the impression that a couple dozen workers were buying Pick 3 tickets almost every day in such a way that there was 1 or 2 winners per month. The simulation illustrates how it is possible to be in a small group of friends where at most points in time several of your friends are winners but nevertheless the group of friends is steadily becoming collectively poorer in time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are interested in the spreadsheet itself, send me an e-mail and I'll send you a copy.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2016-08-26T13:04:10.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11352" PostTypeId="2" ParentId="11330" CreationDate="2016-08-26T21:50:30.780" Score="2" Body="&lt;p&gt;Correctness of algebraic and trigonometric and other identities can usually be tested quite effectively by plugging in random argument-values, and seeing whether the outputs match.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe this doesn't count as &quot;practical&quot; for them, though, ...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-08-26T21:50:30.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11353" PostTypeId="1" AcceptedAnswerId="11354" CreationDate="2016-08-26T23:51:57.503" Score="4" ViewCount="317" Body="&lt;p&gt;Thinking about the counterintuitive &lt;em&gt;Monty Hall Problem&lt;/em&gt; (stick or switch?),&#xA;revisited in &lt;a href=&quot;https://matheducators.stackexchange.com/a/11346/511&quot;&gt;this ME question&lt;/a&gt;,&#xA;I thought I would issue a challenge:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Give in one (perhaps long) sentence a convincing explanation of why &lt;em&gt;switching&lt;/em&gt; is twice as likely to lead to winning as &lt;em&gt;sticking&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Assume the game assumptions&#xA;are pre-stated and clear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The probabilities are not even close, so there should be a convincing explanation after all&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Monty_Hall_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;the discussion of this topic&lt;/a&gt;,&#xA;even though &quot;1,000 Ph.D.&quot;s got it wrong (in 1990 when it first went viral).&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-08-31T23:54:14.970" Title="Monty Hall challenge" Tags="|mathematical-pedagogy|probability|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11354" PostTypeId="2" ParentId="11353" CreationDate="2016-08-27T00:19:51.977" Score="9" Body="&lt;p&gt;If you &quot;stay&quot; then you win when the prize is behind the &lt;em&gt;one&lt;/em&gt; door your originally selected, yet when you &quot;switch&quot; you win when the prize is behind one of the &lt;em&gt;two&lt;/em&gt; doors you originally did not select.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="511" LastEditDate="2016-08-27T00:44:38.563" LastActivityDate="2016-08-27T00:44:38.563" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11355" PostTypeId="2" ParentId="11353" CreationDate="2016-08-27T22:39:10.677" Score="4" Body="&lt;p&gt;My preferred explanation - &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key thing to understand is that MH knows the correct door. Say there were a thousand doors. Your chance of choosing the correct door is 1/1000. Now, MH has 999 doors, and after opening 998, there's one left. In effect, he has reduced all the chance, the .999 to that one door. Now, by switching, your chance of success is 999/1000 because your chance of being right (pre-switch) was always 1/1000, and wrong, 999/1000. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-08-28T14:14:00.670" LastActivityDate="2016-08-28T14:14:00.670" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11357" PostTypeId="2" ParentId="11348" CreationDate="2016-08-29T00:49:45.923" Score="4" Body="&lt;p&gt;I know a research group in the School of Education at the University of Michigan that would be very happy to receive them.  My email address is in my profile.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-08-29T00:49:45.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11358" PostTypeId="1" CreationDate="2016-08-30T18:12:46.267" Score="6" ViewCount="540" Body="&lt;p&gt;I am looking for a good book for junior-senior US math majors on functional analysis.  Some background and detail:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last semester I taught a special topics course in functional analysis to junior-senior level math majors.  It was the first time teaching functional analysis both for me and for my institute as a whole.  I used the book &lt;a href=&quot;http://www.springer.com/us/book/9781848000049&quot; rel=&quot;nofollow noreferrer&quot;&gt;Linear Functional Analysis by Rynne and Youngson&lt;/a&gt;, partly because I like the overall structure and partly because my students can get Springer e-books for free through my institute's library, and as much as possible I try to support and facilitate free, open information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rynne and Youngson is a nice book, but the issue I encountered is that for most US undergraduates, it assumes more preparation than they are likely to have (I think it is really meant for UK undergraduates, who specialize sooner).  For example, it assumes you already know what $L^p$ spaces are and just need a quick reminder.  My students had had one semester of real analysis at most, so I ended up writing a lot of material for them on metric spaces, measure theory, Lebesgue integration, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For another run of the course, I am looking for an alternate book I could use instead, that would include a more detailed treatment of background information.  It would also be good for the book to be somewhat geared toward the application of functional analysis to PDE, though the students last semester loved learning about the more pure aspects of infinite-dimensional Banach spaces, so it doesn't have to be super applied.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recommendations?  I saw &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1696/an-application-heavy-functional-analysis-textbook&quot;&gt;mention on this site&lt;/a&gt; of &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471504599&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kreysig&lt;/a&gt;, which looks good but I am slightly worried about availability after looking on Amazon.  A nice bonus would be if the book has an e-version I can get the library to buy so that students can use it for free.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks so much in advance!!&lt;/p&gt;&#xA;" OwnerUserId="4906" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-12-12T00:59:23.317" Title="Functional analysis text for junior-senior math majors?" Tags="|undergraduate-education|textbooks|functional-analysis|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11359" PostTypeId="2" ParentId="7408" CreationDate="2016-08-31T04:10:31.540" Score="0" Body="&lt;p&gt;Repetition is great for learning arithmetic but rather terrible for learning mathematics in general. Most of mathematics is not inherently algorithmic like arithmetic is, and so rote-style learning is the square peg trying to fit into the round hole. Instead, algebra is very much about relationships between numbers and other things (like graphs), so your pedagogical approach to learning and teaching algebra should be about understanding those relationships. To give an example of what I mean:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the equation y=3x-5. This equation can be said to describe the relationship between the unknown number y and the unknown number x. A solution for this equation is a pair of numbers, an x and a y, for which that (arithmetical) relationship holds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The central difference between arithmetic and algebra is that in arithmetic, you are interested in the particular numbers and the operations being performed on them. In algebra, the particular numbers are no longer so important and instead you interested their arithmetical relationships in general. So your pedagogy (both for learning it yourself as well as teaching it to your son) should be based on trying to identify and understand these relationships.&lt;/p&gt;&#xA;" OwnerUserId="7098" LastActivityDate="2016-08-31T04:10:31.540" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11360" PostTypeId="2" ParentId="11309" CreationDate="2016-08-31T05:01:06.767" Score="4" Body="&lt;p&gt;From my own experiences as a undergrad math major, I would say that good, quality exercises are one of the best tools for teaching math to people. Paul Halmos was right: &quot;The only way to learn mathematics is to do mathematics.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I myself, got by for a long time on just showing up to class and paying attention in college math classes (I even aced vector calculus doing that; grade was all exams, so I didn't do any of the homework). But eventually I got the point in math where that stopped working (and did rather poorly in Advanced Linear Algebra because of it; it was also a rather challenging senior-level class and that was my first year of university after having been at a community college). By my senior year, I was actually working though the homework, and I was acing all of my advanced math classes. I remember the material I actually worked on &lt;em&gt;far better&lt;/em&gt; than the material I didn't actually work though (I don't even know vector calculus very well these days, but I can still do challenging topology proofs even though it's been 3 years since I took that class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what are good, quality exercises? In my opinion, the best kinds of exercises and the ones that present a problem, maybe a new thing related to what they've been learning, and then guide them in exploring that new thing via some prodding questions designed to make the student examine its relation to the things they've been learning about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, suppose you were teaching algebra students about &lt;em&gt;associativity&lt;/em&gt; and &lt;em&gt;commutativity&lt;/em&gt;. A good, quality exercise, might be to introduce 2x2 matrices and their addition and multiplication operations (definition in problem style; as is common in advanced math texts) and then ask them to show that those operations are associative. Then ask if they are commutative and to explain their answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Questions like this are great for several reasons: They force the student to comprehend the definitions presented, as well as requiring them to understand the ideas they have been learning. It requires basic problem solving skills, and implicitly forces them to learn about the new object they are being introduced to (great way to stimulate curiosity in mathematics students). It forces them to work from definitions towards what they want to show, which emphasizes the necessity for justification and the reasoning behind a result.&lt;/p&gt;&#xA;" OwnerUserId="7098" LastActivityDate="2016-08-31T05:01:06.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11361" PostTypeId="2" ParentId="11353" CreationDate="2016-08-31T09:08:27.557" Score="3" Body="&lt;p&gt;The sticking strategy does not use the additional information revealed by the presenter, and thus cannot have more chance of winning than if the presenter would open no door, which is 1/3.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-08-31T09:08:27.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11362" PostTypeId="1" CreationDate="2016-08-31T15:03:57.940" Score="7" ViewCount="53" Body="&lt;p&gt;Hattie &amp;amp; Timperley (2007) mention a meta-analysis about video or audio feedback with a mean effective size of .64 (Table 2 on page 84). They state that this analysis was part of the meta-analysis done by Hattie in 1999. However, I couldn't find a reference to the cited work in Hattie (1999) nor in Hattie &amp;amp; Timperley (2007).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;My question:&lt;/strong&gt; Do you know any research about the effectiveness of feedback given with audio or video (in comparision to face-to-face feedback or written feedback)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What I have found so far:&lt;/strong&gt; Allen et al. (2002) conclude in a meta-analysis that &quot;students find distance learning as satisfactory as traditional classroom learning formats.&quot; (page 93). They also see evidence that distance learning is as effective as classical learning formats. Thus, it wouldn't wonder me, that the same is true for video feedback vs. classical face-to-face feedback.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Allen, M., Bourhis, J., Burrell, N., &amp;amp; Mabry, E. (2002). Comparing student satisfaction with distance education to traditional classrooms in higher education: A meta-analysis. The American Journal of Distance Education, 16(2), 83-97. &lt;a href=&quot;http://www.anitacrawley.net/Resources/Articles/Comparing%20student%20satisfaction%20meta%20analysis.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Hattie, J., &amp;amp; Timperley, H. (2007). The power of feedback. Review of educational research, 77(1), 81-112. &lt;a href=&quot;https://insightplatform-public.sharepoint.com/SiteAssets/feedback-and-reporting/characteristics-of-effective-feedback/power_feedback.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Hattie, J. (1999). Influences on student learning. Inaugural lecture given on August, 2, 1999. &lt;a href=&quot;http://www.collegenet.co.uk/tools/download/hattie%20augural%20address_20_doc.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; I already asked this question &lt;a href=&quot;https://cogsci.stackexchange.com/q/15734/8857&quot;&gt;on http://cogsci.stackexchange.com&lt;/a&gt; nearly two weeks ago. Since I haven't got an answer there I reask this question here. I hope that's okay.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="-1" LastEditDate="2017-04-13T12:56:18.120" LastActivityDate="2016-08-31T15:03:57.940" Title="Effectiveness of audio / video feedback" Tags="|reference-request|technology-in-education|tutoring|interactive-teaching|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11363" PostTypeId="1" CreationDate="2016-08-31T20:56:26.793" Score="11" ViewCount="128" Body="&lt;p&gt;How do you learn math as a person who has no sense of touch and sight? The only sense they can use is hearing, besides smell and taste, which I can't see being applicable in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the question is, how do you learn math as a person whose only sense is auditory? Are there any techniques yet? Are there any barriers to understanding mathematics only through sound?&lt;/p&gt;&#xA;" OwnerDisplayName="hit" LastEditorUserId="143" LastEditDate="2016-09-01T15:38:28.833" LastActivityDate="2016-09-01T15:38:28.833" Title="How do you learn math as a person who has no sense of touch and sight?" Tags="|special-needs-education|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11364" PostTypeId="2" ParentId="11363" CreationDate="2016-08-31T23:13:15.703" Score="1" Body="&lt;p&gt;Not really a question for a math forum and I wonder if it is even possible to completely lose the sense of touch (and I mean COMPLETELY, which would imply that their hearing is impaired too) without being in coma but... It tingles me so... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll assume we are talking about actual kids education, that you're curious and that this is not a mental experiment; in which case I'd say that the question is so complex and requires so many theoretical assumptions involving centuries of endless discussions between the greatest minds in history (plus the latest findings in neurophysiology) that it's naive to even ask it in this context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the dawn of reason some (Plato i.e.) have thought that the basic concepts of math are so easy to grasp that they must be somewhat innate. True or not, while incredibly different cultures originated from different places on earth, mixed up, ruled, disappeared, etc etc... the same basic mathematical concepts started to get used by each single culture (ofc with different levels of development, sofistication and uses) without a common origin to be found. And still survive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some eto/psycho/biologists even claim that animals are capable of different levels of logical-mathematical reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people however - anthropologists and poets especially (sarcasm) - differ. I suggest to ignore them. If innate, genetically predisposed, programmed and destined are all problematic concepts and you don't like them, let's call it X. Given X, every man is capable of exhibiting a contextually logical behaviour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This said (and so assuming that any man with enough time and motivation is capable of developing from zero all the math ever created (whatever this means)) as long as you have a sufficiently complex shared language available I see no problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just forget about geometry and the supposedly critical pedagogic value it has. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[SIDE NOTE: The lack of immediate geometrical constructions for advanced math topics is what causes difficulties to most people : but is it because the topics are &quot;hard&quot; or because - since people relied on geometrical intuition - they never learnt to do the necessary step towards abstraction?]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to link the basic concepts you want them to understand to something relevant for their experience. A song for example (a lot of math was developed under the influence of music theory), or a story, or money. Even try to forget yourself the book, the drawings and your intuition of mathematical concepts. Translate the point into other concepts clear to the kids. Of course it requires a lot patience and commitment and creativity but it is what it is. And if the principal doesn't like what you're doing because you're not sticking to the program... I'm sorry for both you and those kids...&lt;/p&gt;&#xA;" OwnerDisplayName="Lorenzo" LastActivityDate="2016-08-31T23:13:15.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11365" PostTypeId="2" ParentId="11309" CreationDate="2016-09-01T15:08:57.617" Score="1" Body="&lt;p&gt;I keep rather copious statistics on student performance in my classes, but these statistics are unpublished and not peer-reviewed. In my community college remedial algebra courses, doing the assigned homework exercises is, among 40 factors assessed, the single highest-correlated item with the test average, final exam score, and end-of-semester grade ($R^2$ = 0.20, 0.10, 0.15 respectively; N = 149 over the last year-and-a-half). This is self-reported by students (Likert 5-point scale), and despite the homework not being collected nor directly worth any points in the class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This effect is not so pronounced in my sophomore-level statistics class ($R^2$ = 0.02, 0.02, 0.03 on the same measures; N = 208). I might guess that more competent students are able to &quot;see&quot; the principles and concepts immediately in class, whereas weaker students are more dependent on an extended series of rote exercises to develop the expected skills.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-09-01T15:08:57.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11366" PostTypeId="2" ParentId="11302" CreationDate="2016-09-01T17:44:49.777" Score="4" Body="&lt;p&gt;One of the drawbacks to the Common Core Standards is the implied assumption that students will not need revision on previous material. All classes need reteaching on certain topics at times, and teachers will need extra time for such remedial instruction. Clearly if students are lacking foundational material that was supposedly taught in previous years, the teacher will have to fill in these gaps before moving forward. This is even more true as you work with students who are lacking knowledge because of a change in curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Area concepts are extremely important. Americans tend to overemphasize logic and proofs in their geometry courses, sometimes to the detriment of geometry content. (Other countries spread the study of proofs more evenly throughout the math curriculum.) Most geometry students will not go on to do college level math and they should clearly leave the course with sufficient knowledge of geometry to handle basic work requirements and needs from daily life. Clearly, measurement concepts are among the few areas of geometry that most people will need in life. I think this is why the Common Core emphasizes area concepts so much in grades 3 through 7.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that the Common Core does not address area as thoroughly as it should at the high school level. However, one of the main stated aims of the Common Core was to trim curricula, so the standards never repeat topics supposedly already covered. The Common Core does, however, go into a few area concepts besides circle sectors at the high school level. For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;G-GPE 7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★&lt;/p&gt;&#xA;&#xA;&lt;p&gt;G-GMD 1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;G-MG 2:Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For teachers looking to increase their students' familiarity with measurement concepts, the two starred standards (indicating a connection with modeling) would be a good opportunity to allow students to practice their understanding of area through practical exercises or even project-based learning.&lt;/p&gt;&#xA;" OwnerUserId="6782" LastActivityDate="2016-09-01T17:44:49.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11367" PostTypeId="1" AcceptedAnswerId="11371" CreationDate="2016-09-02T07:48:53.213" Score="13" ViewCount="738" Body="&lt;p&gt;In category theory, there's the idea of the product as an object satisfying a particular universal property.&#xA;Can you suggest ways to make the concept of the product intuitive? (So far, &lt;a href=&quot;https://arbital.com/p/product_universal_property/&quot;&gt;my attempt&lt;/a&gt; contains three examples: &quot;product of finite sets&quot; and two instances of &quot;product in a poset category&quot;, and I'm teaching primarily by example.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Background:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've recently been writing a series about the category-theoretic concept of the universal property, over on &lt;a href=&quot;https://arbital.com/&quot;&gt;Arbital&lt;/a&gt; (an open collaborative maths-explanations site). The aim is to make an &lt;em&gt;intuitive&lt;/em&gt; explanation for people who are comfortable enough with mathematical notation that they can read formulae if they have explanations attached, and who are completely comfortable with the notion of a &quot;function&quot;, but who have no exposure to category theory. (For example, fairly advanced students of computer science or engineering, or second-year undergraduate mathematics students. I studied maths, so I know the level of mathematics study I want from the students; my guesses for compsci and engineering are less well-founded.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm running into what I presume are two of the standard problems (certainly the &lt;a href=&quot;http://ncatlab.org&quot;&gt;nLab&lt;/a&gt; has both these problems in spades, because it's not really designed to &lt;em&gt;teach&lt;/em&gt;, but to be a reference work): &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I don't really know whether my explanation would teach someone who didn't already know the content. The only person we can get to dog-food this content is someone who already kind of knows the shape of some basic category theory.&lt;/li&gt;&#xA;&lt;li&gt;Category theory turns out to be quite difficult to explain. (Who knew.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;We've already got &lt;a href=&quot;https://arbital.com/p/empty_set_universal_property/&quot;&gt;Universal Property of the Empty Set&lt;/a&gt;, which I'm pretty pleased with; now I'm trying to make intuitive the notion of the product, on &lt;a href=&quot;https://arbital.com/p/product_universal_property/&quot;&gt;Universal Property of the Product&lt;/a&gt;. The latter is what this Question is about.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The following is not necessary for an answer to this Question, but if you have any suggested specific improvements to the Product page, I'd be very happy to hear them. Similarly, anyone assisting in any way with the rest of the &lt;a href=&quot;https://arbital.com/&quot;&gt;Universal Properties project&lt;/a&gt; would receive twenty Internet points from me.&lt;/p&gt;&#xA;" OwnerUserId="7101" LastEditorUserId="7101" LastEditDate="2016-09-03T14:10:16.750" LastActivityDate="2016-09-05T15:59:55.693" Title="Teaching intuition for the universal property of the product (category theory)" Tags="|concept-motivation|online-instruction|category-theory|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11368" PostTypeId="1" CreationDate="2016-09-02T12:07:14.287" Score="7" ViewCount="2208" Body="&lt;p&gt;&lt;strong&gt;&lt;em&gt;Why do we need fractions such as $\dfrac{3}{-5}$?&lt;/em&gt;&lt;/strong&gt; I need a convincing answer suitable for 8th grade students. Here's what I've already thought of (which don't fully satisfy me!):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Solving the equation $-5x+3=6$, we get to $x=\dfrac{3}{-5}$.&lt;/li&gt;&#xA;&lt;li&gt;In the formula relating distances of image and object from the mirror and the its focal length, sometimes we need to put negative numbers for $f$ or $q$: $$\dfrac{1}{p}+\dfrac{1}{q}=\dfrac{1}{f}$$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;do you have any other suggestion?&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="1229" LastEditDate="2016-09-02T14:06:45.713" LastActivityDate="2016-09-05T19:19:06.030" Title="Negative Denominator in Fractions; Importance and Applications" Tags="|fractions|negative-numbers|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11369" PostTypeId="2" ParentId="11368" CreationDate="2016-09-02T19:33:09.413" Score="0" Body="&lt;p&gt;I'm actually on the student's side here and would say that we never need fractions such as $\dfrac{3}{-5}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like @JohnColeman said, it is more flexible to allow examples such as your first one, but I don't remember seeing anyone use these beyond middle school, except to demonstrate a specific point in a calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I think your second example is misleading, since it's confusing two issues. When we have a rational algebraic expression, the denominator can surely turn out negative, but when you plug in the numbers and write it as a numeric fraction, you wouldn't usually write it with the minus in the denominator, just like you wouldn't usually write the fraction in a non-reduced form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to sum up, in my opinion, there is never a need for these fractions except to direct the readers' gaze to something meta-mathematical. I think a similar example is with writing a decimal with trailing zeros in the sciences (e.g. 13.20mL), where the trailing zero has no mathematical meaning, but is used to demonstrate the precision of the measurement.&lt;/p&gt;&#xA;" OwnerUserId="688" LastActivityDate="2016-09-02T19:33:09.413" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11370" PostTypeId="2" ParentId="11368" CreationDate="2016-09-02T22:10:05.213" Score="8" Body="&lt;p&gt;I would say you're doing your student a disservice if you were to seriously disallow a negative denominator.  A fraction is simply a ratio of two integers (where the denominator is not allowed to be zero). I disagree with @yoniLavi that we &lt;em&gt;never&lt;/em&gt; need such fractions.  Since division by negative numbers makes sense, such a fraction with a negative denominator also does.  (Though without any further context, I would generally say it's bad form to leave the negative in the denominator; however, in certain real world applications, it might be more desirable to leave it there in the denominator.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such fractions come up all the time in algebra and calculus.  For example, in the definition of slope.  We just say given $(x_1,y_1)$ and $(x_2,y_2)$, then&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;m= \frac{y_2 - y_1}{x_2 - x_1}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;we don't define it as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;m= \frac{y_2 - y_1}{x_2 - x_1} \text{ }\text{ if $x_2-x_1 &amp;gt; 0$}&#xA;$$&#xA;and&#xA;$$&#xA;m= \frac{y_1 - y_2}{x_1 - x_2} \text{ }\text{ if $x_2-x_1 &amp;lt; 0$}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, in application problems, a negative number can easily arise in the denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for how to convince an 8th grader?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, I would start by considering the student's background, interests, and (mathematical) proclivities.  If the student has a high interest in the mathematical motivation for allowing a negative denominator, I would start with what I said in the first paragraph above.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regardless of whether the student had a strong interest in mathematics, I would also give a concrete example that the student could relate to.  I would remind the student that fractions can be used not only to express ratios as in the comparison on lengths (like the relative lengths of the sides of a triangle--they'll see this as soon as they're in high school geometry), but that fractions can also be used to measure rates of change:  how one quantity varies with respect to the variation of another quantity.  This is where knowing something about the student's interests and background are relevant.  Come up with something the student is interested in where you're comparing the changes of two quantities and where the second quantity can decrease (ie., it's negative).  I would think that that should be enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having worked with the tail-end of this age group quite extensively the past 8 years, I would be sure to feed their curiosity, congratulate them on the interesting question, and perhaps encourage them to come up with a real life application which makes sense to them where a negative denominator makes sense.&lt;/p&gt;&#xA;" OwnerUserId="3491" LastEditorUserId="3491" LastEditDate="2016-09-02T22:21:41.997" LastActivityDate="2016-09-02T22:21:41.997" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11371" PostTypeId="2" ParentId="11367" CreationDate="2016-09-03T06:11:44.633" Score="3" Body="&lt;p&gt;I'm going to explain how I think of the category-theoretic definition of products.  Unfortunately, this viewpoint won't lead directly to the usual universal property but rather to an equivalent formulation in terms of morphisms into products.  I hope that it can still be of some use even if you'd rather go directly to the universal property.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In most of the familiar categories (e.g., the categories of sets, of topological spaces, of groups) one would like a description of the product to say (among other things) that the elements of $A\times B$ are ordered pairs $(a,b)$ consisting of an element $a\in A$ and an element $b\in B$. From a category-theoretic point of view, there are two problems with this idea. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, we should not attempt to define an object individually but only up to isomorphism, since isomorphic objects have the same category-theoretic properties. So we should rather say that the elements of $A\times B$ are in (canonical) bijection with ordered pairs as above.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, we can't exactly talk about elements in the usual sense; the best we can do in category-theory is to talk about morphisms into an object.  A morphism $X\to Y$ can usefully be regarded as an $X$-indexed family of elements of $Y$ (some people say &quot;generalized element of $Y$&quot; for this), and much of what we want to say about elements can be said instead about such families of elements.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case at hand, this means that we'd describe $A\times B$ by saying that an $X$-indexed family of elements of $A\times B$ corresponds to a pair consisting of an $X$-indexed family of elements of $A$ and an $X$-indexed family of elements of $B$. Formally, this means that the morphisms $X\to A\times B$ are in (canonical) bijection with ordered pairs of a morphism $X\to A$ and a morphism $X\to B$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If, as is usual in category theory, we take &quot;canonical&quot; to mean (at least) natural, then this description is equivalent to the universal property.  Specifically, taking $X$ to be $A\times B$ itself, we get that the identity morphism of $A\times B$ must correspond to a pair of morphisms $p_A:A\times B\to A$ and $p_B:A\times B\to B$, and it follows from naturality that these have the required universal property to serve as the projection morphisms of $A\times B$ to the factors $A$ and $B$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, in the second paragraph above, I said that the (non-categorical) description of $A\times B$ would include the specification of its elements &quot;among other things&quot;. Of course, in the category of groups, those other things would include the group operation on $A\times B$, and in the category of topological spaces they would include the product topology of $A\times B$.  One of the delightful aspects of category theory is that, by using morphisms into $A\times B$ (families of elements) instead of just elements, the definition automatically includes those other things, without having to describe them separately.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2016-09-03T06:11:44.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11372" PostTypeId="2" ParentId="11367" CreationDate="2016-09-03T10:36:05.270" Score="2" Body="&lt;p&gt;In my opinion, the intuition regarding the use of universal property comes most directly through its use in defining ordered pairs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To any pair of morphisms $f : X \to A$ and $g : X \to B$, there is the corresponding ordered pair $(f,g) : X \to A \times B$. Conversely, any morphism $h : X \to A \times B$ is an ordered pair; specifically $h = (\pi_0 \circ h, \pi_1 \circ h)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Described at a higher level, $(\cdot,\cdot)$ is the canonical natural bijection&#xA;$$\hom(-,A) \times \hom(-,B) \to \hom(-, A \times B)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this is not entirely dissimilar from &lt;a href=&quot;https://matheducators.stackexchange.com/a/11371/797&quot;&gt;Andreas Blass's answer&lt;/a&gt;; one could even argue we're saying the same thing in different language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the use of ordered pairs here is important, because it promotes the universal property from being a &quot;description&quot; to an &quot;operation&quot; &amp;mdash; while you can do this in principle with any universal property, the product is especially amenable to being used in calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, note that this emphasizes the &lt;em&gt;morphisms&lt;/em&gt; between the objects involved, rather than the objects themselves.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-09-03T10:45:49.770" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11373" PostTypeId="1" AcceptedAnswerId="11375" CreationDate="2016-09-03T15:52:47.407" Score="4" ViewCount="492" Body="&lt;p&gt;I took pre-calculus in high school, and I did not get to learn about matrices, and conic sections, vectors law of sines and cosines, and etc.  I took geometry as well and found that matrices were also covered in the textbook, yet we did not cover it in class.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why is there some material that does not get covered, yet it is very important to learn, or much yet be exposed to? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I know some students in other high schools in New Jersey who do not get exposed to matrices, and other material.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So is the common core really something normalized for everybody, yet, there is a difference in what is being learned? Sometimes it feels like others have advantages over others based on what is being taught, so what is really the common core doing?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(In my opinion its like a foot race where we all have the same finish line, but have different places to start, some closer to the finish line then others.)&lt;/p&gt;&#xA;" OwnerUserId="7105" LastEditorUserId="262" LastEditDate="2016-09-05T18:25:15.673" LastActivityDate="2016-09-05T18:25:15.673" Title="Common Core Question: What is included and excluded in high school mathematics?" Tags="|mathematical-pedagogy|secondary-education|common-core|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11374" PostTypeId="2" ParentId="11373" CreationDate="2016-09-03T16:06:48.523" Score="8" Body="&lt;p&gt;Your teacher decided it was not as important to learn as the topics they chose. Textbooks publishers put in everything the teacher might possibly want to use. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you never learn matrices until a Linear Algebra course, you'll still be fine. You will learn matrices there. They are a tool for solving systems of equations, among other things.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2016-09-03T21:33:37.403" LastActivityDate="2016-09-03T21:33:37.403" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11375" PostTypeId="2" ParentId="11373" CreationDate="2016-09-03T17:30:43.540" Score="2" Body="&lt;p&gt;The Common Core State Standards for Mathematics can be found here:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://www.corestandards.org/Math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.corestandards.org/Math/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Besides the mathematical standards, see also &lt;a href=&quot;http://www.corestandards.org/about-the-standards/&quot; rel=&quot;nofollow noreferrer&quot;&gt;about the standards&lt;/a&gt;, &lt;a href=&quot;http://www.corestandards.org/what-parents-should-know/&quot; rel=&quot;nofollow noreferrer&quot;&gt;what parents should know&lt;/a&gt;, and whatever other parts of the site that may be of interest. From the about section's &lt;strong&gt;FAQ&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/156IS.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/156IS.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to your [edited] post, you write:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So is the common core really something normalized for everybody, yet, there is a difference in what is being learned? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;Common Core&lt;/strong&gt; is normalized in the sense that the standards are freely available; they are in the first link above, and they can be searched through quite easily using the site. For example, &lt;a href=&quot;http://www.corestandards.org/search/?f=math&amp;amp;t=matrix&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; is a search through the Math Standards for the word 'matrix' to show where and how it appears.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that the issue you are describing is around curriculum; note that in the first excerpt above, it says (emphasis in original) that, &quot;Education standards, like Common Core are &lt;em&gt;not&lt;/em&gt; a curriculum.&quot; Curricular concerns -- picking one, adhering to one, supplementing one, etc -- are major issues facing mathematics teachers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In just the past couple weeks, Matt Larson (current NCTM President) put out a post on &lt;a href=&quot;https://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Matt-Larson/Curricular-Coherence-in-the-Age-of-Open-Educational-Resources/&quot; rel=&quot;nofollow noreferrer&quot;&gt;curricular coherence in the age of open educational resources&lt;/a&gt;; see also the response from Dan Meyer (&lt;a href=&quot;http://blog.mrmeyer.com/2016/nctm-puts-up-a-sign/&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;) who has a pointer to another post by (former student of mine! -- though I take no credit -- and occasional contributor to this site, MESE) Tyler Auer (&lt;a href=&quot;http://www.mathfireworks.com/2016/08/responding-to-larsons-post-on-curriculum/&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, concerns like yours are prominent right now among math educators -- including but not limited to math teachers and curriculum writers. Succinctly, to respond to another excerpt from your post:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why is there some material that does not get covered, yet it is very important to learn, or much yet be exposed to? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One response is, there is more &quot;important to learn&quot; material than could possibly be covered. But another response is, if the Common Core Standards for Mathematics were properly addressed within a curriculum (but whose responsibility is this?) then you &lt;em&gt;would&lt;/em&gt; see, for example, matrices and &lt;a href=&quot;http://www.corestandards.org/search/?f=math&amp;amp;t=conic+section&quot; rel=&quot;nofollow noreferrer&quot;&gt;conic sections&lt;/a&gt;. The discrepancy in &lt;em&gt;how&lt;/em&gt; to meet the Standards' learning goals is an important area of contemporary consideration. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, a digital resource: For a nice online &quot;Common Core Curriculum&quot; to address both the ELA and Mathematics Standards, you might check out the materials freely available at $\text{engage}^\text{ny}$ (&lt;a href=&quot;https://www.engageny.org/common-core-curriculum&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-09-03T17:30:43.540" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11376" PostTypeId="2" ParentId="11211" CreationDate="2016-09-04T18:21:20.047" Score="2" Body="&lt;p&gt;It always makes me cringe when I hear someone present the definition of irrational as a number whose decimal expansion does not repeat.  It does a huge disservice to the history of math, its beauty, and the notion of rational.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A definition of irrational as a decimal number which doesn't repeat lacks beauty in that the word &lt;em&gt;irrational&lt;/em&gt; bears no relation to what you're describing.  The traditional, the Greek definition, of rational as a ratio makes very clear then what &lt;em&gt;ir&lt;/em&gt;rational is talking about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The definition of rational and irrational as inherited from the Greeks is simple, elegant, and beautiful.  It presents a particular view of the world and its nature.  Teaching this definition is an opportunity (a window from a modern classroom looking out onto an ancient world) to understand something about the human desire to comprehend and make sense of the world.  It's also an opportunity to teach the students the degree to which this ancient world view is still present with us.  We &lt;em&gt;expect&lt;/em&gt;, to a certain degree, that the world itself is &lt;em&gt;rational&lt;/em&gt;.  This expectation has played a huge role in the development of mathematics and physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics, as taught in high school, frequently comes across as dry with little or no connection to the world of human drama.  Why not tell of the Pythagoreans and their punishment for anyone who dared reveal that the square root of two cannot be expressed as the ratio of two whole unit lengths.  In various guises this plays out again and again through history.  Move forward in time to Copernicus, Kepler, and Gallileo's discoveries and observations challenging accepted conventional wisdom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The history of the square root of $-1$ itself is fascinating.  Resistance to the idea being revealed in the names &lt;em&gt;imaginary&lt;/em&gt; and &lt;em&gt;real&lt;/em&gt;: such as your &lt;em&gt;imaginary friends&lt;/em&gt; and your &lt;em&gt;real friends&lt;/em&gt;.  But then as non-Euclidean geometry is developed and theories of electro-magnetism worked out, &lt;em&gt;imaginary&lt;/em&gt; numbers gain greater credibility as their &lt;em&gt;complexity&lt;/em&gt; is discovered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to how we teach what the irrational numbers, there are a variety of reasons for resorting to a definition which hinges on the nature of the decimal expansion.  One reason is perhaps laziness.  To understand why $\sqrt{2}$ cannot be expressed as the ratio of two whole numbers requires a degree of logical reasoning that students may be resistant to (or, unfamiliar with).  The teacher may not want to deal with that factor and goes down the path of least resistance.  Of course, the laziness may be impossed upon the teacher by time factors.  To appreciate the Greek's definition requires time to carefully consider and construct a logically coherent argument.  You're not going to accomplish this in one class period--not if you want to get all the students to be on-board with you.  (Other reasons could be the teacher's own lack of understanding, but here let's just assume for the sake of argument that we're talking about someone who is highly qualified to teach mathematics.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it is worth pointing out that there really is nothing simple about defining irrational in terms of a nonrepeating decimal expansion.  It presents an opportunity for a great challenge to give the students:  prove to me that such a number exists.  How do you show that no matter how far out you go in the decimal expansion of a number that it doesn't eventual start repeating itself?  Perhaps the students come up with something clever, but then how do they show that their clever approach really does produce what they claim it produces?  If you can show that such numbers exist, what does that tell you about the nature of numbers?  (Oh, there are so many wonderfully juicy questions to explore.  The tie-ins with philosophy and how we look at the outside world are potentially enormous.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've never heard of anyone taking the following approach.  However, I can't imagine that I'm the only person to have tried this in a high school classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the approach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Define rational numbers as numbers that can be expressed as the ratio of two whole numbers.  Next, show that you can &lt;em&gt;enumerate&lt;/em&gt; the rational numbers:  you don't have to use the word &lt;em&gt;enumerate&lt;/em&gt;, just tell the kids, &quot;Hey, look at this!  We can write out a list of them.&quot;  Then look at the decimal expansion of those numbers; fill in trailing zeros where necessary; don't worry about whether the decimals repeat or not.  Now, use the diagonalization approach to create a new number that's not in that list.  Voila!  A number that is not rational.  [This can easily take two or three days of class time if you do it right.  And, if you do it right, there are lots of places to emphasize the curriculum you are expected to cover.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why approach it this way?  Well, it's a clever argument.  But it's also one that is easier for the students to follow and make sense of.  It's very tangible.  You actually have the try to write out a list of all the rational numbers.  Let them figure out their own approach to doing this.  They're bright enough; they'll figure something out.  And, if they don't come up with what you already have in mind, that's fine.  [This is part of the reason this takes several days to accomplish.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This appoach shows a way to indesputably create an irrational number.  (Students buy into it more because they like decimal better than fractions.)  If the students came up with different ways of listing the rational numbers, then have them compare their answers to see that they all have come up with different irrational numbes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Importantly, this approach has actually taught them something else:  while you can make a list of all the rational numbers, you cannot do the same with all real numbers (since by forming any such list you can create a number not in the list).  And, this conclusion can be part of a wrap up exercise.  [It also creates a great opportunity to talk about other matter such as to why $0.99999$, repeating, is equivalent to $1$.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next you can then have the students show why rational numbers must have repeating decimals.  Conversely, given a repeating decimal, you can teach them to show how to find two whole numbers whose ration will reproduce the desired expansion.  Now you've shown that not only do rational numbers have a repeating decimal, irrational numbers must not have such an expansion.  Into this you can build a whole series of lessons about primes and the natures of remainders upon division.  And there are various cool tangents to go down: such as looking at cyclic primes and the decimal expansions of fractions whose denominators are such primes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then, you can ask the students, &quot;Are there any lengths which we can construct with a compass and ruler that cannot be expressed rationally?  In comes the usual proof of the irrationality of $\sqrt{2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this is quite feasible in a geometry class.  And it would be a lesson suitably taught toward the end of the year after the students have gained plenty of experience building logical arguments.  It could be taught over the course of several weeks with other topics peppers in (or rather the other way around, &quot;irrationality&quot; peppered in among other topics).  But, however you try to go about it, it takes a lot of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given my experience teaching geometry in a high school classroom, the curriculum usually doesn't allow you the luxury to spend that much time building toward a particular topic in a manner that will create a substantial buy-in and appreciation from the students.  It's not necessarily that you can't squeeze in all the various topics the curriculum demands.  If you have the luxury of writing your own lesson plans and structuring out how you will teach what over the year, it's very feasible.  However, that's not the reality for most classroom teachers.  Frequently, the teacher is shackled by a curriculum impossed from outside the four walls of their classroom, or the teacher may be forced to follow a scripted lesson plan, and required to follow lock-step what the other teachers in his/her department are doing.  If the teacher must work with other teachers, either in their own department or across a school district, things get even more difficult.  And, so the teacher sighs, throws up their hands, teaches a definition that lacks elegance, and loses an opportunity to show students some of the inherent of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parting thought.  I love the word &quot;expansion&quot;.  It suggests an unpacking of the ratio into decimal form.  What exactly do we mean then when we say &quot;the decimal expansion of an irrational number&quot;?  I think it's a great question to have your students ponder and philosophize over.&lt;/p&gt;&#xA;" OwnerUserId="3491" LastEditorUserId="3491" LastEditDate="2016-09-04T20:49:21.777" LastActivityDate="2016-09-04T20:49:21.777" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="11377" PostTypeId="1" AcceptedAnswerId="11381" CreationDate="2016-09-04T22:30:35.630" Score="7" ViewCount="483" Body="&lt;p&gt;What are the most important websites where university jobs in Math Education are advertised? The kind of boards I am looking for should preferably be able to reach an international audience.&lt;/p&gt;&#xA;" OwnerUserId="39" LastEditorUserId="1608" LastEditDate="2016-09-10T20:45:46.833" LastActivityDate="2016-09-11T00:56:43.463" Title="Job boards for university positions in Math Education?" Tags="|undergraduate-education|employment|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11378" PostTypeId="2" ParentId="11377" CreationDate="2016-09-04T22:49:23.957" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://www.higheredjobs.com/&quot; rel=&quot;nofollow&quot;&gt;HigherEdJobs.com&lt;/a&gt; is fairly prominent, I think. It has a U.S. focus but does permit a non-U.S. location, for example: &lt;a href=&quot;https://www.higheredjobs.com/search/advanced_action.cfm?JobCat=104&amp;amp;PosType=1&amp;amp;PosType=2&amp;amp;InstType=1&amp;amp;InstType=2&amp;amp;InstType=3&amp;amp;Keyword=&amp;amp;Remote=1&amp;amp;Remote=2&amp;amp;Region=10&amp;amp;SubRegions=75&amp;amp;Submit=Search%20Jobs&quot; rel=&quot;nofollow&quot;&gt;this search&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-09-04T22:49:23.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11379" PostTypeId="2" ParentId="11377" CreationDate="2016-09-05T01:00:52.597" Score="4" Body="&lt;p&gt;In the U.S., &lt;a href=&quot;https://www.mathjobs.org/jobs&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathJobs.org&lt;/a&gt; dominates.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathjobs.org is an automated job application system, sponsored by the &lt;em&gt;American Mathematical Society&lt;/em&gt;. We serve all job applicants with advanced degrees in Mathematics, and employers (academic or industrial) searching for mathematicians. The system is free for applicants.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Despite its U.S. focus, they are international to some extent:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/Qg7Pd.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Qg7Pd.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://www.mathjobs.org/jobs&quot; rel=&quot;nofollow noreferrer&quot;&gt;www.mathjobs.org&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-09-05T01:00:52.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11380" PostTypeId="2" ParentId="11377" CreationDate="2016-09-05T01:23:36.593" Score="1" Body="&lt;p&gt;I used the Chronicle of Higher Education when I looked for jobs. Is it still useful? (They now have a special site for jobs, &lt;a href=&quot;https://chroniclevitae.com/&quot; rel=&quot;nofollow&quot;&gt;chroniclevitae.com&lt;/a&gt;.) &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2016-09-05T01:23:36.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11381" PostTypeId="2" ParentId="11377" CreationDate="2016-09-05T01:37:26.600" Score="3" Body="&lt;p&gt;The main place of which I am aware is the math education listserv run by &lt;a href=&quot;http://ehs.siu.edu/ci/faculty-staff/faculty-all/becker.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Dr. Jerry Becker&lt;/strong&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest that you contact him to ask about listing a position; he will send it out with a subject line that begins &quot;Position Announcement&quot;; early September is not the most common time to list a position, but here is a screenshot of his listings from 27 Oct 2015 - 18 Dec 2015 to give you a sample glimpse:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/d2zgZ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/d2zgZ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Click through to see an enlarged image.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggested as a side-note back in &lt;a href=&quot;https://matheducators.stackexchange.com/a/8203&quot;&gt;&lt;strong&gt;MESE 8193&lt;/strong&gt;&lt;/a&gt; that math educators should also &lt;em&gt;subscribe&lt;/em&gt; to Dr. Becker's listserv; I reiterate that suggestion here!&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-09-05T01:37:26.600" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11382" PostTypeId="2" ParentId="11367" CreationDate="2016-09-05T15:59:55.693" Score="6" Body="&lt;p&gt;Here is my intuition:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, each object can be used to encode some information which is accessible through morphisms. For example, take the $\mathbf{Set}$ category and two objects $A = \{0\}$, $B = \{1, 2\}$. Now you can encode one bit of information by picking one of the morphisms $f_1(0) = 1$ or $f_2(0) = 2$. Now, suppose that we could assign to each object some quantity that represents its potential for encoding information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The product $A\times B$ of $A$ and $B$ is an object that holds exactly the amount of potential for encoding information as both $A$ and $B$ together, not more, not less. To make this more concrete:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Not less:&lt;/strong&gt; If you have some information encoded as $f_A : Y \to A$ and $f_B : Y \to B$, then you can &quot;store&quot; that together in $f_{A \times B} : Y \to A \times B$ and then &lt;em&gt;recover&lt;/em&gt; that information separately using projections $\pi_A$ and $\pi_B$ (separately meaning the information with regard to $A$ is independent from the information with regard to $B$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Not more:&lt;/strong&gt; There is no &lt;em&gt;additional&lt;/em&gt;, hidden information you could recover from $A \times B$ besides what you get via $\pi_A$ and $\pi_B$. In other words, if both morphisms $f_{A \times B}$ and $g_{A \times B}$ get you the same information when used with $\pi_A$ and $\pi_B$, then they have to be the same morphism (otherwise we could use that difference to encode more information).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the categories get more complicated the encoding/decoding gets more complicated too, but I could not find an example for which this intuition breaks (although, note that I'm not a category-theorist).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2016-09-05T15:59:55.693" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11383" PostTypeId="2" ParentId="11368" CreationDate="2016-09-05T19:19:06.030" Score="2" Body="&lt;p&gt;There are numbers that one (or many) may never use, is this a reason to eliminate them.  Is there a reason to have 1 to any non-negative integer power?  When responding to questions of this type &quot;Why do we need fractions such as...? &quot; It is not a question of popularity of use but is a question of &quot;Does this numeral name a well define number&quot;? Then depending upon the the response you may want to elaborate on expected or agreed form. Do we need a VII or are we forced to use '7' or like the Maya :|   Give the student the definition of a fraction and ask &quot;does this example follow the definition.&lt;/p&gt;&#xA;" OwnerUserId="6333" LastActivityDate="2016-09-05T19:19:06.030" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11384" PostTypeId="1" CreationDate="2016-09-05T19:39:06.940" Score="11" ViewCount="6627" Body="&lt;p&gt;I would like to give multiple choice tests with randomized questions and order of the answers that is unique for each test, to be printed and given on paper sheets. Is there a software that would do the randomization?&lt;/p&gt;&#xA;" OwnerUserId="5601" LastActivityDate="2023-10-14T04:39:36.983" Title="Software for randomizing multiple choice tests" Tags="|software|testing|test-design|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11385" PostTypeId="2" ParentId="11048" CreationDate="2016-09-05T21:09:32.553" Score="0" Body="&lt;p&gt;Good opportunity for a lesson.&lt;br&gt;&#xA;For each of the following pictures that you draw on the board ask and discuss with student (or class as the case presents) after you draw each.  &quot;Which is bigger?&quot;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Stick figure of short fat  and tall skinny person.&lt;/li&gt;&#xA;&lt;li&gt;normal size (i.e. same font) 6  and  7&lt;/li&gt;&#xA;&lt;li&gt;Tall Big 6 tiny 7&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;From this will come the need to clarify terms.  And introduce the word &quot;Magnitude&quot; in terms of non-math terms.  i.e. storms, hurricanes  and/or earthquakes.  This will not solve the 'problem' until &quot;magnitude&quot;is defined for the math person..i.e. in terms of distance from zero on the number line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now Discuss the following 'drawings'  i.e. number line with the following points labeled:                 -7, -6, -1, 0, 1,  6  7&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have students make sentences using 2 of the numbers on the line with either the  of words &amp;lt;, &gt;, magnitude.  Do not repeat a number.&lt;/p&gt;&#xA;" OwnerUserId="6333" LastActivityDate="2016-09-05T21:09:32.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11386" PostTypeId="2" ParentId="11384" CreationDate="2016-09-05T22:43:27.587" Score="7" Body="&lt;p&gt;&lt;a href=&quot;http://wpslive.pearsoncmg.com/cmg_instructor_testgen_1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pearson's TestGen&lt;/a&gt; is a crucial tool for my teaching for this reason and others. If you have an instructor account with Pearson, then the software download is free, and so are testbanks for almost all of their textbooks. The testbanks include premade questions of several different types, and of course you can write your own. It has well-constructed formatting for math equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have a test made, it's a one-click process to scramble the order of any multiple-choice options. In addition to that, questions can have variable entities, so that another click randomly changes the visible numerical or text values within certain questions (works somewhat better for testbank multiple-choice items, or self-made short-answer items). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also has an export feature which allows a test to be exported to HTML, Blackboard, or WebCT. For example: I find the test-creation in TestGen so much more elegant than Blackboard that all of my online quizzes are first drafted in TestGen, then exported to Blackboard. Plus my in-class tests use the random values/random ordering function each semester, with some manual edits to more conceptual short-answer/essay items. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;During initial test construction you can insert a number of random questions from a premade testbank this way (from the &lt;a href=&quot;http://wpslive.pearsoncmg.com/wps/media/objects/8722/8931918/TestGen_Tutorial.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;TestGen Tutorial&lt;/a&gt;, p. 6): &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ymf5O.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ymf5O.png&quot; alt=&quot;TestGen interface to insert random questions&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then later on you can scramble the order of any questions and.or answers with this menu button (tutorial p. 10):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LUVca.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LUVca.png&quot; alt=&quot;TestGen interface to randomize questions&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the past I also used &lt;a href=&quot;http://www.swlearning.com/examview/examview_main.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cengage's ExamView&lt;/a&gt; for one semester, but it didn't work nearly as well as TestGen.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-09-06T05:15:20.247" LastActivityDate="2016-09-06T05:15:20.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11387" PostTypeId="2" ParentId="11384" CreationDate="2016-09-06T01:23:41.167" Score="11" Body="&lt;p&gt;If you use &lt;a href=&quot;http://www.latex-project.org/&quot;&gt;LaTeX&lt;/a&gt; to prepare your documents, there are many packages that automatically randomize the order of choices in multiple-choice questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One recently updated package is &lt;a href=&quot;http://www.ctan.org/pkg/esami&quot;&gt;esami&lt;/a&gt;.  (&lt;a href=&quot;http://ctan.mirrors.hoobly.com/macros/latex/contrib/esami/doc/esami-doc-en.pdf&quot;&gt;This documentation&lt;/a&gt; is dated July 27, 2016.)  Its &lt;a href=&quot;http://mirrors.ctan.org/macros/latex/contrib/esami/README&quot;&gt;official description&lt;/a&gt; is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The package allows to write various type of exercises (multiple choiche questions with answers&#xA;  varying in random order, with closed or open answer, matching, problems, and so on). Questions&#xA;  of each group are randomly scrambled across the exam and variants of each exercise are chosen&#xA;  randomly from a file which contains them all and.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-09-06T01:23:41.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11388" PostTypeId="2" ParentId="11384" CreationDate="2016-09-07T02:24:22.633" Score="2" Body="&lt;p&gt;I don't see why you couldn't use &lt;a href=&quot;http://webwork.maa.org&quot; rel=&quot;nofollow&quot;&gt;WeBWorK&lt;/a&gt; for this.  It might be more powerful than you are looking for, if you don't want truly random questions (that is, random numbers for each problem), but it certainly can do this.   Apparently they have just implemented &lt;a href=&quot;http://webworkgoehle.blogspot.com/2016/02/hardcopy-themes.html&quot; rel=&quot;nofollow&quot;&gt;new themes&lt;/a&gt; for hardcopy.  Anyway, people may not know that WeBWorK problems are certainly available for physical exams/homework too (though those won't grade themselves!).&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-09-07T02:24:22.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11389" PostTypeId="2" ParentId="11384" CreationDate="2016-09-08T15:48:35.677" Score="0" Body="&lt;p&gt;&lt;a href=&quot;http://home.gna.org/auto-qcm/&quot; rel=&quot;nofollow&quot;&gt;AutoMultiplechoice&lt;/a&gt; does that, including autocorrection using a scanner (there is a code telling the machine what was the order of the questions and answers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is compatible with LaTeX, but does not require it. It can be quite efficient, but I didn't have much experience with it up to now.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-09-08T15:48:35.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11390" PostTypeId="1" AcceptedAnswerId="11398" CreationDate="2016-09-08T23:40:05.750" Score="7" ViewCount="363" Body="&lt;p&gt;I mostly tutor community college students ranging from beginning algebra to calculus level.  There are several ways in which I explain the domain:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The domain of a function is all the values you can plug in for $x$ and not get an error, such as dividing by zero.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If given a graph, I will say &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The domain is all the numbers on the $x$- axis above or below which you have a piece of the graph&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often times this goes ok, but sometime times the students still don't understand. Is there a better way of explaining it?&lt;/p&gt;&#xA;" OwnerUserId="7075" LastEditorUserId="7075" LastEditDate="2016-09-08T23:56:34.737" LastActivityDate="2016-10-07T17:40:16.070" Title="Explaining the domain of a function to students?" Tags="|algebra|functions|precalculus|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11391" PostTypeId="1" CreationDate="2016-09-09T00:09:02.733" Score="6" ViewCount="1310" Body="&lt;p&gt;I am looking for an article (or other source) that addresses the calculus textbook industry, specifically which calculus textbooks are most commonly adopted. I'm primarily interested in that of the U.S.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've heard that Stewart's calculus texts dominate approximate 2/5 of the calculus textbook market, but I'm looking for a source that supports such a claim (and also discusses the performance of other texts (e.g., Thomas) in the calculus textbook market).&lt;/p&gt;&#xA;" OwnerUserId="7130" LastEditorUserId="143" LastEditDate="2016-09-09T12:32:36.757" LastActivityDate="2016-09-10T00:53:40.073" Title="Most Commonly-Adopted Calculus Textbooks" Tags="|calculus|textbooks|usa|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="11392" PostTypeId="1" AcceptedAnswerId="11394" CreationDate="2016-09-09T00:32:14.880" Score="9" ViewCount="369" Body="&lt;p&gt;I mostly tutor students ranging from beginning algebra to calculus level. I think the explaining the range as &quot;the set of all possible outputs&quot; would not really cut it for someone struggling with math at that level. Usually I default to:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Draw a rough sketch of the graph, including all the shifts, stretches, etc. The range is all the portions on the $y$ axis which have a piece of the graph to the left/right of them.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel that not only is this explanation very good, but many times when I ask the students to draw a rough sketch of the graph, they have no idea how; so now I have to reteach the student how to graph equations. However, I am not a private tutor; I work for my school, and I am not allowed to spend more than $15-30$ minutes with a student at a time, so I do not have time (nor want to) reteach the student half a chapter just so they can solve one problem. How can I explain the range in a better and less time-consuming way? &lt;/p&gt;&#xA;" OwnerUserId="7075" LastActivityDate="2016-09-09T23:39:06.093" Title="How to explain the range to students?" Tags="|algebra|functions|precalculus|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11393" PostTypeId="1" CreationDate="2016-09-09T02:37:04.073" Score="3" ViewCount="317" Body="&lt;p&gt;The undergraduate students are having difficulty in solving problems like algebra and geometry in a classroom. In order to improve their poor performance in mathematics, an intervention (enhancement program) is to be introduced. At first i have to assess their mathematical performance by giving them a pretest. After analyzing the pretest, an enhancement program is introduced by teaching them selected topics in Algebra and Geometry using cooperative learning, peer tutoring, mastery, feedback and reward system. Afterwards, a post-test will be given to see if the enhancement program is effective or not. There will be also an interview to students to know their perception about the enhancement program.&#xA;What is the best conceptual framework for this study?&lt;/p&gt;&#xA;" OwnerUserId="7131" LastEditorUserId="7131" LastEditDate="2016-09-11T12:45:35.160" LastActivityDate="2016-09-11T13:10:33.440" Title="conceptual framework" Tags="|education-research|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11394" PostTypeId="2" ParentId="11392" CreationDate="2016-09-09T16:24:56.107" Score="7" Body="&lt;p&gt;I tend to speak of input and output and sometimes mention vending machines as an analogy. There is a range of products available from a soda machine. Typically it might include Pepsi. It wouldn't include a snickers bar -- not a possible output of a soda machine. An important part of understanding a vending machine is knowing what it vends. Similarly, an important part of understanding a function is knowing what values it produces. This also provides a possible answer to your other question about teaching domains. A vending machine will accept some combinations of coins but not others. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously this doesn't directly help with the problem of e.g. finding domain of a simple polynomial or rational function. But, if the students have a good mental model for what a domain &lt;em&gt;is&lt;/em&gt; then they can perhaps better understand it in a concrete case. After mentioning vending machines, if often use $f(x) = x^2$ as a simple example. Like a vending machine that only gives diet products (as might be found in a gym), this is a function that only gives nonnegative numbers.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastEditorUserId="5563" LastEditDate="2016-09-09T23:39:06.093" LastActivityDate="2016-09-09T23:39:06.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11395" PostTypeId="2" ParentId="11392" CreationDate="2016-09-09T22:30:41.960" Score="6" Body="&lt;p&gt;I think the problem is, in part, that you are coming at this from the wrong perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They have a problem regarding ranges, so you've become fixed on helping them understand ranges. Then, when you discover they have a more fundamental problem... you are still fixated on your goal of teaching them ranges, rather than helping them with their actual problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what they &lt;em&gt;need&lt;/em&gt; is help with their actual problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, if a student can't graph functions, you really do need to help them with graphing functions. And with the aim of learning how to graph functions, rather than how to fake it enough to replicate your solution to the range problem they've been assigned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're good, you can even slip in range-related ideas as you do so &amp;mdash; e.g. commenting how the graph $y = x^2 - 1$ has $y$ start at $0$, dip down to $-1$, and then back to $0$ as $x$ varies over the interval $[-1,1]$.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2016-09-09T22:30:41.960" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11396" PostTypeId="1" CreationDate="2016-09-10T00:14:44.377" Score="37" ViewCount="6366" Body="&lt;p&gt;In the comment thread of &lt;a href=&quot;https://academia.stackexchange.com/questions/76531/do-academic-integrity-pledges-work&quot;&gt;this&lt;/a&gt; academia.SE question, the following generated some strong reactions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My very different (community-college) perspective is that the math discipline will end up as a filter no matter how much the institution desires otherwise. Most of our students will never grok 8th-grade algebra no matter how many times they try, nor under what circumstances. Other disciplines can be made easier in many ways; math has a central unavoidable core of intellectual honesty. Our institution may in fact get rid of any real math as a basic requirement for this reason. E.g., writings of Andrew Hacker.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is there any meaningful, reasonably objective way to determine what fraction of the population is incapable of learning algebra, even with repeated effort and competent instruction? I'm looking for evidence-based answers that cite references, not anecdotes or personal impressions. It would seem obvious to me that the answer has to be greater than zero (e.g., there are developmentally disabled people) and less than one (because some people do learn algebra).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Success rates in high school algebra probably do not even provide an upper or lower bound on the answer. It seems to be common for high school algebra teachers here in the US to pass students who can't do the work but make an effort. And much instruction probably is not competent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;:&#xA;To make the discussion easier for non-U.S. SE participants to follow, note that &quot;GED&quot; is an alternative way of getting a high school diploma in the U.S.  It apparently stands for &quot;General Educational Development&quot; -- although I always thought it stood for General Equivalency Diploma.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="-1" LastEditDate="2017-04-13T12:49:57.670" LastActivityDate="2022-04-01T14:37:15.447" Title="What fraction of the population is incapable of learning algebra?" Tags="|secondary-education|algebra|struggling-students|" AnswerCount="5" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="11397" PostTypeId="2" ParentId="11391" CreationDate="2016-09-10T00:53:40.073" Score="3" Body="&lt;p&gt;As a rough start, I get this from Amazon...&lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/BL1oH.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BL1oH.jpg&quot; alt=&quot;authors&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2016-09-10T00:53:40.073" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11398" PostTypeId="2" ParentId="11390" CreationDate="2016-09-10T11:49:01.953" Score="6" Body="&lt;p&gt;Actually your definition is not right. Take the function $f:\mathbb R^{+}\to\mathbb R:x\mapsto x^2$ for which the domain is $\mathbb R^{+}$ but there are more $x$ values for which you can evaluate $x^2$. Your method does also not work for all functions. What about complex valued functions or the function which assigns to each person its name?!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend teaching the function as a triple of domain, codomain and an assignment of arguments to its functions value. Take the following functions&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$f:\mathbb R\to\mathbb R:x\mapsto x^2$&lt;/li&gt;&#xA;&lt;li&gt;$f:\mathbb R\to\mathbb R^{+}_0:x\mapsto x^2$&lt;/li&gt;&#xA;&lt;li&gt;$f:\mathbb R^{+}_0\to\mathbb R:x\mapsto x^2$&lt;/li&gt;&#xA;&lt;li&gt;$f:\mathbb R^{+}_0\to\mathbb R^{+}_0:x\mapsto x^2$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The assignment rule is always the same but the functions differ in their properties (for example injectivity or surjectivity). This example demonstrates that the domain (and codomain) is needed in the definition of the function. So I would say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A function is an assignment of arguments from a given domain to values of a given codomain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For motivating the concept you may ask the students about functions in their daily life (like the function which assigns to each person its name). It's important that the students do not undergeneralize the concept of a function.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="5097" LastEditDate="2016-10-05T12:08:43.330" LastActivityDate="2016-10-05T12:08:43.330" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11399" PostTypeId="2" ParentId="11377" CreationDate="2016-09-11T00:56:43.463" Score="1" Body="&lt;p&gt;Although MathJobs was mentioned, there is a larger list (or perhaps it &lt;em&gt;was&lt;/em&gt; a larger list) at the &lt;a href=&quot;http://eims.ams.org/jobs/&quot; rel=&quot;nofollow&quot;&gt;Employment Information in the Mathematical Sciences&lt;/a&gt; website, also by the AMS.  It is not an application site, though (as opposed to MJ), and I don't know how many math ed jobs there would be - today it was everything from OR to stats to 'math curriculum designer' (whatever that is) to various postdocs.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-09-11T00:56:43.463" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11400" PostTypeId="2" ParentId="11393" CreationDate="2016-09-11T13:10:33.440" Score="5" Body="&lt;p&gt;A conceptual framework should inform the experimental design, not be something that you add on like a garnish after the entire experiment has been planned.  You say that the students are having trouble with &quot;problems like algebra and geometry&quot;.  What &lt;em&gt;specifically&lt;/em&gt; are they having trouble with?  What do you think is the reason why they are having those problems?  Why do you think the proposed intervention would be useful?  You seem to have planned the entire intervention without giving thought to those questions.  Answer those question &lt;em&gt;first&lt;/em&gt;.  That's your conceptual framework.  Then use the answers to design the intervention.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-09-11T13:10:33.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11401" PostTypeId="2" ParentId="11396" CreationDate="2016-09-11T18:05:48.233" Score="26" Body="&lt;p&gt;I spent some time looking for information that might provide some kind of reliable, evidence-based answer to my own question. I came up with the following, which is not perfect, but I thought it would be worth posting as an answer. I tried two methods of probing this: standardized test scores of students with high socioeconomic status (SES), and heritability of standardized test scores as measured by twin studies.&lt;/p&gt;&#xA;&lt;h2&gt;Standardized test scores of students with high SES&lt;/h2&gt;&#xA;&lt;p&gt;For this method I used data from the US.&lt;/p&gt;&#xA;&lt;p&gt;We can't use success in high school algebra as a way of determining innate ability to learn algebra, for a variety of reasons. Students in some states are required to take algebra as early as 8th grade, but many are &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4151197/&quot; rel=&quot;nofollow noreferrer&quot;&gt;not developmentally ready&lt;/a&gt; at that age, so their failure doesn't necessarily indicate a permanent inability. Anecdotally, many US high school algebra teachers report that they pass students who haven't learned the material but who were punctual, obedient, and turned in homework. Because many high schools have algebra as a graduation requirement, there is intense political pressure to say that students have passed algebra, regardless of whether they really understand it. Many students fail algebra not because of a lack of innate ability but because of a lack of motivation or interest. Many public schools serving a student population with low SES do not provide competent instruction.&lt;/p&gt;&#xA;&lt;p&gt;To get around these problems, I looked for data on SAT math scores among kids with high SES. Among the most affluent families in the US (the top decile of SES), &lt;a href=&quot;http://www.futureofchildren.org/publications/figures-tables/figure_show.xml?fid=145&quot; rel=&quot;nofollow noreferrer&quot;&gt;about 80% of kids go to college&lt;/a&gt; (although, surprisingly, only about 2/3 of those get a degree). The vast majority of these kids were exposed to competent instruction in algebra during high school, and they were typically under intense social and family pressure to get into a good college. Nearly all of them took the SAT, which is required by almost all schools in the US that have selective admissions.&lt;/p&gt;&#xA;&lt;p&gt;For students with the highest SES (more than $200,000 per year of family income), the average SAT math score is about 600. A score of about 600 is also used as a cut-off by many colleges in determining which of their students are allowed to take calculus without having to take a prerequisite course. The math section of the SAT includes a lot of algebra. Therefore it seems reasonable to me to take a score of about 600 as indicating pretty decent mastery of algebra. If the mean of the scores for this group roughly coincides with the median, then about 50% of people in this group have mastered algebra.&lt;/p&gt;&#xA;&lt;p&gt;It seems reasonable to assume that there is no significant difference in innate mathematical ability between the most affluent people in the US and the rest of the population. Therefore the scores of this group can probably be interpreted as a rough measure of what level of achievement you get in math, at this age, given a favorable environment. That is, they may give a reasonable estimate of what nature permits, given the right nurture.&lt;/p&gt;&#xA;&lt;p&gt;The mean score for Asians (regardless of SES) is also about 600. If we assume that there is no innate difference in mathematical ability between Asians and other ethnic groups, then this would seem to support the hypothesis that a mean score of about 600 represents a broad average of &lt;em&gt;potential&lt;/em&gt; mathematical achievement, among college-bound kids at age 18, and that this potential is reached if these students receive competent instruction and grow up with cultural expectations that pressure them to do well in academics.&lt;/p&gt;&#xA;&lt;p&gt;Since about 80% of high-SES students go to college, and about 50% of those get an SAT math score of 600 or higher, my estimate would be that something like 50-60% of high school kids in general are &lt;em&gt;not&lt;/em&gt; capable of learning algebra. This seems to be in the right ballpark to support Daniel Collins's statement that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Most of our students will never grok 8th-grade algebra no matter how many times they try, nor under what circumstances.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Many people who don't master algebra in high school nevertheless go to college, where they will typically be required to take remedial (&amp;quot;developmental&amp;quot;) courses in math. Some succeed in learning algebra at that point, so we should correspondingly reduce our estimate of the percentage of the population unable to learn algebra. However, remediation in math appears to be &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5905/what-is-the-evidence-about-the-effectiveness-of-remediation-in-math&quot;&gt;extraordinarily ineffective&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The most questionable assumption in my estimate seems to me to be that high-SES kids who get less than 600 really tried to learn math in high school. They were probably under strong pressure to succeed in math, but they may have responded to that pressure by doing the minimum required in order to get a certain grade, and doing the minimum may have meant going through the motions of taking a math course without actually learning the math. If there were many students like this, then the percentage of the population lacking the innate ability to learn algebra may be much lower than the above estimate. The size of this bias is not likely to be huge, however, because then one would expect a significant number of these students to succeed when remediated in college, whereas in fact remediation almost never works.&lt;/p&gt;&#xA;&lt;p&gt;Since there are a lot of shaky assumptions in the above estimate, it's interesting to see if we can place any more definitive upper and lower bounds on the figure.&lt;/p&gt;&#xA;&lt;p&gt;As an upper bound, the percentage of high school kids in the US incapable of learning algebra must be no more than about 94%. This can be inferred because out of 4.3 M Americans aged 18, about 1 M take the SAT, and 25% score 590 or higher on the math portion. Thus about 6% of the population demonstrates mastery of algebra every year, when they reach this age.&lt;/p&gt;&#xA;&lt;p&gt;As a lower bound, the percentage incapable of learning algebra seems sure to be at least 3%, since that's roughly the percentage of the population that is considered intellectually disabled.&lt;/p&gt;&#xA;&lt;h2&gt;Heritability of standardized test scores as measured by twin studies&lt;/h2&gt;&#xA;&lt;p&gt;Kids in the UK have to go to school until they're 16, and at that age they take a set of standardized exams called GCSEs, which includes a test of math. The Mathematics GCSE's &amp;quot;higher tier&amp;quot; is 30% based on algebra. The level of the algebra questions is about at the level of a first high-school algebra course in the US, i.e., it overlaps with the math portion of the SAT, but doesn't include the most difficult material you would find on the SAT.&lt;/p&gt;&#xA;&lt;p&gt;Twin studies show that GCSE math scores are &lt;a href=&quot;http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0080341&quot; rel=&quot;nofollow noreferrer&quot;&gt;55% heritable&lt;/a&gt;. &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4210287/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Only about half&lt;/a&gt; of this genetic influence appears to be due to intelligence; the other half seems to be because of other partially heritable personality traits such as &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Self-efficacy&quot; rel=&quot;nofollow noreferrer&quot;&gt;self-efficacy&lt;/a&gt;&amp;quot; and behavior problems.&lt;/p&gt;&#xA;&lt;p&gt;I would like to study this approach more thoroughly, but these findings seem to show that we can't necessarily think of a hypothetical innate inability to learn algebra solely as an innate intellectual shortcoming. For example, a student who refuses to do his algebra homework may be doing so partly because the heritable component of his intelligence tends to make the activity hard and not enjoyable for him, but also partly because a partially heritable component of his personality makes him unwilling to stay on task in order to achieve a future goal.&lt;/p&gt;&#xA;&lt;p&gt;If the twin studies obtained data on SES, then it would be interesting to combine the two approaches.&lt;/p&gt;&#xA;&lt;p&gt;A counterintuitive part of the interpretation of these data is that high heritability of test scores goes hand in hand with a more socially equal, meritocratic society. Suppose we make some change in social policy that makes us more meritocratic, e.g., &lt;a href=&quot;https://en.wikipedia.org/wiki/Head_Start_%28program%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Head Start&lt;/a&gt; programs, or eliminating the practice of giving preferences for college admission to children of alumni. By doing this, we've reduced the amount of variability in the environmental factors affecting education. Therefore the amount of variation in educational achievement will go down. Of this smaller remaining amount of variation, the fixed amount due to genetics will now represent a greater fraction.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="-1" LastEditDate="2022-04-01T14:37:15.447" LastActivityDate="2022-04-01T14:37:15.447" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="11402" PostTypeId="1" AcceptedAnswerId="11403" CreationDate="2016-09-12T18:54:09.287" Score="15" ViewCount="788" Body="&lt;p&gt;It frequently happens that there's some material I have to cover which is, frankly, boring. The subject itself may be boring, or it may be the particular exercises, but in any case I have to get through it. When this happens I try to adopt a positive attitude in the hopes that at least I will not be as boring as whatever it is I'm teaching. This only helps so much, though, especially in front of a class of students that don't particularly care about math a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know from personal experience that when you get bored you stop paying attention. What are some ways of not losing your class during the more tedious bits? Something I've thought about is telling them about the applications of the material, or even replacing the exercise I'm doing by something more &quot;real world&quot;; right now my students are in engineering, so they might appreciate a practical example. This doesn't always work, though, because usually applications of the material are more advanced than the basic stuff I want to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've also considered straight up telling the class that this stuff is boring but we just have to get through it, i.e., take the &quot;accomplice&quot; route. Whether this would work is debatable, and also as a teacher I feel I should try to find the fun in everything I teach instead of &quot;taking sides&quot; with the students.&lt;/p&gt;&#xA;" OwnerUserId="5269" LastActivityDate="2016-09-13T20:14:24.973" Title="How to get through the boring stuff?" Tags="|undergraduate-education|mathematical-pedagogy|student-motivation|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11403" PostTypeId="2" ParentId="11402" CreationDate="2016-09-12T22:08:20.423" Score="8" Body="&lt;p&gt;I will concur with and collect several comments and suggest that one consider &lt;em&gt;not teaching it&lt;/em&gt;, or at least cutting or refining the presentation. I'll say that in my own teaching, I have come to be sensitive to this as a sign that something is off with my lecture. If some part of the class actually seems to be boring &lt;em&gt;me&lt;/em&gt;, then I take that as a signal to inspect and revise that part of my presentation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe that involves mostly cutting the item, and leaving it for student reading outside of class (example: I used to go over the syllabus in detail on the first day; that's been cut). Or it may be reducing or refining the item (ex.: reducing the number of exercises; cutting basic exercises and going straight to moderate-level ones where multiple points of interest are at play). Or it might be totally re-writing the presentation in a totally different structure (ex.: inheriting some presentation from a book that seems cluttered, and finding a much more straightforward and elegant path of my own). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would recommend against calling out an item as &quot;boring&quot; in class. If your life is dedicated to the classroom, then do yourself a spiritual favor and find some way to make it not-boring. Hopefully this will have a secondary effect of making it not-boring for students, and infecting them with your interest. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-09-12T22:08:20.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11404" PostTypeId="2" ParentId="11402" CreationDate="2016-09-13T03:45:22.437" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How to get through the boring stuff?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You could present it in a different way yet still cover the same concepts and procedures.  For example, if your topic is multi-digit multiplication of positive integers, instead of providing the factors and asking for the product, you could have them fill in missing digits from the factors and the product, as shown below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lFMNz.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lFMNz.png&quot; alt=&quot;Multi-digit multiplication&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-09-13T03:45:22.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11405" PostTypeId="1" AcceptedAnswerId="11406" CreationDate="2016-09-13T04:03:14.437" Score="8" ViewCount="228" Body="&lt;p&gt;I'm interested in journals that present challenging problems for high school students &lt;em&gt;and&lt;/em&gt; take answer submissions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I coach some talented high school students and would like to encourage them to pursue such problems and submit their work, earning recognition in the process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's been a while since I've checked it out, but I think the &lt;a href=&quot;http://www.maa.org/press/periodicals/college-mathematics-journal/the-college-mathematics-journal&quot;&gt;College Mathematics Journal&lt;/a&gt; has a section of problems like this. Yet, I wonder if that's the correct level. Do you know of other resources?&lt;/p&gt;&#xA;" OwnerUserId="6662" LastEditorUserId="6662" LastEditDate="2016-09-13T07:03:42.490" LastActivityDate="2016-09-13T22:57:51.720" Title="Journals that feature challenge problems for high school students?" Tags="|reference-request|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11406" PostTypeId="2" ParentId="11405" CreationDate="2016-09-13T13:01:36.913" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://cms.math.ca/crux/&quot; rel=&quot;nofollow&quot;&gt;Crux Mathematicorum&lt;/a&gt; is one that specifically is aimed at this particular cohort.  One particularly interesting feature is that some problems are in both French and English, since this is a Canadian journal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes problems in &lt;a href=&quot;http://www.maa.org/press/periodicals/mathematics-magazine&quot; rel=&quot;nofollow&quot;&gt;Mathematics Magazine&lt;/a&gt; are appropriate, but usually Monthly ones are just too hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, sometimes the &lt;a href=&quot;http://www.pme-math.org/journal/problems.html&quot; rel=&quot;nofollow&quot;&gt;Pi Mu Epsilon Journal&lt;/a&gt; has appropriate problems as well, though I am less familiar with that one and the website seems to be somewhat moribund.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2016-09-13T20:17:06.710" LastActivityDate="2016-09-13T20:17:06.710" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11407" PostTypeId="2" ParentId="11402" CreationDate="2016-09-13T13:26:49.220" Score="3" Body="&lt;p&gt;I want to depart from the stance that boring parts may simply be put aside, and that one should not acknowledge them as boring. I claim that &lt;strong&gt;there are some boring and tedious tasks that are needed to learn mathematics&lt;/strong&gt;. One needs to compute a few integrals, a few Taylor series, a few derivatives to really get what it is all about. One needs to work some proof in tedious details before having the background enabling one to see the big picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Claiming there is no boring part is somewhat misleading and might even be considered rude by the students, who could be bored even by what the teacher finds exciting. Some acknowledgement of boredom is simply honesty. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I usually compare studies (in maths or something else) to high-level sport. You can enjoy tennis. Some train to become good tennis player to enjoy it even more. But training to be a good tennis player needs some boring, tedious (and tiring) stuff: working out, repeating the same gesture again and again to get it right, etc. It does not get right by himself, one needs to put effort in it. I think it &lt;strong&gt;very important to stress student that studying needs efforts&lt;/strong&gt; as much as it can be (in order for it to be) rewarding and enjoyable. No one says to a young person wanting to play tennis: &quot;sure, you can be good at it without running or practicing!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be more constructive, a teacher must try to make material non-maximally boring; and one should certainly take the most boring part out of the main lecture by giving it as assignments: the boring parts are usually the ones student need to do by themselves. One can even take a boring proof which one wants to be part of the curriculum, and give it (suitably divided into questions) as mandatory homework. But students need to do some boring stuff, i don't think it is avoidable and it should be explained.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-09-13T13:26:49.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11408" PostTypeId="2" ParentId="11402" CreationDate="2016-09-13T13:36:22.983" Score="1" Body="&lt;p&gt;Exactly what the dull topic is matters. The following idea works best for things that are more like prerequisites: Take for example, the domain and range. One option would be to devote a class to them so that you can simply refer to the material later. (In my mind this is fairly dull.) Another option is to fold that material into other lectures in a just-in-time fashion. Say, introduce the domain in a example where there is domain where the formula is valid and a domain where the problem is physically well-defined. &lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2016-09-13T15:08:08.720" LastActivityDate="2016-09-13T15:08:08.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11410" PostTypeId="2" ParentId="11402" CreationDate="2016-09-13T20:14:24.973" Score="5" Body="&lt;p&gt;I'm going to restate and elaborate on something I said briefly in the comments:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;If you think something is boring, you need to figure out what is interesting about it.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Boring&quot; is in the eye of the beholder; everything is interesting to &lt;em&gt;someone&lt;/em&gt;, and everything is  &lt;em&gt;potentially&lt;/em&gt; interesting to you and your students.  Part of the teacher's job is to make the content seem interesting to students, and that is not possible for teachers who do not themselves find the content interesting.  So if you don't see what is interesting about a specific topic then it is your &lt;em&gt;professional obligation&lt;/em&gt; to stop and think about it more and find an angle on it that makes it interesting to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have not provided us with any context or specifics (except for the &lt;code&gt;undergraduate-education&lt;/code&gt; tag) so it will be difficult for us to help you with this, but I assure you:  nothing is intrinsically, universally boring. Whatever you are teaching, someone, somewhere thinks it is interesting.  You need to figure out why.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-09-13T20:14:24.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11411" PostTypeId="2" ParentId="11405" CreationDate="2016-09-13T22:57:51.720" Score="4" Body="&lt;p&gt;Here are two journal recommendations, and one website recommendation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, you may consider the calendar problems in the &lt;em&gt;Mathematics Teacher&lt;/em&gt; (&lt;a href=&quot;http://www.nctm.org/publications/mathematics-teacher/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;) which is published by the National Council of Teachers of Mathematics (NCTM). They also tweet some calendar problems that you could check out for free (&lt;a href=&quot;https://twitter.com/mt_at_nctm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;twitter&lt;/strong&gt;&lt;/a&gt;). Although they do not actively solicit student submissions for publication, I think that talented high school students could actually consider publishing &lt;em&gt;articles&lt;/em&gt; in the journal if they have ideas about particular problems. Specifically, these mathematical explorations might be appropriate for the Delving Deeper section of the journal, possibly with your assistance (the journal actively seeks writers who are practitioners i.e. math teachers!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, and more directly in response to your query, is the Mathematical Association of America's (MAA) &lt;em&gt;Math Horizons&lt;/em&gt; (&lt;a href=&quot;http://www.maa.org/press/periodicals/math-horizons&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;), which includes a section in each issue called &lt;strong&gt;The Playground&lt;/strong&gt; for which students can have their answers, as well as their problems, published. They will credit students who submit correct answers by name; in my experience, they are also quite receptive to problem proposals. I've put two into their issues; one of these problems was gamefied by &lt;a href=&quot;http://www.playwithyourmath.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;PlayWithYourMath.com&lt;/strong&gt;&lt;/a&gt; into a worksheet (&lt;a href=&quot;http://www.playwithyourmath.com/pdfs/12-Space%20Race-Worksheet.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;pdf&lt;/strong&gt;&lt;/a&gt;); the other problem I put in was back in 2012:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eG0g7.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eG0g7.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see from the image above that &quot;any type of problem may appear in the Jungle Gym&quot; (one of the problem sections in The Playground) — the solution that I sent in along with this proposal used Catalan numbers, which is outside most high school curricula but was related to an article in that same issue of &lt;em&gt;Math Horizons&lt;/em&gt;. My own belief is that fostering creativity through problem posing is just as important as (or perhaps even more important than!) problem solving.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a non-journal recommendation, 538 has a weekly puzzle section called &lt;strong&gt;The Riddler&lt;/strong&gt; (&lt;a href=&quot;http://fivethirtyeight.com/tag/the-riddler/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;). The one (minor) gripe that I have with these posts is that they are not well-sourced. For example, &lt;a href=&quot;http://fivethirtyeight.com/features/can-you-solve-the-puzzle-of-the-picky-eater/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Picky Eater&lt;/strong&gt;&lt;/a&gt; is Putnam 89/B1 (&lt;a href=&quot;https://mks.mff.cuni.cz/kalva/putnam/putn89.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;) and its extra credit modified to an equilateral triangle was asked in &lt;a href=&quot;https://math.stackexchange.com/q/1688936&quot;&gt;&lt;strong&gt;MSE 1688936&lt;/strong&gt;&lt;/a&gt;. Offhand, I also recall &lt;a href=&quot;http://fivethirtyeight.com/features/can-you-solve-the-puzzle-of-your-misanthropic-neighbors/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Misanthropic Neighbors&lt;/strong&gt;&lt;/a&gt; having a somewhat storied history beginning (as far as I know) with SIAM 62-3 (1964; &lt;a href=&quot;http://www.jstor.org/stable/2028090&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;JSTOR&lt;/strong&gt;&lt;/a&gt;) and continuing with work by D Knuth.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2016-09-13T22:57:51.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11412" PostTypeId="1" CreationDate="2016-09-14T00:24:36.140" Score="19" ViewCount="557" Body="&lt;p&gt;After introducing eight different ways of viewing the derivative of a function&#xA;(infinitesimal, symbolic, logical, geometric, rate, approximation, microscopic),&#xA;Thurston, in his famous essay,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Thurston, William P. &quot;On proof and progress in mathematics.&quot; &lt;em&gt;New Directions in the Philosophy of Mathematics&lt;/em&gt; (1998): 337-55.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;says&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/hsMc8.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hsMc8.jpg&quot; alt=&quot;ThurstonExcerpt&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;His last sentence above suggests that, in many circumstances, one should not&#xA;introduce precise definitions as a start to explaining a mathematical concept, but instead sketch out the different ways of thinking about that concept, and only&#xA;then show how these different ways are reflected in a formal definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My guess is that this is not a widely applied teaching technique, and that&#xA;many teachers would disagree, and believe you cannot convey a mathematical&#xA;concept until those definitions are in place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two questions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Q1&lt;/strong&gt;. Is my hunch correct? Is there a widespread belief among math educators&#xA;  that definitions should precede the concept (to put it in a nutshell)?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Q2&lt;/strong&gt;. If so, what are some counter arguments to delaying the definitions&#xA;  so they don't collapse the different ways of viewing a concept to one way?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-09-14T01:00:17.603" LastActivityDate="2016-09-19T00:07:31.610" Title="Against introducing precise definitions first" Tags="|undergraduate-education|secondary-education|concept-motivation|general-pedagogy|definitions|" AnswerCount="5" CommentCount="27" ContentLicense="CC BY-SA 3.0" />
  <row Id="11413" PostTypeId="2" ParentId="11412" CreationDate="2016-09-14T08:27:17.760" Score="10" Body="&lt;p&gt;At the secondary level, students have not yet mastered formal mathematics and most will need to continue learning concepts before definitions in many cases. The van Hieles (the Dutch educators who developed the Van Hiele Model of how students learn geometry) insisted that definitions should never come first. They claimed that students should first have informal experiences with a new idea until the class has developed a shared language around a concept before introducing the formal vocabulary that goes with the idea. Some educators disagree, insisting students need experience with analyzing new definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the decision of when to introduce formal definitions will often need to be made on a case-by-case basis, and may depend on both the abstractness of the definition and the maturity of the student. Many ideas will need some exposure before the formal definition so that a context exists in which the definition can make sense. But some ideas are transparent enough that students could tackle them without preparation. For example, geometry students can probably understand the formal definition of a quadrilateral without any introduction because they will have already had exposure to that idea in previous classes. Students might be able to understand the definition of a logarithm if it is unpacked immediately, but a bit of preparation beforehand to introduce the definition of logarithm would probably be helpful. Defining a limit or a derivative for the first time will definitely require a considerable amount of conceptual preparation before introducing the formal definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't forget that introducing definitions first may be common to mathematics courses, but in the &quot;real world,&quot; definitions almost always come about after mathematicians have wrestled with an idea for some time. Early definitions are often vague and imprecise. Allowing students to approach new ideas in a similar way is therefore an intellectually honest approach to real mathematics.&lt;/p&gt;&#xA;" OwnerUserId="6782" LastActivityDate="2016-09-14T08:27:17.760" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11414" PostTypeId="1" AcceptedAnswerId="11415" CreationDate="2016-09-14T18:32:00.427" Score="9" ViewCount="150" Body="&lt;p&gt;Context: I am teaching a course on plane geomtery at an mid-size state University in the USA for students who wish to be high school math teachers. They will have to make presentations once per week, and I want to provide them with a list of goals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;able to answer questions from the audience (within reason)&lt;/li&gt;&#xA;&lt;li&gt;able to talk about the material rather than read from notes&lt;/li&gt;&#xA;&lt;li&gt;able to connect this presentation to previous material &lt;/li&gt;&#xA;&lt;li&gt;etc&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Can anyone (recommend resources that) suggest what else I might ask for in a student presentation?&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2016-09-16T11:59:58.280" Title="What do you look for in a student presentation?" Tags="|assessment|presentation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11415" PostTypeId="2" ParentId="11414" CreationDate="2016-09-14T22:00:57.797" Score="8" Body="&lt;p&gt;I have students present (in classes ranging from grades 7 to 12) and use the following rubric. It is not original, and googling various snippets suggests to me that it has been pieced together from several different sources. Perhaps modifying this could yield something useful (note, e.g., that I have the students present in &lt;em&gt;groups&lt;/em&gt; — so if you are having them present individually, then some of the language will certainly need tweaking).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Google Drive &lt;a href=&quot;https://drive.google.com/file/d/0B6w4YIaFBvYUd25WZ2hXaEZ6WWs/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt; to PDF of the file.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Image of main body:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/98TBi.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/98TBi.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-09-14T22:00:57.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11416" PostTypeId="2" ParentId="11414" CreationDate="2016-09-16T11:59:58.280" Score="1" Body="&lt;p&gt;Some skills I find valuable in real life:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Being able to continue the talk without slides, e.g. due to projector or laptop malfunction, etc.&lt;/li&gt;&#xA;&lt;li&gt;Being able to stop the talk in the middle (e.g., having multiple possible stop-points). This is because our time might run out due to some non-related issues like a group of students behaving badly, the principal having an announcement, etc.&lt;/li&gt;&#xA;&lt;li&gt;Being able to stop the talk late, that is, continue for some time after the designated mark. This might be due to things like &quot;oh, we have covered this in physics class&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="42" LastActivityDate="2016-09-16T11:59:58.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11417" PostTypeId="2" ParentId="11412" CreationDate="2016-09-16T15:29:35.303" Score="2" Body="&lt;p&gt;I agree with Scott Eberle that the decision should be made on case-by-case basis, but I would like to argue that most of the established concepts would actually benefit from giving the precise definition first. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic reason is that it gives students something &lt;strong&gt;concrete&lt;/strong&gt; to think and reason about, thus avoids any vague hand-wavy limbo state.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My rule of thumb is to check if there is one, well-established canonical definition of a concept. If there is, I would state it first, and only then consider:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;different intuitions and understandings,&lt;/li&gt;&#xA;&lt;li&gt;how they compare to the given formal definition, &lt;/li&gt;&#xA;&lt;li&gt;how we could use some other definition which currently we have as theorems,&lt;/li&gt;&#xA;&lt;li&gt;how some intuitions &lt;em&gt;fail&lt;/em&gt;, i.e., should not be used as definition (but can be possibly still useful, e.g. they work for some smaller class of objects).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One the other hand, if there is no such clear canonical answer, i.e., different areas use different definitions, or the actual definition is bootstrapped from simpler cases, then I find prefer to start with laying out the various intuitions and generalizations (partial definitions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give some examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There is one, very simple definition of a function: a collection of pairs; only after that I would say anything about representing it as a function-graph or creating such pair collections from formulas.&lt;/li&gt;&#xA;&lt;li&gt;Regular languages can be defined using monoids, automata, grammars/regular expressions; each approach has its own set of intuitions, and I find it beneficial to state those first, before picking option as &lt;em&gt;the&lt;/em&gt; definition and proving others as theorems.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have never taught derivatives, but I think I would like to state the formal definition first, and then show how it accommodates various intuitions we would like it to have.&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2016-09-16T15:29:35.303" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11418" PostTypeId="1" AcceptedAnswerId="11419" CreationDate="2016-09-16T17:34:09.583" Score="8" ViewCount="642" Body="&lt;p&gt;Yes, I've read a number of definitions&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In analytic geometry, an asymptote of a curve is a line such that the&#xA;  distance between the curve and the line approaches zero as they tend&#xA;  to infinity. Some sources include the requirement that the curve may&#xA;  not cross the line infinitely often, but this is unusual for modern&#xA;  authors.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And for most examples, such as $y=1/x$, it seems clear what's going on, and how to discuss the asymptote. Yesterday, a student asked me about this function -&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RtrCF.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RtrCF.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only after I told her it was a discontinuous function, and when drawing it, be certain to show that at x=0, both lines should be an open circle indicating &quot;not a point&quot;, did she tell me the teacher said that $y=1$ and $y=-1$ were asymptotes. I think there's a mistake here as there's nothing approaching the lines, the lines are the equation. Just as a first order binomial, $y=x+2$, is just a line, no asymptotes there. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This prompted the follow on -&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bQ29K.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bQ29K.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And the question of whether today's usage allows crossing to occur. In this example, the distance itself, from first definition, will cross zero, infinitely many times. Looking at the quoted definition, what does &quot;modern authors&quot; mean? Do I need to be concerned that an older book will call the X axis of second graph &quot;Not Asymptote&quot;, but a newer one is fine? &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="6053" LastEditDate="2018-05-07T22:07:11.753" LastActivityDate="2018-05-07T22:07:11.753" Title="Teaching Asymptotes" Tags="|calculus|graphing|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11419" PostTypeId="2" ParentId="11418" CreationDate="2016-09-16T23:23:54.453" Score="6" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Asymptote - if $f(x)-g(x)\to0$ as $x\to a$, then $f(x)$ is asymptotic to $g(x)$ as $x\to a$.  Similarly, $g(x)$ is asymptotic to $f(x)$ as $x\to a$.  (IMO)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Personally, I do not see the problem with intercepting any amount of times.  If you wish to modify the definition of an asymptote to fit this, I would probably use &lt;a href=&quot;https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior&quot; rel=&quot;nofollow&quot;&gt;limit superior and limit inferior.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Modified Asymptotes - if $\limsup\limits_{x\to a}f(x)-g(x)=\liminf\limits_{x\to a}f(x)-g(x)=0$, then $f(x)$ is asymptotic to $g(x)$ as $x\to a$ and vice versa regardless of interceptions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;IMO, I have never seen something marked as non-asymptotic because of interceptions.  If you wanted, you could still squeeze theorem the asymptote using limit superior and limit inferior, so it shouldn't be any problem.  Also note that classifying it as non-asymptotic is probably less helpful than classifying it as an asymptote.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, while you are dealing with the more geometric asymptotes, there are also asymptotes that are more &quot;functional&quot; (as I would put it) in the fields of &lt;a href=&quot;https://en.wikipedia.org/wiki/Asymptotic_analysis&quot; rel=&quot;nofollow&quot;&gt;&quot;Asymptotic Analysis&quot;&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Asymptotic_theory&quot; rel=&quot;nofollow&quot;&gt;&quot;Asymptotic Theory&quot;&lt;/a&gt;, along with the concept of &lt;a href=&quot;https://en.wikipedia.org/wiki/Big_O_notation&quot; rel=&quot;nofollow&quot;&gt;&quot;Big 'O' Notation&quot;&lt;/a&gt;. Such things allow different meanings to &quot;asymptote&quot;, for example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\lim_{x\to\infty}\frac{f(x)}{g(x)}=1\implies f(x)\sim g(x)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A famous example of when the above does not agree with our original definition of asymptote is &quot;Stirling's Approximation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\lim_{n\to\infty}\frac{\sqrt{2\pi n}\left(\frac ne\right)^n}{n!}=1,\lim_{n\to\infty}\sqrt{2\pi n}\left(\frac ne\right)^n-n!=-\infty$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea can be thought of in the much more simple example of comparing $f(x)=10^x+x$ and $g(x)=10^x$.  Clearly $f(6)=10\text{ Meg}+6$, but $g(6)=10\text{ Meg}$ (Meg=million), but what is $6$ in comparison to $10\text{ Meg}$?  It is negligible.  But our original definition of asymptote does not consider the $6$ as negligible in comparison to the ten million...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Big $O$ notation is also useful, for example,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{x^5+7}{x^2-2}=x^3+O(x^2)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;all this is really saying is that the rational function has a dominant $x^3$ growth/behavior, and the rest of the behavior is of $x^2$ order or less.  This can quickly help determine the behavior of extremely complicated functions by extracting the dominant behavior of a function for either limits or approximations, etc.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="1147" LastEditDate="2016-09-18T00:11:05.017" LastActivityDate="2016-09-18T00:11:05.017" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11420" PostTypeId="2" ParentId="11418" CreationDate="2016-09-17T03:16:25.683" Score="3" Body="&lt;p&gt;Mathematically a horizontal line is asymptotic to itself. So for example the horizontal asymptote of the constant function $f(x)=3$ is the line $y=3$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This issue of whether the definition of a horizontal asymptote should place some sort of restriction on the graph of $y=f(x)$ crossing a horizontal asymptote seems to come up frequently. As a mathematician this seems peculiar, and has always puzzled me. I suspect the root of this resides in the etymology of the word &quot;asymptote&quot; which traces back to something like &quot;not together.&quot; Thinking about the etymology in this way might tempt one to think that the mathematical definition of an asymptote perhaps &lt;em&gt;should&lt;/em&gt; include some restriction to rule out the notion of a line being asymptotic to itself. But it should not and does not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we have here is just a peculiar and interesting drift over time from the etymological meaning of a word to a meaning and usage that is quite different. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-09-17T03:16:25.683" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11421" PostTypeId="2" ParentId="11418" CreationDate="2016-09-17T12:55:21.457" Score="3" Body="&lt;p&gt;I don't think that a horizontal line being asymptotic to itself is a &quot;trick question we just never really run into&quot; (as you suggest in the comments). Probability theory gives a natural context in which this sort of thing occurs (albeit in a way which is more than &lt;em&gt;just&lt;/em&gt; a horizontal line). A standard result is that a continuous function $F$ defined on $(-\infty, \infty)$ is a cumulative distribution function of a continuous random variable if and only if it is monotone increasing and is asymptotic to $0$ as it approaches $-\infty$ and $1$ as it approaches $\infty$. For some distributions (e.g. a normal distribution) these are asymptotes as you typically think of them -- you approach $0$ or $1$ but never get there for any finite value. On the other hand, if the distribution is for a random variable with bounded support then the asymptotes $0$ and $1$ will be achieved long before $\pm \infty$. For example, the graph of the cdf of the standard uniform variable on $[0,1]$ looks like &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/N9tUM.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/N9tUM.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the graph consists of horizontal lines which are equal to the asymptotic values. I don't see any reason to consider this a counterexample to the claim that cumulative distribution functions are always asymptotic to $0$ and $1$.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastEditorUserId="6891" LastEditDate="2016-09-17T13:46:30.250" LastActivityDate="2016-09-17T13:46:30.250" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11422" PostTypeId="2" ParentId="11412" CreationDate="2016-09-17T16:10:42.380" Score="6" Body="&lt;p&gt;Teaching undergrads and graduate students, by this point I do &lt;em&gt;not&lt;/em&gt; like &quot;precise definitions&quot; at the outset, for the sort of reasons given by W. Thurston, and also for reasons given by I.M. Gelfand. Namely, a suite of &lt;em&gt;examples&lt;/em&gt; of phenomena and of causal mechanisms to illustrate various aspects of the topic at hand. Arguably, only after looking through the immediate examples can we understand what our terminology needs to cope with and refer to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I've found that most undergrads have cognitive or semantic troubles with &quot;definitions&quot;... since very little else in the world works this way. They often react by &quot;memorizing&quot; the definitions, rather than integrating them with phenomena. Artifacts of formatting and wording are often indistinguishable to them from genuine issues. This seems bad to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the time students are in grad school, they seem to be &quot;reconciled to&quot; checking definitions without questioning the virtues of them. But, indeed, just as with dictionaries of human languages, definitions of complicated things are often not infinitely precise, and meanings or references drift with time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The notion of &quot;function&quot; exemplifies some of these troubles. First, it would be silly to deny that at entry level most functions &lt;em&gt;are&lt;/em&gt; given by formulas. Second, yes, we want a notion of &quot;function&quot; that does not depend entirely on formulas. Third, yes, the set-theoretic epiphany to define a function $f:X\to Y$ as a subset of the cartesian product $X\times Y$ with certain properties was a very nice umbrella. &quot;(E.g., Lebesgue) measurable&quot; as a qualification was/is very useful,... and/but leads to a slight loss of pointwise sense, since now &quot;almost everywhere&quot; turns out to be the useful degree of precision. And, indeed, $L^2$ spaces of &quot;functions&quot; do not have pointwise values, yet are the right things to look at for many purposes. Similarly, &quot;generalized functions&quot; (a.k.a. &quot;distributions&quot;) usually do not have pointwise values, so are certainly not specifiable by &quot;graphs&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... though, yes, the Schwartz Kernel theorem does recover many of the lost items!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, &quot;pointwise convergence&quot; is not usually the real issue, to my mind! Convergence in $L^2$, or in Sobolev spaces, or &lt;em&gt;uniformly&lt;/em&gt; pointwise, sure. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, then, one of my first talking points in teaching graduate-level real analysis this year, with an eye to updating the course content, is to illustrate the various troubles with &quot;pointwise&quot; notions, arguing that we need/want a broader outlook on the notion of &quot;function&quot;, at the very least to be able to accommodate actual practice!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the viewpoint of exposition, too, I do not like definitions. If/when they appear without motivation or examples, why should I accept them? Why should the students? An implicit justification &quot;by authority&quot; is not the kind of reason I want to promote. I want to see phenomena first, and any discussion of modifications of word-usage afterward.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, although I realize that quite a few people like the other-worldliness of mathematics, a great many students have difficulties, and I think with good reason, operating in a context where they are essentially required to suspend their own judgement and play by rules whose significance and origin is effectively kept secret. There may be virtues in the formal exercise of playing such games, but it is not clear to me that this is really the main selling point of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, I do like actual proofs, rather than merely heuristics, and this does require precise &lt;em&gt;references&lt;/em&gt; and referents, but that's not the same thing as &quot;definitions&quot;. Also, proving boring, superficial things very precisely is an odd sort of exercise, of limited interest to many people, or only of interest at very particular moments. A more interesting kind of thing is, for example, legitimizing the mathematical aspects of things physicists (Dirac, Bethe, et al) have done, with observable physical corroboration, to repurpose the mathematics more broadly. In that sort of circumstance, it is not within our power to &quot;define&quot; anything into or out of existence. Instead, to my perception, the issue is of grappling with examples and phenomena. Yes, there is the secondary issue of mutual intelligibility, but it does seem to me secondary.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-09-17T16:10:42.380" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11423" PostTypeId="2" ParentId="7243" CreationDate="2016-09-17T19:06:11.207" Score="3" Body="&lt;p&gt;Give your students a multi-step problem and 5 different &quot;solutions&quot; written by fictitious students (you, really).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Here are five different solutions to this problem. First, rank them from best to worst. Second, discuss your ranking with the student next to you and determine appropriate criteria for assessing solutions. Third, list what you believe the most common mistakes are and how one might prevent or catch them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then it's class discussion time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Include in those solutions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A range in the quality of solutions, from junk to A+ work.&lt;/li&gt;&#xA;&lt;li&gt;The most common 2 or 3 mistakes.&lt;/li&gt;&#xA;&lt;li&gt;A variety of checks, responses to the checks, and reasonableness judgments.&lt;/li&gt;&#xA;&lt;li&gt;A variety of neatness, from practically LaTeX formatted to illegible scrawls. This can lead to a discussion on how to write down neat work and communicate mathematically on paper.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2016-09-17T19:06:11.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11424" PostTypeId="1" AcceptedAnswerId="11475" CreationDate="2016-09-18T11:48:11.830" Score="6" ViewCount="192" Body="&lt;p&gt;I'm currently teaching Calculus I problem solving sessions for college freshmen.&#xA;The problem is that the majority of students have studied the material in high school (what's a function, even/odd, limits, asymptotes...).&#xA;What can I do in this situation to make those who're familiar with the material engaged in the problem solving sessions?&lt;/p&gt;&#xA;" OwnerUserId="7152" LastActivityDate="2016-10-02T01:50:20.217" Title="Calculus I known material, getting students engaged" Tags="|undergraduate-education|calculus|student-motivation|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11425" PostTypeId="2" ParentId="11412" CreationDate="2016-09-18T19:00:21.523" Score="5" Body="&lt;p&gt;In the comments I mentioned using a game called Definitions war. Before presenting a given definition, ask students to come up with their own definitions of an object. Then work on improving it, with some fun. &lt;a href=&quot;http://education.lms.ac.uk/2016/09/alexandra-o-fradkin-exploring-rectangles/&quot; rel=&quot;noreferrer&quot;&gt;Here's an example&lt;/a&gt; of a class attempting to come up with a good definition of 'rectangle.'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we are to 'teach' the definition of derivative, we can start with thinking about a tangent line to a curve (students generally have intuition about this), and explore why that's so hard to 'define.'&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2016-09-18T19:00:21.523" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11426" PostTypeId="2" ParentId="11412" CreationDate="2016-09-19T00:07:31.610" Score="8" Body="&lt;p&gt;One of my most vivid memories of graduate school was working on a problem concerning the Gelfand-Kirillov dimension of certain rings.  I had been puzzling over the definition (which was rather new to me) and had reached the conclusion that for a certain class of rings the GK-dimension was greater than 1.  My advisor told me that I could not possibly be correct, because the dimension of such rings &lt;em&gt;had to be&lt;/em&gt; 1.  I insisted that per the definition of GK-dimension, my result was correct, and began to outline the proof, but he cut me off.  &quot;If what you say is right,&quot; my advisor said, &quot;Then you have the definition wrong.&quot;  But (I protested) that was the definition in the textbook!  &quot;If the definition in the textbook leads to that conclusion,&quot; said my advisor, &lt;strong&gt;&quot;then the definition in the book is wrong, because that conclusion is unacceptable,&lt;/strong&gt; and you need to figure out what the definition &lt;em&gt;ought to be&lt;/em&gt;.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that while from a purely logic-deductive perspective definitions are &lt;em&gt;a priori&lt;/em&gt; stipulations from which conclusions follow, this perspective in many ways fundamentally misrepresents what mathematicians actually &lt;em&gt;do&lt;/em&gt;.  In practice definitions are instrumental:  we construct them so that the properties we want to follow as conclusions will be entailed by them.  (See also:  Imre Lakatos, &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=1n6SFdXCOBQC&amp;amp;printsec=frontcover&amp;amp;dq=proofs%20refutations%20lakatos&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwi-0bmej5rPAhUF9YMKHavoDuUQ6AEIHjAA#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proofs and Refutations&lt;/a&gt;&lt;/em&gt;.)  For this reason I think it's always important for students to first see some examples of the thing to be defined, build some intuition of what kinds of properties they hold, and &lt;em&gt;then&lt;/em&gt; present a definition that actually captures those properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Related:  my answer at &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10620/why-would-you-teach-calculus-before-teaching-real-analysis/10623&quot;&gt;Why would you teach Calculus before teaching Real Analysis?&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-09-19T00:07:31.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11427" PostTypeId="2" ParentId="11396" CreationDate="2016-09-19T16:40:41.353" Score="3" Body="&lt;p&gt;Say student X tries a dozen different teachers and tutors and learning methods to grok algebra and they all fail. Is X incapable or is there another approach that would work better? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question basically cannot be answered decisively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the world's greatest math teachers all combined efforts to find new ways to teach X, would that work? Could any empirical study test them all? [One thing to keep in mind here is that most of us are unaware of how incredibly narrow our math education experiences are and that there are many radically different ways of teaching math.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To know what share of the population is inherently incapable of learning math, we'd have to know ALL teaching methods across ALL time compare that with a PERFECT and COMPLETE understanding of the brain. Ain't gonna happen any time soon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;i.e. In the next thousand years, what improvements will math pedagogy undergo? Maybe X will just learn algebra Matrix-style in 2063.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest other questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What are the best known ways to teach algebra to diverse students?&lt;/li&gt;&#xA;&lt;li&gt;How can we scale up them up?&lt;/li&gt;&#xA;&lt;li&gt;How can improve them further?&lt;/li&gt;&#xA;&lt;li&gt;What else do we need to learn about algebra pedagogy?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2016-09-19T16:40:41.353" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11428" PostTypeId="2" ParentId="8487" CreationDate="2016-09-21T01:58:13.683" Score="4" Body="&lt;p&gt;This indirectly answers a number of implicit or explicit questions, but obviously not this entire huge question; hope it's useful, if you are still trying this project, but certainly to others wishing to try similar projects.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;A number of upcoming (free) textbooks, including at least one &lt;a href=&quot;http://scholarworks.gvsu.edu/books/10/&quot; rel=&quot;nofollow noreferrer&quot;&gt;already-used free calculus book&lt;/a&gt;, are using or switching to the &lt;a href=&quot;https://pretextbook.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;write once, read anywhere&quot; philosophy of PreTeXt&lt;/a&gt;.  It has a definite learning curve, but you may find it a useful way to keep everything together.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another online LaTeX editor (and full Linux environment) is &lt;a href=&quot;https://cloud.sagemath.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;SageMath Cloud&lt;/a&gt;.  Overkill for just TeXing but if you want to start adding computation or easily adding certain graphics then using it with &lt;a href=&quot;http://doc.sagemath.org/html/en/tutorial/sagetex.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;SageTeX&lt;/a&gt; works very nicely.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Perhaps most importantly, if you really want to crowdsource (that wasn't 100% clear in your post) things, you may wish to post your code in an online distributed repository for people to comment on or even submit change requests, such as &lt;a href=&quot;https://github.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Github&lt;/a&gt; or &lt;a href=&quot;https://github.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bitbucket&lt;/a&gt; or &lt;a href=&quot;https://about.gitlab.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gitlab&lt;/a&gt;.  They may seem intimidating (and I wouldn't ask anyone to learn &lt;code&gt;git&lt;/code&gt; as a first revision control system, but apparently nowadays everyone else disagrees) but is very, very useful for open projects (and others, as long as you only let authorized people see the source).  Even trivial typo fixes like &lt;a href=&quot;https://github.com/Max-Hailperin/Operating-Systems-and-Middleware--Supporting-Controlled-Interaction/pull/115&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one I recently noted on a book I've been reading&lt;/a&gt; can be easily crowdsourced this way (though it's no guarantee they &lt;em&gt;will&lt;/em&gt; be!).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;PS: your comment&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In my opinion, if a student is impressed by the high quality, ease of understanding, and straightforwardness of the textbook, then with good teaching, the student would successfully learn the material as opposed to memorizing facts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;may work for very motivated students as yourself, but you will soon find that the &quot;with good teaching&quot; part is the really challenging piece.  And perhaps you will someday be one such great teacher, good luck!&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2019-08-12T17:28:36.640" LastActivityDate="2019-08-12T17:28:36.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11430" PostTypeId="1" CreationDate="2016-09-21T20:23:24.100" Score="7" ViewCount="214" Body="&lt;p&gt;I'm a new graduate student, and am currently TAing a calculus II class (which covers primarily integration).  At my university this involves both marking assignments and leading tutoring sessions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like I have a pretty good grasp on most of how I should mark &amp;amp; tutor, but the one thing which I've heard opposing opinions on is the 'harshness' of marking, e.g. say a person misuses the De Morgan laws, but consistently, throughout an assignment.  How should one mark this?  I've seen two schools of thought:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;hammer them each time they make the mistake so that they will (ideally) never make it again &lt;/li&gt;&#xA;&lt;li&gt;mark off once the first time, but only point out their mistake each subsequent time (provided that their mistake does not make the problem too easy)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I can see advantages and faults in both approaches.  I would love your thoughts on which is better, or to hear of a third way if you have come up with one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;cheers!&lt;/p&gt;&#xA;" OwnerUserId="7163" LastActivityDate="2016-09-28T17:12:23.860" Title="being a good marker as a TA" Tags="|undergraduate-education|grading|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11431" PostTypeId="2" ParentId="11430" CreationDate="2016-09-21T21:25:17.057" Score="2" Body="&lt;p&gt;I personally give homework small weight for such courses. I encourage my TAs to grade fairly harshly, because I want students to be aware when they do not know something, to motivate them to learn the material well. Hopefully for them, that will lead them to study more and increase their subsequent exam performance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would lean to instructing TAs to follow option (1). I'm not opposed to option (2) in cases such as the one you propose, but I find it ends up being a lot of work for minimal practical impact. Because my homework weights are so low, the difference in grades has little appreciable difference on the student's final average.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2016-09-22T04:12:39.577" LastActivityDate="2016-09-22T04:12:39.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11432" PostTypeId="2" ParentId="2623" CreationDate="2016-09-21T21:36:31.420" Score="10" Body="&lt;p&gt;&lt;strong&gt;Update for JMM 2020&lt;/strong&gt;: See the potentially relevant blogpost &lt;a href=&quot;https://blogs.ams.org/jmm2020/2020/01/16/152/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The post begins:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I am interested in how blind people learn mathematics at any level, but particularly before college. &lt;strong&gt;Math is often taught using a lot of visualization; how does this work with blind people?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The following information concerns not the main question, about the use (or nonuse) of visualization in teaching mathematics to blind people, but rather about new findings in how blind people understand mathematics; in particular, with respect to the part of the brain largely responsibly for visualization (i.e., the visual cortex). To this end, I point to an article recently published in PNAS:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kanjlia, Shipra, et al. &amp;quot;Absence of visual experience modifies the neural basis of numerical thinking.&amp;quot; &lt;em&gt;Proceedings of the National Academy of Sciences&lt;/em&gt; (2016): 201524982. &lt;a href=&quot;http://www.pnas.org/content/early/2016/09/14/1524982113&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link 1&lt;/strong&gt;&lt;/a&gt;; &lt;a href=&quot;http://www.pnas.org/content/suppl/2016/09/14/1524982113.DCSupplemental/pnas.201524982SI.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link 2&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here are a couple of findings about mathematical understanding from the aforecited taken out of a mainstream media piece (&lt;a href=&quot;http://www.futurity.org/blind-brain-vision-math-1252282-2/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Futurity&lt;/strong&gt;&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;People blind from birth appear to do math in a part of the brain typically devoted to vision, a study finds.&lt;/p&gt;&#xA;&lt;p&gt;Researchers using functional MRI watched the visual cortex in the brains of congenitally blind people as they solved algebra problems in their heads.&lt;/p&gt;&#xA;&lt;p&gt;The visual cortex didn’t merely respond, the researchers say. The more complicated the math, the greater the activity they saw in the vision center. The same did not happen in the brains of sighted people who—with masks covering their eyes—did the same math exercises as the blind subjects.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The aforelinked continues:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The study contradicted conventional wisdom in another way, says lead author Shipra Kanjlia, a graduate student in psychological and brain sciences. Humans as young as babies have a basic number sense: “My dish has more carrot sticks than apple slices,” or “This picture has more dogs than cats.” Many believe this number sense evolves from seeing the world and trying to quantify all the sights. But seeing has nothing to do with it; the study showed that the brain network behind this sort of numerical reasoning is identical in blind and sighted people.&lt;/p&gt;&#xA;&lt;p&gt;“The number network develops totally independently of visual experience,” Kanjilia says. “These blind people have never seen anything in their lives, but they have the same number network as people who can see.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I reiterate that I do not know how visualization is incorporated into mathematics education for the blind, but I suspect that current/future efforts to create materials in this direction would do well to take into account findings such as the one described above.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-01-16T23:28:08.317" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11433" PostTypeId="1" AcceptedAnswerId="26557" CreationDate="2016-09-23T16:42:43.623" Score="25" ViewCount="817" Body="&lt;p&gt;Most textbooks I've seen (and teachers I've met, myself included) are rather careless about the distinction between variables and functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, when we write $y=f(x)$ we all know that $f$ is &lt;em&gt;the function&lt;/em&gt;, while $x$ and $y$ are variables. We also know that $y\neq f$. Still, we go on to call $y$ &lt;em&gt;a function of&lt;/em&gt; $x$ in front of students, even though $y$ is not a function in the mathematical sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The terminology &quot;$y$ is a function of $x$&quot; has a long tradition that seems to predate the moment when people decided to also call $f$ a function. So there is no hope of changing this. In principle there is also no need of changing terminology, if we all agree never to call $f$ &quot;a function of $x$&quot; nor $y$ &quot;a function&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, people soon drop the &quot;...of $x$&quot; and call $f(x)$ a function, they call $x^2$ a function and they call the temperature a function. Or they write $y(1)$ and $df/dx$, even though the first one is meaningless when $y$ is a variable, while the second would probably be zero in most cases, since $f$ does not depend on $x$. In many applied areas it is common to go as far as writing $y=y(x)$, completely blurring the difference between the function and the dependent variable. Here is a nice example from Dray and Manogue that illustrates the &lt;a href=&quot;http://math.oregonstate.edu/bridge/ideas/functions/&quot;&gt;effect this can have&lt;/a&gt; (click &quot;Next&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I'm teaching introductory calculus to engineers and see their difficulties with function application notation, composition of functions and the chain rule, I sometimes wonder how much of that is caused by this constant blurring of notions by the teachers.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there any research in mathematics education that tries to measure the effect the common blurring between functions and&#xA;  variables has on students, while they are learning&#xA;  the concept of a function?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="590" LastActivityDate="2023-06-19T00:24:15.337" Title="&quot;Function&quot; vs &quot;Function of ...&quot;: how much does it contribute to students difficulties?" Tags="|calculus|education-research|functions|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="11434" PostTypeId="2" ParentId="11433" CreationDate="2016-09-23T16:55:27.507" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://www.maa.org/press/books/making-the-connection-research-and-teaching-in-undergraduate-mathematics-education&quot; rel=&quot;nofollow&quot;&gt;This MAA Notes volume&lt;/a&gt; has a couple of good articles about related topics, including the article &quot;Foundational Reasoning Abilities that Promote Coherence in Students' Function Understanding&quot; by Oehrtmann et al. which I think has some valuable connections.  I think the references for the articles would lead you further - unfortunately I don't have a print copy in front of me, just a few pages I photocopied long ago that reminded me of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think it directly addresses this issue but definitely deals with the issue of function versus variable, which I think is at the root of &lt;code&gt;f&lt;/code&gt; versus &lt;code&gt;y&lt;/code&gt; versus &lt;code&gt;f(x)&lt;/code&gt; or even &lt;code&gt;y(x)&lt;/code&gt; (though the latter is so darn useful in differential equations).&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-09-23T16:55:27.507" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11435" PostTypeId="1" AcceptedAnswerId="11490" CreationDate="2016-09-23T23:44:29.513" Score="10" ViewCount="926" Body="&lt;p&gt;And how to make them intuitive?&#xA;&lt;hr /&gt;&#xA;We are tasked to prove $P \implies Q$. So we assume $P$ and are trying to&#xA;prove $Q$. We assume not-$Q$ ($\neg Q$) and derive a contradiction, establishing $Q$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is something counterintuitive about this. To prove $Q$ is true, we assume $Q$ is false. So we have in mind both $Q$ and $\neg Q$ simultaneously.&#xA;We reason with a falsity to arrive at truth.&#xA;Why is it legitimate to reason with something you know (eventually) is false?&#xA;Couldn't assuming something false lead your reasoning astray?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have any of you had difficulty making this type of reasoning convincing?&#xA;I would appreciate hearing your strategies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;sup&gt;&#xA;I think my question is different from:&#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/q/10349/511&quot;&gt;Why do students like proof by contradiction?&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-10-23T19:32:58.970" Title="Why are proofs by contradiction counterintuitive?" Tags="|mathematical-pedagogy|proofs|logic|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="11436" PostTypeId="1" CreationDate="2016-09-24T13:51:37.250" Score="40" ViewCount="20493" Body="&lt;p&gt;In algebra II, USA, we teach our students complex numbers.  However, after algebra II, they never use complex numbers until pretty much complex analysis.  The whole point of teaching them complex numbers is to find the roots of polynomials... but, that's all we ever do with them.  Sure, we do some algebraic manipulation of them, just to get a feel, but that's it.  Nothing ever too deep, and often times there isn't even a geometric understanding of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And by the time students ever reach a point where they need to use complex numbers... it'll be very far in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as real world applications, it'll probably be finding the real roots of something for anyone who doesn't take complex analysis or similar courses that deal with complex numbers.  And at that point, you probably don't sit down and factor your polynomial.  I would think most would either see it as a quadratic or go straight to a calculator (one that can solve such things with complex numbers or numerical methods)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my question is if its really worth it teaching students about complex numbers.  Most students will forget about complex numbers even, as complex analysis is usually a far way off, if they ever get there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe to be more specific,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why do we teach students about complex numbers if most will never reach a course that uses them?  When do laymen use complex numbers in real world applications?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2022-11-20T14:53:44.783" Title="Why do we teach complex numbers?" Tags="|algebra|teaching|applied-mathematics|applications|complex-numbers|" AnswerCount="14" CommentCount="23" ContentLicense="CC BY-SA 3.0" />
  <row Id="11437" PostTypeId="2" ParentId="11436" CreationDate="2016-09-24T14:20:52.450" Score="29" Body="&lt;p&gt;We owe students a presentation of the &lt;strong&gt;Fundamental Theorem of Algebra&lt;/strong&gt; -- that every nonconstant polynomial has a root; or, equivalently, the marvelous fact that every polynomial of &lt;em&gt;n&lt;/em&gt;th degree has precisely &lt;em&gt;n&lt;/em&gt; roots (including multiplicities). When made, it serves as a capstone and culmination of all the work that the student has done in elementary algebra. Of course, this statement can only be made in the language of complex numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complex numbers are the answer to &lt;em&gt;what's the algebraic completion of polynomial roots&lt;/em&gt;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complex numbers serve as an introduction to higher-degree metric spaces and matrices (i.e., point the way towards linear algebra). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complex numbers are used in many real-world applications (esp. any case where a 2-dimensional measurement is neatly packaged thus): Control theory, fluid dynamics (flow in two dimensions), electrical engineering (impedence), signal analysis, fractals, etc. (&lt;a href=&quot;https://en.wikipedia.org/wiki/Complex_number#Applications&quot;&gt;https://en.wikipedia.org/wiki/Complex_number#Applications&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the Preface to his &lt;em&gt;Visual Complex Analysis&lt;/em&gt;, Tristan Needham writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If one believes in the ultimate unity of mathematics and physics, as I&#xA;  do, then a very strong case for the necessity of complex numbers can&#xA;  be built on their apparently fundamental role in the quantum&#xA;  mechanical laws governing &lt;em&gt;matter&lt;/em&gt;. Also, the work of Sir Roger&#xA;  Penrose has shown (with increasing force) that complex numbers play an&#xA;  equally central role in the relativistic laws governing the structure&#xA;  of &lt;em&gt;space-time&lt;/em&gt;. Indeed, if the laws of matter and space-time are ever&#xA;  to be reconciled, then it seems very likely that it will be through&#xA;  the auspices of the complex numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-09-24T16:32:25.630" LastActivityDate="2016-09-24T16:32:25.630" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11438" PostTypeId="2" ParentId="11436" CreationDate="2016-09-24T16:23:04.563" Score="58" Body="&lt;p&gt;Some of your students will become engineers, and engineers use complex numbers all the time, e.g., to represent impedance. This kind of thing is by far the most common application. Complex numbers are also used in quantum mechanics.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;after algebra II, they never use complex numbers until pretty much complex analysis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I assume you mean &quot;they never use complex numbers &lt;em&gt;in a math course&lt;/em&gt; until...&quot; Many of them will use complex numbers in an engineering or physics course, and they will simply never take complex analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also not convinced that STEM students will never see a complex number in a math course until complex analysis. For example, in freshman calculus they may learn to do an integral like $\int e^x \sin x dx$ using complex numbers. Complex numbers come up in linear algebra as eigenvalues.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When do laymen use complex numbers in real world applications?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm confused by your usage of &quot;laymen&quot; here. When I first read the question, I took this to mean non-mathematicians (i.e., 99.999% of students in Algebra II). But in your comment on Daniel R. Collins's answer, you seem to be talking about &quot;laymen&quot; as including bartenders and truck drivers...? Does &quot;laymen&quot; include engineers? The majority of the population never understands or uses any math beyond arithmetic (even if they are forced to go through the motions of taking a course such as algebra). The reason we force college-bound students to take a course like Algebra II is that it indicates that they have the ability to do abstract thought.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2016-09-24T16:31:07.143" LastActivityDate="2016-09-24T16:31:07.143" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11439" PostTypeId="2" ParentId="11436" CreationDate="2016-09-24T20:04:22.293" Score="1" Body="&lt;p&gt;I think your premise is flawed &amp;quot;Why do we teach students about complex numbers if most will never reach a course that uses them?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;As soon they get a calculator they will test &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;.  They are going to see and answer and ask you what it means.&lt;/p&gt;&#xA;&lt;p&gt;Also, &amp;quot;When do laymen use complex numbers in real world applications?&amp;quot;  Check out this &lt;a href=&quot;https://web.archive.org/web/20171231142107/http://mathforum.org/library/drmath/view/53606.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;answer&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7175" LastEditorUserId="6053" LastEditDate="2022-11-17T22:16:03.123" LastActivityDate="2022-11-17T22:16:03.123" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="11441" PostTypeId="2" ParentId="11436" CreationDate="2016-09-24T21:16:22.863" Score="27" Body="&lt;p&gt;I am not a mathematician, nor a teacher, but I can give an example of how learning complex numbers in school made me fall in love with mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was the first time I realized it was possible to be creative with numbers. &quot;What if we just define sqrt(-1) to be something&quot;.  I remember realising that I could have thought of that. It wasn't hard, it wasn't some professor somewhere thinking up rules, I could have tried that. Up until this point, math had always been a set of rules that were supplied that could never be broken. And this one made me realize that these rules can sometimes be questioned and broken. I could try something and see what happens. It could be a creative, experimental, process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later on, I also realised that certain &quot;what ifs&quot; can make calculations easier. Sometimes the r-theta space makes the numbers easier than the x,y space. Sometimes the Laplace transform is where its at. Sometimes using complex numbers is easier than not using them. Someone, somewhere, said &quot;what if I try this...&quot; to create the mathematical tools I use today as an Engineer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complex numbers is an example of how a very simple &quot;what if&quot; can create a new way of answering previously unanswered questions. It is an example of how math can be a creative experimental thing.&lt;/p&gt;&#xA;" OwnerDisplayName="user2138" LastEditorUserId="7190" LastEditDate="2016-09-26T13:31:58.287" LastActivityDate="2016-09-26T13:31:58.287" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11445" PostTypeId="2" ParentId="11436" CreationDate="2016-09-25T19:45:33.570" Score="16" Body="&lt;p&gt;I am surprised that nobody has mentioned differential equations. If you know about complex numbers and Euler's formula then there is a beautiful unified theory of linear differential equations with constant coefficients in terms of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Characteristic_equation_(calculus)&quot;&gt;characteristic equation&lt;/a&gt;. Without complex numbers the theory becomes somewhat ad-hoc, with different solutions depending on whether or not the polynomial has irreducible quadratic factors. This is often taught to sophomore math/engineering students who haven't had any previous college-level exposure to complex numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, complex numbers are just beautiful. If a student were to ask me why they should study them I would be tempted to quote to them the following passage from Peter Høeg's novel &quot;Smilla's Sense of Snow&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do you know what the foundation of mathematics is? The foundation of&#xA;  mathematics is numbers. If anyone asks me what makes me truly happy, I&#xA;  would say: numbers. Snow and ice and numbers. And do you know why?&#xA;  Because the number system is like human life. First you have the&#xA;  natural numbers. The ones that are whole and positive. The numbers of&#xA;  a small child. But human consciousness expands. The child discovers a&#xA;  sense of longing, and do you know what the mathematical expression is&#xA;  for longing ... The negative numbers. The formalization of the feeling&#xA;  that you are missing something. And human consciousness expands and&#xA;  grows even more, and the child discovers the in between spaces.&#xA;  Between stones, between pieces of moss on the stones, between people.&#xA;  And between numbers. And do you know what that leads to? It leads to&#xA;  fractions. Whole numbers plus fractions produce rational numbers. And&#xA;  human consciousness doesn't stop there. It wants to go beyond reason.&#xA;  It adds an operation as absurd as the extraction of roots. And&#xA;  produces irrational numbers ... It's a form of madness. Because the&#xA;  irrational numbers are infinite. They can't be written down. They&#xA;  force human consciousness out beyond the limits. And by adding&#xA;  irrational numbers to rational numbers, you get real numbers ... It&#xA;  doesn't stop. It never stops. Because now, on the spot, we expand the&#xA;  real numbers with the imaginary square roots of negative numbers.&#xA;  These are numbers we can't picture, numbers that normal human&#xA;  consciousness cannot comprehend. And when we add the imaginary numbers&#xA;  to the real numbers, we have the complex number system. The first&#xA;  number system in which it's possible to explain satisfactorily the&#xA;  crystal formation of ice. It's like a vast, open landscape. The&#xA;  horizons. You head toward them and they keep receding.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2016-09-25T19:45:33.570" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11446" PostTypeId="2" ParentId="11436" CreationDate="2016-09-26T00:22:00.423" Score="14" Body="&lt;p&gt;I think the fundamental tenet of this question is simply false. Here are some of the many encounters that undergraduate college students have in my classes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In Calculus II, as an application of power series, we discuss Euler's identity:&#xA;$$e^{i\theta} = \cos(\theta)+i\sin(\theta).$$&lt;/li&gt;&#xA;&lt;li&gt;Still in Calculus II as an application of the preceding example, we derive formulae for the trig expansions of $\sin(\alpha+\beta)$ and $\cos(\alpha+\beta)$. I further mention this application as the &quot;easy way&quot; to derive these identities any time I need to use one in a class.&lt;/li&gt;&#xA;&lt;li&gt;In Calculus III after polar coordinates, we discuss the $n^{\text{th}}$ roots of a real number, i.e. the solutions of $x^n-a=0$.&lt;/li&gt;&#xA;&lt;li&gt;In linear algebra and differential equations, we discuss the interpretation of complex eigenvalues.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4912" LastEditorUserId="376" LastEditDate="2018-07-17T13:27:30.793" LastActivityDate="2018-07-17T13:27:30.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11447" PostTypeId="2" ParentId="11435" CreationDate="2016-09-26T00:26:13.500" Score="0" Body="&lt;p&gt;I think the context is red herring &amp;mdash; the core issue is the fact that (where $\bot$ denotes contradiction)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$R \implies \bot$ is equivalent to $\neg R$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Similarly, by applying double negation&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\neg R \implies \bot$ is equivalent to $R$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So what you need to do is wrap your head around &lt;em&gt;these&lt;/em&gt; facts. Once you have that, we can apply it to the proof form you're considering:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$P \implies Q$ is equivalent to $P \implies (\neg Q \implies \bot)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2016-09-26T00:48:28.643" LastActivityDate="2016-09-26T00:48:28.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11448" PostTypeId="1" CreationDate="2016-09-26T01:23:56.473" Score="8" ViewCount="1132" Body="&lt;p&gt;I don't know about all algebra courses, but the following is the basic outline of my algebra years:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Alg. I - Lots of linear graphs and solving systems of equations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Alg. II - Lots of quadratics, some harder systems of equations, and light touch on some topics like exponential functions, the meaning of 'functions', complex numbers, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now, I can understand why we teach what we teach in Alg. I, it is easy/good start and marks a foundation to manipulate equations, something you can hardly do with out.  Not to mention, linear graphs are pretty much the simplest graphs, yet you can get fundamental knowledge from them (slope, familiarity with graphs).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, I'm not so sure I understand the Alg. II curriculum.  From my experience, Alg. II &lt;strong&gt;grinds&lt;/strong&gt; quadratics into you so hard that at times I was outright &lt;strong&gt;done&lt;/strong&gt; with parabolas and factoring.  Why is it that they do this?  Sure, quadratics are the next step beyond linear functions, but I most certainly do not feel the amount of time we spend over the subject was worth my sanity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why not do less of the quadratics and more of other topics?  For example, we could inquire more about higher degree polynomials.  In my area (USA), we are taught trigonometry first, then Alg. II, so we could solve cubic polynomials using trig functions, which has a derivation that isn't excruciatingly hard.  We could talk about so many other topics, like inequalities and such...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: &lt;strong&gt;Why not do less of the quadratics and more of other topics?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorDisplayName="user507" LastEditDate="2016-09-26T03:48:11.577" LastActivityDate="2017-12-01T18:41:04.190" Title="Is algebra unnecessarily quadratic intensive?" Tags="|algebra|curriculum|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11449" PostTypeId="2" ParentId="11436" CreationDate="2016-09-26T18:34:33.973" Score="11" Body="&lt;p&gt;Most people are not going to use most of what they use in high school. High school (in the US, at least) is about building a broad foundation for students to be able to jump into any specialization when they go to college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some examples of things I haven't had to do since high school:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Name the 5 (I think 5, maybe it changed?) biological kingdoms &lt;/li&gt;&#xA;&lt;li&gt;Diagram a sentence &lt;/li&gt;&#xA;&lt;li&gt;Explain the meaning of the scaffold in &quot;The Scarlet&#xA;Letter&quot; &lt;/li&gt;&#xA;&lt;li&gt;Discuss the differences between the Indus Valley Civilization&#xA;and Mesopotamian Civilization&lt;/li&gt;&#xA;&lt;li&gt;Etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(Aside: I don't consider having had taken those courses a negative by any means, just that they aren't &quot;useful&quot; in my line of work, and, I assume, in most people's daily lives)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as complex numbers explicitly, they show up again in Trigonometry, which immediately followed Algebra II in my HS curriculum, and appeared from time to time in succeeding courses  in math and physics. However, I really had to start using them in an application setting was in Control Systems, which was junior year of college. By that point you don't want to want to waste time explaining &lt;em&gt;what&lt;/em&gt; complex numbers are, you want to be able jump in and use them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if you're not proficient with complex numbers at that point, you are comfortable with their existence, which is better than getting to that point thinking sqrt(-1) is meaningless and learning that your whole mathematical life is a lie. &lt;/p&gt;&#xA;" OwnerUserId="7199" LastActivityDate="2016-09-26T18:34:33.973" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11450" PostTypeId="1" AcceptedAnswerId="11454" CreationDate="2016-09-26T18:41:34.780" Score="3" ViewCount="225" Body="&lt;p&gt;During K12 I had problems understanding maths because in most cases I could not understand how it would be useful or what its applications are. I passed K12 maths mostly by memorizing the process to get a correct answer. Now at 32 years, it is starting to hinder my growth. I see the beauty of what I have lost and would like to go back and educate myself with these k12 foundations of maths and physics so that i can be ready to tackle the more complicated stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm may be more of a &quot;creative type&quot; but my hobbies, interests and profession do get technical (Architecture, Synthesizers, Music, Computers). So, to aid in my relearning process, Can you recommend resources (preferably books) that will help me?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you and best regards.&lt;/p&gt;&#xA;" OwnerUserId="7195" LastActivityDate="2016-09-27T03:58:05.250" Title="Relearning Mathematics" Tags="|self-learning|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11451" PostTypeId="2" ParentId="11436" CreationDate="2016-09-26T18:52:46.130" Score="6" Body="&lt;p&gt;There are many subjects in High School that are taught which some will never use.  Accounting, Drama, Band, Woodshop etc.  These are entire subjects that students learn and most will never use.  Does that mean we shouldn't teach them? Of course not!  These are valuable subjects and just having exposure to the subjects will cause some to pursue them and others to appreciate them.  It makes them more creative and able to do different things or at least not afraid to try.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't believe High School is only about giving students things that all of them will use.  Students need to be exposed to subjects to figure out what they do want to do in life, whether it's college or trade school or some other path.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complex numbers is just one part of mathematics, and some will be intrigued enough to pursue such topics later.  Maybe THAT's the subject that really drew them in.  No, you won't use complex numbers in shop or accounting (although some firms do use imaginary numbers -- nudge nudge wink wink), and at that age their brains are still developing and exposure to areas they find more difficult is actually beneficial to them. How many people do you know that hated to practice piano and didn't and now regret it?  How many DID practice and are grateful for it now?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing hard things, making them think is good for them both for character and brain development.  These are established results from both psychology and neuroscience.  While I wouldn't stuff High Schoolers full of these types of things, complex numbers are important enough that exposure is necessary for those who will need it later, and common enough in many various fields that merit at least some attention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If instead of complex numbers you were teaching algebraic topology or string theory, I would be in complete agreement.&lt;/p&gt;&#xA;" OwnerUserId="7200" LastActivityDate="2016-09-26T18:52:46.130" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11452" PostTypeId="2" ParentId="6003" CreationDate="2016-09-26T22:51:26.100" Score="1" Body="&lt;p&gt;Here is my &quot;unfinished&quot; attempt that I used in one of my classes last year. Generally speaking, it is based on the idea of &quot;proof-generated definitions&quot; introduced by Lakatos. As such, the question is which “theorem” could justify and motivate the definition of a convergent sequence. In fact, the first theorem that comes to mind works: if a sequence converges, then its limit is unique. Obviously, you might say we need &quot;a definition for a limit of a sequence&quot; to prove this &quot;obvious&quot; statement. That would be the whole point of a proof-generated definition. My unfinished attempt can be found &lt;a href=&quot;http://amirasghari.com/teaching-ideas/proof-generated-definitions/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.     &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2016-09-26T22:51:26.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11453" PostTypeId="2" ParentId="11436" CreationDate="2016-09-27T01:45:12.540" Score="2" Body="&lt;p&gt;Speaking of complex numbers, why do we not teach them Geometric Algebra, or even appreciate it ourselves? In there, we can give a geometric meaning to imaginary numbers, as oriented segments of a plane (bivectors). We also can use them to e.g. rotate vectors conveniently! They are very easy to handle! GA gives a lot of new connections btw physics and math too, like new interpretations of Pauli and Dirac matrices. I was blown away when I first stumbled on it! I'm studying physics at master level, and nobody told me about this...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Intro here:&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=srwoPQfWWS8&amp;amp;list=PLLvlxwbzkr7igd6bL7959WWE7XInCCevt&quot; rel=&quot;nofollow&quot;&gt;https://www.youtube.com/watch?v=srwoPQfWWS8&amp;amp;list=PLLvlxwbzkr7igd6bL7959WWE7XInCCevt&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7204" LastActivityDate="2016-09-27T01:45:12.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11454" PostTypeId="2" ParentId="11450" CreationDate="2016-09-27T03:58:05.250" Score="3" Body="&lt;p&gt;It depends on what you want to learn. Do you want something like a textbook, or something that gets you thinking but doesn't 'cover' all of one subject necessarily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a very personal question. &lt;em&gt;Measurement&lt;/em&gt;, by Paul Lockhart is lovely, but it might be too hard for your first adventure. You might like &lt;a href=&quot;http://wordplay.blogs.nytimes.com/2012/11/26/joy-of-x/?_r=0&quot; rel=&quot;nofollow&quot;&gt;Steven Strogatz's &lt;em&gt;The Joy of X&lt;/em&gt;&lt;/a&gt;. Once you find one book you like, it will help you find others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want a textbook, you might enjoy &lt;em&gt;Mathematics: A Human Endeavor&lt;/em&gt;, by Harold Jacobs. (Older editions will be inexpensive.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2016-09-27T03:58:05.250" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11456" PostTypeId="2" ParentId="11448" CreationDate="2016-09-27T19:06:49.233" Score="4" Body="&lt;p&gt;It is unfortunate that some courses may overemphasize quadratics to the detriment of other topics. Quadratics is a very important topic, but focusing exclusively on them could get needlessly tedious. Personally I have never seen an Algebra I course that did not cover at least the basics of quadratics, but I suppose some curricula might put quadratics off until Algebra II.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the U.S., most states have adopted the Common Core Standards. There are about 50 standards in the Common Core that might be included in a traditional Algebra I/II sequence, of which 6 involve quadratics. If all 6 standards are put off until Algebra II, that could make for up to one fourth of the course involving quadratics. This could be a bit much. If some of those standards are treated in Algebra I, it should make for a balanced Algebra II curriculum.&lt;/p&gt;&#xA;" OwnerUserId="6782" LastActivityDate="2016-09-27T19:06:49.233" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11457" PostTypeId="1" CreationDate="2016-09-27T20:11:16.510" Score="6" ViewCount="461" Body="&lt;p&gt;What do you say to the following way of teaching &amp;quot;if...then&amp;quot; and &amp;quot;the following are equivalent&amp;quot;? Has somebody ever taught it like this?&lt;/p&gt;&#xA;&lt;p&gt;An implication &lt;span class=&quot;math-container&quot;&gt;$(A {\implies} B)$&lt;/span&gt; can be viewed as asserting that &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is at least as true as &lt;span class=&quot;math-container&quot;&gt;$A.$&lt;/span&gt; Thus, if &lt;span class=&quot;math-container&quot;&gt;$(A {\implies} B)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(B{\implies} A),$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; have the same truth value (&amp;quot;at least as true&amp;quot; in both directions).&lt;/p&gt;&#xA;&lt;p&gt;Also, under this interpretation, it is easy to see that it suffices to prove the cyclic chain of implications &lt;span class=&quot;math-container&quot;&gt;$(A_1 {\implies} A_2 {\implies} \dots {\implies} A_n {\implies} A_1)$&lt;/span&gt; in order to show that &lt;span class=&quot;math-container&quot;&gt;$A_1, \ldots, A_n$&lt;/span&gt; are equivalent, that is, have the same truth value.&lt;/p&gt;&#xA;" OwnerUserId="7213" LastEditorUserId="372" LastEditDate="2022-07-15T17:35:39.947" LastActivityDate="2023-12-21T04:28:40.727" Title="Teaching &quot;if...then&quot; and &quot;the following are equivalent&quot;" Tags="|logic|terminology|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11459" PostTypeId="2" ParentId="11435" CreationDate="2016-09-28T12:10:31.230" Score="3" Body="&lt;p&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was looking for modes of though that would produce your question, and I found one, which might help you clarify some things. Still, (I hope not) it might confuse you even more, I don't know. Proceed at your own risk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;On proof by contradiction:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When writing a proof by contradiction of $P \to Q$, we indeed assume $\neg Q$ and try to prove that together with $P$ it implies falsehood $\bot$, that is,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$P \land \neg Q \to \bot.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can think of it, as an &lt;em&gt;enumeration of cases&lt;/em&gt; type of proof. Let me give you an example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose Steve can be in New York or San Francisco (at least one). It is enough to check that he is &lt;em&gt;not&lt;/em&gt; in New York to conclude that he is in San Francisco.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This type of reasoning hinges on the fact that there are at most two cases to check. In other words, proof of contradiction looks like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Assume $P$. Suppose that $Q$ can be either true or false. It is enough to check that $Q$ is not false, to conclude that it is true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It works, because in classical logic we have the law of excluded middle: $Q \lor \neg Q$. However, sometimes $Q$ can have more values, in particular in &lt;a href=&quot;https://en.wikipedia.org/wiki/Intuitionistic_logic&quot; rel=&quot;nofollow&quot;&gt;intuitionistic logic&lt;/a&gt;, where proof by contradiction is not a valid proof. For example, if we would allow third valuation &lt;em&gt;unknown&lt;/em&gt;, then you cannot conclude something is true based on whether it is not false.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A nice everyday example is &quot;not X&quot; language pattern. Many times people prefer to use &quot;not good&quot; instead of &quot;bad&quot;, among others, because it doesn't say how much the person is disappointed/angry/whatever – it could be continued by &quot;a few small tweaks and it will work&quot; or &quot;we won't be able to fix it in a million years&quot;. On the other hand, &quot;not X&quot; has clear meaning if there are only two possible states, e.g. &quot;What are you doing?! The gun's safety is not on!&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar phrases can also mimic the proof of contradiction. For example, observe how drastically the following dialogue changes meaning:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Alice: Where is Steve, is he working?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Bob: He's not in New York.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;if from the context of the situation we know that Steve:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;has a girlfriend in New York, where he spends all the free time he has.&lt;/li&gt;&#xA;&lt;li&gt;he is an international journalist who frequently visits his parents to care for them.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The first one basically says &quot;yes, he's working&quot;, while the other means Bob doesn't have the slightest idea whether Steve embarked on another work-related trip, or is just visiting his parents.&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2016-09-28T12:10:31.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11460" PostTypeId="2" ParentId="11457" CreationDate="2016-09-28T16:45:39.013" Score="2" Body="&lt;p&gt;I don't know that that's the most common way of teaching it, however, the idea that &quot;if A then B&quot; represents an ordering of truth values such that B is at least as true as A is mathematically and conceptually sound.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2016-09-28T16:45:39.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11461" PostTypeId="2" ParentId="11430" CreationDate="2016-09-28T17:12:23.860" Score="1" Body="&lt;p&gt;In your case, as @ff524 suggested, I would ask the lecturer how he wants the problems to be graded. On the other hand, if you would like to dwell on the topic for a few minutes more, here you can find some additional comments...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a number of practical reasons I prefer the grading method to be &lt;strong&gt;composable&lt;/strong&gt;, in particular, you should be able to use scores from different portions of the exam without the knowledge of what caused/produced those scores (this is similar to how abstractions in software engineering should not leak their implementation details).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mostly, because it frequently happens that different people check different parts of the exam (e.g., person A checks tasks 1-10, person B checks tasks 11-20). In fact, in my experience, checking task 1 of everybody, then checking task 2 of everybody, etc. is slower (compared to all tasks of one student, and then all tasks of another student, etc.), but gives more consistent results, and the overall quality of grading is higher. However, when I get to task 2 of the first student, I do not remember what was in his first task, and I don't want to invest additional time to re-read that part (so this is almost as it was a different person checking). Also, I heard stories about how some professor checked (for example) tasks 1 and 2 of all students, and then decided it took too much time, so he checked tasks 3-20 of first 30 students, and then decided that also took too much time and gave all the rest to his grad student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This are some reasons why I think grades should be composable.&#xA;Nevertheless, that doesn't mean you should cut points every time a student repeats the very same mistake. It means that, if your score for each task is represented by a single numerical value, say from $[0,1]$, then the total exam score should be calculated only using those numbers (it doesn't have to be plain sum).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to count an error in some formula only once, then you should make it a part of the score, for example $[0,1]\times\{\mathtt{GoodFormulaForXYZ},\mathtt{BadFormulaForXYZ}\}$, and then you can calculate the total score of the exam using those two values, say sum of the first numbers and $-10$ if there was $\mathtt{BadFormulaForXYZ}$ in any second coordinate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice I don't think I would keep a count for something so specific as De Morgan laws, unless it would be really important for that particular exam. I had used (in computer science course) scores like $(\mathrm{algorithm}, \mathrm{complexity}, \mathrm{proof}, \mathrm{clarity})$. One advantage of this is that I could tweak scoring to reflect hardness of the problems well (meaning each problem would still be worth, say 10 points, but each component could be scaled non-linearly, for example 0.2 alg would give you 1 point, while 0.3 alg could give you 4 points). In a regular math course I could use score with components representing students idea, its execution, and the overall clarity of the solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One more thought, I'm not sure if everyone will see this as benefit (I do), but it makes the scores more transparent – students, when seeing their marked solutions (and potentially challenging the grade), would know better how they got their score (and, perhaps, what they need to improve).&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2016-09-28T17:12:23.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11462" PostTypeId="2" ParentId="11433" CreationDate="2016-09-28T19:48:24.377" Score="2" Body="&lt;p&gt;The issue you point out is an example of &lt;a href=&quot;https://en.wikipedia.org/wiki/Abuse_of_notation&quot; rel=&quot;nofollow&quot;&gt;Abuse of notation&lt;/a&gt;. Quoting Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion). Abuse of notation should be contrasted with misuse of notation, which should be avoided.&#xA;  [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;One encounters, in many textbooks, sentences such as &quot;Let f(x) be a function ...&quot;. This is an abuse of notation, as the name of the function is f, and f(x) denotes normally the value of the function f for the element x of its domain. The correct phrase would be &quot;Let f be a function of the variable x ...&quot; or &quot;Let x ↦ f(x) be a function ...&quot; This abuse of notation is widely used, as it simplifies the formulation, and the systematic use of a correct notation becomes quickly pedantic.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A similar abuse of notation occurs in sentences such as &quot;Let us consider the function $x^2 + x + 1$.&quot;. In fact $x^2 + x + 1$ is not a function. The function is the operation that associates $x^2 + x + 1$ to $x$, that is $x \rightarrow x^2 + x + 1$. Nevertheless, this abuse of notation is widely used as, generally, it is not confusing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course one should tell the students that f is the function and f(x) is the function value at x. But if one can prevent misunderstandings, then abuse of notation is a very useful thing. Without it, we would be like machines who understand everything literally. To quote Bourbaki:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have made a particular effort always to use rigorously correct language, without sacrificing simplicity. As far as possible we have drawn attention in the text to abuses of language, without which any mathematical text runs the risk of pedantry, not to say unreadability.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7221" LastEditorUserId="5563" LastEditDate="2016-09-29T04:32:40.897" LastActivityDate="2016-09-29T04:32:40.897" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11463" PostTypeId="2" ParentId="11457" CreationDate="2016-09-28T19:59:47.607" Score="5" Body="&lt;p&gt;Maybe Terence Tao's logic section of his text &quot;Analysis 1&quot; is an example. He writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One can also think of the statement “if X, then Y ” as “Y is&#xA;  at least as true as X” - if X is true, then Y also has to be true,&#xA;  but if X is false, Y could be as false as X, but it could also be&#xA;  true. This should be compared with “X if and only if Y ”, which&#xA;  asserts that X and Y are equally true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So there he explains why $(A\implies B)\quad\land\quad (B\implies A)$ is the same as saying that $A$ and $B$ have the same truth value. He also gives the following exercise:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose you know that X is true if and only if&#xA;  Y is true, and you know that Y is true if and only if Z is true.&#xA;  Is this enough to show that X, Y, Z are all logically equivalent?&#xA;  Explain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Using the information given in the text (that &quot;If A, then B&quot; means &quot;B is at least as true as A&quot;), the solution you have given is obvious.&lt;/p&gt;&#xA;" OwnerUserId="7221" LastActivityDate="2016-09-28T19:59:47.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11464" PostTypeId="2" ParentId="11433" CreationDate="2016-09-29T01:24:23.267" Score="8" Body="&lt;p&gt;I really appreciate your question. It gets to something I think about often when teaching calculus. You should read about Ed Dubinsky's notion of reflective abstraction, rooted in Piaget.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At one point in mathematical development, after learning to count, positive integers are &quot;encapsulated&quot; by children as primary objects. Later, while learning algebra, variables such as $x$ and $y$  become encapsulated as primary objects, after a struggle which some never surpass. Then when learning calculus, functions like $f$ and $g$ might become encapsulated (abstracted) as primary objects. But it's a messy process, complicated by the convenient, historical &quot;abuse of notation&quot; blurring $f$ and $f(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teaching calculus with computers can help college-aged students encapsulate functions as primary objects. In Dubinsky you can read about the language ISETL. I think teaching calculus using Maple also helped students reach this level of abstraction, but that was when the only input was &quot;classic.&quot; There you set up functions by notation such as $f:=x\rightarrow x^2$. In Church's lambda calculus, this is equivalent to distinguishing between $x^2$ and $\lambda x.x^2$. Once 2-d input was put into Maple with document mode, this pedagogical tool was lost to &quot;disambiguation dialogue boxes.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question you bring up is very important. Keep thinking about it.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-09-29T01:24:23.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11466" PostTypeId="2" ParentId="11436" CreationDate="2016-09-30T15:54:00.707" Score="3" Body="&lt;p&gt;As Ben Cromwell mentioned engineers and physicists use complex numbers. His example of impedance is a common one but here are some another examples along with easily recognized applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Signal and Image Processing: Analzying images and audio signals is commonly done using a technique called a Fourier transform. &lt;a href=&quot;https://en.wikipedia.org/wiki/Fourier_transform&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Fourier_transform&lt;/a&gt;&#xA;Complex numbers are crucial to this and it has a variety of applications. For example, a lot of operations in Photoshop(Gaussian blurs, interpolation) typically use Fourier analysis in some way. When your cell phone transmits a signal, it encodes the outgoing signal on a &quot;carrier&quot; wave in a process called modulation. This also requires complex numbers. DJs help improve the quality of their sound by compensating for something called the &quot;frequency response&quot; of the room, the frequency response is a Fourier transform of the room's acoustic properties. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Controls Theory: Controls theory is closely related to signal processing(using a more generalized form of the Fourier transform called the Laplace transform) and it used in a lot of robotics and automation. Cruise control and aircraft autopilots all use controls theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should also point out that I studied these topics and college and ran into complex numbers as soon as second semester, freshman year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are just the areas I'm very familiar with but there are certainly many more. Any of the engineers working on these projects need complex numbers. As for the rest of the population; there are numerous other responses suggesting that not everybody will use everything we learn in high school, so it doesn't seem fair to select complex numbers as opposed to anything else.&lt;/p&gt;&#xA;" OwnerUserId="7230" LastActivityDate="2016-09-30T15:54:00.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11469" PostTypeId="2" ParentId="11424" CreationDate="2016-10-01T05:28:57.513" Score="1" Body="&lt;p&gt;If your students are anything like my students in Calc I, I'd emphasize a different perspective on what they're doing.  Many first year students have a mechanical/algorithmic understanding of the derivative and don't have a very strong geometric understanding.  A favorite of mine is the inverse function theorem:  if $f$ is differentiable and invertible near $b$, then $(f^{-1})'(b)=\frac{1}{f'(f^{-1}(b))}$.  This has a great geometric justification.  Sure, the strongest ones might be able to prove this algebraically using implicit differentiation, but it'd be nice for them to understand why it's &quot;obvious&quot; geometrically.  (Also, problems involving $(f^{-1})'$ are also hard for them to interpret physically; that's another reason I emphasize this.)&lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="5563" LastEditDate="2016-10-01T15:45:07.057" LastActivityDate="2016-10-01T15:45:07.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11471" PostTypeId="1" AcceptedAnswerId="11473" CreationDate="2016-10-01T13:47:50.240" Score="5" ViewCount="921" Body="&lt;p&gt;Some students understand &lt;em&gt;how&lt;/em&gt; this works, like they know what the theorem means, but, say, imagine some student asks &lt;em&gt;why&lt;/em&gt;, not how. Not really &lt;em&gt;proving&lt;/em&gt; the theorem, rather why does it exist or why is it the way it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a simple way to explain this?&lt;/p&gt;&#xA;" OwnerUserId="7232" LastEditorUserId="7232" LastEditDate="2016-11-09T15:54:30.170" LastActivityDate="2016-11-09T15:54:30.170" Title="Good way to explain fundamental theorem of arithmetic?" Tags="|undergraduate-education|concept-motivation|number-theory|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11472" PostTypeId="2" ParentId="11471" CreationDate="2016-10-01T16:46:57.210" Score="11" Body="&lt;p&gt;To really understand why the integers $\mathbb{Z}$ have unique prime factorization, it helps to understand how unique prime factorization can fail in other settings. For example,&#xA;$$(2 + \sqrt{10}) \cdot (2 - \sqrt{10}) = -6 = -2 \cdot 3,$$&#xA;so prime factorizations in $\mathbb{Z}[\sqrt{10}] = \{a + b\sqrt{10}: a, b \in \mathbb{Z}\}$ aren't unique. On the other hand, prime factorizations in $\mathbb{Z}[\sqrt{d}] = \{a + b \sqrt{d}: a, b \in \mathbb{Z}\}$ &lt;em&gt;are&lt;/em&gt; unique for many values of $d$ (conjecturally, for infinitely many squarefree integers $d &amp;gt; 1$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's the reason for this difference? One perspective is, it's a matter of whether we have a good notion of &quot;greatest common divisor&quot;, since this is what lets us compare any two prime factorizations of the same number and show they're the same (up to reordering and multiplication by units).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, if we have a gcd function, then we can recover the largest power of a prime $p$ dividing an integer $n$ as the largest $k$ such that $\gcd(p^k, n) = p^k$. Assuming we have a well-defined gcd function, a similar argument works in much more generality. (For those familiar with the terminology, a Noetherian commutative ring is a &lt;a href=&quot;https://en.wikipedia.org/wiki/GCD_domain&quot; rel=&quot;nofollow&quot;&gt;GCD domain&lt;/a&gt; if and only if it's a &lt;a href=&quot;https://en.wikipedia.org/wiki/Unique_factorization_domain&quot; rel=&quot;nofollow&quot;&gt;unique factorization domain&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it's easy to see that the integers have a gcd function: the set of all common divisors of two given integers is a bounded set of integers, and bounded sets of integers are finite, so we can literally pick the largest of all the common divisors with respect to the usual way of ordering the integers. (In more general contexts, any of these could fail: the set of common divisors might not be bounded, bounded subsets might not be finite, and there might not be a reasonable notion of &quot;ordering&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to explain this to students depends on the level of the students. Anyway, like Javier said in the comment above, your statement and proof of the FTA aren't right, so you should probably make sure you understand those yourself before trying to explain the deeper connections to students.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastEditorUserId="3505" LastEditDate="2016-10-03T23:24:27.183" LastActivityDate="2016-10-03T23:24:27.183" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11473" PostTypeId="2" ParentId="11471" CreationDate="2016-10-01T17:47:14.237" Score="14" Body="&lt;p&gt;The way I have explained the fundamental theorem of arithmetic in the past is by establishing what it means to be prime (has exactly two positive divisors) and then having students construct factor trees (where the prime factors at the end are circled).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;prime&lt;/em&gt; definition avoids some of the caveat-language otherwise seen (&quot;a number only divisible by itself and 1, but not 1&quot;) and the factor trees are fun to do with a number like 72, after which you can ask different students to draw their trees on the board to compare and contrast.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One important observation to emerge is that when you create a factor tree for a natural number, although different numbers may appear along the way (and although some may not appear at all -- for example, the commonly used approach never reveals the factor of 1) the circled numbers at the bottom of the tree are &lt;em&gt;always the same&lt;/em&gt;. And, in fact, this is the fundamental theorem of arithmetic:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fix a whole number: All of its factor trees have the same numbers at the bottom.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The next approach I take is to give students a more formal definition of the fundamental theorem of arithmetic, and then ask them to think about different metaphors that capture this idea. The most common ones I have seen are DNA and fingerprints, but having them create analogies for a mathematical idea is a powerful tool for instruction and learning. Once students come to see prime numbers as something like building blocks that uniquely define each whole number, I think that the question of &lt;em&gt;why it works like this&lt;/em&gt; can be quite successfully postponed till a much later course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this is to say, I believe the &lt;em&gt;why&lt;/em&gt; is best answered by developing one's intuition around the theorem, and that a more careful answer than that might be best staved off till a course that involves formal proof writing (at which point the existence of prime factorization is shown easily to students who can think inductively -- factor the number, if any number in its factorization is non-prime, then that means it can be written as the product of strictly smaller factors, so continue till you have all primes -- and then use something like Euclid's lemma to show uniqueness). Incidentally, the existence part of the theorem is also a very good example of when you want &lt;em&gt;strong&lt;/em&gt; induction in your toolbox.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly: You mention that some students &quot;know what the theorem means,&quot; and so I thought I'd direct you to a nice paper that discusses this in the context of younger students (ages 11-13):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Griffiths, M. (2013). Intuiting the fundamental theorem of arithmetic. &lt;em&gt;Educational Studies in Mathematics, 82&lt;/em&gt;(1), 75-96. &lt;a href=&quot;http://link.springer.com/article/10.1007/s10649-012-9410-1&quot;&gt;&lt;strong&gt;Springer Link&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(You may be interested in some of its references, as well.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-10-01T17:47:14.237" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11474" PostTypeId="1" CreationDate="2016-10-01T23:13:41.623" Score="3" ViewCount="299" Body="&lt;p&gt;I am seeking advice for understanding how I can improve my abstract reasoning abilities. More specifically, I would like to understand how I can revise my reasoning abilities in mental areas such as spatial-discernment and logical-discernment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the entirety of my lifespan, I have suffered from great difficulty understanding mental abstraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I tried to understand elementary, geometrical concepts, for example, I frequently had to stop concentrating on visualizing the material, because trying to mentally construct and connect the actual structures was an agitating process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have tried to visualize logical patterns in the same manner as during my spatial manipulation, and I experienced the same sort of difficulty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though I have improved somewhat in those regards, I can usually only perform very simple discernment such as using the area/perimeter of a rectangle/square/triangle. As for my logical-reasoning, I can partially understand simple predicates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I would like to improve this ability for my own use. I would like to be able to not only visualize geometric/logical problems and decompose them into their components, yet also construct different, viable &quot;solutions&quot; and deductions from those components.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this moment, I am trying to use jigsaw puzzles and strategy board games such as Chess to progress towards that goal. However, my disabilities are interfering with that process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the actual disabilities, I am not entirely aware of the names. I often develop &quot;shifting thoughts&quot; and become confused during mathematical symbolism/operations(And certain other subjects as well). Once the confusion occurs, the agitation usually follows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, thank you for reading the question.&lt;/p&gt;&#xA;" OwnerUserId="7237" LastEditorUserId="143" LastEditDate="2016-10-02T22:47:20.003" LastActivityDate="2016-10-02T22:47:20.003" Title="How Can I Mitigate My Learning Disabilities And Improve My Abstract-Reasoning Abilities?" Tags="|self-learning|disability|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11475" PostTypeId="2" ParentId="11424" CreationDate="2016-10-02T01:50:20.217" Score="3" Body="&lt;p&gt;One thing you might do is try to incorporate non-routine questions into your Calculus problem solving sessions. Knowing the requisite material is not enough to solve a [non-routine] &lt;em&gt;problem&lt;/em&gt; (i.e., question for which the method of solution is unknown at the outset; used in contradistinction with &lt;em&gt;exercise&lt;/em&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a lovely paper (&quot;technical report&quot;) with five example problems that you might try out:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Selden AN, Selden JO, Hauk SH, Mason AL. (1999). Do calculus students eventually learn to solve non-routine problems? &lt;em&gt;Department of Mathematics, Tennessee Technological University.&lt;/em&gt; &lt;a href=&quot;http://math.tntech.edu/techreports/TR_1999_5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here are the five non-routine problems:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iJX92.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iJX92.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper explores how effectively students did (or did not) use their &lt;em&gt;resources&lt;/em&gt; (in the sense of Schoenfeld, 1985) when attempting these problems. Perhaps the ideas contained in the aforecited/aforelinked can help inform how you approach your own problem solving sessions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To close, here is the final, non-original (Putnam 98/B2) problem foot-noted by the authors:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given a point $(a, b)$ with $0&amp;lt; b&amp;lt; a$, determine the minimum perimeter of a triangle with one vertex at $(a, b)$, one on the $x$-axis, and one on the line $y=x$. You may assume that a triangle of minimum perimeter exists. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(I suggest trying &lt;strong&gt;all&lt;/strong&gt; the problems yourself, first, before reading the solutions!)&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-10-02T01:50:20.217" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11476" PostTypeId="1" AcceptedAnswerId="11481" CreationDate="2016-10-03T15:17:29.563" Score="3" ViewCount="2278" Body="&lt;p&gt;I have a copy of the &quot;Teacher Edition&quot; of a pre-calc textbook my students are using. It's similar to their text, but the pages are reduced a bit to offer room for answers and some commentary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I see a &quot;Solution Key&quot; for this same title. The page count is less than half of the text. Unfortunately, there's no preview feature. I'd have been happy to see 2 random pages from a chapter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What, exactly can I expect from this book? Is it a summary of answers, or is there a comprehensive set of worked out problems? I realize this invites the potential for &quot;How would we know? We don't even know the book title?&quot; But I'm trusting that, as in many areas of life, there's a meaning to the words used, and I'm hoping a member here will have experience with the printed &quot;Teacher edition&quot; vs &quot;Solution Key.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Update - Indeed, Daniel's answer was correct. The solution key had the answers worked out as I hoped. It was published by the same publisher, but had no author credited, for what that's worth. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-10-22T12:58:38.420" LastActivityDate="2016-10-22T12:58:38.420" Title="Teacher Edition vs Solution Key?" Tags="|textbooks|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11477" PostTypeId="2" ParentId="10645" CreationDate="2016-10-03T23:13:28.070" Score="1" Body="&lt;p&gt;Another source:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Friedberg, Solomon. &quot;Teaching mathematics graduate students how to teach.&quot; &lt;em&gt;Notices of the AMS&lt;/em&gt; 52.8 (2005). (&lt;a href=&quot;http://www.ams.org/notices/200508/fea-friedberg.pdf&quot; rel=&quot;nofollow&quot;&gt;PDF download&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This article cites (and quotes from) an earlier book (which I have not examined) by the same author:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Friedberg, Solomon. &lt;em&gt;Teaching Mathematics in Colleges and Universities: Case Studies for Today's Classroom.&lt;/em&gt; Vol. 10. American Mathematical Soc., 2001.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-10-03T23:13:28.070" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11478" PostTypeId="1" AcceptedAnswerId="11479" CreationDate="2016-10-04T00:31:47.953" Score="1" ViewCount="930" Body="&lt;p&gt;Recently I had taken the SATs, and a question came up that involved partial fractions decomposition.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\frac{x^2-4x+5}{x-3}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is not the exact problem but a similar one. If the SAT math is supposed to go up to Algebra 2, in what class are partial fractions taught?&lt;/p&gt;&#xA;" OwnerUserId="7250" LastEditorUserId="7250" LastEditDate="2016-10-04T01:39:30.470" LastActivityDate="2016-10-04T01:39:30.470" Title="When are partial fractions taught?" Tags="|mathematical-pedagogy|" AnswerCount="1" CommentCount="7" ClosedDate="2016-10-09T18:54:02.527" ContentLicense="CC BY-SA 3.0" />
  <row Id="11479" PostTypeId="2" ParentId="11478" CreationDate="2016-10-04T01:01:09.637" Score="8" Body="&lt;p&gt;Sure this isn't a typo, or maybe for the SAT math subject test (once known as SAT II?)  I would be surprised if partial fractions were part of this ... on the other hand, synthetic division of polynomials is quite likely, but then the answer here would be $$(x-1)+\frac{2}{x-3}$$ But usually that's not what we think of as &quot;partial fractions&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-10-04T01:01:09.637" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11480" PostTypeId="1" CreationDate="2016-10-04T14:43:40.697" Score="6" ViewCount="81" Body="&lt;p&gt;I've recently been challenged on my proposal that a reasonable way to estimate a student's future GPA is to look at his/her current GPA.  I am looking for any papers that study the correlation of students' GPAs along their academic career.  In particular, I'd like to know the correlation between first-year grades and second-year grades and if the correlation varies by subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found some documents that look to be internal-use (for example, these &lt;a href=&quot;http://www.baylor.edu/irt/index.php?id=75588&quot;&gt;http://www.baylor.edu/irt/index.php?id=75588&lt;/a&gt; from Baylor University), but nothing that has been peer-reviewed.&lt;/p&gt;&#xA;" OwnerUserId="5978" LastEditorUserId="376" LastEditDate="2016-10-04T18:33:32.600" LastActivityDate="2016-10-04T18:33:32.600" Title="References for persistence of grades" Tags="|reference-request|" AnswerCount="0" CommentCount="2" FavoriteCount="0" ClosedDate="2016-10-06T11:51:21.777" ContentLicense="CC BY-SA 3.0" />
  <row Id="11481" PostTypeId="2" ParentId="11476" CreationDate="2016-10-05T00:37:34.420" Score="1" Body="&lt;p&gt;Former comment, now an answer: Usually these have worked out-solutions, expanding on any answer key in the instructor test, and usually written by a 3rd party author. A secondary clue to this point is the length of it (half the original text). But it could possibly go either way. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-10-05T00:37:34.420" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11482" PostTypeId="2" ParentId="11390" CreationDate="2016-10-05T15:29:28.017" Score="0" Body="&lt;p&gt;There are a few ways to define the domain of a function that you and your students should consider. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The best and clearest way to specify the domain of a function is by the set of values over which the function is initially defined. Ideally, when your students are given a function, they should be told the domain. As Stephen Kulla explains in his answer, a number of properties like bijectivity, surjectivity, and symmetry depend on the domain over which the function is being considered, and his method for defining the function is the most complete. Another example of functions which are commonly given along with a specified domain are piece-wise functions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In common high-school and college algebra and precalculus textbooks, such as Larson's &lt;em&gt;Algebra 2&lt;/em&gt; and Blitzer's &lt;em&gt;College Algebra&lt;/em&gt; and &lt;em&gt;Precalculus&lt;/em&gt;, the domain, when not specified, is considered to be the &lt;em&gt;natural domain&lt;/em&gt; of the function. The natural domain is usually considered to be the maximal set over which the function returns real number values. So, when your students find the domain by excluding the values that make the denominator zero or give even-indexed roots of negative numbers, they are really finding the &lt;em&gt;natural domain&lt;/em&gt; in this vein. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It is crucial to distinguish between the domains you wish to discuss for a given function. My recommendation is to always use the words &lt;em&gt;natural domain&lt;/em&gt; when discussing the domain as in (2) above, because sometimes the domain under discussion will be different from the natural domain. An important example is the sine and cosine functions. In obtaining the arcsine and arccosine functions, we need to consider &lt;em&gt;restricted domains&lt;/em&gt; for the sine and cosine functions on which they are bijective, otherwise their inverses are not well-defined. These restricted domains are different from the natural domains, which are both the set of real numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, if one does not distinguish between the maximal set over which the function returns &lt;em&gt;finite&lt;/em&gt; values, and that over which the function returns &lt;em&gt;finite real&lt;/em&gt; values, additional confusion may occur. I remember being confused why, as a college algebra student, I was required to find complex zeros of a polynomial function but was made to ignore and even avoid complex values of other functions. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastActivityDate="2016-10-05T15:29:28.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11483" PostTypeId="1" AcceptedAnswerId="11484" CreationDate="2016-10-05T20:07:25.587" Score="11" ViewCount="544" Body="&lt;p&gt;I am currently tutoring a student in linear algebra.  She is a very hard worker and does well on computational problems, but struggles to build mathematical intuition.  This struggle is compounded by the fact that she is not working in her native language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Her attitude toward abstract concepts was resilient in the first month of the semester, but her determination seems to be slipping.  Where she once worked to ask targeted questions, she's now resorted to the more typical &quot;this doesn't make any sense.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, she seems to have yet to realize the multitude of connections between bases, span, independence, dimension, rank, nullity, etc.  She can get small pieces, but the big picture is definitely absent.  True and false questions are a massive stumbling block.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've done my best to provide visual illustrations for concepts where I can, but in what ways can I continue to support her in her efforts to learn the more abstract ideas?  What sort of things should I encourage her to do in her own time?  Should I tell her to deliberately spend time away from the material in the hope that we she returns she might have a better perspective?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realize that this is a rather soft question, but any help whatsoever is much appreciated.&lt;/p&gt;&#xA;" OwnerUserId="5988" LastActivityDate="2016-10-05T22:55:46.490" Title="Helping a student exasperated by abstract concepts in linear algebra" Tags="|tutoring|linear-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11484" PostTypeId="2" ParentId="11483" CreationDate="2016-10-05T22:55:46.490" Score="16" Body="&lt;p&gt;&lt;strong&gt;Definitions and other facts&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I find particularly helpful with Linear Algebra is to help the student deal with the definitions in multiple ways. In Linear Algebra there are definitions, and there are properties that things have that are always true but aren't definitions. (They could have been chosen as definitions, but the chosen definition is more useful for proofs.) There are also things that are sometimes true. On top of this, there are definitions, and there are lists of things that are guaranteed to satisfy/not satisfy the definition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, take linear independence/dependence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The definition is that the list of vectors $\mathbf{v}_1$, ..., $\mathbf{v}_r$ is linearly independent when the only solution to the equation $x_1 \mathbf{v}_1 + ... + x_r \mathbf{v}_r = \mathbf{0}$ is $x_1=0$, ..., $x_r=0$. They are linearly dependent when there is at least one solution where at least one $x_i$ is not zero. This definition was chosen to make proofs and also numerical checking easier to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also the fact that the list of vectors $\mathbf{v}_1$, ..., $\mathbf{v}_r$ is linearly dependent when at least one of the vectors is a linear combination of at least one of the other vectors. This is perfectly true and makes more sense geometrically but it is not the definition. It is often easy to tell that the vectors are dependent using this idea, but it is not easy to tell they're independent using this idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, there's a list of things that tell you that a set of vectors is linearly dependent/independent without directly using either of these. For example, if the list contains the zero vector, it's automatically dependent. If the list is made of vectors in $\mathbb{R}^n$ and it has more than $n$ vectors then it's automatically linearly dependent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can make a big table of these things with &quot;linearly dependent&quot; on one side  and &quot;linearly independent&quot; on the other, highlighting things that are definitions and things that are always true and things that only work sometimes. I find this really helps students who are struggling with the multitude of ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly with subspaces. There is the definition with the three laws including closed under scalar multiplication and vector addition. But so often we don't describe what it looks like when something &lt;em&gt;isn't&lt;/em&gt; a subspace. Showing some pictures and some equations that don't satisfy and why they don't is useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then there are things you know must be subspaces: the span of any vectors and the solution to homogeneous linear equations. If the subspace is described by an equation, then if the equation isn't linear then it probably isn't a subspace. If the subspace describes the way to build the points, then if one of the linear combinations is missing it's not a subspace. A list of things that definitely are and things that definitely aren't is a great way of helping students come to terms with these things. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes they need a way of not always focussing on the abstract definition. They need a way of recognising the situation they're in without checking the definition, because when all is said and done, us experts don't check the definition every time either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can encourage her to make similar lists with this and other concepts in her own time. You can also encourage her to every time she sees a word, to attempt to remember the definition and several things that are true about it before doing anything else. This should help her remember the definition (which is important), but it should also give her something to do to stop her panicking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Multiple representations and sets&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find a common issue is that students actually don't perceive the different ways of representing a subspace as different ways of representing the same thing. They also don't understand sets in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most students I see struggling with linear algebra don't understand that the fundamental property of a set is a rule which tells you whether a thing is in the set or not in the set. It doesn't occur to them that they could check if a point is in a subspace or list points in the subspace. They don't see it that way. Thus the &quot;closed under addition&quot; thing is a bit mystical because they don't get the idea of &quot;in&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find it useful to draw out the similarities and differences between sets like this $\{(x,y,z) \in \mathbb{R}^3 | x+ 2y - z = 0\}$ and like this $\{(2a,3a-b,5b+a)|a,b \in \mathbb{R}\}$. The first explicitly lists a rule to check whether a point is in the set, but it's not easy to just list points that are in the set. The second which tells a way to construct all the points in the set easily, but it's not easy to tell if a random point is in the set. (The point of a basis, by the way, is to rewrite your subspace in the second &quot;construct all the points&quot; format.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would encourage her to, every time she is faced with a definition of a set, to make up some points and check if they are in it or not, and also to find several points in the set. Again, it will give her something to do to begin with before she gets overwhelmed. I'd also encourage her to look for multiple representations of the same thing. For example, to explicitly write, once a basis is found, something like $\{(x,y,z) \in \mathbb{R}^3 | x+ 2y - z = 0\}=\text{span }\{(1,0,1),(0,1,2)\}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proofs&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The final issue is with proofs. Most students have not the slightest clue how to start them, or how to finish them, or how to write them. It's a style of maths writing and thinking they've never had to deal with before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find it very useful to look at existing proofs and put coloured circles around the different pieces to show that it has different parts doing different things. It's also useful to look at an example proof and reconstruct it from scratch, focussing in the decisions that were made at each stage of what to do. It's this deciding what to do that is the problem for most students. They're used to having methods to follow and making decisions for themselves is hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, when doing the proof itself, I find it helps to write the information they have at the top of the page, and the goal they want to get to at the bottom. Then the proof goes in the middle. The major decision-making question that can be asked at any point they are stuck is &quot;so what does this mean?&quot;. If your final line should be &quot;$\mathbf{u}+\mathbf{v} \in V$&quot;, then this means that $\mathbf{u} + \mathbf{v}$ should satisfy whatever condition dictates whether a vector is in $V$. This may sound too simple, but believe me it is not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea earlier of spending time investigating what it would mean to list a vector in the set and how you would check if a vector was in a set might help with this too. I could also recommend she looks through her notes herself and finds places where she thinks a decision had to be made so she can ask you/her teacher how the decision was made.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2016-10-05T22:55:46.490" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11485" PostTypeId="2" ParentId="9728" CreationDate="2016-10-06T07:58:29.843" Score="4" Body="&lt;p&gt;Baby Rudin was (and continues?) to be used for the first semester of Harvard's Math 25ab sequence (the &lt;em&gt;second&lt;/em&gt; hardest freshman math class).  We got through the first 7 chapters in Math 25a, before moving on to Halmos &lt;em&gt;Finite Dimensional Vector Spaces&lt;/em&gt; for Linear Algebra and Spivak &lt;em&gt;Calculus on Manifolds&lt;/em&gt; for the second semester, Math 25b. (The (in)famous Math 55ab sequence, reputed to be the hardest undergraduate course in the country, sometimes used Rudin as well, or Loomis and Sternberg as an alternative, as well as some other supplemental texts for abstract algebra and topology).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, the students who made it through these courses constitute the top ~50 or so students of an incoming freshman class of ~1600.  Also, the attrition rate was severe and most (~3/4) of the students in Math 25 and all the students in Math 55 were exposed to proof writing through high school math competitions (often USAMO/IMO winners in the case of Math 55) and/or university courses.  Personally, I had taken Calc III, linear algebra, ODE's and PDE's in high school and was a USAMO participant (although I had already decided against being a math concentrator by the time I entered college).  Nevertheless, I found Math 25a and 25b to be two of the most work-intensive and challenging math classes I had even taken.  I would be shocked if any cohort of students in the history of American education was well-prepared (or talented?) enough to use Rudin as a standard freshman text.&lt;/p&gt;&#xA;" OwnerUserId="7256" LastActivityDate="2016-10-06T07:58:29.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11486" PostTypeId="1" CreationDate="2016-10-06T19:38:08.137" Score="7" ViewCount="96" Body="&lt;p&gt;In an earlier &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5730&quot;&gt;question&lt;/a&gt;, the book &lt;em&gt;Exploring Probability in School&lt;/em&gt; (2005) (&lt;a href=&quot;http://www.springer.com/gp/book/9780387245294&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;) was mentioned. It gives an overview of the research on the teaching and learning of probability up to that point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does there exist a similar book on the same topic with a more recent publication date covering the newest research in the field? If not, what are the important new papers to read on that subject?&lt;/p&gt;&#xA;" OwnerUserId="39" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-10-06T19:49:07.627" Title="Literature on teaching and learning probability" Tags="|education-research|reference-books|probability|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11487" PostTypeId="1" AcceptedAnswerId="11488" CreationDate="2016-10-06T20:09:05.460" Score="3" ViewCount="194" Body="&lt;p&gt;As &lt;code&gt;Lynn Steen&lt;/code&gt; in &lt;code&gt;The Science of Patterns&lt;/code&gt; says, I think &lt;code&gt;Mathematics is the science of pattern&lt;/code&gt; too. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a 10 years old student which is weak in mathematics. It seems he can do arithmetic(addition, multiplication, ...) but cannot understand the meaning of problems and their relation to real word. I want to help him to rebuild foundation of his mathematics and promote his creativity and attention to real world. Now with respect to the role of &lt;code&gt;patterns&lt;/code&gt; in mathematics, I want to start with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know &lt;strong&gt;free&lt;/strong&gt; (probably online) resources like PDF files which have much exercises in various flavors or have ideas for me to make them? I know I can build exercises by paying attention to the world, but the time is important for me and it is better to use ideas of other peoples. Also my approach is to hold enjoyment and so, &lt;code&gt;games&lt;/code&gt; have advantage.&lt;/p&gt;&#xA;" OwnerUserId="5649" LastEditorUserId="5649" LastEditDate="2021-10-11T18:38:12.060" LastActivityDate="2021-10-11T18:38:12.060" Title="Reinforcement mathematics thinking of 10 years old child by teaching &quot;patterns&quot;" Tags="|primary-education|interactive-teaching|reference-books|creativity|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11488" PostTypeId="2" ParentId="11487" CreationDate="2016-10-06T22:58:43.697" Score="2" Body="&lt;p&gt;This article on the Fibonacci numbers could provide you with rich material.&#xA;But you would have to develop an approach suitable for your 10-yr old.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;George Dvorsky.&#xA;  &quot;15 Uncanny Examples of the Golden Ratio in Nature.&quot;&#xA;  (&lt;a href=&quot;http://io9.gizmodo.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature&quot; rel=&quot;nofollow noreferrer&quot;&gt;gizmodo.com link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/kmI7H.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kmI7H.jpg&quot; alt=&quot;Sunflower&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;You might explore &lt;a href=&quot;http://www.wartgames.com/themes/math/fibonacci.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this wartgames.com site&lt;/a&gt;, which has some&#xA;possibly useful links on the Fibonacci numbers&#xA;for children (which I didn't explore carefully).&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-10-06T23:53:48.110" LastActivityDate="2016-10-06T23:53:48.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11489" PostTypeId="2" ParentId="10501" CreationDate="2016-10-07T00:42:39.493" Score="2" Body="&lt;p&gt;Personally, I've found that writing a rubric &lt;em&gt;a priori&lt;/em&gt; to be a nightmare once answers move into the realm of proofs and such--there always seems to be an incorrect answer that ticks off parts of the rubric it shouldn't really deserve. My preferred method is somewhat similar to how I was once taught to grade essays: sort the responses into stacks that correspond to the quality of the answer. For example, I might make stacks of&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Completely correct, good notation throughout&lt;/li&gt;&#xA;&lt;li&gt;Almost correct (perhaps a notation or arithmetic error)&lt;/li&gt;&#xA;&lt;li&gt;Correct approach, but some structural errors (failed to address a minor case, etc)&lt;/li&gt;&#xA;&lt;li&gt;General gist, but flawed argument&lt;/li&gt;&#xA;&lt;li&gt;Deeply flawed argument, wouldn't be correct without major revisions&lt;/li&gt;&#xA;&lt;li&gt;lost in the woods / no clue / can't follow&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and then assign points / letter grades to each stack. Alternatively, I think of making stacks of A/B/C/D/F level work, without exactly quibbling about why something ended up in the specific stack (should have been 2$\epsilon$ instead of 3$\epsilon$ or whatever). Plus, it just makes me feel like Robert Pirsig in &lt;em&gt;Zen and the Art of Motorcycle Maintenance.&lt;/em&gt; This approach also keeps me from getting stuck in sweating weather an answer deserves 7.6 or 7.595 points according to a rubric. If necessary, you also have all of the &quot;A&quot; level answers next to each other if you want to justify why one answer might be 10 points while another is 9.5 points if you want to further refine the grading. &lt;/p&gt;&#xA;" OwnerUserId="7260" LastActivityDate="2016-10-07T00:42:39.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11490" PostTypeId="2" ParentId="11435" CreationDate="2016-10-07T02:07:23.403" Score="9" Body="&lt;p&gt;One reason why proof by contradiction is difficult for students is because mathematical notation (and other written language) does not allow for a &lt;a href=&quot;https://en.wikipedia.org/wiki/Subjunctive_mood&quot;&gt;subjunctive mood&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me elaborate on this:  In English and other &quot;natural&quot; languages, we can distinguish between a state of affairs that &lt;em&gt;is&lt;/em&gt; true, and a state of affairs that can be &lt;em&gt;provisionally thought of as true&lt;/em&gt;.  We can say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I am the President of the United States, and therefore I support this legislation&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;but we can also say&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If I were the President of the United States, then I would support this legislation&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and we understand these two sentences as meaning very different things.  Mathematical notation, however, elides or erases those shades of meaning:  If $P$ is the proposition &quot;I am the President of the United States&quot; and $Q$ is the proposition &quot;I support this legislation&quot; then &lt;em&gt;both&lt;/em&gt; of the two sentences above &#xA;would be symbolized as $P \implies Q$ or written as &quot;If $P$ then $Q$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What does this have to do with proof by contradiction?  Consider how one starts a proof that $x^2=2$ has no rational solutions.  We say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose $x = m/n$ is a solution, with $m, n$ relatively prime whole numbers.  Then $m^2 = 2n^2$, so $m^2$ is even, which means that $m$ is even.  Therefore...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now, in the above proof, consider how you would read aloud the equation $m^2 = 2n^2$.  I strongly suspect that most people would read it as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;$m$ squared &lt;strong&gt;equals&lt;/strong&gt; two $n$ squared&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;rather than as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;$m$ squared &lt;strong&gt;would equal&lt;/strong&gt; two $n$ squared&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is precisely the sticking point:  We read the symbol $=$ as saying (falsely!) that two things &lt;em&gt;are equal&lt;/em&gt;, rather than as saying (correctly!) that two things &lt;em&gt;would be equal&lt;/em&gt; if the initial hypotheses &lt;em&gt;were&lt;/em&gt; true.  The absence of a subjunctive mood in mathematics leads us to say things like &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Suppose $m/n$ &lt;strong&gt;is&lt;/strong&gt; a solution; therefore...&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;when what we really mean is &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Suppose $m/n$ &lt;strong&gt;were&lt;/strong&gt; a solution; then...&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This observation, by the way, has been made before:  A few years ago I came across a very brief article from &lt;em&gt;The Mathematics Teacher&lt;/em&gt;, published more than 60 years ago, that made essentially the same argument.  The article is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Butler, C.  (1955).  &quot;A note on the statements of theorems and assumptions.&quot;  &lt;em&gt;Mathematics Teacher&lt;/em&gt;, 48, 106-107.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Butler article does not explicitly invoke the subjunctive mood by name, but expresses the same idea, as the following excerpt shows (all emphases in the original):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Such apparent discrepancies... do bother immature high school students who are just learning what a demonstration means.  To them it doesn't make sense to say &quot;assume that $\angle x = \angle y$&quot; when they can see from the diagram that angle $x$ actually is not equal to angle $y$, or to conclude that $RS \hskip{0.05 in} || \hskip{0.05 in} PQ$ when those lines can be seen actually to intersect in the diagram.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;On the other hand, most students, even though mathematically immature, would find no difficulty in accepting the statement that &quot;IF angle $x$ WERE equal to angle $y$, then $RS$ WOULD BE parallel to $PQ$&quot; as a reasonable statement even though the diagram does not appear to exhibit these conditions.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;To cite a non-geometrical proposition which is the exact counterpart of the foregoing example, it might easily appear ridiculous to young students for a $5 \frac{1}{2}$ foot man to say &quot;Assume that I am $9$ feet tall; therefore I can touch the ceiling,&quot; but the same students would readily accept from the same man the statement, &quot;IF I WERE 9 feet tall, THEN I COULD touch the ceiling&quot; and regard it as a perfectly normal kind of statement.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Logically there is no difference between saying &quot;Assume this is true; therefore that is true&quot; and &quot;If this were true, then that would be true,&quot; but there is a profound difference in the attitude or feeling induced in immature students by the two forms of statement.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Notice in particular that Butler realizes that a symbolic expression like &quot;$\angle x = \angle y$&quot; would be read  as &quot;angle $x$ is equal to angle $y$&quot;; the only way to invoke the subjunctive mood is to drop the symbolic language altogether and use verbal formulations instead.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-10-07T02:07:23.403" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11491" PostTypeId="1" AcceptedAnswerId="11492" CreationDate="2016-10-07T12:11:37.900" Score="5" ViewCount="245" Body="&lt;p&gt;A student is given the question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Round off each of the following numbers, correct to &lt;strong&gt;two significant figures.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;32.4892&lt;/li&gt;&#xA;&lt;li&gt;8.2673&lt;/li&gt;&#xA;&lt;li&gt;475.98&lt;/li&gt;&#xA;&lt;li&gt;0.078625&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are two marks for each.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student answers with &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;32.0000&lt;/li&gt;&#xA;&lt;li&gt;8.3000&lt;/li&gt;&#xA;&lt;li&gt;480.0000&lt;/li&gt;&#xA;&lt;li&gt;0.07900&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;How would you grade them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that $32.0000=32$ so the student has correctly rounded... although I concede they might have somewhat missed the point.&lt;/p&gt;&#xA;" OwnerUserId="584" LastActivityDate="2016-11-04T16:45:32.400" Title="Grading a Simple Rounding Exercise" Tags="|grading|arithmetic|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="11492" PostTypeId="2" ParentId="11491" CreationDate="2016-10-07T16:22:00.780" Score="6" Body="&lt;p&gt;What is the whole point of rounding values? &quot;For brevity, we approximate long decimals by finding the nearest specified place value.&quot;  (This is copied directly from the slide in my first-day college statistics lecture.) If the expression didn't get shorter, then the student really has missed the whole point, and needs correction. Since it was the same basic error repeated, I would tend to take off the value of one single item (1/4 of the total value for this sequence; i.e, 2 points per your comment). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondarily, while mathematically it's true that $32.0000 = 32$, in a practical scientific context that carries different, distinct information. The first expression is saying &quot;this measurement is accurate to the ten-thousandths place&quot;, whereas the second is saying &quot;we are only confident of the precision to the units place&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This latter issue is something that I'm not sure my students ever really understand; they may just be going through the motions by rote, which doesn't make me super happy, but we don't have extra time for investigation of instrumentation accuracy. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-11-04T16:45:32.400" LastActivityDate="2016-11-04T16:45:32.400" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11493" PostTypeId="2" ParentId="11390" CreationDate="2016-10-07T17:40:16.070" Score="0" Body="&lt;p&gt;When I'm teaching students who are still struggling to get the idea of domain, I refer to the function as a machine. You throw x's into the machine and it spits out y's. Your domain is x's that the machine understands and can compute. Often times it may be easier to figure out what isn't in your domain (e.g. all reals, except for a few numbers). I would explain this to students as x's that when you throw them into your machine (function), they break the machine. It doesn't know how to process those numbers. These are often divide by 0 issues or square root of negative issues. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously this isn't perfect and doesn't take into account all mathematical functions or domains, but it helps students who struggle with the concept.&lt;/p&gt;&#xA;" OwnerUserId="7264" LastActivityDate="2016-10-07T17:40:16.070" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11494" PostTypeId="1" CreationDate="2016-10-07T18:02:01.847" Score="5" ViewCount="129" Body="&lt;p&gt;I have little experience in teaching young kids. Soon, I will present my research to a class of 7 to 16 year old kids. I do some analytical modelling but I mainly do individual-based simulations in population genetics to investigate various evolutionary processes. I also do Approximate Bayesian Computation. At the start I would like to give a good introduction for the interest of numerical simulations generally speaking and why we do numerical modeling in science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How should I go about explaining the general reason why we do numerical modeling in science?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking about starting with the concept of thought experiment and how we can, in our head simulate something without actually performing the action in the real world. Then, I would go toward more complicated thought experiment explaining that there is a point where our mind and intuition aren't powerful enough and we must let a computer do the thought experiment for us. I would say something like &quot;I imagine a world that is a simplification of the reality, chose a set of rules that dictate this world, chose an initial state/action and let the computer make the thought experiment in this simplified world. Assuming our world was close enough to reality, we can make predictions about what would happen in reality&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you give me some advice or even a guideline for the narrative?&lt;/p&gt;&#xA;" OwnerUserId="1204" LastEditorUserId="1204" LastEditDate="2016-10-07T18:07:32.367" LastActivityDate="2016-10-10T17:52:39.797" Title="Why do we do numerical modeling in science? (for 7-16 year old)" Tags="|secondary-education|computer-science|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11495" PostTypeId="2" ParentId="11494" CreationDate="2016-10-07T18:13:51.583" Score="2" Body="&lt;p&gt;A couple of the reasons we solve problems numerically (or approximately) is that they are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) 'Too hard' -- even with Maple, Mathematica, or another computer algebra system, sometimes you will have equations that aren't easily solved (or don't have a single solution)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) 'Too expensive' -- even if a solution exists, it may take too long to get an exact answer out, so we take an approximate solution instead&lt;/p&gt;&#xA;" OwnerUserId="5873" LastEditorUserId="1147" LastEditDate="2016-10-09T16:58:45.903" LastActivityDate="2016-10-09T16:58:45.903" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11496" PostTypeId="1" AcceptedAnswerId="11500" CreationDate="2016-10-08T11:49:53.360" Score="6" ViewCount="903" Body="&lt;p&gt;I need to produce at least 30 test/exam papers a year, plus a few assignments and investigations. I have good source material but need to change numbers and contexts and output original papers that meet our curricula/student requirements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do you, as a high school mathematics teacher, produce mathematics test papers? I’m interested in methods from screenshot-crop-n-paste hacks to use of advanced typesetting programs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most importantly, how do you get consistent-looking graphs and figures? If it were just equations and mathematical statements, Microsoft Word would be sufficient. But figures and graphs are essential.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are so many programs and packages for mathematics and publishing but what works for producing day-to-day high school test papers quickly?&lt;/p&gt;&#xA;" OwnerUserId="7046" LastEditorUserId="39" LastEditDate="2016-10-08T12:29:39.497" LastActivityDate="2016-10-12T14:14:32.837" Title="What is your workflow for producing high school mathematics test papers?" Tags="|secondary-education|test-design|" AnswerCount="7" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11497" PostTypeId="1" AcceptedAnswerId="11502" CreationDate="2016-10-08T12:29:11.243" Score="2" ViewCount="318" Body="&lt;p&gt;A possible explanation is that some yet unsolved advanced functions have a known codomain, but not a known range.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Got any better ideas?&lt;/p&gt;&#xA;" OwnerUserId="7232" LastEditorUserId="7232" LastEditDate="2016-10-09T23:16:47.113" LastActivityDate="2016-10-09T23:16:47.113" Title="How to explain codomain of a function?" Tags="|undergraduate-education|functions|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11498" PostTypeId="2" ParentId="11496" CreationDate="2016-10-08T13:41:23.463" Score="2" Body="&lt;p&gt;For creating graphs, I almost always use Desmos.com to plot the graph and then I screenshot it to paste into Word. Desmos let's you graph pretty much anything you'd want. &lt;/p&gt;&#xA;" OwnerUserId="7264" LastActivityDate="2016-10-08T13:41:23.463" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11499" PostTypeId="2" ParentId="11496" CreationDate="2016-10-08T15:16:26.357" Score="4" Body="&lt;p&gt;As a community-college math lecturer, my primary course-preparation tool is the &lt;a href=&quot;http://wpslive.pearsoncmg.com/cmg_instructor_testgen_1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pearson TestGen&lt;/a&gt; program. Pearson provides pre-made testbanks to accompany all of their textbooks; so you can begin by drag-and-dropping selected questions into a test of your construction. Thereafter you've got a single push-button functionality to randomize values within questions, scramble the order of multiple-choice options, etc. Math formula editing is straightforward. You can even export a test to the Blackboard learning management system, if desired (which is how I make tests and quizzes on Blackboard; easier than BB's own interface). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;TestGen is free but requires an instructor account with Pearson; contact a representative and hopefully you can get one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XdQiz.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XdQiz.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-10-08T15:16:26.357" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11500" PostTypeId="2" ParentId="11496" CreationDate="2016-10-08T17:48:52.780" Score="3" Body="&lt;p&gt;Really great question, I will first start with a general overview of my test-generation workflow and then go into more detail on the specifics below that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;General Workflow For Generating Tests&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first thing that I do when I start to create a test is I go through all of the notes/slides/worksheets/homework from the chapter or unit that I am testing.  From these things, I make a list of what specific skills and types of problems that I think students should be able to complete after learning the chapter (I do this before I start teaching a chapter as well, but I always like to redo the list at the end just incase there was something extra that I got into or something that we skipped, for whatever reason).  Once I have my list of topics/problem types, I start to search for them.  Now, at this point, I do have a pretty large stockpile of worksheets, images, problems, and other assessments all stored digitally so I will usually start digging through them first.  If there is something that I would like to have on the test but don't currently have a copy of, rather than trying to make it all from scratch I will do some Googling to see if I can find something that has already been made or at least partially made. Once I find a problem I am looking for, I quickly screenshot it and paste it into a Google Doc.  At this point, I am not worried about organization, I am more just trying to collect all of the problems that I would like to use in one place, which I will organize and structure later.  Once I have compiled my &quot;collage&quot; of problems, I start to refine them and organize them in Google Docs.  Now this part really depends on the specific types of problems and the quality of the screenshots.  I try to use the raw screenshots as much as possible, but more often than not, they will either be to low quality to withstand photocopying or they will need to be reworded, so I will usually end up using my screenshots as models for problems and then end up retyping/formatting them.  At this point my test is pretty much done.  The last step is inserting graphics/charts/etc, which I will go more into below.  I find that this general workflow is very efficient for me (usually takes no more than an hour to make a test completely from scratch) and always ends up with an organized and fair assessment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Finding Problems&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like I said above, I already have a pretty large collection of worksheets/images/problems/etc to pull from.  My biggest suggestion for finding problems is to &lt;strong&gt;do Google/Google Image searches and just horde anything you can find that is at all relevant on Google Drive&lt;/strong&gt;.  I really like Google Drive for this because it makes it so all of my resources are accessible from anywhere so if I need to make a test over the weekend, I don't have to lug my work laptop home on the train, I can just use my home desktop.  It also integrates really nicely with Google Docs, which I prefer over Word for the same &quot;anywhere-accessiblilty&quot;.  The key to collecting these resources is to keep them very organized, otherwise all the time you save by collecting them will be wasted when you go to try to find a specific problem.  I like to make a main folder for each chapter and then subfolders for each specific type of problems/worksheets. It does take some time to find these resources, but once you have them saved and organized you should never need to go hunting for them again.  One last thing on collecting problems, it is probably worth a look in your school's basement/textbook storage closet to see if there are any old workbooks or textbooks lying around that you could scan and use.  I have found a lot of my best resources digging through the piles and piles of books in the basement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Working With Images&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My main technique for compiling the individual problems into one Google Doc is to use screenshots from worksheets.  For Mac, use Command Option Shift 4 and click and drag over the problem and it will automatically get copied to your clipboard which you can paste directly into a Google Doc (you won't have 500 screenshots saved to your desktop this way!).  For Windows, I use the Snipping Tool.  Just save as image and then drag into Google Docs. If the problem you found was already an image then even better, just drag into Google Docs! Once in Google Docs, I will further manipulate the images to try to get them to work for the test without having to retype the whole problem.  Your best friends will be &lt;strong&gt;Crop&lt;/strong&gt; (right click the image and select Crop Image) &lt;strong&gt;Image Options&lt;/strong&gt; (right click the image and select Image Options).  Cropping is super-useful to get rid of the numbering of problems on worksheets, or to reduce the whitespace around an image so that it will fit better in with the rest of the problems.  I use the Image Options to adjust the brightness and contrast of images to make them more readable.  This doesn't always work, depending on how low-quality the original image was, but it usually is enough to make an almost-readable problem, into a totally-readable problem.  The last trick for working with images in Google Docs is to take advantage of the different image modes: Inline, Wrap, and Break (appear in Google Docs when you click on the image). I use the Wrap Text mode for all of my images, because this allows me to manually position them wherever I want on the page and not have to worry about dealing with tons of spaces and returns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Graphics and Diagrams&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the most part, I try to recycle diagrams and graphics from problems that I have saved in my collection.  Just Crop out any of the text, leave the diagram, and reuse for a different problem.  For instance, if the test is on parallel lines cut by a transversal, no need to make my own diagram.  Just find one already made and tailor whatever problem you want on the test  to fit to the numbering/labelling/etc on the diagram. If I really have to create my own diagram for a test I use one of two applications: either Geogebra or Geometer's Sketchpad.  They both have a &lt;em&gt;slight&lt;/em&gt; learning curve but with about half an hour of practice you should be off and making whatever diagrams you would like.  I have recently been using Geometer's Sketchpad, only because my district has a bunch of licenses, but Geogebra is great as well and I have made a ton of diagrams with it.  What I will usually do is screenshot directly from Geometer's Sketchpad/Geogebra into the test and then save the actual file later just incase I want to tweak the diagram in the future.  If I want to make a graph, I will use Desmos and just screenshot.  You can do graphs in Geogebra but it is a bit annoying.  To organize my custom diagrams, I will usually make a Google Doc for each category of diagram (triangles, quadrilaterals, graphs, etc) and just copy and paste the screenshots into there without worrying about formatting.  This way, I don't have to find and open the original file if I want to reuse the diagram, I just go to the Doc and Copy Paste.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Reusing Tests&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally, after I have made a test and have students take it, I will make notes of bad questions or confusing directions and then immediately update my file with the changes. Then, the next year when I go to administer the same test it will already have the improved language and I won't have to try to remember what didn't work about it.  For the most part I will reuse tests and just change the specific numbers in problems and the order of the problems from year to year.  I have never had an issue with students from a previous year giving current students old tests because even if they do do this, the updated numbers and different order of questions is typically enough to negate having access to the previous versions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps, I will update this answer if I can think of anymore tricks/hacks/techniques that I use.  But for the most part &lt;strong&gt;screenshot, crop, and reuse&lt;/strong&gt; are the words I live by when generating tests.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also side note, Kuta software is an amazing application and I highly suggest trying to get your hands on a copy.  I rarely use it to make chapter or unit tests but I almost always use Kuta for larger cumulative exams like midterms and finals because it is just so fast and is completely customizable and can totally randomize and easily generate many different versions of the same test.  I do use Kuta heavily to generate homework and classwork handouts too, again because of its amazing ease use.  &lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2016-10-08T17:48:52.780" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11501" PostTypeId="2" ParentId="11496" CreationDate="2016-10-08T17:51:13.610" Score="6" Body="&lt;p&gt;Disclaimer: I do not teach high school, and this isn't the easiest startup option.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you don't mind a bit of a learning curve, I would suggest using &lt;a href=&quot;https://www.artofproblemsolving.com/wiki/index.php?title=LaTeX&quot;&gt;LaTeX&lt;/a&gt;, which is very customizable and (key) will allow for easy reuse and renumbering and re-everything.  Some college instructors use &lt;a href=&quot;http://doc.sagemath.org/html/en/tutorial/sagetex.html&quot;&gt;SageTeX&lt;/a&gt; with it for creating graphs or randomization, though one can also use TikZ for that; on nice feature is that you could change some numbers on a harder problem and have it compute the answer key for you as well.  &lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-10-08T17:51:13.610" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11502" PostTypeId="2" ParentId="11497" CreationDate="2016-10-09T09:26:05.023" Score="4" Body="&lt;p&gt;Codomain is a primary notion in the definition of a function, range is something that can be computed given the whole definition, one cannot possibly replace the other. What bothers me is that you seem to imply that you introduced codomain after range, and I don't see how that could be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only possible question left would be &quot;why not always introduce functions that have their range equal to their codomain?&quot;, i.e. &quot;why don't we only define onto functions?&quot; which might be your real question. To that, there are several answers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the one you gave: it might be difficult to determine the range. This is a strong one, and you don't need to involve advanced functions. We want to be able to define the function $x\mapsto x^4-x^3+12x-7$ from $\mathbb{R}$ to $\mathbb{R}$ without having to determine its range!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;in many occasion we &lt;em&gt;want&lt;/em&gt; to have a larger codomain, e.g. when we consider composed function: the codomain of one function should match the domain of the other one; when we consider paths in, say, the plane: we don't want the path to take its value in some subset of the plane, but to consider it as taking its values in the plane itslef; imagine you are modeling a physical problem with a function, say the temperature at a point expressed as a function of the time: you would not like to have to know the range before getting started. This last case points to an important point: domain and codomain really are about the &lt;em&gt;type&lt;/em&gt; of the input and output of the function (in the physics example, both are real numbers but physically one is a time and the other a temperature, in maths a function could have a integer input and real output, or real number input and point output, etc.) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-10-09T09:26:05.023" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11504" PostTypeId="2" ParentId="11496" CreationDate="2016-10-09T11:20:32.153" Score="2" Body="&lt;p&gt;I teach elementary school and also create tests for middle school students. I suspect some of my techniques could be used in high school. I have Smart Notebook software on my computer. This is free because I have Smart Notebook in the classroom.  I have found it is terrific for diagrams, particularly geometric ones. I also use Microsoft Word with MathType by design science which I found sufficient for all equations and easier (for me) than LaTex which is also popular. For graphs, I have used TI Connect software (from Texas Instruments) to take screenshots of my graphing calculator.  If you google screenshots of TI 83, you can find different methods and products. Good Luck.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-10-09T11:20:32.153" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11505" PostTypeId="2" ParentId="11497" CreationDate="2016-10-09T12:51:36.490" Score="2" Body="&lt;p&gt;I'll give a brief answer. Basically, $f: A \rightarrow B$ paired with a rule $f(x) \in B$ for each $x \in A$ defines a function. There are three moving parts, the choice of $A$, the choice of $B$ (the codomain) and the rule $f(x)$. These choices do not exist in isolation. We must have $f(x) \in B$ so the function is &lt;strong&gt;into&lt;/strong&gt; and we must have $f(x)$ is a single element in $B$ so the function is &lt;strong&gt;single-valued&lt;/strong&gt;. One very nice aspect of this definition is it allows us to change the domain without changing the formula and it allows us to change the codomain without changing the formula. Why this matters? Certainly good references to problems in higher mathematics which require the codomain have already been mentioned in the comments. I would point out the terminology allows us to take any function and create a corresponding bijection $\iota \circ f \circ s$ where $s: C \rightarrow A$ is a map which selects just one point in each fiber of the domain and $\iota: f(A) \rightarrow B$ is the inclusion map. If we were to exchange the range $f(A)$ for the codomain $B$ then it would be difficult or impossible to have this discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, I think your answer has the right idea for beginning students. We could say this: $f: A \rightarrow B$ is like planning a trip. The $B$ describes where you may possibly go whereas $f(A)$ is where you actually go. This reflects the view that we may hold a function with partial understanding. In particular, this is a reality for students who when given $f(x)$ do not entirely understand it's content (until they do much analysis of say, $f(x) = \frac{x}{x-2}$ etc.)&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2016-10-09T12:51:36.490" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11506" PostTypeId="2" ParentId="925" CreationDate="2016-10-09T14:48:16.127" Score="1" Body="&lt;p&gt;&lt;em&gt;First answer:&lt;/em&gt; Isn't sequence like a list of numbers that appear as according to function?&#xA;And series is more like the addition of the terms of the sequence.&#xA;I guess There convergence properties differ from one another. &#xA;Can it not be thought like a function happening from N to R (sequence) and then the sum of the resulting numbers up to 'n' or infinity the series?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But can the convergence in sequence have any influence on the convergence of a series?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Second answer:&lt;/em&gt; If I take the function from N to R as 1/n, its sequence (function) will have the numbers 1, 1/2, 1/3 etc listed in its range or will be the 'terms' . Now this particular sequence will converge to 0 b/c I will find some abscilla which is grater then the difference of any term and 0.&#xA;However if I talk bout series(join, addition, summation) , it will be the sum of all the terms of the sequence I mentioned above and that same may not necessarily be smaller than abscilla which is why this &lt;strong&gt;series&lt;/strong&gt; will not converge to a particular value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this is the difference and hope it's elaborate enough.&lt;/p&gt;&#xA;" OwnerUserId="7267" LastEditorUserId="143" LastEditDate="2016-10-09T18:50:37.633" LastActivityDate="2016-10-09T18:50:37.633" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="11507" PostTypeId="1" CreationDate="2016-10-09T15:47:17.160" Score="1" ViewCount="294" Body="&lt;p&gt;I am going over rate of change with my 8&lt;sup&gt;th&lt;/sup&gt; graders tomorrow and I foresee a difficulty. I would like to see different ways to explain this for my students who have trouble with it.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In terms of magnitude, which function has the greater rate of change?&#xA;  $$y=−2x+1 \qquad{\text{or}}\qquad y=2x+1$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7270" LastEditorUserId="5220" LastEditDate="2016-11-02T01:16:05.213" LastActivityDate="2016-11-02T01:16:05.213" Title="Comparing the magnitudes of the slopes of two lines" Tags="|secondary-education|algebra|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11508" PostTypeId="2" ParentId="11487" CreationDate="2016-10-09T16:26:10.767" Score="4" Body="&lt;p&gt;Most elementary students enjoy finding all the patterns in Pascal's triangle. This &lt;a href=&quot;http://www.mathsisfun.com/pascals-triangle.html&quot; rel=&quot;nofollow&quot;&gt;link&lt;/a&gt; at mathisfun covers many different patterns including the Fibonacci numbers (yes, they are in Pascal's triangle) and the Hockey Stick pattern.  The page also explains the connection between Pascal's Triangle and &lt;a href=&quot;http://www.mathsisfun.com/sierpinski-triangle.html&quot; rel=&quot;nofollow&quot;&gt;The Sierpinski Triangle&lt;/a&gt;.  You can also see the connection in this &lt;a href=&quot;https://www.youtube.com/watch?v=wcxmdiuYjhk&quot; rel=&quot;nofollow&quot;&gt;YouTube video&lt;/a&gt;.  Have fun.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-10-09T16:26:10.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11509" PostTypeId="2" ParentId="11507" CreationDate="2016-10-09T18:40:41.127" Score="0" Body="&lt;p&gt;Magnitude implies absolute value. While a vector heading west has the opposite sign of one heading east, the magnitude may still be the same. In your example, 2 &gt; -2 but they have the same absolute values. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-10-09T18:40:41.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11510" PostTypeId="2" ParentId="8193" CreationDate="2016-10-10T01:46:58.840" Score="6" Body="&lt;p&gt;A late answer, because I think none of the existing responses have yet to address the theme of &quot;why particularly in the U.S.?&quot; (laying aside, momentarily, the other solid criticisms of standardized tests in general). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recall that the U.S. has an almost unique &lt;a href=&quot;https://en.wikipedia.org/wiki/Federal_government_of_the_United_States&quot; rel=&quot;noreferrer&quot;&gt;federalist&lt;/a&gt; structure which traditionally reserves lots of power to the states, outside of the national government. In particular, the U.S. constitution does not give any explicit power to the federal government to form or guide any national educational policy or curriculum. This is why educational standards in the U.S. vary so widely between states (or even moreso: between localities, as schools are largely locally-funded through property taxes). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Partly because of this tradition, we have a culture where control over education is rather jealously guarded as a right of the state and local community. As an example, consider the recent &lt;a href=&quot;http://www.motherjones.com/politics/2014/09/common-core-education-reform-backlash-obamacare&quot; rel=&quot;noreferrer&quot;&gt;Common Core&lt;/a&gt; movement: it was considered such a third-rail issue that federal education experts did not dare even express support for this multi-state compact; yet nevertheless a political battle emerged over whether the Obama administration supported it in secret or not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So one part of the answer is that in the U.S., standardized testing (on top of other limitations) smacks of outside influence, control, and surveillance (such as by the U.S. federal government) over what is considered to be a local-control right, and the right of parents to determine what their children learn. That may be unusual among other countries, but part of the culture here (and one might say that this also feeds into the general acceptance of lower-quality schools for minorities who live in poor areas and are limited in ability to fund schools, which normalizes ongoing inequality in the U.S.). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Richard Hofstadter wrote a book, &lt;a href=&quot;https://en.wikipedia.org/wiki/Anti-intellectualism_in_American_Life&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Anti-intellectualism in American Life&lt;/em&gt;&lt;/a&gt;, which won the Pulitzer Prize in 1964, on some of these issues, that you might want to consider as a starting point for the unique U.S. perspective on educational issues. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-10-10T13:57:09.650" LastActivityDate="2016-10-10T13:57:09.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11511" PostTypeId="1" AcceptedAnswerId="11518" CreationDate="2016-10-10T06:07:48.830" Score="8" ViewCount="356" Body="&lt;p&gt;In recent years I have seen several courses in pure math in the undergrad level (year 2, 3, 4) such as real analysis and topology where the entire course consists of:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;notes written during the lecture (with several recommended textbooks that the prof does not follow in detail), and &lt;/li&gt;&#xA;&lt;li&gt;a long list of exercises questions drawn from various sources with no solution provided to the students&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The questions for the exams are directly drawn from the list of exercises questions with modifications. There is absolutely no homework, no grades aside from grades of several exams and solution to exercises are never given aside from hints you can obtain by talking to the professors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From talking to the professors, I get the sense that they are mainly concerned about:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;cheating, using homework solution from online of standard texts thus hindering mathematical development&lt;/li&gt;&#xA;&lt;li&gt;encouraging students to think more about problems &lt;/li&gt;&#xA;&lt;li&gt;providing ways to incentize students to do exercises by putting modified exercise problems on exams&lt;/li&gt;&#xA;&lt;li&gt;weeding out students who are not committed to learning about math&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Has anyone ever went through or conducted such a course? Is this an effective way to deliver a proof based math course? Are there any potential pitfalls and flaws using this approach?&lt;/p&gt;&#xA;" OwnerUserId="3560" LastEditorUserId="3560" LastEditDate="2016-10-12T04:08:19.657" LastActivityDate="2016-10-12T04:08:19.657" Title="Is proof-based exercise-oriented math course without solution an effective way to teach pure math?" Tags="|mathematical-pedagogy|course-design|proofs|teaching|exercises|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11512" PostTypeId="2" ParentId="8193" CreationDate="2016-10-10T11:38:38.063" Score="2" Body="&lt;p&gt;I read through the answers and am surprised no one mentioned that issue of bias against race and lower class.  There is a history of tests being designed for white middle class students.  Students from other backgrounds did poorly because they didn't have the background or context to understand the questions.  The result was tests which gave further advantages to a group that was already perceived to have the advantages; similarly the tests gave educators an excuse to hold students back.  In one glaring example from the 1960-1970's, my mother, a reading teacher, noticed a question: What color is your cheek? An African American student picked brown but of course that was marked wrong because the correct answer was pink.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the US where there are so many minorities,  a goal of equality, and a history of racial inequality, these tests were greatly criticized by many.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of the problems are being addressed today, but they still exist.  An attempt has been made by makers of standardized math tests,  to use simpler vocabulary and to avoid word problems in standardized tests about situations that would be unfamiliar to lower class children.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-10-10T11:38:38.063" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11513" PostTypeId="2" ParentId="11511" CreationDate="2016-10-10T13:01:01.250" Score="4" Body="&lt;p&gt;This sounds vaguely similar to an &quot;inquiry-based learning&quot; course.  So although this isn't what you are describing (notably the lectures), you may find resources at &lt;a href=&quot;http://www.inquirybasedlearning.org&quot; rel=&quot;nofollow&quot;&gt;the Academy of Inquiry Based Learning&lt;/a&gt; helpful.  &lt;a href=&quot;https://www.artofmathematics.org/blogs/jfleron/what-is-inquiry-based-learning&quot; rel=&quot;nofollow&quot;&gt;Here is a separate description&lt;/a&gt; from a related venture.  A most extreme version of this (extreme in a good sense, I think) is the so-called &lt;a href=&quot;http://legacyrlmoore.org/reference/mahavier1.html&quot; rel=&quot;nofollow&quot;&gt;Moore method&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-10-10T13:01:01.250" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11514" PostTypeId="2" ParentId="11511" CreationDate="2016-10-10T14:10:15.127" Score="3" Body="&lt;p&gt;The courses that you mention -- real analysis and topology -- are most often senior-level courses for undergraduate mathematics majors. At this point the program is trying to give you a taste of graduate/professional work, where the overall project is one of generating proofs for as-yet unsolved open problems. It's also pretty universal that textbooks at this level contain no complete solutions at the back (more frequently some rough-sketch hints for select exercises). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Look at this as a good opportunity to decide on whether you want to pursue math further at the next level or not; this is what it's like all the time. As &lt;a href=&quot;https://en.wikipedia.org/wiki/Alfr%C3%A9d_R%C3%A9nyi&quot; rel=&quot;nofollow&quot;&gt;Alfréd Rényi&lt;/a&gt; joked, &quot;A mathematician is a device for turning coffee into theorems&quot;. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-10-10T14:10:15.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11515" PostTypeId="2" ParentId="11494" CreationDate="2016-10-10T17:52:39.797" Score="1" Body="&lt;p&gt;Perhaps look at this new, 3-minute video:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=xHtsuOB-TPw&amp;amp;index=2&amp;amp;list=PLf_ipOSbWC87nFTi6MaB4begv56yPKsOJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;SIAM Video: What is Mathematical Modeling? Part I&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are seven parts; I linked just to Part I. Aimed at college students,&#xA;but certainly comprehensible to high-school students.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/owyQK.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/owyQK.jpg&quot; alt=&quot;SIAM_MM&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-10-10T17:52:39.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11516" PostTypeId="1" CreationDate="2016-10-10T23:28:16.780" Score="10" ViewCount="164" Body="&lt;p&gt;Our department teaches two very large first-year &quot;Mathematical Methods&quot; courses (600-ish students) to Engineering students. The syllabus is dictated by their (future) needs and covers a huge array of topics, but none to any great depth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For resourcing reasons, we cannot put on a separate course for the 10-15% of students who might go on to study Maths or Physics, but we feel that a breakneck tour through a huge number of mathematical methods is not an ideal start for these students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are exploring methods of adding (optional) depth to this course to better serve this small subset of the students. So far we've tried having optional &quot;explore by yourself&quot; sections in the weekly tutorial sheets, and we've tried having out-of-class optional &quot;enrichment lectures&quot;, but neither of these has worked really well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is: given the constraints outlined above, what are possible methods of adding optional mathematical depth to an Engineering maths course to better serve future Maths students?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that this is rather vague and wishy-washy, but I'm really after anecdotal experiences from people who have tried this sort of thing.&lt;/p&gt;&#xA;" OwnerUserId="4795" LastActivityDate="2022-09-08T10:57:06.950" Title="How to incorporate optional higher level mathematical content in an Engineering Maths course?" Tags="|mathematical-pedagogy|engineering-mathematics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11517" PostTypeId="2" ParentId="11516" CreationDate="2016-10-11T00:28:00.547" Score="9" Body="&lt;p&gt;In our department, large introductory math courses, such as calculus, linear algebra, and discrete mathematics, come together with little satellite courses called &quot;advanced investigations in &lt;strong&gt;*&quot;, where *&lt;/strong&gt; is the main course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students who want to explore the subject in depth register for the main course and the satellite course. Example: in main Calculus I, students learn the definition of a continuous function, in Advanced Investigations in Calculus I they also learn the proof of Intermediate Value Theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main course is worth 3-4 credits while the satellite course gives them only 1 credit. Example: if a student gets B+ (4.0) in Calculus I (4 credits) and A- (4.5) in Advanced Investigations in Calculus I, the average grade will be &#xA;$$\frac{4}{5}\cdot 4+\frac{1}{5}\cdot 4.5=4.1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that this arrangement is not perfect because it requires to create an extra course, but it is a little course and it takes little effort to teach it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Will it work for you?&lt;/p&gt;&#xA;" OwnerUserId="7274" LastActivityDate="2016-10-11T00:28:00.547" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11518" PostTypeId="2" ParentId="11511" CreationDate="2016-10-11T01:08:00.760" Score="5" Body="&lt;p&gt;I have taught such courses and I don't think that a long list of exercises without solutions is good idea - at least it didn't work for me. It is because &#xA;1) students are not yet able to tell a correct proof from an incorrect proof. &#xA;2) often students don't know how to get started, but if you give them a hint, they will be able to solve the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What works for me is that they type their solutions into a shared online document, I write comments, and they modify their solutions according to my feedback. Then, in the end of the semester, they have solutions nicely typed, but the solutions are produced by themselves. &lt;a href=&quot;https://www.sharelatex.com/project/57fc3a8a179e38c4552fce05&quot;&gt;Here&lt;/a&gt; is a copy of the actual document that my students typed last week (the real document is shared with permission for public to edit)&lt;/p&gt;&#xA;" OwnerUserId="7274" LastActivityDate="2016-10-11T01:08:00.760" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11519" PostTypeId="1" AcceptedAnswerId="11520" CreationDate="2016-10-11T01:57:10.907" Score="8" ViewCount="601" Body="&lt;p&gt;I'm attempting to put together a list of topics for a general education course centered around &quot;great ideas in mathematics,&quot; and I'm looking for input. Ultimately, I would like topics to be approachable for those with a background only up through precalculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Things I've thought of so far include the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Numeral systems&lt;/li&gt;&#xA;&lt;li&gt;Graph theory&lt;/li&gt;&#xA;&lt;li&gt;Infinite sums&lt;/li&gt;&#xA;&lt;li&gt;Noneuclidean geometry&lt;/li&gt;&#xA;&lt;li&gt;And much more!&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Note: I'm interested in especially influential or revolutionary ideas that tend to fall on the purer side of math.  Measures of fun and/or practicality are of secondary importance to me, as I tend to be better at convincing students that something is interesting, rather than entertaining or useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something like RSA cryptography is great, because it's somewhat fun/practical while also having a really robust theoretical underpinning that allows me to discuss primes, modular arithmetic, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you all think? I'd love to hear any additional ideas.&lt;/p&gt;&#xA;" OwnerUserId="5988" LastActivityDate="2016-10-14T22:38:52.933" Title="Topics for a general education course" Tags="|undergraduate-education|course-design|" AnswerCount="7" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11520" PostTypeId="2" ParentId="11519" CreationDate="2016-10-11T02:35:07.413" Score="4" Body="&lt;p&gt;Two very different curricula which are very much along these lines are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1466554649&quot; rel=&quot;nofollow&quot;&gt;Sally and Herrmann&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.artofmathematics.org&quot; rel=&quot;nofollow&quot;&gt;The Art of Discovering Mathematics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are plenty of others which cover 2 and RSA but finding ones with 1, 3, and 4 that are available for non-majors (especially once you get beyond the &quot;highly selective&quot; colleges) is a bit more challenging.  I suspect that &lt;a href=&quot;http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP002457.html&quot; rel=&quot;nofollow&quot;&gt;The Heart of Mathematics&lt;/a&gt; will have most of what you are looking for buried in it as well.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-10-11T02:35:07.413" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11521" PostTypeId="2" ParentId="11519" CreationDate="2016-10-11T02:37:46.363" Score="5" Body="&lt;p&gt;You should probably read Eves' &lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/great-moments-in-mathematics-before-1650&quot;&gt;&lt;em&gt;Great Moments in Mathematics&lt;/em&gt;&lt;/a&gt; series? (Two volumes, about 40 lectures/moments, and an appendix of 20 more that were left out for space.)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-10-11T02:37:46.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11522" PostTypeId="2" ParentId="11519" CreationDate="2016-10-11T04:46:10.750" Score="3" Body="&lt;p&gt;A few things that are interesting and a bit real world:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;the formula for the area of an irregular polygon in terms of its vertices. For example, &lt;a href=&quot;http://www.mathopenref.com/coordpolygonarea.html&quot; rel=&quot;nofollow&quot;&gt;this website&lt;/a&gt;, but, it's a standard calculus III homework problem you'll find in many text books.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;counting via Burnside. There are some nice examples in Fraleigh's text or Rotman's basic abstract algebra text. For example, how many nonstandard dice you can design using the usual shape and dots. The possibilities are endless in terms of weird counting problems which are best understood via the group theory. I think the basic ideas can be described without getting overly technical.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;group representation in quantum mechanics. Ok, this is harder, but if you could find a toy example this could be very interesting to those students curious about the role symmetry plays in our modern physics. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;fourier analysis and the analogy to finite dimensional vectors. Maybe begin by explaining how any vector in three (or two) dimensions can be decomposed into vector components then... for music we need infinitely many components.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I must stop. There is so much you could try. I wish we had such a course, where math was approached as an art as opposed to a task.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2016-10-11T04:46:10.750" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11523" PostTypeId="1" CreationDate="2016-10-11T06:54:25.853" Score="5" ViewCount="98" Body="&lt;p&gt;I have a new student in math class (high school) since last month, and she's being diagnosed with dyslexia and dyscalculia. We are about to go into conic sections and she should be able to do some basic algebraic manipulation like $1/x + x$, but she says that she cannot do it because of her problem. I don't know how to help her and she is  not able to explicitate her difficulty... any help?&lt;/p&gt;&#xA;" OwnerUserId="6286" LastActivityDate="2016-10-11T06:54:25.853" Title="Resources for teaching algebraic fractions to a dyscalculic pupil" Tags="|secondary-education|algebra|fractions|dyscalculia|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11525" PostTypeId="1" CreationDate="2016-10-11T17:58:44.390" Score="8" ViewCount="416" Body="&lt;p&gt;I'm asking this question right now due to the fact that a lower back problem has made it very difficult for me to do much but lie down for large sections of the day when it flares up, and the fact that it is quite difficult to take notes when lying on your back. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How important would one say taking notes is to the mathematical learning process? What other ways are there of learning? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've always taken notes in the past, but the development of this back problem has made it very painful to stand or sit longer than necessary at times, and during those times much of my sitting time must go towards doing the problem sets. &lt;/p&gt;&#xA;" OwnerUserId="7278" LastEditorUserId="5220" LastEditDate="2019-09-09T21:46:26.167" LastActivityDate="2019-09-09T21:46:26.167" Title="The importance of note taking in mathematics" Tags="|undergraduate-education|lecture-notes|" AnswerCount="0" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="11528" PostTypeId="2" ParentId="11519" CreationDate="2016-10-12T00:04:29.947" Score="3" Body="&lt;p&gt;Here are some of the topics from a great ideas in mathematics class that I am teaching this semester:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Modular arithmetic: parity/UPC/ISBN codes, periodic phenomena like days of the week (E.g. 365 mod 7 = 1)&lt;/li&gt;&#xA;&lt;li&gt;Rational/irrational numbers: What their decimal expansions look like. Why we know $\sqrt{2}$ is irrational.&lt;/li&gt;&#xA;&lt;li&gt;Cardinality: Countable sets &amp;amp; the Hilbert Hotel, Uncountable sets. &lt;/li&gt;&#xA;&lt;li&gt;Tesselations and Platonic solids (plus Penrose tilings, etc)&lt;/li&gt;&#xA;&lt;li&gt;Graph theory: Bridges of Koenigsburg, Hamiltonian &amp;amp; Euler paths&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Our textbook is &quot;The Heart of Mathematics:&#xA;An invitation to effective thinking,&#xA;E. B. Burger and M. Starbird&quot; but personally I don't make a lot of use of it in my class.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2016-10-12T00:04:29.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11529" PostTypeId="2" ParentId="11519" CreationDate="2016-10-12T00:29:23.430" Score="1" Body="&lt;p&gt;I'll just add one more little cute topic that can make a 2-hour lecture: &lt;a href=&quot;https://en.wikipedia.org/wiki/Secretary_problem&quot; rel=&quot;nofollow&quot;&gt;Secretary problem&lt;/a&gt; (probability and random processes). If you're interested, I can share my old power point slides with you.&lt;/p&gt;&#xA;" OwnerUserId="7274" LastActivityDate="2016-10-12T00:29:23.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11530" PostTypeId="2" ParentId="11496" CreationDate="2016-10-12T00:55:55.097" Score="1" Body="&lt;p&gt;Maybe my &lt;a href=&quot;https://www.sharelatex.com/project/57fd854ba359963312c39617&quot; rel=&quot;nofollow&quot;&gt;LaTeX template&lt;/a&gt; will be helpful to someone. It's a relatively short LaTeX code producing 24 different versions for 4 tutorial slots where students sit by 6 at each table. It's a bit tricky to auto-generate solutions, especially for the first question, but let me know if you're interested - I'll figure it out.&lt;/p&gt;&#xA;" OwnerUserId="7274" LastActivityDate="2016-10-12T00:55:55.097" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11531" PostTypeId="2" ParentId="11519" CreationDate="2016-10-12T00:57:37.587" Score="2" Body="&lt;p&gt;As a department chair, I think this is the most difficult and expensive course to staff, unlike college algebra, elementary statistics, calculus, etc. I want the best faculty who can bring their own mathematical interests to teach this class. This course requires an instructor with exceptional content knowledge and teaching skills. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What topic should you include? Find that mathematical interest that truly interests you, and struggle to find a way to pitch it to this audience. From your question, I think you have a good start already!&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-10-12T00:57:37.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11532" PostTypeId="2" ParentId="11496" CreationDate="2016-10-12T14:14:32.837" Score="1" Body="&lt;p&gt;I used Word for most questions, and for formatting. For graphs (in Algebra I and II) I would insert a grid and then draw on the grid with Word's tools or by hand. I think LaTeX is very promising despite its relatively steep learning curve. Other images that I needed were typically drawn or cropped from other sources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I would like to persuade you to work with your local colleagues. Especially important if you are not the only math teacher in your school, but worth cultivating relationships with neighboring schools if you are on your own. You should use a format that is compatible with the people you are most likely to share materials with. Being able to share resources conveniently with other teachers will be more valuable than any particular workflow or format. Furthermore you may find that some of your colleagues may have some great ideas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Working with others will help you refine materials to create better resources as you will have more time and more sets of eyes on each piece. I like using shared resources because they tend to have a more diverse style of questioning than I would write by myself. The process of refining materials is much easier to do if you don't have to create so many of the resources you need.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may have to deal with some inefficiency if some of your colleagues are stuck in their ways, but I have found that to be minor compared to the time saved by being able to share resources. Introducing new programs has had some success, and I think it is more likely to be successful if you're willing to learn your colleagues way of doing it before you ask them to learn yours.&lt;/p&gt;&#xA;" OwnerUserId="4980" LastActivityDate="2016-10-12T14:14:32.837" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11533" PostTypeId="2" ParentId="10501" CreationDate="2016-10-13T03:41:45.297" Score="-1" Body="&lt;p&gt;I haven't taught real analysis, but I've taught plenty of proof-based courses, such as linear algebra, topology, differential geometry etc. After 10 years of teaching, I find that the simplest grading scheme is always the best. If everything is correct, then give the student the full score. If something is wrong, it should be 0. It's the only way to keep grading fair and consistent across a large class with many tutors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I imagine that some students will no be happy if the grading scheme is so strict. There are two ways to work around it by giving them a room for an error:&#xA;1. Let's say that an assignment has 6 questions, each worth 10 points and let's say that G is the sum of scores for individual questions. Then the grade for the whole assignment is min(40, G) instead of G, i.e., students who solve 4 or more problems out of 6 completely get the full score for the assignment.&#xA;2. You let them resubmit if they made a mistake. Let's say that they have one chance to resubmit if it's a handwritten paper and unlimited attempts if it is a shared online document.&lt;/p&gt;&#xA;" OwnerUserId="7274" LastActivityDate="2016-10-13T03:41:45.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11537" PostTypeId="1" AcceptedAnswerId="11541" CreationDate="2016-10-14T03:28:25.937" Score="8" ViewCount="268" Body="&lt;p&gt;I am in my third year of undergraduate math, and now that classes are becoming more proof-based, many of my homework questions are proofs of relatively basic concepts that can be found with a quick Google search.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I do my homework, I always come up with my own solution first, but then many times I will search stackexchange for other proofs and see how mine compares. Not infrequently I will come across something that's more elegant or just generally better than my proof. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am always hesitant to replace my answer with the &quot;better&quot; one, since I feel as though the work that I submit should be mine. On the other hand, the point of homework is to learn, and as long as I understand the process of coming up with the &quot;better&quot; answer, perhaps the real goal of the assignment has been fulfilled. Most of the time I try to use as little of the &quot;better&quot; answer as possible and perhaps just incorporate the main idea into my own proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm curious as to whether there is a consensus on this subject, as I'm sure I'm not the first person to encounter this dilemma.&lt;/p&gt;&#xA;" OwnerUserId="7290" LastActivityDate="2016-10-15T23:33:04.460" Title="Ethics of looking at other proofs before submitting work" Tags="|undergraduate-education|mathematical-pedagogy|proofs|general-pedagogy|homework|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11539" PostTypeId="2" ParentId="11519" CreationDate="2016-10-14T22:38:52.933" Score="2" Body="&lt;p&gt;I teach, essentially, a &quot;math course for non-math students who need to fulfill a college requirement&quot; and have found great success with topics that don't &lt;em&gt;seem&lt;/em&gt; like math to the students but certainly are math. Each semester, I teach 3ish units of self-contained material, and I have listed some of those topics below. We use the text &lt;a href=&quot;http://www.macmillanlearning.com/Catalog/product/forallpracticalpurposes-tenthedition-comap&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;For all Practical Purposes&lt;/em&gt;&lt;/a&gt;, which contains enough material to do each of the units below, although I often supplement the text with extra examples from &quot;the real world&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find this works well because the students are already apprehensive about having to take a math course. Indeed, I presume that many of them had lackluster experiences in high school, either due to poor teaching or lack of confidence/ability. So, I like these topics because (1) I cannot actually assume that any of them have even basic algebra skills or decent numeracy, and (2) I want to, overall, allay their fears and show them that there is a lot more to math than number-crunching, timed tests, and anxiety.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Graph theory in management:&lt;/strong&gt; We learn about Euler and Hamilton circuits in graphs and see how they can be used in city planning. As part of a homework assignment, I have them draft a letter to our city's Department of Public Works to recommend rethinking the way they route their snow plows. We also talk about the Traveling Salesman Problem, and I supplement the text with some information about computational complexity and &quot;the limits of human knowledge&quot;. I then touch on minimal spanning trees, showing them Kruskal's and Prim's Algorithms. (In general, graph theory is a great topic for a course like this because you can go as in depth into one topic as you'd like, depending on your students' backgrounds.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Voting systems and elections:&lt;/strong&gt; We learn about different ways to take individual rankings from voters and make a collective choice: Condorcet's Method, Borda count, Instant-runoff, Plurality, etc. We touch upon Arrow's Theorem and we work through the logic of a proof of a weaker version of it. This leads to lots of discussions, as well, about how no one in the U.S. seems to even question &lt;em&gt;how&lt;/em&gt; we vote (because everyone is fixated on &lt;em&gt;who&lt;/em&gt; to vote for). (Although that may change this year if &lt;a href=&quot;https://ballotpedia.org/Maine_Ranked_Choice_Voting_Initiative,_Question_5_(2016)&quot; rel=&quot;nofollow&quot;&gt;ballot initiative #5 passes in Maine&lt;/a&gt;!) &lt;em&gt;I believe this could fit your course, as well, because Arrow's Theorem is a truly revolutionary result in that field and it's decidedly, purely mathematical in nature.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Cryptography and coding schemes:&lt;/strong&gt; We learn about the ISBN coding scheme and other similar examples. This introduces them to modular arithmetic and, depending on the background of your students, you could go more in depth about algebraic structures (e.g. &quot;For which $n$ is $\mathbb{Z}$ modulo $n$ a group?&quot;) We also learn about using matrix multiplication to send coded messages, which can be used to introduce them to the idea of inverses in a new context (other than simple arithmetic). &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Statistics &amp;amp; data:&lt;/strong&gt; I feel like the students in this course, if it will be the &lt;em&gt;only&lt;/em&gt; course they take in college, should learn some skills about how to interpret &quot;margin of error&quot; of a poll, and to understand the difference between correlation and causation. So, we learn about that kind of stuff, normal distributions and standard deviations and the 68-95-99.7 rule, etc. Again, you can go as in depth into this as you'd like.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="80" LastActivityDate="2016-10-14T22:38:52.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11540" PostTypeId="2" ParentId="11537" CreationDate="2016-10-14T23:26:48.010" Score="3" Body="&lt;p&gt;Of course if a course involves game-playing with (conceivably artificial) rules, ask the rule-maker about the rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, many of us are aware of the existence of the internet, and its useful-though-inevitably-compromised information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to my mind, it is not reasonable to tell people that they &quot;cannot look on the internet&quot;. It's analogous to telling people they cannot go look in the library (loooong ago...). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even long ago, the fact that many &quot;exercises&quot; had become completely standard, and were often proven in textbooks, suggests to me that to try to compel novices to pretend that the &quot;model&quot; solutions do not exist is perverse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, we come to the genuinely subtle point: since all the exercises are done and known on the internet, what should be studied? Well... I'd claim that for people looking forward to genuine mathematics (as opposed to people fulfilling requirements), the point is &lt;em&gt;internalization&lt;/em&gt;. That is, not just &quot;can you walk and chew gum at the same time&quot; _in_principle_, but, ... in practice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, the low friction for (not-necessarily-high-quality) information's travel is very low... and it is (in my opinion) silly to pretend that we/students should pretend things... but/and if/when people have a timed, closed-book exam to explain various things... it is infinitely better to merely be &quot;remembering&quot;, than &quot;problem-solving/thinking&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, I do agree that it is non-trivial to convey the issue, much less convince, ... but I do think that the salient point is less simple than in the past, and some variants of the &quot;compulsion to do &quot; are indeed no more than filters for some other path. Which is demeaning both to mathematics and to the person... but, nevermind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to the literal question: it is good to think about things oneself, and it is good to take the trouble to discover that (many) other people have thought about things before, and to learn from their travails, and ... Duh.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My most pointed comment here would be that many of the traditional mythologies of mathematics and other academic versions of things implicitly deny, or disregard, the palpable fact that most thoughts that occur to us are not at all new. The &quot;external funding&quot; or other administrative strictures seem ... to compel us to declare belief in a &quot;Whig&quot; version of history, in which everything else was just a prelude to our own ... special uniqueness. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, very many ideas that occur to us have occured to the minds of many people before, especially in the last 230 years or so, as an approximation to &quot;modern times&quot;. There is that ol' mythology that seems to tell us that we must/can transcend all the things that long-ago people did. But this is simply unreasonable, since we have scant advantage... apart from the accumulated scholarship, for those who track it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, again, ask what your &quot;rules&quot; are, but/and realize that it is best to learn from the hard-won (by &quot;suffering&quot;) experiences of others, rather than repeating the same suffering.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-10-14T23:26:48.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11541" PostTypeId="2" ParentId="11537" CreationDate="2016-10-15T23:33:04.460" Score="3" Body="&lt;p&gt;You are paying for your own education through your tuition and your time invested. You have an obligation to yourself to make the most out of your time in school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is value both in working through problems alone and in studying others solutions. The former will build problem-solving and tenacity. The latter will often consolidate understanding and show optimal technique. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many instructors will often insist that solutions be entirely your work. I think this is unrealistic to take literally. You should always be willing to try a problem on your own. But in cases where you become stuck, you can often derive more value by looking up the solution than to wallow in confusion. As long as you make an honest attempt to understand the problem and your shortcomings in trying the problem, you are working in an academically honest fashion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider, if in doubt, that after reading the solution, you leave it sit for a few days. Then, write it out as best you understand it at a later date. If you can reproduce the argument from scratch, then you have made inroads towards learning the material.&lt;/p&gt;&#xA;" OwnerUserId="1018" LastActivityDate="2016-10-15T23:33:04.460" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11542" PostTypeId="1" CreationDate="2016-10-17T08:30:18.053" Score="3" ViewCount="760" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;An algebra book requires a different type of reading than a novel or a short story. Every sentence in a math book is full of information and logically linked to the surrounding sentences. You should read the sentences carefully and think about their meaning. As you read, remember that algebra builds upon itself; for example, the method of multiplying binomials that you'll study on page 200 will be useful to you on page 544. Be sure to read with a pencil and paper: Do calculations, draw sketches, and take notes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the first paragraph from the section entitled Reading Your Algebra Book in the American high school textbook Algebra, Structure and Method Book 1 and also Algebra and Trigonometry, Structure and Method Book 2 both written by Brown, Dolciani, Sorgenfrey et. al. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question concerns the relationship between the sentences &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Every sentence in a math book is full of information and logically linked to the surrounding sentences. You should read the sentences carefully and think about their meaning. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My interpretation is that both careful reading and thought about the meaning  should be applied to the information contained in each sentence and the logical links between sentences. Is this interpretation reasonable for a self-learning student?&lt;/p&gt;&#xA;" OwnerUserId="7305" LastEditorUserId="-1" LastEditDate="2016-10-18T08:27:52.510" LastActivityDate="2018-03-25T00:28:30.500" Title="Reading Your Algebra Book" Tags="|secondary-education|textbooks|self-learning|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11543" PostTypeId="1" AcceptedAnswerId="11546" CreationDate="2016-10-17T08:50:37.803" Score="17" ViewCount="1321" Body="&lt;p&gt;A friend of mine just showed me this article about the &quot;how old is the shepherd&quot; problem: &lt;a href=&quot;https://mystudentvoices.com/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff#.d2ko4q2lo&quot;&gt;Link&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I'm shocked by the finding that 75 percent of kids give an answer other than &quot;there isn't enough information.&quot;  But I wonder whether the fustiness of the example may have had something to do with it:  shepherds and flocks of sheep and dogs aren't exactly commonplace for most American schoolchildren. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question: Does anyone know of a similar study that uses terms and concepts that might be more familiar to schoolchildren.  Something like &quot;There are 14 girls and 9 boys in a class. How old is the teacher?&quot; (Maybe that one is even a little behind the times, too.)&lt;/p&gt;&#xA;" OwnerUserId="5969" LastEditorUserId="262" LastEditDate="2016-10-18T00:49:47.927" LastActivityDate="2018-02-13T01:48:59.323" Title="Does the &quot;how old is the shepherd&quot; phenomenon occur for more relatable word problems?" Tags="|reference-request|primary-education|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11544" PostTypeId="2" ParentId="11543" CreationDate="2016-10-17T14:56:43.687" Score="5" Body="&lt;p&gt;Briefly, I'd say, &quot;yes&quot; to your question.  Even for more relatable questions, the students still draw the same sort of invalid answer.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find it a stretch that students can't &lt;em&gt;understand&lt;/em&gt; the shepherd problem.  If they can't &lt;em&gt;relate&lt;/em&gt; to it, then we've done a very poor job of educating them about the world, history, and a lot of other things which have very little to do with mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, I don't fully agree with the analysis of the linked article either.  Ask the students a completely different question, &quot;There are 125 sheep and 5 dogs in a flock. What is the shepherd's name?&quot;  Whether or not 75% will give you an answer, you will still see a disappointingly large percentage of students providing you an answer.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student, &quot;The shepherd's name is John.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teacher, &quot;How did you come to that conclusion?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student, &quot;Well, it could be.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's just ignore that the student didn't answer the teacher's question.  I've encountered this scenario numerous times in the classroom where a question doesn't really have an answer, so the students make something up, and, because the answer has some kind of plausibility to it, the students think they're done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way I would work with these sorts of problems is to pepper them throughout the course.  But also, I would focus on the reasoning skills the students need to solve the problem.  So, you can have a discussion of what's sorts of information you need to determine someone's age.  This can lead to all sorts of interesting answers.  For example, many students will offer, &quot;their birthday.&quot;  The students can relatively quickly figure out that knowing a birthday is insufficient; eventually they'll come up with, &quot;their birth date&quot; or something equivalent.  And then the teacher can ask, &quot;do you need to know their birth date to determine their age?&quot;  This becomes more challenging for them because the previous answer has locked many of their minds into one way of thinking about the problem.  Yet, with some patience and good give-and-take between the teacher and the students, they should be able to see that you can answer such questions as, &quot;My sister was born when my mom was 30 years old.  My mom is now 45.  How old is my sister?&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am personally fond of questions like  &quot;Jackie's brother is five years older than Paul.  Paul was born in 2002.  How old is Jackie's brother?&quot;  This isn't quite like the shepherd problem, but it's similar in that there isn't quite enough information to know with 100% certainty.  However, there are two plausible answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So back to the teacher's question, &quot;how did you come to that conclusion?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students need to be taught how to explain how they're getting their answers.  In an English class, saying something like, &quot;It could be John&quot;, might be perfectly acceptable (I'm not an English teacher so I won't weigh in too strongly about the validity of that answer).  But, in mathematics and often in the sciences, a plausible answer is usually not what we're looking for.  We're usually looking for reasoning that starts for a collection of &lt;em&gt;facts&lt;/em&gt; and draws various conclusions based upon those facts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, it might be plausible that the shepherd is 25 years old.  It might just happen that way.  Great.  But is this supported by the facts?  I would approach this as outlined above:  by stepping back and having the students review what sort of reasoning skills they need to use and how fact relate to the sorts of answers they're trying to come up with.  And then once you've perhaps convinced them that we can't answer a question about the shepherd's age, ask the students, &quot;what sorts of questions could we ask about this shepherd from the facts presented?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, the students should be able to answer a question like, &quot;There are 125 &lt;em&gt;gabbalahs&lt;/em&gt; and 5 &lt;em&gt;tumps&lt;/em&gt;.  What is Margaret's age?&quot;  If the students give up and say, &quot;I don't know.  What's a &lt;em&gt;gabbalah&lt;/em&gt;?  I don't know what a &lt;em&gt;tump&lt;/em&gt; is&quot;, then the students are again missing the point of the question.  Giving up isn't the same thing as saying the question as presented isn't answerable.  In fact, what I'd most want to hear from the students is, &quot;Do &lt;em&gt;gabbalah&lt;/em&gt; and &lt;em&gt;tumps&lt;/em&gt; have anything to do with figuring out someone's age?&quot;  That's a great question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The younger the student, generally, the more a concrete thinker the student is.  It's very challenging for them to think about something abstractly (such as in the &lt;em&gt;gabbalah/tump&lt;/em&gt; question).  But also, I would say that it's not a failure so much of mathematics education that makes the students think they need to come up with an answer.  It's about how we teach the students over all.  Our whole system of education is biased toward asking questions that the students can answer.  (It's not too hard to see why we do that.)  And, this bias in the pedagogy biases the students in how they approach problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I may disagree that the shepherd question is &lt;em&gt;fusty&lt;/em&gt;,  even if it is, that should not be preventing the students from answering the question.  In our pedagogical aim to make problem &lt;em&gt;relevant&lt;/em&gt; to students, we lose sight that we should also be expanding their horizons, broadening their perspectives, and showing and teaching them ways of grappling with the unfamiliar.  (Too often we're encouraging them to stay within their myopic world views by only presenting what is already known and familiar.)  Speaking as a mathematician or a scientist, there is very little of interest done in mathematics (or the sciences) which is commonplace; what is interesting in mathematics and the sciences are those places where we lack knowledge and understanding---where our comfort zone is being challenged.&lt;/p&gt;&#xA;" OwnerUserId="3491" LastEditorUserId="3491" LastEditDate="2016-10-17T15:52:29.523" LastActivityDate="2016-10-17T15:52:29.523" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11546" PostTypeId="2" ParentId="11543" CreationDate="2016-10-17T23:11:05.383" Score="15" Body="&lt;p&gt;&lt;strong&gt;Note (Feb 2018):&lt;/strong&gt; There is an alleged &quot;Chinese math problem&quot; (see, e.g., &lt;a href=&quot;https://www.washingtonpost.com/news/worldviews/wp/2018/01/31/this-chinese-math-problem-has-no-answer-actually-it-has-a-lot-of-them/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WaPo article&lt;/a&gt;) going around about the second example problem below (cited to Reusser 1988, but can be found in Reusser 1986, as I've tweeted &lt;a href=&quot;https://twitter.com/benjamindickman/status/960309120363651072&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). Interesting that it has gone viral without anyone having sourced it.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This study can be easily replicated, and has been: with multiple scenarios, and in multiple languages. One keyword to use in the context of the mathematics education literature is &lt;strong&gt;sense making&lt;/strong&gt;, and one author to whom you might look is &lt;strong&gt;Alan Schoenfeld&lt;/strong&gt;; for example, see &quot;&lt;a href=&quot;https://www.researchgate.net/profile/Alan_Schoenfeld2/publication/265777113_A_modest_proposal/links/54bd6ff20cf27c8f2814b882.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A Modest Proposal&lt;/strong&gt;&lt;/a&gt;&quot; from the &lt;em&gt;Notices of the AMS&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The linked paper provides several examples, including the shepherd one (I believe the original citation for that phrasing is Barsuk (1989); or, at least, that is what I asserted in a &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5877/should-word-problems-be-reasonable#comment10933_5877&quot;&gt;&lt;strong&gt;comment&lt;/strong&gt;&lt;/a&gt; two years ago). You may as well click through to read the write-up, but here is an excerpt to keep this self-contained:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2TiuW.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2TiuW.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2018-02-05T22:50:19.927" LastActivityDate="2018-02-05T22:50:19.927" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11547" PostTypeId="2" ParentId="2429" CreationDate="2016-10-18T18:38:13.860" Score="8" Body="&lt;p&gt;Judson's &lt;a href=&quot;http://abstract.ups.edu/&quot; rel=&quot;noreferrer&quot;&gt;Abstract Algebra: Theory and Applications&lt;/a&gt; is different in that it is an open source textbook that is available at no cost. I haven't used it (yet), but I think it's worth pointing out for the aforementioned reasons. In addition to PDF and source versions, there's a &lt;a href=&quot;http://abstract.ups.edu/aata&quot; rel=&quot;noreferrer&quot;&gt;web version&lt;/a&gt; that has proofs collapsed by default (handy for high-level reading and for students who want to try proving the theorems themselves first) and live &lt;a href=&quot;http://www.sagemath.org/&quot; rel=&quot;noreferrer&quot;&gt;SageMath&lt;/a&gt; cells.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also the &lt;a href=&quot;http://www.maa.org/press/maa-reviews/abstract-algebra-theory-and-applications&quot; rel=&quot;noreferrer&quot;&gt;MAA review&lt;/a&gt; by Christopher Thron.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2016-10-19T07:16:17.353" LastActivityDate="2016-10-19T07:16:17.353" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11548" PostTypeId="1" CreationDate="2016-10-18T21:37:55.213" Score="9" ViewCount="471" Body="&lt;p&gt;I'm seeking the textbooks that were released in the 1960's call Unified Modern Mathematics. I'm aware 3 parts of the course exist online but I would like to use them in hard copy form. I find these books highly intuitive if used right. Also, Archive.org has &lt;a href=&quot;https://archive.org/details/unifiedmodernmat00seco&quot; rel=&quot;nofollow noreferrer&quot;&gt;some courses&lt;/a&gt; but not all 6. Wikipedia mentions that the publisher Addison Wesley released hard copies. If anyone has any information on how to obtain these books, it would be much appreciated.&lt;/p&gt;&#xA;" OwnerUserId="7309" LastEditorDisplayName="user507" LastEditDate="2019-04-06T16:24:46.013" LastActivityDate="2022-07-19T12:14:25.333" Title="Obtaining printed copies of the textbook series Unified Modern Mathematics" Tags="|mathematical-pedagogy|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11550" PostTypeId="1" CreationDate="2016-10-20T10:04:46.323" Score="5" ViewCount="3219" Body="&lt;p&gt;I need a convincing math problem or real-world example, to show that simplifying  algebraic expressions such as $2x+7y-3+6y-9x-2$ is really important and profitable, and it's reasonable to learn how to do it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know any such problem or example?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt; The problem or example should be comprehensible for a student which has just got to know the algebraic expressions of degree one; no familiarity with graphing equations or using algebra for solving challenging problems.&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="1229" LastEditDate="2016-10-20T13:26:59.690" LastActivityDate="2016-10-23T18:03:30.360" Title="Importance of Simplifying Algebraic Expressions" Tags="|algebra|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11551" PostTypeId="2" ParentId="11550" CreationDate="2016-10-21T03:08:40.083" Score="2" Body="&lt;p&gt;Your restrictions are quite limiting.  But never the less, for a linear expression, you can definitely do something.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Approach A&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have had my student write stories for an expression.  So something in like&#xA;$$&#xA;3x + 4y - 5&#xA;$$&#xA;You could say that $x$ represents the number of oranges and $y$ the number of apples.  The students could write something like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Joanne sold $x$ oranges for \$3.00 each, $y$ apples for \$4.00 each, and \$5.00 were stolen from the till&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and for a different expression&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;  6x + 7y -3y +18 -2x -23 &#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;the students could come up with a story like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jim sold $x$ apples for \$6.00 each, $y$ oranges for \$7.00 each, bought another $y$ oranges for  \$3.00 each, found \$18.00 lying on the ground, bought $x$ apples for \$2.00 each, and spent \$23.00 for a new kitchen table &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So maybe they're hoakie.  But now we can ask, &quot;Who has more money? Jim or Joanne?  Explain your answer.&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are all sorts of variations on this.  What I'd stress here isn't an application of simplifying algebraic expressions, I'd stress that expressions can be tied to stories.  The more students play with and create these sorts of stories, the less intimidating word problems are for them.  But it does take a lot of work.  And, it takes a lot of convincing to get them to realize what is and isn't a good story.  At first, many of their stories will be nonsensical (I'm assuming you're teaching high school or junior high school students).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;OK.  So there's an application---contrived as it is---that you could use in your class.  But, if you allow graphing and the students have spent some time working with quadratic equations, then you have a lot more room to show the importance of simplifying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Approach B&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here the context is to talk about scientific exploration/inquiry.  If you develop a theory, you might be inclined to express the relationship between two quantities in one form, but after collecting data, it might be more sensical to present a relation between the two quantities differently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can present a situation where two different people have come up with two different expressions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kate came up with the following equation for one set of data&#xA;$$&#xA;  y = (x+2)(x-4)&#xA;$$&#xA;Ken had a different set of data and wrote out the following equation&#xA;$$&#xA;  y = (x-1)^2 - 9&#xA;$$&#xA;Now provided you've talked about graphing, zeroes of polynomials, and the vertex of a parabola, then you can talk about a context where simplifying can answer questions.  You can ask, &quot;Could the data Ken and Kate collected come from similar experiments?&quot;  Just looking at the data wouldn't suffice since the data sets are different.  But, if the students know how to simplify the expressions, then they &#xA;can answer the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Granted you might feel you want a clean example when first presenting the notion of gathering like-terms and simplying expressions.  But sometimes, you just don't have enough to work with yet to make the point.  It's important to teach your students that there isn't always an immediate answer to &lt;em&gt;why&lt;/em&gt; we do something.  But then promise them that you'll revisit the idea in a few weeks when they've developed some more skills and a few more ideas.  If you're good to your promises, in my experience, the students will often play along.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In all of this, long range planning is important.  You need to know what skills and knowledge base you're starting with in your students; how you plan to build on that; and, where you intend to get.  Sometimes, to create interesting and convincing &quot;real world&quot; applications, you have to be creative in how you present the material in the first place.&lt;/p&gt;&#xA;" OwnerUserId="3491" LastEditorUserId="3491" LastEditDate="2016-10-21T03:13:47.570" LastActivityDate="2016-10-21T03:13:47.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11552" PostTypeId="2" ParentId="11550" CreationDate="2016-10-21T06:54:57.137" Score="4" Body="&lt;p&gt;The question seems to overlook the value of &lt;strong&gt;brevity&lt;/strong&gt; in our writing, mathematical or otherwise. Shorter is better -- it makes more efficient use of our writing and time, makes things easier to read and understand, and also reduces possible errors (due to fewer parts to possibly mis-read or transcribe). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That might bear saying in a class once and should be a fairly obvious point. If one were in an English class and had overly verbose, rambling prose that needed editing down (while keeping the meaning the same), hopefully one would likewise see the value in that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that finding a &quot;real-world example&quot; is unlikely, because this is something done in the pure algebraic expression alone. It does not reflect a change in any outside model or application. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-10-21T06:54:57.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11553" PostTypeId="2" ParentId="11550" CreationDate="2016-10-21T12:33:39.043" Score="5" Body="&lt;p&gt;IS there a number $x$ such that $2x-7−3x+6−9x−2+10x=0$ ? Is there a number $x$ such that $2x-7−3x+6−9x−2+11x=0$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can encapsulate these question in word problems easily. For example: &quot;Alicia has received and made gifts in dollar and bitcoins: she got two bitcoins, then gave 7 dollars, then offered 3 bitcoins, then received 6 dollars, gave 9 bitcoins, gave 2 dollars, and last received 10 bitcoins. Brahimi says the bitcoin rate is unsufficient for her to break even. Is he right? At what bitcoin rate would she break even?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point of the particular two examples is that although they look very similar, their answers are very different. Once simplified, one sees immediately why.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="1147" LastEditDate="2016-10-23T18:03:30.360" LastActivityDate="2016-10-23T18:03:30.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11554" PostTypeId="1" CreationDate="2016-10-21T16:56:57.300" Score="0" ViewCount="457" Body="&lt;p&gt;Suppose I want to prove $\sqrt{7}$ is not a rational number. I suppose it is and it brings me to a contradiction. Here how it goes line be line: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First line. $\sqrt{7}=\frac{m}{n}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second line (after squaring both sides of the equality above). $7n^2=m^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third line. A solution for this equality contradicts the uniqueness part of the fundamental theorem of arithmetics, thus it has no solution.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used the argument above in my number theory class today. It was hard for one of the students to accept it for an interesting reason I had never seen before. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;No solution for $7n^2=m^2$ means $7n^2$ is not a square number, but no solution for $\sqrt{7}=\frac{m}{n}$ means $\sqrt{7}$ is not a rational number. These two statements are not the same, are they? Asked the student!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The lecture that happened to be me had not an answer, or at least, at that moment, and even now, could not find a similar chain of algebraically equivalent statements that each one brings a different meaning to mind, while it is &quot;easy&quot; to see those meanings refer to the same thing.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess my question is do you know an example like the one that I tried to summarize in the title and explain in the body of the post. Or, should I ask, if you were the lecturer what was your answer to the student?      &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2016-10-21T20:36:28.450" Title="When two equivalent algebraic statements have two &quot;different&quot; meanings" Tags="|algebra|proofs|number-theory|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11555" PostTypeId="2" ParentId="11554" CreationDate="2016-10-21T17:29:17.290" Score="3" Body="&lt;p&gt;There may be an issue with quantifiers here.  &quot;No solution for $7n^2=m^2$&quot; does &lt;em&gt;not&lt;/em&gt; mean &quot;$7n^2$ is not a square number&quot;.  It means, perhaps, &quot;there is no $n$ so that $7n^2$ is a square number&quot;.  But it's important to remember that &quot;no solution&quot; means &quot;no simultaneous choice of n and m&quot;.  The phrasing &quot;$7n^2$ is not a square number&quot; suggests that, by separating the variables, the student has gotten confused about the variables, and is now treating n as a constant and is looking for solutions in m.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-10-21T17:29:17.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11556" PostTypeId="2" ParentId="11554" CreationDate="2016-10-21T17:43:58.980" Score="9" Body="&lt;p&gt;The two statements aren't literally &quot;the same&quot;, because as the student observed, they say different things. However, they are &lt;em&gt;logically equivalent&lt;/em&gt;: each implies the other. (Similarly, &quot;4/2&quot; isn't literally the same expression as &quot;5 - 3&quot;, but they provably represent the same number.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we could only prove things by repeating the same statement, we couldn't get anywhere. The whole point is that different statements can &lt;em&gt;imply&lt;/em&gt; each other, so knowing one statement to be true lets us logically deduce the truth of other statements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From another perspective, we could say that the meaning of a statement is &lt;em&gt;determined&lt;/em&gt; by its implications, in which case logically equivalent statements &lt;em&gt;do&lt;/em&gt; mean the same thing — when we look more closely at what &quot;$\sqrt{7}$&quot; and &quot;rational number&quot; mean, we see that &quot;$\sqrt{7}$ is a rational number&quot; really does mean the same thing as &quot;there exist integers $m$ and $n$ with $n \neq 0$ such that $7n^2 = m^2$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student has trouble accepting this, it might be because they aren't in the habit of thinking about what words or expressions mean in such a precise way. They might be taking &quot;square root&quot; and &quot;rational number&quot; as fuzzy notions whose general sense or usage is commonly understood — the way words are used in everyday, nontechnical language — rather than as notions &lt;em&gt;specified by&lt;/em&gt; mathematically precise definitions. This misconception about the nature of mathematical definitions is documented in the literature, even among math majors: see, for example, [Edwards–Ward 2004].&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;B. Edwards and M. Ward, &quot;Surprises from Mathematics Education Research: Student (Mis)use of Mathematical Definitions&quot;. &lt;em&gt;American Mathematical Monthly&lt;/em&gt;, vol. 111, no. 5 (May 2004), pp. 411–424. DOI: 10.2307/4145268 [&lt;a href=&quot;https://ir.library.oregonstate.edu/xmlui/bitstream/handle/1957/17497/Edwards.SurprisesMathEdResearch.pdf?sequence=1&quot;&gt;PDF&lt;/a&gt;] [&lt;a href=&quot;https://www.jstor.org/stable/4145268&quot;&gt;JSTOR&lt;/a&gt;]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3505" LastEditorUserId="3505" LastEditDate="2016-10-21T20:36:28.450" LastActivityDate="2016-10-21T20:36:28.450" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11557" PostTypeId="2" ParentId="11554" CreationDate="2016-10-21T17:55:04.273" Score="2" Body="&lt;p&gt;The two statements are equivalent, assuming care with quantifiers. In the original form $\frac{n}{m}=\sqrt{7}$ we have an equation over the real numbers. We trade this for the second form $7m^2=n^2$ which is an equation over the integers. In this second form, we can make use of integer factorization to attack the problem. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-10-21T17:55:04.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11558" PostTypeId="2" ParentId="11550" CreationDate="2016-10-22T01:43:40.647" Score="0" Body="&lt;p&gt;How about counting proteins, fats, and carbs? For example, 100 g of bread contain&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;total fat - 3 g, including saturated fat - 1 g &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;total carbs - 51 g, including fiber - 2 g, sugars - 4 g &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;proteins - 8 g&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The nutritional value of 1 g of fat is 9 kcal, 1 g of protein - 4 kcal, 1 g of carbs but not dietary fiber - 4 kcal, dietary fiber - 0.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose that $x$ is the weight of one piece of bread in grams. You eat two pieces of bread for breakfast, three pieces of bread for lunch, and one as an afternoon snack. What's the amount of fats, sugars, non-fiber carbs and the total nutritional value of all the bread you eat in the whole day?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later, when you introduce systems of linear equations, you can pose questions like planning daily menu out of, say, bread, eggs and salmon so that the total calorie intake is 2000 with 50% coming from carbs, 30% coming from protein and 20% coming from fats.&lt;/p&gt;&#xA;" OwnerUserId="7274" LastActivityDate="2016-10-22T01:43:40.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11559" PostTypeId="2" ParentId="11330" CreationDate="2016-10-22T13:17:31.467" Score="1" Body="&lt;p&gt;Returns on a stock portfolio. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.moneychimp.com/features/market_cagr.htm&quot; rel=&quot;nofollow&quot;&gt;Moneychimp&lt;/a&gt; will give you data, the S&amp;amp;P 500 return for each year since 1871. The classic exercise is to suggest we start with some amount of money, say $1M, at retirement time. We assume a targeted rate of withdrawal each year, and run the simulation, i.e. randomly choose the return for each year, generally based on the market average 9% and standard deviation of 18.7%, and see if you outlive the portfolio. Then, of course you run it a few thousand times. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I hear the phrase &quot;Monte Carlo simulation&quot; this is exactly what comes to mind. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-10-22T13:17:31.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11560" PostTypeId="1" AcceptedAnswerId="11571" CreationDate="2016-10-22T16:22:23.770" Score="0" ViewCount="514" Body="&lt;p&gt;When we study double integral many Calculus textbooks state that for a region $R$ in the plane  $$\iint_R1\ dA= \text{area bounded by }R   $$&#xA;But double integral actually give the volume of a solid. This happens to be numerically equivalent. That is we get an answer like &lt;em&gt;v cubic metres&lt;/em&gt; which we say is &lt;em&gt;v square metres&lt;/em&gt;. How do physicists view this? Is not this ugly mathematics?&lt;/p&gt;&#xA;" OwnerUserId="548" LastEditorUserId="6053" LastEditDate="2018-05-07T00:32:05.783" LastActivityDate="2018-05-07T00:32:05.783" Title="Double Integral: Area or Volume?" Tags="|calculus|integration|" AnswerCount="4" CommentCount="5" ClosedDate="2018-05-08T12:49:55.307" ContentLicense="CC BY-SA 3.0" />
  <row Id="11561" PostTypeId="2" ParentId="11560" CreationDate="2016-10-22T16:43:15.810" Score="7" Body="&lt;p&gt;You are integrating a function $z=f(x,y)$ but the units of $z$ do not have to be for length. The units could be for mass density, in which case the units of the double integral would be for mass. Or $z$ could be unitless. In this case, the units of the double integral would be for area.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-10-22T16:43:15.810" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11562" PostTypeId="2" ParentId="11560" CreationDate="2016-10-23T17:52:48.933" Score="0" Body="&lt;p&gt;As copied from the comments on an English Language Learners &lt;a href=&quot;https://ell.stackexchange.com/questions/42963#42967&quot;&gt;post&lt;/a&gt; about how mathematical expressions are read aloud:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was taught to use the terms &quot;integral&quot; and &quot;integration&quot; for the various integral signs (such as &quot;∫&quot;, &quot;∬&quot;, &quot;∭&quot;, &quot;∮&quot;, &quot;∯&quot;, and &quot;∰&quot;). Similarly, I was taught to use the d notation for full derivatives with respect to a variable, and the partial notation for partial derivatives with respect to a variable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I often say &quot;double integral&quot; when performing either an integral of an integral (such as ∬ a dx dy ) or an area integral (such as ∫ a dA ) or a surface integral (such as ∯ a dA ). Similarly, I often say &quot;triple integral&quot; or &quot;volume integral&quot; when performing either an integral of an integral of an integral (such as ∭ a dx dy dz ) or a volume integral (such as ∫ a dV ).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was taught to use the term &quot;sum&quot; for discrete sums, such as when the (capital) &quot;sigma&quot; notation ( &quot;&quot; ) is used. I use the term &quot;series&quot; to refer to the sequence generated by evaluating the sum for the various values of n. For example, &quot;1, 2, 3, 4, …&quot; is the sequence of integers. &quot;1, 1/2, 1/4, 1/8, …&quot; is a corresponding geometric sequence. &quot;1, 3/2, 7/4, 15/8, …&quot; is the corresponding series, where each item in the series is the sum of the first i elements of the geometric sequence.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="-1" LastEditDate="2017-04-13T12:55:17.673" LastActivityDate="2016-10-23T17:52:48.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11563" PostTypeId="2" ParentId="11435" CreationDate="2016-10-23T19:27:46.787" Score="-1" Body="&lt;p&gt;A &lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_by_contradiction&quot; rel=&quot;nofollow&quot;&gt;proof by contradiction&lt;/a&gt; starts by assuming the opposite (complement) of your main assumption, and deriving a contradiction with your hypothesis. Since &quot;not A&quot; (the opposite of your assumption) produces an absurd result, by implication, your main assumption must be true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It requires a good understanding of the ideas of &quot;complementarity,&quot; contrapositve and &quot;converse,&quot; which many students don't have, because they haven't studied enough set theory and logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's easy enough to understand for someone who has a good grounding in say, truth tables.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2016-10-23T19:32:58.970" LastActivityDate="2016-10-23T19:32:58.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11565" PostTypeId="2" ParentId="10348" CreationDate="2016-10-24T05:46:03.643" Score="10" Body="&lt;p&gt;In Russian, the Squeeze Theorem (a.k.a. The Pinching Theorem) is called &quot;Теорема о двух милиционерах&quot; &amp;mdash; &quot;Two Policemen Theorem&quot;. The idea is that if two policemen are holding a criminal between them, the bad guy is going to the same place, probably jail or precinct, where the policemen are going. &lt;/p&gt;&#xA;" OwnerUserId="6762" LastActivityDate="2016-10-24T05:46:03.643" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11566" PostTypeId="1" AcceptedAnswerId="11568" CreationDate="2016-10-24T06:31:11.140" Score="29" ViewCount="7475" Body="&lt;p&gt;I'd like to raise the issue of constant misuse of parentheses in the U.S., and I'm wondering if anybody else shares the same feelings, has had the same issues, knows any history behind it, and can offer some thoughts on the subject. Let me explain what I'm talking about.&lt;/p&gt;&#xA;&lt;p&gt;In elementary algebraic notation, i.e. when writing algebraic expressions, parentheses mean two things: order of operations and function notation; &lt;a href=&quot;http://mathworld.wolfram.com/Parenthesis.html&quot; rel=&quot;noreferrer&quot;&gt;http://mathworld.wolfram.com/Parenthesis.html&lt;/a&gt; seems to agree with me. (There are all the other things, such as interval notation, matrices, etc, but I'm not discussing those here.) But here in the U.S. it is used all the time to enclose each factor in multiplication, so &lt;span class=&quot;math-container&quot;&gt;$5\cdot2$&lt;/span&gt; would be written as &lt;span class=&quot;math-container&quot;&gt;$(5)(2)$&lt;/span&gt;. And while it's technically speaking not wrong, as it still means the same thing, it's redundant and pedagogically terribly harmful, imho. This way of writing multiplication is so persistent from early on and then everywhere, that students grow up believing that that's what parentheses mean: that their purpose is to enclose each individual quantity in multiplication. Not for grouping, not for function notation, but mostly just for that. Almost any of my students here (and I've tried asking a whole class, so it's kinda experimentally confirmed) will write the product of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x+1$&lt;/span&gt; as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$(2)x+1$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In my teaching I always try to correct my students' bad writing habits. Here's a recent example. Solving an example on the board during my &amp;quot;College Trigonometry&amp;quot; class, I actually came to the need of setting up a product of something like (don't remember exactly) &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin x+1$&lt;/span&gt;, which I intentionally wrote as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$2\cdot\sin x+1$&lt;/span&gt;&amp;quot;. I turned to my class and said: &amp;quot;By the way, this is wrong. Can you correct what I wrote?&amp;quot; One of the best students in the class immediately said: &amp;quot;It misses parentheses.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Halleluja! My efforts haven't been futile,&amp;quot; I thought to myself... until he continued: &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; must be in parentheses because it's multiplied.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It would have been just a little notational nuisance, had it not been a cause of so many real mistakes. The most obvious one is when people don't distribute multiplication, since there's nothing indicating the need to distribute. I mean, as a quick example, when instead of &lt;span class=&quot;math-container&quot;&gt;$2(x+1)$&lt;/span&gt; someone wrote &lt;span class=&quot;math-container&quot;&gt;$(2)x+1$&lt;/span&gt;, and then they need to plug e.g. &lt;span class=&quot;math-container&quot;&gt;$x=3$&lt;/span&gt;, they'll get &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;. Another one, that I believe stems from the same misuse of parentheses, is misreading function notation. Again, I've done this literally with thousands of my students, and most read &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;&amp;quot; as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; or &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt;&amp;quot; as &amp;quot;sine times &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I only have the experience of working in two countries. In Russia, my homecountry, nobody ever writes &lt;span class=&quot;math-container&quot;&gt;$(2)(5)$&lt;/span&gt;. (And mind you, its educational system has a truckload of problems of its own, but at least this is not one of them.) In the U.S., it's not just about the students, but it's all over the books, textbooks, and in almost everybody's math writing, including teachers and professors — which is probably why all students pick up this habit. I honestly don't know about the rest of the world.&lt;/p&gt;&#xA;&lt;p&gt;So, back to the questions that I asked in the beginning. Can anybody explain where this writing tradition comes from? Does anyone agree that it causes some real problems in students' learning of math?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt; Let me clarify my concerns a little bit. As others pointed out below, &lt;span class=&quot;math-container&quot;&gt;$(2)(5)$&lt;/span&gt; is weird but not wrong. And I agree: it's not wrong mathematically, but it is wrong pedagogically. And although it still irritates me (to put it mildly), that's not the problem in itself. But &lt;span class=&quot;math-container&quot;&gt;$(2)x+1$&lt;/span&gt; is actually wrong when it stands instead of &lt;span class=&quot;math-container&quot;&gt;$2(x+1)$&lt;/span&gt;. I apologize for not being clear enough, but here's what I meant: I believe that seeing &lt;span class=&quot;math-container&quot;&gt;$(2)(5)$&lt;/span&gt; results in students' writing &lt;span class=&quot;math-container&quot;&gt;$(2)x+1$&lt;/span&gt;, and that's wrong, and that's the problem.&lt;/p&gt;&#xA;" OwnerUserId="6762" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-10-29T00:38:10.413" Title="Misuse of parentheses for multiplication" Tags="|students-mistakes|notation|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11567" PostTypeId="2" ParentId="11566" CreationDate="2016-10-24T12:52:48.430" Score="18" Body="&lt;p&gt;I disagree that it is &quot;terribly harmful&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do not prevent them from writing $(2)(5)$.  Instead prevent them from writing things that are actually wrong.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thinking that $\sin x$ is $\sin$ times $x$ can happen (and does happen) even without parentheses.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree $(2)(5)$ looks strange, but if they can write $(4-2)(4+1)$, what is wrong with $(2)(5)$?&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2016-10-24T12:52:48.430" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="11568" PostTypeId="2" ParentId="11566" CreationDate="2016-10-24T13:41:37.413" Score="22" Body="&lt;p&gt;To answer the ultimate question (&quot;Can anybody explain where this writing tradition comes from?&quot;): &lt;strong&gt;It's explicitly taught that way by many U.S. instructors and textbooks.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples: From the otherwise excellent Martin-Gay &lt;em&gt;Prealgebra &amp;amp; Introductory Algebra&lt;/em&gt; (sec 1.5):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The × is called a multiplication sign... The symbols ∙ and () can also&#xA;  be used to indicate multiplication.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Shaum's Outlines &lt;em&gt;Elementary Algebra&lt;/em&gt; (sec 1-4):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The symbols for the fundamental operations are as follows...&#xA;  Multiplication: ×, (), ∙, no sign&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A research presentation for the NCTM actually took &lt;em&gt;not&lt;/em&gt; thinking that parentheses are multiplying as a sign of student misunderstanding (Welder, &quot;Using Common Student Misconceptions in Algebra to Improve Algebra Preparation&quot;): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Misconceptions: Bracket Usage -- Beginning algebra students tend to be&#xA;  unaware that brackets can be used to symbolize the grouping of two&#xA;  terms (in an additive situation) and as a multiplicative operator&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I could go on (and probably more U.S. teachers teach this even more adamantly; I've never met anyone who opined otherwise). To answer the other question (&quot;Does anyone agree that it causes some real problems in students' learning of math?&quot;), then I would say: &lt;strong&gt;Yes, absolutely.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most common mistake I see in this vein is that if one is taught both that &quot;parentheses mean multiplying&quot; and in the order of operations that &quot;parentheses come first&quot;, then that combination results in many students thinking that juxtaposed multiplying is meant to happen first, e.g.: They will simplify $2(3)^2$ as $6^2 = 36$ and so forth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Others will think that parentheses are entirely interchangeable with the dot notation and juxtaposition, e.g: when substituting $x = -2$ in the expression $3x^2$, they're prone to writing things like $3 \cdot -2^2$ or $3 - 2^2$, which they may or may not recover from on the next line. Even just writing $3 \cdot -2$ is nonstandard, of course, but if corrected they'll grouse about it as being easier than using parentheses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As noted in a comment, I've written about this on my blog in the past on &lt;a href=&quot;http://www.madmath.com/2013/10/are-parentheses-multiplication.html&quot;&gt;MadMath&lt;/a&gt;. I think it would be tremendous if we could flip this around and start teaching that it is &lt;em&gt;juxtaposition&lt;/em&gt; that indicates multiplying, not the parentheses themselves (and therefore reserve more focus for parentheses grouping and giving precedence to any operation contained inside, including unary negation). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-10-24T13:41:37.413" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11569" PostTypeId="2" ParentId="11566" CreationDate="2016-10-24T14:51:28.860" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Another one, that I believe stems from the same misuse of parentheses, is misreading function notation&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To me, this is the more serious problem that occurs.  I don't necessarily have people saying &quot;f(x) is f times x&quot; but I do have people that then note $f(3)=6$ deduce that the function is multiplication by two.  Once you start composition of functions (this even happened to me &lt;em&gt;today&lt;/em&gt;), we get the question of $f(g(x))=f(x)g(x)$...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as to a &quot;solution&quot;, I have a different point of view on this.  I agree that the ambiguity is terribly annoying and can be harmful; but natural languages have much more ambiguity, and we somehow get around that.  Because we have a limited number of symbols and ways to indicate typical operations, and because much notation is ingrained due to many years of use, it is often better to teach what people will actually see in &quot;real life&quot; and provide good warnings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a non-parenthesis example, I will take $\frac{df}{dx}$.  No one would suggest (I hope) that, historically speaking, this notation has not been very successful, and that it not only reminds us that derivatives come from difference quotients, but even (properly handled) allow very convenient manipulations for e.g. solving differential equations.  Nonetheless, from the point of view of the question here, this is horrible notation!  And some misconceptions arise from it.  So we teach when you can use this or when not (at least if we are mathematicians and not physicists).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is exponential notation, where I often have students who are quite confused why in $e^{x^2}$ the &quot;inside function&quot; is $x^2$, and exactly how things are related.  Better, from this point of view, is $\exp(x^2)$.  But there are other natural advantages to exponential notation, so we use it, and - hopefully - point out the ambiguity and give good exercises or activities to highlight the &quot;right&quot; answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to bring this back to parentheses, I think that the &quot;right&quot; solution is to acknowledge this ambiguity and try to resolve it.  One way in which I often do this is to explicitly write $\cdot$ or $\times$ as appropriate; another way is to have activities which highlight this difference and then really engage each student individually on it.  This is pretty labor-intensive, and won't work for every situation.  But simply avoiding the parentheses altogether - for now - is worse, because students simply will not interpret $3+2x \cdot x+1$ as anything other than $4+2x^2$, as as you point out grouping is what they are for.  In fact, I usually have the opposite problem of getting them to &lt;em&gt;use&lt;/em&gt; parentheses to indicate this is $(3+2x)\cdot (x+1)$.  So that is what I recommend, in this context.  Seeing $(2)(5)$ on occasion is a small price to pay for them understanding the grouping bit; there isn't really any other interpretation possible, at least.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that said, if you have good ideas of how to resolve this systematically from your previous experience, so that no one writes $(2)(5)$ but still knows what to do with $x^2(x+1)$, it would probably be a good thing.   (And hopefully that doesn't include replacing $2\cdot 5$ with $2\, . 5$ as the British do!)  I'm just not sure how to do that, and I have to teach the students I actually have.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2016-10-25T18:31:17.493" LastActivityDate="2016-10-25T18:31:17.493" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11570" PostTypeId="2" ParentId="11560" CreationDate="2016-10-24T15:47:48.647" Score="2" Body="&lt;p&gt;Integrals do not inherently have units associated with them.  They are mathematical tools which summarize a process of taking a limit of a sum of lots of small things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, the integral $$\int_{-1}^1 \sqrt{1-x^2} d x $$ could represent the area of half of the unit circle, or (if the velocity of some particle was described as $v(t) = \sqrt{1-x^2}$) as the distance traveled by a particle, or as a mass,  or as whatever else.  You have to choose units which are sensible to the problem at hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this particular case, there is a bit of geometric insight:  that one can calculated the area of a plane region as the volume of cylinder of height 1 with that region as the base.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One need not pass through the volume interpretation however:  you can directly view this double integral as summing the areas of lots of small rectangles comprising the region.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-10-24T15:47:48.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11571" PostTypeId="2" ParentId="11560" CreationDate="2016-10-24T17:41:29.267" Score="6" Body="&lt;p&gt;A double integral represents integrating &lt;em&gt;over&lt;/em&gt; an area. If the integrand is a height (i.e. with units of length), the result will be a volume (i.e. units of length$^3$). If the integrand is a flux (e.g. units of power/length$^2$), the result will be in units of  power. If the integrand is unitless, the result will be an area. In any case, the result will have the same units as the integrand times length$^2$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, you don't have to integrate over a geometrical area; it could be time, temperature, pressure, etc., and you'd have to adjust the units accordingly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The easiest way to think about the correct units is to realize the units are simply the units of the integrand times the units of the differentials (dx, dV, dA, dT, etc), just like you would do when multiplying any other values. Keep in mind that the units of a differential are the same as the units of the base value. So dx, dy, dz has units of length, dA has units of  length$^2$, etc. &lt;/p&gt;&#xA;" OwnerUserId="7199" LastEditorUserId="1147" LastEditDate="2016-10-24T18:51:36.537" LastActivityDate="2016-10-24T18:51:36.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11572" PostTypeId="1" AcceptedAnswerId="11574" CreationDate="2016-10-24T18:36:07.743" Score="19" ViewCount="6005" Body="&lt;p&gt;I mainly tutor adults in college algebra classes or lower. Sometimes an expression like $\dfrac {x+5}{5}$ will come up, and the students will say:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&quot;We can cancel out the $5$'s and get $x+1$, right?&quot;&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tell them that this is incorrect, and usually respond with a very ambiguous sentence:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&quot;In order to be able to cancel out factors, everything in the numerator and the denominator must be multiplied by something&quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that the students don't really get what I mean, and I just rely on examples from here. I know that a more correct response would be: &quot;In order to cancel out factors, the numerator and denominator must consist exclusively of a single factor or a product of factors&quot; (Though I'm not even sure of what a simple definition of factor would be here). I am just having a lot of trouble getting this concept into more understandable terms. The next time the students see $\dfrac {5+xy}{zy}$, they will want to cancel the $y$'s, because &quot;they are being multiplied by something&quot;. &lt;/p&gt;&#xA;" OwnerUserId="7075" LastEditorUserId="7075" LastEditDate="2016-10-24T18:58:09.423" LastActivityDate="2016-11-03T19:10:37.420" Title="How to teach students when they can and can't cancel factors in a fraction?" Tags="|undergraduate-education|algebra|fractions|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11573" PostTypeId="2" ParentId="11572" CreationDate="2016-10-24T23:29:48.050" Score="4" Body="&lt;p&gt;Frankly, I don't know how to teach this topic from scratch, but then we're not discussing that here, are we? It sounds like your situation is that of remediation &amp;mdash; your students already have some concept of reducing fractions, but it happens to be wrong. I constantly find myself in a similar situation when my college students, say in a Calculus class, do exactly the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My approach is to ask them the meaning of the two words: &lt;strong&gt;terms&lt;/strong&gt; and &lt;strong&gt;factors&lt;/strong&gt;. Funny thing: most students will say that they know the words, but they can't describe their meaning. (In my book, that means that they don't really know the words, but I digress.) So I give a few examples, leading my students to the following working definition: &lt;strong&gt;&lt;em&gt;terms&lt;/em&gt; are things that are added or subtracted&lt;/strong&gt; and &lt;strong&gt;&lt;em&gt;factors&lt;/em&gt; are things that are multiplied&lt;/strong&gt;. Not perfectly accurate, but in my experience it helps. Then I reiterate that the rule for reducing fractions states that &lt;strong&gt;fractions are reduced by common factors&lt;/strong&gt;, and emphasize that to use it correctly we need to know what &lt;strong&gt;factors&lt;/strong&gt; are, which we now do. There's still more: I show examples that to be a factor of the numerator, this quantity must be factored out, i.e. the numerator must be this quantity times everything else; same for the denominator, of course. That helps to explain why we can't reduce $y$'s from $\frac{5+xy}{zy}$ &amp;mdash; because in the numerator $y$ is only in one of the terms, not a factor of the numerator. Note how at this point knowing the word &quot;term&quot; comes into play!&lt;/p&gt;&#xA;" OwnerUserId="6762" LastActivityDate="2016-10-24T23:29:48.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11574" PostTypeId="2" ParentId="11572" CreationDate="2016-10-25T01:07:51.937" Score="17" Body="&lt;p&gt;Instead of presenting &quot;cancelling&quot; as an arbitrary rule (which is often how students have seen it — or at least how they learned it — before), explain it in a way that shows what's actually going on. So, if we have $\frac{5x + 5}{5}$, instead of just &quot;cancelling the 5's&quot; (which sounds kind of arbitrary and mysterious), write out several steps, like:&#xA;$$\frac{5x + 5}{5} = \frac{5 \cdot (x + 1)}{5 \cdot 1} = \frac{5}{5} \cdot \frac{x + 1}{1} = 1 \cdot \frac{x + 1}{1} = \frac{x + 1}{1} = x + 1.$$&#xA;Of course, you wouldn't want to give this level of detail every time, but I think it'd help to start out with this level of detail, and more gradually start omitting steps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you start with something like $\frac{x + 5}{5}$ instead, the same argument won't work, and I think a student is less likely to think $5(x + 1)$ is equal to $x + 5$ than to &quot;cancel&quot; incorrectly: &quot;cancelling&quot; is something many students do by reflex, without thinking about it, and writing out the steps in more detail leaves no reasonable choice but to think more carefully about what's going on. (You could still get errors like incorrectly factoring $x + 5$ as $(x + 1) \cdot 5$, but this at least reduces the problem to understanding that multiplication distributes over addition.) And if you go through the same process anyway, you get a true statement:&#xA;$$\frac{x + 5}{5} = \frac{5 \cdot (\frac{1}{5} x + 1)}{5 \cdot 1} = \frac{5}{5} \cdot \frac{\frac{1}{5} x + 1}{1} = 1 \cdot \frac{\frac{1}{5} x + 1}{1} = \frac{\frac{1}{5} x + 1}{1} = \tfrac{1}{5} x + 1.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To reinforce this, avoid using mysterious words like &quot;cancel&quot; (and &quot;move&quot;, as in &quot;move to the other side&quot;) in your own explanations, and don't accept &quot;cancelling&quot; as an explanation from a student who's recently made mistakes when &quot;cancelling&quot; — even if they have a correct result, ask them what exactly they're doing when they're &quot;cancelling&quot;. If they can't say, how do they know they can &quot;cancel&quot; like that at all?&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-10-25T01:07:51.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11575" PostTypeId="2" ParentId="11572" CreationDate="2016-10-25T03:22:15.917" Score="4" Body="&lt;p&gt;For what it's worth, here's the slide on that topic from my college algebra class:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zV67P.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zV67P.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I start by checking a counterexample of that common reduction error. In accordance with my theme that the essence of algebraic thought is to think about inverse operations, I try to connect with the intuition that it makes sense that &lt;em&gt;division only cancels with multiplying&lt;/em&gt; (not addition or any other operation). I agree with zipirovich that it's important to be previously drilling students on the names of things, so that we can concisely state the important difference at this point (factors vs. terms). Then I run a few exercises (with answers hidden until another mouse-click by me). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is no silver bullet in my experience; students still commonly have trouble on later days -- perhaps they find fractions so unpleasant that they just shut off that part of the brain whenever rational expressions show up. But connecting with the inverse operations gets a few heads nodding, and perhaps as important, continues the broad theme of the course that hopefully pays off on later days. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-10-25T05:14:37.690" LastActivityDate="2016-10-25T05:14:37.690" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11576" PostTypeId="2" ParentId="562" CreationDate="2016-10-25T04:22:10.643" Score="0" Body="&lt;p&gt;A thought from Steven Krantz's &lt;em&gt;How to Teach Mathematics&lt;/em&gt;, 3rd Ed., published by the American Mathematical Society (sec. 2.12):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If a textbook uses notation or other conventions that you do not like,&#xA;  then don't use that book. You really are obliged to follow the&#xA;  notation and definitions and other paradigms in the text you have&#xA;  chosen. Otherwise all but the gifted students will be lost. If you&#xA;  repeatedly criticize the text as the course proceeds, then you will be&#xA;  sending a confusing message to the students: Why did you choose this&#xA;  book if it is obviously so full of flaws? Isn't it your job to select&#xA;  a text that you can teach from?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Granted that a textbook is unlikely to vary between multiple different notations, then the recommendation here is likely &quot;no&quot;. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-10-25T13:01:24.580" LastActivityDate="2016-10-25T13:01:24.580" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11577" PostTypeId="2" ParentId="942" CreationDate="2016-10-25T04:25:08.133" Score="1" Body="&lt;p&gt;A thought from Steven Krantz's &lt;em&gt;How to Teach Mathematics&lt;/em&gt;, 3rd Ed., published by the American Mathematical Society (sec. 2.12):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If a textbook uses notation or other conventions that you do not like,&#xA;  then don't use that book. You really are obliged to follow the&#xA;  notation and definitions and other paradigms in the text you have&#xA;  chosen. Otherwise all but the gifted students will be lost. If you&#xA;  repeatedly criticize the text as the course proceeds, then you will be&#xA;  sending a confusing message to the students: Why did you choose this&#xA;  book if it is obviously so full of flaws? Isn't it your job to select&#xA;  a text that you can teach from?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-10-25T13:00:45.970" LastActivityDate="2016-10-25T13:00:45.970" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11578" PostTypeId="2" ParentId="11566" CreationDate="2016-10-25T10:18:41.020" Score="30" Body="&lt;p&gt;It is actually wrong to say that parenthesis means multiplication. In $(2)(5)$ it is the lack of an operator between the parenthesis that implies multiplication, NOT the parenthesis. The parenthesis are &quot;needed&quot; because $25$ means the number twenty-five, the parenthesis are purely for grouping.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;No operator means multiplication&quot; is an extremely common convention that is much more useful to learn than &quot;parenthesis means multiplication&quot;, which is just plain wrong, since it only means multiplication when there is either a multiplication operator (obviously) or no operator, which the first rule already covers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So yes, I agree that teaching students something that is not true and unnecessary very likely causes problems for students learning math&lt;/p&gt;&#xA;" OwnerUserId="7334" LastActivityDate="2016-10-25T10:18:41.020" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11579" PostTypeId="2" ParentId="11566" CreationDate="2016-10-25T18:32:27.460" Score="12" Body="&lt;p&gt;This problem starts in elementary school.  In fact, I am one of the elementary school teachers that has started students along this path that you complain about.  I have taught my 5th and 6th grade students that once they start using variables they should no longer use the multiplication sign since it might be mistaken for the variable x.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach my students that while the book might use a dot for multiplication the students should never write $5\cdot2$ since they might confuse it with $5.2$ From experience I can assure you that elementary students will not be able to distinguish between the multiplication dot that they write and the decimal point that they write.  In a printed book it is much clearer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As is standard in textbooks and the literature that I've read, I teach my students that they can use parentheses for multiplication symbols. I have tried to teach order of operations in great depth so that they don't think that $(2)5+1$ is anything other than $10+1$. In fact my students know that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(5)2+1=10+1=11$ because they know that $5\cdot2+1=10+1=11$   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest that a greater effort be made to teach order of operations in depth. Most of the elementary teachers that I know don't understand order of operations and many are still stuck on PEMDAS and are convinced that addition comes before subtraction since A comes before S.  They believe:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$8-3+5=8-8=0$  and not $8-3+5=5+5=10$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also suggest that * be adopted as a multiplication symbol.  It is easy for students to draw and widely used in spreadsheets. It is much easier for students to use than the multiplicative dot and would eliminate the problems of parentheses.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-10-25T18:32:27.460" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11580" PostTypeId="1" CreationDate="2016-10-26T00:24:04.460" Score="18" ViewCount="698" Body="&lt;p&gt;Background: I'm a final-year math student applying for faculty jobs at small schools in the US.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm currently designing a few courses, and this questions has been bothering me for some time now: Why can't I just assign an older edition of a text to students?  It would be an edition I am more familiar with, plus it will be dramatically cheaper for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, consider David C. Lay's &lt;em&gt;Linear Algebra and Its Applications&lt;/em&gt;.  I used the 3rd edition as an undergraduate, but they're now on the 5th.  I don't care about whatever software/codes/etc may or may not come with the text, so the only reasons I can think of are the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It's presumably out of print, so maybe there won't be enough copies for students to easily get one?  Unlikely with popular texts such as Lay's, but I could see it with a less common text. &lt;/li&gt;&#xA;&lt;li&gt;The university book store can't buy an older edition, so I am forced to use a newer one?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In any case, this all seems very unfair to students, especially when there's already so much discussion about the financial accessibility of higher education.&lt;/p&gt;&#xA;" OwnerUserId="5988" LastEditorUserId="5988" LastEditDate="2016-10-26T00:29:12.913" LastActivityDate="2018-01-18T01:43:11.430" Title="What reasons are there preventing me from using an old edition text?" Tags="|course-design|textbooks|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11581" PostTypeId="2" ParentId="11580" CreationDate="2016-10-26T01:04:12.960" Score="11" Body="&lt;p&gt;Some universities (like mine) are accredited by a third party agency.  One of their requirements for a university to be accredited is for it to have an &quot;updated set of reference books&quot; where &quot;updated&quot; means &quot;published within the last five years.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that this makes sense for some fields of study like the natural sciences or computer studies, but for mathematics I believe this requirement does not make sense.  My favorite trigonometry textbook (Paul Rider's &quot;Plane and Spherical Trigonometry&quot;) was first published in the late 1940's (I think), and I like to use the 1965 reprint of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I must add that the book is locally reprinted legally very often so my students can very easily buy a copy of it.)&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-10-26T01:04:12.960" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11582" PostTypeId="2" ParentId="11580" CreationDate="2016-10-26T04:44:39.790" Score="7" Body="&lt;p&gt;I'm in the habit of allowing students to use either the current edition of a text or the preceding one. So far I've been able to keep a homework exercise list synchronized so that both groups of students are working on equivalent problems (possibly renumbered between the texts). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The new book is carried in the bookstore, but, as you note, the old edition is not. For that, students need to go online (say: Amazon, et. al.) and order a used copy of the older edition. Used cost is frequently around $5, saving over a hundred. This works fine for my one section, but even so, I can see the used stock getting depleted over a few semesters. If I were to try to run this strategem across several sections or a whole department, I think that we'd run out of copies of the book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another issue for some of us is that departments can dictate what books are to be used by committee. If you're designing your own course then that's probably not an issue. On the other hand, Steven Krantz writes in &lt;em&gt;How to Teach Mathematics&lt;/em&gt; by the AMS that the one time he was officially disciplined was when he went off-book from his department's dictate. (He also notes that most of us will never design our own courses, so you're already outside the normal curve by a good stretch.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More recently I've taken a step towards using open-source digital books so we can copy them forward freely as much as we want in the future, and sidestep the issue of depleting used stocks. I feel those works (e.g., OpenStax at Rice University) have just in the last year or so crossed a barrier from unusable to acceptably useful. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-10-26T04:44:39.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11583" PostTypeId="1" CreationDate="2016-10-26T12:08:19.770" Score="10" ViewCount="333" Body="&lt;p&gt;So I'm doing a mathematics education extension for my current undergraduate maths course, and for one bit of the final assessment we're asked to create a detailed lesson plan on the (strong) Goldbach Conjecture - that any even number is the sum of two primes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the task is to make it a flexible enough plan that it's fairly independent of age and time allotted. We're also assuming the students know at least what primes are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've got a few ideas (listed below), but I was just wondering if there's any interesting/novel ways of explaining or exploring the Goldbach Conjecture at primary or secondary student level? There seems to be some interesting symmetry/geometric number line intuition one can draw, but I'm not sure how to make that concrete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's what I got so far,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Explain the conjecture, provide some simple examples to begin&lt;/li&gt;&#xA;&lt;li&gt;Visual worksheet to get a 'feel' for it, (eg. Filling in blanks in on a diagram similar to the one on &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Goldbach_partitions_of_the_even_integers_from_4_to_50_rev4b.svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;(For secondary students, having not mentioned whether its proved or not), practice forming an induction argument. Why does it fail? (eg. &lt;a href=&quot;https://math.stackexchange.com/questions/155083/why-proof-by-induction-fails-for-goldbachs-conjecture&quot;&gt;this post.&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Talk a bit about the twin prime conjecture and how it's related&lt;/li&gt;&#xA;&lt;li&gt;(For any age) discuss the difference between showing the conjecture is true for many numbers, and a rigorous proof. (Eg. We know the pythagorean theorem is always true)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Are there any other interesting, mathematical concepts one can easily draw from the Goldbach conjecture?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="7343" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2018-06-28T12:44:46.070" Title="Planning high school workshop on Goldbach Conjecture" Tags="|secondary-education|number-theory|prime-numbers|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11585" PostTypeId="2" ParentId="11580" CreationDate="2016-10-27T15:22:38.727" Score="4" Body="&lt;p&gt;You should be aware of the Higher Educational Opportunity Act of 2008. This does place textbook availability and fair-disclosure requirements on institutions of higher education. I don't think this explicitly prohibits you from requiring an out of print textbook, but be aware that there might be administrative issues that you will need to address.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This might seem like bureaucratic nonsense, but it can be traced to a good intention to protect students. Some for-profit institutions of higher education have been known to require very expensive textbook material that is difficult to locate on the open market, and only reveal these textbook requirements to their students just before classes start, in order to force them to buy from their bookstore. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2016-10-27T20:01:19.317" LastActivityDate="2016-10-27T20:01:19.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11586" PostTypeId="2" ParentId="11572" CreationDate="2016-10-27T16:31:36.260" Score="5" Body="&lt;p&gt;@DanielHast's answer has it, but here is another tack to take, especially for students who aren't yet in a position to be all that 'trustful' of algebraic manipulations, or to comfortably follow the teasing out of common factors in top and bottom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After the student performs such a cancellation, move to some scratch paper and present them with something (adjust the numbers to suit their mental-math capacities) like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Q:&lt;/strong&gt; Add four and nine for me.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;A:&lt;/strong&gt; Uh, thirteen.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Q:&lt;/strong&gt; OK, now divide by four.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;A:&lt;/strong&gt; Um, three... and a quarter. 3.25.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Q:&lt;/strong&gt; Great - now, write out what 'four plus nine' looks like for me.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;A:&lt;/strong&gt; &lt;em&gt;(writes)&lt;/em&gt; '$4 + 9$'&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Q:&lt;/strong&gt; And divide it all by four.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;A:&lt;/strong&gt; &lt;em&gt;(writes)&lt;/em&gt; '$\frac{4 + 9}{4}$'&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now refer back to their cancellation, and carry it out on the expression that they have written. Take pains to show that what you're doing here is the same as what they had done there:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{4 + 9}{4} = 1+9 = 10 \neq 3.25 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We now have a wonderful and approachable contradiction to resolve, which the student is likely to find accessible with some guidance.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2016-10-27T16:31:36.260" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11587" PostTypeId="2" ParentId="11542" CreationDate="2016-10-27T17:43:48.737" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LsX7y.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LsX7y.jpg&quot; alt=&quot;![this is &amp;quot;***The*** Classic&amp;quot;](https://i.stack.imgur.com/QK9Pq.jpg)&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your interpretation is correct. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a self-learner try to put more emphasis on the thinking of the &quot;logical links&quot; between sentences  because of the plural form of the possessive &quot;their&quot; used in &quot;their meaning.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.cornell.edu/~hubbard/readingmath.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;This link&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt; may help. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Educating students on developing the thinking  skills needed to read a math book independently is not considered to be a form of job security, as shown by the down vote. But these skills will be necessarily called upon at the college level, as explained in the link. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These two textbooks also have the same list of 7 &quot;Thinking skills&quot; in their index:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recall and transfer&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analysis &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Spatial perception &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applying concepts&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interpreting &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Synthesis &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reasoning and inferrencing &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These skills definitely go hand-in-hand  with learning how to read a math book. &lt;/p&gt;&#xA;" OwnerUserId="7351" LastEditorUserId="-1" LastEditDate="2017-01-26T20:22:31.733" LastActivityDate="2017-01-26T20:22:31.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11588" PostTypeId="2" ParentId="562" CreationDate="2016-10-28T11:26:39.153" Score="1" Body="&lt;p&gt;I think that the only possible answer is: &lt;strong&gt;it depends&lt;/strong&gt;. Is fluency with various notation a goal of the course? If yes, you have your answer, if no, you would better concentrate on other stuff. Dealing with notation is hard and can be confusing and time-consuming, so don't engage in it unless you have good reasons to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, for engineering students it might make sense to have your course using the notation they will find in other courses, or to emphasize a notation which is more suited to your topics and a notation they will find elsewhere, to help them relate what you teach and the context in which they will have use it.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-10-28T11:26:39.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11589" PostTypeId="1" CreationDate="2016-10-28T16:27:52.683" Score="20" ViewCount="260" Body="&lt;p&gt;I'm trying to find more problems suitable for early college students (students who know algebra and calculus) that involve translating words into mathematical notions.  A nice example is this one: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Show that on the graph of any quadratic polynomial the chord joining the points for which $x=a$ and $x=b$ is parallel to the tangent line at the midpoint $x=(a+b)/2$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I &lt;em&gt;don't&lt;/em&gt; mean the thing that's usually called a &quot;word problem&quot;, where there's some &quot;real world&quot; situation that has to be interpreted into mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is for a &quot;transition to advanced math&quot; course that's supposed to be a step between calculation based calculus and proof based analysis and algebra.  What I really want them to practice is the process of solving problems that use defined notions by replacing those definitions with their meaning, since we find that's something students in the later courses struggle with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem above really caught my eye, because a lot of students at that level seem to have no idea how to start with it - they see a bunch of words that don't pattern match to any problem they've solved before - but as soon as they recognize that &quot;_ is parallel to _&quot; means &quot;the slope of _ is equal to the slope of _&quot;, they know how to calculate the slopes and check equality.  That feature - problems which seem impenetrable when taken as a whole, but which can be reduced to something simple by breaking apart the meaning - is the main thing I'm trying to illustrate.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2017-12-24T20:47:30.057" Title="Problems which require interpreting definitions" Tags="|problem-design|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11590" PostTypeId="2" ParentId="11572" CreationDate="2016-10-28T17:46:52.497" Score="6" Body="&lt;p&gt;I am not a teacher but it seems to me that &quot;cancelling&quot; is sort of obtuse. It should be divide both numerator and denominator by the same thing. Hopefully it is evident that 5 ~= (s+5). I had a friend who taught at the University of Illinois and had a college student answer on a test that cos(x)/x=cos( ).&lt;/p&gt;&#xA;" OwnerUserId="3277" LastActivityDate="2016-10-28T17:46:52.497" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11593" PostTypeId="2" ParentId="1037" CreationDate="2016-10-28T23:02:26.687" Score="2" Body="&lt;p&gt;Great question! I also have the same problem (don't we all?) in absolutely any class I teach. One additional idea that I've recently came to is the following. Our students don't distinguish between solving a problem and writing your solution. Solving can naturally be messy, because you need to think and figure out how to do solve the problem. Of course, even that idea is quite novel to many students, because mostly they view their math classes as training in following the given step-by-step samples. But back to solving: after you figure out how to solve the problem, you need to write a clean narrative for someone, such as your instructor or grader, who will read it. But our students don't realize that: their idea of working on a math problem is to immediately start writing all over the place, scratching and crossing and correcting as necessary hoping to get to the answer, and they believe that that's what expected as their submission.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I recently started telling my students that for any work they do, be it homeworks, quizzes, or even in-class tests, they need two stacks of paper: some scratch paper where they figure out a solution, and one clean paper (for in-class tests, it's the test paper itself where space is provided) for writing a clean version of the solution. I can't say that it actually solved their messy writing issue completely, but quite a few students followed my advice and actually started submitting &lt;strong&gt;nicely written&lt;/strong&gt; works!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides that, I agree with the excellent answers above. I use some grading penalty too, but not always &amp;mdash; depends on the messiness of the paper, difficulty of material, etc.&lt;/p&gt;&#xA;" OwnerUserId="6762" LastActivityDate="2016-10-28T23:02:26.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11594" PostTypeId="2" ParentId="11457" CreationDate="2016-10-29T04:01:38.023" Score="2" Body="&lt;p&gt;It's an interesting idea.  In programmers' terms, if you count 0 as your &quot;truth&quot; value and 1 as your &quot;false&quot; value (which is how it works in some programming languages), then:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A =&amp;gt; B$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;means &quot;A is equal to or greater than B.&quot;&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;A &amp;gt;= B&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if A is 0, then B has to be 0 as well (true) because it can't be anything smaller.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Likewise if B is 1 (false), then A can't be anything greater, so must be equal (also false).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if A is 1, B could satisfy the comparison by being either 0 or 1.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;It's a very interesting (and humorous!) thought experiment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would &lt;strong&gt;NOT&lt;/strong&gt; recommend it as your first pedagogical entry to the subject of logical implication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bring it up after your students are comfortable with the idea of logical implication, and bring it up labeled clearly as an unusual way of looking at it, not generally agreed upon.  (My advice.)&lt;/p&gt;&#xA;" OwnerUserId="5751" LastActivityDate="2016-10-29T04:01:38.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11598" PostTypeId="1" CreationDate="2016-10-30T05:24:46.323" Score="17" ViewCount="667" Body="&lt;p&gt;My school employs undergradates graders for most of the undergraduate math courses, and this year I have been tasked with grading an intro to proof-based linear algebra course. This is students' second exposure to proofs after the $\epsilon$-$\delta$ intro to analysis sequence, so unfortunately they don't know how to do anything but fill the blanks in an $\epsilon$-$\delta$ template, i.e. they haven't thought through a logical proof really. This makes grading a bit tough since there are around 30 students writing horribly long, incorrect proofs for what could be 2 line proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, how can I grade the homework assignments more efficiently? I've tried the 'one problem at a time' method where I grade one problem on every single sheet and repeat, and I've tried the 'one sheet at a time' where I go linearly and grade every single problem on a single set and do this for each student (looks like Fubini's theorem applies here :p) I'm interested in other graders' routines.&lt;/p&gt;&#xA;" OwnerUserId="7362" LastActivityDate="2016-10-31T10:19:59.650" Title="Efficient ways to grade proof-based math courses" Tags="|undergraduate-education|grading|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11599" PostTypeId="2" ParentId="11598" CreationDate="2016-10-30T14:42:27.573" Score="5" Body="&lt;p&gt;For a small class (with, say, ten students), it's easy to sort the papers from &quot;poorest&quot; to &quot;best&quot; answers.  Their scores should thus go from lowest to highest.  For larger classes (like yours with 30 students), it may be difficult to compare all of them at the same time.  You might want to do the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students do mostly good work, then I would use a &quot;subtract points&quot; method, where the score is initially a perfect score, then every error gets a corresponding deduction in points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students do mostly poor work, then I would use an &quot;add points&quot; method, where the score is initially zero, then every correct idea gets a corresponding addition in points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students do work that is in between good and poor, then you might need to use a combination of both methods.  This can lead to problems:  Say the perfect score is 10, and the student made errors worth a total of 6 points, but showed correct ideas worth a total of 6 points.  So is the score 4 or 6?  For cases like these, I would tend to give the average of the two scores (in this case, 5).&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2016-10-30T14:42:27.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11600" PostTypeId="2" ParentId="11598" CreationDate="2016-10-30T16:08:03.120" Score="2" Body="&lt;p&gt;I often use a variation of Joel Reyes Noche's proposition, where for each question I first rank the answer into one of two states: &quot;at best mostly inadequate&quot; and &quot;at least mostly adequate&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the former, I see which ideas can be worth partial points, if any, and never let these partial points exceed one third of the points for the question. If too blatant error are also present, they may induce deduction and a question with a start of an idea but a false reasoning could get zero. As soon as I see enough blatant errors, I set the mark for the question to zero and pass to the next one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the latter, I remove partial points for small errors, usually (but not always) retaining at least two-third of the points the question is worth. In some cases, there are enough errors that I don't want to give more than two third, but enough ideas that I don't want to give less, and I am done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is crude, as summing points for independent questions is not really a sound grading scheme. I sometimes have a separate note for quality of writing, so that I can boost students who did few questions but did them well (I prefer to value quality over speed).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it is a final exam and the student do not see their paper without the grader being present, I refrain to annotate the paper too much as this costs time. If it is a test that I turn back to student, I annotate a lot to help them see what they need to improve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also have given once a final exam where the first exercise was a quizz with 6 question giving a multiplying note applied to the rest of the paper. At 2 correct answers or below, the multiplicative coefficient was zero, which speeds up the grading quite a bit. I kept this part very elementary so that I would not mind at all failing hard a student not answering at least half of them.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-10-30T16:08:03.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11602" PostTypeId="2" ParentId="11598" CreationDate="2016-10-31T10:19:59.650" Score="1" Body="&lt;p&gt;When giving grades to exams I do the following, assuming there is a large number of exams:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Solve the problems, preferably in writing (that can be later shared with the students).&lt;/li&gt;&#xA;&lt;li&gt;Process one question at a time.&lt;/li&gt;&#xA;&lt;li&gt;Read through enough answers so that I am no longer surprised by them.&lt;/li&gt;&#xA;&lt;li&gt;Read all the answers in detail. Write comments and mark errors while reading. Give zero points or full points where obvious.&lt;/li&gt;&#xA;&lt;li&gt;Create a rudimentary grading scheme for the most common types of answers.&lt;/li&gt;&#xA;&lt;li&gt;Go through the answers and grade them according to the scheme. Improve the grading scheme as you go by adding notes such specific examples of what gave how many points, and by writing down how many points you gave to one of the strange answers. Postpone any answers that are difficult to grade.&lt;/li&gt;&#xA;&lt;li&gt;Go through the remaining difficult answers that don't fit nicely into your grading scheme.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The rudimentary grading scheme is specific to a given type of answer (There often are two or three styles of answers to any given question.), though they should of course be comparable in that answers showing equivalent understanding should get an equal number of points.&#xA;I typically separate a number of key points (steps in a proof, common errors that are not committed) and assign points for each. I don't otherwise remove points from errors, but do set a maximum of 5 points out of 6 if the answer contains blatant nonsense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if the task was to solve the inequality $|x-1| &amp;lt; 2$, different types of answers might be using the definition of absolute value, solving $-2 &amp;lt; x-1 &amp;lt; 2$, drawing a picture and using the geometric interpretation, and simply ignoring the absolute value and instead solving $x-1 &amp;lt; 2$. The final method of answering is a special case of earlier ones, and thus does not require different scheme for grading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the first method, the preliminary grading might be: 2 points for using the definition of absolute value correctly, 2 points for solving both of the inequalities, 2 points for presenting a coherent and correct answer. The geometric approach might be worth at most 3 or 4 points if executed correctly (since it is not rigorous, but does lead to a correct answer and does demonstrate understanding).&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2016-10-31T10:19:59.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11604" PostTypeId="2" ParentId="135" CreationDate="2016-10-31T16:25:45.087" Score="3" Body="&lt;p&gt;Since this does not seem to be mentioned in other answers, it should be noted that strictly speaking, the Riemann integral is only formulated for functions that are bounded on a compact interval [a,b]. But the Lebesgue integral does not assume compactness of the function domain, nor does it assume the function is bounded. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-10-31T16:25:45.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11605" PostTypeId="1" AcceptedAnswerId="11607" CreationDate="2016-10-31T18:29:27.600" Score="24" ViewCount="773" Body="&lt;p&gt;Teachers at the university level (at least what I've seen in the US) are responsible for both&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;teaching students the material for a course, and&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;assessing the students' understanding of the course material.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This seems to encourage the creation of exams that assess whether or not students can solve very specific kinds of problems rather than assess a broad understanding of a course's content&lt;sup&gt;[1]&lt;/sup&gt;. Because of this it doesn't seem too unreasonable to have different people in charge or assessing students than are in charge of teaching them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any institutions that have a broad policy like this, where the grading and assessment is more centralized and out of the hands of individual instructors? By &lt;em&gt;broad&lt;/em&gt;, I mean more than just having common finals for large classes with multiple instructors. Maybe even broad to the extreme of having a separate department for designing and grading assessments?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have there been any studies into the whether or not this is a good idea? Have education researchers thought about this idea or written about the pros and cons of this before?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;[1]: To provide a specific example of what inspired this question, I just proctored and graded an intro to calculus midterm (as a teaching assistant). This was one of the problems:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Use the squeeze theorem to evaluate the limit&#xA;  $$\lim\limits_{x\to \infty} \frac{\cos(x-2)+3}{2x}\;.$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Many of the solutions to this problem were identical to what the primary instructor had written up in their grading key. So these students could remember the procedure the professor would go through when answering an example problem like this in lecture, and would themselves write out these steps like a recipe. But this didn't indicate to me that they really &lt;em&gt;knew&lt;/em&gt; what the squeeze theorem meant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea then is that having separate people teaching the material and grading the students would avoid situations like this. The teacher would have to teach calculus much more broadly to prepare students for a wider range of questions/problems that they don't have direct control over.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2016-11-01T04:30:54.430" LastActivityDate="2018-01-05T22:12:13.440" Title="Separating the roles of &quot;teacher&quot; and &quot;assessor&quot;" Tags="|undergraduate-education|reference-request|grading|assessment|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11607" PostTypeId="2" ParentId="11605" CreationDate="2016-11-01T03:22:15.283" Score="13" Body="&lt;p&gt;A question that occurs with a project like this (broader than one department, as you put it) would be: Who is qualified to make those assessments? Probably not any other department at a particular college, certainly -- the one department is, by definition, where all the experts in that subject work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To some degree this actually is done in places, in the context of increasing corporatization of the university. For example: A college or state might outsource required tests to a company like Pearson or the College Board. Or a college might pay for a one-time development of an online course, say (either by resident faculty or an outside party), and then from that point on students learn from those packaged materials and present faculty only give grading and feedback on assignments (or vice-versa). This is particularly done at for-profit schools. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One problem with this is that it takes away faculty governance, control, and academic freedom. Outside agencies might be even &lt;em&gt;less&lt;/em&gt; interested in upholding strong standards in the discipline than professional faculty; indeed, in some contexts the primary goal is strictly higher graduation numbers. The result of that might end up like, e.g., NYC public schools (which currently boast about a 90% graduation rate, and the majority of those students need grammar-school level remediation when they reach college). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gary Rhoades, &quot;We are all contingent&quot;, &lt;em&gt;AFT on Campus&lt;/em&gt;, Summer 2014:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Reduced faculty control is particularly clear in traditional distance&#xA;  education, massive open online courses, and decisions to outsource&#xA;  segments of the general education curriculum to Pearson or other&#xA;  third-parties, often for-profit providers. In each of these cases,&#xA;  academic managers have developed structures and created mechanisms to&#xA;  bypass effective faculty decision-making related to the curriculum.&#xA;  Exceptions can be found, and collective bargaining agreements can&#xA;  build in protections against this. But the pattern is a strong one,&#xA;  and the growth of these high-tech infusions is linked to the&#xA;  furthering of contingency in academe. The model of high-tech&#xA;  employment is a just-in-time, just-in-the-classroom,&#xA;  just-at-the-manager’s-will type of employment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Furthermore, the work of the instructor becomes less rewarding and more stressful in that type of environment. UNH Department of English website, &quot;Stress, Control, and the Deprofessionalizing of Teaching&quot;, October 2009:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is a Faustian bargain. When teachers lose control of&#xA;  decisionmaking—when they prepare students for tests they have no role&#xA;  in designing (and often no belief in), when they must abandon units&#xA;  they love because there is no longer time, when they must follow the&#xA;  plans designed by others, when they are locked in systems of&#xA;  instruction and evaluation they don’t create or even choose—they will&#xA;  not be relieved of stress. Their jobs are not made easier, they are&#xA;  made harder and more stressful. While some find a way to resist,&#xA;  others acquiesce, though they feel, as one teacher put it, that “the&#xA;  joy is being drained out of teaching.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There's actually a particular phrasing in the literature for this decontextualized-division-of-labor, but at the moment it's escaping me. Perhaps someone else can chime in with that wording. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-11-01T03:22:15.283" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11608" PostTypeId="2" ParentId="10933" CreationDate="2016-11-01T23:47:28.597" Score="1" Body="&lt;p&gt;I think that this depends a lot on the context in which you are introducing fractional terms. Since the students haven't been introduced to the idea of a function though, I would definitely favor non-functional notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you need to introduce fractional terms in the language of functions, like if you wanted to talk about the range of the term or you wanted to pull your parameters from an weird domain, then you should really introduce the idea of a function first.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2016-11-01T23:47:28.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11609" PostTypeId="2" ParentId="11507" CreationDate="2016-11-02T00:09:10.780" Score="1" Body="&lt;p&gt;Without first discussing what the word &lt;em&gt;magnitude&lt;/em&gt; means in mathematical context, it's really pointless to ask them this question. How can they think and learn about the magnitude of a rate-of-change if they don't know what magnitude means?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe for 8&lt;sup&gt;th&lt;/sup&gt; graders, you could just say that the magnitude of something is its size/force/power with any information about it's direction thrown away. Talk about speed being the magnitude of velocity. Where velocity can be negative or positive depending on whether you are going forwards or backwards (like a roller-coaster on its tracks), speed doesn't care about direction; it's just how fast the thing is going. &lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2016-11-02T00:09:10.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11611" PostTypeId="2" ParentId="11572" CreationDate="2016-11-03T19:00:20.157" Score="1" Body="&lt;p&gt;I agree with some other answers that &quot;cancelling&quot; is a misleading word. As a non-native speaker I resort to Wikipedia and find that reducing or simplifying would be the preferrable word here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I'd try to make clear to the students in question that &lt;em&gt;reducing means dividing both numerator and denominator by the same value ($\neq 0$)&lt;/em&gt;. If one attempts to reduce a fraction like $\frac{x+5}{5}$ and the divisor $5$ comes to mind, they should check what $5:5$ is and what $(x+5):5$ is and if the result is helpful at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way would be to reverse the &quot;reduction&quot; and multiply the new/wrong numerator and denominator by the value that was used for the &quot;reduction&quot;. They would then (hopefully) get $(x+1) \cdot 5 = 5x+5 \neq x+5$ for the numerator and see their error.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It all boils down to proper application of the distributive law.&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2016-11-03T19:10:37.420" LastActivityDate="2016-11-03T19:10:37.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11614" PostTypeId="2" ParentId="11491" CreationDate="2016-11-04T15:33:29.010" Score="1" Body="&lt;p&gt;$32.4892$ rounded off to two significant digits is $32$. The student's answer i.e. $32.0000$ contains six significant digits and an absolute error $|\epsilon_a|=0.5×10^{-4}$&lt;/p&gt;&#xA;" OwnerUserId="5999" LastActivityDate="2016-11-04T15:33:29.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11615" PostTypeId="2" ParentId="7978" CreationDate="2016-11-04T18:07:41.383" Score="3" Body="&lt;p&gt;For French (my answer is based on the examples of &lt;a href=&quot;http://www.rpi.edu/dept/math/ms_graduate/resources/SayingMath.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.rpi.edu/dept/math/ms_graduate/resources/SayingMath.pdf&lt;/a&gt;, and on my experience as I’m a native speaker) :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$2^2$ deux au carré&lt;/li&gt;&#xA;&lt;li&gt;$2^3$ deux au cube&lt;/li&gt;&#xA;&lt;li&gt;$2^4$ deux puissance quatre&lt;/li&gt;&#xA;&lt;li&gt;$2^n$ puissance n-ième de 2&lt;/li&gt;&#xA;&lt;li&gt;$\sqrt{25}$ racine de 25&lt;/li&gt;&#xA;&lt;li&gt;$a^2+b^2=c^2$ a deux (&lt;em&gt;ou carré&lt;/em&gt;) plus b deux égal c deux&lt;/li&gt;&#xA;&lt;li&gt;$a^2, 2a$ a deux, deux a&lt;/li&gt;&#xA;&lt;li&gt;$1/2, 1/3, 1/4, 1/5$ un demi, un tiers, un quart, un cinquième&lt;/li&gt;&#xA;&lt;li&gt;$x/2$ x sur deux&lt;/li&gt;&#xA;&lt;li&gt;$\sin^2 x$ sin carré (de) x&lt;/li&gt;&#xA;&lt;li&gt;$\tan^{-1} x$ tangente moins un de x &lt;em&gt;ou&lt;/em&gt; arctangente de x&lt;/li&gt;&#xA;&lt;li&gt;$\ln 2$ L N de deux&lt;/li&gt;&#xA;&lt;li&gt;$\exp x$ exponentielle de x&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Finally, your example would be:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Une primitive de x carré est x cube sur trois&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ou plus simplement:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Intégrale de t deux dt [&lt;em&gt;dété&lt;/em&gt;] (de 0 à 1) égale à un tiers&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7388" LastEditorUserId="7388" LastEditDate="2016-11-07T09:44:03.637" LastActivityDate="2016-11-07T09:44:03.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11616" PostTypeId="1" CreationDate="2016-11-04T19:34:23.310" Score="6" ViewCount="135" Body="&lt;p&gt;Is there a website where people can find a mathematical pen pal of sorts? It would be interesting to find someone with similar mathematical interests at a similar level to anonymously keep mathematically up to date with, so you could try to help each other learn, but I don't know any community where I can find someone like this easily. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does such a place to find these people exist?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Email would be the place for this to occur when I say pen-pal. Further, instant messaging doesn't seem appealing, asynchronous communication is strongly preferred)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[No idea how to tag this sorry]&lt;/p&gt;&#xA;" OwnerUserId="7389" LastActivityDate="2016-11-04T19:34:23.310" Title="Finding a mathematical pen pal" Tags="|undergraduate-education|" AnswerCount="0" CommentCount="2" ClosedDate="2016-11-14T16:14:30.883" ContentLicense="CC BY-SA 3.0" />
  <row Id="11617" PostTypeId="1" CreationDate="2016-11-04T23:39:02.000" Score="11" ViewCount="217" Body="&lt;p&gt;Me and my 7th-grade students are going to a 3-day trip. It's an excellent opportunity to have some outdoor math games and activities. What games and activities do you suggest? Do you know any source for finding them?&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2016-11-13T14:31:41.810" Title="Math Outdoor/Camping Games and Activities" Tags="|secondary-education|games|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11618" PostTypeId="1" AcceptedAnswerId="11620" CreationDate="2016-11-05T06:09:39.537" Score="17" ViewCount="1039" Body="&lt;p&gt;I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most students study chemistry, biology or programming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When teaching derivatives I have given the definition, used it once on a linear function, and drawn pictures. I give the formulas for differentiating functions as black boxes. A student asks: Is there an easy geometric way to see why differentiating the exponential function gives the exponential function?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best I can come up with is the differential equation $f = f'$, which is not very geometric. The students are not expected to know differential equations at this point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does there exist a more geometric explanation?&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2016-12-13T22:12:43.130" Title="Geometric intuition for $D(e^x) = e^x$" Tags="|undergraduate-education|calculus|exponential|" AnswerCount="12" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11619" PostTypeId="2" ParentId="11617" CreationDate="2016-11-05T08:31:29.870" Score="1" Body="&lt;p&gt;On one outdoors activity day, we got set a version of the little child's toy with three posts and different size rings made out of numbered tyres. I would consider the reasoning in that a form of maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(When the instructor realised I knew how to do it, he insisted we started again without me saying what to do. So I pointed.)&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-11-05T08:31:29.870" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11620" PostTypeId="2" ParentId="11618" CreationDate="2016-11-05T10:33:47.213" Score="14" Body="&lt;p&gt;Here is a method I use in all my calculus classes to show that it &lt;em&gt;may&lt;/em&gt; be true that the derivative of $e^x$ is itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sketch a graph of $y=e^x$ high up on the board and sketch coordinate axes below that on the board. Say that the lower graph will be the derivative of the upper graph.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start at a point on the far left on the $x$-axis. Ask the class what the slope of $e^x$ is there. They will quickly say that the slope is positive but small, since the graph is increasing slowly there. Plot a point on the lower graph to correspond to this. Progress to the right, and students will see that the slope gets more and more positive. Keep graphing points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end, the graph of the derivative will look very much like the original graph. Explain that mathematicians have shown that the two graphs are in fact identical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works well--no student has ever disputed this, and this is in classes where I give extra credit to students who successfully dispute something I say.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2016-11-05T10:33:47.213" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="11621" PostTypeId="2" ParentId="11618" CreationDate="2016-11-05T11:47:45.650" Score="2" Body="&lt;p&gt;This is not so geometric, but if you have students who can actually do some difference quotients (numerically only, I mean) then just having them do a table where they estimate the derivative (say, $(f(3.001)-f(3))/(3.001-3)$) for various values, and then compare with the actual value of $e^x$, they might get surprised.  No, this is not an original idea at all.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-11-05T11:47:45.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11622" PostTypeId="2" ParentId="11618" CreationDate="2016-11-05T14:07:51.277" Score="3" Body="&lt;p&gt;Bunny rabbit generations: look at the discrete case $P(n)=2^n$ with $P(0)=1$. It is easy to see that $\frac{\Delta P}{\Delta n}=P(n)$ where $\Delta P=P(n+1)-P(n)$ and $\Delta n=1$. Or more generally for $P(n)=b^n$ where $b&amp;gt;1$ we have $\frac{\Delta P}{\Delta n}=cP(n)$ where $c=b-1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This discrete case for exponential growth makes it reasonable to expect that &quot;differentiating an exponential function gives the exponential function back,&quot; at least up to a constant multiplier $c$. As you know, the fact that $e$ corresponds to the case $c=1$ in the continuous variable case is simply the definition of the number $e$.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-11-05T14:07:51.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11623" PostTypeId="2" ParentId="11618" CreationDate="2016-11-05T23:07:38.267" Score="6" Body="&lt;p&gt;It may help to compare several exponential curves,&#xA;e.g., $y=5^x$, $y=e^x$, and $y=2^x$,&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/oXjIj.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oXjIj.jpg&quot; alt=&quot;ThreeExps&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;and calculate slopes at a few points, as in this nice drawing from&#xA;&lt;a href=&quot;https://www.wyzant.com/resources/lessons/math/calculus/derivative_proofs/e_to_the_x&quot; rel=&quot;nofollow noreferrer&quot;&gt;wyzant.com&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/6Rhwj.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6Rhwj.gif&quot; alt=&quot;graph_slopes&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;It becomes plausible that there is some curve between $2^x$ and $5^x$ such that&#xA;the slope at each point is exactly the $y$-coordinate of that point.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-11-06T10:44:21.713" LastActivityDate="2016-11-06T10:44:21.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11624" PostTypeId="2" ParentId="11618" CreationDate="2016-11-05T23:11:48.593" Score="5" Body="&lt;p&gt;But, a &lt;strong&gt;geometric&lt;/strong&gt; definition for the exponential function is the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The function $f$ for which $f(0)=1$ and is such that its value is equal to its slope at each point is &lt;strong&gt;the&lt;/strong&gt; exponential function. Or, in the language of differential equations, it is the unique solution of $\frac{dy}{dx}=y$ for which $y(0)=1$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As usual, the question really is... how are we &lt;strong&gt;defining&lt;/strong&gt; the exponential? If we are to assume the students just &lt;em&gt;know&lt;/em&gt; the exponential by its graph then I suppose Rory's explaination is a good approach.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2016-11-05T23:11:48.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11625" PostTypeId="2" ParentId="11618" CreationDate="2016-11-06T04:52:35.077" Score="3" Body="&lt;p&gt;Here's an approach that can be illustrated with a simple picture, and also gives an actual bound on the derivative of $e^x$ as being, at the very least, &lt;em&gt;fairly close&lt;/em&gt; to $e^x$ (which makes it much more plausible that they're in fact equal).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compare the slope of secant line through $(x - 1, e^{x-1})$ and $(x, e^x)$ with the slope of the secant line through $(x, e^x)$ and $(x + 1, e^{x+1})$:&#xA;$$\frac{e^x - e^{x - 1}}{x - (x - 1)} = e^x - e^{-1} \cdot e^x = (1 - e^{-1})e^x \approx 0.63e^x$$&#xA;and&#xA;$$\frac{e^{x + 1} - e^{x}}{(x + 1) - x} = e \cdot e^x - e^x = (e - 1)e^x \approx 1.72e^x.$$&#xA;The derivative is a limit of slopes of secant lines, and it's not hard to see (from the definition or, informally, by visual inspection of the graph) that the derivative of $e^x$ is strictly increasing. Given these facts, the above slopes give us bounds on the derivative: the derivative of $e^x$ at $x$ is somewhere between $(1 - e^{-1})e^x \approx 0.63e^x$ and $(e - 1)e^x \approx 1.72e^x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, even without knowing &lt;em&gt;exactly&lt;/em&gt; what the derivative of $e^x$ is, we can see it's fairly close to $e^x$, and taking closer secant lines will give tighter bounds, making it even more plausible. You can draw a picture of the tangent line and the two secant lines at a point to show how this bounds the slope of the tangent line. (The correspondence between the above algebra and the geometric picture with secant and tangent lines also nicely illustrates the close relation between algebra and geometry.)&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-11-06T04:52:35.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11626" PostTypeId="2" ParentId="11618" CreationDate="2016-11-06T21:19:36.603" Score="5" Body="&lt;p&gt;As mentioned in other answers, the starting point is the definition of the exponential you are using.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possible definition is as the solution of $f'=f$ taking value $1$ at $0$ (or other equivalent phrasings if you don't want to involve differential equation explicitly) and then there is nothing to explain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One more natural definition is that $\exp(x)=e^x$ where $e$ is some number (then you have to explain powers by a real number, which maybe tricky, but can be done with various level of details). This is the definition I recommend for people that will have to compute with exponential and logarithms, because it embeds the morphism property $e^{x+y}=e^x e^y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The number $e$ is somewhat difficult to justify &lt;em&gt;a priori&lt;/em&gt;, so what you can aim for is explaining why an arbitrary exponential function $x\mapsto a^x$ is always proportional to its derivative (and then it is easily accepted that for some choice of $a$ the proportionality coefficient is $1$, and that makes an excellent definition of $e$). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you aimed at a computational approach, you would write&#xA;$$ \frac{a^{x+h}-a^x}{h} = a^x\frac{a^h-1}{h}$$&#xA;and then one sees immediately that &lt;em&gt;if&lt;/em&gt; $(a^h-1)/h$ has a limit when $h\to 0$ (a small leap of faith), then $x\mapsto a^x$ is indeed proportional to its derivative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But you ask for a geometric approach, so let us make this geometric. Draw a plausible graph for an exponential, and translate geometrically the definition of an exponential: if we move on the graph by adding any value $h$ to the $x$-axis, we obtain a fixed &lt;em&gt;relative&lt;/em&gt; increase, i.e. the $y$-coordinate is &lt;em&gt;multiplied&lt;/em&gt; by an amount that only depends on $h$. This exactly tells you that the slope of a chord from $(x,a^x)$ to $(x+h,a^{x+h})$ depends on the point, but the ratio of the slope to the $y$-coordinate of the leftmost point only depends on $h$. Passing to the limit, you get the desired property.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that if you where to define the exponential with the differential equation, this explanation could help understand why we end up with a function that takes a fixed value to the power the argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, I think the chosen explanation should be an opportunity to manipulate the morphism property of the exponential, because it is tremendously important and often mistreated by students.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-11-06T21:19:36.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11627" PostTypeId="1" AcceptedAnswerId="11632" CreationDate="2016-11-07T13:16:52.847" Score="2" ViewCount="411" Body="&lt;p&gt;Trigonometry has some complex stuff, but sometimes students have trouble with the beginning, the basics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, here's what seems to be a simple question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$2cos(x)=1$$ $$0°≤x≤360°$$&#xA;  &lt;em&gt;Find the possible values of&lt;/em&gt; $x$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For students, this question may not seem as simple to them. Is there a good way I can explain this question, and these classes of questions in general?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can, provide an example question and an example explanation.&lt;/p&gt;&#xA;" OwnerUserId="7232" LastEditorUserId="7232" LastEditDate="2016-11-07T15:32:22.697" LastActivityDate="2017-07-17T00:14:11.480" Title="How to explain this trigonometric problem properly?" Tags="|undergraduate-education|trigonometry|" AnswerCount="3" CommentCount="33" ContentLicense="CC BY-SA 3.0" />
  <row Id="11628" PostTypeId="2" ParentId="11618" CreationDate="2016-11-07T19:03:37.627" Score="3" Body="&lt;p&gt;I would say to first sketch the curve $4^x$ and it's derivative $4^x \ln 4$, and the same for $3^x$ and $2^x$, then it will be easy to see that the case $e^x$ is a particular case. For the first two functions the derivative graph is a little higher than the function, but with $2^x$ the derivative is lower, so there must be a number between 2 and 3 such that the derivative graph is equal to the function itself. After this it will be evident that the responsible for this difference is the logarithm that appears in the derivative, the final stroke is the fact that $\ln e=1$ which makes the two functions identical. &lt;/p&gt;&#xA;" OwnerUserId="6547" LastActivityDate="2016-11-07T19:03:37.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11629" PostTypeId="2" ParentId="11618" CreationDate="2016-11-08T11:25:28.300" Score="2" Body="&lt;p&gt;Geometrically the relation $f^{\prime} = f$ means that the slope at $(x, f(x))$ of the graph of $f$ is equal to $f(x)$. We look for a function $f$ that has this property. Consider $g(x) = 1 + x$. Its graph is $(x, 1 + x)$ and the slope is $1$ at every point. Hence for $x$ very small this function almost has the desired property; it does have the desired property for $x = 0$. So we zoom in at the point $(0, 1)$ by replacing $g(x)$ by $h(x) = 1 + x/n$ for a big $n$. This doesn't work because the slope is $1/n$ not $1$, as we would like for $x$ near $0$. However, since the derivative of $h(x)^{n}$ is $nh(x)^{n-1}h^{\prime}(x)$, and $nh(0)^{n-1}h^{\prime}(0) = 1$, the function &#xA;\begin{equation}&#xA;g_{n} = \left(1 + \frac{x}{n}\right)^{n}&#xA;\end{equation}&#xA;should approximately have the desired property. Indeed,&#xA;\begin{equation}&#xA;g_{n}^{\prime} = \left(1 + \frac{x}{n}\right)^{n-1} = g_{n}^{1-\tfrac{1}{n}},&#xA;\end{equation}&#xA;so the slope of the graph of $g_{n}$ at $(x, g_{n}(x))$ is almost $g_{n}(x)$. This suggests that when $n \to \infty$ there results a function that has the desired property. This in fact works, and yields the exponential function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is nothing special about the initial function $g(x) = 1 + x$. Any $g$ differentiable at the origin and satisfying $g(0) = g^{\prime}(0) = 1$ would work. So the idea can be summarized as follows: we start at a point on a graph that has the desired property, and we zoom in at this point in an appropriate way. The remarkable thing is that we always get the same result.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2016-11-08T11:25:28.300" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11630" PostTypeId="2" ParentId="1447" CreationDate="2016-11-08T20:40:56.097" Score="2" Body="&lt;p&gt;I've also liked using angle measures, like degree measure, to motivate this.  When thinking about rotational symmetry I often ask a question to my students like, &quot;Should a rotational symmetry by 0 degrees be the thought of as the same as a rotational symmetry by 360 degrees or 720 degrees for that matter?  Sometimes I've done it with middle school children by focusing on numbers and units.  For example, consider the expressions 22(4) and 8(11).  These expressions are not the same, 22(4) expresses that there are 22 groups of 4 and 8(11) expresses that there are 8 groups of eleven.  But we note that when converted to a like unit, namely 1's, we can see that they are equivalent, but not the same because they didn't mean the exact same thing in their expression.  This is also true with fractions, &quot;Should we consider 1/2 the same as 2/4 or 3/6?&quot; No, because they express something different from each other (and in certain contexts we prefer one over the other, like in probability I'd rather know the unsimplified fraction first), but their values are equivalent when converted to a like unit.  What is nice about the rotational symmetry context is that students can place points on a polygon which they are choosing to rotate and see that all points land in the same place after doing both a 0 degree rotation (the identity rotation) and a 360 degree rotation.  This however, is not the case for any other rotation regardless of its symmetries.  Therefore, students, after some consideration and playing, have always come to the conclusion that a rotational symmetry by a 360 degree rotation is the not the same in process as a symmetry by a rotation of 0 degrees, but their result/product is indeed equivalent because every point is in the same place as it started under both rotations.  Separating process/meaning from results/products is a necessary idea when thinking about equivalence classes.  We choose to call things equivalent that have the same product even if the process to get them or their strict meaning is different.&lt;/p&gt;&#xA;" OwnerUserId="7398" LastActivityDate="2016-11-08T20:40:56.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11631" PostTypeId="2" ParentId="11618" CreationDate="2016-11-08T22:14:18.737" Score="1" Body="&lt;p&gt;Geometricaly, ln(x) is the area from 1 to x under the graf of y=1/x.&#xA;Geometricaly, derivation of the area under f(x) is f(x).&#xA;Hence, d(ln(x))/dx = 1/x.&#xA;exp(x) is the inverse of ln(x) i.e. if y=ln(x) then x=exp(y).&#xA;Hence, if dy/dx=1/x then dx/dy=x i.e. the derivative of exp is exp. &lt;/p&gt;&#xA;" OwnerUserId="7399" LastActivityDate="2016-11-08T22:14:18.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11632" PostTypeId="2" ParentId="11627" CreationDate="2016-11-09T16:21:22.553" Score="4" Body="&lt;p&gt;It seems like a problem of unclear or inapplicable definitions. As stated in the comments, it sounds like you've given the students the following definition: Given an angle $t$, we consider a right triangle with an angle of measure $t$, and define $\cos(t)$ to be the ratio of the side length of this triangle adjacent to the given angle to the length of the hypotenuse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is, this definition only makes sense when $0 &amp;lt; t &amp;lt; \pi/2$ (or, if you prefer degrees, $0^\circ &amp;lt; t &amp;lt; 90^\circ$. Indeed, a right triangle's other two angles have measure strictly between $0^\circ$ and $90^\circ$. But you've given the students a problem that assumes cosine has been defined for any $t$ between $0^\circ$ and $360^\circ$, which makes no sense with the definition they've been given!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would recommend instead defining cosine as the $x$-coordinate on the unit circle. A simple diagram (drawing an appropriate right triangle inside the unit circle) shows this agrees with the other definition for angles in the first quadrant, but the unit circle definition also clearly makes sense for any angle. Plus, it makes it much easier to reason about this kind of question — just draw the line $x = 1/2$ and see where it intersects the unit circle, then use geometric reasoning to find the angles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm still a bit confused about what you mean by &quot;algebraic methods&quot;, by the way. Both the triangle definition and the unit circle definition are geometric, so you really can't compute values of cosine using these definitions without appealing to geometry.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-11-09T16:21:22.553" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11633" PostTypeId="2" ParentId="1447" CreationDate="2016-11-09T21:36:22.677" Score="9" Body="&lt;p&gt;As a simple illustration of the most basic case, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}/2\mathbb{Z}$&lt;/span&gt;, I sometimes push the light switch of the classroom about a dozen times in a row, too fast for them to count. Then I ask: &amp;quot;how many times did I push the switch?&amp;quot;. They of course can't answer. Then I ask: &amp;quot;did I push the switch an odd or even number of times?&amp;quot;, and then just by comparing the lights before and after, they can tell. I then conclude that as far as the state of the lights is concerned, the number of time does not matter, only its parity. It then makes sense to have a vocabulary to ignore all irrelevant information from a mathematical object given a precise purpose. This is what equivalence classes are for.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2021-07-15T10:41:59.957" LastActivityDate="2021-07-15T10:41:59.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11635" PostTypeId="2" ParentId="11548" CreationDate="2016-11-10T15:20:46.720" Score="3" Body="&lt;p&gt;In order to help you, we should know which country you are from. In Germany, the library of the MPI in Berlin seems to be the only library where you can find these books, see &lt;a href=&quot;http://gso.gbv.de/DB=2.1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://gso.gbv.de/DB=2.1/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2016-11-10T15:20:46.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11636" PostTypeId="1" CreationDate="2016-11-11T08:16:11.267" Score="8" ViewCount="258" Body="&lt;p&gt;I need to come up with a mathematical task for middle school (9th grade), which involves either algebra, functions, probability or statistics (anything but geometry actually). My problem is, that the idea behind the task is to allow students to work together and learn from ideas of each other. Therefore, the task should be such that there are 5 different ways of solving it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will give you an example. In the picture below, you see a sequence of hexagons. If I asked students to calculate the perimeter of the hexagons, and to predict the perimeter of 20 hexagons, they would solve it in various ways. One student would count the perimeter (each edge is 1), find out an arithmetic sequence and derive the formula (4n+2). Another student could count the upper edges, multiply by 2 and reduce 2, leading to the same solution. A third student could multiply the number of hexagons by 6 and reduce the mutual edges. What I am saying, that different approaches lead to the same solution (4n+2).&#xA;&lt;a href=&quot;https://i.stack.imgur.com/DrykF.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DrykF.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I need to come up with another idea where students can come up with different solutions, different ways of thinking, and end up with the same final solution. The problem should not be too hard, middle school, and not concerning geometry (for algebra lesson, which includes the topics I specified above).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example below doesn't count as geometry since it involves sequences. No geometry means no Euclidean geometry. I would appreciate it if you could give me some ideas because I am pretty much lost. Thank you in advance.&lt;/p&gt;&#xA;" OwnerUserId="7408" LastEditorUserId="376" LastEditDate="2016-11-13T09:24:38.123" LastActivityDate="2016-11-14T11:59:29.903" Title="Mathematical Task with Various Solutions" Tags="|secondary-education|algebra|problem-design|probability|middle-school|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11637" PostTypeId="1" CreationDate="2016-11-11T17:02:39.200" Score="15" ViewCount="212" Body="&lt;p&gt;I am often tasked with teaching large enrollment courses, and one thing that becomes obvious quickly is that students attempt to pick up new information and solve problems at much different rates. They also succeed at their tasks with different frequencies. A simple, admittedly naive, classification system is that they exhibit $2 \times 2$ cases:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Students who quickly absorb new information and process it reliably. These students typically have a good background for the course and a strong mathematical aptitude. It is very hard for them not to be successful in the course, though often they will not learn as much because I will not teach the class at a fast enough pace for them.&lt;/li&gt;&#xA;&lt;li&gt;Students who quickly absorb new information, but often process it unreliably. These students typically rush through problems, often overlooking key reasoning steps along the way.&lt;/li&gt;&#xA;&lt;li&gt;Students who slowly absorb new information, but are capable of processing it reliably. These students can easily be overwhelmed by a course that moves too quickly, as they cannot understand the reasoning steps when they are presented too quickly for them to process. But when given adequate time to process what they are learning, they are often able to reach mastery.&lt;/li&gt;&#xA;&lt;li&gt;Students who slowly absorb new information, and are not capable of processing it reliably. Such students often lack adequate background for the course and typically do not have a very strong mathematical aptitude. It is very hard for them to be successful in the course, as there is not adequate time for them to address all the obstacles to success they need to overcome in the short time of the course.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Of course, many students don't fit neatly into one of these categorizations, but to the extent this model gives some approximation to what goes on in the classroom, what are effective methods you have for helping students in cases 2 and 3 to succeed? How do you handle giving lectures to an audience that consists of a mixed group of students that run the spectrum of the four cases. I generally try to encourage students in case 4 to drop the course and, if possible and appropriate, I try to encourage students in case 1 to move to a more challenging course for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I struggle with how to balance the students in the middle two cases. My style preference is to tend toward the slower side of presentation, using the extra time to detail carefully the specific reasoning skills used in each step of the problem and trying to take time to summarize the overall strategy when solving problems that are sufficiently complex. In theory, it works perfectly. It forces the students in case 2 to pay attention to the full depth of the problem, while it gives sufficient coverage so students in case 3 so they can achieve mastery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice, it does not work this way. The students in case 2 often become bored by the slow pace; rather than pay attention to the extra details I discuss, they often disengage from the material. On the other hand, the students in case 3 often become dependent on detailed explanations and typically don't show development in overall problem solving skills. While they do tend to master the specific skills that we spend time in class on, but these skills often don't generalize well beyond narrow application in the context they saw the material. And, of course, a slower pace means that I have to sacrifice some content relative to a faster pace.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is, what factors do you consider when you plan the pacing of your course? Are there ways to use pace in a course to improve performance among some student groups? Is there research suggesting pacing decisions lead to tangible outcome differences?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I typically teach introductory calculus courses, but welcome input from any specific mathematics course at the college or high school level, or general course-independent observations.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="143" LastEditDate="2016-11-12T15:57:41.063" LastActivityDate="2016-11-12T15:57:41.063" Title="Pacing your teaching for a variety of learning speeds" Tags="|course-design|classroom-management|large-lecture|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11639" PostTypeId="2" ParentId="11637" CreationDate="2016-11-12T09:55:09.713" Score="5" Body="&lt;p&gt;I don't know how successful it is, but I use a pattern of&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;short piece of theory and examples&lt;/li&gt;&#xA;&lt;li&gt;practice questions for the students to complete&lt;/li&gt;&#xA;&lt;li&gt;talk through the solutions&lt;/li&gt;&#xA;&lt;li&gt;repeat&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I do this by combining slides with writing on the board.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I think this addresses the question is that students can make use of the 'practice question' time in different ways. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some are still copying down the written material. Some are re-reading the material to process it at a speed they feel happier with. Some are slowly working through the straight-forward questions. Some speed through the easier questions, then ponder the more stretching ones. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the atmosphere is right, they also use each other as learning resources, and practice actually speaking maths. And some take a break, hopefully regaining some concentration as a result. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By asking for their solutions, you also get a bit of feedback about how they are progressing with the material, helping you to judge whether you need to change pace.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-11-12T09:55:09.713" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11640" PostTypeId="2" ParentId="11627" CreationDate="2016-11-12T13:54:27.450" Score="1" Body="&lt;p&gt;For the easy memorizing of special points:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\cos(0^\circ)=\frac{\sqrt4}2\\\cos(30^\circ)=\frac{\sqrt3}2\\\cos(45^\circ)=\frac{\sqrt2}2\\\cos(60^\circ)=\frac{\sqrt1}2\\\cos(90^\circ)=\frac{\sqrt0}2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same thing for $\sin(\theta)$ but going backwards.  Then it should be easy to see that $x=60^\circ$ is one answer.  From there, I would try to show them the unit circle and how it is symmetric.  Using that, it should become obvious that $300^\circ$ is the only other answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beyond that, either you are teaching some pretty advanced trigonometry, it happens to be a special special point, like $\cos(15^\circ)$ using the half-angle theorem, or you give them calculators.  I can't imagine it being any other way, unless you were teaching a numerical analysis type of class.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="6220" LastEditDate="2016-11-12T14:03:28.393" LastActivityDate="2016-11-12T14:03:28.393" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11641" PostTypeId="2" ParentId="11636" CreationDate="2016-11-12T14:30:51.963" Score="0" Body="&lt;p&gt;How about this problem from the Iranian 7th grade math textbook:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;How can we transform the colored octagon into the octagon C? Name the transformation(s).&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are at least 5 options for just one transformation!&#xA;&lt;a href=&quot;https://i.stack.imgur.com/w38Cp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/w38Cp.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; My bad. I didn't notice that geometry is not disired!&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-11-13T07:20:47.333" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11642" PostTypeId="2" ParentId="11636" CreationDate="2016-11-12T15:55:23.783" Score="5" Body="&lt;p&gt;Here is one I enjoyed from middle school.  This was a project:  I think we had a whole week to experiment, and discuss, and come up with a solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider a rectangle a 231 by 84 rectangle which is tiled with 1 by 1 squares.  How many squares does a diagonal of this rectangle pass through?  I think this was phrased in terms of a mouse running from one hole to another along a tiled floor (for cuteness points).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We did a lot of experiments with smaller rectangles.  Students noticed that sometimes the line would pass through a lattice point, and asked if this counted as passing through all 4 adjacent squares or not.  We had a discussion about this and decided, as a group, that we would define &quot;passing through a square&quot; to exclude only touching one of the corners:  you had to pass through the &quot;interior&quot; of the square.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;People had lots of different approaches. The solution involves the common factors of the edge lengths, making this more &quot;number theory&quot; than &quot;geometry&quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-11-12T15:55:23.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11644" PostTypeId="1" AcceptedAnswerId="11942" CreationDate="2016-11-13T08:44:03.877" Score="3" ViewCount="1838" Body="&lt;p&gt;I received the error message &quot;Question Syntax. There is an error in the syntax of this question. One or more variables may be invalid.&quot; when I tried to make the following question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compute $\|\langle [x], [y], [z] \rangle \times \langle [u], [v], [w] \rangle \|$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer I entered was&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\sqrt{(y \times w - z \times v)^2 + (z \times u - x \times w)^2 + (x \times v - y \times u)^2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone help me, please ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: I am using the &quot;blackboard learn&quot; course management software. I am a teacher, making an online quiz for my students.&lt;/p&gt;&#xA;" OwnerUserId="6893" LastEditorUserId="6893" LastEditDate="2016-11-14T13:56:15.610" LastActivityDate="2017-01-26T01:12:28.853" Title="Problem with &quot;Calculated Formula&quot; question in blackboard online test" Tags="|technology-in-education|assessment|online-instruction|" AnswerCount="1" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="11647" PostTypeId="2" ParentId="11617" CreationDate="2016-11-13T14:31:41.810" Score="2" Body="&lt;p&gt;Students can play many estimation games. Students can find objects in nature and then challenge others to estimate the length, weight, and surface area.  These can be checked with measuring tools and whoever is closest gets a point.  Determining who is closest is also a mathematical exercise.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also make a list of things to find (a scavenger hunt) with a range of lengths or weights, e.g. between 5 and 6 inches or between 25 and 30 grams. Give them a time limit to find the items.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my students favorite games which can be played when walking in a group is where I pick a number (e.g. 3,050,322, 3/7,  -25, or 0.55673) and the students have to guess.  I give them no clues to start but they are told higher or lower.  Give the class a time limit to find the number.  Start with reasonable numbers e.g. a whole number in the thousands and work your way to harder numbers.  Always write the number down on a paper in your pocket so they know it's really the number you picked.  If they guess the number in the time limit they get a point and if not you do.  For some reason they adore this game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope these ideas help.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2016-11-13T14:31:41.810" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11649" PostTypeId="2" ParentId="11636" CreationDate="2016-11-14T11:59:29.903" Score="0" Body="&lt;p&gt;One possibility:&#xA;Counting &lt;a href=&quot;https://oeis.org/A000055&quot; rel=&quot;nofollow noreferrer&quot;&gt;the number of &lt;em&gt;unlabled&lt;/em&gt; trees&lt;/a&gt;:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/cXt9m.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cXt9m.gif&quot; alt=&quot;UnlabeledTrees&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No closed formula is known, but the number for each small $n$ is known.&#xA;You'd have to explain&#xA;when two trees are structurally different. &#xA;Aiming for $a(7)=11$ would generate a lot of variety. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Counting the number of &lt;em&gt;labeled&lt;/em&gt; trees leads to Cayley's formula, $n^{n-2}$.&#xA;It may be more satisfying to have a formula, &#xA;but I think it is more difficult to manage the labeling, as the count&#xA;grows so quickly.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-11-14T11:59:29.903" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11650" PostTypeId="1" CreationDate="2016-11-14T15:42:05.593" Score="4" ViewCount="84" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Are there any existing research on the effectiveness of computer simulation and/or interactive visualization for the learning of mathematical concepts?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I ask because:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;There has been some efforts in developing &amp;quot;virtual laboratories&amp;quot; in the physical sciences and part of the justification is that virtual laboratories are better than no laboratories: this is especially the case for things like environmental sciences where it may not always be possible to travel to a destination with the right sort of environment.&lt;/p&gt;&#xA;&lt;p&gt;By extension one may expect there to be certain benefits derived from giving students &amp;quot;hands-on&amp;quot; access to mathematical/geometric objects. But is there any research that actually qualifies/quantifies the benefit?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I've seen lots of people pushing the use of things like &lt;a href=&quot;https://www.desmos.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Desmos&lt;/a&gt;: one of the great things I like about it is that the graphs can interactively depend on adjustable parameters. Intuitively I would say that these kinds of models with visual feedback are more effective than simply a graph drawn on a black board. But is this supported by research evidence?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="125" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-11-15T02:10:29.263" Title="Research on the effects of simulation and interactive visualization" Tags="|undergraduate-education|secondary-education|reference-request|technology-in-education|education-research|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11651" PostTypeId="1" CreationDate="2016-11-14T19:35:19.680" Score="15" ViewCount="794" Body="&lt;p&gt;Mathematics can come across as a sterile, dead subject - a catalogue of techniques long-ago decided, and forever relearned by each successive generation of students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is &lt;em&gt;approximately&lt;/em&gt; true for elementary and secondary mathematics, and for the standard progression of undergraduate courses (eg, Calculus 1,2,3, discrete math / combinatorics, ODE + Vector Calc, Analysis and Algebra).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the subject is alive and kicking, with many thousands of active researchers learning, creating, refining, and publishing every day. But the vast majority of fresh research requires considerable expertise to understand, and are therefore inaccessible to younger students of the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What, then, are some recent results that are interesting and accessible to students at (say) a secondary school level, which might exemplify that the subject remains active?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of examples that come to mind (which could be fleshed out as answers) are the recent progress against the Twin Prime Conjecture, and the surprising observation that primes ending in $X$ 'favor' being followed by a prime ending in $Y$, for various $(X,Y)$ pairs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where it's appropriate, please include links to any media treatment of the result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's roughly define 'recent' as being within the lifetime of some collection of students.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2022-01-13T21:09:05.140" Title="What are some recent, interesting, accessible pieces of mathematics" Tags="|student-motivation|big-list|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11653" PostTypeId="2" ParentId="11651" CreationDate="2016-11-14T22:47:42.223" Score="26" Body="&lt;p&gt;Perhaps: The discovery &lt;strike&gt;a year ago&lt;/strike&gt;&#xA;in 2015 of a new tiling of the plane by&#xA;a convex polygonal tile, found by&#xA;Mann, McLoud, and Von Derau (the latter of whom was an undergraduate at&#xA;the time of the discovery):&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/eMead.png&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Here is a nice article on the discovery in&#xA;&lt;a href=&quot;https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Guardian&lt;/a&gt;,&#xA;by Alex Bellos.&#xA;As Alex says, the problem has been studied for &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; years now,&#xA;since Reinhardt in 1918.&#xA;It remains unknown if the current list of &lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt; such tilings is complete. (But see update below.)&lt;/p&gt;&#xA;&lt;p&gt;Not only is there no algorithm to determine if a candidate tile can indeed&#xA;tile the plane, &lt;a href=&quot;https://cstheory.stackexchange.com/q/32538/337&quot;&gt;it is not even known&lt;/a&gt; that the problem is decidable.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mann, Casey, Jennifer McLoud-Mann, and David Von Derau. &amp;quot;Convex pentagons that admit &lt;span class=&quot;math-container&quot;&gt;$ i $&lt;/span&gt;-block transitive tilings.&amp;quot;&#xA;&lt;a href=&quot;https://arxiv.org/abs/1510.01186&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv abstract&lt;/a&gt; (2015).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;.&#xA;Rao, Michaël (2017), Exhaustive search of convex pentagons which tile the plane, &lt;a href=&quot;https://arxiv.org/abs/1708.00274&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv:1708.00274&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We present an exhaustive search of all families of convex pentagons&#xA;which tile the plane. This research shows that there are no more than&#xA;the already 15 known families. In particular, this implies that there is&#xA;no convex polygon which allows only non-periodic tilings.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2022-01-09T14:28:45.747" LastActivityDate="2022-01-09T14:28:45.747" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11654" PostTypeId="2" ParentId="11650" CreationDate="2016-11-15T02:10:29.263" Score="3" Body="&lt;p&gt;This is not a knowledgeable answer, just one citation, perhaps not the &#xA;most relevant (as it is a decade old):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Rösken, Bettina, and Katrin Rolka. &quot;A picture is worth a 1000 words–The role of visualization in mathematics learning.&quot; &lt;em&gt;Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education&lt;/em&gt;. Vol. 4. 2006.&#xA;  (&lt;a href=&quot;http://files.eric.ed.gov/fulltext/ED496934.pdf#page=465&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I mention this primarily because (a) it reviews related work, and (b) because of this interesting quote in its conclusion:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;These aspects highlight the ambivalence of visualization as Tall (1994, p. 37) points&#xA;  out: “It is this quality of using images &lt;em&gt;without being enslaved by them&lt;/em&gt; which gives&#xA;  the professional mathematician an advantage but can cause so much difficulty for the&#xA;  learner.” &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-11-15T02:10:29.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11656" PostTypeId="2" ParentId="11651" CreationDate="2016-11-15T20:47:53.127" Score="9" Body="&lt;p&gt;The 2016 result about &lt;a href=&quot;https://arxiv.org/abs/1603.03720&quot; rel=&quot;noreferrer&quot; title=&quot;Arxiv Paper&quot;&gt;Unexpected biases in the distribution of consecutive primes&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is really pretty simple to understand - the distribution of primes had been supposed to be unconditionally random, but this interesting bias is suddenly discovered. The amazing thing about it is that the observation could have been made by pretty well any beginner computer programmer in the past 30 years, but only now had anyone noticed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story had a number of treatments in a range of media: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.nature.com/news/peculiar-pattern-found-in-random-prime-numbers-1.19550&quot; rel=&quot;noreferrer&quot;&gt;http://www.nature.com/news/peculiar-pattern-found-in-random-prime-numbers-1.19550&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/&quot; rel=&quot;noreferrer&quot;&gt;https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.scientificamerican.com/article/peculiar-pattern-found-in-random-prime-numbers/&quot; rel=&quot;noreferrer&quot;&gt;https://www.scientificamerican.com/article/peculiar-pattern-found-in-random-prime-numbers/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="308" LastActivityDate="2016-11-15T20:47:53.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11658" PostTypeId="2" ParentId="11651" CreationDate="2016-11-16T14:06:02.397" Score="3" Body="&lt;p&gt;There are elementary results in mathematics being produced regularly either in the sense that the ideas don't need a lot of background to understand them or sometimes that even the &quot;details&quot; of the proofs are comprehensible. An example, is that of the de Bruijn graph, which has ties to the theory of Eulerian circuits in directed graphs and has found many applications, in particular, to assisting with genome sequencing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/De_Bruijn_graph&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/De_Bruijn_graph&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2016-11-16T14:06:02.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11660" PostTypeId="2" ParentId="1447" CreationDate="2016-11-17T00:04:12.730" Score="4" Body="&lt;p&gt;To be honest, I wasn't sure to write this answer or not, since it was hard to choose what to write about! Believe or not, my whole PhD thesis entitled &quot;&lt;a href=&quot;http://wrap.warwick.ac.uk/62104/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Equivalence&lt;/a&gt;&quot; was to provide a ground for answering what you have asked. Part of which, dedicated to equivalence relations, has been published under the title &quot;&lt;a href=&quot;http://rdcu.be/mEf7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Experiencing equivalence but organizing order&lt;/a&gt;&quot;. There you can find a lot of historical information and a task that I used with students. I hope, the paper helps you to see the complexity behind nearly all of the answers suggested so far. I also really suggest a glimpse of Chapter 9 of the thesis where you can find some pedagogical points. There you can see that it is easier to motivate partitions than equivalence classes (though mathematically they are equivalent). More, it is not easy at all to move from partitions to equivalence class.        &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2016-11-17T00:04:12.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11661" PostTypeId="2" ParentId="11618" CreationDate="2016-11-18T22:51:32.890" Score="3" Body="&lt;p&gt;Take the graph of an exponential function such as $2^x$. If, for example, you shift it to the right by 3 units and then stretch it vertically by a factor of 8, you get the same graph back again. But the vertical stretch also increases the slope of the tangent line by a factor of 8. This holds for any shift, and therefore the derivative of this function is proportional to the function itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same argument holds for an exponential function with any base. We define $e$ as the base for which the constant of proportionality is 1.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2016-11-18T22:51:32.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11662" PostTypeId="1" AcceptedAnswerId="11675" CreationDate="2016-11-20T16:20:52.497" Score="6" ViewCount="329" Body="&lt;p&gt;I am looking for a test, paper based, execution time less than one hour, for an initial assessment of the math knowledge level of a secondary school student. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I need to evaluate, very approximately, if the real knowledge of the student is equivalent to the average of first year students or second year students, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are talking about very exceptional cases, individual students that comes to recovery courses with long records of problems in maths learning.&lt;/p&gt;&#xA;" OwnerUserId="7441" LastEditorUserId="7441" LastEditDate="2016-11-21T10:37:32.743" LastActivityDate="2016-11-25T18:27:05.397" Title="Test of level of a student for secondary school" Tags="|secondary-education|assessment|testing|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11663" PostTypeId="1" CreationDate="2016-11-23T01:31:19.633" Score="1" ViewCount="107" Body="&lt;p&gt;My brother is taking the normal GRE (not the math subject test), and I need to help him learn geometry. I know basic geometry myself, but I need a book that will present it well and will have good exercises. Any suggestions are appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me the GRE covers about 50% of U.S. highschool geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Topics in the GRE include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Vertical angles&lt;/li&gt;&#xA;&lt;li&gt;Parallel lines with a transverse&lt;/li&gt;&#xA;&lt;li&gt;Area of &quot;slices&quot; of circles&lt;/li&gt;&#xA;&lt;li&gt;Exterior angles of triangles&lt;/li&gt;&#xA;&lt;li&gt;Similar Triangles&lt;/li&gt;&#xA;&lt;li&gt;Triangles inscribed in circles&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;etc.&lt;/p&gt;&#xA;" OwnerUserId="7075" LastEditorUserId="2083" LastEditDate="2022-10-19T06:48:50.520" LastActivityDate="2022-10-19T06:48:50.520" Title="GRE Geometry Reference Request?" Tags="|reference-request|geometry|usa|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11664" PostTypeId="2" ParentId="11663" CreationDate="2016-11-23T02:08:41.757" Score="5" Body="&lt;p&gt;There are many resources on this topic, most of which are profit-making and at some point require a fee.&#xA;At least one exception is the linking of the GRE topics to the relevant Kahn Academy videos: &lt;a href=&quot;https://support.khanacademy.org/hc/en-us/community/posts/5226512153741-GRE-Prep-According-to-ETS&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;link here&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Also, ETS itself offers a free review package, with sample questions: &lt;a href=&quot;https://www.ets.org/s/gre/pdf/gre_math_review.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;PDF download&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KGFfT.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KGFfT.jpg&quot; alt=&quot;GRE Figs&quot; /&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;Figures from GRE Math Review.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2022-10-19T00:03:41.337" LastActivityDate="2022-10-19T00:03:41.337" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11665" PostTypeId="2" ParentId="11651" CreationDate="2016-11-23T12:03:05.337" Score="7" Body="&lt;p&gt;Not a single answer but rather a resource:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The snapshots of modern mathematics from the mathematical research institute Oberwolfach (&lt;a href=&quot;http://www.mfo.de/math-in-public/snapshots/&quot; rel=&quot;noreferrer&quot;&gt;http://www.mfo.de/math-in-public/snapshots/&lt;/a&gt;) aim to provide pretty much exactly what you ask for. Research mathematicians present some &quot;recent developments&quot; in their field in a way that is understandable for high school students/ math teachers/ undergraduates (the actual level varies quite a bit between the various snapshots).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The snapshots are freely available on the homepage of the institute and cover quite a lot of topics.&lt;/p&gt;&#xA;" OwnerUserId="7448" LastActivityDate="2016-11-23T12:03:05.337" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11666" PostTypeId="1" CreationDate="2016-11-23T17:49:04.083" Score="17" ViewCount="655" Body="&lt;p&gt;Here is the problem that I asked undergraduate students of an introductory number theory course to prove: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Prove that if $N$ is a nonsquare natural number, then $\sqrt N $  is&#xA;  irrational.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Many of them proceeded as follows: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose we can write it as $\frac{p}{q}$ where $p$ and $q$ have no&#xA;  common factor. Thus, $N=\frac{p^2}{q^2}$. This can only happen if&#xA;  $q^2=1$; that means $N=p^2$, contradicting the fact that $N$ is a&#xA;  nonsquare natural number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Any idea of how I could possibly explain what is going wrong in a constructive way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also love to have some similar examples of such &quot;proofs&quot;, though I don't know what a similar example might look like. I am just hoping your answer help me to see a general phenomenon for which the &quot;proof&quot; above is just an example!!  &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="1217" LastEditDate="2016-11-26T09:13:02.297" LastActivityDate="2017-03-10T07:55:59.730" Title="How to explain what is wrong in this &quot;proof&quot; that $\sqrt N$ must be irrational?" Tags="|proofs|number-theory|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11668" PostTypeId="2" ParentId="11666" CreationDate="2016-11-23T22:14:34.983" Score="11" Body="&lt;p&gt;What's wrong is that most of the justification is missing. &lt;em&gt;Why&lt;/em&gt; can $N = p^2/q^2$ only happen if $q^2 = 1$? This can be justified using the fundamental theorem of arithmetic (which states that integers have unique prime factorization), but that justification needs to be made explicit, and they need to make sure that the fundamental theorem of arithmetic has already been proven.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first sentence also assumes that rational numbers can be written as a ratio of &lt;em&gt;coprime&lt;/em&gt; integers, which also should be justified. The usual way to do this is to appeal to the Euclidean algorithm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key principle here is that every claim must be justified with precise and logically valid reasoning. To construct other examples of incomplete proofs like this, just take any valid proof and remove parts of the reasoning, leaving bare assertions in their place.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-11-23T22:14:34.983" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11669" PostTypeId="2" ParentId="8575" CreationDate="2016-11-24T02:34:06.950" Score="2" Body="&lt;p&gt;Finally there is a book &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.crcpress.com/A-MatLab-Companion-to-Complex-Variables/Wunsch/p/book/9781498755672&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.crcpress.com/A-MatLab-Companion-to-Complex-Variables/Wunsch/p/book/9781498755672&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;MATLAB companion to complex variables&quot; published Aug 2016. I can't wait till my library acquire this book.&lt;/p&gt;&#xA;" OwnerUserId="3560" LastActivityDate="2016-11-24T02:34:06.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11670" PostTypeId="2" ParentId="11666" CreationDate="2016-11-24T04:05:48.080" Score="1" Body="&lt;p&gt;Perhaps start by asking if $\frac{p}{q}=M\in Z$ implies $q=1$. This might focus the student on the hypothesis that $p$ and $q$ are relatively prime. Then direct the conversation to relating the factors of $p$ and $q$ to $p^2$ and $q^2$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you raise is an interesting problem in teaching proof. The original statement involves irrational numbers, rational numbers, and integers. Maybe it is better to start by not even bringing up the main statement about $\sqrt{N}$, and not bringing in rational numbers. Just a cold start, saying &quot;Let's look at the (Diophantine) equation $p^2=Nq^2$.&quot; Only &#xA;after analyzing this in the integers do you connect it to the notion of $\sqrt{N}$ being irrational.   &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-11-24T04:05:48.080" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11671" PostTypeId="2" ParentId="11666" CreationDate="2016-11-24T11:11:29.853" Score="5" Body="&lt;p&gt;After writing for the students where they have gone wrong with their given proofs, I realized that it was just me thinking they are wrong, just because I was accustomed to seeing the fundamental theorem of arithmetic explicitly used and stated. To see the point, let me quote the proofs given by J. L. &lt;a href=&quot;http://libarch.nmu.org.ua/bitstream/handle/GenofondUA/11063/64c4116d0ffbd0624a8d4fa5a4183368.pdf?sequence=1&amp;amp;isAllowed=y&quot; rel=&quot;noreferrer&quot;&gt;Lagrange in his Lectures on Elementary Mathematics&lt;/a&gt; (1898) and More than half a century earlier (1831), Augustus De Morgan in his book &lt;a href=&quot;ftp://skipjack.soc.lib.md.us/pub/gutenberg/3/9/0/8/39088/39088-pdf.pdf&quot; rel=&quot;noreferrer&quot;&gt;ON THE STUDY AND DIFFICULTIES OF&#xA;MATHEMATICS&lt;/a&gt; (I've seen these in &lt;a href=&quot;http://www.cut-the-knot.org/proofs/sq_root.shtml&quot; rel=&quot;noreferrer&quot;&gt;cut-the-knot&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Lagrange&lt;/strong&gt;: it's impossible to find a whole number which multiplied by&#xA;  itself will give 2. It cannot be found in fractions, for if you take a&#xA;  fraction reduced to its lowest terms, the square of this fraction will&#xA;  again be a fraction reduced to its lowest terms, and consequently&#xA;  cannot be equal to the whole number 2.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Augustus De Morgan&lt;/strong&gt; : 7 is not made by the multiplication either of any whole number or fraction by itself. The first is evident; the&#xA;  second cannot be readily proved to the beginner, but he may, by taking&#xA;  a number of instances, satisfy himself of this, that no fraction which&#xA;  is really such, that is whose numerator is not measured by its&#xA;  denominator, will give a whole number when multiplied by itself, thus&#xA;  4/3×4/3=16/9 is not a whole number, and so on.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;True, both of them appeal to the fundamental theorem of Arithmetic indirectly. But, both of them are based on what the authors have assumed to be correct previously. This reminds me of the role of the context of a proof as described by Ian Stewart and David Tall in &lt;a href=&quot;http://www.maa.org/press/maa-reviews/the-foundations-of-mathematics&quot; rel=&quot;noreferrer&quot;&gt;The Foundations of Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One situation that is of great concern to students is what constitutes&#xA;  a proof acceptable in an examination. To a certain extent, the answer&#xA;  depends on the examiner, but part of the anxiety is due to uncertainty&#xA;  about the context...in an examination it may not be clear at which&#xA;  level a proof is required. Do all the steps have to be included? What&#xA;  can safely be missed out?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, it seems that I should change all the comments I have written for the students, and instead, ask them to specify where they have used the fundamental theorem of arithmetic! And in the future, If the use of a certain tool/concept/approach matters to me, I should specify it at the outset!! &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2016-11-24T11:11:29.853" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11672" PostTypeId="2" ParentId="11666" CreationDate="2016-11-24T11:35:38.573" Score="15" Body="&lt;p&gt;This is indeed tricky, and it seems to me the most effective way (in far more general, similar situations) is to &lt;strong&gt;show them the problem would be to have them apply their method to another, close problem where the answer is actually opposite.&lt;/strong&gt; Either they will explain why it does not apply there, and you can argue that the difference is subtle enough to warrant more details in their original proof, details which must stress the difference between the examples, or they first believe they can prove the same result in the second case and your point will be made in an even more compelling way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the present case, I suggest to ask whether there are two integers $p,q$ such that &#xA;$$18 = \frac{p^2}{q^3}$$&#xA;If they think there are not, and apply the previous reasoning, then ask them about $p=12$ and $q=2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the more general case, your extra knowledge as a teacher often comes in handy: you would take the most general positive result, remove the most harmless-looking hypothesis, look in the proof where it is needed, and cook up an example where the theorem is not true from there (but beware that in some cases, the hypothesis might not be needed after all if one settles for a more complicated proof, so wide culture is key). This is one of the reasons, by the way, teachers should know &lt;em&gt;way&lt;/em&gt; much than a perfect mastering of the content they teach.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2017-03-10T07:55:59.730" LastActivityDate="2017-03-10T07:55:59.730" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11673" PostTypeId="1" CreationDate="2016-11-24T12:06:40.800" Score="7" ViewCount="361" Body="&lt;p&gt;I’m looking for books of a general interest in maths, like a maths novel, a fun book about maths, a humorous introduction to higher concepts, something outside of tests, something they can read on their free time and appreciate the beauty of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me add more details. I’m working as a TA for the course ‘’Maths and Physics’’ at a university college. The students have chosen the direction of environmental sciences. They basically have only two courses of maths in the entire bachelor program and no physics. They come from different backgrounds, but the majority of them has done some high school mathematics (not deeply) and a very elementary course in statistics. During the one semester course we will cover stuff from precalculus (functions, trigonometry, logarithms, etc.), stuff from introductory calculus (limits, derivatives, integrals), and then vectors and finally physics. Also the lecturer has been introducing them to number theory ideas (pigeonhole principle, golden ratio, induction).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My job is primarily to solve the exercises the lecturer gives them once a week and answer as many of their questions as time permits. I’m not satisfied with only that so I want to make them more interested in maths and eventually physics. Most of them won’t use maths again in another course in their program. So, now you have the whole picture, how can I peak their interest?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, one of my students is interested in the philosophy of maths. I have already given him an introductory article about the incompleteness theorem and another one about the ‘’Unreasonable effectiveness of mathematics’’ by Wigner. Do you have any further suggestions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning all the students it doesn’t have to be only books, any resource in general would be fine. But, maybe they can appreciate books more easily? I was thinking about a book introducing them to the why’s of mathematics or maybe some higher ideas (in an introductory level), or why not maths literature, perhaps a maths novel, a comic about mathematics? What are your ideas?&lt;/p&gt;&#xA;" OwnerUserId="7453" LastEditorUserId="5220" LastEditDate="2016-11-27T18:30:08.870" LastActivityDate="2022-11-10T21:28:30.947" Title="Entertaining and fun books about mathematics for (basically) liberal arts students" Tags="|undergraduate-education|mathematics-in-daily-life|recreational-learning|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11674" PostTypeId="2" ParentId="9679" CreationDate="2016-11-24T21:01:35.243" Score="1" Body="&lt;p&gt;As most colleges and universities are moving their online &lt;a href=&quot;https://en.wikipedia.org/wiki/OpenCourseWare&quot; rel=&quot;nofollow noreferrer&quot;&gt;open course ware&lt;/a&gt; to hosting sites such as &lt;a href=&quot;https://www.edx.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;edX&lt;/a&gt; and &lt;a href=&quot;https://www.coursera.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;coursera&lt;/a&gt; I would search their catalogs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Search results for calculus&lt;br&gt;&#xA; - &lt;a href=&quot;https://www.edx.org/course?search_query=calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;edX&lt;/a&gt;&lt;br&gt;&#xA; - &lt;a href=&quot;https://www.coursera.org/courses?languages=en&amp;amp;query=calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;coursera&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;TL;DR&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To find sites that host open and free online courses check the list of &lt;a href=&quot;https://en.wikipedia.org/wiki/Massive_open_online_course#Notable_providers&quot; rel=&quot;nofollow noreferrer&quot;&gt;notable providers&lt;/a&gt; in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Massive_open_online_course&quot; rel=&quot;nofollow noreferrer&quot;&gt;Massive open online course&lt;/a&gt; Wikipedia article&lt;/p&gt;&#xA;" OwnerUserId="1959" LastActivityDate="2016-11-24T21:01:35.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11675" PostTypeId="2" ParentId="11662" CreationDate="2016-11-25T18:27:05.397" Score="2" Body="&lt;p&gt;You might look at Tailwind placement exams. They align with curriculua in the US so maybe this is not what you want. The enterprise involves a partnership with the University of Wisconsin System.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://tailwindtesting.com/placementexams/math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://tailwindtesting.com/placementexams/math/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-11-25T18:27:05.397" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11676" PostTypeId="2" ParentId="7274" CreationDate="2016-11-26T23:56:38.143" Score="4" Body="&lt;p&gt;This one is somewhat &quot;real-world&quot; in that it involves a question that can be posed without knowing anything about L'Hopital's rule. But I must warn you that it takes three iterations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the point $(\cos\theta,\sin\theta)$ on the unit circle, drop a perpendicular to the $x$-axis, and consider the region bounded by that perpendicular, the $x$-axis, and the arc from $(1,0)$ back to $(\cos\theta,\sin\theta)$. Ask what fraction of the area of that region is between the &lt;b&gt;chord&lt;/b&gt; from $(1,0)$ to $(\cos\theta,\sin\theta)$ and the &lt;b&gt;arc&lt;/b&gt; from $(1,0)$ to $(\cos\theta,\sin\theta)$, and then take the limit of that fraction as $\theta\downarrow0.$&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2016-11-26T23:56:38.143" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11677" PostTypeId="2" ParentId="11673" CreationDate="2016-11-27T17:56:16.737" Score="6" Body="&lt;p&gt;I'll suggest the book &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/048627263X&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Flatland&lt;/em&gt;&lt;/a&gt; by Edwin A. Abbott. It's an interesting read about geometry and thinking in a higher dimension. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The downside is that this book is, well, antiquated to put it nicely. Among other things, only men get to be polygons; women are just line-segments.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2016-11-27T19:28:17.943" LastActivityDate="2016-11-27T19:28:17.943" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11678" PostTypeId="1" AcceptedAnswerId="11684" CreationDate="2016-11-27T20:16:28.250" Score="25" ViewCount="3522" Body="&lt;p&gt;I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in university.  We will be doing a lesson on visual proofs of the Pythagorean theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are at least three that are easy and fun to do using some paper triangles.  They'll be encouraged to vote on their favorite.  But these students are bound to ask why do it three ways?  Why do it &lt;a href=&quot;http://www.cut-the-knot.org/pythagoras/index.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;118 ways?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/10494/different-approaches-to-proofs-that-are-the-same&quot;&gt;This&lt;/a&gt; post, &lt;a href=&quot;https://math.stackexchange.com/questions/1637562/can-a-mathematical-theorem-be-proved-in-infinite-ways&quot;&gt;this&lt;/a&gt; post, and others do a fine job discussing what makes two proofs different and how many proofs a theorem can have, but what I want to know is this:  why do we care?  Is it just a sense of nerdy joy in finding a different argument for the same theorem?&lt;/p&gt;&#xA;" OwnerUserId="4703" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-11-29T18:29:37.013" Title="Why do we care about multiple proofs of the same theorem?" Tags="|proofs|concept-motivation|geometry|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11679" PostTypeId="2" ParentId="11678" CreationDate="2016-11-27T21:14:48.847" Score="20" Body="&lt;p&gt;I'd say different proofs usually employ different techniques, which in turn might be applicable to different sets of other theorems. So the more proofs I know for one theorem, the higher the chances that I'll be able to adapt at least one of them to a similar (or maybe not so similar) theorem I'm trying to prove.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, seeing several techniques applied to a single theorem also helps comparing the techniques, both in objective terms (e.g. is it constructive, is it algebraic) and subjective terms (e.g. do I consider the approach intuitive, can I see how the goal dictates the steps to reach it). If each technique were demonstrated using a different theorem, the differences in these theorems might overshadow the differences between techniques.&lt;/p&gt;&#xA;" OwnerUserId="1248" LastActivityDate="2016-11-27T21:14:48.847" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11680" PostTypeId="1" AcceptedAnswerId="11681" CreationDate="2016-11-27T22:55:07.260" Score="4" ViewCount="125" Body="&lt;p&gt;We know that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\! 2^{2^{2^2}} \ne \left[{\left(2^2\right)}^2\right]^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for some students, this might be confusing. Is there a good way to explain this?&lt;/p&gt;&#xA;" OwnerUserId="7232" LastActivityDate="2016-11-27T23:13:57.697" Title="How to explain this concept in tetration?" Tags="|undergraduate-education|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11681" PostTypeId="2" ParentId="11680" CreationDate="2016-11-27T23:13:57.697" Score="6" Body="&lt;p&gt;The main point here is that $[(2^2)^2]^2\neq 2^{[2^{(2^2)}]}$ --- in other words, order of operations matters.  That shouldn't be too unfamiliar, because they've seen the same in other contexts (and can do the calculations themselves to verify it).  The important thing to realize is that the expression is ambiguous, and we need to do something to resolve the ambiguity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the only thing to learn that there's an arbitrary convention about which one we mean when we don't include parentheses.  (I've usually see the rule explained as &quot;things in a smaller font happen first&quot;, which seems like a reasonable mnemonic.)  Again, this is similar to the order of operations they've learned with other operations.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-11-27T23:13:57.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11683" PostTypeId="2" ParentId="11678" CreationDate="2016-11-28T06:02:27.827" Score="6" Body="&lt;p&gt;The main motivation is that different proofs generalize to different situations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition to that&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Different proofs may require different tools and different theory. Sometimes, it is nice to have an &quot;elementary&quot; proof, which only uses simple tools, but maybe in a difficult or hard-to-understand way. Sometimes, another theory, once developed, might give certain result as an obvious corollary, but to really understand it one has to understand the entire theory.&lt;/li&gt;&#xA;&lt;li&gt;Different proofs often have slightly different assumptions. They might be written in a standard form, but upon inspecting the proof and seeing what assumptions are actually used, some of them might be weaker than what is written down. This is often the case in analysis, where there are several different degrees to which something can be continuous or differentiable.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Neither of these concerns is easy to demonstrate before university-level mathematics, I fear.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;A &lt;strong&gt;didactic reason&lt;/strong&gt; is that different people might understand or remember different proofs, especially if some are visual, others algebraic, and so on. A proof is always a way to reconstruct and remember a theorem.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2016-11-28T07:05:28.623" LastActivityDate="2016-11-28T07:05:28.623" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11684" PostTypeId="2" ParentId="11678" CreationDate="2016-11-28T06:47:26.230" Score="11" Body="&lt;p&gt;Worth noting here is that there is an additional level of certainty in the validity of a statement when you have multiple proofs of said statement. By having multiple ways to solve the same problem, all yielding the same answer, our surity of the statement increases dramatically. In a classroom setting you could perhaps bring up examples of &quot;proofs&quot; which were thought to be groundbreaking but were later proved wrong by subsequent results; on the other hand, you might also consider giving examples of when a theorem was questioned by the mathematical community but was confirmed by independent proofs, causing the theorem to be more widely accepted.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Originally From Comments:&lt;/em&gt;&lt;/strong&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are plenty of examples. For the first case, perhaps start with &lt;a href=&quot;https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-wrong&quot;&gt;this MathOverflow post&lt;/a&gt;. For example, that page lists the false result that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A convergent infinite series of continuous functions is continuous.&#xA;  Cauchy gave a proof of this (1821)...Five years later Abel pointed out&#xA;  that certain Fourier series are counterexamples.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A more trivial example is a tougher question. That same page lists the Euler's Formula $V\text{(ertices)}−E\text{(dges)}+F\text{(aces)}=2$ was long thought to hold for all polyhedra, and a proof was given that was initially accepted for many years until a counterexample was found.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the second case, immediate examples that come to mind include &lt;a href=&quot;https://en.wikipedia.org/wiki/Controversy_over_Cantor&amp;#39;s_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cantor's Set Theory&lt;/a&gt; (though subsequent proofs were largely due to Cantor himself, with some strong avocation from Hilbert) as well as &lt;a href=&quot;https://en.wikipedia.org/wiki/G%C3%B6del&amp;#39;s_incompleteness_theorems#First_incompleteness_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Godel's Incompleteness Theorem&lt;/a&gt; (see the page &lt;a href=&quot;https://en.wikipedia.org/wiki/Rosser%27s_trick&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rosser's Trick&lt;/a&gt; as well as &lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_sketch_for_G%C3%B6del&amp;#39;s_first_incompleteness_theorem#Boolos.27s_short_proof&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boolo's Short Proof&lt;/a&gt; ). &lt;em&gt;Neither of these examples are fantastic, but I am sure a little research could definitely yield some better examples!&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="7464" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2016-11-29T06:12:19.600" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11685" PostTypeId="2" ParentId="11678" CreationDate="2016-11-28T07:43:11.177" Score="3" Body="&lt;p&gt;For me this question brought to mind Bill Thurston's &lt;a href=&quot;https://arxiv.org/pdf/math/9404236v1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Proof and Progress in Mathematics&lt;/a&gt;. There is value in being able to understand the same idea from different viewpoints (that is, the same person adopting different viewpoints, not just different people choosing their own preferred one).&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-11-28T07:43:11.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11688" PostTypeId="2" ParentId="3933" CreationDate="2016-11-28T14:38:26.763" Score="1" Body="&lt;p&gt;I agree with the consensus that the general answer is an emphatic &lt;strong&gt;No&lt;/strong&gt;. As has been pointed, mathematics is fundamentally a study of valid means of reasoning, so the method is much more important than the answer itself. It helps to make this explicit to students. And in order to do that, it is necessary that the instructor think carefully about why this is the case. As the other answers explain this point very well, I won't elaborate further.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, context does matter. When the steps are simple and the method is well-known from the past, it would be absurd to require a student to show all the trivial intermediate steps. For example, in a middle school algebra class, I'd expect a student solving $3x + 7 = 19$ to methodically subtract $7$ from each side, then divide each side by $3$ and only then to arrive at the answer $x = 4$. On the other hand, if a student in calculus course encountered this equation as part of a more complicated problem, I'd have no problem with them immediately deducing $x = 4$ from the equation. That is the sense in which I mean that context matters.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2016-11-28T14:38:26.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11689" PostTypeId="2" ParentId="11678" CreationDate="2016-11-28T15:42:42.013" Score="8" Body="&lt;p&gt;Claude Monet did a famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Haystacks_(Monet_series)&quot; rel=&quot;noreferrer&quot;&gt;series&lt;/a&gt; of over 20 paintings, all of one or two haystacks, all in the the same field, with various types of light and weather conditions.  Here's one example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QGfnD.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QGfnD.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is another, for contrast:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/i1dGu.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/i1dGu.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A mathematical proof is much like a painting.  There is an overall conclusion and  many fine details, and one gets a real sense of beauty from nicely made ones.&lt;/p&gt;&#xA;" OwnerUserId="270" LastActivityDate="2016-11-28T15:42:42.013" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11690" PostTypeId="2" ParentId="11678" CreationDate="2016-11-28T17:10:39.127" Score="3" Body="&lt;p&gt;Sometimes a different proof can provide more insight, or a different kind of insight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a relatively elementary example, you could think of the angle sum formula for sine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first proof I learned used computed the length of a certain line segment in two ways:  using the law of cosines, and using the distance formula.  After a lot of algebra, the formula falls out.  However, I get very little intuition for why the result is true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A proof I discovered much later uses the linearity of rotation as a map from $\mathbb{R}^2 \to \mathbb{R}^2$. The angle sum formulas for both sine and cosine are the result of matrix multiplication of rotation matrices.  Not only is this proof more intuitive, it is so intuitive that I no longer remember the angle sum formulas:  I just take a second to derive them each time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take home message:  a proof is an explanation for why something works.  Sometimes some explanations appeal to you more than others.  So it is worth it to prove the same result in many ways.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-11-28T17:10:39.127" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11691" PostTypeId="1" AcceptedAnswerId="11696" CreationDate="2016-11-28T19:46:17.403" Score="6" ViewCount="1182" Body="&lt;p&gt;I am helping my brother study for the GRE and we have come across some problems like this in my old precalculus textbook:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1) Karen and Betty have been hired to pain a house. Working together, they can paint the house in $\dfrac 23$ the time it would take Karen working alone. Betty could paint the house by herself in $6$ hours. How long would it take Karen to paint the house working alone?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;2) Stan and Hilda can mown the lawn in 40 minutes if they work together. If Hilda works twice as fast as Stan, how long would it take Stan to mown the lawn alone?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The way I advised my brother to do these is to write the &quot;$d=rt$&quot; equation for every case: each person working alone, and both people working together; introduce as many variables as you need. From then, I just told him to try to manipulate the equations to solve for the unknown variable. However, he found this approach difficult. He said that somebody really smart could understand what's going on better and not have to rely on algebraic manipulations. Is there a better way to do these? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is how I solved them:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Suppose the house has $x$ square meters of surface area that needs to be painted. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$r_k =$ rate of Karen (in $m^2$/hour)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$r_B =$ rate of Betty (in $m^2$/hour)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$t_k =$ time it takes Karen to paint the house alone (in hours)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$r_k = 6$ hours (given) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The three $d=rt$ equations are&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = r_k \cdot t_k$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = 6 \cdot r_B$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = \dfrac 23 \cdot t_k \cdot (r_B + r_k)$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are trying to solve for $t_k$. Transform the second equation into $r_B = \dfrac x6$ and substitute in for $r_B$. Distribute $\dfrac 23 t_k$ as well:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = \dfrac 23 \cdot t_k \cdot \dfrac x6 + \dfrac 23 t_k r_k$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Substitute $r_k \cdot t_k = x$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = \dfrac 23 \cdot t_k \cdot \dfrac x6 + \dfrac 23 x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you can cancel the $x's$ and solve for $t_k$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Using similar definitions as above (and $x$ is $m^2$ of lawn now)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = r_S \cdot t_S$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = (2 r_S) \cdot t_H$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = (r_S + 2r_S) \cdot 40$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Setting the first and third equations equal to each other, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$r_S \cdot t_S = (3r_S) \cdot 40$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you can cancel $r_S$ and solve for $t_S$.&lt;/p&gt;&#xA;" OwnerUserId="7075" LastActivityDate="2017-04-14T16:01:30.233" Title="Is there a more intuitive way to solve combined rates of work problems?" Tags="|algebra|teaching|precalculus|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11692" PostTypeId="1" AcceptedAnswerId="11698" CreationDate="2016-11-28T20:36:57.107" Score="31" ViewCount="10240" Body="&lt;p&gt;My six-year old daughter was given &lt;a href=&quot;http://nrich.maths.org/5648&quot; rel=&quot;noreferrer&quot;&gt;this maths problem&lt;/a&gt; for her homework:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given a regular square grid of 4 × 4 dots, how many different triangles with one dot in the middle can you draw?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We were given no additional information other than that stated in the page.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Faced with this, my immediate conclusion was that there was no way of answering the question. Not enough information is given to create a clear problem statement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example it is not stated whether we are limited to equilateral triangles or can use different kinds of triangles. It is also not obvious as to what &quot;different&quot; means in this context: are mirror images and other transpositions of the same triangle on the grid &quot;different&quot; or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because the sample used an equilateral triangle, we followed that and presumed that transpositions were allowed. Now we see the solution, that's not what was required. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me this sort of issue is going to cause confusion for children rather than foster good understanding of maths. Is that correct? Is it fair of me to complain to the teacher that it's a poorly set problem?  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Note, &lt;a href=&quot;http://nrich.maths.org/5648/solution&quot; rel=&quot;noreferrer&quot;&gt;here is a solutions page&lt;/a&gt; discussing what other groups did to approach the problem.&lt;/p&gt;&#xA;" OwnerUserId="833" LastEditorUserId="996" LastEditDate="2016-12-02T23:20:52.617" LastActivityDate="2016-12-10T14:51:26.187" Title="Is this homework problem on counting triangles within a 4x4 grid too vague?" Tags="|geometry|homework|" AnswerCount="10" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="11693" PostTypeId="2" ParentId="11692" CreationDate="2016-11-28T21:21:21.627" Score="5" Body="&lt;p&gt;Think of the pedagogical goal of the problem as being to generate discussion about assumptions being made, just as you are doing. The goal is not to come up with a pre-determined correct number. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein is purported to have said, “The formulation of the problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.”&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-11-28T21:21:21.627" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11694" PostTypeId="2" ParentId="11692" CreationDate="2016-11-28T21:40:55.223" Score="66" Body="&lt;p&gt;This is a bit of a can of worms. Let's unpack a little.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;We&lt;/strong&gt; were given no additional information other than that stated in the page.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is your daughter's homework, not yours. Be careful with this distinction, because coloring a child's question with adult interpretations can lead to trouble (and does in this case).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;it is not stated whether we are limited to equilateral triangles or can use different kinds of triangles&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Your daughter likely makes no distinction between different classifications of triangles, and this isn't something that she would even consider, let alone trip up on, without your help. But for what it's worth, the absence of a restriction implies that there's no restriction - why imagine one?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As an aside, the example triangle isn't equilateral. In fact, it's impossible to make an equilateral triangle on a pegboard like that.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;are mirror images and other transpositions of the same triangle on the grid &quot;different&quot; or not[?]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In this context, &lt;strong&gt;yes&lt;/strong&gt;, they are different. Again, this isn't a point-of-clarification that your daughter is in a position to seek out. Children at that age don't sort objects / shapes by equivalence classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imaging your daughter looking at a page with some squares drawn on it.&#xA;Now imagine asking her to count the squares.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did she answer &quot;there's one square, transposed, rotated, and scaled all over the page&quot;?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;is going to cause confusion for children rather than foster good understanding of maths&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Another aside, but I consider causing confusion to be the hallmark of a good question. There is opportunity here for your daughter to recognize and exploit symmetries, translations, etc, but it question doesn't rely on these faculties either.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Now we see the solution, that's not what was required. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you (and your daughter) take an activity like this as a &lt;em&gt;set of ideas to play with&lt;/em&gt; rather than a piece of &lt;em&gt;answer getting&lt;/em&gt; to do, it's likely to be much more rewarding for her, and less frustrating for you both. Whatever formulation of the question you eventually tackled had value in its own right.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2016-11-28T21:40:55.223" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11695" PostTypeId="2" ParentId="11691" CreationDate="2016-11-28T22:43:22.040" Score="7" Body="&lt;p&gt;I don't have an answer for whether there is a &quot;royal road&quot; to solving such problems (I don't think so), but want to comment on why such problems are not so easy to think about intuitively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://www.springer.com/us/book/9780817636777&quot; rel=&quot;noreferrer&quot;&gt;Gelfand and Shen's &lt;em&gt;Algebra&lt;/em&gt;&lt;/a&gt;, there is the following nice puzzle:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Problem 130&lt;/strong&gt;.  A swimming pool is divided into two equal sections.  Each section has its own water supply pipe.  To fill one section (using its pipe) you need $a$ hours.  To fill the other section you need $b$ hours.  How many hours would you need if you turn on both pipes and remove the wall dividing the pool into sections?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This involves the so-called &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Harmonic_mean&quot; rel=&quot;noreferrer&quot;&gt;harmonic mean&lt;/a&gt;&lt;/em&gt;, which is not a particularly well-known or used mean among high schoolers.  If we had the same intuition for that as the regular (additive/arithmetic) mean, maybe this kind of problem would be obvious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the context of your problems, consider this version:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Betty could paint the house by herself in 6 hours.  Karen could paint it by herself in 5 hours.  How long would it take them together?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This already requires thinking of a problem like $$x(1/6+1/5)=1$$ $$x=\frac{1}{1/6+1/5}=\frac{30}{11}$$  Now the problems you actually state require one to solve a version of this where the individual rates aren't both given.  So unless one had a very intuitive grasp of the harmonic mean &lt;em&gt;and&lt;/em&gt; used at least a little algebra, I don't feel like there would be a nice way for a high schooler to do these (without the methods you or other commenters mention).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I bet that with time and thought one could come up with a nice algorithm that avoided algebra to solve them, and likely such problems are of great enough antiquity that people did come up with them.  The problem then is that you have to memorize some non-algebraic algorithm; the algebraic advantage is that you are directly translating everything and so one can recover the problem from the algebra.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2016-11-29T03:38:58.803" LastActivityDate="2016-11-29T03:38:58.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11696" PostTypeId="2" ParentId="11691" CreationDate="2016-11-28T23:19:44.077" Score="12" Body="&lt;p&gt;By throwing in all of the $r$, $t$, and $d$, not to mention $x$, you're overly complicating things, especially if your brother is algebra-averse. There's really only one unknown, and you can call it $k$. It's the number of hours it would take Karen to paint $1$ house by herself. Everything else can be expressed in terms of $k$. Don't complicate things with $x$ and units of $m^2$; call your units &quot;houses&quot; instead, and then you can talk about $1$ $house$ instead of $x$ $m^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the simplest way I can think to do this one is as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Betty paints $1$ house in $6$ hours. Therefore, Betty paints at a rate of $\frac16$ houses per hour.&lt;/li&gt;&#xA;&lt;li&gt;We are looking for &quot;How long would it take Karen to paint the house working alone&quot;. Let's call this $k$ hours. That is, Karen paints $1$ house in $k$ hours. Therefore, Karen paints at a rate of $\frac1k$ houses per hour.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;At this point, you can see that it's important to understand the inverse relationship between &quot;houses per hour&quot; and &quot;hours per house&quot;, and how to manipulate each quantity to obtain the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Continuing:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Together, assuming they don't get in each other's way, they can paint $\frac1k + \frac16$ houses per hour, so it will take them $\frac{1}{\frac1k + \frac16}$ hours to paint $1$ house.&#xA;But that latter figure, $\frac{1}{\frac1k + \frac16}$ hours, is $\frac23\space *$ &quot;the time it would take Karen working alone&quot; or $\frac23 * k$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So we have $$\frac{1}{\frac1k + \frac16} = \frac23 * k = \frac{2k}{3}$$&#xA;$$\frac1k + \frac16 = \frac{3}{2k}$$&#xA;$$\frac16 = \frac{3}{2k} - \frac{2}{2k} = \frac{1}{2k}$$&#xA;$$6 = 2k$$&#xA;$$3 = k$$&#xA;$$k = 3\space hours$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Checking:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Karen paints a house in $3$ hours, so she paints $\frac13$ of a house per hour. Betty paints a house in $6$ hours, so she paints $\frac16$ of a house per hour.&#xA;Together they paint $\frac16 + \frac13 = \frac12$ of a house per hour.&#xA;Therefore, together they paint a house in $2$ hours.&#xA;$2$ hours = $\frac23$ of $3$ hours, check.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can do the lawn-mowing and tub-filling problems similarly, by simplifying to talking about $1$ lawn and $1$ tub.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Important concepts:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Simplify to a single variable if possible&lt;/li&gt;&#xA;&lt;li&gt;Inverse relationships between &quot;things per time unit&quot; and &quot;time units per thing&quot;&lt;/li&gt;&#xA;&lt;li&gt;Fractional manipulation (add, subtract, multiply, divide) with variables&lt;/li&gt;&#xA;&lt;li&gt;Check your work at the end&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3593" LastEditorUserId="3593" LastEditDate="2017-04-14T16:01:30.233" LastActivityDate="2017-04-14T16:01:30.233" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11697" PostTypeId="2" ParentId="7274" CreationDate="2016-11-29T00:41:26.170" Score="2" Body="&lt;p&gt;At the beginning of a race, the race cars have zero position and velocity, but they will have some non-zero derivative. So have a red car with greater acceleration and lower top speed, and a blue car with lower acceleration and greater top speed, and graph the ratio of distance travelled as a function of time. See if the ratio at early times depends on top speed or what. The position vs time functions can be chosen for convenience (your polynomials are probably good enough if you make a few simple modifications to give zero initial speed and velocity and nonzero acceleration.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the benefit of this is that they are working with concepts they are familiar with (beginning a race instead of $x=0$, starting at the beginning from rest instead of $f(0)=0,\ f'(0)=0$, acceleration instead of $f''(x)$, etc).&lt;/p&gt;&#xA;" OwnerUserId="7479" LastActivityDate="2016-11-29T00:41:26.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11698" PostTypeId="2" ParentId="11692" CreationDate="2016-11-29T01:17:01.063" Score="51" Body="&lt;p&gt;Given the &lt;a href=&quot;http://nrich.maths.org/5648/solution&quot; rel=&quot;noreferrer&quot;&gt;solution they were looking for&lt;/a&gt;, yes this is a &lt;em&gt;very&lt;/em&gt; poorly posed problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The usual convention for problems of this general type is, for example, that all 16 of the translates and rotations of the sample triangle count as different triangles. And any other shape of triangle counts as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A well-posed problem following that convention ought to actually indicate that somehow, but could be given a pass if that detail is left implicit and the reader can be expected to have some awareness of the convention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the website is asking for congruence classes of triangles; it is not following the standard convention for such problems. Even a posteriori, absolutely nothing in the problem indicates that fact.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In isolation, this is wholly unreasonable. It might be reasonable as one of a series of problems where the nonstandard convention has already been established so that this one can be assumed to continue the pattern, but that does not seem to be the case here.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2016-11-29T03:38:07.567" LastActivityDate="2016-11-29T03:38:07.567" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11699" PostTypeId="2" ParentId="11673" CreationDate="2016-11-29T04:18:41.580" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0385477058&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension&lt;/a&gt;, by Michio Kaku. A little long to be light reading, but I love the chapters at the beginning discussing society's fascination with the fourth dimension, and the story behind the acceptance of dimensions higher than three in the math community.  In my experience, undergraduates don't believe or never considered that mathematicians every really fought over these ideas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would recommend &lt;a href=&quot;http://www.flatlandthemovie.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this movie version of Flatland.&lt;/a&gt;  It can be shown in a single class period, with time to spare for some discussion.&lt;/p&gt;&#xA;" OwnerUserId="4703" LastActivityDate="2016-11-29T04:18:41.580" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11700" PostTypeId="2" ParentId="11692" CreationDate="2016-11-29T04:31:19.503" Score="10" Body="&lt;p&gt;I think the key issue is with this part of the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How many &lt;strong&gt;different triangles&lt;/strong&gt; with one dot in the middle can you draw?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which depends on the context of how the subject is being taught&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I looked at the &lt;a href=&quot;http://nrich.maths.org/9288&quot; rel=&quot;noreferrer&quot;&gt;other UK Key Stage 1 level Maths questions&lt;/a&gt; focussing on triangles. Assuming this is an okay overview of the course structure, spatial transformations aren't dealt with at this level. This is meant to be the basic building blocks - i.e.  &lt;strong&gt;what shapes are and how they are constructed&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The learning path following through the problem sets (only triangle shape ones) looks like:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How can you make triangles shapes with straight lines of different lengths?&lt;/li&gt;&#xA;&lt;li&gt;What triangle shapes (constructed with straight lines of varying length) are similar to other triangle shapes?&lt;/li&gt;&#xA;&lt;li&gt;How many unique triangle shapes can you make with x limitations?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In the context of that specific path of learning - the question and solutions do make sense. The question you have posted is &lt;strong&gt;a lateral thinking problem about drawing triangle shapes&lt;/strong&gt;. Which is following on from the previous questions. i.e. using different lengths of straight lines to make triangle shapes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, anyone with any understanding of spatial transformations will get different answers in the current way it's being asked. Apparently it's assumed no-one at KS1 level knows about spatial transformations. Is this a mistake/misleading? Depends... I can totally see a child getting confused if they have knowledge of spatial transformations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A better wording might have been:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How many &lt;strong&gt;triangles that are not the same shape&lt;/strong&gt; (&lt;em&gt;read: different angles and line lengths&lt;/em&gt;) and one dot in the middle can you draw?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;tl;dr&lt;/strong&gt; it's all about context&lt;/p&gt;&#xA;" OwnerUserId="7485" LastEditorUserId="7485" LastEditDate="2016-11-29T04:57:28.563" LastActivityDate="2016-11-29T04:57:28.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11702" PostTypeId="2" ParentId="11673" CreationDate="2016-11-29T17:01:22.470" Score="3" Body="&lt;p&gt;Permit me a possibly unusual recommendation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Alexander, Amir. &lt;em&gt;Infinitesimal: How a dangerous mathematical theory shaped the modern world&lt;/em&gt;. Macmillan, 2014.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Most of the action takes place in Europe in the 16th and 17th&#xA;centuries. A central focus is how (and why) the Jesuits were so adamantly opposed to&#xA;infinitesimals, and how their success hampered Italian mathematics.&#xA;The book is more history, politics, and philosophy than math, but it is very well-researched and I found it an engaging read.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/nRhGb.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nRhGb.jpg&quot; alt=&quot;Cover&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-11-29T17:01:22.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11703" PostTypeId="2" ParentId="11678" CreationDate="2016-11-29T18:29:37.013" Score="1" Body="&lt;p&gt;It is better to solve one problem five different ways, than to solve five problems one way. —George Polya&lt;/p&gt;&#xA;" OwnerUserId="4786" LastActivityDate="2016-11-29T18:29:37.013" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11704" PostTypeId="2" ParentId="11692" CreationDate="2016-11-29T20:06:45.433" Score="5" Body="&lt;p&gt;Having taught gifted math to elementary school students for over 25 years, I was delighted to see that your 6 year old daughter was given this question. However, as I read further, &lt;strong&gt;I discovered that the teacher had given this for homework and was focused on the correct answer of 9&lt;/strong&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student need to be taught how to solve problems like this&lt;/strong&gt;. We want student to develop strategies for solving problems which is why the answers focus on strategies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One method for assigning this question is to give it in class for children to work on in groups.  The teacher could listen in and give &lt;a href=&quot;http://nrich.maths.org/5648/clue&quot; rel=&quot;nofollow noreferrer&quot;&gt;hints&lt;/a&gt; to those that are stuck and/or clarify any questions that arise.  Note that one hint is to work with a partner. In this setting the ambiguities won't matter as much. After the groups work by themselves, the groups should come together as a class and discuss their answers and their assumptions. The next step would be for the teacher to clarify ambiguities.  Groups could then reassess their answers. Only at this point would it be be reasonable to discuss the correct answer.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;If you approach the teacher which I think you should,  I suggest you approach the teacher as a partner in your child's education instead criticizing the question.&lt;/strong&gt; Don't raise your questions (did you mean equilateral triangles?). Instead tell the teacher your daughter's questions and ask how you should have handled the ambiguities. Ask how students are supposed to develop strategies for such explorations which are different then the typical word problem. Did the teacher teach students how to figure this out and what strategies can you reinforce at home. If needed to make your case, you might also point out that the nrich website seems to focus on strategies as much as the answer and suggests under &lt;a href=&quot;http://nrich.maths.org/5648/clue&quot; rel=&quot;nofollow noreferrer&quot;&gt;hints&lt;/a&gt; that students should work with a partner. You might also ask if the students worked on other problems like this in class.   I hope you'll report back on your conversation with the teacher. &lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2016-11-29T20:35:34.040" LastActivityDate="2016-11-29T20:35:34.040" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11705" PostTypeId="2" ParentId="11692" CreationDate="2016-11-30T00:34:08.197" Score="3" Body="&lt;p&gt;Just for amusement: an advanced formulation of the problem is that of reflexive polygons. &quot;Reflexive&quot; means one lattice point on the interior.  A nice expository paper about this problem is found here&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.hmc.edu/~ursula/notes/reflexivepolytopesarticle.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.math.hmc.edu/~ursula/notes/reflexivepolytopesarticle.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Up to GL(2,Z) equivalence there are 16 classes of reflexive polygons in the plane. Five of these classes are triangles. See figure 10. These come in two pairs of dual triangles, and one self-dual. You will see these among the 9 in the answer to the original problem. So some of the 9 in the solution are equivalent under GL(2,Z). So from this formulation, the answer is &quot;5 triangles up to GL(2,Z) equivalence.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that by shears transformations, you can generate reflexive triangles that need arbitrarily large grids to fit. See figure 8. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2016-11-30T00:48:34.917" LastActivityDate="2016-11-30T00:48:34.917" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11706" PostTypeId="2" ParentId="11692" CreationDate="2016-11-30T07:40:55.937" Score="8" Body="&lt;p&gt;I think you're focusing too much on 'the answer'. I suspect the teacher was tired and busy, and quickly needed to come up with an exercise of a suitable level that was worth completing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I was marking this, I wouldn't be looking for the answer '9'. I'd be looking for a solution, and particularly what understanding has been demonstrated by the pupil in their solution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I see it, the question tests different aspects: mathematical creativity to create different shape triangles; understanding of Euclidean isometries to either list all translates/rotations or point out that they don't count; systematic thinking to find &lt;em&gt;all&lt;/em&gt; answers; creation and communication of proof to explain why their answer is complete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A six year old is not going to have perfectly mastered all of these, by any means. But their solution should demonstrate at least some understanding of one or more of these, and help to answer the question 'what level is this pupil working at?', which is the question that should be being applied for a six year old (not 'how many marks did they get?').&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-11-30T07:40:55.937" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11707" PostTypeId="2" ParentId="11692" CreationDate="2016-11-30T11:13:36.620" Score="15" Body="&lt;p&gt;I've come across some creative 10 year olds who produced solutions like &lt;a href=&quot;https://i.stack.imgur.com/IbFnU.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IbFnU.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when there are questions set that seem closed  you can allow it to be opened out and the child's exploration can begin, helping them to understand it's ok to think and not just follow convention.&lt;/p&gt;&#xA;" OwnerUserId="7508" LastEditorUserId="7508" LastEditDate="2016-11-30T11:57:52.693" LastActivityDate="2016-11-30T11:57:52.693" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11708" PostTypeId="1" CreationDate="2016-11-30T18:16:45.427" Score="4" ViewCount="968" Body="&lt;p&gt;Which is the best software to make very quick pictures to illustrate math and physics problems (like geometric pitures, cartesian planes, vectors)?&lt;/p&gt;&#xA;" OwnerUserId="5601" LastActivityDate="2016-12-06T00:43:00.793" Title="Software to make illustrations for exercises" Tags="|software|specific-software|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11709" PostTypeId="2" ParentId="11673" CreationDate="2016-11-30T19:37:00.060" Score="4" Body="&lt;p&gt;The Number Devil: A Mathematical Journey by Hans Magnus Enzensberger is an essential in the &quot;fun math book&quot; category.  Very accessible, great illustrations and actually touches on pretty sophisticated concepts such as limits and infinity.  Very highly recommend this book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0805062998&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Number-Devil-Mathematical-Adventure/dp/0805062998&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3237" LastActivityDate="2016-11-30T19:37:00.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11710" PostTypeId="2" ParentId="11708" CreationDate="2016-11-30T21:42:41.137" Score="4" Body="&lt;p&gt;I think this depends on your kind of math and physics problems, but&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://geogebra.org/&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;GeoGebra&quot;&gt;GeoGebra&lt;/a&gt; and&#xA;&lt;a href=&quot;https://inkscape.org/&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Inkscape&quot;&gt;Inkscape&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;can be used to create graphics that can be exported in vector and pixel formats. If you need a strictly mathematical picture (function plot, definite proportions and distances), GeoGebra is more suitable. If you need something more &quot;artsy&quot;, Inkscape will do the job. I sometimes even use both: create something in GeoGebra, export as SVG and change the axis' arrow tips for example.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2016-11-30T21:42:41.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11711" PostTypeId="1" CreationDate="2016-11-30T22:51:24.973" Score="5" ViewCount="178" Body="&lt;p&gt;I'm studying Mathematics in the UK. Back in Italy, where I am from worked-out problems books are very popular in mathematics and they are always recommended or compulsory at university. Going through some of them I found them extremely helpful and interesting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, I wanted to know if you can give me some lists of great (and possibly modern!) books were the main focus (if not the only, that'd be better!) is to solve problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not looking for books such as &quot;problem solving in mathematics&quot; or &quot;how to think like a mathematician&quot;. I mean books like &quot;Worked Examples in Applied Mathematics&quot; by HART.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know if you know any in the following areas:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Applied Mathematics (Kinematics, Particle Motion, Fluid Dynamics, etc, even more advanced)&lt;/li&gt;&#xA;&lt;li&gt;Probability (Distributions problems, Bernoulli, Poisson, Normal, Gamma, Beta, Erlang, etc)&lt;/li&gt;&#xA;&lt;li&gt;Statistics (Confidence Intervals, Hypothesis Testing, Clinical Trials, etc)&lt;/li&gt;&#xA;&lt;li&gt;Core Mathematics (Analysis, ODEs, PDEs, etc)&lt;/li&gt;&#xA;&lt;li&gt;Vector Calculus/Complex Variable (Stokes, Greens, Divergence, Cauchy, etc..)&lt;/li&gt;&#xA;&lt;li&gt;OR &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I know this is probably the best place for this. However I know the community of StackExchange is very active and prolific, so I am sure anyone will have some recommendations. I will mark as a final answer just the most useful and comprehensive list, which actually will suggest problem-based books.&lt;/p&gt;&#xA;" OwnerUserId="7512" LastActivityDate="2016-12-06T00:33:56.697" Title="Great Worked-out examples books for Mathematics" Tags="|textbooks|self-learning|problem-solving|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11712" PostTypeId="2" ParentId="5626" CreationDate="2016-12-01T03:55:05.610" Score="3" Body="&lt;p&gt;One potato, two potato, three potato, four! The game is mod &lt;em&gt;however many kids are playing&lt;/em&gt;. Board games with spaces around the edges work too, though they have much larger mod values.&lt;/p&gt;&#xA;" OwnerUserId="7515" LastActivityDate="2016-12-01T03:55:05.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11713" PostTypeId="2" ParentId="11691" CreationDate="2016-12-01T04:24:47.980" Score="5" Body="&lt;p&gt;I don't know much about math education, but I can give my thoughts on how to solve this problem. I think the main issue is whether to express the rate of work as a amount of work per unit time or its reciprocal, the amount of time per unit work. When you have two people working together, it is the amount of work per unit time that combines additively, so that is the natural thing to work with. (In another situation where you have one person painting two houses of different sizes, and you were asked how long it would take, you would want to work with the amount of time per unit work.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see how thinking of it in terms of the reciprocal can help, let's do the two problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they can paint the house in 2/3 the time it takes Karen to paint the house, then they can paint 3/2 houses in the time it takes Karen to paint one house. Therefore, when Karen has painted one house, Betty has only painted a half a house. This means Betty is half as fast as Karen. So if it takes Betty 6 hours to paint a house, it takes Karen only 3 hours. (Here taking the reciprocal was the key step that allowed us to compare how many houses each was able to do in a fixed amount of time, and thereby compare how quickly they work.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stan and Hilda mowed a lawn, with Hilda doing twice as much as Stan. This means Stan only mowed 1/3 lawn in forty minutes, so it takes him 3*40 minutes to mow a whole lawn. (Again we decomposed the rate 1 lawn / 40 minutes into the sum of two contributions: 1/3 lawn in 40 minutes and 2/3 lawn in 40 minutes. Notice these are both amounts of work per unit time.)&lt;/p&gt;&#xA;" OwnerUserId="7479" LastActivityDate="2016-12-01T04:24:47.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11714" PostTypeId="2" ParentId="11692" CreationDate="2016-12-01T12:24:00.380" Score="4" Body="&lt;p&gt;The question is:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;How many different triangles with one dot in the middle can you draw?&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;I don't know where is the middle of a triangle --who knows (exept, maybe, an equilateral triangle where the centre seems to be a good candidate). As NiloCK says, you can't draw an equilateral triangle on this 4x4 grid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my answer is 0: no triangle have one dot in the middle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sure my 6 years daughter answers differently to the question.&lt;/p&gt;&#xA;" OwnerUserId="7518" LastActivityDate="2016-12-01T12:24:00.380" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11715" PostTypeId="2" ParentId="5626" CreationDate="2016-12-01T19:05:59.373" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://themathlab.com/natural/mental%20math%20tricks/casting%20nines/casting.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Casting out nines&lt;/a&gt; is a good way to check arithmetic calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most American students are unfamiliar with casting out nines, but it is very easy to teach the method.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2019-03-28T19:47:18.517" LastActivityDate="2019-03-28T19:47:18.517" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11716" PostTypeId="2" ParentId="3933" CreationDate="2016-12-01T23:37:31.863" Score="1" Body="&lt;p&gt;It depends on the goals you have for the assessment for which you are giving credit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some assignments, getting the right answer, or a certain amount of right answers, might be the only real goal you set out for students. For example, if you're giving an assignment to assess a kid's skills with doing multiplication of two-digit integers, you could give the kid 20 problems to do and 20 minutes in which to do them, then grade each one on whether it's right or wrong and don't think about the work. You could do something similar with any rote mechanical skill, e.g. Finding very simple derivatives. In this case, the goal of the assignment is to demonstrate evidence that the kid is sufficiently fluent in the mechanical processes that they can get a reasonable number of answers in a certain period of time. In that case, the work being done is not really all that necessary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other cases, the goal of the assignment has little to do with the answer and everything to do with the process and the explanations. I suspect this is where a lot of our assessments fall -- homework, projects, complex problems on exams, etc. In that case the goal of the assessment is not to get a right answer &lt;em&gt;per se&lt;/em&gt; but to get an answer and then explain why it's right, and you're assessing the explanation and answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both approaches to assessment could reasonably have a place in a math class, and one's not better than the other. They just measure different things. The important thing is to know what your objectives are, communicate these clearly to the students in advance, and then give credit in a way that's consistent with your goals. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2016-12-01T23:37:31.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11717" PostTypeId="1" CreationDate="2016-12-02T04:56:46.157" Score="6" ViewCount="202" Body="&lt;p&gt;I teach physics and math at a community college in California. I got in a politically tinged discussion with a colleague today in which I claimed that one of socialism's weaknesses was its historical unwillingness to throw failed programs in the trash, and I challenged him to name a single positive educational reform that had occurred within our lifetimes in American community colleges and had spread and taken root. Annoyingly, he was able to come up with an example, which was accelerated developmental math education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hadn't known until today that such a thing existed, and it doesn't yet exist at my school. I'm still hazy on what it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What exactly is accelerated developmental math education? How does it work, and is it a good idea?&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2016-12-03T10:50:29.747" Title="What is accelerated developmental math education, and is it a good idea?" Tags="|community-colleges|remedial-courses|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11718" PostTypeId="2" ParentId="11717" CreationDate="2016-12-02T06:48:57.803" Score="7" Body="&lt;p&gt;Accelerated developmental education includes a variety of mechanisms to shorten time students spend in remediation. It can include: Batching together multiple courses of remediation in one semester (say: 6 or 9 credits in one semester), likely with &quot;supports&quot; such as tutors and learning communities; batching together remediation in simultaneous credit-bearing courses (e.g., a 6 credit combined remedial algebra and statistics course); giving short workshops of a few days and then re-taking placement tests in basic skills; or possibly simply eliminating remedial courses altogether. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The motivation is based on the observation that students assigned to remediation are likely to fail and drop out of school -- and an interpretation that anything that minimizes or eliminates remediation is a good thing (i.e., keeps students in school, keeps them attending other courses, increases total credits and chance of graduation). One point that proponents of acceleration like to make is that some percentage of students who &lt;em&gt;pass&lt;/em&gt; a remedial course still drop out before the next math course in the sequence (I've seen 20% thrown around as an estimate; &quot;life happens&quot; is a common attribution). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thinkers at the Community College Research Center, Teachers College, Columbia University have been at the forefront of this movement. Consider Jaggers, et. al., &lt;a href=&quot;http://ccrc.tc.columbia.edu/media/k2/attachments/designing-meaningful-developmental-reform-research-overview.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;Designing Meaningful Developmental Reform&quot;&lt;/a&gt; (CCRC, 2013). They identify three &quot;tensions&quot; in developmental education reform: (1) institutional autonomy vs. system-wide consistency, (2) efficient vs. effective assessment, and (3) supporting student progress vs. maintaining academic standards. On this latter point, they write: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Colleges often feel ambivalent about implementing changes that might&#xA;  improve student progression but could possibly undermine academic&#xA;  quality. Mounting evidence suggests, however, that accelerated&#xA;  developmental models—such as shortening developmental sequences and&#xA;  mainstreaming upper level developmental students into college-level&#xA;  courses with mandatory supports—lead to improved outcomes for these&#xA;  students.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They cite several findings that accelerated remediation leads to increased percentages of students who pass credit-bearing English and Math courses, for example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RT69U.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RT69U.png&quot; alt=&quot;College-Level Course Completion Rates at CCBC&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Critics would include Scherer and Anson in &lt;a href=&quot;http://www.palgrave.com/us/book/9781137336002&quot; rel=&quot;noreferrer&quot;&gt;&quot;Community Colleges and the Access Effect&quot;&lt;/a&gt; (Palgrave Macmillan, 2014). In chapter 1 they make the point that thinking on remediation has made something of a full circle: in the 60's-70's larger number of students were entering unprepared and failing college-level math courses (under a  &quot;'Right to fail' philosophy&quot;); the 80's-90's saw the blossoming of remedial education to address these student's needs; and the 00's-10's are seeing a backlash against remediation. In chapter 3 they write of developmental education:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Enemy. Gated. Broken. Brick wall. Dysfunctional. Disaster. Bridge to&#xA;  nowhere. Burial ground. Senior research analyst for the U.S.&#xA;  Department of Education Clifford Adelman ultimately tied the divisive&#xA;  rhetoric to blind allegiance to the completion agenda and explained&#xA;  that, “As a consequence, what often pours out are scare stories that&#xA;  make for good press and bad policy. The bad data-driven scare story,&#xA;  in fact, has become the preferred narrative.” Developmental education&#xA;  has been especially painted with the broad brush of failure by&#xA;  completion guardians who believe that maintaining open admissions at&#xA;  our nation’s community colleges and freeing low-skilled students from&#xA;  assignment to developmental education constitutes sensible completion&#xA;  strategy. Implicit in the arguments of developmental education critics&#xA;  are many false assumptions, chief among them that most students&#xA;  currently enrolled in developmental education would be much better&#xA;  served by enrolling in college-level coursework with embedded support.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;That assumption, prompted by the realization that life events have a&#xA;  greater chance of derailing completion the longer a student remains in&#xA;  college, has led some completion advocates to herald the “strategy” of&#xA;  racing low-skilled students through higher education as an acceptable&#xA;  way to increase American postsecondary credential award...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Using the example of someone pursuing “a two-year nursing&#xA;  certificate,” Bill Gates declared, “We owe it to every American to&#xA;  make it pretty darn easy for them to get through that system.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(I will point out that the Gates Foundation is one of the primary sponsors of education reform today; note that they funded the CCRC report quoted above.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As one example, the CCRC report provides a pretty good road map for changes to remediation that have occurred at CUNY in the past few years. Institutional autonomy on the issue has been largely sidestepped in favor of central system planning. Students universally taking placement tests for basic arithmetic skills was eliminated a few years back. Most recently, the requirement that all students pass basic writing and algebra tests before graduating has been removed. A new central requirement is that non-STEM majors be provided with a credit-bearing math course &lt;em&gt;sans&lt;/em&gt; algebra prerequisite. The expectation is that graduation rates will rise as a result. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-12-02T15:37:13.270" LastActivityDate="2016-12-02T15:37:13.270" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11719" PostTypeId="1" CreationDate="2016-12-02T07:44:46.213" Score="6" ViewCount="653" Body="&lt;p&gt;I'm searching for an offline portable program that demonstrates the concept of how does a function revolve around X or Y axis It should be similar to this: &lt;a href=&quot;http://www.shodor.org/interactivate/activities/FunctionRevolution/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solid of revolution simulator&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but I want to use it in a place that has no WiFi connection so if any one knows a similar program or maybe how to extract the program from the website I'd be very thankful.&lt;/p&gt;&#xA;" OwnerUserId="7524" LastActivityDate="2016-12-06T20:39:54.007" Title="Solid of revolution simulator?" Tags="|software|functions|graphing|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11720" PostTypeId="2" ParentId="11719" CreationDate="2016-12-02T13:59:22.907" Score="2" Body="&lt;p&gt;Calcplot3d might be a good choice:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://web.monroecc.edu/calcNSF/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://web.monroecc.edu/calcNSF/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-12-02T13:59:22.907" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11721" PostTypeId="1" AcceptedAnswerId="11724" CreationDate="2016-12-02T17:01:51.530" Score="3" ViewCount="296" Body="&lt;p&gt;I was coaching a student on how to approach this problem. 2 equations given and the question was where they intersect. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oF1dR.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oF1dR.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, with a bit of practice on polar coordinates, producing the graph by hand wasn't too tough. I was able to point out the petal in the first quadrant wan't going to intersect the vertical line unless rotated 30 degrees clockwise. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student said the result was expected to be algebraic, i.e. set the 2 equal, manipulate and show no solution. I wasn't able to do this after about 10 minutes of manipulation, I just would up with sine to the 3rd power along with a number of sine*cos terms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not so much looking for this single solution as I am a process how to approach this style of problem. (This is for a pre-calc class, typically with high school juniors)&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-12-16T01:47:11.477" Title="Solving a Polar set of equations algebraically?" Tags="|trigonometry|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11722" PostTypeId="2" ParentId="11721" CreationDate="2016-12-02T17:42:16.653" Score="0" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sin(3\theta)=\sec \theta$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sec \theta=(4\cos^2 \theta-1)\sin \theta$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(4\cos^2 \theta-1)\sin \theta \cos \theta=1$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(4\cos^2 \theta-1)^2(1-\cos^2 \theta) \cos^2 \theta=1$$&lt;/span&gt;&#xA;which is an &lt;a href=&quot;http://www.wolframalpha.com/input/?i=sin(3x)%3Dsec(x)&quot; rel=&quot;nofollow noreferrer&quot;&gt;eighth degree polynomial in &lt;span class=&quot;math-container&quot;&gt;$\cos \theta$&lt;/span&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="4791" LastEditDate="2019-12-16T01:47:11.477" LastActivityDate="2019-12-16T01:47:11.477" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11723" PostTypeId="2" ParentId="11721" CreationDate="2016-12-02T18:06:26.777" Score="2" Body="&lt;p&gt;$R=4\sec(\theta)$ is the same as $x=4$. The polar curve $R=4\sin(3\theta)$ is contained in the circle $R=4$. The only possible intersections of the vertical line and the circle occur when $3\theta=\pm\pi/2,\pm3\pi/2,\cdots$. So the polar curve and the vertical line do not intersect. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2016-12-02T18:06:26.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11724" PostTypeId="2" ParentId="11721" CreationDate="2016-12-02T18:17:35.973" Score="11" Body="&lt;p&gt;This might not quite be an &quot;algebraic&quot; solution in the sense the student was looking for, but I think a more reasonable approach to this sort of problem is using &lt;em&gt;bounds&lt;/em&gt;, not setting things equal from the outset.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The general strategy is to notice that one curve is entirely &lt;em&gt;inside&lt;/em&gt; a circle of radius 4, while the other is entirely outside that circle, so any intersection must occur on the boundary. This is often a good strategy for showing two objects don't intersect: bound them inside larger, simpler regions that can be shown to intersect only in certain very limited ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More precisely, since $\sec(\theta) = 1/\cos(\theta)$, we have&#xA;$$\lvert 4\sec(\theta) \rvert \geq 4$$&#xA;with equality if and only if $\lvert \cos(\theta) \rvert = 1$. However,&#xA;$$\lvert 4 \sin(3 \theta) \rvert \leq 4,$$&#xA;so if $4 \sec(\theta) = 4 \sin(3\theta)$, then $\lvert \cos(\theta) \rvert = 1$, which means $\theta$ is an integer multiple of $\pi$. But that implies $4 \sin(3\theta) = 0$, a contradiction.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2016-12-02T18:17:35.973" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11725" PostTypeId="1" CreationDate="2016-12-02T22:31:15.667" Score="3" ViewCount="474" Body="&lt;p&gt;Hi I hope every one is fine ,&#xA;I am an Electrical Engineer. I asked before about real and complex analysis because I am interested in Signal Processing also I am interested in coding theory and information theory. I felt I am in need of more mathematics&#xA;so I am searching the internet continuously to get the idea how could I find my path to my interest but I was encountered by two classification scheme , I don't know the difference between them ;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Wikipedia :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;quantity&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;structure&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;space&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;change&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;2)Princeton companion : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Algebra&lt;/li&gt;&#xA;&lt;li&gt;number Theory&lt;/li&gt;&#xA;&lt;li&gt;Geometry&lt;/li&gt;&#xA;&lt;li&gt;Algebraic geometry&lt;/li&gt;&#xA;&lt;li&gt;Analysis Logic&lt;/li&gt;&#xA;&lt;li&gt;logic&lt;/li&gt;&#xA;&lt;li&gt;Combinatorics&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;so what is the difference between the two classifying schemes,&#xA;Also what accounts as a foundation and if I want to read about philosophy of Mathematics and Foundation and their evolution pls suggest a good book&lt;/p&gt;&#xA;&#xA;&lt;p&gt;thanks&lt;/p&gt;&#xA;" OwnerUserId="2181" LastEditorDisplayName="user8458" LastEditDate="2017-07-21T20:54:11.557" LastActivityDate="2017-07-23T16:07:38.223" Title="Mathematics Branches and Foundation" Tags="|logic|axioms-foundations|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11726" PostTypeId="2" ParentId="11717" CreationDate="2016-12-03T10:50:29.747" Score="0" Body="&lt;p&gt;My sense is that it is a good idea. I will come back later to write a more complete answer, but much of what I will say can be found at the website of the &lt;a href=&quot;http://accelerationproject.org/Publications&quot; rel=&quot;nofollow noreferrer&quot;&gt;California Acceleration Project&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2016-12-03T10:50:29.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11727" PostTypeId="2" ParentId="11583" CreationDate="2016-12-03T14:23:22.290" Score="5" Body="&lt;p&gt;I like to ask my probability students the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If you pick an integer between 2 and 100 uniformly at random, what's the probability that it's the average of two (not necessarily different) primes?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I like that above question (easily equivalent to Goldbach) because there's no preference for even versus odd numbers as in Goldbach.  It also gets students checking a lot of cases.  On the one hand, they quickly notice the pattern and &lt;em&gt;make the conjecture themselves&lt;/em&gt;, and they also want to argue that the pattern holds for all numbers (that way they don't have to check 100 cases).  But the beauty of it is that the prime numbers that show up feel random, and it very much feels like the pattern keeps working by accident.  It's also a great opportunity to stress the difference between &quot;seems to be true&quot; and &quot;it's always true.&quot; You could strip the probability from it if you like.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You could ask why the conjecture isn't about odd numbers.  What's special about evens versus odds (there aren't that many even primes)?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Asking &quot;are there infinitely many solutions to prime = prime + prime?&quot; is equivalent to the twin prime conjecture and sounds Goldbach-like.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The whole conjecture is obviously false unless there are infinitely many primes.  You could discuss that connection and why there are infinitely many primes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You could say the weak Goldbach-conjecture &quot;every number (greater than 1) is the sum of at most 3 primes.&quot;  You could say why this is implied by regular Goldbach, and you could try to discuss why this doesn't immediately imply regular Goldbach (though that's tricky to do since both statements are likely true.  And you can't really argue that two true statements don't imply each other.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7530" LastEditorUserId="7530" LastEditDate="2016-12-03T14:49:02.587" LastActivityDate="2016-12-03T14:49:02.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11728" PostTypeId="1" AcceptedAnswerId="11729" CreationDate="2016-12-03T10:42:52.533" Score="9" ViewCount="258" Body="&lt;p&gt;I have recently finished working on an article on a subject that falls into the category of &quot;practical mathematical (undergraduate) teaching&quot; -- a set of concrete propositions and examples on a specific issue, that enlighten and help with the teaching of this issue. An example of an article as such can be found &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/0020739X.2016.1153737&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, when I try to publish this article, I can not find the appropriate platform. Most journals on mathematical teaching/education look for more theoretical papers, so they decline my paper; on the other hand, this paper does not contain a breakthrough in mathematics, so I can not submit it to &quot;classical&quot; journals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I am looking for a journal that on one hand deals with mathematical teaching, but on the other can accept a paper with practical subjects. Would realy appreciate any help on this problem.&lt;/p&gt;&#xA;" OwnerUserId="7529" OwnerDisplayName="Mike" LastEditorUserId="7529" LastEditDate="2016-12-04T20:01:56.877" LastActivityDate="2018-10-04T22:53:19.287" Title="Appropriate journal for an article on practical mathematical teaching" Tags="|publishing|journals|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11729" PostTypeId="2" ParentId="11728" CreationDate="2016-12-03T20:40:43.250" Score="11" Body="&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; See also the PDF linked &lt;a href=&quot;https://www.nctm.org/Publications/Journal-for-Research-in-Mathematics-Education/2017/Vol48/Issue4/Journal-Quality-in-Mathematics-Education/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zXAO0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zXAO0.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Wikipedia has a &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_mathematics_education_journals&quot; rel=&quot;nofollow noreferrer&quot;&gt;list of mathematics education journals&lt;/a&gt;; I happen to prefer the list of mathematics education journals compiled &lt;a href=&quot;https://mathedjournals.wikispaces.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, as they are provided along with some basic facts: journal summary, type of articles solicited, audience reached, etc. Some of these facts derive from the following article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Törner, G., &amp;amp; Arzarello, F. (2012). Grading mathematics education research journals. &lt;em&gt;Editorial: Reminiszenzen&lt;/em&gt;, pp. 31-34. &lt;a href=&quot;http://www.didaktik-der-mathematik.de/pdf/gdm-mitteilungen-95.pdf#page=31&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt; (no pay wall). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The second link includes journals that are of generally higher repute in the field of mathematics education. So, for example, although the journal you mention in the OP can be found at the first link (without its own wikipage) and is graded with a B in the Törner and Arzarello article, it is &lt;em&gt;missing&lt;/em&gt; from the second link; so, I might be hesitant about targeting it as a destination for publishing. I know that it has published a couple of &lt;a href=&quot;https://academia.stackexchange.com/q/76653&quot;&gt;questionable articles&lt;/a&gt; in the past.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My advice would be to look through recent issues of some of the journals linked to above (assuming you can access them, e.g., at a library or an academic institution), and see where you can find pieces that resemble the sort of paper you would like to write. You may also wish to look at where/if they were cited later on, or at the relevant journal's impact factor/circulation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example of a journal that is practice-oriented is NCTM's &lt;em&gt;Mathematics Teacher&lt;/em&gt;, which includes a section specifically around mathematical findings called &lt;strong&gt;Delving Deeper&lt;/strong&gt;. A description of the journal can be found in the second link above (click through for larger print, or check link):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2pP95.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2pP95.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The appropriate destination will depend on the level of the mathematics discussed (both in content and in the language that you use) as well as your ability to ensure that it is not &lt;em&gt;just&lt;/em&gt; mathematical findings of too low a caliber to include in a journal of mathematics, but rather that it is connected to &quot;practical mathematical teaching&quot; in a manner that you can describe, justify, and link to the literature on mathematics education.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2018-10-04T22:53:19.287" LastActivityDate="2018-10-04T22:53:19.287" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11730" PostTypeId="5" CreationDate="2016-12-04T00:40:04.407" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2016-12-04T00:40:04.407" LastActivityDate="2016-12-04T00:40:04.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11731" PostTypeId="4" CreationDate="2016-12-04T00:40:04.407" Score="0" Body="For questions about journals that publish work in Mathematics Education" OwnerUserId="39" LastEditorUserId="39" LastEditDate="2016-12-05T00:39:34.260" LastActivityDate="2016-12-05T00:39:34.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11732" PostTypeId="1" CreationDate="2016-12-05T05:13:42.887" Score="7" ViewCount="204" Body="&lt;p&gt;&lt;strong&gt;&quot;It's impossible because I tried but couldn't do it!&quot;&lt;/strong&gt;&#xA;I need situations which shows that this kind of reasoning above is not working!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do have an example, but look for more:&#xA;It's so hard to cover a $6\times 10$ rectangle by pentominos (I hope no student would manage to do it), but actually possible in &lt;a href=&quot;http://puzzler.sourceforge.net/docs/pentominoes.html&quot; rel=&quot;noreferrer&quot;&gt;hundreds of ways&lt;/a&gt;!&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2018-03-07T18:15:19.177" Title="Showing the Necessity of Proving the Impossibilities" Tags="|secondary-education|proofs|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11733" PostTypeId="2" ParentId="11732" CreationDate="2016-12-05T12:25:06.167" Score="5" Body="&lt;p&gt;I will reuse something I already answered another question with. I think the problem you describe will likely come up in a circumstances when we, as teachers, state &lt;em&gt;and prove&lt;/em&gt; something is impossible (usually, an equation has no solution). It is often difficult to have students understand how such a proof is relevant, and the best way I can think of is to design a problem which is very close to the one at hand, with no obvious solution, but yet with some solution. I am not sure a general list is so useful, so I'll just pick one example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine you discussed the irrationality of $\sqrt{2}$. In other words, you discussed the impossibility to solve $p^2=2q^2$ in integers. If some students feel it obvious that there could not be a solution, then propose them the equation (also to be solved in integers) $p^2=12 q^3$. It is not obvious to find a solution when one is not familiar with the decomposition into primes, but of course $(p,q)=(18,3)$ does the job.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-12-05T12:25:06.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11734" PostTypeId="2" ParentId="11732" CreationDate="2016-12-05T13:09:00.303" Score="10" Body="&lt;p&gt;Here is a nice one which I am surprised wasn't posted yet:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\text{Prove }n^{17}+9 \text{ and } (n+1)^{17}+9 \text{ are not relatively prime for all n }$ by giving a counter example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first counterexample is $n=8424432925592889329288197322308900672459420460792433$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That ought to keep your students trying for a while :)&lt;/p&gt;&#xA;" OwnerUserId="7537" LastActivityDate="2016-12-05T13:09:00.303" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11735" PostTypeId="1" AcceptedAnswerId="11766" CreationDate="2016-12-05T15:34:08.980" Score="3" ViewCount="327" Body="&lt;p&gt;Context: a very basic level statistics package computer lab. A scatter plot is produced for one integer variable versus another integer variable. The students are asked why the points form a grid pattern. Students are 1st year undergrads.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problem: I am struggling in my attempts to come up with a clear question, that allows the students to get to see (...by 'thinking'...) that the discrete plot positions are due to the variable's data type. Using terms like 'class of the variable' is currently unlikely to be in their vocabulary. Forming a question that does not obviously contain the answer evades my befuddled brain.&lt;/p&gt;&#xA;" OwnerUserId="7539" OwnerDisplayName="Big Old Dave" LastEditorUserId="7539" LastEditDate="2016-12-05T15:56:40.990" LastActivityDate="2016-12-13T03:33:56.977" Title="How to ask a student a question to get the answer '...integer not continuous...'" Tags="|mathematical-pedagogy|teaching|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11736" PostTypeId="1" CreationDate="2016-12-05T05:48:00.680" Score="20" ViewCount="3831" Body="&lt;p&gt;I am a freshman studying math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everytime I learn Mathematics, I do it the slow-way. I tend to read the definitions until I make sense of what is written in the texts. I can spend a whole day trying to get one page of the textbook. I also do all the theorem proving by myself, even it takes me a lot of time to do that. I always get a good depth of understanding learning this way. I can also do a lot of hard problems given in the textbooks. (Not in a trivial way, it also takes a lot of time to get the jobs done.) The works tend to take a longer time than everybody else, but I do get the feeling I know the material better than other guys in my class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The downside of it is that I don't have enough time to get all of my tasks done in a reasonable amount of time. I always wish to be able to do research in Mathematics, so I need a deep level of understanding. But as a slow runner, I think I have to change somehow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this question is made to ask for advice and tips in learning Mathematics. Also, please share with me how you work with textbooks. Thank you, Stack Exchange.&lt;/p&gt;&#xA;" OwnerUserId="7540" OwnerDisplayName="Michael D Nguyen" LastEditorUserId="376" LastEditDate="2018-07-04T13:26:39.753" LastActivityDate="2020-03-02T04:19:58.437" Title="How to learn math from textbooks in the right way?" Tags="|self-learning|" AnswerCount="6" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="11737" PostTypeId="2" ParentId="11736" CreationDate="2016-12-05T11:52:33.750" Score="4" Body="&lt;p&gt;Jump over the unnecessary parts. The point is that not all of the proofs from every theorem needs to be learned. Ones the theorem is proved, it can be taken for granted. Some proofs must be learned by heart, but those are the ones close to the subject one specializes to. There is simply too much to learn. Building intuition to the theorems is possible, without memorizing the proof. Learning to form the abstract overall picture is also important, because science is pushed forward by proving hypothesis, which in the end are just hunches about how the abstract overall picture could be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematicians tend to trivially see things, that's why there is unsolved problems in a form: proof if you can. This leads to mathematicians building bridges, math is not the same as building a house block by block. Seeing the solution is a great heuristic, even if the hunch is wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rule of thumb is to learn strategies and patterns which can be used for the proofs. Also if you are learning methodology, learn the main method. You only need to understand Principal component analysis (PCA) to be able to learn fast every PCA type of method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: clarified the overall by merging with a comment.&lt;/p&gt;&#xA;" OwnerUserId="7552" OwnerDisplayName="user3644640" LastEditorUserId="7552" LastEditDate="2016-12-07T07:37:35.027" LastActivityDate="2016-12-07T07:37:35.027" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11738" PostTypeId="2" ParentId="11736" CreationDate="2016-12-05T22:52:06.430" Score="16" Body="&lt;p&gt;I tend not to answer the questions that I have no professional knowledge about or the questions that are too general to be answered here. But, your question reminded me of my own undergraduate years when I had the exact same strategy as you! I am writing this to confess that it was not a right strategy. Let me explain, using two examples. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you have just learned the definition of convergence of sequences. It is a hard definition and takes a good time to grasp. Then you have a couple of standard theorems: uniqueness of limit, the limit of the sum is the sum of the limits, and so on. If you are as you described you are, as soon as you see the proof of one of these theorems, you will be able to do the rest (yet, it would not be easy, but it is mainly algebra). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now suppose you want to know whether you can count the real numbers or not and then prove your conjecture (just in case that you do not know the answer, I do not give more away). It was an act of genius to find and prove the answer to this question. Should you spend your time to solve this problem yourself? The answer that I wish someone gave me when I was younger is NO. You need to learn to respect what the others did, and more importantly, learn how they did it and move.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to find a balance between what you are able to do yourself and what you should learn from the others (lecturers, textbooks, friends,history, and so on). At the end, I really suggest that you look at some of the answers given to this question &quot;&lt;a href=&quot;https://mathoverflow.net/q/130883/29316&quot;&gt;Are there proofs that you feel you did not “understand” for a long time?&lt;/a&gt;&quot;  There you see examples of great and well-known mathematicians who were even memorizing some of the proofs when they where undergraduate students. There you see we all need time to understand some of the concepts, definitions, and proofs we read and that understanding only comes when we see those in the light of what we haven't learned yet!    &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2016-12-05T23:13:07.177" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11739" PostTypeId="1" CreationDate="2016-12-06T00:10:31.737" Score="9" ViewCount="2006" Body="&lt;p&gt;Is mathematical education better outside UK and USA? Studying in UK I noticed that the Uk syllabus of 3-4 modules  per area of mathematics, is often easily covered by one module in central Europe, for instance Italy or Germany. Looking online, I found out (link in the comments) that each year students in UK study between 300 and 600 hours less than anyone else in EU. The result is that a second year student in Eu has more skills and knowledge of a Master student in the UK. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Someone might argue it is quality not quantity that matters, however if we compare the exercises or the exam papers, EU exam papers are clearly richer and harder. Also, I challenged my tutor to solve one tricky exercise from a first year module in mathematical physics and he didn't know how to solve it, despite being an internationally renewed professor.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is mathematics really taught in a better (surely different) way in mainland Europe compared to UK? If so, then way is it common knowledge that UK/USA degrees are worth more?&#xA;  Is a maths degree from the UK or USA REALLY equivalent to one from mainland Europe?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is a link to lecture notes of a &quot;Mathematical Physics&quot; module in Italy, I know you won't understand the language, but most words are similar in the titles, so you can have an idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.science.unitn.it/~moretti/runfismatI.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fisica-Trento&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;: Here you can compare some lecture notes they are all from first years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://wwwf.imperial.ac.uk/~kb514/M1F.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Foundation of Analysis - Imperial College&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://wwwf.imperial.ac.uk/~kb514/M1P1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Analysis 1 - Imperial College&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://wwwf.imperial.ac.uk/~kb514/M1M1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Methods 1 - Imperial College&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://wwwf.imperial.ac.uk/~kb514/M1M2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Methods 2 - Imperial College&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.dm.unipi.it/~acquistp/analisi1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;University of Pisa - Analysis 1&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.dmmm.uniroma1.it/~paola.loreti/Dispensa_13-10_01.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;University of Rome - Analysis 1 -FOR ENGINEERS, not even mathematicians!&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: The description of the bounty is wrong, I thought I was assigning the bounty to another question, I apologize, I got confused.&lt;/p&gt;&#xA;" OwnerUserId="7513" LastEditorUserId="7513" LastEditDate="2016-12-16T19:15:01.743" LastActivityDate="2016-12-24T01:27:36.320" Title="Mathematics in UK vs Central Europe" Tags="|undergraduate-education|graduate-education|teacher-preparation|" AnswerCount="3" CommentCount="20" ContentLicense="CC BY-SA 3.0" />
  <row Id="11740" PostTypeId="2" ParentId="11711" CreationDate="2016-12-06T00:24:05.617" Score="3" Body="&lt;p&gt;I can give you some:&#xA;Collection of Problems in probability theory - Leo F. Boron&#xA;One Thousand Exercises in Probability - Grimmett&#xA;Theory and Problems of theoretical mechanics - Spiegel&lt;/p&gt;&#xA;" OwnerUserId="7541" LastActivityDate="2016-12-06T00:24:05.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11741" PostTypeId="2" ParentId="11711" CreationDate="2016-12-06T00:33:56.697" Score="3" Body="&lt;p&gt;There aren't many, however Introduction to Applied Mathematics by Strang is good,McGraw-Hill Introduction to the theory of probability and statistics is great too, and Complex Variables and Applications by McGraw-Hill again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps&lt;/p&gt;&#xA;" OwnerUserId="7543" LastActivityDate="2016-12-06T00:33:56.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11743" PostTypeId="2" ParentId="11708" CreationDate="2016-12-06T00:43:00.793" Score="4" Body="&lt;p&gt;Try using the Python library manim, as described here. 3Blue1Brown is an excellent online mathematics teacher and his videos are very clear! Try watching some of his videos! Python library manin - 3Blue1Brown&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.3blue1brown.com/about&quot; rel=&quot;nofollow noreferrer&quot;&gt;3blue1brown&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7543" LastActivityDate="2016-12-06T00:43:00.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11744" PostTypeId="2" ParentId="11732" CreationDate="2016-12-06T01:58:43.577" Score="3" Body="&lt;p&gt;An old professor of mine made this point by repeated reference to aviation skeptics / skepticism. Up to a certain point in history, engineers were able to point to any number of failed attempts at machines that produced sustained flight, and no successful attempts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student who fails to produce example that satisfies some set of conditions and then decides that &lt;em&gt;no such thing&lt;/em&gt; can satisfy these conditions is like an engineer who has built or observed many crashing airplanes, and decided that a flying machine cannot be built.&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2016-12-06T01:58:43.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11745" PostTypeId="1" AcceptedAnswerId="11752" CreationDate="2016-12-06T06:11:12.637" Score="11" ViewCount="426" Body="&lt;p&gt;(It will be difficult to answer this question without 'advertising' for a retailer, but I've searched for these several times in the past few years, to no avail.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/082186873X&quot; rel=&quot;noreferrer&quot;&gt;Math From Three To Seven (The story of a Mathematical Circle for Preschoolers)&lt;/a&gt;, the author makes frequent reference to a set of plastic shapes, each of which has four attributes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Colour&lt;/strong&gt; (red, green, blue, yellow)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Shape&lt;/strong&gt; (square, circle, triangle)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Size&lt;/strong&gt; (big, small)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Holiness&lt;/strong&gt; (it, uh, has a hole in it. Or it doesn't)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The set is complete - all $(4)(3)(2)(2) = 48$ blocks.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The footnote in the text refers to these as &quot;Dienes Blocks&quot;, which today, are &quot;universally known and easy to obtain&quot;. The problem, though, is that searching for 'Dienes Blocks' returns different products from the same manufacturer / brand / personal legacy of Zoltan Dienes - namely Base-10 blocks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www.zoltandienes.com/about/&quot; rel=&quot;noreferrer&quot;&gt;ZoltanDienes.com&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dienes’ name is synonymous with the Multi-base blocks (also known as Dienes blocks) which he invented for the teaching of place value.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That's great and all, but how do I find these &lt;em&gt;other&lt;/em&gt; blocks?&lt;/p&gt;&#xA;" OwnerUserId="308" LastActivityDate="2016-12-07T14:07:18.530" Title="Where can I find a set of these 'logic' blocks?" Tags="|primary-education|logic|manipulatives|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11746" PostTypeId="2" ParentId="11739" CreationDate="2016-12-06T10:00:18.830" Score="5" Body="&lt;p&gt;First off I would like to point out that at least one of the things you're comparing is a bad comparison. Mathematical Physics (or in my country just mathematics for Physics majors) will have a completely different breadth to any standard math course for mathematicians. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Physics majors don't have to care too much about proofs and theory but they need to get up to speed in calculations and basic understanding of mathematics extremely fast. Pretty much all of Physics assumes familiarity with decently hard differential equations, hard linear algebra and some to lots of numerical analysis and methods. They need to catch up very fast so the physics make any sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember back in my undergrad I was flabbergasted what my friend taking physics knew from Mathematical Analysis. He could solve complex differential equations at a time we were still developing a theory of the real numbers and could hardly do any limits. On the other hand he obviously had no clue what a real number was or how to construct it because it was useless for him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having that out of the way, I can't quite compare the UK, but I have a decent knowledge of the USA and the Czech republic. What I found is that what exactly you learn when you get a BC. in math in the USA is extremely student dependent. You can learn &quot;almost no&quot; math taking something like 16 credits worth of math courses or you can learn tons of math including taking advance MA/PHD levels courses. So how much you learn is often up to you rather than the institution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the Czech republic I ended my MA equivalent with roughly 200 credits of math courses for comparison and most of them would be high level undergrad or low level graduate type courses in the US. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said in the USA there are some very considerable breadth level requirements meaning that you have to take lots of non math courses as well. And at the PHD level the US all of a sudden tightens up a huge amount and the courses are IMO often better (if not by a huge amount) than in Europe. &lt;/p&gt;&#xA;" OwnerUserId="5896" LastActivityDate="2016-12-06T10:00:18.830" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11747" PostTypeId="1" CreationDate="2016-12-06T15:58:25.370" Score="11" ViewCount="462" Body="&lt;p&gt;I am curious, if we define success or failure by grades earned, or pass-rate (pass meaning the student does not need to retake the course) then what is the typical pass rate for students in upper level courses at universities in the US. Of course, this question seems too broad, hence let us refine the question further:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For institutions which have a required course in abstract algebra (at the level of Gallian, Fraleigh etc. ) what percentage of students do not pass the class ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Likewise,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For institutions which have a required course in real analysis (at the level* of Pfaffenberger, Lay, Krantz etc. ) what percentage of students do not pass the class ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Failing either of these classes implies the student is not ready to graduate with a degree in Mathematics. It follows the students either need to retake the course, or, seek a transfer of these courses from a different institution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me offer my own personal anecdote here. When I was an undergraduate in Math and Physics I always saw some subset of my peers fail the courses I was required to take. In retrospect, I think comparatively high fail rates are symptomatic of the course being required as to complete a degree. In contrast, electives have better pass rates because, well, students elect to take them. Getting back to my point, I certainly thought there was danger of getting a bad grade in my core math or physics courses, especially at the junior or senior level. So, as an instructor, the idea that some students don't pass the required courses I teach is not particularly surprising. This brings me to my third and related question which probably should be its own, but, I include it here to complete my thought.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it unreasonable to expect students to retake a major course in their last year ? How should we deal with the students who don't succeed in the higher level courses?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I appreciate both anecdotes and links to data. Thanks!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*admittedly, I am not sure these texts are on the same level, perhaps I should include baby Rudin in this list, anyway, I'm trying to describe what is usually a 400-level, often terminal, course on real analysis for math majors.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2016-12-10T14:24:33.843" Title="Typical rate of success in junior or senior level math courses at US universities" Tags="|undergraduate-education|education-research|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11748" PostTypeId="2" ParentId="11719" CreationDate="2016-12-06T19:00:31.063" Score="1" Body="&lt;p&gt;I imagine you would find this on GeoGebra.org &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can download the software and any community produced .ggb file for offline use, and there's thousands of ready-made interactives. You can even compile them into an ebook with notes and exercises. &lt;/p&gt;&#xA;" OwnerUserId="901" LastActivityDate="2016-12-06T19:00:31.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11749" PostTypeId="2" ParentId="2060" CreationDate="2016-12-06T20:08:27.390" Score="4" Body="&lt;p&gt;re: Senior undergrad, pharmacology major: absolutely no calculus used in biology courses. She actually laughed when I asked her.&#xA;This truly unbelievable. I can find no Pharmacokinetics text that does not use AUC = Area Under the Curve, a Calculus concept if ever there was one. &#xA;How can you be a pharmacologist without knowing about Bioavailability, a concept defined in terms of AUC? My guess that she just didn't realise what AUC actually meant. Sad. &#xA;But this isn't just restricted to Professionals. I have seen inserts in prescription medicines, meant to be read by the uninitiated, referencing &quot;the AUC to Infinity&quot; (!) (I would include a scan but I don't know how to insert an image)&lt;/p&gt;&#xA;" OwnerUserId="7548" LastActivityDate="2016-12-06T20:08:27.390" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11750" PostTypeId="2" ParentId="11719" CreationDate="2016-12-06T20:32:51.813" Score="1" Body="&lt;p&gt;If you have installed Mathematica or  &quot;CDF Player&quot; on your machine, you can download and use these demonstration while offline: &lt;a href=&quot;http://demonstrations.wolfram.com/SolidOfRevolution/&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; &lt;a href=&quot;http://demonstrations.wolfram.com/SolidsOfRevolution/&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="1229" LastEditDate="2016-12-06T20:39:54.007" LastActivityDate="2016-12-06T20:39:54.007" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11751" PostTypeId="2" ParentId="7825" CreationDate="2016-12-07T11:24:36.887" Score="0" Body="&lt;p&gt;What about constructing a double cone from paper. Also make some sections using the details from the paper:&lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/apostol388.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/apostol388.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="998" LastActivityDate="2016-12-07T11:24:36.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11752" PostTypeId="2" ParentId="11745" CreationDate="2016-12-07T14:07:18.530" Score="2" Body="&lt;p&gt;There are several vendors, but I was unable to find shapes that differ in holiness. Instead, they are different in size, color, shape and &lt;em&gt;thickness&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They seem to go by the name &lt;em&gt;attribute blocks&lt;/em&gt; or &lt;em&gt;logic blocks&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They are available at amazon or, as @AmyB said in the comments, here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.minilandeducational.com/en/logical-blocks/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.minilandeducational.com/en/logical-blocks/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2016-12-07T14:07:18.530" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11753" PostTypeId="2" ParentId="11736" CreationDate="2016-12-08T03:30:42.343" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Everytime I learn Mathematics, I do it the slow-way.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Slow is not good.  But superficial is worse.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I tends to read the definitions until I make sense of what is written in the texts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I wish every student did this.  Fantastic, well done, don't stop doing this, ever.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do, however, consider using alternate sources—&lt;strong&gt;multiple&lt;/strong&gt; alternate sources—to look up key definitions.  Think for &lt;em&gt;yourself,&lt;/em&gt; and ensure that what you are reading makes sense to &lt;em&gt;you.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The test of any “truth” is whether it is true for &lt;em&gt;you.&lt;/em&gt; If, when one has gotten the body of data, cleared up any misunderstood words in it and looked over the scene, it still doesn’t seem true, then it isn’t true so far as you are concerned. Reject it. And, if you like, carry it further and conclude what the truth is for &lt;em&gt;you.&lt;/em&gt; After all, &lt;em&gt;you&lt;/em&gt; are the one who is going to have to use it or not use it, think with it or not think with it.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;—&lt;a href=&quot;http://www.thewaytohappiness.org/thewaytohappiness/precepts/be-competent.html&quot; rel=&quot;noreferrer&quot;&gt;The Way to Happiness, section 17-2, &lt;strong&gt;&quot;Learn&quot;&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(The above linked article is highly recommended reading, by the way.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I can spend a whole day trying to get one page of the textbook.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To be frank—to me, this means your textbook is probably badly written.  Sorry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;em&gt;Or,&lt;/em&gt; it means you're studying at the wrong level.  A middle school student given a well-written college math text would spend a long time on each page, too.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I also do all the theorem proving by myself, even it takes me a lot of time to do that.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Good.&lt;/strong&gt;  Keep doing this.  There is no better way to learn math than by actually understanding and working through the concepts yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may want to start differentiating the &lt;em&gt;important&lt;/em&gt; theorems from the &lt;em&gt;unimportant&lt;/em&gt;.  Start considering which theorems will expand your knowledge and understanding of math, or even just which theorems you are interested in.  Which of the theorems have application or relevance to the understandings of math you already have?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not necessary that you perfectly understand &lt;em&gt;every&lt;/em&gt; theorem you &lt;em&gt;ever&lt;/em&gt; read...and even if you do want to attain that, you may attain it more rapidly with &lt;em&gt;important&lt;/em&gt; theorems and proof techniques under your belt.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The works tend to take a longer time than everybody else, but I do get the feeling I know the material better than other guys in my class.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You're probably correct on both counts.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;My advice to you:&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;First, &lt;a href=&quot;http://www.thewaytohappiness.org/thewaytohappiness/precepts/be-competent.html&quot; rel=&quot;noreferrer&quot;&gt;the article I've linked above&lt;/a&gt; contains some excellent tips on learning in general, not limited only to mathematics.  Start with that.  (You can turn off sound if you like, but the one-minute video is actually pretty cool.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, from my own experience in studying mathematics, I will add some specific points:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Ensure you get &lt;strong&gt;clear, accurate and &lt;em&gt;precise&lt;/em&gt; definitions for every mathematical term&lt;/strong&gt; used in your text.  (And every other term, too, for that matter.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have to fumble around mentally to remember what five of the words used in the sentence &lt;em&gt;mean,&lt;/em&gt; you're not likely to catch the meaning of the &lt;em&gt;sentence.&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The same goes for notation.  &lt;strong&gt;Play with newly introduced notation until it is as clear to you as English,&lt;/strong&gt; or maybe even clearer.  The notation &lt;em&gt;itself&lt;/em&gt; must not be a barrier to your understanding.  Play with it, and practice, practice, practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You're learning a language.  Learn it well.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;em&gt;ideas&lt;/em&gt; you are studying must become thoroughly familiar, common ground to you.  You need to &lt;strong&gt;drill on the basics of the branch of math you are studying until you know them cold.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You haven't mentioned any specific branches of mathematics, so I will mention a couple examples myself from my experience as a high school math teacher:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trigonometry students start flying after they are drilled and drilled and drilled and drilled on a simple right triangle diagram with a letter for each acute angle and a letter for each side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;What's sine x?&quot;  &quot;Uh, P over Q.&quot;  &quot;Good.  What's arctan F over P?&quot;  &quot;Uh, angle c.&quot;  On and on and on (ensuring you wait for the correct answer each time no matter how long it takes, and then validate the correct answer when given) until the student gets every question correct without appreciable hesitation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With several right triangles of different sizes drawn in random orientations, and with all different letters, this makes an &lt;em&gt;excellent&lt;/em&gt; drill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Actually it's several drills, all with the same setup of several triangles.  First, just do sines of angles.  Then cosines of angles.  Then tangents.  Then randomly alternate between sin, cos, tan.  Then cover cotangents.  Then secants.  Then cosecants.  Then randomly between all six basic functions.  Then start into the inverse functions: arcsine, arccos, etc.  Always back it up if the student has trouble and make sure you allow the student to &lt;em&gt;succeed&lt;/em&gt;; he should &lt;em&gt;win&lt;/em&gt; at the drill.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a similar set of drills for Calculus.  (Gosh, I should probably write these up in a book....)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that you need to get the &lt;em&gt;basics&lt;/em&gt; of the branch of math perfectly understood without the slightest hesitation, so you can read &lt;em&gt;fluently&lt;/em&gt; whatever is expressed about that branch of math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the most thorough students I ever taught had this as his only shortcoming: He got everything 100%, but he didn't work toward &lt;em&gt;fluency&lt;/em&gt;, so he remained slow.  He didn't see $8!/6!$ and think instantly, &quot;(seven times eight equals) &lt;strong&gt;56&lt;/strong&gt;&quot; too fast to even think the words.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The only other advice I have, is &lt;strong&gt;work everything out on paper or better yet, in your head.&lt;/strong&gt;  If you are using a calculator or computer to &quot;avoid errors,&quot; you shouldn't be using a calculator or a computer yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you are 100% confident in your ability to compute the answers yourself, with pencil and paper, and are &lt;em&gt;solely&lt;/em&gt; using a computer to save &lt;em&gt;time&lt;/em&gt;—then, okay.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, if you really know your stuff, the absence of a computer or a calculator won't bother you much.  It hardly saves any time at all, if you get sufficiently fast and well-practiced.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5751" LastActivityDate="2016-12-08T03:30:42.343" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11754" PostTypeId="2" ParentId="11739" CreationDate="2016-12-08T08:24:18.973" Score="6" Body="&lt;p&gt;Part of the answer to your question is that the statement 'mathematics is considered good in the UK and USA' refers, I believe, to research mathematics, not to undergraduate teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't have direct experience of education on the continent, although my colleagues do. There is a significant difference in standard. The culture and the education system are much more in favour of mathematics. The UK system, and even more so the US system on average, do not generally leave students well prepared to study maths at university level. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I still find it implausible that the difference is as much as you say. That your tutor can't do a first-year exercise raises a red flag for me. He will be an expert in the sense that he is good at using general understanding to address even unfamiliar questions. That he can't do it suggests the solution depends on some particular piece of information or a trick in the method. If you have been shown what to do, the question would be much easier to solve.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2016-12-08T08:24:18.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11755" PostTypeId="1" CreationDate="2016-12-09T12:55:47.763" Score="7" ViewCount="300" Body="&lt;p&gt;I'm teaching math in first years of university (matrix algebra, differential analysis, etc.) since maybe 5 years, and I usually give written (paper) exams to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm looking for ideas to change a little bit my practice: I would like to do more exams, i.e. more tests during the semester to test the continuous work of the students, but without doubling or multiplying by 3 the exam-marking job.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Have you seen good examples of math exams using a computer, or multiple choice questions?&lt;/strong&gt; for example for a basic matrix algebra course (matrix operations, rank, linear systems) &lt;/p&gt;&#xA;" OwnerDisplayName="user7559" LastActivityDate="2016-12-12T21:30:30.307" Title="Design of a math exam using multiple choice or computer" Tags="|course-design|technology-in-education|exams|notation|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11756" PostTypeId="2" ParentId="11755" CreationDate="2016-12-09T13:23:00.513" Score="1" Body="&lt;p&gt;You may be able to construct a multiple choice computer exam using &lt;a href=&quot;http://webwork.maa.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;WeBWorK&lt;/a&gt;.  There certainly are multiple choice ones in its library, but whether there were enough for a given exam would depend a lot upon the subject.  Of course, you could learn to write your own.  It is used for placement exams some places, at any rate.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-12-09T13:23:00.513" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11757" PostTypeId="1" AcceptedAnswerId="12002" CreationDate="2016-12-09T17:08:28.793" Score="6" ViewCount="1586" Body="&lt;p&gt;I'm looking for small data sets ($N\approx10$) that have integral (or even rational) &lt;strong&gt;&lt;em&gt;sample&lt;/em&gt;&lt;/strong&gt; standard deviations. Given a list of observations $\{x_1,\ldots,x_N\}$ with mean $\bar{x}$, the sample standard deviation is given by&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$s=\sqrt{\frac{\displaystyle\sum_{i=1}^N (x_i-\bar{x})^2}{N-1}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice the $N-1$ in the denominator here. With a simple google search I've found an article by Frank J. Dudek giving small data sets with integral &lt;strong&gt;&lt;em&gt;population&lt;/em&gt;&lt;/strong&gt; standard deviations, but the population standard deviation is computed with the formula above except the denominator of the fraction is $N$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm aware of data sets that have the following form:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\{a,\ldots,a,\frac{a+b}{2},b,\ldots,b\}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where the number of $a$'s appearing is equal to the number of $b$'s appearing. The sample standard deviation for this set is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$s=\left|\frac{b-a}{2}\right|$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I'd like an example that is a little more interesting than this one. Thanks in advance for any help.&lt;/p&gt;&#xA;" OwnerUserId="21" LastActivityDate="2017-05-15T11:26:17.080" Title="Small data sets with integral sample standard deviations" Tags="|examples|statistics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11758" PostTypeId="2" ParentId="11747" CreationDate="2016-12-10T14:24:33.843" Score="4" Body="&lt;p&gt;My experience has been that the grade distribution curve in upper-level classes is different, and, I think, rightly so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The incapable/uncaring students were almost certainly &quot;culled&quot; by an earlier class.  Viz. If students come into the school on a gaussian of &quot;ability&quot; (univariate for simplicity, this generalizes to many dimensions), then as time goes by the left tail gets chopped off, and by the time students are in abstract algebra, you only have the right tail of students, and it is reasonable for their grade distribution to be skewed right... On the other hand, if we assign a gaussian of grades for every course, and the bottom x% fail out of principle, then after y courses, only (1-x)^y% remain.  If x is 10%, and y is 12 courses, that is 28% of students.  If your school only graduates 28% of the students it accepts then it needs to reevaluate its acceptance criterion.&lt;/p&gt;&#xA;" OwnerUserId="842" LastActivityDate="2016-12-10T14:24:33.843" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11759" PostTypeId="2" ParentId="11692" CreationDate="2016-12-10T14:51:26.187" Score="1" Body="&lt;p&gt;This is an interesting exercise, as the many different responses show.  Sure, it's ambiguously worded (I thought they meant all the different reflections/rotations which is certainly hard). With the right attitude, this is a wonderful chance for a student to explore and have a good time with you, the parent.  She can find some, you can suggest other avenues and she can find more.  Then she can go to school and find out there were perhaps more you did not consider.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is when the teacher (whether tired, overworked, or just incompetent) makes this about the &quot;correct&quot; answer of 9.  There may be only one answer if the question is worded correctly, but when it's not worded correctly the teacher is just starting a pattern of mental rigidity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When my son was 5, he brought home a worksheet from his teacher, and asked for my help.  There was a picture of 2 frogs in a pond.  Next to it, was a picture of 4 frogs jumping out of the pond.  The question was, &quot;How many frogs are left in the pond?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My son had watched Cyberchase, a TV show on PBS that is wonderful, but intended for older children (maybe 10).  He was precocious, with a good visual and spatial sense.  He turned to me, and asked &quot;-2?&quot;.  He knew that made mathematical, but perhaps not physical sense.  He did not know quantum physics yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I laughed, and said &quot;I can't believe that is what they are asking in a kindergarten class, but put that down and bring it to your teacher.  I think your answer is amazing.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It turns out that we were supposed to consider the two pictures together so that 6 frogs minus 4 = 2.  This was completely unobvious to both of us. But the teacher, rather than say &quot;That's wonderful! They didn't mean that, but your answer shows that you understand negative numbers, bravo!&quot; just wrote an X on his paper, which he then brought home.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was the one who had to tell him that it's ok if they were not looking for that answer, that his answer made sense in its own way, and that he was wonderful, and should keep thinking about math and having fun with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not blaming her (entirely), but today, his interest in math is largely burned out.  It's sad because he showed such amazing early promise.&lt;/p&gt;&#xA;" OwnerUserId="1032" LastActivityDate="2016-12-10T14:51:26.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11760" PostTypeId="1" CreationDate="2016-12-11T12:10:47.163" Score="11" ViewCount="1099" Body="&lt;p&gt;I'm teaching mathematics on my free time for young pupils. Once I have seen that they denote angles like $\angle ABC$. But sometimes I have difficulties to understand whether they mean an angle or its explementary angle, especially in problems containing angles in a circle. Should I worry about this or should I teach them some other notation, like $\angle ABC$ is the explementary angle of $\angle CBA$ or the $\angle ABC$ where the some given point $D$ is between the rays $BA$ and $BC$? And also if some most talented pupils wants to participate in mathematics competitions, should he or she write the answers so that one knows exactly which angle he or she means or is it better to save a few seconds just by leaving the reader to think which one angle he or she means?&lt;/p&gt;&#xA;" OwnerUserId="7560" LastEditorUserId="376" LastEditDate="2016-12-11T13:09:12.027" LastActivityDate="2017-08-28T15:41:12.733" Title="How to denote angle?" Tags="|geometry|notation|teaching|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11761" PostTypeId="2" ParentId="11618" CreationDate="2016-12-11T16:59:14.337" Score="4" Body="&lt;p&gt;here is a way i teach about the natural exponential function. we look at the slope of the graphs $y = 2^x$ and $y = 3^x$ at $x = 0, y = 1$ by evaluating the average rate of change(slope of the secant line) of these functions on $[0, h]$ for small values of $h.$ conclude that there is a number $e$ with the property that the slope of the graph $y = e^x$ is one. we make a small characteristic triangle with one corner at $(0,1)$ and the others at $(1+h, 1), (h, e^h).$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;now we use addition property of the exponential function specialized to moving one unit to the right and what it does to the $y$-value embodied in $e^{1+x} = e e^x.$  that is, the $y$ value is multiplied by the base $e.$ they can, and usually do, see that the characteristic triangle at $(1, e)$ is the vertically stretched version of characteristic triangle at $(0,1).$ that is the side parallel to $y$ axis is multiplied by $e$ while the side parallel to the $x$ stay the same. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;they conclude that the slope of the graph of $y = e^x$ at $(1, e)$ is $e.$ you can now generalize this to idea that the slope of $y = e^x$ at any point is $y$ coordinate of the point. &lt;/p&gt;&#xA;" OwnerUserId="3441" LastEditorUserId="3441" LastEditDate="2016-12-13T22:12:43.130" LastActivityDate="2016-12-13T22:12:43.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11762" PostTypeId="2" ParentId="11739" CreationDate="2016-12-11T23:42:07.040" Score="2" Body="&lt;p&gt;I'm from Hungary and the level of physics education is pretty good compared to the UK (my graduate friends have no problem getting jobs there). However, I'm a bit confused, as A-levels exams are more difficult than our national exam, with the STEP exam much more difficult. I guess Cambridge is stronger than the best Hungarian universities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who is your tutor, if you don't mind?&lt;/p&gt;&#xA;" OwnerUserId="7562" LastActivityDate="2016-12-11T23:42:07.040" CommentCount="20" ContentLicense="CC BY-SA 3.0" />
  <row Id="11763" PostTypeId="2" ParentId="11755" CreationDate="2016-12-12T14:38:35.317" Score="3" Body="&lt;p&gt;Here are some examples of using WeBWorK for homework in matrix linear algebra and ordinary differential equations. (MTH165 at University of Rochester)&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The current MTH165 course is &lt;a href=&quot;https://math.webwork.rochester.edu/webwork2/fall16mth165&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  You can login as guest. You can print out a pdf of any of the assignments.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You can use Moodle as a front end and us WeBWorK to provide the assignments: &lt;a href=&quot;https://devel3.webwork.rochester.edu/moodle/course/view.php?id=21&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here is an example&lt;/a&gt; (you can log in as guest and see the assignment version of the WeBWorK questions). You can also print out pdf versions of the assignments from this version of the course.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To see the pure moodle version of the questions (the &quot;quizzes&quot;) where WeBWorK is completely hidden, you will need to login to the address above with the login visitor and the password visitor.  Moodle does not allow guests to view quizzes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you want to look behind the scenes you can visit hosted2.webwork.rochester.edu, and look at UR101, UR102, etc. Use profa as login and profa as password. These are demo courses and not specifically linear algebra. Don't expect any changes to be saved. :-) Check out the Library browser and look for problems involving matrices or linear algebra.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have a little patience with the free trial signup at webwork.maa.org. I think something failed over the weekend and they haven't noticed the outage yet. It's a busy time for them with the Joint Math meetings coming up soon. &#xA;(Added 12/12/2016 10am) At this time the link at forms.maa.org/r/webworktrial/add.aspx for requesting a free trial was accepting submissions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7565" LastEditorUserId="3237" LastEditDate="2016-12-12T18:55:01.780" LastActivityDate="2016-12-12T18:55:01.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11764" PostTypeId="2" ParentId="737" CreationDate="2016-12-12T17:52:34.393" Score="3" Body="&lt;p&gt;Every instance that I can think of of needing to check that a function is “well defined” actually fits into this general framework:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $\pi: A \to B$ be a surjection. We are attempting to define a function $f: B \to C$. What we have is a function $g: A \to C$. Checking that $f$ is “well defined” means that we need to check that if $\pi(a_1) = \pi(a_2)$ then $g(a_1) = g(a_2)$. In this case, $g$ must factor through $\pi$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I advocate this viewpoint in this question on MathOverflow:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/41955/does-any-textbook-take-this-approach-to-the-isomorphism-theorems&quot;&gt;https://mathoverflow.net/questions/41955/does-any-textbook-take-this-approach-to-the-isomorphism-theorems&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take the example of &quot;defining&quot; $f(\frac{a}{b}) =  a+b$ mentioned by Anschewski.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we are actually doing here is defining a function $g:\mathbb{Z} \times \mathbb{Z}\backslash\{0\} \to \mathbb{Z}$ by $g(a,b) = a+b$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the question we are asking is &quot;does $g$ factor through $\pi:\mathbb{Z} \times \mathbb{Z}\backslash\{0\} \to \mathbb{Q}$ defined by $\pi(a,b) = \frac{a}{b}$?&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see if this is the case, we need to check whether $g$ &quot;respects&quot; $\pi$:  are there $(a_1,b_1)$ and $(a_2,b_2)$ with $\pi((a_1,b_1)) = \pi((a_2,b_2))$ but with $g(a_1,b_1) \neq g(a_2,b_2)$?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer to this question is clearly yes:  $\pi(1,3) = \pi(2,6) = \frac{1}{3}$, but $g(1,3) = 4$ and $g(2,6) = 8$.  So $g$ does not factor through $\pi$, and hence there is no map &quot;$f$&quot; (or you could say &quot;$f$ is not well defined&quot;).&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-12-12T17:52:34.393" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11765" PostTypeId="2" ParentId="11755" CreationDate="2016-12-12T21:30:30.307" Score="0" Body="&lt;p&gt;Concerning the time issue, you can use AutoMultipleChoice to write multiple choice exams in LaTeX, with randomization of questions (or simply randomization of their order and the order of answers), and which can be automatically graded using a simple scanner. It will take some time at first but you should gradually build a database of exercises that should be reusable. These might exist available, but I don't know a source.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning the proper construction of the questions, there are several ways. If you want to test particular mistakes, propose an answer that follows from that mistake (e.g. I ask in almost all my calculus tests to differentiate $f(x)=2^x$ and I always propose $f'(x)=x2^{x-1}$). If you want to test computations of a certain sort, do the computation once for each proposed answer but each time add one mistake you really want your student not to do (this depends on your goal: you may include small silly mistakes if you want them to be flawless, or only include really bad mistakes).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-12-12T21:30:30.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11766" PostTypeId="2" ParentId="11735" CreationDate="2016-12-13T03:33:56.977" Score="3" Body="&lt;p&gt;Make an assignment that has the students find that scatter plot for continuous random variables.  And as a second part of the assignment, make them do the plot you describe.  Ask them to describe qualitative differences in those two plots and explain why this is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, try to make the plots qualitatively similar except for the discontinuities.&lt;/p&gt;&#xA;" OwnerUserId="7530" LastActivityDate="2016-12-13T03:33:56.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11767" PostTypeId="2" ParentId="11312" CreationDate="2016-12-13T04:06:22.337" Score="3" Body="&lt;p&gt;Well you run into an issue here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Namely, (1) you want to say &quot;here's this thing called a probability measure,&quot; which is something you're defining for the students.  But at the same time (2) you want to say why the probability measure should be additive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem of course is that additivity is a property used to &lt;em&gt;define&lt;/em&gt; what a probability measure is in the first place!  So they are additive because we define them to be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So really, you want to argue why this definition is natural.  For this, I would suggest a few routes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) this property holds when we have equally likely outcomes, where it amounts to saying $|A \cup B| = |A| + |B|$ when $A$ and $B$ are disjoint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) it is intuitively clear for a lot of people, but the difficulty [if any] is in notation.  So if you can get them ok with the notation, they'll be ok with the fact.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) You can have them prove some basic stuff like $P(A^c) = 1 - P(A)$ or inclusion/exclusion formula from axioms.  This doesn't give intuition so much as practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) You could have them think from a frequentist perspective.  Say &quot;we looked at 20 rocks pulled up from the ocean floor, and here's the data about them [20 row table.  Each row has columns for weight, color, and shape].&quot;  Then tell the students to predict about what percent of ocean rocks happen to be [gray].  Then ask them to predict what percent are [round].  Then ask them to predict what percent are [blue].  Then predict what percent are [blue or gray].  And predict for [gray or round].  (Et cetera)  The idea is that students will all naturally come up with probability for this, and the additive property is clear.  It also emphasizes that if the events aren't disjoint then additivity fails.  You can get the students to articulate why they used additivity, and why they want their estimates of probability to have this property [even though they don't know the distribution at all].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(5) This is clear if the sample space is finite.  In fact, you could just say that $P(A)$ is short-hand for $\sum_{a \in A} P(\{a\}).$&lt;/p&gt;&#xA;" OwnerUserId="7530" LastEditorUserId="7530" LastEditDate="2016-12-13T12:14:06.803" LastActivityDate="2016-12-13T12:14:06.803" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="11768" PostTypeId="2" ParentId="11312" CreationDate="2016-12-14T10:12:00.350" Score="2" Body="&lt;p&gt;I agree with Pat Devlin answers that the point seems to give an intuition why this part of the definition is natural. I would think that the following example is pretty telling (learned from &lt;a href=&quot;https://plus.google.com/+TimothyGowers0/posts/RCjWvhV6ehk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tim Gowers&lt;/a&gt;) : who will be the replacement to Nico Rosberg as a F1 pilot? Turns out &lt;a href=&quot;http://www.bbc.com/sport/formula1/38185491&quot; rel=&quot;nofollow noreferrer&quot;&gt;some people&lt;/a&gt; gave estimates of probability for several possible pilots, who add up to 5.6 (Fernando Alonso: 6/10 chance; Daniel Ricardo : 6/10 chance; etc.) One interpretation is precisely that these people are unaware that the probability of a union of mutually exclusive events is the &lt;strong&gt;sum&lt;/strong&gt; of their individual probability. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example where people get it right: if you ask what is the probability of a regular dice to turn up the face 4, the answer $1/6$ precisely comes from the combination of additivity and equiprobability (which itself can be justified by symmetry).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, we should insist that the core property related to additivity is &lt;em&gt;disjointness&lt;/em&gt; (and this cannot be stressed too much).&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-12-14T10:12:00.350" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11769" PostTypeId="2" ParentId="11312" CreationDate="2016-12-14T11:38:05.283" Score="2" Body="&lt;p&gt;Here's another crack at an answer.  Let me know if it's closer to what you want.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Bob picks a number from 1 to 100 (uniformly at random).  What's the probability that Bob's number doesn't contain the letter &quot;e&quot; when you spell it out?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This has the advantage that {numbers with that property} are hard to understand, but the event is much easier to view as {2 or 4 or 6 or 30 or 32 or ... or 66}.  And then the answer is just P(2) + P(4) + ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I expect students will naturally solve the problem like this, and after they do, you just have to emphasize that what they did was&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Rewrite the event as a set [skip if this is not an emphasis of the course]&lt;/li&gt;&#xA;&lt;li&gt;Realize that P(this or that) = P(this) + P(that)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Of course, you don't want them to feel like everything is additive.  You could perhaps follow this up by asking if P(2 or 4 or 6) = P(2 or 2 or 4 or 6) and letting them discuss this.  [at some point, have them argue against the claim that P(2 or 2 or 4 or 6) = P(2) + P(2 or 4 or 6)]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you really want to emphasize that they're actually ADDING probabilities [as opposed to merely counting], you could somehow make the distribution for each number different.  Then they would certainly solve it by adding up P(2) + P(4) + ..., but this is of course harder to swing.  Perhaps after they answer for the uniform distribution, you say &quot;Alice also picks a number from 1 to 100, but she does it in a special way.  She first flips a coin, and if it lands H, then she will pick the number 2.  If it lands T, then she will pick...  What's the chance that Alice will pick a number with that property?&quot;  or more to the point might be to say &quot;Alice also picks a number from 1 to 100, but she does it in a weird way.  50% of the time, she picks 7, which is her favorite number.  30% of the time, she picks the number 42.  12% of the time, she picks...&quot;  [maybe some table for this]&lt;/p&gt;&#xA;" OwnerUserId="7530" LastEditorUserId="7530" LastEditDate="2016-12-14T11:51:48.257" LastActivityDate="2016-12-14T11:51:48.257" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11770" PostTypeId="1" CreationDate="2016-12-14T15:28:14.597" Score="8" ViewCount="479" Body="&lt;p&gt;What could be a good &quot;layman&quot; metaphor for illustrating the difference between uniform and pointwise convergence of function series? I am teaching calculus to engineering undergrads; for many of them, the definitions are way too abstract, and mathematical examples are not that illuminating. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking about people &quot;converging uniformly&quot; towards the end of their kindergarten time (in at the age of 3, all out at the age of 7, say) and &quot;converging pointwise&quot; towards the graduation from university (some need 5 years, some need 10...). However, it turns out that life is not that simple even in the kindergarten, at least in the country where my university is, and I am looking for a more precise metaphor.     &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the answers show that my question is a bit too vague. It is &lt;strong&gt;definitely not&lt;/strong&gt; my intention to substitute a mathematical definition by any kind of metaphor, or to start the discussion in the class with such a metaphor, or to use it repeatedly or systematically. I was wondering what can I say &lt;em&gt;en passant&lt;/em&gt; when the definitions are already given, the examples are discussed, and this is the third class about uniform convergence, but some people still stare at me with empty faces. For those people, the mathematically rigorous presentation obviously did not work, and they need something that might motivate them to think over once again. &lt;/p&gt;&#xA;" OwnerUserId="7571" LastEditorUserId="7571" LastEditDate="2016-12-22T14:59:00.347" LastActivityDate="2016-12-28T22:25:38.763" Title="Good metaphor to explain the difference between pointwise and uniform convergence" Tags="|undergraduate-education|calculus|examples|" AnswerCount="9" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11771" PostTypeId="2" ParentId="11770" CreationDate="2016-12-14T17:12:36.997" Score="1" Body="&lt;p&gt;I actually am not sure that there is really any 'lay' way to think about this.  But you could try to modify this to your preference:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose you could compare a race of some kind with a security sweep.  Even in a turtle race (more fun to watch than you think!) everyone might eventually get to the end, though at vastly different times.  But if you are looking for a missing person everyone has to approach the end of the region being searched at roughly the same rate - and arrive together, or you might have missed something.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-12-14T17:12:36.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11772" PostTypeId="2" ParentId="11770" CreationDate="2016-12-14T18:09:47.807" Score="6" Body="&lt;p&gt;My advice is generally to stick to the rigorous definition with plentiful examples of proofs for sequences of functions that converge one way or the other, and skip metaphors all together. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you &lt;em&gt;have&lt;/em&gt; to have a metaphor, you might consider runners in a race, each confined to a lane. The runners all move at different speeds, some perhaps even slowing as they approach the finish line, some very far away or very close at any given time. The runners in each lane need have nothing in common- some runners might even be approaching the finish line from the other direction. Some finish at different times than others, but they all finish at the finish line. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This race is like point-wise convergence, the runner's lane position is like the value of the independent variable, and the position of each runner is like the values of the intermediate functions at each value of the independent variable as they approach the respective value of the limiting function- the finish line. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Uniform convergence, on the other hand, is like all of the runners moving from their starting positions to the the finish line in roughly the same amount of time. Since they are moving in a coordinated fashion, the amount of the race that they have finished is not different from lane to lane, it only depends on how long the race has gone on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this metaphor still has some problems, because it's a metaphor. Instead of using a metaphor, though, I recommend that you simply provide ample examples and explain that showing point-wise convergence is really demonstrating that for every &lt;em&gt;number&lt;/em&gt; $x_0$ in some interval, the sequence of &lt;em&gt;numbers&lt;/em&gt; $(f_n(x_0))$ converges to the &lt;em&gt;number&lt;/em&gt; $f(x_0)$, while uniform convergence is really a statement that the series of &lt;em&gt;functions&lt;/em&gt; $(f_n)$ converges to a &lt;em&gt;function&lt;/em&gt; $f$ on some whole interval. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2016-12-15T16:34:19.193" LastActivityDate="2016-12-15T16:34:19.193" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11773" PostTypeId="2" ParentId="11770" CreationDate="2016-12-14T18:33:48.307" Score="3" Body="&lt;p&gt;How about this: Think of the epsilon in the definition of either convergences as an allowable error. Now imagine you were given the task of filling 100 containers (of various sizes) with a certain prescribed amount (e.g., the $i$th container needs to contain $i^2$ Liters.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Uniform Convergence: When the foreman comes around and checks your work, he will check all the containers at the same time and none of the containers can be off by more than 1 mL. It doesn't matter if that container has 1 L or 10000 L in it; none can be wrong by more than 1 mL.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pointwise Convergence: When the foreman comes around and checks your work, none of the containers can be off by more than 1 mL, but if there are some that you didn't get close enough, he'll come back later and check again on those containers. As long as you eventually get them all within 1 mL, he's happy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analogy decoded: $f_t(i) \to f(i)$ where $f(x)$ is the desired amount of water in container $x$ and $f_t(x)$ is the actual amount of water in container $x$ at time $t$.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2016-12-14T18:33:48.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11774" PostTypeId="1" CreationDate="2016-12-14T19:57:05.387" Score="1" ViewCount="113" Body="&lt;p&gt;My department uses the Major Field Assessment Test in Math (I think we refer to it as the &quot;MFT&quot;) and that is great for its purpose. However, I am currently teaching a capstone course where we spend about half the semester working through multiple choice Math GRE-type questions which run the spectrum of undergraduate math; calculus I, II and III, differential equations, linear algebra, complex analysis, real analysis, elementary topology, basic abstract algebra and some combinatorics. I'd like to give a 2 hour final which is roughly reflective of these topics and of course I can just recycle the multiple choice questions I have at my disposal. But, I'd rather use a short-answer test which runs through a similar range of topics. Hence my question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do you know of a broad spectrum math test at the undergraduate level which is suitable for Math majors who have completed real and abstract algebra ? Either multiple choice or short-answer will be useful for my purpose.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance for your help!&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2016-12-14T19:57:05.387" Title="Comprehensive Assessment Test for Undergraduate Math Program" Tags="|undergraduate-education|test-design|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11775" PostTypeId="2" ParentId="11770" CreationDate="2016-12-14T20:05:54.113" Score="3" Body="&lt;p&gt;I am not sure metaphors are as good a way to explain math as we tend to think, and I am quite convinced that metaphors that are assumed to be grounded in &quot;real life&quot; tend to have their silliness obscure their point. So let me try a go at a metaphor that is clearly not grounded in real life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sophia has an infinite collection of hourglasses, each of which has a duration of $n$ minutes (one hourglass for each positive integer $n$). She also has infinitely many arms, and use them to start all her hourglasses simultaneously. Then we know that each individual hourglass will be finished at some time, but at no time will all hourglasses be finished. In mathematical words, the hourglass all finish, but not in a uniform time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe the most interesting thing with this example is to look at the drawbacks it seems to exhibit. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, the fact that the convergence is quite blunt, since each hourglass not only goes to the state &quot;finished&quot;, but actually reaches it, is rather a simplification: it separates the concept of uniformity (which is pretty general) from the concept of convergence. You should also give other, mathematical examples of uniformity and non-uniformity (it all boils down to inversing quantifiers, and their I like the metaphor: &quot;each girl is liked by some boy, but no boy likes all girls&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, having an infinite number of hourglasses may seem a bit too much. But it is necessary, since finiteness would make uniformity automatic! This is obvious by taking the maximum duration of an hourglasses, and can be used to prove that the convergence of a vector-valued sequence reduces to convergence of each coordinate individually. This is interesting, because the context in which non-uniform convergence occurs, function spaces, is precisely the infinite-dimensional case, where the max argument fails.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2016-12-14T20:05:54.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11776" PostTypeId="2" ParentId="11736" CreationDate="2016-12-14T20:07:34.323" Score="0" Body="&lt;p&gt;I use that way in my studying but that's not a right way. you do not need to make proofs by yourself for every theorem you met, learn how to use theorems is better, read the proofs rewrite the proofs in your words if you want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every time I see a question about mean value theorem I try to prove it and I do not SOLVE the question and when I read the proof,  it used extreme value theorem so I start to think how to prove it and so on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This way in studying is not suitable for students it is slow and you spend much time in it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end we need to know how to drive a car but we do not need to know how the mechanism works.&lt;/p&gt;&#xA;" OwnerUserId="5993" LastActivityDate="2016-12-14T20:07:34.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11777" PostTypeId="2" ParentId="11721" CreationDate="2016-12-14T20:26:36.487" Score="4" Body="&lt;p&gt;What about taking what @whatever got: &lt;span class=&quot;math-container&quot;&gt;$ (4\cos^2(\theta)-1)^2(1-\cos^2(\theta)) \cos^2(\theta)=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$ x = \cos(\theta).$&lt;/span&gt; This gives us a more familiar looking: &lt;span class=&quot;math-container&quot;&gt;$(4x^2-1)^2(1-x^2)x^2=1$&lt;/span&gt;, which we can expand (and clear leading negatives for appearances sake):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$16 x^8 - 24 x^6 + 9 x^4 - x^2 + 1=0$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if your students are capable of doing polar coordinates at the level you're doing, then most assuredly they've been exposed to dealing with polynomials and their roots. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A simple application of the Rational Root Test shows no rational roots. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, applying Descartes's Rule of Signs says: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;4 possible real roots, and hence 4 possible complex solutions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's an even-degree polynomial with positive leading coefficient. Without techniques from calculus or non-familiar things, however, they should do exactly what all of us would: graph that polynomial on their calculators. No zeroes. Therefore the original question is answered, because a polynomial they're familiar with has no (real) roots. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This could be a place to allude to the fact that even though the polar graphs don't intersect in the real plane, they have DO have an intersection -- in the complex plane, just like &lt;span class=&quot;math-container&quot;&gt;$y=-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt; intersect in the complex plane but not the real one.&lt;/p&gt;&#xA;" OwnerUserId="1249" LastEditorUserId="11" LastEditDate="2019-12-15T23:25:39.597" LastActivityDate="2019-12-15T23:25:39.597" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11778" PostTypeId="1" CreationDate="2016-12-14T23:59:03.347" Score="5" ViewCount="342" Body="&lt;p&gt;Prompted by the question,&#xA;&quot;&lt;a href=&quot;https://matheducators.stackexchange.com/q/11760/511&quot;&gt;How to denote angle?&lt;/a&gt;,&quot;&#xA;I am interested to learn when students consider and reason with &#xA;angles $&amp;gt; 180^\circ$. &#xA;For example, when do they reason with an angle of $270^\circ$ representing&#xA;three Cartesian quadrants?&#xA;Perhaps this varies from country to country?&#xA;The issue arises because, e.g., protractors only measure angles in $[0^\circ,180^\circ]$. And the notation $\angle ABC$ seems to mean&amp;mdash;at least in certain&#xA;contexts&amp;mdash;the $\le 180^\circ$ angle defined by the two segments $AB$ and $BC$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own teaching has been primarily at the US college- and graduate-school levels, where angles $&amp;gt; 180^\circ$ are pretty much taken for granted.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-12-16T02:14:48.887" Title="At what educational stage are angles greater than 180 introduced?" Tags="|geometry|notation|teaching|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11779" PostTypeId="1" CreationDate="2016-12-15T07:15:29.390" Score="17" ViewCount="1834" Body="&lt;p&gt;I'm designing a lesson for an &lt;em&gt;Introduction to Integral Calculus&lt;/em&gt; class, and I want to encourage students to evaluate integrals without just going straight for the antiderivative and using the fundamental theorem of calculus. I want them to think geometrically about the situation before diving in with computations. Here are a few general topics I've thought to include so far:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Integration of even and odd functions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Integration of functions with graphs that are familiar shapes. Like $\int_{-2}^2 \sqrt{4-x^2}\;\mathrm{d}x$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maybe integrating a function without an nice antiderivative on the domain of integration, like $\int_{-a}^a |x|\;\mathrm{d}x\,$? But I would want the function so be more interesting that just $|x|$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What are good examples of integrals that illustrate the ideas above and would work well in a lesson? Or better, does anyone have other ideas for topics that I should (or shouldn't) include in this lesson?&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2016-12-24T20:39:31.110" LastActivityDate="2017-07-19T17:06:12.703" Title="Evaluating integrals geometrically, without using the fundamental theorem of calculus" Tags="|undergraduate-education|calculus|examples|integration|" AnswerCount="9" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11780" PostTypeId="2" ParentId="11779" CreationDate="2016-12-15T15:30:42.813" Score="11" Body="&lt;p&gt;A useful trick is the idea of rearranging values of a function. For example, while $\sin^2 \theta$ and $\cos^2 \theta$ have different graphs on $0 \leq \theta \leq \pi/2$ you can tell from the graphs of $\sin \theta$ and $\cos \theta$ that the values which the squared functions take must be equal. In short,&#xA;$$ \int_{0}^{\pi/2} \cos^2 \theta \,  d\theta = \int_{0}^{\pi/2} \sin^2 \theta \,  d\theta $$&#xA;Of course, you can make the same conclusion for other intervals such as $[0, \pi]$ or any $2\pi$-length interval. Use the Pythagorean identity and the fact $\int_{a}^{b} c \, d\theta = c(b-a)$ to derive:&#xA;$$ \frac{\pi}{2} = \int_{0}^{\pi/2} 1 \, d\theta = \int_{0}^{\pi/2} \left(\cos^2 \theta +\sin^2 \theta\right) \,  d\theta  = &#xA;\int_{0}^{\pi/2} \cos^2 \theta \,  d\theta + \int_{0}^{\pi/2} \sin^2 \theta \,  d\theta $$&#xA;Hence, $\displaystyle \int_{0}^{\pi/2} \cos^2 \theta \,  d\theta = \int_{0}^{\pi/2} \sin^2 \theta \,  d\theta = \pi/4$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This trick paired with those which exploit the fact that multiples of a period of a sinusoid integrate to zero can save a lot of work on common trigonometric integrals. More to the point, these are relevant to your lesson because they allow integration without a deep understanding of yet-taught techniques &lt;/p&gt;&#xA;&#xA;&lt;p&gt;( I assume students are largely ignorant of basic trigonometric identities like $\sin^2 \theta = \frac{1}{2} \left( 1 - \cos 2 \theta \right)$, a belief those outside the USA will likely gasp in disbelief, but those within the US will easily recognize as a realist position given current malpractice in our standards of precalculus etc.)&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="688" LastEditDate="2016-12-17T10:26:19.183" LastActivityDate="2016-12-17T10:26:19.183" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11781" PostTypeId="2" ParentId="11770" CreationDate="2016-12-15T17:26:36.550" Score="6" Body="&lt;p&gt;I don't think you need a metaphor at all &amp;mdash; their intuition of convergence is probably &lt;em&gt;already&lt;/em&gt; about uniform convergence, you just need:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To give surprising examples of pointwise convergence, to motivate the need for the more refined notion&lt;/li&gt;&#xA;&lt;li&gt;To give them the algebra they need to explain how uniform convergence is different.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One example of a surprising sequence is that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ f_n(x) = \begin{cases} 0 &amp;amp; x &amp;lt; n&#xA;\\ 1 &amp;amp; n \leq x \leq n+1&#xA;\\ 0 &amp;amp; n+1 &amp;lt; x \end{cases} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;converges pointwise to the zero function, despite the fact the function doesn't &quot;visually&quot; get smaller &lt;em&gt;at all&lt;/em&gt;. We can do this with continuous functions and without going off to infinity as well:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ g_n(x) = \begin{cases} 0 &amp;amp; x \leq 0&#xA;\\ n x &amp;amp; 0 \leq x \leq \frac{1}{n}&#xA;\\ 2-nx &amp;amp; \frac{1}{n} \leq x \leq \frac{2}{n} &#xA;\\ 0 &amp;amp; \frac{2}{n} \leq x \end{cases} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiply both examples by $n$ if you want something even more dramatic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In both of these examples, you can put words to the idea that these 'ought' not converge by observing $\max_x |f_n(x) - 0| = 1$, which is clearly &lt;em&gt;not&lt;/em&gt; decreasing to zero as $n \to \infty$. This translates to the characterization that $f_n \to f$ uniformly iff&#xA;$$ \lim_{n \to \infty} \left( \sup_x | f_n(x) - f(x) | \right)$$&#xA;which makes it easy to see algebraically what's going on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then on the $\epsilon-\delta$ front, demonstrate how these bad examples of pointwise convergence all have $\delta$ depending on the choice of $x$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Depending on their particular engineering background, they may already know practical examples of this; e.g. if they've seen some sort of signal processing, they may know the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gibbs_phenomenon&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gibbs phenomenon&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2016-12-17T11:22:46.603" LastActivityDate="2016-12-17T11:22:46.603" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11782" PostTypeId="2" ParentId="11778" CreationDate="2016-12-15T23:25:11.967" Score="5" Body="&lt;p&gt;Students are introduced to reflex angles in primary school in Colorado.  However, they use them very little.  I teach grades 7-12, and in the Common Core State Standards there are very few standards (if any) that require the use of reflex angles.  So, you are correct that just about every student in the U.S. that sees the $\angle{ABC}$ will take that to mean the angle $\angle{ABC}$ that is $\le180$.&lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2016-12-15T23:25:11.967" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11783" PostTypeId="2" ParentId="11778" CreationDate="2016-12-16T01:54:37.127" Score="2" Body="&lt;p&gt;My wife uses Singapore Math curriculum. She says angles larger than $180^o$ appear in the 4th grade curriculum. In second grade they see right angles. In third grade they see acute angles and obtuse angles.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2016-12-16T02:14:48.887" LastActivityDate="2016-12-16T02:14:48.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11784" PostTypeId="2" ParentId="11779" CreationDate="2016-12-16T03:25:44.887" Score="9" Body="&lt;p&gt;In his &lt;em&gt;Calculus&lt;/em&gt; book, Spivak gives these two exercises before he ever introduces the fundamental theorems of calculus.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Evaluate without doing any computations:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int\limits_{-1}^{1} x^3\sqrt{1-x^2} \,\mathrm{d}x \qquad\qquad\int\limits_{-1}^{1} \left(x^5+3\right)\sqrt{1-x^2} \,\mathrm{d}x$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-12-16T05:10:23.880" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11785" PostTypeId="1" CreationDate="2016-12-16T21:30:57.350" Score="13" ViewCount="299" Body="&lt;p&gt;I am a teaching assistant in mathematics at a university in Continental Europe. The course for which I am an assistant is a third year course, so usually the students are expected to know the basic notions from calculus and linear algebra. However, I often notice that students are not as familiar with the basic material as I would wish them to be$^1$. So, the natural question to ask is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What should be my teaching strategy?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are several possibilities. Each of them has its own pros and cons, which might contradict each other; but in fact, they don't, because you can regard it as a pro or a con, depending on your general attitude towards teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1. Go very slowly and repeat everything.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Pros:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They can actually learn a lot from the exercise class.&lt;/li&gt;&#xA;&lt;li&gt;Everything I do in the exercise class is self-contained; they find everything relevant to a certain topic concentrated on a few sheets of paper. Whence:&lt;/li&gt;&#xA;&lt;li&gt;Learning the material is made easier for the students (at least in a certain way).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Cons:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;People who already know the basic material get bored easily.&lt;/li&gt;&#xA;&lt;li&gt;I have trouble finishing; I lose plenty of time repeating material that should be known.&lt;/li&gt;&#xA;&lt;li&gt;When learning the material, they lose focus on what's important and what's not. This goes hand in hand with:&lt;/li&gt;&#xA;&lt;li&gt;Due to me repeating everything, the density &quot;content per time&quot; seems to be very high for the students, although they should already know a lot of things I say.&lt;/li&gt;&#xA;&lt;li&gt;A funny aspect which I encountered is the following: If I repeat basic material, students do not seem to understand the material, but if I skip the repetition and just use the results (e.g., I say &quot;follows from linear algebra&quot;), they think that they understand. But in fact, they don't, because they do not really understand what happens at a particular step in an exercise.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2. Just assume that they know the material.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Pros:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I can actually finish in time, so they get solutions for every exercise.&lt;/li&gt;&#xA;&lt;li&gt;The students who know the material can follow and don't get bored.&lt;/li&gt;&#xA;&lt;li&gt;The students should look up the material on their own; in fact, one goal of the university is to make people work independently and scientifically.$^2$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Cons:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They don't really understand what happens during a particular step of an exercise.&lt;/li&gt;&#xA;&lt;li&gt;As a TA, I somehow feel bad when using this method, because I know that there are a lot of students who don't understand the material. Of course, I want them to learn as much as possible.&lt;/li&gt;&#xA;&lt;li&gt;At the latest they should be able to apply the material in the exam. If they don't learn it fast, they never will, and they will most likely fail the exam. (See footnote 2: With this method, I don't encourage them as much as with method 1 to really learn the material.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3. A mixture of 1. and 2.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somehow combines a few pros and cons of each 1. and 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Which method do I apply?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually I go by method 1., because my aim is to make the students (all students! - even if this is utopian) fully understand the material. However, I am a bit sceptical because of the cons.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Related questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Should I even care about students as in footnote 1?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Assume that the students would be interested in the material but forgot a lot of basic notions. Does this change the answer to the main question?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Would it be a good idea to go by method 2. and give a sheet of paper with the used basic theorems to the students?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;$^1$ This is actually an euphemism. Unfortunately, the students I talk about do not care about grades or even understanding the material. They just want to pass the exam. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$^2$ ... but this does not work very well in practice. The kind of students mentioned in the footnote above just don't care and blame the TA for giving bad exercise classes. They don't understand that at a university not everything is served up on a plate so that they only have to eat the meal - they also have to cook. But it is a matter of the TA's attitude if one should prepare the ingriedients for them (like in a cooking class) or just give them the full recipe.&lt;/p&gt;&#xA;" OwnerUserId="7578" LastActivityDate="2017-01-03T11:10:19.437" Title="Repeating basic material in exercise class at university" Tags="|teaching|class-participation|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11786" PostTypeId="2" ParentId="11785" CreationDate="2016-12-17T00:41:55.567" Score="8" Body="&lt;p&gt;This is a good question, and I hope my answer is just the first among other (likely better!) ones.  [I'll try not to rant, but I'm not making any promises.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me start with a few general remarks.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This is a good question, and the fact that you are thinking about teaching in this way is (obviously) a very good sign.  Keep reflecting, keep asking, and keep experimenting!  (Try incorporating new ideas into your teaching approach, and see what happens.  Reflect, discuss, experiment, reflect, discuss, ...).&lt;/li&gt;&#xA;&lt;li&gt;In general, there's a dangerous trap in all questions like this.  Namely, we need to avoid the temptation of thinking that there is a best answer.  What works well is highly dependent on the individual students, the instructor, the material, the course format, et cetera.  And even with all things being equal, there are still many different effective approaches.  So going along my first point, take my answer (and any others) with a grain of salt, and be sure to develop a style that works best for you.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now let me address some of your questions one at a time.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Should the course be too fast or too slow?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Given the choice between those, I (like you) prefer to err on the side of too slow; your job is for them to learn, not for you to prevent boredom.  That said, naturally the best is somewhere &quot;just right,&quot; and if you're close enough to that sweet spot, then it wouldn't matter whether you're a little too fast or too slow.  [Opinions differ on this point, but we can all agree that the course should ideally be at just the &quot;right&quot; pace]&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;How do we know how fast is too fast?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I wondered this question for a while until one day I realized the answer.  Although I may have some idea, I ultimately don't really know what's too fast---but the students do!  To say it differently, you may be an expert in the material, but only the students can learn it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the best thing you can do is (1) ask plenty of questions [in class, quizzes, take-home assignments] so that you [and ideally the students!] can discover what the students do and do not know, and (2) let the students dictate the pace.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, you might raise a question for the class and have them work on it individually or discuss it in groups.  You'll be able to see how quickly they work through things, and you can walk around and help out students individually [this lets you get a finer understanding of each student's background, and it also lets you have really personalized instruction].  If you see (perhaps to your surprise) that many students are struggling with some concept that they should have known, then you can address this as a class or give some extra homework on it.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;How do we teach students who don't know the pre-requisites?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;So this is tricky, and depends on many things.  Let me give some scenarios to illustrate what I might do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Scenario 1:&lt;/strong&gt; Say you're teaching differential equations.  I would assume the students are all proficient in the mechanics of first-semester calculus (otherwise I couldn't possibly teach everything), and on the first day I would probably say something to them along the lines of &quot;To do well in this course, you'll need to know topics X, Y, and Z.  Here is a quiz on these topics that you should already know.  [they take quiz, then perhaps we discuss the answers]  I don't really care if you know this material &lt;em&gt;right now,&lt;/em&gt; but I will give you another quiz on this material at the end of the week.  If you don't ace that quiz, you should change out of this class because I literally cannot imagine how you could possibly pass the course otherwise.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Scenario 2:&lt;/strong&gt; Suppose you're teaching some class, and there's material from a few classes back that they maybe forgot [for example, trigonometric identities used to compute integrals like $\cos^2 (x)$].  In this situation, I would spend some time reminding the students of the facts we need, but I wouldn't address it nearly at the level that would be expected in their previous class.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;What about individual students who are falling behind?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;In office hours, if a student asks me to go over section 4, I usually start by asking them a question about section 3.  I then go back in time until we can put our finger on what it is they don't know.  Students almost never like &quot;relearning&quot; things that they &quot;already know,&quot; but they need it.  After you figure out what they need to work on, ask them plenty of questions and let them drive the pace entirely.&lt;/p&gt;&#xA;" OwnerUserId="7530" LastActivityDate="2016-12-17T00:41:55.567" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11787" PostTypeId="2" ParentId="11770" CreationDate="2016-12-17T04:30:11.910" Score="2" Body="&lt;p&gt;I dont think that we can explain any concept in mathematics dealing with infinitely many (like points, sequence) using real life examples, simply because in reality there is nothing infinite (an Avogadro number of particles, after all). The only way  is to make assumptions which are unrealistic (as mentioned in a previous comment, like having infinitely many hands), and then achieve the desired goal.&lt;/p&gt;&#xA;" OwnerUserId="7579" LastActivityDate="2016-12-17T04:30:11.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11788" PostTypeId="2" ParentId="11770" CreationDate="2016-12-17T08:04:28.597" Score="2" Body="&lt;p&gt;For uniform convergence: Consider a &lt;em&gt;tube&lt;/em&gt; around the graph of $f$. No matter how thin the tube is, there exists an $N$ such that graphs of all functions (except possibly the first $N$ of them) completely fit in the tube.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In pointwise convergence there may be no such $N$ for some tube.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/sDtHe.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sDtHe.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="376" LastEditDate="2016-12-17T08:52:29.380" LastActivityDate="2016-12-17T08:52:29.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11789" PostTypeId="1" CreationDate="2016-12-17T18:12:43.383" Score="7" ViewCount="214" Body="&lt;p&gt;In &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387954848&quot; rel=&quot;noreferrer&quot;&gt;Practical Analysis in One Variable&lt;/a&gt;, Donald Estep introduces &lt;a href=&quot;https://en.wikipedia.org/wiki/Lipschitz_continuity&quot; rel=&quot;noreferrer&quot;&gt;Lipschitz continuity&lt;/a&gt; early on, delaying the standard definition of continuity, along with uniform continuity, until the beginning of his more theoretical section on analysis in the final part of the book. According to Estep, Lipschitz continuity makes it easier to give constructive proofs of several key results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lipschitz continuity captures the idea of making the change in output arbitrarily small by making the change in input small. A nonnegative Lipschitz constant, $L$, if it exists, provides an upper bound for the change in output based on the change in input: $|f(x_1)-f(x_2)|\leq L|x_1-x_2|$. That said, $x^{1/3}$ for instance fails to be Lipschitz continuous on any interval containing zero, although it is continuous by the standard definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the pros and cons of early Lipschitz continuity? I can imagine that its quantitative approach of calculating an upper bound might be easier for some students to grasp than using epsilon-delta. If you have any experience with using it, I would be interested to know.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2016-12-18T08:50:35.810" LastActivityDate="2016-12-18T08:50:35.810" Title="Lipschitz continuity before standard definition of continuity" Tags="|undergraduate-education|calculus|mathematical-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11790" PostTypeId="2" ParentId="11789" CreationDate="2016-12-17T19:03:54.087" Score="7" Body="&lt;p&gt;The question doesn't say whether this is for freshman calc or for an upper-division class in which students are being reintroduced to concepts such as continuity that they've already seen. In this answer, I'll assume the former.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need Lipschitz continuity to get uniqueness results for the solutions to certain differential equations. Without it, you can get weirdnesses such as the famous Norton's dome, which, if you take it seriously, means that Newtonian mechanics is nondeterministic. I would have liked it if my freshman calc teacher had told me about these ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, most of us are teaching students who are almost all completely unlike ourselves at the same age, so it sounds like a bad idea to me to introduce Lipschitz continuity in first-semester freshman calculus. Students at that stage in their education are grappling with several very different things: (1) in most cases, they need to remediate their weak skills in abstract/symbolic/formal reasoning and in arithmetic and basic algebra; (2) they're learning a body of algorithmic techniques for differentiation; and (3) they're learning some very deep and abstract notions such as limits and the completeness property of the reals. At my school (a community college in California), roughly half of each class is destined to fail because they can't handle #1, and of the remainder, very few succeed in #3 (which is why they don't get an A). For this reason, teaching multiple definitions of the limit sounds like a bad idea to me, unless this is something like honors freshman calc at MIT.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2016-12-17T19:09:07.020" LastActivityDate="2016-12-17T19:09:07.020" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11791" PostTypeId="2" ParentId="11770" CreationDate="2016-12-17T21:05:20.063" Score="3" Body="&lt;p&gt;Other answers suggest sticking to the definitions and just teasing out their logic carefully. I agree that this is a good method and you should mostly stick to this. @Behzad's answer &lt;a href=&quot;https://matheducators.stackexchange.com/a/11788/80&quot;&gt;with the picture and the notion of a &quot;tube&quot;&lt;/a&gt; around the limiting function's graph is also a good one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm going to suggest an analogy that you might consider using, &lt;em&gt;in addition to&lt;/em&gt; the other suggestions. I don't think it's perfect, and I don't believe anything could be better than simply working through all the details of the formal definitions. However, when you introduce the concept, you may want briefly mention this analogy, and also draw some &quot;tubes&quot; on the board. This way, students at least have some visual intuitions to latch onto when you start diving into the formal epsilontics of the definitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Think about snow falling onto flat ground. (That is, the limiting function is the constant 0 function.) Imagine a sheet of snow sliding off the roof and falling into your yard. This is sorta like uniform convergence: all of the snow is approaching the ground &lt;em&gt;at the same pace&lt;/em&gt;, even it's not at the same height all the time. If you put your eye level at a certain height above the ground, there must be some point in time after which &lt;em&gt;all of the snow is below your eye level&lt;/em&gt;. Some of the snow might be lower than other parts, but surely there won't be some snow that is lagging so far behind the other parts that it's not close to the ground yet. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same situation, but instead think about individual snowflakes falling from the sky to cover the ground. This is sorta like pointwise convergence. You can be assured that every single individual flake will eventually reach ground level. However, each one is falling at its own pace. While some have already reached the ground, some may have just left their clouds and are well above the ground. There cannot be a single point in time after which &lt;em&gt;all&lt;/em&gt; of the flakes are below your eye level. You have to track each individual flake separately. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, I'm aware this is not perfect and I'm sure you could pick apart the analogy. However, I think it may be helpful to start with a brief mention of this idea, &lt;em&gt;then&lt;/em&gt; spend time working with the formal definitions and some examples, and even then returning to this analogy and asking the students, &quot;In what ways was the analogy reasonably correct, and in what ways was it flawed?&quot; If they are able to have a meaningful discussion about not only the original concept but also an analogy meant to illustrate that concept ... well, I think that means they learned something!&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2016-12-17T21:05:20.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11792" PostTypeId="1" AcceptedAnswerId="11793" CreationDate="2016-12-17T22:14:01.500" Score="7" ViewCount="293" Body="&lt;p&gt;I came across many people who believe the below  false implication. I don't know why people believe it true in high school and middle school and also students in university level. Really I would like to know why they believe: $\displaystyle c&amp;gt;0\implies \frac{1}{c}&amp;lt;0$ is true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;: $C$ is a real number &lt;/p&gt;&#xA;" OwnerUserId="7583" LastActivityDate="2016-12-17T23:37:43.843" Title="Why many people believe that: $\displaystyle c&gt;0\implies \frac{1}{c}&lt;0$?" Tags="|mathematical-pedagogy|real-numbers|inverse|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11793" PostTypeId="2" ParentId="11792" CreationDate="2016-12-17T23:37:43.843" Score="4" Body="&lt;p&gt;I don't think anyone believes this in the way you have stated. Perhaps you should be more concrete in how the student is actually being presenting the misconception. Perhaps they think $\frac{1}{2} &amp;lt; 0$ or something similar?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find students will be unsure rather than actively wrong about the relationship between $\frac{1}{2}$ and $0$. But it's very easy to correct. Would you rather have half ($\frac{1}{2}$) a pizza or no ($0$) pizza at all? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This will likely not be the end of the development in the student's understanding of the number line, but it's a strong starting point that you can call upon later.&lt;/p&gt;&#xA;" OwnerUserId="1018" LastActivityDate="2016-12-17T23:37:43.843" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11794" PostTypeId="1" CreationDate="2016-12-18T00:57:55.483" Score="12" ViewCount="452" Body="&lt;p&gt;I came to mathematics via physics, in part because of the reputation of physics as allowing &quot;non-rigorous&quot; reasoning. The subject felt more free and less anal-retentive than mathematics. This is not to say that physics does not require care, but in many physics books, proofs are scarce and derivations plentiful. Arguments are given only so far as needed to convince the student of the truth of something under the &quot;nicest&quot; possible conditions. When I was young, this was appealing. The sense that I only needed to argue to a point where I was convinced led me to revisit concepts and computations far more often than if some taskmaster were standing over me with a ruler waiting for me to make an error I wasn't prepared (read: mature enough) to acknowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, now I am a mathematics professor and I take a far different approach with my students. I require them to understand things that (based on their performance on assignments) they really do not appreciate to a degree I find satisfactory. Their basic skills are often abysmal, and the impression I get is that, if they were not forced to, they would never revisit anything mathematical. Many students see no beauty in this subject, only fear and a need to be right and get &quot;the answer&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This condition, and its contrast with my own rather lax beginnings in the subject, makes me wonder whether we should be doing something vastly different than we are doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Has an approach to mathematics education been tried that explicitly deemphasizes correctness and assesses almost solely based on exploration, interest and beauty? If so, how does one implement such an approach?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, in the end, we want students to appreciate the need to be careful and precise. My implicit claim is that by providing an enjoyable experience for students, they will voluntarily think about topics again and again and therefore will &quot;converge&quot; to correctness via willing and frequent re-exposure. Is this general view valid in any sense? What research supports or critiques this idea?&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2016-12-25T23:22:42.473" Title="Correctness in learning mathematics" Tags="|undergraduate-education|secondary-education|primary-education|creativity|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11795" PostTypeId="2" ParentId="11794" CreationDate="2016-12-18T02:29:19.430" Score="2" Body="&lt;p&gt;I'm sure I've posted this elsewhere, but (in somewhat different ways) two curricula that try for this, I think, are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP002457.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heart of Mathematics&lt;/a&gt; (not free)&lt;/li&gt;&#xA;&lt;li&gt;Discovering the &lt;a href=&quot;https://www.artofmathematics.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Art of Mathematics&lt;/a&gt; (free)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note that there are probably lots of others (and I hope other posters mention them), and also note that I'm not sure that they are an approach to the entirety of undergraduate mathematics education.  Most &quot;&lt;a href=&quot;http://www.inquirybasedlearning.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;inquiry-based&lt;/a&gt;&quot; methodologies &lt;a href=&quot;http://www.jiblm.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;for the entire curriculum&lt;/a&gt; end up emphasizing correctness &lt;em&gt;and&lt;/em&gt; beauty.  (Which I like.  But that wasn't what you wanted, I think.)&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-12-18T02:29:19.430" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11796" PostTypeId="2" ParentId="737" CreationDate="2016-12-18T17:05:06.730" Score="2" Body="&lt;p&gt;Here is an interesting but simple example of something that is not well defined: Let $f$ be a binary function (&lt;em&gt;resembling&lt;/em&gt; exponentiation or repeated multiplication) on the set of natural numbers $N$ (including $0$) such that: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$f(a,2)=a\times a$&lt;/li&gt;&#xA;&lt;li&gt;$f(a,b+1)=f(a,b)\times a$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Question: Is this definition sufficient to uniquely determine the value of $f$ for every ordered pair of natural natural numbers, i.e. is $f$ well defined on $N\times N$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer: Almost. You can determine a unique value for every ordered pair but $(0,0)$ for which &lt;em&gt;any&lt;/em&gt; value will work. Proof left as an exercise to the reader.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Lest you think the function $f$ was indeterminate at $(0,0)$ simply because we started at $2$, consider the binary function $g$ (&lt;em&gt;resembling&lt;/em&gt; multiplication or repeated addition) on $N$ such that:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$g(a,2)=a+a$&lt;/li&gt;&#xA;&lt;li&gt;$g(a,b+1)=g(a,b)+a$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It also starts at $2$, but it is well defined on all of $N\times N$. Proof left as an exercise to the reader.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2016-12-19T22:49:28.683" LastActivityDate="2016-12-19T22:49:28.683" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11797" PostTypeId="1" CreationDate="2016-12-19T00:03:39.907" Score="8" ViewCount="344" Body="&lt;p&gt;I am a high school senior planning to major in mathematics next year at college. My question is this: Are there any fields of mathematics which do not have a wide body of literature (maybe even just a few papers or monographs), yet have potential for further research? Please give some sources for learning about these fields, if you can.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am mainly looking for an area of mathematics where I can reach the frontier of knowledge reasonably quickly, and hopefully even contribute during my undergraduate years. I have looked into graph theory as a possibility (is it a good one?), but I was just wondering if there were any other fields out there which might work as well.&lt;/p&gt;&#xA;" OwnerUserId="7586" LastActivityDate="2023-10-08T21:29:31.770" Title="Are there any undeveloped areas of mathematics accessible (for both understanding and research) to an undergraduate?" Tags="|undergraduate-research|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11798" PostTypeId="2" ParentId="11797" CreationDate="2016-12-19T00:30:20.440" Score="6" Body="&lt;p&gt;In line with your identification of graph theory, I suggest you might look into&#xA;what is now known as &amp;quot;Discrete and Computational Geometry.&amp;quot;&#xA;Although there is much to learn (there is a 1937-page &lt;em&gt;Handbook of Discrete and Computational Geometry&lt;/em&gt;,&#xA;&lt;a href=&quot;https://www.routledge.com/Handbook-of-Discrete-and-Computational-Geometry-Third-Edition/Toth-ORourke-Goodman/p/book/9781498711395&quot; rel=&quot;nofollow noreferrer&quot;&gt;CRC link&lt;/a&gt;, now in its 3rd edition), one can reach frontiers in narrow areas&#xA;without mastering the whole field. In particular, you might find browsing&#xA;the titles in the recent conference collection below.&#xA;The papers include,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;Mario Kart is Hard,&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;Continuous Folding of Regular Dodecahedra,&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;On Evasion Games on Graphs,&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;among other accessible topics.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Akiyama, Jin, Hiro Ito, and Toshinori Sakai, eds. &lt;em&gt;Discrete and Computational Geometry and Graphs: 18th Japan Conference&lt;/em&gt;, JCDCGG 2015, Kyoto, Japan, September 14-16, 2015, Revised Selected Papers. Vol. 9943. Springer, 2016.&#xA;&lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=L9WSDQAAQBAJ&amp;amp;oi=fnd&amp;amp;pg=PR5&amp;amp;dq=Discrete+and+Computational+Geometry+and+Graphs:+18th+Japan+Conference+2015,+2016,+LNCS+994&amp;amp;ots=9V-hNWZpp3&amp;amp;sig=4t0Ni_VMMku7mFV8qXwoJWF87ik#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google books link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/HW40N.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HW40N.jpg&quot; alt=&quot;Akijamap19&quot; /&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Fig.9: &amp;quot;Reversible Nets of Polyhedra,&amp;quot; p.19.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2023-10-08T21:29:31.770" LastActivityDate="2023-10-08T21:29:31.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11799" PostTypeId="2" ParentId="11779" CreationDate="2016-12-19T07:26:49.607" Score="3" Body="&lt;p&gt;Just ask the students to think geometrically about the basic properties of integrals. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Thinking of an integral as &quot;the area under the curve,&quot; write a geometric explanation of why each of the following properties of integrals is reasonable to believe.&#xA;  $$\int\limits_a^b cf(x) \,\mathrm{d}x \;=\; c\!\int\limits_a^b f(x) \,\mathrm{d}x &#xA;\qquad\qquad&#xA;\int\limits_a^b f(x) \,\mathrm{d}x \;=\; \int\limits_0^b f(x) \,\mathrm{d}x - \!\int\limits_0^a f(x) \,\mathrm{d}x\\&#xA;\int\limits_a^b \big(f(x)+g(x)\big) \,\mathrm{d}x \;=\; \int\limits_a^b f(x) \,\mathrm{d}x + \!\int\limits_a^b g(x) \,\mathrm{d}x$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or for more of a challenge, ask the same question for the property&#xA;$$&#xA;\int\limits_0^a f(x) \,\mathrm{d}x = \int\limits_0^a f(a-x) \,\mathrm{d}x&#xA;$$&#xA;If they've covered $u$-substitution it would be reasonable to ask them to prove this formally.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2016-12-19T07:26:49.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11800" PostTypeId="1" CreationDate="2016-12-19T08:43:20.757" Score="0" ViewCount="100" Body="&lt;p&gt;In mathematical expressions, the internationally recognized symbol % (percent) may be used with the SI to represent the number 0.01. Thus, it can be used to express the values of dimensionless quantities. According to Le Système international d’unités/The International System of Units, (Brochure sur le SI/SI brochure), 2006.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, &quot;%&quot;, or the abbreviations &quot;pct.&quot;, &quot;pct&quot;; sometimes the abbreviation &quot;pc&quot; is also used.[1] A percentage is a dimensionless number (pure number). Based on Wikipedia, &lt;a href=&quot;https://en.wikipedia.org/wiki/Percentage&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Percentage&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is percentage considered as a number?, e.i. can I add a number to percentage?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which is correct&lt;/p&gt;&#xA;&#xA;&lt;p&gt;30+10%=30.1&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;30+10%=33&lt;/p&gt;&#xA;" OwnerUserId="7587" LastActivityDate="2016-12-21T11:12:22.840" Title="Percentage nature" Tags="|rational-numbers|" AnswerCount="2" CommentCount="5" ClosedDate="2016-12-21T14:23:14.267" ContentLicense="CC BY-SA 3.0" />
  <row Id="11801" PostTypeId="1" CreationDate="2016-12-19T15:33:12.603" Score="17" ViewCount="412" Body="&lt;p&gt;For example&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\lim\limits_{n\to\infty}\left(1+\frac{1}{2n+1}\right)^{n} =\lim\limits_{n\to\infty}{1}^{n}=1\,.$$&lt;br&gt;&#xA;Here the student has replaced the sub-part $\frac{1}{2n+1}$ with its  limit $0$, but he left the other $n$ at the exponent. I always tell them this is wrong, and give examples, but some of them keep making this errors again and again.  So I'm interested in other ways to make them avoid this error.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there are cases when this doesn't lead to errors, if the sub-part which is replaced with its limit can be separated from the rest of the expression, such as in&#xA;$$\lim\limits_{x\to 1}\frac{\sin(x^2-1)}{x^2+x-2}=\lim\limits_{x\to 1}\frac{\sin(x^2-1)}{x^2-1}\frac{x^2-1}{x^2+x-2}=\lim\limits_{x\to 1}\frac{x^2-1}{x^2 +x-2}\,.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tell them, &quot;here we applied the theorem that says: limit of product $=$ product of limits,&quot; but in the above example there is no theorem that can justify it.&lt;/p&gt;&#xA;" OwnerUserId="5972" LastEditorUserId="5220" LastEditDate="2016-12-20T22:01:59.150" LastActivityDate="2016-12-22T15:17:12.483" Title="Frequent calculus error: replacing interior part of an expression with its limit" Tags="|calculus|students-mistakes|limits|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11802" PostTypeId="2" ParentId="11800" CreationDate="2016-12-19T16:57:02.163" Score="5" Body="&lt;p&gt;This is a hard question to answer formally, but an easy question from an educator's perspective: it is very unlikely that adding a pure number and a percentage is a good idea.  While a percentage has the same units as a pure number, they're usually represent different types, so adding them is unlikely to give a meaningful quantity.  Anyone who finds themselves doing this should probably stop and ask themselves what led them there, and what the combined quantity is supposed to mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unit analysis is as far as most classes get into the details of the typing rules of written mathematics, and the connection of those rules to meaningful computations, but teachers should always be modeling good practice.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-12-19T16:57:02.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11803" PostTypeId="2" ParentId="11801" CreationDate="2016-12-19T18:24:07.957" Score="2" Body="&lt;p&gt;Very good question.  I find it helpful to point out common errors such as this.  If you make a point to show them the common mistake, they are less likely to continue that mistake.  It seems by your post that you've talked to them a little about this, but it may be helpful to go over several common errors that you see in calculus at the beginning of class.  For example, many students make the mistake of dropping the limit notation when simplifying.  They may understand what they are doing, but this is NOT allowable.  I think it is important to talk about mistakes such as these.  As for other ways to make them avoid making this error, I'm not sure.&lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2016-12-19T18:24:07.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11805" PostTypeId="2" ParentId="11770" CreationDate="2016-12-19T20:56:29.017" Score="1" Body="&lt;p&gt;From a non-mathematician (and let's hope I haven't made any elementary mathematical mistake):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They aspire to be engineers, so let's consider a new production process.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being new, there exists a &quot;learning curve&quot;, an initial period where we are not fully efficient in implementing it. Let $x \in (0,1]$ be the &quot;intensity&quot; with which we try to learn and implement the new process.We excluded $0$ from the domain to reflect the fact (or the belief) that we always pay even a little bit of attention. Let $A$ be &lt;em&gt;full-efficiency&lt;/em&gt; output (for a given pre-specified level of inputs used). $n$ denotes the number of times the process runs a full circle. Our production function is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f_n(x) = \left (1-\frac{1}{2+nx}\right)\cdot A$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first term is the one representing the fact that we are gradually approaching full efficiency, based on the times we have run it ($n$), but also on the intensity with which we try to learn ($x$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The production function converges pointwise to $f(x) = A$ for any given &lt;em&gt;fixed&lt;/em&gt; $x \in (0,1]$. This is translated : with even a little but fixed level of learning intensity, we will eventually reach full efficiency output. That's comforting.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the function does not converge uniformly because, say, if we set $x=1/n$ we get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\left |f_n(x) - f(x)\right | = \frac 13 A $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;that does not depend on the value of $n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;DO WE CARE?&lt;/strong&gt;&lt;br&gt;&#xA;Does the difference between pointwise and uniform convergence reflects here something real-world important?&#xA;&lt;em&gt;It does&lt;/em&gt;: that we cannot rely on &quot;automatic learning-by-doing&quot; (the increase in $n$) to compensate for gradual loss ($x=1/n$) of conscious learning intensity, at least not for all rates of such gradual loss.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We examine the journey of a function looking at two possible influencing factors ($n$ and $x$). &#xA;In pointwise convergence, we &quot;commit&quot; one of them to a specific value beforehand, so we examine &lt;em&gt;separate&lt;/em&gt; scenarios of a function that is now influenced only by a single factor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In uniform convergence, we allow both influencing factors to vary at the same time as we move along the sequence -and if uniform convergence obtains, we learn that &lt;em&gt;even varying $x$ &quot;in real time&quot; does not eventually matter&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This, to me, is not a &quot;subtle theoretical difference&quot;: it may be a very important difference, as I think my crude example above indicated. Introduce uncertainty about what the actual production function is, and the conventional wisdom that &quot;sometimes it is better to commit rather than maintain flexibility&quot; gets a good argument in favor (or equivalently, in an uncertain world, perhaps we may satisfy ourselves with pointwise convergence and accept its restrictive requirements as regards our behavior).&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2016-12-28T22:25:38.763" LastActivityDate="2016-12-28T22:25:38.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11807" PostTypeId="2" ParentId="11801" CreationDate="2016-12-20T15:50:32.403" Score="2" Body="&lt;p&gt;Sometimes a gauntlet of true/false questions on specific (simple) manipulations like this gets some traction. Anytime I see a student mistake like this in class I try to turn it into a true/false question on the board so we can check in with the whole class on their understanding. The true/falsity emphasizes that this is not just some notational nicety. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-12-21T00:09:44.180" LastActivityDate="2016-12-21T00:09:44.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11808" PostTypeId="1" CreationDate="2016-12-20T17:47:41.730" Score="5" ViewCount="147" Body="&lt;p&gt;I want to present these three descriptions of a parabolic curve to my precalculus class:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The graph of a quadratic function $f(x) = ax^2+bx+c$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given a line called the &lt;em&gt;directrix&lt;/em&gt; and a point called the &lt;em&gt;focus&lt;/em&gt;, the set of all points in the plane that are equidistant to the directrix and focus.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The intersection of a cone with a plane that is parallel to a tangent-plane to the cone.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Showing students that the first and second descriptions actually give us the same curve isn't too bad: just draw a focus and directrix in the plane, take an arbitrary point that is equidistant to them, and show that the $y$-coordinate is quadratic in the $x$-coordinate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But showing that these first two descriptions are the same as the third seems quite a bit harder. Does anyone know an intuitive way to show that the conic section description of a parabola gives us the same curve as the other two descriptions? I really want something brief but satisfying (so not necessarily thorough) so that my students don't just tune out. But every method I've seen so far has been a bit too involved to present to a precalculus class (like &lt;a href=&quot;https://en.wikipedia.org/wiki/Parabola#Diagram.2C_description.2C_and_definitions&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;), and I don't see an easy way to prune out some details of these methods.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2016-12-20T19:09:29.527" LastActivityDate="2016-12-20T19:59:38.293" Title="Consolidating three descriptions of a parabola in precalculus" Tags="|undergraduate-education|secondary-education|geometry|precalculus|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11809" PostTypeId="2" ParentId="11808" CreationDate="2016-12-20T19:39:50.130" Score="2" Body="&lt;p&gt;This 5-minute video from Creative Learning is effective:&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=HO2zAU3Eppo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conic Section 3D Animation&lt;/a&gt;.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/OSCkg.jpg&quot; height=&quot;360&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;But it might be too long (one of your worries). In which case, &#xA;I would use the Wolfram Demo &lt;a href=&quot;http://demonstrations.wolfram.com/ConicSectionsTheDoubleCone/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conic Sections: The Double Cone&lt;/a&gt;,&#xA;in which you interactively manipulate the plane and rotate the&#xA;image in 3D,&#xA;and can spend as little or as much time as you'd like:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/pMth2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pMth2.jpg&quot; alt=&quot;ConeDemo&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2016-12-20T19:59:38.293" LastActivityDate="2016-12-20T19:59:38.293" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11810" PostTypeId="2" ParentId="11801" CreationDate="2016-12-20T23:33:17.287" Score="5" Body="&lt;p&gt;Perhaps this is not a good answer per-se, insofar as it yet-once-again questions (to a certain degree) the premises of the question... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, indeed, in many propitious situations, everything in sight is continuous, and it is justifiable to replace sub-expressions with their limits. I might claim that the practical success of calculus substantially depends exactly on the fact that &quot;doing the obvious thing&quot; very often succeeds in practice. Indeed!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that context, it is mildly perverse (I do claim) to make beginners worry about bad possibilities that (with luck) do not occur at an entry level. Further, I would claim that the physical intuition that students have (naive as it is) is fairly correct about &lt;em&gt;good&lt;/em&gt; limits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tell my first-year calculus students that if plugging in the limit value produces a sensible value, it is very likely that it is correct. And I also tell them that if their or other peoples' lives depended on it, they'd want more assurance than that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, seriously, what &lt;em&gt;bad&lt;/em&gt; limits will people have reasonable motivation to worry about? I think the difficulty many beginners have in taking &quot;the rules&quot; seriously is that all they see are contrived examples quite-obviously designed to &quot;punish&quot; the non-conformist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, the success of calculus and such is that mostly it &lt;em&gt;does&lt;/em&gt; work as one might imagine, and even better. It was (as I've ranted many places) not a big hit because of invitations to fretting about rigor, but because it decisively answered questions about real things (whether in the external physical world or in the more-internal mathematical world).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, for example, I cannot bring myself to advocate &quot;conformance to rules&quot;, but I can easily advocate &quot;viewpoint that explains things&quot;. Yes, it is not easy to find subtle mathematical issues that have much sense or moment for scientifically naive kids. I conclude that subtle analytic issues should be delayed until they might genuinely matter.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-12-20T23:33:17.287" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11811" PostTypeId="2" ParentId="11800" CreationDate="2016-12-21T11:12:22.840" Score="0" Body="&lt;p&gt;&lt;strong&gt;Correct one:&lt;/strong&gt; 30+10%=30.1&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Corrected one:&lt;/strong&gt; 30 * 110%=33&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Percentage means exactly as you say. 1% = 0.01. It has a few versions: &lt;a href=&quot;https://en.wikipedia.org/wiki/Percent_sign&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Percent_sign&lt;/a&gt; , permille ‰ is often used for alcohol content in blood. The per-cent means per hundred and per-mille per thousand. Because it is a shortening, it requires always a number front of it. % + % * 5 + %$^2$, would not be acceptable.&lt;/p&gt;&#xA;" OwnerUserId="7552" LastActivityDate="2016-12-21T11:12:22.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11813" PostTypeId="1" CreationDate="2016-12-21T13:22:35.683" Score="26" ViewCount="2239" Body="&lt;p&gt;If it is true that &lt;a href=&quot;http://www.maa.org/external_archive/devlin/devlin_03_06.html&quot; rel=&quot;noreferrer&quot;&gt;we first learn formalism&lt;/a&gt;...how to do things that we don't understand, should we regard teaching students mathematics as programming dumb machines with formal rules (to the greatest extent possible) and allow them to eventually incorporate meaning?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am thinking of this as the passage from &lt;a href=&quot;https://ocw.mit.edu/high-school/humanities-and-social-sciences/godel-escher-bach/lecture-notes/MITHFH_geb_v3_5.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;M&quot; mode to &quot;I&quot; mode in Godel Escher Bach.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems to be what we are saying when we set specific learning goals for students to be able to do, like find the derivative of $x^{3}$...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Question:&lt;/em&gt; Is there mathematics education research, perhaps in cognitive load theory, that indicates that the above approach is superior to concept-based teaching, especially at the undergraduate level. Particularly, that mechanics should precede concepts that organize them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I ask this is that our colleagues in physics and engineering departments (see my first link) want to see students proficient in mechanical computation...and cannot understand why two semesters of calculus are needed to train students to do this. We, though, as mathematics professors, try to emphasize and train reasoning and concept. Some of us (like myself) trying very hard to get students to move past mechanical symbol pushing toward metacognition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic starting assumptions of the physics faculty and mathematics faculty seem to differ. They say that students should &quot;at least be able to find integrals&quot; etc. (we all agree), whereas it is likely the case that trying to teach conceptual thinking increases cognitive load of weaker students so much that they cannot handle the symbol pushing as well as if it were emphasized WITHOUT the meaning. Which basic approach is better?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is certainly the case that a differentiated approach will work, so I am not asking about that. What I am asking is whether or not doing what students and physics and engineering departments (and Keith Devlin in the above link) seem to want: blind and correct manipulation of symbols before concepts that organize such calculations, is more sound than teaching students organizing principles that rudder such calculations with meaning first. (To me, the answer seems obvious...if you are taking unjustified steps then you will make serious errors due to being very dumb. I could be wrong, though...hence this question!) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: I think my original version of this may not be as accurate as I wanted. The question is, perhaps, more about whether moving past simplified (perhaps even oversimplified) intuition in calculus classes creates so much cognitive load that we'd be better off sticking with a course that looks pretty much like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Differential calculus:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) The derivative is the slope of a tangent line. Look, instantaneous velocity is an example. Limit means that if x gets really really close to a then f(x) gets arbitrarily close to its limit. (So, don't get into the problems of what we mean by arbitrarily.) In these velocity problems, we end up dividing by zero, so we do algebra to get rid of this problem. (Very little discussion of why or emphasis on definition of derivative beyond saying &quot;change in y over change in x&quot; and talking about &quot;infinitesimal changes&quot; like an 18th century mathematician or physicist.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Here are lots of nifty formulas for computing derivatives. Let's practice the daylights out of them. Sometimes this is hard, so we implicitly differentiate...so let's do that to death, too!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) There are higher derivatives. The second one describes concavity, which acceleration is an example of. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Maxima and minima happen at endpoints or at the tops and bottoms of hills. We don't care so much about anything tricky...let's do a slew of optimization problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) Talk about linear approximation. Do a bunch of mechanical problems that ensure students can do such problems. Maybe even require them to be able to explain with a picture why this works. (No, that's REALLY pushing it...&lt;a href=&quot;http://www.uky.edu/~gmswan3/544/9_ways_to_reduce_CL.pdf&quot; rel=&quot;noreferrer&quot;&gt;produces too much cognitive load&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6) Maybe do some differential equations. Teach them to write $y=e^{kx}$ if they see $\frac{dy}{dx}=ky$ and to plug in initial conditions. Very little discussion of, or emphasis on, the fact that $y=e^{x}$ is a fixed point for differentiation and such things... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Integral calculus:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) The derivative of the area function is the original function (hand wave hand wave)...here's how to take integrals using antiderivatives...practice to death with u substitutions and integration by parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Talk about slicing and Riemann sums and do very basic examples. Don't make too much of a big deal about knowing how to model something by appropriately employing Riemann sums (appropriately slicing things, for example, along directions where things are well-behaved and can be considered locally constant)...just do a bunch of examples so students can model later experience on these. Don't worry about general conceptual development that will allow the use of integrals in any new situation so much as getting the student to be able to recognize that &quot;If the force doesn't vary with x then you can just do force times distance, but if the force varies with x then you have to integrate&quot;. Or, we could aim to get the students to recognize that &quot;When you are working with finite things we can add, but when we move to continuous things we have to trade the sum for an integral&quot;...without emphasizing the nuts and bolts of Riemann sums and various subtleties.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I'm trying to tease out what the difference in emphasis is, here. There is surely a spectrum, here. Experienced mathematicians teaching calculus probably are guilty of trying to move their students toward thinking about calculus in a way that is preparatory for later, more modern, mathematics...whereas physicists want us to churn out great 18th century mathematicians...I don't know, but I'd really like to.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="1385" LastEditDate="2017-04-20T18:32:09.970" LastActivityDate="2021-04-15T09:02:44.023" Title="Should we &quot;program&quot; calculus students, like the physicists seem to want us to?" Tags="|undergraduate-education|reference-request|infinitesimals|" AnswerCount="5" CommentCount="24" ContentLicense="CC BY-SA 3.0" />
  <row Id="11814" PostTypeId="1" CreationDate="2016-12-21T14:22:12.433" Score="2" ViewCount="327" Body="&lt;h2&gt;Free, open-source, customizable, interactive &lt;del&gt;books&lt;/del&gt; games.&lt;/h2&gt;&#xA;&lt;p&gt;Most kids find computer games much more fun than learning math. Many programmers (including some programmers with 5-6 digit monthly salaries) love StackOverflow because it's a &amp;quot;game&amp;quot; that matters.&lt;/p&gt;&#xA;&lt;p&gt;So why not combine them and make learning math a game, where a user will be rewarded shinny useless internet points in the form of coins or badges etc?&lt;/p&gt;&#xA;&lt;p&gt;Why not make it open-source, free and customizable, so that anyone with a great idea can make this world better, by making education more fun and efficient?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Mathematica's CDF&lt;/h2&gt;&#xA;&lt;p&gt;Looks awesome. It's much more easy to use than, e.g., IPython + Sympy + Numpy. However it's not free.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;My initial attempts&lt;/h2&gt;&#xA;&lt;p&gt;Initially I created an android app that helps my students practice on things like:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;-2 + 5 = &#xA;(-0.5) (+2.6) = &#xA;-2.5 +6.1 -0.6 = &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;They get shinny coins when they answer correctly, and I can track their progress.&lt;/p&gt;&#xA;&lt;p&gt;Even an 8 year old played it and said proudly that he had gathered 120 coins!&lt;/p&gt;&#xA;&lt;p&gt;However, such apps, have a very limited scope. They are not easy to make and distribute, and having a single app for each math concept is impossible.&lt;/p&gt;&#xA;&lt;p&gt;Of course this app doesn't substitute teachers; it is simply a tool that generates exercises and checks the answers provided by the student.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Elements to be included&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;History&lt;/strong&gt;, containing wrong answers provided by the user, so that the teacher can figure out what went wrong. Also, shows when exactly each answer was provided in order to determine how much time the student spent, whether he answered extremely fast (indicating cheating) etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Reveal answer&lt;/strong&gt; and generate new question &lt;strong&gt;button&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Shiny coins&lt;/strong&gt; (and perhaps an option to change the reward to candy or flowers etc.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Palette&lt;/strong&gt; for insertion of fractions, powers etc. like Mathematica's &lt;a href=&quot;https://blog.wolfram.com/data/uploads/2009/01/quickstart_palettes.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;palette&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;No wheel reinventing!&lt;/h2&gt;&#xA;&lt;p&gt;Before going any deeper into the colossal project of turning all (high school) math into small little games, I need to make sure I'm in the right track. I need to know which are the best tools for the job. But most importantly I need to know if someone is already working on creating interactive free open-source books that gamefy math learning.&lt;/p&gt;&#xA;&lt;p&gt;I have explored &lt;a href=&quot;https://youtu.be/o7Tb7YhJZR0?t=16m15s&quot; rel=&quot;nofollow noreferrer&quot;&gt;IPython's widgets and interactivity&lt;/a&gt; and they seem like the right way to go, since I'm familiar with Python. It does have its limitations though; &lt;a href=&quot;https://stackoverflow.com/questions/39676933/create-a-popup-widget-in-ipython&quot;&gt;some of its parts are under construction&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;QUESTIONS&lt;/h2&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is including game elements in math learning, through free open-source interactive books a good idea? Would it significantly improve math teaching throughout the world?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there an already existing project that is focused on creating the above books (that includes &lt;em&gt;all&lt;/em&gt; elements mentioned)?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Which are the best tools (IPython, etc.) for the job?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Anything else you consider relevant is welcome.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;sub&gt;Note: The question might be broad. If this is the case, let me know what to remove or improve. Or perhaps how to split it in several questions if needed.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="7025" LastEditorUserId="7025" LastEditDate="2020-11-29T18:15:42.097" LastActivityDate="2020-11-29T18:15:42.097" Title="Free, open-source, customizable, interactive math books with gaming elements" Tags="|general-pedagogy|games|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11815" PostTypeId="2" ParentId="11814" CreationDate="2016-12-21T14:48:46.217" Score="2" Body="&lt;p&gt;Just one narrow point in response to your several questions: &#xA;An alternative to Mathematica's CDF.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Sage is a free, open source alternative to software such as Mathematica and, thanks to its interact function, it is fully capable of producing advanced, interactive mathematical demonstrations with just a few lines of code.  The Sage language is based on Python and is incredibly easy to learn.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.sagemath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sage link&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://wiki.sagemath.org/interact&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sage Interactions&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.sagemath.org/tour-graphics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graphics tour explaining &lt;em&gt;interact&lt;/em&gt;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(Apologies for the somewhat annoying GIF, from last link above.)&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Uu07M.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Uu07M.gif&quot; alt=&quot;SageInteract&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2016-12-21T14:48:46.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11816" PostTypeId="2" ParentId="11814" CreationDate="2016-12-21T14:57:44.983" Score="4" Body="&lt;p&gt;I'm not sure whether this question really belongs here, but if it does, I think you should be able to do something along these lines by combining several APIs.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, first I need to mention one of the more relevant books about this from the past decade, Keith Devlin's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1568814313&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics Education for a New Era&lt;/a&gt;. It is a contentious book, of course! But it has a lot of good thinking about how to approach this question.  I have no idea whether the &lt;a href=&quot;https://profkeithdevlin.org/tag/mathematics-video-game/&quot; rel=&quot;nofollow noreferrer&quot;&gt;actual video game&lt;/a&gt; this spawned is still successful.  You may be interested in &lt;a href=&quot;http://www.maa.org/programs/maa-awards/writing-awards/a-modern-mathematical-adventure-in-call-of-duty-black-ops&quot; rel=&quot;nofollow noreferrer&quot;&gt;a related article&lt;/a&gt; in &lt;em&gt;Math Horizons&lt;/em&gt; about a &quot;mod&quot; in &lt;em&gt;Call of Duty&lt;/em&gt;. Naturally, there is also plenty of literature on this, and it would be worth getting to a good academic library to do a thorough search before you spend potential thousands of hours developing said library of games!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To your question.  I think some combination of the following would do nicely for some of your goals.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://webwork.maa.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;WeBWorK&lt;/a&gt; is a homework platform which is open source, and which already supports not just knowing previous answers, but randomization, and even basic badges and awards (though obviously not in an &quot;app&quot; context).   It has an API which allows at least the problem engine to function - see &lt;a href=&quot;http://mathbook.pugetsound.edu/examples/webwork/sample-chapter/html/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this example&lt;/a&gt;, sponsored by OpenOregon.  I'm not sure it would &quot;know&quot; previous answers in that context.  Realistically you would want some more serious database connections anyway.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;a href=&quot;http://sagecell.sagemath.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sage cell server&lt;/a&gt; is another open source option for verifying mathematical content online.  It doesn't sound like you'd need that much firepower, but I'm not sure what you're envisioning.&lt;/li&gt;&#xA;&lt;li&gt;As you point out, IPython widgets are a good tool.  I could even potentially imagine some combination of the &lt;a href=&quot;http://ipython.org/notebook.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jupyter notebook&lt;/a&gt; and other tools making a potential game.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2016-12-21T14:57:44.983" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11817" PostTypeId="2" ParentId="11813" CreationDate="2016-12-21T22:52:43.053" Score="32" Body="&lt;p&gt;I'm primarily a physicist, but I also teach first-semester freshman calc once in a while. Your characterization of a cultural divide between physicists and mathematicians on this subject does not seem at all accurate to me. If anything, I think the characterizations should be reversed, at least on the average -- but it would only be an average, because different teachers are different.&lt;/p&gt;&#xA;&lt;p&gt;Here is a dialog that I have had many, many times with my physics students in my office hours:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;student: I'm supposed to find the maximum power. How do I do that?&lt;/p&gt;&#xA;&lt;p&gt;me: Think back to your calculus.&lt;/p&gt;&#xA;&lt;p&gt;student: Oh, I can do it using calculus? OK, so ... what do I do?&lt;/p&gt;&#xA;&lt;p&gt;me: It's a function, and we're trying to find an extremum...&lt;/p&gt;&#xA;&lt;p&gt;student: Oh, an extremum! So I set it equal to zero.&lt;/p&gt;&#xA;&lt;p&gt;me: Set what equal to zero?&lt;/p&gt;&#xA;&lt;p&gt;student: The power?&lt;/p&gt;&#xA;&lt;p&gt;me: Er, if you were running a business, and you wanted to maximize your profits, would you set your profits equal to zero?&lt;/p&gt;&#xA;&lt;p&gt;student: Oh, no. Huh. So what do I do?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;At this point, I prompt the student to sketch a function with a maximum and draw a tangent line at the peak. Then they remember that they should be taking the derivative and setting &lt;em&gt;it&lt;/em&gt; to zero. The point of relating this dialog, which I've had dozens of times over the years, is that my students' problems are almost never with computing things using calculus. The problems are with recognizing when they need to use their calculus, and applying it in a way that shows conceptual understanding.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;our colleagues in physics and engineering departments (see my first link) want to see students proficient in mechanical computation...and cannot understand why two semesters of calculus are needed to train students to do this.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As described above, this remark is very hard to reconcile with my experience. I would also point out that in freshman calc and freshman physics classes these days, biology majors are about as numerous as engineering majors. (This varies, e.g., here in California, the UC system requires calc-based physics for bio majors, but the Cal State system doesn't. I teach at a community college, so we serve both populations.) The real problem IMO is that the content of second-semester calc is utterly irrelevant to the biologists. They are simply never going to integrate using a trig substitution or evaluate a Taylor series.&lt;/p&gt;&#xA;&lt;p&gt;I think it's important for calculus students to understand why calculus works, not just how to do computations. However, I find that many of my colleagues who teach calc imagine that the &amp;quot;why&amp;quot; is very narrow. In the case of a derivative, they seem to think of &amp;quot;why&amp;quot; understanding as absorbing the definition of the limit, being able to do epsilon-delta proofs, and applying those skills to computing derivatives, before learning to do derivatives using computational rules. &amp;quot;Why&amp;quot; knowledge should also mean the kind of knowledge lacked by the student in the dialog above. And it should mean things like understanding why the Leibniz notation makes sense, interpreting differentials as small changes, being able to explain why the chain rule makes sense in terms of dimensional analysis, and being able to sketch the graph of the derivative of a function given a graph of the function.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-12-21T22:52:43.053" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11818" PostTypeId="2" ParentId="11801" CreationDate="2016-12-22T01:28:51.430" Score="0" Body="&lt;p&gt;I like to talk about a &quot;tug of war&quot; or &quot;competing tendencies&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, with $\lim_{x \to \infty} (1+\frac{1}{x})^x$, the base of the exponential is going to 1, while the power is tending to infinity.  One tendency is driving the whole to 1, and the other to infinity (since the base is always slightly bigger than 1).  These two competing tendencies must be resolved in some way before we just &quot;substitute&quot; part of the expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand for an expression like $\lim_{x \to 2} \frac{x+1}{x+2}$, there is no issue.  There is no competition here.  You can just go ahead and substitute. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something like $\lim_{x \to \infty} \frac{x+2}{2x+3}$, on the other hand, does have a tug of war.  The numerator is &quot;pulling the fraction to infinity&quot; and the denominator is &quot;pulling it to 0&quot;.  So it would not be fair just to substitute into one part of the expression:  that would be declaring a winner in this contest without investigating who is really stronger.  So you need to do some more algebra to manipulate this into a form where you can &quot;clearly&quot; see the winner.  Rewriting it as $\lim_{x \to \infty} \frac{1+2/x}{2+3/x}$ erases the tug of war.  Now you can plainly see the answer is $\frac{1}{2}$ &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2016-12-22T01:28:51.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11819" PostTypeId="1" AcceptedAnswerId="11821" CreationDate="2016-12-22T10:55:31.983" Score="4" ViewCount="216" Body="&lt;p&gt;I saw a mathematical problem where was given a wrong definition and was asked if some numbers satisfies the definition. Is this solution fine if there is a mistake in the definition?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A real number is algebraic if it is a root of some polynomial with integer coefficients. These polynomials are of the form &lt;span class=&quot;math-container&quot;&gt;$$P(x)=a_0+a_1x+a_2x^2+\cdots + a_nx^n$$&lt;/span&gt; where the degree of the polynomial &lt;span class=&quot;math-container&quot;&gt;$n=1,2,3,\ldots$&lt;/span&gt; and the coefficients &lt;span class=&quot;math-container&quot;&gt;$a_0,a_1,a_2,\ldots,a_n$&lt;/span&gt; are integers.&lt;/p&gt;&#xA;&lt;p&gt;Prove that the numbers &lt;span class=&quot;math-container&quot;&gt;$x=2/3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x=\sqrt 3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x=2+\sqrt{3}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$x=\sqrt{2}+\sqrt{3}$&lt;/span&gt; are algebraic.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My solution is just to take &lt;span class=&quot;math-container&quot;&gt;$P(x)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now the problem seems to be that zero polynomial satisfies the first sentence of the definition but not the second one. So what should a student answer as the first sentence of the definition and its explanation does not corresponds to each others?&lt;/p&gt;&#xA;&lt;p&gt;This question was from the Finnish matriculation examination on autumn 2016.&lt;/p&gt;&#xA;" OwnerUserId="7598" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-12-27T17:01:47.783" Title="What to do if there is a wrong definition in a problem?" Tags="|exams|" AnswerCount="3" CommentCount="7" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11820" PostTypeId="2" ParentId="11801" CreationDate="2016-12-22T15:17:12.483" Score="0" Body="&lt;p&gt;Although they surely can't put it in those terms, this sort of manipulation can be described in terms of multivariable limits; in many situations you &lt;em&gt;can&lt;/em&gt; do what they are doing. For example,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \lim_{x \to \infty} \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}&#xA;= \lim_{(x,y) \to (\infty, \infty)} \frac{1 + \frac{1}{x}}{1 - \frac{1}{y}}&#xA;=  \lim_{x \to \infty} \lim_{y \to \infty} \frac{1 + \frac{1}{x}}{1 - \frac{1}{y}} = \lim_{x \to \infty} 1 + \frac{1}{x} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I won't try and state the conditions under which this sort of thing is valid since I'll surely get the fine detail wrong; my point is just to show that it's a reasonable and even rigorously justifiable manipulation in sufficiently well-behaved cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of this, I believe that simply telling them &quot;that's wrong&quot; is not particularly effective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the right approach is two-fold:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Demonstrate signs that should trigger red flags in their mind; e.g. in the example problem, having a $1^\infty$ form should &lt;em&gt;immediately&lt;/em&gt; trigger skepticism about previous steps, and compel them to use more detail, precision, and rigor when they would otherwise not&lt;/li&gt;&#xA;&lt;li&gt;Emphasize alternative ways of doing the sort of thing they're trying to do.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For example, the &lt;em&gt;usual&lt;/em&gt; trick for rigorously implementing the sort of idea I use above is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \lim_{x \to \infty} \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}&#xA;= \lim_{x \to \infty} \frac{1}{1-\frac{1}{x}} \cdot \left( 1 + \frac{1}{x} \right) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;with the key idea being using factoring to &lt;em&gt;pull out&lt;/em&gt; the part of the limit I wanted to simplify, and then use limit-of-products to justify it. Maybe a less trivial example is how to replace $\sin x$ with $x$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \lim_{x \to 0} \frac{\sin x}{\sqrt{1 + x^2} - 1} &#xA;= \lim_{x \to 0} \frac{\sin x}{x} \cdot \frac{x}{\sqrt{1 + x^2} - 1} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I'm sure I saw these sorts of manipulations a lot in class, it was always &quot;now I do this magical trick that is useful in this case&quot; &amp;mdash; I had to develop the underlying goal-oriented strategy myself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I imagine it would help to explicitly demonstrate it in that fashion. Have an entire lesson or so dedicated to recognizing parts of a limit that one might want to simplify and the algebraic tactics that allow you to enact that simplification.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2016-12-22T15:17:12.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11821" PostTypeId="2" ParentId="11819" CreationDate="2016-12-23T03:50:18.943" Score="4" Body="&lt;p&gt;Yeah it looks like &lt;em&gt;technically&lt;/em&gt; the zero polynomial satisfies the constraints of the problem (see Benjamin Dickman's comment), but just saying to take $P(x) = 0$ misses the point of the examination. You are being assessed, and listing just the zero polynomial doesn't demonstrate that you know what the examiner is testing you for. The point of this question, or any question on any test for that matter, is just to give you a prompt to demonstrate that you know what you are doing. I would imagine that a good response that both demonstrates your aptitude and acknowledges what you pointed out could be worded like,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By the definition of an algebraic number given, each of these numbers is algebraic because they are each a root of the zero polynomial. But each is &lt;em&gt;also&lt;/em&gt; the root of a non-zero polynomial. ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2016-12-27T17:01:47.783" LastActivityDate="2016-12-27T17:01:47.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11822" PostTypeId="2" ParentId="11814" CreationDate="2016-12-23T08:35:47.480" Score="2" Body="&lt;p&gt;I don't think it has been listed yet, so here is my suggestion: &lt;a href=&quot;https://www.khanacademy.org/&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Khan Academy&quot;&gt;Khan Academy&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used it to great effect to refresh and even come to master the basics, and it includes every maths topic taught in mandatory education (grades...?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is effective in its teaching with explanatory videos, hints and reactive questions, and has what I feel is the appropriate amount of gamification: it rewards doing some work regularly, promotes successes without explicitly punishing mistakes, and never locks one out of any content, regardless of level, although it suggests work appropriate to one's skill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also allows a teacher (or a parent) to assign specific work to a pupil, and allows review of the child's performance, even telling you how many seconds they took to complete one specific exercise!&lt;/p&gt;&#xA;" OwnerUserId="6703" LastActivityDate="2016-12-23T08:35:47.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11823" PostTypeId="2" ParentId="11819" CreationDate="2016-12-23T12:47:41.697" Score="5" Body="&lt;p&gt;The zero polynomial doesn't satisfy the degree constraint because the degree of a polynomial is the highest power of $x$ with a &lt;em&gt;nonzero&lt;/em&gt; coefficient.  If a student doesn't know this, it doesn't make $P(x) \equiv 0$ a valid answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem you post says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...where the degree of the polynomial $n=1,2,3,\dots$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To me that sounds like a (bad) way of saying this, and maybe our interpretations of this is where we differ:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...where the degree of the polynomial is $n$ for some integer $n \ge 1$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The zero polynomial doesn't satisfy this constraint.  The zero polynomial is considered to have undefined degree, or degree $-1$, or degree $-\infty$, depending on context.  So the zero polynomial just isn't a legitimate answer to the problem, but I can't tell if this is what you're actually asking.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My solution is just to take $P(x)=0$.  Now the problem seems to be that zero polynomial satisfies the first sentence of the definition but not the second one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then why would you give the zero polynomial as a solution when you know it doesn't satisfy the problem requirements?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So what should a student answer as the first sentence of the definition and its explanation does not corresponds to each others?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm not sure I understand what that means.  If a student's answer doesn't satisfy all the constraints, then it's not a correct answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, what is the mistake that you mentioned a couple times?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is this solution fine if there is a mistake in the definition?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user6648" LastEditorDisplayName="user6648" LastEditDate="2016-12-23T20:02:34.253" LastActivityDate="2016-12-23T20:02:34.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11824" PostTypeId="2" ParentId="11819" CreationDate="2016-12-23T17:13:20.107" Score="3" Body="&lt;p&gt;This is not a wise way for a student to respond to a test question. It seems like you know that the standard definition of a polynomial of $n$th degree requires that $a_n \ne 0$ (even though you didn't explicate that observation in your question). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: You know what the standard definition is. You can picture what is probably on the answer sheet for the test (which will not be the zero function). Relying on a curt, technical &quot;gotcha&quot; due to a typo is neither going to display how much you know, nor synchronize with the real definition, nor match the official answer sheet. There are occasional cases where students have challenged official answers after the fact and succeeded, but those are very, very rare (that being an enormous judicial challenge). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Immensely better if you: (a) point out that you &lt;em&gt;know&lt;/em&gt; what the standard definition is, and (b) write a more sophisticated answer compliant with the the standard definition, and (c) make it easier for the graders to match up with the predefined grading rubric. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2016-12-23T17:13:20.107" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11826" PostTypeId="1" CreationDate="2016-12-24T04:49:58.740" Score="5" ViewCount="726" Body="&lt;p&gt;I am starting out as a teacher and I'm confused about the verbs that we use in Bloom's taxonomy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was looking at a lot of websites that provide verbs for Bloom's level that describe the skill associated at that level.&#xA;For example, this &lt;a href=&quot;http://www.fractuslearning.com/wp-content/uploads/2016/01/blooms-taxonomy-verbs-1280.png&quot; rel=&quot;noreferrer&quot;&gt;chart&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I need more clarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The verb 'identify' is given in the Comprehension level and the Analysis level.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How is that possible?&lt;/li&gt;&#xA;&lt;li&gt;Why is it there in both levels?&lt;/li&gt;&#xA;&lt;li&gt;What is the difference in the skill if it's there in both the levels?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7602" LastActivityDate="2017-01-17T12:41:10.713" Title="Bloom's taxonomy" Tags="|general-pedagogy|teacher-development|transferrable-skills|" AnswerCount="2" CommentCount="3" ClosedDate="2016-12-24T21:34:01.297" ContentLicense="CC BY-SA 3.0" />
  <row Id="11827" PostTypeId="2" ParentId="11826" CreationDate="2016-12-24T07:31:03.083" Score="5" Body="&lt;p&gt;Bloom's taxonomy is not very helpful for maths, as many of the verbs don't make much sense, and ones that do have meanings occur in the wrong order. There are a couple of people (at least) who have written versions more adapted to maths, with more understandable verbs. I don't have the references to hand though.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit to add some references:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Anderson &amp;amp; Krathwohl, 2001, A taxonomy for learning, teaching and assessing (Two dimensional version) (see also a summary &lt;a href=&quot;http://www.depauw.edu/files/resources/krathwohl.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Pointon &amp;amp; Sangwin, 2003, An analysis of undergraduate core material in the light of hand held computer algebra systems (found &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/0020739031000148930&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Darlington, 2013, The use of Bloom's taxonomy in advanced mathematics questions&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="4746" LastEditDate="2017-01-17T12:41:10.713" LastActivityDate="2017-01-17T12:41:10.713" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="11828" PostTypeId="2" ParentId="11779" CreationDate="2016-12-24T20:17:17.770" Score="6" Body="&lt;p&gt;The integrals of sine and cosine have a nice geometric interpretation, if you and your students are comfortable treating the differential of a circular arc as equivalent to a straight hypotenuse (in the limit as $\Delta\theta \rightarrow 0$). This is perhaps more of an example than a simple exercise for the student.  But I think it answers the OP's objective of thinking geometrically about integrals instead of in terms of algebraic antidifferentiation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, $\int_a^b \sin\theta \; d\theta = \cos a - \cos b$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/eSALf.png&quot; width=&quot;300&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One may find the reasoning (which I assume the reader can fill in) a little loose, but it's a true picture that helps give the student a new perspective on sine and cosine.  I prove the ratio of the lengths of an arc and its subtending chord approaches $1$ as the lengths approach zero as well as prove the derivatives in a similar geometric fashion, so it feels more comfortable to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This idea may be found in &lt;a href=&quot;https://books.google.com/books?id=PXi9nl7LKGEC&amp;amp;pg=PA312&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pascal, &lt;em&gt;Traité des sinus du quart de cercle&lt;/em&gt; 1659, Prop. I&lt;/a&gt;: &lt;em&gt;La somme des sinus d’un arc quelconque du quart de cercle est égale à la portion de la base comprise entre les sinus extrêmes multipliée par le rayon.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2017-01-01T00:58:29.610" LastActivityDate="2017-01-01T00:58:29.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11829" PostTypeId="2" ParentId="11826" CreationDate="2016-12-24T21:06:27.903" Score="0" Body="&lt;p&gt;Not to directly address the question per se, but to get at how I view the thing in the context of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One good thing about Bloom's taxonomy is the idea that not all forms of &quot;knowing&quot; are the same, and you can get a lot of leverage by addressing different aspects of the thing in your math classes.  But looking at particular versions or phrasings of the thing don't seem very useful for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I do like the idea of knowledge building on itself, and I like the different flavors of questions students can explore.  For example, I like questions like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Alice thinks X, but Bob thinks Y.  Who's right and why?  What's wrong with either argument?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or another I might ask on a calculus exam.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the fundamental theorem of calculus?  Explain this as if to a friend, and include several diagrams.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;By encouraging students to these &quot;deeper&quot; levels of understanding (whatever you may feel like calling them), I find it's quite effective and a lot more fun to boot.&lt;/p&gt;&#xA;" OwnerUserId="7530" LastActivityDate="2016-12-24T21:06:27.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11830" PostTypeId="2" ParentId="11794" CreationDate="2016-12-25T23:22:42.473" Score="7" Body="&lt;p&gt;Here is my attempt to answer your question, if not directly, then at least in spirit! I am specifically responding to the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Many students see no beauty in this subject, only fear and a need to be right and get &quot;the answer&quot;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Has an approach to mathematics education been tried that explicitly deemphasizes correctness ... ?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What research supports or critiques this idea?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Something that I have done to move away from this need to get the right answer is, rather than directly &quot;deemphasizing correctness,&quot; trying instead to &lt;em&gt;emphasize the importance of mistake-making&lt;/em&gt; in learning (mathematics). For example, I introduce the following &lt;strong&gt;norm&lt;/strong&gt;/expectation on Day 1 of my classes (and have it written out on the corresponding class sites):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Mistake-Making&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I expect that we will all make mistakes as we wrestle with new (and sometimes old) ideas, and will use these mistakes as opportunities for learning. There is a large body of research on mathematics, and other subjects, that says we learn through mistakes — after all, when your work is mistake-free it only demonstrates what you already know! &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I expect we will take intellectual risks and make mistakes in the process, and that we will do our best to embrace the ability to honor, even celebrate, being wrong﻿.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For research to support this idea, I would point you to Jo Boaler (for mathematics and &lt;em&gt;mathematical mindset&lt;/em&gt;, in particular) or Carol Dweck (for &lt;em&gt;mindset&lt;/em&gt; research, more generally). Boaler has, for example, a mention on this &lt;a href=&quot;https://www.youcubed.org/think-it-up/mistakes-grow-brain/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Mistakes Grow Your Brain&lt;/strong&gt;&lt;/a&gt; page of the following paper:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Moser, J. S., Schroder, H. S., Heeter, C., Moran, T. P., &amp;amp; Lee, Y. H. (2011). Mind Your Errors Evidence for a Neural Mechanism Linking Growth Mind-Set to Adaptive Posterror Adjustments. &lt;em&gt;Psychological Science&lt;/em&gt;. &lt;a href=&quot;http://cpl.psy.msu.edu/wp-content/uploads/2011/12/Moser_Schroder_Moran_et-al_Mind-your-errors-2011.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and a mathematics-specific reference to:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Steuer, G., Rosentritt-Brunn, G., &amp;amp; Dresel, M. (2013). Dealing with errors in mathematics classrooms: Structure and relevance of perceived error climate. &lt;em&gt;Contemporary Educational Psychology, 38&lt;/em&gt;(3), 196-210.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The latter is summarized by Boaler as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Gabriele Steuer and her colleagues looked at the climate of math classrooms to consider the impact of “mistakes friendly” or “mistakes unfriendly” environments on students’ reactions to errors and the amount of effort they would put into classes (Steur et al., 2013).  They found that when students perceived their classroom as mistakes friendly – above and beyond other aspects of their classrooms environment – they increased their effort in their work.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Anecdotally, I certainly observe an increased willingness by students to share their work and not obsess over the right answer. When I ask for volunteers to discuss some idea they have just had, or problem they have just attempted, I often hear a comment to the effect of, &quot;I'll come up to the board! I think a made mistake, though,&quot; to which I reply, &quot;Great!&quot; Indeed, sometimes there is a mistake - but then others can chime in. In fact, sometimes a student realizes the way she &lt;em&gt;began&lt;/em&gt; the problem was erroneous, and - rather than saying &quot;whoops, never mind!&quot; - will end up solving it out on the board for the first time (or, at least, re-attempting it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should also note that, personally, I find the philosophy of &lt;em&gt;mistake-making as a plus&lt;/em&gt; to be anxiety reducing. Neither students nor teachers need to obsess over correctness; the notion that getting something wrong can be helpful reduces the burden of initial precision. Still, I caveat that, &lt;strong&gt;one&lt;/strong&gt;, instructors should perhaps be careful about taking this to an extreme (e.g., being unprepared as a teacher and having one's work constantly full of careless errors), and, &lt;strong&gt;two&lt;/strong&gt;, this is &lt;em&gt;not&lt;/em&gt; a way to &quot;deemphasize correctness&quot; permanently - rather, it is the notion that errors play a natural, and helpful, role in facilitating students' &lt;em&gt;convergence to correctness&lt;/em&gt; (to pilfer wording from the OP) as they wrestle with new mathematics and mathematical ideas.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2016-12-25T23:22:42.473" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11831" PostTypeId="2" ParentId="11785" CreationDate="2016-12-26T02:49:19.337" Score="1" Body="&lt;p&gt;As a student, I think I can give an answer that could possibly be useful and complementary to the other ones. It is mainly based on my personal experience, so it could be irrelevant. If people think so, don't hesitate to downvote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I understand your question, you are more concerned about prerequisites not being well known by the students rather than them not knowing the material taught in the lectures your exercise sessions are intended to illustrate, if so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, I believe it depends a lot on what is the &lt;em&gt;philosophy&lt;/em&gt; in your institution. In mine, exercise sessions are not mandatory and they are specifically addressed to students who are motivated to understand the material taught during the lectures. I personally used to not go to the lectures neither the exercise sessions and this was probably an awful decision at the time, but it's not the point. The point is that when I decided to go to these sessions, I did not expect the teaching assistant to remind me the properties I didn't remember because of lack of work as I was perfectly aware that I &lt;em&gt;should&lt;/em&gt; have known them. Instead, the fact that the teaching assistant did rely on the assumption that the students should know the prerequisites, acted like a wake-up call, sending the important message that the students should work if they want to keep up.&#xA;Of course, if on the contrary your institution makes the exercise sessions mandatory, this changes the all perspective, but in that case, I agree with the fact that reminding prerequisites may still lead the students think they know while they don't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, as you say, repeating prerequisites slows the session down and may bother the students who know them and may give a false impression of understanding that will surely backfire sooner or later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, despite being convinced that you should not repeat what should be known, I think you could say things like &quot;I use theorem X you learned last year. You should check if the hypotheses are respected&quot;. Another solution used by some of my professors is to devote the first exercise session of the year/semester to quick exercises on concepts and results considered as important (but reasonably complex) prerequisites, as it is useful for all students in general (even the best ones, but it is only based on my (narrow) observations and they could naturally be biased) and addresses the problem in an intellectually interesting way, without slowing down the more important sessions to come.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a final note, I think that the exercise session covering a new material X should always be preceded by some short reminder of the main new (or recent) definitions and results useful for the material X. For example, if your exercise session is about the introduction of Lebesgue integration, I think it is useful to quickly remind the students how the integral is constructed and what are the main tools (e.g. Lebesgue dominated convergence theorem,...) even if they have seen these in the lectures the exercise session is intended to cover.&#xA;It may seem obvious but not all the teaching assistants I have had began by doing so and it can be very unproductive, in my humble opinion.&lt;/p&gt;&#xA;" OwnerUserId="5809" LastEditorUserId="5809" LastEditDate="2016-12-26T16:48:48.807" LastActivityDate="2016-12-26T16:48:48.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11832" PostTypeId="1" AcceptedAnswerId="11834" CreationDate="2016-12-26T21:21:49.600" Score="5" ViewCount="928" Body="&lt;p&gt;I recently read in a book about a proof that Archimedes did. I don't remember the exact details and I don't have the book on me right now, but it involved proving an equality. So let's say proving that $A=B$. In order to do this, he first assumed that $A&amp;gt;B$ and found a contradiction. He then assumed that $A&amp;lt;B$ and once again, found a contradiction. Therefore, he concluded, it must be that $A=B$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had never thought of this proof method or encountered it, so I thought it was very neat. Of course you're doing twice as much proving, if you like, but I feel like proving an inequality can often be easier than an equality because you can 'neglect' certain terms which don't effect the inequality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know if it is just the education I have had, but I am an undergraduate now and, like I said, I've never encountered this proof. I feel this is a shame because it would be a handy tool to know for any student. So my question is, why is this type of proof not used more often and taught?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book was &lt;em&gt;Journey through genius by William Dunham&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user7607" LastActivityDate="2016-12-28T08:25:12.737" Title="Why is this type of reductio ad absurdum not taught more?" Tags="|undergraduate-education|proofs|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11833" PostTypeId="2" ParentId="11832" CreationDate="2016-12-26T21:33:19.530" Score="5" Body="&lt;p&gt;Direct proofs are generally favored over proof by contradiction, or at least that's what I've been taught more than once.  I think it's because it's usually simpler to do a direct proof, but of course each problem is different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think I've ever seen a proof by contradiction as you described to show that $A=B$ but I have seen several times the very similar approach of showing $A\le B$ and $A\ge B$ directly.&lt;/p&gt;&#xA;" OwnerDisplayName="user6648" LastActivityDate="2016-12-26T21:33:19.530" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11834" PostTypeId="2" ParentId="11832" CreationDate="2016-12-26T22:06:18.563" Score="9" Body="&lt;p&gt;This form of proof is quite common, especially in real analysis.  Any time you are proving something about ordered fields -- and therefore have access to a law of trichotomy -- this form of argument is likely to come up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just to take one example, consider the set $S = \{x \in \mathbb{Q}^+ | x^2 \le 2 \}$.  One proves that this set is bounded, but has no least upper bound; this demonstrates that $\mathbb{Q}$ is not complete.  Then we move to $\mathbb{R}$, which has the completeness property, so the same set $S$ has a least upper bound $r \in \mathbb{R}$.  We want to prove that $r^2 = 2$.  So, first we assume that $r^2 &amp;gt; 2$, and derive a contradiction, which eliminates that possibility; then we assume $r^2 &amp;lt; 2$, and derive a contradiction, which eliminates that possibility.  The only option left is that $r^2 = 2$.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-12-26T22:06:18.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11835" PostTypeId="2" ParentId="11779" CreationDate="2016-12-27T06:32:47.397" Score="3" Body="&lt;p&gt;Based on a discovery of &lt;a href=&quot;https://books.google.com/books?id=xMBhhRtupQsC&amp;amp;pg=PT34&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gregory of St. Vincent, &lt;em&gt;Opus geometricum quadraturae circuli et sectionum coni&lt;/em&gt; (1647), Props. CVIII-CIX, 585-586&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consider integrating the hyperbola $y = 1/x$:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/wqpR0.png&quot; alt=&quot;Mathematica graphics&quot;&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If the $x$ coordinates of a partition of the interval from $1$ to $b=2$ are multiplied by $a=3$, the partition is transformed to one of the interval from $a = 3$ to $ab=6$. Furthermore, all the rectangles in a Riemann sum over $[1,2]$ are transformed to equal rectangles in a Riemann sum over $[3,6]$, since the bases are multiplied by $a=3$ and the heights multiplied by $1/a=1/3$.&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;p&gt;Why does this show $\int_1^2 {dx \over x} = \int_3^6 {dx \over x}$?&#xA;  Argue similarly that $\int_1^b {dx \over x} = \int_a^{ab} {dx \over x}$ for any $a,b&amp;gt;0$).&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Let $L(a) = \int_1^a {dx \over x}$. Show that $L(ab)=L(a)+L(b)$. Such a function is called a logarithmic function. (First argue that $\int_1^{ab} {dx \over x} = \int_1^{a} {dx \over x} + \int_a^{ab} {dx \over x}$.)&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Given that $L(a)$ is a logarithm, estimate its base. (First recall what the value of $\log_B(B)$ is.)&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2016-12-28T05:50:31.613" LastActivityDate="2016-12-28T05:50:31.613" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11836" PostTypeId="1" AcceptedAnswerId="11840" CreationDate="2016-12-27T12:50:29.470" Score="9" ViewCount="879" Body="&lt;p&gt;Day 1 in my &quot;Into to Pure Maths&quot; class...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to have a very simple set of axioms defining something, not necessarily a useful thing, but a system that is suitable for making short deductions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just want to convey the idea of proving something using the axioms and then revealing an actual instance and confirming that the newly-proved fact does indeed hold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I seem to remember hearing something years (decades) ago that used something geometric - perhaps even finite projective planes - for this purpose, but I cannot find it online..&lt;/p&gt;&#xA;" OwnerUserId="4795" LastActivityDate="2017-01-23T06:48:20.793" Title="Is there a toy example of an axiomatically defined system/ structure?" Tags="|undergraduate-education|concept-motivation|" AnswerCount="8" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11837" PostTypeId="2" ParentId="11836" CreationDate="2016-12-27T15:15:26.673" Score="3" Body="&lt;p&gt;One example from Euclidean geometry in planes. Consider the following three axioms:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) There exist three non-collinear and pairwise different points in the plane. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Every 2 distinct points can be connected by a unique line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) For every line $g$ in the plane and every point $A$ not on $g$ there exists a unique line $h$ through $A$ (i.e. $A \in h$) with $g \cap h =\emptyset$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you assume these axioms to be true, then you can easily prove that the plane consists of at least 4 points in the plane:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you have a plane. Axiom 1) yields that there are at least 3 non-collinear pairwise different points $A$, $B$ and $C$. According to axiom 2) there is a unique line $g$ through $A$ and $B$. Yet, $C \not \in g$ since $A$, $B$ and $C$ are not collinear due to 1). Now axiom 3) tells you that there is a line $h$ through $C$ with $g \cap h =\emptyset$. According to 2) $h$ is a unique line through (at least) 2 points with one being $C$ by construction. Hence, there has to be a fourth point $D$ in the plane with $D \in h$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this example helps a bit. I tried to keep it simple and did not use any definitions for a line for example. In a more scientific context this should, however, be done. Since you asked for a toy example the above example should be sufficient.&lt;/p&gt;&#xA;" OwnerUserId="7608" LastEditorUserId="7608" LastEditDate="2016-12-28T07:55:28.093" LastActivityDate="2016-12-28T07:55:28.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11838" PostTypeId="1" CreationDate="2016-12-27T16:29:50.900" Score="3" ViewCount="223" Body="&lt;p&gt;I came across some interesting mistakes in many area of mathematics with my students and do not let me also to tell you for  university students level, I would like to know How do i  deal with students who make these  mistakes :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Mistakes&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;1) For  ‎$x,y &amp;gt;0 , \log (x+y)=\log x+\log y ‎$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) For  ‎$z\in {C}  ,\cos²z+\sin²z \neq 1 ‎$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) ‎$1 ‎$ is a  prime number because   ‎$1 ‎$ divides  ‎$1 ‎$ and itself &lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) For ‎$x, y \in {R},\sqrt{(x²+y²)}=x+y ‎$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) For  ‎ ‎$g(x)\neq 0, \displaystyle \int_{a}^{b}\frac{f(x)}{g(x)}dx=\frac {\int_{a}^{b} f(x)}{\int_{a}^{b}g(x)}dx ‎$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6) ‎${0}^{\infty}‎$ is indeterminate case .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some Teachers in mathematics believe that the claim 6 is true .&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;My question here is :&lt;/strong&gt; How do i  deal with students who make these  mistakes?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I edited the question to be clear and according to the below comments &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you for any help !!!&lt;/p&gt;&#xA;" OwnerUserId="7583" LastEditorUserId="7583" LastEditDate="2016-12-28T18:58:11.427" LastActivityDate="2016-12-28T18:58:11.427" Title="How do i deal with students who make these mistakes?" Tags="|secondary-education|students-mistakes|integration|number-theory|complex-numbers|" AnswerCount="0" CommentCount="9" ClosedDate="2016-12-28T19:50:21.693" ContentLicense="CC BY-SA 3.0" />
  <row Id="11839" PostTypeId="2" ParentId="11836" CreationDate="2016-12-27T17:05:49.357" Score="7" Body="&lt;p&gt;Howard Eves has a nice example in &lt;em&gt;Great Moments in Mathematics (After 1650)&lt;/em&gt;, Lecture 36: a set $K$ of elements and a relation $R$, with the following 4 postulates:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If $a \ne b$, then either $aRb$ or $bRa$.&lt;/li&gt;&#xA;&lt;li&gt;If $aRb$, then $a \ne b$.&lt;/li&gt;&#xA;&lt;li&gt;If $aRb$ and $bRc$, then $aRc$.&lt;/li&gt;&#xA;&lt;li&gt;$K$ consists of exactly four distinct elements. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;He then proves seven specific theorems (with more in the exercise list), and also discusses changes if some postulates are altered. Then he provides applications of those theorems in a variety of fields (arithmetic, geometry, and genealogy). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2016-12-27T20:06:41.833" LastActivityDate="2016-12-27T20:06:41.833" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11840" PostTypeId="2" ParentId="11836" CreationDate="2016-12-27T18:52:37.233" Score="10" Body="&lt;p&gt;This may go beyond what you are asking for, but there is a wonderful book called &lt;em&gt;Introduction to the Foundations of Mathematics&lt;/em&gt; by Raymond L. Wilder. I provided its axioms and an example of how they could be satisfied using SET cards in &lt;a href=&quot;https://matheducators.stackexchange.com/a/2530&quot;&gt;&lt;strong&gt;MESE 2528&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is again a list of the axioms:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9Dxhg.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9Dxhg.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The full book is available online (&lt;a href=&quot;https://archive.org/details/IntroductionToTheFoundationsOfMathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;open access&lt;/strong&gt;&lt;/a&gt;) and I have derived quite some joy over the years from proving the various theorems that follow the above-mentioned axioms. Wilder himself begins his discussion of these five axioms by showing that they can be &quot;satisfied&quot; with an interpretation around people as members of clubs, and remarks that it is achievable with just four people; so, he continues, this cannot be a full description of (e.g.) Euclidean geometry, since we know there are many more than four points in the plane!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The theorems that Wilder uses begin as innocuous; according to my own notes, the first one is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Theorem 1&lt;/strong&gt; Every point is on two distinct lines&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and become increasingly more difficult; e.g., given: $n$ total points; let $m$ be the number of lines containing a given point - this has already been shown as well-defined in an earlier theorem; and let $r$ be the number of points on a line - also shown well-defined in an earlier theorem. Then one has:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Theorem 10&lt;/strong&gt; The total number of lines is $m \cdot r$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The book is from 1952 (the link above goes to its reprinted second edition from 1965) but I think it holds up quite well. Even if the full book turns out not to be ideal for your present purposes, I would recommend this as a reference; there are some nice problems and explanations scattered throughout.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2016-12-27T18:52:37.233" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11841" PostTypeId="2" ParentId="11836" CreationDate="2016-12-27T20:12:55.577" Score="3" Body="&lt;p&gt;This approach below does not work equally well in all countries, but examples based on changing the rules of a familiar sport/game I think can help some students with regard to ideas related to axiomatics. In this column (basically about the taxicab plane) that I wrote some years ago for the Feature Column of the American Mathematical Society I have a brief discussion of this idea in the context of American baseball. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Scroll down to the section on &quot;axomatics.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/samplings/feature-column/fcarc-taxi&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.ams.org/samplings/feature-column/fcarc-taxi&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usual baseball - (3,4,9)-baseball - has 3 strikes for an out, 4 balls for a walk, and 9 innings. What happens when one changes these parameters?&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2016-12-27T20:12:55.577" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11842" PostTypeId="2" ParentId="11836" CreationDate="2016-12-28T03:23:55.833" Score="5" Body="&lt;p&gt;For my money the best possible &quot;toy&quot; examples are the &lt;strong&gt;MU-System&lt;/strong&gt; and the &lt;strong&gt;pq-System&lt;/strong&gt; introduced and defined in Chapters 1 and 2 of Douglas Hofstadter's &lt;em&gt;Gödel, Escher, Bach: An Eternal Golden Braid&lt;/em&gt;.  The latter, in particular, is extremely useful for stressing the delicate balance between syntax and semantics:  on the one hand, having an interpretation of the words and axioms in your system can play an important role in helping anticipate which theorems are likely to be true and which ones false; on the other hand, one's reasoning should not &lt;em&gt;depend&lt;/em&gt; on that interpretation, because there may be other interpretations that are equally valid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like this example better than some of the geometric ones people are suggesting because it's hard for beginners to suppress their intuition about what the words &quot;point&quot;, &quot;line&quot; and &quot;plane&quot; mean.  (Hilbert's apocryphal quote about &quot;tables, chairs and beer mugs&quot; fits in well here.)  In Hofstadter's &lt;strong&gt;pq-System&lt;/strong&gt;, the reader genuinely doesn't know what &lt;strong&gt;p&lt;/strong&gt; and &lt;strong&gt;q&lt;/strong&gt; mean (or might mean) until &lt;em&gt;after&lt;/em&gt; they have figured out some of the theorems and non-theorems of the system.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-12-28T03:23:55.833" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11843" PostTypeId="2" ParentId="11832" CreationDate="2016-12-28T08:25:12.737" Score="2" Body="&lt;p&gt;The argument I think you're referring to is a common &quot;low level&quot; argument &amp;mdash; for practical purposes its applications are usually subsumed by &quot;higher level&quot; idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, one might use that sort of argument to prove the theorem&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $\small f(x) \leq g(x)$ for all $x$, then $\small \sup_x f(x) \leq \sup_x g(x)$. If the limits exist, then $\small\lim_{x \to a} f(x) \leq \lim_{x \to a} g(x)$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and this theorem is convenient to use for typical applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example of applying this idea is the approach to the fundamentals of integration by upper and lower Riemann sums. Then once you have the theory of integration, a wide class of problems where one might have used that argument form one instead sets up an integral.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2016-12-28T08:25:12.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11844" PostTypeId="1" CreationDate="2016-12-28T10:05:39.630" Score="5" ViewCount="396" Body="&lt;p&gt;I hope this is a good place to ask the question. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My issue is that I have finished high school a number of years ago, and although I did some (non-advanced) math courses then, I did not do an exam in maths for the final high school matriculation exam.&#xA;Now, years later, I would like to enter a university course for computer science, but all of the programmes of interest to me require sufficient knowledge of maths as an admissions criterion. Theoretically I could go back to supplement my matriculation exam with an advanced math grade, but this is quite a bureaucratic and logistical hurdle since I live abroad and it would take a couple of years to officially pass all the courses. I am confident in being able to self-study the curriculum with support from online resources, but the issue is how to prove my knowledge to the admissions people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would really appreciate it if someone could let me know whether there are any certificates or qualifications for high school maths, accepted either internationally or in Europe, so that I could attend just one big exam instead of trying to organise how to sit all the 10+ individual courses and the big exam in a high school in my home country. &lt;/p&gt;&#xA;" OwnerUserId="7611" LastActivityDate="2017-01-18T07:29:39.770" Title="Are there any math certificates/qualifications on high school maths that you can do after high school?" Tags="|secondary-education|exams|self-learning|prerequisites|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11845" PostTypeId="2" ParentId="11836" CreationDate="2016-12-28T12:44:50.193" Score="0" Body="&lt;p&gt;Instead of making a toy set of axioms, I would go to some axioms taken from an established set and prove something trivial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would try with the field axioms of real numbers (or just some of them) and then prove that 2+2=4 or 3*2=6 - of course you would need to define 2, 3, 4 and 6 first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, although this approach would fit a mathematical analysis course, it might be not the best for a &quot;pure math&quot; course.&lt;/p&gt;&#xA;" OwnerUserId="7178" LastActivityDate="2016-12-28T12:44:50.193" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11846" PostTypeId="2" ParentId="11844" CreationDate="2016-12-29T10:06:13.250" Score="0" Body="&lt;p&gt;In Canada, universities and colleges offer non-credit courses covering grade 12 subjects required for offered programs. I believe they can also be taken in parallel with program courses. There are also reduced entry requirements for mature students, that is, for individuals who have been in the work force for 4-5 years. This seems to be cheating a bit, but I remember reading that the success rate for mature students is quite high; apparently mature students tend to be more highly motivated than students entering university straight out of high school.&#xA;There is strong feeling that people who for one reason or another dropped out before graduating high school should be actively encouraged, (including subsidized education and living for individuals who have been on unemployed for a while) to get qualified for something, get a job, or better job, start a career, so they are drawn into the mainstream: get married/coupled, buy a house, have a family etc. In the long term this is cheaper for society as a whole. I believe refugees are also encouraged to get into training programs to get themselves economically established as soon as possible. &lt;/p&gt;&#xA;" OwnerUserId="6202" LastActivityDate="2016-12-29T10:06:13.250" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11847" PostTypeId="1" CreationDate="2016-12-29T21:39:46.727" Score="18" ViewCount="6839" Body="&lt;p&gt;Are complex numbers taught in high school in other countries? I am from Germany and complex numbers are next to never touched in high school with the exception of extra-curricular activities, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess complex numbers also won't be discussed in high schools in other countries and are simply neglected and never mentioned. However, I think it is important to at least let students know that complex numbers exist.  I know that discussing complex numbers in full detail would require a lot of time which is often not available. Yet, without ever mentioning complex numbers, a teacher is leaving out an important piece of information. For example, when students are learning to find roots of polynomials they will usually be searching for them in the &quot;wrong&quot; set ($\mathbb{R}$). Very often examples are picked where a polynomial will have only real roots which guarantees the teacher to avoid the complex numbers. However, I think that it is also important to make students aware that things might not always be as easy as in the examples picked by the teacher and that a polynomial might very well have complex roots. The fundamental theorem of algebra, for example, is a theorem which is very easy to state and to discuss in school. The theorem will require some knowledge about the existence of complex numbers, yet, I think it is well worth discussing in high school: as the theorem's name suggests: it is fundamental to algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are your thoughts about the topic? Should complex numbers be discussed in high school? Or are there even countries where they are part of the curriculum?&lt;/p&gt;&#xA;" OwnerUserId="7608" LastActivityDate="2017-01-02T03:36:02.530" Title="Complex numbers in high school" Tags="|secondary-education|mathematical-pedagogy|mathematical-analysis|complex-numbers|" AnswerCount="10" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="11848" PostTypeId="2" ParentId="11847" CreationDate="2016-12-29T21:49:12.840" Score="16" Body="&lt;p&gt;In the United States, complex numbers are a standard part of the typical high school Algebra 2 course.  Students (normally in grades 10 or 11, corresponding approximately to ages 15-17) learn to add, multiply, and divide complex numbers; to solve quadratic equations with no real roots; and to find all $n$ roots of an $n$th degree polynomial (usually, carefully chosen so that the rational roots theorem gets you most of the way, and the quadratic formula gets you the rest of the way).  They also learn the statement of the Fundamental Theorem of Algebra.  The next year, if they continue on to a Precalculus course, they typically also learn the representation of complex numbers as points in a plane.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2017-01-01T00:22:46.987" LastActivityDate="2017-01-01T00:22:46.987" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="11849" PostTypeId="2" ParentId="11847" CreationDate="2016-12-30T01:03:06.540" Score="19" Body="&lt;p&gt;Although there is a tradition in the U.S. of nominal mention of complex numbers in the high school curriculum, in my observation it is invariably superficial, and complex numbers are not mentioned subsequently. In particular, there is no mention of Euler's identity expressing sine and cosine in terms of complex exponentials, and, therefore, no mention of the palpable fact that all trig identities are deducible &quot;by algebra&quot; from properties of the exponential function (which itself is not treated in any way that would allow understanding of Euler's identity, unfortunately).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since high schools in the U.S. do not mention solvability of cubics in radicals, there will be no mention of the &quot;irreducible case&quot;, where the discriminant is complex, despite there being three real roots. So the &lt;em&gt;quadratic&lt;/em&gt; story can easily be dismissed (in the minds of the students, as in the minds of many over the centuries) by saying that in some cases there simply &quot;are no roots&quot; (meaning no real roots).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, both in my direct recollection from decades ago, and from observation in more recent times, kids in high school are not led to take complex numbers seriously, but only to view them as yet one more menu-item on the laundry list... Sort of an &quot;option&quot;, or something that might be on the final, ... but is not genuine or manifest in the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, mathematics grad students at my R1 university tend to default into a similar attitude, that &quot;complex analysis&quot; is just a pre-specialty thing that presents a hurdle to them, for who-knows-what reason, but is not relevant to much of anything. This is dismaying... as are many things to cranky old people, I guess! :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2016-12-30T01:03:06.540" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11850" PostTypeId="2" ParentId="11847" CreationDate="2016-12-30T02:04:48.090" Score="8" Body="&lt;p&gt;Complex numbers are also taught in China. I observed a mathematics classroom in Nanjing, Jiangsu Province, China in 2008-09 for students in their penultimate year of secondary school (their high schools have 3 years, whereas U.S. high schools have 4 years) and the end of the book discussed complex numbers. This was a standard public school in a second-tier Chinese city.&lt;/p&gt;&#xA;&lt;p&gt;The specific book they used, in case you want to track down a physical copy, was the 2005 edition of:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;普通高中课程标准实验教科书&lt;/p&gt;&#xA;&lt;p&gt;选修2-2&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(The remaining parts are my own translations.)&lt;/p&gt;&#xA;&lt;p&gt;Section 3 is entitled, &lt;strong&gt;Generalizing Number Systems and an Introduction to Complex Numbers&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Its three subsections are (pp. 103-118):&lt;/p&gt;&#xA;&lt;p&gt;3.1 &lt;em&gt;Generalizing Number Systems&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;3.2 &lt;em&gt;Arithmetic Operations with Complex Numbers&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;3.3 &lt;em&gt;Geometric Interpretation of Complex Numbers&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Below are three pictures from the textbook (of which I have a copy). The first is of the introduction; the second is of four sample problems (that I will translate); and the third is a fleeting mention of Euler's Identity (which &lt;strong&gt;paul garrett&lt;/strong&gt; laments is lacking from many US secondary classes).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Intro:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eOT5l.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eOT5l.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problems:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MHwjk.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MHwjk.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Translation errors my own!)&lt;/p&gt;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; is an imaginary number, and let &lt;span class=&quot;math-container&quot;&gt;$w = z + \frac{1}{z}$&lt;/span&gt;. Prove: &lt;span class=&quot;math-container&quot;&gt;$|z| = 1$&lt;/span&gt; is both a necessary and sufficient condition to conclude that &lt;span class=&quot;math-container&quot;&gt;$w \in \mathbb{R}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given &lt;span class=&quot;math-container&quot;&gt;$z_1, z_2, \in \mathbb{C}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$|z_1| = |z_2| = 1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$|z_1 + z_2| = \sqrt{3}$&lt;/span&gt;, compute &lt;span class=&quot;math-container&quot;&gt;$|z_1 - z_2|$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given an ellipse with major axis of length &lt;span class=&quot;math-container&quot;&gt;$2a$&lt;/span&gt;, and the coordinates of its foci &lt;span class=&quot;math-container&quot;&gt;$F_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F_2$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$z_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z_2$&lt;/span&gt;, respectively, pick any point &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; on the ellipse, and write out the relationship satisfied by &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$z_1$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$z_2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; be an imaginary number, and let &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\bar{z}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{z}$&lt;/span&gt; correspond to the vectors &lt;span class=&quot;math-container&quot;&gt;$\vec{OA}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\vec{OB}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\vec{OC}$&lt;/span&gt;. Write out:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;(1) the relation between &lt;span class=&quot;math-container&quot;&gt;$\vec{OA}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec{OB}$&lt;/span&gt;;&lt;/p&gt;&#xA;&lt;p&gt;(2) the relation between &lt;span class=&quot;math-container&quot;&gt;$\vec{OB}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec{OC}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Euler's Identity:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YNhWl.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YNhWl.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;These are essentially historical notes (the chapter begins with a quotation from Leibniz and a mention of Cardano) and do not have corresponding problems. I am not sure whether Euler's Identity would be pursued in the students' next, and last, year of secondary school mathematics.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-12-30T05:19:17.307" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11852" PostTypeId="2" ParentId="11847" CreationDate="2016-12-30T10:45:42.707" Score="5" Body="&lt;p&gt;In Norway, it's part of the curriculum for the optional high school course &quot;&lt;a href=&quot;http://www.udir.no/kl06/MAT2-01/Hele/Kompetansemaal/mathematics-x?lplang=eng&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matematikk X&lt;/a&gt;&quot;, together with some elementary number theory and probability with continuous distributions (especially the normal distribution). Unfortunately, it's not a course many take, since if you do, it will still come on top of the normal mths courses, and most people who take math prioritize physics and chem&lt;/p&gt;&#xA;" OwnerUserId="7624" LastActivityDate="2016-12-30T10:45:42.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11853" PostTypeId="1" CreationDate="2016-12-30T10:50:37.737" Score="14" ViewCount="1867" Body="&lt;p&gt;To my mind, there seem to be two main paths to differential geometry. There is the classical approach, focusing on curves and surfaces in $\mathbb{R}^n$, especially $\mathbb{R}^3$. Prerequisites tend to be multivariable calculus and linear algebra. Some exposure to real or complex analysis and metric spaces or point-set topology is helpful, but not strictly required. A couple of titles following this approach include Pressley's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/184882890X&quot; rel=&quot;noreferrer&quot;&gt;Elementary Differential Geometry&lt;/a&gt; and Tapp's recent &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/3319397982&quot; rel=&quot;noreferrer&quot;&gt;Differential Geometry of Curves and Surfaces&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern approach defers much of the geometry and studies manifolds first. An example of this path would be Lee's trilogy: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B0182QAD1E&quot; rel=&quot;noreferrer&quot;&gt;Introduction to Topological Manifolds&lt;/a&gt;, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1441999817&quot; rel=&quot;noreferrer&quot;&gt;Introduction to Smooth Manifolds&lt;/a&gt; and &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387983228&quot; rel=&quot;noreferrer&quot;&gt;Riemannian Manifolds: An Introduction to Curvature&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider a scenario in which upper-division undergraduates study classical differential geometry and then return to the subject at graduate level, the latter with a manifolds-first approach. Would the classical introduction help or hinder?&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2017-01-03T09:56:34.720" LastActivityDate="2017-01-03T10:30:17.400" Title="Does learning classical approach to differential geometry before modern approach help or hinder?" Tags="|undergraduate-education|graduate-education|differential-geometry|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11854" PostTypeId="2" ParentId="11847" CreationDate="2016-12-30T11:32:37.970" Score="4" Body="&lt;p&gt;In Italy complex numbers are commonly taught in science and technically oriented high schools, like the &lt;a href=&quot;https://en.wikipedia.org/wiki/Liceo_scientifico&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;liceo scientifico&lt;/em&gt;&lt;/a&gt; or the various types of technical institutes (e.g., on electronics, electrotechnics, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the technical institutes, the teaching of complex numbers if fundamental for the understanding of circuit theory in the AC regime (phasors).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complex numbers are usually taught during the 3rd year (16-year old students), and common topics are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Representations of complex numbers&lt;/li&gt;&#xA;&lt;li&gt;Operations on complex numbers&lt;/li&gt;&#xA;&lt;li&gt;Euler's identity&lt;/li&gt;&#xA;&lt;li&gt;Powers and roots of complex numbers&lt;/li&gt;&#xA;&lt;li&gt;Complex exponential&lt;/li&gt;&#xA;&lt;li&gt;Complex logarithm&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5367" LastActivityDate="2016-12-30T11:32:37.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11856" PostTypeId="2" ParentId="11853" CreationDate="2016-12-30T16:41:17.067" Score="6" Body="&lt;p&gt;Looking through the Pressley book, it seems to me that a student could study the whole thing, and at the end of the semester they would believe that all of differential geometry happens on curves and 2-surfaces embedded in Euclidean 3-space, and that space is endowed by its creator with a certain natural coordinate system and a certain natural system of measurement. These are all intuitively appealing beliefs, and all of them will have to be unlearned if that student is going to go on to study general relativity or differential geometry on a manifold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student would imagine themself to be knowledgeable in differential geometry, but would never have heard the term &quot;metric&quot; (which is absent from the index). Nor have they ever heard of tensors, much less the Riemann tensor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, the whole treatment has a very 19th-century flavor. I read an old-fashioned differential geometry book of this type, by do Carmo, when I was a grad student before I took general relativity, thinking it would be helpful to develop geometric intuition in a context that was more visualizable. It did me no good at all.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2016-12-30T16:41:17.067" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11857" PostTypeId="2" ParentId="11847" CreationDate="2016-12-30T17:11:59.130" Score="4" Body="&lt;p&gt;Here is a screenshot from the 11th grade textbook Chapter 6 in India. Sometimes complex numbers are also needed for some physics problems so students actually needed to know complex numbers a little earlier.&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, the syllabus has been simplified and math reduced compared to when I was a student. The following topics are included in the CBSE class XI syllabus &lt;a href=&quot;http://www.cbsesyllabus.in/class-11/mathematics-class-11-syllabus&quot; rel=&quot;nofollow noreferrer&quot;&gt;quoted from here&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;2. Complex Numbers and Quadratic Equations&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Need for complex numbers, especially √1, to be motivated by inability to solve some of the quardratic equations. Algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system. Square root of a complex number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oAzup.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oAzup.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Based on this knowledge, students appear in entrance examinations for engineering colleges roughly after two years of learning above. The following is a sample question that they must answer in less than a minute.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QFWNb.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QFWNb.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7628" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2016-12-30T23:15:32.210" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11858" PostTypeId="2" ParentId="11847" CreationDate="2016-12-30T22:35:36.790" Score="4" Body="&lt;p&gt;In most Israeli high schools, mathematics can be taken on one of 3 levels. The highest level, known as &quot;5 units&quot; and taken by about 10% of students in the country (generally those planning a career in science/engineering) includes the topic of complex numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These students are then tested on complex numbers in the matriculation exam (&lt;a href=&quot;https://en.wikipedia.org/wiki/Bagrut&quot; rel=&quot;nofollow noreferrer&quot;&gt;bagrut&lt;/a&gt;), but the questions are usually quite easy (compared to the other topics on the exam), requiring not much beyond addition and multiplication of such numbers, and being able to convert between the rectangular and polar forms with a bit of algebra and trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the &lt;a href=&quot;http://kibinimatika.co.il/Upload/%D7%A9%D7%90%D7%9C%D7%95%D7%A0%D7%99%20%D7%A7%D7%99%D7%A5%202016%20%D7%9E%D7%95%D7%A2%D7%93%20%D7%91/math_summer_807.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2016 summer bagrut (807 questionnaire)&lt;/a&gt; includes the following section (comprising a sixth of the questionnaire's total marks):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given the complex number $z$, $$z = \frac{(\cos\frac{\pi}{9}+i\sin\frac{\pi}{9})^3}{(\cos\frac{\pi}{12}-i\sin\frac{\pi}{12})^2}$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(1) Find $|z|$ and the argument (angle) of $z$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(2) Find all values $n$ ($n$ is a natural number) for which $z^n$ is a&#xA;  purely imaginary number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="688" LastEditorUserId="688" LastEditDate="2017-01-02T03:36:02.530" LastActivityDate="2017-01-02T03:36:02.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11859" PostTypeId="2" ParentId="11847" CreationDate="2016-12-31T14:45:58.150" Score="1" Body="&lt;p&gt;In the United Kingdom complex numbers are taught to pupils who stay in school from age 16 to 18 to do their GCE A or AS levels in maths. This may help: &lt;a href=&quot;https://www.heacademy.ac.uk/system/files/pre-university-maths-guide.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.heacademy.ac.uk/system/files/pre-university-maths-guide.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7637" LastActivityDate="2016-12-31T14:45:58.150" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11861" PostTypeId="2" ParentId="11847" CreationDate="2017-01-01T02:35:48.017" Score="3" Body="&lt;p&gt;I am a computer engineering student from Egypt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Egypt, you choose first what branch you want to study in the high school: maths, science (biology), or arts (history, philosophy, etc). Complex numbers are taught to 3rd year math students (the year just before the university). You choose this branch only if want to study engineering, computer science, mathematics, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an engineering student, I find complex numbers very important. They're used in engineering maths and physics. They're heavily used in, for example, AC circuit analysis, and Control theory. Fourier series and transform are used a lot in signal processing courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complex numbers are a must for many engineering departments.&lt;/p&gt;&#xA;" OwnerUserId="7645" LastActivityDate="2017-01-01T02:35:48.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11862" PostTypeId="2" ParentId="11847" CreationDate="2017-01-01T03:45:42.470" Score="1" Body="&lt;p&gt;I'm a college student in China. As Benjamin has mentioned, complex numbers are taught in China. But in a rather implicit way.We are taught that they exit, but we don't have enough background knowledge to understand the reason of introducing such an weird &quot;i&quot;...lol.What we're taught is to do the calculation as accurate as possible as well as understanding and using the geometric property of the complex plane which is frequently associated with vectors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'm trying to state is that it's hardly possible to dipict an explicit picture of complex numbers to senior high students.Think about this question: If you just tell them that for some practical reasons, we have to define an i as the sqrt of -1,what if they ask you why we can't define a number which yield 1 when it's multiplied by 0?How will you answer that question?Such thing did happen,and as far as I am concerned,the teacher simply can't answer that question.In fact it's not until that I went to college that I know the answer to that using abstract algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So there're some problems when you introduce such a weird thing to students,especially curious ones.My point is:It's good to introduce it,but we have to figure out a better way to define it well.&lt;/p&gt;&#xA;" OwnerUserId="7648" LastActivityDate="2017-01-01T03:45:42.470" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11864" PostTypeId="2" ParentId="10751" CreationDate="2017-01-01T18:45:24.923" Score="3" Body="&lt;p&gt;Actually the most 'intuitive' proof of the Pythagoras theorem involves cutting the big square into only 3 pieces and reassembling them into the two smaller squares (using merely translations). Absolutely no subtraction is involved! (See the last section below for why subtraction of area is a nontrivial issue.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;For students&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UXl9Bm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UXl9Bm.png&quot; alt=&quot;pythagoras-mincut&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Cut the green square along the red line and reassemble into the yellow and blue squares.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;For advanced students&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that the proof is still not that simple. I think the easiest way is as follows. Start with the yellow and blue squares, then construct the point at the top and the 4th point at the bottom by translations of the appropriate points. Now use translations to prove that the green shape has equal sides, and use the four congruent triangles to prove that it has equal angles, and hence it is a square. The four congruent triangles also show that the area of the green square is the same as the total area of the yellow and blue squares.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;For teachers&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;All that is used here is &lt;a href=&quot;https://en.wikipedia.org/wiki/Scissors_congruence&quot; rel=&quot;nofollow noreferrer&quot;&gt;scissors-congruency&lt;/a&gt;, which for polygons turns out to be equivalent to having the same area. (It is not equivalent for polytopes in general.) But &lt;strong&gt;what is area&lt;/strong&gt; in the first place? In modern mathematics we could just use the Lebesgue measure, but arguably there is a more natural definition that extends the idea of scissors-congruency, which turns out to yield the &lt;a href=&quot;https://en.wikipedia.org/wiki/Jordan_measure&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jordan measure&lt;/a&gt; and yet is not hard for a bright student to grasp.&lt;/p&gt;&#xA;&lt;p&gt;Specifically we say that &lt;span class=&quot;math-container&quot;&gt;$A \precsim B$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; can be cut into finitely many pieces along &lt;a href=&quot;https://en.wikipedia.org/wiki/Simple_curve#Definition&quot; rel=&quot;nofollow noreferrer&quot;&gt;simple curves&lt;/a&gt; and reassembled by rigid motions (translation or rotation or reflection) to fit within &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; such that none of the pieces overlap (except perhaps at boundaries). Now for each non-negative real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; choose some fixed rectangles &lt;span class=&quot;math-container&quot;&gt;$R_x$&lt;/span&gt; with length &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and width &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. Finally we define that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; has area &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; iff ( both &lt;span class=&quot;math-container&quot;&gt;$R_y \precsim A$&lt;/span&gt; for any non-negative real &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$y &amp;lt; x$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$A \precsim R_z$&lt;/span&gt; for any non-negative real &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$x &amp;lt; z$&lt;/span&gt; ).&lt;/p&gt;&#xA;&lt;p&gt;(Remark: If we naively try to define that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; has area &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$R_x \precsim A \precsim R_x$&lt;/span&gt;, then it will non-trivially &lt;a href=&quot;https://en.wikipedia.org/wiki/Bolyai%E2%80%93Gerwien_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;work for polygons&lt;/a&gt; but will even more non-trivially &lt;a href=&quot;http://link.springer.com/article/10.1007/BF02759727&quot; rel=&quot;nofollow noreferrer&quot;&gt;fail for the circle&lt;/a&gt;! That is why we have to relax the condition via the notion of limits.)&lt;/p&gt;&#xA;&lt;p&gt;It is now easy to establish all the standard properties of area, except for one, namely that &lt;span class=&quot;math-container&quot;&gt;$R_x$&lt;/span&gt; has area &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$x=y$&lt;/span&gt;. This property is essentially the same as the fact that if two shapes have the same area and we remove a region of the same shape from each then the two resulting shapes still have the same area. Since many proofs of the Pythagoras theorem use this fact, there is thus a &lt;strong&gt;legitimate&lt;/strong&gt; reason to prefer a proof that does not use it.&lt;/p&gt;&#xA;&lt;p&gt;Once we have this last property, we can easily show that any region that has area has a unique area. To see why, take any region &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; that has areas &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$x &amp;lt; y$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$z = \frac12(x+y)$&lt;/span&gt;. Then trivially &lt;span class=&quot;math-container&quot;&gt;$R_t \precsim R_z$&lt;/span&gt; for any non-negative real &lt;span class=&quot;math-container&quot;&gt;$t &amp;lt; x$&lt;/span&gt;. Also &lt;span class=&quot;math-container&quot;&gt;$R_z \precsim A \precsim R_t$&lt;/span&gt; for any non-negative real &lt;span class=&quot;math-container&quot;&gt;$t &amp;gt; x$&lt;/span&gt;, the first inequality because &lt;span class=&quot;math-container&quot;&gt;$z &amp;lt; y$&lt;/span&gt;. Thus &lt;span class=&quot;math-container&quot;&gt;$R_z$&lt;/span&gt; has area &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and hence a contradiction.&lt;/p&gt;&#xA;&lt;p&gt;For completeness, here is a sketch of a proof of that last property. Take any measurable shape &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; that has area &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$x \le |A|$&lt;/span&gt;, by additivity and monotonicity of the Lebesgue measure since there is a finite partition of &lt;span class=&quot;math-container&quot;&gt;$R_x$&lt;/span&gt; into measurable regions that fit within &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; without any overlap. Hence &lt;span class=&quot;math-container&quot;&gt;$R_x$&lt;/span&gt; cannot have area &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$x &amp;lt; y$&lt;/span&gt;. Maybe this is cheating a bit, but I do not see an easier way.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="1550" LastEditDate="2021-12-24T06:43:01.840" LastActivityDate="2021-12-24T06:43:01.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11865" PostTypeId="2" ParentId="10751" CreationDate="2017-01-01T20:31:35.013" Score="14" Body="&lt;p&gt;I think an important aspect of this question -- one that I don't think has been mentioned yet in the other answers -- is the verb &quot;see&quot;, as in &quot;Should my 8th grader &lt;em&gt;see&lt;/em&gt; a proof&quot;.  While &quot;Explain a proof of the Pythagorean Theorem and its converse&quot; is indeed one of the Common Core Standards (and thanks to &lt;a href=&quot;https://matheducators.stackexchange.com/users/117/steven-gubkin&quot;&gt;Steven Gubkin&lt;/a&gt; for providing the link in &lt;a href=&quot;https://matheducators.stackexchange.com/a/10753/29&quot;&gt;his answer&lt;/a&gt;) it's important to notice that the Standards describe what &lt;em&gt;students should be able to do&lt;/em&gt;, not what students should &lt;em&gt;see the teacher do&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mention this because it is common for teachers (at all levels!) to think that if we &lt;em&gt;show&lt;/em&gt; student a proof, they now &lt;em&gt;know&lt;/em&gt; the proof. I think that is a fallacy and a dramatic oversimplification of how students actually learn the skills of proof and reasoning.  If you want your students to learn to prove the Pythagorean Theorem, you have to engage them in the activity of proving so that &lt;em&gt;they&lt;/em&gt; construct the argument themselves.  That requires scaffolding and careful planning that goes beyond the question of &quot;What proof do I show?&quot;&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-01-01T20:31:35.013" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11866" PostTypeId="1" CreationDate="2017-01-02T00:06:27.297" Score="3" ViewCount="281" Body="&lt;p&gt;I happen to be a student in America taking AP Calculus BC, or Calculus II, and recently, I had the following problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Determine whether the following integral converges and evaluate it if it does:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_0^\infty xe^{-x}\ dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And my response:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int_0^\infty x^se^{-x}\ dx$&lt;/span&gt; converges absolutely for &lt;span class=&quot;math-container&quot;&gt;$\text{Re}(s)&amp;gt;-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\int_0^\infty x^se^{-x}\ dx=\Gamma(s+1)\stackrel{s\in\mathbb N}=s!$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Reference: en.wikipedia.org/wiki/Gamma_function&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int_0^\infty xe^{-x}\ dx=1!=1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This question happened to show up around the section of integration by parts and improper integrals, and as I used neither, I only got one point on this question for the apparently correct answer.&lt;/p&gt;&#xA;&lt;p&gt;Of course, I knew this would most likely happen, but as it was, I wasn't particularly worried about getting it right or wrong since the teacher and I both know I am sufficient in my Calculus, but I was wondering what you guys though about this scenario.&lt;/p&gt;&#xA;&lt;p&gt;The question is basically that an integral like this is &amp;quot;well-known&amp;quot; to me, and though I know the rubric like the back of my hand, I wonder what stance something well-known has when looking at a problem, like whether or not introducing special functions far beyond the scope of the course could/should be reasonable.&lt;/p&gt;&#xA;&lt;p&gt;Indeed, as some more context, I am in the class only for credits and GPA and such, and I passed a mock final exam with flying colors.&lt;/p&gt;&#xA;&lt;p&gt;For example, I post answers &lt;a href=&quot;https://math.stackexchange.com/questions/2079801/evaluating-the-integral-int-0-infty-fracx3eax-1dx/2079808#2079808&quot;&gt;like this one&lt;/a&gt; all the time on MSE, so something like the above is fairly elementary for me.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-01-02T01:17:42.087" Title="How should I deal well-known versus the obvious rubric?" Tags="|calculus|course-design|grading|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="11867" PostTypeId="2" ParentId="11866" CreationDate="2017-01-02T00:25:29.377" Score="11" Body="&lt;p&gt;Quoted from the first part of my answer on &lt;a href=&quot;https://academia.stackexchange.com/questions/80898/should-a-student-be-penalized-for-using-a-theorem-outside-of-the-curriculum/&quot;&gt;https://academia.stackexchange.com/questions/80898/should-a-student-be-penalized-for-using-a-theorem-outside-of-the-curriculum/&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... the point of an exam is to assess mastery of basic knowledge&#xA;  covered in the course. If one uses a more powerful outside theorem,&#xA;  then the steps that they've skipped likely contained important&#xA;  concepts or techniques, that there is now no proof that the student&#xA;  has mastered. So the professor needs to ping the student to&#xA;  demonstrate those basic techniques before progressing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;More complete answer at the Academia SE link above. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="-1" LastEditDate="2017-04-13T12:49:57.670" LastActivityDate="2017-01-02T00:25:29.377" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11868" PostTypeId="1" CreationDate="2017-01-02T01:36:07.513" Score="28" ViewCount="1146" Body="&lt;p&gt;We tell undergraduate students that they should study two to three hours for every hour they spend in class. We know that many students don't follow through with this nearly to the degree that they should. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In response to this, faculty I know do everything from constructing elaborate daily homework exercises graded with various degrees of rigor, to simply giving students weekly quizzes and midterm exams...since if the students aren't going to take responsibility for the feedback given to them, there really is no point in sinking many, many extra hours grading homework. In fact, many mathematics faculty report very little gain from grading homework, which may be related to the sort of damping phenomenon found in certain answers to this helpful &lt;a href=&quot;https://matheducators.stackexchange.com/questions/712/is-it-worth-grading-calculus-homework&quot;&gt;question on grading homework&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, depending on the type of institution, the answers to my following question will differ. That said, there must be some research to help us guide our efforts to build a sense of responsibility and autonomy of study in our undergraduate students. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What is an appropriate amount of responsibility for an undergraduate student in mathematics courses, roughly classified by student year (Freshman, Sophomore, Junior and Senior). Please discuss in particular Freshman service courses like Calculus and Finite Mathematics. What I mean by responsibility is, essentially, how much and what sort of professor feedback foster optimal student performance? What, &lt;em&gt;according to mathematics education research&lt;/em&gt;, is an appropriate amount of responsibility to put on undergraduate students in mathematics courses?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The purpose of this question is, roughly, to address the difficulty that comes from giving either too much or too little feedback to one's students. Qualitatively, colleagues have noticed that when faculty over-organize student work, students continue to expect this and don't take on adequate responsibility for their mastery in later coursework, whereas too little guidance in early courses simply leads to disastrous outcomes since students seem to leave high school with very little sense of how to appropriately study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of this question is to provide some sort of standard guidelines for determining when it is best to &quot;remove training wheels&quot; etc. for undergraduate students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally try to use in &lt;a href=&quot;http://www.inquirybasedlearning.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inquiry Based Learning&lt;/a&gt; approach as early on as possible that puts a great deal of responsibility on the students, but this may not be best, scientifically. &lt;strong&gt;EDIT:&lt;/strong&gt; I should flesh this out a bit. Students seem to come to college without the idea that they both need to teach themselves AND we need to teach them. The IBL approach puts a ton of responsibility on students to create knowledge for themselves, but requires a lot of energy from the instructor to guide that knowledge creation. The general idea is that student homework is the content that drives an IBL course, and the professor evaluates this content production. This, in contrast to an instructor meticulously grading homework and passing it back in order for it to be glanced at and filed away or thrown in the bin. IBL puts enormous responsibility on the student, and a lot of work on the professor. I'm wondering how much of such responsibility is best, according to research.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2017-01-20T04:45:48.423" Title="The Undergraduate Responsibility Gradient" Tags="|undergraduate-education|mathematical-pedagogy|calculus|assessment|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="11869" PostTypeId="1" CreationDate="2017-01-02T08:35:35.797" Score="19" ViewCount="1493" Body="&lt;p&gt;I was watching the result of &lt;a href=&quot;https://www.imo-official.org/results.aspx?column=year&amp;amp;order=desc&amp;amp;language=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;IMO 2014&lt;/a&gt; where I found that China was on Top and USA, Russia were following China.  This is not the exceptional case, you can pick the results of each IMO in some of the previous years and you will find that these countries have ranked in that order from a long time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question is what makes these countries so great in this field&lt;/strong&gt; and &lt;strong&gt;What's the reason behind poor performance of globally known countries such as India, Brazil.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I shall be thankful if you guys compare the programs running for this purpose in all of the above countries and any advantage of being an IMO participant in high scoring and low scoring countries a participant have.&lt;/p&gt;&#xA;" OwnerUserId="7654" LastEditorUserId="7654" LastEditDate="2017-03-01T19:05:19.903" LastActivityDate="2017-03-01T19:05:19.903" Title="Reason behind the dominance of particular countries in the IMO (International Mathematical Olympiad)" Tags="|education-research|math-contests|cultural-differences|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="11872" PostTypeId="2" ParentId="11869" CreationDate="2017-01-03T02:12:52.407" Score="7" Body="&lt;p&gt;The book by Masha Gessen on Grigory Perelman has a&#xA;fascinating description of the Russian math camps,&#xA;specialized schools (e.g., Specialized Mathematics School Number 239 in Leningrad), and the instructors who&#xA;excel in training the future IMO candidates.&#xA;One such instructor was Valery Ryzhik (School 239), but the most&#xA;influential was Sergei Rukshin:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;...the care and teaching of Perelman became the thing that gave&#xA;  meaning to Rukshin's life.&quot; (p.29)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Only countries with such a focussed culture and institutions&#xA;surrounding the IMO and similar&#xA;competitions can excel year after year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/sW2SQ.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sW2SQ.jpg&quot; alt=&quot;Perelman&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-01-03T13:49:31.317" LastActivityDate="2017-01-03T13:49:31.317" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11873" PostTypeId="2" ParentId="1325" CreationDate="2017-01-03T10:09:43.967" Score="6" Body="&lt;p&gt;This fallacy is probably less well-known than others: &lt;em&gt;large samples always mean better confidence&lt;/em&gt;. This turns out to be false in the presence of even the slightest bias.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine an experiment to determine if a subject can read another's mind. The experimenter picks a card in a random (uniform) deck of blue and red cards, looks at it, and the subject guesses the color. Then one determines the probability to achieve at least the observed success rate if the subject had no supernatural power (i.e. guess chance $50\%$) and determines a $p$ value. Now, this experiment is subject to very small bias (the experimenter might lead the subject by slightly different reactions depending on the color, even unconsciously). If the raw guess chance is indeed $50\%$ but the bias improves it to $50.01\%$, then using a huge enough sample the observed guess rate will approach $50.01\%$ sufficiently to exclude the $50\%$ hypothesis with large confidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here the sample size needed would be really big (of the order of the hundreds of millions, since the spread decreases as the square root of the sample), but with more important bias even smaller sample will give the illusion of great confidence. But however small the bias is, and however strong the confidence interval is taken, there is a sample size above which the fallacy of large numbers will drop in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The take away is that &lt;strong&gt;when confronted with a study that uses a huge sample size and has a tiny $p$-value, one should be concerned about the effect size.&lt;/strong&gt; A small effect size might mean the $p$-value is driven by the bias and the large sample rather than by an intrinsic effect.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-01-03T10:09:43.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11874" PostTypeId="2" ParentId="11853" CreationDate="2017-01-03T10:30:17.400" Score="6" Body="&lt;p&gt;For students who mean to continue in mathematics after graduating, I would advise a combination of the old and modern approaches. The motivation is that it might be confusing later on to rely too strongly on the underlying Euclidean structure, but that you don't want your student to end you course without understanding why there is no perfectly accurate plan map of any part of the Earth or thinking all this is abstract nonsense. Differential geometry is both quite heavily formalized and deeply entrenched in physical reality, and we should convey both aspects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of combination would be to consider curves in the plane, surfaces in the space, curves on surfaces (with intrinsic and extrinsic curvatures), simple intrinsic surfaces (e.g. flat tori and Riemann sphere constructed by gluing two complex lines by $z\mapsto 1/z$, so as to make connection with complex analysis) and some very simple intrinsic $3$-manifolds (flat $3$-tori, round $3$-sphere constructed by stereographical projection taking inspiration from the $2$-dimensional one).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be honest, in France this could be done only in the very top institutions - even the basics of curves and surfaces are hopelessly impossible to teach to undergraduates in most if not all universities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, a reminder that I always feel compelled to make in such occasion: in the definition of a manifold, you &lt;strong&gt;do not want&lt;/strong&gt; to neglect the Hausdorff nor the paracompact assumptions. If you do, search about leaf spaces of foliations and about the long line and the wild world of non-metrizable surfaces. You'd be amazed. Back to the point, this means you may prefer to give a few example without general definition of a manifold in an undergraduate course.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-01-03T10:30:17.400" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11875" PostTypeId="2" ParentId="11836" CreationDate="2017-01-03T10:39:47.790" Score="1" Body="&lt;p&gt;To follow up Pere's answer and address your comment, you could work on some simple finite field or group in disguise, giving arbitrary names to elements. You take as undefined terms one or two operations (maybe denoted by $\square$ and and $\triangle$) and the objects (say $A,B,C$ if you work on $\mathbb{Z}/3\mathbb{Z}$), and as axioms the defining property of a group or field and part of the operation's tables. Then you seek to complete the tables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This might be a bit tedious, though. Maybe the most interesting case is with $4$ elements, so that you can ultimately relate this introduction with the fact that there are two (isomorphism classes of) groups with $4$ elements, but only one field.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-01-03T10:39:47.790" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11876" PostTypeId="2" ParentId="11785" CreationDate="2017-01-03T11:10:19.437" Score="2" Body="&lt;p&gt;First: &lt;em&gt;you should coordinate and discuss this with the teacher giving the lecture.&lt;/em&gt; You still have some freedom, but you should at least know what he or she expects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second: even if the precise students, level, and institution matter, I would broadly advise to mix &lt;strong&gt;1.&lt;/strong&gt; and &lt;strong&gt;2.&lt;/strong&gt; but with more emphasis on &lt;strong&gt;1.&lt;/strong&gt; in the earlier sessions, much more emphasis on &lt;strong&gt;2.&lt;/strong&gt; on the latter sessions, some pointers to learning material all along the way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third: make all you can to make very very clear that the prerequisite &lt;strong&gt;must&lt;/strong&gt; be worked out, that it is absolutely ok (for you) if they need to work &lt;strong&gt;a lot&lt;/strong&gt; for this, and that failing to master them as they should will give them an &lt;strong&gt;automatic fail&lt;/strong&gt;. This is hard, especially if you feel uncomfortable with the &quot;full metal jacket&quot; model of teaching. If you have a say on the final exam, you can introduce a punitive part on the prerequisites and warn students abundantly. Maybe punitive is not the best word, but to explain what I mean, here is what I did once in a Freshman course (I was in charge of the lectures and the exam, not the exercises sessions). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The exam started with 6 (very simple) multiple choice questions on the prerequisites. They gave a small amount of points, but also a &lt;em&gt;multiplicative&lt;/em&gt; coefficient to the grade, from &lt;em&gt;1.1&lt;/em&gt; for 6 correct answer down to &lt;strong&gt;zero&lt;/strong&gt; for 2 or less correct answers (no negative points for incorrect answers to avoid stress-motivated non-answers, but answering randomly would have given the coefficient 0 with large probability anyway). To limit cheating, I used AutoMultipleChoice to mix randomly the order of questions and answers (I had one pdf file of exam sheets consisting of more than 300 different versions - one for each student). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It turned out ok, but I didn't had the opportunity to run this on several years due to changes in my teaching. Also, one should make a dozen version only to be able to check them thoroughly: my pdf has been ill-printed and some part went missing in about on sheet out of ten, complicating things to an absurd point.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-01-03T11:10:19.437" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11877" PostTypeId="1" CreationDate="2017-01-03T16:55:57.553" Score="4" ViewCount="86" Body="&lt;p&gt;I'll be teaching a U.S. college, freshman-level, pre-proofs discrete math course (for future teachers) this semester and one of the topics I like to cover is determining if a system of (usually linear) Diophantine equations has a solution via modular reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example: Consider the system: $$3x+5y=7$$ $$9x-7y=12$$&#xA;A quick consideration modulo 2 (or thinking about odds and evens) shows that this does not have solutions. Some examples I use in class require other moduli. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem I have is that I cannot find a good exposition (print, video, or web) of this topic at the students' level. Everything I find is mired in a more advanced exposition on linear Diophantine equations suitable for a proof-based Number Theory course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone else cover this type of reasoning with this level of student? Do you know of any (print, video, or web) materials that might be helpful to share with my students?&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2017-01-05T20:32:12.010" LastActivityDate="2017-01-05T20:32:12.010" Title="Solvability of Systems of Diophantine Equations using Modular Reasoning" Tags="|discrete-math|resource-request|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="11878" PostTypeId="1" CreationDate="2017-01-03T20:14:51.053" Score="5" ViewCount="97" Body="&lt;p&gt;I am teaching a &quot;computation&quot; course, where we get students to explore mathematics and statistics via a high-level programming language.   So the course is less about mathematics proper, and more about computation, and the process of learning about mathematics.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for good examples of the form where the main ideas behind the techniques are easy to motivate, and yet the technique (aided with a computer) produces otherwise difficult-to-obtain results.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been hoping to find a good example from the MCMC family of ideas.  Perhaps it's because I'm not terribly familiar with this branch of stats myself that I haven't developed good basic examples.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps this is a stretch, but in these family of ideas would there be a good, simple way of estimating the volume of an icosahedron?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't need examples quite that sophisticated.  But I would like the examples to be intuitive-enough to stand on their own, but flexible enough so the student could have some idea of how to apply the technique to a wide variety of problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These students know linear algebra and multi-variable calculus, and a little bit about programming languages. &lt;/p&gt;&#xA;" OwnerUserId="202" LastActivityDate="2017-01-05T10:56:35.173" Title="Looking for examples from the MCMC family of ideas" Tags="|technology-in-education|statistics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11879" PostTypeId="2" ParentId="11878" CreationDate="2017-01-04T01:58:43.740" Score="6" Body="&lt;p&gt;Perhaps the &quot;Metropolis Ball-Walk&quot; algorithm for computing the volume of a polyhedron might be a good example?&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/DqVRV.png&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DqVRV.png&quot; alt=&quot;BallWalk&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;I found two sets of lecture notes on the topic, neither of which may be ideal, but...&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Alistair Sinclair (2009): &lt;a href=&quot;https://people.eecs.berkeley.edu/~sinclair/cs294/n16.pdf&quot;&gt;PDF download&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Shuchi Chawla (2007): &lt;a href=&quot;http://pages.cs.wisc.edu/~shuchi/courses/880-S07/scribe-notes/lecture26-1.pdf&quot;&gt;PDF download&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The snapshot above is from Sinclair's notes.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-01-04T01:58:43.740" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11880" PostTypeId="1" CreationDate="2017-01-04T03:20:43.720" Score="3" ViewCount="443" Body="&lt;p&gt;We'd like to create  a course that covers beginning and intermediate algebra in one course (8 hours a week), getting students ready to succeed in precalc and calculus. I am at a community college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like textbook recommendations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not 100% sure yet of the topics. I think we'll do a tiny bit of trig, because that's the hardest thing in precalc. Something like: &#xA;1 - Equations&#xA;2 - Graphing&#xA;3 - Multiplying and Factoring, finding zeros&#xA;4 - Rational Expressions&#xA;5 - Roots and Quadratics (also absolute value functions)&#xA;6 - Exponential Functions and Logs&#xA;7 - Geometry and Trig&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2017-01-04T21:59:40.540" LastActivityDate="2017-01-04T21:59:40.540" Title="Good Textbook for combined Beginning and Intermediate Algebra Course" Tags="|undergraduate-education|algebra|textbooks|community-colleges|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11881" PostTypeId="2" ParentId="7660" CreationDate="2017-01-04T03:53:00.407" Score="2" Body="&lt;p&gt;I've been put in the same situation as you.  I wound up &quot;hijacking&quot; the course and teaching it along the lines of the Millman and Parker intro to DG book.  It went really well -- perhaps the highest teaching evaluations I've ever received.  I'm not sure if that means much, though!  The class had a population of about 20 or 22, and there were 2 or 3 education students in the course.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I viewed the course as something of an opportunity to cement the students' understanding of calculus and linear algebra.  For those that did not know linear algebra, it was a &quot;purposeful&quot; first pass. &lt;/p&gt;&#xA;" OwnerUserId="202" LastActivityDate="2017-01-04T03:53:00.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11882" PostTypeId="2" ParentId="7660" CreationDate="2017-01-04T04:19:54.423" Score="6" Body="&lt;p&gt;I think you may be looking for &lt;strong&gt;Geometry from a Differentiable Viewpoint&lt;/strong&gt; 2nd Edition by John McCleary. I'm not sure it's the right level for your course, but, here is the advertisement:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The development of geometry from Euclid to Euler to Lobachevsky, Bolyai, Gauss, and Riemann is a story that is often broken into parts - axiomatic geometry, non-Euclidean geometry, and differential geometry. This poses a problem for undergraduates: Which part is geometry? What is the big picture to which these parts belong? In this introduction to differential geometry, the parts are united with all of their interrelations, motivated by the history of the parallel postulate. Beginning with the ancient sources, the author first explores synthetic methods in Euclidean and non-Euclidean geometry and then introduces differential geometry in its classical formulation, leading to the modern formulation on manifolds such as space-time. The presentation is enlivened by historical diversions such as Hugyens's clock and the mathematics of cartography. The intertwined approaches will help undergraduates understand the role of elementary ideas in the more general, differential setting. This thoroughly revised second edition includes numerous new exercises and a new solution key. New topics include Clairaut's relation for geodesics, Euclid's geometry of space, further properties of cycloids and map projections, and the use of transformations such as the reflections of the Beltrami disk.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If this is not what you want, it at least should give some ideas. It seems to have both axiomatic geometry and the basic differential geometry in a somewhat classical fashion.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-01-04T04:19:54.423" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11883" PostTypeId="2" ParentId="11880" CreationDate="2017-01-04T07:31:39.960" Score="4" Body="&lt;p&gt;I've recently switched to using OpenStax open education resources in my college algebra courses. That is, they are: Free of cost, free to redistribute, free to edit if desired. For many years I thought the quality of OER materials was unacceptably low, but in the last year or two they've crossed the threshold of usability for me. They're digital in format (web view, PDF, and other options) so I can carry them on a tablet, and also throw them on an overhead projector when desired. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://openstax.org/details/algebra-and-trigonometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax Algebra and Trigonometry&lt;/a&gt; has what you describe. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If not that, then I've also liked Sullivan's &lt;em&gt;Algebra and Trigonometry&lt;/em&gt;, but it's a massive tome (1034 pages plus answer section and index in the 7th edition). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-01-04T07:31:39.960" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11884" PostTypeId="1" CreationDate="2017-01-04T20:27:31.693" Score="7" ViewCount="3710" Body="&lt;p&gt;I was having a discussion with a colleague a few weeks ago, and she opted to use the term &quot;Summation Notation&quot; whereas I had a preference for &quot;Sigma Notation&quot;. We didn't have the time to tangentially discuss the differences, so I put it on the back burner. After revisiting it, I've done some cursory research, and there doesn't appear to be a preferred or agreed upon name. To me, &quot;Summation Notation&quot; is cumbersome sounding, so I've always gone with the former, but this could just be because I learned it as such.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is there a general consensus amongst mathematicians which term to use?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2017-01-05T10:57:50.767" Title="&quot;Sigma Notation&quot; vs. &quot;Summation Notation&quot;" Tags="|vocabulary|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11886" PostTypeId="2" ParentId="11884" CreationDate="2017-01-04T21:22:02.787" Score="8" Body="&lt;p&gt;Probably there is no &lt;em&gt;consensus&lt;/em&gt;... as usual.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there is a small argument in favor of calling it &quot;summation notation&quot;, even if the character is literally an upper-case sigma, since the name &quot;summation...&quot; suggests its function, as opposed to merely its appearance. For that matter, any instance of it is literally a sum (even if infinite), so it makes sense to refer to &quot;the sum&quot; rather than the symbols or characters (or font!) that denote the sum.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-01-04T21:22:02.787" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11887" PostTypeId="2" ParentId="11868" CreationDate="2017-01-05T07:35:11.990" Score="4" Body="&lt;p&gt;(I thought of answering your question, but then I realized that it was more of a comment. As I began typing, it turned into be too big to fit into the comment section, so take this as a grain of salt.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My first thought was that this needs to be solved piecewise. STEM majors are very different from Liberal Arts majors; students at four year Universities are different from students working on an Associate's degree. However, quantifying your question with the phrase &quot;Freshman service courses like Calculus...&quot; probably means you are interested in those on the STEM track, so I'm assuming that's what you're interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, my only experience with the types of courses I think you're interested in is as a student, so I'm unqualified to answer your question. However, to aid anyone who is looking to answer, early Google Scholar searching yields one relevant book on Student Responsibility by Davis and Murrell (available for free download &lt;a href=&quot;https://eric.ed.gov/?id=ED372703&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). Though it is not mathematics specific, it might be a helpful starting point.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2017-01-05T07:35:11.990" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11888" PostTypeId="2" ParentId="11868" CreationDate="2017-01-05T08:31:10.643" Score="5" Body="&lt;p&gt;&lt;strong&gt;Gradient Limitations:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the United States, scientists and engineers are trained in how to handle long-term projects as part of their technical writing classes, laboratory classes, design classes, and thesis-writing classes.  They start with writing 1-5 page papers, with executive summaries.  They work up to 20 page laboratory write-ups, design documents, and business plans.  By their senior year of college, they write 30 - 50 page theses.  Master's theses and Ph.D. theses are much longer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their math classes are supposed to provide a tool-kit, and lots of practice using the tool-kit.  Their math classes are not expected to teach them how to break down projects that take longer than an hour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My Experience:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have taken math classes (and math-heavy classes) in a wide range of formats.  These formats included independent study, large lectures with recitation sections, small seminars, and individual tutoring.  I have taken classes in public schools, community colleges, MIT departments, and MIT's Experimental Study Group.  Fortunately, I did not take any math classes from MIT's Math Department, though I knew many students who did.  I personally have performed and published an applied math project that took several months -- but this project was not part of my undergraduate education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Effective Freshman/Sophomore Format for Math Classes for Science and Engineering Students&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my undergraduate math courses (including Calculus through surface and volume integrals, Linear Algebra, Differential Equations, and Introductory Statistics), the most effective courses had the following expectations:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Classes met two or three times per week.&lt;/li&gt;&#xA;&lt;li&gt;Classes were small enough that large colleges might call them &quot;recitation sections&quot; or &quot;seminars&quot;.&lt;/li&gt;&#xA;&lt;li&gt;Students doing passingly well were expected to spend about twice as much time on preparation as in class.&lt;/li&gt;&#xA;&lt;li&gt;Students were expected to attend all or nearly all classes.  (In California's community college system, this was the basis for the college's funding.)  &lt;/li&gt;&#xA;&lt;li&gt;Homework was assigned after each class, except there was not homework immediately before mid-terms.&lt;/li&gt;&#xA;&lt;li&gt;Students were expected to perform a Check-By-Substitution at the end of each problem, to self-check whether they had gotten the answer correct.&lt;/li&gt;&#xA;&lt;li&gt;Students could look up answers to their homework problems in the back of the textbook.&lt;/li&gt;&#xA;&lt;li&gt;Students were expected to read the material for each class session ahead of time.&lt;/li&gt;&#xA;&lt;li&gt;At the beginning of each class (other than mid-term sessions), students were expected to say which problems they had trouble with.&lt;/li&gt;&#xA;&lt;li&gt;The first third of each class was dedicated to going through the homework problems that caused the most trouble for students, or the exam questions from the previous session that caused the most trouble for students.&lt;/li&gt;&#xA;&lt;li&gt;The remainder of each class was used to cover that session's topic.&lt;/li&gt;&#xA;&lt;li&gt;Mid-terms every couple of weeks, on coherent chunks of the course.  Mid-terms were held during class time.  Mid-term questions were similar to homework questions.&lt;/li&gt;&#xA;&lt;li&gt;Different courses had different final exam formats.  Merced College's math instructors used 50-question multiple choice questions graded by Scan-Tron.  MIT's courses often had original questions that would be used as future homework questions.  Both formats worked well.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Homework / Exam Summary:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Homework is for the benefit of the student.  (And as a real-time feedback mechanism for the instructor.)&lt;/li&gt;&#xA;&lt;li&gt;Students should check their own homework.&lt;/li&gt;&#xA;&lt;li&gt;Homework difficulties provide &quot;teachable moments&quot;.&lt;/li&gt;&#xA;&lt;li&gt;Students should be learning from their textbook, in class, and while doing homework.  Students should not be cramming for mid-terms.&lt;/li&gt;&#xA;&lt;li&gt;Whether students have done their homework should be obvious to their classmates; failure to learn from homework will result in doing badly on mid-terms.&lt;/li&gt;&#xA;&lt;li&gt;Frequent mid-terms can provide the bulk of the grading.&lt;/li&gt;&#xA;&lt;li&gt;Final exams are a sanity check that the material was solidly learned earlier in the course.  Students should not be cramming for finals.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2017-01-20T04:45:48.423" LastActivityDate="2017-01-20T04:45:48.423" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11889" PostTypeId="2" ParentId="11884" CreationDate="2017-01-05T08:52:02.300" Score="4" Body="&lt;p&gt;I do not think that you could speak of a consensus there. People will probably use the term they prefer. In fact, Columbia University, for example, uses both terms &lt;a href=&quot;http://www.columbia.edu/itc/sipa/math/summation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; (my emphasis):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Often mathematical formulae require the addition of many variables &lt;strong&gt;Summation or sigma notation&lt;/strong&gt; is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, if we have a look at some numbers, the term &quot;summation notation&quot; appears to be more frequent. A full text search of &lt;a href=&quot;https://arxiv.org/archive/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv&lt;/a&gt; for &quot;summation notation&quot; returns 200 entries while the same search for &quot;sigma notation&quot; returns only a number of 109 articles.&#xA;Other search enginges return similar results. For example, &lt;a href=&quot;http://projecteuclid.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Project Euclid&lt;/a&gt; lists 36 results for &quot;summation notation&quot; but only 23 for &quot;sigma notation&quot;. Last but not least &lt;a href=&quot;https://scholar.google.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Scholar&lt;/a&gt; finds 4750 results containing &quot;Summation notation&quot; while only returning 1310 results containing &quot;Sigma notation&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, &quot;Summation notation&quot; is the term that is used more frequently.&lt;/p&gt;&#xA;" OwnerUserId="7608" LastActivityDate="2017-01-05T08:52:02.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11890" PostTypeId="2" ParentId="11878" CreationDate="2017-01-05T10:56:35.173" Score="4" Body="&lt;p&gt;To sample a point uniformly inside a convex set (e.g. to evaluate an integral, for example if you want to know the average distance from the center in your icosahedron), to my knowledge the most efficient algorithm is the &quot;hit-and-run&quot;: from a point in the convex, you take a uniform random direction (easily constructed from a radially symmetric Gaussian random vector), draw the maximal segment in the convex generated by the current location and the drawn direction, a jump to a uniform random point on that segment. Then you iterate.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-01-05T10:56:35.173" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11891" PostTypeId="2" ParentId="11884" CreationDate="2017-01-05T10:57:50.767" Score="3" Body="&lt;p&gt;I'd argue for not using either of the two, but if I have to chose I'd much prefer summation notation.&lt;/p&gt;&#xA;&lt;p&gt;The symbol &lt;span class=&quot;math-container&quot;&gt;$\sum$&lt;/span&gt; is the &lt;strong&gt;summation symbol.&lt;/strong&gt;  While it is derived from a capital sigma,  usually one does not even literally use a capital sigma to typeset it. Observe &lt;span class=&quot;math-container&quot;&gt;$\sum_{i=1}^n i^2$&lt;/span&gt; versus &lt;span class=&quot;math-container&quot;&gt;$\Sigma_{i=1}^n i^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;One can then denote a sum using the summation symbol. One might say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Express the sum of the first 230 odd natural numbers using the summation symbol.&lt;/p&gt;&#xA;&lt;p&gt;Rewrite &lt;span class=&quot;math-container&quot;&gt;$1 -2 + 3 -4 + \dots +99-100$&lt;/span&gt; using the summation symbol.&lt;/p&gt;&#xA;&lt;p&gt;In this lecture we will recall how to use the summation symbol to express sums in a convenient form.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And if one says some formula like &lt;span class=&quot;math-container&quot;&gt;$\sum_{i=1}^{20}$&lt;/span&gt; I would argue for &amp;quot;sum from one to twenty&amp;quot; and not &amp;quot;sigma from one to twenty.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I do not really see the need for a more dedicated term. But perhaps I am missing something.&lt;/p&gt;&#xA;&lt;p&gt;If one does it like this, one then has in analogy &lt;span class=&quot;math-container&quot;&gt;$\prod$&lt;/span&gt; the product symbol, derived from a capital pi &lt;span class=&quot;math-container&quot;&gt;$\Pi$&lt;/span&gt;. If one has used summation notation, one can call this  product notation. Yet,  if one has called it sigma notation one would call it pi notation? I find this not optimal.&lt;/p&gt;&#xA;&lt;p&gt;There are various related symbols and notations, and I find it of considerable advantage to insist on the meaning rather than the appearance or even etymology of the symbol when referring to it.&lt;/p&gt;&#xA;&lt;p&gt;Finally, while it is a arguably a bit farther away, consider the fact  that the integral symbol is also derived from a letter, namely a long s, but we do not talk about the long s notation.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-01-05T10:57:50.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11892" PostTypeId="2" ParentId="11868" CreationDate="2017-01-05T15:56:05.633" Score="7" Body="&lt;p&gt;I think that your point here is spot on: &quot;Qualitatively, colleagues have noticed that when faculty over-organize student work, students continue to expect this and don't take on adequate responsibility for their mastery in later coursework, whereas too little guidance in early courses simply leads to disastrous outcomes since students seem to leave high school with very little sense of how to appropriately study.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach high school mathematics, and I adjunct for lower level math classes at a Junior College.  It is my opinion that as soon as students enter the collegiate level, they are primarily responsible for their learning.  We must maintain rigor, or a college degree will become essentially worthless.  Students should be held to a high level of accountability.  I do not think that professors should ponder this question because we are not here to coddle students.  Every professor is different in the &quot;load&quot; they give, but our primary responsibility is to teach our subject matter at the collegiate level.  Learning at this level is totally dependent on the student.  I do not hold this same view in teaching high schoolers, but college is a different game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;This would serve better as a comment.  I understand I've not referenced any mathematics education research, but the comment box would not hold this response.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2017-01-05T15:56:05.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11893" PostTypeId="2" ParentId="11868" CreationDate="2017-01-05T20:59:30.397" Score="12" Body="&lt;p&gt;Here's another too-long comment posted as an answer:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We tell undergraduate students that they should study two to three&#xA;  hours for every hour they spend in class. We know that many students&#xA;  don't follow through with this nearly to the degree that they should.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've become pretty unconvinced that raw amount of study time is a significant factor to mastery of mathematical material. Frankly, some students just &quot;get it&quot; on the first presentation in class, while others do not. If the mean amount of study time is 2-3 hours per week, that could be true; but the variation by student is enormous, between 0 and 6 hours per credit being likely (and sharply right-skewed). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I keep a rather sick level of data on all my courses, including pre-course diagnostics and mid-semester study surveys of my design (none of it published or IRB'd). The self-reported average in my college algebra courses is about 1 hour per week per credit, in statistics about 2 hours per week per credit (N ~ 100 and 200). A few students have said they're studying 20 hours per week, and one reported 30 at one point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the correlation of study time with final results (score on final exam, and quality points from letter grade in the course) is effectively zero in all of my courses. I don't think that's a novel observation, because there's even an exercise in the statistics book I use that expresses that. On the other hand, Likert responses to &quot;Agree that you've done all the homework&quot; are significantly correlated with final results (R^2 ~ 0.15 in college algebra for me). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I'm wary of berating students that they should be studying a particular amount of time each week; the variance is so high that 68% of the class may think I'm full of BS because it obviously takes much less, while 7% thinks I'm totally unrealistic because it obviously takes much more (numbers pulled from my statistics class self-responses on study time). And I don't see any way to fix that right-skewed distribution of study time without creating distinct assignments for every individual which (without any graders or TA's at my institution) seems logistically and grading-wise infeasible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think I even knew this for myself as an undergraduate/early graduate student; if I just paid rapt attention during a lecture, and avoided taking notes, I could basically digest the material in one go. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-01-06T02:12:04.793" LastActivityDate="2017-01-06T02:12:04.793" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11894" PostTypeId="1" CreationDate="2017-01-05T23:49:07.727" Score="11" ViewCount="202" Body="&lt;p&gt;I wanted to ask the following on Mathematics stack exchange, but checking the help centre, it appears that this site is more relevant: &quot;&lt;strong&gt;Mathematical education&lt;/strong&gt;, especially about teaching and pedagogy: Mathematics Educators&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Throughout my undergraduate degree, it felt that for any course that I would need to study, I could find a textbook that covered all of the content needed and more - with this in mind, I would simply linearly work through the textbooks, and whenever I didn't understand something from one, I would move to another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that I am in graduate school, this model of education does not seem to work. It seems that any textbook I find on more advanced topics only barely will touch on the thing I am interested, and will otherwise be largely off topic. Is a student meant to transition into a new state of study, where 1) I simply read a few paragraphs from this textbook, and a few from another, and so on?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2)Is it better that I give up on textbooks and simply look through the literature for the original papers? It seems that doing so would decrease the breadth of my study, but perhaps this is necessary. &lt;/p&gt;&#xA;" OwnerUserId="2019" LastActivityDate="2017-01-06T19:42:43.733" Title="Transitioning from 'Read textbook to learn' to 'Read original paper/Read paragraphs of textbooks'" Tags="|textbooks|graduate-education|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11895" PostTypeId="2" ParentId="11894" CreationDate="2017-01-06T02:42:10.617" Score="3" Body="&lt;p&gt;The answers largely depend on which areas of math you're asking about and how close to the frontiers of research they are, but here's what I'd say in general.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1) I simply read a few paragraphs from this textbook, and a few from another, and so on?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If that's what works best for you, then by all means do it.  In graduate school, especially after the first year or two in a PhD program, more of a focus is placed on independent learning, so it's really up to you how you do that.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;2)Is it better that I give up on textbooks and simply look through the literature for the original papers?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Not necessarily.  I would say learn from as many different sources as possible.  Original papers are better if you're learning something closer to the frontiers of research.  But if you're talking about, say, a graduate-level introduction to numerical analysis, then original papers may be overkill.  Not that overkill is necessarily a bad thing in this context, but at the early stages of a graduate program you'll likely be more focused on passing qualifying exams, etc.&lt;/p&gt;&#xA;" OwnerDisplayName="user6648" LastActivityDate="2017-01-06T02:42:10.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11896" PostTypeId="1" AcceptedAnswerId="11897" CreationDate="2017-01-06T14:04:07.907" Score="4" ViewCount="235" Body="&lt;p&gt;The following are all logical equivalences&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$p \wedge q \Rightarrow r \;=\; p \Rightarrow (q \Rightarrow r)$&lt;/li&gt;&#xA;&lt;li&gt;$p \Rightarrow q \wedge r \;=\; (p \Rightarrow q) \wedge (p \Rightarrow r)$&lt;/li&gt;&#xA;&lt;li&gt;$p \vee q \Rightarrow r \;=\; (p \Rightarrow r) \wedge (q \Rightarrow r)$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Do they individually have names?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are corresponding laws of exponentiation. As far as I know they don't have individual names.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$r^{pq} \;=\; (r^p)^q$&lt;/li&gt;&#xA;&lt;li&gt;$(qr)^p \;=\; q^p \, r^p$&lt;/li&gt;&#xA;&lt;li&gt;$r^{(p+q)} \;=\; r^p \, r^q$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="527" LastActivityDate="2017-01-08T16:51:07.167" Title="Names for laws involving implication and/or exponentiation" Tags="|logic|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11897" PostTypeId="2" ParentId="11896" CreationDate="2017-01-06T15:09:58.263" Score="2" Body="&lt;p&gt;For exponentiation, we might say that&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;$(qr)^p \;=\; q^p \, r^p$&lt;/li&gt;&#xA;&lt;li&gt;$r^{(p+q)} \;=\; r^p \, r^q$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;are &lt;strong&gt;distributive laws&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And (perhaps) that&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$r^{pq} \;=\; (r^p)^q$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;is an &lt;strong&gt;associative law&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-01-06T15:09:58.263" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11899" PostTypeId="2" ParentId="11894" CreationDate="2017-01-06T19:42:43.733" Score="2" Body="&lt;p&gt;I'd say that it is not &quot;linearly&quot; vs.  &quot;nonlinearly&quot;  or &quot;whole book&quot; vs. &quot;paragraphs&quot; but rather you should change your focus when starting to read. When you transition to research, you should not start reading to learn some piece of theory, but to solve a problem or to understand a certain property. So the motivation should be different, and this usually leads to a different reading style. Also note that there are books that are not really meant to read cover to cover. &lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2017-01-06T19:42:43.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11900" PostTypeId="2" ParentId="11896" CreationDate="2017-01-07T04:29:43.303" Score="4" Body="&lt;p&gt;The names for your exponentiation laws are pretty standardized in elementary algebra texts; for example, from Martin-Gay, &lt;em&gt;Prealgebra &amp;amp; Introductory Algebra&lt;/em&gt;, Sec 10.1 (ordering as per the book):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Product Rule (for Exponents): $a^m \cdot a^n = a^{m+n}$&lt;/li&gt;&#xA;&lt;li&gt;Power Rule (for Exponents): $(a^m)^n = a^{mn}$&lt;/li&gt;&#xA;&lt;li&gt;Power of a Product Rule: $(ab)^n = a^nb^n$ &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Looking at other sources: Bittinger's &lt;em&gt;Intermediate Algebra&lt;/em&gt; (Sec. R.7): (1) Product Rule, (2) Power Rule, (3) Raising a Product to a Power. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ratti &amp;amp; McWaters, &lt;em&gt;Precalculus: A Right Triangle Approach&lt;/em&gt; (Sec. P.2): (1) Product Rule of Exponents, (2) Power Rule for Exponents, (3) Power-of-a-Product Rule. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, it's common to call standard distribution of real numbers $a(b+c) = ab + ac$ by the fully-formed name of the &quot;Distributive Property of Multiplication Over Addition&quot; (e.g., Martin-Gay does this); so I think it's also fair to call (3) the &quot;Distribution Property of Exponents Over Multiplication&quot;, and I've seen something like that in places. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my college algebra classes, I consolidate these rules into what I call the &quot;General Distribution Rule&quot;: operations distribute over any operation one line lower in the order-of-operations. But that's a &lt;a href=&quot;http://www.madmath.com/2012/07/power-rules.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;phrasing of my own invention&lt;/a&gt;, as far as I know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Looking at a significantly older text, Rietz and Crathorne, &lt;em&gt;Introductory College Algebra&lt;/em&gt; (1933), Ch. 1, they say of $a^ma^n = a^{m+n}$, &quot;This is often called the first law of exponents&quot;. Names for the other rules are not clearly given. I doubt this is common usage today; but for what it's worth, the &lt;em&gt;ordering&lt;/em&gt; in all of these books referenced is identical, with the Product Rule indeed always given first.)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-01-08T16:51:07.167" LastActivityDate="2017-01-08T16:51:07.167" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="11901" PostTypeId="1" AcceptedAnswerId="11903" CreationDate="2017-01-07T17:56:30.007" Score="9" ViewCount="750" Body="&lt;p&gt;I'm an &lt;a href=&quot;https://math.stackexchange.com/users/88078/nero&quot;&gt;nntaleb&lt;/a&gt; fan so I'm glad I learned about the Mandelbrot set, but I notice that said topics are not in &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0073383171&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brown-Churchill&lt;/a&gt; or '&lt;a href=&quot;http://math.sfsu.edu/beck/complex.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;A First Course in&#xA;Complex Analysis&lt;/a&gt;' while they are in Coursera's '&lt;a href=&quot;https://www.coursera.org/learn/complex-analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Complex Analysis&lt;/a&gt;'.&lt;/p&gt;&#xA;&lt;p&gt;If so:&lt;/p&gt;&#xA;&lt;p&gt;Why? Why are they not in those books?&lt;/p&gt;&#xA;&lt;p&gt;If not:&lt;/p&gt;&#xA;&lt;p&gt;Why? How could they be useful in some introductory complex analysis classes such as the one in Coursera?&lt;/p&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2022-02-28T07:11:40.410" LastActivityDate="2022-02-28T07:11:40.410" Title="Are the following topics usually in an introductory Complex Analysis class: Julia sets, Fatou sets, Mandelbrot set, etc?" Tags="|undergraduate-education|mathematical-analysis|complex-numbers|fractals|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11902" PostTypeId="2" ParentId="11901" CreationDate="2017-01-07T19:17:53.123" Score="13" Body="&lt;p&gt;A brief stab at an answer before someone more knowledgeable comes along: My brief experience says that: no, fractal sets are not usually a topic in complex analysis (they're not in any of the 3 textbooks that I'm familiar with). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We might take a guess and say that they're in the Coursera MOOC because it provides a fluffy, visual, easy-to-promote introduction to their mass-market class. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-01-07T23:04:00.433" LastActivityDate="2017-01-07T23:04:00.433" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11903" PostTypeId="2" ParentId="11901" CreationDate="2017-01-07T20:51:05.327" Score="17" Body="&lt;p&gt;No, these topics are not usually included in courses on complex analysis, for several reasons, which I will explain below. At the same time, it is easier to understand why relatively old textbooks did not include such topics: things about iteration of mappings in the complex domain is a much younger topic (as old as it is by this year) than most of the rest of what is called &quot;complex analysis&quot;, by almost 100 years. Thus, textbooks either created 60 years ago, or emulating such, would not have mentioned such things... and still would be overflowing with important ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This comes to a less artifactual point: the &quot;traditional&quot; topics in complex analysis are highly useful across much of mathematics and other parts of science, which is why complex analysis is considered one of the pillars of the standard curriculum, especially for graduate students in math, but also for physicists and many engineers. The dynamics of iterated maps is interesting, but less universally useful, certainly at a basic level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen a few attempts to include the dynamics of iterated maps in otherwise &quot;standard, required&quot; graduate complex analysis courses. There are some conflicts in this scenario, although these are partly artifactual, and not an argument against trying something &lt;em&gt;similar&lt;/em&gt;. Namely, if the standard &quot;required&quot; course is to prepare grad students for prelims, and the latter don't include things about iterated maps, probably something that &lt;em&gt;is&lt;/em&gt; included was omitted, and this disserves the students. Yet, arguably, the stuff is interesting, and lends itself to actual contemporary research far more than the old-timey (but incredibly practical-useful) complex analysis. Really, I'd argue that these are &quot;different subjects&quot;, in the same way that basic X is typically utterly different from research-in-distant-descendent-of-X. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a related vein, it is not easy to arrange meaningful &quot;tests&quot; on iterated maps, any more than it is easy to arrange appropriate &quot;tests&quot; on any kind of still-alive mathematics. This is not a good reason to not teach such things, but it does complicate matters in the face of the inertia of structural expectations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, as @DanielRCollins speculates, situations in which the goals are popularization and promotion (for better or for worse), colorful graphics are obviously a winner. Cool pictures that nowadays can be easily programmed or at least viewed naturally achieve broader popularity than proving theorems and such.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-01-07T20:51:05.327" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11904" PostTypeId="2" ParentId="11901" CreationDate="2017-01-08T08:46:56.877" Score="6" Body="&lt;p&gt;It seems to me there is a fundamental disconnect between what one might call &quot;traditional complex analysis&quot; and the study of fractals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Traditional complex analysis, as used in physics and engineering, is primarily about analytic and meromorphic functions, whose most important property is that they are infinitely differentiable except at a set of isolated points (their poles).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand fractals are essentially continuous but non-differentiable functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, &quot;fractals which happen to be easy to define using complex numbers&quot; are a subset of fractals in general. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, transplanting an &quot;introduction to fractals&quot; into a different branch of math doesn't seem a very useful idea. I don't see anything wrong with the notion that dynamical systems, fractals, chaos theory, etc could and/or should be added to the &quot;core applied math&quot; curriculum, but they deserve to form a separate topic in their own right, and/or linked to practical applications such as wavelet transforms in signal processing, the general behaviour of nonlinear differential equations, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as others have said, pretty pictures do sometimes generate motivation!&lt;/p&gt;&#xA;" OwnerUserId="7677" LastActivityDate="2017-01-08T08:46:56.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11905" PostTypeId="2" ParentId="11779" CreationDate="2017-01-10T02:24:08.227" Score="3" Body="&lt;p&gt;From&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/QK4dR.png&quot; alt=&quot;Mathematica graphics&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_0^x \sqrt{a^2-t^2}\;dt = {1 \over 2}\,x\,\sqrt{a^2-x^2} + {1\over2}\,a^2 \arcsin\left({x \over a}\right)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Exercise:&lt;/em&gt;&#xA;Show that the derivative of the right-hand side is $\sqrt{a^2-x^2}$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Similarly, from&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/e8XuC.png&quot; alt=&quot;Mathematica graphics&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_0^t \arcsin x \; dx = t\,a - \int_0^a \sin y \; dy$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $a = \arcsin t$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This one may be best kept for when the students know how to integrate $\sin y$.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2017-01-10T02:24:08.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11907" PostTypeId="2" ParentId="224" CreationDate="2017-01-11T14:36:01.467" Score="2" Body="&lt;p&gt;I think Coq is not the perfect tool to teach &quot;introduction to proof&quot; courses at entrance of university. Using Coq, impose to learn the syntax and tactics which is too demanding for introduction to proof course. For a course about logic, where you have time to focus on the formal rules of let's say natural deduction, then it is possible. I have done it with second year student. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the &quot;introduction to proof&quot;, I think the Edukera.com system (which is based on Coq) is more adapted, as it has a purely point and click user interface, it does not require to learn a syntax. I tried it with first year and fourth year students both could do their first proofs few minutes after starting. The drawback compared to Coq is that the teacher can not build his own exercices, and the student can not give his own definitions.&lt;/p&gt;&#xA;" OwnerUserId="7685" LastEditorUserId="7685" LastEditDate="2017-01-25T09:37:15.890" LastActivityDate="2017-01-25T09:37:15.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11908" PostTypeId="2" ParentId="8352" CreationDate="2017-01-11T14:49:16.783" Score="3" Body="&lt;p&gt;There is no system which takes as input the pdf or .tex of a proof in natural language and check it.&#xA;But there are interactive proof systems which can check some student proofs. For example there are the Lurch and Edukera.com systems.&#xA;In Lurch, the student type his proof and annotate it to explain to the system where are the facts and justifications.&#xA;In Edukera, you have a point and click user interface, the student can chose which theorem or logical rule to apply and the system update the current state of the textual proof.&lt;/p&gt;&#xA;" OwnerUserId="7685" LastEditorUserId="7685" LastEditDate="2017-01-16T08:39:30.410" LastActivityDate="2017-01-16T08:39:30.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11913" PostTypeId="1" CreationDate="2017-01-16T03:18:26.900" Score="7" ViewCount="142" Body="&lt;p&gt;I am teaching an undergraduate course which consists entirely of exploratory projects on any interesting mathematical topic. I'm looking for projects where a first-year student who has only taken calculus can make discoveries within an hour, and an advanced undergraduate thinking about graduate school can continue even after twenty hours. I prefer projects where students can do short investigations, make conjectures, and then try to prove them (or run computations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far I have found the book &lt;a href=&quot;http://press.princeton.edu/titles/10824.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Joy of SET&lt;/a&gt;, which I think is ideal for such projects. I am also thinking about projects on prime numbers, cellular automata, and lattice paths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like additional resources that I can work from, where the authors have already thought out how to prompt students to explore topics with particular low-hanging fruit in mind, and tons more fruit beyond it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be clear, I'm not looking for textbooks for students to read themselves as in &lt;a href=&quot;https://matheducators.stackexchange.com/q/4118/7707&quot;&gt;Recommendations for inquiry based/aided discovery textbooks&lt;/a&gt;, although I got some ideas there (&lt;a href=&quot;http://en.wikipedia.org/wiki/Indra%27s_Pearls_%28book%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Indra's Pearls&lt;/a&gt;). I also got some ideas from &lt;a href=&quot;https://matheducators.stackexchange.com/a/8294/7707&quot;&gt;Topics for Discovery-based Projects&lt;/a&gt;, such as &lt;a href=&quot;http://jrmf.org/problems.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Julia Robinson Mathematics Festivals&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="7707" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-01-24T12:41:55.283" Title="Resources for Inquiry-based Projects with Undergraduates" Tags="|undergraduate-education|undergraduate-research|inquiry-based-learning|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11914" PostTypeId="2" ParentId="11913" CreationDate="2017-01-16T18:55:53.070" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0883855038&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Student Research Projects in Calculus&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.maa.org/press/books/cameos-for-calculus-visualization-in-the-first-year-course&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Cameos For Calculus&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I particularly like the first one because the authors include with each project a description of how long it may take a student, any issues they've encountered, and (sometimes) how it can be explored further. &lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2017-01-16T18:55:53.070" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11915" PostTypeId="2" ParentId="11844" CreationDate="2017-01-17T12:46:23.610" Score="2" Body="&lt;p&gt;The UK A-levels are widely studied outside the UK. The new syllabus is designed to have all the exams at the end, rather than the modular system that was in place before. So while there might be more than one exam (I don't remember the details), they could be taken in one block.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="4746" LastEditDate="2017-01-18T07:29:39.770" LastActivityDate="2017-01-18T07:29:39.770" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11916" PostTypeId="2" ParentId="4118" CreationDate="2017-01-17T22:22:14.557" Score="2" Body="&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://math.dartmouth.edu/news-resources/electronic/kpbogart/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Combinatorics Through Guided Discovery&lt;/a&gt;&lt;/em&gt; by the late Kenneth Bogart is a great introduction to combinatorics through a guided set of problems and is freely available for download at the link given above.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2017-01-17T22:22:14.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11917" PostTypeId="2" ParentId="8352" CreationDate="2017-01-18T02:20:04.860" Score="0" Body="&lt;p&gt;One type of answer is &lt;a href=&quot;http://webwork.maa.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;WeBWorK&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it cannot check proofs yet.  But it can check an answer when it is a formula, rather than just a number.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-01-18T02:20:04.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11918" PostTypeId="2" ParentId="11913" CreationDate="2017-01-18T23:41:34.880" Score="3" Body="&lt;p&gt;Nice question and I should say Inquiry-based Project is taking my attention. So, I really like these books:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Tanton, J.(2001). &lt;em&gt;Solve this: math activities for students and clubs&lt;/em&gt;.&#xA;  Cambridge University Press.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Cofman, J. (1990). &lt;em&gt;What to Solve?: Problems and Suggestions for Young&#xA;  Mathematicians&lt;/em&gt;. Oxford University Press, New York.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I hope these resources could help you. &lt;/p&gt;&#xA;" OwnerUserId="1789" LastEditorUserId="1789" LastEditDate="2017-01-24T12:41:55.283" LastActivityDate="2017-01-24T12:41:55.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11919" PostTypeId="2" ParentId="11542" CreationDate="2017-01-19T02:43:38.967" Score="2" Body="&lt;p&gt;These two textbooks also have the same list of 7 &quot;Thinking skills&quot; in their index:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recall and transfer&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analysis&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Spatial perception&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applying concepts&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interpreting&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Synthesis&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reasoning and inferrencing&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These skills definitely go hand-in-hand with learning how to read a math book.&lt;/p&gt;&#xA;" OwnerUserId="7711" LastActivityDate="2017-01-19T02:43:38.967" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11920" PostTypeId="1" AcceptedAnswerId="11927" CreationDate="2017-01-20T02:20:06.750" Score="17" ViewCount="409" Body="&lt;p&gt;This question comes from the perspective of an undergraduate math major who feels that much (although not all) of the mathematical discipline is a liberal art, rather than a science, and should be presented as such. I regularly interact with bright, creative peers, whom I think could make terrific mathematicians with training, but who immediately dismiss the study of math. This is often due to poor experiences with math courses in primary and secondary school, or &quot;I'm just more of a humanities person.&quot; My interest is more in the latter response: I &lt;em&gt;too&lt;/em&gt; am &quot;more of a humanities person&quot;! I would much rather spend a semester doing nothing but literature study than nothing but biology; mathematics is for me and many in my department (and, I'm sure, many people on this forum and math.SE and mathoverflow) artistic, creative, and self-cultivating in the way humanities studies are. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How, then, can we present the mathematical discipline, in its &quot;true form&quot;, to students who may otherwise cast it aside as &quot;hard,&quot; &quot;cold,&quot; and &quot;calculating&quot;? Certainly, there are students who just won't like math, and that's perfectly fine. However, I worry often that there are many students who would flourish while studying math but are not shown the breadth of study that the field encompasses. Perhaps I'm just rambling now, but this does interest me. Thoughts?&lt;/p&gt;&#xA;" OwnerUserId="7714" LastActivityDate="2017-01-26T01:16:09.970" Title="How to invite humanities students to study mathematics?" Tags="|undergraduate-education|mathematical-pedagogy|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11921" PostTypeId="2" ParentId="10163" CreationDate="2017-01-20T10:07:16.870" Score="1" Body="&lt;p&gt;I recently pulled out my old &lt;a href=&quot;https://en.wikipedia.org/wiki/Dice#Polyhedral_dice&quot; rel=&quot;nofollow noreferrer&quot;&gt;polyhedral dice&lt;/a&gt; from my role-playing days and &lt;a href=&quot;https://en.wikipedia.org/wiki/Gamification_of_learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;gamified&lt;/a&gt; multiplication tables with my seven-year-old twins: roll two dice and multiply them. How many can you do correctly in five minutes? We are up to two ten-sided dice (2d10) now, but I still have a couple d12 and d20. And you can always use more than two dice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, there isn't really all that much mathematical wonder in multiplication tables, I'll admit that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I took the d4 and had my daughter (the son had gone off) count the number of vertices $V$, the number of edges $E$ and the number of faces $F$, then had her calculate $V-E+F$. Then we did the same for the d6, the d8 and so forth. (The little tricks I explained to calculate $V$ and $E$ for the d12 and the d20, by noticing the regularity of the faces and accounting for double-counting, went a bit over her head.) She was fascinated by the fact that the result of $V-E+F$ was the same for all dice, got her brother and explained this to him. (With a little help from daddy.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My kids now know the &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler_characteristic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler formula for convex polyhedra&lt;/a&gt;. And they seem to like it; especially my daughter brings it up every other day or so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No, I haven't gone into proofs or CW complexes yet. Time enough for that when they get into high school.&lt;/p&gt;&#xA;" OwnerUserId="5845" LastActivityDate="2017-01-20T10:07:16.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11922" PostTypeId="1" AcceptedAnswerId="11935" CreationDate="2017-01-20T15:02:29.177" Score="3" ViewCount="198" Body="&lt;p&gt;Which is correct terminology: &quot;A Cartesian plane&quot; or &quot;The Cartesian plane&quot;? (As in the directions for a section of homework being, &quot;Plot a point on ______ Cartesian plane.&quot; In that context, I feel that one of the two should be used consistently, but find there's little agreement on Google for which is appropriate.&lt;/p&gt;&#xA;" OwnerUserId="6675" LastEditorUserId="143" LastEditDate="2017-01-29T19:20:49.883" LastActivityDate="2017-01-29T19:20:49.883" Title="&quot;A&quot; or &quot;The&quot; Cartesian plane?" Tags="|geometry|terminology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11923" PostTypeId="1" CreationDate="2017-01-20T19:54:38.953" Score="11" ViewCount="896" Body="&lt;p&gt;Sometimes students will contact me, as my email is visible. This time, an undergraduate in Sri Lanka has no number theory courses available and is self-studying.&lt;/p&gt;&#xA;&lt;p&gt;My own experience is that it helps to run computer experiments; that is how I learned the topic. I took a course in C++ some twenty years ago, and that has served me well. Steep learning curve, though. It helped a good deal when I found I could use the GMP library for large integers. My impression is that many current computer languages have large integers built in.&lt;/p&gt;&#xA;&lt;p&gt;What computer languages might one recommend for, say, investigations in number theory?&lt;/p&gt;&#xA;&lt;p&gt;As far as books, she is currently reading Burton. I have requested that and Burn, &lt;em&gt;A Pathway into Number Theory&lt;/em&gt;, from my library. The one thing I would add to early number theory is quadratic forms, maybe just binary, before attempting quadratic fields. I like that Burns does the automorphism group of a binary quadratic form; he may not call it a group, not sure yet.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/4118/recommendations-for-inquiry-based-aided-discovery-textbooks&quot;&gt;Recommendations for inquiry based/aided discovery textbooks&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3564" LastEditorUserId="376" LastEditDate="2020-09-11T12:17:35.560" LastActivityDate="2020-09-11T12:27:37.087" Title="Number theory for self-study students: books and computer languages" Tags="|number-theory|inquiry-based-learning|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11924" PostTypeId="2" ParentId="11923" CreationDate="2017-01-20T23:49:05.567" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What computer languages might one recommend for, say, investigations in number theory?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I find &lt;a href=&quot;http://www.wolfram.com/mathematica/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematica&lt;/a&gt; ideal, &#xA;e.g.:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;&lt;a href=&quot;https://mathoverflow.net/q/191518/6094&quot;&gt;Mod sequences that seem to become constant; and the number 316&lt;/a&gt;&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;&lt;a href=&quot;https://mathoverflow.net/q/240392/6094&quot;&gt;Does 53 diverge to infinity in this Collatz-like sequence?&lt;/a&gt;&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But: (a) there is a huge start-up learning curve, and (b) Mathematica is not free. Because of the latter, I recommend &lt;a href=&quot;http://www.sage.com/us&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sage&lt;/a&gt; / &lt;a href=&quot;http://www.sagemath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;SageMath&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Cjo6P.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Cjo6P.png&quot; alt=&quot;MultiGraph&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;Multi-edge graph: &quot;All pairs of characters in the sentence 'I am a cool multiedge graph with loops.'&quot;&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.020" LastActivityDate="2017-01-21T03:18:26.403" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11925" PostTypeId="2" ParentId="11923" CreationDate="2017-01-20T23:53:44.573" Score="11" Body="&lt;p&gt;I would recommend Python combined with &lt;a href=&quot;http://www.sagemath.org/&quot;&gt;SageMath&lt;/a&gt;, as already recommended by Joseph O'Rourke, or rather SageMath and Python comes naturally. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Python is a modern, and widely used, interpreted language (no compilation needed) it supports big integers via the bignum type. (But using SageMath I think this is tangential, I mention it for completeness mainly.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SageMath is a free CAS  whose user language is essentially Python. &#xA;SageMath has lots of number theory functionality; its founder William A. Stein is a number theorist. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He also  wrote a nice book &lt;a href=&quot;http://wstein.org/ent/&quot;&gt;&quot;Elementary Number Theory: Primes, Congruences, and Secrets&quot;&lt;/a&gt; that uses this software; you can check it out on his site. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you mentioned PARI/gp you might be interested to know that PARI is an integral part of SageMath.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="511" LastEditDate="2017-01-21T03:19:20.970" LastActivityDate="2017-01-21T03:19:20.970" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11926" PostTypeId="2" ParentId="11922" CreationDate="2017-01-21T00:53:33.153" Score="4" Body="&lt;p&gt;Why do you think they should be used consistently?  I think there are situations appropriate for each of these choices.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-01-21T00:53:33.153" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11927" PostTypeId="2" ParentId="11920" CreationDate="2017-01-21T11:21:01.973" Score="13" Body="&lt;p&gt;I am going to answer this question from the perspective, ironically, of a biology major who was convinced to look into mathematics after some poor experiences in both early undergrad and secondary school. But I was a humanities-heavy biology student:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Set a realistic objective - odds are you're not going to get very many people to switch to math whole-hog. Rather, go for something closer to &quot;appreciation&quot; and a few math classes, with the hope of something more.&lt;/li&gt;&#xA;&lt;li&gt;Don't push &quot;Math is Beautiful&quot; too heavily. This was an approach a lot of people took with me, and to be frank, it really didn't resonate. It took a long time until I started appreciating elegance in mathematical concepts, and in the timeframe you're talking about, you're facing the problem that they probably consider their field to be beautiful &lt;em&gt;right now&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Instead, I'd suggest showing how it can be relevant to their studies. How even a little bit of mathematical background can help push things forward. I'll give two examples of this, but for me, &quot;This can make Biology cooler...&quot; was the entry, rather than &quot;Math is amazing in its own right...&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Example 1: I met a medieval historian whose background was in civil engineering. Her project was to take the semi-hysterical accounts of the number of Viking ships involved in the invasion of Ireland, typically in the form &quot;ALL THE SHIPS! THOUSANDS!&quot;, and try to actually put realistic bounds on those numbers, based on things like the population of Viking settlements, how many ships could have been supported, etc. That's not terribly difficult math, but it was an entirely different way of looking at a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example 2: &lt;em&gt;The Three-Body Problem&lt;/em&gt; by Cixin Liu and Ken Liu. This is an amazing novel spanning the Cultural Revolution to the modern age, all built around the problem in the title. It's a &lt;em&gt;good&lt;/em&gt; book without knowing what the Three Body Problem is and just using &quot;It's hard math&quot; as a standin. But it's actively more enjoyable if you get the math behind it - and again, the math isn't particularly advanced.&lt;/p&gt;&#xA;" OwnerUserId="7718" LastActivityDate="2017-01-21T11:21:01.973" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11928" PostTypeId="2" ParentId="11836" CreationDate="2017-01-23T06:48:20.793" Score="1" Body="&lt;p&gt;In the vein of other answers that would prefer a &quot;real&quot; axiomatic system, my suggestion would be...point-set topology. I know, crazy. Hear me out though...&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Only three axioms to give! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- arbitrary unions of open sets are open&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--  finite intersections of open sets are open&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- the empty set and universal set are open&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There are some background definitions (what is a set? intersection/union of sets? what is the complement of a set?), but students taking a &quot;pure maths&quot; course likely have a reasonable intuition of what these are. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;The start of point-set topology, with its three axioms, is probably simple enough to come up with a &quot;toy system&quot; for if you like the bait and switch idea. Replace &quot;open/closed&quot; with any other pair of descriptors you like...good/bad, left/right, red/blue, ...&lt;/li&gt;&#xA;&lt;li&gt;It gives an example of a &quot;real&quot; bit of mathematics where axiomatization is vitally important. Axiomatizing arithmetic/numbers/planar geometry can feel a bit hair-splitting at times. &lt;/li&gt;&#xA;&lt;li&gt;Within short order, you get to discuss the inevitable claim that &quot;since this set is not open, it is closed!&quot; In my mind, this is one of the primary reasons that we need axioms: the need to have a level playing field, so that we can separate logical truths from intuition.&lt;/li&gt;&#xA;&lt;li&gt;You can axiomatize point-set topology in terms of closed sets instead: finite unions and arbitrary intersections of closed sets are closed, the empty set and universal set are closed. You can then show how these two collections of axioms are equivalent and lead to the same mathematical universe.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are certainly some cons, which are largely connected to the mathematical sophistication of the class. Keep in mind that calculus students will be familiar with open and closed sets while multivariable calculus students will likely have seen simply connected domains. &lt;/p&gt;&#xA;" OwnerUserId="7260" LastActivityDate="2017-01-23T06:48:20.793" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11930" PostTypeId="2" ParentId="11542" CreationDate="2017-01-24T13:08:14.880" Score="2" Body="&lt;p&gt;Yes you are correct. I would understand the statement in the following way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most Math requires the student to:&#xA;a. Remember definitions (Knowledge - e.g. what is the definition of area, the formula for area)&#xA;b. Understand the meaning of the concept (e.g what does &quot;area&quot; denote in real life)&#xA;c. Apply the formula (Application - e.g. use the formula to calculate area in a specific case)&#xA;and &#xA;d. finally bring all these together to answer a real / word problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A math text book outlines the first three aspects - namely definitions and formula, some examples of how the concept is used,  and some problems solved using the formula.&#xA;A student would need to understand all these three aspects and then how they fit together to be able to look at a related problem and solve it. For example, if a student were to just remember the formula for area without understanding what area is, he or she would not be able to answer a problem because they are not able to identify that they need to use the area formula!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence it is important for students to read each sentence and understand it, and then see how it links to other sentences on that topic. That way they can build a full picture of the concept (area in our example) and have the tools to use it. &lt;/p&gt;&#xA;" OwnerUserId="7726" LastActivityDate="2017-01-24T13:08:14.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11931" PostTypeId="2" ParentId="11923" CreationDate="2017-01-24T19:44:16.143" Score="4" Body="&lt;p&gt;For beginning number theory, &lt;a href=&quot;http://artofproblemsolving.com/school/course/intro-numbertheory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Art of Problem Solving has an online course&lt;/a&gt;. The textbook used with it, &lt;em&gt;&lt;a href=&quot;https://artofproblemsolving.com/store/item/intro-number-theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Number Theory&lt;/a&gt;&lt;/em&gt; by Matthew Crawford, can be used alone for self-study. (I have not used the course. I have used the textbook.) I tutored a very advanced 9-year-old using this book, and enjoyed it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see they also have an &lt;a href=&quot;http://artofproblemsolving.com/school/course/intermediate-numbertheory&quot; rel=&quot;nofollow noreferrer&quot;&gt;intermediate number theory course&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-01-24T19:44:16.143" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11932" PostTypeId="2" ParentId="11923" CreationDate="2017-01-24T19:53:01.600" Score="4" Body="&lt;p&gt;Seconding/complementing other answers: Python (and/or Python as a part of Sage) has a command-line interface (on Linux/Unix and on Mac OS) that &lt;em&gt;does&lt;/em&gt; allow defining variables, pre-loading files that set things up, and so on. Python (and, thus, Sage) has built-in large integers that are easier to use than C++ large integers (in my opinion). And freely available.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: ... and (I forgot to mention) it is quite easy to use the built-in graphing and graphical capabilities of Sage (especially in the &quot;notebook&quot; mode).&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2017-01-24T20:03:04.500" LastActivityDate="2017-01-24T20:03:04.500" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11933" PostTypeId="2" ParentId="11920" CreationDate="2017-01-24T20:18:52.337" Score="7" Body="&lt;p&gt;Many humanities students find it unexpected that mathematicians think about fairness questions - cost sharing, fair elections, fair games, fair division questions of different kinds, etc. Many of these areas can be discussed without much background other than arithmetic, and students find both the questions and the mathematics rather appealing. If you are not familiar with this &quot;domain&quot; you might look at the book:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;H.P. Young, Fairness in Theory and Practice, Princeton U. Press.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2017-01-24T20:18:52.337" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11934" PostTypeId="2" ParentId="11920" CreationDate="2017-01-25T00:35:58.757" Score="2" Body="&lt;p&gt;A natural point of contact with the humanities is found in philosophy. Many philosophical questions are rooted in problems of the philosophy of mathematics. For example, philosophical pragmatism, which in present times is often applied as general cultural critique, has roots in the work of Charles Sanders Peirce...son of Benjamin Peirce, a leading Harvard mathematician of his day. This is more of a comment, but I think that talking about the associated philosophical problems can lead to an interest in the mathematics...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Questions about the nature of human thought and the mind can be connected to the nature of proof, a la Hofstadter's &quot;Godel, Escher, Bach&quot;...&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2017-01-26T01:16:09.970" LastActivityDate="2017-01-26T01:16:09.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11935" PostTypeId="2" ParentId="11922" CreationDate="2017-01-25T10:27:33.843" Score="5" Body="&lt;p&gt;I think from a philosophical point of view, both terms may be used. From an educational point of view, I would generally use the term &quot;the&quot; Cartesian plane in school context as every two Cartesian planes are isomorphic and should most of the time be considered the same object. This may reduce cognitive complexity. Only in cases where you have several distinct objects that might be called Cartesian plane, I would change the terminology towards &quot;a&quot; Cartesian plane. This could happen in a university course on geometry, where you may cut out Cartesian planes from projective spaces, for example.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2017-01-25T10:27:33.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11936" PostTypeId="2" ParentId="1234" CreationDate="2017-01-25T10:29:04.447" Score="2" Body="&lt;p&gt;If you are looking for a system to perform  interactive proof exercise in logic, building natural deduction trees for example, I can cite:&#xA;coqweb, Lurch and edukera.com. They are interactive proof systems which can check some student proofs. &#xA;In Lurch, the student type his proof and annotate it to explain to the system where are the facts and justifications. &#xA;In Edukera, you have a point and click user interface, the student can chose which theorem or logical rule to apply and the system update the current state of the textual proof and even a graphical proof tree in the upcoming version.&lt;/p&gt;&#xA;" OwnerUserId="7685" LastActivityDate="2017-01-25T10:29:04.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11937" PostTypeId="1" AcceptedAnswerId="11939" CreationDate="2017-01-25T12:42:09.267" Score="0" ViewCount="83" Body="&lt;p&gt;I'm in my second year of Mathematics. I would like to take a PhD and if possible also a master. However, I was thinking to take a master in Machine Learning, work for a bit in that industry (5-10 years) and then go back to use Machine Learning in a PhD in Mathematical Physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I would like to know if any of you had the experience of , or knew anyone with that experience, taking a PhD after say 5 or 10 years from taking the master or BSc. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Would it be possible to be accepted? Would it be possible to get funding?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am talking especially in UK, but also the rest of EU and America are fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are indeed an educator, would you accept someone with that background?&lt;/p&gt;&#xA;" OwnerUserId="7513" LastActivityDate="2017-01-25T14:29:03.793" Title="Can I do a PhD after many years I've graduated and can I get funding?" Tags="|graduate-education|" AnswerCount="2" CommentCount="3" ClosedDate="2017-01-25T20:30:15.743" ContentLicense="CC BY-SA 3.0" />
  <row Id="11938" PostTypeId="2" ParentId="11937" CreationDate="2017-01-25T13:49:05.863" Score="0" Body="&lt;p&gt;&lt;strong&gt;Yes&lt;/strong&gt;, we have many old consultants in applied math as PhD. Our professors have been at the industry and it is only a good thing that you have some actual experience how things really work instead of blindly staring at those Big O:s, which do make a good article but in reality fail because they do not scale for parallel or memory operations are too slow.&lt;/p&gt;&#xA;" OwnerUserId="7552" LastActivityDate="2017-01-25T13:49:05.863" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11939" PostTypeId="2" ParentId="11937" CreationDate="2017-01-25T14:29:03.793" Score="0" Body="&lt;p&gt;There is no absolute rules on age for doing a Ph.D. The underlying ideas are the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You need a subject for your thesis&lt;/li&gt;&#xA;&lt;li&gt;You need a thesis advisor (that can support your thesis problematic)&lt;/li&gt;&#xA;&lt;li&gt;You need funding&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you manage to get all of these, there is &lt;em&gt;a priori&lt;/em&gt; no obstacle for the doctorate school to accept your application.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The origin of the funding is (normally) not a concern for the university. Of course, while not being part of a continuous scholarship, you probably disqualify for applying to the common funding opportunities. There will be other opportunities opened to you though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is, in some cases, possible to do a Ph.D while still a working full-time job. This might require a bit of work with the head of the doctoral school, to clearly define how things are going to work, how long etc. But provided you are getting a pay at least for yourself, there should be opportunities.&lt;/p&gt;&#xA;" OwnerUserId="7728" LastActivityDate="2017-01-25T14:29:03.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11941" PostTypeId="1" CreationDate="2017-01-25T20:39:14.087" Score="2" ViewCount="86" Body="&lt;p&gt;I just tried to pitch a &lt;a href=&quot;http://area51.stackexchange.com/proposals/106276/math-recommendations&quot;&gt;Math Recommendations&lt;/a&gt; site, and it was shot down because it is too broad and that Stack Exchange sites operate on a separation of concerns principle; which makes sense to me but doesn't solve a problem I've been looking to solve since I went to school taking math classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was in school as I remember it the issue wasn't that I didn't like math, it was actually that the problems in the math book didn't apply to anything that I was actually interested in at the time; and therefore I didn't realize how important math actually was to everyday life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it seems to me that one of the issues in educating people about math comes from the fact that there isn't a place where they can take their real world problems (i.e. things they care about) and find out what specific mathematical concepts they need to learn to fix their particular issue, or apply it to something they care about.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did eventually find something that applied to me in the form of creating video games in computer programming that made me care about math, but I'm sure there are other people out there different from myself that have similar needs but are unaware of it, and I was wondering if there was a site or community of a similar nature to what I am proposing where answers to these Math Recommendation questions are what types of math you should learn (including pre-requisites) and examples of using it to solve the stated problem; this would then allow the question asker to study up and eventually understand the mathematical concepts they need to solve a problem that they care about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you recommend such a site, or a tag related to this within the existing Stack Exchanges?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. This may be more of a meta question, but I don't see a meta site, and nobody would answer this question on the Mathematics Stack Exchange.&lt;/p&gt;&#xA;" OwnerUserId="7731" LastActivityDate="2017-01-25T20:39:14.087" Title="Finding the correct mathmatics based on application as opposed to the other way around?" Tags="|student-motivation|concept-motivation|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11942" PostTypeId="2" ParentId="11644" CreationDate="2017-01-25T21:39:48.747" Score="3" Body="&lt;p&gt;The problem is that the question was written using the Blackboard graphical formula editor, which cannot recognize the square brackets as variable definitions. For that you need to put the square-bracket variables in normal Blackboard text, not the formula editor. (Note the UI directions on the top of the screen: &quot;Add &lt;em&gt;question text&lt;/em&gt; that includes variables in square brackets [x] corresponding to variables in the formula that will be used to calculate the answer to this question.&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having just tested it in Blackboard: The answer formula is fine. If I write the question using a Blackboard formula, then I get the same error as you. But if I instead write the question using basic text (like &quot;Compute ||&amp;lt;[x], [y], [z]&gt; x &amp;lt;[u], [v], [w]&gt;||&quot;), then that is accepted without the error. Obviously that text version isn't perfectly formatted, so you might need to play around with fonts or special characters to get it looking right. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, you could write the question in natural language, as I think that feature is intended, i.e., &quot;Find the norm when multiplying vectors &amp;lt;[x], [y], [z]&gt; and &amp;lt;[u], [v], [w]&gt;&quot;. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-01-26T01:12:28.853" LastActivityDate="2017-01-26T01:12:28.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11944" PostTypeId="1" AcceptedAnswerId="11945" CreationDate="2017-01-26T19:23:32.977" Score="8" ViewCount="439" Body="&lt;p&gt;I am teaching a Trigonometry class, and every year we get to this point my students start asking a lot of questions. And for good reason.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is my issue:  We teach that $\arccos\frac{1}{2}=\frac{\pi}{3}$ and this is the only answer because our range is restricted to $0\le{y}\le{\pi}$  We've been over that cosine and sine are not one to one functions which is why we restrict the domain to find the inverses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, when we start solving trigonometric equations and solve problems such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Find the solutions of $2\cos x=1$ in the interval [$\pi,2\pi]$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We solve by dividing by 2, and we have $\cos x=\frac{1}{2}$.  Now, the book states that we take $\arccos(\frac{1}{2})$ and we find that equals $\frac{5\pi}{3}$.  The problem I run into is that students object that $\arccos(\frac{1}{2})=\frac{5\pi}{3}$ because it does not fit within the range of $0\le{y}\le{\pi}$.  Is it incorrect notation to use arccos in this situation as the book does?  Because I agree with students that this does not fall within the range of arccos.  It confuses students every year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, is it more proper to leave the equation $\cos x=\frac{1}{2}$ and find the value of $x$ without showing the arccos step?&lt;/p&gt;&#xA;" OwnerUserId="7550" LastEditorUserId="7550" LastEditDate="2017-01-27T16:30:25.877" LastActivityDate="2017-01-27T18:57:55.380" Title="Applying inverse trigonometric functions" Tags="|trigonometry|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11945" PostTypeId="2" ParentId="11944" CreationDate="2017-01-26T19:42:19.497" Score="5" Body="&lt;p&gt;You use arcos to get pi/3. Then you use that as a reference angle, and find the angle in your range with the same reference angle. If the angle is not from a special triangle (30-60-90 or 45-45-90), you will definitely need the arccos step.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-01-26T19:42:19.497" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11946" PostTypeId="2" ParentId="11944" CreationDate="2017-01-26T20:21:55.410" Score="2" Body="&lt;p&gt;It's really not the same function, unless you start talking about Riemann surfaces and multivalued functions (which I assume you won't be).  &lt;a href=&quot;http://sagecell.sagemath.org/?z=eJwrKMrMK1Eozs8pS9Uw0krOL9ao0LS1NdSp0CnJjy_Iz6mMB8vZqqflFyWnqmvychWAdSSClBrqG2kCAOzjFcI=&amp;amp;lang=sage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Computer algebra systems distinguish&lt;/a&gt; between solving an equation and evaluating inverse trig functions, too.  Personally, I would avoid using the arccosine language when solving, other than the specific step.  Inverse trig functions are their own beasts, useful in their own ways (notably solving useful integrals).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, when solving, draw a picture of the function (here, $2\cos(x)$ or $\cos(x)$) and see what they can expect in terms of the symmetry.  For my money, solving to get $\pm \pi/3$ and then pointing out the $2\pi$ repetition, and &lt;em&gt;then&lt;/em&gt; finally narrowing to the interval in question is the best bet.  However, that requires enough time to have a fairly leisurely discussion, unless you feel like they really &quot;get&quot; that periodicity. Since your students are complaining that it's wrong instead of just writing things down numbly, you may be in good shape to have these discussions!&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-01-26T20:21:55.410" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11947" PostTypeId="1" CreationDate="2017-01-27T01:04:26.533" Score="0" ViewCount="213" Body="&lt;p&gt;How does substitution work as an alternative to the balance model in introducing solving equations? My biggest worry is that, lacking a concrete representation, is too abstract for middle schoolers. However, the balance model has enough limitations that I want to pursue using teaching students how to solve equations using solely substitution. Maybe someone here can stop me from making a horrible mistake!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The balance model has two main limitations&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It presupposes that there is only one possible value for $x$. &lt;/li&gt;&#xA;&lt;li&gt;It masks the most fundamental meaning of the $=$ sign: $a=b$ iff $a$ and $b$ are are different names for the same object.&lt;/li&gt;&#xA;&lt;li&gt;It only works when the objects on both sides of the equation are numbers. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Albeit more computationally intense, Substitution allows one to use language that makes clear to students that there could very well be more than one possible value for $x$. This allows the equality in $y=2x+4$ to be instantly fused with the equality in $x+4=2x$, instead of being treated separately, to be fused at a later date (more often never). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am thinking of starting by using teaching students to evaluate an expression by substitution of a variable, and slowly introduce larger substitutions (as in, first evaluating some $f(a)$, and then using the result to evaluate some $g(f(a))$ where $f(x)$ appears in the definition of $g(f(x))$ multiple times. We would spend some time establishing what equivalence means when dealing with expressions with variables (that they have the same value no matter what $x$ is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One and two step equations would be introduced purely as a guess and check / undoing procedure, without a formal way to write out steps. This is purely so that students understand that finding $x$ means ending with a statement $x=\_\_$ We will only require that for tackling equations for which the undoing model does not make sense - particularly equations with variables on both sides. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When given an expression such as $x+1=2x$ and asked to solve for it, guess and check is still an available method. We would use this until positing an equation such as $10x+4=2x-1$ where guessing and checking is unlikely to help one arrive at an answer. Problem: there is an $x$ on both sides! We will explain that a purposed substitution can take care of this problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can point out that $2x- 1 -2x = 2x -1 -2x$. We can combine this with the premise that $10x+4=2x-1$ to substitute $10x+4$ for $2x-1$ on one side of the equation, giving us $10x+4-2x=2x-1-2x$. We then can write $10x+4-2x=8x+4$ and $2x-1-2x=1$. Substituting back into both sides, we have reduced the problem to a previously solved problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As students get used to this, we could introduce shorthand notation of performing an opertation to both sides under the equation. The end result of the process of solving an equation would look the same, but students would come away with an exact understanding of what all these variables and equations mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own evaluation: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pros: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Teaches students the full meaning of the equal sign.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Exposes students to more rigorous mathematical thinking. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Allows teachers and students to freely use the powerful tool of substitution in other contexts, such as geometry.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Well unified progression of concepts&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;This thing is the same as this thing, so I can interchange them&quot; is an easier concept to grasp than &quot;This thing has the same value as this thing, so if I add the same thing to both of them, their values will stay the same&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Gives students the opportunity to problem solve when asked to solve an equation (&quot;What substitution should I use&quot;), rather than having to be told that one is allowed to divide both sides of an equation by the same number&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Cons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;More computationally intense.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Relies on many abstract concepts.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Requires building a strong foundation on equivalence in expressions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Takes more classroom time to build up to.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7734" LastEditorUserId="7734" LastEditDate="2017-01-27T01:27:03.853" LastActivityDate="2017-01-27T02:38:56.517" Title="Using Substitution in place of the balance model" Tags="|algebra|course-design|curriculum|middle-school|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11948" PostTypeId="2" ParentId="11947" CreationDate="2017-01-27T02:38:56.517" Score="2" Body="&lt;p&gt;Solving equations using the balance model is a necessary computational skill that students need for all levels of higher mathematics.  I have the unique opportunity to teach all students in grades 7-12 in our rural school district, and I don't think that high school students have a difficult time grasping that x can take on more than one value when they get to Algebra I.  I think the pros of using the balance model far outweigh the limitations.  Math is meant to build upon itself.  They need to become proficient in basic computational skills in junior high to be successful in higher level math classes.&lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2017-01-27T02:38:56.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11950" PostTypeId="1" CreationDate="2017-01-27T06:29:18.150" Score="15" ViewCount="1347" Body="&lt;p&gt;&lt;sub&gt;&lt;a href=&quot;https://www.quantamagazine.org/20160303-michael-atiyahs-mathematical-dreams/&quot; rel=&quot;noreferrer&quot;&gt;Source 1:&lt;/a&gt; Siobhan Roberts, &lt;em&gt;Mathematical Beauty: A Q&amp;amp;A with Fields Medalist Michael Atiyah&lt;/em&gt;, Quanta Magazine, 2016/3/9.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;Is there one big question that has always guided you?&lt;/h3&gt;&#xA;&lt;p&gt;I always want to try to understand why things work. I’m not interested in getting a formula without knowing what it means. &lt;strong&gt;I always try to dig behind the scenes, so if I have a formula, I understand why it’s there. And understanding is a very difficult notion.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;People think mathematics begins when you write down a theorem followed by a proof. &lt;strong&gt;That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way.&lt;/strong&gt; But at the end, just as a composer has to put it down on paper, you have to write things down. &lt;strong&gt;But the most important stage is understanding. A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;a href=&quot;http://superstringtheory.com/people/atiyah.html&quot; rel=&quot;noreferrer&quot;&gt;Source 2:&lt;/a&gt; &lt;em&gt;Sir Michael Atiyah on math, physics and fun&lt;/em&gt;.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;  [...] You've got to have a lot of input and material from somewhere, you've got to have ideas coming from physics, concepts from geometry. You've got to have imagination, you're going to use intuition, guesswork, vision, like a creative artist has. In fact, proofs are usually only the last bit of the story, when you come to tie up the... dot the i's and cross the T's. Sometimes the proof is needed to hold the whole thing together like the steel structure of a building, but sometimes you've stopped putting it together, and the proof is just the last little bit of polish on the surface.&#xA;  &lt;strong&gt;So the most time mathematicians are working, they're concerned with much more than proofs, they're concerned with ideas, understanding why this is true, what leads where, possible links. You play around in your mind with a whole host of ill-defined things.&lt;/strong&gt;&lt;br /&gt;&#xA;  &lt;span class=&quot;math-container&quot;&gt;$\color{#009900}{\Large{[2.]}}$&lt;/span&gt; And I think that's one thing the field can get wrong when they're being taught to students. They can see a very formal proof, and they can see, this is what mathematics is. My story I can tell. When I was a student I went to some lectures on analysis where people gave some very formal proofs about this being less than epsilon and this is bigger than that. Then I had private supervision from a Russian mathematician called Bessikovich, a good analyst, and he'd draw a little picture and say, this -- this is small, this -- this is very small. Now that's the way an analyst thinks. None of this nonsense about precision. &lt;strong&gt;Small, very small. You get an idea what is going on. And then you can work it out afterwards. And people can be misled, if you read books, textbooks or go to lectures, and you see this very formal approach and you think, gosh that's the way I gotta think, and they can be turned off by that because that's not an interesting thing, mathematics, you see. You aren't thinking at that point imaginatively.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there a term that describes for what I bolded above (about ex ante presentiment)? 'Intuition' can be ex ante or ex poste, and so is ambiguous.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Assumption per 2 above: Too many textbooks and lecturers state only theorems and proofs, and fail to explain  the presentiment.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Then how can a maths student who is not a maths genius, learn presentiment:&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;for a specific (proven) theorem and proof?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;for solving unseen problems or proving unseen theorems?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-09-16T04:07:22.413" Title="How does a student learn to 'dig behind the scenes' or 'feel' math like a Fields Medallist?" Tags="|self-learning|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="11951" PostTypeId="2" ParentId="11944" CreationDate="2017-01-27T09:26:19.997" Score="5" Body="&lt;p&gt;I certainly disagree strongly with your book: there should be one unique and consistent definition of each function. As is usual, you defined $\arccos$ as the map sending $x\in[-1,1]$ to the unique $\theta\in[0,\pi]$ such that $\cos\theta=x$, and this definition should not vary even by a nanometer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To solve your problem, you could use (and prove) that the unique solution $\phi\in[\pi,2\pi]$ of $\cos\phi=x$ is $2\pi-\arccos x$ (it is easily checked that this number is in the correct range, and that its cosine is indeed $x$; then uniqueness is basically a change of coordinates). Using a trigonometric circle as mentioned by Michael E2 also has an important part of understanding issue, you don't want your students to memorize a bunch of raw formulas without any understanding or way to check them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Oh, and by the way: $\arccos(1/2)$ is $\pi/3$, not $2\pi/3$.)&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-01-27T09:26:19.997" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11952" PostTypeId="2" ParentId="11944" CreationDate="2017-01-27T18:57:55.380" Score="4" Body="&lt;p&gt;I like to draw the analogy to the square-root function vs. solving a square. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we want to calculate $\sqrt{9}$ then there is only one answer (especially if we're using a calculator) and it is $3$. We &lt;strong&gt;&lt;em&gt;choose&lt;/em&gt;&lt;/strong&gt; to take the positive root. However, when solving $x^2=9$, we typically want all $x$ solutions and so we say $x=\pm\sqrt{9}=\pm 3$. It may help to remind them that the graph of $y=\sqrt{x}$ is only half of the graph of $y^2=x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Likewise, when we want to calculate $\arccos(1/2)$ there is only one answer (and this is the one the calculator gives) and it is $\pi/3$. This, too, is a &lt;strong&gt;&lt;em&gt;choice&lt;/em&gt;&lt;/strong&gt;. So, again, when solving $\cos(x)=1/2$, we want all those choices. For $\arccos$ we have more than just the $\pm$, we also have to consider $x=\pm \pi/3+2\pi k$ (for $k$ an integer). (And $\arcsin$ gets even more difficult since $\pm$ isn't the right thing.) Again, remember that the graph of $y=\arccos(x)$ is only a small piece of $\cos(y)=x$ that gets repeated over and over again.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2017-01-27T18:57:55.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11954" PostTypeId="1" CreationDate="2017-01-28T13:25:30.047" Score="3" ViewCount="468" Body="&lt;p&gt;There exist a large number of reasons why &amp;quot;mathematics is difficult&amp;quot;. If one exclude &amp;quot;subjective reasons&amp;quot; such as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;math anxiety, math fear,...&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and education factors as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;inadapted  math programs, books, teaching level, ...&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and focus only on &lt;strong&gt;mathematical cognitive process&lt;/strong&gt; as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;difficulties in the memorization process.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;difficulties linked to the misunderstanding of terminology.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What other reasons (in one line please) can we give?&lt;/p&gt;&#xA;" OwnerUserId="7738" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-02-23T15:08:12.517" Title="Mathematical difficulty" Tags="|mathematical-pedagogy|meta-cognition|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11955" PostTypeId="1" CreationDate="2017-01-28T18:34:27.590" Score="6" ViewCount="298" Body="&lt;p&gt;I am having a great deal of trouble with the definitions used throughout the book so far - i.e. linear, homogenous, non-homogenous, etc. I am not sure why exactly they are useful to know. I am having trouble seeing any reason to label one as a particular form, if all that is needed is to see the form and know what procedure might be used to solve it. Other than perhaps years from now, looking up &quot;how to solve x-type definition problem&quot; by the definition? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, in calculus, if I see a problem that must be solved with a particular procedure, it is because I have recognized a form. So I go look up the procedure (ex: integration by parts), or the Heaviside cover up procedure. Most of the  techniques are named after procedures, if I recall, like there wasn't a definition for trigonometric substitution forms, you just knew the form and the procedure for it, with no additional label layer in between needed. I don't recall such a huge emphasis on definitional labeling in mathematics before, but also I am still an undergraduate! :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any recommendations on how to understand these definitions better? Most everything online and in my books keeps referring to long symbol salad strings and I am a bad memorizer. I remember the motions of the procedures and the shapes they take and know why I do such a thing such a way, and I am scared this is going to be an achilles heel for me in a huge way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it is relevant for a response, I am a returning student working towards a BS in Physics and I am in my differentials/linear equations/physics 2 part of my educational path.&lt;/p&gt;&#xA;" OwnerUserId="7739" LastActivityDate="2017-01-30T16:45:42.407" Title="Differential equations - definitions" Tags="|definitions|differential-equations|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="11956" PostTypeId="2" ParentId="11955" CreationDate="2017-01-28T22:47:27.303" Score="3" Body="&lt;p&gt;Definitions &quot;linear&quot;, &quot;homogeneous&quot;, etc. are things that you need to know to carry out the solutions of these differential equations.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since most differential equations cannot be solved in closed form, it is useful to know cases that can.  (And in the past, a lot of effort has been spent on developing the methods for this.)&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-01-28T22:47:27.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11957" PostTypeId="2" ParentId="11955" CreationDate="2017-01-29T02:24:44.840" Score="6" Body="&lt;p&gt;One reason that calculus solution concepts didn't necessarily have labels of this kind is that such things are fairly ad-hoc.  But all of the terms you listed - &quot;linear&quot;, &quot;homogeneous&quot;, etc. - have deep meaning in the algebra of solutions.  (Indeed, given your current state of studies, you should be asking about the relationship between differential equations and linear algebra, which might clarify this for you).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, a homogeneous equation has the very nice property that if $f$ is a solution, so is any constant multiple of $f$.  That seems convenient, I hope!  (Not to mention the connection to linear algebra, but again I'm not sure if you have encountered that yet.)  So it's not just the way in which you solve it, but the relationship the solutions have to each other.  As &lt;a href=&quot;http://sagecell.sagemath.org/?z=eJxLU7BVSCvNSy7JzM_TUE9T19So0OTlcnEFCqdkpqVppOlUaOqm8XIVZ-SXa6SkFufnlKVquLjqpGlqYgjqGgGF9YozcwtyMtMq49NKc3I0NDUBoYAeJw==&amp;amp;lang=sage&quot;&gt;this example shows&lt;/a&gt;, both types might have an infinite number of solutions with just one constant, but clearly the second one can't be manipulated in the same way as the first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And it's going to be the same with most other terminology now that you are past calculus.  But it's not bad, it's actually good; structure is awesome and useful!  Good luck.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-01-29T02:24:44.840" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11958" PostTypeId="2" ParentId="11955" CreationDate="2017-01-29T12:54:00.827" Score="5" Body="&lt;p&gt;I'll give you the big picture of the differential equations course I've taught for a few years, I'm not sure what the curriculum is at your university, but, odds are we have much in common.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, our general goal is to solve differential equations. Depending on the problem, that may mean we seek a function whose derivatives satisfy the given DEqn, but, it may also mean we are happy to find an equation whose differential consequence is the DEqn. For example, $\frac{dy}{dx} = y$ has &lt;strong&gt;explicit solutions&lt;/strong&gt; $y=ce^x$, whereas $(6x^5+2xy^2)dx+(5y^4+2x^2y)dx$ has &lt;strong&gt;implicit solutions&lt;/strong&gt; in the form of the level curves $x^6+x^2y^2+y^5=c$. So, in the implicit case it's not practical to solve for $y$ as a function of $x$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Writing a differential equation in the form $Mdx+Ndy=0$ is known as &lt;strong&gt;Pfaffian Form&lt;/strong&gt;. Pfaff, a teacher of Gauss, pioneered studying differential equations as equations in differentials. There is a theorem of Pfaff which states that many DEqns of the form $Mdx+Ndy=0$ can be multiplied by $I$ such that $IMdx+INdy = dF$ where $dF = F_xdx+F_ydy$ is the total differential you likely learned in multivariate calculus. Well, this is great, because $IMdx+INdy=0$ then has $F(x,y)=c$ as a solution. Let me give you a specific instance of this,&#xA;$$ \frac{dy}{dx}+py = q $$&#xA;is the &lt;strong&gt;linear first order differential equation in standard form&lt;/strong&gt;. We can write that also as:&#xA;$$ dy = (q-py)dx \ \ \Rightarrow \ \ (py-q)dx+dy =0 $$&#xA;this is not an exact equation because we cannot find $F=F(x,y)$ for which $dF= (py-q)dx+dy$. However, if we multiply by $I = I(x)$ then&#xA;$$ I(py-q)dx+Idy = 0 $$&#xA;is exact provided $\partial_x I = \partial_y (Ipy-Iq)$ or,&#xA;$$ \frac{dI}{dx} = Ip  \ \ \Rightarrow \ \ \frac{dI}{I} = pdx \ \ \Rightarrow \ \ I = \exp \left( \int pdx \right). $$&#xA;Perhaps you've seen this as a device to solve $\frac{dy}{dx}+py=q$. Anyway, my point here is that the term &lt;strong&gt;integrating factor&lt;/strong&gt; is used for studying exact equations and also for linear 1st-order DEs and at first glance these techinques do not seem common. Yet, the linear problem's integrating factor is actually a particular instance of the more general idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ok, so, for first order problems, mostly the game is to either use calculus or algebra to wrangle the problem into something we can just integrate to see the solution. This is no longer true for higher order problems. For example,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;to solve the $n$-th order homogeneous differential equation with &lt;strong&gt;constant coefficients&lt;/strong&gt; we write down its characteristic polynomial, factor it, and read off the solution (this should be derived and explained, but at the end of it we have a simple prescription for solving the DE with not need for integration). For example, to solve $y''+3y'+2y=0$ write $r^2+3r+2 = (r+1)(r+2)=0$ hence $y=c_1e^{-x}+c_2e^{-2x}$ (solved, almost too easy, yes?)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;to solve the $n$-th order nonhomogeneous differential equation we take the fundamental solution set of the corresponding homogeneous DEqn and run it through the variation of parameters technique. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;to find that fundamental solution set of the $n$-th order homogeneous problem with &lt;strong&gt;possibly nonconstant coefficient functions&lt;/strong&gt; we use power series techniques (however, in my experience, no one has patience to teach more than second order problems here)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To solve systems of differential equations we really need some insights from linear algebra, but, in a nutshell, the matrix exponential is where it's at. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then, there are DEqns with discontinuous forcing functions, for these you want Laplace transforms. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Differential equations with singular points (say the regular type), then the Frobenius method is the way to go (this leads to many of the important special functions of mathematical physics)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I have bad news, we're still not to physics. Most of physics is PDEs (partial differential equations, this means more than one independent variable, like space and time in Maxwell's Equations or your namesake's equation). That's a whole other story. But, to understand it, you need a firm grounding in ordinary DEqns. I recommend Habermann's PDE book to get a computational grounding there. For DEqns, it's nice to have a copy of Nagle Saff and Snider's text, get the 3rd or 4th edition, they're inexpensive and have more to say than more recent editions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, the labels in DEqn have to be more sophisticated than in calculus, because, we have so many different kinds of problems to solve. Moreover, as Gerald Edgar indicated, &lt;strong&gt;most&lt;/strong&gt; differential equations we encounter in real world applications cannot be solved via the closed form methods I mention in this answer (or for the 100's of techniques I have not even touched here). Instead, a numerical technique is used to study the DEqns. Simulation plays a big role in what &lt;em&gt;some&lt;/em&gt; physicists believe these days. If you can approximately build something on a computer, numerically solving relevant PDEs etc... then many will believe that's a physical possibility. &lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2017-01-29T20:51:09.760" LastActivityDate="2017-01-29T20:51:09.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11959" PostTypeId="2" ParentId="11954" CreationDate="2017-01-29T13:04:08.510" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;difficulty in understanding the interrelation of mathematical constructs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;difficulty of fitting a mathematical problem on just one line&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;difficulty in compressing the problem to a communicable slogan&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;etc.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-01-29T13:04:08.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11960" PostTypeId="2" ParentId="11955" CreationDate="2017-01-30T02:43:28.340" Score="2" Body="&lt;p&gt;You gave one excellent example yourself for knowing correct names for things: Needing to search for the subject later, either online or in an index of some sort. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, be aware that mathematical practice is always a shared endeavor. &lt;em&gt;You need to be able to read, write, and talk to other people about your practice.&lt;/em&gt; This may be somewhat masked in the environment up to undergraduate work, where the focus is on being tested in isolation. But along the way you may need to: Read or listen to future presentations, ask questions to professors, understand their answers, work with students in a study group, receive help from a tutor, help another student with a critique or correction, work as a tutor or instructor yourself, read or write a paper discussing new research findings, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need the shared language to be able to communicate in these contexts. In theory, it might be possible to communicate in writing purely via algebraic expressions; but it would be tedious and fragile and is not commonly done. More pressingly, when speaking verbally it's effectively impossible to precisely express algebraic notation, and you need to know the definitions of the natural-language terminology to make that give-and-take efficient. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-01-30T02:43:28.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11961" PostTypeId="1" AcceptedAnswerId="11972" CreationDate="2017-01-30T14:28:24.797" Score="4" ViewCount="185" Body="&lt;p&gt;Which of these is the correct ordering:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a polynomial multiplied by a monomial&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;a monomial multiplied by a polynomial&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;if what I want to achieve is something like 3x(x+5)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are my initial thoughts:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we use the phrase &quot;divided by,&quot; order must stay the same, as in 9 divided by 3 = 9/3, or a polynomial divided by a monomial would imply polynomial/monomial. Thus the ordering of 9 multiplied by 3 should be 9(3), and monomial multiplied by polynomial is correct?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, if I have a person hit by a truck, it is the truck that did the hitting, not the person. So 9 multiplied by 3 implies that the 3 is doing an action to the 9, thus 3(9), and polynomial multiplied by monomial implies that the monomial should be distributed to each term of the polynomial, so polynomial multiplied by monomial is correct?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which is technically correct? Which phrase paints the picture the correct way in the mind of the typical student?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="6675" LastEditorUserId="6675" LastEditDate="2017-01-30T14:34:48.400" LastActivityDate="2017-02-01T12:46:47.607" Title="Proper ordering of phrase &quot;multiplied by&quot;" Tags="|secondary-education|terminology|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="11962" PostTypeId="2" ParentId="11955" CreationDate="2017-01-30T16:45:42.407" Score="0" Body="&lt;p&gt;One of the important meta-lessons you will learn in differential equations is how to deal with problems you cannot solve completely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For almost all differential equations, no one knows how to write a closed form formula for the solution.  Nevertheless, this doesn't mean you know nothing about the solution.  Frequently, one can learn about many properties, either of individual solutions, or about how different solutions are related to each other, &lt;strong&gt;without&lt;/strong&gt; knowing formulas for the solutions.  The goal is to extract as much information as possible about the solution even if you cannot solve the equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The terms you are learning relate to various properties of the solutions or the relationship between different solutions.  Sometimes these properties can be combined with each other to actually solve the equation, but often you won't be able to solve the differential equation, and these properties will still give you useful information.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2017-01-30T16:45:42.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11963" PostTypeId="1" CreationDate="2017-01-30T19:32:54.370" Score="17" ViewCount="22187" Body="&lt;p&gt;What are some good topics for an undergraduate Math seminar?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for topics which are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Approachable for at least second or third year students and beyond (The students have taken all of the introductory Math courses, Logic, Real Analysis, Discrete, Algebra, etc)&lt;/li&gt;&#xA;&lt;li&gt;outside of standard course material.&lt;/li&gt;&#xA;&lt;li&gt;tangentially connected to things the students already know.&lt;/li&gt;&#xA;&lt;li&gt;presentable from 45 to 90 minutes&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Not all of the above points are required, but as long as it satisfies most of the above it should be relevant. Above all else, if you think it is a good topic idea please share it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: If you have books or resources for the seminar topic, that is also very helpful.&lt;/p&gt;&#xA;" OwnerUserId="7742" LastEditorUserId="7742" LastEditDate="2017-01-31T18:21:55.320" LastActivityDate="2023-02-15T17:58:23.457" Title="topics for an undergraduate Math seminar" Tags="|undergraduate-education|presentation|seminar|" AnswerCount="11" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="11964" PostTypeId="2" ParentId="11961" CreationDate="2017-01-30T21:56:32.867" Score="3" Body="&lt;p&gt;Consider the article at the MAA website by Keith Devlin (Executive Director of the Human-Sciences and Technologies Advanced Research Institute at Stanford University and The Math Guy on NPR's Weekend Edition), &lt;a href=&quot;http://www.maa.org/external_archive/devlin/devlin_01_11.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;What Exactly is Multiplication?&quot;&lt;/a&gt;. In it, he writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For instance, the mathematician's concept of integer or real number&#xA;  multiplication is commutative: M x N = N x M. (That is one of the&#xA;  axioms.) The order of the numbers does not matter. Nor are there any&#xA;  units involved: the M and the N are pure numbers. But the&#xA;  non-abstract, real-world operation of multiplication is very&#xA;  definitely not commutative and units are a major issue. Three bags of&#xA;  four apples is not the same as four bags of three apples. And taking&#xA;  an elastic band of length 7.5 inches and stretching it by a factor of&#xA;  3.8 is not the same as taking a band of length 3.8 inches and stretching it by a factor of 7.5...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;With multiplication you have a &lt;em&gt;multiplicand&lt;/em&gt; (written second)&#xA;  multiplied by a &lt;em&gt;multiplier&lt;/em&gt; (written first). The unit for the&#xA;  multiplier has to be sets of the unit for the multiplicand. For&#xA;  example, if you have 3 bags each containing 5 apples, then you can&#xA;  multiply to give &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[3 BAGS] x [5 APPLES PER BAG] = 15 APPLES.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So while Devlin makes a distinction and naming convention (that I've seen elsewhere) between different factors, unfortunately he doesn't clarify the natural-language ordering.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's look at a few mass-market elementary algebra books to see how the section titles are phrased for the simplification of something like $3x(x+5)$:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Martin-Gay, &lt;em&gt;Prealgebra &amp;amp; Introductory Algebra&lt;/em&gt; (Ch. 10): &quot;Multiplying Monomials by Polynomials&quot;&lt;/li&gt;&#xA;&lt;li&gt;Bittinger, &lt;em&gt;Intermediate Algebra&lt;/em&gt; (Ch. 4): &quot;Multiplying Monomials and Binomials&quot;&lt;/li&gt;&#xA;&lt;li&gt;Ratti &amp;amp; McWaters, &lt;em&gt;Precalculus: a right triangle approach&lt;/em&gt; (Ch. P): &quot;Multiplying a Monomial and a Polynomial&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So based on that short survey, it seems like your second formulation is more customary. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-01-30T21:56:32.867" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11965" PostTypeId="2" ParentId="11963" CreationDate="2017-01-30T23:30:22.740" Score="4" Body="&lt;p&gt;Here are three books I have used in various ways for such a course, focused on mathematics history.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471500305.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Journey Through Genius&lt;/a&gt; - lower level in some ways but easy to supplement for any background&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.math.nmsu.edu/~history/#Expeditions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Expeditions&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.math.nmsu.edu/~history/#Capstone&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Masterpieces&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Any book by Eli Maor is also definitely game.  Again, supplementing for higher levels - but there is a surprising amount that they won't know!  So much math, so little time.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-01-30T23:30:22.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11967" PostTypeId="2" ParentId="11961" CreationDate="2017-01-31T02:32:59.650" Score="6" Body="&lt;p&gt;Drawing an analogy to the multiplication of matrices (where order does matter):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I read &quot;$A$ multiplied by $B$&quot; I would probably think of $AB$ first, but probably I would realize that multiplication of matrices is composition of linear maps, and I would than wonder if it could be $BA$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, my conclusion is, that I would not write &quot;$A$ multiplied by $B$&quot; at all, but rather &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$A$ times $B$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;for $AB$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In more a general non-commutative context, e.g. group theory, there are specialized notions, e.g. $gh$ means to &quot;multiply $h$ from the left with $g$&quot; or &quot;multiply $g$ from the right by $h$. Also an element $g$ may &quot;act from the left&quot; or &quot;act from the right&quot; which seems to indicate that neither the left nor the right position is canonically more passive or active than the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an obviously commutative  situation like to multiplication of polynomials I would find it disturbing to read an additional &quot;left&quot; or &quot;right&quot; when referring to the multiplication and would ask myself if I missed some subtlety.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="3398" LastEditDate="2017-02-01T12:46:47.607" LastActivityDate="2017-02-01T12:46:47.607" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11968" PostTypeId="2" ParentId="11950" CreationDate="2017-01-31T18:23:35.647" Score="9" Body="&lt;p&gt;This question makes me think of the James Gleick quote on two kinds of genius: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;There are two kinds of geniuses: the 'ordinary' and the 'magicians'. An ordinary genius is a fellow whom you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what they've done, we feel certain that we, too, could have done it. It is different with the magicians. Even after we understand what they have done it is completely dark. Richard Feynman is a magician of the highest calibre.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know how to learn to think like a Fields's Medalist. Those are the magicians.  But helping students get plain old &lt;em&gt;mathematical insight&lt;/em&gt; might be something us ordinary mortals can shoot for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;em&gt;How Children Fail&lt;/em&gt;, John Holt describes (as anecdotes) his experiences where he has seen children make the switch from rote processing to having insight into the meaning of the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The text of the book is available online at &lt;a href=&quot;http://www.schoolofeducators.com/wp-content/uploads/2011/12/HOW-CHILDREN-FAIL-JOHN-HOLT.pdf&quot;&gt;http://www.schoolofeducators.com/wp-content/uploads/2011/12/HOW-CHILDREN-FAIL-JOHN-HOLT.pdf&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, the online PDF does not include page numbers. (!!!)  But some sections do have dates.  If you run a search for &quot;March 11, 1961&quot; you will find the story of Dorothy who, through sheer repetition &quot;got a feel&quot; for determining whether a number was evenly divisible by 2, 3, 4 etc:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;People to whom I have described this child's work have found it all but&#xA;impossible to believe. They could not imagine that even the most wildly&#xA;unsuccessful student could have so little mathematical insight, or would use&#xA;such laborious and inefficient methods to solve so simple a problem. The&#xA;fact remains that this is what the child did....&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also search for the date &quot;October 1, 1959&quot; to ready the story of Dr. Gattegno's demonstration of children with learning disabilities. Holt describes the demonstration:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I thought, what must it be like to have so little idea of the way the world works. So little feeling for the regularity, the sensibleness of things? . . . Then as I watched, the dark haird boy &lt;em&gt;saw&lt;/em&gt;! Something went &quot;click&quot; inside his head. And for the first time, his hand visibly shaking with excitement, he reached without trial and error for the right rod...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I first read that story close to 30 years ago, and it still moves me.  Unfortunately, these are mere anecdotes. Holt isn't able to definitively answer how to catalyze the transformation from trial and error to genuine insight.  But I find his accounts incredibly moving. Maybe you can determine from his stories what the &quot;essential ingredients&quot; are.&lt;/p&gt;&#xA;" OwnerUserId="5465" LastEditorUserId="5465" LastEditDate="2017-01-31T18:32:30.423" LastActivityDate="2017-01-31T18:32:30.423" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11969" PostTypeId="2" ParentId="11963" CreationDate="2017-01-31T18:47:57.993" Score="2" Body="&lt;p&gt;Many fine probability topics could be suitable, depending on background and interest.  A few such topics:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Random walk, culminating with the law of the iterated logarithm&lt;/li&gt;&#xA;&lt;li&gt;The law of large numbers, or the central limit theorem&lt;/li&gt;&#xA;&lt;li&gt;If not covered: The Laplace approximation for integrals&lt;/li&gt;&#xA;&lt;li&gt;Extreme value theory&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Some other topics:&lt;/p&gt;&#xA;&lt;p&gt;5: Matrices: The Perron-Frobenius theorem for nonnegative matrices, with application &amp;quot;the Google matrix&amp;quot;&lt;br /&gt;&#xA;6: Simulation:  Some examples of mcmc ...&lt;br /&gt;&#xA;7: Error-correcting codes ...&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="122" LastEditDate="2023-02-14T14:29:54.323" LastActivityDate="2023-02-14T14:29:54.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11970" PostTypeId="2" ParentId="11950" CreationDate="2017-01-31T19:36:53.660" Score="6" Body="&lt;p&gt;Take a look at this article:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The roles of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra (Tracy Goodson-Espy, 1998)&#xA;&lt;a href=&quot;http://The%20roles%20of%20reification%20and%20reflective%20abstraction%20in%20the%20development%20of%20abstract%20thought:%20Transitions%20from%20arithmetic%20to%20algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://link.springer.com/article/10.1023/A:1003473509628&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So much has been written about the construction of a mathematical &quot;edifice,&quot; by the building of abstraction upon abstraction, that I'm not even going to try to cite a source.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic premise of the article, as I understand it, is this: Students need to work (e.g., solve problem with) with an abstract idea until it begins to feel concrete to them.  But once the abstract notion has been reified, students (or mathematicians, for that matter) can use it in the construction of even more abstract ideas.  (&quot;Reify,&quot; by the way, comes from Latin &lt;em&gt;res&lt;/em&gt;, meaning &quot;thing.&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, some time ago I read John Derbyshire's &lt;em&gt;Prime Obsession&lt;/em&gt; on the Riemann Hypothesis. The hypothesis is easily stated (from Wikipedia):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is, to understand the Riemann Hypothesis requires understanding the Riemann Zeta Function.  And understanding the Riemann Zeta Function requires layer upon layer of abstraction.  The problem is I read the book like a novel, instead of textbook.  Once I thought I understood a particular abstraction, I moved on to the next chapter.  But understanding the words in a sentence is a far cry from understanding at a gut level.  In order to truly understand the Riemann Hypothesis, I would probably have to spend &lt;em&gt;years&lt;/em&gt; reading and digesting Derbyshire's book, along with a textbook on number theory and plenty of practice problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think that is part of the answer to your question: Don't go too fast.  Don't move on to a more abstract notion until the ones that it builds upon are deeply ingrained in you (or your students). That takes time, practice, trial and error, and a certain &lt;em&gt;capacity&lt;/em&gt; for tolerating abstract thought.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT (3/11/2017): Here's a related quote, taken from Feynman's &lt;em&gt;What Do You Care What Other People Think&lt;/em&gt;, that also refers to the need to base new ideas on firmly rooted old ideas:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It's natural to explain an idea in terms of what you already have in&#xA;  your head.  Concepts are piled on top of each other: this idea is&#xA;  taught in terms of that idea, and that idea is taught in terms of&#xA;  another idea, which comes from counting, which can be so different for&#xA;  different people!  I often think about that when I'm teaching some&#xA;  esoteric technique, such as integrating Bessel functions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5465" LastEditorUserId="5465" LastEditDate="2017-03-11T19:28:21.350" LastActivityDate="2017-03-11T19:28:21.350" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11971" PostTypeId="1" AcceptedAnswerId="11975" CreationDate="2017-01-31T21:28:11.473" Score="9" ViewCount="14587" Body="&lt;p&gt;I always was non math background student and programming is my hobby. I was attempting to program code instruction &lt;a href=&quot;https://stackoverflow.com/questions/8587047/support-resistance-algorithm-technical-analysis/8590007#8590007&quot;&gt;given here&lt;/a&gt;. Since I don't know calculus I'm stuck. I would like to know what are the things I need to learn before I start learning calculus? I can solve some basic algebra problems.&#xA;Could anyone please guide me? Thanks.&lt;/p&gt;&#xA;" OwnerUserId="7751" LastEditorUserId="5220" LastEditDate="2017-12-18T16:12:10.337" LastActivityDate="2023-10-29T16:07:41.467" Title="Requirements to learn calculus" Tags="|precalculus|" AnswerCount="10" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="11972" PostTypeId="2" ParentId="11961" CreationDate="2017-02-01T09:11:11.930" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;a monomial multiplied by a polynomial&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is best. Maybe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is important for students to understand the &lt;a href=&quot;https://en.wikipedia.org/wiki/Commutative_property&quot; rel=&quot;nofollow noreferrer&quot;&gt;commutative property&lt;/a&gt; of multiplication (and other binary operators) and become procedurally fluent in using it with numbers, variables, monomials, polynomials and all other forms of algebraic expressions. However, after reading your comments carefully, I see what you really mean and I think there is an answer for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we write a monomial, we try to follow the convention of writing numbers before letters. For example, by convention we write $4x$ not $x4$. Both are mathematically correct. But we are human and some ways a clearer than other ways. Also, using conventions reduces interpretive effort, allowing us to get more done in less time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$4x^2+4x$ factorises to $4x(x+1)$. We prefer $4x(x+1)$ over $(x+1)4x$. Therefore, by convention, not mathematical rule, we put monomials before polynomials in a factorised expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That’s fine for mathematical expressions, but how about the word expression “a monomial multiplied by a polynomial”? Conventions get a little muddy here. If conventions get muddy, it is probably because there isn’t really a need for a convention. You could argue that the order in which things should appear in a sentence is the order in which they appear in the mathematical expression. So, for $4x$ we would say “$4$ multiplied by $x$” and not “$x$ multiplied by $4$”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some will argue that because &lt;strong&gt;$x$ multiplied by $4$&lt;/strong&gt; means $4$ acts on $x$, the phrase “multiplied by” is not commutative. However, when used in the context of mathematics, the binary multiplication operator is commutaive and the product of $4$ acting on $x$ is the same as the product of $x$ acting on $4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, do you strictly apply the language syntax order that matches the desired conventional order in the mathematical expression?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have recently been reading Steven Pinker (&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Sense_of_Style&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Sense of Style&lt;/a&gt;) and what he emphasises in writing is “flow”. Consider the flow of your writing. Not just a touchy-feely “flow” but an actual cognitive flow. If students are trying to visualise mathematical expressions as you speak or write, then it would be choppy to say B before A if you then by convention write A before B, unless they are so proficient with the syntax of English language that they would view this as an unforgivable travesty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you intend to write the monomial before the polynomial in a mathematical expression (e.g. $3x(x+5)$) then you should say $3x$ multiplied by $x+5$, or &lt;strong&gt;a monomial multiplied by a polynomial&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the sake of consistency (and to avoid confusion with your students, especially those for whom English is not their native language), &lt;strong&gt;use the word order that matches the expression order – from left to right!&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="7046" LastEditorUserId="7046" LastEditDate="2017-02-01T09:48:45.323" LastActivityDate="2017-02-01T09:48:45.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11973" PostTypeId="2" ParentId="11963" CreationDate="2017-02-01T15:26:15.603" Score="6" Body="&lt;p&gt;Almost any chapter of this wonderful book could serve as a seminar topic.&#xA;Each chapter is pretty much self-contained.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Matoušek, Jiří. &lt;em&gt;Thirty-three miniatures: Mathematical and Algorithmic applications of Linear Algebra&lt;/em&gt;. Vol. 53. American Mathematical Soc., 2010.&#xA;  (&lt;a href=&quot;http://bookstore.ams.org/stml-53&quot; rel=&quot;noreferrer&quot;&gt;AMS link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/JzrmZ.jpg&quot; height=&quot;300&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-02-01T15:26:15.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11974" PostTypeId="2" ParentId="11963" CreationDate="2017-02-01T23:23:09.117" Score="5" Body="&lt;p&gt;(1) I am shocked that no one has mentioned the so-called Monty Hall problem/paradox. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) The point that &quot;probability&quot; does not directly describe &quot;randomness&quot;, e.g., although the sequence of heads-tails HHHHHHHHHH seems &quot;unlikely/not-random&quot;, it has the same probability ($2^{-10}$) as &quot;random-seeming&quot; sequences such as HHTHTTTHHT.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) &quot;The other quadratic formula&quot; (this one is more shallow, but fun): if $ax^2+bx+c=0$, and $c\not=0$, then also $a+b(1/x)+c(1/x)^2=0$, and the quadratic formula for $1/x$, inverted, gives an unexpected formula for $x$ itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) Using linear algebra of 2x2 matrices to get the formula for the $n$th Fibonnaci number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(5) Liouville's theorem (really just a corollary of the mean value theorem!) that an irrational real number that &quot;can be approximated too well&quot; by rationals must be &lt;em&gt;transcendental&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-02-01T23:23:09.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11975" PostTypeId="2" ParentId="11971" CreationDate="2017-02-02T03:18:51.047" Score="14" Body="&lt;p&gt;When I teach people calculus, the big reasons they don't succeed tend to be problems with arithmetic and very basic algebra. For example, students won't know how to compute 2/(3/4), or they'll try to simplify $1/(x+y)$ to $1/x+1/y$. If you can handle this kind of stuff, then you're better prepared to learn calculus than most of my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is some material often taught in a trig or precalculus course that may also be useful, but it's not critical. For example, it may be helpful to know what a function is; to know how to manipulate exponentials, e.g., $e^{a+b}=e^ae^b$; and to know trigonometry. If you don't know about trigonometric functions and exponential functions, then you won't be prepared to do calculus on them.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2017-02-02T03:18:51.047" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11979" PostTypeId="1" CreationDate="2017-02-02T20:08:19.487" Score="19" ViewCount="29249" Body="&lt;p&gt;I am 26 now. I left study after completing school. Now I am thinking of doing a Bachelor's degree in mathematics. I purchased a basic &lt;strong&gt;Calculus&lt;/strong&gt; book and I found it interesting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I took some advice from a PhD scholar. His straight answer was that I am not gonna be successful. He quoted some words from a mathematician, &quot;Mathematics is young man's game&quot;.&lt;/p&gt;&#xA;" OwnerUserId="7756" LastEditorUserId="376" LastEditDate="2017-02-04T12:29:29.143" LastActivityDate="2017-02-05T03:40:56.880" Title="Is it too late for me to start learning mathematics?" Tags="|self-learning|" AnswerCount="9" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="11980" PostTypeId="2" ParentId="11979" CreationDate="2017-02-02T20:20:59.887" Score="33" Body="&lt;p&gt;Certainly someone your age (or even much older) can learn Calculus, even get a degree in mathematics.  And get a good job afterward.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That &quot;young man's game' quote refers to doing mathematical research at the highest level.  There is some validity to it, but there are also well-known counterexamples.  (And even most young men never make it there.)&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-02-02T20:20:59.887" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11981" PostTypeId="2" ParentId="11979" CreationDate="2017-02-02T20:24:37.117" Score="10" Body="&lt;p&gt;For sure you are not too old. If you are really interested and if you are motivated to work hard, then you can have a shot. Maybe you try some of the exercises in the book to get an impression, how well you are in teaching yourself maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A friend of mine is a philosopher and he started with maths, when he was older than 30 and it worked quite well. &lt;/p&gt;&#xA;" OwnerUserId="7758" LastActivityDate="2017-02-02T20:24:37.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11982" PostTypeId="2" ParentId="11979" CreationDate="2017-02-03T02:27:15.457" Score="3" Body="&lt;p&gt;It is never to late to expand and exercise your mind. The stimulation alone will keep your brain young. Math is exercise for the brain. How strong you want to be is largely up to you. The more you learn the more potential you have to learn. &lt;/p&gt;&#xA;" OwnerUserId="7760" LastActivityDate="2017-02-03T02:27:15.457" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11983" PostTypeId="2" ParentId="11979" CreationDate="2017-02-03T07:43:47.357" Score="7" Body="&lt;p&gt;It really depends on what your goal is. If you want to be the best Mathematician, it may be hard no matter when you start. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess the major reason why you want to study Mathematics is just to enrich your way of thinking. No one is too late for that purpose. Also there are many benefits of studying math, like logical thinking and abstraction.&lt;/p&gt;&#xA;" OwnerUserId="7764" LastActivityDate="2017-02-03T07:43:47.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11984" PostTypeId="2" ParentId="11979" CreationDate="2017-02-03T09:20:38.037" Score="3" Body="&lt;p&gt;A good answers is the personal story of Barbara Oakley who change from Language to engineering after 20.&lt;br&gt;&#xA;There is a TED talk : &lt;a href=&quot;https://www.youtube.com/watch?v=O96fE1E-rf8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=O96fE1E-rf8&lt;/a&gt;&#xA;and a MOOC&#xA;Learning how to learn :&#xA;&lt;a href=&quot;https://www.coursera.org/course/learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.coursera.org/course/learning&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7765" LastActivityDate="2017-02-03T09:20:38.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11985" PostTypeId="2" ParentId="11979" CreationDate="2017-02-03T10:12:03.563" Score="5" Body="&lt;p&gt;Just a little example on this topic - when I got into studies(at the age of 18), I was top student in my generation along with a student who was 40 years old, decided to give it a try at that age and who has family. He got top scores in all math subjects (algebra, analysis etc.), so it means you can definitely succeed if you are interested enough. I find math beautiful and enjoy every bit of it, so as long as you are interested enough and have even a slight talent for it, you should give it a try.&lt;/p&gt;&#xA;" OwnerUserId="7766" LastActivityDate="2017-02-03T10:12:03.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11987" PostTypeId="2" ParentId="11979" CreationDate="2017-02-03T12:05:08.287" Score="3" Body="&lt;p&gt;You can start learning to any subject at any age of your life, as long as you are interested toward the subject age doesn't matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quote you mentioned here &quot; &lt;em&gt;Mathematics is young man's game&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where ; YOUNG word is not always concern with your body age, &#xA;If you are looking forward to learn something your &lt;strong&gt;mind&lt;/strong&gt; and &lt;strong&gt;brain&lt;/strong&gt; is young always, no matter how old you are.&lt;/p&gt;&#xA;" OwnerUserId="7769" LastActivityDate="2017-02-03T12:05:08.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11988" PostTypeId="2" ParentId="11971" CreationDate="2017-02-03T19:39:06.630" Score="3" Body="&lt;p&gt;Naturally, it's what they call &lt;em&gt;Precalculus&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;.. where &quot;they&quot; is whoever's calculus materials you'll be using.  While there is some variety, I expect it to include algebra to a few steps beyond the most basic problems, some geometry, and at least the basics of trigonometry.  Some curricula also introduce limits in precalculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In algebra, you should be comfortable using basic identities including those in @BenCrowell's answer, solving quadratic equations, arithmetic on polynomials, raising polynomials to small exponents, dividing and factoring polynomials, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In geometry you should understand cartesian coordinates, be familiar with graphing functions, and know the equations for basic shapes.  It may be helpful to get the gist of parametric equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In trigonometry you should be comfortable solving algebra and geometry problems involving trig functions or which require trig functions for their solutions, and using the common trig identities to rewrite expressions.  It may be helpful to know something about the relationship with complex exponents and/or about hyperbolic functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that's for learning Calculus overall, and I've been somewhat inclusive to hedge against variety in calculus curriculum.  If you're studying on your own you can probably backfill as needed to a large degree.  If you're only interested in this one problem and have access to knowledgable people (or can hire a tutor) you can probably focus pretty narrowly and learn a small subset just sufficient for this one problem.  Glancing at the problem, it touches on techniques from numerical analysis and statistics, so at some point it might be better to consult a knowledgeable person and target specifics rather than to try to (in effect) take all of the courses involved.  (That might come close to getting a math minor - which I wouldn't discourage, but it depends on your goals.)&lt;/p&gt;&#xA;" OwnerUserId="1019" LastActivityDate="2017-02-03T19:39:06.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11989" PostTypeId="2" ParentId="11971" CreationDate="2017-02-03T22:58:43.490" Score="5" Body="&lt;p&gt;Traditional calculus study also requires trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays you will also find some &quot;watered down&quot; calculus texts with no trig functions, intended ironically* for business students.&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt;*You would think business students would be interested in cyclical phenomena, wouldn't you?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-02-03T22:58:43.490" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="11990" PostTypeId="2" ParentId="11971" CreationDate="2017-02-04T00:38:04.450" Score="5" Body="&lt;p&gt;From Velleman's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486809366&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Calculus: A Rigorous First Course&lt;/strong&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Bodho.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Bodho.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to the content of the first chapter, it includes (but is not limited to) the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;decimal notation, integers, rational numbers, irrational numbers&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;sets, subsets, elements, unions, intersections, intervals&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;expressions, equations, inequalities&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;triangle inequality, Pythagorean Theorem (and distance formula)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;functions, domain, independent and dependent variables, composition of functions&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;absolute value function, square root function, linear functions (slope-intercept and point-slope forms), quadratic functions, polynomials, rational functions, trigonometric functions (along with how to use radians, the unit circle, etc)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="262" LastActivityDate="2017-02-04T00:38:04.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11991" PostTypeId="2" ParentId="11979" CreationDate="2017-02-04T06:48:49.270" Score="3" Body="&lt;p&gt;Firstly, check out &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Edward_Witten&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ed Witten&lt;/a&gt;; ok, he was 22 not 26 when he switched to maths and physics, but I think it is still a reference point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The young man's game thing is a common prejudice. I have wondered whether any truth in this is more on the lines of many people do peak early in their maths careers and given typical age profiles this means many have success ata a particular age.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not think this is the same thing as saying someone in their mid twenties is too old. What is necessary is the ability to immerse yourself in your subject.&lt;/p&gt;&#xA;" OwnerUserId="3587" LastActivityDate="2017-02-04T06:48:49.270" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11992" PostTypeId="2" ParentId="11963" CreationDate="2017-02-04T19:19:05.373" Score="4" Body="&lt;p&gt;How about a talk on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Continuum_hypothesis&quot; rel=&quot;nofollow noreferrer&quot;&gt;continuum hypothesis&lt;/a&gt; (CH) and related topics?  It's a fascinating subject and can be tailored for basically any level depending on how detailed you get.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Full disclosure, I'm biased because I gave such a talk aimed at MS students and upper-level undergrads on this topic almost 9 years ago.  And I think this meets all four of your criteria listed as of the time I write this, since they're very similar to what I was trying to do.  Here's an outline of my presentation as best as I can remember it; feel free to draw from it as you wish:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Different sets of numbers:  $\Bbb N, \Bbb Z, \Bbb Q, \Bbb R$.&lt;/li&gt;&#xA;&lt;li&gt;$\Bbb N$ and $\Bbb Z$ have the same size.&lt;/li&gt;&#xA;&lt;li&gt;More surprisingly, $\Bbb Q$ has the same size as both $\Bbb N$ and $\Bbb Z$.&lt;/li&gt;&#xA;&lt;li&gt;But the size of $\Bbb R$ is strictly larger.  (Cantor's diagonalization argument.)&lt;/li&gt;&#xA;&lt;li&gt;Is there a set whose size is strictly larger than $\Bbb N$ and strictly smaller than $\Bbb R$?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This question in the last bullet point above is the perfect way to bring up the CH (since the question is basically, &quot;Is the continuum hypothesis false?&quot;).  And what's interesting about the CH is the answer is independent of ZFC set theory, meaning we can take the answer to be &quot;yes&quot; or &quot;no&quot; and either way we won't get a contradiction in ZFC set theory.  We know this from work done independently by Kurt Gödel and Paul Cohen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think there were some slightly related topics and/or generalities I discussed, such as cardinal numbers and transfiniteness, etc.  It's been too long to say for sure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was my main reference:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0743422996&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity&lt;/a&gt; by Amir Aczel.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Oh, and bonus points if you take it to the next level &lt;a href=&quot;https://i.stack.imgur.com/nKpvn.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;like I might have&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt;  Another idea for a talk is how linear algebra can be used to determine solvability of a &lt;a href=&quot;https://en.wikipedia.org/wiki/Lights_Out_%28game%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;lights out game&lt;/a&gt;.  A good starting resource is &lt;a href=&quot;http://mathworld.wolfram.com/LightsOutPuzzle.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this site on Wolfram MathWorld&lt;/a&gt;.  I expanded on it &lt;a href=&quot;https://puzzling.stackexchange.com/questions/47291/putting-up-the-lights-two/47357#47357&quot;&gt;in this post on Puzzling.SE&lt;/a&gt;.  I'm sure the paper referenced in that Wiki article would also be a good resource.&lt;/p&gt;&#xA;" OwnerDisplayName="user6648" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:46.390" LastActivityDate="2017-02-04T20:51:24.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11993" PostTypeId="2" ParentId="11979" CreationDate="2017-02-05T03:40:56.880" Score="9" Body="&lt;p&gt;&lt;strong&gt;&lt;em&gt;No&lt;/em&gt;&lt;/strong&gt;, it is not too late.&lt;br&gt;&#xA;Not even close.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I failed at my first attempt at college, went back at 28, initially only part-time, and eventually ended up with bachelors degrees in math, physics, and CS.  Now I'm in the Ph.D program in CS.  You might not be as prepared/ambitious/indecisive/whatever as I was, but it's not too late for you to learn math.  I also studied with quite a few classmates who were coming back for another degree or who switched majors late, or who had worked in industry for years (or decades in a few cases).  If anything, the older ones were more interested and more focused.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure what your Ph.D. scholar friend had in mind; depending on why you're interested and how it goes it's not unlikely that getting a Ph.D. in math won't be for you.  But if a Ph.D. isn't for you it won't be because of your age.  If you can spend the time and money and are interested in the subject, you can learn it.  And you don't even have to be a full-time student.&lt;/p&gt;&#xA;" OwnerUserId="1019" LastActivityDate="2017-02-05T03:40:56.880" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="11994" PostTypeId="1" CreationDate="2017-02-05T15:52:35.010" Score="2" ViewCount="174" Body="&lt;p&gt;Assuming that I remember only basic arithmetic, what (if any) mathematics must I learn before tackling Boas' &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0471198269&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Methods in the Physical Sciences&lt;/a&gt;&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerDisplayName="user7783" LastActivityDate="2017-02-06T16:08:24.813" Title="Preparatory text/s for Boas' Mathematical Methods in the Physical Sciences?" Tags="|textbooks|" AnswerCount="1" CommentCount="0" ClosedDate="2017-02-08T19:33:58.460" ContentLicense="CC BY-SA 3.0" />
  <row Id="11995" PostTypeId="2" ParentId="11963" CreationDate="2017-02-06T05:17:43.717" Score="4" Body="&lt;p&gt;Personally I would discuss Graph theory, in a diverse and conceptual way with a few deeper dives. Students of that level should be familiar of weighted graphs, have an intuition for the size of functions (which could be leveraged in a discussion of travelling salesman, big O notation, how quickly networks can grow, Ramsey Theory R(3,3), chromatic polynomials, etc). There is so much novelty in graph theory, and it draws many areas of maths together in novel ways. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a minimum, I'd include Erdös Renyi Random Graph models, and the emergance of a large connected cluster. It is an accessible mention of asymptoic analysis, simple graph enumeration, and probability distributions(as the edges per node modelled via Poisson is appreciably different to edges as randomly selected pairs of nodes). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depends on time and how quickly you move through content, but you could try to mention dynamical systems, such as predator-prey models, and how graph theory can be applied in the spread of diseases and informtion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this topic is well suited, because it is an example of three important chracteristics of matha: Interconnected, Rich in application, and the sheer enormity of open problems. E.g. P=NP, Graph Homomorphism, Ramsey (5,5,), etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At 2nd year ans above, it is likely many already have heard of Monty Hall,C&#xA; Cardinal Infinities, and Mandelbrot(although this could be included if you hedge your bet with a split between dynamical systems and graph theorg). &lt;/p&gt;&#xA;" OwnerUserId="3252" LastActivityDate="2017-02-06T05:17:43.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11996" PostTypeId="2" ParentId="11963" CreationDate="2017-02-06T15:05:22.597" Score="2" Body="&lt;p&gt;If these are not present in course work:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Galois Theory: an overview of what it accomplished with an emphasis on history.&lt;/li&gt;&#xA;&lt;li&gt;Group actions and counting if not emphasized too deeply in the standard course.&lt;/li&gt;&#xA;&lt;li&gt;Interaction between the geometry of principal fiber bundles and the physics of Yang Mill's theory. Weyl's crazy idea and how it led to our modern theory of classical fields.&lt;/li&gt;&#xA;&lt;li&gt;What is representation theory? &#xA;&lt;em&gt;(I wish I knew all, but I'll settle for more, of the answers to this question)&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;What are spinors? On the representation of the Lorentz group and how a bit of imagination finds supersymmetry.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Really most graduate math classes condensed into a best-of talk without proofs can be very interesting. Also, this might serve to motivate students to take such classes if that is an option &lt;/p&gt;&#xA;&#xA;&lt;p&gt;( &lt;em&gt;or for non-graduate math possessing institution, such talks can help give students a better sense of the big world of math beyond the core course work. Naturally,attending a JMM or similar math meet is even better in this regard&lt;/em&gt; )&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-02-06T15:05:22.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="11997" PostTypeId="2" ParentId="11994" CreationDate="2017-02-06T15:53:37.807" Score="3" Body="&lt;p&gt;You would need to work your way through high school algebra (typically, 2 years of coverage, such as the two Mary P. Dolciani texts &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0395142555&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Algebra, Book 1&lt;/strong&gt;&lt;/a&gt; and &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B000J4LEJS&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Algebra, Book 2&lt;/strong&gt;&lt;/a&gt;) and precalculus (another year of coverage, such as Mary P. Dolciani's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0395286972&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Modern Introductory Analysis&lt;/strong&gt;&lt;/a&gt;), then a standard 3-semester calculus text (diff. calculus, integral calculus, multivariable calculus -- such as Thomas/Finney's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0201531747&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Calculus and Analytic Geometry&lt;/strong&gt;&lt;/a&gt;). That might be minimally enough, but better would be to also cover a one-semester course in ordinary differential equations (such as Abell/Braselton's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0124172199&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Introductory Differential Equations&lt;/strong&gt;&lt;/a&gt;) and a one-semester course in elementary linear algebra (such as Howard Anton's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0470559918&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Elementary Linear Algebra&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2017-02-06T16:08:24.813" LastActivityDate="2017-02-06T16:08:24.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11998" PostTypeId="2" ParentId="10910" CreationDate="2017-02-06T19:18:04.320" Score="7" Body="&lt;p&gt;I think that you might find useful Harel &amp;amp; Kaput's chapter in Tall's &lt;em&gt;Advanced Mathematical Thinking&lt;/em&gt; (1991). You can find it for free download at &lt;a href=&quot;http://www.math.ucsd.edu/~harel/publications/Downloadable/Conceptual%20Entities.pdf&quot;&gt;http://www.math.ucsd.edu/~harel/publications/Downloadable/Conceptual%20Entities.pdf&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarize, they identify three roles for &quot;conceptual entities&quot; (including reified objects):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;em&gt;Alleviating working memory or processing load&lt;/em&gt; when concepts involve multiple constituent elements.&lt;/li&gt;&#xA;  &lt;li&gt;Facilitating &lt;em&gt;comprehension&lt;/em&gt; of complex concepts: the cases of &quot;uniform&quot; operators, &quot;point-wise&quot; operators, and &quot;object-valued&quot; operators.&lt;/li&gt;&#xA;  &lt;li&gt;Assisting with the &lt;em&gt;focus of attention&lt;/em&gt; on appropriate structure in problem solving.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you have not read this chapter before, you may find it helpful.&lt;/p&gt;&#xA;" OwnerUserId="7789" LastActivityDate="2017-02-06T19:18:04.320" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11999" PostTypeId="2" ParentId="10910" CreationDate="2017-02-07T01:41:27.980" Score="4" Body="&lt;p&gt;Not a high-quality answer, since I can only refer to some decades' observation, rather than systematic/deliberate study of the phenomenon:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having observed maybe 1,000 pretty-darn-good grad students, some of them very good, I find that it is not typical among mathematically-interested, mathematically-talented people (at least 20-30 yrs.) to have a good instinct for &lt;em&gt;abstraction&lt;/em&gt;, which is surely a part of &lt;em&gt;reification&lt;/em&gt;. I see this throughout basic grad courses: to write Fourier transform as a linear mapping is vaguely disturbing... as is declaring &quot;integral&quot; a continuous linear mapping (rather than a limit of Riemann sums, or something described by the fundamental theorem of calculus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, in my observation (of others and myself), people become very attached to particulars, to specifics of a landscape, and it is jarring to find them declared irrelevant or illusory. This attitude to &quot;progress/abstraction&quot; can be manifest in particular to objection to processes becoming objects, since &quot;object&quot; does substantially connote inertness (though it's not so clear why it should). Actors versus acted-upon? (Stress over the second dual of a topological vector space having an imbedded copy of the original?!?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that leads me to believe that reification (and various other processes) are not widely natural even among &quot;mathematically talented&quot; people, whatever that may say about our specific psychology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people successfully broaden or enhance their world-view to embrace this idea, while others balk. It is possible to survive while rejecting (to a significant degree) various abstraction-methodologies, in my observation, though, to my taste, much of this seems like just killing time. Tastes vary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, I think that reification and other comparable abstraction processes are not immediately natural even for people who have sympathy for and talent for mathematics. Hence the resistance and difficulties.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-02-07T01:41:27.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12000" PostTypeId="2" ParentId="10910" CreationDate="2017-02-07T08:24:53.707" Score="4" Body="&lt;p&gt;The literature lacks a clear mechanism, that is why theories on the process-object-duality are criticized sometimes. Anna Sfard's &lt;em&gt;reification&lt;/em&gt;, Dubinsky's &lt;em&gt;APOS&lt;/em&gt; and Tall's &lt;em&gt;procept&lt;/em&gt; may help describing the situation, but hardly give specific advice what to do. (Note that Anna Sfard had suddenly stopped her work on reification and moved to commognition, a socio-constructive view on learning.) The clear strenght of these theories is, however, that they can well explain, why difficulties with seemingly simple problems occur, which build on object views. Think of power-sets or indexed variables which use the same concept twice (set of sets, variable variable) thus enforcing to think on object level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal feeling from the literature and my teaching and learning experience would give you two advices: First, do not think of a clear dichotomy of process or object view. Students often can handle something more or less like an object, maybe depending on the context.&#xA;Second, give &lt;em&gt;reification&lt;/em&gt; or &lt;em&gt;encapsulation&lt;/em&gt; a chance. You cannot force it, but try to find contexts in which an object view would make the situation much easier. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take the example of natural numbers. (Here, a missing object-view is often associated to discalculia.) Children start counting, this a process obviously. It is rather like saying a poem while pointing on physical objects. If they manage to not point on two objects while saying &quot;se-ven&quot;, you may let them count two objects on each step like 2, 4, 6, ... or even three objects. Still, it is a process but the role of the numbers slightly change. You may also want to look at the reverse process like counting 5, 4, 3, ... and so on. Then you treat numbers as if they were objects, asking what is 5+2? The process-view says it is counting on two more stepts, starting from 5. Then ask, what is 2+5? As a process, it is something completely different from 5+2, but identifying the results may help students recognizing that they just coordinate counting processes which are somewhat &quot;the same&quot;. You could go on asking for patterns (like what is 9+1, 8+2, 7+3, ...) so you essentially treat the numbers as objects but the students may still start working with a proces view. Slightly, you may move to questions like &quot;how many numbers are there between 4 and 8?&quot; treating numbers as objects while still students might start the counting process in answering.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can do very similar things on higher content like in undergraduate programs. Dubinsky et al. had great success in making their students programming on computers. A function, for example, needs to be distinguished from its values and treated like an object. Similarly to kindergarten, you may just play around, chaining functions, inversing functions (if possible, discussing the impossibility if not), restricting functions, extending functions, making functions become input or/and output of functions (operators!) and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will still find the &quot;resistance to thinking about objects rather than processes&quot; mentioned in your question. It is natural and abstraction will always take time. But without challenging this view in demanding tasks, reification/encapsulation is very unlikely to occur.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2017-02-07T08:24:53.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12001" PostTypeId="2" ParentId="11963" CreationDate="2017-02-07T08:35:17.190" Score="2" Body="&lt;p&gt;Basic topological ideas and knot theory are surprisingly accesible because they treat objects concretely representable in three space, via planar diagrams, and via actual physical models. (Basic ideas like isotopy can be illustrated with computer generated simulations.) Combinatorial methods in knot theory (knot polynomials, grid diagrams) can be learned and used without understanding their (sometimes quite deep) topological content. Students who know a bit of group theory can be shown connections with the braid group. There are numerous popular and undergraduate level accounts of knot theory, and most are full of examples that could be adapted to a classroom presentation. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2017-02-07T08:35:17.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12002" PostTypeId="2" ParentId="11757" CreationDate="2017-02-07T20:00:23.657" Score="5" Body="&lt;p&gt;Sure, try this data set:&#xA;$-2, -2, -1, -1, -1, 0, 0, 0, 3, 4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unless I've fudged things up, it has $\overline{x} = 0$, $\sum\left(\overline{x}-x_i\right)^2=36$, $N-1=9$.&lt;/p&gt;&#xA;" OwnerUserId="7789" LastEditorUserId="7789" LastEditDate="2017-02-08T13:38:30.543" LastActivityDate="2017-02-08T13:38:30.543" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12003" PostTypeId="1" CreationDate="2017-02-07T20:05:36.853" Score="5" ViewCount="258" Body="&lt;p&gt;I've got an issue: from time to time I have to teach some math to people who either avoided it, or got through by only knowing a few working algorithms. The only thing that unites all those people: they are sure that math is extremely boring and there's no sense an learning anything except for how to solve the tasks they're given.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to be sure that my lessons not only provide primitive formulas or concepts, but motivate students for learning more about the topic, thus I need to show how all that 'pointless knowledge' may turn out to be not so pointless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing that I usually find myself troubled by lacking proper examples of why a sophisticated theory can actually turn out to be benificial for anyone who isn't pursuing a career of a scholar, programmer, nuclear physicist etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for now I'm limited to usual stuff:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Showing how some analysis tricks can make computations easier (L'Hopital's rule and all that) or even possible (in case our only option for a function is approximating it with a series)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Show how grasping visual ideas can make it much easier to use all monstrous formulas — case of basic linear algebra (determinants, linear transformations, matrix multiplication etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Explaining how treating multiplication as area can allow to derive all neede polynomial formulas (including solving quadratics) without knowing anything else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Refering to how going beyond the scope of a usual math course can make life easier while dealing with polynomials — e.g. knowing the Fundamental theorem and rules of polynomial division, rational roots test etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a desperate search of inspiration I've watched some video courses (the best one yet is Revisiting Calculus by Herb Gross) as well as popular math youtubbers (Numberphile, Mathologer, 3Blue1Brown and others), but still feel like I can use a lot of help from people of the Math community who have way more experience in the field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't teach anything fancy: basic elementary math, some trigonometry, calculus, and basic linear algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks kindly for your insights and suggestions.&lt;/p&gt;&#xA;" OwnerUserId="7794" LastEditorUserId="238" LastEditDate="2019-11-02T23:53:22.460" LastActivityDate="2019-11-03T01:43:25.253" Title="Benefits of knowing theory" Tags="|undergraduate-education|secondary-education|examples|intuition|" AnswerCount="1" CommentCount="12" ClosedDate="2019-11-05T04:08:07.503" ContentLicense="CC BY-SA 4.0" />
  <row Id="12004" PostTypeId="2" ParentId="11971" CreationDate="2017-02-08T03:40:10.627" Score="2" Body="&lt;p&gt;The answer depends on the level you want to learn calculus at. I'll assume you want to learn it the way most North American students do, with little emphasis on proofs and theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that case, the content of Serge Lang's &lt;em&gt;Basic Mathematics&lt;/em&gt; is more than enough preparation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, Marsden and Weinstein's &lt;em&gt;Calculus I&lt;/em&gt; has a self-test section at the beginning to tell you if you can start learning calculus directly, if reading their review chapters is enough (and which ones), or if you should go back and learn from a precalculus book.&lt;/p&gt;&#xA;" OwnerUserId="7797" LastActivityDate="2017-02-08T03:40:10.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12006" PostTypeId="1" AcceptedAnswerId="12007" CreationDate="2017-02-08T20:23:30.053" Score="9" ViewCount="494" Body="&lt;p&gt;Recently I was asked to tutor some students individually on a basic course of Linear Programming, centered on all the &quot;typical&quot; introductory subjects like graphical methods, Simplex Algorithm and Transfer Pricing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, this is not the first time I teach someone on a University level, but it is the first time that I was asked-as part of my job as a tutor-to teach a subject I don't like and which in my years as a student spend a very limited time studying. Basically I studied for a couple of weeks or so before the exams and very rarely attended class. I only picked it as a course because it seemed interesting at first, but when I became disenchanted I considered it a matter of principle to pass it-and that I did. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as you guess my understanding of it is rather limited and although I am confident that I can quickly bring myself to a competent enough level to teach it, I am beset by doubt.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Even if one has a good grasp of the material he is supposed to teach,how does one overcome his dislike of this material? &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Is it perhaps better to simply refuse to teach a subject you don't like?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;br&gt;&#xA;I am less worried about any implications this will have for me on a professional level and more worried about doing the &quot;right thing&quot;. &lt;/p&gt;&#xA;" OwnerUserId="7697" LastActivityDate="2017-02-11T18:19:25.473" Title="Should I teach a subject I don't like?" Tags="|undergraduate-education|general-pedagogy|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12007" PostTypeId="2" ParentId="12006" CreationDate="2017-02-08T23:47:46.443" Score="13" Body="&lt;p&gt;This is a contentious and highly individual thing, of course, so all answers in this should be taken with a grain of salt.  But, if this is your &lt;em&gt;job&lt;/em&gt; then:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;YES.&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why?  First, some practical reasons that apply to any job.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You are not an island.  Some topics and/or classes are distasteful to everyone, or at least to everyone who is on the list to teach it.  Part of having a real job is sometimes taking on distasteful tasks for good reasons.&lt;/li&gt;&#xA;&lt;li&gt;In a well-functioning department, usually one is not asked again and again to do something really personally not preferred, at least not without being excused from something else bad.  Roommates all have to pay rent, but maybe one person pays less but then has to take out the garbage.&lt;/li&gt;&#xA;&lt;li&gt;Jobs are not just about what you are doing, they are about who you are doing them with.  (At least sometimes.)  If you have a motivated student to tutor, or one where even while tutoring something boring, you get to see their eyes light up with, &quot;Oh, that's how it works!&quot; then the intrinsic rewards may be substantial.  That's a lot different from writing homework problems for LP for a textbook.&lt;/li&gt;&#xA;&lt;li&gt;You could get fired, or blacklisted.  Let's hope not, but that is a reality.  In the wider academic universe, you could be denied tenure (at least, outside the rarified atmosphere of Research I universities).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There is another reason which is important, but which you may not yet be at a point to internalize if you are early in your math teaching/tutoring career.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes we learn, in teaching something we find boring or distasteful, that there is a rich depth we didn't anticipate.  I was not particularly enamored of analysis any of the &lt;em&gt;three&lt;/em&gt; times I took it at various levels.  But teaching it now, I see how it is really about topology and the very nature of the reals - even if I can't always say that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may be wishing you were tutoring someone in real analysis.  Fine.  But linear programming and optimization in this sense also have huge depth.  Go crazy - find real-life data to optimize off of, look into the burgeoning literature on operations research for disaster relief, start learning about integer programming versus &quot;regular&quot; linear programming and polytopes... the world is your oyster.  I wish I would have understood this when I taught linear programming in a very similar situation to yours, I think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, this may not convince you, and it may indeed be right that you shouldn't teach this.  I shouldn't have taught a stats course in the first five or six years I was teaching, for instance, and I would still not be overly enthusiastic about it.  Maybe you have a bad manager.  If it is really &lt;em&gt;that&lt;/em&gt; distasteful, maybe you need to find a different job (if possible).  And of course it wouldn't hurt to be looking elsewhere if you are pretty sure that this will be the &lt;em&gt;most&lt;/em&gt; enticing topic you will be tutoring in the near future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that sounds like a bit of a stretch from your description.  If you are in a well-functioning situation where you otherwise would like to stay and where you will not perpetually be asked to do things you don't like, and perhaps even where there is support for professional development, then just do it.  If having to teach something we find boring once in a while sounds bad, then &lt;a href=&quot;http://existentialcomics.com/comic/98&quot;&gt;see the 13th panel in this amusing webcomic&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-02-08T23:47:46.443" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12008" PostTypeId="2" ParentId="2074" CreationDate="2017-02-09T02:52:22.717" Score="6" Body="&lt;p&gt;I am a high school math teacher and whenever I teach the geometry class, I always do full proofs.  I use the geometry class to show kids how we construct all the ideas of geometry from a single point and I use proofs to explain to them the idea of mathematical truths.  I teach at an IB school so this preps them for the TOK course they'll take in a few years.  But proofs in the Geometry unit really gives me the opportunity to wax philosophical and some the kids from countries where they are just expected to regurgitate facts really appreciate learning that there is an answer &quot;why is this true?&quot; beyond &quot;because I said so.&quot;, behind everything they have learned.&lt;/p&gt;&#xA;" OwnerUserId="7800" LastEditorUserId="5563" LastEditDate="2017-02-14T00:30:45.987" LastActivityDate="2017-02-14T00:30:45.987" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12009" PostTypeId="1" CreationDate="2017-02-09T14:14:10.910" Score="5" ViewCount="208" Body="&lt;p&gt;Do anybody have a program which shows a computer graphing a function pointwise in stages? For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At stage 1, it graphs all the integer x values&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stage 2, all the multiples of 1/2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stage 3, all the multiples of 1/4&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and so on until we have a full picture of a continuous function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A single continuous animation of random points being plotted until it fills in a continuous graph would be just as good. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comment if you are interested in this. I might try my hand at making one if nobody has already. &lt;/p&gt;&#xA;" OwnerUserId="7734" LastActivityDate="2017-02-13T00:06:28.687" Title="Animation of graphing a function" Tags="|graphing|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12010" PostTypeId="2" ParentId="12006" CreationDate="2017-02-09T15:53:54.067" Score="3" Body="&lt;p&gt;Tell the person who asked you that you are willing to do some extra preparation on your own to carry out this assignment, but frankly they'd be better off giving this assignment to someone else, because you and linear programming didn't &quot;click&quot; when you took the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they can't find anyone else, they'll ask you again, but they'll be forewarned, and your accepting the assignment will give you a brownie point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can take a few hours over the next few days, it might be helpful to do some reading on the subject to find out more about what this assignment would entail.&lt;/p&gt;&#xA;" OwnerUserId="4956" LastActivityDate="2017-02-09T15:53:54.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12011" PostTypeId="2" ParentId="2074" CreationDate="2017-02-10T12:10:00.317" Score="9" Body="&lt;p&gt;Since no one here seems to be speaking against the teaching of Euclidean Geometry in highschool I guess I'll explain a few structural problems I have with EG.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My main complaint is what Toscho sees as an good feature. In particular:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Loss of Calculations&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ok, fair enough, standard EG following in the steps of the long dead Euclid proceeds from Axioms to Theorem with out so much as a number in sight. I studied it in high school, admittedly the construction of perpendicular bisectors etc. by compass and straight-edge alone is fascinating. But, I ask, is this loss of calculation natural in our modern age ? Have students already been exposed to math which is over a millenia beyond Euclid &lt;strong&gt;before&lt;/strong&gt; the take EG ? The answer to my question is quite obvious since about the 17th century. From a student's perspective, what I think is a more accurate portrait of EG is this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Artificial&lt;/strong&gt; loss of calculations&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ok, so, I can't trisect an angle. &lt;strong&gt;Really&lt;/strong&gt;, because if I use a ruler and a compass we can trisect an angle. Or, to be more direct, if I have a protractor and the idea of dividing real numbers by $3$ then trisecting an angle is not a big deal. In short, students already know the logical successor of EG, in particular &lt;strong&gt;analytic geometry&lt;/strong&gt; is woven throughout our modern discussion of Mathematics and its application to geometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue, students would be better served by studying proofs which involve calculation. One of the great absurdities, created in part by the existence of the &quot;transition to proofs&quot; class, is this idea that there is some conceptual divide between math involving equations and math involving proofs. &lt;strong&gt;Certainly not.&lt;/strong&gt; In fact, what we have is merely a dichotomy in &lt;strong&gt;standards.&lt;/strong&gt; On the one hand, the student has an algebra course where to find the vertex of a parabola they simply use an equation, often with no expectation of supporting argumentation or explanation, just the plug-and-chug method so to speak. Then that same student in their EG course is asked to prove something which is frankly &lt;strong&gt;obvious&lt;/strong&gt; with a 5 line column proof where they have to refer to each Axiom and step with more precision than we require in some proof classes at the undergraduate level. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't say what is best to replace EG with, but, I would tend to agree with Burke that is not actually about curriculum. The real struggle here is about &lt;strong&gt;standards&lt;/strong&gt;. Why do we insist on having standards in EG, but, not so much in other areas of math. Students should be expected to explain &lt;strong&gt;why&lt;/strong&gt; they are calculating in a particular way. The over-emphasis of &lt;strong&gt;problem-solving&lt;/strong&gt; verses the coherence of theory is the true root of the problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't blame teachers most places, there is simply no intellectual will-power or structural room to build good mathematics. We're too busy meeting SOL's and worrying about inane federal guidelines and such. The idea that students should be held to a higher standard is at odds with every other societal direction, so, I don't think there is a global fix. The solution is at most local, with particular educators investing their vision and coherent understanding of Math where proof and critical analytical thinking are woven into &lt;strong&gt;every&lt;/strong&gt; math class, not just the EG course. This is why I am a strong advocate for every math teacher having at least an undergraduate degree in math.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-02-10T12:10:00.317" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12012" PostTypeId="2" ParentId="12009" CreationDate="2017-02-10T18:35:03.263" Score="8" Body="&lt;p&gt;This is pretty straightforward in Mathematica:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;f[x_] := x^2;&#xA;xrangeleft = -2;&#xA;xrangeright = 2;&#xA;yrange = {0, 4};&#xA;xrange = {xrangeleft,xrangeright};&#xA;frames = Table[ListPlot[Table[{x, f[x]}, {x, xrangeleft, xrangeright, 1/2^i}], &#xA;         PlotRange -&amp;gt; {xrange, yrange}], {i, 0, 10, 1}]&#xA;Export[&quot;FunctionPlot.gif&quot;, frames]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;(You might need to use &lt;code&gt;Directory[]&lt;/code&gt; to find out where it exported your file.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This creates the following: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ftYqX.gif&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ftYqX.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2017-02-10T18:41:29.013" LastActivityDate="2017-02-10T18:41:29.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12013" PostTypeId="2" ParentId="12009" CreationDate="2017-02-11T09:50:42.593" Score="3" Body="&lt;p&gt;Here is the code in Matlab for the sine function:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;for k = 0:6&#xA;n=2^k          % stage number, k=0: integer values, k=1: stage 2, ...&#xA;x=0:1/n:3*pi;&#xA;y = sin(x);    % sine function&#xA;pause(1);      % pause duration &#xA;plot(x,y,'b.')&#xA;axis ([0 3*pi -1.5 1.5])&#xA;grid on&#xA;end&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="7738" LastActivityDate="2017-02-11T09:50:42.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12014" PostTypeId="2" ParentId="12006" CreationDate="2017-02-11T18:19:25.473" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Should I teach a subject I don't like?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It depends on other factors, but don't automatically &lt;strong&gt;not&lt;/strong&gt; teach a subject just because you don't like it.  I'm currently teaching a course that combines linear algebra with differential equations and I really can't stand linear algebra at the introductory level.  It's so dry and often times tedious and it takes a while before any of the fun applications (I know that's subjective) can be covered.  But I've actually found that teaching it is more enjoyable than when I had to learn it.  You may end up feeling the same way about linear programming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I noticed you also said this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But as you guess my understanding of it is rather limited and although I am confident that I can quickly bring myself to a competent enough level to teach it, I am beset by doubt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you have any doubt that you can teach the material, then you shouldn't teach it.  At least wait until you can clear up this doubt.  Doing otherwise is a disservice to yourself, your students, and the institution.  And this is (or should be) a perfectly acceptable reason to give your employers for not agreeing to teach a course.  When I interviewed for my current teaching job (part-time side gig at a local comm college) they asked in the interview if I could also teach statistics.  I said if I really had to I suppose I could but I would never choose it since I just don't feel comfortable enough with it.  They didn't seem to mind and that was before I was even employed by them.&lt;/p&gt;&#xA;" OwnerDisplayName="user6648" LastActivityDate="2017-02-11T18:19:25.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12015" PostTypeId="2" ParentId="1598" CreationDate="2017-02-11T20:35:25.007" Score="4" Body="&lt;p&gt;I agree that the selected best answer for this question is excellent. I'd like to add that in Stephen Krantz's &lt;em&gt;How to Teach Mathematics&lt;/em&gt; (AMS, 3rd Ed.), the primary thesis of the entire book is basically &lt;em&gt;make sure not to do this&lt;/em&gt;. (In fact, conceivably the person who posed the question heard about it from this book.) Quotes from Sec. 1.2, &quot;Prepare&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Some people rationalize not teaching well by saying (either to&#xA;  themselves or to others), &quot;My time is too valuable. I am not going to&#xA;  spend it preparing my calculus lecture. I am so smart that I can just&#xA;  walk into the classroom and wing it. And the students will benefit&#xA;  from watching a mathematician think on his feet.&quot; (As a student, I&#xA;  actually had professors who announced this nonsense to the class on a&#xA;  regular basis. And, as you can imagine, these were professors who&#xA;  royally botched up their lectures on a regular basis.)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It is true that most of us can walk into the room most of the time and&#xA;  mostly wing it. But most of us will not be very successful if we do&#xA;  so. Thirty minutes can be sufficient time for an experienced&#xA;  instructor to prepare a calculus lecture... Make sure that you have&#xA;  the definitions and theorems straight. It is a good idea to have a&#xA;  single page of notes containing the key points and also briefly&#xA;  listing your motivating ideas...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I cannot emphasize too strongly the fact that preparation is of the&#xA;  utmost importance if you are going to deliver a stimulating class.&#xA;  However it is also true that the more you prepare the more you lose&#xA;  your spontaneity. You must strike a balance between (i) knowing the&#xA;  material cold and (ii) being able to &quot;talk things through&quot; with your&#xA;  audience.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-02-12T03:32:50.423" LastActivityDate="2017-02-12T03:32:50.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12016" PostTypeId="2" ParentId="11396" CreationDate="2017-02-12T12:37:14.473" Score="1" Body="&lt;p&gt;Your question will not have a research-based answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You assume that there is a determinable &quot;fraction of the population&quot; that &quot;is incapable of learning algebra, even with repeated effort and competent instruction&quot;. Philosophically, I wonder if this fraction is well-defined. In your own answer, you mention the SES reflecting the society as an important factor. Would you assume the society as given in your mental model or potentially being a subject to change? What about neuroscientific methods that could be invented? The &quot;fraction&quot; you are looking for is very fictional.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, assuming you solve this problem, how could you practically determine this fraction? You may test all people of the world, but tests only give estimates of people's abilities, not their potential. What evidence could you give to make us believe that a person really cannot reach a given level? Would you assume, for example, that people from the 5th century were unable to learn complex numbers just because no one did?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even assuming you will find a way to estimate peoples potential, what would the pedagogical implications be? Still you would try to help them learn as much as they can. The learning support, however, should rather be based on what they actually have learnt and not potentially could learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the question may evoke some substantial concerns. Both teachers' and students' thinking of people having fixed abilites has proven to hinder learning as the &quot;dumb&quot; ones may seem to deserve less care or effort. Moreover, what if someone is able to learn large parts of stochastics, geometry and calculus but not algebra? Wouldn't you want to know this? So, how much measuring would be enough? Shouldn't we rather use our ressources for something of more practical use? Thinking of fixed abilities, in addition, is likely to produce distinct worlds in the educational system. A world for the &quot;good&quot;, a world for the &quot;dumb&quot;, maybe another world for the &quot;very gifted&quot; students, disregarding many problematic effects of such separations (e.g., focussing on &quot;good&quot; mechanisms of evaluating their potential instead of focussing on learning; focussing on learning algebra at the cost of social learning). We know people have different abilites, but should this knowledge lead our decisions? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is just to explain why I strongly believe that you won't find &quot;evidence-based answers that cite references, not anecdotes or personal impressions&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to add that from a philosphical point of view, I still find your question interesting. In some &quot;applied research&quot;, however, progress may not only be given by answers to questions but also by changing questions and rethinking the situation and what you find interesting and why. This question is a candidate for this, I think.&lt;/p&gt;&#xA;" OwnerUserId="199" LastActivityDate="2017-02-12T12:37:14.473" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12017" PostTypeId="2" ParentId="12009" CreationDate="2017-02-13T00:06:28.687" Score="2" Body="&lt;p&gt;If I may suggest a tiny variation on Aeryk's nice code, to slow it down,&#xA;and join the dots:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ddYuc.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ddYuc.gif&quot; alt=&quot;PlotGIF&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;f[x_] := x^3;&#xA;xrangeleft = -2;&#xA;xrangeright = 2;&#xA;yrange = {-8, 8};&#xA;xrange = {xrangeleft, xrangeright};&#xA;frames = Table[&#xA;   ListPlot[Table[{x, f[x]}, {x, xrangeleft, xrangeright, 1/2^i}]&#xA;    , Joined -&amp;gt; True&#xA;    , Mesh -&amp;gt; Full&#xA;    , PlotRange -&amp;gt; {xrange, yrange}]&#xA;   , {i, 0, 3, 1}];&#xA;Export[&quot;FunctionPlot.gif&quot;, frames, &quot;DisplayDurations&quot; -&amp;gt; 3.0]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-02-13T00:06:28.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12021" PostTypeId="2" ParentId="11950" CreationDate="2017-02-14T22:47:55.677" Score="10" Body="&lt;p&gt;In addition to other insightful answers/comments/remarks, apart from the issue of &quot;degree&quot; and &quot;what kind of genius&quot;, I think a large part of the problem is exactly that mathematics is mostly portrayed as a school subject, in which a large part of &quot;success&quot; is measured by a certain conformity to &quot;the teacher&quot;'s expectations, and conformity with the worldview manifest in the textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I claim that this problem exists at all levels of schooling and mathematics. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, kids and other people are mostly led to believe that they will/must learn mathematics in some sort of school, whether accelerated or whatever, and that their success will be measured by some form of external approval. This is not toooo ridiculous, but given the tendency to commodify, the goals for the teachers also become externalized and commodified (and thereby corrupted), to meet &quot;no-one left behind&quot;, &quot;graduation rate&quot;, and other administratively imposed mis-measures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, to really understand mathematics or anything else, probably people should first &quot;be allowed&quot; to trust their own judgement. Few school-math scenarios even remotely come close to encouraging this viewpoint. It's not that teachers try to &lt;em&gt;persuade&lt;/em&gt; students, but &quot;command&quot;, from a position of authority. Authority is not the same as expert-ness. An expert can persuade by showing how wonderfully a thing can be done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, sure, if one is genuinely curious, and not worried about invalidation or failure, one behaves very differently than the opposite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if the goal is to make this happen in the framework of &quot;school&quot;, then there is a fundamental conflict, I do seriously think, since the measure of success in &quot;school&quot; has almost nothing to do with one's curiosity or standards. Oop.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-02-14T22:47:55.677" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12022" PostTypeId="1" AcceptedAnswerId="12023" CreationDate="2017-02-15T15:41:42.907" Score="12" ViewCount="444" Body="&lt;p&gt;I will soon tutor someone I know in math. Because of the bad results on tests, combined with the feedback from the teacher, he has a very bad sense of achievement. Which in turn, has given him a low motivation for the subject. I think the problem is a lack of basic understanding in some very early topics in math, which has given him problems keeping up in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How should I go about finding the holes in his math understanding? Finding what it is he is struggling with? When I ask, he gives me very general answers, usually just the current topic. I'm thinking that a test would not be the best way, but I might be wrong. Do you have any other suggestions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anybody has experience with a similar situation, any other advice would also be appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for the help.&lt;/p&gt;&#xA;" OwnerUserId="7822" LastEditorUserId="7822" LastEditDate="2017-02-15T15:53:29.067" LastActivityDate="2017-02-22T21:50:30.727" Title="Best way to find out what math topics a person struggles with when tutoring" Tags="|algebra|tutoring|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12023" PostTypeId="2" ParentId="12022" CreationDate="2017-02-15T17:19:25.703" Score="16" Body="&lt;p&gt;Since you didn't mention what level your student is, I'll try to make my answer as general as possible, leaning more toward secondary math students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience tutoring and teaching secondary level mathematics, I have observed that many students who struggle most have a number of skill domains that are &lt;em&gt;far&lt;/em&gt; below grade level, and that these deficiencies hinder progress, in some cases completely. If students have such deficiencies and are having trouble factoring quadratic expressions, for example, discussing quadratic expressions and laying out procedures and completing examples and exercises of quadratic expressions for hours and hours will produce very little, if any, improvement. So spending a lot of time assessing secondary topics or procedural knowledge of their current topics, or even topics from &quot;earlier this year&quot; is not your best choice. One of the biggest mistakes we can make as educators is to assume our students understand something just because it is simple, or remember something just because they have been exposed to it before. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find that the most effective use of assessment during tutoring is to start with very, very basic maths. You might be surprised what you find. Here are some examples of elementary skills you should assess:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Multiplication tables&lt;/li&gt;&#xA;&lt;li&gt;Factoring composite numbers and writing the prime factorization&lt;/li&gt;&#xA;&lt;li&gt;Multiplying and dividing multi-digit numbers manually&lt;/li&gt;&#xA;&lt;li&gt;Multiplying and dividing multi-digit numbers mentally&lt;/li&gt;&#xA;&lt;li&gt;Adding and Subtracting multi-digit numbers manually&lt;/li&gt;&#xA;&lt;li&gt;Adding and Subtracting multi-digit numbers mentally&lt;/li&gt;&#xA;&lt;li&gt;Adding and subtracting small positive and negative numbers mentally&lt;/li&gt;&#xA;&lt;li&gt;Computing squares of integers and roots of perfect squares&lt;/li&gt;&#xA;&lt;li&gt;Solving one-step and two-step linear equations mentally&lt;/li&gt;&#xA;&lt;li&gt;Ordering rational numbers&lt;/li&gt;&#xA;&lt;li&gt;Simplifying rational numbers&lt;/li&gt;&#xA;&lt;li&gt;Identifying Order of operations and simplification mistakes. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To compose a more comprehensive list, find a source of primary education standards in mathematics and jot down learning outcomes from 1st to 6th or 7th grade. Whether your student is a 35 year old struggling in a college calculus course or a 16 year old struggling in high school algebra, the best place to begin looking for holes is primary education. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thankfully, though these are elementary skills, secondary students are usually ready to discuss numbers as objects, so you can work on strengthening these skills without having to use primary education methods, such as number tiles, but instead focus on the &quot;structure of numbers&quot;, which is why factoring numbers is at the top of the list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you determine which of these skills is lacking, it should be easy to determine a course of action. You should not spend the time &lt;em&gt;drilling&lt;/em&gt;, but instead teaching the student and their parent drills that they can do daily at home to strengthen these fundamental skills. And give them access to sources of exercises, like the infinite math software or deltamath.com. You should convince the parents that the one hour or so that you spend with them each week is meaningless unless they spend a little bit of time practicing each day. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition to elementary skills, you should assess neatness, organization, and study skills. Look at the student's homework. Is it easy to identify the task they were given? Is it well formatted on the page and easy to follow? Are similar problems done in a similar manner each time? Even the most casual assignment should be done with attention to detail. This will allow students to identify their own errors, a skill that you should teach them as well. It's possible that the student has the background to understand the material, but is not able to follow their own work or identify when they have completed the task because they lack organization skills. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2017-02-22T21:50:30.727" LastActivityDate="2017-02-22T21:50:30.727" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12024" PostTypeId="1" CreationDate="2017-02-13T20:34:44.090" Score="7" ViewCount="9345" Body="&lt;p&gt;Before you downvote this question, I actually want an answer to this. Is the calculator going to give me my derivative? No. Is it going to give me my integral? No. It can sure give me the answer to my integral, but will it give me the calculations? and steps? No. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In linear algebra, will the calculator give me the matrix reduced form? No. Will it give me the answer to a set of equations? Maybe, but without work obviously. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teachers mark the &quot;work&quot;, and award part marks for each correct piece of work you show. That work cannot be done by a calculator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something that frustrates me is that I am in post-secondary, and that my level of math right now is not basic. So, like many, I tend to forget the basics of math, like adding fractions, reducing fractions to lowest form fast, subtracting big numbers. Sure, I can do all those, but I would be more likely to make a mistake doing these and loose marks then if I were to just use a calculator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teachers want to make sure we &quot;understand&quot; the concept, but I ask, does &quot;understanding the concept&quot; really have to do with our ability to &quot;add, multiply&quot;? Are they testing whether we can do basic arithmetic that we have so long been used to performing with a calculator?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When your working with integration and derivatives, you will get ugly fractions, and adding them, finding denominators, is really a waste of time and effort for me, when the calculator can just do that for me. In the real world I will have a calculator, so why not here? As I said before, what are they testing? Are they testing how I can do basic math, or how I can compute the limit, evaluate the integral, calculate the deriviative? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why?&lt;/p&gt;&#xA;" OwnerUserId="13282" OwnerDisplayName="K Split X" LastEditorUserId="143" LastEditDate="2017-02-15T21:08:56.917" LastActivityDate="2024-02-05T04:24:37.590" Title="Why are calculators not allowed in post-secondary exams?" Tags="|calculus|calculators|" AnswerCount="10" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12025" PostTypeId="2" ParentId="12024" CreationDate="2017-02-13T20:40:57.193" Score="41" Body="&lt;p&gt;As a professor/teacher I have some insight. You just answered your own question: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;my level of math right now is not basic. So, like many, I tend to forget the basics of math, like adding fractions, reducing fractions to lowest form fast, subtracting big numbers&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By avoiding using a calculator you'll strengthen the basics.&lt;/p&gt;&#xA;" OwnerUserId="8559" OwnerDisplayName="BobaFret" LastActivityDate="2017-02-13T20:40:57.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12026" PostTypeId="2" ParentId="12024" CreationDate="2017-02-13T20:51:12.607" Score="24" Body="&lt;p&gt;In addition to @BobaFret's answer I'd like like to point out following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Before you downvote this question, I actually want an answer to this. Is the calculator going to give me my derivative? No. Is it going to give me my integral? No. It can sure give me the answer to my integral, but will it give me the calculations? and steps? No.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is not true, it really depends on the calculator, and there is a whole range of them, from those that cannot even find square roots to programmable calculators with a full CAS. So it would be unfair to give some people an advantage just because they use a &quot;better&quot; calculator. And just restricting certain types of calculators is also very difficult as some might still have some more advanced functions than others. Another solution would be forcing everyone to use the exact same calculator which is also a bad idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead it is just a lot easier to disallow all forms of calculators. This makes you really strengthen the basic skills (look at it as a positive thing, since you already mentioned that you forgot how to do some of them) and it also makes you think of elegant ways to solve problems instead of memorizing &quot;brute force&quot; attacks. &lt;/p&gt;&#xA;" OwnerUserId="1936" OwnerDisplayName="flawr" LastActivityDate="2017-02-13T20:51:12.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12027" PostTypeId="2" ParentId="12024" CreationDate="2017-02-13T20:55:09.593" Score="4" Body="&lt;p&gt;One major concern is cheating. People can enter information into their calculators that may give them an unfair advantage on the exam, in essence using the calculator as a &quot;cheat sheet&quot;. Having a professor verify that a couple hundred students don't have any information stored in their calculators isn't feasible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also an economic argument: a calculator, especially a graphing calculator, is expensive enough that not all students may be able or willing to buy one. If calculators are useful then they should therefore be banned so that wealthier students don't have an unfair advantage, and if they aren't useful then why allow them?&lt;/p&gt;&#xA;" OwnerDisplayName="carmichael561" LastActivityDate="2017-02-13T20:55:09.593" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12028" PostTypeId="2" ParentId="12024" CreationDate="2017-02-13T21:03:52.510" Score="4" Body="&lt;p&gt;When I taught, we permitted students to have a symbolic calculator at all times. The implication for students of this is that the questions in the exam take this fact into account. When such a decision is taken, the questions in an exam become more theoric and/or demand more imagination.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in an exam on optimisation, the focus would be on setting up the right function, not on deriving it or solving $f'(x)=0$. Once the possible optimums were found, the reason for accepting or rejecting them was the important part. The presentation of the solution was also important. We did sometimes have exams not using the calculator, for example on derivating or integrating functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example: I used to have a 2-hour exam which consisted in 1) the class helping me to find the formula for the $n$-th derivative of $x\cdot e^x$ and proving the formula by induction (about 20-30 minutes). Then 2) the students had to find and prove a formula for the $n$-th derivative of $x^2\cdot e^x$ working in teams of 2. &lt;/p&gt;&#xA;" OwnerUserId="503" OwnerDisplayName="Bernard Massé" LastActivityDate="2017-02-13T21:03:52.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12029" PostTypeId="2" ParentId="12024" CreationDate="2017-02-13T21:06:13.793" Score="12" Body="&lt;p&gt;I'm going to add onto flawr's answer, which added onto BobaFret's answer. :-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sympathetic, because I know plenty of mathematicians who are not particularly good at arithmetic.  However, I think that if a test is not intended to test someone's ability at arithmetic in the first place, and is written well in the second place, it should not unduly tax your arithmetic skills.  Expressions should &quot;clean up nicely&quot; in the mathematics exam world.  When I write an exam, I tend to try to make it so that as little arithmetic computation as possible is required.  That being said, &lt;em&gt;some&lt;/em&gt; level of arithmetic is needed.  That's unavoidable, but probably not to the level of requiring a calculator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's more, as amd's comment implies, there is more to mathematics than understanding the concepts, as important as that understanding is.  One should learn not just to apply concepts correctly, but to understand when they are being applied incorrectly, whether it was you who made the mistake or someone else.  In &quot;real life&quot; (assuming facts not in evidence), I might be asked to look over someone else's work, either as a colleague or as a supervisor.  The ability to perform a quick sanity check to see whether it makes sense that the area of a circle $5$ cm in diameter is $79$ cm$^2$, &lt;em&gt;and to identify the likely cause of any error&lt;/em&gt;, is invaluable.  (You shouldn't need a calculator to figure out what likely happened in this case.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exams in other quantitative subjects, like physics or chemistry, may well allow calculators, because the practical aspects of problem-solving are weighted more heavily.  I'm not convinced that the same applies to a mathematics course.&lt;/p&gt;&#xA;" OwnerDisplayName="Brian Tung" LastActivityDate="2017-02-13T21:06:13.793" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12030" PostTypeId="2" ParentId="12024" CreationDate="2017-02-13T21:09:23.570" Score="18" Body="&lt;p&gt;One thing that really freaks me out about students on this site is an apparent inability, at least an unwillingness, to draw ordinary graphs on graph paper, maybe $y = x^3 - 3 x + 2$ or the like. If I comment about the desirability of doing this, I generally add a link from which graph paper can be downloaded (as a pdf) and printed, &lt;a href=&quot;https://www.printablepaper.net/category/graph&quot;&gt;https://www.printablepaper.net/category/graph&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From what I can see, no MSE user has ever drawn a graph by hand owing to a suggestion of mine. When a reason is given sometimes it is &quot;I'm not allowed a calculator in exams,&quot; as though the only possibility for getting a graph is a machine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The end of the world is nigh. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pFGsN.jpg&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pFGsN.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3564" OwnerDisplayName="Will Jagy" LastActivityDate="2017-02-13T21:30:12.750" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12032" PostTypeId="1" CreationDate="2017-02-16T21:36:11.693" Score="9" ViewCount="486" Body="&lt;p&gt;I live in Brazil and here we have some problems with teaching mathematics in High School.  In some point of the students' life (I think it happens in the 5th grade), they start &quot;hating mathematics&quot; and, at least here, it's totally acceptable.  So students start thinking they are not capable enough of learning mathematics and simply stop studying.  It becomes dull and they think it's useless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My colleagues and I discuss a lot about how we can motivate them to keep enthusiastic about learning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This topic is for us to discuss some ideas!&lt;/p&gt;&#xA;" OwnerUserId="7829" LastEditorUserId="77" LastEditDate="2017-03-31T04:48:37.630" LastActivityDate="2017-04-01T20:33:01.160" Title="Motivation in School" Tags="|secondary-education|student-motivation|primary-education|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12033" PostTypeId="2" ParentId="12022" CreationDate="2017-02-17T11:34:57.360" Score="1" Body="&lt;p&gt;As an experienced elementary school teacher and high school tutor, I would like to  suggest some modifications of the  list of elementary skills in the accepted answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the skills on the list have no bearing on the ability to do high school math such as multiplying and dividing multi-digit numbers mentally.  Furthermore many students have a calculator that they are allowed to use in their high school math classes and it would be good to verify if that is the case here.  Even if they can use a calculator, knowing there multiplication and division tables through 10 by 10 is essential knowledge. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the most important skills that teachers complain about is their students lack of understanding of fractions.  Therefore operations with fractions (addition, subtraction, multiplication, division, simplification) should certainly be on the list. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I certainly agree that neatness counts and have dealt with too many students who couldn't read what they wrote in the middle steps of a problem and messed up because they thought the 6 was a 0 etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I would like to suggest that you can work on the current material and see what gaps there are and then go back.  I have had much success with this method since students tend to be more motivated when they know how the basics connect with what they are currently learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2017-02-17T11:34:57.360" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12034" PostTypeId="2" ParentId="12032" CreationDate="2017-02-17T13:26:06.877" Score="3" Body="&lt;p&gt;Thanks for bringing up a valid issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Myself have come across many situations when students doesn't take math seriously just because they think &quot;they can't learn it&quot; But sooner I have made a serious change to my &quot;way of teaching&quot;. The change did not show a direct change but it did it, gradually.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The change was to make the class interactive, and also relating it will real time example. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's take Primary - 5 math problem: &#xA;So the group of you are going for movie(Iron Man) this weekend, but the cashier asked only once person can pay the money to get 10% discount upon purchase of 10 tickets. If the ticket cost $10, &#xA;what is the amount need to give to the cashier? (Multiplication, Understanding Percentage)&#xA;what is the price each one has to contribute after discount? (Division, rounding off)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This way, going about with real-time example, they will know how important math is and how easy it could be!&lt;/p&gt;&#xA;" OwnerUserId="7832" LastActivityDate="2017-02-17T13:26:06.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12035" PostTypeId="2" ParentId="12032" CreationDate="2017-02-18T02:11:03.717" Score="3" Body="&lt;p&gt;Just one example, that frustratingly to me, is rarely broached in algebra (in U.S. secondary schools) when covering polynomials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you print to a printer, or look at rendered text on a screen (say, PDF),&#xA;you are likely using either &#xA;Postscript fonts, which are drawn by cubic (Bézier) polynomials, or TrueType fonts, which are drawn by quadratic polynomials.&#xA;So polynomials are used literally daily by the very students learning polynomials, without them ever realizing that the dry topic they may be&#xA;avoiding by browsing is actually being used to support their browsing.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/IYD90.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IYD90.jpg&quot; alt=&quot;Q&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;The letter &lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt; drawn with Bézier cubics.&#xA;Image from &lt;a href=&quot;http://learn.scannerlicker.net/2014/04/16/bezier-curves-and-type-design-a-tutorial/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-02-18T02:11:03.717" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12036" PostTypeId="2" ParentId="12024" CreationDate="2017-02-18T05:48:34.563" Score="12" Body="&lt;p&gt;I very much agree with already posted great answers (I do share most of the same opinions), but I'd like to add something else. Talking about math instruction we're not talking about math only &amp;mdash; there's a psychological dimension to it too. And from this perspective, from a pedagogical point of view, I see a lot of harm done by calculators, or rather by thoughtlessly relying on them. In schools where calculator usage is rampant, students are lead to believe that mastering math is equivalent to memorizing which buttons to press on a calculator for each task. So discouraging calculator use is intended to break this misconception and force our students to actually learn math. The sad truth is that for some of them it will be the first time they would actually be forced and encouraged to think with their own heads in a math class! So for pedagogical reasons, we have to do something about it, and forbidding calculators on tests is actually a reasonable solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may sound like too strong an opinion, but I see this kind of thing on a daily basis. Often when I see a student in a Calculus class who doesn't have any conceptual knowledge of pretty much any Precalculus topic, and when I talk to such a student to find out more about their background, I would find out that he/she transferred from some other &quot;calculator-reliant&quot; college or school, where they &quot;did everything on a calculator&quot; (this is an actual quote).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Truth be told, I don't walk my talk. We allow non-graphing calculators on tests in our classes. (Long story...) And inevitably I see students wasting half of their time incessantly punching buttons even to compute things like &quot;$-3+3$&quot; and even &quot;$0*5$&quot; &amp;mdash; and I'm not making this up! (Let's not even start on things like &quot;$\sin\frac{\pi}{2}$&quot;, which is kinda funny, in a sad way, when their calculator is in degree mode.)&lt;/p&gt;&#xA;" OwnerUserId="6762" LastActivityDate="2017-02-18T05:48:34.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12037" PostTypeId="2" ParentId="12032" CreationDate="2017-02-18T10:12:54.577" Score="3" Body="&lt;p&gt;A short time ago I read the following article about motivation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ambrose, S. A., Bridges, M. W., DiPietro, M., Lovett, M. C., &amp;amp; Norman, M. K. (2010). What factors motivate students to learn. How learning works: Seven research-based principles for smart teaching, 66-90. Available under &lt;a href=&quot;https://sites.temple.edu/bett/files/2014/01/Ambrose-Chapter-3-What-motivates-students1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://sites.temple.edu/bett/files/2014/01/Ambrose-Chapter-3-What-motivates-students1.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It generally discuss the factors of students' motivation and how a teacher can enhance it. Thereby Ambrose et al. (2010) mention three dimensions which influence students' motivation:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;students’ goal value&lt;/li&gt;&#xA;&lt;li&gt;students’ efficacy expectancies&lt;/li&gt;&#xA;&lt;li&gt;environment: supportive or unsupportive&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Ambrose et al. argued that only in the case, when all three dimensions are fulfilled, the students are motivated. In total they see the following reaction of a student to the three dimensions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AtERI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AtERI.png&quot; alt=&quot;image&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you can ask yourself which of the above dimensions are not met and how you can enhance them. Ambrose et al. also give a couple of concrete strategies which you find in &lt;a href=&quot;https://sites.temple.edu/bett/files/2014/01/Ambrose-Chapter-3-What-motivates-students1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;their article&lt;/a&gt;. I also summarized the article &lt;a href=&quot;http://www.doc.ma.edu.tum.de/kulla/blog/post/what_factors_motivate_students_to_learn/&quot; rel=&quot;nofollow noreferrer&quot;&gt;in a blog post&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="5097" LastEditDate="2017-02-20T08:24:41.630" LastActivityDate="2017-02-20T08:24:41.630" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12038" PostTypeId="2" ParentId="12022" CreationDate="2017-02-18T10:59:25.780" Score="2" Body="&lt;p&gt;This is an addition to &lt;a href=&quot;https://matheducators.stackexchange.com/a/12023/5097&quot;&gt;Andrew's answer&lt;/a&gt; which is to long for a comment. :-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your question suggests that your student shows a low level of efficiency expectancy which results in a low motivation (as you describe it). It's important that your student thinks he is able of showing good results in order to be motivated. So I would recommend you to enhance his self esteem in mathematics and his belief in his ability to successfully solve mathematical problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is mainly influenced in the way he attributes past actions and their (positive or negative) outcome. For example successes in mathematics need to be attributed internal and stable (&quot;I got a good grade because I am good in mathematics&quot;) while negative outcomes shall be attributed external and/or unstable (&quot;I got a bad grade because the exam was to hard&quot; / &quot;I got a bad grade because I didn't learn enough but I can change this the next time&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ambrose et al. (2010) recommend the following strategies for enhancing the students' efficiency expectancy:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Identify and create an appropriate level of challenge&lt;/li&gt;&#xA;&lt;li&gt;Provide early success opportunities&lt;/li&gt;&#xA;&lt;li&gt;Articulate your expectations&lt;/li&gt;&#xA;&lt;li&gt;Provide constructive feedback&lt;/li&gt;&#xA;&lt;li&gt;Be fair&lt;/li&gt;&#xA;&lt;li&gt;Educate students how to attribute success and failure&lt;/li&gt;&#xA;&lt;li&gt;Describe effective study strategies&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this will help you. I guess it will be a long way to change your student's perspective on his mathematics aptitude but it's worth it. I also just gave an &lt;a href=&quot;https://matheducators.stackexchange.com/a/12037/5097&quot;&gt;answer on how to enhance students' motivation&lt;/a&gt; which you might find helpful. Besides I recommend you the article by Ambrose et al. (2010) where the above strategies are detailedly described.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;References&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ambrose, S. A., Bridges, M. W., DiPietro, M., Lovett, M. C., &amp;amp; Norman, M. K. (2010). What factors motivate students to learn. How learning works: Seven research-based principles for smart teaching, 66-90. Available under &lt;a href=&quot;https://sites.temple.edu/bett/files/2014/01/Ambrose-Chapter-3-What-motivates-students1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://sites.temple.edu/bett/files/2014/01/Ambrose-Chapter-3-What-motivates-students1.pdf&lt;/a&gt; &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2017-02-18T10:59:25.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12039" PostTypeId="2" ParentId="12032" CreationDate="2017-02-18T11:28:28.183" Score="13" Body="&lt;p&gt;I will try to give a research-related answer. There are several suggestions from the literature and you may have to take a deeper look at them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, a decrease in motivation is also observed in other countries, not just Brazil (Frenzel, Goetz, Pekrun, &amp;amp; Watt, 2010).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In recent psychological theories (Krapp, 2002), (Krapp, 2005), (Renninger, 2009), interest development is based on the satisfaction of psychological needs, especially the needs for perceived competence, autonomy and social relatedness from self-determination theory (Deci &amp;amp; Ryan, 1985), (Deci &amp;amp; Ryan, 2002). Students should have these needs satisfied in your teaching as often as possible. But how should this work?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Empirical studies on mathematics interest confirm factors like competence support (Rakoczy, Harks, Klieme, Blum, &amp;amp; Hochweber, 2013) and autonomy support (Valås &amp;amp; Søvik, 1994), but also work on different conceptual levels, e.g. teaching methods (Bikner-Ahsbahs &amp;amp; Halverscheid, 2014), (Lerkkanen u. a., 2012), (Prendergast &amp;amp; O’Donoghue, 2014), (Rowan-Kenyon, Swan, &amp;amp; Creager, 2012), (Schukajlow u. a., 2011), (Sonnert, Sadler, Sadler, &amp;amp; Bressoud, 2015), classroom management (Rowan-Kenyon u. a., 2012), task design (Schukajlow &amp;amp; Krug, 2014) or the big-fish-little-pond effect (Frenzel u. a., 2010), which all more or less explicitly relate to need satisfaction (see also (Carmichael, Callingham, Watson, &amp;amp; Hay, 2009)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, however, that the motivation role of applications tends to be over-estimated by teachers since students may also have interest in pure aspects of mathematics (Rellensmann &amp;amp; Schukajlow, 2016).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best framework for the design of classroom activities that support the psychological needs and foster students’ motivation is the one by Bikner-Ahsbahs and Halverscheid (2014). In brief, they plan student activities based on exploration and discovery, group work and valuing of (nearly) any student observation and comment. These activities cannot be found for each session (e.g., not for the fostering of calculation routines) but may really help for deep connections of the mathematical content and most of all they may substantially improve students’ motivation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am sorry I cannot give a short and simple answer. If we had such an answer, it would have been spread around the world already. Anyway I hope my post may help you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Bikner-Ahsbahs, A., &amp;amp; Halverscheid, S. (2014). Introduction to the&#xA;Theory of Interest-Dense Situations (IDS). In A. Bikner-Ahsbahs &amp;amp; S.&#xA;Prediger (Eds.), Networking of Theories as a Research Practice in&#xA;Mathematics Education (pp. 97–113). Cham: Springer International&#xA;Publishing. Retrieved from&#xA;&lt;a href=&quot;http://link.springer.com/10.1007/978-3-319-05389-9_7&quot; rel=&quot;noreferrer&quot;&gt;http://link.springer.com/10.1007/978-3-319-05389-9_7&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Carmichael,   C., Callingham, R., Watson, J., &amp;amp; Hay, I. (2009). Factors influencing   the development of middle school students’ interest in statistical   literacy. SERJ - Statistics Education Research Journal, 8(1), 62–81.&lt;/li&gt;&#xA;&lt;li&gt;Deci, E. L., &amp;amp; Ryan, R. M. (1985). Intrinsic Motivation and&#xA;Self-Determination in Human Behavior. New York: Plenum Press. &lt;/li&gt;&#xA;&lt;li&gt;Deci,&#xA;E. L., &amp;amp; Ryan, R. M. (2002). Handbook of Self-Determination Research:&#xA;Theoretical and Applied Issues. Rochester, NY: University of&#xA;Rochester Press. &lt;/li&gt;&#xA;&lt;li&gt;Frenzel, A. C., Goetz, T., Pekrun, R., &amp;amp; Watt, H.&#xA;M. G. (2010). Development of Mathematics Interest in Adolescence:&#xA;Influences of Gender, Family, and School Context. Journal of Research&#xA;on Adolescence, 20(2), 507–537.&#xA;&lt;a href=&quot;https://doi.org/10.1111/j.1532-7795.2010.00645.x&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1111/j.1532-7795.2010.00645.x&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;Krapp, A. (2002).&#xA;An Educational-Psychological Theory of Interest and Its Relation to&#xA;SDT. In E. L. Deci &amp;amp; R. M. Ryan (Eds.), Handbook of&#xA;self-determination research (pp. 405–427). Rochester, NY: University&#xA;of Rochester Press. &lt;/li&gt;&#xA;&lt;li&gt;Krapp, A. (2005). Basic needs and the&#xA;development of interest and intrinsic motivational orientations.&#xA;Learning and Instruction, 15(5), 381–395.&#xA;&lt;a href=&quot;https://doi.org/10.1016/j.learninstruc.2005.07.007&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1016/j.learninstruc.2005.07.007&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Lerkkanen,&#xA;M.-K., Kiuru, N., Pakarinen, E., Viljaranta, J., Poikkeus, A.-M.,&#xA;Rasku-Puttonen, H., … Nurmi, J.-E. (2012). The role of teaching&#xA;practices in the development of children’s interest in reading and&#xA;mathematics in kindergarten. Contemporary Educational Psychology,&#xA;37(4), 266–279.&lt;/li&gt;&#xA;&lt;li&gt;Prendergast, M., &amp;amp; O’Donoghue, J. (2014). “Students&#xA;enjoyed and talked about the classes in the corridors”: pedagogical&#xA;framework promoting interest in algebra. International Journal of&#xA;Mathematical Education in Science and Technology, 45(6), 795–812.&#xA;&lt;a href=&quot;https://doi.org/10.1080/0020739X.2013.877603&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1080/0020739X.2013.877603&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;Rakoczy, K., Harks,&#xA;B., Klieme, E., Blum, W., &amp;amp; Hochweber, J. (2013). Written feedback in&#xA;mathematics: Mediated by students’ perception, moderated by goal&#xA;orientation. Learning and Instruction, 27, 63–73.&#xA;&lt;a href=&quot;https://doi.org/10.1016/j.learninstruc.2013.03.002&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1016/j.learninstruc.2013.03.002&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Rellensmann, J.,&#xA;&amp;amp; Schukajlow, S. (2016). Does students’ interest in a mathematical&#xA;problem depend on the problem’s connection to reality? An analysis of&#xA;students’ interest and pre-service teachers’ judgments of students’&#xA;interest in problems with and without a connection to reality. ZDM,&#xA;1–12. &lt;a href=&quot;https://doi.org/10.1007/s11858-016-0819-3&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1007/s11858-016-0819-3&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Renninger, K. A.&#xA;(2009). Interest and Identity Development in Instruction: An&#xA;Inductive Model. Educational Psychologist, 44(2), 105–118.&#xA;&lt;a href=&quot;https://doi.org/10.1080/00461520902832392&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1080/00461520902832392&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Rowan-Kenyon, H. T.,&#xA;Swan, A. K., &amp;amp; Creager, M. F. (2012). Social Cognitive Factors,&#xA;Support, and Engagement: Early Adolescents’ Math Interests as&#xA;Precursors to Choice of Career. The Career Development Quarterly,&#xA;60(1), 2–15. &lt;a href=&quot;https://doi.org/10.1002/j.2161-0045.2012.00001.x&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1002/j.2161-0045.2012.00001.x&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Schukajlow, S., &amp;amp; Krug, A. (2014). Do Multiple Solutions Matter?&#xA;Prompting Multiple Solutions, Interest, Competence, and Autonomy.&#xA;Journal for Research in Mathematics Education, 45(4), 497–533.&#xA;&lt;a href=&quot;https://doi.org/10.5951/jresematheduc.45.4.0497&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.5951/jresematheduc.45.4.0497&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Schukajlow, S.,&#xA;Leiss, D., Pekrun, R., Blum, W., Müller, M., &amp;amp; Messner, R. (2011).&#xA;Teaching methods for modelling problems and students’ task-specific&#xA;enjoyment, value, interest and self-efficacy expectations.&#xA;Educational Studies in Mathematics, 79(2), 215–237.&#xA;&lt;a href=&quot;https://doi.org/10.1007/s10649-011-9341-2&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1007/s10649-011-9341-2&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Sonnert, G., Sadler, P.&#xA;M., Sadler, S. M., &amp;amp; Bressoud, D. M. (2015). The impact of instructor&#xA;pedagogy on college calculus students’ attitude toward mathematics.&#xA;International Journal of Mathematical Education in Science and&#xA;Technology, 46(3), 370–387.&#xA;&lt;a href=&quot;https://doi.org/10.1080/0020739X.2014.979898&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1080/0020739X.2014.979898&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;Valås, H., &amp;amp; Søvik, N.&#xA;(1994). Variables affecting students’ intrinsic motivation for school&#xA;mathematics: Two empirical studies based on Deci and Ryan’s theory on&#xA;motivation. Learning and Instruction, 3(4), 281–298.&#xA;&lt;a href=&quot;https://doi.org/10.1016/0959-4752(93)90020-Z&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1016/0959-4752(93)90020-Z&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="199" LastActivityDate="2017-02-18T11:28:28.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12040" PostTypeId="1" CreationDate="2017-02-18T17:16:56.823" Score="14" ViewCount="1353" Body="&lt;p&gt;I've encountered classes of 25 or so undergraduates in which more than half -- maybe two-thirds -- of the students claim to have had a high-school geometry course but none has seen a proof of the Pythagorean theorem or the proof of any theorem in geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It used to be that some students who didn't remember that circles or triangles or lines or planes were ever mentioned in geometry nonetheless remembered that geometry is the subject where you prove things. That's the principal content of the course. So it was when I was in high school, and so it is in Euclid, which was the standard textbook before and during the time of the Roman Empire and from the middle ages until the middle of the 20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have the impression that some high school courses present a bunch of formulas about areas and volumes in a dogmatic way and call it &quot;geometry&quot;. And some others do intelligent things, e.g. thinking about groups of rotations, reflections, and translations, or combinatorial things about convex hulls or polyhedra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there some kind of standard secondary-school geometry curriculum today?&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2017-02-19T00:07:59.563" LastActivityDate="2017-04-08T02:45:49.300" Title="What is currently called geometry in high school?" Tags="|geometry|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12041" PostTypeId="2" ParentId="12040" CreationDate="2017-02-19T00:04:06.810" Score="6" Body="&lt;p&gt;I teach physics and a little calculus at a community college in California. My own kids went to local public high school, which is unusually good, and took the IB curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes in my physics classes, when discussing a topic like Newton's laws or DC circuits, I will use proof-based high school geometry as a reference point for the kind of formal reasoning and rigorous logic that is required. When I ask for a show of hands, about half my students say that they did two-column proofs in high school geometry. (This is usually accompanied with groans.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My kids used a high school geometry textbook that did include a heavy dose of theorem proving, although proof was not emphasized as much as it would be if one was taking one's agenda directly from Euclid. To my taste, the approach was a little ugly and baroque, but the beautiful simplicity of Euclid was definitely embedded somewhere in there. The real number system was treated as something separate from but connected to geometry -- a strange and ugly approach, IMO.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know if you're familiar with the US system. Our schools and curricula have traditionally been under completely local control, but over the last 40 years or so, control has gradually been shifting to the states and the federal government. We have something called Common Core, which is a set of national standards, which showed up around the same time that a well intended but sometimes farcical national political initiative called No Child Left Behind began to collapse under the weight of its impossible promises. Depressingly, Common Core &lt;a href=&quot;http://www.corestandards.org/Math/Content/HSG/introduction/&quot; rel=&quot;noreferrer&quot;&gt;says&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you try to parse this closely, it really doesn't make much sense. How does one do &quot;careful proofs&quot; without starting from &quot;a small set of axioms?&quot; It smells like language that was worked out as a compromise by a committee, taking into account the fact our educational system is neither able to provide rigorous educational opportunities for all, nor willing to make the attempt.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2017-02-19T00:14:03.537" LastActivityDate="2017-02-19T00:14:03.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12042" PostTypeId="2" ParentId="12040" CreationDate="2017-02-19T04:38:52.560" Score="12" Body="&lt;p&gt;I'm going to say mostly the same thing as Ben Crowell, from just a slightly different perspective. I'm at an urban community college on the U.S. East Coast, teaching college algebra, statistics, etc. To the question, &quot;Is there some kind of standard secondary-school geometry curriculum today?&quot;, then yes, the best answer currently is &lt;a href=&quot;http://www.corestandards.org/Math/Content/HSG/&quot; rel=&quot;noreferrer&quot;&gt;Common Core (Geometry)&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Addressing a few of your specific items:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A proof of the Pythagorean Theorem is an expected standard for earlier in the 8th grade (&lt;a href=&quot;http://www.corestandards.org/Math/Content/8/G/B/6/&quot; rel=&quot;noreferrer&quot;&gt;8.G.B.6&lt;/a&gt;: &quot;Explain a proof of the Pythagorean Theorem and its converse.&quot;). &lt;/li&gt;&#xA;&lt;li&gt;Proofs of theorems are an expected part of high-school geometry (&lt;a href=&quot;http://www.corestandards.org/Math/Content/HSG/&quot; rel=&quot;noreferrer&quot;&gt;HSG&lt;/a&gt;: &quot;Prove geometric theorems&quot;, &quot;Prove theorems involving similarity&quot;, &quot;Prove that all circles are similar&quot;, etc.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In this regard, I'm modestly optimistic about what I've seen in Common Core mathematics, and that it's something of a re-alignment to solid, traditional principles. However: It's hotly debated, not implemented in all states, not yet fully implemented in the states that have signed on in the last few years, possibly not rigorously tested, not yet representative of what graduating high school students would know, etc. To the extent that schools drifted in a bunch of different directions, it would (at best) take time for Common Core to wrangle them back on a sensible path. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will say this: Within the last year I've been giving a 1st-day introduction to all my courses including bullet points such as &quot;This class will include proofs&quot;. As part of that I've been asking, &quot;In what high school course did you talk about proofs a lot?&quot;, and gratifyingly every time so far I've gotten a common group shout-out of &quot;Geometry&quot;. So for me that's actually been a bright spot recently, and a point from which I can start making connections and deeper understanding. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-02-19T15:56:26.847" LastActivityDate="2017-02-19T15:56:26.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12044" PostTypeId="2" ParentId="12040" CreationDate="2017-02-19T21:02:37.630" Score="5" Body="&lt;p&gt;I can share with you my experience. When common-core started to take off, I had just started Geometry. We were expected to prove things regularly, triangle congruency, similarity, angle-sum, parallelograms, Pythagorean  theorem, etc. And we were graded on our reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a list of what I remember we went through, proofs found there way in every part except transformations and constructions though we were expected to still explain what we were doing and why:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Parallel lines cut by a transversal&lt;/li&gt;&#xA;&lt;li&gt;Triangle congruency, and proving theorems about triangles&lt;/li&gt;&#xA;&lt;li&gt;Transformations in the plane &lt;/li&gt;&#xA;&lt;li&gt;Constructions &lt;/li&gt;&#xA;&lt;li&gt;Triangle similarity (This is when we started to prove Pythagorean theorem)&lt;/li&gt;&#xA;&lt;li&gt;Trigonometry (sin,cos,tan,law of cosines, law of sines)&lt;/li&gt;&#xA;&lt;li&gt;Coordinate Geometry&lt;/li&gt;&#xA;&lt;li&gt;Proving and using theorems involving quadrilaterals &lt;/li&gt;&#xA;&lt;li&gt;Proving and using theorems about circles&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5384" LastEditorUserId="5384" LastEditDate="2017-04-07T04:01:52.717" LastActivityDate="2017-04-07T04:01:52.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12045" PostTypeId="1" AcceptedAnswerId="12047" CreationDate="2017-02-20T06:31:46.433" Score="12" ViewCount="3140" Body="&lt;p&gt;The following question appeared in &lt;a href=&quot;https://www.khanacademy.org/math/basic-geo/basic-geo-volume-sa/basic-geometry-surface-area/e/surface-areas&quot; rel=&quot;noreferrer&quot;&gt;this Khan Academy exercise&lt;/a&gt; in January 2017, and the actual data on the ten most frequent incorrect responses is shown below the question. &lt;strong&gt;What mistakes did students make to get to each of these wrong answers?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8YsIA.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8YsIA.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of student incorrect responses...&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;177 occurred ~8% of the time&lt;/li&gt;&#xA;&lt;li&gt;126 occurred ~7% of the time&lt;/li&gt;&#xA;&lt;li&gt;100 occurred ~7% of the time&lt;/li&gt;&#xA;&lt;li&gt;156 occurred ~6% of the time&lt;/li&gt;&#xA;&lt;li&gt;155 occurred ~4% of the time&lt;/li&gt;&#xA;&lt;li&gt;166 occurred ~3% of the time&lt;/li&gt;&#xA;&lt;li&gt;111 occurred ~2% of the time&lt;/li&gt;&#xA;&lt;li&gt;89 occurred ~2% of the time&lt;/li&gt;&#xA;&lt;li&gt;122 occurred ~2% of the time&lt;/li&gt;&#xA;&lt;li&gt;138 occurred ~2% of the time&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7842" LastActivityDate="2017-02-23T00:11:40.637" Title="How are students messing up in this Khan Academy surface area problem? (right triangular prism: 3-4-5 triangular base, height of 11)" Tags="|students-mistakes|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12046" PostTypeId="2" ParentId="12045" CreationDate="2017-02-20T19:06:46.070" Score="6" Body="&lt;p&gt;From everyone's comments, here's what we've got so far (see @benjamin's analysis for more details):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align}&#xA;177&amp;amp;=\underbrace{3\times55}_{\text{3 times SA big rect}} + \underbrace{6+6}_{\text{area of triangles}}&#xA;\\\\&#xA;126&amp;amp;=\underbrace{55+44}_{\text{2 rects correct}}+\underbrace{(15)}_{\text{3rd rect wrong (3*5 instead of 3*11)}}+\underbrace{6+6}_{\text{area of triangles}}&#xA;\\\\&#xA;100&amp;amp;=\underbrace{144-44}_{\text{missing a rect}}&#xA;\\\\&#xA;156&amp;amp;=\underbrace{121}_{\text{area of 3 rects}}+\underbrace{(3)(4)+(3)(4)}_{\text{didn't divide by 2 for triangles}} &#xA;\\\\&#xA;155&amp;amp;=\underbrace{144+11}_{\text{messed up on a rect somehow}}&#xA;\\\\&#xA;166&amp;amp;=\underbrace{144+22}_{\text{messed up on a rect somehow}}&#xA;\\\\&#xA;111&amp;amp;=\underbrace{144-33}_{\text{missing a rectangle}} \qquad \text{OR} \qquad \underbrace{3\times33 + 6+6}_{\text{similar mistake as for 177}}&#xA;\\\\&#xA;89&amp;amp;=\underbrace{144-55}_{\text{missing a rectangle}}&#xA;\\\\&#xA;122&amp;amp;=\underbrace{144-22}_{\text{messed up on a rect somehow}}&#xA;\\\\&#xA;138&amp;amp;=\underbrace{144-6}_{\text{missing a triangle}}&#xA;\end{align}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: As @michael pointed out, these most common wrong answers tend to be in the ballpark of the correct answer (144).&lt;/p&gt;&#xA;" OwnerUserId="7842" LastEditorUserId="7842" LastEditDate="2017-02-20T22:29:58.720" LastActivityDate="2017-02-20T22:29:58.720" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12047" PostTypeId="2" ParentId="12045" CreationDate="2017-02-20T19:53:08.427" Score="12" Body="&lt;p&gt;One quick way to guess at wrong answers is just to forget a face.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Omitting units, the correct answer is $144$ and the faces have surface areas of: $6$, $6$, $33$, $44$, and $55$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Missing one face: $144-6 = 138$, $144-33 = 111$, $144-44 = 100$, and $144-55 = 89$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, each of $138$, $111$, $100$, and $89$ appears in the list of incorrect responses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose some might have multiple mistakes; for example, forgetting one of the rectangular faces, and counting both of the other two as $5 \times 11$ for a total of $110$. In this scenario, you could then add on the triangular faces for a total of $122$, which also appears in the list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, even an erroneous approach could produce the correct answer. For example, one might compute the base rectangle as $4 \times 11$ and mistakenly believe all three of the rectangular faces have surface area $44$, for a total of $132$. Adding on the two triangular faces yields $144$, which is the right answer - arrived at in the wrong way!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Meanwhile, computing each rectangular face as $3 \times 11$ yields $99$, which adds with the triangular faces to (again) produce $111$. So, incorrect answers can arise in more than way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, as always, there is the simple possibility of altogether not knowing what to do and haphazardly combining numbers. For example, adding two of the edge lengths, $4$ and $5$, for $9$; then adding the other two edge lengths, $3$ and $11$, for $14$. But that results in two numbers, $9$ and $14$, and the final answer is supposed to be just one number. Okay - multiply them together: $9 \times 14 = 126$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note 1.&lt;/strong&gt; It is a bit tough to guess how an error has occurred, but if one believes the last situation is possible - haphazardly adding two edge lengths, then adding the other two edge lengths, then multiplying their respective sums - then we might expect this to occur for other pairings of edge lengths. Since it does not, I am guessing that the common $126$ error arose in another way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compute everything except the back face correctly. This gives a surface area of $$55 + 44 + 6 + 6 = 111$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, mid-solution, forget which label corresponds to which edge, and misread the diagram as labeling the back face as $3 \times 5$, for an additional contribution of $15$. This yields $126$, as (un)desired.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note 2.&lt;/strong&gt; For completeness, observe that many of the incorrect answers are off by a multiple of $11$; i.e., $177$, $166$, $155$, $122$, $111$, $100$, and $89$. Each of these can be explained by mis-computing around the rectangular faces, all of which have a side length of $11$. It is marginally interesting (to me) that $133$ did not appear as a frequent incorrect answer. All wrong responses are now accounted for.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="1147" LastEditDate="2017-02-23T00:11:40.637" LastActivityDate="2017-02-23T00:11:40.637" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12048" PostTypeId="1" CreationDate="2017-02-21T11:57:13.573" Score="1" ViewCount="4709" Body="&lt;p&gt;I have searched the web but not found too much that fits what I want. So, I thought I would try asking here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for projects to use as PBL for grade 10 math. Specifically for the topics of:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Functions&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Exponents/Roots&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The projects ideally would be worked on by students in teams of 2-3, using about 8-12 hours of team time over a month.  The results derived would lend themselves to making a short presentation or PowerPoint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most things I find are too simplistic where the product is simply a calculation to complete a worksheet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for any ideas.&lt;/p&gt;&#xA;" OwnerUserId="7850" LastActivityDate="2021-02-11T16:42:34.983" Title="Project Based Learning (PBL) ideas for secondary math topics" Tags="|secondary-education|functions|exponential|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12049" PostTypeId="2" ParentId="12048" CreationDate="2017-02-21T16:26:54.953" Score="2" Body="&lt;p&gt;The North Carolina School of Science and Mathematics has always done a great job of sharing projects.  If you go &lt;a href=&quot;https://www.ncssm.edu/residential-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; you'll find links to various courses and the modules that they use in them.  I think the Algebra 2 and AFM courses are probably most relevant to you.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-02-21T16:26:54.953" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12050" PostTypeId="2" ParentId="11954" CreationDate="2017-02-22T09:08:43.407" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Gaps in basic knowledge which make progressing difficult&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2017-02-22T09:08:43.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12051" PostTypeId="2" ParentId="12032" CreationDate="2017-02-22T09:19:30.993" Score="4" Body="&lt;p&gt;I introduced a program for students in 2nd through 6th grade which made:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;math cool in my school&quot;.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There were two components:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students who excelled in math had an opportunity to take enrichment at lunchtime.  These students designed activities and assemblies for their peers.  Assemblies included math fairs, original math games, math plays, treasure hunts etc. Additional activities included math newspapers, magic shows, and scavenger hunts. My enrichment students were proud to present, the attending students enjoyed the presentations, and many attending students worked harder at math to get into my program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second component was given at Friday lunchtime.  Anyone who was invited (usually about 40 students), could come and eat lunch with us and play math games. I had puzzles from think fun, math books, and a variety of math games.  This was very successful in piquing additional interest in math. Students who were in enrichment were invited and were allowed to bring a friend.  This made the enrichment program even more appealing (and more students worked harder at math). Additionally students who worked hard in math but weren't strong might be invited by me - giving additional motivation to students to work at math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any program which gives the successful math students special privileges (related to math) will increase motivation and give the students the feeling that math is cool!&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2017-02-22T09:19:30.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12052" PostTypeId="2" ParentId="4535" CreationDate="2017-02-22T23:49:19.593" Score="1" Body="&lt;p&gt;Most of the time, technical considerations are what allow us to make our ideas &lt;em&gt;more&lt;/em&gt; general. For example, the radial definition of a circle allows us to consider &quot;circles&quot; in generalized metric spaces, while the less technical &quot;round and symmetric&quot; definition does not. For the theory to be unable to capture the ideas in full generality, as with functions, is very rare. When this happens, it is usually due to technical reasons that can be sidestepped with even more technical constructions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a function that takes sets as inputs, here are some ways to do so:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Considering a function which maps a set to it's size is fine, as you can always consider a set of finite sets. We can even allow the set of all finite sets to be a set, or even the set of all &lt;em&gt;countable&lt;/em&gt; sets to be a set! As long as there is a set that is larger in size than the sets we are considering, it's just fine.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even when we actually want a function whose domain is &lt;em&gt;any&lt;/em&gt; set (no matter how large), we still have a trick to talk about them. They are called &lt;em&gt;class functions&lt;/em&gt;, and can be formally expressed through formulae. For example, if we want to talk about the function $a \to \{a\}$, we can let the formula $&quot;\phi(a)&quot; =\  &quot; \exists_{x}x=\{a\}&quot;$. Then, if we want to say $\phi(a)=\{\varnothing\}$, we would just substitute $\exists_{x}x=\{a\}$ for $\phi(a)$, to get &quot;$\exists_{x}x=\{a\}\wedge x=\{\varnothing\}$&quot;. Thus we can always talk about these functions even if they aren't sets. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;While I am not sure how suitable such theories are to actually doing mathematics, there are &quot;weak&quot; set theories where the set of all sets is also a set! &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7734" LastEditorUserId="7734" LastEditDate="2017-02-25T20:17:01.073" LastActivityDate="2017-02-25T20:17:01.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12053" PostTypeId="2" ParentId="11954" CreationDate="2017-02-23T15:08:12.517" Score="2" Body="&lt;p&gt;Difficulty to keep the thinking abstract e.g. using symbols instead of &quot;real&quot; world objects&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2017-02-23T15:08:12.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12054" PostTypeId="2" ParentId="12040" CreationDate="2017-02-23T17:50:43.543" Score="4" Body="&lt;p&gt;This is really a comment on some of the other answers but was too long to add as a comment to other answers, specifically about &quot;short&quot; axioms systems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that &quot;proofs&quot; tend to be given too much attention in lower grades but let me say something about &quot;axiomatics.&quot; In America when people look to do Euclidean geometry &quot;rigorously&quot; they turn to Hilbert's axioms. The first version of Hilbert's axioms were not independent but in the often used independent version there are 20 axioms, which is not so small a list. And then there are his 6 primitives. Also, Hilbert axiomatizes 3-dimensional geometry rather than plane geometry and for the plane geometry one has to &quot;worry about&quot; non-Desarguesian planes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_axioms&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Hilbert%27s_axioms&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one one wants to give the &quot;spirit&quot; of axiomatic systems and proving theorems within them one can select a small list of &quot;more powerful&quot; axioms and/or ignore issues that Hilbert calls attention to. I would rather show samples of proofs about what for me are more interesting geometric phenomena. Two of my favorites are Euler's traversability theorem for connected graphs (Euler circuits) and Euler's polyhedral formula ($V + F - E = 2$). I feel both the proof aspects here, and the &quot;content&quot; of the geometry is more important than a lot of the proofs done in axiomatic geometry contexts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;${{}}$&lt;/p&gt;&#xA;" OwnerUserId="1865" LastEditorUserId="205" LastEditDate="2017-04-08T02:45:49.300" LastActivityDate="2017-04-08T02:45:49.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12055" PostTypeId="1" CreationDate="2017-02-23T21:35:25.873" Score="5" ViewCount="357" Body="&lt;p&gt;Different people use different mental models, such as Webb's DoK (depth of knowledge), CCSM (Common Core State Standards) aspects of rigor, and Bloom's Taxonomy. At a high level, here's how I'm thinking about how they relate to each other (the mapping is obviously imperfect):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GwuCC.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GwuCC.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Which mental model is most useful when thinking about math education: Webb's DoK, CCSM's Aspects of Rigor, or Bloom's Taxonomy? Are there others?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="7842" LastEditorUserId="5220" LastEditDate="2017-12-24T17:12:48.270" LastActivityDate="2017-12-24T17:12:48.270" Title="Which mental model, Webb's DoK, CCSM's Aspects of Rigor, or Bloom's Taxonomy, is most useful?" Tags="|general-pedagogy|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12061" PostTypeId="1" CreationDate="2017-02-24T23:05:55.077" Score="8" ViewCount="457" Body="&lt;p&gt;I have just turned 18 and am in calculus BC (calc 1 &amp;amp; calc 2). I most certainly grasp and understand the concepts of calc 1 however every once in a while a I seem to struggle with the calc 2 work. More often than not I stay after school and end up understanding the concepts and/or my mistakes and why I made them. I'm a very anxious person so I if I don't understand something I worry that I cannot become a professor (this is a huge goal I would love to achieve). Is there any advice for self confidence and any advice at my chances of working at an University later in my life.&lt;/p&gt;&#xA;" OwnerUserId="7859" LastEditorUserId="-1" LastEditDate="2021-11-19T13:31:39.900" LastActivityDate="2021-11-19T13:31:39.900" Title="I'm worried that my struggles with calc 2 mean I won't be able to become a professor later" Tags="|calculus|student-motivation|struggling-students|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12063" PostTypeId="2" ParentId="4535" CreationDate="2017-02-25T07:41:19.913" Score="2" Body="&lt;p&gt;This kind of interaction is one reason why I'm personally &lt;em&gt;not&lt;/em&gt; fond of the classroom pattern of presenting examples first, and then prompting students to intuit the definition as an exercise (perhaps this a pedagogical anti-pattern?). The one advantage to that approach that I can think of is that it somewhat resembles the investigatory practices of mathematicians; but even then you're more frequently working with pre-existing definitions. A major downside is that you're spending lots of time on an exercise that confuses students as to what end-goal of the course is; likely they are &lt;em&gt;not&lt;/em&gt; going to be given assignments or tests that ask them to invent new mathematical definitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would recommend avoiding these situations entirely by simply presenting the definitions (to some workable level of formality) as the very first thing in a presentation; and then subsequently exploring motivations, affirmative and negative examples, patterns and theorems around those constructs, etc. Advantages:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It maximizes the time that the proper definition is hitting the students' visual receptors. &lt;/li&gt;&#xA;&lt;li&gt;It models professional mathematical practice of laying out definitions first by fiat in a paper, and then exploring their ramifications. &lt;/li&gt;&#xA;&lt;li&gt;It avoids the frustration and confusion of students thinking that maybe the primary activity they'll be responsible for is creating new definitions.&lt;/li&gt;&#xA;&lt;li&gt;It permits more time on questions and skills that students &lt;em&gt;will&lt;/em&gt; be responsible for; like recognizing what entities do and do not qualify for that definition, resulting theorems, proofs, applications, etc. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Now, I &lt;em&gt;will&lt;/em&gt; say that one of my most memorable classes in my undergraduate program was the day in senior-level intro to analysis when the professor challenged us to precisely describe what it means for a function to be continuous. That was a useful exploration of the possible pitfalls, for which all the parts of the epsilon-delta definition really are required. But: We all actually had seen the formal definition prior to this. And I don't think it would be a good use of time to do that every day. Usually, my recollection as a student is to be very frustrated and feel like it was a waste to be spending time on examples or being asked for speculate on answers before a definition-of-terms was presented for the topic in question. Most of the time it really feels like flailing around to no good purpose. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Coincidentally, I was just at an academic conference a few days ago where this happened in numerous sessions -- presenters wanted to do a &quot;think-pair-share&quot; exercise &lt;em&gt;before&lt;/em&gt; presenting their findings, and in most cases there was an interesting pushback from the professional academic audience. One attendee said, &quot;You're the expert, I'd prefer to spend more time hearing from you&quot;, which I think is a cogent analysis. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-02-27T23:35:44.227" LastActivityDate="2017-02-27T23:35:44.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12064" PostTypeId="1" AcceptedAnswerId="12066" CreationDate="2017-02-25T11:08:24.523" Score="5" ViewCount="321" Body="&lt;p&gt;In the country I'm living we learn how to draw the graph of quadratic equations such as $y=x^2$ or $y=x^2+5x+3$ before knowing calculus (in fact we don't even learn calculus until we begin the undergraduate studies) using the fact they are parabolas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me this is just a memorization. Are there some standard recognized methods to show students quadratic equations are parabolas?&lt;/p&gt;&#xA;" OwnerUserId="5236" LastActivityDate="2017-02-28T05:00:23.780" Title="How to teach to draw graphs of quadratic equations without knowing calculus?" Tags="|teaching|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12065" PostTypeId="2" ParentId="12064" CreationDate="2017-02-25T13:53:50.903" Score="7" Body="&lt;p&gt;This is one of the most celebrated applications of completing the square.  $$y=ax^2+bx+c \Longrightarrow y=a\left(x+\frac{b}{2a}\right)^2+\left(c-\frac{b^2}{4a}\right)$$  Since a square (in your non-complex context) must be nonnegative, and will have a minimum when it is zero, the vertex of the parabola will be at $$(x,y)=(-b/2a,c-b^2/4a)$$  If this equation doesn't look like the stretched parabola $ax^2$ shifted by this vertex, what would?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, you can find the places it hits the $x$-axis by solving for $y=0$ using the quadratic formula or factoring.  The $y$-intercept would be $c$, of course.  Hopefully with these three pieces of information you could plot as much as you want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, this isn't really a matheducators.SX type answer, but I hope it leads you to thinking about the many problems typically assaulted with calculus that may not need it, proving that &lt;a href=&quot;http://www.jstor.org/stable/10.4169/074683410x475074&quot; rel=&quot;noreferrer&quot;&gt;dogs don't need calculus&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-02-25T13:53:50.903" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12066" PostTypeId="2" ParentId="12064" CreationDate="2017-02-25T14:21:07.410" Score="5" Body="&lt;p&gt;Start with the definition of a parabola as a &lt;a href=&quot;https://en.wikipedia.org/wiki/Parabola#Definition_of_a_parabola_as_locus_of_points&quot; rel=&quot;noreferrer&quot;&gt;locus of points&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A parabola is a set of points, such that for any point $P$ of the set the distance $|PF|$ to a fixed point $F$, the focus, is equal to the distance $|Pl|$ to a fixed line $l$, the directrix: $\{P\mid|PF|=|Pl|\}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then locate the parabola in a rectangular coordinate system.  Start with the simplest case of a parabola with its focus $F$ at coordinates $(0,f)$ (with $f&amp;gt;0$) and its directrix at $y=-f$.  For a point $P$ (with coordinates $(x,y)$) on the parabola, we get from $|PF|^2=|Pl|^2$ the equation $x^2+(y-f)^2=(y+f)^2$.  This results in the quadratic equation $y=\frac{1}{4f}x^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Give more complicated examples until you get to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Parabola#General_case&quot; rel=&quot;noreferrer&quot;&gt;general case&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2017-02-25T14:21:07.410" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12067" PostTypeId="2" ParentId="12061" CreationDate="2017-02-25T16:01:19.120" Score="22" Body="&lt;p&gt;A lot of students seem to make it through high school and well into college with the idea that school is supposed to be easy, and that having to work hard, or being confused at times, or struggling with some topics means that something has gone wrong.&lt;/p&gt;&#xA;&lt;p&gt;This is nonsense. Struggling with material is a normal part of learning. Indeed, it's an &lt;em&gt;essential&lt;/em&gt; part of learning, because everyone eventually hits a point where they struggle, and learning how to deal with is an important skill.&lt;/p&gt;&#xA;&lt;p&gt;Relatedly, you almost certainly &lt;em&gt;don't&lt;/em&gt; understand the concepts of calc 1. That's not a criticism; it's basically impossible to fully understand the concepts of calculus 1 when you've just taken it. Learning math is a cyclical process, where you learn concepts, use them as a basis to learn additional concepts, and then return to the old concepts and learn more about them in light of what you now know.&lt;/p&gt;&#xA;&lt;p&gt;I bring this up because one thing you should expect while you learn calc 2 is to have to revisit things you think you were done with and realize new things about them.&lt;/p&gt;&#xA;&lt;p&gt;To answer your second question, we  have no information about your chances of working at a university later. &lt;em&gt;No&lt;/em&gt; information. Neither the fact that you breezed through calc 1, nor the fact that you're struggling a bit more in calc 2, is a particularly strong indicator, especially since calc BC is very unlike the sort of math professors do.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2021-11-08T17:54:09.220" LastActivityDate="2021-11-08T17:54:09.220" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12068" PostTypeId="2" ParentId="12061" CreationDate="2017-02-25T20:01:01.417" Score="8" Body="&lt;p&gt;Henry is completely right that what your struggle is normal. I want to add that your reaction to it extremely positive. Keep doing what you are doing! Knowledge cannot be directly transmitted from your teacher's head to yours; it must be translated into a communication medium such as words or pictures. There is always something lost in this process. That is why people make mistakes or form misconceptions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By staying after and carefully considering every mistake and misconception, you are rebuilding what was lost when the knowledge was first communicated. Consistently doing this both in and out of class is the key to truly learning from the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You started this practice motivated by your desire to stay competitive for an academic position in the future. Stay committed to your this approach to learning. If you continue building the habit of consistently filling all the holes you find in your knowledge, you will develop confidence that you thoroughly understand all of the material presented in class. As you continue doing this, I think you will find that desire to be confident in the material will overtake anxiety as the motive force for your studying. &lt;/p&gt;&#xA;" OwnerUserId="7734" LastActivityDate="2017-02-25T20:01:01.417" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12069" PostTypeId="2" ParentId="12061" CreationDate="2017-02-27T11:45:48.547" Score="5" Body="&lt;p&gt;I'd also like to add that if you intend to be a teacher, your struggles now will most likely make you a better teacher because you will have a better sense of what your students go through and you'll be able to share your coping strategies with them. In fact, if being a good teacher is one of your goals, I suggest you watch how your fellow students cope as well for further ideas to help your future pupils.&lt;/p&gt;&#xA;" OwnerUserId="7870" LastActivityDate="2017-02-27T11:45:48.547" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12070" PostTypeId="1" AcceptedAnswerId="12091" CreationDate="2017-02-28T03:59:57.857" Score="10" ViewCount="1025" Body="&lt;p&gt;I'm a college math/science tutor and I'm really interested in STEM education. I'm currently starting work on a project I hope to present in a couple of months at a tutoring conference and I was wondering if you could point me in the direction of useful research. The project is about mathematical intuition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm aware of at least a couple of ways to define the thing:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;As the innate, untutored sense some have for certain mathematical&#xA;concepts. I'm aware (in passing) of research that claims children&#xA;have an innate sense of number, for example. As my students are&#xA;college-age, I'm not sure this is the definition I want to focus on. &lt;/li&gt;&#xA;&lt;li&gt;As the sense, usually gained from experience, of the basic nature of the&#xA;solution to a problem, without having formally attempted to solve it.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It is this later definition that piqued my curiosity. In my experience, while many of my students can do calculations just fine, they struggle to decide whether or not their calculation makes sense. In Chemistry for example, students have a hard time figuring out which units are bigger than which, frequently making errors like:&#xA;$$10\,miles &amp;lt; 50\,meters$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or:&#xA;$$number\,of\,atoms\,in\,5\,moles\,of\,F &amp;lt; 10^{23}$$&#xA;As far as mathematics goes, I frequently have to remind students that variables follow the same rules &quot;normal&quot; numbers follow, for example that:&#xA;$$\frac{x}{2x}+\frac{x}{x^{2}}$$&#xA;Would be solved in a similar way as&#xA;$$\frac{1}{3}+\frac{3}{5}$$&#xA;is solved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I find this one of the most frustrating parts of my job, as it suggests we (both the professors and myself) have issues communicating both conceptual understanding and a successful problem-solving heuristic, and instead are training students in symbol manipulation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This second definition of mathematical intuition brings both Polya and Fermi to mind, as they both were at least partly focused on making plausible approximations. Further, I'm also interested in the &quot;bad&quot; intuitions students bring to a problem. Here I am thinking of things like the gambler's fallacy which is so prevalent in beginning statistics classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My search for relevant articles has turned up little that relates to higher education, and more that relates to either intuition as it applies to professional mathematicians or primary/secondary education. My questions are as follows:&#xA;Do you know of any research on the common mathematical misconceptions and intuitions of college students/college courses, where they come from and how to combat them?&#xA;Do you know of any research on strategies to develop helpful intuitions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also be very curious to know if you have encountered &quot;bad&quot; intuitions, and about what you personally do to cultivate &quot;good&quot; intuitions.&lt;/p&gt;&#xA;" OwnerUserId="7872" LastActivityDate="2017-03-05T01:17:43.327" Title="&quot;Good&quot; and &quot;Bad&quot; student intuitions when teaching and learning mathematics" Tags="|undergraduate-education|reference-request|education-research|students-mistakes|tutoring|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12071" PostTypeId="2" ParentId="12064" CreationDate="2017-02-28T05:00:23.780" Score="0" Body="&lt;p&gt;I'd say this is a highly visual problem. I'd agree with Mr. Noche in that the best way to approach this is from a geometry perspective. In addition. I'd use something like &lt;a href=&quot;https://www.desmos.com/calculator&quot; rel=&quot;nofollow noreferrer&quot;&gt;Desmos&lt;/a&gt;, to show students how the graph of a line changes when we &quot;multiply by x&quot; (how, for example, the equation $y=ax^{2}+b$ seems to &quot;open and bend&quot; as a gets progressively closer to 0 and the equation gets closer to $y=b.$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like Mr. Noche suggests, start slowly, adding complications as students get more comfortable the equation's structure. Here's how I would approach this thing:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Refresh students on linear variation. &lt;/li&gt;&#xA;&lt;li&gt;Numerically compare linear variation to quadratic variation.&lt;/li&gt;&#xA;&lt;li&gt;Have them plot $y=x$ (by hand or in Desmos) &lt;/li&gt;&#xA;&lt;li&gt;Have them plot $y=x^{2}.$ Discuss similarities and&#xA;differences.&lt;/li&gt;&#xA;&lt;li&gt;Ask something like: &quot;What do you think will happen if I&#xA;add some number $b$ to this equation?&quot; After some discussion, show&#xA;them what happens.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask them something like &quot;What do you think will happen if I make this number $b$ bigger/smaller?&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask them something like: &quot;What would happen if I&#xA;multiply this $x^{2}$ by some number $a$?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Build progressively, until students are playing around with the general quadratic.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7872" LastActivityDate="2017-02-28T05:00:23.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12072" PostTypeId="2" ParentId="10510" CreationDate="2017-02-28T05:40:05.103" Score="9" Body="&lt;p&gt;In my view, Khan Academy should be used as one would (ideally) use a textbook: To provide students with an overview of the material and algorithmic drills so the teacher has more free time to discuss the nuances. I think Khan Academy is brilliant if used in this way. I personally learned to handle basic probability thanks to Sal's explanations. The trouble (as with any technology) arises when a teacher becomes over reliant on the method. Khan Academy has its problems:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sal has a policy of not editing out digressions and initial errors from his videos. While this is commendable on the one hand (because it shows math and science as experimental rather than dogmatic) it may be confusing to students who are learning a topic initially. It's the teacher's responsibility to keep an eye out for those difficult bits in the presentation.&lt;/li&gt;&#xA;&lt;li&gt;The exercises, while vastly improved, still focus heavily on procedural understanding. Again, this is not a problem if we use Khan Academy as a textbook of sorts, but if we don't supplement we run the risk of leaving holes in the students' understanding of a topic.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7872" LastActivityDate="2017-02-28T05:40:05.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12074" PostTypeId="1" CreationDate="2017-02-28T13:26:02.990" Score="16" ViewCount="291" Body="&lt;p&gt;One idea for mid-level college math classes (i.e. after calculus, before highly proof based courses) which I've seen people speak very positively of is to have a portion of the course consist of students producing conjectures on their own by coming up with examples and counterexamples until they have a viable conjecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This sounds like a great fit with, for instance, a discrete math course, where engaging with lots of examples is important to grasping the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However every time I've tried this, I find myself serving as an oracle, because there's no &quot;stopping point&quot; if they get the answer right: if there's a counterexample, they might be able to find it if they looked harder, but once they have a right conjecture, the only tool they have for verifying it is asking me whether or not it's right.  So every time they have a new conjecture, they ask me if it's right.  (And, at its worst, we alternate between me giving a counterexample and them producing a new conjecture taking into account that one additional counterexample and then asking again.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I like about the concept is that it starts to move away from &quot;professor as arbitrary decider of right and wrong&quot; which many of my students have internalized. But when the definition of correct conjecture is &quot;professor says you can stop now&quot;, it ends up reinforcing that notion instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is how to avoid this: for students who can't be asked to prove their conjectures, how do they know when to stop looking for counterexamples?&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-01-02T19:41:01.303" Title="Student Conjectures without Oracular Professor" Tags="|undergraduate-education|inquiry-based-learning|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="12075" PostTypeId="2" ParentId="352" CreationDate="2017-02-28T20:50:58.657" Score="19" Body="&lt;p&gt;Let me add my 2 cents in. As someone who grew up in Latin America (where the tables were drilled into us, or else) and who now tutors mathematics, more than once I've had the experience of students were AMAZED that I could recall simple multiplication faster than they could type it in their calculator. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally like to think we memorize the tables for the same reason we learn to drive, or for the same reason Feynman reportedly advised students to be so familiar with derivatives they should be able to do them in their sleep. If we were always hesitant about driving, if we were always thinking about the individual tasks driving requires, we would never enjoy the actual purpose of driving (getting to nice places). Similarly, if we don't memorize the times tables we will be too lost in the trees to see the actual mathematical forest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is my honest opinion that math reforms swing the pendulum too far away from drill. Drill and memorization have their place in mathematics. It is only after we have become so proficient at a task that it becomes automatic that we can begin to fully appreciate it and tinker with it in creative ways.&lt;/p&gt;&#xA;" OwnerUserId="7872" LastActivityDate="2017-02-28T20:50:58.657" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12076" PostTypeId="2" ParentId="12074" CreationDate="2017-02-28T22:15:41.763" Score="5" Body="&lt;p&gt;If you absolutely don't want to introduce your students to proof methods yet, I think you should set a clear timeline. Give them a day or two to work on their conjectures (preferably in groups) before bringing them to class. You could look at this as an iterative design problem, where the testing-redesign cycle is out of whack. While an engineer ideally wants their product to be perfect, they'll often settle for something workable if time-to-market is an issue. I'd say give them something workable to shoot for (like: &quot;your fellow students shouldn't be able to find counterexamples within a specified time frame&quot;) and something that's a &quot;moonshot&quot; (like: &quot;Your professor shouldn't be able to find counterexamples within a specified timeframe&quot;) and then let them loose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That way, you take yourself somewhat out of the equation and you avoid the &quot;time-sinkiness&quot; that is in my opinion the greatest pitfall of an inquiry-based mathematics approach. To have this as a multi-day group project would also allow the students to engage with their own &quot;mathematical community&quot; or even with larger communities like SX. You could even offer rewards (like extra credit points) for students who are able to stump you or their fellow students.&lt;/p&gt;&#xA;" OwnerUserId="7872" LastActivityDate="2017-02-28T22:15:41.763" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12078" PostTypeId="2" ParentId="12070" CreationDate="2017-03-01T06:00:32.130" Score="3" Body="&lt;p&gt;Tversky and Kahneman's work showing that people rely on heuristics might be in the right direction. Their research showed that peoples intuition for statistics is often bad due to our heuristics that can ignore critical information about a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think mathematical intuition especially with college level concepts on up is the second definition you used, that &quot;good&quot; mathematical intuition is mostly a matter of practice and being familiar with a problem and &quot;bad&quot; mathematical intuition is not enough experience or heavily biasing one piece of information i.e: &lt;img src=&quot;https://latex.codecogs.com/gif.latex?2%5E%7B2%5E%7B2%5E%7B2%5E%7B2%7D%7D%7D%7D&quot; title=&quot;2^{2^{2^{2^{2}}}}&quot; /&gt; looks like a small number because it's made of 2s but if you know that stacking exponents make numbers incredibly large it's not surprising to find out that it's in the order of &lt;img src=&quot;https://latex.codecogs.com/gif.latex?10%5E%7B19728%7D&quot; title=&quot;10^{19728}&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore symbol manipulation is great for problems that can't be understood concretely. Few people can intuit a 4 dimensional surface and following a rote procedure can allow a mathematician to manipulate that space even if they don't &quot;understand&quot; it. &lt;/p&gt;&#xA;" OwnerUserId="7881" LastEditorUserId="-1" LastEditDate="2017-03-10T09:42:42.233" LastActivityDate="2017-03-01T06:00:32.130" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12080" PostTypeId="1" AcceptedAnswerId="12092" CreationDate="2017-03-01T17:12:53.403" Score="10" ViewCount="1359" Body="&lt;p&gt;There was previously a &lt;a href=&quot;https://matheducators.stackexchange.com/q/12077/3593&quot;&gt;question/rant&lt;/a&gt; here on MESE about why so many are still using the PEMDAS/BODMAS/BIDMAS/BEDMAS mnemonics to teach order of operations. The question was deleted (still viewable by 10K+ users), but there were some comments and an answer that had a link to an &lt;a href=&quot;http://www.madmath.com/2009/03/pemdas-terminate-with-extreme-prejudice.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;argument against PEMDAS&lt;/a&gt;, from which an interesting and useful question can be extracted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like the &lt;a href=&quot;https://matheducators.stackexchange.com/users/5563/daniel-r-collins&quot;&gt;author&lt;/a&gt; of the linked argument, I, who attended government-run schools in the suburban US in the 1970s, also had never heard of PEMDAS until I was an adult. In fact, I probably first encountered the mnemonic while reading edu-blog posts and/or MESE or MSE. I don't remember how we were taught, but I know that I have internalized the rules so I don't have to think about them, whereas I encounter adults who have to write down &quot;PEMDAS&quot; before they can begin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realize this may be several questions, but they are interrelated: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why, and less importantly when and where, did mathematics educators begin to use mnemonics to teach order of operations?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: there is a &lt;a href=&quot;https://matheducators.stackexchange.com/q/10981/3593&quot;&gt;related question&lt;/a&gt; about why the mnemonic rule works, but that doesn't get into the justification for teaching order of operations in this way.&lt;/p&gt;&#xA;" OwnerUserId="3593" LastEditorUserId="3593" LastEditDate="2018-10-22T01:20:28.320" LastActivityDate="2023-04-21T01:04:24.687" Title="What is the pedagogical justification and history for using mnemonics to teach order of operations?" Tags="|mathematical-pedagogy|arithmetic-operations|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12081" PostTypeId="2" ParentId="12032" CreationDate="2017-03-01T19:38:08.443" Score="2" Body="&lt;p&gt;Well... I really want to answer this question. Maybe two days earlier, I was watching a &lt;em&gt;Numberphile&lt;/em&gt; video by Edward Frenkel. He discussed the question &lt;strong&gt;Why So many people hate mathematics??&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The video start with an interesting example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edward said: Suppose you are gonna take an art class and you are given work of painting fence (This thing is quite boring if you don't like it initially) but were never shown the paintings of great artists. When you will grew older, you will say Man! I was really bad at art,I hated it but what you would really be saying is that &lt;strong&gt;I was bad at painting the fence&lt;/strong&gt; and that's what actually happens in Mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember that once professor &lt;em&gt;Walter Lewin&lt;/em&gt; said that if you are bad at physics, you didn't had a good physics teacher. I would say that this applies on Maths too. This is certainly fault of Maths teachers. Actually teachers are paid to make students love their subject because if they will love it, they will become good at it themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, Imho the only thing you should do is to relate Mathematics to daily life of students. If they will see it around them, they will try to explore it and hence they will love it.&lt;/p&gt;&#xA;" OwnerUserId="7654" LastActivityDate="2017-03-01T19:38:08.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12082" PostTypeId="2" ParentId="11971" CreationDate="2017-03-01T23:28:28.510" Score="0" Body="&lt;p&gt;Explain exactly why you need to learn calculus in order to program an algorithm. If you can identify a specific need, then you can focus on that specific requirement or need. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus usually consists of 3 general topics - differential calc, integration and vector calc?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which one of these do you require for solving your programming problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regards&lt;/p&gt;&#xA;" OwnerUserId="7886" LastActivityDate="2017-03-01T23:28:28.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12083" PostTypeId="2" ParentId="10923" CreationDate="2017-03-02T02:47:57.283" Score="6" Body="&lt;p&gt;One suggestion would be to take an approach similar to that used for describing the $52!\approx 8.063*10^{67}$ ways to arrange a standard deck of playing cards as outlined in &lt;a href=&quot;http://czep.net/weblog/52cards.html&quot; rel=&quot;noreferrer&quot;&gt; Scott Czepiel's Blog&lt;/a&gt; and popularized in this vsauce &lt;a href=&quot;https://www.youtube.com/watch?v=ObiqJzfyACM&amp;amp;ab_channel=Vsauce#t=14m40s&quot; rel=&quot;noreferrer&quot;&gt;youTube video @ 14:40&lt;/a&gt;. The example given in the blog post is to set a timer for $8.06*10^{67}$ &lt;em&gt;seconds&lt;/em&gt; and pass the time by&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...picking your favorite spot on the equator. You're going to walk around the world along the equator, but take a very leisurely pace of one step every billion years. Make sure to pack a deck of playing cards, so you can get in a few trillion hands of solitaire between steps. After you complete your round the world trip, remove one drop of water from the Pacific Ocean. Now do the same thing again: walk around the world at one billion years per step, removing one drop of water from the Pacific Ocean each time you circle the globe. Continue until the ocean is empty. When it is, take one sheet of paper and place it flat on the ground. Now, fill the ocean back up and start the entire process all over again, adding a sheet of paper to the stack each time you’ve emptied the ocean.&#xA;  Do this until the stack of paper reaches from the Earth to the Sun. Take a glance at the timer, you will see that the three left-most digits haven’t even changed. You still have 8.063e67 more seconds to go. So, take the stack of papers down and do it all over again. One thousand times more. Unfortunately, that still won’t do it. There are still more than 5.385e67 seconds remaining. You’re just about a third of the way done.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Given that you want to emphasize how large $10^{220}$ is, you can stack all sorts of other counters on top of those mentioned (the linked blog post / video also gives the example of dealing a 5-card hand, buying a single lottery ticket, putting a grain of sand in the grand canyon, etc.). The key to all this is that multiplying numbers with the same base results in the exponents adding (this can be a good exercise for students, verifying that the calculations are correct). &lt;/p&gt;&#xA;" OwnerUserId="7260" LastEditorUserId="7260" LastEditDate="2017-03-02T03:02:48.687" LastActivityDate="2017-03-02T03:02:48.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12084" PostTypeId="2" ParentId="10510" CreationDate="2017-03-02T16:03:27.463" Score="6" Body="&lt;p&gt;Khan Academy has been very helpful as a &lt;em&gt;supplement&lt;/em&gt; to my instruction in the classroom, especially for developmental students in higher education. I agree it can be confusing for students at times, as some topics are hard to find, but I occasionally set up a Khan Academy shell for my classes (especially over the Summer when my courses are online) to help direct them through the correct videos. Many students - especially the older or &quot;non-traditional&quot; students in my classes - have commented that the website is fantastic as a supplemental aid to the main tools I use to deliver instruction. &lt;/p&gt;&#xA;" OwnerUserId="7888" LastActivityDate="2017-03-02T16:03:27.463" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12085" PostTypeId="1" AcceptedAnswerId="12086" CreationDate="2017-03-02T16:36:35.537" Score="12" ViewCount="363" Body="&lt;p&gt;I have instructed for community colleges and adjuncted for major universities for just over fifteen years now. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In all my time doing this job, I've found that the worst manifestation of so-called &quot;mathphobia&quot; inevitably appears in my Statistics courses when we move to the material dealing with inferential methodology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have used many different source materials (different textbooks, websites, Youtube videos, example problem writeups posted in addition to lecture, etc.) but it invariably follows the same pattern each semester: one or two of the more vocal students will eventually interrupt me mid-lecture and will claim that they do not understand anything that's going on. For reasons that are probably psychological and similar to &quot;giving up,&quot; they almost always claim they haven't understood anything in the class at all, &lt;em&gt;including&lt;/em&gt; very elementary concepts like measures of center and introductory empirical probability (e.g. from small tables that give counts for just two binomial variables). This will invariably snowball into the whole class claiming the same lack of understanding, even including those who are earning the highest grades in the section. Among all the math classes I've taught, this part of this course presents the toughest psychological barriers I have to help students overcome, and, ironically, since spreadsheet formulas usually crunch the numbers for us on almost all the problems we encounter, this part also represents the &lt;em&gt;least&lt;/em&gt; computation-focused part of any of my courses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After facing this scenario again yesterday, I have reached two possible contributors to this phenomenon: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The processes of confidence intervals and hypothesis testing is difficult and counterintuitive for students learning it for the first time. Even though I've attempted to use analogies (e.g. the &quot;innocent until proven guilty&quot; analogy for hypothesis testing), students have a hard time understanding the concept of using a sample to draw inferences about parameters, and do not understand why error is needed, even if I show them that they already implicitly understand that even the most representative sample still leads to error. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students are looking at the increasingly involved equations needed to compute standard error, z or t-scores for levels, and for test statistics, and math panic sets in even if I tell them that they will far more often lean on technology for these computations and instead should focus more on actual interpretation. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These issues inevitably pop up to form mass confusion and delayed lecturing schedules, even if I convey multiple times that patience and disciplined practice is the true teacher in learning inferential statistics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A great example of this phenomenon is the example I used yesterday comparing two population proportions: one student just would not stop arguing with me that .51 and .5 (results for two sample proportions I compared in an example) were different numbers and therefore had to mean that the two proportions were different; the student simply would not accept my explanation that the margin of error for the difference implied that a true parameter difference could in fact be zero with the .01 difference appearing out of acceptable error due to the fact that we are sampling. The idea wouldn't sink in even after I slowly reviewed the idea of inference and error with her to her agreement; when we revisited the problem, she just slammed her fist on the table, declared a .01 difference meant two different numbers again in anger, and (as is usually the case) demanded the age-old question of &quot;where she would ever have to use this stuff.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I apologize for venting a bit, but I guess my central point is to ask if any other math educator finds inferential statistics to be one of the top points of difficulty with students, especially math-phobic ones, and what you do to help students get past the frustration and the tendency to discount the presence of error and the fact that we are estimating (confidence interval) or testing a population claim (hypothesis testing) by use of a representative but much smaller sample. Either I have been consistently teaching the subject wrong for years, or else this should be an ordeal many other teachers face when teaching these concepts to students for their first time. &lt;/p&gt;&#xA;" OwnerUserId="7888" LastActivityDate="2018-07-23T16:07:39.190" Title="Teaching new stats students confidence intervals, hypothesis testing, and other general techniques for inference" Tags="|student-motivation|statistics|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12086" PostTypeId="2" ParentId="12085" CreationDate="2017-03-02T17:06:22.560" Score="13" Body="&lt;p&gt;I've been teaching introductory statistics for the same amount of time at a large urban community college. I have never had this response from a class &lt;em&gt;in toto&lt;/em&gt;. Last semester I did have one student say that privately. Every semester I definitely have sharp students who do in fact &quot;get it&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know exactly what may cause this, but here's a few tidbits from my practice:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I use Weiss &lt;em&gt;Introductory Statistics&lt;/em&gt; as a textbook, and follow it very closely. I think it's clear and readable. Linking closely to a textbook, I think, communicates that I'm not just personally crazy and making this stuff up any given day. The exercises are almost entirely from real-world data. &lt;/li&gt;&#xA;&lt;li&gt;I actually do not rely on, or usually mention, any advanced technology in the body of the course; everything is in fact hand-computed (which is against current trends). I think this is worthwhile for a first course, to really dig into the specifics, develop some intuition, and avoid yet another layer of novelty (I do mention &amp;amp; give some examples on the last day). &lt;/li&gt;&#xA;&lt;li&gt;Every day I'm machine-gunning conceptual question at the class (e.g., &quot;Can a mean be negative? Can a standard deviation? Why? How can you check such calculations?&quot;) and will discuss/return to anything that some of the class trips over. &lt;/li&gt;&#xA;&lt;li&gt;In particular, from the first day to the last, I'm constantly pressing students to recognize the difference between population and sample statements. (&quot;Is the first sentence regarding population or sample? Is the second sentence? Is this table? Is this number? What's the probability that the sample mean is equal to the population mean? Does the population need to be normally distributed for our procedure to be valid?&quot;). This is key.&lt;/li&gt;&#xA;&lt;li&gt;Early on I present a paper from the JAMA full of C.I.'s and P-values, to emphasize that this is everyday scientific practice. &lt;/li&gt;&#xA;&lt;li&gt;Compared to other instructors, I eliminate a large swath of unnecessary probability content, so as to have plenty of time in the schedule to focus on the inferences (because: that really is the new, hard, important part). Total lecture time spent on discrete probability in my course is 90 minutes. &lt;/li&gt;&#xA;&lt;li&gt;In the latter part of the course, I run a concrete example of C.I.'s with playing cards (5 groups, each gets 4 cards, form a C.I. for the mean at 80% confidence; how many C.I.'s should contain the population mean? Check). I do think this generates some real intuition. (&quot;Is the sample mean always the same as the population mean? Could we have gotten A-A-A-A? Is it likely?&quot;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-07-23T16:07:39.190" LastActivityDate="2018-07-23T16:07:39.190" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12087" PostTypeId="1" CreationDate="2017-03-03T03:55:18.023" Score="13" ViewCount="2299" Body="&lt;p&gt;Ever since I learned about &lt;a href=&quot;https://projecteuler.net/&quot; rel=&quot;noreferrer&quot;&gt;Project Euler&lt;/a&gt;, I have been astonished and wondering about how Colin Hughes (the creator of Project Euler) manages to come up with such problems at such a rapid pace (once a week, except for Summer). The criteria for PE is very clearly and narrowly defined.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The problem has to be challenging.&lt;/li&gt;&#xA;&lt;li&gt;The problem has to be impossible to brute force.&lt;/li&gt;&#xA;&lt;li&gt;The problem has to require some mathematical insight.&lt;/li&gt;&#xA;&lt;li&gt;But even with the insight, you need some programming skills.&lt;/li&gt;&#xA;&lt;li&gt;An efficient program can always be written which will solve the problem in under a minute.&lt;/li&gt;&#xA;&lt;li&gt;The problems are recreational and fun.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note that not all PE problems satisfy all of these criteria strictly. The problems do range in difficulty. Obviously not all problems are &quot;fun&quot; for everyone. An easy problem can be brute forced in a reasonable amount of time, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being a math educator myself, I also, very frequently, have a defined objective. But I find it very difficult and time consuming to come up with good math problems that satisfy my objective, for my students. Many others have this problem and related questions have been asked on &lt;a href=&quot;https://matheducators.stackexchange.com/questions/tagged/problem-design&quot;&gt;this forum&lt;/a&gt; multiple times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;How can Project Euler come up with so many good problems with a frequency (that I think is) so high?&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am genuinely curious about this. Is there a big group of researchers/educators which works on this? Is there an army of grad students which does the research and finds the solution before a problem is posted? Or is it simply talent which cannot be taught and passed-on? If it is a skill, I am hoping that it'll help myself and others with the problem of problem-designing.&lt;/p&gt;&#xA;" OwnerUserId="342" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2018-07-28T13:08:53.077" Title="How does Project Euler come up with such good problems so rapidly?" Tags="|mathematical-pedagogy|problem-design|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12088" PostTypeId="2" ParentId="10923" CreationDate="2017-03-03T19:30:11.340" Score="0" Body="&lt;p&gt;I suggest coupling your Mandelbrot set zoom with the classic Eames &lt;em&gt;Powers of Ten&lt;/em&gt; movie.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.eamesoffice.com/the-work/powers-of-ten/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.eamesoffice.com/the-work/powers-of-ten/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Powers_of_Ten_(film)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Powers_of_Ten_(film)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4984" LastActivityDate="2017-03-03T19:30:11.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12089" PostTypeId="1" CreationDate="2017-03-03T20:26:18.327" Score="12" ViewCount="390" Body="&lt;p&gt;I see a lot of my students (I am in the US), usually Indian, write the limits of integration to the LEFT of the integral sign rather than customary top or right. The formula will look for example like ${}_5^7 \int \sin(t)dt$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this standard in India? &lt;/p&gt;&#xA;" OwnerDisplayName="user5108" LastEditorUserId="143" LastEditDate="2017-03-04T00:50:33.230" LastActivityDate="2017-03-08T18:55:24.227" Title="Why do some of my, usually international/Indian students, write limits to the left of the integral?" Tags="|notation|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12090" PostTypeId="2" ParentId="12085" CreationDate="2017-03-04T05:42:34.313" Score="5" Body="&lt;p&gt;I have been tutoring stats for a couple of years, and I find that there are a couple of things that students generally find difficult in stats (as compared to other courses):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The symbols&lt;/strong&gt;: Students seem to have trouble grasping the concept that the calculations are basically the same (save for standard deviation) for both parameters and statistics, and get stressed out by the sheer number of different symbols. I talk myself hoarse every semester stressing the importance of keeping a table of symbols and what they mean. Not being fluent with the Stats symbols is like trying to think critically about a French novel without knowing any French. I also think that, while well-intentioned, efforts to &quot;simplify&quot; the symbols (like those in several Social Science Stats classes) are misguided. All we end up with are two different tables of symbols, which mean the same things but make both the student and the tutor's job much harder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The tedium&lt;/strong&gt;: While I understand that handling large data sets is kind of required for the inference part of Stats, I wonder if something could be done about it. Dealing with large data sets results in what I think is a lose-lose situation. On the one hand, large data sets encourage calculator use, which can be a double edged sword, because you can use the reliance on tech to have more time to tackle the concepts, but it also leaves students feeling like the formulas are some sort of magic box, without mathematical meaning. More than once I've had students that by their own admission have no idea what the standard deviation is about, until I take them step by step through the formula, saying things like: &quot;this $x-\bar{x}$ bit comes from us looking for the distance between our data points and the mean.&quot; On the other hand, if we try to have them do the calculations by hand, large data sets lead to automatic behavior, and huge amounts of frustration when a sign is not inputted correctly or a decimal point is missing somewhere and the student has to start the calculation all over again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The &quot;word problems&quot;&lt;/strong&gt;: Now, this is a problem for the entirety of math education, but it is specially prevalent with Stats. Students have a mighty difficult time figuring out what the interpretation questions are asking of them. It doesn't help that the wording of the problems (specially on platforms like MyMathlab) seems to be designed to trip them up. Again, this is a much bigger problem involving both mathematics and reading comprehension, but I'm sure you are familiar with it in Stats.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The difference between samples, populations, and sampling distributions&lt;/strong&gt;: This is specially evident once the Central Limit Theorem is covered and the familiar z-score formula suddenly exchanges the standard deviation for the standard error of the mean. In my honest opinion this disconnect prevents students from seeing how the test statistic formulas are related, and this results in them compartmentalizing them separately, instead of realizing that, for instance:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$Z=\frac{x-\mu}{\sigma}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is really asking you to do the same sort of thing that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$t=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1-\mu_2)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is asking you to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The confidence intervals:&lt;/strong&gt; More to your question, I find that they don't give my students as much trouble as I expected. Whenever we discuss them with my students, we play the following game: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I tell them that I'll give them a 5 bucks to guess my age. They invariably give me a single number. I ask them how confident they are that this is my true age, and I also ask them if it is likely that, among all the ages I could possibly be, they picked the correct number. &lt;/li&gt;&#xA;&lt;li&gt;I explain to them that this is their point estimate, and proceed to ask them to give me a range, reminding them that in English we have phrases like &quot;give or take a few.&quot; &lt;/li&gt;&#xA;&lt;li&gt;They (after some prodding) give me a range centered around their previous point estimate. &lt;/li&gt;&#xA;&lt;li&gt;I ask them again if they feel confident in their answer. Usually, after they realize that addition of the range increases their chances of winning, they up their &quot;confidence level.&quot; &lt;/li&gt;&#xA;&lt;li&gt;I  then ask them to give me a much bigger range, and this comes with a corresponding increase in their confidence of winning the game. I also ask if it is either fair or useful to have such a big range, if their objective is to guess my age.&lt;/li&gt;&#xA;&lt;li&gt;I explain to them that this is their margin of error, and that while having a large marging of error does increase confidence, it can get to the point that it is no longer useful.&lt;/li&gt;&#xA;&lt;li&gt;I also ask other people close to the student to guess my age, and explain that this is analogous to having a larger sample size, in that it can increase your confidence in the answer while narrowing down the margin of error.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;They usually walk away with 5 dollars in hand and a better understanding of Stats.&lt;/p&gt;&#xA;" OwnerUserId="7872" LastActivityDate="2017-03-04T05:42:34.313" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12091" PostTypeId="2" ParentId="12070" CreationDate="2017-03-05T01:12:03.393" Score="3" Body="&lt;p&gt;By no means a direct or definitive answer, just three references: the 1st on estimating&#xA;in college-level engineering, the 2nd more speculative at the research level&#xA;in mathematics and theory of computation, the 3rd just for fun&#xA;(at middle-school level).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Rebecca Bourn, Sarah Baxter. &quot;Developing Mathematical Intuition by Building Estimation Skills.&quot; 2013 &lt;em&gt;ASEE Annual Conference &amp;amp; Exposition&lt;/em&gt;. &lt;a href=&quot;https://www.asee.org/public/conferences/20/papers/6112/download&quot; rel=&quot;nofollow noreferrer&quot;&gt;Download link&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By thinking critically about the validity and purpose of particular&#xA;  numbers and applying them outside the bounds of their standard usage, students show mastery of&#xA;  comprehension and move on to the pedagogical realms of application and creation. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) Richard Lipton (blog). &lt;a href=&quot;https://rjlipton.wordpress.com/2010/10/01/mathematical-intuition-what-is-it/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Intuition—What Is It?&lt;/a&gt;.&#xA;Oct. 2010.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Open Problems&lt;/strong&gt;.&#xA;  Is intuition simply built up by learning more and more about an area? Or is intuition something that is separate from just being an expert in an area? Can you be quite strong in an area and still have weak intuition, or is that impossible?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(3) Robert Kaplinsky, 2013: &lt;a href=&quot;http://robertkaplinsky.com/work/drug-money/&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Much Money IS That?!&lt;/a&gt;.&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/eYJAD.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eYJAD.jpg&quot; alt=&quot;Cash&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;Assume all US\$100 bills.&#xA;&lt;br /&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; About $220,000,000, i.e., nearly a quarter-billion US dollars.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-03-05T01:17:43.327" LastActivityDate="2017-03-05T01:17:43.327" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12092" PostTypeId="2" ParentId="12080" CreationDate="2017-03-05T19:35:33.060" Score="6" Body="&lt;p&gt;One source I found is the &lt;a href=&quot;https://web.archive.org/web/20180111045655/http://mathforum.org/library/drmath/view/52582.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;following from Dr. Math at Drexel&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I suspect that the concept, and especially the term &amp;quot;order of&#xA;operations&amp;quot; and the &amp;quot;PEMDAS/BEDMAS&amp;quot; mnemonics, was formalized only in&#xA;this century, or at least in the late 1800s, with the growth of the&#xA;textbook industry. I think it has been more important to text authors&#xA;than to mathematicians, who have just informally agreed without&#xA;needing to state anything officially.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20180310183355/http://mistermcintoshsays.org:80/2010/08/14/gemdas-not-pemdas/comment-page-1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Another link&lt;/a&gt; from Mr. McIntosh discusses GEMDAS&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the aforementioned department meeting, Ms. Hertzog, a math teacher at Challenger, said something about GEMDAS being superior to PEMDAS because with PEMDAS some learners get it stuck in their heads that parentheses are the only grouping symbols that need to be taken into account, or else they get confused when some other grouping symbol is used instead of parentheses. She made no claim to inventing GEMDAS, but apparently heard about it at a workshop somewhere.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Additional history is cited in &lt;a href=&quot;https://people.math.harvard.edu/%7Eknill/pedagogy/ambiguity/&quot; rel=&quot;nofollow noreferrer&quot;&gt;a paper from Harvard&lt;/a&gt; about the ambiguity of order of operations which in turn cites &lt;a href=&quot;https://slate.com/technology/2013/03/facebook-math-problem-why-pemdas-doesnt-always-give-a-clear-answer.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;a Slate article by Tara Haelle&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Internet rumors claim the American Mathematical Society has written &amp;quot;multiplication indicated by juxtaposition is carried out before division,&amp;quot; but no original AMS source exists online anymore (if it ever did). Still, some early math textbooks also taught students to do all multiplications and then all divisions, but most, such as this &lt;a href=&quot;http://books.google.com/books?id=hS47AQAAIAAJ&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q=multiplications&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;1907 high-school algebra textbook&lt;/a&gt;, this &lt;a href=&quot;http://books.google.com/books?id=QxgAAAAAYAAJ&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q=multiplications&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910 textbook&lt;/a&gt;, and this &lt;a href=&quot;http://books.google.com/books?id=HxkAAAAAYAAJ&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=snippet&amp;amp;q=fundamental%20and%20operations&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912 textbook&lt;/a&gt;, recommended performing all multiplications and divisions in the order they appear first, followed by additions and subtractions. (This convention makes sense as well with the Canadian and British versions of PEMDAS, such as BEDMAS, &lt;a href=&quot;http://www.mathsisfun.com/operation-order-bodmas.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;BIDMAS, and BODMAS&lt;/a&gt;, which all list division before multiplication in the acronym.) The most sensible advice, in a 1917 edition of &lt;em&gt;Mathematical Gazette&lt;/em&gt;, recommended using parentheses to avoid ambiguity. (Duh!) But even noted math historian Florian &lt;a href=&quot;http://books.google.com/books?id=7juWmvQSTvwC&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q=arithmetical%20or%20algebraical%20term%20contains%20&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori wrote in &lt;em&gt;A History of Mathematical Notations&lt;/em&gt;&lt;/a&gt; in 1928-29, &amp;quot;If an arithmetical or algebraical term contains &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;, there is at present no agreement as to which sign shall be used first.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="15884" LastEditDate="2023-04-21T01:04:24.687" LastActivityDate="2023-04-21T01:04:24.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12093" PostTypeId="2" ParentId="10949" CreationDate="2017-03-06T04:36:31.017" Score="2" Body="&lt;p&gt;Andrew's answer is very good. However, I'm going to describe abstraction slightly differently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we're doing mathematics, we often notice similarities in different subfields. Abstraction can be thought of ignoring the differences. We define a new concept in terms of the similarities only and give it a name so that we can talk about it independently of any particular instance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The hard part is that it's sometimes not clear what the similarities are or how we can express them independently. It took half a century, for example, to come up with category theory, which is probably the most powerful abstraction language yet found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is my excuse for answering this question. Since you're self-learning, I wanted to give you just a little picture of just how powerful abstraction can be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, let's consider sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose that $A$ and $B$ are finite sets. Then the number of functions with domain $A$ and codomain $B$ is $|B|^{|A|}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's think about the empty set $\emptyset$ for a moment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For any non-empty finite set $A$, how many functions are there with domain $A$ and codomain $\emptyset$? Well, it's just $0^{|A|}$, which is $0$. There are no functions $f$ such that $f : A \rightarrow \emptyset$ if $A$ is nonempty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What about the other way? How many functions are there with domain $\emptyset$ and codomain $A$? Well, it's $|A|^0$, which is $1$. There is exactly one function $f$ such that $f : \emptyset \rightarrow A$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the above argument also applies to infinite set, however we also need to consider the case $f : \emptyset \rightarrow \emptyset$. By convention, when doing combinatorics, $0^0 = 1$, and it turns out (if, for example, you consider the graph of the function) it makes sense to do this too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The empty set is the only set with the following property:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For any set $A$, there is &lt;em&gt;exactly&lt;/em&gt; one function $f$ such that $f : \emptyset \rightarrow A$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now, let's think about logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For logic, I'm going to use some notation that may be unfamiliar at the high school level. Given two propositions $A$ and $B$, we will denote $A \vdash B$ (pronounced &quot;$A$ entails $B$&quot;, or &quot;$A$ yields $B$&quot;) to mean that given the premise $A$, you can conclude $B$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proposition $\hbox{False}$ has the interesting property that from that premise, you can prove anything; this is known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Principle_of_explosion&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;ex falso quodlibet&lt;/em&gt;, or the principle of explosion&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But more than that, if you formalise the notion of what constitutes a &quot;distinct proof&quot; very carefully, you can show that $\hbox{False}$ is the only proposition with this property:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For any proposition $A$, there is &lt;em&gt;exactly&lt;/em&gt; one proof of $\hbox{False} \vdash A$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There's an abstraction to be discovered here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In category theory, an object with this property is known as an &quot;initial object&quot;. There is also the dual notion of a &quot;terminal object&quot;, that is, an object $T$ such that for any other object $A$, there is exactly one &quot;arrow&quot; such that $A \rightarrow T$. It won't surprise you to learn that $\hbox{True}$ is the unique terminal object in logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also a terminal object in sets, and it's actually the &quot;singleton set&quot;, that is, the set with only one element in it. Of course, I say &quot;the&quot; singleton set as if there's only one, when you can probably think of lots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In category theory terms, we say that all singleton sets are &lt;em&gt;isomorphic&lt;/em&gt;. They all have the same &quot;shape&quot;, much like in Andrew's group theory example, where all the groups have the same &quot;shape&quot;. You can think of them as different &quot;physical&quot; groups, or you can think of them as the same abstract group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Informally, you can't tell the difference between any two singleton sets without looking inside the set to see what the element is. That's the price you pay for abstraction: the actual element inside the set is an irrelevant detail that you're abstracting away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what you gain is that it's possible to prove a theorem which is true of both sets-and-functions &lt;em&gt;and&lt;/em&gt; propositions-and-proofs. An example of such a theorem is: &quot;All initial objects are identical up to isomorphism.&quot; That's a very abstract statement, but you should now have an idea just how powerful that statement is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, since group theory has already come up, the &quot;trivial group&quot; (i.e. the group with one element) is both an initial object &lt;em&gt;and&lt;/em&gt; a terminal object, which makes it very interesting indeed. That's a topic for another time.&lt;/p&gt;&#xA;" OwnerUserId="7896" LastEditorUserId="7896" LastEditDate="2017-03-06T04:44:18.540" LastActivityDate="2017-03-06T04:44:18.540" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12094" PostTypeId="1" CreationDate="2017-03-06T08:14:17.887" Score="14" ViewCount="426" Body="&lt;p&gt;&lt;em&gt;This question was asked to Math SE at first but it seems like it is more appropriate to ask it  here.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;While learning new concepts in algebra it is quite helpful to check some examples which includes the groups that we are familiar with, like &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}/6\mathbb{Z}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$GL(2,\mathbb{R}), \dots$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For example, while learning quotient groups, we can choose a normal subgroup &lt;span class=&quot;math-container&quot;&gt;$H=\{\bar{0},\bar{2},\bar{4}\}$&lt;/span&gt; of the group &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}/6\mathbb{Z}$&lt;/span&gt; to understand &lt;em&gt;what is going on&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I know my question seems subjective, but, I think that some of you have the groups that you know well and use them often. So, what groups -or rings, fields-  can you suggest? Thanks!&lt;/p&gt;&#xA;" OwnerUserId="7895" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-03-09T12:30:58.277" Title="What are the &quot;best&quot; groups to use as examples while learning new concepts in algebra?" Tags="|abstract-algebra|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12095" PostTypeId="2" ParentId="12094" CreationDate="2017-03-06T12:47:10.013" Score="14" Body="&lt;p&gt;Symmetry groups of basic regular polygons and polyhedra.  For one thing, they are noncommutative; for another, sometimes they coincide with other &quot;known&quot; groups; for a third, there can be physical meaning to some concepts in group theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And they are very &quot;hands-on&quot; - it can be fun to make sure students understand how to generate the symmetry group for the cube with as few elements as possible by building a toothpick model.  (This kind of thing can even work with non-major classes, if you wanted to introduce them to group theory - presumably somewhat more slowly than for an upper-level US group theory course.)&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-03-06T12:47:10.013" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12096" PostTypeId="1" CreationDate="2017-03-06T20:45:35.200" Score="17" ViewCount="318" Body="&lt;p&gt;I have a few students that have difficulties writing. To generalize, these students have nearly illegible handwriting, they take a very long time to write, they become anxious or fatigued when writing large amounts or writing in a timed task, and they also have trouble formatting their work logically on the paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically in a high school algebra course, I want to help these students develop strategies to improve their writing or manage the demands of their math classes, so that their writing does not become a major hindrance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Right now, I allow these students to write very large characters on un-lined or quad-ruled paper, I encourage them to try large pencil grips, and we work on strategies to plan and format work before putting it on the page. I am very familiar with Latex as a computational tool for writing math, but the learning curve is a bit steep for these students. I am curious what remediation strategies and accommodating tools are available to these students, especially computer software and therapeutic exercises. &lt;/p&gt;&#xA;" OwnerUserId="5690" LastActivityDate="2017-06-23T06:35:21.927" Title="What are some strategies to remedy and accommodate dysgraphia?" Tags="|secondary-education|writing|special-needs-education|disability|specific-software|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12097" PostTypeId="2" ParentId="12094" CreationDate="2017-03-07T00:22:51.963" Score="7" Body="&lt;p&gt;You mentioned the linear groups over the real numbers, which I think is a useful example, with the potential drawback that it is infinite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To &quot;remedy&quot; this one can consider the linear groups over finite fields and even rings, yet for concreteness just over $\mathbb{Z}/n\mathbb{Z}$ for some particular small $n$ should suffice for a start. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Computing with low-dimensional matrices is relatively efficient, and the fact that one is dealing with matrices gives some obvious structure to the group. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are plenty of subgroups (diagonal matrices, upper triangular matrices, triangular with all ones on  the diagonal, the special linear group, etc). Some of them normal, some not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To determine the center of these groups is manageable but still interesting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The isomorphism of $S_3$ and $GL(2, \mathbb{Z}/2\mathbb{Z})$ is curious. &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2017-03-07T00:22:51.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12098" PostTypeId="1" CreationDate="2017-03-07T23:18:39.297" Score="5" ViewCount="196" Body="&lt;p&gt;What are some guidelines on erasing students' work such as on a chalkboard/whiteboard in a classroom or on paper in a private tutorial class?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually this is for the parts of maths that involves drawing or geometry rather than algebra I guess&lt;/p&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2017-03-08T15:22:05.833" LastActivityDate="2017-03-08T15:22:05.833" Title="Erasing students' work - etiquette/guidelines?" Tags="|mathematical-pedagogy|general-pedagogy|interactive-teaching|classroom-interaction|psychology|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12099" PostTypeId="2" ParentId="12098" CreationDate="2017-03-08T14:22:01.483" Score="4" Body="&lt;p&gt;To answer your actual question about guidelines, I would say this.  There aren't really any &quot;official&quot; guidelines that I'm aware of.  Institutions may have their own policies regarding this, although to me that would be borderline micromanagement if they did.  I've never been affiliated with such a place as far as I know, so I've always just gone by this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If it isn't graded work, there's no harm in erasing anything as long as you aren't interfering with their note-taking or learning process in general.  And preferably don't make them feel stupid in the process.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If it's graded work that they did then erasing it (or modifying it at all aside from corrections that are clearly part of grading) is a big no-no.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;With that aside, I really wonder, why erase it for them?  On a board I can understand if the student has already returned to their seat but I don't see why they can't erase their own work if it's a private tutorial.  Having them correct their own work could be a better learning experience.  Another option is to leave what they did alone, and then draw a correct version so the student can compare and see what was incorrect about their attempt.&lt;/p&gt;&#xA;" OwnerDisplayName="user6648" LastActivityDate="2017-03-08T14:22:01.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12100" PostTypeId="2" ParentId="12089" CreationDate="2017-03-08T18:55:24.227" Score="9" Body="&lt;p&gt;First, Cajori's &lt;a href=&quot;http://www.jstor.org/stable/1967725?seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;The History of Notations of the Calculus&lt;/a&gt; presents a rather unproblematic history of various notions suggested for the definite integral and none has the limits of integration written on the left. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, I checked the textbooks published by &lt;a href=&quot;http://www.ncert.nic.in/index.html&quot; rel=&quot;noreferrer&quot;&gt;The [Indian] National Council of Educational Research and Training (NCERT)&lt;/a&gt;. They publish textbooks in English, Hindi, and Urdu. The &lt;a href=&quot;http://ncertbooks.prashanthellina.com/class_12.Mathematics.MathematicsPartII/Integral%20ch_7%2006.11.06.pdf&quot; rel=&quot;noreferrer&quot;&gt;English&lt;/a&gt; and the &lt;a href=&quot;http://ncertbooks.prashanthellina.com/class_12.Mathematics.GanitII/chapter%207.pdf&quot; rel=&quot;noreferrer&quot;&gt;Hindi&lt;/a&gt; versions of Year 12 textbook where we can find definite integrals write them as &lt;em&gt;we&lt;/em&gt; do. I couldn't find the Urdu version. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third, and this is the most interesting of all, I checked a number of Youtube videos of some of &quot;famous&quot; (according to the number of viewers) Indian mathematics teachers. They mostly write the lower limit inside on the RIGHT of the integral sign and the upper integral outside on the LEFT of the integral sign. Here are two snapshots of two different teachers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9rUvQ.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9rUvQ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2017-03-08T18:55:24.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12101" PostTypeId="2" ParentId="12094" CreationDate="2017-03-09T12:30:58.277" Score="6" Body="&lt;p&gt;In addition to the dihedral groups, the quaternion group $Q_8$, the symmetric and alternating groups, and small matrix groups, I have always found the following examples useful:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Automorphism groups of graphs:&lt;/strong&gt; I usually introduce these at around the same time I introduce permutations.  This gives a wealth of interesting examples of groups, and also gives a simple example of how groups can be used to understand other mathematical structures.  There are lots of small graphs (e.g. trees) that provide simple examples of quotients and normal subgroups, and you can also use graphs to give nontrivial applications of various group action concepts.  For example, I have a homework problem where students compute the order of the automorphism group of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Petersen_graph&quot; rel=&quot;noreferrer&quot;&gt;Petersen graph&lt;/a&gt; using the orbit-stabilizer theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Unit groups:&lt;/strong&gt; Gallian's book places an emphasis on the unit groups $U(n)$, i.e. the group of units in the integers modulo $n$.  I really like the emphasis on these groups, because their structure is not obvious, so students really have to dig into them to figure them out.  For example, listing the subgroups of $U(15)$ or $U(24)$ is an interesting exercise, as is computing the automorphisms of either of these.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Symmetry groups of polyhedra:&lt;/strong&gt; Others have already mentioned these so I won't go into too much detail here, but I should point out that lots of polyhedra beyond the platonic solids and prisms have interesting symmetry groups.  For example, one of my homework problems asks the students to compute the symmetry group of the tetrahedron with vertices at $(8,8,1)$, $(8,-8,-1)$, $(-8,8,-1)$, and $(-8,-8,1)$.  I'm also fond of &lt;a href=&quot;https://en.wikipedia.org/wiki/Antiprism&quot; rel=&quot;noreferrer&quot;&gt;antiprisms&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2017-03-09T12:30:58.277" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12102" PostTypeId="2" ParentId="10348" CreationDate="2017-03-09T14:15:57.253" Score="5" Body="&lt;p&gt;Reading a paper from an English student in England I came across this sentence: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To expand brackets in Algebra, they were taught the &quot;&lt;strong&gt;crab claw&lt;/strong&gt;&quot;&#xA;  method, a method that I was used to.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Since I had personally never heard of the term &quot;crab claw&quot; in Algebra, I thought it would be beneficial to add it here. &#xA;&lt;a href=&quot;https://i.stack.imgur.com/4YkUO.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4YkUO.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;  &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="1217" LastEditDate="2017-03-09T14:21:34.807" LastActivityDate="2017-03-09T14:21:34.807" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12103" PostTypeId="1" CreationDate="2017-03-10T04:59:11.653" Score="10" ViewCount="3957" Body="&lt;p&gt;What are some real-life applications of gcd? I am looking for a motivating way of introducing this topic in an elementary number theory course. &lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="77" LastEditDate="2017-03-14T12:52:48.193" LastActivityDate="2017-05-16T18:11:44.917" Title="Greatest common divisor applications" Tags="|teaching|number-theory|applications|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12104" PostTypeId="2" ParentId="4045" CreationDate="2017-03-10T05:20:18.863" Score="2" Body="&lt;p&gt;Coming late to this question ... but hopefully with something useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I find the historical reason plenty, as well as ease of proof, there is something to be said for approaching not perfect numbers, but rather the summation function/aliquot divisors function, as the key concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, if we start with the reasonable function $\tau(n)=\sum_{d\mid n}1$ and try to find patterns in it, such as when it is even or odd, one might also try to find patterns in the related $\sigma(n)=\sum_{d\mid n} d$.  And also $\phi(n)$, etc.  What is one pattern you might seek?  Well, do you ever get a perfect (!) multiple of your input as output?  (As well as whether it is multiplicative, and lots of other questions.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in an introductory number theory course, there is lots of room to look for interesting patterns in &lt;em&gt;many&lt;/em&gt; arithmetic functions, and then the perfect number question is just a nice example of one we can actually answer - but surely not the only one, or even a particularly important one from this point of view.  (Personally I enjoy exploring asymptotic behavior of arithmetic functions.)  I like all the answers here, but this is one approach to take for motivation if for some reason you didn't like those answers.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-03-10T05:20:18.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12105" PostTypeId="1" CreationDate="2017-03-10T12:51:33.467" Score="11" ViewCount="138" Body="&lt;p&gt;In a discussion with a physics lecturer he mentioned that one major area where students fail is understanding assumptions - for example, if we are interested in two objects hitting each other and then bouncing off, we can probably ignore gravity, since the interesting even happens very fast. That is, we may assume gravity to be zero for the calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematics also there are plenty of assumptions. They tend to be more explicit then in physics. Students still have problems with them - confusing what should be proven with what may be assumed, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has this phenomenon (of students failing in using assumptions) been studied, and if yes, what are the major discoveries, or where should I start reading to find out more about this?&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2017-03-10T12:51:33.467" Title="Literature on student understanding of assumptions" Tags="|undergraduate-education|reference-request|proofs|physical-sciences|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12106" PostTypeId="2" ParentId="12103" CreationDate="2017-03-11T05:31:29.653" Score="6" Body="&lt;p&gt;Here's a word problem for the greatest common divisor:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;12 boys and 15 girls are to march in a parade.  The organizer wants them to march in rows, with each row having the same number of children, and with each row composed of children with the same gender.  What is the largest number of children per row that satisfies these constraints?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There should be $\operatorname{gcd}(12,15)=3$ children per row ($4$ rows of boys and $5$ rows of girls).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I know that you didn't ask for it, but when I teach the gcd I also teach the least common multiple.  Here's a word problem for the lcm.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A child is having a birthday party and knows that either 12 or 15 guests would attend it.  The child's father wants to buy candies for the guests, with each guest having the same number of candies, but doesn't want to buy more candies than is necessary.  What is the minimum number of candies that the father should buy?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The father should buy $\operatorname{lcm}(12,15)=60$ candies, so that if $12$ friends come, they each get $5$ candies, and if $15$ come, they each get $4$.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2017-03-11T05:31:29.653" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12107" PostTypeId="2" ParentId="12103" CreationDate="2017-03-11T08:05:44.527" Score="4" Body="&lt;p&gt;I think for the most elementary level of introduction Joel and Benjamin has provided good examples. I think I should try too.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$1.$ When we have a linear diophantine equation like $ax+by=c$ where $a,b \ and \ c$ are integers, we get integer solutions in $x,y$ iff $(a,b)|c$. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$2.$  As a rearrangement of a well theorem of divisibility, we know that for two numbers $a,b$ we have $[a,b]=\frac{|ab|}{(a,b)}$. So, ultimately knowing G.C.D. of two numbers give us their L.C.M and vice versa.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$3.$ This one may overflow for a high school student because the way it is designed, consider the question:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Two regular polygons are inscribed in the same circle. The first polygon have $1982$ sides and second have $2973$ sides. If polygons have any numbers of common vertices, how many such vertices there will be??&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The solution is simply the common roots of the equation $z^{1982}-1=0$ and $z^{2973}-1=0$ which in turn equals gcd$(1982,2973)=991$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7654" LastActivityDate="2017-03-11T08:05:44.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12108" PostTypeId="1" CreationDate="2017-03-11T08:29:30.007" Score="7" ViewCount="160" Body="&lt;p&gt;I am a high school student who will be entering college in one and a half year (It is so long though).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, Mathematics primarily means Number Theory. My interest in Number Theory always motivates me to go and take a look at those wiki articles which are not really discussed in classic or even modern books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, while studying about prime numbers, I found many things not specifically discussed in books. Some of them are Legendre conjecture, Prime gaps, Mills constant etc. I am confused whether I should keep studying these things which authors don't dare to discuss or follow the patterns of books instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For your kind information, I would mention that I frequently use NZM along with G. H. Hardy's &lt;code&gt;Introduction to Number Theory&lt;/code&gt; sometimes. For problems, I often do UC Berkeley's online assignments.&lt;/p&gt;&#xA;" OwnerUserId="7654" LastEditorUserId="8198" LastEditDate="2017-05-16T08:30:10.430" LastActivityDate="2017-05-16T13:00:25.933" Title="What specifically should I include to my self study notes?" Tags="|self-learning|number-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12109" PostTypeId="2" ParentId="2074" CreationDate="2017-03-11T16:43:22.110" Score="8" Body="&lt;p&gt;I am generally favorable to Euclidean geometry and proofs, within limits, but we do need to understand why learning proofs is so difficult that many countries have reduced such content considerably. Only by appreciating why learning proofs in high school is problematic can we hope to overcome the difficulties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A considerable amount of research, especially in the 1980s, has shown that students are not as successful with geometric proofs as they seem. For example, in the US, where proofs are still usually taught, teachers developed a system of &quot;two-column&quot; proofs to make the process easier to grade and more &quot;rigorous&quot;. Unfortunately two-column proofs often very predictably go through similar processes which teachers manage to teach in such a way that many students learn to successfully imitate them without any deep understanding of what they are actually doing. For example, a great many proofs students learn to do can be solved with a two-column proof by identifying the given information in a triangle, identifying the appropriate theorem (ASA, SAS, SSS, etc.) and continuing to the conclusion. The whole question of &lt;em&gt;why&lt;/em&gt; we are doing this in the first place, or what it all means, is given little emphasis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One large study (Usiskin, Z. [1982]. &lt;em&gt;Van Hiele levels and achievement in secondary school geometry&lt;/em&gt;. Chicago, IL: University of Chicago) found that for many, perhaps most, students who had completed a full year of EG, not only did they not understand what they were doing, but they did not yet even have the prerequisites to &lt;em&gt;begin&lt;/em&gt; a study of geometric proofs. There has been a considerable attempt to fill in the gaps found in these studies (both in geometric content and logical reasoning) in earlier grades, but it is not still clear if today's students are ready to fully appreciate EG, especially when taught all at once in a one-year course, as is done in the US.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There have been similar findings in Europe (e.g. Gutiérrez, Á., &amp;amp; Jaime, A. [1998]. On the assessment of the Van Hiele levels of reasoning. &lt;em&gt;Focus on Learning Problems in Mathematics&lt;/em&gt;, 20(2/3), 27-46.)&lt;/p&gt;&#xA;" OwnerUserId="6782" LastActivityDate="2017-03-11T16:43:22.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12110" PostTypeId="1" CreationDate="2017-03-12T10:10:26.483" Score="7" ViewCount="258" Body="&lt;p&gt;I have just started to teach Calculus to freshmans and sophomores who study non-mathematical subjects, e.g., international relations, psychology. They have to take few mathematics classes -including Calculus- in order to graduate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The course is comprehensive, we gave the $\epsilon-\delta$ definiton of a limit and find the derivative of functions using the definition of derivative. So they can not just memorize the 'formulas' and solve problems, instead, they have to think and spend some effort.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to give motivation in the beginning. Before giving the $\epsilon-\delta$ definition of a limit, I tell a story about adding some salt while cooking. Like if you add $8$ mg. of salt, the food becomes the most delicious according to your taste. I played with the amount of salt, e.g., $(7.9,8.1)-\{8\}$, and after I see the light in some of the students eyes, I gave the formal definiton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know how hard for some of the students to take this class. Maybe, they have chosen their departments because of maths. Does not matter how much I try so hard to explain in a more simple way, sometimes I feel the emptiness in their minds, the pain and suffer they have. This bothers me a lot and instead of being reckless, I want to help them to understand or at least to pass the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What can I do, so that they can understand the subject and pass the class with a preferably high grade? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="7895" LastActivityDate="2017-03-21T01:55:37.710" Title="How to catch students from different subjects' interest to math?" Tags="|undergraduate-education|mathematical-pedagogy|calculus|student-motivation|mathematical-analysis|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12111" PostTypeId="2" ParentId="12103" CreationDate="2017-03-12T15:49:55.640" Score="11" Body="&lt;p&gt;Another classic is the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A rectangular floor measures $300 \text{ cm} \times 195 \text{ cm}$. What is the largest square tiles that can be used to cover the floor exactly? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="39" LastActivityDate="2017-03-12T15:49:55.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12112" PostTypeId="2" ParentId="12103" CreationDate="2017-03-14T00:49:45.670" Score="3" Body="&lt;p&gt;The special case of $gcd(a,b) = 1$ (i.e. $a,b$ are relatively prime) is in its own right an important motivation for $gcd(a,b)$, especially if you link it with the (extended) Euclidian algorithm. Students tend to like the idea of secret codes. You can connect the notion of being relatively prime with the notion of a number $a$ being invertible mod $b$ and show how the Euclidean algorithm can efficiently compute this modular inverse, and then point out how this is the crucial step in computing private/public key pairs in RSA encryption. &lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2017-03-14T00:49:45.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12113" PostTypeId="1" CreationDate="2017-03-14T03:09:31.420" Score="12" ViewCount="954" Body="&lt;p&gt;I'm teaching an introductory statistics class at a community college, and we've just finished a unit on discrete probability. At the moment, the students' conception of the probability of an event A is &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$P(A)=\frac{\text{# of outcomes in A}}{\text{# of total outcomes}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Supposing all outcomes of the random experiment are equally likely to occur). We can use this basic understanding of probability to derive probability mass functions for discrete random variables, and graph these functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do I help the students jump from the discrete to the continuous? Beginning with the basic understanding of probability as a ratio (required to understand the above formula), how do I move to the notion that the probability for a range of continuous values can be represented as the area under a curve called the probability density function?&lt;/p&gt;&#xA;" OwnerUserId="21" LastEditorUserId="5987" LastEditDate="2018-05-02T22:36:15.177" LastActivityDate="2018-05-02T22:36:15.177" Title="Moving from discrete probability distributions to continuous ones" Tags="|concept-motivation|statistics|probability|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12114" PostTypeId="1" CreationDate="2017-03-14T10:28:04.363" Score="20" ViewCount="323" Body="&lt;p&gt;I retired after 25 years of teaching and moved to Israel a year ago. My Hebrew is okay, but before moving here, I had no experience talking about math in Hebrew.  I have been learning Hebrew math vocabulary by reading math textbooks and taking an online math course in Hebrew. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently started volunteering in an after school program to help Hebrew-speaking students prepare for tests in mathematics.  These tests (called bagruyot) are standardized tests taken at the end of high school and used to determine entrance to college.  I have no trouble understanding the problems on the practice exams, but I see that  it is difficult for me to teach in Hebrew.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Currently my strategies (when I can't explain clearly in Hebrew) include: acting out problems with students, writing out solutions in math symbols, and finding similar problems with solutions in the textbooks  This is not how I taught in my native language!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in any tips to becoming a better teacher in a language that is foreign to me but native to my students.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2017-03-14T18:28:11.290" LastActivityDate="2017-05-26T18:16:28.290" Title="Tips for Teaching Math in a Language Foreign to the Teacher" Tags="|tutoring|language-use|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12115" PostTypeId="2" ParentId="12113" CreationDate="2017-03-14T13:49:28.040" Score="7" Body="&lt;p&gt;This is an uncomfortable moment, mathematically, in a non-calculus-based statistics course; frankly, we simply need to steal the calculus concept and hope that students trust us about it, without formal grounding. It's somewhat degenerate mathematics but it's the position we're required to deal with. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said: I find that students do a pretty good job of picking up the idea by just flat-out being explicitly told (&quot;the area under a curve is the probability of getting an outcome in that range&quot;) and then practicing some with a normal-curve table. Start with a very simple case: normal curve with mean 9, standard deviation 2 (sketch); what is the probability of getting a value above 9? Every class I've taught, at least one student has intuited that the answer is 50%; and I can say, yes, because half of the area is above 9. Run more exercises and in every case sketch the curve and check for reasonability. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach out of Weiss &lt;em&gt;Introductory Statistics&lt;/em&gt; and it works perfectly fine. Up through the 8th Edition there wasn't any text on continuous probability distributions in the general sense at all; it just started using a normal curve table like this in Ch. 6. As of the 9th edition they added a 2-page introduction on density curves in general, but I don't find that any more illuminating; I just skip it. In fact -- I even skip the whole chapter on discrete probability distributions (Weiss marks it as optional). Due to the conceptual gap here between discrete/continuous distributions, I don't think the discrete case increases understanding of the continuous case, and it delays getting to the crown jewels of the course: inferences with confidence intervals and P-values for means and proportions for large samples. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-03-14T13:49:28.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12116" PostTypeId="2" ParentId="4409" CreationDate="2017-03-14T15:22:21.083" Score="3" Body="&lt;p&gt;The controversy between English and Continental mathematicians about who should get credit for inventing Calculus, as described (in a journalistic sort of way) in the book &quot;The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time&quot; by Jason Socrates Bardi. This brings out the human element in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some applications of calculus of types not covered in most textbooks might be interesting. For example, few students growing up in the digital age realize just how sophisticated analog computers had become by the end of World War II. The German rocket scientists used an analogue integrator of acceleration to obtain speed called an &lt;a href=&quot;https://en.wikipedia.org/wiki/PIGA_accelerometer&quot; rel=&quot;nofollow noreferrer&quot;&gt;integrating accelerometer&lt;/a&gt; to control engine shut-off for their V2 rockets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For Calc III, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Planimeter&quot; rel=&quot;nofollow noreferrer&quot;&gt;planimeter&lt;/a&gt; (an analogue device for calculating area by tracing around a perimeter) can be introduced in the context of discussing Green's Theorem (an idea that I got from the third edition of Stephen Krantz's wonderful book &quot;How to Teach Mathematics&quot;).&lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2017-03-14T15:22:21.083" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12117" PostTypeId="2" ParentId="12103" CreationDate="2017-03-14T16:32:16.533" Score="9" Body="&lt;p&gt;Going back to Euclid, I have found questions such as `Given a large supply of rods of length $15$ and $21$, what lengths can be measured?' can appeal to students. This also motivates the result $\gcd(a,b) = ra+sb$ of the (extended) Euclidean Algorithm.&lt;/p&gt;&#xA;" OwnerUserId="640" LastActivityDate="2017-03-14T16:32:16.533" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12118" PostTypeId="1" CreationDate="2017-03-14T22:42:22.470" Score="23" ViewCount="1147" Body="&lt;p&gt;As part of my duties at a GTA, I spend several hours per week in our department's drop-in tutoring center. The center is open to all students enrolled in 100- and 200-level math courses, with the majority coming from Calc 1 &amp;amp; 2. A typical load would be 20-50 students with a roving staff of 5-10 tutors (graduate students/adjuncts/junior-senior undergraduate math majors). As with any good tutoring, our stated policy is to provide help with learning material by explaining concepts, finding and explaining errors, etc; we are explicitly &lt;em&gt;&lt;strong&gt;not&lt;/strong&gt;&lt;/em&gt; a &amp;quot;what's the answer to problem #3&amp;quot; service (although difficulties with problem #3 are a great starting point).&lt;/p&gt;&#xA;&lt;p&gt;I find the work generally enjoyable and have had nearly universally positive interactions with students. However, there is one student that I have come to dread seeing every week.&lt;/p&gt;&#xA;&lt;p&gt;Said student is enrolled in Calc 1 and is extremely difficult to deal with, to the extent that my armchair diagnosis would place them somewhere on the autism spectrum. Interactions with them are by far the most frustrating I've ever had teaching mathematics, out of literally thousands of students I've tutored over the years. I have gotten so frustrated with helping them that I've had to &amp;quot;tap out&amp;quot; and go for a walk around the building after trying to help them. I'm not the only one either--I've seen every other tutor become similarly exasperated by this student.&lt;/p&gt;&#xA;&lt;h2&gt;Sample Interaction&lt;/h2&gt;&#xA;&lt;p&gt;For an idea of a sample interaction with this student, consider the problem:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$f(x) = x e^{2x}$&lt;/span&gt;, what's &lt;span class=&quot;math-container&quot;&gt;$f^{(30)}(x)$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;My usual approach for walking a &amp;quot;completely stuck&amp;quot; student through a problem like this would go like:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Can you compute &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; for me? (Possibly leading to a reminder discussion of the product and chain rules)&lt;/li&gt;&#xA;&lt;li&gt;How about &lt;span class=&quot;math-container&quot;&gt;$f''(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f'''(x)$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;li&gt;Do you notice any patterns? Can you predict what &lt;span class=&quot;math-container&quot;&gt;$f^{(4)}(x)$&lt;/span&gt; will be? Does your prediction hold when we compute said derivative?&lt;/li&gt;&#xA;&lt;li&gt;Can you formulate the pattern you're noticing in mathematical notation?&lt;/li&gt;&#xA;&lt;li&gt;Profit.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Usually the problem clicks around step 2 or 3 and they can take it from there. In comparison, here's how an interaction with this student goes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Student: [raises hand]&lt;/p&gt;&#xA;&lt;p&gt;Me [walk over]: How's it going? What are we working on?&lt;/p&gt;&#xA;&lt;p&gt;Student [computer open to online homework problem, blank sheet of paper in front of them]: What's the answer to this problem?&lt;/p&gt;&#xA;&lt;p&gt;Me: What have you tried so far? (Again, they have a blank sheet of work in front of them)&lt;/p&gt;&#xA;&lt;p&gt;Student: I don't know how to compute the 30th derivative&lt;/p&gt;&#xA;&lt;p&gt;Me: How would you compute the second derivative?&lt;/p&gt;&#xA;&lt;p&gt;Student: Prime of the prime [we'll let that slide, bigger fish to fry]&lt;/p&gt;&#xA;&lt;p&gt;Me: Right! Can you go ahead and find &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; for me?&lt;/p&gt;&#xA;&lt;p&gt;Student: I know how to do that. The problem wants the 30th derivative.&lt;/p&gt;&#xA;&lt;p&gt;Me: True! I think we should try to compute the first couple derivatives and see if we can find any patterns&lt;/p&gt;&#xA;&lt;p&gt;Student: [Lough sigh/uggh, audible across the entire room] Fine. [starts computing &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;, nowhere close]&lt;/p&gt;&#xA;&lt;p&gt;Me: Hold up a sec--is that derivative correct?&lt;/p&gt;&#xA;&lt;p&gt;Student: What do you mean.&lt;/p&gt;&#xA;&lt;p&gt;Me: Well, there's an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &lt;em&gt;times&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$e^{2x}$&lt;/span&gt; --&lt;/p&gt;&#xA;&lt;p&gt;Student: So?!&lt;/p&gt;&#xA;&lt;p&gt;Me: ...so we need to use the product rule, right?&lt;/p&gt;&#xA;&lt;p&gt;Student: [crickets]&lt;/p&gt;&#xA;&lt;p&gt;Me: When we want to take the derivative of a product, we use the product rule. [write &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx} f(x) \cdot g(x) = f'(x) g(x) + f(x) g'(x) $&lt;/span&gt;]&lt;/p&gt;&#xA;&lt;p&gt;Student: DUUHH. I know this. Why are you explaining stuff I already know; I need to know how to find the 30th derivative.&lt;/p&gt;&#xA;&lt;p&gt;Me: Well, it doesn't look like you used product rule or chain rule to compute &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; there. Can you try to fix what you have for &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Student: How?&lt;/p&gt;&#xA;&lt;p&gt;Me: [more detailed explanation of product/chain rules. Look up to see that they have opened a tab for facebook on their computer] Dude...do you want me to help you or -&lt;/p&gt;&#xA;&lt;p&gt;Student: I know that stuff. I NEED TO KNOW HOW TO COMPUTE THE 30TH DERIVATIVE! [entire room looks up]&lt;/p&gt;&#xA;&lt;p&gt;Me: [fighting to be calm] Please don't shout. I'm trying to explain how to find the 30th derivative to you. There isn't a magic &amp;quot;30th derivative formula&amp;quot;---we find it by computing the first few derivatives and finding some patterns. [I go ahead and write down &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f''(x)$&lt;/span&gt;]&lt;/p&gt;&#xA;&lt;p&gt;Student: Why are you doing this? That's exactly what I have written.&lt;/p&gt;&#xA;&lt;p&gt;Me: Well...notice how I have &lt;span class=&quot;math-container&quot;&gt;$1 \cdot e^{2x} +  x\cdot  e^{2x}\cdot 2$&lt;/span&gt; and you have just [point at their work] &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Student: So what?&lt;/p&gt;&#xA;&lt;p&gt;...[15 more minutes of Abbott &amp;amp; Costello]&lt;/p&gt;&#xA;&lt;p&gt;...I give up and just give them the answer to escape from this nightmare. Other students are waiting for help.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have noticed that this student wears a medical bracelet, and there's a few other behaviors as well---things like no sense of personal space, bad personal hygiene, a pronounced facial tic, not looking at me when I'm talking (or looking at anything I'm writing down), etc.&lt;/p&gt;&#xA;&lt;p&gt;So my question is, in my best Edward James Olmos meets Eric Cartman,&lt;/p&gt;&#xA;&lt;h2&gt;How do I reach [this] kid?&lt;/h2&gt;&#xA;&lt;p&gt;Myself and the other tutors have come to dread any interaction with this student, and have reached the point that we more or less just give up after a minute and write down the answer in order to placate them for the next 20-30min so we can provide actual help to other students. I honestly struggle to understand how they've made it to the point in math of studying calculus, and sincerely wonder if they've just learned this behavior of being obstinate until they're given the answer. This approach is demonstrably wrong as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They aren't learning anything, but are getting credit for it.&lt;/li&gt;&#xA;&lt;li&gt;They clearly need help with basic calculus concepts, but absolutely refuse to be provided such help.&lt;/li&gt;&#xA;&lt;li&gt;Just giving this student the answer belittles the work of all their other classmates&lt;/li&gt;&#xA;&lt;li&gt;Our attempts to earnestly help this student seem to wastes everybody's time, including the other students waiting for help.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7260" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2021-12-11T17:00:19.307" Title="Tutoring a recalcitrant/awkward/exasperating student---special needs?" Tags="|undergraduate-education|calculus|tutoring|special-needs-education|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12119" PostTypeId="2" ParentId="12113" CreationDate="2017-03-15T02:50:26.620" Score="5" Body="&lt;p&gt;For discrete distributions, it's helpful to look at coinflips. You can find the probability of getting m heads/tails out of n flips. (Though a normal distribution may be simpler to start with than a binomial distribution.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for continuous distributions, it may be helpful to look at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maxwell-Boltzmann Distribution&lt;/a&gt;, which can tell you the probability of finding a particle traveling at velocity v (where v is the random continuous variable). Since the total probability is one (100%), it means that the area under the curve described by the function is also one, which can be exploited to normalize the function.&lt;/p&gt;&#xA;" OwnerDisplayName="user7915" LastEditorDisplayName="user7915" LastEditDate="2017-05-19T19:10:01.637" LastActivityDate="2017-05-19T19:10:01.637" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12120" PostTypeId="2" ParentId="12118" CreationDate="2017-03-15T06:42:42.463" Score="22" Body="&lt;p&gt;This is a student who doesn't understand social cues. He only knows that if he is rude you keep trying.  If acts disruptive, he will get the answer without working for it. As long as you reward him for being difficult by giving him the answer, he'll have no motivation to change behavior. You are essentially letting him walk all over you because you want to help and don't know how.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest that before you can reach him, you have to teach him to behave. Make a plan with the other tutors. Sit him down with at least two tutors and tell him before you can help him, there are new rules for him to follow because he has been rude and disruptive. Explain what is acceptable behavior, e.g. no Facebook while talking to you, no shouting, etc.  Tell him the consequences of not following the behavior - eg the tutor will walk away and help another student if he looks at Facebook, he will be asked to leave if he shouts. Make sure he understands that he can come back when he behaves.  If you and the other tutors stick to this plan he will start to behave differently and then you can try to reach him.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2017-03-15T15:34:50.697" LastActivityDate="2017-03-15T15:34:50.697" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12122" PostTypeId="2" ParentId="4409" CreationDate="2017-03-15T12:21:31.807" Score="2" Body="&lt;p&gt;One of my favorites is this quote of Abel, identifying the notion of &lt;em&gt;analytic function&lt;/em&gt;. (emphasis mine)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My work in the future must be devoted entirely to pure mathematics in its abstract meaning. I shall apply all my strength to bring more light into the tremendous obscurity which one unquestionably finds in analysis. It lacks so completely all plan and system that it is peculiar that so many have studied it. The worst of it is, it has never been treated stringently. There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. &lt;strong&gt;Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has led to do few of the so-called paradoxes. It is really interesting to seek the cause.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In analysis, one is largely occupied by functions which can be expressed as powers. As soon as other powers enter—this, however, is not often the case—then it does not work any more and a number of connected, incorrect theorems arise from false conclusions. I have examined several of them, and been so fortunate as to make this clear. ...I have had to be extremely cautious, for the presumed theorems without strict proof... had taken such a stronghold in me, that I was continually in danger of using them without detailed verification.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiquote.org/wiki/Niels_Henrik_Abel&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikiquote.org/wiki/Niels_Henrik_Abel&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-03-15T12:21:31.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12123" PostTypeId="2" ParentId="12118" CreationDate="2017-03-15T23:49:03.073" Score="29" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How do I reach [this] kid?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Let me be blunt: You probably don't. This is a person who is so intransigent that you effectively need to &lt;a href=&quot;https://en.wikipedia.org/wiki/Triage_tag&quot; rel=&quot;noreferrer&quot;&gt;black-tag&lt;/a&gt; them. A hard lesson is that you can't save everyone. At this point the priority is to make some kind of defense so you aren't overly stressed, psychologically damaged, or burnt out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple personal reactions to the example interaction: I would not let any minor points slide by. Part of the job is to communicate clearly when someone does &lt;em&gt;not&lt;/em&gt; have the adequate basics. When they said &quot;prime of the prime&quot; I would totally stop and ask them for the correct name. When they couldn't find the first derivative correctly, I would absolutely go no further until they had corrected that. When they said, &quot;I know this&quot;, I would say clearly, &quot;Let's be honest: you do &lt;em&gt;not&lt;/em&gt; know this, because your first derivative calculation was incorrect.&quot; (Actually, I pretty much said exactly that each of the last two days.) Points like these are perfect exit spots; &quot;I'll let you work on that and I'll come back&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should also have the capability of refusing service if someone is rude. For me, if I'm answering a question someone has asked, and they look down at a phone-tablet-Facebook -- then I'm out, game over, the end. (Usually accompanied with an &quot;If you're not interested in my answer, then I'm moving on to someone else...&quot; exit).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like the idea above of setting an explicit time limit for the interaction and possibly setting a watch or timer to enforce it (hence my &quot;related&quot; link in the comments above). You may need to do that uniformly with other students while the student is in the room so you're not accused of bias (unfortunately). Indeed, other students deserve your time at least as much as this one. E.g., in class I try to limit my interactions to one or two problem-resolutions per person at most (so that I can get to everyone). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On that note, be prepared for the student to complain to some higher authority. Possibly get a shared plan with the other assistants to confirm your experiences if possible. Best of all, see if you can get one faculty member to engage the student and make recommendations to you after. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have also had (in a decade+ of teaching) maybe one single interaction where a student became completely outraged by my responding to a question by asking a follow-up question (so as to gauge where their understanding started and ended). &quot;You're the teacher, you're supposed to answer questions, not ask them!!&quot; and stormed out of the room. If someone doesn't even &lt;em&gt;get the concept of how teaching should work&lt;/em&gt; as a give-and-take, then that's outside the scope of your responsibilities (or maybe anybody's) to fix. Even Socrates himself couldn't deal with that. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-03-18T02:23:14.240" LastActivityDate="2017-03-18T02:23:14.240" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12124" PostTypeId="2" ParentId="10842" CreationDate="2017-03-16T02:28:02.683" Score="3" Body="&lt;p&gt;Late to the party to answer, but hopefully this will be helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say that the best source of incorrect proofs can be your students themselves! My undergraduate real analysis professor basically built the course around this idea in a pseudo-Morse methodology. The breakdown would go like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Assigned set of ~15 proofs from the text were due at the start of class every day.&lt;/li&gt;&#xA;&lt;li&gt;The first ~5 minutes of class were devoted to students writing one of their proofs on the boards around the classroom (worked well for a class of roughly 25 students with ample boardroom). Wasn't a particular grade for &quot;thou shalt write X problems up each week,&quot; but the prof would encourage regular contributions from all the students.&lt;/li&gt;&#xA;&lt;li&gt;We would then go over each of the proposed proofs as a class and decide if they worked (~5-10 minutes). If not, we would decide what was necessary to fix them or if we needed to reject them outright. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A key point is that none of this was graded (FERPA!) and semi-anonymous. The focus was always on &quot;is this a valid argument&quot; and not &quot;did Joe fail to prove something yet again.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a student, the first week was certainly a touch awkward. Once we all settled into the routine and got over the stage fright, it ended up being one of the most effective courses I had. Just like learning to edit is a key component of becoming a competent writer, learning to evaluate (other people's) arguments is a key to becoming a competent mathematician. &lt;/p&gt;&#xA;" OwnerUserId="7260" LastActivityDate="2017-03-16T02:28:02.683" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12125" PostTypeId="1" AcceptedAnswerId="12128" CreationDate="2017-03-16T02:59:30.883" Score="21" ViewCount="3502" Body="&lt;p&gt;As mathematics educators, we all have come across students using mathematical notation incorrectly (looking at you, $\frac{d}{dx}$ vs $\frac{dy}{dx}$ or $\frac{\infty^2}{\infty}$). My question focuses on &quot;verbal notation.&quot; For example, my hackles go up when I hear the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;take the prime of $f$&quot; or &quot;$d$-$dx$ the function&quot; or &quot;derive the function&quot; instead of &quot;compute the derivative of $f$&quot; or &quot;find $f'(x)$&quot; (&lt;em&gt;edit&lt;/em&gt;: or &quot;differentiate the function&quot;). Double chalkboard-fingernails for &quot;the prime of the prime&quot; and it's ilk.&lt;/li&gt;&#xA;&lt;li&gt;&quot;anti-derivative the function&quot; instead of &quot;integrate the function&quot; or (even better) &quot;find the indefinite integral of the function&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;minus/minusing $a$ from $b$&quot; instead of &quot;subtract $a$ from $b$&quot; or &quot;compute $b$ minus $a$&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;plus/plussing $a$ and $b$&quot; or instead of &quot;add $a$ and $b$&quot; or &quot;find $a$ plus $b$&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;take the inverse of a fraction&quot; instead of &quot;take the reciprocal of a fraction&quot; (debatable, the &quot;multiplicative inverse of a fraction&quot; does appear in sources)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The list goes on from there--I would be curious to hear your pet peeve phrases! My question is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is it overly picky and pedagogical to correct such phrases?&lt;/strong&gt; If it is appropriate to correct these phrasings, is it situation dependent (tutoring/recitation/lecture) and how would you do so?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to emphasize that this is a question &lt;em&gt;specifically about phrasing&lt;/em&gt; and verbalizing mathematical operations. Assume that the hypothetical student is generally performing the correct operations, is capable of reasonably proper written notation, and &quot;plussing $a$ and $b$&quot; would be the correct step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit: Running list of other phrasing&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;vertexes&quot; vs. &quot;vertices&quot;, probably applying to pluralizations of many other words as well (axises vs axes, ...). Credit to kcrisman&lt;/li&gt;&#xA;&lt;li&gt;Opposite of the above, &quot;vertices&quot; or &quot;vertice&quot; to refer to a single object (c.f. $x$-axes etc). Credit to Andreas Blass&lt;/li&gt;&#xA;&lt;li&gt;Misuses of mathematical verbs such as &quot;Solve 16 + 58&quot; or &quot;Prove the integral.&quot; Credit to Jack M&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7260" LastEditorUserId="7260" LastEditDate="2017-03-18T20:37:38.417" LastActivityDate="2022-11-04T19:48:19.797" Title="Grating mathematical phrases---How to correct?" Tags="|students-mistakes|notation|" AnswerCount="5" CommentCount="21" ContentLicense="CC BY-SA 3.0" />
  <row Id="12126" PostTypeId="2" ParentId="12125" CreationDate="2017-03-16T03:17:19.597" Score="3" Body="&lt;p&gt;The standard verb is &quot;(anti)differentiate&quot;, right?  That's quite a mouthful.  Probably okay to correct but with a light heart - make it into a joke, if the context is right.  It is useful to be able to use standard terminology, so I hear you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, I had a graph theory class once where one student consistently said &quot;vertexes&quot; rather than &quot;vertices&quot; - I never once had to correct him after the first week, another student and he made it into a running game.  For all I know he tells this same story in his career as a jazz musician (not kidding!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you shouldn't do is find ways to shame students who are struggling with the computations, let alone concepts.  (I'm not suggesting you are doing this.  But it's easy to come across this way, as many of us have experienced.)  Bonus points for first person to use correct terminology?  Or pie for the first one to come up with a reason why &quot;prime the prime&quot; would be ambiguous?  That last one seems pretty unambiguous to me, by the way - it's more annoying because it focuses too much on algebra than the idea of acceleration than because of the wording.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note, &quot;verbing the noun&quot; seems to be more and more common, and is probably a normal linguistic change within English in general.  This discussion may seem quaint a hundred years from now (imagine smiley emoji/emoticon here).&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-03-16T03:17:19.597" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12127" PostTypeId="2" ParentId="12125" CreationDate="2017-03-16T03:20:46.617" Score="3" Body="&lt;p&gt;Here's my stab at a self-answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think we would all agree on the need for precise written notation is important within mathematics. Unless the context is specifically reverse polish notation, a student writing $+~2~~ 2$ would be bizarre and incorrect. As such, I feel that it is also important to emphasize precision when verbalizing mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the analogy of mathematics as foreign language, it would be strange to learn French with strict emphasis on proper spelling and grammar but to never have pronunciation corrected. While mathematics is primarily a written language, more emphasis is justly placed on written notation. However, I feel that we should also place value on spoken mathematics by correcting such phrasings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal approach follows advice of what I've heard to do when a colleague is using a fancy vocab word incorrectly: try to use the same word in a proper context as soon as possible, rather than a direct &lt;a href=&quot;https://www.youtube.com/watch?v=G2y8Sx4B2Sk&amp;amp;ab_channel=bagheadinc&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;I do not think it means what you think it means.&quot;&lt;/a&gt; My goal is to point out the mistake while not coming off as nit-picky. My personal approach also tries to be sensitive---humiliating a student, even unintentionally, in front of their peers can be quite damaging.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if a student used one of these phrasings while offering a suggestion or asking a question during class, I would try to parrot the statement back correctly and placing a slight emphasis on the correct phrasing:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;Good---in order to find the critical points, we'll need to &lt;strong&gt;compute the derivative of $f$&lt;/strong&gt; and ...&lt;/li&gt;&#xA;&lt;li&gt;&quot;I agree, &lt;strong&gt;subtracting&lt;/strong&gt; $b$ from both sides of the equation will ...&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7260" LastActivityDate="2017-03-16T03:20:46.617" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12128" PostTypeId="2" ParentId="12125" CreationDate="2017-03-16T04:15:46.403" Score="28" Body="&lt;p&gt;Personally, I don't think we attend to this sufficiently in lower-level mathematics (where it's actually needed most). Students need that vocabulary to interface with books, future teachers, tutors, other students, etc. I run questions on it in weekly quizzes; and if I had my druthers, it would be a major component of all tests (in addition to application-level stuff). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, you've got to jump on that stuff as directly, firmly, and as soon as possible to make a difference. Really lead by example that it's a priority for you that students know how to interface with that language for their next step. I never let it go by if it comes up in class; I always address the class with, &quot;Can anyone help me? What's the correct word for this?&quot;. At least by the level of college algebra and above my students definitely respond positively to this, and it gets better rapidly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of my non-native English speakers express outright fear the first day when it becomes clear that this is the emphasis, but I do try to reassure them that in some sense we're all in the same boat, and prior students in that situation have rapidly improved and done extremely well. They're usually thankful for that emphasis by the end of the semester. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-03-17T02:43:46.063" LastActivityDate="2017-03-17T02:43:46.063" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12129" PostTypeId="2" ParentId="9869" CreationDate="2017-03-16T04:22:41.743" Score="1" Body="&lt;p&gt;In my own lectures, there are a couple things that I have found to be useful for addressing notation issues in math (especially calculus). Personally, I do a minimum of &quot;correcting&quot; while grading exams outside of what will be necessary to reconstruct why points were taken off. For example, I might just circle the $\infty$ stretch and put &quot;-3&quot; next to it. The students will presumably get an answer key and correcting such mistakes really requires an interaction rather than a paragraph written on an exam. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Make it clear from the start that notation is important, just as spelling and grammar are important for writing. The time for correcting notation is not on the exam proper, but rather assessments leading up to the exam and class discussions following the exam. There does, however, need to be an acknowledgement of some leeway. For example, the posed question on the exam is grammatically incorrect---there's no period at the end of the sentence. How many points should the exam writer lose?&lt;/li&gt;&#xA;&lt;li&gt;Be thoroughly meticulous on the board. It would be hard to learn a foreign language if the instructor mis-spelled many words. This means that every instance of $\lim$ is accompanied by bounds, bounds don't vanish from integrals, etc. Every. Single. Time. I've even taken this to the extent of &quot;$2+2=4.$&quot;, as there needs to be a period at the end of mathematical sentences. This is not to say that the instructor is to be blamed, but rather that a high level of rigor allows you to be beyond reproach when taking points off for notation.&lt;/li&gt;&#xA;&lt;li&gt;Emphasize why notation matters by analogy with more familiar topics. Perhaps I've noticed some of my students writing $\lim_{x \to \infty} =0$ or $\frac{\mathrm{d}}{\mathrm{d}x} = 2x$. I might start a class with writing $+ = 3$ on the board and asking them to solve. Once they've agreed that such a string of symbols is ridiculous, circle back to calculus and emphasize that the calculus-y symbols are operators.&lt;/li&gt;&#xA;&lt;li&gt;The particular answer you have given is problematic as we run into the issue of the correct answer with incorrect reasoning. Taking points off (as you should) will invariably lead to the complaint &quot;But I got the right answer!&quot; If possible, write questions where $\frac{\infty}{\infty^2}$ type notation will lead to the wrong answer.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7260" LastActivityDate="2017-03-16T04:22:41.743" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12130" PostTypeId="2" ParentId="12125" CreationDate="2017-03-16T07:41:16.550" Score="14" Body="&lt;p&gt;I wish to give a slightly different answer compared to the others.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Strict and Standardized Notations is Very Important&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;They not only help us communicate better, they also help us think. They prime us to remember things and understand things better. For example, if I see &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2$&lt;/span&gt;, I think Pythagoras Theorem and right angle triangles. If I see &lt;span class=&quot;math-container&quot;&gt;$k^2 + y^2 = t^2$&lt;/span&gt;, I don't.&lt;/p&gt;&#xA;&lt;p&gt;It also allows you to be more accurate, and make sure your logic is not flawed.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Non-standard Notation Could also be helpful&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Though rare, not using standard notations could help with thinking about a problem in another way, or coming up with a different sub-field of math.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;There are Different Standards/Language is Evolving&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Like any other language, mathematical language is evolving. If enough people uses a phrase, it is a correct phrase. Different mathematical papers uses different standards.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Commonly used Standard Notation could be suboptimal&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The first notation for a subfield is usually made by the guy who ventures into this subfield. Being the first, he is exploring in unfamiliar territory, and his standards ends up suboptimal. Then more people come in, and each tries to invent a better standard, or a more universal notation, and it ends up like xkcd 927.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, it should be noted that different notations are more useful in at different times.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Verbal Notations are often much more flexible than written ones&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;People often don't speak in completely correct sentences. Things are shortened. Words are changed.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Verbal Math is often an attempt to translate a formula to English?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;How would you say &lt;span class=&quot;math-container&quot;&gt;$(2a + b) \times c$&lt;/span&gt;? There is no guide to speaking formula. Do you say &amp;quot;The product of c and the sum of two-a and b&amp;quot;? That clearly got the multiplicands in reverse, and what is a two-a? Or do you say &amp;quot;Open bracket, two times a plus b, close bracket, times c.&lt;/p&gt;&#xA;&lt;p&gt;As another example, take &amp;quot;three x plus four b over seven all over nine&amp;quot;. What does that mean?&lt;/p&gt;&#xA;&lt;p&gt;So I'd say saying &amp;quot;d d x&amp;quot; or &amp;quot;d over d x&amp;quot; is perfectly fine. And if we can say that &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; is f prime; why can't be say that it is the prime of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, or that the action of differentiating is taking the prime of f?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Not Everyone Uses the Standard Notations/the same standard notations as you do&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Unless your students will only be talking to you/other people that strictly follow the standards, they'll need to be flexible.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Are you sure you are right?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Are you sure that the things you find incorrect are actually incorrect, and not just using a particular standard?&lt;/p&gt;&#xA;&lt;p&gt;Are you sure vertexes is not a allowable pluralization of vertex? Are you sure that the word vertices is not an appendage that is being/has been phased out? Will you insist that data must be plural, and one must use datum for the singular?&lt;/p&gt;&#xA;&lt;p&gt;Are you sure you are pronouncing &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; correctly. Do you pronounce it like &lt;a href=&quot;https://youtu.be/4Pf4LP7qikM?t=1m17s&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; or &lt;a href=&quot;https://www.youtube.com/watch?v=VLt_GRe1Xj8&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;? The former is more Greek, and is often used by people from certain areas in Europe (And, sometimes if taken to the extreme, sounds like &amp;quot;OH MY GOD&amp;quot;). Something like the latter is more used in America. The common pronounciation for me and my peers is something slightly different from the latter.&lt;/p&gt;&#xA;&lt;p&gt;Are you sure the word &amp;quot;derive&amp;quot; cannot be used to mean &amp;quot;differentiate&amp;quot;? I cannot support this with evidence, but I remember some sources using derive in that manner, and some sources claiming that derive can indeed mean &amp;quot;differentiate&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;You've already mentioned the debatability of inverse. I'm going to claim that using inverse of a fraction instead of reciprocal is perfectly allowed. And I would argue that &amp;quot;minus a from b&amp;quot; is perfectly allowed as well. &amp;quot;Plus a and b&amp;quot; is slightly more awkward. However, without consulting a mathematical grammar guide and dictionary, can you tell me why &amp;quot;plus&amp;quot; can not be used that way in math?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The Bottom Line&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Everything considered, notations are important. You should seek to introduce your students to the different types of verbal notations. They definitely should be able to fluently use the word &amp;quot;differentiate&amp;quot;. You should impress that some notations are more proper than others, and should be used most of the time. If called upon, they should be able to use proper notations.&lt;/p&gt;&#xA;&lt;p&gt;However, it is also important for them to understand and use other &amp;quot;less proper&amp;quot; notations. In general, it is fine to use these &amp;quot;less proper&amp;quot; verbal notations. However, if it leads to a situation where the students are unable to use proper notation, or when the usage of certain verbal notation is hindering communication or thought, proper notation should be emphasized.&lt;/p&gt;&#xA;&lt;p&gt;Finally, you can simply use the &amp;quot;correct&amp;quot; notation in your speech, and in general, the students will follow. You can also explicitly note, every so often, that while &amp;quot;derive&amp;quot; can be used sometimes, there are other notations, and &amp;quot;differentiate&amp;quot; is in a generally better and more clear word.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I feel the need to add onto this answer.&lt;/p&gt;&#xA;&lt;p&gt;I would first like to draw attention to &lt;a href=&quot;https://math.stackexchange.com/questions/264610/why-is-abuse-of-notation-tolerated&quot;&gt;this question&lt;/a&gt;, which has great answers.&lt;/p&gt;&#xA;&lt;p&gt;To quote some of the quotes given:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The student of mathematics has to develop a tolerance for ambiguity.&#xA;Pedantry can be the enemy of insight.&amp;quot; - Gila Hanna&lt;/p&gt;&#xA;&lt;p&gt;As far as possible we have drawn attention in the text to abuse of&#xA;language, without which any mathematical text runs the risk of&#xA;pedantry not to say unreadability. - Bourbaki&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also linked in the answers to that question is an &lt;a href=&quot;https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;article by Terence Tao&lt;/a&gt;, who describes the progression of mathematical education in three stages: &amp;quot;pre-rigorous&amp;quot;, &amp;quot;rigorous&amp;quot;, and &amp;quot;post-rigorous&amp;quot;. I'd argue that any sub-field in math is learnt kind of in this manner. I would say that the student should be only be steered toward correct notation in the pre-rigorous stage, and that if notation is to be emphasized, it should be during the &amp;quot;rigorous&amp;quot; stage.&lt;/p&gt;&#xA;" OwnerUserId="7919" LastEditorUserId="13864" LastEditDate="2021-04-18T18:47:31.953" LastActivityDate="2021-04-18T18:47:31.953" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="12132" PostTypeId="1" CreationDate="2017-03-17T13:24:07.670" Score="10" ViewCount="301" Body="&lt;p&gt;Some people talk about visual thinkers and non-visual thinkers, but I am interested in a contrast within styles of visual thinking.  There are people who readily visualize complicated flow charts and other diagrammatic graphics for information, but do not readily picture, say, a  Möbius strip or an octahedron (let alone 4 dimensional objects). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I get this from the history of mathematics where some great examples (Richard Dedekind, Emmy Noether, Alexander Grothendieck) at first seem to be non-visual thinkers.  They would seem to be algebra people rather than geometry people in terms of this question: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10777/evidence-for-or-against-the-claim-that-some-students-are-algebra-people-and-ot&quot;&gt;Evidence for or against the claim that some students are &amp;quot;algebra people&amp;quot; and others are &amp;quot;geometry people&amp;quot;&lt;/a&gt;  Yet each made great, explicit, deliberate contributions to geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I wonder if they are better described as visual in a different way than spatial visualization. Certainly Grothendieck made vast use of diagrams which are not really spatial diagrams (albeit they can be drawn on a blackboard).  They are diagrams of conceptual relations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics educators may know a lot more about this than I do, so I will not try to be more precise, unless people ask for that in comments. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: where can I go to learn more about this distinction between different visual thinking/learning styles?&lt;/p&gt;&#xA;" OwnerUserId="3390" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.590" LastActivityDate="2017-03-24T23:42:30.813" Title="Styles of visualization in geometry" Tags="|reference-request|visual-proof|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12133" PostTypeId="1" CreationDate="2017-03-17T22:53:45.960" Score="1" ViewCount="116" Body="&lt;p&gt;I'm preparing a workshop about  ODEs and connected rates of change for pre-college students (smart and interested in mathematics). I'd like to include some fun parts as well - even if these are standard exercises but in unexpected context. Could you share your experience in teaching these subjects?&lt;/p&gt;&#xA;" OwnerUserId="7936" LastEditorUserId="7936" LastEditDate="2017-03-18T18:46:14.533" LastActivityDate="2018-04-08T15:47:14.170" Title="Workshop about ODEs and connected rates of change (A-levels)" Tags="|secondary-education|differential-equations|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12134" PostTypeId="2" ParentId="12132" CreationDate="2017-03-17T23:34:16.297" Score="6" Body="&lt;p&gt;It seems the &lt;em&gt;Stanford Encyclopedia of Philosophy&lt;/em&gt; has a useful article that at least&#xA;tangentially addresses your interesting question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Can visual thinking lead to discovery of an idea for a proof in more advanced contexts? Yes. Carter (2010) gives an example from free probability theory.&#xA;  [...] Reflection on a diagram such as Figure 9 does the work.&quot;&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;  &lt;a href=&quot;https://i.stack.imgur.com/kMQx6.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kMQx6.jpg&quot; alt=&quot;Permut&quot;&gt;&lt;/a&gt;&#xA;  &lt;hr /&gt;&#xA;  Giaquinto, Marcus. &quot;The Epistemology of Visual Thinking in Mathematics.&quot; The &lt;em&gt;Stanford Encyclopedia of Philosophy&lt;/em&gt;, Winter (2015).&#xA;  &lt;a href=&quot;https://plato.stanford.edu/entries/epistemology-visual-thinking/#ProDis&quot; rel=&quot;nofollow noreferrer&quot;&gt;HTML link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;His last sentence:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Also, visual thinking accompanying a proof may deepen our understanding of the proof, giving us an awareness of the direction of the proof so that, as Hermann Weyl put it,&lt;sup&gt;1&lt;/sup&gt; we are not forced to traverse the steps blindly, link by link, feeling our way by touch.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Incidentally, the author of the encyclopedia entry, Marcus Giaquinto, wrote a book entitled, &lt;em&gt;Visual Thinking in Mathematics&lt;/em&gt; (Oxford University Press, 2007).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;Weyl, H., 1995 [1932], “Topology and abstract algebra as two roads of mathematical comprehension”, &lt;em&gt;American Mathematical Monthly&lt;/em&gt;, 435–460 and 646–651.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-03-18T17:39:47.620" LastActivityDate="2017-03-18T17:39:47.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12135" PostTypeId="2" ParentId="11106" CreationDate="2017-03-19T00:42:56.667" Score="3" Body="&lt;p&gt;When I think of a geometric sequence, I think of something where the initial input value = 1, not 0.  Most interest problems would start at time = 0, so I would exclude these unless you said something like &quot;let x = $ in bank at beginning of each year&quot;.  Then your first input value would be 1.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, geometric sequences have a domain of only natural numbers (1,2,3,...), and a graph of them would be only points and not a continuous curved line. So again, a problem about earned interest might not be a perfect example, since you can withdraw your money at any instant and not only at whole number year values.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best one I have come up with is tile values in the game 2048.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;tile 1 = 2 &lt;/li&gt;&#xA;&lt;li&gt;tile 2 = 4 &lt;/li&gt;&#xA;&lt;li&gt;tile 3 = 8&lt;/li&gt;&#xA;&lt;li&gt;tile 4 = 16&lt;/li&gt;&#xA;&lt;li&gt;tile 5 = 32 &lt;/li&gt;&#xA;&lt;li&gt;tile 6 = 64&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and so on...&lt;/p&gt;&#xA;" OwnerUserId="7937" LastActivityDate="2017-03-19T00:42:56.667" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12136" PostTypeId="2" ParentId="11106" CreationDate="2017-03-19T02:34:53.600" Score="6" Body="&lt;p&gt;This is a nice demonstration that&#xA;$$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}\cdots&#xA;= 2 \;.$$&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/EZ0XG.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EZ0XG.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Geometric_progression#Infinite_geometric_series&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article on Geometric progression&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-03-19T02:34:53.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12137" PostTypeId="1" CreationDate="2017-03-19T20:07:43.107" Score="3" ViewCount="407" Body="&lt;p&gt;How much of introductory calculus can be learned without using analytic geometry or for that matter any algebraic notations but simple euclidean geometry? Are there any resources(new ones not the old ones of Newton's times) that treat calculus topics like derivative, integrals, maxima and minima using pure euclidean geometry? The genuine problems that I face is how to understand and explain polynomial curves,trigonometric and logarithmic functions and their derivatives or integrals without use of algebra.But I still believe that a high school kid will have more affinity towards the pure geometric visualization rather than the algebraic one. So do suggest some good resources(preferably books) on this kind of approch( I found one going by the name Calculus without Analytic Geometry by Barry Mitchell, but struggled to get its contents).&lt;/p&gt;&#xA;" OwnerUserId="6560" LastActivityDate="2017-03-20T02:51:06.890" Title="calculus without analytic geometry" Tags="|secondary-education|calculus|geometry|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12138" PostTypeId="2" ParentId="12133" CreationDate="2017-03-20T02:06:39.800" Score="0" Body="&lt;p&gt;A problem I often given in calculus III goes roughly like this: give a car is driving on a hill with equation $x^2+3y^2+z^2=1$ if you know the speed in the $x$-direction and the speed in the $y$-direction then find the speed in the $z$-direction at a given point. Also find the speed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The total differential gives $2xdx+6ydy+2zdz=0$ then divide by $dt$ to obtain&#xA;$2x \frac{dx}{dt}+6y \frac{dy}{dt}+ 2z\frac{dz}{dt} = 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, we can use the same math to do things like find curves which line up with vector fields. First, understand $F(x,y)=c$ is a solution to $dF = \partial_x F dx+ \partial_yF dy = 0$ and $\nabla F = \langle \partial_x F , \partial_yF \rangle$ points directly away from the curve $F(x,y)=c$. Thus, to find a curve which lines-up with $\vec{G}  = \langle M , N \rangle$ we want to find leve curves which have a gradient which is perpendicular to $\vec{G}$. That means, solve $Ndx- Mdy = 0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, to find integral curves to $\vec{G} = \langle y, -x \rangle$ we solve $-xdx-ydy = 0$ hence $-ydy = xdx$ or $-\frac{1}{2}y^2 = \frac{1}{2}x^2+C$ which is better expressed as $x^2+y^2 = R^2$. If you picture $\vec{G}$ the fact that circles are integral curves is expected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea that you can calculate with total differentials then simply divide by $dt$ to obtain relations between rates involved is fascinating. A victory of notation. Students who understand nothing of the implicit function theorem are able to successfully implement it through little more than symbol pushing.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-03-20T02:06:39.800" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12139" PostTypeId="2" ParentId="12137" CreationDate="2017-03-20T02:51:06.890" Score="3" Body="&lt;p&gt;There are some calculus courses for kids that focus on the splitting up into slices and that sort of thing. They don't go as far as you'd probably like, and they aren't Euclidean, really. This is a course that was offered last year: &lt;a href=&quot;http://naturalmath.com/2016/02/newsletter-february-1-2016/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://naturalmath.com/2016/02/newsletter-february-1-2016/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-03-20T02:51:06.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12140" PostTypeId="1" AcceptedAnswerId="12142" CreationDate="2017-03-20T07:13:46.337" Score="4" ViewCount="136" Body="&lt;p&gt;I'm teaching a history of mathematics class for master students in an integrated teacher training program. They have calculus and linear algebra plus a &quot;proof&quot; course and a geometry course, but most of them don't have analysis. I want to cover topics that are &quot;related&quot; to school mathematics. It doesn't have to be things they will use when teaching, but things that will give them a better understanding to things they will be teaching. I've tried both Katz and Stilwell. Both are excellent books, but not quite right for my audience and my class. I liked Stillwell because it was more mathematical, but I would have liked to have seen more applied mathematics, too.  I'm thinking about trying Mathematics in Civilization by Resnikoff and Wells or Journey through Genius by Dunham. Has anybody tried them? Or any other suggestions for similar books?&lt;/p&gt;&#xA;" OwnerUserId="3566" LastActivityDate="2017-03-20T13:30:36.473" Title="Has anybody used Mathematics in Civilization by Resnikoff and Wells or Journey through Genius by Dunham for a history of math class?" Tags="|history-of-math|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12142" PostTypeId="2" ParentId="12140" CreationDate="2017-03-20T13:30:36.473" Score="6" Body="&lt;p&gt;There are a number of good options for such a class, but it depends wildly on what your objectives are.  I have only used Journey Through Genius in an actual class setting, so I have a few relevant comments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Advantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Very readable&lt;/li&gt;&#xA;&lt;li&gt;Many interesting personality anecdotes&lt;/li&gt;&#xA;&lt;li&gt;Often connects to less well-known characters&lt;/li&gt;&#xA;&lt;li&gt;Low prerequisites&lt;/li&gt;&#xA;&lt;li&gt;Useful material not often covered in &quot;normal&quot; curriculum outside a math history course&lt;/li&gt;&#xA;&lt;li&gt;Fun results&lt;/li&gt;&#xA;&lt;li&gt;Could easily use other Dunham books to supplement without jarring stylistic changes&lt;/li&gt;&#xA;&lt;li&gt;Very reasonable price and easy to obtain used copies&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Disadvantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Couldn't possibly be a solo text, needs considerable supplement in terms of time periods and coverage even if you have a &quot;short course&quot;&lt;/li&gt;&#xA;&lt;li&gt;Low prerequisites mean anything beyond infinite series (including calculus) can't be covered properly with just this&lt;/li&gt;&#xA;&lt;li&gt;Definitely not a textbook, so some fairly noticeable gaps if you were to intend it used in this way (e.g. very heavy on Greek math and number theory)&lt;/li&gt;&#xA;&lt;li&gt;You'll have to think of your own homework and/or quiz material&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The way I've used it is as a cheap primary text with lots of stuff interesting to students who otherwise might see a math history class as a very tedious hoop to jump through, and then I supplement considerably with material from &quot;missing&quot; sections and more advanced results.  I've found many MAA journals and the Notices of the AMS often have good material along those lines, and the &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk&quot; rel=&quot;noreferrer&quot;&gt;MacTutor&lt;/a&gt; history site is great even if the interface is a bit dated.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-03-20T13:30:36.473" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12143" PostTypeId="2" ParentId="12110" CreationDate="2017-03-21T00:15:12.693" Score="5" Body="&lt;p&gt;I know that this is a difficult issue that many of us struggle with. By the time a typical student reaches college, they have had over a decade of math classes and teachers. As such, most of your students will have likely decided whether or not they're a &quot;math person&quot; by the time they're sitting in your classroom. You have to confront that an individual might be capable of calculation and dislike math, be poor at computation and appreciate math, or be poor at computation and dislike math (I would argue that there is a correlation). Regardless of the specifics, you're fighting against phobias.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to maintain a level of empathy for your students---as a lecturer in math, I would wager that math was always an easy/easier subject for you throughout school. Even if you didn't necessarily fall in love with math until you took a class from Prof. X or read a book by [famous mathematician], you've likely always been good at pushing symbols around and solving for $x$. Don't take these skills for granted in your students. Try not to combine algebraic steps on the board, give examples like $\sqrt{3^2 + 4^2} \overset{?}{=} \sqrt{3^2} + \sqrt{4^2}$, try to write questions that don't hinge on a clever simplification step. Furthermore, encouraging students to work and study in groups can be supremely beneficial: hopefully they'll be able to correct each other's mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the motivation question, why do &lt;em&gt;you&lt;/em&gt; think that undergraduates studying liberal arts should learn calculus? Personally, I would argue that the discovery of calculus is perhaps the greatest intellectual achievement of humanity over the last millennium. What do you want your students to know about calculus at the end of the course? Once they graduate? In ten years? Try to strike a balance between needing to learn rote computation, motivation, and learning the sorts of problems that mathematics can solve. For example, having your students learn enough mathematics to recognize that there might be a mathematics question lurking somewhere and &lt;em&gt;hint hint&lt;/em&gt; hire a mathematician to help them out can be a useful outcome. Try to keep them from unnecessarily  &lt;a href=&quot;https://academia.stackexchange.com/questions/9602/rediscovery-of-calculus-in-1994-what-should-have-happened-to-that-paper&quot;&gt;reinventing the trapezoid rule&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The history of calculus is rife with stories and anecdotes that can be incredibly compelling to students of history, literature, anthropology, and other liberal arts. Consider, e.g., the historical rivalry between Newton and Leibniz, the dysfunctional Bernoulli family, the tragic lives of Abel &amp;amp; Galois, the autodidactism of Ramanujan, the prejudice that Sophie Germain (&lt;em&gt;Monsieur LeBlanc&lt;/em&gt;) had to fight. You don't need to provide an entire history of mathematics for every topic, but even just an offhand mention can help engage students: &quot;Did you know that a mathematician named Evariste Galois proved in the 1800s that there isn't a thing like the quadratic equation [point to where you just used this on the board] doesn't exist for polynomials of degree five or higher? The legend goes like this: after challenging another man to a duel, he realized that he was probably going to die in the morning (again, he was a mathematician) and stayed up all night producing about ten pages of mathematics that graduate students still study to this day. He died at the age of twenty.&quot; Give them a reason to see calculus as a human endeavor and not just an edifice of facts. I have even gone as far as having calculus students write essays about historical figures. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you mentioned with your analogy of using salt in cooking, try to motivate definitions and theorems with everyday experiences. It's difficult to imagine our modern understanding of the universe without the influence of calculus. Mean Value Theorem? Average speed over a road trip. Linearization? What was the weather like when you walked into the building...predict if it's nicer out now. Exponential functions? Account balance under interest. Derivative? How can you figure out your speed if the speedometer is broken. Integration? How can you figure out how far you've driven if the odometer is broken. One tip that I've found is to explicitly &lt;em&gt;not&lt;/em&gt; use introductory examples from physics. While we might find the notion of a swinging pendulum or accelerating rocket intuitive, it is surprising how many students struggle with these analogies. Even the example of throwing a ball under the effect of gravity trips people up (although you should absolutely address this misconception!). Most students have poor intuition when it comes to science, e.g.,  &lt;a href=&quot;https://www.theatlantic.com/education/archive/2014/02/what-americans-dont-know-about-science/283864/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article from The Atlantic&lt;/a&gt; or &lt;a href=&quot;http://news.harvard.edu/gazette/story/2013/04/understanding-student-weaknesses/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article from the Harvard Gazette&lt;/a&gt; or &lt;a href=&quot;http://newyorkscienceteacher.com/sci/pages/miscon/phy.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;this list of Physics misconceptions by New York Science Teacher&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_common_misconceptions#Science_and_technology&quot; rel=&quot;nofollow noreferrer&quot;&gt;this list of misconceptions on Wikipedia&lt;/a&gt;. Being able to correct even a few of these is also a good objective for a calculus course. I personally find that economics/money or driving a car to be the most accessible areas to pull motivating examples from, before addressing physics examples with your newfound machinery. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a parting thought, try pondering the flipped situation: you presumably had to take some liberal arts courses as an undergraduate for your presumably math-or-related degree. What did you get out Foreign Language / Art / Music / Literature / Psychology / Economics 101? What do you remember? Why was it important that you take those courses? Did you have any particularly effective instructors?&lt;/p&gt;&#xA;" OwnerUserId="7260" LastEditorUserId="-1" LastEditDate="2017-04-13T12:49:57.670" LastActivityDate="2017-03-21T00:31:56.497" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12144" PostTypeId="1" CreationDate="2017-03-21T01:15:49.680" Score="8" ViewCount="73" Body="&lt;p&gt;I have been searching this for a long time, but most of what I find spends more time on cultural aspects and what is expected from the teacher, etc. Perhaps this is really trivial to find, but I am Brazilian and maybe I am googling for it with the wrong terms (sometimes I think that my translation suits adequately but end up discovering that no one uses that expression for what I tried to say). I tried: &quot;International mathematics curricula comparison&quot; and similar stuff, but what I found was not very helpful.&lt;/p&gt;&#xA;" OwnerUserId="272" LastActivityDate="2017-03-21T01:15:49.680" Title="Where can I find a comparison of mathematical subjects taught for primary and secondary school around the world?" Tags="|curriculum|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12145" PostTypeId="2" ParentId="12110" CreationDate="2017-03-21T01:24:59.007" Score="4" Body="&lt;p&gt;I am not here to bring hopeful news, but a harsh dose of reality. There is no simple way to make most students grasp the $\varepsilon$-$\delta$ definition of a limit in a first calculus course and I think it is a mistake to force this on them. Doing so feels to me to be analogous to requiring everyone to be able to repair an internal combustion engine  before they're allowed to learn how to drive a car. The rigorous limit concept is extremely subtle (it took over 150 years after Newton/Leibniz for it to be formalized, and the need for it escaped Euler), and most people starting out in calculus don't have sufficient mathematical maturity to grasp it. Let me quote from the introduction to Serge Lang's textbook A First Course in Calculus: &quot;Any student is ready to accept as intuitively obvious ... limits and their basic properties. Experience shows that the students do &lt;em&gt;not&lt;/em&gt; have the proper psychological background to accept a theoretical study of limits, and resist it tremendously.&quot; &#xA;I think what Lang is saying is quite accurate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I took calculus in high school I learned the $\varepsilon$-$\delta$ definition of a limit (it was part of the AP calculus curriculum in America) as a kind of mantra without really being able to see what it meant and could apply it to the (in retrospect, stupid) case of linear functions, but the textbook's way of applying the definition of a limit to quadratic polynomials made the whole business look horribly complicated. It all made very little sense, and I was a very &lt;em&gt;good&lt;/em&gt; student in math; it must have been really meaningless to the majority of the class. Two years later I was a college freshman in a course based on Rudin's &lt;em&gt;Principles of Mathematical Analysis&lt;/em&gt; and figured out how to use the $\varepsilon$-$\delta$ language comfortably, although I only truly mastered it when I took topology and learned how to express the idea of a limit without saying $\varepsilon$ and $\delta$ at all! Why was I unable to grasp the $\varepsilon$-$\delta$ definition in my first calculus course? I think, as Lang put it, I just was not psychologically ready. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In continental Europe, I've been told there is no calculus, college math courses start with real analysis, and almost everyone flunks that course. (Are you in the UK? You wrote maths rather than math.) Colleagues of mine from Spain and Greece told me that only a handful of their classmates survived the first year math program during their university education. So I think the difficulties of conveying the idea and use of the $\varepsilon$-$\delta$ definition of a limit are quite serious and not likely to be overcome anytime soon. No amount of storytelling about the personalities in the history of math is going to help anyone solve math problems, so while it may be interesting I don't see how supplementing your lectures with such anecdotes will have an impact on the grades of those in your class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admire your desire to pass on to the students your hard-won knowledge of limits rather than making it their fault that they can't figure this out, but I don't see any way to make most of the class understand the rigorous definition of a limit when they don't even know what calculus can do for them in a pre-rigorous form.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2017-03-21T01:55:37.710" LastActivityDate="2017-03-21T01:55:37.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12146" PostTypeId="1" CreationDate="2017-03-21T09:18:01.173" Score="11" ViewCount="1730" Body="&lt;p&gt;I'm a recent graduate in my first year of teaching. I teach secondary students, and I have found that some low-level students are not listening in class. They seem to have given up and stopped trying. I feel it's hard to make a difference or help them because they don't seem to care. How can I engage these students?&lt;/p&gt;&#xA;" OwnerUserId="7487" LastEditorUserId="238" LastEditDate="2019-12-09T09:02:22.800" LastActivityDate="2019-12-09T09:02:22.800" Title="How to engage low-level students?" Tags="|secondary-education|student-motivation|struggling-students|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12147" PostTypeId="2" ParentId="12125" CreationDate="2017-03-21T11:22:57.367" Score="1" Body="&lt;p&gt;Personally I feel like there are much more important issues for all students I've encountered while TA'ing/teaching in the US than whether they verbalize math correctly. I've generally taught lvl 300 courses (basic calc) and I've uniformly found that the students have incredibly poor notation with very few exceptions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While verbalizing math badly isn't great, unless your students are drastically different to the ones I've seen, it feels like focusing on that is a bit like making sure the aspiring cook who doesn't know how to turn on the stove is great at naming the ingredients for beef wellington.&lt;/p&gt;&#xA;" OwnerUserId="5896" LastActivityDate="2017-03-21T11:22:57.367" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12148" PostTypeId="2" ParentId="12113" CreationDate="2017-03-21T19:12:39.767" Score="6" Body="&lt;p&gt;This is treason, but anyway:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your students can jump from &quot;ratio of outcomes in $A$ over all possible outcomes&quot; to &quot;ratio of length of interval, over total feasible length&quot;, then the answer why probability can be represented as the area under a curve could half-jokingly (but only half-) be&#xA;&lt;em&gt;&quot;for convenience, since we set the height of the curve at a value that it will replicate our &quot;ratio&quot; approach&quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0ncgfm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0ncgfm.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;ratio approach&quot; would require that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$P([a,b]) = \frac {length[a,b]}{length[a,c]} = (b-a)\cdot \frac {1}{c-a}$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of doing a division in one dimension, we do a multiplication in two dimensions, by setting the probability density curve at height $1/(c-a)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With any other distribution, one has to move from simple multiplication of the sizes of a rectangle, to integration to find the area under a curve.&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2017-05-19T01:23:30.220" LastActivityDate="2017-05-19T01:23:30.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12149" PostTypeId="2" ParentId="12146" CreationDate="2017-03-21T19:35:34.593" Score="2" Body="&lt;p&gt;In my opinion a very important tool for the teacher is a lesson plan for every lesson. I was a math tutor in a math center for 8 years and I conducted multiple worshops in a classroom setting with over 20 students at a time. Techniques that worked for me were: inviting students to ask questions during the lesson , polite responses to questions , eye contact with the students , doing a worked out example on the board and then writing a similar problem beside it and asking the students to solve it at their desk while I walked around and looked at their work and gave helpful hints if the student seemed stuck. Also I would have students work in groups of 4 or 5 so they could exchange ideas with their peers. You have to mix it up a little so some students don't get bored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps :)&lt;/p&gt;&#xA;" OwnerUserId="6972" LastActivityDate="2017-03-21T19:35:34.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12150" PostTypeId="2" ParentId="12132" CreationDate="2017-03-22T18:38:34.067" Score="4" Body="&lt;p&gt;You might find the essay by Marjorie Senechal, &quot;Visualization and visual thinking.&quot; &lt;em&gt;Geometry's Future&lt;/em&gt; (1991): 15-21, published by the Consortium for Mathematics (COMAP), of interest. In particular she addresses the distinction between &quot;visual thinking&quot; and &quot;visualization.&quot; (You can also find articles that cite this work by searching on &quot;Google Scholar&quot;; currently $24$ articles cite it.)&lt;/p&gt;&#xA;" OwnerUserId="1865" LastEditorUserId="511" LastEditDate="2017-03-24T23:42:30.813" LastActivityDate="2017-03-24T23:42:30.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12153" PostTypeId="1" CreationDate="2017-03-25T09:56:38.790" Score="14" ViewCount="323" Body="&lt;p&gt;I am interested in educators real-world experience or being pointed to any research in this area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a student whose arithmetic skills are weak for his/her age. The student counts on his/her fingers when attempting to multiply, struggles to add two integers with a total greater than 20, cannot compare the relative size of fractions with different denominators, etc. These are skills which I understand &quot;most&quot; students would hope to master several years earlier than this student. I believe this student has fallen into a comfortable routine of (i) finger-counting or (ii) guessing or (iii) giving up on a question. Not to be critical of finger-counting, it is just that even the most basic of exam questions are inaccessible to this student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have used a number of tools such as Cuisenaire rods (which I think are extremely powerful and flexible teaching aids) to develop his/her number sense and skills in basic arithmetic. However, I decided to enter the student for an exam where basic electronic calculators are permitted, i.e. there are no non-calculator papers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My sense/ instinct is that if a &quot;number/arithmetic sense&quot; is not developed then handing a student a calculator is pointless because the student has not feel for expressions and questions like &quot;the ratio of the lengths is 2:5&quot; or &quot;this cost has increased by 20%&quot; or &quot;the sale discount is 15% what is the selling price?&quot; or &quot;how much should we distribute £1m grant money if we want to distribute it equally between 7 organisations?&quot; etc. etc. My sense is that &quot;teaching how to answer the question&quot; is a very &quot;fragile&quot; form of learning since a question can be rephrased slightly and the student can feel he/she hasn't been taught how to answer that question. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So do people have evidence that routinely using electronic calculators helps or harms development of basic/fundamental mathematical problem-solving skills?&lt;/p&gt;&#xA;" OwnerUserId="7954" LastActivityDate="2017-03-26T05:41:40.433" Title="Do electronic calculators inhibit mathematical thinking?" Tags="|calculators|arithmetic|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12154" PostTypeId="2" ParentId="12133" CreationDate="2017-03-25T13:17:51.083" Score="1" Body="&lt;p&gt;I would like to comment immediately below the original question, but am currently blocked from doing so.&#xA;@Paula. Why do you feel the &quot;standard&quot; connected rates of change questions e.g. relating the rates of volume, surface area and linear dimensions of a 3D object such as a cylinder, sphere or cone where one of the rates is constant, are lacking? For example a standard question, liquid pours into an empty right cone at a fixed rate but leaks out at a rate proportional to the depth of fluid in the cone, then what is height of fluid at time = t? Does the height reached a fixed value? In what way are you looking for questions to go &quot;beyond&quot; this?&lt;/p&gt;&#xA;" OwnerUserId="7954" LastActivityDate="2017-03-25T13:17:51.083" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12155" PostTypeId="1" CreationDate="2017-03-26T02:17:02.607" Score="17" ViewCount="871" Body="&lt;p&gt;I recently gave an exam problem that roughly 2/3 of the class made much more difficult by using the identity $1/x+1/y=(x+y)/xy$. Their work would have been much simpler if they hadn't done this. It seems like a symptom of the type of intellectual laziness that causes people to do things by pattern recognition rather than because they have a logical reason for doing them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When students are taught this identity, are they told a reason why they might want to use it? What would the reason be?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will ask them why they did it when I return the exam, but I suspect I'll get answers to the effect of &quot;that's how you do that kind of problem in algebra.&quot; Maybe this is similar to stuff like multiplying out an expression such as $(x+y+z)(a+b+c+d)$, which I think they do because they think of the notation as a set of orders that they need to carry out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this reflex similar to the one that compels them to rationalize the denominator in an expression such as $1/\sqrt{2}$? That one at least I think I understand the original reason for. If you're doing decimal arithmetic and performing division using long division (rather than log tables, a slide rule, or an electronic calculator), then rationalizing the denominator is a big win, so that would have sort of made sense before slide rules became common. Would the justification for habitually doing $1/x+1/y=(x+y)/xy$ be that division is a more expensive operation than multiplication if you don't have a calculator, slide rule, or log tables?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or maybe math teachers want answers like $1/\sqrt2$ in a canonical form so that it's easier to check students' work? (But this seems like an anachronism now that we have WeBWorK et al.) But I don't see how $(x+y)/xy$ is more canonical than $1/x+1/y$, and in general it's preferable not to have the same variable appearing more than once, e.g., if we want to reason about the dependence on that variable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(1) Why is this identity taught?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(2) Why do students use it in inappropriate situations, and what can we do to cure this problem?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2017-03-27T02:17:45.197" LastActivityDate="2017-04-12T12:27:49.763" Title="Mindless use of &quot;antisimplifications&quot; such as $1/x+1/y=(x+y)/xy$ and $1/\sqrt{2}=\sqrt{2}/2$" Tags="|algebra|" AnswerCount="4" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="12156" PostTypeId="2" ParentId="12155" CreationDate="2017-03-26T04:34:18.747" Score="12" Body="&lt;p&gt;For students at that level, routine algebraic calculations are an easy step, while thinking about what steps to take is quite difficult. So they'll often prefer to do familiar calculations - even long and tedious ones - and only turn to more extreme measures like stepping back to think about the big picture as a last resort. They also have an extreme aversion to backtracking (most of them think of trying an unsuccessful solution as a pure failure, rather than as progress towards finding the right one), so it's not surprising that, having backed themselves into a corner with an unhelpful algebraic manipulation, they push forward rather than trying something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One factor to consider is that students don't usually know how hard a problem should be: they can't tell the difference between &quot;this problem get too messy, I should try something else&quot; and &quot;this problem just needs some more effort&quot;. (This works in both directions: students persevering through awful calculations, and also students who redo an easy problem repeatedly because they can't believe the answer isn't an integer.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2017-03-26T04:34:18.747" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12157" PostTypeId="2" ParentId="12153" CreationDate="2017-03-26T05:41:40.433" Score="3" Body="&lt;p&gt;I am going to venture an answer here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try the calculator. I have had students whose minds were opened by less likely tools, and you never know. You shouldn't be restricted by thinking logically and linearly, viz. foundation prepares for scraping the sky. Neurology is more complicated than that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen my students' eyes opened by other &quot;tools.&quot; Also, the calculator may act as a much-needed crutch and allow this student to lean a bit. This comfort may just replace the comfort of finger counting, but it may help them reach a breakthrough moment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As was mentioned above, there are papers and books on the subject, but my gut says that the calculator should be included in your strategy. It will probably not be clear initially, or ever, if it &quot;helped&quot; or &quot;harmed,&quot; as you put it. But, maybe, learning to use the calculator (not as trivial as one might expect, especially for a struggling student) could help push this student to that next level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember, using a calculator requires one to mentally translate mathematics from one form to another, via mental effort.&lt;/p&gt;&#xA;" OwnerUserId="5410" LastActivityDate="2017-03-26T05:41:40.433" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12158" PostTypeId="1" CreationDate="2017-03-26T11:17:36.393" Score="0" ViewCount="633" Body="&lt;p&gt;I have always taught that, in case of two independent events P(1) and P(2), probability of P(1) AND P(2) = P(1) x P(2)&#xA;and that probability of P(1) OR P(2) is P(1) + P(2).&#xA;But consider this scenario:&#xA;Bag A contains 1 red and 7 blue balls.&lt;br&gt;&#xA;Bag B contains 1 black and 7 green balls.&#xA;Select a ball from each bag.&#xA;What is probability of choosing a blue ball from Bag A OR a green ball from Bag B?&#xA;Probability of choosing blue from Bag A = 7/8. Probability of choosing green from Bag B = 7/8.&#xA;So probability of choosing a blue ball from Bag A OR a green ball from Bag B = 7/8 + 7/8 = 14/8 which is more than 1 and clearly incorrect.&#xA;Where am I going wrong, please? &lt;/p&gt;&#xA;" OwnerUserId="7956" LastActivityDate="2017-03-26T11:27:06.237" Title="Probability AND/OR" Tags="|secondary-education|" AnswerCount="1" CommentCount="6" ClosedDate="2017-03-27T13:30:30.917" ContentLicense="CC BY-SA 3.0" />
  <row Id="12159" PostTypeId="2" ParentId="12158" CreationDate="2017-03-26T11:27:06.237" Score="7" Body="&lt;p&gt;$P(A\text{ and }B) = P(A)P(B)$   is correct for independent events.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P(A\text{ or }B) = P(A) + P(B)$   is correct for mutually exclusive events.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you were right about the first one.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, there is an &quot;inclusion–exclusion&quot; rule like this&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P(A\text{ or }B) = P(A) + P(B) - P(A\text{ and } B)$ for any events $A, B$.  They need not have a special relation like &quot;mutually exclusive&quot; or &quot;independent&quot;.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-03-26T11:27:06.237" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12160" PostTypeId="2" ParentId="12146" CreationDate="2017-03-27T02:56:20.867" Score="1" Body="&lt;p&gt;The answer isn't the same for every educator or every group (or every combination thereof). I find that the key is to consider myself constantly trying to find this answer. In other words, from day 1 to the final second of your time with these students, you are trying to find the &quot;holy grail,&quot; so to speak. ...or at least get closer... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, you are trying to reach every student, every small group of students (as they exist), and the larger group (class as a whole). Therefore, I try to connect with each individual and each group as authentically as possible (a little fake-it-til-you-make-it can help here) If it feels forced, it is... but don't give up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a strict lesson plan is authentic to you, then do that, but make sure you PAY ATTENTION TO EXACTLY &lt;em&gt;WHO&lt;/em&gt; YOU ARE TEACHING. Connect with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you manage to understand them better, you may also find ways to articulate the lessons differently, ultimately framing them in a more relatable way. A very simple example: you find a certain cross-section of students are soccer players, so all of your momentum lessons (physics application here!) have to do with soccer balls. Just be careful you don't get too hokey ;-) ...you don't want the students rolling their eyes too hard. But thy really do appreciate the effort if you're being sincere and totally authentic.&lt;/p&gt;&#xA;" OwnerUserId="5410" LastActivityDate="2017-03-27T02:56:20.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12161" PostTypeId="2" ParentId="12155" CreationDate="2017-03-27T07:13:40.627" Score="15" Body="&lt;p&gt;First, I would expect this identity &lt;em&gt;not to be taught&lt;/em&gt;, but to be obtained by students as a particular case of the method of reducing fractions to the same denominator, which is sufficiently useful not to need an explanation why it is taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, the only way I see to take care of this problem (and many others) would be to teach less standardized exercises, and induce more sense-making. In particular, we should teach &lt;em&gt;tactics&lt;/em&gt; and &lt;em&gt;strategy&lt;/em&gt; of manipulating an algebraic expression in order to reach a certain goal (if we want to find roots of an expression, let it factorized; if we want to sum several expressions, a distributed form might be better; etc.) The problem is that teaching this is very difficult and very time-consuming (we cannot stick to a few exercises, we need several similar-looking exercises that needs different methods, to be able to point out which subtle differences could be used to construct different strategies), and needs more efforts from the students. It may feel like much effort for little gain, compared to focusing on a few standard exercises and have student turn out decent tests. Seeing student fail again and again when we try to test that kind of understanding is pushing us all toward standardized exercises and tests, and make it yet more difficult to change our ways. We should still try hard, because ultimately it is understanding and sense-making have value incommensurable with having mastered a handful of meaningless methods, but we should be warned that we will feel like Sisyphus all along the way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A last remark (&lt;strong&gt;edit:&lt;/strong&gt; which actually &lt;em&gt;is&lt;/em&gt; a remark, and do not claim to be an answer about why &lt;em&gt;these&lt;/em&gt; students would use some form over another): concerning $\sqrt{2}/2$ versus $1/\sqrt{2}$, one important purpose of normalized forms is to help recognize when two numbers are equal. There is even a lot of theory of normal forms in various mathematical settings, where one expect to be able to prove that two objects are equal if and only if their normal forms are equal, and to devise algorithms to produce a normal form from an arbitrary expression. &lt;em&gt;Term rewriting&lt;/em&gt; is very much about formalizing this (but the same ideas appear in various subfields).&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2017-03-28T11:24:30.823" LastActivityDate="2017-03-28T11:24:30.823" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12162" PostTypeId="2" ParentId="11143" CreationDate="2017-03-28T13:16:30.670" Score="8" Body="&lt;p&gt;I have always found this to be an extremely hard proposition in the US system of grading (hw + 2-3midterms + final=Grade). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of the systems I've seen were imperfect, but some were better than others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find it is essential to correctly structure the final exam in the first instance. You want to make sure that the students performance on it actually in some way measures their mastery of the subject. This means making sure most things are covered and there is enough granularity to distinguish knowing almost nothing, knowing a decent amount and actually having mastered the material. This also means making sure that you cover most of the material, though we usually tend to be more focused on the latest material learned. This is supported both by the fact that the other material has been covered by midterms and the fact that the latest material tends to often presuppose knowledge of older material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regards to grading usually the Final will count more than the midterms but not overwhelmingly so. Something like 10% hw, 20% each midterm, 30% final.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is always the question what to do if someone seems do well in general and then flunks the final. This might be due to a random occurrence and should be considered. Usually we would do our best to give people the benefit of the doubt and not let their grade suffer too terribly but there's only so much you can do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The opposite situation where a person does bad on midterms and than great on the final is not great either since really you should care what they know at the end, but I tend to chalk that up to a lesson in consistency being important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Altogether I'm a much bigger fan of oral exams that can be retaken multiple times as is the custom in many EU institutions. As a professor you can actually gauge the knowledge of the student instead of guessing, a fluke day doesn't ruin your chances and it's much easier to just tell the student to pack up and come back when they've actually learned something.&lt;/p&gt;&#xA;" OwnerUserId="5896" LastEditorUserId="5896" LastEditDate="2017-04-14T11:15:31.057" LastActivityDate="2017-04-14T11:15:31.057" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12163" PostTypeId="1" AcceptedAnswerId="12605" CreationDate="2017-03-28T16:31:41.980" Score="7" ViewCount="647" Body="&lt;p&gt;I have taught College Algebra several times and will teach it again in the next semester. College Algebra, according to the the catalogue of my college, is described as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This course provides students an opportunity to gain algebraic&#xA;  knowledge needed in many different fields such as engineering,&#xA;  business, education, science, computer technology, and mathematics.&#xA;  Graphical, numerical, symbolic, and verbal methods support the study&#xA;  of functions and their corresponding equations and inequalities.&#xA;  Students will study linear, quadratic, rational, exponential,&#xA;  logarithmic, inverse, composite, radical, and absolute value&#xA;  functions; systems of equations and inequalities modeling applied&#xA;  problems; and curve fitting techniques. There will be extensive use of&#xA;  graphing calculators.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have been using College Algebra with Modeling &amp;amp; Visualization; Hardcover; Author ‑ Gary K. Rockswold as the textbook for the past few semesters. This book was not chosen by myself; it is the &quot;default value&quot; determined by other teachers when I started teaching. One issue that I have with this textbook (as with many other textbooks) is that newer edition will be available every several years (I think for this particular one, the sixth edition has been published), which increased my burden in preparation. I am searching for a &quot;steady&quot; textbook such as Serge Lang's &lt;a href=&quot;http://www.springer.com/us/book/9780387967875&quot; rel=&quot;noreferrer&quot;&gt;Basic Mathematics&lt;/a&gt;, the latest edition being published in 1986 and since the author has deceased, unlikely there will be a newer edition every few years. However, the Basic Mathematics may not be a suitable College Algebra textbook, based on the description above. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am here asking for a &quot;stable&quot; College Algebra textbooks that preferably satisfies the following conditions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It must fit the description of the course mentioned above.&lt;/li&gt;&#xA;&lt;li&gt;Newer editions will very unlikely be published frequently. This might imply that the textbook is very old, but this is okay with me as long as it is still available and fit the description.&lt;/li&gt;&#xA;&lt;li&gt;I do not need fancy textbooks. I prefer plain and cheap ones (similar to &quot;Serge Lang's &lt;a href=&quot;http://www.springer.com/us/book/9780387967875&quot; rel=&quot;noreferrer&quot;&gt;Basic Mathematics&lt;/a&gt;&quot; in style.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I wrote the question under the assumption that Serge Lang's &lt;a href=&quot;http://www.springer.com/us/book/9780387967875&quot; rel=&quot;noreferrer&quot;&gt;Basic Mathematics&lt;/a&gt; is not suitable for this course. But perhaps by teaching from selected chapters from this book, this book will be a tolerable choice for College Algebra?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2017-03-28T19:42:20.153" LastActivityDate="2017-07-16T05:02:48.967" Title="Looking for a &quot;Stable&quot; College Algebra Textbook" Tags="|undergraduate-education|algebra|textbooks|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12164" PostTypeId="2" ParentId="12163" CreationDate="2017-03-28T22:49:19.600" Score="9" Body="&lt;p&gt;I've recently started using the &lt;a href=&quot;https://openstax.org/details/college-algebra&quot; rel=&quot;noreferrer&quot;&gt;OpenStax College Algebra&lt;/a&gt; text for my courses. I can be pretty picky, but it's &quot;good enough&quot; for me to work with. Advantages include (1) it being openly accessible, (2) you can trust that students have access to it on day one, (3) cost is zero, (4) students can share it with others, (5) students have no reason not to keep it as a reference permanently (i.e., no resale at end of semester), (6) digital format allows one to put it up on an overhead when desired, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For your purposes, another advantage is that due to the open licensing, you never have to update if you don't want to. Keep a copy and just redistribute it forever if you like. There may be some changes, but there's no financial incentive for them to do so, so I don't expect them to be frequent or radical. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-03-28T22:49:19.600" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12165" PostTypeId="1" CreationDate="2017-03-29T17:16:11.590" Score="18" ViewCount="1603" Body="&lt;p&gt;A few times now I've found myself in a situation where I want to give a precalculus-level undergraduate student something to think about that's kinda fun and that's well outside of their coursework. But at the same time, I want to ask them a question they can figure out just by thinking about it, or pondering over it, rather than having to work out some calculations or necessarily write things down. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far my go-to question has been this one:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose you take a pen and mark five points on a ball. I claim that no matter where you draw those points, I can find a closed hemisphere of the ball that contains exactly four of those points. Is this true?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I like this question because there's very little you can write down, and the geometric set up is simple enough that a student should be able to play with it in their head (or use an actual ball). But I'm not perfectly happy with this question because I have to explain to them about what I mean by a &lt;em&gt;closed hemisphere,&lt;/em&gt; and I usually have to discuss some ideas of spherical geometry with them. But I feel like this gives them the impression that this problem would never have been tractable without my guidance. I want a problem that they evidently can think about and figure out without some necessary guidance, and possibly without having to write anything down or draw out any examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have better ideas for questions like this I can ask students?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I'm asking for such questions is that I want students to get over their fear of &quot;doing the wrong thing,&quot; so I want to give them problems where there is nothing to write down and they &lt;em&gt;have to&lt;/em&gt; mentally grapple with the question. This fear seems to spring up more easily if figuring out a question involves writing (maybe because I can see and judge their work?). They'll often sit there with a blank page and just look to me for guidance: they want me to tell them what the &quot;right thing to do&quot; is.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-03-13T16:29:04.190" LastActivityDate="2023-01-31T10:10:28.603" Title="Good Examples of Questions to have Students Ponder Over Without Paper" Tags="|undergraduate-education|problem-design|exercises|big-list|" AnswerCount="9" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12166" PostTypeId="2" ParentId="12165" CreationDate="2017-03-29T18:58:45.590" Score="10" Body="&lt;p&gt;A possibility, requiring one definition: What is a tiling of the plane with an infinite supply of congruent copies of a single tile (technically,&#xA;a &lt;em&gt;monohedral&lt;/em&gt; tiling). This can go as deep as you'd like, perhaps in stringing together several mini-sessions.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Can every triangle tile the plane? (&lt;em&gt;Yes&lt;/em&gt;.)&#xA;Form parallelograms, then argue that a parallelogram tiles the plane.&#xA;&lt;br&gt; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/lyLe3.png&quot; alt=&quot;TriTiles&quot;&gt;&#xA;&lt;br&gt; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;(Image from &lt;a href=&quot;http://demonstrations.wolfram.com/AnyTriangleCanTile/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematica Demo&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Can every quadrilateral tile the plane? (&lt;em&gt;Yes&lt;/em&gt;.)&#xA;This is not as easy to see. Easier if restricted to convex quads.&#xA;&lt;br&gt; &amp;nbsp; &amp;nbsp; &amp;nbsp;&lt;img src=&quot;https://i.stack.imgur.com/cGbvD.png&quot; alt=&quot;QuadTiles&quot;&gt;&lt;br&gt; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;(Image from &lt;a href=&quot;http://demonstrations.wolfram.com/AnyQuadrilateralCanTile/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematica Demo&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;br&gt;&#xA;Cited in &lt;a href=&quot;http://mathworld.wolfram.com/QuadrilateralTiling.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathworld&lt;/a&gt;, with this reference:&#xA;Wells, D. &lt;em&gt;The Penguin Dictionary of Curious and Interesting Geometry&lt;/em&gt;. London: Penguin, pp. 177-179, 208, and 211, 1991.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Can every pentagon tile the plane? (&lt;em&gt;No&lt;/em&gt;.) The regular pentagon cannot. This is at least easy to see if the tiling is edge-to-edge.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You get to mention a recent advance at the frontiers of math:&#xA;&lt;br&gt; &amp;nbsp; &amp;nbsp; &amp;nbsp;&lt;img src=&quot;https://i.stack.imgur.com/PZGYj.png&quot; alt=&quot;PentTiling&quot;&gt;&lt;br&gt; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;(Image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Pentagonal_tiling&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You get to mention &lt;a href=&quot;https://cstheory.stackexchange.com/q/32538/337&quot;&gt;an unsolved problem&lt;/a&gt;, beyond the frontier: Is there an algorithm&#xA;for deciding if a given, single tile can tile the plane?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt; (&lt;em&gt;11 Jun 2017&lt;/em&gt;).&#xA;The pentagon tiling problem is just settled: The list of $15$ types is&#xA;complete. See the &lt;a href=&quot;https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Natalie Wolchover article in &lt;em&gt;Quanta&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-12-14T01:22:37.923" LastActivityDate="2017-12-14T01:22:37.923" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12167" PostTypeId="2" ParentId="11779" CreationDate="2017-03-29T19:09:30.013" Score="4" Body="&lt;p&gt;Notice through a geometric argument that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_0^af(x)\ dx=\int_0^af(a-x)\ dx$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_0^af(x)\ dx=\frac1b\int_0^{ab}f(x/b)\ dx$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second which may follow through a squeezing of the integrals or Riemann sums.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now compute the following integrals:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_0^{\pi/2}\frac{\sin^\pi(\theta)}{\sin^\pi(\theta)+\cos^\pi(\theta)}\ d\theta$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_0^{\pi/2}\ln(\sin(\theta))\ d\theta$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the first integral:&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; \begin{align}\int_0^{\pi/2} \frac{\sin^\pi(\theta)}{\sin^\pi(\theta)+\cos^\pi( \theta)}\ d\theta &amp;amp;=\int_0^{\pi/2}\frac{\sin ^\pi( \frac\pi2-\theta)}{\sin^\pi(\frac\pi2-\theta) +\cos^\pi(\frac\pi2-\theta)}\ d\theta\\&amp;amp; =\int_0^{\pi/2}\frac{\cos^\pi( \theta)}{\sin^\pi(\theta)+\cos^\pi( \theta)}\ d\theta\\ \therefore I+I=\int_0^{\pi/2} \frac{\sin^\pi(\theta) + \cos^\pi(\theta)}{\sin^\pi(\theta)+\cos^\pi( \theta)}\ d\theta&amp;amp;=\int_0^{\pi/2}1\ d\theta = \frac\pi2\\\therefore\int_0^{\pi/2} \frac{\sin^\pi(\theta)}{\sin^\pi(\theta)+\cos^\pi( \theta)}\ d\theta &amp;amp;=\frac\pi4\end{align}&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For the second integral:&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; \begin{align} \int_0^{\pi /2} \ln(\sin(\theta))\ d\theta = \int_0^{\pi /2} \ln(\cos(\theta))\ d\theta\end{align} \begin{align} \therefore I+I=\int_0^{\pi /2}\ln(\sin(\theta))+\ln(\cos(\theta)) \ d\theta &amp;amp;= \int_0^{\pi/2} -\ln(2)+\ln(2\sin(\theta)\cos(\theta)) \ d\theta \\&amp;amp;= -\frac\pi2\ln(2) +\frac12\int_0^\pi \ln(\sin(\theta))\ d\theta \\&amp;amp;= -\frac\pi2\ln(2) +I\end{align} \begin{align}\therefore \int_0^{\pi /2} \ln(\sin(\theta))\ d\theta =-\frac\pi2\ln(2)\end{align}&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="6220" LastEditDate="2017-03-29T20:42:02.213" LastActivityDate="2017-03-29T20:42:02.213" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12168" PostTypeId="2" ParentId="12165" CreationDate="2017-03-30T00:33:05.000" Score="2" Body="&lt;p&gt;Ask them to give you a number between $0$ and $100$. The number they give you must be as close as possible to $2/3$ of the average of all the class' numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We could argue that this is more psychology than math, but it fits the no wrong answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Guess_2/3_of_the_average&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; for some history of this.&lt;/p&gt;&#xA;" OwnerUserId="805" LastActivityDate="2017-03-30T00:33:05.000" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12169" PostTypeId="2" ParentId="12165" CreationDate="2017-03-30T02:09:16.890" Score="5" Body="&lt;p&gt;Perhaps logic puzzles would work in this case. Some classic examples are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;blockquote&gt;&#xA;  &lt;p&gt;You're traveling along a road and arrive at a fork. Two guides are posted, but one always lies and the other always tells the truth, but you don't know which one is which. What one question can you ask to find out which path you should take?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;blockquote&gt;&#xA;  &lt;p&gt;Three boxes are labeled &quot;Apples&quot;, &quot;Oranges&quot; and &quot;Apples and Oranges&quot; but each box is labeled incorrectly. How can you tell what each box really is by only picking one fruit from one box?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;blockquote&gt;&#xA;  &lt;p&gt;You know you have ten white socks and ten black socks in a drawer. In the darkness of the morning, how many do you have to pull blindly out before you are guaranteed to have a pair? What if there's only two white socks and one-hundred black socks? What if there are also some grey socks? What if you are an alien and need three matching socks?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What I like about these is that they're pretty easy to understand but nontrivial to solve and guessing is worthless because the real answer comes with a reason/proof/explanation.&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="5220" LastEditDate="2017-03-31T17:34:06.183" LastActivityDate="2017-03-31T17:34:06.183" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12171" PostTypeId="2" ParentId="7920" CreationDate="2017-03-31T02:17:42.787" Score="2" Body="&lt;p&gt;The answer might be simpler than this. For whatever reason, their prior teachers may have required them the produce &quot;exact&quot; answers, leaving pi or square roots. I'm not going to contradict student's current teachers if that's what they want, but I do remind them that stuff isn't measured that way. You buy things by the lb, ft, ton, etc, and these measurements are to a nearest decimal, 7.48 instead of the square root of 56. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-03-31T02:17:42.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12174" PostTypeId="2" ParentId="12165" CreationDate="2017-03-31T17:12:13.803" Score="1" Body="&lt;p&gt;Here's another one that I like.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose you and a friend are sitting at a circular table and you each have a large stack of pennies, so you start to play this game where you take turns placing one penny flat on the table. At each turn you may not place a penny over any other penny (because the pennies must be &lt;em&gt;flat&lt;/em&gt; on the table), no pennies may lay over the edge of the table, and once a penny has been placed it cannot be moved. The last person to be able to place a penny on the table wins.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It's your turn to go first. Do you have a winning strategy?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(And then after the student answers the first question) What if the table isn't circular? On what other shaped tables will your strategy work?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There's isn't an obvious procedure to solve this problem, and since the situation just involves placing pennies on a table (so placing little circles into a bigger circle), a student should be able to fiddle with the problem in their head easily. A student should at least be able to come up with a response that will prompt a discussion without having to write something down first.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-02-22T03:52:35.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12178" PostTypeId="1" CreationDate="2017-04-01T22:40:55.803" Score="5" ViewCount="136" Body="&lt;p&gt;The grading scheme of almost all of the mathematics courses in my university is such that a substantial amount (e.g., $ \geq 20 \ \text{percent}$) of a student's grade is based on homework assignments. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In most upper-level mathematics courses, the grade is based almost exclusively on homework (indeed, there are some courses in which the mark is based entirely on homework). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there is a clear problem with this: nowadays, any student can make a math.stackexchange account and receive an answer to any mathematical question at the undergraduate (and even graduate) level within hours at most and reproduce whatever answers he or she receives on his or her homework assignment. Thus, in principle, a student can pass the course, or even get an $A$, without understanding anything. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, in theory, homework questions are frowned upon on the math.se website, but in practice, an answer is almost always given.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my question is, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there any solution to this problem besides relegating the role of homework in university-level mathematics? Has the existence of this website affected how instructors choose to allocate marks in mathematics courses?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6155" LastActivityDate="2017-04-01T22:40:55.803" Title="Math.stackexchange and Homework" Tags="|homework|" AnswerCount="0" CommentCount="6" ClosedDate="2017-04-02T07:10:24.793" ContentLicense="CC BY-SA 3.0" />
  <row Id="12180" PostTypeId="1" AcceptedAnswerId="12181" CreationDate="2017-04-04T17:59:33.557" Score="5" ViewCount="153" Body="&lt;p&gt;Calc 2 (integration) courses often begin by introducing the idea of approximating the area under curves by rectangles, drawing pictures like this one of $y=\sqrt{x}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ftLRi.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ftLRi.png&quot; alt=&quot;Integral approximation with rectangles&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here we can approximate the area under the curve by the sum (yellow rectangles):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$0.2 (\sqrt{0.2} + \sqrt{0.4} + \sqrt{0.6} +\sqrt{0.8} +\sqrt{1}) \approx 0.75 $ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing these sums by hand gets tiresome very quickly, so when teaching this material I often resort to a spreadsheet for calculations. The drawback of using a spreadsheet is that it can't easily draw pretty pictures like the one above. I imagine this would be a good use case for Mathematica or Matlab, but I'd prefer not to use such expensive software for a brief topic like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there free or open source software that would allow me to demonstrate and explore, both graphically and numerically, approximating integrals by finite sums such as these? I don't mind if a bit of programming is required on my part. It would be nice it was easy to share my workbook/code with students.&lt;/p&gt;&#xA;" OwnerUserId="182" LastActivityDate="2017-04-06T17:07:32.420" Title="Notebook software for exploring integral approximation with finite sums?" Tags="|calculus|software|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12181" PostTypeId="2" ParentId="12180" CreationDate="2017-04-04T18:41:13.940" Score="8" Body="&lt;p&gt;I do this kind of thing using &lt;a href=&quot;http://www.sagemath.org&quot; rel=&quot;noreferrer&quot;&gt;SageMath&lt;/a&gt; every time I teach it.   In this case, there are even pre-made interactive things for this.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Here is one for &lt;a href=&quot;https://wiki.sagemath.org/interact/calculus#Numerical_integrals_with_the_midpoint_rule&quot; rel=&quot;noreferrer&quot;&gt;the midpoint rule&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Here is one &lt;a href=&quot;https://wiki.sagemath.org/interact/calculus#Numerical_integrals_with_various_rules&quot; rel=&quot;noreferrer&quot;&gt;for various rules&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can even embed them in webpages, use them in the &lt;a href=&quot;http://sagecell.sagemath.org/?z=eJytVlFv2zYQfg-Q_3BwkYlKmFh281RYwR66DgO6Ac3Ql8quQFmUzYGSHJKOZRj57zuSsqy48RJsM2CYOt59_Hjf3cmPTJGgCcLzs59FZbhic3N-lvMCSpGvajSRAmIQ1Wpt0qxuCG6xtTRo06IizfdxCFcwpmC2K47GP-9Den4G-HFoj0zGilULnmopcq5IRGFkvxRaoNiabkMKkmVcxoPf2rDBHqdalxlXaV2kep3l4lFoUVfanu8REWr8HPG2A_vDxUJdAFLnusPklb9bqtbSstZc8rmpFUmC39trBxSCz7xwv_disXSLr6sVV26n3uBiRqFS9UbHo-7EX1pksMiDMPxwfuaPZBQyPKlkK_LlC-2yE_pdm89Hq4QJWkuxruZoLJg26ZxJhJecFKSJDebq_uMnav11nJhZG5A36P7tG8muWTjE35fz5p19RGOzmMz8w7b_UNQKHpAjOO1OQX3wzk7s4jincZu-npM_84ZhEqucPFwi4ytg4cGByxdxOklexcJv3gzHr0N6Rd-G9zqaL4tjNKueqPJU1qhfWrJGlOuSWF0p7A_oVm0uktHsFKc38PBV-QoPUf0_PHzBuDpFmyuZxpYMVlU_1i8Vtpf1_1Wx1VLMNQn_ZaE1JaJ0XIdjy7a3i5uNTh56BLZo2j43eTJXMUhRcZIkTXltoWg0o91669ZXP6yjGTa9WmTzWiL1GHD8RDQKwxfQ_fQkDYVt-A8xKEhqSeKvHYWnpGI0C5No3-xYTT6INcdB_To7BPmwd7A0pSTBZPn-DmcjVwJD3CxaKCY1bIRZglnybvi7MTYZovt-LullvSErWZvDCaiBuzRFdVDK2JZTgzxwleHlvc1d0z45u-PfIrbjC185cxyOTV_td7BSlkVdyS12Y8E3kIuFwPziSF_hoa5Ano2hBiYw4te3R12guFmrCqK-Ts5yz5n8JLjMyTgKu-LW6zK1qUKqgwsN03leG5hOJU60BKxhquwEmQ3gAkiOEgeYhODmL8wZSWBQkAsd2j17I-GbQ-ybYy989Z_O0EYRj96D3_bgu0O8yimrNE4H9y73cqfdXicltbo44W2zjqPIVk_7ytRGlMzw_ACETXhpM4VsXIe9sZsP5YgKGrNNncbEVaYKgsDvTTAA5pJpHQ_w3OXgrr3RNOMLUe2YFIvq8mlvtHNwx56-77In2BV2IE3xD0HzBD8hUZfM6d7VarsT8QidL7T3ToXzn37k0jB4KeqU4QamOBQ7NpMh0r7DS8Dz8jt8UMtjTeiJ_zi0K0PaFQv9QYgwDP8GSinNpg==&amp;amp;lang=sage&quot; rel=&quot;noreferrer&quot;&gt;Sage cell server&lt;/a&gt; (try it!), as well as in notebooks locally or online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: Geogebra should also be able to make very nice things for this, but I haven't used them in quite some time and their license is no longer as free as in speech as it used to be.  But it's certainly also an appropriate tool for this, certainly.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-04-04T18:41:13.940" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12182" PostTypeId="2" ParentId="12180" CreationDate="2017-04-05T19:16:48.880" Score="3" Body="&lt;p&gt;In Geogebra you can use the commands UpperSum[function, x-start, x-end, n] and LowerSum[function, x-start, x-end, n] to get both a visualization and the computed area of the rectangles. You can also add a slider for $n$, so students can interactively see how the visualization and computed area get closer to the desired area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EVz2T.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EVz2T.png&quot; alt=&quot;Geogebra UpperSum screenshot&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These commands however require you to work with intervals of the form $[x_0+k \Delta x, x_0+(k+1)\Delta x]$. If you need to, you can do some programming yourself (or let your students do it) to get different intervals.&lt;/p&gt;&#xA;" OwnerUserId="493" LastEditorUserId="182" LastEditDate="2017-04-06T17:07:32.420" LastActivityDate="2017-04-06T17:07:32.420" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12183" PostTypeId="2" ParentId="12180" CreationDate="2017-04-06T02:55:47.740" Score="3" Body="&lt;p&gt;Desmos has the capability to do this and &lt;a href=&quot;https://www.desmos.com/calculator/ej5s4ecohf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; is a worksheet that has done this pretty successfully.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-04-06T02:55:47.740" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12184" PostTypeId="2" ParentId="36" CreationDate="2017-04-07T16:10:41.830" Score="4" Body="&lt;p&gt;My students and I have been using&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.monroecc.edu/faculty/paulseeburger/calcnsf/CalcPlot3D/&quot; rel=&quot;nofollow noreferrer&quot;&gt;CalcPlot3D&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It supports most of the visualization you would want in a multivariable calculus course:  3D graphs, vector fields, contour maps, parameterized curves and surfaces, etc.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorDisplayName="user7990" LastEditDate="2020-08-30T20:17:26.657" LastActivityDate="2020-08-30T20:17:26.657" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12185" PostTypeId="1" CreationDate="2017-04-07T16:17:26.113" Score="46" ViewCount="4453" Body="&lt;p&gt;I asked this question once on math.se, but don't follow the link unless you want to risk biasing your own response: &lt;a href=&quot;https://math.stackexchange.com/questions/444696/how-to-respond-to-solve-this-equation-in-a-basic-algebra-class&quot;&gt;https://math.stackexchange.com/questions/444696/how-to-respond-to-solve-this-equation-in-a-basic-algebra-class&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is really a question that belongs here though. Maybe I posted it there before I knew about this site. I'd like to see responses from this community. So here it is, restated, somewhat edited.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine yourself teaching a basic algebra class: maybe to grade schoolers or to high schoolers, or in my case, to adults ages 18 and up in a community college. You will encounter &quot;problems&quot; like the following, where for now I am intentionally leaving out words:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$2x+3=6−x$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;answer&quot; to a question like this somehow communicates that $1$&#xA;is the only solution, that $x$ needs to equal $1$, that the solution set is $\{1\}$, or $\{x\mid x=1\}$, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of my colleagues feel that if the task was to &quot;solve this equation&quot;, that &quot;$x=1$&quot; is not an acceptable final response from a student. They say that &quot;$x=1$&quot; is an &quot;equivalent equation&quot; to the original equation, because it has the same solution set. They say that to &quot;solve this equation&quot;, to the exclusion of other ways a student might respond, is to write a set as part of an English statement. They are happy with: &quot;The solution is $1$&quot;, &quot;The solution set is $\{1\}$&quot;, or &quot;The solution set is $\{x\mid x=1\}$&quot;. But to them, &quot;$x=1$&quot; or &quot;The solution to the equation is that $x=1$&quot; cannot count as fully correct responses. Answers like these may not get full credit on their exams, and always get commentary on written homework. These colleagues would say the student has not learned the difference between an &quot;equivalent equation&quot; and the solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I support answers of the form &quot;$x=1$&quot;, even better as &quot;The solution to the equation is that $x=1$&quot;, because it is a whole statement that explicitly states &lt;em&gt;what&lt;/em&gt; is equal to $1$, instead of relying on the implication that since $x$ is the only variable, that is what we must intend for the $1$ to take the place of. I counter the idea that &quot;$x=1$&quot; has to be interpreted as an &quot;equivalent equation&quot; by saying that sometimes &quot;$x=1$&quot; is an assertion/assignment rather than an equation; I'm asserting that $x$ has to equal $1$ for the equation to be true. I point to any number of programming languages, where you have one symbol for testing equality, and another for assigning a value (sometimes &quot;==&quot; and &quot;=&quot;, sometimes &quot;=&quot; and &quot;:=&quot;) so these are two different ideas but mathematics tends to overload them onto one symbol, &quot;=&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question to this community is how do you feel about formatting answers for questions like this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has important (to me) implications. For one, I code problems for WeBWorK, and I strive to make the experience simulate the pencil-and-paper experience with feedback as close as possible to how a human teacher/grader would respond. I can do a lot with responding conditionally to this format or that, but I need to know what formatting people want most. Second, I am working on a massive OER for content at this level (&lt;a href=&quot;http://spot.pcc.edu/math/orcca/orcca.html&quot; rel=&quot;noreferrer&quot;&gt;draft, ever in progress&lt;/a&gt;), and its expository sections and WeBWorK homework should be consistent with each other and format answers the &quot;right&quot; way. It's most important that I get my own colleagues to be OK with how this OER treats this issue, but I also have an eye toward the broader community.&lt;/p&gt;&#xA;" OwnerUserId="767" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:36.660" LastActivityDate="2018-09-19T21:57:08.420" Title="How to respond to “solve this equation” in a basic algebra class" Tags="|algebra|textbooks|" AnswerCount="4" CommentCount="19" ContentLicense="CC BY-SA 3.0" />
  <row Id="12186" PostTypeId="2" ParentId="12185" CreationDate="2017-04-07T16:56:43.377" Score="71" Body="&lt;p&gt;This is a really interesting question, because similar issues---the question of how demanding to be about formatting of answers---come up a lot, at all levels, and the answers aren't always straightforward.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case I think your colleagues are straightforwardly wrong, for the following reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They're wrong about what students who write &quot;x=1&quot; think.  This is easily testable: if students actually can't tell the difference between an equivalent equation and a solution, they should think that, say, $x=3-2$ is an equally acceptable solution.  I'm comfortable predicting that they don't.  In fact, the notation &quot;x=1&quot; probably &lt;em&gt;mostly&lt;/em&gt; means &quot;1 is the solution&quot; to them, and they'd have to stop to think about the fact that it's also an algebraic equation they could manipulate.&lt;/li&gt;&#xA;&lt;li&gt;They're wrong about what most students who write &quot;the solution set is {1}&quot; think.  It's very unlikely that students who are writing that are actually expressing a meaningfully different thought than the ones who don't; it's much more likely that they've memorized the particular format that this teacher is demanding.&lt;/li&gt;&#xA;&lt;li&gt;They're wrong about convention.  Writing &quot;the solution is x=1&quot; is standard throughout mathematics, and students will see it in later classes (including weirder things like &quot;x=2,-3&quot; as the solution to a quadratic, which, while notationally weird, is easily understood and widely used, even if &quot;$x\in \{2,-3\}$&quot; would theoretically be preferred).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Putting those last two together means that I think they're actively doing harm by being doctrinaire about this.  They're not wrong that these are different concepts, but they're probably not succeeding at teaching students to distinguish them.  Instead, what the students are probably learning is that math is mostly about doing arbitrary steps to please their teacher, and that you can get a problem totally right and lose points anyway because your teacher made up some rules.  (It doesn't help that, in this case, the rules really are made up, in that no one else, including other mathematicians, actually follows them.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2017-04-07T16:56:43.377" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12187" PostTypeId="2" ParentId="12185" CreationDate="2017-04-08T07:58:29.303" Score="6" Body="&lt;p&gt;These &quot;colleagues&quot; carry pedantry to asinine levels. How many of your kids could/would &quot;solve&quot; the problem: $x = 1$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a lot of problems. First is the expression of the problem isn't explicit. It doesn't state what domain x may come from. It's &lt;em&gt;not&lt;/em&gt; necessarily true that x=1 even &lt;em&gt;has&lt;/em&gt; a solution. But I see little value in teaching this subtle point to adult learners (unless they are going quite a bit further in their math education than algebra and differential calculus). I suspect that some of this &quot;theory&quot; is taught in K-12 education, but for adult learners? Why?!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When someone gives me a test which says &quot;Solve the following equations&quot; and then there are a list of equations, it does not require the explicit directions for me to reduce the equations to such a form where the values, if any, for which the equations are true are made explicit. (Technically, one doesn't &quot;solve&quot; an equation, one solves a problem (and the problem may be to determine the values for which the equation is true, but may also include other things, such as domain, range, etc.) So, would it hurt to explain this? I don't think so, but I'd bet it will be useless for 99% of your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which gets me back to the question that you didn't address, but requires an answer prior to answering your question: what is the purpose of not only your class, but the educational tracks these students are on? Is learning silly pedantry concerning formal mathematics likely to be useful? It seem to me so blindingly &lt;em&gt;obvious&lt;/em&gt; that this is a complete waste of time and energy, and demotivating as well, as to beg the question: do you really have to ask?&lt;/p&gt;&#xA;" OwnerUserId="7998" LastEditorUserId="2083" LastEditDate="2017-04-13T16:02:39.473" LastActivityDate="2017-04-13T16:02:39.473" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12188" PostTypeId="2" ParentId="12185" CreationDate="2017-04-08T12:57:04.493" Score="27" Body="&lt;p&gt;Ask your pedantic colleagues to reconcile their formatting expectations with the context&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;solve $F=m\cdot a$ for $a$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would they prefer to see just $\frac{F}{m}$, or maybe this expression jazzed up with set brackets somehow? This formatting of the answer looks jarring to me. I think most educators and scientists expect to see the answer presented as $a=\frac{F}{m}$.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2017-04-08T12:57:04.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12189" PostTypeId="2" ParentId="12155" CreationDate="2017-04-11T19:51:40.720" Score="3" Body="&lt;p&gt;I certainly agree that there is a lot of &quot;mindless&quot; emphasis on various aspects of teaching algebra. However, to the extent that the mathematics community wants to encourage students to have a more &quot;playful&quot; way of looking at mathematics and to see the way that mathematics is the science of patterns, perhaps this &quot;identity&quot; could be used opportunistically to look at the phenomenon of Egyptian Fractions - the issue of writing p/q (p, q positive integers) as the sum of fractions of the form 1/x with distinct denominators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So perhaps a student who added (1/3) + (1/10) in the spirit of (1/x) + (1/y) and found the answer to be 13/30 might find looking at (1/3) + (1/10) = 13/30 a source of &quot;new&quot; mathematical thoughts:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a. Can 13/30 be written in other ways as the sum of two unit fractions? as the sum of three unit fractions? What is the smallest denominator of unit fractions with distinct denominators that sum to 13/30?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b. Can any fraction p/q be written as the sum of unit fractions with distinct denominators?  (Tradeoff between number of fractions involved and the largest denominator that appears?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c. Study the solutions to: (1/x) + (1/y) = 1/z (e.g. (1/3) +(1/6) = 1/2)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So lots of current curriculum is not optimal, but if teachers have the freedom to do a bit &quot;extra&quot; and have a sufficient knowledge base perhaps some additional &quot;positive&quot; outcomes are possible.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2017-04-11T19:51:40.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12190" PostTypeId="2" ParentId="12155" CreationDate="2017-04-11T22:33:05.693" Score="4" Body="&lt;p&gt;I think that  &quot;rationalizing numbers&quot; (clearing the denominators of radicals) is an anachronism:  it was useful at one time, because if you want to evaluate $\frac{1}{\sqrt{2}}$, it is much easier to perform the division $\frac{\sqrt{2}}{2}$ if you are obtaining $\sqrt{2}$ by a table lookup.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it was a useful skill at one time, but now it is just being passed along by rote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I feel this way about a lot of math.  For instance, I think most of what we teach students in a &quot;Calc 2&quot; course is similarly anachronistic.  &lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2017-04-12T12:27:49.763" LastActivityDate="2017-04-12T12:27:49.763" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12191" PostTypeId="1" AcceptedAnswerId="12193" CreationDate="2017-04-11T23:40:24.210" Score="7" ViewCount="947" Body="&lt;p&gt;I am going to teach a 400-level topics class on knot theory at an American, mid-sized, public university. Prerequisites include multivariable calculus, linear algebra, and a proof course, but no exposure to topology or abstract algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know of resources for such a course? Notes, exercises, etc?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I have so far:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Books: Livingston, Adams, Crowell/Fox, Prasolov&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Visualization aid: &lt;a href=&quot;http://www.knotplot.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.knotplot.com/&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="98" LastEditorUserId="376" LastEditDate="2020-06-01T12:01:45.423" LastActivityDate="2020-06-01T12:01:45.423" Title="Resources for undergraduate knot theory" Tags="|undergraduate-education|resource-request|topology|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12192" PostTypeId="2" ParentId="12191" CreationDate="2017-04-12T02:20:15.773" Score="7" Body="&lt;p&gt;You may find the &lt;a href=&quot;http://doc.sagemath.org/html/en/reference/knots/sage/knots/knot.html&quot; rel=&quot;noreferrer&quot;&gt;SageMath knot and links capabilities&lt;/a&gt; useful for computation and visualization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://katlas.org/wiki/Main_Page&quot; rel=&quot;noreferrer&quot;&gt;Knot Atlas&lt;/a&gt; might be a bit more comprehensive than you are looking for but is certainly a reference to be quite aware of.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-04-12T02:20:15.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12193" PostTypeId="2" ParentId="12191" CreationDate="2017-04-12T08:14:24.160" Score="6" Body="&lt;p&gt;I don't know much about knot theory but I know that Meike Akveld taught knot theory at both high school and university level. Here's a bibliography of one of her courses at ETH Zürich:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www2.math.ethz.ch/education/bachelor/lectures/fs2015/math/knot/bibliography_FS2015.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www2.math.ethz.ch/education/bachelor/lectures/fs2015/math/knot/bibliography_FS2015.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It includes Englisch and German books both for high school and university level. I only list English books for university level:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;General books about knots - accessible to (under-)graduate students:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;M.A. Armstrong, &lt;em&gt;Basic Topology&lt;/em&gt;, Undergraduate Texts in Mathematics, Springer-Verlag, 1983 - Chapter 10 is devoted to knots.&lt;/li&gt;&#xA;  &lt;li&gt;G. Burde, H. Zieschang, &lt;em&gt;Knots&lt;/em&gt;, Walter de Gruyter &amp;amp; Co., Berlin, 1985.&lt;/li&gt;&#xA;  &lt;li&gt;A.Kawauchi, &lt;em&gt;A Survey of Knot Theory&lt;/em&gt;, Birkhäuser Verlag, Basel, 1996.&lt;/li&gt;&#xA;  &lt;li&gt;W.B.R. Lickorish, An Introduction to Knot Theory, Springer-Verlag. New York, 1997.&lt;/li&gt;&#xA;  &lt;li&gt;C. Livingston, &lt;em&gt;Knotentheorie für Einsteiger&lt;/em&gt;, Vieweg, 1995 (also available in English).&lt;/li&gt;&#xA;  &lt;li&gt;K. Murasugi, &lt;em&gt;Knot Theory &amp;amp; Its Applications&lt;/em&gt;, Chapters 5 and 6, Birkhäuser Boston, 2008.&lt;/li&gt;&#xA;  &lt;li&gt;J. Roberts, &lt;em&gt;Knots Knotes&lt;/em&gt;, unpublished lecture notes, 2010, &lt;a href=&quot;http://math.ucsd.edu/~justin/Roberts-Knotes-Jan2015.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://math.ucsd.edu/~justin/Roberts-Knotes-Jan2015.pdf&lt;/a&gt;.&lt;/li&gt;&#xA;  &lt;li&gt;D. Rolfsen, &lt;em&gt;Knots and Links&lt;/em&gt;, AMS Chelsea Publishing, 2003.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are still old exercises and solutions from the class available online &lt;a href=&quot;https://www2.math.ethz.ch/education/bachelor/lectures/fs2015/math/knot/uebungen.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. They usually get deleted a year or two after the class, so I'd download them just in case. Plus, you can find some hand-drawn examples on the &lt;a href=&quot;https://www2.math.ethz.ch/education/bachelor/lectures/fs2015/math/knot.html&quot; rel=&quot;noreferrer&quot;&gt;lecture website&lt;/a&gt; itself.&lt;/p&gt;&#xA;" OwnerUserId="493" LastActivityDate="2017-04-12T08:14:24.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12194" PostTypeId="1" CreationDate="2017-04-12T14:15:55.553" Score="1" ViewCount="138" Body="&lt;p&gt;I would like to learn the subject &lt;strong&gt;'complex numbers'&lt;/strong&gt;. My goal is to study this on my own. Are there any good tips, books, sites to study this?&lt;/p&gt;&#xA;" OwnerUserId="8031" LastActivityDate="2017-04-13T01:39:39.570" Title="Complex numbers" Tags="|self-learning|complex-numbers|" AnswerCount="2" CommentCount="7" ClosedDate="2017-04-13T08:16:18.410" ContentLicense="CC BY-SA 3.0" />
  <row Id="12195" PostTypeId="2" ParentId="12194" CreationDate="2017-04-12T20:46:19.033" Score="2" Body="&lt;p&gt;May I recommend what I did in the MESE question &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/a/4501/511&quot;&gt;Complex analysis (Applied versus pure)&lt;/a&gt;&#xA;(and in this earlier MSE question, &quot;&lt;a href=&quot;https://math.stackexchange.com/a/85244/237&quot;&gt;Visual research problems in geometry&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/CXmuw.jpg&quot; alt=&quot;Needham cover&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;img src=&quot;https://i.stack.imgur.com/HpbKO.png&quot; alt=&quot;Figure 13&quot;&gt;&lt;hr /&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.077" LastActivityDate="2017-04-12T20:46:19.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12196" PostTypeId="2" ParentId="12194" CreationDate="2017-04-13T01:39:39.570" Score="2" Body="&lt;p&gt;I suspect that the question may be starting from the point of &quot;what are complex numbers&quot; (far below the level of complex analysis). For that purpose, &lt;a href=&quot;https://en.wikipedia.org/wiki/Complex_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia: Complex Number&lt;/a&gt; is a perfectly acceptable article with plenty of information. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're more of a video-type person then &lt;a href=&quot;https://www.khanacademy.org/math/algebra2/introduction-to-complex-numbers-algebra-2/the-complex-numbers-algebra-2/v/complex-number-intro&quot; rel=&quot;nofollow noreferrer&quot;&gt;Khan Academy: Intro to Complex Numbers&lt;/a&gt; might be satisfying for your purposes. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-04-13T01:39:39.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12197" PostTypeId="2" ParentId="36" CreationDate="2017-04-13T05:18:07.390" Score="4" Body="&lt;p&gt;Sounds like you want them to learn something, so I would recommend the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Start students working with 3D plots in &lt;a href=&quot;https://geogebra.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;GeoGebra&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Set up &lt;a href=&quot;https://cocalc.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;CoCalc&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Introduce CoCalc to students having them create markdown documents pasting images of 3D plots and writing up problems, maybe using some LaTeX if you're comfortable.&lt;/li&gt;&#xA;&lt;li&gt;Ease off the GeoGebra and move to Jupyter notebooks where you will move back and forth between using Python's matplotlib 3D plotting and R's ggplot.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7003" LastEditorDisplayName="user7990" LastEditDate="2020-08-31T01:44:19.167" LastActivityDate="2020-08-31T01:44:19.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12198" PostTypeId="2" ParentId="11143" CreationDate="2017-04-13T19:47:16.710" Score="3" Body="&lt;p&gt;Assuming the comprehensive exam is well-constructed, it should be the most valid data there is, and therefore the easiest to justify weighting highly. As many ways as students have to game the system, I trust a good exam much more than I do homework to tell me what a student really knows. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When students do well on earlier tests or quizzes and then crash and burn on the final, that tells me one of two things. One possibility is that students were using short-term methods such as cramming to achieve the temporary illusion of mastery. The other is that they have previously seen only one unit's worth of material at a time, so they've never learned to select the right concept from an entire semester's material. Either way, their success on earlier tests does not indicate that they truly understand the material at the end of the course. Therefore, I see little to entitle them to a good grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, I also believe that some traditional course structures inadvertently enable these weaknesses. That means that before we tell students that only what they know at the end truly matters, we need to do a better job of preparing them for that final comprehensive exam. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth, the way I have chosen to address this is to make all of my tests contain a lot of material from earlier units. That means that students don't forget as much since they keep using old concepts, and they spend the entire course learning to select the right ideas from more than just the most recently learned concepts. It also means that the final is really just a slightly larger version of what they've been doing all along. That makes it much less likely that conscientious students will do poorly on the final just because they didn't know what to expect. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few years ago, I abandoned the traditional approach of testing one unit at a time (with a comprehensive final) in favor of the approach I just described, and I've been very happy with the results. The final exam scores now match students' previous test scores pretty closely. Under the previous system, all but the best students' exam scores tended to lag the unit test scores by at least a letter grade. (If it matters to you, I teach high school calculus and pre-calculus.)&lt;/p&gt;&#xA;" OwnerUserId="5417" LastActivityDate="2017-04-13T19:47:16.710" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12199" PostTypeId="2" ParentId="11542" CreationDate="2017-04-13T22:21:31.403" Score="1" Body="&lt;p&gt;This is how I would group the advice given in this paragraph; along with the rubrics and the so called&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;7 thinking skills&quot;:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;$$\Huge \color{navy}{\text{Reading Your Algebra Book}}$$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An algebra book requires a different type of reading than a novel or a short story. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Every sentence in a math book is full of information&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\Large \color{red}{Vocabulary}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Important words whose meanings you'll learn are printed in &lt;strong&gt;heavy type.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Recall and transfer&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;logically linked to the surrounding sentences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\Large\color{red}{Symbols}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You must be able to read these symbols in order to understand algebra&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Analysis &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;$\Large\color{red}{Diagrams}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They contain information that will help you  understand the concepts under discussion. Study the diagrams carefully as you read the text that accompanies them. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Spacial perception &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You should read the sentences carefully&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;think about their meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\Large\color{red}{\text{Displayed Material}}$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Be sure to read and understand the material in the these boxes.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Applying concepts &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;$\Large\color{red}{\text{Reading Aids}}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They will help you become a more effective reader and problem solver.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;interpreting &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;$\Large\color{red}{Exercises,}$&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Synthesis &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;$\Large\color{red}{Tests,}$&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Reasoning and inferencing &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;$\Large\color{red}{{\text{and Reviews}}}$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;a Chapter Summary that lists important ideas from the chapter.&lt;/p&gt;&#xA;" OwnerUserId="216" LastEditorUserId="216" LastEditDate="2018-03-24T17:45:39.837" LastActivityDate="2018-03-24T17:45:39.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12200" PostTypeId="1" AcceptedAnswerId="12202" CreationDate="2017-04-14T00:05:18.117" Score="14" ViewCount="999" Body="&lt;p&gt;My brother has not taken a math class in &lt;span class=&quot;math-container&quot;&gt;$10-15$&lt;/span&gt; years. He is studying for the GRE so I have been teaching him a chapter or two from my precalculus book. So far, he has learned (and excelled at) basic algebra skills such as factoring quadratics, solving equations and inequalities in one variable, graphing lines, etc. But now the section on functions is coming up.&lt;/p&gt;&#xA;&lt;p&gt;I am having a hard time finding a good balance between between a rigorous and an informal definition of a function. If I make it too rigorous (a function is a subset of &lt;span class=&quot;math-container&quot;&gt;$A \times B$&lt;/span&gt; with blah blah blah propreties), the topic will seem useless and very separated from the mathematics he has done so far. If I make it too informal (a function is just a rule that assigns members from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to members of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, with certain restrictions), there will be a lot of confusion (&amp;quot;So is the function the actual arrows from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;?&amp;quot; &amp;quot;Is the equation &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt; the function?&amp;quot; Or is &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; the function?&amp;quot; &amp;quot;Is &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; the function?&amp;quot; &amp;quot;Is &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; a function?&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;My current plan is to get the best of both worlds, but I would like to know if there is a better way to teach it, and I would also like to know what I should teach after going over the definition.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Current Plan:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A function from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is a triplet &lt;span class=&quot;math-container&quot;&gt;$A, B, f$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$A, B$&lt;/span&gt; are sets and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is some sort of rule that assigns a unique value &lt;span class=&quot;math-container&quot;&gt;$b \in B$&lt;/span&gt; to each &lt;span class=&quot;math-container&quot;&gt;$a \in A$&lt;/span&gt;. The rule coult be a bunch of arrows, an equation, among other things.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Using this method I think I will still get some of the confused questions mentioned above, but I think using this definition I can defend the position that the function is the triplet &lt;span class=&quot;math-container&quot;&gt;$A, B, f$&lt;/span&gt; and the book is being lazy if they say something like &amp;quot;Find the zeroes of the function &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt;&amp;quot;. I would still like some advide about answering the &amp;quot;confused questions&amp;quot; though if you have it.&lt;/p&gt;&#xA;&lt;p&gt;I still think that my brother will find questions such as &amp;quot;Find the zeroes of &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt;&amp;quot; as useless and fluffed up ways of saying &amp;quot;Solve &lt;span class=&quot;math-container&quot;&gt;$2x = 0$&lt;/span&gt;. How can I alleviate his concerns?&lt;/p&gt;&#xA;" OwnerUserId="7075" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-02-19T21:02:28.707" Title="How should I introduce the concept of a function to a precalculus student?" Tags="|teaching|functions|precalculus|teacher-preparation|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12201" PostTypeId="2" ParentId="12200" CreationDate="2017-04-14T11:59:29.213" Score="7" Body="&lt;p&gt;Try applying physics. If you throw a ball up in the air, there are functions to describe the balls position, speed, and acceleration - all as functions of time. You input the time, and the function returns the an output like the position, speed, or acceleration. If you have a graph, you input x in the function and it returns the output y. &lt;/p&gt;&#xA;" OwnerDisplayName="user7915" LastEditorDisplayName="user7915" LastEditDate="2018-06-05T11:18:23.017" LastActivityDate="2018-06-05T11:18:23.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12202" PostTypeId="2" ParentId="12200" CreationDate="2017-04-15T22:23:56.783" Score="9" Body="&lt;p&gt;When introducing functions to a student, I usually give thought to two main methods, each with its pros and cons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Method 1:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use the set definition of the function. This is what you're attempting to do at the moment. The set definition of the function states that a function&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;is a relation between a set of inputs and a set of permissible outputs&#xA;  with the property that each input is related to exactly one output&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Using this definition, saying something like $f(x) = x^2$ becomes just a shorthand method to express a real valued function $f : \mathbb R \to \mathbb R$ that maps the domain $x$ to the range $x^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method is very strong because it gives the formal definition of the function, and, while confusing now, prevents all sorts of latter confusion when studying set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's hard to quickly move to explaining why zeros are useful or what you'll get out of graphing a function on coordinate planes if you use this definition. It's too abstract, and the examples you can use when teaching it are little. However, it does a good job at explaining the properties of a function, and the difference between a function and a relation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Method 2:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Defining a function the way a programmer would define one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method sacrifices (or just delays, really) the formal definition of the function for something that's easier to comprehend and work with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It defines a function as something that takes in an input and produces an output. You would see a lot of people who use this definition draw something like &lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Function_machine2.svg/1200px-Function_machine2.svg.png&quot; rel=&quot;noreferrer&quot;&gt;this diagram&lt;/a&gt;. They'd say that $x$ is just a dummy variable representing whatever input we have in the function.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You put something in, you get something out&#xA;  Its output only depends on its input (it does not produce new information).&#xA;  For every input, you get only one output (but multiple inputs can have the same output)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You'd see them identifying the function using the words &quot;the function $f$ as a function of $x$&quot; or &quot;the function $f$ of the input $x$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method is easy because you can throw lots of examples for it that involve physics or some observable phenomenon.&#xA;For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the velocity $v$ of a moving car &lt;em&gt;as a function of time&lt;/em&gt; t being $v(t) = v_i+at$&lt;/li&gt;&#xA;&lt;li&gt;the probability $p$ of the event $A$ being $p(A) = 1/8$&lt;/li&gt;&#xA;&lt;li&gt;the length of the hypotenuse $h$ of a right triangle as a function of the lengths of its sides, $h(a,b) = \sqrt{a^2+b^2}$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This definition allows you to quickly demonstrate why zeroes, graphs, and functions of more than 1 variable are important, and how they relate to our world. It is also easier to link this definition to trigonometry and differential calculus later on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this method does not make the concepts of 1-to-1 functions and onto functions, and the fact that a function has to map each domain to exactly one range very clear. It also does not explain the concept of domains, co-domains and ranges as clearly as the other method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the most important selling point of this method is that it allows you to quickly introduce function tranformations, like $u(x+4)$, $g(x)=ch(x)$, and $V(t) = V_0 - V_c(t)$...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;well, given both methods, which do I choose? I generally choose both. I start with method 2 to be able to give the student an intuitive idea of what a function is supposed to be, then I move on to formally defining it once they have a good idea of how to do calculations that involve functions later on in their calculus course (when it's time to deal with vectors or $\mathbb R^3$ or something).&lt;/p&gt;&#xA;" OwnerUserId="3568" LastActivityDate="2017-04-15T22:23:56.783" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12203" PostTypeId="2" ParentId="12200" CreationDate="2017-04-16T02:14:13.073" Score="8" Body="&lt;p&gt;A simple but rigorous definition is that a function is a graph in the plane that passes the vertical line test: a vertical line never intersects more than one point on the graph. (A graph is simply a set of points. &quot;The plane&quot; means the Euclidean plane.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This avoids the problems associated with defining a function as a rule. Defining it as a rule raises the question of how to define &quot;rule.&quot; Defining it as a rule is also inconsistent with the standard definition of a function, since there can only be countably many rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also avoids all the possible student confusion associated with heavy use of set-theoretic ideas and notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It isn't as general as the mathematician's general definition of a function, but that's a side issue, and the generalization can easily be carried out later.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2017-04-16T02:20:54.567" LastActivityDate="2017-04-16T02:20:54.567" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12205" PostTypeId="2" ParentId="11159" CreationDate="2017-04-17T02:41:23.237" Score="3" Body="&lt;p&gt;The Math Stack Exchange has some great ideas for &lt;a href=&quot;https://math.stackexchange.com/questions/2237243/what-are-good-math-habits-that-have-improved-your-mathematical-practice&quot;&gt;math habits that improve your mathematical practice&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the ideas mentioned:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Know the items in your toolkit.&lt;/li&gt;&#xA;&lt;li&gt;When learning about a new concept, visualize examples of the concept, and visualize examples of things that are not-quite-the-concept.&lt;/li&gt;&#xA;&lt;li&gt;When learning about theorems, try to come up with counter-examples -- and understand which piece of the theorem's assumptions disqualifies each counter-example.&lt;/li&gt;&#xA;&lt;li&gt;Simplify a problem until you can solve the simplified problem, then modify your solution to deal with the complication(s) you removed.&lt;/li&gt;&#xA;&lt;li&gt;Make notes about what is confusing.&lt;/li&gt;&#xA;&lt;li&gt;It is better to make an obvious mistake -- especially if you label it obviously -- than to rely on a concept you do not understand.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some other questions on this Stack Exchange have ideas that can help, as well:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Add some &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10786&quot;&gt;easy-to-learn tools&lt;/a&gt; to your toolkit.&lt;/li&gt;&#xA;&lt;li&gt;If possible, draw a picture and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10206#10208&quot;&gt;label your variables&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Always sanity-check your work.&lt;/li&gt;&#xA;&lt;li&gt;If possible, check-by-substitution.&lt;/li&gt;&#xA;&lt;li&gt;Keep track of your most &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7243&quot;&gt;common mistakes&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2017-04-17T02:48:19.063" LastActivityDate="2017-04-17T02:48:19.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12206" PostTypeId="1" CreationDate="2017-04-17T18:40:01.180" Score="8" ViewCount="295" Body="&lt;p&gt;Today I had a discussion on how to introduce the basic concept of variables in math using real life examples. We came up with ideas of using boxes containing matches, or M&amp;amp;Ms representing the variables and little piles of the actual items so e.g. 4 boxes with 2 M&amp;amp;Ms would equal a small pile of M&amp;amp;Ms. Problems of giving candy to kids aside, is that a proper way to introduce the concept of a variable? Are there better real life examples which they can perform at school?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have any ideas how to teach this to 13 year olds with a very mixed performance in school (really slow ones to some fast learners).&lt;/p&gt;&#xA;" OwnerUserId="8046" LastEditorUserId="2083" LastEditDate="2017-04-18T16:57:24.317" LastActivityDate="2017-04-19T13:16:47.040" Title="Introducing the concept of variables to kids" Tags="|algebra|primary-education|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12207" PostTypeId="2" ParentId="12206" CreationDate="2017-04-19T13:16:47.040" Score="2" Body="&lt;p&gt;I'm sticking with food math here. The &quot;recipe&quot; for cooking rice is a 2:1 ratio of water to rice. You can show that W=2R and start the discussion by telling students that the box says to use, say 2 cups of rice and 4 cups water. But you realize that there are 3 cups of raw rice and you don't want to waste what's left, nor run to the store for more. Since W=2R, you see that W=2*3 = 6, and use all the rice with 6 cups water. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can create other problems with pizza slices. &quot;Knowing that, on average, boys eat 3 slices and girls, 2 (pls forgive the sexism, this data is pretty accurate) create an equation to show how many slices are needed for the next soccer meet.&quot; The 8 to a pie also is great to show fractions. I resort to this with high school students who need help there. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-04-19T13:16:47.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12208" PostTypeId="2" ParentId="10962" CreationDate="2017-04-19T15:58:06.763" Score="5" Body="&lt;p&gt;The answer to your question whether there is such research is affirmative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the approach adopted at my university and used to train over 400 students over the past three years, the role of infinitesimals is not to &lt;em&gt;replace&lt;/em&gt; epsilon-delta definitions, but rather to &lt;em&gt;prepare&lt;/em&gt; the students for such definitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students react positively to such an approach, as reported in &lt;a href=&quot;http://u.cs.biu.ac.il/~katzmik/infinitesimals.html#17h&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; in &lt;em&gt;Journal of Humanistic Mathematics&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We use a modification of Keisler's approach in his book &lt;a href=&quot;https://www.math.wisc.edu/~keisler/calc.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Elementary Calculus&lt;/em&gt;&lt;/a&gt;. We strengthen the epsilon-delta component but only after the students are already familiar with the basic concepts of the calculus like derivative and continuity via more intuitive infinitesimal definitions &lt;em&gt;a la Cauchy&lt;/em&gt;.  Once the students understand the concepts, it is easier for them to relate to the epsilon-delta &lt;em&gt;paraphrases&lt;/em&gt; of their definitions.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-04-20T07:55:36.940" LastActivityDate="2017-04-20T07:55:36.940" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12209" PostTypeId="2" ParentId="11813" CreationDate="2017-04-19T16:00:52.617" Score="3" Body="&lt;p&gt;With regard to your comment &quot;talking about 'infinitesimal changes' like an 18th century mathematician or physicist&quot;. The implication of this comment is that we &lt;em&gt;should not&lt;/em&gt; teach students this way, because this way of teaching is outdated.  However, today we can combine intuition and rigor in a course in infinitesimal calculus; see &lt;a href=&quot;http://u.cs.biu.ac.il/~katzmik/infinitesimals.html#17h&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;.  Infinitesimals are no longer things of the 18th century, but rather the cutting edge of both mathematical research and mathematics education.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-04-20T07:05:59.160" LastActivityDate="2017-04-20T07:05:59.160" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12211" PostTypeId="1" CreationDate="2017-04-20T07:05:47.353" Score="3" ViewCount="713" Body="&lt;p&gt;I am comfortable with explaining to my high school students that for an event $A$ we have that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P(A) + P(A^C) = 1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what is the best way to help students realize that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P(A \mid B) + P(A^C \mid B) = 1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;remains true when you have the additional assumption of the event $B$ occuring. To be clear; I understand why this is so, I am just afraid of my explanation being too technical.&lt;/p&gt;&#xA;" OwnerUserId="6527" LastActivityDate="2020-09-17T06:06:55.690" Title="How to best explain that the sum of conditional probabilities still sum to 1" Tags="|probability|" AnswerCount="8" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12212" PostTypeId="2" ParentId="12211" CreationDate="2017-04-20T07:36:21.167" Score="5" Body="&lt;p&gt;If your mother gives you a candy, either you will eat it or you won't.  Sure thing.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2017-04-20T07:36:21.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12213" PostTypeId="2" ParentId="11159" CreationDate="2017-04-20T12:00:13.267" Score="2" Body="&lt;p&gt;As was pointed out, you have millions of way of improving problem solving  and logical thinking skills. No answer could be comprehensive. Thus I insert my 5 cents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a course &lt;a href=&quot;https://www.edx.org/course/effective-thinking-through-mathematics-utaustinx-ut-9-01x-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;“Effective Thinking Through Mathematics”&lt;/a&gt; by Michael Starbird from the University of Texas at Austin. It offers several rather easy puzzles. A special feature of this course is that it demonstrates more general problem solving methods using these puzzles as examples. Students are told to observe their process of thinking and solving so they notice which strategy turned out useful. As a result, they can even invent their own strategies tailored to their way of thinking.&lt;/p&gt;&#xA;" OwnerUserId="7992" LastActivityDate="2017-04-20T12:00:13.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12214" PostTypeId="2" ParentId="7702" CreationDate="2017-04-20T13:32:17.980" Score="3" Body="&lt;p&gt;The idea that historical infinitesimals were self-contradictory is prevalent among historians (see e.g., the accepted &lt;em&gt;answer&lt;/em&gt; above) and also many mathematicians, but it has been challenged in the recent literature; see e.g., &lt;a href=&quot;http://u.cs.biu.ac.il/~katzmik/infinitesimals.html#13f&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; in &lt;em&gt;Erkenntnis&lt;/em&gt; as well as &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=3053644&quot; rel=&quot;nofollow noreferrer&quot;&gt;its MathSciNet review&lt;/a&gt;. Furthermore &lt;a href=&quot;http://u.cs.biu.ac.il/~katzmik/infinitesimals.html#17h&quot; rel=&quot;nofollow noreferrer&quot;&gt;this recent study&lt;/a&gt; presents the results of a questionnaire that indicate that students themselves overwhelmingly prefer infinitesimal definitions &lt;em&gt;a la Cauchy&lt;/em&gt; of basic calculus concepts like continuity and derivative. The reason epsilon-delta &lt;em&gt;paraphrases&lt;/em&gt; still dominate in teaching are due to a history of prejudice.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-04-21T11:02:43.677" LastActivityDate="2017-04-21T11:02:43.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12216" PostTypeId="2" ParentId="12211" CreationDate="2017-04-21T02:50:48.740" Score="5" Body="&lt;p&gt;The key lies in reiterating the definition of probability as&#xA;$$\frac{\rm favorable {\ } cases}{\rm possible {\ } cases}$$&#xA;The conditional probabilities are computed with &lt;em&gt;less&lt;/em&gt; possible cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, say we roll two dice, one after the other (so we can differentiate them), and consider the following statements:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A: &quot;The second die is bigger than or equal to the first&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B: &quot;The sum of rolls is 6&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To compute $P(A)$ and $P(A^c)$, we consider 36 possible cases, with 21 and 15 favorable cases respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the conditional probabilities $P(A|B)$ and $P(A^c|B)$ only consider as &lt;em&gt;possible&lt;/em&gt; those cases in which $B$ occurred, namely 5. Of these (and only these) cases, 3 are favorable to statement $A$, and 2 to its negation.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2017-04-21T02:50:48.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12217" PostTypeId="2" ParentId="12211" CreationDate="2017-04-21T04:20:35.300" Score="3" Body="&lt;p&gt;You can realize probability as the relative area between a figure and an interesting part of that figure. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, if you have a calendar of last year, and you marked all the days where it rained in blue, the probability during the year that it will rain is defined as the number of blue days to the number of days total&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But different months rain different amounts. If we look only at the month of April, likely more will be blue that month than on any other month. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conditional probability here would be linked to looking &lt;em&gt;only&lt;/em&gt; at the month of April. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The total probability, then, is the number of days it rained (days marked in blue) and the number of days it didn't (not marked) added together, divided by the number of total days. Of course, there are 30 days in April, and hopefully each day, it either rained or it didn't rain, since any other possibility would be hard to imagine.&lt;/p&gt;&#xA;" OwnerUserId="1018" LastActivityDate="2017-04-21T04:20:35.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12219" PostTypeId="2" ParentId="993" CreationDate="2017-04-21T11:53:23.643" Score="1" Body="&lt;p&gt;Yes, I have taught a course on this and am currently teaching a course on this, including microcontinuity you mentioned; see &lt;a href=&quot;http://u.cs.biu.ac.il/~katzmik/88-826.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  An important distintion here is between &lt;em&gt;infinitesimal calculus&lt;/em&gt; and &lt;em&gt;infinitesimal analysis&lt;/em&gt;.  In the former, it is inappropriate to dwell on constructions of number systems, whether real or hyperreal.  In the latter of course this is eminently appropriate. The ultrapower approach is more accessible to the broader mathematical public than Robinson's original approach via the compactness theorem of first order logic, in my experience.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2017-04-21T11:53:23.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12220" PostTypeId="1" CreationDate="2017-04-22T13:53:07.393" Score="4" ViewCount="446" Body="&lt;p&gt;My son is taking pre-algebra in 6th grade. He mastered algebra 1 in detail more than what school can teach. How to prove to school that he mastered Algebra 1? He wants to start Algebra 2 in 7th grade. We are in PA.&#xA;Are there any authorized professional assessors available outside school? If I can get my son assessed by outside professionals, would school accept it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;thanks.&lt;/p&gt;&#xA;" OwnerUserId="8061" LastActivityDate="2017-05-05T17:02:26.983" Title="how to prove my 6th grade son knows algebra 1?" Tags="|algebra|self-learning|assessment|middle-school|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="12221" PostTypeId="1" CreationDate="2017-04-22T18:18:37.357" Score="6" ViewCount="1816" Body="&lt;p&gt;I'm currently planning on taking AP Calculus BC and AP Statistics as a freshman in high school next year, and am looking to go into the computer science field. I was wondering what track I should continue on, going into discrete mathematics, linear algebra, probability, etc. What courses should I take in the duration of high school, with dual enrollment available, and what should I plan to take in college? I've also looked at my high school's options; should I take the offered Discrete Mathematics &amp;amp; Probability course after calculus, or even at all?&lt;/p&gt;&#xA;" OwnerUserId="8062" LastActivityDate="2017-12-21T22:18:15.060" Title="What should I learn after calculus?" Tags="|secondary-education|computer-science|course-sequence|" AnswerCount="8" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12222" PostTypeId="2" ParentId="12221" CreationDate="2017-04-22T20:32:43.043" Score="10" Body="&lt;p&gt;Take your time Andrew, don't worry so much about classes.  Learn Python and solve your calculus problems with it.  Learn R and solve your Stats problems with it.  If you can do this by the end of your freshman year you've accomplished an incredible amount.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next year, if you can take a discrete class that includes some basic work with matrices this would be ideal.  Also, take all your other AP classes and beast them.  Work to understand how you can solve problems in Psychology, Economics, Biology, Chemistry, History, and Literature using the computer in addition to your pencil. These are free college credits, no need for dual enrollment in classes that you don't know will transfer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Breadth young brother.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Listen to a lot of music, absorb everything around you, have fun.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-04-22T20:32:43.043" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12223" PostTypeId="2" ParentId="12221" CreationDate="2017-04-23T01:19:31.080" Score="4" Body="&lt;p&gt;This same question has arisen at my current institution even earlier (i.e., after AP Calculus AB). The traditional next course after the Calculus is Real Analysis (constructing the real numbers in one of the four ways - Dedekind cuts, infinite decimal expansions, rational sequences modulo Cauchy, or with the axiom of completeness: usually via one of the last two), which can be followed by a second course on Real Analysis (e.g., broaching Lebesgue Measure Theory for which I recommend the &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0120839717&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;book&lt;/strong&gt;&lt;/a&gt; of H.S. Bear as a friendly introduction) or a course in Complex Analysis. But our mathematics department has opted for a different route, which I think might be appropriate for someone interested in computer science, as well:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We teach a course on non-routine problem solving, which is roughly in the spirit of Putnam A1/B1 (approaching from below) geared towards students without experience in mathematics competitions. The topics covered include mathematical induction and the well-ordering principle, the pigeonhole principle, combinatorics (e.g., stars and bars), graph theory (e.g., Euler's polyhedral formula), and a healthy dose of problem &lt;em&gt;posing&lt;/em&gt;, i.e., working towards student-generated problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a couple of examples of problems we used early in the year:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UnwkQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UnwkQ.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;There is no official textbook for the class, as I try to formulate the problems - or have the students do so - based on their interests. For example, a recent &lt;a href=&quot;https://www.nytimes.com/2017/04/15/opinion/sunday/the-worlds-most-beautiful-mathematical-equation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;opinion piece&lt;/strong&gt;&lt;/a&gt; in the NYTimes mentioned an older study that found Euler's identity, $e^{i \pi} + 1 = 0$, to be widely considered the &quot;most beautiful equation&quot; in mathematics. Even with Calc 1 knowledge, a few assumptions about power series (e.g., that they exist, that differentiation/integration work like you might hope) is enough to figure out how $e^{ix}$ can be written, and then this identity is not tough to derive in a class period.$^{\dagger}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answer is, in part, my advocating for other schools to consider such a route for students interested in mathematics as an alternative to more and more Calculus/Analysis. But for completeness with respect to the OP, here are two specific books that I would recommend if you are looking to put together an independent study of this nature (or for teachers who are considering creating a similar course).&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0470458380&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Applied Combinatorics&lt;/strong&gt;&lt;/a&gt; by Alan Tucker.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/B004SH7AOA&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Stanford Mathematics Problem Book&lt;/strong&gt;&lt;/a&gt; by George Polya and Jeremy Kilpatrick.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;$\dagger$ I wrote up a brief description of how one can derive Euler's Identity with a couple of assumptions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/HX9MC.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HX9MC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2017-12-20T03:54:02.777" LastActivityDate="2017-12-20T03:54:02.777" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12224" PostTypeId="2" ParentId="7631" CreationDate="2017-04-23T02:01:07.947" Score="3" Body="&lt;p&gt;One software that is both easy-to-use (does not require scripting) and generates beautiful high-res plots with real-time rendering then is &lt;a href=&quot;http://www.runiter.com/graphing-calculator/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graphing Calculator 3D&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/D9HSJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/D9HSJ.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2214" LastActivityDate="2017-04-23T02:01:07.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12225" PostTypeId="2" ParentId="4347" CreationDate="2017-04-23T11:08:22.200" Score="4" Body="&lt;p&gt;Regarding your parenthetical comment &quot;And it turns out that limits are not the only way to do so. Non-standard analysis uses infinitesimals in a logically rigorous way&quot; I would like to comment that the opposition limits &lt;em&gt;vs&lt;/em&gt; infinitesimals implied here is not entirely accurate.  &lt;em&gt;Limits&lt;/em&gt; are present in both approaches; the true opposition is between epsilon-delta methods in the context of an Archimedean continuum, and infinitesimal methods in the context of a Bernoullian (i.e., infinitesimal-enriched, as practiced by Bernoulli) continuum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Robinson's framework, the limit is defined in terms of the standard part function.  Thus, $\lim_{x\to 0} f(x)$ is the standard part of $f(\epsilon)$ when $\epsilon$ is a nonzero infinitesimal.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2017-04-23T11:08:22.200" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12226" PostTypeId="1" CreationDate="2017-04-23T16:26:48.003" Score="11" ViewCount="696" Body="&lt;p&gt;What is the best way to introduce Fermat’s Little Theorem (F$l$T) to students?&#xA;What can I use as an opening paragraph which will motivate and have an impact on why students should learn this theorem and what are the applications of F$l$T? Are there any good resources on this topic?&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="8039" LastEditDate="2017-04-27T07:59:54.677" LastActivityDate="2018-05-12T12:30:59.570" Title="What is most motivating way to introduce Fermat's Little Theorem" Tags="|concept-motivation|teaching|number-theory|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12227" PostTypeId="2" ParentId="12226" CreationDate="2017-04-23T18:12:00.133" Score="8" Body="&lt;p&gt;I wrote an article for NCTM's grades 8-14 journal, &lt;em&gt;The Mathematics Teacher&lt;/em&gt;, which is about building towards ${\rm F}\ell{\rm T}$ by thinking through ways in which the following statement can be generalized:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If you subtract a natural number from its square, then the result is even.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I wrote more about this in some earlier StackExchange posts, e.g., &lt;a href=&quot;https://matheducators.stackexchange.com/a/908&quot;&gt;&lt;strong&gt;MESE 907&lt;/strong&gt;&lt;/a&gt;. Since then, the article has appeared; its citation and link are:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dickman, B. (2017). Enriching divisibility: multiple proofs and generalizations. &lt;em&gt;The Mathematics Teacher&lt;/em&gt;, &lt;em&gt;110&lt;/em&gt;(6), pp. 416-423. &lt;a href=&quot;https://www.researchgate.net/publication/303920376_Enriching_Divisibility_Multiple_Proofs_and_Generalizations&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt; (without pay-wall).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The write-up is less concerned with specific applications, and has as its goal, instead, to indicate how the process of generalizing a routine number theoretic proposition can lead to theorems like ${\rm F}\ell{\rm T}$.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="163" LastEditDate="2018-05-10T23:19:41.247" LastActivityDate="2018-05-10T23:19:41.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12228" PostTypeId="1" CreationDate="2017-04-24T07:32:26.247" Score="6" ViewCount="227" Body="&lt;p&gt;What is the most motivating way to introduce Wilson’s Theorem? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is Wilson’s theorem useful? With Fermat’s little Theorem we can say that working with residue 1 modulo prime p makes life easier but apart from working with a particular (p-1) factorial of a prime what other reasons are there for Wilson’s theorem to be useful?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any good resources on this topic?&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="143" LastEditDate="2017-04-24T10:11:19.677" LastActivityDate="2017-04-25T22:01:23.563" Title="How to introduce Wilson's Theorem?" Tags="|mathematical-pedagogy|concept-motivation|number-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12229" PostTypeId="2" ParentId="12226" CreationDate="2017-04-24T12:21:43.503" Score="4" Body="&lt;p&gt;Beauty.  I really enjoy the &lt;a href=&quot;https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_theorem#Necklaces&quot; rel=&quot;nofollow noreferrer&quot;&gt;combinatorial necklace proof&lt;/a&gt;, which alone could motivate it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way to think of it is a representation which &lt;a href=&quot;http://math.gordon.edu/ntic/ntic/section-powers-mod-n.html#subsection-visualize-powers-mod-n&quot; rel=&quot;nofollow noreferrer&quot;&gt;I&lt;/a&gt; first learned of in &lt;a href=&quot;http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470412151.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wagon and Bressoud&lt;/a&gt;, where you look at a table of &lt;em&gt;all&lt;/em&gt; powers of the integers (or units) modulo $p$ and color-code entries by least nonnegative residue.  Then the last column of the table is all one color, just like the first column; the first column is $a^0\equiv 1$, but the last is $a^{p-1}\equiv 1$; so we have a visualization of FlT (among many other results).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://sagecell.sagemath.org/?z=eJwlzEEKwjAQBdB9oXfIrhmIYBcqCIEeoXvRMm0nNdIkQxKoeHqrmdXn8f901meKOOW6mskIDhvFIeO40sBryJK1vCiO1tEQ0S8kT0cAuNaV2K_XDnO071ItWfKhVTcXZomK4TEKE6JAYb0o-3bXv40_SwUZ7qAmh6ybF-UGyvv0DJvslbFLsh_SZ_gCbpQ3ew==&amp;amp;lang=sage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Try it out&lt;/a&gt;! (Quick Sage code below, you may wish to relabel the rows or columns.)&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;@interact&#xA;def power_table_plot(p=(7,prime_range(50))):&#xA;    P=matrix_plot(matrix(p-1,[mod(a,p)^b for a in range(1,p) for b in srange(p)]),cmap='jet')&#xA;    show(P,figsize=6)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-04-24T12:21:43.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12230" PostTypeId="2" ParentId="12228" CreationDate="2017-04-24T13:05:04.737" Score="7" Body="&lt;p&gt;Wilson's Theorem is very powerful for &lt;a href=&quot;http://math.gordon.edu/ntic/ntic/section-sqrt-mod-n-lemma.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;proving things related to quadratic residues&lt;/a&gt;, and I think also in general a good theoretical tool.  Motivating it by just saying &quot;hey let's multiply everything together&quot; seems very reasonable and even playful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is this theoretical?  It is basically saying that if you multiply all units together you get a specified result (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Wilson%27s_theorem#Gauss.27s_generalization&quot; rel=&quot;nofollow noreferrer&quot;&gt;the generalization due to Gauss&lt;/a&gt;; there are others, too).  But &lt;a href=&quot;http://math.gordon.edu/ntic/ntic/section-wilson-flt.html#theorem-wilsons-theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;I don't necessarily&lt;/a&gt; feel the need to motivate it further than that there are beautiful things that we can say about congruences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: as another example of the theoretical power, you can use it to prove this amazing (see Hardy and Wright Appendix) formula for the prime counting function: $$\pi(n) = -1 + \sum_{j=3}^n \left((j-2)!-j\left\lfloor \frac{(j-2)!}{j}\right\rfloor\right)$$&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2017-04-25T18:47:21.443" LastActivityDate="2017-04-25T18:47:21.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12232" PostTypeId="2" ParentId="12200" CreationDate="2017-04-25T16:00:56.577" Score="2" Body="&lt;p&gt;One reason students sometimes have trouble with the notation of a function is that the examples they see are so predominately couched in &quot;formula&quot; terms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to overcome this and show them how fruitful the function concept is involves showing them the connection between the process of conducting elections and ideas about functions. The discussion below is designed to be intuitive rather than formal but I think goes in the direction of helping students think about an &quot;input/output&quot; approach to functions in a way that expands their understanding of the function idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose one has n voters each of whom produces an ordinal ballot involving m candidates (choices) where each voter ranks all of the candidates, including the possibility of ties (indifference in the voters mind between two or more candidates. (One can also think about the case where each voter produces a full ranking without indifference of all of the candidates as well.) The goal is to construct a ranking (or perhaps a single winner) for society based on the rankings of the individual voters. So the domain consists of n-tuples of preference schedules and the codomain consists of a set of preference schedules of the choices, ties allowed (or perhaps list of candidates when a single winner is involved). (Single winner is in some sense less &quot;clean&quot; because of the need to deal with issues of the possibility of &quot;ties&quot; when the voting is very &quot;symmetrical.&quot;  One would like to pick a &quot;President&quot; based on the voters ballots but for some sets of ballots some methods might result in ties, unless a tie breaking system is built into the election method.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Different methods of deciding elections based the ballots correspond to different functions which take ballots of individuals and turn them into a choice for society.  So systems include plurality, run-off, sequential run-off, Borda Count, and many other proposed election methods (functions).  One is also interested in &quot;fairness&quot; conditions that different functions here might obey.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is in this framework that the recently (2017) deceased Kenneth Arrow formulated his famous theorem that when there are at least three candidates there is no election method that obeys all of a small list of &quot;fairness&quot; conditions.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2017-04-25T16:00:56.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12233" PostTypeId="2" ParentId="12228" CreationDate="2017-04-25T19:03:39.600" Score="4" Body="&lt;p&gt;My opinion: powerful and effective problems will increase interest and motivation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, We can prove Wilson's Theorem with using Fermat's little theorem. Therefore (If preferred) we can take Wilson's theorem an appication of Fermat's. More specifically,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When $p=2$ easily $(2-1)!\equiv -1\pmod{2}$. Let $p&amp;gt;2$ be a odd prime. If we solve&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x^{p-1} - 1 \equiv 0 \pmod{p} \tag{1}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;by Fermat's theorem $x=1,2,\dots , p-1$ satisfy $(1)$. Each root is a factor of $x^{p-1} - 1$ polynomial and $$ x^{p-1} - 1 \equiv (x-1)(x-2)\cdots (x-(p-1))\pmod{p} \tag{2}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we put $x=0$ in $(2)$ then $1\cdot 2 \cdots (p-1) \equiv -1 \pmod{p}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, if we find direct applications/examples of Wilson's Theorem, (I hope that) they may increase the motivating. I can give two Wilson's Theorem applications:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) $\dfrac{10!}{13}+x$ is a positive integer then, what is mimimum positive real value of $x$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) If $16!$ divided by $323$, what is the remainder? (You may give a hint: $323= 17\cdot 19$.)&lt;/p&gt;&#xA;" OwnerUserId="6372" LastEditorUserId="6372" LastEditDate="2017-04-25T22:01:23.563" LastActivityDate="2017-04-25T22:01:23.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12234" PostTypeId="1" CreationDate="2017-04-18T08:40:00.280" Score="19" ViewCount="1595" Body="&lt;p&gt;I am teaching probability to high school students. The material we are covering is pretty standard and includes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Introducing how to calculate the probability of events, e.g. coin flips, card draws, lottery tickets.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The sum of the probabilities of all events is $1$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Independent Events: $P(B \mid A) = P(B)$ and $P(A) \cdot P(B) =P(A \cap B)$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Venn Diagrams and $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Conditional probability: $P(B \mid A) = \frac{P(A \cap B)}{P(A)}$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bayes Rule: $P(B \mid A) \cdot P(A) = P(A \mid B) \cdot P(B)$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Product Rule: $P(B \cap A) = P(A) \cdot P(B \mid A)$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Total Probability: $P(B) = P(A) \cdot P(B \mid A) + P(\bar{A}) \cdot P(B \mid\bar{A})$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Binomial Trials&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am looking for examples of common misconceptions students have, or common mistakes that they do, when solving problems. Trivial examples are welcome. A preferred answer should also suggest ways in which one can prevent students to have said misconception.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;An example of a common mistake would be:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Problem:&lt;/em&gt; &#xA;You draw a card from a deck. What is the probability that it is red and has value J,Q,K or A?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Answer:&lt;/em&gt;  $\frac{26}{52} + \frac{16}{52}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My best suggestion for explaining to a student that this is wrong might be to change the problem to say &lt;strong&gt;red and black&lt;/strong&gt; instead of simply red, and thus following the same logic getting a probability greater than $1$. But having students make similar thoughts  seems unrealistic to me, and I am lost at how to best proceed. &lt;/p&gt;&#xA;" OwnerUserId="6527" OwnerDisplayName="Improve" LastEditorUserId="143" LastEditDate="2017-04-25T21:33:08.737" LastActivityDate="2023-02-05T18:03:30.910" Title="Common misconceptions in high school probability curriculum" Tags="|secondary-education|students-mistakes|probability|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12235" PostTypeId="2" ParentId="12234" CreationDate="2017-04-18T08:57:57.663" Score="10" Body="&lt;p&gt;Here are some things I occasionally encounter in the first few tutorial sessions as a TA for an undergraduate introduction to probability theory/statistics course.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why &quot;and&quot; corresponds to intersection, while &quot;or&quot; corresponds to union. Also, why &quot;not&quot; corresponds to the complement.&lt;/li&gt;&#xA;&lt;li&gt;The difference between $P(A \cap B)$ and $P(A \mid B)$, and which natural language statements to translate into which.&lt;/li&gt;&#xA;&lt;li&gt;That $P(A \cap B) = P(A)P(B)$ holds only for independent events $A,B$, while $P(A \cap B) = P(A\mid B)P(B)$ always holds; and that this corresponds to the fact that $P(A \mid B) = P(A)$ is another way to define independence.&lt;/li&gt;&#xA;&lt;li&gt;The fact that if $A = A_1 \sqcup A_2 \sqcup \cdots \sqcup A_n$, then $P(A) = \sum_{i=1}^n P(A_i)$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Most of them would probably give the right answer if presented with the question explicitly and directly, but many of them made errors in the context of larger problems, or failed to think of the rule when it was natural to use it to solve a particular problem.&lt;/p&gt;&#xA;" OwnerUserId="8066" OwnerDisplayName="Mees de Vries" LastActivityDate="2017-04-18T08:57:57.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12236" PostTypeId="2" ParentId="12234" CreationDate="2017-04-18T09:15:36.267" Score="5" Body="&lt;p&gt;Regarding your example, a tree diagram might help to explain it properly. Also, basic stochastics often comes down to counting problems, that is we have to rephrase the problem such that every event gets the same probability (in your example, we are looking for four eight events out of 52, all having the same probability). If you can make your students understand this, you will help them a lot. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an other example, take the question: &quot;Throwing two dices, what is the chance of having the same number on each of them?&quot;. To properly solve it and more importantly develop the corresponding theory, one should look at all $36$ possible outcomes and from there deduce that the chance is $6/36 = 1/6$. There are of course other ways to see it (e.g. &quot;fix one dice, what is the chance that the other gives the same?&quot;), but in my opinion, turning it into a Laplace experiment is the one that can be used most often in such problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have control over the exercises, make your students always write down the total set of possible outcomes (maybe paraphrased if too big) and the set they are interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Btw, a little advice on your way of explaining it with the probability bigger than one: This helps the student to see that there is something wrong there. Depending on their level of understanding, they will either say &quot;ok, stupid rule with the one, whatever&quot; or they will agree and understand that that can not be, however, they will still not be any wiser. You can use such an example with a very intelligent student to show him that he is wrong, but the general &quot;I fu***** hate math!&quot; student might just get demotivated by &quot;BAM, you are wrong!&quot;, so a constructive approach to show them how to properly do it might be better. (I know I am exaggerating here and you most likely don't do it like that.^^)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A nice thing my own high school teacher did was the following:&#xA;Every Friday, he gave us two different experiments (with dices, drawing balls,...). Then he left the room and gave us about 10min to discuss them, we where to compute the probability. Then he came back and we did the experiment we, as a class, chose. If we &quot;won&quot; - that is, we succeeded for example to through two times the same number with dices, we got no homework for the weekend. The chances for success where always between 10% and 30%, so we really had to figure out which one to choose if we wanted good chances. :)&#xA;Of course for something like that, you need to know your class, you need the people to be motivated and you might not want to leave 30+ students alone to discuss (in this case maybe a majority vote for one of the systems might be better), but having such active experiments might always help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another nice thing he did (now I am really starting to remember^^): In the very first class, he wanted to show us that we had a natural understanding for Laplace experiments, all outcomes having the same probability. So he brought a dice, said we will through it 100 times and we should assume how often we will see a one. Turns out, we had 92x a one in the end, because it was a wooden dice and he filled the six holes opposite to the one with some heavy metal... xD&#xA;However, we afterwards discussed why we, without knowing anything about the dice, assumed to see about $100/6$ ones and this was a great and fun start into the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But to come back to your question of other examples: I think the most difficult thing is to find the right model for a problem. Most errors I myself made or encountered were of the form &quot;damn, I should have counted that without repetition, with order,...&quot;. So I claim that the most difficult thing in basic stochastics is not to do counting, not to compute binomial coefficients but rather to translate an exercise into a model. It is also often not clear from the text if they are considering repetitions, etc., so you might want to make sure that your students not only write down computations or results but also a little explanation on which model they chose and why.&lt;/p&gt;&#xA;" OwnerUserId="8296" OwnerDisplayName="Bemte" LastActivityDate="2017-04-18T09:40:51.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12237" PostTypeId="2" ParentId="9835" CreationDate="2017-04-25T23:17:09.420" Score="6" Body="&lt;p&gt;You can disambiguate the two senses of ‘linear’ as ‘homogeneous linear’ and ‘affine linear’.  You can then say that in this class, ‘linear’ means ‘affine linear’ by default but in other situations it will mean ‘homogeneous linear’ instead.&lt;/p&gt;&#xA;" OwnerUserId="408" LastActivityDate="2017-04-25T23:17:09.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12238" PostTypeId="2" ParentId="12226" CreationDate="2017-04-26T19:33:12.090" Score="1" Body="&lt;p&gt;Generally We enter the subject(FLT) with a problem like $a^m \equiv ? \pmod{n}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After the training like these:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$3^{100}\equiv ? \pmod{5}$, $2^{80}\equiv ? \pmod{7}$, ... etc&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students are important to 'We need to smallest positive $m$ that $3^{m}\equiv 1 \pmod{5}$ or $2^{m}\equiv 1 \pmod{7}$, ... etc. (in fact $m$ is order of $3$ in $\mod 5$.) But sometimes we are in difficulty for finding the order. Especially for bigger $p$ prime numbers in $ \mod{p} $. If we ask to students $2^{100}\equiv ? \pmod{101}$, they will spend a lot of time. Thus, we can give FLT to students. If students do not use FLT, they will conclude that it will be very difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, if $\gcd (a,p)=1 $ then $a^{p-1}\equiv 1 \pmod{p}$ but $p-1$ coudn't be order of $a$ in $\mod p$. So, we can say $\text{ord}(a) \mid {p-1}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last of all; because of The World need to the order(!), FLT is important for us:) &lt;/p&gt;&#xA;" OwnerUserId="6372" LastEditorUserId="6372" LastEditDate="2017-05-26T20:14:24.990" LastActivityDate="2017-05-26T20:14:24.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12240" PostTypeId="2" ParentId="9835" CreationDate="2017-04-27T12:52:32.810" Score="4" Body="&lt;p&gt;It isn't just in elementary mathematics that functions of the form $f(x) = mx+b$ are called &lt;em&gt;linear&lt;/em&gt;. The term &quot;linear regression&quot; is used almost universally in statistics to describe the method of fitting a line to a data set. I don't remember ever once seeing the term &quot;affine regression&quot; used instead. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though mathematics can be modeled as a formal language/system such as ZFC, it is actually done in a formal language/ natural language hybrid, one which is not immune from the sort of etymological quirks that characterize all natural languages. &lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2017-04-27T12:52:32.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12241" PostTypeId="1" AcceptedAnswerId="12242" CreationDate="2017-04-27T17:49:33.120" Score="6" ViewCount="1807" Body="&lt;p&gt;My old question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1506/the-rearranging-approach-to-teaching-logarithms&quot;&gt;here&lt;/a&gt; was about logarithms, but as I teach more and more (precalculus) algebra, I've generalized the question a bit in my mind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should students &lt;strong&gt;do the same thing to both sides&lt;/strong&gt;,  or &lt;strong&gt;know the inverse function well enough to move symbols around&lt;/strong&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll try to illustrate what I mean with some simplified examples.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Here are two possible solutions to the question &quot;Solve for $x$: $x + 5 = 12$.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Subtract $5$ from both sides. The $+5$ cancels the $-5$ on the left so we get $x = 12 - 5$. So $x = 7$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Move the $5$ to the right side to get $x = 12 - 5$. So $x = 7$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the first case, the student applies the inverse of a one-to-one function to both sides of an equation to get a new equation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the second case, the student appeals to the core definition of what subtraction means: the whole idea of subtraction is that if $x + 5 = 12$ then $x = 12 - 5$. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Similarly here are two possible answers to the question &quot;Solve for $x$: $log_5 x = 3$.&quot; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Exponentiate both sides with a $5$ to get $5^{log_5 x} = 5^3$. The $5$ and the $log_5$ cancel on the left so we get $x = 5^3$. So $x = 125$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $log_5 x = 3$, then $x = 5^3$. So $x = 125$. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the first case, the student applies the inverse of a one-to-one function to both sides of an equation to get a new equation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the second case, the student appeals to the core definition of what a logarithm means: the whole idea of the logarithm is that if $log_5 x = 3$ then $x = 5^3$. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Advanced students know that you can &quot;move symbols&quot; in certain ways to rearrange the statement in an equation, so an advanced student is more likely to successfully implement the second strategy. Additionally, it seems to me that math faculty tend to favor the second strategy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, novice students are more likely to succeed with the first strategy and are more likely to be implementing the core mathematical plan &quot;do the same thing to both sides&quot; throughout mathematics. &lt;strong&gt;Which of the two strategies should we favor, when, and why?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;One more example: &quot;Solve for $x$: $(\pi - 1)x = 12$.&quot; Would you tell a student to &quot;divide both sides by $\pi - 1$?&quot; Or would you ask them to move the symbols around, hoping that they are correctly appealing to the definition of division? Is your answer different if the objects involved are more or less complicated?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: Jim Belk's answer to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1506/the-rearranging-approach-to-teaching-logarithms&quot;&gt;the question linked at the top&lt;/a&gt; is very insightful and I am hoping for more discussion of that type, or at least a reference-request for &lt;strong&gt;what is this pedagogical concept called&lt;/strong&gt;?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-05-01T15:21:39.063" Title="Apply the inverse operation on both sides, or know the inverse function?" Tags="|reference-request|algebra|inverse|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12242" PostTypeId="2" ParentId="12241" CreationDate="2017-04-27T18:21:54.303" Score="9" Body="&lt;p&gt;I'd say both: they must know the inverse function, and that it's being applied to both sides. This then connects up with the fundamental properties of equality. As a teacher of many algebra courses (elementary to college level), students who think of &quot;moving&quot; terms around can almost never generalize to the next level. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-04-27T18:21:54.303" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12243" PostTypeId="2" ParentId="12241" CreationDate="2017-04-27T22:29:30.317" Score="1" Body="&lt;p&gt;The fundamental concept is always: You can do the same thing to both sides, as long as it does not involve an illegal operation (like dividing by zero, or taking the logarithm of a negative number). But there are an infinite number of choices of &quot;things&quot; you can do to both sides, most of which don't help solve the problem. The student needs to know what the relevant inverse operation is, in order to choose which &quot;thing&quot; to do to both sides.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2017-04-27T22:29:30.317" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12244" PostTypeId="1" CreationDate="2017-04-28T01:59:19.937" Score="3" ViewCount="243" Body="&lt;p&gt;If mathematicians and math teachers and textbooks in all cultures in all eras say that a certain theorem has been proved, then the reasonable layperson trusts them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, a student who wants to understand mathematics works each day on trying to understand independently of such authorities why things in mathematics are as they are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How widespread is knowledge of the point in the second paragraph above among those who have finished secondary-school mathematics? I.e. how many students know that pursuing independent understanding is the principal activity among those activities that constitute learning mathematics? &lt;b&gt;Has anyone attempted to measure this objectively?&lt;/b&gt;&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2017-04-30T21:02:48.207" LastActivityDate="2017-04-30T21:02:48.207" Title="How many secondary-school students are working on achieving independent understanding?" Tags="|education-research|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="12245" PostTypeId="2" ParentId="12241" CreationDate="2017-05-01T12:58:26.780" Score="4" Body="&lt;p&gt;As a practicing mathematician, I don't even &lt;em&gt;distinguish&lt;/em&gt; between the two options you describe. You're not presenting two different &lt;em&gt;options&lt;/em&gt;, but two descriptions of the &lt;em&gt;same&lt;/em&gt; option.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Moving&quot; the $5$ to the other side (or, more accurately, getting rid of its effect on $x$) is an effect I want to achieve. Subtracting 5 from both sides is the mechanism by which I achieve this effect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, I think I would also use the two descriptions of the same option mostly as as synonyms. So, I would be wary of assuming that people who sound like they speak in terms of one description are making an actual choice between the two.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So, my unqualified&lt;sup&gt;*&lt;/sup&gt; opinion is that you need both &amp;mdash; having students recognize what &lt;em&gt;goal&lt;/em&gt; they are trying to achieve is just as important as having students recognize what &lt;em&gt;technique&lt;/em&gt; to apply to a problem. And, of course, to learn how to discern which techniques can be used to achieve which goals and when.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;*: I am not an educator&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The flip side of your concerns, I think, are the students that don't develop a goal-oriented approach to problems, so when faced with a problem that doesn't match one of their recipes and so they are lost when they have to develop their own goals for manipulating formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or worse, then the goal for a specific problem is different than the 'usual' goal one would have for the formula; e.g. calculus problems where one wants to manipulate an equation to make $x-1$ appear someplace, rather than the 'usual' goals of simplification.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2017-05-01T15:21:39.063" LastActivityDate="2017-05-01T15:21:39.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12246" PostTypeId="1" AcceptedAnswerId="12247" CreationDate="2017-05-01T15:06:17.700" Score="14" ViewCount="3628" Body="&lt;p&gt;I was reading answers to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1/encouraging-class-participation/5#5&quot;&gt;this question&lt;/a&gt; today when I realized we never had this followup question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How do you effectively call on students by name in a math class&lt;/strong&gt;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a way to encourage class participation and class preparation, many of my colleagues in other disciplines swear by the practice of randomly selecting students to ask questions of them during class. In my mind, the obvious issues are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The practice is stressful for students (but wait, this is the primary alleged benefit of the practice).&lt;/li&gt;&#xA;&lt;li&gt;It is different-levels-of-stressful for different students or groups of students.&lt;/li&gt;&#xA;&lt;li&gt;It needs to be okay for a student to say they do not know something, but it can't always be okay, or you are defeating the purpose.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Do any of you regularly ask questions of non-volunteering students in your math classes in a way that avoids the pitfalls above? I'm personally interested in undergraduates but would be interested in answers from other perspectives as well.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-06-15T22:32:43.320" Title="Calling on students by name (who do not volunteer) in math classes" Tags="|undergraduate-education|class-participation|" AnswerCount="7" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="12247" PostTypeId="2" ParentId="12246" CreationDate="2017-05-01T15:24:47.683" Score="10" Body="&lt;p&gt;This isn't much of an answer, but I think it goes a long ways toward what to think about.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Know your class really well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Different questions will work for different students.  Your third point is a good one, and so you need to know the class well enough to know what sort of question will work for each student.  This doesn't obtain in a huge class, though there one might be able to get away with this technique for other reasons.  A corollary:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Have a backup plan.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There always needs to be something you can use to extricate yourself from a bad situation or a truly confused student.  That could be to validate the student's feeling (see e.g. &lt;a href=&quot;http://blogs.ams.org/matheducation/2017/05/01/i-am-so-glad-you-made-that-mistake/&quot; rel=&quot;noreferrer&quot;&gt;today's blog post by Allison Henrich&lt;/a&gt; which has a number of very useful points), or something else - not just to be able to retract the question.  But there has to be a way to deal with the issue of when despite your best preparation you misread a student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own personal preference is to only ask &lt;strong&gt;very&lt;/strong&gt; softball questions of completely random people, or questions whose answers are literally before their eyes.  But I've seen it &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Sally&quot; rel=&quot;noreferrer&quot;&gt;done effectively&lt;/a&gt; to every single student, given the right personality and expectations for the class.  Also, in a less consumer-centric educational context than the USA (or in law school, I hear), this strategy might be perfectly appropriate all the time.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-05-01T15:24:47.683" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12248" PostTypeId="1" AcceptedAnswerId="12250" CreationDate="2017-05-01T17:17:31.687" Score="22" ViewCount="5333" Body="&lt;p&gt;I was reviewing percentages with my son (US equivalent of 7th grade) and the more I dug into explanations, the less I could understand &lt;strong&gt;why&lt;/strong&gt; they are taught. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand how they technically work but using them introduces, I think, a complexity layer which does not help in the actual computation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some subjective thoughts:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$20\,\%$ is $0.2$ of something. In order to get $20\,\%$ of $123\,€$, I need to first realize that $20\,\% = 0.2$, and then compute $123·0.2$. So the existence of a percentage did not help in the calculation.&lt;/li&gt;&#xA;&lt;li&gt;the fact that $20\,\%$ seems easier to grasp than $0.2$ is, I believe, just a matter of being used to it. After some time $0.2$ would seem natural as well.&lt;/li&gt;&#xA;&lt;li&gt;percentages being on a scale of $0\,\%$ to $100\%$, things like $130\,\%$ may seem weird. $1.3$ is better, it shows that there is one whole, and then $0.3$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Is there a specific reason percentages are a thing?&lt;/p&gt;&#xA;" OwnerUserId="8076" LastEditorUserId="-1" LastEditDate="2017-05-02T20:44:14.693" LastActivityDate="2017-05-03T23:47:08.840" Title="Why are percentages part of the curriculum?" Tags="|proportional-reasoning|" AnswerCount="7" CommentCount="21" ContentLicense="CC BY-SA 3.0" />
  <row Id="12250" PostTypeId="2" ParentId="12248" CreationDate="2017-05-01T17:57:41.790" Score="55" Body="&lt;p&gt;Percentages are widely used throughout society - in news, scientific publications, and so on.  Learning to understand them is a necessary literacy skill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Percentages don't have a particular significance in mathematics, and as you've pointed out, they're basically a small extra layer on top of the actual calculations.  If mathematicians ruled the world, we &lt;em&gt;wouldn't&lt;/em&gt; have percentages: everyone would just use decimals, for exactly the reasons you're describing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the question is why percentages are so popular outside of mathematics, I suspect the answer is that they're easier for people who aren't especially mathematically literate, because they keep to whole numbers.  The issues calculating with them are irrelevant for people who don't need to calculate, and the additional difficulties just aren't very big.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2017-05-01T17:57:41.790" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="12251" PostTypeId="2" ParentId="12248" CreationDate="2017-05-01T18:52:34.623" Score="12" Body="&lt;p&gt;Percentages provide a &quot;common unit&quot; for expressing changes that are relative to a base amount, and/or are compoundable.  They simplify comparisons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many processes can have simple rules, which can be expressed in terms of fractions or percentages.  For example, the old Filene's Basement used to incrementally mark down sale goods by 1/4 of the original price each week.  When the price reached zero, the goods were given away to charity.  As another example, the monthly interest on a mortgage might be 11/3200 of the outstanding principal.  Or a baseball player might have gotten &quot;on base&quot; an average of 2 out of 5 of their &quot;plate appearances&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now suppose you want to compare these processes.  How much has the dress been discounted?  (Not, what is the price of the dress?)  Intuitively comparing 1/4, 1/2, and 3/4 is not as easy as comparing 25%, 50%, and 75%.  Which is the better deal -- a mortgage with a monthly interest rate of 1/384, or 11/3200?  It is much easier to compare an &quot;annual percentage rate&quot; (APR) of 3.125% &lt;em&gt;versus&lt;/em&gt; an APR of 4.125%.  Which season was Ricky Henderson more likely to get on base -- the year he succeeded 261 out of 656 times, or the year he succeeded 82 out of 222 times?  It is much easier to compare an &quot;on base percentage&quot; (&quot;OBP&quot;) of 0.398 &lt;em&gt;versus&lt;/em&gt; an OBP of 0.369.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Baseball statistics are usually compared using 3-digit decimals (technically, millages).  Most other rate statistics (in this order of magnitude) are compared using percentages.  (As a side note, the terms &quot;millage&quot; and &quot;permillage&quot; were coined in the late 1800s, well after American professional baseball became popular.  &quot;Percentage&quot; was coined in the late 1700s.  Baseball statistics are traditionally called &quot;percentages&quot; even though they are actually &quot;millages&quot;.)&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2017-05-03T15:14:10.023" LastActivityDate="2017-05-03T15:14:10.023" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12252" PostTypeId="2" ParentId="12246" CreationDate="2017-05-01T20:45:30.603" Score="4" Body="&lt;p&gt;As a teacher, I tend to avoid this because even when I ask generic questions to the whole class the discomfort of the students is often patent. With mathematics, also, you are frequently  trying to get the students to understand a new idea, and it usually takes time and practice to get it; which means that being partially lost is a common ocurrence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a student, I never obtained any benefit from a teacher following this practice. In highschool, it made me really uncomfortable and it made me rebel and not study. In university, if a teacher had done this, I would have simply stopped attending class.&lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2017-05-01T20:45:30.603" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12253" PostTypeId="2" ParentId="12248" CreationDate="2017-05-01T20:58:01.537" Score="8" Body="&lt;p&gt;Somewhat as others have said: Historically (pre-decimalization), the alternative was fractions such as 3/5, 2/7, 1/2, etc. Comparisons are not entirely trivial; many people would not be able to reliably answer which of those are the largest. So by regulating partial calculations (as for taxes, interest, etc.) to parts of 100 made those comparisons much easier for common people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Decimalization solves the issue of fractions with much the same idiom, but it is a surprisingly recent development (e.g., prices on the U.S. stock market didn't switch from fractions to decimals until the year 2001). But even with decimals, comparisons can be tricky. Which is larger: 0.2 or 0.08? Many people will answer that incorrectly because of the hidden place value. Percentages make the like place values explicit and solve the problem for more people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if we take a proportion like 0.2 and say that it's easy to multiply 123 by 0.2, that's not trivial for all people. What is multiplying by a decimal? There's a multiply and also a decimal shift which some people will get incorrect. Actually: 0.2 with its one decimal place may be one of the only examples you can pick which is arguably easier than the percentage. Other stuff like 0.75, 0.125, 1.05, etc., with any other number of decimal places shows little advantage in the operation, and a different number of shifts in each case. Even for myself I find it easier to read the % as a consistent &quot;divide by 100&quot; (i.e., always a 2-place shift), and then separately multiply by the magnitude of the percentage. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I quote Alfred North Whitehead from &lt;em&gt;An Introduction to Mathematics&lt;/em&gt; (1911):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Probably nothing in the modern world would have more astonished a&#xA;  Greek mathematician than to learn that... a large proportion of the&#xA;  population of Western Europe could perform the operation of division&#xA;  for the largest numbers. This fact would have seemed to him a sheer&#xA;  impossibility... Our modern power of easy reckoning with decimal&#xA;  fractions is the almost miraculous result of the gradual discovery of&#xA;  a perfect notation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-05-02T03:59:56.813" LastActivityDate="2017-05-02T03:59:56.813" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12254" PostTypeId="2" ParentId="12246" CreationDate="2017-05-01T23:12:58.083" Score="5" Body="&lt;p&gt;I emphasize from the beginning of the year that &lt;strong&gt;mistake-making&lt;/strong&gt; is to be expected; the specific language that I use can be found in &lt;a href=&quot;https://matheducators.stackexchange.com/a/11830&quot;&gt;&lt;strong&gt;MESE 11794&lt;/strong&gt;&lt;/a&gt;. I believe that by ensuring this is a &lt;em&gt;norm&lt;/em&gt; from the beginning of the academic school year, one is better positioned to call on students who are not volunteering, which is something that I do frequently. I feel very strongly that much of the stress, or anxiety, of being cold-called can be averted by engaging in this practice deliberately and often, and weaving unsureness, or outright errors, into something productive as pertains to the topic at hand.&lt;/p&gt;&#xA;&lt;p&gt;I do not, however, call on students &amp;quot;randomly&amp;quot;: I know that some teachers use various methods to do this, e.g., spinning a name wheel or drawing popsicle sticks with student names out of a cup (possibly without replacement so that different people are called) or just circling around the room with one's questions. In my own estimation, part of the instructor's teaching expertise should be reflected in their choices around which student is called on, and when. Moreover, I believe that participation -- like mathematical tasks -- can be &amp;quot;low floor high ceiling&amp;quot; in the sense that, the initial student voicing can be very simple, and follow ups can be scaffolded towards the very difficult.&lt;/p&gt;&#xA;&lt;p&gt;As an example of a low floor participatory task, I might say something like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Chris, can you read problem 3 on page 11?&lt;/p&gt;&#xA;&lt;p&gt;[Chris reads problem]&lt;/p&gt;&#xA;&lt;p&gt;And do you, or does anyone else, have any ideas about how we can get started on this?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would probably stay with the reader for a moment, to give them a chance to say something on their mind; otherwise, I will call on others who are volunteering, but I might do something slightly subtle and re-summarize at some point by using the reader's name. In the above example, this could mean, e.g., after Alice chimes in with an idea about how to proceed, I might say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Okay, so Chris asked [problem 3], and Alice had an idea about how we could broach this. She suggested we [suggestion]. Does anyone else want to weigh in, either by building on what's been said, or by asking for something to be clarified further?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I will not make a claim that this is paradigm-shifting in its efficacy, but I believe that keeping track of names and who did what -- whether it is reading a problem aloud, or suggesting a (possibly incorrect) method -- is a good way to maintain a sense of participation by multiple individuals.&lt;/p&gt;&#xA;&lt;p&gt;Although describing in full how I try to manage classroom discourse would make for a far too lengthy post, there is more about productive talk moves &lt;a href=&quot;http://www.mathsolutions.com/documents/9780941355537_ch2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;, which I generally co-sign.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Succinctly:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Establish a norm from the beginning of the year that making mistakes is expected and valued.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Establish a norm from the beginning of the year that you will call on students who have not volunteered.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Recognize that there are many forms of participation -- ranging from having one student asking another to clarify their reasoning, to giving a full and complete explanation for a problem, to simply reading aloud from a handout -- and value each of these forms, repeating contributions with attribution as you (the teacher) talk through the class' interactions around a problem/lesson.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-05-02T04:30:29.053" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12255" PostTypeId="2" ParentId="12248" CreationDate="2017-05-02T05:28:54.067" Score="-3" Body="&lt;p&gt;Percentages are taught so the those who consider them unnecessary will better understand the nature of the numeric grade they've earned in that course.&lt;/p&gt;&#xA;" OwnerUserId="8083" LastActivityDate="2017-05-02T05:28:54.067" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12256" PostTypeId="2" ParentId="8356" CreationDate="2017-05-02T07:24:18.367" Score="1" Body="&lt;p&gt;The tool GCLC by Predrag Janicic offer a textual language to describe geometric figures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The tool can export to tikz and pstricks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See: &lt;a href=&quot;http://poincare.matf.bg.ac.rs/~janicic//gclc/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://poincare.matf.bg.ac.rs/~janicic//gclc/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The language is described in the following paper:&#xA;&lt;a href=&quot;https://link.springer.com/article/10.1007/s10817-009-9135-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/article/10.1007/s10817-009-9135-8&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7685" LastActivityDate="2017-05-02T07:24:18.367" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12258" PostTypeId="2" ParentId="12246" CreationDate="2017-05-02T13:55:32.233" Score="6" Body="&lt;p&gt;&lt;strong&gt;1. Be aware of your goals for calling upon students.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As teachers, it is vitally important that we are goal oriented. As such, we should establish goals not only for learning outcomes but for classroom management and maintaining an atmosphere that is beneficial to learning. Whether or not the method for calling on students is &lt;em&gt;effective&lt;/em&gt; depends on whether it &lt;em&gt;effectively&lt;/em&gt; supports &lt;em&gt;your&lt;/em&gt; teaching goals. Do you want to make sure the students are paying attention? Do you want to generate formative discussion? Do you want to assess understanding? Set appropriate goals, then gear classroom questioning toward accomplishing them. Periodically, evaluate how effectively the method supports your goals, then adapt the method if necessary to better suit your needs.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2. Make question-and-answer a class activity, not an individual one.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Asking students questions, when done incorrectly, can actually be like giving the class permission to check out. Avoid creating situations that only require one person to think. If I ask, &quot;Johnny, what is the square root of 9?&quot; anyone not named Johnny knows they can relax and they miss out on the practice or the thinking that the question was meant to provoke. Instead, direct questions to the whole class. I often phrase questions similar to the following manner, &quot;Here is a question that I'd like you to answer: what is the square root of 9? I'll give you ten seconds.&quot; This way, students know what the question is and how much time they have to think about it. If I ask, &quot;Okay class, what is the value of $x$ in this diagram? I'll give you 40 seconds.&quot; Students know that they have enough time to think carefully, work out the answer on paper, and perhaps even consult a classmate- and they don't expect that they should know the answer immediately. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clearly, it's important for this method that students are not shouting out answers when they arrive at them, but waiting until the time expires with their hand raised or their pencil down, and that the teacher actually gives enough time to think about the question before calling on a student volunteer or by name. This way the student who takes longer to think or has to write intermediate steps has time to arrive at an answer and is able to meaningfully participate. This makes a huge difference in student confidence when answering questions in front of the class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3. Ask questions with open answers&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't ask questions that allow for guessing, like &quot;yes or no&quot; questions, like &quot;is this a right triangle?&quot;, or even worse, &quot;This is a right triangle, isn't it?&quot; Even if the student answers correctly, it's not as helpful for determining if they understand or are paying attention than an open ended question that must be answered in a complete sentence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4. Follow through by asking about reasoning&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regardless of whether the answer is correct, ask the student how they arrived at their answer. This will allow other students who arrived at the same answer to understand common errors, which develops an atmosphere where it's okay to give a wrong answer, or allow students who are making errors to observe how to get to the right answer. I'll often ask for a second student to try to explain the reasoning of the first in the event that the first answer is correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5. Make student selection fair.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unlike some comments on the OP, I am not of the opinion that asking open questions is unfair to introverts. On the other hand, allowing students to volunteer answers, and never calling upon students, will destroy a learning atmosphere. Students will learn that they simply don't need to try to think of an answer if they don't feel like it or they find it difficult. Then it's tempting to call on students as a disciplinary action- to ask them when you know they haven't paid attention. This is detrimental to a positive learning atmosphere, causes disdain for the teacher, and hinders learning by wasting time with poor guesses and &quot;uh, can you repeat the question?&quot;. Relying on frequent volunteers can result in students feeling that the teacher favors students who volunteer regularly or that the students who do not volunteer are inferior to students who are able to answer- &quot;She just get's it, because she's a math wiz. I just can't do that&quot;. In the event that a regular hand-raiser does get an answer incorrect, the atmosphere is not one that encourages or accepts incorrect answers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my classroom, each student has a number on their assigned desk. After I ask the question and time has expired or I see that the class has finished reasoning, I roll a large die and call on the student that it picks. This keeps the selection above criticism of unfairness and the distribution of students called upon relatively uniform. I often allow students to roll the die, which adds excitement and some humor. One can find twenty and thirty sided dice from a number of retailers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;6. Get multiple answers before revealing the correct one.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When a student answers a question, before discussing the answer, I often ask another student or two for their response. When the students have multiple answers, I get excited- it's a perfect opportunity to have them demonstrate their work or reasoning on the whiteboard. I keep extra credit tickets that I offer to students who go to the board, regardless of whether or not they are correct, so students are usually excited for the opportunity to get a bonus point on a quiz or test. This helps create a classroom atmosphere where giving the wrong answer is okay, and where being wrong can actually benefit the student directly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, find a question method that works for you, works for your students,  and allows your students the greatest advantage for learning. More discussion of these ideas in addition to a wealth of helpful suggestions about classroom management can be found in David R. Johnson's book, &lt;em&gt;Every Minute Counts: Making Your Math Class Work.&lt;/em&gt; &lt;/p&gt;&#xA;" OwnerUserId="5690" LastEditorUserId="5690" LastEditDate="2017-05-03T01:11:50.217" LastActivityDate="2017-05-03T01:11:50.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12259" PostTypeId="2" ParentId="12248" CreationDate="2017-05-02T15:51:51.083" Score="6" Body="&lt;p&gt;Math is for real people, not just mathematicians. It is useful. Real people think about and use percentages in many areas of life on a daily basis. All of my kids learned percentages long before they dealt with them in their curriculum, because they were useful to them. For example, when my daughter sees a blouse on a 60% off sale, she knows she will pay 40% of the listed price. She automatically moves the decimal in her head and multiplies by four. There were many ways she could have prolonged the mathematical computation, but the real life way that the store described the sale will always probably be a certain percent off the regular or another sale price. It is part of our language and culture. Many kids say about math: &quot;When will I ever use this?&quot; Percentages are math that they will use daily.&lt;/p&gt;&#xA;" OwnerUserId="8094" LastActivityDate="2017-05-02T15:51:51.083" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12260" PostTypeId="2" ParentId="7954" CreationDate="2017-05-02T16:16:42.337" Score="3" Body="&lt;p&gt;I find this to be an interesting question. It seems to propose that the math is language dependent. I believe that math itself is a language and we learn it alongside our other languages. Certain things are more precisely expressed in certain languages, and some languages have poetic simplicity yet need more clarification in explaining certain ideas not often expressed in that language. As a language, math is used without other languages guiding it, which is why when we are confronted with a problem in our language we need to translate it into  mathematical sentences. For an anecdote, when I was a young girl my parents went to Japan. We were there for three weeks and I had to bring my school work with me. I was learning Algebra at the time, and there was a young college student where we were staying who noticed me struggling one day. He struggled speaking English and I only knew a few phrases in Japanese. But when he sat down to help me he had no trouble guiding me through my difficulties. I learned how to recognize when he said numbers or that I had lost my sign. It was he who stressed the importance to me of keeping my work neat and orderly marching down the page to my final conclusion.&lt;/p&gt;&#xA;" OwnerUserId="8094" LastActivityDate="2017-05-02T16:16:42.337" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12261" PostTypeId="2" ParentId="4347" CreationDate="2017-05-02T17:45:22.717" Score="9" Body="&lt;p&gt;It has become almost a dogma that the math curriculum should teach technical prerequisites to what will be covered later. The consequence is that zillions of high-school students learn algorithms for doing partial-fractions decompositions and will never take later courses in which that is used. In effect the broad public learns that mathematics consists of learning to apply meaningless algorithms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;High-school teachers don't know how to alter the curriculum to something that makes more sense, and math professors don't want to do things for students who are so weak that they think mathematics consists of learning to apply meaningless algorithms, which is exactly the lesson that the high-school curriculum designed by math professors taught them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students know (and, as this applies to most students, they are right) that if they just learn the meaningless algorithms they will get good grades, but if they heed any hints that there something else to math besides that, that won't help their grades and takes time away from things that would. The professors fail to realize that what they perceive as stupidity among the students is in fact deliberate strategic stupidity; it is in fact a good strategy for what the students want to do. So both the students and the professors are deliberately stupid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is why following conventional curricula &lt;em&gt;merely&lt;/em&gt; because they are conventional is contemptible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stewart's textbook covers&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad$&lt;b&gt;(1)&lt;/b&gt; $\displaystyle\lim_x \frac{f(x)}{g(x)}$ where $f(x),g(x)\text{ (both)} \to0$ and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad$&lt;b&gt;(2)&lt;/b&gt; $\displaystyle \lim_x \frac{f(x)}{g(x)}$ where $g(x)\to\ell\ne0$ and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad$&lt;b&gt;(3)&lt;/b&gt; $\displaystyle\lim_x \frac{f(x)}{g(x)}$ where $f(x),g(x) \text{ (both)}\to\infty$ and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad$&lt;b&gt;(4)&lt;/b&gt; $\displaystyle\lim_x (f(x) - g(x))$ where $f(x),g(x) \text{ (both)} \to\infty$ and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad$&lt;b&gt;(5)&lt;/b&gt; one sided limits of artificial piecewise defined functions and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad$&lt;b&gt;(6)&lt;/b&gt; infinite limits and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad$&lt;b&gt;(7)&lt;/b&gt; various ways in which limits fail to exist&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and a lot of other possibilities. &lt;b&gt;Guess what students learn from this?&lt;/b&gt; They do &lt;b&gt;not&lt;/b&gt; learn that case &lt;b&gt;(1)&lt;/b&gt; above plays a central role in differential calculus. They write on the final exam that in a particular problem the limit does not exist because the numerator and denominator both approach $0$, and the fact that the whole of differential calculus could not exist if that were true does not occur to them; after all point &lt;b&gt;(1)&lt;/b&gt; above; they do &lt;b&gt;not&lt;/b&gt; learn that limits justify formulas about derivatives, since as far as they know, math is dogmatic rather than having justifications. &lt;b&gt;The practice of teaching technical prerequisites first is hardly, if at all, distinguishable from teaching that mathematics is dogmatic.&lt;/b&gt; It is wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably at least $99.99\%$ of those who would tell you to cover limits first have no reason for saying that except that they've never given the question an instant's thought after observing that the definition of &quot;derivative&quot; in its most usually seen form relies on a concept of limits.&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2017-05-03T23:34:16.330" LastActivityDate="2017-05-03T23:34:16.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12262" PostTypeId="2" ParentId="12248" CreationDate="2017-05-02T18:04:45.783" Score="4" Body="&lt;p&gt;Percentages are, with currency, one of the most common units one face in daily life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you go past some shop you can read &quot;20% discount&quot;, when you read the notes on anything in grocery store you can read that the stuff in your hands is stuffed with 60% of raw meat, 1% of salts, etc. When you pick a bottle of some spirit there is &quot;40 vol.%&quot; written somewhere on the label. In the bank they adveritse loans with 6% interest rate...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One need to know what that fancy circles close to a slash means. And how to operate with them, obviously.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When calculating, say 20%, of anything it is just multiplying by two and moving the decimal comma/dot one digit to the left. There is, actually, no additional layer, when you are used to read them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aditionally, the most common percentages to be calculated &quot;in your head&quot; are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;0% - nothing,&lt;/li&gt;&#xA;&lt;li&gt;12.5% - 1/8. (social and healthcare &lt;strike&gt;ransom&lt;/strike&gt; tax in CZR)&lt;/li&gt;&#xA;&lt;li&gt;25% - a quarter&lt;/li&gt;&#xA;&lt;li&gt;33.3% - roughly a third.&lt;/li&gt;&#xA;&lt;li&gt;50% - a half&lt;/li&gt;&#xA;&lt;li&gt;66.7% - roughly two third&lt;/li&gt;&#xA;&lt;li&gt;75% - three quarters&lt;/li&gt;&#xA;&lt;li&gt;100% - everything.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Other values are just to be compared between each other (6% &amp;lt; 7,5%) and between the whole (75% &amp;lt; 100% &amp;lt;130%). And one need to know that 133.3% means the whole PLUS another third extra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's the meaning of a percent for a average Joe/Lucy. For those who will stick in math more it is just another training of the brain for the future. Just like learning to ride a bike if you want to ride a motorbike.&lt;/p&gt;&#xA;" OwnerUserId="8099" LastActivityDate="2017-05-02T18:04:45.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12263" PostTypeId="2" ParentId="4347" CreationDate="2017-05-02T18:55:33.543" Score="2" Body="&lt;p&gt;You ask why to cover limits prior the derivatives when it would be easier to cover derivatives first and limits later; to show why are limits good to know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why shall we cover derivatives prior without [anything that uses derivatives]?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it is easier to define derivatives when the students are &quot;familiar&quot; with limits. Same for integrals with the knowledge of derivatives. What really matters is HOW does the teacher present the part that is taught. One shall show the students an idea where the course is about to go and why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I accidentaly passed all the exams in a highschool and thought math is one of the most boring and useless subjects. Man, how wrong I was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the university I was lucky to have the teacher I had. He followed the syllabus in terms:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Setting up the &quot;vocabulary&quot; (lemma, proof, set,...) and &quot;tools&quot; (proof,...)&lt;/li&gt;&#xA;&lt;li&gt;Operations on sets and basics of boolean logic.&lt;/li&gt;&#xA;&lt;li&gt;Sequence.&lt;/li&gt;&#xA;&lt;li&gt;Limit of a infinite sequence. (What happens at the end of the universe?)&lt;/li&gt;&#xA;&lt;li&gt;Series.&lt;/li&gt;&#xA;&lt;li&gt;Sum of infinite series. (What happens when we try to sum infinite ammount of numbers?)&lt;/li&gt;&#xA;&lt;li&gt;Function.&lt;/li&gt;&#xA;&lt;li&gt;Limits of a function. (What happens at the end? And what happens rally closeto some point?)&lt;/li&gt;&#xA;&lt;li&gt;Derivatives.&lt;/li&gt;&#xA;&lt;li&gt;Integrals.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;He taught us the way that it was interesting and fun. I cannot forget his &quot;mutated definitions&quot; that were even in tests. He randomly changed quantificators and relation in a definition and ask us to find a sequence/seires/function that fits in. He forced us to think out of the box and use any tool available.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have students' passion, you can either lead them trough blind phase (limits prior derivatives) and they will follow you or directly to the finish (derivatives) and explain all the tools behind later.&lt;/p&gt;&#xA;" OwnerUserId="8099" LastActivityDate="2017-05-02T18:55:33.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12264" PostTypeId="1" CreationDate="2017-05-03T03:19:36.343" Score="15" ViewCount="657" Body="&lt;p&gt;I am teaching Calculus I and will be teaching it again. To me, the $\varepsilon$-$\delta$ definition of limit is one of the key ideas of Calculus; learning calculus without learning $\varepsilon$-$\delta$ well is like building a tall building without a solid foundation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My problem is, most students in my class have never received any formal training in logic, whether in Pre-Calculus or in other courses. They are unfamiliar with the quantifiers and some basic ideas in logic, which makes learning the $\varepsilon$-$\delta$ definition very difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possible solution that I have never tried is: Spending one or two or three classes on logic in my Calculus class, before teaching the $\varepsilon$-$\delta$ definition. After some training in logic, it is my hope that my students will understand the concept better. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is having a few classes on logic in a Calculus class a good idea?&lt;/li&gt;&#xA;&lt;li&gt;If yes, any recommendation on materials? My goal in to cover the very basics of logic in no more than say four classes.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="205" LastEditDate="2017-07-14T02:57:05.073" LastActivityDate="2017-10-18T15:41:45.940" Title="Is it a good idea to have one or two or three classes on basic logic before teaching $\varepsilon$-$\delta$ in Calculus?" Tags="|calculus|course-design|teaching|logic|" AnswerCount="6" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="12266" PostTypeId="2" ParentId="12264" CreationDate="2017-05-03T07:36:32.047" Score="17" Body="&lt;p&gt;Your assumption that teaching calculus needs to be backed by the $\varepsilon$-$\delta$ definitions could be challenged, but since it is not your question I won't do that here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My recent experience about a few logic classes first has been disappointing. It took much hard work, and the outcome seemed good at first, but vanished as soon as we got to the main topics. While I don't have much more to build on than my experience, I tend to think that it is in fact preferable to insert small bits of logic and reasoning principles at the point of use, rather than put them aside at the beginning of the course. The student tend to keep things apart in their mind, and to fight this it seems that not separating them in lecture is a sound strategy.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-05-03T07:36:32.047" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12267" PostTypeId="2" ParentId="12264" CreationDate="2017-05-03T15:38:04.557" Score="6" Body="&lt;p&gt;It is well known that learning epsilon-delta definitions is difficult and is the intellectual equivalent of jumping over a tall &lt;em&gt;wall&lt;/em&gt; in order to join the enlighted ones on the other side, a feat never accomplished by at least three-quarters of the student population in calculus (a conservative estimate).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is clear that students need some preparation that would serve as a ladder to help them over the proverbial &lt;em&gt;wall&lt;/em&gt;.  The use of alternating quantifiers needs to be prepared.  This can be done either by giving a series of examples of formulas of increasing quantifier complexity, or by providing a more intuitive approach to the key concepts like the derivative and continuity first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://dx.doi.org/10.5642/jhummath.201701.07&quot; rel=&quot;nofollow noreferrer&quot;&gt;One recent proposal in the literature&lt;/a&gt; is to teach them rigorous infinitesimals as a way of providing a ladder so as to climb the &lt;em&gt;Epsilontik&lt;/em&gt; wall. Once the students understand the basic concepts like &lt;em&gt;derivative&lt;/em&gt; and &lt;em&gt;continuity&lt;/em&gt; via their intuitively appealing definitions exploiting infinitesimals (essentially following &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-017-9534-y&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy's approach&lt;/a&gt;), they are in a better position to scale the long-winded epsilon-delta &lt;em&gt;paraphrases&lt;/em&gt; of those definitions.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-10-18T15:41:45.940" LastActivityDate="2017-10-18T15:41:45.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12268" PostTypeId="2" ParentId="12246" CreationDate="2017-05-03T18:16:34.873" Score="5" Body="&lt;p&gt;I have the opportunity to teach both high school students and undergraduate students in College Algebra and College Trigonometry.  In my high school classes, I call on each student at least once per class.  This has always been a practice of mine as it encourages participation and engagement.  I strongly disagree with calling on students in a college class because it is not my job for students to be engaged in a college class.  Students should be self motivated in college or they won't succeed.  If they can't pay attention to a lecture, then they probably won't and shouldn't be successful in college.  Self motivation has to be a factor to success or a college degree will continue to become more and more watered down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your question: An effective way to call on students is, as mentioned above, to know your students.  Do not start immediately randomly calling on students.  Only call on students after you've gotten a few formal assessments graded.  Then, you know Johnny struggles, so you throw him your softball questions.  And, you know Sally is doing very well.  So, you may throw her some questions that require critical thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Be sure to have follow up questions that could lead a student to an answer.  This is important because some students will sit silently or say they don't know until you call on someone else.  Guide the student to what you're looking for by asking simpler follow up questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Affirm students whether they are correct or incorrect.  Using words like &quot;No&quot; will make students more anxious and less likely to be willing to participate in the future.  Instead, you might say &quot;You're on the right track... Let me ask you this follow up question....&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I don't agree with the practice of calling on students in a college math class, these are a few approaches you may use.&lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2017-05-03T18:16:34.873" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12269" PostTypeId="2" ParentId="12248" CreationDate="2017-05-03T23:47:08.840" Score="2" Body="&lt;p&gt;Percent literally means &quot;per hundred&quot;. In other words - x% is a shorthand for the fraction $\frac{x}{100}$. In fact, the % signal is an abbreviation of /100 (lose the 1 to get /00, move the slash to get 0/0, now shrink it and you have %). In the days when accounting was done with quill and parchment (did you know that writing was initially developed for bookkeeping, and that literature and story telling came much later?), compact ways of writing things were very important. So the percentage sign became a shorthand for a convenient fraction. Why was it convenient?&#xA; A tax (excise) was often expressed as &quot;pennies on the dollar&quot; (or whatever your favorite currency was). When currency was divisible by 100, a tax that was also divisible by 100 made for easy math. You need to pay 5% on 35 dollars, that will be 5x35 cents please.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That makes for easy visualization as well: of every dollar, you give me 5 cents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally - fractions are usually taught before decimals. Intuitively, fractions are simpler - and when you use the same denominator every time, they are simpler yet. By comparison, the &quot;magic&quot; of moving a decimal point around is something that is not at all obvious - as demonstrated by the rather sobering statistics quoted in &lt;a href=&quot;http://www.psy.cmu.edu/~siegler/2015-LF-etal.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So percentages are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Easy to write&lt;/li&gt;&#xA;&lt;li&gt;Easy to calculate&lt;/li&gt;&#xA;&lt;li&gt;Easy to visualize&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The implication is that (integer) percentages are something you can learn about, and manipulate, without even knowing what a decimal point is. This also helps with learning about relative size of things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper I quoted above shows ample evidence that pupils have a really hard time with magnitude of answers. The percentage, as a simple fraction of a constant denominator, should provide a pathway to help estimating results of calculations without getting bogged down in manipulation. If it is taught in that way, I believe it has a great deal of usefulness in the early formation of the quantitative skills every adult needs to have.&lt;/p&gt;&#xA;" OwnerUserId="8112" LastActivityDate="2017-05-03T23:47:08.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12270" PostTypeId="2" ParentId="7954" CreationDate="2017-05-04T09:23:20.613" Score="3" Body="&lt;p&gt;The idea that mathematics is natural language-dependent is a tempting one, and would be interesting to see an illustration.  The particular example you give, namely an alleged relation between (1) emphasis on existence and uniqueness issues and (2) the existence of definite and indefinite articles in many Western languages, fails to convince because in Russian for example there are no definite or indefinite articles yet there is no evidence that the Russians are any less preoccupied by issues of existence and uniqueness than English speakers.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2017-05-04T09:23:20.613" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12271" PostTypeId="1" AcceptedAnswerId="12272" CreationDate="2017-05-04T21:06:07.353" Score="20" ViewCount="5256" Body="&lt;p&gt;I adjunct for a local community college teaching College Algebra and College Trigonometry.  Every year, the community college math department insists on students memorizing each of the trig identities.  I can understand why students should have the pythagorean identities memorized and the unit circle memorized.  But, why should a student have to memorize double angle, half angle, product to sum formulas, etc?  I'm trying to get a better grasp on why the college requires this and will not allow students to have a notecard for these.  I don't remember the product to sum formula being a pre-requisite for Calculus or any higher level courses.  Does anyone have any insight on this, or is it just an odd request from the college?  &lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2020-01-05T23:25:16.350" Title="Memorizing Trig Identities" Tags="|trigonometry|" AnswerCount="9" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12272" PostTypeId="2" ParentId="12271" CreationDate="2017-05-04T21:28:13.043" Score="17" Body="&lt;p&gt;I agree this memorization is not necessary.&lt;/p&gt;&#xA;&lt;p&gt;If students understand how the trigonometric functions are defined (unit circle) and know several basic identities, everything else can be derived. I think the pythagorean identity and the sine and cosine of sum of angles are sufficient.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin^2 \alpha + \cos^2 \alpha = 1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Math is not about memorization. If students are taught the basics and learn how to derive whatever they need, everybody will be happier.&lt;/p&gt;&#xA;" OwnerUserId="8119" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-05-04T23:10:30.510" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12273" PostTypeId="2" ParentId="12271" CreationDate="2017-05-04T21:33:12.223" Score="10" Body="&lt;p&gt;If you want to integrate $\sin^m x$ or $\cos^m x$ for even $m$, you need to reduce the powers, which requires some trigonometric identities beyond pythagorean identities (in this case, $\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$). If the students are struggling with the application of trigonometric identities at this stage, then they are in trouble when they are asked to perform a &quot;simplification&quot; like the following before the calculus even begins:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\sin^4 \theta = \left(\sin^2\theta\right)^2 = \left(\frac{1-\cos2\theta}{2}\right)^2 = \frac{1 - 2\cos2\theta + \cos^22\theta}{4} = \frac{1 - 2\cos2\theta + \frac{1+\cos4\theta}2}{4}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A very very large number of students in my calculus classes do not get the &quot;$4\theta$&quot; at the end there, and the ones who are having trouble applying the identities in this way certainly never get the factor of $\frac14$ that comes from integrating such a thing, because they are buried in arcane symbols.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I can imagine, at least for &quot;double angle&quot; or &quot;half angle&quot; formulas, that this line of reasoning would justify students committing these things to memory. If our calculus students see this &quot;power-reducing&quot; identity as &quot;easy stuff I learned a long time ago,&quot; then they can much more easily reach the goals I've set out above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I don't know where to draw the line on memorization. But I can easily support drawing the line a little further out than just pythagorean identities and the unit circle.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-05-04T21:33:12.223" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12274" PostTypeId="2" ParentId="7954" CreationDate="2017-05-04T21:56:36.860" Score="2" Body="&lt;p&gt;I think the point is that quantifiers often roughly translate into more-or-less colloquial English in a way that can be expressed by choice of articles. That is, there is a not-so-bad mapping from mathematics to colloquial English. That's a good thing. However, the point that other natural languages do not have the details of English, and thus would not allow the same rough mapping to natural/colloquial language, certainly does not entail any lack of mapping.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For that matter, if we go so far as to ask about mapping of first-order logic expressions of mathematics to natural language, I think we are asking for needless trouble, insofar as it is my feeling that the mapping of mathematics to first-order logic is imprecise and does not capture everything we'd want... despite the benefits of the mapping landing us in first-order logic, which achieves (by some simplisticness, etc) a seeming clarity that has admitted appeal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is surely worth noting that in natural-language settings, rarely do existence and uniqueness play a significant role, because they're either irrelevant or strongly implicit. So it would be inaccurate to say that colloquial English really makes that distinction. Rather, mathematicians speaking English have overloaded the colloquial language with responsibility for making distinctions that would be incomprehensible to almost all native English speakers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, just on the grounds of these disconnects, one shouldn't think that natural languages with different grammar would have trouble mapping mathematics into themselves. After all, again, the alleged distinctions made by the mapping in English are incomprehensible to most native speakers. It's just that the mapping would be different.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-05-04T21:56:36.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12275" PostTypeId="2" ParentId="12271" CreationDate="2017-05-05T04:00:19.150" Score="8" Body="&lt;p&gt;If you have lots of time then you can surely derive most of the identities from the basics but basically it slows down your learning when these identites are required. Also not memorising these will give you a huge disadvantage in exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another reason: sometimes it is required to know the form of the result of applying an identity quickly. Like while doing integration probably you will want to know that $\sin(x)\cos(y)=\text{some constant}*\text{sum of trig functions}$. You cannot just keep deriving identities and checking weather they will work or not. In other words you must know which identity to use and when and how.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly I want to tell you my approach for memorising these:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(i) Everyday take one identity to be memorised&lt;br&gt;&#xA;  (ii) Derive it once in the beginning&lt;br&gt;&#xA;  (iii) Solve many questions related to it&lt;br&gt;&#xA;  (iv) If you forget it in between, rederive it&lt;br&gt;&#xA;  (v) repeat from step (iii) until it is memorised.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The advantage of this approach is that it will give you a solid understanding of the identity and also you will know where and when to apply it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact I had never lerned the identities in the trig class ..  i somehow got away with it. But when i went further and studies coordinate geometry, conic sections, integration, inverse trigonometry, etc. then I realised the importance of memorising them.&lt;/p&gt;&#xA;" OwnerUserId="1289" LastActivityDate="2017-05-05T04:00:19.150" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12276" PostTypeId="2" ParentId="12264" CreationDate="2017-05-05T08:05:01.090" Score="9" Body="&lt;p&gt;I had the same thought this year. My suspicion was that many students get anxious about suddenly dealing with quantifiers and they also struggle with understanding how the ordering of them can affect the logical statement. To try to remedy this, I spent a short amount of time (less than 2 classes) introducing quantifiers and practicing parsing simple logical statements about real numbers starting from 1 variable (e.g. $\forall x,\thinspace x^{2}+1&amp;gt;x$) to 3 variables (e.g. $\forall x\exists y\forall z,\thinspace x-\left|y\right|\leq\left|1-z\right|$). I will say that it was helpful in that I was able  to catch some misunderstandings before we got to the calculus concepts and it was very easy to resolve them at that stage. As for this approach's effectiveness and the ideal materials to use, I am not sure and I am also interested in hearing others' experiences.&lt;/p&gt;&#xA;" OwnerUserId="8121" LastEditorUserId="8121" LastEditDate="2017-05-05T08:18:38.027" LastActivityDate="2017-05-05T08:18:38.027" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12277" PostTypeId="2" ParentId="12220" CreationDate="2017-05-05T17:02:26.983" Score="2" Body="&lt;p&gt;You can find information about how Pennsylvania handles acceleration in education at this address.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.accelerationinstitute.org/Resources/Policy/By_State/Show_Policy.aspx?StateID=45&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.accelerationinstitute.org/Resources/Policy/By_State/Show_Policy.aspx?StateID=45&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you truly want your son placed in Algebra II, you may need to consider educational alternatives if the local school board is not accommodating. Considering the schedule for Algebra II may conflict with the rest of his classes, it might be more difficult for them than just the evaluation, which in itself should not be difficult at all.&lt;/p&gt;&#xA;" OwnerUserId="8094" LastActivityDate="2017-05-05T17:02:26.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12278" PostTypeId="1" CreationDate="2017-05-06T04:15:57.233" Score="5" ViewCount="102" Body="&lt;p&gt;As the title, I want to ask for some examples showing necessity and advantage of emperical definition of probability in teaching high school students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We knew that the classical definition has some disadvantages, i.e, the sample space has to be finite and all cases are equally possible. Those are motivations for mathematicians to come up with the emperical definition. I would like to know some simple examples (to teach high school students) which we cannot apply the classical definition to calculate the probability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you very much.&lt;/p&gt;&#xA;" OwnerUserId="7241" LastActivityDate="2017-05-06T19:26:21.753" Title="Examples showing necessity and advantage of emperical definition of probability" Tags="|secondary-education|probability|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12279" PostTypeId="1" CreationDate="2017-05-06T05:20:11.510" Score="13" ViewCount="214" Body="&lt;p&gt;I have a question about formulating problems and exercises in Mathematics. When attempting to create a problem of Number Theory or Real Analysis, for example, in this process, is the problem first established completely and only then is its solution or do the solution and formulation of the problem happen simultaneously or even, is the solution  first thought, and then the problem is established ? What is the order?&lt;/p&gt;&#xA;" OwnerUserId="6569" LastEditorUserId="11" LastEditDate="2017-05-12T16:23:15.797" LastActivityDate="2017-05-12T16:23:15.797" Title="Question about the process of creation of problems and exercises in Mathematics" Tags="|undergraduate-education|graduate-education|problem-design|problem-solving|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12280" PostTypeId="2" ParentId="12278" CreationDate="2017-05-06T05:53:38.750" Score="4" Body="&lt;p&gt;Consider any complicated real-world event. How likely is it that a given airplane crashes? Will Russia invade more parts of its neighbouring countries within 10 years? How far will an ant walk in a minute?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The classical definition utterly breaks in all of them. The third can be treated easily by repeated experiments (even as classroom activity, if one is so inclined, though maybe with a shorter time period). The second may not. The first one is a borderline case; should you use all airplanes or only the airplanes that share particular qualities?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I presume the ant example, or some other where the space of outcomes is continuous but easily observed, is what you are looking for. The others are more advanced and overshoot your goals.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2017-05-06T05:53:38.750" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12281" PostTypeId="2" ParentId="12278" CreationDate="2017-05-06T19:26:21.753" Score="3" Body="&lt;p&gt;Sports statistics should give plenty of examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, what is the probability of a successful free throw in college basketball?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a newspaper article discussing how this number has stayed remarkably consistent over the years: &lt;a href=&quot;http://www.nytimes.com/2009/03/04/sports/basketball/04freethrow.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;For Free Throws, 50 Years of Practice Is No Help&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2017-05-06T19:26:21.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12282" PostTypeId="1" AcceptedAnswerId="12283" CreationDate="2017-05-07T11:41:07.477" Score="6" ViewCount="199" Body="&lt;p&gt;My students haven't learned about ratio yet, but in the test, there are questions that relate to ratio. I'm not sure how to explain this kind of problem without talking about ratio. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;every 100g soup contains 150kj energy, one serve of the soup contains 450kj energy, what is the mass of one serve of the soup?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A 8mm * 8mm square computer chip is represented by a 4cm * 4cm graph, what scale is used in the drawing?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;How would you explain this kind of problem without talking about ratio? Thank you!&#xA;&lt;/p&gt;&#xA;" OwnerUserId="7487" OwnerDisplayName="EmmaXL" LastEditorUserId="8132" LastEditDate="2017-05-07T21:56:33.980" LastActivityDate="2017-05-09T12:14:00.657" Title="How can I explain ratio problems without talking about ratio?" Tags="|secondary-education|word-problems|proportional-reasoning|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12283" PostTypeId="2" ParentId="12282" CreationDate="2017-05-07T14:23:03.983" Score="8" Body="&lt;p&gt;Have your students covered division and multiplication yet?  To me, these problems can all be solved using the &quot;meaning of division&quot; and the &quot;meaning of multiplication&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Meaning of multiplication&quot;:  If I have n groups of equal size m, then I have n*m total objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Meaning of division&quot;:  There are actually two!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a) If I have m objects which are partitioned into n equally sized groups, then I have m/n objects in each group (this is the &quot;how much in each group&quot; interpretation of division).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) If I have m objects, and I put these objects into equally sized groups where each group has n objects in it, then I have m/n groups of objects (this is the &quot;how many groups&quot; interpretation of division).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are the answers to your questions from this perspective:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1.&#xA;Every 100g of soup contains 150kj of energy.  Call 100g of soup a &quot;portion&quot;.  I have a &quot;serving&quot; of soup which contains 450 kj of energy.  So, if I want to see how many 100g portions of soup are in one serving, I need to see how many groups of 150kj there are in 450kj. The answer to this question (by &quot;the meaning of division&quot;) is 450/150 = 3 portions.  Then, since each of these 3 equally sized portions of soup is 100g, (by &quot;the meaning of multiplication&quot;) I have 3*100g = 300g of soup.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2.&#xA;4cm represents 8mm.  So I have 4 centimeters, which, in total, represent 8 millimeters.  How many millimeters does one centimeter represent? Each centimeter represents an equal sized group of millimeters.  So I have 4 equally sized groups of millimeters.  Thus each group has 8/4 = 2 millimeters in it.  So each centimeter represents 2 millimeters.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2017-05-09T12:14:00.657" LastActivityDate="2017-05-09T12:14:00.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12284" PostTypeId="2" ParentId="12264" CreationDate="2017-05-08T01:47:03.997" Score="7" Body="&lt;p&gt;Generally speaking, it would be nice to have a foundations class at the initiation of the Math major. Some of my colleagues envision this course centered around teaching &lt;em&gt;college algebra&lt;/em&gt;. Well, to be more precise, algebra done with care, algebra beyond symbol pushing and problem solving. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Foundations course we currently run in our major requires Calculus II as a prerequisite, but, this is mostly for maturity. Logically, it ought to go at the start. If the proofs course was first then more students would have a better view to what &lt;em&gt;major&lt;/em&gt; course work looks like in Math. Calculus is still very much a general education requirement, so, the inculcation of students into native math culture really only begins in the proofs course at our institution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, it's not really this simple. In fact, we do have epsilon delta proofs in our first semester calculus. Indeed, we also study convergence and divergence of sequences, sometimes the least upper bound property and other bits a pieces of analysis in the second semester. However, these are not the true flavor of the calculus sequence. They're deviations from the main story of calculation and application. In my personal opinion, the real analysis needed for second semester calculus is far more sophisticated than mere epsilon delta proofs. For example, the proof that the derivative of a power series is the derivative of a power series is quite beyond the students. I'm certain of this as none of them ever complained the many years I failed to appreciate the need of the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We introduce integrals in calculus I in terms of limits of sequences! Many texts do this. So, this is after we cover sequential limits, right? Wrong. The limit appearing in the Riemann integral is merely intuitively framed by justifying approximations. Now, don't misunderstand, I understand the place of providing an intuitive version of calculus before rigor sets in. But, I am aware of the hypocrisy of carefully covering epsilon deltas in the same semester I fail to properly define limits of sequences, much less the integral itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, why study epsilon deltas in calculus I? It doesn't seem practical to maintain the level of detail throughout the calculus sequence. We must admit there are tools we wish to use, but, not construct; how many major theorems do we have time to present the proof? I have a lot of time here, but, even so, it's not possible to cover everything. I can tell you from experience, the students only appreciate so much honesty in a given semester. So, why epsilon deltas?&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;it gives students a much needed chance to study how inequalities work.&lt;/li&gt;&#xA;&lt;li&gt;it can be a great chance to better understand how graphs can organize analytical thought&lt;/li&gt;&#xA;&lt;li&gt;the logic isn't that complicated if you do a half-dozen examples. Keep it simple, basic functions like linear, quadratic, reciprocal all are helpful.&lt;/li&gt;&#xA;&lt;li&gt;it allows you to be precise in the meaning of a limit not existing, what we mean when we say the limit point need not be considered in the evaluation of a limit&lt;/li&gt;&#xA;&lt;li&gt;it does make introduction of sequential limits a natural continuation of an existing discussion&lt;/li&gt;&#xA;&lt;li&gt;the students who can think can do it. It's not that big a deal.&lt;/li&gt;&#xA;&lt;li&gt;for students who are math majors at heart, it's a breath of fresh air in an otherwise logically questionable development. I have had the topic gain the major new students. In fact, my best student to date came to Math from a different major in large part because I was overly formal in Calculus I. It can make a difference.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Ok, so, how many students don't get it? Lots. Just like lots of students don't get a number of topics in calculus. Convergence, divergence, the chain rule, the idea that you plug the number into the derivative to find the equation of a tangent line, this list is endless really. There is a risk and a possible reward, you have to decide on the basis of your audience whether it's worth the risk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, do you need a logic class beforehand? No, not in my experience.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-05-08T01:47:03.997" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12285" PostTypeId="1" CreationDate="2017-05-09T07:18:46.327" Score="29" ViewCount="1293" Body="&lt;p&gt;Here is an overview of the practice of &quot;gateway testing&quot;, which explains it much better than I could:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://sites.lsa.umich.edu/michigan-math-in-action/2015/09/24/25-years-gateway-testing-at-michigan/&quot; rel=&quot;noreferrer&quot;&gt;https://sites.lsa.umich.edu/michigan-math-in-action/2015/09/24/25-years-gateway-testing-at-michigan/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for completeness I'll quote the basic idea:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What’s a gateway test? Our definition is that it is a test of basic skills. These may be skills that are prerequisite for success in a course, or may be skills which every student in a course should develop.... However, while these skills are intrinsic to or essential for success in the courses, they are not our educational focus. Our courses focus on conceptual understanding... Our gateway tests are 7 or 10 question tests, administered on-line... which students may take multiple times and which they must complete with almost no errors to pass.... Students may practice the tests as many times as they like, but to get credit for having passed the test they must take the test in a proctored lab where their identity is verified and where they are not allowed to use outside resources.... completion of the test is not a part of the students’ course averages; instead, any student who doesn’t pass a gateway by the specified deadline... has her/his final grade reduced by 1/3 to a full letter grade at the end of the semester.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The above link includes a bit of evidence that the gateway tests work at Michigan, at least that they correlate with students learning the gateway material.  But I would like to know if any more serious data analysis has been done on gateway tests (anywhere)?  In particular, I would like to know whether implementing gateway tests has a positive effect on &lt;em&gt;overall&lt;/em&gt; student performance.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2018-11-01T22:42:23.390" Title="Do &quot;gateway tests&quot; work?" Tags="|undergraduate-education|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="12286" PostTypeId="1" CreationDate="2017-05-09T19:01:50.613" Score="13" ViewCount="468" Body="&lt;p&gt;I would like to know if there is a simple real-world problem which requires knowing a closed form for $\displaystyle \sum_{i=1}^n i$ and/or the sum of the first $n$ even/odd numbers.  The only application I can think of is when doing Riemann Sums, but this student has not taken calculus. &lt;/p&gt;&#xA;" OwnerUserId="7075" LastActivityDate="2017-12-04T16:14:48.047" Title="Is there a simple real-world problem I can use to motivate a formula for $\displaystyle \sum_{i=1}^n i $?" Tags="|concept-motivation|examples|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12287" PostTypeId="1" AcceptedAnswerId="12295" CreationDate="2017-05-09T20:49:33.743" Score="6" ViewCount="540" Body="&lt;p&gt;Colleagues. In a few weeks I will be interviewed for a position at a community college. I got selected for an interview and the teaching demonstration is as follows: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume you are teaching a Multivariable Calculus class (Calculus III):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose  $(a, b)$ is a critical point of $f(x,y)$.  Introduce the Second Derivative Test with an emphasis on why it is valid for finding local maxima, local minima, and saddle points.&#xA;(A rigorous proof is not necessary.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's what I have planned:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Introduce the second derivative test talking about the input points where the gradient is zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Show a graphical representation of the points where the tangent plane on the graph of f is flat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Talk about the second partial derivative test and what defines a max/min, saddle etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Reasoning behind the validity for finding local max, min and saddle points - &lt;strong&gt;here's where I need help. A rigorous proof is not necessary so how deep should I go if I only have 15 minutes to present?&lt;/strong&gt; &lt;/p&gt;&#xA;" OwnerUserId="2198" LastActivityDate="2017-05-24T15:49:13.353" Title="Calculus 3 Teaching Demonstration for Community College Teaching Position" Tags="|mathematical-pedagogy|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12288" PostTypeId="2" ParentId="12286" CreationDate="2017-05-09T20:58:08.247" Score="7" Body="&lt;p&gt;I am not quite sure if this answers the question in the body of your post, but here is how I have motivated the search for a closed form regarding these summations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summing the first $n$ counting numbers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used this formula when I demoed for my current teaching position, and think it is a good way to start a new year off, too: To learn student names, I have each of them say their own name, and then the name of each of their predecessors. So, the first person says just her own name ($1$ name), the second person says her sole predecessor's name as well as her own ($2$ more names), the third person says each of the first two's names and then her own ($3$ more names), and so forth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end, I ask the students whose name was said the most times, whose name was said the fewest times, and how many total name utterances there were. Generally this is solved in one of three (related) ways: the Gaussian summation trick of duplicating the total, evaluating it as $n(n+1)$, and dividing by $2$ for the answer; pairing the number of utterances by the first and last speaker (i.e., $1$ and $n$), the second and penultimate speaker ($2$ and $n-1$) etc., with a small caveat around when there are an odd number of students; or multiplying the average number of utterances by the number of students (to my surprise, a group of eighth graders devised this method when I actually demoed!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to follow-up and ask a question such as, e.g., for a class of $16$ students: Would the total number of utterances be greater, fewer, or the same, if instead of having them go through the names as above, we broke into two groups of $8$ students and had each group proceed with this name game? (This can be computed as $16(17)/2$ versus $2 \cdot 8(9)/2$, or answered without computation...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you devise a method of summing the first $n$ counting numbers and realize it as $n(n+1)/2$, then the sum of the first $n$ even numbers is (clearly) twice this, i.e., $n(n+1) = n^2 + n$. And once you know that, you can sum the first $n$ odd numbers by summing the first $n$ even numbers and subtracting $1$ from each of the $n$ addends, which gives a total of $(n^2 + n) - n = n^2$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An application of the fact that the first $n$ odds sum to $n^2$ arises when one broaches quadratic functions. Specifically, students in my algebra classes will write a quadratic expression in either vertex form, $a(x-h)^2 + k$, or standard form, $ax^2 + bx + c$, and then graph them by finding the vertex, first, and then (when the quadratic is monic, i.e., $a=1$) going over one, up $1$; over another, up $3$, over another, up $5$, etc. When the quadratic has a coefficient $a \not = 1$, then you can take these same rises and multiply through by $a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the quadratic function defined by $x \mapsto 2(x-3)^2 - 8$ can be graphed by first noting that the vertex is at $(3, -8)$, and then going over one (to the left and right) and up $2 \cdot 1 = 2$; over another, up $2 \cdot 3 = 6$; over another, up $2 \cdot 5 = 10$, etc. This is a quick way to graph the quadratics that we are dealing with, but making sense of &lt;em&gt;why this works&lt;/em&gt; relies on the closed formula for the sum of odds. (So, any problem you have around quadratics that benefits from a graphical representation can be readily connected to $\sum$odds as described above.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2017-12-02T16:01:48.797" LastActivityDate="2017-12-02T16:01:48.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12289" PostTypeId="2" ParentId="12286" CreationDate="2017-05-09T21:50:15.573" Score="9" Body="&lt;p&gt;If $n$ people meet and everyone shakes hands with everyone else, how many hand shakes were there?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could do this combinatorially, e.g. $n$ choose 2, but another way to think about it is that the first person shakes $n-1$ times, the next $n-2$, ... until the second to last person only shakes the last person's hand. This leads to the summation.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2017-05-09T21:50:15.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12290" PostTypeId="2" ParentId="12286" CreationDate="2017-05-09T22:21:30.760" Score="8" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You are able to buy cubical wooden blocks of unit size, and you want to build a staircase to get over a wall of height $n$ units. How many blocks are required?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A gold mining operation has two shifts. The day shift bores a tunnel that is deeper by an additional depth $\delta$ each day. The night shift comes in and works the whole accumulated length of the tunnel every day, extracting gold worth $g$ dollars per unit of depth. Find the gross revenue after $n$ days.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The population of a town grows over time as 1, 2, 3, ... Each person produces one unit of garbage per year, which fills up a landfill.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;[Re #3, from comments:]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is that model close to the actual way populations grow? I don't want to make the example sound too artificial.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would approach this from the point of model building. Models don't need to be perfectly realistic in order to be informative. Also, linear population growth is likely to be a pretty decent approximation over an appropriate time interval (say a decade).&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2017-05-09T22:27:00.483" LastActivityDate="2017-05-09T22:27:00.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12291" PostTypeId="2" ParentId="12286" CreationDate="2017-05-09T23:58:54.700" Score="10" Body="&lt;p&gt;Students sometimes find applications to computer programming interesting. The sum in question often pops up in determining the complexity of various algorithms. For a simple example, &lt;a href=&quot;https://en.wikipedia.org/wiki/Selection_sort&quot; rel=&quot;noreferrer&quot;&gt;selection sort&lt;/a&gt; sorts a list of $n$ items, by first scanning over it, selecting the first element , then scanning over the remaining $n-1$ items to find the second element of the sorted list, etc. The total amount of scanning is $1 + 2 + ... + n$. Alternatively, it involves $1 + 2 + ... + (n-1)$ pairwise comparisons in the course of those scans. In either case, the formula for the sum shows that selection sort is $O(n^2)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, this sort of analysis is applicable to any algorithm with nested for-loops where the outer loop has its index (say $i$) ranging from $1$ to $n$ and the inner loop has its index (say $j$) ranging from $1$ to $i$. &lt;/p&gt;&#xA;" OwnerUserId="6891" LastEditorUserId="6891" LastEditDate="2017-05-10T00:04:09.310" LastActivityDate="2017-05-10T00:04:09.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12292" PostTypeId="2" ParentId="12287" CreationDate="2017-05-10T02:14:43.210" Score="5" Body="&lt;p&gt;In my opinion, this is a really difficult topic because the &quot;appropriate technology&quot; for understanding this stuff is at a higher level than is appropriate for most Calc 3 students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway here is how I think about it:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $f:\mathbb{R}^n \to \mathbb{R}$ be a sufficiently nice function (let's say 3 times differentiable).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the second order Taylor expansion of $f$ at a point $p$ is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;f(p+\vec{v}) \approx f(p)+\nabla f\big|_p \cdot \vec{v} + \vec{v}^\top H\big|_p \vec{v}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $H\big|_p$ is the Hessian matrix of $f$ at $p$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So at a critical point&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;f(p+\vec{v}) \approx f(p)+\vec{v}^\top H\big|_p \vec{v}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the Hessian is symmetric (by Clairaut's theorem), there is an orthonormal basis which diagonalizes this matrix, by the real spectral theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second derivative test comes down to looking at the definitness of the Hessian, which you can read off from the eigenvalues.  If the eigenvalues are all positive, the term $\vec{v}^\top H\big|_p \vec{v}$ is also positive, so $f(p)$ is a local min.  If they are all negative then you get a local max.  Some positive some negative leads to a saddle.  A zero eigenvalues means you cannot tell, and you probably have to look at higher terms of the Taylor polynomial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally we only treat this the case that $f:\mathbb{R}^2 \to \mathbb{R}$.  The reason is because they do not know about eigenvectors/eigenvalues yet.  In the two dimensional case, you can use a trick:  determinant is the product of eigenvectors.  So pos determinant means they both have the same sign, negative means a saddle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This story is not really told well anywhere that I have seen.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To translate this into some implementable advice, I would say focus on the idea of &quot;best quadratic approximation&quot;.  Investigate what the graphs of homogeneous quadratic functions in two variables look like.  Pay particular attention to understanding things like $\pm(ax+by)^2\pm (cx+dy)^2$.  If you can understand these, you should be able to understand anything by completing the square, but completing the square is harder with two variable polynomials if you do not know some linear algebra (unless someone can educate me in the comments about how to do it!)&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2017-05-10T02:14:43.210" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12293" PostTypeId="2" ParentId="12286" CreationDate="2017-05-10T18:34:34.277" Score="9" Body="&lt;p&gt;Perhaps you can make the topic &quot;applicable&quot; by comparing it to some popular social media posts of the form, &quot;How many triangles are in this figure? &lt;em&gt;Only geniuses&lt;/em&gt; get this right!&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mZO0O.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mZO0O.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take a $5\times 1$ rectangular grid, for example, and ask the students: How many rectangles &lt;em&gt;of any size&lt;/em&gt; can be found in this figure? Surely, some of them will just start counting right away and find that the answer is 15. Then, prod them further by asking, &quot;Okay, of those 15 rectangles, how many of them are $1\times 1$? How many are $2\times 1$? etc. up to $5\times 1$&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students will see that there are 5, 4, 3, 2, and 1 such rectangles, respectively. Hopefully, they will then realize how to &lt;em&gt;generalize&lt;/em&gt; the problem: if we start with an $n\times 1$ rectangle, the total number of rectangles will be $n+(n-1)+(n-2)+\cdots+3+2+1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, ask them what the answer to this problem would be if $n=100$. See how many students started punching calculator buttons or doing mental/paper arithmetic. Stop them immediately and say, &quot;Surely, there must be a better, &lt;em&gt;more efficient&lt;/em&gt;, method than just adding 100 numbers!&quot; This &lt;em&gt;motivates&lt;/em&gt; the necessity of a closed-form solution, and you can use this to jump into finding such a formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Addendum:&lt;/em&gt; Depending on the audience and time available, you could also use that problem to introduce binomial coefficients, since ${n+1\choose 2}$ is the solution to that problem. You can then segue into a combinatorial &quot;proof without words&quot; for that same fact. The image below demonstrates a bijection between the $1+2+\cdots+n$ elements in the upper triangle and &lt;em&gt;pairs&lt;/em&gt; of elements in the row of $n+1$ elements in the bottom row (&lt;a href=&quot;https://math.stackexchange.com/questions/2260/proof-for-formula-for-sum-of-sequence-123-ldotsn&quot;&gt;source via math.se&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tSuyY.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tSuyY.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, you could follow this up by exploring a similar 2-dimensional problem: how many squares can you find in an $m\times n$ rectangular grid? The solution is ${m\choose 2}\cdot{n\choose 2}$. &lt;a href=&quot;https://math.stackexchange.com/questions/2260/proof-for-formula-for-sum-of-sequence-123-ldotsn&quot;&gt;See this link for more&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2017-05-10T18:56:53.137" LastActivityDate="2017-05-10T18:56:53.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12294" PostTypeId="2" ParentId="12287" CreationDate="2017-05-11T16:10:18.570" Score="4" Body="&lt;p&gt;Here is a more down to Earth approach, which incorporates the same ideas as above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this topic really deserves several days to be done right, not just 15 minutes!  So you might have to summarize rather than actually try to fit it all in.  Or just do the first parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A.&lt;/strong&gt; Start with some motivation:  we want to get to a second derivative test for functions of two variables.  This is not as easy as in 1 variable!  Reteach second derivative test from single variable using a new perspective:  the 2nd degree Taylor polynomial says function &quot;looks like&quot; $f(p)+f'(p)(x-p)+ f''(p)(x-p)^2$ near a point $p$.  So when $f'(p)=0$ we have a chance at having a max or min.  Because we know $(x-p)^2$ is always greater than zero, we can tell whether max or min by sign of $f''(p)$.  If $f''(p) = 0$, we get no info and would have to climb higher up on the Taylor series.  We want to do this in two variables now.  However, that involves 2nd degree Taylor approximation near $(a,b)$: $$ f(x,y) \approx f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)+ \frac{1}{2}f_{xx}(a,b)(x-a)^2+f_{xy}(x-a)(y-b)+ \frac{1}{2}f_{yy}(a,b)(y-b)^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to tell a similar story with multiple variables, we will need to have a really good understanding of expressions like $ax^2+bxy+cy^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;B.&lt;/strong&gt; First get students to graph some basic quadratic forms.  Examples, in increasing order of difficulty:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$z=x^2$&lt;/li&gt;&#xA;&lt;li&gt;$z=y^2$&lt;/li&gt;&#xA;&lt;li&gt;$z=2x^2+3y^2$&lt;/li&gt;&#xA;&lt;li&gt;$z=-2x^2-y^2$&lt;/li&gt;&#xA;&lt;li&gt;$z = 4x^2-y^2$&lt;/li&gt;&#xA;&lt;li&gt;$z=7y^2-2x^2$&lt;/li&gt;&#xA;&lt;li&gt;$z=3(x+y)^2+4(x-2y)^2$&lt;/li&gt;&#xA;&lt;li&gt;$z = 2(x-y)^2-(x+y)^2$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;C.&lt;/strong&gt; Make sure that they understand when these quadratic forms are nonnegative, nonpositive, indefinite, and when they are only semidefinite.  You do not have to use these words, but get the concepts across.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;D.&lt;/strong&gt; Get them to graph transformed versions of these.  For example, what does $z = (x-1)^2+2(y-3)^2$ look like?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;D&lt;/strong&gt; Now give them an arbitary quadratic form like $x^2+3xy+2y^2$ and have them put it into the form $(ax+by)^2+(cx+dy)^2$.  For fuller explanation of this see &lt;a href=&quot;http://tricki.org/article/Complete_the_square&quot; rel=&quot;nofollow noreferrer&quot;&gt;tricki.org/article/Complete_the_square&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;E&lt;/strong&gt; Return to the Talyor approximation.  Give them some functions and have them compute the quadratic approximation.  Then have them complete the square to put this into an &quot;understandable&quot; form. Play with lots of examples.  Understand how definiteness, semidefiniteness, and indefiniteness play into this. Especially, see some cases where a semidefinite quadratic form has all kinds of different behaviour (based on what happens in the &quot;flat&quot; direction of the form).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;F&lt;/strong&gt; Only after all of this, complete the square on a general quadratic.  Show that the &quot;second derivative test&quot; follows. &lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2017-05-11T16:39:16.883" LastActivityDate="2017-05-11T16:39:16.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12295" PostTypeId="2" ParentId="12287" CreationDate="2017-05-11T18:18:43.427" Score="4" Body="&lt;p&gt;I'm going to focus heavily on the constraints of the original question (15 minutes, community college interview committee) rather than on the mathematical basis for the answer. I think the answer below might actually fit into 15 minutes, if practiced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The short version of my answer would be &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Recall why the second derivative test is valid in one variable. Show a two-variable function and point out there are multiple &quot;second derivatives.&quot; Precisely show how to execute the test. Explain why it works in a simple case by an analogy to the single-variable case. Complete a simple example.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My detailed plan&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;(3 minutes) Quickly execute the second derivative test on three or four single-variable functions. Conclude that at a critical point, $y'' &amp;gt; 0$ demands a minimum, $y'' &amp;lt; 0$ demands a maximum, and $y'' = 0$ means anything goes, including maximums, minimums, and weird stuff. Do not say &quot;inflection point;&quot; you don't have time to muddy the water and engage calculus lecturers' pet peeves.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(1 minute) Note that in a two-variable situation, there are three (four? More than one, anyway. Don't get side-tracked, this detail is not important in this setting. While teaching a 200-level class, can you avoid being side-tracked by things that are above the level of the course or tangential to the topic at hand? This is part of the point of the interview) second derivatives, so things will be slightly more complicated, but still similar to the &quot;easy case&quot; (Do not say the word &quot;easy;&quot; that's useless and doesn't reassure students).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(2 minutes) Introduce a specific, very simple two-variable function and its critical point that will be your focus for the explanation -- one where $f_{xy} = 0$. Maybe like the point $(1, 0)$ on the function $f(x, y) = 4x^2 - 8x + 3y^2$. Point out that if someone asks for &quot;the second derivative&quot; of this function, that is not enough information, because there are lots of second derivatives. Find them.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(4 minutes) Carefully write the &quot;formula / process&quot; of the second derivative test starting with something like &quot;Given a critical point of the function $z = f(x, y)$, calculate $H = f_{xx}f_{yy} - f_{xy}^2$ at that point.&quot; and continuing until the end. Make sure all the gory details are precisely correct here and in an easy-to-reference box even though the rest of your explanation is not going to be very (mathematically) precise. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(4 minutes) Explain that the reason this works is complicated, but it makes sense in simple situations, namely when $f_{xy} = 0$. When $f_{xy} = 0$, the &quot;formula / process&quot; (Do not say the word &quot;Hessian.&quot; Do you have a Ph.D.? If so, part of the interview is checking whether you actually want this job or if this is something crappy you are settling for.) just checks whether $f_{xx}$ and $f_{yy}$ &lt;em&gt;agree&lt;/em&gt; about there being a maximum or a minimum. When they agree, you get the single-variable answer, when they don't agree, you get a saddle point, and when either one is zero, anything goes. These pictures are easy to sketch, or you can do them with your fingers by making U shapes with the thumb and forefinger of both hands and combining them together.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YH3ZR.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YH3ZR.png&quot; alt=&quot;Quick diagrams and captions&quot;&gt;&lt;/a&gt;l &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;(1 minute) Go back to your originasimple (Don't say &quot;simple!&quot;) two-variable function and calculate H, showing that its critical point is a local minimum.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(After time expires) Break character; talk to the committee. Reassure the mathematicians that you know $f_{xy}$ is important and that this logic isn't completely right, but that in the time constraints given, this would be your approach. Reassure everyone in the room that your next step would be to complete a more thorough example problem in detail. Reassure the administration in the room that in an actual classroom, you would use technology to show more accurate diagrams, but that the sketches are also valuable since it shows the students that the idea is in their grasp. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2017-05-24T00:18:40.130" LastActivityDate="2017-05-24T00:18:40.130" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="12296" PostTypeId="1" CreationDate="2017-05-12T00:58:28.093" Score="7" ViewCount="365" Body="&lt;p&gt;I have been trying to find a standardized exam for statistics, but have not had any luck. For example, there is an AP statistics exam for high schoolers to take, but I am looking for something that graduating undergraduates would take as a sort of exit exam to test proficiency. Neither is the Mathematics GRE Subject Test what I am looking for, since it is mostly about pure math and not specifically statistical methods. Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="8142" LastActivityDate="2017-10-14T01:48:04.843" Title="Standardized Exam for Statistics?" Tags="|exams|statistics|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12297" PostTypeId="2" ParentId="12279" CreationDate="2017-05-12T04:21:40.270" Score="5" Body="&lt;p&gt;I will take a stab at an answer though clarifying what level of education we are talking about would help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have never created problems for things like Qual exams (essentially masters exams) so I can't really speak to those. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For almost everything else I would tend to start with a very well formed idea of what the problem would be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You always know what it is you want to teach/test/train. From that you get the framework of your problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Testing if they know the quotient rule? I probably want them to differentiate something of the form $f(x)/g(x)$ and it would be a good idea if it didn't simplify, so no $\frac{x^2-1}{x+1}$ but $\frac{x^2}{x+1}$ might work, $\frac{\sin(x)}{x+1}$ is even better. There is very little more to do here since the problems are very easy. I don't need to do any backtracking since the answers will be reasonably &quot;nice&quot;. I would check how long it takes and consider how I might split up the grading on it, but since it's a very straight forward problem I probably don't care much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's take something more complex though still very easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to have the students train calc type graphing. The standard type of problem would be, &quot;Graph function $f(x)$ using calculus techniques.&quot; It's a bit boring so I might consider wrapping it into a word problem, but that makes the problem a lot harder so let's leave it like that for the moment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here I want a reasonable function. $f(x)=x^2$ is too easy. On the other hand $f(x)=\log(\sin(x^3+e^x))$ is too hard. Depending on what we've seen in class chances are I will go for a rational function. $f(x)=\frac{x^3+x+1}{x^4-4x+5}$ maybe? This looks like it's gonna a be pain though. Neither the top nor the bottom factors nicely and taking (and simplifying enough to use) the first derivative will be awful, nevermind the second. So here I want to start much closer to the answer. I want both the top and the bottom to factor. Possibly even cancel partially so I might go for $\frac{(x-1)^2(x+2)}{(x^2-1)(x+3)}$. The second derivative will still be nasty though and even the first is nothing to write home about, I might be ok with that or do more work engineering the problem to get all the numbers to come out right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This last example is quite common in my experience for common complexity problems. It works the same for polynomial factoring or root finding, elliptic curve point addition and Markov chain models. Once the problems are complex enough you usually want the students to get decent answers and so you tweak or work backwards from a chosen answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then there are proof based problems. At that point you are usually back to the basic approach and don't need to bother tweaking anything much. That's due to the the fact that for most standard complexity proof based problems, (prove that an associative quasigroup has a unit element,prove that MA implies $\mathfrak{d}=\mathfrak{c}$) there are no numbers to bother about. As to how you come up with these very often it is something you thought about at some point or it's a &quot;trivial result&quot; noone has bothered to write up. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course here I'm discounting the most common way of assigning problems which is check the exercise section of the book you're teaching from.&lt;/p&gt;&#xA;" OwnerUserId="5896" LastActivityDate="2017-05-12T04:21:40.270" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12298" PostTypeId="1" CreationDate="2017-05-12T09:15:58.760" Score="32" ViewCount="12842" Body="&lt;p&gt;My 11 year old child recently took an important numeracy test. One of the questions required her to know that M = 1000 in Roman numerals. This made me very angry: I could not see how this relatively obscure piece of trivia was a useful way of measuring a child's numeracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I complained about it on social media, and the complaint gained a great deal of traction. My opinions on this - and other aspects of testing - are now being sought by the national media. Being someone who believes in informed opinion where possible, especially at that level, I wanted to make sure I was correct in assuming this was a somewhat superfluous part of the curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some defenses I've seen offered are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The linguistics of &quot;M&quot; have lead to it being used as part of the mili- and mega- prefixes in metric measurement. But units of measurement and conversion between them are already taught (and tested) separately. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It helps children understand that there are number systems other than decimal. But surely very basic algebra (which is already taught and tested) also demonstrates this? And it would also seem possible to understand (and test) different number systems without this precise piece of information?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is useful general/cultural knowledge. Well fine: I'm not going to dispute that. But that doesn't make it a useful aspect of measuring a child's numeracy.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Are my assumptions on this correct? Or have I overlooked some other value in testing this knowledge? &lt;/p&gt;&#xA;" OwnerUserId="833" LastEditorUserId="64" LastEditDate="2017-05-12T20:40:20.503" LastActivityDate="2020-04-21T21:10:01.433" Title="What is the mathematical value of children learning and being tested on Roman numerals?" Tags="|testing|test-design|" AnswerCount="13" CommentCount="26" ContentLicense="CC BY-SA 3.0" />
  <row Id="12299" PostTypeId="2" ParentId="12298" CreationDate="2017-05-12T14:39:48.643" Score="37" Body="&lt;p&gt;As I said in the question this has attracted quite a bit of attention. And after a lot of nonsense, I did finally get an answer from a British maths teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although measurements and conversions are taught, schools are also supposed to pass on the direct equivalences between these and Roman numerals. i.e. to teach that C and M are the basis for centi-, mili- and that each case where it's used is an operation on 100 and 1000 respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While each piece of knowledge by itself may not be particularly useful, the key thing is that learning the connection between them aids understanding and recall. It becomes a contiguous piece of information where each bit supports the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect a lot of schools are perhaps not making this connection explicit, or that it's not being properly understood by the children. But it's good to see the reasoning here. &lt;/p&gt;&#xA;" OwnerUserId="833" LastActivityDate="2017-05-12T14:39:48.643" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12300" PostTypeId="2" ParentId="12298" CreationDate="2017-05-12T19:36:07.557" Score="28" Body="&lt;p&gt;One other little defense of roman numerals: a lot of students see the &quot;place value&quot; notation we have for numbers as inevitable, since they've used it from before they can remember. But of course it took hundreds of years to invent. Try multiplying CXI by X, for example -- imagining that you're a Roman who can't just translate these into Arabic numerals, multiply, and translate back. It quickly helps you realize what a powerful technology we have hidden in the notation.&lt;/p&gt;&#xA;" OwnerUserId="1784" LastActivityDate="2017-05-12T19:36:07.557" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12301" PostTypeId="1" AcceptedAnswerId="12304" CreationDate="2017-05-12T19:38:56.227" Score="11" ViewCount="526" Body="&lt;p&gt;As the title says, do you learn math faster by working on problems in groups, where you can discuss your solutions or by doing it all by yourself? I can't see any advantage of working in groups because if you get stuck on a problem, you can just ask the smartest person in group how he or she solved the problem which means you won't have to work as hard. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to know what professional mathematicians think. Are people (like me) who prefers to work by themselves missing out?&lt;/p&gt;&#xA;" OwnerUserId="7837" LastActivityDate="2017-06-15T01:01:58.820" Title="Is it better to study in groups or by yourself?" Tags="|self-learning|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12302" PostTypeId="2" ParentId="12298" CreationDate="2017-05-12T19:42:51.410" Score="16" Body="&lt;p&gt;Aside from your own answer, I also am a firm believer in providing our children with a good educational background.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Knowledge in itself is value, even if you cannot always use it for any practical reason, and even if it stands pretty much on its own. In the case of the roman numerals, you cannot do much with them, they do not lead on to other insights, and they have nothing leading up to them. They are remarkably on their own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your child will encounter such numerals plenty, in his future. Kings,  year-of-print in books, dice, street signs, clocks, etc.. At least this is the case in my country. And while I certainly do not remember every last of their letters, and would probably struggle a bit to decode or encode larger number spontaneously, I am still pretty happy that I know, in principle, what they're about, how the Romans thought about it, and how I could, in principle, work out how to decode one, if I really had the dire need to do so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same goes for everything else. Does a child really need to know what a minuet is, whether stalactites grow up or down, how many planets our solar system has or how a worm digests its nutrients? You could do away with an awful lot of our (higher level) education with that thinking. I know people who made it all the way through to pension age without &lt;em&gt;ever&lt;/em&gt; doing a &lt;em&gt;single&lt;/em&gt; mathematical calculation except basic &lt;code&gt;+&lt;/code&gt;, &lt;code&gt;-&lt;/code&gt; and &lt;code&gt;*&lt;/code&gt; (seldomly &lt;code&gt;/&lt;/code&gt;) in the low digit range. I'm not sure even if they can really do &lt;code&gt;*&lt;/code&gt; and &lt;code&gt;/&lt;/code&gt; or if they cheated with &lt;code&gt;x*2 = x+x&lt;/code&gt; and &lt;code&gt;x/2&lt;/code&gt; = &quot;try numbers until their double is &lt;code&gt;x&lt;/code&gt;&quot;. And still we teach our kids all kinds of maths for many years. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in summary, I would probably not care that much if my child did not understand the roman numerals, or forgot about them soon enough. But I am certainly not getting angry by the fact that he is taught the topic. I would be pretty sad if I learned that they were not taught anymore, though.&lt;/p&gt;&#xA;" OwnerUserId="8146" LastEditorUserId="8146" LastEditDate="2017-05-12T23:33:34.240" LastActivityDate="2017-05-12T23:33:34.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12303" PostTypeId="2" ParentId="12298" CreationDate="2017-05-12T22:00:06.280" Score="70" Body="&lt;p&gt;Learning about different numeral system develops the critical thinking and understanding. Arabic numerals are so ubiquitous that most of us take them for granted. Seeing other numeral systems and analyzing them creates a lot of opportunities for thinking about why did we pick one over the other and that there was a pick at some time, not a given (somebody had to come up with the design of characters, somebody had to create the numerical system and somebody decided that Arabic is better).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of this understanding comes from doing operations and experiencing first hand how cumbersome it can be in other number systems. It also enforces the idea that digits are just symbols to express ideas (i. e. numbers). Actually, the whole purpose of a numerical system is communication - it provides a way of communicating abstract concepts (numbers) and perform operations on them. Thus, they can see that 3 and Ⅲ are two different ways of expressing the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find it interesting that the numerical symbols are so embedded in our brains that it becomes very hard to switch them or even be aware of it. This also helps when learning new numerical bases that might use different symbols by keeping their mind open.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And also, knowing Roman numerals has a practical application. Every once in a while, I come across a year or a quite big number written in Roman numerals.&lt;/p&gt;&#xA;" OwnerUserId="8147" LastEditorUserId="-1" LastEditDate="2017-05-15T13:55:11.637" LastActivityDate="2017-05-15T13:55:11.637" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12304" PostTypeId="2" ParentId="12301" CreationDate="2017-05-13T03:22:43.540" Score="11" Body="&lt;p&gt;There is evidence that group studying in college is &lt;em&gt;negatively&lt;/em&gt; correlated with improvements in critical thinking, as measured by a test called the CLA (Collegiate Learning Assessment). This is described in  Arum and Roksa, Academically Adrift: Limited Learning on College Campuses, p. 100.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2017-05-13T03:22:43.540" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12305" PostTypeId="2" ParentId="12298" CreationDate="2017-05-13T09:38:35.397" Score="25" Body="&lt;p&gt;You forget, that school is not only about &lt;em&gt;knowledge&lt;/em&gt;, but also about &lt;em&gt;thinking&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Learning Roman numerals and how they work teaches child, that the same thing may be expressed in different ways. And that we may define new system with a new set of rules and now we have to use these different rules to work with this system - hell, the whole maths is just about it! Roman numerals are perfect way to slowly introduce child to this specific way of thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a master's degree in mathematics - I studied on Wrocław's University in Poland. Currently I'm working for 10 years as a programmer and I don't use much &lt;em&gt;knowledge&lt;/em&gt; I gained while studying. But even now, knowing that, I definitely would still chose mathematics as my studies, because I utilize &lt;em&gt;way of thinking&lt;/em&gt; that studies taught me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm having an impression, that nowadays people tend to change the educational system to teach only &lt;em&gt;practical&lt;/em&gt; things. And such system produces robots, who can go straight, but when problems gets more complicated, they can't think outside the box. In our company we have trouble with hiring &lt;em&gt;good&lt;/em&gt; programmers, because a lot of candidates are just like that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe, instead of getting angry and going to teacher and telling, that he's teaching your child stupid things, it would be wiser to widen horizon of your child even more and show him Mayan numeric system, which is 20-based or Indian one, which uses separators in different way than western systems? Maybe it is a good occasion, to tell your child about binary or hexadecimal system and how it is used to transmit data today? Or sit down and invent your own numeric system and try to use it? Did you think about it?&lt;/p&gt;&#xA;" OwnerUserId="5122" LastEditorUserId="5122" LastEditDate="2017-05-13T17:10:57.013" LastActivityDate="2017-05-13T17:10:57.013" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12306" PostTypeId="2" ParentId="12298" CreationDate="2017-05-13T10:21:57.987" Score="44" Body="&lt;h1&gt;Common knowledge&lt;/h1&gt;&#xA;&lt;p&gt;Roman numerals are part of the common knowledge in Western cultures for representing numbers. That's why it's valuable for children growing up in these societies to learn them. It's the same as the reason we teach children Arabic numerals.&lt;/p&gt;&#xA;&lt;p&gt;If you don't know how to read Roman numerals, you'll be effectively illiterate—or &amp;quot;innumerate&amp;quot;—whenever you encounter one. Roman numerals function like &amp;quot;italics for numbers&amp;quot;: a distinctive, alternative way of writing the same thing. For example, they're used for page numbers in long prefaces to distinguish them from the main page numbers of the book, for items in hierarchies (such as essay outlines) to distinguish different levels, for the roots of musical chords where Arabic numerals denote intervals, for groups in the periodic table, etc. They denote ordinal numbers of monarchs (e.g. Henry VIII), sporting events (Super Bowl LI), yearly formal gatherings, world wars, version numbers, movie sequels, &lt;a href=&quot;https://en.wikipedia.org/wiki/Coagulation#Regulators&quot; rel=&quot;noreferrer&quot;&gt;clotting factors in the fibrin cascade&lt;/a&gt;, etc. And they represent years or hours in very formal contexts such as stone inscriptions, sundials, and clocks meant to last a long time. When people write Roman numerals, they expect that any member of our society with even a minimal education will understand them. They're even used in slang: a &amp;quot;C-note&amp;quot; is a hundred-dollar bill.&lt;/p&gt;&#xA;&lt;p&gt;That's what I mean by &amp;quot;common knowledge&amp;quot;: knowledge or conventions that are shared by most people in the culture &lt;em&gt;and&lt;/em&gt; that are known by most people to be shared. When something is common knowledge, we can depend on other people knowing it, other people knowing that we know it, etc. The classic example of common knowledge is language: without that, we couldn't communicate much at all. Roman numerals are just part of our common language of numbers.&lt;/p&gt;&#xA;&lt;p&gt;If you're thinking that numeracy, since it relates to mathematics, should be independent of culture, that's an error. By that reasoning, Arabic numerals would also be excluded. From the standpoint of pure mathematics, the fact that the symbol 6 stands for the same abstract quantity that we refer to by the word &amp;quot;six&amp;quot; is an obscure piece of trivia. &amp;quot;Numeracy&amp;quot; is a vague concept, but I understand it to emphasize practical matters, and that includes knowing your culture's conventions for representing numbers—even a less-often-used convention that exists alongside the most common one.&lt;/p&gt;&#xA;&lt;h1&gt;Tangential mathematics and history&lt;/h1&gt;&#xA;&lt;p&gt;Some people argue that Roman numerals should be taught because they illustrate things like multiple ways to represent a number (a kind of flexibility that you need for algebra and anything beyond), or the value of positional notation, or the value of a symbol for zero (since you really come to appreciate something by trying to do without it), or that the C and M &lt;a href=&quot;https://latin.stackexchange.com/q/4389/118&quot;&gt;stand for&lt;/a&gt; the Latin roots of our prefixes &lt;em&gt;centi-&lt;/em&gt; and &lt;em&gt;milli-.&lt;/em&gt; Roman numerals provide an &lt;em&gt;opportunity&lt;/em&gt; to teach all those things, and children may learn them as a happy side-effect even if the teacher doesn't point them out, but the related topics aren't &lt;em&gt;the reason&lt;/em&gt; for teaching Roman numerals. Kids need to learn Roman numerals so they can participate in the common knowledge of the adult world—the same reason that nearly everything in elementary school through high school is taught. Notice that if learning the related topics were sufficient reason to teach Roman numerals, then Roman numerals should be taught even in Mongolia.&lt;/p&gt;&#xA;&lt;p&gt;By the way, the numerals used by the ancient Romans did not exactly follow the conventions that today we call &amp;quot;Roman numerals&amp;quot;. They were less standardized and the forms changed over time. For example, one convention represented 500 as IƆ; 18 might be represented as IIXX. More &lt;a href=&quot;https://en.wikipedia.org/wiki/Roman_numerals#History&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;—but those sorts of things are relatively obscure trivia, not common knowledge. While the Roman numerals taught in elementary school today have deep roots in our past, they are part of our present-day culture.&lt;/p&gt;&#xA;" OwnerUserId="8149" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-01-19T12:51:19.960" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12307" PostTypeId="2" ParentId="9842" CreationDate="2017-05-13T12:06:14.953" Score="2" Body="&lt;p&gt;I would like to suggest a (English) language based approach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I went to the store three times.&quot;  (Should be totally clear).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Three times I went to the store.&quot;  (Still clear).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I went to the store times three.&quot;  (Maybe understandable, but rather odd).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Three I went to the store times.&quot;  (Doesn't make much sense).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Upshot: The number modifying 'times' wants to go before rather than after 'times'.  Thus the number before times says how many occurences; while the other element says what is repeatedly occurring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Three times five&quot;...five is occurring three times: $5 + 5 + 5$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;When teaching this idea (to preservice teachers), I sometimes suggest that we can translate 'times' as 'groups of' (or 'group of').  Then &quot;Three times five is three groups of five ($5 + 5 + 5$).  This later extends to fractions with $\frac 12 \times 6$ becomes half of a group of $6$.&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2017-05-13T12:06:14.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12308" PostTypeId="1" AcceptedAnswerId="12310" CreationDate="2017-05-13T13:06:08.613" Score="24" ViewCount="3669" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wZ50y.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wZ50y.jpg&quot; alt=&quot;cone volume problem&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was with a 2nd year high school class, preparing for our (US) state's standardized test. I asked the class how they would solve this, and they flipped through the sheets to find&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$V=\frac{1}{3}\pi r^2h$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(To be clear, there is a 'formula' page that's given with the exam.) As they fumbled with their calculators, I asked them what the radius was. 3.5. and I wrote the numbers on the board.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$V=\frac{1}{3}\pi (3.5)^2(9)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I attempted to show them how 3.5 squared is close enough to 12. Divide by 3 to get 4, and we are left with 36pi. Since 36x3 is 108, the only answer that can make sense is 115. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my lecture, I offered 3 benefits of this process. (1) The speedy approach can save precious time on this type of question so more time remains for those requiring more thought and time. (2) If you still use the calculator, this is a good double check to be sure the answer makes sense, that you didn't hit the wrong key. (3) If you forget your calculator, the proctor won't always have an extra, and the whole exam can be done by hand. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Out of the 12 questions we did as a group, a full half lent themselves to this method, as number were easy to round, and the answers were different enough so cumulative rounding errors didn't inch results too far away from the true answer. After my second attempt, on the next appropriate question, to explain this approach, I realized it wasn't helpful to this class and I stopped. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recall that as a student, estimating was a skill that we were taught as part of the regular math class. Is this no longer considered to be valuable? &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-07-23T01:06:10.297" LastActivityDate="2021-04-01T16:33:16.983" Title="Is 'estimating' still considered a valuable skill?" Tags="|secondary-education|algebra|" AnswerCount="10" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="12309" PostTypeId="2" ParentId="12298" CreationDate="2017-05-13T14:58:56.097" Score="6" Body="&lt;p&gt;Maybe it's not purely a maths thing, but maths comes into other subjects, and the crossover is both ways. As a former teacher in U.K. I can't believe that the national curriculum contains only 'M' and 'C'. Actually I can! Usual half-cock. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a maths standpoint, as already discussed here, it's good that children understand other number systems - decimal being one of very many. I've taught base 5 to 11 yr olds, and it helps them understand place value. Then moved on to base two - binary - which is in everyday use. The French have a strange way of counting, so maybe that ought to be addressed as well - normal till 60, then 70 is 60+10, but 80 is called 4x20s, with 90 = 4x20+10. Just another &lt;em&gt;interesting&lt;/em&gt; bit of maths, that could spark the attention of bored little Willy at the back of the class...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At 11, I could read the burial dates in graveyards, and the BBC still dates films etc in R.N. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, algebra normally gets used in base 10, so no big deal there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer, there are many many things that get tested at school, at 11 - and younger and older - and the vast majority of them can be, and are questioned by parents and pupils, and sometimes by the teachers themselves.Certainly by me! So, if that's the only issue you have with the testing regime, you're very lucky!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To finish - why doesn't someone tell crossword compilers that 99 is not IC?&lt;/p&gt;&#xA;" OwnerUserId="8151" LastActivityDate="2017-05-13T14:58:56.097" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12310" PostTypeId="2" ParentId="12308" CreationDate="2017-05-13T15:01:30.060" Score="25" Body="&lt;p&gt;Yes, of course it is. You've identified many of the important reasons in your question. In practice in quantitative fields, we are all estimating as a first-pass on whether problems are soluble all the time. Estimation is incorporated throughout Common Core standards (e.g., &lt;a href=&quot;http://www.corestandards.org/Math/Content/3/MD/A/2/&quot; rel=&quot;noreferrer&quot;&gt;Grade 3&lt;/a&gt;, &lt;a href=&quot;http://www.corestandards.org/Math/Content/4/OA/A/3/&quot; rel=&quot;noreferrer&quot;&gt;Grade 4&lt;/a&gt;, &lt;a href=&quot;http://www.corestandards.org/Math/Content/7/EE/B/3/&quot; rel=&quot;noreferrer&quot;&gt;Grade 7&lt;/a&gt;; there are more). And it was part of official state standards long before that (&lt;a href=&quot;http://educationnext.org/the-common-core-sanity-check-of-the-day-estimation-is-not-a-fuzzy-math-skill/&quot; rel=&quot;noreferrer&quot;&gt;link&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I am concerned that in some locations it may not in fact be taught. When I teach remedial arithmetic courses at my community college, the entire classroom of students will claim to have &lt;em&gt;never heard the term before&lt;/em&gt; (either &quot;estimate&quot; or &quot;approximate&quot;, etc.). It is, in fact, utterly bewildering to them. Therefore, in that course I make it the topmost priority and try to work estimation skills into the exercises every single day. Nevertheless, at the end of a semester, many students will say something like, &quot;I never got this estimation thing&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often when I share this with other people I get great skepticism (&quot;but my child is learning that in the third grade right now&quot;). We might theorize how it came to be: Maybe my students come from specially broken-down school systems. Or maybe many of the students have intellectual disabilities and were taught to rely on the calculator in the absence of basic numeracy. Or maybe they've just never built anything practical such that they saw the value in double-checking for errors. At any rate, it's that kind of gap for which Common Core is trying to give guidance, I think. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-05-13T15:01:30.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12311" PostTypeId="2" ParentId="12308" CreationDate="2017-05-13T15:51:38.703" Score="16" Body="&lt;p&gt;I can't speak for a broad consensus, but I personally certainly think that estimation is still a valuable skill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, here's my rough, loose, quick way of solving the problem, using only easy mental arithmetic, no pencil, paper, or calculator. Seeing this will make my reasons below clearer.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;This problem would be a whole lot easier if it were a rectangular prism. The base would be 7 ⨉ 7 = 49 in&lt;sup&gt;2&lt;/sup&gt;, which is almost 50. Multiplying that by the height of 9 would yield a volume of 450 in&lt;sup&gt;^2&lt;/sup&gt;. Now we just have to &quot;shave off&quot; everything that isn't part of the cone. Notice that this already shows that answer D is hopeless.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;OK, let's shave it down to a cylinder. That's pretty easy. The base is a circle, which has an area of a little more than 3/4 of the square with the same &quot;diameter&quot; (i.e. a circumscribed square). 3/4 of 49…ecch, let's make this easier and say 3/4 of 48…that's 36. So the cylinder has a volume of a little more than 9⨉ that…10⨉36 = 360, subtract 36, that's 324.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;em&gt;idea&lt;/em&gt; of the volume formula for the cone is just that it's 1/3 of the cylinder. So, to shave away the part of the cylinder that's not the cone, just divide by 3. 324/3 = 300/3 + 24/3 = 100 + 8. So the volume of the cone is a little more than 108 in&lt;sup&gt;3&lt;/sup&gt;. Looks like answer B.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Now here are some reasons why being able to estimate is a valuable skill:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;As you said, the estimate is &lt;strong&gt;a double-check on the precise calculation&lt;/strong&gt;. Really, you should never do a precise calculation without also doing an estimate. Precise calculations, especially if a calculator is involved, can easily be off by orders of magnitude if you make a tiny mistake, like a wrong keypress. You should always know the &quot;ball park&quot; before you do the calculation, or your common sense can't function. Notice how considering a rectangular prism set an upper bound on reasonable answers. Just from looking at the picture, you should be thinking that the right answer is a little less than a third of that. Estimates can be wrong, too, but the point is: when the calculation disagrees with your estimate, you know &lt;em&gt;something&lt;/em&gt; is wrong. Without an estimate, you can't tell.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When students are taught math as a bunch of formulas and procedures to memorize, they often can't think fluidly or flexibly about it. When they meet a problem for which they don't know a formula or procedure, they're instantly stumped and give up. &lt;strong&gt;Estimating gives the student easy practice in &lt;em&gt;thinking&lt;/em&gt; mathematically.&lt;/strong&gt; Most of mathematical reasoning consists of seeing the same thing as it is involved in different relationships, each of which sheds some light on it or constrains it in some way. Estimating gives you practice in feeling around for helpful relationships, and in distinguishing between important and unimportant. It's easier than usual because you don't need to be exact. It gives you practice in the ubiquitous mathematical technique of solving a much easier problem than you were given and then figuring out how to correct for the difference. And it shows you that &lt;strong&gt;mathematics makes sense&lt;/strong&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Estimating teaches through direct experience that there is not just one right way to solve a problem.&lt;/strong&gt; Notice that your way of reaching the same estimate was more efficient than mine. That's to be expected. I don't mess with geometry much. I just fumbled around a bit, but I was still able to find the answer completely in my head. I don't &lt;em&gt;need&lt;/em&gt; to be boned up on solid geometry to find a passable way to solve the problem. I can cook something up that's good enough, without being an expert. There are many, many paths, and they all lead to the same answer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Consequently, &lt;strong&gt;estimating builds confidence&lt;/strong&gt;. The more you estimate, the more you see math as a system of relationships that tend to guide you to the thing you're looking for. If you miss one relationship, no big deal, math is abundant in relationships. And if you're off by a little bit, no big deal. You see that math is best approached loosely, lightly, playfully. This is the opposite of the feeling that students usually get: that math is a minefield—make one false step and it's over.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In real life, you almost never need an exact answer. &lt;strong&gt;Estimating is the main practical skill of mathematics&lt;/strong&gt; beyond arithmetic. In real life, outside of fields like accounting, usually you can't even get very precise measurements. The first couple digits and the order of magnitude are usually all you need to know—often, all you can know. Also in real life, you seldom encounter problems fully spelled out. Usually you have to figure out what available information would enable you to make a good estimate. You can't really do that effectively without experience estimating.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For inspiration, read about &lt;a href=&quot;http://www.hk-phy.org/articles/caesar/caesar_e.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Enrico Fermi&lt;/a&gt;. He is said to have been able to solve pretty much any physics problem in his head to one significant digit and the order of magnitude. And yes, he &lt;a href=&quot;https://www.grc.nasa.gov/www/k-12/Numbers/Math/Mathematical_Thinking/fermis_piano_tuner.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;taught this skill&lt;/a&gt; to his students.&lt;/p&gt;&#xA;" OwnerUserId="8149" LastEditorUserId="8149" LastEditDate="2017-05-16T05:02:47.647" LastActivityDate="2017-05-16T05:02:47.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12312" PostTypeId="2" ParentId="12264" CreationDate="2017-05-13T16:56:24.577" Score="6" Body="&lt;p&gt;No. This is the same kind of pedagogical fallacy that led to the &quot;new math&quot; of the 1960s, when they tried to teach elementary school students deep concepts of abstract algebra as an introduction to arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, think of epsilon-delta proofs as an introduction to mathematical logic. If you're worried that students can't understand calculus without them, consider Newton, Leibniz, the Bernoullis, Euler, and Fourier. They all did pretty well with calculus despite not knowing a single epsilon-delta proof. There's a reason why epsilon-delta proofs didn't emerge until the 19th century: before then, they weren't needed—but at that time, the need for them became pressing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The pedagogical fallacy is thinking that teaching a deeper, more abstract, more refined concept first makes it easier for the students to learn a concept that can be mathematically deduced from it. That's not how learning works—or even logic in the everyday sense. To grasp an idea, you need to anchor it in some way. For example, we learn basic concepts of quantity by comparing tangible quantities, like longer and shorter things, and heavier and lighter things, before we learn real analysis. We learn numbers by counting things, not by deducing from the real-number axioms. We start with one, not zero. We start with the concrete and build progressively more abstract ideas upon it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A frequent complaint that chemistry professors make about math professors is that students who've been taught linear algebra can't solve stoichiometric equations. This is because the students were taught linear algebra as a bunch of axioms and derivations unanchored in anything more concrete and understandable. That way of teaching leads the student to see mathematics as a symbol game disconnected from reality. Of course they can't see how linear algebra applies to stoichiometric equations—who thinks of chemistry when they're told the rank-nullity theorem? This manner of teaching teaches most students to hate linear algebra. They have no idea of its vast applicability. That's what math teachers do to students when they try to &quot;front-load&quot; ever more abstract ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Epsilon-delta proofs are a clever way to put calculus into a framework that avoids various paradoxes of infinity. They're one of the great intellectual achievements of human history, and calculus students should get some exposure to them. But they're nearly impossible to appreciate until you've seen the troubles that arise without them. See &lt;a href=&quot;http://www.maa.org/press/books/a-radical-approach-to-real-analysis&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;A Radical Approach to Real Analysis&lt;/em&gt;&lt;/a&gt; by David Bressoud for a nice explanation of the need for them and how the approach was worked out. Basically, it's hard to tell if trigonometric series converge or not without them. I don't think you need such a hard example to motivate epsilon-delta proofs, but you need &lt;em&gt;some&lt;/em&gt; conceptual difficulty that's meaningful to the students. Maybe just a couple series for which it's hard to tell whether they diverge or converge. Before you can lead them to such an impasse, they need a fair amount of experience with plain ol' calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Starting them off with negations of double quantifiers will just pile confusion on top of confusion. But once they've struggled with epsilon-delta proofs a while, oh will they be ready to learn about negating quantifiers!&lt;/p&gt;&#xA;" OwnerUserId="8149" LastEditorUserId="8149" LastEditDate="2017-05-14T02:20:22.610" LastActivityDate="2017-05-14T02:20:22.610" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12313" PostTypeId="2" ParentId="12308" CreationDate="2017-05-13T19:22:23.323" Score="7" Body="&lt;p&gt;I have found that estimating is a very valuable skill, particularly for finding orders of magnitude of solutions without having to use a calculator. It comes in very useful for helping students work out problems on the fly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, I have found that most students now have no feel for this skill, opting instead to use a calculator for any calculation, even those that could easily be worked out exactly in one's head. When we get into calculations involving quantities with uncertainties, this lack of understanding of really how precisely one can express an answer becomes even more pronounced. Students seem to always think the correctness of the answer corresponds to the number of decimal places that are written down, and they seemingly have no feel for whether a possible answer is feasible or not without working it out with a calculator. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think we should sometimes require students to write exams without a calculator. That's how many of us learned not to require one for every little calculation. Estimating the answer in their head beforehand would also help students to realize when an answer from their calculator might be questionable.&lt;/p&gt;&#xA;" OwnerUserId="8155" LastActivityDate="2017-05-13T19:22:23.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12314" PostTypeId="2" ParentId="12308" CreationDate="2017-05-13T19:29:04.527" Score="6" Body="&lt;p&gt;In general, yes, estimation is a great way to get a quick sanity check.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this particular example it looks like they chose values such that using $22/7$ as an approximation for $\pi$ lets you quickly cancel stuff out,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{eqnarray*}&#xA;V = {\frac{1}{3}}\times{\frac{22}{7}}\times{\frac{7}{2}}\times{\frac{7}{2}}\times9&#xA;= {\frac{11\times7\times3}{2}}&#xA;= 115.5&#xA;\end{eqnarray*}$$&lt;/p&gt;&#xA;" OwnerUserId="8156" LastEditorUserId="8156" LastEditDate="2017-08-04T14:17:28.113" LastActivityDate="2017-08-04T14:17:28.113" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12315" PostTypeId="2" ParentId="12298" CreationDate="2017-05-14T00:15:37.063" Score="-4" Body="&lt;p&gt;Roman numbers should be taught to show the historical failure of math in the ancient west - both Greece and Rome. It is unfortunate that most of the math used today be it numbers, counting, concept of zero, infinite series, basic algorithms almost all come from India. Yet if you google mathematicians some imaginary pictures of Aristotle and Euclid come up, like they invented something. BTW modern numbers are NOT arabic numbers - they are the Indian place value system.&lt;/p&gt;&#xA;" OwnerUserId="8161" LastActivityDate="2017-05-14T00:15:37.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12316" PostTypeId="2" ParentId="8193" CreationDate="2017-05-14T05:38:34.540" Score="1" Body="&lt;p&gt;Kenny LJ failed to say when he took those tests.  As another person pointed out, some of the ones mentioned were possibly less subject to cultural bias in the first place but bias really surfaced as an issue in the 60's and 70's (1974 episode “The IQ Test” of the sitcom Good Times and appeared in scholarly journals in the 80's (Williams, Teresa Scotton. &quot;Some Issues in the Standardized Testing of Minority Students.&quot; Journal of Education, v 165 n2 pgs 192-208, Spring 1983).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest mention I found was an article from The Atlantic about the bias question which went into some history. [&quot;Lemann found an item in the diary of Henry Chauncey, the first president of ETS, showing that he had read an article in the April 1948 issue of The Scientific Monthly arguing that tests like the SAT could be biased against low-income students. But Chauncey dismissed it as a 'radical point of view.'&quot;]  So mentioned in 1948 as maybe a problem but dismissed as radical by a major testing organization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There appear to be lots of articles in science and education journals in the 90's so I would guess bases on the information here and my parent's discussions as educators, that the issue of removal of cultural bias really gathered steam in the 70's and 80's with later decades being measurement of the results of removal and/or efforts to make sure it wasn't creeping in again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, as other responders have mentioned, a key component of the resistance to standardized testing has been our rabid demand for local control of schools because you never know what those other people will want to teach your kids (like evolution or sex ed or using Harry Potter books with witches in the classroom or any of numerous other hot-button issues).&lt;/p&gt;&#xA;" OwnerUserId="8163" LastActivityDate="2017-05-14T05:38:34.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12317" PostTypeId="2" ParentId="12308" CreationDate="2017-05-14T07:02:36.940" Score="5" Body="&lt;p&gt;First, an even easier estimate:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$V = \frac{1}{3} \pi r^2 h \approx \frac{1}{3} \frac{22}{7} \left( \frac{7}{2} \right)^2 9 = \frac{2 * 11 * 7^2 * 3^2}{3 * 7 * 2^2} = \frac{1}{2}*11*7*3=115.5.$$&lt;/span&gt;&#xA;Never underestimate the power of &amp;quot;engineer's &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;&amp;quot; ! Admittedly, some of the cancellations make this problem even easier.&lt;/p&gt;&#xA;&lt;p&gt;I would whole-heartedly agree with the other answers that YES estimation is vitally important. I'll try to flush out a few more reasons:&lt;/p&gt;&#xA;&lt;h3&gt;Daily Life&lt;/h3&gt;&#xA;&lt;p&gt;While the old teacher reply that &amp;quot;you might not have a calculator with you&amp;quot; has been somewhat invalidated by most people carrying around a computer in their pocket, many daily problems just don't require precise answers or have vaguely defined inputs. For example,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I'm throwing a party. How many pizzas to order / cases of ...soda...to buy / pounds of ice? Especially since I have a few flaky friends that might not show up or might bring along dates.&lt;/li&gt;&#xA;&lt;li&gt;I'm driving on the interstate and have a quarter tank left. Do I need to pull off at the next exit or can I make it to the next town that's 100 miles down the road? What if the gas at the next exit is overpriced?&lt;/li&gt;&#xA;&lt;li&gt;I'm painting my house. Do I need 5 gallons of paint or 10? I got a quote from a professional painter...is their price reasonable, considering that I would spend the entire weekend doing it myself?&lt;/li&gt;&#xA;&lt;li&gt;I'm splitting a meal at a restaurant with some friends. How much do I need to throw in (including tip) if I don't want to spend 20 minutes breaking down everyone's exact costs?&lt;/li&gt;&#xA;&lt;li&gt;Is my water bill lower if I take a 10-minute shower every day or a bath every other day? If I buy the more energy-efficient (but expensive) LED light bulbs with redtooth interweb connectivity, how long will it take to recoup that investment?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The list goes on and on. Most of these sorts of problems are related to basic numeracy, of which estimation is a huge component.&lt;/p&gt;&#xA;&lt;h3&gt;Preparation for Calculus&lt;/h3&gt;&#xA;&lt;p&gt;At its core, I would argue that calculus is the study of making a sequence of &lt;em&gt;approximations&lt;/em&gt; that become &amp;quot;infinitely accurate.&amp;quot;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How can I derive that &lt;span class=&quot;math-container&quot;&gt;$V = \frac{1}{3} \pi r^2 h$&lt;/span&gt; if I don't know how to estimate the volume of a cone to begin with? Will increasing the radius or height of this cone increase the volume quicker? This problem states that the height is &lt;em&gt;exactly&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$9in$&lt;/span&gt; and the diameter is &lt;em&gt;exactly&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$7in$&lt;/span&gt;. If you actually measured a similar cone with a ruler, would the answer you got out of your calculator likely be higher or lower than the true volume? What if you spent a lot of time measuring &lt;em&gt;really carefully&lt;/em&gt;?&lt;/li&gt;&#xA;&lt;li&gt;The speedometer in my car is broken (or rather, was until I replaced it). How did I manage to not get pulled over for speeding on the highway? What if the odometer was broken instead of the speedometer? Speaking of which, how does your smartphone give you a current speed?&lt;/li&gt;&#xA;&lt;li&gt;How can I find the volume of water in a lake or volume of water moving down a river, when they aren't convenient geometric shapes? If you live close to said lake and it rains for forty days, do you need sandbags or an ark?&lt;/li&gt;&#xA;&lt;li&gt;Do I need to pedal fast, really fast, or &lt;em&gt;really really fast&lt;/em&gt; to clear this jump on my bike? If I ski off this cliff, is it going to hurt or &lt;em&gt;hurt a lot&lt;/em&gt;?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h3&gt;Science &amp;amp; Engineering&lt;/h3&gt;&#xA;&lt;p&gt;Never underestimate the power of a good estimation or toy problem to sanity check a result. Even with &amp;quot;hardcore applied math,&amp;quot; one of the first steps is to define your assumptions for modeling a physical system---which assumptions are reasonable and which are more likely to introduce errors? Is it the identical spherical cows or the flat and infinite grazing pasture that's the problem?&lt;/p&gt;&#xA;&lt;p&gt;Since other answers have already touched on Fermi estimations and such, I'd like to give one of them Real World Examples™. A calculator (or computer) does a fine job of returning a Number™, but doesn't tell you if that Number™ is the Answer™.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My uncle works as a lead airframe engineer on fighter jets. He once tasked a junior engineer with figuring out what the fuel economy of their current design was, i.e., how many gallons of fuel the jet would use in an hour while flying at cruising speed. The junior engineer went off and did a week of computation and computer modeling before proudly returning with a Number™. Said number seemed a bit high to my uncle, but he was busy at the time and couldn't look over the report until the end of the day. A second look showed said number to be...really big. A bit of estimation later, he found that the junior engineer had proudly reported that the jet would use almost 2 Pentagons of fuel per hour of flight (yes, as in the building used as a unit of volume). The junior engineer had apparently flubbed a unit conversion at the outset and had &lt;em&gt;absolutely no idea&lt;/em&gt; how absurd the number they got out of the program was, at least until they were called into my uncle's office the next day.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7260" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-05-14T11:53:54.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12318" PostTypeId="1" CreationDate="2017-05-14T08:19:26.477" Score="6" ViewCount="635" Body="&lt;p&gt;Is it better to write everything in words as the concepts themselves should be known? Or will some teachers in some countries prefer to be able to choose questions which also test the student's knowledge of the notation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tlzfs.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tlzfs.png&quot; alt=&quot;Mathematical Notatios&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\overline{AB}$ is parallel to $\overline{CD}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\overline{AB}$ is perpendicular to $\overline{CD}$&lt;/p&gt;&#xA;" OwnerUserId="7587" LastActivityDate="2017-05-15T11:53:15.973" Title="Wording VS mathematical notations" Tags="|geometry|notation|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12319" PostTypeId="1" CreationDate="2017-05-15T00:52:19.197" Score="3" ViewCount="102" Body="&lt;p&gt;For almost 50 years, I've been looking for a textbook I had as a 12th grader in Abe Lincoln HS (Brooklyn) in 1969; a book that opened my eyes up to math- which resulted in me going into actuarial work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All I remember is that it was a big, heavy book, yellowish-greenish cover, with quadratic equations on the cover- - (b^2-4ac). If I had it in 1969 it must have been printed in the few years before then.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please, if anyone remembers the title or likely author please e-mail Gary Rothy at actuarylaw@yahoo.com&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you so much.  &lt;/p&gt;&#xA;" OwnerUserId="8174" LastActivityDate="2017-05-15T00:52:19.197" Title="Advanced algebra circa 1969" Tags="|algebra|" AnswerCount="0" CommentCount="2" ClosedDate="2017-05-16T12:23:26.553" ContentLicense="CC BY-SA 3.0" />
  <row Id="12320" PostTypeId="2" ParentId="12308" CreationDate="2017-05-15T04:32:07.537" Score="3" Body="&lt;p&gt;My calculator will return 13 digits. Not nearly enough to approximate my release point to slingshot around Saturn on my way to Betelgeuse. But way more than I need to slice an apple pie for the kids. Estimation, iteration, approximation. All tools in your toolbox for answering questions and solving problems. Don't handicap yourself or your students.&lt;/p&gt;&#xA;" OwnerUserId="8176" LastActivityDate="2017-05-15T04:32:07.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12321" PostTypeId="2" ParentId="12308" CreationDate="2017-05-15T04:43:50.107" Score="3" Body="&lt;p&gt;Estimating is absolutely crucial for high level success in mathematics, not because of a specific application (though there are plenty of uses for mental estimation in daily life), but because constantly making estimates is helpful exercise to make the brain more adept at processing numerical calculations and connecting them to context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why does a soccer (i.e. football, if you're not American) player run sprints during practice? Because they may want to race a teammate later on? Because they will be sprinting in an exact straight line for a set distance during their upcoming matches? No, but they will be executing more complicated runs that involve additional skills such as cutting, sliding and kicking all while trying to track the motion of the ball and/or key players on the pitch. But in order to do that effectively, they need to be able to run very quickly, and that can only be developed over time through extensive practice. Sprinting isn't the end goal during practice; it's a way to develop an important skill that will be used to accomplish a greater goal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to solve challenging mathematical problems, you have to be willing to run through many different ideas and a first start is trying to do easy calculations, which often will require you to do some estimation in order to keep the calculations simple. If you don't develop a good number sense through practicing exercises that cover, among other topics, estimation, then you will be missing a key tool needed to deal with more advanced problems. Estimation likewise is not an end goal; it's a way to develop an important skill that will be used to accomplish a greater goal.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2017-05-15T04:43:50.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12322" PostTypeId="2" ParentId="12318" CreationDate="2017-05-15T07:00:47.240" Score="2" Body="&lt;p&gt;Testing students' ability to identify notation is a perfectly sensible thing to do if learning the notation is an intended learning outcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the notation you show is not entirely standard. For example, the 'ray' notation would more commonly denote a vector. Some of them, such as parallel and perpendicular, are really shorthand rather than mathematical notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematical writing, symbols should be used when symbols are appropriate, but not as shorthand when words are appropriate. Understanding the distinction is not easy, and is probably beyond all pupils and most teachers at school level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When writing test questions, the question to consider along these lines is 'what mark would I want a student to get if they didn't understand the notation but otherwise knew the concepts?' If the wording of a question would stop an otherwise knowledgeable pupil from starting the question, you need to be comfortable that that is the aim of the question. Context will play a part in deciding how appropriate that is. An internal test written by one school teacher is much lower stakes than a nation-wide standard exam.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2017-05-15T07:00:47.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12323" PostTypeId="2" ParentId="12318" CreationDate="2017-05-15T10:40:58.060" Score="1" Body="&lt;p&gt;Mathematical notation is there to convey concepts in a very compressed manner, as opposed to the verbosity of the natural language when it comes to expressing the same ideas. Being able to read and understand it fast is a key skill in any profession with require working with anything related to mathematics, such as engineering. However, this is a learned ability and it takes time to learn and master.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From personal experience, I felt (still feel) learning to read maths as learning a new language. At the beginning I read it loud in my mind and then link the concepts to understand the meaning of the expression and, after a while, I could understand the meaning directly from the symbols. I felt the experience quite similar to learning to read music.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering this, I think the tests should be designed in such a way that they introduce slightly more mathematical notation every time such that students are exposed more and more to the mathematical language and less to the natural language.&lt;/p&gt;&#xA;" OwnerUserId="8147" LastActivityDate="2017-05-15T10:40:58.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12324" PostTypeId="2" ParentId="11757" CreationDate="2017-05-15T11:26:17.080" Score="3" Body="&lt;p&gt;These are easy to just make as needed:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt; 1 2 3                   has sd 1  (n=3)&#xA; 6 8 8 8 8 9 9           has sd 1  (n=7)&#xA; 1 2 3 4 4 5 6 7         has sd 2  (n=8)&#xA; 2 3 3 4 5 5 6 7 7 8     has sd 2  (n=10)&#xA; 1 1 3 3 4 5 5 5 6 7     has sd 2  (n=10)&#xA; 2 3 5 7 7 7 11          has sd 3  (n=7)&#xA; 1 3 5 5 6 6 6 7 12      has sd 3  (n=9)&#xA; 1 1 1 1 1 1 2 3 5 8 9   has sd 3  (n=11)&#xA; 0 2 3 3 3 5 5 6 9 14    has sd 4  (n=10)&#xA; 3 3 3 4 4 5 5 7 11 15   has sd 4  (n=10)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Of course you can shift these up or down as desired.&lt;/p&gt;&#xA;" OwnerUserId="755" LastActivityDate="2017-05-15T11:26:17.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12325" PostTypeId="2" ParentId="12318" CreationDate="2017-05-15T11:53:15.973" Score="6" Body="&lt;p&gt;I apologize in advance if my answer will be long, but this is a very interesting question and I'll try to give it the answer it deserves. I will refer not exclusively to notation in geometry but in mathematics in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The very short answer is: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;it is fundamental to use &lt;strong&gt;some&lt;/strong&gt; notation, or we'd be back to the painful description of equations of early algebra, but use too much and you'll cruelly add a burden to your students.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Now, for the long answer: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematical notation can have on students, especially pre-university or non-maths students (meaning, people that did not choose willingly to attend maths lessons), a double-sided effect:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;it can be extremely arousing for us nerds, as it becomes the key to understanding that mathematics is, after all, a way of manipulating concepts as a language per se, and as a language it has its own &quot;alphabet&quot; (forgive the poor wording in terms of linguistics). It is also like giving us new toys to play with, and we then start having fun with them in external contexts - as when we use existential quantifiers in everyday's life. &#xA;Furthermore, it speeds up explanation significantly, which means gifted students will be more stimulated;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;it can be extremely discouraging for the self-diagnosed-mathematically-impaired, which see in symbols one further obstacle that the teacher/system/Illuminati have made up with the purpose of making this senseless subject even less understandable.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Because of this, as with many things in teaching, it's not intrinsically better to choose to use as much or as little notation notation as possible, but balance is key:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;use too much notation and you'll make things harder for the second group;&lt;/li&gt;&#xA;&lt;li&gt;use too little notation and you'll be missing on a very useful tool, and possibly bore the best students to death.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Given all this, let me then give you some, hopefully useful, practical tips for introducing more notation with minimum damage:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is &lt;strong&gt;fundamental for every student to understand&lt;/strong&gt; is that no notation has been given to us, perfect and immutable, by some divinity: &lt;strong&gt;notation is an agreement between two parts communicating&lt;/strong&gt;, and different parts (e.g., texts, fields, teachers..) can give to the same symbol different meanings. Mathematics is more alive than one thinks, &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_mathematical_notation#Symbolic_stage&quot; rel=&quot;noreferrer&quot;&gt;as well as constantly evolving&lt;/a&gt;! Therefore agree explicitly with your students: &lt;em&gt;if you don't mind, from now on I'd use symbol x instead of writing every time [meaning of x]&lt;/em&gt;. This will also make the whole teaching feel less an unidirectional flow of pre-compiled stuff from your head to their notebooks;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;A bit at a time&lt;/strong&gt;: add new symbols progressively, trying to avoid your students an indigestion for receiving too much new information in one go;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;stress out the importance of minor differences&lt;/strong&gt;, sometimes students tend to quotient over little graphical things thinking they are only quirks while they actually carry a meaning;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;set a &quot;minimum level&quot; of notation&lt;/strong&gt; and use it in your questions, but &lt;strong&gt;encourage your students to use more&lt;/strong&gt;, for example by providing a little extra credit to those who can also formulate their answers in a concise way rich of notation - potentially, notation-only!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8181" LastActivityDate="2017-05-15T11:53:15.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12326" PostTypeId="2" ParentId="12298" CreationDate="2017-05-15T12:48:15.853" Score="2" Body="&lt;p&gt;Successfully completing one's schooling requires clearing many hurdles, some of which have are very useful in later schooling and career, and others that may not add much value.  Roman numerals mostly falls into the latter category.  Many books still use i, ii, iii for forwards and older films may list the copyright as MCMXLIX.  You can survive without this knowledge (or google it when needed), but it is useful cultural knowledge.  In any case, this particular hurdle should not be too hard to clear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The testing described in the question sounds more like a cultural than numeracy test.  In mathematics, Roman numerals are one example of a different ancient numeral system.  Did this same test ask about Mayan or Babylonian numeral systems?  As a base 60 system, the Babylonian system is very interesting from a mathematical point of view. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I have a BA in Mathematics and have been a software engineer for many years.&lt;/p&gt;&#xA;" OwnerUserId="8187" LastActivityDate="2017-05-15T12:48:15.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12327" PostTypeId="2" ParentId="12308" CreationDate="2017-05-15T13:29:30.787" Score="3" Body="&lt;p&gt;It's a skill that we have to encourage physics postgrads to practice and use.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My physics degree included a course on estimation/approximate methods (the only time I've considered a spherical mammal on an infinite plane in a vacuum).  I wish it was taught more.  But I suggest it's more like a practical skill than say finding the roots of a quadratic.  Certainly I've done this sort of estimation more in practical problems (How many tins of paint to paint a room? How much cable should I buy? Have I got enough petrol for the whole journey?) than in any classroom&lt;/p&gt;&#xA;" OwnerUserId="8180" LastActivityDate="2017-05-15T13:29:30.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12328" PostTypeId="2" ParentId="12298" CreationDate="2017-05-15T13:47:41.403" Score="5" Body="&lt;p&gt;In my country, many buildings (city halls and other official buildings) have their construction date, sculpted upon the building, in Roman numerals.&#xA;Imagine my standing next to a 12-year old child, before such a building, I ask him/her when this building has been built and (s)he has no clue about the meaning of the &lt;code&gt;MCMxxx&lt;/code&gt; letters, I'd be very disappointed in this child's education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As stated before, it's up to schools to pass general knowledge to their students, and the mathematics lectures should be no exception on that.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-05-15T13:47:41.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12329" PostTypeId="2" ParentId="12298" CreationDate="2017-05-15T16:12:03.423" Score="3" Body="&lt;p&gt;The Roman numerals are used. It is useful to be able to read them, isn't it?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They are used because of aesthetics. vsz noted Star Wars, episode IV versus Star Wars, episode 4. Old buildings in Europe have them as designation when they were built.&lt;/li&gt;&#xA;&lt;li&gt;They are used to emphasize different (level of) enumeration. Chapter I, section 1, for example. Or you can use Roman numerals to count Contents or Attachments and Arabic numerals to count the &quot;true&quot; pages.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It is also good to train in levels of abstraction. You have five particular apples on the desk, then we move to abstract number five described with particular symbol 5 or V. It can make another level of mathematical abstraction a bit easier, the step where a number five is replaced by a variable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the difference between systems of writing down the numbers - I, II, III, IV, V, VI, VII, VII, IX, X, ... - compared to the common decimal and background binary, octal and hexadecimal systems - 0 1 2 3 10 11 12 13 20, ... (base 4) - is The Lesson to be taught. And the &quot;M for 1000&quot; is the mandatory step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rule: If the numerals are in ascending order the smaller one is substracted, if they are in descending order they are sumarized forces us to use nontrivial &quot;decoder&quot; to handle Roman numbers. It is a good training to have agile mind.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Side note: In Czech, there is dummy sentence &quot;Lidé Co Dávají Málo&quot; (people who give a few) which resembles in the Roman numerals for 50 and above (&lt;em&gt;I V X&lt;/em&gt; &lt;strong&gt;L C D M&lt;/strong&gt;). I think it is easy to create simillar dummy sentence for any language.&lt;/p&gt;&#xA;" OwnerUserId="8099" LastActivityDate="2017-05-15T16:12:03.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12330" PostTypeId="1" CreationDate="2017-05-15T17:03:19.163" Score="17" ViewCount="446" Body="&lt;p&gt;I am looking for ways I can correct fundamental math mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am currently tutoring someone taking a course which is a cross between first year calculus and grade 12 functions. In high school he learned math by memorizing a bunch of rules and then matching them to different types of questions. This works ok for simple questions but now if he sees something he does not instantly recognize for he has no idea what to do and attempts to solve by doing things that look similar to the rules he knows. As a result the only math he can do is plug n' chugging numbers into memorized solutions with no real understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some examples of the mistakes he makes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\frac{x+5}{y+5} = \frac{x}{y}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He has a rule that says if you see the same thing on the top and bottom of a fraction it equals 1 and can be crossed out but he fails to understand that this rule only applies for multiplication not addition.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$x^2 - x = x$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The difference between $x^2$ and $2x$ is not clear to him.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\frac{15}{3x} * 2x = \frac{15*2x}{3x * 2x}$  or sometimes $\frac{15}{3x}*2x = \frac{15}{3x*2x}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He simply can't wrap his head around fractions he would have no issue with this example if it was presented in the form $\frac{15}{3x} * \frac{2x}{1}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He also struggles with basic algebra such as solving a linear systems, 2 equations 2 unknowns. He can not put together the solution and gets lost just moving numbers around until he forgets what the goal of the question is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I see him make a mistake like this I get him to stop the question and then give him a small example like #1 with dummy numbers and get him to solve it the right way and the wrong way to make it clear that they are not equal. Then I let him return to the question and he corrects the mistake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for anything else I can try to fix his misunderstandings with fundamental mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also I am not a professional tutor I am just trying to help this guy out.&lt;/p&gt;&#xA;" OwnerUserId="8192" LastEditorUserId="11" LastEditDate="2017-05-16T16:56:52.380" LastActivityDate="2017-05-16T16:56:52.380" Title="Addressing fundamental math errors" Tags="|students-mistakes|teaching|struggling-students|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12331" PostTypeId="2" ParentId="12330" CreationDate="2017-05-15T21:15:49.720" Score="9" Body="&lt;p&gt;It sounds like he is not prepared for a course in Calculus.  I would encourage him to go back and retake College Algebra which reinforces factoring and canceling out terms.  Sadly, it is going to be an uphill battle for someone in a Calculus course that makes mistakes that most Algebra 1 students don't make.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When my students make these mistakes, the only thing you can do is reteach the concept to them.  For example, for a student that is struggling to multiply $\frac{1}{3}\times3$ I would reteach multiplying fractions and representing whole numbers as fractions.  Students make the mistake of canceling out 5 in $\frac{x+5}{y+5}$ frequently.  You might ask them to plug in a number for x and y and see if $\frac{x+5}{y+5}=\frac{x}{y}$.  Then they may come to realize that you may not cancel a value unless it is being multiplied to the numerator and denominator.&lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2017-05-15T21:15:49.720" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12332" PostTypeId="2" ParentId="12301" CreationDate="2017-05-16T09:35:32.550" Score="3" Body="&lt;p&gt;Let me answer your question from my own experience as a student, several years ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Studying alone had the disadvantage that I could take a time-off whenever I felt like it, which is bad for a person with less self-discipline.&lt;br/&gt;&#xA;Studying in group had the disadvantage that I could ask for explanation about the course, which could reduce my ability to solve my own problems, a skill which is quite crucial while doing a math exam :-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence I ended up with the following procedure: I studied together in group, as much as possible, but there was the firm criterion that I only did this when none of the other people was studying the same matter I was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a result, I ended up studying with futur lawyers, agriculture engineers, nurses, architects, ..., and even mathematicians, but the latter only when they were following other classes than me :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-05-16T09:35:32.550" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12333" PostTypeId="1" AcceptedAnswerId="12334" CreationDate="2017-05-16T12:13:22.580" Score="0" ViewCount="93" Body="&lt;p&gt;Do you know a (simple) inverse modulo calculator somewhere on the Internet?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to explain simply inverse modulo? (when the concept of modulo is already a bit known)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to explain someone how to find a calculation resulting with a given modulo? For instance, finding a calculation with a modulo = 5.&lt;/p&gt;&#xA;" OwnerUserId="7575" LastActivityDate="2017-05-16T12:21:31.527" Title="How to explain inverse modulo?" Tags="|secondary-education|self-learning|calculations|" AnswerCount="1" CommentCount="2" ClosedDate="2017-05-20T17:41:40.340" ContentLicense="CC BY-SA 3.0" />
  <row Id="12334" PostTypeId="2" ParentId="12333" CreationDate="2017-05-16T12:21:31.527" Score="2" Body="&lt;p&gt;For the first, there are a number of apps out there, though I will recommend that you go somewhat non-simple and use &lt;a href=&quot;http://sagecell.sagemath.org/?z=eJxLVLBV8M1P0TDWMdfk5SooyswrUdBNhLES43QNAZxGCVk=&amp;amp;lang=sage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sage&lt;/a&gt;.  Are you looking for additive or multiplicative inverse?  In either case, it is built-in.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;a = Mod(3,7)&#xA;print -a&#xA;print a^-1&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;This gives &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;5&lt;/code&gt;, respectively.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-05-16T12:21:31.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12335" PostTypeId="2" ParentId="12301" CreationDate="2017-05-16T12:53:26.470" Score="6" Body="&lt;p&gt;Keep in mind that this is a very personal question. Some people simply work better in groups and some people work better alone. We are all different, and we should embrace that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a more objective point of view, both of them have their advantages and disadvantages. What I noticed from people studying in groups is that they usually forget one basic thing: learning is an interior process, not an exterior one. It is you who has to take the information and internalize it, mix it in the already built web of knowledge from your mind. Study groups are useful only as long as they support this process. However, most of the times, study groups tend to shift the focus from each individual's gain to the overall gain, thus making it more inefficient for some.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another aspect of the group study is fun. For example, I get very frustrated at times when I can't comprehend something or I don't manage to find a solution. Speaking to somebody once in a while helps relieve this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sharing is another advantage of study groups. The most helpful teachers for me were not those who taught me something, but those who opened doors for me by briefly explaining a concept/mentioning a resource/etc. I think this is the ultimate goal of a study group: study on your own, share your experiences, build upon other's experience. Unfortunately in practice doesn't always work this way.&lt;/p&gt;&#xA;" OwnerUserId="8147" LastActivityDate="2017-05-16T12:53:26.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12336" PostTypeId="2" ParentId="12108" CreationDate="2017-05-16T13:00:25.933" Score="4" Body="&lt;p&gt;Congratulations for your passion you have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not a mathematician so I can't give you specific advice, but I would say go with what fuels your passion and curiosity. There is not a one-time decision, you can mix reading standard textbooks with reading some more &quot;exotic&quot; resources. I would suggest you combine textbooks/college courses (which are meant to give a foundation) with research papers in the field, which are usually more in depth (and also more time consuming) than textbooks. Actually, textbooks are merely a literature review over older (now considered basic) papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bottom line is: dare to follow your curiosity, it can't hurt, but don't forget about the basics.&lt;/p&gt;&#xA;" OwnerUserId="8147" LastActivityDate="2017-05-16T13:00:25.933" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12337" PostTypeId="1" AcceptedAnswerId="12339" CreationDate="2017-05-16T15:00:48.693" Score="8" ViewCount="283" Body="&lt;p&gt;I am struggling with my students who can think only in concrete terms, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;they can compute with concrete numbers but are not able to think in terms of e.g. functions on natural numbers and come up with a general solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This cognitive skill would be needed when I teach them writing a function in a programming language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to search for teaching abstraction but that gave me mostly hits about programming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to find math teaching articles that would help me teaching this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the name of this subdiscipline in math education?&lt;/p&gt;&#xA;" OwnerUserId="8210" LastActivityDate="2017-05-16T22:26:13.170" Title="What is the name of this discipline in mathematics education?" Tags="|algebra|functions|number-theory|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12338" PostTypeId="2" ParentId="12103" CreationDate="2017-05-16T18:11:44.917" Score="8" Body="&lt;p&gt;Some nice geometrical applications arise in the analysis of periodical curves such as &lt;a href=&quot;http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node34.html&quot; rel=&quot;noreferrer&quot;&gt;Roulettes&lt;/a&gt; (Spirograph curves), &lt;a href=&quot;http://en.wikipedia.org/wiki/Star_polygon&quot; rel=&quot;noreferrer&quot;&gt;Star Polygons&lt;/a&gt;, etc. Concrete experience with implementations in toys like &lt;a href=&quot;https://en.wikipedia.org/wiki/Spirograph&quot; rel=&quot;noreferrer&quot;&gt;Spirograph&lt;/a&gt; also provides excellent motivation for more abstract concepts such as cyclic groups. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WknW7.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WknW7.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2017-05-16T18:11:44.917" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12339" PostTypeId="2" ParentId="12337" CreationDate="2017-05-16T22:26:13.170" Score="8" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the name of this subdiscipline in math education?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In one of the comments, &lt;strong&gt;Dave L Renfro&lt;/strong&gt; has a reference to Piaget, whose work was primarily done with younger schoolchildren. With respect to extending this work into the older years of one's education, a potentially good place to look would be &lt;strong&gt;APOS Theory&lt;/strong&gt;, which is due to Ed Dubinsky and collaborators. For an introduction, see &lt;a href=&quot;https://pdfs.semanticscholar.org/6850/b01648bf43e15cbdbcf871b215c3cf44825d.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research&lt;/strong&gt;&lt;/a&gt; (pdf) by Dubinsky and Michael McDonald. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, you might search out additional information on APOS theory using &lt;a href=&quot;https://scholar.google.com/scholar?hl=en&amp;amp;q=APOS+Theory&amp;amp;btnG=&amp;amp;as_sdt=1%2C33&amp;amp;as_sdtp=&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;google scholar&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As pertains to your description:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;they can compute with concrete numbers but are not able to think in terms of e.g. functions on natural numbers and come up with a general solution&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;These students are probably at either the &lt;strong&gt;A&lt;/strong&gt;ction level, or, if they are carrying out these computations in an algorithmic, procedural way, then they may be at the &lt;strong&gt;P&lt;/strong&gt;rocesses level. (The final two letters stand for &lt;strong&gt;O&lt;/strong&gt;bjects and &lt;strong&gt;S&lt;/strong&gt;chemas; more can be found in the first link, above, as well as in its references.)&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2017-05-16T22:26:13.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12340" PostTypeId="1" CreationDate="2017-05-17T20:55:25.347" Score="13" ViewCount="314" Body="&lt;p&gt;This is a more general pedagogical question than specifically teaching mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a student, I have always been frustrated that some things were kept from me and not thought deeply. For example, in school, I was taught matrix multiplication as &quot;this is how we multiply matrices&quot;, after a bit of discussion about what is a matrix. However, there is a beautiful world behind the simple algorithm we mindlessly applied all over again that attaches a lot of meaning to every operation and I have to wait years until I got to find about it. The same goes for determinants, limits and much more concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One might argue that there is not enough time for all this. But there is. Between the tens and hundreds of matrix multiplication exercises we've done (by we I mean me and my colleagues), there was time for a few tens of minutes of explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some time ago I started tutoring/mentoring students on an individual basis in maths and computer science (more of the latter than the former, but the question and the problem is still the same). What I found is that my requirement to go and understand the fundamentals before doing anything seems to frustrate and demotivate them. They won't have the patience of looking at the results and trying to get the meaning behind them, and my attempts were received with frustration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After a while, those who &quot;survive&quot; get to thank me for this treatment, but I would like to make this process more enjoyable for them. How to keep their passion and curiosity and guide them to go for the meaning rather than for the result?&lt;/p&gt;&#xA;" OwnerUserId="8147" LastActivityDate="2017-05-25T04:20:07.657" Title="How to keep passion in students?" Tags="|tutoring|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="12341" PostTypeId="1" AcceptedAnswerId="12344" CreationDate="2017-05-18T15:35:04.490" Score="4" ViewCount="1789" Body="&lt;p&gt;This is a question about elementary geometry, so I think it belongs on this site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $d$ be the length of a diagonal in a rectangle, and let $m$ be half the perimeter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jNKEo.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jNKEo.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then a formula for the area of the rectangle is given by $$A=\frac 1 2 (m^2-d^2).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can easily prove the formula using algebraic manipulations, but I feel it should be possible to see directly from the diagram that the size of the rectangle is half the difference between the sizes of those squares.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is a simple geometric (and visual) proof of the formula?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="39" LastEditorUserId="39" LastEditDate="2017-05-18T19:07:11.697" LastActivityDate="2017-05-20T03:32:13.747" Title="A formula for the area of a rectangle" Tags="|geometry|visual-proof|" AnswerCount="3" CommentCount="5" ClosedDate="2017-05-23T13:43:12.093" ContentLicense="CC BY-SA 3.0" />
  <row Id="12343" PostTypeId="2" ParentId="12341" CreationDate="2017-05-18T18:47:33.820" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ZW8AE.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZW8AE.png&quot; alt=&quot;Let m = a + b. Then d^2 = a^2 + b^2. Remove that from m^2 and you have twice the original rectangle.&quot;&gt;&lt;/a&gt;&#xA;$                                  $&lt;/p&gt;&#xA;" OwnerUserId="8149" LastActivityDate="2017-05-18T18:47:33.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12344" PostTypeId="2" ParentId="12341" CreationDate="2017-05-18T18:57:39.580" Score="10" Body="&lt;p&gt;First copy your rectangle like this to make a big square of side-length m with a square of side-length d drawn inside it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WtNIm.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WtNIm.png&quot; alt=&quot;four rectangles to make some squares&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The big square minus the small square leaves four half-rectangles (coloured pink in this picture).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xTv9e.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xTv9e.png&quot; alt=&quot;four half rectangles&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Half of the pink area is two half-rectangles, and so is the area of one whole rectangle.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2017-05-18T18:57:39.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12345" PostTypeId="2" ParentId="5730" CreationDate="2017-05-19T01:13:32.087" Score="4" Body="&lt;p&gt;The 13-year old is &quot;right&quot; -it is the question that is wrong. It is a classic example of an attempt to create &quot;tangible situations&quot; in order to present abstract concepts, forgetting in the process that &quot;tangible situations&quot; introduce aspects that have been abstracted out in order for the abstract concept to work, &lt;em&gt;but only by those that are already familiar with the abstract concept.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the 13-year old is not &quot;already familiar&quot; with the concept, and so it uses plain logic, still unaffected by what we are now trying to teach him.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it does not think of abstract numerical ratios  $2/1$ and $20/10$, or $2$-in-$3$ and $20$-in-$30$. Instead, it pictures one bag with 3 marbles, and another bag with 30 marbles,with colors distributed as mentioned. Probably the &lt;em&gt;last&lt;/em&gt; thing it will pay attention to is &lt;em&gt;the relation between the ratios or the proportions&lt;/em&gt;. Other things may arise in its thinking: for example, the size of the bags. Are they of equal volume? If yes the one will be much more empty than the other... will this affect its chances if it puts its hand in? Is anybody telling the 13-year-old that this aspect of the &quot;tangible situation&quot; &lt;em&gt;won't&lt;/em&gt; affect its chances? And if this is the case, &lt;em&gt;why&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are the bags such that they really allow for homogeneous layers of marbles? If the bag is made of soft leather, the one filled with the 50 marbles will tend to create an asymmetric heap, with larger base than tip... does this affect the &lt;em&gt;actual real-world chances of drawing a black marble from the bag?&lt;/em&gt;   This aspect is also intended to be &quot;abstracted out&quot;, but did anyone tell that to the 13-year old? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because its logic will likely try to assess all these aspects of the &quot;tangible situation&quot; in order to answer the question, an then... all bets are off, as regards what it will appear as the correct answer to its mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, when creating such examples, we must pay great care so that they are as &quot;neutral&quot; as possible, leaving the information we want the answer to be based upon (here the ratios/proportions) as the main piece of information coming out of the example, so that we increase the chances that the student will indeed focus on them, and not on all these other unintended aspects our example may introduce.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not saying this is easy, on the contrary it is hard to very-hard, because we have to temporarily put aside our own familiarity with the concepts we are trying to teach.&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2017-05-19T01:18:44.700" LastActivityDate="2017-05-19T01:18:44.700" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12348" PostTypeId="1" CreationDate="2017-05-20T01:31:50.810" Score="8" ViewCount="140" Body="&lt;p&gt;I'm a student majoring in maths. The sole reason I chose mathematics is that I &lt;em&gt;like&lt;/em&gt; mathematics. I'm sure I do not belong to that &quot;gifted&quot; category. I haven't participated in maths competitions either. I don't know if the question I'm about to ask is a consequence of lack of formal training when it comes to problem solving and the like. The following is my question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've experienced that the problems I do as exercises have a predisposition to not stay in my mind. For example if a problem I did in the last month is given to me today I have to go through my thinking process anew. I don't know if it is normal or not. I don't know if this issue is rather psychological either. But I &lt;em&gt;believe&lt;/em&gt; it is not. How would you remedy this? What are the tips that one could propose so that I may &lt;em&gt;appraise and retain the important or the key points of an exercise&lt;/em&gt;? Please help.&lt;/p&gt;&#xA;" OwnerUserId="8233" LastActivityDate="2017-05-20T12:26:28.137" Title="How to retain the key points of an exercise?" Tags="|undergraduate-education|student-motivation|problem-solving|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12349" PostTypeId="2" ParentId="12341" CreationDate="2017-05-20T03:32:13.747" Score="0" Body="&lt;p&gt;Here's what I like about this problem, as a math educator. It brings together lots of different aspects of math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that two pieces of information (length and width) determine the size of a rectangle, giving both its area and perimeter. If I know two independent pieces of information about the rectangle, I should be able to find the rest. This related to systems of equations (beginning algebra) and linear algebra. &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-05-20T03:32:13.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12350" PostTypeId="2" ParentId="12348" CreationDate="2017-05-20T07:08:03.210" Score="5" Body="&lt;p&gt;For me, the process is as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Do the exercise.&lt;/li&gt;&#xA;&lt;li&gt;Do the exercise again. This is probably faster than the first time, since I have a vague feeling of what I should be doing, and maybe remember some dead ends that I can avoid.&lt;/li&gt;&#xA;&lt;li&gt;Repeat step number 2. At some point, if I have to do the exercise often enough, I learn it by heart. This is rare.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In practice, this rarely happens unless I am teaching and encounter the problem often there, or I do research on a given subject and have to think through the same lemma several times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, the problem solving heuristics and patterns of thought that are repeated often are what stick to mind. When encountering a familiar problem, a problem of familiar type, I have a feel for the problem and know which common technique to try first, and I know what the likely steps in solving the problem are. I still have to do the work to be sure the attempted proof works (unless I have done the same argument several times recently).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;It is, in general, okay to not remember solutions by heart. If you can repeat the argument, even if it feels like fumbling, that is already good.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most elementary context in which I can observe this is multiplication; say, the powers of two. How much is $2^7$? I can remember $2^6 = 64$ (because I have calculated this so frequently) so $2^7 = 128$. If I want to be sure this is the right answer, I have to check it. If I do this calculation often enough, then I can guess the right answer immediately, but must still check it to be certain. Before I remembered $2^6$, I had to start with $2^5 = 32$, and before that, I had to start from $2^3$, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar process happens with less elementary problems, though the steps that happen by heart or from strong intuition are more interesting.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2017-05-20T07:08:03.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12351" PostTypeId="2" ParentId="12348" CreationDate="2017-05-20T12:26:28.137" Score="4" Body="&lt;p&gt;Any exercise is just one example of a family of similar exercises. You can explore those similar exercises by writing and solving your own variations of the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be more concrete, I assume that you have had at least first semester calculus so have seen something like the sliding ladder problem as an exercise. Suppose that you had the following exercise:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A 10 ft. ladder is leaning against a wall. The bottom of the ladder is&#xA;  being pulled away from the wall at the rate of 2 fps. How fast is the&#xA;  top of the ladder sliding down when it is 8 ft. high?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After you have solved the problem, start by simply changing the numbers. For example change 10 to 12 or 8 to 5. You will quickly notice that the original problem used a Pythagorean triple ($6,8,10$ satisfy $6^2 + 8^2 = 10^2$) which is convenient for a numerically nice answer but not essential -- radicals present little problem. Once you solve the same problem numerous times with different numbers, see if you can solve it once and for all with symbols like $L$ replacing specific numbers like $10$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, start making more substantial changes. Push the ladder towards the wall rather than away from it. Change the given information so that it gives the rate that the top of the ladder sliding down the wall and asks about the rate at which the bottom of the ladder is moving. Maybe even get a bit creative -- what if it is an extension ladder which is extending (or contracting) at a given rate in the course of the sliding? Or -- what if the wall isn't perpendicular to the floor (an implicit assumption)? Convince yourself that you can still do it with the law of cosines replacing the Pythagorean theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you get to the point that you not only know how to solve a given problem but are able to create similar problems on your own, then you will have a deeper understanding of the relevant key points, an understanding which is unlikely to be so quickly forgotten.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2017-05-20T12:26:28.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12352" PostTypeId="2" ParentId="12113" CreationDate="2017-05-20T20:14:57.553" Score="0" Body="&lt;p&gt;I present just a couple of ideas that might make a bridge between the discrete and continuous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, one thing I notice with many students is they don't have an idea that the probabilities of all the outcomes add up to 1. This stops them being able to do a lot of calculations for continuous distributions later. Moreover, it means they don't get the fundamental idea of distribution which is a description of &lt;em&gt;all&lt;/em&gt; the possibilities and how likely they &lt;em&gt;all&lt;/em&gt; are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that we should probably never ask for the probability of any event in isolation but always also ask for probabilities for a set of events that complete it to the whole sample space. For example, don't just ask for the probability of getting a 7 on two dice, but also less than 7 and more than 7. At the very least, always ask for the probability of A and not A. My instinct is that to start with, students will calculate both based on the ratios, and maybe later will realise there's an easier way. But at least we'll drive home the idea that events never exist without other events to fill them out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, perhaps a road to asking about continuous probabilities, is to ask about a discrete set of objects with continuous measurements on them. If the measurements are written with decimals it will be even more obvious that they are supposed to be continuous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, take 14 trees with their heights measured in feet:&lt;br&gt;&#xA;60.92 63.39 64.10 59.07 63.05 59.64 60.07 60.69 60.28 61.62 58.49 56.81 56.43 59.49 (from the Loblolly dataset in R).&lt;br&gt;&#xA;Then you could ask questions like the probabilities of choosing a tree with height in the ranges 55 to 57.5, 57.5 to 60, 60 to 62.5 and 62.5 to 65 including drawing a graph showing these probabilities. Asking various different collections of zones might help to necessitate the axis being marked out on a scale rather than in discrete labels. Choosing zones that aren't equal might make it easier to necessitate a way to show the amount of probability in a way other than height so that the graph isn't so misleading.&#xA;A dataset with 100 numbers in it might make this even more obvious and lead towards the probability distribution which has infinitely many numbers.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2017-05-20T20:14:57.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12353" PostTypeId="1" AcceptedAnswerId="12358" CreationDate="2017-05-20T21:50:31.977" Score="5" ViewCount="177" Body="&lt;p&gt;Could anyone recommend me some software that helps me teach about the &lt;code&gt;centroid&lt;/code&gt; of an object?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something that was dynamic on the computer screen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Destined for 18 years of age.&lt;/p&gt;&#xA;" OwnerUserId="8236" LastEditorUserId="8236" LastEditDate="2017-05-21T23:21:37.150" LastActivityDate="2017-05-25T20:00:11.987" Title="How to teach mass center for young people" Tags="|geometry|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12354" PostTypeId="1" CreationDate="2017-05-21T13:39:37.870" Score="6" ViewCount="188" Body="&lt;p&gt;The CollegeBoard website lists many &lt;a href=&quot;http://www.collegeboard.com/html/apcourseaudit/courses/calculus_bc_textbook_list.html&quot; rel=&quot;noreferrer&quot;&gt;AP calculus BC references&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it also mentions that &quot;The materials on this List range in alignment from 59% to 100%.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, which of them is aligned the most with the college board specification of the AP calculus?&lt;/p&gt;&#xA;" OwnerUserId="8241" LastEditorUserId="11" LastEditDate="2017-05-22T15:33:12.763" LastActivityDate="2017-05-22T15:33:12.763" Title="Which calculus textbook is aligned the most with the CollegeBoard course description?" Tags="|calculus|textbooks|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12355" PostTypeId="2" ParentId="12113" CreationDate="2017-05-21T19:42:20.363" Score="1" Body="&lt;p&gt;Draw a square on the board and shade in half of it.  Ask your students, &quot;If you throw a dart and hit the board, what's the probability that you hit the shaded region?&quot;  Cut the region down to a quarter of the square and ask the question again.  Lead this into a discussion of how the probability is the ratio of the area of the region to the total area and tie this back to the discrete probability formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now ask the same question about a point.  A point has &quot;0 area&quot; so the probability of hitting it is 0.  When there are infinitely many possibilities, i.e. there are infinitely many points inside the square versus, for example, a finite number of die rolls, we can't talk about the probability of individual events - just ranges of events/results.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2017-05-21T19:42:20.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12357" PostTypeId="1" AcceptedAnswerId="12359" CreationDate="2017-05-22T06:52:08.073" Score="8" ViewCount="394" Body="&lt;p&gt;For example,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\displaystyle \lim_{(x,y,z)\rightarrow(0,0,0)} \frac{x^2y^2z^2}{x^2+y^2+z^2}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This question is from 8th edition of Stewart Calculus textbook. My fellow graduate student TAs and myself recently got many students ask this question in office hour/by email.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because this is a freshman level calculus class, most instructors choose to only briefly explain this topic and (probably) do not expect students to write a full proof of such a problem on the exams. In our single variable calculus course syllabus the $\epsilon-\delta$ definition of limit is also omitted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, for a few people who do want to know how to solve this problem, what's the best way to explain?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some thought&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Convince them they must know the rigorous limit definition for such a problem. (Any better approach?)&lt;/li&gt;&#xA;  &lt;li&gt;Bound the function by 0 and $(x^2+y^2+z^2)^2$ and apply squeeze theorem. (Almost as hard as 1.?)&lt;/li&gt;&#xA;  &lt;li&gt;Change to spherical coordinate. (They won't learn this until they get to the multivariable integration chapter.)&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user2139" LastEditorDisplayName="user507" LastEditDate="2017-05-23T13:31:33.540" LastActivityDate="2017-05-25T19:01:01.920" Title="What's the best way to explain multivariable limit problems to students who are not familiar with $\epsilon-\delta$ proofs?" Tags="|calculus|limits|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12358" PostTypeId="2" ParentId="12353" CreationDate="2017-05-22T13:32:37.577" Score="3" Body="&lt;p&gt;Perhaps this simple &lt;a href=&quot;http://www.physicsclassroom.com/Physics-Interactives/Balance-and-Rotation/COM-Builder/Center-Of-Mass-Interactive&quot; rel=&quot;nofollow noreferrer&quot;&gt;Center of Mass Builder&lt;/a&gt;&#xA;is what you had in mind?&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/IDAhf.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IDAhf.jpg&quot; alt=&quot;Snapshot&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;As one click-adds (blue) mass to an object, the center of mass updates dynamically. Then you can pin a swivel point with the (green) pin, and the shape&#xA;swings around that point under the influence of gravity.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/2uoqG.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2uoqG.jpg&quot; alt=&quot;Swivel&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-05-22T16:35:22.807" LastActivityDate="2017-05-22T16:35:22.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12359" PostTypeId="2" ParentId="12357" CreationDate="2017-05-22T13:36:55.903" Score="10" Body="&lt;p&gt;Showing why there is a need for good definitions here is probably more important than showing students why a difficult definition &quot;works&quot; via plug-in hocus-pocus.  (Though if you have the time to do the proper definition properly that is even better; I certainly don't in multivariate calc.  Squeeze theorem, as commenter John Coleman points out, is a good compromise there, as long as you make it clear you know it's a deus ex machina for them at this point.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With modern software this becomes very easy.  You can do it with just about anything; I use &lt;a href=&quot;http://www.sagemath.org&quot; rel=&quot;noreferrer&quot;&gt;SageMath&lt;/a&gt;, and also find &lt;a href=&quot;http://web.monroecc.edu/manila/webfiles/pseeburger/CalcPlot3D/&quot; rel=&quot;noreferrer&quot;&gt;CalcPlot3D&lt;/a&gt; a good (free) option.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what I use in this situation (modifying your example slightly in the first instance since visualizing yours would require students &quot;seeing&quot; how contour surfaces work).  You can surely use whatever single graphic is in the text and modify it in your language of choice.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://sagecell.sagemath.org/?z=eJyN0L0OgjAQwPHdxHcgLIA9QlPFjXfga4Y00CAR26agoW9v0Q4EGdz-w_WXu76o8t0ZtBscD6mTOHIQ07n154qcdEWiJZCJAPwZwhhiE9rGMlpL0fNpTAjGBsgWgCr6YJPqm9pavnkPGnR1jUxePtyXIWDgRgxCJZ5irQfTrW_unI1jEhsu3-fCvzyhKO_Yliz2yRkw4NWN1pBPJYcfo9w3sNkJr77HGp1ijG-JFGUoRwUq37dCc7E=&amp;amp;lang=sage&quot; rel=&quot;noreferrer&quot;&gt;Example where the limit exists&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://sagecell.sagemath.org/?z=eJxtz70OgjAUhuHdxHsgLBR7CLXK2HvgZ4c00Ggj0qZU0969JTAYZXuH8z3JeXODYgc-To-HMmKRHpW9DMidfEtz5FqKfXtNATnICihC-C2Ww04rOdmZUULCvFrm3PCnsEb23SahAIGHc07XMYUQvRqVYYkRQwL2LvvHJOaZFQGp95FsVbJdRhk-3cSv1OxLDgiQr4c2Q7-MHv-MEle4xs0HdUtXjw==&amp;amp;lang=sage&quot; rel=&quot;noreferrer&quot;&gt;Example where the limit doesn't exist&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-05-22T13:36:55.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12360" PostTypeId="2" ParentId="12357" CreationDate="2017-05-22T13:37:30.123" Score="10" Body="&lt;p&gt;You can make this $\epsilon/\delta$ proof easy enough that an interested student should find the argument believable without experience with $\epsilon/\delta$ proofs.&#xA;Divide both the numerator and denominator by $x^2 y^2 z^2$ to get&#xA;$$\frac{1}{ \frac{1}{y^2 z^2} + \frac{1}{x^2 z^2} + \frac{1}{y^2 z^2}}$$&#xA;Then if $|x|,|y|,|z| &amp;lt; \epsilon$ then we have&#xA;$$ \frac{1}{y^2 z^2} + \frac{1}{x^2 z^2} + \frac{1}{y^2 z^2} &amp;gt; \frac{3}{\epsilon^4}.$$ Thus&#xA;$$\frac{1}{ \frac{1}{y^2 z^2} + \frac{1}{x^2 z^2} + \frac{1}{y^2 z^2}} &amp;lt; \frac{\epsilon^4}{3}$$&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2017-05-22T13:37:30.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12363" PostTypeId="2" ParentId="12357" CreationDate="2017-05-23T01:31:23.437" Score="2" Body="&lt;p&gt;Let's consider a path $$\left( x(t), y(t), z(t) \right)$$ with $\left( x(0), y(0), z(0) \right) = (0,0,0)$.  For small $t$, we can approximate this path with a straight line, in which case $$\left( x(t), y(t), z(t) \right) = (at, bt, ct)$$ for some constants $a,b,c$.  This is admittedly non-rigorous, but it should seem intuitively plausible that a smooth curved path looks straight if you zoom in close enough -- that, after all, is a fundamental idea in Calculus 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So now substitute that into the formula:  We need to compute&#xA;$$\lim_{t \to 0} \frac{(at)^2(bt)^2(ct)^2}{(at)^2+(bt)^2+(ct)^2}$$&#xA;which is a limit they can calculate by single-variable methods.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2017-05-23T01:31:23.437" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12364" PostTypeId="2" ParentId="12357" CreationDate="2017-05-23T13:47:42.637" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;$\displaystyle \lim_{(x,y,z)\rightarrow(0,0,0)} \frac{x^2y^2z^2}{x^2+y^2+z^2}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Even if this were a course where students were required to write epsilon-delta proofs, I would still imagine that anyone competent would approach this problem by first thinking informally about why this limit is expected to exist and be zero. I would split this into three levels of rigor:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1)  As you approach the origin, each of the three coordinates gets very small. The numerator is sixth order and the denominator second order, so we expect the numerator to get small much faster than the denominator. The first goal should be to get students to the level where this mode of reasoning seems natural to them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Next, I would worry about whether this argument fails for some special path leading to the origin. The only obvious possibilities that might have such &quot;special&quot; behavior are paths lying along the axes -- these are the only paths that seem obviously to have been singled out by the structure of the expression. We check that the argument in #1 still holds for these paths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) Do something more rigorous, such as the possibilities you suggested in the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the level at which your course is being taught, #3 doesn't seem like a realistic expectation on a homework problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it is more important for students to be competent enough to do #1 than for them to be able to cook up some tricky approach that applies specifically to this problem.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2017-05-23T13:47:42.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12365" PostTypeId="2" ParentId="12287" CreationDate="2017-05-24T15:49:13.353" Score="1" Body="&lt;p&gt;Remember that you are interviewing for a position at a community college, not Harvard.  That is not to denigrate community college students at all--they are just a different &quot;audience.&quot;  I assume that the interviewers already know why the second derivative test is valid for testing the points.  I would focus on reminding my listeners exactly what a second derivative is (a recipe for the slope of the slope) and give an understandable explanation from that point of view.   My guess is that your interviewers already know that you know it--they just want to see if you can teach it to THEIR students.  Good luck!&lt;/p&gt;&#xA;" OwnerUserId="8255" LastActivityDate="2017-05-24T15:49:13.353" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12366" PostTypeId="2" ParentId="12298" CreationDate="2017-05-24T16:14:06.630" Score="2" Body="&lt;p&gt;In this high-tech world, your child will eventually find it useful to understand Binary, Hexadecimal and other number systems that are not Base Ten.  (By the way, &quot;decimals&quot; are included in our base-ten system.)  As an understanding of exponents is needed to learn these systems, it makes sense to begin learning a new number system (Roman numerals) that does not require any complex mathematics.  &lt;/p&gt;&#xA;" OwnerUserId="8256" LastActivityDate="2017-05-24T16:14:06.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12367" PostTypeId="2" ParentId="12340" CreationDate="2017-05-25T04:20:07.657" Score="4" Body="&lt;p&gt;The problem is that you assume that people want to gain understanding. That is not what many students want. Many students simply want it to be over as soon as possible. The idea that there is something deeper, or, that they can understand more than is asked of them is completely foreign to their thinking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Star Trek terms, you're violating the &lt;em&gt;Prime Directive&lt;/em&gt;. Your assumption is that your culture should be their culture. The audacity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I jest, I think the best thing is to be passionate. Share what you know, be positive, and it may wear off. They may find out that they can do more than just mimic math. They can actually &lt;em&gt;understand&lt;/em&gt; it. That's the dream.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, the people who are annoyed that you do more than just explain how to do problems are exactly the people you should probably never let guide your way of being. There will always be critics. Rarely are their motives worthwhile.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-05-25T04:20:07.657" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12368" PostTypeId="1" AcceptedAnswerId="12388" CreationDate="2017-05-25T10:11:46.860" Score="19" ViewCount="848" Body="&lt;p&gt;Although @Gareth Shepherd recently posted &lt;a href=&quot;https://matheducators.stackexchange.com/questions/12330/addressing-fundamental-math-errors&quot;&gt;Addressing fundamental math errors&lt;/a&gt; close to the issue, I experienced my problem of misunderstanding in class, where two good K10 students were asked to present at blackboard the reduction of $\frac{12-3a^2}{3(2+a)}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Self-invented &quot;rule&quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students factored 3 in to denominator, then applied a &lt;strong&gt;self-invented rule&lt;/strong&gt; of &quot;&lt;em&gt;divide numbers by numbers, symbols by symbols&lt;/em&gt;&quot;, transforming $\frac{12-3a^2}{6+3a}$ into &#xA;$\frac{12}{6} - \frac{3a\cdot a}{3a}=&#xA;\underline{\underline{2-a}}$,&#xA;where - of course - the right hand side (RHS) is correct, but the left hand side (LHS) represents a faulty operation on a fraction in focus here,&#xA;as the RHS is &lt;strong&gt;not  a transformation&lt;/strong&gt; of the LHS (despite the &lt;em&gt;result&lt;/em&gt; is that LHS$=$RHS, as pointed out in &lt;a href=&quot;http://www.madmath.com/2010/06/stuff-that-shouldnt-work.html&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Blog mentioned by Daniel in his comment below&quot;&gt;the blog of Daniel R Collins&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In class, I pose as a proponent for documenting ones (mathematical) actions with rules, and I have engaged in a negotiation with my students, who are used to emphasizing on results only from earlier school experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Building knowledge on mistakes&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I expect to continue teaching the class after summer, and see the mistake by two relatively strong students asn opportunity to build a faults-are-necessary-culture in he math lessons for this class. &#xA;Thus, I revisited Lakatos' &lt;a href=&quot;https://math.berkeley.edu/~kpmann/Lakatos.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proofs and Refutation&lt;/a&gt; where both Teacher and Pupils attempt, refute, &lt;strong&gt;and improve&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for suggestions as to &lt;strong&gt;how I can use the two students' mistake constructively&lt;/strong&gt;.&#xA;Deployment of the misconceptions should meet the following criteria.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Both the two mistaken students and their classmates &#xA;must feel encouraged rather than intimidated by me &#xA;bringing up our discussion in the public of the class &#xA;(thus exposing their misunderstanding). &#xA;This can be done by not mentioning the discussion, &#xA;but then I would run the risk of the two students feeling&#xA;my approach surreptitious - sneaking in on their weaknesses.&#xA;Hence, I prefer an approach, where I take responsibility for the discussion,&#xA;and the entire class sees the mistake &lt;strong&gt;as an opportunity&lt;/strong&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The explanation / treatment should simple enough to be briefly presented, &#xA;yet sufficiently stringent to state the point that the introduced, self-invented rule was based on an invalid conjecture&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The 'proof' tag is mainly intended to the students' &lt;strong&gt;lack&lt;/strong&gt; hereof in their deployment of the self-invented rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit 1&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There &lt;em&gt;are&lt;/em&gt; no counterexamples, so the comment of @Amir Asghari is taken into account by deleting, what mentioned a counterexample.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit 2&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of &quot;indicating&quot; that a transformation is not generally valid by the use of $\neq$, I acknowledge the insight of @DRF -- Students may have a hard time interpreting the symbols, including $=$ and $\neq$, in their original, intended meaning. &#xA;Thus, it is counterproductive to use (any) sign in class to &quot;indicate&quot;&#xA;anything apart from what these signs are actually meant to signify.&#xA;I have thus reformulated my &quot;temptation&quot; to use that sign from symbolic to worded.&lt;/p&gt;&#xA;" OwnerUserId="3502" LastEditorUserId="8571" LastEditDate="2018-06-27T23:45:33.723" LastActivityDate="2018-06-29T03:33:21.977" Title="Constructive refutation of student misconception" Tags="|secondary-education|mathematical-pedagogy|proofs|students-mistakes|fractions|" AnswerCount="3" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12369" PostTypeId="2" ParentId="7243" CreationDate="2017-05-25T14:06:09.393" Score="1" Body="&lt;p&gt;A pair of fresh eyes may see goodies and errors alike, that you cannot see yourself after writing the assignment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, I experiment with letting students peer-review (an anonmized fraction of) their upcoming assignment using the same rubrick as I intend to use when asessing their final submission. For this purpose, I have a class on Moodle (my highschool uses a local Danish LMS not supporting peer review).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moodle, being open source and widely supported, &#xA;has many innovative users worldwide, &#xA;as for example the fine folks at UNSW Sydney, &#xA;who have an excellent bunch of pages on teaching &#xA;&lt;a href=&quot;https://teaching.unsw.edu.au/peer-assessment&quot; rel=&quot;nofollow noreferrer&quot;&gt;Students using Peer Review&lt;/a&gt;, &#xA;supported with several video comments. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience tells that crediting students &#xA;for their effort reviewing and assessing peers' work &#xA;&lt;em&gt;and&lt;/em&gt; &#xA;inviting them to reflect on the benefits of actually doing&#xA;some assessment themselves, &#xA;increases retention and thus reduces the number of errors, &#xA;as you were looking for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://moodlecloud.com/en/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moodlecloud&lt;/a&gt; &#xA;offers a commercial-sponsored 50-user licence for trying it out.&#xA;You need to sysadmin a bit: Creating users and a course, &#xA;enrolling users, before students have access to the course's ressources.&lt;/p&gt;&#xA;" OwnerUserId="3502" LastActivityDate="2017-05-25T14:06:09.393" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12371" PostTypeId="1" CreationDate="2017-05-25T17:48:09.107" Score="2" ViewCount="462" Body="&lt;p&gt;I realize that trig tables are somewhat out-of-fashion, but I think my question still makes sense in the computer age: why do trig tables treat the case of a right triangle instead of an arbitrary triangle?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Qt1T7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Qt1T7.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Choose a triangle with altitude 1. Label the base angles a and b.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Sin a = 1/y   Sin b = 1/z&#xA;Tan a = 1/w   Tan b = 1/x&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Solve each equation for the variable in the denominator:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;y = 1/Sin a   z = 1/Sin b&#xA;w = 1/Tan a   x = 1/Tan b&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;This analysis leads to the creation of a trig table for general triangles. Where two angles are of measure a and b, the three sides will be in the proportion y : z : (w+x). If we divide these three parts by (w+x), the result is a proportion of two parts (usually) not equal to 1, and a third part always equal to 1 and therefore needless to specify.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mEJyy.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mEJyy.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that this statement of the proportion is easier to understand than the usual version. Is the table accurate? Does it make any difference whether we consider an acute, right or obtuse triangle? When have trig tables been done like this before, and why is this version not familiar? If the right triangle format is preferred because it is smaller and fits better on one page, perhaps that preference should be reconsidered in an era in which trig is almost always done on a calculator (or another computer)?&lt;/p&gt;&#xA;" OwnerUserId="8258" LastActivityDate="2017-05-27T14:49:25.050" Title="Trig Tables and Right Triangles" Tags="|trigonometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12372" PostTypeId="2" ParentId="12353" CreationDate="2017-05-25T20:00:11.987" Score="3" Body="&lt;p&gt;If you or your student(s) are aquainted with Geogebra, you can define geometrical objects in 2D and subject the object to the function &lt;a href=&quot;https://wiki.geogebra.org/en/Centroid_Command&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Documentation, note that GeoGebra is translated to many languages, if you need that&quot;&gt;Centroid&lt;/a&gt;, which provides you with a point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may want to check out the community's pertinent ressources on &lt;a href=&quot;https://www.geogebra.org/search/perform/search/centroid&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geogebra.org/search/perform/search/centroid&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3502" LastActivityDate="2017-05-25T20:00:11.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12373" PostTypeId="2" ParentId="12114" CreationDate="2017-05-25T20:26:31.943" Score="6" Body="&lt;p&gt;If your students are willing to take the time, I would say you can add a lot of value to their understanding and skills by aproaching the challenge from a &quot;socratic&quot; point of view.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Facilitate conceptualization through &quot;concept cards&quot;: &#xA;I have my students make a mindmap with the math concept in the center.&#xA;On one side, 2-5 examples.&#xA;On the other hand, the definition(s) needed to handle this math object.&#xA;Below 2-4 examples of the math in use.&#xA;Above the center, a few negations: What this math concept is not.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Within a specific field, glossary lists may help the student (and you!)&#xA;demarcating -- or at the least characterizing -- the field and starting your students (and you) on the linguistic&#xA;exploration of the math field (of the week...)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Certainly, the tacit, read vocabulary recommended by @KCd. &#xA;But do not underestimate, what &lt;a href=&quot;https://www.theguardian.com/education/2015/aug/02/sugata-mitra-school-in-the-cloud&quot; rel=&quot;noreferrer&quot;&gt;Sugatha Mitra&lt;/a&gt; dubbed &#xA;the 'granny method': Stand behind your pupils and adore &#xA;&lt;em&gt;whatever&lt;/em&gt; they are doing.&#xA;Your mere presence offers to the students your professional skillset,&#xA;there is a task for them to go grab it, which should be capitalized.&#xA;Math, after all is a pretty international language, just as is esperanto.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let students explain to other Hebrew speaking students.&#xA;Even if your Hebrew is insufficient, expressing oneself will help &#xA;to grasp the concept, and the fellow Hebrew-speaker may know&#xA;stuff that supplements what the speaker does not understand --&#xA;or you can contribute at this point.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3502" LastActivityDate="2017-05-25T20:26:31.943" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12375" PostTypeId="2" ParentId="12114" CreationDate="2017-05-26T18:16:28.290" Score="1" Body="&lt;p&gt;Well, I was in just such a situation when I had to learn Spanish and learn to teach in Spanish simultaneously.  What I did was sitting every evening with the books and dictionaries and translating, and writing down, word for word, what I had to say next day. Timeconsuming, yes, but it worked.  It did help to have some students in class which just knew some english and could help. &lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2017-05-26T18:16:28.290" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12376" PostTypeId="2" ParentId="1635" CreationDate="2017-05-27T09:47:14.703" Score="5" Body="&lt;p&gt;I was once a member of an email discussion list (about some long forgotten statistics software).  Once somebody answered a question, some other pointed out he did some silly arithmetic error in that response.&lt;/p&gt;&#xA;&lt;p&gt;He responded that, yes, he had dyscalculia, and, for that reason, had very early on decided that he had to study mathematics, because then he only needed to calculate with letters, not numbers, thus avoiding his dyscalculia problems!&lt;/p&gt;&#xA;&lt;p&gt;The point being, of course, that dyscalculia in itself does not necessarily say anything about mathematical talent or potential. It might be just an isolated problem with numerals.&lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="12255" LastEditDate="2021-06-09T14:54:15.270" LastActivityDate="2021-06-09T14:54:15.270" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12377" PostTypeId="1" CreationDate="2017-05-27T12:22:59.517" Score="10" ViewCount="2990" Body="&lt;p&gt;Is there a difference between &lt;em&gt;numeracy&lt;/em&gt; and &lt;em&gt;number sense&lt;/em&gt;, or are they synonymous? In my language they are often both translated to the same word (&lt;em&gt;tallforståelse&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm thinking that perhaps numeracy describes a competency, while number sense is more about having a &quot;feel&quot; for numbers or understanding relations between numbers. Is this how the terms are used in education literature? &lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2020-11-16T01:43:31.400" Title="What is the difference between &quot;numeracy&quot; and &quot;number sense&quot;?" Tags="|education-research|terminology|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12378" PostTypeId="2" ParentId="12371" CreationDate="2017-05-27T14:49:25.050" Score="3" Body="&lt;p&gt;1) Are trig tables out of fashion, no. we just use trig tables that are hard wired into silicon these days.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) the right angle triangle is by far the most important. whenever you split a vector into orthogonal components, for summation, you are using a right angle triangle. Splitting vectors into non-orthogonal components is generally far less useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) use the sine or cosine rules for dealing with the few % of cases where someone actually cares about an arbitrary triangle. This was trivial even in the old days when slide rules were still in fashion, let alone with todays computers. &lt;/p&gt;&#xA;" OwnerUserId="1615" LastActivityDate="2017-05-27T14:49:25.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12379" PostTypeId="1" AcceptedAnswerId="12401" CreationDate="2017-05-27T16:04:02.290" Score="3" ViewCount="7493" Body="&lt;p&gt;Clearly, &lt;strong&gt;whole numbers&lt;/strong&gt; specify &lt;em&gt;how many&lt;/em&gt; elements there are in a collection while &lt;strong&gt;decimal numbers&lt;/strong&gt; specify &lt;em&gt;how much&lt;/em&gt; of a substance there is in a lump---but only after a &lt;strong&gt;unit&lt;/strong&gt; of that substance has been chosen. But then whether we use a &lt;em&gt;whole number&lt;/em&gt; or a &lt;em&gt;decimal number&lt;/em&gt; depends on the chosen unit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0.004 Kilometers is a whole number of meters, namely 4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0.00004 Kilometers is a whole number of centimeters  namely 4,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it gets worse: While &lt;/p&gt;&#xA;&#xA;&lt;p&gt;0.00004 KiloDollars is a whole number of centiDollars, namely 4?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;can we really say that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;0.004 KiloPeople is a whole number of Peoples, namely 4,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then what about&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0.00004 KiloPeople is a whole number of centiPeople, namely 4?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where do you draw the line between &lt;em&gt;whole&lt;/em&gt; and &lt;em&gt;decimal&lt;/em&gt; and how do you &lt;em&gt;explain&lt;/em&gt; it to &lt;em&gt;very raw beginning&lt;/em&gt; students who want to &lt;em&gt;understand&lt;/em&gt;?  (Saying that, &lt;em&gt;here&lt;/em&gt;, 4 is really the decimal number 4.0 does not really help.)&lt;/p&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="4815" LastEditDate="2017-05-28T17:40:03.060" LastActivityDate="2017-06-08T00:24:45.887" Title="Difference between whole numbers and decimal numbers" Tags="|mathematical-pedagogy|" AnswerCount="3" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="12380" PostTypeId="2" ParentId="12377" CreationDate="2017-05-27T20:33:33.947" Score="4" Body="&lt;p&gt;This answer could be totally wrong. It just describes the shades of meaning that the terms have for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, &quot;number sense&quot; feels more specific and has the connotation of describing someone who habitually and competently makes sense out of numerical relationships. For example, I teach my students to do order-of-magnitude estimates (Fermi problems), and once I posed a problem of estimating the amount of blood sucked by a flea. One of my students estimated a liter. This student lacked number sense. He wasn't in the habit of thinking about whether numbers made sense, and/or he wasn't competent enough to tell that this particular number didn't make sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example of lacking number sense is when someone writes the result of an experiment as $0.03798\pm 0.00213$, not understanding that it doesn't make sense to express the result to the 100,000ths place when the uncertainty is in the thousandths place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Numeracy&quot; feels to me more like a broad term that describes facility with numbers. For example, someone who is numerate should understand that a hamburger with a 1/3 pound patty contains more meat than one with a 1/4 pound patty. (There was a famous ill-fated advertising campaign that didn't anticipate how many people figured 1/4 pound to be more meat, since 4 is greater than 3.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A numerate person can estimate, without reaching for a calculator, that \$12 is about 1% of \$1000.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A numerate person may not have memorized what 7x6 is, but if necessary they can reconstruct it in a few seconds by doing (7x3)x2. If someone lacks numeracy, then not only do they not have 7x6=42 memorized, but they can't reconstruct that fact.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2017-05-27T20:49:11.737" LastActivityDate="2017-05-27T20:49:11.737" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12381" PostTypeId="2" ParentId="12379" CreationDate="2017-05-29T00:54:21.327" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Why can't we say &amp;quot;0.04 People&amp;quot; since we can say &amp;quot;0.04 KiloPeople&amp;quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some quantities (e.g., people) are discrete quantities and some (e.g., meters, dollars) are continuous quantities.&lt;/p&gt;&#xA;&lt;p&gt;The following discussion is from &lt;a href=&quot;https://joelreyesnoche.wordpress.com/2015/05/27/theory-of-quantity/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  (I've emphasized the words &amp;quot;natural number&amp;quot; and &amp;quot;decimal.&amp;quot;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Classification of quantities&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A quantity is either &lt;em&gt;discrete&lt;/em&gt; or &lt;em&gt;continuous&lt;/em&gt;. A discrete quantity is the magnitude of a countable set (one whose elements are “mutually separated and individually distinct”). Its numeral value is a &lt;strong&gt;natural number&lt;/strong&gt; (“division into a quantity less than a unit cannot be considered”) and its unit is clear at the start. An example of a discrete quantity is “three boys.”&lt;/p&gt;&#xA;&lt;p&gt;A continuous quantity is the magnitude of a “continuum” (“a continuous entity which can be divided into any number of smaller parts” such that “any two such entities can be combined into a larger one”). Its numeral value (a &lt;strong&gt;decimal&lt;/strong&gt; or a fraction) and its unit “have not been determined &lt;em&gt;a priori&lt;/em&gt;.” An example of a continuous quantity is “three dollars.”&lt;/p&gt;&#xA;&lt;p&gt;A continuous quantity is either &lt;em&gt;extensive&lt;/em&gt; or &lt;em&gt;intensive&lt;/em&gt;. The former expresses breadth or magnitude (such as area or weight); the latter expresses quality or intensity (such as density or speed). An extensive quantity has additivity: the attribute of the union of two bodies is equal to the sum of the attributes of the two bodies. An intensive quantity does not have additivity. For example, the weight of two bodies is necessarily the sum of their weights, but the speed of two bodies is not necessarily the sum of their speeds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The text is written for mathematics educators, but it can be reworded to be more easily understood by beginners.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;My original answer (included here for context) which the OP pointed out did not address the intended question:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some quantities, such as, say, &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; liter, have decimal representations (&lt;span class=&quot;math-container&quot;&gt;$0.\overline{3}$&lt;/span&gt; liters) but no whole number representations.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-05-30T08:16:47.893" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="12382" PostTypeId="1" CreationDate="2017-05-29T01:03:08.580" Score="8" ViewCount="192" Body="&lt;p&gt;I am an undergraduate applied math student. In about a month, I will be teaching two nine-hour math courses (one precalculus, one calculus) to a small group of motivated high school students. My broad objective is to refresh their existing skills and to give them an opportunity to present and work interesting problems together. This is why I am planning on using the problem sequences from &lt;a href=&quot;http://jiblm.org/downloads/dlitem.php?id=114&amp;amp;category=jiblmjournal&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mahavier's Trigonometry and Calculus notes&lt;/a&gt;. In addition, I will encourage them to see me during some regular office hours that I will hold. I prefer this style of teaching given that we only meet for an hour two times a week and also because I really enjoyed a full math class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I think now I face the conflict of using these resources with them and not knowing how far we will get, or perhaps switching over to a more lecture-based format and being more certain about this. Then there is the issue of assessment, because I have spoken to the former teachers of these classes and they have typically devoted one class for a traditional exam and another at the end for a cumulative exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, does my preferred way of running the class fit in with the short time we have with each other?&lt;/p&gt;&#xA;" OwnerUserId="8263" LastEditorUserId="5220" LastEditDate="2020-06-04T17:21:05.007" LastActivityDate="2020-06-04T17:21:48.500" Title="How can I deal with the time pressure of teaching a short course?" Tags="|mathematical-pedagogy|calculus|teaching|precalculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12383" PostTypeId="2" ParentId="978" CreationDate="2017-05-29T05:39:20.390" Score="-2" Body="&lt;p&gt;I am no psychologist but I see nothing wrong with your friend's  behavior. In my opinion, these are the defense mechanisms of an &lt;em&gt;honest&lt;/em&gt; person to whom nobody has explained what a fraction is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, try the following.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Do not use &quot;naked numbers&quot; but only &quot;number phrases&quot; such as &lt;strong&gt;3 Apples&lt;/strong&gt; where &lt;strong&gt;3&lt;/strong&gt; is a &lt;em&gt;numerator&lt;/em&gt; and &lt;strong&gt;Apple&lt;/strong&gt; is a &lt;em&gt;denominator&lt;/em&gt;. (&lt;strong&gt;3&lt;/strong&gt; does not correspond to anything in the &lt;em&gt;real world&lt;/em&gt; but &lt;strong&gt;3 Apples&lt;/strong&gt; does: &lt;strong&gt;3&lt;/strong&gt; &lt;em&gt;enumerates&lt;/em&gt; the elements in a &lt;em&gt;real world collection&lt;/em&gt; and &lt;strong&gt;Apple&lt;/strong&gt; &lt;em&gt;denominates&lt;/em&gt; them. You might call that a model-theoretic viewpoint.)&lt;/li&gt;&#xA;&lt;li&gt;Similarly, do not use &quot;naked fractions&quot;. Use for instance &lt;strong&gt;7/3 Apple&lt;/strong&gt; which you then read as &quot;&lt;strong&gt;7 of which it takes 3 to get an Apple&lt;/strong&gt;&quot; where &lt;strong&gt;7&lt;/strong&gt; is the &lt;em&gt;numerator&lt;/em&gt; and where &quot;&lt;strong&gt;of which it takes 3 to get an Apple&lt;/strong&gt;&quot; is the &lt;em&gt;denominator&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Without telling your friend anything more, ask her what is &lt;strong&gt;7/3 Apple&lt;/strong&gt; equal to. Once she decides to read the fraction as in 2. rather than try to give you the &quot;right answer&quot;, she will quickly get &lt;strong&gt;2 Apples and 1/3 Apple&lt;/strong&gt;. But she absolutely &lt;em&gt;must&lt;/em&gt; read the fraction as in 2. So, keep telling her to do so and don't tell her anything else.&lt;/li&gt;&#xA;&lt;li&gt;Once that is established, she will have no problem dealing with equivalent fractions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="4815" LastEditDate="2017-05-31T17:20:00.117" LastActivityDate="2017-05-31T17:20:00.117" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12384" PostTypeId="1" AcceptedAnswerId="12386" CreationDate="2017-05-29T22:26:30.457" Score="8" ViewCount="212" Body="&lt;p&gt;I'm tutoring an eighth grader in math, and while he is particularly bright he is not very good at taking the time to show his work and write things down. He likes to try to solve a problem in his head, write down the answer, and move on. Oftentimes he's correct, but I'm afraid that down the road if he keeps this up he will eventually land in a class that he can't approach this way and won't have anything to fall back on. He just finished algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically, I'd like to help him learn to solve problems on paper, and also to help him learn to explain his solutions clearly. I have two thoughts I'd appreciate feedback on, and if anyone has any suggestions of other directions I could go I would appreciate that as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My two ideas at the moment are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Make some worksheets of problems similar to what he saw in algebra, in which he is asked to solve/explain the problems in stages, e.g. 'What are we trying to accomplish?', 'What information is known?', 'Try X solution method and show all work', 'Explain your solution', etc.&lt;/li&gt;&#xA;&lt;li&gt;Introduce him to some basic parity or divisibility proofs, and emphasize being able to explain things clearly and completely.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thoughts?&lt;/p&gt;&#xA;" OwnerUserId="8266" LastActivityDate="2017-05-30T15:14:51.400" Title="How can I help an eighth grader learn to write mathematically?" Tags="|secondary-education|algebra|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12385" PostTypeId="2" ParentId="12384" CreationDate="2017-05-30T01:55:12.827" Score="2" Body="&lt;p&gt;Good for you for emphasizing this. I would recommend partly role-playing all the future situations in which he'll need to convince a skeptical colleague. How will he explain it to: a fellow student, someone he was tutoring, a confused fellow tutor, a mistaken teacher, a professional colleague, a boss on the job, a student when he becomes the teacher? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Look for exercises in books (usually near the end of a section) that ask for explanations/justifications and work on those (as evidence that it's not something you've made up). Possibly sell it as an extension that only advanced students are expected to perform (not to say that's strictly true). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start working on proofs like establishing known identities in an algebra/trigonometry course. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-05-30T15:14:51.400" LastActivityDate="2017-05-30T15:14:51.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12386" PostTypeId="2" ParentId="12384" CreationDate="2017-05-30T08:51:42.093" Score="4" Body="&lt;p&gt;Based on your description of the student (particularly bright,  not showing his work, often answers correctly) my guess is that he feels the problems he's doing are too easy in general. In addition, he feels that showing he can do things quickly in the head is the best way to show his skill/understanding of the material (I think that way when I was at his age, and I feel like many kids around this age think the same way.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll continue by assuming both your observation and my guess are correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your ideas are fine. However, if you're going to do 1., I'll recommend making the problem complicated enough such that these things actually help/are necessary, rather than forcing him to explain the detail for a simple problem. For example,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$2x+1=5x-11\Rightarrow x=4$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Don't force him to explain adding 11 on both sides, subtracting $2x$ on both sides, and dividing 3 on both sides if he is able to get $x=4$ in his head quickly. He might feel you're insulting his intelligent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, make something like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$8x+9\{6x+7[5x+4(2x+3)]\}=1700x-63$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;where I doubt he can compute everything in his head. In this case, writing down the detail steps is probably valuable if not necessary to get the final answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bottom line is, don't force him to do the work that he feels unnecessary, instead give examples that shows such a work (of writing things in detail) is valuable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think 2. is a great idea, and probably more important than 1. if you want to compare.&lt;/p&gt;&#xA;" OwnerDisplayName="user2139" LastActivityDate="2017-05-30T08:51:42.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12387" PostTypeId="2" ParentId="7243" CreationDate="2017-05-30T11:52:45.300" Score="2" Body="&lt;p&gt;One of the crucial points here is that verifying a result must be different than repeating the calculation, let me explain:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Verify that 221/12 = 18, rest 5:&#xA;Instead of doing the calculation all over, calculate 12x18+5, which indeed equals 221.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Verify that -2 and 3 are solutions of x^2-x-6=0:&#xA;Fill in the values: (-2)^2-(-2)-6 = 4+2-6 = 0, and 3^2-3-6 = 0&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Verify that &lt;code&gt;(x-1)^2+(y-2)^2=4&lt;/code&gt; is a circle with center in (1,2) and radius 2 (bad example, I know, it's just for the sake of the argument). This means (for example) that the graph can't contain a point with a negative y-coordinate. Just try to put y=-1, then your equation becomes:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;(x-1)^2+((-1)-2)^2=4&#xA;(x-1)^2+9=4&#xA;(x-1)^2=-5 =&amp;gt; a square can't be negative, so no x-coordinate, hence no point with y=-1&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-05-30T11:52:45.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12388" PostTypeId="2" ParentId="12368" CreationDate="2017-05-30T12:43:43.960" Score="6" Body="&lt;p&gt;&lt;strong&gt;Mathematics&lt;/strong&gt;. This seemingly faulty transformation is trickier than it looks. First, notice that it is not completely wrong and in a sense, it is even correct! As Daniel has mentioned in his comment, this is somehow related to a difference-of-squares. Let's have a closer look. $(a-b)(a+b)=a^2-b^2$. A student's version for getting from the LHS to the RHS would be something along this line: a times a (that is $a^2$), minus times plus (that is $-$), b times b (that is $b^2$). Having that version in mind, it would be quite &quot;meaningful&quot; to simplify $\frac{a^2 - b^2}{a-b}$ or $\frac{a^2 - b^2}{a+b}$ by dividing $a^2$ by $a$, sign by sign, and $b^2$ by $b$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This allows us to link another pair of students' version of doing algebra, one of them being the most infamous algebraic misunderstanding: $(a+b)^2=a^2+b^2$. See how this is supported by the student's way of thinking mentioned above. $(a+b)^2$ is $(a+b)(a+b)$ that &quot;is&quot; $a$ times $a$, plus times plus, $b$ times $b$. &quot;Accordingly&quot;, $\frac{a^2+b^2}{a+b}=a+b$! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pedagogy.&lt;/strong&gt; This is a great opportunity to build knowledge on a &lt;strong&gt;network&lt;/strong&gt; of mistakes and takes. First of all, noticing the difference between two cases above, we have to qualify &quot;not a transformation&quot; (bolded in your question) to &quot;&lt;strong&gt;generally&lt;/strong&gt; not a transformation&quot;. From here you can transform the &quot;mistake&quot;  into an investigative task. The question for students is &quot;when that self-invented rule works?&quot; As such, instead of giving them plenty of examples, you encourage them to generate their own examples while they are &lt;strong&gt;consciously working with the mistakes rather than being told about the mistakes.&lt;/strong&gt; (Please also see my answer to a relevant question &lt;a href=&quot;https://matheducators.stackexchange.com/a/4103/1217&quot;&gt;here&lt;/a&gt;)     &lt;/p&gt;&#xA;" OwnerUserId="1217" LastActivityDate="2017-05-30T12:43:43.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12389" PostTypeId="1" CreationDate="2017-05-30T16:39:23.043" Score="5" ViewCount="139" Body="&lt;p&gt;I'm trying to find some online questions and training material related to this sort of question. It's not something that I can identify the name of:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A raffle has these prizes (ending 00 @ £12, ending 5 at £1.5), and tickets cost 50p. If all the winning tickets from 750 are sold (and some non-winning tickets), the total profit is £650. How many non-winning tickets were sold?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The solution goes along the lines of&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;calculate total winning tickets&lt;/li&gt;&#xA;&lt;li&gt;calculate total prize money from #tickets&lt;/li&gt;&#xA;&lt;li&gt;calculate total revenue&lt;/li&gt;&#xA;&lt;li&gt;calculate total # tickets &lt;/li&gt;&#xA;&lt;li&gt;subtract willing tickets&lt;/li&gt;&#xA;&lt;li&gt;final answer. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I understand how to solve this specific problem, but is it a standard type of question, or just a one-off type of question? I'm hoping that there is a simple strategy for explaining this sort of question.&lt;/p&gt;&#xA;" OwnerUserId="6520" LastEditorUserId="6520" LastEditDate="2017-06-09T08:34:11.780" LastActivityDate="2017-06-11T06:23:49.747" Title="How is this problem categorised?" Tags="|secondary-education|problem-solving|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12390" PostTypeId="1" CreationDate="2017-05-31T03:43:22.410" Score="12" ViewCount="1379" Body="&lt;p&gt;Every Pre-Calculus I have examined starts with functions in general, then polynomial and rational functions, followed by exponential and logarithmic functions and Trigonometry, and ending with sequences, summations, probability, and limits. Some have vectors and binomial theorem thrown in towards the end. I would rather begin with number theory, sequences, summations and binomial theorem rather than with polynomials. Is there a pedagogical or theoretical reason why this order is so common?&lt;/p&gt;&#xA;" OwnerUserId="7051" LastActivityDate="2017-06-15T01:05:25.763" Title="The Order in Pre-Calculus Textbooks" Tags="|textbooks|precalculus|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12391" PostTypeId="2" ParentId="12390" CreationDate="2017-05-31T04:16:07.527" Score="11" Body="&lt;p&gt;Mitchell,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Great question.  I think that precalculus is usually a wasteland for some random classical techniques that used to be required to study calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that you can take a completely different approach if you're interested and incorporate data and matrices in place of a bunch of algebraic manipulation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just taught a precalculus class and rather than the standard topics, we focus on using functions to model data and some introductory linear algebra.  I believe this is much more relevant in the computer age. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're interested I'm happy to share more(here's a &lt;a href=&quot;http://nbviewer.jupyter.org/github/DubMathematics/jupyterday_philly_precalc/blob/master/jupyterday_Philly.ipynb&quot; rel=&quot;noreferrer&quot;&gt;link&lt;/a&gt; to a talk I just gave on the class), but we used the MAA textbook by Sheldon on &quot;Functions, Data, and Models&quot; as a foundational text to solve problems with Python in Jupyter notebooks.  The material was broken into a few sections:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Introductory work with functions&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Key Features of linear, quadratic, and exponential functions as recursive sequences through population models&lt;/li&gt;&#xA;&lt;li&gt;Relate recursive forms to closed forms&lt;/li&gt;&#xA;&lt;li&gt;Intro to harmonic motion and key features&lt;/li&gt;&#xA;&lt;li&gt;Use trig to model some basic periodic phenomena&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Intro with data&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;measures of center&lt;/li&gt;&#xA;&lt;li&gt;linear and polynomial regression&lt;/li&gt;&#xA;&lt;li&gt;non-linear regression and logarithms&lt;/li&gt;&#xA;&lt;li&gt;machine learning and regression&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Matrices&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;images as matrices through dimension and definition of matrix&lt;/li&gt;&#xA;&lt;li&gt;add, subtract, scalar multiply matrices and application to image manipulation&lt;/li&gt;&#xA;&lt;li&gt;systems of equations and matrices&lt;/li&gt;&#xA;&lt;li&gt;Transpose and regression again&lt;/li&gt;&#xA;&lt;li&gt;Trig and rotation by matrix multiplication&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Either way, I think that using functions(linear, quadratic, higher polynomials, exponentials, trigonometric) to model real world situations, a little on data analysis, and some introductory linear algebra and applications is an appropriate precalculus course.  Take it further depending on how advanced your students are.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastEditorUserId="7003" LastEditDate="2017-05-31T13:38:47.920" LastActivityDate="2017-05-31T13:38:47.920" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12392" PostTypeId="1" AcceptedAnswerId="12399" CreationDate="2017-05-31T15:03:38.690" Score="9" ViewCount="2075" Body="&lt;p&gt;When looking at a given picture to be used in a geometric proof, often times single, double, or triple &quot;slashes&quot; mark off equal line segments or arcs. What is the correct term for these? I've seen hatch marks, hash marks, tick marks, tic marks, slash marks, etc. Wikipedia uses the term hatch marks &lt;a href=&quot;https://en.wikipedia.org/wiki/Hatch_mark#Congruency_notation&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, but I'd rather not just take their word for it. &lt;/p&gt;&#xA;" OwnerUserId="6675" LastActivityDate="2017-06-02T16:39:14.353" Title="What is the term for the marks used to show congruence in geometric figures?" Tags="|geometry|terminology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12393" PostTypeId="2" ParentId="899" CreationDate="2017-05-31T18:01:48.090" Score="-1" Body="&lt;p&gt;The difference between &lt;em&gt;math students&lt;/em&gt; and &quot;&lt;em&gt;the rest of us&lt;/em&gt;&quot; is quite real and ought to be acknowledged: the mathematician's goal is to &lt;em&gt;prove&lt;/em&gt; a statement from already proven statements while the rest of us try to &lt;em&gt;understand&lt;/em&gt; the statement sufficiently well to see what it says about the &lt;em&gt;real world&lt;/em&gt;. (You might call that a &lt;em&gt;model-theoretic viewpoint&lt;/em&gt; -- not to be confused with, and as opposed to, the &lt;em&gt;modelling approach&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The math-teaching community, though, is loath to acknowledge said distinction, let alone to do anything about it. For possibly why, see, for instance, Keisler's comments in Section 13. &lt;strong&gt;Institutional Difficulties with Teaching Infinitesimals&lt;/strong&gt; of Vinsonhaler's &lt;a href=&quot;http://scholarship.claremont.edu/jhm/vol6/iss1/17/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching Calculus with Infinitesimals&lt;/a&gt;. And for another possible approach, &lt;strong&gt;Laurent Polynomial Approximations&lt;/strong&gt;, a special kind of &lt;strong&gt;Asymptotic Expansions&lt;/strong&gt;, see my &lt;a href=&quot;https://www.researchgate.net/publication/281444563_To_Calculate_In_Calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;To Calculate in Calculus&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="4815" LastEditDate="2017-05-31T21:57:32.880" LastActivityDate="2017-05-31T21:57:32.880" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12394" PostTypeId="2" ParentId="12390" CreationDate="2017-06-02T01:10:42.873" Score="4" Body="&lt;p&gt;Precalculus is not just a course that can be taken before calculus. Almost half a century ago, Precalculus, an American invention, started to replace College Algebra as a preparation for Calculus and in fact, if College Algebra seems to have made a comeback, Precalculus remains almost invariably a &lt;em&gt;requirement&lt;/em&gt; for calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that case then, given that, for instance, &quot;Functions of various kinds&quot; are presented by &lt;a href=&quot;https://en.wikipedia.org/wiki/Function_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; as &quot;&quot;the central objects of investigation&quot; in most fields of modern mathematics&quot;, given that the calculus is the calculus of &lt;em&gt;functions&lt;/em&gt;, and given that the calculus is the first tool in the &quot;hard&quot; sciences, it seems to me that the best way to prepare students for the calculus is indeed with a study of power functions followed by &quot;polynomial and rational functions, followed by exponential and logarithmic functions&quot; and followed by trigonometric functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem &lt;em&gt;I&lt;/em&gt; have with Precalculus is that it is a bag of tricks. However, this need not be the case if one is willing to use &lt;strong&gt;Laurent Polynomial Approximations&lt;/strong&gt; (i.e. asymptotic expansions using power functions as gauges) to investigate said functions.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2017-06-02T01:10:42.873" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12395" PostTypeId="2" ParentId="1883" CreationDate="2017-06-02T12:26:14.037" Score="2" Body="&lt;p&gt;I want to jump in because I see so many answers and comment debating about what is &lt;em&gt;the&lt;/em&gt; correct primary meaning of complex numbers. If I where an electrical engineer, I would probably claim that complex numbers are really meant to describe impedance of electrical circuits - with a magnitude describing the resistance and an argument describing the phasing effect on periodic signals - and that the laws of addition and product are what is needed to correctly describe the impedance of several circuits in series or in parallel. That would be just as wrong as saying that &lt;em&gt;the&lt;/em&gt; true meaning of complex number is in their geometrical interpretation, or in their algebraic construction, or in their use to compute antiderivatives of $x\mapsto \cos^{12}(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a given that there are several ways to approach complex numbers, and the fact that they appear consistently in so many ways is in my opinion the most important and amazing thing about them. But one cannot reach every such aspects and connexion in a course, and therefore I will conclude something obvious: &lt;strong&gt;choose an approach that fits the goals of the course and the future need of the students&lt;/strong&gt;. Depending on this, you may choose one or several aspects, and some or all connections between them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, you may construct complex numbers algebraically in chapter 1, to solve the issue of missing square roots; then in chapter 2 consider similarities of the plane and try to express them all in terms of their action on coordinates, observe that two parameters are sufficient and look at what happens when we compose too of them. You would then find out in chapter 3 that both points of view reconcile through complex numbers. That is something that is meaningful for undergraduates majoring in maths, to teach abstract construction and to show how connected maths are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a more advanced level, one can relate complex analysis and the geometry of conformal transformations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a more basic level, one can try to relate the algebraic meaning of complex numbers with trigonometry; etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among the awesomeness that seems to have been overlooked in previous answers and comments, is the fact that adding roots to quadratic polynomials actually gives roots to all polynomials, even those with coefficients in the new field (that is far from true with $\mathbb{Q}$!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that seems clear to me, from a French perspective, is that the geometrical interpretation is usually overlooked, and thus feels foreign even to many advanced students. This is why I would try to put more geometry rather than less (it also fits my personal taste, to be honest). Unless you only have a very specific need for complex numbers as a tool in a certain context, in which case the angle of approach should be clearly dictated, teaching them insisting on only one single aspect is probably a wrong choice - irrespective of this aspect being algebraic or geometric.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-06-02T12:26:14.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12396" PostTypeId="1" AcceptedAnswerId="12406" CreationDate="2017-06-02T16:01:17.677" Score="7" ViewCount="748" Body="&lt;p&gt;This is one of these questions that &lt;strong&gt;students&lt;/strong&gt; ask and for which I have never found an answer that &lt;strong&gt;students&lt;/strong&gt; would accept as &lt;em&gt;convincing&lt;/em&gt;. Here is an instance:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student&lt;/strong&gt;: 2•3 is a number, namely 6. 2+3 is a number, namely 5. 1/2•1/3 is a number, namely 1/6. 1/2+1/3 is a number, namely 5/6. √2•√3 is a number, namely √6. So why isn't √2+√3 a number?&lt;/p&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="143" LastEditDate="2017-06-04T12:16:54.790" LastActivityDate="2017-06-20T14:23:13.653" Title="What number is the sum of two roots" Tags="|undergraduate-education|real-numbers|" AnswerCount="4" CommentCount="20" ContentLicense="CC BY-SA 3.0" />
  <row Id="12397" PostTypeId="2" ParentId="1635" CreationDate="2017-06-02T16:25:51.993" Score="4" Body="&lt;p&gt;Possibly relevant:  In the first chapter of &lt;em&gt;Where Mathematics Comes From: How The Embodied Mind Brings Mathematics Into Being&lt;/em&gt;, Lakoff &amp;amp; Núñez discuss a few examples of individuals who have suffered some kind of neurological damage that partially or completely disables their ability to perform numerical calculations.  With respect to this question, the most interesting example is probably this one:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Not only is rote calculation localized separately from basic arithmetic abilities but algebraic abilities are localized separately from the capacity for basic arithmetic. Dehaene (1997) cites a patient with a Ph.D. in chemistry who has &lt;em&gt;acalculia&lt;/em&gt;, the inability to do basic arithmetic. For example, he cannot solve $2\cdot 3$, $7-3$, $9\div 3$, or $5\cdot 4$. Yet he can do abstract algebraic calculations. He can simplify $(a \cdot b) / (b \cdot a)$ into $1$ and $a \cdot a \cdot a$ into $a^3$, and could recognize that $(d/c) + a$ is not generally equal to $(d + a) / (c + a)$. Dehaene concludes that algebraic calculation and arithmetic calculation are processed in different brain regions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The citation to Dehaene (1997) refers to the book &lt;em&gt;The Number Sense&lt;/em&gt;.  The relevant passage from that book seems to be p. 199:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Up to now, this book has been concerned only with elementary arithmetic. But what about more advanced mathematical abilities, such as algebra? Should we postulate yet other neuronal networks dedicated to them? Recent discoveries by the Austrian neuropsychologist Margarete Hittmair-Delazer seem to suggest so. She has found that acalculic patients do not necessarily lose their knowledge of algebra. One of her patients, like Mrs. B, lost his memory of addition and multiplication tables following a left subcortical lesion. Yet he could still recalculate arithmetic facts by using sophisticated mathematical recipes that indicated an excellent conceptual mastery of arithmetic. For instance, he could still solve $7 \times 8$ as $7 \times 10 - 7 \times 2$. Another patient, who had a Ph.D. in chemistry, had become acalculic to the point of failing to solve $2\times 3$, $7-3$, $9\div 3$, or $5\times 4$. He could nevertheless still execute abstract formal calculations. Judiciously making use of the commutativity, associativity,and distributivity of arithmetic operations, he was able to simplify $\frac{a \times b}{b \times a}$ into $1$ or $a\times a\times a$ into $a^3$, and he recognized that the equation $\frac{d}{c} + a = \frac{d+a}{c+a}$ is generally false. Although this issue has been the matter of very little research to date, these two cases suggest, against all intuition, that the neuronal circuits that hold algebraic knowledge must be largely independent of the networks involved in mental calculation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Regarding these passages, a couple of comments are in order:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I am not sure if &lt;em&gt;acalculia&lt;/em&gt; and &lt;em&gt;dyscalculia&lt;/em&gt; are the same phenomenon, or what exactly the relationship is between them.  Nor is it clear from this brief excerpt whether the patient (the one with the Ph.D in Chemistry) developed acalculia  in adulthood, for example as the result of some injury, or whether it was a condition he had all along.  (Hittmair-Delazer's first patient suffered from a &quot;left subcortical lesion&quot;, but the Ph.D in Chemistry is &quot;another patient&quot;, of whom no details are provided.)  The distinction seems significant, as there is (probably?) a big difference between someone learning algebraic reasoning &lt;em&gt;despite&lt;/em&gt; suffering from acalculia, on the one hand, and someone &lt;em&gt;retaining&lt;/em&gt; algebraic reasoning that they learned &lt;em&gt;prior to&lt;/em&gt; the onset of acalculia.&lt;/li&gt;&#xA;&lt;li&gt;Algebraic reasoning and manipulation is one thing, but proof-based advanced-level mathematics at the level of an undergraduate analysis course is (maybe?) something else.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The primary source for this is probably one of the following two papers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;HITTMAIR-DELAZER, M., SAILER, U., and BENKE, T. Impaired arithmetic facts but intact conceptual knowledge – a single case study of dyscalculia. &lt;em&gt;Cortex&lt;/em&gt;, 31: 139-147, 1995.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;HITTMAIR-DELAZER, M., SEMENZA, C., and DENES, G. Concepts and facts in calculation. &lt;em&gt;Brain&lt;/em&gt;, 117: 715-728, 1994.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both of these papers are cited in the references of yet another paper:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dehaene, S., &amp;amp; Cohen, L. (1997). Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. &lt;em&gt;Cortex&lt;/em&gt;, 33(2), 219-250.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect any of these three papers might have additional references that could be helpful in understanding these phenomena.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2017-06-04T03:18:46.253" LastActivityDate="2017-06-04T03:18:46.253" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12398" PostTypeId="2" ParentId="12396" CreationDate="2017-06-02T16:33:53.940" Score="27" Body="&lt;p&gt;First of all, $\sqrt{2}+\sqrt{3}$ most certainly &lt;em&gt;is&lt;/em&gt; a number. It is a real number, approximately equal to $3.14626$ Perhaps what you're asking is why the sum of two simple radicals isn't also a simple radical, when the sum of two integers is an integer, and the sum of two fractions is a fraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roots of integers are examples of algebraic numbers - numbers that are the roots of polynomial equations with integer coefficients. The sum of two algebraic numbers is an algebraic number, but it doesn't have to be as simple as the addends are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something like this is already true with fractions: $\frac12$ and $\frac13$ are both &lt;em&gt;unit fractions&lt;/em&gt;, but their sum is a more complicated fraction, with larger numbers in both the numerator and denominator than the fractions we started with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, the numbers $\sqrt2$ and $\sqrt3$ are roots of the polynomials $x^2-2$ and $x^2-3$, respectively. Their sum, which is most simply expressed as $\sqrt2+\sqrt3$, is a root of the polynomial $x^4-10x^2+1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This particular algebraic number can also be written as $\sqrt{5+2\sqrt{6}}$, where you can see both $2+3$ and $2\times 3$ playing a role. (To see this, write $x=\sqrt2+\sqrt3$, square both sides, and combine integers.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just like you can't expect reciprocals of integers to be closed under addition, you can't expect roots of integers to be closed under addition. That's because we're involving more complicated operations than addition, namely, division and taking of roots. On the other hand, you can expect the sum of two rational numbers (fractions) to be rational, and you can expect the sum of two algebraic numbers (roots of integer polynomials) to be an algebraic number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does this help at all?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As requested in the comments, I'll put this in the voice I would use to address a high school or middle school student asking this question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;First of all, $\sqrt2 + \sqrt3$ is definitely a number. Here, let's calculate its value... [calculator].... as you can see, it's not a very pretty one, but we can see a few decimal places: $3.14626\ldots$. Huh, it's kind of close to $\pi$, but a little bigger.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Anyway, let's see if we can express this number in a nicer form:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align}&#xA;x &amp;amp;= \sqrt2 + \sqrt3\\&#xA;x^2 &amp;amp;= (\sqrt2 + \sqrt3)^2\\&#xA;x^2 &amp;amp;= 2 + 2\sqrt2\sqrt3 + 3\\&#xA;x^2 &amp;amp;= 5 + 2\sqrt6\\&#xA;x &amp;amp;= \sqrt{5+2\sqrt6}&#xA;\end{align}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I'd talk through the steps of that algebra, making sure it's clear after each line.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Ok, so it's a square root, but it's a square root of something more complicated that what we started with. I guess that's fair. After all, when you add two fractions like $\frac12$ and $\frac13$, which are pretty simple, you end up with $\frac56$, which is more complicated - it's not just a $1$ on top, and both numerator and denominator are bigger than what we started with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Actually, it's pretty interesting, that the numbers $5$ and $6$, which are $2+3$ and $2\times 3$, both show up in the fraction $\frac56$ and in the radical $\sqrt{5+2\sqrt{6}}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The reason adding fractions is more complicated than adding integers, and adding radicals more complicated still, is that fractions are made of division, and radicals are made of roots, both of which are more complicated than addition and subtraction in the first place.&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Second edit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One more run at this, just to see how succinctly I can get the main point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;To see what something is the square root of, square it:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(\sqrt2 + \sqrt3)^2 = 2 + 2\sqrt6 + 3 = 5 + 2\sqrt6$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;As you can see, we don't get a whole number, because in FOIL*, we have middle terms giving us the $2\sqrt6$ part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;It's different from a fraction, because if you look at the sum $\frac12 + \frac13$, there is a common demonimator $6$ you can multiply by that makes it the sum of two whole numbers: another whole number. No cross terms arise, because there's no FOIL going on.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(* FOIL = distributive rule applied to binomials; mnemonic for &quot;First terms, Outside terms, Inside terms, Last terms&quot;)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Also instructive are the cases where it does work. For example, $\sqrt2 + \sqrt8 = \sqrt{18}$. You can &quot;FOIL&quot; it out and see why.&lt;/p&gt;&#xA;" OwnerUserId="2034" LastEditorUserId="2034" LastEditDate="2017-06-05T13:23:57.353" LastActivityDate="2017-06-05T13:23:57.353" CommentCount="29" ContentLicense="CC BY-SA 3.0" />
  <row Id="12399" PostTypeId="2" ParentId="12392" CreationDate="2017-06-02T16:39:14.353" Score="6" Body="&lt;p&gt;While I doubt very much that there is an official or canonical name for these markings, the most detailed analysis of the way geometric diagrams marshal semiotic resources is probably in an article by Justin Dimmel and Pat Herbst, &quot;The Semiotic Structure of Geometry Diagrams: How Textbook Diagrams Convey Meaning&quot;, published in JRME in 2015.  (&lt;a href=&quot;http://www.jstor.org/stable/10.5951/jresematheduc.46.2.0147&quot; rel=&quot;noreferrer&quot;&gt;http://www.jstor.org/stable/10.5951/jresematheduc.46.2.0147&lt;/a&gt;).  Dimmel &amp;amp; Herbst refer to those marks as &quot;hash marks&quot; (p. 166), and identify them as examples of a larger system of notational devices that they call &quot;geometric diacriticals&quot;.  These, in turn, are part of a subsystem of &quot;relational attributes&quot;, which belong to a larger &quot;attributes system&quot;.  The relevant passage is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Relational markings&lt;/em&gt; are the symbols that convey geometric properties when they are applied to parts in the geometry diagram. This set of symbols includes the small squares (i.e., quadrants) that indicate right angles, small arcs that indicate congruent angles, hash marks that indicate congruent segments, and sets of arrows that indicate sets of parallel lines. These markings are resources through which the diagram directly communicates geometric properties without any supporting literal or symbolic statements. Thus, an angle marked by a quadrant is a right angle, regardless of whether it is visually perceived as a right angle (see Gal &amp;amp; Linchevski, 2010). We refer to relational markings as &lt;em&gt;geometric diacriticals&lt;/em&gt; because they signal geometric properties of objects in diagrams analogously to the way that diacritical markings (e.g., accents, tildes, umlauts) signal phonetic properties of letters in words.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(This is a really good paper, by the way.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2017-06-02T16:39:14.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12400" PostTypeId="2" ParentId="12390" CreationDate="2017-06-02T16:43:18.517" Score="3" Body="&lt;p&gt;I think the main pedagogical reason is this: Calculus students need to be familiar with the elementary functions, and with the idea of functions and their graphs in general. Not every precal class gets through the desired curriculum, due to various reasons. Thus, if something is going to get cut off at the end, because of snow days or time lost for whatever reason, most teachers would rather miss out on sequences or probability than on trigonometry. Students can succeed in Cal I just fine without sequences, but if they miss out on trig, they're up a creek.&lt;/p&gt;&#xA;" OwnerUserId="2034" LastActivityDate="2017-06-02T16:43:18.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12401" PostTypeId="2" ParentId="12379" CreationDate="2017-06-03T02:24:16.807" Score="7" Body="&lt;p&gt;&quot;Counting&quot; (leading to counting numbers) is a special case (with ambiguities) of &quot;measuring&quot;, where the role of &quot;the measure/unit&quot; is more visible. Obviously (I think) the natural unit implied in &quot;counting&quot; situations is some relevant-atomic unit (such as &quot;whole, operational person&quot;, rather than not-so-functional smaller-part of a person).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, counting implicitly measures with unit the smallest/atomic feasible/operational measure (often so universally implicit that it is beyond discussion).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A fancier analogue arises when more advanced undergrads are first exposed to the idea that infinite sums (a.k.a. &quot;series&quot;) fall under the umbrella of &quot;integrals&quot;, but with &quot;counting measure&quot;... and that discrete sets have at least one natural regular, positive Borel measure, namely, the counting measure.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-06-03T02:24:16.807" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12403" PostTypeId="2" ParentId="4484" CreationDate="2017-06-04T00:39:03.383" Score="4" Body="&lt;p&gt;Your desire for some beautiful integrated thematic math person version of course shows not thinking about two important things:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Your students and what they need.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How people learn best.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For number 1, they are mostly engineering and physics students.  They benefit from exposure to the tricks of the trade so that when they have derivations and problems in their courses (fluids, EE, etc.) they have some mathematical familiarity with the tools they are using in the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For number 2, the beautiful sleek presentation is not always the easiest way to learn complex and new abstract material.  It tends to make more sense for a second exposure than a first.  Or to someone who is a professor than a student.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of practical needs:  basic linear first order dffy qs with integrating factors and then the second order equation for harmonic oscillator is what people need the most.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The hard thing about diffyqs is that it basically should be close to a full year course (even at a beginner level).  And I don't mean making it long with Sobolev spaces and other advanced concepts.  I just mean even the basic book just can't be completed in a semester.  This is very different than calc 1/2/3 or any previous math courses where the course pretty much involves going over the whole text (I mean stereotypical texts).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students hate having the feeling of not covering a whole text.  Professors don't care.  But students do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you want a reasonably short and sweet good ODEs course, look at the first 5 chapters of Kreyszig Advanced Engineering Mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The good thing about Kreysig is that it is reasonably short. That it still covers a lot of topics. That it selects those of most importance to an engineer or physicist. Also, the order of the topics seemed to make sense. Pretty much the classical order but with efficient coverage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other than that, I worked through it and just felt it very easy to grok. Sorry, I am straining brain, but that is the best I can explain. Common book and worth a look. I had 5th edition FWIW.&lt;/p&gt;&#xA;" OwnerUserId="8280" LastEditorUserId="143" LastEditDate="2017-06-05T10:25:07.757" LastActivityDate="2017-06-05T10:25:07.757" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12406" PostTypeId="2" ParentId="12396" CreationDate="2017-06-05T04:22:28.310" Score="21" Body="&lt;p&gt;I think G Tony Jacobs's answer is an excellent one, and I admit I don't quite understand what the OP's objection to it is all about.  But I am going to take a stab at trying to explicate what I &lt;em&gt;think&lt;/em&gt; might be underlying the objection that &quot;$\sqrt{2}+\sqrt{3}$ is not a number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking over the examples that precede the question, we can notice that they all have a kind of grammatical structure in common:  namely, an expression that initially consists of a binary operation performed on two &quot;single&quot; object is evaluated or simplified to produce another &quot;single&quot; object.  That is, they all have the structure of&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[thing 1] (operation) [thing 2] = [thing 3]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The sum of $\sqrt{2}$ and $\sqrt{3}$ resists this grammatical structure, since there is no way to combine the two &quot;things&quot; into something that &lt;em&gt;looks like&lt;/em&gt; a single term.  This is probably related at some deep conceptual level to the well-documented phenomenon of students perceiving an equals sign as a command to &quot;do something&quot;, separating an unperformed computation on the left from its result on the right.  In this sense, the sum $\sqrt{2}+\sqrt{3}$ looks like a &lt;em&gt;question&lt;/em&gt;, not an &lt;em&gt;answer&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For another instance of this, consider the following problem, common in first-year Algebra:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Simplify each of the following expressions:&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;$5x^2 + 3x^2$&lt;/li&gt;&#xA;  &lt;li&gt;$5x^2 \cdot 3x$&lt;/li&gt;&#xA;  &lt;li&gt;$5x^2 + 3x$&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Many students find it &lt;em&gt;extremely frustrating and counterintuitive&lt;/em&gt; that while the first question can be answered as a single monomial (&quot;Combine like terms to get $8x^2$&quot;) and the second question can be answered as a single monomial (&quot;Multiply the coefficients and add the exponents to get $15x^3$&quot;), the third problem &lt;strong&gt;cannot be simplified&lt;/strong&gt; any further than it already is.  &lt;em&gt;We&lt;/em&gt; know that $5x^2 + 3x$ can be regarded as a single quantity -- and we use the word &quot;polynomial&quot; to describe such quantities -- but to (some) students, it looks like a question that has not yet been answered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to address this?  In the algebraic case I described above, I usually begin by turning the question around on the student:  Why &lt;em&gt;should&lt;/em&gt; we be able to combine those two terms into a single term?  For that matter, why is it possible to simplify the &lt;em&gt;other&lt;/em&gt; expressions?  This can lead into fruitful discussions about (for example) the distributive property, the associative property, the meaning of exponents, and so forth.  But just because &lt;em&gt;some&lt;/em&gt; expressions can be simplified into a single term does not mean that &lt;em&gt;all&lt;/em&gt; expressions can be simplified into a single term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, in the case of the OP, I might begin by asking: &quot;Well, what do you think $\sqrt{2} + \sqrt{3}$ &lt;em&gt;ought to be&lt;/em&gt; equal to?&quot;  In many cases what the student really wants to know is something like:  &quot;Why isn't it $\sqrt{5}$?&quot;  Once they have articulate what they think the sum &lt;em&gt;ought to be&lt;/em&gt;, we can then explain why it &lt;em&gt;isn't&lt;/em&gt; that:  because when you square it, you don't get $5$.  At this point I would probably follow G Tony Jacobs's line of reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Possibly relevant: &lt;a href=&quot;https://matheducators.stackexchange.com/q/10226/29&quot;&gt;Why unlike terms cannot be simplified?&lt;/a&gt; and the second-to-last paragraph of my answer at &lt;a href=&quot;https://matheducators.stackexchange.com/a/1058/29&quot;&gt;https://matheducators.stackexchange.com/a/1058/29&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2017-06-05T04:22:28.310" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12408" PostTypeId="5" CreationDate="2017-06-05T15:45:54.940" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2017-06-05T15:45:54.940" LastActivityDate="2017-06-05T15:45:54.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12409" PostTypeId="4" CreationDate="2017-06-05T15:45:54.940" Score="0" Body="For questions related to the teaching of integral calculus" OwnerUserId="39" LastEditorUserId="39" LastEditDate="2017-06-07T11:44:28.277" LastActivityDate="2017-06-07T11:44:28.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12410" PostTypeId="1" AcceptedAnswerId="12427" CreationDate="2017-06-06T14:25:17.627" Score="10" ViewCount="2544" Body="&lt;p&gt;My daughter is in 6th grade. She likes mathematics, i.e., she likes to learn mathematics at school, is also good at what she has learned at school and is rather quick at picking new things. However, she is currently only limited to text books, which I feel are not inspirational, and don't make you fall in love with mathematics. I am looking for some books that may do exactly this for her age group. (i.e., it should be fun to read while showing the real beauty of mathematics.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, topics like the golden ratio, and symmetry are simple enough and fun to explain, but at the same time can be used to show how mathematics is all around us or to show even beauty can be to some extent quantified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have looked at some other questions, some are really for &lt;a href=&quot;https://matheducators.stackexchange.com/questions/238/what-are-some-great-books-for-inspiring-children-to-explore-mathematics&quot;&gt;kids&lt;/a&gt; who are too young, or &lt;a href=&quot;https://matheducators.stackexchange.com/questions/294/what-are-some-great-books-for-exploring-mathematics-not-kids-books-and-not-te?noredirect=1&amp;amp;lq=1&quot;&gt;adults&lt;/a&gt;.&#xA;(Books in German or with German translation are a plus).&lt;/p&gt;&#xA;" OwnerUserId="8287" LastEditorUserId="77" LastEditDate="2017-12-13T13:44:09.500" LastActivityDate="2021-11-12T20:46:05.743" Title="Inspirational Mathematics Books for Teenager" Tags="|reference-books|applied-mathematics|mathematics-in-daily-life|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12411" PostTypeId="2" ParentId="12410" CreationDate="2017-06-06T15:04:01.993" Score="11" Body="&lt;p&gt;If you looked at those other topics, you saw the links to the books page at my blog, &lt;a href=&quot;http://mathmamawrites.blogspot.com/p/math-books.html&quot; rel=&quot;noreferrer&quot;&gt;Math Mama Writes&lt;/a&gt;. There are a number of books there that would work for a teen. One of my favorites is &lt;strong&gt;&lt;em&gt;Carry On, Mr. Bowditch&lt;/em&gt;&lt;/strong&gt;, by Jean Lee Latham, the (slightly fictionalized) biography of Nathaniel Bowditch, who modernized navigation in the 1700s. But maybe that has too many U.S. connections.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;The Man Who Counted&lt;/em&gt;&lt;/strong&gt;, by Malba Tahan, is full of mathematical puzzles in the delightful short stories told.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;How to Count Like a Martian&lt;/em&gt;&lt;/strong&gt;, by Glory St. John, deals with different number systems in a delightful way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another favorite is &lt;strong&gt;&lt;em&gt;Math Girls&lt;/em&gt;&lt;/strong&gt;, by Hiroshi Yuki, though the math in it gets pretty hard. It reminds me of anime or manga. The characters are a bunch of high school students. The narrator is a boy who admires a number of girls who share his joy in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;In Code&lt;/em&gt;&lt;/strong&gt;, by Sarah Flannery, tells of her own experiences getting involved with math. I would have really liked it at your daughter's age. (But then, I'd have liked all of these.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When picking books for a teen, it helps to know their other interests. If you want to edit the question to include that, I might think of other books.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-06-06T15:04:01.993" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12412" PostTypeId="2" ParentId="12410" CreationDate="2017-06-07T01:17:55.013" Score="7" Body="&lt;p&gt;&quot;(Books in German or with German translation are a plus)&quot;&#xA;and &quot;she is particularly interested in geometry.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The OP's notes suggest:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ziegler, Günter M. &lt;em&gt;Do I Count?: Stories from Mathematics&lt;/em&gt;. CRC Press, 2013.&#xA;  &lt;a href=&quot;https://www.crcpress.com/Do-I-Count-Stories-from-Mathematics/Ziegler/p/book/9781466564916&quot; rel=&quot;nofollow noreferrer&quot;&gt;CRC link&lt;/a&gt;.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/jez7l.jpg&quot; width=&quot;200&quot; /&gt;&#xA;  &lt;hr /&gt;&#xA;  Ziegler, Günter M.: &lt;em&gt;Darf ich Zahlen? Geschichten aus der Mathematik&lt;/em&gt;. Piper, München, 2014. Taschenbuchausgabe, 3. Auflage.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is originally in German and now translated to English.&#xA;Günter's specialty is discrete geometry.&#xA;His stories and short, easy to read, entertaining, and often dwell on what it&#xA;means to really &lt;em&gt;do&lt;/em&gt; mathematics. I thoroughly enjoyed it myself.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-06-07T15:46:27.260" LastActivityDate="2017-06-07T15:46:27.260" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12414" PostTypeId="1" AcceptedAnswerId="12418" CreationDate="2017-06-07T17:48:51.467" Score="16" ViewCount="284" Body="&lt;p&gt;I'm teaching a graduate applied mathematics qualifying exam review course this summer, and I have 9 males and 1 female.  My co-instructor and I are also both male.  While I don't necessarily see this being an issue, I'm trying to be more sensitive to such &quot;classroom environment&quot; issues, and I've been wondering if there is anything specific I can try to do to help keep the atmosphere welcoming.  I imagine this is a general issue that others have thought about at the upper division and graduate level, so I welcome any suggestions or references to articles, books, etc.   &lt;/p&gt;&#xA;" OwnerUserId="227" LastActivityDate="2017-06-08T07:17:53.237" Title="How to improve atmosphere in male-dominated courses" Tags="|graduate-education|classroom-interaction|gender|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12415" PostTypeId="2" ParentId="12379" CreationDate="2017-06-08T00:24:45.887" Score="1" Body="&lt;p&gt;I think the confusion is largely a consequence of the fact that many people find the prefixes of the metric system (&lt;em&gt;kilo-&lt;/em&gt;, &lt;em&gt;centi-&lt;/em&gt;, etc.) unfamiliar, and find decimals (even terminating ones) less intuitive than the &quot;vulgar fractions&quot; they represent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If somebody asked me &quot;How can 0.004 Kilometers, a decimal number, be the same as 4 meters, a whole number&quot;? (as the OP mentions in the comments below his question), I would respond with something like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are you also bothered by the fact that $1/2$ a dozen eggs, a fraction, is the same as 6 eggs, a whole number?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What would come next depends on the questioner's response.  But let's assume they respond with something like:  &quot;Okay, I guess I get that.  But why can I say '0.04 kilopeople' but I can't say '0.04 people'?&quot;  In that case, I would respond with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are you also bothered by the fact that you can boil half a dozen eggs, but you can't boil half an egg?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The point of these responses, to be clear, is not to shut down the conversation with a zinger, but rather to bring to the surface what the underlying issues are:  &quot;1 kilopeople&quot; means the same as &quot;1000 people&quot;, and you can have half of a thousand people in just the same way that you can have half of a dozen eggs.  On the other hand you &lt;em&gt;can't&lt;/em&gt; have $1/7$ of a thousand people, in exactly the same way that you can't have $1/7$ of a dozen eggs.  &lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2017-06-08T00:24:45.887" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12416" PostTypeId="2" ParentId="12396" CreationDate="2017-06-08T02:25:56.083" Score="2" Body="&lt;p&gt;I like GTonyJacobs and mweiss's answers. It really boils down to the core fact that in addition, you can only combine like terms. However (to my frequent frustration), that is not actually taken as fundamental in a basic algebra course. Rather: The distributive property (of multiplication over addition) is. So I'd like to try an alternative answer that speaks directly in terms of that axiomatic basis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most likely misapprehension is that $\sqrt{2} + \sqrt{3} = \sqrt{5}$; in other words, that $\sqrt{2} + \sqrt{3}$ = $\sqrt{2 + 3}$. But that implies a belief that radicals distribute over addition, which is false. For a whole-number counterexample, consider: Is it true that $\sqrt{9 + 16} = \sqrt{9} + \sqrt{16}$? No, it's obviously not true, because by the order-of-operations (and grouping under a vinculum), the left-hand side simplifies to $5$, while the right-hand side simplifies to $7$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, distribution (and hence combining) does work for multiplication-over-addition; but it does not work for radicals-over-addition. Different operations work differently. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-06-08T02:25:56.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12418" PostTypeId="2" ParentId="12414" CreationDate="2017-06-08T07:17:53.237" Score="18" Body="&lt;p&gt;I'm a female who was often 'the only one' and later became a teacher in classes with 'only one' or very few females.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was a student in a normal (say 100th-ranked university), I just worked hard (and screwed up the grading curve for everyone hehehe) and didn't notice the gender ratio or whatever.  Then I moved to grad school at a really elite school and from engineering to math-ish so probably a steeper boy-girl ratio, and there I noticed that the boys played mental games a lot, for instance pretending to be so much better than you when they really were not.  The professors did that too.  A lot of acting and less substance than lowly hardworking me.  I confirmed that in my first year.&lt;br&gt;&#xA;Oh yeah by the way I'm black too lol.&lt;br&gt;&#xA;And I noticed heavy levels of prejudice that were a big burden for me.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My comments are that it's nice that you care, and I'm sure it's appreciated.  If, as you coach these people towards their exams and in maths in general, you can offer a helpful comment that isn't awkward, then go ahead and please do so.&lt;/strong&gt;&lt;br&gt;&#xA;I ended up quitting.  Nobody's fault, really.  And I truly appreciate the people that were intentionally helpful and the people that were not.  C'est la vie.  Some people will be ###holes, right?&lt;br&gt;&#xA;May I add that I did not quit just because I wasn't doing very well or because the 'atmosphere' was not very good, but also because the purpose, long-term, of the career just didn't really seem to fit right, so I couldn't see myself really in many of the successful leaders in the fields I tried, not exactly; and also because the career itself seemed to have an emotional problem and things would not get better even five years or ten years after successfully graduating.  There just seemed to be too high a rate of non-success and unhappiness and for too deep into the career path.   So let's not put too much weight on prejudice :) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Years later, I've found that as a teacher I'm sometimes harder on the (few) young women than many of the young men.  So I try my best to create standards that ensure I'm more blind and more fair.  For example I'll try to grade blindly to make sure I'm not letting the bias (on various dimensions, not just gender, say 'attitude', ...) through in this supposedly objective measure.  It's not easy to do, even in classes with many dozens of students, because I can feel a lot of subjective elements through their exams, so sometimes I know who I'm grading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ah what can I say, I'm still going to be a mathematician someday.  And if the system had been better they would have kept me without things having to be so hard.  Can't blame any one person.  And beyond it all, I'm happy.  Broke but happy.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was your question again?  :D&lt;br&gt;&#xA;&lt;strong&gt;Watch Hidden Figures.&lt;br&gt;&#xA;Don't worry too much about it.  And if you can't stop worrying about it, get into advocacy part-time.&lt;/strong&gt;  &lt;/p&gt;&#xA;" OwnerUserId="8295" LastActivityDate="2017-06-08T07:17:53.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12419" PostTypeId="1" CreationDate="2017-06-08T22:30:48.477" Score="5" ViewCount="102" Body="&lt;p&gt;If I studied laplace transforms, fourier analysis, multivariable calculus, differential equations, partial differential equations by 9th grade, does that mean I have talent for math or for physics. Most of the topics I studied are applied mathematics meaning they are mostly applied in the sciences. &lt;/p&gt;&#xA;" OwnerUserId="8297" LastActivityDate="2017-06-08T22:30:48.477" Title="Should I pursue math or physics as a career?" Tags="|student-motivation|" AnswerCount="0" CommentCount="3" ClosedDate="2017-06-11T18:04:19.953" ContentLicense="CC BY-SA 3.0" />
  <row Id="12420" PostTypeId="1" CreationDate="2017-06-08T23:23:11.943" Score="6" ViewCount="412" Body="&lt;p&gt;I'm a rising sophomore in high school. So far, I've taken Algebra One, Two, and Geometry in school. I want to learn higher math such as precalculus/trigonometry, calculus, linear algebra, and more, so I can go into topics such as cryptography, advanced computer science, and possibly take the AMC and other olympiad tests (I'm not too interested in that).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only problem, though, is that my abilities in problem solving and other stuff in math aren't that good. I do pretty well in my classes (high As) but that doesn't mean anything. The U.S. system doesn't seem too good in actually teaching math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I can do whatever is on my homework or tests. But, if I'm given a more difficult problem than usual concerning a topic I learned (say logarithms or something), I can't solve it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like this is going to be a hindrance to me learning higher math, doing well in more difficult subjects like calculus and linear algebra, doing well on olympiad tests, and going into math-heavy fields like computer science and cryptography.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, how can I change all of this and improve my skills? Are there any books that teach problem-solving, mathematical thinking, and higher math (or something like precalculus)? Again, I want to better these skills so I can do well not only in math, but other fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any help is really appreciated.&lt;/p&gt;&#xA;" OwnerUserId="8298" LastActivityDate="2017-06-11T12:32:02.143" Title="How can I improve my problem solving/critical thinking skills and learn higher math?" Tags="|mathematical-pedagogy|self-learning|general-pedagogy|precalculus|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12421" PostTypeId="1" AcceptedAnswerId="12422" CreationDate="2017-06-09T00:27:52.323" Score="7" ViewCount="491" Body="&lt;p&gt;This is not really a new question, more a revisiting of @vonbrand's &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/q/99/511&quot;&gt;Any suggestions on how to approach recursion and induction&lt;/a&gt;?&quot; In an introductory programming class this past year, I asked the students to create a Mondrian-like image&#xA;via recursion.&#xA;Here is a typical submission:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/M3WkH.png&quot; width=&quot;400&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;These 1st-yr college students had little difficulty understanding&#xA;the recursion necessary to produce such images. Most of them had&#xA;little beyond US high-school mathematics&amp;mdash;maybe calculus, no discrete math, no proofs.&#xA;And yet at the same time I taught more advanced students induction in the&#xA;theory of computation, and it was a mental hurdle for most of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Despite the informative answers to &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/q/10021/511&quot;&gt;Why are induction proofs so challenging for students?&lt;/a&gt;,&quot; I am still&#xA;uncertain. The Mondrian experience has made me wonder if it might not be&#xA;better pedagogically to introduce recursion first, nail that down thoroughly, and then teach&#xA;induction&amp;mdash;the apparently more difficult concept&amp;mdash;by connecting to &#xA;and building upon their&#xA;understanding of recursion. This may work even for students who have no programming experience, and are taught recursion &quot;theoretically.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-06-22T05:39:53.847" Title="Is induction or recursion easier to understand?" Tags="|undergraduate-education|proofs|induction|recursion|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12422" PostTypeId="2" ParentId="12421" CreationDate="2017-06-09T08:56:52.910" Score="9" Body="&lt;p&gt;One thing that you have to keep in mind here, is that you don't need to understand recursion to implement it. There is a big difference between &quot;we were taught to do it like that, I implement it and it works&quot; and really understanding the concepts and why it works as it does. With induction, on the other hand, you are supposed to write down a complete, formal proof, so you need at least better understanding of the concepts, if possible you should understand completely why it works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding your question if you should teach recursion or induction first, I would suggest this order:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Teach recursion and let the students program recursive functions.&lt;/li&gt;&#xA;&lt;li&gt;After the students saw that it worked, focus on the theoretical concepts. For example give them a short recursive program and ask them to prove what it computes on a given input. Or go back to the problems programmed before and ask the students to prove that they really do what they are supposed to.&lt;/li&gt;&#xA;&lt;li&gt;Show the students that for such proofs, it is easier to build them up from the bottom, i.e. not showing &quot;on input $n$, we do this and that and use the function on $n-1$. Here, on input $n-1$, we do...&quot; but rather starting with $n=1$.&lt;/li&gt;&#xA;&lt;li&gt;Let them train induction and recursion parallel. For example assume that the students should show&#xA;$$\sum_{i=1}^n i = \frac{n(n+1)}{2}.$$&#xA;Let them write a recursive program, computing the left hand side (that should be a rather easy program...), then apply the methods learned above to show that this program actually computes the right hand side.&lt;/li&gt;&#xA;&lt;li&gt;Gradually drop the programing tasks and focus more on the proofs.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In this way, the students don't just get induction as a strange, abstract concept, they have a practical component in learning it. They can learn that a proof is not something strange, it's just a formalized way to show something (in this case: &quot;my program does what I claim it does&quot;), show them that this formalism is not something to fear but is rather making it easier to show the correctness of their recursive programs and statements. In this way, you might actually be able to get them to understand that a formal proof is something good; this knowledge will be useful far beyond your course on induction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer: I am assuming an optimal class. In practice, some of these things might not work so well due to time limits or unmotivated students.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2017-06-09T08:56:52.910" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12423" PostTypeId="2" ParentId="12421" CreationDate="2017-06-09T10:53:55.317" Score="4" Body="&lt;p&gt;As a disclaimer, I am a CS teacher, so I teach both concepts within that context.  However, there is no doubt in my mind that induction is far harder for students to grasp.  I have not been able to figure out why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the problem with induction is that there are simply so many moving parts.  The inductive step can be quite complex (particularly in my field, where we also need to teach about &lt;strong&gt;loop invariants&lt;/strong&gt;), and at the start, it always seems to me that the student think that we are somehow trying to prove that $n = n+1$.  It takes a long time for that particular fog to lift.  And, most importantly, none of it makes sense until the entire thing &lt;em&gt;gels&lt;/em&gt;, so partial understandings don't lead to any real understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By contrast, I can often get my kids to develop recursion on their own by asking them how they would figure out the size of a directory.&lt;/p&gt;&#xA;" OwnerUserId="8274" LastActivityDate="2017-06-09T10:53:55.317" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12424" PostTypeId="2" ParentId="12096" CreationDate="2017-06-09T22:14:21.743" Score="7" Body="&lt;p&gt;(Quick note: I am not a math teacher.  I am a CS teacher with a bit of dysgraphia myself.  However, my insights here actually came from when I used to be a &lt;strong&gt;high school music teacher&lt;/strong&gt;, where I was practically &lt;em&gt;forced&lt;/em&gt; by the subject matter to tackle this problem with my beginning theory, AP Music Theory, and advanced theory classes.  Also, there are some nice insights about dysgraphia &lt;a href=&quot;https://www.understood.org/en/learning-attention-issues/child-learning-disabilities/dysgraphia/understanding-dysgraphia#item7&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would have a number of these students every year.  But musical staves are small, and the writing must be legible to be useful.  &lt;em&gt;(You can only imagine the pain of trying to sight-read at the piano, in real time, what dysgraphic students wrote for me!)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the motor issues that I encountered were really cognitive issues.  Students with really terrible writing did not understand why it mattered, and thus had never put serious attention and care into figuring out how to write things neatly when they needed to.  They didn't really know how to write the letters of the alphabet, and (more importantly to me), they were incapable of simply looking at my treble clefs, or my eighth notes, or my quarter rests, and organically figuring out how to imitate them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a result, I actually created worksheets, much like we all got in early grammar school, with every key symbol broken down into small, instructional writing pieces.  &lt;em&gt;(Treble clef: First draw the dot on the second line. Try it three times here: Then, make a quarter loop up to the third line like so.  Try it three times here:  Now, continue the loop down to the first line like this.  Try it three times here: ...)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Breaking down how to write the symbols into tiny steps this way was &lt;strong&gt;absolutely tedious&lt;/strong&gt;, but paid off handsomely for me in all of the cleaner writing I received.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For students who needed extra help, I would sit down with them, explain why handwriting mattered in this context, and ask them to copy a page of very nicely done work by another student, with attention paid to making everything precise and neat.  I gave them a small amount of extra credit for this work. I didn't mind providing this, as I was asking them to do extra work.&lt;/p&gt;&#xA;" OwnerUserId="8274" LastEditorUserId="8274" LastEditDate="2017-06-09T22:22:08.807" LastActivityDate="2017-06-09T22:22:08.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12425" PostTypeId="2" ParentId="12420" CreationDate="2017-06-10T00:13:01.340" Score="8" Body="&lt;p&gt;&quot;For example, I can do whatever is on my homework or tests. But, if I'm given a more difficult problem than usual concerning a topic I learned (say logarithms or something), I can't solve it.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did you mean you can't solve it or that you can't solve it right away?  It's been my experience teaching college math that students often become quickly frustrated if they can't get the correct answer in a relatively short period of time.  I have a masters in math and, when I was in grad school, I would struggle with problems for hours or sometimes days before they finally started to make sense.  This kind of persistence is an important part of both advanced learning and research.  So, to answer your question, I would start with persistence.  There's nothing wrong with struggling with problems - that's a sign that learning is happening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a more immediate note, don't be shy about asking questions.  If you ask your teachers, just ask if they can help you get started.  Don't look for a complete solution to the problem or even for them to help you walk through the problem.  Again, it's been my experience that this is what students often expect, i.e. that they should walk away from a conversation with a complete solution.  Instead, get just enough help to get you moving again then go back to working on the problem on your own.  The more of the answer you develop yourself, even if it takes time, the better your understanding is going to be.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2017-06-10T00:13:01.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12426" PostTypeId="2" ParentId="7274" CreationDate="2017-06-10T10:13:12.157" Score="5" Body="&lt;p&gt;In statistics we use the Box-Cox transform &#xA;$$&#xA;  y^{(\lambda)} = \begin{cases} \frac{y^\lambda -1}{\lambda},   &amp;amp;       \text{if $\lambda \not= 0$}       \\  &#xA;\ln y, &amp;amp; \text{if $\lambda=0$} &#xA;\end{cases}&#xA;$$&#xA;as a continuous extension of the power transforms that include the natural logarithms as a limiting case when $\lambda=0$.  To show the limit when $\lambda \rightarrow 0$ use L'hopitals rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example from statistics is the following. The $\alpha$-divergence between two probability densities defined on the same space is&#xA;$$&#xA;   d_\alpha  (f || g) = \frac1{\alpha-1} \ln \int f(x)^\alpha g(x)^{1-\alpha}\; dx&#xA;$$&#xA;The limit when $\alpha \rightarrow 1$ is the Kullback-Leibler divergence:&#xA;$$&#xA;   D_{KL}(f || g) = \int f(x) \ln\frac{f(x)}{g(x)} \; dx&#xA;$$&#xA;which can be calculated by L'hopital. &lt;/p&gt;&#xA;" OwnerUserId="122" LastEditorUserId="122" LastEditDate="2017-06-13T18:01:39.530" LastActivityDate="2017-06-13T18:01:39.530" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12427" PostTypeId="2" ParentId="12410" CreationDate="2017-06-10T14:48:47.750" Score="5" Body="&lt;p&gt;For your particular case, I highly recommend &quot;The Joy of X&quot; by Dr. Steven Strogatz.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And although you didn't request it, for higher level students and adults I also recommend &quot;Mathematics for Non-Mathematicians&quot; by Morris Kline which serves a similar purpose of opening the world of mathematics to a much broader audience and without the feel of a textbook, which it isn't.&lt;/p&gt;&#xA;" OwnerUserId="7893" LastActivityDate="2017-06-10T14:48:47.750" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12428" PostTypeId="1" CreationDate="2017-06-10T23:01:39.380" Score="-2" ViewCount="99" Body="&lt;p&gt;Here is the problem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If Bill takes 3 hours to paint a room and James takes 5 hours to paint a room, how long&#xA;will it take to paint a room if they work together? Show your working.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank You.&lt;/p&gt;&#xA;" OwnerUserId="8307" LastActivityDate="2017-06-10T23:16:37.460" Title="How do I solve this ratio problem?" Tags="|problem-solving|" AnswerCount="1" CommentCount="2" ClosedDate="2017-06-11T01:55:10.087" ContentLicense="CC BY-SA 3.0" />
  <row Id="12429" PostTypeId="2" ParentId="12428" CreationDate="2017-06-10T23:16:37.460" Score="2" Body="&lt;p&gt;This question should be migrated, and the answers will follow. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider Bills rate is 1/3 rooms per hour. James is 1/5 rooms per hour. Together, 8/15 rooms per hour. so 15/8 hours for one room. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-06-10T23:16:37.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12430" PostTypeId="2" ParentId="12389" CreationDate="2017-06-11T06:23:49.747" Score="2" Body="&lt;p&gt;Answering the indirect question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math-aids.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math-aids.com/&lt;/a&gt; generates practice questions at a range of levels. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't identified which (if any) sheets you want, but then I'm not entirely sure what you consider equivalent to the original question.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2017-06-11T06:23:49.747" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12431" PostTypeId="2" ParentId="12420" CreationDate="2017-06-11T07:11:53.333" Score="0" Body="&lt;p&gt;You probably have reasonable facility if you get high As, even in courses that are not super hard. The simple answer to your question is to work more problems.  Try to do some harder ones, but don't jump off the deep end.  Have some sense of progression (from easy, to medium, to mind stretcher).  Some students overestimate their mastery of the basics and then it holds them back on the trickier problems. Look at other books and work problems in them.  Check out Schaum's Outlines or look for older texts where there is more emphasis on learning lots of tricks, I mean techniques.  Hart College Algebra is a good one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math is a lot like sports.  You need to build your muscles up.  Do problems.  Don't try lifting 1000 pounds if you are just learning to squat.  But do have some sense of progression so you are increasing the weight on the bar.  And get the reps in.  It helps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, look at where your issues are, where you are weaker (logs more the issue than exponents, etc.)  And target some exercises where they are needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also add that there is more and more research on the brain showing that we are not wired for math the way we are for reading.  For a computer it is normal to calculate and to hold abstractions.  But evolution has trained our minds to have much more facility for the nuances of the verbal than of the logical.  This means that math can not be learned passively.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like complicated gymnastics, you will never just grow up around math stuff and have it rub off on you and start doing it. You have to work the progressions and learn to do things that are possible but not something we are so much wired for.  In contrast some physical tasks are things we pick up easily (gait for instance is amazingly complex from a mechanical standpoint, much more than a backflip).  Also just exposure somewhat pleasantly and observationally to verbalisms can lead to ready incorporation.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But we are complicated and have lots of capabilities.  So you can learn to do tricky problems, even building some ability to do new ones or ones that are combinations, just from learning other tricks along the way.&lt;/p&gt;&#xA;" OwnerUserId="8308" LastEditorUserId="143" LastEditDate="2017-06-11T12:32:02.143" LastActivityDate="2017-06-11T12:32:02.143" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12433" PostTypeId="2" ParentId="11021" CreationDate="2017-06-11T07:23:56.217" Score="2" Body="&lt;p&gt;Schaum's Outlines are great.  First Year College Math covers algebra 2, trig, analytic geometry and precalculus.  I would hit algebra 1 and geometry first though.  Also, older books are often better if you get the classics.  Usually they have all the answers, so self study is aided.  I don't know Khan Academy, but if it is basically programmed learning, I endorse it.&lt;/p&gt;&#xA;" OwnerUserId="8310" LastActivityDate="2017-06-11T07:23:56.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12434" PostTypeId="2" ParentId="10504" CreationDate="2017-06-11T07:29:38.570" Score="1" Body="&lt;p&gt;Programmed learning is the answer.  I have used it for learning accounting during a transatlantic flight before having to do a hardcore finance class (knowing nothing on business).  Also fastest way I know to learn nautical rules of the road.  Great English text that is a hidden treasure for naval officers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same holds in math.  Check out Stroud's books.  He really did it right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ken_Stroud&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ken_Stroud&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8311" LastActivityDate="2017-06-11T07:29:38.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12435" PostTypeId="2" ParentId="899" CreationDate="2017-06-11T07:41:44.327" Score="0" Body="&lt;p&gt;I think a standard sort of tricks and techniques course is fine for the math people.  A few proofs can be included but more for motivation and understanding than...well proof.  Or even for learning how to prove.  After all, the math majors will have theoretical calc later (real analysis). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also numbers wise, there are way more engineers and scientists than math majors.  Their needs should be taken care of more.  And then students move around anyways amongst majors.  So a typical Granville style calculus class is fine.&lt;/p&gt;&#xA;" OwnerUserId="8312" LastActivityDate="2017-06-11T07:41:44.327" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12436" PostTypeId="2" ParentId="82" CreationDate="2017-06-11T10:35:31.073" Score="1" Body="&lt;p&gt;I would use a quincunx (also known as Galton board).  I realize that has been answered before, but here I will give some images of how a quincunx could look like.  First a photo (by myself) of a very large quincunx in the science museum &lt;a href=&quot;http://www.universum.unam.mx/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.universum.unam.mx/&lt;/a&gt; in the large UNAM camp in south Mexico City: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/dQVyW.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dQVyW.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is very large, about three meters long if I remember right, and works by being tipped like a childrens seesaw.  Below another image showing better the principle: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WEQsU.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WEQsU.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="122" LastActivityDate="2017-06-11T10:35:31.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12439" PostTypeId="2" ParentId="1922" CreationDate="2017-06-11T13:12:38.497" Score="5" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think approaching the teacher is the answer to your problem. Sure, the teacher may be mediocre.  But still your boy should know how to do sin(x+y).  He could have gotten that from books, from drilling himself, etc.  Just take any book and work all the problems and you will learn.  Also, a kid who is sooper smart in math complaining about the teacher not explaining stuff is not the answer.  He should be able to nuke it out from books.  I suspect that he is either not as bright as you, not putting in as much effort, or is just too accelerated.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a world of gap between can't remember sin (x+y) and those theorems you mentioned.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As someone else mentioned, what is the point of talking to the teacher?  Are you going to change them into something they are not?  Really?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;These two things show me you are not really thinking strategically and shrewdly about what you want to do and how to get there.  Go back and think a little more about what you want and how to get it.  Even think on downsides.  I think you are putting a little too much of what you want, what you would want into what your son does.  Based on the remarks about your Ph.D. and the super theorems.  I mean who cares in the grand scheme?  This is about him, not you.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Oh and you don't need to learn linear algebra before trigonometry.  I didn't, Feynman didn't, Lisa Randall didn't, etc. etc.  The answer to this kid learning his chops is not that he has not been doing enough theory.  He has not even learned things in the bag of tricks sense.  And he could.  Just grab some decent (old) books and do it. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Probably even the books now that he has are fine. One VERY practical tip that I have for him is to do all the homework problems. I mean ALL. Not just the assigned ones. But everything in the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was a bright kid but struggled in school as I did not do all the HW. Going from doing now HW to doing assigned homework took me from a C to a low A. Doing all the homework took me to a very high A and outperforming students smarter than I. It actually got me super interested in the subject as well, not just drill. I ended up sort of self teaching myself AP calculus working each section ahead of time before class. and then the classes were literally (and not like a millennial says it) review, for me&lt;/p&gt;&#xA;" OwnerUserId="8315" LastEditorUserId="143" LastEditDate="2017-06-11T13:24:50.607" LastActivityDate="2017-06-11T13:24:50.607" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12442" PostTypeId="2" ParentId="10052" CreationDate="2017-06-11T13:28:36.433" Score="0" Body="&lt;p&gt;A lot of baby set theory was introduced into math in the 50s, 60s (New Math) but it did not have a good outcome for students IN GENERAL and so we went back to the traditional methods of 1900 or so.  Which actually themselves had a lot of hard earned pedagogical evolution behind them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Practically speaking rather than pushing abstract math to all kids, I think the answer could be little books or side activities or the like.  But that said, it is still for most people easier to have a base of traditional skills and appliable classes (calculus and trig for physics, etc.) before getting into proofs and more abstract, useless math topics.  Math maturity and all that.  And even then only a very small group will really like the more pure math.&lt;/p&gt;&#xA;" OwnerUserId="8318" LastActivityDate="2017-06-11T13:28:36.433" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12443" PostTypeId="2" ParentId="11104" CreationDate="2017-06-11T13:32:33.367" Score="-2" Body="&lt;p&gt;If you want to be better at Olympiad work old Olympiad problems.  This is the most efficient (deliberate practice) advice.  If you lack the facility for some of the things in those problems, than study the courses needed to have that facility.  Note that becoming super good at Olympiad problems is not the same as mastering courses which is not the same as facility for research.&lt;/p&gt;&#xA;" OwnerUserId="8319" LastActivityDate="2017-06-11T13:32:33.367" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12444" PostTypeId="2" ParentId="12410" CreationDate="2017-06-11T14:51:02.027" Score="1" Body="&lt;p&gt;Men of Mathematics by Eric Temple Bell&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Men_of_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Men_of_Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note for PC police, there is a woman in the &quot;Men&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Book is not 100% historically accurate.  But too much is made of the minor dramatizations by critics.  Book motivated many young students into math or sciences.  Comes up a lot on famous scientist reading lists while young (that and Microbe Hunters).&lt;/p&gt;&#xA;" OwnerUserId="8320" LastActivityDate="2017-06-11T14:51:02.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12445" PostTypeId="1" CreationDate="2017-06-11T16:27:57.387" Score="10" ViewCount="752" Body="&lt;p&gt;A &lt;strong&gt;8th grade&lt;/strong&gt; student I was working with recently was faced with squaring 37 in a problem that they were working on. The student stared down the numbers for some time, and then came up with some slightly incorrect answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I put my own pencil to the margins of the page and did the multiplication by hand, and asked why the student had attempted such a large multiplication in their head rather than making some scratch work of it. The answer surprised me:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We're not allowed to do multiplication by 'stacking'.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is this practice wide spread? What motivates it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;edit: by 'multiplication by stacking', I mean this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/h0kQk.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/h0kQk.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="308" LastEditorUserId="308" LastEditDate="2017-06-11T23:22:53.527" LastActivityDate="2017-06-29T13:16:47.940" Title="Arguments against multiplication by 'stacking'" Tags="|secondary-education|multiplication-table|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12446" PostTypeId="2" ParentId="12445" CreationDate="2017-06-12T00:42:00.807" Score="8" Body="&lt;p&gt;Some context is missing here. But I'll go ahead and speculate scenarios for why a math teacher would (temporarily?) prohibit multiplication by stacking [and in writing? --- nothing stops the student from doing stacked multiplication in their head]. In no particular order:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt; Sometimes, there is too much 'reflex' and not enough thinking. For example, $15 \times 200$ can be done by stacking, but it might be overkill. [cf Ben I's and James S. Cook's comments] &lt;/li&gt;&#xA;&lt;li&gt; Sometimes, a course textbook or an educational standard emphasizes different methods and the teacher may want to motivate other methods by first asking students to multiply numbers but removing the [perhaps] one and only method they know how. &lt;/li&gt;&#xA;&lt;li&gt; Since the question gives an example involving multiplying squares, I can imagine some Algebra related build-ups via $(a + b)^2$ [again, James Cook's comment] &lt;/li&gt;&#xA;&lt;li&gt; Specifically with perfect squares and the previous bullet point, consider this mental method. $(35 + 2)^2 = 1225 + 140 + 4$ ... the &quot;mental&quot; part is recognizing a little 'trick' with multiplying perfect squares that end in $5$ --- namely, and by example, $35\times35$ is the concatenation of $3 \times 4$ and $25$. Another example, $105^2$ is the concatenation of $10\times 11$ and $25$ --- $11025$. So you can take any $x$ and to find $x^2$ it's a matter of computing $((x-k) + k)^2$ where $k$ is the smallest non-negative integer such that $x-k$ is divisible by $5$. For example, $89^2 = (85 + 4)^2 = 7225 + 680 + 16$ --- we can argue that $(80 + 9)^2$ is easier and so be it. I'm speculating about why a certain method is restricted. [Of course, $83^2$ might be more easily handled as $(80 + 3)^2$ rather than $(85 - 2)^2$] &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6898" LastActivityDate="2017-06-12T00:42:00.807" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12447" PostTypeId="1" CreationDate="2017-06-12T00:48:49.707" Score="8" ViewCount="180" Body="&lt;p&gt;This post is largely copied from &lt;a href=&quot;https://mathoverflow.net/a/271611/6094&quot;&gt;a response&lt;/a&gt; to the &#xA;&lt;a href=&quot;https://mathoverflow.net/q/271608/6094&quot;&gt;17 Camels Trick&lt;/a&gt; MO posting.&#xA;Here I ask: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. Does anyone (in any country) teach this &#xA;  binary method of decimal multiplication? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It strikes me as easy as (if not easier than)&#xA;the traditional method of multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a method for multiplying two decimals, say $13 \times 27$.&#xA;Form two columns headed by $13$ and $27$.&#xA;Halve $13$ and discard any remainder, and double $27$.&#xA;Continue halving the first column and doubling the second&#xA;until the first column reaches $1$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{array}&#xA;\mbox{13} &amp;amp; 27 \\&#xA;{\color{red}{6}} &amp;amp; {\color{red}{54}} \\&#xA;3 &amp;amp; 108 \\&#xA;1 &amp;amp; 216&#xA;\end{array}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now discard every row for which the first column is even ($\color{red}{6}$ above).&#xA;Sum the remaining elements of the second column: &#xA;$$27 + 108 + 216 = 351 = 13 \times 27 \;.$$&#xA;This is of course using the binary representation $13 = 2^0 + 2^2 + 2^3$, and excluding the $\color{red}{2^1}$ row &#xA;$\color{red}{6} \;\; \color{red}{54}$.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-08-21T14:47:52.970" Title="Multiplying two decimals using (camouflaged) binary representation" Tags="|secondary-education|primary-education|multiplication-table|" AnswerCount="1" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="12448" PostTypeId="2" ParentId="12445" CreationDate="2017-06-12T13:08:26.023" Score="3" Body="&lt;p&gt;Maybe instead of the &quot;stacking&quot; method we learned in the last century, they are supposed to use the &lt;a href=&quot;https://signalvnoise.com/archives2/thinking_diagonally_with_multiplication_grids.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;diagonal&quot; method&lt;/a&gt; like this&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/R3SwO.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/R3SwO.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-06-12T13:08:26.023" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12450" PostTypeId="1" CreationDate="2017-06-13T01:57:27.050" Score="11" ViewCount="867" Body="&lt;p&gt;I've been looking through the Common Core State Standards (&lt;a href=&quot;http://www.corestandards.org/Math/&quot; rel=&quot;noreferrer&quot;&gt;http://www.corestandards.org/Math/&lt;/a&gt;) and have been surprised to find very little reference to exponential functions and logarithms.  Specifically, as far as I can tell, none of the following are included in the Standards:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Formal properties of logarithms, e.g. &lt;span class=&quot;math-container&quot;&gt;$\log(ab)=\log(a)+\log(b)$&lt;/span&gt;, etc.&lt;/li&gt;&#xA;&lt;li&gt;Exponential functions written with the base &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, i.e. functions of the form &lt;span class=&quot;math-container&quot;&gt;$f(x)=Ae^{kx}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Continuously compounded interest — or in fact anything about compounding interest.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As a matter of fact, the only standards I &lt;em&gt;can&lt;/em&gt; find that mention logarithms or the number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; are the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;F-IF.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior&lt;/p&gt;&#xA;&lt;p&gt;F-BF.5:  Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.&lt;/p&gt;&#xA;&lt;p&gt;F-LE.4: For exponential models, express as a logarithm the solution to&#xA;&lt;span class=&quot;math-container&quot;&gt;$a\cdot b^{ct} = d$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; are numbers and the base &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;; evaluate the logarithm using technology.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I know that the CCSS are not intended to be a comprehensive listing of absolutely everything that students are expected to learn, but it still seems bizarre to me that they do not include basic facts about &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; or the fundamental properties of logarithms.  Am I just missing them?  Are they in there somewhere?&lt;/p&gt;&#xA;&lt;p&gt;(Note:  I do &lt;em&gt;not&lt;/em&gt; intend or hope for this question to prompt an opinion-based discussion about whether the Standards are good, bad, or otherwise.  I just want to know what, if anything, the Standards say about exponential and logarithmic functions, other than what I have already listed.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-01-30T19:15:45.313" Title="What do the Common Core Standards expect secondary students to learn about logarithms or the number $e$?" Tags="|secondary-education|common-core|logarithm|exponential|" AnswerCount="1" CommentCount="4" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12451" PostTypeId="1" CreationDate="2017-06-13T18:34:31.393" Score="32" ViewCount="30889" Body="&lt;p&gt;I am curious why it is necessary for people to learn how to use compasses and straightedges in geometry, and not just rely on a drawing program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a couple ideas, but I might be missing something or have a gap, so any opinions supported by facts or credible sources is great!&lt;/p&gt;&#xA;" OwnerUserId="8328" LastActivityDate="2018-10-16T21:30:54.143" Title="Why should kids learn how to use a compass and straightedge, and not rely on a drawing program?" Tags="|geometry|" AnswerCount="11" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="12452" PostTypeId="2" ParentId="12451" CreationDate="2017-06-13T22:26:37.350" Score="19" Body="&lt;p&gt;Of course there is the deep (~2000 yrs) &lt;a href=&quot;https://en.wikipedia.org/wiki/Compass-and-straightedge_construction#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;history of compass/straightedge constructions&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another more modern alternative is origami constructions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Robert Lang, &quot;Origami and Geometric Constructions,&quot; 1996.&#xA;  &lt;a href=&quot;https://www.langorigami.com/files/articles/origami_constructions.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In particular, one can trisect an angle via origami folding&#xA;(under, e.g., &lt;a href=&quot;http://www.langorigami.com/article/huzita-justin-axioms&quot; rel=&quot;nofollow noreferrer&quot;&gt;Huzita's origami axioms&lt;/a&gt;):&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/qHaEf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qHaEf.png&quot; alt=&quot;Trisection&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;Figure from &lt;a href=&quot;http://www.gfalop.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Geometric Folding Algorithms: Linkages, Origami, Polyhedra&lt;/em&gt;&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-06-18T18:10:37.210" LastActivityDate="2017-06-18T18:10:37.210" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12453" PostTypeId="2" ParentId="12451" CreationDate="2017-06-13T22:32:25.850" Score="46" Body="&lt;p&gt;The point of compass-and-straightedge constructions is for the students to get experience reasoning about axiomatic systems. The accuracy of the actual drawings is basically irrelevant (as long as it's not so sloppy that it impedes visualization) — the point is that they can &lt;em&gt;prove&lt;/em&gt; the accuracy of the idealized constructions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method of axiomatic construction was first studied for historical reasons that perhaps aren't as relevant now — having to do with the &quot;synthetic&quot; style of axiomatic geometry used by the Ancient Greeks, as opposed to the &quot;analytic&quot; style more popular in modern mathematics (e.g., Cartesian coordinates). However, geometric constructions can still serve useful educational purposes now, because it's a simple axiomatic system that can be a good way for students to learn to connect geometric intuition with careful, precise reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, once students have learned how to construct bisections of angles, one can ask them to try to trisect an angle, and use this to introduce the concept of an impossibility proof. This could also be used to illustrate how subtle changes to axioms can have a major impact on what's provable/constructible (like how doubling the cube is impossible with compass and straightedge, but becomes possible as soon as you can mark lengths on the straightedge).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are just a few specific ideas that came to mind — the general idea is to use the constructions as a tool to teach logical and geometric reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a case study in how &lt;em&gt;not&lt;/em&gt; to teach geometric constructions — which may highlight some pitfalls to avoid — see &lt;a href=&quot;http://jwilson.coe.uga.edu/TiMER/Schoenfeld%20(1988)%20Good%20Teach%20Bad%20Results-2.pdf&quot; rel=&quot;noreferrer&quot;&gt;Schoenfeld 1988, 'When Good Teaching Leads to Bad Results: The Disasters of &quot;Well-Taught&quot; Mathematics Courses'&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2017-06-13T22:32:25.850" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12454" PostTypeId="2" ParentId="12451" CreationDate="2017-06-14T00:37:23.157" Score="14" Body="&lt;p&gt;In addition to other good observations in comments and answers, I think (based on my own arc, and on observations of many others) that the physical/tactile issues about &quot;lines and circles&quot; do indeed provide a physically reasonable, but explicitly limited, way to interact with &quot;geometric reality&quot; as well as &quot;axiom-generated/axiom-mediated reality&quot;. That is, in particular, I think this is not only an artifact of history, but a fairly natural product of human interaction with the geometric world, as well as humans' interaction with the formality of first-order predicate logic ... sorts of things. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sufficiently able kids immediately observe that the constraints are a bit artificial, yet interesting. This does echo the 2,000-year-old attitude, too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But/and of course the limitations of simple/natural things are of meta-interest in themselves...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-06-14T00:37:23.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12455" PostTypeId="2" ParentId="2074" CreationDate="2017-06-14T01:32:48.717" Score="6" Body="&lt;p&gt;When I went to high school in the early 80s, I studied geometry and did formal mathematical proofs.  I then studied economics undergrad and business grad school. When I decided to return in my 30s to school to pursue a master's degree in mathematics, I did not need an undergraduate math degree to apply to the graduate programs that interested me, but I did need to have completed certain courses (like Elementary Number Theory). When I started taking these upper division math classes, some required that I write proofs. Without my high school experience, I would have been lost.  Further, whether proofs are introduced in geometry or in some other way in hgih school, learning and practicing logical thinking is foundational to a number of undergraduate disciplines. Thus proofs seem to be an essential college preparatory skill.  &lt;/p&gt;&#xA;" OwnerUserId="8331" LastActivityDate="2017-06-14T01:32:48.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12456" PostTypeId="2" ParentId="2074" CreationDate="2017-06-14T08:01:14.267" Score="4" Body="&lt;p&gt;&quot;Should Euclidean geometry be taught to highschool students?&quot; Certainly, yes. The whole idea that it shouldn't is a misguided mistake with identifiable historical sources and based on identifiable philosophical errors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Searching through both the &lt;em&gt;question&lt;/em&gt; and the existing &lt;em&gt;answers&lt;/em&gt; I notice that there has been no mention made until now of the so-called &lt;em&gt;New Math&lt;/em&gt;. I think this should be mentioned is because the basic impetus behind Bourbaki, Dieudonne &quot;death to triangles&quot; call, and &lt;em&gt;New Math&lt;/em&gt; is the same, the philosophical errors are the same, and the reasons for failure are the same.  For a detailed analysis of the history of the &lt;em&gt;New Math&lt;/em&gt; educational debacle see &lt;a href=&quot;http://www.journals.uchicago.edu/doi/10.1086/678170&quot; rel=&quot;nofollow noreferrer&quot;&gt;this publication&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More specifically, the philosophical errors involved are traceable to Piaget.  Piaget was interested in analyzing the structures of the learner's mind.  Piaget claimed an affinity between such structures and the structures of &quot;modern&quot; mathematics.  Such an identification was a non-sequitur but it influenced people like Begle who were in key positions in the US to affect pre-university education.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-06-20T06:49:18.797" LastActivityDate="2017-06-20T06:49:18.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12457" PostTypeId="2" ParentId="12451" CreationDate="2017-06-14T08:55:33.540" Score="10" Body="&lt;p&gt;I would like to express here a personal complaint, which will also answer your question: as a child I learnt to work with a compass and straightedge, but there was a great drawback: I learnt that a compass is used to draw circles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This, obviously, is right, but there also is another use of the compass, which is to estimate a distance between two points: this can be seen in movies about ancient ships, where the captain holds a ruler in one hand, and uses the compass to measure multiples of a certain distance along that ruler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I had not learnt to use the compass in that way, I always thought that a circle is something round, and during later education, I first needed to pass via that &quot;round&quot; comprehension before realising that the definition of a circle is a list of points at an equal distance from a central point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I had learnt to use the compass in two ways (drawing a circle and measuring distances), I might have realised earlier that a circle is a set of equidistant points directly, without needing to pass via the &quot;round&quot; comprehension, which is for a student something which reduces the understanding of the matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now to answer your question: if you don't use a compass but instead you use a computer program to draw circles and arcs, how will you then pass the most important property of a circle to your students: not the fact that a circle is round but the fact that it consists of equidistant points?&lt;/p&gt;&#xA;" OwnerUserId="8190" LastEditorUserId="-1" LastEditDate="2017-06-14T23:15:57.973" LastActivityDate="2017-06-14T23:15:57.973" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12458" PostTypeId="2" ParentId="2074" CreationDate="2017-06-14T11:16:42.597" Score="2" Body="&lt;p&gt;While talking about proofs, I notice that nobody is mentioning the conditions: while performing mathematics you can't just do what you want: all conditions need to be fulfilled before you can actually do something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When explaining a proof to a class, it's very important that the teacher mentions the conditions, and while writing the proof, the teacher should clearly indicate at what point the mentioned conditions are used (e.g. while proving Pythagoras' theorem, the teacher should clearly indicate why this only applies to right-angle triangles).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This way of working learns the students why the conditions are so important and automatically they will watch their steps while setting up complicated reasonings (not only in math, by the way).&lt;/p&gt;&#xA;" OwnerUserId="8190" LastEditorUserId="1385" LastEditDate="2017-06-16T07:23:15.643" LastActivityDate="2017-06-16T07:23:15.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12459" PostTypeId="2" ParentId="1952" CreationDate="2017-06-14T11:55:13.257" Score="1" Body="&lt;p&gt;If I could I would post as a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just to be the exception to your experience, I didn't even know (or absolutely don't remember) about similar triangles and I find the relation to Thales' theorem (that I do remember well) not that much obvious. For details, I'm 20 and probably been taught Thales's theorem around 11 so I may have simply forgot.&lt;/p&gt;&#xA;" OwnerUserId="8337" LastActivityDate="2017-06-14T11:55:13.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12461" PostTypeId="2" ParentId="2262" CreationDate="2017-06-14T19:16:23.980" Score="4" Body="&lt;p&gt;After teaching calculus for a long time, I have found that actually doing math during class is absolutely essential. By this I mean that you should justify as much as you possibly can, and tell a story about how these ideas arise. The other major way I have seen math taught is through a slide show where concepts are listed and then applied, as if they are magically true and the only interesting thing to do in math is applying these concepts to specific problems. When math is presented as a sequence of concepts that are applied to solve problems, students do not experience math as a coherent language that itself leads to new concepts derived from familiar ones. Under this approach, I cannot define ln(x) until one can integrate functions, knows the mean value theorem, and of course can use limits. If f(x)=ln(x) is introduced before integrals, how do you define it? As the inverse to g(x)=e^x? OK, what is the definition of e^x, and how do derivatives and inverses relate? By going down the path of early transcendentals,  you can find more derivatives earlier in your college career (by 2 months or so), but you can't integrate until later in your college career, and also don't know the definitions of many of the functions that you are using, nor how they are related. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Early transcendentals leads the professor to justify the concepts like all of the facts about ln(x) by saying &quot;because I said so&quot; which I think is unacceptable at the college level. This is OK in grade school when math is taught by people whom are not experts in math. However, the fact that this is OK in grade school contributes to the US scoring behind many countries in math proficiency, since math is taught as a sequence of disconnected and arbitrary concepts to be memorized for the test and then discarded.&lt;/p&gt;&#xA;" OwnerUserId="8346" LastActivityDate="2017-06-14T19:16:23.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12462" PostTypeId="2" ParentId="12451" CreationDate="2017-06-14T23:22:26.073" Score="6" Body="&lt;p&gt;I took drafting and CAD in HS. As a carpenter, I've used my drafting knowledge here and there. The only thing CAD has helped me do is build &lt;em&gt;Doom&lt;/em&gt; game levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's my answer to the title, because we never used anything else but a pencil in geometry. As to why you teach a student anything that a tool can do for you, it's to understand the concept; how the ins-and-outs of it all work. That's the real reason, it's not, &lt;em&gt;'But what if you don't HAVE a calculator?'&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="8148" LastEditorUserId="8148" LastEditDate="2017-06-15T00:24:56.727" LastActivityDate="2017-06-15T00:24:56.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12463" PostTypeId="2" ParentId="12221" CreationDate="2017-06-15T00:50:24.637" Score="0" Body="&lt;p&gt;Take calc 3 (multivariable) and calc 4 (ODEs).  This is standard and will work with almost all things you want to go into.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Real analysis (theoretical calc) is only useful if you go into math.  Even theoretical physicists don't need it.&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-15T00:50:24.637" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12464" PostTypeId="2" ParentId="12246" CreationDate="2017-06-15T00:59:30.597" Score="1" Body="&lt;p&gt;I was routinely called on in high school and college.  I think it is a fine practice.  I would not obsess too much on how to do it perfectly or some big fear od doing it imperfectly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still remember snapping awake at the Naval Academy when the Indian ODEs prof asked me some ODE needing an integrating factor to solve it. I somehow woke instantly from total sleep, saw the board and spit out the answer xsqetothex or whatever it was.  Not too simple though.  He was amazed and commented how he always wished he could do that as a student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keep 'em awake, keep 'em guessing.  Don't baby the ball!&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-15T00:59:30.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12465" PostTypeId="2" ParentId="12301" CreationDate="2017-06-15T01:01:58.820" Score="3" Body="&lt;p&gt;I think solo studying is way more effective. I would keep a little sheet or back pages of the notebook for prof questions, but would often figure them out by the end of the self study session.  In the event I didn't, it was easy to ask the prof the next day.&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-15T01:01:58.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12466" PostTypeId="2" ParentId="12390" CreationDate="2017-06-15T01:05:25.763" Score="2" Body="&lt;p&gt;I think the topics you want to start with are not that important to having the algebraic chops to handle calculus.  I suspect you like them for theoretical reasons.  But that is different from what your students need to learn.&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-15T01:05:25.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12467" PostTypeId="2" ParentId="12221" CreationDate="2017-06-15T15:16:37.297" Score="2" Body="&lt;p&gt;That depends on what you're interested in.  If you're thinking about biology or chemistry as a career, differential equations would be a good choice.  If you're interest is in computer science then I would go with the discrete math option.  For electrical engineering, you might think about a class in complex analysis.  (Complex refers to complex numbers not the difficult of the class.)  Other kinds of engineering like mechanical or aeronautic would benefit from a third semester of multi-variable calculus or vector analysis.  If you're thinking about math as a major then you're probably going to have to take the multi-variable calculus course so that would be a good next choice unless you're getting a little tired of calculus in which case any of the other options would be good.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2017-06-15T15:16:37.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12468" PostTypeId="2" ParentId="12451" CreationDate="2017-06-15T19:06:30.897" Score="3" Body="&lt;p&gt;In primary school We used to do mental calculations, never touched a calculator back then.. in college, some programming lessons with all the IDE's help deactivated. &#xA;At university I found myself relaying on papers &amp;amp; pencils because I knew what I was doing &amp;amp; feeling confident about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To resume, I think it's because they want to teach kids how not to rely on such programmable helps because they might find themselves in situations with no such tools available that require speed of execution &lt;/p&gt;&#xA;" OwnerUserId="8355" LastEditorUserId="8571" LastEditDate="2018-07-17T00:03:55.043" LastActivityDate="2018-07-17T00:03:55.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12469" PostTypeId="2" ParentId="12246" CreationDate="2017-06-15T22:17:12.003" Score="6" Body="&lt;p&gt;If the question really is &quot;how (best) to call on students?&quot;, then I'd say I don't know. If the question is &quot;do I call on students?&quot;, then I'd say &quot;no, I do not&quot;. I do not do so in undergrad or grad classes, required or elective. I &lt;em&gt;do&lt;/em&gt; encourage questions, even if frivolous or humorous, and usually there are several students (whether academically strong or not...) who, by personality, are happy to ask the question that everyone else is thinking but too inhibited to ask.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, I tend to structure &quot;lectures&quot; not really as lectures, but as me-doing-examples (which, on one hand, are of the sort the students know they will be &quot;graded on&quot;, but/and, also, are what I consider to be &lt;em&gt;important&lt;/em&gt; scientifically/intellectually). Students are encouraged to ask clarification questions, background questions, please-repeat-what-you-just-said questions, or small-detail questions. Anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most that I'd inclined to do is, when I hear a sharp inhalation, or see a perplexed facial expression, that I might ask a student whether they were unhappy with something...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think that asking non-volunteer students to ask questions is rather aggressive, and cultivates an adversarial/game-ish atmosphere, if only due to the naive perceptions of young students (sub-25?), regardless of the good intentions of the teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For that matter, it is a non-trivial issue about whether one can easily, extemporaneously, formulate a coherent statement about a thing. Since I generally &lt;em&gt;can&lt;/em&gt; do so, it took me a long time to understand that &lt;em&gt;incapacity&lt;/em&gt; in this regard is &lt;em&gt;not&lt;/em&gt; a sign of stupidity or lack of understanding or earnestness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, last, whether or not we declare that students &quot;should not feel embarrassed/humiliated&quot;, immature people do manage to feel humiliated for reasons that don't seem reasonable to older people, etc. And, then, will be resentful, and distracted. The idea that they &quot;should&quot; behave otherwise is as impractical and untenable as the idea that they &quot;should&quot; have a different attitude in their approach to mathematics, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of &quot;trust&quot;, the only time I ask students questions, expecting or hoping to be answered in real time, is with my research students in rehearsals of their presentations. This presumes many, many months of acquaintance, and a negotiated &quot;trust&quot; situation, and is absolutely not &quot;public&quot;. And it is understood that &quot;failure&quot; is not literal failure, but is &lt;em&gt;educational&lt;/em&gt;. But, even then, people need some reflection to understand that is indeed &quot;how it could be&quot;. And usually needs substantial prior one-on-one discussion about &quot;what the game is&quot;, and that we are not adversaries, etc. Even so, this does not seem to work well with immature grad students.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2017-06-15T22:32:43.320" LastActivityDate="2017-06-15T22:32:43.320" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12471" PostTypeId="1" CreationDate="2017-06-16T16:36:26.930" Score="9" ViewCount="166" Body="&lt;p&gt;TL;DR: do you know of an undergrad textbook that covers numeracy with an emphasis on genuine context (as opposed to the artificial word problems that are all too common)? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Less concisely:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am an instructor in the US, teaching in a large, research  university. I am looking for a textbook that teaches numeracy at the college-level (so above the level of algebra 2) to students who need math courses for their degree, but have limited interest in the subject. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More specifically: I am looking for a book that teaches the students different skills (logical, statistical, etc) for analyzing the math that appears in the newspaper. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware of books like &quot;A Mathematician Reads the Newspaper,&quot; and I would be happy to find other books like this, but am hoping to discover a &lt;strong&gt;textbook&lt;/strong&gt; on or near this topic.   &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2017-06-17T01:00:46.227" Title="Textbook search: learning the math required to read the newspaper" Tags="|undergraduate-education|textbooks|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12472" PostTypeId="2" ParentId="12471" CreationDate="2017-06-16T16:53:59.457" Score="6" Body="&lt;p&gt;I've heard good things about Bolker and Mast's &lt;a href=&quot;http://www.maa.org/press/books/common-sense-mathematics&quot; rel=&quot;noreferrer&quot;&gt;Common Sense Mathematics&lt;/a&gt;, though I haven't used it myself.  I can't imagine the MAA wouldn't send you a review copy in your situation.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-06-16T16:53:59.457" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12473" PostTypeId="2" ParentId="12471" CreationDate="2017-06-17T01:00:46.227" Score="3" Body="&lt;p&gt;Not direct answers (not textbooks), but there have been several books published recently on estimation that might serve as background:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Weinstein, Lawrence, and John A. Adam. &lt;em&gt;Guesstimation: Solving the world's problems on the back of a cocktail napkin&lt;/em&gt;. Princeton, NJ, USA: Princeton University Press, 2009.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/ghpw5.jpg&quot; width=&quot;200&quot; /&gt;&#xA;  &lt;hr /&gt;&#xA;  Weinstein, Lawrence. &lt;em&gt;Guesstimation 2.0: Solving today's problems on the back of a napkin&lt;/em&gt;. Princeton University Press, 2012.&#xA;  &lt;hr /&gt;&#xA;  Mahajan, Sanjoy, and Carver A. Mead. &lt;em&gt;Street-fighting mathematics: The art of educated guessing and opportunistic problem solving&lt;/em&gt;. MIT Press, 2010.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-06-17T01:00:46.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12474" PostTypeId="2" ParentId="12451" CreationDate="2017-06-17T11:32:12.450" Score="4" Body="&lt;p&gt;There is an idiom in my country.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It depends on how you handle a fool and scissors.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This mean&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Everything comes in handy when used right.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and how you handle a scissors has a bit difficult. Even if it's a collapsed scissors,&#xA;it works by how to use power and method. Therefore it is useful that everyone know how to use some tools.&#xA;Also it is fundamental work to create image with compass and straightedge on geometry, and that makes us clever, since the geometrical algorithm is some complex. &#xA;Even now the easy shapes which can be created by those tools are targets for present mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="8365" LastActivityDate="2017-06-17T11:32:12.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12475" PostTypeId="2" ParentId="12451" CreationDate="2017-06-17T15:01:54.480" Score="4" Body="&lt;p&gt;because you learn more when you build something with your hands.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The human brain is not a computer.  It learns by training, by doing, not by listening or watching or being given a long complicated set of correct steps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I don't really get the point of geometry and proof exposure.  Seems like a diversion from the algebra through calculus track.  But that is a different issue.&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-17T15:01:54.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12476" PostTypeId="1" CreationDate="2017-06-17T22:29:48.073" Score="30" ViewCount="8808" Body="&lt;p&gt;Here is the concept:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Student is presented with a problem.  He/she may not even understand what is being asked, or may attempt.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students reads a solution to the problem.  In it there may be explanations about concepts - so there is a &amp;quot;content&amp;quot; part, but as part of an actual problem being solved, at the level the student is currently at.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If answer is correct, moves on to the next problem.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When a set of problems of a given topic is consistently correct (say 10 out of 10 consecutive questions), student is given the next topic, or the next difficulty level.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Point being:  At no point is there a proper &amp;quot;lesson&amp;quot;. What are your views about such an approach, compared with lesson-solve, lesson-solve, etc.&lt;/p&gt;&#xA;" OwnerUserId="7960" LastEditorUserId="5220" LastEditDate="2021-04-06T16:07:50.863" LastActivityDate="2021-04-12T10:46:30.460" Title="Can mathematics be learned by ONLY solving problems?" Tags="|mathematical-pedagogy|problem-solving|" AnswerCount="9" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="12477" PostTypeId="1" CreationDate="2017-06-17T23:38:40.630" Score="5" ViewCount="242" Body="&lt;p&gt;I'm going into highschool. Thanks to an acceleration program at my district, I've already taken Algebra I and Geometry. I am very much a math and physics nerd (take a look at my activity over on MSE and PSE if you wish) and spend a lot of time on my own reading and learning - for example, I've taught myself a decent amount of the basics of single-variable calculus (take a look at my notes on github - my github account is linked in my profile) and some linear algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to take some math classes over the summer (online) in a way that my highschool would be likely to accept them - i.e., I want to take Algebra II this summer, but I don't really want to take it again this fall =) I'd also prefer the course to not be super expensive, as I will probably be paying for it myself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks, and I apologize if this is not the right site, but I saw similar-ish questions here.&lt;/p&gt;&#xA;" OwnerUserId="8366" LastActivityDate="2017-06-21T01:40:22.593" Title="Online course for Algebra II" Tags="|secondary-education|algebra|online-instruction|resource-request|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12478" PostTypeId="2" ParentId="12476" CreationDate="2017-06-18T04:00:55.453" Score="6" Body="&lt;p&gt;This approach would be fundamentally a violation of the entire axiomatic idiom of mathematical understanding and proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular: Mathematics starts with careful definitions of terms. The proposed learning process contains no mention of definitions. I tell my students all the time that one of the essential strengths and advantages of the mathematics discipline is the use of unambiguous definitions for words. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a person doesn't have that, then they'll be forced into piecing together a relational, approximate understanding of terminology. Like in natural languages. Which is essentially un-mathematical. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Plus theorems and concepts will not be proven. Essentially you'd be promoting not real mathematics, but &lt;a href=&quot;https://en.wikipedia.org/wiki/Cargo_cult&quot; rel=&quot;noreferrer&quot;&gt;cargo-cult&lt;/a&gt; mathematics. Faith-based math. Fake math. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-06-18T04:00:55.453" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="12479" PostTypeId="2" ParentId="12476" CreationDate="2017-06-18T05:59:11.807" Score="41" Body="&lt;p&gt;Such an approach seems designed to force (or at least, strongly encourage) students to learn by pattern-matching from examples. This is one of three modes of student learning in mathematics described in &lt;a href=&quot;https://intranet.math.vt.edu/people/fquinn/education/techkiller.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article by Frank Quinn&lt;/a&gt;; it is the least powerful, most fragile, and most error-prone of the three. To quote the relevant passage:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There are (roughly) three levels of student work in mathematics:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Follow patterns inferred from examples;&#xA;2.  Use systematic methods and algorithms that, among other things, account&#xA;for the patterns in examples; and&#xA;3.  Exploit the mathematical structures that lie behind algorithms.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Each level provides substantially more flexibility, range and accuracy in applications than the one before it.  Learning at each level is faster, more powerful, and more transferrable  than  at  the  one  before  it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As Daniel Collins points out in their answer, a pattern-in-examples approach misses the fundamental way mathematics is structured and often facilitates deep conceptual misunderstandings. I find the &amp;quot;fundamental principles of mathematics&amp;quot; in &lt;a href=&quot;https://math.berkeley.edu/%7Ewu/NoticesAMS2011.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hung-Hsi Wu's article &amp;quot;The Miseducation of Mathematics Teachers&amp;quot;&lt;/a&gt; to be a helpful way to think about this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Every concept is precisely defined, and &lt;strong&gt;definitions&lt;/strong&gt; furnish the basis for logical deductions. [...]&#xA;2. Mathematical statements are &lt;strong&gt;precise&lt;/strong&gt;. At any moment, it is clear what is known and what is not known. [...]&#xA;3. Every assertion can be backed by logical &lt;strong&gt;reasoning&lt;/strong&gt;. [...]&#xA;4. Mathematics is &lt;strong&gt;coherent&lt;/strong&gt;; it is a tapestry in which all the concepts and skills are logically interwoven to form a single piece. [...]&#xA;5. Mathematics is goal-oriented, and every concept or skill in the standard curriculum is there for a &lt;strong&gt;purpose&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The approach you describe would probably violate the first three of these fundamental principles: problems would be assigned that use terms that haven't been precisely defined, it would be unclear what theorems were considered &amp;quot;known&amp;quot; and available for students to use at any given time, and the solutions might make assertions whose reasoning isn't fully explained.&lt;/p&gt;&#xA;&lt;p&gt;To avoid this, one could fully define all terms in the problems, include full proofs of all assertions in the solutions, and require students to give full proofs of all claims (allowing citations to previous solutions' proven theorems) in their own work. However, this would &lt;em&gt;significantly&lt;/em&gt; increase the difficulty of the course, as students would have to develop all the theorems for newly stated definitions on their own. Unless the questions are of a very basic nature, this would essentially put the students in the position of researchers.&lt;/p&gt;&#xA;&lt;p&gt;Now that I think about it, this is starting to sound pretty similar to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Moore_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moore method&lt;/a&gt;, where the students are given lists of definitions and theorems and expected to prove them on their own and present proofs to the class. An advantage of this method is that it can lead to much deeper student learning of the content, and builds mathematical maturity by teaching research skills. Closely related difficulties are that it requires much more mathematical maturity and discipline (including on the instructor's part), and it's not practical to cover nearly as much content unless the students are very advanced and willing to put a whole lot of time into it (and maybe not even then).&lt;/p&gt;&#xA;&lt;p&gt;More broadly, inquiry-based learning (IBL) methods are becoming increasingly widespread and have been used in many classrooms with great success. It might be worth looking into various IBL models to see if one of those could accomplish what you're going for here without drawing students into a &amp;quot;pattern-in-examples&amp;quot; mindset.&lt;/p&gt;&#xA;&lt;p&gt;(As an unfortunate footnote, I feel I shouldn't mention Moore's method without noting that its namesake, Robert Lee Moore, &lt;a href=&quot;http://www.math.buffalo.edu/mad/special/RLMoore-racist-math.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;was a virulent racist&lt;/a&gt; who abused his position of power to overtly and systematically discriminate against Black students.)&lt;/p&gt;&#xA;" OwnerUserId="3505" LastEditorUserId="3505" LastEditDate="2021-04-06T16:27:22.690" LastActivityDate="2021-04-06T16:27:22.690" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12480" PostTypeId="2" ParentId="12477" CreationDate="2017-06-18T12:29:45.667" Score="4" Body="&lt;p&gt;The &lt;a href=&quot;https://www.flvs.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Florida Virtual School&lt;/a&gt; offers high school level classes in all subjects, including Algebra 2, and is very well regarded.  They're considered a school district within the Florida school system so I would expect a course from them to be considered acceptable by another public or private high school.  If you &lt;a href=&quot;https://www.flvs.net/options&quot; rel=&quot;nofollow noreferrer&quot;&gt;don't live in Florida&lt;/a&gt;, you can still take classes there but I believe they charge a fee.  I'm not sure how much it is or if it would be in your price range.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2017-06-18T12:29:45.667" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12481" PostTypeId="2" ParentId="12476" CreationDate="2017-06-18T16:48:15.353" Score="12" Body="&lt;p&gt;Daniel Hast's answer is great, but I want to add one thing:&#xA;What kind of mathematical ability do you want your students to learn?&#xA;Are you measuring that ability or something else?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen way too many students who can do some formal manipulations (solving equations, differentiating functions, or the like) but who do not understand what any of it means.&#xA;Sure, it is useful to be able to differentiate $x^{\sin(\log(x))}$, but that without any understanding of what it means to differentiate something with respect to something has little value &amp;mdash; although it is often enough to pass exams with decent grades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ability to produce correct solutions is not exactly the same things as understanding.&#xA;If a student can explain their answer so that every symbol has a meaning and every intermediate step is a meaningful statement, then understanding is likely.&#xA;But if only the final number or formula or something is correct and nothing else is evaluated or checked, be careful about your conclusions.&#xA;The key question is: &lt;strong&gt;What is a sufficient or correct solution?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might be able to embed lectures into your questions or your answers.&#xA;But trying to avoid any explicit lecturing and using only exercises does not seem easy or worthwhile.&#xA;I have never seen any material written in such a coy way, and I doubt it's possible.&#xA;(I'll be gladly proven wrong, though!)&#xA;There is also the issue that students may neglect the &quot;story&quot; and proceed with their exercises; if the answer stated on the last line is the same as they got, why read what's in between?&#xA;If you want all students to focus on theory/lecture/(whatever you call it), I strongly recommend reserving time for it explicitly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And please &lt;strong&gt;make your students not only solve problems, but also explain their solutions&lt;/strong&gt;.&#xA;To themselves, to each other, and to you.&#xA;If you can include that in your concept, it sounds far better to me.&#xA;Problem solving in any kind of isolation sounds dangerous and inefficient.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastActivityDate="2017-06-18T16:48:15.353" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12482" PostTypeId="2" ParentId="12476" CreationDate="2017-06-18T19:49:09.550" Score="14" Body="&lt;p&gt;I have been teaching students for the past 6½ years- in all levels of college undergraduate math (decent bit of physics too).  I have found that analyzing learning and all the ways to understand mathematics, in particular, is a very necessary first step.  (See above posts!)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To the OP: I believe your conclusion is correct, the process you outlined alone doesn't constitute a proper lesson.  One commonality I see that most students crave but don't always realize, is a &lt;strong&gt;narrative&lt;/strong&gt;.  Some do, and ask: how do we know these things?, why are we doing this?, etc. Narrative may facilitate a crucial step in teaching after all the analyzing of how we present mathematics is done.   After we have given the types of lectures, assignments and one-on-one advice that we think are the most effective,  we may have missed packaging it all up in a narrative.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on the context; giving an arc of where we have been and why we must do it this way, to get us to where we are going next, is a good way to motivate a desire to delve deeper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More explicitly,  you want students to be able to do the algorithmic process and also understand how or why it must be done that way.  That way ultimately they may view mathematics more axiomatically as they see a need for consistent definitions and theorems.  It is then that they will understand this is necessary in being able to know if one is correct or not logically.  I am often quite up-front in saying this is how you will learn this material for now.  I tell algebra, trig, and calculus students the things they should simply memorize for the time.  Usually with the backing: [Of course these things must not be taken for granted, but can be proven rigorously using only logic.  We will not be doing that for this course or even in your later courses.  However, at a certain level, after you have learned the &lt;em&gt;how&lt;/em&gt; you must treat with the &lt;em&gt;why&lt;/em&gt;.  This is where you will go in math.] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When there is particular confusion or curiosity I will engage in a small discussion about abstract ideas.  The further students are in math the more I think they will feel comfortable hearing that what they are doing now is just a very small piece of the whole of mathematics.  Often these abstract connections help students and serve as glue to get them engaged in what they are doing now.&lt;/p&gt;&#xA;" OwnerUserId="8369" LastEditorUserId="-1" LastEditDate="2017-06-20T07:34:49.380" LastActivityDate="2017-06-20T07:34:49.380" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12483" PostTypeId="2" ParentId="12476" CreationDate="2017-06-19T01:28:16.683" Score="9" Body="&lt;p&gt;I am an alumni of &lt;a href=&quot;https://en.wikipedia.org/wiki/Fazekas_Mih%C3%A1ly_Gimn%C3%A1zium_(Budapest)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fazekas Mihály Gimnázium (Budapest)&lt;/a&gt; and I can attest to the fact that we were educated in a problem solving manner -- although not exactly as OP describes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For four years, all we did we solved problems. We did nothing else. There were no lectures as such. The teacher provided guidance or stated definitions and theorems based on the problems we just solved. But problem solving was the backbone of all we did. On the grade 9-12 level we are talking about this worked perfectly. I know anecdotally this was extended later to grade 5-12. To emphasize, OP said &quot;He/she may not even understand what is being asked, or may attempt.&quot; -- this never was the case. The problems were definitely solvable -- provided, of course, you remembered everything you learned so far and managed to apply in just the correct way. By no means was this easy but it was doable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, you could posit a problem where you have a right triangle ABC with A having the right angle and draw the altitude AD. Now provide some lengths and ask about others. This will lead pupils to recognize the triangles ABC, DBA, and DAC are similar and in turn, this will lead to &lt;a href=&quot;http://www.cut-the-knot.org/pythagoras/&quot; rel=&quot;nofollow noreferrer&quot;&gt;proof #6 of the Pythagoras theorem&lt;/a&gt;. Disclaimer: this is not to say this was how we have proven Pythagoras for the first time -- I don't remember exactly as it was more than 25 years ago now. But it could've been this one. And we definitely had one more than proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every other theorem can be formulated as a problem in a similar fashion. I have four A4 sized several hundred pages thick exercise books to show it can be done :) If I ever wanted to go into teaching (I have a math teacher masters but I do not teach) I would use these as an example to create my lessons. I treasure these so much that I FedExed them over at considerable cost when I immigrated (with just two suitcases so these didn't fit).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more, see Bruce R Vogeli: Special Secondary Schools for the Mathematically Talented: An International Panorama which has the 1995-1996 curriculum from our school in Appendix A1.&lt;/p&gt;&#xA;" OwnerUserId="1850" LastEditorUserId="1850" LastEditDate="2017-06-20T15:07:01.297" LastActivityDate="2017-06-20T15:07:01.297" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12484" PostTypeId="2" ParentId="12477" CreationDate="2017-06-19T04:36:22.697" Score="5" Body="&lt;p&gt;This won't help the OP for the summer of 2017, but let me put this here for future readers of this site:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.mivhs.org/high-school-courses&quot; rel=&quot;noreferrer&quot;&gt;Michigan Virtual High School&lt;/a&gt; is (as I write these words) developing its own online Algebra 2 curriculum.  The curriculum, which is expected launch in September 2017, consists of two separate half-courses (Algebra 2A and Algebra 2B) which can be taken independently.  I think very highly of the Algebra 2B curriculum, as I happen to the person writing it. &lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2017-06-19T04:36:22.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12485" PostTypeId="2" ParentId="12476" CreationDate="2017-06-19T08:45:39.370" Score="2" Body="&lt;p&gt;I'm seeing a drawback in this kind of approach: imagine you are teaching quadratic equations to a class, and you give them following examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;x^2-5x+6=0 (solution 2 and 3)&lt;/li&gt;&#xA;&lt;li&gt;x^2-4x+4=0 (solution 2 appears twice)&lt;/li&gt;&#xA;&lt;li&gt;x^2-4x+5=0 (no real solutions)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Keep out: the second one is an equation with two solutions (the multiplicity of the solution equals 2).&lt;/p&gt;&#xA;&lt;p&gt;I believe that students will think that a quadratic equation has either zero real solutions, either one, either two, but in fact, quadratic equations have either zero real solutions or two, but the two solutions might coincide.&lt;/p&gt;&#xA;&lt;p&gt;I don't think that students are capable of inventing the notion of solution multiplicity by themselves (although this becomes very important while working with higher level polynomial equations and complex numbers), therefore I'd say that a minimum level of theory is always required.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastEditorUserId="8190" LastEditDate="2021-04-12T10:46:30.460" LastActivityDate="2021-04-12T10:46:30.460" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12486" PostTypeId="1" CreationDate="2017-06-19T08:51:44.140" Score="2" ViewCount="99" Body="&lt;p&gt;I am working on a project where I have to group/classify calculus problems. Now with most the calculus topics, it's usually obvious how it's divided in various textbooks, but when it comes to related rates, all the textbooks I checked don't divide it further.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking of classifying the problems depending on whether it's &quot;related time rate problem&quot;, &quot;related rate problem&quot;, and on the geometrical shape/rule involved in the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I was wondering if there is a more standardized classification, a classification based on a textbook or if you have better suggestions than the ones I mentioned.&lt;/p&gt;&#xA;" OwnerUserId="8241" LastActivityDate="2017-06-19T08:51:44.140" Title="what is the standard subdivision or classification of calculus related rates problems?" Tags="|mathematical-pedagogy|calculus|concept-maps|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12487" PostTypeId="2" ParentId="6031" CreationDate="2017-06-19T10:44:45.100" Score="3" Body="&lt;p&gt;This is yet another variant of the ac method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3x^2 -14x - 5 = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solve the &quot;companion&quot; quadratic ($ax^2+bx+c=0 \to x^2+bx+ac=0) $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x^2 -14x - 15 = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(x-15)(x+1) = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x \in \{15, -1\}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roots of original polynomial are above roots divided by $a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{Roots are &#xA;        $\left\{\dfrac{15}{3}, -\dfrac{1}{3}\right\} &#xA;        = \left\{5,  -\dfrac{1}{3}\right\}$}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reconstruct factors&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(x-5)(3x+1)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;PROOF&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$ax^2 + bx + c = 0$ has roots $r=\dfrac{-b + \sqrt{b^2-4ac}}{2a}$ and&#xA;$s=\dfrac{-b - \sqrt{b^2-4ac}}{2a}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$x^2 + bx + ac = 0$ has roots $r'=\dfrac{-b + \sqrt{b^2-4ac}}{2}$ and&#xA;$s'=\dfrac{-b - \sqrt{b^2-4ac}}{2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that $r = \dfrac 1ar'$ and $s = \dfrac 1as'$&lt;/p&gt;&#xA;" OwnerUserId="5144" LastEditorUserId="5144" LastEditDate="2017-06-19T19:07:38.620" LastActivityDate="2017-06-19T19:07:38.620" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12489" PostTypeId="2" ParentId="12476" CreationDate="2017-06-19T20:17:30.260" Score="5" Body="&lt;p&gt;What looks to be missing is teacher interaction.  The student is interacting with a workbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So where is the learning occurring?  The student &lt;em&gt;may&lt;/em&gt; learn something while exploring the problem.  However, learning the difficult things this way is pretty difficult.  Few people &quot;discover calculus&quot; on their own because some problems were put in their way.  So that means most of the learning is occurring during the phase where they are reading the solution and related material.  For this to be a &quot;good&quot; learning phase, you would have to somehow have constructed the material presented to cover the things the student needs to know, despite not having interacted with the student before that part of the workbook was written.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The approach does remind me greatly of the Mr. Miyagi approach to teaching:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Wax some cars I needed waxed&lt;/li&gt;&#xA;&lt;li&gt;Paint a fence I needed painted&lt;/li&gt;&#xA;&lt;li&gt;Stain a deck I needed stained&lt;/li&gt;&#xA;&lt;li&gt;Now I'll show you that what you did was Karate&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;However, the Eastern approach to teaching martial arts is very different from that of Western school teaching!  The concepts that are taught in school don't always transfer well to an alien form of teaching.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2017-06-19T20:17:30.260" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12491" PostTypeId="2" ParentId="12396" CreationDate="2017-06-20T13:47:00.283" Score="2" Body="&lt;p&gt;One could hazard the guess that this question is precisely the one that led to Book X of Euclid's &lt;em&gt;Elements&lt;/em&gt;, the famous &lt;em&gt;crux mathematicorum&lt;/em&gt; (i.e. the &quot;Cross of the Mathematicians&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I do agree with the rejoinder that &quot;$\sqrt{2}+\sqrt{3}$ is most definitely a number&quot;, and it is a very useful thing to tell the student this, it's still not entirely right in a pedagogic sense. First off, a student is not wrong to wonder about the fact that, apparently, $\sqrt{2}+\sqrt{3}$ can't be simplified. My guess would be that, subconsciously, most students just &lt;em&gt;assume&lt;/em&gt; that $\sqrt{2}+\sqrt{3}$ can be simplified, until set straight that is, just as all comparable types of expressions encountered up to that point (like $\sqrt{2} \times \sqrt{3}$) can be simplified. The moment where a student asks why $\sqrt{2}+\sqrt{3}$ &quot;is not a number&quot;, that's exactly the moment the student realizes he or she has been making this assumption all along. This is a valuable moment in the learning process, and one shouldn't throw it away by simply &quot;correcting their mistake&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here's what I would not do: I would not suggest to the student that he or she was wrong to say $\sqrt{2}+\sqrt{3}$ is not a number (after all, a precise definition of &quot;number&quot; usually hadn't been given anyway, although students would probably agree that the sum of two numbers is again a number), and I would also not go so far as to say or suggest that what he or she &lt;em&gt;should&lt;/em&gt; have asked was something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The latter point is especially important to my taste. It is not easy to give a precise formulation to one's deepest thoughts straight-away, mathematical or otherwise; on the contrary, the most interesting observations can grow out of the stupidest-sounding questions. If a student is discouraged to give voice to their intuitions (which could easily happen after meeting with the remark that &quot;what you really meant to ask was X&quot;, which is usually said for the benefit of the teacher anyway, who gets to show off their own unusual talent for reframing a pointless question into a meaningful one), simply out of a fear to be corrected, then the learning process is stifled.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Perhaps I should also add what my answer to your student would be, because it is such an excellent question (even if it isn't phrased in the best of all possible ways): &quot;Why is $\sqrt{2}+\sqrt{3}$ not a number?&quot; I think I would start by saying that there is no why to it, it is simply that way and no other. It's just a matter of biting the bullet. (Asking for a reason, they are running into the fact that they were expecting something that they didn't have a reason to expect -- you don't have to &lt;em&gt;tell&lt;/em&gt; them this, but this is what they will take from it.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also think it is very useful to square the expression, just as G Tony Jacobs suggested, to show that it isn't a square root of something simple. But this approach is also kind of limited, because it doesn't exactly show there isn't some other simple form that it could take.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there are other difficulties here, and the main problem is that this is the type of question that opens up such a gigantic can of worms, that you simply can't do justice to it in the space of say ten minutes, or even a whole hour. Because one could equally well ask: why can't we write $\sqrt{2}$ any simpler? And there you've run into the irrationality of square roots of non-square integers. And if you want to prove that $\sqrt{2}+\sqrt{3}$ is not of the form $a+\sqrt{b}$, with $a$ and $b$ integers, that's a similar type of problem, but only much more messy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would go so far as to say that the only really satisfying way to answer this is to phrase these facts in the language of (basic) algebraic number theory. And you simply can't do that at their level. So you will have to treat the question as a moment where the student is discovering something that they won't be able to &lt;em&gt;really&lt;/em&gt; get their head around unless they're going to do some serious mathematics. And it's not a bad thing if they decide not to do that. But it is good to leave them with a sense of wonder about the issue, and not take that away with some kind of facile reponse, because it is only right that they should feel this wonder.&lt;/p&gt;&#xA;" OwnerUserId="8387" LastEditorUserId="8387" LastEditDate="2017-06-20T14:23:13.653" LastActivityDate="2017-06-20T14:23:13.653" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12492" PostTypeId="2" ParentId="12477" CreationDate="2017-06-21T01:40:22.593" Score="5" Body="&lt;p&gt;The folks at &lt;a href=&quot;https://artofproblemsolving.com/school/course/intermediate-algebra&quot; rel=&quot;noreferrer&quot;&gt;Art of Problem Solving&lt;/a&gt; have what you need. It's not cheap ($559), but they work with many students like you, and are highly recommended. I think you will find it worth what it costs, if you can afford that. (And if not, ask them for a scholarship. I don't know if they'd do it or not...)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-06-21T01:40:22.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12493" PostTypeId="1" CreationDate="2017-06-21T21:42:35.603" Score="6" ViewCount="465" Body="&lt;p&gt;I'd like to hear some ideas for problems or projects related to this summer's solar eclipse, in particular for the high school or undergrad level (algebra through calculus).&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="2083" LastEditDate="2017-06-22T09:06:30.230" LastActivityDate="2017-06-22T19:13:31.567" Title="What are good solar eclipse questions/projects for students?" Tags="|undergraduate-education|secondary-education|homework|applications|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12494" PostTypeId="2" ParentId="12493" CreationDate="2017-06-21T22:38:45.973" Score="2" Body="&lt;p&gt;This could be on the difficult side, but the story of &lt;a href=&quot;https://www.wired.com/2009/05/dayintech_0529/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eddington's 1919 confirmation of Einstein's general theory of relativity&lt;/a&gt; comes to mind. This was done on the basis of measurements made during a solar eclipse, using the distorted apparent positions of stars near the sun. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are relatively elementary accounts of this, and there are multiple angles. (For example, there is a story in the literature that the measurement error was too large for the experiment to yield any meaningful conclusions as to the validity of GR; it would be fun to run a check on this -- if this is at all feasible in a student project of course.)&lt;/p&gt;&#xA;" OwnerUserId="8387" LastActivityDate="2017-06-21T22:38:45.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12495" PostTypeId="2" ParentId="12476" CreationDate="2017-06-21T23:12:54.113" Score="4" Body="&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can one learn math only by solving problems? Technically yes, practically it depends. Is it a good idea? That cannot be answered without considering why one wants to learn math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Babbling warning&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have been warned, proceed at your own risk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Disclaimer I:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is another counterpoint to the (currently) highest voted answers. It should come as no surprise that I am heavily biased in favor of the problem-based approach, because that was how I learned math as a kid. Please adjust the size of your &quot;pinch of salt&quot; accordingly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Disclaimer II:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was mulling over the current answers and I have to admit (I guess mostly due to my bias) that a few rub me the wrong way. Someone is bad-mouthing good ol' problem-approach that pushed forward math understanding for hundreds of years – I had to do something. Still, when I tried to find any concrete data that would concradict them, nothing came up. But then, why I am so confident that they are wrong? Then I realized that perhaps we understood the OP's question differently and that ambiguity is the source of disagreement. The rest is not actually an answer, but a number of related comments that the reader (I hope) will still find useful in the context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Two kinds of math work:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a gross oversimplification, but that distinction will be useful here, so let us split the mathematicians into two groups:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Mathematicians that apply already-known math to some applications (accountants, engineers).&lt;/li&gt;&#xA;&lt;li&gt;Mathematicians that invent new math results (math researchers).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Of course, realiy is more complex and one kind could not exist without the other, but I am doing this to emphasize difference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;On the meaning of progress:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me ask a question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is our ultimate goal of teaching math?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are lots of possible answers, to give a few:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To make the world better (or worse).&lt;/li&gt;&#xA;&lt;li&gt;To push human civilization forward (backward).&lt;/li&gt;&#xA;&lt;li&gt;To make people happier (or more miserable).&lt;/li&gt;&#xA;&lt;li&gt;To improve probability of human survival (or delay our inevitable demise).&lt;/li&gt;&#xA;&lt;li&gt;To undersand the world better (or just the theory rather than the world).&lt;/li&gt;&#xA;&lt;li&gt;To create something beatiful (or too complicated/abstract/meta).&lt;/li&gt;&#xA;&lt;li&gt;To earn a living (or take some jobs away from people).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Whatever our personal goal is, the best proportion of math work (math application vs. math research) differs. We can't answer which learning approach is better until we know what our ultimate goal is, and depending on that goal the problem-based approach of learning could be good or completly bad.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;On &quot;lectures&quot; in problem-based learning:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To confuse the matter even more, let me observe that even in problem-based learning there are &quot;content-giving&quot; phases like explaining the solution, etc., which in extreme sense might become plain lectures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, many math courses end with exams and so the whole thing could be viewed as problem-based approach (the problems were on the exam) with too long explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, the distinction is far from precise and some common sense is needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;On problem-based learning:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since I remember math for me was always a tool to do something – this is how I learned math. Behind everything I have learned there was some problem to solve, even in normal highschool math classes I would solve problems from math circle. Only at the university level I got some non-problem-based education in form of lectures. Many friends I had at the university or from math circle and various competitions had similar experience. This is only anecdotal data, but I will state something stronger: every single person I know that produces good math research (and I know anything of how that person learned) had learned math mostly on problems. Furthermore, it might be the best way to teach theoretical computer science (at least it got me the best results of everything I've tried).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please observe that with math research things rarely are precisely stated, many concepts may lack definitions (creating good definitions is a crucial step) and instead of reasoning we rely on intuition. First drafts of the proofs are usually not coherent and lots of research starts without clearly stated goals. Also, in math research using known algorithms not always yields results and mathematicians have to infer patterns from (sometimes abstract or only partially-formed) examples. These and similar soft skills are essential in math research and should be trained. For me they are more important than just knowledge, which can be picked-up along the way with a deeper understanding than non-problem-based approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I want to say is that problem-based learning seems excellent if we want to train a research mathematician. It probably is one of the things that caused me to do math research. It might be worse for non-research math, but I have yet to see a conclusive results in this matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;On math metrics:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One problem with evaluating teaching method quality is what qualiy metric to use. First, it depends on our &quot;ultimate math-teaching goal&quot;, and even if that was agreed on, the right metric might not exist or be impossible to measure. Do we take imperfect test scores? Or try to gaguage deep understanding? Math fluency after 10 years? Number of succesful math applications? Number of theorems our society deem great in the next century?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see lots of reasons why non-problem-based approach could be potentially better for people who will not become research mathematicians.&#xA;However, gathering any data that would truly be convincing seems impossible. Similarly, proving superiority of problem-based teaching also looks hopeless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;On some drawbacks of problem-based teaching:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be obvious, but let me emphasize it here: a good sequence of problems is crucial for the problem-based teaching to work. In particular each student may need a different sequence of problems to reach deep understanding or even succeed. Frequently I found that I have to adjust problems on-the-fly to make them both interesting and managable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, one has to anticipate students' mental models to provide them with tasks that will help them learn, which is hard without years of experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problem-based approach can also be really frustrating and exhausting for both the student and the teacher – strong motivation (or encouragement) is necessary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problem-based teaching works really well for students that are naturally curious esp. if they reached a higher level of mathematical maturity. Unfortunately, according to my experience, without curiosity (or if attention is on something else) it works much worse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taking that into account, any research that would evaluate problem-based approach needs to get all the above issues right, but then it becomes even more impossible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own experience suggest that problem-based teaching is excellent in producing research mathematicians. Still:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It is hard to teach using problems well (probably not every teacher can do it).&lt;/li&gt;&#xA;&lt;li&gt;It does not work well for every student.&lt;/li&gt;&#xA;&lt;li&gt;Teaching people to be research mathematicians when they don't need it seems like a waste of time and effort.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Can math be learned only by solving problems? Technically yes, but that might be a really bad choice. Among others that depends on:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How rigid your &quot;only&quot; is and how efficient with your learning you want to be.&lt;/li&gt;&#xA;&lt;li&gt;Your ultimate goal for learning math and whether you want to become a research mathematitian.&lt;/li&gt;&#xA;&lt;li&gt;Whether you have patience and strong motivation.&lt;/li&gt;&#xA;&lt;li&gt;Whether you have a problem set that is tailored to you and dynamically adapts with your growth (read: whether you have a teacher that is willing to invest a lot of time and effort in teaching you).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope this helps $\ddot\smile$&lt;/p&gt;&#xA;" OwnerUserId="42" LastActivityDate="2017-06-21T23:12:54.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12496" PostTypeId="2" ParentId="12493" CreationDate="2017-06-22T01:00:50.677" Score="3" Body="&lt;p&gt;I would myself be interested to know what is the algebraic form of the equation&#xA;of the curve the shadow follows.&#xA;Perhaps in a spherical coordinate system centered at the Earth's core?&#xA;Or in some convenient coordinate system.&#xA;How far does the path deviate from a geodesic?&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/e63vg.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/e63vg.jpg&quot; alt=&quot;Epath&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;To more directly address your question: What is the great-circle path between&#xA;the shadow's West-coast and East-coast endpoints, Portland to Charleston?&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-06-22T01:00:50.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12497" PostTypeId="2" ParentId="12421" CreationDate="2017-06-22T05:39:53.847" Score="2" Body="&lt;p&gt;In the context of programming, it seems more natural to distinguish recursion from iteration than from induction, as iteration is the algorithmic realization of iteration. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the context of mathematical proof, recursion becomes something like the method of infinite descent (e.g. the classic proof of the irrationality of $\sqrt{2}$), and is quite natural for proving something like the Euclidean algorithm for the greatest common divisor. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In either context, students struggle with the notion that requires consideration of a potentially infinite process. The principle of induction asserts something psychologically difficult, that showing that case $i$ implies case $i+1$ shows something for all $n$, no matter how big. The basis of recursion is that some sort of backwards iteration cannot continue infinitely downwards. Students are accustomed to considering a single step, and nothing more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether it makes more sense to emphasize descent/recursion or induction/iteration surely depends on the educational context and goals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The recursive approach to understanding Euler's method may be important conceptually for understanding the mathematics of unique factorization in a principal ideal domain. My experience in basic programming classes is that students initially have a lot of difficulty programming it recursively. Their naive approach is the brute force iterative check (to find the gcd of $a &amp;lt; b$, start dividing both by $a$, then $a-1$, etc.). The iterative approach is more accessible, and for many students it probably needs to be tried first, before the recursive approach can be understood. It's important for students to digest that such apparently different approaches resolve the same problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a programming context, recursion requires understanding that a function (or some similar construct) can call itself and this is difficult for many students who have never (well) understood the idea of functions in any context. An iterative procedure is by its nature step-by-step, and this is how most students have been taught. More fundamentally, it requires (in practice) no operational functional concept, just repetition of a set of instructions with a change of a parameter. Also, in practice recursion can involve dealing with issues of memory use (the stack) that require more sophistication (from a programming point of view). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I wonder if in mathematically oriented programming contexts recursion is sometimes over emphasized (probably because of its aesthetic appeal). From a mathematical point of view the issues of computational efficiency and complexity are important, both pedagogically and practically, and often (always?) iterative implementations can both be made faster in execution and more parsimonious with memory than recursive implementations. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2017-06-22T05:39:53.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12498" PostTypeId="2" ParentId="12493" CreationDate="2017-06-22T19:13:31.567" Score="4" Body="&lt;p&gt;I highly recommend watching Terry Tao's lecture:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=7ne0GArfeMs&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=7ne0GArfeMs&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It contains several ideas which could be used to combine astronomical observations with basic trig and geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first experiment (finding the radius of the Earth) would be a bit hard, but you could certainly have them time how long it takes the moon to set, how long it takes the sun to rise, how long a lunar eclipse lasts (if you are lucky enough to have one), how long a solar eclipse lasts, observing that (by pure coincidence!) the moon almost perfectly covers the sun in a solar eclipse, etc.  Then you could use these facts to determine the distance to the moon, the radius of the moon, etc.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2017-06-22T19:13:31.567" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12499" PostTypeId="1" AcceptedAnswerId="12505" CreationDate="2017-06-22T23:02:14.970" Score="6" ViewCount="243" Body="&lt;p&gt;For research about cognitive information retention, I'm trying to find a scientific reference where they explain the difference between the capability to apply real mathematical skills (entailing symbols, patterns, the works basically) and the capability to follow simple recipes as in normal arithmetic or calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I'm not talking about arithmetics or calculus, but more like anything that proves which kind of skills or insights you need in order to be good at &quot;real&quot; mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that I keep bumping into articles where people just build on assumptions and I really need to find it soon. Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="8395" LastEditorUserId="11" LastEditDate="2017-06-23T17:46:56.113" LastActivityDate="2017-06-25T00:54:22.723" Title="Need scientific source to prove the difference between arithmetics/calculus and real mathematical skills" Tags="|reference-request|education-research|reference-books|meta-cognition|resource-request|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12500" PostTypeId="2" ParentId="8586" CreationDate="2017-06-23T02:01:39.693" Score="3" Body="&lt;p&gt;In addition to the other good answers... and as they say... &quot;motivation&quot; and &quot;application&quot; are not necessarily the same, at all, so to accidentally/implicitly impose on students/listeners the idea that they &lt;em&gt;are&lt;/em&gt; is confusing and dis-crediting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that, one might (as I would) further attempt to debunk/disclaim the apparent distinction between a mathematical idea and its &quot;application&quot;. Although the attempt at such a distinction has historical and philosophical interest, I do seriously think it is misguided (although sanctified in many standard mythologies of mathematics and its history). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(And then there is the non-trivial issue of bad teaching, bad communication, ... The trouble this causes is endless. It might be useful for any specific individual to try to ascertain the extent to which their attitudes and tastes about (external? Objective?) mathematics was in fact strongly influenced, or entirely determined, by good/bad, hostile/friendly teachers in related courses.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After some of that noise is sorted through, I'd like to make the claim that there is no meaningful distinction between &quot;theory&quot; and &quot;applications&quot;, although there are many reasons that people would like to claim so. That is, although at a naive level it is plausible that there could be significant distinctions between &quot;applications-oriented&quot; and &quot;theoretical&quot; mathematics, upon closer inspection the claimed distinctions are ... content-less... unless one grants the reality of one of several possible wildly simplistic world-views. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-06-23T02:01:39.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12501" PostTypeId="2" ParentId="12096" CreationDate="2017-06-23T04:35:36.323" Score="3" Body="&lt;p&gt;I have cerebral palsy, which makes me almost unable to write legibly by hand. (I can do a bit of scratch work, but that's it.) Discovering &lt;a href=&quot;http://www.lyx.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;LyX&lt;/a&gt; changed my life. LyX is basically a user-friendly front end for LaTeX that doesn't require you to memorize commands (you memorize what you use a lot, but that's it) and shows you your equations as you type them. I've used it to write papers and large parts of a textbook in mathematical biology. It's awesome, free, and will run on any operating system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For drawing and graphing, &lt;a href=&quot;https://www.geogebra.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;GeoGebra&lt;/a&gt; is terrific. What I particularly like is that you can go back and forth between drawing and using commands. It's also free and available for all operating systems. There are even mobile versions. This is both an adaptive technology and a teaching tool. I wish this kind of software had been available when I was a kid!&lt;/p&gt;&#xA;" OwnerUserId="4627" LastEditorUserId="4627" LastEditDate="2017-06-23T06:35:21.927" LastActivityDate="2017-06-23T06:35:21.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12502" PostTypeId="2" ParentId="12499" CreationDate="2017-06-23T15:32:14.410" Score="-1" Body="&lt;p&gt;I think there is a strong belief in the people who comment here that proof skills are more important than calculational skills.  But I am not so clear that this is really true, is not completely how the human mind works.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mean does understanding Dedekind cuts (or whatever they are called) really help you do arithmetic?  If you learned that and never learned long division, versus someone who did the converse, who would be more powerful if you had to calculate something?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some manipulative skill can be very helpful when doing trials or developing new mathematics.  And the algorithms and methods developed over the years for arithmetic, algebra, and calculus have a lot of power in them.  Represent a lot of distilled experience and effort to make things clear and simple and useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally you should have both.  Understanding of concepts and manipulative skill.  And move back and forth between them.  And have them inform each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember there a lot more people who need to understand math like Feynman did so they can follow physics derivations or do engineering problems than there are who need to play Bourbaki.  And even for the Bourbaki soldiers, some combat with manipulation will make them better mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This doesn't mean there is no value to concepts either.  They can help with remembering methods, can help with understanding them, etc.  But having a toolbag of tricks is valuable too.  Don't look down on it.  &lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-23T15:32:14.410" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12504" PostTypeId="2" ParentId="11126" CreationDate="2017-06-23T17:15:06.293" Score="2" Body="&lt;p&gt;I think you are doing too much and too different.  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;My advice is to do a standard diffyqs survey.  the class will be mixed, but mostly engineers.  They need to do some manual calculations so they really get a feel for things before disappearing into Mathematica world.  This will help them in upper division engineering or physics courses where a lot of time spent on derivations on the blackboard or where they need to show work in homework.  (If they end up being a very particular sort of design engineer, not most of them, or doing Ph.D. research, they may use more Mathematica then.  But this is a good chance to have a feel for things.  Every painter needs to grind his pigments once...)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For Laplace, I would just show them the basics and use a table.  It gives an exposure to all the engineers.  Only the EEs (or maybe systems Es) need to go in depth on Laplace.  But they will get that more detail in their courses.  For this first exposure, I would keep it gentle.  And the table helps to have some feel for things more than a computer black box.  Yes, they may not get the conceptual blabla from the table.  But they won't get that from the black box either.  Won't get it even if you lecture on it.  Keep things simple and make a basic exposure.  For the few who need deeper understanding (on concepts, letalone methods), they can get that later in a more tailored fashion.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-23T17:15:06.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12505" PostTypeId="2" ParentId="12499" CreationDate="2017-06-25T00:54:22.723" Score="3" Body="&lt;p&gt;I think the OP asks a difficult question that will not have a succinct answer.&#xA;Permit me to point to one publication,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Schoenfeld, Alan H., ed. &lt;em&gt;Assessing Mathematical Proficiency&lt;/em&gt;. Mathematical Sciences Research Institute Publications. &lt;strong&gt;53&lt;/strong&gt;. Cambridge University Press, 2007. (&lt;a href=&quot;http://www.msri.org/people/staff/levy/files/Book53/book53-all.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download of book&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;which addresses the question of what constitutes &lt;em&gt;mathematical proficiency&lt;/em&gt;&#xA;in at least two chapters:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;James Milgram &amp;quot;What is Mathematical Proficiency?&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Alan Schoenfeld. &amp;quot;What Is Mathematical Proficiency and How Can It Be Assessed?&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Let me quote one passage from Milgram's chapter (p.33):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Realistically, in describing what mathematics is, the best we can do is to&#xA;discuss the most important characteristics of mathematics. I suggest that these&#xA;are:&lt;br /&gt;&#xA;   (i) Precision (precise definitions of all terms, operations, and the properties of&#xA;these operations).&lt;br /&gt;&#xA;   (ii) Stating well-posed problems and solving them. (Well-posed problems are&#xA;problems where all the terms are precisely defined and refer to a single universe&#xA;where mathematics can be done.)&#xA;&lt;br /&gt;&#xA;It would be fair to say that virtually all of mathematics is problem solving in&#xA;precisely defined environments, and professional mathematicians tend to think&#xA;it strange that some trends in K–12 mathematics education isolate &lt;em&gt;mathematical&#xA;reasoning and problem solving&lt;/em&gt; as separate topics within mathematics instruction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A very crude summary is that the OP's &amp;quot;capability to apply real mathematical skills&amp;quot; requires an appreciation and understanding of &lt;em&gt;precision&lt;/em&gt;, and a facile skill at &lt;em&gt;problem solving&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-06-25T00:54:22.723" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12506" PostTypeId="2" ParentId="4119" CreationDate="2017-06-25T05:51:29.733" Score="1" Body="&lt;p&gt;Here's a good example to show how beautiful math is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://touch-geometry.karazin.ua/sitemap&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://touch-geometry.karazin.ua/sitemap&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/00rAn.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/00rAn.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8236" LastActivityDate="2017-06-25T05:51:29.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12507" PostTypeId="1" CreationDate="2017-06-25T07:06:04.950" Score="5" ViewCount="431" Body="&lt;p&gt;Concerning the structure of the learner's mind, psychologist Piaget claimed that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There exists, as a function of the development of intelligence as a whole, a spontaneous and gradual construction of elementary logico-mathematical structures and that these 'natural' ('natural' the way that one speaks of the 'natural' numbers) structures are much closer to those being used in 'modern' mathematics than to those being used in traditional mathematics. (p. 79 in Piaget 1973). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Piaget appears to postulate an affinity between, on the one hand, the structures of the mind and, on the other, the structures of modern mathematics (mainly following Bourbaki). The essay in question is &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Piaget, J. &quot;Comments on Mathematical Education,&quot; in A. G. Howson, ed., Developments in Mathematical Education: Proceedings of the Second International Conference on Mathematical Education, 79--87, Cambridge: Cambridge University Press, 1973 &lt;a href=&quot;http://search.proquest.com/openview/fde80c4bce085d4925c0bafb81853b39/1?pq-origsite=gscholar&amp;amp;cbl=1816594&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Piaget's postulated affinity has apparently been challenged by some scholars in the context of the &lt;em&gt;New Math&lt;/em&gt; controversy. Is there a source that provides a detailed analysis of such a postulation? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note that I am &lt;em&gt;not&lt;/em&gt; looking for general sources on the &lt;em&gt;New Math/Modern Math&lt;/em&gt; controversy, but rather for an analysis of this particular identification). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just came across a book that might be relevant: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;All Positive Action Starts with Criticism: Hans Freudenthal and the Didactics of Mathematics. By Sacha la Bastide-van Gemert. Springer, 16 Jan 2015 &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here the author quotes Freudenthal as follows on page 211: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It thus did not begin with the Sputnik shock. It had already begun in the early 1950s. They had even managed to convince Piaget, who did not understand anything of it except for the fact that the word &quot;structure&quot; appealed to him. With Piaget's name on the billboard they felt confident of the support of psychology. What now, psychology! Mathematics is ruled by a logical order and he who teaches mathematics is easily seduced to sacrifice the psychological, the educational order to the logical order. I have done my utmost to avoid this and in my 'fragment Rekendidactiek' of 1942, if not earlier, I wanted to warn others. But what was now happening before my very eyes? A logical order brought to ecstasy, a systematic of mathematics as a whole--that is how mathematics should be taught. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is clear from this that Freudenthal was sceptical of these developments but unfortunately he does not elaborate the details of his objections.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-06-27T13:47:12.343" LastActivityDate="2017-06-30T02:01:27.160" Title="Source for analysis of identification of structures in learner's mind and mathematical structures?" Tags="|education-research|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12508" PostTypeId="2" ParentId="995" CreationDate="2017-06-25T12:45:46.503" Score="1" Body="&lt;p&gt;Answer:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can do something graphical where they use &quot;box counting&quot; to determine the integral.  For instance an electrical current that has a set of triangles on top of a base current.  (for every t unit increment, I goes from 1 to 2 in a line and then drops immediately to 1.  [the average current is 1.5 per t.]  Obviously they won't know a formula for that weird sawtooth graph.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want something &quot;different&quot; to the normal applications, do some box counting with a peak oil bell curve shape.  See the 1955 or maybe 1952 paper by Hubbard for a nice discussion.  (And FWIW, I'm not pushing any peak oil slant.  If you want to show failure of peak oil predictions give them the US natural gas prediction and then production to date.  But the nice thing about the Hubbard paper is the explication of the math.  Even someone who does not know calculus can follow it super easy.  And just the boxes are super well drawn...I know it sounds silly, but it makes it seem simple.)  The one thing to watch out for with this sort of answer though is you need to get an estimate versus the triangle graph where it can be exact.  Maybe tell the kids &quot;to the closest TCF&quot; (or whatever the unit is).  Also knowing the range can help your grading later!  Also, this is a little more sophisticated as you ask them to estimate, show a Gaussian and they think a little more about needing a formula.  Than the sawtooth, where it is a quicker insight to count boxes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comment:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think having the kids learn formulas, bag of tricks, etc. is also useful.  I would not replace the &quot;toolkit&quot; approach with an emphasis on integration fundamental.  Instead make the sort of thing that you want additive and enriching.  But the bag of tricks and knowing how to find the hammer for the nail and screwdriver for the screw and even to combine tools when needed is also powerful.  And important for engineers and physicists.&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-25T12:45:46.503" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12509" PostTypeId="2" ParentId="12507" CreationDate="2017-06-25T19:03:25.217" Score="5" Body="&lt;p&gt;This is not on general New Math controversy but more on Piaget.  I suggest reading a few articles in general about issues with his assertions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a simple overview, but several more are available by Googling Piaget and criticism.  Some of the comments may indirectly or directly relate to your questions on development of the mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See here:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.psychologicalscience.org/observer/jean-piaget#.WVAHDGfrvIU&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.psychologicalscience.org/observer/jean-piaget#.WVAHDGfrvIU&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The second criticism of Piaget concerns the nature of the stages themselves. Stage theories of development have fallen out of favor in developmental research. Although stages are seen as useful heuristics for describing the trajectory of human behavior, several problems have pushed stage theories aside. One problem is that stages often fail to capture the complexities of intraindividual and interindividual variation in development. Furthermore, in the case of Piaget’s theory, intelligence is now viewed more as a modular system than as a unified system of general intelligence. In Piaget’s system, once an individual has mastered an overall intellectual operation, it should apply across all domains of the mind, as a “structured whole” (or “structure d’ensemble,” to use Piaget’s phrase). Contemporary developmental approaches to intelligence consider modules of the mind to be largely independent, and do not assume a structured whole.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest that consideration will show that it is common that learners or even yourself does not immediately apply an intellectual structure (say in EE) to a problem in a domain that seems different (say econ).  Of course we do have the ability to do this at times.  But it is an aspect of general intelligence and not infallible.  This is one reason why word problems are trickier than non-word problems.  &lt;/p&gt;&#xA;" OwnerUserId="8403" LastEditorUserId="1385" LastEditDate="2017-06-27T15:48:13.860" LastActivityDate="2017-06-27T15:48:13.860" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12510" PostTypeId="2" ParentId="4119" CreationDate="2017-06-25T19:12:00.377" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Octacube_(sculpture)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Octacube_(sculpture)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also check out the woodcuts for De re Metallica.&lt;/p&gt;&#xA;" OwnerUserId="8404" LastActivityDate="2017-06-25T19:12:00.377" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12511" PostTypeId="2" ParentId="4119" CreationDate="2017-06-25T20:00:38.387" Score="3" Body="&lt;p&gt;Since this question has been bumped up, I recently learned (via an AMS &lt;a href=&quot;https://twitter.com/amermathsoc/status/877617646635593728&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;tweet&lt;/strong&gt;&lt;/a&gt;) about the work done by Crockett Johnson -- best known as the creator and illustrator of &lt;em&gt;Harold and the Purple Crayon&lt;/em&gt; -- at the intersection of &quot;artistic works&quot; and mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More information can be found in the write-up from &lt;a href=&quot;http://www.atlasobscura.com/articles/crockett-johnson-math-art-paintings-harold-purple-crayon&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Atlas Obscura&lt;/strong&gt;&lt;/a&gt;; here are two sample excerpts, along with images, that I found particularly interesting. The first is on the Pythagorean Theorem, and the second is on an alternative proof [to, among others, that of Archimedes] that the regular heptagon is constructible, which was published in a 1975 issue of &lt;em&gt;The Mathematical Gazette&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Johnson, C. (1975). A construction for a regular heptagon. &lt;em&gt;The Mathematical Gazette, 59&lt;/em&gt;(407), 17-21.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/aPGK0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aPGK0.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OBC6l.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OBC6l.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2017-06-25T20:00:38.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12512" PostTypeId="2" ParentId="4119" CreationDate="2017-06-25T20:12:36.487" Score="1" Body="&lt;p&gt;Popular fractal images (paisley looking things.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also this is maybe more applied math or data science, but a lot of the Edward Tufte book images are compelling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Edward_Tufte#/media/File:Minard.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Edward_Tufte#/media/File:Minard.png&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-25T20:12:36.487" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12513" PostTypeId="2" ParentId="995" CreationDate="2017-06-26T04:59:28.743" Score="2" Body="&lt;p&gt;Integration is everywhere there is &lt;strong&gt;superposition&lt;/strong&gt;. One very nice class of examples comes from &lt;em&gt;electrostatics&lt;/em&gt;. In particular, if we set the zero for voltage at $\infty$ then the tiny voltage $dV$ at $\vec{r}$ due to a tiny charge $dQ$ at $\vec{r}'$ is given by the &lt;strong&gt;Coulomb Potential&lt;/strong&gt;&#xA;$$ dV = \frac{dQ}{4\pi \epsilon_o \| \vec{r} - \vec{r}' \|}$$&#xA;If we're given a bunch of $dQ$'s over some reasonably nice geometric object then we can use integration to find the net voltage for such a collection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example: ring of charge&lt;/strong&gt; if we have total charge $Q$ around ring of radius $R$ in the $xy$-plane (aka $z=0$ plane) then at the origin it's not bad to calculate $V$. We have $\vec{r} = (0,0,z)$ and using $s = \sqrt{x^2+y^2}$ and $\theta$ for the usual standard angle, the ring of charge is at $s=R$ and $z=0$ and &#xA;$$ dV =\frac{dQ}{4\pi \epsilon_o\| \vec{r} - \vec{r}'\|} = \frac{dQ}{4\pi \epsilon_o \sqrt{R^2+z^2}} $$&#xA;since every $dQ$ is distance $\sqrt{R^2+z^2}$ from $(0,0,z)$. Thus adding up the bits of voltage due to each $dQ$ distributed over the loop which I have labeled $C$,&#xA;$$V = \int dV = \int_C \frac{dQ}{4\pi \epsilon_o \sqrt{R^2+z^2}} = \frac{1}{4\pi \epsilon_o \sqrt{R^2+z^2}} \int_C dQ = \frac{Q}{4\pi \epsilon_o \sqrt{R^2+z^2}}$$&#xA;Then, by symmetry the electric field must point along the $z$-axis and we can derive its magnitude from the voltage just calculated: $E_z = -\frac{dV}{dz}$;&#xA;$$ E_z = -\frac{d}{dz}\frac{Q}{4\pi \epsilon_o \sqrt{R^2+z^2}} = \frac{zQ}{4\pi \epsilon_o (R^2+z^2)^{3/2}}$$&#xA;As $z \rightarrow \infty$ we obtain $E_z \sim \frac{Q}{4\pi \epsilon_o z^2}$. Intuitively this makes sense, from very far away a ring of charge with charge $Q$ is the same as a point charge $Q$ at the origin. We find the &#xA;&lt;em&gt;Coulomb field&lt;/em&gt; in the appropriate limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example: plane of charge with uniform density $\sigma = \frac{dQ}{dA}$&lt;/strong&gt;. Building off what we just did, if we divide the plane into rings of radius $s$ with thickness $ds$ then each such ring has charge $dQ = \sigma dA = 2\pi s  \sigma ds$ hence the voltage at $(0,0,z)$ due to the ring of charge is (using my first example with $R=s$ and $Q$ replaced with appropriate $dQ$):&#xA;$$ dV = \frac{2\pi s  \sigma ds}{4\pi \epsilon_o \sqrt{s^2+z^2}}$$&#xA;The voltage due to a disk of radius $s=R$ is thus, assuming $z&amp;gt;0$,&#xA;\begin{align} V &amp;amp;= \int_{0}^R \frac{\pi s  \sigma ds}{2\pi \epsilon_o \sqrt{s^2+z^2}} \\ &#xA;&amp;amp;= \frac{ \sigma }{2 \epsilon_o}\int_{0}^R \frac{ s ds}{\sqrt{s^2+z^2}} \\&#xA;&amp;amp;= \frac{ \sigma }{2 \epsilon_o}\sqrt{s^2+z^2} \bigg{|}_0^R \\&#xA;&amp;amp;= \frac{ \sigma }{2 \epsilon_o} \left( \sqrt{R^2+z^2} - z \right)&#xA;\end{align}&#xA;Thus, again by symmetry the electric field must be of the form $\vec{E} = \langle 0,0, E_z \rangle$ at $(0,0,z)$ and we calculate: for $z &amp;gt;0$,&#xA;$$ E_z = -\frac{dV}{dz} = -\frac{ \sigma }{2 \epsilon_o}\left( \frac{z}{(R^2+z^2)^{3/2}} - 1\right) $$&#xA;As $R \rightarrow \infty$ for fixed $z&amp;gt;0$ the expression above simply yields $E_z \sim \frac{ \sigma }{2 \epsilon_o}$. Now, we usually derive this result using a little cylinder and Gauss' Law, but, it's nice to see it can be derived directly by potential methods. Also, we can stop before taking $R \rightarrow \infty$ and we have formulas for the potential due to a disk of charge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typically we teach how to find electric fields by direct superposition of the Coulomb fields due to the point charges in the source distribution. However, that is harder than what I show here in that the integration is over vectors. In contrast, the potential is a scalar function. In both calculation schemes, if we diverge from standard geometries then the actual integrals get really tough. For example, if we get away from the axis in my example then I think Bessel functions are needed. Much of the study of electrostatics is centered around how to set up these sort of integrals. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-06-26T04:59:28.743" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12514" PostTypeId="2" ParentId="12507" CreationDate="2017-06-26T23:07:58.670" Score="2" Body="&lt;p&gt;Many decades ago, c. 1965 and subsequently, I read Piaget's stuff with interest, as it was (seemingly) a fresh viewpoint. From my naive viewpoint at the time, various of his assertions seemed plausible. With some decades' hindsight, his &quot;insights&quot; with regard to &quot;mathematics&quot; were of an unfortunately superficial and cliched type, much as a very intelligent but essentially ignorant person might present today. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, yes, an ill-informed person (or an idiot) can say (apparently profound) things which are correct. There is no rule that prevents... Yet, this is not an argument in favor of pronouncements of intelligent people about things they actually are ignorant of. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, for the sake of not wasting time, we are not obliged to debunk every ill-informed assertion of otherwise-very-intelligent people. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, I'd suspect (based only on general considerations, not on specific information) that people've not been very concerned on rebutting what turned out to be somewhat-fanciful speculations. (This is considerably in the vein of the more recent non-necessity of rebutting every wrong thing that is asserted on the internet. Woof.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, at best, this may be a meta-answer to the meta-issue of whether anyone currently cares about Piaget's opinions. They were certainly interesting (to a naive kid like me, anyway, and apparently others), but their eventual naivete may not have been construed as needing explicit refutation in scholarly literature. I don't know, but that would be my guess.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="1385" LastEditDate="2017-06-28T07:11:19.680" LastActivityDate="2017-06-28T07:11:19.680" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12516" PostTypeId="1" AcceptedAnswerId="12517" CreationDate="2017-06-27T13:55:55.570" Score="6" ViewCount="815" Body="&lt;p&gt;In comments &lt;a href=&quot;https://academia.stackexchange.com/a/91318/43544&quot;&gt;here&lt;/a&gt; there seemed to be significant difference of opinion on how much credit to give to a short exam question with a single critical error in it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the example of student work below, from a community-college College Algebra exam. The only real mistake is the pair of sign errors in the first line of work:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LJOnD.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LJOnD.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a percentage basis, how much credit should be awarded to this student on this problem? And, related: How many discrete gradations of quality are warranted for a problem of this extent (e.g., as they might appear in a &lt;a href=&quot;https://en.wikipedia.org/wiki/Rubric_%28academic%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;grading rubric&lt;/a&gt;)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(P.S. It may be helpful to note that this kind of mistake is probably the most common source of errors for students in all the courses that I teach.)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-06-30T03:25:46.467" LastActivityDate="2018-09-19T18:38:54.583" Title="How much credit to give a short exam question with one error?" Tags="|students-mistakes|grading|" AnswerCount="8" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12517" PostTypeId="2" ParentId="12516" CreationDate="2017-06-27T14:46:07.990" Score="14" Body="&lt;p&gt;I'll try to make this answer a little more general than just telling how many points I would give for this particular error (if interested: I'd give 5/10 at most, most likely less).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For that, let's discuss three different kinds of computation errors (i.e. not including logical errors, wrong proofs, etc.). The given error in your image falls in the third category and thus should lose lots of points...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Careless mistake, e.g. due to missing concentration.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the most common mistake, the one you might have thought about when asking this question. The student simply put a wrong sign/a wrong digit/..., but did all the right computations. Here, the general rule is: Give almost full points, unless the problem got a lot easier through the mistake.&#xA;As an example, consider the task to differentiate &#xA;$$\frac{5x^4 + 7x^2-3x}{x+3}.$$&#xA;If a student forgets to copy the three from the question sheet and instead differentiates&#xA;$$\frac{5x^4 + 7x^2 - 3x}{x} = 5x^3 + 7x^2 - 3,$$&#xA;this is a lot easier and can only give few points, even if it was an honest mistake of just forgetting to copy this one digit. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;2&quot;&gt;&#xA;  &lt;li&gt;Results that can be verified.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are many examples of exercises where the results can be verified quite simply, e.g. computing zeros of a polynomial, computing the kernel of a matrix, etc. If there is a small error here that goes unnoticed, it should give more points off. On the other hand, if a student writes something along the line of &quot;I put this value into the polynomial and it turns out to not be a root, however, I can't find where my computation went wrong&quot;, then doing this test should be rewarded by only taking off points according to case 1. This is also important in other classes (physics, etc.), if you for example make an error in computing the speed of an object, it should at least give partial points if you notice that this car, driving at about twice the speed of light, is highly unlikely. :)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;3&quot;&gt;&#xA;  &lt;li&gt;Mistakes due to not understanding the topic.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the gravest situation, and the situation you have in your image. Thus, it should give the most points off. Let's take your image as an example of what I mean. The question to write this as a single polynomial asks the student to perform two steps:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Use the distributive law to eliminate parentheses.&lt;/li&gt;&#xA;&lt;li&gt;Use the distributive law to collect the powers of $x$ together.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Note that collecting the powers of $x$ can be done without understanding the distributive law. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now this student did not understand this law and failed at the first step. It might look like only two wrong signs, but looking more closely, the student simply dropped the parentheses without doing anything else. Thus, this error is just as bad as saying&#xA;$$5(x^2+3x+7) = 5x^2+3x+7.$$&#xA;As the distributive law seems to be the current topic in class, there should have been quite a few examples discussed before the test and this student simply didn't understand this topic. Thus, you should only give few points here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that it is sometimes difficult to distinguish between the first case and the third one, especially if only looking at a single question, e.g. you might consider it case one instead of three if the student did all other problems with distributive law correctly... Furthermore, it also depends on the current topic taught in class: if your given mistake happens in a graduate course in mathematics, it might be considered case one; whereas in a first year course, discussing the distributive law, it is a very, very bad mistake.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;edit: Regarding the comment(s), I'll try to make the argumentation for how many points to give short(er than it was before). When deciding on points for an exercise, you have to see both the whole exam and make sure that points are fair, and you have to look at each single exercise and make sure that it can be properly graded. If you give not enough points, then you might be forced to go down to half points or even quarters when grading, because &quot;There is an error, but still, most is correct...&quot;. If you give too many points, you might end up giving points to a completely wrong exercise, because you can only take off so much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at the exercise you posted, I'd give 4 points.&#xA;There are two major tasks here, as explained above: Removing the parentheses and collecting the $x$. For each step, two points is a good number. This will allow to take one point off for minor mistakes (e.g. $5+4 = 8)$ and two points off for major mistakes (e.g. $2x^2 + x^3 = 2x^5$). &lt;/p&gt;&#xA;" OwnerUserId="8296" LastEditorUserId="8296" LastEditDate="2017-07-04T06:34:18.740" LastActivityDate="2017-07-04T06:34:18.740" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12518" PostTypeId="2" ParentId="12516" CreationDate="2017-06-27T15:00:35.173" Score="2" Body="&lt;p&gt;I suppose this all depends on what type of expectations are set up for the class and the type of instructions that are given for the exam. Personally, I've never cared for grading problems individually devoid of the performance on the rest of the exam. And this is the type of expectation I set up for the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the exam questions are to be graded independently on a problem-by-problem basis, then go ahead nick a few points [3 out of 10, let's say] because presumably there are numerous other errors [combining like terms, eg] that can be made and they are all worth something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if there were an &lt;em&gt;exam wide&lt;/em&gt; rubric where one of the things to check for was understanding how the parentheses are handled in the presence of a minus sign, then it's a matter of seeing what the performance on the rest of the exam was like. If the student does not demonstrate an understanding of how to reconcile $-(ax^m - bx^n - c)$ (or similar structures) on other problems, then they should get a few points taken off [however many were attributed to the skill / concept / mechanic] for the entire exam [or proportionally per problem]. However, if this was the only mistake of this type on the entire exam, I would lean towards a minimal deduction, perhaps even zero depending on how the rest of the exam looks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a nutshell, I try to be qualitatively cognizant of the topic-wise correlation inherent in exam problems. I tend to not want to repeatedly penalize a student for the same mistake throughout the exam, especially if the exam is testing on a diverse set of topics [unlike, say a basic arithmetic exam].&lt;/p&gt;&#xA;" OwnerUserId="6898" LastEditorUserId="6898" LastEditDate="2017-06-27T17:33:17.013" LastActivityDate="2017-06-27T17:33:17.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12519" PostTypeId="2" ParentId="12516" CreationDate="2017-06-27T20:48:59.383" Score="5" Body="&lt;p&gt;Without knowing anything else about the exam (which class it was, when in the term, etc.), my vote is 7/10 points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this is a basic/intermediate algebra class, then I see at least four things being assessed here:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Identifying and combining like terms&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Signed number arithmetic&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Distributive property (or &quot;subtraction being the addition of the opposite&quot;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Writing a polynomial in standard form&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As given, the student's answer demonstrates three of these four items. Whether these are &quot;equal&quot; in importance is debatable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As instructors, focusing on one error is tempting, because we might look at a problem worth 10 points and say &quot;I can't give this problem 70% -- it definitely doesn't show a &lt;em&gt;passing&lt;/em&gt; knowledge of the material.&quot; However, if I take a good look at any one problem I assign and ask &quot;exactly what knowledge or skills are necessary here&quot;, I may find that students are having to show a lot more understanding on skills I didn't plan to assess. Therefore, I would focus on what things students must do correctly, and then award points based on a correct demonstration of those discrete skills necessary to get the right answer (&quot;points earned&quot; vs. &quot;points lost&quot;).&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2017-06-27T20:48:59.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12520" PostTypeId="2" ParentId="12516" CreationDate="2017-06-27T21:41:18.697" Score="3" Body="&lt;p&gt;I've seen it go both ways. &quot;The answer is wrong. No credit&quot; at one extreme. Or as little as 2/10 off for failing to distribute the -1 before the parenthesis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When students complain to me that their teacher was unfair, having given zero for this, I explain (via a story from my own HS experience) that a doctor can potentially kill a patient for the misplaced decimal. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also remind them SATs are scored either right or wrong for each answer. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-07-02T12:27:27.980" LastActivityDate="2017-07-02T12:27:27.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12521" PostTypeId="2" ParentId="12516" CreationDate="2017-06-28T01:50:05.663" Score="7" Body="&lt;p&gt;I like to grade questions like this on a five-point rubric. The main aspect of this problem is distributing the negative sign into the second group. Presumably students had practice problems similar to this. I would score this as 3/5 or 60%. It is D work. It is not satisfactory. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2017-06-28T01:50:05.663" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12522" PostTypeId="1" AcceptedAnswerId="12523" CreationDate="2017-06-28T08:08:08.457" Score="0" ViewCount="88" Body="&lt;p&gt;Im triying to explain this delta-epsilon problem, but I didnt find a way to attack effectively this rigorous demonstration &#xA;&lt;a href=&quot;https://i.stack.imgur.com/a9IXF.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/a9IXF.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I actually i tried a lot of inequalities (Cauchy-Schwarz etc), but nothing came up with this demostration, Im stuck here, its a difficult solution or Im just dump??&lt;/p&gt;&#xA;" OwnerUserId="8413" LastActivityDate="2017-06-28T08:17:53.413" Title="Multivariable limit problem" Tags="|vector-calculus|limits|" AnswerCount="1" CommentCount="2" ClosedDate="2017-06-28T16:43:20.680" ContentLicense="CC BY-SA 3.0" />
  <row Id="12523" PostTypeId="2" ParentId="12522" CreationDate="2017-06-28T08:17:53.413" Score="4" Body="&lt;p&gt;Welcome to SE. As you are asking on a side for math educators, I assume that you want to explain this to students. My advice: don't. Start with much, much easier examples to get the students accustomed to such problems, then come back to it later; if at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you, as the teacher/professor, are not able to find a solution for this problem, it is too hard for students who are new to the topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore,  you have to really understand all the concepts behind it. Therefore, reciting an answer you might get here on how to do it might not be convincing and thus bad for your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case that this is a homework problem or you are looking for how to answer it for different reasons than teaching it, you might want to try at Math.SE.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2017-06-28T08:17:53.413" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12524" PostTypeId="2" ParentId="12516" CreationDate="2017-06-28T18:01:03.260" Score="3" Body="&lt;p&gt;In the particular problem posed there are essentially two tasks. The first is to distribute the sign over the second term (this implicitly requires understanding that negation is involutive). The second is to correctly pair the coefficients of like powers of $x$. The student has failed completely at the first task, and has completed the second task successfully. How much credit to give boils down to how much importance one places on each of these tasks. The first seems slightly more difficult (however, I have limited experience teaching at this level), so I'd be inclined to give slightly less than half credit on this exercise. The perhaps more difficult half of the exercise is botched and the final answer is wrong (the student should be encouraged to evaluate and compare the initial and final form for some simple values of $x$ such as $0$ and $1$ as a way of checking the correctness of the response).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The precise point values to assign depend on the grading scale and system in use. This feels like a failing answer, so if 1/2 is minimum passing, I'd view this as roughly 1/3 (so in the US, where 7/10 is minimum passing, maybe it would be scored apparently higher), but saying anything precise about numerical values requires knowing well the local context, grading system, and standard practices. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2017-06-28T18:01:03.260" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12525" PostTypeId="2" ParentId="12445" CreationDate="2017-06-28T18:15:35.193" Score="3" Body="&lt;p&gt;Although anecdotes are usually badly reported samples of size one, I'll answer with one. Trying to diagnose his problems with multiplication, I tried giving my grade school son variants of the same problem, written in &quot;stacking&quot; form, e.g. $123 \times 456$, $456 \times 123$, $1230 \times 456$, $4560 \times 123$, $1230 \times 4560$, etc. My hope was that after the first he would realize that rest could be deduced from its result with minimal computation. Instead I discovered that he had not learned to multiply well by &quot;stacking&quot; (he doesn't write neatly, and his columns are like the one in Pisa, so to obtain the final sum he adds digits from different columns; usually each row of the answer is correct, but the final sum is wrong). He did in fact perceive what I had hoped - he recognized that his five answers (four of them incorrect) were inconsistent, e.g. that the first two had to be the same and his answers weren't, and so forth, but this recognition didn't help him decide which answer was correct or where his mistakes were! The moral (for me at any rate) is simple (although perhaps this is just confirmation bias in operation) - for a student who doesn't have operational mastery of the basic rote operations it's silly to worry much about interpreting the operations being performed. Approaches that emphasize such interpretation are important and useful, but they can't substitute for learning some effective and efficient algorithm for reliably performing multiplications correctly. Rather they should accompany and follow the exercise/practice/drill that is their necessary precursor.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2017-06-29T13:16:47.940" LastActivityDate="2017-06-29T13:16:47.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12526" PostTypeId="2" ParentId="12507" CreationDate="2017-06-28T18:50:54.983" Score="2" Body="&lt;p&gt;For Piaget, &lt;em&gt;'the key tool of the constructivist is the group'&lt;/em&gt; (yes, the mathematical notion of a group!) and a child is understood as part of a feedback loop embedded in their environment consistent with much of the other cybernetic discourse, Norbert Weiner in particular.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nothing in these books targets the actual teaching of mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Read and enjoy:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Genetic Epistemology&lt;/em&gt; by Piaget&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Structuralism&lt;/em&gt; by Piaget &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To begin his work on Genetic Epistemology, Piaget attempts to problematize the contemporary visions for scientific knowledge.  Piaget believes that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Scientific thought, then, is not momentary; it is not a static instance; it is a process. More specifically it is a process of continual construction and reorganization.&quot;&lt;/em&gt; - GE, 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He offers two examples to support this.  First, is Louis de Broglie who shifted from adhering to Niels Bohr's view of indeterminism.  de Broglie later changed his mind, hence his knowledge is constantly in the process of being constructed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second example is Bourbaki.  The mother structures weaknesses have been exposed, and Bourbaki's architectures were giving way to Eilenberg and MacLane's use of categories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;As a result, today part of the Bourbaki group is no longer orthodox but is taking into account the more recent notion of categories.&quot;&lt;/em&gt;- GE, 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, knowledge is not static, it is actively in construction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, logical developments in children are subdivided into figurative and operative aspects.  Figurative refers to imitative behavior whereas operative represents a transformation from one state to another. These operative modes are what matters if knowledge is in fact an active construction, and the actions on things are what matter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;it includes actions themselves, which transform objects or states and it also includes the intellectual operations, which are essentially systems of transformation...to know is to assimilate reality into systems of transformations...knowing an object does not mean copying it-it means acting upon it. It means constructing systems of transformations that ccan be carried out on or with this object...knowledge, then, is a system of transformations that become progressively adequate.&quot;&lt;/em&gt; - GE, 15.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Piaget believes then that he has argued &quot;the roots of logical and mathematical structures are to be found in the coordination of actions, even before the development of language...&quot;(GE, 21)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Operating on something for Piaget means that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;An operation is an action that can be internalized &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An operation is reversible &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An operation always supposes some conservation&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;No operation exists alone &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thus, for Piaget, &quot;every operation is related to a system of operations, or to a total structure as we call it.&quot;- GE 22.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Bourbaki's had three 'mother structures'--algebraic, order, and topological.  An example of an algebraic structure would be a group, ring, or field.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The group is Piaget's favorite tool.  In mathematics, a group is a structure that consists of a collection of things and some action(s) on these things.  Similarly, a group has its four axioms of closure, associativity, identity, and existence of inverses.  These map closely to Piaget's description of operative knowledge, and it was deliberate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Piaget described an interaction with the Bourbaki member Dieudonne at a meeting on 'Mathematical Structures and Mental Structures' to emphasize the connection:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Dieudonne gave a talk in which he described the three mother structures.  Then I gave a talk in which I described the structures that I had found in children's thinking, and to the great astonishment of us both we was that there was a very direct relationship between these three mathematical structures and the three structures of children's operational thinking.  We were, of course, impressed with each other, and Dieudonne went so far as to say to me: &quot;This is the first time that I have taken psychology seriously.  It may also be the last, but at any rate it's the first.&quot;&lt;/em&gt;-GE, 26&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For Piaget, algebraic structures are readily observable through the logic of classes, i.e. classification.  If you give children some different shapes, according to Piaget, around age 7 or 8 they will be able to classify them operationally as described above.  Otherwise, Piaget noticed a range of pre-operational thinking in younger children.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the lowest level was figurative behavior where children would make shapes of similar shapes, i.e. a triangle out of triangles, circle of circles.  Later, children dismiss the figurative aspect and piles the shapes.  Piaget claims that this child does not yet understand class inclusion however.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;he cannot deduce, for instance, that the total class must necessarily be as big as, or bigger than , one of its constituent subclasses. A child of this age will &#xA; agree that all ducks are birds and that not all birds are ducks. But then, if he is asked whether out in the woods there are more birds or more ducks, he will say, &quot;I don't know; I've never counted them.&quot;&lt;/em&gt;&quot;- GE, 27.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the child is still not evincing all the characteristics of operational thinking.  This relationship of class inclusion is what gives rise to &lt;strong&gt;THE OPERATIONAL STRUCTURE OF CLASSIFICATION&lt;/strong&gt;.  This structure is something like an algebraic structure, though not exactly as distributivity is not a feature. (birds + birds = birds, hence (bird + bird) - bird = 0 but bird + (bird - bird) = 0)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note the similarity to how he is describing the changing characteristic of a child's knowledge based on group like behavior.  Only when the child engages with classification tasks in such a way as to demonstrate all four axioms, they are demonstrating 'pre-operational' thinking, and these different forms are what he bases his stages of development on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other structures are there too.  Piaget goes on to describe the example of an ordering structure through the problem of seriation, and that of topological through children's use of either topological, euclidean, or projective frames in geometric contexts.  Piaget believed children think like topologist's first, and you can find more on this in his geometry book, &lt;em&gt;The Child's Conception of Geometry&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you get a sense for how he uses structures as models for cognition here though, and if you're interested I suggest reading &lt;em&gt;Genetic Epistemology&lt;/em&gt; and &lt;em&gt;Structuralism&lt;/em&gt;.  They're not long and very interesting to see how involved Piaget was with using mathematics and physics to build his system. &lt;/p&gt;&#xA;" OwnerUserId="7003" LastEditorUserId="7003" LastEditDate="2017-06-30T02:01:27.160" LastActivityDate="2017-06-30T02:01:27.160" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12527" PostTypeId="2" ParentId="12516" CreationDate="2017-06-28T20:39:39.547" Score="3" Body="&lt;p&gt;On a 10 point scale, I'd roughly break up the work as 4 points for correctly distributing the minus sign, 4 points for combining like terms correctly, and 2 points for correctly performing the basic arithmetic inherent in these steps. Since this student correctly performed the latter two tasks, but not the first, I would award 6/10 points.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2017-06-28T20:39:39.547" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12528" PostTypeId="2" ParentId="12445" CreationDate="2017-06-28T21:54:39.567" Score="1" Body="&lt;p&gt;I've never heard of 'stacking', the method shown appears the standard way to multiply digit by digit. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;37 squared. For those who wish to do this in their head, I'd go up to 40, and the same 3 down to 34. 34 doubles to 68, and again to 136, then x10 to 1360. Now add back the square of 3, the 9 we lost by taking (a+b)(a-b). &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-06-29T10:07:14.033" LastActivityDate="2017-06-29T10:07:14.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12529" PostTypeId="1" CreationDate="2017-06-29T15:46:56.013" Score="14" ViewCount="305" Body="&lt;p&gt;I am planning to do &quot;online office hours&quot; for students in math courses (undergrad level). My motivation is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To give the students the possibility to ask questions remotely&lt;/li&gt;&#xA;&lt;li&gt;To give the students the possibility to ask questions anonymously&lt;/li&gt;&#xA;&lt;li&gt;To give feedback in realtime&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Hence, I am looking for a software with the following features:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I can create a chat room that I can open to a specific group of people (e.g. by sharing a link)&lt;/li&gt;&#xA;&lt;li&gt;chat rooms can handle about 100 people&lt;/li&gt;&#xA;&lt;li&gt;chat rooms are not public&lt;/li&gt;&#xA;&lt;li&gt;students can enter anonymously (e.g. with a self chosen  user name)&lt;/li&gt;&#xA;&lt;li&gt;simple LaTeX can be rendered&lt;/li&gt;&#xA;&lt;li&gt;it has a smartphone app and a browser interface.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you don't know of a software that fulfills all criteria, I would also like recommendations for tools that only fulfill some of them.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="3398" LastEditDate="2018-05-24T10:17:08.667" LastActivityDate="2018-05-24T10:17:08.667" Title="Chat software for &quot;online office hours&quot;" Tags="|undergraduate-education|online-instruction|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="12530" PostTypeId="1" CreationDate="2017-06-29T17:02:11.123" Score="6" ViewCount="487" Body="&lt;p&gt;After such amazing answers I got here for a related question (link at the end if someone still wants to share with me their views)...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the concept:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you were to create the ULTIMATE Calculus syllabus - a tree of topics - from pre-requisites to fundamentals to basic to intermediate to advanced to very advanced... and beyond...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would this tree of topics look like?  When I say a &quot;tree&quot; I mean a hierarchy of subjects, not just a list of topics.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;UPDATE:  Reading some comments (thank you @schremmer) - I would definitely appreciate also a narrower question, as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What MIGHT a tree of topics look like for a &quot;Calculus I&quot; course that's required for most science and engineering undergrads?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;(Here is my previous question - regarding teaching only by problem-solving:  &lt;a href=&quot;https://matheducators.stackexchange.com/questions/12476/can-mathematics-be-learned-by-only-solving-problems&quot;&gt;Can mathematics be learned by ONLY solving problems?&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you :)&lt;/p&gt;&#xA;" OwnerUserId="7960" LastEditorUserId="7960" LastEditDate="2017-07-03T08:52:30.820" LastActivityDate="2017-07-08T01:28:56.857" Title="What is the ULTIMATE Calculus syllabus" Tags="|mathematical-pedagogy|calculus|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12531" PostTypeId="2" ParentId="12529" CreationDate="2017-06-29T21:03:51.597" Score="7" Body="&lt;p&gt;One of my students recommended using &lt;a href=&quot;https://piazza.com/&quot; rel=&quot;noreferrer&quot;&gt;https://piazza.com/&lt;/a&gt; for my classes. Basically, it gives you a way of holding online office hours and harnesses some of the utility of the stack exchange model without being public. I haven't personally used it, but, I think it might do pretty much everything you want and then some. Check it out.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-06-29T21:03:51.597" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12532" PostTypeId="2" ParentId="890" CreationDate="2017-06-29T22:24:36.973" Score="2" Body="&lt;p&gt;I took AP BC calculus in early 80s in a top suburban public school.  The sequence there was Algebra, Geometry, Algebra 2, Trig (1 sem), Functions (1 sem), Analytical Geometry (1 sem).  Then AP Calc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the GT kids took &quot;Algebra 2, trig&quot; (2 semesters instead of 3).  The Functions class was pretty cool: inequalities, precalc (basic differentiation and integration), other basic properties of functions and relations.  Analytical geometry was pretty much the standard thing with translation and rotation of coordinates (what a pain in the ass), conic sections, etc.  I would note that it did do a good job on how to analyze and graph strange, difficult expressions (looking for critical points, using calc for that, intercepts, asymptotes, etc.).  This is pretty standard, but some (even older, classical) texts do not do a good job on curve visualization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For AP calc We used Thomas Finney, 4th edition.  Designed for AP Calculus.  We did cover epsilon-delta.  If not in AP Calc, we hit it in &quot;Functions&quot;.  I sorta think that is how we did it.  Also remember doing Archimedes method for parabola.  I think that was all year before, actually in Functions class.  So we covered some of it to get exposed.  But we definitely did not jump down the rathole of &quot;baby real analysis&quot; like Caltech does.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had ZERO problems handling higher level classes at a decent (not Caltech, but not community college) school afterwards.  Calc 3, 'zoics, EE, etc.  Actually the school was the US Naval Academy.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read that AP Calc has had some minor changes to water it down since I took it.  (They do a little less rotational solids integration methods, they do a little less series methods but still some, and they cut second order constant coefficient diffyqs.)  And they do calculator games (most colleges still don't do this).  What can I say, that is ETS, being ETS.  They have gotten a little captured by the ed reformers and more interested in moving how things are done than in just copying college (for AP).  But still way less damage in Calc than in Chem or Bio.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I really expect that for most people calc is a pretty standard thing (you can just read Granville from 100 years ago and it is almost like a template for how things are done now, still.)  I suspect the much bigger factor in the differences amongst your students is native ability along with how many homework problems they did in high school.  Calc is calc.  Sure there are differences in a course.  But student variability is the bigger factor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[edit]  Just read more of your questions asking for differences.  Well obviously one is high school and one is college.  More hours in the room in HS.  More homework.  More interaction and feedback.  More grading.  More going to the blackboard.  More frequent testing.  Basically (IMO) a superior pedagogical situation than college.  Then again, I went to a trade school for college and they taught like high school (and did so because they were more interested in training people than a laid back attitude of sink or swim).  But other than that, same content man. Math is math.&lt;/p&gt;&#xA;" OwnerUserId="8348" LastEditorUserId="8348" LastEditDate="2017-06-30T01:28:09.950" LastActivityDate="2017-06-30T01:28:09.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12533" PostTypeId="2" ParentId="890" CreationDate="2017-06-29T23:40:36.593" Score="2" Body="&lt;p&gt;Although I've not taught in 4-year universities, I've taught in a small variety of up-scale &quot;research&quot; universities for some decades. Even with that disclaimer, I have &lt;em&gt;observed&lt;/em&gt; the math-sociology of nearby 4-year colleges lo' these last 30+ years. In particular, I'd wager that the dominant issue in appraising reaction of calculus students in 4-year colleges is sociological, with the corresponding issue in high schools being (with random perturbations depending on school district) somewhat less so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As in other comments and answers, and as might be clear for many other reasons, it's not that &quot;calculus&quot; has changed much, ... er, at all, ..., in the last 150, or, really, maybe 300 years. True, our collective official opinions have drifted, and the Cauchy-Weierstrass mythology has been dominant for many decades. Still, that doesn't necessarily mean anything about peoples' understanding of calculus and how to use it in amazing ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As in my comment above, in high schools in the U.S., to take &quot;AP calculus&quot; is a significantly sociological thing... and, at the same time, is (in my direct observation) often degraded to a sort of &quot;you've earned an easy ride, based on previous work&quot;, so that kids do not actually learn much. (And, sadly, invidiously, I do speculate that, statistically, mathematics is the least-competently-taught high school subject. I say this despite both my parents' being high school math teachers. It's not meant to impune anyone's ethics, etc, either...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;In college&quot;, although quite a few kids don't manage to achieve this minimum, &quot;showing up for the midterms and final&quot; no longer garners high praise. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The chief psychological symptom is the expected: kids who were once &quot;the best&quot;, have managed to be admitted to situations where &lt;em&gt;everyone&lt;/em&gt; was &quot;best in their high-school&quot;, and this can be confusing/comforting, depending.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to teaching of calculus: I do seriously claim that it is a special case of that very general sociological situation and its corollaries. That is, seriously, it's not about calculus per se.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-06-29T23:40:36.593" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12534" PostTypeId="2" ParentId="10092" CreationDate="2017-06-29T23:46:54.877" Score="1" Body="&lt;p&gt;Edit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Work your way through Schaum's Outline First Year College Mathematics by Frank Ayres, Jr.  I prefer the earlier (1958) edition to later versions.  Also, get a used copy not a reprint or electronic one.  (print quality suffers on later ones).  You can get it quite cheap on Amazon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This will cover algebra (what used to be called college algebra and is called algebra 2 in high schools now, but don't worry it covers everything in first year high school algebra also), trigonometry, analytic geometry, and precalculus.  You would need to get geometry elsewhere.  And it is lacking much on vectors (but don't let perfect be the enemy of better!)  I would hold off on calculus (even worrying about a text) until you have the basics down solid.  But this one book will really help you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that Schaum's is written for busy people, is simple, is great for review, is not a logorheic doorstopper like the texts that are sold to schools.  Just WORK PROBLEMS, not passively read.  And you will be fine.  Oh and all the answers are included.&lt;/p&gt;&#xA;" OwnerUserId="8348" LastEditorUserId="8348" LastEditDate="2017-07-01T19:23:04.077" LastActivityDate="2017-07-01T19:23:04.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12535" PostTypeId="2" ParentId="10421" CreationDate="2017-06-29T23:58:45.880" Score="1" Body="&lt;p&gt;My advice is to take complex analysis, but a more applied version (weak on proofs, strong on contour integration).  Just based on your question and what you said about yourself, I think you will enjoy this more.  And you can take more theoretical analysis versions later (if you choose).  But at least you will have some exposure to a fundamental field.  &lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-29T23:58:45.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12536" PostTypeId="2" ParentId="10585" CreationDate="2017-06-30T00:18:25.143" Score="2" Body="&lt;p&gt;I would keep it down the middle (accelerated, not enriched).  Think that gives you more option value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Frosh: Calc 1 and 2 (standard version)&#xA;Soph: Calc 3 (multivariable), ODEs &lt;/p&gt;&#xA;&#xA;&lt;p&gt;ODEs is the first place where you may have issues as almost any textbook covers more material than is comfortable to do in a semester (unless you treat it by doing double hours).  Whereas calc, you can just work through a text.  My recc is to either double up hours or take a little longer.  But run all the way through Tenenbaum.  Most colleges handle this by just selecting topics.  But it is different from calc where you can work straight through most texts. Kreyszig has a very clean, short diffyqs approach, but it is inside a fat book of other stuff. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Junior: PDEs, Complex.  (You will have some similar issues on PDEs but not as bad as ODEs since it is mostly about the standard three problems.  I would lean towards using the easier Farlow text that Dover has for PDEs.  Can come back for harder later.  For complex, there are some good simple applied semester long texts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Senior: Lot of choices, really.  You are kind of done with the standard engineer/science math route.   I would say linear algebra.  (Actually now that I think of it, do it earlier. Little break from calculus track.  Choose an easy text.)  maybe a math methods course to hit the other semester.  There are some texts that get recommended a lot for self studiers.  But you could pick something else too.  (simple probes and shafts course for instance.)&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-06-30T00:18:25.143" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12537" PostTypeId="1" AcceptedAnswerId="12538" CreationDate="2017-06-30T03:50:42.493" Score="4" ViewCount="306" Body="&lt;p&gt;A question related to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/12516/how-much-credit-to-give-a-short-exam-question-with-one-error&quot;&gt;this one&lt;/a&gt;, on an equivalent problem on a community-college College Algebra exam. In the prior question most readers observed the error as a critical conceptual issue. Now consider this example of student work:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/g15EH.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/g15EH.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, the distribution has been done correctly, and two of the three combinations of like terms. But in the very last term, a signed-arithmetic error has been made; the student replaced $-3-14$ with an incorrect $-11$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a percentage basis, what is the most appropriate amount of credit to award in a case like this? Assume that the problem is worth an appropriately granular number of points (not necessarily 10). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-06-30T05:42:49.727" Title="How much credit for a single arithmetic mistake?" Tags="|students-mistakes|grading|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12538" PostTypeId="2" ParentId="12537" CreationDate="2017-06-30T05:42:49.727" Score="5" Body="&lt;p&gt;The correct realization of the operations prior to the error indicates that the student does understand how to take sums and differences of negative numbers, how to negate a negative numbers, and how to realize more complicated operations, such as the distribution of the negative sign over a sum, and this suggests that the mistake in the final step is due to carelessness. If one accepts this reasoning, only a minor deduction should be made. Exactly what it should be depends on the context. Were this a final step in the solution of a problem in a university differential equations class, I might not even deduct points (were everything else perfect), but in a class dedicated to teaching precisely this sort of operation, some deduction is in order, on the order of $1/8$ of total value (depending on how one counts there are $6-10$ operations that have to performed, and one of them has been performed incorrectly, and the evidence supports the conclusion that the student understands what to do, simply has made a careless mistake).&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2017-06-30T05:42:49.727" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12539" PostTypeId="2" ParentId="890" CreationDate="2017-06-30T07:22:11.233" Score="2" Body="&lt;p&gt;I think it is important to avoid lumping all &quot;colleges&quot; (in the broad sense) into a single category but distinguish between small (and expensive) colleges, on the one hand, and large universities, on the other. I had a conversation with a budding dentist a few years ago in Chicago who engaged me in a conversation and I ended up telling him about my interest in teaching calculus with infinitesimals. His reaction was completely unexpected.  He said: &quot;How can you teach calculus &lt;em&gt;without&lt;/em&gt; infinitesimals?&quot;  Apparently at his small (expensive) college in California they have been using infinitesimals all along (without letting the big shots in the AMS know what they are doing).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say that the difference between highschool and small college calculus teaching, on the one hand, and university teaching, on the other, is that the former is more likely to be practical-minded whereas the latter is more likely to be ideologically committed to what research mathematicians view as the &quot;right&quot; way of doing mathematics since Weierstrass.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2017-06-30T07:22:11.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12540" PostTypeId="2" ParentId="12529" CreationDate="2017-06-30T21:26:52.050" Score="8" Body="&lt;p&gt;There aren't many choices that hit all the bullet points you listed, but there are some approximations that are quite good: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Piazza (mentioned in another answer) is good. I've used it several times for classes. It does handle LaTeX very well. However, it's a threaded discussion board and not chat software -- i.e. it does not do things in real time (unless you count refreshing the browser). &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://slack.com&quot; rel=&quot;noreferrer&quot;&gt;Slack&lt;/a&gt; is a group messaging app that was designed for software development teams. I have also used this for a class before. Messages are posted in real time, and stored in channels so they can serve as asynchronous discussions. You can have office hours chats with individuals or groups either in a public channel or through private direct messages. I am not sure of the upper limit on how many people can be included in one of those. Drawbacks for you: There's no anonymous posting and no LaTeX. (There is a plugin that is supposed to handle LaTeX but IMO it's not good.) &lt;/li&gt;&#xA;&lt;li&gt;A third option is &lt;a href=&quot;https://gitter.im/&quot; rel=&quot;noreferrer&quot;&gt;Gitter&lt;/a&gt; which is a chat/networking tool related to GitHub (Gitter = &quot;GitHub&quot; + &quot;Twitter&quot;). It's a very simple chat environment that handles LaTeX and Markdown syntax very nicely. But, you cannot separate out discussions into groups or private channels, and users need a GitHub or Twitter account to log in. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of these three, I think Slack is the best -- there is a bit of a learning curve but my students really liked using it, and to work around the LaTeX issue we just agreed to enter math syntax using &quot;Wolfram|Alpha speak&quot;, i.e. $\sqrt{x^2 + y^2}$ was entered sqrt(x^2 + y^2) and that worked well enough. &lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2017-06-30T21:26:52.050" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12541" PostTypeId="2" ParentId="12529" CreationDate="2017-06-30T23:35:51.713" Score="4" Body="&lt;p&gt;At my institution, some profs use &lt;a href=&quot;https://www.adobe.com/products/adobeconnect/trial.html?gclid=CJWlk7Xg5tQCFdVLDQodJi4PzA&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Adobe Connect&lt;/em&gt;&lt;/a&gt;. &lt;a href=&quot;https://blackboard.jhu.edu/bbcswebdav/institution/EP/GeneralCourseContent/AdobeConnectInformation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here is a Johns Hopkins University link&lt;/a&gt; explaining its use for online office hours, and &lt;a href=&quot;https://canvas.uw.edu/courses/721562/pages/adobe-connect-faq&quot; rel=&quot;nofollow noreferrer&quot;&gt;here is a similar link from University of Washington&lt;/a&gt;. I have experimented with it, but not used it in a course myself. (I use &lt;a href=&quot;https://piazza.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;piazza&lt;/a&gt; in all my classes, but not for live online office hours.) I doubt Adobe Connect satisfies all your criteria; certainly not LaTeX.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One prof here who does use it regularly (in Organic Chemistry, ~75 students) says it works great, once you master the technical ideosyncracies. He likes that it off-loads some daytime pressure to past his children's bedtime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One aspect I found interesting is that, apparently, many students would join the office hours and then only kinda half-listen. They didn't have specific questions or confusions, but they wanted to hear whatever was explained to those who did. So they &quot;lurked.&quot; This is not necessarily bad, just interesting.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-07-01T21:45:09.553" LastActivityDate="2017-07-01T21:45:09.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12542" PostTypeId="2" ParentId="9975" CreationDate="2017-07-01T16:32:09.720" Score="-3" Body="&lt;p&gt;I agree with Jessica B. You are giving names an importance they do not and should not have. Let's say you want to illustrate naïve set theory with examples. Just keep using different names as you go. Then you can use all sorts of names, using &quot;alphabets that go in directions other that left-to-right, and languages that use other forms of writing.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I disagree with her &quot;Pedagogically that would be bad&quot;: the names we write between { } and or in Venn Diagrams are just markers which we recognize in a holistic manner. They could even be pictures or photos. But indeed only &lt;em&gt;as long as there are only a few of them&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then if you begin to need too many for them to be immediately distinguishable, you have a situation which calls for a better way and you can propose this as a problem to your students.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2017-07-01T16:32:09.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12543" PostTypeId="1" AcceptedAnswerId="12544" CreationDate="2017-07-01T18:38:52.123" Score="13" ViewCount="2226" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EtC1F.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EtC1F.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was helping a home-schooled student with her homework when we came upon several images of figures that we were supposed to find the perimeter of. In several of the figures, some of the lengths were not given; yet, with some thought, the perimeter could still be figured out. For example, the perimeter of the above figure is 28. She could not &quot;see&quot; this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Her arithmetic skill are admirable. She is neat, organized, and is capable of looking over her work and finding where she made her mistake. Yet, drawing the square and showing what equalled what, did not impress her. I longed for a box of toothpicks. I got the feeling that she was getting area and perimeter confused in some way but I ran out of time before we could resolve her problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any ideas what conceptual problems she wasn't resolving?&lt;/p&gt;&#xA;" OwnerUserId="5144" LastEditorUserId="5144" LastEditDate="2017-07-01T19:12:13.853" LastActivityDate="2017-07-16T23:04:21.427" Title="Problem &quot;seeing&quot; the perimeter of a figure" Tags="|mathematical-pedagogy|meta-cognition|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12544" PostTypeId="2" ParentId="12543" CreationDate="2017-07-01T19:26:12.490" Score="9" Body="&lt;p&gt;This is pure speculation — I don't have enough information to know what the student was thinking — but they may have only known how to compute the perimeter in one way: taking the lengths of each edge and adding them up. (They may have even unconsciously &lt;em&gt;identified&lt;/em&gt; the perimeter as exclusively being the result of that procedure.) However, in the given figure, we only know two of the six edge lengths; the other four edges have to be split up into two pairs for which we only know the &lt;em&gt;sum&lt;/em&gt; of the edge lengths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, it's plausible that the missing observation was something like this: to compute the perimeter of a polygon, it's enough to know the sums of edge lengths for any partition of the edges; it's not necessary to know each edge length individually. Once stated outright, it's not hard to see this is true — it follows immediately from associativity of addition — but many observations that are obvious after the fact aren't always obvious beforehand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Putting this observation into practice also requires more reasoning than adding up a bunch of known edge lengths, since you have to think about which edges can be grouped together so that their total length can be computed. It's a problem that can't be resolved by purely mechanical application of a fixed procedure; one typically has to play around with it a bit and try out a few things. Students who are less experienced or comfortable with problems that require use of reasoning, trying out ideas that aren't guaranteed to be successful, and don't conform to a fixed procedure will often have difficulty with such problems.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2017-07-01T19:26:12.490" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12545" PostTypeId="2" ParentId="12530" CreationDate="2017-07-01T19:29:30.327" Score="3" Body="&lt;p&gt;Hopefully the short and sweet is best.  Here's what I want out of a first semester class regardless of background.  I think depending on the individual student they will be ready to engage with these ideas at different levels of rigor and complexity.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Calculus I&lt;/strong&gt;: Concept of Differentiation and Integration, connection through simple ODE's.  Students should be able to use these ideas to solve real problems, and constantly be moving back and forth between approximate and exact solutions, discussing accuracy of models and solutions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Next, depending on the context the first is a more traditional approach(service class) or an alternative more focused on modeling with calculus.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Calculus II&lt;/strong&gt;: Representing functions with series and doing calculus on these.  Basically, a complexifying ideas from first semester through tasks that necessitate representing functions with Taylor series.  Light connections to work in ODE and PDE problems to motivate what would likely be a class or two in Diff Eq's.  Finally, students should experience some work with vectors and higher dimensional problems.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Calculus II&lt;/strong&gt;(alternative): Introduce Taylor polynomials and series through solving differential equations problems, spend half the course here focusing on both solving problems and dealing with the rigor of series conversion.  Spend the second half focused on vector calculus and solving some good problems in 3D.  I prefer this, and would love if students had an introductory understanding of solving physical problems so important to the emergence of calculus, particularly something of Newton's laws and their use in modeling things like a wave. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In any of these I would want a student to use a computer well to help solve the problems.  Python is the best from my perspective in terms of doing calculus with students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means there are only two concepts I suppose: Integration and Differentiation.  These concepts are related through the FTC, which describes their inverse character.  Students need to understand this first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, we can't differentiate or integrate most functions that we would want to.  As such, we need approximation techniques which come in the form of Taylor series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third, all of these problems above can occur in dimensions higher than 2, and students should understand some of the operational complexity that comes when moving to a higher dimension with the derivative and integral, but also recognize their utility in situations much like those from first semester.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-07-01T19:29:30.327" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12546" PostTypeId="2" ParentId="12530" CreationDate="2017-07-01T20:19:29.690" Score="2" Body="&lt;p&gt;A contrarian answer:  I believe the standard curriculum is optimal.  Think Granville.  Syllabus should emphasize drill and problems that help a general technical track.  This gives best options for most students who don't know their future major.  I think incorporating baby real analysis or the like is not optimal.  Many students don't need that ever.  And those who do can better get it later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;LATER EDIT:  Still not the soup to nuts spread you were asking for (as I think real analysis is better done as a recursion into calculus instead of same time), but one guilty pleasure I would have loved as a student would be to work all the way through Edwards (especially the mammoth Tripos tricks filled Integral volumes).  You would be a badass at doing integrals if you did all those problems!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/treatiseonintegr01edwauoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/treatiseonintegr01edwauoft&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8348" LastEditorUserId="8348" LastEditDate="2017-07-02T14:45:24.900" LastActivityDate="2017-07-02T14:45:24.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12547" PostTypeId="2" ParentId="12543" CreationDate="2017-07-01T22:49:23.627" Score="5" Body="&lt;p&gt;Perhaps a targeted illustration, a &quot;proof without words,&quot; might suffice to&#xA;break the grip of algebra and computations, which might be confusing her...?&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/eb9ml.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eb9ml.jpg&quot; alt=&quot;Perimeter&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-07-01T22:49:23.627" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12548" PostTypeId="2" ParentId="12543" CreationDate="2017-07-02T03:41:49.410" Score="3" Body="&lt;p&gt;&quot;Adding up the lengths of the edges&quot; really is an adequate definition for the perimeter of a figure. The concept she might be missing is that it doesn't matter how long the individual edges are, the perimeter is always the same. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would try explaining this to her using a challenge. Give her some grid paper (if she's used to it), tell her the length of all the edges except two, and ask her to find all the possible perimeters for the figure. That isn't a rigorous mathematical proof, but it might help convince her. &lt;/p&gt;&#xA;" OwnerUserId="8421" LastActivityDate="2017-07-02T03:41:49.410" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12549" PostTypeId="2" ParentId="12543" CreationDate="2017-07-02T08:55:59.940" Score="3" Body="&lt;p&gt;Your approach sounds a bit stiff to me. Put a picture inside the shape. &quot;Adding up the lengths of the edges&quot; might be &quot;How far is it round the edge?&quot; for a start. It doesn't take long before the picture can be dropped and better mathematical language can be introduced, unless your pupil has limited learning skills. &lt;/p&gt;&#xA;" OwnerUserId="8423" LastActivityDate="2017-07-02T08:55:59.940" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12550" PostTypeId="2" ParentId="12543" CreationDate="2017-07-03T01:10:15.050" Score="5" Body="&lt;p&gt;Give her a collection of 5 such shapes on graph paper, all of which have the same perimeter, and have her compute.  Then see if you can talk about why these shapes all have the same perimeter.  See if she can draw another shape on her own which has the same perimeter.  Make sure that all of these use &quot;the same trick&quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2017-07-03T01:10:15.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12551" PostTypeId="1" CreationDate="2017-07-03T08:55:15.133" Score="6" ViewCount="194" Body="&lt;p&gt;I am searching for a book on combinatorics and/or mathematical puzzles for very beginners in easy English. The book should not contain detail mathematical expressions, will contain easy understanding logic behind the theory instead. I don't need problems/exercises and solutions as the book is not for mathematics student. I need the book for mathematics lovers who want to enjoy the theory maybe with some history and story. This should be a story book which one can read in leisure mood and can enjoy the thrill of mathematics. &lt;/p&gt;&#xA;" OwnerUserId="8428" LastEditorUserId="8428" LastEditDate="2017-07-03T09:26:59.693" LastActivityDate="2017-11-22T20:59:46.457" Title="Easy and good book on combinatorial problems" Tags="|mathematical-pedagogy|reference-books|applied-mathematics|mathematics-in-daily-life|math-puzzle|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12552" PostTypeId="1" AcceptedAnswerId="12557" CreationDate="2017-07-03T13:45:08.037" Score="5" ViewCount="311" Body="&lt;p&gt;A colleague of mine asked an interesting question reproduced below with his permission. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - even if one accepts that, all things being equal, it is an easier language to learn - when the vast majority of the mathematical and scientific community still speak in the &quot;Weierstrassian&quot; language. It is possible to teach them both, but this may come at a certain expense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When one wonders whether or not to teach &lt;a href=&quot;https://www.math.wisc.edu/~keisler/calc.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;calculus a la Keisler&lt;/a&gt;, it is not because it is or isn't the right to do, but rather a matter of personal preference. A typical mathematician grew up with Weierstrass and Cauchy, feels at home with standard analysis, and is comfortable teaching it. He knows where the pitfalls are, where to appeal to intuition and where to warn students to be wary, the right examples to grab onto for support, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He doesn't have the same comfort level with infinitesimals, because of how he learned to think about things and look at them, and therefore could not convey it properly to a classroom, certainly not with the same confidence and enthusiasm, that is important in these classes. Given enough time, he could probably gain that same level of comfort with infinitesimals, but it would take a lot more time and effort than he is willing to take on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to summarize, is it really to the students' advantage to learn the language of infinitesimals? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is requested is a reasoned response (based on reliable sources rather than personal opinions) on (1) historical, (2) mathematical, and (3) philosophical aspects of the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question was posed a while ago at &lt;a href=&quot;https://math.stackexchange.com/q/1648838/72694&quot;&gt;MSE&lt;/a&gt; where some of the editors felt that ME is more appropriate.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1608" LastEditDate="2017-07-03T14:30:40.923" LastActivityDate="2017-07-04T17:35:56.667" Title="Is it to the students' advantage to learn the language of infinitesimals?" Tags="|infinitesimals|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="12553" PostTypeId="2" ParentId="12543" CreationDate="2017-07-03T16:53:54.243" Score="2" Body="&lt;p&gt;Is it just a simple matter of pointing out two pairs of congruent segments? It is evident that the perimeter of the rectangle is 4x7=28.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TNom6.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TNom6.jpg&quot; alt=&quot;convex&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2017-07-03T16:53:54.243" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12554" PostTypeId="2" ParentId="12552" CreationDate="2017-07-03T18:33:41.800" Score="5" Body="&lt;p&gt;Yes, Karl-Dieter is referring to myself and a colleague.  My current long-term project is to produce a modern full-color calculus textbook using infinitesimals, making use of my own definitions, notation, etc.  I currently plan to classroom-test the first-semester portion of the book this fall, although I have already taught the infinitesimal material in all three semesters of calculus (anywhere from two to five semesters each).  In regards to the original question, my plan is to make the text as easy to use as possible by instructors unfamiliar with the methods, mostly through ample comments in the margins.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In many ways, the text I am writing is similar in scope to Stewart and other popular calculus textbooks.  The same topics are covered in a very similar order, with the main difference being the use of infinitesimals.  I have incorporated a few other differences from a pedagogical standpoint, based upon decades of experience with algebraically-weak modern calculus students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a labor of love and I have not yet spent much time seeking a publisher.  As I am at a heavy teaching load university myself, it will take a few more years to finish.&lt;/p&gt;&#xA;" OwnerUserId="8432" LastActivityDate="2017-07-03T18:33:41.800" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12555" PostTypeId="2" ParentId="2318" CreationDate="2017-07-03T22:26:46.550" Score="1" Body="&lt;p&gt;I indicate the site &lt;a href=&quot;https://www.wolframalpha.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.wolframalpha.com&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this site you can make several types of calculations, such as derivatives, integrals and use of numerical symbology&lt;/p&gt;&#xA;" OwnerUserId="8236" LastActivityDate="2017-07-03T22:26:46.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12556" PostTypeId="2" ParentId="9783" CreationDate="2017-07-03T22:32:28.330" Score="2" Body="&lt;p&gt;You can find lots of information on this site for mathematics:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.webmath.com/sn_convert.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.webmath.com/sn_convert.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8236" LastActivityDate="2017-07-03T22:32:28.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12557" PostTypeId="2" ParentId="12552" CreationDate="2017-07-04T00:10:16.583" Score="6" Body="&lt;p&gt;This is an interesting question...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think there is a volatile bifurcation at the very outset: certainly students who will (one way or another) be filtered/tested on the Cauchy-Weierstrass viewpoint would not immediately benefit from being made aware that there were other &lt;em&gt;viable&lt;/em&gt; viewpoints. &quot;Might confuse them&quot;?!?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet, ironically, many physics and engineering groups (in my direct observation) &lt;em&gt;say&lt;/em&gt; that they want &quot;rigor&quot;, but in fact operate in the happy, optimistic world where there are infinitesimals. I think that what it amounts to is that there is filtering not on the literal mathematical content, but on ... well, obedience? Assimilation of cultural mythology?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, mixed conclusion at this point. It is entirely plausible to me that we (collectively) could arrange that students be better able to answer operational questions (rather than questions intended to gauge their orthodoxy) by using infinitesimals. If I had to bet, I'd wager that epsilon-delta is just tooooo top-heavy to survive. That is, it was not inevitable, and is not the unique &quot;foundationalization&quot; of calculus... supposing that we needed a foundation (Bishop Berkeley nevertheless...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me possible that there would be almost-completely-good reasons to introduce non-standard analysis at a graduate-math level, as a counterpoint to the usual &quot;real analysis&quot;, and as a counterpoint to (often too-cliched) &quot;functional analysis&quot;. Indeed, by this point, the people might be mature enough to appreciate that there truly is no mandate to &quot;go Cauchy-Weierstrass&quot;... even while one is &quot;doing mathematics&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast, I'd worry that 18-year-olds might not care to, or be able to, appreciate the nuances of such a discussion. Perhaps I'm too pessimistic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A last comment: apart from orthodox math people, no one will ever object to reasoning based on infinitesimals. :) It's more persuasive than the epsilon-delta thang. Duh. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: in regard to @DSF's and @BenCrowell's comments: yes, it is certainly possible to rigorously justify manipulation of infinitesimals, but that's not literally what people &lt;em&gt;do&lt;/em&gt;, operationally. Rather, in my observation, they learn some &quot;safe&quot; manipulations, and mostly can avoid trouble by only doing the &quot;safe&quot; things... without necessarily knowing &lt;em&gt;why&lt;/em&gt; they are safe.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2017-07-04T17:35:56.667" LastActivityDate="2017-07-04T17:35:56.667" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12558" PostTypeId="1" CreationDate="2017-07-05T08:16:36.520" Score="4" ViewCount="214" Body="&lt;p&gt;When are operator priorities for addition, subtraction, multiplication and division typically taught in school? This this obviously differs by geography, I'm trying to understand how common it is for computer programming languages (JavaScript, more precisely) to be taught before operator priorities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know some forms of programming (e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Logo_(programming_language)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Logo&lt;/a&gt;) may be taught quite early, but I doubt they're taught before the fact that &lt;code&gt;*&lt;/code&gt; and &lt;code&gt;/&lt;/code&gt; have higher priority than &lt;code&gt;+&lt;/code&gt; and &lt;code&gt;-&lt;/code&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8436" LastActivityDate="2017-07-05T13:41:18.023" Title="When are arithmetic operator priorities typically taught in school, in relation to computer programming?" Tags="|computer-science|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12559" PostTypeId="2" ParentId="12558" CreationDate="2017-07-05T13:41:18.023" Score="3" Body="&lt;p&gt;One data point here ... order of operations, 5th grade (age approx 11).  I don't know what (if any) programming instruction occurs by that age.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-07-05T13:41:18.023" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12560" PostTypeId="1" CreationDate="2017-07-05T15:09:16.337" Score="15" ViewCount="318" Body="&lt;p&gt;As far as I know (and here I am refering to my own math education), operator priorities of &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cdot$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt;, power and parenthesis are taught via some simple phrases like&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;pointy&amp;quot; operators (&lt;span class=&quot;math-container&quot;&gt;$\cdot,\div$&lt;/span&gt;) before &amp;quot;dashy&amp;quot; operators (&lt;span class=&quot;math-container&quot;&gt;$+,-$&lt;/span&gt;),&lt;/li&gt;&#xA;&lt;li&gt;powers before both,&lt;/li&gt;&#xA;&lt;li&gt;parentheses before everything.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But for many students a formula like &lt;span class=&quot;math-container&quot;&gt;$1 + 3 \cdot 4 \div 7 - 2^2$&lt;/span&gt; is nothing more than a confusing sequence of symbols. I (as a mathematician and partially computer scientist) think more in terms of trees when seeing something like this. My brain automatically inserts the parenthesis and I have a feeling for how to type this into my calculator to get the result that is implied by these symbols.&lt;/p&gt;&#xA;&lt;p&gt;However, we all know that students often struggle with applying these rules. My conjecture here is (and please correct me) that they see the above formula as a linear object instead if a tree like structure:&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/CkXI9.png&quot; width=&quot;250&quot; /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What do you think about this approach of teaching operator priorities, or the structure of a formula in general? Could it be helpful when student are taught to draw these trees like they were initially taught schemes for addition, multiplication, etc.?&lt;/li&gt;&#xA;&lt;li&gt;Is such an approach already in use? If not, do you think it would be useful? If yes, does it help?&lt;/li&gt;&#xA;&lt;li&gt;At which stage of learning math do you think it is important to start teaching this? Maybe elementary school is too early, maybe university is too late.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8437" LastEditorUserId="15290" LastEditDate="2022-07-01T17:20:11.853" LastActivityDate="2022-07-05T07:13:42.903" Title="Learning operator priorities by drawing trees" Tags="|undergraduate-education|arithmetic-operations|arithmetic|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12561" PostTypeId="2" ParentId="10094" CreationDate="2017-07-05T16:39:51.650" Score="4" Body="&lt;p&gt;This question simply turns on the definition of the word &quot;identity&quot;. Michael E2's answer is currently the best one, because he actually references a definition. Any college algebra or precalculus book that I can touch has such a definition, and every one that I see is effectively equivalent. In addition to Michael E2's examples, I can add:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ratti &amp;amp; McWatters, &lt;em&gt;Precalculus: A right triangle approach&lt;/em&gt;, 2E (Sec 1.1) [2011]:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An equation that is satisfied by every real number in the domain of&#xA;  the variable is called an &lt;strong&gt;identity&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Sullivan, &lt;em&gt;Algebra &amp;amp; Trigonometry&lt;/em&gt;, 7E (Sec 1.1) [1987-2005]:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An equation that is satisfied for every value of the variable for&#xA;  which both sides are defined is called an &lt;strong&gt;identity&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Rietz &amp;amp; Crathorne, &lt;em&gt;Introductory College Algebra&lt;/em&gt;, Revised (Ch. II, Art. 18) [1923-1933]:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The two members of an identity are equal for all values of the symbols&#xA;  for which the expressions are defined.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I include dates and the latter example to highlight that the customary definition of an &quot;identity&quot; has been stable and consistent for at least the last century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The OP should refer the student to the definition in use in the course's textbook and/or lecture notes (which hopefully matches all of the preceding examples). In particular, an identity only needs values to match when both sides are defined; so it is not foiled by looking at values outside the domain of the expressions. This answers the student's question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a similar example that one should be preparing students to look very carefully at their definitions, consider the fact that the &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10600/why-is-continuity-only-defined-on-its-domain&quot;&gt;definition of a continuous function similarly looks only at points in the domain&lt;/a&gt; (so that, e.g., $1/x$ qualifies as a continuous function). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-07-05T16:39:51.650" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12562" PostTypeId="2" ParentId="5586" CreationDate="2017-07-05T22:00:45.580" Score="4" Body="&lt;p&gt;I feel that the excellent answers given already may not do enough to discourage the idea that you &lt;em&gt;need&lt;/em&gt; to be perfect. While I do understand your aim towards a perfect grade, you need to embrace the possibility that this may not happen and to not let that deter you from continuing your math education with the same passion you've shown so far. Grades are an arbitrary measure, which have their place in education, but they must not become what defines you; you are not your grade. It is from failure that we build the most astounding successes. Mathematical perfection in a student is beautiful in the way the smell of a flower makes the tree beautiful: will you reject the refreshing taste of its fruit, the intricate pattern of its leaves, the impressive strength of its root because its flower didn't smell how you expected? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Embrace rigor, and never settle for less than your best, but seeking perfection during your mathematical exploration above all else may cloud the wonderful growth it may spur in you whenever you meet failure, the same failure even your teachers met, as they were preparing to guide you through the wonderful journey ahead.&lt;/p&gt;&#xA;" OwnerUserId="7893" LastActivityDate="2017-07-05T22:00:45.580" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12564" PostTypeId="2" ParentId="5586" CreationDate="2017-07-06T01:15:22.713" Score="1" Body="&lt;p&gt;The best way to do well in math is to solve problems.  (Not to deeeeply understand things.)  Deeply understanding things is fine.  Even beneficial.  but skill in problems is more important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Per the actual question:  yes, Schaum's Outlines.  Perfect for what you want which is to drill the A.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oh...and maybe it is just me, but I find myself realizing things about the concepts AS I DRILL.&lt;/p&gt;&#xA;" OwnerUserId="8441" LastActivityDate="2017-07-06T01:15:22.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12565" PostTypeId="2" ParentId="238" CreationDate="2017-07-06T18:33:10.123" Score="2" Body="&lt;p&gt;These suggestions are more from experience during vacations back from my childhood.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.goodreads.com/book/show/6242815-mathematics-can-be-fun&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics can be fun&lt;/a&gt; by Yakov Perelman&lt;/li&gt;&#xA;&lt;li&gt;Puzzles to puzzle you by Shakuntala Devi&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.goodreads.com/book/show/13591377-the-sherlock-holmes-puzzle-collection&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sherlock Holmes puzzle collection&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8442" LastActivityDate="2017-07-06T18:33:10.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12566" PostTypeId="2" ParentId="238" CreationDate="2017-07-06T22:38:20.897" Score="2" Body="&lt;p&gt;Chaos&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Flatland&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Men of Mathematics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anything by Martin Gardner, but especially his columns in IASFM (not sure if ever collected).&lt;/p&gt;&#xA;" OwnerUserId="8443" LastActivityDate="2017-07-06T22:38:20.897" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12567" PostTypeId="2" ParentId="12530" CreationDate="2017-07-08T01:28:56.857" Score="2" Body="&lt;p&gt;As already pointed out, there is no ULTIMATE calculus syllabus but, considering that a large majority of students in the Precalculus-Calculus sequence are not prospective mathematicians or even prospective physicists or prospective engineers, the following might be worth giving it some thought before reaching for one's gun:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Arithmetic-Algebra&lt;/strong&gt;. A major emphasis on &lt;strong&gt;decimal numbers&lt;/strong&gt;. In particular, look at &lt;em&gt;fractions&lt;/em&gt; and &lt;em&gt;roots&lt;/em&gt; as just code for algorithms generating decimals. Then, show how the four operations on &lt;strong&gt;Laurent polynomials&lt;/strong&gt; are a mere continuation, sometimes simpler (No carryover, borrowing, etc), of the four operations on (finite) decimal numbers. Forego just about everything else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Precalculus-Differential Calculus&lt;/strong&gt;. As a continuation of the above and in place of a rehash of Algebra and a bad introduction to trigonometry followed by avoidance maneuvers around limits, bring in the differential calculus as the &lt;strong&gt;calculus of approximations&lt;/strong&gt; rather than as the &quot;calculus of change&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, study algebraic functions by looking at the &lt;strong&gt;Laurent Polynomial Approximations&lt;/strong&gt; of $f(x_{0}+h)$. Define $f^{(n)}$ as the function which outputs the coefficient of $h^{n}$ (divided by $n!$ to permit recursion.). Study similarly the transcendental functions defined from the differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Integral Calculus&lt;/strong&gt;. Replace by &lt;strong&gt;Dynamical Systems&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Notes&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A treatment such as the above has a much better chance of being &quot;convincing&quot;---as in Stephen Toulmin's &lt;em&gt;The uses of Argument&lt;/em&gt; with non---mathematics majors---and for the late mathematical bloomer offers a serious basis on which to ask serious questions about rigor. For more details, see &lt;a href=&quot;http://www.ams.org/notices/201303/rnoti-p340.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Content Matters&lt;/a&gt; and &lt;a href=&quot;https://www.researchgate.net/publication/281444563_To_Calculate_In_Calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;To Calculate In Calculus.&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also, concerning the use of decimals, see Timothy Gowers: ``&lt;em&gt;physical measurements are not real numbers. That is, a measurement of a physical quantity will not be an exactly accurate infinite decimal. Rather, it will usually be given in the form of a finite decimal together with some error estimate:&lt;/em&gt; x=3.14 +/- 0.02 &lt;em&gt;or something like that&lt;/em&gt;.'' &lt;a href=&quot;https://www.dpmms.cam.ac.uk/~wtg10/continuity.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;One way to arrive at the definition of continuous functions&lt;/a&gt; and &lt;a href=&quot;https://www.dpmms.cam.ac.uk/~wtg10/decimals.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;What is so wrong with thinking of real numbers as infinite decimals?&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2017-07-08T01:28:56.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12568" PostTypeId="1" CreationDate="2017-07-08T04:14:06.653" Score="4" ViewCount="151" Body="&lt;p&gt;One is often drawn to offbeat mathematical ideas and how they could revolutionize mathematics or at least make maths more easy to learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Current examples are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://web.maths.unsw.edu.au/~norman/Rational1.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rational Trigonometry&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://tauday.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tau&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.dozenal.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dozenal&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/052171916X&quot; rel=&quot;nofollow noreferrer&quot;&gt;Category Theory&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Non-standard_calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Non-Standard Calculus&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of course, such innovation can be seen in a historical perspective where all math concepts were at some time radical (irrational numbers, logarithms or hyperbolic geometry).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, these innovations can be seen to reduce cognitive load (division in base 12 is easier), simplify notation (replace 2 \pi with \tau), enrich our vocabulary (sets + morphisms = category) or clarify important concepts (triangle geometry vs circle geometry).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But is it worth pursuing these ideas and their actual use in the classroom? Or do they just get in the way and we should just accept things for what they are?&lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="998" LastEditDate="2017-07-08T04:24:39.167" LastActivityDate="2017-07-08T13:17:41.260" Title="What is the role of the efforts to change the fundamentals of maths?" Tags="|technology-in-education|notation|" AnswerCount="2" CommentCount="2" ClosedDate="2017-07-15T17:33:14.933" ContentLicense="CC BY-SA 3.0" />
  <row Id="12569" PostTypeId="2" ParentId="12560" CreationDate="2017-07-08T11:08:06.537" Score="7" Body="&lt;p&gt;It is nice, but I suspect nice for an adult or at least a much older child who is moroe used to these graphical structures.  I'm sorry but sometimes ust having extra ways of showing something is not helpful, is actually the opposite. And this would be my take on the operator priority since you are making kids learn complicated logic trees at the same time they learn the basic syntax.  I think instead having them draw in the parentheses is more intuitive and readable.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer might be different for an adult looking at the order of operations in Excel, which annoyingly is different from standard order (for no good reason just MSFT messed up and now there is too much code written to change back).&lt;/p&gt;&#xA;" OwnerUserId="8449" LastActivityDate="2017-07-08T11:08:06.537" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12570" PostTypeId="2" ParentId="12568" CreationDate="2017-07-08T11:11:32.583" Score="4" Body="&lt;p&gt;It may or not be worth pursuing them yourself (often not).  But even if worth it for yourself, it is a very bad idea for the students.  these are kids who are new to normal trigonometry.  It is enough of  rush of strangeness just to cover that.  And it is hard for them.  And it has high utility in other math courses and in physics.  Concentrate on the immediate task, which is plenty hard enough, to teach normal math.&lt;/p&gt;&#xA;" OwnerUserId="8450" LastActivityDate="2017-07-08T11:11:32.583" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12571" PostTypeId="2" ParentId="12568" CreationDate="2017-07-08T13:17:41.260" Score="3" Body="&lt;p&gt;I would strongly advise against teaching any of these topics except possibly non-standard calculus. (Even then you should spend some time to mention limits.) A large part of early math education is learning the usual language so that your students can communicate with others. Using radically non-standard methods robs them of the enormous benefits of communication without a benefit of compensating size. (i.e. Base-12 arithmetic buys you easier division by 3, which is a pretty mild benefit. More usefully, learn base-10 to communicate with people and base-2 to communicate with computers.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Non-standard calculus should, however, be fine. After all, it looks exactly like standard calculus other than in theorems involving limits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Category theory is also deserves its own separate mention. It just doesn't have any obvious benefit unless the objects you are dealing with are fairly abstract.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. Are your completely sure that rational trig is even true? That website seems a bit ... off.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2017-07-08T13:17:41.260" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12572" PostTypeId="2" ParentId="12560" CreationDate="2017-07-08T21:56:23.477" Score="7" Body="&lt;p&gt;It is rather interesting to note how parallel this suggestion runs to the idea of diagramming sentences in English grammar classes at around the same age, historically. I say &quot;historically&quot; because (as noted in comments) my understanding is that diagramming sentences is very much in disrepute, no longer taught, and mostly unknown by current education-school graduates. More generally, even the very idea of dedicated grammar lessons has been officially frowned upon for the last few decades. It is claimed that a multitude of studies show no relation between diagramming sentences/grammar lessons and educational outcomes. Some articles with summaries and links to such meta-studies:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.theatlantic.com/education/archive/2014/02/the-wrong-way-to-teach-grammar/284014/&quot; rel=&quot;noreferrer&quot;&gt;The Wrong Way to Teach Grammar&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://ttlearning.com/blog/why-dont-us-schools-teach-grammar/&quot; rel=&quot;noreferrer&quot;&gt;Why Don't U.S. Schools Teach Grammar?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://radicalscholarship.wordpress.com/2014/08/24/diagramming-sentences-and-the-art-of-misguided-nostalgia/&quot; rel=&quot;noreferrer&quot;&gt;Diagramming Sentences and the Art of Misguided Nostalgia&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Among the most common criticisms are (very similar to the answer here by user guest): &quot;... On a less emotional level, they point out that it involves mastering not one skill but two: the rules of grammar and syntax and the making of diagrams, which is something like baking a loaf of bread and cutting it into fancy shapes before you can make a sandwich.&quot; (This by sentence-diagramming enthusiast &lt;a href=&quot;https://opinionator.blogs.nytimes.com/2012/06/18/taming-sentences/&quot; rel=&quot;noreferrer&quot;&gt;Kitty Burns Florey&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally speaking, I think this is a loss, and personally I refer to the twinned removal of grammar/algebra skills around the junior-high level as part of the &lt;a href=&quot;http://www.madmath.com/2013/06/the-war-on-structure.html&quot; rel=&quot;noreferrer&quot;&gt;&quot;War on Structure&quot;&lt;/a&gt;. However, before pushing for this particular CS search-tree-like method, you should at least be aware that such methods are explicitly frowned upon in many educational circles, lack any analog that students are familiar with from grammar lessons, and supposedly have hundreds of studies showing that they do not benefit students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally I do drill students on the order-of-operations by asking them to &lt;strong&gt;write out in a linear list&lt;/strong&gt; the order of the operations as they will be performed. To my mind, this does address somewhat more directly the fact that operations have an order that will occur one at a time. For example, it's how the quiz on my drill site &lt;a href=&quot;http://www.automatic-algebra.org/orderofoperations.htm&quot; rel=&quot;noreferrer&quot;&gt;Automatic Algebra&lt;/a&gt; is structured. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-07-09T01:48:50.690" LastActivityDate="2017-07-09T01:48:50.690" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12573" PostTypeId="2" ParentId="12560" CreationDate="2017-07-09T00:28:11.233" Score="6" Body="&lt;p&gt;I wonder if you can make an inroad by pointing out that every calculator they use employs some order of operations (e.g., BODMAS), and actually builds the tree and then evaluates the expression following its structure. So you could mix the math with the &quot;how-does-a-calculator-work&quot; discussion. Maybe that would be more impactful than just introducing trees as a convenient mental technique&amp;mdash;tree expressions are actually used by their phone calculators.&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/5lQ1s.png&quot; width=&quot;500&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;Image from &lt;a href=&quot;https://www.mathsisfun.com/algebra/operations-order-calculator.html&quot; rel=&quot;noreferrer&quot;&gt;Order of Operations Calculator&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-07-09T00:54:59.807" LastActivityDate="2017-07-09T00:54:59.807" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12574" PostTypeId="1" CreationDate="2017-07-09T14:12:47.470" Score="-3" ViewCount="706" Body="&lt;p&gt;At the end of the mathematical high-school education I usually introduce the easiest facts of set theory: counbtability and Cantor's proof as the basis of modern mathematics. Now my brightest student has opposed by mentioning that set theory contradicts the limit notion of analysis. His argument: In every interval (0, n] the ratio of not enumerated fractions and natural numbers is 0. We can consider this as a sequence: 0, 0, 0, ... which in analysis can only have the limit 0. According to set theory the limit is $\infty$ since no fraction in the interval (0, $\omega$) is not enumerated. This makes set theory useless as a basis of analysis. I could not disagree at the moment. What is a valid counter argument?&lt;/p&gt;&#xA;" OwnerDisplayName="user8455" LastActivityDate="2017-07-14T03:15:30.107" Title="How can I convince my brightest student of Cantor's theory?" Tags="|mathematical-analysis|set-theory|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12575" PostTypeId="2" ParentId="12574" CreationDate="2017-07-09T14:34:04.283" Score="14" Body="&lt;p&gt;Not every property is preserved by limits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a more basic situation in which the same reasoning is used:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For each natural number $n$, there are only finitely many natural numbers in the interval $[0,n]$.  However, letting $n \to \infty$, there are infinitely many natural numbers in the interval $[0,\infty)$.  There is clearly no contradiction here, but the argument is of the same form as offered by the student.  The function which assigns the word &quot;finite&quot; or &quot;infinite&quot; to sets is not continuous under limits.  So why should his function be continuous?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also point out that the statement that &quot;set theory is useless as a basis for analysis&quot; is just plain false.  It is the basis for analysis.  He just isn't using it correctly.  It is similar to saying that ``hammers are useless for driving nails'' after attempting once to drive a nail into a piece of wood by hitting it with the handle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wouldn't recommend being this blunt with a student though.  For constructive advice would be to focus on the point about ``continuity''.  &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2017-07-09T14:34:04.283" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12576" PostTypeId="2" ParentId="9728" CreationDate="2017-07-09T18:22:53.637" Score="0" Body="&lt;p&gt;I don't see why a freshman should jump right into Rudin.  Why not learn calculus in terms of manipulation with the basic explanations that are usually done in courses.  And go all the way through Calc 1, 2, 3, and ODEs.  Then junior year is fine to do a recursion into theoretical calculus.  The vast amount of engineers will never ever need real analysis.  For the few who do need it, most of them will learn it better by recursion than by trying to do everything at once.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But somehow people here think it is better to learn it as a freshman.  Would you teach Dedekind cuts for arithmetic students also?  Even many math majors who go into anything applied (stats, actuarial, OR, finance, etc.) say there is little value from the theoretical calc course.  Conversely being able to handle some algebraically complex (no pun intended) problems is highly useful to physicists and engineers in derivations in their courses and in their homework problems. I don't even really think the more real analysis you pack into a calc course the better it is.  The more tricks and techniques, the better it is.&lt;/p&gt;&#xA;" OwnerUserId="8456" LastActivityDate="2017-07-09T18:22:53.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12579" PostTypeId="2" ParentId="8" CreationDate="2017-07-10T02:38:37.053" Score="4" Body="&lt;p&gt;Here is what Richard Feynman advised CALTECH students regarding mathematics (section 1-3 of Feynman Lectures Physics):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, this guy comes into my office and asks me to try to make everything straight that I taught him, and this is the best I can do. The problem is to try to explain the stuff that was being taught. So I start, now, with the review. I would tell this guy, “The first thing you must learn is the mathematics. And that involves, first, calculus. And in calculus, differentiation.” &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, mathematics is a beautiful subject, and has its ins and outs, too, but we’re trying to figure out what the minimum amount we have to learn for physics purposes are. So the attitude that’s taken here is a “disrespectful” one towards the mathematics, for sheer efficiency only; I’m not trying to undo mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we have to do is to learn to differentiate like we know how much is 3 and 5, or how much is 5 times 7, because that kind of work is involved so often that it’s good not to be confounded by it. When you write something down, you should be able to immediately differentiate it without even thinking about it, and without making any mistakes. You’ll find you need to do this operation all the time—not only in physics, but in all the sciences. Therefore differentiation is like the arithmetic you had to learn before you could learn algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, the same goes for algebra: there’s a lot of algebra. We are assuming that you can do algebra in your sleep, upside down, without making a mistake. We know it isn’t true, so you should also practice algebra: write yourself a lot of expressions, practice them, and don’t make any errors. Errors in algebra, differentiation, and integration are only nonsense; they’re things that just annoy the physics, and annoy your mind while you’re trying to analyze something. You should be able to do calculations as quickly as possible, and with a minimum of errors. That requires nothing but rote practice—that’s the only way to do it. It’s like making yourself a multiplication table, like you did in elementary school: they’d put a bunch of numbers on the board, and you’d go: “This times that, this times that,” and so on—Bing! Bing! Bing!&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;P.s. Having some rote skills makes it easier to work on other aspects of problems.  That is not same as saying having no intuitions is helpful either.  Especially for things you don't grind into muscle memory.  But for others better to have almost iconic recall, like &quot;bsq-4ac&quot;; do you really want to be completing the square because you don't remember the quadratic as you go through all of upper math, chemical equilibrium problems, physics everywhere, or EE or control systems (characteristic equation of harmonic oscillator ODE).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.  I actually find that some ideas come to me AS I DRILL, about the substance of what I am working on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.s.  When working a homework set (on a new concept), it can be useful to have some easier, drill type problems for the new equation.  Then progress to harder problems with proofs or word problems or combining concepts. Because you have a little more feel for things than just reading the text would give you, before you dive into the nutcracker questions.  This is actually a major failing of harder classes, graduate school etc. in that the easier running start problems are omitted.  Than again these classes seem to have a very cavalier attitude towards actually training people.&lt;/p&gt;&#xA;" OwnerUserId="8460" LastActivityDate="2017-07-10T02:38:37.053" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12580" PostTypeId="2" ParentId="12574" CreationDate="2017-07-10T05:50:25.880" Score="9" Body="&lt;p&gt;The problem is not one of reconciling analysis and set theory at all; it's one of understanding analysis properly (and how things go sideways when dealing with the infinite). One of the first and primary questions of analysis is, &quot;when can limits and operators be interchanged?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see why this is not trivial, consider the following exampe, which is a standard demonstration of the need for the dominating function in the Dominated Convergence Theorem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $f_n(x)=n\chi_{_{(0,1/n]}}$. Then $$\int{f_n}=1$$ for every $n$ whence $$\lim\int f_n =1$$&#xA;$\textrm{}$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But now let's consider the pointwise limit of our sequence of functions. Clearly we have that $$\lim f_n=0$$ which means that $$\int\lim f_n=0$$ and thus we can conclude that $$\lim\int f_n=1\ne 0= \int\lim f_n$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider that the argument in play from your student is analogous. The claim is that the limit and the taking of the ratios commute, i.e. the limit of the ratio is the same as the ratio of the limit (the &quot;limit&quot; of the sets is itself a measure theory notion, not simply a set theoretic one!). But there is no justification for that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a side note (and perhaps I've misinterpreted the tone of things), this could be a teachable moment for your student. While bright, the knowledge and understanding that they possess is but a sliver of the knowledge that any one of thousands of mathematicians across the world hold at this moment in time, to say nothing of those throughout history. Topics like these are not new and have been fleshed out through over a hundred years of painstaking work. We want our students to ask questions and to think critically, but when it comes to fairly settled topics in math, they will likely learn much more with a, &quot;my argument leads me to a seemingly unreasonable result; where did I go wrong?&quot; than with the thought that they have shown those who have come before to be wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; The original claim itself wasn't formally written out, so I'm adding this as an addendum so that we can see the claim made explicit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let the set $A_n=\mathbb{Q}\cap (0,n]$ and $B_n=\mathbb{N}\cap (0,n]$. Then define $f(n)=\frac{\lvert B_n\rvert}{\lvert A_n\rvert}$ (note that in order for this function to be defined, we are working with the extended real numbers). Then $$\lim_{n\to\infty} f(n)=\lim_{n\to\infty} 0=0$$ But $$f\left(\lim_{n\to\infty}n\right)=f(\omega)=1$$ The claim is then that $$\lim_{n\to\infty} f(n)=f\left(\lim_{n\to\infty}n\right)$$ which would indicate a contradiction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Except for the limit commuting with the operator (the operator is $f$) is not &lt;em&gt;a priori&lt;/em&gt; allowed. In this case, the property allowing for the commutation would be continuity (as noted in Steven Gubkin's answer), for which there has been no justification.   &lt;/p&gt;&#xA;" OwnerUserId="6949" LastEditorUserId="6949" LastEditDate="2017-07-14T03:15:30.107" LastActivityDate="2017-07-14T03:15:30.107" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12582" PostTypeId="2" ParentId="12574" CreationDate="2017-07-10T20:21:28.170" Score="8" Body="&lt;p&gt;Here is another attempt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the function $f:\mathbb{N} \to \mathbb{R}$ defined by $f(n) = \textrm{ The number of elements in the set $\{n,n+1,n+2,...2n\}$}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words $f(n) = n+1$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clearly $$\lim_{n \to \infty} f(n) = \infty$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the limit of the sets $\{n,n+1,n+2,...2n\}$ is the empty set, since no element would be a member of such a limit.  So the &quot;set theoretic'' limit would be $0$.  Is mathematics in jeopardy because of this &quot;contradiction''?  Or perhaps there is no contradiction at all.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2017-07-10T20:21:28.170" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12583" PostTypeId="1" AcceptedAnswerId="12584" CreationDate="2017-07-11T04:21:56.407" Score="12" ViewCount="376" Body="&lt;p&gt;I'm teaching a remedial algebra class, and I recently put a radical equation on a quiz. At this point, the students had only solved polynomial equations by factoring, so the equation had to turn out nice (plus, there would be issues checking whether a potential solution like $1 + \sqrt{2}$ is extraneous).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My Problem:&lt;/strong&gt; Find $(a, b, c)$ so that $a\sqrt{x + b} = x + c$ has solutions that can be found by squaring and factoring a quadratic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked a computer (&lt;a href=&quot;https://sagemath.org&quot; rel=&quot;noreferrer&quot;&gt;Sage&lt;/a&gt;) to try various to run through combinations of $a, b, c$, weeding out those that didn't have at least one integer solution. It worked perfectly well, although it required some legwork to get the &quot;program&quot; running. I initially tried some algebraic things with discriminants (making the brute force more efficient, but it ended up taking less than a second either way), but I don't know much about Diophantine equations or perfect squares and had to resort to brute force all the same, just slightly differently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there any way number theory could help me out here? Has anyone personally benefited from a bit of number theory, creating problems like this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realize now that just picking two points on a graph like $x = a(y - k)^2 + h$ and connecting them with a line is probably a &lt;em&gt;much&lt;/em&gt; more efficient approach to generate equations, but might require more manual labor to avoid equations that are more complicated to solve than I'd like. Even if that is the case...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;...this is mostly a question of curiosity; I don't see a pressing need to churn out equations like this, and brute force works just fine. But I've always wondered whether knowing more (or &lt;em&gt;any&lt;/em&gt;, at this point) number theory would be at all useful, creating problems for an algebra course like this.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Sidenote: I went through a similar process the last time I taught the class. Then, I wanted a quadratic equation whose solutions, using the quadratic formula, would need simplifying by reducing the radical and then cancelling a common factor of the numerator and denominator. I ended up stumbling around quadratic fields. Although I got my examples, nothing really came of it.&lt;/p&gt;&#xA;" OwnerUserId="5153" LastActivityDate="2017-07-11T14:38:03.783" Title="Can number theory help me create equations with nice solutions?" Tags="|algebra|problem-design|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12584" PostTypeId="2" ParentId="12583" CreationDate="2017-07-11T14:09:20.907" Score="5" Body="&lt;p&gt;I think that the number theory in this case is pretty light. Let's suppose that we want some integer solution for $x$ and that $a,b,c$ are also integers. Then $x+b$ must be a perfect square, so let $N^2=x+b$. The radical equation can then be rewritten as $$N^2-aN-b+c=0.$$ Note: I'm assuming $N\geq 0$ because it doesn't matter. The quadratic equation then gives $$N=\frac{a\pm\sqrt{a^2+4(b-c)}}{2}.$$ As $N$ is an integer, we need the part under the radical to again be a perfect square. So let $$M^2=a^2+4(b-c).$$ Again, we may as well suppose $M\geq 0$. Then $N=\frac{a\pm M}{2}$. Clearly $a$ and $M$ must have the same parity as the difference of their squares is divisible by 4, so we don't have to worry about the fraction being a half integer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, choose any integers $a$ and $b$. Then choose a (non-negative) integer $M$ with the same parity as $a$ and for which at least one of $a\pm M =2N\geq 0$. Then $c=\frac{a^2-M^2}{4}+b$. (Which is an integer since $a$ and $M$ have the same parity.)&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2017-07-11T14:09:20.907" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12585" PostTypeId="2" ParentId="12583" CreationDate="2017-07-11T14:38:03.783" Score="5" Body="&lt;p&gt;Begin with factored quadratic $(u-r_1)(u-r_2)=0$ so $u^2-(r_1+r_2)u+r_1r_2=0$. Choose one root positive, say $r_1&amp;gt;0$ whereas $r_2&amp;lt;0$ then we solve for $u$ using positive root,&#xA;$$ u = \sqrt{(r_1+r_2)u-r_1r_2} $$&#xA;Let $u=x-C$ thus,&#xA;$$ x-C = \sqrt{(r_1+r_2)(x-C)-r_1r_2} $$&#xA;thus,&#xA;$$ x = C+\sqrt{(r_1+r_2)x-C(r_1+r_2)-r_1r_2} $$&#xA;whose solution $u=r_1$ means $x-C=r_1$ or $x=C+r_1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me try it out. Let $r_1=3$ and $r_2=-2$ then&#xA;$$ x = C+\sqrt{x-C+6} $$&#xA;Set $C=2$ to find $x=2+\sqrt{x+4}$. Now, if my reverse engineering worked, we should find $x=3+2$ and $x=-2+2=0$ as solutions of which $x=0$ is extraneous. Isolate the radical,&#xA;$$ x-2 = \sqrt{x+4} \ \ \Rightarrow \ \ (x-2)^2= x+4 \ \ \Rightarrow \ \ x^2-4x+4 =x+4 $$&#xA;thus $x(x-5)=0$ so we find $x=0$ or $x=5$. However, $0=2+\sqrt{0+4}$ suggests that $2+\sqrt{4}=0$ which is absurd whereas $x=5$ will check as a genuine solution to the constructed problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, basically, I give a template to set-up problems of the form $x = C+\sqrt{x+B}$. Notice I am missing the desired $A$-factor. It's easy to multiply my set-up by a constant to get problems of the form $mx =mC+ m\sqrt{x+B}$. For example, no one should be surprised that $3x = 6+3\sqrt{x+4}$ has solution $x=5$. To make it more exciting, pull the $3$ inside the root, $3x=6+\sqrt{9x+36}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to set-up a nice revered engineered version for your stated problem, but, getting the $A$ to appear there without fractions requires certain conspiracies between the constants which are difficult to arrange.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, I think this is a great question. The problem of constructing nice problems is a hard problem. I would not be surprised if someone could improve on my bruteforce answer here.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-07-11T14:38:03.783" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12586" PostTypeId="1" CreationDate="2017-07-11T15:48:03.887" Score="10" ViewCount="904" Body="&lt;p&gt;A trigonometry text like Sullivan's &lt;em&gt;Algebra &amp;amp; Trigonometry&lt;/em&gt; often has a prohibition like this (Sec. 7.3):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;WARNING:&lt;/strong&gt; Be careful not to handle identities to be established as if they were conditional equations. You &lt;em&gt;cannot&lt;/em&gt; establish an identity&#xA;  by such methods as adding the same expression to each side and&#xA;  obtaining a true statement. This practice is not allowed, because the&#xA;  original statement is precisely the one that you are trying to&#xA;  establish. You do not know until it has been established that is, in&#xA;  fact, true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'll look at the first such example in Sullivan. Establish the identity: $\csc \theta \cdot \tan \theta = \sec \theta$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The given solution is: $\csc \theta \cdot \tan \theta = \frac{1}{\sin \theta} \cdot \frac{\sin \theta}{\cos \theta} = \frac{1}{\cos \theta} = \sec \theta$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But let's consider using the prohibited operations that generate equivalent equations. We may write: $\csc \theta \cdot \tan \theta = \sec \theta \iff \frac{1}{\sin \theta} \cdot \frac{\sin \theta}{\cos \theta} = \frac{1}{\cos \theta} \iff \sin \theta = \sin \theta$. [Multiplying both sides by $\sin \theta \cdot \cos \theta$ in the last step.] Now, personally, this does persuade me that the original equation is an identity, because it is equivalent to an obviously-true equation (excepting any values not in the domain of one side, which are identical to the official answer above). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we want to be completely rigorous, then I think we could just reverse the sequence of statements above. Starting with the reflexive property of equality: $\sin \theta = \sin \theta \iff \frac{1}{\sin \theta} \cdot \frac{\sin \theta}{\cos \theta} = \frac{1}{\cos \theta} \iff \csc \theta \cdot \tan \theta = \sec \theta$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The official book proof is slightly shorter, and it can be written down directly without the reversal, but it seems mostly like a stylistic point -- fighting with students to abandon their equivalent-equation techniques is a great burden, and right now I'm not entirely seeing the necessity for it. Does anyone permit this alternative as an acceptable proof in their trigonometry class? If not, what is the compelling reason for Sullivan's dire warning? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit: This question presumes that any applied operations &lt;em&gt;do correctly generate equivalent equations&lt;/em&gt; (noting gaps in the domain as required). The question is whether Sullivan's &quot;cannot establish an identity by such methods as adding&quot; statement is strictly true.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-07-12T20:59:56.197" LastActivityDate="2017-09-03T22:39:31.070" Title="May we permit identities to be established by equivalent equations?" Tags="|mathematical-pedagogy|proofs|trigonometry|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12587" PostTypeId="2" ParentId="12586" CreationDate="2017-07-11T16:57:06.807" Score="6" Body="&lt;p&gt;For nearly any identity that we will run across in a standard trig class, I believe that the method that you described (the rigorous one) would be sufficient. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the problem with the logic is that not all modifications that we do to equations are invertible, so one would need to take great care in some situations. Consider the incorrect identity $\sin(\theta)=\sqrt{1-\cos^2(\theta)}$ which we could &quot;prove&quot; by the following: $$\begin{align}\sin(\theta)&amp;amp;=\sqrt{1-\cos^2(\theta)}\\ \iff\sin^2(\theta)&amp;amp;=1-\cos^2(\theta)\\ \iff \sin^2(\theta)+\cos^2(\theta)&amp;amp;=1\\ \iff 1&amp;amp;=1\end{align}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can of course see where this is wrong and make appropriate adjustments and notes, but our students would not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general though, I think that the idea that your presenting is useful for finding the correct thought process much the same way that we find the appropriate $\varepsilon$ in analysis by working backward. It can be used as part of the &quot;figuring out&quot; process, but in my opinion there is still a value for teaching a more rigorous approach (logically speaking).&lt;/p&gt;&#xA;" OwnerUserId="6949" LastActivityDate="2017-07-11T16:57:06.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12588" PostTypeId="2" ParentId="12586" CreationDate="2017-07-11T17:02:16.923" Score="10" Body="&lt;p&gt;Well, &lt;em&gt;if&lt;/em&gt; their &quot;equivalences&quot; are in fact equivalences,&#xA;then everything is fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem seems to be mostly a psychological one:&#xA;When students want to verify an equivalence, they check&#xA;the left-to-right implication, but they often seem to take&#xA;the right-to-left one for granted. As a result, they perform&#xA;operations like multiplying both sides of an equation&#xA;with some value (that is not guaranteed to be non-zero!),&#xA;and they put, quite routinely, an equivalence sign between&#xA;both formulas – which is wrong.&#xA;Unfortunately, when they&#xA;start with the property to be proved, the left-to-right implication&#xA;is pointless, and the right-to-left implication is the important one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I want to demonstrate a proof on the blackboard and start with the property to be proved, I'd rather use&#xA;a right-to-left implication sign than an equivalence sign;&#xA;just because it makes students focus on the important one&#xA;of the two directions.&lt;/p&gt;&#xA;" OwnerUserId="8472" LastActivityDate="2017-07-11T17:02:16.923" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12589" PostTypeId="2" ParentId="12586" CreationDate="2017-07-12T07:00:30.390" Score="8" Body="&lt;p&gt;When you solve an algebraic equation like $2x - 4 = 0$, what you're really doing is finding assignments for the variables &lt;em&gt;assuming&lt;/em&gt; the given equation is true. Doing the same thing to both sides of an equation is meant to take a &lt;em&gt;true&lt;/em&gt; equation and turn it into another true equation. That is, it can be used to form correct &lt;em&gt;implications&lt;/em&gt;, like $2x - 4 = 0 \implies 2x = 4$ and $2x = 4 \implies x = 2$. Those implications can be chained to get $2x - 4 = 0 \implies x = 2$, which is the &quot;answer&quot;. This implication is all that the problem is asking you to show, that assuming $2x - 4 = 0$ is true, $x = 2$ is also true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you're trying to &lt;em&gt;prove&lt;/em&gt; an identity, you don't assume that the identity is true, and an implication doesn't serve as an answer. Applying the same sort of steps used above to $\csc \theta \cdot \tan \theta = \sec \theta$ gets you $\csc \theta \cdot \tan \theta = \sec \theta \implies \sin \theta = \sin \theta$. But you need to show $\csc \theta \cdot \tan \theta = \sec \theta$ by itself. You can't use the implication you just derived because it goes in the wrong direction (and like Uwe pointed out in their answer, this direction of implication isn't used in the problem at all, and could be distracting to include). You instead need $\sin \theta = \sin \theta \implies \csc \theta \cdot \tan \theta = \sec \theta$; then, since you already know $\sin \theta = \sin \theta$, you'd get just $\csc \theta \cdot \tan \theta = \sec \theta$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, the particular algebraic steps you used can be inverted to get this implication. The biggest problem is that some steps aren't invertible, which rnrstopstraffic mentioned in their answer. Another problem, IMO, is that students wouldn't understand the correct logical steps of their answer if they wrote it like solving an equation. They probably wouldn't recognize that the correct proof uses the inverses of the operations they wrote down. It'd be clearer to them if they wrote down such inverse steps explicitly if they were proving it this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book provides that warning because it wants to point out that solving it like an equation wouldn't be using correct logic (because it's a textbook, a place where such a detail should be mentioned). The statement is also stated generally to be applicable for proofs in general (like those found in upper-level college math courses). If you're more concerned about students just doing the algebra correctly, then you don't need to mark your students off for doing it like this, but I wouldn't solve it like an equation for examples you show during class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I prefer how the book solved it, by just simplifying the expressions on one (or both) sides of the identity. It's simpler, and the expression on each side of the equal sign remains equivalent to the ones in the original identity, so it's clearer what was actually shown: $\frac{1}{\cos \theta} = \frac{1}{\cos \theta}$ proves that $\csc \theta \cdot \tan \theta = \sec \theta$ because the respective left sides are equal to each other, and same for the right sides. You didn't need any equivalent equations at all, just equivalent expressions in the same equation.&lt;/p&gt;&#xA;" OwnerUserId="8473" LastActivityDate="2017-07-12T07:00:30.390" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12593" PostTypeId="1" CreationDate="2017-07-13T01:09:24.670" Score="10" ViewCount="493" Body="&lt;p&gt;Standard calculus textbooks begin by introducing limits, including&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;limits of a fraction as the numerator and denominator approach $0,$&lt;/li&gt;&#xA;&lt;li&gt;limits of a fraction as the numerator and denominator approach $\infty,$&lt;/li&gt;&#xA;&lt;li&gt;limits of continuous functions as their argument approaches a point in their domain,&lt;/li&gt;&#xA;&lt;li&gt;limits of piecewise defined functions,&lt;/li&gt;&#xA;&lt;li&gt;limits at $\infty$ of things like the arctangent function,&lt;/li&gt;&#xA;&lt;li&gt;things like $\lim\limits_{x\,\to\,-\infty} 2^x,$&lt;/li&gt;&#xA;&lt;li&gt;infinite limits,&lt;/li&gt;&#xA;&lt;li&gt;limits of functions whose very artificial-looking graphs are given,&lt;/li&gt;&#xA;&lt;li&gt;limits of a difference as both terms approach $\infty,$&lt;/li&gt;&#xA;&lt;li&gt;limits of areas or lengths as geometric figures approach specified shapes,&lt;/li&gt;&#xA;&lt;li&gt;et cetera.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;On the final exam, students are asked to find something like $\displaystyle \lim_{x\to1} \frac{x^2-1}{x-1}$ and reply that it's undefined because it's zero over zero. If that were right, then derivatives would not exist. They miss the central point, which is that in differential calculus limits are there primarily to deal with limits cases where the numerator and denominator approach zero, because that's what derivatives are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For students who are not there for the purpose of becoming mathematicians, comprehensively covering &lt;b&gt;all&lt;/b&gt; topics involving limits that they are able to understand is a mistake: It distracts them from the main event. Are there textbooks written in a way that is consistent with that fact?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This could applied to many other areas of mathematics as well.)&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="262" LastEditDate="2017-10-16T22:44:17.677" LastActivityDate="2017-10-16T22:44:17.677" Title="Surrounding a subject and strangling it to death versus concentrating on the main point" Tags="|mathematical-pedagogy|calculus|textbooks|curriculum|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="12594" PostTypeId="2" ParentId="10021" CreationDate="2017-07-13T03:12:05.763" Score="10" Body="&lt;p&gt;As already touched on here, perhaps proof by induction should &lt;em&gt;not&lt;/em&gt; be the first real method of proof that students learn. (The two-column proofs of geometry common in North American schools don't really count somehow.) Perhaps it ought to be the culmination of an introductory course on the basic methods of proof. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the self-study, interactive tutorial that comes with my &lt;a href=&quot;http://www.dcproof.com&quot; rel=&quot;noreferrer&quot;&gt;DC Proof&lt;/a&gt; software (a free, downloadable proof checker), I present other more elementary methods first: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Direct proof &lt;/li&gt;&#xA;&lt;li&gt;Proof by contradiction &lt;/li&gt;&#xA;&lt;li&gt;Proof by contrapositive &lt;/li&gt;&#xA;&lt;li&gt;Proof by cases &lt;/li&gt;&#xA;&lt;li&gt;Proof of biconditionals &lt;/li&gt;&#xA;&lt;li&gt;Manipulating Quantifiers &lt;/li&gt;&#xA;&lt;li&gt;Very basic set theory &lt;/li&gt;&#xA;&lt;li&gt;Very basic number theory (an introduction to Peano's Axioms) &lt;/li&gt;&#xA;&lt;li&gt;Finally, proof by induction &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Each worked example is followed by exercises with hints and full solutions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first proof presented is about as simple as it gets: Prove $P \land Q \implies Q \land P$ (i.e. if propositions $P$ and $Q$ are true, then propositions $Q$ and $P$ are true). Baby steps compared to proof by induction.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2017-07-13T21:28:13.030" LastActivityDate="2017-07-13T21:28:13.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12595" PostTypeId="2" ParentId="10021" CreationDate="2017-07-13T03:51:53.167" Score="9" Body="&lt;p&gt;I will add my own rough theory here. Since American students are not trained in basic logic, I think the critical fact is that they have no familiarity or understanding about implication statements $P \Rightarrow Q$. In many cases they'll be learning about the form of an implication statement, as well as induction, all at the same time, and they can't distinguish between the two issues. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there was more teaching of basic logic, and completely separate exercises on proving implication statements alone, then I would expect the difficulty of induction to be at least somewhat reduced. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-07-13T04:28:57.910" LastActivityDate="2017-07-13T04:28:57.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12596" PostTypeId="2" ParentId="12593" CreationDate="2017-07-13T07:39:14.727" Score="3" Body="&lt;p&gt;It is well known that Abraham Robinson's framework occurs in &lt;em&gt;standard&lt;/em&gt; mathematics (ZFC, etc) and Keisler's &lt;a href=&quot;https://www.math.wisc.edu/~keisler/calc.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;famous textbook&lt;/a&gt; based on rigorous infinitesimals is well aware of the problem you mentioned.  Therefore it starts with derivatives before continuity and continuity before limits.  Then limits can be defined as a paraphrase of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Standard_part_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;shadow&lt;/a&gt;. In my experience teaching over 400 undergraduates over the past 4 years using this approach, students don't make that kind of mistake when they learn true infinitesimal calculus using this &lt;em&gt;standard&lt;/em&gt; textbook.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2017-10-15T12:16:44.183" LastActivityDate="2017-10-15T12:16:44.183" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12597" PostTypeId="2" ParentId="1817" CreationDate="2017-07-13T13:32:57.057" Score="5" Body="&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Uhlenbeck_Karen.html&quot; rel=&quot;noreferrer&quot;&gt;Karen Uhlenbeck&lt;/a&gt;&lt;/strong&gt; ought to be mentioned. She made deep &lt;a href=&quot;http://www.ams.org/mathscinet/search/publications.html?pg1=INDI&amp;amp;s1=175775&quot; rel=&quot;noreferrer&quot;&gt;contributions&lt;/a&gt; in the theory of minimal immersions (more generally, harmonic maps), gauge theory of Yang Mills equations (the work of Taubes and Donaldson on four manifolds uses her work), wave maps, integrable systems (e.g. solitons, instantons), etc. She is one of the most important researchers in geometrically and physically motivated partial differential equations of her generation. Although her work as such is probably almost completely inaccessible to any but the most exceptional high school students, the study of soap films and minimal surfaces can be presented to high school students, and some part of her work treats themes that can be seen (in some distant way) as emerging from that source. Maybe something could also be said about solitons, which have applications in telecommunications (although Uhlenbeck's work is not oriented towards applications).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although her mathematics is not the most accessible, she is a model example of a contemporary woman mathematician of the highest level intellectually and professionally. She has also been &lt;a href=&quot;http://www.math.ias.edu/wam/about/note&quot; rel=&quot;noreferrer&quot;&gt;active&lt;/a&gt; in promoting the participation of women in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2017-07-13T13:32:57.057" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12598" PostTypeId="2" ParentId="10919" CreationDate="2017-07-13T19:54:24.693" Score="1" Body="&lt;p&gt;After three years of intensive work with the new math standards (as part of the writing team for a new, national common core-based mathematics curriculum), my partner and I began consulting for school districts, training their teachers in the very thing you inquired about....HOW best to teach this new math.  Almost without exception the teachers we worked with asked one common question...&quot;How can we help our students' parents?&quot;  It seemed as though nothing existed (yet) that would work with ANY curriculum to simplify the process of TEACHING it....to basically just help parents navigate these new waters in a way that made them feel confident.  After much searching, for a resource we could recommend, we found we needed to create it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In response to all of the parents and teachers (and PARENTS WHO TEACH!!) we created the &quot;Thinking Math Differently&quot; series of guide books. Each level-specific book will work alongside any curriculum you are using, enabling you to 1. gain your own understanding of each topic and 2. develop your own style of teaching it that will foster deep understanding in your child.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These guides, available on Amazon.com, are a true resource for teaching elementary mathematics!  Please feel free to reach out with any specific questions you may still have! Our email address is conceptuallearningassociates@gmail.com.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://smile.amazon.com/Thinking-Math-Differently-Essential-Teachers/dp/0997789263/ref=sr_1_4?ie=UTF8&amp;amp;qid=1499974944&amp;amp;sr=8-4&amp;amp;keywords=Thinking+Math+Differently&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://smile.amazon.com/Thinking-Math-Differently-Essential-Teachers/dp/0997789263/ref=sr_1_4?ie=UTF8&amp;amp;qid=1499974944&amp;amp;sr=8-4&amp;amp;keywords=Thinking+Math+Differently&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8478" LastActivityDate="2017-07-13T19:54:24.693" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12599" PostTypeId="1" AcceptedAnswerId="12610" CreationDate="2017-07-14T10:40:45.473" Score="5" ViewCount="270" Body="&lt;h2&gt;Course background&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I am planning on studying MSc Mathematics through the UK's Open University (distance learning). The course comprises of pure mathematics (number theory, complex analysis, coding theory etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am planning on taking the course over 3 years. The academic year is October to July.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;My background&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I graduated 8 years ago with a BSc Mathematics and Statistics (equal credit weighting in pure, applied and stats) with a 2:1. Since then I have worked in market research (so reasonably quantitative). I am a bit nervous about going back to studying and especially taking exams again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What tips are there for a person returning to study?&lt;/li&gt;&#xA;&lt;li&gt;What study tips are there for someone someone studying an online MSc?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user8481" LastEditorUserId="11" LastEditDate="2017-07-14T14:01:56.843" LastActivityDate="2017-07-16T22:51:50.980" Title="Are there any tips for studying an MSc Pure Mathematics online?" Tags="|self-learning|online-instruction|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12600" PostTypeId="1" AcceptedAnswerId="12601" CreationDate="2017-07-14T11:37:09.330" Score="4" ViewCount="196" Body="&lt;p&gt;I have an idea to help with self-study and recall of definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is to create a crossword where the clues are the definitions and the words to solve are the concepts being defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't come across this study technique before, so my question is &lt;strong&gt;is there any evidence that solving crosswords helps improve declarative memory for recalling maths definitions?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user8481" LastActivityDate="2017-07-19T16:48:22.757" Title="Does solving crosswords help with recall of definitions?" Tags="|definitions|math-puzzle|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12601" PostTypeId="2" ParentId="12600" CreationDate="2017-07-14T23:22:49.807" Score="5" Body="&lt;p&gt;I had a teacher friend who learned in college that word searches were worthless in most respects, but crosswords should have some intrinsic value because it requires you to read the definition and connect it with the proper vocabulary word. So, I did some research and found two things worth sharing:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This study reports 90% of students said crossword puzzles enhanced their learning. &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2972511/&quot; rel=&quot;noreferrer&quot;&gt;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2972511/&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best research I uncovered is below. They not only did research on the topic itself, but cite many other papers that show that crossword puzzles do indeed have many specific educational benefits. However, the results of the study itself were mixed when educational outcome was specifically looked at. With one group benefitting from the crossword puzzle and one group seemingly suffering poorer grades after using the crossword puzzle for test review, more research seems necessary to definitively answer your question. &#xA;&lt;a href=&quot;http://www.uncw.edu/jet/articles/vol9_3/davis.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.uncw.edu/jet/articles/vol9_3/davis.pdf&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="8484" LastActivityDate="2017-07-14T23:22:49.807" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12603" PostTypeId="2" ParentId="12593" CreationDate="2017-07-15T14:58:19.607" Score="5" Body="&lt;p&gt;I think pedagogically sometimes approaching something with a lot of examples actually makes it easier and gets you there faster than an approach geared only to the final insight (or to a point of generality).  Bleecker expresses this well (with data from his teaching experience) in the preface to his book on partial differential equations.  He discusses how it is better not to get into Sturm Louisville immediately but approach it indirectly via several examples or smaller problems than the general case.  I suspect the same to be true about limits which are really a kind of new breathtaking concept when the student hits them at first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I understand your point about the students not knowing the answer to your question at the end of the year.  But to be fair one would want to look at an alternate approach and if it does any better.  Or what is lost if you just concentrate on that one question.  Not even making a statement on what the result would be, but just don't be sure it would go your way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally I don't think missing that question is the most important learning for a calculus student.  One can forget some of the theoretical details on calculus and still do more important results (for the mass of the population taking calculus) like finding minima and maxima to curves--key concept and practice in engineering, physics, chemistry, biology and business.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I struggle with questions like this that have a huge predicate (relevant to pedagogy and worthy of discussion) and then a simple final request.  Will the Q and A police come down on me because I discussed the predicate, not gave a specific textbook answer to the question at the very end.&lt;/p&gt;&#xA;" OwnerUserId="8477" LastEditorUserId="11" LastEditDate="2017-07-16T20:25:50.210" LastActivityDate="2017-07-16T20:25:50.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12604" PostTypeId="2" ParentId="12593" CreationDate="2017-07-15T15:50:31.133" Score="4" Body="&lt;p&gt;Perhaps you can restructure the final exam question as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is the case that $\displaystyle{\lim_{x\to1}\frac{x^2-1}{x-1}=2}$. Establish this fact three ways:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;by simplifying $\displaystyle{\frac{x^2-1}{x-1}}$ algebraically, and using limit properties&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;by identifying the limit as the difference quotient for a derivative $f'(a)$. Be sure to clearly identify&#xA;$f(x)$ and $a$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;by using L'Hospital's rule (perhaps omit this case). &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In this way you give the student enough clues to orient their thinking on the exam and show they can connect the limit with various stages of the course.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2017-07-15T15:50:31.133" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12605" PostTypeId="2" ParentId="12163" CreationDate="2017-07-16T05:02:48.967" Score="0" Body="&lt;p&gt;I would think there must be an implicit other condition in your list (that book is currently being published so bookstore can stock it). The only book I can think of that is being republished but is stable like you want (but is quite good) is Schaum's Outline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Otherwise, if you are comfortable telling the students to buy a book that they need to pick up used on Amazon, than I am big fan of Hart's College Algebra.  In particular the one used for WW2 Education Manual (EM315).  &lt;/p&gt;&#xA;" OwnerUserId="8477" LastActivityDate="2017-07-16T05:02:48.967" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12606" PostTypeId="2" ParentId="4482" CreationDate="2017-07-16T15:38:34.657" Score="3" Body="&lt;p&gt;Would be interested to hear the &quot;after action&quot; report on the text that you used.  Also interested to hear if you decided to require a drill book like Schaum's and how that went.  Even if it did not work out 100% to your liking or if the kids did not appreciate what you were trying to do for them (or even better if it did).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would love to see colleges using the better Dover paperbacks.  (I think this is cheap enough so that it is virtually free.  And you were considered to require a Schaum's so I don't see your cost avoidance stance as absolute.)  For example Tenanbaum for ODE.  I think in this case, I would definitely try for the ones that have the answers in the back as an instructor or solutions manual may not be an option.  So you can't automatically pick one.  But there are many good ones available.  I'm sure 90% of the students will appreciate your lowering the semester cost of textbook from 200 to 20.  (Imagine what that is like for the kids when you add all their classes.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think a paper format is desirable as it is still much easier to skip around in a book than a digital format.  I just think it is a more convenient tool than a screen.&lt;/p&gt;&#xA;" OwnerUserId="8477" LastActivityDate="2017-07-16T15:38:34.657" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12607" PostTypeId="2" ParentId="11284" CreationDate="2017-07-16T16:08:12.987" Score="1" Body="&lt;p&gt;Like Mwiess, I would give an example from social sciences or sports or the like of a function with several inputs.  I think this is the critical first idea to get across.  After this and only after this, you can get to discussing things like interactions of the factors.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would NOT start with a saddle point as in one answer.  This requires thinking in 3D Cartesian space and puts people's back up who are less mathy.  Not saying it is not useful example and would use it in a class (it has value, which is why texts use it all the time).  But realize than that even with weak students, you are dealing with people who are in a process of learning and used to these structures. With a layman, would not refer to such a structure at first.  If you absolutely have to use a visual, just try something like a topo map or talk about mountains.  (But even here, I worry that the reference to a Cartesian system (directions and altitude) will bother some people.  But worth a shot.  Definitely do that before the saddle point though!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once, you have gotten into useful discussion of the concept of multivariable functions, you have give the person the most useful insights for daily life.  Even though you haven't gotten into the diffyq part of it at all.  If the person responds well and you want to carry on further, you could mention that PDEs are a method of going from evidence about how the factors change to try to unravel the equation.  If the person knows integral calculus, refer to that.  Solving an ODE or PDE is kissing cousins to &quot;going backwards&quot; in integration.  Asuming they don't, don't refer to it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other thing that is extremely useful for a layman is to give them some context of where PDEs occur in a curriculum and who needs to learn them.  (Something that I find lacking at times on Stack.)  Mention how it is really part of the &quot;calculus track&quot;.  That it is like &quot;harder calculus&quot;.  and comes after 3 semesters of calculus and one of ODEs (just say simple DEs if you want).  That it is used by engineers and physicists (and of course mathematicians).  But I would emphasize the place in the technical curriculum over the math track...after all look at the numbers of students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. I will even admit to not knowing PDEs well.  Have encountered them at times but never really mastered them.  But knowing where they &quot;fit in the universe&quot; is still useful.  So I have some sympathy for the layman and some feel for what is useful for him to understand.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is why I struggle with Wikipedia articles about math concepts or math subjects which take a definitional approach (with lots of rigor) as the very first step to understanding.  Rather than saying &quot;a lion is sort of like a cat but much bigger.  It is yellow and has big teeth.  The males have a bunch of hair on their head called a mane.  They live in Africa and can kill big animals like deer or zebra and can kill a man no problem.&quot;  This helps me out way more than some formal taxonomy (with massive amounts of recursions or lookups...in math with an amount of lookups that basically requires learning courses to understand even an overview encyclopedia article!) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An easy example would be Lie groups.  I have no idea what they are but would like to know.  I don't have a deep math background but way more than the average cat (1580 on old style SATs, math through ODE and some engine math, lots of engineering and science and even some baby group theory in the context of crystallography and IR spectroscopy.)  So while perhaps not ready to learn Lie algebra, I am WAY more knowledgeable than a complete layman (say a smart NYT reader who &quot;hated math&quot; and was an English major in college).  Yet I can't even just get a basic orientation.  So in your discussion with the layperson on PDEs, I would try to give him some orientational description o the field.  NOT the first lecture in a course and NOT an attempt to teach most of it an hour (to a person who even lacks prereqs).  He just wants to know what kind of animal the PDE is!&lt;/p&gt;&#xA;" OwnerUserId="8477" LastActivityDate="2017-07-16T16:08:12.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12608" PostTypeId="2" ParentId="6124" CreationDate="2017-07-16T16:23:23.770" Score="1" Body="&lt;p&gt;Life requires prioritization.  You are emphasizing some very intricate backgrounds (group theory to quantum to semiconductor physics to EE).  And implicitly ignoring the huge amount of times that students will need basic calculus or ODEs in derivations or homework problems.  More of C is less of A.  The list is prioritized, your remarks are essentially &quot;why not more C&quot; so don't even show the benefit of comparison.  Of stating what you would lower on the list if you raise something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I actually get a little bit the impression you know a lot about math but not that much about EE.  I'm not an EE, but took two solid semesters of junior level EE as part of engineering requirement and also had training in it at nuke school. And even worked in a firm that had ME and EEs.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, I have a pretty solid quantum background for a chemist and did a Ph.D. in solid state that involved a lot of semiconductor physics (but light and experimental for a physicist, but still way more at least exposure than the average cat).  And I never had my learning hurt by the lack of theoretical group theory for quantum (it was more diffyq based).  Heck, even if you do look at quantum theory in terms of group theory, the average physicist would just use the parts of group theory he needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And the idea that a BS EE needs that level of quantum theory?  Noooo.  (There are only so many hours in the day and EEs already bust ass.  They have to learn about a gazillion things from op amps to squirrel cage induction motors and Y-D transformers.)&lt;/p&gt;&#xA;" OwnerUserId="8477" LastActivityDate="2017-07-16T16:23:23.770" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12609" PostTypeId="2" ParentId="7431" CreationDate="2017-07-16T16:52:57.003" Score="1" Body="&lt;p&gt;If you are collecting and grading homework, can't you also check that they used right technique to get the answer?  NWNC (no work no credit, acronom from Navy nuke school).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are not collecting and grading homework (I wouldn't, just let the students drill themselves, try to pick texts with all the answers like Thomas Finney), than who cares?  Just use frequent tests or quizzes to drive performance.  When I took AP calculus, we had a full period test every Friday (first period and the teach had them graded during her practice period and available before day end).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Granted in college you don't have as many seat hours with the kids, but you could still do a test every second Friday (assuming MWF schedule).  Really a lot of research shows that practice or tests (tests are emotionally intense practice episodes) are more helpful to learning than lecture anyhow.  So don't feel bad about getting out of an hour of stage (or guide on the side) time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jaime Escalante did daily quizzes.  (very short at beginning of class.  No reason you can't do the same, even have the kids exchange papers and grade them so now work for you.  Don't worry about cheating as it is minor and if you keep the point value low, it will still motivate kids but not be that much of a differential determiner.)&lt;/p&gt;&#xA;" OwnerUserId="8477" LastActivityDate="2017-07-16T16:52:57.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12610" PostTypeId="2" ParentId="12599" CreationDate="2017-07-16T22:51:50.980" Score="2" Body="&lt;p&gt;The only advice I can give is from my own experience: I've studied 3 of their MSc modules, at a rate of one per year whilst working full time. It's a shame their undergraduate courses are now so expensive as some would be ideal (re)preparation for the MSc. It's useful/important to know that the &quot;routes&quot; they describe are just suggestions: you can study any combination of the modules, as long as the dissertation is included. I've taken variational principles, coding theory and fractal geometry and really enjoyed them all. Will probably move to number theory next. The student reviews are interesting to read: I think I'll avoid the numerical course that uses Maple. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most courses have a set text which you could buy in advance and try and make some progress through, for example during the summer months. The OU provide very helpful online tutorials and feedback on assignments is also generally very useful. There are also online forums for discussion/clarification of anything and everything in the course. Specimen and past papers help to prepare for the exams. Moreover each course comes with a kind of glossary which you can annotate and take into exams: develop the skill of condensed writing and then these can be a very reassuring resource in the exams. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of more general study skills, you just need to plan how to fit the studying into your life: for me as a teacher, I would do big chunks at half term and in holidays etc. Others prefer to set aside a couple of hours every week. Particularly when you study two modules at the same time, there can be stressful points in the year if they each have an assignment due around the same time. The exams typically take place early June. &lt;/p&gt;&#xA;" OwnerUserId="8489" LastActivityDate="2017-07-16T22:51:50.980" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12611" PostTypeId="1" AcceptedAnswerId="12632" CreationDate="2017-07-16T22:53:42.090" Score="10" ViewCount="655" Body="&lt;p&gt;Recently, a friend of mine who plans to work as a &quot;social-impact consultant&quot; (she is currently a College senior with a background in intro statistics) requested that I offer some kind of curriculum that targets &quot;math for social justice.&quot; I took this to mean a math class that offers enough background to understand &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;the mathematics already used in the literature (probably things like basic numerical literacy, statistics, growth rates etc. are included in this.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given some bulk data etc., have enough mathematical background to interpret it for herself.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In light of Chris' comment: maybe it would be worthwhile to know enough math to tell when mathematics is being used poorly in politics/law/social issues (for example, police algorithms, sentencing algorithms etc. mentioned in the book.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I think the question (at least abstractly) is roughly equivalent to the math required for any kind of anthropological/sociological considerations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet, I'm interested if it is known whether or not there is already a curriculum available that uses data and statistics relevant to those interested in advocating for minority groups (particularly in the United States.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found some kind of meta &quot;how-to-teach math for social justice&quot; (&lt;a href=&quot;http://math.arizona.edu/~cemela/english/content/workingpapers/SACNAS_SocialJusticeTalk3.pdf&quot; rel=&quot;noreferrer&quot;&gt;such as this&lt;/a&gt;) but was hoping for something more direct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, are there known college courses with this goal in mind? What would be additional features one would hope to capture in this field? I've seen a possible list &lt;a href=&quot;http://www.radicalmath.org/categories.php&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, but this looks more like a high school curriculum. &lt;/p&gt;&#xA;" OwnerUserId="6276" LastEditorUserId="6276" LastEditDate="2017-07-18T22:38:48.273" LastActivityDate="2017-12-07T03:35:03.920" Title="Math for Social justice curriculum" Tags="|undergraduate-education|reference-request|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12612" PostTypeId="2" ParentId="12543" CreationDate="2017-07-16T23:04:21.427" Score="2" Body="&lt;p&gt;I often see a &quot;shift&quot; among my fourth graders when we move into geometry units.  The students who have been quite successful with computation and even algebra concepts begin to struggle to &quot;see&quot; things in geometry...similar to what you describe.  What's helped is putting the area and perimeter piece on hold until we've done a thorough investigation of the attributes of two-dimensional figures.  Once the students are truly comfortable identifying and classifying figures based on their attributes they have a much easier time &quot;seeing&quot; with problems like this.&lt;br&gt;&#xA;The go-to strategy most of my students would use here would be decomposition of this shape into two rectangles.  They'd then be able to apply what they know about congruent opposite sides of rectangles to &quot;see&quot; that the lengths of the two parallel unknown sides must equal 7. &lt;/p&gt;&#xA;" OwnerUserId="8478" LastActivityDate="2017-07-16T23:04:21.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12613" PostTypeId="2" ParentId="12476" CreationDate="2017-07-16T23:23:23.873" Score="3" Body="&lt;p&gt;I'm a little surprised not to see any reference to the 'Harkness' approach in the replies so far. I won't write at great length, but will include some references. Can students learn ONLY from solving problems? Almost, but there are crucial discussion-based lessons taking place in between each problem set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Harkness teaching began at Philips Exeter and they share their mathematics problem sets online: &lt;a href=&quot;https://www.exeter.edu/mathproblems&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.exeter.edu/mathproblems&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wellington College in the UK have successfully implemented this approach for A level Maths (equivalent to US grades 11/12), among other subjects. They share their mathematics resources here: &lt;a href=&quot;http://maths.wellingtoncollege.org.uk/resources/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://maths.wellingtoncollege.org.uk/resources/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8489" LastActivityDate="2017-07-16T23:23:23.873" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12614" PostTypeId="2" ParentId="11779" CreationDate="2017-07-17T00:03:19.123" Score="4" Body="&lt;p&gt;There are many nice problems involving integrals of the &quot;integer floor&quot; function, which are evaluated as series. Eg questions 2 and 3 here: &lt;a href=&quot;https://maths.org/step/sites/maths.org.step/files/assignments/assignment3.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://maths.org/step/sites/maths.org.step/files/assignments/assignment3.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8489" LastActivityDate="2017-07-17T00:03:19.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12615" PostTypeId="2" ParentId="11627" CreationDate="2017-07-17T00:14:11.480" Score="1" Body="&lt;p&gt;If students are struggling to understand a relatively simple trig equation like the one given, I find that graphs often help. We can sketch y=2cos(x) and y=1, and get a feeling for how many solutions to expect in the given interval. Moreover symmetries of the graphs can help determine the full set of solutions, much as the unit circle approach does. Interestingly, I believe in the UK the unit circle development of trig functions isn't such a common approach but many teachers use the (IMO awful) 'CAST' diagram for solving trig equations, devoid of deeper understanding. &lt;/p&gt;&#xA;" OwnerUserId="8489" LastActivityDate="2017-07-17T00:14:11.480" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12616" PostTypeId="1" AcceptedAnswerId="12617" CreationDate="2017-07-17T00:16:26.537" Score="2" ViewCount="341" Body="&lt;p&gt;Should multiplication be taught before addition and subtraction?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The obvious answer for most people is 'no'. However, I think there are a few valid points that could change the way students approach and respond to higher level mathematical concepts.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The human brain is inherently logarithmic. We perceive light and volume in orders of magnitude, i.e. candela and decibel. Sources include Weber Fechnel law, and &lt;a href=&quot;https://www.scientificamerican.com/article/a-natural-log/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.scientificamerican.com/article/a-natural-log/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This would help students gain an intuitive sense of numbers, which would help them gain a deeper understanding of advanced concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplication rules are easy. . It is easy for students to learn that anything times 1 is itself, everything times 0 is 0, everything times 10 is the same thing followed by a 0 (moved a decimal place over), etc. In contrast, when 100 is added to something it must be added to the hundreds place, which is further abstraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though children are taught that multiplication is iterated addition, there are other methods of teaching it such as geometry or counting rectangular arrays.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;When students reach algebra, logarithms could have more meaning. Rather than being taught at the end and hardly understood, logarithms could be a valuable tool to teaching concepts such as order of operations, exponents, powers, and more.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The skills and intuition developed by multiplication tend to be more used in daily life (when am I going to us this) and are more beneficial as a whole.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;However, there are reasons why not to. For example, addition can be done on fingers so it is simple and then addition can be used to help teach multiplication (though I dislike this because students use this as a crutch through secondary education instead of actually memorizing the dang multiplication table, though I don't know if this would make multiplication any less labor intensive or accessible)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me know your thoughts, just be sure to include examples and backing instead of isolated opinions :)&lt;/p&gt;&#xA;" OwnerDisplayName="user8490" LastActivityDate="2017-07-19T18:44:40.873" Title="The Order in Which Arithmetic Operators are Taught" Tags="|primary-education|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="12617" PostTypeId="2" ParentId="12616" CreationDate="2017-07-17T07:34:04.167" Score="5" Body="&lt;p&gt;One thing you have to keep in mind is the size of numbers. You don't start your first graders off with five digit numbers, no, you start with one digit, then increase to numbers below $20$, then slowly increase to one hundred. Here, you will run into problems with multiplication, as you can only build the (nontrivial) products $2 \cdot 2$, $2 \cdot 3$, $2 \cdot 4$ and $3 \cdot 3$ at first, while you haven't introduced multiple digits yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we look a little further ahead and discuss the rules taught for adding or multiplying big numbers, you may notice that most multiplication rules use addition, so you will also get into trouble teaching kids how to multiply big numbers, if they don't know how to add them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last, but not least, let's talk about the brain. Even if we are assuming that we generally think in a logarithmic scale, there is something far more natural and more important: counting. The first thing taught to kids is counting to do basic addition of small numbers. And I really don't see how multiplication can be taught in any way that is more intuitive than counting fingers (and toes, once the numbers get too big :) ).&lt;br&gt;&#xA;On the other hand, we do teach logarithmic scale by looking at the number of digits, but we don't start off with it before doing addition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oh, and by the way, I don't think that one single ancient Amazon tribe would be enough to change such fundamental basics of teaching...&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2017-07-17T07:34:04.167" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12618" PostTypeId="1" CreationDate="2017-07-17T23:50:48.570" Score="7" ViewCount="590" Body="&lt;p&gt;I know it's common for high school teachers to use software (such as Geogebra) to formulate geometry problems for their students, so I wonder: Do professors of multivariable calculus use softwares (like MATLAB, Maple or any other that plot surfaces and curves) to formulate exercises and problems for their students? Can anyone give me examples? &lt;/p&gt;&#xA;" OwnerUserId="6569" LastEditorDisplayName="user507" LastEditDate="2017-07-18T00:37:35.333" LastActivityDate="2018-08-13T15:40:55.380" Title="The use of software to formulate problems in multivariable calculus" Tags="|undergraduate-education|software|vector-calculus|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12619" PostTypeId="2" ParentId="12611" CreationDate="2017-07-19T13:19:30.830" Score="5" Body="&lt;p&gt;Statistics.  If there is a &quot;statistics for sociology&quot; course than that is already right on target.  [Concepts are not that much different than general statistics for ag or bio, but examples will be chosen from population surveys or crime or education or business.]  Also, emphasis should be on getting tools and using tools, NOT on theoretical background.  IOW, minimum math prereqs, non calculus, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0415805015&quot; rel=&quot;noreferrer&quot;&gt;https://www.amazon.com/Social-Statistics-Basics-Sociology-Re-Wired/dp/0415805015&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cod.edu/course/information/SOCIO2205/index.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.cod.edu/course/information/SOCIO2205/index.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-07-19T13:19:30.830" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12620" PostTypeId="2" ParentId="12600" CreationDate="2017-07-19T16:48:22.757" Score="1" Body="&lt;p&gt;When geometry was suddenly added to my elementary school curriculum my students were overwhelmed with all of definitions that they had to learn at once.  A group of 5th grade students voluntarily created crosswords for the rest of the class to practice the vocabulary.  The students who created the crosswords learned the most. The rest of the class enjoyed practicing vocabulary much more this way. Any method that gets students to practice will help them learn and so I reused the crosswords every year. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.discoveryeducation.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discovery Education&lt;/a&gt; has a great &lt;a href=&quot;http://puzzlemaker.discoveryeducation.com/CrissCrossSetupForm.asp?campaign=flyout_teachers_puzzle_crisscross&quot; rel=&quot;nofollow noreferrer&quot;&gt;site&lt;/a&gt; for creating crosswords that both I and my students used.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2017-07-19T16:48:22.757" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12621" PostTypeId="2" ParentId="11779" CreationDate="2017-07-19T17:06:12.703" Score="-1" Body="&lt;p&gt;I like the explication and just the example in this paper (really makes you think about an accumulation and a rate).  Very nice graphics also (for box counting).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://home.hiroshima-u.ac.jp/yhiraya/er/ZR11_Z_03.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://home.hiroshima-u.ac.jp/yhiraya/er/ZR11_Z_03.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, he ended up underestimating hydrocarbon production but the thought process is still very nice.  And I think doing something topical like this makes it relevant.  Not just another cannonball.&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-07-19T17:06:12.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12622" PostTypeId="2" ParentId="12616" CreationDate="2017-07-19T18:44:40.873" Score="1" Body="&lt;p&gt;I will argue that no such sequence can be logically coherent. The standard definition of multiplication in natural numbers is based on addition being defined first (e.g., Peano axioms for a formal treatment). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In comments, the OP has been asked a few times for an alternative definition for multiplication, and hasn't been able to provide one. Exercises, e.g., skip counting, are equivalent to addition, just under a different name, with which the OP agrees in comments. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-07-19T18:44:40.873" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12623" PostTypeId="1" AcceptedAnswerId="12625" CreationDate="2017-07-19T21:54:01.867" Score="5" ViewCount="365" Body="&lt;p&gt;In my university, in Italy, most professors want students (at least the undergraduate ones) to handwrite what they are saying in an oral exam. If I understood correctly, this is somewhat due to needing an official document for the oral exam. Does this occur in other countries too?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not uncommon for students to have a not so clear handwriting (I know many), one has to put care for the text to be clear, it ends up being kind of distracting. Maybe that's not to the point of being a huge problem, but nowadays LaTeX is anyway an important instrument for a mathematician, would it not be feasible to offer a LaTeX course in the first semester of the first undergraduate year, and then have LaTeX-assisted orals?&#xA;I think the speed of handwriting can be partially undermined by the need to put that (probably extra) care, resulting, in average, comparable to the speed of LaTeX-typing after an entire dedicated course.&#xA;&lt;strong&gt;Is there a good reason not to give the students a keyboard and a monitor, provided that they have been taught LaTeX?&lt;/strong&gt; In the end, the printed document would be manually signed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Added stuff from the comments)&lt;/p&gt;&#xA;" OwnerUserId="6536" LastEditorUserId="6536" LastEditDate="2017-07-21T07:11:42.833" LastActivityDate="2017-07-21T07:11:42.833" Title="Are there advantages to make students handwrite (instead of LaTeX typing) what they say in an oral exam?" Tags="|undergraduate-education|exams|oral-exams|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12624" PostTypeId="2" ParentId="12611" CreationDate="2017-07-20T01:55:12.217" Score="4" Body="&lt;p&gt;It turns out that the &quot;service-learning&quot; side of this question (which is NOT the entirety of your question, just one piece of what might fall under your question) is poorly documented when it comes to math.  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://math.gordon.edu/~kcrisman/ServiceLearning/&quot; rel=&quot;nofollow noreferrer&quot;&gt;I have a collection of links to various resources and talks&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://slmath.com/about/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Josh Wilkerson has a nice dedicated site with more links to theoretical aspects&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-07-20T01:55:12.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12625" PostTypeId="2" ParentId="12623" CreationDate="2017-07-20T08:27:05.070" Score="3" Body="&lt;p&gt;There are two types of oral exams I'm aware of: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A Q&amp;amp;A-style, where the student is supposed to show knowledge and learned concepts by giving definitions or answering short questions.&lt;/li&gt;&#xA;&lt;li&gt;A talk-style, where the professor discusses a problem with the student and the student is supposed to show that he understands the concepts taught and can apply them to the given problem.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Most professors are familiar with both styles and adjust according to the student: If things are going well, you talk, if the student seems to be completely lost, you fall back to Q&amp;amp;A.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, especially in the second case of exam, you want some written document. Not (only) for official documentation, but rather to have something to talk about and discuss. The student is most likely nervous, and if you want to come back to a prior point, if you want to point the student in the right direction, he might have well forgotten what exactly he said two minutes ago. Furthermore, many students (especially undergraduates) are not yet accustomed to a mathematical way of talking. This will result in many situations of the sort &quot;didn't I essentially just say that?&quot; If you have the student write down his definitions or arguments, you can ask him to look at it again closely, if he is sure that everything is correct with it.&lt;br&gt;&#xA;Such a situation can get really frustrating, if you are trying to get a student to say something he thinks he already said and it might get troublesome to prove that he actually didn't, as long as you haven't got any written documents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last but not least, writing things down takes some momentum off the whole thing. It gives a nervous student time to think, time to order his thoughts, time to write them down properly. If you just talk with no end, you would have to fail many students not for being bad but just for being overly nervous.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2017-07-20T08:27:05.070" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12626" PostTypeId="2" ParentId="12623" CreationDate="2017-07-20T11:53:56.780" Score="0" Body="&lt;p&gt;That is overkill and too close to a written exam.  Just have a form for the prof to put a grade and a couple paras in what was covered and what was done well or poorly.  &lt;/p&gt;&#xA;" OwnerUserId="8499" LastActivityDate="2017-07-20T11:53:56.780" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12627" PostTypeId="2" ParentId="12611" CreationDate="2017-07-20T23:41:03.063" Score="3" Body="&lt;p&gt;Two pointers, both perhaps at earlier educational levels than is your main focus:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Perhaps &lt;a href=&quot;http://www.radicalmath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;RadicalMath&lt;/a&gt;, aimed at&#xA;high-school education, could help suggest analogous resources at the&#xA;college level:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Wo4Fa.gif&quot; width=&quot;200&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/4nBDE.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4nBDE.gif&quot; alt=&quot;radmth&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) Here is an opinion article in the GradHacker blog,&#xA;emphasizing pre-service mathematics teachers: &#xA;&quot;Equity and Social Justice… in Mathematics Methods Courses?&quot;&#xA;Juan Gerardo.&#xA;&lt;a href=&quot;https://www.insidehighered.com/blogs/gradhacker/equity-and-social-justice%E2%80%A6-mathematics-methods-courses&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inside Higher Ed&lt;/a&gt;, Apr 2017.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-07-20T23:41:03.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12628" PostTypeId="2" ParentId="4367" CreationDate="2017-07-21T13:41:30.507" Score="11" Body="&lt;p&gt;When teaching binary to any age group I always start by taking a set of kitchen weights into the classroom. 2lb (32oz) 1lb (16oz) 8oz 4oz 2oz 1oz. Then make a table asking what weights yo1u would use to make  up certain values. If a weight is used you show it with a &quot;1&quot; if it is not used you show it with a &quot;0&quot;.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;                                    32    16    8    4    2    1&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;What weights would you use for 1oz                                 1&#xA;                               2oz                            1    0&#xA;                               3oz                            1    1&#xA;                               4oz                       1    0    0&#xA;                               5oz                       1    0    1&#xA;                               6oz                       1    1    0&#xA;                               7oz                       1    1    1&#xA;                               8oz                  1    0    0    0&#xA;                               9oz                  1    0    0                 &lt;/p&gt;&#xA;" OwnerUserId="8502" LastActivityDate="2017-07-21T13:41:30.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12629" PostTypeId="1" CreationDate="2017-07-21T14:30:59.000" Score="4" ViewCount="188" Body="&lt;p&gt;Good Day!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I need help in finding a free validated and reliable tool in assessing (elementary or high school) students' Mathematics Self-Efficacy.&lt;/p&gt;&#xA;" OwnerUserId="8410" LastEditorUserId="122" LastEditDate="2017-07-22T10:46:43.430" LastActivityDate="2017-09-21T09:49:05.067" Title="Mathematics Self-Efficacy Questionnaire" Tags="|mathematical-pedagogy|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12630" PostTypeId="2" ParentId="11725" CreationDate="2017-07-21T21:52:31.713" Score="2" Body="&lt;p&gt;The OP is looking at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article on Mathematics in general&lt;/a&gt;. In the first line it defines mathematics as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”)&#xA;  is the study of topics such as quantity (numbers), structure, space,&#xA;  and change.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You should be aware that this specific terminology is not used in mathematics practice itself as defined terms (at least, none that I've ever seen). Rather, this article is written at the outset from a hypothetical perspective of a reader who's never heard of any mathematics at all, and in that respect, I'd say it's a halfway clever way of encapsulating what the discipline is about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further down in the article there are links provided to more fully developed articles on a variety of topics, and those links associate these (undefined) terms to more normal mathematical topic terminology, to wit:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Quantity → Arithmetic&lt;/li&gt;&#xA;&lt;li&gt;Structure → Algebra&lt;/li&gt;&#xA;&lt;li&gt;Space → Geometry&lt;/li&gt;&#xA;&lt;li&gt;Change → Calculus&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Each of the second items in the list above is a full article on Wikipedia, a common course in schools, and one of many books you could find on that subject. Of course, the opening Wikipedia definition uses the qualifier &quot;such as&quot;, so this list is not represented as being complete. The article also has links to other topics, such as: Foundations and philosophy (logic, set theory, category theory, computation), applied mathematics, statistics, etc. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-07-21T21:52:31.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12632" PostTypeId="2" ParentId="12611" CreationDate="2017-07-22T17:51:38.070" Score="5" Body="&lt;p&gt;So while I don't know I am aware of what is a comprehensive social justice curriculum, there are many pieces that can be used from some existing sources.  I think there are a few different ways of understanding social justice however, and a few directions in the literature depending.  &lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Quantitative Literacy&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;At it's most basic, there is the quantitative literacy discourse, pretty much a traditional take on content dressed in some kind of supposed everyday context.  These courses tend to emphasize social justice as making the student more critical of information presented with numbers.  Some examples here are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Bolker and Mast &lt;em&gt;&lt;a href=&quot;http://commonsensemathematics.net/&quot; rel=&quot;noreferrer&quot;&gt;Common Sense Mathematics&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Statway and Quantway from Carnegie Learning &lt;a href=&quot;https://www.carnegiefoundation.org/in-action/carnegie-math-pathways/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h3&gt;Social Justice through Mathematics&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;This is what I think of with the Radical Math work.  Here, students usually are using mathematics to speak to some kind of social justice issue.  The most comprehensive work I'm aware of here is &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Rico Gutstein's &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0415950848&quot; rel=&quot;noreferrer&quot;&gt;Reading and Writing the World with Mathematics: Toward a Pedagogy for Social Justice&lt;/a&gt;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This book was the result of Rico's class in a Chicago school and references many activities from the class and while getting old can still give good ideas for what to do in a social justice setting.  There was also a series of activities that were later produced along the lines of the book that is a collection of articles on teaching mathematics with social justice here:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0942961552&quot; rel=&quot;noreferrer&quot;&gt;Rethinking Mathematics: Teaching Social Justice by the Numbers&lt;/a&gt;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h3&gt;Ethnomathematics&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Another strand associated with social justice discourse in mathematics education is that of ethnomathematics.  Again, I don't think a comprehensive curriculum exists, but you can certainly glean a number of activities from:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Marcia Ascher's &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0412989417&quot; rel=&quot;noreferrer&quot;&gt;Ethnomathematics: A Multicultural View of Mathematics&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ron Eglash &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0813526140&quot; rel=&quot;noreferrer&quot;&gt;African Fractal's: Modern Computing and Indigenous Design&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Eglash also has a TED talk &lt;a href=&quot;https://www.ted.com/talks/ron_eglash_on_african_fractals&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;All of these areas have professional journals and conferences associated with them, however as your question was more about a full curriculum the above mentioned resources are the most comprehensive depending on need/interest.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-07-22T17:51:38.070" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12633" PostTypeId="2" ParentId="11725" CreationDate="2017-07-22T19:12:29.913" Score="3" Body="&lt;p&gt;(1) The American Mathematical Society (AMS) maintains a &#xA;&lt;a href=&quot;http://www.ams.org/msc/msc2010.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics Subject Classification&lt;/a&gt;&#xA;(MSC)&#xA;for the mathematics literature. Here is a graph of the topics&#xA;built by Andrius Kulikauskas:&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/56pPk.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/56pPk.png&quot; alt=&quot;MSC Tree&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;Image by &lt;a href=&quot;https://ncatlab.org/nlab/show/AndriusKulikauskas&quot; rel=&quot;nofollow noreferrer&quot;&gt;AndriusKulikauskas&lt;/a&gt;.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;And here is a more readable detail (from roughly the center of the chart):&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ivJKE.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ivJKE.png&quot; alt=&quot;detail&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) And here is another chart, more idiosyncratic, emphasizing&#xA;physics, from this paper:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Tegmark, Max. &quot;Is 'the theory of everything' merely the ultimate ensemble theory?&quot; &lt;em&gt;Annals of Physics&lt;/em&gt; 270.1 (1998): 1-51.&#xA;  (&lt;a href=&quot;https://arxiv.org/pdf/gr-qc/9704009v2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/0txkx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0txkx.png&quot; alt=&quot;Max&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;Image by Max Tegmark.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-07-23T16:07:38.223" LastActivityDate="2017-07-23T16:07:38.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12634" PostTypeId="2" ParentId="12308" CreationDate="2017-07-23T00:50:06.163" Score="1" Body="&lt;p&gt;IT is common in many technical fields, in business consulting and in engineering in the field.  I've also seen it in the Navy in nuclear power (&quot;radcon math&quot;), in TMA (maneuvering boards), and in submarine TMA (where you lack range information and have only bearing, assuming passive sonar).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure it needs to be stressed in math class if it is addressed in physics, chemistry, etc.&lt;/p&gt;&#xA;" OwnerUserId="8504" LastActivityDate="2017-07-23T00:50:06.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12635" PostTypeId="2" ParentId="8534" CreationDate="2017-07-23T03:04:14.777" Score="2" Body="&lt;p&gt;In If you are feeling difficulty in SSC then &lt;a href=&quot;http://www.gocareer.in/search/label/ssc&quot; rel=&quot;nofollow noreferrer&quot;&gt;SSC exam&lt;/a&gt; there is no method till now made by officials, they do not use normalisation method.&#xA;But if in future SSC decides to use normalisation method to take into consideration the difficulty level variations in the question papers across different sessions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The normalization process is based on a simple formula and based on a few  parameters as discussed below, the  Exam organizing committee arrives at the following formula for calculating the normalized marks for the multi session papers.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/6d2ZO.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6d2ZO.png&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once the answers are checked and assessed, the normalized marks of a candidate will be calculated corresponding to the actual marks secured by the candidate in the examination and the Score card will be devised based on the normalized marks.&lt;/p&gt;&#xA;" OwnerUserId="8505" LastEditorUserId="8505" LastEditDate="2017-07-23T16:53:41.980" LastActivityDate="2017-07-23T16:53:41.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12636" PostTypeId="2" ParentId="8534" CreationDate="2017-07-23T14:05:27.917" Score="3" Body="&lt;p&gt;Again, not a &quot;standard method&quot; but one that is gaining traction is comparative judgement, as explained on the No More Marking website. &lt;a href=&quot;https://www.nomoremarking.com/aboutcj&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nomoremarking.com/aboutcj&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has been used in studies by the regulators for UK exams (Ofqual) to determine factors such as if difficulty has increased over time and if different exam boards are producing examinations of comparable standards of difficulty. See for example  &lt;a href=&quot;https://ofqual.blog.gov.uk/2015/05/21/gcse-maths-final-decisions/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ofqual.blog.gov.uk/2015/05/21/gcse-maths-final-decisions/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My understanding is that each individual question would be uploaded to the site. These are then allocated randomly in pairs for 'experts' to make a series of judgements on the basis of one focussed question. (For example, &quot;Which of these questions is the more difficult to answer?&quot;.) Once enough judgments have been submitted by enough experts, a ranking of the questions is produced. However, you would then still need to decide how to apply that ranking to compare the overall difficulty of each shift's examination.&lt;/p&gt;&#xA;" OwnerUserId="8489" LastActivityDate="2017-07-23T14:05:27.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12637" PostTypeId="2" ParentId="8534" CreationDate="2017-07-23T18:11:07.673" Score="1" Body="&lt;p&gt;I have a practical suggestion which is to not have one person write a test for one shift and the other write it for the other.  Instead pool your questions and just do a random sample of them for the group.  Still some danger that luck will drive a difference but probably much less than the way you are doing it now.  This will also ensure coverage is more even (e.g. algebra versus geometry).&lt;/p&gt;&#xA;" OwnerUserId="8510" LastActivityDate="2017-07-23T18:11:07.673" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12639" PostTypeId="2" ParentId="9806" CreationDate="2017-07-23T19:42:35.000" Score="0" Body="&lt;p&gt;This is a hard question because it starts with a debatable position and than says how to implement it.  Are we not allowed to comment meaningfully on the debatable position?  E.g. what if more CAS in the curriculum leads to student confusion or worse learning of calculus.  Maybe it doesn't, but it seems like this is a meaningful discussion to be had, if we care about the students, rather than just accepted as if A, then how B.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To seriously address the question, I think the best way to handle the use of CAS is in higher level classes (particularly non math technical classes, e.g. I used Maple in grad level thermo for phase diagram PDE analysis).  In the lower level classes better to let the students learn classically.  So this is not &quot;not being in the same century&quot;, but just using a tool as a tool after mastering the fundamentals earlier.  Certainly having basic knowledge and familiarity with analytical solutions, for me, made it easier to use Maple on a tricky applied problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good example to me is use of the calculator.  Having an intuitive and mechanical ability with arithmetic makes the rest of math and science much easier.  But there is no reason why 11th grade chemistry students need to do long division or row multiplication when doing the frequent stoichiometry calculations.  But it is still useful to have the background rather than only have a feeling for arithmetic of what numbers to punch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So...my answer to the &quot;if A, then B&quot; is to wait.  And let them use CAS in higher level classes.  I worry that they won't learn the techniques and then will struggle when trying to read a physics paper or follow a chalkboard derivation in engineering or physics classes otherwise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally for students that will end their math with calculus, they will have lost an opportunity to get an intuitive feeling for what is going on if it is just feed stuff into a black box and let it do the work.  The other students may recover or be strong enough that it doesn't hurt them.  but you will do a real disservice to the more basic ones.&lt;/p&gt;&#xA;" OwnerUserId="8512" LastActivityDate="2017-07-23T19:42:35.000" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12640" PostTypeId="2" ParentId="8195" CreationDate="2017-07-23T20:46:00.310" Score="0" Body="&lt;p&gt;If you replace computations with proofs (since the students don't like it, haven't had it), than you are substituting beets for spinach.  ;-) (Worse for bad, if you didn't get it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students did not validate the course than they have problems with the material.  For those who can easily learn the computations, fine, it will be an easy A.  But I would be much more wary of kids who NEED more practice since they struggled the first time through.  Your counsel to them should be very much on the need to practice, to stay up to speed, and to NOT get overconfident because it is easy at first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In class and after class drill is what the students need.  If they are going to be engineers or physicists or chemists, they need to just be able to think of calculus as no big deal within a derivation or some homework problem in fluids, electrostatics, or P chem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This isn't just me being a hardass, this is the counsel from Feynman or lots of physics grad students.  Lots of classroom and after class drill is the best medicine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;FYI, you admit your lack of experience with what general technical students need, but I can honestly tell you that I have broad background here (I probably don't know what the math student needs!)  general engineering at collged, chemistry through Ph.D., mechanical engineering (in the field), officer Navy nuclear power school, applied psychology (heavy stats), etc. And that included classes (but not concentrations) in EE and control systems.  You need the basic calculus building blocks to be comfortable when there are derivations or homework problems in technical classes (and many students will at least need to get through ODE.  If you aren't solid on integrals, good luck with ODE.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teach the kids basic classical Granville.  A few proofs along the way are fine.  As motivation, examples, etc.  But not as the reason.  Please don't jump down the &quot;real analysis during calculus&quot; path.  There is no reason kids can't go back and do that later.  And lots of very high level theoretical physicists say it doesn't help them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I encourage you to expand your horizons and get some general technical education.  It will make you a better teacher of engineers or the like.  One example of a very powerful mathematician who still had an understanding of what technical students needed, and even a sympathy of wanting them to learn it, was Kreyszig.  He wrote a very classical mathy (but pedagogical) Advanced Engineering Math text.  But he also wrote books on functional analysis and differential geometry.  So it's not like he didn't know the fancy stuff.&lt;/p&gt;&#xA;" OwnerUserId="8513" LastActivityDate="2017-07-23T20:46:00.310" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12643" PostTypeId="2" ParentId="2428" CreationDate="2017-07-23T22:46:14.117" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think trying to get into some sort of philosophical meaning is a mistake.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even worse is to lard in the baby real analysis.  You are the only one feeling the lack from that.  Not the kids.  And making things tougher when the kids are already having issues with prior prep makes no sense at all.  It is going the wrong direction.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think you should emphasize tricks and drill and practice.  There is a real feeling of mastery from learning the different tricks and how to use them.  Look at what Jaime Escalante did in Stand and Deliver.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8515" LastActivityDate="2017-07-23T22:46:14.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12644" PostTypeId="1" CreationDate="2017-07-24T15:41:03.153" Score="10" ViewCount="2487" Body="&lt;p&gt;After searching for applications of calculus for biology students, I've found that many of the results are all either contrived exercises, or are way over the heads of students that are seeing calculus for the first time (like activities for a computational biology or biological modelling course). The results that are at an appropriate level all seem to center around differential calculus, and especially &lt;a href=&quot;https://matheducators.stackexchange.com/q/1164/5220&quot;&gt;related rates&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some good activities to give to biology students in a one hour discussion section in an &lt;em&gt;integral&lt;/em&gt; calculus course? I would appreciate either specific activities or problems, or just good resources for activities. Here are a few good sources I've found so far:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Integration Applications&lt;/strong&gt; &lt;a href=&quot;http://www.brynmawr.edu/math/people/vandiver/documents/Integration.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;(dead link)&lt;/a&gt; by R. Vandiver, although these exercises read like, &quot;&lt;em&gt;here's some biology background, now perform this mildly related calculation just because,&lt;/em&gt;&quot; which I don't think is very engaging.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mike Mesterton-Gibbons' &lt;a href=&quot;http://www.math.fsu.edu/~mesterto/biocalc.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lectures on calculus for life science majors&lt;/a&gt;, which I haven't read through too thoroughly yet.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2019-08-03T19:37:49.747" LastActivityDate="2019-08-08T05:09:56.077" Title="Activities for biology undergraduates taking integral calculus" Tags="|undergraduate-education|calculus|concept-motivation|biology|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12645" PostTypeId="1" CreationDate="2017-07-24T16:18:45.370" Score="10" ViewCount="365" Body="&lt;p&gt;I teach at a small liberal-arts college and advise our Math Club. In past years, I have talked a few students into taking the Putnam Exam. I've found that I pretty much have to &quot;coerce&quot; them into taking it at all, let alone prepare for it adequately. In the month or so leading up to the exam, I hold evening sessions after classes where I go over past &quot;easy&quot; problems and remind them to really focus on problems 1 and 2 in each session since those are more doable. This proves helpful for some, but other students just don't bother to attend.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd also like to encourage our students to work on problems posed in journals, like the &lt;em&gt;Pi Mu Epsilon&lt;/em&gt; journal, as well as the MAA's &lt;em&gt;Mathematics Magazine&lt;/em&gt; and &lt;em&gt;College Math Journal&lt;/em&gt;. Working in groups on these problems can really foster the students' &quot;ownership&quot; of their mathematical expertise. This can even lead to a mention in a journal, which can boost their confidence and make a nice addition to their CV. Even &lt;a href=&quot;https://projecteuler.net/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Project Euler&lt;/em&gt;&lt;/a&gt; problems would be great, as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, I want to encourage our math majors to work on problems &lt;em&gt;outside of their course work&lt;/em&gt;, for the reasons mentioned above as well as just to strengthen their education. &lt;strong&gt;What are some effective methods for fostering students to work on challenging problems, like those mentioned above, in groups outside of their standard course work?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested in hearing about your personal experiences with this, and would love to hear about methods that have/haven't worked. I'm also interested in any research about intrinsic student motivation and how to encourage that, because I imagine that's germane here. (If students &lt;em&gt;want&lt;/em&gt; to solve problems outside of their courses, then they'll make time for it.)&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2017-07-24T20:00:54.513" LastActivityDate="2017-07-27T23:15:48.103" Title="Drumming up interest in journal-problem-solving and competition prep" Tags="|undergraduate-education|student-motivation|problem-solving|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12646" PostTypeId="2" ParentId="2428" CreationDate="2017-07-24T16:19:08.523" Score="5" Body="&lt;p&gt;I second Alexander's suggestion to focus on approximation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I particular:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In applications of integrals, you can make a big deal about the transition from finite approximation to the definite integral.  In each case you have some desired quantity (like area, volume, work, mass, etc) which you cannot compute directly.  However, you can break the problem into millions of tiny pieces, each piece can be approximated in an &quot;easy&quot; way.  The sum of these approximations gives a Riemann sum for an integral.  Taking the limit gives an exact answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since most functions do not have elementary antiderivatives, you often cannot get an &quot;exact&quot; answer by using the fundamental theorem of Calculus.  So ask your students to bound the error.  How many terms do I need in my approximating sum to get to within a desired degree of accuracy of the final answer?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar story plays out for infinite series.  Often, even if a series is summable you cannot find a nice expression for the exact sum.  Bounding the tail end of the series by a geometric series (or by a suitable improper integral) can give you precise error estimates.  They learn the skills of comparison test, integral test, alternating series test, etc in the context of approximation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use Taylor series to solve problems (like finding solutions to physically relevant Diff EQs) and have them bound the error.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think all of this can be highly motivated from the perspective of the future engineer (you need to know how precise your answers are from an engineering perspective) or the mathematician (this is very good practice with epsilonics).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree it is hard to motivate getting good at techniques of integration.  For these, I let them know that the skill itself is not so useful (given that only a limited class of functions yields to such methods), but that the process of learning it will make them very good at algebra.  It is really the meta level skills which are valuable here.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2017-07-24T16:19:08.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12647" PostTypeId="2" ParentId="12644" CreationDate="2017-07-24T23:17:24.907" Score="2" Body="&lt;p&gt;You could study the geometry of a &lt;a href=&quot;https://en.wikipedia.org/wiki/Nautilus&quot; rel=&quot;nofollow noreferrer&quot;&gt;nautilus&lt;/a&gt;, whose shape is&#xA;close to a logarithmic spiral.&#xA;The nautilus uses its volume ratio of air/water buoyancy to descend and rise.&#xA;Calculating a nautilus's volume could be an interesting integral calculus exercise.&#xA;Of course you could start in 2D with the enclosed area.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/9wyDY.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9wyDY.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Logarithmic_spiral&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;And there is quite a bit of (more advanced) calculus in understanding&#xA;buoyancy, deriving &lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedes%27_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedes' principle&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-07-24T23:58:00.477" LastActivityDate="2017-07-24T23:58:00.477" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12648" PostTypeId="2" ParentId="12645" CreationDate="2017-07-25T00:04:15.737" Score="4" Body="&lt;p&gt;Not a high-quality answer, but based on some decades of observation: among the kids' &quot;peers&quot; (and you have to figure out what that means), participation in the program needs to be positioned to enhance status (as they say). Otherwise, you will only (potentially) get &quot;loners&quot;, etc. Of course, the &quot;loner&quot; demographic is not the worst population to look into for high-performance Putnam Exam takers, but I think this is sorta the opposite of your question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my own direct experience in trying to generate student (undergrad or grad) enthusiasm, free food (pizza...) will get you great mileage. After that, you can do your spiel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I've seen the perhaps-silly success of pizza-as-bait in the grad program at my own R1 uni, where more-senior grad students lured-in beginning grad students to prelim-prep activities simply by pizza... and game-show-styled formats! People like &quot;winning&quot;, apparently.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-07-25T00:04:15.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12649" PostTypeId="2" ParentId="12644" CreationDate="2017-07-25T00:50:54.647" Score="0" Body="&lt;p&gt;My advice would be to think broadly about what is relevant for biologists and include some things that are environmental, chemical or even geological.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also some biological problems that are purely biological (anything involving depletion or accumulation, biofermenters, toxins, etc. are a good example).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, emphasize examples from human physiology as most biologists are pre-med, pre-pharma, etc.  And it is just intrinsically interesting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some quick Google research results:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.brynmawr.edu/math/people/vandiver/documents/Integration.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.brynmawr.edu/math/people/vandiver/documents/Integration.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=integral+calculus+biology+problem&amp;amp;sourceid=ie7&amp;amp;rls=com.microsoft:en-US:IE-Address&amp;amp;ie=&amp;amp;oe=&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=integral+calculus+biology+problem&amp;amp;sourceid=ie7&amp;amp;rls=com.microsoft:en-US:IE-Address&amp;amp;ie=&amp;amp;oe=&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note, I was going to post the actual results but...there is a social justice problem stopping guests from posting more than 2 links.)&lt;/p&gt;&#xA;" OwnerUserId="8518" LastActivityDate="2017-07-25T00:50:54.647" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12650" PostTypeId="2" ParentId="12645" CreationDate="2017-07-25T01:42:46.123" Score="2" Body="&lt;p&gt;One reason students might not be motivated is that the Putnam is too hard and the other options come with no tangible rewards. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might want to consider setting up a competition within your school with even token prizes. Perhaps a set of weekly problems which would be graded, with a regularly updated high scores table and prizes awarded to the top n people at the end of the semester. Once you have a small group of people going at this, you could set up your group and hopefully these students will now be more receptive to doing maths in their free time.&lt;/p&gt;&#xA;" OwnerUserId="8519" LastActivityDate="2017-07-25T01:42:46.123" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12651" PostTypeId="2" ParentId="12644" CreationDate="2017-07-25T12:13:18.073" Score="6" Body="&lt;p&gt;AUC (&lt;a href=&quot;https://en.wikipedia.org/wiki/Area_under_the_curve_(pharmacokinetics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Area Under the Curve&lt;/a&gt;) plays an important role in pharmacokinetics. The basic ideas should be both accessible and interesting to biology students. There is a potentially interesting tidbit associated with this. The trapezoidal rule is often used to estimate this integral, but the much-cited paper which helped to popularize the trapezoidal rule in pharmacokinetics was written by a researcher who didn't realize that they were &lt;a href=&quot;https://academia.stackexchange.com/q/9602/72815&quot;&gt;reinventing a wheel&lt;/a&gt; which had been known for centuries.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastEditorUserId="6891" LastEditDate="2017-07-25T18:41:04.553" LastActivityDate="2017-07-25T18:41:04.553" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12652" PostTypeId="1" CreationDate="2017-07-26T15:05:28.713" Score="10" ViewCount="132" Body="&lt;p&gt;Have there been any studies done into the effect of having fewer classes per term on a student's comprehension of their mathematics course material? Also are there any examples of schools that have shorter class terms with fewer classes per term?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was an undergraduate at a California State University where there are two 15-week terms per year, and each student takes five or six classes per term. Now I'm a graduate student at University of California campus where there are three 10-week terms per year, and students take about four classes per term. I've noticed that my own study habits and the attitudes of the students seem different at these two schools. I was wondering if this is just my perception, or just a result of the different culture at these two Universities, or if there is something more general that can be said here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically for mathematics, I think that among the classes in a typical undergraduate's schedule, their mathematics class is probably their least favorite. Then if an undergraduate has many classes per term, I'm worried that their least favorite class will get put on the back-burner and the student will tend to do the bare minimum to pass that class in favor of studying for classes they like more. &lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2017-07-26T21:52:22.313" Title="Studies into the effects of having fewer classes per term" Tags="|undergraduate-education|reference-request|education-research|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12653" PostTypeId="2" ParentId="12645" CreationDate="2017-07-26T15:18:57.527" Score="5" Body="&lt;p&gt;Our MAA section (North Central) has an annual team math competition for undergraduates.  It has proven more attractive to the students than the Putnams (which we haven't tried for many years).  The reasons for this are: (1) It's just a Saturday morning (9:00-12:00), rather than the 6-hour Putnam format.  (2) The problems are more accessible, although some are quite challenging. (3) The students seem to like working in groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a link to the competition website, including links to past tests.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://sections.maa.org/northcen/teamcomp.html&quot; rel=&quot;noreferrer&quot;&gt;http://sections.maa.org/northcen/teamcomp.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. It also doesn't hurt to offer milk, juice, and doughnuts to 'bribe' the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.P.S. Some professors offer extra credit in their classes for participation; although this sometimes brings in some not so serious students.&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2017-07-26T15:18:57.527" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12654" PostTypeId="2" ParentId="9806" CreationDate="2017-07-26T15:49:17.537" Score="2" Body="&lt;p&gt;Great Question!  I think that you should target the use of technology that students can use in additional spaces and will have easy access to.  Here, I would recommend Python or R or even just using Google Sheets to explore so that they have connections to other work they will do in the future.  Producing high quality presentations and written work is a skill important to both our mathematical and non-mathematical students.  As simple recursion and plotting is all that is needed for calculus, I think any of the three choices make sense.  Google sheets will be most familiar, R will produce the nicest output with RMarkdown, and Python with the Jupyter notebooks is great and easily moves from computational exploration to paper to slides to web to etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been using Jupyter and Python with my students who are liberal arts undergrads and don't have substantial mathematics backgrounds.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Central to my class is recourse to the difference calculus aka the use of sums and differences to motivate the important concepts of integration, differentiation, and differential equations.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that recent curricular forms have started to push a larger existence for finite approaches within algebra and precalculus, but simply understanding linear, quadratic, and exponential sequences defined recursively and the patterns that emerge from looking at sums and differences of these things are best explored on a machine. The syntax required to do so is minimal in Python.  For example, if you have an area approximation problem with four rectangles width 1 and heights 1, 2, 3, 4, we can express this with lists and operations on that list as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;&#xA;w = 1 &#xA;h = [1, 2, 3, 4]&#xA;areas = w*h&#xA;tot_area = sum(areas)&#xA;&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can easily vary the structure of the list to have more rectangles, and there are very easy functions to build lists quickly.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typically, I want to do a few small simple problems like this and turn to the computer to demonstrate how we deal with larger cases and see if there are recognizable patterns that emerge.  If we had ten rectangles, this might be a bit difficult for timely work in class except for the computer.  The images below are easily produced and values representing the areas are easily computed with python.  There are also existing libraries in both SymPy and SciPy to evaluate expressions symbolically and numerically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/pRHYw.png&quot; alt=&quot;](https[![areas in matplotlib&quot;&gt;]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's an example of some basic area problems that I made:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.dubmathematics.com/htmls/0.5_calc_summations_and_areas.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.dubmathematics.com/htmls/0.5_calc_summations_and_areas.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not aware of a solid holistic text that does a good job interweaving accessible technology, but I am working up the rest of my class materials with Python if you're interested and I'm happy to share.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-07-26T15:49:17.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12656" PostTypeId="1" AcceptedAnswerId="12667" CreationDate="2017-07-27T02:40:02.737" Score="36" ViewCount="13037" Body="&lt;p&gt;This is just the third semester I've been teaching, but I've been tutoring for many years. At the moment I'm teaching to community college students a &quot;Business Calculus&quot; course whose curriculum is attempting to cram an entire semester of calculus into eleven four-hour sessions. I've always had problems taking timed exams myself and find the anxiety crippling, so for this reason, coupled with the brevity and density of the course, I prefer to administer take-home exams. They seem to go over well with the students, and they don't take precious time away from class. And usually people do not perform &quot;unexplainedly better&quot; on my take-homes versus in-class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I've run into a snag. One of my students handed in his take-home exam based on the algebra review in the beginning of the course, and did fairly well on it. But his homework is atrocious. Tonight I helped him through a basic problem in front of the class, and I realized as I watched him cluelessly push the marker around the board that he has no idea what he is doing even on a rudimentary level. And I suspect that he had a great deal of help from someone on that initial test. Disappointing to say the least.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then my question is this: do I need to switch to in-class exams because of this one guy? The take-homes work for everyone else and I much prefer them; they allow for longer, more comprehensive testing of the material, and they don't take time out of a class into which I'm struggling desperately to fit the entire curriculum. Can anything else be done?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any words of wisdom would be greatly appreciated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Thank you all so much for your consideration and thoughtful responses. They have definitely provided me with invaluable insight, and given me a lot to consider.&lt;/p&gt;&#xA;" OwnerUserId="8529" LastEditorUserId="8529" LastEditDate="2017-07-29T03:55:09.663" LastActivityDate="2017-08-07T13:33:25.233" Title="Should I change my take-home exam policy because of one suspected cheater?" Tags="|exams|cheating|" AnswerCount="15" CommentCount="19" ContentLicense="CC BY-SA 3.0" />
  <row Id="12657" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T05:07:48.637" Score="28" Body="&lt;p&gt;It's not just one student. As mentioned in comments, this is just the tip of the iceberg. Various studies gauge the percent of college students who cheat at &lt;a href=&quot;https://www.psychologytoday.com/blog/the-campus/201202/academic-dishonesty-prevalent-preventable&quot; rel=&quot;noreferrer&quot;&gt;somewhere between 75% and 98%&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would recommend that you have in-class, proctored exams. That's the only way to ensure that one's math course is not, essentially, a fraud. You may even want to check photo IDs at exams, since sending in &quot;ringers&quot; in place of registered students is also a known strategy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The community college where I teach has an Accessibility office which can specially proctor tests for students who demonstrate documented reasons why they may need more time on exams. Hopefully yours does, too. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-07-27T05:07:48.637" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12658" PostTypeId="2" ParentId="2428" CreationDate="2017-07-27T09:23:11.650" Score="3" Body="&lt;p&gt;For me as an educator it was always about how the material is delivered in the classroom \ lecture hall that made the difference. So perhaps in terms of the broad range of topics you posted, so in the interests of being &lt;em&gt;useful and imteresting&lt;/em&gt; could I offer these nuggets...&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Calculus card matching: &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;On card (no bigger than a letter envelope write a set of mathematical symbols, say $3x^2, 4x^3, 12x^3, \ldots$ and on other cards write their anti-derivatives (whether you want to write the constant after integration is up to you) . &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of this task is to allow students to visually match the cards by &lt;em&gt;seeing&lt;/em&gt; the derivative / integral in their minds without the use of a pencil and paper. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would often sidestep the need for writing $\frac{d}{dx}$ by giving the students a square of red card that 'fits' in between the two cards to emphasise which card has the derivative on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a slightly advanced technique, a blue card could represent $\frac{d^{2}}{dx^{2}}$ and get the students to match a card with $\frac{x^{6}}{3}$ on it with $10x^{4}$ on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have sets of cards (with red-square 'first deriviatves') between $5-10$ as a lesson Starter or Blue-square (second derivative) as a main bulk of a lesson with $10-15$ cards. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good one to have is $\ln x$ and $x^{-1}$ and variations that may imply that one is a derviative of another, to test the students' reasoning. random cards as red herrings are also useful, with $e^{-2}$ and $\infty$ on it are also fun to see the students reason with each other where they go, perhaps having a piile for red-herrings off the main bulk of matches.&lt;/p&gt;&#xA;" OwnerUserId="8532" LastActivityDate="2017-07-27T09:23:11.650" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12659" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T11:16:55.517" Score="6" Body="&lt;p&gt;My school math teacher was very smart and I'll tell about his solution of (any) cheating problem. If anybody is suspected in cheating just call him to the desk and ask about doing same(or very similar) exercises. If one fails to do something, one has already done, it's cheating and if not, there is no reason to punish the person, because the one has got the knowledge that is required.&lt;/p&gt;&#xA;" OwnerUserId="8533" LastActivityDate="2017-07-27T11:16:55.517" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12660" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T11:22:23.820" Score="13" Body="&lt;p&gt;My own experience is to not give any &lt;em&gt;high-stakes&lt;/em&gt; take-home exam-like content (I'm not speaking of a paper, of course) in a lower-level course.  There is too much incentive for even well-meaning students to cheat, and not in such egregious ways as you are mentioning.  It's key to note that many students who do this either do not recognize it as cheating or self-justify without ill intent, so it is partly up to us to create an environment with fewer incentives to cheat.  That's not right, but it's reality. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With an upper-level class you know well that might not be an issue, and I have a lot of take-home material in those.  Also, James M. Lang has an entertaining and useful book called &lt;a href=&quot;http://www.hup.harvard.edu/catalog.php?isbn=9780674724631&quot; rel=&quot;noreferrer&quot;&gt;Cheating Lessons&lt;/a&gt; which has a lot of practical tips and historical context on this issue.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-07-27T11:22:23.820" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12661" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T13:06:33.137" Score="6" Body="&lt;p&gt;Some students will cheat. (So do some grownups.) In my experience it's not a large fraction, despite the studies that seem to say otherwise. I think I caught most of what went on in my classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was unwilling to forego the educational benefit of take home exams, group work and well thought out projects that could be honed and reassigned just to catch the few cheaters. You seem to understand that. So my answer to your explicit question &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;do I need to switch to in-class exams because of this one guy?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;would be &quot;no&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course you need some checks, some in class work. It helps if there's school policy, and an administrative structure to back you up if you want to bring a case. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the handout I've given my students over the years: &lt;a href=&quot;http://www.cs.umb.edu/~eb/honesty/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.cs.umb.edu/~eb/honesty/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4984" LastEditorUserId="4984" LastEditDate="2017-07-27T17:34:33.740" LastActivityDate="2017-07-27T17:34:33.740" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12662" PostTypeId="2" ParentId="12645" CreationDate="2017-07-27T14:31:43.687" Score="2" Body="&lt;p&gt;I would recommend adding some &lt;strong&gt;structure&lt;/strong&gt; since many students respond well to an activity if it makes it onto a transcript or resume, due to being trained this way in high school. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The way they got me into the competition scene as an undergraduate was to offer a 1-credit course called a &quot;problem seminar&quot; that was explicitly a fun time to work on hard competition-type problems. Then all math majors were encouraged to enroll in it. Once we registered, we had external motivation to show up, try, and accomplish hard things and then it was natural to try the Putnam afterwards.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At my current institution we proctor a national math contest but have the prep for that contest tied up in student clubs (the Math &amp;amp; Engineering Club) and we offer a prize for the highest scorer at our school even if they do not place nationally. I had some luck getting the Software &amp;amp; Tech Club to complete some Project Euler problems as well by making it a weekly meeting; see also &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2265/project-euler-for-a-casual-collaborative-student-group&quot;&gt;this&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2145/how-to-build-a-class-based-on-project-euler&quot;&gt;this&lt;/a&gt; for more on Project Euler.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The common thread here is that convincing students to do things on their own time due to intrinsic motivation doesn't work so well. Instead, setting up structures so that students put themselves into a situation with extrinsic motivators seems effective in my experience. Good luck!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-07-27T14:31:43.687" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12663" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T14:55:24.393" Score="3" Body="&lt;p&gt;I agree with much of what's already been said here already, but looking at the question from a different angle, you said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I've always had problems taking timed exams myself and find the&#xA;  anxiety crippling, so for this reason...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Do you not think that this is a really good reason to get the students used to operating under these conditions? The &quot;real&quot; exams presumably will be so they should get used to it and practice on ones which are arguably less important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You wouldn't teach someone to drive on an empty airfield, without any experience on a real, busy road and then suddenly tell them to take the real test in a scary situation that they have never experienced before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the above reasons, regardless of the possible cheating (which others have commented on), I'd 100% swap at this point in time. It might be a little &quot;unfair&quot; on the students who have been honest and were expecting exams at home, but life it not fair and things out of your control change - this is another good lesson for the students to learn.&lt;/p&gt;&#xA;" OwnerUserId="8534" LastActivityDate="2017-07-27T14:55:24.393" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12664" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T16:57:51.813" Score="5" Body="&lt;p&gt;What if a day or two after the take-home test was turned in you gave a short in-class test that covered some of the same subject matter (very similar questions).  You could use this to weight the take-home test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, you could say that you get the lesser of the grade on the take home and in-class tests.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose to be fair you might further say that if they missed similar answers on both tests causing the in-class to have a lower percentage score then they could still use the overall take-home score since you're really only trying to penalize cases where they get the answer at home but can't duplicate the work in the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea of this would be to give more than enough time to answer the few questions in class but still cover the subject material adequately.&lt;/p&gt;&#xA;" OwnerUserId="8536" LastActivityDate="2017-07-27T16:57:51.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12665" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T18:12:34.667" Score="3" Body="&lt;p&gt;I had a professor who had dealt with a similar situation, although in computer science, and dealt with it like this. The assigned homework was considerably more difficult than the tests, and group work was permitted for the homework. The tests were fairly trivial compared to the homework, not testing much more than &quot;can you do the absolute bare minimum&quot;. The professor would then look for students with very high homework grades and very low exam grades, and decide if they believed those students were cheating, or simply very bad at test, or etc. and handled those students on a case by case basis.&lt;/p&gt;&#xA;" OwnerUserId="8537" LastActivityDate="2017-07-27T18:12:34.667" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12666" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T18:52:10.990" Score="6" Body="&lt;p&gt;I think timed exams are a better way of assessing mastery of material than take homes.  If you can't do it fast, under pressure, and with limited assistance*, than you don't really know the stuff the way you should.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*And take homes are fraught with all kinds of aspects of outside assistance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  And you ruin their nights and weekends by giving take homes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.  Drill (and testing is EXCELLENT drill) is a much more valuable use of time than lecture.  So I go totally the opposite on the question of &quot;sparing the time, so more lecture can be done&quot;.  And really you are eating more into the students outside time also.  Not efficient for them.&lt;/p&gt;&#xA;" OwnerUserId="8348" LastActivityDate="2017-07-27T18:52:10.990" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12667" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T19:59:22.430" Score="3" Body="&lt;p&gt;My opinion here is that 44hrs is plenty of time for the course to be taught (based on most higher academic institution course lengths), but you are doing them a favour based on your experience, and unfortunately sometimes one person can ruin it for everyone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;****Putting that aside...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only other option you'd have (instead of changing to in-class exams), is to pull the student aside and say something like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I noticed that you did very well on the exam, but I find the drastically increased grade suspicious based on your homework before the exam and your performance since the exam. I don't mean to discourage you; if you truly completed the exam without cheating, that is terrific and is an accomplishment to be proud of. However, if this cannot be explained to my satisfaction, I will have to take steps forward in [the institution's name]'s academic dishonesty policy.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After saying this tell the student you are available to help them if they need it, but emphasize that &lt;strong&gt;&lt;em&gt;cheating is not to be taken lightly, and has significant repercussions&lt;/em&gt;&lt;/strong&gt;. Also emphasize that you are not targeting that student, but you have an obligation to ensure each student earns their grades and responsibility to prevent academic dishonesty.&lt;/p&gt;&#xA;" OwnerUserId="8538" LastActivityDate="2017-07-27T19:59:22.430" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12668" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T20:16:23.550" Score="2" Body="&lt;p&gt;This is quite a simplistic approach to the one individual so forgive me if its too simple to be workable - but how about calling him in and putting the exact same exam paper in front of him and asking him to re do it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general though, from a uni student's perspective, I recently received a take home exam from our online portal and although I didn't cheat myself (what's the point, Tutors and Examiners have seen it all before) I do know a fair few people who did and I was a bit taken back that this was even allowed really. &lt;/p&gt;&#xA;" OwnerUserId="8539" LastEditorUserId="8539" LastEditDate="2017-07-27T20:33:10.923" LastActivityDate="2017-07-27T20:33:10.923" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12669" PostTypeId="2" ParentId="12264" CreationDate="2017-07-27T23:15:12.543" Score="2" Body="&lt;p&gt;Just as a supplement to other answers: of course, in the first place, configuring &quot;a calculus course&quot; depends on the larger purpose of the thing. I do think it is too naive to say that it is literally to teach exactly the mathematical content of the course... since, as the other answers show, there seem to be larger philosophical issues in play. At the same time (e.g., in the U.S., most familiar to me) there are dominant sociological issues in the situation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After those platitudes, my main point would be that I don't think that mathematics is fundamentally about &quot;logic&quot;, nor that calculus (or even &quot;analysis&quot;) is fundamentally (!) about epsilons and deltas (and certainly not about order of quantifiers). That is, yes, while certain immediate issues can be &lt;em&gt;formalized&lt;/em&gt; as being those things, first, attempts to change things by directly addressing &quot;(first-order predicate) logic&quot; will mostly fail, because FOPL is not reflected in the ambient culture, and, second, there is some artificiality (and convention) involved in deciding that first-order predicate logic is &quot;the true logic&quot;, &lt;em&gt;and&lt;/em&gt; deciding that epsilon-delta is &quot;the true analysis&quot;. Neither of these is clearly true, although simple, popular choices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I myself do like genuine proofs of interesting things, but I am less enthusiastic about implicit meta-hypotheses that force me/you/us into technically awkward/ponderous arguments. My fave is the intermediate value theorem: obviously true, without &quot;definition&quot; of continuity or the real numbers. Why does it even need proof? (Yes, I know, ...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the &lt;em&gt;syntax&lt;/em&gt; of legitimate mathematical argument should (for stability) resemble ordinary language as much as possible. Yes, ordinary language has many (sometimes unfortunate) assumptions that are contrary to the &quot;truths&quot; of first-order predicate logic. There's the point: do we (people explaining/teaching mathematics) want to tell students that the ambient culture (&lt;em&gt;their&lt;/em&gt; culture) is &quot;wrong&quot; (&quot;kewl&quot;, sure, but silly...), or, rather make a higher-level transition from the not-easily-changed vernacular, and explain mathematics &lt;em&gt;in&lt;/em&gt; &lt;em&gt;those&lt;/em&gt; &lt;em&gt;terms&lt;/em&gt; !?!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, I don't think we all have to learn Esperanto before we can do the amazing things that calculus (and other good mathematics) allows us. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I know I have a bit-extreme viewpoint... )&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-07-27T23:15:12.543" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12670" PostTypeId="2" ParentId="12645" CreationDate="2017-07-27T23:15:48.103" Score="2" Body="&lt;p&gt;I would like to suggest there is a significant psychological difference between solving Putnam-like competition problems, and working on open problems.&#xA;For example, the &lt;a href=&quot;http://2017.cccg.ca/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Canadian Conference on Computational Geometry&lt;/a&gt; has an annual open-problem session that poses problems accessible to undergraduates:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://2017.cccg.ca/proceedings/CCCG2017.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download of full proceedings&lt;/a&gt;,&#xA;  Open Problems p.73ff.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To get a flavor without downloading that (large file), see the MO posting,&#xA;&lt;a href=&quot;http://2017.cccg.ca/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sofa in a snaky 3D corridor&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Working on these problems requires faculty guidance, but it has a different feel than &#xA;competition problems. My experience is that some students get totally hooked on this frontier of the unknown in a way that differs from their reaction to&#xA;a competition.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-07-27T23:15:48.103" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12671" PostTypeId="2" ParentId="12656" CreationDate="2017-07-27T23:38:05.033" Score="11" Body="&lt;p&gt;As a student currently taking college Programming Fundamentals II and Pre-Calculus II, but having the experience of the world (33 years old, 8 year army vet), I can say that a student has significant incentives to cheat, if provided an opportunity. To fail and retake a course would be costly, due to the additional work, the tuition, and the books. I find that at my college books range from \$100 to \$250. To make that worse, just because you retake the &quot;same class&quot; doesn't mean you'll get the same teacher, which means you might need a different book. If I sucked at math and my teacher was giving me take homes, and I knew that I didn't need math for my degree like others do....I would certainly take that test home and Symbolab and google the crap out of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically I'm saying that if you allow students to take home a test, you should expect some cheaters. The point of a test is to check their understanding, and if you allow them to utilize their understanding of Google instead of their understanding of math then I feel you are failing your students academically. Online classes have proctored tests for a reason.  &lt;/p&gt;&#xA;" OwnerUserId="8540" LastEditorUserId="996" LastEditDate="2017-07-29T09:14:54.637" LastActivityDate="2017-07-29T09:14:54.637" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12672" PostTypeId="2" ParentId="12656" CreationDate="2017-07-28T01:09:33.643" Score="3" Body="&lt;p&gt;In my Business Calculus classes, I have three standard in-class tests and four to five projects that they have one week to complete.  The projects test the kind of more creative thinking that I think you feel you can better assess with your take home tests and the in-class tests evaluate more procedural, textbook-type questions.  There's always the possibility of cheating on any &quot;home&quot; assignment but this is an issue that every subject faces.  I try to always grade the projects in one sitting so that any similarities between papers stand out and I regularly search the Internet for key phrases from the assignments so that I can see if they're on any &quot;homework help&quot; websites.  Having a combination of in and out of class assignments helps to mitigate the affects of cheating and still lets me evaluate both of the types of skills that I think are important.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2017-07-28T01:09:33.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12673" PostTypeId="2" ParentId="805" CreationDate="2017-07-28T01:25:34.117" Score="3" Body="&lt;p&gt;I suggest nearly-unlimited time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a student, I don't suffer from the anxiety suggested by other posters, but I do have trouble understanding &lt;em&gt;how thorough&lt;/em&gt; I need to &lt;em&gt;not&lt;/em&gt; be in presenting my work.  I will also sometimes get stuck in &quot;algebra hell&quot; attempting to manipulate the expression towards a solution, and methodical reworking after a wrong-turn takes time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I acknowledge that a student who doesn't fully understand the material may suffer diminishing returns on their score to time ratio.  However, up through Calc 1, I haven't personally hit that point when a teacher calls &quot;pencils down&quot;.  I nearly always still have several problems I haven't even gotten to yet, even if they're easy points, because I started at the beginning and kept working.&lt;/p&gt;&#xA;" OwnerUserId="8541" LastActivityDate="2017-07-28T01:25:34.117" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12674" PostTypeId="1" CreationDate="2017-07-28T04:41:20.870" Score="3" ViewCount="270" Body="&lt;p&gt;Question for a research project:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the standard way of quantifying a student's skill in arithmetic ranging from having to look up numbers on a times-table to computing large sums in their head in a few seconds?&lt;/p&gt;&#xA;" OwnerUserId="8542" LastEditorUserId="1537" LastEditDate="2017-07-31T01:16:57.313" LastActivityDate="2017-07-31T01:16:57.313" Title="Quantifying arthmetical skill" Tags="|reference-request|arithmetic|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12676" PostTypeId="1" CreationDate="2017-07-28T08:54:05.333" Score="6" ViewCount="122" Body="&lt;p&gt;I have to write some practice questions, or toy problems for some students in the age range 11 to 14. This is for a workshop I will be running. The students are encouraged to work on these questions in downtime, at points that they are not able to progress with their core project (Robot building). This is likely to happen when programs are downloading or they are waiting for a tool to become available. They are well motivated and smart kids. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have never written this sort of thing before. I'm not really sure what the do's and don'ts are. Here is one of my attempts:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7dQKD.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7dQKD.png&quot; alt=&quot;A question with answer box&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Its supposed to be on topic and self contained. The maths should be very easy, the challenge is to figure out how the formula should be used. I think they have probably seen this formula before. The story is crudely robot themed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next question is of a similar nature and follows the story of the first:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KOdcn.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KOdcn.png&quot; alt=&quot;question 2&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do these questions seem reasonable to you? How would you improve them? Are there any question writing rules I should keep in mind? &lt;/p&gt;&#xA;" OwnerUserId="8154" LastActivityDate="2017-08-27T03:08:56.520" Title="What are the fundamentals of writing practice questions for early teen students?" Tags="|problem-design|problem-solving|physical-sciences|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12678" PostTypeId="2" ParentId="805" CreationDate="2017-07-28T10:22:55.443" Score="0" Body="&lt;p&gt;A student with excellent insight and understanding, A-level, mastery, should finish the test comfortably.  A weaker student will feel time pressure.&lt;/p&gt;&#xA;" OwnerUserId="8518" LastActivityDate="2017-07-28T10:22:55.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12679" PostTypeId="2" ParentId="12656" CreationDate="2017-07-28T14:37:27.853" Score="2" Body="&lt;p&gt;One course I took on University had a simple approach on this: Everyone had to explain one of their answers in front of the class, and failure to do that appropriately resulted in the student loosing &lt;em&gt;twice&lt;/em&gt; the points awarded for the question (thus a question omitted or only half-answered meant less points to be potentially lost, rewarding honesty). Of course this was a small group and it concerned weekly tasks such that statistically speaking it wouldn't be unfair.&lt;/p&gt;&#xA;" OwnerUserId="1120" LastActivityDate="2017-07-28T14:37:27.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12681" PostTypeId="1" AcceptedAnswerId="12683" CreationDate="2017-07-28T19:34:48.850" Score="6" ViewCount="225" Body="&lt;p&gt;Background: I teach Calculus at first year Engineering degrees in a large public University in Spain. I am introducing more and more active activities (no pun intended) so, to begin with, I want the students to do practical exercises. This is the right content to train since, in fact, the exam consists of several problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I ask a student to write an exercise on the chalkboard (or even worse, I write it myself) I feel that I am teaching both the wrong format and the wrong skills, let alone having a terrible time management. So I think it is better if they all do the exercise on paper and then I could project one of them on the screen and grade/discuss it for everyone to see.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is not about the comparative learning effectivity of this technique (although any comments are appreciated) but I ask you to address my two insecurities:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the students, will they laugh at me as a complete technological dinosaur, as if I had intended to use the overhead projector (which, by the way, &lt;em&gt;is&lt;/em&gt; still in the classroom)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the management that should authorize the expense, can I use the argument that this technology is often used in top-tier Universities? Because I have no knowledge of anyone using it &lt;strong&gt;in the classroom&lt;/strong&gt;, at least in my School.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So are document cameras often used &lt;em&gt;in the classroom&lt;/em&gt; for projecting students' work?&lt;/p&gt;&#xA;" OwnerUserId="8545" LastActivityDate="2017-07-29T03:10:34.733" Title="Are document cameras widely used at college level?" Tags="|technology-in-education|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12683" PostTypeId="2" ParentId="12681" CreationDate="2017-07-29T03:10:34.733" Score="5" Body="&lt;p&gt;I cannot speak to European usage, but document cameras are definitely used in many US classrooms.  They are in fact quite flexible from the standpoint of not requiring any particular specific technology of students, and in principle one can even take &quot;screenshots&quot; with them and save to a zip drive.  (They are very inflexible in other ways, of course.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Honestly, it is quite good to use them in the same way as some of the best simple math videos (see &lt;a href=&quot;http://mooculus.osu.edu&quot; rel=&quot;noreferrer&quot;&gt;Mooculus&lt;/a&gt;, for instance).  That is, step by step writing or unveiling of things.  For some reason students need to see that step-by-step process, and especially in a large room, or when having them write it out by hand (chalkboard or elsewhere) is too time-consuming, the document camera is a very good option.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The downside is when an entire solution is presented in one visual view and no one thinks they have any questions - so be prepared for thinking of something to ask to keep the class on their toes!  But of course that is not about the document camera itself.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-07-29T03:10:34.733" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12684" PostTypeId="2" ParentId="12644" CreationDate="2017-07-29T20:12:52.573" Score="2" Body="&lt;p&gt;I think a common example is &lt;em&gt;cardiac output&lt;/em&gt;.  Here, with the dye dilution technique, you have a monitor that measures dye pushed through the bloodstream.  Maybe, for example, you would like to investigate Turkey's heart rates as they run on treadmills.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/5TiGU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5TiGU.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The monitor would measure a concentration in the blood at certain intervals.  We can consider the cardiac output as the total volume of dye measured divided by the time as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$R=V/T$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, we can express this as the amount of dye(D) over the volume(CT) as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$R=D/CT$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, in our example above, the CT is the sum of the concentrations of dye in the blood, which would be found by integration.  I use these examples early in integration before much work with a definite integral.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A specific example could have 5 liters of dye injected with the following concentrations measured every second:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$c = [0, 0.1, 0.2, 0.6, 1.2, 2.0, 3.0, 4.2, 5.5, 6.3, 7.0, 7.5, 7.8, 7.9, 7.9, 7.9, 7.8, 6.9, 6.1, 5.4, 4.7, 4.1, 3.5, 2.8, 2.1, 2.1, 2.2]$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our cardiac output would be found by:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$R = \frac{5}{\sum_{i=1}^{25} c_i} = \frac{5}{45.43}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Graphically, you can interpret this as area approximations under the curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KNDLq.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KNDLq.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-07-29T20:12:52.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12685" PostTypeId="2" ParentId="12644" CreationDate="2017-07-30T01:37:54.660" Score="5" Body="&lt;p&gt;For many years Dorothy Wallace has taught a course at Dartmouth College&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.dartmouth.edu/~m4s17/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://math.dartmouth.edu/~m4s17/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;that gives bio majors and pre-med folks an opportunity to contribute to the mathematical modeling of biological systems literature (predator-prey models, population models, disease models, pharmacokinetics).  For many of the projects and assignments in the course, all the calculus a student needs is an intuitive understanding of the derivative, an appreciation for exponential growth and decay and a willingness to use and understand Euler's method. The latter is not typically a Calculus I topic (maybe it should be), but I find that it is not hard for students to pick up.  The projects in the course often take the form of &quot;here's an interesting paper or model in the literature, let's consider this tweak to it...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The course is for students who have completed a semester of calculus, but the population modeling and pharmacokinetics models I feel should be accessible to students taking a first course in calculus.  For example: the drug nifurtimox is used to treat chagas, a tropical parasitic disease.  You could give them the simple system of differential equations that represents the way the drug is processed by the body (and ask them to explain it) and then ask them which is better:  a three dose a day regimen or a four dose a day regimen (the Centers for Disease Control says both regimens are possible but doesn't say which is &quot;better&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Full disclosure:  Dorothy and I have a book on this topic: &quot;Applications of Calculus to Biology and Medicine: Case Studies from Lake Victoria&quot; published by World Scientific.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="2083" LastEditDate="2019-08-08T05:09:56.077" LastActivityDate="2019-08-08T05:09:56.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12687" PostTypeId="2" ParentId="12656" CreationDate="2017-07-31T05:23:57.777" Score="2" Body="&lt;p&gt;Most in-class exams I have seen are indeed too long and, as it happens, essentially highly redundant. So, how about giving shorter exams which require a bit of thinking ---as opposed to &quot;example duplication&quot;. The downside, of course, is that the instructor her/himself must not ... duplicate examples while teaching and that in fact the whole approach requires what &lt;strong&gt;Hestenes&lt;/strong&gt; called &quot;restructuring contents&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note to commentators: I am speaking of the real world, not of, e.g. the world of twenty pounds textbooks.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="4815" LastEditDate="2017-07-31T05:29:40.530" LastActivityDate="2017-07-31T05:29:40.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12688" PostTypeId="2" ParentId="9667" CreationDate="2017-08-01T15:15:08.360" Score="2" Body="&lt;p&gt;The answer of this question might be found in the question: &quot;When are multiplications used in primary education?&quot;, and for answering that question, let's have a look at following example: the multiplication of numbers containing multiple digits, like in this example: 789*567.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are performing this multiplication as follows:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;9*7=63&#xA;80*7=560&#xA;700*7=4900&#xA;=&amp;gt; result 5523&#xA;9*6(0) = 54(0)&#xA;80*6(0) = 480(0)&#xA;700*6(0) = 4200(0)&#xA;=&amp;gt; result 47340&#xA;9*5(00) = 45(00)&#xA;80*5(00) = 400(00)&#xA;700*5(00) = 3500(00)&#xA;=&amp;gt; result 394500&#xA;Add everything : 5523 + 47340 + 394500&#xA;3 =&amp;gt; result 3&#xA;80+40 = 120&#xA;600+300+500 = 1400&#xA;5000+7000+4000 = 16000&#xA;40000+90000 = 130000&#xA;300000 = 300000&#xA;=&amp;gt; result =  447363&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Total amount of calculations (how many times did we calculate (add, multiply, ...) something?):&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;9 multiplications&#xA;6 additions&#xA;=&amp;gt; 15 calculations&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Now do the same but replacing a*b by a+a+...+a (b times)&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;9*7 = 9+9+9+9+9+9+9 = 63&#xA;80*7=80+80+80+80+80+80+80 = 560&#xA;700*7=700+700+700+700+700+700+700 = 4900&#xA;=&amp;gt; result 5523&#xA;9*6(0) &amp;lt;- 9+9+9+9+9+9  = 54, add a (0) = 540&#xA;80*6(0) &amp;lt;- 80+80+80+80+80+80 = 480, add a (0) = 4800&#xA;700*6(0) &amp;lt;- 700+700+700+700+700+700 = 4200, add a (0) = 42000&#xA;=&amp;gt; result 47340&#xA;9*5(00) &amp;lt;- 9+9+9+9+9 = 45, add two (0) = 4500&#xA;80*5(00) &amp;lt;- 80+80+80+80+80 = 400, add two (0) = 40000&#xA;700*5(00) &amp;lt;- 700+700+700+700+700 = 3500, add two (0) = 350000&#xA;=&amp;gt; result 394500&#xA;Add everything : 5523 + 47340 + 394500&#xA;3 =&amp;gt; result 3&#xA;80+40 = 120&#xA;600+300+500 = 1400&#xA;5000+7000+4000 = 16000&#xA;40000+90000 = 130000&#xA;300000 = 300000&#xA;=&amp;gt; result =  447363&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Total amount of calculations (how many times did we calculate (add, multiply, ...) something?):&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;0 multiplications&#xA;53 additions&#xA;=&amp;gt; 53 calculations&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;In other words, in case the kik doesn't know the multiplications by heart, (s)he will need to do 3 times the amount of calculations, just for multiplying two numbers of three digits. This, in its term, will ensure that (s)he will need 3 times more time for finishing his/her homework, tests and exams, leading to disappointing school results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess this simple example clearly shows that, once the kid will need to multiply numbers of at least three digits, (s)he needs to know the multiplication tables by heart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck&lt;/p&gt;&#xA;" OwnerUserId="8190" LastEditorUserId="8190" LastEditDate="2017-08-02T10:31:50.077" LastActivityDate="2017-08-02T10:31:50.077" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12689" PostTypeId="1" AcceptedAnswerId="12691" CreationDate="2017-08-02T09:45:28.857" Score="6" ViewCount="241" Body="&lt;p&gt;Recently I have started Tutoring First Year University Mathematics 2, as opposed to Mathematics 1, at a local Hall of Residence at my University. As I am being paid quite a bit, I would love to get some tips on what to do, what not to do, and some common tricks of the trade so that I can be a better maths tutor. The topics that the students are covering in their course are the following;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sequences and Series (Geometric, P-series, taylor, etc)&lt;/li&gt;&#xA;&lt;li&gt;A bit more tricky integration (still single variable though.)&lt;/li&gt;&#xA;&lt;li&gt;Some differential equations&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Algebra:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A bit more advanced,but still rather basic, vector algebra.&lt;/li&gt;&#xA;&lt;li&gt;Solving Systems of linear equations using Matrices.&lt;/li&gt;&#xA;&lt;li&gt;Finding the inverse of matrices bigger than 2x2.&lt;/li&gt;&#xA;&lt;li&gt;Linear transformations&lt;/li&gt;&#xA;&lt;li&gt;basic counting principles.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This mathematics course is also not very formal yet as it is more of an advanced introduction to more complicated mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My &quot;class&quot;, if you will, is quite small, only three students. My students are also quite bright, they ask deeper questions such as &quot;Why can we express all differentiable  functions as a taylor series&quot;, or &quot;what are the implications of matrix multiplication being non-commutative&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am sorry if this question is a duplicate, but I can't seem to find a similar question dealing with Uni level maths tutoring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My exact job as a tutor for this Hall of Residence is just to be there to answer any questions that the student's might have. Close to tests or the exam period at the end of semester I will also be there to give them tips on how to answer the exam and maybe fill in some of the necessary gaps in there knowledge to help them do well in the course.&lt;/p&gt;&#xA;" OwnerUserId="8566" LastEditorUserId="1905" LastEditDate="2017-08-03T00:15:35.087" LastActivityDate="2017-08-03T03:47:10.023" Title="Tutoring Mathematics at University" Tags="|undergraduate-education|tutoring|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12690" PostTypeId="1" CreationDate="2017-08-02T21:55:35.093" Score="12" ViewCount="332" Body="&lt;p&gt;How can I refer to a number written out in its decimal expansion (e.g., 1.25) or binary expansion (e.g., 1.01) to distinguish it from a number expressed as a fraction? I am teaching students to use different bases so do not want a term referencing a specific base, such as &quot;decimal expansion&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In computer architecture, the term &quot;floating point&quot; (or &quot;fixed point&quot;, depending on the implementation) is used, but I don't think this is a mathematical term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case, my question is unclear, I want to complete this analogy: $\frac{1}{2}$ is to fraction as .5 is to ?&lt;/p&gt;&#xA;" OwnerUserId="8573" LastActivityDate="2021-10-27T10:54:49.833" Title="Is there a base-independent term for numbers written out with decimal/binary points?" Tags="|terminology|rational-numbers|" AnswerCount="2" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="12691" PostTypeId="2" ParentId="12689" CreationDate="2017-08-03T03:47:10.023" Score="5" Body="&lt;p&gt;Here's a simple tip: ask questions! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask questions that lead them to the answer. The student will more likely learn something than had you just given them the answer (unless it's exam time and you're tutoring hordes of students). I used this approach when I tutored back in university and still use it now, years later, as a professor. &lt;/p&gt;&#xA;" OwnerUserId="8559" LastActivityDate="2017-08-03T03:47:10.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12692" PostTypeId="2" ParentId="12690" CreationDate="2017-08-03T04:24:04.883" Score="4" Body="&lt;p&gt;The point is definitely called, in general, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Radix_point&quot; rel=&quot;nofollow noreferrer&quot;&gt;radix point&lt;/a&gt; (as stated in a comment by @user52817). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not familiar with, nor succeeding at a search for, a general name for the representation method. I would be comfortable calling one an &quot;$n$-ary representation&quot;, following the term &lt;a href=&quot;https://en.wikipedia.org/wiki/Decimal_representation&quot; rel=&quot;nofollow noreferrer&quot;&gt;decimal representation&lt;/a&gt; (similar to a comment by @mweiss). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-08-03T04:24:04.883" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12694" PostTypeId="5" CreationDate="2017-08-03T17:53:01.727" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2017-08-03T17:53:01.727" LastActivityDate="2017-08-03T17:53:01.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12695" PostTypeId="4" CreationDate="2017-08-03T17:53:01.727" Score="0" Body="For questions related to creating, categorizing and identifying appropriate problems as well as the benefits of problem solving and related pedagogy including motivation." OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2017-08-05T16:14:11.520" LastActivityDate="2017-08-05T16:14:11.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12696" PostTypeId="2" ParentId="12221" CreationDate="2017-08-05T22:04:15.363" Score="1" Body="&lt;p&gt;I think you should focus on learning practical and interesting skills that will help you to get a coding job. For example, how about learning how to code in C#, Java or even Javascript? &lt;br/&gt;&#xA;Coding is a practical and real world skill. It's not a specialty reserved for research scientists anymore. &lt;br/&gt;&#xA; I suggest you begin by focusing on real world situations before jumping into specialized academic approaches that hardly anyone uses. &lt;br/&gt;&#xA; Also, complex math just isn't used much in real world coding situations. &lt;br/&gt;&#xA;There are a few specialized companies that do scientific types of coding where it's useful. &lt;br/&gt; But even there, we have pre-built libraries for handling all of that stuff.&lt;br/&gt; A few nights (or so) of study is all you'll need to make use of specialized tools like that.&lt;br/&gt;&#xA; I strongly suggest you avoid an impractical and academic approach to learning to code. &lt;br/&gt;&#xA;That approach will not lead to a job, and isn't useful for most of the coding that you will need and want to do.&lt;/p&gt;&#xA;" OwnerUserId="990" LastActivityDate="2017-08-05T22:04:15.363" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12697" PostTypeId="2" ParentId="12221" CreationDate="2017-08-06T00:00:32.443" Score="3" Body="&lt;p&gt;What you likely will not have experienced after Calculus and Statistics is: proofs. At most institutions, both Discrete Math and Linear Algebra introduce, and may even concentrate on, proofs. The same is true of Theory of Computation in computer science. Multivariate Calculus (sometimes called Calc III) often does not emphasize proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So that can serve as a &lt;em&gt;disideratum&lt;/em&gt;: Choose a course that exposes you to rigorous proofs.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-08-06T00:00:32.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12698" PostTypeId="2" ParentId="12221" CreationDate="2017-08-06T14:42:39.610" Score="2" Body="&lt;p&gt;As a number theorist I am biased, but I think a conjecture-proof study of elementary number theory could be a good place to start.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many summer programs for mathematically-inclined high school students (e.g., the Ross Program, the Hampshire College Summer Studies in Math, PROMYS) focus on number theory and these have a long track record of producing mathematicians who go into academia.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing number theory in this way combines many aspects of the other responses:  (1) interesting problems with little technical background, (2) development of programming skills as you formulate and verify conjectures, (3) development of proof-writing and (4) applications (cryptography, mostly). Books such as &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/press/ebooks/number-theory-through-inquiry&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/ebooks/number-theory-through-inquiry&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://bcs.wiley.com/he-bcs/Books?action=index&amp;amp;itemId=0470412151&amp;amp;bcsId=4878&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://bcs.wiley.com/he-bcs/Books?action=index&amp;amp;itemId=0470412151&amp;amp;bcsId=4878&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;might be good places to start.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="1537" LastEditDate="2017-08-07T21:57:46.567" LastActivityDate="2017-08-07T21:57:46.567" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12699" PostTypeId="1" CreationDate="2017-08-07T00:34:42.423" Score="11" ViewCount="456" Body="&lt;p&gt;Some schools in the UK are &lt;a href=&quot;https://www.nytimes.com/2017/08/05/world/asia/china-textbooks-britain.html&quot; rel=&quot;noreferrer&quot;&gt;adopting&lt;/a&gt; English translations of a grade school math textbook from Shanghai. The book appears to be designed to work with a specific teaching approach, the only specifics given in the NY Times article being:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Students explain concepts to other students.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The class doesn't move forward until everyone understands.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any more detailed info, such as sample pages? Opinions or data on how well this works compared to standard modes of instruction in the UK and US? Is it similar to what we refer to at the college level as &quot;active learning&quot; or &quot;active engagement?&quot;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2017-08-07T00:34:42.423" Title="Shanghai math -- what is it, and how good is it?" Tags="|textbooks|primary-education|interactive-teaching|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12700" PostTypeId="2" ParentId="12656" CreationDate="2017-08-07T13:33:25.233" Score="1" Body="&lt;p&gt;I'm a very firm believer of the idea that everybody needs to do his or her own work, whether it be homework, tests, exams or anything else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's start talking about homework:&lt;br/&gt;&#xA;Nowadays many teachers are giving homework exercises, which students can &quot;make&quot; at home by filling in something using their computer.&lt;br/&gt;&#xA;Yeah, right!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When using such a system, you risk very smart students actually making the homework, and other students only need to push on three buttons (Ctr+C and Ctrl+V, copy/paste, you see?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admit, when you give homework in writing, they can still copy from the smartest students, but at least they will need to write everything themselves, which is already a bigger effort than a simple copy/paste.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now about the exams:&lt;br/&gt;&#xA;You are saying that you let students take their exam home and you hope they will nicely make it themselves, without any form of cheating, and you really believe that only one of them is cheating? Did you already consider the fact that some of your students might be gathering together and make the exercises in group? It's not as bad as copy/paste, but it also does not check the problem solving capabilities of the students (which is in essence the reason behind mathematics classes, isn't it?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore I'm advising you not to let students make their exams at home, but don't stop this for the sake of just one cheating student, but for the sake of all possible cheating students you might not have discovered yet.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-08-07T13:33:25.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12703" PostTypeId="1" AcceptedAnswerId="12706" CreationDate="2017-08-08T01:32:51.877" Score="12" ViewCount="1375" Body="&lt;p&gt;I'm trying to teach some secondary school students on how to complete the square.  The goal is to rewrite:&#xA;$$y = ax^2 + bx + c \ \ \Rightarrow \ \ y = a(x-h)^2 + k$$&#xA;The first thing I did was to ask them to confirm, via FOIL:&#xA;$$\left( x + \tfrac{b}{2} \right) \left( x + \tfrac{b}{2} \right) \ \ = \ \ x^2 + bx + \tfrac{b^2}{4}$$&#xA;With this as a guide, they were able to rewrite something like:&#xA;$$y = x^2 + 6x - 10 \ \ \Rightarrow \ \ y = (x + 3)^2 -19$$&#xA;Problems occur when the leading coefficient does not equal one.  So far, I am not able to explain why I need to reduce the leading coefficient to $1$ before completing the square.  (Moreover, some of them are having problems with distributions.)  For example:&#xA;\begin{align}&#xA;y \ \ &amp;amp; = \ \ 2x^2 + 8x - 9 \\&#xA;&amp;amp; = \ \ 2(x^2 + 4x - 4.5) \\&#xA;&amp;amp; = \ \ 2(x^2 + 4x + 4 - 8.5) \\&#xA;&amp;amp; = \ \ 2(x^2 + 4x + 4) - 17 \ \ = \ \ 2(x + 2)^2 - 17&#xA;\end{align}&lt;/p&gt;&#xA;" OwnerUserId="8586" LastEditorUserId="39" LastEditDate="2017-08-12T21:19:55.830" LastActivityDate="2017-08-13T13:48:43.387" Title="Teaching Completing the Square" Tags="|secondary-education|algebra|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12704" PostTypeId="1" AcceptedAnswerId="12705" CreationDate="2017-08-08T12:27:28.697" Score="10" ViewCount="1355" Body="&lt;p&gt;I will be teaching assistant for an analysis course next semester: I'll present the solutions of the exercices to the class. Syllabus is sequences and series of functions, Riemann and Lebesgue integrals, L^2 spaces, etc. Solutions of the harder exercices can get lenghty and technical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to use colors to liven up the board, highlight the structure of the demonstrations and make technical manipulations easier to follow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have examples/tips of what to do and not to do?&lt;/p&gt;&#xA;" OwnerUserId="8330" LastActivityDate="2017-08-10T01:11:57.770" Title="How to effectively use colors on a whiteboard?" Tags="|presentation|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12705" PostTypeId="2" ParentId="12704" CreationDate="2017-08-08T14:03:50.987" Score="5" Body="&lt;p&gt;When teaching complex numbers I used one colour for imaginary parts of the numbers and another for the real parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, whenever you have a clearly distinguishable object, you might to assign it a colour. If a sequence of functions is approaching a limit, you might make the limit blue. The sequence might be red.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few points to keep in mind:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Be absolutely consistent. Come up with a silly justification for the colours, or use some other mnemonic, and refer to previous notes.&lt;/li&gt;&#xA;&lt;li&gt;Green-red colourblindness is not that uncommon, I think. Check this.&lt;/li&gt;&#xA;&lt;li&gt;The colours should be a useful aid, and nothing should rely on them for understanding. This is for the previous two reasons, and also because you might not always have the colours available.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2017-08-08T14:03:50.987" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12706" PostTypeId="2" ParentId="12703" CreationDate="2017-08-08T15:10:20.810" Score="6" Body="&lt;p&gt;This may not really answer your question, and, it may be inappropriate to the level of your class, but, I find the idea of looking at an algebraic manipulation algebraically is at times helpful. The whole point of doing algebra is to engage in the kind of thinking I give below so I think there is something to gain from taking this sort of &quot;abstract&quot; approach. The idea here is to discover how arbitrary $h$ and $k$ and indeed $A$ must relate to $a,b,c$ in order that &#xA;$$ y = ax^2+bx+c = A(x-h)^2+k. $$ &#xA;This is itself and algebra problem. Do algebra to do algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider &#xA;$$ ax^2+bx+c = A(x^2-2xh+h^2)+k $$&#xA;or&#xA;$$ ax^2+bx+c = Ax^2-2hAx+Ah^2+k $$&#xA;requires that these be the same polynomial. Hence, equating coefficients, &#xA;$$ \begin{array}{cc} x^2: &amp;amp; a=A \\ x: &amp;amp; b=-2Ah \\ 1: &amp;amp; c = Ah^2+k \end{array} $$&#xA;So, the $h$ and $k$ we seek are given by:&#xA;$$ A = a, \ \ h = -\frac{b}{2a}, \ \ k = c - \frac{b^2}{4a}. $$&#xA;Therefore,&#xA;$$ ax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a} $$&#xA;Then, give numeric examples galore until it sinks in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's likely as or more important to explain &lt;strong&gt;why&lt;/strong&gt; it is important to complete the square. Of course the danger of my method, and it is a &lt;strong&gt;real danger&lt;/strong&gt; is that students may black-box this solution. Heaven forbid teachers just say well, you set $h = b/2a$ and $k = c-\frac{b^2}{4a}$ and that's it. The universe in which I present this to a class is the same universe in which the students are also held responsible for reproducing the abstract result in addition to the expectation they do numerically specific examples with ease. Ease born of practice.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-08-08T15:10:20.810" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12707" PostTypeId="2" ParentId="12703" CreationDate="2017-08-08T16:19:39.387" Score="4" Body="&lt;p&gt;I don't know if the following will be of much use to your specific case (our audiences are quite different---I teach this to college freshmen), here is my usual approach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, we typically start the quarter by getting a good handle on geometric transformations of graphs of functions.  Given some function $f:\mathbb{R}\to\mathbb{R}$ with a known graph, the graph of the function defined by&#xA;$$ x \mapsto B f\left( \frac{x-h}{A} \right) + k $$&#xA;is the same as the graph of $f$&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;shifted right $h$ units (where $h&amp;lt;0$ is a shift to the negative right, or left),&lt;/li&gt;&#xA;&lt;li&gt;scaled horizontally by a factor of $A$,&lt;/li&gt;&#xA;&lt;li&gt;reflected across the $y$-axis if $A &amp;lt; 0$,&lt;/li&gt;&#xA;&lt;li&gt;scaled vertically by a factor of $B$, &lt;/li&gt;&#xA;&lt;li&gt;reflected across the $x$-axis if $B &amp;lt; 0$, and&lt;/li&gt;&#xA;&lt;li&gt;shifted up $k$ units (again, down is negative up).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;We can then consider the humble parabola defined by $x \mapsto x^2$.  This lovely function has a zero at zero, goes through the points $(1,1)$ and $(-1,1)$, and has inverses on the domains $[0,\infty)$ and $(-\infty,0]$ given by the positive and negative square roots, respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We might first note that if we transform the graph of $x\mapsto x^2$, the two multiplicative constants can be combined:&#xA;\begin{equation}&#xA;  B\left( \frac{x-h}{A} \right)^2 + k&#xA;  = \left( \frac{B}{A^2} \right)(x-h)^2 + k,&#xA;\end{equation}&#xA;so we may fairly conclude that any transformation of the graph will be given by a function of the form&#xA;\begin{equation*}&#xA;  x \mapsto C(x-h)^2 + k,&#xA;\end{equation*}&#xA;where $C$ represents some kind of scaling (we may as well understand it as a vertical scaling, but it combines both the horizontal and vertical scalings into one gooey mess), and $h$ and $k$ are translations, as usual.  In particular, the vertex of this graph is the point $(h,k)$, and the vertex will represent a minimum or maximum for the function, depending on whether $C &amp;gt; 0$ or $C &amp;lt; 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can also expand this mess in order to obtain&#xA;\begin{align}&#xA;C\left( x-h \right)^2 + k&#xA; &amp;amp;= C (x^2 - 2hx + h^2) + k \\&#xA; &amp;amp;= \left(C\right)x^2 + \left( -2hC \right)x + \left( h^2C + k\right) \\&#xA; &amp;amp;= ax^2 + bx + c,&#xA;\end{align}&#xA;which is a more familiar expression to most of my students, as they have mostly seen the material before in high school.  The next &quot;obvious&quot; (for certain values of &quot;obvious&quot;) question is:  can we go the other way?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, if we are told that&#xA;$$ f(x) = ax^2 + bx + c,$$&#xA;can we understand the graph of $f$ as the graph of our basic parabola subject to some elementary transformations?  That is, given arbitrary $a,b,c$ (with $a\ne0$), are there $C,h,k$ such that&#xA;$$ f(x) = ax^2 + bx + c = C(x-h)+k?$$&#xA;The answer is &quot;Yes!&quot;, with the details as provided by James S. Cook's answer to this question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this exposition, the idea is that the &lt;em&gt;geometry&lt;/em&gt; motivates exploration of the problem.  We want to know how to &quot;complete the square&quot; so that we can find the vertex of the transformed parabola, or so that we can determine an inverse on some domain (i.e. solve a quadratic equation).&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2017-08-08T16:19:39.387" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12708" PostTypeId="2" ParentId="12704" CreationDate="2017-08-09T01:07:17.300" Score="4" Body="&lt;p&gt;Last semester I purchased lots of different colors, and used them liberally.&#xA;I could not be as disciplined and as careful as Tommi Brander's impressive control.&#xA;But certainly distinct parts of geometric drawings,&#xA;labels of vertices and labels of edges, any objects of distinct&#xA;categories, were easy choices for different colors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of my students took notes using colored pencils. Some took phone-photos of&#xA;the more complicated constructions.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/WeyHx.png&quot; width=&quot;200&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;Incidentally, 8% of males and fewer than 1% of females are color blind.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-08-10T01:11:57.770" LastActivityDate="2017-08-10T01:11:57.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12710" PostTypeId="2" ParentId="12703" CreationDate="2017-08-09T15:48:42.227" Score="2" Body="&lt;p&gt;My Approach:&#xA;\begin{align}&#xA; y \ \ &amp;amp; = \ \ 2x^2 + 8x - 9 \\ &#xA;&amp;amp; y+9 = \ \ 2(x^2 + 4x + \underline{\quad}\  ) \\&#xA;&amp;amp; y+9+8 = \ \ 2(x^2 + 4x+4) \\&#xA;&amp;amp; y+17 = \ \ 2(x^2 + 4x + 4) \\ &#xA;&amp;amp; y+17 = \ \ 2(x + 2)^2 \\&#xA;\end{align}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notes - Moving the constant to the left in the first step avoids the errors some students might make dividing this number. Now, when you add the '4' to complete the square, you ask &quot;What did we add to the right side?&quot; &quot;4&quot; &quot;don't forget the number to multiply, here, '2', so we added 8 to the right, and we'll add 8 to the left.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last, the format of that last line is the one I prefer. It's the &quot;Vertex Form&quot; and students can quickly see (-2,-17) is the vertex. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-08-13T13:48:43.387" LastActivityDate="2017-08-13T13:48:43.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12711" PostTypeId="2" ParentId="12703" CreationDate="2017-08-09T16:41:25.290" Score="-2" Body="&lt;p&gt;I think it is easier to practice doing this without the y (with the equation set to zero).  Of course it is same thing but for someone just learning, doing it as just a single equation is easier and concentrates the mind on roots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also try to do it very mechanically, writing out each step (like when you are in pre-algebra and first learn how to sort things out with first order equations in X only.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just some thoughts.&lt;/p&gt;&#xA;" OwnerUserId="8591" LastActivityDate="2017-08-09T16:41:25.290" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12712" PostTypeId="2" ParentId="12703" CreationDate="2017-08-09T17:59:18.297" Score="2" Body="&lt;p&gt;I present completing the square as a recipe.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given $x^2 + bx$, add $\big(\frac{b}{2}\big)^2$ to both sides. You obtain a different expression, but your new expression can be written neatly as a perfect square:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;\begin{align*}&#xA;x^2 + bx &amp;amp;\xrightarrow{\text{add }\left(\frac{b}{2}\right)^2} x^2 + bx + \big(\frac{b}{2}\big)^2 = \big(x + \frac{b}{2}\big)^2&#xA;\end{align*}&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;At some point, I emphasize how important it is that we started with $x^2$ without a (non-one) coefficient: the recipe produces a new expression of the form $(x + \text{something})^2$, which, when expanded, will &lt;em&gt;never&lt;/em&gt; produce anything whose $x^2$ term has a non-one coefficient. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, maybe you can really play up that it's a &quot;faithful, but simple&quot; method that &lt;em&gt;only&lt;/em&gt; works for expressions of the form $x^2 + bx$: The recipe deals exclusively with quadratics whose leading term is $1$, end of story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: You face similar problems getting students to factor $2x^2 - 3x + 5$ using the &quot;ac method&quot; or whatever you prefer to call it. Students are used to the process &lt;em&gt;find a pair of numbers that multiply to this, and add to that&lt;/em&gt; from the &quot;easy&quot; quadratic case. But, unlike the &quot;easy&quot; case, here we don't just get to jump right to a factorization. Why? Because $(x + \text{thing1})(x + \text{thing2})$ will never produce quadratics whose leading coefficient isn't $1$, and we finally need that. So, we have to add another step or two to the old method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I will make a bit of a big deal about the jump from expressions like $x^2 + 6x + 1$ to $2x^2 + 6x + 1$: Our version of completing the square &lt;em&gt;requires&lt;/em&gt; us to have just $x^2$. Is all hope lost; do we need a brand new version of completing the square?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is where I take the approach suggested by &lt;a href=&quot;https://matheducators.stackexchange.com/questions/12703/teaching-completing-the-square/12710#comment30927_12703&quot;&gt;DRF's comment&lt;/a&gt;: We reduce a superficially new problem to a known, already solved problem. By adding the minor modification of factoring out the leading coefficient, we get the much vaunted $x^2 + [\text{something}]x$ on which our recipe relies.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I'll add that there is a &lt;em&gt;super&lt;/em&gt; slick, more general version of completing the square that can handle non-one leading coefficients without factoring/division, given &lt;a href=&quot;https://math.stackexchange.com/a/49243/120540&quot;&gt;here&lt;/a&gt; by André Nicolas (and I think the linked answer is the first place I saw it). It's more appropriate for solving equations by completing the square (instead of rewriting quadratic expressions), but it's so nice that I think it deserves to be more widely-known.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main idea is that $(2ax + b)^2 = 4a^2x^2 + 4abx + b^2$, so given an expression like $ax^2 + bx$, you'll multiply by $4a$ then add $b^2$, at which point you can write your new expression as a perfect square. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have yet to mention this for a remedial algebra / precalculus class, but occasionally I'll get a math ed student and show them this method in addition to the usual approach.&lt;/p&gt;&#xA;" OwnerUserId="5153" LastActivityDate="2017-08-09T17:59:18.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12714" PostTypeId="2" ParentId="12703" CreationDate="2017-08-09T21:15:51.873" Score="6" Body="&lt;p&gt;The goals of my approach are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Emphasize that the purpose of completing the square is a process of writing something that is &lt;em&gt;equal to the original expression&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Emphasize that there are &lt;em&gt;choices&lt;/em&gt; to make, differentiating between a choice being &lt;em&gt;useful&lt;/em&gt; versus it being &lt;em&gt;true&lt;/em&gt; (avoiding the words &quot;&lt;em&gt;correct&lt;/em&gt;&quot; or &quot;&lt;em&gt;right&lt;/em&gt;&quot; which conflate useful and true).&lt;/li&gt;&#xA;&lt;li&gt;Avoid unnecessary algebra and quantifiers; my students have a hard enough time with $x$; adding in $b$ and $c$ obscures my previous two goals.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Complete the square: $x^2 + 8x + 19$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;First ask yourself: &quot;is it a perfect square?&quot; If it was, you would be able to find two numbers that add up to 8, multiply to 19, and &lt;em&gt;are the same number&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If not, ask yourself what the two numbers would have to be to add to 8 and be the same number. Then recognize that the constant 19 is the &lt;em&gt;wrong constant&lt;/em&gt;. Ask yourself: What is the constant you want it to be? Then don't let your dreams be dreams, just write it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\begin{align*} &amp;amp; x^2 + 8x + 19 \\ = &amp;amp; x^2 + 8x + 16 \end{align*}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Notice that this is &lt;em&gt;useful&lt;/em&gt; (it factors now) but not &lt;em&gt;true&lt;/em&gt; (they aren't equal). Fix what you've written so that it is also &lt;em&gt;true&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\begin{align*} &amp;amp; x^2 + 8x + 19 \\ = &amp;amp; x^2 + 8x + 16 &amp;amp; + 3\end{align*}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then factor the part of the result that you have engineered to work out.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\begin{align*} &amp;amp; x^2 + 8x + 19 &amp;amp; \\ = &amp;amp; x^2 + 8x + 16 &amp;amp; + 3 \\ = &amp;amp;(x + 4)^2 &amp;amp; + 3\end{align*}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I have tried a variety of approaches, and I like the abstract skills that this approach allows students to practice. For example, when a student does this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\begin{align*} &amp;amp; x^2 + 8x + 19 \\ = &amp;amp; x^2 + 8x + 4 + 15 \end{align*}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I can ask students if that was a &lt;em&gt;good idea&lt;/em&gt; and tease out the distinction that there is a difference between something being &lt;em&gt;true&lt;/em&gt; and &lt;em&gt;useful&lt;/em&gt;. Since one of the primary goals of algebra courses is convincing students that they have the freedom to do whatever they want when it is true (sure, multiply both sides by 10 if you want man; worst case you can undo it later), this synergizes well with my approach to the rest of algebra.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2017-08-09T22:44:46.153" LastActivityDate="2017-08-09T22:44:46.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12715" PostTypeId="1" AcceptedAnswerId="12719" CreationDate="2017-08-09T21:18:04.730" Score="2" ViewCount="89" Body="&lt;p&gt;Do the following sentences express 5-2?&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;a. 5 fewer 2 is 3&#xA;b. 2 fewer than 5 is 3&#xA;c. 5 less 2 is 3&#xA;d. 5 gave 2 is 3&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;I also saw online that &quot;shared&quot; can also mean &quot;minus&quot;, can someone please give me an example?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you! &lt;/p&gt;&#xA;" OwnerUserId="7487" LastEditorUserId="1905" LastEditDate="2017-08-10T00:25:57.753" LastActivityDate="2017-08-10T15:09:49.580" Title="How to use these actions words for subtraction?" Tags="|language-use|arithmetic-operations|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12716" PostTypeId="1" AcceptedAnswerId="12725" CreationDate="2017-08-09T21:29:57.220" Score="-1" ViewCount="242" Body="&lt;p&gt;I saw this phrase/sentence in a worksheet, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;subtract 4 from 3 times X&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The question asks the students to write the statement using numbers and symbols. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the correct answer is 3X - 4, but how can I explain to the students it's not (3 - 4)X? Thank you!&lt;/p&gt;&#xA;" OwnerUserId="7487" LastEditorUserId="7487" LastEditDate="2017-08-15T08:27:50.557" LastActivityDate="2017-08-15T10:01:41.127" Title="What is &quot;subtract 4 from 3 times X&quot; and why?" Tags="|algebra|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12717" PostTypeId="2" ParentId="12716" CreationDate="2017-08-10T00:03:58.360" Score="1" Body="&lt;p&gt;Seems more of a grammar question than a math education question. There is only one way to parse the sentence. &quot;Subtract&quot; is a transitive verb which requires two objects. These are &quot;4&quot; and &quot;3 times X&quot;. Thus it reads as $3*X - 4$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The alternative reading doesn't make sense since &quot;subtract 4 from 3&quot; isn't a noun phrase which can be used as an object for &quot;times&quot;, especially since &quot;times&quot; isn't even a verb (it is a preposition).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I am using parts of speech rather than precedence rules to parse the sentence. Natural languages don't have well-defined rules of precedence. A phrase like &quot;3 times X minus 4&quot; is simply ambiguous and could be interpreted in different ways depending on how the speaker stresses the words and times their pauses. &quot;3 times [pause] x minus 4&quot; would seem to mean $3*(X-4)$ but &quot;3 times X [pause] minus 4&quot; would seem to mean $3*X - 4$. &lt;/p&gt;&#xA;" OwnerUserId="6891" LastEditorUserId="1147" LastEditDate="2017-08-11T04:34:32.177" LastActivityDate="2017-08-11T04:34:32.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12718" PostTypeId="1" CreationDate="2017-08-10T01:46:41.107" Score="13" ViewCount="724" Body="&lt;p&gt;This question concerns only proven statements. I don't know if research papers do, but most math textbooks don't. Counterarguments:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Space?&lt;/strong&gt;&lt;br&gt;&#xA;1.1. The increased length from explaining the discovery is justified; if proofs' discoveries aren't explained and revealed, how can they be learned and discovered, especially to solve &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics&quot; rel=&quot;noreferrer&quot;&gt;unsolved questions&lt;/a&gt;?&lt;br&gt;&#xA;1.2. The discovery may be more vital than the proof: easier is deducing the latter from the former, rather than searching the latter for the former. So if space is a problem, why not focus on the discovery and leave the proof to the reader, instead of vice versa that is the status quo?  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Ignorance of the discovery?&lt;/strong&gt; Maybe these &lt;a href=&quot;https://en.wikipedia.org/wiki/Flash_of_genius&quot; rel=&quot;noreferrer&quot;&gt;flashes of genius&lt;/a&gt; cannot be described? This seems unlikely for statements proven long ago. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Desire for students to toil, and painstakingly discover the discovery themselves?&lt;/strong&gt; But this is inefficient; if certain theories expended centuries for a famous mathematician, how can a math student discover within a day? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Suppression of knowledge?&lt;/strong&gt; This is least likely, and paranoid; maybe authors don't wish students to learn the discoveries easily, because then future students would outperform them?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;E.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality#Second_proof&quot; rel=&quot;noreferrer&quot;&gt;Cauchy-Schwarz Inequality&lt;/a&gt; is often proven &lt;a href=&quot;https://math.stackexchange.com/q/182808/53259&quot;&gt;without its discovery explained&lt;/a&gt;. Textbooks don't explain how a student would divine $ \bf{0} \le \|\bf{u} - \cfrac{\langle u, v \rangle}{\|v\|^2}\bf{v}\|^2$.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2022-11-28T12:31:32.270" Title="Why don't textbooks explain proofs' discovery?" Tags="|proofs|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12719" PostTypeId="2" ParentId="12715" CreationDate="2017-08-10T04:37:24.810" Score="4" Body="&lt;p&gt;&lt;strong&gt;From your list&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;a) 5 fewer 2 is 3          is never used&lt;/p&gt;&#xA;&lt;p&gt;b) 2 fewer than 5 is 3     is used&lt;/p&gt;&#xA;&lt;p&gt;C) 5 less 2 is 3           Can be used but a better usage is 2 less than 5 is 3&lt;/p&gt;&#xA;&lt;p&gt;d) 5 gave 2 is 3           Should be rewritten as 5 gave 2 leaves 3&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Additional items (not all are action words - verbs)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;e) 2 units &lt;strong&gt;(shorter/slower/lighter) than&lt;/strong&gt; 5 units is 3 units&lt;/p&gt;&#xA;&lt;p&gt;f) How many units &lt;strong&gt;(longer/faster/heaver) than&lt;/strong&gt; 2 units is 5 units? It is 3 units longer&lt;/p&gt;&#xA;&lt;p&gt;g) Bob has 5 apples and &lt;strong&gt;shares&lt;/strong&gt; 1 with each of his 2 sisters.  He has 3 apples &lt;strong&gt;left&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;h) Bob has 5 apples. Lisa has 2 apples. Lisa has 3 &lt;strong&gt;fewer&lt;/strong&gt; apples than Bob.&lt;/p&gt;&#xA;&lt;p&gt;i) The &lt;strong&gt;difference of/between&lt;/strong&gt; 5 and 2 is 3&lt;/p&gt;&#xA;&lt;p&gt;j) &lt;strong&gt;Subtract&lt;/strong&gt; 2 from 5 and the answer is 3&lt;/p&gt;&#xA;&lt;p&gt;k) 5 &lt;strong&gt;minus&lt;/strong&gt; 2 is 3&lt;/p&gt;&#xA;&lt;p&gt;l) Bob has 5 apples and eats 2.  There are 3 &lt;strong&gt;remaining&lt;/strong&gt; apples. OR There are 3 &lt;strong&gt;left&lt;/strong&gt;. OR There 3 apples &lt;strong&gt;left over&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;m) 5 d&lt;strong&gt;ecreased by&lt;/strong&gt; 2 is 3.&lt;/p&gt;&#xA;&lt;p&gt;For more words you can search for&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Arithmetic subtraction key words&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-08-10T15:09:49.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12720" PostTypeId="1" AcceptedAnswerId="12724" CreationDate="2017-08-10T07:32:54.983" Score="9" ViewCount="1772" Body="&lt;p&gt;What are the effects of allowing students take one paper, say of A4 size, full of whatever they want, into an exam? It might be called a cheat sheet or something similar. Students might write it by hand, they might copy one from someone else, they might use $LaTeX$, etc. Anything goes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have taken such an exam twice in physics studies, but would be interested in the experiences of people who have assigned such exams, and literature on the subject, if any.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am particularly interested in university-level mathematics courses, which probably include proofs and maybe calculations.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2017-08-11T03:32:12.537" Title="A4 paper of notes in an exam" Tags="|undergraduate-education|exams|graduate-education|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12721" PostTypeId="2" ParentId="12718" CreationDate="2017-08-10T13:05:52.200" Score="11" Body="&lt;p&gt;I'm not exactly sure if I understand what you are asking, but I will try to give at least a partial answer. I think there are a couple reasons why an explanation, of how a proof was first arrived at, is not given:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It isn't known&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For a lot of proofs not even the author can exactly explain how he arrived at the given proof. &quot;Well I thought about how to prove the problem, thought about what my intuition said, tried to make that rigorous, found the places where I couldn't quite get enough rigor, played around with some other ideas, and finally came up with the proof when I realized that I could just reduce this to a coloring problem which was much easier to solve.&quot; And that's one of the more specific examples. Often people spends weeks or months on a problem trying many dead ends before they come up with the correct approach or as happened to me figure out that they are trying to prove the wrong thing and prove the exact opposite.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;The original proof was quite ugly&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This is very often the case for many harder problems. When you first work on a hard problem you might be starting a new direction. The example that comes to my mind is the proof of independence of CH. The original Cohen paper is quite hard to read since when he did it, the technique he invented wasn't formalized and smoothed. He and others went on to refine the technique into something that almost a baby can use (for the easy problems anyway), but this took some time. Explaining how he came up with the original proof would probably be hard and wouldn't really help much.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;It is time consuming&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Explaining how a proof was first arrived at will either be very time consuming (explaining all the dead ends, ideas that didn't work, refinements etc.) or will leave out 3/4 of the important stuff (all the dead ends, ideas that didn't work, refinements etc.). Usually in Mathematics classes (especially at the lower levels) students have more than enough trouble just grasping the refined and cleaned up proofs and ideas, much less being able to follow the complex process that led to their creation.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;We don't know/ lack of rigor of the original proofs&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A lot of especially the original proofs are old. Cauchy lived in the late 18th early 19th century and I doubt he wrote down how he came up with the proof. Further more though quite a bit of older mathematics wasn't proved to current standards. Some of Newton's and Leibniz's results were such and while the results hold in essence the actual founding theory which makes them work nicely wasn't created until much later. &lt;/p&gt;&#xA;" OwnerUserId="5896" LastActivityDate="2017-08-10T13:05:52.200" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12722" PostTypeId="2" ParentId="12718" CreationDate="2017-08-10T13:28:56.577" Score="18" Body="&lt;p&gt;In many cases, standard theorems in a theory were first proved as special cases before the theory itself was even invented. For example, &lt;a href=&quot;https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)&quot; rel=&quot;noreferrer&quot;&gt;Lagrange's theorem&lt;/a&gt; in group theory predated the invention of group theory. He proved it as a result involving certain sets of permutations (of variables in polynomial equations) which were only much later recognized to form a group in the modern sense. A book in abstract algebra will typically &lt;em&gt;begin&lt;/em&gt; with the axioms of group theory -- axioms which were not clearly spelled out in Lagrange's life time. To explain what Lagrange himself actually did you would need a long detour into the theory of polynomial equations, one which would have only an indirect link to the modern result known as Lagrange's theorem (which was only proved decades after Lagrange's original result). This is only a single, somewhat preliminary result in group theory. If you tried to do it for all results in group theory textbooks would blossom into thousand-page tomes that would lose almost all students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are some text books that have a somewhat narrow focus and which do have a more historical approach. For example, Harold Edward's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/038790980X&quot; rel=&quot;noreferrer&quot;&gt;Galois Theory&lt;/a&gt; does get into what Lagrange actually did. While such books are valuable, and all math students could benefit from at least occasionally working through such things, they aren't a very good way to get a good understanding of the modern theory. Mathematics is hard enough without overburdening its exposition with excessive history.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastEditorUserId="5563" LastEditDate="2017-08-10T14:10:27.737" LastActivityDate="2017-08-10T14:10:27.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12723" PostTypeId="2" ParentId="12720" CreationDate="2017-08-10T14:23:22.780" Score="3" Body="&lt;p&gt;I would recommend that the instructor give everyone a premade formula sheet. This keeps things more fair and focused, and is common practice on standardized tests. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Everyone is working from the same resource.&lt;/li&gt;&#xA;&lt;li&gt;Students don't waste time on the drafting/formatting of their sheet.&lt;/li&gt;&#xA;&lt;li&gt;Students who can write smaller or have better vision are not advantaged.&lt;/li&gt;&#xA;&lt;li&gt;Many textbooks come with a formula card like this included.&lt;/li&gt;&#xA;&lt;li&gt;The common sheet can be practiced and referenced in class and homework.&lt;/li&gt;&#xA;&lt;li&gt;If a student fails to bring a sheet, then a standard copy can be provided.&lt;/li&gt;&#xA;&lt;li&gt;If there are special items that the instructor feels must be memorized by heart (e.g., Pythagorean theorem), then this can be clarified and enforced. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-08-10T17:55:22.930" LastActivityDate="2017-08-10T17:55:22.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12724" PostTypeId="2" ParentId="12720" CreationDate="2017-08-10T16:14:07.357" Score="12" Body="&lt;p&gt;I believe allowing students to prepare notes for use on an exam is a valuable way to help them focus their exam studying. I do not see the creation of the sheet as a waste of time. To make the notes, they need to reflect on the course and think about what was important, then summarize the information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I allow a &lt;strong&gt;handwritten notecard&lt;/strong&gt; on exams rather than a possibly-typed full sheet of paper, because I am worried about the two issues below:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If allowed to photocopy or type the note sheet, students will tend to ask their most-competent friend to make the notes, then use them. These students lose the study benefits of creating the notes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If allowed too much space, students will tend to copy down entire solutions to problems from solutions manuals or notes, hoping that they will see a similar problem on the exam which will allow them to copy the outline of the correct solution and change the numbers. This leads to solutions being turned in on exams that look like cheating attempts, because they are very well-written but completely incorrect or irrelevant. Confronting students in these situations is very awkward, so I prefer to prevent them from happening.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-08-10T16:14:07.357" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12725" PostTypeId="2" ParentId="12716" CreationDate="2017-08-10T18:32:41.053" Score="2" Body="&lt;p&gt;I think this is a fine education question, particularly because teachers and students should be communicating clearly and correctly to each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect &quot;subtract 4 from 3 times X&quot; is ambiguous &lt;em&gt;enough&lt;/em&gt; for most readers that we might choose to avoid the operation words (subtract, times, etc.) and use noun phrases. If the arithmetic expression you're after is $3X - 4$, then a clearer option might be &quot;The difference of $3X$ and $4$&quot;. Here, you're more clearly separating the the objects (the terms) being subtracted by using the word &quot;and&quot;. Similarly, if you want to describe $3(X-4)$, then you could say &quot;the product of $3$ and the difference of $X$ and $4$&quot; -- Kind of awful to have to say, but not ambiguous. The factors (parts of the product) are each clear objects themselves: $3$ and $X-4$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answering this from a teacher's perspective, I would avoid saying anything like this out loud (if at all possible) without also showing the relevant mathematical expression, unless it was something extremely simple (such as &quot;5 plus 2&quot;). After teaching a section on this topic (translating Math to-and-from English), my students &lt;em&gt;always&lt;/em&gt; come away thankful that we can actually use symbols instead of having to be so wordy. However, whenever we happen to read a mathematical expression out in words, I am always taking the time to (try to) make the phrase unambiguous, or I'm pointing to the particular symbols.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2017-08-10T18:32:41.053" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12726" PostTypeId="2" ParentId="12718" CreationDate="2017-08-10T23:56:56.333" Score="5" Body="&lt;p&gt;In addition to other good points already made, there is also the mythology that there are &quot;ultimate&quot; proofs, and that the messy human processes that occur along the way to development of &quot;the correct&quot; proof are best ignored... insofar as those human activities are not the &quot;platonic&quot; ideal of mathematics. Timeless, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, many peoples' affection for &quot;formality&quot; of presentation would not allow them to give examples prior to a definition (which was, in reality, only conceived long after the examples were well understood, and motivated the definition). In my opinion, this is a stylistic conceit that does &lt;em&gt;not&lt;/em&gt; help people understand mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Said in another way, there is an extreme &quot;Whiggishness&quot; in accounts of mathematics, as though (cf. &quot;Whig account of history&quot;) people in the past were pathetic and benighted, and everything has been improving toward an ideal, which is represented most closely by the current situation. If we buy that, then there's not much interest in examining the sad, ignorant flailing-about of old-timers like Newton, Leibniz, the Bernoullis, Euler, Lagrange, and so on. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More seriously, the sophistication required to understand the complexity of mathematical developments probably is not available to most 18-year-olds, or 22-year-olds (in grad school). So maybe a bad investment of time and effort?&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-08-10T23:56:56.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12727" PostTypeId="2" ParentId="12720" CreationDate="2017-08-11T03:32:12.537" Score="-1" Body="&lt;p&gt;I think for most courses, it is not a good idea.  Not only should the students be familiar with the material, they should be practiced at using it.  How practiced are you at doing the problems, if you don't remember the formula?&lt;/p&gt;&#xA;" OwnerUserId="8591" LastActivityDate="2017-08-11T03:32:12.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12728" PostTypeId="1" CreationDate="2017-08-11T14:10:48.993" Score="13" ViewCount="1029" Body="&lt;p&gt;That title is a quote from Fred Roberts: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fred Roberts. &quot;Is Calculus Necessary?&quot;&#xA;  &lt;em&gt;Proceedings of the Fourth International Congress on Mathematical Education&lt;/em&gt;.&#xA;  1980. p.52ff.&#xA;  &quot;Calculus is not necessary. That will be my premise in this debate.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There was much discussion in the 1980's around the issue of whether or not&#xA;calculus might be displaced with some form of discrete mathematics, e.g.:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lynn Arthur Steen.&#xA;  &quot;Developing Mathematical Maturity.&quot; In:&#xA;  Ralston, Anthony, and Gail S. Young, eds. &lt;em&gt;The Future of College Mathematics&lt;/em&gt;. New York: Springer-Verlag, 1983. p.104.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Steen says, &quot;The gradient from elementary to advanced mathematics is far easier along the&#xA;road of discrete mathematics than it is along the highway of calculus.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I am not finding a sustained literature on this debate since those early days.&#xA;A more recent advocacy is Art Benjamin's TED talk:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Art Benjamin, &quot;Teach Statistics before Calculus!&quot; &#xA;  TED video. 2009. &lt;a href=&quot;https://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education&quot; rel=&quot;noreferrer&quot;&gt;Video link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He says,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;The world has moved from analog to digital, and it is time for our mathematics&#xA;  curriculum to change from analog to digital&amp;mdash;from the more classical continuous&#xA;  mathematics to the more modern discrete mathematics.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He believes students should still learn calculus, but calculus should not be the&#xA;pinnacle high-school course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. Where does this debate stand today? &#xA;  Have some schools implemented Benjamin's suggestion?&#xA;  Is there current, thoughtful literature on the topic?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-08-16T10:38:43.560" LastActivityDate="2018-08-07T15:52:40.237" Title="Is Calculus Necessary?" Tags="|calculus|reference-request|curriculum|discrete-math|" AnswerCount="4" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="12729" PostTypeId="1" AcceptedAnswerId="12758" CreationDate="2017-08-11T17:22:15.340" Score="4" ViewCount="580" Body="&lt;p&gt;Is it &lt;strong&gt;regrouping&lt;/strong&gt;? Upon googling it seems regrouping is borrowing or carrying over collectively. What's the word for not borrowing and carrying over? It's supposedly to train mental computation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 1:&lt;/strong&gt;&#xA;When evaluating $21-9$, one can perform borrowing or do the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$21-9$$&#xA;$$=10+11-9$$&#xA;$$=10+2$$&#xA;$$=12$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently the students who do this first build up mastery of addition and then subtraction facts like 11-9 so the idea is to teach students that they can  &lt;a href=&quot;https://www.reddit.com/r/Jokes/comments/5u6y8s/an_engineer_physicist_and_mathematician_have_been/dds4s9d/&quot; rel=&quot;nofollow noreferrer&quot;&gt;reduce&lt;/a&gt; 21-9 to 11-9 to train mental computation instead of borrowing 10 from 20.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 2:&lt;/strong&gt;&#xA;For $12+9=21$, one can perform carrying over or do the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$12+9$$&#xA;$$=12+8+1$$&#xA;$$=20+1$$&#xA;$$=21$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea is that students are supposed to have mastered different ways to make 10 (1+9,2+8,3+7, etc) and then to make 20 (11+9,12+8,13+7,etc) so students would want to extract from 9 that which they can use to make 20 with 12, which is 8 to train mental computation instead of carrying over 10 from 2+9=11.&lt;/p&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2017-08-18T05:20:05.110" LastActivityDate="2017-08-18T05:20:05.110" Title="What's the word for addition and subtraction without borrowing or carrying over?" Tags="|primary-education|arithmetic-operations|preschool-education|mental-math|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12730" PostTypeId="2" ParentId="12728" CreationDate="2017-08-11T18:58:43.303" Score="2" Body="&lt;p&gt;I actually have some sympathy for this point of view in that there is an awful lot of useful insight that can be gathered from algebra, exponential curves, trig, stats, investment calculations, nomographs, maneuvering boards, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, anyone heading into engineering, physics, or chemistry still needs standard (Granville) calculus.&lt;/p&gt;&#xA;" OwnerUserId="8591" LastActivityDate="2017-08-11T18:58:43.303" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12731" PostTypeId="1" AcceptedAnswerId="12738" CreationDate="2017-08-13T18:10:16.443" Score="6" ViewCount="1376" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/11182/what-is-the-proper-way-to-ask-a-find-the-domain-question&quot;&gt;Possibly related&lt;/a&gt;, though of a different flavour.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Background&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;In most of the precalculus texts with which I am familiar, readers/students are given a crash course in set theory, handed the definition of a relation, then told that a function is a special kind of relation that associates only one element of the codomain to each element of the domain.  For example, a relevant screenshot from Sullivan's &lt;em&gt;Precalculus&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hMRoO.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hMRoO.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is, of course, entirely correct.  A function is defined by its domain, its codomain, and how it associates elements of the domain with elements of the codomain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem I have with this standard approach is that students are often asked to answer questions of the type&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f(x) := \sqrt{x+3}$.  What are the the domain and range $f$?.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems that students are meant to implicitly understand that $f$ is a real-valued function of a real variable.  However, as neither is actually specified, there are two reasonable answers:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The domain is $\mathbb{R}$.  In this case, the codomain is $\mathbb{C}$, at which point the question of the range becomes quite difficult, as I imagine we don't want to talk about branches of complex functions in a precalculus class.  Unfortunately, most of my college freshman precalculus students have been exposed (if only briefly) to complex numbers in their high school algebra classes, so they often want to argue that $\sqrt{-4} = -2i$, and therefore there is no issue with negative real numbers.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The domain is $[-3,\infty)$, in which case I would argue that the question is ill-posed, and might better be written&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f(x) := \sqrt{x+3}$.  What is the largest set of real numbers on which this formula defines a function with codomain $\mathbb{R}$?  What is the range of this function?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This seems reasonable, but there is something about it that just &quot;tastes&quot; off to me.  I can't really put my finger on the discomfort, but  I feel like this approach causes some confusion when we describe restricting domains in order to define inverse functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am also concerned that this approach elides the importance of specifying the domain before defining a function---a function without a specified domain doesn't even make sense, so what is the notation $f(x)$ meant to represent?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;h3&gt;My Solution&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Instead of taking the traditional route, I am considering the introduction of a slightly modified definition of a function:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A &lt;em&gt;function&lt;/em&gt; $f : X \to Y$ is a relation that associates to each element of $X$ at most one element of $Y$.  The set $X$ is called the &lt;em&gt;natural domain&lt;/em&gt;, and the set $Y$ is called the &lt;em&gt;natural range&lt;/em&gt; (or &lt;em&gt;codomain&lt;/em&gt;).  If $x\in X$ and there is some $y\in Y$ such that $f$ associates $y$ to $x$, then we write $x\mapsto y$ and say that $y$ is the &lt;em&gt;image&lt;/em&gt; of $f$ at $x$.  The set of all $x\in X$ such that $x\mapsto y$ for some $y\in Y$ is called the &lt;em&gt;domain&lt;/em&gt; of $f$, and the set of all $y$ such that $x \mapsto y$ for some $x\in X$ is called the &lt;em&gt;range&lt;/em&gt; of $f$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here the question of the domain and range of $f(x) := \sqrt{x+3}$ becomes straightforward:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Define a function $f : \mathbb{R} \to \mathbb{R}$ by $f(x) := \sqrt{x+3}$.  What are the domain and range of $f$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Additionally, we continue to emphasize the fact that the collection of possible inputs and outputs is specified &lt;em&gt;in advance&lt;/em&gt;, thus dealing with the potential observation that imaginary numbers exist---we've already ruled them out as outputs by specifying the codomain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think that this approach is entirely unreasonable, and it even has some moderate precedent.  For example, when dealing with unbounded operators on Banach spaces, we understand that the operators naturally live on the Banach space, but must be restricted to the domain where they actually make sense.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Questions&lt;/h3&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In writing $f : X \to Y$, we are saying that $X$ is the largest set of values that we want to consider as inputs, rather than the actual set of inputs.  The emphasis is on &lt;em&gt;first&lt;/em&gt; determining the universe of possible inputs, &lt;em&gt;then&lt;/em&gt; deciding on which inputs are actually valid (as opposed to the reverse).  Does this seem like a reasonable approach?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This is nonstandard, and actively conflicts with the usual notation.  However, most of my precalculus students are unlikely to take higher math (indeed, for many of them precalculus is a terminal math class), and I think that those going on to calculus or even proofs-based math classes should be just fine.  As such, am I causing any harm if I introduce the concepts in this way?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Am I missing anything important?  Is there any obvious drawback to this approach?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2018-03-30T16:04:54.810" Title="Functions, Domains, and Ranges in Precalculus" Tags="|undergraduate-education|functions|precalculus|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12732" PostTypeId="2" ParentId="12731" CreationDate="2017-08-14T15:43:29.013" Score="2" Body="&lt;p&gt;I say yes reasonable approach.  Stewart-Redlin-Watson write &quot;We usually consider functions for which the sets &lt;em&gt;A&lt;/em&gt; and &lt;em&gt;B&lt;/em&gt; are sets of real numbers.&quot; in their precalculus text.  In a footnote on page 46 of 10th edition Sullivan writes the same:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The sets &lt;em&gt;X&lt;/em&gt; and &lt;em&gt;Y&lt;/em&gt; will usually be sets of real numbers, in which case a&#xA;  (real) function results. The two sets can also be sets of complex&#xA;  numbers, and then we have defined a complex function. In the broad&#xA;  definition (proposed by Lejeune Dirichlet), &lt;em&gt;X&lt;/em&gt; and &lt;em&gt;Y&lt;/em&gt; can be any two&#xA;  sets.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also, you are reasonable because on page 52 of 10th edition Sullivan writes (as Daniel R. Collins mentions above):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Often the domain of a function &lt;em&gt;f&lt;/em&gt; is not specified; instead, only the&#xA;  equation defining the function is given. In such cases, we agree that&#xA;  the domain of &lt;em&gt;f&lt;/em&gt; is the largest set of real numbers for which the value&#xA;  &lt;em&gt;f(x)&lt;/em&gt; is a real number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In short, your approach is consistent with the approach of your text.  Perhaps you could give a similar speech to the effect of &quot;hey, when we math people ask each other questions, sometimes there's a lot going on behind the scenes, like assumptions that we are all making together, maybe we could talk about that for a few minutes.&quot; I've always lectured &quot;First of all, functions in this class will have real inputs because you are preparing to take calculus of a single real variable.  Calculus with complex variables is a thing, but don't worry about that right now.  That means your response to solving x^2+1=0 should be 'no real solutions' instead of plus or minus &lt;em&gt;i&lt;/em&gt;.  Second, the domain of a function is what I say it is. I get to pick it. If you write a function, you get to pick the domain, every single time.  If I fail to pick a domain or don't say anything, you must assume that it's the natural domain, or the largest possible domain.&quot;&lt;/p&gt;&#xA;" OwnerUserId="4703" LastActivityDate="2017-08-14T15:43:29.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12733" PostTypeId="1" AcceptedAnswerId="12734" CreationDate="2017-08-14T16:32:20.117" Score="3" ViewCount="2799" Body="&lt;p&gt;Related: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/4473/where-does-the-word-roots-come-from-when-talking-about-zeros/4474#4474&quot; title=&quot;Where does the word “roots” come from when talking about zeros&quot;&gt;Where does the word “roots” come from when talking about zeros&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student recently wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The positive root of 3 sin x = x is near 2.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am questioning the student's use of of the word root.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stewart-Redlin-Watson do not define the root of an equation or expression at all.  However, in the section titled &quot;Real Zeros of Polynomials,&quot; after spending most of the section referring to the zeros of polynomials, an exercise appears stating:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Show that the equation (fifth degree polynomial) = 0 has exactly one rational root...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They are careful with the definition of a zero:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If &lt;em&gt;P&lt;/em&gt; is a polynomial function, then &lt;em&gt;c&lt;/em&gt; is called a zero of &lt;em&gt;P&lt;/em&gt; if &lt;em&gt;P(c)=0&lt;/em&gt;. In other words, the zeros of &lt;em&gt;P&lt;/em&gt; are the solutions of the polynomial equation &lt;em&gt;P(c)=0&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Sullivan's Precalculus has a very similar approach, and further the term &lt;em&gt;root&lt;/em&gt; is essentially defined as a synonym to the term &lt;em&gt;zero&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So polynomial functions have zeros, or roots, so we could also say polynomial expressions have zeros or roots if we say &quot;well just regard that polynomial as a function.&quot;  But do equations have zeros or roots? It seems that equations just have solutions, and graphs have &lt;em&gt;x&lt;/em&gt; intercepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Followup question: is it just polynomials or algebraic expressions that have roots?  Certainly, the history of the term comes from the case of looking for &lt;em&gt;n&lt;/em&gt;th roots of numbers.  Do transcendental functions have zeros, roots, both, or neither?&lt;/p&gt;&#xA;" OwnerUserId="4703" LastActivityDate="2019-10-01T21:11:11.637" Title="Definition of root of equation/expression" Tags="|terminology|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12734" PostTypeId="2" ParentId="12733" CreationDate="2017-08-14T17:30:38.570" Score="6" Body="&lt;p&gt;Equations have &lt;em&gt;solutions&lt;/em&gt;, not zeros or roots. A solution to an equation is an assignment of values to each indeterminate in the equation that makes the equation true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Functions have &lt;em&gt;zeros&lt;/em&gt;, also called &lt;em&gt;roots&lt;/em&gt;. A zero of a function is an element of the domain of the function such that the corresponding value of the function is zero. &quot;Roots&quot; is most commonly used for polynomial functions, but I don't see any reason not to talk about roots of functions more generally, as a synonym of &quot;zeros&quot;. (Similarly, polynomial expressions have roots/zeros, by looking at the corresponding polynomial function.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would correct a student on using &quot;zero&quot; or &quot;root&quot; for a solution to an equation. I think it's perfectly fine to use &quot;zero&quot; and &quot;root&quot; interchangeably for functions, whether they're polynomial or not. In fact, I'd explicitly define them as synonyms from the outset; I can't think of a situation where it's useful to distinguish between the two terms.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2017-08-14T17:30:38.570" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12735" PostTypeId="2" ParentId="12733" CreationDate="2017-08-14T18:03:23.003" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Do transcendental functions have zeros, roots, both, or neither?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The expression &quot;roots&quot; is also used for transcendental functions, and means the same thing as &quot;zeros.&quot; Namely, a root of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is an &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; (in an explicitly or implicitly specified domain) such that &lt;span class=&quot;math-container&quot;&gt;$f(x_0)=0$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, like you quoted, one might talk about the roots of an equation &lt;span class=&quot;math-container&quot;&gt;$f(x)= 0$&lt;/span&gt; to refer to the set of its solutions. This is possibly a bit old-fashioned, but still not uncommon. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some authors even extend the usage to expressions of the form &lt;span class=&quot;math-container&quot;&gt;$f(x)=g(x)$&lt;/span&gt;, that is equations. At least in the case where the right-hand side is a constant or the situation is otherwise concrete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus to answer your question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But do equations have zeros or roots?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some authors refer to solutions of an equation as roots of an equation. It is possibly a bit old-fashioned and usually confined to certain contexts, which would however encompass the particular one your mentioned.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, to talk about the zeros of an equation in general would not make sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that historically it is not the case that this usage of &quot;root&quot; was extrapolated from the usage in square-root and alike. It is rather the other way around. It was generically used somewhat like an unknown and one might rather think of &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;-th root of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, as short for a root of the equation &lt;span class=&quot;math-container&quot;&gt;$x^7=2$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arguably, by now this general usage of root is possibly outdated and could cause confusion. I do not advocate for it, but would stop short of calling it wrong.   &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="238" LastEditDate="2019-10-01T21:11:11.637" LastActivityDate="2019-10-01T21:11:11.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12736" PostTypeId="1" CreationDate="2017-08-14T19:57:36.027" Score="11" ViewCount="637" Body="&lt;p&gt;In a course for non-math-majors at a liberal arts college, I would like to give a few lectures and activities about groups and symmetry. I think it's straightforward to explain the group axioms and why they apply to, for instance, the set of symmetries of a regular polygon like a square. However, I would like to motivate these axioms by also presenting examples of algebraic structures that violate one, or some combination, of the standard group axioms: closure, associativity, identity, and invertibility. I also want to explain the importance of commutativity, although simple examples of noncommutative operations abound (e.g. subtraction of integers). However, the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Non-associative_algebra#Examples&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia page for non-associativity has only technical examples&lt;/a&gt; that will be incomprehensible to, for instance, a History major. But that doesn't mean they can't appreciate symmetry, so ... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q: Can you suggest particular examples of &quot;algebraic structures&quot;, with a preference for &quot;real life&quot; examples, that violate some of the standard group axioms and related algebraic properties? It would be even better if you could suggest in-class activities that would make the concepts especially clear to a non-mathematically-trained student.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2017-08-23T13:39:03.133" Title="Simple examples that violate group axioms" Tags="|algebra|examples|abstract-algebra|" AnswerCount="9" CommentCount="7" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12738" PostTypeId="2" ParentId="12731" CreationDate="2017-08-15T01:05:14.830" Score="5" Body="&lt;p&gt;The notion you are considering is called a &lt;em&gt;partial function&lt;/em&gt;. When working with such things, ordinary functions are called &lt;em&gt;total functions&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(aside: the category of sets and partial functions turns out to be equivalent to the category of pointed sets and point-preserving functions)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should probably emphasize that you are not teaching the ordinary notion of function, but instead a &lt;em&gt;generalization&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a mathematician who has no experience teaching this stuff (aside from being on mse and the like) but has spent some amount of time thinking through what doing math with partial functions rather than functions would be like, I think there are two features that probably make this a bad idea for an &lt;em&gt;isolated&lt;/em&gt; introductory class. (which is a shame, because I think many low level classes really want to work with partial functions instead of functions)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The first is that, IMO, it strongly engenders a subtly different notion of what an &quot;object&quot; is. Probably the simplest way to put it is as follows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $I$ be a set with one element. The notion of &quot;an element of the set $X$&quot; is basically the same thing as &quot;a (total) function $I \to X$&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When doing algebra in terms of partial functions, one should be working with what I would call &quot;a partial element of the set $X$&quot;, which is basically the same thing as &quot;a &lt;em&gt;partial&lt;/em&gt; function $I \to X$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While this has some appealing features &amp;mdash; e.g. we could introduce the term &quot;undefined element&quot; to mean the partial element corresponding to the partial function with empty domain &amp;mdash; my impression is that its nuances would be extremely difficult to teach to someone who didn't already have a good grasp of functions and mathematical grammar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question then, I suppose, is how confused a student that doesn't grasp the nuances between the two notions of element will be if they learn one way and everything else they encounter is the other way.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The other is an ambiguity in basic logic. Suppose we define the square root as a partial function from the reals to itself; in particular, negative numbers are not in its domain. Consider the equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \sqrt{-1} = 1 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this &lt;em&gt;false&lt;/em&gt;? Or does it have an &lt;em&gt;undefined&lt;/em&gt; truth value? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And consider&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \sqrt{-1} = \sqrt{-2} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could plausibly argue all three possibilities of true, false, and undefined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is, I think, that there isn't a right answer: there are at least two good answers. And I think you need &lt;em&gt;both&lt;/em&gt; of these subtly different notions of what an equation means in order to reason properly. &lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2018-03-30T16:04:54.810" LastActivityDate="2018-03-30T16:04:54.810" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12739" PostTypeId="2" ParentId="12736" CreationDate="2017-08-15T04:13:33.817" Score="1" Body="&lt;p&gt;Here's a &quot;real life&quot; example of non-commutativity: walking through a grid-like city. Starting at a point in the city, walking a block then turning left then right has you arrive at a different point from walking a block (in the same direction) then turning right then left. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For non-associativity (besides subtraction, mentioned above), division. I'm pretty sure division is useful in &quot;real life&quot;.&lt;/p&gt;&#xA;" OwnerUserId="8559" LastEditorUserId="8559" LastEditDate="2017-08-15T05:00:06.697" LastActivityDate="2017-08-15T05:00:06.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12740" PostTypeId="2" ParentId="12736" CreationDate="2017-08-15T05:03:20.750" Score="4" Body="&lt;p&gt;No identity: mathematics before the discovery of zero. Addition always meant getting something new.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No inverses: burning a stick. There's no going back.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not associative: the spread of an STD amongst three people. It certainly matters in what order you &quot;operate&quot;.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2017-08-15T05:03:20.750" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12742" PostTypeId="2" ParentId="12736" CreationDate="2017-08-15T12:26:21.300" Score="7" Body="&lt;p&gt;For something that's not commutative, try rotations about axes through the origin in 3D. A simple rectangular book can illustrate this very well. Try rotating the book in 90 degrees around vertical axis then 90 degrees around North-South axis, then go back to start and try in the opposite order. (Interestingly, the fact that these aren't commutative is related to the fact that multiplication in the quaternions isn't commutative.)&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2017-08-15T12:26:21.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12743" PostTypeId="2" ParentId="12736" CreationDate="2017-08-15T13:13:24.820" Score="16" Body="&lt;p&gt;Combining colored paint is an interesting example of a non-associative operation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Define $Paint_1 * Paint_2$ to be the paint obtained by mixing the two paints in a $1:1$ ratio. It is easy to see that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(Red * Blue) * White \neq Red * (Blue * White).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They are different shades of purple. I haven't tried it, but it should be easy enough to make a classroom lesson out of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a computer implementation. It is obtained by simply averaging the red, green, and blue values of the colors in 24-bit RGB color space and then using Excel VBA's &lt;code&gt;RGB()&lt;/code&gt; function to color the interiors of various cells. I'm not sure that it completely corresponds to paint mixing, but it adequately illustrates the idea:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mhScX.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mhScX.gif&quot; alt=&quot;color mixes&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6891" LastEditorUserId="6891" LastEditDate="2017-08-15T16:35:57.863" LastActivityDate="2017-08-15T16:35:57.863" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12744" PostTypeId="2" ParentId="12736" CreationDate="2017-08-15T16:52:29.597" Score="3" Body="&lt;p&gt;Some simple examples of noninvertibility:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Projection. Given the object, you can determine what its shadow would look like, but the opposite is not true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Another visual example would be pixelating someone's face in an image to protect someone's identity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) Dividing a natural number by 2 and throwing away the remainder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) Playing musical chairs. Each round is a cyclic permutation, which is invertible, composed with eliminating one player, which is not (if we assume that the description of the state gives only a picture of the people who are still in play).&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2017-08-15T16:58:33.400" LastActivityDate="2017-08-15T16:58:33.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12745" PostTypeId="2" ParentId="12736" CreationDate="2017-08-16T01:57:24.350" Score="11" Body="&lt;p&gt;I really like the example of the game &lt;a href=&quot;https://en.wikipedia.org/wiki/Rock%E2%80%93paper%E2%80%93scissors&quot; rel=&quot;noreferrer&quot;&gt;Rock-Paper-Scissors&lt;/a&gt;, thinking of it as a binary operation $\star$ on the set {Rock, Paper, Scissors}.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rules are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;  &lt;li&gt;Rock$\star$Paper $\mapsto$ Paper&lt;/li&gt;&#xA;  &lt;li&gt;Rock$\star$Scissors $\mapsto$ Rock&lt;/li&gt;&#xA;  &lt;li&gt;Paper$\star$Scissors $\mapsto$ Scissors&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The operation is certainly commutative, but it is &lt;em&gt;not&lt;/em&gt; associative. For example, $$(Rock \star Paper)\star Scissors = (Paper)\star Scissors = Scissors$$&#xA;but $$Rock \star (Paper \star Scissors) = Rock \star (Scissors) = Rock$$&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2017-08-16T01:57:24.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12746" PostTypeId="1" AcceptedAnswerId="12767" CreationDate="2017-08-16T04:01:02.173" Score="7" ViewCount="1070" Body="&lt;p&gt;This summer is the first time I am tutoring a class of middle-school students (about 20).  I have done individual tutorings before, but was not formally trained as a teacher.  So, for the first time, I am learning how to manage a class, and more importantly, where my own boundaries are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What grates on my nerve most is when a student screams out loud in the middle of the class that he or she does not understand something.  The material is not too difficult.  Rather, it is brand new and the students are not familiar with it.  I explain that it is okay if they don't get it; just watch and follow what I do.  But two seconds later, some one is going to scream out load again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Correct me if I am wrong.  Such student expects that he or she going to get it right away, and refuses to do the hard work.  Or, the student is just looking for an excuse to be disruptive.  Now, this tutoring center allows me to send out students for discipline issues.  My question is how to kick out someone and still protect myself from unfair accusations.  You know what the student is going to say next, &quot;I don't understand something and you just send me out ....&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One idea is to tell the students to be specific about what they don't understand.  If they cannot give me a coherent question, then they have to hold their judgments and follow what I do.  For example, 1 and 2 are coherent questions, but 3 is not.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How do you go from line 5 to line 6?&lt;/li&gt;&#xA;&lt;li&gt;Why are you interested in this value?&lt;/li&gt;&#xA;&lt;li&gt;What the hell is this?  (Yes, I got this before.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Now, what should I do if someone is trying to argue with me about what is a coherent question?  Clearly, a student like that has to go, but I'd like to know how to protect myself also.&lt;/p&gt;&#xA;" OwnerUserId="8586" LastActivityDate="2017-08-22T02:58:46.420" Title="Dealing with Resisting Students" Tags="|tutoring|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12747" PostTypeId="2" ParentId="5644" CreationDate="2017-08-16T04:12:28.003" Score="5" Body="&lt;p&gt;Dieudonné's text is very good,but it's definitely &lt;strong&gt;not&lt;/strong&gt; for beginners. In fact, when he wrote the book's original lectures in the late 1950's, the book was intended &lt;strong&gt;as a graduate textbook.&lt;/strong&gt; I doubt it could be used that way today without extensive supplementing-too much progress has been made in analysis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;During the Bourbaki era, there was a lot of chatter among the acolytes of that movement that &lt;em&gt;Foundations&lt;/em&gt; would quickly replace Rudin as the standard undergraduate analysis text for serious undergraduates. Despite the golden age of mathematics education and research that was occurring then,I doubt any but the most gifted undergraduates could seriously learn analysis from it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if you like Dieudonné's approach to analysis on normed-particularly Banach-spaces, then I have several excellent recommendations for you. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The definitive treatment of calculus on Banach spaces was authored by Henri Cartan in the mid-1960's in his famous &lt;em&gt;Cours de calcul différentiel&lt;/em&gt; at the University of Paris at Sorbonne. Cartan later wrote up the course as 2 volumes that were later translated into English: &lt;em&gt;Differential Calculus&lt;/em&gt; and &lt;em&gt;Differential Forms&lt;/em&gt;.  Together, the 2 books present a concise,rigorous course on differential and integral calculus on Banach spaces. Cartan writes beautifully and it's a terrific,deep and informative course of study. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a long time, both books were out of print and very difficult to obtain without spending a king's ransom. &lt;em&gt;Forms&lt;/em&gt;, the second half on integration on Banach spaces, was republished by Dover in 2006 and is readily and cheaply available-but it's very difficult to read and use without the first half since there aren't a lot of pre-graduate level books on the subject. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm happy to report that the first volume has just been republished via Createspace in both an inexpensive paperback and Kindle e-book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do I know this? Well, because I'm the publisher. Go,me.........lol &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ok, all kidding aside,I'm serious that my book and its sequel could be just what you're looking for. I'm not only really proud to make this book available again cheaply and readily, I've also included a new detailed preface that describes the historical background of the book in depth and-more importantly for you,I think-a detailed bibliography that suggests not only how to use the books in a course or self-study, but how to use the books as supplements to standard analysis textbooks like Rudin. It also suggests several other books to use in concert with Cartan's books for a complete analysis course on normed spaces. I think you may find several excellent alternatives to Dieudonné's text suggested therein-many of them very inexpensive! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The website of the book can be found with a lot more information &lt;a href=&quot;http://www.tuloomath.com/my-account/text-cartan-dcalculusonnormspaces-2nd-ed/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. I hope you'll find it very helpful.These books are classics and should be available to everyone cheaply.      &lt;/p&gt;&#xA;" OwnerUserId="7033" LastActivityDate="2017-08-16T04:12:28.003" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12748" PostTypeId="2" ParentId="10620" CreationDate="2017-08-16T04:53:30.630" Score="1" Body="&lt;p&gt;This whole conversation is fascinating to me because as those who are old enough or are fluent in the history of mathematics education know, there actually WAS an attempt to structure elementary mathematics education around the abstract theory-the so-called &quot;new math&quot; of the 1960's. Set theory was taught to grade school students,group theory and linear algebra were taught in high school geometry courses and $\epsilon$-$\delta$ formulations of limits alongside Dedekind cuts were regularly taught in high school and university calculus courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The results were predictably disastrous, for many of the reasons mweiss gave above (below?). The human mind just isn't wired for abstraction and there's a long conditioning process from concrete examples to generality for most students before they can begin thinking like mathematicians. This has been proven out also in cognitive science experiments-you can search online for many of those specific studies. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So part of the answer to your question is we don't teach calculus as real analysis-except to the most gifted students, of course-because it doesn't work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there's another reason-and that's because the formal foundations of calculus only tell half the story of calculus.The applications of calculus are as integral part of calculus as the structure of the real number system is. Determining the work done over a mechanical system via an integration is just as important as understanding the Riemann sums that go into constructing that calculation. Being able to calculate the acceleration of the one dimensional motion of an object is just as important as understanding the 2 successive limit operations that define the second derivative. So we really aren't wasting students' time by concentrating on the applications of calculus; this is really just as important as the underlying theory and it's a lot easier and more accessible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said-I am quite sympathetic to your question and viewpoint as someone who cares greatly about mathematical rigor. I believe both aspects of calculus are equally important and both should be discussed in a serious university calculus course. I &lt;strong&gt;do&lt;/strong&gt; think colleges today have gone too far in the other direction and thrown the baby out with the bathwater.I think it's important to try and put the guts back into calculus, as John Stillwell says. But we have to do it in a manner that integrates careful theory with the more intuitive aspects of the subject. I hope to discuss this in depth in a future book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In closing, I do want to inform you of something you may find quite exciting and right up your alley.There actually was a very famous attempt to create a calculus course for freshman that was entirely rigorous and abstract-it was created by Henri Cartan at the University of Paris at the Sorbonne. According to Cartan himself, this was an attempt to build a calculus course from the Bourbakian perspective. The course discussed differential and integral calculus &lt;strong&gt;on Banach spaces&lt;/strong&gt;!  Yes,it had all the standard topics of a university calculus course done at this incredible level of abstraction. In essence, it was a calculus course of precisely the type your question suggested. While the course is beautiful, it's far too difficult for any but the very best students and I think when you see it, you'll see why. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The course is currently available as 2 inexpensive paperbacks and Kindle E-books. The first half was out of print for many years and difficult to obtain-it has recently been reprinted by Createspace in an inexpensive paperback. The book's website-with much background on the book-can be found &lt;a href=&quot;http://www.tuloomath.com/my-account/text-cartan-dcalculusonnormspaces-2nd-ed/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Both books can be bought very cheaply. I think you'll find them both quite interesting and they're by a master in Cartan. &lt;/p&gt;&#xA;" OwnerUserId="7033" LastActivityDate="2017-08-16T04:53:30.630" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12749" PostTypeId="2" ParentId="12746" CreationDate="2017-08-16T06:45:33.457" Score="3" Body="&lt;p&gt;Well, first of all I hope you exaggerate and you don't really have people screaming at you all the time. If they really scream, that is reason enough to make them leave the class. If they ask in a civilized manner, you might have a chance to deal with it. For that, I have two suggestions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Plan specific times for questions. Communicate to you students that you will first present a topic and after 5-10min you will make a break for questions and discussions. You might even use these breaks to ask questions of your own, to see if they understood what you did so far. It might be hard to tell students &quot;no&quot; at first, but if you stay true to your word and really make the promised breaks at appropriate points, they will most likely learn to wait with questions. Hopefully they will see that doing it this way is more efficient.&lt;/li&gt;&#xA;&lt;li&gt;Talk to colleagues and superiors about the problem. It might be that some of your students just love to disrupt and you are not the first one to encounter this problem. Especially if you are planning to exclude students from the class, you should have some sort of consent from above first, exactly to avoid a situation like the one you described (the student complaining about you, that you don't want him to learn).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2017-08-16T06:45:33.457" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12750" PostTypeId="2" ParentId="10620" CreationDate="2017-08-16T06:54:09.530" Score="-3" Body="&lt;p&gt;&quot;Let's assume our students are actual aspiring mathematicians.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is an illogical statement for any realistic setting.  The total number of students who need real analysis is far lower than that who need calculus.  And by and large students have not selected themselves out at the age of taking first calculus class (often in high school or first year uni).  It gives way more option value to take calculus first and analysis later than the converse (if someone switches majors).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Plus calculus is easier so you get a learning progression.  (And it's more useful.  HA!)&lt;/p&gt;&#xA;" OwnerUserId="8591" LastActivityDate="2017-08-16T06:54:09.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12751" PostTypeId="2" ParentId="12746" CreationDate="2017-08-16T06:56:12.317" Score="4" Body="&lt;p&gt;If you have the ability to kick them out of class, I would do that.  The behavior you are talking about is disrespectful.  And disruptive.&lt;/p&gt;&#xA;" OwnerUserId="8591" LastActivityDate="2017-08-16T06:56:12.317" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12752" PostTypeId="2" ParentId="12746" CreationDate="2017-08-16T14:33:21.120" Score="2" Body="&lt;p&gt;Sorry to tell you this, but at this particular moment you seem not to be fit to deal with such a situation just by yourself. This is OK, lots of people have authority issues, especially when starting teaching. Also, the fact that you don't have authority now, does not mean that you can't learn how to get it, but you need to work on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I propose you to get help, either from other teachers (from the same school, very important), either from the principal(s).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case you get the support you ask for, don't be afraid to follow their advise, even if it means that you need to punish a student harder than you would do yourself.&lt;br/&gt;&#xA;In case you don't get supported, either by fellow teachers, either from the principal(s), get out of there and find another job. It seems hard, but it's the &#xA;best way to deal with such a situation.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-08-16T14:33:21.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12753" PostTypeId="1" AcceptedAnswerId="12760" CreationDate="2017-08-16T17:16:59.337" Score="9" ViewCount="1382" Body="&lt;p&gt;My question is similar to this one: &lt;a href=&quot;https://math.stackexchange.com/questions/1833459/math-without-pencil-and-paper&quot;&gt;https://math.stackexchange.com/questions/1833459/math-without-pencil-and-paper&lt;/a&gt; Except I'm not physically disabled.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pencil and paper have been used for centuries to calculate things. It works, but it is not environmentally friendly and storing paper is a hassle. Doing it using desktop computers isn't efficient because the keyboard and mouse limits you. But nowadays, we have tablets, styluses, kindles and all sorts of small high dpi devices to play with. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore I wonder if any mathematician or student have found something  that beats pencil and paper? I suspect the answer is (still)  &lt;strong&gt;no&lt;/strong&gt;, but I wonder if someone has tried? How far off are we until we can go fully digital? With far off I mean what were the limiting factors? Low dpi, stylus sensitivity, low display contrast or something else?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: Even though this question is opinion-based I like @JoeTaxpayer's answer. The writing in his screenshot looks a little smushy... Not as &quot;detailed&quot; as you can write on paper. :) So if the iPad is the best for handwriting,  I will have to wait many more years before I can replace my pen &amp;amp; paper with an electronic device.&lt;/p&gt;&#xA;" OwnerUserId="7837" LastEditorUserId="7837" LastEditDate="2017-08-19T19:01:48.787" LastActivityDate="2020-07-17T09:00:50.410" Title="Electronic devices to replace pencil and paper" Tags="|self-learning|writing|mathematics-in-daily-life|" AnswerCount="5" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12754" PostTypeId="2" ParentId="12746" CreationDate="2017-08-16T17:41:22.537" Score="6" Body="&lt;p&gt;A student behaves disruptively for the first time:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jane, in my classroom, if you have a question I expect you to raise your hand. [Smile.] Now if there's something you want to ask about, could you try to explain what it is that I could try to clarify?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The same student repeats the behavior on another day:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jane, we talked about this on Tuesday, and I made my expectations clear. You need to raise your hand rather than shouting or screaming. I'm going to ask you to leave class and go to the principal's office and talk to her about what just happened. If she's not busy, I think you should be able to get that done in about 10 minutes, so I'm hoping to see you back here by 11:20. I want to help you learn, but if we're going to make that happen you're going to have to stop repeating this behavior.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2017-08-16T17:41:22.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12755" PostTypeId="2" ParentId="12728" CreationDate="2017-08-16T19:42:23.350" Score="10" Body="&lt;p&gt;Society always benefits when new mathematical tools are developed to get insights into the world around us. There is little doubt that Calculus has  helped change the quality of the lives we can lead over the hundreds of years since it was born and Calculus has evolved in many unexpected ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real question should be to what extent it is worth distorting the mathematical education of all students for the benefit of those who eventually need and use a particular a mathematical tool whether it be Calculus, queuing theory, sorting algorithms, or what have you? Engineering and many other professions need Calculus but many careers don't &quot;need&quot; Calculus. However, knowing what Calculus can do is also valuable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own position is that K-12 mathematics education should be much broader than the curiously narrow CCSS-M (now to some extent collapsed and replaced by each of 50 states' curriculum often using the CCSS-M in modified form) or even, from my vantage point, the somewhat better NCTM Standards. Why emphasize depth over breadth for mathematics in K-12 as many in the mathematics community did when they cut back on breadth to put together the CCSS?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Currently there are technical topics involving algebra and trigonometry that students are taught, which prepare them for Calculus but have relatively little value for a majority of students. Current curriculum emphasizes techniques (roots of quadratic equations, trig functions etc.) over more thematic topics such as optimization, growth and change, codes, information, risk, etc. This emphasis on mastering highly technical technique skills is part of what turns so many otherwise intellectually curious and accomplished students into mathophobes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mathematics that leads to Calculus is much more hierarchical than what constitutes many of the nifty topics one can study that belong to discrete mathematics (graph theory, recursion and difference equations, social choice theory, elections, fairness models, etc.). And, one could treat Calculus in K-12 as a nifty topic (not covering what you need to know to compute the integral of sec x) but where students could get the idea of the difference between instantaneous speed and average speed, as well as understand what marginal revenue  and costs mean, etc. and how sometimes one needs to go beyond the Bolyai-Gerwien-Wallace Theorem. The fact that discrete mathematics builds on important ideas that are not as hierarchical as topics that lead up to Calculus is another big advantage to doing more with discrete topics in K-12. We certainly need more discrete mathematics in K-12.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2017-08-16T19:42:23.350" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12756" PostTypeId="2" ParentId="12753" CreationDate="2017-08-17T16:47:41.970" Score="10" Body="&lt;p&gt;&lt;em&gt;Taking a break from a convoluted computation...&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me the key limiting factor is space. I can spread out several sheets over the table and have various bits and pieces directly visible. I cannot do this to the same extent on my tablet or my computer. Therefore I also use loose sheets not something that is bound together.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One might argue I could set up a system with one or even several larger screens and that would solve this problem, but than this is still  costly and maybe a hassle to get right (though I think much less so than it used to be until somewhat recently). And then, I would be bound to the place where I have this set up. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not know how it actually would compare, but the production of all the screens and the energy they consume should be non-negligible, too. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You asked for alternatives. It may not be feasible for you, but consider a blackboard. It's rather the opposite to being an innovation, but in some mathematicians' opinion still the best medium. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A drawback is that it is hard to save one's work. But there modern innovations can enter the workflow. Take a photo before erasing. &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2017-08-17T16:47:41.970" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12757" PostTypeId="2" ParentId="12729" CreationDate="2017-08-17T22:06:18.910" Score="1" Body="&lt;p&gt;When I studied those things, in Spanish it's was called &quot;suma algebraica&quot;, i. e., algebraic sum. Numbers in a term could be positive or negative.&lt;/p&gt;&#xA;" OwnerUserId="6338" LastActivityDate="2017-08-17T22:06:18.910" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12758" PostTypeId="2" ParentId="12729" CreationDate="2017-08-18T00:12:34.483" Score="4" Body="&lt;p&gt;This topic seems to be called &quot;Bridging 10s&quot; or &quot;Making 10s&quot;: &lt;a href=&quot;http://www.helpingwithmath.com/by_subject/addition/making-10-1oa6.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.helpingwithmath.com/by_subject/addition/making-10-1oa6.htm&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure that you will find many discussions about the utility of this in particular, but many of the debates around &quot;Common Core mathematics&quot; include disagreements on whether it is beneficial for students to learn alternate methods of computing the same sum. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found a description of a number of different addition strategies here: &lt;a href=&quot;https://www.whatihavelearnedteaching.com/models-strategies-for-two-digit-addition-subtraction/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.whatihavelearnedteaching.com/models-strategies-for-two-digit-addition-subtraction/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2017-08-18T00:12:34.483" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12760" PostTypeId="2" ParentId="12753" CreationDate="2017-08-18T03:51:51.380" Score="6" Body="&lt;p&gt;Here's a sample of the iPad's resolution. The original is a page wide (8in) but clipped a bit short. The writing tends to be a bit bigger than I'd have with pencil and paper. The graph was just a snapshot, using Desmos. This ability to mix an image onto a page has value depending on the use case. There are times that when drawing graphs by hand, the graph paper image comes in handy. Last, editing after the fact, to correct a mistake or just to add lines to better show some process, is very handy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Jfiqv.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Jfiqv.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-08-18T11:11:07.123" LastActivityDate="2017-08-18T11:11:07.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12761" PostTypeId="2" ParentId="12753" CreationDate="2017-08-18T08:35:23.687" Score="3" Body="&lt;p&gt;Your question is just an opinion call, so I need not cite any authoritative sources. ☺&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding desktops, writing LaTeX is sometimes more convenient than writing by hand with a stylus or pen. A limiting factor is UI. LaTeX is just a language, and text is typed into a text editor which is good for natural languages, but not for formal languages. IMHO, a visual modal editor for abstract syntax tree with ergonomic keyboard would be ideal. Emacs is close to this, but it is still not visual, and it is not modal. The prices of real ergonomic keyboards are about 500 USD. Just for comparison, I rarely write text on paper because keyboard is more convenient. Keyboard has its advantages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding styluses, limiting factors are screen cost, UI, and physical differences. I am a poor man living in a poor country, so I bought a device with a screen of size 133 mm by 75 mm. I desperately need a way to split writing into sheets and arrange them on a higher level. Actually, this need will show up independently of screen size. No matter how big a screen is, it is not able to hold all writing. Stylus feels different from pen: there is no friction between stylus and screen. Maybe it is not a problem, but it is kind of unusual.&lt;/p&gt;&#xA;" OwnerUserId="7992" LastActivityDate="2017-08-18T08:35:23.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12762" PostTypeId="2" ParentId="5644" CreationDate="2017-08-18T14:29:07.173" Score="0" Body="&lt;p&gt;According to the author &amp;quot;This volume is an outgrowth of a course intended for first year graduate students or exceptionally advanced undergraduate students ...&amp;quot;.&#xA;The book cannot be recommended for students who don't know the stuff and want to learn it. And it cannot be recommended from general principles: In modern education we endavour to include the student's whole person, i.e., all is senses and capabilities by applying a diversity of media. Dieudonné however does refrain &amp;quot;deliberately from introducing any diagram in the book&amp;quot; to expell geometric intuition.&#xA;This shows that the author is mentally very unbalanced and one-sided and obviously unable to understand that diagrams as well as movies or spoken text are as well a means of communication and therefore of teaching and thinking as written language. Now, a printed book is excluding spoken text and movies but not diagrams! It is possible to understand him, but it is stupid to rely solely on written text. To think that only the latter is suitable for sharp and axiomatic thinking is somehow as if you willfully blinded yourself in order to adhere solely to some artificial language which is believed to be particularly precise (but is not).&lt;/p&gt;&#xA;&lt;p&gt;In order to judge the praised Bourbaki approach we should not trust in voices of mathematicians who are as one-sided and restricted in their means of communication as Dieudonné and his collaborators. Here are some voices concerning &amp;quot;Bourbaki&amp;quot; (what a childish idea and name already):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... the Bourbakian abstract nonsense leaves you with such a bitter taste that it feels more like Hell. (Doron Zeilberger)&lt;/p&gt;&#xA;&lt;p&gt;...  mercifully, the Bourbaki plague is dying out. (Murray Gell-Mann)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Only if you are a very one-sided student with no visual abilities this book might be of some help for you.&lt;/p&gt;&#xA;" OwnerUserId="8557" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-08-18T14:35:24.663" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12763" PostTypeId="2" ParentId="12736" CreationDate="2017-08-18T17:45:33.927" Score="7" Body="&lt;p&gt;I can't believe no one has mentioned (positive, say) integers under exponentiation as a non-associative operation. It's by far my favorite, though of course this example is also not commutative and has no inverses ... well, at any rate it shows why associativity isn't &quot;obvious&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$2^{(2^3)}\neq (2^2)^3$$&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-08-18T17:45:33.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12764" PostTypeId="2" ParentId="12736" CreationDate="2017-08-21T14:35:03.037" Score="2" Body="&lt;p&gt;In the game of Set, &lt;a href=&quot;https://www.setgame.com/set&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.setgame.com/set&lt;/a&gt;, you can define a binary operation on the cards by Card1 * Card2 gives the card that completes the 'set'.  (In the case that the two cards are equal, define Card1*Card1 = Card1, as this would give a 'set' according to the rules; although, in the commercially-available version of Set, there are no duplicated cards.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This operation is commutative; is nonassociative; has no identity element, and hence inverses can't be defined.  However, cancellation holds: if Card1*Card2 = Card1*Card3, then Card2 = Card3.  Also if Card1*Card2 = Card3, then Card1*Card3 = Card2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Activities for this game might include practicing with the operation; convincing oneself that it gives a well-defined binary operation on the cards; and verifying (some of) the above-mentioned properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a fair amount of mathematical literature on this game. (Try googling Mathematics of the game of Set).&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2017-08-21T14:35:03.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12765" PostTypeId="2" ParentId="12447" CreationDate="2017-08-21T14:47:52.970" Score="3" Body="&lt;p&gt;I am someone (in a country), and I teach it in certain (college) classes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a discrete math class I teach it and then use the fact that if you multiply $n\times 1$ with this method (halving $n$ and doubling $1$), you get $n$ as a sum of powers of $2$.  This then justifies a common algorithm for converting a base-10 representation of an integer to binary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a number theory class, I take these ideas further and use them to develop the the successive squaring and reducing algorithm for computing $a^b \pmod{m}$ efficiently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also teach it in a math for elementary teachers class as an alternative multiplication algorithm.  (Some of the students in this class seem amazed and delighted, and are willing to work through why it works.)&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2017-08-21T14:47:52.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12766" PostTypeId="1" CreationDate="2017-08-21T18:45:20.387" Score="3" ViewCount="128" Body="&lt;p&gt;In Australian math curriculum, four strands of mathematical proficiency are defined: : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Understanding&lt;/li&gt;&#xA;&lt;li&gt;Fluency&lt;/li&gt;&#xA;&lt;li&gt;Problem Solving&lt;/li&gt;&#xA;&lt;li&gt;Reasoning&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Are these items hierarchical - I mean, for example, is reasoning a &quot;higher&quot; proficiency than understanding?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt; &#xA;In Australian curriculum,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Understanding refers to a deep understanding of the mathematical principles and patterns that underpin classroom learning as well as the connections between concepts.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Fluency refers to building students’ content, basic skills, speed and accuracy in routine questions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Problem-Solving refers to having students attempt never-before tried problems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Reasoning is the proficiency strand that requires students to prove that their thinking is mathematically valid or that someone else’s thinking is not mathematically valid.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Reference: &lt;a href=&quot;https://www.backtofrontmaths.com.au/uncategorized/what-do-the-proficiency-strands-mean&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.backtofrontmaths.com.au/uncategorized/what-do-the-proficiency-strands-mean&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1229" LastEditorUserId="1229" LastEditDate="2017-08-22T15:13:32.477" LastActivityDate="2017-08-22T17:46:40.287" Title="Are Proficiency Strands Hierarchical?" Tags="|curriculum|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12767" PostTypeId="2" ParentId="12746" CreationDate="2017-08-22T02:58:46.420" Score="3" Body="&lt;p&gt;When I started teaching I had trouble with classroom management.  It is natural, and you will get better as you get more experience.  The number one thing you have to do is set your expectations.  Even if you're only tutoring one or two students at a time, you have to let them know what your expectations are.  After detailing the expectations with your students, let them know the consequences for not meeting those expectations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I have a few rules in my classroom.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Arrive to class on time.&lt;/li&gt;&#xA;&lt;li&gt;Be prepared.&lt;/li&gt;&#xA;&lt;li&gt;Respect your classmates and teacher.&lt;/li&gt;&#xA;&lt;li&gt;Be responsible for your actions.&lt;/li&gt;&#xA;&lt;li&gt;No eating, drinking, or phones (out of pocket) in class.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I explain these expectations thoroughly.  I would explain that blurting out an answer without raising their hand is violating rule 3 because they are not being respectful to their teacher (blurting out is a pet peeve of mine).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After explaining these expectations, let them know the consequences for not meeting these expectations.  My consequences are as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student violates one of the rules (such as blurting out) I give them a warning.  On their second offense in a week they have to stay after school for detention.  At the end of the week we start over and every student gets one warning before having detention for breaking a rule.  If you set these expectations and are consistent (this is KEY!), then you will have very few discipline problems.  I know that detention may not be at your disposal, but you can alter it to fit your needs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many teachers are starting to use positive reinforcement, but this simply does not work for middle and high school students.  It also does not model the real world.  Students need to learn that their are consequences for their actions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, before teaching anything in your next meeting I would go over my expectations (whatever you outline them to be) with your students.  You will be glad you did because you will find that once they are being respectful, you will have a lot more time to teach them.&lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2017-08-22T02:58:46.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12768" PostTypeId="2" ParentId="12766" CreationDate="2017-08-22T17:35:53.180" Score="3" Body="&lt;p&gt;Based on the fuller descriptions in the links, I think the first 3 are hierarchical and the last one is separate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS1. fluency-basic test problems &lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS2. understanding-word problems (same math concept as before, but able to decode and model)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS3. problem solving-able to extend previous concepts to next step &lt;/p&gt;&#xA;&#xA;&lt;p&gt;W1. reasoning-able to describe the concepts &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue the first three, while having some differences, are all about problem solving and it is unlikely to show skill in the higher strand and not in the lower.  The last strand is really separate from problem solving.  You probably need to train on it specifically.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, you do need at least some modicum of ability to explain something, but probably not even very good &quot;fluency&quot;.  There is a big difference in description versus execution.  (But on the other hand, in some cases you need to comprehend the concept much more particularly to describe it, than what is needed to execute any problems.)  IOW, to a certain extent it is possible to be a good problem solver (by which I mean the general meaning, not the strand name) and a poor explainer and visa versa.  Thus the need for separate practice in both problems and description.  [If you buy that description is important...many people would think it is more minor than execution ability.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;FWIW1, I do think &quot;problem solving&quot; (the strand specifically) and to an extent &quot;understanding&quot; (the strand) is more about student intelligence than training.  IOW, yes, you need to have the basic concepts down to extend them.  Or to weed though the chaff of a word problem.  But it's kind of hard to practice the ability to take the next step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;FWIW2:  I think the strands are poorly named.  Even writing here it is hard as problem solving and understanding are really more general concepts in the English language than what the Aussie edubabble labels describe.  Something more like the following would be more helpful:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Basic problem solving fluency&lt;/li&gt;&#xA;&lt;li&gt;Complicated problem solving &lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ability to extend concepts&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Descriptive ability&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8618" LastEditorUserId="8618" LastEditDate="2017-08-22T17:46:40.287" LastActivityDate="2017-08-22T17:46:40.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12769" PostTypeId="1" CreationDate="2017-08-23T01:59:33.387" Score="2" ViewCount="190" Body="&lt;p&gt;I am starting my PhD this Fall in Math Education.  I would like to either teach math at a university or work in a curriculum and instruction department doing research and teaching for a university.  For this reason, I have included many graduate level math classes in my program of study along with the required curriculum and instruction courses.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, when I get done with my PhD I would like to have as good of prospects as possible to find a job at a university.  I want to research something I'm passionate about, but I also want ideas on hot topics in math education research currently.  If my dissertation research aligns with what some others in the field are researching then this may increase my chances of future employment.  Currently I'm interested in studying the detrimental effects of using a calculator on mathematical thinking.  I don't know how interesting this is to others in my field so I am looking for some suggestions.  What are some hot topics in math education research currently?&lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2017-08-26T06:51:30.677" Title="Math Education Research Areas" Tags="|education-research|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="12770" PostTypeId="1" CreationDate="2017-08-23T10:33:57.920" Score="-1" ViewCount="294" Body="&lt;p&gt;My original phrasing was &quot;how many constants are there in...&quot;. I am trying to determine if this is more clear. This is not intended for an exam, just a basic question to get intro-level algebra students thinking about the difference between variables and constants.  Thank you.  &lt;/p&gt;&#xA;" OwnerUserId="6259" LastEditorUserId="6259" LastEditDate="2017-08-23T19:37:52.300" LastActivityDate="2017-08-23T21:44:48.423" Title="As it is appears here, how many constants do you see in $2x^3 + y^4 = \sqrt 5$?" Tags="|algebra|" AnswerCount="3" CommentCount="23" ContentLicense="CC BY-SA 3.0" />
  <row Id="12771" PostTypeId="2" ParentId="12770" CreationDate="2017-08-23T10:43:51.257" Score="1" Body="&lt;p&gt;As the term &quot;constant&quot; is mostly used in polynomials, I would look at this equation as the polynomial&#xA;$$2x^3 + y^4 - \sqrt{5} \in \mathbb{R}[x,y].$$&#xA;Thus, there is exactly one constant, the $\sqrt{5}$.&lt;br&gt;&#xA;Depending on your definition, the number could of course differ.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I can't really see how this has something to do with mathematical education, care to explain?&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2017-08-23T10:43:51.257" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12772" PostTypeId="2" ParentId="12770" CreationDate="2017-08-23T12:43:05.563" Score="7" Body="&lt;p&gt;The question&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As it is appears here, how many constants do you see in $2x^3+y^4=\sqrt{5}$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is quite vague.  For example, are we to consider $x$ and $y$ as variables?  Or perhaps they are also constants?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also a problem with &quot;invisible&quot; constants.  For example, a quadratic equation is often defined as an equation of the form $ax^2+bx+c=0$, where $x$ is a variable and $a$, $b$, and $c$ are constants.  $x^2-x=0$ is a quadratic equation, and one can argue that it has three constants: $a=1,b=-1,c=0$.  (Or it could have four constants if we consider the $0$ at the right hand side of the equation.)  Perhaps your equation has some &quot;invisible&quot; constants that should also be included in the count?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me also repeat John Coleman's comment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I would say 4 constants (although there is only 1 &lt;em&gt;constant term&lt;/em&gt;). It of course depends on your definition of &quot;constant&quot;. My answer 4 is from a programmer's perspective, where &quot;constant&quot; is roughly the same as &quot;literal&quot;. That expression has 4 literal integers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In short, I feel that your question is too vague if it is to be intended to be asked by a teacher in, say, an exam.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2017-08-23T12:43:05.563" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12773" PostTypeId="2" ParentId="12736" CreationDate="2017-08-23T13:39:03.133" Score="0" Body="&lt;p&gt;You can start with the basic operations on, say, relative integers. Multiplication is not invertible (not even when we add rationals, because of $0$), and as mentioned in a comment subtraction is not associative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a real-life examples (or rather non-technical close analogies, maybe) of non-commutativity, ask the students if putting on their shoes before their socks would result in a nice result (or if you are into super-heroes and not too shy, ask about putting their underwear after their pants). I am still looking for something similar about non-associativity.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-08-23T13:39:03.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12774" PostTypeId="1" CreationDate="2017-08-23T14:44:26.720" Score="8" ViewCount="955" Body="&lt;p&gt;When we have discussions about which technology to include in our classrooms today, we are often somewhat conflicted with many standard arguments and worries being presented on both sides. To help inform this discussion, &lt;strong&gt;I'm looking for sources that chronicle math educators' opinions about the adoption of the scientific calculator in the math classroom.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be very interested in seeing things like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;even-handed discussions of pros and cons written in the 1970s United States, or similar transition periods in other places&lt;/li&gt;&#xA;&lt;li&gt;&quot;pro-calculator&quot;/&quot;anti-calculator&quot; statements from professional organizations from a time when calculators were just becoming widespread&lt;/li&gt;&#xA;&lt;li&gt;rants from angry instructors or students written during a transition&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am not interested in anecdotes from instructors or students who saw the transition happen first-hand unless you can refer to notes or writings that illustrate what you were thinking &lt;strong&gt;at that time.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example if anyone has access or links to the sources mentioned below (from &lt;a href=&quot;http://hackeducation.com/2015/03/12/calculators&quot; rel=&quot;noreferrer&quot;&gt;&quot;A Brief History of Calculators&quot;&lt;/a&gt;), I would love to see them:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1975, the National Advisory Committee on Mathematical Education (NACOME) issued a report on calculators, suggesting that those in eighth grade and above should have access to them for all class work and exams. Five years later, the National Council of Teachers of Mathematics (NCTM) recommended that “mathematics programs [should] take full advantage of calculators … at all grade levels.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-08-25T14:42:56.920" Title="What did math educators think about the transition to widespread classroom use of calculators?" Tags="|reference-request|technology-in-education|history|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12775" PostTypeId="2" ParentId="12774" CreationDate="2017-08-23T16:11:35.217" Score="6" Body="&lt;p&gt;Googling &lt;strong&gt;&quot;NACOME&quot; 1975 calculators&lt;/strong&gt; seemingly leads to the report that you mention: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hill, S. (1975). Overview and analysis of school mathematics, grades K-12 (NACOME Report). In &lt;em&gt;Washington, DC: National Advisory Committee on Mathematical Education, Conference Board of the Mathematical Sciences&lt;/em&gt;. &lt;a href=&quot;http://www.mathcurriculumcenter.org/PDFS/CCM/originals/school_math_k12_report.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt; (pdf). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The section on &lt;em&gt;Calculators&lt;/em&gt; begins on PDF 40 (just after the section on &lt;em&gt;Computers&lt;/em&gt;). You may also link chase using google scholar to see who cited this report, and then who cited them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically, five citations of the NACOME report can be found &lt;a href=&quot;https://scholar.google.com/scholar?cites=12231746492927091277&amp;amp;as_sdt=5,33&amp;amp;sciodt=0,33&amp;amp;hl=en&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;, after which you can click through the &quot;Cited by&quot; to find lots more. For example, clicking through the 1977 piece by Bell leads &lt;a href=&quot;https://scholar.google.com/scholar?cites=4638616385912972223&amp;amp;as_sdt=5,33&amp;amp;sciodt=0,33&amp;amp;hl=en&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;, and clicking through the 1980 piece by Roberts leads &lt;a href=&quot;https://scholar.google.com/scholar?cites=13264502942197425406&amp;amp;as_sdt=5,33&amp;amp;sciodt=0,33&amp;amp;hl=en&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. And so on, and so forth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a final note of interest, you may find that &lt;a href=&quot;https://en.wikipedia.org/wiki/Zalman_Usiskin&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Zalman Usiskin&lt;/strong&gt;&lt;/a&gt; shares in your curiosity, as remarked in the recent(ish) interview:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Karp, A., &amp;amp; Roberts, D. L. (Eds.). (2014). &lt;em&gt;Leaders in mathematics education: Experience and vision&lt;/em&gt;. Springer. &lt;a href=&quot;https://books.google.com/books?id=_OSCBAAAQBAJ&amp;amp;pg&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt; (google books).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From pp. 180-181:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kcf9y.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kcf9y.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2017-08-23T16:11:35.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12776" PostTypeId="1" AcceptedAnswerId="17792" CreationDate="2017-08-23T20:28:42.700" Score="14" ViewCount="361" Body="&lt;p&gt;As far as I can find, the major resources available to the layteacher about APOS (Action, Process, Object, Schema) theory refer to &quot;undergraduate&quot; concepts such as group theory and vector spaces (such as &lt;a href=&quot;https://link.springer.com/chapter/10.1007/0-306-47231-7_25&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in whether people have implemented or investigated APOS theory at the high school or lower-undergraduate level, and whether there are resources for the high school math teacher that can be used concretely, for example, to design a unit. So far I have found &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B00EBWKXMI&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; book, but this seems unnecessarily technical to reach a larger audience of teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, when I described APOS theory to my coworkers, they immediately connected to it and it helped them understand the struggles that their students were facing in particular problems, such as transitioning from seeing &quot;sine and cosine&quot; as &quot;find the ratio between pairs of sides&quot; to &quot;functions.&quot; However I was unable to find good reading for them that referenced mathematics that they used frequently.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2020-01-26T21:57:51.950" Title="Resources for high school teachers about APOS theory" Tags="|secondary-education|reference-request|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12777" PostTypeId="2" ParentId="12770" CreationDate="2017-08-23T21:44:48.423" Score="2" Body="&lt;p&gt;The question is too ambiguous to be answered unequivocally, at almost all possible levels of sophistication. As in other answers and comments: from one viewpoint there are visibly 4 &quot;literals&quot;, which might be what is intended by &quot;constant&quot;. From another viewpoint, &quot;x&quot; and &quot;y&quot; might be names for constants, we have no idea. The fact that more than a single pair of possible values for x,y satisfy the equation does not give us any actual information about what they refer to. For that matter, for all we know the asserted equation is &lt;em&gt;false&lt;/em&gt;... (this is possible because it is easily possible to make grammatically correct but factually false assertions). In a related vein, a problem is that the expression &quot;$... = \sqrt{5}$&quot; has two &quot;free&quot; &quot;variables&quot; (in a logic sense), $x$ and $y$. And so on.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2017-08-23T21:44:48.423" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="12778" PostTypeId="2" ParentId="12774" CreationDate="2017-08-24T06:29:26.643" Score="3" Body="&lt;p&gt;[edited 25AUG, see comments]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chris:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Not meant to be a putdown, just some basic instruction in how to search.]  Here is a Google Scholar search with certain keywords and restricted to 1950 to 1980.  You can consider to open it through 1990 or later (see anecdote).  You can add or remove keywords.  If you want to make the search more restrictive, put the keywords &quot;teacher&quot; or &quot;reaction&quot; in quotes (that makes the search require them.)]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://scholar.google.com/scholar?q=calculator+math+education+teacher+reaction&amp;amp;hl=en&amp;amp;as_sdt=1%2C47&amp;amp;as_ylo=1950&amp;amp;as_yhi=1980&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://scholar.google.com/scholar?q=calculator+math+education+teacher+reaction&amp;amp;hl=en&amp;amp;as_sdt=1%2C47&amp;amp;as_ylo=1950&amp;amp;as_yhi=1980&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is first paper I found that hits your needs (first page of Google Scholar window):  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1980.tb09656.x/full&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1980.tb09656.x/full&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would consider to open up the search a little later into post 1980 (calcs really weren't in many math classes before then...experiments sure, but not wide pattern of use.)  Would also consider to just read some of the reports on efficacy and then skim them to look for teacher reaction...since that is more your interest than efficacy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Anecdotal segue, but it is to frame discussion at least:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to be careful to distinguish use of calculators in science number crunching and in math class.  The former was very rapidly introduced and noncontroversial.  The latter was very slow and remains controversial--like ask yourself, do kids have a hard time learning to punch buttons so they need to practice it--it was not a problem in chem class to use it for stoichiometry.    I went to school in the 70s and 80s and scientific calculators were VERY fast in introduction into the sciences.  There was zero controversy about replacing slide rules or log tables (I never used a slide rule, but I used log tables and was just a couple years past routine use of slide rules.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally don't remember any controversy about calculators in math class either...as it just wasn't done.  I went to a competitive grad school in the late 1990s (in the hard sciences) and I don't recall seeing calculator use in undergrad or upper grad courses and many difficult algebra manipulations were all done by hand (e.g. Jackson E&amp;amp;M, Tinkham superconductivity...lots of Bessel function craziness).  I remember being forced to use Maple once, but it was to learn how to make that beast print and set up a complex phase diagram and associated diffyQs and a numerical solver.  Not because of needing the CAS.  I suspect a lot of physicists still expect and get manipulation (see comments to this effect on Physics Forum) and there are many colleges where calculators are not used in a normal calculus class (most of the problems don't have arithmetic number crunching like chemistry and they want you to learn the methods of integration and manipulation and the like, not have a CAS do it for you.)]&lt;/p&gt;&#xA;" OwnerUserId="8618" LastEditorUserId="8618" LastEditDate="2017-08-25T04:42:07.220" LastActivityDate="2017-08-25T04:42:07.220" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12779" PostTypeId="2" ParentId="12728" CreationDate="2017-08-24T18:57:15.507" Score="4" Body="&lt;p&gt;I don't believe that there is much of debate about Calculus today.  Major universities continue to push the overly rigorous vision embodied in texts like Stewart or Hughes Hallet.  The AB/BC Calculus curricula have seen little to no change in decades.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems lost on many arguing for Stats over Calc, that they are one and the same.  Statistics developed from Physics and Astronomy using many of the typical ideas from first semester calculus.  Despite this architecture, too often we find people arguing for statistics &lt;em&gt;or&lt;/em&gt; calculus, or even stats &lt;em&gt;over&lt;/em&gt; calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not suggesting that when you are introducing measures of center and spread in 9th grade algebra class you need to fully develop centers of mass in the language of calculus.  I am acknowledging that historically, statistics was developed using the tools of calculus.  All this is to say that the stats vs. calc division is an imaginary one that is reinforced rather than remedied by the arbitrary battle lines so often drawn in teaching and learning mathematics that wish to favor a certain stylistic application of the tools of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many examples of groups who have tried to take the stats over calc path.  A large example would be the Statway work from Carnegie.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue that the discrete and continuous should have a presence in all mathematics coursework; particularly the undergraduate and high school curriculum.  I don't fully agree that the CCSS is a step backwards here, as there is an explicit effort to include functions with discrete domains throughout the high school work with functions.  To improve this wedding, the CCSS should have included the matrix and vector concepts throughout rather than as optional additions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a calculus class, if we are discussing integration we should include centers of mass and understand how this plays a role in Pearson's r.  These ideas have also  emerged as important in the field of machine learning and classification algorithms like k-means. Students can understand remedial examples of these topics at much earlier levels. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to differentiation we should use first difference to motivate a conceptual understanding.  Optimization lends itself to such an interpretation, move to the limiting case to discuss continuous situations.  Now we can engage with Nash equilibrium and with an understanding of differentiation with respect to a given variable we can derive least squares formulas for linear, polynomial, and exponential situations using our knowledge of minimizing functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose what I'm arguing is that it should not be a question of one or the other but our efforts should target unification rather than division.  Discrete mathematics deserves its place in the curriculum but I don't believe this needs to be at the expense of the continuous.  Instead, these domains reinforce one another and would support a broader learning experience for students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My favorite writing on this subject would be Felix Klein's &lt;em&gt;Elementary Mathematics from an Advanced Standpoint&lt;/em&gt; books I and II or any of Tom Apostol's writing on Calculus including his textbooks.  Klein argues beautifully for a larger role of the difference calculus in the curriculum.  Also, Leibniz's &lt;em&gt;History and Origins&lt;/em&gt; is a great read where he describes the motivation of his work through sums and differences from the familiar Pascal's triangle.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-08-24T18:57:15.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12780" PostTypeId="2" ParentId="12774" CreationDate="2017-08-25T14:42:56.920" Score="3" Body="&lt;p&gt;I want to preface the response with the fact that calculators are not in wide use in high schools in the United States of America.  I also think that we should not use calculators anymore now that we have access to tools like Jupyter notebooks and languages like R and Python with substantial libraries that crush anything possible on a handheld calculator.  Further, nobody I've met carries a graphing calculator around with them except math students and teachers.  When people grow up they use computers to solve these problems and it's about time we did the same in the mathematics classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article Usiskin is referencing in his answer comes from the NCTM's &lt;em&gt;History of School Mathematics&lt;/em&gt;, a two volume set that has some relevant articles including a history of technology in the mathematics classroom.  Here's a link to the books on Amazon. (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0873534719&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/History-School-Mathematics-Volumes/dp/0873534719&lt;/a&gt;)&#xA;There is also the Ackerberg Hastings text on technology in the mathematics classroom.(&lt;a href=&quot;https://jhupbooks.press.jhu.edu/content/tools-american-mathematics-teaching-1800%E2%80%932000&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://jhupbooks.press.jhu.edu/content/tools-american-mathematics-teaching-1800%E2%80%932000&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of research that discusses the use of calculators, you may be interested in the volumes edited by M. Kathleen Heid on technology in mathematics education that include a variety of works dealing with calculators. (&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1931576181&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Research-Syntheses-M-Kathleen-Heid/dp/1931576181&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The NCTM has many different articles across their journals that have taken on calculators and have published position papers on the issue and give these references as support:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Ellington, A. J. (2003). A meta-analysis of the effects of calculators on students' achievement and attitude levels in precollege mathematics classes. Journal for Research in Mathematics Education, 34, 433–463.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;National Council of Teachers of Mathematics. (2015). Strategic use of technology in teaching and learning mathematics: A position of the National Council of Teachers of Mathematics. Reston, VA: Author.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Reys, B. J., &amp;amp; Arbaugh, F. (2001). Clearing up the confusion over calculator use in grades K–5. Teaching Children Mathematics, 8, 90–94&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you are interested in the computing work rather than calculators you might find the Seymour Papert or James Kaput work interesting.  Also, there are even early FORTRAN courses from the SMSG (a group in the 'new math' of the 50's and 60's) that tried to introduce computing in elementary, middle, and high school classrooms.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-08-25T14:42:56.920" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12781" PostTypeId="2" ParentId="12769" CreationDate="2017-08-26T06:51:30.677" Score="2" Body="&lt;p&gt;I hope you won't mind if I suggest a different approach.  Rather than worrying about what will be trendy 5 years from now, identify a professor in your department whose advisees consistently graduate, whom you find approachable and easy to talk to, and see if you can get (genuinely) interested in that person's research program.  Find out what you can about your candidate from some of the more experienced graduate students before making a commitment!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you choose a boring dissertation topic, the top-tier schools may not be interested in hiring you -- but if you choose a bad advisor, you will not finish your Ph.D.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Background: I have a Ph.D. in math, and regularly serve on hiring committees for tenure-track faculty positions in mathematics at a public university in the United States.]&lt;/p&gt;&#xA;" OwnerUserId="6696" LastActivityDate="2017-08-26T06:51:30.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12782" PostTypeId="2" ParentId="11106" CreationDate="2017-08-26T07:54:49.483" Score="9" Body="&lt;p&gt;I like to explain why arithmetic and geometric progressions are so ubiquitous. Using the examples other people have given.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometric progressions happen whenever each agent of a system acts independently. For example population growth each couple do not decide to have another kid based on current population. So population growth each year is geometric. Each radioactive atom independently disintegrates, which means it will have fixed decay rate. In other words that is why there is &quot;half-life&quot; of a radioactive element, in a fixed amount of time it becomes half. Email chains, Interest rate, etc are more examples of the same kind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other end global/singular decisions give arithmetic progressions. If you add a fixed amount to your piggy bank each week that is arithmetic progression. The child who swings extra each time is likely to give only a constant extra force each time, so it is not likely for that to be geometric, it will be an arithmetic progression. There are exceptions of course like the ball bouncing is geometric even though it is singular because of coefficient of restitution. In general singular decisions can be anything - but typically arithmetic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In reality, these are ideal cases, most of the natural phenomenon will have both global and local influencers. Making it somewhere in between arithmetic and geometric progressions.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If the population is already huge having another kid might not be so conducive. So the population growth will stop when overall resources get limited. Thomas Malthus wrote that all life forms, including humans, have a propensity to exponential population growth when resources are abundant but that actual growth is limited by available resources.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Tumour growth, the growth rate is exponential unless it becomes so large that it cannot get food to grow effectively. So it starts of exponentially and stops completely. A more precise statement is known as Gompertz Law of Mortality - &quot;rate of decay falls exponentially with current size&quot;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Even radioactive decay is not really immune, there is something called Quantum anti-Zeno effect if you wanna go wiki hopping.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Malthusian_growth_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Malthusian_growth_model&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gompertz_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Gompertz_function&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8632" LastEditorUserId="8632" LastEditDate="2017-12-05T10:49:21.420" LastActivityDate="2017-12-05T10:49:21.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12783" PostTypeId="2" ParentId="12753" CreationDate="2017-08-26T17:17:33.477" Score="1" Body="&lt;p&gt;Pencil and paper will always been unbeatable for many applications, but sometimes electronic devices are more useful to me. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Wacom tablet + Microsoft OneNote + Movavi Screen Capture is what I use to create instructional videos on a MAC. The whole setup is inexpensive and the pressure sensitivity creates &quot;realistic&quot; writing that is a bit more aesthetic than electronic-writing that has a constant-thickness.  (the tablet is like 80 bucks, OneNote is free, Movavi is not free but was worth purchasing since it has quite a bit more functionality than Quicktime's screen capture), &lt;/p&gt;&#xA;" OwnerUserId="6259" LastActivityDate="2017-08-26T17:17:33.477" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12785" PostTypeId="2" ParentId="12676" CreationDate="2017-08-27T03:08:56.520" Score="1" Body="&lt;p&gt;Few thoughts (with practice, short bursts of time and question writing rules in mind):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;First question is text heavy. It need not be too explanatory unless you do want to teach a few things through the question.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;No need to mention anything about acceleration in the first question.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;First question can be further broken down into two questions. First part can be just about calculating velocity to go to front desk and come back. Scaffolding is always a good idea.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Not sure what is the scope of this workshop. Maybe you can think of gradually taking the question towards introducing velocity and acceleration as vectors (why the negative sign will matter).&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2017-08-27T03:08:56.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12786" PostTypeId="2" ParentId="11760" CreationDate="2017-08-27T03:36:35.660" Score="8" Body="&lt;p&gt;The answer is:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;You should not worry about it.&lt;/li&gt;&#xA;&lt;li&gt;Yes, you should let the reader interpret the angle according to the context.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There is no defined convention for notation of explementary angles. Because it's not of much practical importance. However if you skim through some papers, you'll notice that generally, angles are taken to be the ones less than 180 i.e., unless explementary is explicitly mentioned. In other words explementary angles are often taken to be greater than 180.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, when it's imperative to explicate which angle is being referred to, it's better to label the corresponding arc, and mention the angle as the one spanning through the specified arc.&lt;/p&gt;&#xA;" OwnerUserId="2191" LastEditorUserId="2191" LastEditDate="2017-08-28T15:41:12.733" LastActivityDate="2017-08-28T15:41:12.733" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12787" PostTypeId="2" ParentId="12753" CreationDate="2017-08-27T08:53:02.453" Score="1" Body="&lt;p&gt;You may wish to consider Rocketbook. The basic idea of their products is to write on paper but easily convert the output to electronic storage. I found the android app rather unsuccessful, but I am told it works well on apple.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2017-08-27T08:53:02.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12788" PostTypeId="1" AcceptedAnswerId="12789" CreationDate="2017-08-28T18:39:14.947" Score="10" ViewCount="1180" Body="&lt;p&gt;I am tutoring a 9th grade student. And he is terribly weak in mathematics. He doesn't remember the multiplication tables, can't divide efficiently. Doesn't know how to proceed with solving a mathematical expression. He says he is not interested in the subject and lacks motivation as well. Also, he is not serious about studies; a month from now he has exams. It really takes all my energy when trying to teach him a topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Enough of my vent, I just want to know how to deal with this situation? I don't want him to rote learn and try passing the exam. It is only going to get worst. He never had a tutor before and seems like he had been just getting passed by rote learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What should I do?&#xA;When his mathematical base is extremely weak, and he himself is not serious. I like teaching in depth. But it takes a whole lot of time. And I can't devout it.&lt;/p&gt;&#xA;" OwnerUserId="9003" LastEditorUserId="9003" LastEditDate="2017-08-29T11:49:26.340" LastActivityDate="2018-04-18T04:50:43.303" Title="How to teach a weak student?" Tags="|primary-education|tutoring|struggling-students|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12789" PostTypeId="2" ParentId="12788" CreationDate="2017-08-28T21:55:11.823" Score="10" Body="&lt;p&gt;Weaker students need drill and rote.  Make flashcards and drill the heck out of him.  That will help him more than any kind of detailed explanations or motivation talks.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just treat it like a sports practice and drill, drill, drill.  As he starts to master the drill, that will help him to become more confident (justifiably).  You need to be less of a thoughtful explainer and more of a sports coach or drill sergeant.  &lt;/p&gt;&#xA;" OwnerUserId="8618" LastActivityDate="2017-08-28T21:55:11.823" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12790" PostTypeId="2" ParentId="12788" CreationDate="2017-08-29T13:09:32.753" Score="7" Body="&lt;p&gt;As you mentioned:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;he is not interested in the subject and lacks motivation&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As a simple tutor you don't have the authority to deal with such a situation, so the best thing to do is to stop working as his tutor, and inform the parents about your reasons to stop (his lack of interest and motivation):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two things you should be aware of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Children should obey older people, so he must listen to you very carefully. In case (s)he doesn't, the parents need to intervene.&lt;/li&gt;&#xA;&lt;li&gt;When you keep on tutoring this child and (s)he fails, you'll be blamed for not having succeeded supporting the poor child, and the parents will be mad at you for having &quot;stolen&quot; their money (I imagine you're being paid for tutoring that child?).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I can very well imagine this is not easy (you want to succeed) but sometimes admitting defeat (especially when it's not your fault) is the best thing to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-08-29T13:09:32.753" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12791" PostTypeId="2" ParentId="12788" CreationDate="2017-08-29T14:29:58.590" Score="4" Body="&lt;p&gt;It seems like the lack of motivation is what is really holding the student back. You don't have much way of telling how hard he finds the maths if he isn't even trying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You (or rather, he and his parents) need to decide what the aim is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With an exam coming up soon, they best option at this point is probably to do the rote learning. You are not going to reach a good understanding in a short time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If instead (or afterwards) you want to focus on solving the problem longer term, I believe you should take a big step back. Find out why he's so resistant to learning maths. Is he trying to hide real difficulties? Is it too uncool to be seen to try? Is he just that uninterested in learning in general, or maths specifically? If the latter, perhaps try looking at applications, without trying to learn the details yet.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2017-08-29T14:29:58.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12792" PostTypeId="2" ParentId="11106" CreationDate="2017-08-29T15:39:59.013" Score="5" Body="&lt;p&gt;They might be interested to know about both Moore's Law and &lt;a href=&quot;https://en.wikipedia.org/wiki/Jakob_Nielsen_(usability_consultant)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Nielsen's Law&quot;&lt;/a&gt;.  You've probably heard about &lt;a href=&quot;https://en.wikipedia.org/wiki/Moore%27s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moore's Law&lt;/a&gt;, where computer complexity doubles about every two and a half years.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Internet bandwidth seems to also have a doubling time, in this case that doubling time is 21 months. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ldBgI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ldBgI.png&quot; alt=&quot;Exponential Growth of Bandwidth, Seen on a Log Scale&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nngroup.com/articles/law-of-bandwidth/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nngroup.com/articles/law-of-bandwidth/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8642" LastEditorUserId="8642" LastEditDate="2017-09-02T20:15:43.957" LastActivityDate="2017-09-02T20:15:43.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12793" PostTypeId="2" ParentId="11265" CreationDate="2017-08-29T17:46:02.483" Score="4" Body="&lt;p&gt;My slant is more tactical than the general idea of modeling.  An ODE course is useful for students in engineering and physics because they will need the concepts to follow derivations and work homework problems in their majors courses.  In mechanics, E&amp;amp;M, circuits, heat transfer, and many, many more.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also a good idea but not as vital for chemists (helps with p chem class).  I'm sure someone will point out the pretty predator prey problem of biology, but this is a sidelight IN BIOLOGY.  Biology has some systems to it, but is much more descriptive.  Obviously English majors and the like don't need it!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &quot;need it&quot;, I mean in a very utilitarian manner.  Not &quot;my life was enriched&quot;, but &quot;if I didn't have this, it would be hard to do my majors courses&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The benefit of having a course on the math involved in a topic you are learning (say in physics or engineering) is that you can concentrate more on the topic and less on the math.  It can become tough to learn a new topic, and new math (and perhaps new notation) all at the same time.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, math courses tend to teach more with x's and y's, which is just a less complicated fashion than learning math with a bunch of constants and variables signifying particular quantities (energy, force, etc.)  The ability to move between different ways of looking at the same thing is helpful for pedagogy and practice.  So being able to understand something in terms of just x and y equation solving as well as in the context of (say) thermodynamics is helpful in coming to grasp with the thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A simpler (than ODE) example would be knowing basic algebra through the quadratic and logs and exponentials prior to chemistry course where you work stoichiometry, rate, and equilibrium problems.  Those problems are bad enough with all the unit analysis, chemical concepts, quantities (many not simple numbers).  You don't want to be having to learn the math itself at the same time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also note that some engineering profs really don't want you pushing a lot of modeling at the kids.  That can be done fine in the engineering courses themselves.  But get the basic concepts, tricks, etc. across to the kids in terms of x and y, so they have a better time of surviving their engineering courses (which can be tough on the kids).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oh...and nail the second order linear constant coefficient (homogenous and nonhomo).  That thing comes up everywhere.  As the quadratic equation is to &quot;ICE&quot; chemical equilibrium problems, so is the harmonic oscillator for basic engineering and physics.&lt;/p&gt;&#xA;" OwnerUserId="8618" LastActivityDate="2017-08-29T17:46:02.483" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12794" PostTypeId="1" AcceptedAnswerId="12809" CreationDate="2017-08-29T18:40:41.840" Score="8" ViewCount="515" Body="&lt;p&gt;Let's say you're a university lecturer who regularly teaches remedial and/or college algebra courses. A standard textbook for such a course usually starts out with a series of facts about real number properties, but doesn't distinguish between facts that are definitions, versus facts that can be proven from more basic assumptions. A small number of examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is the definition of subtraction and division in use?&lt;/li&gt;&#xA;&lt;li&gt;Is the fact that $(-1)x = -x$ a definition, or a theorem?&lt;/li&gt;&#xA;&lt;li&gt;Is the fact that $a^0 = 1$ a definition or a theorem?&lt;/li&gt;&#xA;&lt;li&gt;What about the fact that $a/b = c/d \Leftrightarrow ad = bc$?&lt;/li&gt;&#xA;&lt;li&gt;Is the fact that we can add a value to both sides of an equation and maintain equivalence a definition, or provable from simpler assumptions?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Over the years I've spent a good amount of time disentangling these issues, but I'm wondering if there is an existing book that I could recommend to others. Say: We start by taking the real numbers as a roughly undefined term, and define the equality relation clearly. Explicate the definitions of the various operations in use, and the order of the operations. Establish the properties of commutativity, association, and distribution as axioms. From that point forward, clarify what facts are definitions, versus what are theorems, and present explicit proofs for the latter. Even if we don't present the information to our students exactly this way, I think it would be a relief for a mathematically-trained instructor to know what really must come first in the logical sequence, and what can really be proven to an inquisitive student, versus the few things that are effectively &quot;just because&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there such a resource that formally develops topics from the start of a basic algebra course? &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-08-30T03:00:41.603" LastActivityDate="2017-09-19T23:13:41.367" Title="Is there a resource that formally develops the topics of elementary algebra?" Tags="|undergraduate-education|algebra|axioms-foundations|resource-request|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="12795" PostTypeId="2" ParentId="1649" CreationDate="2017-08-29T19:16:37.670" Score="1" Body="&lt;p&gt;Let me give you some insight how I calculate things by heart:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First we start with the simple formula: $a^2-b^2=(a-b)*(a+b)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can be used for two things:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;perform a multiplication&lt;/li&gt;&#xA;&lt;li&gt;calculate a square&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;1) Perform a multiplication is the easy one : &lt;br/&gt;&#xA;In order to multiplicate two numbers, calculate their average and the difference to that average, and substract their squares, as you can see here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$17*23$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;the average is $20$, the difference to the average is $20-17=23-20=3$&lt;br/&gt;&#xA;So:&lt;br/&gt;&#xA;$17*20 = (20-3)*(20+3) = 20^2-3^2 = 400-9 = 391$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) calculate a square, based on :&lt;br/&gt;&#xA;$a^2-b^2 = (a-b)*(a+b) =&amp;gt; a^2 = (a-b)*(a+b) + b^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, how to calculate a square of a number, close to 50,100,200, ...?&lt;br/&gt;&#xA;Just rely on the distance to that number, as you can see here:&#xA;$56^2$&lt;br/&gt;&#xA;56 is close to 50, the difference is 6, so:&lt;br/&gt;&#xA;$56^2 = (56-6)*(56+6) + 6^2$&lt;br/&gt;&#xA;$     = 50*62+36$&lt;br/&gt;&#xA;$  = 3100+36$&lt;br/&gt;&#xA;$  = 3136$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next one: how to go from one square to the next one?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$(a+b)^2=a^2+2*a*b+b^2$&lt;/li&gt;&#xA;&lt;li&gt;$(a-b)^2=a^2-2*a*b+b^2$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Let's do this for b=1 and b=2, this gives:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$(a+1)^2=a^2+2*a+1$&lt;/li&gt;&#xA;&lt;li&gt;$(a-1)^2=a^2-2*a+1$&lt;/li&gt;&#xA;&lt;li&gt;$(a+2)^2=a^2+4*(a+1)$&lt;/li&gt;&#xA;&lt;li&gt;$(a-2)^2=a^2-4*(a-1)$ or $a^2-4*a+4$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;How much are now the squares of 21, 29, 42 and 58?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$21^2=20^2+2*20+1=400+40+1=441$&lt;/li&gt;&#xA;&lt;li&gt;$29^2=30^2-2*30+1=900-60+1=841$&lt;/li&gt;&#xA;&lt;li&gt;$42^2=40^2+4*(40+1)=1600+164=1764$&lt;/li&gt;&#xA;&lt;li&gt;$58^2=60^2-4*60+4=3600-240+4=3360+4=3364$ (subtracting by 4*59 might prove difficult by heart)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Combine this with the square of a number, ending by 5:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$57^2=55^2+4*56=3025+224=3249$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(or if you prefer the method in the first point)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$57^2=(57-7)*(57+7)+7^2=50*64+49=3249$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next one: how do we divide by numbers, close to natural numbers?&#xA;(this one has been very handy while converting currencies, before the Euro was introduced)&lt;br/&gt;&#xA;Imagine the time of the Belgian and the French currency, both were called &quot;fran(c)(k)&quot;, and at the end their rate was 5.7.&#xA;How to divide by 5.7? It comes down to:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1/5.7 = 1/(6-0.3)$&lt;br/&gt;&#xA;$      = 1/6*(1/(1-1/20))$&lt;br/&gt;&#xA;$   = 1/6*(1+(1/20)+(1/20)^2+(1/20)^3+...)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;can be simplified by:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$     1/6*(1+1/20+...)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, let's do this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$600/5.7$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First divide by $6$ =&gt; first result = $100$&#xA;Divide this by $20$ =&gt; $5$,&lt;br/&gt; and add this to the first result =&gt; $result = 105$&lt;br/&gt;&#xA;Divide the $5$ again by $20$ ($0.25$), &lt;br/&gt;and add this to the result =&gt; $result = 105.25$&lt;br/&gt;&#xA;... (you can go on as far as you want, but one or two steps is impressive enough)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And indeed : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$600/5.7 = 105.263...$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last one:&lt;br/&gt;&#xA;how to take a square root of a number?&#xA;(this one is based on $(a+b)^2=a^2+2*a*b+b^2$)&#xA;Let's do this for, e.g. 328.&#xA;You need to know the first 20-30 squares by heart, and so you know that this is close to $324=18^2$, hence:&lt;br/&gt;&#xA;$(18+a)^2=328$, hence:&lt;br/&gt;&#xA;$324+36*a+a^2=328$, now let go of $a^2$ (it's very small with regards to the other numbers):&#xA;36*a is very near to 328-324, or $a=4/36=1/9$, so, next approximation:&#xA;$sqrt(328)$ is near to $18.11...$ (indeed, it's 18.11077...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Sorry for the bad formatting, I'll deal with this later)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-08-29T19:16:37.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12796" PostTypeId="1" AcceptedAnswerId="12801" CreationDate="2017-08-29T21:08:10.370" Score="20" ViewCount="2754" Body="&lt;p&gt;The slope-intercept form of the equation of a line is often presented in textbooks (in the US) as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$y = mx + b\,,$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is the slope of the line and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept. How did  &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; become the standard variables used for the slope and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept? What should we tell students &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; represent to help them make sense of the roles of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; in that equation?&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2020-01-21T15:28:25.123" LastActivityDate="2020-01-21T15:28:25.123" Title="Why are $m$ and $b$ used in the slope-intercept equation of a line?" Tags="|algebra|notation|terminology|history|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="12797" PostTypeId="1" AcceptedAnswerId="12799" CreationDate="2017-08-30T10:28:43.823" Score="1" ViewCount="153" Body="&lt;p&gt;I have to deliver a lecture for secondary school, about one of these topics: integers, Euclid's Elements, polyhedra, prime numbers, non-Euclidean geometry, arithmetic functions or graphs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It should be something interesting and easy to understand for high-school students, and 45 minutes long.&#xA;My ideas are: platonic solids, something about prime numbers (but I don't know what exactly), golden ratio. They should be good but they seem to me not very original.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have any suggestion? Maybe an interesting problem solved using one these mathematical tools? Or a particular aspect of one of these that could catch the eye?&lt;/p&gt;&#xA;" OwnerUserId="8647" LastEditorUserId="2083" LastEditDate="2017-08-30T13:34:47.450" LastActivityDate="2017-08-30T18:04:04.617" Title="Interesting math lesson on integers, Euclid's Elements, polyhedra, prime numbers, non-Euclidean geometry, arithmetic functions or graphs" Tags="|secondary-education|geometry|number-theory|presentation|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12798" PostTypeId="2" ParentId="12796" CreationDate="2017-08-30T13:29:03.830" Score="11" Body="&lt;p&gt;I taught low level algebra for a bit and those students really struggled with knowing what variables were and that they actually stood for numbers. They would see $y=mx+b$ and just not have any idea what it meant or stood for. I've heard the &quot;monter&quot; explanation before, but the students didn't find it helpful (they didn't speak french after all...). What I ended up doing, is replacing a lot of the variables in the class with blanks. So I would write $y=\_\_\,x + \_\_$ and then ask what went in each blank and they would respond with &quot;slope&quot; and &quot;intercept&quot; so I would write $$y=(\text{slope})\,x+(y\text{-intercept})$$ and they seemed to follow that a bit better since they themselves came up with the words to use and didn't have to memorize &quot;m for slope&quot; and &quot;b for y-intercept&quot; since those seem more disconnected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;b&quot; for &quot;begin&quot; and &quot;m&quot; for &quot;move&quot; explanation seems like it would have worked for this level of student, I wish I could have tried that.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastEditorUserId="5220" LastEditDate="2017-08-30T21:01:49.300" LastActivityDate="2017-08-30T21:01:49.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12799" PostTypeId="2" ParentId="12797" CreationDate="2017-08-30T13:43:46.433" Score="2" Body="&lt;p&gt;One thing I like to do is to look at the Numberphile Youtube channel. They have good bits of deeper math but presented in a way that most high school students can understand. One idea that jumps out to me from your list (because I actually used it when I taught an Honors Sophomore class) was to play the game &lt;a href=&quot;https://www.youtube.com/watch?v=OAss481FfAQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brussels Sprouts&lt;/a&gt;. I won't tell you my exact layout and lesson plan, as it is up to you to come up with that. But keep in mind what others have said: Any topic can be fun or boring, it depends on your energy and presentation as well as how active you let your students be in the lesson. Are they sitting and listening for 45 minutes? Are they investigating some task on their own? Are they solitary or in groups? There are a ton of factors to consider when planning a lesson.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2017-08-30T13:43:46.433" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12800" PostTypeId="1" CreationDate="2017-08-30T13:46:13.300" Score="8" ViewCount="452" Body="&lt;p&gt;I'm posting this here as it's off-topic for the Math stack exchange, and I'm hoping there will be some educators here that can point me in the right direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I enjoyed watching the YouTube series &lt;a href=&quot;https://www.youtube.com/channel/UCoxcjq-8xIDTYp3uz647V5A&quot; rel=&quot;noreferrer&quot;&gt;Numberphile&lt;/a&gt;, as it makes math accessible to me in a way it hasn't been before. Which is awesome; I always sucked at math. Anyway, there's an episode where &lt;a href=&quot;https://www.youtube.com/watch?v=6Lm9EHhbJAY&amp;amp;feature=youtu.be&amp;amp;t=90&quot; rel=&quot;noreferrer&quot;&gt;they mention that in Euclid's time, they would learn about shapes and geometry as the basic elements of mathematics and numbers were simply the result of that&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That made me think about the one time I really shined in math classes. Geometry. I was always slow to learn math concepts, as I found the procedural &quot;Just do it like this and you'll get the answer&quot; teaching boring and rote. But not geometry! You can &lt;em&gt;see&lt;/em&gt; the relationships and you do things a certain because they make &lt;em&gt;sense&lt;/em&gt;. I mean, I'm sure those procedures they were teaching before also made sense in a manner that wasn't yet accessible, but geometry just naturally made sense. Other aspects of mathematics just don't click until I reach a certain point of intuitive understanding (which is why I love Numberphile so much, as it's full of visuals that illustrate relationships between numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, where can I find resources to learn more in this visual relationship-based manner? I took a pre-calc class in college, but had to drop it when the concepts just weren't clicking. Everything was procedure-based, and I need to find resources that help me to understand the why before teaching the how.&lt;/p&gt;&#xA;" OwnerUserId="8648" LastActivityDate="2017-09-06T22:20:41.830" Title="Learning Math like Euclid" Tags="|geometry|self-learning|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12801" PostTypeId="2" ParentId="12796" CreationDate="2017-08-30T14:53:27.320" Score="26" Body="&lt;p&gt;This is also borderline not-an-answer, but it might be a nice broadening of your students' worldview to know that the &quot;$m$&quot; and the &quot;$b$&quot; are not universally accepted. Showing them this map (even though I do not know its original source, so it may not be accurate in its details) could help their thinking out a bit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/aJIzM.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aJIzM.png&quot; alt=&quot;Map of linear equation letters&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Substantial edit: I now no longer believe that the above map has very many if any of its details correct, although the general sentiment (&quot;$mx + b$ is not a worldwide convention&quot;) is still correct. I went through all the languages of Wikipedia that had an article for Linear_equation and found the first mention of &quot;[variable]x + [variable]&quot; in the article. I kept the same absurd color-coding as the map, even though most of my results disagree with the map, even in Europe where languages map more precisely to countries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;First &quot;$mx+b$-type expression&quot; in each language's Wikipedia page for Linear_equation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4fPsA.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4fPsA.png&quot; alt=&quot;Linear Equation by Language in Wikipedia&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So anyway, one way or another, even though the map is probably-wrong, there definitely isn't much worldwide agreement on which variables to use in which slot.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2017-08-31T23:27:34.433" LastActivityDate="2017-08-31T23:27:34.433" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12802" PostTypeId="2" ParentId="9835" CreationDate="2017-08-30T15:32:07.100" Score="-2" Body="&lt;p&gt;Electronic engineers describe a linear amplifier or other linear device as one obeying Output = Input x Gain (or loss),  so if input = 0 then output  = 0,  as in &quot;direct variation&quot; above used to describe a line that goes through the origin, i.e. Y=mX.&lt;/p&gt;&#xA;" OwnerUserId="8649" LastActivityDate="2017-08-30T15:32:07.100" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12803" PostTypeId="2" ParentId="12797" CreationDate="2017-08-30T18:04:04.617" Score="1" Body="&lt;p&gt;I would personally choose Platonic Solids, every time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In her book &lt;em&gt;Geometry&lt;/em&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Mich%C3%A8le_Audin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michèle Audin&lt;/a&gt; says,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In this section, I want to prove that there are five types of regular polyhedra in 3-dimensional space.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;with the footnote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In my opinion, this statement is part of the cultural heritage of mankind, as are the &lt;em&gt;Odyssey&lt;/em&gt;, the sonatas of Beethoven or the statues of Easter Island (without mentioning the Pyramids) and I cannot imagine how a citizen, &lt;em&gt;a fortiori&lt;/em&gt; a maths teacher, could not know it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think she has a wonderful (if somewhat scathing!) viewpoint on this particular facet of geometry, and sadly, I don't know how often students meet polyhedra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 45 minutes, you should have time to&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Introduce the Platonic Solids, although you really should have paper or wire-frame (e.g. Zometools) models for this.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove that at most five Platonic Solids exist (by examining how much of $360^\circ$ is taken up by various numbers of regular polygons, around any vertex). It's &lt;em&gt;very&lt;/em&gt; elementary, although requires some explanation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;with time to spare. Good additional topics are to&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Count faces of each dimension (probably not combinatorially, just by inspection. I've had students who have a hard time with the icosahedron, due to all the symmetry!) and verify &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler_characteristic#Polyhedra&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler's famed $V - E + F = 2$ formula&lt;/a&gt; (which of course holds more generally than just for Platonic Solids).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Verify &lt;a href=&quot;http://mathworld.wolfram.com/DescartesTotalAngularDefect.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Descartes' Total Angular Defect Theorem&lt;/a&gt; (add up all the angles in a given vertex; that's the angular defect of that vertex. The sum of these defects, over each vertex of the polyhedron, is $720^\circ$), which also holds more generally than for just Platonic Solids.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I like these last topics because they show us that the world has some very cute, but not at all obvious, rules about what is physically possible; restrictions by the very nature of three-dimensional space!&lt;/p&gt;&#xA;" OwnerUserId="5153" LastActivityDate="2017-08-30T18:04:04.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12804" PostTypeId="1" CreationDate="2017-08-30T21:01:15.633" Score="6" ViewCount="379" Body="&lt;p&gt;I am going to teach a two-quarter sequence on abstract algebra at a mid-size american public university. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, this course would introduce groups, rings, and fields, then end with some applications. I would like to get to the impossibility of squaring the cube, trisecting an arbitrary angle with a ruler and compass, etc but I don't think that there will be time. (If you disagree with this attitude, please point me in the direction of an appropriate text book that gets from nothing to minimal polynomials in less than 20 weeks.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am therefore brain-storming possible two-term abstract algebra classes, ideally containing both theory and applications (to mathematics or to other disciplines), and ideally being more deep than broad (the second term should use the first). Ideas I have had so far include:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;try to get to groups, rings, and fields, at the probable cost of ending before we've arrived at any rewarding application&lt;/li&gt;&#xA;&lt;li&gt;all group theory: the basics (homomorphisms, quotients, etc), then study group actions enough to solve the Rubik's cube or prove Sylow's theorems.&lt;/li&gt;&#xA;&lt;li&gt;all ring &amp;amp; field theory: homomorphisms, quotients, prime and maximal ideals, polynomial rings, factorization, and extension fields&lt;/li&gt;&#xA;&lt;li&gt;(your ideas here, please)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Please note that the students taking this class will be mostly working on secondary education degrees, and relatively unlikely to be planning to study mathematics in grad school. This discourages me from suggesting something like group theory 1st quarter, intro to representations in the 2nd quarter. (Feel free to disagree with me here, too) &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2017-09-02T01:36:52.987" Title="Content for a two-quarter class on abstract algebra" Tags="|undergraduate-education|abstract-algebra|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12805" PostTypeId="2" ParentId="12804" CreationDate="2017-08-30T22:54:58.197" Score="5" Body="&lt;p&gt;Given your clientele, I think focusing mostly on groups and very concrete rings is good.  Or skip more advanced group theory to do the trisection, though perhaps you are right that you just need too much group theory to get there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, my own idea is that I don't see why you couldn't do quite a bit of number theory (I will admit I am biased) from the groups standpoint.  It fits in very nicely.  Likewise things like frieze/crystallographic groups.  So basic groups, a bit of rings, then lots of topics.  Not as comprehensive, but probably better in the long run - ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, there are several textbooks out now addressing these groups specifically ... here are some random links.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://sl2x.aimath.org/book/aafmt/&quot; rel=&quot;noreferrer&quot;&gt;Elementary Abstract Algebra&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.maa.org/press/books/learning-modern-algebra-from-early-attempts-to-prove-fermats-last-theorem&quot; rel=&quot;noreferrer&quot;&gt;Learning Modern Algebra&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.maa.org/press/maa-reviews/an-introduction-to-abstract-algebra-with-notes-to-the-future-teacher&quot; rel=&quot;noreferrer&quot;&gt;An Introduction to Abstract Algebra with Notes to the Future Teacher&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-08-30T22:54:58.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12806" PostTypeId="2" ParentId="12804" CreationDate="2017-08-31T00:12:45.583" Score="6" Body="&lt;p&gt;Abelian group theory emphasis with application being point symmetry (playing cards, stars, polyhedra, buildings, molecules, etc..  I wouldn't go into IR stretching modes, but you could at least mention that point symmetry is very practical and important to chemists and biochemists doing spectroscopy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would try to really take it easy with the abstractness and the rigor but give them some feeling for why they should care about it.  None of them will teach it in high school so it is not so important that they are proofed to death.  But if they at least know the story (even without the details) that group theory helped show the quantic is not solvable, etc.  That will inform them about what it is about.  And can come out in HS algebra class as a remark.&lt;/p&gt;&#xA;" OwnerUserId="8618" LastActivityDate="2017-08-31T00:12:45.583" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12807" PostTypeId="2" ParentId="12804" CreationDate="2017-08-31T00:52:58.640" Score="5" Body="&lt;p&gt;I would like to recommend you at least look at Nathan Carter's &lt;em&gt;Visualizing Group Theory&lt;/em&gt; (&lt;a href=&quot;https://www.maa.org/press/ebooks/visual-group-theory&quot; rel=&quot;noreferrer&quot;&gt;MAA link&lt;/a&gt;), even&#xA;if you don't use it as a textbook (although I think it would make a fine textbook, especially for education-degree students).&#xA;You could supplement with exploring crystallographic groups, as &lt;em&gt;kcrisman&lt;/em&gt; suggests, perhaps culminating in&#xA;Conway's &quot;Magic Theorem&quot; (discussed, e.g., in &lt;em&gt;The Symmetries of Things&lt;/em&gt;).&#xA;This could easily fill two quarters.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/0yl8B.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0yl8B.jpg&quot; alt=&quot;VGT&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-08-31T00:52:58.640" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12809" PostTypeId="2" ParentId="12794" CreationDate="2017-09-01T18:40:41.770" Score="2" Body="&lt;p&gt;&lt;strong&gt;Short answer:&lt;/strong&gt; (migrated from comments by request)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hung-Hsi Wu (Berkeley) has a &lt;a href=&quot;https://math.berkeley.edu/~wu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;home page&lt;/strong&gt;&lt;/a&gt; full of links to writings of this ilk. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two specific examples that may fit the bill are ones I mentioned back in &lt;a href=&quot;https://matheducators.stackexchange.com/a/1858&quot;&gt;&lt;strong&gt;MESE 1857&lt;/strong&gt;&lt;/a&gt; (April 2014); in particular, links to a text on Pre-Algebra (&lt;a href=&quot;http://math.berkeley.edu/~wu/Pre-Algebra.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;pdf&lt;/strong&gt;&lt;/a&gt;) and an Intro to School Algebra (&lt;a href=&quot;http://math.berkeley.edu/~wu/Algebrasummary.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;pdf&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Further comments:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of H.H. Wu's materials are intended more for pre-service/in-service teachers to bolster their mathematical content knowledge. The question of how to enact these ideas in one's classroom practice is a tough one. As an example, I have seen (simple) &lt;strong&gt;fractions&lt;/strong&gt; presented as pairs of numbers with a horizontal line (vinculum aka fraction bar) between them, for which the top number is an integer and the bottom number is a non-zero integer. One can then define two fractions $\frac{a}{b}$ and $\frac{c}{d}$ to be equivalent under (some condition) and then define the &lt;strong&gt;rational numbers&lt;/strong&gt; as being the fractions modulo this equivalence relation (whatever that means). Of course, the notation using a horizontal line is merely a convention; one can also think of rational numbers as being ordered pairs in $\mathbb{Z}\times\mathbb{Z}_{\neq 0}$ modulo the appropriate equivalence relation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mention this example since the post above contains:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Say: We start by taking the real numbers as a roughly undefined term, and define the equality relation clearly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Incidentally, I recently encountered a secondary school mathematics teacher who defined two fractions $\frac{a}{b}$ and $\frac{c}{d}$ to be equivalent if there exists an $r \in \mathbb{R}$ for which $ar = c$ and $br = d$. This definition is plausible (in some sense) if, as in the quotation pull above, one begins by allowing for something called the &quot;real numbers.&quot; Formally, though, it can turn out to be problematic: Usually the development of $\mathbb{R}$ proceeds in some way from the development of $\mathbb{Q}$, so defining the latter by using an element of the former becomes circular. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The standard condition given for fraction equivalence is, indeed, one that is also contained in this post: equality under cross multiplication, i.e., $a/b = c/d$ if and only if $ad = bc$. Another approach is to talk about &quot;simplifying fractions,&quot; and then one can say that two fractions are equivalent if, when fully simplified, each yields the same ordered pair of integers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, motivating this level of formality is an expected challenge irrespective of the population with whom one works. I do not know of a mathemagical way to excite everyone around distinguishing between theorems and definitions but, as remarked in the comments above (in response to another comment), I think we should avoid concluding that an enrollee in a remedial course on college algebra will find these ideas un-interesting, or that such courses are devoid of &quot;eager, well-motivated math students.&quot; It is my sense that students can end up in these courses because, all too frequently, they have experienced mathematics in earlier years as a nebulous collection of assertions that lack the cohesion/interconnectedness that may be more apparent when the formal definitions are clearly articulated. And the corresponding lack of sense-making in previous math courses can, indeed, take its toll on one's eagerness/motivation to pursue mathematical understanding. Under such conditions, I sincerely doubt that continuing to teach by fiat is the right response.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2017-09-01T18:40:41.770" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12810" PostTypeId="2" ParentId="12794" CreationDate="2017-09-02T01:21:54.283" Score="1" Body="&lt;p&gt;The best careful textbook you can find on elementary algebra that I'm aware of is &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0817636773&quot; rel=&quot;nofollow noreferrer&quot;&gt;the classic by Gelfand&lt;/a&gt;. It is beautifully written and completely careful-it's the book you wish you'd had in high school or grade school. I think you'll find it VERY helpful for both yourself and your students. &lt;/p&gt;&#xA;" OwnerUserId="7033" LastActivityDate="2017-09-02T01:21:54.283" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12811" PostTypeId="2" ParentId="12804" CreationDate="2017-09-02T01:36:52.987" Score="3" Body="&lt;p&gt;As someone who learned algebra from the old classic by Herstein-probably too difficult for your students, but a jewel worth browsing by any instructor-I think it's important to cover group theory as thoroughly as possible The reason for this is that subsequent algebraic structures-rings, fields,modules-are all most directly constructed by adding structures to a group. For example,a vector space can most easily be visualized as a structure where the Abelian group of scalars in a field acts on an Abelian group who's elements we call vectors. Another example is how the structure theorem for finitely generated modules over a principal ideal domain is a straightforward generalization of the classification theorem of finite Abelian groups. I would cover as much group theory as possible and then build ring theory on top of it. Not only will it go much faster since most of the basic results will be familiar from group theory,it will be much pedagogically clearer to the students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also try and add as much geometric content and applications as I can at this level. The use of symmetry groups and the decomposition of modular arithmetic gives 2 very important applications using group and ring theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck! &lt;/p&gt;&#xA;" OwnerUserId="7033" LastActivityDate="2017-09-02T01:36:52.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12812" PostTypeId="1" CreationDate="2017-09-02T03:20:26.797" Score="18" ViewCount="6826" Body="&lt;p&gt;I'm going to soon review the 3rd edition of Saunders MacLane And Garrett Birkoff's &lt;em&gt;Algebra&lt;/em&gt; at my blog soon and this is the first time I'm really carefully reading it. While I'm really enjoying the book,I've got a real dilemma gnawing at me. What kinds of students can this book be used for in 2017?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand from professors who were students at Harvard that MacLane himself would use the book-in all it's versions,from the first mimeographed drafts in the early 1960's to the 3rd edition he used in his last teaching days in the mid-1990's-for both undergraduate and graduate courses in algebra depending on the strength of the students, which would vary enormously from class to class. Usually,for strong undergraduate courses, he'd supplement &lt;em&gt;Algebra&lt;/em&gt; with examples drawn from he and Birkoff's classic undergraduate text, &lt;em&gt;A Survey Of Modern Algebra&lt;/em&gt;. For a first year graduate course-which at Harvard,usually included a number of undergraduates-he'd use the book straight up and cover most of it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question: What level students today can &lt;em&gt;Algebra&lt;/em&gt; be used for? I think it would definitely be too difficult for any but the very best undergraduates and I think there are several books that would be better for that purpose: Herstein's &lt;em&gt;Topics in Algebra&lt;/em&gt; or Vinberg's &lt;em&gt;A Course in Algebra&lt;/em&gt; come to mind. I don't think hitting students with category theory unless they've already had considerable exposure to algebra is a good idea. I doubt they'll be able to fully appreciate the enormous generalization and unification it provides without a giant stock of examples (which is why I love Emily Riehl's &lt;em&gt;Category Theory In Context&lt;/em&gt; as the definitive introduction to category theory for mathematics students. She gets this.) But at the same time, since the book contains very little homological algebra and no representation theory beyond the definition,can it really be used as graduate course today? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would like to hear feedback on this question. &lt;/p&gt;&#xA;" OwnerUserId="7033" LastActivityDate="2019-06-04T09:58:05.867" Title="Is MacLane and Birkoff's &quot;Algebra&quot; suitable today as either an undergraduate or graduate text in abstract algebra?" Tags="|undergraduate-education|textbooks|abstract-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12814" PostTypeId="2" ParentId="1649" CreationDate="2017-09-03T04:01:42.780" Score="2" Body="&lt;p&gt;Multiplying whole numbers times $25$ is counting quarters.  Many American students find this easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Getting a better that whole number approximation of square roots is possible using &#xA;$$\left( a+\frac{1}{2}\right)^2 = a(a+1)+\frac{1}{4}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\left( a+\frac{1}{2}\right)^2 = a^2+a+\frac{1}{4}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can quickly see that $\sqrt{72}$ is between $8$ and $9$, but can we do better?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$8.5^2 = 8\cdot9+0.25 = 72.25$.  So $8.5$ is a pretty good approximation for $\sqrt{72}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not a terribly useful trick, but it is a nice use of the algebra behind it. This is essentially the same trick as that cited by the OP on squaring numbers ending in $5$.&lt;/p&gt;&#xA;" OwnerUserId="8658" LastEditorUserId="8658" LastEditDate="2017-09-04T22:03:03.477" LastActivityDate="2017-09-04T22:03:03.477" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12815" PostTypeId="2" ParentId="12586" CreationDate="2017-09-03T04:42:58.147" Score="3" Body="&lt;p&gt;I believe that the point of having students prove trigonometric identities is to give them practice rewriting expressions involving trigonometric equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The actual problems they are preparing to solve involve, for example, two people working the same problem independently and then comparing answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:  If I got $\csc{x}\cdot\cot{x}$ and you got $\cos{x}$, did we get the same answer?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that this is the motivation for the kind of quote offered in the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, the method in the quote teaches students to think in terms of chains of equal expressions rather than chains of equivalent equations.  I see some value in that as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I also have not much difficulty with the idea of teaching students to 'prove' an identity by solving the equation.  If the solution set includes all numbers in the domains of both sides of the original equation, then the equation if an identity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That seems more difficult to teach and likely to be more difficult for students to understand than the approach in the text quoted above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit:  I am not ignoring anyone.  The rules do not permit me to comment until I have more reputation points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I doubt that every textbook author takes the same view I do.  I would like to see more open questions in textbooks.  In trigonometry , there should be more:  &quot;Are these two expressions equivalent when both are defined?&quot; questions than I have seen in the books I have been assigned.&lt;/p&gt;&#xA;" OwnerUserId="8658" LastEditorUserId="8658" LastEditDate="2017-09-03T22:39:31.070" LastActivityDate="2017-09-03T22:39:31.070" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12816" PostTypeId="1" CreationDate="2017-09-03T04:44:01.890" Score="9" ViewCount="249" Body="&lt;p&gt;I have autistic spectrum disorder / Asperger's syndrome (diagnosed by a psychiatrist). Also I have ADHD.&lt;/p&gt;&#xA;&lt;p&gt;I have trouble processing information in an abstract way. I can only think visually. I am good at programming. I am bad at math.&lt;/p&gt;&#xA;&lt;p&gt;I want to do computer science, but I am hesitant to do so because of the mathematics required, despite me being proficient in algorithms / programming in general.&lt;/p&gt;&#xA;&lt;p&gt;I even have trouble processing questions. I have to go over the question 4 or 5 times to be able to understand it. This is probably also because of ADHD, and also when I am reading a question I have to have it visualised in my brain very clearly.&lt;/p&gt;&#xA;&lt;p&gt;A lot of things in maths are abstract and cannot be visualised, but if they cannot be visualised I simply cannot learn them. I need to understand every tiny detail so I can internalise it in a logically consistent way. The most I've done is pre-calculus, and that was before I was diagnosed with ADHD, and I struggled in it. I am 20 right now, I think my IQ component relevant to numeracy is low.&lt;/p&gt;&#xA;&lt;p&gt;In school, I would complete tasks in 1-2 minutes with ease, and the rest of the class may spend a whole lesson on them (e.g. in subjects such as economics). However, in maths, I always struggled. The other problem is that I have a short working memory but an excellent long term memory. If I like the subject, I get hyperfocused and excel in it a lot, but unfortunately I do not think I have ever felt that way with math.&lt;/p&gt;&#xA;" OwnerUserId="8659" LastEditorUserId="7365" LastEditDate="2024-03-07T19:43:18.357" LastActivityDate="2024-03-07T19:43:18.357" Title="Is it possible for me to learn math if I cannot process information in an abstract way?" Tags="|disability|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12817" PostTypeId="2" ParentId="12816" CreationDate="2017-09-03T07:59:27.913" Score="5" Body="&lt;p&gt;'Visual' and 'abstract' are by no means opposites. I would be tempted to argue they are independent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It depends on what you mean by 'doing maths', and also a bit on what you mean by 'thinking visually'. I tend to think in a visual way, and my research usually features lots of pictures, but there are other pictures in my head that are far too abstract to put on paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some areas of maths are more naturally visual than others. Geometric topology, and some areas of geometry, can definitely be pictured (and are probably better done visually). You might like to look at knot theory, as some of that is very diagram-based. Other areas can be understood using visual means. For example, you can think about groups using Cayley graphs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know exactly what maths you will need for computer science, but if you are ok with algorithms I would think you could learn the material, provided you find the right way to go about it &lt;em&gt;for you&lt;/em&gt;. Talk to the admissions tutor, get to talk to someone who actually knows about the teaching in the subject, and see if they are willing to work with you. You may well need to base your choices on where there are people who have the right attitude.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The two things I know of are likely to come up are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;geometry using matrices (pretty visual in 2d/3d; there might be a need to think in more dimensions);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;databases using relations (can be thought of visually, at least to some extent, or just in terms of examples).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2017-09-03T07:59:27.913" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12818" PostTypeId="2" ParentId="1649" CreationDate="2017-09-03T15:09:09.743" Score="1" Body="&lt;p&gt;To multiply two numbers from $11 - 19$, written in base ten as $1a$ and $1b$, we can use,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(10 + a)(10 + b) = 100 + (a + b)10 + ab$   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It could be remembered as '&lt;em&gt;One add multiply&lt;/em&gt;'. For example, $ 12 \times 13 = 156$  from $100 + 50 + 6$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we go higher than the products $12 \times 14$ or $13^2$ care is needed in carrying over,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$13 \times 15 =100 + 80 + 15 = 195$  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$17^2 = 100 + 140 + 49 = 289$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps this is the trick mentioned in the &lt;em&gt;Notices of the AMS&lt;/em&gt; article honoring &lt;em&gt;&lt;a href=&quot;http://www.ams.org/notices/201407/rnoti-p706.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedrich Hirzebruch&lt;/a&gt;&lt;/em&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where his daughter Ulrike writes, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;' he often surprised us with beautiful simple problems, for example, with the trick of how to quickly multiply two numbers between 10 and 20 (he had learned this trick from his father, a math teacher)'&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="8660" LastActivityDate="2017-09-03T15:09:09.743" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12819" PostTypeId="2" ParentId="12794" CreationDate="2017-09-03T22:13:00.003" Score="0" Body="&lt;p&gt;This is a possibly arch answer but I think it warrants being documented here. The formal development of the topics of elementary algebra is embedded in, and historically the motivation for, any text in abstract algebra. And perhaps also the principal reason why high-school math educators should take such a course. The first few sections of a typical elementary algebra text are effectively presenting the &lt;a href=&quot;https://en.wikipedia.org/wiki/Field_%28mathematics%29#Definition&quot; rel=&quot;nofollow noreferrer&quot;&gt;field axioms&lt;/a&gt; for the real numbers, and then proceeding to topics that can be proven from those. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks to comments by beroal and DRF for making me think about that. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-09-19T23:13:41.367" LastActivityDate="2017-09-19T23:13:41.367" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12820" PostTypeId="2" ParentId="1649" CreationDate="2017-09-04T22:11:12.700" Score="1" Body="&lt;p&gt;It's not really mental math, but it is more useful that one might expect.&#xA;And it encourages students to see factoring and grouping possibilities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Evaluate $P(x) = x^3-7x^2+6x-8$ at $x = 3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$P(3) = (3)^3-7(3)^2+6(3)-8$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= 3(3^2)-7(3^2)+2(3^2)-8$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= (3-7+2)(9)-8$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= -2(9)-8 = -18-8=-26$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It comes up more often than one might expect in textbook exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would use it for $P(5)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$P(5) = (5)^3-7(5)^2+6(5)-8$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= 25(5)-35(5)+6(5)-8$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= (-10+6)(5)-8$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$= -20-8 = -28$$&lt;/p&gt;&#xA;" OwnerUserId="8658" LastActivityDate="2017-09-04T22:11:12.700" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12821" PostTypeId="2" ParentId="12812" CreationDate="2017-09-05T10:07:30.010" Score="18" Body="&lt;p&gt;&lt;em&gt;Preface.&lt;/em&gt; Birkhoff &amp;amp; Mac Lane's &lt;em&gt;Algebra&lt;/em&gt; is a brilliant book. I should probably spend some time with it again, actually. Also, I apologize for such a long response. I think too much about algebra pedagogy and textbooks. The short version is I think the book can be used for either undergraduates or graduates with some success, but I think it is less than ideal in both cases.&lt;/p&gt;&#xA;&lt;h1&gt;Preliminary Thoughts&lt;/h1&gt;&#xA;&lt;p&gt;In both undergraduate courses and graduate courses students' abilities and background are quite variable; that much has not changed in the past few decades. There are still many overqualified students and many underqualified students, and I can provide a constructive proof of that fact. Students of the first type would benefit from a more challenging text, whereas students of the second type would benefit from an easier one.&lt;/p&gt;&#xA;&lt;p&gt;I do not believe students are significantly less capable today than they were several decades ago, as you seem to suggest. In fact, they may be better when it comes to algebra. Algebra was almost unheard of at the undergraduate level not too long ago, whereas now an undergraduate program would be remiss not to require students take at least an introductory course covering the basic theories of groups, rings, and fields. On the other hand, calculus has been pushed into high school, and introductory courses are substantially easier and less rigorous than in the days of Apostol, Courant, and Spivak. In this way, I do not expect today's students are substantially better than past students, either. Also keep in mind that Dummit &amp;amp; Foote is fairly commonly used at the undergraduate level, and not just at top schools or in honors classes. I would not consider that presentation substantially easier than Birkhoff &amp;amp; Mac Lane, albeit perhaps a bit.&lt;/p&gt;&#xA;&lt;h1&gt;Potential Courses&lt;/h1&gt;&#xA;&lt;p&gt;I think the text &lt;em&gt;could&lt;/em&gt; be used for either undergraduate or graduate students with decent results, but I do not think it is the best choice as a primary text. Anyway, let me say a bit about my thinking for how this book would work for either audience.&lt;/p&gt;&#xA;&lt;h2&gt;Undergraduates&lt;/h2&gt;&#xA;&lt;p&gt;Birkhoff &amp;amp; Mac Lane is ill-suited for standard courses at schools with typical undergraduates; however, schools where the average undergraduate is exceptional could probably make do and similarly for honors classes at other schools.&lt;/p&gt;&#xA;&lt;p&gt;As Mac Lane did years ago, it is best to supplement the text with something easier, something intended for undergraduates. You mention Herstein and Vinberg, which are good; Artin or Gallian also work, for example. (I have come to like Gallian quite a bit since first giving it a real chance to see what my school teaches.)&lt;/p&gt;&#xA;&lt;p&gt;It is unlikely you will cover the entire book, either, even in a one year course. A typical first undergraduate course may cover group theory through the isomorphism theorems and the structure theorem for finite abelian groups,possibly including group actions and the Sylow theorems (c.f. Mac Lane Chapters I, II, VII), basic ring theory (c.f. Chapter III), and some field and vector space theory (c.f. Chapters VI, VII). As aforementioned, some category theory, especially an awareness of universal properties and basic definitions (&lt;em&gt;e.g.,&lt;/em&gt; categories, functors), can be good, and it seems to be becoming more common, too (c.f. Chapter IV). A second course may cover a bit more group theory (c.f. Chapter VII), more field theory and linear algebra (c.f. Chapters IX-XI), and Galois theory (c.f. Chapter XIII).&lt;/p&gt;&#xA;&lt;p&gt;Mac Lane has an enduring reputation as a great expositor. Indeed, I find his writing quite agreeable. That the basic objects of groups and rings are presented early on and come back to several times, rather than spending one-hundred pages only talking about groups, then changing to only talking about rings, should also help students see how algebraic structures are related; it also makes the beginning easier, and provides an opportunity for students to save themselves if they fall behind early on.&lt;/p&gt;&#xA;&lt;p&gt;The biggest problem is that Birkhoff &amp;amp; Mac Lane use categorical algebra all from Chapter III onward, which reflects the fact that they intended readers to be fairly mature. Although not enough to reconcile this problem, their exposition on categorical notions is quite clear, at least, albeit maybe not as good as some treatments that have since come. They do not avoid using universal properties, and they do not always bother to give students something concrete to hold on. Young students can handle abstraction. Some people mistakenly believe they cannot. Being able to handle abstraction does not mean students should not learn many concrete, basic examples, however, nor does it mean they should learn things at the most abstract possible level and be expected to figure out the less abstract consequences on their own. Maximum generality entailing sophisticated machinery can seem efficient in the abstract, but it rarely works so well in practice. You do not define a group as a groupoid with one object, at least not in an introductory course. (This is not to say Mac Lane and Birkhoff do this!)&lt;/p&gt;&#xA;&lt;p&gt;This is one of the primary reasons an undergraduate course ought to supplement this book. In fact, I think this book works better as the supplement.&lt;/p&gt;&#xA;&lt;p&gt;The truth, though, is that undergraduates are fairly unlikely to read their textbook. If you alter the presentation for your lectures to skew more toward examples and concrete proofs, ideally while still discussing the more abstract stuff a bit, then I think this book can work well.&lt;/p&gt;&#xA;&lt;p&gt;Or, maybe you have a truly prodigious bunch who have all seen algebra before or who are as mature as most incoming graduate students at good schools, in which case following the text more closely ough to work without much issue.&lt;/p&gt;&#xA;&lt;h3&gt;Suggestions&lt;/h3&gt;&#xA;&lt;p&gt;Again, I think Gallian's &lt;em&gt;Contemporary Abstract Algebra&lt;/em&gt; is better as a primary text. Motivation, examples, clear writing, reasonable exercises, they are all there. Artin (2e, &lt;em&gt;not&lt;/em&gt; 1e) is also good and emphasizes linear algebra and geometric intuition, which is good considering how often students will need things from linear algebra and how often they will find themselves ignorant of those things. Vinberg seems similar but more intense, so I imagine it would work well, too. Dummit &amp;amp; Foote is not good enough for ring theory and too encyclopedic to be used as the primary text for an undergraduate course, in my opinion; the group theory is &lt;em&gt;so&lt;/em&gt; good, though. Aluffi is really something special, but it is probably a bit too category theory-centric for students new to the subject in the sense that they will not have enough familiarity with the lower-level stuff. Judson is good all around. If you want an older book written by a master expositor and mathematician, then I think Herstein works better than Mac Lane. It has a lot of linear algebra, which is good, and it is not too hard, but it requires some work. It also introduces the student to modules, but it does not insist on working with modules instead of vector spaces whenever possible, which is probably good, because modules often serve to slightly confuse without adding anything more than a bit of generality.&lt;/p&gt;&#xA;&lt;p&gt;Some combination of the approaches of the above books ought to make an excellent undergraduate course.&lt;/p&gt;&#xA;&lt;h2&gt;Graduates&lt;/h2&gt;&#xA;&lt;p&gt;I agree that having no real homological algebra or representation theory is a major drawback of Birkhoff &amp;amp; Mac Lane. But, hey, at least there are lattices! (I am joking, if it is not clear.) I also think there should probably be more commutative algebra. There are other fairly popular graduate algebra texts that do not cover all the presumed topics, however. Hungerford's &lt;em&gt;Algebra&lt;/em&gt; is a pretty good book, but the author includes little about homological algebra, and the only time you see the word &amp;quot;representation&amp;quot; is when discussing category theory. Rotman's &lt;em&gt;Advanced Modern Algebra&lt;/em&gt; is pretty light on the Galois theory (but he gets points for having a chapter on algebras). Aluffi (Is &lt;em&gt;Chapter 0&lt;/em&gt; an undergraduate or graduate text, or is it neither?) also fails to include much Galois theory while completely ignoring commutative algebra and representation theory (but he gets points for recognizing theses omissions and planning a &lt;em&gt;Chapter 1&lt;/em&gt;). It is also worth noting Mac Lane &lt;em&gt;does&lt;/em&gt; cover multilinear algebra pretty well, which is frequently forgotten.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps not every course or book needs to cover all of these topics in detail, though, as much as the books by Lang and Dummit &amp;amp; Foote would have one believe. I am not sure. Certainly, algebraists expect everyone has seen some homological algebra and representation theory by the time they get their Ph.D., but plenty of people do not need either, both topics are often taught in separate courses, and so forth.&lt;/p&gt;&#xA;&lt;p&gt;I think you could cover the entire book, minus perhaps Chapter XIV on lattices, which are not typically emphasized and are partly historical, and possibly Chapter VIII, which is a bit odd, in my opinion. Supplementing this with another book to get some coverage of representation theory and homological algebra is probably ideal. Dummit &amp;amp; Foote or Lang are the obvious places to go, but I do not think those books' best parts cover these topics. Unfortunately, I am not sure what the best supplements are. I quite like Etingof's &lt;em&gt;Introduction to representation theory&lt;/em&gt;, which is available at his &lt;a href=&quot;http://math.mit.edu/%7Eetingof/replect.pdf&quot; rel=&quot;noreferrer&quot;&gt;site&lt;/a&gt;, but it takes two chapters before it gets to the case of finite groups, which is usually what algebra professors focus on in a first graduate course. I do not think using algebras is a problem, but three whole chapters is probably a bit much, so some would have to be cut. I think taking parts from Rotman's &lt;em&gt;An Introduction to Homological Algebra&lt;/em&gt; may work in a similar vein, albeit with much more cutting. The advantage of not falling back on a general reference as a supplement is you can use the really great parts of these excellent books, and the students know where to go for much more; plus, both of these are written at a reasonably low level. Adding in some applications may be good, too; I do not remember many being in Mac Lane. Robert Ash's &lt;em&gt;Algebra: The Basic Graduate Year&lt;/em&gt; includes some good applications to algebraic geometry and algebraic number theory, for example.&lt;/p&gt;&#xA;&lt;h3&gt;Suggestions&lt;/h3&gt;&#xA;&lt;p&gt;I am less sure what makes a really excellent graduate course in terms of extant texts. I love Lang, especially for things like Galois theory, but it is too hard, too fast, too big, too encyclopedic, and, dare I say, too modern for most graduate students. Certainly, one can survive it, but it is probably suboptimal for most. Lots of people cannot stand Lang's writing, too. Dummit and Foote is great for group theory, but it suffers in other parts and may also too big and encyclopedic. I often found it dry, too. Similarly, books like Cohn, Grillet, and Jacobson can be too advanced or too focused on being references. Rowen has been talked about a good bit, which is deserving for its extensive presentation on algebras and many applications, but I am not sure starting with modules is a good idea, for example.&lt;/p&gt;&#xA;&lt;p&gt;I think graduate courses should use category theory pretty openly. This is when students should come face-to-face with having to understand universality, or else. Yet, I also think a course should start with basic material. Perhaps a lecture reviewing elementary set theoretical notions, then cover some linear algebra (introducing basic category theory after seeing direct sums), then cover some ring theory, then plenty of group theory, then modules and advanced linear algebra, followed by field and Galois theory, representation theory (using algebras and specializing quickly to groups), commutative algebra (including some applications to algebraic geometry and the like), and finally homological algebra, with some advanced or extra topics at the end, if possible (&lt;em&gt;e.g.,&lt;/em&gt; a word on universal algebra or further category theory). Linear algebra is first, because students have the best intuition there; ring theory is next, because the examples and applications are nicer there than in groups and the quotient construction is easier. Logic suggests the standard groups, rings, fields, modules, vector spaces, &lt;em&gt;etc.&lt;/em&gt; sequence, but it does not demand it.&lt;/p&gt;&#xA;&lt;p&gt;The idea is to a.) cover all of the absolute basics of graduate algebra (&lt;em&gt;e.g.,&lt;/em&gt; basic modules, Jordan-Hölder), b.) cover many of the further topics (&lt;em&gt;e.g.,&lt;/em&gt; inverse limits, infinite Galois theory), c.) get some glimpses into truly modern topics, even research (&lt;em&gt;e.g.,&lt;/em&gt; deformation rings), and d.) see some applications (&lt;em&gt;e.g.,&lt;/em&gt; the fundamental group, valuations, schemes). Because there is quite a bit, and because students are assumed to be mature and to be getting used to &lt;em&gt;reading&lt;/em&gt; math, not just going to lecture, one can easily assign boring lemmas and the like as exercises or reading (but only truly easy ones as exercises!). The lecture ought to focus on the main ideas, examples, and results, and the book ought to clearly show what is important, very important, &lt;em&gt;etc.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Getting across the rich motivation, ideas, analogies, history, and so forth should be an explicit goal, but it will not always be possible. Ideally, the course book would have lots of this and the lecture somewhat less. There should also be specific portions of book and lecture that are intentionally plain, definition-theorem-proof type things, but where all the richness is introduced later down the line, as to not snub the students. This will get everyone to do the work they will need to do to read many papers, which are not uniformly well written and often take having a mature audience as an excuse not to motivate or show any work.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, there should be parts where the students are discovering things or work from ideas, trying to work out the formality. Developing the ability to take some vague notion and work out the details is key to graduate education. Again, eventually the students should be given this stuff to them. We must not risk them never learning it.&lt;/p&gt;&#xA;&lt;p&gt;I think these principles in lecture and text, along with hard work by the students, lots of student-student communication (including some forced working together), some student-professor communication (including some forced communication, &lt;em&gt;e.g.,&lt;/em&gt; an oral exam or a special, discussion-based lecture), good homework (comprising most of the grade and including a good number of routine exercises, perhaps even a few utterly obvious ones, a few intermediate ones, and a few truly hard ones, perhaps in many steps to resemble research or requiring some additional reading), and reasonable exams would make for a nearly perfect course. Theoretically, less-prepared students and more-prepared students would not only survive but also get from this something substantial. I think the needs of the graduate student are more complex, in a sense, than the undergraduate's needs, because many are less obvious than in the undergraduate case. For undergraduates, we obviously need to ensure they see examples, learn the basic theory, be able to write proofs, &lt;em&gt;etc.&lt;/em&gt;, and those are hard things to do, but it is easier to recognize what those things are and there are fewer of them than for graduate students.&lt;/p&gt;&#xA;&lt;h1&gt;Where it Belongs&lt;/h1&gt;&#xA;&lt;p&gt;This is one of many books I believe is best used as a supplement or for review/reference. At both the graduate and undergraduate level, it seems there are books that are better in parts or in whole. This book also does not match a clear (unorthodox) vision of how the subject should be taught, unlike, say, Aluffi or Adkins. Those are perhaps worth substantial time, even if they are somewhat too odd for most courses. It is a good, once standard book, but there are texts that better suit the needs of modern students.&lt;/p&gt;&#xA;" OwnerUserId="8665" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-09-05T10:15:17.567" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12822" PostTypeId="5" CreationDate="2017-09-05T17:38:47.353" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2017-09-05T17:38:47.353" LastActivityDate="2017-09-05T17:38:47.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12823" PostTypeId="4" CreationDate="2017-09-05T17:38:47.353" Score="0" Body="Questions about teaching Mathematics in English for students who are not yet fluent in the language" OwnerUserId="39" LastEditorUserId="39" LastEditDate="2017-09-06T19:10:23.123" LastActivityDate="2017-09-06T19:10:23.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12824" PostTypeId="1" CreationDate="2017-09-05T22:39:20.070" Score="13" ViewCount="636" Body="&lt;p&gt;Many of my students arrive in college believing that lines are (in some way) made out of points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They also believe that points have no length.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They want to know how a bunch of zero length points make a line of length 1?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit&#xA;I did ask this question in math.stackexchange, but because it is about teaching it was declared off topic and closed.  I changed to question there in hopes that if I understood it better, then I might be able to explain it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have asked my original question here in mathematics education in hopes of hearing teaching advice. I see it as a question about geometry and teaching geometry although it comes up in a class on ideas of infinity.&lt;/p&gt;&#xA;" OwnerUserId="8658" LastEditorUserId="8658" LastEditDate="2017-09-07T12:33:30.850" LastActivityDate="2017-12-21T19:46:24.017" Title="Can we explain to undergraduates how points make a line?" Tags="|undergraduate-education|geometry|infinity|" AnswerCount="6" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="12825" PostTypeId="2" ParentId="12271" CreationDate="2017-09-06T16:52:25.567" Score="5" Body="&lt;p&gt;If you see it filtering out students differentially by race or gender, you could discuss it in your department and college as an equity issue. I have taught trig dozens of times, and still do not have those memorized. (I can derive them easily.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math courses being used to filter students out is quite problematic, and equity concerns are one way we are making math courses fairer.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2017-09-15T15:14:35.213" LastActivityDate="2017-09-15T15:14:35.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12826" PostTypeId="1" CreationDate="2017-09-06T18:18:33.650" Score="6" ViewCount="345" Body="&lt;p&gt;It's clear that a polynomial has a &quot;degree&quot;. For instance our $$x^2 + 2x + 3$$ is a of degree two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can we apply the &quot;degree&quot; terminology to the coefficients? Here we have the coefficient tuple $[1\ 2\ 3]$. Can we call $2$ the &quot;degree one coefficient&quot; by synecdoche (meaning it's the coefficient of the degree one term of the polynomial)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If not, how else do we pinpoint a particular coefficient position? In particular, without depending on the order of the syntax?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if the formula is given non-canonically as $2x + 3 + x^2$, how can we fill in &quot;the ___ coefficient&quot; in a way which unambiguously refers to the $2$?&lt;/p&gt;&#xA;" OwnerUserId="6819" LastActivityDate="2017-09-17T09:17:52.920" Title="Terminology: degree of coefficient?" Tags="|terminology|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12827" PostTypeId="2" ParentId="12800" CreationDate="2017-09-06T19:50:19.157" Score="3" Body="&lt;p&gt;For more geometry, you might enjoy Paul Lockhart's book, &lt;em&gt;Measurement&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a very common sense approach to a number of math topics, you might enjoy Steven Strogatz's book, &lt;em&gt;The Joy of X&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a list of my favorite math books &lt;a href=&quot;http://mathmamawrites.blogspot.com/p/math-books.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;at my blog&lt;/a&gt;, and you might find others that are appealing to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, precalculus has a few main themes. One is understanding how equations and graphs go together, a very visual topic. Another is trigonometry, also a visual topic. For the first, play around with graphing different functions on &lt;a href=&quot;https://www.desmos.com/calculator&quot; rel=&quot;nofollow noreferrer&quot;&gt;desmos&lt;/a&gt;. For the second, there are &lt;a href=&quot;http://www.businessinsider.com/7-gifs-trigonometry-sine-cosine-2013-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;lots of cool gifs online&lt;/a&gt; that can help you see relationships.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-09-06T19:50:19.157" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12828" PostTypeId="2" ParentId="12800" CreationDate="2017-09-06T22:20:41.830" Score="8" Body="&lt;p&gt;There are a number of efforts to utilize the power of visualization&#xA;to convey various aspects of mathematics, often far from geometry.&#xA;Permit me to make four recommendations, three of which are repost details:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/85244/237&quot;&gt;(1: MSE post)&lt;/a&gt; Tristan Needham's&#xA;&lt;a href=&quot;http://usf.usfca.edu/vca//&quot; rel=&quot;noreferrer&quot;&gt;Visual Complex Analysis&lt;/a&gt;, 1999, Oxford Univ. Press.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/CXmuw.jpg&quot; alt=&quot;Needham cover&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/12807/511&quot;&gt;(2: MESE post)&lt;/a&gt; Nathan Carter's &lt;em&gt;Visualizing Group Theory&lt;/em&gt;, 2009&#xA;(&lt;a href=&quot;https://www.maa.org/press/ebooks/visual-group-theory&quot; rel=&quot;noreferrer&quot;&gt;MAA link&lt;/a&gt;).&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/0yl8B.jpg&quot; width=&quot;200&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/a/86775/6094&quot;&gt;(3: MO post)&lt;/a&gt; Bryant and Sangwin's &lt;a href=&quot;http://press.princeton.edu/TOCs/c8624.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;How Round Is Your Circle&lt;/em&gt;&lt;/a&gt;, 2008:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/3KQQx.jpg&quot; width=&quot;300&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(4)&#xA;Martin H. Weissman's&#xA;&lt;em&gt;An Illustrated Theory of Numbers&lt;/em&gt;, 2017,&#xA;(&lt;a href=&quot;http://bookstore.ams.org/mbk-105&quot; rel=&quot;noreferrer&quot;&gt;AMS link&lt;/a&gt;):&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/m03eg.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/m03eg.jpg&quot; alt=&quot;#th&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-09-06T22:20:41.830" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12829" PostTypeId="2" ParentId="11326" CreationDate="2017-09-07T01:44:56.543" Score="4" Body="&lt;p&gt;Yes, however there aren't lots of great resources.  For instance, I have a friend who taught a combinatorial game theory course that turned into research for many students, but I don't think he wrote anything up as a resource for &quot;how to do it&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I don't have any great insight into this, but there are a some papers in PRIMUS you may find useful if you can get access to them.  (There is apparently even a special issue on undergrad research, perhaps some of those articles touch on class-based ones.)  Note that they to some extent only indirectly touch on your question, though I think they all have some good models. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/10511970.2013.870264&quot; rel=&quot;nofollow noreferrer&quot;&gt;Experimental Mathematics for the First Year Student&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.tandfonline.com/doi/full/10.1080/10511970.2016.1143899?src=recsys&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching the Inquiry Process Through Experimental Mathematics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/10511970.2016.1183247&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cultivating a Culture of Undergraduate Research at a Public Comprehensive University&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/10511970.2013.764365&quot; rel=&quot;nofollow noreferrer&quot;&gt;A First-Year Course That Teaches Research Skills&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Full disclosure; I am currently an associate editor for this journal.  But I'm posting these because they're the only ones I'm aware of - perhaps CMJ or another MAA journal would have some good similar articles?  I'd love to see people post more answers to this important question.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-09-07T01:44:56.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12830" PostTypeId="2" ParentId="12826" CreationDate="2017-09-07T02:51:46.867" Score="2" Body="&lt;p&gt;In your example, I would call 2 the coefficient of the linear term, or the linear coefficient for short. Then 1 would be the quadratic coefficient. And 3 would be the constant.&lt;/p&gt;&#xA;" OwnerUserId="8670" LastActivityDate="2017-09-07T02:51:46.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12831" PostTypeId="1" CreationDate="2017-09-07T04:03:04.720" Score="9" ViewCount="231" Body="&lt;p&gt;In the spring I'll be teaching introductory operations research at an undergraduate liberal arts college in the US.  The target audience consists of applied math majors and students in our interdisciplinary mathematics/economics major.  The students are required to have taken a semester-long probability course and I am required to cover queueing theory in addition to various programming methods (linear, nonlinear, dynamic, e.g.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My background is in computational mathematics and so I'd like a book that makes early use of simulation and programming in assignments.  The instructor in the past has used &lt;a href=&quot;http://Winston-Operations%20Research-Applications%20and%20Algorithms&quot; rel=&quot;noreferrer&quot;&gt;Hillier-Lieberman&lt;/a&gt; and it covers all the right topics but doesn't get to simulation until Chapter 20. &lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="15884" LastEditDate="2021-07-17T04:45:32.570" LastActivityDate="2021-07-17T04:45:32.570" Title="Recommendation for an introductory operations research book" Tags="|textbooks|resource-request|interdisciplinary|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12832" PostTypeId="2" ParentId="12824" CreationDate="2017-09-07T08:38:58.103" Score="6" Body="&lt;p&gt;This is what I came up with thinking about your question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would start by exploring what it means to say that a line is 'made up of' points, because I think that is a really important thing that doesn't get taught in schools. Students know 'the equation of a line', but usually have no real concept of what that means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Admittedly, that will be somewhat tricky if they have not done an introduction to sets. I would perhaps try using plotting points:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Imagine using a computer to plot '$y=3x+2$' on the screen by looking at each pixel at a time. Pick one pixel. It has an $x$ coordinate and a $y$ coordinate. We check whether these points satisfy the equation. That is, if we put the two numbers into the equation '$y=3x+2$', do we end up with a statement that is true, or a statement that is false? If it's true, that means the pixel is on the line, and we colour it red. It it's false, that means it is not on the line, so we colour it blue. Once we've done this for all the pixels, we'll have our picture of the line.'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'd then look at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Zeno%27s_paradoxes&quot; rel=&quot;noreferrer&quot;&gt;Paradoxes of motion&lt;/a&gt;. The ideas go a bit beyond what most of the students will need, but the starting point is within reach, and it's reasonably easy for students to find out more if they are interested. If nothing else, it might make some of them think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd end by saying that really understanding the answer to the original question means studying measure theory. There are two reasons I see for pointing this out. Firstly, it assures the students that it's ok that they do not understand, because what they are asking is more advanced than the maths they have done. But it also points out that is isn't magic, or made up. There is a real answer out there, that they could reach if they really put their minds to it.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2017-09-07T08:38:58.103" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12833" PostTypeId="2" ParentId="12826" CreationDate="2017-09-07T12:45:10.287" Score="7" Body="&lt;p&gt;Personally, I would use &quot;the leading coefficient&quot; to refer to the coefficient of the highest degree term and &quot;the constant term&quot; to refer to the constant term.  Aside from that, I am not sure that the intermediate coefficients occur often enough in computation to require anything less verbose that &quot;the coefficient of the $n$-th degree term&quot; or &quot;the coefficient of $x^n$&quot; (which only costs two extra syllables over &quot;the $n$-the degree coefficient&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, I can't see any harm in using &quot;the $n$-th degree coefficient,&quot; as long as one is clear that this is non-standard terminology, which may momentarily confuse folk who have not been in your classroom.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2017-09-07T12:45:10.287" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12834" PostTypeId="2" ParentId="12831" CreationDate="2017-09-07T15:38:41.940" Score="3" Body="&lt;p&gt;If you are teaching a class for the first time, there is something to be said about using the same text as the last instructor. But if you are interested in trying something new, you might like &quot;Operations Research: A Practical Inroduction&quot; by Carter and Price. It covers all of the topics that you listed, but more succinctly than Hillier. It doesn't get into the theory behind the algorithms in any depth, which is a weakness from some points of view and a strength in others. If you are really going to hit upon all of those topics in a single semester you will necessarily have to be somewhat hand-wavy about the theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was originally going to say that the book is somewhat dated (my copy is from 2001 and no other editions are in print) but Amazon indicates that there will be a second edition coming out in just a couple of months.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer: I haven't used this as a textbook in Operations Research. The text that I have used several times is &quot;Deterministic Operations Research&quot; by Rader (which is one of my favorite textbooks). Your requirements rule that out. Still, I have found the Carter-Price text to be quite readable and have occasionally used it as a source of ideas when preparing for lectures.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2017-09-07T15:38:41.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12835" PostTypeId="2" ParentId="12271" CreationDate="2017-09-07T23:07:51.677" Score="3" Body="&lt;p&gt;Students cannot be forced to memorize trig identities unless they are tested orally.  Some of the students will and should learn to quickly derive some of them.&lt;br&gt;&#xA;The ability to derive identities quickly  as needed is a valuable skill.  But tell them honestly that they can answer the exam questions correctly by either memorization or derivation of identities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students who study calculus will benefit from recognizing parts of identities.  This is especially evident in learning integration techniques.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach mathematics.  I am not in any of the other professions that use trigonometry, so I don't know which particular identities are needed in those fields But I do know that many use trigonometry in their jobs.  I assume that they have need for some identities.&lt;/p&gt;&#xA;" OwnerUserId="8658" LastEditorUserId="8658" LastEditDate="2017-09-08T16:18:08.220" LastActivityDate="2017-09-08T16:18:08.220" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12836" PostTypeId="2" ParentId="12271" CreationDate="2017-09-07T23:56:57.853" Score="-4" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think there may be some differences in the right approach for a community college student versus for a math grad student.  We are talking about very different intrinsic skill levels.  Maybe the smarter students can derive more and the du...less smart students can derive less.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Not sure exactly how memorization is enforced.  Verbal recital?  If it is on problems, what is to stop from just fast deriving it (for the kids that can).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Myself, I found some identities not worth remembering (any of the squared ones, can pretty much be derived from sinsq plus cossq.  But I did find a few identities, that it just seemed better to memorize (maybe the sum of sins and cosine angles).  From that, you can get to double angles or differences of angles (just a different case of what you sum).  And I had seen the deriviation of the initial identity.  But the deriviation didn't stick as much, so I just committed the sum angles to memory. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I actually don't think it is the end of the world to memorize a page of trig identities.  It comes pretty natural if you do it in concert with heavy drill to use the identities.  And the student can kind of figure out per (3), which ones to memorize and which to re-derive during exams.  I guess, basically I am more to the derive bias than the memorize, but it is not an all or nothing.  And the key in either case, is heavy drill.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8673" LastActivityDate="2017-09-07T23:56:57.853" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12837" PostTypeId="2" ParentId="12271" CreationDate="2017-09-08T11:21:34.837" Score="2" Body="&lt;p&gt;I think that it's an odd request from the college, for several reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I cannot remember any situation where I've needed these formulas, except perhaps for one or two integration exercises. (And then it's again the question: How many people are still solving their integrals manually in practice?)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If I need them, I can find them in any math formula collection.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If I need them and if I don't have a math formula collection available, I can easily derive them from the identities $e^{i\theta} = \cos\theta + i \sin\theta$ and $\sin^2 \theta + \cos^2 \theta = 1$. (These are the two formulas that one should memorize!)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The formulas may be more important for some people working in physics or electrical engineering, but these people need complex numbers anyhow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I really believe that there are more important things that students should learn in mathematics, say, mathematical proof techniques.&lt;/p&gt;&#xA;" OwnerUserId="8472" LastEditorUserId="8472" LastEditDate="2017-09-08T15:02:48.847" LastActivityDate="2017-09-08T15:02:48.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12839" PostTypeId="2" ParentId="12796" CreationDate="2017-09-09T14:53:14.540" Score="7" Body="&lt;p&gt;I would suggest not focusing on the notation &quot;m&quot; and &quot;b&quot; and just explain why they are different in terms of their relationships to x and y. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What's in a name? That which we call a rose by any other name would&#xA;  smell as sweet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The slope has to interact with `x' because it is describing the relationship between how much y changes when x changes. If x changes by a lot, the slope passes this information on to y.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The intercept sits out there by itself because it is an absolute location on the line. If x changes by a lot, the y-intercept is still just the y-intercept. It doesn't care how x is changing.  &lt;/p&gt;&#xA;" OwnerUserId="6259" LastEditorUserId="6259" LastEditDate="2017-09-09T15:06:09.943" LastActivityDate="2017-09-09T15:06:09.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12840" PostTypeId="2" ParentId="12271" CreationDate="2017-09-10T02:48:48.923" Score="7" Body="&lt;p&gt;Knowledge is power. The more trigonometry you can derive without needing to consult Google etc the better you can do on tests and, more importantly, the faster you can follow later conversations in calculus and differential equations and so forth. Here I assume the student embraces the idea that mathematical knowledge does not exist in isolation. Every new idea should be contrasted and compared and fit into the tapestry of their existing knowledge. In other words, I encourage the radical idea that college students should be &lt;strong&gt;scholarly&lt;/strong&gt;. I know, this is at odds with the ever encroaching idea that we should make college as &lt;strong&gt;easy&lt;/strong&gt; as possible for customers (&lt;em&gt;students&lt;/em&gt;). Of course, if we allow students to look up every basic fact of trigonometry then it does make the course friendlier, but, when the course is about trigonometry it is (in my opinion) natural to expect the students to learn all the basic trig. identities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me discuss here a method to derive such identities, brevity being the soul of wit, draw your own conclusions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to see us teach how $e^{i\theta} = \cos \theta + i \sin \theta$ naturally encodes just about every trig. identity you run into as a mere consequence of algebra. For example,&#xA;$$ e^{ia}e^{ib} = e^{i(a+b)} $$&#xA;is tantamount to the adding angles formulas for sine and cosine. It's not hard to derive $\cos x = \frac{1}{2}(e^{ix}+e^{-ix})$ and $\sin x = \frac{1}{2i}(e^{ix}-e^{-ix})$ and use them to calculate things like:&#xA;\begin{align}&#xA;\sin(a)\cos(b) &amp;amp;= \frac{1}{2i}(e^{ia}-e^{-ia})\frac{1}{2}(e^{ib}+e^{-ib}) \\&#xA;               &amp;amp;= \frac{1}{2i}\frac{1}{2}(e^{i(a+b)}-e^{-i(a+b)}+e^{i(a-b)}-e^{-i(a-b)}) \\&#xA;               &amp;amp;= \frac{1}{2}\underbrace{\frac{1}{2i}(e^{i(a+b)}-e^{-i(a+b)})}_{\sin(a+b)} +&#xA;\frac{1}{2}\underbrace{\frac{1}{2i}(e^{i(a-b)}-e^{-i(a-b)})}_{\sin(a-b)}  &#xA;\end{align}&#xA;Therefore, $\sin(a)\cos(b) = \frac{1}{2}\sin(a+b)+ \frac{1}{2}\sin(a-b)$. Of course, this is just a token example, there is so much more you can do if you learn this way of thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My Modern Physics professor (Stephen Reynolds) mentioned that he would like to show us imaginary exponentials, but, he was either forbidden or discouraged (I forget, it's been a few years) from doing it, so, instead he'd just use real trigonometry. In retrospect, I really wish he had ignored whoever had given that advice. Imaginary exponentials make trigonometry into &lt;strong&gt;algebra&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that insightful application of the adding angles identities for sine and cosine can also derive very many things, but, it seems to me that technique is far more clever than the one I outline here. To use the adding angles formulas you have to come towards your goal in a sort-of sideways fashion. For example,&#xA;$$ \cos(x+x) = \cos x \cos x - \sin x \sin x = \cos^2 x-(1- \cos^2 x) $$&#xA;gives us a path to derive $\cos^2 x = \frac{1}{2}(1- \cos (2x))$. In contrast, to  derive this with imaginary exponentials I just begin with my target $\cos^2 (x)$, do algebra, and find $\cos^2 x = \frac{1}{2}(1- \cos (2x))$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, &lt;strong&gt;why are we afraid of imaginary exponentials?&lt;/strong&gt; I think the fear is real, but, is it rational? I wish we would all take Gauss' advice and get over this terminology &quot;imaginary&quot;. Accept $\mathbb{C}$ as an integral and important part of basic school mathematics. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-09-10T02:48:48.923" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12841" PostTypeId="2" ParentId="12831" CreationDate="2017-09-10T15:51:57.387" Score="0" Body="&lt;p&gt;As an undergraduate, I learned operations research out of Winston's book (cited below).  Operations research is not my field but, at the time, it seemed like a fairly solid text.  The exercises are (generally) illuminating, the text is readable, and it seems to cover the content that you require.  On the downside it is quite heavy (as in, it weighs several pounds), and expensive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Winston, Wayne L.&lt;/em&gt;, Operations research. Applications and algorithms., Andover, Hempshire: Duxbury Press. 1100 p. (1994). &lt;a href=&quot;https://zbmath.org/?q=an:0867.90079&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL0867.90079&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2017-09-10T15:51:57.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12842" PostTypeId="1" CreationDate="2017-09-10T23:15:11.627" Score="5" ViewCount="284" Body="&lt;p&gt;This semester is my first time to teach Calculus II in a classroom which contains around 25 students. (In fact, this is a discussion class.) I didn't have much experience about teaching. For instance, I thought I explained something quite clearly in the class, but the students still did not do very well in the weekly quiz. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have any useful suggestions for how to start with teaching and being a good teacher?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(By the way, I am not a native English speaker. My accent might confuse them, however, I wrote down most of what I said on the blackboard.)&lt;/p&gt;&#xA;" OwnerUserId="8680" LastEditorUserId="2083" LastEditDate="2017-09-11T15:03:59.600" LastActivityDate="2017-09-13T15:31:36.637" Title="How to become a good teacher in the classroom when first starting to teach?" Tags="|undergraduate-education|teaching|language-use|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12843" PostTypeId="2" ParentId="12842" CreationDate="2017-09-11T01:25:55.600" Score="6" Body="&lt;p&gt;I highly suggest reading the book &lt;em&gt;How to Teach Mathematics&lt;/em&gt; by Steven G. Krantz, published by the American Mathematical Society (AMS). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/1470425521&quot; rel=&quot;noreferrer&quot;&gt;https://www.amazon.com/How-Teach-Mathematics-Steven-Krantz/dp/1470425521&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/press/maa-reviews/how-to-teach-mathematics&quot; rel=&quot;noreferrer&quot;&gt;https://www.maa.org/press/maa-reviews/how-to-teach-mathematics&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-09-11T01:25:55.600" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12844" PostTypeId="2" ParentId="12842" CreationDate="2017-09-11T13:03:03.677" Score="2" Body="&lt;p&gt;Two things you need to realise: if you say things completely wrong, then nobody will have understood it, so if there are questions not one student answers during a quiz it most probably means that you did not explain/pronounce it correctly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as &quot;Which questions should I ask?&quot;, I can only advise you to go to another math teacher in your school, and ask him/her about &quot;Are the questions too easy/too difficult/ ...&quot;? In case you don't do that, you risk ending up in my situation: I had a very bad student who came to me after the exams and told me &quot;Thank you, sir! I never had such good grades for mathematics!!!&quot;, and this gave him the impression that even not paying attention during my class, he could still pass the exams, so he became completely unmanageable. Apparently my questions were far too easy.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-09-11T13:03:03.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12845" PostTypeId="1" CreationDate="2017-09-11T22:21:17.130" Score="31" ViewCount="11507" Body="&lt;p&gt;I've been either a student or an instructor in Precalculus or Calculus 1 at about 6 institutions now, and teaching the definition of even functions (where $f(-x) = f(x)$) and odd functions (where $f(-x) = -f(x)$) has been universal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But why? I don't see how these concepts are so useful that they need to be in the courses that are taught to everyone. I don't see how they lay the stage for understanding calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mean, seeing how it works graphically is nifty. But it seems like a disproportionate emphasis is placed on these classifications, in every curriculum I've seen.&lt;/p&gt;&#xA;" OwnerUserId="748" LastEditorUserId="64" LastEditDate="2017-09-16T01:38:12.387" LastActivityDate="2021-09-11T01:20:02.703" Title="Why do we teach even and odd functions?" Tags="|undergraduate-education|functions|precalculus|" AnswerCount="10" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12846" PostTypeId="2" ParentId="12845" CreationDate="2017-09-11T23:28:08.420" Score="42" Body="&lt;p&gt;One of the major themes of precalculus is what I call “connecting geometry to algebra”.  Being able to translate between an algebraic statement like $f(x)= f(-x)$, and the geometric statement that the graph of $f$ is symmetric about the vertical axis is a great instance of this.  This is just one more way to practice reinforcing function concepts, and the connection with graphs.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="-1" LastEditDate="2017-09-12T13:00:34.087" LastActivityDate="2017-09-12T13:00:34.087" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12847" PostTypeId="2" ParentId="12845" CreationDate="2017-09-11T23:36:32.020" Score="14" Body="&lt;p&gt;Learning to think about functions abstractly should be one goal in precalculus, and function symmetry helps. Also suppose we carefully protected a student from knowing anything about function symmetry. Upon learning about flux in vector calculus, would this student be able to quickly see that the flux of the vector field ${\bf F}(x,y,z)= y^2{\bf j}$ through the unit sphere is 0?&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2017-09-12T16:59:35.757" LastActivityDate="2017-09-12T16:59:35.757" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12848" PostTypeId="2" ParentId="12845" CreationDate="2017-09-12T02:05:31.980" Score="18" Body="&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/438691/how-does-knowing-a-function-as-even-or-odd-help-in-integration&quot;&gt;Here&lt;/a&gt; you can see that knowing if the function is even or odd can help you when you are integrating over the interval $[-a, a]$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can reduce really-hard-to-look-at integrals to zero just by knowing this. As an example, to calculate $E(Z)$ where $Z \sim N(0, 1)$, the standard normal distribution, you have:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\displaystyle E(z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}ze^{-z^2/2}dz$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which is immediately reduced to zero as the inner function in the integral is odd, and you are integrating over $(-\infty, \infty)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may not be too difficult, but knowing this property about odd functions lets you generalize this to all the odd moments of the standard normal: $E(Z), E(Z^3), E(Z^5), ...$ all of them equal zero.&lt;/p&gt;&#xA;" OwnerUserId="8687" LastEditorUserId="8687" LastEditDate="2017-09-14T11:54:37.760" LastActivityDate="2017-09-14T11:54:37.760" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12849" PostTypeId="2" ParentId="11326" CreationDate="2017-09-12T04:40:55.240" Score="2" Body="&lt;p&gt;For years Bela Bajnok has been teaching such a course at Gettysburg College:  go &lt;a href=&quot;http://www.d.umn.edu/~jgallian/PURMweb/Bela.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; for his description of the (series of) course(s) he teaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A new MAA-sponsored program called &lt;a href=&quot;https://www.maa.org/pic-math&quot; rel=&quot;nofollow noreferrer&quot;&gt;PIC Math&lt;/a&gt; might also be of interest to you.  Here the research problems come from industry.  A list of faculty who have been involved in the teaching of such a course is &lt;a href=&quot;https://www.maa.org/node/675534/&quot; rel=&quot;nofollow noreferrer&quot;&gt;available&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-09-12T04:40:55.240" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12850" PostTypeId="1" CreationDate="2017-09-12T06:56:35.677" Score="2" ViewCount="209" Body="&lt;p&gt;The way that naïve set theory is usually presented in undergraduate education is via very concrete examples of sets, often involving non-mathematical elements. When power sets are treated, having a set as an element of some other set is just another new and exciting [YMMV] thing to consider. I saw a recent example that discussed how $\{\emptyset\}$ is both an element &lt;em&gt;and&lt;/em&gt; a subset of $\{\emptyset,\{\emptyset\}\}$. To a traditionally trained set theorist this is perfectly natural. However, there are &lt;a href=&quot;https://golem.ph.utexas.edu/category/2012/12/rethinking_set_theory.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;other axiomatisations of sets&lt;/a&gt; that do not allow sets to be elements of sets. In this setup the power set of $X$, which &lt;em&gt;surely&lt;/em&gt; must have subsets of $X$ as elements, seems to be a little tricky to think of. But not really: one just has to realise that elements of the power set &lt;em&gt;code&lt;/em&gt;, or are &lt;em&gt;labels for&lt;/em&gt;, or &lt;em&gt;name&lt;/em&gt; subsets of $X$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering whether there has been any treatment at the level of introductory (and/or naïve) set theory that takes this tack, or if people have had any experience using such a trick in introducing the power set.&lt;/p&gt;&#xA;" OwnerUserId="1426" LastEditorUserId="1426" LastEditDate="2017-09-12T07:19:00.010" LastActivityDate="2017-09-12T07:19:00.010" Title="Introduction of the power set as a collection of *labels* or *names* for subsets" Tags="|undergraduate-education|mathematical-pedagogy|reference-request|discrete-math|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12851" PostTypeId="2" ParentId="12845" CreationDate="2017-09-12T09:29:22.257" Score="16" Body="&lt;p&gt;Besides applicability in topics like integration and Fourier analysis, it also &lt;strong&gt;connects algebra to calculus&lt;/strong&gt; at least in the way that multiplication of even/odd &lt;em&gt;functions&lt;/em&gt; behaves like addition even/odd &lt;em&gt;numbers&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Multiplying two even functions gives an even function.&lt;/li&gt;&#xA;&lt;li&gt;Multiplying two odd functions gives an even function, too.&lt;/li&gt;&#xA;&lt;li&gt;Multiplying an even&#xA;and an odd function gives an odd function.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also, you can decompose every function as a sum of an even and an odd function as&#xA;$$&#xA;f(x) = \frac{f(x)+f(-x)}2 + \frac{f(x)-f(-x)}2&#xA;$$&#xA;(which is a very useful concept an the same as writing a matrix as the sum of a symmetric and an antisymmetric one as $A = \tfrac12(A+A^T) + \tfrac12(A-A^T)$).&lt;/p&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="3398" LastEditDate="2017-09-12T11:47:57.853" LastActivityDate="2017-09-12T11:47:57.853" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12852" PostTypeId="2" ParentId="12845" CreationDate="2017-09-12T12:41:03.317" Score="12" Body="&lt;p&gt;Even and odd parity are probably the simplest examples of function symmetries. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In applied mathematics, the general observation of function symmetries allows to simplify calculations (as stated by others) and to produce more meaningful graphs. In physics, symmetrical parts of a function are sometimes associated to different physical phenomena.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If you have a function which is invariant under inversion (a more complex symmetry), that is, &#xA;$$f(1/x) = f(x),\ x&amp;gt;0,$$&#xA;it is better to plot the function by using a logarithmic $x$ axis because $f(a^x)$ is even.&lt;/li&gt;&#xA;&lt;li&gt;The example given by &lt;a href=&quot;https://matheducators.stackexchange.com/a/12851/5367&quot;&gt;Dirk&lt;/a&gt; on the decomposition of a matrix in the sum of a symmetric and antisymmetric part is useful in circuit theory: the symmetric part of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Impedance_parameters&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;impedance&lt;/em&gt;&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Admittance_parameters&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;admittance&lt;/em&gt;&lt;/a&gt; matrices is associated to the average power dissipated by an electrical network when subjected to sinusoidal excitations.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5367" LastActivityDate="2017-09-12T12:41:03.317" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12853" PostTypeId="2" ParentId="12845" CreationDate="2017-09-12T16:49:59.973" Score="0" Body="&lt;p&gt;I agree that it is disproportionate.  Ben Crowell (in comments) says that is helpful for applications.  Well, I have a very strong background in natural sciences and engineering, and it is not a big building block.  Very few derivations in physics rely on it for instance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know this is (-1) controversial, but I think the appeal comes from the ease of doing tricky questions based on it.  Sort of ETS style questions.  A similar thing would be the vertical angle theorem (or whatever it is called, where a line crosses a couple parallel lines) is big on the SAT.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(-2) I also think that many pure math people prefer things that are definitional and classificational.  And that in preference to being able to do multistep mechanical problems like a maximization or power series expansion.&lt;/p&gt;&#xA;" OwnerUserId="8697" LastActivityDate="2017-09-12T16:49:59.973" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12854" PostTypeId="2" ParentId="12842" CreationDate="2017-09-12T17:01:35.153" Score="1" Body="&lt;p&gt;I would do more drill, recitation, games, etc.  IOW practice.  Students have limited time and work ethic.  If you use the time to practice rather than sage on the stage ing, than they will actually get some traction with the problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say this in general, but even more so for someone who  is a first time instructor and with a foreign accent.  Lecture can be torture.  You can probably much quicker move to a practice model than to being a fluent enjoyable lecturer.  And even with the most fluent and entertaining lectures, I still learn more from practice than from listening and watching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Watch the movie Stand and Deliver.&lt;/p&gt;&#xA;" OwnerUserId="8697" LastActivityDate="2017-09-12T17:01:35.153" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12855" PostTypeId="2" ParentId="2164" CreationDate="2017-09-12T17:58:54.900" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The exercises and answers ARE the most important part of the book.  Math is a practical, almost physical, art.  You learn by doing.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The value of the exposition comes mostly in use to help with doing the problems.  The only case, where it makes sense for the students to properly work through the exposition is if they are pre-reading the material before lecture.  I am highly supportive of that.  But if they aren't, it's not that important to read it afterwards.  Better to dive into the problems.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;2.5.  If you want to encourage pre-reading consider to do very simple one question quizzes on the reading assignment. 5 minutes at beginning of class and have the students exchange papers to grade them.  The idea is to promote the drill, but do so at minimum complication factor.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;The reference material is the least important aspect.  (Least means lowest of the three, not zero.)  This is because even for review, the best method is to work problems.  For those students who use the technique of preparing for exams by writing a formula sheet (not for use during the exam but for prep), than the reference aspect has some limited value.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Extra credit:  The one practical tip I have for incorporating the textbook (in a useful manner) into the course is to TEACH FROM THE BOOK.  And tell the students ahead of time you are doing that.  And test and quiz from the book.  (Not the exact problems, but completely analogous ones.)  Many teachers instead want to be &quot;enriched&quot; and assume the students master the book and then the give some alternate perspective (usually right out of another book, I know you sneaky teachers!)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this only serves a TINY fraction of students who are already mastering the given text (and this question has as predicate that we know that doesn't happen enough) and want more deep info.  For the vast amount of students, it confuses them rather than helps them to have the twin perspectives along with the invariable differences in notation and the like.  And I understand where the students are coming from then, when they say screw the book and just follow lecture notes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simple hint:  Do a few practice problems right out of the book in class.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simpler hint:  Pick a user friendly book with all the answers and very simple exposition.  There are WAY too many professors who write books based on comments from other professors rather than to help the students.&lt;/p&gt;&#xA;" OwnerUserId="8697" LastActivityDate="2017-09-12T17:58:54.900" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12856" PostTypeId="2" ParentId="12845" CreationDate="2017-09-13T04:59:52.567" Score="1" Body="&lt;p&gt;To express functions as sum of even and odd functions&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f(x) = f_{even}(x) + f_{odd}(x)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And look at the properties of their graphs.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teach Fourier transform and say it becomes easy to compute when $f(x)$ is an odd function. And the integral becomes zero. &lt;/p&gt;&#xA;" OwnerUserId="8704" LastEditorUserId="-1" LastEditDate="2017-09-13T19:17:32.877" LastActivityDate="2017-09-13T19:17:32.877" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12857" PostTypeId="2" ParentId="12824" CreationDate="2017-09-13T08:35:33.187" Score="3" Body="&lt;p&gt;I agree with the premise, lines are (in some way) made out of points, and points have no length.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If, restricting ourselves to a straight line, we can consider these as subsets of the real numbers $\Bbb R$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A line is a subset of this with the property, if $a, b \in \Bbb L$ then any real number between $a, b$ is in $\Bbb L$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calling elements of a line points, we can define the length of the line between the two points (on a straight line) say $a, b$  as the distance between them, and this distance is $\left|b - a\right|$. The length here being a property of two points, on a line, rather than the sum of a property of all individual points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We could mention subsets of the real numbers, even with uncountably many elements, where ways to measure a length gives zero, such as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cantor_set&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cantor set&lt;/a&gt;, and these do not contain line segments.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;They want to know how a bunch of zero length points make a line of length 1?  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I find this tough. We could consider a line $[0,1]$ of length $1$ as divided into $n$ segments of length $\frac{1}{n}$.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The total length of these segments is $\frac{n}{n} = 1$ while as $n \rightarrow \infty$ the length of the segments $\rightarrow 0$&lt;/p&gt;&#xA;" OwnerUserId="8660" LastActivityDate="2017-09-13T08:35:33.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12858" PostTypeId="2" ParentId="12842" CreationDate="2017-09-13T15:31:36.637" Score="6" Body="&lt;p&gt;I'll give you the answer I got from several professors when I started teaching:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;talk slow and pause&quot; and &quot;write what you say and say what you write&quot;&lt;/p&gt;&#xA;" OwnerUserId="8559" LastActivityDate="2017-09-13T15:31:36.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12859" PostTypeId="2" ParentId="12296" CreationDate="2017-09-13T20:11:30.193" Score="0" Body="&lt;p&gt;I found a nice group of general stat final exams at UNC Charlotte, they may be just what you are looking for!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://math.uncc.edu/undergraduate-programs/course-outlines-and-past-common-final-exams/stat-1222-introduction-statistics&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://math.uncc.edu/undergraduate-programs/course-outlines-and-past-common-final-exams/stat-1222-introduction-statistics&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8708" LastActivityDate="2017-09-13T20:11:30.193" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12861" PostTypeId="2" ParentId="12296" CreationDate="2017-09-13T22:35:15.577" Score="1" Body="&lt;p&gt;Some of the &lt;a href=&quot;http://www.beanactuary.org/exams/preliminary/?fa=preliminary-exams&quot; rel=&quot;nofollow noreferrer&quot;&gt;actuarial exams&lt;/a&gt; emphasize statistics and probability, specifically SOA Exam P and CAS exam S. This might be too broad or too specific, but they are indeed standardized exams which test more-advanced statistical knowledge.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2017-09-13T22:35:15.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12862" PostTypeId="2" ParentId="12824" CreationDate="2017-09-14T01:12:26.573" Score="1" Body="&lt;p&gt;This question alludes to a deep developmental issue. Your students are starting to think hard about the continuum, even if they do not realize it. A countable collection of points (for example rational numbers) in [0,1] has Lebesgue measure zero. Yet the collection of all points in [0,1] has Lebesgue measure 1. But of course the notion of Lebesgue measure was not fully developed until the early 20th century. Perhaps an undergraduate would get to this understanding in an upper division analysis course, but most do not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another perspective is from projective plane geometry. Here points and lines are both undefined terms, and they are on the same footing. By projective duality, ranges of points (points incident with a given line) are logically the same as pencil of lines (lines incident with a given point). In other words, instead of thinking as points as fundamental and deriving lines as ranges of points which is what we tend to do, we could take lines as the fundamental objects and ``derive'' points by considering pencils of lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I point out these two notions not because they will be of immediate help to younger undergraduate mathematics students, but to indicate to the OP just how deep the question is.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2017-09-14T01:12:26.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12863" PostTypeId="2" ParentId="12824" CreationDate="2017-09-14T22:49:36.103" Score="2" Body="&lt;p&gt;Often the fact that words have connotations in common parlance which are at odds with the more technical way these words are used within mathematics causes tensions in getting ideas across to students. But the notion of point and line are especially subtle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid gives definitions for the words point and line but by the time Hilbert and others came along, point and line were left as undefined in axiom systems that were developed to understand the Euclidean plane and other kinds of geometry. So one can construct &quot;models&quot; that satisfy a collection of axioms that involve point and line as undefined terms and give an interpretation to these words in the model. If one picks any of Hilbert's axioms, say A, one can construct a model where the interpretations of point and line satisfy each of the axioms other than A and where A fails to hold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To help show what can go on I like to show students finite geometries. Some students get the &quot;point&quot; here but most just don't accept the fact that the &quot;lines&quot; of these finite geometries are &quot;really&quot; lines! You might find this brief introduction to finite geometries of interest:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/samplings/feature-column/fcarc-finitegeometries&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.ams.org/samplings/feature-column/fcarc-finitegeometries&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then there is the model where points are pairs (x,y) where x and y are rational numbers and lines are linear equations with rational coefficients - a geometry which is not the Euclidean plane but has lines with lots of points between any pair of points, but lots of &quot;holes,&quot; too.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2017-09-14T22:49:36.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12864" PostTypeId="2" ParentId="12845" CreationDate="2017-09-14T23:14:16.300" Score="-2" Body="&lt;p&gt;Learning mathematics isn't always about applying things to further mathematics - it can also be about actually applying that knowledge to other areas. So let me bring you a completely different answer to your question, coming from the background of someone that does programming and some web design.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can easily apply even/odd functions to things like tables: one row has a dark background, the other has a lighter background. A simple enough requirement, that a good programmer might understand not only on the surface, but more deeply such as yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I apologize if this answer is not relevant to the mathematics community directly, but I felt the need (seeing this question in a suggested list) to remind people that math isn't just about math ;)&lt;/p&gt;&#xA;" OwnerUserId="8715" LastActivityDate="2017-09-14T23:14:16.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12865" PostTypeId="1" CreationDate="2017-09-14T23:21:46.917" Score="3" ViewCount="72" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: Do you have any suggestions on professional software that implements project allocation with preferences algorithms (short SAP-P)? Do you know any schools where they use professional software to allocate hours to TAs?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Our particular problem is assigning courses to 250 teaching assistants based on their preferences on courses and hours range (eg. 100-150 hours for semester).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Operating system:&lt;/strong&gt; any will do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Budget:&lt;/strong&gt; price can be quite flexible because as of now multiple people are hired to do this task. Ideally we want to hear what software brands are generally available that solve this problem and then think about price.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Features:&lt;/strong&gt; The input would be a a)list of prioritized courses (eg. size 10) from 250 students (TAs) and b)the preferred work hours for the year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only important constraints are how many TAs and hours are needed for each course. So if a course requires more students, then some of the TAs' preferences will be ignored.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this software will at least get the job done. But ideally there are other things it can do:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a)It can use the TAs' course history, to correct for giving them courses in the past that were not high in their list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b)It can use the TAs' teaching score from student reviews, to match them with the courses they actually did well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you for your help&lt;/p&gt;&#xA;" OwnerUserId="8714" LastEditorUserId="8714" LastEditDate="2017-09-15T16:56:52.417" LastActivityDate="2017-09-15T16:56:52.417" Title="Assigning TA hours: Professional software implementing Student-project allocation with preferences algorithms" Tags="|software|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12866" PostTypeId="2" ParentId="12845" CreationDate="2017-09-16T00:46:38.200" Score="0" Body="&lt;p&gt;Well symmetry helps investigate functions. It's the first things I need to know about the function I deal with. It helps as noted, evaluating integrals that are otherwise difficult to integrate. And it helps visualizing functions. I was a bit disapointed by the few examples given in class ass aplications of these concepts.&lt;/p&gt;&#xA;" OwnerUserId="8717" LastActivityDate="2017-09-16T00:46:38.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12867" PostTypeId="2" ParentId="12845" CreationDate="2017-09-16T01:48:00.480" Score="1" Body="&lt;p&gt;As with many items in math that have a name, if we didn't name it, it would still exist. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When manipulating and equation that contained Cos(-x) we are able to observe that this term is the same as Cos(x). On the other hand, when we see Sin(-x) and want to manipulate to form a positive argument, it's equal to -Sin(x). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't feel that we spend that much time on the concept. It's introduced in algebra, a parabola possibly being even, a third degree possibly odd, and then again in Trig with the examples I gave. Your experience may be different, more focus than I've observed. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-09-16T01:48:00.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12868" PostTypeId="1" AcceptedAnswerId="12872" CreationDate="2017-09-16T02:14:51.540" Score="5" ViewCount="1383" Body="&lt;p&gt;For instance in the following problem:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;   _____&#xA;48)4368  &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;To determine an initial &lt;code&gt;9&lt;/code&gt; for the first number in the Quotient, you have to look at how many times does &lt;code&gt;48&lt;/code&gt; go into &lt;code&gt;436&lt;/code&gt; because; &lt;code&gt;48&lt;/code&gt; doesn't go into &lt;code&gt;4&lt;/code&gt; and it doesn't go into &lt;code&gt;43&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then one might start trying to multiply &lt;code&gt;48&lt;/code&gt; by increments of &lt;code&gt;1&lt;/code&gt; or increments of &lt;code&gt;5&lt;/code&gt; by hand until they find two numbers multiplied together to determine which number being multiplied gets one closest to the number without going over it; (this is awfully tedious and time consuming), for example:&#xA; - 48*1 = 48, 48*2=96, 48*5=240 ... 48*9=432&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The inefficiency of this method (by incrementing by one) can be displayed in a graph and spreadsheet (though when required to do these calculations by hand no such oversight is available):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FQWfl.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FQWfl.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now one does not, in this example get hung up somewhere in the middle, since &lt;code&gt;9&lt;/code&gt; as the first digit in the quotient is the end of the line, but if it was not, and the correct first digit of the quotient was something like &lt;code&gt;6&lt;/code&gt;, one would have to at the very least do the calculation for the first digit of the quotient &lt;code&gt;7&lt;/code&gt; and also do the calculation to determine the first digit of &lt;code&gt;6&lt;/code&gt;, to determine that &lt;code&gt;6&lt;/code&gt; is the correct number to use to reach the nearest result, since &lt;code&gt;7&lt;/code&gt; multiplied by the divisor would be greater than the result.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;   __9_&#xA;48)4368  &#xA;  -432&#xA;  -----&#xA;     48&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;etc...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What tools / methods can one use to estimate which two digits you need to try this with...without guessing / taking a shot in the dark, or calculating each and every possibility linearly?&lt;/p&gt;&#xA;" OwnerUserId="7731" LastEditorUserId="7731" LastEditDate="2017-09-16T02:36:15.710" LastActivityDate="2022-12-06T04:54:14.777" Title="Determining the first digit of the Quotient using hand long division efficiently?" Tags="|intuition|calculations|division|mental-math|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12869" PostTypeId="1" CreationDate="2017-09-15T19:55:19.757" Score="6" ViewCount="408" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; the equation below, writing the universe set and the solution set:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{a}{b}+\frac{b}{a}-\bigg(\frac{a^2}{bx}+\frac{b^2}{ax}\bigg)=1$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Let's choose &lt;span class=&quot;math-container&quot;&gt;$U=\{x\in \mathbb R;x\neq 0\}$&lt;/span&gt; as the universe set.&lt;/p&gt;&#xA;&lt;p&gt;Using this universe set, the equation above is equivalent to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(a^2+b^2-ab)x=a^3+b^3$$&lt;/span&gt; which is the same as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(a^2+b^2-ab)x=(a+b)(a^2+b^2-ab)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Therefore&lt;/p&gt;&#xA;&lt;p&gt;if &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2-ab=0$&lt;/span&gt;, the solution set is &lt;span class=&quot;math-container&quot;&gt;$S=U$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If if &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2-ab\neq0$&lt;/span&gt;, the solution set is &lt;span class=&quot;math-container&quot;&gt;$S=\{a+b\}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I was teaching my niece in this way, and she said that her teacher said this question has only one solution and it was simpler than mine (she didn't remember very well how he solved).&lt;/p&gt;&#xA;&lt;p&gt;Maybe, the teacher didn't mention that &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2-ab$&lt;/span&gt; could be zero, since the school teachers in my home country aren't very formal and sometimes they didn't know how to solve correctly even simple questions.&lt;/p&gt;&#xA;&lt;p&gt;In my opinion if he solved like this, he is wrong. So my question is how should I teach her? forgetting or pointing out these &amp;quot;small&amp;quot; details? am I too formal and rigorous?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Remark&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The real problem is I'm afraid she gets boring and thinks that mathematics is very complicated. Besides that maybe some other teacher or student gives her a simpler wrong solution and finds me a bad teacher who makes things worse. (she doesn't have the mathematical maturity to know which solution is right or wrong).&lt;/p&gt;&#xA;" OwnerUserId="5236" OwnerDisplayName="user26832" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-09-19T17:58:35.910" Title="Am I too formal teaching this middle school student?" Tags="|middle-school|" AnswerCount="4" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="12870" PostTypeId="2" ParentId="12869" CreationDate="2017-09-15T20:49:26.507" Score="5" Body="&lt;p&gt;It is not incorrect to think through all of the cases, and it is not unreasonable to make even a middle schooler think through the possibilities, but you do have to be careful about it, and keep track of the assumptions that have been made.  If you are working over the real numbers, then the set of all possible solutions is $\mathscr{U} = \mathbb{R}\setminus \{0\}$, as you noted.  However, &lt;em&gt;you&lt;/em&gt; seem to be choosing this universe---it is not made explicit in the problem.  If you were to choose a different universal set (say $\mathscr{U} = \mathbb{C}\setminus \{0\}$, or $\mathscr{U} = \mathbb{N}$), then the results would be different).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, once you have fixed your universe, you are correct that you need to consider the two cases:  either (1) $a^2 - ab + b^2 \ne 0$, or (2) $a^2 - ab + b^2 = 0$.  Your analysis of case (1) is essentially correct, but your analysis of case (2) is incomplete.  To wit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $a^2 - ab + b^2 = 0$, then we have&#xA;\begin{align} &amp;amp;a^2 - ab + b^2 = \left(a-\frac{b}{2}\right)^2 + \frac{3}{4} b^2 = 0 \\&#xA;&amp;amp;\qquad \implies \left(a - \frac{b}{4}\right)^2 = -\frac{3}{4}b^2 \\&#xA;&amp;amp;\qquad \implies a - \frac{b}{4} = \sqrt{-\frac{3}{4}b^2}.&#xA;\end{align}&#xA;If we are assuming that $a$ and $b$ are real (which is something that we have not explicitly done at any point, but our universe of solutions is real, so we have probably done this implicitly), then  $b^2$ is nonnegative, and so the quantity on the right will either be pure imaginary, or zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the expression on the right is zero, then $a=b=0$.  But if this is so, then the original equation is undefined.  There are two possible conclusions:  either we live in a universe where such a choice is not possible (i.e. we impose the additional hypothesis that at least one of $a$ and $b$ is nonzero), or the equation is vacuously solved by all possible values of $x$, since a false statement (i.e. $1 = \text{something undefined}$) implies any conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if the expression on the right is pure imaginary, then we run into other problems, as it would imply that at least one of $a$ and $b$ must be a non-real complex number.  Do we allow $a$ and $b$ to be complex, or must they be real?  We've made no assumption about what $a$ and $b$ are, but if we want to start with the assumption that the universe of possible $x$ is real, then I think it is safe to assume that $a$ and $b$ are real.  In this case, we never conclude that $S = \mathscr{U}$, as we cannot find $a,b\in\mathbb{R}$ such that $a^2-ab+b^2=0$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Since this is now on the Math Ed SE, let me expand on the first paragraph a bit.  Personally, I think that it is important to be precise when doing mathematics, and doubly important when teaching mathematics.  I don't think that it is appropriate to tell little children that you &lt;em&gt;can't&lt;/em&gt; subtract 5 from 3 (you can, you just get a negative number; better to say &quot;We don't know how yet.&quot;), nor do I think that it is okay to say that $x^2 + 2 = 0$ has no solutions (it does, it just doesn't have any &lt;em&gt;real&lt;/em&gt; solutions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, I think it is wholly appropriate to work through the problem presented above in detail with a middle schooler, particularly if that middle schooler already has enough set theory under their belt to be worried about a universal set of possible solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, I would not present the solution $S = \mathscr{U}$ without also being very careful about determining &lt;em&gt;when&lt;/em&gt; this happens.  You can't stop at $a^2 - ab + b^2 = 0$.  You have to explore what it would mean for this equation to be true, and note that it typically will lead to some kind of contradiction to the assumptions that have already been made.&lt;/p&gt;&#xA;" OwnerUserId="8571" OwnerDisplayName="Xander Henderson" LastEditorUserId="8571" LastEditDate="2017-09-16T13:57:44.180" LastActivityDate="2017-09-16T13:57:44.180" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12871" PostTypeId="2" ParentId="12869" CreationDate="2017-09-16T02:32:13.533" Score="1" Body="&lt;p&gt;$$a^2-ab+b^2=\frac{3}{4}a^2+\left(\frac{a}{2}-b\right)^2.$$&#xA;Thus, if $a^2-ab+b^2=0$ then $a=0$ and $\frac{b}{2}-a=0$, which is impossible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, $a^2-ab+b^2\neq0$ and we can divide both sides by $a^2-ab+b^2$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an easier way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since our equation has unique root (if it has root) and for $x=a+b$ we obtain&#xA;$$\frac{a}{b}+\frac{b}{a}-\frac{\frac{a^2}{b}+\frac{b^2}{a}}{a+b}-1=\frac{a}{b}+\frac{b}{a}-\frac{\frac{a^3+b^3}{ab}}{a+b}-1=$$&#xA;$$=\frac{a}{b}+\frac{b}{a}-\frac{a^2-ab+b^2}{ab}-1=0,$$&#xA;we see that $a+b$ is the root.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the domain gives that $x\neq0$, which gives the answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $a=0$ or $b=0$ or $a=-b$ our equation has no solutions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $a\neq0$, $b\neq0$ and $a\neq-b$ then $\{a+b\}$.&lt;/p&gt;&#xA;" OwnerDisplayName="Michael Rozenberg" LastActivityDate="2017-09-16T02:32:13.533" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12872" PostTypeId="2" ParentId="12868" CreationDate="2017-09-16T20:02:24.057" Score="8" Body="&lt;p&gt;The awkwardness of &quot;guessing&quot; in the division algorithm is an artifact of the base-ten representation of numbers. If you represent in binary, then your only possible &quot;guess&quot; is 1. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In binary, your problem is to divide the six-bit number 110000  into the thirteen-bit number 1000100010000. Scanning the bits of the second number(the dividend) from left to right, there is no guessing needed when deciding where to put the leading 1 when forming the quotient above the dividend.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2017-09-16T20:02:24.057" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12873" PostTypeId="2" ParentId="12826" CreationDate="2017-09-17T09:17:52.920" Score="2" Body="&lt;p&gt;In the following textbook, terms “$n$-th coefficient” and “$n$-th term” are used. A quote.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This ring is usually denoted by $R[[X]]$ and its elements are called&#xA;  &lt;strong&gt;formal series in the indeterminate&lt;/strong&gt; $X$ &lt;strong&gt;with coefficients in&lt;/strong&gt; $R$. If $f=(f_n)_{n\in\mathbb{N}}\in R[[X]]$, for every&#xA;  $n\in\mathbb{N}$, $f_n$ is called $n$&lt;strong&gt;-th&lt;/strong&gt; &lt;strong&gt;coefficient&lt;/strong&gt;&#xA;  and $f_n X^n$ the $n$&lt;strong&gt;-th&lt;/strong&gt; &lt;strong&gt;term&lt;/strong&gt; of the formal power series $f$. In&#xA;  particular $f_0 = f_0 X^0$ is called its &lt;strong&gt;constant term&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Menini, Claudia, and Freddy Van Oystaeyen. &lt;em&gt;Abstract Algebra: A Comprehensive Treatment.&lt;/em&gt; Marcel Dekker, 2004.&lt;/p&gt;&#xA;" OwnerUserId="7992" LastActivityDate="2017-09-17T09:17:52.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12874" PostTypeId="1" CreationDate="2017-09-18T05:53:23.917" Score="4" ViewCount="178" Body="&lt;p&gt;I've had to try to explain the following problem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $PQ$ be a line segment, given $P(x_1, y_1)$ and the midpoint $M(x_2, y_2)$, find the coordinates of $Q$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always draw a diagram and draw $\Delta x$ and $\Delta y$ and explain why they are $x_2 - x_1$ and $y_2 - y_1$ respectively. I then explain that you have to add the differences to move from the midpoint to the other endpoint. This always confuses the kids. Should I introduce the idea of a vector? What would you suggest to help them understand this procedure?&lt;/p&gt;&#xA;" OwnerUserId="8722" LastActivityDate="2017-09-18T19:48:18.240" Title="How to explain how to get an end point of a segment from the other end point and it's midpoint?" Tags="|geometry|tutoring|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12875" PostTypeId="2" ParentId="12874" CreationDate="2017-09-18T10:51:24.837" Score="4" Body="&lt;p&gt;Let $Q(x_3,y_3)$. Using the midpoint formulas $x_2=\frac{x_1+x_3}{2}$ find $x_3$.&lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2017-09-18T10:51:24.837" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12876" PostTypeId="2" ParentId="12874" CreationDate="2017-09-18T10:52:02.487" Score="2" Body="&lt;p&gt;Vectors are a natural way to think about the problem, but if the students are struggling with this problem then explicitly introducing a whole new abstraction might confuse them even more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could perhaps sneak in a vector intuition without explicitly defining vectors: instead of adding the differences to the midpoint, add &lt;em&gt;twice&lt;/em&gt; the differences to the starting point, describing it as going twice as far from the starting point in the same direction as before. Even without a formal notion of vectors, students should have some understanding that slope determines direction. You could even draw arrows in the diagram to illustrate the situation. As an added benefit, this might prime the pump for subsequent work with vectors.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2017-09-18T10:52:02.487" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12877" PostTypeId="2" ParentId="12869" CreationDate="2017-09-18T12:08:21.303" Score="4" Body="&lt;p&gt;Yes.  You are messing up now.  You can change though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should teach the basics first and the caveats, concerns, specialties last.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teach the kid how to isolate x and get an answer.  And THEN double check for some divide by zero concerns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you have to realize that to you the &quot;solve for x&quot; aspect is trivial, while the &quot;watch out for divide by zero&quot; is interesting.  But for someone who is new to algebra, the &quot;solve for x&quot; is still new and interesting on its own. [I even feel stronger and think it is the main thing to learn.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do change, it will help the kid and she will enjoy her time with you more also.  I think you will like that human interaction in the end also.  Give it a shot.&lt;/p&gt;&#xA;" OwnerUserId="8697" LastActivityDate="2017-09-18T12:08:21.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12878" PostTypeId="2" ParentId="12874" CreationDate="2017-09-18T15:36:00.113" Score="7" Body="&lt;p&gt;Putting aside the two-dimensional question for a moment, &lt;strong&gt;consider the corresponding one-dimensional question&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You are given two numbers $p$ and $q$. $p$ is $5$. The average of the two numbers is $8$. What is $q$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you ask the class (or yourself), you might get multiple different approaches. For example, you might go with&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Well, it is $3$ units from $5$ to the middle. So it must be $3$ more units from the middle to the other side. So the answer is $q = 8 + (8-5) = 11$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or perhaps:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I don't know what $q$ is, but I know how to find the average of $p$ and $q$. The average is $8 = \frac{p + q}{2}$. Plugging in $5$ for $p$ lets me solve for $q$, and $q$ comes out to 11.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then you can decide between:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Choose the first method as the official method and explain its 2D analog to the class.&lt;/li&gt;&#xA;&lt;li&gt;Choose the second method as the official method and explain its 2D analog to the class.&lt;/li&gt;&#xA;&lt;li&gt;Express the idea that both methods are fine, there is no such thing as an official method, and show both 2D analogs, and let everyone do what they think is easiest as long as they show what they are doing. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I don't know which of the three will work best in your classroom environment, but &lt;strong&gt;whatever your approach is to the easier 1D problem, you should go with the same thing in 2D&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-09-18T15:36:00.113" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12879" PostTypeId="2" ParentId="12874" CreationDate="2017-09-18T19:48:18.240" Score="3" Body="&lt;p&gt;Assuming the students know the principle of similar triangles, one might use the following approach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3SijA.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3SijA.png&quot; alt=&quot;Midpoint &amp;amp; similar triangles&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$  \triangle PMR\sim\triangle PQS$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We know that $\dfrac{PQ}{PM}=2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore by the property of similiar triangles we know that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \frac{PS}{PR}=2 \text{ and }\frac{QS}{MR}=2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \frac{x_3-x_1}{x_2-x_1}=2 \text{ and }\frac{y_3-y_1}{y_2-y_1}=2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solving these two equation for $x_3$ and $y_3$ gives&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ x_3=2x_2-x_1 \text{ and }y_3=2y_2-y_1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can be checked by verifying that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \frac{x_1+x_3}{2}=x_2 \text{ and }\frac{y_1+y_3}{2}=y_2$$&lt;/p&gt;&#xA;" OwnerUserId="6736" LastActivityDate="2017-09-18T19:48:18.240" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12880" PostTypeId="2" ParentId="12869" CreationDate="2017-09-19T17:58:35.910" Score="0" Body="&lt;p&gt;You wrote: if a&lt;sup&gt;2&lt;/sup&gt;+b&lt;sup&gt;2&lt;/sup&gt;2−ab=0, the solution set is S=U&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But your condition cannot be true, which was explained in two ways, in one comment and one answer. It makes some sense to point out that a real mathematician checks this. But the teacher's answer is correct.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-09-19T17:58:35.910" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12881" PostTypeId="1" AcceptedAnswerId="12886" CreationDate="2017-09-19T19:24:25.500" Score="7" ViewCount="2949" Body="&lt;p&gt;Endeavoring to bring some flavor of &quot;real world&quot; application to each topic in my community college precalculus class, I find myself struggling to provide some non-geometric motivation for perpendicular lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applications of parallel lines are a dime-a-dozen, as the idea of rate-of-change is easy for students to discuss when there are units involved. The discussion of the relevant units makes the conversation meaningful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, is there some level-appropriate, non-geometric application of perpendicularity? I am having trouble thinking of something that doesn't strictly rely on the angle between the functions (&quot;find the equations for the sides of this square&quot;), tangent lines (e.g. finding the gradient), or a special case with the units ignored (&quot;this thing goes up by 1 each year, and the other thing goes down by 1 each year&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have a go-to example of a non-geometric application of perpendicularity for precalculus students? In your example, do the units tell something useful? If so, I'd love to hear about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, here's what I'm thinking of as an easy-to-build application of parallel functions for my students:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Two people begin working at the same job, and each year they will both see a 2000 dollar-per-year raise. Person A comes with experience, beginning at 55000 dollars per year, and person B starts at 45000 dollars per year.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2017-09-19T21:14:25.850" LastActivityDate="2017-10-01T17:49:23.657" Title="Application of perpendicular lines" Tags="|applications|" AnswerCount="7" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12882" PostTypeId="2" ParentId="12881" CreationDate="2017-09-19T20:04:28.913" Score="6" Body="&lt;p&gt;Builders use plumb lines and levels to determine vertical and horizontal directions (which are perpendicular) in order to ensure that floors are horizontal and walls are vertical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When dealing with vectors (in the plane or in space), which are used constantly in statics and dynamics, one often represents each vector as the sum of a pair of perpendicular vectors. Flip through any introductory engineering statics or dynamics book for many such examples.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2017-09-19T20:04:28.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12883" PostTypeId="2" ParentId="12881" CreationDate="2017-09-19T20:59:59.147" Score="8" Body="&lt;p&gt;Given a set of sites determine which points are closer one of the sites than any of the others? If closer is measured by Euclidean distance then given the distinct points A and B the points equidistant from A and B lie on the perpendicular bisector of the segment A and B. The regions one obtains are known as the Voronoi diagram associated with the sites, and pieces of lines perpendicular to the lines joining pairs of sites play an important role. The sites may be post offices or schools so the &quot;cells&quot; of the Voronoi diagram are the postal districts or school districts. Here is a primer for these ideas: &lt;a href=&quot;http://cs.brown.edu/courses/cs252/misc/resources/lectures/pdf/notes09.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://cs.brown.edu/courses/cs252/misc/resources/lectures/pdf/notes09.pdf&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2017-09-19T20:59:59.147" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12884" PostTypeId="2" ParentId="12881" CreationDate="2017-09-20T00:10:42.053" Score="4" Body="&lt;p&gt;There are many applications of perpendicular lines in high school Physics. My students must often figure out the perpendicular to a given line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Normal_force&quot; rel=&quot;nofollow noreferrer&quot;&gt;normal force&lt;/a&gt; is perpendicular to the surface on which an object is resting or sliding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Snell%27s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Snell's Law&lt;/a&gt;, which governs how a light ray behaves when it crosses the boundary between two media (such as air and glass), requires the line perpendicular to the boundary.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2017-10-01T17:49:23.657" LastActivityDate="2017-10-01T17:49:23.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12885" PostTypeId="2" ParentId="12881" CreationDate="2017-09-20T00:22:50.400" Score="4" Body="&lt;p&gt;This is a bit fanciful, and likely too geometric for your purposes, but if your students have encountered trig...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Road intersections are safest if the roads cross at right angles.&#xA;Suppose the roads are $\sin$ and $\cos$ functions, say $\sin x$ and&#xA;$\cos(x+\pi)$. Do they cross at right angles? No. Suppose you scale up the&#xA;second road to the steeper $s \cos(x+\pi)$. What $s&amp;gt;1$ leads to crossing at right angles?&#xA;Of course to solve this they need the derivative, but perhaps this could motivate&#xA;slopes.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/7YyhC.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7YyhC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;$\sin x$ and $2 \cos(x+\pi)$ cross orthogonally at $x=-\frac{1}{4}\pi,\, \frac{3}{4}\pi$.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-09-20T00:22:50.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12886" PostTypeId="2" ParentId="12881" CreationDate="2017-09-20T06:56:27.703" Score="13" Body="&lt;p&gt;There's a reason why you can't find a good non-geometric example: when the dimensions of the axes on a graph are distinct, perpendicularity is units-dependent.  There is thus no natural aspect ratio with which to draw the graph.  If you change the scale for just one axis, you destroy any perpendicularity that was present, thus demonstrating that it was an accident.&lt;/p&gt;&#xA;" OwnerUserId="8729" LastEditorUserId="8729" LastEditDate="2017-09-20T13:06:44.270" LastActivityDate="2017-09-20T13:06:44.270" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12887" PostTypeId="2" ParentId="11963" CreationDate="2017-09-20T13:28:16.440" Score="8" Body="&lt;p&gt;We run a weekly seminar at my university where undergrads give math talks to other undergrads. To encourage people to give talks we collected a list of suggested topics which may be useful to you -- &lt;a href=&quot;https://uwseminars.com/potential-topics/&quot; rel=&quot;noreferrer&quot;&gt;https://uwseminars.com/potential-topics/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: As suggested by Joel, I'll add a few of my favourite ones from the list (there are 83 topics now, but the number will increase this term).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;em&gt;Continued Fractions and Hyperbolic Geometry&lt;/em&gt; (Farey tesselation etc): &#xA;   Example Reference: &lt;a href=&quot;http://homepages.warwick.ac.uk/~masbb/HypGeomandCntdFractions-2.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://homepages.warwick.ac.uk/~masbb/HypGeomandCntdFractions-2.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;em&gt;Optimal Stopping Theory and the Secretary Problem:&lt;/em&gt;&#xA;   Example reference:&#xA;&lt;a href=&quot;https://www.math.upenn.edu/~ted/210F10/References/Secretary.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.math.upenn.edu/~ted/210F10/References/Secretary.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &lt;em&gt;Penrose tilings (and possible connections to the dynamics of tiling):&lt;/em&gt;&#xA;   Example reference:&#xA;&lt;a href=&quot;https://people.maths.ox.ac.uk/ritter/masterclasses.html&quot; rel=&quot;noreferrer&quot;&gt;https://people.maths.ox.ac.uk/ritter/masterclasses.html&lt;/a&gt;,&#xA;Miles of Tiles by Charles Radin &lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) &lt;em&gt;The Rental Harmony Theorem&lt;/em&gt;&#xA;    Example reference:&#xA;    &lt;a href=&quot;http://www3.math.tu-berlin.de/combi/wp_henk/wp-content/uploads/2011/05/rentsu.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www3.math.tu-berlin.de/combi/wp_henk/wp-content/uploads/2011/05/rentsu.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) &lt;em&gt;Representation Theory and Voting Theory&lt;/em&gt;&#xA;   Example reference: &lt;a href=&quot;https://arxiv.org/abs/1508.05891&quot; rel=&quot;noreferrer&quot;&gt;https://arxiv.org/abs/1508.05891&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6) &lt;em&gt;Ultrafilters and Nonstandard analysis:&lt;/em&gt; Example reference: &lt;a href=&quot;https://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Davis.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Davis.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This one is not exactly on that list but occurs elsewhere on the site&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7) &lt;em&gt;Automated Verification of Computatational systems&lt;/em&gt;: Example reference: &lt;a href=&quot;http://www.cmi.ac.in/~madhavan/papers/pdf/resonance-jul2009.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.cmi.ac.in/~madhavan/papers/pdf/resonance-jul2009.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;8) &lt;em&gt;Differential Equations on Fractals&lt;/em&gt;: Example reference: Differential equations on fractals, R. Strichartz&lt;/p&gt;&#xA;&#xA;&lt;p&gt;9) &lt;em&gt;Fuzzy Logic&lt;/em&gt;: Example reference: &lt;a href=&quot;http://www.francky.me/doc/course/fuzzy_logic.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.francky.me/doc/course/fuzzy_logic.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;10) &lt;em&gt;General Secure Multi-party computation from any Linear Secret Sharing Scheme&lt;/em&gt;: Example reference: &lt;a href=&quot;https://www.iacr.org/archive/eurocrypt2000/1807/18070321-new.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.iacr.org/archive/eurocrypt2000/1807/18070321-new.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sorry for the slant towards some topics but there's a &quot;tag&quot; option on the webpage which allows you to see topics by area.&lt;/p&gt;&#xA;" OwnerUserId="8732" LastEditorUserId="8732" LastEditDate="2017-09-21T06:07:48.913" LastActivityDate="2017-09-21T06:07:48.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12889" PostTypeId="2" ParentId="11963" CreationDate="2017-09-20T21:14:04.890" Score="2" Body="&lt;p&gt;Linear optimization as used in oil refineries (or even midstream movements and trading).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anything dealing with statistics in business.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Matching algorithms (of mates or medschool students).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additional to the above:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;molecular or crystallographic symmetries.  (BTW, if you approach it from the chemical or visual standpoint, rather than abstract algebra, it will be easier to understand and more interesting/applied.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's another cool topic, gambling skill:  &lt;a href=&quot;https://www.youtube.com/watch?v=658xlubwnDc&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=658xlubwnDc&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And another (more fluffy, but fun) is the fourth dimension in art:  &lt;a href=&quot;https://en.wikipedia.org/wiki/Fourth_dimension_in_art&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Fourth_dimension_in_art&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bottom line, keep it fun.  DON'T be afraid to be relevant or to have some non pure math in there.  You are not trying to teach a course in a seminar.  You are trying to get some interest, that maybe pays of later.&lt;/p&gt;&#xA;" OwnerUserId="8734" LastEditorUserId="143" LastEditDate="2017-09-21T19:25:36.097" LastActivityDate="2017-09-21T19:25:36.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12892" PostTypeId="2" ParentId="5877" CreationDate="2017-09-21T23:04:34.143" Score="9" Body="&lt;p&gt;Physically (or even economically) unreasonable answers are confusing.  The student needs to be shown that math is a tool that can solve practical problems.  Answers that are unreasonable send the wrong message.  (That math is a logic game versus a commercially relevant skill.)  Why do that, when there are so many easy problems to construct that don't make this mistake?  There is no need.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Heck, I have to wonder if a problem has an unreasonable answer that the math teacher lacks basic science/engineering skills.  I think for any service course, teachers should have basic competency in college physics, chemistry, and a smattering of engineering.  This is not so they can teach those courses, but so they can understand the connection between the subjects while teaching students (the vast majority of whom will use the tools from math class in these subjects).  I think this sort of basic understanding gives a little bit of sympathy, empathy and just understanding for what the students are doing.  This ends up helping the instruction.&lt;/p&gt;&#xA;" OwnerUserId="8745" LastActivityDate="2017-09-21T23:04:34.143" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12893" PostTypeId="2" ParentId="12881" CreationDate="2017-09-22T00:25:01.287" Score="2" Body="&lt;p&gt;In calculus I class today: Find the point on y=x&lt;sup&gt;2&lt;/sup&gt; that is closest to (1,0). (Actually, approximate it, using Newton's Method.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest point makes a line segment perpendicular to the tangent line. [True for a point and a line. Is this true for a point and a curve by definition? It is just what came to me in class, when a student asked me to do this problem. I had assigned it but never thought about it carefully. I see that using distance gets me to the same place algebraically, but I haven't yet established the perpendicularity to my satisfaction.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student had tried to minimize distance, which is much harder. Using the perpendicular made the problem relatively straightforward.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2017-09-22T01:02:36.027" LastActivityDate="2017-09-22T01:02:36.027" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12894" PostTypeId="1" CreationDate="2017-09-22T00:27:34.477" Score="6" ViewCount="410" Body="&lt;p&gt;OK, this is not exactly a Math Education question, but in lieu of a functioning Science Education site and because in middle schools Math and Science are often taught by the same teachers, I'll ask the question here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My daughter just started her Middle School, and it turns out that in our school district there are no Science textbooks, despite the daily Science class. Apparently the curriculum changed, but the corresponding textbooks won't be ready for the next two years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The kid is curious about things and is very good at Math. Therefore I'd like to get her a book (or a set of books) that would be simultaneously readable for a Middle School kid and informative on the Science subjects they cover.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you recommend such book(s)?&lt;/p&gt;&#xA;" OwnerUserId="674" LastActivityDate="2017-10-05T15:59:05.300" Title="What science books would you recommend a curious middle school student to read at home?" Tags="|textbooks|physical-sciences|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="12896" PostTypeId="2" ParentId="12894" CreationDate="2017-09-22T15:55:53.867" Score="2" Body="&lt;p&gt;You could just go ask what the previous books used were, or look at what some gifted schools use for texts in 7th, 8th grade science. I'm not sure how fun/useful such a text is, especially if disconnected from the course, but nothing wrong with trying.  Just because something only MIGHT work, is not a reason not to do it.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My 7th and 8th grade G/T experience was a course in &quot;environmental science&quot; that was descriptive ecology.  Nothing wrong with it and fun and fascinating.  Then a course in &quot;introductory physical science&quot;, which was baby chem/physics with an emphasis on measurements and units (and some fun experiments like making fudge...and methanol).  I don't really recall the texts used and the classes really were more instructor led.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two books that many great physicists say inspired them as children are Microbe Hunters and Men of Mathematics.  Both are early 20th century but still resonate today.  (MoM has one woman in it, also, if that is a concern.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also suggest the books by Richard Feynman: Surely You're Joking, Mr. Feynman and What Do You Care What Other People Think.  Very readable and fun.  The last half of the latter book is about the first Space Shuttle crash investigation and is eerie in how prescient it was of the second crash.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also sign up for Things of Science.  They mail you a little blue box with science experiments in them.  I tried doing it in 5th grade, at my dad's urging, but was too young for it (was too hard). But smart middle schooler lacking textbook would be perfect customer.  [If that doesn't exist any more, consider to buy her some other kits.]&lt;/p&gt;&#xA;" OwnerUserId="8745" LastEditorUserId="8745" LastEditDate="2017-09-22T16:07:45.697" LastActivityDate="2017-09-22T16:07:45.697" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12899" PostTypeId="2" ParentId="12894" CreationDate="2017-09-24T01:30:58.507" Score="1" Body="&lt;p&gt;I found this book a delightful mix of geometry, history, &#xA;and engineering. Your daughter might have to skim of sections that&#xA;employ trigonometry, but skimming in this book would not lose her&#xA;the thread.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;John Bryant &amp;amp; Chris Sangwin. &lt;a href=&quot;https://press.princeton.edu/titles/8624.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How Round Is Your Circle&lt;/em&gt;&lt;/a&gt;, &#xA;  Princeton U Press, 2008.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;  &lt;img src=&quot;https://i.stack.imgur.com/3KQQx.jpg&quot; width=&quot;400&quot; /&gt;&#xA;  &lt;hr /&gt;&#xA;  The book is especially good on linkages. Other topics: packing,&#xA;  dissections, and of course, constant-width bodies (cover image).&#xA;  See the &lt;a href=&quot;https://press.princeton.edu/titles/8624.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Princeton link&lt;/a&gt; for a detailed table of contents.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-10-05T15:59:05.300" LastActivityDate="2017-10-05T15:59:05.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12901" PostTypeId="2" ParentId="5885" CreationDate="2017-09-24T22:54:55.043" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;Maybe try to use progression?  Don't give them the hardest word problems to start.  (Or if some kids can handle the regular word problems, keep a mix of difficulties.)  By hard, I don't mean math hard only, I mean amount of word translation needed.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;9 - 5 =?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nine minus five =?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is left when you subtract five from nine?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Johnny has nine pencils and gives five away.  How many does he have left?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How many good pencils does Johnny have at the end of the day, if he had nine at the beginning of the day and during the day, he broke five pencils?&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Also consider to lower math difficulty while raising word difficulty.  Think about learning bra ket notation in quantum mechanics.  When I have to learn that new, strange looking, notation, I prefer to do so with a very easy QM problem I have already solved with conventional notation.  Not a hard old one. And definitely not a new problem (combining new content with new notation for the first smack in the face).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Love the comment about money.  Emphasize that a lot.  It is tactical, students have experience, it makes it matter, and it helps them in daily life.  Same exact problem with money is probably easier than with physical dimensions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Agree with the 3 reads concept and with teaching a process of translation.  You got to take the WP, translate it into an equation, solve equation, translate back.  Just teach this (step by step).  Not as a concept.  But actual step by step practice.  (And per 2, do this with a crushingly easy example.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Drill, drill, drill.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8745" LastActivityDate="2017-09-24T22:54:55.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12904" PostTypeId="2" ParentId="12881" CreationDate="2017-09-26T21:19:07.930" Score="0" Body="&lt;p&gt;I don't know if you're doing anything with matrices, but I think this would be a nice opportunity to show the difference in approaches to statistical models. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Linear Regression is easily demonstrated using a transpose and matrix multiplication.  This model assumes noise in both the $x$ and $y$ variables.  An alternative situation involves noise in only one measure, and now you would want to minimize the distance perpendicular to the line of fit rather than the strict vertical distance as in linear regression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is Principal Component Analysis, and also can be understood through some basic matrix operations.  Of course, there is geometry involved in the 2D case that we can visualize but higher than 3D doesn't quite give way to images.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a site with nice visualizations and a fun example about eating in the UK.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://setosa.io/ev/principal-component-analysis/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://setosa.io/ev/principal-component-analysis/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, see Wikipedia for a more substantial mathematical presentation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Principal_component_analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Principal_component_analysis&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-09-26T21:19:07.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12906" PostTypeId="2" ParentId="12894" CreationDate="2017-09-27T01:38:35.020" Score="1" Body="&lt;p&gt;Not mathy, but I loved &lt;em&gt;T Rex and the Crater of Doom&lt;/em&gt;, by Walter Alvarez (a scientist), about why the dinosaurs went extinct. His explanation of the history of the question gives some great insight into the process of scientific discovery.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-09-27T01:38:35.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12908" PostTypeId="2" ParentId="5877" CreationDate="2017-09-27T07:12:52.067" Score="3" Body="&lt;p&gt;I think trying to make &quot;real-world&quot; word problems is often posing more problem than it solves. One expects using maths in real-world problem would make them more appealing and show their usefulness, but honestly in most cases the problems are either ridiculous, or artificial, or boring to students, or all of these. Then the whole idea backfires, and what you describe is a good example of a much more general issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, this does not literally answer your question, but the best option might be to make &quot;fantasy-world&quot; word problem instead of &quot;real-world&quot; ones. They will be more fun, less misleading, and you won't have to care too much about the result being reasonable (if you compute the size of a Gargumash from its shadow, 8 feet seems just as reasonable as any other (positive) result given no one nows what a Gargumash is).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This goes further than maths: I know a biology teacher who did not want to use fake data for real species in an exam in genetics, but wanted to design it freely to tune the difficulty. She chose to make the test about unicorns.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-09-27T07:12:52.067" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12909" PostTypeId="1" CreationDate="2017-09-27T12:56:20.600" Score="4" ViewCount="116" Body="&lt;p&gt;I posted a related question on the Math.SE, but was directed here where I'm asking an similar but different question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been tasked with helping to redesign a math curriculum for an enrichment program and I have very few restrictions on what the curriculum should include or how it should flow. The curriculum is meant to be an enrichment curriculum (supplementary to the existing one here in Ontario).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wanted to find some resources on proven and effective ways to design and teach math so that students have a deeper, more rigorous and more intuitive understanding of math - an approach that prepares them to tackle math abstractly and ultimately prepares them for higher level math. The curriculum starts at grade 4 and continues on to grade 11.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm aware this is a broad topic, but where can I start to get information that hasn't been contradicted or proven to not be effective? &lt;/p&gt;&#xA;" OwnerUserId="8616" LastActivityDate="2017-09-27T12:56:20.600" Title="Research on how to teach math to children - what proven approaches are there to teaching math effectively?" Tags="|secondary-education|mathematical-pedagogy|reference-request|general-pedagogy|teaching|" AnswerCount="0" CommentCount="4" ClosedDate="2017-10-03T10:52:31.217" ContentLicense="CC BY-SA 3.0" />
  <row Id="12910" PostTypeId="1" AcceptedAnswerId="12913" CreationDate="2017-09-27T13:33:41.627" Score="7" ViewCount="219" Body="&lt;p&gt;I need to do a presentation on my thesis, which is in arithmetic geometry. This presentation is meant for all students of mathematics, but I will assume some knowledge of abstract algebra (i.e. groups, rings, fields). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first part of my presentation, I want to explain what Faltings' Theorem is and why it is important (and nice). Defining a curve $C$ is not too hard to do (although I skip over some details) and defining $C(k)$ (or I might just stick with $C(\mathbb{Q})$) for $k$-rational points on the curve. I then introduce some examples of interesting curves, such as the circle, an elliptic curve, and hyperelliptic curves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm having difficulty however, to explain in simple terms the concept of the genus $g(C)$ of the curve $C$. Most intuitive might be too think of it topologically &quot;as the number of holes&quot; and use the Weierstrass $\wp$ function to explain how we can think of the elliptic curve $\mathcal{E}$ as some torus $\mathbb{C}/\Lambda$. This takes up quite a lot of time, though. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an easier, shorter, way to explain the genus $g(C)$ or might I be able to skip the whole genus and explain Faltings' in some other way?&lt;/p&gt;&#xA;" OwnerUserId="8774" LastActivityDate="2017-09-29T15:12:01.887" Title="Explaining genus to students" Tags="|concept-motivation|abstract-algebra|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12911" PostTypeId="1" AcceptedAnswerId="12912" CreationDate="2017-09-27T14:00:50.523" Score="2" ViewCount="187" Body="&lt;p&gt;I've been using my tablet for years teaching in a high school. Now I moved to another school teaching middle school. I spoke to the principal and he said there's no problem in using a tablet as long as it doesn't have a SIM card or any distracting apps (social networks, chatting, etc.). After the first week of school, they are telling me that &quot;students don't accept this tablet in class&quot; and they consider it &quot;no different from a cellphone&quot;. I explained to them that all my books and courses are on it and it isn't easy to print all of them and that, after a little time, students won't notice it anymore. I also explained to the students that this tablet contains their course and exercises and that I only use it as a reader but that didn't help much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, is it normal to ban a tablet in school? If so, it seems I only have two choices: either print all the courses or stop working at this school. What should I do about it?&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="501" LastEditDate="2017-09-28T08:33:57.183" LastActivityDate="2017-09-28T08:33:57.183" Title="Can't use samsung tablet in class?" Tags="|technology-in-education|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="12912" PostTypeId="2" ParentId="12911" CreationDate="2017-09-27T16:49:15.693" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;(edit)  I just reread the question.  I would just stick to what you are doing and say the principle approved it.  It is not clear who &quot;they&quot; is in the context of the complaints.  If the principal changes his mind and gives you a direct order, then comply with that new order. But don't get dragged into debate with colleagues or kids. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;My advice is to consider this an issue between you and the administration.  The issue of the students is not the key concern.  Yes, I get that the admins refer to this and I get the kids chip in on it, but at the end of the day, students get used to what happens.  I would just not engage them in debate.  It's not up to them.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As far as the admin:  Either you just do what they tell you or you get them to allow you to do things how you prefer. It is hard for me to understand how much of a big deal it is or not.  Or how important you are to them.  My gut would be just to follow policy.  But it may be possible that they can/will bend.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I get that you are used to this tool and it makes you more efficient, but perhaps there is some workaround.  Transferring things to a cheap used laptop (is that allowed)?  Or just some efficient paper based system--I'm old enough so that I remember teachers using those little attendance/grading books that look like a ledger.  Perhaps you are using a lot of worksheets or the like, but could convert to using the textbook more or the chalkboard (for drill)?  Also, transparencies and overhead projection can be very easy (faster than writing chalk and erasing).  And you can save the content (use permanent ink, use those 3 hole sheet protector sleeve things to put the acetate sheets in).  Donno, and not saying it will be as nice as your system on your tablet.  But think about it and how important the job is.  [And at the end of the day, don't stress too much.  Life goes on and every job has its little BS.  How do the other teaches get their classes done?  Why can't you do what they do?]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8745" LastEditorUserId="8745" LastEditDate="2017-09-27T17:03:48.070" LastActivityDate="2017-09-27T17:03:48.070" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12913" PostTypeId="2" ParentId="12910" CreationDate="2017-09-29T15:12:01.887" Score="6" Body="&lt;p&gt;As a fellow arithmetic geometer working on some problems related to Faltings' theorem, I've given a fair number of explanations of algebraic curves to non-specialists, so I can tell you how I usually do it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I &lt;em&gt;never&lt;/em&gt; do unless I'm talking to number theorists or algebraic geometers is get into the details of complex tori, the Weierstrass $\wp$-function, or anything like that. What is usually do for explaining the genus is something like this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Describe the genus as the &quot;number of holes&quot; without giving a precise definition.&lt;/li&gt;&#xA;&lt;li&gt;Draw a few examples on the board and say what their genus is. I usually do a sphere, a torus, and a genus two or three curve. (Drawing tip: draw the holes first, then the visual boundary.)&lt;/li&gt;&#xA;&lt;li&gt;Briefly mention that what I'm drawing here is the set of &lt;em&gt;complex&lt;/em&gt; solutions to the defining equations of the curve, not just the real solutions, which is why it looks like a surface rather than a curve. (For a non-mathematician audience, I might leave this out entirely and only bring it up if someone asks why my drawings look 2-dimensional. If the audience knows complex analysis, I'll mention these are just compact Riemann surfaces.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Most people can visualize &quot;number of holes&quot; just fine once you've drawn a few examples. If you wanted to spend more time on the concept of genus, you could give examples of how to construct a genus two curve by gluing sides of an octagon, like here: &lt;a href=&quot;https://plus.maths.org/content/making-double-torus-octagon&quot; rel=&quot;noreferrer&quot;&gt;https://plus.maths.org/content/making-double-torus-octagon&lt;/a&gt; (I've never taken this approach in a talk, but it seems like a reasonable thing to do.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for actually motivating why higher genus curves behave differently, I like to handwave a bit about the fundamental group: I'll note how any loop on a sphere can be contracted to a point, loops on a torus can't always be contracted but still have a simple description, and loops on a higher genus curve have a much more complicated structure (I'll say &quot;nonabelian&quot; here if they know groups). Since my research is on Kim's non-abelian Chabauty method, which is motivated by anabelian geometry, this is a natural perspective for me, but may be less natural if you're viewing it from Faltings' or Vojta's perspectives (which I don't think directly use the fundamental group).&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2017-09-29T15:12:01.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12914" PostTypeId="1" AcceptedAnswerId="12923" CreationDate="2017-09-29T17:46:34.907" Score="6" ViewCount="404" Body="&lt;p&gt;I'm not a math educator but I'm curious about this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this always true, or are there some schools/educational programs where students are shown negative numbers before fractions? Why is it done in this order? Is it because rationals are more &quot;real&quot; or useful than negatives?&lt;/p&gt;&#xA;" OwnerUserId="8780" LastActivityDate="2017-10-01T19:04:42.313" Title="Why are fractions taught before negative numbers?" Tags="|negative-numbers|rational-numbers|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12915" PostTypeId="1" CreationDate="2017-09-29T21:11:21.407" Score="6" ViewCount="1490" Body="&lt;p&gt;I am a teacher and want to start on online website with videos to make work easy for both parents and teachers in mathematics for children. I saw this video and was wondering if someone could help me identify the software or techniques or better still tools used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/results?search_query=Mphoweh+Jude+Nzembayie&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/results?search_query=Mphoweh+Jude+Nzembayie&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please feel free to provide your suggestions which may have better options.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="8781" LastEditorUserId="11" LastEditDate="2017-09-29T21:13:15.970" LastActivityDate="2017-10-15T15:35:02.943" Title="Software and techniques for creating short maths animation videos for kids" Tags="|technology-in-education|tutoring|software|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12916" PostTypeId="2" ParentId="12915" CreationDate="2017-09-30T00:09:06.513" Score="1" Body="&lt;p&gt;Maybe start here:&#xA;&lt;a href=&quot;https://elearningindustry.com/the-top-6-animated-video-software-in-the-elearning-market&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Top 6 Animated Video Software In The eLearning Market&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;GoAnimate&lt;/li&gt;&#xA;&lt;li&gt;CrazyTalk&lt;/li&gt;&#xA;&lt;li&gt;iClone &lt;/li&gt;&#xA;&lt;li&gt;Toon Boom Harmony&lt;/li&gt;&#xA;&lt;li&gt;After Effects&lt;/li&gt;&#xA;&lt;li&gt;PowToon&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Few of these are free; the link above gives pricing. Here is a PowToon image:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/VcTJf.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VcTJf.jpg&quot; alt=&quot;PowToon&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;For totally free software, search for &quot;free software to make simple educational animations.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-09-30T00:09:06.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12918" PostTypeId="1" CreationDate="2017-09-30T09:54:59.433" Score="6" ViewCount="71" Body="&lt;p&gt;I am going to carry out a small scale action research on ratio problems. I would ask the same questions in different ways and see whether wordings would affect students' performance. However, past research showed that numerical values would affect questions' difficulty. Say comparing (1:2 and 1:4) and (3:7 and 5:11), the latter one is obviously more difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I control for the numerical factors so that I can make inferences on &quot;problem wordings&quot;?&lt;/p&gt;&#xA;" OwnerUserId="8783" LastActivityDate="2017-09-30T09:54:59.433" Title="How to control for numerical factor in mathematics education research?" Tags="|education-research|undergraduate-research|word-problems|rational-numbers|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12922" PostTypeId="1" CreationDate="2017-10-01T16:47:17.677" Score="2" ViewCount="155" Body="&lt;p&gt;European universities are running the so called &quot;Bologna plan&quot; which includes the recommendation of keeping some percentage of the final grades to be filled with various activities during the lessons period. Like everything in life, it has pros and cons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will tell you the cons I have seen (in the particular case of mathematics related students), then I will appreciate a list of pros (because I don't see any of them).&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The world isn't perfect, so, although mathematics is really time consuming, there are people who work while they study, so they physically can't attend classes, which usually is one of the activities taken into account directly (signature lists) or indirectly (correcting problems in the blackboard). So they begin with a clear disadvantage compared to their classmates (they de facto lose a fraction of their grades).&lt;/li&gt;&#xA;&lt;li&gt;Again, the world is not perfect, there are teachers whose lessons simply aren't worth, and, for sure, a student who had spent the lesson time in the library reading a book about the subject will know more than his lesson-attending mates, and, probably, get a worse grade, despite having a way better done exam. In some (but not infrequent) cases passing the exam without attending lessons is formally impossible.&lt;/li&gt;&#xA;&lt;li&gt;Over again, the world isn't a wonderland, and every student has lots of subjects, every one of them with lots of continuous evaluation stuff (mandatory lesson attending, non-liberating partial exams, sets handing over,...), so, students tend (and sometimes are kind of forced) to do the work of each subject as fast as they can without really understanding what they are doing (really sad, even more in math-related stuff). I think university is about learning, not about getting constantly evaluated. With continuous evaluation student have $n$ weeks getting evaluated and $0$ weeks learning while the ideal will be $n$ weeks learning and maybe $3$ weeks getting evaluated.&lt;/li&gt;&#xA;&lt;li&gt;I have heard that continuous evaluation produces better overall results. I must disagree. I have seen really brilliant students getting a $7$ out of $10$ as a final grade with a perfect final exam, as well as people who &lt;strong&gt;really doesn't understand the subject&lt;/strong&gt; getting a $6$ (pass) while failing the final exam with a $3$. Also I have seen people with a good exam $6$ or $7$ out of $10$ having failed the subject with a $3$ or a $4$. Is this fair?&lt;/li&gt;&#xA;&lt;li&gt;In general, this dedicated fraction of the grading (usually around a 30 %) is kept for the extraordinary calls. ¿Isn't it nonsense?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I know some students like continuous evaluation, some of them because it allows them to pass the subject without really understanding it deeply, some others because they say it helps them to study. So, why not let students decide? If some of them don't want this system, just evaluate them by the final exam while the system keeps running for the others. Maybe another solution is to make a good use of the $\max$ function?&lt;/p&gt;&#xA;" OwnerUserId="8785" LastEditorUserId="2083" LastEditDate="2017-10-03T13:10:31.257" LastActivityDate="2017-10-04T07:20:34.667" Title="Continuous evaluation/Participation points" Tags="|grading|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12923" PostTypeId="2" ParentId="12914" CreationDate="2017-10-01T19:04:42.313" Score="3" Body="&lt;p&gt;In the US, we have the controversial Common Core.&#xA;A summary of it is available &lt;a href=&quot;http://www.corestandards.org/Math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In Grade 6, instructional time should focus on four critical areas:&#xA;  (1) connecting ratio and rate to whole number multiplication and&#xA;  division and using concepts of ratio and rate to solve problems; (2)&#xA;  completing understanding of division of fractions and extending the&#xA;  notion of number to the system of rational numbers, which includes&#xA;  negative numbers&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the first place where negative numbers are introduced, clearly after fractions. A comment on the question suggested that &quot;What we need here is actual publications on this, not merely opinions.&quot; I agree, to a point. The documents suggest an order, but don't go in to any satisfying detail about why the order should be that way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This citation confirms that, at least in the US, this is the order. In my opinion, fractions are very natural. A child old enough to talk will quickly see the unfair nature of splitting something any way but 1/2 and 1/2. Money literally (again, in the US) has a coin called a quarter. And when I am talking fractions to teens, they can relate to our circular pizza which the laws of both God and man dictate be cut into 8 equal sections through the center.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The negative numbers are a bit tougher to grasp, again as a commenter suggested, I typically offer an example of money owed. You have \$2, and spend \$3, one of which is borrowed. Your 'balance sheet' says you have -$1. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-10-01T19:04:42.313" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12924" PostTypeId="1" AcceptedAnswerId="12933" CreationDate="2017-10-01T21:40:56.220" Score="17" ViewCount="441" Body="&lt;p&gt;To try to find out if students knew what the gradient was, after the computational questions, I asked the following question on an exam:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f(x, y) = 5 - x - y$&lt;/span&gt;. Why doesn't it make sense to find &lt;span class=&quot;math-container&quot;&gt;$\nabla(\nabla f)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My students almost universally failed to give a good answer to the question. The most common answers are things like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\nabla f$&lt;/span&gt; already points in the direction of steepest ascent, so doing it again gives you the same thing.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Perpendicular to &lt;span class=&quot;math-container&quot;&gt;$\nabla f$&lt;/span&gt; is a level curve.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These kind of answers show a lack of precise knowledge of what kind of object the gradient is. This kind of &amp;quot;data type understanding&amp;quot; is at the core of a lot of new student problems in vector calculus.&lt;/p&gt;&#xA;&lt;p&gt;A correct answer should be something like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\nabla f$&lt;/span&gt; is a vector; it doesn't make sense to find the gradient of a vector.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I'm interested in your approaches to &lt;strong&gt;help students focus on the different data types involved in multivariable calculus&lt;/strong&gt; (some objects are vectors, some are scalars, many operations turn vectors into scalars or pairs of vectors into one or the other). Would you recommend writing things like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\nabla : $&lt;/span&gt; scalar-valued functions of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; variables &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-dimensional vectors&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;when we introduce new objects in this course? That's pretty complicated. Or something else? What can I do to better emphasize data types?&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2022-04-26T14:20:46.143" Title="How can we focus students on the various data types in multivariable calculus?" Tags="|functions|vector-calculus|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="12926" PostTypeId="1" CreationDate="2017-10-02T08:09:26.873" Score="5" ViewCount="255" Body="&lt;p&gt;One of my private students is a 13year old girl who started school at age 5 (instead of the regular 6, in my country).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Testimony from parents show she had no sense of number at the time, would not grasp even quantities nor could she relate them to physical objects (like recognizing 3 or 4 pencils as being 3 or 4).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She was then entrusted to a person (private tutor) who decided to spend the paid time making small puppets and painting together. By the time the parents found out, she was on the 3rd grade struggling with addition and subtraction (unable of learning division and multiplication).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays, she has not appropriated multiplication, nor division, yet. 3 months ago, she could not even relate the concepts &quot;double&quot; and &quot;half&quot; to their real as well as to their mathematical expressions. One year ago, she could not read numbers with 3 or more digits, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The max time I can get her attention on math work is precisely 1 hour and not 1 second more, even less if she is a bit ill or too tired from school that day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've spent the last 2,5 months creating specific exercises/problems based on the real world, using her personal &quot;youtuber heros&quot; as a motivational characters, to build some experience and familiarity with: patterns and regularities, addition, subtraction, equations, notion of variable, reference numbers, multiple representations of numbers, etc. (Not to mention the previous year, prior to hollidays, working on the very basics from primary school).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was some progress, but meanwhile school started and the time we have (3 times a week) is only enough to go through what she is being taught at school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is obvious to me that she is smart and can reason very well, she has been progressing well in Probabilities, but there is always somewhat of a barrier when we deal with fractional/decimal numbers or expression manipulation (she still ads up instead of using multiplication, on most occasions, and she still doesn't really know how to subract if the result has to go into negative territory, or if she has to operate with negative numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point I believe she will eventually fill up the gaps of what was left behind while we work on present day chores, I think I'm starting to see that happening (for example, the notion of probability, which she took in well, is helping to &quot;understand&quot; division, maybe working as &quot;an entrance door&quot; to it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone dealt with such situations? What is your point of view on this matter? Possible approaches?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you&lt;/p&gt;&#xA;" OwnerUserId="8792" LastActivityDate="2019-04-04T17:12:36.883" Title="Course of action with 13 year old with weak sense of number and operations" Tags="|mathematical-pedagogy|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="12927" PostTypeId="1" AcceptedAnswerId="12930" CreationDate="2017-10-02T00:00:35.490" Score="12" ViewCount="986" Body="&lt;p&gt;I have heard contradicting views that both parties say are research based. Specifically, I am looking for research on which teaching methods (direct instruction or student centered approaches) are most effective in high school mathematics (grades 9-12). I went to a workshop about a year ago that said direct instruction is proving to be more effective, but every curriculum and instruction department in academia focuses on teaching pre-service math teachers student-centered approaches. What does the research say? Searching google scholar does not seem to yield any useful results.&lt;/p&gt;&#xA;" OwnerUserId="7550" OwnerDisplayName="Tye Campbell" LastEditorUserId="143" LastEditDate="2017-10-02T10:45:20.420" LastActivityDate="2019-10-14T18:48:20.123" Title="What does the research say about direct instruction vs. student centered approaches" Tags="|secondary-education|education-research|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12928" PostTypeId="2" ParentId="12926" CreationDate="2017-10-02T13:39:37.137" Score="3" Body="&lt;p&gt;Your student's difficulties could be the result of early neglect of arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, if you don't see improvement, you may need to consult a specialist.  There is something called &quot;dyscalculia&quot; which is (roughly) an inability to understand numbers and/or number symbols.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that it is rare, but I have seen diagnosed in college students it a couple of times over the years.&lt;/p&gt;&#xA;" OwnerUserId="8658" LastEditorUserId="8658" LastEditDate="2017-10-03T16:10:08.130" LastActivityDate="2017-10-03T16:10:08.130" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12929" PostTypeId="2" ParentId="12926" CreationDate="2017-10-02T15:03:10.110" Score="1" Body="&lt;p&gt;In my view, the worst thing you can do is go back and only focus on elementary topics. I've taught students grades 7-12 for the past several years, and many 7th graders come in not knowing their multiplication facts, struggling to add and subtract negative integers, and struggling with operations on fractions. However, I've found that students will eventually learn these elementary concepts if you move on with learning at their appropriate grade level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interesting side note: The use of calculators has been found to have no negative effects on student computational skills. In fact, it generally aids in learning computational skills. See article here: &lt;a href=&quot;http://www.nctm.org/uploadedFiles/Research_and_Advocacy/research_brief_and_clips/2011-Research_brief_18-calculator.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.nctm.org/uploadedFiles/Research_and_Advocacy/research_brief_and_clips/2011-Research_brief_18-calculator.pdf&lt;/a&gt;. So, I would allow her to use a calculator which may catch her up to speed on some elementary topics. &lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2017-10-02T15:03:10.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12930" PostTypeId="2" ParentId="12927" CreationDate="2017-10-02T17:34:24.007" Score="13" Body="&lt;p&gt;Abstract from Alfieri, Brooks, Aldrich, &quot;Does Discovery-Based Instruction Enhance Learning?&quot;, Journal of Educational Psychology, 2011:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Discovery learning approaches to education have recently come under&#xA;  scrutiny (Tobias &amp;amp; Duffy, 2009), with many studies indicating&#xA;  limitations to discovery learning practices. Therefore, 2&#xA;  meta-analyses were conducted using a sample of 164 studies: The 1st&#xA;  examined the effects of unassisted discovery learning versus explicit&#xA;  instruction, and the 2nd examined the effects of enhanced and/or&#xA;  assisted discovery versus other types of instruction (e.g., explicit,&#xA;  unassisted discovery). Random effects analyses of 580 comparisons&#xA;  revealed that outcomes were favorable for explicit instruction when&#xA;  compared with unassisted discovery under most conditions (d = –0.38,&#xA;  95% CI [–.44, –.31]). In contrast, analyses of 360 comparisons&#xA;  revealed that outcomes were favorable for enhanced discovery when&#xA;  compared with other forms of instruction (d = 0.30, 95% CI [.23,&#xA;  .36]). The findings suggest that unassisted discovery does not benefit&#xA;  learners, whereas feedback, worked examples, scaffolding, and elicited&#xA;  explanations do.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Clark, Kirschner, Sweller, &quot;Putting Students on the Path to Learning: The Case for Fully Guided Instruction&quot;, American Educator, Spring 2012, also make the following points: (1) less-skilled students learn more from guided instruction, more-skilled from self-guided discovery (although if allowed to choose, each group will pick as a preference the type they learn &lt;em&gt;less&lt;/em&gt; from), and (2) per Richard Mayer, the cycle of self-guided instruction has been disabused, and then resurrected under a different name every decade or so since the 1950's (and each generation of proponents seems unaware/disinterested in the findings that came before). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-10-02T17:34:24.007" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12931" PostTypeId="2" ParentId="12927" CreationDate="2017-10-02T18:30:55.517" Score="3" Body="&lt;p&gt;In the recent MAA publication &quot;&lt;a href=&quot;https://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/national-studies-college-calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Insights and Recommendations from the MAA National Study of College Calculus&lt;/a&gt;&quot; active learning is one of the 7 characteristics of successful Calculus programs.  I would probably start with Chapters 3 and 8 and the references therein.  For example, on page 97 you can find:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A meta-analysis of 39 studies in multiple STEM fields found that&#xA;  small-group learning had a positive impact on achievement,&#xA;  persistence, and student attitudes (Springer, Stanne, &amp;amp; Donovan, 1999)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;These studies have consistently found that students from [calculus] reform courses developed stronger conceptual understanding and were more likely to persist in STEM fields while showing little or no negative impact on procedural fluency (Chappell &amp;amp; Killpatrick, 2003; Hurly, Koehn, &amp;amp; Ganter, 1999; Joiner, Malone, &amp;amp; Haimes, 2002).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-10-02T18:30:55.517" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12932" PostTypeId="2" ParentId="12924" CreationDate="2017-10-03T02:24:34.697" Score="3" Body="&lt;p&gt;Two things stand out about your question &quot;Why doesn't it make sense to find $\nabla \left(\nabla f\right)$?&quot;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What you're really doing is testing whether students understand the domain of the map $\nabla$. I assume you would spend lots of time in a college algebra class, testing their understanding of the domain of a function of a single variable, whether through context (&quot;x represents the number of years after 1990&quot;) or admissible input values (&quot;x cannot be 3&quot;). In your problem, the data type is the issue, and this may be a new concept to explore for your class. This could be the time to add this to the &quot;list of ways a relationship may fail&quot;. You may be introducing this issue for the first time in this course, so you could couch the data-type issue in terms of what they've already done with the domain of a real-valued function.&lt;/li&gt;&#xA;&lt;li&gt;Problems of this type require practice, so it may have caught some of them off guard if they hadn't been prepped to think along those lines. I think students sometimes read the same meaning into the phrases &quot;Why doesn't it make sense to...&quot; and &quot;Why can't you...&quot;, where I might have meant &quot;redundant&quot; in the first case and &quot;impossible&quot; in the second. Those subtle differences tend to skew where their minds go, when all I wanted to know from them was whether they knew that &quot;$\nabla$ doesn't act on vectors&quot;, or whatever.&#xA;Thought: What if you had given them a list of options for the &quot;best reason&quot; and let them choose? Maybe giving multiple choice practice problems a la &quot;choose the best reason&quot; would relieve them of having to come up with perfect wording, but tell you who knew what was really going on.&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As for writing things like $\nabla: f(x_1,x_2,\cdots,x_n) \mapsto \vec{x}$, I don't think this is necessarily too complicated (with practice). Vector calculus students are sophisticated enough to handle the new &quot;larger&quot; idea and notation about data types. Isn't it common practice for Calc books to say things like: $\frac{d}{dx}:\text{Quadratic Polynomials} \rightarrow \text{Linear Polynomials}$?&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2017-10-03T02:24:34.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12933" PostTypeId="2" ParentId="12924" CreationDate="2017-10-03T09:09:49.173" Score="12" Body="&lt;p&gt;I might seem picky, but &lt;strong&gt;I would first refrain from saying that $\nabla f$ is a vector&lt;/strong&gt;. It is a vector &lt;em&gt;field&lt;/em&gt;. This might be considered a common abuse of vocabulary, but using it amounts to assuming that student can fix it up routinely. The problem you are faced with shows very convincingly they don't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I bet we all have been confronted to someone who, asked to compute the derivative of say $\cos^2x$ at $x=\pi$, wrote &#xA;$$(\cos^2(\pi))'=((-1)^2)'=(1)'=0$$&#xA;Even if your students seem to have passed this level of confusion once they reached the multivariate calculus class, they very probably can fall in it again when confronted with the more sophisticated material of this class. Therefore you need much greater care and rigor than will be strictly necessary later on, with students who passed the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now to the heart of your problem. I don't pretend I have a complete solution, it is a prevalent and very difficult problem; my propositions are partial, and are what I try to do (I do not follow them as strictly as I should myself, even if I speak first person below). I would advise to&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;discuss the question extensively in class,&lt;/li&gt;&#xA;&lt;li&gt;regularly make quizzes on the nature of objects,&lt;/li&gt;&#xA;&lt;li&gt;make clear that the understanding of the nature of the object will be tested.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;h3&gt;1. discuss the question extensively in class&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Multivariate calculus is probably one of the most ill-understood by student at the point of curriculum it is considered. It is quite easy to get students able to make a number of computations, but very difficult to get them to understand the meaning of the computed objects. One should constantly discuss the natural set to which belongs each object: $f$, $\nabla f$, $df$, $\nabla f(x_1,\dots,x_n)$, $Df(x_1,\dots,x_n)$, $Df(x_1,\dots,x_n)\bullet(v_1,\dots,v_n)$, etc. Doing so at first is in my experience not sufficient, one has to discuss this at each lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One should also discuss in class the content of your question. Present a vector field and discuss whether it makes sense or not to take its gradient. Explain why it does make sense to write $\nabla\big(\langle \nabla f, \nabla g\rangle \big)$; everything that gives you opportunities to discuss the nature of more or less complex objects.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;2. regularly make quizzes on the nature of objects&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Student need to be tested regularly, to assess their level of understanding in order to adjust the lecture, but also to convince them to work the s*** out of this mess. It is sometimes ungrateful work, but it is important work if multivariate calculus is to be taught at all (and rewarding once one gets it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some quizzes should closely match what has been discussed in class, some can be given in advance to show student they did not understand what they thought they did, to introduce some explanations they might be reluctant to give attention to at first. For the anecdote, here is my favorite kind of question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consider $F:\mathbb{R}^3\to\mathbb{R}$ defined by $F(x,y,z)=xyz-x^2-y^2-z^2$ &lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Compute the differential $DF$ of $F$.&lt;/li&gt;&#xA;  &lt;li&gt;Compute $DF(1,2,3)\bullet(4,5,6)$, i.e. the differential of $F$ at the point of coordinates $(1,2,3)$, evaluated on the vector of coordinates $(4,5,6)$.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I usually have 100% correct answers to 1. but no correct answer to 2. To make my point clear I grade this two questions together, with &lt;strong&gt;no&lt;/strong&gt; partial credit if 2. does not have a correct answer. After nailing my students with this, I can expect them to answer similar questions correctly on the final test. It took me some time assuming they would answer this correctly before realizing I had to ask them, and ask them hard.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;3. Make clear that the understanding of the nature of the object will be tested.&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;There is a more general point here. Students have a way of finding out when we are bluffing. Each time we mention something is really important, but it seems quite difficult to them, and we don't test it for fear they will fail, they register the event and realize even more that we don't really mean it when we say it is important. I thus try not to say something is important if I don't want to test it, and I try to test everything that I mentioned as important. If I don't have the time to test everything, I give priority with what students would not want to be on the test. The goal is not to make them miserable, so I tell them in advance how I work for them to be warned. They usually think I BS them, until one or two tests go by, and then they know I am serious and start taking me (and, more importantly, their need to work) seriously.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-10-03T09:09:49.173" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12934" PostTypeId="1" CreationDate="2017-10-03T08:56:28.833" Score="7" ViewCount="382" Body="&lt;p&gt;My government has told me that I must teach 17 year olds how to invert a 3 by 3 matrix using adjoints/cofactors, without using any technology.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there any reason why you would want to know how to do this by hand? (As opposed to on calculator)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When we went through the step-by-step recipe (&quot;And now, you change the signs of these entries...&quot;) the students wanted to know &lt;strong&gt;why&lt;/strong&gt; it works. I am loath to hand over a magic recipe without justifying, but am unsure of what to say. What could I say?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8797" OwnerDisplayName="Jezza Judge" LastActivityDate="2017-10-03T23:14:49.623" Title="Why bother calculating the inverse of 3 by 3 matrix?" Tags="|inverse|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12935" PostTypeId="2" ParentId="12934" CreationDate="2017-10-03T13:56:34.240" Score="0" Body="&lt;p&gt;It shows why every matrix with non-zero determinant has an inverse, and vice-versa, every invertible matrix has non-zero determinant.  So there is a quick calculation to decide whether three simultaneous equations have a unique solution.&lt;/p&gt;&#xA;" OwnerUserId="8798" LastActivityDate="2017-10-03T13:56:34.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12936" PostTypeId="2" ParentId="12934" CreationDate="2017-10-03T23:03:28.183" Score="3" Body="&lt;p&gt;In computer graphics, the &lt;em&gt;view matrix&lt;/em&gt; is the inverse of the &lt;em&gt;camera matrix&lt;/em&gt;.&#xA;This is needed, e.g., in &lt;a href=&quot;https://www.3dgep.com/understanding-the-view-matrix/&quot; rel=&quot;nofollow noreferrer&quot;&gt;game programming&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, matrices are used to convert from coordinate system A to coordinate system B, and the inverse converts in the opposite direction, from B to A. &lt;a href=&quot;https://gamedev.stackexchange.com/a/43930/13295&quot;&gt;There are circumstances where both are needed&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/qQamR.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qQamR.jpg&quot; alt=&quot;ProjMat&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from UC Santa Barbara, CS180 Intro Comp Graphics.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Generally the matrices used in computer graphics are $4 \times 4$&amp;mdash;homogenous coordinates. But that is a minor difference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this motivates the need for the inverse (&quot;Why bother calculating the inverse ...&quot;?). You could not program this without knowing how to do it &quot;by hand&quot; (1). So a motivation for understanding it at that level is that detailed knowledge is needed to actually implement the inverse. A motivation for understanding the concept is that one cannot become a game programmer without understanding matrix inverses.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2017-10-03T23:14:49.623" LastActivityDate="2017-10-03T23:14:49.623" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12937" PostTypeId="1" AcceptedAnswerId="12946" CreationDate="2017-10-03T23:10:44.053" Score="8" ViewCount="288" Body="&lt;p&gt;How do you explain straightness to a 5-year-old?&#xA;I am having a hard time trying to explain straightness to little kids. I have been asked, &quot;How do you know that the ruler is straight?&quot; Are there any physical activities that can be done to demonstrate that something is straight?&lt;/p&gt;&#xA;" OwnerUserId="8800" LastEditorUserId="77" LastEditDate="2017-10-04T01:31:00.207" LastActivityDate="2017-10-10T19:22:05.783" Title="How do you explain straightness to a 5-year-old?" Tags="|geometry|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12938" PostTypeId="2" ParentId="12922" CreationDate="2017-10-03T23:22:08.610" Score="1" Body="&lt;p&gt;I will use the term &quot;formative assessment&quot; to refer to the &quot;activity&quot; points you describe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some positives to awarding points in this way:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;b&gt;You can give students the opportunity to practice math “under pressure”. &lt;/b&gt;Not every “formative assessment” has to be called out as such before it is given. I tell my students that there will be an assessment each-and-every class period. It turns out that half the time it’s simply a “practice” quiz, and students grade themselves (everyone gets the 5 points if they were present) – the other half are “real” quizzes with just a few questions, and I grade those for correctness. Taking a timed assessment is something that requires practice – many students need to actually learn how to take one, so this is something we practice. I find they get better and more comfortable with them over the term. Because I think this is important, I will award points for it.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;You can give immediate feedback.&lt;/b&gt; Students want to know if they’re keeping pace with the learning, and there is no harm in giving them that feedback. I remember taking classes where, until the midterm exam (5 weeks in), my instructor had no way of knowing if the class was “getting it”. Similarly, we students didn’t really know how we would perform on an exam, because we didn’t know the instructor’s style of question-writing, expected depth of understanding, etc. It’s great to get and give feedback, and it’s great if students can learn something in the process. Again, because I think it is important, I award points for it.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;You can tailor subsequent assessments to previous performance.&lt;/b&gt; If your students fail a particular question on a quiz, then you can change how that particular question is asked on the next assessment – driving at the real issue: Was the question poorly worded, or did they just not understand the concept, etc.? &lt;b&gt;Example&lt;/b&gt;: I gave a formula for a function $f(x)$ and asked students to “Solve the equation $f(x) = 5$.” I got a variety of answer types: $x = 7$, or the point $(5, 7)$, or some evaluating $f(5)$. It occurred that some knew what the question meant, but needed prompting that they were just finding an $x$-value. So on the next day’s assessment, I wrote a new function and said “Find $x$ if $f(x) = 13$. Give your answer in the form $x = $.” I used it to instruct them that the two quiz questions were identical in meaning, and that they should be wording their answers to both questions as they were prompted to in the 2nd version. Because I want students to practice what they'll encounter on the exam, I will award points for it.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;You get more data points for an overview of a student’s progress.&lt;/b&gt; There is something about actually being present for lessons (provided they aren’t terrible) that is what we’re selling in higher ed. I wouldn’t want to pass a student if they skipped the whole term and just showed up to ace the final. There are larger discussions they need to engage in, problem solving skills they need to practice, pitfalls they need to learn to avoid – there is more going on than just the things I ask on the final exam. A student can come to class on the first day with all of the knowledge necessary to pass the final exam (I see this every term), but they haven’t “taken the class” for all it is worth.&lt;/li&gt;&lt;/ol&gt;&#xA;" OwnerUserId="470" LastActivityDate="2017-10-03T23:22:08.610" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12939" PostTypeId="2" ParentId="12937" CreationDate="2017-10-04T01:28:12.000" Score="6" Body="&lt;p&gt;You can define a straight line as something that looks the same after you flip it (either parallel to itself or perpendicular to itself), rotate it, or move it along itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see if an edge of a ruler is straight, first draw a line with it.  Then flip it over (one way) and draw a new line over the old one.  Do it again different ways (flip over the other way, rotate, move along itself).  If they look different, then the ruler isn't straight.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2017-10-04T01:37:10.983" LastActivityDate="2017-10-04T01:37:10.983" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12940" PostTypeId="2" ParentId="12927" CreationDate="2017-10-04T07:02:09.220" Score="1" Body="&lt;p&gt;I think you need to consider what is really under each of those terms (student-based versus direct instruction).  For example one of the first answers here assumes that student based equates to discovery method.  If I self-study a textbook (reading the chapter, doing homework problems, grading myself, etc.), is that student based?  (yes).  But is it discovery method (no).   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are more than two simple choices for how to teach a subject.  To even have a good hypothesis to test, a little more effort is needed on definition of the alternatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do like that you have restricted the subject (HS math), but I would also consider that different methods may work better with different populations, especially student ability.&lt;/p&gt;&#xA;" OwnerUserId="8802" LastActivityDate="2017-10-04T07:02:09.220" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12941" PostTypeId="2" ParentId="12924" CreationDate="2017-10-04T07:09:50.903" Score="1" Body="&lt;p&gt;I think the panapoly of curves, vectors and surfaces is what makes 3rd semester calculus a bit of a mine field. There is a reason why there was a book written called &quot;Div, Grad, Curl, and All That.&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe it is just a little bit like QM, where you just have to get used to the strangeness (single variable calculus on the other hand seems very intuitive).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in addition to your test question, I would just show some sort of table that has the different terms and concepts and then which are vectors or curves or what have you. You can have the formula in the table, the name, the notation, what it is useful for (application), physical analogy, other?. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I seem to remember such a table used by Feynman in his lectures for exactly this topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Show the table, drill on it.  Occasionally ask questions on it.&lt;/strong&gt; I think this is the only way to get used to Div, Grad, Curl and All That. &lt;/p&gt;&#xA;" OwnerUserId="8803" LastEditorUserId="11" LastEditDate="2017-10-04T20:24:28.863" LastActivityDate="2017-10-04T20:24:28.863" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12942" PostTypeId="2" ParentId="12922" CreationDate="2017-10-04T07:20:34.667" Score="0" Body="&lt;p&gt;In addition to 138's points, it can serve as a motivation to push some work getting done by the student.  Not everyone is a perfectly rational time utilizer balancing work and study.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This doesn't mean I agree that all the pros overcome the cons.  I do think there can be some busy work on being in a class versus self studying.  But also, there can be students that just never get the kick in the butt needed until the final (failure) who would be better caught with intermediate interventions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My preference for math would be intermediate testing steps rather than class participation though.  One of the best courses I ever took was a 5-day a week class that had every Friday as an hour test.&lt;/p&gt;&#xA;" OwnerUserId="8804" LastActivityDate="2017-10-04T07:20:34.667" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12943" PostTypeId="1" AcceptedAnswerId="13012" CreationDate="2017-10-04T13:27:17.157" Score="1" ViewCount="274" Body="&lt;p&gt;I'm using a textbook but in my course the chapters are numbered differently. For example, the book has the following chapters:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Real Numbers&lt;/li&gt;&#xA;&lt;li&gt;Square Roots&lt;/li&gt;&#xA;&lt;li&gt;Polynomial in a single variable (they don't say indeterminate!)&lt;/li&gt;&#xA;&lt;li&gt;Circles&lt;/li&gt;&#xA;&lt;li&gt;Quadrilaterals&lt;/li&gt;&#xA;&lt;li&gt;System of Equations&lt;/li&gt;&#xA;&lt;li&gt;Fractional Expressions (that is rational fractions)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;whereas the first chapter in my course is &lt;em&gt;Real Numbers&lt;/em&gt; which contains chapters 1 and 2 of the book, and my chapter 2 is &lt;em&gt;Algebraic Expressions&lt;/em&gt; which contain chapters 3 and 7 of the book since they are closely related.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now there's a problem in doing this in middle school. Doing this at high school is fine but in middle school, students resist changing the order of chapters and see it as a way to make the subject harder for them not the other way. Also they are not rational enough to understand the arguments I give for this. &lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="470" LastEditDate="2017-10-04T16:42:09.273" LastActivityDate="2017-10-20T19:08:44.827" Title="The sequence of chapters" Tags="|textbooks|course-sequence|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="12945" PostTypeId="1" CreationDate="2017-10-04T23:25:09.453" Score="5" ViewCount="615" Body="&lt;p&gt;Why is it difficult to find really challenging vector analysis problems         (problems about Green's, Stokes' and Gauss' theorems in a Calculus 3 course) in Calculus books? Most of the problems are elementary, at least that's the impression I have(I could be wrong). Is it really difficult to formulate new and interesting problems in this part(Vector Analysis), or do the authors try to go very slowly on this part of the subject?&lt;/p&gt;&#xA;" OwnerUserId="6569" LastActivityDate="2018-06-06T01:41:21.260" Title="A question about Vector Analysis problems" Tags="|undergraduate-education|problem-design|vector-calculus|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12946" PostTypeId="2" ParentId="12937" CreationDate="2017-10-04T23:41:17.427" Score="7" Body="&lt;p&gt;I have done this with my five year old:  If you put two points on a piece of paper and ask them to draw the shortest path from one point to the other.  It's good fun watching them work out it out, but for her, at least, she got the idea pretty easily.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-10-04T23:41:17.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12947" PostTypeId="2" ParentId="12894" CreationDate="2017-10-04T23:53:23.977" Score="0" Body="&lt;p&gt;I think there are a number of memoirs/biographies that are appropriate a young future scientist:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- In Code: A Mathematical Journey by Sara Flannery (it's been a while since I've read this one)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- Lab Girl by Hope Jahren (I'm reading this one now)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a few compilations of mini biographies of women scientists (I've heard good things about Headstrong: 52 Women Who Changed Science — And The World by Rachel Swaby) but I haven't looked at them personally.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-10-04T23:53:23.977" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12948" PostTypeId="2" ParentId="12937" CreationDate="2017-10-05T12:53:46.110" Score="4" Body="&lt;p&gt;For a ruler: hold it up, perpendicular to your face, adjust it so the entire edge of the ruler seems to be a single point.  If some parts of the edge bulge this way or that, you will be able to see that it is not straight.  (Of course this pushes the question to something else: does light travel in a straight line...)&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-10-05T12:53:46.110" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12949" PostTypeId="1" CreationDate="2017-10-05T14:33:05.757" Score="6" ViewCount="7008" Body="&lt;p&gt;Comparing an &lt;a href=&quot;https://i.stack.imgur.com/HsDcz.png&quot; rel=&quot;noreferrer&quot;&gt;old&lt;/a&gt; french &lt;a href=&quot;https://i.stack.imgur.com/ClpgG.png&quot; rel=&quot;noreferrer&quot;&gt;mathematics&lt;/a&gt; book for grade 9 and a new one, one obvious difference is the ordering of the chapters. In the old book, all the algebra chapters are grouped together then all the geometry chapters so the teacher either should finish all the arithmetics and algebra before moving to geometry then to calculus or the school should have separate weekly periods for algebra and geometry. The new book mixes algebra and geometry chapters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it better to finish algebra chapters before geometry or to mix them? I prefer the first one, I don't like to have geometry chapters between the polynomial chapter and the rational fractions for example or to have algebra chapters between similar triangles and trigonometry.&lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2018-01-13T22:14:51.877" Title="Should we finish algebra before starting geometry?" Tags="|curriculum|education-research|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="12950" PostTypeId="2" ParentId="12949" CreationDate="2017-10-05T17:39:36.850" Score="3" Body="&lt;p&gt;I don;t really know definitely but maybe the following is helpful.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Discussion of same question:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://forums.welltrainedmind.com/topic/245197-algebra-ii-or-geometry-after-algebra-idoes-it-really-matter/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://forums.welltrainedmind.com/topic/245197-algebra-ii-or-geometry-after-algebra-idoes-it-really-matter/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;My personal experience was that I probably had the brains to go on to Alg2 right after Alg1, I lacked the maturity (too be specific, I was too lazy).  Geometry ended up being a good break before slamming into logarithms and the like.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;2.5. Also, geometry is a weird beast anyway you cut it (versus the algebra to calculus track).  But there is maybe a tiny value in getting exposure to proofs before doing induction in algebra 2.  [I think this is a miniscule positive, but including because it occurred to me.]&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;In the USA, there used to be some benefit in getting Alg1 &amp;amp; Geometry before hitting the SATs.  I think there may be some Alg 2 content in there now, but still not much.  So you are maybe a little marginally better prepared (earlier) to take the SAT and practice it, if you take geometry earlier. That test loves the darned vertical angle theorem.  Um...and adding stuff to 180.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think in the US and Anglo countries, the more typical fashion is algebra 1, geometry, algebra 2.  So there are more books, school systems, etc. geared to it.  VHS versus beta.  Or maybe there was a reason why they converged to that, not sure.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;4.5.  I guess traditionally schools (even quite good ones with smart students) didn't cover more than algebra 1 and geometry at all.  Algebra 2 was &quot;college algebra&quot;.  So maybe there is some argument that if you are only going to get two things done, get algebra 1 and geometry.&lt;/p&gt;&#xA;" OwnerUserId="8813" LastActivityDate="2017-10-05T17:39:36.850" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12951" PostTypeId="2" ParentId="12945" CreationDate="2017-10-05T17:49:06.900" Score="-1" Body="&lt;p&gt;I suspect calc 3 suffers a little from not having enough time to cover all the topics and really go in some depth on them.  A compromise is made where students get at least an exposure (but not mastery) to Stokes and the like, towards the end of their class.  I don't know a good improvement for this given the time available and the content that needs to be covered and that while the topic may be easy for math profs it is always new and hard for the endless newly birthed students.  By analogy same issue exists with Laplace transforms in diffyscrews, where you just do an exposure since there is not time for more and at least that way, they've heard of it (and can learn more in an EE or ME controls class if needed).&lt;/p&gt;&#xA;" OwnerUserId="8814" LastActivityDate="2017-10-05T17:49:06.900" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12952" PostTypeId="2" ParentId="12945" CreationDate="2017-10-05T17:55:42.230" Score="-4" Body="&lt;p&gt;FWIW, physics students get a lot more exposure, practice, and harder problems in junior year classical E&amp;amp;M.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just check out the first 40 pages of Wangsness. (And that's the review, the rest of the whole book is just practicing more vector calc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.scribd.com/document/342675118/Roald-K-Wangsness-Electromagnetic-Fields-pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.scribd.com/document/342675118/Roald-K-Wangsness-Electromagnetic-Fields-pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[combine please]&lt;/p&gt;&#xA;" OwnerUserId="8815" LastActivityDate="2017-10-05T17:55:42.230" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12953" PostTypeId="2" ParentId="12937" CreationDate="2017-10-05T18:16:47.847" Score="1" Body="&lt;p&gt;I best do this by placing any object at the centre of a neatly drawn (perfect) circle and tell them that the relative distance between the object and any point on the circle is straight. So if it is a ruler, I keep one end fixed and rotate the other edge about the fixed point to draw a circle and the tell them that any distance the ruler creates from centre to circumference of circle qualifies as STRAIGHT.&lt;/p&gt;&#xA;" OwnerUserId="8816" LastActivityDate="2017-10-05T18:16:47.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12954" PostTypeId="1" CreationDate="2017-10-05T19:14:28.873" Score="0" ViewCount="131" Body="&lt;p&gt;May I get some recommendation on text book on real analysis for undergrad in statistics? &#xA;Thank you. &lt;/p&gt;&#xA;" OwnerUserId="6370" LastActivityDate="2017-10-05T23:54:15.983" Title="Text book on real analysis for undergrad in statistics" Tags="|textbooks|mathematical-analysis|statistics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12955" PostTypeId="2" ParentId="12954" CreationDate="2017-10-05T23:54:15.983" Score="1" Body="&lt;p&gt;Maybe this would be more useful &quot;for [an] undergrad in statistics&quot;&#xA;than a text on pure real analysis? &#xA;A review in an ASA journal says it is &quot;user-friendly&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Adams, Malcolm Ritchie, and Victor Guillemin.&#xA;  &lt;em&gt;Measure theory and probability&lt;/em&gt;. Boston: Birkhäuser, 1996.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;a href=&quot;https://i.stack.imgur.com/ZmiCD.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZmiCD.jpg&quot; alt=&quot;cover&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;…the text is user friendly to the topics it considers and should be very accessible…Instructors and students of statistical measure theoretic courses will appreciate the numerous informative exercises; helpful hints or solution outlines are given with many of the problems. All in all, the text should make a useful reference for professionals and students.&quot;&amp;mdash;&lt;em&gt;The Journal of the American Statistical Association&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-10-05T23:54:15.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12956" PostTypeId="1" AcceptedAnswerId="12957" CreationDate="2017-10-06T09:57:18.410" Score="4" ViewCount="3727" Body="&lt;p&gt;Is there a specific example when the analytic form of a derivative $\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ is preferred to the numerical form $\frac{f(x+h)-f(x)}{h}$, $h \ll 1$? Are there cases when the numerical approach fails?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm trying to figure out why in the secondary school we focus on the analytic form and not on the numerical one. In my opinion, the numerical form is much easier to understand since it doesn't require a difficult mental transition from a secant to a tangent. In addition, the numerical approach provides good accuracy for calculating the derivative. Finally, calculators and computers use the numerical approach to plot derivatives so that they can be easily visualized.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, we can say that the analytic form is important for definition of continuity, Taylor series etc. These topics, however, are higher level mathematics and can be left for the classes with advanced math or even college.&lt;/p&gt;&#xA;" OwnerUserId="6306" LastActivityDate="2017-11-25T22:46:47.023" Title="When analytic form of derivatives is preferred over numerical form?" Tags="|calculus|derivative|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12957" PostTypeId="2" ParentId="12956" CreationDate="2017-10-06T12:38:38.307" Score="7" Body="&lt;p&gt;I think maybe if you try to write down precisely what you mean by $h&amp;lt;&amp;lt;1$ you will end up writing the definition for $\lim_{h \to 0}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From this point of view, there is really no difference between the two approaches; what you are calling the numerical approach is the same thing as the analytical approach.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For example, try to compute the derivative of $\frac{1}{(x-2.99)(x-3.01)}$ at $x = 3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The numerical approach will initially use $h = 0.1$, but you will notice that when you switch to $h = 0.01$, something qualitatively changes, so you go with $h = 0.001$ instead, and then $h = 0.0001$, and then you are convinced that things are going well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you are doing is finding the limit of a sequence, letting $h \to 0$ by using the sequence $h_n = 10^{-n}$.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-10-06T12:38:38.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12958" PostTypeId="2" ParentId="12956" CreationDate="2017-10-06T13:53:06.697" Score="4" Body="&lt;p&gt;I'm reading the difference between your two formulae as the fact that, in the first, you have a limit and, in the second, you compute the difference quotient for a single, small, value of $h$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we talk about the analytic derivative we usually are in the situation that $f$ has some definite, known, formula. That often allows you compute the limit algebraically. Students do encounter a few more cases like $e^x$ and the trig functions which require additional lemmas but in either case, you end up with collection of rules which transforms a formula to another formula. This is good because we get the value of the derivative everywhere at the cost of a fairly small amount of work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Numerical differentiation is an approximate version which is useful in the case where we have some function, but we don't know a formula for it. For example, the function is known at a finite sample of points; like data from an experiment. Since you don't know a formula for the function or its value at most places, you &lt;em&gt;cannot&lt;/em&gt; take a limit. You don't get as much out of the numerical derivative and it will not be perfectly accurate, but it is all you have available to you when you are working with discrete data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Numerical_differentiation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Numerical_differentiation&lt;/a&gt; for some error estimates and some multi-point numerical methods with greater accuracy.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2017-10-06T13:59:16.500" LastActivityDate="2017-10-06T13:59:16.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12961" PostTypeId="1" AcceptedAnswerId="12967" CreationDate="2017-10-07T05:53:13.570" Score="0" ViewCount="174" Body="&lt;p&gt;In a few weeks i'll be interviewing at a community college in California for a tenured Mathematics position. They've asked me to present the following in only 12 minutes. I'd appreciate any constructive feedback or advice for the plan of action that I have:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teaching Demonstration&lt;/strong&gt;: You will be asked to give a twelve-minute teaching demonstration, during which the interview panel will play the role of your Statistics class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please motivate/introduce type I and/or type II errors. This is an algebra-based introductory level statistics course, and you may assume you have already presented the basics of hypothesis testing, so you are showcasing a twelve-minute sample from your typical class session.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The room in which you will give this teaching demonstration is equipped with an LCD projector, with both VGA and HDMI connections, and there is also a document camera and a desktop computer running Windows 7 and Office 2016 attached to this projector.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;My approach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;**1) Motivate the Type I and Type II error using the example below and say &quot;There's always room for error when doing H testing&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Type I: When we reject a valid claim&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Type 2: When we support an invalid claim**&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2) Remind students about our &quot;H-Testing Graphic Organizer for Population Proportion P&quot;&lt;/strong&gt;:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/m1J4K.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/m1J4K.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3FKvx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3FKvx.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3) Model an example and use the projector to walk students through my built in TI-84 on my computer to help guide them on the TI instructions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LaWai.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LaWai.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CeC3R.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CeC3R.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4) Model how a Type I error can occur in this example.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5) Model another example where a Type 2 error can occur&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2198" LastEditorUserId="2198" LastEditDate="2017-10-07T17:39:17.203" LastActivityDate="2017-10-09T20:04:30.283" Title="Advice for Community College Interview Statistics Demonstration" Tags="|mathematical-pedagogy|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="12962" PostTypeId="2" ParentId="2164" CreationDate="2017-10-07T19:39:01.417" Score="3" Body="&lt;p&gt;Phase 1. Students scour the new chapter, extract key information and write it in their notebooks. In our textbook series, key information is conveniently presented in blue boxes. They don’t need to understand it yet so this is low cognitive load work. But knowledge is slowly starting to go in anyway. This activity distills the great unknownness of the chapter into a small unknownness which can serve as the reference point for reading the expository parts later. It’s easy to push students along here. But a boring way to start a Unit. I generally throw in a few schemata-supporting activities and games first. And at least one good challenging question (ye olde opening problem) which the students can at least understand the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Phase 2. Consider, reflect, discuss the material from Phase 1 as a class. Be careful not to get caught up correcting and teaching here. The stuff distilled from the textbook needs to become shared knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Phase 3. Examples and exercises. Mathematics is learned best by just doing it. So, students look at the next problem. If they can do it, they do it. If they can’t do it, they refer first to their notes from phase 1. If that helps, they move on. If it doesn’t, they know where to go hunting in the section/chapter for more details (because at least they remember what page they copied that stuff from). Provided you have given students a list of exercises to complete, they can get started and work at their own pace. Our textbook series has too many questions, so I publish a minimum list at the start of the year/Unit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Phase 4. Step back from the exercises and discuss what we can now do. What have we learned? Skills, concepts, facts. This is where we focus on the processes and concepts. And we start to consider how they could be used in other contexts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Phase 5. Explorations, investigations, challenging problems. If there is still time left in the Unit (it depends on the students) we spend time tackling bigger things and learning how to communicate using our new mathematical language and skills. The textbook sometimes has bits for this but often I have to draw on other resources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That’s basically how I approach using a textbook well and making sure students get the most benefit out of the expensive books they’ve purchased (although my current school loans them to students from the library). Plus, I don’t enjoy reinventing the wheel. We have good books so I want to use them as best we can.&lt;/p&gt;&#xA;" OwnerUserId="7046" LastEditorUserId="7046" LastEditDate="2017-10-07T19:44:51.257" LastActivityDate="2017-10-07T19:44:51.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12963" PostTypeId="2" ParentId="5877" CreationDate="2017-10-07T20:02:18.467" Score="2" Body="&lt;p&gt;Yes. Making sure a solution makes sense in the context of a real-life situation is an essential part of applied mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it’s not just enough to have “real-life” problems with solutions that make sense in the context of real-life situations. When a ridiculous context is forced onto a pure mathematics problem, you reinforce the pervasive belief that mathematics is not actually useful in real life. Real-life tasks/problems have to be authentic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the worst thing we do as mathematics teachers is invent real-life contexts to make pure mathematics questions seem more relevant. I think the results are lame 99% of the time.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Robin and Pat are planning a wedding banquet. The cost per guest, $G$ dollars, is modelled by the function $G(n)=95e^{(-0.02n)}+40$, for $20&amp;lt;n&amp;lt;200$, where $n$ is the number of guests.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From a real examination question!&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“… and they say pure mathematics has no real-world applications.” (Professor Farnsworth, Futurama S06E10)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Robin and Pat are the sort of people who scare others away by applying mathematics indiscriminately and are unlikely to have friends numbering in the range given.&lt;/p&gt;&#xA;" OwnerUserId="7046" LastEditorUserId="7046" LastEditDate="2017-10-07T20:14:02.013" LastActivityDate="2017-10-07T20:14:02.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12964" PostTypeId="1" CreationDate="2017-10-08T14:34:16.073" Score="11" ViewCount="434" Body="&lt;p&gt;It seems there is an overall agreement that Geometry is the right place to introduce proof in Basic School. However, number theory (arithmetic) looks like to be a more simple environment (consider, for instance, a sentence like this one: &quot;if &lt;em&gt;n&lt;/em&gt; is integer and multiple of 2, then &lt;em&gt;n&lt;/em&gt; is multiple of 4&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any references (books, scientific articles, thesis) that had studied this question, I mean, the use of number theory instead geometry to introduce proof in Basic School?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2019-05-07T04:37:23.963" Title="Using number theory instead geometry to introduce proof in Basic School?" Tags="|proofs|geometry|number-theory|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="12965" PostTypeId="2" ParentId="12961" CreationDate="2017-10-08T23:06:47.227" Score="2" Body="&lt;p&gt;I would say one of the most important things to show off would be your ability to manage time and realistically plan lessons. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whatever you decide to do, especially if it is the outline you have showed us above, please practice it and time it. Leave time for interaction with the class.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-10-08T23:06:47.227" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12966" PostTypeId="2" ParentId="12961" CreationDate="2017-10-09T06:47:50.940" Score="1" Body="&lt;p&gt;You have 12 minutes. You need to make every one count. Thus, I would not spend any time doing anything not directly in service of adequately and effectively teaching the two errors. This means that the walkthrough of the entire hypothesis test is probably not going to score you any points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would motivate things by revisiting the idea of a p-value decision and reinforcing that we're still playing a probability game and hence could be wrong. Once we establish that we can be wrong, it's off to the races to classify our errors. Once classified, we put our knowledge into practice with a few examples consisting of, &quot;in test X, researchers concluded Y about claim Z; what type of error is possible?&quot; and the like.  By that time, you've already blown past your 12 minutes unless you've been extremely concise. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I've never met an interview committee that was disappointed by someone who could use a white board and marker(s) effectively. While a solid use of (unobtrusive) technology can be an aid, you would probably be better off to focus on creating a well-crafted low-tech presentation than to risk all the things that could go wrong (both technically and content-wise) by doing a computer aided presentation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="6949" LastActivityDate="2017-10-09T06:47:50.940" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12967" PostTypeId="2" ParentId="12961" CreationDate="2017-10-09T20:04:30.283" Score="2" Body="&lt;p&gt;*Please motivate/introduce type I and/or type II errors.*&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having been on hiring committees at community colleges and 4-year schools, I always read the directive to &quot;motivate/introduce&quot; here as an opportunity for candidates to really step take a back and show how they would use non-mathematical language (first) to help students care about some new thing. This is often done through questioning them on a current event or something from the news, or a hypothetical where students are asked to bring their opinions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of *motivating* type I and/or type II errors, I think your example would be improved by having students first think about the severity of the two situations: &quot;10% of students smoke, but we end up rejecting this&quot; or &quot;Some other percent of students smoke, but we don't reject the idea that 10% smoke.&quot; Yours might be a tricky example with which to do this (since it is a two-tailed test), but getting students to understand which situation could be &quot;worse&quot; could help them to *want* to identify these situations by name. Maybe ELAC is studying this in order to choose a plan of action -- If $H_0$ is rejected (in favor of $p$ being greater than 0.10, say), then the administration will pour resources into reducing the number of smokers. If $H_0$ is *not* rejected, then the administration will use the funds for something else needed on campus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A question to students could be: &quot;Which situation would you find more objectionable: There really are only 10% that smoke, but the test causes us to reject this number, and the administration pours money into helping reduce the number of smokers&quot;, or; &quot;The percent of smokers is higher than 10%, but the test causes us to *not* reject the assumed number of 10%, and the administration saves the money for something else.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I like examples like yours, but given you only have &lt;b&gt;12 minutes&lt;/b&gt;, I might focus on having students understand what the errors mean in context first. Then, after they could describe/identify the errors in a few different settings, explore the math involved. As for the work you plan to show, this is what I would expect of an interviewee to show us for the &quot;follow-up&quot; discussion. [The best interview demos begin with a here's-what-we've-done (usually just printed and handed out), then the demo, and then a where-we-will-go-next (also printed and handed out) to show the interview committee that you actually have a plan and you know where the lesson fits into the whole term.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;tl;dr version as my opinion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;li&gt;Spend most of the demo on meaning first, instead of jumping to calculations.&lt;/li&gt;&#xA;&lt;li&gt;Build your calculations into a discussion of where you would go with the class &lt;em&gt;after&lt;/em&gt; the demonstrated material.&lt;/li&gt;&#xA;&lt;li&gt;Make sure to develop a full plan for before-during-after the demo, and give this to the committee so they know.&lt;/li&gt;&#xA;&lt;li&gt;Overall, keep the introduction light, meaningful and active for students. Ask them questions in plain language first, so they won't have to immediately jump a math/notation hurdle before understanding what's going on.&lt;/li&gt;&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2017-10-09T20:04:30.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12968" PostTypeId="1" CreationDate="2017-10-09T21:51:52.273" Score="11" ViewCount="399" Body="&lt;p&gt;I am teaching Calculus 2 and 3 this semester from a common text, and a significant number of my students are handing in perfect homework assignments that are clearly copied from Chegg, et al.  As homework grades are a minuscule part of their overall grade, I'm not concerned that some of them are &quot;getting away&quot; with this, because they'll get washed on quizzes and exams if they can't do the work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I am concerned about what cheating does to their willingness to ask good questions in class or come to office hours.  Students are already hesitant to ask questions of professors, but I believe cheating on homework creates two additional roadblocks:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Students may be worried that asking questions will make it more obvious that they have no idea what they're doing, despite the fact their work looks perfect.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students may be worried that coming to office hours will allow them to be cornered by me, since their lack of understanding will be evident in a one-on-one conversation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;While they have chosen their own path by deciding to cheat, I would like to have a better understanding of what I can do to remedy this situation.  The question of the cheating itself has already been addressed in considerable detail &lt;a href=&quot;https://matheducators.stackexchange.com/questions/6/how-to-assign-homework-when-answers-are-freely-available-or-attainable-online/&quot;&gt;here&lt;/a&gt;, but I am more interested in how best to solicit more questions from students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any thoughts or experience with this situation?&lt;/p&gt;&#xA;" OwnerUserId="5988" LastActivityDate="2017-10-09T23:48:50.003" Title="How cheating on homework affects student questions" Tags="|homework|student-question|cheating|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="12969" PostTypeId="2" ParentId="12968" CreationDate="2017-10-09T23:48:50.003" Score="5" Body="&lt;p&gt;&lt;p&gt;Imagine a scenario where you don't have a hypothesis about your students' unwillingness to ask questions in class or during office hours. Your question may just boil down to: &quot;How do you get students to interact with you in class, without expecting them to initiate?&quot;Absent your hypothesis, you might:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&lt;li&gt;Ask them to (anonymously?) submit their answers to a general question, such as &quot;What is something you wish we spent more time on in class?&quot;&lt;/li&gt;&#xA;&lt;li&gt;Have them work on a problem in groups of four students and write up, as part of their solution, what they though was the hardest part of the problem, and one thing they still aren't sure of. Read the responses to the class, and comment on them as necessary.&lt;/li&gt;&#xA;&lt;li&gt;Make &quot;asking a question&quot; part of their homework. Give them examples of &quot;good questions&quot; you might want to see.&lt;/li&gt;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;We all know that some students cheat. I open the internet and show my classes Wolfram|Alpha and the Mathematics stack exchange, and that they can find answers to most of their questions in relatively little time. I have a serious conversation with them, warning them that they will most-likely fail if they can't figure out how to work on their own. I make myself available. However, I know that many students have just been brought up (scholastically speaking) with this as their mode of learning. They will not be bothered to physically come to my office, during a certain window of time, and ask me stuff. And I completely understand this!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I think the best questions can be asked in class, and I have a greater chance of hearing them if I make them (questions) part of the assignment.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2017-10-09T23:48:50.003" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="12970" PostTypeId="2" ParentId="12937" CreationDate="2017-10-10T19:22:05.783" Score="3" Body="&lt;p&gt;Hold a piece of string at the two ends of the ruler (just above). Does the ruler follow the path of the string?&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-10-10T19:22:05.783" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12971" PostTypeId="1" CreationDate="2017-10-11T03:32:28.827" Score="2" ViewCount="405" Body="&lt;p&gt;Is it advisable to set questions of multiple-choice, true/false, and fill-in-the-blanks in the bachelor's exam papers of courses on  topology, analysis, functional analysis, and abstract algebra? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this practice in vogue at any reputed universities in the USA, UK, Canada, Australia, or anywhere else in the world's more advanced countries? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can somebody please send me some relevant links to the webpages of some top-ranked universities where past exam papers of these math courses are available? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In which situations are these types of questions preferred over the longer-solution 'subjective type&quot; questions? &lt;/p&gt;&#xA;" OwnerUserId="5400" LastActivityDate="2017-10-11T15:53:45.843" Title="How About MCQs, True/False, and Fill-In-The-Blank Questions in Undergrad-Level Topology, Analysis, And Abstract Algebra Exams?" Tags="|exams|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12972" PostTypeId="2" ParentId="12924" CreationDate="2017-10-11T05:00:49.040" Score="4" Body="&lt;p&gt;This is an excellent and important question! After a few years away from it, I'm teaching Multivariate Calculus this semester again, and I'm struggling with the same issues. (Well, don't we all?) The answer above from @BenoîtKloeckner is absolutely amazing, but I'd like to add one more thing. In fact, this comes straight from my experience teaching this course this semester.&lt;/p&gt;&#xA;&lt;p&gt;The thing is that I found myself talking about data types from day one. In Multivariate Calculus this becomes an issue to be addressed from the very beginning — as soon as we have both vectors and scalars around. Actually, even earlier on, there's inevitably an opportunity — and I use it! — to talk about &amp;quot;data types&amp;quot; when students set up the vector between points &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\overrightarrow{PQ}=P-Q$&lt;/span&gt;&amp;quot;. This immediately prompts a dialogue like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;– Why did you take off points here?&lt;/p&gt;&#xA;&lt;p&gt;– What are &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;? What kinds of objects are they?&lt;/p&gt;&#xA;&lt;p&gt;– They are points.&lt;/p&gt;&#xA;&lt;p&gt;– And is subtraction an operation on points? Did we ever define subtraction of points, or did we define subtraction for something else?&lt;/p&gt;&#xA;&lt;p&gt;– Oh…&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And then next we get into dot and cross products, where emphasizing the types of objects should be a daily routine. So my answer to your question is: &lt;strong&gt;Yes, of course it's extremely important, but in Multivariate Calculus this should be emphasized from the very beginning of the semester in a variety of contexts, so that by the time you get to gradients students already know that understanding data types is a big deal&lt;/strong&gt; (and gradients is just yet another context where it's again important).&lt;/p&gt;&#xA;&lt;p&gt;In fact, even in earlier classes there are some opportunities for emphasizing data types. For example, one of the common answers to the question thrown into the audience in Calculus I: &amp;quot;How do we interpret geometrically the derivative at a point?&amp;quot;, is that &amp;quot;It is the tangent line&amp;quot;, to which I retort that &amp;quot;No, it's not. The tangent line is a geometrical object, while the derivative at a point is a number, a different kind of object. It is related to the tangent line, but it is not the tangent line.&amp;quot; But if in earlier courses I find myself addressing this only sporadically, in Multivariate Calculus it is a really big deal!&lt;/p&gt;&#xA;" OwnerUserId="6762" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-10-11T05:00:49.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12974" PostTypeId="2" ParentId="12971" CreationDate="2017-10-11T15:53:45.843" Score="1" Body="&lt;p&gt;I am not from a top ranked department, but I usually have a few T/F questions just to check understanding of basic definitions and theorems.  I do require a brief justification.  Some examples from a test I gave today &quot;If the linear dipphantine equation ax+by=c has a solution, then gcd(a,b)=c&quot;, &quot;If a|c and b|c then (a+b)|c&quot;, and &quot;0 divides 0&quot;. &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2017-10-11T15:53:45.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12976" PostTypeId="1" AcceptedAnswerId="12979" CreationDate="2017-10-11T20:06:56.043" Score="7" ViewCount="140" Body="&lt;ul&gt;&#xA;&lt;li&gt;preferably areas of algebra of value and interest to computing practitioners.&lt;/li&gt;&#xA;&lt;li&gt;any level from introductory to (say) Grobner bases.&lt;/li&gt;&#xA;&lt;li&gt;preferably using open source computer algebra software.&lt;/li&gt;&#xA;&lt;li&gt;no preference as to Windows, Mac, Linux, etc.&lt;/li&gt;&#xA;&lt;li&gt;in English or French.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thanks for your input.&lt;/p&gt;&#xA;" OwnerUserId="8847" LastActivityDate="2017-10-12T05:04:25.950" Title="Looking for online abstract algebra courses making use of computer algebra systems" Tags="|abstract-algebra|online-instruction|computer-programs|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12977" PostTypeId="1" CreationDate="2017-10-12T00:05:22.783" Score="5" ViewCount="716" Body="&lt;p&gt;I'm in Algebra II this year, and I have to admit, it's kind of boring. The only new thing we've touched on so far this year is how to graph piecewise functions, and those are really easy to graph. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As it is, since the teacher puts the problems for the week on the backboard, I work ahead based on the homework for the next couple of days (there were a couple of days where I was the full week ahead, so I just read in the back of the class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I'm investigating options to generally accelerate the class (maybe try to finish it in a semester instead of two?) I know the likelihood of those options working out is very, very low, so I'm asking this question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What sorts of extension or substitute activities could be done based on the Algebra II curriculum in the United States? For whatever it's worth, I've done a decent amount of math on my own - some linear algebra and calculus, a general awareness of some of the topics in abstract algebra, and a heavy interest in physics.&lt;/p&gt;&#xA;" OwnerUserId="8366" LastActivityDate="2017-12-16T22:28:39.883" Title="Extension activities in Algebra II" Tags="|algebra|gifted-students|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12978" PostTypeId="2" ParentId="12977" CreationDate="2017-10-12T02:30:42.137" Score="6" Body="&lt;p&gt;&lt;strong&gt;Edit [12.16.17]:&lt;/strong&gt; A write-up concerning a few of the problems below was accepted for publication in NCTM's grades 8-14 journal, &lt;em&gt;The Mathematics Teacher&lt;/em&gt;. It is slated for publication in Fall 2018, but a pre-pre-print can be found &lt;a href=&quot;https://www.researchgate.net/publication/318100979_Root_Problems_Non-routine_Algebra_Tasks&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; (RG link to PDF, 8pp). &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I taught two sections of Algebra 2 (ninth and tenth grade, US) last year. The materials we use are internally created, and while there is not (yet?) a public repository that makes all of them freely available, I paste below a problem set that I personally wrote for students to complete. I'm not sure if this is the sort of thing that you are looking for, but the problems are (i) related to Algebra 2; (ii) original hence cannot be found elsewhere; and (iii) nonstandard and, I think, reasonably challenging.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;1) As before, let us write $i$ to denote $\sqrt{-1}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) Is $\sqrt{5+i}$ a real number or an imaginary number? Explain how you know. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Hint:&lt;/strong&gt; Find a polynomial $f$ for which $\sqrt{5+i}$ is a root, i.e., for which $f(\sqrt{5 + i}) = 0$. What does the graph of $f$ indicate about &lt;em&gt;all&lt;/em&gt; of the roots of $f$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;b) Is $\sqrt{i}$ a real number or an imaginary number? Explain how you know, using a method similar to your approach in 1(a).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) Let $\alpha$ be the number $\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$. Carefully calculate $\alpha^2$. Does this support or contradict your answer in part (b)? Explain!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Suppose $f$ is a polynomial for which the sum of the coefficients is $0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) Give an example of a cubic polynomial $f$ that satisfies the above criterion, and find &lt;em&gt;all&lt;/em&gt; of its roots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) Explain why &lt;em&gt;any&lt;/em&gt; polynomial whose coefficients sum to $0$ has, in its graph, at least one $x$-intercept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) Give an example of a polynomial $f$ with degree $5$, whose coefficients sum to $0$, which has &lt;em&gt;only&lt;/em&gt; one $x$-intercept. Explain clearly how you know that your example polynomial has only one $x$-intercept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Suppose $f$ is the following piecewise defined function:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f(x) =$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$2x \text{ when } x \leq -3;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(x-h)^2 - 6 \text{ when } -3 &amp;lt; x \leq 0; \text{ and}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$4x-h \text{ when } x &amp;gt; 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) For what value of $h$ is the piecewise defined function above &lt;em&gt;continuous&lt;/em&gt; (i.e., the graph has no &quot;jumps&quot;)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) Using the value of $h$ that you found in 3(a), sketch a graph of $f$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) Does $f$ have an inverse function? If not, explain clearly why not; if so, explain why (but you do not need to &lt;em&gt;find&lt;/em&gt; its inverse).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Let $f$ be the polynomial defined by $f(x) = -x^5 + Ax^4 + Bx^3 + Cx^2 + Dx + 2$, where each of $A, B, C,$ and $D$ is an integer. Suppose that the graph of $f$ has &lt;em&gt;at least&lt;/em&gt; four distinct $x$-intercepts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explain clearly why $f$ must have &lt;em&gt;at least&lt;/em&gt; two &lt;strong&gt;irrational&lt;/strong&gt; roots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) Let $f$ be a quadratic polynomial that can be written in either vertex form or standard form, i.e.,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f(x) = a(x-h)^2 + k \text{     or     } f(x) = ax^2 + bx + c$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose further that the domain of $f$ is $[s, \infty)$ for $s \in \mathbb{R}$ minimal such that $f$ is invertible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) Find the inverse of $f$ when $f$ is written in vertex form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) Find the inverse of $f$ when $f$ is written in standard form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) What is $s$, and how do you know?&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2017-12-16T22:28:39.883" LastActivityDate="2017-12-16T22:28:39.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12979" PostTypeId="2" ParentId="12976" CreationDate="2017-10-12T05:04:25.950" Score="6" Body="&lt;p&gt;There's &lt;a href=&quot;http://abstract.ups.edu/aata/&quot; rel=&quot;noreferrer&quot;&gt;Abstract Algebra: Theory and Applications&lt;/a&gt; by Judson and Beezer.  It doesn't get to Grobner bases but does cover the standard material (and some non-standard stuff, too).  It has Sage code and exercises throughout.  I will also add that it includes chapters on coding theory, cryptography and lattices and boolean algebras, all of interest to people in computer science.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-10-12T05:04:25.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12980" PostTypeId="2" ParentId="12977" CreationDate="2017-10-12T13:45:03.293" Score="1" Body="&lt;p&gt;My advice is to accelerate.  Just think you are better off moving on and continuing your math education since you feel you are ready.  Get through the material and start working on trig and pre-calc.  If you want to come back and enrich at some later time, so be it.  But just think you will be better off moving on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personal experience:  When I went to school, it was normal for Algebra 2 to be combined with Trig (i.e. 3 semesters in 2) for the kids on the GT track (who had taken algebra 1 in 8th grade.)   Also, I did the Alg2/trig in summer before 10th so got more accellerated.  (this isn't to say yeah me, but that if you sense you can handle the acceleration, go ahead and power through the material and move on to a real precalc course).&lt;/p&gt;&#xA;" OwnerUserId="8850" LastActivityDate="2017-10-12T13:45:03.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12981" PostTypeId="2" ParentId="12977" CreationDate="2017-10-13T00:10:02.797" Score="1" Body="&lt;p&gt;Perhaps you would find this extends Algebra II in interesting&#xA;and challenging directions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Kendig, Keith. &lt;em&gt;A guide to plane algebraic curves&lt;/em&gt;. Vol. 46. MAA, 2011.&#xA;  (&lt;a href=&quot;https://www.maa.org/press/ebooks/a-guide-to-plane-algebraic-curves&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/MwlZp.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MwlZp.gif&quot; alt=&quot;DOL&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;From the publisher's summary:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;a friendly introduction to plane algebraic curves. It emphasizes geometry and intuition, and the presentation is kept concrete. You'll find an abundance of pictures and examples to help develop your intuition about the subject, which is so basic to understanding and asking fruitful questions.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/miWDT.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/miWDT.jpg&quot; alt=&quot;Fig1.10&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;An 8-petal rose morphing to a Lissajous figure [p.19].&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-10-13T00:10:02.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12982" PostTypeId="2" ParentId="12977" CreationDate="2017-10-13T00:18:13.727" Score="2" Body="&lt;p&gt;You might consider look at &lt;a href=&quot;https://www.exeter.edu/mathproblems&quot; rel=&quot;nofollow noreferrer&quot;&gt;Exeter's books of math problems&lt;/a&gt; or the &lt;a href=&quot;https://artofproblemsolving.com/store&quot; rel=&quot;nofollow noreferrer&quot;&gt;Art of Problem Solving&lt;/a&gt; texts.  They are targeted at students around your age but look very different from your standard Algebra II presentation.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-10-13T00:18:13.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12983" PostTypeId="2" ParentId="12296" CreationDate="2017-10-14T01:48:04.843" Score="0" Body="&lt;p&gt;There are the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/DSST_(standardized_test)&quot; rel=&quot;nofollow noreferrer&quot;&gt;DSST&lt;/a&gt; tests.  Syllabus and sample questions can be found &lt;a href=&quot;http://getcollegecredit.com/assets/pdf/dsst_fact_sheets/DSST_PRINCIPLES_OF_STATISTICS_2016-2017_Fact_Sheet.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  These exams are designed to give students college credit and according to Wikipedia &quot;are comparable to the final or end-of-course examinations in undergraduate college courses&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-10-14T01:48:04.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12984" PostTypeId="1" CreationDate="2017-10-14T17:37:57.663" Score="17" ViewCount="1102" Body="&lt;p&gt;I am teaching an intro to ring theory, and after grading the first quiz, I realize most of my students are under the assumption that rings must be commutative. I have given them the example of matrices over the reals, but clearly we need to spend a little more time on non-commutative rings.&lt;/p&gt;&#xA;&lt;p&gt;What are the most basic examples of non-commutative rings? (Most basic in the sense that I don't want to spend all day proving that this thing is indeed a ring.) Specifically, are there interesting quotients of &lt;span class=&quot;math-container&quot;&gt;$M_2(\mathbb{R})$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Please note: this is a rings-first abstract algebra class, so the group ring of a non-abelian group will take us a little far afield.&lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="376" LastEditDate="2021-01-30T16:09:40.980" LastActivityDate="2021-01-30T16:09:40.980" Title="Examples of basic non-commutative rings" Tags="|undergraduate-education|examples|abstract-algebra|" AnswerCount="8" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12985" PostTypeId="2" ParentId="12984" CreationDate="2017-10-14T17:39:53.123" Score="15" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Quaternion&quot; rel=&quot;noreferrer&quot; title=&quot;quaternion ring&quot;&gt;quaternion ring&lt;/a&gt; is a pretty simple example of a non-commutative ring (a skew-field, even).&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2017-10-14T17:39:53.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12986" PostTypeId="1" CreationDate="2017-10-14T19:02:21.093" Score="3" ViewCount="983" Body="&lt;p&gt;Is there any difference between saying that say $\pi \approx 3.14$ and $\pi = 3.14(2dp)$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A Chinese student has asked me this. They claim that the latter version doesn't exist in Chinese. They always use the $\approx$ notation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone explain whether these two notations have any difference and if so what is it?&lt;/p&gt;&#xA;" OwnerUserId="8859" LastActivityDate="2019-07-21T14:08:31.497" Title="Is there any difference between saying that say $\pi \approx 3.14$ and $\pi = 3.14(2dp)$?" Tags="|secondary-education|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="12987" PostTypeId="2" ParentId="12984" CreationDate="2017-10-14T19:54:38.397" Score="5" Body="&lt;p&gt;The Weyl algebra (the ring of differential operators with polynomial coefficients). It admits a quite concrete description and there are many interesting things to say about it.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2017-10-14T19:54:38.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12988" PostTypeId="2" ParentId="12984" CreationDate="2017-10-14T20:02:41.190" Score="8" Body="&lt;p&gt;How about $\mathbb{Z}[i]$ together with conjugation, thought of as a ring element? Conjugation and multiplication by $i$ do not commute. &lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2017-10-14T20:08:48.500" LastActivityDate="2017-10-14T20:08:48.500" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="12989" PostTypeId="2" ParentId="12984" CreationDate="2017-10-14T20:21:35.573" Score="10" Body="&lt;p&gt;Why not turn $M_2(\mathbb{R})$ into a multiplication on $\mathbb{R}^4$? Here's a fun, somewhat weird, ring:&#xA;$$ \left[ \begin{array}{cc} a &amp;amp; b \\ c &amp;amp; d \end{array} \right] \left[ \begin{array}{cc} x &amp;amp; y \\ z &amp;amp; w \end{array}\right] = \left[ &#xA;\begin{array}{cc} ax+bz &amp;amp; ay+bw \\ cx+dz &amp;amp; cy+dw \end{array}\right]$$&#xA;hence&#xA;$$ (a,b,c,d) \star (x,y,z,w) = (ax+bz, ay+bw, cx+dz, cy+dw)$$&#xA;So, $(\mathbb{R}, \star, +)$ gives a fairly unfamilar ring which is a fairly concrete example. Addition is as usual, but, multiplication is a bit surprising. The unity here is $(1,0,0,1)$. Of course, you see right through that, but, to students this would be shrouded in mystery for a little while.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2017-10-15T04:26:17.800" LastActivityDate="2017-10-15T04:26:17.800" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12990" PostTypeId="2" ParentId="12986" CreationDate="2017-10-14T20:50:18.567" Score="7" Body="&lt;p&gt;The difference in my mind is that the former can be used more widely yet is less precise. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance one might also write $\pi \approx 22/7$ to indicate that $\pi$ and $22/7$ are approximately equal, where there is no precise meaning to &quot;approximately.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, I would also consider it as correct to say that $\pi \approx 3.15$ as the quantities are approximately equal. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Others might not as possibly they only use the notation (in this context) if all the digits written out are 'correct.' But I would not be aware of this being a universal convention. &lt;/p&gt;&#xA;" OwnerUserId="143" LastActivityDate="2017-10-14T20:50:18.567" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12991" PostTypeId="2" ParentId="12986" CreationDate="2017-10-15T03:04:09.743" Score="9" Body="&lt;p&gt;As a U.S. person, I have never seen this second notation. My first instinct was that it was 3.14 multiplied by some error factor(s).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, there tends to be a difference in notation between pure math courses and other scientific or statistical fields. In math we would tend to write $\pi \approx 3.14$. In a statistics or programming text I tend to see $\pi = 3.14$, where the significance is implied by the number of decimal digits, or in narrative text, or in the context of a certain section. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I think it's legitimate to write &quot;Rounded to the hundredths place, $\pi = 3.14$&quot;, or, &quot;The values in this table are accurate to the 4th decimal place&quot; or somesuch. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-10-15T03:04:09.743" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="12992" PostTypeId="2" ParentId="1550" CreationDate="2017-10-15T10:37:14.337" Score="2" Body="&lt;p&gt;Since you didn't explicitly ask for problems that students should be able to solve, here are some of interest to many engineers: Which shape of the wing of an F1-car maximises downward force? Or which shape of wing of given span maximise the lift of an airplane. Which shape of a rear-view mirror for a car reduces drag? etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These problems are not constrained by algebraic equations but by PDEs, and the objective quantity is given an integral. These examples are from mechanical engineering, but I'm sure there are many similar problems in electrical or civil engineering. Engineers often solve these by discretising the PDEs and then applying the usual methods taught in constrained optimization. But they are also very interested in staying directly at the level of PDEs and applying Lagrange multipliers there. Part of this is known as the &quot;continuous adjoint method&quot; among engineers.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2017-10-15T10:37:14.337" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12993" PostTypeId="2" ParentId="12986" CreationDate="2017-10-15T13:16:15.353" Score="5" Body="&lt;p&gt;My understand is as follows. When using the $\approx$ notation one can state $\pi \approx 3.14$, $\pi \approx 3.142$ or even $\pi \approx 3.0$ the latter approximation is useful for doing calculations in ones head. The $\approx$ notation does not need to be rounded correctly. For example $\pi \approx \frac{22}{7} = 3.142857(6dp)$ but $\pi \ne 3.142857(6dp)$ because $\pi = 3.141593(6dp)$. The $dp$ notation must be rounded correctly. Finally, sure, one can use words to express $dp$ but to write $3.14(2dp)$ as 3.14 rounded to two decimal places seems to me to be similar to writing $1+1=2$ as one plus one equals two. &lt;/p&gt;&#xA;" OwnerUserId="8860" LastActivityDate="2017-10-15T13:16:15.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12995" PostTypeId="2" ParentId="12915" CreationDate="2017-10-15T15:35:02.943" Score="0" Body="&lt;p&gt;I personally started to make Math videos early this year with a Mac. My videos look a bit &quot;rustic&quot; because they are mostly hand-drawn, but they have some character and I think that makes them interesting (similar to an XKCD comic if you're familiar with that). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use Microsoft OneNote (a free application) to write within. I use a Wacom Tablet (like $80) for writing on the screen (You hold a pen and OneNote records what you're doing. It is pressure sensitive too, which gives the writing a nice realistic quality). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use software called &quot;Movavi Screen Capture 4&quot; to record what I'm writing. (Mac has a free screen capture included with Quicktime but Movavi lets you set a specific resolution, which I've found useful).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I will edit in &quot;Final Cut Pro&quot;. (This is definitely optional and a bit of a cost/time expenditure). I bought an external Microphone to increase the quality of my voice as well.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The end result isn't a professional as what you're looking at, but it allows you to relatively quickly make a video to get some point across, which I think is extremely beneficial to students since they can watch/rewatch/rewind on their own time whenever they want. &lt;/p&gt;&#xA;" OwnerUserId="6259" LastActivityDate="2017-10-15T15:35:02.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12997" PostTypeId="2" ParentId="12593" CreationDate="2017-10-15T22:09:40.617" Score="7" Body="&lt;p&gt;I recommend a newer Calculus textbook, which was published earlier this year: DJ Velleman's text, &quot;Calculus: A Rigorous Approach&quot; (&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0486809366&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;; also mentioned in my answer to &lt;a href=&quot;https://matheducators.stackexchange.com/a/11990&quot;&gt;&lt;strong&gt;MESE 11971&lt;/strong&gt;&lt;/a&gt;). In the introduction, there is a &quot;note to the instructor&quot; as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/G1LIk.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/G1LIk.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the &quot;novel approaches&quot; in this text is its presentation and notation of and around limits:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/q67q8.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/q67q8.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say that from the perspective of this text, the &quot;main point&quot; as concerns limits fits into a broader focus around which the topic of Calculus is structured: that which Velleman calls the &lt;em&gt;paradox of precision through approximation&lt;/em&gt;. Here is one final excerpt in that direction, which is followed in the text by concrete, motivating examples (and section 2.2 is entitled &lt;strong&gt;What Does &quot;Limit&quot; Mean?&lt;/strong&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CUU6f.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CUU6f.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that the subtle but important change(s) in notation, along with the structuring of this text around broader themes in Calculus, make it a worthy candidate for the sort of treatment of limits (and beyond) about which you ask. &lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2017-10-15T22:09:40.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="12999" PostTypeId="2" ParentId="12984" CreationDate="2017-10-16T01:41:25.170" Score="5" Body="&lt;p&gt;Another elementary ring to consider is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Exterior_algebra&quot; rel=&quot;noreferrer&quot;&gt;Exterior Algebra&lt;/a&gt; of a vector space. That has the advantage of having both commutative and non-commutative parts, and having a relationship to linear algebra via determinants.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2017-10-16T01:41:25.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13001" PostTypeId="2" ParentId="12984" CreationDate="2017-10-16T10:55:15.873" Score="3" Body="&lt;p&gt;Group rings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students are already familiar with some non-commutative groups (e.g. the symmetric groups) and you take a field to go with it that is familiar to the students (e.g. $\mathbb{R}$ or $\mathbb{F}_2$), then these rings should be rather natural and the multiplication and addition easy to understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, you can choose any group and any ring to start with, so you won't run out of examples that soon. :)&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2017-10-16T10:55:15.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13002" PostTypeId="1" AcceptedAnswerId="13010" CreationDate="2017-10-16T16:13:40.993" Score="2" ViewCount="139" Body="&lt;p&gt;I heard that there is evidence that a &quot;connected sage-on-the-stage&quot; teaching style is at least as effective as guide-on-the-side / flipped classrooms. I have been unable to find significant references for &quot;connected sage-on-the-stage&quot;, so I may have the term slightly wrong. Does anyone know a reference for this idea?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a variant of sage-on-the-stage that calls for heavy classroom interaction and participation but without radical change like flipping the classroom.&lt;/p&gt;&#xA;" OwnerUserId="7707" LastActivityDate="2017-10-19T10:57:54.307" Title="Reference or term for connected sage-on-the-stage?" Tags="|reference-request|general-pedagogy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13003" PostTypeId="1" AcceptedAnswerId="13009" CreationDate="2017-10-17T04:25:29.113" Score="13" ViewCount="779" Body="&lt;p&gt;This question is about mathematics only. My forethought is that courses are usually based on textbooks, but they have less content than the textbooks. There are less problem sets, and chapters are skipped. Is it true that you can learn more by self-study with textbooks rather than college courses? Given that one could find help in the internet.&lt;/p&gt;&#xA;" OwnerUserId="7540" LastEditorUserId="2083" LastEditDate="2017-10-18T00:10:06.083" LastActivityDate="2019-08-11T18:54:36.587" Title="Without a constraint of time, is self-study with math textbooks better than college courses?" Tags="|textbooks|self-learning|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13004" PostTypeId="2" ParentId="13003" CreationDate="2017-10-17T10:10:32.383" Score="4" Body="&lt;p&gt;&lt;em&gt;It is a mistake to think that one learning method is best for all students.&lt;/em&gt;  There may be some students who learn more by self-study.  There may be other students who learn more when taking a structured course with the help of an instructor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;added&lt;/strong&gt;&lt;br&gt;&#xA;Some students, if studying on their own, will soon quit.  But a regular lecture, with homework and deadlines and the threat of a failing grade may be enough to keep them going.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a similar way, some people hire fitness coaches to force them to do their exercises, since when they are left on their own they don't do it.  Or don't do enough of it.  &lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2017-10-18T00:12:37.993" LastActivityDate="2017-10-18T00:12:37.993" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13006" PostTypeId="1" CreationDate="2017-10-18T00:12:08.927" Score="6" ViewCount="175" Body="&lt;p&gt;I think I've done a decent job with teaching my students limits and derivatives so far in elementary calculus -- they were particularly intrigued with how easy and how accurate a first-order, linear approximation can be. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We'll soon start on integration, and I am wondering if I could give them an alternative way of thinking about the Riemann integral other than that &quot;it is the area under the curve&quot;.  What way of thinking about the Riemann integral in one variable would surprise them a bit?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(They are mostly freshmen and have seen calculus in some form in their high school days, it seems, but I don't think any of them would have any numerical methods background for me to discuss numerical integration with them.) &lt;/p&gt;&#xA;" OwnerUserId="8915" OwnerDisplayName="D.Hutchinson" LastActivityDate="2017-10-18T07:51:38.443" Title="Alternative ways of thinking about the one-variable Riemann integral for elementary calculus," Tags="|calculus|integration|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13007" PostTypeId="2" ParentId="13006" CreationDate="2017-10-18T00:20:09.680" Score="3" Body="&lt;p&gt;Perhaps start with Riemann sums to find distance covered when velocity is known but you can't guess an antiderivative - something they might appreciate if they are fond of derivatives. Gives you a head start on the fundamental theorem of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can point out that the same formalism finds areas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can finesse the numerical methods part by demonstrating a simple (easy to read Python) program or spreadsheet that finds Riemann sums with the number of division points as an input parameter.&lt;/p&gt;&#xA;" OwnerUserId="4984" OwnerDisplayName="Ethan Bolker" LastActivityDate="2017-10-18T00:20:09.680" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13008" PostTypeId="2" ParentId="13006" CreationDate="2017-10-18T01:36:29.770" Score="2" Body="&lt;p&gt;I find people are amazed, when they see it, that Euler's method works.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a way it is intuitive in a similar manner to how first order approximation works, and makes it less focused on abstract limits and convergence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, there is a hit movie that centers on it (&lt;em&gt;Hidden Figures&lt;/em&gt;)&lt;/p&gt;&#xA;" OwnerUserId="23002" OwnerDisplayName="eSurfsnake" LastActivityDate="2017-10-18T01:36:29.770" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13009" PostTypeId="2" ParentId="13003" CreationDate="2017-10-18T15:43:06.933" Score="7" Body="&lt;p&gt;&lt;em&gt;Without a constraint of time...&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you mean &quot;because self-study can be done over a longer period than a typical college quarter or semester&quot;, then this is as relevant as the other points you make. You don't elaborate on it in the body of your question, but a time-constraint can work both ways -- &lt;em&gt;not&lt;/em&gt; having one can be great, because you can work at your own pace and not miss things along the way. [I actually consider the typical term length to be a &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0553418815&quot; rel=&quot;nofollow noreferrer&quot;&gt;weapon of math destruction&lt;/a&gt;, particularly because students who can't assimilate concepts at the prescribed rate often find themselves in the fail-retake spiral, never completing their degree.] However, working &lt;em&gt;with&lt;/em&gt; a time-constraint can be good, particularly if it builds motivation to keep at it. I know I &lt;em&gt;could&lt;/em&gt; learn to paint on my own, but having an instructor give me deadlines really keeps me working.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you mean &quot;let's not consider the constraints of time in this discussion&quot;, assuming person &lt;em&gt;A&lt;/em&gt; takes the course face-to-face and person &lt;em&gt;B&lt;/em&gt; self-studies their way through the material, then I would bet that person &lt;em&gt;A&lt;/em&gt; learns more, as I'll explain below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;...courses are usually based on textbooks, but they have less content than the textbooks. There are less problem sets, and chapters are skipped.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mainstream commercial textbooks try to appeal to a wide audience, so they often include more information than any course requires or has time for. So in this sense, you're right -- a student who self-studies could certainly take on more information by just finishing the book. But wouldn't this also apply to students in a face-to-face course? Every few terms, I have a student who just decides to work through that last chapter because she's interested in the subject, but this outside of the norm &lt;em&gt;by far&lt;/em&gt;. My experience, and research I've read, says that students don't read textbooks anyway, except when mimicking a worked example for a homework problem. This may be one downfall of face-to-face classes where instructors explain every last thing in detail  -- students are conditioned to &lt;em&gt;not&lt;/em&gt; read the text because nothing bad ever happens when they don't, just as people in general are conditioned to &lt;a href=&quot;https://techtalk.pcpitstop.com/2012/06/12/it-pays-to-read-license-agreements-7-years-later/&quot; rel=&quot;nofollow noreferrer&quot;&gt;not read End User Licensing Agreements&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Given that one could find help in the internet.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;If&lt;/em&gt; you're an independent learner who knows how to search for content, &lt;em&gt;and&lt;/em&gt; you plan to work through a book at a matching level of difficulty, &lt;em&gt;and&lt;/em&gt; you set aside time to actually do it, then you could definitely cobble together your own course and learn more than is required of you in a typical college course. You will find enough help on the internet to address nearly every potential issue that could come up with misunderstanding the material. However, a procedure for diagnosing and treating this misunderstanding will be costly time-wise, either for the learner or for the person they found to help them online. One thing a teacher can do better than a machine is to ask leading questions to determine what a student really knows, and then build that toward learning a new thing. If you're on your own and you really misunderstand something, it may be quite difficult to figure out the root of the problem by searching online. Getting on a discussion forum (such as the Mathematics stack exchange) could be the fastest way to get this feedback, but the answers there can vary so widely in their scope that figuring out which one to focus on will be a new challenge, every time you ask a question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, I would say &quot;yes&quot; to your question, in theory. However, I don't see students reading the book on their own, even when they're in my classes. I think having a deadline in an actual course provides motivation for students to actually keep working on the material, and having a human teacher with whom to interact will provide trained, tailored feedback to address misunderstandings and prompting. Finally, a community of other students is a great environment for learning. Having/getting to explain what you're thinking to someone next to you helps solidify your own ideas. This may never happen with self-study.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="8875" LastEditDate="2017-10-19T14:35:26.590" LastActivityDate="2017-10-19T14:35:26.590" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13010" PostTypeId="2" ParentId="13002" CreationDate="2017-10-19T00:49:18.047" Score="3" Body="&lt;p&gt;Some references, with no particular claim of their significance:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.researchgate.net/publication/280572090_People_Matter_The_Role_of_Peers_and_Faculty_in_Students%27_Academic_Engagement&quot; rel=&quot;nofollow noreferrer&quot;&gt;Plett, Melani &amp;amp; Wilson, Denise &amp;amp; Bates, Rebecca &amp;amp; Allendoerfer, Cheryl &amp;amp; Carlson Jones, Diane &amp;amp; Floyd-Smith, Tamara &amp;amp; Veilleux, Nanette &amp;amp; Wasilewski, Caitlin. (2014). People Matter: The Role of Peers and Faculty in Students’ Academic Engagement. ASEE Annual Conference and Exposition, Conference Proceedings.&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://cornerstone.lib.mnsu.edu/cgi/viewcontent.cgi?article=1050&amp;amp;context=jur&quot; rel=&quot;nofollow noreferrer&quot;&gt;Painter, Sarah (2014) &quot;Modeling Student Engagement in the Classroom,&quot; Journal of Undergraduate Research at Minnesota State University, Mankato: Vol. 14 , Article 6.&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/87567555.1993.9926781&quot; rel=&quot;nofollow noreferrer&quot;&gt;King, Alison (1993) &quot;From Sage on the Stage to Guide on the Side&quot;, College Teaching, Vol. 41, No. 1, 30-35&lt;/a&gt; [mistakenly given as &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/87567555.1993.9926781&quot; rel=&quot;nofollow noreferrer&quot;&gt;2010&lt;/a&gt; in my comment, 2010 being the date of online publication].&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2017-10-19T10:57:54.307" LastActivityDate="2017-10-19T10:57:54.307" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13011" PostTypeId="2" ParentId="12943" CreationDate="2017-10-19T18:09:26.857" Score="1" Body="&lt;p&gt;In my experience, this hasn't been an explicit reason for students finding the subject harder. In fact, it was quite common to teach subjects (not just maths) in a different order than what is given in the textbook in my own education. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd suggest the following for you (with the notion of encouraging students to get used to a different order from middle school itself):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Counsel the students to understand what exactly is the basis for their resistance. This should not only help you understand the issue but will help them realize that their fear is just psychological.&lt;/li&gt;&#xA;&lt;li&gt;Give them a broad overview of why the order in which you're going to teach is particularly more engaging and interesting than the order given in the textbook. This is to hook them to the chapters that you're going to teach first.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For middle school students I would rely more on practical and engaging dimensions of the chapters rather than rational arguments to justify my order. Get them excited about the activities that they'll be able to get their hands on sooner than later!&#xA;You can even consider using reverse psychology or lament about difficulties faced by students in the past - if students are still resistant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, what you're describing seems to be an institutional problem (of the culture) which will take some time for them to get adapted to. You can't expect them to change overnight, but sooner this is inculcated the better.&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2017-10-19T18:09:26.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13012" PostTypeId="2" ParentId="12943" CreationDate="2017-10-20T11:39:36.397" Score="2" Body="&lt;p&gt;Seem you have two problems.  The students resistance and the misunderstandings of which chapter you are referring to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To overcome the student's resistance, suggest that they (the students) are part of an experiment to improve the order of the chapters in the book. Tell them when you finish, the class will help you write to the publisher why your order is better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To avoid the misunderstandings of which chapter you should involve the students. Ask the students in the beginning how to handle this.  Have a few suggestions and discuss the pros and cons.  The student will feel part of the decision and they will have heard the cons (eg we will only use page numbers but these are hard to remember so you must write them down - will help them remember to write them down. Here are some suggestions to discuss with the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A.  Only assign page numbers and topics.  Avoid discussion of chapters&#xA;    and refer to the topics and page numbers.  The con is that the&#xA;    students have to write down the page numbers.  It is a lot easier to&#xA;    remember Chapter 3, than pages 89 - 98 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;B.  Tell students that chapters&#xA;    assigned will be from the book but remind them that you won't be&#xA;    going in order. The con is that the students might mix up what&#xA;    chapter they are up to. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;C.  Tell students that you are going discuss&#xA;    topics. Of course the problem is that it may be clear where the&#xA;    topics aer in the book, especially if the topic is in more than one&#xA;    chapter and you have only learned part of a topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;D.  Let the students brainstorm for other solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.  Use a combination of these methods&lt;/p&gt;&#xA;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2017-10-20T11:39:36.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13013" PostTypeId="2" ParentId="12943" CreationDate="2017-10-20T13:46:18.670" Score="-2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Stop referring to your topics as &quot;chapters&quot;.  That is confusing.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Consider to just go in the same order as the book.  I'm serious.  I don't like teachers mixing and matching (and I am postgraduate).  The textbook is actually the more critical and important teaching tool than the teacher.  Find a book and follow it.  You will get more accomplished by following a clear written plan well than to make an optimal plan (in some abstract sense or just in your prejudice) but which is hard to follow.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8889" LastActivityDate="2017-10-20T13:46:18.670" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13014" PostTypeId="2" ParentId="12984" CreationDate="2017-10-20T15:04:32.220" Score="4" Body="&lt;p&gt;Since it's an intro to rings course, all the students will be familiar with functions. Also, they should have the sense already that function composition is non-commutative. So how about an example of a ring of functions where the addition is just the usual $+$ and the multiplication is given by composition? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something like $\operatorname{End}(G)$, the ring of endomorphisms of an abelian group $G$ will work. Or if you don't want to talk about groups at all, you can still use $\operatorname{End}(G)$ for a specific $G$ that students will be familiar with. &lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2017-10-21T17:53:36.520" LastActivityDate="2017-10-21T17:53:36.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13015" PostTypeId="1" CreationDate="2017-10-11T15:21:28.217" Score="3" ViewCount="171" Body="&lt;p&gt;I'm looking to put together a 1 hour talk about monodromy aimed at undergraduate math and physics majors. In terms of prerequisites, I only want to assume the students have seen multivariable calculus, intro ODE's, and intro linear algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to have an example from physics (preferably mechanics) that I can refer to throughout the talk, but I'm having trouble thinking of one. Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="344" OwnerDisplayName="Avi Steiner" LastActivityDate="2017-10-20T20:58:36.850" Title="Monodromy examples for undergraduates" Tags="|differential-equations|differential-geometry|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13016" PostTypeId="2" ParentId="12943" CreationDate="2017-10-20T19:08:44.827" Score="1" Body="&lt;p&gt;You use the phrase &quot;chapter[s] in my course&quot; (as opposed to the book), which makes no sense linguistically. I would recommend that you find some other way of expressing what you're trying to say. This itself may be the major point of confusion for your students. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-10-20T19:08:44.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13017" PostTypeId="2" ParentId="13015" CreationDate="2017-10-20T19:09:53.893" Score="1" Body="&lt;p&gt;How about the Aharanov–Bohm effect?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Aharanov–Bohm effect is then a manifestation of the fact that a connection with zero curvature (i.e. flat), need not be trivial since it can have monodromy along a topologically nontrivial path fully contained in the zero curvature (i.e. field free) region. By definition this means that sections that are parallelly translated along a topologically non trivial path pick up a phase, so that covariant constant sections cannot be defined over the whole field free region.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2017-10-20T19:09:53.893" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13018" PostTypeId="2" ParentId="13015" CreationDate="2017-10-20T20:58:36.850" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I appreciate your desire for a practical application, especially one from mechanics (more conceptual).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I wasn't able to find any from that field, exactly.  ;-(  [But you said that was nice to have, not strict requirement.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Google does show a lot of examples from solid state physics.  Essentially monodromy has to do with periodicity and crystal lattices are...well...periodic.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=monodromy+solid+state+physics&amp;amp;sourceid=ie7&amp;amp;rls=com.microsoft:en-US:IE-Address&amp;amp;ie=&amp;amp;oe=&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=monodromy+solid+state+physics&amp;amp;sourceid=ie7&amp;amp;rls=com.microsoft:en-US:IE-Address&amp;amp;ie=&amp;amp;oe=&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Given your target audience and the prereq  level and then the inherent difficulty of the topic, I recommend to try to make your talk conceptual rather than rigourous.  Diagrams, not theorems.  &quot;This is what it is about and what it is good for&quot; rather than &quot;this is a proof&quot;.  Do that and maybe it motivates someone to learn the topic properly.  Or at least to have conceptual awareness of what the heck it is.  So even if they never learn the topic, they have an idea of what the topic is about.  I don't think you can do more than that, in an hour, given the students, and their lack of group theory prereqs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Good luck.  Fight the good fight.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8889" LastActivityDate="2017-10-20T20:58:36.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13019" PostTypeId="1" CreationDate="2017-10-21T06:45:12.830" Score="6" ViewCount="718" Body="&lt;p&gt;Some teachers teach multiplication between integers with the following rules:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;plus with plus gives plus&lt;/p&gt;&#xA;&lt;p&gt;plus with minus gives minus&lt;/p&gt;&#xA;&lt;p&gt;etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So for example in order to deal with the multiplication &lt;span class=&quot;math-container&quot;&gt;$2\cdot(-3)=-6$&lt;/span&gt;, they uses the second rule above.&lt;/p&gt;&#xA;&lt;p&gt;I've never liked this approach, for me it's like a memorization which the students don't know the really meaning of this multiplication.&lt;/p&gt;&#xA;&lt;p&gt;So my question is how to teach students about multiplication between integers for the first time?&lt;/p&gt;&#xA;" OwnerUserId="5236" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-10-23T14:20:29.740" Title="How to teach multiplication between integers for the first time" Tags="|arithmetic-operations|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13020" PostTypeId="2" ParentId="13019" CreationDate="2017-10-21T10:09:40.020" Score="2" Body="&lt;p&gt;There are two useful practical situations where the interaction of signs appears. I recall using the temperature analogy when in school.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Money&lt;/h3&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Adding money makes you richer; + and + gives +.&lt;/li&gt;&#xA;&lt;li&gt;Taking away money makes you poorer: + and - gives -.&lt;/li&gt;&#xA;&lt;li&gt;Adding debt makes you poorer: - and + gives -.&lt;/li&gt;&#xA;&lt;li&gt;Removing debt makes you richer: - and - gives +.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h3&gt;Temperature&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Zero degrees (Celsius) is the temperature where water freezes. Around where I used to live, winter temperatures were often much colder than that. This approach is physically suspect, though.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Adding heat (hot water, say) makes something hotter.&lt;/li&gt;&#xA;&lt;li&gt;Adding cold (cold water, say) makes something colder.&lt;/li&gt;&#xA;&lt;li&gt;etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Both money and temperature and familiar concepts to school children and can be used to illustrate the double negative.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2017-10-21T10:09:40.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13022" PostTypeId="2" ParentId="13019" CreationDate="2017-10-21T16:07:51.347" Score="2" Body="&lt;p&gt;The best way to teach multiplication of integers is to use two sided counters.&#xA;This is clearly explained in this &lt;a href=&quot;https://www.youtube.com/watch?v=Yhoz1g35alw&quot; rel=&quot;nofollow noreferrer&quot;&gt;YouTube Video&lt;/a&gt;  I call this the best way because it gives a clear conceptual picture of what's going on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some clips from the video with my comments.&#xA;Integers are represented by the yellow side for positive, the red side for negative and equal numbers of red and yellow for zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Wz8nV.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Wz8nV.jpg&quot; alt=&quot;Representing integers&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When multiplying a positive times a negative such as 4 times -2, start with zero and add 4 sets of -2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VJWZc.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VJWZc.jpg&quot; alt=&quot;4 times -2&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When multiplying a negative times a negative you will want to take away sets.  You can take away sets by starting with pairs of positive and negative that add up to zero. Here is an example where we are multiplying -4 times -2.  We know we will have to take away 4 sets of -2.  So we start with 0 made up pairs of -2 and + 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kBfR7.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kBfR7.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this second step of -4 times -2, we circle the 4 sets of -2 that we are taking away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gubXp.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gubXp.jpg&quot; alt=&quot;removing four sets of -2&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We then take the four sets away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/43DlA.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/43DlA.jpg&quot; alt=&quot;-4 times -2 after the four sets are removed&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can count what's left or remove the extra zero pairs and then count.  Either way -4 times -2 leave +8.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/dJJ1o.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dJJ1o.jpg&quot; alt=&quot;-4 times -2 leaves +8&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The video continues with additional examples including -4 times 2 in which you take 4 copies of 2 away and leave -8.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary.&#xA;a) When multiplying by a +, start with zero and add that many sets of yellow (+), or red (-)&#xA;b) When multiplying by a -, start with zero (in this case pairs of chips that are equal to zero) and take away that many set of yellow (+), or red (-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Acting this out helps make sense of the rules in the beginning of your question and students will soon discover the rules and be able to use the rules without the chips.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2017-10-21T16:07:51.347" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13023" PostTypeId="2" ParentId="13019" CreationDate="2017-10-21T18:17:02.120" Score="1" Body="&lt;p&gt;One way uses a number line.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&#xA;  |----|----|----|----|----|----|----|----|----|----|----|----|----|----|&#xA; -6   -5   -4   -3   -2   -1    0    1    2    3    4    5    6    7    8        &#xA;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Taking a concrete geometrical point of view, short joinable rods made of straws could be used.&lt;br&gt;&#xA;A number of rods each of length 1, 2, 3, ... can be made and the following  done on a flat page.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the origin place rods on the number line, either right or left. Given a placed rod we may apply the negative sign to it. The negative sign rotates a rod a half-turn around the origin.&lt;br&gt;&#xA;Note: The negative sign &quot;$-$&quot; itself has half-turn rotational symmetry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&#xA;  2 x  3 =  6 : start at the origin with a rod of length 3 and join a second end-to-end. &#xA;&lt;/pre&gt;&#xA;&#xA;&lt;pre&gt;&#xA;  2 x -3 = -6 : first place the rod to the left so it is length -3 and join a second. &#xA;&lt;/pre&gt;&#xA;&#xA;&lt;pre&gt;&#xA; -2 x  3 = -6 means first place the rod of length 3 to the right,  &#xA;              then join a second of these making length 6 to the right,  &#xA;              then apply the negative sign which is a half-turn.  &#xA;&#xA;It may be better if eventually this rotation and scaling could be done in one operation.  &#xA;&lt;/pre&gt;&#xA;&#xA;&lt;pre&gt;&#xA; -2 x -3 = 6 means place the rod to the left so it is length -3,  &#xA;             join a second of these making length -6 to the left,&#xA;             then apply the negative sign which is a half-turn, ending up 6 to the right!&#xA;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;&lt;br&gt; &#xA;This would useful later if introducing complex numbers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By this stage they should be familiar with angles and multiplying  pairs of now abstract numbers where there is no difference in type between the first and second number (which doesn't have to represent a quantity from physics like a rod of length 3cm). Studying examples like above, they may see that multiplying pairs of numbers means summing their angles and scaling their lengths.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(They could begin this using 1 and -1, remembering +1 is one to the right and -1 is one to the left.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is then quite a puzzle to see where a unit length must be placed from the origin, so that when multiplying it by itself (two equal rotations), it ends up as -1. The number which satisfies this no longer lies on the number line but, is somewhat between -1 and 1 in resemblance, and is called i.     &lt;/p&gt;&#xA;" OwnerUserId="8660" LastActivityDate="2017-10-21T18:17:02.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13024" PostTypeId="2" ParentId="13019" CreationDate="2017-10-21T23:55:49.177" Score="2" Body="&lt;p&gt;Many of these rules are the way they are for the sake of consistency. When negative numbers were created (or discovered, if you like), we had to work out the rules such that multiplication would still be consistent with arithmetic, otherwise it would lead to contradictions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While intuitive analogies have their place in teaching, and certainly modeling and applications as well, the only way I see it to get past the  &lt;em&gt;memorization&lt;/em&gt; stage in learning new topics in mathematics, is to know enough of the connected pieces such that the rule makes sense as it is, for the sake of consistency. Obviously, in the beginning, that isn't possible because the student is learning, so naturally, the students must rely on memorization until more of that connected web builds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any event, you might want to start showing students some of the inconsistencies that occur IF we apply the wrong rule. For example, if (-25) times 1 was (25), then 25 divided 1 would be (-25), but we know that 25 divided by 1 is 25. Thus, (-25) times 1 must be -25 and -25 divided by -25 must 1, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, memorization gets a bad rap, but it is a natural part of the process early on. You want to pay attention to the progression of sophistication in your students' thought and see that they progress from the rote stages to the knowing stages.&lt;/p&gt;&#xA;" OwnerUserId="8894" LastEditorUserId="8894" LastEditDate="2017-10-22T00:05:03.210" LastActivityDate="2017-10-22T00:05:03.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13025" PostTypeId="2" ParentId="13019" CreationDate="2017-10-22T02:21:56.520" Score="15" Body="&lt;p&gt;My preferred model of multiplication with integers involves motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine you are recording a video of a car driving at a certain velocity.  If the car is going forward, the velocity is positive; if the car is going backward, the velocity is backward.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now when you play the video, you can choose whether to play it at normal speed, or at 2x, 3x or 4x normal; you can also choose whether to play the video backward at any of those speeds, and backward playback speeds are treated as negative numbers (-2x, -3x, -4x...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you watch the video, the car appears to be driving at a speed that is the product of its actual velocity and the playback speed.  That is, if the car is driving at 10 miles per hour and you play the recording at 3x speed, it appears to be driving at 30 miles per hour.  If the car is driving at 10 miles per hour and you play the recording at a -2x speed, the car will appear to be driving at  -20 miles per hour  (i.e. it will seem to be going backward).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now it should be clear that a negative times a negative is a positive:  a car driving backward, watched in reverse, will appear to be going forward.  All of the other multiplication rules for integers also make sense in this context.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2017-10-22T02:21:56.520" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13026" PostTypeId="1" AcceptedAnswerId="13027" CreationDate="2017-10-22T04:08:35.713" Score="5" ViewCount="489" Body="&lt;p&gt;Let's say we're running a basic algebra course and we're committed to showing proofs of everything we reasonably can. The development of equations of lines seems most straightforward in this sequence: (1) the standard (defining) formula &lt;span class=&quot;math-container&quot;&gt;$Ax + By = C$&lt;/span&gt;, (2) the definition of slope &lt;span class=&quot;math-container&quot;&gt;$m = \frac{y_2 - y_1}{x_2 - x_1}$&lt;/span&gt;, (3) the point-slope formula &lt;span class=&quot;math-container&quot;&gt;$y - y_1 = m(x - x_1)$&lt;/span&gt;, and then (4) the slope-intercept formula &lt;span class=&quot;math-container&quot;&gt;$y = mx + b$&lt;/span&gt;. That is: the definition in (2) after a few steps of manipulation leads to (3), which itself after a few steps of manipulation leads to (4).&lt;/p&gt;&#xA;&lt;p&gt;However, if we consider presenting things in this sequence, there seems to be a dearth of interesting exercises we can assign at the point when we have formula (3) but not (4). We could just give a point and a slope and ask students to substitute them, but that is trivial/uninteresting. The half-dozen textbooks I've surveyed only ever use formula (3) as part of exercises (say, starting from two points) that ultimately lead to expressing the answer in the form of (4).&lt;/p&gt;&#xA;&lt;p&gt;So: Are there any interesting (partially challenging, enlightening) exercises that make use of the point-slope form (3) without any reference or use of the slope-intercept form (4)? If not, then I suppose I should just present both formulas immediately together and then proceed to those exercises that convert from one to the other.&lt;/p&gt;&#xA;&lt;p&gt;As a negative example, consider a sample exercise set from OpenStax College Algebra (Section 2.2). &amp;quot;For the following exercises, find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.&amp;quot; (Various point-slopes and pairs of points are given.) That's not useful in our present case because we're at a moment when slope-intercept is not available. Likewise, &lt;strong&gt;the goal of this question is not to find applications or problem-solving uses&lt;/strong&gt; (which are, somewhat embarrassingly, not part of the curriculum in question), but rather to directly exercise writing and manipulating the point-slope formula itself algebraically.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2022-10-02T01:07:28.220" LastActivityDate="2023-05-19T05:33:00.910" Title="Are there any good exercises on point-slope form of a linear equation by itself?" Tags="|undergraduate-education|secondary-education|algebra|exercises|graphing|" AnswerCount="7" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13027" PostTypeId="2" ParentId="13026" CreationDate="2017-10-22T04:24:34.383" Score="1" Body="&lt;p&gt;One possibility: Given a point and fractional slope for a line, use the point-slope formula, and then write the line in &lt;em&gt;standard&lt;/em&gt; form, and with no fractional coefficients. While not commonly seen in textbooks, this at least highlights the facts that: (a) any line described by slope and point can in fact be written in terms of the defining formula, (b) one should practice clearing fractions from equations, and (c) the attraction of standard form that any equation with rational coefficients can be expressed entirely with integers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example:&lt;/strong&gt; Write the equation for the line through &lt;span class=&quot;math-container&quot;&gt;$(4, 8)$&lt;/span&gt; and with slope &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; in standard form with all integer coefficients. &lt;em&gt;Solution:&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$y - 8 = \frac{1}{2}(x - 4) \Leftrightarrow 2y- 16 = x - 4 \Leftrightarrow x - 2y = -12$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now, several years after initially writing this answer, I discovered that this exercise is included in Martin-Gay, &lt;em&gt;Intermediate Algebra&lt;/em&gt;, Section 3.5, which earns them extra points in my book.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-05-19T05:33:00.910" LastActivityDate="2023-05-19T05:33:00.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13028" PostTypeId="2" ParentId="13026" CreationDate="2017-10-22T04:55:10.717" Score="0" Body="&lt;p&gt;Maybe I'm not sure what you mean by &quot;interesting&quot; but is something like (from &lt;a href=&quot;https://brilliant.org/wiki/linear-equations/#linear-equations-word-problems&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Henry has \$125 in his pocket, but he is spending \$20 per hour at the arcade.  If he leaves in 2.5 hours, how much money will he have left?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;what you mean?  It can be solved in a variety of ways but I think using the point slope form is most natural.  There are other problems on the linked site that might be of interest, too.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For something a little more challenging, there are some AMC problems that might be interesting.  For example Problem #11 on the 2010 AMC B Test, you'll find&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A line with slope 3 intersects a line with slope 5 at the point $(10,15)$.  What is the distance between the $x$-intercepts of these two lines?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A google search for &quot;AMC problems point slope&quot; might lead you to more such problems.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-10-22T04:55:10.717" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13029" PostTypeId="2" ParentId="13026" CreationDate="2017-10-22T11:18:01.850" Score="5" Body="&lt;p&gt;The Fahrenheit and Celsius temperature scales are linearly related.  A change of one degree Celsius is a change of 1.8 degrees Fahrenheit.  The freezing point of water is 0° Celsius or 32° Fahrenheit.  Find the formula for computing the Fahrenheit temperature &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; given the Celsius temperature &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;[Probably most useful as an exercise in the US, where the students have heard of Fahrenheit.]&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2023-04-20T06:18:45.857" LastActivityDate="2023-04-20T06:18:45.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13031" PostTypeId="1" CreationDate="2017-10-22T18:49:11.507" Score="8" ViewCount="287" Body="&lt;p&gt;First question in this stack. Here it goes:&lt;/p&gt;&#xA;&lt;p&gt;So, I had this 2 or 3 year old intuos tablet and I said why not? And started doing my own videos. I've been making my own videos for my students last couple weeks and I notice that are way more powerful than when I just assign one to them from a random channel in YouTube. In their own words the way I talk and the words that I use are more approachable to them.  However my hand writing using the tablet its awful.&lt;/p&gt;&#xA;&lt;p&gt;So my question is: What kind of software can I used to make my digital writing as sharp as this one?&#xA;&lt;a href=&quot;https://i.stack.imgur.com/PG9DY.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PG9DY.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Because this is my best result:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/233L3.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/233L3.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And even tho my hand writing is not made in heaven is not as awful as that second picture.&lt;/p&gt;&#xA;&lt;p&gt;Any ideas? My table is a Wacon Intuos Creative pen &amp;amp; touch tablet. If some of you have this same model it would be awesome if you can share any ideas or software to use it at full potential.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="8853" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-11-24T14:47:04.533" Title="How can I get a better result making math videos with a tablet?" Tags="|secondary-education|technology-in-education|teaching|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13032" PostTypeId="1" CreationDate="2017-10-22T18:52:32.037" Score="17" ViewCount="766" Body="&lt;p&gt;When asked to show a math problem has a unique solution, students sometimes think that if an algorithm leading to a solution has unambiguous instructions at each step (no need to make choices at any point) then the solution they find has to be the only solution.  Loosely speaking, if each step in reaching the solution is &quot;unique&quot; then the end result has to be unique (the only possible solution).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not true, and I think it would be nice to have a list of examples at different levels (of undergraduate mathematics) showing why this idea is mistaken.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if asked to solve $55x+32y  = 1$ in integers then Euclid's algorithm for computing $\gcd(55,32)$ followed by back-substitution (reversing the steps of Euclid's algorithm) is a procedure where each step is completely determined by the previous ones and leads to the definite answer $(x,y) = (7,-12)$, but the original equation has infinitely many integral solutions: $(x,y) = (7+32t,-12-55t)$ for integers $t$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What other examples can people offer? I am not interested only in computational problems. &lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2023-12-17T15:58:08.993" Title="Unique steps leading to a non-unique answer" Tags="|undergraduate-education|proofs|" AnswerCount="7" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13033" PostTypeId="2" ParentId="13032" CreationDate="2017-10-22T19:16:13.273" Score="-5" Body="&lt;p&gt;I might think in any diofantine equation as the one you propose. Also a lot of trigonometric equations. Something like $\sin\theta=\frac{1}{2}$.&lt;/p&gt;&#xA;" OwnerUserId="8853" LastActivityDate="2017-10-22T19:16:13.273" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13034" PostTypeId="1" CreationDate="2017-10-22T23:42:56.570" Score="11" ViewCount="1289" Body="&lt;p&gt;I am a European postdoc who recently teaching at a large public university in the United States. I will have to teach a course for undergraduate students that introduces them to proving and reasoning in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students have possibly no exposure to mathematical reasoning in general. At the end of the course, they should be able to read and write proofs, and use the standard logical/set-theoretical notation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What books or lecture notes can you recommend (or dis-recommend) for such a course? I am particularly interested in material that gets them as close as possible to being able to read non-American textbooks. &lt;/p&gt;&#xA;" OwnerUserId="8895" LastEditorUserId="39" LastEditDate="2017-10-23T07:34:48.970" LastActivityDate="2020-08-24T23:00:31.870" Title="Book request: teaching proving and reasoning at an American university" Tags="|textbooks|proofs|logic|axioms-foundations|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="13035" PostTypeId="2" ParentId="13034" CreationDate="2017-10-22T23:48:17.253" Score="6" Body="&lt;p&gt;Some things we're currently considering for a similar course at a large urban community college:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Epp, Discrete Mathematics with Applications&lt;/li&gt;&#xA;&lt;li&gt;&lt;strike&gt;Artin, Algebra&lt;/strike&gt;&lt;/li&gt;&#xA;&lt;li&gt;Gilbert, Elements of Modern Algebra&lt;/li&gt;&#xA;&lt;li&gt;Lay, Analysis with an Introduction to Proof&lt;/li&gt;&#xA;&lt;li&gt;Wade, An Introduction to Analysis&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And also some OER (open educational resources) options:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sundstrom, Mathematical Reasoning: Writing and Proof (formerly a mass-market textbook) &lt;a href=&quot;http://scholarworks.gvsu.edu/books/9/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://scholarworks.gvsu.edu/books/9/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Hammack, Book of Proof &lt;a href=&quot;http://www.people.vcu.edu/~rhammack/BookOfProof/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.people.vcu.edu/~rhammack/BookOfProof/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Hutchings, Introduction to Mathematical Arguments (27 pages; I esp. like the &quot;Logic in a nutshell&quot; table on p. 9) &lt;a href=&quot;http://math.berkeley.edu/~hutching/teach/proofs.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://math.berkeley.edu/~hutching/teach/proofs.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Turner, A Brief Introduction to Proofs (9 pages; meant as a primer to any course involving proofs) &lt;a href=&quot;http://persweb.wabash.edu/facstaff/turnerw/Writing/proofs.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://persweb.wabash.edu/facstaff/turnerw/Writing/proofs.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Edit: Struck out Arten's Algebra from list above due to consensus in comments.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-02-08T18:42:12.167" LastActivityDate="2018-02-08T18:42:12.167" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13036" PostTypeId="2" ParentId="13032" CreationDate="2017-10-23T01:11:41.713" Score="8" Body="&lt;p&gt;To find a basis for the row space of a matrix, perform elementary row operations to get to reduced row-echelon form. The nonzero rows of the row-reduced matrix are a basis for the row space of the original matrix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course many other bases are possible (with an exception of the zero matrix).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edited to address a commenter's request: It is true that the steps to row reduce a matrix are not unique (or uniquely ordered); however,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) It is possible to give an algorithmic specification of row reduction to reduced row-echelon form for matrices (i.e., no choices); and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) The reduced row-echelon form of a matrix is unique.  So if we use row reduction as a method to get a basis for the row space of the original matrix, we will all get the same reduced row-echelon form, and hence the same basis, even if some of us do different steps (or steps in a different order) than others of us.&lt;/p&gt;&#xA;" OwnerUserId="5884" LastEditorUserId="5884" LastEditDate="2017-10-24T20:04:37.400" LastActivityDate="2017-10-24T20:04:37.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13037" PostTypeId="2" ParentId="13034" CreationDate="2017-10-23T02:32:56.327" Score="6" Body="&lt;p&gt;Another free option is &lt;a href=&quot;https://web.archive.org/web/20191221162938/https://courses.csail.mit.edu/6.042/spring17/mcs.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lehman, Leighton, and Meyer's Mathematics for Computer Science&lt;/a&gt;. It's written for an MIT introductory discrete math course that emphasizes training students in proof-writing.&lt;/p&gt;&#xA;" OwnerUserId="8898" LastEditorUserId="11" LastEditDate="2020-08-24T23:00:31.870" LastActivityDate="2020-08-24T23:00:31.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13038" PostTypeId="2" ParentId="13034" CreationDate="2017-10-23T04:05:47.770" Score="3" Body="&lt;p&gt;A book we often use in my department is &lt;a href=&quot;https://link.springer.com/book/10.1007%2F978-1-4419-9479-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reading, Writing and Proving&lt;/a&gt; by Gorkin and Daepp.  One feature of this book that distinguishes it from others mentioned in other answers is that it has chapters on $\mathbb{R}$, its completeness, the convergence of secuences in $\mathbb{R}$, and the Cantor-Schröder-Bernstein theorem.  In other words, its focus is more on preparing students for a course in real analysis and less on discrete mathematics.  Considering that the South American universities I have taught at often start with the completeness of the reals in first semester calculus, this might be the kind of book that will get students reading &quot;non-American&quot; textbooks quickly.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-10-23T04:05:47.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13039" PostTypeId="2" ParentId="13034" CreationDate="2017-10-23T11:52:25.563" Score="6" Body="&lt;p&gt;I know no better book for reasoning than&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0273728911&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Thinking Mathematically, by John Mason, Leone Burton and Kaye Stacey&lt;/strong&gt;.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;It is superb in inspiring action and instilling methods of reasoning.&lt;/strong&gt; Quoting from the introduction:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Experience in working with students of all ages has convinced us that mathematical thinking can be improved by&lt;/p&gt;&#xA;&lt;p&gt;● tackling questions conscientiously;&lt;/p&gt;&#xA;&lt;p&gt;● reflecting on this experience;&lt;/p&gt;&#xA;&lt;p&gt;● linking feelings with action;&lt;/p&gt;&#xA;&lt;p&gt;● studying the process of resolving problems; and&lt;/p&gt;&#xA;&lt;p&gt;● noticing how what you learn fits in with your own experience.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The authors delineate the process of thinking mathematically, presenting a rubric for attacking problems. This rubric does not mean a recipe, but a set of words designed to get you thinking and prompt you toward action (in the spirit of Polya's questions). However, the book does not explicitly deal with problems to prove, and for that I recommend two other books:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0521675995&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;How to Prove It, by Daniel Velleman&lt;/strong&gt;&lt;/a&gt;,&lt;/p&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/052171978X&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;How to Think like a Mathematician, by Kevin Houston&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;These two books combined will help students discuss and learn:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Logic (Sentential and Quantitative) - Velleman&lt;/li&gt;&#xA;&lt;li&gt;Proof strategies (negations, conditionals, quantifiers, existence and uniqueness, contradiction, induction, etc) - Velleman and Houston&lt;/li&gt;&#xA;&lt;li&gt;Relations - Velleman and Houston&lt;/li&gt;&#xA;&lt;li&gt;Functions - Velleman and Houston&lt;/li&gt;&#xA;&lt;li&gt;How to read definitions and theorems - Houston&lt;/li&gt;&#xA;&lt;li&gt;Divisors, the Euclidean algorithm and modular arithmetic - Houston&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I believe the combination of these three more will result in an overhaul in your students' thinking habits. Their abilities of reasoning will vastly improve.&lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-10-23T11:52:25.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13040" PostTypeId="2" ParentId="13034" CreationDate="2017-10-23T12:15:22.073" Score="6" Body="&lt;p&gt;Daniel Solow, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0471680583&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;How To Read and Do Proofs&lt;/em&gt;&lt;/a&gt;&lt;br&gt;&#xA;Intended for abject beginners, unlike some of those other answers.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-10-23T12:15:22.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13041" PostTypeId="2" ParentId="13032" CreationDate="2017-10-23T12:41:51.523" Score="7" Body="&lt;p&gt;A &quot;non-mathematical&quot; answer:  If you are in need of a doctor, then one unambiguous algorithm for finding one is to call everyone in the phone book, in order, and ask them if they are a doctor.  Eventually you will find one.  However, this does not imply there is only one doctor in your city.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2017-10-23T12:41:51.523" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13042" PostTypeId="2" ParentId="13019" CreationDate="2017-10-23T14:20:29.740" Score="4" Body="&lt;p&gt;I always liked fill in the following pattern:&lt;br&gt;&#xA;$$(3)(4)=12$$&#xA;$$(3)(3)=9$$ &#xA;$$(3)(2)=6$$&#xA;$$(3)(1)=3$$&#xA;$$(3)(0)=0$$&#xA;$$(3)(-1)=$$&#xA;$$(3)(-2)=$$&#xA;The students should see that when the second multiplication number decreases by 1, then the product decreases by 3. Then when they hear &quot;negative times a positive = negative&quot; they can actually see it isn't a random rule, but a logical rule that was decided in order to make this pattern consistent. Once they know a negative times a positive equals a negative, then you can use&lt;br&gt;&#xA;$$(-3)(3)=-9$$ &#xA;$$(-3)(2)=-6$$&#xA;$$(-3)(1)=-3$$&#xA;$$(-3)(0)=0$$&#xA;$$(-3)(-1)=$$&#xA;$$(-3)(-2)=$$ &#xA;And then they see here that when the second number is decreased by 1, now the answer increases by 3. So the &quot;negative times a negative = positive&quot; rule isn't a random rule for no reason, again it was decided in order to make this pattern consistent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps using this approach, you can get the students to create the rules themselves without you telling them, and that way they will more likely be able to recall those rules since they &quot;invented&quot; the rule themselves instead of it being something they were told to memorize.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2017-10-23T14:20:29.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13043" PostTypeId="2" ParentId="13031" CreationDate="2017-10-24T01:07:15.530" Score="1" Body="&lt;p&gt;Alright. So, I've been looking around all day and yesterday and I found this great option: smoothdraw 4 beta. It really let me customize the pen and the flow of it improving my digital hand writing so is kinda what I was looking for. Someone told me that I should try photoshop and that the results are insanely better than anything else and I would try it in the near future.  So if anyone had this same question this could be your answer ;). &lt;/p&gt;&#xA;" OwnerUserId="8853" LastActivityDate="2017-10-24T01:07:15.530" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13044" PostTypeId="2" ParentId="12984" CreationDate="2017-10-24T02:34:51.163" Score="1" Body="&lt;p&gt;Related to your example of $M_2(\mathbb{R})$ is to consider the ring $UT_2(\mathbb{R})$ of upper triangular $2 \times 2$ matrices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting nontrivial quotient is the map $UT_2(\mathbb{R}) \to \mathbb{R} \times \mathbb{R}$ which sends a matrix to the pair of entries on its diagonal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The kernel $I$ of this quotient is the ideal of all matrices whose only entry is in the top-right corner. This is a particularly interesting ideal since $I^2 = 0$.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2017-10-24T02:57:23.213" LastActivityDate="2017-10-24T02:57:23.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13045" PostTypeId="2" ParentId="13032" CreationDate="2017-10-24T10:54:24.743" Score="2" Body="&lt;p&gt;Another example from Euclid (in the proof of Proposition 20, Book IX):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given a finite list of primes $p_1,p_2,\cdots,p_n$, there is a prime&#xA;  not included in the list.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Construction: Let $N=p_1 p_2 \cdots p_n +1$. Let $p$ be the smallest (to make the construction unique, Euclid didn't specify this) prime factor of $N$. Then $p$ is not equal to any of the primes $p_1,p_2,\cdots,p_n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are infinitely many other solutions to the problem of finding such a prime.&lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2017-10-24T10:54:24.743" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13049" PostTypeId="1" AcceptedAnswerId="13057" CreationDate="2017-10-25T13:40:40.540" Score="11" ViewCount="440" Body="&lt;p&gt;I've tutored math SAT prep for some years now, and have developed a routine with certain texts and exercises that I have found to be fairly successful in improving scores overall. I recognize that many students face a challenge in translating word problems into algebraic expressions, and am usually able to make progress toward this problem with most of my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I am currently tutoring a student for whom this issue presents a greater challenge than I've encountered. I say &quot;greater challenge&quot; because my metric is the disparity between his comprehension of and familiarity with the underlying algebra (which is fairly strong) and his ability to put word problems into algebraic terms (which is almost non-existent). I've looked online for instances of this disparity occurring at such a remarkable level, but have only found literature on the general difficulty faced by many children.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any experience with this sort of problem, and are there any techniques, exercises, methods of explanation, etc. that have exhibited notable success toward remedying such an issue?&lt;/p&gt;&#xA;" OwnerUserId="8529" LastEditorUserId="2083" LastEditDate="2017-10-26T07:30:04.387" LastActivityDate="2018-09-28T18:54:03.363" Title="Difficulty with word problem interpretation" Tags="|secondary-education|word-problems|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="13052" PostTypeId="2" ParentId="13049" CreationDate="2017-10-26T02:39:00.477" Score="10" Body="&lt;p&gt;I suggest you break it down into smaller parts. In fact, don't start on equations. Start on translation of terms:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Double of a number, translation: $2x$&lt;/li&gt;&#xA;&lt;li&gt;Sum of two numbers, translation: $x+y$&lt;/li&gt;&#xA;&lt;li&gt;The triple of a number after adding five units, translation: $3(x+5)$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Next combine translations into equations:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The double of a number is equal to the triple of another, translation: $2x = 3y$&lt;/li&gt;&#xA;&lt;li&gt;The sum of two numbers is equal to the triple of the first after adding five units, translation: $x+y = 3(x+5)$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Make sure you work through many standard terms before moving on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A table may be helpful: one column for the word terms and another for the algebraic translation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Treat each line of the table as one translation step as you read a word problem: start with &quot;The double of a number...&quot;, and the second column reads $2x$; then continue with &quot;...minus 30...&quot;, and the second column reads $2x-30$; complete with &quot;...equals the number.&quot;, and the second column reads $2x-30=x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sounds tiring, but worth the effort if your student understands and internalizes the process to the point of no longer needing such device.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2017-10-26T02:39:00.477" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13053" PostTypeId="1" CreationDate="2017-10-26T11:09:33.667" Score="7" ViewCount="316" Body="&lt;p&gt;My 12-year-old got interested on topology puzzles and thought it would make a great presentation for his school assignment. I myself am not familiar with topology so am of little help. Basically, he wants to do an introduction to topology that he and his age group could easily grasp and appreciate. Would you have suggestions on what topics and puzzles he can include in his 20-minute presentation? So far, we're thinking he can introduce puzzles in the beginning, make the introduction to topology (definition, genus, etc.) with some slides, explain some real-life applications of topology, and then&lt;/p&gt;&#xA;" OwnerUserId="8909" LastActivityDate="2017-10-30T18:37:19.690" Title="Topology presentation for middle school" Tags="|topology|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13054" PostTypeId="2" ParentId="13053" CreationDate="2017-10-26T16:47:06.243" Score="3" Body="&lt;p&gt;One thing that your son might consider talking about is the winding number. One example that I sometimes give involves garden hoses. When the hose is nicely coiled on the ground and you try to drag it (staying along the ground) to the garden, you develop kinks in the hose. One way of interpreting this is that the winding number is being preserved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there are ways of unkinking the hose, but they involve 3 dimensional movement. (In particular that loops on the 2-sphere can be contracted --- $\pi_1(S^2)=0$ --- unlike the case of the unit circle.)&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2017-10-26T16:47:06.243" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13057" PostTypeId="2" ParentId="13049" CreationDate="2017-10-27T12:12:34.870" Score="7" Body="&lt;p&gt;Recently, I got to teach a new high school class called &quot;Essentials of College Math.&quot; It sounds like hard content, but it was basically just a transition course from Algebra II to Precal. Anyways, &lt;a href=&quot;http://www.sreb.org/sites/main/files/math_ready_teacher_guide_student_manual_v3.pdf&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; is the teacher's manual for the course (which also includes all the student handouts). The first unit was called &quot;algebraic expressions&quot; and dealt with this topic of translating words to equations quite a bit with some good activities, especially the card sorting activities. I found Task #5 (&quot;the swimming pool&quot;) to be quite useful. In fact, I did a whole 2 or 3 days after that task just devoted to dot patterns (from &lt;a href=&quot;http://www.visualpatterns.org/&quot; rel=&quot;noreferrer&quot;&gt;visualpatterns.org&lt;/a&gt;) and having students create their own expressions for the patterns. We proved that two different expressions generated the same pattern. I would also give them a list of different solutions and ask them where they see the different aspects of the expression in the visual picture. (similar to this &lt;a href=&quot;https://www.illustrativemathematics.org/content-standards/tasks/21&quot; rel=&quot;noreferrer&quot;&gt;task&lt;/a&gt;, I found Levi's comment on the task particularly insightful).&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2017-10-27T12:12:34.870" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13058" PostTypeId="1" CreationDate="2017-10-27T23:09:08.990" Score="11" ViewCount="859" Body="&lt;p&gt;I've seen the previously-asked questions about the effectiveness of multiple choice problems, but I want to know about design. What are best practices in multiple choice problem design? The &quot;common&quot; understanding I've come to is based on perennial complaints -- &quot;hard to write well&quot;; &quot;punishes students for small mistakes (like sign errors)&quot;; &quot;great when well-written&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know of studies related to the design of questions so they address particular outcomes/skills, particularly in mathematics? Is there a manual on the topic?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back Story: Our department has decided to face an impending assessment edict by trying a new thing: common multiple-choice section on all final exams. Therefore, limit suggestions to either general theory on writing MC questions, or resources somehow focusing on writing MC in summative (not formative) assessments.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2017-11-23T18:21:39.797" Title="How to design multiple choice questions" Tags="|exams|assessment|community-colleges|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13060" PostTypeId="1" CreationDate="2017-10-28T05:14:35.547" Score="0" ViewCount="724" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/2386/ideal-undergraduate-sequence&quot;&gt;Ideal Undergraduate Sequence&lt;/a&gt;&#xA;&lt;strong&gt;Main question:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I looked above and what I'm interpreting out of it is that one should be able to do proofs after studying some intro to proofs class, calculus multivariable+vector calculus, and differential equations linear algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it possible that one won't be able to do proofs after finishing doing calc 3/vector calc and differential equations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Background: (context)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I really want to be a pure mathematician and not some high school teacher in the future. I suck at other subjects. math's all I got left, i.e. I can't write, I can't code, manual labor or mcdonald's job sounds possible future.  I'm taking calculus 3 and differential equations right now and I'm wondering whether there's no hope in me becoming a mathematician if I fail to be able to do proofs like the ones in algebra or analysis like weistrass M-test and something banach after finishing calculus 3 and differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot prove theorems at this point independently and can't study some class like real analysis or number theory right now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Optional questions:&lt;/strong&gt; (ignore if you're busy)&#xA; Is what the link above says that if you can't do proofs after calc 3 and differential equations, you'll never be a mathematician?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does finishing taking calculus 3/vector calculus and differential equations transform your mind to being able to do proofs? I mean why did we suggest (in the hyperlink above) that people should take multivariable calculus and differential equations before a serious proof class? Are calc 3/ differential equations special? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are math teachers in high school the people who took calculus 3 and differential equations but just didn't have the intelligence or weren't transformed from such an experience to be able study proof writing? &lt;/p&gt;&#xA;" OwnerDisplayName="user8788" LastEditorDisplayName="user8788" LastEditDate="2017-10-28T16:17:47.240" LastActivityDate="2017-10-29T15:26:01.833" Title="Is it possible to have taken intro to proofs, calculus 3 and differential equations and still lack the ability to do proofs?" Tags="|undergraduate-education|proofs|curriculum|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13062" PostTypeId="1" CreationDate="2017-10-28T11:46:41.657" Score="4" ViewCount="129" Body="&lt;p&gt;I'm looking for sites, or books, for bright 5th graders or above, with math problems and respective solutions.&lt;/p&gt;&#xA;" OwnerUserId="2029" LastEditorUserId="122" LastEditDate="2018-01-02T08:25:49.640" LastActivityDate="2018-01-02T08:25:49.640" Title="Sites/books with math problems for bright 5th graders" Tags="|textbooks|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13063" PostTypeId="2" ParentId="13058" CreationDate="2017-10-28T12:33:30.803" Score="9" Body="&lt;p&gt;Personally in my own math courses, I have found that the gold-standard is to have an advance cycle of short-answer responses, document the most common student responses, and then turn those into the multiple-choice options in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Disclaimer:&lt;/em&gt; In general I frown upon multiple-choice testing, since the math discipline is inherently about explaining/justifying things in writing. However, my department has dictated in certain courses that the final exams be all multiple choice, for reasons similar to those expressed by the OP. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do believe that in practice, the most common way to write math multiple-choice questions is for the writer to basically flip some of the signs in the work at some point (this is certainly a very common source of errors). However, I think that multiple-choice defenders overlook how multitudinous the misunderstandings can really be, and these are masked unless one gives short-answer questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in the spring of 2015 on a remedial algebra test I posed the short-answer problem, &quot;Factor completely: $50a^4 - 18b^2$&quot; and tallied 21 distinct incorrect answers (from among 54 students). Other problems may have more distinct responses than I had time or patience to distinguish (e.g., solving an equation by factoring, or finding a linear equation given two points). This brings me back to my original thesis: If multiple-choice tests must be given, and the required time is made available, then an advance cycle of short-answer testing is an objective way of witnessing the most common misunderstandings at least once. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-10-28T12:33:30.803" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13065" PostTypeId="1" CreationDate="2017-10-28T18:54:31.490" Score="5" ViewCount="516" Body="&lt;p&gt;Yesterday in my precalculus class, I decided to teach students how to find the formula for an inverse function in a new way (to me).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this curriculum, they have already used the notation $f^{-1}(x)$ to evaluate &lt;em&gt;numerically&lt;/em&gt;, with the understanding that $x$ is the output of the function $f$, so evaluating $f^{-1}(3)$ means &quot;find the input value for $f$ if the output is $3$. They have practiced this with a function's formula by setting it equal to $3$ and solving; they have also practiced this graphically and interpreted it in the context of verbally-defined functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, they have used $f(g(x))$ numerically, graphically and verbally, and recently they have composed two formulas together and simplified. They have numerically composed two inverse functions together, understanding that what one function does, the other undoes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an attempt to build on this function composition notation, I have strayed from the typical &quot;Switch&quot; method (exchange $x$ and $y$, solve for $y$, and then call this $f^{-1}(x)$), instead giving students a sequence of problems, such as:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Find a function $f(x)$ that satisfies the equation $5f(x)+x = 4$. [Hint: Use basic algebra to isolate $f(x)$.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let $g(x) = x^3-5$. Find a function $h(x)$ so that $g(h(x)) = x+2$. [Hint: Simplify the left-hand side of this equation by using the function $g$, where the unknown function $h(x)$ is the input.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then, returning to numerical understanding:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $P(x) = 4x-3$ and $Q(x)=0.25x+0.75$, compute the following:&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;$P(Q(0))$&lt;/li&gt;&#xA;&lt;li&gt;$P(Q(4))$&lt;/li&gt;&#xA;&lt;li&gt;$Q(P(2.5))$&lt;/li&gt;&#xA;&lt;li&gt;$Q(P(14.8))$&lt;/li&gt;&#xA;&lt;li&gt;What is the relationship between the functions $P$ and $Q$?&lt;/li&gt;&#xA;&lt;li&gt;Simplify $P(Q(x))$.&lt;/li&gt;&#xA;&lt;li&gt;Simplify $Q(P(x))$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;With the stage now set, and after a brief conversation about what the result should be when composing a function and its inverse, they try the following problem:&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;10&quot;&gt;&#xA;&lt;li&gt;Find the inverse of the function $f(x) = 6-8x$. [Hint: Remember what the composition $f(f^{-1}(x))$ should equal? Write this as an equation, and then simplify the left-hand-side as in problem 2. Use algebra to solve for the function $f^{-1}(x)$.]&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Now, this is perhaps a longer introduction than the Switch method found in many books, but it seems worth it in a few respects. First, it &lt;strong&gt;reinforces&lt;/strong&gt; the notion and notation of composition -- students are taught to recite this each time they begin solving for an inverse. Second, they are getting a first glimpse that functions are objects which may be manipulated with algebra, something they will see again in differential equations. Finally, and not the most compelling reason, it causes students to use the new notation we've been working with all term -- for most of the functions we're dealing with (until logarithms), just switching $x$ and $y$ and solving is something we could have asked them to do in an intermediate algebra class, without any thought of functions or inverse processes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this method may downplay the notion that a function and its inverse have switched roles in terms of input and output. The Switch method reinforces that idea each time, where students literally exchange input for output between the functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:  Are my proposed benefits worth it for students, either in the long or short run? For those who have tried this method, did your students come away better off in any way?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:  Am I missing any other ways in which the &quot;switch $x$ and $y$ and solve&quot; method is preferable?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regard to Question 2, one situation I am anticipating for my class is that, given a formula for $f(x)$, they may write $f^{-1}(f(x)) = x$ instead of $f(f^{-1}(x)) = x$ and then have no idea what to do. The Switch method imposes no order of composition on the process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for taking the time with this long question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;update&lt;/strong&gt;: Removed a third part of this question as it was too broad in scope.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="11" LastEditDate="2017-10-30T15:41:04.590" LastActivityDate="2017-10-30T21:51:20.367" Title="When discussing inverse functions, how can our notation and methods reinforce student understanding?" Tags="|mathematical-pedagogy|precalculus|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13068" PostTypeId="2" ParentId="13065" CreationDate="2017-10-29T11:27:18.010" Score="6" Body="&lt;p&gt;In classes I've taught (at least since the mid 1980s), I always put the &quot;switch $x$ and $y$&quot; part where I think it belongs --- at the end of the calculation. For example, $f(x) = 2x+1$ becomes $y=2x+1$ and then $x=\frac{1}{2}(y-1)$ and then $f^{-1}(y)=\frac{1}{2}(y-1),$ at which point we CHANGE TO USUAL FUNCTION NOTATION (i.e. use $x$'s for the independent variable) and write $f^{-1}(x)=\frac{1}{2}(x-1).$ Note that the two function-inverse function identities, $f^{-1}(f(x))=x$ and $f(f^{-1}(y))=y$ are easy to prove using $y=f(x)$ and $x=f^{-1}(y)$ (i.e. when you don't switch $x$'s and $y$'s).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In motivating the change to usual function notation at the end, I remind students that $f^{-1}$ is the name of the rule whereas $f^{-1}(y)=\frac{1}{2}(y-1)$ is simply one (algebraic) way to describe the rule. The same rule can be described by $f^{-1}(u)=\frac{1}{2}(u-1)$ and $f^{-1}(P)=\frac{1}{2}(P-1)$ and $f^{-1}(z)=\frac{1}{2}(z-1).$ Thus, if the question asks you to give $f^{-1}(x),$ then you answer with $\frac{1}{2}(x-1)$ rather than with $\frac{1}{2}(y-1).$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I never understood why books, and teachers who have taught this material more than a few times, put the &quot;switch $x$ and $y$&quot; part at the beginning. Doing it that way seems to me to reinforce the belief in some students that mathematics consists of ad hoc procedures that one has to memorize.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2017-10-29T11:27:18.010" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13069" PostTypeId="2" ParentId="13065" CreationDate="2017-10-29T13:52:04.497" Score="0" Body="&lt;p&gt;I'll answer &lt;strong&gt;question 2&lt;/strong&gt;, providing an opportunity you will miss by avoiding the &quot;switch $x$ and $y$&quot; approach. My goal is the same as Dave L Renfro's: try to minimize the number of ad hoc procedures that students are expected to memorize.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students internalize the fact that the inverse function &lt;strong&gt;switches inputs with outputs&lt;/strong&gt; in every case, then it provides a mental framework for each of the following questions, tying them together into a neat conceptual package:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Find a formula for the inverse of the function $f(x) = 2x + 3$.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Find a table for the inverse of the following function:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;x   f(x)&#xA;2   6&#xA;3   8&#xA;4   13&#xA;&lt;/code&gt;&lt;/pre&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Find a graph of the inverse of the following function:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wMz4m.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wMz4m.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The graphical example is I think the best argument in favor of &quot;switch the inputs and outputs,&quot; because it frees the students from their destructive instinct to find an equation for everything first.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Finally, I have an in-class demonstration I do that helps with the graphical form of &quot;switch inputs and outputs&quot; that I think is the real payoff. It goes like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Using a thick marker of one color, draw the x-axis along the center of a square piece of paper on both sides of the paper. Label it &quot;$x$&quot; on both sides.&lt;/li&gt;&#xA;&lt;li&gt;Using a thick marker of another color, draw the y-axis along the center of the square piece of paper on both sides of the paper. Label it &quot;$y$&quot; on both sides.&lt;/li&gt;&#xA;&lt;li&gt;Fasten a dowel to the paper with tape along the line $y=x$.&lt;/li&gt;&#xA;&lt;li&gt;Display the paper to the class holding it by the line $y=x$, and ask what you do to find the inverse function (&quot;switch inputs and outputs,&quot; or &quot;switch $x$ and $y$&quot; as the target answers).&lt;/li&gt;&#xA;&lt;li&gt;Twirl the paper quickly 180 degrees, keeping the dowel in the same spot. It makes a satisfying snap sound, and reflects the paper over the line $y=x$. The $x$ and $y$ axes trade places!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This demonstrates that the transformation &quot;reflect over the line $y=x$&quot; is not another separate ad hoc magic rule to memorize, but is actually just another form of the overarching concept &quot;switch the inputs and outputs to get the inverse function.&quot;&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-10-29T13:52:04.497" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13070" PostTypeId="2" ParentId="13060" CreationDate="2017-10-29T14:30:30.190" Score="3" Body="&lt;p&gt;I think you are reading a lot into the answers to that question which is not there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether vector calculus and diff EQ prepare you for proof based courses is entirely dependent on the content of the particular course you have taken.  These can be quite rigorous courses, or they can be &quot;cookie cutter&quot; pattern matching types of courses depending on the instructor.  There is no special power of Calc 3 and Diff EQ to transform people into proof-ready mathematics students.  In fact, I would say that it is much more typical that Calc 3 and Diff EQ are the last &quot;service courses&quot; to other departments.  These courses are incredibly valuable from the perspective of real world applications, because the vast majority of mathematical problems in the real world involve multiple variables, and often involve relationships between quantities and rates of change in those quantities.  They absolutely should be included in the training of any mathematician for that reason.  They are placed first in these lists because they have less prerequisites, and because they often ``weed out'' students who are not willing to put in hard work mastering difficult content.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, learning how to do proofs requires a lot of hard work.  It doesn't happen all at once.  You can expect to get better and better throughout your entire career!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not knowing you beyond this question, there is no reason to think that you couldn't become a decent mathematician.  There is also little harm in trying and failing:  a math degree is pretty valuable for building your logical reasoning capability, and this is useful for lots of careers outside of being a professional mathematician.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for your final question:  There are lots of incredibly gifted people who pursue math education to the exclusion of math research not because they ``couldn't hack'' pure math, but because they are more motivated to teach than to do research.  You will find many such people on this website.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2017-10-29T14:30:30.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13071" PostTypeId="2" ParentId="13060" CreationDate="2017-10-29T15:26:01.833" Score="4" Body="&lt;p&gt;It seems to me that you don't really understand how mathematics education works (possibly education in generally). You don't magically acquire the ability to do anything by taking a course. You acquire the skills by practising it yourself, which hopefully you will do in the process of taking (at least some) courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, you learn to write proofs by getting on and trying to write them. You won't be able to do analysis proofs without studying some analysis, because you won't have a clue what you are trying to prove or how to go about doing it. But an Intro to Proofs course should give you a good idea of how to go about proofs in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is certainly not true that everyone who takes Calc 3 will be able to write proofs. That happens all the time. Calc 3 is not particularly intended to teach rigorous proof. That will happen in the algebra and analysis courses. But I would expect someone who is going to go on to be a pure mathematician to be showing signs of it even in non-proof-based courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that being a pure maths academic is extremely competitive (academia in general is, and it's harder to move to industry from pure maths than from many other areas of science). As well as generally requiring long hours, it restricts how much say you can have over where you live (often to essentially none). People might reasonably choose to teach at school level instead because they want more flexibility, or because they actually enjoy doing so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you should be wary of saying you definitely want to be a pure mathematician, given you have done very little pure maths yet. Also, some of the skills you seem to have given up on are necessary for that as a career path. As someone has mentioned, writing is &lt;em&gt;very&lt;/em&gt; important as a researcher. Without writing papers, you will not have a job. The style is different to what you are probably referring to, and learning more maths will help, but it is writing none-the-less. Coding is not essential, but understanding algorithms and logic are pretty important. And for the vast majority of academics, teaching makes up a large part of the job.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the apparent underlying question: it sounds like you are feeling rather down at the moment. It might be a good idea to visit your careers service, and find out about other jobs that are open to you. There is no reason to think that the options are only academic, teacher or McDonalds. There will be lots of jobs you haven't heard of yet. It might take some work to acquire the right skills/work experience, but that isn't a reason to give up. It may also be that you can do more than you think, and it is feeling down that is limiting your achievement.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2017-10-29T15:26:01.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13072" PostTypeId="2" ParentId="11396" CreationDate="2017-10-29T15:30:32.953" Score="3" Body="&lt;p&gt;In Jean Piaget's theory of cognitive development, the &quot;formal operational stage&quot; was the last one identified, achieved between adolescence and adulthood (ages 11 to 15-20), in which &quot;intelligence is demonstrated through the logical use of symbols related to abstract concepts&quot; (&lt;a href=&quot;https://en.wikipedia.org/wiki/Piaget%27s_theory_of_cognitive_development#Formal_operational_stage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One overview of research on this topic was presented by Mary Carol Day, &quot;Thinking at Piaget's Stage of Formal Operations&quot; (&lt;em&gt;Educational Leadership&lt;/em&gt;, October 1981, &lt;a href=&quot;http://www.ascd.org/ASCD/pdf/journals/ed_lead/el_198110_day.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;). She writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Piaget's position has often been used to predict that one who can&#xA;  think in a formal operational manner will always do so. However, only&#xA;  about 50 percent of those over 12 years of age who are presented with&#xA;  tests of formal operations perform in what would be considered a&#xA;  formal operational manner. This is true of college-educated adults as&#xA;  well as adolescents. In addition, even people who use formal&#xA;  operational skills on one task may not use them on another.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;These unexpected results have prompted a variety of responses. One is&#xA;  that Piaget was wrong; formal operations is not a stage attained by&#xA;  everyone. A second response came from Piaget himself. He maintained&#xA;  that all individuals attain formal operations, but perhaps only in&#xA;  areas with which they have had much experience (Piaget, 1972). A third&#xA;  approach has been to reexamine the meaning of the term &quot;stage.&quot; A&#xA;  stage can be viewed as a description of all cognitive activity that&#xA;  occurs during the stage (or cognitive &lt;em&gt;performance&lt;/em&gt;) or it can be viewed&#xA;  as a description of the highest level cognitive activity of which one&#xA;  is capable (cognitive &lt;em&gt;competence&lt;/em&gt;).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The available data tend to support the second and third alternatives&#xA;  rather than the first...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that the figure of &quot;about 50%&quot; of students who perform formal operations when tested is very much in line with Ben Crowell's answer here and other similar estimates. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-10-14T19:48:21.753" LastActivityDate="2018-10-14T19:48:21.753" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13073" PostTypeId="1" CreationDate="2017-10-29T16:30:39.763" Score="14" ViewCount="1314" Body="&lt;p&gt;Having read several undergraduate textbooks in complex analysis (Stein-Shakarchi, Gamelin, etc.), I find that some &amp;quot;hand-waving&amp;quot; arguments are frequently used. An example (the proof of the Cauchy integral formula in Stein-Shakarchi) is given in the appendix regarding what I mean for &amp;quot;hand-waving&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Hand-waving arguments in students' homework may be sometimes (usually?) due to a certain degree of &amp;quot;dishonesty&amp;quot;: one wants to gloss over a part that one does not fully understand to get credits. If one is brutally honest to oneself when doing homework, then one might not use such arguments at all &lt;em&gt;or&lt;/em&gt; one would explicitly admit that one is using such arguments so that the readers should be cautious in advance.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, textbooks written by masters are another story.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Questions&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;What are possible reasons for making hand-waving arguments in textbooks of undergraduate analysis? (Some are deliberately left as helpful exercises for the readers but some are not. I should be referring to those non-exercise ones.) A hand-waving argument in algebra is very different from one in analysis and I would like to restrict my question in the context of &lt;em&gt;analysis&lt;/em&gt; here.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In the views of students, how should one deal with such arguments for learning?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;[An example of hand-waving arguments in Stein-Shakarchi.]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/z0BIn.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/z0BIn.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/17gwV.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/17gwV.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7lZ0l.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7lZ0l.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/0PvKB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0PvKB.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user5897" LastEditorUserId="376" LastEditDate="2023-07-18T08:37:22.433" LastActivityDate="2023-07-24T05:58:49.703" Title="Why are hand waving arguments made in textbooks of undergraduate analysis and how should readers deal with them?" Tags="|undergraduate-education|textbooks|self-learning|mathematical-analysis|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13074" PostTypeId="2" ParentId="13073" CreationDate="2017-10-29T18:19:48.693" Score="19" Body="&lt;p&gt;A proof is meant to convince a reader of the truth of some statement.  When a mathematician is communicating an argument to another mathematician, you only include the level of rigor that you need so that the other mathematician is convinced that they could (in principle) give a fully rigorous argument.  Even that isn't quite accurate because no one (short of people producing fully formal computer verified proofs) actually produces totally formal arguments ever.&lt;/p&gt;&#xA;&lt;p&gt;In your example, while you could write down equations and inequalities for this keyhole shape, doing so would take up a large amount of room and distract from the clarity of the argument.&lt;/p&gt;&#xA;&lt;p&gt;If you are not currently at a level where the argument is clear, then it is your job to fill in the details to a level of rigor you feel comfortable with.  This is a nice additional exercise.  In fact, every textbook (even one which is very rigorous) will have frequent small gaps in logic which you need to fill in on your own.  Reading a math book is a creative endeavor in which you create the mathematics, and refine your own arguments, with a lot of assistance and guidance from the book.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="376" LastEditDate="2023-07-24T05:58:49.703" LastActivityDate="2023-07-24T05:58:49.703" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13075" PostTypeId="1" AcceptedAnswerId="13086" CreationDate="2017-10-29T20:40:06.277" Score="5" ViewCount="219" Body="&lt;p&gt;A few weeks ago, I started tutoring a student in Discrete Mathematics (a subject I took a year ago). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have previously tutored both pre-calculus and calculus, but never a proof based class. I have been approaching the job by helping the student work through the homework and helping her figure out content that she couldn't follow during class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am afraid that I might actually be doing the student more harm than good, as Discrete Mathematics is a proof based course (the first of this kind for her), and thus I am afraid that she will begin relying on me to get intuition for problem solving and writing proofs in general. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking about transitioning to a structure where she proves everything that they proved in class again by herself (with help where needed). The idea is inspired by &lt;a href=&quot;http://calnewport.com/blog/2008/11/25/case-study-how-i-got-the-highest-grade-in-my-discrete-math-class/&quot; rel=&quot;noreferrer&quot;&gt;this blog post&lt;/a&gt; I found last night. Does anyone with experience tutoring a proof based math course have any advice for me?&lt;/p&gt;&#xA;" OwnerUserId="8922" LastEditorUserId="5367" LastEditDate="2017-10-29T22:47:34.013" LastActivityDate="2017-10-31T04:29:53.867" Title="Tutoring Discrete Mathematics" Tags="|proofs|tutoring|discrete-math|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13076" PostTypeId="2" ParentId="13062" CreationDate="2017-10-30T01:41:07.253" Score="2" Body="&lt;p&gt;Check out the three books by James Fixx (Solve It, Games for the Super-intelligent, and More Games for the Super-intelligent).  I don't think they are super difficult, but of course please examine and make sure they are not too tough for your advanced 5th graders.  (If so, look for some other puzzle books a little easier.)&lt;/p&gt;&#xA;" OwnerUserId="8850" LastActivityDate="2017-10-30T01:41:07.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13077" PostTypeId="2" ParentId="11396" CreationDate="2017-10-30T02:22:16.110" Score="-2" Body="&lt;p&gt;You're asking a big question.  These kinds of things are not really settled yet to allow some pretty numerical answer.  Instead of pushing so hard for a nice pretty answer, ALLOWING open discussion, opinions, debate, would be actually be more revealing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For that matter the terms of the question are not precise (like how much intensive education are you willing to posit?  60 years of one on one tutoring?).  I don't mean this in some neener, neener be precise manner.  Just that even defining the terms is worthwhile and leads to a more manageable question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think to start with, try some easier questions:  what percent of the population passes* 9th grade algebra on first try?  (That's not the answer to your ultimate question, but it's informative to start thinking about the topic.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe as a second question, you could look at what percent of students eventually pass basic algebra.  Restrict it to a state perhaps if that makes it easier to evaluate.  (Make sure you account for dropouts of course.)  That would tell you the impact of SOME higher intervention (repeating courses).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*ideally you define this with an achievement test, but if these are not nation-wide, use grades.  Or you could look at some state(s) that do have an achievement test and consider them indicative of the overall population (maybe not precisely, but approximately).&lt;/p&gt;&#xA;" OwnerUserId="8850" LastActivityDate="2017-10-30T02:22:16.110" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13079" PostTypeId="2" ParentId="13049" CreationDate="2017-10-30T15:37:05.233" Score="1" Body="&lt;p&gt;You need to make sure he really understands what kind of actions each of the basic operations serve as a model.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So before algebraization of problems, try to work on &lt;strong&gt;identify the actions involved&lt;/strong&gt;. Next, &lt;strong&gt;Singapur bar models&lt;/strong&gt; can be useful as an intermediate way of modelling what's happening, specially on the relevance they put un identifying the total involved.&lt;/p&gt;&#xA;" OwnerUserId="8927" LastActivityDate="2017-10-30T15:37:05.233" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13080" PostTypeId="2" ParentId="13053" CreationDate="2017-10-30T18:08:51.157" Score="2" Body="&lt;p&gt;I find the &quot;Topology for Beginners&quot; youtube videos by Richard Southwell easy enough for middle-school-age kids to understand.&lt;/p&gt;&#xA;" OwnerUserId="8909" LastActivityDate="2017-10-30T18:08:51.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13081" PostTypeId="1" AcceptedAnswerId="13083" CreationDate="2017-10-30T18:13:06.607" Score="3" ViewCount="207" Body="&lt;p&gt;Problems like &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is 30 students in the class. $\frac13$ of the students study biology, $\frac15$ of the students study physics and the rest study chemistry. What part of the class studies chemistry?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What I would explain is that first we calculate what part of the class doesn't study chemistry $$\frac13 + \frac15 = \frac{8}{15}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, for finding the fraction we need I would explain that we need to do $$\mathbf{1\,-\,}\frac{8}{15} = \frac{7}{15}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the student gets confused because there are $30$ students, no $1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe there is an easier method for resolving these problems in a test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry if the question title isn't right, I'm not an English native speaker. Thanks for the help!&lt;/p&gt;&#xA;" OwnerUserId="5628" LastEditorUserId="5628" LastEditDate="2017-10-31T19:15:39.893" LastActivityDate="2017-11-22T17:18:40.950" Title="How do you explain the whole integer and fractions subject to a kid in 6th grade?" Tags="|fractions|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13082" PostTypeId="2" ParentId="13053" CreationDate="2017-10-30T18:37:19.690" Score="1" Body="&lt;p&gt;Keep it light and visual (not algebraic) and very, very basic.  Moebius strip.  Klein bottle.  Coffee cup to donut.  7 bridges of Konigsberg.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  It would help to advise you to know how long the talk is.&lt;/p&gt;&#xA;" OwnerUserId="8850" LastActivityDate="2017-10-30T18:37:19.690" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13083" PostTypeId="2" ParentId="13081" CreationDate="2017-10-30T20:43:13.927" Score="9" Body="&lt;p&gt;You could try keeping 30 as the denominator throughout, that is, observing that $\frac{1}{3} = \frac{10}{30}$ and $\frac{1}{5} = \frac{6}{30}$, so the portion of the class that doesn't study chemistry is&#xA;$$\frac{10}{30} + \frac{6}{30} = \frac{16}{30},$$&#xA;and the portion that does study chemistry is&#xA;$$\frac{30}{30} - \frac{16}{30} = \frac{14}{30}.$$&#xA;Once they understand this line of reasoning, you can point out that these fractions can be expressed in terms of smaller denominators.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2017-10-30T20:43:13.927" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13084" PostTypeId="2" ParentId="13081" CreationDate="2017-10-30T21:02:40.700" Score="9" Body="&lt;p&gt;I think this problem is much easier to handle if you refer throughout to the &lt;em&gt;number&lt;/em&gt; of students who study each subject, rather than the &lt;em&gt;fraction&lt;/em&gt;, only expressing the final answer as a fraction at the very end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\frac{1}{3}$ of the $30$ students study Biology — this is $10$ students, because $\frac{1}{3} \times 30 = 10$.&lt;/li&gt;&#xA;&lt;li&gt;$\frac{1}{5}$ of the $30$ students study Physics — this is another $6$ students, because $\frac{1}{5} \times 30 = 6$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Since 16 of the 30 students are accounted for, the remaining 14 must study Chemistry.  Therefore the fraction that studies Chemistry is $\frac{14}{30}$, or (in lowest terms) $\frac{7}{15}$.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2017-10-30T21:02:40.700" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13085" PostTypeId="2" ParentId="13065" CreationDate="2017-10-30T21:51:20.367" Score="1" Body="&lt;p&gt;This idea is a little unusual, but you might consider making the inverse function part of a bigger picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If&#xA;$$f(x)=2x+1$$&#xA;then two applications of the function gives us&#xA;$$f^{(2)}(x)=f(f(x))=2(2x+1)+1=4x+3$$&#xA;and similarly three applications&#xA;$$f^{(3)}(x)=f(f(f(x)))=2(4x+3)+1=8x+7$$&#xA;Once that has been mastered, then mention that no applications of the function gives us&#xA;$$f^{(0)}(x)=x$$&#xA;and then you can introduce the idea that we can do the reverse.&#xA;$$f^{-1}(x)=f^{(-1)}(x)=\frac{x-1}{2}$$&#xA;&lt;hr&gt;&#xA;If you wanted, you could even find general formulas for the applications that are consistent with the inverse.&#xA;$$f^{(n)}=2^n (x+1)-1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then, if you have students with a passion for math, from there you could even introduce the concept that a function could be applied a fractional number of times.&#xA;$$f^{(\frac{3}{2})}(x)=2^{\frac{3}{2}} (x+1)-1=2x \sqrt 2+2 \sqrt 2-1$$&lt;/p&gt;&#xA;" OwnerUserId="8642" LastActivityDate="2017-10-30T21:51:20.367" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13086" PostTypeId="2" ParentId="13075" CreationDate="2017-10-31T04:29:53.867" Score="5" Body="&lt;p&gt;I have found the best thing to do in tutoring a proof based class is to constantly ask the questions &quot;why is that true?&quot; or &quot;what is the definition of that?&quot;. These questions help them build relations between definitions and highlight the connections to proofs as filling in the details relating things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, on occasion you might have to do an example of a proof technique. I remember induction being a challenge myself until I saw it in action a few times. But if they are working a problem themself, start with asking &quot;what is the definition of the beginning object?&quot; and &quot;are there any theorems or propositions we can use relating to it?&quot;. Ask &quot;what is our goal?&quot; and &quot;what do you think we need to show to get there?&quot;. Correct them on their mathematical grammar and clarify that they write (not just say!) their implications correctly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your goal as tutor is to demonstrate good proof technique and to further train them on knowing the definitions and statements of things that they need to know. Questions serve as training their guidance to when to use their statements. Yeah, they'll probably struggle a bit, but you need to help build their confidence and logical thinking skills.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2017-10-31T04:29:53.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13090" PostTypeId="2" ParentId="10866" CreationDate="2017-11-01T17:26:02.417" Score="1" Body="&lt;p&gt;In the following I will try to answer the question asked in the headline and to provide support that is equally applicable to all three styles distinguished in the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on my own experience as a student as well as during a lifetime as a teacher (mathematics and mathematical sciences) I have become convinced that writing is an important tool to digest learnt contents and to fix it in the memory. This may not concern every student but certainly more than 80 %. In order to stimulate this behaviour I have allowed my students to use one page as a crib during exam. And I have told them in the first lesson about this chance to gain an advantage. The only condition: The sheet must be originally handwritten by the student herself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The optimal result is this. The student will write much of the stuff immediately after the lesson. During the semester he will have to drop more and more because of limited space. Therefore she will repeat to write the most important and most difficult things again and again. In addition this method is also practically relevant because also in their later professional life the students will be allowed to consult literature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is common wisdom that writing supports memorizing. But there are also written sources supporting this standpoint. Google shows about 10 million hits for memorizing by writing like those given below:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The very act of writing things down helps to get them lodged into your long-term, ... Memorizing facts for tests by writing them out several times.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Memorize abstracts. If you're memorizing abstractions, such as the value of pi, write down the individual numbers or steps on each flash card. Then personalize ...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8557" LastEditorUserId="8557" LastEditDate="2017-11-01T21:00:19.643" LastActivityDate="2017-11-01T21:00:19.643" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13091" PostTypeId="1" AcceptedAnswerId="13093" CreationDate="2017-11-02T12:23:15.973" Score="13" ViewCount="1785" Body="&lt;p&gt;I am a Ph.D student in computer science. I am TAing one course this semester, which requires the basics of abstract algebra like rings, fields, ideals, and basic theorems about them. I have done two courses in the mathematics department related to algebra (abstract algebra and commutative algebra).&lt;/p&gt;&#xA;&lt;p&gt;Working as a TA, once a week I have to teach some basic algebra objects and their theorems to these students. This is the first time I am teaching things related to algebra. I feel like it is hard to explain these things to students. This is how I teach them: I first write the definition and then give them one or two examples. Many times I find out that these students say that they have understood the definition, but when I ask them the next day, they are blank. I want to simplify things, but I don't know if this will be a good way to teach algebra.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Question: How do I teach algebra for the first time?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8938" LastEditorUserId="376" LastEditDate="2021-01-30T16:14:33.387" LastActivityDate="2021-01-30T16:14:33.387" Title="How to teach abstract algebra for the first time?" Tags="|undergraduate-education|teaching|interactive-teaching|abstract-algebra|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13092" PostTypeId="2" ParentId="13091" CreationDate="2017-11-02T13:15:03.630" Score="8" Body="&lt;p&gt;I think that you probably need to give them some hands-on experience before any of it will sink in. I assume that they are CS students? After you give them the definition and a few examples, you could have them work out -- for themselves -- the addition and multiplication tables for the field with $2^n$ elements, expressed as strings of bits. (Keep $n$ small, maybe $n\leq 4$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can, have them implement some ring in whatever programming language you or they prefer. The ring axioms are very much like a interface specification, after all.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2017-11-02T13:15:03.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13093" PostTypeId="2" ParentId="13091" CreationDate="2017-11-02T15:56:24.920" Score="11" Body="&lt;p&gt;I second Adam. Keep it with strong with examples if you can. There is a &lt;a href=&quot;https://mathoverflow.net/questions/13320/cool-problems-to-impress-students-with-group-theory&quot;&gt;mathoverflow answer&lt;/a&gt; with a ton of group theory examples. You might want to force them to practice doing more &quot;rote&quot; exercises that often come in an intro abstract algebra book to help gather examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also it should be mentioned that not all CS students have a strong background with proof based mathematics. If the course requires the use of proofs (as abstract algebra often does), you might want to highlight proof techniques as well. Any definition you give should come with a few examples, plus I like to include a non-example. This will help build their intuition. Also, theorems and the like should be presented as relating definitions or noticing trends in your now plethora of examples. Proofs are just there as a formal way to explain why things are true. I find that describing proofs this way removes the intimidation factors of having to do a proof from many students.&lt;/p&gt;&#xA;" OwnerUserId="201" LastActivityDate="2017-11-02T15:56:24.920" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13094" PostTypeId="2" ParentId="13091" CreationDate="2017-11-02T22:10:22.593" Score="3" Body="&lt;p&gt;Some years ago I worked as a &quot;CA&quot; (Course Assistant) for preservice/inservice teachers taking a first course on Abstract Algebra (mostly group theory, with a bit on rings and fields towards the end). After the weekly CA sessions concluded, I TeXed a summary of what was covered. I found that having this extra document was very helpful for students (especially those whose schedules precluded them from attending at all!) to refer back to, and I also made an effort to write out a sample midterm, some sample solutions, and a sample final. When discussing answers (and sometimes in the TeXed notes) I made an additional effort to give insight, when I could, into how I came up with certain ideas. Perhaps this becomes more difficult, or less relevant, when one's understanding of a subject has deepened, but I thought it was helpful (and still think it can be helpful!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify the sort of notes that I am talking about, and just in case it might be useful, I have uploaded all of these 2012 notes to dropbox: &lt;a href=&quot;https://www.dropbox.com/sh/qnl0ovm6u3gqjj8/AAAWY3A9ZOryHXTDvHm8Gmy0a?dl=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Folder Link&lt;/strong&gt;&lt;/a&gt;. There are references to various problems in these notes for which I do not have the original materials; still, I hope making these available will be a net positive. (As always: Be on the lookout for errors!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do not feel like digging through dropbox materials, then I can point you to at least one answer I put on MSE, about group theory, in which I tried to describe my thinking in a manner that could be helpful to someone learning the rudiments of the subject: &lt;a href=&quot;https://math.stackexchange.com/a/1215264&quot;&gt;&lt;strong&gt;MSE 1214719&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2017-11-02T22:10:22.593" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13095" PostTypeId="1" CreationDate="2017-11-03T07:01:33.467" Score="4" ViewCount="540" Body="&lt;p&gt;I want to explain Chinese remainder theorem to master level computer science students. There are two versions of CRT one is number theoretic and second requires the definition of ideals, groups etc. Although students know about these definitions, but I think I should explain these definitions again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is from where I should start and how to motivate them. Should I start with examples or first definition and then example. What are the good examples I should give them. I don't want to confuse my students with lots of symbols and notation. Please note that target audience is master level computer science students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The name of the course is Algebraic Computation, it is mostly related computational group theory etc.&lt;/p&gt;&#xA;" OwnerUserId="8938" LastEditorUserId="8938" LastEditDate="2017-11-04T06:15:05.877" LastActivityDate="2019-04-26T11:53:12.083" Title="How to explain Chinese remainder theorem?" Tags="|algebra|teaching|terminology|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13096" PostTypeId="1" CreationDate="2017-11-03T12:12:34.357" Score="4" ViewCount="146" Body="&lt;p&gt;I started tutoring IGCSE Mathematics. In preparation for just a 1 hour class for 1 student, I noticed at least 3 mistakes (confirmed with their replies to me) in Chapter 21 of the Haese 0607 book (quadratic equations and functions). I even noticed a mistake in the errata (&lt;a href=&quot;https://public-koi-snowflake-haesemathematics-com-au.s3.amazonaws.com/2016/02/17/03/39/55/120/igcse.PDF&quot; rel=&quot;nofollow noreferrer&quot;&gt;last updated almost 2 years ago though I found the 3 mistakes in just 1 day&lt;/a&gt;) for a different chapter (probability).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dumb question: How are these mistakes going to be disseminated?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mean if I've noticed these mistakes, I wonder about all the other readers on Earth (students, teachers, tutors, parents, etc) who notice mistakes. I think it would be tiresome for both the authors to reply to the same mistakes and the readers to type up emails instead of having some dynamic errata (forum i guess).&lt;/p&gt;&#xA;" OwnerUserId="5987" LastActivityDate="2017-11-05T20:45:17.650" Title="Dissemination of mistakes in international texts such as the IGCSE" Tags="|textbooks|functions|reference-books|graphing|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13097" PostTypeId="1" AcceptedAnswerId="13098" CreationDate="2017-11-03T14:20:35.450" Score="13" ViewCount="1424" Body="&lt;p&gt;I very often see students writing, for example, $1/3x$ when they mean&#xA; $\frac 13x$. I used to tell them not to write $1/3x$ beause it looks like $\frac{1}{3x}$ until I realized that, according to PEMDAS, it really does mean $\frac 13x$. I feel that using the &lt;em&gt;vinculum&lt;/em&gt; (e.g. $\frac 13$) is easier on the mind than using the &lt;em&gt;solidus&lt;/em&gt;, (e.g. $1/3$) but telling a student that seems to be the equivalent of &quot;because I said so.&quot; Does anyone have a better reason for using one over the other?&lt;/p&gt;&#xA;" OwnerUserId="5144" LastEditorUserId="5144" LastEditDate="2017-11-04T06:38:45.383" LastActivityDate="2017-12-06T21:58:46.400" Title="Writing Fractions &quot;Correctly&quot;" Tags="|notation|" AnswerCount="5" CommentCount="22" ContentLicense="CC BY-SA 3.0" />
  <row Id="13098" PostTypeId="2" ParentId="13097" CreationDate="2017-11-03T15:16:50.900" Score="24" Body="&lt;p&gt;I think that, depending on the maturity level of the students, you could just talk to them about why writing $\frac{1}{3}x$ rather than $1/3x$ makes it much clearer what you mean. They should understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remind them that the purpose of writing anything, including mathematics, is to clearly convey an idea to the reader, and the notation $1/3x$ is ambiguous: some people might read it as $(1/3)x$, but many others will read it as $1/(3x)$. The most straightforward way to avoid that lack of clarity is to just write $\frac{1}{3}x$ instead.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2017-11-03T15:16:50.900" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13099" PostTypeId="1" CreationDate="2017-11-03T15:21:30.387" Score="6" ViewCount="1331" Body="&lt;p&gt;Right now I’m teaching precalculus in high school and I want to propose a project to my students about polynomial functions. They already know enough about quadratic functions and we study variation models recently. We start working with polynomials this week and I really want to do a 2 weeks project for them with this topic but I’m falling short in ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I already have a bacteria growth project prepared for them in exponentials functions but I want also to make something that meaningful to them in this topic too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any ideas?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you in advance.&lt;/p&gt;&#xA;" OwnerUserId="8853" LastActivityDate="2017-11-20T00:33:53.633" Title="Ideas for a 2 weeks project focused in polynomial functions" Tags="|secondary-education|calculus|precalculus|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13100" PostTypeId="2" ParentId="13097" CreationDate="2017-11-03T22:59:20.847" Score="2" Body="&lt;p&gt;This is an issue of what appears to be &lt;a href=&quot;https://en.wikipedia.org/wiki/Concatenation_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;concatenation&lt;/a&gt;, but what we assume is multiplication. That is, placing $x$ next to the $3$ typically means &quot;multiply&quot;, so we give it a place it in the regular order of operations. That being the case, the assumption is that what the author means by $1/3x$ is $1 \div 3 \times x$, which would be done from left-to-right. This is equivalent to the use of the vinculum: $\frac{1}{3}x$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there are definitely times where we concatenate but don't follow the order of operations. Consider the expression $$15 ft^2 \div 5 ft$$ where (I bet) the assumed meaning is $\frac{15 ft^2}{5 ft} = 3 ft$. But considering that the concatenation $5 ft$ could mean $5 \times ft$ or $ft \times 5$, then the original expression (obeying the order of operations) is ambiguous: $$15 ft^2 \div 5 ft = 15 \times ft \times ft \div 5 \times ft = 3 ft^3$$&#xA;Or...&#xA;$$15 ft^2 \div 5 ft = 15 \times ft \times ft \div ft \times 5 = 75 ft$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, this is just the situation you are trying to prevent by cautioning your student to write $\frac{1}{5}x$ instead of $1/5x$. The notation they use should help clarify their meaning. [Incidentally, this issue isn't particular to units -- some &lt;a href=&quot;https://math.stackexchange.com/questions/126351/casio-calculators-give-2-different-answers&quot;&gt;calculators behave differently&lt;/a&gt; when concatenating a number with a parenthesis, and others when concatenating two variables (type &quot;xy/x&quot; on a TI-89).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The short answer I give my students is to treat concatenation with care. If you mean &quot;multiply&quot;, then it is often best to use the symbol. We break this rule when simplifying algebra, writing $6x$, but always treating it as multiplication. I never model something like $1/6x$ without using a multiplication symbol, such as $1/6\cdot x$. So, if a student comes to me with $10/3x$, I will tell them that I honestly cannot tell what this means, and that they need to include a multiplication symbol, or parentheses or both.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2017-11-03T22:59:20.847" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13101" PostTypeId="1" AcceptedAnswerId="13106" CreationDate="2017-11-03T23:26:42.270" Score="12" ViewCount="436" Body="&lt;p&gt;I'm interested in learning about the New Math movement from a historical perspective. I've located some secondary sources about the topic, mainly parodies, highly critical restrospective articles, or histories seeking to explain the failure of the movement. I'm not interested in these.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I would like to find falls into three categories:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What I would most like to read is something like a teacher's manual that addresses changes in class format or teaching technique, beyond just the changes to material content. In other words, instructions on how the idea of new math was to be implemented in practice.  How was the new system actually deployed?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Which standard contemporary textbooks illustrate the specific changes to the curriculum and material content?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Journal articles and other scholarly discussion preceding the movement that discussed the need for and development of New Math. Records of government proposals to implement the changes. What did the developers intend, how did they plan to accomplish it?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Does anyone know where I might get ahold of the above materials?&lt;/p&gt;&#xA;" OwnerUserId="263" LastActivityDate="2017-11-04T11:33:38.247" Title="Where can I find primary sources from the New Math movement in the 60s?" Tags="|reference-request|history|new-math|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13102" PostTypeId="2" ParentId="2074" CreationDate="2017-11-04T02:37:48.803" Score="9" Body="&lt;p&gt;One difficult issue with Euclidean geometry is that Euclid's approach, although it was an important step historically, is highly non-rigorous by modern standards.  This is especially clear in the definitions and basic notions (what does &quot;a line is breadthless length&quot; mean?).  But there's a much bigger problem that already comes up in Prop 1, which is that there's no way to see from the axioms that those two circles intersect at all!  Indeed $\mathbb{Q}^2$ satisfies all of Euclid's axioms, but the circles won't intersect since they share no rational points.  As &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_axioms&quot; rel=&quot;noreferrer&quot;&gt;Hilbert and others&lt;/a&gt; explained, there are ways to fix this issue by adding a lot more axioms.  But at that point you're almost doing analysis anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think number theory, group theory, analysis and graph theory are all better ways to introduce proofs than Euclidean geometry.&lt;/p&gt;&#xA;" OwnerUserId="345" LastActivityDate="2017-11-04T02:37:48.803" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13103" PostTypeId="2" ParentId="13097" CreationDate="2017-11-04T04:56:52.920" Score="5" Body="&lt;p&gt;$1/3x = \frac{1}{3} x$ is the standard.  The advantage of following the standard is that other people who also know the standard will understand what you write, and (most) calculators will do what you expect. Also, this is consistent with how addition and subtraction work, so there's less cognitive overload.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1/3x = \frac{1}{3x}$ has the advantage that the typesetting better aligns with our spatial recognition abilities; i.e. the &lt;em&gt;spacing&lt;/em&gt; delineates how the terms group. Also, in unformatted text, it can be a convenient shorthand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you &lt;em&gt;should&lt;/em&gt; tell the students is &quot;don't use this notation, no matter which way you mean it&quot;, since the goal of writing is to communicate your meaning, and you're doing a poor job of that if you choose to write in a way that is known to be commonly misunderstood.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2017-11-04T04:56:52.920" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13104" PostTypeId="2" ParentId="13095" CreationDate="2017-11-04T05:31:43.360" Score="0" Body="&lt;p&gt;I assume that these CS students know how to encrypt using RSA.  It can be shown that combining RSA with the Chinese Remainder Theorem can speed up the decryption of messages.  This might be good motivation with a CS focus.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-11-04T05:31:43.360" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13106" PostTypeId="2" ParentId="13101" CreationDate="2017-11-04T11:33:38.247" Score="10" Body="&lt;p&gt;The following three books are, I believe, the most significant of the earlier treatments of new math, and I suspect you can find much in them that will direct you towards literature for your questions #1 and #3. Also, since it is very likely that at least one of these 3 books will be cited by any reasonably researched publication, you can google their titles for many other sources of secondary literature (which then should direct you towards more literature, for all of your questions).&lt;/p&gt;&#xA;&lt;p&gt;Of the following three books, &lt;strong&gt;[1]&lt;/strong&gt; is freely available on the internet and has a 24-page bibliography of 346 items. I’ve looked through &lt;strong&gt;[1]&lt;/strong&gt; quite a lot, and further below I’ve included some notes that I have on &lt;strong&gt;[1]&lt;/strong&gt; (table of contents and text of its introduction). I haven’t looked at &lt;strong&gt;[2]&lt;/strong&gt; in several years and I don’t have any notes to myself about it, but &lt;strong&gt;[2]&lt;/strong&gt; is available in most any U.S. college or university library. As far as I can recall, I have never looked at &lt;strong&gt;[3]&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Robert W. Hayden, &lt;a href=&quot;http://lib.dr.iastate.edu/rtd/7427/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;A History of the &amp;quot;New Math&amp;quot; Movement in the United States&lt;/strong&gt;&lt;/a&gt;, Ph.D. Dissertation (under William Brown Rudolph), Iowa State University, 1981, v + 271 - 2 pages. [Note: The last numbered page of this dissertation is labeled &amp;quot;271&amp;quot;. However, in this dissertation there are two instances where a single sheet is labeled with two page numbers---a single sheet labeled &amp;quot;222-223&amp;quot; and a single sheet labeled &amp;quot;263-264&amp;quot;.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Philip S. Jones (editor), &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/9999140688&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;History of Mathematics Education in the United States and Canada&lt;/strong&gt;&lt;/a&gt;, 32nd Yearbook, National Council of Teachers of Mathematics. &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/a-history-of-mathematics-education&quot; rel=&quot;noreferrer&quot;&gt;review by Anthony V. Piccolino  of the 2002 2nd printing&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; William Wooton, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0006BMQOG&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;SMSG. The Making of a Curriculum&lt;/strong&gt;&lt;/a&gt;, Yale University Press, 1965. &lt;a href=&quot;http://science.sciencemag.org/content/150/3693/202.1&quot; rel=&quot;noreferrer&quot;&gt;review by Harry M. Gehman in &lt;strong&gt;Science&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;TABLE OF CONTENTS of [1]&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I. Preface&lt;/strong&gt; (pp. 1-4).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;II. The Mathematical Background of the &amp;quot;New Math&amp;quot;&lt;/strong&gt; (pp. 5-46). subtitled: &lt;em&gt;A. Introduction&lt;/em&gt; (pp. 5-7); &lt;em&gt;B. The Foundations of Geometry and the Reformation of Analysis&lt;/em&gt; (pp. 7-17); &lt;em&gt;C. The Development of Set Theory&lt;/em&gt; (pp. 17-20); &lt;em&gt;D. Non-Euclidean Geometry&lt;/em&gt; (pp. 20-24); &lt;em&gt;E. Modern Abstract Algebra&lt;/em&gt; (pp. 24-27); &lt;em&gt;F. The Spread of Modern Mathematics&lt;/em&gt; (pp. 27-36); &lt;em&gt;G. The Impact of Modern Mathematics&lt;/em&gt; (pp. 36-45); &lt;em&gt;H. Conclusion&lt;/em&gt; (pp. 45-46).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;III. Educational Reform Prior to the &amp;quot;New Math&amp;quot;&lt;/strong&gt; (pp. 47-71).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;IV. The Second World War and Mathematics Education&lt;/strong&gt; (pp. 72-99). subtitled: &lt;em&gt;A. The Importance of Mathematics to Society&lt;/em&gt; (pp. 72-74); &lt;em&gt;B. New Uses for Mathematics&lt;/em&gt; (pp. 74-75); &lt;em&gt;C. The Impact of the War on Applied Mathematics&lt;/em&gt; (pp. 76-80); &lt;em&gt;D. The Impact of the War on Society's Support of Mathematics&lt;/em&gt; (pp. 80-83); &lt;em&gt;E. The Impact of the War on School Mathematics&lt;/em&gt; (pp. 83-87); &lt;em&gt;F. The Impact of the War on College Mathematics&lt;/em&gt; (pp. 87-99).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;V. Secondary School &amp;quot;New Math&amp;quot;&lt;/strong&gt; (pp. 100-172). subtitled: &lt;em&gt;A. The University of Illinois Committee on School Mathematics (UICSM)&lt;/em&gt; (pp. 100-107); &lt;em&gt;B. The University of Maryland Mathematics Project (UMMaP)&lt;/em&gt; (pp. 107-109); &lt;em&gt;C. The Commission on Mathematics of the College Entrance Examination Board (CEEB)&lt;/em&gt; (pp. 109-116); &lt;em&gt;D. Sputnik&lt;/em&gt; (pp. 116-120); &lt;em&gt;E. The School Mathematics Study Group (SMSG)&lt;/em&gt; (pp. 120-139); &lt;em&gt;F. Other Programs&lt;/em&gt; (pp. 139-143); &lt;em&gt;G. Dissemination&lt;/em&gt; (pp. 143-157); &lt;em&gt;H. Reaction&lt;/em&gt; (pp. 157-172).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;VI. Elementary School &amp;quot;New Math&amp;quot;&lt;/strong&gt; (pp. 173-234). subtitled: &lt;em&gt;A. The Problem of Teacher Training&lt;/em&gt; (pp. 173-176); &lt;em&gt;B. The Early Elementary School &amp;quot;New Math&amp;quot; Programs&amp;quot;&lt;/em&gt; (pp. 176-189); &lt;em&gt;C. Training Teachers to Teach the &amp;quot;New Math&amp;quot;&lt;/em&gt; (pp. 190-202); &lt;em&gt;D. Dissemination&lt;/em&gt; (pp. 202-206); &lt;em&gt;E. The Conflict Between Elementary School &amp;quot;New Math&amp;quot; and the Progressive Tradition in Education&lt;/em&gt; (pp. 207-222/223); &lt;em&gt;F. The Neo-progressives&lt;/em&gt; (pp. 222/223-234).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;VII. Conclusion&lt;/strong&gt; (pp. 235-245).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;VIII. References&lt;/strong&gt; (pp. 246-269; 346 items listed).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;IX. Acknowledgments&lt;/strong&gt; (pp. 270-271).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;I. Preface of [1] (pp. 1-4):&lt;/strong&gt; [underlining in the original indicated by italics here]&lt;/p&gt;&#xA;&lt;p&gt;The 1950s and 1960s saw a major upheaval in the content and viewpoint of school mathematics. The changes that took place during those years were part of a coherent movement to reform mathematics education. The various reforms advocated by the leaders of this movement became known collectively as the &amp;quot;new math.&amp;quot; The present work is a history of the &amp;quot;new math&amp;quot; movement in the United States.&lt;/p&gt;&#xA;&lt;p&gt;This movement brought about change in the school mathematics curriculum on a scale and at a rate unknown before---or since. The 1970s and 1980s have seen a drift in the opposite direction. For example, there have been demands to go &amp;quot;back to the basics.&amp;quot; While it is not always clear what is basic, such demands clearly challenge recent reforms and call for a return to a more traditional curriculum. Unfortunately, such reaction against the &amp;quot;new math&amp;quot; has often caused us to lose sight of its entirely valid criticisms of &amp;quot;old math.&amp;quot; What is needed is not to go back, but to go forward more wisely. Part of that wisdom can come from examining the history of the &amp;quot;new math.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;new math&amp;quot; did more than simply criticize the old. It also provided answers to some of the problems of mathematics education. It did not answer every question, nor was every question that it answered correctly, but it did give &lt;em&gt;some&lt;/em&gt; answers. These answers can be relevant to the educational problems of today. For example, while writing this history, the author was teaching a remedial college mathematics course in beginning algebra. The chosen text was written as if the &amp;quot;new math&amp;quot; had never existed. A successful effort was made to have the text changed the next time the course was offered. The new text had as one of its coauthors a leader in the &amp;quot;new math&amp;quot; movement. The new text was not a &amp;quot;new math&amp;quot; book, but its authors had &lt;em&gt;learned&lt;/em&gt; from the &amp;quot;new math,&amp;quot; and what they learned made theirs a much better text. One of the purposes of this history is to prevent the lessons learned in the &amp;quot;new math&amp;quot; experience from being lost. Another purpose is to keep alive an awareness of basic issues in mathematics education that were raised by the &amp;quot;new math.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;There have been no previous attempts to write a history of the &amp;quot;new math.&amp;quot; It is true that many articles, pamphlets, and books explaining &amp;quot;new math&amp;quot; appeared during the 1960s. These often contained a sketch of the historical background of the movement. However, none of these sketches pretended to be serious scholarly studies. They represented instead the impressions of those involved in the movement in the nature of its roots.&lt;/p&gt;&#xA;&lt;p&gt;The one serious scholarly work that treats the &amp;quot;new math&amp;quot; is the Thirty-second Yearbook of the National Council of Teachers of Mathematics, &lt;em&gt;A History of Mathematics Education in the United States and Canada&lt;/em&gt; (1970). Although this work is well-documented, it naturally only gives a part of its attention to the &amp;quot;new math.&amp;quot; In addition, it puts too much emphasis on what was &lt;em&gt;said&lt;/em&gt; about mathematics education, by various experts and commissions, and too little emphasis on what was &lt;em&gt;done&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A final earlier work relevant to the present study is a history of the largest &amp;quot;new math&amp;quot; group written by one of its members (Wooton, 1965). While invaluable in its own area, the book is far too narrow to give an adequate appreciation of the entire movement and its origins.&lt;/p&gt;&#xA;&lt;p&gt;In writing the present work, an attempt has been made to make it as &lt;em&gt;useful&lt;/em&gt; as possible. This attempt has influenced the final work in two main ways. First, to make it readable by as wide an audience as possible, explanations of the mathematical background of the &amp;quot;new math&amp;quot; are given in terms understandable to the educated layman. Second, the most space has been devoted to topics that have not been adequately covered in other works. Thus, the activities of the School Mathematics Study Group are sketched only with enough fullness to maintain continuity, since a detailed history of this group is available (Wooton, 1965). On the other hand, the influence of World War II and of educators and educational philosophy on the &amp;quot;new math&amp;quot; have been largely neglected. Here, they are treated in some detail. Nearly every discussion of the &amp;quot;new math&amp;quot; includes mention of its roots in changes that took place in mathematics in the nineteenth century. Less common is an explanation of those changes, and a description of how those changes gradually diffused over the span of a century to influence the school curriculum. Both of these omissions are corrected here. Our story begins with an account of the effect that mathematicians and changes in mathematics had on the &amp;quot;new math.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="745" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-11-04T11:33:38.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13107" PostTypeId="2" ParentId="13095" CreationDate="2017-11-04T13:52:49.523" Score="2" Body="&lt;p&gt;I would start with motivation first. Since these are CS students, it should be an application. You can use anything that would broadly be considered useful. (i.e. not minutiae of your research.) Motivate why you would like a theorem like the CRT before you state it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do a few toy cases visually. For example, here are residues modulo 3 and 5:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/otW6P.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/otW6P.png&quot; alt=&quot;residues modulo 3 and 5&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and here are residues modulo 3 and 6:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/toBBn.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/toBBn.png&quot; alt=&quot;residues modulo 3 and 6&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that not all of the latter's lattice points are hit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that, you can dive into computational aspects. Avoid the version with ideals until the end; if you can get them to understand the usual version of the CRT, the principle ideal version should be a homework exercise. i.e. once you know what it means for two ideals to be relatively prime, the proof looks identical to the usual one.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2019-04-26T11:53:12.083" LastActivityDate="2019-04-26T11:53:12.083" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13108" PostTypeId="1" CreationDate="2017-11-04T14:35:46.927" Score="3" ViewCount="426" Body="&lt;p&gt;Last year, I taught GCD and LCM and then gave my students word problem relating to these concepts (&quot;Two runners with given different speeds; when will they meet again?&quot;, &quot;Having three kinds of flowers with different numbers, we want to make similar bunches of flowers. What is the size of the largest one?&quot;, etc.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the students didn't &quot;see&quot; GCD and LCM in these problems and they solved them without reference to these concepts. How can I help them?&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2017-12-29T19:20:35.343" Title="&quot;Seeing&quot; GCD and LCM in Word Problems" Tags="|number-theory|problem-solving|word-problems|middle-school|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13109" PostTypeId="1" AcceptedAnswerId="13113" CreationDate="2017-11-05T20:03:19.277" Score="6" ViewCount="561" Body="&lt;p&gt;A discussion with a frustrated 10th grade student sent me here. &#xA;I had provided two linear function expressions, &#xA;$f(x)=2x+2$ and $g(x)=-\frac{1}{2}x-2$, &#xA;now find the intersection of the two lines!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student seemed clear that there was the same $y$-value&#xA;for both $f$ and $g$, thus the equation $f(x)=g(x)$&#xA;made sense, he suggested.&#xA;But planning his work with this, he failed to understand,&#xA;how he could use the equation, where he &lt;strong&gt;knew for sure&lt;/strong&gt;&#xA;that the $y=f(x)$ indeed was the same as the $y=g(x)$:&#xA;Why couldn't he finish the task by solving&#xA;$2x+2=-\frac{1}{2}x-2$, getting $x=-\frac{8}{5}$?&#xA;So where did the $y$ value go? &#xA;He just &lt;em&gt;used&lt;/em&gt; it, but it is not there...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I find interesting, is that he -- in a manner of speaking --&#xA;realises only one side of a dual problem.&#xA;But this duality was hard to get to the students understanding.&#xA;Too hard, actually...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, my question may be phrased: What problems may build &#xA;students' understanding of the duality of arguing for identity &#xA;and establishing value?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the example, the student was perfectly clear, &#xA;that the two lines had same height in intersection.&#xA;He failed, however to see that the height, or $y$ value,&#xA;was not yet established.&lt;/p&gt;&#xA;" OwnerUserId="3502" LastEditorUserId="3502" LastEditDate="2017-11-05T21:34:06.077" LastActivityDate="2017-11-06T15:48:49.993" Title="Stating identity is not the same as knowing value" Tags="|secondary-education|algebra|problem-solving|language-use|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13110" PostTypeId="2" ParentId="13099" CreationDate="2017-11-05T20:36:50.620" Score="1" Body="&lt;p&gt;Perhaps curve fitting of experimental data with more and more terms in a polynomial?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;first term: average, constant&lt;/p&gt;&#xA;&#xA;&lt;p&gt;second term: slope, line&lt;/p&gt;&#xA;&#xA;&lt;p&gt;third term: curvature, quadratic&lt;/p&gt;&#xA;&#xA;&lt;p&gt;fourth term: rate of change of curvature, cubic&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(and maybe up to 4th order)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can show that (depending on the data) that most of the explanation can come from very simple models.  Also gives a little intuition about value of a constant (average) versus line (increasing/decreasing quantity), etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This might be more than what they need or are ready for, but it is an idea.  Definitely don't do something to be cool that is too much for them.  (I wonder if any projects are needed.  But if you do some, make sure they work for the students.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  You don't need any fancy stats modeling programs, you can do it in excel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.statisticshowto.com/excel-multiple-regression/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.statisticshowto.com/excel-multiple-regression/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8953" LastActivityDate="2017-11-05T20:36:50.620" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13111" PostTypeId="2" ParentId="13096" CreationDate="2017-11-05T20:45:17.650" Score="2" Body="&lt;p&gt;I think contacting the author is the way to go.  I was self-studying a text in the oil business (very recently for a consulting engagement). I kept a simple excel file with two tabs (one for mistakes and one for questions).  The questions I mostly got resolved as I read further or looked up on the web.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I sent the file to the publisher and he sent it to the author who fixed the issues in next edition and sent me a warm note in thanks.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I really don't see it as much extra work.  Actually reading that text and working the problems one by one was vastly more time spent than the time spent to document the issues and send the file.  If anything doing this sort of fit into the whole conscientious self study mode I had.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I would definitely just do a point to point email.  Who knows maybe it leads to collaboration or the like in future.&lt;/p&gt;&#xA;" OwnerUserId="8953" LastActivityDate="2017-11-05T20:45:17.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13112" PostTypeId="2" ParentId="13109" CreationDate="2017-11-05T20:49:11.507" Score="4" Body="&lt;p&gt;I'm not entirely sure I understand what the student can do and cannot do, but I believe something I sometimes used to do might help. You have two lines, one given by $y = 2x + 2$ and the other given by $y = -\frac{1}{2}x – 2,$ and you want to find the point that is on both lines. To keep from getting confused with &quot;$x$ as a variable&quot;, and &quot;$x$ as a value that is fixed by an equation&quot;, and other things, let $(a,b)$ be the coordinates of the point that is on both lines. Then what we want to find are the values of $a$ and $b.$ (See &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=7363805&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 20 January 2011 ap-calculus post&lt;/a&gt; at Math Forum for a situation where using this method is even more helpful.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since $(a,b)$ is on the line $y = 2x + 2,$ we have $b = 2a + 2.$ Since $(a,b)$ is on the line $y = -\frac{1}{2}x – 2,$ we have $b = -\frac{1}{2}a – 2.$ Now we have two equations for the two unknowns. Solving this system of simultaneous equations gives us the values of $a$ and $b.$&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2017-11-05T20:49:11.507" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13113" PostTypeId="2" ParentId="13109" CreationDate="2017-11-05T22:02:23.627" Score="5" Body="&lt;p&gt;I had a similar interaction with a student this weekend. I tend to walk it back to talk about these objects are just ways we invented to talk about number and quantity. They translate into a language. So many students get focussed on the machine process of manipulation of quantity to get results. In that they lose the idea these expressions represent something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case you have I would add some mechanical process to their work but explaining the what it means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In cases of equal signs I use $y=f(x)$ means wherever we see $y$ we can replace it with $f(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would have the student set up what they know first:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f(x)=2x+2$&#xA;$g(x)=-\frac{1}{2}x-2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student identified that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$y=f(x)$&#xA;$y=g(x)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus: $f(x)=g(x)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here I would tell the student if $f(x)=2x+2$ means wherever I see $f(x)$ I can replace it with $2x+2$. Until they can produce:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$2x+2=g(x)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would repeat the argument for $g(x)$ to produce:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$2x+2= -\frac{1}{2}x-2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you would want to reflect that this statement says the same thing as $y=y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From here the discussion will be that this equality tells us the $x$ value where the $y$ values are equal, even though we don't know what the $y$ value is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once the $x$ value is found the question is well what was that $y$ value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe even presenting a problem of this nature and emphasizing what equality lets us know on each step, while tracking what still isn't known. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do this in tutoring often. It helps students also avoid the habit of ending problems early if they can track what each step resolves for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an extension in my Physics class I have some problems where we find a value for $v^2$ in on one step, and in the next step we can substitute $v^2$ directly in. Many student will solve for $v$ and square again. I try to emphasize that equality statements allow you to swap expressions no matter if it has been isolated to one variable or not. &lt;/p&gt;&#xA;" OwnerUserId="8952" LastActivityDate="2017-11-05T22:02:23.627" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13114" PostTypeId="1" CreationDate="2017-11-06T06:56:47.790" Score="7" ViewCount="143" Body="&lt;p&gt;I read pages of the book &quot;Elementary and Middle School Mathematics: Teaching Developmentally&quot; by &#xA;John A. Van de Walle, Late of Virginia Commonwealth University&#xA;Karen S. Karp, Johns Hopkins University&#xA;Jennifer M. Bay-Williams, University of Louisville&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reading this extract:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qX4ni.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qX4ni.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;I remember an idea that I still don´t use, so... I would like to use it in this starting week,... but the library where I read the book does not have access to the online company resources (where I suppose that any activity pages about this issue are available)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1.- Do you have any activity pages with the strategy &quot;find the error&quot; about this issue (powers, and fraction operations)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and secondarily,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2.- Do you know papers about this strategy &quot;Find the error&quot;&lt;/p&gt;&#xA;" OwnerUserId="1760" LastEditorUserId="77" LastEditDate="2018-05-08T00:08:40.700" LastActivityDate="2018-05-08T00:08:40.700" Title="Searching activities with &quot;Find the error&quot; strategy for learning maths" Tags="|exponent|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13116" PostTypeId="1" AcceptedAnswerId="16578" CreationDate="2017-11-06T15:05:43.833" Score="5" ViewCount="723" Body="&lt;p&gt;Just as we distinguish between an equation and an identity (or equality), why don't we distinguish between an inequation and an inequality? We solve an inequation and we prove an inequality. In French they are clearly distinguished. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We say &lt;strong&gt;solve this equation&lt;/strong&gt; and &lt;strong&gt;prove this identity&lt;/strong&gt;, but (in English only AFAIK) we say &lt;strong&gt;prove this inequality&lt;/strong&gt; and &lt;strong&gt;solve this inequality&lt;/strong&gt; even though the term &lt;em&gt;inequation&lt;/em&gt; already exists in English and is used in some document but the majority prefer to not use it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example the following is a question in an exam (in French) and the translated version, which doesn't seem to be a good translation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ous5j.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ous5j.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CAKzi.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CAKzi.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;«Utiliser l'inégalité &lt;span class=&quot;math-container&quot;&gt;$(x+1)^2\geq 0$&lt;/span&gt; pour résoudre l'inéquation &lt;span class=&quot;math-container&quot;&gt;$\left(x^2+2x+1\right)(2x-5)\leq 0$&lt;/span&gt;» &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comparing the ngrams of inequation in English and in French, we see a big decline of its use between 1960 and 1977 followed by a small increase in the 80s. In French the frequency is almost the same between 1965 and 1985. Is there a reason for the decline between 1960 and 1977?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IocnQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IocnQ.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ENyIy.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ENyIy.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been told that the term &quot;inequation&quot; is widely used in the middle east and some Arab countries. It's used in some standardized exams in middle school, high school and technical/vocational schools. See for example the &lt;a href=&quot;https://www.crdp.org/curr-content-desc?id=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;center of educational research and development&lt;/a&gt; in Lebanon.&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="11" LastEditDate="2019-10-05T00:53:02.547" LastActivityDate="2019-10-05T19:08:45.953" Title="Why isn't the term *inequation* widely used in English?" Tags="|terminology|" AnswerCount="4" CommentCount="24" ContentLicense="CC BY-SA 4.0" />
  <row Id="13117" PostTypeId="1" CreationDate="2017-11-06T15:21:02.180" Score="7" ViewCount="192" Body="&lt;p&gt;I was watching (as a fellow teacher) a lecture on arithmetic progression. It so happened that a formula was rapidly introduced and used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The formula was $a_i = a_1 + (i-1)r$ (I trust the terms meanings to be easily guessable, but they were, of course, explained in the lecture)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could not shake the feeling that it would have been more productive for the students to do some exercises before this abstraction, to &quot;get a feel&quot; for the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does your &lt;strong&gt;expericence as a teacher&lt;/strong&gt; agree with my intuition? Is there some &lt;strong&gt;research&lt;/strong&gt; on the matter? (I don't mean just for progressions, but in general. Is it reasonable to do some exercises without the formalization to get a hang of things?)&lt;/p&gt;&#xA;" OwnerUserId="1662" LastEditorUserId="1662" LastEditDate="2017-11-06T17:46:48.390" LastActivityDate="2017-11-07T04:50:28.120" Title="Should basic exercises be solved before formulas are introduced?" Tags="|education-research|teaching|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13118" PostTypeId="2" ParentId="13116" CreationDate="2017-11-06T15:31:21.703" Score="1" Body="&lt;p&gt;So equations are statements of equality that evaluate as true or false.&#xA;Inequality statements are statements of inequality that evaluate true or false.&#xA;At the point I introduce inequality statements the symbol in the middle becomes part of a larger set of the language. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So equations are a subset of statements where the truth comparison is done on equality. &#xA;When we start introducing inequality statements we are building the larger set, though we may not explicitly state it the beginning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do try to emphasize that statements are a test and the symbol is the type of test you are doing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this may the reason we don't have a separate word in use for inequality statements. At the point they are introduced we have taken the idea of equality statements and are talking about the more general concept of statements. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though I would say we do imply it more than state this idea. So it may be good to focus language on statement and the type of statements we create.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am really interested in ideas on language choice and pedagogy. Thus I am very interested in the the ideas behind this question and what others have seen and done.&lt;/p&gt;&#xA;" OwnerUserId="8952" LastActivityDate="2017-11-06T15:31:21.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13119" PostTypeId="5" CreationDate="2017-11-06T15:38:49.667" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2017-11-06T15:38:49.667" LastActivityDate="2017-11-06T15:38:49.667" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13120" PostTypeId="4" CreationDate="2017-11-06T15:38:49.667" Score="0" Body="This is a special tag that is designed to be used only by the system for questions that have had all of their other tags removed. Do not add this tag to existing questions, it is reserved for use by the system to identify posts with no valid tags." OwnerUserId="6053" LastEditorUserId="6053" LastEditDate="2017-11-07T00:03:52.330" LastActivityDate="2017-11-07T00:03:52.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13121" PostTypeId="2" ParentId="13109" CreationDate="2017-11-06T15:48:49.993" Score="2" Body="&lt;p&gt;(I assume the student found x but not y. Was that the case? Where did the student get stuck?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd return to the concrete model (i.e.: the cartesian plane).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take some squared paper, and show some points (this is point (3,4), this is point (8,-1))&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, draw some lines that have an integer intersection, and ask for the point in question. (possibly marking the values of x and y points on the axis and drawing an dotted line between the intersection and each axis)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To end, draw the lines of the problem, show the student the x already discovered, and draw the dotted line to the y axis.&lt;/p&gt;&#xA;" OwnerUserId="1662" LastActivityDate="2017-11-06T15:48:49.993" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13123" PostTypeId="2" ParentId="13117" CreationDate="2017-11-06T22:45:43.433" Score="3" Body="&lt;p&gt;My experience is exactly the opposite: that the abstract formula should be displayed as soon as possible, then justified, then exercised. My philosophy on this has been developed by the idea that I want the formula in the student's visual field for as much time as possible, in the hopes that it will sink in mentally. Note that this is sympatico with the requirement in many locations that the teacher clearly state the goal of the lesson at the start of a meeting. Plus: The cycle of theorem-proof is simply traditional mathematical presentation and writing style (for exactly this reason, I think), and students should get to experience and expect that style of presentation as a &quot;real&quot; math class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I always get weirded out when I see instructors doing the opposite. They seem to take most of the presentation time doing these warm-up exercises, and wind up squeezing the ultimate goal in the last few minutes of class (and not actually exercising the formula itself). As both a teacher and a student myself, I feel that we all get confused about what the &quot;point&quot; of the lesson is, what the real take-away skill is, and how it will be assessed in the future. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that many of us wish that we could lead all of our courses and students in &quot;discovery&quot; style lessons where they take personal ownership for all the new material. But as teachers this is simply infeasible granted the limited time in class; and particularly so from my perspective in the college classroom (even if much of my career is spent remediating topics from middle school). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I was reading material by Hung-Hsi Wu recently and he does this always-concrete-warm-up cycle, and while his material is otherwise excellent, I find this distracting and it forces me to flip back-and-forth a lot to uncover the &quot;real&quot; proofs.) &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-11-07T02:28:48.793" LastActivityDate="2017-11-07T02:28:48.793" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13124" PostTypeId="2" ParentId="13117" CreationDate="2017-11-07T04:50:28.120" Score="4" Body="&lt;p&gt;I think it varies.  Advantage of the more intuitive approach is that you get a feel for the topic before the formula.  Advantage of the formula is you see where things are headed.  I would say if the formula is complicated (TO THE STUDENTS) than the intuitive approach is better.  If the exercises are difficult and the formula reasonable than the formula firs approach is better.  In this case, I would prefer the examples first.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't agree with &quot;theorem proof is how everything is going to be&quot;.  Don't think this applies to the vast majority of students that learn math to use in science, engineering, business, medicine, etc. (rather than abstract math majors.)  IOW, that's a bug, not a feature...&lt;/p&gt;&#xA;" OwnerUserId="8956" LastActivityDate="2017-11-07T04:50:28.120" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13125" PostTypeId="2" ParentId="12945" CreationDate="2017-11-07T05:40:52.110" Score="4" Body="&lt;p&gt;You are correct. There are not nearly enough interesting problems typically given for the major theorems of vector calculus. For the most part, the problems I find are not that interesting and mostly amount to &quot;verify the theorem&quot; or some such thing. Of course, there is more. Where to find it? I'm not sure I know a good answer to that off hand, I'm always working to find more such problems. In any event, this is symptomatic of a larger problem with the last third of mainstream calculus texts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What commonly passes for multivariate calculus in the standard textbooks is horrible. There is far too little attention paid to the parametric approach. Hardly any of the books do non-cartesian coordinates justice. We get only 20 pages or so for the crowning jewels of Green, Stokes and Divergence. Deformation theorems are given little coverage. Earlier in the course, many books give little attention to Frenet frames and the geometry of curves. All of these are helpful to understand how we actually apply vector calculus to physics. Furthermore, what is even less likely to find, the foundations of potential theory including Green's 2nd and 3rd identities and Helmholtz Theorem. I haven't even begun to scratch the surface here, there is tons missing for differentiability as well. So, what to do?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simple. Ignore the required text and make your own path. Refuse to accept the sad watered down fake calculus III which is pushed on us by the standard texts where calculus III is essentially an afterthought. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One refreshing counterpoint to my complaints here would be Susan Colley's excellent Vector Calculus text. It actually has the same sort of detail for calculus III as is easily found for Calculus I and II in many texts. I really wish my school split off the text for Calculus III because I am universally displeased with the standard texts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, my displeasure stems not initially from my mathematics. Rather, it was as a physics student it became apparent that much of the real calculational techniques I needed were (for reasons indefensible to me) simply removed from my education which was ostensibly complete. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-11-07T05:40:52.110" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13126" PostTypeId="2" ParentId="1351" CreationDate="2017-11-07T13:20:13.937" Score="0" Body="&lt;p&gt;I like this ...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;from &lt;em&gt;Spontaneous Phenomena&lt;/em&gt;, by F. Topsoe, Academic Press 1990:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;THESIS 22: Those who seek a phenomenon which exactly follows a&#xA;             mathematical model, seek in vain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-11-07T13:20:13.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13127" PostTypeId="1" CreationDate="2017-11-08T02:21:49.483" Score="5" ViewCount="431" Body="&lt;p&gt;I am currently starting to teach math to highschool kids who don't do well on their own, and I'm teaching a guy about notable points and lines of a triangle (incenter, centroid, circumcenter, perpendicular bisector, etc) and since we already finished with all the definitions and properties that he needs for his test, I want him to be able to solve problems related to it in case he faces them in the test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I gave him an excercise about a basketball court with three backboards, in which he had to choose the optimal place to get as many points as possbile in each one without moving, and one about a triangular room in which a pool was going to be placed, and he had to figure out the maximum size possible for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both of these examples seem a bit unlikely to happen in reality and after searching a bit I was unable to find practical uses for these tools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any sample excercise you can come up with will be much appreciated.&lt;/p&gt;&#xA;" OwnerUserId="8963" LastActivityDate="2017-11-09T00:53:12.907" Title="Applications of notable points and lines of a triangle" Tags="|geometry|applications|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13128" PostTypeId="1" AcceptedAnswerId="13142" CreationDate="2017-11-08T08:48:46.857" Score="5" ViewCount="264" Body="&lt;p&gt;I saw a video 3 or 4 years ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The video is about the idea that arithmetic will disappear in the future and only will be a sport, like hunting that passed from a need to a sport.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the video there was a little extract of interviews with three people (I think that one was Salman Khan from Khan academy) and it's about the actual model of education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The video was posted &lt;a href=&quot;http://www.videoeneducacion.com/educacion/el-futuro-de-la-educacion-entrevista-con-tres-innovadores-de-la-educacion&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The video had Spanish subtitles, at &lt;a href=&quot;https://www.scoop.it/t/aprendizaje-innovacion/p/4018407087/2014/03/26/aprendizaje-del-futuro-y-los-innovadores-de-la-educacion&quot; rel=&quot;nofollow noreferrer&quot;&gt;this address&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you help me in finding this video? Sorry there's so little info!&lt;/p&gt;&#xA;" OwnerUserId="1760" LastEditorUserId="6053" LastEditDate="2018-07-03T23:25:42.370" LastActivityDate="2018-07-04T17:23:52.993" Title="Looking for a video about arithmetic disappearing in a few years" Tags="|arithmetic|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13129" PostTypeId="1" AcceptedAnswerId="13130" CreationDate="2017-11-08T17:14:24.100" Score="5" ViewCount="310" Body="&lt;p&gt;I want to talk about the Fibonacci sequence in my Linear Algebra class. So I tried to look online for examples where the sequence appears naturally. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the most often mentioned is that of the bees, where male bees have only a mother, and female bees have a mother and a father. Every resource I found states the above, runs through five or six generations to get 1, 1, 2, 3, 5, 8, 13, and immediately states &quot;so, we have the Fibonacci sequence&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that's not a proof, and I struggled to justify formally that the number of individuals in the next generation has to be the sum of the number of individuals in the previous two. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does anyone know a formal argument to show that the number $s_n$ of individuals after $n$ generations satisfies the recursion $s_{n+1}=s_n+s_{n-1}$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="475" LastActivityDate="2017-11-08T20:49:35.450" Title="Motivation for Fibonacci: Bees" Tags="|concept-motivation|sequences|recursion|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13130" PostTypeId="2" ParentId="13129" CreationDate="2017-11-08T19:11:14.563" Score="2" Body="&lt;p&gt;You can see it by breaking the numbers $s_n$ into parts: $s_n=f_n+m_n$, which represent the number of female and male bees, respectively, at each level of the family tree.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To find $f_{n+1}$, we note that each female bee at level $n+1$ is either the mother of a female bee at level $n$, or else the paternal grandmother of a female bee at level $n-1$. This is a one-to-one correspondence between female bees at level $n+1$ with female bees at the two previous levels, so we have $f_{n+1}=f_n+f_{n-1}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a similar argument for male bees: any male bee at level $n+1$ is either the father-in-law of a male bee at level $n$ (if his daughter's offspring is female), or else the maternal grandfather of a male bee at level $n-1$ (if his daughter's offspring is male). Thus again, $m_{n+1}=m_n+m_{n-1}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align}&#xA;s_{n+1}=f_{n+1}+m_{n+1} &amp;amp;= (f_n+f_{n-1})+(m_n+m_{n-1})\\&#xA;&amp;amp;=(f_n+m_n)+(f_{n-1}+m_{n-1})\\&#xA;&amp;amp;= s_n+s_{n-1}&#xA;\end{align}$$&lt;/p&gt;&#xA;" OwnerUserId="2034" LastActivityDate="2017-11-08T19:11:14.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13131" PostTypeId="2" ParentId="13129" CreationDate="2017-11-08T20:49:35.450" Score="2" Body="&lt;p&gt;After looking at Tony Jacobs argument, here is an argument of my own. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with a single male. At each generation, let $m_n$ and $f_n$ be the total number of males and females respectively, and let $s_n=m_n+f_n$. We have &#xA;$$&#xA;f_0=0, f_1=1, m_0=1, m_1=0. &#xA;$$&#xA;Since each male comes from a female from the previous generation, we have $$ \tag{1}m_{n+1}=f_n,\ \ \ \ n\in\mathbb N.$$&#xA;Since each female comes either from a female or a male, we have &#xA;$$\tag{2}&#xA;f_{n+1}=m_n+f_n,\ \ \ \ n\in\mathbb N.&#xA;$$&#xA;Combining the two formulas we have &#xA;$$\tag{3}&#xA;f_{n+1}=m_n+f_n=f_{n-1}+f_n&#xA;$$&#xA;and&#xA;$$\tag{4}&#xA;m_{n+1}=f_n=m_{n-1}+f_{n-1}=m_{n-1}+m_n. &#xA;$$&#xA;By $(3)$ and $(4)$, &#xA;$$&#xA;s_{n+1}=f_{n+1}+m_{n+1}=(f_{n-1}+f_n)+(m_{n-1}+m_n)=s_{n-1}+s_n.&#xA;$$&lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2017-11-08T20:49:35.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13132" PostTypeId="1" CreationDate="2017-11-08T21:26:58.500" Score="5" ViewCount="424" Body="&lt;p&gt;I'm preparing a design/teaching experiment for my Curriculum Design Course right now. I've decided to cover L'Hopital's rule with a student I've been working with for a year, so skill wise he's ready to go. And I'm probably only going to focus on the form 0/0 in my tasks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm trying to utilize an APOS &amp;amp; RME framework to guide my tasks - but I'm having an issue coming up with the necessity for L'Hopital's rule. &#xA;I know we need it to completely evaluate indeterminate forms - but I'm sure my student will just be fine leaving it and saying he's done. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any insight would be appreciated, and if I'm posting this is the wrong spot or if there's a better place to go I would like to know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="8965" LastActivityDate="2017-11-14T20:25:53.177" Title="L'Hopital's Rule: Why do we need it?" Tags="|graduate-education|teaching|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13133" PostTypeId="2" ParentId="13127" CreationDate="2017-11-09T00:53:12.907" Score="6" Body="&lt;p&gt;I love this example. Suppose you want to cut out an irregular triangle from the &#xA;center of a piece of paper. You can do it with one straight cut,&#xA;first folding flat along angle bisectors:&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/4Ot2f.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4Ot2f.jpg&quot; alt=&quot;Fig5.7&quot;&gt;&lt;/a&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;hr /&gt;&#xA;And of course the angle bisectors meet at the center of the incircle&#xA;(Proposition 4, Book IV of Euclid):&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/yZGjH.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yZGjH.jpg&quot; alt=&quot;Fig5.8&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;I have found it quite effective to hand out sheets &#xA;of paper with a large irregular triangle&#xA;predrawn,&#xA;and have students fold &amp;amp; cut with scissors.&#xA;It makes the incenter tangible.&#xA;&lt;hr /&gt;&#xA;These figures are from&#xA;&lt;a href=&quot;http://howtofoldit.org/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-11-09T00:53:12.907" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13134" PostTypeId="2" ParentId="13132" CreationDate="2017-11-09T12:29:48.067" Score="9" Body="&lt;p&gt;This is an answer to the title. Defining APOS &amp;amp; RME framework would make answering the question easier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Massimo Ortolano mentioned in a comment, l'Hôpital's rule is one tool in a box. Maybe you use it often, maybe rarely, but it is very nice when you can use it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just the other day, when trying to understand how ultrasound mediated electrical impedance tomography works, I invoked l'Hôpital. Certainly I could have derived the proper limit in another way, but l'H is easy to remember and intuitive and easy to use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a bonus my colleague (with an engineering background) knew the rule and so could follow the calculation with fair ease. Generalizing, l'H is good to know because many people in technical fields know it, so it makes communication smoother.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2017-11-09T12:29:48.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13135" PostTypeId="2" ParentId="13132" CreationDate="2017-11-09T18:22:33.243" Score="7" Body="&lt;p&gt;When teaching calculus I like to include L'Hospital's Rule because it exemplifies how the subject is an impressive arsenal of calculational methods, and because it is easy to explain why this rule is not a arbitrary trick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The form $\displaystyle{\frac{f'(a)}{g'(a)}}$ looks magical on the surface, but when viewed as $\displaystyle{\frac{f(a)+f'(a)(x-a)+O(x-a)^2}{g(a)+g'(a)(x-a)+O(x-a)^2}}$ where $f(a)=g(a)=0$, one understands how it is a simple application of the calculus concept &quot;linearization.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The efficacy of calculus as a subject is that it is a toolbox of shortcuts that streamline calculations. L'Hospital is a lesson that focuses on one particular tool that is widely used, difficult to forget, yet is easily grounded in the concept of linearization.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2017-11-09T18:22:33.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13136" PostTypeId="1" AcceptedAnswerId="13138" CreationDate="2017-11-09T20:00:28.793" Score="9" ViewCount="2348" Body="&lt;p&gt;Reference angles are most useful for limited trigonometric tables with values from 0 to $\frac{\pi}{2}$. They can also help illustrate the periodicity of trigonometric functions, but I feel this is done more effectively with graphing. Should we still teach them, and if so how to make them more relevant?&lt;/p&gt;&#xA;" OwnerUserId="5080" LastEditorUserId="511" LastEditDate="2017-11-10T00:01:21.580" LastActivityDate="2018-06-13T22:47:34.167" Title="Why teach reference angles?" Tags="|undergraduate-education|trigonometry|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13137" PostTypeId="2" ParentId="13136" CreationDate="2017-11-09T20:51:34.360" Score="13" Body="&lt;p&gt;I disagree with your statement on the usefulness of reference angles, they have more use in the plane of the unit circle. So to answer the question specifically, yes we should still teach them and one idea on relevance is below in a Physics I problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trigonometric functions in a complete unit circle are not one to one so the concept of a reference angle is needed to orient the inverse functions to the coordinate plane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teaching physics primarily and mathematics secondary when using the trigonometric functions to find components in the forward direction this idea is not needed, but when solving for the angle and using the inverse trigonometric functions the understanding of a reference angle is very important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I have 4 forces &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$F_1= 12N@30^{\circ}$, $F_2=12N@150^{\circ}$, $F_3=12N@210^{\circ}$ and $F_4=12N@300^{\circ}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The resultant force can be found by adding the components in each co-ordinate direction:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\Sigma F_x=12 N \cos{30^{\circ}}+12N \cos{150^{\circ}}+12N \cos{210^{\circ}}+12N\cos{300^{\circ}}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\Sigma F_y=12 N \sin{30^{\circ}}+12N \sin{150^{\circ}}+12N \sin{210^{\circ}}+12N\sin{300^{\circ}}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This solves nicely with the large angles keeping track of the positive and negative values of the components nicely:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\Sigma F_x \approx -4.4 N$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\Sigma F_y \approx -4.4 N$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Magnitude of the Force is found:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$F_R=\left[(F_x)^2+(F_y)^2\right]^{\frac{1}{2}}$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$F_R\approx 6.2 N$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we want to find the direction of this vector we use:&#xA;$\theta = \arctan{\frac{F_y}{F_x}}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\theta = 45^{\circ}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a reference angle. The student needs to apply the concept of the reference angle and recognize that because both components of the vector were negative that this would be in the Third Quadrant and the angle relative to the positive x axis is $\theta=225^{\circ}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So because of the nature of the inverse trigonometric functions when mapping from an answer to an $\arcsin$, $\arccos$ and $\arctan$ students need to understand the concept. It is worth spending time on the concept especially as the concept of angle is increased beyond its initial geometric definitions.&lt;/p&gt;&#xA;" OwnerUserId="8952" LastEditorUserId="8952" LastEditDate="2017-11-23T18:19:17.230" LastActivityDate="2017-11-23T18:19:17.230" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13138" PostTypeId="2" ParentId="13136" CreationDate="2017-11-09T20:55:31.090" Score="7" Body="&lt;p&gt;For students who will &lt;em&gt;only&lt;/em&gt; be using right triangle trig, reference angles are definitely important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I (sometimes) teach a one-credit course on trig for forestry students, and their main application is moving along (or describing) a path connecting a series of points on a map. Note that they do not measure angles counterclockwise, relative to the positive $x$-axis as we do in math, but rather &lt;em&gt;clockwise&lt;/em&gt;, relative to North (called an &lt;a href=&quot;https://en.wikipedia.org/wiki/Azimuth&quot; rel=&quot;nofollow noreferrer&quot;&gt;azimuth&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, a movement such as &quot;travel $68$ feet at $213.5^{\circ}$ boils down to their understanding of a reference angle to make a right triangle. Of course, this angle would be in reference to any convenient cardinal direction so they can determine, say, a northing or easting for that leg of the path.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, it would be awkward to try to force these students to learn a new convention for angles, as azimuths are the standard in their field. So, I stay in their world, having them sketch their particular path on the plane and using a reference angle to draw a right triangle, etc.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2017-11-10T21:32:53.410" LastActivityDate="2017-11-10T21:32:53.410" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13139" PostTypeId="2" ParentId="13099" CreationDate="2017-11-10T08:59:22.410" Score="3" Body="&lt;p&gt;Given a circle, cut it with 1 straight line. You obtain 2 pieces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now cut it with 2 straight lines. If you pick the second line correctly, you obtain 4 pieces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the largest number of pieces you can obtain with $n$ lines?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is a cubic (&lt;a href=&quot;//oeis.org/A000125&quot; rel=&quot;nofollow noreferrer&quot;&gt;OEIS A000125&lt;/a&gt;, &lt;a href=&quot;//mathworld.wolfram.com/CakeNumber.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathWorld:Cake Number&lt;/a&gt;), although you might not manage to prove that to the students' satisfaction. It can be used to explore identifying probable polynomials using iterated finite differences, and extrapolation of polynomials using finite differences. I remember that aspect of it making for the most interesting project we did in secondary school maths.&lt;/p&gt;&#xA;" OwnerUserId="8005" LastActivityDate="2017-11-10T08:59:22.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13140" PostTypeId="2" ParentId="13099" CreationDate="2017-11-10T12:14:24.310" Score="0" Body="&lt;p&gt;This may not be easy to make clear without computation, but the intersection of&#xA;two ellipses leads to a 4th-degree polynomial.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/9LuEk.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9LuEk.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://elliotnoma.wordpress.com/2013/04/10/a-closed-form-solution-for-the-intersections-of-two-ellipses/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elliet Noma&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;See Dave Eberly's &quot;Intersection of Ellipses&quot; (&lt;a href=&quot;http://www.math.unm.edu/~vageli/papers/FLEX/IntersectionOfEllipses.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;)&#xA;for the (formidable) details.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-11-10T12:14:24.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13141" PostTypeId="2" ParentId="13099" CreationDate="2017-11-10T14:05:21.210" Score="1" Body="&lt;p&gt;I think doing something with &lt;a href=&quot;https://en.wikipedia.org/wiki/Polygonal_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;polygonal numbers&lt;/a&gt; might be fun and instructive.  You could ask them to write down the sequence of triangular numbers $1, 3, 6, 10, 15, \dots$ and have them see that the first difference increaseas by one and the second difference is constant.  This suggests that the sequence is given by a quadratic.  They have three points and so the could solve for the coefficients of the quadratic.  The square numbers are easy.  The pentagonal numbers can be handled in the same way as the triangular numbers.  The might even by able to figure out the formula for the $k$-gonal numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They could then maybe explore a famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Fermat_polygonal_number_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;theorem&lt;/a&gt; due to Fermat that says every positive integer $n$ can be written as the sum of $k$ $k$-gonal numbers (if we allow 0 to be a $k$-gonal number for all $k$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among other things, it will reinforce the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;figuring out patters&lt;/li&gt;&#xA;&lt;li&gt;solving systems of equations&lt;/li&gt;&#xA;&lt;li&gt;multiplication as area&lt;/li&gt;&#xA;&lt;li&gt;there is a unique degree $n$ polynomial through $n+1$ points (as long as the points have different $x$-coordinates&lt;/li&gt;&#xA;&lt;li&gt;math has these things called theorems that can be pretty cool&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-11-10T14:05:21.210" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13142" PostTypeId="2" ParentId="13128" CreationDate="2017-11-10T22:34:40.437" Score="2" Body="&lt;p&gt;Using the &lt;a href=&quot;http://web.archive.org/web/20121226090036/http://www.videoeneducacion.com/educacion/el-futuro-de-la-educacion-entrevista-con-tres-innovadores-de-la-educacion&quot; rel=&quot;nofollow noreferrer&quot;&gt;Web Archive&lt;/a&gt;, I was able to find on that page a link to a YouTube video (in English) called &lt;a href=&quot;https://www.youtube.com/watch?v=qC_T9ePzANg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Future Learning&lt;/a&gt;. It was posted in 2012 and the description says: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Students are the future, but what's the future for students? To arm them with the relevant, timeless skills for our rapidly changing world, we need to revolutionize what it means to learn. Education innovators like Dr. Sugata Mitra, visiting professor at MIT; Sal Khan, founder of Khan Academy; and Dr. Catherine Lucey, Vice Dean of Education at UCSF, are redefining how we engage young minds for a creatively and technologically-advanced future. Which of these eduvators holds the key for unlocking the learning potential inside every student?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3593" LastActivityDate="2017-11-10T22:34:40.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13143" PostTypeId="2" ParentId="11384" CreationDate="2017-11-11T03:52:23.820" Score="0" Body="&lt;p&gt;I would really like to find something that would work well in this regard. I have used both &lt;a href=&quot;http://www.tamarack-software.com/products/testgen7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tamarack TestGen&lt;/a&gt; and &lt;a href=&quot;http://www.brownstone.net/publishers/products/dip6.asp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wimba Diploma&lt;/a&gt; pretty extensively, but both have pros and cons.&#xA;&lt;br&gt;&lt;br&gt;&#xA;TestGen has a horrible interface for typesetting mathematics, and very limited ability to refresh the numbers in questions (to create different versions). The quality of the typesetting is sometimes poor, so that it can be hard for students to read. For example, a complicated expression in an exponent won't be superscripted sufficiently so that it is clearly an exponent. The function which you use to change the numbers in a question, as I understand it, is basically ordered lists, so that if you had a question with three different values you wanted to change, you would support that question with three different &quot;variables&quot; that would pull values sequentially from the list linked to each variable. Different versions of the question would use the first value from each list, and then the second value from each list, and so on. So you are limited to the number of values entered into the list.&#xA;&lt;br&gt;&lt;br&gt;&#xA;Diploma was last updated in 2012, so even though it is pretty user-friendly and the typesetting is good (it uses Equation Editor from MS Word, or MathType), and it has some power to generate many versions of a question, it looks like it won't be updated. Wimba, the parent company, was bought by Blackboard, and publishers seem to be deserting the software. I don't know how much longer it will be operating, which is unfortunate as I have spent many hours writing questions in Diploma banks for use in my work.&#xA;&lt;br&gt;&lt;br&gt;&#xA;WeBWorK also seems to be slowly dying, and so far as I understand-- I used it about a decade ago-- has no ability to generate multiple versions of an exam with a key.&#xA;&lt;br&gt;&lt;br&gt;&#xA;An editor from a major publisher described the considerable money being spent by said publisher to allow online delivery of assessments. This is driving everything right now. Is it too much to expect software that does both paper-based and online exams well?&lt;/p&gt;&#xA;" OwnerUserId="8972" LastActivityDate="2017-11-11T03:52:23.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13145" PostTypeId="2" ParentId="1351" CreationDate="2017-11-11T18:52:38.067" Score="-2" Body="&lt;p&gt;There are many who would say the same as the author of the following paper: &quot;Pitowsky's attempts of hidden variables rather seem to reaffirm the old intuition: &quot;if nontrivial set theory, non-constructive mathematics or a non-measurable set is used in an essential way, it cannot be physically relevant'.&quot; [Jakob Kellner: &quot;Pitowsky's Kolmogorovian models and super-determinism&quot;, arXiv (2016)]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would support this opinion, even if that makes set theorists angry. But in order to comfort them there are also applications like the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Banach-Tarski Gyroscope is an intricate mechanism believed to have been constructed using the Axiom of Choice. On each complete rotation counterclockwise, the Banach-Tarski Gyroscope doubles in volume while maintaining its shape and density; on rotating clockwise, the volume is halved. When first discovered, fortunately in the midst of interstellar space, the Banach-Tarski Gyroscope was tragically mistaken for an ordinary desk ornament. Subsequently it required a significant portion of the available energy of the contemporary galactic civilization to reverse the rotation before nearby star systems were endangered; fortunately, the Banach-Tarski Gyroscope still obeys lightspeed limitations on rotation rates, and cannot grow rapidly once expanding past planetary size. After the subsequent investigation, the Banach-Tarski Gyroscope was spun clockwise and left spinning. [Eliezer Yudkowsky: &quot;Banach-Tarski gyroscope&quot;, Hacker News (28 Dec 2008)]&lt;/p&gt;&#xA;" OwnerUserId="8557" LastActivityDate="2017-11-11T18:52:38.067" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13146" PostTypeId="1" CreationDate="2017-11-11T22:30:38.343" Score="3" ViewCount="184" Body="&lt;p&gt;I taught a two-hour introductory class in Econometrics today for high-school and middle-school students (it is a Splash program at my University, and I am a student myself).  We started with about 12 students and ended with 5, it was about even girls and boys starting out, so there is clearly interest by both sides in learning this material, but by the end &lt;em&gt;all&lt;/em&gt; of the girls left before the lecture (and practical in R) finished. I don't know if the material was just too difficult for them or they simply did not find it interesting.  Economics is a male dominated field so I am sad to see these girls dis-engaging from the field before they have even started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I want to start a conversation.  What can we do to engage these girls in our economic classes so that they may continue in the field and lessen this gender gap?&lt;/p&gt;&#xA;" OwnerUserId="8973" OwnerDisplayName="Sunhwa" LastActivityDate="2017-11-13T05:59:12.733" Title="Engaging Women in Economics" Tags="|gender|" AnswerCount="1" CommentCount="9" ClosedDate="2017-11-14T08:46:21.327" ContentLicense="CC BY-SA 3.0" />
  <row Id="13147" PostTypeId="2" ParentId="13097" CreationDate="2017-11-12T10:00:45.867" Score="4" Body="&lt;p&gt;Ask them the meaning of $x^2/3x$. What is its value when $x=2$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of notation is to express an idea so that we or others can later understand it. By using notation in the example above, we have utterly defeated that purpose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Either we meant $(x^2/3)x$ or we meant $x^2/(3x)$, so writing with a &lt;em&gt;solidus&lt;/em&gt; requires brackets regardless. Note that the inclusion of a multiplication sign does not help, since that is implied by the convention of concatenation already, but it is unlikely that you or the students use the convention that concatenation implies multiplication and simultaneous brackets over the concatenated expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using a &lt;em&gt;vinculum&lt;/em&gt; makes it clear immediately: $\frac{x^2}{3}x$ versus $\frac{x^2}{3x}$.&lt;/p&gt;&#xA;" OwnerUserId="6724" LastActivityDate="2017-11-12T10:00:45.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13148" PostTypeId="2" ParentId="13146" CreationDate="2017-11-12T17:45:48.707" Score="-7" Body="&lt;ol&gt;&#xA;&lt;li&gt;You could look at the math content.  There should have been almost none.  PQ curves is it (and without equations, just pictures).  And use $ for the price (not the too theoretical guns and butter). Note, I'm not saying this to dumb it down, I'm saying this because if you had a bunch of math, you are hiding the ideas.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Would say the same about 90%+ intro stats classes in that calculus should not be included.  You don't need it to learn to &quot;think statistically&quot; as George Box would say.  Similarly on econ and math content.  You want your students to be able to &quot;think economically&quot; (recognize sunk cost fallacies, understand the different of a supply or demand shift of the curve versus movement along it, etc.)  This is not a rare thing per se, but unfortunately poor economic thinking is also not rare even at the highest levels in companies (see confusion of economics and accounting all the time).  Teach the kids to think economically and you are doing God's work (figuratively, no blasphemy intended).&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Class should be all micro, no macro.  This is a foundation you need before macro.  And macro still tends to have a lot of places where it is fuzzy on how it works (and even politically charged).  Supply and demand has been pretty clear since the Roman Empire.  Your kids are not going to be implementing tax or Fed financy policy (in all likelihood).  But almost everyone in their daily life will experience things where they need to make investment decisions (fix the AC or buy a new one, add a receptionist to your one man doctor's office or don't or join a group, etc. etc.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;R project?  R project?  [Said in the tone of Jim Mora's &quot;playoffs?&quot;.]  You don't need that.  I wouldn't bother with it at a 100% boys school.  (This is not cutting F=ma out of physics.  It is cutting a not needed distraction.)  But it's really not helping the spread the news cause here.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you have to have a project, make it more like a Harvard business case study or something out of the newspaper.  Or group projects or set of different ones.  Or what have you.  But not the R.  Just no.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember in quantum mechanics (good) intro texts first teach things with normal equation writing BEFORE getting into bra ket operator funny looking things.  It is hard to learn a new concept AND new notation at the same time.  And you're gonna make them do R?  When kids are dropping like flies from your class?  Just no.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know R NOW and I function just fine in the business world doing strategy, valuations for M&amp;amp;A, etc. etc.  I could learn it if I needed to, but I feel no &quot;lack&quot; from not having it.&lt;/p&gt;&#xA;" OwnerUserId="8956" LastActivityDate="2017-11-12T17:45:48.707" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13150" PostTypeId="1" CreationDate="2017-11-13T14:47:22.100" Score="12" ViewCount="3876" Body="&lt;p&gt;I think almost every middle school student in my country has learned sum of two fractions in this non reflexive way (I'm included when I was kid), doing the following steps:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;They calculate the lcm.&lt;/li&gt;&#xA;&lt;li&gt;They draw a long bar above the lcm.&lt;/li&gt;&#xA;&lt;li&gt;They divide the lcm by the denominator of the first fraction and multiply the result by the numerator. The result of this operation they put in the numerator above the long bar.&lt;/li&gt;&#xA;&lt;li&gt;They do the same with the second fraction.&lt;/li&gt;&#xA;&lt;li&gt;They sum the two numbers above the long line&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So how is the better way to teach sum of fractions to students? I want they know what they are doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Y3D8m.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Y3D8m.gif&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5236" LastEditorUserId="2083" LastEditDate="2017-11-13T18:19:02.573" LastActivityDate="2019-10-10T00:48:09.920" Title="How to teach sum of fractions to students?" Tags="|secondary-education|primary-education|fractions|" AnswerCount="8" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13151" PostTypeId="2" ParentId="13150" CreationDate="2017-11-13T16:00:11.700" Score="1" Body="&lt;p&gt;A fundamental thing that students need understand to really know what they're doing when they add fractions is why we have to find the LCM of the denominators, or why we need to make the fractions have &quot;&lt;em&gt;like denominators&lt;/em&gt;.&quot; I would set up the problem up like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say you've got to add the fractions $\frac{2}{3}$ and $\frac{4}{5}$. I would immediately rewrite this problem, and say that we need to add $2 \text{ thirds}$ and $4 \text{ fifths}$. Writing out the denominators as units should make it much clearer to students why we cannot add them immediately (then say something about comparing apples and oranges if you really need to). The idea is that we need to convert thirds and fifths into a common unit so that we can compare the two quantities and add them together. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure the best way to approach teaching how to find this common unit though. I definitely would not use the term LCM. Maybe just tell the kids after they are comfortable adding fractions, &quot;this number is commonly  called the LCM.&quot; But I wouldn't even focus on the &quot;&lt;em&gt;least&lt;/em&gt;&quot; aspect of finding a common multiple: it's way more important to just find a common unit for the two fractions. But it shouldn't be hard to show them how to convert units in general. For example, you can show that them that $1 \text{ third}$ is the same as $2\text{ sixths}$ by just drawing a pie chart subdivided into thirds, shading a single slice, and then subdividing each slice once more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing this conversion for a bit, and looking back to the original problem, I would hope that students would see that we can talk about both fractions in term of $\text{fifteenths}$. We can convert $2\text{thirds}$ to $10\text{ fifteenths}$, and we can convert $4\text{ fifths}$ into $12\text{ fifteenths}$. Then comparing apples to apples, we have $22\text{ fifteenths}$, which we may write as $\frac{22}{15}$ if we'd like. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Reading back, this answer is poorly written. But I trust that my meaning is clear. I'll try to write it better later.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2017-11-13T16:00:11.700" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13152" PostTypeId="2" ParentId="13150" CreationDate="2017-11-13T17:30:31.353" Score="14" Body="&lt;p&gt;First, I would have them really understand equivalent fractions. There are a lot of ways to write the number represented by the fraction $\frac23$. We can call it $\frac23,\frac46,\frac{20}{30},\frac{-2}{-3},$ etc., etc. Similarly, there are many ways to write the number represented by the fraction $\frac45$: as $\frac8{10},\frac{20}{25},$ etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once that's clear, they should practice adding &quot;like fractions&quot;, i.e., fractions with the same denominator. We can add $2$ sevenths and $3$ sevenths to obtain $5$ sevenths for the same reason we can add $2$ dogs and $3$ dogs to obtain $5$ dogs: they are things of the same kind. Calling them &quot;like fractions&quot; also has the advantage of preparing them for adding &quot;like terms&quot; when they get to algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, in order to add fractions, we must first write them as &quot;like fractions&quot;, which requires finding a &quot;common denominator&quot;. Whether it is the &lt;em&gt;least&lt;/em&gt; common denominator, or some other common denominator, is not very important at this point. The can practice figuring out what a common denominator might be for various pairs of denominators, and this practice can be done independently of any adding or subtracting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now they are ready to &quot;add unlike fractions&quot;. First, find a common denominator, then write each fraction as an equivalent fraction with that denominator, and then finally add the resulting like fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Putting it all together for this example, we start out by noting that we have unlike fractions: thirds and fifths. A common denominator for $3$ and $5$ could be $15$ (that's the first one most students will name). Thus we write:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{2}{3}=\frac{?}{15} \,\,\,\,\,\,,\,\,\,\, \frac45=\frac{?}{15}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using what we know about equivalent fractions, we decide to multiply the first numerator by $5$, and the second one by $3$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{2}{3}=\frac{10}{15} \,\,\,\,\,\,,\,\,\,\, \frac45=\frac{12}{15}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, having obtained like fractions, add them:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{10}{15}+\frac{12}{15}=\frac{22}{15}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a final step, if the resulting fraction is not in lowest terms, we can reduce it. In this case, no reduction is necessary.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I like this method because it can be taught as distinct skills, each of which is independently meaningful:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Recognize like fractions vs. unlike fractions&lt;/li&gt;&#xA;&lt;li&gt;Identify a common denominator for two different denominators&lt;/li&gt;&#xA;&lt;li&gt;Write fractions as equivalent fractions with a given denominator&lt;/li&gt;&#xA;&lt;li&gt;Add and subtract like fractions&lt;/li&gt;&#xA;&lt;li&gt;Determine if a fraction is in lowest terms and reduce it if necessary&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;If I were to teach any mechanical algorithm for adding fractions, I'd probably go straight to the &quot;bowtie&quot; method, also called the &quot;butterfly&quot; method. This is illustrated in many videos and webpages; here is &lt;a href=&quot;https://www.youtube.com/watch?v=vqgAYlnY3Fw&quot; rel=&quot;noreferrer&quot;&gt;one example&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2034" LastActivityDate="2017-11-13T17:30:31.353" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="13153" PostTypeId="2" ParentId="13150" CreationDate="2017-11-13T19:49:21.683" Score="3" Body="&lt;p&gt;When teaching, I always try to relate the topic back to something the students understand and experience with.  I most often default to money but depending on the grade level that may or may not work.  When it comes to fractions, I first like to talk about the terminology to help the students know what a fraction really means.  I note that the word &quot;denominator&quot; has the same root as &quot;denomination&quot;, as in the denomination of a dollar bill.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For fractions, the numerator tells you how many things you have but the denominator tells you how big each one is.  Once students get a core understanding of this, you can then start to talk about how you compare two different fractions, both with matching and non-matching denominators.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also like to use pizza.  So 2/3 of a pizza is a pizza sliced into 3 equal pieces and you have two pieces left.  Similarly for 4/5 of a pizza.  How much total pizza does that make?  Help them realize that the 6 slices are different sizes so they cannot be compared directly.  But, would it be possible to cut them into thinner slices so that they match?  This leads to the idea of the least common denominator.  Then cut the pieces and count them up!&lt;/p&gt;&#xA;" OwnerUserId="8979" LastActivityDate="2017-11-13T19:49:21.683" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13154" PostTypeId="2" ParentId="13150" CreationDate="2017-11-13T21:20:20.533" Score="2" Body="&lt;p&gt;I can't say I have much pedagogical advice, but I think it is worth thinking about fractions from a much older point of view, before &quot;fractions&quot; and &quot;division&quot; were unified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of the vinculum was grouping, and it was one of the competitors to parentheses.  By putting a number under the vinculum, you could specify the denomination against which the top number was reckoned.  Using the terminology in Mike Pierce's answer, it gave the units.  It's not the fractions &lt;em&gt;per se&lt;/em&gt; that are being added, but rather numbers are added together in a way that respects their denominations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some strategies I might consider:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Write the denominator smaller than the numerator, strengthening the idea that it is a different kind of number. Maybe also temporarily write the plus sign at the level of the numerators.&lt;/li&gt;&#xA;&lt;li&gt;Write fractions like $1/3$ instead of vertically.  Or temporarily use a new notation like $1/3^{\text{rd}}$, $1\text{ third}$, $1\text{ of }3$, $\frac{1}{\text{thirds}}$ (which should be understood as a vinculum indicating grouping, not division. Example: $\frac{1}{\text{thirds}}+\frac{1}{\text{fifths}}=\frac{8}{\text{fifteenths}}$).&lt;/li&gt;&#xA;&lt;li&gt;Use units everywhere, though it can be awkward to get right! Define an eighths-pizza to a pizza with eight slices, etc.  Then &#xA;$\frac{1\text{ slice}}{\text{eighths-pizza}}$ and $\frac{1\text{ slice}}{\text{tenths-pizza}}$ is in total $\frac{9\text{ slices}}{\text{fortieths-pizza}}$.  It's worth realizing that it is &lt;em&gt;not&lt;/em&gt; actually an equality, though, because the slices will probably be cut differently, so it should be pointed out that we only care about how much pizza we have.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These strategies are based on a pedagogical theory I once heard that is OK to slightly misrepresent material if it will help students reach a point where they may properly master it.  The misrepresentation here is changing the commonly used notation, which I think is forgivable, since the notation is not the concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My hope would be that a notation that doesn't suggest a fake addition rule would cause students to think a little harder, ideally with the effect that they invent common denominators for themselves.  Changing denominations would be be the lesson before this one.&lt;/p&gt;&#xA;" OwnerUserId="8981" LastActivityDate="2017-11-13T21:20:20.533" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13155" PostTypeId="1" CreationDate="2017-11-13T21:26:54.253" Score="14" ViewCount="2263" Body="&lt;p&gt;As of last year, I began teaching a course in theoretical computer science, and one of the topics that we cover is sets.  In particular, we go over:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the actions $\cup$, $\cap$, and $-$, &lt;/li&gt;&#xA;&lt;li&gt;cardinalities&lt;/li&gt;&#xA;&lt;li&gt;Infinite sets in $\aleph_0$, and proving that sets are (or are not) in that set.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I thought that the last part would be tricky (particularly since we not only teach the diagonal graph proof, but also use the proof that hinges an element that points to the set of all elements that don't point to sets that contain themselves... I don't remember the name of that proof, but it is certainly cognitively tricky.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hit these last two subjects hard, doing quite a lot of practice and review.  In the final examination, students did very well with the second two topics, but struggled mightily over (what I believed to be) simple Venn diagram questions like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/q9SWW.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/q9SWW.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Write an expression that represents the region in gray.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some number of kids definitely got it, but for an uncomfortably large number of students, the answers they gave were almost comically convoluted, such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$((\mathbb{U} \cap (C  \cup A)) - ((A - B) \cup (C - B))) \cup (A \cap B \cap C)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... or were simply flat-out wrong.  So, it's clear that I need to teach this better, and I will in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question, then, is thus: &lt;strong&gt;what are the cognitive traps that a teacher should be aware of when teaching union, intersection, and subtraction in sets?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="8274" LastEditorUserId="8274" LastEditDate="2017-11-14T19:27:55.850" LastActivityDate="2017-11-15T10:38:59.257" Title="Cognitive traps in very early set theory" Tags="|set-theory|" AnswerCount="8" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="13156" PostTypeId="2" ParentId="13155" CreationDate="2017-11-14T01:20:07.200" Score="3" Body="&lt;p&gt;I don't think I can address cognitive traps, but&#xA;perhaps this tool could be used for student practice. There is a &quot;Wolfram Alpha Widget&quot; for exploring set Venn diagrams. The widget code is available at &#xA;&lt;a href=&quot;https://www.wolframalpha.com/widgets/gallery/view.jsp?id=eb7ef0469ad23a2c5782e8770da04529&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt;.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/NKxzW.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NKxzW.jpg&quot; alt=&quot;VennW&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Submitted:(A Union B) Intersect (Complement C)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;I hope this links to the widget itself for experimentation:&#xA;&lt;a href=&quot;https://www.wolframalpha.com/widget/widgetPopup.jsp?p=v&amp;amp;id=eb7ef0469ad23a2c5782e8770da04529&amp;amp;title=Venn%20Diagrams%20for%20Sets&amp;amp;theme=red&amp;amp;i0=(A%20Union%20B)%20Intersect%20(Complement%20C)&amp;amp;podSelect=&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-11-14T01:20:07.200" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13157" PostTypeId="2" ParentId="13155" CreationDate="2017-11-14T03:43:02.010" Score="8" Body="&lt;p&gt;I think your specific situation with the exam question is similar to a pre-Calc class that spends a good amount of time on solving trig equations, but then asks on the exam for the student to write down the equation to describe a given scenario. These are just different skills. In a similar Discrete Math course I learned the hard way (as you have now) that students have trouble creating mathematical descriptions (a.k.a. modeling) and adjusted subsequent semesters to include more examples like this in lecture and homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another &quot;cognitive trap&quot; they struggle with is drawing degenerate Venn diagrams, e.g. a Venn diagram where $C \subseteq A \cup B$ or some other condition that reduces the number of regions and changes the way the ovals overlap.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2017-11-14T03:43:02.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13158" PostTypeId="2" ParentId="13155" CreationDate="2017-11-14T05:23:28.243" Score="6" Body="&lt;p&gt;I think the issue here is really more one of problem solving skill or abstraction ability.  You could see the same thing in various problems in computer science or in college algebra where some students can solve an apparently complex problem simply and others get caught up in the complexity.  Or you see the same thing on problems that contain extraneous information where some are able to ignore it and others think they need to incorporate it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my main point is that you are probably thinking of this wrong as some special issue of set theory.  It is more of a common human failing.  And the way to deal with it is probably practice and discussion.  Yes there are some students who can learn the basic rules and then just apply them in a new context.  But this is rare(er).  Probably very common in the sort of people that teach math.  But rare in the student population.  So just live with it and train students on the problems you want them to solve and don't worry so much that not everyone is super smart.&lt;/p&gt;&#xA;" OwnerUserId="8986" LastActivityDate="2017-11-14T05:23:28.243" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13159" PostTypeId="2" ParentId="13155" CreationDate="2017-11-14T14:09:40.997" Score="1" Body="&lt;p&gt;I think it might be helpful to teach for your students to train ways to simplify and transform such longer equations of sets. Since it is a course in computer science, you might even teach them one of the classic algorithms for minimal DNF, which can be easily turned into a problem of sets instead of variables. This should help them to get a feel for things and even the worst students might in the end be able to answer such a question with an overly long but correct union of intersections.&lt;/p&gt;&#xA;" OwnerUserId="8988" LastActivityDate="2017-11-14T14:09:40.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13160" PostTypeId="2" ParentId="13150" CreationDate="2017-11-14T15:15:27.910" Score="0" Body="&lt;p&gt;I am going to try to add to the excellent answers here with a point I like to make and use for many of the techniques that we apply. This point is to emphasize the properties of real numbers and how they appear in our notation and techniques. This conversation happens with the students early in my college algebra class as the reasoning appears at many points later in the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of the great techniques we use break down to an application of the properties of real numbers applied to our problem at hand. I try to emphasize this with students as often as possible so that they have a justification for parts of the process. Also as we develop techniques it can help organize thought.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The overall concept I want to teach is when we have something we don't have a technique for, one of our jobs is to make it look like something we already have a technique for. But we are restricted in that we can not change the value of the quantity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use common denominators as an early example of this idea, when we are going over properties of real numbers to emphasize the power of multiplying by one. Also this emphasizes what the properties of real numbers mean when they are used. The discussion is not meant to be the technique the students use to do this but it is meant to be a justification for the techniques they have been using. A paraphrase the discussion below and skip some points focusing on the things I want to highlight in the discussion. It should be noted the notation I am using in latex is not how exactly how it comes out on the board but is close. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of fractions we are applying distribution and identity multiplication. So the first understanding the student needs to internalize is that for fractions to add or subtract they need a common denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An Example of the reasoning:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{3}{8}+\frac{7}{8}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can write these as factors:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{1}{8}\cdot3+\frac{1}{8}\cdot7$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here we want to note that this is the RHS of the distribution property.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a(b+c)=ab+ac$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we can go from the RHS to LHS. With what we were given we get.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{1}{8}\left(3+7\right)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This completes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{1}{8}\cdot10=\frac{10}{8}=\frac{5}{2}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we extend the idea &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{3}{8}+\frac{5}{6}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We again make factors &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{1}{8}\cdot 3 + \frac{1}{6}\cdot 5$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at what we did earlier we want to make this look like the problem we solved before. This requires the fractions to be equal. We say it would be nice if the the fractions were the same. If multiply 8 by something are there any numbers divisible by 6. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point we discuss the techniques for finding the least common multiple.  That is not my focus here and this discussion with the class is longer than these few sentences. But we find 24 would be a nice denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here we have to dig into another property of real numbers and that is the multiplicative identity:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a \cdot 1 = a$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We want to change the $\frac{1}{8}$ so that we have our nice denominator of 24. But we can not do it in a way that changes the value of the quantity. So I have to multiply my fraction by the number $1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{1}{8}\cdot \frac{?}{3}$  What number replaces the question mark so that we are multiplying this by the number 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We complete this argument and arrange our original problem like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\left(\frac{1}{8}\cdot\frac{3}{3}\right)\cdot3+\left(\frac{1}{6}\cdot\frac{4}{4}\right)\cdot5$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We pause and talk about our goal was to have the fraction common to both terms. We wanted $\frac{1}{24}$ so we organize a little bit more. Here the discussion talks about how fraction multiplication works and justifying the next rearrangement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\left(\frac{1}{8}\cdot\frac{1}{3}\right)\left(\cdot3\cdot3\right)+\left(\frac{1}{6}\cdot\frac{1}{4}\right)\left(\cdot4\cdot5\right)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finishing:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{1}{24}\cdot9+\frac{1}{24}\cdot20$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{1}{24}\left(9+20\right)=\frac{1}{24}\cdot29=\frac{29}{24}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to note the point isn't to give a technique for the students to use, but to develop the reasoning for why the technique is going to work. It sets the stage for later techniques where you need to find other forms of the number 1 or you need to add by zero (complete the square). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember in graduate electrodynamics facing the dreaded Jackson textbook and this horrible problem that none of us were getting after hours of work and pages of attempts. The instructor in one line simplified the problem by multiplying a unit vector divided by a unit vector, the vector multiplication transformed the entire problem to something familiar and the person in front of me exclaimed in agitation we just multiplied by 1. Setting the stage for this type of thinking can extend the concepts of arithmetic into the concepts of mathematics and pattern manipulation and problem solving. Somewhere in the early 80s when I was in 5th or 6th grade I remember having exercises for a few years where you could only solve the problems by using the properties of real numbers and you had to state what you did on each step. I think the idea was develop this idea of building the techniques up. Though it was lost to me at the time I do think it affected how I thought of the process.&lt;/p&gt;&#xA;" OwnerUserId="8952" LastActivityDate="2017-11-14T15:15:27.910" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13161" PostTypeId="2" ParentId="13155" CreationDate="2017-11-14T16:48:22.383" Score="0" Body="&lt;p&gt;I can not see why someone can not see the simple solution. However their approach may be more scalable.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;To solve the over complex problem.&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Anyone that produces a solution that complex can not be a good computer programmer. Therefore help them to see it more clearly, or help them to simplify it. Like we simplify other mathematical expressions.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;What is it (set theory) like? It is like boolean algebra, which in turn is like high-school algebra. So if they are confident in high-shool algebra, then transform to regular high-school algebra.&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\mathbb{U}$ → $1$&lt;/li&gt;&#xA;&lt;li&gt;$\cup$ → $\lor$ → $+$&lt;/li&gt;&#xA;&lt;li&gt;$\cap$ → $\land$ → $\cdot$&lt;/li&gt;&#xA;&lt;li&gt;$A - B$ → $A \cap \bar{B}$ → $A \land \lnot B$ → $A \bar{B}$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And simplify.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;algebra rules&lt;/li&gt;&#xA;&lt;li&gt;plus additional rules:&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$1 = 2$ e.g. &#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$2A = A$, &lt;/li&gt;&#xA;&lt;li&gt;$A^2=A$, &lt;/li&gt;&#xA;&lt;li&gt;$A+1=1 or 2 = 1$, &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;$A\bar{A}= 1\times0 = 0$&lt;/li&gt;&#xA;&lt;li&gt;$A+\bar{A}=1+0=1$&lt;/li&gt;&#xA;&lt;li&gt;$\bar{\bar{A}}=A$&lt;/li&gt;&#xA;&lt;li&gt;de Morgan’s Theorem&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I got $B(A+C)$ → $B\cap(A\cup C)$ &lt;/p&gt;&#xA;" OwnerUserId="5452" LastActivityDate="2017-11-14T16:48:22.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13162" PostTypeId="2" ParentId="13155" CreationDate="2017-11-14T17:35:01.240" Score="-3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;what are the cognitive traps that a teacher should be aware of when teaching union, intersection, and subtraction in sets?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That's your job to figure out. You should be giving your students exercises during class, seeing what they fail at, and talking with them to understand what they're having trouble with. You should be giving them homework problems, and the homework problems should be designed so that when the students get a problem wrong, you have some idea what concepts they're having trouble with. You should have quizzes every week (if not more frequently) that again are giving you feedback about they are understanding (this is one of the purposes of partial credit: you should make it as clear as possible that the more information your students give you about what they do and don't understand, and WHY they are getting a problem wrong, the more points they get). You shouldn't be getting to the final and being surprised at what your students don't understand.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2017-11-14T17:35:01.240" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13163" PostTypeId="2" ParentId="13132" CreationDate="2017-11-14T20:25:53.177" Score="2" Body="&lt;p&gt;If you'd like to graph y=x*lnx, it would be useful to know where the function is headed as x-&gt;0 (from the right). L'H can be used here, though not the 0/0 form.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-11-14T20:25:53.177" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13164" PostTypeId="2" ParentId="13155" CreationDate="2017-11-14T21:11:38.237" Score="4" Body="&lt;p&gt;I may be wrong, but from personal experience I think the other answers are missing a crucial point, what it is like to be taking the test itself, and how people attempt to learn material. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students come up with convoluted, but correct answers, that typically doesn't mean they are &quot;bad at abstract thinking&quot;.  Students come up with convoluted answers to simple problems on tests for many reasons, but if many are doing the same thing then its probably one or many of these:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Your test is time restrictive in some way, in the same way say some sort of entry exam would be.   Its misleading to use the metric of students getting answers correct, when the mental time load for students to actually answer those questions takes a long time.  It might simply be a matter of time for them to get the question correct, but now they have less time to look at other problems.  When they encounter something like this, they might even see something that looks easy.  But when the answer isn't immediately obvious in a situation where it &lt;em&gt;looks&lt;/em&gt; obvious, student panic, because &lt;em&gt;they already spent so much time on the other problems and this one should be easy&lt;/em&gt;. Students will then take the path of least resistance which means mental brute forcing in some form.  If this was one of the few &quot;easy&quot; problems in the test, it might hint further that time restriction was part of the problem, not that the students necessarily lack understanding of intersections and unions. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There was no brush up for the students on these concepts, no matter how simple.  People don't know what they don't know until they encounter a problem that shows them they don't know something &lt;strong&gt;&lt;em&gt;even if that thing was something they used to know&lt;/em&gt;&lt;/strong&gt;.   People don't learn by confirming their biases, they learn by seeing challenges to their mental model.  They will remember when they were &lt;em&gt;wrong&lt;/em&gt; far more often that when they were right.  So if you don't provide material to test their knowledge with and have them verify independently what they got wrong, you are taking a large risk in assuming understanding.  Don't just assume students know things they give you the impression they do with out first verifying it. Even if they do know how to do this stuff on a proficient level, if not enough practice has been given to them, their lack of confidence will be their downfall on the test. It will always take them longer to complete a question correctly they have less confidence in than one they do.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Your test includes many orthogonal concepts, or concepts which can not be applied in a personally meaningful practical cohesive manner until outside the context of the class.  When you include orthogonal concepts on a test, the time to switch how you think, or change your mental model increases (for a variety of reasons like definition overloading).  If students were faced with previously completely different questions, for example &lt;em&gt;those last two subjects&lt;/em&gt;, they would have a harder time adjusting to &lt;em&gt;how&lt;/em&gt; to answer the next question you just presented, especially if you didn't touch on recently (or at all!) the kind of reasoning needed for your vendiagram question.  Their lack of expectations for such a question will cause a huge slow-down in the ability to answer, or hinder ability to come up with a reasonable answer.   &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One way to fix it is to give a hint on &lt;em&gt;how to think about the problem&lt;/em&gt;, ie &lt;em&gt;how difficult the problem should be to solve&lt;/em&gt;.  Don't confuse this with how difficult it should be &lt;em&gt;for the student&lt;/em&gt; to solve, but how many steps it should take and/or how simple the answer should be.   Students may still panic, but will be able to use that as a crutch to understand how they need to think about the problem, with out you actually given them any real help on how to answer vendiagram set question in general. When they start thinking of complicated answers they can quickly short cut all the time wasting bad solutions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give a hint you just need to say something a long the lines of &quot;The solution should be simple&quot;. You may want to put a number of statements max hint (something large enough so it doesn't actually effect answers for students who don't actually know how to do these), but I'm not sure that is the best precedent to set for your questions at large. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way to help is to provide homework for the problem concept. This would be graded (probably recorded for a completion grade, but all wrong solutions actually marked), then the work for those problems is posted on a place they have ready access too.  After the first assignment is due, give them a second assignment covering the same topics but with different questions in different order.  Finally, grade that assignment and cover what students still got wrong in class, still providing the worked out solutions, but not providing more homework assignments on that subject after that point.  These assignments should not be alien to the associated test problems they encounter, or else you have the same orthogonality issues from before. What alien means is subjective, so use your best judgement, they should at least not be &lt;em&gt;surprised&lt;/em&gt; by what they see on the test or how its formatted. &lt;/p&gt;&#xA;" OwnerUserId="8994" LastEditorUserId="8994" LastEditDate="2017-11-14T22:51:03.083" LastActivityDate="2017-11-14T22:51:03.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13165" PostTypeId="2" ParentId="13155" CreationDate="2017-11-15T10:38:59.257" Score="1" Body="&lt;p&gt;There seems to be something very wrong here, not in the mind of the student, but in yours.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You describe the mentioned answer as comical, but there's nothing comical about it: it is the answer of a hardworking student, who understands set theory very well and who has given you a correct answer to a question that was undoubtly very difficult for him/her.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I saw the answer my first reaction was (just like yours): &quot;Jezus, what is this???&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then I started investigating the answer and after some time I must admit that the answer looked correct and revealed true creativity and problem solving skills.&#xA;I admit, there are easier ways to solve the question, but there's where calculation rules come in: using calculation rules you can simplify the answer from the student and like this (s)he might understand that another approach might have simplified his/her work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This latter is also enormously important: when starting to learn mathematics a student starts learning that there is exactly one way to solve a question (which is the way the teacher is doing it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only after some time the student might learn that different approaches for the same question might up given a correct answer (this is crucial in the development of the mind of a student!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of judging that answer as a wrong one, I would advise you to use that answer as a starting point for learning calculation rules about set operations, and from the result of that calculation/simplification you might invite the student to derive from that an easier response to your question, but please, please, never treat the creative result of a hard working student as something comical anymore!&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-11-15T10:38:59.257" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13166" PostTypeId="1" AcceptedAnswerId="13355" CreationDate="2017-11-15T19:46:17.823" Score="8" ViewCount="204" Body="&lt;p&gt;I taught IT in an engineering school during three years in &lt;a href=&quot;https://en.wikipedia.org/wiki/Problem-based_learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;problem based learning&lt;/a&gt; (PBL) only. Now I teach maths to pupils between 10 and 15 years old who have a lot of educational difficulties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm thinking to use PBL to increase their motivation and, by the way, increase their learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, I have a question: is PBL adapted to pupils who have already know a lot of failures (in school and elsewhere)?&lt;/p&gt;&#xA;" OwnerUserId="8777" LastEditorUserId="143" LastEditDate="2017-11-16T15:42:57.077" LastActivityDate="2017-12-23T19:42:34.093" Title="Problem based learning (PBL) for pupils with educational difficulties" Tags="|secondary-education|student-motivation|problem-solving|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13167" PostTypeId="1" AcceptedAnswerId="13171" CreationDate="2017-11-16T13:38:54.030" Score="7" ViewCount="242" Body="&lt;p&gt;Can anyone recommend online or printed sources on anything related to teaching university level math tutorials (that accompany a lecture course taught by someone more senior, but I'd also be happy about material for the lecturer himself)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for general strategies/insights/reflections on anything ranging from exciting students for a subject (presenting extracurricular material can motivate but also intimidate), to effectively presenting proofs (being detailed can reassure but also bore), to pointing out patterns, to structuring homework assignments, to grading assignments, to structuring the classes (solve homework in front of students? hand out solutions/hints? after or before they hand their solutions in?), to repeating old material versus anticipating future material, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should it matter, I am going to teach mostly students of mathematics (as opposed to business, engineering, medicine,..) from freshman to graduate level and am mostly doing analysis (PDE, functional- and complex analysis, numerics).&#xA;However, I am not looking for course book recommendations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that I should have made up my mind about these things back when I sat through tutorials myself, and I have -- partly -- but I figure it won't do any harm to double check and profit from other people's actual teaching experience, and I found it quite difficult to find anything substantial on this. &lt;/p&gt;&#xA;" OwnerUserId="6395" LastActivityDate="2019-04-24T16:30:48.347" Title="Material on tutoring university level math classes" Tags="|reference-request|tutoring|reference-books|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13168" PostTypeId="2" ParentId="1351" CreationDate="2017-11-16T14:48:14.607" Score="0" Body="&lt;p&gt;Tell them that this seems like a cheap way of increasing the food supply by reproducing grain on a massive scale, but experience shows that the resulting loaves taste no better than sawdust so in practice world-wide hunger continues.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2017-11-16T14:48:14.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13169" PostTypeId="2" ParentId="1351" CreationDate="2017-11-16T16:03:47.500" Score="1" Body="&lt;p&gt;I'm baffled that no one has referenced the very entertaining book &lt;a href=&quot;https://www.crcpress.com/The-Pea-and-the-Sun-A-Mathematical-Paradox/Wapner/p/book/9781568813271&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Pea and the Sun&lt;/em&gt;&lt;/a&gt;, written for undergraduates and the possibly mythical &quot;interested laypeople&quot;.  Chapters 6 and 7 explicitly address this issue of physical (ir)relevance, leaving it to the reader in the end as to whether to imagine BTT in hadron behavior (not in making a pea into the sun, though).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, the book also has more or less a proof in terms of words in groups (yes, it actually works written at this level), which would lead me to give the answer to a student that it's as much about that kind of abstraction as about nonmeasurable sets.  Can you touch a rotation or flip in $D_4$? More likely you can only perform it.  Same problem here.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-11-16T16:03:47.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13170" PostTypeId="2" ParentId="13166" CreationDate="2017-11-16T22:30:43.843" Score="2" Body="&lt;p&gt;I cannot provide any research results supporting my claim, but intuitively, I'd assume that it can work out if the level of the exercises is low enough to provide success experience. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I had a pupil aged about 15 who had really big problems in maths (and also other subjects because she attended a Montessori school), failed entry exams for a regular school and had to go to a private school in the end. I gave her exercises which were intended for primary school kids, that is, required no prior knowledge but at the same time required some thinking and trying around. I had the impression that they fitted her level quite well and increased her motivation. Also, they motivated the introduction of some mathematical tools (say, systems of linear equations) quite nicely. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course, if you are dealing with a group of pupils rather than a single one, it is nearly impossible to fit everybody's level so you might bore some and overwhelm others.&lt;/p&gt;&#xA;" OwnerUserId="3542" LastActivityDate="2017-11-16T22:30:43.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13171" PostTypeId="2" ParentId="13167" CreationDate="2017-11-17T03:23:03.250" Score="9" Body="&lt;p&gt;I am very fond of the book by Steven G. Krantz, &lt;em&gt;How to Teach Mathematics&lt;/em&gt;, published by the American Mathematical Society (AMS), now in its 3rd edition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book is mostly aimed at professors and lecturers of undergraduate mathematics. However, I think that much the advice would be broadly applicable. There is a section addressed specifically to TA's, but it's only a single page in length.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/press/maa-reviews/how-to-teach-mathematics&quot; rel=&quot;noreferrer&quot;&gt;A review is here on the MAA site.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2017-11-18T15:18:04.687" LastActivityDate="2017-11-18T15:18:04.687" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13172" PostTypeId="2" ParentId="13058" CreationDate="2017-11-17T07:28:27.917" Score="3" Body="&lt;p&gt;Consider how a student can solve the problems you present.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Identify the potential errors they may commit - forgetting to carry the tens or acting on addition before multiplication, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For each error, work out the result from making this error. It is helpful to make a list of answers and the &quot;cause&quot; of that answer - remember to put the correct one in this list as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now include the answers that come from the errors you want to find and fix. Depending on the question, this can be anything from using algorithms wrongly to not grasping an appropriate intuition for the concepts needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that it is multiple choice, you won't always fit every wrong option into the list presented to students. However, you can prioritise for the errors that are most commonly made, for those that best make a point you want to emphasise, or that force a student to compound errors or find and fix their own errors in order to answer.&lt;/p&gt;&#xA;" OwnerUserId="6724" LastActivityDate="2017-11-17T07:28:27.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13173" PostTypeId="2" ParentId="13150" CreationDate="2017-11-17T17:17:16.167" Score="0" Body="&lt;p&gt;I would suggest to give names to operations and objects. If you want to add two fractions, first find their &lt;strong&gt;common denominator&lt;/strong&gt;. This can be the lcm but sometimes other options are more suiting (for prime denominators it's just the product, no need to involve the lcm, sometimes a power of ten as common denominator is better than the lcm, ... - let them be flexible here). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you need to &lt;strong&gt;expand&lt;/strong&gt; each of the fraction to the common denominator by multiplying the nominator and the denominator by the suiting &lt;strong&gt;expansion factor&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you need to &lt;strong&gt;combine&lt;/strong&gt; the fraction to one by taking the common denominator as denominator and the sum of the nominators as nominator, so you write a &lt;strong&gt;big fraction line&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, those are the same steps as mentions in some of the other answers, but I would like to emphasize the importance of naming things to give them meaning and give the algorithm more flexibility and structure - you then will have distinct steps which make sense by their own, not only as part of some algorithm.&lt;/p&gt;&#xA;" OwnerUserId="3542" LastActivityDate="2017-11-17T17:17:16.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13174" PostTypeId="2" ParentId="13099" CreationDate="2017-11-17T17:48:57.913" Score="3" Body="&lt;p&gt;You can identify polyhedra with polynomials. Let $a_0$ be the number of vertices, $a_1$ the number of edges and $a_2$ the number of faces, then the polynomial for a 3D polyhedron is $p_3(x)=-x^3+a_2x^2-a_1x+a_0$, for a 2D polyhedron (an n-gon) it is $p_2(x)=x^2-a_1x+a_0=x^2-nx+n$, a 1D polyhedron (a line) $p_1(x)=-x+a_0=-x+2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to get a prism with an n-gon base, multiply the base with the line segment: $$p_\text{prism}(x)=(x^2-nx+n)(-x+2)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want a cube, take the third power of a line: $$p_\text{cube}(x)=(-x+2)^3$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also explore higher dimensional polyhedra (polytopes) and read off the number of $k$-dimentional cells from its polynomial. Say, how many edges does a fourdimensional cube have? No problem, just compute $(-x+2)^4$ and read off the first power coefficient. Is there some way to compute a pyramid with a given basis like it worked for a prism? Explore examples!&lt;/p&gt;&#xA;" OwnerUserId="3542" LastActivityDate="2017-11-17T17:48:57.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13175" PostTypeId="2" ParentId="13150" CreationDate="2017-11-18T04:34:06.627" Score="1" Body="&lt;p&gt;I know I'm late to the discussion; but like to share a thought that I think complements the answers from above:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way G Tony Jacobs's answer suggests to progress seems to be common ground. In order to explain the „why“ – as opposed to the „how“ - for adding fractions, this thread only mentions the pizza model. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, to me, if using a model (like the pizza-model) in this context, there are two questions I usually ask: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How does the model explain what the numbers are?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A good model for numbers explains the numbers as objects. For fractions, thats not an easy task! But I guess this is where G Tony Jacobs's answer starts, when aiming at students understanding like and unlike fractions.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Does the model explain &lt;em&gt;all&lt;/em&gt; elementary operations (addition, subtraction, multiplication, division). &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In a recent  article, &lt;a href=&quot;http://www.nctm.org/Publications/mathematics-teaching-in-middle-school/2013/Vol19/Issue3/The-Whole-Story_-Understanding-Fraction-Computation/&quot; rel=&quot;nofollow noreferrer&quot;&gt;J. Dixon and J. Tobias (2013)&lt;/a&gt; do a nice job explaining how to use the pizza-model for all operations. But getting to division, all over the sudden the pizza gets a different shape: it’s not round but rectangular! &#xA;But then, it could have been rectangular to begin with...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now the pizza model explains fractions as „part of a whole“. But there are other aspects of fractions, that need to be considered: Fractions in measurement situations, fractions from sharing, fractions as rates used for comparison, fractions as intensive quantities (like density)...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, creating links between procedures and conceptual understanding, I use more then the pizza model: you can use paper strips and actually fold fractions (see e.g. &lt;a href=&quot;https://youtu.be/d1b_cEA-lTU&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paper Folding to Model Addition of Fractions...&lt;/a&gt; on YouTube), use the number line, and so on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, talking about different models and what addition in these models look like, discussing the transition from one model to another, provides a good background in order to follow the procedures with more understanding.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastEditorUserId="9004" LastEditDate="2017-11-18T08:54:30.900" LastActivityDate="2017-11-18T08:54:30.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13176" PostTypeId="2" ParentId="13167" CreationDate="2017-11-18T17:00:27.827" Score="3" Body="&lt;p&gt;Talking about the old classics: I like the writings of Paul Halmos a lot. A list of his papers can be found at the &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Extras/Halmos_popular_papers.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;history.mcs - Site&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular the two papers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.jstor.org/stable/2319737?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;The problem of learning to teach&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://math.cmu.edu/~bwsulliv/halmos-what-is-teaching.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;What is teaching&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9004" LastEditorUserId="9004" LastEditDate="2017-11-18T17:41:16.607" LastActivityDate="2017-11-18T17:41:16.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13177" PostTypeId="2" ParentId="13081" CreationDate="2017-11-19T02:16:00.777" Score="1" Body="&lt;p&gt;They do not grasp that $\frac{7}{15}$ means &quot;seven out of every separate group of fifteen are being counted&quot;. Additional work on equivalent fractions is needed for them to understand the connection between your seven of fifteen and the fourteen of thirty that they are trying to determine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They are also unclear about the meaning of $1$ as a fraction itself being $\frac{1}{1}$, that is, one of every item is being considered, i.e. when writing the $1-...$ we mean that we are starting with all of them and then taking away the ones we don't want.&lt;/p&gt;&#xA;" OwnerUserId="6724" LastActivityDate="2017-11-19T02:16:00.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13179" PostTypeId="2" ParentId="10510" CreationDate="2017-11-19T08:46:53.967" Score="1" Body="&lt;p&gt;I think that Khan does a great job as a resource for kids to check out concepts that they may have forgotten in the past. For my students the website did not provide enough areas for students to apply what they have learned. My kids enjoy Beestar which has provided them with learning the different concepts and applying them to their assignments. There is also a great reward system and it helps the students stay focused as they enjoy seeing their names on the honor rolls. The kids also get a chance to join their online math competition and end up being placed in GT programs at their school. Having a wide variety helps kids in learning different concepts and applying them accordingly. &lt;/p&gt;&#xA;" OwnerUserId="9010" LastActivityDate="2017-11-19T08:46:53.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13180" PostTypeId="2" ParentId="13099" CreationDate="2017-11-20T00:33:53.633" Score="0" Body="&lt;p&gt;Since a precalculus is, by its nature, designed to prepare students for calculus, I recommend gently introducing students to the concepts of calculus using the tools they already have. In this case, you can use quadratic functions to introduce students to the concept of a tangent line and a limit, without really using that terminology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the first project (&quot;How fast does the pumpkin fly?&quot;) in the link below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.dropbox.com/s/60qdz1npspwu7o0/Precalculus%20Projects%20-%20Brendan%20W%20Sullivan.pdf?dl=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.dropbox.com/s/60qdz1npspwu7o0/Precalculus%20Projects%20-%20Brendan%20W%20Sullivan.pdf?dl=0&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To celebrate autumn, I bought a a small trebuchet with which to launch&#xA;  pumpkins and investigate their flight paths. I am able to control the&#xA;  initial launch height and velocity of the pumpkin, but I need your&#xA;  help to determine where it will hit the ground and how fast it will be&#xA;  traveling at that moment.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I put my trebuchet atop a hill so that the pumpkin is launched from&#xA;  exactly 80 feet above ground level. The launch imparts a vertical&#xA;  velocity of 64 feet per second and a horizontal velocity of 40 feet&#xA;  per second. Because I know the strength of gravity on Earth, the&#xA;  height h of the pumpkin, t seconds after launch, is given by this&#xA;  function: $h(t) = -16t^2 + 64t + 80$.&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Use factoring or the quadratic formula to determine when the pumpkin returns to ground level. For the purposes of the rest of the&#xA;  project, let's say your answer is $T$ seconds after the launch.&lt;/li&gt;&#xA;  &lt;li&gt;Use the given horizontal velocity to identify where the pumpkin will hit the ground. (You may assume there are no effects on the&#xA;  flight path like friction or wind, so the horizontal velocity is&#xA;  constant.)&lt;/li&gt;&#xA;  &lt;li&gt;Your next goal is to find the speed of the pumpkin as it hits the ground. Suppose $d &amp;gt; 0$ is a small positive value, and consider the&#xA;  height of the pumpkin at time $t = T - d$ and at time $t = T$. Explain&#xA;  why the ratio $\frac{h(T)-h(T-d)}{d}$ represents the average velocity&#xA;  of the pumpkin over the time interval $T-d\leq t\leq T$. (Possible&#xA;  hint: Think about the units of measurement of the top and bottom of&#xA;  that fraction.)&lt;/li&gt;&#xA;  &lt;li&gt;Use a calculator to estimate that ratio for the values $d = 1, d = 0.1, d = 0.01$, and $d = 0.001$. Make a prediction for what you think the vertical velocity of the pumpkin is as it hits the ground.&lt;/li&gt;&#xA;  &lt;li&gt;Now, instead of using specific numbers, use the given function $h(t)$ to simplify that expression, $\frac{h(T)-h(T-d)}{d}$, as much&#xA;  as possible so that it no longer involves division. Then, substitute&#xA;  $d = 0$ to find the instantaneous vertical velocity of the pumpkin at&#xA;  that moment. Compare to your prediction.&lt;/li&gt;&#xA;  &lt;li&gt;In fact, the overall speed of the pumpkin as it hits the ground is $S = \sqrt{V^2+H^2}$, where $V$ is its vertical velocity and $H$ is&#xA;  its horizontal velocity. Find $S$.&lt;/li&gt;&#xA;  &lt;li&gt;Find the equation of the line that passes through $(T, 0)$ and has slope $V$. Graph this line together with the function $h(t)$ itself on&#xA;  the domain $0\leq t\leq T$ to verify that you found the tangent line&#xA;  to the graph of $h$ by finding that instantaneous velocity.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Extra Credit&lt;/em&gt; Follow a procedure similar to the one used above to find the instantaneous vertical velocity at the moment where $t = 2$.&#xA;  Graph $h$ and the tangent line together. Explain why your result makes&#xA;  sense physically.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Note: $t=2$ is where the maximum height is reached so velocity is 0.)&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2017-11-20T00:33:53.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13181" PostTypeId="1" CreationDate="2017-11-20T03:48:31.530" Score="8" ViewCount="401" Body="&lt;p&gt;This question may be a bit overly-broad for MESE, but I am hoping to find some responses that can help to fill in my understanding of two similar forms of instruction that had their heyday in the 1960s and 1970s, but seem to have then diverged significantly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first instructional form I'm interested in is &lt;strong&gt;Personalized System of Instruction (PSI)&lt;/strong&gt;, also called the &quot;Keller Plan&quot;.  According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Keller_Plan&quot; rel=&quot;noreferrer&quot;&gt;the Wikipedia article on the Keller Plan&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Keller Plan, also called the Personalized System of Instruction&#xA;  (PSI), was developed by Fred S. Keller with J. Gilmour Sherman,&#xA;  Carolina Bori, and Rodolpho Azzi in the middle 1960s as an innovative&#xA;  method of instruction for the then-new University of Brasília. PSI was&#xA;  conceived of as an application of Skinner's theories of learning,&#xA;  grounded in operant conditioning strategies of behaviorism...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;While traditional teaching is &quot;same pace, different learning&quot;, a key distinguishing factor of PSI is that it instead advocates &quot;different pace, same learning&quot;. A traditional course might have all students follow the same weekly lectures, exercises, etc., and then sit an end-of-course exam at the same date — but possibly with huge variation in learning outcomes (e.g., 95% achievement for a strong student, but just 55% for a weak one). In a course run according to PSI, all students must pass a high threshold of achievement on each module within the course (for instance 90%). The difference between weak and strong students would then be that the stronger ones be able to finish the course quicker, while the weaker ones would need more time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I became aware of the Keller Plan as an undergraduate at the University of Michigan in the 1990s, when it was still being offered as an option for students taking 1st and 2nd semester Calculus-based Physics.  While I did not take the Keller Plan Physics course myself, I did work as a peer tutor in the &quot;Keller room&quot; as an upperclassman, and gained a real appreciation for how well the program can work for highly motivated, disciplined, self-directed learners.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I know, the Keller Plan had its widest adoption in Physics and Astronomy; I don't know if there ever was a Keller Plan / PSI Calculus or Statistics, for example.  (That's one of the reasons why I'm not 100% sure this is on-topic for MESE.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Michigan stopped offering Keller Plan Physics around 2007 or so, but a bit of Googling suggests that it was still being offered at some other Universities more recently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second instructional form I'm interested in is &lt;strong&gt;Individually Prescribed Instruction (IPI)&lt;/strong&gt;, which saw a brief period of wide adoption in the late 1960s/early 1970s.  According to &lt;a href=&quot;http://blog.mathed.net/2011/07/rysk-erlwangers-bennys-conception-of.html&quot; rel=&quot;noreferrer&quot;&gt;one summary of the program&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;IPI was designed for students to &quot;proceed through sequences of&#xA;  objectives that are arranged in a hierarchical order so that what a&#xA;  student studies in any given lesson is based on prerequisite abilities&#xA;  that he has mastered in preceding lessons&quot; (Lindvall and Cox, as cited&#xA;  in Erlwanger, 1973, p. 51). To measure that mastery, IPI relied&#xA;  heavily on assessments that were checked by the teacher or an aide,&#xA;  who would then have the opportunity to conference with the student and&#xA;  check for understanding.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the basic structures of PSI and IPI were not that different: students would work individually in &quot;programmed&quot; materials, complete assessments that were meant to demonstrate mastery, and get individualized help only if and when they needed it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unlike PSI, which was mostly taken up at the college level, IPI was implemented at the elementary level in many progressive schools.  The program eventually earned a terrible reputation, however, due in large part to a notorious case study by Stanley Erlwanger, who documented a student who -- despite having demonstrated successful learning in an IPI curriculum -- had learned a bizarre array of incorrect and self-contradictory &quot;rules&quot; for operating with fractions.  Erlwanger's landmark &quot;Benny&quot; study demonstrated that the IPI curriculum was not capable of measuring the difference between &lt;em&gt;mere performance&lt;/em&gt; and &lt;em&gt;actual understanding&lt;/em&gt;.  By the late 1970s, IPI had essentially disappeared from the landscape of American education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With all of that as background, here are my questions.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Was PSI ever used for college-level mathematics, as it was for Physics -- and if so, what kind of success or failure did it achieve?&lt;/li&gt;&#xA;&lt;li&gt;Are there any online, open-source repositories of PSI or IPI materials?&lt;/li&gt;&#xA;&lt;li&gt;Were there any structural differences between the PSI and IPI programs that might account for the different trajectories these two programs followed?  (One obvious difference was that PSI was mostly used at the post-secondary level; I'm wondering whether there were other issues that had to do with the design of the materials themselves.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="29" LastActivityDate="2017-11-20T18:54:05.690" Title="&quot;Personalized System of Instruction&quot; (PSI) vs. &quot;Individually Prescribed Instruction&quot; (IPI)" Tags="|undergraduate-education|textbooks|primary-education|curriculum|self-learning|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13182" PostTypeId="2" ParentId="13181" CreationDate="2017-11-20T18:42:01.180" Score="2" Body="&lt;p&gt;I've done a lot of reading on the topic of remedial math education in college (i.e., topics up to elementary algebra), and the one and only thing that seems firmly established is this: independent work for such students has catastrophic results. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example is the attempt to run remedial math courses via online systems like MOOCs, which has been terminated many times, wherever it was tried. One analyst concluded that &quot;The failure rates were so high that it seemed almost unethical to offer the option&quot; (&lt;a href=&quot;http://www.madmath.com/2013/06/online-remedial-courses-considered.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper I was most recently reading by Kristina P. Corey Legge, &lt;em&gt;Does Mandatory Supplemental Instruction Work in Developmental Math Education?&lt;/em&gt; (doctoral thesis, Temple University, 2010) studied elementary algebra students in three treatments: (1) standard lecture, (2) lecture plus mandatory supplemental instruction with peers, (3) individualized pacing with computer software and professorial coach. While groups (1) and (2) were statistically indistinguishable (as is usually the case in such studies), the individualized group (3) had a 20% greater failure/dropout rate. This follows similar findings in other literature. (&lt;a href=&quot;http://digital.library.temple.edu/cdm/ref/collection/p245801coll10/id/85619/&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;; see results Table 4.23 on p. 59)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, many of us involved in academics are self-driven and welcome opportunities to learn on our own. I could hold out hope that people holding baccalaureate degrees are more generally in this category. But for lower-level learners (perhaps the majority of college students today), the one solid piece of the literature is that individualized learning is a disaster. Instead, approaches that have shown promise in recent years have been things like learning communities and so forth. (&lt;a href=&quot;https://www.mdrc.org/publication/what-have-we-learned-about-learning-communities-community-colleges&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-11-20T18:42:01.180" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13183" PostTypeId="2" ParentId="13181" CreationDate="2017-11-20T18:54:05.690" Score="2" Body="&lt;p&gt;In 1982 I was a student in a Keller plan Mathematical methods course. It was the worst grade I ever got at university as it required persistence and planning, rather than ability to answer questions under pressure.&#xA;Then in the early 2000s I developed a PSI course &quot;Quantitative methods for business&quot;. With online testing, it was a lot simpler to administer, though our university still required supervised testing as well. For some students it was miraculous, and they overcame their lifelong (these were often mature students) fear or anxiety about maths. For others, their lack of discipline meant they failed to complete in time.&lt;/p&gt;&#xA;" OwnerUserId="9013" LastActivityDate="2017-11-20T18:54:05.690" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13184" PostTypeId="1" CreationDate="2017-11-21T10:19:03.740" Score="9" ViewCount="1219" Body="&lt;p&gt;I perceive that among the general population the belief that mathematics is some sort of highly intellectual esoteric subject is still quite prevalent. After studying math proper I've discovered that even though the notation might look weird sometimes, the concepts that are dealt with are very natural and some of it can be observed in almost any real-world situation. Therefore mathematics is not esoteric, but is related to our surroundings (which everyone should be able to observe).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why do people fear maths, even though they can see a lot of math around them all the time. It's just not notated, but the phenomena are there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, surely it'd be better for mathematics itself, if it wasn't perceived to be esoteric, but accessible to everyone?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_anxiety&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Mathematical_anxiety&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9016" LastEditorUserId="64" LastEditDate="2017-12-12T02:28:50.727" LastActivityDate="2017-12-13T12:59:56.240" Title="What are the reasons for “math fear”?" Tags="|student-motivation|anxiety|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13185" PostTypeId="1" CreationDate="2017-11-21T15:18:52.287" Score="8" ViewCount="293" Body="&lt;p&gt;Are the math syllabi of these three exams comparable? Which syllabus' scope is wider and deeper? Which helps students to be better prepared for math in undergraduate level? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that A level is deeper and has wider scope but Gao Kao of China is famous for its difficulity. Also I heard my friends from Malaysia that Math in STPM is not easy.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot find any comparisons of the math syllabi in these three exams in the internet. If anyone here has some experience about math in these three exams, I am really looking forward to hear about your thoughts about the math syllabus of these three exams. Thanks. And also sorry if this question is too opinion based or off topic as I do not exactly know where should I post this type of question.&lt;/p&gt;&#xA;" OwnerUserId="9017" LastEditorUserId="8557" LastEditDate="2017-11-23T10:33:31.183" LastActivityDate="2017-11-23T10:33:31.183" Title="Compare depth and scope of math syllabus between Malaysia's STPM, Gao Kao and A level" Tags="|secondary-education|curriculum|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13186" PostTypeId="2" ParentId="13184" CreationDate="2017-11-21T16:36:26.373" Score="8" Body="&lt;p&gt;Certainly not a final answer. But to me, maybe the most important aspect is the behavior of the teacher. The way he or she understands mathematics and the beliefs that are hold affect the learning of the students - a triviality...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But once you go on and read the classic by Stella Baruk (1971) &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/2020028301&quot; rel=&quot;noreferrer&quot;&gt;Echec et maths&lt;/a&gt;, you get more than a glimpse on how math anxiety may develop. You may then appreciate, that besides being knowledgeable about math and teaching, why it is so important to hold a &quot;&lt;a href=&quot;http://www.growthmindsetmaths.com&quot; rel=&quot;noreferrer&quot;&gt;Growth Mindset&lt;/a&gt;&quot; belief and care about all your students as human beings. Within math teaching, Jo Boaler's work with &lt;a href=&quot;https://www.youcubed.org/resource/growth-mindset/&quot; rel=&quot;noreferrer&quot;&gt;YouCubed&lt;/a&gt; provides many examples that I find inspiring.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2017-11-21T16:36:26.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13187" PostTypeId="2" ParentId="13058" CreationDate="2017-11-21T19:22:43.477" Score="8" Body="&lt;p&gt;Use them for what they're best at testing, like conceptual understanding questions.&lt;/p&gt;&#xA;&lt;p&gt;One way to do this is by having students evaluate statements they would never be expected to come up with on their own, but should be able to understand the truth or falsehood of. This can also be useful for questions that have many different way of getting to an answer that have nonequivalent difficulty.&lt;/p&gt;&#xA;&lt;p&gt;For example, consider the following question given to a first year calculus class that has discussed concavity.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Assume that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; has a critical point at &lt;span class=&quot;math-container&quot;&gt;$x=c$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$f''(x) &amp;gt; 0$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f''(x)&amp;lt;0$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;c$&lt;/span&gt;. What could be the behavior around the critical point?&lt;/p&gt;&#xA;&lt;p&gt;(a) &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; has a local minimum at &lt;span class=&quot;math-container&quot;&gt;$x=c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(b) &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; has a local maximum at &lt;span class=&quot;math-container&quot;&gt;$x=c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(c) &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is non-increasing with a stationary point at &lt;span class=&quot;math-container&quot;&gt;$x=c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(d) &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is non-decreasing with a stationary point at &lt;span class=&quot;math-container&quot;&gt;$x=c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you just asked the question students would be completely lost, and wouldn't know how to express an answer (unless you'd basically explained the question to them in class). However, they should be able to think critically about such a question and determine what of the answers is plausible.&lt;/p&gt;&#xA;&lt;p&gt;I think such questions get at whether or not the students have a conceptual understanding of the material. Most students (at the college level anyway) have been trained to do computations but have never really been forced to think about the conceptual side of things. I think one reason for this is that instructors don't want a student losing all points on a question because of a conceptual error at the beginning. Multiple choice can provide a less punishing place to determine students conceptual understanding.&lt;/p&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-11-23T18:21:39.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13188" PostTypeId="1" CreationDate="2017-11-21T19:37:47.427" Score="10" ViewCount="449" Body="&lt;p&gt;I am interested in looking at any design resources or &quot;guiding principles&quot; on the distribution of different types of question difficulties on evaluative examinations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can use Item Response Theory and other data analysis techniques to get a fairly good idea of how discriminatory and difficult questions on our test are, so I was curious in moving a step forward into test design: given a set of questions with different discrimination indexes and difficulty indexes what should the distribution of questions on an actual exam be? Should you just go for highest discrimination and call it a day? This would seem to ignore &quot;A&quot;-maker and &quot;F&quot;-maker questions that have high discrimination on one end of the spectrum, possibly giving you more granular data about the students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of any research on test design that addresses distribution of difficult and or discrimination on exams? Or even if there is no concrete research any documents outlining guiding principles?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I've done a literature search and so I'm going to edit this question with what I've found. I hope that this is a useful jumping off point for anyone trying to understand modern testing theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;TLDR: Read “&lt;a href=&quot;http://psycnet.apa.org/record/1994-08136-001&quot; rel=&quot;nofollow noreferrer&quot;&gt;Comparison of Classical Test Theory and Item Response Theory and Their Applications to Test Development&lt;/a&gt;” by Hambleton and Jones and look at &lt;a href=&quot;https://www.ets.org/Media/Research/pdf/RR-03-16.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Evidence Centered Design&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most modern testing theory comes from psychology, where tests are the primary tool of research. There are two main schools of thought on modern testing metrics: &lt;strong&gt;Classical Test Theory&lt;/strong&gt; (CTT) and &lt;strong&gt;Item Response Theory&lt;/strong&gt; (IRT). The goal of CTT is to create reliable tests (ie tests with reproducible results) while the goal of IRT is to create reliable items (questions) by understanding their relationship to an underlying measure of aptitude called ability. These theories have substantively different definitions of discrimination and difficulty.&#xA;CTT:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Discrimination&lt;/strong&gt;: Pearson correlation between item and total test score&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Difficulty&lt;/strong&gt;: Percentage of students who answered question correctly. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;IRT: In IRT, a function called an Item Response Function (IRF) $P_i(\theta)$ is fitted to the $i$'th question. This function takes a number $\theta$ called “ability” which is a measure of a students knowledge collected from assignments, interviews, etc. The function returns the percentage chance that a student will get the question correct. Since a student with low ability should have a low percentage change of answering a question correctly and a student with high ability should have high chance, this function (for dichotomous data) should interpolate between 0 at the low end and 1 at the high end. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The IRF may depend on many other variables, for a great rundown of different IRF's and their uses see “&lt;a href=&quot;https://www.metheval.uni-jena.de/irt/VisualIRT.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Visual Guide to Item Response Theory&lt;/a&gt;” by Ivailo Partchev. For IRT, &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Discrimination&lt;/strong&gt;: The maximum slope of the IRF. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Difficulty&lt;/strong&gt;:  The ability level at which a student has a .5 chance of answering correctly. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note: These metrics can be very different between CTT and IRT. For a description and comparison of the models and their uses see “Comparison of Classical Test Theory and Item Response Theory and Their Applications to Test Development” (&lt;a href=&quot;http://psycnet.apa.org/record/1994-08136-001&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://psycnet.apa.org/record/1994-08136-001&lt;/a&gt;) by Hambleton and Jones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So now the question: given a set of questions we want to make a test. How should we proceed and what effected does Difficulty and Discrimination have? From Hambleton and Jones and others we have the following criteria:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(CTT) High discrimination, difficulty right in the middle. This is to maximize variance giving the final score more meaning. This is also thought to increase reliability, as was partially vindicated in “The relationship between the distribution of item difficulties and test reliability” by Feldt, although he also showed that while restricting item difficult to the middle of the spectrum did increase reliability, it did so almost imperceptibly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(IRT) In general, high discrimination. However, IRT has a device called an Item Information Function which allows you to roughly view the amount of information your test will gather about students at different ability levels. You can add in items to tune this function to test an ability range you are interested in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(ECD) The third model was developed by ETS and is called Evidence Centered Design. For their introduction see their &lt;a href=&quot;https://www.ets.org/Media/Research/pdf/RR-03-16.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;research report&lt;/a&gt;. The idea is that a test should measure competency in certain areas, and that you as the administrator want to prove on some level that is it. ECD used IRT to gives guidelines to construct and evaluate test items that measure student ability on specific topics in a verifiable way. Although it requires multiple rounds of testing using the same items.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conclusion: Are you trying to &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Maximize meaningfulness of the score? Use Classical Test Theory, questions in middle difficulty (.3-.8) and high discrimination. This maximizes variance in the score while preserving reliability. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Tests discriminating between certain ability levels? Use Item Response Theory, matching your test items to the desired Test Information Function. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are you trying to test specific knowledge, accurately across time, while allowing yourself to reuse questions on different populations? Use the Evidence Centered Design process to provide evidence that test actually evaluates desired area of knowledge. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;While all of the above is useful to understand how one thinks about test construction, I have yet to find any guidelines specific to mathematics. Although that might be fodder for another question here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anyone is interested in digging deeper and answering more specific questions, the best place to find articles are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://eric.ed.gov/?&quot; rel=&quot;nofollow noreferrer&quot;&gt;Institute of Educational Sciences&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.tandfonline.com/toc/hame20/current&quot; rel=&quot;nofollow noreferrer&quot;&gt;Journal of Applied Measurement in Education&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Articles mentioned here:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First place to go to understand CTT and IRT methodology: &lt;a href=&quot;http://psycnet.apa.org/record/1994-08136-001&quot; rel=&quot;nofollow noreferrer&quot;&gt;Comparison of Classical Test Theory and Item Response Theory and Their Applications to Test Development&lt;/a&gt;, Hambleton and Jones.&lt;/li&gt;&#xA;&lt;li&gt;A great introduction to the nuts and bolts of IRT: &lt;a href=&quot;https://www.metheval.uni-jena.de/irt/VisualIRT.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Visual Guide to Item Response Theory,&lt;/a&gt; Ivailo Partchev.&lt;/li&gt;&#xA;&lt;li&gt;A review of CTT Test Design with empirical results: &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1207/s15324818ame0601_3?journalCode=hame20&quot; rel=&quot;nofollow noreferrer&quot;&gt;The relationship between the distribution of item difficulties and test reliability&lt;/a&gt;, Feldt.&lt;/li&gt;&#xA;&lt;li&gt;Introduction to Evidence Centered Design &lt;a href=&quot;https://www.ets.org/Media/Research/pdf/RR-03-16.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Brief Introduction to&lt;br&gt;&#xA;Evidence-centered Design &lt;/a&gt;, Mislevy, Almond, Lukas&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.washington.edu/assessment/scanning-scoring/scoring/reports/item-analysis/&quot; rel=&quot;nofollow noreferrer&quot;&gt;University of Washington's Guidelines on item level analysis&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="9019" LastEditDate="2017-12-08T18:47:41.670" LastActivityDate="2017-12-08T18:47:41.670" Title="What is known about discrimination and difficulty in test questions?" Tags="|reference-request|test-design|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13189" PostTypeId="2" ParentId="13184" CreationDate="2017-11-21T21:48:22.323" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;We are not computers and the human mind did not evolve to do math problems.  We are much better at looking for the snake in the grass or recognizing our children's faces in a crowd than we are at solving for X (let alone proving epsilon delta).  Our minds are actually stronger at speaking and hearing than they are at reading and writing.  But even reading and writing is much more suitable to our minds than math.  That doesn't mean we can't do it at all.  Obviously we can.  But it is not as natural.  That's just life.  We are not as agile in the trees as monkeys or squirrels either, even though we can do some tricks with enough training.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For a lot of people, lack of skill/understanding drives this fear.  I think this is logical and understandable.  (A lot of people have fear of physical tasks that they are not good at also.)  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think that nomenclature and pedagogy always helps either.  I get more out of a PDE book that starts with units I know than if it is a blizzard of Fraktur and Greek symbols (and not alpha, talking &quot;squiggle&quot; and &quot;other squiggle&quot;).  Or consider quantum mechanics.  Lowe's graduate QC text makes the good argument in preface that it is better to teach the basics of the concept first before hitting the students with bra and ket.  It is just hard to learn too many new things at once.  You can get a lot of basic benefits from learning the basics of Schroedingers Equation for the hydrogen atom first in normal PDE land before getting hit with bra and ket and postulates.  And already knowing some of the basics, makes it easier to have some ground to learn these new concepts on).  Sometimes how the topics are taught or described in texts is poor as well (lack of motivation, not enough progression, not enough &quot;small wins&quot; or drill.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I would put the issue as more on 1 and 2 (seems like same thing after I wrote it) than on 3.  Even if you have wonderful teachers and texts, it does not make all the problems go away.  And it is a long, long, loooong fallacy of educational reform that if we just changed this (no THIS) than everyone would be an A student.  But sometimes the &quot;harden up and handle your Rudin&quot; attitude does do the student a disservice.&lt;/p&gt;&#xA;" OwnerUserId="9020" LastActivityDate="2017-11-21T21:48:22.323" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13191" PostTypeId="2" ParentId="13081" CreationDate="2017-11-22T17:18:40.950" Score="3" Body="&lt;p&gt;Unfortunately, I think you may be asking the wrong question. When it comes to fractions, a huge share of students have &lt;em&gt;no idea&lt;/em&gt; what they even mean. They have likely not internalized the idea that fractions are numbers, which makes all the fractional arithmetic you do completely rote (in their minds).  In other words, it's prior knowledge gaps that are holding them back. Here's how I would assess for those prior knowledge gaps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ge them a list of numbers such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{11}{7}, \frac{8}{2}, 5, \frac{3}{4}, 1.6$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, with no hints, ask them to draw their own number line from scratch and place everything appropriately. &lt;a href=&quot;https://www.carnegiefoundation.org/wp-content/uploads/2013/05/stigler_dev-math.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A massive share of high school graduates can't do this&lt;/a&gt;, so it's very likely your grade 6 students can't either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also ask them to do $3-\frac{4}{5}$ and see if they can answer that using &quot;common sense&quot; instead of &quot;common denominators&quot;. Have them draw a picture to explain their intuition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, ask them to determine if the answers to the following are bigger or smaller than &lt;strong&gt;700&lt;/strong&gt; and how they know. See if they can explain their reasoning without doing the &quot;standard algorithm&quot; calculations. (All exercises involve 701.27 to discourage exactly that and to encourage thinking about meaning and estimation instead.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{3}{8}\times701.27=\_\_\_\_$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{8}{3}\times701.27=\_\_\_\_$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{701.27}{3}\times8=\_\_\_\_$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{8}{3}\div701.27=\_\_\_\_$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$701.27\div\frac{8}{3}=\_\_\_\_$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$701.27\div\frac{3}{8}=\_\_\_\_$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{3}{8}\div701.27=\_\_\_\_$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$Two\ thirds\ of\ 701.27\ is\ \_\_\_\_$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$Two\ thirds\ times\ 701.27\ is\ \_\_\_\_$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$701.27\ groups\ of\ two-thirds\ is\ \_\_\_\_ $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, I'd suggest you draw a picture like this to represent the students, where P represents a physics students, B represents a biology student, and C represents a chemistry student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P P P P P P&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B B B B B B B B B B&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C C C C C C C C C C C C C C&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given this picture, are they able to describe the class correctly using fractional vocabulary and equivalent fractions? Can they use those fractions to combine the fractions of physics and biology students? A shocking share of students can't. And if they can't, there's no way they're ready to tackle word problems in which &lt;em&gt;they&lt;/em&gt; must determine and/or draw and/or visualize the situation.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2017-11-22T17:18:40.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13192" PostTypeId="2" ParentId="13184" CreationDate="2017-11-22T17:44:53.757" Score="5" Body="&lt;p&gt;I believe it is a combination of the following, among other things.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The &lt;a href=&quot;https://math.berkeley.edu/%7Ewu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;writings of Hung-Hsi Wu&lt;/a&gt; have plenty of examples of the gibberish and pseudo math that goes on in many math classrooms. This practically guarantees that only a tiny minority of students benefit from their &quot;math&quot; experiences in school.&lt;/li&gt;&#xA;&lt;li&gt;A huge share of &lt;a href=&quot;https://www.thestar.com/yourtoronto/education/2016/05/13/for-many-teachers-math-just-doesnt-add-up.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;elementary teachers, victims of pseudo math themselves, hate math&lt;/a&gt;. At least they hate what they believe math to be. As I've said elsewhere, an art hater becoming a children's art teacher is obviously stupid. But having math haters teach math to children is the norm. Negative attitudes towards math are passed on from teachers to students.&lt;/li&gt;&#xA;&lt;li&gt;Many passionate math educators constantly tell themselves to defeat their own &lt;a href=&quot;https://en.wikipedia.org/wiki/Curse_of_knowledge&quot; rel=&quot;nofollow noreferrer&quot;&gt;curse of knowledge&lt;/a&gt;, they obsess over empathizing with students. And they fail. A lot. I include myself in this group. I look back at learning activities from a few years ago that I thought - no, &lt;em&gt;I knew&lt;/em&gt; - were good... and now I feel like I'm one of those &lt;a href=&quot;https://www.google.ca/search?q=doctors%20advertising%20cigarettes&amp;amp;safe=off&amp;amp;client=firefox-b-ab&amp;amp;dcr=0&amp;amp;source=lnms&amp;amp;tbm=isch&amp;amp;sa=X&amp;amp;ved=0ahUKEwjq9Nma29LXAhUH52MKHbSdC6EQ_AUICigB&amp;amp;biw=1857&amp;amp;bih=1008#imgrc=_&quot; rel=&quot;nofollow noreferrer&quot;&gt;doctors who used to recommend cigarettes to their breathing-impaired patients&lt;/a&gt;. We educators overload our students all the time but students blame themselves when they're overloaded and get wrong answers.&lt;/li&gt;&#xA;&lt;li&gt;Traditional math assessment is extremely algorithmic, to the exclusion of logic and meaning, to the point where you can memorize the algorithms and earn an A in your decimal calculations test... while believing $7.30$ is greater than $7.3$. &quot;Isn't 30 bigger than 3? Who cares? That's not on the test so it has nothing to do with math so I don't care and I don't need to know!&quot; After 10 years of rote &quot;learning&quot;, a student gets to a teacher who asks conceptual questions and the only way to even begin to understand those questions is to learn 10 years of concepts in time for the high-stakes test tomorrow.&lt;/li&gt;&#xA;&lt;li&gt;Math is extremely cumulative. In the early grades, it's obvious that one must learn to count to 10 and beyond before learning how to determine $8+6=\_\_\_\_$. You wouldn't really care what was on the test; you'd have the teacher teach the child how to count. But in high school, students, parents, and teachers complain angrily if I suggest the student should understand that $\frac{4}{3}&amp;gt;1$ before tackling $y=-\frac{4}{3}x-12$. &quot;Tomorrow's test isn't on comparing pies vs numbers. It's on graphing!&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;All of this reinforces the belief that math is esoteric, that only a tiny share of people can understand it, that there is no beauty behind it, that math is just fearful rules to memorize.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2017-12-13T06:23:41.050" LastActivityDate="2017-12-13T06:23:41.050" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13193" PostTypeId="2" ParentId="13058" CreationDate="2017-11-22T18:07:04.383" Score="0" Body="&lt;p&gt;This is very hard to do, but I would suggest a simple rule: Multiple choice questions should be trivial to answer if you understand the concept and very difficult otherwise. I can't think of a great example off the top of my head, but when I do, I'll post here!&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2017-11-22T18:07:04.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13194" PostTypeId="2" ParentId="12085" CreationDate="2017-11-22T19:37:53.573" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;... they almost always claim they haven't understood anything in the class at all... including those who are earning the highest grades in the section.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This implies a problem with your assessments: False positives. It is, apparently, possible to get high grades in your class while understanding very little content.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The world of medical testing has internalized the idea of both positive and negative results being either true or false. But the world of math education - nearly all of education, really - has not, so assessments are not designed or interpreted with this in mind. Many students can get high grades (positive test result) while mastering nothing (negative reality). Concepts are hard to teach and assess, while procedures are easy. So most assessments focus on procedures and memorizing algorithms. &quot;If that number is 0.04, then reject, and it's obviously false...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another issue that bottlenecks post-secondary math courses is that even many high school graduates have ~zero understanding [I do mean &lt;em&gt;~zero&lt;/em&gt;] of rational numbers... but that's a whole other issue.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;she just slammed her fist on the table... [and] demanded the age-old question of &quot;where she would ever have to use this stuff.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Consider checking out the first two chapters of &quot;Statistics&quot; by David Freedman. The introductory chapters are basically cases of statistical inference from controlled vs uncontrolled research. It's mostly relatable, relevant (real data, useful questions), and qualitative, which is what students want to hear when asking &quot;When will I ever use this stuff?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, at the beginning of the semester, give your students a simple hypothetical case of social workers assigned specifically to combat poverty. Data is collected and...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pfDWv.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pfDWv.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then pose the following task: &quot;Generate ten different hypotheses for what you see in the table. For each hypothesis, define the minimal evidence it would take to prove or disprove it. Then state how you should collect that evidence and analyze it.&quot; I'd have them do this individually first, then compare with a peer or two, then share with the class. This would probably have to be spread over a week or two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, if they're in an introductory stats course, their answers to that task will likely be intuitively appealing but statistically terrible because we all have terrible statistical intuitions until we are trained otherwise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can spend the rest of the semester continually revisiting their answers and they should see their thinking improve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course... it doesn't have to be social workers and poverty, it could be whatever context you wanted. :)&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2017-11-22T19:37:53.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13195" PostTypeId="2" ParentId="12551" CreationDate="2017-11-22T20:59:46.457" Score="3" Body="&lt;p&gt;This book is unfortunately out of print, but it is still accessible: &lt;em&gt;The Penguin Book of Curious and Interesting Puzzles&lt;/em&gt; by David Wells. A majority of the problems are of a discrete or combinatorial flavor, and he avoids getting into mathematics in the form of mathematics in the answer. In addition, the book essential doubles as a history of recreational puzzles which is the original reason I came across it. &lt;/p&gt;&#xA;" OwnerUserId="668" LastActivityDate="2017-11-22T20:59:46.457" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13196" PostTypeId="2" ParentId="12085" CreationDate="2017-11-22T21:40:28.307" Score="2" Body="&lt;p&gt;I thought Dan had some great response for you.  My advice would be to cover LESS and to cover more intuitively.  Think of another topic:  fluid mechanics.  I have worked as an engineer and operator a LOT in many industries involving piping.  The basic intuitions of &quot;fatter pipe will flow more&quot; and &quot;hard to flow around corners&quot; are actually more important and powerful than Bernoulli's equation with PDEs.  I mean I know it sounds  trivially simple.  But you would not believe how many times I was able to figure something out on the job by having &quot;physical intuition&quot; rather than a derived knowledge or an equation.  (And let's be real, everyone uses nomographs anyways!)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just the way you worded your question was &quot;hard&quot; to me.  And I LOVE statistics (at least intuitive statistics.  Also, I got the sense that you were trying to build up justifications by derivation rather than intuition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't mean to say all the problems will go away.  Let's be real.  Your students are not rocket scientists or McKinsey consultants (and even some of these struggle at times!)  And it is not the only think important to them.  They need to be able to count the pills correctly, get the right ones, enter the chart correctly.  And maybe be a little nice to the patients at times.  And a bunch of other stuff.  But still.  (I have dated a lot of nurses, including one who came crying up to my room after a long time patient died, one who misprescribed, etc.)  But if you can get across to them some semblance of realization that stats can be tricky (not to trust every paper), that bigger sample sizes are better, and a couple other things that will help in the end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they just remember stats class as a formula blizzard, that will be bad.  And I remember my &quot;probes and shafts&quot; course as a formula blizzard.  And it was relatively gentle!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Look at this video.  Not to use it (it is not exactly what you are teaching).  It's mfg engineers and execs that it is targeted at.  (Like the audience for The Goal by Goldratt).  But look at it for pedagogical motivation.  This is George Box!  But look at how easy he makes things.  And how he concentrates on things that are &quot;easy&quot;.  Isn't that a lot more fun to watch than a formula blizzard?  Won't more people get things out of it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=qF0dhl61Bdo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=qF0dhl61Bdo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  If you got any cute caring nursing students on the market. Well...  But those qualifiers are probably redundant!&lt;/p&gt;&#xA;" OwnerUserId="8986" LastActivityDate="2017-11-22T21:40:28.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13197" PostTypeId="2" ParentId="13031" CreationDate="2017-11-24T14:47:04.533" Score="1" Body="&lt;p&gt;I use the app Xournal (with a Thinkpad X200 Tablet as hardware, currently available at about 200€ in very good used state) and get output like this: &lt;a href=&quot;https://i.stack.imgur.com/5H1By.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5H1By.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My handwriting could be better, but I think, it is quite smooth in terms of software. Xournal is free software, available at: &lt;a href=&quot;http://xournal.sourceforge.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://xournal.sourceforge.net/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3542" LastActivityDate="2017-11-24T14:47:04.533" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13198" PostTypeId="1" CreationDate="2017-11-24T18:22:27.780" Score="12" ViewCount="183" Body="&lt;p&gt;I am looking for resources that have tasks such as the one below that encourage argumentation. I want tasks that 8th graders could do but would also be appropriate for high school students. I want to do some kind of research on the mathematical practice &quot;construct viable arguments and critique the reasoning of others&quot;. I've seen several good exploration tasks such as this one, but they are all scattered. Are there any good books or resources that include these types of tasks?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8Gs9Z.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8Gs9Z.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fendel, D., Resek, D., Alper, L., &amp;amp; Fraser, S. (1996). Interactive mathematics program year 1 - unit 2: The game of pig (p.99). Emeryville, CA: Key Curriculum Press.&lt;/p&gt;&#xA;" OwnerUserId="7550" LastEditorUserId="5220" LastEditDate="2017-11-24T18:39:16.697" LastActivityDate="2017-11-27T13:13:23.253" Title="Tasks that encourage argumentation" Tags="|undergraduate-education|secondary-education|reference-request|proofs|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13199" PostTypeId="2" ParentId="13198" CreationDate="2017-11-24T19:18:30.927" Score="5" Body="&lt;p&gt;Statistics cases are usually a fertile source of arguments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some believe that laptops in math classrooms help learning. Some believe laptops harm learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Task:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Consider your own math classroom, teacher, and final exam and think about the impact that laptops in the classroom would have on your class's average final exam score. Pick a number on a scale from -5 to 5, where -5 means &quot;Laptops definitely decrease that average score&quot; and 5 means &quot;Laptops definitely increase that average score&quot; and 0 means &quot;totally uncertain&quot;. Explain your pick.&lt;/li&gt;&#xA;&lt;li&gt;Then, name the minimum threshold of evidence it would take to make you pick a different number. Explain your threshold.&lt;/li&gt;&#xA;&lt;li&gt;Compare your answers with others. Make changes to your prior answers as necessary.&lt;/li&gt;&#xA;&lt;li&gt;[Class discussion and perhaps direct teaching on data collection, design of experiments, statistical inference, net effects and counterfactuals, linear vs non-linear relationships, variance, etc.]&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2017-11-24T19:33:43.013" LastActivityDate="2017-11-24T19:33:43.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13200" PostTypeId="2" ParentId="13198" CreationDate="2017-11-24T19:48:50.273" Score="1" Body="&lt;p&gt;When I taught high school, a then-colleague of mine developed this great project to be used in a high school geometry class.  The project was to be used when students were first learning how to write proofs:  in particular, when they were understanding the differences between the givens, the conclusion, the logical steps in between, etc.  The students were asked to find an editorial in a newspaper and try to take it apart into given, conclusions, logical steps, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students were also being taught the two-column format of proof writing and so they were asked to turn the editorial into a two-column proof.  Then, when they started writing their own proofs, they could compare the rigor of their proofs to the rigor (and long list of givens) of the editorials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To really promote discussion and argumentation, you could have the students work in pairs.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-11-24T19:48:50.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13203" PostTypeId="2" ParentId="13114" CreationDate="2017-11-25T08:19:52.163" Score="7" Body="&lt;p&gt;As to question 1: I do have a german speaking background, so please excuse the source actually is in german:&#xA;There is a really nice journal called &lt;a href=&quot;http://www.wurzel.org&quot; rel=&quot;noreferrer&quot;&gt;Wurzel&lt;/a&gt;, the german word for &quot;square root&quot;. In there, Attila Furdek has a section called &quot;Schlaue Leute werden durch die Fehler von anderen klug&quot; (trans. &quot;Clever people become wise through the mistakes of others&quot;). The structure of the tasks is always the same: The Problem is posed, along with two or more answers, all getting to a different conclusion. The question then is, which solution is correct and why. &lt;a href=&quot;http://www.wurzel.org/werkstatt/&quot; rel=&quot;noreferrer&quot;&gt;Sample problems&lt;/a&gt; can be accessed online, Attila Furdek also collected the problems along with solutions in a &lt;a href=&quot;https://www.amazon.de/Fehler-Beschw%C3%B6rer-Typische-Fehler-L%C3%B6sen-Mathematikaufgaben/dp/3831121109&quot; rel=&quot;noreferrer&quot;&gt;book&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way to get these questions is similar to &lt;a href=&quot;https://matheducators.stackexchange.com/a/13063/9004&quot;&gt;the way distractors for multiple choice questions are generated&lt;/a&gt;: Furdek would collect his students work to a problem. If there are different solutions among them, he would pick two or three different solutions ans give them back to the class, together with the question which one would be the correct one and why. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This way, the different alternatives are actual mistakes, which makes the problems very authentic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used translations of the book in my own classes, but more over, collected &lt;em&gt;my&lt;/em&gt; students solutions in order to use the same style of task. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On question 2: I have to admit, I haven't &quot;Google scholared&quot; this topic as I should.  My experiential knowledge is, that this type of task, put right, can be a great learning opportunity. Similar to tasks using &lt;a href=&quot;https://www.doodlemaths.com/custom-content/uploads/2017/01/DM_teachingtools_conceptcartoon_Jan17.pdf&quot; rel=&quot;noreferrer&quot;&gt;concept cartoons&lt;/a&gt;, it stimulates classroom discussions not only about the &quot;what&quot; and the &quot;how&quot;, but also the &quot;why&quot; of a solution. This way, I've the impression my students were less relying on &quot;solutions from the book&quot; and went towards more confidence in their answers. Opposite of the worries of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13114/searching-activities-with-find-the-error-strategy-for-learning-maths#comment32033_13114&quot;&gt;DRF (see comments)&lt;/a&gt;, in my experience this tasks did not reinforce wrong procedures, to the contrary, they helped my students a lot in &quot;making sense&quot; of what they do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much in the sense of a saying by Groucho Marx: &quot;Learn from the mistakes of others. You can never live long enough to make them all yourself.&quot;&lt;/p&gt;&#xA;" OwnerUserId="9004" LastEditorUserId="9004" LastEditDate="2017-11-25T08:26:37.150" LastActivityDate="2017-11-25T08:26:37.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13204" PostTypeId="2" ParentId="12956" CreationDate="2017-11-25T22:46:47.023" Score="2" Body="&lt;p&gt;To stress a point in Adam's answer: try to study the variations of $x\mapsto x^3-2x^2+x-1$ without the analytic form of derivative. You will do a large amount of computation and most student will likely make an error somewhere. With the analytic form of differentiation, this is very simple.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as numerical computation go, most mathematician should stop assuming they know how computers do get to compute things, because most of the time they don't. I believed that characteristic polynomial was useful to diagonalize matrices, while in fact it is the other way round: characteristic polynomial are used to compute numerically roots of polynomial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of derivative, to compute a derivative by the ratio $\frac{f(x+h)-f(x)}{h}$ to a given precision, you need a crazy precision on the computation of $f$, this is really slow. I guess many methods have been studied, but when your function is actually computed by a processor, then a beautiful method exists. It consist in taking the raw definition of the function, which ultimately reduces a (complicated) combination of the few operations that a processor can perform on floats. Then you replace all these basic operations by their equivalent in first-order calculus, where numbers are replaced by pairs $(x,dx)$. For example the product $(x,y) \mapsto x\times y$ is replaced by $((x,dx),(y,dy))\mapsto (x\times y, x\times dy+y\times dx)$ (here I use the notation $dx$ but this is a float). Once this substitution is done, the algorithm that got $x$ as entry and gave $f(x)$, now gives $(f(x),f'(x))$ whenever it is fed $(x,1)$. Voilà! The beauty of it is that it runs only a factor more slowly than the original algorithm, while using the slope approximation takes a completely different order of magnitude.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2017-11-25T22:46:47.023" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13207" PostTypeId="1" AcceptedAnswerId="13208" CreationDate="2017-11-26T17:44:40.747" Score="9" ViewCount="3592" Body="&lt;p&gt;Why do some math (and science) textbooks/solution books only include answers to odd- or even-numbered problems?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some textbooks have a separately purchased answer book with the missing answers, while others only include odd or even-numbered answers &lt;em&gt;in the solutions book&lt;/em&gt; (and none in the textbook). Why is this the case and how do publishers decide this stuff?&lt;/p&gt;&#xA;" OwnerUserId="9028" LastActivityDate="2018-01-18T01:18:24.533" Title="Odd or Even-Numbered Answers in Textbooks" Tags="|textbooks|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13208" PostTypeId="2" ParentId="13207" CreationDate="2017-11-26T21:05:00.693" Score="24" Body="&lt;p&gt;Having answers to all the problems in a book is often inconvenient for teachers.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;For very elementary topics in which not much work needs to be shown by students (such as what is the domain of $f(x) = \frac{x}{x^2 – 4}),$ allowing students to have the answers makes it more difficult to grade homework. For example, suppose a student has the correct answer and the student work is not clear, but the work &lt;em&gt;might be&lt;/em&gt; on the right track. How much partial credit do you give? If the students are not given the answers, then you can lower your &quot;show your work&quot; bar in deciding on how much credit to give.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For more difficult problems, especially when an intuitive “trick” is needed, if a student has the answer, then the student might be able to reverse engineer the answer to come up with a solution. Counting problems (i.e. combinatorics) can sometimes be tricky, but if you know the answer, then you might be able to make educated guesses as to whether permutations or combinations are used, and whether additional modifications are needed (such as when certain objects treated as identical).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If students have all the answers, then they will not get practice in making safety checks to see if their answers make sense, a skill especially important when the answer sought is very important, which is often the case in real life situations, such as how much is it going to cost the company if such-and-such policy is put in place.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If answers are given to all the problems in the book, then for certain topics at least, this can prevent teachers from using these questions on tests, which makes the book less useful for the teacher than a book that doesn’t give answers to all the problems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="745" LastEditorUserId="5220" LastEditDate="2017-11-27T04:21:54.877" LastActivityDate="2017-11-27T04:21:54.877" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13209" PostTypeId="1" AcceptedAnswerId="13211" CreationDate="2017-11-27T07:46:19.203" Score="6" ViewCount="241" Body="&lt;p&gt;Consider the problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find $f(x)$ if $f’(x)=4x$ and $f(3)=12$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have always done this, and taught it, as a two-step problem: First, find the general anti-derivative, $f(x)=2x^2+C$, and then plug in the given value, $12=2(3)^2+C$, to solve for $C$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently saw a student do it in one step: $$f(x)=12+\int_3^x 4t \,dt$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had not seen this before, and I find it kind of brilliant but kind of bizarre. Do many people do such problems this way? I like its directness, but it also seems a bit less accessible, somehow. Does anyone have experience with teaching these problems both ways? How do students do with these two methods? What are the respective advantages and pitfalls?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance for any insights.&lt;/p&gt;&#xA;" OwnerUserId="2034" LastActivityDate="2018-02-20T19:38:31.800" Title="Simple initial value problems - pros and cons of different methods" Tags="|undergraduate-education|calculus|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13210" PostTypeId="2" ParentId="13209" CreationDate="2017-11-27T09:45:38.183" Score="7" Body="&lt;p&gt;Look in your textbook where the Fundamental Theorem of Calculus is discussed. You might see the formula $$f(x)\; =\; f(a) + \int_a^x f'(t) \,dt$$ or a very close version of this, such as $$f(x) - f(a) \; = \;\int_a^x f'(t) \,dt.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally have not seen a student do this, but I can easily imagine a student who wasn't paying attention in class solving this by chancing upon these general equations, which are usually displayed quite prominently in texts. You might want to ask this student how he/she came up with this method, and maybe share it with the class, with credit being given to this student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for what to do in the future, I recommend that you continue to teach it as you have been, which is pretty much universal now and for at least the last 150 years (based on my extensive experience in looking at old textbooks), because it is less formularic and it is the method used in many other contexts --- solving differential equations, curve-fitting, definitions by recursion, etc. in which step 1 is to obtain a general solution (or identify the general form a solution takes) and step 2 is to solve for the specific constants that are involved in the general solution.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2017-11-27T10:22:21.520" LastActivityDate="2017-11-27T10:22:21.520" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13211" PostTypeId="2" ParentId="13209" CreationDate="2017-11-27T10:41:59.483" Score="4" Body="&lt;p&gt;Differential equations are employed to construct models of physical phenomena. In particular, consider the very simple linear non-autonomous differential equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y'(x) = g(x)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;with initial condition $y(x_0) = y_0$. Linear differential equations of this type usually describe a physical system forced by the term $g(x)$ which starts from a certain initial state.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your two-step solution does not give a formal solution valid for any integrable $g(x)$ and any initial condition $y_0$. It seems that any $g(x)$ and any $y_0$ should be treated in a special way, and there's no way to speak of the general properties of a solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one-step solution is instead a particular case of&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y(x) = \int_{x_0}^x g(t) \,\mathrm{d}t + y(x_0),$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a form which is valid for &lt;em&gt;any&lt;/em&gt; integrable $g(x)$ and $y_0$. But this, in turn, is a particular case of the convolution integral&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y(x) = \int_{x_0}^x h(x-t)g(t) \,\mathrm{d}t + y(x_0),$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which represents the response of any general linear time-invariant system — represented by $h(x)$ — with one input forced by the term $g(x)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These forms allow to discuss general properties of the solutions (e.g. natural and forced response, eigenfunctions and eigenvalues, frequency response, stability), something that it would not be possible by treating each differential equation as a special case. Of course, one should not give the impression that any differential equation possesses solutions that can be written explicitly in this way, but when this is possible there are many advantages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The solution that you find bizarre is actually quite common in every application of ordinary differential equations.&lt;/p&gt;&#xA;" OwnerUserId="5367" LastEditorUserId="5367" LastEditDate="2018-02-20T19:38:31.800" LastActivityDate="2018-02-20T19:38:31.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13212" PostTypeId="1" CreationDate="2017-11-27T12:22:04.637" Score="1" ViewCount="100" Body="&lt;p&gt;I've developed an &lt;a href=&quot;https://play.google.com/store/apps/details?id=com.sl.cells&quot; rel=&quot;nofollow noreferrer&quot;&gt;android math game called MathLab&lt;/a&gt; that has started to gain popularity. It's a simple fill-in-the-grid puzzle with variable cell sizes and locking cells. My daughter's math teacher got very excited about it, so she asked me to add age/grade information to each difficulty setting so that she can recommend it to parents. I'd be very thankful to anyone who can provide me with that kind of rating info.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, does this seem suitable for the Google Play for Education program?&lt;/p&gt;&#xA;" OwnerUserId="9032" LastEditorUserId="5220" LastEditDate="2017-11-27T16:05:47.457" LastActivityDate="2017-11-27T16:05:47.457" Title="What's the age recomendation for my app?" Tags="|math-puzzle|games|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="13213" PostTypeId="2" ParentId="13198" CreationDate="2017-11-27T13:13:23.253" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://www.illustrativemathematics.org/content-standards&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.illustrativemathematics.org/content-standards&lt;/a&gt;&lt;br&gt;&#xA;Select the grade level and the content standards you are looking for. It lists several open ended questions and lists which standards and mathematical practices it aligns with. It also gives out the solutions with some commentary on how to discuss different parts of the problem, different points of views, common mistakes, etc.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2017-11-27T13:13:23.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13214" PostTypeId="2" ParentId="7394" CreationDate="2017-11-27T18:31:05.663" Score="1" Body="&lt;p&gt;Check out the &lt;a href=&quot;http://www.bcamt.ca/weeklymathtasks/&quot; rel=&quot;nofollow noreferrer&quot;&gt;British Columbia Association of Math Teachers Rich Task List&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2017-11-27T18:31:05.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13215" PostTypeId="2" ParentId="951" CreationDate="2017-11-27T18:59:55.217" Score="11" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Is the teacher's confirmation bias a serious problem?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes and this is not specific to math class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example of Confirmation Bias in Assessment&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On Monday, the teacher thinks: &quot;If my students have mastered addition and subtraction of of fractions, then they will be able to calculate, say $\frac{2}{5}+\frac{7}{10}=\_\_\_\_\_\_\_$ and $\frac{1}{5}-\frac{2}{11}=\_\_\_\_\_\_\_$ using an efficient algorithm. So, if they can get the answer and use the LCDs of 10 and 55, they've mastered adding fractions.&quot; Students then do very well on the assessment scoring an average of 90%. Everyone is pleased and the teacher feels the notion of &quot;My class has mastered the addition and subtraction of fractions&quot; to be &lt;strong&gt;confirmed&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On Tuesday, the teacher asks the class to compute $7-\frac{1}{2}=\_\_\_\_\_\_$.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Student A says it can't be done because you can't take pizzas away from numbers. &lt;/li&gt;&#xA;&lt;li&gt;Student B says the answer is $3\frac{1}{2}$ because you're taking half of $7$ away from $7$. &lt;/li&gt;&#xA;&lt;li&gt;Student C says the question is impossible because you have to have common denominators to add or subtract fractions and $7$ doesn't even have a denominator. &lt;/li&gt;&#xA;&lt;li&gt;Student D says it depends what you're taking half of, so there are many correct answers.&lt;/li&gt;&#xA;&lt;li&gt;Most of the class stares silently with an obviously confused look on their faces.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What happened?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Explanation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A teacher will frequently ask themselves: &quot;If my students have mastered concept X, then they will be able to do task Y.&quot; They then &lt;em&gt;infer&lt;/em&gt; that &quot;If my students can do task Y, then they have mastered concept X.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mastering the addition and subtraction of fractions implies you'll do well on a test of $\frac{2}{5}+\frac{7}{10}=\_\_\_\_\_\_\_$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But doing well on a test of $\frac{2}{5}+\frac{7}{10}=\_\_\_\_\_\_\_$ emphatically does &lt;em&gt;not&lt;/em&gt; imply mastery of addition and subtraction of fractions. It does not imply that students have, at any point, incorporated any aspect of fractional number knowledge with prior whole number knowledge. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many people are highly confused by the notion that fractions can be greater than one whole, yet they passed all their fraction tests in school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Valid reasoning and intuitive reasoning are miles apart... even if they feel like they should overlap.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How could I make some particular teacher aware of it (as a parent, or as a student)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Teachers and students need to ask themselves:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Can students excel on my assessments and have no idea what's going on? One way to test this is to create math assessments that involve &lt;em&gt;no&lt;/em&gt; calculations, only estimates, drawings, discussion, comparing and contrasting, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Can students excel on my assessments and then forget everything they've learned the next day? Will they be able to build on it in a year with no further review?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have they fully integrated all the new content (fractional arithmetic) with old content (whole number arithmetic)?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When doing math, can I explain why every step is true? Why it's useful? Or am I just following instructions mindlessly? i.e. Am I making sense of this or just answer-getting?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Ultimately, we are all biased and we all intuitively seek evidence of what we want, expect, and believe. To counter that bias, we must deliberately seek evidence that students have &lt;em&gt;not&lt;/em&gt; mastered what they appear to know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For those ready for formal study of the counterintuitive nature of valid inferences from assessments, they should study &lt;a href=&quot;https://en.wikipedia.org/wiki/Sensitivity_and_specificity&quot; rel=&quot;noreferrer&quot;&gt;sensitivity and specificity and related topics&lt;/a&gt;. This is mainstream in medicine. Education desperately needs it to become mainstream as well.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How could we make the teachers aware of it (not as frequently, but even colleagues from university fall for this)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've linked to this many times but here it is again... &lt;a href=&quot;https://www.carnegiefoundation.org/wp-content/uploads/2013/05/stigler_dev-math.pdf&quot; rel=&quot;noreferrer&quot;&gt;Many high school graduates struggle with $0.03=0.030$&lt;/a&gt;. They've been passing assessments like the one above for years and thus consider their notions of being &quot;good enough&quot; at math as &lt;strong&gt;decisively confirmed&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who should have disconfirmed such notions? When? And how? Probably not with traditional calculation-based assessments...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In first-year calculus, many students &quot;confirm&quot; their mastery by getting a high grade. But if you ask them to explain how we know the fundamental theorem of calculus is true, you get a blank stare. You'd get a similar blank stare if you asked them why the power rule works. You might even get a blank stare if you asked them what you can do with calculus that you can't do with prior math.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2017-12-16T18:30:46.777" LastActivityDate="2017-12-16T18:30:46.777" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13216" PostTypeId="1" AcceptedAnswerId="13219" CreationDate="2017-11-27T19:55:43.200" Score="-5" ViewCount="557" Body="&lt;p&gt;I just read an interesting book: &quot;Classical and nonclassical logics&quot;, Princeton Univ. Press (2005) by Eric Schechter. On p. 208 he writes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Also for simplicity of notation, we have chosen an alphabet that is only countably infinite. That alphabet is adequate for most applications of logic, but some logicians prefer to allow uncountable alphabets as well. (Imagine an even larger infinite computer keyboard, with real numbers written on the key caps!)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My question: How could the manufacturer write a real number except the few which have their own names like $2$ or $\sqrt3$ or $1/4$ or $\pi$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student would ask me, I really could not answer since the real numbers written on the key caps have to be individuals, i.e., it is not sufficient to distinguish each one from some &quot;given&quot; real numbers but each one must differ from &lt;em&gt;all&lt;/em&gt; other real numbers. How can that be possible by finite strings of symbols on the key caps? I assume consent that infinite strings of symbols don't carry any information that could be with sufficient completeness conveyed to the typist.&lt;/p&gt;&#xA;" OwnerUserId="8557" LastActivityDate="2017-11-28T22:47:22.830" Title="How to write an individual real number?" Tags="|logic|" AnswerCount="4" CommentCount="11" ClosedDate="2017-11-28T18:06:50.113" ContentLicense="CC BY-SA 3.0" />
  <row Id="13217" PostTypeId="2" ParentId="13216" CreationDate="2017-11-27T20:50:26.367" Score="1" Body="&lt;p&gt;Well if you are imagining a keyboard with an infinite amount of keys, you would want to imagine an infinite amount of space to write the number on each key. So the unnamed real numbers would contain infinite space to write. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But how would the typist distinguish them?&#xA;Well how does the mathematician distinguish them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By their position. We don't read off the digits but locate them in the place where they belong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So each key and its neighbor would have the ability to have a key between them. And likewise for those keys.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we are imagining; here this is what this means. If each key were a distinct entity holding position to another key with nothing between them, it would be countable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in our imagining we have to come up with a mechanism to make the numbers infinitely dense. So using the definition of this in real numbers we will place a key inbetween two other keys as needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not sure if this is the point of the analogy or the imagining exercise, but as analogy it breaks down; if we don't add room for that infinite density of real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And so in the imagining we create a contraption that looks like a keyboard from afar but if we stare at the spaces between the keys is different. Much like real numbers look like integers, until we look more closely and find we have to squeeze more and more in the spaces between each named thing.&lt;/p&gt;&#xA;" OwnerUserId="8952" LastActivityDate="2017-11-27T20:50:26.367" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13218" PostTypeId="2" ParentId="13216" CreationDate="2017-11-27T21:15:58.363" Score="5" Body="&lt;p&gt;Schechter is talking about infinitary logic (see the &lt;a href=&quot;https://en.wikipedia.org/wiki/Infinitary_logic&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; and &lt;a href=&quot;https://plato.stanford.edu/entries/logic-infinitary/&quot; rel=&quot;noreferrer&quot;&gt;Stanford Encyclopedia of Philosophy&lt;/a&gt; entries), although Schechter's &quot;infinite computer keyboard&quot; comment is meant to be picturesque and not mathematically precise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also Keisler's 1970 paper &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0003484370800055&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Logic with the quantifier &quot;there exist uncountably many&quot;&lt;/em&gt;&lt;/a&gt; and David Marker's Fall 2007 &lt;a href=&quot;http://homepages.math.uic.edu/~marker/inf.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;A Primer on Infinitary Logic&lt;/em&gt;&lt;/a&gt; and Dickmann's 1975 book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0720422817&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Large Infinitary Languages&lt;/strong&gt;&lt;/a&gt; (book reviewed &lt;a href=&quot;https://projecteuclid.org/euclid.bams/1183538661&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;). Incidentally, Dickmann's book deals with &lt;a href=&quot;https://en.wikipedia.org/wiki/Large_cardinal&quot; rel=&quot;noreferrer&quot;&gt;large cardinal&lt;/a&gt; infinitary languages, in case you were wondering just how far down the uncountable rabbit hole logicians have gone with this stuff. (Answer: Further than you can probably imagine.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the google search &lt;a href=&quot;https://www.google.com/search?q=infinitary+languages&quot; rel=&quot;noreferrer&quot;&gt;infinitary languages&lt;/a&gt; brings up a lot of relevant hits.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2017-11-27T21:15:58.363" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13219" PostTypeId="2" ParentId="13216" CreationDate="2017-11-27T22:07:24.227" Score="3" Body="&lt;p&gt;So, here are two assumptions that I think are built into your question. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The manufacturer is writing on the keys with only a finite, or perhaps countable alphabet of symbols. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The manufacturer can only fit a finite number (maybe unboundedly large) of symbols on each key.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If assumption one fails, then you could just give each real number a symbol, and be done with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If assumption two fails, you could name each real number by its perhaps infinitely long decimal expansion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If both assumptions hold, then the answer is &quot;They can't&quot;. This is because, as Cantor proved a while back, there are uncountably many real numbers. But there are only countably many finite strings of symbols over a given finite (or countable) alphabet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The manufacturer's best bet, if they want to do this systematically, might be to write on each key a computer program that prints out the decimal representation of the real number that key is supposed to produce. This would get you the &lt;a href=&quot;https://en.wikipedia.org/wiki/Computable_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;computable numbers&lt;/a&gt;, a larger set of reals than just algebraic numbers plus common transcendentals like π. But, there remain some reals with names that can't be represented this way, like Chaitin's Ω, so this still isn't optimal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, &lt;em&gt;there is no optimal solution&lt;/em&gt;. Suppose there were. Then we could add a key which said &quot;the real whose first digit is the first digit of the alphabetically least real from the old keyboard, plus one (or minus one, if the first digit is 9), whose second digit is the same as the second digit of the alphabetically second-least real from the old keyboard, and so on&quot;, and in that way add a key which denotes a real not denoted by any key on the first keyboard (it can't be denoted by the first key, because it differs in the first digit from that real, it can't be denoted by the second, because it differs in the second digit...). So, since we've added a key, the original keyboard was not optimal. Contradiction, and our assumption must have been false.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm slightly fudging things for simplicity in the previous paragraph by pretending that decimal representations are unique. But you could run the same argument but on a scheme where they &lt;em&gt;are&lt;/em&gt; unique by using e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Continued_fraction&quot; rel=&quot;nofollow noreferrer&quot;&gt;continued fractions&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5625" LastEditorUserId="5625" LastEditDate="2017-11-28T22:47:22.830" LastActivityDate="2017-11-28T22:47:22.830" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13221" PostTypeId="2" ParentId="13216" CreationDate="2017-11-28T00:25:44.583" Score="5" Body="&lt;p&gt;I assume you mention the real numbers because you are considering the case where the set of real numbers is taken to be your alphabet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, the &quot;keyboard manufacturer's&quot; job is simple; every key is labeled with a &lt;em&gt;single&lt;/em&gt; symbol.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point you're missing is that, if you take you take the real numbers to be your alphabet, they are &lt;em&gt;the very symbols you use to write with&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;It may help to understand &lt;em&gt;why&lt;/em&gt; one might want to use an uncountable alphabet. One might get the impression that strings over an alphabet are meant to model human writing, and while that's &lt;em&gt;one&lt;/em&gt; application, it is by no means the &lt;em&gt;only&lt;/em&gt; one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A different example would be to axiomatically define the notion of a &quot;real vector space&quot;. A typical method would be to define a language consisting of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The usual logical symbols&lt;/li&gt;&#xA;&lt;li&gt;The constant symbol $\vec{0}$ (the &quot;zero vector&quot;)&lt;/li&gt;&#xA;&lt;li&gt;The binary function symbol $+$ (the operation of &quot;vector addition&quot;)&lt;/li&gt;&#xA;&lt;li&gt;For every real number $r$, the unary function symbol &quot;$r \cdot {}$&quot; (the operation of &quot;scalar multiplication by $r$&quot;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and formally write out the axiom schema for the real vector spaces in this language:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The axiom $a+(b+c) = (a+b)+c$&lt;/li&gt;&#xA;&lt;li&gt;For every real numbers $r,s,t$ with $r+s=t$, include the axiom $t \cdot a = r \cdot a + s \cdot a$&lt;/li&gt;&#xA;&lt;li&gt;... and so forth&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One's first tendencies might be to try and include in the axioms a minimalist axiomatization of the real numbers, but that's missing the point &amp;mdash; the real numbers here are a &lt;em&gt;given&lt;/em&gt;, and the point of this theory is to define what a &quot;real vector space&quot; is, not what a &quot;real number&quot; is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, there are important mathematical reasons for having the theory presented in this particular form; e.g. it shows that &quot;real vector space&quot; is an example of a thing called a &lt;em&gt;universal algebra&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2017-11-28T18:16:17.483" LastActivityDate="2017-11-28T18:16:17.483" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="13222" PostTypeId="2" ParentId="951" CreationDate="2017-11-29T00:25:15.590" Score="0" Body="&lt;p&gt;Here is a second example of devastatingly terrible and common confirmation bias. [It's probably more important than my first post...]&lt;/p&gt;&#xA;&lt;h2&gt;Example: Flight&lt;/h2&gt;&#xA;&lt;p&gt;Imagine it's 1902, a year before the Wright Brothers' first success. A skeptic might say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If it's impossible for a heavier-than-air machine to fly, then the Wright Brothers will fail to make that machine over and over and over again. And so will everyone else who tries. Since everyone has, in fact, failed over and over and over again, it must be impossible to make a heavier-than-air machine.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is obviously stupid &lt;em&gt;because we already know the conclusion is wrong empirically&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But it's also stupid &lt;em&gt;because the argument itself is invalid&lt;/em&gt; (i.e. the premises of prior failure do not imply the conclusion of impossibility). But it could feel right to the person who says it because intuitive interpretations of evidence frequently suck and we all suffer from confirmation bias.&lt;/p&gt;&#xA;&lt;p&gt;The idea that someone could find a better way to make these machines that no one has thought of yet? Doesn't cross the skeptic's mind.&lt;/p&gt;&#xA;&lt;h2&gt;Example: Education&lt;/h2&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If it's impossible for student Y to learn X, then everything I try to get them to learn it will fail over and over and over again. All of their prior teachers will have failed too. Since those prior teachers have failed and I've failed over and over and over again, it must be impossible for student Y to learn X.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This feels right.&lt;/p&gt;&#xA;&lt;p&gt;But it's stupid.&lt;/p&gt;&#xA;&lt;p&gt;It's an &lt;em&gt;objectively invalid argument&lt;/em&gt;. If the student doesn't get X, it emphatically does &lt;em&gt;not&lt;/em&gt; imply the student's inherently incapable of learning X. Perhaps there is a better teacher somewhere in the world that could teach the student X. Maybe the teacher is overestimating the variety of pedagogy he or she has tried so far. Maybe all the teachers come from a school that has traumatized the student and they've only tried teaching the student in that school. Maybe the student simply needs more time. Maybe the student is totally capable of learning X but needs to learn prior knowledge U, V, and W first.&lt;/p&gt;&#xA;&lt;p&gt;It's very easy to &amp;quot;confirm&amp;quot; incapability as a teacher, but it's actually impossible to prove. How do you know that &lt;em&gt;nothing&lt;/em&gt; could ever help the student learn? That there is &lt;em&gt;no&lt;/em&gt; teacher anywhere that could do a better job? How do you prove that &lt;em&gt;no&lt;/em&gt; learning environment could make a big difference?&lt;/p&gt;&#xA;&lt;p&gt;Now... How much human potential do we throw away because of confirmation bias? Probably some tragic amount...&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-11-29T00:25:15.590" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13223" PostTypeId="2" ParentId="11436" CreationDate="2017-11-29T15:08:31.570" Score="2" Body="&lt;p&gt;I &lt;em&gt;highly&lt;/em&gt; suggest you obtain a (used) copy of the CME Project &lt;strong&gt;Algebra 2&lt;/strong&gt; (orange) textbook and look at its section on complex numbers. They should be thought of as very geometric objects - they apply to computer graphics, optimization, and all sorts of other areas of mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applied Mathematics (which is what my graduate degree is in) is not just about &quot;the real world.&quot; It is about using the power of mathematics to model whatever scenarios come to mind. You don't have to model &quot;real world&quot; things -- you can do a linear programming situation based on goblins and unicorns, for example. You can model social interactions at a party. You can use complex numbers to model/design PacMan. Whatever you want. &#xA;One person's application is another person's dull bore. That's why it's not about the -specific- applications you are teaching. You should be teaching a mindset, an approach, a worldview. I find all of the examples listed using physics as reasons for complex numbers completely exhausting and couldn't care less; however, I don't need to care about those -- I need to understand why they exist and why physicists must employ complex numbers to model the situations they encounter. THAT is what we're teaching, not just specifically complex numbers to find zeroes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm still with most people that I don't agree with your assumptions nor what it means to be a &quot;laymen.&quot; &lt;/p&gt;&#xA;" OwnerUserId="1249" LastActivityDate="2017-11-29T15:08:31.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13224" PostTypeId="2" ParentId="11436" CreationDate="2017-11-29T20:45:29.107" Score="7" Body="&lt;p&gt;Just another question from a math guy showing his ignorance of math as a service course.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;2nd order diffyQ with constant coeffiecients (most important diffyQ for applications) has complex roots in the characteristic equation.  [IOW very standard part of standard ODE course; OFTEN part of even a second semester calculus course during the diffyQ survey...was when I took AP, part of Granville, etc.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Complex numbers are normal part of AC circuits.  REQUIRED part of survey course for ALL engineers (not just EEs).  Sometimes part of calculus based intro physics as well.  Also for physicists needed for E&amp;amp;M junior year course.  (Note:  all these uses occur even if complex analysis is NEVER studied).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For college chemistry solution of the quadratic equation is normal part of many equilibrium, rate, and stoichiometry problems.  While real roots are only ones usually needed, having familiarity with complex versus real makes it easier to grasp for students when we exclude some solutions of the quadratic.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Solutions of harmonic oscillator (i.e. the second order constant coefficients diffyQ above) is normal part of required courses for junior year quantum (all physics majors), P-Chem (all chem majors), weapons systems controls (all military academy students), and probably a bunch of required courses for engineers (fluids, etc.) which I don't know in detail.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9064" LastEditorUserId="9064" LastEditDate="2017-11-29T20:51:38.153" LastActivityDate="2017-11-29T20:51:38.153" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13225" PostTypeId="2" ParentId="11448" CreationDate="2017-11-29T22:42:24.497" Score="3" Body="&lt;p&gt;Quadratic formula is key part of most chemical equilibrium and rate problems (big part of college frosh chem).  Many kinematics problems (intro college physics) also involve it (acceleration is a second derivative of position).  It is also needed to solve the characteristic equation of the second order diffyQ with constant coefficients (most important ODE for applications, comes up in EE, quantum mechanics, controls/systems, etc.)  Also helpful wrt some trig derivations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now actually answering if it is &quot;too much&quot; attention involves knowing what percentage, x, of the algebra 2 course is spent on it, and if x is &quot;too much&quot;.  But I sense you aren't strong on applications, so sharing the above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course classes will differ. I remember it not being covered that heavily when I took it, but I was in an advanced class and most people just got it quickly.  Lot of other material to spend time on since we did alg2/trig in a year and you have to hit exponents, logs, etc. polynomial division, etc. also.  Perhaps weaker classes spend more time on the quadratic and less time on other &quot;college algebra&quot; topics. &lt;/p&gt;&#xA;" OwnerUserId="9020" LastActivityDate="2017-11-29T22:42:24.497" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13226" PostTypeId="2" ParentId="10521" CreationDate="2017-11-29T22:53:32.870" Score="2" Body="&lt;p&gt;&quot;While calculus has applications, I'm sure other branches of math have applications just as well.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Educating yourself on the applications might answer your question.  And make your teaching more informed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been a chemist, engineer, military officer, and finance professional and dabbled in applied psychology and economics.  Calculus is needed for that stuff.  Number theory is not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math courses are arranged (in high school and first couple years of college) to service the mass of students who ARE NOT math majors.  Learning about their courses and what math is involved in their HW problems, derivations, etc. would help to inform you on why certain classes are part of a standard curriculum and others are not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. Richard Feynman (pretty good math guy, won the Putnam with no study as last minute team member at MIT) started out as a math major.  He ditched it because 'the only use of the things you learn are to become a math professor and teach more math majors'.  He went to EE instead and then to physics.&lt;/p&gt;&#xA;" OwnerUserId="9020" LastActivityDate="2017-11-29T22:53:32.870" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13227" PostTypeId="2" ParentId="11448" CreationDate="2017-11-30T01:07:14.300" Score="6" Body="&lt;p&gt;Here are some brief [and admittedly inchoate] thoughts, as I'm not quite ready to invest in writing a formal essay about your very important question with respect to what an Algebra 2 course &lt;em&gt;should&lt;/em&gt; contain. (I am also purposefully avoiding a discussion over whether an Algebra 2 course &lt;em&gt;should&lt;/em&gt; be required for students to complete secondary school, or even whether it should be a college/university requirement.) That said/unsaid, my thinking right now is roughly as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that the study of linear equations is important, and that linear algebra is an area that can be pursued with theoretical interest and/or for its practical applications. So, I think that an Algebra 1 and Geometry experience that really builds an understanding of lines and systems of linear functions is important. For Algebra 2, I like the idea of segueing from lines to piecewise linear functions to a specific example of a piecewise linear function: the absolute value function. I think that a lot of analytical thinking and reasoning can be built into a rigorous approach to absolute value functions, equations, and inequalities, and that this can lead nicely to quadratics, which have analogies with absolute values (concavity; x-intercepts; vertex; y-intercept; defined by three parameters/constants) and can pave the way towards the more general study of polynomials. (I also think that  insufficient attention to absolute values is unfair for students who will pursue Calculus courses that use delta-epsilon proofs, or go on to major in mathematics and take courses on, for example, Real Analysis.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Through solving quadratics, one can get some wonderful experience with rigorous mathematical thinking (including the broaching of complex numbers). I wrote of a problem set that I used last year in my Algebra 2 class in my answer to &lt;a href=&quot;https://matheducators.stackexchange.com/a/12978&quot;&gt;&lt;strong&gt;MESE 12977&lt;/strong&gt;&lt;/a&gt;, which should give a rough impression of the sort of questions that I think an Algebra 2 student can be prepared to answer. In my own experience, it is possible to pursue quadratic functions, equations, and inequalities in a way that sets students up to launch into polynomials, or complex numbers, or rational functions, or limits (&quot;end behavior&quot;), or a variety of other topics. However, I feel that there is a complete overfocus on &lt;em&gt;factoring&lt;/em&gt; quadratics (and, for what it is worth, would prefer to see a greater number theoretical focus on factoring whole numbers at other points in a school curriculum).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As an addendum (just for fun) here is a blitz treatment of quadratics that I think can be informative: Consider the two primary ways of writing quadratic functions.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;standard form: $f(x) = ax^2 + bx + c$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;vertex form: $f(x) = a(x-h)^2 + k$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;We can move from one representation to the other by expanding the latter and matching coefficients. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, in my classes, we note that $x=h$ corresponds to the line of symmetry for a parabola, on which the $c$ from standard form has no effect; thus, in particular, we set $c=0$ to find roots at $x=0$ and $x=-b/a$, so that a line of symmetry between them occurs at $h = -b/2a$. Since $k$ is the $y$-coordinate for the vertex that has $x$-coordinate $h$, we note $k = f(h)$. This is enough information to move from standard form to vertex form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To move in the other direction, we rearrange to find $b = -2ah$ and observe that $c = f(0)$ is simply the $y$-intercept of $f$. Bearing this in mind, we derive the quadratic formula rapidly from vertex form:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a(x-h)^2 + k = 0 \implies x = h\pm\sqrt{\frac{-k}{a}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which yields the quadratic formula in terms of $a$, $b$, and $c$ by using the above substitutions of $h = -b/2a$ and $k = f(h)$, although the latter is a bit tedious to compute. All of this is to say, I find the idea of bazillions of quadratic factoring exercises to be a suboptimal use of time, since one can really get into some deep mathematical thinking by experimenting with graphs (see also the end of my answer to &lt;a href=&quot;https://matheducators.stackexchange.com/a/12288&quot;&gt;&lt;strong&gt;MESE 12286&lt;/strong&gt;&lt;/a&gt;) and then diving into algebra in ways such as the derivation above.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2017-11-30T01:07:14.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13228" PostTypeId="2" ParentId="11448" CreationDate="2017-11-30T10:24:23.807" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;In a certain sense quadratic equations are the simplest that have TWO solutions. Students might believe equations have only one solution.&lt;/li&gt;&#xA;&lt;li&gt;The very important No-Zero Divisors Theorem is first seen when solving quadratic equations and so...&lt;/li&gt;&#xA;&lt;li&gt;Quadratic Equations provide a motivation for factoring.&lt;/li&gt;&#xA;&lt;li&gt;In a certain sense the simplest function with a local max/min is a quadratic function.&lt;/li&gt;&#xA;&lt;li&gt;They are required later. Students SHOULD be able to factor in their sleep. Learning later takes time away.&lt;/li&gt;&#xA;&lt;li&gt;In a certain sense they are the simplest non-affine functions for students to graph.&lt;/li&gt;&#xA;&lt;li&gt;They exhibit axial symettry about their extremum.&lt;/li&gt;&#xA;&lt;li&gt;Their discriminant introduces students to the problem of a square root of a negative number.&lt;/li&gt;&#xA;&lt;li&gt;Word problems involving the area of a rectangle, usually lead to quadratic equations.&lt;/li&gt;&#xA;&lt;li&gt;They test students on B|E|MD|AS&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="584" LastActivityDate="2017-11-30T10:24:23.807" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13229" PostTypeId="1" AcceptedAnswerId="13231" CreationDate="2017-11-30T15:53:54.887" Score="2" ViewCount="443" Body="&lt;p&gt;My Geometry class is doing triangle congruency proofs these days. In general, we find three pairs of congruent parts (sides or angles) in two triangles; we show that these congruencies reveal that the triangles are either congruent or similar; and we conclude that further parts are congruent (in the case of congruent triangles) or proportional (in the case of similar triangles).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the column of justifications, therefore, the last two justifications are usually SSS (or SAS or ASA or AAS or AAA) and then either CPCTC (Corresponding Parts of Congruent Triangles are Congruent) or Definition of Similarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’ve done pretty much the same thing each time I’ve taught Geometry, but this year we’re using a text by Pearson in which this topic appears in Chapter 4. After doing this a zillion times, I have begun to wonder something about this nomenclature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is that where triangles are CONGRUENT, we DO refer to CPCTC, and we DO NOT refer to a definition; but where triangles are SIMILAR, we DO NOT refer to CSSTP, but we DO refer to a definition?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/t9iOC.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/t9iOC.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8258" LastEditorUserId="8258" LastEditDate="2017-11-30T16:09:55.963" LastActivityDate="2017-11-30T19:47:27.683" Title="Is the Nomenclature of Triangle Congruency Proofs Consistent?" Tags="|proofs|geometry|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13230" PostTypeId="2" ParentId="13229" CreationDate="2017-11-30T17:54:59.383" Score="4" Body="&lt;p&gt;In a mathematical proof, any logically valid justification is permissible. In particular, you can cite any definition, assumption, or previously proved theorem at any time. Also, strictly speaking, you never &lt;em&gt;have&lt;/em&gt; to appeal to a theorem, since a theorem is just a consequence of the definitions; the point of packaging reasoning in theorems is to avoid duplication of effort, organize concepts, and make proofs shorter and easier to read.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to directly answer your question, for both similarity and congruence, you can either prove something straight from the definition or cite a relevant theorem. If your textbook seems to prefer one over the other, that's probably just an arbitrary choice by the author(s), not anything logically necessary. If your textbook makes it seem like only one way is &lt;em&gt;acceptable&lt;/em&gt;, that's a serious flaw in the textbook, since it'd be reinforcing the harmful misconception that there's only one way to prove a given theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the specific context of congruence, by the way, I don't know why you wouldn't just teach the following more general theorem: &lt;em&gt;Rigid motions of the plane preserve measures of angles and lengths of line segments.&lt;/em&gt; The proof is exactly the same—assuming you've defined &quot;congruent shapes&quot; in the usual way, that there's a rigid motion of the plane taking one to the other—but the statement doesn't unnecessarily require the line segments and angles to occur as part of triangles. Likewise for similarity: &lt;em&gt;Uniform scaling of the plane preserves measures of angles and ratios of lengths of line segments.&lt;/em&gt; I can understand restricting to special cases when the general case is more complicated, but here, there's no extra complexity in general, and I'd be concerned that students would think it's something special about triangles rather than completely general properties of congruence and similarity.&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2017-11-30T17:54:59.383" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13231" PostTypeId="2" ParentId="13229" CreationDate="2017-11-30T19:47:27.683" Score="7" Body="&lt;p&gt;Put me in the camp of people who would prefer to say &quot;Definition of Congruent Triangles&quot; rather than &quot;CPCTC&quot;.  However, even if we agree that CPCTC is too well-established in the vernacular to get rid of, there are still good reasons not to use CSSTP: it is too specific and does not cover all of the different uses for which triangle similarity proofs are used.  Consider the following triangle similarity arguments:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Use two pairs of congruent angles to prove triangles similar; then conclude that the sides are pairwise-proportional.  (&quot;Corresponding sides of similar triangles are proportional&quot;, or CSSTP.)&lt;/li&gt;&#xA;&lt;li&gt;Use two pairs of pairwise-proportional sides and one pair of congruent included angles to prove triangles similar; then conclude that the third pair of sides is in the same ratio as the others.  (&quot;Corresponding sides of similar triangles are proportional&quot;, or CSSTP.)&lt;/li&gt;&#xA;&lt;li&gt;Use two pairs of pairwise-proportional sides and one pair of congruent included angles to prove triangles similar; then conclude that the other two pairs of angles are pairwise-congruent.  (&quot;Corresponding angles of similar triangles are congruent&quot;, or CASTC.)&lt;/li&gt;&#xA;&lt;li&gt;Use three pairs of pairwise-proportional sides to prove triangles similar; then conclude that all three pairs of angles are pairwise-congruent.  (&quot;Corresponding angles of similar triangles are congruent&quot;, or CASTC.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&quot;Definition of Similarity&quot; encompasses both CSSTP and CASTC.  The distinction is not necessary to make when discussing triangle congruence, because &quot;congruent angles&quot; and &quot;congruent sides&quot; are described using the same word (even though they are technically different relations).&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2017-11-30T19:47:27.683" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13232" PostTypeId="2" ParentId="11448" CreationDate="2017-11-30T23:24:36.563" Score="4" Body="&lt;p&gt;IMHO, the problem is not overemphasis on quadratics, but rather that many students completely miss the point of algebra. They think algebra means &quot;Solve for $x$&quot;. Really, they should be thinking about algebra as the generalization of arithmetic, that algebra allows us to reason with a class or classes of numbers rather than known constants. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main problem with a lot of algebra classes isn't the overemphasis on quadratics but the underemphasis on algebraic generality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a sample of how weak &lt;a href=&quot;https://www.carnegiefoundation.org/wp-content/uploads/2013/05/stigler_dev-math.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;algebraic reasoning is at a community &lt;em&gt;college&lt;/em&gt;&lt;/a&gt;, i.e. a place where all students have graduated high school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/sSWYr.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sSWYr.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see, overemphasis on quadratics may have harmed them, but far more conceptual &quot;damage&quot; was done by missing the most basic properties of algebra. These students have not yet learned the generality of grade &lt;em&gt;1&lt;/em&gt; arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;THEY HAVEN'T GENERALIZED GRADE &gt;&gt;ONE&amp;lt;&amp;lt; ARITHMETIC!!&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But they were good enough at following fixed math &lt;em&gt;procedures&lt;/em&gt; to graduate high school, though. I bet they applied the quadratic formula with fidelity...&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2017-12-01T18:41:04.190" LastActivityDate="2017-12-01T18:41:04.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13233" PostTypeId="1" CreationDate="2017-12-01T02:25:04.640" Score="8" ViewCount="1076" Body="&lt;p&gt;I wish to teach myself how to do math calculations–like $99\times 58$ or $2048+1296$ or $506+998$–inside my head.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know there already exist two methods–&lt;a href=&quot;https://en.wikipedia.org/wiki/Abacus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abacus&lt;/a&gt; and &lt;a href=&quot;http://mathlearners.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vedic Maths&lt;/a&gt;. I don't know of more, nor I have had experience with any.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has someone had experience with one or more of such techniques? Which one works best for a person with background of being a high school passout?&lt;/p&gt;&#xA;" OwnerUserId="9088" LastEditorUserId="8571" LastEditDate="2017-12-05T01:41:46.853" LastActivityDate="2023-06-19T12:08:40.283" Title="What's the best technique to do math calculations in my head?" Tags="|arithmetic|" AnswerCount="7" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13234" PostTypeId="2" ParentId="13233" CreationDate="2017-12-01T03:25:50.473" Score="6" Body="&lt;p&gt;One technique that I like is &quot;chunking&quot;, breaking a calculation into easier pieces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$99 \times  58 = (100 - 1) \times 58 = 5800 - 58=5742$$&lt;/p&gt;&#xA;" OwnerUserId="8559" LastEditorUserId="8559" LastEditDate="2017-12-01T05:05:20.300" LastActivityDate="2017-12-01T05:05:20.300" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13235" PostTypeId="2" ParentId="13233" CreationDate="2017-12-01T07:53:33.777" Score="6" Body="&lt;p&gt;For the calculation at the hand, the technique by Boba Fret is probably the way to go. In general, there are different techniques.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A source that I like is the book by &lt;a href=&quot;https://www.math.hmc.edu/~benjamin/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arthur Benjamin&lt;/a&gt; and Mike Shermer called &quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0307338401&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;The secrets of mental math&lt;/a&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For completeness: Next to the Abacus and Vedic Maths you mention, one could add the &lt;a href=&quot;https://en.wikipedia.org/wiki/Trachtenberg_system&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trachtenberg system&lt;/a&gt;; some resources provided at the end. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since this is a question in MESE, I'd dare to add: When teaching mental math to students, don't miss the opportunity to explain the techniques based on the understanding about place value and (pre)-algebra. Unfortunately I don't see this done in many resources. For example, the resources on Vedic Math usually just state their &quot;sutras&quot; as rules to be used, like the &quot;vertical and crosswise&quot;, or &quot;all from nine, last from ten&quot;. While this may be great for memorization, it should also be discussed why they work.&#xA;For example, the distributive rule helps, together with an understanding of place value system, to explain a technique to square numbers ending in 5's, which is simple for 2-digit numbers: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The how: To square, for example &lt;span class=&quot;math-container&quot;&gt;$65$&lt;/span&gt;, increase the tenth-digit by one and multiply, then append 25: &lt;span class=&quot;math-container&quot;&gt;$6 \times 7 = 42$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$65^2 = 4225$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The why: Write the number in place values, and use the distributive rule to rearrange: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*} 65^2 &amp;amp;= (60 + 5)(60+5) \\ &#xA;&amp;amp;= 60 \times 60 + 5 \times 60 + 5 \times 60 + 5 \times 5\\&#xA;&amp;amp;= 60 \times(60 + 5 + 5) + 25\\&#xA;&amp;amp;= 6 \times (6+1) \times 100 + 25&#xA;\end{align*}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the last line can be read to explain the technique to square numbers ending in 5.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This way, techniques for mental calculation can be based on conceptual understanding, as opposed to &quot;mystic rules&quot; from ancient India.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt; as requested, some resources:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Resources for the Trachtenberg System:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The classic is a book by Ann Cutler and Rudolph McShane: &quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0285629166&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Trachtenberg Speed System of Basic Mathematics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.speed-math.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;speed-math.com&lt;/a&gt;&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Resources for Vedic Math: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.vedicmaths.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;vedicmaths.org&lt;/a&gt; provides plenty of Material, also some &lt;a href=&quot;http://www.vedicmaths.org/courses/introductory-course-free-version&quot; rel=&quot;nofollow noreferrer&quot;&gt;free courses&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9004" LastEditorUserId="238" LastEditDate="2019-12-16T13:24:32.217" LastActivityDate="2019-12-16T13:24:32.217" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13236" PostTypeId="1" CreationDate="2017-12-01T10:07:50.930" Score="3" ViewCount="405" Body="&lt;p&gt;I am thinking about the links between SUVAT equations (constant acceleration), and equations for motion when higher-order measurements are constant (for example, when jerk is constant, or snap is constant). See &lt;a href=&quot;https://physics.info/kinematics-calculus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; for an excellent discussion of how to derive the jerk equations using calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am attempting to teach this topic first without Calculus, and then to revisit it later once the students are aware of differentiation and integration. They understand informally that calculating the gradient of velocity function gives you acceleration, and finding the area underneath undoes this process, but have no specific knowledge about how to differentiate or integrate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, we can derive the SUVAT equation below by calculating the area of the relevant trapezium on the velocity-time graph (see below)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$s = ut + \frac{1}{2}at^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any way that derives the third jerk equation, without using formal calculus?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$s = ut + \frac{1}{2}a_u t^2 + \frac{1}{6}jt^3$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this equation, $s$ is displacement, $u$ is initial velocity, $t$ is time, $a$ is initial acceleration, and $j$ is constant jerk.&lt;/p&gt;&#xA;" OwnerUserId="8797" LastEditorUserId="5220" LastEditDate="2017-12-01T15:46:20.897" LastActivityDate="2017-12-01T15:46:20.897" Title="Deriving Jerk Equations without using Calculus" Tags="|calculus|precalculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13237" PostTypeId="2" ParentId="13236" CreationDate="2017-12-01T12:05:40.353" Score="4" Body="&lt;p&gt;I've never seen the jerk discussed in any non-calculus physics course. An easy way to do it without using geometry is the following. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, the variation of $t^2$ is $\Delta t^2=(t+h)^2-t^2\approx 2th$ and $\Delta t^3\approx 3t^2h$ for small $h$. Second, the velocity is to the displacement what the acceleration is to the velocity and what the jerk is to the acceleration. Third, in your last equation $\frac{\Delta s}{\Delta t}=u+at+\frac12 jt^2$ which is identical to your first equation. &lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2017-12-01T12:05:40.353" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13238" PostTypeId="2" ParentId="13233" CreationDate="2017-12-02T09:05:32.227" Score="3" Body="&lt;p&gt;Adding to the proposal of BobaFret: I would first calculate the big part of the number, then the small part. For example: $$2048+1296=?$$&#xA;First do the big part: $$2000+1200=3200$$&#xA;Then the small part: $$48+96=44+100=144$$&#xA;So in total it is $$2048+1296=3200+144=3344$$&#xA;where in the small part calculation I transferred a 4 from  48 to 96 to get to a nice number. This transferring is quite helpful as well, actually:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$506+998=504+1000=1504$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for doing the big part first is that it is usually the easier part (3200 is easier to memorize than 144) and also it gives you a rough estimate of the result which makes it easier to memorize as well (3200 feels like it is of the right order of magnitude for the result of 2048+1296 while 144 is just some random number appearing in the calculation).&lt;/p&gt;&#xA;" OwnerUserId="3542" LastActivityDate="2017-12-02T09:05:32.227" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13239" PostTypeId="2" ParentId="13233" CreationDate="2017-12-02T10:39:12.787" Score="4" Body="&lt;p&gt;When I do math in my head, I usually combine a few approaches:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Always keep track of units.&lt;/li&gt;&#xA;&lt;li&gt;Try to re-use math I have memorized (such as the multiplication tables, powers of 2, powers of 3, common square roots, common trig ratios, &lt;em&gt;et cetera&lt;/em&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Use &quot;carrying&quot; to split each number into a round number and an &quot;error&quot; term.  Often, the first operations will be on the number(s) that are closest to round numbers.&lt;/li&gt;&#xA;&lt;li&gt;An &quot;error term&quot; can either be a number (such as 2 or -4 or 0.216), or a percentage of the round number.  For example, 0.3048 is 1.6% more than 0.3.  In the example below, I convert &quot;0.3048&quot; to &quot;0.3 (+ 1.6%)&quot;.  A more standard way of writing this would be &quot;0.3 * (1 + 1.6%)&quot;.&lt;/li&gt;&#xA;&lt;li&gt;Use memorized values to handle round number inputs to square roots or trig functions, and interpolation to adjust for the &quot;error&quot; terms.&lt;/li&gt;&#xA;&lt;li&gt;Chunk four-digit numbers into pairs of digits.&lt;/li&gt;&#xA;&lt;li&gt;Use my fingers to store values as Roman numerals.  (This allows storing a pair of digits on my hands:  Right fingers = Is; right thumb = V; left fingers = Xs; left thumb = L.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;99 * 58 = (100 - 1) * 58 = 5800 - 58 = 5700 + 100 - 58 = 5742&lt;/li&gt;&#xA;&lt;li&gt;2048 + 1296 = 2048 + 1300 - 4 = 2044 + 1300 = 3344&lt;/li&gt;&#xA;&lt;li&gt;506 + 998 = 506 + 1000 - 2 = 504 + 1000 = 1504&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Since the original poster asked (in a comment) about square roots:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \sqrt{\frac{2 \cdot 0.3048 \text{ meters}}{(9.80665 \frac{\text{meters}}{\text{second}^2})}}  $$&#xA;$$= \sqrt{\frac{2 \cdot 0.3048 \text{ second}^2}{9.80665}} $$&#xA;$$= 1 \text{ second} \sqrt{\frac{2 \cdot 0.3048}{9.80665}} $$&#xA;$$\approx 1 \text{ second} \sqrt{\frac{2 \cdot 0.3 (+ 1.6\%)}{10 (- 2\%)}} $$&#xA;$$\approx 1 \text{ second} \sqrt{\frac{0.6}{10} (+1.6\% +2\%)}$$&#xA;$$= 1 \text{ second} \sqrt{0.06 (+1.6\% +2\%)}$$&#xA;$$= 0.1 \text{ second} \sqrt{6 (+3.6\%)}$$&#xA;$$= 0.1 \text{ second} \sqrt{6.216}$$&#xA;$$= 0.01 \text{ second} \sqrt{621.6}$$&#xA;$$\approx 0.01 \text{ second}\cdot (\sqrt{625}-\frac{3.4}{625-576})$$&#xA;$$\approx 0.01 \text{ second}\cdot (25-\frac{3.4}{50})$$&#xA;$$= 0.01 \text{ second}\cdot (25-\frac{6.8}{100})$$&#xA;$$\approx 0.01 \text{ second}\cdot (25-0.07)$$&#xA;$$= 0.01 \text{ second}\cdot 24.93$$&#xA;$$= 0.2493 \text{ seconds}.  $$&#xA;(Actual value is 0.249322 seconds, at 45&amp;deg; latitude on Earth.)&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2017-12-05T06:49:56.493" LastActivityDate="2017-12-05T06:49:56.493" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13240" PostTypeId="1" AcceptedAnswerId="13243" CreationDate="2017-12-01T05:07:01.840" Score="21" ViewCount="729" Body="&lt;p&gt;I'm currently a research mathematician, getting involved in more and more outreach activities. One of these involves delivering lessons for elementary school math teachers (K-6). The purpose is two-fold: to motivate them about the coolness of math, and to give them more fodder for their own classroom lessons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have taught a lot of different types of audiences before, but not this one. I am wondering what would be safe to assume as far as the mathematical background of someone who teaches math in a US public school, grades kindergarten through 6th grade. What would be the minimum highest level math course a person in that profession would have to have taken? What would they be likely to remember, mathematically, from when they were in college?&lt;/p&gt;&#xA;" OwnerUserId="9091" OwnerDisplayName="j0equ1nn" LastEditorUserId="39" LastEditDate="2017-12-02T18:08:00.610" LastActivityDate="2023-01-15T07:09:42.730" Title="What is the required mathematical background of a US elementary school math teacher?" Tags="|teachers|usa|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="13241" PostTypeId="2" ParentId="13240" CreationDate="2017-12-01T05:20:22.110" Score="17" Body="&lt;p&gt;Based on my experience in the US, most school districts teachers teach all subjects including mathematics in the elementary schools.  There typically aren't &quot;mathematics teachers&quot; at this level although there are some school districts that have designated mathematics teachers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should probably not assume anything beyond basic arithmetic- they might have taken a traditional college algebra course, but many have them have forgotten whatever they might have learned in such a course.  Many of them will have taken specialized courses in mathematics for teachers and in math education, but the actual mathematical content of these courses varies tremendously. Furthermore, many of these teachers have been traumatized by their own experience in mathematics courses and are afraid to engage with the subject- getting them to participate in any kind of problem-solving activity is extremely challenging.   &lt;/p&gt;&#xA;" OwnerUserId="5841" OwnerDisplayName="Brian Borchers" LastEditorUserId="9091" LastEditDate="2017-12-03T23:29:22.790" LastActivityDate="2017-12-03T23:29:22.790" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13242" PostTypeId="2" ParentId="10289" CreationDate="2017-12-02T16:35:01.763" Score="1" Body="&lt;p&gt;Kinematics is easiest conceptual example of derivative and something everyone has visceral physical experience with (acceleration, speed, position).  Optimization (particularly business or manufacturing type problems) is something everyone can understand as a practical use of calculus.  And is a KEY concept, use of calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something I learned about giving academic science talks is to always mention impact of science knowledge on humanity, usually some commercial impact (could cure cancer, make computer chips cheaper, etc.)  It is motivation to listen to some of the boring science talk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disagree with answer above pushing calculus of variations (for weak students in intro calc course?)  This is perfect example of the opposite of &quot;getting it&quot; in terms of the question.  I would consider to avoid trig functions also (for semester long course).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eschew integration by parts and partial fractions and u substitution.  Cut related rates, diffyQ survey, and series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Give the kids the understanding of what you will cover/won't cover at beginning of course and in individual lectures.  I find mathematicians tend to trust people to go along with them on a winding road of explication but without knowing where it is heading.  This is poor pedagogy.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if you put it in concept &quot;what calculus is about&quot; it is actually useful knowledge even if the kids can't do partial fractions.  Even knowing what the typical courses are and what other majors require them.  I've never had some courses but I know what they are about and where they are needed.  This is a practical man's way of thinking about a topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would mix in a little &quot;history of calculus&quot; (the Newton, Leibniz rivalry; how/when the topic started to be used by Prussian artillerists and Voltaire [I am making that last one up, but you get point]).  Humanists want to know what something is about and how it influenced people even if they don't know it in detail.  (E.g. famous books you have not read but still know about, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. Jasper had some important questions (e.g. one semester, no credit for moving on or prereq to physics, would think so?) that you did not answer.&lt;/p&gt;&#xA;" OwnerUserId="9064" LastActivityDate="2017-12-02T16:35:01.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13243" PostTypeId="2" ParentId="13240" CreationDate="2017-12-02T17:23:35.110" Score="16" Body="&lt;p&gt;Supporting the answer by Brian Borchers, and also the comment by aparante001: K-6 elementary teachers in the U.S. will know effectively zero math, or even a negative amount of math in many cases. It's well established that for over a century in the U.S., the very weakest college students, and in particular the ones with the highest level of math anxiety, self-select for K-6 education (often presuming that no math is required for that):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.madmath.com/2016/02/hembree-on-math-anxiety.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.madmath.com/2016/02/hembree-on-math-anxiety.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As one case-study described in &lt;a href=&quot;https://www.psychologytoday.com/intl/blog/freedom-learn/201003/when-less-is-more-the-case-teaching-less-math-in-school&quot; rel=&quot;nofollow noreferrer&quot;&gt;Psychology Today&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In an article published in 2005, Patricia Clark Kenschaft, a professor&#xA;of mathematics at Montclair State University, described her&#xA;experiences of going into elementary schools and talking with teachers&#xA;about math. &lt;strong&gt;In one visit to a K-6 elementary school in New Jersey,&#xA;she discovered that not a single teacher, out of the 50 that she met&#xA;with, knew how to find the area of a rectangle.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Taking CUNY as an example, K-6 majors are not required to take intermediate or college-level algebra (as of next year, they will not even need to demonstrate proficiency in elementary, i.e., 8th-grade, algebra). Consider a case study at CUNY Brooklyn college -- requirements for major and the two math courses in question are linked below.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.brooklyn.cuny.edu/web/academics/schools/education/undergraduate/childhood/majors_details.php?major=253&amp;amp;div=U&amp;amp;dept_code=30&amp;amp;dept_id=527&amp;amp;mode=data&quot; rel=&quot;nofollow noreferrer&quot;&gt;Requirements for K-6 education major&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.brooklyn.cuny.edu/courses/ShowCourse.do?redirect=/acad/course_info.jsp&amp;amp;dsc=MATH.&amp;amp;crs_num=1401&amp;amp;div=U&quot; rel=&quot;nofollow noreferrer&quot;&gt;MATH 1401 Elementary Mathematics from an Advanced Standpoint&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.brooklyn.cuny.edu/courses/ShowCourse.do?redirect=/acad/course_info.jsp&amp;amp;dsc=MATH.&amp;amp;crs_num=1406&amp;amp;div=U&quot; rel=&quot;nofollow noreferrer&quot;&gt;MATH 1406 Mathematics in Education&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The two required courses are described as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;*&lt;em&gt;MATH &lt;em&gt;1401 Elementary Mathematics from an Advanced Standpoint&lt;/em&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Mathematics content needed for teaching major strands in the early&#xA;childhood and elementary school mathematics curriculum: Problem&#xA;Solving; Sets; Number Systems; Geometry; Probability and Statistics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;*&lt;em&gt;MATH &lt;em&gt;1406 Mathematics in Education&lt;/em&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Concepts and principles of mathematics underlying the elementary school curriculum. Taught in&#xA;coordination with Education 3206 [44]. a. Early childhood education&#xA;section: emphasis on topics relevant to teaching children from&#xA;prekindergarten to grade 3. b. Elementary, bilingual, and special&#xA;education section: emphasis on topics relevant to teaching children&#xA;from kindergarten to grade 6.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What is somewhat obscured is that the MATH 1401 course may or may not be taken under CUNY's &amp;quot;Pathways&amp;quot; policy. That is: If a student starts at one of the community colleges, they may take an entirely different liberal-arts math appreciation course which satisfies that same requirement.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-01-15T07:09:42.730" LastActivityDate="2023-01-15T07:09:42.730" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13244" PostTypeId="2" ParentId="13240" CreationDate="2017-12-02T19:13:47.370" Score="5" Body="&lt;p&gt;Academic, applied, and research math is the world where you come from. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Edit: Much of] Elementary &quot;math&quot; is a whole different world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to know why quotation marks are literally necessary in the prior sentence, check out &lt;a href=&quot;https://math.berkeley.edu/~wu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the writings of Hung Hsi Wu&lt;/a&gt;. Then, prepare for something radically worse than that when you ask students to explain their thinking aloud.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then prepare for something radically worse than &lt;em&gt;that&lt;/em&gt; when you ask teachers to explain &lt;em&gt;their&lt;/em&gt; thinking aloud.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Despite the variance of teachers' math preparation, the mean, median, and mode are &lt;em&gt;terrifying&lt;/em&gt;. For the record, I am not blaming teachers; my mother was a teacher in a high-needs elementary school for much of her career. I consider them, and many others, &lt;em&gt;victims&lt;/em&gt; of an incredibly stupid &quot;math&quot; education system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teacher certification is similar in Canada and the US so this &lt;a href=&quot;https://www.thestar.com/yourtoronto/education/2016/05/13/for-many-teachers-math-just-doesnt-add-up.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Toronto Star article&lt;/a&gt; should give you a taste of what you may be in for. Short version is that many teachers-in-training :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Cry when asked about their own experience with elementary school math.&lt;/li&gt;&#xA;&lt;li&gt;Believe that a quotient of 1 with a remainder of 3 means the same thing as a 1.3.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So, with this in mind, what do you plan to do with those elementary school teachers?&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2017-12-05T08:49:03.910" LastActivityDate="2017-12-05T08:49:03.910" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13245" PostTypeId="2" ParentId="13240" CreationDate="2017-12-03T09:14:50.480" Score="13" Body="&lt;p&gt;Despite what the negative answers, there are elementary teachers who are good at math. I count myself among them.  The problem is that there are a broad range of abilities and attitudes among those who will you be talking to.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My suggestion is that you look for cool things about math that don't require a knowledge of higher math.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example might be &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10163/imbuing-a-six-year-old-with-a-sense-of-mathematical-wonder/10182#10182&quot;&gt;my and other's answers to Imbuing a six year old with a sense of mathematical wonder&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is the many cool patterns in Pascal's Triangle or some properties of Fibonacci numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also direct you to books by Marilyn Burns on mathematical education as well as many NCTM publications.  These will give you a starting point for approaching teachers who don't have a math background.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2017-12-03T09:14:50.480" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13246" PostTypeId="2" ParentId="13240" CreationDate="2017-12-03T17:39:23.727" Score="8" Body="&lt;p&gt;Here are the requirements for Teacher Certification in Massachusetts:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.teachercertificationdegrees.com/certification/massachusetts/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Massachusetts Teaching and Certification Resource&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://teach.com/states/become-a-teacher-in-massachusetts/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Become a Teacher in Massachusetts&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;An image pasted from the latter link:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VAcPK.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VAcPK.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The examination in Massachusetts is called the MTEL. To prepare for it sufficiently (especially in teaching &quot;elementary&quot; as opposed to &quot;early elementary&quot;) I can speak to the two course sequence at a particular teacher-education program, that of the Boston University School of Education (where I previously worked as a postdoc and have taught both of these courses):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wHWH9.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wHWH9.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xBpUK.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xBpUK.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even in Massachusetts, there are a range of alternative paths towards teaching (depending, for example, on the needs of a particular school or school district). Looking to the United States as a whole, the number of possible paths to teaching elementary school is finite, but barely so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may have better luck homing in on a particular school, and investigating how the teachers presently at that institution were prepared (as well as the sort of ongoing professional development required/attended). Your final two questions, if answered literally, will do little to provide insight into the planning/implementation of outreach activities.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What would be the minimum highest level math course a person in that profession would have to have taken? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Across all possible paths: None.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What would they be likely to remember, mathematically, from when they were in college?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This will vary tremendously, as some elementary school teachers were math majors, some were math education majors, some were education majors with a focus on math, some were education majors who did not focus on math (and may not have completed any relevant coursework), and some may have majored in something unrelated to math or education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest that you interact directly with a potential site for outreach and the teachers who are working there, and go in with an open mind and a sincere belief that the teachers want what is best for students - even if, in practice, this leads to actions that may appear as irrational from the perspective of someone outside of the institution of k12 education and its associated structures/requirements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more information about how some elementary school teachers are honing their mathematical craft, search Twitter for the &lt;strong&gt;MTBOS&lt;/strong&gt; (Math Twitter Blog-O-Sphere) or &lt;strong&gt;iteachmath&lt;/strong&gt; or &lt;strong&gt;tmwyk&lt;/strong&gt; (Talking Math With Your Kids) hashtags, or check out &lt;a href=&quot;https://twitter.com/TracyZager&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Tracy Zager&lt;/strong&gt;&lt;/a&gt;'s book, &quot;Becoming the Math Teacher You Wish You'd Had: Ideas and Strategies from Vibrant Classrooms&quot; (&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/157110996X&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Amazon&lt;/strong&gt;&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=cRm7DQAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Google Books&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2017-12-03T17:39:23.727" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13247" PostTypeId="1" CreationDate="2017-12-03T21:01:48.957" Score="14" ViewCount="368" Body="&lt;p&gt;I'm teaching a college course in discrete math, where students are writing their first proofs.  One of the phrases we use a lot in mathematical English is &lt;em&gt;such that.&lt;/em&gt;  I notice that students don't seem to know when it should be used and sprinkle it randomly in their proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's an example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For a and b to be relatively prime, gcd(a,b) = 1 = ax + by s.t. x, y are integers. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My first comment is that two equivalent conditions are being asserted at the same time, which seems sloppy, but the major issue is with &lt;em&gt;such that $x$ and $y$ are integers.&lt;/em&gt;  I wrote, &quot;'such that' is a phrase that modifies a verb of being, e.g., 'exist'.  Putting it after 'equals' doesn't really sound right.  You could say, '$1 = ax+by$ &lt;em&gt;where&lt;/em&gt; $x$ and $y$ are integers,' or '$1 = ax + by$ &lt;em&gt;for&lt;/em&gt; integers $x$ and $y$.'&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think many of my students know a lot about English grammar, let alone the roughly half of the class who are ELLs.  So other than modeling proper writing, what be useful ways to explain this?&lt;/p&gt;&#xA;" OwnerUserId="446" LastActivityDate="2017-12-09T16:02:53.683" Title="Teaching the proper syntax of “such that”" Tags="|writing|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13248" PostTypeId="1" CreationDate="2017-12-03T22:47:39.820" Score="2" ViewCount="116" Body="&lt;p&gt;What are the best methods for learning proofs? I'm tasked with learning two dozen proofs about the properties of continuous functions and real numbers in a week well enough to be able to present them. Like Taylor's Formula, the fundamental theorem of calculus, and Bolzano-Weierstrass theorem. What is the best way to do it? My main problem is that there are no exercises to practice on.&lt;/p&gt;&#xA;" OwnerUserId="7837" LastEditorUserId="8571" LastEditDate="2017-12-04T18:59:27.963" LastActivityDate="2017-12-04T18:59:27.963" Title="Strategies for learning proofs" Tags="|proofs|self-learning|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13249" PostTypeId="2" ParentId="12286" CreationDate="2017-12-04T16:14:48.047" Score="1" Body="&lt;p&gt;In my experience, this most commonly pops up in terms of averages of a uniform distribution. What is the average face value of a standard die (6-sided)? Of a polyhedral 4, 8, 12, or 20-sided die? What is the average of the numerical cards in a standard deck (2-10)? While the answer may seem trivial, many students find it totally opaque on first encounter. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2017-12-04T16:14:48.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13251" PostTypeId="1" CreationDate="2017-12-04T19:57:59.117" Score="4" ViewCount="231" Body="&lt;p&gt;Does anyone know of any cases of students being adversely affected by cranks or trolls in online math forums (e.g. sci.math)? Are they even a minor problem? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dan&lt;/p&gt;&#xA;" OwnerUserId="6103" LastActivityDate="2017-12-05T17:13:09.607" Title="Are students being adversely affected by cranks or trolls in online math forums?" Tags="|secondary-education|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="13252" PostTypeId="1" CreationDate="2017-12-04T20:10:39.180" Score="6" ViewCount="390" Body="&lt;p&gt;Please forgive me if this is naive. I am only an &lt;em&gt;aspiring&lt;/em&gt; educator after all. Why do we still teach estimation when there are easily accessible/teachable exact techniques? Why do statisticians teach $p$-value estimation instead of exact computation. Why do calculus professors teach Riemann sums alongside integration? &lt;/p&gt;&#xA;" OwnerUserId="9153" LastEditorUserId="5220" LastEditDate="2017-12-05T16:18:18.363" LastActivityDate="2017-12-20T14:35:47.537" Title="Why do we teach estimation in Statistics and Mathematics?" Tags="|calculus|statistics|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13254" PostTypeId="2" ParentId="13252" CreationDate="2017-12-05T01:18:16.720" Score="15" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Number Sense:&lt;/strong&gt; At the elementary level, estimation helps students to develop number sense.  As Daniel R. Collins notes, order of magnitude estimates can be quite important.  Anecdotally, I once rented an apartment.  I moved out of the apartment about halfway through the month, and the landlord offered to prorate the rent.  I said &amp;quot;Great!&amp;quot;  However, when she operated her calculator, she determined that the prorated rent was &lt;em&gt;more than&lt;/em&gt; the monthly rent.  When I asked how this could be right, she conceded that it was confusing, but the number on the calculator don't lie.  Better number sense could have saved us all some time.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Emphasizing Uncertainty:&lt;/strong&gt;  At least one of the &amp;quot;exact&amp;quot; techniques that you mention is rarely exact.  To wit, most of the probability functions used in statistics don't have nice antiderivatives in terms of elementary functions.  This means that any attempt to integrate them is almost certainly going to have to rely on numerical integration.  Getting estimates for &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-values (for example, via the &amp;quot;empirical rule&amp;quot;) once again provides a check against technology.  It &lt;em&gt;also&lt;/em&gt; can be used to emphasize that we lack certainty.&lt;/p&gt;&#xA;&lt;p&gt;For example, if I have a table for a &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; distribution that only gives entries for multiples of ten degrees of freedom, then I am going to have difficulty coming up with an &amp;quot;exact&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-value for a hypothesis test involving 47 degrees of freedom.  I have to make a choice:  should I use 40 or 50 degrees of freedom?  In this context, I want use the number that gives me &lt;em&gt;less&lt;/em&gt; certainty, so I work with 40 d.f.  This is still an issue with computers.  If a computer gives me a &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-value that rounds to my level of significance, what should I do?  Understanding that there are estimates throughout the process can help me to make that decision.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Emphasizing Theory:&lt;/strong&gt;  I believe that we should be teaching mathematics not just in service to other fields, but also in service to the field of mathematics itself.  You may not have that many future pure math Ph.D.s in your class, but you still need to make sure that you are preparing those students as well as all of the others.  If integration or linear algebra is taught as a bunch of computational techniques, the math majors are going to be in a very poor position when they start taking &amp;quot;real&amp;quot; math classes.&lt;/p&gt;&#xA;&lt;p&gt;To use your example, Riemann sums are emphatically &lt;em&gt;not&lt;/em&gt; an estimation technique, but the actual tool that is necessary to build the Riemann integral.  If you can't work with Riemann sums, how are you ever going to work through a proof of the fundamental theorem of calculus?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-12-05T01:18:16.720" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13255" PostTypeId="2" ParentId="13247" CreationDate="2017-12-05T05:23:41.203" Score="0" Body="&lt;p&gt;It is admittedly confusing for ELLs. But nothing to worry too much as students would pick it up automatically with enough exposure and practice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Meanwhile you can try showing them a few examples to differentiate between &quot;for&quot;, &quot;such that&quot;, &quot;there exists&quot; etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More often than not, &lt;em&gt;such that&lt;/em&gt; will be used as a &lt;em&gt;condition&lt;/em&gt; on a &lt;em&gt;proposition&lt;/em&gt;. So, crudely speaking (for beginners), a mathematical statement follows the structure:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Proposition) &lt;em&gt;s.t&lt;/em&gt; (condition) &lt;em&gt;for&lt;/em&gt; (domain)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;There exists&quot; would fall under the proposition itself.&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2017-12-05T05:23:41.203" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13256" PostTypeId="1" CreationDate="2017-12-05T06:24:17.320" Score="21" ViewCount="806" Body="&lt;p&gt;I have read and heard from some other instructors that they attempt to encourage students who find math hard by saying &quot;math is hard for me too, in fact it's hard for everyone!&quot;  I have tried this a few times, but it always rings false to me, because it seems clear to me that math is &lt;em&gt;not&lt;/em&gt; hard for me and never was, at least not in the same way that it's hard for the students in question.  And it seems to me that the students must realize that too and conclude that I'm lying to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a way that I can understand and explain &quot;math is hard for everyone&quot; in such a way that both the students and I can agree and believe that it applies to me as well as to them?  Or, failing that, is there something better I can say?&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2017-12-06T03:20:19.577" Title="Should I tell my students that math is hard for me?" Tags="|student-motivation|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13257" PostTypeId="2" ParentId="13252" CreationDate="2017-12-05T07:14:45.893" Score="5" Body="&lt;p&gt;Most real-world problems are only approximately described by nice mathematical formulas.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on the situation, it can be either silly or dangerous to assume that an &quot;exact&quot; result of a mathematical calculation exactly describes a real-world situation that caused you to perform the calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, to what extent does an &quot;exact&quot; result of a p-value calculation predict what would happen the next time you randomly tried to use the baseline hypothesis to get a value?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When performing real-world integrations, often an approximate answer (which resembles a Riemann-sum) is both more practical to determine, and more accurate, than an &quot;exact&quot; calculation based on a function that more-or-less describes the real-world situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, teaching the approximate method teaches an easy-to-remember rule-of-thumb, which has several advantages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Some people find it easier to remember.&lt;/li&gt;&#xA;&lt;li&gt;It might help the student know when the exact method is likely to be useful.&lt;/li&gt;&#xA;&lt;li&gt;It might help the student remember the logic of how the exact method was derived.&lt;/li&gt;&#xA;&lt;li&gt;It provides a way to sanity check results from the exact method.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2017-12-05T07:35:51.067" LastActivityDate="2017-12-05T07:35:51.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13258" PostTypeId="2" ParentId="10788" CreationDate="2017-12-05T07:18:58.290" Score="2" Body="&lt;p&gt;Probably the most important advantage that the definition &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \exp(x) = \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;has over the other forms you consider is that the only prior notion of exponentiation required is the case of positive integer exponents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The usual track for rigorous development from foundations is to define $\exp$ and $\log$ &lt;em&gt;before&lt;/em&gt; general exponentiation, which is then defined by the identity&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ x^y = \exp(y \log(x)). $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To use the other forms, you have to already have general exponentiation as a given.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this is all moot. It's about doing exercises in how to give formal definitions and working their properties out from scratch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I really get the impression that you are in a setting where you really should be approaching this as an issue of calculating limits &amp;mdash; e.g. proving the theorems that all of these related limits are given by exponential functions.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Incidentally, for the purposes of doing calculations with complex numbers, you don't want the limit above anyways; a more convenient theorem to use is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \lim_{y \to \infty} \left( 1 + \frac{x}{y} \right)^y = \exp(x)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that the limit variable need not be restricted to integers. (I assume you take the principal value of the exponentiation)&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2017-12-05T07:36:18.123" LastActivityDate="2017-12-05T07:36:18.123" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13259" PostTypeId="2" ParentId="13256" CreationDate="2017-12-05T08:39:04.353" Score="5" Body="&lt;p&gt;First, I'd want to figure out &lt;em&gt;why&lt;/em&gt; math is so much easier for you than it is for them. Often it's simply background knowledge and work habits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, you can model the type of struggle you expect from your students. What do you expect them to do when stuck? Do you actually make mistakes in front of them and then catch them with a convincing check? Do you ever solve a problem very inefficiently and then look back and ask &quot;Hey, is there a better way we could have done this? How early on in the process could we possibly have known this? Or that?&quot; Do you explain not only how you knew to do X do, but how you knew to do X at this exact point in time, and how you knew W, Y, and Z were not relevant?&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2017-12-05T08:39:04.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13260" PostTypeId="2" ParentId="13256" CreationDate="2017-12-05T12:07:25.450" Score="3" Body="&lt;p&gt;Two quick thoughts that I started writing as a comment, but maybe they're better as an answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, I'd avoid saying anything to the effect that math is hard for you, as doing this will likely come across as contrived, and even if true in some other sense such as having difficulties with research, it's likely to sound hollow (like a multimillionaire complaining about tax shelter difficulties to someone homeless).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, if there's something you're really bad at (and I mean truly pathetic), then you might want to share those difficulties one-on-one (or in a very small group meeting). However, you'll want to avoid something that completely defeated you to the extent that there's no way to spin it as something you managed to persevere through. In my case, it was Foreign languages. I was extremely lucky to eventually land in (after 3 previous programs) a graduate program that required only one Foreign language and lucky again that I managed to (quite surprisingly, and not just to myself) pass the language exam (in French, the only language my high school offered and which in high school I barely scraped by in). I also had great difficulties as an undergraduate, when my problems caused two delays in graduation, the second being when I wound up transferring to a less selective university (which at that time had no university-wide Foreign language requirement) after several of my attempts at the (supposedly entrance level) language competency test were so far from acceptable and after a couple of failures of the first semester French class, that I didn't see any other alternative. (Those who know of my interest in math history, and wonder how I manage when I’m so bad with Foreign languages, may be interested in my posts in the March 2005 sci.math thread &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=3687763&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the cardinal of the Cantor set&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2017-12-05T12:07:25.450" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13261" PostTypeId="2" ParentId="13256" CreationDate="2017-12-05T15:32:18.397" Score="8" Body="&lt;p&gt;In the framework of tutoring at a community college (Remedial Math to Differential Equations) and teaching College Algebra and Physics at a Liberal Arts College. Most of my interactions are with people who think they can not do math or where not made to do math. My philosophy is that it is time spent working on problems, the use of that time that make the difference. I also distinguish between arithmetic and mathematics. Arithmetic you can buy a $15 calculator that can do for you, mathematics is the language and patterns we use arithmetic in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tell my students, math is hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a t-shirt that says &quot;math is hard, so is life, get over it.&quot; I wear that occasionally in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also tell the story of why I think math is &quot;easier&quot; for me.&#xA;I had older aunts and uncles. My Uncle John taught me the times tables, but not memorizing he said if you had 3+3 that is adding 3 twice, this is the same as 2 times three. In kindergarten class we were counting and I recited this like a parrot. The teacher noticed and complimented me on my math. I liked the pat on the back, so every time math came up I tried hard. I put more time into it. And I put my time into when I was young. I exercised the muscle often, in my free time. In doing that the progression in mathematics was &quot;easier&quot; for me. But if we add up the time and effort spent studying and learning it wasn't &quot;easier&quot; I just put that time in at a different place in my life and liked it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I explain to my students that we are learning algebra. This is a language. A language that takes the idea of counting and removes it from the object we were counting and looks at the pattern of quantity. When you learn a language you can think in a language. The goal of algebra is that you have the language of quantity as part of your skill set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you think about how you learn language, you spend the first 7 to 8 years of your life learning your native language in bits and pieces. Imitating the words, phrases and sentences around you. Learning how to attach them to objects, actions, things you want, things that don't exist yet. By the time you are in elementary school you begin to study the language itself and how you can use it to express more.  This is hard, it takes lots of work and practice, but we do it early and it surrounds us, so we do it. With this skill we can now rap, sing, speak, debate, question, write poetry, write technical manuals, fiction, document life and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math is similar, but we already have language and we are adding to the tool set, so we borrow some of the knowledge from spoken language and start learning this new language.  For some parts of it we are like an infant repeating sounds and trying to attach them to objects. This frustrates our older selves, that do not expect to be infant like in learning.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the hard part:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics talks about the pattern of quantity, but quantity does not directly exist. The number three is a noun, but I can not give you a three, I can only give you a three of something. This abstraction we have to form a place for in our thoughts.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the hard part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The big secret is everything is hard at some point in learning it, it is new and different and there is a risk. Recognizing that, analyzing the risk, understanding our feelings and managing them is a skill that work for any part of life that is hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And with any skill practice, making mistakes, correcting mistakes and learning from the mistakes is part of making that particular skill less hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a friend that uses a video on juggling. Juggling is hard. But you can watch a video on juggling. Now you know how to juggle. You still need to practice the skill, make the mistakes and correct them. The practice for some will be easier than others, but this is about you not them. Practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=kCt1bmSASCI&quot; rel=&quot;noreferrer&quot;&gt;Juggling 3 Balls&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also give the students these documents with the syllabus:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ctl.ua.edu/CTLStudyAids/StudySkillsFlyers/Miscellaneous/mathmyths.htm&quot; rel=&quot;noreferrer&quot;&gt;Math Myths&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I talk honestly about the parts of mathematics I found easy. I talk about the parts I found hard, and how I got past the harder parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And sometimes I found things other people thought were really hard easy - Tensor Calc, why because I have developed more geometrical thinking by the time I took it and it made sense in the framework of what I was studying.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compared to statistics, I had a philosophical bias against the idea of chance. To get past that I had to do the entry level problem more often, I had to develop a sense of random when I was studying it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think being human and explaining mathematics as a human invention we use to model and make predictions is a better reference point also then the philosophy that mathematics is the truth. By making it a human invention, looking at the rise from 1 a few, many, too much, to counting, to the creation of sets as human answers to the questions they had at the time a great framing tool in the discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end I tell them it comes down to time spent doing mathematics, a willingness to make mistake and not expecting a perfect answer immediately and correcting mistakes when made. If this mindset is added to the habits, the hard part can be dealt with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what you want to do is look back at when you spent time learning it. Integrate that time, add it up, that is where it was hard for you. Hard is effort, you just did it at a different time and with a different attitude. The difficulty doesn't change, but the perception of that difficulty does.&lt;/p&gt;&#xA;" OwnerUserId="8952" LastEditorUserId="8952" LastEditDate="2017-12-05T15:57:38.540" LastActivityDate="2017-12-05T15:57:38.540" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13262" PostTypeId="2" ParentId="13251" CreationDate="2017-12-05T17:13:09.607" Score="6" Body="&lt;p&gt;I had a couple but one in particular sticks out: years ago I was teaching a summer program for middle school students, one of the other instructors suggested I show my students the youtube video &quot;Imagining the 10th Dimension.&quot; So I watched the first 5 minutes and it looked like a nice flatland-like explanation of higher dimensional space, so I let my TA show it while I was out of town. It turns out after about 6 minutes of flatland it runs off into a really misguided attempt justify the conformal anomaly (sometimes understood as 10 spacial dimensions) in string theory in terms of &quot;all the possible version of actions in the universe.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a great example of bad mathematics/physics (although the author doesn't really understand the difference). It slips from well established mathematics to incorrect and incoherent speculation without batting an eye, and the narrator has an easy authority that says &quot;just trust me.&quot; My poor TA luckily figured out something was seriously wrong and told the students that they shouldn't take it seriously. We took it as a teaching opportunity and had an hour long lesson about evaluating claims, thinking for yourself, and the difference between speculation and proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had a conversation about it with the person who suggested me the video, but he didn't really seem to understand the problem...&lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2017-12-05T17:13:09.607" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13263" PostTypeId="2" ParentId="13256" CreationDate="2017-12-06T00:07:01.640" Score="3" Body="&lt;p&gt;Do &lt;em&gt;you&lt;/em&gt; think this is a good idea to share with students? That is, &quot;I have read and heard from some other instructors&quot; could be reason enough to consider an idea, but why do &lt;em&gt;you&lt;/em&gt; want students to hear this message that &quot;math is hard&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it's because you want to sympathize with them that learning math is work, then maybe talking about the merits of hard work in general would free you up from having to fib about your own math abilities. You could discuss how you have seen others (students, colleagues) find success learning hard topics in math. Have you seen students struggle and have an &quot;a ha&quot; moment? Maybe ask your current students if they've pushed through a hard topic and how they made it to the other side. Do they find that particular topic &quot;hard&quot; anymore? I suspect they'd say &quot;well, it was until I realized (such and such).&quot; The point is, an individual may not always find math difficult, though math may always remain hard for people on average.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll assume you &lt;em&gt;don't&lt;/em&gt; want to tell them it's alright to just be bad at math, but I know many students already carry that message with them. For some students, if their parents were bad at math, and told them so, then they feel resigned to a similar fate. If you, the instructor, are telling them that math is &quot;hard&quot;, the message is probably more like &quot;math is hard for many people, which is why we have so many different math classes. These classes build on each other, so I routinely see students who come in with low skills and eventually reach the higher levels. It can be a very hard subject, and some students won't make it to the end. But I have been through it, and I want you to succeed&quot;, etc.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2017-12-06T00:07:01.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13264" PostTypeId="1" AcceptedAnswerId="13265" CreationDate="2017-12-06T00:23:07.533" Score="5" ViewCount="318" Body="&lt;p&gt;let's target kids at most of primary school, what are the surprising things that would let them wow ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mobius strip would be one. first, it has only one side; then, as &lt;a href=&quot;https://www.youtube.com/watch?v=SXHHvoaSctc&amp;amp;index=1&amp;amp;list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ&quot; rel=&quot;noreferrer&quot;&gt;Dr Tadashi Tokieda shows&lt;/a&gt;, cutting through the centre line will end up as one circle, not two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Magic_square&quot; rel=&quot;noreferrer&quot;&gt;Magic Square&lt;/a&gt; will be another one, it's amazing that all rows, columns and diagonals have the same sum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pls suggest other ideas that sounds surprising and fun to kids, assuming no more knowledge than primary school level?&lt;/p&gt;&#xA;" OwnerUserId="9059" LastActivityDate="2017-12-12T08:08:18.680" Title="what are some surprising result of math for a kid?" Tags="|big-list|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13265" PostTypeId="2" ParentId="13264" CreationDate="2017-12-06T02:51:07.260" Score="7" Body="&lt;p&gt;I had a lot of luck when running a primary school math club (ages 9-10) with the game of Nim. It's not much harder than tic-tac-toe to play, but there is a lot more math underneath that sounds &quot;hard&quot; but doesn't actually have any prerequisites.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not as quick of a payoff as your ideas in the question, but the outline goes like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Show them how to play the game&lt;/li&gt;&#xA;&lt;li&gt;Learn some reeally basic strategy like &quot;don't leave just one pile&quot;&lt;/li&gt;&#xA;&lt;li&gt;Notice that matching piles basically might as well not be there at all (so &quot;3 + 3 = 0&quot;)&lt;/li&gt;&#xA;&lt;li&gt;Notice that piles of 1, 2, and 3 together might as well not be there at all (so &quot;1 + 2 + 3 = 0&quot;)&lt;/li&gt;&#xA;&lt;li&gt;Develop the whole addition table for Nim with piles up to size 7 by just playing the game&lt;/li&gt;&#xA;&lt;li&gt;If mathematical maturity allows, go into binary so that the addition table's structure can get some sense to it, but this isn't really necessary; just building an addition table that says 3 + 5 = 6 is pretty awesome and fun for a group of kids, especially when it isn't just made up, it has an application (the game)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If they get really into it, you can teach them Kayles next and really there is no end to the things you can do; see Winning Ways and their Mathematical Plays by Conway, Berlekamp, Guy. &lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-12-06T02:51:07.260" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13266" PostTypeId="2" ParentId="13256" CreationDate="2017-12-06T03:20:19.577" Score="10" Body="&lt;p&gt;I've long read - perhaps misread - the following comment from Einstein as lacking &lt;a href=&quot;https://www.youtube.com/watch?v=1Evwgu369Jw&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;empathy&lt;/strong&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/buCuJ.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/buCuJ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(What use is: Oh? You have a headache? I can assure you, mine is worse!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel as if sometimes teachers' phrasings around this shared difficulty in mathematics suffer similarly. The route that I prefer is an attempt to shift students' views about what &quot;doing mathematics&quot; consists of, and, in particular, emphasizing that doing mathematics is about being &lt;strong&gt;stuck&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This idea is well-illustrated in Ben Orlin's &lt;a href=&quot;https://mathwithbaddrawings.com/2017/09/20/the-state-of-being-stuck/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;MathWithBadDrawings&lt;/strong&gt;&lt;/a&gt; (&quot;The State of Being Stuck&quot;) recollection of an interaction with Andrew Wiles. The first time that I, personally, recall thinking of mathematics in this way came from it being framed as such by &lt;a href=&quot;https://en.wikipedia.org/wiki/Neil_R._Grabois&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Neil Grabois&lt;/strong&gt;&lt;/a&gt;. Students sometimes harbor a conception of mathematicians as sitting down to work, proving their theorems for the day, and then moving on; whereas, the way that mathematics actually proceeds is somewhat &lt;a href=&quot;https://mathoverflow.net/a/7207&quot;&gt;&lt;strong&gt;different&lt;/strong&gt;&lt;/a&gt;. Discussing mathematics as an area in which practitioners are, by the very nature of the discipline, often &lt;strong&gt;stuck&lt;/strong&gt; can (I think) help students to reconsider what it means to find the subject &quot;hard.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, I believe that the &lt;em&gt;expectation&lt;/em&gt; of getting stuck, eventually unstuck, and then stuck again can reduce anxiety among students. After all, this is a natural part of the process of doing mathematics, and we can discuss - without needing to stake claims over whose &quot;problems are greater&quot; - the frustration (or other feelings) that comes with getting stuck, feeling as if we are making headway and realizing we are not, or even figuring something out only to move on and immediately get stuck again. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(To hear me literally voice this feeling on mathematics and being stuck, my school has a &lt;a href=&quot;https://www.hewittschool.org/academics/steam-maker-education&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;video&lt;/strong&gt;&lt;/a&gt; up in which I speak about exactly this for a blitzed 10 seconds, between 2:18 and 2:28!)&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2017-12-06T03:20:19.577" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13268" PostTypeId="2" ParentId="13252" CreationDate="2017-12-06T09:25:33.687" Score="7" Body="&lt;p&gt;I'd say&lt;/p&gt;&#xA;&lt;h2&gt;a good approximation is often better that an exact result.&lt;/h2&gt;&#xA;&lt;p&gt;This may sound counterintuitive, but as the phrase is vague anyway, here is a longer explanation what I mean: An &amp;quot;exact result&amp;quot; is often a formula, and often it's a complicated formula and one needs at least a calculator to evaluate is. Since calculators use finite arithmetic there is, most likely, some approximation going on in the background (imagine, that the answer is &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\pi}$&lt;/span&gt;…). An &lt;strong&gt;approximation&lt;/strong&gt; often comes a sequence of approximations and thus, a kind of &lt;strong&gt;algorithm&lt;/strong&gt; alongside with an error bound. For example the series representation &lt;span class=&quot;math-container&quot;&gt;$e = \tfrac{1}{0!} + \tfrac{1}{1!} + \tfrac{1}{2!}+\cdots$&lt;/span&gt; gives you 1) a way to compute an approximation of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, 2) gives an idea that &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is somewhere near &lt;span class=&quot;math-container&quot;&gt;$2.5$&lt;/span&gt;, 3) can be augmented with an error bound to &lt;em&gt;prove&lt;/em&gt; that &lt;span class=&quot;math-container&quot;&gt;$e\approx 2.7181 \pm 0.0001$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Some further aspects:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Statistics&lt;/strong&gt; (in contrast to stochastics) is not about calculating certain probabilities, but &lt;em&gt;inferring&lt;/em&gt; them from experiments. And inference is approximation/estimation.&lt;/li&gt;&#xA;&lt;li&gt;In &lt;strong&gt;analysis&lt;/strong&gt;, a good estimate if often good enough for an exact result. An example is the &amp;quot;epsilon of room&amp;quot;-strategy: Instead of proving &lt;span class=&quot;math-container&quot;&gt;$a=0$&lt;/span&gt;, prove that &lt;span class=&quot;math-container&quot;&gt;$a\leq \epsilon$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$a\geq -\epsilon$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$\epsilon&amp;gt;0$&lt;/span&gt; (by using estimates) and then conclude &lt;span class=&quot;math-container&quot;&gt;$a=0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;In &lt;strong&gt;applications&lt;/strong&gt; (real world applications, I mean), approximation are much more important than exact computations. This is due to many reasons, one is that data is often uncertain, and an exact result based on uncertain data is worse that a good approximation (no one would do an exact data fit for noisy data) and another is, that approximations are often much quicker to get and still good enough for all practical purposes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-12-06T09:25:33.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13270" PostTypeId="2" ParentId="13264" CreationDate="2017-12-06T15:42:34.243" Score="3" Body="&lt;p&gt;Another math circle activity that is fine for kids (warning: I have not tried this specifically on primary school, so the patience required might be too high):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modular arithmetic modulo small numbers is required, but kids are more than happy to accept it. Write on a piece of paper that (only on this piece of paper) 3 = 0. Then talk about it; if 3 = 0, then for sure also 4 = 1. And 11 = 8 = 5 = 2. When they have to do 2 + 2, they get 4, but you say &quot;wait a minute, on this paper, what is a simpler way to write 4?&quot; and you get 2 + 2 = 1 mod 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you &lt;strong&gt;build Pascal's Triangle modulo 3&lt;/strong&gt;, or modulo 4, or whatever modulus they want to try. The algorithm for building the next row is easy, and the patterns you get out of this are pretty awesome. For some visuals in advance, check out &lt;a href=&quot;https://www.maa.org/press/periodicals/loci/joma/patterns-in-pascals-triangle-with-a-twist-first-twist-what-is-it&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/periodicals/loci/joma/patterns-in-pascals-triangle-with-a-twist-first-twist-what-is-it&lt;/a&gt;  . You can get these patterns by just not writing the 0's in Pascal's triangle, or by writing each number in a different color.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For maximum prettiness of the pictures, you will want to use &lt;strong&gt;hexagonal grid paper&lt;/strong&gt; with very light lines so that kids can perfectly space the numbers. At the time of this post, &lt;a href=&quot;https://incompetech.com/graphpaper/hexagonal/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://incompetech.com/graphpaper/hexagonal/&lt;/a&gt; still lets you generate free hexagonal graph paper and choose the weight of the lines. If too much time has passed, hopefully some other kind internet soul has some for you.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2017-12-06T15:42:34.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13271" PostTypeId="2" ParentId="13247" CreationDate="2017-12-06T17:56:17.800" Score="2" Body="&lt;p&gt;I suggest having a class discussion about formal logical phrases and symbols, and how to translate those into natural English. For example, your student wrote this&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For a and b to be relatively prime, gcd(a,b) = 1 = ax + by s.t. x, y are integers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Surely, their intended message was this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\forall a,b\in \mathbb{Z}\large{\textbf{.}}\normalsize (a\text{ and }b\text{ are relatively prime}) \iff  \exists x,y\in\mathbb{Z}\large{\textbf{.}}\normalsize ax+by=1$ &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Show them both and have the class discuss the ways in which the two versions are similar and dissimilar. The point of this discussion is not to clarify that the student's version is &quot;wrong&quot;, nor that the other version is &quot;more correct&quot;. Rather, the point is to have students think &lt;em&gt;at all&lt;/em&gt; about how their mathematical writing comes across to others who read it, and to recognize that there are often many ways of saying the same thing, and some can be clearer or less ambiguous than others. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I teach an intro to proofs course, the first month or so is a combination of content (sets, proof techniques) and an introduction to quantifiers and logical symbols. We practice translating between &quot;wordy sentences&quot; and &quot;mathy/symbolic sentences&quot;, like the two examples above. In the process of translating symbolic sentences into natural English language sentences, we collectively realize what I wanted them to learn all along: &lt;strong&gt;&quot;such that&quot; comes after an existential quantification&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another suggestion, &quot;such that&quot; is perfectly synonymous with the wordier but perhaps more natural phrase &lt;strong&gt;&quot;with the property that&quot;&lt;/strong&gt;. Perhaps you can exclusively use that one with your students for a while until they use it comfortably and correctly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Addendum:&lt;/em&gt; In fact, I have a $\LaTeX$ macro for the &quot;large dot&quot;, and the shorthand is \st for &quot;such that&quot;: &lt;code&gt;\newcommand{\st}{{\text{\huge {.}}}\;}&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Edit:&lt;/em&gt; If you have not used the logical symbols $\forall$ and $\exists$ and they are not part of your course learning objectives, you can replace them with the words &quot;for every&quot; and &quot;there exists&quot;. Again, the motivation behind comparing the two versions is not &quot;formal vs. informal&quot; necessarily, but rather striving for correctness, clarity, and a lack of ambiguity.&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2017-12-06T18:09:05.453" LastActivityDate="2017-12-06T18:09:05.453" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13272" PostTypeId="2" ParentId="13247" CreationDate="2017-12-06T19:06:59.197" Score="1" Body="&lt;p&gt;One possibility is to avoid &quot;such that&quot; when it does not directly translate the $|$ symbol in set builder notation, such as $\{n^2 \mid n\in \mathbb{Z}\}$.  There it is more or less unambiguous.  For statements of theorems, one could say, &quot;Suppose $a$ and $b$ are integers with the property $a^2+b^2$ is an integer.&quot;  Or, &quot;Consider pairs of integers $\{(a,b)\mid a^2+b^2\in \mathbb{Z}\}$,&quot; where s.t. fits in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Upon further reflection, I don't do this myself, but maybe I should aspire to it; it might help in other ways too.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2017-12-06T19:06:59.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13273" PostTypeId="2" ParentId="13247" CreationDate="2017-12-06T21:43:44.013" Score="0" Body="&lt;p&gt;I would simply say that &quot;such that&quot; introduce a restriction or additional information. You need to give several examples and explain the differences.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Let $x$ be a real number such that $x^2&amp;gt;1$. Then $x&amp;gt;1$ or $x&amp;lt;1$&quot;: we introduce a variable and assume some condition on it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;For all real number $x$ such that $x^2&amp;gt;1$, $x&amp;gt;1$ or $x&amp;lt;1$&quot;: we restrict the universal quantifier (this is only a slight variant of the previous case).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;There exists some real number $x$ such that $x^2&amp;gt;1$.&quot;: we claim an &lt;em&gt;additional&lt;/em&gt; property (this makes sense but may look backward compared to the universal quantifier).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;$\{x\in \mathbb{R} \mid x^2&amp;gt;1 \}$&quot;: we restrict the numbers we consider by using a condition.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added in edit after comment:&lt;/strong&gt; the above kept implicit an important point: &quot;such that&quot; is used when variables are introduced, and on top of the words that introduce them. In &quot;$1 = ax+ by$ such that $x$ and $y$ are integers&quot;, the main problem is that the variables are &lt;em&gt;not&lt;/em&gt; introduced in the first place (or rather, &quot;such that&quot; is used to introduce them, while it is not its purpose, making the phrase senseless).&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2017-12-09T16:02:53.683" LastActivityDate="2017-12-09T16:02:53.683" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13274" PostTypeId="2" ParentId="13097" CreationDate="2017-12-06T21:51:26.317" Score="-1" Body="&lt;p&gt;Division symbol ÷ Or / mean only 1 thing&#xA;Left of operator is numerator.&#xA;Right of operator is denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus in 1/3x it is clear that denominator is product 3x.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However if one wants to write fraction(1/3)*x. They must simply put () around fraction as I have written or simply write x/3. This clears any ambiguity in the expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3x is a product (3)(x) no matter where it is in expression. So you were initially right to correct your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;1/2 of x&quot; cannot be written as 1/2x But as x/2 As x must be in numerator and not in denominator. For the product x * 1/2 is evaluated correctly as x/2. Especially if brackets are not used. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;3/4 of x&quot; as x * 3/4 or 3x/4 or use brackets (3/4)x Or  (x)3/4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of these can be successfully evaluated using PEMDAS/ BODMAS without ambiguity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One rule while writing fractions is to keep numerator in numerator and denominator in denominator and not mix them up.&lt;/p&gt;&#xA;" OwnerUserId="9174" LastEditorUserId="9174" LastEditDate="2017-12-06T21:58:46.400" LastActivityDate="2017-12-06T21:58:46.400" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13275" PostTypeId="2" ParentId="894" CreationDate="2017-12-06T22:55:40.637" Score="10" Body="&lt;p&gt;Studies such as &lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED171573.pdf&quot; rel=&quot;noreferrer&quot;&gt;Suydam (1979)&lt;/a&gt; indicate that there is not an adverse effect.  However, most of the cited studies on the effects go back to the late 70's or early 80's.  Anecdotally, as a high school teacher, I think that the early results are no longer valid.  Additional studies need to be carried out using today's technology.  The early calculators required a considerable mathematical knowledge to operate correctly.  Today's calculators (especially with CAS) do not.  As a result, many high school students require a calculator for items as simple as $-3 \times 22$, or for decomposing numbers.  Many of my Algebra II students (11th Grade) no longer remember how to conduct long division, because they have relied upon calculators.  They have lost fluency in many calculations that should be done with automatically.  Often, I can solve problems much faster than the students because I do not need to use a calculator for fairly simple calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, there are many studies, but I do not think that their results are valid for today's electronics.&lt;/p&gt;&#xA;" OwnerUserId="9175" LastEditorUserId="5220" LastEditDate="2017-12-07T16:27:42.753" LastActivityDate="2017-12-07T16:27:42.753" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13276" PostTypeId="1" CreationDate="2017-12-07T01:36:37.993" Score="3" ViewCount="126" Body="&lt;p&gt;I have been doing volunteer high-school math tutoring in person.  But nowadays, I think there should also be an online way of connecting a student in need of help, with someone willing to help them.  This could be in real time, or via message.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any phone apps or websites that will do this?  They should really include a &quot;blackboard&quot; where freehand drawing could take place, on both ends.  Thanks for any information and/or ideas!&lt;/p&gt;&#xA;" OwnerUserId="9176" LastActivityDate="2018-01-11T20:36:07.843" Title="Phone app or webpage to link HS students to free math tutor?" Tags="|secondary-education|tutoring|interactive-teaching|online-instruction|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13277" PostTypeId="1" CreationDate="2017-12-07T02:36:49.397" Score="4" ViewCount="141" Body="&lt;p&gt;Are there classes dedicated to understanding the work of a particular mathematician?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen courses dedicated to a theorem (I saw for example one that sought to prove and understand the Atiyah-Singer Index theorem, or the prime number theorem.) But I'm wondering about a mathematician-- which sounds difficult given that the prerequisites may be sporadic, but I'm curious if sometimes a course may not culminate in just one theorem, but just cover the work of a prominent mathematician. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose conversely, some courses are dedicated to a particular point of view, such as Algebraic Geometry courses that seek to follow the &quot;Grothendieck philosophy,&quot; but this seems difficult from following EGA or something like that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify, I'm not asking about the pedagogical merits of such a course (although that is an interesting question in and of itself.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the course curriculum is available, I would also love access to it.&lt;/p&gt;&#xA;" OwnerUserId="6276" LastActivityDate="2017-12-07T18:00:26.040" Title="Is there a class curriculum that studies the work of a mathematician?" Tags="|reference-request|course-design|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13278" PostTypeId="2" ParentId="12611" CreationDate="2017-12-07T03:35:03.920" Score="3" Body="&lt;p&gt;In this book two colleagues and I give examples of lessons where we connect math to social and political issues in our courses for future teachers. Much of the content could be adapted to the classroom as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.infoagepub.com/products/Reflecting-the-World&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reflecting the World: A Guide to Incorporating Equity in Mathematics Teacher Education&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9177" LastActivityDate="2017-12-07T03:35:03.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13280" PostTypeId="2" ParentId="13277" CreationDate="2017-12-07T18:00:26.040" Score="10" Body="&lt;p&gt;I taught a class based on the work of Bertrand Russell.  It was essentially devoted to set theory, but via his life and emerging interest in mathematics, philosophy, and ultimately logic (and its allied emerging fields of set theory and transfinite numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The text I used was the graphic textbook/biography called Logicomix.  That alone got a number of students to sign up.  They thought they were getting away with something, but actually wound up learning a lot of Mathematics and Mathematical Philosophy, while also rubbing shoulders with Frege, Cantor, the World's Fair of 1900, Russell and his Paradox, David Hilbert's Problems for the new century, and--last but not least--Alice in Wonderland and Lewis Carroll.&lt;/p&gt;&#xA;" OwnerUserId="9183" LastActivityDate="2017-12-07T18:00:26.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13281" PostTypeId="2" ParentId="13276" CreationDate="2017-12-07T21:52:41.230" Score="0" Body="&lt;p&gt;You certainly will not find a free one, but I worked for tutor.com. It is a good site to recommend to students. The tutors are verified and readily available whenever the student needs help. In my experience as a tutor and mentor, the tutors make the students do the work and do not give them the answers. I'm not sure the cost for the student, but I'm sure it is pretty significant.&lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2017-12-07T21:52:41.230" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13282" PostTypeId="2" ParentId="7" CreationDate="2017-12-08T17:30:31.607" Score="4" Body="&lt;p&gt;You are getting a lot done--feel good.  Your approach is very thoughtful and the pedagogy well explained.  Can't think of major changes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are battling a couple disadvantages (only having an hour, laziness).  I was disappointed that so many answers did not really discuss how to help students given the disadvantages, but just opined in a vacuum on how to be more real analysis-y (for ideal students).  [But the comment about flaws in calculus was helpful, although would be more so if combined with idea on how it alters the actual tutor session.]  I also think your council on exam prep makes sense and that the counters were not realistic (not considering students are not already A students).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One small tip:  charge for your time.  It makes both you and the student care more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. General &quot;you&quot;, I realize this is old Q.  But any Q&amp;amp;A site is meant to help others in same situation.&lt;/p&gt;&#xA;" OwnerUserId="9020" LastActivityDate="2017-12-08T17:30:31.607" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13283" PostTypeId="1" AcceptedAnswerId="13297" CreationDate="2017-12-08T18:03:23.233" Score="11" ViewCount="1500" Body="&lt;p&gt;I'm looking for a good abstract linear algebra text (i.e., not matrix crunching) for students who have completed a Strang-level linear algebra course plus exposure to a proof writing (e.g., induction and contradiction are not foreign concepts).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The course is for students who want to revisit core/essential linear algebra concepts from an abstract viewpoint (e.g., for preparation for functional analysis or understanding theories in machine learning) but only so far as this abstract viewpoint supports better applications (I'm not going the Halmos or Lang route, for example). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student mix will likely be statisticians, operations research, compsci and engineering, physics, and perhaps some math majors (applied, most likely).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I perused Lax's book and I generally like its contents, especially the inclusion of a dynamics chapter and lots of cool appendices that show how the abstract viewpoint is not just elegant, but very useful. However, it's also rather thin (which is actually nice!) but makes me wonder how idisyncratic Lax's choices were for the material in the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone used this book in a class -- if so, how did you and your students like it? Did most seem to think the price of the book was worth it and planned to keep it for reference?&lt;/p&gt;&#xA;" OwnerDisplayName="user9185" LastActivityDate="2017-12-11T23:02:07.820" Title="Is Lax's Linear Algebra and its Applications comprehensive or idiosyncratic?" Tags="|textbooks|linear-algebra|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="13284" PostTypeId="1" CreationDate="2017-12-09T04:59:57.803" Score="6" ViewCount="229" Body="&lt;p&gt;I would like some advises which contents to teach in a 45 minutes class to first year undergraduates about the mean value theorem. I'm thinking to do the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Motivation&lt;/li&gt;&#xA;&lt;li&gt;The theorem and the demonstration&lt;/li&gt;&#xA;&lt;li&gt;applications and examples&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So my questions are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Do you think this is the best approach? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you taught  this class, what would you teach in the items 1 and 3 above? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5236" OwnerDisplayName="user26832" LastActivityDate="2017-12-09T16:42:42.260" Title="What to teach in a class of 45 minutes about the mean value theorem" Tags="|calculus|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13285" PostTypeId="2" ParentId="13284" CreationDate="2017-12-09T12:32:47.023" Score="3" Body="&lt;p&gt;The main uses of the mean value theorem are the proofs of many of the other theorems about derivatives.  Presumably those will come some time after this 45-minute class.  In the &quot;motivation&quot; include my first sentence above.  In &quot;examples&quot; draw some graphs, and observe that there are (one or more) tangent lines parallel to the secant line.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-12-09T12:32:47.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13286" PostTypeId="2" ParentId="13284" CreationDate="2017-12-09T14:03:02.830" Score="4" Body="&lt;p&gt;I would assume Rolle's Theorem had been proved in a previous class. Then,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;motivate from physics of motion: the mean value theorem simply says the average speed must match the instantaneous speed at some time during the motion.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I like to say what it doesn't say as well, for instance, if you tell the police that your average speed was 30mph since it takes 10 minutes to travel 5 miles to your house, they probably don't care to hear that when they stop you going 50mph in a particular 30mph zone. Silly motivations aside, then I would&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;prove the mean value theorem&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To do this carefully, I think you can do all of the above in about 15-20 minutes if you prepare. Then in the remaining time:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;give an example where you actually calculate $c$ in $(a,b)$ for which $f'(c) = \frac{f(b)-f(a)}{b-a}$. I usually try to make it clear that the MVT itself does not find $c$ it just tells us that our search is not futile. It is there somewhere to find.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Then, to my taste, one important application is to graphing. To prove $f$ is increasing on $(a,b)$ we need $a&amp;lt;x_1 &amp;lt;x_2&amp;lt;b$ implies $f(x_1) \leq f(x_2)$. Notice,&#xA;$$ \frac{f(x_2)-f(x_1)}{x_2-x_1} = f'(c) \ \ \Rightarrow \ \  \boxed{f(x_2)-f(x_1) = f'(c)(x_2-x_1)} $$&#xA;thus if $f'(c) &amp;gt;0$ we find&#xA;$$ f(x_2)-f(x_1) &amp;gt;0 $$&#xA;which shows $f(x_1)&amp;lt; f(x_2)$. In other words, if $f'(x) &amp;gt; 0$ for $x \in (a,b)$ then we can show $f$ is strictly increasing on $(a,b)$. Naturally, this discussion is limited to differentiable function. If we extend the analysis to piecewise differentiable, or just continuous functions then the derivative need not be defined and we may not be able to use calculus to analyze the graph of such a function. For those examples we have to rely on the definition of increasing function which is based on inequalities. In any event, the $\boxed{ equation}$ or it's close cousin $f(x_2) = f(x_1)+f'(c)(x_2-x_1)$ is how we often use the mean value theorem to linearize the function with it's derivative. Even when we can't find $c$ explicitly, the existence alone suffices to prove many theorems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, just &quot;prove&quot; what I say above with some appropriate pictures paired with the MVT. Pictures are obviously important in this lecture.  &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2017-12-09T14:03:02.830" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13287" PostTypeId="2" ParentId="13276" CreationDate="2017-12-09T14:04:56.510" Score="3" Body="&lt;p&gt;Why not use Google Hangouts? It enables screen sharing both ways -  effectively turning your screen into a blackboard. As others pointed out there's probably no free and yet good site/app that aligns with what you propose. However, functionality-wise hangouts would suffice. Having a dedicated website would certainly help in creating a good tutor community.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even Skype will suit your needs and falls in the same league.&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2017-12-09T14:04:56.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13289" PostTypeId="2" ParentId="13284" CreationDate="2017-12-09T16:42:42.260" Score="7" Body="&lt;p&gt;I think your proposed outline is fine.  For these theorems in Calc I (also, e.g., the Intermediate Value Theorem), I try to start with a step 0 in which you pose a question to the students to get them to start thinking about the theorem before you state it formally (maybe this is your &quot;Motivation&quot; step but I interpret that more as why you're taking the time to state and prove this theorem in class.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.math.cornell.edu/~GoodQuestions/&quot; rel=&quot;noreferrer&quot;&gt;Cornell Good Questions Project&lt;/a&gt; has questions of this type.  See page 36 of this &lt;a href=&quot;http://www.math.cornell.edu/~GoodQuestions/GQbysubject_pdfversion.pdf&quot; rel=&quot;noreferrer&quot;&gt;pdf&lt;/a&gt; for example.  Question 5, in particular, is how I might get the students thinking before getting all formal.  It might also be useful to get them to try Question 6 before stating the theorem so that they can really see why the hypothesis is needed.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2017-12-09T16:42:42.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13292" PostTypeId="2" ParentId="11358" CreationDate="2017-12-10T19:50:40.973" Score="6" Body="&lt;p&gt;I am, perhaps, late to the party, but this question has no answer, so here are a couple of my own suggestions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Stein, Elias M.; Shakarchi, Rami&lt;/em&gt;, Functional analysis. Introduction to further topics in analysis, Princeton Lectures in Analysis 4. Princeton, NJ: Princeton University Press (ISBN 978-0-691-11387-6/hbk; 978-1-400-84055-7/ebook). xv, 423 p. (2011). &lt;a href=&quot;https://zbmath.org/?q=an:1235.46001&quot; rel=&quot;noreferrer&quot;&gt;ZBL1235.46001&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There is a five volume set by Stein and Sharkarchi that covers nearly all of the material that a group of clever undergraduates might hope to see before graduate school.  I am most familiar with the books on real and Fourier analysis, but those are both quite good.  The one complaint that I have is that the books are a little Janus-headed:  there is some excellent exposition, and there are some great problems, but the text and exercises often don't seem to gel entirely.  It is sometimes as though one author wrote the text and the other the exercises, without much communication between them.  Still, I would recommend the series highly.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Kreyszig, Erwin&lt;/em&gt;, Introductory functional analysis with applications, New York etc.: John Wiley &amp;amp; Sons. XIV, 688 p. (1978). &lt;a href=&quot;https://zbmath.org/?q=an:0368.46014&quot; rel=&quot;noreferrer&quot;&gt;ZBL0368.46014&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is the book that I worked out of as an undergraduate.  It is a little old, but still quite nice.  The exposition is careful and thorough, but not to to point of being pedantic.  It is also available fairly cheaply, though you have to look for it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Reed, Michael; Simon, Barry&lt;/em&gt;, Methods of modern mathematical physics. 1: Functional analysis, New York-London: Academic Press, Inc. XVII, 325 p. (1972). &lt;a href=&quot;https://zbmath.org/?q=an:0242.46001&quot; rel=&quot;noreferrer&quot;&gt;ZBL0242.46001&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I would hold this text up as the gold standard in functional analysis.  The exposition is clear and concise without being overly terse, and the exercises are generally quite good.  It may be a bit advanced for an undergraduate audience, but I would still recommend checking it out.  I could imagine teaching a very satisfactory quarter (or semester) long course out of the first three or four chapters.  That being said, you should gauge your audience carefully.  I don't think that it requires that much background, but it starts of fairly quickly with some measure theory and moves on from there.  Your students would need to be fairly motivated and committed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Rudin, Walter&lt;/em&gt;, Functional analysis, McGraw-Hill Series in Higher Mathematics. New York etc.: McGraw-Hill Book Comp. XIII, 397 p. (1973). &lt;a href=&quot;https://zbmath.org/?q=an:0253.46001&quot; rel=&quot;noreferrer&quot;&gt;ZBL0253.46001&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There is always Rudin.  Personally, I wouldn't use this text.  I find Rudin's texts incredibly helpful and useful to have around as references—they are quite encyclopedic in that regard—but I wouldn't want to teach a class from any of his books.  Rudin's style is terse to the point of perversity, and he seems fond of pulling rabbits out of his hat (I can't think of anything in the functional text off the top of my head, but the proof of the mean value theorem in Baby Rudin is, to my mind, particularly egregious).  Moreover, Rudin works in a much more general setting than is likely to be reasonable for undergraduates (as with Reed and Simon).  On the other hand, if you have a particularly ambitious group of students or if you are willing to put in a lot time to prepare them, Rudin can be a good reference or secondary text to supplement lectures notes.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-12-10T19:50:40.973" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13293" PostTypeId="2" ParentId="11358" CreationDate="2017-12-11T07:43:31.160" Score="1" Body="&lt;p&gt;Worth a mention is Pons' &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/1461496373&quot; rel=&quot;nofollow noreferrer&quot;&gt;Real Analysis for the Undergraduate: With an Invitation to Functional Analysis&lt;/a&gt;. While it is primarily an introductory real analysis text, the final section of each chapter is a functional analysis topic. For students who have already had a light introduction to real analysis, you could still make use of these final sections and possibly chapters 8 and 9 on Lebesgue measure and Lebesgue integration respectively as a bridge to further material. Furthermore, the book is in the Springer e-books collection and therefore has the same advantage as Rynne &amp;amp; Youngson in being available for free to your students. &lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2017-12-11T07:43:31.160" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13295" PostTypeId="1" CreationDate="2017-12-11T21:39:00.923" Score="2" ViewCount="344" Body="&lt;p&gt;Might it be helpful for students to have two different symbols for subtraction (-) and negation ( _ )? Subtraction, after all is a binary operation (with 2 operands). Negation is a unary operation (with 1 operand). It also usually has a higher order of precedence than subtraction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The order of operations could be taught using the nmonic BENDMAS with N standing for negation or negative. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Order of Operations (Precedence)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;B: Brackets &lt;/li&gt;&#xA;&lt;li&gt;E: Exponents (right to left ), e.g. $2^{3^4} = 2^{(3^4)}$&lt;/li&gt;&#xA;&lt;li&gt;N: Negative (right to left), e.g. _ _ $2 =$ _$($ _ $2)=2$&lt;/li&gt;&#xA;&lt;li&gt;D: Division, M: Multiplication (left to right)&lt;/li&gt;&#xA;&lt;li&gt;A: Addition, S: Subtraction (left to right)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Note that _ $x^2 =$ _$(x^2)$ as in most algebra textbooks. To be consistent  and not have to make exceptions for constants, you would require _ $2^2 =$ _ $4$ and $($_ $2)^2 = 4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would save converting negative signs to multiplication by minus 1, and other embarrassing patches on something as fundamental as the order of operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dan&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2017-12-13T13:36:01.257" LastActivityDate="2017-12-13T13:36:01.257" Title="Might it be helpful for students to have different symbols for subtraction (-) and negation ( _ )?" Tags="|secondary-education|algebra|middle-school|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13297" PostTypeId="2" ParentId="13283" CreationDate="2017-12-11T23:02:07.820" Score="7" Body="&lt;p&gt;I'd like to thank all the commenters for their insights. In the end, I have decided to go with Lax's Linear Algebra and its Applications, primarily due to its insistence on &lt;em&gt;abstract&lt;/em&gt; vector spaces (i.e., the field is not assumed to be complex or real numbers) and the focus on linear transformations as opposed to matrices, which will set up the students nicely for functional analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, my concerns about his &quot;applied&quot; examples have been allayed, as I think the students will find them interesting and not the typical &quot;plug and chug&quot; applications they get elsewhere. The focus on analysis-related topics will also help tie in some aspects of what they learned in calculus (e.g., $e^A$ as a Taylor series...cool connection between linear algebra and calculus!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I doubt I'll go through the whole book for a semester -- despite it's brevity, it's very, very dense on knowledge (so in terms of $/bit, it's quite well priced ;-). I think that covering Chapters 1-7 is a must as core abstract material.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Fundamentals&lt;/strong&gt; (basic definitions of abstract linear spaces and quotient spaces)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Duality&lt;/strong&gt; -- I'm including this because I want to hit on &lt;em&gt;spaces of functions&lt;/em&gt; at the end wrt machine learning, where the idea of a dual space is important.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Linear Mappings&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Matrices&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Determinants and Trace&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Spectral Theory&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Euclidean Structure&lt;/strong&gt; (lots of important and very widely applied concepts here)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Also, given the interest in machine learning among students this group, I also think that Chapters 8,9,10, and 16 should be delved into, since symmetric and/or positive matrices play a role in many machine learning algorithms. &lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;8&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Spectral Theory of Self-Adjoint Mappings&lt;/strong&gt;: Covers the common case of symmetric matrices (e.g., in PCA and aspects of SVD), especially the ubiquitous quadratic forms that underlie KKT conditions and many optimization methods.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Calculus of Vector and Matrix-Valued Functions&lt;/strong&gt;: Views these objects as vector-objects, not sets of scalar equations. Very useful for understanding convergence of optimization algorithms and getting a new view on multivariate calculus (e.g., be able to calculate $\frac{d}{dt}A(t)B(t)$). Does not really cover &lt;em&gt;Matrix Calculus&lt;/em&gt;, such as $\frac{d\mathbf{F}}{\mathbf{X}}$, but that is not a huge concern for me as it's mainly conventions for bookkeeping of variables.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Matrix Inequalities&lt;/strong&gt;: Pop up all the time when dealing with constraints. Also covers the extremely useful Singular Value Decomposition.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;(Chap 16) &lt;strong&gt;Positive Matrices&lt;/strong&gt;: Perron-Frobenius Theorem -- enough said!&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;After that, I thought that Chapter 11 on Dynamics (to pull it together) and my own intro to Reproducing Kernel Hilbert Spaces (the conceptual world of kernel-based learning methods) would be good ways to end the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, thanks all for your input! Very much appreciated the community here.&lt;/p&gt;&#xA;" OwnerDisplayName="user9185" LastActivityDate="2017-12-11T23:02:07.820" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13298" PostTypeId="2" ParentId="13184" CreationDate="2017-12-11T23:46:00.073" Score="1" Body="&lt;p&gt;First, we need to stop insisting that &quot;math is all around us!&quot; (usually said with Carl Sagan-like starry-eyed wonder ;-) (jk)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a STEM major, I had a natural interest in things that math talked about. However, I was not quite so enthralled with Chaucer, The Stamp Act, Art, or most of the humanities (I've gradually come to enjoy them too). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I sympathize with students who are not interested in a STEM career -- unlike, say, history, writing/literature, or even science, mathematics &lt;strong&gt;is&lt;/strong&gt; quite removed from daily life. And it's not just because most people will never have to solve the quadratic equation. It's that its supposed &quot;knock-on&quot; benefits appear oversold to most non-STEM folks. Here's a possible view from their perspective:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take &lt;strong&gt;history&lt;/strong&gt; -- I'll never write a history article or scour the archives, yet, knowing history helps put &lt;em&gt;everyday life&lt;/em&gt; into perspective. History directly affects our lives and we constantly talk about it in politics and our own lives. &lt;strong&gt;Relevant&lt;/strong&gt;!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, take &lt;strong&gt;English/writing&lt;/strong&gt; -- I'm not going to be an author or journalist, but I need not tell anyone here that being able to write clearly and coherently is probably one of the biggest determinants of success in your career. From cover letters, to reports, to presentations, if you cannot express yourself, you're limiting yourself. &lt;strong&gt;Relevant&lt;/strong&gt;!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No, onto our sister discipline -- &lt;strong&gt;science&lt;/strong&gt;. I'm not going to be a scientist or engineer (well, OK...&lt;strong&gt;I&lt;/strong&gt;  did become one, but that's beyond the point!) but science is about &lt;em&gt;tangible&lt;/em&gt; things and is responsible for our technical wealth. Not to mention global warming, stem cells, cancer treatment, space exploration, and others are things many people are interested in talking about or having an opinion on. &lt;strong&gt;Relevant&lt;/strong&gt;!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, on to our collective favorite -- math. While the above subjects give knowledge that is quite useful outside the academic discipline, math gives us points, lines, planes, quadratic formulas, and calculus. I've found that &quot;everyday examples&quot; of these ideas are quite contrived, as most people will never need to write a formula down or solve anything but the simplest equation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is the other fields are geared towards being useful to the &lt;em&gt;general citizen&lt;/em&gt; while math classes appear to be attempting to groom the next generation of STEM majors. I'd say, leave that to college. To people who aren't interested in solving equations or proving theorems, math class is just one puzzle after another, with no relevance (and, guess what, they're right!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I'm not being down on math. I think there is hope, but it's not by making non-STEM-focused students love the quadratic equation, trigonometry, or integrals by making up examples of cannonballs falling, bridge building, or leaky buckets. It's by copying from the playbook of History, English, and, to a lesser degree, science. We need to focus on areas where complex mathematical concepts creep up in &lt;em&gt;actual&lt;/em&gt; daily life. In my experience, this area is &lt;strong&gt;statistics&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Statistics pops up &lt;em&gt;everywhere&lt;/em&gt; in life, and even if you're not going to be a statistician, the quantitative reasoning skills (and knowing the logical traps) required to understand statistical arguments is extremely useful in everyday life. Politics, sports, theology (Bayesian arguments for/against god .. Ok, hot button issue, but still ;-), medicine, gambling...these are things &lt;em&gt;ordinary people&lt;/em&gt; actually do and actually could be helped by some numeracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Until the curriculum is changed, I think we'll be stuck with contrived examples from algebra, trig, and calculus. Also, I agree that we need some basic arithmetical numeracy like finding areas/volumes, and understanding arithmetic operations. However, I think that Algebra/Trig/Calculus sequence is really best left for students who will be doing STEM. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd really, really, like the general ed math to focus more on &lt;em&gt;asking quantitative questions&lt;/em&gt; and &lt;em&gt;being skeptical of numbers&lt;/em&gt;. I think that number-phobia is manifested by people either trusting or mis-trusting an argument &lt;em&gt;simply because&lt;/em&gt; it has numbers. They've never been given the confidence to &quot;challenge the teacher or the numbers&quot;, which is what you need for correct quantitative reasoning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine students leaving after spending four years honing their ability to deconstruct a quantitative argument in a newspaper, to question a politicians statements, to be able to appreciate clinical research, to incorporate uncertainty and risk properly in their lives...that would be something!&lt;/p&gt;&#xA;" OwnerDisplayName="user9185" LastActivityDate="2017-12-11T23:46:00.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13299" PostTypeId="1" CreationDate="2017-12-11T23:52:35.313" Score="2" ViewCount="56" Body="&lt;p&gt;I'm looking for an app that can be used when the teacher gives a set of exercises to his class to show for each exercise how many students solved it, how many students tried to solve it, etc. The students may use their smartphone to input data in real time. Of course I would also like to be able to check for each student what exercises he has been able to solve. &#xA;Does anyone know an app that can do something like this?&lt;/p&gt;&#xA;" OwnerUserId="9197" LastEditorUserId="9197" LastEditDate="2017-12-12T00:50:07.723" LastActivityDate="2017-12-13T01:25:50.373" Title="App to show exercises solved by students in real time" Tags="|secondary-education|interactive-teaching|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13300" PostTypeId="2" ParentId="13295" CreationDate="2017-12-12T00:56:14.350" Score="-4" Body="&lt;p&gt;Really bad idea.  This is sacrificing the 99 kept sheep to save the 1 lost sheep.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A.  It's not that hard.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;B. Everything else they read will be different from how you teach.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;C.  What makes you think you should use a teaching position to drive a difference in usage (versus having teaching follow usage).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;D. (And Chris will kill me for saying this but &quot;Ich kann nicht anders&quot;) Don't you have something else more useful to improve?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oh...and 99% of people don't get the subtle differences.  Me included.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. Sorry, cable reset my IP again.  Hence the low post count.&lt;/p&gt;&#xA;" OwnerUserId="9198" LastActivityDate="2017-12-12T00:56:14.350" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13302" PostTypeId="2" ParentId="41" CreationDate="2017-12-12T02:54:49.710" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/sX49W.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sX49W.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tend to loop my ζ so it doesn't look like a Z and I also put the lines on the z so it is not a 2.I also make my 1 like shown in the picture so it is not an I. I make my lowercase g fancy so it is not a 9. I put lines on my Θ so they are not θs. I also make my α short so they are not a ∝. I do not make my Phis like a plain old circle with a line. I tend to make a loop and then come down.  My μs have a loop on the side. &lt;/p&gt;&#xA;" OwnerUserId="9199" LastActivityDate="2017-12-12T02:54:49.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13303" PostTypeId="2" ParentId="13295" CreationDate="2017-12-12T04:35:00.560" Score="3" Body="&lt;p&gt;It is perhaps worth noting that on most graphing calculators there &lt;em&gt;are already&lt;/em&gt; different symbols for subtraction and negation, instantiated on different keys.  For example on the TI-84 Plus (see image below) there is a key labeled (-) at the bottom right corner, for negation, and a separate key along the right-hand side for subtraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QpoIA.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QpoIA.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this &quot;better&quot; than having a single key that handles both functions?  Personally I doubt it.  More often than not the need to distinguish between the two keys leads to syntax errors -- as, for example, when a student wants to set the &lt;code&gt;XMIN, XMAX, YMIN, YMAX&lt;/code&gt; variables for a viewing window and accidentally uses the subtraction symbol instead of the negation symbol.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2017-12-12T04:35:00.560" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13304" PostTypeId="2" ParentId="13264" CreationDate="2017-12-12T08:08:18.680" Score="2" Body="&lt;p&gt;Guessing the result of $V-L+F$ of a solid figure without hesitation leaves them pretty surprised, usually...&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2017-12-12T08:08:18.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13305" PostTypeId="2" ParentId="13295" CreationDate="2017-12-12T11:23:52.747" Score="4" Body="&lt;p&gt;I do recall some elementary texts that do this.  Subtraction&#xA;$$6 - 5$$ &#xA;written in a different way than a negative number&#xA;$$&#xA;{}^-8&#xA;$$&#xA;so we can do calculations&#xA;$$&#xA;5 - {}^-2  = 5 + 2 = 7&#xA;$$&#xA;Presumably at some point, the students are switched to the conventional notation.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2017-12-12T11:23:52.747" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13306" PostTypeId="1" CreationDate="2017-12-12T13:07:53.123" Score="7" ViewCount="154" Body="&lt;p&gt;In the final throes of the quadratic formula, you reduce a fraction. Consider the following two examples.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$y = 6x^2 + 11x + 3$; the quadratic formula reveals the roots $x = -4/12$ or $x = -18/12$. The parts of the first fraction (4 and 12) have a common factor of 4, so the fraction reduces to -1/3. The parts of the second fraction (18 and 12) have a common factor of 6, so the fraction reduces to -3/2. The common factor of all three parts (4, 12 and 18) is 2.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$y = 42x^2 + 77x + 21$; the quadratic formula reveals the roots $x = -28/84$ or $x = -126/84$. The parts of the first fraction (28 and 84) have a common factor of 28, so the fraction reduces to -1/3. The parts of the second fraction (126 and 84) have a common factor of 42, so the fraction reduces to -3/2. The common factor of all three parts (28, 84 and 126) is 14.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Now. Compare the two solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We multiplied by 7 the right side of Equation 1 to produce Equation 2. As a result, the common factors of the equations were all multiplied by 7, and after reducing the fractions we discover the same roots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is whether that step, reducing the fraction, reveals anything about the numerical factors of the equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that any common numerical factor of a, b and c (like 7 in Equation 2) will also be a common factor of the parts of the fractions, and cancel when we reduce the fraction. But the parts of the fraction can have other common factors (like 2 in both Equations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when we come to that step, of reducing the fraction, does the common factor reveal anything about the numerical common factors of a, b and c?&lt;/p&gt;&#xA;" OwnerUserId="8258" LastEditorUserId="8258" LastEditDate="2017-12-12T13:21:29.427" LastActivityDate="2017-12-16T00:33:07.380" Title="What can we learn as we reduce the fraction at the end of the quadratic formula process?" Tags="|algebra|number-theory|" AnswerCount="2" CommentCount="4" ClosedDate="2017-12-16T20:01:45.050" ContentLicense="CC BY-SA 3.0" />
  <row Id="13307" PostTypeId="2" ParentId="13306" CreationDate="2017-12-12T16:29:43.560" Score="2" Body="&lt;p&gt;This would be a good process to introduce the rational root test in quadratic form. Seeing rational answers and comparing to the possible roots from the rational root test. Thus when you extend using the rational root test to higher order polynomials the seeds of the idea would already be there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using your example the rational root test says the possible solutions are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Factors of $c = \pm 1, \pm 3$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Factor of $a = \pm 1, \pm 2, \pm 3, \pm 6$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possible rational solutions are: $\pm \frac{1}{6}, \pm  \frac{1}{3}, \pm \frac{1}{2}, \pm 1, \pm \frac{3}{2}, \pm 3$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The solutions are there and the you can see the source of the common factors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also you can illustrate the limits of the rational root test by introducing irrational answers that don't meet the test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: $ x^2+8x-7=0$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rational root test says this would have rational roots $\pm 1, \pm 7$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quadratic formula gives $-4\pm\sqrt{23} \approx 0.796 \text{and} -8.796$.  Looking at the rational possible roots of $1$ and $-7$ we see it is near, relatively. Which would be an easy entry to some of the approximation techniques on polynomials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like this as an approach to add earlier in my presentation of quadratics to prepare for higher order polynomials solutions. I am adding that into my notes for the quadratic formula for next year.&lt;/p&gt;&#xA;" OwnerUserId="8952" LastEditorUserId="8952" LastEditDate="2017-12-12T18:59:25.020" LastActivityDate="2017-12-12T18:59:25.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13308" PostTypeId="2" ParentId="12165" CreationDate="2017-12-12T22:50:26.240" Score="4" Body="&lt;p&gt;Since &lt;a href=&quot;https://matheducators.stackexchange.com/users/262/benjamin-dickman&quot;&gt;Benjamin Dickman&lt;/a&gt; mentioned &lt;em&gt;The Tokyo Puzzles&lt;/em&gt; in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/12165/good-examples-of-questions-to-have-students-ponder-over-without-paper#comment28362_12165&quot;&gt;the comments&lt;/a&gt;, I'll include a couple of the questions from that book here that I thought fit the prompt nicely; neither requires a pen and paper to think about, and a student can just ponder them mentally.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;56&quot;&gt;&#xA;  &lt;li&gt;Two brothers decided to run a 100-meter race. The older brother won by 3 meters. In other words, when the older brother reached the finish, the younger brother had run 97 meters. They decided to race again, this time with the older brother starting 3 meters behind the starting line. Assuming that both boys ran the second race at the same speed as before, who do you think won? &lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;90&quot;&gt;&#xA;  &lt;li&gt;Between noon today and noon tomorrow, how many times does the long hand on the clock pass the short hand? &quot;Pass&quot; means that one hand follows, overtakes, and goes ahead of the other. Since both hands are at the same spot at noon, the long hand does not pass the short hand at twelve o'clock, the starting time.  &lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2017-12-12T22:50:26.240" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13310" PostTypeId="2" ParentId="13184" CreationDate="2017-12-13T08:28:19.097" Score="0" Body="&lt;p&gt;some months ago, on the Flemish radio, somebody mentioned another problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It's cool to be bad at mathematics&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And face it, it's true: when people are coming on television and they are asked about their skills or knowledge, when they need to admit being bad at history, geography, languages or other basic knowledge, people openly feel bad about it.&lt;br/&gt;&#xA;But when admitting being bad at mathematics, people feel proud about it! In a television show, I remember having seen once following dialog between a locally famous person and the television host:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Famous person: &quot;I was actually not that bad at mathematics&quot;&lt;/li&gt;&#xA;&lt;li&gt;TV host: &quot;Really???&quot; (as he meant &quot;Aren't you ashamed for that???&quot;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;When popular persons (singers, artists) would start showing another attitude towards mathematics (not saying they were bad at it, or in case they were, showing real honest shame for it, or, when they were good at it, showing real pride for it), that would change the whole mindset about mathematics, the mentality around it might change and side-effects like math anxiety might be reduced.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-12-13T08:28:19.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13311" PostTypeId="2" ParentId="13295" CreationDate="2017-12-13T09:07:29.327" Score="1" Body="&lt;p&gt;It's a common mistake that negative numbers are just the result of an operator on a positive number: they have a right to exist and the &lt;code&gt;-&lt;/code&gt; sign in front of them is just for recognition purposes, so next to subtraction and negation, there also is indication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me explain you:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two kinds of numbers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;positive ones : 0, 1, 2, ...&lt;/li&gt;&#xA;&lt;li&gt;negative ones : 0, -1, -2, ... (indication)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Negation is the function which flips positive numbers into their negative equivalent and vice versa:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;code&gt;- (1) = -1&lt;/code&gt; (negation leads to indication)&lt;/li&gt;&#xA;&lt;li&gt;&lt;code&gt;- (-2) = 2&lt;/code&gt; (negation leads to removal of the indication)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There is an operator (the subtraction) between two numbers, which is the addition of the first number with the negation of the second number:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;code&gt;5 - 3 = 5 + (-(3)) = 5 + (-3)&lt;/code&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;code&gt;6 - (-4) = 6 + (-(-4)) = 6 + 4&lt;/code&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the example of &lt;code&gt;5 - 3&lt;/code&gt;, we have the following usage of the &lt;code&gt;-&lt;/code&gt; sign:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;5 - 3 =&gt; subtraction operator&lt;/li&gt;&#xA;&lt;li&gt;5 + (-(3)) =&gt; negation operator&lt;/li&gt;&#xA;&lt;li&gt;5 + (-3) =&gt; indication that the second number is negative (no operator)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Whether or not it's a good idea to use different characters for the different meanings, I think you can only succeed if you do it for the three, not just the two you mentioned.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-12-13T09:07:29.327" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13312" PostTypeId="2" ParentId="41" CreationDate="2017-12-13T09:59:18.050" Score="2" Body="&lt;p&gt;I don't understand how you can mix up a &lt;code&gt;t&lt;/code&gt; and a &lt;code&gt;+&lt;/code&gt;-sign:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the letter &lt;code&gt;t&lt;/code&gt; contains a curl at the bottom and the bar does not pass left of the vertical line&lt;/li&gt;&#xA;&lt;li&gt;the &lt;code&gt;+&lt;/code&gt;-sign consists only of straight lines.&lt;a href=&quot;https://i.stack.imgur.com/wyi4I.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wyi4I.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;the letter &lt;code&gt;y&lt;/code&gt; starts with a curl and has a hole at the bottom&lt;/li&gt;&#xA;&lt;li&gt;the digit &lt;code&gt;4&lt;/code&gt; consists only of straight lines.&lt;/li&gt;&#xA;&lt;li&gt;the letter &lt;code&gt;z&lt;/code&gt; is written in the French way, the hole is quite long.&lt;/li&gt;&#xA;&lt;li&gt;the digit &lt;code&gt;2&lt;/code&gt; has a small but clear hole. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2017-12-13T09:59:18.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13313" PostTypeId="2" ParentId="13184" CreationDate="2017-12-13T12:59:56.240" Score="1" Body="&lt;p&gt;Another reason, I think, is how mathematics is typically graded.  Do 20 problems, get red Xs through 7 of them, you earn a D.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every time a student exposes some misunderstanding, they are punished.  This is backwards:  students should be rewarded for showing that they do not understand something, because this gives them an opportunity to grow.  Since exposing misunderstanding is punished, students hide (even from themselves!) what they do not know.  This leads to a death spiral of mathematical knowledge.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2017-12-13T12:59:56.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13314" PostTypeId="2" ParentId="12824" CreationDate="2017-12-13T13:50:00.727" Score="4" Body="&lt;p&gt;They don't. You need &lt;em&gt;additional information&lt;/em&gt; to organize a set of points into anything resembling geometry; a bare set of points is simply lacking the context to allow anything more interesting than counting them (i.e. the cardinality of the set).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A typical form of context boils down to &quot;remembering&quot; how the points are embedded in a Euclidean plane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below are two answers I've given to similar questions at &lt;a href=&quot;https://math.stackexchange.com&quot;&gt;https://math.stackexchange.com&lt;/a&gt;. Both questions have a variety of other answers to browse through as wel.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/1084093/14972&quot;&gt;https://math.stackexchange.com/a/1084093/14972&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/1805439/14972&quot;&gt;https://math.stackexchange.com/a/1805439/14972&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2017-12-13T13:55:22.953" LastActivityDate="2017-12-13T13:55:22.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13315" PostTypeId="1" AcceptedAnswerId="13969" CreationDate="2017-12-13T17:36:34.817" Score="8" ViewCount="631" Body="&lt;p&gt;I am an instructor at a mid-sized American university, preparing to teach a two-quarter geometry course for junior and senior math majors. My plan is to use Hartshorne's &quot;Geometry: Euclid and Beyond,&quot; which is designed to be read in parallel with Euclid's elements. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like the pace of the class to be a little more relaxed than the pace Hartshorne sets (for example, it appears chapter 1 takes Hartshorne three weeks to cover, and this involves reading the first four books of the Elements.) In particular, I am looking for problems that could be used as homework or in-class group work to solidify the students understanding of Euclid. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there other companions to Euclid's Elements that are more leisurely paced?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do you know of a bank of geometry problems that contains problems appropriate for someone working through the Elements?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="98" LastActivityDate="2018-04-19T18:18:12.273" Title="Resource to supplement to Euclid's Elements" Tags="|undergraduate-education|geometry|resource-request|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13316" PostTypeId="2" ParentId="12165" CreationDate="2017-12-13T22:18:59.983" Score="1" Body="&lt;p&gt;There are some great puzzles &lt;a href=&quot;http://www.math.utah.edu/~cherk/puzzles.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here on Andrej Cherkaev's website&lt;/a&gt;, most of which he attributes to Vlad Mitlin. I'll list paraphrased versions of a few of them here which I thought most fit the requirement that they may be pondered over without any suggestion that they require calculations to answer. Note that I've reworded these questions rather dramatically.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;blockquote&gt;&#xA;  &lt;p&gt;A monk needs to meditate for exactly forty-five minutes, but - living in an abbey - he doesn't have a watch or a clock with which to time himself. All he has is two incense sticks, which he knows each take exactly one hour to burn. Unfortunately, being hand-made, the incense sticks aren't identical to each other, and they're imperfectly shaped so that he can't rely on a stick burning at the same rate all the time (so if he were to break an incense stick in half, there's no guarantee that would burn for exactly half an hour). Using these incense sticks and some matches, how can the monk arrange for exactly forty-five minutes of meditation?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;blockquote&gt;&#xA;  &lt;p&gt;You throw a party and invite only your $n$ closest friends. Everyone arrives at the party, but then you realize that none of your $n$ closest friends have ever met one another! You frantically start introducing them, when you realize that you can make a game out of this situation. Can you introduce your friends in such a way that after you've finished, no three of your friends will have met the same number of people at the party?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;blockquote&gt;&#xA;  &lt;p&gt;The military is preparing to do a training exercise at night on a large flat open field. They need to illuminate the entire field for the exercise, so four planes fly overhead and each airdrops a giant lamps somewhere onto the field. Each lamp is very powerful, but they were designed to be placed in corners, so they're each only able to illuminate a quadrant, an area spanned by ninety degrees, of the region around them. Upon trying to move the lamps to the corners of the field though, the soldiers on the ground realize that the lamps are too heavy to move. The lamps however can be rotated in place. Is there way that the lamps can be rotated so that when they are turned on, they can still illuminate the entire field?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-04-19T23:11:06.387" LastActivityDate="2018-04-19T23:11:06.387" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13317" PostTypeId="1" AcceptedAnswerId="13322" CreationDate="2017-12-13T23:24:45.283" Score="12" ViewCount="2020" Body="&lt;p&gt;I will give a talk to undergrad students about Markov chains. I would like to present several concrete real-world examples. However, I am not good with coming up with them beyond drunk man taking steps on a line, gambler's ruin, perhaps some urn problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to have more. I would favour eye-catching, curious, prosaic ones. Specially if there has been a paper about it. For example, I remember reading years back a paper, based on actual data, about the probability that a football player will change teams, or something like that.&lt;/p&gt;&#xA;" OwnerUserId="9206" LastEditorUserId="77" LastEditDate="2017-12-26T16:10:44.843" LastActivityDate="2017-12-29T23:23:15.383" Title="Real-world Markov chains" Tags="|undergraduate-education|examples|probability|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13318" PostTypeId="2" ParentId="13317" CreationDate="2017-12-14T09:56:50.640" Score="8" Body="&lt;p&gt;I'm not sure if you consider the board game &lt;em&gt;Monopoly&lt;/em&gt; as a real-world example, but it is often used to explain Markov Chains to laypeople.  Ian Stewart has a couple of &lt;em&gt;Mathematical Recreations&lt;/em&gt; articles about it in &lt;em&gt;Scientific American&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first (in the April 1996 issue) has the title &quot;How Fair is Monopoly?&quot;  (A copy it can be found &lt;a href=&quot;http://www.math.harvard.edu/archive/21b_fall_05/Articles/howfairismonopoly.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second (in the October 1996 issue) has the title &quot;Monopoly Revisited&quot;  (A copy of it can be found &lt;a href=&quot;http://www.math.harvard.edu/archive/21b_fall_05/Articles/monopolyrevisited.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2017-12-14T09:56:50.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13319" PostTypeId="2" ParentId="13317" CreationDate="2017-12-14T18:39:39.213" Score="-1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Hydrology: classic area, very easily intuitively to think of water filling, draining areas (but with variation in the input).  It is very graphical and physical.  This is the classic example!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Markov chains in economics.  Little less physically intuitive, but huge area in terms of relevance to Mankind.  And for smart students and/or world aware ones, won't be that bad (not like some weird physics, chem problem).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Climate data (tree rings, etc.).  The math is a little trickier and examples a little less pure, but may be of relevance to your students.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9198" LastEditorUserId="9198" LastEditDate="2017-12-15T03:43:56.930" LastActivityDate="2017-12-15T03:43:56.930" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13320" PostTypeId="2" ParentId="3950" CreationDate="2017-12-14T18:47:01.697" Score="1" Body="&lt;p&gt;Some older diffyQ books may be of interest to you also (Boole, Taylor (Treatise, NOT Theory), Piaggio).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that older books were much more commonly containing answers.  This is an area where modern practice has become LESS liberal, not more so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might look at Schaum's Outlines as they contain a lot of solved problems (sometimes answers, sometimes whole problem).  I don't know they are &quot;hard&quot;.  But you can pick through them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thomas and Finney prior to early 80s had all the answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may also consider to look on the web for old (prior to 1930) Cambridge Tripos exams.  &lt;/p&gt;&#xA;" OwnerUserId="9198" LastActivityDate="2017-12-14T18:47:01.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13321" PostTypeId="2" ParentId="535" CreationDate="2017-12-14T19:12:03.340" Score="4" Body="&lt;p&gt;If you test them (and without any ability to reach the Internet, no phones, computers) than it will be an action that makes it hard to over-rely on WA.  If you do graded homework, it is the opposite.  Thus, I would suggest more tests, quizzes (in class) and less graded homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I think for a very strong student (or a working professional) there is nothing wrong with using WA or even just a handbook.  After you are done with a course.  But there is a disadvantage in being someone who only can use WA and can not do anything himself.  I would include Wolfram, the actual man, and many others in this group.  They learned it first analytically and it informs their use of WA, their development of it, their fluency reading physics or engineering derivations, and their analysis of problems in the real world.&lt;/p&gt;&#xA;" OwnerUserId="9198" LastEditorUserId="9198" LastEditDate="2017-12-15T17:57:57.970" LastActivityDate="2017-12-15T17:57:57.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13322" PostTypeId="2" ParentId="13317" CreationDate="2017-12-14T19:20:24.787" Score="6" Body="&lt;p&gt;The Markov Chains I work with are usually called in the epidemiological and in the chemistry literature &quot;compartmental models&quot;. The most famous (from an epidemiological viewpoint) is the SIR model for infectious diseases. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;SIR model&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main idea is that individuals can be classified into three groups: Susceptible (people that can get the disease), Infected (people that have the disease and are contagious), and Removed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wdY14.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wdY14.png&quot; alt=&quot;Diagram of the SIR model&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main idea is that almost all individuals start &lt;em&gt;Susceptible&lt;/em&gt; (they can catch the disease) with a handful of &lt;em&gt;Infected&lt;/em&gt;. Considering discrete time (such as days) some infected individuals will either get better or die (hence becoming &lt;em&gt;Removed&lt;/em&gt;) and some &lt;em&gt;Susceptible&lt;/em&gt; will become &lt;em&gt;Infected&lt;/em&gt;. The interesting part is how interactions between the compartments give rise to different models. Also, there are &lt;em&gt;a lot&lt;/em&gt; (and I mean &lt;em&gt;a lot&lt;/em&gt;) of infectious disease models that use this idea and add additional compartments (boxes in the picture or &lt;em&gt;states&lt;/em&gt; of your chain). For example you can divide the infected into symptomatic and asymptomatic, you can divide the susceptible into vaccinated and unvaccinated, etc. A more complete account on this models can be found in &lt;a href=&quot;https://link.springer.com/book/10.1007/978-3-540-78911-6&quot; rel=&quot;noreferrer&quot;&gt;Mathematical Epidemiology&lt;/a&gt; by Brauer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a plot of what the model looks like (the simulation is done in R as explained in &lt;a href=&quot;https://www.r-bloggers.com/a-discrete-time-markov-chain-dtmc-sir-model-in-r/&quot; rel=&quot;noreferrer&quot;&gt;this blogpost&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ztd95.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ztd95.png&quot; alt=&quot;SIR model plot&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Chronic disease models&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chronic disease models do not usually have interaction between compartments (think diabetes which is usually not contagious).The main idea is that everyone is Healthy, then gets the disease and finally dies. This models are usually done to evaluate policies (&lt;em&gt;what would happen if we give medicine to everyone?&lt;/em&gt; or &lt;em&gt;what would happen if we improve everybody's diet?&lt;/em&gt;). Again you can add more compartments by differentiating sexes, or age categories, body mass index, etc.  You can see the paper &lt;a href=&quot;https://www.ispor.org/workpaper/Modeling_Methods/State-Transition_Modeling-3.pdf&quot; rel=&quot;noreferrer&quot;&gt;State-Transition Modeling: A Report of the ISPOR-SMDM Modeling&#xA;Good Research Practices Task Force-3&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Connections between SIR and physics&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The SIR model and most infectious-disease models are related to compartmental models in physics. The associated physics models are called interactive particle systems and are quite an interesting area of research. From a physics standpoint you can read van Kampen or Gardner (which might be closer to an advanced undergraduate). A book on how the connections are made is &lt;em&gt;Quantitative Sociodynamics&lt;/em&gt; and a graduate level (classical) book on this systems is  &lt;em&gt;Interactive Particle Systems&lt;/em&gt; by Liggett.&lt;/p&gt;&#xA;" OwnerUserId="5922" LastEditorUserId="5922" LastEditDate="2017-12-15T00:53:19.843" LastActivityDate="2017-12-15T00:53:19.843" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13323" PostTypeId="2" ParentId="13306" CreationDate="2017-12-16T00:33:07.380" Score="2" Body="&lt;p&gt;Let $y=a d x^2+ bd x + cd$ with $a,b,c,d \in \mathbb{Z}$ with $a,d \neq 0$, i.e., suppose the coefficients of the quadratic share a common factor of $d$. Then&#xA;\begin{align*}&#xA;x &amp;amp;= \frac{ - bd \pm \sqrt{(bd)^2 - 4 ad cd}}{2 ad} \\&#xA;  &amp;amp;= \frac{-bd \pm \sqrt{d^2 (b^2 - 4ac)}}{d2a} \\&#xA;&amp;amp;=  \frac{-bd \pm |d|\sqrt{b^2-4ac}}{d2a} \tag{1}.&#xA;\end{align*} &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two possibilities: either $d &amp;gt;0 $ or $d&amp;lt;0$. If $d&amp;gt;0$, equation (1) reduces to &#xA;$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ &#xA;and we have the exact same roots as $y=ax^2 + bx + c$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $d&amp;lt;0$, then $|d| = - d$. Using this in equation (1) gives&#xA;\begin{align*}&#xA;x&amp;amp;=   \frac{-bd \pm -d\sqrt{b^2-4ac}}{d2a} \\&#xA; &amp;amp;= \frac{-b \mp \sqrt{b^2 - 4ac}}{2a} \\&#xA;&amp;amp;= \frac{-b \pm \sqrt{b^2-4ac}}{2a}&#xA;\end{align*}&#xA;and again we have the same roots as $y=ax^2 + bx + c$. Thus rescaling all the coefficients doesn't change the roots, although it should be noted that this isn't necessarily the only cancellation that can happen when simplifying the roots! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An easier way to see this result is to recall the fundamental theorem of algebra, wherein any quadratic can be uniquely factored (up to ordering) as&#xA;$$y= a (x- r_1)(x - r_2)$$&#xA;where $r_1, r_2$ are the roots. Changing the factor of $a$ to $a'$ doesn't affect the roots. &lt;/p&gt;&#xA;" OwnerUserId="7260" LastActivityDate="2017-12-16T00:33:07.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13325" PostTypeId="2" ParentId="7718" CreationDate="2017-12-16T20:34:25.587" Score="3" Body="&lt;p&gt;Sorry if this sounds simple, but I think the biggest consideration is efficiency (maybe a bit related to the missing Y issue).  If you study books that are too hard, it may take you longer than a more progressive approach would get you (even including eventually using the tougher text).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, the danger of giving up is not to be trivialized.  I think there is a balance between the joy of working through difficulty and the value of a kinder, progressive training.   In addition, even if the book is &quot;easy&quot; one can still get joy from it by covering it faster.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I find way more value in topics I master than those I don't.  Courses I got A+ in stick with me for decades.  B or C classes (I went to a school that still gives those) never &quot;sunk in&quot;.  So if you master an easy text, you may have more benefit than if you just have partial success in a tougher one.&lt;/p&gt;&#xA;" OwnerUserId="9198" LastActivityDate="2017-12-16T20:34:25.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13326" PostTypeId="2" ParentId="74" CreationDate="2017-12-16T21:01:53.723" Score="1" Body="&lt;p&gt;All of the courses I remember taking (science and math) from grade school through graduate were very liberal about allowing any solution method to be OK.  And I personally tend to be a bit (not extreme but a bit) of a reductionist.  Why learn the derivative quotient rule when you can use the multiplication rule and a raised to the minus 1?  Why learn the two point formula of a line, when y=mx+b is so much simpler to remember and you can still derive what you need?  (These are real examples, even now.)  But it is a bit of a pain in the butt when I find myself doing 2 point line derivations (in real life business analysis) and only remember y=mx+b.  In converse, the quadratic equation is drilled into my headflesh and I don't need to complete the square!  bsquared-4 ac is almost an icon.  And if you do chemistry, physics, engineering, quadratics are all over the place.  Ready facility is useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So as I get older (different learnings, but many physical tasks), I find myself more valuing deliberate practice of specific methods.  That's fine that you like to block karate chops with &quot;wax on, wax off&quot; but you need to learn &quot;paint the fence&quot; too.  You'll be better off having both in repertoire. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So even though it is artificial, I think deliberate practice in specific methods is good training.&lt;/p&gt;&#xA;" OwnerUserId="9198" LastActivityDate="2017-12-16T21:01:53.723" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13327" PostTypeId="2" ParentId="189" CreationDate="2017-12-16T21:13:22.480" Score="5" Body="&lt;p&gt;If you are not going to formally split the classes (with different exams, class names, etc.), I would advise just teaching to the basic level.  [But I like Adam's idea to add ungraded extra hard problems for the students who just elect to do that for edification and enjoyment.]  There is also nothing wrong with just letting the advanced kids know that this will be a bit easier for them, don't goof off and let down though as there is new content later, and maybe put some of the time they are saving from course X being easy into their other classes that need help more.  If the students are really advanced advise them to take a final exam to validate the class and move on.&lt;/p&gt;&#xA;" OwnerUserId="9198" LastActivityDate="2017-12-16T21:13:22.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13329" PostTypeId="2" ParentId="12824" CreationDate="2017-12-17T15:09:23.733" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;They want to know how a bunch of zero length points make a line of&#xA;  length 1?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It may help to think of the Euclidean plane as a set, each element of it being a point in the plane. Then any geometric figure (e.g. a line, or a circle) would be a subset of the Euclidean plane, each element of that subset being a point in the plane. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 1:&lt;/strong&gt; A subset of the plane P is a circle iff there exists a point C in P and a distance r such that every point in that subset is a distance r (the radius) from the point C (the center). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\{ X \in P : |CX| = r\}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 2:&lt;/strong&gt; A subset of the plane P is a straight line iff there exists a pair of distinct points A and B in P such that every point in that subset is equidistant from both A and B.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\{ X\in P : |AX| = |BX|\}$$&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2017-12-21T19:46:24.017" LastActivityDate="2017-12-21T19:46:24.017" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13330" PostTypeId="2" ParentId="11971" CreationDate="2017-12-17T23:42:47.690" Score="1" Body="&lt;p&gt;The Calculus is all about limit concepts. So, you need to understand some basic computations with all type of functions like polynomials, exponentials, logarithmic, trigonometric, inverse trigonometric, hyperbolic function... and more.&lt;/p&gt;&#xA;&lt;p&gt;In order to visualize the Calculus concepts, you need to know the geometric shapes in 2D and 3d and its properties.&lt;/p&gt;&#xA;&lt;p&gt;If you want to optimize something, you have model/function and you need to find the critical points of the function/model. so, to find those critical points or something. You need to solve the equations, sometimes systems. So, you would know Algebra.&lt;/p&gt;&#xA;&lt;p&gt;so, to start learning Calculus. My suggestion is you would get through Algebra ( Basic, Intermediate), Geometry( 2D and 3D) and Trigonometry.&lt;/p&gt;&#xA;&lt;p&gt;Also, you must familiar with all types of coordinate systems, rectangular, polar, cylindrical and sphere.&lt;/p&gt;&#xA;&lt;p&gt;if you know all the concepts well, before start learning Calculus. Then, you enjoy the Calculus. It is great fun.&lt;/p&gt;&#xA;" OwnerUserId="9219" LastEditorUserId="2083" LastEditDate="2023-10-29T08:47:59.583" LastActivityDate="2023-10-29T08:47:59.583" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13331" PostTypeId="1" CreationDate="2017-12-18T03:19:33.163" Score="9" ViewCount="1411" Body="&lt;p&gt;A beginner in math, reading Badiou, I found the following quote on set theory in &lt;em&gt;Being and Event&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The axiomatization consists in fixing the usage of the relation of belonging, $\in$, to which the entire lexicon of mathematics can finally be reduced, if one considers that equality is rather a logical symbol.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If Badiou and so many others have such high esteem for the fundamental importance of set theory, why is it not being taught at the very outset of math education? I understand Badiou is, after all, a philosopher. I am just an idiot without a teacher, wanting to comprehend. &lt;/p&gt;&#xA;" OwnerDisplayName="Ken" LastEditorUserId="8571" LastEditDate="2017-12-20T18:01:00.420" LastActivityDate="2021-02-25T03:29:42.400" Title="Why is set theory not taught at the outset of math education?" Tags="|curriculum|set-theory|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="13332" PostTypeId="2" ParentId="13331" CreationDate="2017-12-18T04:34:18.743" Score="13" Body="&lt;p&gt;&lt;em&gt;Not a complete answer (could there be one?), but too long for a comment.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;why is it [&lt;em&gt;Set Theory&lt;/em&gt;] not being taught at the very outset of math education?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It has been tried, most likely still is in certain places at different degrees. For some history and background, lookup the New Math of the '60s, possible keywords Belgium,  Willy Servais, Georges Papy. Unfortunately, many of the relevant papers are behind paywalls now, but quoting from one which is not (&lt;a href=&quot;https://www.researchgate.net/publication/273143778_Belgian_contributions_to_the_New_Math_movement_in_Europe_The_role_of_international_organizations_and_the_influence_of_the_periphery&quot; rel=&quot;noreferrer&quot;&gt;Geert Vanpaemel - Belgian contributions to the New Math movement in Europe ...&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Servais  emphasized  the  concept  of  a &lt;em&gt;pédagogie  ouverte&lt;/em&gt;,  an  open  approach  to  the  learning&#xA;  process. [...] He wrote with enthusiasm about the choice of set theory as the foundation of mathematical&#xA;  education, and in particular lauded Papy’s use of Venn diagrams and arrow-graphs, which were, in&#xA;  his opinion, ideally suited to the mind of young children. He also endorsed their attempts, criticized&#xA;  by  some, to  bring  the  New  Math  to  primary  schools. [...] Servais was convinced that the basic unity, acquired by the use of set theory,&#xA;  provided the solution to the pedagogy that he had in mind: as the goal of mathematics teaching was&#xA;  to activate the mind of the child towards grasping the mathematical structures in the world around&#xA;  him,  it  was  necessary  to  define  these  structures  and  to  make  them  the  backbone  of the  whole&#xA;  syllabus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9221" OwnerDisplayName="dxiv" LastActivityDate="2017-12-18T04:34:18.743" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13333" PostTypeId="2" ParentId="11971" CreationDate="2017-12-18T05:52:41.113" Score="3" Body="&lt;p&gt;You should consider (and the answerers) how much you really need/want to learn.  Perhaps only some basics, perhaps only a specific problem or two are really what you need for the task(s) you have in front of you.  If you want to learn calculus as a real topic than that will be a bit more work and you should make sure you want that enough.  Even so, given what you tell us, I strongly urge you to work with &quot;easier&quot; books than hard ones.  You will get more out of something you don't give up on.  (Can always go back and do it harder if that is a need, later.)&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Take a look at &quot;Calculus Made Easy&quot; or &quot;Calculus for the Practical Man&quot;.  Both are written in a nonpompous style and are relatively easy in excluding some harder topics.  They are available as free (legit) pdfs; Google search for best download.  After you get one of those (I quite like the Thompson one in being almost fun to read), take a look at some of the book, see if you can learn from it, and consider how it pertains to your programming tasks.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The suggestions about Precaculus and being up on algebra (you sounded weak) were very good ones.  Not only is it hard to work on calculus with weak algebra but these are topics that are useful themselves (and perhaps even more applications than calculus; for example exponentials and rational functions are common in oil EUR programming on Tableau, Spotfire and Excel)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A cheap, good text in this area is Frank Ayres Schaum's Outline First Year College Mathematics which covers everything up to Calculus other than Geometry (which you don't need for Calculus) and even has a little intro to Calculus (which might be all you need or at least help you before you make the jump to a calculus text.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a link but you can try other booksellers.  I recommend the original 1958 version (lots of used versions available on the net).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/B0007DPVM2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Theory-problems-first-college-mathematics/dp/B0007DPVM2/ref=sr_1_1?s=books&amp;amp;ie=UTF8&amp;amp;qid=1513575322&amp;amp;sr=1-1&amp;amp;keywords=Schaum%27s+Outline+of+first+year+College+Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;If you have worked through the Ayres, you could look at &quot;normal textbooks&quot; like Thomas Finney, Swokowski, Stewart, etc.  But I worry that they are a little too formal (they get sold to professors or committees that select them and are used when a teacher is available to support with lots of lectures).  Better off with one of the suggestions from point 1 or perhaps some Dummies brand book or another Schaum's Outline just on calculus.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Disagree with the Velleman text.  It's not as bad as it sounds, are some good aspects to it.  But it's not a good suggestion for someone who self identifies as non mathy, mature, and needing calculus for work.  Would suggest it instead for a precocious math student self studying or even for a regular (strong) class.  There are also some places where it really emphasizes precision on limits and introduces new notation even.  Just not the right thing to worry about with someone who didn't make it to calc when he could have in school.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also look at Khan academy.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.khanacademy.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.khanacademy.org/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are both video and problem assistance and the training is pretty supportive and clear and gentle.  It may also appeal to you since you are into programming and it is a little techie in terms of the interfaces (some video game aspects of the problem solving, how the lectures are done on YT with an etchasketch).  Even just watching a quick video here might help you get a little motivated or intrigued to learn more.&lt;/p&gt;&#xA;" OwnerUserId="9198" LastActivityDate="2017-12-18T05:52:41.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13334" PostTypeId="1" CreationDate="2017-12-18T18:43:14.843" Score="1" ViewCount="508" Body="&lt;p&gt;Sources: 1 &lt;a href=&quot;https://www.theguardian.com/commentisfree/2016/mar/11/maths-isnt-problem-curriculum-lacking-imagination&quot; rel=&quot;nofollow noreferrer&quot;&gt;by Tim Gowers&lt;/a&gt;.&#xA;2 by &lt;a href=&quot;https://www.theguardian.com/commentisfree/2015/apr/29/people-hate-maths-my-plan-make-it-work-for-them&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marcus du Sautoy&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This question assumes that pre-calculus, probability &amp;amp; statistics ought still be taught in high school, and involves only those who won't need university-level math. Can studying Logic also teach you the non-quantitative benefits cited below (e.g. thinking, pattern searching, problem solving)?  If not, why are they exclusive to studying math?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;[1.]&lt;/h3&gt;&#xA;&lt;p&gt;Mathematics should be a tool for increasing one’s thinking power but for many children it is just a set of rather pointless rules for manipulating symbols.&lt;br /&gt;&#xA;[...]&lt;br /&gt;&#xA;It is therefore good for the health of a country if its population has high standards of mathematical literacy: without it, people are swayed by incorrect arguments, make bad decisions and are happy to vote for politicians who make bad decisions on their behalf.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;[2.]&lt;/h3&gt;&#xA;&lt;p&gt;But why should maths be privileged above learning a foreign language or history? Does everyone need to know what a &lt;a href=&quot;https://www.mathsisfun.com/algebra/trig-cosine-law.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;cosine&lt;/a&gt; is if the UK is to have a brighter future? Does the success of our economy depend on every citizen feeling confident factorising a &lt;a href=&quot;https://www.theguardian.com/science/video/2013/feb/01/solve-linear-quadratic-equations-tutorial-video&quot; rel=&quot;nofollow noreferrer&quot;&gt;quadratic equation&lt;/a&gt;? It may come as a surprise to you that I don’t think so, but I’m still a big believer in teaching maths to 18. What will be important is making sure that the maths we expose students to is both relevant to their future and the future of our country.&lt;/p&gt;&#xA;&lt;p&gt;What many are not aware of is that maths is so much more than the technical cogs that currently form the backbone of the curriculum. It is about pattern searching, extended analytical and logical thinking, problem solving. I am just embarking on making a new programme for the BBC about the beauty of algorithms. Many of the best algorithms contain no numbers or equations at all, but are full of mathematical thinking. And it is those algorithms that are creating efficient approaches to a whole range of business solutions, from the distribution of goods from supermarket warehouses to decisions about flight schedules at Heathrow airport.&lt;/p&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;p&gt;What about those humanities students or creative artists or vocational students who might argue that they will never need more maths? I believe that even these students, if exposed to the right curriculum, will recognise the benefits of more maths. I am doing an event with the Booker-winning novelist Ben Okri at the Hay festival next month about the connections between mathematical proof and literary narrative. As a novelist, Okri is the first to recognise the importance of a logically consistent narrative to the success of a novel – but also the wonderful benefit that a mathematical sensitivity to pattern and structure can give novelists as they create a narrative arch. From musical composition to carpentry, from street art to journalism, a mathematical mindset potentially gives one an edge.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://old.reddit.com/r/changemyview/comments/ccckve/cmv_math_after_a_middle_school_level_in_high/etn3a3d/&quot; rel=&quot;nofollow noreferrer&quot;&gt;This Reddit comment&lt;/a&gt; affirms that math is logic:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And mathematics is philosophy, it's just a lot less &amp;quot;wordy&amp;quot; and rather brought to the meta level. I mean just look how complicated you have to explain a valid and a sound argument and how nicely you can calculate with them in the boolean framework.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="15884" LastEditorDisplayName="user155" LastEditDate="2023-02-09T17:16:26.890" LastActivityDate="2023-02-21T05:55:44.323" Title="What does math teach students who won't need university-level math, that Logic can't?" Tags="|logic|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13335" PostTypeId="2" ParentId="13331" CreationDate="2017-12-18T23:08:14.817" Score="5" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Because many worthwhile topics in math don't require set theory to learn them and apply them.  You can get a lot of good stuff done with arithmetic, algebra, trig, calc, etc. that is uninformed by set theory.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Because set theory is hard for 5 year olds just because of the development of their intellect.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Because it is easier to learn set theory already having some experience in other areas of math.  (E.g. real, rational, complex numbers)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9198" LastActivityDate="2017-12-18T23:08:14.817" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13336" PostTypeId="2" ParentId="11950" CreationDate="2017-12-18T23:47:39.737" Score="1" Body="&lt;p&gt;I suggest taking a look at Polya's two volume set on induction:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematics_and_plausible_reasoning&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Mathematics_and_plausible_reasoning&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, the preface and first chapter have some good comments on learning by guessing, guessing well.  And on the differences between formal proof of ideas as opposed to derivation of them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Polya's spirit, I would actually try a few problems in the first chapter!  (He said even induction as a CONCEPT should be studied VIA INDUCTION.  Brilliant!)  But you really won't need to study the whole two volumes to get Oolya's main points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think there are some questions one could ask to chip around this question.  First do all Fields medalists think similarly?  What are the major commonalities and differences.  [The format of the question and even the Siobhan/Atiyah remarks seems to assume that they mostly think similarly and hey let's learn about it.  Could even be right.  But I wouldn't assume it from outset.  Even trying to look for differences amongtst the population maybe reveals similarities, even similarities that are different from the general population.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, perhaps writings of more famous &quot;magicians&quot; could be revealing.  Ramujan is an iconic example.  Of interest, it is how much he enjoyed and was inspired by math contest-y, manipulation style things like Edwards Calculus or that famous Tripos crib book.  [Things that Hardy disliked!] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third and this is a deeper (harder) area to evaluate but one to consider:  what is the difference between intelligence and creativity?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess other questions to ask the Fields guys are &quot;did you always think this way or did something change over time&quot;.  Consider Greg LeMond who said &quot;it never gets easier, you just go faster&quot;.  Is it like that?  Or different. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing to ask the Fieldsers could be &quot;do you see differences in how you think&quot; (versus math students, versus lesser professional mathematicians). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Polya has some interesting comments in Solve It about problem solving.  Even for very simple problems, it is possible for students (non Fields caliber!) to make some jump of intuition.  He gives the example of cube diagonal formula derivation.  And even for students that don't figure it out on own (hard), you can get them SOME of the feeling and process of mental &quot;jumping&quot; by using hints and gradual ones, rather than going straight to explaining it or to giving too much of a hint.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder if this in some ways is similar to the self discovery process involved in teaching by Socratic dialog (something I tend to do almost subconsiously...rather than just explaining how to do the car tire problem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;is there anything in this week's chapter that applies?&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Uh...maybe the ideal gas law?  But...blabla...don't know how to apply it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Well what does the ideal gas law say&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Recites it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;WEll from the problem statement, do we have ANY of those quantities listed&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Uh...OK.  I got this and that but missing X and Y.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;So is there anything similar to X?  Maybe not exactly same but transformable to X?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hmm...well this thingie (more help given if needed)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(more questions to push towards standard molar volume if that is the needed conversion factor)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Ok plug it in and see how it goes&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Does)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Now check your answer in the back&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Does.  Happiness.)&lt;/p&gt;&#xA;" OwnerUserId="9198" LastActivityDate="2017-12-18T23:47:39.737" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13340" PostTypeId="1" CreationDate="2017-12-19T15:50:24.240" Score="6" ViewCount="412" Body="&lt;p&gt;I am not sure this is the best or correct place to ask this question, but I thought I would give it a shot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am currently deployed overseas (from the US) and prior to this I was attending a community college and taking math courses (just finished Calc I).  Unfortunately, my schools requires a proctor for online courses and there is not anybody qualified at my location, which prevents me from taking courses while I am deployed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been searching for accredited mathematics course online (Calc II, Discrete Mathematics and Linear Algebra) but I not found anything worthwhile.  Any advice would be greatly appreciated.   &lt;/p&gt;&#xA;" OwnerUserId="9224" LastEditorUserId="122" LastEditDate="2017-12-23T01:56:02.170" LastActivityDate="2018-06-24T03:25:28.067" Title="Online Accredited Mathematics Course" Tags="|undergraduate-education|online-instruction|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="13341" PostTypeId="1" CreationDate="2017-12-19T18:46:35.047" Score="5" ViewCount="137" Body="&lt;p&gt;I'm trying to find a program, or set of programs, that will allow easy markup of student assignments submitted as PDFs.  I'd like to include more complicated math with this markup.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've tried pen tools, like the one in Acrobat DC, but they tend to be too messy to read.  (I have an old Bamboo pen tablet which isn't terribly sensitive.)  Right now my best setup is typing in Word using MathType for the math and using the snipping tool to transfer it to the PDF in Acrobat.  It's pretty clunky.&lt;/p&gt;&#xA;" OwnerUserId="9225" LastActivityDate="2017-12-21T16:32:00.617" Title="Marking PDF Files" Tags="|grading|software|online-instruction|" AnswerCount="1" CommentCount="4" ClosedDate="2017-12-23T03:54:44.027" ContentLicense="CC BY-SA 3.0" />
  <row Id="13343" PostTypeId="2" ParentId="13252" CreationDate="2017-12-20T06:12:23.203" Score="3" Body="&lt;p&gt;From the perspective of a professional statistician, I often find myself asking the opposite question: why do we spend so much time learning about closed form solutions in our calculus classes?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I say this is that for the majority of applied statistics problems that require any sort of integration (such as the 95% of the field of Bayesian statistics), there is no closed form solutions and numerical methods are required. Similar for optimization problems (i.e. maximum likelihood estimation): some simple problems have closed form solutions (i.e. linear regression)...but that vast majority of problems are solved iteratively. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So from &lt;em&gt;my&lt;/em&gt; professional perspective, we spent a lot of time learning closed form solutions to various problems...but when it comes time to do the math behind any new statistical methods, 99.9% of the time we end up using iterative methods instead.&lt;/p&gt;&#xA;" OwnerUserId="9227" LastEditorUserId="9227" LastEditDate="2017-12-20T14:35:47.537" LastActivityDate="2017-12-20T14:35:47.537" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13344" PostTypeId="1" AcceptedAnswerId="13366" CreationDate="2017-12-20T12:46:11.350" Score="-1" ViewCount="161" Body="&lt;p&gt;I was looking for a good explanation text on single-variable calculus, especially limits, introduction and examples. I found this &lt;a href=&quot;http://samples.jbpub.com/9780763749651/59957_CH02_CalcCONFIRMING.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pdf&lt;/a&gt; and I really like it and would like to read the whole book. I was looking into it, but couldn't find the author or the textbook name.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know it?&lt;/p&gt;&#xA;" OwnerUserId="9228" LastActivityDate="2017-12-26T16:04:37.910" Title="What textbook is this chapter from?" Tags="|calculus|textbooks|" AnswerCount="1" CommentCount="6" ClosedDate="2017-12-28T12:14:29.663" ContentLicense="CC BY-SA 3.0" />
  <row Id="13345" PostTypeId="2" ParentId="13341" CreationDate="2017-12-20T14:48:52.290" Score="2" Body="&lt;p&gt;I don't think it's perfect for your situation, but check out &lt;a href=&quot;https://crowdmark.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;crowdmark&lt;/a&gt;. When grading you can type comments in MathJax (LaTeX-like math markup). However, from how I've used it, their intended setup is a bit different than yours. Crowdmark's idea is that you make the exams on the crowdmark website, print them out and give those to the students, and then scan and reupload to crowdmark. I'm not sure if you can just upload your own PDFs to their site. &lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2017-12-21T16:32:00.617" LastActivityDate="2017-12-21T16:32:00.617" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13346" PostTypeId="1" CreationDate="2017-12-20T17:13:25.367" Score="10" ViewCount="171" Body="&lt;p&gt;I'm looking for a theoretical framework to classify a task based on its cognitive demand. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I only have the Smith and Stein's (1998) proposal and PISA framework such as my principal references. In particular, the PISA categories which help us to classify items in standarized tests: Reproduction, Connection and Reflection. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you in advance. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Smith, M. S., y Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. &lt;em&gt;Mathematics Teaching in the Middle School, 3(5)&lt;/em&gt;, 344-350.&lt;/p&gt;&#xA;" OwnerUserId="1789" LastActivityDate="2017-12-21T09:38:33.690" Title="Cognitive demands of a mathematical task" Tags="|mathematical-pedagogy|assessment|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13347" PostTypeId="2" ParentId="13346" CreationDate="2017-12-20T22:05:47.860" Score="4" Body="&lt;p&gt;One theoretical framework for designing actives and assessments is ETS's &lt;a href=&quot;https://www.ets.org/Media/Research/pdf/RR-03-16.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Evidence Centered Design&lt;/a&gt; framework, although it requires long term analysis of a task or series of tasks in conjunction with other methods of measurement of ability. The basic idea is to use  rigorous assignment/activity/test design process that allows more concrete conclusions to be drawn about the skills actually being tested. Item Response Theory is used to classify the difficulty and discrimination of actives. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said I'm not sure if this is what you're looking for, it's more a design process framework than a framework for understand the cognitive components of a task. &lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2017-12-20T22:05:47.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13348" PostTypeId="2" ParentId="13346" CreationDate="2017-12-21T09:38:33.690" Score="5" Body="&lt;p&gt;Extending the answer from Nate Bade: Within ECD (Evidence Centered Design), there are quite a few possible models for 'cognitive demand'. Probably the widest use within LSA (large scale assessment) such as PISA, but also for example &lt;a href=&quot;http://www.smarterbalanced.org&quot; rel=&quot;noreferrer&quot;&gt;Smarter Balanced&lt;/a&gt; or &lt;a href=&quot;https://parcc-assessment.org&quot; rel=&quot;noreferrer&quot;&gt;PARCC&lt;/a&gt;, models are based on Webb's concept of DOK (Depth of Knowledge). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;General resources of Webb's concept can easily be found on the internet, so I won't even start listing resources. Specifically for Math, I find the conception of &lt;em&gt;cognitive rigor&lt;/em&gt; by &lt;a href=&quot;http://www.karin-hess.com/cognitive-rigor-and-dok&quot; rel=&quot;noreferrer&quot;&gt;Kathrin Hess&lt;/a&gt; pretty enlightening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, one can start discussing the relevance of these models. This is not a new question, but discussed (or at least commented...) on MESE:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/q/12055/9004&quot;&gt;Which mental model is most useful when thinking about math education: Webb's DoK, CCSM's Aspects of Rigor, or Bloom's Taxonomy? Are there others?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2017-12-21T09:38:33.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13349" PostTypeId="2" ParentId="6135" CreationDate="2017-12-21T19:45:53.180" Score="11" Body="&lt;p&gt;If you do not wish to push for optimal hypothesis (on the function), or for optimal conclusion (i.e., you can live with the &quot;error bound&quot;), I think what I'm giving below would be a perfectly fine approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with the Fundamental Theorem of Calculus&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ f(x) = f(a) + \int_a^x f'(x_1)\,dx_1 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and substitute the integrand by&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ f'(x_1) = f'(a) + \int_a^{x_1} f''(x_2)\,dx_2 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;so that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \begin{align} f(x) &amp;amp;= f(a) + \int_a^x \left( f'(a) + \int_a^{x_1} f''(x_2)\,dx_2\right)\,dx_1 \\&#xA;&amp;amp;= f(a) + f'(a)(x-a) + \int_a^x\int_a^{x_1} f''(x_2)\,dx_2\,dx_1. \end{align} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if $|f''(x_2)|\leq M$ for all $a&amp;lt;x_2&amp;lt;x$, we have a simple bound for the remainder:&#xA;$$ \left|\int_a^x\int_a^{x_1} f''(x_2)\,dx_2\,dx_1 \right| \leq \int_a^x\int_a^{x_1} M\,dx_2\,dx_1 = \frac{M}{2}(x-a)^2. $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly we can readily deduce (the &quot;Taylor polynomial&quot; and) the remainder $R_n$ as an iterated integral, which then yields the Lagrange error bound&#xA;$$ |R_n| \leq \frac{M}{n!}(x-a)^n $$&#xA;($M$ being a bound for $f^{(n)}$ on the interval). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With a little bit of multi-variable thinking, we can even get the Lagrange remainder:&#xA;$$ R_n = \frac{f^{(n)}(\xi)}{n!}(x-a)^n \quad\text{for some}\quad \xi\in(a,x)$$&#xA;(a simple application of extreme value theorem and intermediate value theorem). The hypothesis for all this to work is that $f\in C^n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I probably should ask a separate question that I am puzzled by: Where do we truly need the actual remainder, in any of the forms — exact but with an unknowable $\xi$)? It seems to me that the error bound suffices for all applications, such as showing convergence of Taylor series). I understand the Mean Value Theorem is used in a crucial step in the Fundamental Theorem of Calculus, but where does the Lagrange remainder (for $n\geq 2$) show up other than to deduce the error bound (which as I just showed is a much simpler result)?&lt;/p&gt;&#xA;" OwnerUserId="9235" LastEditorUserId="9235" LastEditDate="2017-12-26T02:12:04.570" LastActivityDate="2017-12-26T02:12:04.570" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13350" PostTypeId="2" ParentId="12221" CreationDate="2017-12-21T20:09:35.643" Score="1" Body="&lt;p&gt;Also consider linear algebra.  Given your interests one emphasizing matrix manipulation rather than theory is preferable, for now at least.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. In general, when accelerating in high school try to take courses that are mainstream in both topic (used in many majors) and treatment (how commonly taught) as that keeps your options most open. Also consider time efficiency. &lt;/p&gt;&#xA;" OwnerUserId="9236" LastEditorUserId="11" LastEditDate="2017-12-21T22:18:15.060" LastActivityDate="2017-12-21T22:18:15.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13352" PostTypeId="2" ParentId="377" CreationDate="2017-12-22T00:16:32.207" Score="8" Body="&lt;p&gt;Please refer to &lt;a href=&quot;http://web.math.rochester.edu/people/faculty/rarm/the_new_math.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Whatever became of the New Math?&lt;/em&gt;&lt;/a&gt;, a series written by Professor Raimi of the Department of Mathematics of the University of Rochester, and also &lt;a href=&quot;http://www.americanheritage.com/content/whatever-happened-new-math-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Whatever Happened To New Math?&lt;/em&gt;&lt;/a&gt;. Together with PSSC in physics education, SMSG (most popular of the New Math) actually was initiated from professional mathematicians, started to produce high school textbooks (not the elementary school).&lt;/p&gt;&#xA;&lt;p&gt;I found many articles online (or even a book) details the New Math, but there are not much postmodem on PSSC.&lt;/p&gt;&#xA;&lt;h1&gt;Some excerpts from &lt;em&gt;What happened to new math?&lt;/em&gt;&lt;/h1&gt;&#xA;&lt;p&gt;The differences in experience and outcomes were as varied as the nation’s geography and, most important, only as good as the teachers. But what new math became is not what it was intended to be. In fact, there never was just one new math.&lt;/p&gt;&#xA;&lt;p&gt;Popular sentiment was beginning to shift, and with it congressional enthusiasm for the financing of educational reform. Good teaching and good textbooks could not be separated from bad. And for every bright student with a thirst for math, there was one who had trouble figuring the charges on his paper route. New math got no credit for the enthusiasm and all the blame for the ignorance, even in those school districts where it was never seriously adopted.&lt;/p&gt;&#xA;&lt;p&gt;Where it had been successful, it lingered on in the teaching techniques of individual instructors and in watered-down new-math textbooks, which are still evident in elementary and high schools today. Because it put such stock in creativity and the abstract, new math appealed most to the brighter, college-bound students, who, some have argued, probably would have done well anyway. Average and marginal students could suffer dearly at the hands of an uninspired teacher and a poor textbook, and frequently did. But that spirit-crushing combination could exist independently of new math too.&lt;/p&gt;&#xA;&lt;p&gt;There is a excellent speech by French mathematician Laurent Lafforgue &lt;a href=&quot;http://web.math.rochester.edu/people/faculty/rarm/whyschool.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why the school&lt;/a&gt;. In his speech, he talked about the purpose of schooling and what should be taught in school in general. he stressed the school should teach only knowledge, not thinking, for thinking cannot be taught. The teaching must nourish. Does it mean we still need to teach Euclid's plane geometry?&lt;/p&gt;&#xA;" OwnerUserId="9238" LastEditorUserId="2083" LastEditDate="2024-02-14T08:40:31.080" LastActivityDate="2024-02-14T08:40:31.080" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13353" PostTypeId="1" CreationDate="2017-12-22T23:20:52.737" Score="44" ViewCount="2473" Body="&lt;p&gt;&lt;strong&gt;tl;dr:&lt;/strong&gt; Some students expect to be told &quot;what's on the test&quot;, to memorize and then move on. What can be done to change &lt;em&gt;how&lt;/em&gt; they learn while teaching them &lt;em&gt;what&lt;/em&gt; to learn?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Context:&lt;/strong&gt; Introductory, foundational course for undergraduates, like College Algebra or Precalculus or even Calculus I, where most of the students are science majors and must complete some portion of the Calculus sequence for their department.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Observation:&lt;/strong&gt; It seems as though many younger undergraduate students have difficulty learning mathematics because of the way they were taught in high-school, where the focus was more on rote, procedural knowledge and &quot;teaching to the test&quot;. In college/university, the focus is much more on conceptual understanding, and it is assumed that students will work independently outside of class to enhance their understanding. In addition, students will be assessed on that conceptual understanding and will not merely be asked to regurgitate memorized formulas or solve problems identical to some model problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Main question: What can I do, as a teacher, to help students break this mold and get used to learning mathematics in a deeper way (and being assessed on that deeper understanding), while simultaneously actually teaching them the material?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note: &lt;strong&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/2098/what-to-do-with-students-who-think-they-already-know-it-but-actually-dont&quot;&gt;MESE 2098&lt;/a&gt;&lt;/strong&gt; already addressed issues of students who feel like they already understand a course because they took it in high-school. I am not concerned here with those issues about students' content knowledge. I am focused on assessment, expectations, and learning behavior.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Expectations:&lt;/strong&gt; A good answer will share suggestions for activities or assessments or discussion topics to use that will teach students how to learn mathematics and teach them about the expectations of college-level mathematics. I'd prefer to keep any discussion about &lt;em&gt;why&lt;/em&gt; this &quot;high-school attitude&quot; persists to the comments, unless you find such a discussion germane to your answer. I'm more interested in &lt;em&gt;what can be done now&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Motivation:&lt;/strong&gt; I have noticed this kind of behavior in some students in the past, but I recently dealt with a course where this attitude was widespread. Students complained to me (and even other teachers) about quizzes and exams that assessed their conceptual understanding. For example, after a unit on equations, functions, and graphs, a midterm exam included questions like the following. (Parentheticals are what I expected as an answer and to which I would give full credit, even with such brevity.)&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If $f(x)$ is a function and I know its graph, explain how to find the graph of $-3\cdot f(x-2)$. (&quot;Flip it across the $x$-axis, stretch it vertically by a factor of 3, shift to the right 2 units.&quot;)&lt;/li&gt;&#xA;&lt;li&gt;If $g(x)$ is a function, explain how to determine whether it has even symmetry, &lt;em&gt;without&lt;/em&gt; knowing its graph. (&quot;Determine whether $g(-x)=g(x)$ for every input $x$.&quot;)&lt;/li&gt;&#xA;&lt;li&gt;If $h(x)$ is a function and I know its graph, explain how to find the graph of $h^{-1}(x)$. (&quot;Reflect it across the line $y=x$.&quot; Or, &quot;take every point $(x,y)$ and swap the coordinates to be $(y,x)$.&quot; I would have even given extra credit if they added, &quot;First use the Horizontal Line Test to see if it is 1-to-1 and therefore invertible.&quot;)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;After this exam, I gave an (anonymous) survey to solicit feedback from students about how the course was going. Several students complained about those kinds of questions. The quote below is not the only one, but it exemplifies the issue behind my main question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;It's really hard to explain how to do math problems or the concepts&#xA;  behind them on tests and quizzes. The time crunch makes it hard to&#xA;  think when you're ready to complete problems, not explain something.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Many other students said something to the effect of, &quot;You should tell us what's gonna be on the tests and quizzes so we know what to study.&quot; This is despite the fact that I created an assignment containing lots of practice problems that reflect the content they should know, in addition to making a list of major topics. So, while part of me says, &quot;Okay, I guess that this may be the first time you're being asked questions like this,&quot; another part of me says, &quot;What do you want me to do, tell you &lt;em&gt;exactly&lt;/em&gt; what I'm going to ask in advance?&quot; And I fear that, yes, this is (almost) what they expect because it's what they are used to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did discuss this in class with the students: I tried to explain why conceptual understanding is essential, and I said that unless you can explain a concept to someone else then your actual understanding of that concept is superficial or fuzzy at best. However, I fear that this only demotivated the students, and that what they heard was not, &quot;You need to be better about this,&quot; but rather, &quot;You're not good and will never be good at this.&quot; I would like them to understand that conceptual understanding is important and I want them to strive for that deeper understanding. How can I help them see that as a goal, in the first place, and then guide them towards it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Justification:&lt;/strong&gt; I think this question belongs on MESE because I imagine this behavior is more prevalent in mathematics courses than in those of other subjects. I am sure that teachers in all disciplines lament poor background skills or learning habits of their students, but I doubt there are students in a History class complaining that they had to write an essay response on an exam.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2021-11-10T00:31:52.047" Title="Teaching undergraduates who expect a high-school-like learning environment" Tags="|undergraduate-education|student-motivation|assessment|struggling-students|" AnswerCount="4" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="13355" PostTypeId="2" ParentId="13166" CreationDate="2017-12-23T19:42:34.093" Score="3" Body="&lt;p&gt;PBL, especially if you have a lot of students that struggle with mathematics, is definitely something that you will have to design carefully; ordinary project based approaches can sometimes exacerbate already existing difficulties.  At the same time, it is important not to lower the rigor or expectations for those who struggle, and so the standard recommendation that folks might offer of differentiated instruction, where you give instruction at different levels and with different emphases to different students, is probably ill-advised, particularly if you are hoping to have all students work together on projects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having reviewed the literature in this area, I think there is one study that really stands out as an example of what you would ideally be striving for. Bottge et. al (2007) demonstrated the effectiveness of a method called anchored instruction with students with learning disabilities.  In anchored instruction, students are presented with a video case (or &quot;anchor&quot;) that serves as a source of data for a project-based lesson.  More basic mathematical skills are presented to the students in the context of the problem, on an as-needed basis, rather than being taught prior to engaging in the project.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You do mention that you worry about your students being on different levels and trying to keep them together working on the same material.  You might consider using an instructional system known as Complex Instruction (Cohen and Lotan, 2014).  In Complex Instruction, teachers carefully design both intellectually rigorous and accessible tasks for students to work on in groups and then carefully observe (and sometimes intervene) to make sure that all students are actively participating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bottge, B. A., Rueda, E., LaRoque, P. T., Serlin, R. C., &amp;amp; Kwon, J. (2007). Integrating Reform-Oriented Math Instruction in Special Education Settings. Learning Disabilities Research &amp;amp; Practice, 22(2), 96–109. &lt;a href=&quot;https://doi.org/10.1111/j.1540-5826.2007.00234.x&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1111/j.1540-5826.2007.00234.x&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cohen, E. G., &amp;amp; Lotan, R. A. (2014). Designing Groupwork: Strategies for the Heterogeneous Classroom Third Edition. Teachers College Press.&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2017-12-23T19:42:34.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13356" PostTypeId="2" ParentId="13353" CreationDate="2017-12-23T22:00:55.053" Score="5" Body="&lt;p&gt;Is there less proof of conceptual understanding if you've shown them each type of question beforehand? I have some questions for calculus I that I think address conceptual understanding, and we practice those sorts of questions before the test. Students may have understood the material, but just not known how to respond to those questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also allow each test to be taken at least twice. My tests are split up into mini-tests which each center on one big idea. The ones I can easily make many versions of, they can retake in my office. For the others, we find a time they can all meet, outside of classtime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've laid it out in &lt;a href=&quot;http://mathmamawrites.blogspot.com/2013/01/calculus-mastery-tests-sbg.html&quot; rel=&quot;noreferrer&quot;&gt;this blog post&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-12-23T22:00:55.053" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13357" PostTypeId="2" ParentId="13353" CreationDate="2017-12-24T01:00:48.553" Score="8" Body="&lt;p&gt;I find class voting/clickers a good way to get students used to thinking about conceptual stuff.  In fact, I often frame my lectures around three progressively more challenging conceptual questions.  Sometimes I write my own, but &lt;a href=&quot;https://web.archive.org/web/20170107104719/https://math.cornell.edu/%7EGoodQuestions/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://mathquest.carroll.edu&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; are some models of good, interesting questions.&lt;/p&gt;&#xA;&lt;p&gt;For a good overview of the depth of learning that can come from applying this technique in the classroom, there's a nice &lt;a href=&quot;https://www.youtube.com/watch?v=CltyNirNaxk&amp;amp;t=749s&amp;amp;index=1&amp;amp;list=WL&quot; rel=&quot;nofollow noreferrer&quot;&gt;video&lt;/a&gt; by Derek Bruff.&lt;/p&gt;&#xA;&lt;p&gt;The typical cycle that I use is that students are asked thought provoking questions, asked to vote on an answer, asked to discuss the question with a partner and then asked to vote on the question again.  Then, we discuss it as a class.  These mini assessments, I find, help them get ready for the kinds of conceptual questions that I want to put on an exam.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="14134" LastEditDate="2021-11-10T00:31:52.047" LastActivityDate="2021-11-10T00:31:52.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13358" PostTypeId="2" ParentId="12055" CreationDate="2017-12-24T17:11:06.563" Score="1" Body="&lt;p&gt;I think that any response to this question would just be a matter of taste. It's hard to say anything objective in the way of one model being &lt;em&gt;more useful&lt;/em&gt; than another. I would recommend abandoning the thought that there is a most useful model. Instead, read about these three models, and about any other models you find, looking only to &lt;em&gt;understand why&lt;/em&gt; the authors decided that their model represents how math should be taught or learned. Then from this understanding, form your own model of how to teach mathematics. I think that you can safely disregard these models as immutable rules, but instead think of them only as scaffolding to build your own philosophy on teaching mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something does need to be said in support of designing models like these in the first place, though. Teaching feels very natural to some people, and to these people models like these feel silly, like a &lt;a href=&quot;https://food52.com/blog/20083-the-scientific-10-step-way-to-eat-a-sandwich&quot; rel=&quot;nofollow noreferrer&quot;&gt;formalization of how to eat a sandwich&lt;/a&gt;. But teaching doesn't come so naturally to everyone. Or at the very least not everyone has developed a coherent philosophy of how to teach yet. I think this is especially true of some early-career mathematics educators, or of math education students who have so far thought about teaching mathematics very little in their lives. Having models like these provide scaffolding, a jumping-off point, for people like this to begin developing their own thoughts on teaching. They provide a solid base to stand on while maturing as a mathematics teacher.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2017-12-24T17:11:06.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13359" PostTypeId="2" ParentId="11589" CreationDate="2017-12-24T20:47:30.057" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I'm trying to find more problems suitable for early college students (students who know algebra and calculus) that involve translating words into mathematical notions, [...] problems which seem impenetrable when taken as a whole, but which can be reduced to something simple by breaking apart the meaning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am not aware of a specific resource for problems like this, but I also wish there were one because it does seem that students (even advanced math majors) struggle with this aspect of problem-solving. I trust that we all try to model this problem-solving behavior when we work out problems for students (assessing known and unknown information, deciding what to try next, reassessing from results) but it seems like students have trouble developing these skills. It would indeed be nice to have a workbook of problems that target this skill of problem-solving. In addition, it may serve as a way to teach students to synthesize their knowledge from various courses, which can often feel to them like discrete, disconnected islands in the vast sea of mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I do think it would not be too challenging for you to create such problems on your own by taking some known fact and &quot;translating/encoding&quot; the ideas appropriately into definitions. (I do think this will be &lt;em&gt;time-consuming&lt;/em&gt;, but not necessarily challenging.) Essentially, do the work that you expect the student to do, but in reverse!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below, I will show a few examples of what I have in mind to not only provide example problems but also show one may create many of their own such problems. Each example is titled with the essential fact upon which it is based. (I created this list of examples by trying to conjure interesting facts and seeing what came to my mind.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 1: Tangent line to a circle is perpendicular to its radius&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a well-known fact that students likely prove in a high-school geometry course but may never revisit. The following problem has them use calculus and basic algebra to see this fact in a new light.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consider the circle defined by the equation $x^2+y^2=r^2$, and let $P=(u,v)$ be any point on the circumference of that circle. Use implicit differentiation to find the slope of the tangent line at $P$, and then show that the tangent line is perpendicular to the line from the origin to $P$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You could even be less explicit about expectations by removing &quot;to find the slope of the tangent line&quot; and expecting students to identify that &quot;perpendicular&quot; means &quot;opposite reciprocal slopes&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could even design the problem to assess their skill in dealing with arbitrariness by not providing the equation $x^2+y^2=r^2$ and instead saying: &quot;Consider a circle of radius $r$ and any point $P$ on its circumference&quot;, and replacing &quot;origin&quot; later with &quot;center of the circle&quot;. This new version relies on the student choosing an appropriate coordinate system and notation.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 2: The symmetry of a parabola across its vertex&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Algebra and pre-calculus students are taught completing the square and &quot;vertex form&quot;, but I find that students later in calculus do not appreciate (or think to use) the symmetry of a parabola. This problem shows them that symmetry using calculus.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f$ be an arbitrary quadratic polynomial. If $x=a$ and $x=b$ are two inputs that are equidistant from the $x$-coordinate of the vertex, show that (i) $f(a)=f(b)$ and (ii) the tangent lines at the points on the graph of $f$ for which $x=a$ and $x=b$ intersect at a point whose $x$-coordinate is the same as that of the vertex.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You could change (ii) a little bit as follows: &quot;the tangent lines to the points $(a,f(a))$ and $(b,f(b))$ and the vertical line through the vertex intersect at one common point.&quot; However, I find that recognizing that &quot;the point on the graph of $f$ corresponding to the input $x=a$&quot; means &quot;the point $(a,f(a))$&quot; is occasionally not at all obvious to a student(!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This problem also allows for multiple approaches. Will the student take an arbitrary function $f(x)=Ax^2+Bx+C$ and work with it twice to find the tangent lines and find their intersection point? Or will they complete the square to &quot;encode&quot; the symmetry the problem gets at in the form of the function itself, thereby making the rest of the problem easier?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 3: The derivative of an inverse function at some input is the reciprocal of the derivative of the original function at its corresponding input&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Okay, this idea is stated much better in theorem form (as seen in a Calculus 1 course):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $f$ is invertible and differentiable in an interval containing $x_0$, and $x_0$ satisfies $f'(x_0)\neq 0$, then $f^{-1}$ is differentiable at $y_0=f(x_0)$ and $(f^{-1})'(y_0)=\frac{1}{f'(x_0)}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can construct a problem from this in many ways, by perhaps giving a specific function and having the student find its inverse as part of the problem; or, more generally, by providing a class of functions that are invertible and asking for the same steps (like $f(x)=\frac{ax+b}{cx+d}$, provided some conditions on $a,b,c,d$, that you may even ask the student to find). Here's an example of the simpler variety:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consider $f(x)=x^3$. Show that $(2,8)$ is on the graph of $f$, and $(8,2)$ is on the graph of $f^{-1}$. Then, show that the tangent lines at those points have reciprocal slopes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Other ideas:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Ask about generalizations.&lt;/strong&gt; For example, take the problem you posed in the question and follow it up with, &quot;Does the same property hold for cubic functions? Quartic functions?&quot; This will hopefully encourage students to ask these kinds of questions on their own in the future.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Ask for examples &amp;amp; counterexamples.&lt;/strong&gt; I find that creating (counter)examples to be an essential skill in my own learning and understanding of mathematics, and I find this skill is too often given short shrift in actual course-work. An example problem might be, &quot;For each situation, give an example or explain why it is not possible: a function $f$ and a tangent line at the point $(a,f(a))$ on the graph of $f$ that intersects the graph of $f$ (i) 0 times, (ii) exactly once, (iii) exactly twice, (iv) infinitely many times.&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="80" LastActivityDate="2017-12-24T20:47:30.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13360" PostTypeId="2" ParentId="535" CreationDate="2017-12-24T20:53:57.060" Score="3" Body="&lt;p&gt;I think that the computer is an important thing to consider in an introductory mathematics class, and this may mean simply interacting with the internet and wolfram alpha, or using computational tools to solve mathematics problems.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I were you, I would be incredibly excited to not have to spend weeks in the drudgery of differentiation techniques.  I would ask myself what kind of calculus questions would you feel comfortable about presenting if you were to allow your students full access to wolfram alpha. WA cannot answer most important calculus questions, however it can do some basic computation and symbolics.  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;To &lt;strong&gt;eliminate dishonesty&lt;/strong&gt; simply ask your own questions, and encourage students to use technology and the internet, including supporting the use of sites like stack overflow!  Have them ask questions here, or use a class Piazza board.  There is nothing wrong with talking about problems.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Farm the mechanical stuff out to a machine and let the students use WA or something else to answer it.  Nothing wrong with software for problem delivery if used wisely. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have students present their work and expect the work to be polished.  This will show the students why to use technology.  Demand that they incorporate &lt;em&gt;words, tables, graphs, symbols,&lt;/em&gt; and other diagrams to make their arguments, and have them present these to others regularly.  I'm amazed we still ask students in college to scribble work with paper and pencil simply because it's mathematics class.  I begin my semester having students bring in scratch work and using short writing labs to work on turning these into more polished (typed!) solutions.  I use $\LaTeX$, Python and Jupyter notebooks, but you could do this with google docs and wolfram alpha if you wanted. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-12-24T20:53:57.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13361" PostTypeId="1" CreationDate="2017-12-24T22:39:26.857" Score="15" ViewCount="378" Body="&lt;p&gt;I teach exercise sessions for a Linear Algebra course for 1st semester students in Europe. Students have to prepare some exercises at home. In class, I call on students to present their solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One big problem that I see is that in exercises of the form &quot;prove or disprove&quot;, students' disproofs usually ends with an equation with variables instead of a concrete counterexample. For example, to prove that a certain statement does not hold true, a student proof may go like this: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If the statement does hold, then [...] And in the end, we find that, for all real numbers $x,y$, that $x=y$. This is clearly not true, so the original statement is false.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When I tell my students, &quot;This proof is okay, but a concrete counterexample would be nicer&quot;, they usually say, &quot;But from this equation I can see it just as clearly&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see three main reasons to advocate for concrete counterexamples instead:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;They force you to actually think about the existence of a counterexample, or to recognize that the resulting equation is &quot;vacuous&quot;. (However, with some easy Linear Algebra examples, this does not seem to be an issue.)&lt;/li&gt;&#xA;&lt;li&gt;The proof looks simpler and more convincing for the reader. (But admittedly, not so for my students. It seems they would they have to derive the equation, find a concrete counterexample, and then write the disproof again with their newly discovered counterexample.)&lt;/li&gt;&#xA;&lt;li&gt;It is the usual, accepted style.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;However, I tell my students these reasons repeatedly and it seems to have no effect. A related issue is that they often &quot;prove&quot; a universally quantified statement via an example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Are there any easy examples where you could easily find an empty equation and no counterexample which I could show to my students?&lt;/li&gt;&#xA;&lt;li&gt;Has there been any reasons about why thinking in counterexamples is hard for fresh(wo)men?&lt;/li&gt;&#xA;&lt;li&gt;What can I do to improve the situation?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9240" LastEditorUserId="80" LastEditDate="2017-12-29T03:25:25.047" LastActivityDate="2017-12-29T03:25:25.047" Title="How to teach students the value of concrete counterexamples?" Tags="|proofs|logic|writing|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13362" PostTypeId="1" CreationDate="2017-12-25T09:13:42.973" Score="3" ViewCount="130" Body="&lt;p&gt;I volunteered abroad in rural Ethiopia and a village in a remote mountainous area of China. I plan to return to both places and teach for as long as I can make it work. I want to focus on STEM, but first I would like to reteach myself math and science from pre-school all the way to college-level math, such as Calculus, Physics, etc., so that I am confidently able to teach them. Almost none of them speak English, so I will be teaching them that as well along the way, since I can speak Amharic, Mandarin and Cantonese.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;BUT I would like to teach them from a &lt;strong&gt;real-world approach where I am able to immediately show them the practical usefulness of what I'm teaching them&lt;/strong&gt;. I've seen the website &quot;Mathalicious&quot; and it is inspiring. But I would like to get a set of textbooks that are widely used in academia to get them from a pre-school level of understanding of these subjects, where the only required prerequisite is to understand English, to the advanced levels. And I have also seen websites like Khan Academy but I would like something that is more official and mimics the curriculum progression flow that is standard in public/private school settings. Eventually, I hope to find a way to get at least a few children here somehow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I've found so far: &lt;a href=&quot;http://www.doe.mass.edu/frameworks/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.doe.mass.edu/frameworks/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This website lists objectives and goals, but doesn't list any textbooks to follow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.hawaiipublicschools.org/TeachingAndLearning/StudentLearning/CommonCoreStateStandards/Pages/home.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.hawaiipublicschools.org/TeachingAndLearning/StudentLearning/CommonCoreStateStandards/Pages/home.aspx&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the only state website I've seen so far that includes state-approved textbook materials that I can go through. It only lists Reading and math so far and the list is not really detailed at all though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to be as specific as possible, I am searching for a list of good (text)books (based on your professional experience) that I can work through to arrive at advanced math and science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully, I can get an answer like, &quot;Follow this progression of books &quot;xxx&quot;. Ask here if you need help figuring out how to apply it to the 'real world'.&quot;&lt;/p&gt;&#xA;" OwnerUserId="9243" LastEditorUserId="9243" LastEditDate="2017-12-25T22:42:09.880" LastActivityDate="2017-12-27T16:39:56.433" Title="books to teach K-12 math/science via real-world approach?" Tags="|textbooks|self-learning|reference-books|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13364" PostTypeId="2" ParentId="13361" CreationDate="2017-12-25T14:44:06.763" Score="6" Body="&lt;p&gt;Perhaps &quot;thinking in counterexamples is hard&quot; because that requires thorough understanding, a grasp of the boundaries where a claim holds, and where it does not. For example, is matrix multiplication commutative, i.e., does $AB = BA$?&#xA;This hypothesis is best refuted by a counterexample, say, 3D rotations. But then it raises the question of what conditions ensure that matrix multiplication is commutative. $\textrm{SO}(2)$ is commutative, as is the subset of $\textrm{SO}(3)$ representing rotations about a common axis. Exploring the boundary between truth and falsity leads to robust understanding of the involved concepts. &#xA;For matrix commutativity, one might be led to simultaneous diagonalization.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2017-12-25T14:44:06.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13365" PostTypeId="2" ParentId="13361" CreationDate="2017-12-25T17:09:31.450" Score="5" Body="&lt;p&gt;If you're speaking about counterexamples in general, the following aphorism really struck me:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In order to understand what something is, you must also understand what it is not.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One of the main values for knowing lots of counterexamples is that they give examples of what something is not.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastEditorDisplayName="user797" LastEditDate="2017-12-25T17:15:42.233" LastActivityDate="2017-12-25T17:15:42.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13366" PostTypeId="2" ParentId="13344" CreationDate="2017-12-26T16:04:37.910" Score="0" Body="&lt;p&gt;Jefferey Thompson provided this &lt;a href=&quot;http://www.jblearning.com/catalog/9780763759957/preview/&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt; to Calculus: Early Transcendentals, Fourth Edition.&#xA;&lt;a href=&quot;https://www.google.ca/search?q=Calculus%3A%20Early%20Transcendentals%2C%20Fourth%20Edition&quot; rel=&quot;nofollow noreferrer&quot;&gt;Googling this&lt;/a&gt; title shows various online purchase options for the various editions.&lt;/p&gt;&#xA;" OwnerUserId="9250" LastActivityDate="2017-12-26T16:04:37.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13367" PostTypeId="1" CreationDate="2017-12-26T16:46:06.253" Score="11" ViewCount="929" Body="&lt;p&gt;I am now tutoring a student in &lt;strong&gt;Grade 9&lt;/strong&gt;, who falls behind in math study. He lacks the basic understanding of operations and inverse operations, and have trouble dealing with negative numbers and fractions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a consequence, he knows how to solve an equation like $2x+3=7$, but fails to solve equations like $2x+3=1$. When I asked him what number times 2, and then plus 3 equals 1, he hesitated and answered: there is no such number. Then I told him to pay attention to the last operation performed on the LHS and use the inverse operation, he then figured out the right answer. Now try $7-3x=10$. He found that the last operation is subtraction, so he added $7$ to both sides. I then told him $7-3x$ actually means $7+(-3x)$ and we need to subtract, but he seemed confused. He didn't know  subtraction is addition of the opposite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same thing goes for $\dfrac{3}{2}x=-5$. It seems to me that he is not familiar with real numbers and used to seeking for solutions only within the set of natural numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, he is not well prepared for algebra, since he's not familiar with the number properties. When he combines like terms, not only does he add or subtract the coefficients but also does the same thing to the exponents. (Sometimes he even fails to identify the like terms). I told him that was not right, and explained how the distributive property works here. Next time, he made the same mistake. He is easily confused and shows poor understanding of abstract concepts and expressions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Now he is learning geometry (not the advanced class that requires two-column proof) and his school teacher commented he was weak in mathematical reasoning.&lt;/strong&gt; I really want to help him build a solid foundation in algebra (since he can follow the geometry course in school, I don't need to teach him much geometry), fill up the gaps, but we don't have too much time and therefore need a clear and efficient plan. I don't know where to get started, since there is too much to learn and it takes him too much time to understand, internalize and solidify the knowledge and skills. Many times, I decide to start somewhere only to find that he lacks something even more basic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any suggestion to help him into the gate of high school algebra? To start with the most basic knowledge (like G1~G6 math) or just start where he seems to struggle the most? Is pre-algebra a good choice for him?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Additional information&lt;/strong&gt;: I have only 2 hours with him each week, and now I'm trying to get more time. Maybe on holiday, I can make and carry out some teaching plan.&lt;/p&gt;&#xA;" OwnerUserId="9022" LastEditorUserId="9022" LastEditDate="2017-12-28T15:10:35.570" LastActivityDate="2024-01-28T06:10:27.627" Title="How to teach a student algebra who misses too much previous knowledge?" Tags="|secondary-education|mathematical-pedagogy|algebra|course-design|tutoring|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13368" PostTypeId="2" ParentId="13367" CreationDate="2017-12-26T19:54:15.780" Score="9" Body="&lt;p&gt;It seems he has a lot of gaps.  It would be good to teach him each gap separately.  For example, don't expect to teach him about adding negative numbers and identifying like terms by exponents at the same time, although you may be able to work on more than one gap in a lesson.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Identify a gap and then work on it with him.  He should have a notebook just for working on his background unrelated to class.  Introduce him to Khan Academy.  There he can drill on what you've taught him.  If he identifies you as his teacher, you can assign topics for him to work on and see his progress.  You can also see what he's done.  That means if you tell him to practice for a certain amount of time (e.g. 30 minutes 3 times a week or 10 minutes everyday), you can see if he's done what was assigned.  The computer will give him drills until he gets a certain number right and then text him on the same concepts later in master drills.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You seem to have a clear handle on where his gaps are.  It doesn't seem like you need someone to tell you what to work on, just how to get him to move forward.  I hope this advice will help.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2017-12-26T19:54:15.780" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13369" PostTypeId="2" ParentId="10181" CreationDate="2017-12-26T21:15:43.353" Score="3" Body="&lt;p&gt;Maybe consider a sinusoid(ish) curve superimposed on a general trend.  A place where you can see this is in business (e.g. volumes of production) where there is some general trend over time but there is SEASONALITY of the business.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some examples: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;ND oil production from 2009-2014 (low during winters because of weather issues but growing over time with development of the field at high prices).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Wal-mart expansion over last couple decades (maybe they have slowed recently, donno) but if you look at monthly data, obviously it is higher in the holidays and lower in January.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;[But I struggle with why you would want to find max/min.  I guess you can come up with some rationales like hiring for Walmart or takeaway/service support for oil business.  Not a perfect example, but maybe there is some insight in there, that is developable.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  The mechanical examples to use are higher order polynomials. Easy for them to work on and has an added benefit of getting them to think about the analytical geometry issues of how order relates to number of min/max points.  But I am struggling to think of a good applied example of a cubic or higher, especially with noticeable wiggles.  But still, they get value even from just crunching the math even if it is hard to think of an application.&lt;/p&gt;&#xA;" OwnerUserId="9236" LastActivityDate="2017-12-26T21:15:43.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13370" PostTypeId="2" ParentId="13367" CreationDate="2017-12-26T21:33:02.073" Score="6" Body="&lt;p&gt;In addition to the suggestions about remediation and the like, consider to just teach the fellow to mechanically solve for x.  Not even thinking about &quot;what number what get me this&quot;...I think this is more how gifted kids or at least average kids without gaps might think about the problem.  But it is going to be super hard sledding (and just not work) for someone who is bad at double negatives and fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider to just teach him to mechanically &quot;do everything to each side of the equation&quot; in order to solve for x.  Every step written out explicitly for each side of the equation. Perhaps this gives him some progress at least with the immediate objective and over time helps him to realize the concept of negatives and fractions and the like.  Some people have to learn that way.&lt;/p&gt;&#xA;" OwnerUserId="9236" LastActivityDate="2017-12-26T21:33:02.073" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13371" PostTypeId="2" ParentId="130" CreationDate="2017-12-26T23:39:14.043" Score="-3" Body="&lt;p&gt;I think the HS teachers have a point.  It seems like you love the topic and want to find a justification for it (examples why the teachers are wrong) rather than really considering the question openly if maybe they are right.  Maybe the reason you strain so hard for a rationale is the the math teachers have a point!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider:  I like gymnastics as a sport, but it is a stretch to say that working on it will help a high school football player get better at football or be a sound use of his time.  (Wrestling on the other hand, especially for defensive players has some noticeable benefit.  But is still not a requirement.)&lt;/p&gt;&#xA;" OwnerUserId="9236" LastActivityDate="2017-12-26T23:39:14.043" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13372" PostTypeId="1" CreationDate="2017-12-27T01:15:12.400" Score="2" ViewCount="260" Body="&lt;p&gt;I edited and shortened &lt;a href=&quot;https://betterexplained.com/articles/understanding-the-birthday-paradox/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the following's origin&lt;/a&gt;. Is learning to intuit exponentiality, implausible (e.g. like hearing &lt;a href=&quot;https://en.wikipedia.org/wiki/Iannis_Xenakis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Xenakis&lt;/a&gt;'s gratingly dissonant music as Mozart's tonal consonant melodies)?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h2&gt;Problem 1: Exponents aren’t intuitive&lt;/h2&gt;&#xA;&lt;p&gt;Math and statistics, aren't natural. Here’s an example: What’s the chance of getting 10 heads in a row when flipping coins? The untrained brain might think like this:&lt;/p&gt;&#xA;&lt;p&gt;“Well, getting one head is a 50% chance. Getting two heads is twice as hard, so a 25% chance. Getting &lt;strong&gt;ten&lt;/strong&gt; heads is probably 10 times harder… so about 50%/10 or a 5% chance.”&lt;/p&gt;&#xA;&lt;p&gt;And there we sit, smug as a bug on a rug. No dice bub.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;After pounding your head with statistics&lt;/strong&gt;, you know not to divide, but use &lt;strong&gt;exponents&lt;/strong&gt;. The chance of 10 heads is not &lt;span class=&quot;math-container&quot;&gt;$.5/10$&lt;/span&gt; but .5&lt;sup&gt;10&lt;/sup&gt; &lt;span class=&quot;math-container&quot;&gt;$ \approx  .001$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But despite training, we still remain caught again. At 5% interest we’ll double our money in 14 years, rather than the “expected” 20. Did you naturally infer the Rule of 72 when learning about interest rates? Probably not. Understanding compound exponential growth with our linear brains is hard.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-12-30T22:26:44.550" Title="How can a layperson learn to intuit exponential growth?" Tags="|exponential|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13373" PostTypeId="1" CreationDate="2017-12-27T02:23:32.807" Score="8" ViewCount="392" Body="&lt;p&gt;I have a very clever student in Grade 7. Due to carelessness and English (which is not his native language), he failed the placement test and can't learn Grade 8 math (algebra 1) in school. School math is easy and bores him, so he chooses me to teach him in advance. He is quick in thought, but a little careless (which is not a problem here since we find a way to fix that). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He attends the math club, and tries to understand problems designed for students in Grade 10 or above. And indeed, he can understand the problem well and ask me the right questions. I'm willing to teach him for he trusts me and follows me well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My thought is to teach what is related to his school math, but deeper and more challenging. Now he's studying algebraic expressions in school, so I start with polynomials, which works well. Now I've just finished teaching him multiplying polynomials (Pascal's triangle was also introduced and he really likes it) and plan to teach him factorization in the next lesson. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many topics in polynomials (division, rational root theorem, GCF/LCM), and many more are related to it (polynomial functions, solve polynomial equations, rational expressions and functions), from algebra 1 all the way to precalculus, even higher algebra (Eisenstein's criterion). So I have to make a choice. Since he also wants to participate in AMC, there are more things to learn, number theory, combinatorics, Euclidean geometry (maybe not now), etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem I face is, I now have two goals: one is to teach him something new (from higher grade math), the other one is to make it interesting and challenging, by introducing him to more advanced topics and more difficult problems. I can't merely follow the easy textbooks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Jumping from a topic to another is not a good idea, and I want to design a curriculum for him: what to teach and in what order, and to what degree, and what problems are suitable to assign to him so that he can try to solve them by himself and we can then discuss in the next class.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Any suggestion on the course design and reference books is welcome. And we have 3 hours on Friday each week, how can I structure the class?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may not be a good idea to let him read some books by himself, and discuss with me what he read, since he prefers listening to reading (and his English level is not good enough).&lt;/p&gt;&#xA;" OwnerUserId="9022" LastEditorUserId="9022" LastEditDate="2017-12-28T16:08:44.393" LastActivityDate="2017-12-29T22:32:56.583" Title="What can I teach a talented student who is enthusiastic about math?" Tags="|secondary-education|course-design|tutoring|math-contests|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13374" PostTypeId="2" ParentId="13361" CreationDate="2017-12-27T09:40:08.103" Score="3" Body="&lt;p&gt;Students are not accustomed to speaking/writing precisely. When presented with an affirmation, such as &lt;em&gt;a homogeneous linear system of equations has always a nontrivial solution&lt;/em&gt; a student may realize that the statement is false for some particular overdetermined system without realizing that the existence of a particular counterexample invalidates the affirmation. This occurs because in colloquial speech one customarily qualifies affirmations, so it is habitual to speak of something as true when it is generally true. In the opposite direction, a student might consider the affirmation true because it is true for all underdetermined systems and systems ... We have to teach students the difference between mathematical (or, more generally, careful, precise, technical) speech and colloquial speech. This is perhaps the most important reason we teach linear algebra to so many students who will not later use its results in an instrumental way.&lt;/p&gt;&#xA;&lt;p&gt;The only way to teach this, so far as I can see, is to force students to explain carefully the reasoning in a number of illustrative examples. One way to do it is to make an affirmation and ask the student to decide whether it is true or false, providing justification in the case of truth, and a counterexample in the case of falsity. Often it helps to put side by side superficially (that is, verbally) similar problems. For example consider the following affirmations:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;1. A matrix is orthogonally diagonalizable if and only if it is symmetric.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;2. A matrix is diagonalizable if and only if it is symmetric.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Many students will not distinguish between &amp;quot;diagonalizable&amp;quot; and &amp;quot;orthogonally diagonalizable&amp;quot; so will not see any difference between the affirmations. Putting the two affirmations side by side signals to a mildly attentive student that there is a difference between them. This will be true even for students who have already seen in class an example of a &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; diagonalizable matrix that is not symmetric, and had it pointed out to them explicitly that this matrix is diagonalizable but not orthogonally diagonalizable. So one has to explain the reasoning in detail.&lt;/p&gt;&#xA;&lt;p&gt;At the bottom is the (quite general) confusion between necessary and sufficient conditions. Most people do not distinguish well between the two until taught to do so. This has to do with what Polya called plausible inference. While it is true mathematically that just because A implies B it need not be the case that B implies A, in scientific, and less technical contexts, the truth of B is evidence in favor of A. This mode of thinking is natural to thoughtful and observant people, and useful in many contexts, although it is not correct in purely mathematical contexts. It is the math teacher's job to indicate how mathematicians use language differently from others, and why it is useful to do this. This is part of why we fuss so much about questions such as whether every conservative vector field has a potential.&lt;/p&gt;&#xA;&lt;p&gt;This is my sense of what V. I. Arnol'd meant when he wrote (in his usual intentionally tendentious style) on p. 740 of &lt;a href=&quot;https://link.springer.com/article/10.1134/S1560354714060100&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; (although I do not share Arnol'd's antagonism to Bourbaki):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now it became possible to apply the techniques developed in the problem of adiabatic invariants. As soon as I accomplished that,&#xA;Kolmogorov suggested that I should submit the paper on perpetual&#xA;adiabatic invariance to ZHETF, the main physical journal in the USSR.&lt;/p&gt;&#xA;&lt;p&gt;A few weeks later, M. A. Leontovich (who was, as far as I remember, a&#xA;deputy to the editor-in-chief of ZHETF) invited me to his home (near&#xA;the Atomic Energy Institute of the USSR Academy of Sciences) to&#xA;discuss the manuscript. Having fed me, as usual, by boiled buckwheat&#xA;and calling me, as usual, “Dimka” (M. A. called me in such a way until&#xA;his death), Mikhail Aleksandrovich explained to me that the paper&#xA;could not be published in ZhETF due to the following reasons.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The manuscript contained the words “theorem” and “proof” forbidden in ZhETF.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The manuscript claimed that “A implies B” while every physicist knew examples showing that B does not imply A.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The manuscript used the unintelligible terms “Lebesgue measure”, “invariant tori”, “Diophantine conditions”.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Mikhail Aleksandrovich therefore proposed that I should rewrite the&#xA;paper.&lt;/p&gt;&#xA;&lt;p&gt;Now I realize how right he was in defending a physical journal&#xA;from the Bourbaki-like mathematical jargon.&lt;/p&gt;&#xA;&lt;p&gt;For instance, indeed,&#xA;while claiming that “A implies B” the author must point out explicitly&#xA;whether the converse holds, otherwise any reader not spoiled by the&#xA;mathematical slang would understand the claim as “A is equivalent to&#xA;B”.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="672" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2017-12-27T09:40:08.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13376" PostTypeId="2" ParentId="13362" CreationDate="2017-12-27T16:39:56.433" Score="2" Body="&lt;p&gt;Biased though I am, because I am a co-author, I think the book For All Practical Purposes (now in 10th edition) does a good job of connecting foundational arithmetic skills to the varied ways that mathematics is applied outside of mathematics. Many resources related to mathematical modeling are available on the COMAP web site:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.comap.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.comap.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Including being able to download the current issue of Consortium (which comes out twice a year) for free.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also look at:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://qubeshub.org/groups/mmhub/resources_page&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://qubeshub.org/groups/mmhub/resources_page&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2017-12-27T16:39:56.433" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13377" PostTypeId="2" ParentId="13367" CreationDate="2017-12-27T19:56:58.733" Score="12" Body="&lt;p&gt;My guess is that he depends on &quot;following the rules&quot;, and there are now too many rules for him, because none of the 'rules' makes any sense to him. I believe he needs to see things differently to succeed. I don't know for sure without meeting him. How old is this student? Did he hire you, or did his parents or school? If the latter, it may be an unworkable situation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If he cares about getting it, I might start out with puzzles meant to intrigue him. It will not bear fruit quickly.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-12-27T19:56:58.733" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13378" PostTypeId="1" AcceptedAnswerId="13381" CreationDate="2017-12-27T21:20:41.047" Score="27" ViewCount="4114" Body="&lt;p&gt;In an exam we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question (5points)&#xA;find $\lim_{x\to\infty}(x-\sqrt{x})$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student answered:  $\lim_{x\to\infty}(x-\sqrt{x}) =\lim \sqrt{x}(\sqrt{x}-1)=\infty \cdot\infty=\infty$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: how many point you will give the student? A professor in our department says it should be zero or one out of five, arguing that the student does not understand what are limits for and treat them as real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think 4/5 is fair. Because he almost did everything except that he did not mention that the product of two infinite limits is infinite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I am talking about the first course in the university which is calculus.&lt;/p&gt;&#xA;" OwnerUserId="9255" LastEditorUserId="9255" LastEditDate="2017-12-28T04:25:11.023" LastActivityDate="2018-01-02T22:28:57.127" Title="Grading a limit problem" Tags="|calculus|grading|" AnswerCount="16" CommentCount="28" ContentLicense="CC BY-SA 3.0" />
  <row Id="13379" PostTypeId="2" ParentId="13378" CreationDate="2017-12-28T03:16:10.663" Score="3" Body="&lt;p&gt;I’d give 4/5, i.e., a B grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A generous reading has the student mentally glossing over some of the finer points of limit work. Does the student know what’s they’re glossing over? From their work, we can’t tell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of what we teach in math is notation, and part of what we teach is attention to detail. These parts are &lt;em&gt;as important&lt;/em&gt; as just “knowing math”. Why don’t I then suggest 2/5, or something closer to a third? Well, I say the first 2/5 is for making an honest attempt, and for the valid steps they did write down. However, they should know that this is a shorthand, and what it's short for.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; I'm leaving my previous answer up, for accountability, but I've been persuaded by the answers saying the student should receive full credit. If there's anything wrong with their thinking, a different question should be used to determine that. &lt;/p&gt;&#xA;" OwnerUserId="2034" LastEditorUserId="2034" LastEditDate="2017-12-31T00:54:30.633" LastActivityDate="2017-12-31T00:54:30.633" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13380" PostTypeId="2" ParentId="13378" CreationDate="2017-12-28T04:01:01.880" Score="33" Body="&lt;p&gt;If this is calc I, that deserves a 5/5. If this is analysis, it depends on what you taught them. Don't you set up a grading rubric ahead of time? What do the 5 point answers look like? What do other not-so-great answers look like?&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-12-28T04:01:01.880" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13381" PostTypeId="2" ParentId="13378" CreationDate="2017-12-28T05:17:46.277" Score="54" Body="&lt;p&gt;The student changed something which was indeterminate ($\infty-\infty$) into something which was not ($\infty\cdot \infty$). How does that not merit a perfect score? Changing indeterminate expressions into determinant ones is, generally speaking, the point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the professor had some other solution in mind, then they made a mistake. They should have chosen a question without an easy out. It happens; I've certainly had students provide solutions which short-cut what I had planned. That, however, &lt;em&gt;does not make the answer incorrect&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2017-12-28T05:17:46.277" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13382" PostTypeId="2" ParentId="13373" CreationDate="2017-12-28T07:54:43.870" Score="7" Body="&lt;p&gt;The Art of Problem Solving have designed exactly such a &lt;a href=&quot;https://artofproblemsolving.com/store&quot; rel=&quot;noreferrer&quot;&gt;curriculum&lt;/a&gt;, which is also meant to prepare students for the AIME and AMC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also very warmly recommend the Gelfand series of books, starting with &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0817636773&quot; rel=&quot;noreferrer&quot;&gt;Algebra&lt;/a&gt;; which was originally designed for his correspondence school for talented math students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/1895997283&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; little book on problem solving by Ravi Vakil is a gem.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;PS) You've mentioned the math-club at school. There might also be math circles in your location, which can be very good experience.&lt;/p&gt;&#xA;" OwnerUserId="9258" LastActivityDate="2017-12-28T07:54:43.870" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13383" PostTypeId="2" ParentId="13378" CreationDate="2017-12-28T10:00:39.463" Score="11" Body="&lt;p&gt;The answer seems ok to me, in that it shows that the student understands what the limiting behavior of $x - \sqrt{x}$ is as $x \to \infty$. If the questioner wants to see a formal justification of that, then the word &quot;find&quot; should be replaced by some more precise indication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The interpretive problem is that that the exercise could be better posed. &#xA;A better way to ask the same question is simply to ask: &lt;em&gt;what is the limiting behavior of $x - \sqrt{x}$ as $x \to \infty$?&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the classical sense the limit $\lim_{x \to \infty}x^{2}$ does not exist. What is meant by writing $\lim_{x \to \infty}x^{2} = \infty$ is different than what is meant by $\lim_{x \to 2}x^{2} = 4$, because $\infty$ is not a real number. The expression $\lim_{x \to \infty}f(x) = \infty$ is a shorthand that means that $f(x)$ is eventually (on the positive real line) larger than any positive real number. It is used to distinguish the behavior of $\log{x}$ and $\cos{x}$ as $x \to \infty$. Neither has a limit as $x \to \infty$ in the usual sense, but their limiting behaviors are different, in the sense that one function grows unboundedly while the other is oscillatory, although bounded. This can be indicated by assigning the putative &quot;value&quot; $\infty$ to $\lim_{x \to \infty}\log{x}$ and declaring that $\lim_{x \to \infty} \cos{x}$ &quot;does not exist&quot;, even though $\infty$ to $\lim_{x \to \infty}\log{x}$ also does not exist in the usual sense (it cannot be assigned a value in $\mathbb{R}$). So one is using the same notation to indicate a limit that exists in the usual sense and a limit that, although it does not exist in the usual sense, can be given sense in that the limiting behavior of the argument function has a well-defined character ... This is akin to the unfortunate and sloppy conflation of integrals and primitives that one often encounters in first calculus courses; apparently identical notation is used to indicate operationally and/or conceptually distinct entities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because the relevant distinction is a subtle for students, it seems to me a pedagogical error to treat $\lim_{x \to \infty}x^2 = \infty$ and $\lim_{x \to 2}x^{2} = 4$ on the same level. Their meanings and interpretations are different. I would try to avoid writing the former expression, or would make a serious effort to explain that it is &lt;em&gt;formal&lt;/em&gt; notation indicating something different than the well-defined notation in the latter expression (one encounters similar issues when treating integrals with infinite limits or integrals of unbounded functions). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When one asks &quot;find $\lim_{x\to \infty}f(x)$&quot; it is implicit in the question being well posed that $f(x)$ has a well-defined limiting behavior that can be summarized by a single notational expression (so is not oscillatory), but &quot;find&quot; is really the wrong word. &lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2017-12-29T15:30:40.363" LastActivityDate="2017-12-29T15:30:40.363" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13384" PostTypeId="1" CreationDate="2017-12-27T20:55:16.047" Score="1" ViewCount="86" Body="&lt;p&gt;In an exam we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question (5points)&#xA;find $\lim_{x\to\infty}(x-\sqrt{x})$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student answered:  $\lim_{x\to\infty}(x-\sqrt{x}) =\lim \sqrt{x}(\sqrt{x}-1)=\infty \cdot\infty=\infty$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: how many point you will give the student? A professor in our department says it should be zero or one out of five, arguing that the student does not understand what are limits for and treat them as real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think 4/5 is fair. Because he almost did everything except that he did not mention that the product of two infinite limits is infinite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="9255" OwnerDisplayName="Muath Karaki" LastActivityDate="2017-12-28T12:16:36.763" Title="Treating infinity as numbers in exams," Tags="|calculus|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2017-12-28T12:18:53.047" ContentLicense="CC BY-SA 3.0" />
  <row Id="13385" PostTypeId="2" ParentId="13378" CreationDate="2017-12-27T21:20:10.630" Score="1" Body="&lt;p&gt;I think context is everything here.  When I teach calculus, one of my stock phrases is &quot;infinity is shorthand for a limit.&quot;  This is an elementary (Calc I) limit, so at this level, the whole point is whether the student understands the background rigor. Here, he is working the problem backwards, making something less rigorous when I told him to make it more rigorous. I would accept the above calculation in a DE course, but in Calc I, probably 0 points out of 5.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't really care if the answer is right or if it makes sense or if a &quot;real&quot; scientist/engineer would work it that way.  I need to know what's going on in your head, and this answer short-circuits this.  It may as well be a multiple-choice question, otherwise.&lt;/p&gt;&#xA;" OwnerUserId="19450" OwnerDisplayName="B. Goddard" LastActivityDate="2017-12-27T21:20:10.630" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13386" PostTypeId="2" ParentId="13378" CreationDate="2017-12-27T21:46:41.853" Score="14" Body="&lt;p&gt;When I taught math, I used to follow the principle &quot;do no harm&quot;, or &quot;innocent until proven guilty&quot;. In other words:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Maybe here we deal with a student who is treating infinity as a number. (&quot;Guilty&quot;)&lt;/li&gt;&#xA;&lt;li&gt;Maybe we are dealing with someone who knows exactly what they are doing and have found a good shortcut (&quot;Innocent&quot;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Because I cannot tell, based on the answer, I would mark it as 5/5 and be careful to next time ask a better question, which would hopefully let me better distinguish the &quot;guilty&quot; from &quot;innocent&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's me. YMMV.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;: In &lt;em&gt;this&lt;/em&gt; case, my guess is that the student is &quot;innocent&quot;. A &quot;guilty&quot; student is, I guess, far more likely to write that the solution is $\infty-\infty=0$, or $\infty-\infty=\text {indeterminate} $ or something like that.&lt;/p&gt;&#xA;" OwnerDisplayName="user8734617" LastActivityDate="2017-12-27T21:58:04.423" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13387" PostTypeId="2" ParentId="13378" CreationDate="2017-12-27T21:47:04.633" Score="-2" Body="&lt;p&gt;That depends on the notation used in class. I think your professor wants you to say, 'The limit does not exist' or something similar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in proper context it makes sense to use the notation $$\lim_{x\to\infty}{f(x)}=\infty$$ to mean that the function $f$ diverges. But one should not &lt;em&gt;never&lt;/em&gt; say $\infty\cdot\infty=\infty$ in this context, for it implies a sort of extended arithmetic, which is not the usual interest of analysts, but people like set theorists, say.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to answer the question. I would mark the student down for saying $\infty\cdot\infty=\infty$ but otherwise note that they understand that the function diverges. That is sufficient. So, a $2.5$?&lt;/p&gt;&#xA;" OwnerUserId="9204" OwnerDisplayName="Allawonder" LastActivityDate="2017-12-27T21:47:04.633" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13388" PostTypeId="2" ParentId="13378" CreationDate="2017-12-27T22:29:55.553" Score="8" Body="&lt;p&gt;It really depends on how this was taught in the course. When working with limits it is useful to define the extended real number system $\overline{\mathbb{R}}:=\mathbb{R}\ \cup \{\infty,-\infty\}$. &#xA;We call $U$ a neighbourhood of $\infty$ ($-\infty$) if there exists a $K&amp;gt;0$ such that $(K,\infty) \subseteq U$ ( $(-\infty, -K)\subseteq U$ ). Now some sequences which diverge in $\mathbb{R}$ converge in $\overline{\mathbb{R}}$, namely those which &quot;diverge to infinity&quot; (although not all properties of convergent sequences hold). Then it is possible to define addition and multiplication on $\overline{\mathbb{R}}$ by looking at the addition and multiplication of the limits of sequences which converge to $\pm\infty$. For example, if  $(x_n)_{n\in\mathbb{N}}\subseteq\mathbb{R}$ and $(y_n)_{n\in\mathbb{N}}\subseteq\mathbb{R}$ both tend to infinity for $n\rightarrow \infty$, then $(x_n+y_n)_{n\in\mathbb{N}}$ goes to $\infty$ as well, thus we define $\infty + \infty = \infty$. In a similar fashion one obtains $\infty \cdot \infty = \infty$. Note that for instance $0 \cdot \infty$ has to be left undefined because $\lim_{n\rightarrow\infty}\frac {1}{n}\cdot n  =1$, but $\lim_{n\rightarrow\infty} 0 \cdot n = 0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this sense of the real extended number system $\infty$, contrary to some other answers here, becomes a number (although $\overline{\mathbb{R}}$ doesn't form a field) and it is &lt;em&gt;perfectly valid to argue like the student did&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user9260" LastActivityDate="2017-12-28T00:05:28.737" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13389" PostTypeId="2" ParentId="13353" CreationDate="2017-12-29T02:27:57.053" Score="2" Body="&lt;p&gt;This may seem strange to offer an answer to my own question after just a week, but I did find a useful resource that may be helpful in addressing the issues described in my question. This week, I have been reading the recently-posted &lt;strong&gt;&lt;a href=&quot;https://www.maa.org/programs/faculty-and-departments/ip-guide&quot; rel=&quot;nofollow noreferrer&quot;&gt;Instructional Practices Guide&lt;/a&gt; from the MAA&lt;/strong&gt; (&lt;a href=&quot;https://www.dropbox.com/s/xpvkni52tkf0wgt/MAA_IP_Guide_V1-1.pdf?dl=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;direct link to pdf&lt;/a&gt;). Throughout the document, they cite evidence-based research to advocate for &lt;em&gt;active engagement&lt;/em&gt; in mathematics courses. A section in the Design Practices chapter addresses some challenges in designing and implementing such a classroom, and provides a link to an in-class activity to be done on the first day that may help students better understand the instructor's motivations and adjust their attitude towards the course. I will share excerpts from the MAA guide and the blog post they cite, both interspersed with and followed by some comments about how I think this may address my main question.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;DP.3. Challenges and opportunities&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Successful implementation of engaging classroom components (a) necessitates a shift in the instructor’s role relative to traditional lecture, and (b) tends to alter the time and effort cost for students (Lee, Y., Rosenberg, J., Robinson, K., et al., 2016). In this section we discuss challenges and opportunities related to designing an active engagement environment.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[MAA, p. 137]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;DP.3.2. Other Challenges&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Buy-in&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In order to successfully implement a course design, instructors must have the support of both administrators and students when employing a new instructional approach. Instructors should set realistic expectations and create an atmosphere where students can be successful. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Most students will expect and be comfortable in a traditional mathematics classroom environment. They understand the traditional implicit contract between students and instructor: the instructor delivers clear lectures with appropriate examples, and the students listen to the lecture and do the assigned problems.&lt;/strong&gt; Students may not hold up their end of the contract, but they understand the rules.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;An instructor implementing new instructional approaches with different underlying agreements must make the new expectations explicit.&lt;/strong&gt; Students will need time to come to terms with the intentional coherence of the methods and messages in the course. This process will require struggle and might elicit rebellion and frustration before students are comfortable with the revised contract. In order to facilitate this process an instructor can explain to students on the first day of class that this will not be a lecture-based class and ask them to read Dana Ernst’s blog post &lt;a href=&quot;http://danaernst.com/setting-the-stage/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://danaernst.com/setting-the-stage/&lt;/a&gt;. Using new instructional approaches may require new tools to help students succeed in the class such as offering additional office hours, meeting with struggling students early in the term, and coordinating tutoring opportunities.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[MAA, p. 140, emphasis mine]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This section resonated with my experiences last semester in some ways, especially the emphasized sentences above. However, I think the main message of the section applies to any instructional method that bucks a students' expectations, even if it's as simple as different kinds of homework or exam problems (per the examples in my question). That is, the suggestion of making expectations explicit applies to any kind of &quot;new&quot; activity or assessment that students may not be familiar or comfortable with.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The linked blog post provides an example of an activity to engage with students and explain the motivation behind a course and its components, or to make more explicit some new expectations. The post describes how the author uses this activity to discuss with students why the course will follow an &lt;em&gt;inquiry-based learning&lt;/em&gt;, but I believe this activity can be modified appropriately for other teaching styles and assessment methods. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Whenever I’m teaching via inquiry-based learning (IBL), it is important to get student buy-in. I often refer to this as “marketing IBL”. My typical approach to marketing involves having a dialogue with my students, where I ask them leading questions in the hope that at the end of our discussion the students will have told me that something like IBL is exactly what we should be doing.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Directions to the Students:&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Get in groups of size 3–4.&lt;/li&gt;&#xA;  &lt;li&gt;Group members should introduce themselves.&lt;/li&gt;&#xA;  &lt;li&gt;For each of the questions that follow, I will ask you to:&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Think about a possible answer on your own.&lt;/li&gt;&#xA;  &lt;li&gt;Discuss your answers with the rest of your group.&lt;/li&gt;&#xA;  &lt;li&gt;Share a summary of each group’s discussion.&lt;/li&gt;&#xA;  &lt;/ol&gt;&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Questions&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;￼What are the goals of a university education?&lt;/li&gt;&#xA;  &lt;li&gt;How does a person learn something new?&lt;/li&gt;&#xA;  &lt;li&gt;What do you reasonably expect to remember from your courses in 20 years?&lt;/li&gt;&#xA;  &lt;li&gt;What is the value of making mistakes in the learning process?&lt;/li&gt;&#xA;  &lt;li&gt;How do we create a safe environment where risk taking is encouraged and productive failure is valued?&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;http://danaernst.com/setting-the-stage/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://danaernst.com/setting-the-stage/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The blog post contains a brief discussion of common answers to the questions and the author's experiences with it in the past, as well as a slide presentation to display in class during the activity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As written, this activity may be more involved than is necessary to simply discuss with students that learning mathematics in a college/university environment is different than learning in high school. However, I do think a version of this activity can be very useful to engage students with the course and the instructor, to serve as a focal point in the future (&quot;Remember when we discussed this?&quot;), and to show the students that the instructor has (hopefully) good reasons for teaching the course and assessing students in the manner they have chosen.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I also envision that this activity could be modified to &quot;teach&quot; students about types of assessment they may encounter in the course and simultaneously serve as a tool for both the instructor and students to assess prerequisite knowledge. For example, let's take the context of a Calculus 1 course made of mostly 1st and 2nd year students. The instructor may be dealing with students who took an AP Calculus class in high school but performed poorly on the exam (or the school's placement assessment) as well as students who took a Precalculus course at the college the previous semester. The instructor can pose several mock exam questions to the students that assess similar background knowledge in different ways, such as: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Consider $f(x)=x^2$. Graph $f$. Then, simplify the expression $-3\cdot f(x-2)$ and graph that function, as well.&lt;/li&gt;&#xA;&lt;li&gt;The image provided is the graph of a function $f$. On the axes provided, sketch a graph of $-3\cdot f(x-2)$.&lt;/li&gt;&#xA;&lt;li&gt;If $f(x)$ is a function and I know its graph, explain how to find the graph of $−3\cdot f(x−2)$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Have students work to answer these three questions on their own for two minutes. Then, have them discuss their answers with their neighbors. As a class, discuss the correct answers. (You even wish to discuss with students the grading of incorrect or partially correct answers to these questions, if this comes up.) Then, have the students compare and contrast the question styles in small group discussions, leading to a larger discussion as a class. Explain to students how there can be various levels of understanding of a concept, and that we instructors care about many of them. Emphasize that students will be asked questions of each variety, just like the examples provided.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The actual examples chosen can be tailored to the course in question. Or, you could pick a very basic concept to ensure that all students will already understand the essential content, so that discussion can focus on the styles of assessment questions, instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not tried such an activity in my own courses, but I do plan on trying something like this in the near future. Perhaps, it can be followed up and revisited throughout the semester, with survey questions or brief personal reflection assignments to remind students about the importance of conceptual understanding.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2017-12-29T02:27:57.053" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13390" PostTypeId="1" CreationDate="2017-12-29T02:55:58.733" Score="9" ViewCount="210" Body="&lt;p&gt;&quot;[Myth] that exams are objectively graded. Daniel Stark and Edward Elliot sent two English essays to 200 high school teachers for grading. They got back 142 grades. For one paper, the grades ranged from 50 to 99; for the other, the grades went from 64 to 99. But English is not an &quot;objective&quot; subject, you say. Well, they did the same thing for an essay answer in mathematics and got back grades ranging from 28 to 95. Though most of the grades they received in both cases fell into the middle ground, it was evident that a good part of any grade was the result of who marked the exam and not of who took it&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://www.nyu.edu/projects/ollman/docs/why_exams.php&quot; rel=&quot;noreferrer&quot;&gt;https://www.nyu.edu/projects/ollman/docs/why_exams.php&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The study the author is referring to is: &quot;Starch D, Elliott EC. Reliability of grading work in mathematics.&quot; &lt;a href=&quot;http://www.jstor.org/stable/1076246?seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;http://www.jstor.org/stable/1076246?seq=1#page_scan_tab_contents&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is pretty consistent with my experience, both as a grader and a student. The cited experiment was done in 1913, however. I am curious about more recent, broader studies about testing the variance of grading - especially for math exams such as those we give in college algebra or calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same studies are referenced here: &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4041495/&quot; rel=&quot;noreferrer&quot;&gt;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4041495/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only recent reproduction I have found was for the Starch/Elliot experiment about english papers, not math exams: &lt;a href=&quot;http://pareonline.net/getvn.asp?v=16&amp;amp;n=17&quot; rel=&quot;noreferrer&quot;&gt;http://pareonline.net/getvn.asp?v=16&amp;amp;n=17&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8029" LastActivityDate="2017-12-30T04:00:23.053" Title="Experimental results about the variability of grades on a math exam" Tags="|reference-request|exams|education-research|grading|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13391" PostTypeId="2" ParentId="13378" CreationDate="2017-12-29T04:13:00.453" Score="-5" Body="&lt;p&gt;I'm not a math teacher, but I would give a zero because that's never how I was taught to work with indeterminate forms. What does factoring accomplish, aside from making it look like the student did some work?&lt;/p&gt;&#xA;" OwnerUserId="9269" LastActivityDate="2017-12-29T04:13:00.453" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13392" PostTypeId="2" ParentId="13062" CreationDate="2017-12-29T06:41:17.530" Score="2" Body="&lt;p&gt;Aimed at students or teachers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the latter, &lt;a href=&quot;https://www.cut-the-knot.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cut-the-knot&lt;/a&gt; is quite good, with many resources at this level. For the former, &lt;a href=&quot;https://www.khanacademy.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Khan Academy&lt;/a&gt; is free, and &lt;a href=&quot;https://beastacademy.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beast Academy&lt;/a&gt; from the Art of Problem Solving is good (only workbooks for now, but online classes are supposedly coming soon).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In terms of puzzle books, I can add Smullyan's books to the other recommednations. Those are all logic puzzles about truth-tellers and liars (which he calls knights and knaves) and are very entertaining to reason through together with children. &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/048647027X&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Lady or the Tiger&lt;/a&gt; is a good one to start with.&lt;/p&gt;&#xA;" OwnerUserId="9258" LastActivityDate="2017-12-29T06:41:17.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13394" PostTypeId="2" ParentId="13378" CreationDate="2017-12-29T16:28:09.620" Score="7" Body="&lt;p&gt;I will add yet another answer saying that this deserves full score, because I feel that two cents are still missing.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The student obviously knows what he does. Do you want to punish such a student?&lt;/strong&gt; Do you really think it's correct to deduct score points for not mentioning things such as &lt;em&gt;&quot;limit of a product is a product of limits given these limits exist and the product is well defined&quot;&lt;/em&gt;? In my opinion, these things should not be sought for while testing practical computations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;If you want to test knowledge of calculus related theorems, there are much much better ways.&lt;/strong&gt; And testing practical calculus should be separated from testing theoretical knowledge.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Moreover, if you want to test, in practical calculus, things that depend on precise knowledge of the related theorems, you should propose corresponding problems, whose solution without this knowledge is impossible.&lt;/p&gt;&#xA;" OwnerUserId="818" LastActivityDate="2017-12-29T16:28:09.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13395" PostTypeId="2" ParentId="13372" CreationDate="2017-12-29T17:04:02.800" Score="2" Body="&lt;p&gt;In the 1970s in the US, there was a &lt;a href=&quot;https://www.youtube.com/watch?v=brC_jK6stBs&quot; rel=&quot;nofollow noreferrer&quot;&gt;commercial&lt;/a&gt; for Faberge Organics shampoo that illustrated the power of doubling. The story was that a woman liked her new shampoo so much that she told two friends about it, and those two friends each told two friends, and so on... With each additional &quot;and they told two friends&quot;, there was a doubling of the number of identical women shown.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was &lt;a href=&quot;https://www.youtube.com/watch?v=JA7CKvoKEmE&quot; rel=&quot;nofollow noreferrer&quot;&gt;spoofed&lt;/a&gt; in the 1992 movie &lt;a href=&quot;http://www.imdb.com/title/tt0105793/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wayne's World&lt;/a&gt; where the Wayne character is trying to explain how guys spread information.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Wayne is talking to a woman.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;Well, you know how these things start. One guy tells another guy some'n', an' he tells two friends...&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Two Waynes are shown&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;and they tell two friends...&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Four Waynes&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;and they tell their friends...&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Eight Waynes&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;and so on&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Sixteen Waynes&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;and so on&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Thirty-two Waynes&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;and so on&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Back to Wayne and the woman&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;You know how these things go.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9272" LastEditorUserId="3593" LastEditDate="2017-12-30T22:26:44.550" LastActivityDate="2017-12-30T22:26:44.550" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13396" PostTypeId="2" ParentId="13390" CreationDate="2017-12-29T17:21:17.963" Score="3" Body="&lt;p&gt;Eductational Testing Service has done a lot of work on essay grading for AP exams and similar (SATs when they've had essays, etc.)  They put a fair amount of work into coming up with a detailed key and having support mechanisms: essays are graded twice, large variances examined, essays graded together with a supervisor able to deal with new questions that are generated (e.g. unusual solution method, questions of the graders).  They used to have a lot of methodology articles on their website.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From personal experience, having a detailed key with partial credit spelled out helps to drive better results.  Maybe more easily justified for a monster exam with many graders (e.g. TA's for a college chemistry exam).  But probably still good practice even for an individual teacher grading a 30 person section.  [Like war planning, even if you don't anticipate everything, the effort to plan out a key helps when running the grading and dealing with questions that arise as you grade exams/fight the enemy.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  How about instead of (or at least in addition to) giving us the range, giving us the standard deviation or the 90% interval? While much of mathematics is logical and rigorous, much of EDUCATION is statistical in nature and even Bayesian (in the sense that we don't know all the variables or the actual pdf being sampled).  Saying &quot;OMG, what a range&quot; is not even how to think about educational methodology, which is a practical art with cost benefit tradeoffs and uncertainties (not a Euclidean proof).  INstead thinking about the typical difference (e.g. SD/mean) is a more intutive way to think about the scale of the issue.&lt;/p&gt;&#xA;" OwnerUserId="9272" LastEditorUserId="9272" LastEditDate="2017-12-30T04:00:23.053" LastActivityDate="2017-12-30T04:00:23.053" CommentCount="24" ContentLicense="CC BY-SA 3.0" />
  <row Id="13397" PostTypeId="2" ParentId="13108" CreationDate="2017-12-29T19:20:35.343" Score="3" Body="&lt;p&gt;The premise of your question seems to indicate an undesirable precedent. When the students are able to solve a problem correctly it's immaterial what approach they followed. In fact, giving positive feedback to those who conjure creative methods (instead of following conventional approach taught in the class) is imperative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, to be fair, your question might be pertaining to just the pedagogy of teaching these 2 concepts. I suggest the following in that case:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Appreciate the student for solving the question without using LCM/GCD&lt;/li&gt;&#xA;&lt;li&gt;Ask the student to solve the same question by applying LCM/GCD to let him see by himself the application of LCM/GCD&lt;/li&gt;&#xA;&lt;li&gt;Whatever approach the student came up with, must be the same as LCM/GCD. &lt;strong&gt;Facilitate&lt;/strong&gt; the student to realise this.&lt;/li&gt;&#xA;&lt;li&gt;Unless the student's approach is more efficient (which I doubt it will be) let the student appreciate the conventional approach.&lt;/li&gt;&#xA;&lt;li&gt;As a last resort frame the word problem to force the student to use LCM. For example, in the question &quot;Two runners with given different speeds; when will they meet again?&quot;, I figure some students must be writing down the sequences and looking for the next match. In this case you can simply increase the number of runners and the speed as well.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;To acheive your goal of letting the students see LCM/GCD, ultimately, they need a lot of practice on problems on these same concepts (preferably varying question types). This is how anyone gains intuition of concepts in maths.&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2017-12-29T19:20:35.343" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13398" PostTypeId="2" ParentId="13378" CreationDate="2017-12-29T22:04:51.697" Score="0" Body="&lt;p&gt;Sometimes educators like to think about test questions as measuring specific things. What are you trying to assess with the question? Depending on what you’re trying to get, you may want to give full credit and some comment about being careful with statements like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\infty \times \infty = \infty$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want this to test the knowledge of symbolics and operations with infinity I’d give him zero, although I would feel bad about it.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastEditorUserId="7003" LastEditDate="2017-12-30T17:19:05.940" LastActivityDate="2017-12-30T17:19:05.940" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13399" PostTypeId="2" ParentId="13373" CreationDate="2017-12-29T22:32:56.583" Score="2" Body="&lt;p&gt;In addition to the perfect suggestion for the Art of Problem Solving, I would recommend getting your hands on the mathcounts materials, some other contest problem books, and old AMC exams.  You may also be interested in books by the USA Olympiad coach Titu Andrescu or the new Coach Po-Shen Loh, at his website expii.com.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-12-29T22:32:56.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13400" PostTypeId="2" ParentId="130" CreationDate="2017-12-29T22:48:38.633" Score="1" Body="&lt;p&gt;I don’t think that you can explain this to the students. Perhaps instead, you should better familiarize yourself with the high school standards and draw the connections.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A high school mathematics teacher will teach material that heavily overlaps with all undergraduate coursework, and is only a slight extension of most ideas.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While a high school mathematics teacher may spend their time teaching a bunch of 9th grade algebra, there is much to connect in their linear algebra and/or modern algebra courses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They should also be prepared to teach through BC calculus or HL IB mathematics, which are not superficial. Further, AP stats requires a solid understanding of introductory statistics.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would recommend these students work related to making some of these curricular connections and how they could be creative about teaching them. Incorporate the wealth of research on teaching and learning advanced mathematics, discuss math teams, research opportunities for high schoolers, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;High school teaching is a very demanding profession that requires deep understanding of formal mathematics at a high level of you are going to be a true professional and find continuing opportunities for yourself. There should be no argument about this, just look at what high school mathematics students are expected to do in terms of material content through precalc in the CCSS and the AP or IB exams.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2017-12-29T22:48:38.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13401" PostTypeId="2" ParentId="13317" CreationDate="2017-12-29T23:23:15.383" Score="3" Body="&lt;p&gt;A bit late to the party, but PageRank is partially built off of this notion (if I'm on page x, what is the probability I go to page y? What does this say about the importance of pages?) There's a paper by Larry Page &lt;a href=&quot;http://ilpubs.stanford.edu:8090/422/1/1999-66.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link here&lt;/a&gt; about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's also quite a bit of work on things like context-based autocompletion (using Markov chains with smoothing) as well as other predictive works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the world of Numerical Linear Algebra, you could view eigenvalue computations (especially sparse eigenvalue computation) using power iteration as a Markov-like process.&lt;/p&gt;&#xA;" OwnerUserId="9274" LastActivityDate="2017-12-29T23:23:15.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13404" PostTypeId="1" CreationDate="2017-12-30T01:58:21.383" Score="17" ViewCount="552" Body="&lt;p&gt;I teach at a community college, almost all physics but also a math course now and then. My current practice is that I assign homework worth a very small number of points (1 point per problem, as opposed to about 20-25 points per problem on exams), and I withhold solutions until students have turned in their problem sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an different set of practices which I think is more common in math courses than physics courses. Full solutions are available (e.g., in the book) for all problems. Problem sets are assigned but not collected or graded. At the beginning of each class, there is a short quiz, in which the students are asked to solve one problem that is of the same flavor as the ones on the problem set. I'm toying with the idea of switching to this system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone give pros or cons based on their experience with such a system? Are there common pitfalls in implementing such a system?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this system more likely to work well at a school with selective admissions, where students are more likely to be willing to put in many hours of work every week without immediate grade incentives? Or, conversely, does this system work better in the type of course that does not require much critical thinking or higher-level reasoning, so that students are simply differentiating $2x^3+1$ on the homework and then differentiating $2x^4+1$ on a quiz?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[EDIT] Some clarifications on what I'm asking: (1) I'm not talking about changing the intellectual level of the course. There would still be the same high level of critical thinking and higher-level reasoning that I've been requiring for the last 22 years. (2) I'm not talking about abdicating what I see as my responsibility to provide written feedback to students on their work. The written feedback would just be on their quizzes rather than their homework. (3) My question is about what outcomes people have experienced with students when they tried such a system, their impressions of pros, cons, and pitfalls, and how it works out in terms of different students' levels of motivation, response to rewards, and levels of maturity. (4) I'm not asking for recommendations of some other system besides the one described in the question. (5) I'm asking for descriptions of people's personal experience with using this system, which is specifically one in which &lt;em&gt;all solutions are freely available.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2018-01-01T20:06:09.337" LastActivityDate="2018-01-03T21:06:35.653" Title="Making solutions to all problems freely available" Tags="|grading|homework|community-colleges|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13405" PostTypeId="2" ParentId="2536" CreationDate="2017-12-30T04:12:12.383" Score="2" Body="&lt;p&gt;In addition to recitation (excellent suggestion by ncr), I would do more pop quizzes (graded in class by paper passing), in class drill (ungraded), and more period-long tests. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would not bother grading homework other than to verify it was done/not done.  Just not worth the effort.  I would stick with the no late work policy.  In addition to not bucking the system, it is also really sending a better message to the students that REGULAR practice and KEEPING UP are the ways to learn rather than cramming.  &lt;/p&gt;&#xA;" OwnerUserId="9272" LastActivityDate="2017-12-30T04:12:12.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13406" PostTypeId="2" ParentId="9636" CreationDate="2017-12-30T04:18:23.777" Score="0" Body="&lt;p&gt;I always felt that I got better mastery from courses with a lot of drill and testing.  I think graded homework appeals to professors but that says more about how professors think than if it is best pedagogy. In particular, I found that (in grad school sciences) that graded homework tended to be very hard and that then not enough progression and easy problems were completed.  Can lead to big gaps on mastery.  And can be time inefficient if all you are doing is b$%|-buster problems.&lt;/p&gt;&#xA;" OwnerUserId="9272" LastActivityDate="2017-12-30T04:18:23.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13407" PostTypeId="2" ParentId="13062" CreationDate="2017-12-30T05:04:31.500" Score="2" Body="&lt;p&gt;A slightly different approach is to watch short videos like those in &lt;a href=&quot;https://www.youtube.com/user/numberphile/videos&quot; rel=&quot;nofollow noreferrer&quot;&gt;Numberphile&lt;/a&gt; together with the child, to stop often and discuss the material. This requires more involvement from the adult, but is a much more open-ended learning experience. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic idea comes from research about how children interact with educational tv programs; and it (i.e., my recollection of the take-away conclusion) seems to suggests that it's not very effective in general &lt;em&gt;unless&lt;/em&gt; accompanied with parental (or equivalent) interaction around the material. I think Numberphile videos are particularly suitable, as they are short and often contain very simple proof ideas which lend themselves to further examples, discussion, and generalisations.&lt;/p&gt;&#xA;" OwnerUserId="9258" LastActivityDate="2017-12-30T05:04:31.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13408" PostTypeId="1" CreationDate="2017-12-30T15:58:47.980" Score="20" ViewCount="1433" Body="&lt;p&gt;Alert: I am not a math educator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that multiplication is first taught using the cross notation, for example $3\times 5=15$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First question - is that even correct? Maybe not all schools in every country even teach this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later on in education and work this notation is almost never used - by grade six it's $3 \cdot 5=15$ and soon after you only use any sign if you multiply naked numbers (it's $3.1\cdot 10^{-7}$) but mostly you only write that you want three x'es, like $3x=15kg$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why confuse people and introduce them to notation that is only used briefly and soon overtaken by other notation while the x becomes unknown instead of multiplication for further confusion?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do understand that the $\times$ is used on advertisements, cashier calculators and some other material aimed at general public, but that shouldn't be the cause for such choice. It's a consequence.&lt;/p&gt;&#xA;" OwnerUserId="766" LastEditorUserId="5220" LastEditDate="2017-12-31T16:26:57.917" LastActivityDate="2023-03-02T17:45:12.787" Title="Why is multiplication taught using cross notation at first?" Tags="|primary-education|notation|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13409" PostTypeId="2" ParentId="13408" CreationDate="2017-12-30T16:27:14.743" Score="13" Body="&lt;p&gt;In the U.S. at least, we do still use the $\times$ in scientific notation, so your example would be $3.1 \times 10^{-7}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think, for young children, the dot looks too much like the period we use (in the U.S.) for a decimal point. Even adults will sometimes misinterpret. (They see $3\cdot7$ and think it's $3.7$.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="4912" LastEditDate="2018-02-15T19:34:48.857" LastActivityDate="2018-02-15T19:34:48.857" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13410" PostTypeId="2" ParentId="2536" CreationDate="2017-12-30T16:38:19.137" Score="1" Body="&lt;p&gt;Switching to online marking for mechanical and repetitive practice is a very good idea.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are the alternatives to the &quot;practice, practice, practice&quot; method to gain good skills in computations?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Though I still recommend large amounts of repetition, you can dramatically increase the &lt;em&gt;quality&lt;/em&gt; of those repetitions in a variety of ways.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;For a given exercise that requires a multi-step solution, generate five &quot;answers&quot; of varying quality from five fictitious students. Have your actual students individually rank them by quality, then discuss their rankings in groups, with the aim of establishing criteria of assessment. This is called &quot;Assessment &lt;em&gt;as&lt;/em&gt; learning&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Instead of grading large volumes of students' work, switch toward grading a much smaller amount of it and giving them extremely concrete comments on how they should improve next. This is &quot;Assessment &lt;em&gt;for&lt;/em&gt; learning&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use retrieval practice. A typical task might be &quot;From memory, create examples and solutions for the ten most likely types of questions you expect on the test.&quot; Let them try first try this from memory, i.e. just pencil and paper and nothing else. Then they can pull out a pen, open their textbooks, and mark their work and/or their peers' work. They might form a consensus on those top 10 exercises. You can provide feedback. Then they can go through the cycle again. Of course, you'll get a better handle on whether your students are aware of what the big concepts even are, how difficult they expect the test to be, if they can distinguish among all the skills, etc. Read more at &lt;a href=&quot;http://www.learningscientists.org/blog/2016/6/23-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Learning Scientists&lt;/a&gt; and &lt;a href=&quot;https://theeffortfuleducator.com/2017/09/14/color-coding-recall-attempts-to-assess-learning/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Effortful Educator&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Check out &lt;a href=&quot;http://bentilly.blogspot.ca/2009/09/teaching-linear-algebra.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this homework policy&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2017-12-30T16:49:36.163" LastActivityDate="2017-12-30T16:49:36.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13411" PostTypeId="2" ParentId="13378" CreationDate="2017-12-30T21:49:36.160" Score="18" Body="&lt;p&gt;Instead of arguing with other people's answers in the comments I thought it might be more productive to present my own point of view.  I find myself completely unable to understand why anyone would take off points for this student's answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just to be clear, this isn't because I'm being somehow lax or generous as a grader.  My opinion is that &lt;strong&gt;this is a model solution to the problem&lt;/strong&gt;, written clearly and well, and I can imagine writing exactly what this student wrote as part of homework solution or exam solution that I distribute to a class.  In the context of Calculus I, it's also how I would do this problem on the board during class if a student asked me about it.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;On the Status of Infinity&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Some of the other calculus teachers here have mentioned that they teach their students that &quot;infinity isn't a number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find this statement very strange, and I suppose that my position is that &lt;strong&gt;infinity is a number&lt;/strong&gt;.  It certainly isn't a  &lt;em&gt;real&lt;/em&gt; number, since it's not included in the usual real number system.  But neither is the imaginary unit $i$, and I don't think many people would argue that $i$ isn't a number.  The number $i$ is included in the system of &lt;em&gt;complex numbers&lt;/em&gt;, and the number $\infty$ is included in the system of &lt;em&gt;extended real numbers&lt;/em&gt;, which is the set $\mathbb{R}\cup\{-\infty,\infty\}$.  I don't see the difference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there's no standard definition of &quot;number&quot; in mathematics, so there's no objective truth either way.  This is part of why it strikes me as so odd that a teacher would say that &quot;$\infty$ isn't a number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's possible that what they mean is that &quot;you can't do arithmetic with $\infty$&quot;.  But of course &lt;strong&gt;you can do arithmetic with $\infty$&lt;/strong&gt;.  For example,&#xA;$$&#xA;\infty + \infty = \infty,\qquad \infty \cdot \infty = \infty,\qquad\text{and}\qquad 3\cdot \infty = \infty.&#xA;$$&#xA;These definitions are absolutely standard in mathematics, and I would feel free to use them in a conference talk or journal article without comment.  I would hope that most calculus students would know how to do basic arithmetic with $\infty$ by the end of a first calculus course, but apparently this varies by instructor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also arithmetic operations involving $\infty$ that are undefined, such as&#xA;$$&#xA;\infty - \infty,\qquad \frac{\infty}{\infty},\qquad\text{and}\qquad 0\cdot\infty.&#xA;$$&#xA;The last is sometimes defined to be zero (e.g. in the theory of Lebesgue integration), but in the context of calculus it's better to leave it undefined.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I know, all of this is completely standard, and in my experience arithmetic involving $\infty$ and $-\infty$ is commonly used by mathematicians without further explanation or comment.  I've seen lots of examples of this, but to cite a specific one it's certainly the case that Rudin's &lt;i&gt;Real &amp;amp; Complex Analysis&lt;/i&gt; textbook (an extremely standard choice for a graduate analysis course) uses the extended real number system throughout.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;On the Student's Answer&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The student's answer depends primarily on the following theorem&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; &lt;em&gt;Let $f\colon \mathbb{R}\to\mathbb{R}$ and $g\colon\mathbb{R}\to\mathbb{R}$ be functions, and let $a\in [-\infty,\infty]$. If&lt;/em&gt;&#xA;$$&#xA;\lim_{x\to a} f(x) = L\qquad\text{and}\qquad \lim_{x\to a} g(x) = M&#xA;$$&#xA;&lt;em&gt;for some $L,M\in[-\infty,\infty]$ and the product $LM$ is defined, then&lt;/em&gt;&#xA;$$&#xA;\lim_{x\to a} f(x)\,g(x) = LM.&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a well-known and standard theorem in analysis.  In the context of this theorem, the student's work constitutes a perfectly good proof of the fact that&#xA;$$&#xA;\lim_{x\to\infty} \bigl(x-\sqrt{x}\bigr) = \infty.&#xA;$$&#xA;It is no more or less correct than something like&#xA;$$&#xA;\lim_{x\to 0} \frac{x\sin x + 2 \sin x}{x} = \lim_{x\to 0} \,\bigl(x+2\bigr)\!\left(\frac{\sin x}{x}\right) = (2)(1) = 2.&#xA;$$&#xA;I don't see why this proof would require any more explanation or rigor, in either a calculus or real analysis course, and I feel the same way about the student's proof.  I suppose it &lt;em&gt;might&lt;/em&gt; be reasonable for an analysis professor to always require students to cite the theorems that they are using, as opposed to using theorems implicitly as part of a calculation.  I &lt;em&gt;certainly&lt;/em&gt; don't think this would be a reasonable requirement for student answers in a calculus course.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Should we teach arithmetic with infinity to calculus students?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I do, and I would certainly hope that most other calculus instructors do as well. Dealing with the concept of infinity is a major theme of calculus, and the rules for arithmetic involving infinity ultimately derive from the idea of a limit.  How does it help to avoid talking about this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, it seems to me that it would be difficult to cover the idea of an &quot;indeterminate form&quot; without covering this material.  I guess at least some of the teachers here manage to avoid saying that &quot;infinity plus infinity equals infinity&quot; by always saying &quot;the sum of two quantities that are both approaching infinity again approaches infinity&quot;, but what's the purpose of being so obtuse?  If there's a simple way to say something, just say it that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And in any case, the reality is that you &lt;em&gt;can&lt;/em&gt; do arithmetic with infinity.  Saying that $\infty+\infty$ is undefined or indeed anything other than $\infty$ is just wrong, both at an intuitive level and from the point of view of standard notation and terminology.  Students will figure out that it's true on their own, and will try to guess what other arithmetic rules you're not telling them.  If you tell students that $\infty + \infty$ isn't $\infty$, you lose your credibility, and they won't believe you later when you tell them that $\infty - \infty$ isn't $0$.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Okay, but should we mark the student wrong?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Even if you don't talk about arithmetic involving infinity in your calculus class, the fact remains that it is absolutely standard mathematical notation.  Students often seek help from mathematics tutors, other math professors, online videos, and so forth, and any one of those sources might be teaching your students about how to use infinity in this fashion. Can you really justify deducting points from students who don't write their mathematics the way that &lt;em&gt;you&lt;/em&gt; want it written?  I feel like one of the most basic principles of grading is that &lt;strong&gt;correct answers should receive full credit&lt;/strong&gt;, unless the answer explicitly violates the instructions for the question.  This student's answer is completely correct, and in my opinion giving it anything less than 5/5 is just arbitrary and unfair.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2017-12-30T21:49:36.160" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13412" PostTypeId="2" ParentId="13378" CreationDate="2017-12-31T03:06:31.957" Score="8" Body="&lt;p&gt;I understand starting a discussion violates etiquette, but I cannot resist: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was the desired solution?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was it:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\begin{align*}&#xA;\lim_{x \to \infty} x - \sqrt{x} &amp;amp;= \lim_{x \to \infty} (x - \sqrt{x}) \frac{x+\sqrt{x}}{x+\sqrt{x}}\\&#xA;&amp;amp;=\lim_{x \to \infty} \frac{x^2-x}{x+\sqrt{x}}\\&#xA;&amp;amp;= \infty \textrm{ by leading terms theorem}&#xA;\end{align*}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or something like this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students solution is very clever.  What would they have had to say to get full points for their solution?  Maybe quote some theorem that if $\displaystyle \lim_{x \to \infty} f(x)$ and $\displaystyle \lim_{x \to \infty} g(x)$ both tend to infinity, then their product does as well?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, what do you expect students to do when confronted with a ``determinate form'', such as $\frac{0}{\infty}$ or $\infty \cdot \infty$ or $0^\infty$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am genuinely curious what the OP (and others who agree with him) would like to see.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2017-12-31T19:22:20.283" LastActivityDate="2017-12-31T19:22:20.283" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13414" PostTypeId="2" ParentId="13408" CreationDate="2017-12-31T09:52:38.607" Score="3" Body="&lt;p&gt;In the UK, we never use the dot notation in education up to 18. Instead, you use the x sign until you start to do algebra and want to save space by using bracket instead like $3(5) = 15$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is done to keep things more consistent so that students don't get confused by a basic symbol changing part way through their education. Even in my advanced maths class, we use $\times$ to denote multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a British student, the first time I encountered the dot was on an exchange in Belgium, where I was rather confused until I realised it was another sign for multiplication.&lt;/p&gt;&#xA;" OwnerUserId="9283" LastActivityDate="2017-12-31T09:52:38.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13415" PostTypeId="2" ParentId="13408" CreationDate="2017-12-31T15:58:10.917" Score="10" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;So why confuse people and introduce them to notation that is only used briefly and soon overtaken by other notation [...] ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think that the $\times$ symbol is used to teach multiplication the first time it's seen because that symbol relates multiplication to area. I recall learning both of these concepts &lt;em&gt;around&lt;/em&gt; the same time. So when you see the notation $3 \times 5$, instead of &quot;&lt;em&gt;three times five&lt;/em&gt;&quot; read it as &quot;&lt;em&gt;three by five&lt;/em&gt;&quot;. Some children may already be familiar with this language since it is common in stuff &quot;&lt;em&gt;aimed at the general public&lt;/em&gt;&quot; as you mention. It's also used in work environments (&lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/2_By_4_Clue_Stick.jpg/1024px-2_By_4_Clue_Stick.jpg&quot; rel=&quot;noreferrer&quot;&gt;dimensions of lumber&lt;/a&gt;), and is very important while playing with Legos&lt;sup&gt;®&lt;/sup&gt;. But in general, instead of talking about repeated addition, you could teach a child to think of $3 \times 5$ as the number of small ($1 \times 1$) squares in a rectangle that measures three by five. This $\times$ notation can help students make this connection between multiplication and the concept of area.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I do understand that the $\times$ is used on advertisements, cashier calculators and some other material aimed at general public, but that shouldn't be the cause for such choice. It's a consequence.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I disagree. That &lt;em&gt;should&lt;/em&gt; be cause for such a choice, and whether it's a good precedent is beside the issue. When young'uns are first learning multiplication, I think the goal of getting them to learn &lt;em&gt;what multiplication is&lt;/em&gt; is more important than setting a good precedent for their future math and science education. Simply getting them to think about multiplication correctly is the most important thing. And if it helps them learn it by being able to relate what they're seeing in class to examples they've seen on calculators and in advertisements, then that's for the better.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2017-12-31T17:33:02.843" LastActivityDate="2017-12-31T17:33:02.843" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13416" PostTypeId="2" ParentId="13378" CreationDate="2018-01-01T01:26:11.300" Score="11" Body="&lt;p&gt;Calculus classes are taught at an 18th century standard of rigor, and analysis classes at a 20th century standard of rigor.  It doesn't make much sense to try to invent some arbitrary combination of the two.  So if you aren't expecting proofs written in sentences with all proofs eventually going back to epsilon-delta definitions, then you should accept anything that Euler would have written.  Would Euler have written this?  Of course he would have!  Full points.&lt;/p&gt;&#xA;" OwnerUserId="345" LastActivityDate="2018-01-01T01:26:11.300" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13417" PostTypeId="1" CreationDate="2018-01-01T09:54:03.967" Score="6" ViewCount="154" Body="&lt;p&gt;I'm interested in playing a role in determining mathematics curriculum and goals for K-12 students. I'm currently a college student. How do I even begin down this path? Math? Political Science? Education?&lt;/p&gt;&#xA;" OwnerUserId="9288" LastActivityDate="2018-01-11T20:54:53.420" Title="How do I work in creating education standards?" Tags="|secondary-education|curriculum|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13418" PostTypeId="2" ParentId="13404" CreationDate="2018-01-01T19:43:08.287" Score="11" Body="&lt;p&gt;This is an &lt;em&gt;opinion&lt;/em&gt;, but perhaps worth sharing:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I assign that I expect students (particularly lower division students, such as those in calculus or precalculus classes) to do on their own time, I assume that they are going to collaborate, ask the Google, and otherwise use all of the resources that are available to them.  Demanding that they do the work on their own is a fool's errand, and can be rather paternalistic.  As such, I don't like to make homework a huge proportion of the final grade in a class, as I want a student's grade to reflect their individual ability and learning, rather than their ability to find the answers on the internet.  I would prefer to base a student's grade on work that they do in class, under better controlled circumstances, e.g. exams or presentations (with moderately intense questioning).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, most students cannot learn and internalize new material without having had a good deal of practice.  Homework provides that practice.  As such, homework is a vital part of class, and cannot be ignored.  In an ideal world, I would assign homework, and my students would be mature and well-motivated enough to recognize that working through the problems is to their benefit.  In this ideal world, it would be entirely unnecessary to grade (or even collect) this work&amp;mdash;the effort that the students put into it would be reflected in their exam scores.  Since we don't live in an ideal world, and not all students are mature or well-motivated enough to do work on their own without some kind of Pavlovian reward, I typically give some minimal credit for homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Returning to my ideal world, once I have assigned homework, I would collect it, and give feedback to all of my students in detail.  However, if I have a large class, there are not enough hours in the day to give that kind of feedback.  As such, there are a few ways that I can assign work (this is not meant to be exhaustive, but should hit the major points):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Use a computer driven system, such as MyMathLab, WebWork, or ALEKS, then using the scores returned by the system for the homework component of my gradebook.  Such a system has the advantage of requiring very little input from me (once the class is set up), and gives students instantaneous feedback.  On the downside, these systems are only good for low-level drill-and-kill style questions, and tend to do a poor job of forcing students to use higher order thinking.  I think that the questions also have a tendency to break down problems too much, and give students the idea that there are different problem &quot;types&quot; that, once mastered, give them mastery over the material.&lt;/li&gt;&#xA;&lt;li&gt;Assign homework with answers (but not solutions) given, and grade that on the basis completion.  Here, the students know if they have obtained the correct answer (since the answers are given), and my workload is tractable, since I only have to check off the assignments.  I can also give deeper questions in the homework.  The disadvantage is that the students get very little feedback, unless they choose to come and discuss the problems during office hours.  I have also observed that students seem not to value work that is graded on the basis of completion&amp;mdash;they do the minimum necessary to get the points, then move on.  This hurts them, come exam time.&lt;/li&gt;&#xA;&lt;li&gt;Assign problems with no answers given, then grade either entirely on the basis of completion, or pick a few problems on which to give feedback and give completion credit for the rest.  From the point of view of giving good feedback, this is likely the best that can be done for a large class.  Again, office hours are good for addressing problems.  On the downside, students are only getting feedback on a few problems, and may have serious misconceptions that are never addressed.  This can be somewhat ameliorated by providing solutions once the assignments are handed in.&lt;/li&gt;&#xA;&lt;li&gt;As you suggest, assign work but do not collect or grade it.  You can then assess their homework performance on the basis of short quizzes (I wouldn't give them answers in advance, as this defeats the assessment).  My strategy is to give them 5&amp;ndash;10 minutes to answer one or two questions taken more-or-less directly from the homework.  Sometimes I even let them keep their homework out to answer these questions (if they did it right, it is a simple copy-pasta).  I then grade these quizzes fairly harshly, with the justification being that they had plenty of time to discusses troublesome problems during office hours.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Note that all of this is for a &lt;em&gt;lower division&lt;/em&gt; mathematics class, where there is a often a significant requirement that students be able to perform basic computations, which may or may not require deeper understanding, but which &lt;em&gt;does&lt;/em&gt; require a fairly large number of problems for practice (perhaps not quite drill-and-kill, but close).  As courses get more abstract and the students become more mature (both mathematically and as self-motivated learners), a greater emphasis can be placed on written work (e.g. nicely written or typed work in a proofs-based course).&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2018-01-01T23:55:14.517" LastActivityDate="2018-01-01T23:55:14.517" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13419" PostTypeId="1" AcceptedAnswerId="13421" CreationDate="2018-01-01T20:08:56.293" Score="2" ViewCount="311" Body="&lt;p&gt;I would like to attempt this, as I want to place out of geometry next year. It is imperative that I do it now. How should I go about doing this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a copy of &lt;a href=&quot;https://www.cengage.com/c/elementary-geometry-for-college-students-6e-alexander&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Elementary Geometry for College Students, 6th Edition&lt;/em&gt;&lt;/a&gt; by Dan Alexander and Geralyn Koeberlein.&lt;/p&gt;&#xA;" OwnerUserId="9289" LastEditorUserId="5220" LastEditDate="2018-01-02T17:04:18.837" LastActivityDate="2018-01-02T17:04:18.837" Title="Learning High School Geometry in Ten Days" Tags="|geometry|self-learning|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13420" PostTypeId="2" ParentId="13404" CreationDate="2018-01-01T20:55:02.380" Score="5" Body="&lt;p&gt;I teach a highly selective Liberal Arts College.  If you don't assign a component of the final grade to the homework, the majority students in a typical introductory class just won't do it or, if they do it, they'll do it carelessly.  They'll rely on their previous knowledge for the quizzes and it won't go well for them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my classes I assign and collect daily homework through WebWork and make it worth 5% of the final grade.  This encourages them to do it, makes them prepared for class and class discussions go better.  In my experience, since WebWork problems are randomized, students seem to have a harder time finding the solutions on line.  Also, WebWork has no solutions manual (as far as I know), making it harder to find online.  Also, the students know if they got things right because WebWork will indicate whether or not their answer is correct.  In summary, then, I find it better to assign daily homework with neither solutions nor answers available.  The students who want to work, get a lot out of it and the students who don't are no worse off.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I give weekly quizzes based on the previous week's homework.  I find that making the quizzes weekly and the homework daily gives the students a chance to review a whole week's worth of material at a time, something that would be missing in your proposed scheme.  I've given daily quizzes before but believe that students who did the homework were just relying on muscle memory to complete the quizzes rather than thinking more broadly or deeply about the material.  I make the total of these quizzes about 10% of the final grade.  This is the mechanism by which students get feedback on their work (I also almost-weekly Team Homework assignments where I give extensive feedback on their writing and argumentation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;b&gt;Edit:&lt;/b&gt; This doesn't seem that different to @Xander Henderson's response, so maybe I misunderstood the question.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2018-01-01T20:55:02.380" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13421" PostTypeId="2" ParentId="13419" CreationDate="2018-01-01T23:01:33.897" Score="7" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Read the text and work the homework problems.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Design a self study plan based on the time you have available (80-100 hours).  I'm not sure that is enough, would estimate 180 needed if this is material you have never seen before.  But design a plan (hours per section and such) and see how it goes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use a notebook or compiled looseleaf.  Develop some system (sections of notebook or icons at top of page to differentiate self notes, homework, questions (with room for answers!), and self tests.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;3.5. Use the section for questions to &quot;parking lot&quot; things you don't understand as you go.  You can then refer these to the Internet or someone (parent?  older student?)  But I suspect many of them you will figure out yourself.  But still is good discipline to work through as you teach yourself, to write things down you have questions and then put the resolution down (even if it is you after working a few problems or reading next section).&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;For time efficiency, I would attempt the HW problems that have answers in the back and/or reserve some for self testing.  Make sure you actually WORK the problems.  NOT say &quot;oh I get it&quot;.  But really work problems.  And really test yourself.  Anything you get wrong on a HW problem or self test has to be reworked (even if a silly mistake).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Check the book suitability.  You say you want high school geometry but that is some random college book.  Is the material fitting your needs?  Can you check with an upperclassman what book they use or even borrow it for the holidays?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This would be an easier question to answer if you did a little more work yourself in terms of devising an initial plan, describing it and then asking for feedback.  Just showing up with a terse question does not show much investment by you.  And makes it harder to advise you on specifics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, this site is full of community college instructors who love real analysis.  Not sure they can really advise you.  Talking to a HS math teacher (maybe look for a good blog or such) would do more for you.&lt;/p&gt;&#xA;" OwnerUserId="9272" LastEditorUserId="9272" LastEditDate="2018-01-01T23:10:48.563" LastActivityDate="2018-01-01T23:10:48.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13422" PostTypeId="2" ParentId="13378" CreationDate="2018-01-02T06:49:01.190" Score="-1" Body="&lt;p&gt;I would give this a full grade plus a bonus point.  Fantastically elegant solution.&lt;/p&gt;&#xA;" OwnerUserId="9291" LastActivityDate="2018-01-02T06:49:01.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13423" PostTypeId="1" AcceptedAnswerId="13426" CreationDate="2018-01-02T11:11:21.910" Score="19" ViewCount="1226" Body="&lt;p&gt;I know that some instructors collect homework and &quot;grade that on the basis of completion&quot; (e.g., item #2 on &lt;a href=&quot;https://matheducators.stackexchange.com/a/13418/5563&quot;&gt;this answer&lt;/a&gt;). In fact, I tried this myself for several years, based on advice from my mentor in graduate school. But I found it to be terribly frustrating, laborious, and unproductive, and I had to abandon it some years ago for my own sanity. That said, I'm still curious about how others manage to make it work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the main thing I never resolved: What should the criteria be for &quot;checking off&quot; (pass/fail, I presume) an acceptable performance on homework? Here are some examples that I encountered when attempting to apply this process:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Student lists final answers only as shown in back of book; no work is shown whatsoever.&lt;/li&gt;&#xA;&lt;li&gt;Student has what appears to be work, but it is totally illegible. &lt;/li&gt;&#xA;&lt;li&gt;Student has some kind of work, but problem identifiers (labels/numbers) cannot be found that match the assignment.&lt;/li&gt;&#xA;&lt;li&gt;Student has work and answers that don't remotely resemble the correct solutions (e.g., exercises in simplifying variable expressions allege finding a value for the variable; work involving fractions or radicals has no such expressions at the end). &lt;/li&gt;&#xA;&lt;li&gt;Student's work starts with the answer and only checks that answer throughout, without exercising any real solution method (e.g., substituting solutions into an equation without solving it; multiplying factors back together without factoring a polynomial). &lt;/li&gt;&#xA;&lt;li&gt;Student's work exactly matches work on the next submitted page.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These are just some of the examples of what seem to be critical problems which are apparent at a glance, that I am loathe to give credit for. In fact, it seems to send a confusing and perilous signal to give students a passing grade (&quot;you're doing okay&quot;) on work that is inherently flawed like this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: What criteria should be applied for minimally-acceptable performance when &quot;checking homework for completion only&quot;?&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-01-02T12:42:10.910" LastActivityDate="2018-01-03T05:22:08.370" Title="What are the minimum criteria when checking homework for completion only?" Tags="|undergraduate-education|secondary-education|homework|" AnswerCount="3" CommentCount="16" ContentLicense="CC BY-SA 3.0" />
  <row Id="13424" PostTypeId="2" ParentId="13404" CreationDate="2018-01-02T11:25:35.217" Score="2" Body="&lt;p&gt;I'm not sure that I understand your question well: you talk about replacing homework by a small quiz at the beginning of each class, based on variations on freely available solved problems?&lt;br/&gt;&#xA;One of the reasons is your experience that some students don't do their homework themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have teached in a technical school and I can tell you that, in that particular environment, your system would fail heavily:&lt;br/&gt;&#xA;Imagine you start each class by a short quiz, then this would automatically take away some teaching time (an average class takes 50 minutes, if you start with a written quiz you easily lose 15 minutes, or 30%). In order to compensate this, you might need to speed up, with a high risk of weaker students (who already have problems following the course), to get even more into problems.&lt;br/&gt;&#xA;Imagine you start each class by a short oral quiz (just ask one or two questions and let the class find an answer), then you risk that the highly motivated students (who generally are the smartest ones) give the answers very quickly, and the weaker ones won't even bother looking for the answer as they know that they can't win/lose points whether or not they know the answer to the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also don't like the whole idea: you say that there are two ways to learn mathematics:&#xA; - listen to the teacher and then, at home, try to do it yourself (the homework way)&#xA; - listen to the teacher, read it and try to make variations (the quiz way)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't believe the last way can work: the whole idea of mathematics is to learn how to reason, and how difficult this may seem for some students, the only way to learn it is by doing it themselves. You are right that in some cases some students are just copying from other ones, but sometimes they ask to each other &quot;But how did you even think of doing that?&quot;, and in that case, just by getting it explained by someone of their own age, is enough to get them understand it.&lt;br/&gt;&#xA;The most common argument against this is, as you say yourself, that students just copy from each other or just get it from Google (which is most probably the reason for you to propose this new method), but in my opinion there is one simple &quot;solution&quot; for this: don't let students write their homework via a computer, but they need to do it in handwriting. I know, it is old-fashioned and they might dislike it, and oké, some students might still look up via Google or via their friends, but at least they will be forced to write the solutions down them-selves, and simply by doing this, even if they are just copying, they might actually learn from it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2018-01-02T11:25:35.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13425" PostTypeId="2" ParentId="13423" CreationDate="2018-01-02T18:18:58.350" Score="6" Body="&lt;p&gt;It depends on what you are assigning the homework for. The learning objectives of an assignment should determine your rubrics for assessment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to ask yourself why you are assigning the homework, what is its point in the curriculum and the goal of completion of exercises for the students. You then need to ask yourself is homework part of the assessment of knowledge or is it a tool in the learning, or some combination of both. Design your rubric to match those learning goals and realize students use points and grades as an economy and your grading will shape some of the student behavior.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am going to illustrate this with an example from one of my classes, this is not the format I use in all classes my upper level classes I rely on the students desire to learn the material being greater:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my College Algebra Class I have three assignment types.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Homework&lt;/li&gt;&#xA;&lt;li&gt;Notebook Problems&lt;/li&gt;&#xA;&lt;li&gt;Project&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The purpose of each assignment is different so the rubrics and assessments are different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The homework problems are due each class period and consist of highlighted problems from the odd problem sets in the textbook. The answers are in the back of the book. The purpose of the assignment is for the students to practice the concepts introduce in the lecture and do enough problems that they can find where their questions, ask the questions, to begin to see patterns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is graded on a 3 point system. &#xA;Was the problem attempted with sufficient work.  Full marks for the problem.&#xA;Was the problem attempted but insufficient work. 2/3 marks.&#xA;Was the problem attempted by little on no work. 1/3 marks.&#xA;And of course no attempt at the problem no marks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the grades are full marks. If there are attempts at getting to a solution they get the marks. Insufficient work would be the appearance of work but it is not relevant steps. Little or no work is when you get students copying the answers from the back on large sections of problems, usually when they run out of time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point though is the points come from attempting to solve a problem. My goal is to have the students get past the I do not know what to do hurdle, and to do problems. This is to open the door to dialogue on mistakes. Using the answers in the back of the book they have the ability to know if they are correct or not and can self correct or ask questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To test the knowledge from a homework set they will have a quiz on the day the homework is due on material that was assigned in the homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This gives an incentive for the students to do the homework, honestly and to ask questions when they are stuck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The notebook is due every Monday. It is a choice of 5 problems from the homework sets. The students are required to rewrite the problem. Explain each step of their mathematics. And restate the answer in English. (Natural language of the classroom.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rubric is &#xA;5 points for format&#xA;5 points for the answer&#xA;5 points for Reasoning - clarity&#xA;5 points for Reasoning - logic&#xA;5 points for Reasoning - mathematics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason this detail is different than the homework is my reason for this assignment is not to practice and begin the dialogue around particular problem sets but for the student to show me the understand the methods and reasoning used in solving particular problems with mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The project is an extended semester long project that uses datasets to explore all the ideas of the semester with messy real world data. And I won't discuss the rubric here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point is the assignments are structured and graded as such because I am addressing specific issues in the classroom.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Historically students did not self practice, I need to assign homework for them to practice, even if I quiz often.&lt;/li&gt;&#xA;&lt;li&gt;Historically students would stop if they did not know how to begin a problem.&lt;/li&gt;&#xA;&lt;li&gt;Some students when given odd problems would just copy the answers from the back and argue they did the work.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My rubric is designed to address these issues and specifically to highlight the need to practice, self correct, ask for help and communicate in the process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also the students see the importance of practice in the homework with minimal risk. They don't have to be perfect in the practice phase they have to start &quot;to do&quot;. But with the quizzes they have to take the practice seriously it has to lead to an understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also the knowledge accumulates and they have the notebook problems to reflect and explain. To begin to own the thought process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are my goals with the assignments and I try to have my assessment reflect this. Yours need to meet your goals. And thus to answer your question the minimum criteria depends on where you expect your students to be with the assignments and how you use points/grades to motivate or encourage.&lt;/p&gt;&#xA;" OwnerUserId="8952" LastEditorUserId="8952" LastEditDate="2018-01-02T22:01:36.313" LastActivityDate="2018-01-02T22:01:36.313" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13426" PostTypeId="2" ParentId="13423" CreationDate="2018-01-02T19:02:56.463" Score="8" Body="&lt;p&gt;This is going to look a lot like Jeffery Thompson's answer, with some slight modifications (I was working on this answer when his appeared).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First off, I agree entirely that you have to decide what you want students to get out of the homework first, then decide how you are going to grade those assignments.  As I said in the &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13404/making-solutions-to-all-problems-freely-available/13418#13418&quot;&gt;answer to which you linked&lt;/a&gt;, I think that primary goal of homework in a lower division class is to give students a chance to practice the kinds of problems that I think are important, as this will help them to learn and master the material.  Ideally, the students would be mature and engaged, and would do the homework and come to office hours to discuss problems even if &lt;em&gt;no&lt;/em&gt; credit were given for homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, the kinds of students who are typically in a college-level precalculus or introductory calculus class are often not mature enough to do the work without the promise of a reward.  Thus, I collect the homework and assign marks, because if I don't, the students probably won't bother.  This is why I give any grade at all to homework&amp;mdash;I'm playing psychological mind games with my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next question is one of time; specifically, how much time do I have to grade?  In a large lower division class (100+ students), there simply isn't going to be enough time grade all of the work in detail and give meaningful feedback.  I would love to comment on all of the work, but my wife would probably kill me if I spent that much time in the office.  Thus I will typically choose to grade on the basis of &quot;completion,&quot; where completion basically means that the student wrote their name on the top of the page and wrote down solutions (note:  not &lt;em&gt;answers&lt;/em&gt;, but &lt;em&gt;solutions&lt;/em&gt; which show work) to all of the questions.  Typically, I'll make each assignment worth 1 point, and either give a 1 or 0.  Either it is done or not.  In your examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Answers, but no work:  I would give no points.  The answers were given.  It is up to the student to do the work.&lt;/li&gt;&#xA;&lt;li&gt;Illegible work: I would give points on the first couple of assignments, and make the comment that it is illegible, then stop giving credit after two or three assignments have been returned.&lt;/li&gt;&#xA;&lt;li&gt;Wrong problems:  I would give no credit.  If the student doesn't do the assigned problems, then they didn't do the work, and deserve no credit.&lt;/li&gt;&#xA;&lt;li&gt;Poor quality or incorrect work:  I would give credit, but might ask the student to come to office hours.  That being said, I might not even notice that the answers are wrong, since the whole point of grading for completion is that I don't have to check their work.&lt;/li&gt;&#xA;&lt;li&gt;Poor techniques: I would treat this the same as 4.&lt;/li&gt;&#xA;&lt;li&gt;Identical work to another student:  Both students get full credit.  Ideally, they were working together and therefore have the same answers.  More realistically, one student copied off of the other, and was therefore &quot;cheating.&quot;  However, I assume that students will (to put it generously) &quot;collaborate&quot; on anything outside of class&amp;mdash;I have no control over that, and don't want to arbitrate those kinds of disputes.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;That being said, I rather like Jeffery Thompson's rubric, and my start adopting something like that (0 points for blanks, 1 point for the attempt, 2 points for correct solutions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that all of the above applies to the kind of homework assignments that I call &quot;Computational Practice.&quot;  These kinds of assignments (similar to Jeffery Thompson's &quot;Homework&quot;) are generally very computational in nature, and not terribly deep.  They are meant to get students to master algorithms or computational tricks.  These are not far from &quot;drill-and-kill&quot; (though I seem to recall from my ed certification program that there is some fancy jargon for this... &quot;deliberate practice,&quot; maybe?  that has the right ring to it...), and typically involve a large number (15&amp;ndash;20) of problems that should all take very little time (2&amp;ndash;3 minutes each).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to specifically address this point:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In fact, it seems to send a confusing and perilous signal to give students a passing grade (&quot;you're doing okay&quot;) on work that is inherently flawed like this.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have had concerns about that, but I try very hard to make it clear to my students what the point of the work is.  An example of the exact statement on my syllabus (from a stats class last quarter) is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Homework will be assigned daily. Assignments will be due on the Monday after they are assigned, at the beginning of class. All assignments will be posted on Blackboard. The goal of the homework assignments is to give you a chance to get some practice with the kinds of problems that might appear on an exam and to build your understanding of the concepts discussed in class. However, I do not have the time to give in-depth written feedback to every student, so your work be graded on the basis of completion. Answers to odd numbered exercises are in the back of the text, and I am available during office hours or via email to discuss any problems that you might have.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I then repeat something like that every time a hand back homework:  &quot;Remember, I am just grading your homework on the basis of completion.  You should not assume that, because you got credit for the work, you are in good shape in this class.  The exams will be much more important.  If you are having any difficulty, please see me during office hours.&quot;  This, coupled with the fact that homework is worth only 5&amp;ndash;10% of the final grade, &lt;em&gt;seems&lt;/em&gt; to get the point across.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As an addendum, I also tend to give longer &quot;Written Assignments,&quot; but the goal of these if different, and they are worth a larger portion of the student's grade.  The assignments typically consist of two or three more involved problems, and the solutions are graded in detail, with an eye toward good mathematical writing (this is inspired by the &quot;writing across the curriculum&quot; mandate that was all the rage when I taught high school&amp;mdash;the goal is to (approximately) learn how to write in mathematician-ese).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I typically give seven or eight of these written assignments in a 10 week quarter, though it depends on the size of the class&amp;mdash;it is reasonable to grade something on the order of 50 such assignments in a week, spending 2-3 minute per; in larger classes, I would assign fewer written assignments.  The assignments are given out, and students have a week to turn in a rough draft.  I then offer comments, and students have another week to turn in a final draft (though, to be honest, the final drafts have a &quot;floating&quot; deadline).  I then award points as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;one point for turning in a rough draft on time&lt;/li&gt;&#xA;&lt;li&gt;one point for turning in a final draft on time&lt;/li&gt;&#xA;&lt;li&gt;one point for coming to office hours and discussing corrections to the rough draft (these three points represent the &quot;completion&quot; portion of the grade)&lt;/li&gt;&#xA;&lt;li&gt;three points for writing mathematically correct solutions&lt;/li&gt;&#xA;&lt;li&gt;three points for writing in legible, grammatical mathematical English, including the proper use of notation&lt;/li&gt;&#xA;&lt;li&gt;one &quot;bonus&quot; point for TeX-ing up their work.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are fewer of these, and each contains fewer problems.  Even with a large number of students, it is possible to grade a couple of stacks of these each week (particularly since much of the feedback is offloaded to office hours, which is allocated time already).&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2018-01-02T23:41:30.270" LastActivityDate="2018-01-02T23:41:30.270" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13427" PostTypeId="2" ParentId="13378" CreationDate="2018-01-02T22:28:57.127" Score="2" Body="&lt;p&gt;There are two mottoes I recite whenever I feel doubt in grading. They are:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;&quot;If it's not wrong, then it's right.&quot;&lt;/li&gt;&#xA;  &lt;li&gt;&quot;Was I explicit in what I wanted them to do?&quot;&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would give them a 5/5, if I weren't explicit. I would give them a 4/5, if I said justify any limit rules. I would give a 0/5 if I explicitly said I wanted a $\delta-\epsilon$ proof. Any slip ups in phrasing is purely the instructor's fault. Any punitive action against the student because of an instructor's oversight is a sign of a small-minded instructor.&lt;/p&gt;&#xA;" OwnerDisplayName="user1916" LastActivityDate="2018-01-02T22:28:57.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13428" PostTypeId="2" ParentId="13404" CreationDate="2018-01-03T02:58:08.473" Score="1" Body="&lt;p&gt;I think the problem with homework generally is that when we assign problems from almost any book published by a mainstream publisher there are &quot;solutions freely available&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of my colleagues assign problem from the textbook and give weekly quizzes based on that homework for the homework grade. Unfortunately, there are just too many solution manuals etc. freely available and all too often we get mindless copying of &quot;answers&quot;. Similar problems exist for WebAssign, or WebWork etc. again these have the added undesirable trait of de-emphasizing the importance of cataloging work. I think the process of them assembling their own catalog of solutions is as or more important than us giving them feedback. They need to learn to assess their own progress if they are to thrive long term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once tried posting required homework for physics which was then quizzed so the could copy down their solutions. What I found was that there was still a significant portion of the students trying to do the problems in the 3-5 minutes I gave them. The others often had not really thought out the problems for themselves either. In short, to answer your question I did not find this motivated the class better than when I made them use Web Assign. But, it was not a significant increase in apathy. If I had to put a number on it, I'd say I experienced an apathy level of 50 whereas the usual level was 40.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only solution I know which is reliable is to write your own problems each semester and find human graders to either comprehensively or selectively grade the work. To my knowledge, this is the gold standard of homework which gets the most genuine effort from many of the students. I just spent about 4 days crafting 162 mostly new problems for my linear algebra class. It will take me another 30-40 hours to write solutions as the semester unfolds. Fortunately I have a grader to help. It certainly does improve the general aptitude on tests.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2018-01-03T21:06:35.653" LastActivityDate="2018-01-03T21:06:35.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13429" PostTypeId="2" ParentId="13423" CreationDate="2018-01-03T05:22:08.370" Score="3" Body="&lt;p&gt;A high school teacher told me he stamped their homework at the students' desks, so he didn't have to collect it. I liked that, and for years used that system. (I bought a stamper at Staples.) At each test, they turned it all in, and I counted the stamps. It worked for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But apparently students didn't like it, which I heard about this summer from a friend who tutors. So I decided to try collecting homework, and spending as little time as possible grading it. I grade one problem, and the rest of the credit (maybe 3/4 or 4/5) is for completion. Only if they seem to be showing the work and doing the proper problems. Only if turned in on time. If I see someone working on it in class, I will remind them it was due at the beginning, and if it were to continue, I would stop accepting it. I did not accept late homework. I told them it was just too hard for me to keep track of. (True.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It added 3 hours to my work week, I think. Bummer. But in two of my three classes, I think students worked harder than they have in the recent past for me. So, I will keep doing this.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-01-03T05:22:08.370" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13430" PostTypeId="2" ParentId="12949" CreationDate="2018-01-05T01:33:21.777" Score="4" Body="&lt;p&gt;To many students, geometry is very different than algebra.  They see geometry as a bunch of facts (we call them theorems) to be memorized (we prefer that they understand).  I feel the same things happens with trigonometry.  Many students who are successful in algebra struggle with trig and geometry because, I think, it feels so different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I think, that blending the two could help.  Blurring the artificial distinction between the two could help students prepare for advanced courses (e.g., in calculus related rates problems and integrals-as-sums-of-tiny-slices problems require geometry) by showing students that they are all related.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is based on my experience teaching high school geometry for a couple of years as well as high school calculus.  I feel that this blending helped students at the school.  For examples of the kind of blending I mean, see the first few pages of &lt;a href=&quot;https://www.exeter.edu/sites/default/files/documents/Math2_2017.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As an aside, I also did this blending with algebra.  As kids were learning how to solve equations, I asked them to do it in a two column format, where they had to justify the steps they were carrying out)&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2018-01-05T01:33:21.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13431" PostTypeId="1" CreationDate="2018-01-05T01:52:22.003" Score="7" ViewCount="4467" Body="&lt;p&gt;As an undergraduate, I clearly remember learning and using &quot;hat notation&quot; to describe unit vectors.  That is, if $\vec{v}$ is any vector (in 2 or 3 dimensions) then $\hat{v}$ denotes the unit vector in the direction $\vec{v}$, i.e.&#xA;$$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;hat&quot; was also used for the standard unit vectors in the direction of the $x$-, $y$-, and $z$-coordinate axes:&#xA;$$\hat{i}=\langle 1,0,0 \rangle,\quad \hat{j}=\langle 0,1,0\rangle, \quad \hat{k}=\langle 0,0,1\rangle$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that I am teaching multivariable calculus for the first time, I see no use of this notation in our textbook (Stewart, 8th Ed.), and I am wondering if this notational convention is something I picked up from my undergraduate Physics coursework.  (Just to avoid misunderstanding:  Stewart uses boldface $\bf{i}$,$\bf{j}$,$\bf{k}$ for the unit vectors but does not use the hat accent.)  So my questions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is &quot;hat notation&quot; for unit vectors (including the standard unit vectors, but not limited to them) widely used in teaching multivariable calculus, or is this something only physicists use?  Are there math textbooks that use it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edited to add:&lt;/strong&gt; I thought I would mention the main reason I'm interested in this notation:  it makes certain formulas involving projection much simpler.  For example, if $\vec{u}$ and $\vec{v}$ are any two vectors then&#xA;$$\operatorname{proj}_{\vec{u}}\vec{v} = \vec{v} \cdot \hat{u}$$&#xA;$$\operatorname{comp}_{\vec{u}}\vec{v} = \left(\vec{v} \cdot \hat{u}\right) \hat{u} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(compare $\frac{\vec{v} \cdot \vec{u}}{|\vec{u}|}$ and $\frac{\vec{v} \cdot \vec{u}}{|\vec{u}|^2}\vec{u}$, respectively).  Likewise the angle between any two vectors $\vec{u}$ and $\vec{v}$ is given by $\cos(\theta) = \hat{u} \cdot \hat{v}$.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="8571" LastEditDate="2018-01-05T14:53:50.720" LastActivityDate="2018-01-27T23:42:46.783" Title="Is &quot;hat notation&quot; for unit vectors commonly used in mathematics?" Tags="|notation|vector-calculus|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="13432" PostTypeId="2" ParentId="12949" CreationDate="2018-01-05T02:59:31.387" Score="2" Body="&lt;p&gt;I would say yes.  I also worry about my (college) students who take time off from math classes, e.g. not taking one their senior year of high school.  You can forget a lot in a single year, especially with technical information that you aren't using some other way.  The same applies to a two year algebra sequence.  It makes sense to move directly from the first year to the second rather than taking a year off to do something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the US, however, this isn't the usual approach.  It's usually Algebra 1 -&gt; Geometry -&gt; Algebra 2.  The reason for this has more to do with teaching to the test, in this case the SAT, rather than any pedagogical reason.  The SAT has geometry so students need to complete geometry before their junior year.  Since many students don't take Algebra 1 until their freshman year, this means sliding geometry in between the two algebra courses.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2018-01-05T02:59:31.387" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13433" PostTypeId="2" ParentId="13431" CreationDate="2018-01-05T06:26:16.840" Score="5" Body="&lt;p&gt;I wouldn't say it's common (none of the calculus books on my shelf use it).  It's in some math books (e.g., &lt;a href=&quot;https://books.google.com/books?id=HVSVAwAAQBAJ&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; one, and &lt;a href=&quot;https://books.google.com/books?id=Xlc0DwAAQBAJ&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; one and &lt;a href=&quot;https://books.google.com/books?id=bXzAlkODwa8C&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; one) but it's mostly in physics books.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2018-01-05T06:26:16.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13435" PostTypeId="2" ParentId="11605" CreationDate="2018-01-05T21:44:54.883" Score="3" Body="&lt;p&gt;&lt;strong&gt;1 AP&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An easy example is AP Calculus (or chemistry or etc. if I can be allowed the sneaky broadening to general academia).  This is an example where the teacher is trying to prepare students for the test but doesn't know what will be on it, nor design it themselves.  Of course instructors will have to do their own tests for the school grades.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience at a (pretty good) public school in the 80s was that calc (and chem and etc.) teachers tried to teach for general excellent insight and understanding into the topic.  Even to the extent of a little coverage outside the requirements, as class was capable of it.  (well...in chem/bio etc. not in calc.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But all the subjects, even calc, I had the impression teachers were trying to teach a basic course content, regardless of the student sitting for the exam.  Only in week or two before the test was there a little practice on old tests to see the format and get used to style of them (which differed from normal tests during school year.)  However, the textbooks (Thomas Finney Calc, M, S, and S Chem were designed for AP usage (said so in preface), maybe with a few clearly asterisked extra topic chapters.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lately, I get the impression, from &quot;expert teacher&quot; blogs in calc and chem, (with growth of AP and such) that there is a bit more gaming the system, teaching very strictly to the curriculum and eschewing topics that &quot;aren't on the test&quot;.  [Even in some cases to the detriment of the student versus a college course.  Has been some reduction of AP coverage versus standard state school coverage...less diffyQs and rotation volumes in calc, less algebraic chemistry problems...even though college courses still cover this.  IOW, AP moved before rather than following college.  I have tried urging teachers to teach what they think a normal college class should be (not saying Cal Tech, but Perdue, maybe).  But the gaming aspect has gotten strong with teachers and even students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2 Multi-section classes&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience at military academy (still true today) is that they don't really do lecture hall classes.  Even for subjects like chem, calc, physics, navigation, EE etc. that all 1000 kids in a year group take (that many colleges would choose to do in large lecture halls).  So you get a bunch of profs teaching many 30 kid classes (like high school).  But the overall content is prescribed AND the tests are common to the overall group.  (done by some committee or master teacher or such.)  The point of all this exposition is that teaching is very much divorced from assessment.  And you can feel it and it comes across (teachers even saying they haven't seen the test when pushed for hints).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;P.s.  However, all that said, I think there is an assumption in the question that divorcing assessment from testing will lead to more thoughtful questions.  I disagree with this as it is possible to combine the roles and have tests that are thoughtful.  Or to divorce them and have more mechanical questions. This isn't just a theoretical point...I really think these are very independent variables, not much correlated.  I think it is a course design issue, not a who designs tests issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.  I don't completely agree either with the implicit praise for concept questions versus learning how to do typical manipulative problems.  This can be argued rather than assumed.  Just because you are super sharp on definitions and answering tricky concept questions does not mean you will succeed at several step integrations (calc) or stoichiometry problems (chem).  These are important skills for future use (thus worthy of learning) and one needs to practice solving these problems and be tested on them to develop the skill.  I'm sure there is a balance (some concepts, some tricks and skills).  But I would just be wary of assuming the balance needs to shift to more concept.  It sounds fancier especially from a teacher point of view but may not be optimal objective, particularly for math service courses.&lt;/p&gt;&#xA;" OwnerUserId="9298" LastActivityDate="2018-01-05T21:44:54.883" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13436" PostTypeId="2" ParentId="8081" CreationDate="2018-01-06T02:37:50.487" Score="2" Body="&lt;p&gt;Chris, interesting questions and some good responses and your approximation is thoughtful.  Just a few other soft thoughts to consider.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I wouldn't do a test on the kid.  I just think that gets too much into the weeds and kind of breaks a little the feeling of support role of an advisor.  If you really want to stealthily diagnose it, I would just talk to the fellow (&quot;where did you have problems?&quot;  &quot;what did you feel good about--what did you feel bad about&quot;) and try to get a feel for the person's own confidence about their level of knowledge.  EVen just the form of how they describe their knowledge level will give you some feel for their ability.  But I would not quiz them with specific problems.  Keep it in more of a friend and supportive coach role.  Maybe you could look at their course tests if they want to show them, but even that is maybe pushing it.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You should advise the student to talk to the instructor of the next up class (if you are not very familiar with the course yourself or if the content varies significantly by instructor.)  That person is best able to judge what is needed for their class and probably more suitable for that person to be putting the student on the hot spot with a little verbal or even written diagnosis of skills.  (And it also starts to get some dialogue between the teacher and the student for next semester, sets them up for success.)  [The one case, I am leaving out is if you taught them last semester.  In which case, you should have know the student's level of knowledge well and have a gut feel.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;May vary based on grading scale (I guess some places are harder than others).  I have progressed with Cs and even a D (high school algebra!)  Maybe in general, I think your progression scale is a little harsh (basically saying a B is passing).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some other soft factors to consider: skill of the student (smart but lazy or distracted kid can probably succeed more in next class than a slower student who needs a good foundation or he is lost.)  Personal needs of the student:  are they OK with hanging out or do they feel the need to move on.  What their major or career goal is.  (There are some that need stronger math skills than others.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(minor) You can maybe be a little practical when considering areas of weakness.  If the kid says (and you believe him), I got all the calculus except the series convergence stuff, I would progress him as that is something he won't need (comparatively!) unless he is a math major.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9298" LastEditorUserId="9298" LastEditDate="2018-01-06T02:47:33.943" LastActivityDate="2018-01-06T02:47:33.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13437" PostTypeId="1" CreationDate="2018-01-07T18:45:21.067" Score="5" ViewCount="172" Body="&lt;p&gt;I am preparing a unit on operations with algebraic fractions; But, particularly addition/subtraction of algebraic fraction such as &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{3}{b} + \frac{2}{a} ; \frac{x+1}{3}- \frac{2x+3}{4};  \frac{2}{x+1}+ \frac{2+x}{x+4} + \frac{x}{2x}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am researching on realistic word problem for a performance assessment related to those for a student level equivalent of US middle school or 1st year high school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking: considering that 5 girls in the class like basketball and 3 boys like basketball, evaluate the fraction of students who like basketball in the class if we have &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Situation 1: 10 girls and 15 boys&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Situation 2: x girls and 10 boys&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Situation 3: x girls and  3x boys&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Situation 4: there are twice more boys than girls&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Situation 5: we have 10 more girls than boys&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What could be other interesting word problem, situation that could involve the addition or subtraction of algebraic fraction?&#xA;Any other suggestion greatly appreciated&lt;/p&gt;&#xA;" OwnerUserId="9111" LastEditorUserId="9111" LastEditDate="2018-01-08T01:36:02.373" LastActivityDate="2018-01-15T11:11:07.957" Title="Situation involving the application of addition or subtraction of algebraic fraction" Tags="|secondary-education|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13438" PostTypeId="1" AcceptedAnswerId="13444" CreationDate="2018-01-08T17:47:03.847" Score="30" ViewCount="8759" Body="&lt;p&gt;One time I watched a student solve the equation $0 = (x-2)^2-9$ for $x$ like this. &#xA;$$\begin{align*}&#xA;      0 &amp;amp;= (x-2)^2-9&#xA;    \\0 &amp;amp;= (x^2-4x+4)-9&#xA;    \\0 &amp;amp;= x^2-4x-5&#xA;    \\0 &amp;amp;= (x+1)(x-5)&#xA;    \\\\x &amp;amp;= -1 \text{ or } x = 5&#xA;\end{align*}$$&#xA;It would have been quicker, and a bit more natural, to instead just start moving things over the equals sign (add nine to both sides, take a square root, etc). I'll occasionally see students do something similar to this example: what they do isn't &lt;em&gt;wrong&lt;/em&gt;, but there is a way to do it more efficiently or more naturally. In cases like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Should I stop the student while they are writing out the calculation to point out there is a more natural way?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Should I wait for them to finish the calculation, or even finish the entire exercise, before I casually tell them there was a more natural way to work out that part?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Should I say nothing at all, and instead be satisfied that the student understands what they are doing? These details are often independent of the students' conceptual understanding after all.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Maybe the answers to these questions depend on how strong the student is, or on the specific problem they are working on? I expect that there is no good all-encompassing answer to this question, but I would appreciate any general guidelines or advice anyone has about this.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-01-09T03:06:36.403" LastActivityDate="2024-03-11T15:29:27.720" Title="How should a student's inefficient calculation be pointed out?" Tags="|secondary-education|mathematical-pedagogy|algebra|students-mistakes|teaching|" AnswerCount="11" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="13440" PostTypeId="2" ParentId="13438" CreationDate="2018-01-08T18:29:30.407" Score="28" Body="&lt;p&gt;Foremost: It depends on what the lead-in lesson/topic/direction was. If this was the essential point being exercised, then I would interrupt ASAP and refocus them on the lesson/direction that just occurred. Otherwise I would let them complete their process (unless obviously totally infeasible) and then compare to a faster way afterward.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-01-08T18:29:30.407" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13441" PostTypeId="2" ParentId="13438" CreationDate="2018-01-08T21:52:04.857" Score="2" Body="&lt;p&gt;I suggest that you put aside a bit of time to show the student(s) how to solve the problem both ways.  I'm assuming that this is part of the lesson plan, so you can just write-up both solutions manually ahead of time and present them to the class (which shouldn't take much time out of the lession since people read much faster than write).  I'd even argue that the way you have presented here is something that they &lt;em&gt;should&lt;/em&gt; learn how to do, since it's a near-necessity for polynomials of higher orders (e.g. quartics, quintics, etc.).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this is more than just a pedantic issue, you might be able to give a hint before the student(s) start(s) to solve the problem, such as &quot;try solving the problem directly, you shouldn't need to do anything fancy&quot; or &quot;work smarter, not harder&quot; (something that a teacher of mine has used).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interrupting someone while they're working is generally not considered polite either, so I'd advise against it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;DISCLAIMER: I am not an instructor.  &lt;/p&gt;&#xA;" OwnerUserId="9304" LastActivityDate="2018-01-08T21:52:04.857" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13442" PostTypeId="2" ParentId="13438" CreationDate="2018-01-08T22:25:53.883" Score="3" Body="&lt;p&gt;This might be a good time to have them working in groups, so they can compare with their peers. Great luck if both methods are done in one group. What are the pros and cons of each method? Or just ask students who did each to come to the board and write their steps, and then ask the class the pros and cons of each. (What percentage of quadratics are factorable? Not many.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-01-08T22:25:53.883" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13443" PostTypeId="2" ParentId="13438" CreationDate="2018-01-08T23:16:05.177" Score="6" Body="&lt;p&gt;I will talk about the cases when I ask one of the students to come to the board and solve an exercise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the solution is fine (not accidental, not wrong) I let the student complete the solution. The solution that you posted is a good one. It is more general than what you suggest and shows technique that is applicable to more problems. Besides students probably are just learning these techniques - the pracitce won't harm. Perfectly they would pick up the alternative routes after mastering the main one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your case I would wait for the completion and tell &quot;Thank you, your solution is correct. You may take your place now.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then take the marker myself and say to class &quot;I will remind you of a hack that you could use some specific cases including this to save some time. See that the brackets are squared and 9 is also a square? You all know that $a^2-b^2=(a-b)(a+b)$. So you could rewrite the right side to (x-2-3)(x-2+3) to obtain (x-5)(x+1) faster.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would try to add the note in a manner that wouldn't offend the original solution but provide an additional tool that might be useful for them.&lt;/p&gt;&#xA;" OwnerUserId="766" LastActivityDate="2018-01-08T23:16:05.177" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13444" PostTypeId="2" ParentId="13438" CreationDate="2018-01-09T00:36:31.537" Score="22" Body="&lt;p&gt;I like your second option the best:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...wait for them to finish the calculation, or even finish the entire exercise, before I casually tell them there was a more natural way to work out that part?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Instead of just mentioning the easier way, however, you could first applaud them for using good algebra to find the right answers, then remind them of the new method and work through it together, and then up the ante by having them solve $$(x-1)^3-64=0$$ and then $$(x-5)^4-16=0$$ by a method of their own choosing. You might even suggest they try it their way so they feel the pain of having just one tool for every job.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Test them with $$(2x-5)^9+4=0.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you're not just selling them a new technique that they can take or leave, but something they really should practice since doing it &lt;em&gt;their&lt;/em&gt; way could be difficult or impossible.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2018-01-09T02:48:55.550" LastActivityDate="2018-01-09T02:48:55.550" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13445" PostTypeId="2" ParentId="13438" CreationDate="2018-01-09T02:28:35.020" Score="8" Body="&lt;p&gt;An answer regarding your last idea:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Should I say nothing at all, and instead be satisfied that the student understands what they are doing?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;No. I'll explain:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember that by the time I got to calculus, I made this kind of silly mistake all the time. In my case, I tended to get overwhelmed with all of the formulas and procedures I'd learned by this point in my education. I often wouldn't recognize the difference between a nail and a screw, so to speak, and would apply the wrong tool.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you pointed out, using a suboptimal approach can still lead to the best answer as long as the student isn't actually making any mathematical mistakes. &lt;em&gt;However&lt;/em&gt;, &lt;strong&gt;using a suboptimal approach during a test will often lead to the student spending too long on that problem and not having time to finish the test.&lt;/strong&gt; This means that the student isn't able to adequately demonstrate what he's learned. This happened to me with at least one problem on nearly every test. In your example, the student's suboptimal approach probably adds less than half a minute to the solution. However, in later courses where problems are much more complicated than simple algebra, suboptimal paths can cost the student many valuable minutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because tests are almost always timed and slow work leads to poor grades, I think it is very important to point out to a student when they are using an inefficient strategy. Help them recognize the optimal approach and learn to think efficiently. You don't want them to think &quot;what is any way I can solve this problem?&quot; You want them to think &quot;what is the most efficient way I can solve this problem?&quot;&lt;/p&gt;&#xA;" OwnerUserId="9308" LastEditorUserId="9308" LastEditDate="2018-01-09T02:35:51.513" LastActivityDate="2018-01-09T02:35:51.513" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13446" PostTypeId="1" AcceptedAnswerId="13452" CreationDate="2018-01-09T03:32:19.687" Score="1" ViewCount="2331" Body="&lt;p&gt;In my first-year exam for the BA Maths at the University of Cambridge in June 2017, I was the only one in my College to score a shameful 3rd Class Honours (in the past 20 years). My College warned that they'll permanently expel me on grounds of failure of academic progress, if I don’t improve. I:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;hanker to score a First. But I don’t know how or what to improve!  I'm becoming pessimistic and fatalist: the competition’s too crafty, and I appear too dumb, dim-witted. Perhaps I haven’t enough free will over my intelligence, and can’t smarten up!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;love the probability and statistics papers of my course, but I like the abstract papers less. But this shouldn't be a snag, as after first year, I can pick courses to fit my interests.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;attend all my lectures and supervisions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;study at least 8 hours daily. I feel that I’m already working at 110%! In order of priority, I read my textbooks and lecture notes, solve some of the questions in my example sheets, and then solve more problems from other books or past Cambridge exams.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;eschew all non-academic, social events to focus on my studies. I sleep at least 8 hours daily.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;spend at least 2 hours daily preparing and eating my vegan meals and fruits. My College has a dining hall, but the portions are always piddling. Even after eating there, I’m still hungry. I don’t eat there if the vegetarian option of the day is shoddy as it often is.&lt;/p&gt;&#xA;&lt;p&gt;I agree with my Directors of Studies, Supervisors that:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I answer the basics (i.e. first few parts of an exam question) perfectly. I remember my lecture notes and can do anything (e.g. theorems and proofs) previously seen. I don't have much trouble with computations.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But I can never prove anything novel (i.e. previously unseen), or calculate anything if it demands novel &amp;quot;tricks&amp;quot;. Thus I didn't complete most second parts of questions. I'll spend a lot of time trying different ways to prove something or to apply previously seen theorems/techniques, but I  never solve the question.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;E.g., in my first-year exam, I was often to asked prove something not by induction. I spent 10 mins. trying one proof method (e.g. proof by contradiction), but failed to progress anywhere. Then I spent another 10 mins. trying a 2nd proof method (e.g. by contrapositive), but got stuck again. Then I spent the final 10 mins. trying a 3rd proof method (e.g. direct proof), got stuck again, and finally ditched the question to try something else, but never had time left to return to it.&lt;/p&gt;&#xA;" OwnerDisplayName="user9309" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-01-11T19:29:15.027" Title="How can I (failing Cambridge's BA Math) improve my grade to a 1st from a 3rd-class honours?" Tags="|advising|" AnswerCount="3" CommentCount="6" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13447" PostTypeId="2" ParentId="13438" CreationDate="2018-01-09T09:53:50.707" Score="1" Body="&lt;p&gt;This could honestly be due to both cleverness or due to weakness.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Maybe the answers to these questions depend on how strong the student is&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Very much so. In fact, it's not clear to me that this was a suboptimal way to solve the problem:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Taking square-roots &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_fallacy#Positive_and_negative_roots&quot; rel=&quot;nofollow noreferrer&quot;&gt;needs to be done with extra care&lt;/a&gt;.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Factoring, on the other hand, can't really go wrong.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you believe the student did this due to lack of mastery of the &quot;obvious&quot; method, then by all means, follow what the other answers on this page tell you to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if the student is relatively strong, this might have been due to brilliance, not ignorance. Why spend a second worrying about caveats when you don't have to?&lt;/p&gt;&#xA;" OwnerUserId="1423" LastActivityDate="2018-01-09T09:53:50.707" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13449" PostTypeId="2" ParentId="13446" CreationDate="2018-01-09T18:47:28.843" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It's possible that you are at low end of the ability spectrum but given the thought in your post, I suspect that you can do better than 3rd.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I love probability and statistics (and plausable reasoning, inference, consulting, etc.) much more than abstract proofs also.  Think Box, Hunter, Hunter.  These are areas with huge practical application and in non research settings: biomedical, oil and gas, retail, etc.  Try to vector through the BA thing you started (don't bail unless they boot you, almost ALWAYS better to finish things).  But emphasize stuff you like.  And then carve a path away from all the definition sharpness games and towards curious pattern finding.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;OK, but try to emphasize being prepared before them, getting more use out of them.  (see 4.5)  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;8 hrs is plenty.  Make sure you are productive, mind not wandering.  Consider to take 5 minute break every half hour.  Walk or just relax during the 5 minutes off.  Don't do chores AND DON'T read or hit the net.  You can eat or hit the can though.  And don't compress all 8 hours into same time.  Divide it out.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;4.5.  Try to read and work problems BEFORE your lectures.  The lecture then becomes review.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to be efficient when reading the texts.  Don't let mind wander--stay engaged (this can be mentally tough, but you can build up to it).  Don't skim, but at the same time, don't lollygag.  Also be engaged in the reading of the text but also don't get so super strict that you try to derive stuff out of the text.  Work all example problems AS YOU GO, within the reading (not the hw problems at end of section but the atual examples in the reading.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My guess is you are probably going too slow in the reading.  Do it intense but fast. You are hungering to get to the problems (but at the same time, you will be taking them in quasi test practice mental frame, so making sure to understand the text...but move along.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the texts are ba$#-busters (&quot;mature&quot;), get some easier ones to use as side resources.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Write down questions from your reading (or hw) (use the back of your notebook, working backwards, leave space for answers).  These questions should be asked and answered during your interactions with profs.  (Some you will answer yourself as you read further or work problems).  Write down the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to work all the hw problems in the text, night before class.  Check with a solution manual if available.  Re-work any problems you mess up, from the start, treating it as an all new attempt.  (Even if a dumb error.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schedule specific meeting with prof if you just totally get a block in an area (come with written specific questions, in notebook, and with written evidence of trying to work the matter out).  After you get the help, don't just say &quot;I got it now&quot; but do more problems right after the session to prove it to yourself (get extra problem sources).  If you still didn't get it after thinking you did, go right back.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sounds like you are doing a fair amount of problems, but I still get a little bit the flavor of you doing a bit too much reading and not enough drill. Parsing lecture notes is generally not wise for instance.  If the course is following the text, your lecture notes really should just be for unusual things:  &quot;watch out for this&quot;, &quot;cool trick&quot;, etc.  [Remember you have pre-read the text and worked the problems, before the lecture, so you are just reviewing in the lecture and learning nuances, answering questions.]&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;Think you are pushing too hard--don't usually say that, but you aren't doing well and are pushing very hard.  Ergo!  Add a Sabbath (non religious is fine) but a day off. Sleep is good.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Try to add 30-60 minutes exercise period per day.  This can be light (doesn't need to be stress cardio...at max capacity)  but some light walking or biking (or running, but not too  harsh).  Take it easy as you start.  I recommend daylight and outdoors.  Doesn't matter if it rains or snows, enjoy the outdoors.  Get some vitamin Dshine.  We are physical animals.  Yes, we are.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;2hours per day on food prep and eating?  And you have a food plan?  Way too much time.  Reminds me of a Navy XO who told us JOs when we complained about the POD, not leaving enough time for sleep: &quot;if I had a choice between sleep and eat, it would be a no brainer&quot;.  You can wolf the food during class breaks, bring it with you (doesn't mean it needs to be junk or get fat:  finger salad veggies are very easy and filling).  Use lunch hour for exercise.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you are vegan for ethical reasons, stick with it but make sure you get enough whole protein, vitamins (take a multi), and essential fatty acid.  If you are doing it for health or brain reasons, ditch that.  Humans have been omnivores for millions of years.  It will be much simpler and time efficient to get your nutrition and you run less danger of deficiencies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7&amp;amp;8 (and general testing):  This is tough.  I would say, try to find some homework and do practice tests that (a) capture the feeling of novelty and (b) just expose you to a longer trick list.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, the aspect of self testing (put a light pressure on yourself and think of it really as a practice test, not low stakes homework, drill or learning) can help with test anxiety.  This is common in sports:  &quot;4th and goal, last play of the Super Bowl&quot; yelled at the two sides of a scrimmage during practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I do this even for the homework at the end of a section.  Not supreme self-pressure or frantic-ness, but a feeling that it is like a test or at least a sport or game.  Not just slow drill.  (this also helps time efficiency.)  If you do it right, it creates enough pressure so the homework can be fun (game-like) rather than pure drudgery.  And it gives a little bit of test situation prep.   It still won't be total joy...but it helps.  Note also (!) that the practice problems at the end of a reading section will have tricks or things that are NEW TO YOU at that time.  (Even the simplest math texts have some problems that are trickier, the starred ones or the higher numbered ones.  And for a Tripos smartie, I bet MANY of them are hard for you at first.  So consider this as same situation you have during exams.)  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Good luck, man.  You are doubtless a guy with a lot of talent so will have great jobs in life even if you flunk out of Tripos-land.  But try to get through this thing and get the degree you are started on.  It will definitely be something you are proud of in the end--the personal perseverance and guts, not the thing itself.  Again, good luck!&lt;/p&gt;&#xA;" OwnerUserId="9298" LastEditorUserId="9298" LastEditDate="2018-01-09T19:04:58.860" LastActivityDate="2018-01-09T19:04:58.860" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13450" PostTypeId="2" ParentId="13438" CreationDate="2018-01-09T19:23:23.273" Score="2" Body="&lt;p&gt;I would just mention it as an alternate solution method and not make a huge hairy deal out of it.  I don't think your way is that much better/faster but even if it were, I would still be low key about it when giving an alternate faster way (if it is so much better, that will be evident).  But definitely here it's like 5.7 versus half dozen to mangle a saying.  Also, they will encounter times later, like ODEs, where keeping things on one side is the better way. I would think of it more as different methods to catch a fish versus the all best fishing method.&lt;/p&gt;&#xA;" OwnerUserId="9298" LastActivityDate="2018-01-09T19:23:23.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13451" PostTypeId="2" ParentId="13438" CreationDate="2018-01-09T21:41:03.227" Score="1" Body="&lt;p&gt;My first thought is that the third option is the worst one, I guess. When you think your student is doing something wrong, or even inefficiently, you should tell him or her. Even in the case you are wrong, you'll have enough time to reformulate your explanation and/or reaccept his/her answer.&lt;/p&gt;&#xA;&lt;p&gt;Now, the better option depends on the subject and the context. If the student is writing the formula on a blackboard, what I would do is to let him/her finish - unless he/she is wasting a lot of time - and then explain that there is a more efficient way. On the other hand, if it is a writing exercise, I would note to him/her the inefficiency if I was asked, else I would not to say anything - because I consider an inefficiency as not a big problem. Obviously, while correcting an inefficient answer it can be penalized as much as the problem needs - I would be harder in maths than in physics, for instance.&lt;/p&gt;&#xA;" OwnerUserId="9320" LastEditorUserId="7365" LastEditDate="2024-03-07T19:43:35.490" LastActivityDate="2024-03-07T19:43:35.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13452" PostTypeId="2" ParentId="13446" CreationDate="2018-01-10T00:11:01.547" Score="4" Body="&lt;p&gt;Ok so I suppose if your College was at all typical then after your exams they suggested to you that you might want to transfer to a different university/subject. And as you are still doing maths at Cambridge you told them you would like to continue and presumably you have them some kind of indication that you would be changing your ways. As lent term is about to start I’m guessing that you are not making sufficient progress.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First up the bad news (although perhaps you already know this): Transfers are much easier after first year than after second year. If you fail this year (or according to your question get a 3rd) then you won’t be in Cambridge next year. It’s also likely harder to transfer to another university. It’s basically too late to switch to natsci or economics so if you want to stay, you’ll have to do it through maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now some advice:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, lent term will start soon and in second year that means CATAM. It’s a good opportunity to pick up marks and those marks can push you over from a fail to 3rd or a 3rd to a 2.ii. Doing CATAM May also help refresh your memory about courses you’ve already done. On the other hand, much of CATAM requires one to have the ability to prove some things so that may put you at a disadvantage but you should still be able to pick up a decent amount of marks. If you haven’t started yet, please, please do start now (and don’t leave the Easter term one to the last minute—you want to be revising at the start of term, not doing CATAM).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next up, some advice for maths: You say that  your problem is in proving things. Unfortunately almost all questions involve some proof. Pure questions definitely involve proof. Stats questions likely involve proofs and applied questions usually involve some element of slightly informal proof. The first year &lt;em&gt;Numbers and Sets&lt;/em&gt;, &lt;em&gt;Groups&lt;/em&gt; and &lt;em&gt;Analysis&lt;/em&gt; (and &lt;em&gt;Met and Top&lt;/em&gt; if you count that) are meant to be your introduction into pure maths/doing proofs. I would suggest you go back to proofs/exercises from lectures/example sheets from Analysis (and maybe also Numbers and Sets, and Groups). By studying these perhaps you can get a better idea as to how to do proofs. In Analysis a key idea to get into your head is that almost every proof actually only has one idea and every other step is “you look at what you have and there is only one reasonable thing you can write down, so write it down.” The hard part is building the intuition as to what “reasonable” means. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you write a proof you need to avoid the mindset of “I need to figure out what formula to apply and then plug things in.” That works for A-Levels but it almost never works at Cambridge. Therefore don’t have a checklist “First I’ll try proving by contradiction, then if that doesn’t work I’ll try proving by contrapositive, otherwise I’ll try induction, and so on” because you’ll spend all your time flailing around not knowing what to do. What one should do is this&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;work out what it is your trying to prove (or maybe disprove)&lt;/li&gt;&#xA;&lt;li&gt;think about the statement and conditions&lt;/li&gt;&#xA;&lt;li&gt;maybe a counterexample exists. Think about what a counterexample might look like. Maybe you find one or maybe you think of something close and learn something about the problem by doing it. I think this is an important point. There’s no point trying to prove something that’s clearly false&lt;/li&gt;&#xA;&lt;li&gt;think if any theorem you know might be relevant to the statement (in Cambridge maths exams that ask you to prove something, this is almost always the case but it isn’t necessarily obvious which theorem to apply or how or both. You build intuition about this by doing related problems). If you think of a relevant theorems, think what you might need to apply them. Think about what you would need to prove to be able to apply the theorem. Maybe you can use the proof of the theorem that you know as a skeleton for the proof you want to write. Maybe thinking about this gives you a plan to prove the thing you want to prove.&lt;/li&gt;&#xA;&lt;li&gt;if you still don’t know what to do, try negating the statement, (ie proving the contrapositive/by contradiction), or just look at your assumptions, look at what you want to prove and write down everything you know. Now you have some other things to look at so apply the above steps again.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It may be worth getting hold of a copy of &lt;em&gt;How to Solve It&lt;/em&gt; as that might help you with proving things. &lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Courses&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;In second year the only stats/probability courses are &lt;em&gt;Statistics&lt;/em&gt; and &lt;em&gt;Markov Chains&lt;/em&gt;. You might have some luck with stats. If you can’t learn to prove anything (and especially if you never really learnt v&amp;amp;m last year) then you’ll probably struggle with markov chains. You probably know that anyway. In exams you might be able to pick up a few algorithmic questions in markov chains but it’s a short course so there won’t be too many questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My recommendation is that you should definitely do &lt;em&gt;Methods&lt;/em&gt; and &lt;em&gt;Complex Methods&lt;/em&gt;. If you can learn those well and understand them, you should be able to pick up a decent amount of alphas in exams—certainly enough to pass. They don’t rely much on proofs and if you can learn the algorithms alone you should get some alphas from it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might be able to get something from linear algebra if you can learn the algorithms but you need to have an idea about how to do proofs to write answers to these questions (excepting some betas maybe but it isn’t worth learning a huge course for a few betas)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have the right mindset you might be able to get some marks in fluids but don’t be afraid of dropping the course if it’s not going well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don’t know if you did VP/Optimisation last year. Both of those are worth a shot (although perhaps only if you feel up to doing a course in exam term). Optimisation is just learning algorithms and doesn’t need so much proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quantum mechanics is variable. You might get lucky and get some questions that just need calculus but you need to learn the methods, have some intuition about (quantum) physics but be able to turn it off, and sometimes you’re just flailing around with weird questions about operators and commutators which a prelude to PQM in part II. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It’s too late for you to take up GRM now.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Other Advice&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Talk to your DoS. They will have good advice on which courses to take and how to strategise to gain marks in exams. Don’t focus on stats/Markov chains if they aren’t going to get you marks in tripos. You should be able to pick up some marks from doing lots of short questions—they just need you to know the definitions and maybe a few main theorems. You don’t get many marks from trying and failing to do a long question. Read the ones for courses you’ve studied, you should have a rough plan for how each step of the question will go and how hard it will be. Start with the easiest question and continue until you finish it or get completely stuck (in which case move on and maybe come back). It is inadvisable to begin a question if you haven’t a clue as to how it will end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Talk to your tutor. They may be able to help you and give you advice as to what things you might do next. If you do poorly in exams it’s much better for you if you had appeared to be looking for help/struggling. If your tutor suggests intermitting you should seriously consider this (SFE will pay for up to four years towards a single degree at a single university. The degree doesn’t have to be specified which allows one to switch subjects at a single university and for ones “undergraduate” BA to be merged into a MMath to do part III without extra funding. I would &lt;em&gt;strongly suggest&lt;/em&gt; that you do not rule out intermitting for reason of wanting to be able to get a loan for doing part III)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe it’s worth talking to someone in your JCR about this although I can’t think who this would be or how they might help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Believe it or not, maths is a social discipline (even at Cambridge) so I’d also suggest talking to the other people in your college about maths (or not). This can help internalise the knowledge, turning over definitions and theorems in your head and getting a feel for how they feel. This may be as little as talking g to people on the walk back from lectures (if you walk) or going to hall and talking to people over lunch (if they go to hall after lectures), even if you don’t want to eat the food there. You may have heard about some high-performing mathmos who seem to lock themselves into their rooms and come out with top marks in tripos but do not be fooled: most of these people talk amongst themselves and thus may improve their intuition and knowledge from the ideas of others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make sure you have a good diet. Being vegan makes this difficult. You won’t be able to think properly if you aren’t getting proper nutrition. Ideally you shouldn’t need two hours prep time for food either. If you really do then maybe there are some other vegan students also being disadvantaged by this prep time and you could pool you’re resources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make sure you get a chance to relax/socialise. I’m not suggesting you should go out and join the boat club (or indeed any society) but if you spend all your time stressed about your poor performance then you won’t be in a good state to learn. Perhaps there is some society relevant to your interests. Many are relaxed and provide light activities and light socialisation and don’t care if you don’t turn up all the time. You could also go to society talks from the Archimedeans or TMS or scisoc. If I remember rightly the Archimedeans will have their traditional pizza/board games night in the CMS after the CATAM deadline. It’s worth being on the mailing lists even if you don’t think you’ll be interested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the CMS on Saturdays there’s a “maths cafe” which you can go to. Just turn up with some work and there’ll be a long table with people and some water and stuff sat around. You can do your work here and there’ll be people who you can talk to about your problems and what you are stuck on. They should be able to help. Even if (think) you don’t want to talk to people I suggest you just turn up anyway, just in case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should get notes from your supervision (or your partner). Hopefully you can use them with what you remember from the supervision to be able to get the questions you didn’t have previously. If it’s too hard to remember what was said then you might want to ask your supervisor if you can record the supervision (use your phone as a voice recorder). You probably won’t learn much from merely going to a supervision.&lt;/p&gt;&#xA;" OwnerUserId="5235" LastActivityDate="2018-01-10T00:11:01.547" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13454" PostTypeId="2" ParentId="13438" CreationDate="2018-01-10T15:04:37.333" Score="1" Body="&lt;p&gt;Don't stop the student, let them finish. Also, do not refrain from interjecting, especially if you are an educator. Students seek to learn from you. If they learn your methodology from someone else in the future, what you've called the &quot;natural way&quot;,  they will wonder why they never learned that from you and it may invalidate your credential or standings with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see a lot of complicated answers here. I'll make it simple for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In life, there are many different ways to perform a task. Sometimes there are more efficient and effective ways to do something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My advice is to offer the more efficient way as a &lt;em&gt;suggestion&lt;/em&gt;. Let them know that this is the method you employ and that it is faster for you, but also make it clear that it is not the only way. Explain the reasoning behind your methodology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student understands and adopts your method, then that means they are capable of mirroring your thought process. This will allow you to make subsequent suggestions. If they continue doing it using their method, then so be it. The way people think varies from person to person and you can't expect someone to do something your way unless you know what's going on in their head.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do this with my wife all the time ;)&lt;/p&gt;&#xA;" OwnerUserId="9326" LastActivityDate="2018-01-10T15:04:37.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13455" PostTypeId="2" ParentId="12949" CreationDate="2018-01-11T18:58:40.230" Score="4" Body="&lt;p&gt;The question may well be reversed in places not so far away. Should we finish Geometry before Algebra? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real news is that once begun in either there is never any finishing. Both topics are vast. You will not need even simple algebra in early geometry but you need to master the theorems. Mastery is why they resort to memorization, the only way I could have managed. Eventually you will have geometry problems that require algebra to solve. These are what got me addicted. That there could be any link between two such odd disciplines, - any, try dozens and dozens - is an awakening you've yet to experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I learned very late that math subjects do not truly belong in the layers we find them in school. It's just handy to know algebra before you need it in geometry or you will have to go back and learn it, or go back and learn trig to move ahead in calculus. So long as you can get through all the classes whatever order the teachers are used to using will serve you as well as it has those that have gone before you. &lt;/p&gt;&#xA;" OwnerUserId="9330" LastActivityDate="2018-01-11T18:58:40.230" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13456" PostTypeId="2" ParentId="13446" CreationDate="2018-01-11T19:29:15.027" Score="1" Body="&lt;p&gt;I can hardly improve on the detailed answers here but just wanted to add a few things. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen fellow students lock themselves way for the requisite eight hours of study and fail straight out of pre-med. Hours alone will not make it for you. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As in sports or physical activities it is not practices that makes perfect, but Perfect practice that makes perfect. More simply; hours perfecting the wrong grip for the wrong backhand will not yield the desired result. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Accessing other students, some more successful, some also struggling, will give you the latitude you need to spot weaknesses and problem areas you cannot yet report. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot speak to vegan anything except to recommend you get a great deal of protein by any and all means. You will pass nothing without it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In working on proofs. Knowing intellectually each of the approaches does not volunteer the answer. It is seeing the endpoint clearly that will show the path. This sounds impractical as a process to follow but it's the truth. To get such a high level view you need more exposure to other people and the way they think about things, including math. Not even the best biography of mathematicians will let you share their insights. But a drinking buddy, a cooking group, a PhD bartender (I had one) will let your mind develop in ways that mastery of the text never will.&lt;/p&gt;&#xA;" OwnerUserId="9331" LastActivityDate="2018-01-11T19:29:15.027" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13457" PostTypeId="2" ParentId="13276" CreationDate="2018-01-11T20:36:07.843" Score="1" Body="&lt;p&gt;There are a few options, and they are free.  First, google hangouts and google classroom are fine for this.  If you want, you can just use drawing in a google doc or something for the pictures.&lt;/p&gt;&#xA;&lt;p&gt;If you want a little higher level functionality, you can checkout some kind of courseware like Moodle, Blackboard, or Canvas which will let you do things for free but isn't immediately obvious for students.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I would highly recommend CoCalc.com .  Here, you can use a variety of computing languages and tools but there are also very nice chatting capabilities including video conferences.  I tend to use Jupyter notebooks a lot with my classes, so I can pop one here and have a conference with students online including a video chat, but they can also always drop me a note in the chat and I'll be able to see this even if we're not chatting live (this is what you see on right side of screen).  Also you can use a &lt;code&gt;sagechat&lt;/code&gt; and have a classwide document for chatting.  All will let you write mathematics with &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UhiCN.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UhiCN.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Last, you may want to checkout etherpads (etherpad.org).&lt;/p&gt;&#xA;" OwnerUserId="7003" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-01-11T20:36:07.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13458" PostTypeId="2" ParentId="13417" CreationDate="2018-01-11T20:54:53.420" Score="4" Body="&lt;p&gt;As somebody who has been involved in state and national standards work, I think there are a few ways in.  &lt;strong&gt;Most important, become a mathematics teacher and gain some experience in your public schools&lt;/strong&gt;.  From here, you will have opportunities to engage in larger conversations around district and state policies and figure out who's who in that world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Seek out opportunities to be around the people in your district who are a part of these conversations, gain experience, and eventually there will be district level positions or state positions around curriculum and instruction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Along the way, you should complete higher degrees, specifically studying mathematics education.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no quick road, fortunately, and you should expect years of classroom teaching before considering a seat at the table. This is only fair.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2018-01-11T20:54:53.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13459" PostTypeId="2" ParentId="11216" CreationDate="2018-01-12T21:22:16.130" Score="1" Body="&lt;p&gt;First, the hard evidence is obvious -- our standardized mathematics assessments have continued to show a discrepancy between race and class.  This should tell us there is something problematic about the assessment, however usually people who talk the test read this as problems with students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, I feel Ben Dickman's suggestion is completely valid, however, for a different taste, you might want to check out these examples, and follow the larger threads you'll find within.  There is a massive literature of critical mathematics education:  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Valerie Walkerdine- &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0415006961&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Mastery of Reason&lt;/a&gt; - basically redid Piaget's experiments and notes the problems with the way these investigations took place in light of student difficulty&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Robyn Zevenbergen - &lt;a href=&quot;https://www.jstor.org/stable/3482936?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Constructivism as a Liberal Bourgeois Discourse&lt;/a&gt; or &lt;a href=&quot;https://www.jstor.org/stable/4150265?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Technologizing Numeracy: Intergenerational Differences in Working Mathematically in New Times&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Robert Moses - &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0807031275&quot; rel=&quot;nofollow noreferrer&quot;&gt;Radical Equations&lt;/a&gt;, discusses some alternative ideas for engaging students with contexts familiar to their own lives.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jean Lave(all) - Investigates individuals working on mathematics problems in different contexts; situational cognition.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2018-01-12T21:22:16.130" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13460" PostTypeId="1" CreationDate="2018-01-13T06:57:36.913" Score="10" ViewCount="818" Body="&lt;p&gt;there're some students, who belive that &lt;span class=&quot;math-container&quot;&gt;$$\frac10 = \infty $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I need to teach them that this is not true and &lt;span class=&quot;math-container&quot;&gt;$\frac10 $&lt;/span&gt; is undefined, mathematically and give a good picture (for their minds)&lt;/p&gt;&#xA;&lt;p&gt;what is the proper way to teach them without to telling them to memorize this&lt;/p&gt;&#xA;" OwnerUserId="9187" LastEditorUserId="60" LastEditDate="2021-02-14T16:34:30.177" LastActivityDate="2021-02-14T16:34:30.177" Title="teach that $\frac10$ not defined properly" Tags="|calculus|concept-motivation|students-mistakes|teaching|teacher-preparation|" AnswerCount="6" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13461" PostTypeId="2" ParentId="13460" CreationDate="2018-01-13T07:55:46.923" Score="21" Body="&lt;p&gt;What is $\frac 1 a$? It is the unique (real) number such that $a\cdot \frac 1 a=1$. Does there exist a real number that multiplied by $0$ gives $1$? No. Why is this? Because if $0\cdot b=0$ which ever is $b$. This is about not being defined. Still... why is $\frac 1 0=\infty$ not so completely wrong? Because they can see that the smaller is $a$ then the bigger is $\frac 1 a$ (and I would make them appreciate this on the real line, maybe letting $a=\frac 1 n$) and therefore... (vague explanation but your question is vague since you do not specify the level of your students).&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2018-01-13T07:55:46.923" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13462" PostTypeId="2" ParentId="13460" CreationDate="2018-01-13T20:01:52.813" Score="3" Body="&lt;p&gt;If the students can think about graphs, you can graph y=1/x. So if 1/0 = ∞, this should approach ∞ as x -&gt; 0. It does on one side. But on the other, it approaches -∞. Since it doesn't approach ∞ from both sides, we must say it's undefined.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-01-13T20:01:52.813" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13464" PostTypeId="1" CreationDate="2018-01-14T18:04:37.233" Score="9" ViewCount="475" Body="&lt;p&gt;Can someone truly love math, and master and remember discovered calculations, counterexamples, proofs; but still fail to invent anything new (e.g. incapacity to prove anything unseen, calculate something that demands new tricks, or discover counterexamples)? I ask not about &lt;a href=&quot;https://www.reddit.com/r/changemyview/comments/70tfpt/cmv_all_smart_people_are_capable_of_doing_well_in/dn6c10f/&quot; rel=&quot;noreferrer&quot;&gt;dyscalculia&lt;/a&gt; that appears irrelevant, as I'm not asking about arithmetic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I bolded the relevant parts from &lt;a href=&quot;https://www.reddit.com/r/changemyview/comments/70tfpt/cmv_all_smart_people_are_capable_of_doing_well_in/dn5vtav/&quot; rel=&quot;noreferrer&quot;&gt;the Reddit Post&lt;/a&gt; beneath that's anecdotal: is there any evidence?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although I am usually not a full-time teacher, I've taught hundreds of&#xA;  students over the last 35 years.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Students fall into five categories:&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;p&gt;A tiny number of brilliant students who will teach you something.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;A moderate number of pretty smart students who would probably do an OK job if you just handed them the textbook and left.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;A large number of average students for whom your technical class is serious work, but will get through with help from you and study.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;A moderate number of people who don't like or care about math and/or aren't particularly talented who might pass if they put in&#xA;  the work but probably won't.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;A small number of people who will never understand the material, no matter what.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;When I was young I refused to admit that that last category existed. I&#xA;  put a lot of time into a few people who worked really hard but tried&#xA;  to memorize and fake their way through, or just couldn't get it.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;And then something happened to convince me otherwise.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I had a friend who started Math in University at the same time that I&#xA;  did, but he was about five years older. He'd already had two&#xA;  successful careers - he had been a journalist and then quit that to&#xA;  run a campaign for a politician who won his election - but he'd always&#xA;  loved math.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;He did perfectly well in first year - not exceptionally but fine. But&#xA;  in second year, the trouble started. He could not create new proofs -&#xA;  no, not at all. He understood the material quite well, with some gaps,&#xA;  but writing proofs was a bridge he could not see how to cross.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I spent hours with him, but he started to do worse and worse.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Eventually he vanished. I finally got in touch with his brother who&#xA;  said, &quot;I really appreciate your having contacted us, but I'm not going&#xA;  to tell him you called. He had a nervous breakdown and his doctor said&#xA;  he should be kept away from any memory of what happened.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Now don't get me wrong. My guess is that at least half the people who&#xA;  fail math do so because they just aren't interested enough to put the&#xA;  time in, and if they really wanted to, they would succeed.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;But there's a core of people, often otherwise smart people - people&#xA;  who are even really interested - who simply don't think that way, and&#xA;  all the coaching in the world won't fix this problem.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user155" LastEditDate="2018-01-14T19:22:01.890" LastActivityDate="2021-01-18T16:55:19.783" Title="Can some lovers of math truly never create something previously unseen?" Tags="|proofs|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="13465" PostTypeId="1" CreationDate="2018-01-14T22:45:00.153" Score="5" ViewCount="241" Body="&lt;p&gt;The title is based on &lt;a href=&quot;https://www.reddit.com/r/changemyview/comments/5gj3s9/cmvi_believe_people_who_are_exceptionally_good_at/dat0rcs/&quot; rel=&quot;noreferrer&quot;&gt;this Reddit post&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It's 1:30am so I'm not going to go dig up the study, but what it showed was that, on average, a person who performs well in mathematics usually performs better in other subjects compared to people who do not perform well in mathematics. The conclusion of the study was that someone who excels at mathematics, pretty much can excel at everything else. Anecdotally, I'm sure many people can confirm this as well. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2018-01-22T07:01:25.700" Title="Is there a study that proves that math proficients usually perform better in other subjects vs. people who perform shoddily in math?" Tags="|education-research|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13466" PostTypeId="2" ParentId="13437" CreationDate="2018-01-15T11:11:07.957" Score="4" Body="&lt;p&gt;Addition in reciprocals (I would call this &lt;em&gt;harmonic addition&lt;/em&gt; or &lt;em&gt;harmonic sum&lt;/em&gt;, by generalisation from &lt;em&gt;harmonic mean&lt;/em&gt;, but both of those terms seem already to be used) comes up in some situations in physics.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A. Goodier has mentioned the simple lens formula $\frac{1}{f} = \frac{1}{s} + \frac{1}{o}$. You can place two lenses in a row to complicate things if you think this won't lead to too much focus on the physics to the detriment of the mathematics. Basic practical zoom lenses use three in a row, but the algebra starts getting  hairy.&lt;/li&gt;&#xA;&lt;li&gt;Resistor networks: resistors $R_1$ and $R_2$ in series behave like a single resistor with resistance $R_1 + R_2$. Resistors $R_1$ and $R_2$ in parallel behave like a single resistor with resistance $\left(\frac{1}{R_1} + \frac{1}{R_2}\right)^{-1}$. You can combine the two to create expressions of arbitrary complexity.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8005" LastActivityDate="2018-01-15T11:11:07.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13467" PostTypeId="1" CreationDate="2018-01-15T14:33:05.320" Score="4" ViewCount="318" Body="&lt;p&gt;I will present two problems alongside solutions, student is doing problems of type I like a cakewalk but has several issues with the problems of type 2;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Type I&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consider an experiment of rolling two dice:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Sample space $$ S = \{(1,1),(1,2),(1,3), \cdots, (6,6) \} $$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Let $A$ be the event of getting 6 as sum on two dice:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Event $$A = \{(1,5),(2,4),(3,3),(4,2),(5,1)\}$$ &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Let $B$ be the event of getting 4 on first die:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Event $$B = \{(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)\}$$ &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Now, the probability of getting sum of 6 on two dice given that first&#xA;  die appears as 4 is given by&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$p(A \mid B) = \dfrac{p(A\cap B)}{p(B)} = \dfrac{p\{(4,2)\}}{p(B)} =&#xA; \dfrac{1}{6}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Type II&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $C$ be the event of having cancer and $p(C) = 1/2$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The probability of having both tumor and cancer is $p(C\cap T) = 1/6$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Then the probability of having tumor given cancer is given by&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$p(T\mid C) = \dfrac{p(T\cap C)}{p(C)} = \dfrac{1}{3}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Students are understanding type I problems and type 2 problems well, but some students are asking to present $C, T, C\cap T$ in terms of sets. I tried to convince them using Venn diagrams. But they are asking for either roster form (for discrete sets) or set builder form(for any set).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to do like follows &lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align}&#xA;\text{Sample space} = S &amp;amp;= \{x \mid \text{$x$ is a living being} \} \\&#xA;C &amp;amp;= \{x \mid \text{$x$ has cancer}\} \\&#xA;C\cap T &amp;amp;= \{x \mid \text{$x$ has both tumor and cancer}\} \\&#xA;\end{align}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it right way to do? Students are facing difficulty and they are asking every event inform of set, since definition of event is that it is a subset of sample space (which is a set).&lt;/p&gt;&#xA;" OwnerUserId="8052" LastEditorUserId="8052" LastEditDate="2018-01-24T05:02:41.760" LastActivityDate="2018-01-24T05:02:41.760" Title="How to convince the following probability problem to highschool student?" Tags="|probability|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13468" PostTypeId="1" CreationDate="2018-01-16T23:09:26.153" Score="5" ViewCount="403" Body="&lt;p&gt;I am recently going to make a series of videos about real analysis and measure theory. I wonder if anyone can give me some suggestions on how to arrange the material of the course. Should I introduce abstract concepts(such as measure, $\sigma$-field, metric space, etc.)first, or can I start with some interesting examples?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My intended audience includes those who have already studied basic analysis course and know something about theoretical physics.&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="9339" LastEditDate="2018-01-17T02:50:32.740" LastActivityDate="2018-01-26T22:29:54.627" Title="How to teach real analysis?" Tags="|undergraduate-education|real-numbers|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13469" PostTypeId="2" ParentId="13468" CreationDate="2018-01-17T07:33:16.310" Score="6" Body="&lt;p&gt;This comment from MSE by Kimchi Lover is fantastic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples every time.  Can you think of a target end example result and work the course up to understanding it?  If I were teaching it, I'd make the goal something like &quot;Brownian motion paths are continuous and nowhere differentiable&quot; or &quot;most numbers are normal numbers&quot; and construct measure theory and real analysis to be able to state &amp;amp; prove this rigorously.&lt;/p&gt;&#xA;" OwnerUserId="9339" LastActivityDate="2018-01-17T07:33:16.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13470" PostTypeId="1" AcceptedAnswerId="13471" CreationDate="2018-01-17T12:22:02.217" Score="4" ViewCount="388" Body="&lt;p&gt;I would like to practice a large quantity of exercises from limit calculation, derivatives, sequences and series and finaly integrals.&lt;/p&gt;&#xA;" OwnerUserId="9342" LastActivityDate="2018-01-18T18:26:20.500" Title="Could you recommend a book for studying calculus 1?" Tags="|calculus|textbooks|exercises|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13471" PostTypeId="2" ParentId="13470" CreationDate="2018-01-17T19:11:43.033" Score="6" Body="&lt;p&gt;I recommend &lt;a href=&quot;https://openstax.org/subjects/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax&lt;/a&gt; open-resource digital books from Rice University for all levels of college math, from prealgebra through calculus and statistics. For the student, they're free, can be shared and copied without restriction, carried on mobile devices, etc. From the instructor's perspective, one can rely on immediate access by all students, show the content on overhead projectors, remix the content into presentations and handouts, etc. In my opinion they're at the same level as any professional print text, after several years of development, and boast extensive lists of professorial authors, editors, and reviewers (e.g. &lt;em&gt;Calculus&lt;/em&gt; primary author Gilbert Strang is from MIT). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus comes in three volumes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://openstax.org/details/books/calculus-volume-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax Calculus Volume 1&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://openstax.org/details/books/calculus-volume-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax Calculus Volume 2&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://openstax.org/details/books/calculus-volume-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax Calculus Volume 3&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-01-18T18:26:20.500" LastActivityDate="2018-01-18T18:26:20.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13472" PostTypeId="2" ParentId="13470" CreationDate="2018-01-17T23:51:13.377" Score="0" Body="&lt;p&gt;A.  Granville is a good source of classic problems.  Free copy on web: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't be put off by the age.  It is the first modern text.  Almost all the problems have answers in back.  It was actually designed to have double the problems needed (so that instructors could assign a &quot;full&quot; suite for at home work or a full suite for in class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B. Also there are some popular drill books (10-25 bucks for bound copy) available at Amazon or book stores. I would recommend a hard copy as I hear bad things about digitized math books for legibility.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Schaum's 3000 Solved Problems&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Schaum's Outline of Calculus by Frank Ayres (earlier used editions preferred but all OK). &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Humongous Book of Calculus Problems&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;C.  Also check out Khan Academy for free drill problems online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;D.  This is a bit more obscure but I am a fan of Thomas Finney&#xA;Elements of Calculus and Analytic Geometry 4th Edition:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0201076640&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Elements-Calculus-Analytic-Geometry-George/dp/0201076640&lt;/a&gt; (used copies for under $10)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may recognize the cover from the movie Stand and Deliver (I used it in HS also).  Escalante said nice things about its problems as well in a research article.  Note that it has ALL the answers in the back, ideal for the self-learner.&lt;/p&gt;&#xA;" OwnerUserId="9298" LastEditorUserId="9298" LastEditDate="2018-01-17T23:57:19.013" LastActivityDate="2018-01-17T23:57:19.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13475" PostTypeId="2" ParentId="11580" CreationDate="2018-01-18T01:43:11.430" Score="4" Body="&lt;p&gt;A.  &quot;I feel your pain&quot; but issues 1 and 2 are non-trivial.  Haven't you answered your own question?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B.  For smaller classes, you could consider to buck the trend.  But even here, would suggest getting tenure first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C.  One practical suggestion is to use Dover or Springer or some of the better Schaum's editions.  &lt;/p&gt;&#xA;" OwnerUserId="9298" LastActivityDate="2018-01-18T01:43:11.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13476" PostTypeId="2" ParentId="13470" CreationDate="2018-01-18T15:51:50.830" Score="0" Body="&lt;p&gt;&lt;a href=&quot;http://www.springer.com/la/book/9783319154275&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus Problems&lt;/a&gt; by Baronti, De Mari, van der Putten &amp;amp; Venturi might be worth a look. It has problems on all the topics you mention, along with basic ordinary differential equations. There is also a final chapter containing problems whose solution may require techniques from throughout the book.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2018-01-18T15:51:50.830" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13477" PostTypeId="2" ParentId="13467" CreationDate="2018-01-18T20:15:14.353" Score="4" Body="&lt;p&gt;To use set builder notation you need to carefully state to which set every element you want to discuss belongs. In general it goes &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P = \{x\in\text{some set} : x\text{ satisfies some truth evaluable proposition}\}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's how I would do it in your case:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $P$ be the set of all humans, let $C=\{x\in P: x\text{ has cancer}\}$ and let $T=\{x\in P: x\text{ has a tumor}\}$. Then $C\cap P$ is the set of humans that both have cancer and a tumor, or $C\cap P = \{x\in P: (x\text{ has a tumor})\wedge (x\text{ has cancer})\}$. In the previous, the wedge notation is the logical 'and'. &lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2018-01-18T20:15:14.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13479" PostTypeId="2" ParentId="13464" CreationDate="2018-01-20T23:49:56.267" Score="8" Body="&lt;p&gt;I feel like the question you are interested in is whether there is a &amp;quot;kind of person that can produce mathematics&amp;quot; and a &amp;quot;kind of person that can't.&amp;quot; I don't know of any conclusive research on this subject and lot of the interest in it is heavily politicized. That said, it would seem to me that the evidence is overwhelmingly that there are &lt;em&gt;not&lt;/em&gt; two kinds of people.&lt;/p&gt;&#xA;&lt;p&gt;The main reason I offer is that we as humans are shockingly good at education. Every time we move the goal posts a new generation of students stands up and dutifully learns how to do what we've required of them. For example, we now have nursing students diligently passing calculus courses because colleges have decided to require it for premed. Twenty years ago that would be unthinkable and people would have smugly said &amp;quot;of course the type of person who becomes a nurse couldn't do calculus, there are some people that can do math and some people cant.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So then, why do some people struggle with proofs? Well, the current goal of mathematics education isn't (in general) to produce students who can write proofs. The goal instead is to produce students who can apply theorems and perform calculations. If we revised this goal, I am confident that all the students would follow us into real analysis, just as they've followed us into calculus.&lt;/p&gt;&#xA;&lt;p&gt;In addition to my beliefs above, there are a few things I want to point out about the text quoted in the question.&lt;/p&gt;&#xA;&lt;p&gt;First, I really chafe against the equivocation of a student not being in a position to pass a course &lt;em&gt;right now&lt;/em&gt; and somehow being &lt;em&gt;not the kind of person who can do math&lt;/em&gt;. There will always be students that struggle, sometimes it's their way of thinking about the material, sometimes it's their professor, sometimes (very often) it's external stressors. Sure, every semester I will have a student that can't seem to &amp;quot;get&amp;quot; the material no matter how long we work at it together. Sometimes these students try everything and fail, but sometimes that student gets a tutor or makes new friends, stops showing up to office hours, and passes the class. It's very cocky to draw the conclusion &amp;quot;this student will never have any capacity to understand math&amp;quot; from &amp;quot;I can't teach this student this subject in math &lt;em&gt;right now&lt;/em&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The second thing I want to point out is that the conclusion of the anecdote seems horribly flawed to me. Student stress and its effects on mental health are well documented and one of the symptoms is lower cognitive function. I have unfortunately watched mental breakdowns happen to my peers in graduate school, they are debilitating, life destroying events that usually have a long run up to them. They are exacerbated by the fact that the stress they cause make it monumentally harder to do cognitive work. Going from &amp;quot;my friend had a mental breakdown the caused him to be removed from the college he was attending&amp;quot; to &amp;quot;gee, he must be really bad at math!&amp;quot; makes much less sense to me than &amp;quot;my friend must have been going through some serious shit to have a mental breakdown, I wonder if stress was part of why he was having so much trouble doing math.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I think the phenomena that are being described here are much more accurately accounted for by thinking about what factors might make a person struggle in a particular situation, rather than just positing that &amp;quot;some people can't do math.&amp;quot; It seems to me that every time we assume that people as a whole &lt;em&gt;can&lt;/em&gt; do something and proceed to try to educate them to do so we are proven correct.&lt;/p&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="15268" LastEditDate="2021-01-16T14:51:54.123" LastActivityDate="2021-01-16T14:51:54.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13480" PostTypeId="2" ParentId="13467" CreationDate="2018-01-21T19:40:15.743" Score="1" Body="&lt;p&gt;The question is very confusingly stated.  (Hopefully it is more clearly done in native language.)  I can't help with the logic barrier for the students, but if they are having a hard time with the logic and then you add on confusing language, this makes things worse.  Or may be more the problem than the logic barrier itself.     &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Let T be the event of having cancer and p(C)=1/2&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is T tumor or cancer??? And why have you combined such separate ideas (definition of T tumor and probability of C cancer) into one joined compound sentence?  [You are lacking a comma, too.  Wouldn't normally nit about this--make mistakes myself--but it adds to the bafflement.]  For that matter what does &quot;cancer&quot; and &quot;tumor&quot; mean?  (Note how I add the term &quot;detectable&quot; below to help with the huh factor of people equating C and T.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me try:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;C is the fraction of the population that has cancer.  &lt;/li&gt;&#xA;&lt;li&gt;T is the fraction with a detectable tumor.*  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Given:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A.  Probability of cancer C in the population is 0.5&#xA;B.  Probability of having both cancer C and a detectable tumor T is 1/6.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question: What is the probability of having a detectable tumor T, given having cancer C?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer: (your equation)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  This is just a student math problem but mistakes in the medical literature are easy to make and common.  In some cases even affecting patient treatment.  So try to be super precise.  I'm not perfect either probably have some mistakes myself.  But I do know the importance of clear communication in medical work! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Not clear to me if you are considering benign detectable tumors (i.e. possible to have T sans C).  This has implications to how to think about the problem in real world.  For instance, you could have a person who has C (cancer) and has T (detectable tumor) but the detected tumor is benign.  Thus examination, biopsy, surgery etc. might not detect or treat the person's cancer.  I don't think this affects your math problem, but just be careful.&lt;/p&gt;&#xA;" OwnerUserId="9298" LastActivityDate="2018-01-21T19:40:15.743" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13481" PostTypeId="2" ParentId="13465" CreationDate="2018-01-22T07:01:25.700" Score="5" Body="&lt;p&gt;&quot;Other subjects&quot; is a gross generalisation and it certainly is wrong to say that proficiency in mathematics correlates with proficiency in ALL other subjects. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Numerous studies have focused on correlating math ability with English language learning. Generally a significant correlation is attributed between these two (&lt;a href=&quot;https://files.eric.ed.gov/fulltext/EJ1118452.pdf&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt;) but some evidence points against it too (&lt;a href=&quot;https://link.springer.com/article/10.1007/BF00386046&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A stronger correlation is found between math with spatial ability or STEM courses. At the same time, the correlation is weak with NON-STEM courses (&lt;a href=&quot;http://www.freepatentsonline.com/article/Academy-Educational-Leadership-Journal/219062371.html&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2018-01-22T07:01:25.700" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13482" PostTypeId="1" CreationDate="2018-01-22T10:33:59.403" Score="6" ViewCount="252" Body="&lt;p&gt;All summer programs I know for high-school students focuses on number theory, combinatorics, graph theory, logic, and all kinds of topics in discrete mathematics. (I am mainly interested in UK, US, Canada. Any summer program that uses English Language.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone recommend some more &lt;em&gt;continuous&lt;/em&gt; summer programs? That is to say, are there summer schools that offers course in Analysis, Abstract algebra, Topology, and/or set theory? I am also interested in special functions, like &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\zeta$&lt;/span&gt; functions and elliptic integrals: just things more closely related to undergraduate level study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many thanks.&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="238" LastEditDate="2019-12-02T22:05:50.587" LastActivityDate="2019-12-02T22:53:53.400" Title="Is there any high school level summer program that teaches Analysis?" Tags="|secondary-education|reference-request|mathematical-analysis|gifted-students|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="13483" PostTypeId="2" ParentId="13467" CreationDate="2018-01-23T00:03:38.173" Score="1" Body="&lt;p&gt;Since they can do Type-1 problems, I'd begin by trying to make your Type-2 problem look as similar to Type 1 as possible, despite the fact that, in the Type-2 problem, we don't actually know the size of the sample space. So I'd temporarily assume some specific size, say $6$ (to make the arithmetic easy).  So we have $6$ people of whom $3$ have cancer, and one of those $3$ also has a tumor. Let them work out the conditional probability in this case, which I hope they can do because this version of the question is very similar to Type 1.  Then, re-do the problem with a sample space of size, say, 24. And again with 60 or 600, etc., until it becomes clear to them that the information in the problem is enough to determine the answer, regardless of the size of the sample space.  The formula for conditional probability in terms of absolute probabilities summarizes the results of these calculations.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2018-01-23T00:03:38.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13484" PostTypeId="1" AcceptedAnswerId="13485" CreationDate="2018-01-23T13:19:33.107" Score="9" ViewCount="411" Body="&lt;p&gt;I have been presenting compound inequalities like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$3 &amp;lt; x &amp;lt; 7$ &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;as being a shorter way of saying&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$3 &amp;lt; x$ and $x &amp;lt; 7$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From this point of view, though, I end up having to admit that it is okay to write&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$7 &amp;gt; x &amp;lt; 4$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;even though it &quot;simplifies&quot; down to just $x &amp;lt; 4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any better way to formalize compound inequalities that would rule out ever writing &quot;$7 &amp;gt; x &amp;lt; 4$,&quot; or should I embrace the fact that maybe this kind of weird inequality is a good exercise for students to see strange things and unpack the definitions? &lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2020-05-14T02:08:32.127" Title="Framework for Compound Inequalities" Tags="|notation|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13485" PostTypeId="2" ParentId="13484" CreationDate="2018-01-23T14:03:27.930" Score="17" Body="&lt;p&gt;Just because you've defined a meaning for $a &amp;lt; b &amp;lt; c$ does not mean that any mishmash of other relational operators becomes equally well-defined as notation. Stick with what you've defined for a chained equality and don't permit arbitrary, nonstandard, off-track jaunts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, here's the treatment in Sullivan, &lt;em&gt;College Algebra&lt;/em&gt;, Sec. 1.5. Note that each permitted combination is explicitly described, and mixed-direction symbols are prohibited in the last line:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LWigx.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LWigx.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-01-23T17:33:56.640" LastActivityDate="2018-01-23T17:33:56.640" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13486" PostTypeId="1" AcceptedAnswerId="13488" CreationDate="2018-01-23T21:23:27.053" Score="33" ViewCount="6187" Body="&lt;p&gt;Students fresh out of high school are often under the impression that mathematics is a discipline based entirely in recognizing the type of problem and applying an algorithm or cookbook method. These students think there are finitely many types of problems, and each type of problem corresponds to a method. From this perspective, &quot;doing math&quot; is about, in order,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Identifying the type of the problem &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Remember&lt;/strong&gt;ing the method that corresponds to the problem &lt;/li&gt;&#xA;&lt;li&gt;Executing the method&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;However, one of the main goals of quality math courses is to carry students up to the next level of Bloom's Taxonomy, where they might &lt;strong&gt;comprehend&lt;/strong&gt; instead of just &lt;strong&gt;remember&lt;/strong&gt; what is going on. In fact, we hope they will comprehend strategies or facts that transcend specific &quot;types&quot; of remembered problems. I always hope to chip away at the idea of every problem being classifiable into memorized and  cookbook categories.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I am interested in which topics in College Algebra or Precalculus best &lt;strong&gt;reward&lt;/strong&gt; a student who ventures up a level in Bloom's Taxonomy in this way. These should be topics where a cookbook/&lt;strong&gt;remember&lt;/strong&gt;ing student will be able to solve the problem, but a &lt;strong&gt;comprehend&lt;/strong&gt;ing student will be able to solve the problem more easily.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="5321" LastEditDate="2018-01-26T07:20:11.960" LastActivityDate="2018-01-26T16:29:32.853" Title="Math topics that reward going beyond cookbook methods" Tags="|precalculus|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13487" PostTypeId="2" ParentId="13486" CreationDate="2018-01-23T21:23:27.053" Score="11" Body="&lt;p&gt;One really good one is solving &lt;strong&gt;absolute value inequalities&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Students can memorize the following pattern:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When you have &lt;span class=&quot;math-container&quot;&gt;$|X| \geq Y$&lt;/span&gt;, the setup is &lt;span class=&quot;math-container&quot;&gt;$X \geq Y$&lt;/span&gt; &lt;strong&gt;or&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$X \leq -Y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Students with more comprehension will have the following more universal strategy:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When you have an inequality, start by asking whether it is impossible, or always true, or sometimes true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So, if you ask the students to complete:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;:  &lt;span class=&quot;math-container&quot;&gt;$|4x + 17| \geq 0$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You will get the following (correct!) solution from the pattern-recognizer:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$4x + 17 \geq 0$&lt;/span&gt; &lt;strong&gt;or&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$4x + 17 \leq 0$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$4x \geq -17$&lt;/span&gt; &lt;strong&gt;or&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$4x \leq -17$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x \geq -\frac{17}{4}$&lt;/span&gt; &lt;strong&gt;or&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x \leq -\frac{17}{4}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;[number line with everything shaded in]&lt;/p&gt;&#xA;&lt;p&gt;All real numbers are solutions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But the comprehending student will give the following, better solution:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The absolute value of a number is always greater than or equal to 0.&lt;/p&gt;&#xA;&lt;p&gt;So all real numbers are solutions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-01-24T13:24:15.850" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13488" PostTypeId="2" ParentId="13486" CreationDate="2018-01-23T23:23:14.913" Score="20" Body="&lt;p&gt;One answer would be: any real-world word/application problems; insofar as one avoids making them in a repetitive template by &quot;just changing the numbers&quot;. I find that most any textbook has an &quot;Extensions&quot; and/or &quot;Real-World Applications&quot; block of exercises in each section, after the repetitive symbol-manipulation exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I would caution about pushing this too hard. To the extent that Bloom's Taxonomy is a legitimate analytical tool, I've got a thesis, to wit: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.madmath.com/2014/10/blooms-taxonomy-and-math-education.html&quot; rel=&quot;noreferrer&quot;&gt;Math is hard because by its nature it's taught at TOO HIGH a level compared to other classes.&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When most (non-math) teachers gripe about &quot;Level 1: Remembering&quot; their typical examples are &quot;Recite a policy. Quote prices from memory to a customer. Recite the safety rules.&quot; (&lt;a href=&quot;http://www.nwlink.com/~donclark/hrd/bloom.html&quot; rel=&quot;noreferrer&quot;&gt;link&lt;/a&gt;) As soon as we in the math discipline even &quot;change the numbers&quot;, we're already operating a higher level, e.g., &quot;Level 3: Applying&quot; (examples from link: &quot;Use a manual to calculate an employee's vacation time. Apply laws of statistics to evaluate the reliability of a written test.&quot;). Do we usually quiz students on whether they've studied our foundational definitions, or do we just take that for granted? I might posit that we should do more at a &lt;em&gt;lower&lt;/em&gt; level, because our normal practice is blind to the difficulty at even that level. Arguably a problem where even the idea of conceptual pattern-matching is designed to fail/be insufficient, and a brand-new structure or pattern must be created for the one problem, is at the very highest level of the hierarchy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not like &quot;pattern recognition&quot; is bad: many thinkers consider that to be the very definition of mathematics! For example, Ivars Peterson wrote in &lt;em&gt;Islands of Truth&lt;/em&gt; (quoting from Harold Jacobs):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematics is really the science of patterns. Mathematical discovery&#xA;  begins with a search for patterns in data -- perhaps numbers or&#xA;  scientific measurements...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For many of us, having our students be able to recognize patterns regularly would count as an epic victory. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-01-24T03:16:13.963" LastActivityDate="2018-01-24T03:16:13.963" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13491" PostTypeId="2" ParentId="13486" CreationDate="2018-01-24T08:36:17.303" Score="7" Body="&lt;p&gt;Instead of giving precise instructions and objectives, open the question up to whatever kind of inference and creativity the students can generate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Traditional Closed Area Exercise&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A carpet is 4m wide and 6m long. What is its area?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Open Area Exercise&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three rectangular carpets have the same area but have different dimensions. What might the dimensions of those rectangles be? Find a simple triplet of rectangles that works, then a medium hard triplet, then a triplet that nobody else in the class will think of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Traditional Closed % Word Problem&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An investment of $5000 earns 6% interest per year for 10 years. How much will the investment be worth in 10 years?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Open % Word Problem&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some investment prospectuses. [Hand them fictitious and simplified investment prospectuses.] They are all trying you to persuade you that something will happen to your investment over the next 10 years. What is that something? What if they're wrong by a little? What if they're wrong by a lot? Generate examples with numbers you consider plausible to illustrate your thinking.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Many math questions can be opened up like this. It's much harder to assess and grade, but I don't think there's any question as to which type of question requires more reasoning, mastery, and understanding.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-01-24T08:36:17.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13492" PostTypeId="2" ParentId="13486" CreationDate="2018-01-24T09:05:51.060" Score="6" Body="&lt;p&gt;I would think problems on &lt;strong&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/tagged/proof-verification+trigonometry?sort=votes&amp;amp;pageSize=50&quot;&gt;proving trigonometric identities&lt;/a&gt;&lt;/strong&gt; would be well up that alley.  These are the problems I think the &quot;memorize&quot; students tend to find most difficult, as such problems often require the use of a healthy variety of identities and algebraic tools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess these days there are apparently &lt;a href=&quot;http://www.brighthubeducation.com/homework-math-help/72034-proving-trig-identities-how-to-solve/&quot; rel=&quot;noreferrer&quot;&gt;list of steps&lt;/a&gt; to solving them, but a student who can take a step back and try to map out a rough path based upon the unique terms in a given question will usually find things easier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The common false starts and occasional dead ends are perhaps more reminiscent of complex physics proofs/derivations rather than other typical concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure precalculus courses spend much time doing such problems, particularly when accelerated to a college setting.  I think it's often one of the first subjects to be cut back on, given its limited short-term utility.  But having spent an extra semester doing them in a trigonometry course in high school, the practice at comprehending and cognitively applying tools proved mighty useful down the road to me.&lt;/p&gt;&#xA;" OwnerUserId="6994" LastEditorUserId="6994" LastEditDate="2018-01-24T09:14:11.427" LastActivityDate="2018-01-24T09:14:11.427" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13493" PostTypeId="2" ParentId="13486" CreationDate="2018-01-24T12:33:45.887" Score="1" Body="&lt;p&gt;If they are used to smaller fraction calculations they may solve a more complcated version such as,&#xA;$$\large{\frac{\frac{1}{2} - \frac{1}{3}}{\frac{1}{4} + \frac{1}{6}}}$$&#xA;&lt;br&gt;&#xA;by working out the details of the top and bottom separately as they have practised these before:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;taking common denominators,&#xA;  $$\large{\frac{\frac{3-2}{6}}{\frac{3 + 2}{12}}}$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;now simplifying the top and bottom,&#xA;  $$\large{\frac{\frac{1}{6}}{\frac{5}{12}}}$$  &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;then turning the bottom upside down and multiplying to give the answer,&#xA;  $$\frac{1}{6} \cdot \frac{12}{5} = \frac{2}{5}$$   &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is faster to look at the whole fraction: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;remove the fractions from the top and bottom line first using the lcm of all the denominators,&lt;br&gt;&#xA;  $$ \frac{12}{12} \cdot \large{\frac{\frac{1}{2} - \frac{1}{3}}{\frac{1}{4} + \frac{1}{6}}}$$ &#xA;  now simplifying the top and bottom lines, &#xA;  $$\frac{6 - 4}{3 + 2} = \frac{2}{5}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8660" LastActivityDate="2018-01-24T12:33:45.887" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13494" PostTypeId="1" CreationDate="2018-01-24T13:02:51.900" Score="12" ViewCount="360" Body="&lt;p&gt;I'm teaching an introductory analysis course, and I am seeking some feedback on how proofs should be written on the board in class in order to maximize learning. I realize that there is an opinion-based component to this question, but would appreciate any citations of research on the subject (not necessary) in addition to arguments on how student learning might be impacted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question is regarding the actual style that the proof is written on the board including but not limited to the following points&#xA;:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Should theorems and proofs always be written on the board in complete English sentences or is it ok to use abbreviations and symbolic shorthand?&lt;/li&gt;&#xA;&lt;li&gt;Is it ok to use slightly different notation than in the textbook?&lt;/li&gt;&#xA;&lt;li&gt;Is it ok to use slightly (or even completely) different arguments than what is used in the textbook?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm including a couple theorems from Wade's introduction to analysis as examples. I'll type up the theorem statement and proof exactly as in the text and then how I might write it on the board in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a &lt;em&gt;nearly&lt;/em&gt; word for word as the theorem and proof appear in the book:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2.12 Theorem.&lt;/strong&gt; Suppose that $\{x_n\}$ and $\{y_n\}$ are real sequences and are convergent, then&#xA;$$\lim_{n\rightarrow\infty}(x_n+y_n)=\lim_{n\rightarrow\infty}x_n+\lim_{n\rightarrow\infty}y_n.$$&#xA;&lt;strong&gt;&lt;em&gt;Proof.&lt;/em&gt;&lt;/strong&gt; Suppose $x_n\rightarrow x$ and $y_n\rightarrow y$ as $n\rightarrow\infty.$ Let $\epsilon&amp;gt;0$ and choose $N\in\mathbb N$ such that $n\geq N$ implies $|x_n-x|&amp;lt;\epsilon/2$ and  $|y_n-y|&amp;lt;\epsilon/2.$ Thus $n\geq N$ implies&#xA;$$|(x_n+y_n)-(x+y)|\leq |x_n-x| + |y_n-y|&amp;lt;\frac\epsilon2+\frac\epsilon2=\epsilon. \quad \blacksquare$$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Now here is how I might actually write it on the board in class:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Thm 2.12&lt;/strong&gt; $\quad x_n\rightarrow x, y_n\rightarrow y &#xA;\quad  \Longrightarrow \quad  x_n+y_n\rightarrow x+y.$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Proof.&lt;/em&gt;&lt;/strong&gt; Given $\epsilon&amp;gt;0$, choose $N$ s.t. $|x_n-x|&amp;lt;\epsilon/2$, $|y_n-y|&amp;lt;\epsilon/2$  $\ \forall n&amp;gt;N.$&#xA;$$&#xA;\begin{aligned}&#xA;|(x_n+y_n)-(x+y)|&amp;amp;\leq |x_n-x| + |y_n-y|\\&#xA;&amp;amp;&amp;lt;\frac\epsilon2+\frac\epsilon2=\epsilon. \qquad\qquad \blacksquare&#xA;\end{aligned}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The main differences is that the text is more &quot;wordy&quot;, and I will write everything out more &quot;symbolically&quot;. Note that I do stay consistent with the numerical labeling of theorems (e.g. 2.12) in the book so that students can more easily reference the text to compare/study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also sometimes use slightly different arguments than in the text, add in details that are left out of the book or leave out details that are explained in the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.g.:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2.8 Theorem.&lt;/strong&gt; Every convergent sequence is bounded.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Proof.&lt;/em&gt;&lt;/strong&gt; Assume $x_n\rightarrow a.$ Given $\epsilon=1,$ there is an $N\in\mathbb N$ such that $n\geq N$ implies $|x_n-a|&amp;lt;1.$ Hence by the triangle inequality, $|x_n|&amp;lt;1+|a|.$ On the other hand, if $1\leq n \leq N,$ then&#xA;$$|x_n|\leq M:=\max\{|x_1|,|x_2|,\ldots,|x_N|\}.$$&#xA;Therefore, $\{x_n\}$ is dominated by $\max\{M,1+|a|\}. \quad \blacksquare$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Now here is how I might actually write it on the board in class:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Thm 2.12&lt;/strong&gt; $\quad x_n\rightarrow x &#xA;\quad  \Longrightarrow \quad  x_n$ bdd.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Proof.&lt;/em&gt;&lt;/strong&gt; Given $\epsilon&amp;gt;0$ we can find $N$ s.t. $\forall n&amp;gt;N,$ $|x_n-x|&amp;lt;\epsilon.$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So $|x_n|\leq |x|+\epsilon$ when $n&amp;gt;N.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $M=\max\{|x_k|; k\leq N\}.$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore $|x_n|\leq\max\{M,|x|+\epsilon\}$ for any $n. \qquad \blacksquare$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Of course, I also verbally explain each step of the work and will sometimes draw diagrams to help them informally/intuitively understand the argument, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My justification is that there is no need to copy it exactly as it is in the text as the students can simply read that, and that it might be beneficial for them to see it written up slightly differently so that they can become accustomed to different styles of mathematical writing.&lt;/p&gt;&#xA;" OwnerUserId="9365" LastEditorUserId="9365" LastEditDate="2018-01-24T13:17:33.463" LastActivityDate="2018-01-25T09:17:32.397" Title="How to write proofs on the board in the classroom" Tags="|mathematical-pedagogy|proofs|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13495" PostTypeId="1" CreationDate="2018-01-24T17:29:33.777" Score="10" ViewCount="600" Body="&lt;p&gt;I am an high school math teacher and want to develop a math course that is totally focused on sports.  It would be a senior level course and would focus on the math and analytics of sports.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just wanted to see if anyone out there has created a math course like this and if they would be willing to talk about it.  I'm starting from scratch and would like to plan it for next year.  I just need to plan it and get the school board to approve the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyways, if you've done something like this drop me a line.&lt;/p&gt;&#xA;" OwnerUserId="9367" LastEditorUserId="2083" LastEditDate="2018-01-25T14:35:46.777" LastActivityDate="2018-02-02T01:01:27.687" Title="High School Math Course focused on Sports" Tags="|secondary-education|applications|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13496" PostTypeId="1" CreationDate="2018-01-24T18:14:01.333" Score="6" ViewCount="253" Body="&lt;p&gt;I want to give my students the coordinates of three points and ask them to find the coordinates of the orthocenter. This is a fairly long problem that can involve finding the equations of the altitudes and then finding their intersection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A long problem needs to run fairly smoothly, because if we hit a bump we might not sort it all out before the dismissal bell. And then, I think, the time was not very productive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can pick the original six coordinates so that the equations of all three sides of the triangle have intercepts that are integers. But so far I have not figured out how to make the coordinates of the orthocenter integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently I asked such a question here, and some helpful person responded that anyone teaching my subject ought to be able to answer such a question himself. So I would remind this helpful person that I have not despaired of finding an answer myself. But life is short, and I hope that someone out there (probably a real math teacher) has already done the work.&lt;/p&gt;&#xA;" OwnerUserId="8258" LastActivityDate="2018-02-01T18:52:22.437" Title="Tips for choosing coordinates of three points such that the coordinates of the orthocenter are integers" Tags="|analytic-geometry|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13497" PostTypeId="2" ParentId="13496" CreationDate="2018-01-24T19:52:22.977" Score="2" Body="&lt;p&gt;The coordinates will be rational numbers, so you just have to pick arbitrary points, then scale by the LCD. This &lt;a href=&quot;https://www.quora.com/What-is-the-orthocentre-of-a-triangle-when-the-vertices-are-x1-y1-x2-y2-x3-y3&quot; rel=&quot;nofollow noreferrer&quot;&gt;quora&lt;/a&gt; question has several answers for what the formula for the orthocenter is; I haven't verified any of them. If you do have a closed form formula, you can then find a formula for the LCM of all denominators, and then look for numbers that yield one.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2018-01-24T19:52:22.977" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13498" PostTypeId="2" ParentId="13486" CreationDate="2018-01-24T19:53:17.423" Score="8" Body="&lt;p&gt;I fear you will do your students a disservice if you focus too much on avoiding “pattern recognition”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, most mathematical “discoveries” at the level taught in school (as opposed to later on, and even then to some degree) were found through people observing patterns and investigating them. The archetypal example being Gauss and summing series :o)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, it is a great benefit to anyone wishing to &lt;em&gt;apply&lt;/em&gt; mathematical methods to &lt;em&gt;as quickly as possible&lt;/em&gt; recognise if any of the toolkit they have at hand is useful “out of the box” rather than having to spend a lot of time working out what to do. This particularly applies to areas which use mathematical techniques in a routine manner — branches of engineering, physics, chemistry, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What &lt;em&gt;does&lt;/em&gt; help, a lot, is simple practice at spotting patterns in progressively “harder” situations, i.e. moving from spoon-feeding algebraic methods, differentiation, integration etc. towards the more “real-life” examples, although I think that it can’t hurt to slip in (say) a few puzzles stolen from Gardner &lt;em&gt;et al&lt;/em&gt; or the many, many web sites devoted to such things by enthusiasts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’d venture to say in fact that the main benefit of “understanding” the foundations of any method is that it becomes easier to recognise novel situations which might benefit from it.&lt;/p&gt;&#xA;" OwnerUserId="9368" LastActivityDate="2018-01-24T19:53:17.423" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13499" PostTypeId="2" ParentId="13486" CreationDate="2018-01-24T21:43:39.837" Score="5" Body="&lt;p&gt;This is not a direct answer, but too long for a comment. As some of the comments mentioned, the problem isn't patterns, it is blindly applying cookbook solutions. In fact I would argue that mathematics is mostly about patterns, but not about applying them but about finding them. Use this to your advantage, teach them how to do mathematics instead of how to use mathematics, let them make the patterns their own. I would suggest that you try something like the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Give them a few problems with a similar pattern to solve. I assume since you ask about topics that work against them, you know enough topics that have them. But then don't stop there. Instead ask them, if they see the pattern. Maybe give them another similar problem, but ask them not to solve it, but instead to tell you how they would solve it without performing the steps. Then help them to formalize those steps a bit. If this works well, you could even ask them to do the same steps, but with constants instead of numbers. Finally congratulate them on having effectively discovered and proven a mathematical theorem. It might not be a big one, but we all started small.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If done right, this will lead to a far better understanding of what they are doing then solving another dozen of the same problems. The downside for you of course is that its not knowledge you can check directly on a test and it might be a bit more labour-intensive, but I think it could be worth a try.&lt;/p&gt;&#xA;" OwnerUserId="8988" LastEditorUserId="11" LastEditDate="2018-01-26T15:31:19.900" LastActivityDate="2018-01-26T15:31:19.900" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13500" PostTypeId="2" ParentId="13496" CreationDate="2018-01-25T01:34:36.903" Score="7" Body="&lt;p&gt;An explicit formula for the orthocenter as a function of the three&#xA;vertices $(x_i,y_i)$, $i=1,2,3$, can be found on &#xA;&lt;a href=&quot;https://www.quora.com/What-is-the-orthocentre-of-a-triangle-when-the-vertices-are-x1-y1-x2-y2-x3-y3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quora&lt;/a&gt;,&#xA;derived by Shambhu Bhat. The orthocenter coordinates are&#xA;$(o_x,o_y)$, where $o_x=x_{num}\,/\,denom$ and $o_y=y_{num}\,/\,denom$.&#xA;Here the three quantities $x_{num},\, y_{num},\, denom$ are each $2 \times 2$ determinants. The key for integer solutions is the common denominator:&#xA;$$denom \;=\;&#xA;\left|&#xA;\;&#xA;\begin{array}{cc}&#xA; {x_3}-{x_2} &amp;amp; {y_3}-{y_2} \\&#xA; {x_3}-{x_1} &amp;amp; {y_3}-{y_1} \\&#xA;\end{array}&#xA;\;&#xA;\right| \;.&#xA;$$&#xA;In the example below,&#xA;$x_{num},\, y_{num},\, denom \;=\; -240,\, -210,\, -80$,&#xA;yielding $(o_x,o_y) = (3,\frac{21}{8})$.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/Ou6UZ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ou6UZ.jpg&quot; alt=&quot;Denom80&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Vertex coordinates: $(0,0), (10,0), (3,8) \rightarrow (3,\frac{21}{8})$.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Because the denominator is $80$, scaling the coordinates by $80$ is guaranteed&#xA;to yield integer coordinates for the orthocenter. Of course&#xA;often there are cancellations. In this case, multiplying by $8$ would suffice,&#xA;leading to vertices $(0,0), (80,0), (24,64)$ and orthocenter $(24,21)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: Choose three integer-coordinate vertices. Compute $denom$, scale&#xA;the coordinates by $denom$.&#xA;Or, if you want the smallest scaling, go through the calculations&#xA;to compute the orthocenter exactly&#xA;(see the link for the numerator determinants), and then use the LCM of the denominators&#xA;of the orthocenter coordinates, just as user Acccumulation suggests.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-01-25T23:48:33.673" LastActivityDate="2018-01-25T23:48:33.673" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13501" PostTypeId="2" ParentId="13494" CreationDate="2018-01-25T09:17:32.397" Score="5" Body="&lt;p&gt;I very strongly think that &lt;strong&gt;theorems and proofs should be written on the board in complete sentences&lt;/strong&gt;. It is OK to use symbols such as $\implies$ or $\forall$, but only inside a formal assertion. I could for example write&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $(x_n)_{n\in\mathbb{N}}$ be a sequence of real numbers. It holds :&#xA;  $$ (x_n)_{n\in\mathbb{N}} \text{ converges} \implies (x_n)_{n\in\mathbb{N}}\text{ is bounded}.$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;even if that would not be my preferred way to state the result. But it is in my opinion very important to be a model for student's expected answers to test, which should include &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;to write in full sentences and always introduce all variables (too often, one reads statements $x=\dots$ with no way to determine whether the statement is claimed, is to be proved, or is to be refuted; with no way to determine whether there is an $x$ with the claimed property or the claim is for all $x$),&lt;/li&gt;&#xA;&lt;li&gt;to correctly type all variables; in particular you need not to write $x_n$ for the sequence, because $x_n$ represents a number and overloading it can cause great deals of confusion. There is a perfectly fine way to write the sequence, namely $(x_n)_{n\in\mathbb{N}}$ or even $(x_n)$ if you want to shorten a bit, and this writing is pretty clear, very directly related to the sequence being a collection of numbers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Confusion about the type of mathematical object, the status of variables are a primary cause of difficulties among students. As teachers, we need to pave the way as clearly as possible, and that measn be as explicit and precise as we can. It will not be sufficient, but it will be necessary for many students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To illustrate this claim of mine, which I fear might seem pedantic to some, let me present an example which I guess every experienced math teacher has met in one form or another.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt; Let $k$ be a positive integer and let $f$ be the function defined for each $x\in \mathbb{R}$ by $f(x) = k \sin(x/k)$. &lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Show that $f(0)=0$.&lt;/li&gt;&#xA;  &lt;li&gt;Compute $f'(0)$&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Student's answer&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;$f(0) = k \sin(0/k)= k\sin(0) = 0$&lt;/li&gt;&#xA;  &lt;li&gt;$f'(0) = 0' = 0$&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This comes from a wrong typing: there is a confusion between the function and its value at some point. There is no way to debug this kind of problem if one is not very rigorous in its writing.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2018-01-25T09:17:32.397" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13502" PostTypeId="2" ParentId="13495" CreationDate="2018-01-25T23:43:10.270" Score="3" Body="&lt;p&gt;(1) There is an edX course entitled &lt;a href=&quot;https://www.edx.org/course/math-sports-notredamex-mat150x&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Math in Sports&lt;/em&gt;&lt;/a&gt;, with the following prereqs:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Participants will be expected to have a basic knowledge of high school math up to Algebra II, exposure to some probability, statistics, and geometry will be helpful, but is not required.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It could be worthwhile to enroll in this course.&#xA;Taught by a professor of Math and a professor of Physics, Notre Dame.&#xA;&lt;a href=&quot;https://www.youtube.com/watch?time_continue=7&amp;amp;v=lepXR9mBRb0&quot; rel=&quot;nofollow noreferrer&quot;&gt;YouTube promotional video&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) &lt;em&gt;Sports Math&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Minton, Roland B. &lt;em&gt;Sports Math: An Introductory Course in the Mathematics of Sports Science and Sports Analytics&lt;/em&gt;. CRC Press, 2016.&#xA;  (&lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=pB0NDgAAQBAJ&amp;amp;oi=fnd&amp;amp;pg=PP1&amp;amp;dq=+Sports+Math:+An+Introductory+Course+in+the+Mathematics+of+Sports+Science&amp;amp;ots=ApVUcUQE6b&amp;amp;sig=Vggb6RCfrYz-3aFPlBbh-XNbQQk#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google books&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here from &lt;em&gt;Mathematical Reviews&lt;/em&gt;, 2017:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The book is written at a level that is accessible to a large audience. It contains a small number of applications that make use of calculus; otherwise, only a high school level mathematics background is required. Furthermore, one can easily skip over those sections that require calculus and still have plenty of accessible material to read.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Sports Math is well written and easy to read. The book should appeal to anyone interested in the quantitative aspects of athletics. Each chapter of the books ends with a fairly large number of exercises and also pointers to further reading. Thus, the book could be used not only as a textbook for a course but also as a nice resource for student projects.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/eZzYf.png&quot; width=&quot;200&quot; /&gt;&#xA;  &lt;br /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;sup&gt;&#xA;  Projectile Motion, p.9.&#xA;  &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Caveat: I have not used either of these resources.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-01-29T00:31:41.003" LastActivityDate="2018-01-29T00:31:41.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13503" PostTypeId="2" ParentId="13486" CreationDate="2018-01-26T15:59:27.293" Score="-1" Body="&lt;p&gt;A. Defense of recipes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even with your modified question wording, I don't quite share your disdain for algorithms (methods, tools, tricks, etc.).  Much of human effort was spent developing these tools, they have wide utility and students benefit from learning them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though, to you, this stuff is simple and intellectually boring that does not mean it is so for the students.  To whom the methods are new and to whom they open large arenas of problem solving that previously were closed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, even though you list the three steps as mechanical (identify, remember, calculate), I don't think they really are.  It still takes some mental ability and exercise to execute these three.  Even for simple problems.  (If the methods are new to the student.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;B. Possible solutions&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think there can be space to develop some extra difficulty and sophistication.  For example &quot;identify&quot; can be made more difficult in some problems by wording or choice of boundary cases.  Or problems may require multiple methods (making all three steps harder.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, don't make it digital: recipes versus hard problems.  But include some tough problems at the end of tests or as extra credits or the like.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;C.  Caveat:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But even with B, you need to make sure the students have the basics first.  If I've never done exponent manipulation before, let me learn it and practice it a while before you subject me to tricky sophisticated problems (boundary cases, etc.)  This is basic pedagogy.  Let me perfect the back flip for a while before you tell me to do a double back.  Even the single flip is new to me!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, look at the class level of performance on the recipe-based cooking.  Don't make things tougher if they are struggling with the basics.  Get that first.  But if your kids are kicking butt, fine, introduce some trickier problems (gently and watching not to do too much.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, look at the time you have, the objectives of the students/course, and the skill level of the students.  It doesn't make sense to go off the Campbridge Tripos deep end with nursing students at a community college.  It just doesn't.  (And I'm not saying Tripos is &quot;better&quot;--I like nurses!) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Plus they have different goals and needs. You should have some sympathy interest in their track, in their struggles with chem, etc.  Not just wanting everyone to be a math major.  &lt;/p&gt;&#xA;" OwnerUserId="9298" LastEditorUserId="9298" LastEditDate="2018-01-26T16:29:32.853" LastActivityDate="2018-01-26T16:29:32.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13504" PostTypeId="2" ParentId="13460" CreationDate="2018-01-26T19:25:19.593" Score="6" Body="&lt;p&gt;A little late to the party, but I wanted to add my two cents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students should understand that they are looking at the value of &lt;span class=&quot;math-container&quot;&gt;$1/0$&lt;/span&gt; as a limit. This is good but it's not the entire picture, and using that value in normal computation will break algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the following:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;0 = 0 \\&#xA;0\cdot{1} = 0\cdot{2} \\&#xA;\frac{0\cdot 1}{0} = \frac{0\cdot 2}{0} \\&#xA;1 = 2&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that there are many other issues with the &quot;proof&quot; beyond just dividing by zero but the point is to challenge students with the implication of this result to convince them that a statement like &lt;span class=&quot;math-container&quot;&gt;$1/0 = \infty$&lt;/span&gt; is loaded.&lt;/p&gt;&#xA;" OwnerUserId="9383" LastEditorUserId="9383" LastEditDate="2018-11-21T09:15:32.107" LastActivityDate="2018-11-21T09:15:32.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13505" PostTypeId="2" ParentId="13495" CreationDate="2018-01-26T20:55:18.200" Score="2" Body="&lt;p&gt;If your course is statistics, apparently &lt;a href=&quot;https://www.maa.org/press/books/teaching-statistics-using-baseball&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching Statistics Using Baseball&lt;/a&gt; is good enough to have warranted a second edition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same caveat for me: I have not used this (or taught a stats class).&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2018-01-26T20:55:18.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13506" PostTypeId="2" ParentId="13468" CreationDate="2018-01-26T22:29:54.627" Score="1" Body="&lt;p&gt;I do not know of any systematic treatment of the curricular sequencing issues for real analysis.  James Propp wrote an article that discusses different axiomatic approaches to real analysis and the pedagogical implications of each.  Lara Alcock also has a book chapter about mathematics pedagogy and real analysis that discusses pedagogical strategies as they relate to the content of the course.  Another good source for ideas on presenting the material could be William Bauldry's real analysis textbook for prospective secondary mathematics teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alcock, L. (2010). Interactions Between Teaching and Research: Developing Pedagogical Content Knowledge for Real Analysis. In Learning Through Teaching Mathematics (pp. 227–245). Springer, Dordrecht. &lt;a href=&quot;https://doi.org/10.1007/978-90-481-3990-3_12&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/978-90-481-3990-3_12&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bauldry, W. C. (2011). Introduction to Real Analysis: An Educational Approach. John Wiley &amp;amp; Sons.  Preview available at: &lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=eClOhyJmxDkC&amp;amp;oi=fnd&amp;amp;pg=PR7&amp;amp;dq=%22real+analysis%22+pedagogy&amp;amp;ots=h8GulTVPTb&amp;amp;sig=Cbky5PZp4y2Wg3O1f7cOOwNzsk8#v=onepage&amp;amp;q=%22real%20analysis%22%20pedagogy&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=eClOhyJmxDkC&amp;amp;oi=fnd&amp;amp;pg=PR7&amp;amp;dq=%22real+analysis%22+pedagogy&amp;amp;ots=h8GulTVPTb&amp;amp;sig=Cbky5PZp4y2Wg3O1f7cOOwNzsk8#v=onepage&amp;amp;q=%22real%20analysis%22%20pedagogy&amp;amp;f=false&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Propp, J. (2013). Real Analysis in Reverse. American Mathematical Monthly, 120(5), 392–408. &lt;a href=&quot;https://doi.org/10.4169/amer.math.monthly.120.05.392&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.4169/amer.math.monthly.120.05.392&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1798" LastActivityDate="2018-01-26T22:29:54.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13507" PostTypeId="2" ParentId="12450" CreationDate="2018-01-27T09:39:39.430" Score="4" Body="&lt;p&gt;The standards that you identify actually do cover the things you assume are not covered.  The formal properties of logarithms, for example, are proved using exponents, thus:&#xA;F-BF.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exponential functions of base e are covered by:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For exponential models, express as a logarithm the solution to&#xA;  $a\cdot b^{ct} = d$ where $a$, $c$, and $d$ are numbers and the base $b$ is $2$, $10$, or $e$; evaluate the logarithm using technology.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And although CCSS doesn't often reference finance based applications, it continually reiterates real-world problems and applications, of which finance is a major category.  This makes sense, since CCSS does not spell out every application a teacher might utilize, but rather leaves it open as to what real-world situations are used for each topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The CCSS focuses less on procedures and rules, and more on analysis of functions.  So there is a lot more attention given to the graphical and analytic properties of logarithms and exponents than the rules for manipulating them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The CCSS is also intended to be parsimonious rather than comprehensive, in that it does not give a laundry list of topics to &quot;cover&quot; but rather tries to get at the core of what needs to be known. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, remember that CCSS only goes up through algebra two, with an occasional + standard hinting at what might be covered in pre-calc.  The idea is to create a basic level of what needs to be learned before going on to college and careers, not to be exhaustive about everything a potential STEM major might want to learn in high school.&lt;/p&gt;&#xA;" OwnerUserId="1798" LastEditorUserId="80" LastEditDate="2018-01-30T19:15:45.313" LastActivityDate="2018-01-30T19:15:45.313" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13509" PostTypeId="1" CreationDate="2018-01-27T19:48:37.487" Score="4" ViewCount="55" Body="&lt;p&gt;What are some teacher resources on teaching elementary school children (e.g., 2&lt;sup&gt;nd&lt;/sup&gt; graders) how to calculate with an abacus?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did a search on WorldCat for related books, and surprisingly I was only able to find Ruth Merttens's &lt;em&gt;Abacus&lt;/em&gt; series, which I don't think has much to do with abacuses.&lt;/p&gt;&#xA;" OwnerUserId="7971" LastEditorUserId="1147" LastEditDate="2018-01-28T04:10:42.703" LastActivityDate="2018-01-28T04:10:42.703" Title="Teacher resources on abacus for elementary school children" Tags="|resource-request|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13510" PostTypeId="2" ParentId="13431" CreationDate="2018-01-27T23:42:46.783" Score="0" Body="&lt;p&gt;Echoing other comments and @ncr's answer: this notational convention is not unknown, but is not common in mathematical contexts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps more to the point, although relatively experienced math people might guess what is intended, they'd need to &lt;em&gt;ask&lt;/em&gt; to proceed with confidence, so whatever time was thought to be saved by the convenient notation might be lost by having to answer questions about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the very-very-common convention to use &quot;hat&quot; for Fourier transform should give the Fourier-transform use vast precedence over a relatively trivial use of giving the unit vector corresponding to a vector, in my opinion. (The fact that people typically talk about vectors for a while before learning about Fourier transforms doesn't alter my opinion. It's not good to have &lt;em&gt;temporary&lt;/em&gt; notation that becomes wildly misleading soon after.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-01-27T23:42:46.783" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13511" PostTypeId="1" AcceptedAnswerId="13513" CreationDate="2018-01-28T12:19:26.780" Score="7" ViewCount="767" Body="&lt;p&gt;Recently I was watching &lt;a href=&quot;https://youtu.be/uQgcpzKA5jk?t=1h2m35s&quot; rel=&quot;nofollow noreferrer&quot;&gt;this interview of Andrew Wiles&lt;/a&gt; where a secondary school teacher asked this question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How do you teach square roots? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He doesn't answer the question so I'd like to ask it here?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is another example concerning the widespread confusion about this question; &lt;a href=&quot;https://youtu.be/MQXwExo847w?t=27m&quot; rel=&quot;nofollow noreferrer&quot;&gt;in this clip&lt;/a&gt; from University Challenge, he asks &quot;What are the two possible answers to the calculation 5 plus the square root of 4?&quot; &lt;/p&gt;&#xA;" OwnerUserId="216" LastEditorUserId="216" LastEditDate="2020-02-05T15:21:20.010" LastActivityDate="2020-02-05T15:21:20.010" Title="The Way We Teach Square Roots" Tags="|secondary-education|roots|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13512" PostTypeId="2" ParentId="2164" CreationDate="2018-01-28T14:52:24.857" Score="6" Body="&lt;p&gt;I have been making mathematical reading skills a large part of my abstract algebra class this semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A full 20 percent of their grade comes from writing reflections on the reading the text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I gave them a handout in which I analyze a small piece of mathematical writing.  I gave them the definition of &quot;injective&quot;, and the theorem that the composite of two injective functions is again injective.  Then I give different categories of things they should do, together with examples.  For instance, they should look for examples/nonexamples of the definition.  They should try to rephrase the definition into a logically equivalent form.  They should form connections to other definitions they have seen, etc.  I have the students do &quot;self explanations&quot; of the proofs they read following Lara Alcock's paper mentioned above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students have to write a few pages once a week on their reading, and class time is primarily spent resolving questions which have arisen during their reading.  A lot of the resolution comes from other students.  Another 20% of their grade comes from writing proofs of a few theorems each week, and they spend some class time exchanging papers and critiquing each others work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is going extremely well so far.  The engagement with the material is at a much deeper level that in previous semesters. I will update later in the semester.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-01-28T14:52:24.857" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13513" PostTypeId="2" ParentId="13511" CreationDate="2018-01-28T15:23:56.347" Score="13" Body="&lt;p&gt;As he says, this is a convention, and you just have to make a choice and stick with it.  The usual convention (which makes the square root a function) is to take the positive branch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The teacher seems a bit confused.  He seems to think that there is a contradiction between saying $\sqrt{9} = 3$ on one hand, and on the other hand writing $x^2=9 \implies x=\pm3$ on the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we stick to the convention that the square root takes only positive values, then there are two sensible ways to approach the quadratic equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One is to recognize that $\sqrt{x^2} = |x|$, not $x$.  So $x^2 =9 \implies |x|=3 \implies x = \pm 3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other way is to factor $x^2=9 \implies x^2-9=0 \implies (x-3)(x+3) = 0 \implies x = \pm3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In either case, the $\pm$ does not need to be part of the definition of the square root, and including it makes life more complicated because then square root is not a (single valued) function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As mentioned in the comments, it is worth making the distinction between square roots and principle square roots.  We can define $y$ to be a square root of $x$ if and only if $y^2=x$. Then every positive number $x$ has two square roots, $0$ has only $0$ as its square root, and negative numbers have no real square roots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The relation $y^2=x$ is not a function of $x$, but if we throw out all the pairs with negative $y$ values we do obtain a function, called the &quot;principle square roots function&quot;.  This takes a nonnegative real number $x$ as input and returns its nonnegative square root as output.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another observation:  this choice is nice for many reasons, not the least of which is that it is consistent with the definition $x^a = e^{a \log(x)}$ which can be used to define any positive power function (assuming the exponential and logarithm have been defined).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This might help to clarify my remarks above the edit.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2018-01-29T17:06:21.820" LastActivityDate="2018-01-29T17:06:21.820" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13514" PostTypeId="1" CreationDate="2018-01-29T00:05:24.687" Score="8" ViewCount="905" Body="&lt;p&gt;Until a few days ago, every calculus textbook I have seen uses $u$ as the default variable for integration by substitution (a.k.a. integration by change of variables). It was brought to my attention recently that there is an exception: the calculus textbook by Hughes-Hallett et al. uses $w$ as its default letter for integration by substitution. The reason behind this is presumably to distinguish between the variables used in integration by substitution and in integration by parts. While that is an admirable idea, in practice I'd never seen anything other than &quot;$u$-substitution&quot; before.&#xA;(As a graduate student I did teach with a much earlier edition of Hughes-Hallett than the current one but I do not recall the change of variables in the book being $w$, so I don't know if this $w$-substitution was present from the first edition of the book or not.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know calculus books besides Hughes-Hallett that discuss integration by substitution with a default letter other than $u$, or do you teach it with a default letter other than $u$ even if the class textbook uses $u$? (Edit: I do &lt;em&gt;not&lt;/em&gt; have in mind trigonometric substitutions, where it is natural to use the new variable $\theta$ rather than $u$ for traditional geometric reasons. That also has a different flavor than what happens with other instances of integration by substitution in a calculus class, since standard substitutions take the form $u = f(x)$ while trig substitution is $x = f(\theta)$, so I am not asking about trigonometric substitution here.) &lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2018-01-29T02:23:17.350" LastActivityDate="2018-01-29T02:37:51.247" Title="Integration by substitution not using letter u" Tags="|integration|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13515" PostTypeId="2" ParentId="13514" CreationDate="2018-01-29T02:18:54.127" Score="5" Body="&lt;p&gt;Small but not cherry-picked sample.  Books I have in library.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Granville (40s edition but goes back earlier for first editions): Uses $z$ for the $\tan(z/2)$-substitution and for the $z^n$ substitution.  And for the random extra substitutions.  But note also discusses integrals of forms like $\sqrt{a^2-u^2}$.  In other words, using $u$ instead of $x$ for these tables of random integrals.  (So not preserving it for $u\,dv$ parts integration only.)  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Thomas Finney, c. 1980.  Similar to Granville with using $z$ as preferred new variable and other variable names.  Does have one example with a double substitution and in this case, they use $z$ to start and then $\theta$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Swokowski, early 80s.  Uses $\theta$ for most of the random trig substitutions.  Uses $u$ for the hyperbolic trig substitutions.  Uses $z$ for the $\tan(z/2)$ master-blaster substitution.  Uses both $u$ and $z$ for the random other substitutions.  Uses both $a^2-u^2$ and $a^2-x^2$ in discussion special forms (i.e., not protecting $u$, but also sometimes  just using $x$).  The integration by parts is still normal $u\,dv$.  Seems to use $z^n$, for that substitution, but not sure if I found that correctly.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Net/net:  seems like $z$ and $\theta$ like to makes some appearances also. &lt;/p&gt;&#xA;" OwnerUserId="9298" LastEditorUserId="9298" LastEditDate="2018-01-29T02:37:51.247" LastActivityDate="2018-01-29T02:37:51.247" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13516" PostTypeId="1" CreationDate="2018-01-29T07:10:11.777" Score="2" ViewCount="246" Body="&lt;p&gt;I want to understand and teach Mathematics for Data Science. I know for Data Science I need to understand:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Probability&lt;/li&gt;&#xA;&lt;li&gt;Statistics&lt;/li&gt;&#xA;&lt;li&gt;Linear Algebra&lt;/li&gt;&#xA;&lt;li&gt;Calculus&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But I want a quick refresh of everything from the very basics and then go to the above mentioned concepts in some order. But I don't know in what order I can go. I need something like an ordered checklist to ensure that I understood the fundamentals before going on to advanced concepts.&lt;/p&gt;&#xA;" OwnerUserId="9390" LastEditorUserId="511" LastEditDate="2018-01-29T21:02:26.570" LastActivityDate="2018-01-29T21:02:26.570" Title="Ordered checklist for learning Mathematics for Data Science" Tags="|self-learning|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13517" PostTypeId="2" ParentId="4326" CreationDate="2018-01-29T08:17:47.437" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Agree with the draw a triangle advice from Thorne.  This is actually not so uncommon in books, though--just looked at a few.  Maybe, a good pedagogy is to push it with students though, so they don't have as much memorizing rules.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Looked through a few books and most of the problems were just straight symbols are rather artificial word problems (volumes of revolution for given shapes). &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One real area that they are useful is in hydraulics, especially for open channels (not filled pipes).  Granville book has a short chapter on various hydraulics problems. Also Paul's online notes give an example.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Paul's:  &lt;a href=&quot;http://tutorial.math.lamar.edu/Classes/CalcII/HydrostaticPressure.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://tutorial.math.lamar.edu/Classes/CalcII/HydrostaticPressure.aspx&lt;/a&gt; (towards bottom)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Granville:  Sorry,no link.  The free pdf doesn't have the same chapter...grr.  My copy is 1944 edition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are several electrostatics problems requiring trig substitution.  (Google search will show this, but I have a copy of Wangness in hand.)  And electrostatics as taught in physics classes, not design work using FE, does a lot of analytical solutions in problems.  Physicists like students to be able to work out some of the problems too--not just hand it to Mathematica.  I would suspect that various applied problems in other engineering courses also occasionally have an integral needing trig substitution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that said, my advice would not be to teach this by physical examples.  It will probably just make it harder.  Treat it as another trick in the bag of tricks.  If there are questions about application, you can cite some of these (read up on them a little bit so you are not just parroting something I say, but know it yourself  Or just tell the kids, that there are a lot of integrals coming at them in physics and engineering courses and they need to have a decent toolkit.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[I think exposure to some decent variety of tougher integration methods, partial fractions and all that, makes students more confident in  using integral tables also...at least they know they have seen a lot of stuff and worked with a lot of stuff.  Not just blindly consulting the CRC.  But this would be for kids who are taking a solid STEM course.  Not a reduced calc class.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Honest, I think part of the benefit of learning some things in math class is getting some of the learning in a more abstract fashion (with x's and y's) instead of physical variables.  Let's the student learn it in that context.  Then when they see it in physical science course or engineering, they learn it that way also. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can be good to see things a couple different ways.  Even if it is a year later, when they randomly see a trig substituation in a derivation or homework problem of a sci/eng class and they have forgotten it, they will remember they learned it once.  So the discussion in sci/eng class will be easier than if this was first time.  And the subsequent exposure may help solidify the concept or at least briefly refresh it and leave the student confident of knowing he could look up the method in his text or in a table of integrals.&lt;/p&gt;&#xA;" OwnerUserId="9298" LastActivityDate="2018-01-29T08:17:47.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13518" PostTypeId="2" ParentId="13516" CreationDate="2018-01-29T15:05:43.570" Score="3" Body="&lt;p&gt;This link (compiled by Wale Akinfaderin) may help: &#xA;&lt;a href=&quot;https://towardsdatascience.com/the-mathematics-of-machine-learning-894f046c568&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;The Mathematics of Machine Learning&quot;&lt;/a&gt;.&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/1tQos.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1tQos.png&quot; alt=&quot;PieChart&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;The MIT course &lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/18-657-mathematics-of-machine-learning-fall-2015/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Mathematics of Machine Learning&quot;&lt;/a&gt; (Philippe Rigollet) illustrates what might be called advanced math for machine learning,&#xA;pretty much assuming as prerequisites what is in the pie chart above.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-01-29T15:05:43.570" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13519" PostTypeId="2" ParentId="1545" CreationDate="2018-01-30T20:39:57.850" Score="5" Body="&lt;p&gt;Bankruptcy problems ask for how to &quot;fairly&quot; distribute \$E to honest claimants whose claims exceed the amount \$E. For example, $A$ claims \$30, $B$ claims \$50 and $C$ claims \$120 and there is only \$160 to distribute. There are two methods dating back to &quot;medieval&quot; times associated with Moses Maimonides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a. Try to equalize the amount given (gain) to each claimant but without giving the claimant more than the claimant asks for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b. Try to equalize the loss to each claimant but without asking the claimant to subsidize the settlement by adding money to E to make this possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each of these approaches to being fair leads to a constrained optimization problem. There are other approaches to being fair here in addition to the two approaches above, for example, one could give each claimant an amount proportional to his/her claim.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastEditorUserId="98" LastEditDate="2018-01-30T20:47:14.803" LastActivityDate="2018-01-30T20:47:14.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13520" PostTypeId="1" AcceptedAnswerId="13523" CreationDate="2018-01-31T00:12:07.850" Score="22" ViewCount="688" Body="&lt;p&gt;While teaching multi-variable calculus for the first time in a while, I came across a tricky notational point in our textbook (Thomas' calculus - I'm not sure how widespread this notation is).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When $\mathbf{r}(t)$ is a vector-valued function, our book writes the arc length parametrization as $\mathbf{r}(s)$, with the same name for the function, changing only the variable.  From the right perspective, this makes a lot of sense: one thinks of $\mathbf{r}$ as a polymorphic function that interprets its inputs in the uniquely sensible way, so when passed a value of $t$ it returns $\mathbf{r}(t)$, and when passed an arc length returns $\mathbf{r}(s)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, this &lt;em&gt;isn't&lt;/em&gt; the perspective my students (mostly first-year college students) have on functions.  It violates the way functions normally behave---if $\mathbf{r}(t)=\langle t,t^2\rangle$ then $\mathbf{r}(s)$ &lt;em&gt;doesn't&lt;/em&gt; equal $\langle s,s^2\rangle$.  It creates some ambiguities---what does $\mathbf{r}(2)$ mean?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming I'm stuck with this notation (it's the one our textbook uses, and I share a final with other people using the same book), what are good practices to work with and explain this to minimize how confusing it is?  (And maybe help my students get the most out of this new perspective on functions.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-03-24T18:37:12.423" Title="Polymorphic functions in vector calculus" Tags="|undergraduate-education|vector-calculus|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13521" PostTypeId="2" ParentId="13496" CreationDate="2018-01-31T18:52:55.103" Score="1" Body="&lt;p&gt;I often create problem sets using Excel. I wrote the follow suggestion as a recipe easily developed in Excel. Wherever the digit 1, 2 or 3 follows a letter immediately, the intention is a subscript. The sole exponent in this entire answer is indicated by a caret followed by a 2 like this: ^2. Here I define some of the subscripted symbols:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;x1, y1: The x and y coordinates of Point 1.&#xA;x2, y2: The x and y coordinates of Point 2.&#xA;x3, y3: The x and y coordinates of Point 3.&#xA;m1, b1: The slope and intercept of the altitude to Point 1&#xA;m2, b2: The slope and intercept of the altitude to Point 2.&#xA;m3, b3: The slope and intercept of the altitude to Point 3.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Pick k1, k2, k3. Do not pick k1=1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let k4=(k2)^2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pick d.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let e=k3*(1-k1)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let f=[(1+k1)/(1-k1)]*e&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Define the vertices of the triangle as follows:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;        x        y&#xA;Point 1 k4(f)   k2(d)&#xA;Point 2 k4(e+f) k2(d+e)&#xA;Point 3 k4(e)   k2(d+e+f)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The altitudes through these vertices can be computed as follows.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Altitudes through… y = m                 x + b&#xA;Point 1            y = (x2 – x3)/(y3-y2) x + (y1 – x1(x2 – x3)/(y3-y2))&#xA;Point 2            y = (x1 – x3)/(y3-y1) x + (y2 – x2(x1 – x3)/(y3-y1))&#xA;Point 3            y = (x1 – x2)/(y2-y1) x + (y3 – xC(x1 – x2)/(y2-y1))&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Orthocenter’s x-coordinate: (b1-b2)/(m2-m1)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If all picks are integers, then all of following will also be integers: the coordinates of the vertices, the altitudes’ slopes, the altitudes’ intercepts, and the coordinates of the orthocenter. The slopes of the altitudes through Points 1, 2 and 3 will respectively be (k2), (k1*k2), and (-1*k2). Therefore if k2=1, the triangle will be a right triangle, the altitudes to Points 1 and 3 will be perpendicular, and the orthocenter will be Point 2.&lt;/p&gt;&#xA;" OwnerUserId="8258" LastEditorUserId="8258" LastEditDate="2018-02-01T18:52:22.437" LastActivityDate="2018-02-01T18:52:22.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13522" PostTypeId="2" ParentId="13520" CreationDate="2018-01-31T18:57:26.893" Score="4" Body="&lt;p&gt;When I was a student learning the difference, I found the unit-speed interpretation to be more helpful. Try using an animation in Matlab, Python, or Mathematica to illustrate this for simple curves by simultaneously tracing out &lt;span class=&quot;math-container&quot;&gt;$r(t) = (\cos{2t}, \sin{2t})$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r(s)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\tilde{r}(t) = (t, t^2)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tilde{r}(s)$&lt;/span&gt;. Seeing the curve being traced out in its &quot;normal&quot; parametrization and its arclength parametrization will help show that the parameters are describing the same curve but at different speeds (the two parameters for the curve corresponding to &lt;span class=&quot;math-container&quot;&gt;$\tilde{r}$&lt;/span&gt; show this especially well since the standard parametrization is not constant speed). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think students will understand that implicitly, the arclength parametrization is just hiding the composition with a function involving the inverse of the arclength integral and doing so allows you to travel at unit speed. In other words, &lt;span class=&quot;math-container&quot;&gt;$r(s)$&lt;/span&gt; really just means &lt;span class=&quot;math-container&quot;&gt;$r(t(s))$&lt;/span&gt;. The technicalities may be better suited for a differential geometry course (see the first chapter of &lt;em&gt;Elementary Differential Geometry&lt;/em&gt; by Andrew Pressley), as concepts like regularity of a curve and the Inverse Function Theorem are required. &lt;/p&gt;&#xA;" OwnerUserId="9394" LastEditorUserId="9394" LastEditDate="2019-03-24T18:37:12.423" LastActivityDate="2019-03-24T18:37:12.423" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13523" PostTypeId="2" ParentId="13520" CreationDate="2018-02-01T02:47:19.227" Score="14" Body="&lt;p&gt;By coincidence, I'm  teaching multi-variable Calculus for the first time this semester, and have given some thought to how to handle this precise issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems to me to be closely related to an example I first encountered at &lt;a href=&quot;http://math.oregonstate.edu/bridge/ideas/functions/&quot; rel=&quot;noreferrer&quot;&gt;http://math.oregonstate.edu/bridge/ideas/functions/&lt;/a&gt;, which I &lt;strong&gt;strongly&lt;/strong&gt; encourage you to visit before you read the rest of this answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This example is discussed in some detail in a paper by Edward F. Redish and Eric Kuo, published at &lt;a href=&quot;https://link.springer.com/article/10.1007/s11191-015-9749-7&quot; rel=&quot;noreferrer&quot;&gt;https://link.springer.com/article/10.1007/s11191-015-9749-7&lt;/a&gt; (free preprint version at &lt;a href=&quot;https://arxiv.org/pdf/1409.6272&quot; rel=&quot;noreferrer&quot;&gt;https://arxiv.org/pdf/1409.6272&lt;/a&gt;).  The key insight concerns what the authors call &lt;em&gt;Corinne's Shibboleth&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One of your colleagues is measuring the temperature of a plate of metal placed above an outlet pipe that emits cool air. The result can be well described in Cartesian coordinates by the function&#xA;  $$T(x,y) = k(x^2 + y^2)$$&#xA;  where $k$ is a constant. If you were asked to give the following function, what would you write?&#xA;  $$T(r,\theta) = \textrm{ ?}$$&#xA;  The context of the problem encourages you to think in terms of a&#xA;  particular physical system. Physicists tend to think of $T$ as a&#xA;  physical function – one that represents the temperature (in whatever&#xA;  units) at a particular point in space (in whatever coordinates).&#xA;  Mathematicians tend to consider $T$ as a mathematical function – one&#xA;  that represents a particular functional dependence relating a value to&#xA;  a pair of given numbers.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;As a result, physicists tend to answer that $T(r,\theta) = kr^2$&#xA;  because they interpret $x^2 + y^2$ physically as the square of the&#xA;  distance from the origin. If $r$ and $\theta$ are the polar&#xA;  coordinates corresponding to the rectangular coordinates $x$ and $y$,&#xA;  the physicists' answer yields the same value for the temperature at&#xA;  the same physical point in both representations. In other words,&#xA;  physicists assign meaning to the variables $x$, $y$, $r$, and $\theta$&#xA;  – the geometry of the physical situation relating the variables to one&#xA;  another.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Mathematicians, on the other hand, may regard $x$, $y$, $r$, and&#xA;  $\theta$ as dummy variables denoting two arbitrary independent&#xA;  variables. The variables $(r, \theta)$ or $(x,y)$ don't have any&#xA;  meaning constraining their relationship. Mathematicians focus on the&#xA;  mathematical grammar of the expression rather than any possible&#xA;  physical meaning. The function as defined instructs one to square the&#xA;  two independent variables, add them, and multiply the result by $k$.&#xA;  The result should therefore be $T(r,\theta) = k(r^2 + \theta^2)$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Typically, a physicist will be upset at the mathematician's result.&#xA;  You might hear, &quot;You can’t add $r^2$ and $\theta^2$! They have&#xA;  different units!&quot; The mathematician is likely to be upset at the&#xA;  physicist's result. You might hear, &quot;You can’t change the functional&#xA;  dependence without changing the name of the symbol! You have to write&#xA;  something like $$T(x,y) = S(r,\theta) = kr^2.$$ To which the physicist&#xA;  might respond, &quot;You can’t write that the temperature equals the&#xA;  entropy! That will be too confusing.&quot; (Physicists often use $S$ to&#xA;  represent entropy.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Parenthetically, coming up soon in my multivariable Calculus course -- and presumably in the OPs course as well -- are functions expressed in polar, cylindrical, and spherical coordinates, so the particulars of this example may be salient in that context as well.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The relationship between this example and the one in the OP is that the convention of using the notations $\mathbf{r}(t)$ and $\mathbf{r}(s)$ to refer to two different parametrizations of the same path seems (to me) to be in accordance with the &quot;Physicist's interpretation&quot;:  the variables $t$, $s$ and $\mathbf{r}$ are understood to have physical meaning (respectively:  time, arc length, and position), so that $\mathbf{r}(t)$ means &quot;the position corresponding to time $t$&quot; while $\mathbf{r}(s)$ means &quot;the position corresponding to arc length $s$&quot;.  This is perfectly coherent and reasonable if the variables are understood as &quot;physical variables&quot;, but it is at odds with the mathematician's understanding of a function as a mapping that assigns a value to an arbitrary input represented by a dummy variable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What to do about it?  Here I think one strategy that can be helpful is to use &lt;em&gt;less&lt;/em&gt; functional notation, rather than more.  Consider the example of an object moving in a circular path of radius $5$, i.e. $\mathbf{r}(t)=\left&amp;lt;5\cos(t),5\sin(t)\right&amp;gt;$.  I would propose actually eliding the variable from the left-hand side entirely, and writing it this way:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consider a particle whose position at time $t$ is given by&#xA;  $$\mathbf{r} = \left&amp;lt;5\cos(t),5\sin(t)\right&amp;gt;$$ &#xA;  Then the velocity is&#xA;  $$\mathbf{r}'  = \left&amp;lt;-5\sin(t), 5\cos(t)\right&amp;gt;$$&#xA;  so the speed of the object is $$|\mathbf{r}'| = \sqrt{25\cos^2(t) + 25\sin^2(t)} = 5 $$ &#xA;  and therefore the distance traveled between $t=0$ and $t=T$ is $$ \int_0^T 5 dt = 5T$$&#xA;  This gives us the relationship $s = 5t$, where $s$ is the distance traveled by time $t$.  Inverting this relationship we have $t = \frac{s}{5}$, which allows us to express the position of the object when it has traveled a distance $s$ as  $$\mathbf{r} = \left&amp;lt;5\cos\left(\frac{s}{5}\right),5\sin\left(\frac{s}{5}\right)\right&amp;gt;$$ &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Notice that in this exposition functional notation has been almost completely avoided.  An expression like $\mathbf{r}(2)$ (which, as the OP notes, is ambiguous) would be handled by using a descriptive phrase:  either &quot;$\mathbf{r}$ when $t=2$&quot;  or &quot;$\mathbf{r}$ when $s=2$&quot;, depending on which is intended.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you think about it, this is really no different from the common single-variable usage of writing something like $y = 5x^3 +2x$, which also avoids functional notation.  And just as we do not hesitate to write $\frac{dy}{dx}$, I think it is perfectly reasonable to write $\frac{d\mathbf{r}}{dt}$ and $\frac{d\mathbf{r}}{ds}$ to refer to the respective derivatives.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-02-01T02:47:19.227" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13525" PostTypeId="2" ParentId="13520" CreationDate="2018-02-01T14:36:33.620" Score="7" Body="&lt;p&gt;I don't have much to add to mweiss' nice answer in terms of teaching suggestions. But I would like to add my own point of view on what the abuse of notation $\mathbf{r}=\mathbf{r}(s)=\mathbf{r}(t)$ &lt;em&gt;means&lt;/em&gt; and where it comes from historically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first person to do this abuse of notation (implicitly) seems to have been &lt;a href=&quot;https://mathoverflow.net/a/278927/745&quot;&gt;Jacobi around 1830&lt;/a&gt; (I suggest you read that link &lt;em&gt;after&lt;/em&gt; reading the rest of what I'll say). Note that 1830 is more than 100 years after Bernoulli and Euler introduced the notation $y=f(x)$ and more than 140 years after Leibniz startet to talk of functions. In the period between Leibniz and Jacobi we find luminaries like Euler, Lagrange, Laplace, Fourier, Bolzano, Cauchy and Gauss who, as far as I can tell, never did this. So it's not the case, as some people believe, that mathematicians and physicist have been writing $y=y(x)$ ever since the invention of functions. Even after Jacobi there are many people, like Riemann, Peano or Planck who apparently didn't do it. So it's also not the case that physicists invented $y=y(x)$ or somehow need it more urgently than mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why did Jacobi start doing it? In the above link you can read what he himself had to say about it (I highly recommend it to anyone teaching multivariable calculus), but his own words actually don't explain it completely. To understand it better we first need to understand how the word &lt;em&gt;function&lt;/em&gt; was used prior to about 1900. Most mathematicians seem unaware of the dramatic change of meaning this word underwent during the period 1900-1920. And it's a non-trivial task for a modern mathematician with a set theoretic perspective to make sense of what it meant prior to 1900. But let's try.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you open &lt;em&gt;any&lt;/em&gt; calculus textbook written after Leibniz and before at least 1910 (a ~200 year period) you'll find that the word &quot;function&quot; is &lt;em&gt;always&lt;/em&gt; defined as &lt;em&gt;function &lt;strong&gt;of&lt;/strong&gt; something&lt;/em&gt;. Here is a typical example from the end of that period taken from Peano's &lt;a href=&quot;https://archive.org/stream/calcolodifferen00peangoog#page/n44/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Calcolo differenziale e principii di calcolo integrale&lt;/em&gt;&lt;/a&gt; 1884, p.3: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Among the variables  there are those to which we can assign&#xA;  arbitrarily   and successively different values, called &lt;em&gt;independent&#xA;  variables&lt;/em&gt;, and others whose values depend on the values given to the&#xA;  first ones. These are called &lt;em&gt;dependent variables&lt;/em&gt; or &lt;strong&gt;&lt;em&gt;functions of&#xA;  the first ones&lt;/em&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;We shall first treat functions of a single independent variable, and&#xA;  we shall say that: &lt;em&gt;a function $y$ of $x$ is given in an interval&#xA;  $(a, b)$, if to any value of $x$ in between $a$ and $b$ corresponds a&#xA;  unique and determinate value for $y$&lt;/em&gt; - whatever the means of&#xA;  determining it.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;So for example $x^{2}$ is a definite function of $x$ for any value of&#xA;  $x$, and is hence given in any interval; $\sqrt{x}$ understood as the&#xA;  arithmetic root of $x$, is given for all positive values of $x$ ;&#xA;  while  $\frac{1}{1} + \frac{1}{2} +\frac {1}{3} +\cdots + \frac{1}{x}$&#xA;  is a function of $x$ defined only for integer and positive values of&#xA;  the variable, etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So a function, like $x^2$ or $y$, is a variable quantity, just like $x$, only that it satisfies some additional property. If you feel uncomfortable with &quot;$y$ is a function of $x$&quot; because you are so used to modern functions, maybe paraphrasing it as &quot;$y$ depends on $x$&quot; helps a bit. (Certainly if $f:\mathbb{R}\to \mathbb{R}$, no modern mathematician would say that $f$ depends on $x\in \mathbb{R}$.) Hence whenever they called something a function, they had to add &lt;em&gt;of what&lt;/em&gt; that thing is a function. But often it was clear from the context or the notation, so they would soon drop the &quot;of $x$&quot; and simply talk of functions. So for example on p.13 we find Peano writing&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One says that a function $f(x)$ becomes maximal relative to an&#xA;  interval $(a,b)$ when $x=x_0$...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To be correct he should have said something like: &quot;One says that a function &lt;em&gt;of $x$&lt;/em&gt;, $f(x)$, becomes ...&quot;. But it was clear form the notation. (Observe that nothing prevents $f(x)$ from being a function of something else too. For example when $x$ is itself a function of $t$, then $f(x)$ would also be a function of $t$.)  We inherited the phrase &quot;$f(x)$ is a function&quot; &amp;mdash;which we find in every modern calculus textbook (and btw. you used the question)&amp;mdash; from this pre 1900 period, even though according to our modern convention we should correctly say &quot;$f$ is a function&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To emphasise again the difference between &quot;function of ...&quot; and our modern &quot;functions&quot;: when $y=f(x)$ they called $y$ and $f(x)$ a function, while we call $f$ a function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they called $f(x)$ the function, how did they call $f$? After all the notation $f(x)$ existed since Bernoulli ~1718. Didn't they also call $f$ a function? No, not officially. The first ones (Bernoulli, Euler, Lagrange) called $f$ the &lt;em&gt;characteristic of the function $f(x)$&lt;/em&gt;. I interpret this as saying: its just a character used to distinguish one function of $x$, say $f(x)$, from another, say $g(x)$. But mainly people before 1900 didn't call $f$ anything at all! Peano for example doesn't in his calculus book. In fact, no one treated $f$ as a mathematical object on its own, before Dedekind, Peano, Cantor and Frege (to a certain extent independently) &lt;a href=&quot;https://hsm.stackexchange.com/questions/6104&quot;&gt;changed that around 1890&lt;/a&gt;. (There are some surprising historical bits in that thread. For example: Dedekind, Peano and Cantor all gave $f$ a new name, calling it resp. &quot;map&quot;, &quot;prefix sign for a function&quot; and &quot;allocation&quot;. They all preserved the original use of &quot;function&quot; for $f(x)$! Moreover Dedekind formulated his notion of &lt;em&gt;map&lt;/em&gt; nine years before realising that he could identify it with the $f$ appearing in their functions $f(x)$. Only Frege (unfortunately) suggested calling $f$ a function, and somehow it became standard. We would probably have less difficulty communicating today, if $f$ had not gotten the name function.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Returning to $y=y(x)$ and why Jacobi started that. Besides the good reasons about partial derivatives he gives in De Determinantibus 1840, I suspect that he simply wanted a notation to indicate &lt;em&gt;of what&lt;/em&gt; a certain variable is to be considered a function. I used to believe that in the equation $y=y(x)$ the $y$ on the left and on the right were objects of different types which by abuse where given the same name (the right one being of type $\mathbb{R}\to\mathbb{R}$, the left one of type $\mathbb{R}$.) But I now think that this is &lt;strong&gt;not&lt;/strong&gt; what Jacobi intended and how physicist would like to use it. Instead, we should literally think of the $y$ on the right and left as the &lt;em&gt;same&lt;/em&gt; object (of type &quot;variable quantity&quot;), and what is being abused is the notation for &lt;em&gt;function application&lt;/em&gt; $f(x)$. In other words: had Jacobi chosen another notation like square brackets $y[x]$ to annotate that $y$ is considered as function of $x$, instead of using the already existing notation $f(x)$ for &quot;application of a map to a variable&quot;, we might have less trouble with it now. I suspect one could formalise this idea similarly to how computer scientist formalise &quot;ascription&quot;. See for example &lt;a href=&quot;https://flylib.com/books/en/4.279.1.65/1/&quot; rel=&quot;noreferrer&quot;&gt;Pierce, Types and Programming Languages&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said that, I don't think that by doing that we would gain much. Writing $\mathbf{r}=\mathbf{r}[t]$ seems mainly an aide for the reader, and we might as well do without it, as mweiss suggested. The harder problem seems to be to correctly formalise the notion of &quot;variable quantity&quot; and surrounding things like the notation $dy/dx$.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2018-02-01T14:36:33.620" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13526" PostTypeId="2" ParentId="13495" CreationDate="2018-02-02T01:01:27.687" Score="4" Body="&lt;p&gt;A variety of ideas in discrete mathematics/graph theory can be used for a course related to mathematical ideas for use in the area of sports. For example, one can study tournament graphs, complete directed graphs on n-vertices. When there is a &quot;round robin&quot; tournament one records using a edge from the vertex representing team A to the vertex representing team B that team A beat team B. How can one rank the teams in the tournament using the information in this graph? One can also study ways to fairly schedule home and away games for teams which play several games against each other in a tournament. For more details see this &quot;survey:&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/publicoutreach/feature-column/fcarc-sports&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.ams.org/publicoutreach/feature-column/fcarc-sports&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which has many references at the end.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2018-02-02T01:01:27.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13528" PostTypeId="2" ParentId="13511" CreationDate="2018-02-03T06:10:00.497" Score="2" Body="&lt;p&gt;&lt;strong&gt;Original Answer&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I first define that a square root of a number $a$ is a number $x$ such that $x^2 = a$. I then give some examples like: What are the square roots of 4? $2$ and $-2$ because $2^2=4$ and $(-2)^2=4$. I then state that positive real numbers have two square roots, one positive and one negative. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what does $\sqrt{a}$ mean? There are two choices, right?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Define that the symbol $\sqrt{a}$ always denotes the positive square root of $a$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I given a couple of examples: The square roots of 4 are $\sqrt{4}=2$ and $-\sqrt{4}=-2$. The square roots of 2 are $\sqrt{2}$ and $-\sqrt{2}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$n$-th roots are defined in a similar way, as solutions of $x^n=a$. &#xA;Example: $-2$ is cube root of $-8$ because $(-2)^3=-8$&#xA;To entice students, I like to mention that there are actually three cube roots of $-8$: $-2$ and two complex roots (which they will see in their future)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what does $\sqrt[3]{a}$ mean? There are three choices? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer: The symbol $\sqrt[3]{a}$ always denotes the real $3$-rd root of $a$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: $\sqrt[3]{-8} = -2$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The symbol $\sqrt[n]{a}$ denotes the real $n$-th root of $a$, unless there are two real $n$-th roots, in which case pick the positive $n$-th root.  Alternatively, break it up according to whether $n$ is odd or even. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added in response to a question in the comments:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The formulas $(\sqrt{x})^2 = x$ and $\sqrt{x^2} = |x|$ and often confuse students. Here's how I approach it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question to students: $(\sqrt{x})^2$ = ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By definition, $\sqrt{x}$ is the non-negative number such that $(\sqrt{x})^2 = x$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So $(\sqrt{x})^2 = x$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:  $(\sqrt{5})^2 = 5$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question to students: $\sqrt{x^2}$ = ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some students will say $\sqrt{x^2}$ = x$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's test this formula. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: $\sqrt{2^2} = \sqrt{4} = 2$. That works. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: $\sqrt{(-2)^2}$ = ?. Since $(-2)^2 = 4$, we have $\sqrt{(-2)^2} = \sqrt{4} = 2$. But the suggested formula $\sqrt{x^2} = x$ says that $\sqrt{(-2)^2} = -2$. The formula doesn't work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some student will probably now suggest $\sqrt{x^2} = |x|$. If not, you should suggest it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I explain why it works. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By definition, $\sqrt{x^2}$ is the non-negative number such that $(\sqrt{x^2})^2 = x^2$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So either $\sqrt{x^2} = x$ (because $x^2 = x^2$) or $\sqrt{x^2} = -x$ (because $(-x)^2 = x^2$). How to choose? By definition, pick the non-negative one. If $x \geq 0$, then $\sqrt{x^2} = x$. If $x &amp;lt; 0$, then $\sqrt{x^2} = -x$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's write this as a piecewise function:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\sqrt{x^2} = &#xA;\left\{&#xA;\begin{array}{ll}&#xA;x &amp;amp; \text{if } x \geq 0\\&#xA;-x &amp;amp; \text{if } x &amp;lt; 0\\&#xA;\end{array}&#xA;\right.&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does this function seems familiar? This exactly the formula for $|x|$. Therefore &#xA;$$&#xA;\sqrt{x^2} = |x|. &#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;NOTE: I am assuming here I have introduced the piecewise formula for the absolute value $|x|$.  &lt;/p&gt;&#xA;" OwnerUserId="9398" LastEditorUserId="9398" LastEditDate="2018-02-17T20:25:40.543" LastActivityDate="2018-02-17T20:25:40.543" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13530" PostTypeId="1" AcceptedAnswerId="13531" CreationDate="2018-02-04T07:35:04.277" Score="-1" ViewCount="272" Body="&lt;p&gt;I am looking for a book, which has different many different types of functions and their graphs (like, Weierstrass function, Takagi function).&lt;/p&gt;&#xA;" OwnerUserId="8881" LastEditorUserId="2083" LastEditDate="2018-02-04T17:29:06.707" LastActivityDate="2018-02-05T09:53:14.743" Title="Which book has functions and their respective graphs?" Tags="|reference-request|functions|book|" AnswerCount="3" CommentCount="11" ClosedDate="2018-02-05T16:34:09.253" ContentLicense="CC BY-SA 3.0" />
  <row Id="13531" PostTypeId="2" ParentId="13530" CreationDate="2018-02-04T18:39:23.497" Score="1" Body="&lt;p&gt;Here is one I got by Google search. It is Dover (therefore CHEAP).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0486425649&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, minor, but look at the free versions of Granville on the web (archive copies).  There is a chapter with graphs of cycloids and the like.  Little harder than &quot;this is a sin wave&quot;, but not as hard as Weirstrass stuff.  It is from a different era when people were better at learning from graphs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also recommend to look at some basic old school analtical geometry sketching.  Look for roots, critical points, asymptotes, etc.  Way I learned. Before TI time games.  Powerful understanding of relation of formula to shape--more than you get if you let the calculator &quot;grind the pigments&quot; for you.  Lot of good videos and class stuff on the net.  Old way is still very common.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some videos, but there are lot of written class examples that are good also:  &lt;a href=&quot;https://www.google.com/search?q=curve+sketching+analytic+geometry&amp;amp;client=firefox-b-1-ab&amp;amp;source=lnms&amp;amp;tbm=vid&amp;amp;sa=X&amp;amp;ved=0ahUKEwiFgdfg8ozZAhWGt1kKHfL0ALgQ_AUICygC&amp;amp;biw=1366&amp;amp;bih=654&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=curve+sketching+analytic+geometry&amp;amp;client=firefox-b-1-ab&amp;amp;source=lnms&amp;amp;tbm=vid&amp;amp;sa=X&amp;amp;ved=0ahUKEwiFgdfg8ozZAhWGt1kKHfL0ALgQ_AUICygC&amp;amp;biw=1366&amp;amp;bih=654&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9298" LastActivityDate="2018-02-04T18:39:23.497" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13532" PostTypeId="2" ParentId="13530" CreationDate="2018-02-04T20:05:52.423" Score="1" Body="&lt;p&gt;The &lt;a href=&quot;https://dlmf.nist.gov/&quot; rel=&quot;nofollow noreferrer&quot;&gt;NIST Digital Library of Mathematical Functions&lt;/a&gt; has plenty of graphs for most functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another interesting book is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;K. B. Oldham,‎ J. Myland,‎ J. Spanier, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0387488065&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;An Atlas of Functions: With Equator, the Atlas Function Calculator&lt;/em&gt;&lt;/a&gt;, Springer, 2008.&lt;/p&gt;&#xA;" OwnerUserId="5367" LastActivityDate="2018-02-04T20:05:52.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13534" PostTypeId="2" ParentId="8110" CreationDate="2018-02-04T21:25:58.430" Score="0" Body="&lt;p&gt;Have her use tables for a set of exercises.  It is sort of a halfway solution.  You don't have the total black box of the calculator (see numbers next to each other in the book).  Also, using the tables requires you to do that whole mantissa and characteristic.  You get a bit more feel for the things.&lt;/p&gt;&#xA;" OwnerUserId="9298" LastActivityDate="2018-02-04T21:25:58.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13535" PostTypeId="2" ParentId="13530" CreationDate="2018-02-05T09:53:14.743" Score="2" Body="&lt;p&gt;Most of the items below were selected based on your comment &quot;&lt;em&gt;I'm also searching at &quot;some basic old school analtical geometry sketching including roots, critical points, asymptotes&lt;/em&gt; $[\cdots]$&quot;. The Eversole thesis is probably not very well known, the &quot;Weierstrass function&quot; stack exchange question/answer is very limited in scope, and I &quot;stole&quot; the Lockwood book's .pdf file URL from @guest. The others are items I happen to have on my bookshelves that seem especially relevant to classical curve sketching methods (e.g. I have original hardback copies of Gibson/Pinkerton and Osgood/Graustein on my bookshelves). Incidentally, the El-Milick book is quite amazing but appears at present hard to locate. I managed to get a copy (&lt;a href=&quot;https://en.wikipedia.org/wiki/Philip_J._Davis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philip J. Davis’&lt;/a&gt; copy, in fact) back in the early days of internet book buying (2005, I believe), before certain rare but high “general interest” items got snapped up.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Graphs of Exotic Functions&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Ruth Eversole, &lt;a href=&quot;https://mospace.umsystem.edu/xmlui/handle/10355/16194&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A Collection of Graphs to Accompany Certain Topics in the Study of Function Theory of a Real Variable&lt;/strong&gt;&lt;/a&gt; (1913 Masters thesis)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/3058/who-first-drew-the-weierstrass-function&quot;&gt;Who first drew the Weierstrass function?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Classical Surveys of Curve Sketching&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;R. Howard Duncan, &lt;a href=&quot;https://archive.org/details/practicalcurvet00duncgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Practical Curve Tracing with Chapters on Differentiation and Integration&lt;/strong&gt;&lt;/a&gt; (1910)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Percival Frost, &lt;a href=&quot;https://archive.org/details/elementarytreati00fros&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;An Elementary Treatise on Curve Tracing&lt;/strong&gt;&lt;/a&gt; (1918) &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0486495876&quot; rel=&quot;nofollow noreferrer&quot;&gt;recently reprinted by Dover Publishers&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;William Woolsey Johnson, &lt;a href=&quot;https://archive.org/details/curvetracinginc02johngoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Curve Tracing in Cartesian Coordinates&lt;/strong&gt;&lt;/a&gt; (1884)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J. Dennis Lawrence, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0486602885&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A Catalog of Special Plane Curves&lt;/strong&gt;&lt;/a&gt; (1972)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edward Harrington Lockwood, &lt;a href=&quot;http://www.aproged.pt/biblioteca/ABookofCurvesLockwood.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A Book of Curves&lt;/strong&gt;&lt;/a&gt; (1961)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;El-Milick, &lt;a href=&quot;https://www.okazii.ro/elements-d-algebre-ornementale-el-milick-paris-dumod-1936-111-p-a171248243&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Éléments d'Algèbre Ornementale&lt;/strong&gt;&lt;/a&gt; (1936)&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Especially Comprehensive Textbooks on Analytic Geometry&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;George Alexander Gibson and Peter Pinkerton, &lt;a href=&quot;https://catalog.hathitrust.org/Record/000358096&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Elements of Analytical Geometry&lt;/strong&gt;&lt;/a&gt; (1911)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;William Fogg Osgood and William Caspar Graustein, &lt;a href=&quot;https://archive.org/details/planeandsolidan02graugoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Plane and Solid Analytic Geometry&lt;/strong&gt;&lt;/a&gt; (1921)&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2018-02-05T09:53:14.743" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13536" PostTypeId="1" CreationDate="2018-02-05T19:06:35.717" Score="6" ViewCount="181" Body="&lt;p&gt;If someone asks you &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Tell me the next number (add one) after the number &lt;br&gt; &#xA;  one million two hundred thirty-one thousand ninety-nine,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;do you known if it is a common error that the first number that comes to mind, their answer, is &quot;one million two hundred thirty-two thousand&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, do you know any document or reference for usual errors when counting numbers? All references I've found are not applicable to numbers greater than ten.&lt;/p&gt;&#xA;" OwnerUserId="7441" LastEditorUserId="8571" LastEditDate="2018-02-08T08:24:16.260" LastActivityDate="2018-02-08T08:24:16.260" Title="Adding one to numbers bigger than ten" Tags="|reference-request|students-mistakes|arithmetic|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13538" PostTypeId="1" CreationDate="2018-02-07T02:02:18.200" Score="6" ViewCount="317" Body="&lt;p&gt;Some professions have more math phobia than others. Few engineers hate math, but many &lt;a href=&quot;https://www.thestar.com/yourtoronto/education/2016/05/13/for-many-teachers-math-just-doesnt-add-up.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;teachers&lt;/a&gt; and &lt;a href=&quot;http://www.niemanlab.org/2013/11/matt-waite-how-i-faced-my-fears-and-learned-to-be-good-at-math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;journalists&lt;/a&gt; do. This means that university departments of education and journalism would likely have a high concentration of adults in need of people like, well, us. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know of other niches where lots of adults require help with math?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit: The link about journalism is a source to back up the proposition that many journalists do not like or excel at math. I &lt;em&gt;totally&lt;/em&gt; disagree with his notion that simply trying harder will work for all people and all students. Often the problem is prior knowledge and/or non-academic issues. The author could experience this first-hand by jumping into a 4th-year level math course and seeing what difference trying really hard makes.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2018-02-08T18:00:33.757" LastActivityDate="2018-04-09T19:34:47.057" Title="Concentrations of Adult Math Phobia" Tags="|arithmetic|remedial-courses|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13539" PostTypeId="1" AcceptedAnswerId="13582" CreationDate="2018-02-07T15:39:06.070" Score="0" ViewCount="184" Body="&lt;p&gt;Hi There I am new to Mathematics Educators so please bear with me if my question is not up to mark?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in mathematics and have gotten comfortable with basic proofs with Daniel J Velleman's &lt;em&gt;How to Prove it&lt;/em&gt; and am currently working through Sheldon Axler &lt;em&gt;Linear Algebra Done Right&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering given my level of Mathematical-Maturity whether it would be wise to study Linear Algebra, Number Theory, and Abstract Algebra concurrently and if so can you recommend some references? Thank you!&lt;/p&gt;&#xA;" OwnerUserId="9336" LastEditorUserId="5220" LastEditDate="2018-02-08T16:30:11.197" LastActivityDate="2018-02-12T18:24:47.787" Title="Good Combination for Self Study for a Mathematical Novice" Tags="|self-learning|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13540" PostTypeId="2" ParentId="11543" CreationDate="2018-02-07T18:47:31.723" Score="2" Body="&lt;p&gt;Just to add an historical note to the discussion: the original problem was phrased by Flaubert in a letter to his sister and was asking &quot;the age of the captain&quot;: &lt;a href=&quot;https://en.wikipedia.org/wiki/Age_of_the_captain&quot; rel=&quot;nofollow noreferrer&quot;&gt;Age of the captain&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays both in Italy and France such kind of nonsense math problem are named as &quot;age of the captain problem&quot;. &lt;a href=&quot;https://fr.wikipedia.org/wiki/Stella_Baruk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stella Baruk&lt;/a&gt;, a French researcher, published a whole book with this title: &quot;L'âge du capitaine&quot;. De l'erreur en mathématiques. Paris, Seuil, 1985. I haven't checked whether it was translated into English.There is also a math book by &lt;a href=&quot;https://en.wikipedia.org/wiki/Hans_Freudenthal&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hans Freudenthal&lt;/a&gt;: Revisiting mathematics education, Kluwer (1991), in which this kind of educational issues in math are adressed.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2018-02-07T18:47:31.723" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13541" PostTypeId="1" CreationDate="2018-02-07T18:52:40.713" Score="3" ViewCount="156" Body="&lt;p&gt;People use degrees and radians to measure angles and though degree measurement is acceptable and is widely used in everyday life, it is not in the International System of Units and mathematically it is much less convenient to use than radians (as an example, think about the derivative of the sine function if the variable is measured in degrees). Since the main purpose for the course Pre-Calculus is to get the student prepared for Calculus, in which the unit &quot;degree&quot; is seldom used, should the measurement in degrees be avoided starting Pre-Calculus, if not at an earlier course, in the first place? In my mind the measurement in degrees is only good for elementary mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same argument applies to the Common Logarithm (Logarithm using $10$ as the base). How useful is the Common Logarithm compared to the Natural Logarithm in calculus, or in mathematics in general?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="5839" LastEditDate="2018-02-12T22:56:53.047" LastActivityDate="2018-02-12T22:56:53.047" Title="Should Measurement of Angles Using Degree (and perhaps Common Logarithm as well) be Avoided in Pre-Calculus?" Tags="|terminology|precalculus|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13542" PostTypeId="2" ParentId="13541" CreationDate="2018-02-07T19:25:39.623" Score="5" Body="&lt;p&gt;No.  You lose a lot of the intrinsic value of the course.  People need to be able to deal with degrees in physics class (even one not using calculus) to deal with forces and velocities and artillery and such.  This includes both students who go on to calculus AND students who don't.  Your question ignores the value of the course for many students who never take calc, but even the calc taking ones need facility dealing with angles in multiple units.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, there is nothing to be scared about dealing with different units and it is one of the most practical important things people will need to deal with in daily life in any technical feild (to include medicine!)  Psia, psig, atmospheres, Torr, inches of mercury, inches of water, mm of Hg, Pascals, etc. effing etc!  Don't avoid this difficulty, learn to handle it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Chemistry class is great for this because even with most of the work being SI, you are forced to do conversions because of different molar masses.  But even in chem that is mostly &quot;metric&quot;, there is still an expectation to handle different gas laws.  And of course in engineering, it is unavoidable to do unit conversions.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, students are familiar with degrees for angles from pre math class awareness.  Forcing them to learn trig and such only in radians means learning two new things at once.  This is bad pedagogy.  &lt;/p&gt;&#xA;" OwnerUserId="9298" LastEditorUserId="9298" LastEditDate="2018-02-08T15:14:00.013" LastActivityDate="2018-02-08T15:14:00.013" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13543" PostTypeId="2" ParentId="13538" CreationDate="2018-02-07T19:45:39.377" Score="2" Body="&lt;p&gt;Hatred is not equal to phobia (fear). Not meant to nit, but being precise helps in the issue analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that those who are better at math, like it more (hate it less) and fear it less.  [OK, point one has some sub correlation! ;-)]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Different jobs require different amounts of math, regardless of whether the people fear/hate math.  An engineer needs a certain math skill (at least for his classes...in the field much less, but even then some in the field.)  Journalists do not.  You may say they would benefit from it, but so what...we could all benefit from learning Greek...but it's not a PRIORITY.  It's a lot more important that journalists have writing skills.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;But to actually answer your question!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one practical thing I would say for you is that many MBA students (and even some McKinsey consultants!) come back to learning basic finance and econ at age 30 or so and are intimidated by math.  It's a subset.  The engineers and scientists think the stuff is a breeze and wish it were more mathy.  But the English majors and such are intimidated by it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that some MBA schools have little summer prep programs for the weaker math students to help them out (even very prestigious schools).  And McKinsey's &quot;mini MBA&quot; (crash MBA for non MBA Ph.D.s) has a prep week of math for those with weak math backgrounds.  I would also add that these are relatively high paying environments and students are relatively well behaved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a textbook for this sort of work:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0415267846&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Basic-Mathematics-Economists-Mike-Rosser/dp/0415267846/ref=sr_1_2?s=books&amp;amp;ie=UTF8&amp;amp;qid=1518032580&amp;amp;sr=1-2#customerReviews&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(for amusement see the &quot;too easy&quot; complaint of the &quot;buckle up and read your Rudin&quot; math snob.)&lt;/p&gt;&#xA;" OwnerUserId="9298" LastActivityDate="2018-02-07T19:45:39.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13544" PostTypeId="2" ParentId="13541" CreationDate="2018-02-07T23:57:41.470" Score="1" Body="&lt;p&gt;I think it's good to realize and accept that units of measure are (generally) arbitrary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Radians are usually introduced via the length $s$ of the arc of a circular sector. This arc length will be proportional to both the radius $r$ and the angle $\theta$, so that $s = k \cdot r \theta$, for some constant $k$. We choose radians to make this constant $1$, but I couldn't really argue with anyone who thinks choosing $k = 1$ is still arbitrary (although I do feel it's the &lt;em&gt;least&lt;/em&gt; arbitrary). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While formulas and calculations generally look nicer using radians, everything could still be made to work whether we choose to divide a full rotation into $360$ units, $2\pi$ units, or any other number of units. So really, &lt;em&gt;measuring angles&lt;/em&gt; is useful, no matter what you choose as your yardstick. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But probably more valuable than appreciating a &quot;nice&quot; unit of measure, is the ability to transition between units of measure. Because we'll always have different ways of measuring the same thing, there will always be a need to reconcile these superficial differences. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is an important idea: As long as you are talking about the same thing, there's always a way to transition from one viewpoint to another, and express the same idea in equivalent ways. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Granted, I don't spend very much time in any precalc class on changing between different bases of a logarithm, or converting between degrees and radians, primarily because there's not a lot to say about either.&lt;/p&gt;&#xA;" OwnerUserId="5153" LastActivityDate="2018-02-07T23:57:41.470" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13548" PostTypeId="1" CreationDate="2018-02-08T14:02:54.807" Score="7" ViewCount="1100" Body="&lt;p&gt;We know that in writings in mathematics, variables are usually italicised. Now the question is, in handwriting on the board, how should we write the variables differently?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, say I want to write the following:    &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The following statement is true for all real number $x$:&#xA;  $$\sin^2 x+\cos^2 x=1.$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here are some possibilities:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Write the variable $x$ as x without any changes in the font.&lt;/li&gt;&#xA;&lt;li&gt;Purposefully write $x$ using a &quot;handwritten italic&quot; font.&lt;/li&gt;&#xA;&lt;li&gt;Write $x$ as x but with a different colour.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Is there any preferred/suggested practice on this?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2018-02-15T09:32:26.553" Title="How to write variables on the board, which are supposed to be italicised?" Tags="|writing|board-use|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13549" PostTypeId="2" ParentId="13548" CreationDate="2018-02-08T14:21:19.967" Score="5" Body="&lt;p&gt;I choose (1).  On the board, write an x normally. There is no distinction between roman and italic in this setting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back in the Olden Days (before personal computers) I would accosionally need to denote italic in a special way.  In instructions to a typesetter where my manuscript would be prepared for printing I would denote italic x in a special way (conventionally an underline).&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2018-02-08T14:21:19.967" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13550" PostTypeId="2" ParentId="13548" CreationDate="2018-02-08T19:20:17.987" Score="2" Body="&lt;p&gt;My preferred choice is (2), which is also probably the most used in my country.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can use (1) if you don't have to perform numerical calculations too, for otherwise the x can be confused with the multiplication cross (and avoid the dot for the multiplication symbol between numbers, because on the blackboard is barely readable).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I'd avoid (3) because frequent colour switches disrupt the writing flow.  &lt;/p&gt;&#xA;" OwnerUserId="5367" LastActivityDate="2018-02-08T19:20:17.987" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13551" PostTypeId="2" ParentId="13548" CreationDate="2018-02-08T21:57:51.620" Score="3" Body="&lt;p&gt;One letter probably causes more confusion than all the others put together:  &lt;em&gt;x&lt;/em&gt;, because of its similarity to $\times$ (multiplication as taught in schools).  What I was taught in the UK (20 years ago) and still use routinely is (with LaTeX equivalents) as follows:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0UV1Vm.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0UV1Vm.jpg&quot; alt=&quot;x, times, and X&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is also how I write &lt;em&gt;x&lt;/em&gt; in normal handwriting, but that's by no means universal.  The capital is noticeably taller than $\times$ when written on the same line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe, from what I've seen in a university physics department, that this approach is still very common (among students and academics of all ages). We tend to use $\times$ for multiplication more than mathematicians, but even if you use $\cdot$ there's always the cross product of vectors to consider.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is close to your option 2, but (i) specific to &lt;em&gt;x&lt;/em&gt;, and (ii) not really italic, just different.&lt;/p&gt;&#xA;" OwnerUserId="8180" LastActivityDate="2018-02-08T21:57:51.620" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13552" PostTypeId="1" CreationDate="2018-02-09T00:28:15.987" Score="20" ViewCount="6313" Body="&lt;p&gt;I've TAed and tutored calculus for years and of the hundreds of students I've interacted with, it is always a shock when I tell them to change the limits of integration when they do substitutions. When I tell them to do that, they are always confused and act like they never heard about that. I remember being taught that. I might think that one or two teachers might not cover it but it's standard in calculus. Even students in &lt;em&gt;Calculus 3&lt;/em&gt; are shocked. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why do students not know this? &lt;/p&gt;&#xA;" OwnerDisplayName="user5108" LastEditorUserId="5220" LastEditDate="2018-02-09T04:52:37.503" LastActivityDate="2018-02-11T22:18:12.017" Title="Why do no students know to change the limits of integration when doing substitutions?" Tags="|undergraduate-education|calculus|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13553" PostTypeId="2" ParentId="13552" CreationDate="2018-02-09T03:33:33.760" Score="25" Body="&lt;p&gt;Some students are instead taught to change the substitution variable back into the original variable before evaluating the antiderivative at the bounds.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int\limits_0^2 2x\cos(x^2) \;\mathrm{d}x = \int\limits_{x=0}^{x=2} \cos(u) \;\mathrm{d}u = \sin(u) \Big|_{x=0}^{x=2} = \sin(x^2) \Big|_{x=0}^{x=2} = \sin(4) \\\text{versus}\\&#xA;\int\limits_0^2 2x\cos(x^2) \;\mathrm{d}x = \int\limits_{u=0}^{u=4} \cos(u) \;\mathrm{d}u = \sin(u) \Big|_{u=0}^{u=4} = \sin(4) $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recall doing it the former way the first time I was taught calculus, so I'm not sure how standard changing the bounds is. I think whether or not an &lt;em&gt;instructor&lt;/em&gt; does this or instead changes the bounds is just a matter of preference (although I do worry that instructors gloss over the fact that if you don't change the bounds that they are still $x$-bounds, and so students don't really appreciate that and just change the variable back to $x$ because that's the procedure they've been taught). Now whether it is better to teach one or the other, or whether we should teach both, may be an interesting discussion. &lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-02-09T03:47:20.753" LastActivityDate="2018-02-09T03:47:20.753" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="13554" PostTypeId="2" ParentId="13548" CreationDate="2018-02-09T04:21:04.523" Score="1" Body="&lt;p&gt;If the increased clarity of &quot;math mode&quot; is worth it in print, I don't see why it wouldn't be worth it on the board. So I'm on choice (2); my current whiteboard font has a &quot;math mode&quot; $f, x, y$ that look different from normal f, x, y. I'm rendering the sentence&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$y = f(x)$ is foxy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;as approximately&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JfNLD.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JfNLD.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-02-09T04:21:04.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13555" PostTypeId="2" ParentId="13552" CreationDate="2018-02-09T05:39:31.427" Score="2" Body="&lt;p&gt;Following on Mike's example, you might insist on explicitly specifying the variables being integrated over in the limits as well as the substitution used:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\int\limits_{x=0}^{x=2} 2x\cos(x^2) \;\mathrm{d}x = \int\limits_{u=0}^{u=4} \cos(u) \;\mathrm{d}u = \sin(u) \Big|_{u=0}^{u=4} = \sin(4) \space\space$ where $u=x^2, \space du=2x\space dx$&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2018-02-09T05:48:42.337" LastActivityDate="2018-02-09T05:48:42.337" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13556" PostTypeId="2" ParentId="13552" CreationDate="2018-02-09T05:57:32.597" Score="13" Body="&lt;p&gt;Zach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a standard part of how the subject is taught and is in every textbook that I have seen.  It's also an easy thing to miss.  Unless the students are well drilled, they won't internalize it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of being exasperated, just see it as one of those little things that people need reminding of.  (Look at first year rookies in the NFL who forget you have to be &quot;touched down&quot; unlike the college game.  Even though told it, they don't always internalize it.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, realize that you as a TA are in the right end of the bell curve of calculus prowess.  Your trainees will generally not be as strong as you are. Have some empathy and just help them get better.  Don't be surprised they are not already better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nothing has changed.  It was always a tricky thing and will always be.  Every new batch of students brings new ignorance.  That is how teaching and coaching are.  &lt;/p&gt;&#xA;" OwnerUserId="9298" LastActivityDate="2018-02-09T05:57:32.597" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13557" PostTypeId="2" ParentId="13552" CreationDate="2018-02-09T06:17:19.293" Score="8" Body="&lt;p&gt;My guess is this &lt;em&gt;may&lt;/em&gt; be due to practicing indefinite integration before definite integration, where such a thing does not come up. I imagine most people learn definite integration as &quot;indefinite integration, followed by plugging in numbers&quot;, which lacks a substitution step.&lt;/p&gt;&#xA;" OwnerUserId="1423" LastActivityDate="2018-02-09T06:17:19.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13558" PostTypeId="2" ParentId="13552" CreationDate="2018-02-09T17:05:17.813" Score="6" Body="&lt;p&gt;At my institution (and I think this is common in semester systems, including the calculus AP curriculum), substitution of definite integrals is the very last subject of the first semester of calculus, so it's both the thing most likely to get shortchanged if the course runs behind schedule, and there's just less time to practice.  When students pick up in the next semester, they usually jump in with &quot;methods of integration&quot;---integration by parts, trig substitution, and partial fractions---which reinforce the method of u-substitution, but are less likely to reinforce thinking about the bounds in definite integrals.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-02-09T17:05:17.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13561" PostTypeId="2" ParentId="353" CreationDate="2018-02-10T12:39:06.563" Score="0" Body="&lt;p&gt;This may add to the detailed discussions and explanations above.&#xA;I had problems with mathematical implication until I read Keith Devlin's &quot;Introduction to Mathematical Thinking&quot;. Hopefully what follows is a faithful summary of some of the ideas in the book.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Implication is not the conditional&lt;/li&gt;&#xA;&lt;li&gt;Truth tables can help&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;1 We learn and experience that most things happen because of causality, or our interpretation of it. $A$ happens then $B$ happens. This gets expressed in different ways: $A$ happens so $B$ &lt;strong&gt;will&lt;/strong&gt; happen, &lt;strong&gt;if&lt;/strong&gt; $A$ happens &lt;strong&gt;then&lt;/strong&gt; $B$ &lt;strong&gt;will&lt;/strong&gt; happen, etc. These different expressions all get conflated and some people learn or are taught they all are represented by the expressions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;if $A$ then $B$&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A \implies B$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of the above are wrapped up in the the idea of &lt;strong&gt;implication&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We all feel comfortable, or we internalise that when $A$ happens, $B$ &lt;strong&gt;will certainly&lt;/strong&gt; happen because $A$ &lt;strong&gt;caused&lt;/strong&gt; $B$. This is implication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, we are left with an uncertainty about what happens when $A$ doesn't happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To deal with this uncertainty in mathematics we need to abandon the idea of causality and fall back to a reduced idea of the conditional.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Devlin's terms&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;implication is the conditional with causality&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or the equivalent&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;the conditional is implication without causality&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So what?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well we can &lt;strong&gt;choose to define&lt;/strong&gt; the conditional for every possible (4 ways) of combining $A$ and $B$ being True or False. I will break off point 1 here then come back.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For point 2 I understand these things in truth table terms, not linguistic terms.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;In parallel, if we think of simple logic (I am not sure of the exact terminology - &lt;em&gt;propositional logic?&lt;/em&gt;) we can combine two statements $A$ and $B$ with a variety of operations: and, or. We learn (usually passively) even if we can't express the ideas formally, about the meaning of compound statements $A$ OR $B$, $A$ AND $B$ whose truth value depends on the truth value of $A$ and $B$ and we can define these in truth tables. In more concrete terms, most people are comfortable with TRUE OR FALSE = TRUE and TRUE AND FALSE = FALSE and so on for all possible combinations.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Again, so what?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, $A \implies B$ is also a compound statement whose truth value depends upon the truth value of $A$ and $B$. We can now &lt;strong&gt;define&lt;/strong&gt; the truth table values for False $\implies$ False to &lt;strong&gt;be&lt;/strong&gt; TRUE. That's weird and uncomfortable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two ways to deal with this first uncomfortable weirdness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Either, &lt;strong&gt;claim&lt;/strong&gt; that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A \implies B$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is equivalent to &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\neg A \lor B$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;derive the truth table values and demonstrate no contradictions,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;alternatively (and this is how I deal with the situation - Devlin does not write this), we do not think about the result (truth value) of a compound statement for various &quot;input&quot; values as being &lt;strong&gt;True&lt;/strong&gt; or &lt;strong&gt;False&lt;/strong&gt;, but just ask whether the logical argument is &lt;strong&gt;valid&lt;/strong&gt; (in place of true) or &lt;strong&gt;invalid&lt;/strong&gt; (in place of false).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example if A: x = 3, is &lt;strong&gt;False&lt;/strong&gt; and B: x &amp;lt; 7 is &lt;strong&gt;False&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;then&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A \implies B$ in this case is a valid argument. In this case: &#xA;if x $\neq$ 3 then x $\geq$ 7 is &lt;strong&gt;valid&lt;/strong&gt; and hence &lt;strong&gt;true&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than the truth table values being some arbitrary definition of the &quot;meaning&quot; of $A \implies B$, the conditional &lt;strong&gt;contains and is consistent with&lt;/strong&gt; our hazier understanding of implication or if ... then, where A &quot;causing&quot; B is wrapped up in our understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So back to 1. My understanding is this&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A \implies B$ is the conditional without causality, and all written or spoken expressions such as &quot;implies&quot; or &quot;so&quot; or &quot;when&quot; or &quot;sufficent&quot; or &quot;necessary&quot; I translate into the form $A \implies B$ ($A \to B$ ???) and work from there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would recommend Devlin's book (I have no commercial interests or connection) and the free Stanford/Coursera course by Devlin that covers the same ground.&lt;/p&gt;&#xA;" OwnerUserId="7954" LastEditorUserId="7954" LastEditDate="2018-02-10T12:55:14.880" LastActivityDate="2018-02-10T12:55:14.880" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13562" PostTypeId="2" ParentId="8138" CreationDate="2018-02-10T14:53:08.410" Score="0" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The purpose is to contribute to your education by acquainting you with one of the great intellectual, scientific, cultural, and aesthetic achievements of humankind and with the kind of thinking that is used in this field.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The whole thing sounds counterproductive - you can't &lt;em&gt;make&lt;/em&gt; people have an aesthetic appreciation of something, and trying to do so by making it a required prerequisite of something else is just going to cause them to associate the subject with annoyance. Just as studying a &quot;great book&quot; in a mandatory class will leave half the audience with a distaste for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This goes double if the students are adults paying for the course, directly or through accumulated debt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It sounds like what is really needed is to disguise the course in the language and structure of &quot;History and Philosophy of Science&quot;. Teach people about the context, Newton and Leibnitz, and the problems it was developed to tackle, and have them actually do the worked examples, proofs and derivations almost as a side effect.&lt;/p&gt;&#xA;" OwnerUserId="6599" LastActivityDate="2018-02-10T14:53:08.410" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13563" PostTypeId="2" ParentId="13552" CreationDate="2018-02-10T17:03:21.587" Score="1" Body="&lt;p&gt;I found that it was safer for weaker students to evaluate the indefinite interval, back substitute, and the evaluate the limits once.  Multiple evaluations just gave them additional additional steps in which to make errors and didn't really improve their somewhat limited grasp of what it all meant.&lt;/p&gt;&#xA;" OwnerUserId="9426" LastActivityDate="2018-02-10T17:03:21.587" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13566" PostTypeId="1" CreationDate="2018-02-11T00:47:14.957" Score="2" ViewCount="196" Body="&lt;p&gt;&lt;a href=&quot;http://www.thecrimson.com/article/2006/12/6/burden-of-proof-at-1002-am/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Source:&lt;/a&gt; Logan R. Ury. &lt;em&gt;Burden of Proof&lt;/em&gt;. Dec 6, 2006. &lt;a href=&quot;https://en.wikipedia.org/wiki/Math_55&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; on Math 55. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While Harrison ultimately chooses to remain in the class, such conferences motivate several more students to drop, including the only two females: &lt;a href=&quot;https://scholar.harvard.edu/cook-stuntz&quot; rel=&quot;nofollow noreferrer&quot;&gt;[Elizabeth A] Cook ['10]&lt;/a&gt;, who had looked forward to taking the class, and Laura P. Starkston ’10.&#xA;  “The problem was that I wasn’t prepared to think that abstractly,” Cook says. “Gaitsgory pointed it out to me in our private conference. Eventually I just got the picture.” &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What exactly does 'abstract' signify here? What makes the thinking 'that abstract'? &lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user155" LastEditDate="2018-02-12T06:04:03.160" LastActivityDate="2018-02-12T06:04:03.160" Title="What exactly does 'abstract' signify in a course like Harvard's Math 55 (Honors 1st yr)?" Tags="|critical-thinking|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13567" PostTypeId="2" ParentId="13566" CreationDate="2018-02-11T01:18:57.173" Score="7" Body="&lt;p&gt;One definition of &quot;abstract&quot; is &quot; disassociated from any specific instance&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematics, we &quot;abstract&quot; by finding properties which underlie a class of examples.  For instance, the concept of a &quot;group&quot; is an abstraction of many different concrete instances of groups which were important to mathematicians:  composing symmetries of spaces (such as the isometries of the cube, or automorphisms of a vector space, or biholomorphisms of the unit disk), various bookkeeping devices (&quot;homology classes&quot; as bookkeeping device for domains of integration), etc.  In all of these cases, we saw a common structure arising.  It makes sense to study this structure &quot;disassociated from any concrete instance&quot;, or &quot;in the abstract&quot;, and so we began to study groups as their own abstract structure. &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-02-11T01:18:57.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13568" PostTypeId="2" ParentId="13552" CreationDate="2018-02-11T03:52:05.533" Score="2" Body="&lt;p&gt;I'm surprised this hasn't been mentioned yet, but it seems likely that the OPs experience is due in part to selection bias.  That is: the students that one encounters when tutoring, or in office hours, are more likely to be precisely the students that are struggling to understand something.  There are probably lots and lots of students who are learning this skill when it is taught to them; you aren't seeing them, because they don't need to come to you for help.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-02-11T03:52:05.533" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13569" PostTypeId="2" ParentId="13552" CreationDate="2018-02-11T20:01:47.657" Score="4" Body="&lt;p&gt;I suspect that students don't learn to change bounds of integration well because they have no motivation to do so. Whether we agree with them or not, it is evidently simpler for them conceptually to evaluate a definite integral by first evaluating the corresponding indefinite integral and then applying the fundamental theorem to that result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a teacher, there are a couple of things that I do to combat this. First, I devise problems where the whole point of the problem is to change bounds of integration in the first place. For example: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Express $\displaystyle \int_{-2}^{3} f(2x+1)\,dx$ as a definite integral involving $f(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Evaluate $\displaystyle \int_0^{\pi} \sin\left(e^{\sin(x)}\right)\cos(x)\,dx$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, I try to impress upon them that there are plenty of practical situations where this translation from one definite integral to another arises. For example, when exploring probability theory (a natural application of integration in Calc II), we often need to translate a normal integral to a standard normal integral. This amounts to showing that&#xA;$$\int_a^b e^{-(x-\mu)^2/(2\sigma^2)} dx = \int_{(a+\mu)/\sigma}^{(b+\mu)/\sigma}  e^{-x^2/2} dx.$$&#xA;Thus, a rule that many of them learn in elementary statistics arises as a change of bounds in a definite integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As another example, that's nice if you do numeric integration, you might show that&#xA;$$\int_0^1 \frac{\sin(1/x)}{x} \, dx = \int_1^{\infty} \frac{\sin(x)}{x} \, dx.$$&#xA;The point is that the second integral is more palatable from the point of view of numerical integration (rather than from the point of view of symbolic integration, as calc students are accustomed to). The reason is that the higher derivatives of the integrand don't explode over the domain of integration the way they do for the first.  You can even illustrate this using a &lt;a href=&quot;https://sagecell.sagemath.org/?z=eJwrKMrMK1HIK81NLcpMTsyJB_JS04sSczSKM_M0DPUrNPUrdBQMdAw1ebkK8CmFKDTUMTQwMNAEABDjHFU=&amp;amp;lang=sage&quot; rel=&quot;nofollow noreferrer&quot;&gt;little sage code&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;print numerical_integral(sin(1/x)/x, 0,1)&#xA;print numerical_integral(sin(x)/x, 1,1000)&#xA;&#xA;# Out: &#xA;# (-1.9426263726635902, 42.80850345186512)&#xA;# (0.6241500516015874, 5.034861416675085e-14)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The first computation is completely wrong and has an absurd error bound. The second computation is (not surprisingly) correct to three decimal places.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that, in all four of these examples, the integrand &lt;em&gt;cannot&lt;/em&gt; be integrated symbolically; a change of bounds is the whole point.&lt;/p&gt;&#xA;" OwnerUserId="4912" LastEditorUserId="4912" LastEditDate="2018-02-11T22:18:12.017" LastActivityDate="2018-02-11T22:18:12.017" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13570" PostTypeId="1" CreationDate="2018-02-11T20:04:25.793" Score="4" ViewCount="133" Body="&lt;p&gt;At a local Math Circle, I loved some problems worked out through a hinged mirror to illustrate symmetry. I bought a hinged mirror from hand2mind.com, and am looking for some material, ideally books about symmetry that encourage use of the mirror, but .pdf diagrams as well. Does anybody have any pointers? Thanks.&lt;/p&gt;&#xA;" OwnerUserId="9384" LastActivityDate="2018-02-19T03:18:01.480" Title="Books and worksheets on symmetry" Tags="|geometry|examples|applications|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13571" PostTypeId="1" AcceptedAnswerId="13579" CreationDate="2018-02-11T21:40:07.230" Score="3" ViewCount="133" Body="&lt;p&gt;Is there software that allows to combine slides and drawing on top of your slides (like Microsoft Paint)?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found &lt;a href=&quot;https://www.wikihow.com/Draw-Using-PowerPoint&quot; rel=&quot;nofollow noreferrer&quot;&gt;this website&lt;/a&gt; which says that you can draw (just like MS Paint) on top of your own PowerPoint slides and verified this does work.  But I was wondering if there are other software that do this (perhaps better)? &lt;/p&gt;&#xA;" OwnerUserId="9435" LastEditorUserId="8571" LastEditDate="2018-02-15T19:34:58.360" LastActivityDate="2018-02-15T19:34:58.360" Title="What software tools allow an educator to display slides but also draw on the slides like MS Paint?" Tags="|software|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13572" PostTypeId="2" ParentId="13571" CreationDate="2018-02-11T21:52:56.333" Score="0" Body="&lt;p&gt;Print the slides on acetate.  Show them on an OHP.  Use a grease pencil to draw on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not just trolling you.  It's actually more interactive with the students.  Removes some technology barrier.  [And it accomplishes mission of your question.]&lt;/p&gt;&#xA;" OwnerUserId="9428" LastActivityDate="2018-02-11T21:52:56.333" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13573" PostTypeId="2" ParentId="13570" CreationDate="2018-02-11T23:52:24.963" Score="1" Body="&lt;p&gt;It depends what level you want. There are mirror activities for young kids in &lt;a href=&quot;https://naturalmath.com/moebius-noodles/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moebius Noodles&lt;/a&gt; (by my publisher).&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-02-11T23:52:24.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13574" PostTypeId="1" CreationDate="2018-02-12T06:55:29.837" Score="8" ViewCount="362" Body="&lt;p&gt;&lt;a href=&quot;https://www.ox.ac.uk/admissions/undergraduate/courses-listing/mathematics-and-philosophy?wssl=1&quot; rel=&quot;noreferrer&quot;&gt;An undergraduate double major in math and philosophy&lt;/a&gt;, my cousin learns philosophy well, but she's failing her &lt;a href=&quot;https://www.maths.ox.ac.uk/members/students/undergraduate-courses/teaching-and-learning/handbooks-synopses&quot; rel=&quot;noreferrer&quot;&gt;abstract proof-based courses&lt;/a&gt; (Group Theory, Real Analysis and Linear Algebra). &#xA;Presume that she studies each subject at least 3 hours daily 6 days of the week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why can she learn philosophy, but not abstract math?  What can explain her snag: she can calculate or prove something only if it nears something previously solved? On her exam, she failed to calculate, prove, or construct new counterexamples for, anything unseen. &lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2018-02-14T02:18:02.930" Title="What explains why a student can learn philosophy well, but fail in abstract math?" Tags="|critical-thinking|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13578" PostTypeId="2" ParentId="13574" CreationDate="2018-02-12T15:14:38.843" Score="2" Body="&lt;p&gt;Obviously it's hard to tell from a distance. Here are some explanations:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;She may be more motivated to think about the material in the philosophy course. This may be because she can relate to the topics that they analyze. Like one commenter suggests, if you do not know the basic examples or motivation for group theory, it can be hard to feel motivated to learn the methods. (An intervention that may help here is talking about symmetries of Platonic solids; e.g. show her how to quickly count the number of elements in these groups, how to prove some isomorphisms like $S_4$ isomorphic to group of rotations of the cube, $S_3$ isomorphic to symmetries of the triangle, etc. A fun exercise is to find subgroups of $S_4$ using stabilizers of subsets of the cube.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;She may have some degree of math anxiety. See the paper &lt;a href=&quot;https://hpl.uchicago.edu/sites/hpl.uchicago.edu/files/uploads/Maloney%252c%20E.A.%252c%20Schaeffer%252c%20M.W.%252c%20%26%20Beilock%252c%20S.L.%252c%20%25282013%2529.%20Mathematics%20anxiety%20and%20stereotype%20threat.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics anxiety and stereotype threat: shared mechanisms,&#xA;negative consequences and promising interventions&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's possible that she has never learned how to study for a mathematics course at this level. There are books and guides for this. Learning group theory requires doing problems every day for a while; it's a new language. Similarly with analysis.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8029" LastEditorUserId="5220" LastEditDate="2018-02-12T16:24:20.730" LastActivityDate="2018-02-12T16:24:20.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13579" PostTypeId="2" ParentId="13571" CreationDate="2018-02-12T17:04:36.470" Score="1" Body="&lt;p&gt;A couple of solutions come to mind:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A solution similar to guest's answer is to print your slides out on paper, then use an overhead camera to project them.  You can annotate the paper using normal human tools (like pens and pencils, rather than a stylus for a tablet, or a greasepaint pencil or wet-erase markers).  I think that this is a better option than an overhead projector—my experience with overheads is that they are finicky to focus (though this can be true of cheap document cameras, as well), hard to read in a large room, block the view of students behind the overhead (since the overhead itself is tall &lt;em&gt;and&lt;/em&gt; the instructor must stand next to it in order to write), and kind of loud (they get hot and have loud fans).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are in a classroom with a projector and a whiteboard, and the projector happens to project onto the whiteboard, the solution is (I hope) obvious—project your slides onto the whiteboard, and annotate them there.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The solution that I have been using recently is a tablet and &lt;a href=&quot;http://gingerlabs.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Notability&lt;/a&gt;.  Notability is not perfect, but it does a pretty good job.  The general workflow is to create your slides and save them as a .pdf, then import them into Notability.  As you go through your presentation / lecture, you can annotate the slides (the toolset is somewhat limited—you have pens and highlighters in the form of various sized circles to work with), and the annotated work can be exported and distributed to students.  Notability works best on a tablet with a stylus, but it also runs on desktop machines and works reasonably well with a mouse and keyboard.  On the downside, it only runs under Apple operating systems (iOS or Mac OS), so far as I know.&lt;/p&gt;&#xA;&lt;p&gt;While this is not entirely on topic, I have also been using a combination of Notability and &lt;a href=&quot;http://zoom.us&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zoom&lt;/a&gt; to conduct office hours this quarter.  This seems to work quite well, especially when there are large numbers of students—everyone can see what is going on equally well, and my tiny office doesn't get crowded.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are using Microsoft's software already, I think that they have something that plays nice with their stuff (a quick Googling indicates that the software is called &lt;a href=&quot;https://www.onenote.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;OneNote&lt;/a&gt;.  I haven't used this software, but it seems to fit the bill.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-02-12T17:04:36.470" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13580" PostTypeId="1" AcceptedAnswerId="13583" CreationDate="2018-02-12T17:50:02.973" Score="6" ViewCount="1209" Body="&lt;p&gt;How can I teach calculus effectively to non-math students (chemical engineering students, for example)? Sorted by priority, the problems I'm experiencing are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Students cannot keep their voice down.&lt;/strong&gt; This hinders class productivity. Something that can be explained in 60 minutes takes 90 minutes. Frequently telling them to &lt;em&gt;quiet down&lt;/em&gt; is not working. Threatening to lower their grades is also not working.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Low grades.&lt;/strong&gt; What has been done: giving homework each week such that next week the students can write solutions in front and get points.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The main Lecturer gives Math-major level (sometimes also olympiad level) problems in the exam, while they are not from math background.&lt;/strong&gt; (I only assist in explaining and practicing on problems and solutions)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Poor high-school math education.&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These problems has been troubling for two straight semesters. Their grades are low compared to past years calculus students. If it is important to know, the class is a first-year calculus course (techniques of integration, etc). This is the first time I've taught at the University level, and it's my second semester.&lt;/p&gt;&#xA;" OwnerUserId="9439" LastEditorUserId="5220" LastEditDate="2018-02-13T03:20:50.657" LastActivityDate="2018-02-18T17:59:24.763" Title="How to teach calculus effectively to non-math (and noisy) students?" Tags="|calculus|classroom-management|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13581" PostTypeId="2" ParentId="13574" CreationDate="2018-02-12T18:18:59.587" Score="7" Body="&lt;p&gt;&lt;strong&gt;I was also a double major in mathematics and philosophy as an undergraduate&lt;/strong&gt; (at a state university in the U.S.). I think that both are incredibly important, and I'm happy to have both under my belt. Actually: philosophy was my primary major, mathematics secondary. (This confused a lot of my instructors in higher-level math courses looking at the course registration roster.) Now I'm a lecturer in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The truth is, anything in a STEM discipline is simply harder and less forgiving of errors than in courses in the humanities, and this is one example of that. STEM courses like math have problems which are practical and have answers that are clearly correct, or clearly incorrect. In graduate school I once had a four-page proof returned with simply the word &quot;No&quot; in red at the top. The humanities, like philosophy, generally accept a wide variety of answers as long as some background justification is given; and grading is generally much more generous (for a variety of institutional and supply/demand reasons). There's no paper I could possibly write in philosophy for which the response is simply &quot;No&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, proof-based math courses are the big leagues. Some people do advanced proof math competitions all through their youth and teenage years before turning professional. Many colleges have a sophomore-level course such as &quot;Introduction to Proof&quot; or &quot;Bridge to Advanced Mathematics&quot; or the like. Fortunately, I had a very good such course where I went to college (and there the professor was &lt;em&gt;delighted&lt;/em&gt; to have an apparent philosophy major on the first day), and it at least gave me a fighting chance in later courses like Abstract Algebra and Analysis (love the former, not so much the latter). Maybe such a transition course does not exist at Oxford? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand: Yes, you can BS philosophy. Not that I'd personally want to. Some might say that's the very essence of philosophy -- but, of course, others would disagree. Hence the point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider also the &lt;a href=&quot;https://academia.stackexchange.com/questions/76531/do-academic-integrity-pledges-work&quot;&gt;comments to this question&lt;/a&gt; which in turn &lt;a href=&quot;https://matheducators.stackexchange.com/questions/11396/what-fraction-of-the-population-is-incapable-of-learning-algebra/&quot;&gt;spawned this question&lt;/a&gt;. Everywhere we turn, at every level, no matter how much anyone doesn't want it to be, math winds up being the bottleneck/filter that distinguishes between people who can really do it, and those who can't. I personally refer to this as the &lt;strong&gt;brutal honesty of math&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: If it's true that someone is struggling with the transition to proof-based math courses, and lacks a transition course, then my top recommendation would be to look at Richard Hammack's &lt;a href=&quot;http://www.people.vcu.edu/~rhammack/BookOfProof/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book of Proof&lt;/a&gt; (free, open educational resource) -- ideally before their next course starts.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-02-13T15:05:44.037" LastActivityDate="2018-02-13T15:05:44.037" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13582" PostTypeId="2" ParentId="13539" CreationDate="2018-02-12T18:24:47.787" Score="1" Body="&lt;p&gt;If you are also quite new in Mathematics, I would recommend the book :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;The Art and Craft of Problem Solving&quot;&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;by &lt;strong&gt;Paul Zeitz&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although it does not talk about specific math topic, it teaches you how to &quot;behave&quot; effectively and strategically when encounter math problems and ideas. This is useful for strengthen the basic, and therefore it can be applied when you learn other specific math topics.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In addition, you may want to learn Linear Algebra or Number theory first, before learning Abstract Algebra. The former is more concrete than the latter, and &lt;em&gt;Abstract Algebra contains both of them.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;A final addition : Do not study in a hurry. I may refer to a quote &lt;strong&gt;&quot;A pupil in such hurry, learns slowly.&quot;&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="9439" LastActivityDate="2018-02-12T18:24:47.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13583" PostTypeId="2" ParentId="13580" CreationDate="2018-02-12T19:00:55.633" Score="8" Body="&lt;p&gt;&lt;strong&gt;Precomment 1:&lt;/strong&gt;   You mention under point 3, testing, that you are the assistant, not the course leader.  So, to be practical, many &quot;fixes&quot; in the class will not be things you (as an individual, versus the archetypal you) can implement.  We can discuss them in general (for other instructors).  And you can think how to do better when you are in charge.  But don't kill yourself over not being able to fix everything.  That's just life and the World.  Work on what is in your control--plenty to do there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Precomment 2:&lt;/strong&gt;  Teaching engineering students is so much more prevalent than teaching math students that you should mentally think of it as the base case.  Not &quot;how to do it different from teaching math students&quot;.  But how to do it.  Then worry about how to change it for a math majors class!  And if the class is mixed (likely), the math majors will be a tiny minority, and you should ignore special things for them.  Don't worry, they will have analysis as a majors course, later.  And they have other classes like chem or English to keep them occupied also.  Not the end of the world if they have a standard calc course.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;First bullet, noise:&lt;/strong&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't make threats, you don't carry out.  If you say you will lower grades, then just do it.  Really, even a very tiny point deduction (1 point from semester of 500+ points) will change student behavior.  Have seen it with lab google rules in chem lab.  Don't even have to raise voice or debate.  Just let them know the sword fell.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another idea is just to eject a noisy student from class.  Only one, very occasionally, will be dramatic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other than that, I recommend to discuss behavior at beginning of class (too late now for first lesson.  But make it a point of a few minutes at start of a class, now.)  The key thing to emphasize is that it is not fair to classmates and that it is wasting everyone's time.  Don't open to debate or ask for suggestions.  Just tell them, it's a problem and to stop it.  You don't have to be angry or make threats.  Just be direct and in control.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I think having a loud voice is helpful.  In the Navy it is called &quot;command voice&quot;.  It doesn't involve screaming, is not unpleasant in that matter--it is just using projection like an actor or a singer.  (It's easy...just try it.)  In addition, be animated, move around, command presence and keep it interesting!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Second bullet, low grades:&lt;/strong&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't completely understand your point, but I think you are saying the students are performing poorly on tests and you are using homework to help increase the scores.  (I bet TA gets stuck grading those, haha.)  In general, you need to try to make the tests more reasonable (point 3) and ALSO to improve the level of ability so the students do better on the exams.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Different points of view on grading homework (pro/con) but if you are going to do it, I would not do it for the purpose of raising grades in the way you are here.  Instead do it to drive practice and drill (for the purpose of improving test performance).  (Probably a low point value, more frequent turnins--every class for instance, and rapid returns.)  Really, I think it becomes too much of a logistical hassle, especially in college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also find college classes have too few tests (major stakes tests only, which encourages goofing off and cramming).  If you can do more tests during the semester, better.  Yes, this loses some lecture time, but it is highly valuable practice--not like the lectures are perfect anyways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Third bullet, exams too hard and mathy:&lt;/strong&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cut that out.  Use a reasonable text like Thomas Finney and then make the test questions close to the book.  These engineers need to get experience taking derivatives and doing integrals and lots of algebra as well.  That helps them with their homework in fluids class and with following derivations in engineering majors classes and textbooks. Spending time on baby real analysis is a waste if they don't have the basics yet.  Same thing with super hard trick problems if they don't have the basics yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One little practical trick is to literally make the exam questions from the unassigned homework problems.  (It encourages extra drill by the students.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Richard Feynman:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;So, this guy comes into my office and asks me to try to make everything straight that I taught him, and this is the best I can do. The problem is to try to explain the stuff that was being taught. So I start, now, with the review.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I would tell this guy, “The first thing you must learn is the mathematics. And that involves, first, calculus. And in calculus, differentiation.”&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Now, mathematics is a beautiful subject, and has its ins and outs, too, but we’re trying to figure out what the minimum amount we have to learn for physics purposes are. So the attitude that’s taken here is a “disrespectful” one towards the mathematics, for sheer efficiency only; I’m not trying to undo mathematics.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;What we have to do is to learn to differentiate like we know how much is 3 and 5, or how much is 5 times 7, because that kind of work is involved so often that it’s good not to be confounded by it. When you write something down, you should be able to immediately differentiate it without even thinking about it, and without making any mistakes. You’ll find you need to do this operation all the time—not only in physics, but in all the sciences. Therefore differentiation is like the arithmetic you had to learn before you could learn algebra.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Incidentally, the same goes for algebra: there’s a lot of algebra. We are assuming that you can do algebra in your sleep, upside down, without making a mistake. We know it isn’t true, so you should also practice algebra: write yourself a lot of expressions, practice them, and don’t make any errors.&#xA;Errors in algebra, differentiation, and integration are only nonsense; they’re things that just annoy the physics, and annoy your mind while you’re trying to analyze something. You should be able to do calculations as quickly as possible, and with a minimum of errors. That requires nothing but rote practice—that’s the only way to do it. It’s like making yourself a multiplication table, like you did in elementary school: they’d put a bunch of numbers on the board, and you’d go: “This times that, this times that,” and so on—Bing! Bing! Bing!&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[IOW, have some sympathy to building the muscles needed for getting them through chemE majors courses.  Rates of reactions, bleed and feed mixing, fractional distillation, etc.  You don't need to do examples using their course concepts...that makes it harder, since THEY don't know them yet and it is all word problems.  But some sympathy and awareness of what is coming at them--and how calc class helps them--will help inform your teaching.  You may even find some motivation leaking through to your students...]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Fourth bullet, poor HS prep:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can't fix HS prep.  You also can't fix native intelligence.  You play with the cards given.  But I actually think things are not as bad as you think.  In general HS students are getting more math prep than several decades ago.  College algebra and trig are common in HS as is some calculus exposure itself. Yes, there are some things that may have changed for the worse (TI time, atrophy of analytic geometry).  But in general kids are getting more HS math exposure.  Also, we send a lot of kids to college now.  But if they are chemEs, they have the brains to get through a calc class if they buckle down and if it is taught in a practical (i.e. pedagically kind) manner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One practical suggestion is SHOW THE ALGEBRA.   I had this tiny crit of a comment for Mike Pierce where he did a great job showing with a u substitution, the endpoints change (in writing), but he glided over the change of the differential.  I.e. du=2xdx if u=x².  Really these kids can get a lot of practical exposure and practice in algebraic manipulation in calc class.  I was a kid who had poor algebra background, but all the drill of calculus class ended up building up my algebra chops also.  Didn't really need a remediation, just doing the homework problems (a gazillion of them) ended up building my algebra muscles as I did calculus.  So show the algebra.  When you are Carl Bender teaching advanced methods for seniors and grad students, you can glide over some things (but I actually still find him relatively kind).  But if you are teaching calc 101?  Show the manipulation!&lt;/p&gt;&#xA;" OwnerUserId="9428" LastEditorUserId="1147" LastEditDate="2018-02-18T17:59:24.763" LastActivityDate="2018-02-18T17:59:24.763" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13584" PostTypeId="1" AcceptedAnswerId="13586" CreationDate="2018-02-12T21:53:16.780" Score="9" ViewCount="202" Body="&lt;p&gt;I am scheduled to teach a graduate-level course in engineering whose basis is in the solution of ODE’s and PDE’s, and thus is about halfway between a math course and an engineering course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We introduce methods for solving these equations (perturbation expansions, self-similarity, series expansions, and so on) that they may not have had in taking ODE’s as an undergraduate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Between that and the discrepancies in undergraduate curricula, are there guidelines to follow in figuring out how much material I should review from undergrad ODE’s in the graduate-level course?&lt;/p&gt;&#xA;" OwnerUserId="5839" LastEditorUserId="5839" LastEditDate="2018-02-13T00:27:29.707" LastActivityDate="2018-02-13T00:41:47.843" Title="How much prior math should I review in teaching a graduate-level course?" Tags="|graduate-education|differential-equations|review|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13585" PostTypeId="2" ParentId="13584" CreationDate="2018-02-12T23:10:24.487" Score="-1" Body="&lt;p&gt;Review basic ODEs and PDEs (Tenenbaum and Farlow, or the two Schaum's or Kreyszig.)  Take a pre-look at Bender and Orszag text and at the Carl Bender videos on YT.  That's enough.  Umm...little bit of Taylor and McLaurin series might be wise if that has been a while.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  This is a learning math question, not teaching math, but in before the close!&lt;/p&gt;&#xA;" OwnerUserId="9428" LastActivityDate="2018-02-12T23:10:24.487" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13586" PostTypeId="2" ParentId="13584" CreationDate="2018-02-13T00:41:47.843" Score="8" Body="&lt;p&gt;Even for math grad students, I'd forcefully review much more than many traditions seem to indicate. That is, I would &lt;em&gt;not&lt;/em&gt; presume perfect recall of the standard curriculum, especially either in detail or in &quot;big picture&quot;. Further, in my experience, even very smart people with unusually good memories greatly benefit from repetition. It's not &quot;one and done&quot;, ... except for those occasional potentially misleading situations where one's thought processes had arrived at a juncture ripe for an epiphany.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet further, I've found that it is unfortunate to too aggressively assume that there's surely little need for review... after finding that people have supposedly assimilated various fancy things, but cannot do simple things (that are the background and/or foundation for the fancy things). I'm not advocating &lt;em&gt;not&lt;/em&gt; mentioning fancier things, but, rather, advocating going back-and-forth between more sophisticated things and their more elementary antecedents (and examples).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, I am no longer embarrassed about talking about as-simple-as-possible motivating/explanatory examples.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-02-13T00:41:47.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13588" PostTypeId="2" ParentId="13408" CreationDate="2018-02-13T14:58:56.547" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;First question - is that even correct?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;“$\times$” denotes another similar operation, the Cartesian product of sets. For finite sets, cardinality maps the Cartesian product of sets to the multiplication of natural numbers. So it is definitely correct, I mean, it is a sensible choice of a sign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Denoting multiplication by juxtaposition is the worst choice. Every operation is denoted by a sign, so why the multiplication is “better” than the others? Well, the power is denoted by the superscript, but this is also a bad notation because a small font is less legible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many mathematical notations are hardly distinguished in handwriting. I suppose that this is a problem of handwriting.&lt;/p&gt;&#xA;" OwnerUserId="7992" LastActivityDate="2018-02-13T14:58:56.547" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13589" PostTypeId="2" ParentId="13570" CreationDate="2018-02-13T16:22:37.570" Score="1" Body="&lt;p&gt;You could do platonic solids via their reflection groups. There is some material here:&#xA;&lt;a href=&quot;http://www.cutoutfoldup.com/818-platonic-solids-kaleidoscope.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.cutoutfoldup.com/818-platonic-solids-kaleidoscope.php&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's also Conway et. al.'s &lt;em&gt;The Symmetries of Things&lt;/em&gt; which has some mirror related material in the beginning. However, I think that would be something to build material out of rather than use directly.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-02-13T16:22:37.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13593" PostTypeId="2" ParentId="13570" CreationDate="2018-02-13T19:06:04.953" Score="2" Body="&lt;p&gt;I've done wonderful activities from &lt;a href=&quot;https://store.aimsedu.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;AIMS&lt;/a&gt;, including &lt;a href=&quot;https://store.aimsedu.org/item/da10119/make-a-kaleidoscope/1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;making kaleidoscopes&lt;/a&gt;.  You can search &lt;a href=&quot;https://store.aimsedu.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;their site&lt;/a&gt; for other PDF's and books by using the search words mirrors or symmetry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math-salamanders.com/symmetry-activities.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Salamanders&lt;/a&gt; also have some free worksheets about symmetry and mirrors.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2018-02-13T19:06:04.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13594" PostTypeId="1" AcceptedAnswerId="13596" CreationDate="2018-02-13T23:31:41.207" Score="33" ViewCount="1754" Body="&lt;p&gt;This question is motivated by student responses to homework and quiz problems I have recently posed in an undergraduate real analysis course. I will share some examples and observations first, to present the main ideas, before posing a question.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A homework problem asked students to consider an arbitrary set $A\subseteq\mathbb{R}$ that is nonempty and bounded below. From that set, define $B=\{b\in\mathbb{R}\mid b\text{ is a lower bound for }A\}$. The students were asked to prove that $\sup(B)$ must be equal to $\inf(A)$. I noticed two kinds of mistakes. They were not widespread, but they were common enough that I made comments about them during class time.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Proof by example,&lt;/em&gt; such as: &quot;If $A=(2,\infty)$ then $B$ must be $(-\infty,2]$ and notice that $\inf(A)=2$ and $\sup(B)=2$.&quot; This is a gross misunderstanding of &lt;strong&gt;arbitrary&lt;/strong&gt; versus &lt;strong&gt;specific&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Referring to &quot;$b$&quot; without instantiation,&lt;/em&gt; such as: &quot;$b$ is a lower bound for $A$ so all $a$s are upper bounds for $B$.&quot; This contains a kernel of truth (indeed, &lt;em&gt;every&lt;/em&gt; element of $B$ is a lower bound for $A$) but it demonstrates a misunderstanding of a &lt;strong&gt;universal assertion&lt;/strong&gt; versus a &lt;strong&gt;specific instance&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A quiz problem asked the following: &quot;Suppose $A\subseteq\mathbb{R}$ is nonempty and bounded above. Let $s=\sup(A)$, and let $\varepsilon &amp;gt; 0$ be arbitrary. What can we say about the interval $(s-\varepsilon,s]$? How many elements of $A$ lie in that interval? None? 1? At least 1? Infinitely many? What can we say &lt;em&gt;with certainty&lt;/em&gt;? Consider illustrating with examples.&quot;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I expected all students to at least point out that there cannot be &lt;em&gt;zero&lt;/em&gt; elements of $A$ in the interval $(s-\varepsilon,s]$; if so, that would violate the fact that $s$ is the &lt;em&gt;least&lt;/em&gt; upper bound for $A$.&lt;/li&gt;&#xA;&lt;li&gt;I expected the better students to further point out that $s$ may be the &lt;em&gt;only&lt;/em&gt; element of $A$ in that interval. For example, if $A=\{2\}$ is a singleton, then this is true regardless of $\varepsilon$.&lt;/li&gt;&#xA;&lt;li&gt;I also expected the better students to further point out that there may be &lt;em&gt;infinitely many&lt;/em&gt; elements of $A$ in that interval. For example, if $A=[0,2]$, then this is true regardless of $\varepsilon$. &lt;/li&gt;&#xA;&lt;li&gt;Instead, &lt;em&gt;more than half of the class&lt;/em&gt; submitted an answer that amounted to nothing more than saying &quot;there are infinitely many real numbers in any non-trivial interval.&quot; Zero consideration or even literal mention of that arbitrary set $A$. They treated this as if the original question was: &quot;Does an interval have infinitely many elements?&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This worries me for several reasons. One of the main reasons is that I think I have an &quot;expert blind spot&quot; regarding the ability to consider and work with an arbitrary mathematical object because that practice is so natural to me. I'm truly baffled why a student would consider an example as a reasonable solution to problem #1, yet I'm sure there was a time in my educational career where I would have easily made the same error without yet understanding the big issue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But problem #2 worries me the most. Is there something about the way I posed the question that obscured its intent so much that students could have genuinely thought I was cryptically assessing them on their knowledge of the infinitude of the reals? I think that they lack strong skills in mentally working with an arbitrary set, that they need to consider specific examples to make mental progress, and that they are not yet in the habit of responding to a question like this by &lt;em&gt;creating their own examples&lt;/em&gt; to consider while reflecting on the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question: How can I, as an instructor, help students to work with problems like the examples above, knowing that the students do not have good, practical habits for working with arbitrary objects? Beyond simply telling them to do so in the moment, how can I encourage them to create and test examples for themselves, to not &quot;over/under assume&quot; about what they're given, or otherwise just to correctly interpret given information about an arbitrary object?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Upon reflection, I seem to have addressed this in the past by just trying to &quot;model good behavior&quot; when presenting in-class examples and when working with students one-on-one. However, I wonder whether there are particular activities I could use, or problem types to assign and assess, that would better promote the kind of thinking and behavior that I want my students to develop.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good answer will contain suggestions for activities or assessment tools. If there is any research about this, I would love to know about it, as well. If it helps narrow the scope, I am especially interested in undergraduate math majors learning to write proofs in their advanced courses (e.g. real analysis, linear algebra, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Meta comment:&lt;/em&gt; I could not find a good tag that properly encapsulates the main issue of this question. Would &quot;abstraction&quot; be a reasonable tag to create? This question is mostly about a student's ability to abstract from specific cases to general concepts, and I'm sure there are and will be other questions related to that ability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Followup comment:&lt;/strong&gt; I want to give a shoutout to #2 above as a really good problem to give students when learning about suprema before learning about sequences. Today, in my class, we proved the Monotone Convergence Theorem by defining the proposed limit to be the supremum of the set of the terms of the sequence. Given an arbitrary $\varepsilon&amp;gt;0$, I asked the students: &quot;What can we say about the interval $(s-\varepsilon,s]$?&quot; Some students laughed upon realizing this is precisely the quiz question from a few weeks ago and said, &quot;There must be &lt;em&gt;at least one&lt;/em&gt; element in there!&quot;&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2018-02-26T22:57:46.677" LastActivityDate="2020-08-27T12:26:12.890" Title="Inability to work with an arbitrary mathematical object" Tags="|undergraduate-education|mathematical-pedagogy|proofs|real-numbers|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13596" PostTypeId="2" ParentId="13594" CreationDate="2018-02-14T16:25:09.620" Score="22" Body="&lt;p&gt;I'll focus on question 2 from a perspective of &quot;maybe the right thing to think about is: what happens in the students' minds while they read this question?&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;When you say &quot;Suppose $A⊆R$ is nonempty and bounded above,&quot; the following thing happens in my mind &lt;strong&gt;before I continue reading&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/N06vq.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/N06vq.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This picture takes up very little working memory for me because I am familiar with how I can mess with it. This may not happen in their minds. They may think of $A$ as a set that follows arcane rules. After all, the set is arbitrary. It may feel pointless to think about the set at this point. Have the students practiced how to read a quantified phrase like this? Maybe you are assuming they know this from a previous course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you say &quot;Let $s=sup(A)$, and let $ε&amp;gt;0$ be arbitrary.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My new picture is this, and it's important that even though $\epsilon$ is a real number, I haven't placed it on the real line, because the use of the letter $\epsilon$ implies to an expert that this would be the wrong way to think about this specific real number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YGShj.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YGShj.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you say &quot;What can we say about the interval $(s−ε,s]$?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are deep into multiple objects at this point. A weaker student's picture may look as poor as this right now:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TDfUG.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TDfUG.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student is suspicious that they have missed something (and they have; they didn't notice that the way $s$ was introduced is important) but their mental picture is highly deficient, so they said the best thing they could with what they were working with: &quot;there are infinitely many real numbers in any non-trivial interval.&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Basically I agree those answers are terrible. But &lt;strong&gt;investigating more explicitly with the students what should happen in their brains when they read a sentence with multiple quantifiers&lt;/strong&gt; would be the in-class exercise I suggest. They may not realize that every time a variable is quantified, they should now be able to think about it or even have a good schematic picture of it.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-02-14T16:25:09.620" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13597" PostTypeId="2" ParentId="13594" CreationDate="2018-02-14T19:06:20.547" Score="3" Body="&lt;p&gt;If you buy into the notion of &quot;learning outcomes&quot; over &quot;content,&quot; then maybe &quot;develop ability to work with arbitrary mathematical objects&quot; is one of the learning outcomes of this class. Your colleagues in the department will understand this, even if administrator-level assessment gurus don't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my mind this developmental learning goal is one of the reasons we teach analysis. It's just messy and hard, and your question accents this. This is a challenging and important class. Some obvious mechanical checks: does the class have a helpful prerequisite, for example a course in linear algebra with some abstraction and where some proof is introduced, and is class size reasonable? Instructor time with individual students is crucial in this course, so I don't think class size should exceed 15. Something like 10 students is better. Then the usual list of strategies should be employed depending on instructor judgement: students presenting solutions/proofs to one another in class, structured study groups, rewriting proofs and problem solutions after initial review by instructor, etc. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2018-02-14T19:06:20.547" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13598" PostTypeId="2" ParentId="13594" CreationDate="2018-02-14T20:15:36.630" Score="3" Body="&lt;p&gt;Believing that they can infer that something is true for all real numbers based on a few examples points to lack of understanding of basic methods of proof, specifically how we go about making generalizations in mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;May I humbly suggest my interactive, self-study tutorial in the basic methods of proof and accompanying proof checker that provides students with &lt;em&gt;immediate feedback&lt;/em&gt;. Not a single invalid line of proof will be accepted by the software. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my tutorial, I introduce students to the following basic methods of proof with simple, to-the-point worked examples plus exercises with hints and full solutions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Direct Proof&lt;/li&gt;&#xA;&lt;li&gt;Proof by Contradiction&lt;/li&gt;&#xA;&lt;li&gt;Proof by Cases&lt;/li&gt;&#xA;&lt;li&gt;Proof of Biconditionals&lt;/li&gt;&#xA;&lt;li&gt;Manipulating Quantifiers -- including making universal and existential generalizations.&lt;/li&gt;&#xA;&lt;li&gt;Proof by Induction&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Very&lt;/em&gt; Elementary Set Theory (&lt;em&gt;not&lt;/em&gt; ZFC)&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Very&lt;/em&gt; Elementary Number Theory&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Visit my &lt;a href=&quot;http://www.dcproof.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;homepage&lt;/a&gt; for more information, testimonials, and a free, fully-functional download including the above mentioned tutorial.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2018-02-14T22:27:27.010" LastActivityDate="2018-02-14T22:27:27.010" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13599" PostTypeId="2" ParentId="13570" CreationDate="2018-02-15T08:26:32.287" Score="0" Body="&lt;p&gt;I understand that this is an English-language based platform but I thought it could be of some interest to share with you some links to projects about symmetry in Italian primary schools. After all math is an universal language, isn't it? Well, well,... at least I hope that pictures are self-explanatory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://iccupra-ripatransone.gov.it/wp-content/uploads/sites/193/presentazione-percorso-di-geometria.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Material from a primary school&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://specchi.mat.unimi.it/matematica/specchi/Specchi_27_1_11.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A University web-site producing activities for primary schools&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a personal level let me add that I personally tried, with some fun, having kids explore symmetries in Escher tessellations &lt;a href=&quot;http://www.tessellations.org/eschergallery1thumbs.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;Escher&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2018-02-15T08:26:32.287" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13600" PostTypeId="2" ParentId="13548" CreationDate="2018-02-15T09:32:26.553" Score="0" Body="&lt;p&gt;Surely mathematicians would just use the triple bar identity symbol, and it doesn't have to be defined. I was introduced to it in later grade school (O-levels). Special notations are not conducive to the proper training of students in standard notations perfectly acceptable in exams.&lt;/p&gt;&#xA;" OwnerUserId="6202" LastActivityDate="2018-02-15T09:32:26.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13601" PostTypeId="1" CreationDate="2018-02-15T12:17:28.620" Score="8" ViewCount="362" Body="&lt;p&gt;So my question isn't directly related to math, but I think you might be able to help me. I'm a 17 year old, last year at school. My class specializes in math/physics (that's our educational system in Lebanon. Last year classes are divided). I'm really good at math/physics and I can take any given problem thrown at me in those two fields and figure out a solution (with what i know apparently). I honestly dream of becoming a theoretical physicist, knowing that i enjoy maths and physics a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is here: Our educational system here in Lebanon, forces us to take a four-hour mathematics exam (there's an official exam given at the end of the year to all students). As I said, I'm good at maths, but when it comes to that exam, my brain seems to stop functioning after the first hour or hour and a half. And I can't solve simple transformations or simple conics problems. This is getting into me, as my grades are not reflecting my true abilities, and it's depressing. The exam is very long to those who may recommend taking breaks, and it can't be finished if I took breaks, but I'm not able to finish it also because I get tired and my brain shuts down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry for the long, not so related question, but I hope some of you mathematical minds could have some solution.&lt;/p&gt;&#xA;" OwnerUserId="9449" LastEditorUserId="1147" LastEditDate="2018-02-15T15:30:01.503" LastActivityDate="2018-02-19T02:22:20.123" Title="Mathematics exam -- How to keep brain functioning through a four-hour exam?" Tags="|exams|assessment|gifted-students|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13602" PostTypeId="2" ParentId="13594" CreationDate="2018-02-15T12:58:11.963" Score="5" Body="&lt;p&gt;Can all of your students correctly explain what the supremum and infimum of a set are? If not, they will be unable to &lt;em&gt;reason&lt;/em&gt; about these concepts. It would be like expecting a student to do something with the order of an element in an abstract (arbitrary) group if they can't define what the order if an element is (e.g., thinking &quot;order of $g$ is $n$&quot; means only that $g^n = 1$).&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2018-02-15T12:58:11.963" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13603" PostTypeId="2" ParentId="13601" CreationDate="2018-02-15T15:27:51.690" Score="7" Body="&lt;p&gt;Having one's mental abilities fall off after about 90 minutes is consistent with having eaten 20 - 30 grams of sugar before the exam started, but not having eaten any starch at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you last an extra hour if you also eat about 40 - 60 grams of starch before the exam? Can you last the full exam length if you bring a snack (with another 20 - 30 grams of sugar and 40 - 60 grams of starch) to eat halfway through the exam? To prevent tooth decay, American dentists recommend protein in addition to (or instead of) the sugar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have you talked to a doctor about whether a test for diabetes-like syndrome(s) might be appropriate?&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2018-02-15T15:27:51.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13604" PostTypeId="2" ParentId="1550" CreationDate="2018-02-15T17:22:13.510" Score="1" Body="&lt;p&gt;Why not maximum likelihood estimation (from stats, if you're not familiar with this) in the one-parameter case? You don't need to provide all of the statistical jargon to them. Simply present a formula and ask them to find the value of the parameter which maximizes the parameter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I recently finished teaching Calc. I to a friend of mine. Here's my memory of how I worded the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose you have $n$ known numbers $Y_1, \dots, Y_n$ from a normal&#xA;  distribution (which you can simply explain is a &quot;bell-shaped curve&quot;)&#xA;  with known standard deviation $\sigma$ but &lt;strong&gt;unknown&lt;/strong&gt; mean $\mu$. (These terms are not too hard to explain visually.) In&#xA;  statistics, we have to somehow figure out a way to estimate $\mu$. One&#xA;  method entails finding a formula for $\mu$ which maximizes  $$L(\mu) =&#xA; \dfrac{1}{(2\pi\sigma^2)^{n/2}}\exp&#xA; \left[-\dfrac{1}{2\sigma^2}\sum_{i=1}^{n}(Y_i - \mu)^2&#xA; \right]$$ where $\exp(x) = e^{x}$. Find the formula for $\mu$ which maximizes $L$, and&#xA;  be sure to justify that it maximizes $L$ through a second-derivative&#xA;  test. (&lt;em&gt;Hint&lt;/em&gt;: you  may use the fact that $L(\mu) &amp;gt; 0$. Also, why does the resulting formula for $\mu$ makes intuitive sense?)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course, I wouldn't provide this problem to a traditional Calc. I course - it is an intimidating question with all of its symbols. But I appreciated being able to give my friend a taste of how calculus is &lt;em&gt;actually&lt;/em&gt; used in higher-level math, and I feel that not enough attention is paid to statistics in a lot of math courses. Most actual applications of calculus are not as simple as the exercises that are used in calculus textbooks.&lt;/p&gt;&#xA;" OwnerUserId="1907" LastActivityDate="2018-02-15T17:22:13.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13605" PostTypeId="2" ParentId="13594" CreationDate="2018-02-15T18:23:30.647" Score="2" Body="&lt;p&gt;Suppose the conversation begins with the secret goal of tricking students into discovering sup for themselves, to own the idea, and apply it usefully at will.&lt;/p&gt;&#xA;&lt;p&gt;A good 6th grader might understand a number as a thing that has a decimal representation, possibly with infinitely many digits to the right of the decimal place, but only finitely many to the left.&lt;/p&gt;&#xA;&lt;p&gt;S/he can already calculate a bit, and perhaps grasps, at least implicitly, that subtraction is trying to measure the geometric distance between two numbers.&lt;/p&gt;&#xA;&lt;p&gt;Now majoring in math, perhaps our students have at least a very basic grasp of the basic basics of sets. A set of real numbers partners with a bunch of dots on the real line.&lt;/p&gt;&#xA;&lt;p&gt;Does every finite set of numbers have a largest element?&lt;/p&gt;&#xA;&lt;p&gt;( Of course! Just write them all down and pick the biggest one, or pick the rightmost dot in the picture!)&lt;/p&gt;&#xA;&lt;p&gt;Does every infinite set of numbers have a largest element?&lt;/p&gt;&#xA;&lt;p&gt;( Of course not! 1,2,3,...)&lt;/p&gt;&#xA;&lt;p&gt;Nice! But that's too easy! What if all the numbers in the set are at most 10?&lt;/p&gt;&#xA;&lt;p&gt;(Wheels turning....Ah! The set 9, 9.9 ,  9.99,... has no largest element!)&lt;/p&gt;&#xA;&lt;p&gt;Great! So what's going on here? How can we talk about the ``largest element'' when it might not belong to the set?&lt;/p&gt;&#xA;&lt;p&gt;(Now you have tricked them into caring about the idea of sup... they are ready to try to parse the official definition...and they can begin to teach themselves... and each other.... )&lt;/p&gt;&#xA;&lt;p&gt;The mathematical upshot is that students can already &amp;quot;see&amp;quot; that the least upper bound axiom should true.&lt;/p&gt;&#xA;&lt;p&gt;( Indeed there is something like an algorithm for &amp;quot;finding&amp;quot; sup of a set with an upper bound that a 6th grader could in principle parse  1) Start with the original numbers, for each number throw away everything to the right of the decimal and pick the biggest number among the new numbers 2) Start with the original numbers and throw away everything to the right of the 1/10 place, and pick the biggest new number. Of course the sequence we're building is piling up on the sup).&lt;/p&gt;&#xA;&lt;p&gt;The previous paragraph thinks it's a computer program that can &amp;quot;calculate&amp;quot; sup(A) for any bounded set. This could serve as a crutch for students who are used to thinking of mathematics as all about calculation.&lt;/p&gt;&#xA;&lt;p&gt;In general `Getting good at proofs'' is hard, a special case of which is related to the question at hand ``getting good at exercises related to material involving inf and sup.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;``Getting good at proofs'' is hard for most of us. Here's one reason, perhaps a perpetual reason, why.&lt;/p&gt;&#xA;&lt;p&gt;Given some mathematical assumptions and hoped for conclusion, there is a human tendency to try and &amp;quot;classify&amp;quot; every instance of the hypotheses and then &amp;quot;check&amp;quot; every case. If the starting data is a set of real numbers, that could be a lot of &amp;quot;classifying&amp;quot; and &amp;quot;checking&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But there is an antidote. I remember once as an undergraduate, after failing miserably to really understand and especially classify the hypotheses, I was startled to see that one could still reach the desired conclusion, just by using the definitions and assumptions in the hypothesis!&lt;/p&gt;&#xA;&lt;p&gt;There was no need to try to imagine what every possible case ``looks like''.&lt;/p&gt;&#xA;&lt;p&gt;Having said that, it is of course an extremely useful tactic to try and conjure up specific examples which satisfy a given hypotheses --- it can make it much easier to see the REASON a given conclusion might hold more generally.&lt;/p&gt;&#xA;&lt;p&gt;False starts, gratuitous detours, leaps of faith posing as logic, will happen to all students -- especially the good students, since they are at least trying.&lt;/p&gt;&#xA;&lt;p&gt;In general mastery of abstraction is hard, even when the material is ultimately elementary. Yoneda Lemma anyone?&lt;/p&gt;&#xA;&lt;p&gt;The important thing for the instructor is to be ceaselessly positive,encouraging, and tolerant, and to be able to see where/why a given student might be trapped, and offer a way out, at least pointing towards a door.&lt;/p&gt;&#xA;&lt;p&gt;The important thing for the student is to never...ever give up.&lt;/p&gt;&#xA;" OwnerUserId="9450" LastEditorUserId="8571" LastEditDate="2020-08-27T12:26:12.890" LastActivityDate="2020-08-27T12:26:12.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13608" PostTypeId="1" CreationDate="2018-02-16T18:58:09.427" Score="8" ViewCount="2830" Body="&lt;p&gt;How can I convincingly and mathematically explain the reason behind difference between $(-1)^2$ and $-1^2$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used to add &quot;negation&quot; to the order of operations, in the same row as multiplication and division, so that in $-1^2$ we should first carry on the exponentiation and then the negation. Hence the answer is $-1$. But recently I've asked myself if there's a better explanation, more conceivable and convincing.&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2022-11-21T21:12:01.000" Title="Exponents with Negative Base; with or without Parentheses" Tags="|arithmetic-operations|negative-numbers|middle-school|exponent|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13609" PostTypeId="2" ParentId="13608" CreationDate="2018-02-16T19:15:06.607" Score="6" Body="&lt;p&gt;You might explain that BEDMAS is not the whole story when it comes to the order of operations. There is an operation called negation. It reverses the sign on numerical quantifies. It gives the additive inverse of any number, i.e. for all $x \in R$, we have $x + (-x) = 0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, most textbooks use the same symbol for both subtraction and negation. Some calculators use different symbols ( - for negation, $-$ for subtraction.) It should be fairly easy with a bit of practice to tell from the context whether a '$-$' symbol is being used as a subtraction operator or a negation operator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The negation operator usually has a precedence that is less than that of exponentiation but greater than than of multiplication. So, we now have &lt;strong&gt;BENDMAS&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the expression $-1^2$, we have two operations: negation and exponentiation. BENDMAS tells us to do the exponentiation first, then the negation to get a result of $-1$.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2018-02-16T21:52:25.690" LastActivityDate="2018-02-16T21:52:25.690" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13610" PostTypeId="2" ParentId="13608" CreationDate="2018-02-16T20:54:32.997" Score="4" Body="&lt;p&gt;If your students already understand that exponents precede multiplication, and that multiplying by $-1$ is the &quot;negation&quot; operator, then you should be able to convince them that $$-5^2 = -1*5^2 = -1*25 = -25$$ is a reasonable way to interpret this expression.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-02-16T20:54:32.997" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13611" PostTypeId="2" ParentId="1032" CreationDate="2018-02-17T01:17:36.777" Score="2" Body="&lt;p&gt;Give the student a choice. Either: a final grade based on the all the other material submitted, or set a new exam just for the student and let them sit that. &lt;/p&gt;&#xA;" OwnerUserId="9454" LastActivityDate="2018-02-17T01:17:36.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13612" PostTypeId="2" ParentId="13608" CreationDate="2018-02-17T01:19:36.143" Score="0" Body="&lt;p&gt;If you're main concern is students' writing, you could take the route of avoiding the issue altogether and tell students that $-1^2$, regardless of what it should be equal to according to BEDMAS, is just bad notation and that it can be avoided. The purpose of writing mathematics, of writing anything really, is to communicate effectively to a reader. The notation $-1^2$ is ambiguous since it could mean $(-1)^2$ or $-(1^2)$, and this ambiguity can easily be avoided by just writing one of $(-1)^2$ or $-(1^2)$ instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But unfortunately this $-1^2$ notation does appear in textbooks (I'm almost certain I've seen it before). For that, I would just tell my students that &quot;&lt;em&gt;the textbooks authors could have been clearer, but they are just using the general convention that ...&lt;/em&gt;&quot; Students might even like that there is something we have to take as convention. It's a break from the rigidity they see in the rest of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-02-17T18:18:16.990" LastActivityDate="2018-02-17T18:18:16.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13613" PostTypeId="2" ParentId="13608" CreationDate="2018-02-17T04:41:17.940" Score="5" Body="&lt;p&gt;I tutored a student who had a hard time understanding this, and the way that helped him to understand it was this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any time there is a negative sign on a number, we can read it as $(-1)$ . So $-5 \equiv (-1)5$ and $-3^2 \equiv (-1)3^2 = (-1)9 \equiv -9$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case you mention then, $-1^2 \equiv (-1)1^2 = (-1)1 \equiv -1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key to helping him understand it was always wrapping the negative 1 that the minus sign implies in parenthesis, from which he could use BEMDAS on everything.&lt;/p&gt;&#xA;" OwnerUserId="9457" LastActivityDate="2018-02-17T04:41:17.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13614" PostTypeId="2" ParentId="13608" CreationDate="2018-02-17T18:30:16.427" Score="6" Body="&lt;p&gt;I like the presentation on the &lt;a href=&quot;https://web.archive.org/web/20180128165341/http://mathforum.org/library/drmath/view/53194.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;NCTM Math Forum/Dr. Math&lt;/a&gt; website:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We don't usually list unary operators in PEMDAS because they're&#xA;thought of as being implied by the rules for binary operations. You&#xA;can think of the minus sign as either subtraction&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt; -3^2 = 0 - 3^2 = 0 - 9 = -9&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;or multiplication&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt; -3^2 = -1 * 3^2 = -1 * 9 = -9&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;and in either case it has lower precedence than exponentiation, so it&#xA;gives the same result. I can't think of any case where it would matter&#xA;which of these two latter interpretations we give, that is, any case&#xA;where giving &amp;quot;-&amp;quot; an additive precedence or a multiplicative precedence&#xA;will make a difference in the result; commutativity and distributivity&#xA;seem to take care of that.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would observe that the &lt;a href=&quot;https://nrich.maths.org/5961&quot; rel=&quot;nofollow noreferrer&quot;&gt;historic development of negative numbers&lt;/a&gt; was closely related to subtraction (i.e., finding the closure of various number sets under subtraction), so arguably the most straightforward interpretation of negation is &lt;span class=&quot;math-container&quot;&gt;$-x = 0 - x$&lt;/span&gt;. We want &lt;span class=&quot;math-container&quot;&gt;$-1^2$&lt;/span&gt; to have the same value and order (up to the final subtraction) of &lt;span class=&quot;math-container&quot;&gt;$0 - 1^2$&lt;/span&gt;, which implies the convention.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="6053" LastEditDate="2022-11-21T21:12:01.000" LastActivityDate="2022-11-21T21:12:01.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13615" PostTypeId="2" ParentId="13601" CreationDate="2018-02-17T23:18:25.880" Score="5" Body="&lt;p&gt;While it is true (as @Jasper suggests) that your issue might be partly metabolic/biochemical, &lt;em&gt;also&lt;/em&gt; it is simply the case that human beings cannot operate in a very intense mode for very long. In teaching graduate courses in mathematics and giving seminars and so on, I feel myself becoming incoherent after a little more than 60 minutes unless I can take a break mentally and re-orient myself. (E.g., check email, stop thinking about math for 5 minutes.) Also, there is a contrast between what one can maintain as a &quot;sprint&quot; versus a longer &quot;run&quot;: pace.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-02-17T23:18:25.880" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13616" PostTypeId="2" ParentId="2226" CreationDate="2018-02-17T23:42:34.003" Score="1" Body="&lt;p&gt;I myself am not very interested in seeing whether students (whether exceptionally talented or not) can reproduce long, complicated proofs. Or do long, complicated computations. I &lt;em&gt;am&lt;/em&gt; interested in their &lt;em&gt;awareness&lt;/em&gt; of what is true, and of the relative difficulty of proof, and of the prerequisites for a good proof. And awareness of the utility of the fact(s). To my mind, &quot;the standard results&quot; are the context in which one does mathematics. Re-establishing the standard context, as opposed to demonstrating awareness, is not very interesting or useful.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-02-17T23:42:34.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13617" PostTypeId="2" ParentId="13601" CreationDate="2018-02-18T15:33:57.917" Score="7" Body="&lt;ol&gt;&#xA;&lt;li&gt;Take some periodic breaks.  Yes, you can't finish it if you take breaks, but you can't finish it if you don't take breaks.  Think of it as an optimization problem.  Volume is rate times time.  If your rate is 50% higher when you use 10% of the time for breaks, you are still better off.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(100%)(90%) &gt; (50%)(100%)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They don't need to be long breaks but can be grabbing some water or hitting the head.  Or just staring into space for a little while.  Use your judgment when you think you need it, based on fatigue.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Do the exam in a nonlinear manner.  (Easiest problems first.)  Just the moving around will refresh you.  This also allows back of your mind to work on the toughies, also builds confidence, also some actual results/methods of one easy problem may be a subpart of a more complex problem.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also, move around to different parts of the exam if you get stuck or bored in an area (will refresh you).  [For both 2 and 3, it is a judgment call how much to do this, but some amount will help.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Train yourself for endurance by practice.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Improve your skill level.  If you are smart enough to be able to crush the problems, you won't have an issue taking a few minutes to recharge in the exam.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Net, net:  do more homework problems.  You will find your capacity and speed improve.  4 hours is brutal for anyone but a lot less so for those who have the material mastered and beaten into submission.  I suspect you do not and are a little overconfident.  (Common student...and human...failing.)  Tests and math are like sports.  You need to have those skills honed, not just performed occasionally/slowly.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Couple other considerations:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;This is  a high stakes performance given it is yearly.  But for high stakes performances, it is advisable to overtrain.  Both because of the importance and because of the need for more confidence.  Nothing is perfect, but I have always found in work, school, and sports that overtraining for high stakes performances paid off for me.  I might still have a little angst before the game started, but once it got going, I always felt good--the overtraining had made me ready for the &quot;test&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Things MAY (I stress MAY) be a little more harsh, less liberal, in your Levantine school system.  But don't think that Westerners don't have some challenges like this as well.  US college finals can be several hours long.  Not to mention AP tests.  Or grad school qualifiers.  Also, many professional exams (medical, law, engineer) are long and intense.  I've also had 4 to 8 hour exams in the military.  So, you might have it slightly tougher than the average 17 year old. But it's not insane.  (And for that matter at 17, you are moving into adulthood.)  Do your best and figure out how to develop the skill and endurance needed--could even be a benefit if you are &quot;stronger&quot; and then compete in Western schools versus those without same forced training.  [And even if this weren't true, you need to master the darned exams anyways, so try to come up with the positive motivation to trick yourself!]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;P.s.  I looked at the discussion on your locked mathSE thread.  Although a little tougher, the comments from your teachers and the commenters are similar to mine.  And way less verbose.  Really man, you need to do more homework every night.  Do it fast, too.  Not frantic.  But also not slow and pondering.  Get used to working faster and make it part of the game.&lt;/p&gt;&#xA;" OwnerUserId="9428" LastEditorUserId="9428" LastEditDate="2018-02-19T02:22:20.123" LastActivityDate="2018-02-19T02:22:20.123" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13618" PostTypeId="2" ParentId="13601" CreationDate="2018-02-18T16:56:17.983" Score="3" Body="&lt;p&gt;I have had to work on 4- to 6- hours math an physics tests for four years in total in my curriculum, and I train student or 5-hours long math tests. I will not attempt to be as thorough as other answer and just insist on a few points that seem central to me:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;have some food (cereal bars) and water on top of eating adequately before the test,&lt;/li&gt;&#xA;&lt;li&gt;train progressively (first do a 2 hours homework test, then increase 1/2 hour at a time until you get to 4 hours),&lt;/li&gt;&#xA;&lt;li&gt;do not hesitate to take small breaks (empty your head, take a deep breath, a sip of water, relax 1mn, then start a new problem/question).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="187" LastActivityDate="2018-02-18T16:56:17.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13619" PostTypeId="2" ParentId="13601" CreationDate="2018-02-18T17:13:19.290" Score="2" Body="&lt;p&gt;If the claim is that your home and in-class performance is radically different than performance on this test, to a degree that shocks your teacher, then it sounds like you may be suffering from some amount of &lt;a href=&quot;https://en.wikipedia.org/wiki/Test_anxiety&quot; rel=&quot;nofollow noreferrer&quot;&gt;test anxiety&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Advice for dealing with test anxiety is not altogether different from other answers here: Over-practice. Keep a regular schedule with exercise. Get a good night's sleep the night before. Eat well that morning and bring water/snacks. Stretch and release tension during the test. Move around the questions to refresh and get traction. In extreme cases, speaking with a counselor/therapist for relaxation techniques is recommended. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.studygs.net/tstprp8.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Overcoming test anxiety&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.testprepreview.com/test_anxiety.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Coping with test anxiety&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-02-18T17:13:19.290" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13620" PostTypeId="2" ParentId="13511" CreationDate="2018-02-18T20:41:37.313" Score="1" Body="&lt;p&gt;The answer in the video was a bit unsatisfying for you, and I understand why. You expect a firm answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;IRL, when I get this question, I need to quickly determine the level the student is at in her studies. When square roots (or nth roots for that matter) are first discussed, we are only concerned about the positive value. After all, my square with an area of 4 square units has a side that’s a positive number. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then the students move on the the square root as part of the quadratic equation solving, and we must be mindful of both positive and negative roots. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At some point, the 4th root of 16 has 4 solutions, as the +/-2i is also a root. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It’s for the adult math teacher to understand what level the class is at, and whether that level contains the negative or imaginary root as a solution. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-02-18T20:41:37.313" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13621" PostTypeId="2" ParentId="13570" CreationDate="2018-02-19T03:18:01.480" Score="0" Body="&lt;p&gt;The &lt;a href=&quot;http://www.math.ucla.edu/~radko/circles/&quot; rel=&quot;nofollow noreferrer&quot;&gt;UCLA math circle&lt;/a&gt; has several activities involving symmetry.  For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.math.ucla.edu/~radko/circles/lib/data/Handout-778-883.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Parity, Symmetry and Fun Problems&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.math.ucla.edu/~radko/circles/lib/data/Handout-817-915.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Symmetry, Mirrors and Fun&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.math.ucla.edu/~radko/circles/lib/data/Handout-803-905.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mirrors and Kaleidoscopes&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2018-02-19T03:18:01.480" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13622" PostTypeId="1" CreationDate="2018-02-19T12:09:46.893" Score="3" ViewCount="78" Body="&lt;p&gt;In Pakistan, we have until now had 2-year B.A./B.Sc. and 2-year M.A./M.Sc. university degrees after our 12th grade F.A./F.Sc. qualifications. Thus the former was a 14-year academic qualification, whereas the latter was the 16-year qualification. In these degrees, the assessment was based on either end-of-degree aggregate examinations of all the subjects taught or at most annual examinations. This was the system introduced in the subcontinent by the British government. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm wondering if this kind of system still exists in the UK today, or in other countries of Western / Northern Europe, North America, Australia, Singapore, Hong Kong, China, or Japan. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And have all the above-mentioned countries completely switched to a semester/trimester/quarter-based system of continuous assessment? Or, does the annual system still exists anywhere? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, in Pakistan, universities would often send their papers to teachers of other universities for checking / grading in order to ensure transparency / accountability, but later on only papers of the affiliated colleges were sent to the respective academic departments of their affiliating university. Is this practice still in vogue anywhere else as well? &lt;/p&gt;&#xA;" OwnerUserId="5400" LastActivityDate="2018-02-19T12:09:46.893" Title="Is the annual system still in vogue anywhere in Europe, North America, Australia, or East Asia? And what about the 2-year B.A. / B.Sc. degree?" Tags="|undergraduate-education|exams|assessment|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13624" PostTypeId="1" AcceptedAnswerId="13626" CreationDate="2018-02-19T16:55:01.500" Score="13" ViewCount="330" Body="&lt;p&gt;Are there any good pedagogical resources or articles that you would recommend to math graduate student teaching assistants (TAs)? Is there any sweeping advice that you would give a TA to improve their effectiveness in discussion? I'm asking this because TAs usually play a very different role in teaching than the primary instructor, so a lot of general resources for math teachers/instructors won't apply to them.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;TAs don't have much control over the content of the discussion section: it just depends on what the students are learning in lecture that week. Also TAs have varying control over what they do in the discussion section: some primary instructors are hands-off while others will make a worksheet or quiz or something for discussion.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Typically graduate students are much more interested on their research than learning to teach, so they don't want to spend much time reading about it (so recommending full books to read is not the best idea).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Despite these limitations, there must be something that would be beneficial for a graduate TAs to read and think about to improve their effectiveness as a TA.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2018-02-26T04:54:15.353" Title="Pedagogical advice/articles for graduate student teaching assistants" Tags="|undergraduate-education|mathematical-pedagogy|reference-request|general-pedagogy|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13625" PostTypeId="1" CreationDate="2018-02-19T18:07:23.017" Score="1" ViewCount="119" Body="&lt;p&gt;I'm in search of a good book that I can read --- and recommend to my proteges to read --- along with each one of the following books. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Topology&lt;/em&gt; by James R. Munkres, 2nd edition&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Introductory Functional Analysis With Applications&lt;/em&gt; by Erwine Kreyszig&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Principles of Mathematical Analysis&lt;/em&gt; by Walter Rudin, 3rd edition&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For each one of the above three books, I would like to have a book (or a couple of books) that discusses the same material as in this book and which fills the gaps in the presentations of the topics covered herein by illustrating further some or most of the topics discussed rather cursorily in this book, providing more examples on the topics of this book, and even providing the solutions to the exercise problems in this book as text examples. In short, in each case, I would like to have a book (or a couple of books) that reinforces one's understanding of the contents of this book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, last but not least, I would also like an introductory-level, and preferably recently written / updated, functional analysis textbook that is accessible to a reader with the real analysis background from &lt;em&gt;Introduction To Real Analysis&lt;/em&gt; by Bartle &amp;amp; Sherbert, or from Baby Rudin's first seven chapters. &lt;/p&gt;&#xA;" OwnerUserId="5400" LastActivityDate="2018-02-19T18:07:23.017" Title="Which book to use concurrently with each of these mathematics texts?" Tags="|reference-request|textbooks|mathematical-analysis|topology|functional-analysis|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13626" PostTypeId="2" ParentId="13624" CreationDate="2018-02-19T20:46:32.270" Score="8" Body="&lt;p&gt;The MAA has quite a long &lt;a href=&quot;https://www.maa.org/programs/students/student-resources/a-handbook-for-mathematics-teaching-assistants&quot; rel=&quot;nofollow noreferrer&quot;&gt;guide for new TA's&lt;/a&gt;, it's a little too much to take in at once but as a supplement it's very good, it's got a section on everything you could hope for: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree with &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13624/pedagogical-advice-articles-for-graduate-student-teaching-assistants#comment33535_13624&quot;&gt;Brendan&lt;/a&gt; that conversations with other graduate students is the most useful, and I also generally advocate for some kind of weekly meeting for new TA's, preferably with their instructor and older graduate students (and preferably interactive, not just someone talking at them). A lot of graduate student are a bit embarrassed about having to ask for help from the faculty and so will avoid professional development that might make them look incapable in the eyes of the faculty. &lt;/p&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="9019" LastEditDate="2018-02-26T04:54:15.353" LastActivityDate="2018-02-26T04:54:15.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13627" PostTypeId="2" ParentId="13624" CreationDate="2018-02-20T05:35:10.000" Score="1" Body="&lt;p&gt;Check out the work of Eric Mazur and Carl Wieman, both physicists who are working to reform post-secondary science education. Much of what their work applies to math as well. Both rail against lecturing and both have had tremendous success with more interactive methods of teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of their techniques are applicable to TA situations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Downside: Their methods of instruction do require more prep time for teachers.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-02-20T05:35:10.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13628" PostTypeId="1" CreationDate="2018-02-20T08:03:16.490" Score="34" ViewCount="6637" Body="&lt;p&gt;I am teaching Linear Algebra this semester with the textbook &lt;a href=&quot;http://www.springer.com/us/book/9780387962054&quot; rel=&quot;noreferrer&quot;&gt;Introduction to Linear Algebra&lt;/a&gt; by Serge Lang and most (perhaps all?) my students are not majoring in mathematics. As I was carefully introducing the vector-space axioms and proving the propositions and theorems, a student asked me the following question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Shall we do less proving and learn some real-world applications?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I was shocked by the question. To me, I am teaching pure mathematics, not applied mathematics and by proving the results, I am aiming at the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To make the course logically self-contained. At least this is how I learnt mathematics: definitions, examples, theorems and corollaries, proofs, etc.&lt;/li&gt;&#xA;&lt;li&gt;To train the students to think critically and logically.  I believe that students should learn that they should not take anything for granted or by “blind faith”.&lt;/li&gt;&#xA;&lt;li&gt;To build help the students to build a solid mathematical foundation “like a wise man who built his house on the rock”.&lt;/li&gt;&#xA;&lt;li&gt;To follow the topics in the textbook. There is almost no real-world applications in the textbook that I have selected, which I believe is an excellent textbook for beginners in linear algebra.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;About real-world applications? There are courses on topics such as mathematical modelling that are designed for those purposes. Now my question is: &lt;strong&gt;Could/Should courses such as linear algebra, calculus, differential equations be taught theoretically without introducing any real-world applications?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: Thanks to everyone who commented or answered. What would the answer be if we consider other mathematical courses such as general topology, or functional analysis, or real analysis, etc.? Not sure if there are many real-world applications to those courses.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2018-03-05T23:28:16.497" LastActivityDate="2020-08-20T18:59:26.707" Title="Should college mathematics always be taught in such a way that real world applications are always included?" Tags="|undergraduate-education|linear-algebra|applications|applied-mathematics|" AnswerCount="12" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13629" PostTypeId="2" ParentId="13628" CreationDate="2018-02-20T09:20:11.497" Score="41" Body="&lt;p&gt;I have worked with a lot of students coming out of courses such as yours who:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;passed the course by blindly memorising proofs, theorems, and algorithms;&lt;/li&gt;&#xA;&lt;li&gt;learnt nothing (lasting) except solving some calculating exercises, a very vague idea of some terminology;&lt;/li&gt;&#xA;&lt;li&gt;had no idea what they just learnt, why they learnt it, and how it relates too their field of study; some had the idea that the entire point of the course was to teach them performing numerical calculations (which is nonsense, any computer is better at this than they are);&lt;/li&gt;&#xA;&lt;li&gt;had solidified or newly acquired math phobia;&lt;/li&gt;&#xA;&lt;li&gt;even if they learnt something, were unable to translate their knowledge in any way.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Courses such as these heavily rely on motivation, and most of your students have no motivation to do math for its own sake. This does not mean that you should only teach them applications or ditch your goals, but you should at least mention the applications (and motivate the general structure of mathematics).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, for vector-space axioms, you can motivate:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The vast majority of what they will learn during your course is based on those axioms. This means that once they can show that some regular structure fulfils these axioms, they have the entire arsenal of linear algebra at their disposal.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Many relevant structures can be described as vector spaces, e.g., geometrical positions, quantum-mechanical states, solutions to differential equations, Internet search engines, etc. Depending on the particular application and audience, you may briefly explain them or just leave it at mentioning them. But even the latter can be relevant, e.g., even students of physics who haven’t had any courses on quantum mechanics will likely be aware that it is a very central thing in many parts of modern physics and it may motivate them to view and learn linear algebra the abstract way and not just as a way of doing geometry.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;By the way, mentioning applications can even be beneficial to mathematicians. I once sat in a course on differential geometry full of mathematicians who didn’t know that the determinant can be used to quantify volume.&#xA;Moreover, a good mathematician should be able to notice whether their finding is relevant on the next layer of application, whether it is inner-mathematical or in a different field – because the people in those fields will probably not notice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another sidenote: If you don’t know good (educational) applications of a certain piece of mathematics yourself, you can &lt;a href=&quot;https://matheducators.stackexchange.com/questions/ask&quot;&gt;ask on this very site&lt;/a&gt; or &lt;a href=&quot;https://math.stackexchange.com/questions/ask&quot;&gt;Mathematics SE&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2018-02-21T15:11:37.097" LastActivityDate="2018-02-21T15:11:37.097" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13631" PostTypeId="2" ParentId="13628" CreationDate="2018-02-20T13:34:11.857" Score="4" Body="&lt;p&gt;&lt;em&gt;Should&lt;/em&gt; they be taught without any applications? (Note the lack of the over-limiting adjective &quot;real world&quot;.) I would say no; at the very least, applications give us the answer to the question &quot;Why these axioms and not some others?&quot;&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-02-20T13:34:11.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13632" PostTypeId="2" ParentId="13628" CreationDate="2018-02-20T14:09:19.200" Score="1" Body="&lt;p&gt;I think the emphasis should be on calculational manipulation, not on proofs.  This is true for linear algebra, but especially true for diffyQs.  The vast majority of students are not math majors and needs these tools to follow derivations and do homework in physics and engineering classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proofs should be included more along the line of derivations than proofs.  To motivate and justify the calculational tools learned.  Not for their own lofty purpose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some simple, common applications should be included.  The majority of the class though should just be calculation manipulation.  (As word problems are harder.)  The kids will get a lot of practice with equations for circuits, harmonic oscillator, feedback control, etc. later.  But at this point, it is nice that they get some familiarity with just the x/y manipulations. That way it is less a challenge when they learn new technical topics and have some math chops, versus only learning it in math class.  Also some techniques (e.g. second order diffyQ with constant coefficients) have key applications in multiple areas.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But a few applications should be included--emphasize those that are more easy to conceptualize.  So NOT quantum mechanics.  But dilution or simple circuits.  (I am giving diffyQ examples as I know the topic better than linear algebra.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  The tone of your question makes me wonder if you even KNOW the applications of linear algebra or diffyQs.  Maybe there is a gap in your own technical training.  Also, I find a lack of empathy in that you don't think of the motivations of your students and how they may differ from yours.  (Not even if you or they are right, but different strokes for different folks.)        &lt;/p&gt;&#xA;" OwnerUserId="9428" LastEditorUserId="9428" LastEditDate="2018-02-20T14:20:19.493" LastActivityDate="2018-02-20T14:20:19.493" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13633" PostTypeId="2" ParentId="13624" CreationDate="2018-02-20T14:14:29.833" Score="2" Body="&lt;p&gt;I recommend Stephen Krantz's &lt;a href=&quot;https://www.maa.org/press/maa-reviews/how-to-teach-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to Teach Mathematics&lt;/a&gt;. There's a fairly short section aimed at TA's, but if this will be your profession then you'll need everything in there, and the sooner the better. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-02-20T14:14:29.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13634" PostTypeId="2" ParentId="13628" CreationDate="2018-02-20T16:22:04.180" Score="22" Body="&lt;p&gt;At my University, there are four different first-semester Linear Algebra courses taken by Undergraduates:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Math 214, &lt;em&gt;Applied Linear Algebra&lt;/em&gt;, is &quot;an introduction to matrices and linear algebra... The emphasis is on concepts and problem solving. The sequence 214-215 is not for math majors. It is designed as an alternate to the sequence 215-216 for engineering students who need more linear algebra and less differential equations background.&quot;&lt;/li&gt;&#xA;&lt;li&gt;Math 217, &lt;em&gt;Linear Algebra&lt;/em&gt;, is for &quot;Mathematics majors and other students who have some interest in the theory of mathematics&quot;, and is &quot;explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved...  Throughout there will be an emphasis on the concepts, logic, and methods of theoretical mathematics.&quot;&lt;/li&gt;&#xA;&lt;li&gt;Math 417, &lt;em&gt;Matrix Algebra I&lt;/em&gt;, is &quot;an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory.  Diversity rather than depth of applications is stressed. This course is not intended for mathematics majors, who should elect Math 217.&quot;&lt;/li&gt;&#xA;&lt;li&gt;Math 419, &lt;em&gt;Linear Spaces and Matrix Theory&lt;/em&gt;, &quot;covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary.&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Students may only receive credit for &lt;em&gt;one&lt;/em&gt; of these courses, and they are designed to appeal to students with different needs and interests:  most Engineering students take 417, but those majoring in IOE or EECS take 214; most math majors take 217, but Honors math majors may take 419 instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice, however, that &lt;em&gt;all&lt;/em&gt; of those course descriptions include some reference to applications.  Indeed, the textbook for Math 217 -- the course intended for most math majors -- is &lt;em&gt;Linear Algebra with Applications&lt;/em&gt; by Otto Bretscher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To claim that &lt;em&gt;all&lt;/em&gt; mathematics courses must have real-world applications is obviously absurd, and is a straw man.  But perhaps &lt;em&gt;this&lt;/em&gt; mathematics course should.  In general, I think it's really important to ask:  Who is taking the course, and why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edited to add:&lt;/strong&gt;  A few further thoughts.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;On re-reading the OP it strikes me that the 4th reason given for avoiding applications (&quot;To follow the topics in the textbook&quot;) is really quite circular, as the OP &lt;em&gt;chose the textbook&lt;/em&gt;, despite the fact that it avoids applications, because it is (in the OPs opinion) &quot;an excellent textbook for beginners&quot;.  If the OP thought that beginners should encounter applications, presumably a different textbook would have been selected, and then the same reasoning could be used to argue that one &lt;em&gt;must&lt;/em&gt; include applications in order &quot;to follow the topics in the textbook&quot;.&lt;/li&gt;&#xA;&lt;li&gt;What counts as an &quot;application&quot; is highly subjective.  I know people who regard the theory of linear ODEs as a very practical application of linear algebra, and others who regard that as a purely theoretical subject in its own right.  Similarly, projective geometry can be taught as an &quot;application&quot; of linear algebra, but many of your students might regard it as purely abstract and theoretical.&lt;/li&gt;&#xA;&lt;li&gt;The idea that pure logic is the foundation on which mathematical applications are built is ahistorical.  The reason why linear algebra flourished in the 20th century is precisely because people (mainly Physicists, but also Computer Scientists and others) found it extremely useful.  Without that &quot;foundation&quot; of useful applications, it is possible that it would have remained a marginal topic of interest only to a few specialists; instead it has become a cornerstone of mathematical thinking.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2018-02-20T16:59:12.803" LastActivityDate="2018-02-20T16:59:12.803" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13635" PostTypeId="2" ParentId="13628" CreationDate="2018-02-20T20:03:05.770" Score="12" Body="&lt;p&gt;I challenge the assertion that students need to see applications in everything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I first started teaching I labored under the delusion that I should explain connections to physics whenever I could (in calculus, DEQns, linear algebra etc.). Now, I think my efforts do have an audience, but not the main audience that I find in my classes. Personally, I've always had a fairly big majority of engineering students with a sprinkle of math, math-education, comp-sci, various science majors. It turns out, the fear of math is only outdone by the fear of physics. I've learned to go easy on those comments in recent years. If I was to teach an audience of math-physics undergraduates then you would hardly recognize the course in comparison to what I teach now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My approach is simple. In math class we talk about math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will make time for applications in maybe 10% of the time, but I refuse to let it take too much time. If you pick a particular application then that audience will love it, but the rest of the kids are then bored, frightened or annoyed. The common thing all students have in taking a general education math course is the need to study math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To my thinking, homework is the place for applications. How can they use math from class to solve those beloved real world problems? If this homework is recommended and not required then the audiences which express this supposed love of applications can pursue their passion individually.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is plenty of theory in linear algebra to fill the semester. I think by in large homework is the place to play with applications. For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;in class I teach the method of row reduction&lt;/li&gt;&#xA;&lt;li&gt;in homework I ask them to find polynomials to fit data &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now, I will talk applications in passing, but not as motivation. Motivation is by in large given from mathematical questions. It might be born of an example, but not an application. Motivating a general mathematical topic from a particular application can give a very skewed origin story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this said, if I was teaching &quot;Applied Linear Algebra&quot; then my focus is quite different. So, the question is, what is the course description?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, the assertion that math should be taught always without applications is just as absurd as the incessant drumbeat to include applications at the cost of technique and analysis. Balance, moderation, academic freedom for a better future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: after reading some answers I noticed I missed a major point in responding to your question. By your account, you were in the middle of Lecture stating the central core definition of the course when you were interrupted with this purportedly genuine question from the student. It occurs to me it doesn't actually matter if the question is sincere or not. The fact is that this student has violated a basic part of the student/professor relationship. Asking you to change the direction of a lecture mid-stream because he is not particularly excited about the core of a class? What if I did the same in an applications class? How about in the middle of my physics professor working an example I interupt and ask why we are not focusing more on theorems of physics which apply to all problems as opposed to this particular problem which has no bearing on the larger picture? Is it ok for me to demand more theory in the midst of a carefully prepared difficult example the professor has labored to prepare? Of course not. That would show a fundamental disrespect for the professor. I'm not much for holding up professors in terms of academic rank, but for the student/professor relationship to work it is also necessary for students to consider the humanity of professors. In short, I think the best response to such a student when the question happens is: &quot;you are welcome to visit office hours, I'd be happy to discuss applications with you there. Our focus here is theory and in time I'll show you how this theory allows endless application to applications...&quot; something like that. &lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2018-03-16T03:37:01.257" LastActivityDate="2018-03-16T03:37:01.257" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13636" PostTypeId="2" ParentId="13628" CreationDate="2018-02-21T05:48:01.760" Score="15" Body="&lt;p&gt;I believe you need to listen beyond what your student is saying.  Your student is not saying &quot;I want to do some applications in class.&quot; What your student is really saying is &quot;I'm bored and lost and this is rapidly becoming a waste of time for me, so I'm making this suggestion because I care enough about my own learning and trying to connect with you is how I am taking charge of my own education.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your student saying this is a good thing, because your students want to connect with you and sense that you will listen if they make suggestions for how they'd like to learn or say that they aren't learning.  That is better than the alternative.  I bet you the student who asked you this is not the only student who feels this way.  That's how classroom dynamics often works.  Teaching is hard and when done well is a two way street.  This is an opportunity for you to learn from your students and discover new ways to connect the material to them.  I can tell your goals are born out of a desire to see them as independent, critical thinkers.  That's very wise and caring of you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They want to see how your and their goals align, and that's not unreasonable.  In fact, they are already critically thinking.  The difference is that your teaching is the topic.  Part of critical thinking is reflection and they are doing that.  By asking these questions, they are testing &lt;strong&gt;you&lt;/strong&gt; because they do not want to have to take the material you are presenting with the &quot;blind faith&quot; that it'll be useful to them someday.  Those are already some excellent students you have to be so thoughtful!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, know that these are your unique students and random people on the Internet can't tell you what they need to hear (and neither can your textbooks.)  You can't connect with them if you don't know who they are or what they know.  Talk with them, not us.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. I'd reframe your question.  Instead say this: Will the students in my math course this semester succeed more if real world applications are included?&lt;/p&gt;&#xA;" OwnerUserId="9479" LastEditorUserId="9479" LastEditDate="2018-02-21T06:24:54.163" LastActivityDate="2018-02-21T06:24:54.163" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13637" PostTypeId="2" ParentId="13624" CreationDate="2018-02-21T06:10:23.673" Score="2" Body="&lt;p&gt;One part of teaching students is often speaking to them one-on-one and I have an acronym I use with my tutoring staff to help them focus on actions they can use with students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of the strategies here are also useful for leading group discussions and for lecture-style presentations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is our acronym:&#xA;SQWIGLES&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each letter stands for the following actions:&lt;br&gt;&#xA;S: &lt;strong&gt;Speak&lt;/strong&gt; your thinking when you read or write.&lt;br&gt;&#xA;Q: Ask &lt;strong&gt;questions&lt;/strong&gt;, especially open-ended questions.&lt;br&gt;&#xA;W: Encourage the students to &lt;strong&gt;write&lt;/strong&gt;.&lt;br&gt;&#xA;I: Help the students to find &lt;strong&gt;information&lt;/strong&gt;.&lt;br&gt;&#xA;G: &lt;strong&gt;Guide&lt;/strong&gt; the students’ problem-solving (as opposed to doing it for them).&lt;br&gt;&#xA;L: &lt;strong&gt;Listen&lt;/strong&gt; to the students’ thoughts and feelings.&lt;br&gt;&#xA;E: &lt;strong&gt;Explain&lt;/strong&gt; concepts a different way.&lt;br&gt;&#xA;S: Help the students to &lt;strong&gt;sum up&lt;/strong&gt; what they have learned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a link to a blog post I wrote about it with more comprehensive details &lt;a href=&quot;https://blogs.adelaide.edu.au/maths-learning/2016/09/20/sqwigles/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://blogs.adelaide.edu.au/maths-learning/2016/09/20/sqwigles/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2018-02-21T06:32:58.710" LastActivityDate="2018-02-21T06:32:58.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13638" PostTypeId="2" ParentId="13628" CreationDate="2018-02-21T22:02:21.703" Score="7" Body="&lt;p&gt;I'd like to expand a bit on The Chef's answer. Specifically, there's no need to require any kind of emphasis on &quot;real world&quot; applications. That is: Generally &quot;real world&quot; applications refers to some kind of industry mechanism in which some piece of mathematics is directly used (circuits, GPS, etc.). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student uninterested in a particular aspect of mathematics won't suddenly gain interest if one relates said aspect to a microchip an ATM—&lt;em&gt;unless the student were already interested in microchips or ATMs&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think applications of mathematics are always worth while, but an application doesn't necessarily mean some industry is using it. An application can be something as simple as &lt;em&gt;why&lt;/em&gt; this is a cool piece of mathematics. Does it have any other mathematical or even philosophical implications? Is there some crafty riddle that can be written or solved using this? Are there any non-obvious relationships this elucidates? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;(The following is a bit of a philosophical note, but it's the reason I put this answer down: I fear that it's all too common to want to justify the study of mathematics by appealing to some kind of industrial &lt;em&gt;production&lt;/em&gt; of something. I'd like more of an emphasis on the beauty of mathematics—its art and its cleverness. When a student asks &quot;Why am I learning this?&quot; he/she isn't asking &quot;What can I manufacture with this proof?&quot; or &quot;What market can I enter with this equation?&quot; but &quot;Why should I be interested in this?&quot; )&lt;/p&gt;&#xA;" OwnerUserId="9487" LastActivityDate="2018-02-21T22:02:21.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13639" PostTypeId="1" AcceptedAnswerId="13646" CreationDate="2018-02-22T00:06:17.337" Score="9" ViewCount="1731" Body="&lt;p&gt;I am teaching a real analysis class. Students in the class have inconsistent high school algebra skills. They now have a complete but tenuous understanding of $\varepsilon$-$\delta$ limits. I want to give them problems on which to consolidate their understanding of this definition and how to prove that a given proposed limit satisfies it. I want more examples to try out with them that are interesting but do not overly tax their algebra skills or demand too much sophisticated inequality-based reasoning, which is entirely new to them. I am looking for problems. Answers to this question could consist of problem suggestions, or pointing me to sources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To calibrate what I'm looking for, I will go in detail through an example I tried with them last class that was too hard. (It is not very hard if you have certain analysis skills, but these are precisely the skills they don't have.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We worked through an example like $\lim_{x\to -5}10x$; they felt they understood everything that was going on. I threw $\lim_{x\to 4} 3x^2$ at them, thinking it was just one level up, but it was way too much. They attacked it like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The limit is 48, by substitution. We need to find the constraint on $x$ that causes $|3x^2 - 48|$ to be $&amp;lt;\varepsilon$. So we need&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$-\varepsilon &amp;lt; 3x^2 - 48 &amp;lt; \varepsilon$$&#xA;i.e.&#xA;$$-\varepsilon + 48 &amp;lt; 3x^2 &amp;lt; \varepsilon + 48$$&#xA;i.e.&#xA;$$ -\frac{\varepsilon}{3} + 16 &amp;lt; x^2 &amp;lt; \frac{\varepsilon}{3} + 16$$&#xA;and so&#xA;$$ \sqrt{-\frac{\varepsilon}{3} + 16} &amp;lt; x &amp;lt; \sqrt{\frac{\varepsilon}{3} + 16}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well and good (subject to $\varepsilon \leq$ something), but now they needed a $\delta$ such that $0&amp;lt;|x-4|&amp;lt;\delta$ will force this last inequality. Using a taylor series approximation for the square root is not something they would have ever thought of. I proposed working with the previous inequality prior to the square root. My idea was to try to get them to square $4-\delta &amp;lt; x &amp;lt; 4 + \delta$ and then choose $\delta$ to force $-\varepsilon / 3 + 16 &amp;lt; (4-\delta)^2 &amp;lt; x^2 &amp;lt; (4+\delta)^2 &amp;lt; \varepsilon / 3 + 16$. First of all, I had to hand them this move. Secondly, finding the necessary $\delta$ was still hard, once I had handed them this setup. They went:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$-\frac{\varepsilon}{3} + 16 &amp;lt; (4 - \delta)^2 = \delta^2 - 8\delta + 16$$&#xA;so we need&#xA;$$ -\frac{\varepsilon}{3} &amp;lt; \delta^2 - 8\delta$$ &#xA;or&#xA;$$ 8\delta - \delta^2 &amp;lt; \frac{\varepsilon}{3}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and at this point started trying to remember how to solve a quadratic equation. I stopped them and pointed out that $\delta^2$ is positive, therefore this inequality is guaranteed as soon as $8\delta &amp;lt; \varepsilon / 3$, and they immediately concluded that $\delta$ had to be $&amp;lt;\varepsilon / 24$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we looked at the right inequality, $(4 + \delta)^2 &amp;lt; \varepsilon/3 + 16$. They got as far as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\delta^2 + 8\delta &amp;lt; \frac{\varepsilon}{3},$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and again I had to hand them a trick: pointing out that it never hurts to take a smaller delta, we can assume $\delta \leq 1$, implying $\delta^2 \leq \delta$, so that $\delta^2 + 8\delta \leq 9\delta$. From here, they concluded we need $\delta &amp;lt; \varepsilon / 27$. So in the end they chose $\delta =\min(1,\delta / 28)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of all the tricks I had to hand them, and handling the sides separately, it was hard for them to see the big picture at once, so in the end, I don't feel the problem consolidated their understanding of the definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I want is several problems that are not all the same (i.e. not just a bunch of $\lim_{x\to c} f(x)$ for $f(x)$ linear) but that do not put up the types of algebraic/analytic roadblocks I described above: places where you either have to (a) do some slightly heavier HS algebra, like solve a quadratic and then reason clearly about how its solution plays into these inequalities, or (b) use the analyst's trick of exploiting the slack afforded by the inequalities, as above when I pointed out to them we could take $\delta \leq 1$ in order to have $\delta^2 \leq \delta$, which is its own separate skill I would prefer to work on with them once we already have the essential goals of the $\varepsilon$-$\delta$ proof mastered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, I am asking either for your own suggested problems, or suggestions about good sources for such problems. Thanks in advance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;ADDENDUM:&lt;/strong&gt; In addition to Brendan's nice suggestion of degree 1 rational functions, I found some other nice ideas in &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0883857340&quot; rel=&quot;nofollow noreferrer&quot;&gt;Exploratory Examples for Real Analysis&lt;/a&gt;, by Snow and Weller:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Piecewise-linear functions, where you're taking the limit at a joint.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Things like $(x-5)\sin x$ as $x\to 5$, where the key is just to notice that $|\sin x|\leq 1$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I haven't tried them with my students yet to see how they will work, but these are just the types of ideas I was looking for.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="140" LastEditDate="2018-02-23T13:48:59.870" LastActivityDate="2018-02-23T19:03:03.483" Title="Interesting but very easy epsilon-delta problems?" Tags="|mathematical-analysis|problem-design|limits|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13640" PostTypeId="2" ParentId="13628" CreationDate="2018-02-22T01:21:37.293" Score="3" Body="&lt;p&gt;I studied pure and applied math at A level nearly 30 years ago, and I can honestly say that although I could do the exercises, I never had it explained to me the usefulness of what was being taught.  Applied was easy - there were applications for that branch.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pure math?   Well... the domain of the abstract is purely abstract.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A syllabus tells the teacher what students are expected to know / exhibit knowledge / &quot;understand&quot; / &quot;utilise&quot; / &quot;solve&quot; a problem, in order to pass a test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thats a box tick, right?  Well, kind of... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually anchoring the syllabus to concrete use-cases seems to be a rare teaching skill in itself.   The &quot;what&quot; is easy for any teacher.  The &quot;why&quot; may have even eluded them as students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Differential equations?  Factorisation? Integration? Turning points in graphs? Polynomials?   If only the explanations given to me actually resonated... I mean, why would anyone look at a graph and say ah, this is an X^2 behaviour, or say this is a polynomial.    WHY would I need to know where the turning points are?  What's that got to do with anything?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, the context of the explanations were so narrow, I simply couldn't connect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My advice to those teaching Mathematics: Real world examples please, and have a number of them at the ready - you never know which one will resonate with the student, and when the penny drops, it will make it all worth it!&lt;/p&gt;&#xA;" OwnerUserId="9490" LastActivityDate="2018-02-22T01:21:37.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13641" PostTypeId="1" CreationDate="2018-02-22T10:52:39.823" Score="6" ViewCount="343" Body="&lt;p&gt;I think it happens to a lot of students that (at least it feels like) really understanding the material does not really contribute to having a decent grade. It seems to me that quite often exams are more a test of general intelligence than a test whether a student really mastered the material. Of course its  usually the smarter students that also master the covered material more deeply than the rest, but still. It seems to miss the point of a final exam, which is to test whether the students mastered the material.&#xA;How do you guys handle the problem of designing exams that avoid this pitfall?&#xA;Moreover do you think that testing mastery of the material is the right goal to aim for?&lt;/p&gt;&#xA;&lt;p&gt;EDIT:&#xA;One example I could give are proofs that only require knowledge of a few basic operations that one is allowed to perform, but demand that one sees some sort of trick. Almost no (deep) understanding of the material is required and almost no advantage is gained from understanding the ideas behind the concepts. Someone smart without deep knowledge will easily outperform someone less talented who spent a lot of time understanding the concepts. But not only does it not reward understanding, but tricks that are really hard to see are heavily dependent on luck. I might see the trick one day but I won`t on another.&#xA;Of course both factors are impossible to rule out completely but the might well be mitigated if the teacher would try. If exams are designed like the one I complain about, they put too much focus on the performence of students under exam situations while a lot of other qualities are neglected.&lt;/p&gt;&#xA;&lt;p&gt;Take the following question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Determine the radius of convergence for the following infinite sum:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{k=1}^\infty \frac{\pi^k}{k^{1/4}}(e-x)^k$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One day I might do the right transformations and on another day I might not. But I am not sure whether questions like this test deep understanding of mathematical concepts.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand I think that there are also exam questions that are too easy. For example if one wants to test whether the students understood the jordan canonical form, I think it is a very poor test to simply require the students to calculate the matrix and cernels etc. . This can be done by any student who memorizes the required algorithm without understanding what it actually means.&lt;/p&gt;&#xA;" OwnerUserId="9358" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-02-25T10:46:13.393" Title="To which degree should exams test intelligence vs mastery of the material" Tags="|exams|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13642" PostTypeId="1" AcceptedAnswerId="13643" CreationDate="2018-02-22T14:49:07.480" Score="1" ViewCount="85" Body="&lt;p&gt;Which course is harder for students? Complex variables designed for engineers/ scientists&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;odes+linear algebra course for Engineers/Scientists?&lt;/p&gt;&#xA;" OwnerDisplayName="user9495" LastActivityDate="2018-02-22T18:31:50.067" Title="Complex Variables for Engineers versus Ordinary Differential Equations with Linear Algebra difficulty" Tags="|advising|course-sequence|" AnswerCount="2" CommentCount="1" ClosedDate="2018-02-22T17:58:01.410" ContentLicense="CC BY-SA 3.0" />
  <row Id="13643" PostTypeId="2" ParentId="13642" CreationDate="2018-02-22T16:01:13.383" Score="1" Body="&lt;p&gt;The difficulty of the courses depends on the content and execution of the courses in question, and as such, the question does not have an answer.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2018-02-22T16:01:13.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13645" PostTypeId="2" ParentId="13642" CreationDate="2018-02-22T17:56:40.607" Score="-1" Body="&lt;p&gt;Totally guessing but I think the specific ODE course might be a little harder since it is already a big topic without matrices and then you add in LA (so a littler tougher than just a classical ODE course). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I would do the ODE course first.  It is much more fundamental than contour integration is to engineering.  Almost everything is a diffyQ in engineering.  Use of complex variables is actually rather rare in normal engineering homework.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(By use complex variables, I mean full on complex analysis.  Of course you use imaginary numbers and phase angles in EE circuit analysis.  But that is more like basic algebra 2 type stuff.)&lt;/p&gt;&#xA;" OwnerUserId="9428" LastEditorUserId="9428" LastEditDate="2018-02-22T18:31:50.067" LastActivityDate="2018-02-22T18:31:50.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13646" PostTypeId="2" ParentId="13639" CreationDate="2018-02-22T19:45:37.937" Score="6" Body="&lt;p&gt;I suggest using rational functions. Students are used to evaluating limits of rational functions because such examples are prevalent in most calculus courses. Moreover, I think the work required to prove such a limit is a little more challenging than that for a linear function but mostly in terms of the algebra and a little bit of logic. But, by following the thought process I outline below, I think students can learn to see the choice of $\delta=\min\{blah,yadda\}$ as a natural, intuitive choice based on their understanding of limits, and not some mystically dark mathemagic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Claim:&lt;/strong&gt; $\lim_{x\to 0}\frac{2}{x+1}=2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Scratch work for proof:&lt;/strong&gt; Let $\varepsilon&amp;gt;0$ be arbitrary and fixed. The goal is $2-\varepsilon &amp;lt; \frac{2}{x+1} &amp;lt; 2+\varepsilon$ so we'll work backwards from there. We want to eventually see something of the form $|x|&amp;lt;\delta$ since $x\to 0$ is the limit. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiply through by $x+1$ everywhere: $2x+2-x\varepsilon-\varepsilon &amp;lt; 2 &amp;lt; 2x+2+x\varepsilon+\varepsilon$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did we just multiply through by a negative and need to switch the inequality signs? Well, no, as long as $x+1&amp;gt;0$. In other words, as long as $x&amp;gt;-1$, we're fine. Since $x\to 0$, the intuition is that $x$ is &lt;em&gt;eventually close to 0&lt;/em&gt;, and so certainly greater than $-1$. So, let's leave this aside for now and worry about it later if we need to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Subtract 2 from everywhere: $2x-x\varepsilon-\varepsilon&amp;lt;0&amp;lt;2x+x\varepsilon+\varepsilon$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Factor out the $x$ where we can, since we're hoping to see $|x|&amp;lt;\delta$ sometime soon: $x(2+\varepsilon)-\varepsilon &amp;lt; 0 &amp;lt; x(2+\varepsilon)+\varepsilon$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We also want to see $x$ in the middle, so let's subtract that factored $x$ expression everywhere: $-\varepsilon &amp;lt; -x(2+\varepsilon) &amp;lt; \varepsilon$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Divide by $(2+\varepsilon)$ everywhere: $-\frac{\varepsilon}{2+\varepsilon} &amp;lt; -x &amp;lt; \frac{\varepsilon}{2+\varepsilon}$ (Did we just divide by a negative? Surely nope, since $\varepsilon&amp;gt;0$ so $2+\varepsilon&amp;gt;2$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we have something like $|x|&amp;lt;\delta$. Regardless of $x$'s sign, as long as its magnitude is less than $\frac{\varepsilon}{2+\varepsilon}$, then we have what we want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, we would like to set $\delta=\frac{\varepsilon}{2+\varepsilon}$. Then, we could start with $|x|&amp;lt;\delta$ and work all of these steps backwards to get to $\left|\frac{2}{x+1}-2\right|&amp;lt;\varepsilon$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there was that one iffy step when we needed $x&amp;gt;-1$. When writing our proof, working that particular step backwards, we would be dividing through by $x+1$ everywhere. At that point, all we would know about $x$ is that it's between $-\delta$ and $+\delta$. What if $\delta$ were too big? What if $\delta \geq 1$? Then it's possible for $x\leq -1$. Oops! How do we avoid this? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only way that would happen is if the $\varepsilon$ we were given made it so that $\frac{\varepsilon}{2+\varepsilon}\geq 1$. However, algebraically solving this yields $0\geq 2$, so that is impossible. Indeed, $2+\varepsilon&amp;gt;\varepsilon$, so the ratio must be smaller than 1. That's it! The $\delta$ we choose will work for any $\varepsilon$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could even point out here that, if we hadn't noticed whether or not $\frac{\varepsilon}{2+\varepsilon}\geq 1$ were possible, we could have just effectively said, &quot;Make sure $\delta\geq 1$ is never possible. If our formula, in terms of $\varepsilon$, would make that happen, instead just make $\delta$ something smaller, like $\frac12$.&quot; This motivates a definition like: $\delta=\min\{\frac12,\frac{\varepsilon}{2+\varepsilon}\}$. In this example, you don't even &lt;em&gt;need&lt;/em&gt; to do that, but you can use it as an occasion to point out why you might in other situations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Final note: I initially misread your post as requesting examples of limits with sequences, taking an $\varepsilon$ and choosing a sufficiently large $N$. I based the above example on a sequence example I used in my Real Analysis course just last week: $\lim_{n\to\infty}\frac{2n}{n+1}=2$. I converted that example into the one above by setting $x=\frac{1}{n}$. You can do this to create your own similar examples.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Addendum:&lt;/em&gt; I do like user Santiago Canez's comment about bounding $|x|$ versus bounding $|x+1|$. I suggest presenting that &lt;em&gt;after&lt;/em&gt; this method and having a discussion with the students comparing the two. Ultimately, the same proof is written, but it will be instructive for them to compare the different modes of thinking to create that proof.&lt;/p&gt;&#xA;" OwnerUserId="80" LastEditorUserId="80" LastEditDate="2018-02-23T19:03:03.483" LastActivityDate="2018-02-23T19:03:03.483" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13647" PostTypeId="2" ParentId="13641" CreationDate="2018-02-22T19:56:53.193" Score="8" Body="&lt;p&gt;I think majority of the questions on an exam should be pretty straight, not tricky.  Remember all the material is new to the students so in that sense it is already a &quot;trick&quot;.  You can still have a few questions that are harder by doing ones that are complex in terms of manipulation or that combine two methods or that are word problems.  But even that I would go easy on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is unless you are teaching a Putnam class or the like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The smarter kids will still do better because they learn the material better and have better memories for the overall class (as usually more than one idea is tested).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I think there is a bad tendency to want to be cute with tricky problems and then have to curve the exams.  And not even really test well if the basic concepts are learned.  This appeals to people who already know the material and want to do tricky stuff with it (most instructors) but is not sound pedagogy.  It would be like an NFL coach teaching super complicated plays to a group of players who need to master the center-quarterback exchange.&lt;/p&gt;&#xA;" OwnerUserId="9498" LastActivityDate="2018-02-22T19:56:53.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13649" PostTypeId="1" AcceptedAnswerId="13653" CreationDate="2018-02-23T09:12:16.340" Score="17" ViewCount="12721" Body="&lt;p&gt;When teaching implicit differentiation in freshman calculus I lack good examples which might help students relate the theory to applications in other sciences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I'm looking for (relatively simple) examples from physics/engineering/life science/economics etc. of two quantities $x,y$ that depend on each other in such a way, that neither $x$ is a function of $y$, nor $y$ is a function of $x$. In other words: $x,y$ should fulfil some equation, such that there exists an $x$ value for which the equation has at least two solutions in $y$ and vice versa. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Equations are great, but if you have some examples where the relation between $x$ and $y$ is not easily expressed by an equation, but instead by a graph or some other data that is also fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My students are mainly first year engineers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&#xA;There is a related question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/22&quot;&gt;here&lt;/a&gt;, but it's aimed for advanced analysis courses and doesn't ask for examples from applied areas. It has at least one applicable answer with isolines, but to explain it with equations in the firs semester single variable calculus, I would probably have to introduce functions of several variables.&lt;/p&gt;&#xA;" OwnerUserId="590" LastEditorUserId="590" LastEditDate="2018-02-23T15:35:28.590" LastActivityDate="2022-11-01T16:04:05.337" Title="&quot;Real world&quot; examples of implicit functions" Tags="|calculus|examples|functions|" AnswerCount="8" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13651" PostTypeId="2" ParentId="13649" CreationDate="2018-02-23T13:20:02.867" Score="4" Body="&lt;p&gt;How about &lt;strong&gt;Kepler's equation&lt;/strong&gt; (has to do with orbits of planets)&#xA;$$&#xA;M = E - e\sin E&#xA;$$&#xA;($M$ is the mean anomaly, $E$ is the eccentric anomaly, and $e$ is the eccentricity)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For fixed $M$, this defines $E$ implicitly as a function of $e$, but we cannot solve it for $E$ explicitly.&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2018-02-23T20:33:22.983" LastActivityDate="2018-02-23T20:33:22.983" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13652" PostTypeId="2" ParentId="13641" CreationDate="2018-02-23T14:22:35.977" Score="2" Body="&lt;p&gt;Have a large portion of your test consist of written explanations of the concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, here is a typical plug and chug exam problem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Consider the region bounded by $y=x^2+x+10$, $x=1$, $x=3$ and $y=2$. Find the volume of the solid obtained by revolving this region about the $x$-axis &quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students will get this correct by memorizing a particular form of the solution.  You could make it &quot;tricky&quot; by having them revolve about some line other than the $x$-axis (if they have not memorized a formula for that as well).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This same problem can be transformed into a test of conceptual understanding by adding the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Give a sum (using sigma notation) which approximates the volume of this solid by using $N$ slices.  Draw a picture of an arbitrary slice and explain why each part of your formula makes sense.  Use your calculator to evaluate this sum for $N=10$, $N=100$, and $N=1000$.  Finally, write the limit of this sum as a definite integral and evaluate using the Fundamental Theorem of Calculus.  Does this exact answer agree with your approximations?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly, you cannot include as many questions on a test if this is the kind of depth you are looking for.  However, it doesn't require exceptional intelligence:  just hard work and dedication to understanding the concepts involved.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-02-23T14:22:35.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13653" PostTypeId="2" ParentId="13649" CreationDate="2018-02-23T15:22:58.237" Score="11" Body="&lt;p&gt;Many linkages have this sort of behavior and are, perhaps, relevant to a mechanical engineer. For example, you could analyse a bicycle suspension—a type of four-bar linkage.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Linkage_(mechanical)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia: Linkage (mechanical)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Four-bar_linkage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia: Four-bar linkage&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="5220" LastEditDate="2022-09-29T21:46:22.590" LastActivityDate="2022-09-29T21:46:22.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13654" PostTypeId="2" ParentId="13649" CreationDate="2018-02-23T15:40:48.027" Score="5" Body="&lt;p&gt;In a conservative system in one dimension we have $E(x,v) = \frac{1}{2}mv^2+U(x)$ where $-\frac{dU}{dx} = F_{net}$. In other words, we have $E = KE+ PE$ where $PE=U$ and $KE = \frac{1}{2}mv^2$. Now, to study the rate of change in KE or PE we can relate it to the rate of change of the other, or $x$ or $v$. You can give a Poincare Plot of position verses velocity with energy level curves plotted. No equation for $E(x,v)$ need be given, but they could still reason various related rates depending on what you gave them.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-02-23T15:40:48.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13655" PostTypeId="2" ParentId="13639" CreationDate="2018-02-23T17:10:00.650" Score="0" Body="&lt;p&gt;What I would in this situation as instructor is to rewrite $3x^2$ for them as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$3x^2=48+24(x-4)+3(x-4)^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although this will seem like pulling a rabbit out of hat to students, verification is simple algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From here, students should be able to focus on the analysis involved in estimating $|3x^2-48|$ in terms of $|x-4|$.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2018-02-23T17:10:00.650" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13656" PostTypeId="2" ParentId="13649" CreationDate="2018-02-23T17:49:37.277" Score="6" Body="&lt;p&gt;For circular motion, you have x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt; =  r&lt;sup&gt;2&lt;/sup&gt;, so except for at the ends, each x has two y solutions, and vice versa. Harmonic motion is in some sense analogous to circular motion. For instance, for Hooke's law, if you set y = $\sqrt{k/m}v$, then x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt; = constant; if you know the position of a pendulum, you know the speed, but not the direction, and given a velocity, you know the magnitude of the displacement, but not the direction.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2018-02-23T17:49:37.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13657" PostTypeId="2" ParentId="13649" CreationDate="2018-02-23T18:17:57.263" Score="9" Body="&lt;p&gt;Economics has a ton of applications since they are interested in relationships between variables, not functions per se. Two simplish ones that come to mind are&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Price elasticity of demand: Supply curves are just a relation between&#xA;supply and demand, it need not be functional. You can derive the&#xA;price elasticity in this setting. &lt;a href=&quot;https://courses.soe.ucsc.edu/courses/ams11a/Fall12/01/attachments/13719&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's are some good notes from Y.&#xA;Katznelsons course at UCSC&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A typical problem looks like: Price &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and quantity &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; are related by &lt;span class=&quot;math-container&quot;&gt;$p^2 - pq + q^2 = 400$&lt;/span&gt;. When the price is \$4, estimate the percent change in profit when the price is raised 1%.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If your students are particularly bright, you can use &lt;span class=&quot;math-container&quot;&gt;$p^2 - 2pq + q^2 = 400$&lt;/span&gt; and ask them to show you that the analysis is the same for &lt;span class=&quot;math-container&quot;&gt;$p - q = 20$&lt;/span&gt;, which is an equivalent equation since &lt;span class=&quot;math-container&quot;&gt;$p,q \geq 0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To generalize the above, comparative statics uses implicit differentiation  to study the effect of variable changes in economic models. &lt;a href=&quot;https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/dif/t&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's a decent introduction with example problems.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Preference bundles, utility and indifference curves. You have to gloss over some machinery but you're essentially doing calculus on level curves. If this looks interesting peek into the first few chapters of Hal Varian's Intermediate Microeconomics.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A sample problem would look like: Kim has \$10 to buy flour and GI Joe's. A bag of flour is \$.50 and a GI Joe is \$2. If the expected utility of &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; bags of flour and &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt; GI Joes is &lt;span class=&quot;math-container&quot;&gt;$U(x_1,x_2) = \sqrt{x_1^2 + x_2^2}$&lt;/span&gt;, what point along the budget curve maximized utility? &lt;em&gt;Sol: Find the point where tangent line to is the budget line.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="20529" LastEditDate="2022-11-01T16:04:05.337" LastActivityDate="2022-11-01T16:04:05.337" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13658" PostTypeId="2" ParentId="13639" CreationDate="2018-02-23T18:55:41.533" Score="-2" Body="&lt;p&gt;You are making this WAAAAAY too complicated. One simplification is you should be centering around the value you're taking the limit at. So in this case, if you define d = x-4, then x = 4+d and 3x&lt;sup&gt;2&lt;/sup&gt; = 3(16+8d+d&lt;sup&gt;2&lt;/sup&gt;). Since we want an upper limit, for small d we can replace d&lt;sup&gt;2&lt;/sup&gt; with 8d, so we have 48+48d. So if we want that within $\epsilon$ of 48, we just have to have d&amp;lt;$\epsilon$/48. Simple.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is generally how to do these proofs:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Define d = x-x&lt;sub&gt;0&lt;/sub&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Find D(d) = f(x&lt;sub&gt;0&lt;/sub&gt;+d)-f(x&lt;sub&gt;0&lt;/sub&gt;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Find an upper bound B(d) on D(d) that's easily invertible.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Set $\delta$ = B&lt;sup&gt;-1&lt;/sup&gt;($\epsilon$)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If the function has a nice derivative, the process is even simpler. The derivative of 3x&lt;sup&gt;2&lt;/sup&gt; is 6x. Take a neighborhood around 4, say 0 to 10, and the derivative is at most 60. So this gives $\delta = \epsilon/60$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So while this isn't an answer to your question &quot;How do I get simpler problems?&quot;, I think this is the wrong question to be asking, because the problem isn't that your problems are to complicated, it's that your techniques are too complicated. I don't see much point in asking &quot;What are some problems that can still be solved even if you're going about them completely the wrong way?&quot;&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2018-02-23T18:55:41.533" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13659" PostTypeId="2" ParentId="13649" CreationDate="2018-02-24T03:14:12.327" Score="4" Body="&lt;p&gt;Probably the simplest algebraic examples of what you're looking for are given by the elliptic curves. In Weierstrass, Jacobian quartic, and Jacobian intersection form (respectively) such curves are described by the equations&#xA;$$\begin{align}(x,y)&amp;amp;:\mathbb{R}^2 &amp;amp;y^2 &amp;amp;=4x^3-g_2x-g_3 &amp;amp;\mathrm{\delta}t&amp;amp;\equiv \frac{\mathrm{d}x}{y}\\&#xA;(x,y)&amp;amp;:\mathbb{R}^2 &amp;amp;y^2 &amp;amp;=(1-x^2)(1-mx^2)&amp;amp;\mathrm{\delta}t&amp;amp;\equiv \frac{\mathrm{d}x}{y}\\&#xA;(s,c,d)&amp;amp;:\mathbb{R}^3 &amp;amp;s^2+c^2=1&amp;amp;\quad ms^2+d^2=1&amp;amp;\mathrm{\delta}t&amp;amp;\equiv \frac{\mathrm{d}s}{cd}&#xA;\end{align}$$&#xA;In physics, these are the nondimensionalized constraint equations for a particle in a cubic potential, a particle in a certain quartic potential, and the pendulum, respectively; $\mathrm{\delta}t$ is a nondimensionalized time increment that features in the respective physical problem. (For more details on the physics, see the references in the DLMF below.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admit that I've abstracted a bit from the physics and that these examples fit into situations that other answers have described. Still, I think these implicit equations deserve special mention for a few reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Unlike the implicit equations that determine conic sections, it is provably impossible to describe these curves using a rational-function parametrization—you can't &quot;cheat&quot; and use an elementary substitution.&lt;/li&gt;&#xA;&lt;li&gt;The special structures that such curves exhibit make them prototypes for extensive families of &lt;em&gt;exactly solvable models&lt;/em&gt; in physics, generalizing the way we use conic sections (see the references in the DLMF for the &lt;a href=&quot;https://dlmf.nist.gov/22.19&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacobian&lt;/a&gt; and &lt;a href=&quot;https://dlmf.nist.gov/23.3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weierstrass&lt;/a&gt; elliptic functions).&lt;/li&gt;&#xA;&lt;li&gt;For special values of the parameter $m$ or $(g_2,g_3)$, these algebraic curves have rational-number solutions that have been &lt;a href=&quot;http://www.lmfdb.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;extensively catalogued&lt;/a&gt;—useful if you want to &quot;plug in numbers&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9510" LastActivityDate="2018-02-24T03:14:12.327" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13660" PostTypeId="2" ParentId="13649" CreationDate="2018-02-24T06:41:23.110" Score="4" Body="&lt;p&gt;I want to mention  $$\tag1x^2+y^2=1, $$that has already appeared in other answers. We were all led to think (maybe involuntarily) by our teachers that this can easily be solved as $$\tag2y=\sqrt {1-x^2} , $$ or the same with a minus in front.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point is to question the &quot;easily&quot; above. We all became familiar with the square root from an early age, but when you think about it our only access to it was in the case of some easy numbers, or through a magic black box (the calculator).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The equation $(1) $ allows us to analyze the function  $(2) $ in an easier and more direct way (for instance, think about sketching the graph).&lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2018-02-24T06:41:23.110" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13661" PostTypeId="1" CreationDate="2018-02-26T07:19:37.700" Score="18" ViewCount="4145" Body="&lt;p&gt;This is a common way to write the steps during solving equations:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/r090K.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/r090K.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in GeoGebra the steps are shown this way (the highlighted part):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gbX04.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gbX04.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm going to use GeoGebra to teach equations. Is it OK to let the students write the steps just like GeoGebra (I mean, with paranthesis)?&lt;/p&gt;&#xA;" OwnerUserId="1229" LastActivityDate="2018-03-04T19:33:28.993" Title="How to Write Steps of Solving Equations?" Tags="|algebra|writing|middle-school|" AnswerCount="10" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13662" PostTypeId="2" ParentId="13661" CreationDate="2018-02-26T08:52:02.323" Score="10" Body="&lt;p&gt;The problem with&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$(4x+7=6x+2)-6x$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is that there is no subtraction operation that involves subtracting a term from an equation.  Subtraction involves subtracting a term from a term.  So the correct notation is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$(4x+7)-6x=(6x+2)-6x$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-02-26T08:52:02.323" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13663" PostTypeId="2" ParentId="13661" CreationDate="2018-02-26T13:26:52.500" Score="13" Body="&lt;p&gt;I wouldn't do that. The parenthesis in use are also used for legal expressions within equations. So you can end with one line containing the same parenthesis meaning different things, what looks like a seed of confusion to me.&lt;/p&gt;&#xA;&lt;p&gt;If you prefer, you can replace the GeoGebra parenthesis with some others i.e. &amp;lt;&amp;gt;, or [] etc. to keep the notation similar to Geogebra but to remove (reduce) the potential for the confusion.&lt;/p&gt;&#xA;&lt;p&gt;Or what about writing steps like this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;4x + 7 = 6x + 2&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt; -6x = -6x&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;-2x + 7 = 2&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This way you also get a bit of sensibility for systems of equations later.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-02-26T13:26:52.500" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13664" PostTypeId="2" ParentId="13661" CreationDate="2018-02-26T16:56:14.317" Score="10" Body="&lt;p&gt;Speaking as someone who has taught college precalculus several times, I have an intense dislike for the way that Geogebra writes this step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, it is very important to emphasize to students that we are subtracting 6x from &lt;strong&gt;both&lt;/strong&gt; sides of the equation, which means of course that we are subtracting 6x from &lt;strong&gt;two different&lt;/strong&gt; expressions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we write &quot;-6x&quot; only once, it may cause students to have the unfortunately common misconception that subtracting 6x is something you can &quot;just do&quot;. (You can, of course, do it to &lt;strong&gt;both&lt;/strong&gt; sides of a statement that says that &lt;strong&gt;two&lt;/strong&gt; expressions are equal.)&lt;/p&gt;&#xA;" OwnerUserId="9249" LastActivityDate="2018-02-26T16:56:14.317" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13665" PostTypeId="1" CreationDate="2018-02-26T21:00:09.587" Score="4" ViewCount="276" Body="&lt;p&gt;Do you know any good sources of free/open application-style &lt;em&gt;problems&lt;/em&gt; for the precalculus level?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to use an OER precalculus book, but the few I am most happy with seem to lack (what we would call) application problems. There might be a few here and there in a text, but skill questions dominate the homework section.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I am not looking for recommendations for OER precalculus texts, but rather just sources of good problems. Do these exist without the standard expository content found in textbooks? Do feel free to suggest a book if it contains lots of good application/concept problems and less &quot;factor these trinomials&quot; kind of skill problems.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-03-01T22:19:45.370" Title="Resources for precalculus applications" Tags="|geometry|word-problems|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13666" PostTypeId="2" ParentId="13661" CreationDate="2018-02-26T23:33:05.827" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I'm going to use GeoGebra to teach equations. Is it OK to let the students write the steps just like GeoGebra (I mean, with parenthesis)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would &lt;em&gt;not&lt;/em&gt; allow this in my class, but it would depend what you think is good/clear/consistent notation for your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The developers at GeoGebra made a choice for showing algebra steps, but that doesn't mean we have to accept it as a norm. If the developers decided to allow something like $$5x+3=15x$$ $$sub/div (5x+3=15x)$$ $$x=\frac{3}{10}$$&#xA;to indicate a &quot;subtract first, then divide&quot; procedure, would you allow students to write this? I doubt it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that I am fighting &lt;em&gt;all the time&lt;/em&gt; to get my students to write in a clear and consistent manner. They'll write something like $$4x+3=2x+11$$ $$3-4x+3 = 2x+11-3$$ $$4x=2x+8$$ explaining that they've &quot;subtracted 3 from both sides&quot;. They feel that because &lt;em&gt;they&lt;/em&gt; knew what they were doing, that I should be okay with how they wrote. Of course, I am not okay with it, and part of my job is to correct their use of notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, I would not let GeoGebra's shorthand become part of your students' practice. If you do, their future teachers will be justifiably confused, and your students will have to spend valuable time relearning how to show algebraic steps.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-02-26T23:33:05.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13667" PostTypeId="2" ParentId="13661" CreationDate="2018-02-27T00:40:11.253" Score="-1" Body="&lt;p&gt;Geogebera is bad here.  Stop trying to make something bad work.  Do it the old fashioned way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kids have a hard tie with this stuff.  Show every step and emphasize that it is an equality and you perform same operation to each side (NOT move stuff from one side to other).  Once kids have been drilled hard they don't need to show every step.  But when they are new, drill it in detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ax + b = cx - d&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ax + b - cx = cx - d - cx&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ax -cx + b = -d&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a-c)x = -d - b&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a-c)x = -(d + b)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;x = -(d+b)/(a-c)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;x = (d+b)/(c-a)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;x = [evaluate fraction numerically]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And don't just have them solve it generally.  Have them solve numerical examples over and over.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Drill, drill, drill.  If you think that is too tedious for our young computer generation, don't complain to me (or have their higher course math and science teaches complain) when they mess up algebra, get signs reversed or ratios upside down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Pedant protection, making a point here.  If I messed up my algebra, it's kharma.]&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-02-27T00:40:11.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13668" PostTypeId="2" ParentId="13661" CreationDate="2018-02-27T02:47:34.047" Score="3" Body="&lt;p&gt;If I had to communicate the solution to this problem to someone else in writing, I'd probably write&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...gives&#xA;  $$4x+7=6x+2\text{.}$$&#xA;  Subtracting $6x$ from both sides of this equation gives&#xA;  $$-2x+7 = 2\text{.}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If the equality were one in a long chain, I'd write&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\begin{split}&amp;amp;\ldots \\ 4x+7&amp;amp;=6x+2 \\ -2x+7&amp;amp;\stackrel{\text{(a)}}{=}2 \\ &amp;amp;\ldots \end{split}$$&#xA;  We arrive at (a) by subtracting $6x$ from both sides...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If I were just typing up notes for myself, I'd write&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\begin{split}&amp;amp;\ldots \\ 4x+7&amp;amp;=6x+2 &amp;amp;&amp;amp; \\ -2x+7&amp;amp;=2 &amp;amp; &amp;amp;\text{subtract }6x\text{ from both sides} \\ &amp;amp;\ldots \end{split}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and in my calculation notebook I'd probably write&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\begin{split}&amp;amp;\ldots \\ 4x&amp;amp;+7&amp;amp;=\,\,\, 6x+&amp;amp;2\\ \underline{-6x}&amp;amp; &amp;amp;\quad\underline{-6x} \\ -2x&amp;amp;+7&amp;amp;=&amp;amp;2 \\ &amp;amp;\ldots \end{split}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems that, in practice, I use words, not notation, to convey the idea of applying the same function to both sides of an equality to get a new equality. Flipping through a few textbooks suggests that most other mathematics users are largely the same in this respect. The preference for rhetorical instead of symbolic description and the lack of consensus on a notation suggest that having a notation for this idea is not that useful for most people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, people who work with proof assistants &lt;em&gt;do&lt;/em&gt; need such an idiom, because doing the same thing to both sides of an equality to get a new equality is an important proof technique that needs to be formalized. For instance, &lt;a href=&quot;https://github.com/idris-lang/Idris-dev/blob/2eec9992994fd9853bae86bdaeb7d633d4c277e6/libs/prelude/Prelude/Basics.idr#L48&quot; rel=&quot;nofollow noreferrer&quot;&gt;Idris&lt;/a&gt; and &lt;a href=&quot;https://github.com/agda/agda-stdlib/blob/23625bed78e774bdf9986717fb46ac7369a54987/src/Relation/Binary/PropositionalEquality.agda#L29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Agda&lt;/a&gt; have &lt;code&gt;cong&lt;/code&gt;:&lt;/p&gt;&#xA;&#xA;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;cong : {f : t -&amp;gt; u} -&amp;gt; (a = b) -&amp;gt; f a = f b&#xA;cong Refl = Refl&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;cong : ∀ {a b} {A : Set a} {B : Set b}&#xA;   (f : A → B) {x y} → x ≡ y → f x ≡ f y&#xA;cong f refl = refl&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;I expect that this necessity is the motivation that GeoGebra's designers had in mind.&lt;/p&gt;&#xA;" OwnerUserId="9510" LastActivityDate="2018-02-27T02:47:34.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13669" PostTypeId="1" AcceptedAnswerId="13680" CreationDate="2018-02-27T03:42:11.300" Score="14" ViewCount="1151" Body="&lt;p&gt;Sometimes when we discuss limits verbally, we may say that a variable $x$ being &quot;very small&quot; (assuming that $x$ is a real number). But this could mean any one of the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The number $x$ is a positive number with very small absolute value (for example, $x=10^{-10}$ may be considered as &quot;very small&quot;).&lt;/li&gt;&#xA;&lt;li&gt;The number $x$ is a negative number with very large absolute value (for example, $x=-10^{10}$ may be considered as &quot;very small&quot;).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Of course in writing the meaning is usually clear from the context. However, how should one avoid verbal ambiguity when teaching a class? I have heard people using the following terminologies to describe a negative number with large absolute value:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;very small.&lt;/li&gt;&#xA;&lt;li&gt;very negative.&lt;/li&gt;&#xA;&lt;li&gt;very negative large.&lt;/li&gt;&#xA;&lt;li&gt;negative with very large absolute value.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The fourth one is probably most clear but is also more or less verbose. Now the question is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is the most recommended terminology for such a number?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2018-02-27T10:57:23.420" LastActivityDate="2020-05-21T04:31:45.697" Title="What is a recommend way to describe a negative number with large absolute value?" Tags="|terminology|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13670" PostTypeId="2" ParentId="13661" CreationDate="2018-02-27T07:01:54.013" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/13663/9522&quot; title=&quot;Thinkeye&amp;#39;s answer&quot;&gt;Thinkeye's&lt;/a&gt; answer is good in that it easily extends to dividing two related equations and similar, more advanced operations. On the other hand, for the sake of brevity, I would suggest the way I have been taught to explicitly specify operations:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;6x + 14y = 4x + 12y \quad | -4x&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A vertical line to separate the equation from the intended operation. Such vertical lines are frequently used to indicate &quot;a divides b&quot; as in $a|b$, but I think that there is little risk of confusion here. Same is true for absolute value delimiters.&lt;/p&gt;&#xA;" OwnerUserId="9522" LastActivityDate="2018-02-27T07:01:54.013" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13671" PostTypeId="1" AcceptedAnswerId="13677" CreationDate="2018-02-27T07:41:59.973" Score="15" ViewCount="533" Body="&lt;p&gt;By &quot;polynomial factorization over the integers&quot;, I mean problems and solutions like the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Problem:&lt;/strong&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Find a factorization into irreducible polynomials for&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;$24x^2 +x - 10$ and&lt;/li&gt;&#xA;  &lt;li&gt;$5x^3 -2x^2-x+24$.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Suppose a factorization $24x^2+x-10=(ux+v)(u'x+v')$ where $0&amp;lt;u\leq u'$, $uu'=24$, and $vv'=-10$, and look for tuple $(u,v,u',v')$ satisfying $uv'+vu'=1$. By evaluating $uv'+vu'$ at all 32 guesses we find $24x^2+x-10 = \boxed{(3x + 2)(8x-5)}$.&lt;/li&gt;&#xA;  &lt;li&gt;$5x^3 -2x^2-x+24$ has a linear factor if and only if they have a rational root whose numerator divides the $24$ and whose denominator divides the $5$. By evaluating the polynomial at all 32 guesses, we find that $-\tfrac{8}{5}$ is a root. Through polynomial long division, the quotient of the original polynomial by $5x+8$ is found to be $x^2-2x+3$. Since the latter is irreducible, we have $5x^3 -2x^2-x+24=\boxed{(5x + 8)(x^2 - 2x + 3)}$.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This schema is set apart from other polynomial problems and techniques in that it&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;fundamentally relies on integer factorization (of the constant term and leading coefficient),&lt;/li&gt;&#xA;&lt;li&gt;requires making a potentially large number of guesses (depending on the number of factors found for the coefficients), and&lt;/li&gt;&#xA;&lt;li&gt;tends to produce reducible polynomials and linear factors at a frequency much higher than if random integer coefficients were chosen (because of the choices made by the authors of such problems).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;That background aside, I'm asking two questions: one historical, one normative.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Who introduced polynomial factorization over the integers into secondary school curricula, and what was &lt;em&gt;their&lt;/em&gt; justification for doing so? I've had trouble finding historical documents and secondary sources that would answer this question—any references pointing in the right direction would be greatly appreciated.&lt;/li&gt;&#xA;&lt;li&gt;Why &lt;em&gt;should&lt;/em&gt; secondary-school students be taught about polynomial factorization over the integers? I'm looking for evidence—personal experience, studies in mathematical pedagogy, applications in science/technology/engineering—that teaching this topic furthers some goal of the educational system.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9510" LastActivityDate="2018-02-28T05:59:39.580" Title="Why is polynomial factorization over the integers part of secondary school curricula?" Tags="|secondary-education|reference-request|curriculum|history|solving-polynomials|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13672" PostTypeId="1" CreationDate="2018-02-26T18:08:10.210" Score="9" ViewCount="584" Body="&lt;p&gt;I am a graduate teaching assistant at a larger state university teaching a college algebra class. Today we begin our decent into polynomials and one of the facts we will soon get to is the connection between roots of a polynomial and linear factors. We do not teach polynomial long division (explicitly avoid it in fact), which is normally how I could convince students that $x - a$ is a factor of a polynomial $p(x)$ if and only if $p(a) = 0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is how can I convince my students of this fact without appealing to division? Keep in mind that these are largely college freshman who do not necessarily care for explanations. While they would take my word on something and memorize it if I told them it was necessary, I still like to convince them this is indeed mathematical fact and not something that has come out of thin air. &lt;/p&gt;&#xA;" OwnerUserId="9884" OwnerDisplayName="Oiler" LastActivityDate="2018-05-09T19:34:18.427" Title="How to show $(x - a)$ is a factor of a polynomial $p(x)$ if and only if $p(a) = 0$ (without division)" Tags="|roots|" AnswerCount="7" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="13673" PostTypeId="2" ParentId="13672" CreationDate="2018-02-26T18:11:16.930" Score="-3" Body="&lt;p&gt;We have by division algorithm that $$f(x)=(x-a)Q(x)+R$$where $R$ is the real residue of division therefore $$f(x)=R$$which means that $$R=0\leftarrow\rightarrow f(a)=0$$&lt;/p&gt;&#xA;" OwnerDisplayName="Mostafa Ayaz" LastActivityDate="2018-02-26T18:11:16.930" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13674" PostTypeId="2" ParentId="13672" CreationDate="2018-02-26T18:28:49.233" Score="6" Body="&lt;p&gt;This answer presumes that the students know polynomial multiplication. In particular, &#xA;$$(x-a)(x^{n-1}+ax^{n-2}+\cdots+a^{n-1})=x^{n}-a^{n}.$$&#xA;Since &#xA;$$p(x) = c_{n}x^{n}+c_{n-1}x^{n-1}+\cdots+c_0=(x-a)q(x),$$&#xA;and&#xA;$$p(a) = c_{n}a^{n}+c_{n-1}a^{n-1}+\cdots+c_0,$$&#xA;we have&#xA;$$p(x)-p(a) = c_n (x^{n}-a^{n})+c_{n-1}(x^{n-1}-a^{n-1})+\cdots+c_1(x-a)=(x-a)r(x).$$&#xA;Consequently,&#xA;$$p(a) = p(x)-(x-a)r(x)=(x-a)(q(x)-r(x)).$$&#xA;The L.H.S. is a constant, whereas the R.H.S. contains $x$. So $q(x)-r(x)=0$, and $p(a)=0$.&lt;/p&gt;&#xA;" OwnerUserId="9526" OwnerDisplayName="Math Lover" LastActivityDate="2018-02-26T18:28:49.233" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13675" PostTypeId="2" ParentId="13672" CreationDate="2018-02-26T18:35:33.550" Score="4" Body="&lt;p&gt;Let $\,p(x) = c_n x^n + c_{n-1} x^{n-1} + \ldots +c_1 x + c_0\,$, and expand it in powers of $\,(x-a)\,$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\begin{align}&#xA;p(x) &amp;amp;= c_n\big((x-a)+a\big)^n+c_{n-1}\big((x-a)+a\big)^{n-1}+\ldots+c_1\big((x-a)+a\big)+c_0 \\&#xA; &amp;amp;= c_n\left((x-a)^n+\binom{n}{1}(x-a)^{n-1}\cdot a+ \ldots+\binom{n}{n-1}(x-a)\cdot a^{n-1} + \color{red}{a^n}\right) \\&#xA; &amp;amp;\quad + c_{n-1}\left((x-a)^{n-1}+\binom{n-1}{1}(x-a)^{n-2}\cdot a+ \ldots+\binom{n-1}{n-2}(x-a)\cdot a^{n-2} + \color{red}{a^{n-1}}\right) \\&#xA; &amp;amp;\quad \cdots \\[5px]&#xA; &amp;amp;\quad + c_1\big((x-a) + \color{red}{a^1}\big) \\&#xA; &amp;amp;\quad + c_0 \cdot \color{red}{a^0} \\[5px]&#xA; &amp;amp;= (x-a)q(x) + \color{red}{c_n a^n}+\color{red}{c_{n-1}a^{n-1}}+ \ldots + \color{red}{c_1 a} + \color{red}{c_0} \\[5px]&#xA; &amp;amp;= (x-a) q(x) + p(a)&#xA;\end{align}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It then follows that $\,(x-a) \mid p(x) \iff p(a) = 0\,$.&lt;/p&gt;&#xA;" OwnerUserId="9221" OwnerDisplayName="dxiv" LastActivityDate="2018-02-26T18:35:33.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13676" PostTypeId="2" ParentId="13672" CreationDate="2018-02-26T19:12:32.467" Score="1" Body="&lt;p&gt;If you could first convince them of the fundamental theorem of algebra (and if they know of complex numbers) then one writes $p(x)=c\prod_i(x-\alpha_i)$, substituting $a$ will have to convince them from that point that there is $i$ such that $\alpha_i=a$. &lt;/p&gt;&#xA;" OwnerDisplayName="M. Van" LastActivityDate="2018-02-26T19:12:32.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13677" PostTypeId="2" ParentId="13671" CreationDate="2018-02-27T10:22:14.100" Score="9" Body="&lt;p&gt;&quot;&lt;em&gt;any references pointing in the right direction would be greatly appreciated&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference &lt;strong&gt;[1]&lt;/strong&gt; below is probably where you want to look. A few years ago I tried to obtain a copy of &lt;strong&gt;[1]&lt;/strong&gt;, but I was not able to find one. However, I do not have access to interlibrary loan. If you have access to a university’s interlibrary loan, then you might be able to get a copy --- perhaps &lt;a href=&quot;https://vufind.carli.illinois.edu/all/vf-uiu/Record/774215&quot; rel=&quot;nofollow noreferrer&quot;&gt;from here&lt;/a&gt;, for example. Just now I found a copy of &lt;strong&gt;[1]&lt;/strong&gt; &lt;a href=&quot;https://managementbooks.me/browse/concerning-the-evolution-of-the-topic-of-factoring-in-textbooks-of-elementary-algebra-published-in-england-and-the-united-states-from-1631-to-1890/&quot; rel=&quot;nofollow noreferrer&quot;&gt;at this web page&lt;/a&gt;, but I don’t know anything about the legitimacy of this source (legal and/or internet-safety).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few years ago I looked through a copy of &lt;strong&gt;[2]&lt;/strong&gt; that a nearby university library has (some notes I made on this book can be found on pp. 2-6 &lt;a href=&quot;http://mathforum.org/kb/servlet/JiveServlet/download/206-2648082-9590977-1156409/biblo-on-old-textbooks.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;), but I don’t remember to what extent &lt;strong&gt;[2]&lt;/strong&gt; includes anything that you are specifically asking about. Nonetheless, I think it is likely that &lt;strong&gt;[2]&lt;/strong&gt; would be in the bibliography of anything you’re interested in (that is reasonably thorough), so I’ve included &lt;strong&gt;[2]&lt;/strong&gt; for literature search purposes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because neither of these references is very well known, I’ve included the most complete bibliographic information I have for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Hobart Franklin Heller (1901-1981), &lt;a href=&quot;https://vufind.carli.illinois.edu/all/vf-uiu/Record/774215&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Concerning the Evolution of the Topic of Factoring in Textbooks of Elementary Algebra Published in England and the United States from 1631 to 1890&lt;/strong&gt;&lt;/a&gt;, Keystone Publishing Company (Berwick, Pennsylvania), 1940, vi + 165 pages.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This is the published version of Heller's 1940 Ph.D. dissertation (under William David Reeve, 1883-1961) at Columbia University.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Amy Olive Châteauneuf (1901?-1988?), &lt;a href=&quot;https://catalog.hathitrust.org/Record/000423469&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Changes in the Content of Elementary Algebra Since the Beginning of the High School Movement as Revealed by the Textbooks of the Period&lt;/strong&gt;&lt;/a&gt;, Ph.D. Dissertation (under John Harrison Minnick, 1877-1966), University of Pennsylvania, 1929, x + 191 pages.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This also exists as a book published by Westbrook Publishing Company (Philadelphia, Pennsylvania) in 1929 (x + 191 pages). &lt;strong&gt;Reviewed by:&lt;/strong&gt; Lao Genevra Simons (1870-1949) in &lt;a href=&quot;http://www.jstor.org/stable/27951306&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Mathematics Teacher&lt;/strong&gt; 24 #1 (January 1931), pp. 58-59&lt;/a&gt;; Ernst Rudolph Breslich (1874-1966), &lt;a href=&quot;http://www.jstor.org/stable/1080492&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The School Review&lt;/strong&gt; 39 #3 (March 1931), pp. 229-230&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2018-02-27T20:25:11.543" LastActivityDate="2018-02-27T20:25:11.543" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13678" PostTypeId="2" ParentId="11106" CreationDate="2018-02-27T12:30:28.480" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Arithmetics Examples&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Time on clock, each minute hand that the second hand covers is 5 seconds.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;5, 10, 15, 20, 25, 30, ..., 60&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The point at which a runner passes the finish line in a 3000 metre race.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;200m, 600m, 1000m, ..., 2600m, 3000m&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Sound waves or waves in the sea are sinusoids, so they can repeat their pattern for the range of the sinusoid.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Geometric examples&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Half life of carbon or any element. You can then show how all the carbon 14 is depleted over thousands of years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A graph where logs is used is easy to read and can be almost linear, whereas if there is a geometric increase you can't even plot it on paper. Kids love this one, and understand it very quickly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Puzzle: If a frog is 1 metre from a door and jumps halfway, and then jumps halfway again continuing to half its jump each time, will it ever reach the door? This displays limits and geometric decreasing functions and the idea that decreasing jumps results in infinitesimally small results over time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Showing kids how much they pay on a mortgage at 5% interest rate for 10 years, 20 years and 30 years is very insightful and clearly displays geometric increase.&lt;/p&gt;&#xA;" OwnerUserId="9524" LastActivityDate="2018-02-27T12:30:28.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13679" PostTypeId="2" ParentId="13669" CreationDate="2018-02-27T13:21:25.737" Score="2" Body="&lt;p&gt;I tend to say &quot;extreme negative number&quot; for numbers like -10^10&lt;/p&gt;&#xA;" OwnerUserId="9249" LastActivityDate="2018-02-27T13:21:25.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13680" PostTypeId="2" ParentId="13669" CreationDate="2018-02-27T14:21:02.257" Score="16" Body="&lt;p&gt;It depends a little on context and how careful I am trying to be:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If I were being very careful, I might call such a number &quot;a negative number with a large absolute value&quot; or &quot;a negative number with large magnitude.&quot;&lt;/li&gt;&#xA;&lt;li&gt;If I am being a little less careful (which I often am when discussing asymptotic behaviour of, for example, rational functions), I'll generally say &quot;a very large negative number&quot; or &quot;a large number to the left of zero&quot; (like I said, we're probably graphing, so left and right make sense).&lt;/li&gt;&#xA;&lt;li&gt;Even more sloppily, it is just &quot;large number&quot; (maybe the sign doesn't matter at all!) or &quot;a number that is far from zero&quot; or &quot;far to the left of zero.&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;That being said, I think that the more important distinction is between &quot;small&quot; numbers and &quot;large&quot; numbers.  I try to be careful to &lt;em&gt;always&lt;/em&gt; use &quot;large&quot; to refer to numbers of great magnitude (i.e. numbers far from zero, in any direction&amp;mdash;this generalizes nicely to vector spaces), and &quot;small&quot; to refer &lt;em&gt;only&lt;/em&gt; to numbers that are near zero (again, this continues to work well in vector spaces).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other place this comes up is when dealing with inequalities.  One might reasonably say that $-10$ is smaller than $-9$, but I would try to avoid that language.  Instead, I would describe $-10$ as being less than $-9$ (and refer to neither as being &quot;smaller than&quot; the other).  Since the inequality $-10 &amp;lt; -9$ is read &quot;$-10$ is less than $-9$,&quot; there should be no ambiguity.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2018-02-27T14:21:02.257" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13682" PostTypeId="2" ParentId="13671" CreationDate="2018-02-27T22:05:47.737" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Historical development: No clue.  Dave's answer studly as usual.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Rationale:  Based on personal experience and intuition (which seems to even be allowed this time!)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;a.  This fits into the &quot;why should we learn this&quot; now approaching a category of questions, deserving a tag.  It's a legitimate question of course, but non-trivial.  It's not as if everything has had some sort of robust in-use double-blind statistical proof of impact.  Why do we learn Shakespeare?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, I am NOT dismissing the question--it's still relevant even for Shakespeare.  But it's non-trivial. Perhaps the real answer is that it was just &quot;something we could do&quot;, therefore we did it, even though it appealed to teachers and writers, but did not benefit students and wasted their time.  But also perhaps we could make at least a thought experiment of eliminating this topic (or other topics in math or Shakespeare) and then find out years later that there was a damage from the students not learning the topic and the cost/benefit of time versus benefit would pay for itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b.  Sometimes learning a tool does more to build muscles than perhaps to be the tool itself.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance we force doctors to take calculus, sometimes with tricky integration methods like integration by parts.  They don't really generally need it but it does force them to learn some algebra and comfort with mathematical topics in general.  That maybe pays off when they have some feel for ratios (amount of drug per pound of patient), etc. even though it is a simpler topic.  Or tangentially to reading papers where some statistics is discussed (note, I'm not even saying calculus measure theory relation to stats, but just having to deal with some Greek looking math goop on the page for once and feeling more confident when dealing with some in the paper.  And one can easily make the argument that dealing with stats itself might be more efficient.  But still it is a collateral benefit.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Learning to factor and play with equations in MATH is probably building some general muscles that will be helpful when there are manipulations involved in partial fractions or integration by parts.  (And yes, you can make the argument we could just build the muscles when we do these topics...and maybe that is right or maybe some muscle building ahead of time makes that stuff go down easier.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;c.  Also, maybe this is minor but in the sciences (and business) it is common to rearrange and equation and get some insights out of it.  For instance, the quadratic function allows you to calculate the total (given x), but separating into factors might have some physical relationship involved in the multiplication (like number of wells times average production giving total production...or whatever.)  So, it's not uncommon that moving some terms around in a physics, engineering or even business analysis problem gives you an insight.  So a little factoring muscle might be useful.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-02-27T22:05:47.737" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13683" PostTypeId="2" ParentId="13671" CreationDate="2018-02-27T23:38:43.623" Score="3" Body="&lt;p&gt;Here's a response somewhere between the historical and normative questions. Recall that most math courses have an idiom of definitions/theorems/proofs to be presented, and then support by way of homework exercises. Arguably, we sometimes lose sight that the exercises are really just verifying that the actual mathematical content (definitions/theorems) have been learned. In some cases, it is a challenge to extract exercises relevant to the concepts being presented. Also, math courses lose coherency and become more garbled the more that proofs are taken out of the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that light, factoring polynomials in integers is a best-we-can-do response to giving intuition and examples relevant to topics such as the Remainder Theorem, the Factor Theorem, and the Fundamental Theorem of Algebra. It also gives examples of a crucial step in Completing the Square (&lt;a href=&quot;http://bulldog2.redlands.edu/fac/beery/math115/m115_activ_complsq.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;example in Al-Khwarizmi's &lt;em&gt;Al-jabr&lt;/em&gt;&lt;/a&gt;), used to prove the Quadratic Formula. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's true that in practice the techniques may seem very limited, and that is so. But we might also ask: How &lt;em&gt;else&lt;/em&gt; can we give any examples of solutions to equations other than the forms $ax + b = 0$ or $ax^2 + b = 0$? How &lt;em&gt;else&lt;/em&gt; can we build intuition regarding the machinery needed to prove the preceding theorems?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if a course is taught on the &quot;faith-based math&quot; idiom that is more widespread nowadays, then the proofs of the Remainder/Number of Roots theorems may fade into the background, and the factoring machinery that they use may not seem as essential. But ultimately the Fundamental Theorem of Algebra (etc.) is intimately tied to the idea of factoring, and it's hard to think of how that understanding would be otherwise scaffolded.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-02-28T05:59:39.580" LastActivityDate="2018-02-28T05:59:39.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13684" PostTypeId="2" ParentId="13671" CreationDate="2018-02-28T00:25:08.273" Score="5" Body="&lt;h3&gt;If we can do something with integers, we can do it with polynomials too&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Things like adding, subtracting, multiplying, dividing, factoring. At least, that's how I framed these kinds of topics when I taught remedial algebra classes that focused heavily on algebraic manipulations (polynomials, rational expressions, radical expressions). I'm by no means an expert in algebra or number theory (nor do I expect more than a tiny faction of students to study these things), but in Knapp's (basic) &lt;a href=&quot;http://www.math.stonybrook.edu/~aknapp/download.html&quot; rel=&quot;noreferrer&quot;&gt;algebra book&lt;/a&gt;, he says one unifying theme is &quot;the analogy between integers and polynomials in one indeterminate over a field.&quot; In some sense, polynomials are just &quot;nice&quot; and our mathematics has evolved around this fact.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;We have a &lt;em&gt;very&lt;/em&gt; limited toolset for solving equations analytically&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;We can use our trusty arithmetic operations to solve linear equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For nonlinear things, we're in luck if we happen to have access to the inverse $f^{-1}$ of some function $f$, and our equation happens to be linear in $f(x)$ or, more generally, linear in &quot;$f$ of something linear&quot; (still more generally, when our equation arises from the composition of any number of functions with known inverses). For example, $3e^{x - 2} + 1 = 5$ is linear in $e^{x - 2}$, and we can invert exponential functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have exact formulas for 2nd, 3rd, or 4th degree polynomials, essentially based on on clever rewriting (&quot;completing the square&quot;, for example, or &quot;depressing the cubic&quot;) and the inverse idea above. Unfortunately, we don't get formulas for 5th degree polynomials and beyond.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, we're also in luck if we can factor a polynomial, &lt;em&gt;precisely because it converts a nonlinear equation into linear equations&lt;/em&gt;. We can find solutions to $24x^2 - x + 10 = 0$ because we know $24x^2 - x + 10 =(3x+2)(8x−5)$, and that a product is $0$ when at least one of its factors is $0$; we just need to solve $3x + 2 = 0$ and $8x - 5 = 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students learn that &quot;factoring leads to solving&quot; for polynomials, and it helps with (nice) expressions that aren't polynomials. Just today I asked a calc class to find all $x$-values for which the curve $y = e^x \sec(x)$ has horizontal tangents, which leads to solving $e^x\sec(x) + e^x \sec(x) \tan(x) = 0$. Nobody puts forth any ideas for solving the latter equation, until I ask how they normally solve &lt;em&gt;polynomial&lt;/em&gt; equations. Now that &quot;factoring&quot; is in the air, we factor, and get our solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarize this second theme, factoring is basically one of the only two tools we have for solving equations with one indeterminate (I can't think of a third). Because of this, a large percentage of equations that a student will be expected to solve analytically rely on factoring to some extent. If nothing else, factoring polynomials over the integers is a good introduction to factoring in general, and this way of thinking.&lt;/p&gt;&#xA;" OwnerUserId="5153" LastActivityDate="2018-02-28T00:25:08.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13685" PostTypeId="2" ParentId="1002" CreationDate="2018-02-28T14:04:59.467" Score="1" Body="&lt;p&gt;I use logic in my lab to understand and model how cancers grow and develop. More generally, there is a large group of people across the world who use methods in logic (executable biology, logical biology) to understand how tissues grow and develop.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach a course which shows how we use logic to model systems; the documents (which can be run in jupyter notebook) are available here: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://github.com/hallba/WritingSimulators&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://github.com/hallba/WritingSimulators&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've also written a wider set of examples on how to use one specific tool, Z3, to answer problems using logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://github.com/hallba/Z3Tutorials&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://github.com/hallba/Z3Tutorials&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whilst this is an old question, I hope this response will be useful to other who find it!&lt;/p&gt;&#xA;" OwnerUserId="9534" LastActivityDate="2018-02-28T14:04:59.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13686" PostTypeId="1" CreationDate="2018-02-28T15:35:17.897" Score="1" ViewCount="106" Body="&lt;p&gt;I'm taking a number theory course and don't know whether it's worth it. I currently can't understand algebra and real analysis and decided to take # theory to see whether this would help me prove and understand set theory and cardinality and sequences and the prerequisites for analysis and galois theory. Should I expect to get better at proving after this course? &lt;/p&gt;&#xA;" OwnerDisplayName="user9495" LastActivityDate="2018-03-01T05:50:35.597" Title="Does studying elementary number theory improve one's proof skills and ability to understand algebra and analysis?" Tags="|proofs|number-theory|" AnswerCount="1" CommentCount="1" ClosedDate="2018-03-03T11:31:22.607" ContentLicense="CC BY-SA 3.0" />
  <row Id="13688" PostTypeId="2" ParentId="13665" CreationDate="2018-02-28T21:52:44.800" Score="2" Body="&lt;p&gt;I don't know of a good problem book but I found &lt;a href=&quot;https://sites.math.washington.edu/~m120/&quot; rel=&quot;nofollow noreferrer&quot;&gt;D.H. Collingwood, K.D. Prince and M. M. Conroy's OER precalculus book&lt;/a&gt; to be a quite good source of well thought out word problems.&lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2018-02-28T21:52:44.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13689" PostTypeId="2" ParentId="13686" CreationDate="2018-02-28T22:43:40.903" Score="3" Body="&lt;p&gt;At the very least, elementary number theory (e.g., the notion of &quot;congruence&quot;, the notion of &quot;prime&quot;, rings of integers-mod-$n$, Fermat's Little Theorem, Wilson's Theorem, and so on) gives a very &lt;em&gt;tangible&lt;/em&gt; practice scenario for reading and writing proofs. At the same time, most of the elementary number theory results are indeed prototypes and examples for various basic results in abstract algebra (group theory, commutative rings). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arguably, the logical complexity of assertions and proofs in elementary number theory is less than that of basic rigorous analysis, where the nesting of quantifiers is just slightly deeper in many cases.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-02-28T22:43:40.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13690" PostTypeId="2" ParentId="13669" CreationDate="2018-03-01T01:07:54.497" Score="2" Body="&lt;p&gt;I use number 2, very negative.  Concise and clear.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just small related FYI.  Realize that in some contexts, the wording takes care of the issue (loss versus profit, export versus import).&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-01T01:07:54.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13691" PostTypeId="2" ParentId="1002" CreationDate="2018-03-01T01:33:12.090" Score="0" Body="&lt;p&gt;Not a &quot;good answer&quot; since I don't have a problem repository but wanted to share what I could.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;GRE and LSAT have lots of logic problems.  I believe GRE used to have a whole section of logic problems but now it is just a subset of a section. (Took it decades ago.)  You can look at a Kaplan or Princeton Review book (or a sample test from ETS/CB).  I believe they are artificial in the manner you worry about but perhaps looking through them you find some that are more &quot;real world&quot;.  Also, at least it is another context where logic skills are valued.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Quite a few business cases (for admissions or even on studies) use something called an issue tree.  I don't know that this is always rigidly logical in the manner of a Boolean circuit (can sometimes find nuances to violate the logic) but in general (or as an approximation), they are logical or at least one attempts to formulate a structure that will make it so.  See the attached tree for the &quot;profitability case&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.preplounge.com/en/bootcamp.php/case-cracking-toolbox/identify-your-case-type/profitability-case&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.preplounge.com/en/bootcamp.php/case-cracking-toolbox/identify-your-case-type/profitability-case&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Root cause analysis in engineering failures or accidents (NTSB, Space shuttle, etc.):&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.preplounge.com/en/bootcamp.php/case-cracking-toolbox/identify-your-case-type/profitability-case&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.preplounge.com/en/bootcamp.php/case-cracking-toolbox/identify-your-case-type/profitability-case&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Not always a perfect simple 20 questions style of problem solving as you may have multiple causes or interactions but often branches of the issue analysis have a logical structure.)&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Many diagnostic medical programs (already commonly used by a minority of physicians, I had an appointment with a guy using one) have a logical structure.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Qualitative analysis in chemistry (solubility rules):&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=solubility+rules+flowchart&amp;amp;hl=en&amp;amp;source=lnms&amp;amp;tbm=isch&amp;amp;sa=X&amp;amp;ved=0ahUKEwiB2bOP_cnZAhVDm-AKHelxCwcQ_AUICigB&amp;amp;biw=1366&amp;amp;bih=654&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=solubility+rules+flowchart&amp;amp;hl=en&amp;amp;source=lnms&amp;amp;tbm=isch&amp;amp;sa=X&amp;amp;ved=0ahUKEwiB2bOP_cnZAhVDm-AKHelxCwcQ_AUICigB&amp;amp;biw=1366&amp;amp;bih=654&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-01T01:33:12.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13692" PostTypeId="2" ParentId="13665" CreationDate="2018-03-01T03:09:43.600" Score="1" Body="&lt;p&gt;NASA has some interesting problems.  They're maybe more like projects rather than short homework problems, but maybe you'd find them interesting. You can see them &lt;a href=&quot;https://www.nasa.gov/audience/foreducators/exploringmath/precalculus/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, the North Carolina School for Science and Mathematics has various links to various problems they use in their &lt;a href=&quot;https://ncssm.instructure.com/courses/1067&quot; rel=&quot;nofollow noreferrer&quot;&gt;courses&lt;/a&gt; on &lt;a href=&quot;https://ncssm.instructure.com/courses/806&quot; rel=&quot;nofollow noreferrer&quot;&gt;modeling&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A number of proponents of problem based learning (PBL) have mapped interesting problems on to various common core standards.  Maybe you'll find some interesting problems &lt;a href=&quot;https://emergentmath.com/my-problem-based-curriculum-maps/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2018-03-01T03:09:43.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13693" PostTypeId="1" CreationDate="2018-03-01T07:16:18.980" Score="10" ViewCount="499" Body="&lt;p&gt;There has been many posts here and in MSE about different notations of differentiation. See for example &lt;a href=&quot;https://mathoverflow.net/questions/73492/how-misleading-is-it-to-regard-fracdydx-as-a-fraction&quot;&gt;this&lt;/a&gt;, &lt;a href=&quot;https://matheducators.stackexchange.com/questions/4284/why-are-we-so-careful-in-saying-that-dy-dx-is-not-a-fraction&quot;&gt;this&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio/21209#21209&quot;&gt;this&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, those questions only deal with the common misunderstanding about Leibniz notation. But none of them really tell how to avoid getting the wrong idea. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how to teach these different notions about differentials in a way that this misunderstanding doesn't arise? Or should one even care about this because it seems to be a pretty common practice not to (at least in physics)?&lt;/p&gt;&#xA;" OwnerUserId="9535" LastEditorUserId="238" LastEditDate="2019-12-16T14:48:17.457" LastActivityDate="2019-12-16T14:48:17.457" Title="How to teach Leibniz and Newton's notation" Tags="|undergraduate-education|calculus|teaching|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13694" PostTypeId="2" ParentId="13669" CreationDate="2018-03-01T16:43:23.590" Score="8" Body="&lt;p&gt;As far as the OP liked my comment, I put this as an answer even if it is only opinion based:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;option 2: very negative&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I call the number &lt;em&gt;very negative&lt;/em&gt; or &lt;em&gt;very positive&lt;/em&gt; (number) if it has large absolute value. And I call number a &lt;em&gt;tiny positive&lt;/em&gt; or &lt;em&gt;tiny negative&lt;/em&gt;, if the absolute value is very small.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2018-03-01T16:43:23.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13695" PostTypeId="1" AcceptedAnswerId="13721" CreationDate="2018-03-01T18:38:51.943" Score="8" ViewCount="242" Body="&lt;p&gt;Whenever I teach multivariable calculus I find students really struggle with both finding critical points and the method of Lagrange multipliers. I think that the reason is the same: solving systems of polynomial equations requires different tactics and much more care than solving one variable polynomials, or linear systems. As far as I know, the students have no practice until they see it in the calculus context and even worse none of the books that I've seen even mention it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any books/resources/modules that deal with solving systems of polynomial equations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also be interested in anyone who teaches algebra/precalculus weighing on if/how solving systems of polynomial equations is presented.&lt;/p&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="376" LastEditDate="2018-03-02T12:03:26.533" LastActivityDate="2018-03-04T16:06:26.073" Title="Resources on solving systems of polynomial equations in multivariable calculus setting" Tags="|undergraduate-education|reference-request|algebra|vector-calculus|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13696" PostTypeId="2" ParentId="13695" CreationDate="2018-03-01T19:55:22.130" Score="1" Body="&lt;p&gt;Solving systems of polynomial equations is hard in general.  The examples in the textbook are specially cooked up to be possible.  I am not sure that developing skill at solving such systems is a good use of ones time, especially in a course with as much conceptual content to master as Multivariable Calculus.  Maybe let them get approximate solutions by graphing?&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-03-01T19:55:22.130" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13697" PostTypeId="2" ParentId="13665" CreationDate="2018-03-01T22:19:45.370" Score="-1" Body="&lt;p&gt;Don't have a good answer but consider the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Free physics textbooks (non calc based, use the problems, not the book).  Here is one, but are others on the web:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://cnx.org/contents/Ax2o07Ul@9.99:HR_VN3f7@3/Introduction-to-Science-and-th&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cnx.org/contents/Ax2o07Ul@9.99:HR_VN3f7@3/Introduction-to-Science-and-th&lt;/a&gt;  (You will have to skip through to the problem pages)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Linear kinematics is useful for basic algebra.  2-d kinematics (shooting the cannonball) is useful for vectors and trig.  Simple AC circuits (reactance, resistance, impedance) is useful for vectors and trig and complex numbers.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;For exponents, I think the key application is investment economics.  Uses the concepts, easy to understand without a lot of other physical concepts to understand and one sees the application to something that people care about, $$.  William Hart College Algebra has a nice chapter on them but it is either still in copyright (c. 1940s) or no one has put up a pdf yet even if rights lapsed.  There are some sites on the web (including Khan) with similar problems but I didn't find any with large amounts.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Would be helpful if you can describe a little more the nature of your precalc class.  I have seen that term used loosely to mean everything from &quot;Algebra 2&quot; (logs and partial fractions) to trig to analytic geometry to &quot;functions&quot; (inequalities and functions) to &quot;baby calculus&quot; itself.  The applications for different areas will be different and it is not clear to me how advanced your precalc class is (closer to algebra 2 or closer to &quot;baby calculus&quot;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;P.s.  In general, you don't want problems that assume the person has studied the physics or the like but that explain the law or such in the problem statement (or are clearly intuitive).  But I would not hold back from problems that state something technical in the problem statement (e.g. &quot;Scientists have modeled the effect of carbon dioxide as a logarithmic proportionality.  If 200 ppm of CO2 gives 0 change in world temperature and 400 ppm of CO2 gives 2 degrees increase in temperature, what will 600 ppm give?&quot;  [You don't need to understand the chem-physics-earth science to just solve this as a word problem given the model was given to you in problem but is reasonable and applied.]&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-01T22:19:45.370" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13698" PostTypeId="2" ParentId="13693" CreationDate="2018-03-01T22:25:34.913" Score="4" Body="&lt;p&gt;My advice is to not worry too much about this but to teach the course and concepts and tricks and such instead.  You have to have some nonrigorous understanding of something (in general) to understand it rigorously later.  There is a reason we teach kids to count first versus Dedekind cuts. Richard Feynman wrote a nice essay on this issue of pedagogy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just mention the issue with the notation but don't super belabor it or distract from teaching the calculus itself.  When you get to a problem where it matters to not have this confusion (as otherwise an error occurs), you can discuss it in the context of solving that problem or of correcting a false solution.  But you are better off not spending super amounts of time &quot;clarifying&quot; this issue for people who don't even know how to take a derivative yet at all.  &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-01T22:25:34.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13699" PostTypeId="2" ParentId="13695" CreationDate="2018-03-01T22:41:42.823" Score="2" Body="&lt;p&gt;My college algebra text has  section of solving systems of quadratics (intersections of ellipses, intersection of line and parabola, etc.)  There is both calculational method and graphing method (numerical approximation).  It is &quot;starred&quot; though and I don't really remember using it much in my life, even though I freaking covered it!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not an iconic method like the quadratic equation itself or like systems of linear equations.  My advice is to just plow through it.  There is so much else going on within the grabbag of Calc 3.  Div, grad, curl and all that.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The kids who are manipulational whizzes will do well (even if they don't remember their starred section of the College Algebra text, they can just handle it in context of new work) and those who aren't won't but what can you do.  Lots of that going around anyways.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-01T22:41:42.823" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13700" PostTypeId="1" CreationDate="2018-03-02T13:44:59.933" Score="13" ViewCount="1225" Body="&lt;p&gt;Most contemporary curricula define the word &amp;quot;rectangle&amp;quot; inclusively, so that all squares are automatically rectangles.  Are there curricula in which this convention is not followed?  That is, are there curricula in which &amp;quot;rectangle&amp;quot; is defined so as to explicitly exclude squares?  International and cross-cultural perspectives, in particular, would be appreciated. Also, I would like to know if this is age-dependent, like when it comes to kindergartners.&lt;/p&gt;&#xA;&lt;p&gt;Devil's advocate: Would you teach that the sun is white? (Analogy: Apparently, 'The Sun is yellow' is like squares are not rectangles.')&lt;/p&gt;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://math.stackexchange.com/questions/2249741&quot;&gt;Why do we have circles for ellipses, squares for rectangles but nothing for triangles?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some context: See previous revisions.&lt;/p&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2021-01-24T23:54:59.630" LastActivityDate="2021-01-24T23:54:59.630" Title="In what curricula are &quot;rectangles&quot; defined so as to exclude squares?" Tags="|geometry|curriculum|definitions|preschool-education|culture|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13705" PostTypeId="1" AcceptedAnswerId="13708" CreationDate="2018-03-02T21:45:32.157" Score="79" ViewCount="20208" Body="&lt;p&gt;Many counting problems start with the assumption that we have a certain number of men and women or a certain number of couples, with the assumption (often unstated) being that that gender is binary (only men or women) and couples are only heterosexual.  (See for example &lt;a href=&quot;https://math.stackexchange.com/questions/766359/different-ways-of-picking-a-committee-of-12-women-and-10-men?rq=1&quot;&gt;this&lt;/a&gt; or &lt;a href=&quot;https://math.stackexchange.com/questions/1180714/how-many-different-committees-with-at-least-one-man-and-woman-and-no-spouses?rq=1&quot;&gt;this&lt;/a&gt; problem or think for example of Hall's Marriage Lemma.)  Can anyone suggest a good replacement for the concept of couples in these sorts of math problems that better reflects the societal norms of the latest generation of students?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, I'll routinely replace &quot;men and women&quot; with &quot;undergrads&quot; and &quot;grad students&quot; or something similar depending on the class make up.  I'm currently racking my brain for two distinct sets of objects S1 and S2, where we might naturally think of pairing objects in S1 to objects in S2, and where there is one most natural pairing (i.e. replacing men and women, where we compare any heterogeneous couple to pairing men and their wives).  Anything I've thought of  makes for a ridiculously long word problem.  Any suggestions appreciated.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Mod note: If you answer the question please answer the question &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What's a replacement for &quot;married couples&quot; in combinatorics problems?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Please, do not provided answers that say there is no need to do so. This is beyond the scope of this question. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;" OwnerUserId="9542" LastEditorUserId="-1" LastEditDate="2018-03-12T16:08:49.650" LastActivityDate="2021-12-27T16:18:26.040" Title="What's a replacement for &quot;married couples&quot; in combinatorics problems?" Tags="|mathematical-pedagogy|examples|mathematics-in-daily-life|" AnswerCount="17" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13707" PostTypeId="2" ParentId="13705" CreationDate="2018-03-02T22:44:43.420" Score="68" Body="&lt;p&gt;A few possibilities off the top of my head:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students and chairs.  How many ways are there for $n$ students to sit in $k$ chairs.  The game of musical chairs might be fun to play around with.  One can also consider natural restrictions, such as myopic students who need to sit near the front.&lt;/li&gt;&#xA;&lt;li&gt;Replace &quot;men&quot; and &quot;women&quot; with faculty from different departments.  For example, you might need a curriculum committee made up of mathematicians and physicists.&lt;/li&gt;&#xA;&lt;li&gt;In the senate, there are two senators from each state.  How many ways are there of making a committee of 6 senators, no two of which are from the same state?  This really only works in the US, but I am sure that there are similar phrasings that could work elsewhere.  One could also consider party affiliation here, which could lead to more interesting problems.&lt;/li&gt;&#xA;&lt;li&gt;True story:  My daughter, who is in kindergarten, has a 5th grade &quot;reading buddy.&quot;  If there are $n$ kindergarteners and $k$ 5th graders, how many ways are their of pairing up students?  If you want to appeal to an older audience, maybe we are pairing up tutors with students for individual instruction?&lt;/li&gt;&#xA;&lt;li&gt;This is totally not what you want, but still an interesting problem: how many injective functions exist from a set with $n$ elements to a set with $k$ elements?  How about surjective functions?  How does this change with $n$ and $k$?&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/2674513/in-how-many-ways-can-14-tenth-graders-and-10-ninth-graders-be-arranged-in-a-line&quot;&gt;This question on MSE&lt;/a&gt; uses 9th and 10th graders.  &quot;Juniors&quot; and &quot;Seniors&quot; could easily replace these if you wanted a college aged example.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2018-03-03T04:29:07.130" LastActivityDate="2018-03-03T04:29:07.130" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13708" PostTypeId="2" ParentId="13705" CreationDate="2018-03-02T23:24:13.133" Score="91" Body="&lt;p&gt;I've been using &quot;pets&quot; and &quot;owners&quot; (as in: possible pet-shelter adoptees) in recent years. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-03-03T02:24:03.293" LastActivityDate="2018-03-03T02:24:03.293" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13709" PostTypeId="2" ParentId="13705" CreationDate="2018-03-03T02:26:57.903" Score="8" Body="&lt;p&gt;I'm just thinking out loud here, but one possible option is to keep it as men and women, but have it be about ballroom dancing, or figure skating, instead of dating or marriage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That way, it might have fewer undertones of &quot;everybody in the world needs to be married off to someone of the opposite sex&quot;, and might have more connotations of &quot;this is a specialized hobby where we want to pair men with women for aesthetic reasons, but we're not claiming that the world as a whole needs to be this way.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such examples could begin with &quot;In a ballroom dancing club, there are 12 women and 10 men...&quot; That wording, at least ideally, leaves open the possibility that the larger world could have nonbinary people. It's more like &quot;this &lt;em&gt;particular&lt;/em&gt; club has 12 women and 10 men.&quot;&lt;/p&gt;&#xA;" OwnerUserId="9249" LastActivityDate="2018-03-03T02:26:57.903" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="13711" PostTypeId="2" ParentId="13705" CreationDate="2018-03-03T03:15:15.990" Score="34" Body="&lt;p&gt;Try objects that often occur in pairs but are distinct from each other:  forks and spoons (or forks and knives), left and right shoes, salt and pepper shakers, and so on (where each fork has an obvious partner spoon, perhaps sharing the same color or design, and so on).&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-03-03T03:15:15.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13712" PostTypeId="1" CreationDate="2018-03-03T12:44:03.903" Score="8" ViewCount="1555" Body="&lt;p&gt;Students tend to mix up signs when shifting function graphs around: consider $y=x^2$. To shift it one unit upwards (&quot;increasing $y$&quot;), you write $y=x^2+1$, to shift it to the right (&quot;increasing $x$&quot;), you write $y=(x-1)^2$. On the one hand, it's $+$, on the other hand you have to use $-$ for somewhat &quot;the same thing&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It recently came to my mind that the interpretation of the function graph as all points fulfilling the relation $ \mathbb R^2\supset R  = \{(x, y) ~|~ y=x^2\}$ might be more helpful for this matter:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To shift the graph vertically, you use $R_\text{v}  = \{(x, y) ~|~ y+a=x^2\}$.&lt;/li&gt;&#xA;&lt;li&gt;To shift horizontally: $R_\text{h}  = \{(x, y) ~|~ y=(x+a)^2\}$&lt;/li&gt;&#xA;&lt;li&gt;Negative values for $a$ shift in increasing direction for both cases.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Is there any literature or experience on this approach?&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2018-04-24T23:05:58.523" Title="Shifting function graphs: writing vertical offset on the y-side?" Tags="|algebra|functions|graphing|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13713" PostTypeId="2" ParentId="13705" CreationDate="2018-03-03T18:37:08.637" Score="84" Body="&lt;p&gt;In the stable marriage problem, you can introduce the problem as it is. But then you ask your students how things change if you assume there are not only heterosexual but also gay and lesbian people (assuming that a heterosexual person will never marry a person of the same gender, and a gay or lesbian person will never marry a person of the opposite gender). It's actually an interesting problem that requires a completely different solution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Add bisexual people, and your student's heads will explode - hopefully not because of the bisexuality, but because suddenly the problem gets a lot lot harder. On the positive side, people being transgender doesn't affect the problem at all. &lt;/p&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2018-03-03T18:37:08.637" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13715" PostTypeId="2" ParentId="13661" CreationDate="2018-03-03T21:28:53.447" Score="5" Body="&lt;p&gt;Since I recently had a math teacher from another institution observe me during an Algebra 2 lesson and comment positively on my notation (&amp;quot;I'm going to steal that!&amp;quot;) here is a worked example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Find all values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that satisfy &lt;span class=&quot;math-container&quot;&gt;$3(x-4)^2 + 8 = 23$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt; &lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$-8: \hspace{20 mm}3(x-4)^2 = 15$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\div3:  \hspace{20 mm} (x-4)^2 = 5$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt {}: \hspace{20 mm} x-4 = \pm \sqrt{5}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$+4:  \hspace{20 mm} x = 4 \pm \sqrt{5}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Sometimes I will use other symbols on the left hand side such as &lt;span class=&quot;math-container&quot;&gt;$\square$&lt;/span&gt; for &amp;quot;square both sides&amp;quot; or individual letters such as &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; for &amp;quot;collect like terms&amp;quot; or &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; for &amp;quot;factor [at least one side of the equation].&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The general approach can be modified to one's liking, but the key component is indicating clearly how to get from each line of one's work to the next; I had found that students were carrying out some operations in ways that made it tough for me to decipher what they had done because various steps were mixed together along with cancellations or crossed out coefficients/variables. It was also more difficult for me to point with specificity to where an approach had gone awry.&lt;/p&gt;&#xA;&lt;p&gt;I find that the above-notation suitably merges mathematical rigor with pedagogical clarity (especially around following student work) with its main drawback being speed, which I am happy to sacrifice (to some extent) as students cement basic arithmetic/algebraic fluency.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-03-03T21:28:53.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13716" PostTypeId="2" ParentId="13661" CreationDate="2018-03-03T22:17:05.657" Score="2" Body="&lt;p&gt;I encoded my preferred notation in this web-app for practicing equation solving:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://thewessens.net/ClassroomApps/Main/equations.html?topic=algebra&amp;amp;path=Main&amp;amp;id=2&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://thewessens.net/ClassroomApps/Main/equations.html?topic=algebra&amp;amp;path=Main&amp;amp;id=2&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ltvgy.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ltvgy.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has served me quite well.  The picture shows a completed solution - at each step the student enters an operation on each side (unless it is expansion which is one side only) and the result is shown. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note the use of colour, the clarity in choice of an &quot;opposite&quot; operator, and the emphasis on &quot;doing the same thing to both sides&quot;.&lt;/p&gt;&#xA;" OwnerUserId="9557" LastActivityDate="2018-03-03T22:17:05.657" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13717" PostTypeId="1" AcceptedAnswerId="13722" CreationDate="2018-03-04T05:55:09.930" Score="6" ViewCount="739" Body="&lt;p&gt;I am wondering which research communities use the notation $\mathbf i$, $\mathbf j$, $\mathbf k$ for the three-dimensional unit vectors. The calculus textbook I have to use (Stewart) uses that notation. But I am a mathematician, so maybe there is a physics or engineering community out there where that notation is actually used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notations that I am well acquainted with include &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$e_1$, $e_2$, $e_3$&lt;/li&gt;&#xA;&lt;li&gt;$e_x$, $e_y$, $e_z$&lt;/li&gt;&#xA;&lt;li&gt;$\mathbf e_1$, $\mathbf e_2$, $\mathbf e_3$&lt;/li&gt;&#xA;&lt;li&gt;$\mathbf e_x$, $\mathbf e_y$, $\mathbf e_z$&lt;/li&gt;&#xA;&lt;li&gt;...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have never seen $\mathbf i$, $\mathbf j$, $\mathbf k$ in any research paper ever. Can you point out a recent example?&lt;/p&gt;&#xA;" OwnerUserId="8895" LastEditorUserId="376" LastEditDate="2021-02-04T16:25:35.193" LastActivityDate="2021-02-04T16:25:35.193" Title="Who actually uses $\mathbf i$, $\mathbf j$, $\mathbf k$ for the standard unit vectors?" Tags="|notation|linear-algebra|vector-calculus|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13718" PostTypeId="1" AcceptedAnswerId="13720" CreationDate="2018-03-04T07:39:24.887" Score="5" ViewCount="999" Body="&lt;p&gt;The Cauchy criterion is used to prove the convergence of sequences $(a_k)$ with unknown or irrational limit: If for every $\epsilon &amp;gt; 0$ there is a $k$ such that for $m, n &amp;gt; k$ we have $|a_n-a_m|&amp;lt;\epsilon$ then the sequence converges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question: What functions are best suited to show undergraduates that this criterion is useful? That means: I am looking for functions without an easily recognizable convergence that can easily be shown convergent by the Cauchy criterion.&lt;/p&gt;&#xA;" OwnerUserId="8557" LastEditorUserId="8557" LastEditDate="2018-03-04T10:53:02.950" LastActivityDate="2018-03-04T19:36:24.163" Title="An application of the Cauchy criterion for undergraduates?" Tags="|mathematical-analysis|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13719" PostTypeId="2" ParentId="13717" CreationDate="2018-03-04T12:59:39.610" Score="-3" Body="&lt;p&gt;Who?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Crystallographers sometimes in my experience.  But, first page of a Google Scholar search gave 9/10 results from data science statistics, PCA and such.  Some in an application of statistics like a specific medical paper, others just stats or computing in general).  There was only one paper that was crystal-related (lattice vibrations, but in a physics context, not structure determination).  So presumably space vectors or lattice vectors (perhaps not Cartesian).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://scholar.google.com/scholar?hl=en&amp;amp;as_sdt=0%2C47&amp;amp;q=i+j+k+vectors&amp;amp;btnG=&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://scholar.google.com/scholar?hl=en&amp;amp;as_sdt=0%2C47&amp;amp;q=i+j+k+vectors&amp;amp;btnG=&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(You may see slightly different specific results from your search, Google being Google, but no matter, still shows you some people using ijk)&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-03-04T13:06:06.660" LastActivityDate="2018-03-04T13:06:06.660" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13720" PostTypeId="2" ParentId="13718" CreationDate="2018-03-04T13:30:15.213" Score="3" Body="&lt;p&gt;Here are a couple of examples that I believe are fairly standard for illustrating the Cauchy convergence criterion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 1:&lt;/strong&gt; Let $a &amp;lt; b$ be real numbers and define the sequence $\{x_n\}$ of real numbers using $2$-term recursion as follows: $x_1 = a$ and $x_2 = b$ and, for each integer $n \geq 3,$ let $x_n = \frac{1}{2}(x_{n-2} + x_{x-1}).$ This sequence is not monotone, and thus methods using least upper bounds and greatest lower bounds are difficult to apply. For instance, the odd-numbered terms form an increasing sequence that is bounded above by $b$ (and thus converges), and the even-numbered terms form a decreasing sequence that is bounded below by $a$ (and thus converges), but how do we know these two subsequences converge to the same limit? However, it is easy to see that this is a Cauchy sequence --- for each integer $N \geq 2,$ if $m,n \, \geq \, N,$ then $x_m$ and $x_n$ both belong to the same $(N-2)$-bisected subinterval of $[a,b]$ and hence $|x_n – x_m| \, \leq \, \frac{1}{2^{N-2}}(b-a)$ --- and therefore this sequence converges. Incidentally, the limit of this sequence is $\frac{1}{3}(a+2b),$ which can be found by recasting the sequence as the partial sums of&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a + (b-a) - \frac{1}{2}(b-a) + \frac{1}{4}(b-a) - \frac{1}{8}(b-a) + \cdots + (-1)^k2^{-k}(b-a) + \cdots  $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and noticing that this series is geometric beginning with the second term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 2:&lt;/strong&gt; We can show that the harmonic series $\sum\limits_{k=1}^{\infty}\frac{1}{k}$ diverges by showing that its sequence of partial sums is not a Cauchy sequence. For instance, letting $S_n = \sum\limits_{k=1}^{n}\frac{1}{k},$ we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$  S_{2n} \; - \; S_n \;\; = \;\;  \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} \;\; \geq \;\; n\left(\frac{1}{2n}\right) \;\; = \;\; \frac{1}{2}     $$&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2018-03-04T13:30:15.213" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13721" PostTypeId="2" ParentId="13695" CreationDate="2018-03-04T15:20:50.020" Score="1" Body="&lt;p&gt;I have found the following online resource from West Texas A&amp;amp;M University: &lt;a href=&quot;http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut52_nonlinear_sys.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;College Algebra Tutorial 52: Solving Systems of Nonlinear Equations in Two Variables&lt;/a&gt;. It gives a step-by-step approach with examples, using either substitution or elimination. This could be useful for your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(For a more advanced perspective, see books such as &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/3319167200&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ideals, Varieties, and Algorithms&lt;/a&gt; by Cox, Little &amp;amp; O'Shea and &lt;a href=&quot;https://math.berkeley.edu/~bernd/cbms.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solving Systems of Polynomial Equations&lt;/a&gt; by Sturmfels.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2018-03-04T16:06:26.073" LastActivityDate="2018-03-04T16:06:26.073" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13722" PostTypeId="2" ParentId="13717" CreationDate="2018-03-04T16:26:16.863" Score="6" Body="&lt;p&gt;Search arxiv.org for &lt;strong&gt;&quot;unit vectors i, j, k&quot;&lt;/strong&gt; and you will find examples of research papers where the notation is used. Many are from physics communities investigating phenomena in three-dimensional space. Clearly we should not expect to find this notation in research relating to general dimensionality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the image see &quot;unit vectors i, j, k&quot; at the bottom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pmEey.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pmEey.png&quot; alt=&quot;Search Field on arXiv&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2018-03-04T18:48:27.467" LastActivityDate="2018-03-04T18:48:27.467" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13723" PostTypeId="2" ParentId="13661" CreationDate="2018-03-04T19:33:28.993" Score="1" Body="&lt;p&gt;I think it‘s important to represent steps as happening between equations, linking one to the next. I do this similarly to kallikak‘s notation, only with arrows on either side. This shows that steps can be reversed, and also that some operations split one equation into two, such as taking a root or the zero-product rule. Arrow notation also mirrors functional notation and thinking.&lt;/p&gt;&#xA;" OwnerUserId="9577" LastActivityDate="2018-03-04T19:33:28.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13724" PostTypeId="2" ParentId="13718" CreationDate="2018-03-04T19:36:24.163" Score="1" Body="&lt;p&gt;One application is infinite &lt;em&gt;series&lt;/em&gt;, since they are characterized by their &lt;em&gt;sequences&lt;/em&gt; of partial sums. First, use the Cauchy Criterion for Sequences to establish a similar result for series. Then, use this criterion to more easily prove facts about series (e.g. Comparison Tests, Absolute Convergence Test, etc.) That is, for the students, you can frame the Cauchy Criterion as a way to make later proofs about series more direct and revealing. Rather than using the same technique to prove all those results, let's identify that common technique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem (Cauchy Criterion for Series):&lt;/strong&gt; The series $\sum_{k=1}^\infty a_k$ converges if and only if, for any $\varepsilon&amp;gt;0$, there exists an $N\in \mathbb{N}$ such that, whenever $m&amp;gt;n\geq N$, it follows that $|a_{m+1}+a_{m+2}+\cdots +a_n|&amp;lt;\varepsilon$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Proof:&lt;/em&gt; Notice that the partial sums $s_n=a_1+a_2+\cdots+a_n$ and $s_m=a_1+a_2+\cdots+a_m$ make $|s_n-s_m|=|a_{m+1}+a_{m+2}+\cdots +a_n|$. Apply the Cauchy Criterion for Sequences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem (Comparison Tests):&lt;/strong&gt; Assumie $(a_k)$ and $(b_k)$ are sequences with $0\leq a_k\leq b_k$ for all $k$. Then:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If $\sum_{k=1}^\infty b_k$ converges, then so does $\sum_{k=1}^\infty a_k$. &lt;/li&gt;&#xA;&lt;li&gt;If $\sum_{k=1}^\infty a_k$ diverges, then so does $\sum_{k=1}^\infty b_k$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These can be proven using the Cauchy Criterion for Series.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2018-03-04T19:36:24.163" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13725" PostTypeId="2" ParentId="13717" CreationDate="2018-03-04T22:24:30.833" Score="7" Body="&lt;p&gt;I'm not sure this a research setting, but the cross product relations $i \times j = k$, $j \times k = i$, etc are nice for illustrating the connection between quaternions (where $ij=k, jk=i$, etc) and the cross product.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically, the Lie group consisting of unit length quaternions has Lie algebra $\mathbb{R}^3$ with Lie bracket given by cross product.&lt;/p&gt;&#xA;" OwnerUserId="9579" LastActivityDate="2018-03-04T22:24:30.833" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13726" PostTypeId="2" ParentId="13717" CreationDate="2018-03-05T02:16:21.390" Score="1" Body="&lt;p&gt;This notation is completely standard in all Physics classes at the undergraduate level.  I doubt you would find any Physicist anywhere who does not instantly recognize that notation -- but then, I would also have said the same thing about mathematicians, as the notation is completely standard in multivariable calculus at the undergraduate level in the United States.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-03-05T02:16:21.390" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13727" PostTypeId="2" ParentId="13705" CreationDate="2018-03-05T08:49:07.760" Score="7" Body="&lt;p&gt;Nuts and bolts? You'll need to specify bolts which are only long enough to accept one nut each, but that's brief enough to do.&lt;/p&gt;&#xA;" OwnerUserId="9587" LastActivityDate="2018-03-05T08:49:07.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13729" PostTypeId="2" ParentId="13705" CreationDate="2018-03-05T10:23:13.137" Score="4" Body="&lt;p&gt;Cut out the middle man and make it about sexual reproduction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need precisely one sperm to meet one egg.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any other combination would be unviable either due to the inability for one sperm to affect multiple eggs or due to multiple sperm affecting a single egg (i.e. polyspermy) creating an unviable zygote.&lt;/p&gt;&#xA;" OwnerUserId="957" LastActivityDate="2018-03-05T10:23:13.137" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13730" PostTypeId="2" ParentId="13705" CreationDate="2018-03-05T11:06:34.803" Score="37" Body="&lt;p&gt;The issue is not making problems about heterosexual married couples. The issues are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Implicitly making the assumption that all married couples are heterosexual.&lt;/li&gt;&#xA;&lt;li&gt;Making problems about heterosexual marriages but not about other kinds of couples.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Both points are unengaging for people following other types of marriage, but they can easily be solved while keeping the problem clear and interesting. We just need to make all assumptions explicit (thus acknowledging that they aren't universal while making the statement unambiguous) and making a wider array of problems (which can be useful to teach different tools).&lt;/p&gt;&#xA;&lt;h2&gt;An example&lt;/h2&gt;&#xA;&lt;p&gt;There are 100 single people in a village. How many different marriages can be performed among them in the following situations? Please beware that situations depicted are just simplifications to keep this problem easy and solvable.&lt;/p&gt;&#xA;&lt;h3&gt;2018 situation&lt;/h3&gt;&#xA;&lt;p&gt;All people are free to marry any other villager of their choice.&lt;/p&gt;&#xA;&lt;h3&gt;1990 situation&lt;/h3&gt;&#xA;&lt;p&gt;There are 55 men an 45 women in the village. A valid marriage must include a man and a woman.&lt;/p&gt;&#xA;&lt;h3&gt;1940 situation (somewhere)&lt;/h3&gt;&#xA;&lt;p&gt;There are 30 white men, 28 white women, 21 black men and 21 black women in the village. Local laws only allow a person to marry another person of the same race and different sex.&lt;/p&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://www.umanitoba.ca/faculties/arts/anthropology/tutor/case_studies/yanomamo/yano-trm.html&quot; rel=&quot;noreferrer&quot;&gt;Yanomamo kinship system&lt;/a&gt;&lt;/h3&gt;&#xA;&lt;p&gt;There are two clans in the village: clan A (30 men, 28 women) and clan B (21 men and 21 women). People are only allowed to marry somebody of different sex and different clan.&lt;/p&gt;&#xA;&lt;h3&gt;Caste system&lt;/h3&gt;&#xA;&lt;p&gt;There are four castes in the village: caste A (15 men, 15 women), caste B (15 men, 13 women), caste C (10 men, 11 women) and caste D (11 men, 10 women), being A the highest caste and D the lowest caste. A man is allowed to marry a woman of his caste or any lower caste, while a woman is allowed to marry a man of her caste or any higher caste.&lt;/p&gt;&#xA;&lt;h3&gt;Arthur C. Clarke's Rendezvous with Rama system&lt;/h3&gt;&#xA;&lt;p&gt;There are 100 people in the village. Any arbitrary group of people of any size can form a marriage if all they agree.&lt;/p&gt;&#xA;&lt;h2&gt;End note&lt;/h2&gt;&#xA;&lt;p&gt;Please notice that the examples don't endorse any marriage system, and make explicit all assumptions. For example, the 1940 system problem with explicit assumptions is just historical, but the same problem with implicit assumptions would be plainly racist.&lt;/p&gt;&#xA;" OwnerUserId="7178" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-03-05T11:06:34.803" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13731" PostTypeId="2" ParentId="13693" CreationDate="2018-03-05T14:31:58.193" Score="13" Body="&lt;p&gt;The reason why so many people get the wrong idea about differentials is that they aren't really taught what the notation means.  They are merely taught &quot;this is what the notation &lt;em&gt;is&lt;/em&gt;, and please don't ask any deep questions.&quot;  This is a recipe for misusing the notation.  Additionally, some of the standard notations (like for the second derivative) are flat-out &lt;em&gt;wrong&lt;/em&gt;, but we will get to that later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To start out with, you should think of $d()$ as a function. Therefore, $dy$ is actually shorthand for $d(y)$.  The differential function can be applied multiple times, such as $d(d(y))$, which is normally written as $d^2y$. &#xA; So, when you see a notation that says $d^2(y)$ you should think $d(d(y))$ and when you see a notation that says $dx^2$ you should think $(dx)^2$. &#xA; This alone clears up a LOT of confusions that people have in dealing with the notation.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With this explanation in hand, it becomes obvious and clear why $d^2y$ and $dy$ don't cancel.  It's the same reason why you can't cancel with $\sin(\sin(y))$ and $\sin(y)$.  In fact, for this reason, when I'm doing technical writing, I usually write the &quot;d&quot; with non-italic text, like $\mathrm{d}y$, so that it is typographically evident that &quot;d&quot; and &quot;y&quot; are playing different roles (I don't do that here, because I have a bunch of LaTeX macros that help me out that I don't have here).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I'm going to go against mainstream wisdom and say that $\frac{dy}{dx}$ absolutely &lt;em&gt;should&lt;/em&gt; be treated as a fraction.  Doing so simplifies Calculus in a number of ways, which I will show:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It is more clear to the student what is going on&lt;/li&gt;&#xA;&lt;li&gt;It removes the distinction between explicit and implicit differentiation&lt;/li&gt;&#xA;&lt;li&gt;It removes the distinction between single-variable and multivariable differentiation&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In other words, you get a single, unified, standardized process for all of these types of differentiation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way that I teach it is to NEVER take derivatives initially.  ONLY take differentials.  Then, you can SOLVE for whatever derivative you want.  But first, you have to convert your derivative rules into differential rules.  So, for a few rules (hopefully you can figure out the rest):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$d(C) = 0$&lt;/li&gt;&#xA;&lt;li&gt;$d(u) = du$&lt;/li&gt;&#xA;&lt;li&gt;$d(nu) = n\,du$&lt;/li&gt;&#xA;&lt;li&gt;$d(u^n) = n\,u^{n-1}\,du$&lt;/li&gt;&#xA;&lt;li&gt;$d(uv) = u\,dv + v\,du$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now, let's take the equation $z = xy - g$, where all of these letters are actual variables.  Let's say we want to find $\frac{dg}{dy}$.  Well, the first step NO MATTER WHAT derivative you want to find, is to take the differential of both sides:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;d(z) = d(xy - g) \\&#xA;d(z) = d(xy) - d(g) \\&#xA;dz = x\,dy + y\,dx - dg&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, to find the derivative $\frac{dg}{dy}$ we just algebraically manipulate the equation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;dz = x\,dy + y\,dx - dg \\&#xA;dg = dz - x\,dy - y\,dx \\&#xA;\frac{dg}{dy} = \frac{dz}{dy} - x\frac{dy}{dy} - y\frac{dx}{dy} \\&#xA;\frac{dg}{dy} = \frac{dz}{dy} - x - y\frac{dx}{dy}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the &lt;em&gt;real&lt;/em&gt; problem comes with the second derivative.  You can't treat that as a fraction.  However, this is &lt;em&gt;not&lt;/em&gt; because it doesn't work, but because the notation that has been handed down is simply wrong (or, technically, it is only right within a particular context).  The second derivative is usually given as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{d^2y}{dx^2}$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is only true &lt;em&gt;when x is the independent variable&lt;/em&gt;.  If you later say that $x$ depends on some other variable, such as $t$, the notation will break down.  The full notation for the second derivative (which &lt;em&gt;does&lt;/em&gt; allow for treating it as fractions) is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can be easily shown by just applying the quotient rule to the first derivative and simplifying:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;d\left(\frac{dy}{dx}\right) = \frac{dx\,d(dy) - dy\,d(dx)}{dx^2} \\&#xA;= \frac{dx\,d^2y - dy\,d^2x}{dx^2} \\&#xA;= \frac{dx\,d^2y}{dx^2} - \frac{dy\,d^2x}{dx^2} \\&#xA;= \frac{d^2y}{dx} - \frac{dy}{dx}\frac{d^2x}{dx} \\&#xA;$$&#xA;That was just the differential--to get the second &lt;em&gt;derivative&lt;/em&gt; you also have to divide by $dx$, yielding:&#xA;$$ = \frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, as you can see, using basic differential rules, most (all?) of the &quot;problems&quot; with treating the derivative as a fraction go away, and, in its place, you have students who not only know what the notation means, they can use it effectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more information on this notation for the second derivative, see &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer: I am currently working on a book on Calculus, the reason which I wrote it is largely to answer questions like these - most (all?) Calculus books have very awful discussions of differentials and what they mean.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2018-03-05T14:31:58.193" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="13732" PostTypeId="2" ParentId="13705" CreationDate="2018-03-05T18:31:32.483" Score="10" Body="&lt;p&gt;I like several of the answers which found clever sets that avoid assumptions about genders and sexes.  Just to provide an alternative, what about keeping the assumption about sexes, but changing species and turning it into a free biology lesson as well.  Many birds mate for life, such as bald eagles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would also open up the door for some other interesting math problems.  Emperor Penguins are serial monogomists.  They pick a mate and stay with them for a whole year, but rarely pick the same mate year after year (15%), so you could put together some interesting combination problems with that!  They also have the neat fact that the male takes care of the egg, balancing on his feet as the males huddle in brutal cold winds.  The females return later, when the eggs are hatching.  This gives the penguin chicks a head start over other creatures which weren't able to manage this brutal feet.  Mind you that might not help the math any, but I think it's freaking awesome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also be able to play some really interesting combinatorics with bees.  Bees have fascinating caste systems, especially regarding how new queens are dealt with.  (The queen has to make sure nobody usurups her, but if the queen dies for any reason, the hive needs to vote a new queen into office by feeding her royal jelly).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's also the &quot;other&quot; sex systems.  We are used to XX/XY, because that's what humans have.  Many of the arguments regarding not accepting the other genders or sexuality stem around the universality of this system.  Well it's not the only one out there.  Bees use X0/XX.  Many lizards us ZZ/ZW.  A quick biology lesson might start minds turning.  Evolution has had several goes at this concept.  While they, themselves, are not going to be straightforward substitutions into combinations problems, they are a fascinating chance to learn something other than math in math class.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2018-03-05T18:31:32.483" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13733" PostTypeId="1" AcceptedAnswerId="13736" CreationDate="2018-03-05T19:57:31.873" Score="35" ViewCount="7139" Body="&lt;p&gt;When I was a student, it was very much frowned upon to ask for help without making an effort, like how math.stackexchange.com operates (for the most part).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the high school where I work, it is common for teachers to go through full solutions with students who want help despite having made no effort themselves. I know that for questions that are very simple, it takes little effort to go through a full solutions with a student. But for a question like &quot;For which values of $m$ is the line $y=mx+6$ tangent to $x^2+y^2=9$?&quot;, I do not feel like helping students who claim that they have no idea where to start.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there much value in going through full solutions with students? Or is it better to dismiss students by giving them a hint and telling them to go away and attempt it seriously? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related question: I also often have students asking me to check their work, expecting me to find an arithmetic slip or sign error they have made, when for example they are computing a derivative or integral. Can anyone advise how they handle these sorts of questions? Tell them to check it on Wolfram-Alpha?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this is not too opinion-based.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastEditorUserId="8900" LastEditDate="2018-03-08T18:15:11.260" LastActivityDate="2018-03-12T17:12:27.003" Title="Helping students who make no effort to figure things out for themselves" Tags="|secondary-education|" AnswerCount="8" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13735" PostTypeId="2" ParentId="13733" CreationDate="2018-03-05T21:32:42.220" Score="3" Body="&lt;p&gt;A lot of it depends on context.  However, one thing that may help both sides (you with your time and them with their problems) out is to, instead of working the problem for them, make them work it in front of you.  You can prod them with questions until they see how it works.  Sometimes you have to get very close to just giving them the answer, but, since they have to think about it themselves, they won't go to you as a &lt;em&gt;replacement&lt;/em&gt; for thinking about it themselves.  Make them fumble through the book looking for an answer.  Just keep asking questions until they figure it out.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they are truly having trouble, this will equip them with the tools they need to succeed.  If they are trying to skip the &quot;hard work&quot; step, this can actually require more time and work from them, as they have to jump through every question you ask, right there in front of you.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2018-03-05T21:32:42.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13736" PostTypeId="2" ParentId="13733" CreationDate="2018-03-05T22:13:04.913" Score="24" Body="&lt;p&gt;My advice is to use Socratic dialogue and lead them through the process.  Get some kind of interaction, engagement.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a balance.  You can go all the way to a boot camp sergeant sending the maggot away to someone who shows the whole solution.  Some students who are used to more explanation may find Socratic dialogue uncomfortable and resist but even then if you explain that you are trying to get them involved, it is good enough.  [If they still won't engage, let them sink.  I mean it.  Life will be harder than school for sure.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do a Socratic explanation or even a full explanation MAKE SURE to give a follow-on problem (or five!) so that they can drill the learning, prove they can use it, etc.  Humans are not computers.  Just showing us a method is not good enough so that we can use it.  We need to practice it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  It is absolutely in zone as a valid question.  Deep in the pedagogy center.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-05T22:13:04.913" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13737" PostTypeId="2" ParentId="13733" CreationDate="2018-03-05T22:35:40.280" Score="11" Body="&lt;p&gt;In most cases just giving the solution is easy but not fruitful. Student learn by their own work. But many student can be honestly stuck, not in a position to do anything if one simply tell them to &quot;really try&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience is that student who do not know how to start an exercise actually do not &lt;strong&gt;know&lt;/strong&gt; what &quot;really trying&quot; means. They do not need a solution (which they would more often than not try to learn by heart without understanding it), but a method. And this needs some thinking (at least, it took me years to get to answers I find approximately satisfying)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, today I was asked how to prove that, given three subspaces $A,B,C$ of a vector space $E$, the following formula holds true:&#xA;$$ (A\cap B)+ (A\cap C) \subset A\cap (B+C) $$&#xA;I went through the following points:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;realize that the central symbol of the formula is $\subset$: we want to prove an inclusion (this is not obvious to all students!),&lt;/li&gt;&#xA;&lt;li&gt;one should check in one's lecture notes or book the definition of $+$ if the meaning is not obvious (sums of sets should trigger a warning, but of course the lecture has defined this for subspaces); although they had more than a week to prepare this exercise, only one seemed to know the definition of $B+C$,&lt;/li&gt;&#xA;&lt;li&gt;one should remember how an inclusion is usually proved, e.g. by looking into previous exercises of the sort,&lt;/li&gt;&#xA;&lt;li&gt;then one can actually solve the exercise: one considers an arbitrary element $\vec{u}$ in the left-hand-side of the inclusion, and show that it belongs to the right-hand-side; then use the definition of $+$ to exhibit a decomposition of $\vec{u}$, etc. You got the idea.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The actual solution is the last point, one fourth of my answer. In retrospect, I should probably have left it out, and reassign the exercise for the next session to avoid too many students focusing on it and forgetting the first three points, which are actually much broader in scope and much more important to their studies.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2018-03-05T22:35:40.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13738" PostTypeId="2" ParentId="13705" CreationDate="2018-03-05T23:06:46.457" Score="14" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Protons and electrons (form hydrogen atoms)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Or cations and anions (form salts), e.g. Na&lt;sup&gt;+&lt;/sup&gt; and Cl&lt;sup&gt;-&lt;/sup&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Pens and pen-caps&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Bottles and bottle caps, etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Textbooks (for the course being taught) and students&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Light bulbs and light sockets&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Power cords and electrical outlet sockets&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cars and parking spaces&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Seats and attendees, e.g. students in the classroom and classroom seats&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mimes and invisible boxes&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Numerators and denominators&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Boxes of cereal and toys-that-go-at-the-bottom-of-boxes-of-cereal&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I'm currently racking my brain for two distinct sets of objects S1 and S2, where we might naturally think of pairing objects in S1 to objects in S2, and where there is one most natural pairing&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Just for fun, $\TeX$'d up a table:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;{\begin{array}{c|c|c|c} &#xA;&amp;amp; \textbf{S}_\textbf{1} &amp;amp; \textbf{S}_\textbf{2} &amp;amp; \textbf{Natural Pairing} \\ \hline&#xA;\small{1} &amp;amp; \text{proton} &amp;amp; \text{electron} &amp;amp; \text{Hydrogen} \\ \hline&#xA;\small{2} &amp;amp; \text{anion} &amp;amp; \text{cation} &amp;amp; \text{salt} \\ \hline&#xA;\small{3} &amp;amp; \text{pen} &amp;amp; \text{pen cap} &amp;amp; \text{capped pen} \\ \hline&#xA;\small{4} &amp;amp; \text{bottle} &amp;amp; \text{bottle cap} &amp;amp; \text{capped bottle} \\ \hline&#xA;\small{5} &amp;amp; \text{student} &amp;amp; \text{textbook} &amp;amp; \text{prepared student} \\ \hline&#xA;\small{6} &amp;amp; \text{student} &amp;amp; \text{seat} &amp;amp; \text{seated student} \\ \hline&#xA;\small{7} &amp;amp; \text{power cord} &amp;amp; \text{power outlet} &amp;amp; \text{plugged-in cord} \\ \hline&#xA;\small{8} &amp;amp; \text{light bulb} &amp;amp; \text{light socket} &amp;amp; \text{light} \\ \hline&#xA;\small{9} &amp;amp; \text{car} &amp;amp; \text{parking spot} &amp;amp; \text{parked car} \\ \hline&#xA;\small{10} &amp;amp; \text{mime} &amp;amp; \text{invisible box} &amp;amp; \text{boxed mime} \\ \hline&#xA;\small{11} &amp;amp; \text{screw} &amp;amp; \text{screw hole} &amp;amp; \text{inserted screw} \\ \hline&#xA;\small{12} &amp;amp; \text{left glove} &amp;amp; \text{right glove} &amp;amp; \text{gloves} \\ \hline&#xA;\small{13} &amp;amp; \text{numerator} &amp;amp; \text{denominator} &amp;amp; \text{fraction} \\ \hline&#xA;\small{14} &amp;amp; \text{Na}^{+} &amp;amp; \text{Cl}^{-} &amp;amp; \text{NaCl} \\ \hline&#xA;\small{15} &amp;amp; \text{thing} &amp;amp; \text{name} &amp;amp; \text{named thing}  \\ \hline&#xA;\small{16} &amp;amp; x &amp;amp; y &amp;amp; \left(x,~y\right)  \\ \hline&#xA;\small{17} &amp;amp; \text{longitude} &amp;amp; \text{latitude} &amp;amp; \begin{array}{c} \text{possible site of} \\ \text{buried treasure} \end{array}&#xA;\end{array}}_{\huge{.}}&#xA;$$&lt;/p&gt;&#xA;" OwnerUserId="8943" LastEditorUserId="8943" LastEditDate="2018-03-06T00:37:00.683" LastActivityDate="2018-03-06T00:37:00.683" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="13739" PostTypeId="2" ParentId="13705" CreationDate="2018-03-06T00:06:12.893" Score="28" Body="&lt;p&gt;When I taught a class about the stable marriage problem last week, I replaced &quot;men&quot; and &quot;women&quot; with &quot;medical students&quot; and &quot;hospitals&quot;: the classical instance in which the Gale-Shapley algorithm is used in real life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition to the gender issues already mentioned, this has the benefits that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;We avoid envisioning a dystopian future where everyone's preferences are fed into a computer and the computer tells everyone who they should marry. That's just kind of weird.&lt;/li&gt;&#xA;&lt;li&gt;The very similar-looking &quot;men propose&quot; and &quot;women propose&quot; versions of the algorithm now get very different flavor text to distinguish them. In one version, medical students all travel to their next top hospital, and each hospital tells all but one of the visiting students to go home, game-show style. In the other version, hospitals send offers to their favorite student, and each student rejects all but one offer, provisionally accepting the best.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I also took this opportunity to give the participants of my examples the names of characters and hospitals from &quot;House&quot; and &quot;Scrubs&quot;. This amused me but appears to have flown over the heads of today's undergrads. Kids these days.&lt;/p&gt;&#xA;" OwnerUserId="9609" LastActivityDate="2018-03-06T00:06:12.893" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13740" PostTypeId="2" ParentId="13733" CreationDate="2018-03-06T00:09:38.523" Score="11" Body="&lt;p&gt;I'm going to add in here as the mother of one those kind of students, and say in addition to the advice you've already gotten, also consider the student and what actual difficulties they may have (whether they have an IEP, 504, or not) and if it will help them or improve your relationship with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My kid has times when she truly is befuddled with these kind of things (not just in math), and she has to already trust the teacher &lt;em&gt;just to even ask for help&lt;/em&gt;. In math it might take a couple times for her to get it, but once she gets it, &lt;em&gt;she gets it&lt;/em&gt;. Thoroughly and completely. She's in Algebra II now and is tutoring a kid taking Algebra I (this is a huge social advance for her). If she had not had teachers that gained her trust and were willing to work with her, I can guarantee you this would not have happened.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having teachers she trusts that are willing to work with her makes a huge difference in her overall effort in a class, and her grades prove it in their ups and downs. If she doesn't trust enough to ask for help, and get that help, she gives up. Completely. Doesn't matter what the subject is. She thinks she's stupid because her classmates don't understand her jokes, but the majority of people who get her jokes have a master's degree or better. She just has issues she was born with, and any decision you make to help (or not help) a student can ripple out for years and affect many lives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every student is different - learns differently, needs different motivations, different help. I would ask you to not pick a blanket way to approach this issue, but to consider each student and their needs and strengths before making a decision.&lt;/p&gt;&#xA;" OwnerUserId="9610" LastActivityDate="2018-03-06T00:09:38.523" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13741" PostTypeId="2" ParentId="13733" CreationDate="2018-03-06T00:15:44.743" Score="2" Body="&lt;p&gt;Your students are in high school, which means that there are limits to how  much responsibility you can expect them to take for their learning. And you're already most of the way through this year, so it will be difficult to apply this to this year's class. So what follows is the extreme version, and you'll have to decide how strictly to apply it, and you'll probably have to wait until next year with a new class to establish that This Is How It Is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should teach your students how to do problems. When your students need to do a problem, your lessons should show them how to do it. If they are having trouble applying your lessons to the problem, they should be able to show what parts of your lesson they have applied, what part they are having trouble applying, and why they are having trouble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student wants to ask you for help on a problem, they should have a list of key concepts for the problem, lecture notes from when you went over those concepts, what work they've done applying those notes, and questions about how to apply them. If they don't have that ready, walk away. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I said, this is an extreme version, and you'll probably want to not go quite this far, especially at the beginning. But you should be asking them to show effort on their own, and refuse to help if they aren't putting any effort forth. Ask leading/Socratic questions, and as the school year goes on, ask fewer, less leading questions, and demand the students do more and more on their own (not necessarily more and more of the actual math work, but more and more of the meta-learning stuff: identifying what the key concepts in a question are, looking in the book/lecture notes/internet/etc. for those concepts, writing down lists of &quot;things I'm told&quot; and &quot;things I'm supposed to find&quot;, identifying applicable formalae/algorithms/etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In education, there is the metaphor of &quot;scaffolding&quot;: if you're building a building, you may need a bunch of outside supports as you're building, but if the building still needs those supports at the end, you haven't done your job. Your job as a teacher is to take away support, slowly enough that your students aren't left adrift, but fast enough that by the end they can do the work on their own. Getting the balance right is one of the major challenges of teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they want to check their work, an option is to ask another student (assuming you don't have concerns of students copying each other's work). Other options would be giving them the answer, or, if you're worried about them just taking the answer and not doing the work, giving them a list of answers (that is, the answers from all the problems, plus several other decoy answers, with the order scrambled) so they can check whether the answer they got is on the list, or giving them a &quot;hash&quot; of the answer (some property that is unlikely to be shared with an incorrect answer, but is not enough to get the answer). But if this is homework, they shouldn't be stressing out on getting the right answer in the first place. You shouldn't be taking away a significant points for getting homework answers wrong, especially if they are easy to miss errors (if they get an answer that's obviously wrong and don't notice, &lt;em&gt;maybe&lt;/em&gt; that deserves some points taken off). The whole point of homework is to have a chance to make mistakes before the test.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2018-03-06T00:15:44.743" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13742" PostTypeId="2" ParentId="13705" CreationDate="2018-03-06T01:22:14.373" Score="2" Body="&lt;p&gt;Use terms from literature, TV, movies, theater, and pop and folk culture in general. With a large variety of pairings to choose from, questions can be adapted for a local culture without much difficulty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;e.g.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Klingons and Romulans&lt;/li&gt;&#xA;&lt;li&gt;Montagues and Capulets&lt;/li&gt;&#xA;&lt;li&gt;Sharks and Jets&lt;/li&gt;&#xA;&lt;li&gt;Hatfields and McCoys&lt;/li&gt;&#xA;&lt;li&gt;Jews and Greeks (cf. Gal. 3:28)&lt;/li&gt;&#xA;&lt;li&gt;Commies and Alt-Rights&lt;/li&gt;&#xA;&lt;li&gt;Sailors and Landlubbers&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Using some of these can also make your examples/questions &lt;em&gt;fun&lt;/em&gt;! Come up with some weird satirical scenario and make the example stick in peoples' brains. Forsooth, come up with a problem involving matching Montagues and Capulets and mix in Shakespeare-sounding language. Or, pick Sailors and Landlubbers, matey, and be includin some of that stereotypical pirate lingo, arr.&lt;/p&gt;&#xA;" OwnerUserId="7481" LastEditorUserId="7481" LastEditDate="2018-03-06T01:27:31.107" LastActivityDate="2018-03-06T01:27:31.107" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13743" PostTypeId="2" ParentId="13705" CreationDate="2018-03-06T04:49:56.767" Score="12" Body="&lt;p&gt;An idea I find interesting is to use &lt;strong&gt;abstract objects&lt;/strong&gt;: &lt;em&gt;&quot;given X squares and Y circles, in how many ways it's possible to pair one circle with one square?&quot;&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the students are at kindergarten or primary school level, maybe you can give the children squares and circles made with paper, plastic or wood, each with a different color, so they can try these combinations with the objects:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LfMsc.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LfMsc.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally I find this way easier and less distracting than using real life examples. Of course in some exercises you will want the student to be able to examine a real life situation and transform it in a mathematical problem, so in this case abstract objects will not work so well (and there are many good answers with interesting suggestions on meaningful pairings), but when you want to focus only on the mathematics involved, this can be useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I recommend you to avoid using topics which attract controversial and hot discussions (as gender/sexuality issues) as examples in a &lt;em&gt;math&lt;/em&gt; class, as some answers suggested. Firstly, these kind of topics can be confusing to most small children and &lt;em&gt;very&lt;/em&gt; distracting for older students, who possibly already have political opinions. I would be very distracted if I feel that the teacher is trying to &lt;a href=&quot;http://lesswrong.com/lw/gw/politics_is_the_mindkiller/&quot; rel=&quot;noreferrer&quot;&gt;punch &quot;these damned X&quot;&lt;/a&gt; inside a math problem - specially if &lt;em&gt;I am&lt;/em&gt; one of these &quot;X&quot;! Also bringing a political topic can even &lt;a href=&quot;https://grist.org/politics/science-confirms-politics-wrecks-your-ability-to-do-math/&quot; rel=&quot;noreferrer&quot;&gt;reduce the ability of properly solve mathematical problems&lt;/a&gt;! Besides that, parents with a different political leaning may not like to see a teacher injecting political topics on non-political subjects, and if the example implicitly supposes a side is the correct one, this only makes things worse. These parents can cause a problem to you, as trying to process you or the school you teach, for example (if they would be right or wrong on doing that is another question, but I think you want to avoid problems). Basically, just be sure that your replacement is socially acceptable - not only to you, but to other people too!&lt;/p&gt;&#xA;" OwnerUserId="9613" LastActivityDate="2018-03-06T04:49:56.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13744" PostTypeId="2" ParentId="13733" CreationDate="2018-03-06T08:04:47.633" Score="0" Body="&lt;p&gt;I'd understand if this was at a higher level, but we're talking high school here. You are expected to equip them with at least the bare minimum of knowledge needed to pass, because the vast majority are not going to care much about pure mathematics past HS but the topic is sadly mandatory for an unnecessarily long period of time in most places in the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That in itself makes going through full solutions worthwhile - at the very least, this gives those students who don't know where to start (the very same you pointlessly and erroneously deride, as, again, you severely overestimate the importance of your topic for most students) a full guideline that they can then study for the one and only time when knowing it actually matters. Until this changes and you only have to deal with students who are there primarily for maths, no other solution is acceptable, and you're only going to find heavily biased solutions here that do not fit the context you are in.&lt;/p&gt;&#xA;" OwnerUserId="9617" LastActivityDate="2018-03-06T08:04:47.633" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13745" PostTypeId="1" CreationDate="2018-03-06T09:45:34.213" Score="-4" ViewCount="580" Body="&lt;p&gt;According to my experience many students get in school a wrong mental picture of the limit as something that is &lt;em&gt;realized&lt;/em&gt; after infinitely many steps. They think that $0.999...$ and $\sum\limits_{n=1}^\infty\frac{1}{2^n}$ somehow reach $1$ without knowing (before I tell them) that only limits are concerned but that the limit-symbol sometimes (here for 0.999...) is left out by convention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try to convince them that the limit $a_0$ of the sequence $(a_n)$ is simply a number that is approximated better and better and can be proved by the usual criterion $\forall \varepsilon &amp;gt; 0$ $\exists k$ such that $\forall n &amp;gt; k: |a_0- a_n|&amp;lt; \varepsilon$ but in cases of strictly increasing or decreasing sequences is &lt;em&gt;never&lt;/em&gt; reached since every term of the sequence is followed by another one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My best explanation of this notion is the &lt;a href=&quot;https://nl.wikipedia.org/wiki/Paradox_(logica)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Manhatten paradox&lt;/a&gt;: Approximate the diagonal of the unit square by rectangular stairs of equal height and width. Double the number of stairs, i.e., halve their size. Repeat this procedure again and again. Then, according to the criterion,  the limit of the length of the curve approximating the diagonal is the limit of the sequence 2, 2, 2, ..., namely 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If however a &lt;em&gt;last term&lt;/em&gt; could be obtained such that for every stair its lenght and height became zero, represented by a single point, then the curve would be the diagonal with total length $\sqrt{2}$ and no points side by side in vertical or horizontal direction. Since then the cardinal number of stairs would be $\lim\limits_{n\rightarrow\infty} 2^n = \aleph_0$ this would be in disagrement with with the uncountably many points of the diagonal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately my prime example has the disadvantage that the limit $2$ is identical with all terms of the sequence and therefore its existence as a last or final term is hard to exorcize. Therefore I am looking for a better one.&lt;/p&gt;&#xA;" OwnerUserId="8557" LastEditorUserId="8557" LastEditDate="2018-03-23T15:26:04.297" LastActivityDate="2019-01-19T01:10:05.227" Title="How to correct a wrong mental picture of the limit?" Tags="|limits|" AnswerCount="7" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13746" PostTypeId="2" ParentId="13745" CreationDate="2018-03-06T13:23:01.883" Score="7" Body="&lt;p&gt;If your students really think that the limit is something that is realized after &lt;em&gt;infinitely&lt;/em&gt; many steps, I think that you are doing them a disservice trying to dissuade them. Taking the limit is effectively a transfinite operation which does just that. (My students sometimes seem to get the impression that you evaluate finitely many terms then round, especially when we talk about  $0.\overline{9}$.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One reason your diagonal approximation argument does not work is that the staircase sequence that you describe does not limit to the diagonal &lt;em&gt;in the appropriate function space&lt;/em&gt;. That is, the limit exists as a $C^0$ (continuous) function. However, you need some notion of differentiability to define the length of a curve. In any of the function spaces which include such information, your staircase sequence diverges. Since there is no limit of the curves, you can't pass to their lengths and expect to get a sensible answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, there is no reason why the number of stairs should equal the cardinality of the interval. It is entirely possible to decompose the interval into countably many intervals. It would be the number of endpoints of intervals which is countable, not the total number of points.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2018-03-06T13:42:08.273" LastActivityDate="2018-03-06T13:42:08.273" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="13748" PostTypeId="2" ParentId="13705" CreationDate="2018-03-06T16:30:40.037" Score="6" Body="&lt;p&gt;Simply state that the question is purely in the context of purely heterosexual couples right out of the gate, then make reference to homosexual couples who are also pairing off but are not included in the context of the problem, using the exact same language to refer to both groups but specifying that the question only refers to the heterosexual group. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This approach has the advantage of maintaining the original content of the problem, while re framing the context slightly to be more inclusive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And &lt;strong&gt;you should maintain the classical problem as it is constructed&lt;/strong&gt; because classical problems, rather in logic or computing or math are useful intuitive tools that are commonly understood across time and culture. Changing the problem to be &lt;em&gt;more&lt;/em&gt; culturally relatable to American students because they have a different definition of marriage might well force you to use an example that is &lt;em&gt;less&lt;/em&gt; relatable for students from Middle Eastern/African/Asian countries who have different definitions of marriage and might have encountered the problem as it was classically constructed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The overwhelming majority of people across time and culture understand/understood the concept of heterosexual monogamous pair bonding (in that they acknowledge it's existence) even if they have/had differing attitudes towards homosexual monogamous pair bonding and monogamy in general. If you try to find an alternative way of framing the problem, you run the risk of using a far less relatable example. Others have suggested using an example from biology; what if students are ignorant of that? How about pop culture references...well what if a student is from somewhere where they didn't have that thing you are referencing? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is why I argue it is better to append appropriate context to the original problem to make it more inclusive, rather than changing it radically in a way that completely removes the concept of heterosexual pair bonding in humans. By doing that you are making more of an effort to connect your teaching with people from other times and cultures who think differently from you but nonetheless sought to use the same problem to teach the same concept. And you can do that while also making the problem more inclusive in your specific classroom. &lt;/p&gt;&#xA;" OwnerUserId="9632" LastEditorUserId="9632" LastEditDate="2018-03-06T18:58:08.857" LastActivityDate="2018-03-06T18:58:08.857" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13749" PostTypeId="1" AcceptedAnswerId="13791" CreationDate="2018-03-06T16:50:23.247" Score="7" ViewCount="628" Body="&lt;p&gt;I'd like to get suggestions for neat topics or problems to include in an undergraduate, upper division Introduction to Probability class. Many people have taught probability for many years and I'm sure that, over the years, they have collected many gems of probability that aren't as well known as they should be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example would be discussing the Google PageRank algorithm as an application of Markov chains. But I'm interested in any neat topics or insights of any flavor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be especially nice to give students a taste of how probability is applied in practical situations, in a way that makes students feel empowered -- as if they now have a powerful tool at their disposal. For example, Monte Carlo methods for numerically evaluating integrals could be a good topic. Another idea could be to include a taste of coding theory (that is, methods to reliably transmit a sequence of bits over a noisy communication channel, where each bit has a certain probability of being flipped).&lt;/p&gt;&#xA;" OwnerUserId="9633" LastEditorUserId="8571" LastEditDate="2018-03-14T18:23:10.370" LastActivityDate="2022-02-28T07:16:30.237" Title="Neat topics or problems to include in a probability class" Tags="|student-motivation|probability|" AnswerCount="6" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13750" PostTypeId="2" ParentId="13733" CreationDate="2018-03-06T17:37:49.890" Score="0" Body="&lt;p&gt;Just personal experience, don't know if it's a recognized method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When my father used to help me, he would show me the full working out and answer on a blackboard. When he was satisfied that I understood how he obtained the answer, he would erase it all, and then thee me it was my turn to work it out.&lt;/p&gt;&#xA;" OwnerUserId="9638" LastActivityDate="2018-03-06T17:37:49.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13751" PostTypeId="2" ParentId="13700" CreationDate="2018-03-06T17:40:12.993" Score="9" Body="&lt;p&gt;There is a model of how people progress towards abstract reasoning through the subject of geometry called the &lt;em&gt;Van Hiele&lt;/em&gt; model. The model describes five levels: visualization, analysis, abstraction, deduction, rigor.  It dates to the 1950's, and continues to influence curricula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the analysis level, children do not allow overlaps in categories, and will insist that a square is not a rectangle. So your question is at the heart of a developmental matter. Given that educators wish for their students to progress beyond the analysis level, I would be surprised to learn about a curriculum that rigidly defined squares to not be rectangles. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2018-03-06T17:40:12.993" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13752" PostTypeId="2" ParentId="13745" CreationDate="2018-03-06T18:15:23.080" Score="1" Body="&lt;p&gt;I don't think the issue is that they are thinking of them as where you &quot;reach&quot; after an infinite number of steps, as much as they need to understand that that phrasing has a lot of subtlety.  Do remember that one of the major rationales for calculus is that it can use limits to solve Zeno's Paradox, and can do so in a sufficiently rigorous manner (rigorous enough to put spaceships in orbit!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they're thinking of limits of sequences, consider teaching them how strange infinities can behave.  The &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert Hotel&lt;/a&gt; is, in my opinion, an excellent test case showing that infinities act in ways that fundamentally challenge your assumptions that come from finite numbers.  Once they see how &quot;ill behaved&quot; naked infinities can be, they may appreciate the rigorous constraints of limit notation.  It has the advantage of being a very bold series of problems.  There's no subtlety here, just a set of infinities acting very much unlike you expect them to at first glance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They may also appreciate the idea that the real numbers $1$ and $0.\overline{9}$ are actually the same number, expressed with two different sequences, but they probably need to appreciate how wonky infinities can behave before they appreciate the subtlety there. (given the typical meaning of an overbar in a decimal, you need a concept of infinitesimals to tell $1$ and $0.\overline{9}$ apart)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: From the comments below, it seemed useful to post a proof that $0.\overline 9 = 1$.  I have chosen this particular proof because contains nothing but simple algebra, packaging up all the interesting stuff into two assumptions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Assumption 1:&lt;/strong&gt; $10\cdot0.\overline 9 = 9.\overline 9$&lt;br&gt;&#xA;&lt;strong&gt;Assumption 2:&lt;/strong&gt; $9.\overline 9 = 9 + 0.\overline 9$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proof&lt;/strong&gt;:&lt;br&gt;&#xA;$\text{Let }x = 0.\overline 9$&lt;br&gt;&#xA;$10x = 10\cdot0.\overline 9$&lt;br&gt;&#xA;$10x = 9.\overline 9$ &lt;em&gt;(using assumption 1)&lt;/em&gt;&lt;br&gt;&#xA;$10x - x = 9.\overline 9 - x$&lt;br&gt;&#xA;$9x = 9.\overline 9 - x$&lt;br&gt;&#xA;$9x = 9.\overline 9 - 0.\overline 9$&lt;br&gt;&#xA;$9x = (9 + 0.\overline 9) - 0.\overline 9$ &lt;em&gt;(using assumption 2)&lt;/em&gt;&lt;br&gt;&#xA;$9x = 9 + (0.\overline 9 - 0.\overline 9)$&lt;br&gt;&#xA;$9x = 9 + (0.\overline 9 - 0.\overline 9)$&lt;br&gt;&#xA;$9x = 9 + (0)$&lt;br&gt;&#xA;$9x = 9$&lt;br&gt;&#xA;$x = 1$&lt;br&gt;&#xA;$0.\overline 9 = 1$&lt;/p&gt;&#xA;" OwnerUserId="5787" LastEditorUserId="5787" LastEditDate="2018-03-08T08:04:55.993" LastActivityDate="2018-03-08T08:04:55.993" CommentCount="19" ContentLicense="CC BY-SA 3.0" />
  <row Id="13753" PostTypeId="2" ParentId="13745" CreationDate="2018-03-06T18:28:58.713" Score="2" Body="&lt;p&gt;I disagree with your premise. That is, I think (as in Zeno's paradoxes) that one should imagine that the limit &lt;em&gt;is&lt;/em&gt; reached, albeit not necessarily in finitely-many steps. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, otherwise, I fail to see the interest in the notion of &quot;limit&quot;...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-03-06T18:28:58.713" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13756" PostTypeId="2" ParentId="13749" CreationDate="2018-03-07T01:01:07.110" Score="3" Body="&lt;p&gt;I suggest &lt;strong&gt;the probabilistic method in combinatorics and graph theory&lt;/strong&gt;, for a few reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;This will introduce the students to other mathematical concepts that they may not be aware of already, but you can also quickly teach them enough of the basics to apply topics from your probability course to a new problem.&lt;/li&gt;&#xA;&lt;li&gt;This is a somewhat surprising application of probability, in the sense that it will be used to non-constructively demonstrate something exists without exhibiting a specific object. &lt;/li&gt;&#xA;&lt;li&gt;If you extend the discussion just a little bit, you can bring the students to the edge of current human understanding and pose some open problems. This will hopefully excite them, as I assume they are used to being taught known results and are under the impression that math is somehow &quot;solved&quot;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Specifically, I recommend discussing &lt;strong&gt;Ramsey Numbers&lt;/strong&gt;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Probabilistic_method#First_example&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Wikipedia entry for the &lt;em&gt;Probabilistic Method&lt;/em&gt; has a good explanation of how this method can provide a lower bound on the Ramsey number $R(r,r)$&lt;/a&gt; (defined as the smallest $n$ such that any 2-coloring of the edges of $K_n$ must yield a $K_r$ subgraph whose edges are all the same color). Specifically, it can be shown that $R(r,r)\geq n+1$ as long as ${n \choose r}2^{{1-{r \choose 2}}}&amp;lt;1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That quantity on the left-hand side of the inequality is the expected value of the random variable that counts the number of monochromatic $K_r$ subgraphs in a 2-coloring of $K_n$. Since that random variable only takes on integer values, if that expected value is less than 1, then there must be some instance where the variable takes on the value 0. In such an instance, there are &lt;em&gt;no&lt;/em&gt; monochromatic $K_r$ subgraphs, meaning it is &lt;em&gt;possible&lt;/em&gt; to avoid having a monochromatic $K_r$ by 2-coloring $K_n$ in a particular way. Thus, $R(r,r)&amp;gt;n$ must be true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this argument does not show at all how to achieve such a coloring! Indeed, this is still a genuinely hard problem, in general. After this presentation, you can share some further facts with the students about Ramsey Numbers, especially how we know $R(3,3)$ and $R(4,4)$ exactly, but even $R(5,5)$ is unknown. (Note: The upper bound on $R(5,5)$ was just improved recently in 2017.) &lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2018-03-07T01:01:07.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13757" PostTypeId="2" ParentId="13749" CreationDate="2018-03-07T04:20:01.840" Score="4" Body="&lt;ol&gt;&#xA;&lt;li&gt;A lighthearted one is the Caltech sweepstakes hijinks:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://hoaxes.org/archive/permalink/the_caltech_sweepstakes_caper&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://hoaxes.org/archive/permalink/the_caltech_sweepstakes_caper&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(alluded to in the movie Real Genius, some fun clips you could fair use)&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;The old 2 people in class birthday stunt:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://betterexplained.com/articles/understanding-the-birthday-paradox/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://betterexplained.com/articles/understanding-the-birthday-paradox/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Financial options (can get complicated--wouldn't teach them per se but can be discussed)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Insurance (same as 3)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Many military or search problems from OR have a probability foundation (again not just some toy problems or to try to teach a whole OR class but to mention the topic and that it is practical.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Experimental derivation of pi (I did it for a science project in HS geometry):&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Buffon%27s_needle#Estimating_%CF%80&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Buffon%27s_needle#Estimating_%CF%80&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-03-07T05:45:14.070" LastActivityDate="2018-03-07T05:45:14.070" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13758" PostTypeId="2" ParentId="13705" CreationDate="2018-03-07T04:31:22.300" Score="1" Body="&lt;p&gt;Horses and jockeys.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also consider problems with not just 1:1 pairings.  5 person basketball teams, etc.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-07T04:31:22.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13759" PostTypeId="2" ParentId="13749" CreationDate="2018-03-07T04:42:45.797" Score="3" Body="&lt;p&gt;If you just wanted to do a fun, surprising, and easy one, I would do the Montey Hall problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you wanted to overlap some social issues, you could use Bayesian statistics to talk about whether or not screening well people for rare diseases is a good policy.  You could talk about what the prevalence of the disease would have to be compared to the efficacy of the test to make the test worthwhile (for many screening tests in sufficiently rare diseases, if you are tested positive the most likely reason for a positive test is a false positive).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you wanted to do an information theory one, you could talk about Hubert Yockey's critique of RNA World from the perspective of Shannon information theory.  &quot;Origin of life on earth and Shannon's theory of communication&quot;, Computers and Chemistry, 2000. &#xA; &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0097848500800108&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.sciencedirect.com/science/article/pii/S0097848500800108&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2018-03-07T04:42:45.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13760" PostTypeId="2" ParentId="13749" CreationDate="2018-03-07T05:35:41.600" Score="3" Body="&lt;p&gt;There are some ideas &lt;a href=&quot;https://mathoverflow.net/questions/231045/what-are-fun-elementary-subjects-in-probability&quot;&gt;here&lt;/a&gt;:  these include links to various paradoxes (e.g., Simpson's, Bertrand's, and exchange paradoxes) random walks; random graph; ping pong on a Galton board, and others.   There is also a pretty general misunderstanding of what &quot;randomness&quot; looks like:  &lt;a href=&quot;http://www.abc.net.au/science/surfingscientist/pdf/lesson_plan19.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; activity is similar to something I've done in class.  The idea is you ask students to write down the results of 100 coin flips and a sequence of &quot;random&quot; H and T they generate.  The human-generated list will have no long runs of H or T, while the one generated by coin flips will. &lt;/p&gt;&#xA;" OwnerUserId="1537" LastEditorUserId="1537" LastEditDate="2018-03-07T16:59:30.547" LastActivityDate="2018-03-07T16:59:30.547" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13761" PostTypeId="2" ParentId="11732" CreationDate="2018-03-07T17:36:11.557" Score="1" Body="&lt;p&gt;The question reminds of the story told about Gauss and his discovery of the constructibiliy of the 17-gon. As a young person, Gauss made the discovery and approached his professor. The professor (Kastner) dismissed Gauss because after all, the subject of straightedge and compass construction was settled since the time of the ancient Greeks. If more could be known, then over the centuries, smart people would have made the discoveries. So the construction must be impossible. What could a teenager possibly contribute? &lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2018-03-07T18:15:19.177" LastActivityDate="2018-03-07T18:15:19.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13762" PostTypeId="2" ParentId="13749" CreationDate="2018-03-07T22:15:10.570" Score="4" Body="&lt;p&gt;If the target audience includes applied mathematics, physicists or engineers (is not a pure math course) you could also give a shout-out to some easy Markovian models used in science and engineering.  Many of these can be introduced pretty easily with little background.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance the 2D-Ising model of phase transitions is one of the simplest models demonstrating a thermodynamic phase transition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few others that come to mind is the field of percolation theory which studies the critical fraction of 'edges' on a graph where systems transition from having isolated small clusters to a spanning cluster on the order of system size. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Markov Chain Monte Carlo algorithms for integral solving are actually the basis for almost all of statistical mechanics and are some of the earliest examples of computer simulations (See &lt;a href=&quot;https://en.wikipedia.org/wiki/Equation_of_State_Calculations_by_Fast_Computing_Machines&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Rosenbluth 1953&quot;&gt;Rosenbluth 1953&lt;/a&gt;). These computer simulations are used extensively to sample the &lt;a href=&quot;https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)#Canonical_partition_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;partition function&lt;/a&gt; of the system, which is an high dimensional integral over the positions and momenta of every particle in the system of the negative exponential of the Hamiltonian operator (energy).  Designing it into a homework project might be a bit hard without requiring at least some programming.  Plug:  I used to do research in this area, and I'll point out that our simulator &lt;a href=&quot;http://hoomd-blue.readthedocs.io/en/stable/package-hpmc.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;HPMC&lt;/a&gt; is open source.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the very least to introduce statistics driven simulation that may be fun for students is a basic model used to describe  mosh pits.  The paper also includes a browser-based &lt;a href=&quot;https://mattbierbaum.github.io/moshpits.js/&quot; rel=&quot;nofollow noreferrer&quot;&gt;simulation&lt;/a&gt;, and their model can capture and study differences in what causes the mosh pit to be gas-like (random) or have circular motion.  It's fun, the model is simple, and it describes thermodynamics.&lt;/p&gt;&#xA;" OwnerUserId="9668" LastActivityDate="2018-03-07T22:15:10.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13763" PostTypeId="1" CreationDate="2018-03-07T22:51:15.857" Score="15" ViewCount="898" Body="&lt;p&gt;I am a physics grad student and several of my professors have stated that they are against the idea of posting answer keys (i.e., worked solutions) for homework and/or tests (&lt;em&gt;after&lt;/em&gt; the assignment has been completed by the student, of course). Their argument is that having an &quot;answer cheat sheet&quot; discourages the student from thinking critically about the problem and presents the opportunity for students to &lt;em&gt;feel&lt;/em&gt; like they understand how to solve a problem without &lt;em&gt;actually&lt;/em&gt; going through the rigor themselves. In fact, the entire department apparently takes the same stance with regards to posting past qualifying exams online: they post the past exams to use for studying, but not answer keys.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question:&lt;/strong&gt; Does any published education research (specifically for the math/physics fields) examine the pedagogical benefits and downsides of posting answer keys/worked solutions for students? I tried searching for this online, but had little success finding anything. If anyone could point me toward legitimate research on this topic, I'd appreciate it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should add that I was a high school physics teacher for two years, and within that arena it seemed &lt;em&gt;unanimously accepted&lt;/em&gt; that making solutions and answer keys available was the right educational strategy. Hence my skepticism of my professors'/department's rationale. But I'm willing to see what the research says!&lt;/p&gt;&#xA;" OwnerUserId="9649" LastActivityDate="2018-03-08T17:10:05.313" Title="Does education research support the idea that answer keys are bad?" Tags="|mathematical-pedagogy|physical-sciences|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13764" PostTypeId="2" ParentId="13628" CreationDate="2018-03-07T23:44:41.687" Score="1" Body="&lt;p&gt;Much of the mathematics in undergraduate use have direct application to physical applications, and I would highly recommend using them as homework examples. When they fit in naturally, it's quite beneficial to the learning of many students to link the mathematics they see to physical problems and concrete examples.  This, however, is most appropriate in math courses that are given to non-math majors as preparation for physics/engineering.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal of most of the lower-division math courses is to get students to understand the theory, but at a bare minimum be able to use it.  And on the &quot;solve something&quot; problems I think it's natural to set up or describe the equations or mathematics use, and then tell the student to solve it.  I'd also take some pains that the problem is solvable without any of the physical descriptions (here's an equation, solve it).  These lends itself probably most naturally in calculus and differential equations, but in almost every course there are generally natural applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In matrix algebra I do remember that many of the examples and connections to real world use are forced, and the course is fundamentally mostly one in theory.  But for the nuts/bolts problems where students are introduced a topic and asked to &lt;em&gt;compute something&lt;/em&gt; to demonstrate knowledge of the basics I think homework problems or examples can be given that are practical.  In the very early parts of the course when students are first asked to actually perform a matrix multiplication it could be good to note that these can be used for scaling, shearing, inverting or rotating objects.  When students are first taught to do determinants, you could give a homework problem that asks them to compute a 2x2 and 3x3 determinant that happen to be the Jacobian matrix for polar and spherical coordinates (you could also if desired mention that the matrices are Jacobian matrices for transformations for interested students to look up, but these are not the focus of the course).  To mathematicians I suppose this wouldn't be the point, but for the future engineering student this is literally what they need to know.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But at some point in a course in linear algebra it descends into honestly, a course in theory, and there will be fewer and fewer direct computation, and concrete examples will start requiring domain specific knowledge and distract from the point of the exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keep in mind a lot of courses at the first and second  year (and sometimes third) are taken by scientists/engineers, and the goal is mostly to teach enough mathematics that they can perform calculations.  While these are math courses, they're used to serve their needs as well.&lt;/p&gt;&#xA;" OwnerUserId="9668" LastEditorUserId="9668" LastEditDate="2018-03-15T19:31:58.000" LastActivityDate="2018-03-15T19:31:58.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13765" PostTypeId="2" ParentId="13763" CreationDate="2018-03-08T04:21:57.033" Score="2" Body="&lt;p&gt;I would be shocked if there were any research on this, either supporting the practice or against it.  It does not seem like the kind of research question people study -- simultaneously too narrow to be of much interest to the math ed research community, and too broad to answer (which students? in which classes? etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said that, it is certainly the case that many universities have exactly the opposite practice, at least at the &lt;em&gt;undergraduate&lt;/em&gt; level.  For instance, my university offers several years' worth of past exams with answer keys for undergraduate Calculus.  (See &lt;a href=&quot;http://www.math.lsa.umich.edu/courses/115/PastExams/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.lsa.umich.edu/courses/115/PastExams/index.html&lt;/a&gt;, &lt;a href=&quot;http://www.math.lsa.umich.edu/courses/116/Exams/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.lsa.umich.edu/courses/116/Exams/index.html&lt;/a&gt;, &lt;a href=&quot;http://www.math.lsa.umich.edu/courses/215/17exampractice/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.lsa.umich.edu/courses/215/17exampractice/index.html&lt;/a&gt;.)  This is not limited to undergraduate studies, either:  the same department also hosts past qualifying review examination papers with solutions:  &lt;a href=&quot;http://dept.math.lsa.umich.edu/graduate/qualifiers/index.html#PastExams&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://dept.math.lsa.umich.edu/graduate/qualifiers/index.html#PastExams&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, I doubt very much that there has been any research on the effect of this, but the conventional wisdom here seems to be that this is helpful.  I think what you are encountering is simply a local custom, part of the culture of your department, based on intuition and opinion.  Unfortunately most of what happens in undergraduate and graduate education fits that description.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-03-08T04:21:57.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13766" PostTypeId="2" ParentId="13700" CreationDate="2018-03-08T04:32:14.960" Score="36" Body="&lt;p&gt;I cannot answer the OP's question about cross-cultural/international perspectives, but here is a historical perspective that may be helpful.  The issue here (whether the category &amp;quot;rectangles&amp;quot; includes or excludes the category &amp;quot;squares&amp;quot;) is one aspect of a larger question having to do with whether the classification of quadrilaterals should be &lt;strong&gt;partitional&lt;/strong&gt; or &lt;strong&gt;hierarchical&lt;/strong&gt;.  It turns out that until relatively recently (say, 120 years ago) &lt;strong&gt;partitional definitions were the norm&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, Euclid’s own definitions were unequivocally partitional:  in the &lt;em&gt;Elements&lt;/em&gt;, Book I, Def. 22 he wrote (translation is Fitzpatrick’s 2007 edition from Heiberg’s Greek text; emphasis added):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And of the quadrilateral figures: a square is that which is&#xA;right-angled and equilateral, a rectangle that which is right-angled&#xA;&lt;strong&gt;but not equilateral&lt;/strong&gt;, a rhombus that which is equilateral &lt;strong&gt;but not&#xA;right-angled&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus, for Euclid a square was neither a rectangle nor a rhombus.  Not surprisingly textbooks that followed Euclid closely tended to adopt this classification up through the middle of the 19th century.  For example the 1849 edition of Playfair’s &lt;em&gt;Elements of Geometry&lt;/em&gt; defined &amp;quot;square&amp;quot;, &amp;quot;rhombus&amp;quot; and &amp;quot;oblong&amp;quot; (the word &amp;quot;rectangle&amp;quot; does not appear at all) as follows (emphasis added):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Of four sided figures, a square is that which has all its sides equal&#xA;and all its angles right angles...  An oblong is that which has all its&#xA;angles right angles, but has &lt;strong&gt;not all its sides equal&lt;/strong&gt;…  A rhombus is&#xA;that which has all its sides equal, but &lt;strong&gt;its angles are not right&#xA;angles.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Similarly Brewster’s translation (1834 and 1851 editions) of Legendre’s &lt;em&gt;Elements of Geometry&lt;/em&gt; defined the quadrilaterals as follows (emphasis added):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Among the quadrilaterals, we distinguish:  The square, which has its&#xA;sides equal, and its angles right angles.  The rectangle, which has&#xA;its angles right angles, &lt;strong&gt;without having its sides equal&lt;/strong&gt;.  The&#xA;rhombus, or lozenge, which has its sides equal, &lt;strong&gt;without having its&#xA;angles right angles&lt;/strong&gt;.  And lastly, the trapezoid, &lt;strong&gt;only two of whose&#xA;sides are parallel&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Notice that in this last passage, in addition to defining &amp;quot;rectangle&amp;quot; and &amp;quot;rhombus&amp;quot; in an exclusionary way, the word &amp;quot;trapezoid&amp;quot; is also defined so as to exclude parallelograms.&lt;/p&gt;&#xA;&lt;p&gt;However, in the second half of the 19th century this tradition of partitional definitions began to give way to the hierarchical classifications currently in use.  The 1872 edition of Legendre’s &lt;em&gt;Elements&lt;/em&gt; from the same publisher replaced the exclusive definition of rectangle quoted above with an inclusive one, defining a square as an &amp;quot;equilateral rectangle&amp;quot;.  Despite this, &amp;quot;rhombus&amp;quot; was still defined so as to exclude right-angled figures, and &amp;quot;trapezoid&amp;quot; was defined as &amp;quot;a quadrilateral which has only two of its sides parallel.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;From ca. 1850 through 1910 texts were all over the place with respect to the definition and classification of quadrilaterals:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Hill's (1884) &lt;em&gt;A Geometry for Beginners&lt;/em&gt; used exclusive definitions in all cases: &amp;quot;The sides of a parallelogram may be either equal or unequal, and the angles may be either right or oblique ; so there are in all four kinds of parallelograms : the oblique unequal-sided parallelogram or RHOMBOID..  the oblique equal-sided parallelogram or RHOMBUS…  the right unequal-sided parallelogram or RECTANGLE… and the right equal-sided parallelogram or Square.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Hall &amp;amp; Stevens' (1891) &lt;em&gt;A Text-Book of Euclid’s Elements for the Use of Schools&lt;/em&gt;  defined &amp;quot;rhombus&amp;quot; so as to &lt;em&gt;exclude&lt;/em&gt; squares, but &amp;quot;rectangle&amp;quot; so as to &lt;em&gt;include&lt;/em&gt; squares.  (In other words it followed the same conventions as the 1872 English-language edition of Legendre’s text.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;By 1900 or so most textbooks had settled on inclusive definitions for &amp;quot;rectangle&amp;quot; and &amp;quot;rhombus&amp;quot;, so that a square was a species of both.&lt;/p&gt;&#xA;&lt;p&gt;However even after this the convention persisted (and still persists!) to use an exclusive definition for &amp;quot;trapezoid&amp;quot;.  For example, Wells &amp;amp; Gart’s (1915) &lt;em&gt;Plane and Solid Geometry&lt;/em&gt; explicitly stated that &amp;quot;a rectangle is a special parallelogram&amp;quot; and &amp;quot;The square is a special rectangle and also a special rhombus&amp;quot;; however, it continued to define &amp;quot;A Trapezoid is a quadrilateral which has one and only one pair of parallel sides&amp;quot;.  Similarly Auerbach's (1920)  &lt;em&gt;Plane Geometry&lt;/em&gt; had &amp;quot;A trapezoid is a quadrilateral with one, and only one, pair of parallel sides.&amp;quot;  The SMSG (1961) &lt;em&gt;Geometry&lt;/em&gt; defines a trapezoid as &amp;quot;a quadrilateral in which two, and only two, opposite sides are parallel&amp;quot;, and the two modern Glencoe-McGraw Hill &lt;em&gt;Geometry&lt;/em&gt; textbooks that I have access to follow suit.&lt;/p&gt;&#xA;&lt;p&gt;So, to make a long story short:  the hierarchical definitions that are now commonplace first began to appear in the mid-19th century and became standardized in the late 19th and early 20th centuries, but the transition was a gradual and somewhat messy one, and is still incomplete today, at least with respect to trapezoids.&lt;/p&gt;&#xA;&lt;p&gt;Having said all of this, it would not surprise me at all if there are places in the world where partitional definitions for rectangle are still in use – indeed in a setting where Euclid's &lt;em&gt;Elements&lt;/em&gt; is still taught, I would expect that to be the case.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2021-01-24T14:45:58.240" LastActivityDate="2021-01-24T14:45:58.240" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13767" PostTypeId="2" ParentId="13763" CreationDate="2018-03-08T05:34:28.787" Score="0" Body="&lt;p&gt;There is some broader research to the effect that feedback is helpful in learning (anything).  And comparing (after effort) to an answer key is feedback.  Even if you do it all right, there is the video game reward of seeing it.  I think there is a lack of practical psychology not to understand that small rewards help drive behavior--students are not computers or machines or monks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, correction of errors.  Also, in some cases alternate methods.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Over the last 20 years, video has become much more prominent in coaching (even the lowest youth levels--coaches use their phones).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My speculation is that some of this attitude is driven by professors who don't want to be bothered and are academicians, not teachers.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-08T05:34:28.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13769" PostTypeId="2" ParentId="13408" CreationDate="2018-03-08T13:07:10.007" Score="4" Body="&lt;p&gt;In my experience (Catalonia), we use × as the multiplication sign until we introduce equations, where the cross can cause confusion with the letter x. In fact, &amp;quot;×&amp;quot; is the multiplication sign, and it seems that the natural question is why we stop using them in advanced courses.&lt;/p&gt;&#xA;" OwnerUserId="7178" LastEditorUserId="7178" LastEditDate="2023-03-02T17:45:12.787" LastActivityDate="2023-03-02T17:45:12.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13772" PostTypeId="2" ParentId="13745" CreationDate="2018-03-08T16:23:25.817" Score="1" Body="&lt;p&gt;Based on the comments on my first answer, I think I can take a stab at this and help out more.  There are &lt;em&gt;two&lt;/em&gt; issues here.  One is the issue of the student's understanding of the need for the concept of limits, and the other is your understanding of the notation that is being used.  Your interpretation of $0.\overline 9$ and $\sum\limits_{n=1}^\infty\frac{1}{2^n}$ is not the standard interpretation of those expressions, and that non-standard interpretation is causing additional confusion.  Accordingly, I will avoid using those notations until late in the answer, so that we can have agreement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The key concept that I believe your students are missing is that there is no ultimate (final, last, etc.) element to an infinite series.&lt;/strong&gt;  I believe your students have a sense that there is an ultimate element which is, as you say, &quot;realized.&quot;  Finite series have an ultimate element, but infinite ones do not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the challenge is to convince them of that.  Rather than using complex approaches like the Manhattan paradox, stick to the simplest and most famous: Zeno's Paradox.  To run a race, you must first reach halfway to your goal.  From that point, you must reach halfway to the goal again, and again, and again.  Zeno argued that you never reach the goal, because you always have at least one more step in this process.  But obviously every person alive believes you can run somewhere, so there is clearly a problem with our thinking: our thinking claims it is impossible to do something which we know to be possible!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we formalize this paradox, let $d$ be a series which represents how far has been traveled, such that the $n$th element of the series is how far we have traveled after we have reached $n$ halfway points:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\{\frac{1}{2}, (\frac{1}{2}+\frac{1}{4}), (\frac{1}{2}+\frac{1}{4}+\frac{1}{8}), \ldots\}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, your argument is the same as Zenos.  You argue that this &quot;process&quot; can continue forever.  If your students believe they can achieve $1$, then ask them which element of the set is equal to $1$.  The goal here is to get them to phrase it as something like &quot;the last element.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they say something like &quot;the infinity-th element&quot; then you will have to work with them to understand that infinity is not a natural number.  This is obviously true from set theory, but you may not want to go into set theory.  You may have to approach them in an intuitive sense.  If the students go down this approach, they have a concept of what infinity is, and how it behaves.  This would be a key concept to teach them how to unlearn.  I'd query them as to what the result of some expressions are.  For example, most people I've worked with who have learned an informal concept of infinity have the idea that $2\infty =\infty$.  If they have that assumption, then demonstrating the difficulty of reconciling $\frac{2\infty}{\infty}=\frac{\infty}{\infty}=1$ versus $\frac{2\infty}{\infty}=2\frac{\infty}{\infty}=2$ may be enough to demonstrate that infinity is, indeed, not acting like any number.  If your students have a different concept of infinity, you as a teacher will need to learn what that concept is and provide the correction.  Perhaps they will be intrigued by the puzzle of whether infinity is odd or even, given that all natural numbers are either odd or even.  My expectation is that this will be an easier issue to resolve than the original issue in your question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So once we get the students to use the phrasing &quot;ultimate element&quot; or something like that, then we can start to cut away at the concept that there &lt;em&gt;is&lt;/em&gt; an ultimate element of an infinite series.  I think the easiest example is the case we discussed on my other answer, $\{0.9, 0.99, 0.999, 0.999, 0.9999, \ldots\}$.  As you have pointed out, this construction consists only of the digits $0$ and $9$.  There is no way &lt;em&gt;any&lt;/em&gt; element of this series can contain other digits, but if an ultimate element or last element did exist, its very easy to adapt my proof of \$0\overline 9=1$ to show that if you let &quot;S&quot; equal the &quot;last element of that infinite series,&quot; my proof works.  This creates a contradiction: we can prove that every element in the series consists of only 0's and 9's, but the last element (if you claim one exists) must equal 1.  This contradiction will hopefully help the students abandon the idea that a last element of an infinite series exists at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe this is the intuitive challenge that your students need to overcome.  They must become comfortable with the idea that there are series that have no ultimate element.  I think this is the root cause of the issue you are trying to correct in your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have that, the final step is limits.  We all know that we can run places.  We know Zeno's claim about the world is false, but it appears to be true about the math.  How do we resolve that?  The answer, of course, is what you want to teach: limits.  The concept of the limit was invented because everyone had the intuitive idea that we can go places, so we should invent a concept which lines up with that intuitive understanding of reality.  I'm sure many solutions were invented, but the concept of the limit proved to be the one that made the most sense and was most consistent.  Eventually that was formalized with epsilon-delta proofs in 1821, but it was used in varying forms up to that point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Newtonian physics, there are implicit limits everywhere.  We can easily take the limit of the series $d_n$, which is a finite series whose final element contains the distance traveled after reaching $n$ halfway points, as $n\to \infty$, and see that that limit equals 1.  This matches up with our intuition, satisfies the mathematics, and if you look at the mathematics for an object traveling with a constant velocity, and do Riemann integration, you see that it can be visualized as a set of boxes very similar to Zeno's fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So that's the selling point.  The selling point is that the intuitive concepts which we try to use to resolve paradoxes like Zeno's paradox depend on either infinity being a number or on there being an ultimate element of an infinite series.  But both of those assumptions bring about disastrous contradictions, so both must be false.  But we all &lt;em&gt;know&lt;/em&gt; we can run somewhere.  The solution to this was the invention of the limit, which tied this mathematical problem back to reality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, you will note that I explicitly wrote out every sequence in that explanation.  I didn't rely on any shorthands.  When you want to introduce notations like $\sum\limits_{n=1}^\infty\frac{1}{2^n}$, this is where &lt;em&gt;you&lt;/em&gt; will need to be precise, or you will confuse the &lt;em&gt;daylights&lt;/em&gt; out of your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take the summation symbol, $\sum\limits_{n=A}^B$.  In that notation, $A$ and $B$ must be natural numbers, and n basically counts from A to B. But in the glyphs above, there is an $\infty$ sign where $B$ was.  Since $\infty$ is &lt;em&gt;not&lt;/em&gt; a natural number, this is &lt;em&gt;not&lt;/em&gt; a valid summation, by the above definition.  If we wish to use that notation, we must extend its meaning.  And that is exactly what mainstream mathematics has done.  We &lt;em&gt;define&lt;/em&gt; $\sum\limits_{n=A}^\infty f(n) \equiv \lim_{B\to\infty}\sum\limits_{n=A}^B f(n)$.  That definition reached mainstream acceptance &lt;em&gt;because&lt;/em&gt; it so closely matches the intuition of most people.  So thusly we can say \sum\limits_{n=1}^\infty\frac{1}{2^n} = 1$ because the concept of the limit is fundamentally baked into the use of infinite summation notation.  This notation has worldwide acceptance, so your students should learn it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Likewise, the meaning of the overbar notation for repeated decimals is defined to have a limit in it.  If I have the symbol $0.\overline{XYZ_n}$, where $XYZ_n$ is a finite series of n digits, then we define $0.\overline{XYZ_n} \equiv \lim_{m\to \infty}\sum\limits_{p=1}^{m}XYZ_n\frac{1}{10^{np}}$.  For our example of $0.\overline 9$, this means $0.\overline 9=\lim_{m\to \infty}\sum\limits_{p=1}^{m}9\frac{1}{10^p}=1$  Again, the concept of the limit was baked into the definition of the symbology.  This was done because it helped peoples intuition line up with the precise mathematics, which included a limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you teach them that $0.\overline 9 \neq1$, then you put them in a bit of a bind as students.  The greater mathematical community has &lt;em&gt;widely&lt;/em&gt; accepted the above definition for what the symbols &quot;$0.\overline 9$&quot; mean.  It is almost universal.  Instead of challenging that, work with the students to teach them what that symbology &lt;em&gt;means&lt;/em&gt;, and show that the limit appears inside the meaning of that symbology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just remember, your student's intuition that $\sum\limits_{n=1}^\infty\frac{1}{2^n}$ &lt;em&gt;should&lt;/em&gt; equal $1$ is not wrong.  It's not bad.  That intuition is actually what drove several thousand years of mathematical efforts through arithmetic and set theory to cause us to finally reach the concept of limits.  The point of limits is to satisfy that intuition.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastEditorUserId="5787" LastEditDate="2018-03-08T21:58:37.910" LastActivityDate="2018-03-08T21:58:37.910" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13773" PostTypeId="2" ParentId="13763" CreationDate="2018-03-08T17:10:05.313" Score="6" Body="&lt;p&gt;In the paper Research on teaching and learning Mathematics at the Tertiary level, by Biza, Victor-Giraldo, Hochmuth, Khakbaz, Rasmussen, recently published by Springer (ICME 13 Topical Surveys) you find at least some evidence supporting your professors' attitude. It is said there, in fact, that analyzing the way students use mathbooks, and in particular exercises, it was found that the large majority would use solved problems to look for procedures that could be line by line transported to the problem they were working on; which is exactly the opposite of thinking about the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In all new experiences of IBL the emphasis, for example, is not on the answer but on working through it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So i would say that the &quot;conventional wisdom&quot; that solutions re useful does not stand on a more solid ground than your professors approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a personal experience I do not provide worked out solutions to my students, but I am encouraging them to come to office hours and in such case I do not refrain from a line to line analysis of their solutions. &#xA;A procedure which is much more time consuming and tiring for me then posting solutions, so that &quot;guest&quot; speculation is, at least in my case, completely misguided.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2018-03-08T17:10:05.313" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13774" PostTypeId="2" ParentId="13745" CreationDate="2018-03-08T19:27:46.747" Score="6" Body="&lt;p&gt;I am genuinely surprised that in a SE site populated by (among others) math education researchers, nobody has yet mentioned Tall &amp;amp; Vinner's landmark paper:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Tall, D., &amp;amp; Vinner, S. (1981). Concept image and concept definition in&#xA;  mathematics with particular reference to limits and continuity.&#xA;  &lt;em&gt;Educational studies in mathematics&lt;/em&gt;, 12(2), 151-169.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Full paper &lt;a href=&quot;https://link.springer.com/article/10.1007/BF00305619&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)  Tall &amp;amp; Vinner use the phrase &lt;em&gt;concept image&lt;/em&gt; to refer to &quot;the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes.  It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures.&quot;  As such, the concept image is more than a definition, and more than even a single metaphor or mental image.  As they authors point out, &quot;As the concept image develops it need not be coherent at all times.  The brain does not work that way...  different stimuli can activate different parts of the concept image, developing them in a way which need not make a coherent whole.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the main purpose of Tall &amp;amp; Vinner's paper is to introduce and elaborate on the notion of &quot;concept image&quot;, they develop their ideas in the specific context of limits and continuity, which makes this research particularly relevant for the OP's question.  Some tidbits from the paper:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As we shall see shortly the concept images of limit and continuity are quite likely to contain factors which conflict with the formal concept definition.  Some of these are subtle and may not even be consciously noted by the individual but they can cause confusion in dealing with the formal theory...  For instance, the verbal definition of a limit &quot;$s_n \to s$&quot; which says &quot;we can make $s_n$ as close to $s$ as we please, provided that we take $n$ sufficiently large&quot; induces in many individuals the notion that $s_n$ cannot be equal to $s$ (see Schwarzenberger and Tall, 1978).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To the specific question of whether $0.999\dots = 1$, the authors note that in one study&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;fourteen out of 36 students claimed [that] $$\lim_{n\to\infty} \left( 1 +\frac{9}{10}+ \frac{9}{100} + \cdots + \frac{9}{10^n} \right)=2$$ but $0.\bar{9}&amp;lt;1$....  Clearly the two questions evoked different parts of the concept image of the limiting process.&#xA;  In a subsequent test the same students were asked to write the following ecimals in fractions:  $$.25, .05, .\bar{3}=.333\dots, .\bar{9}=.999\dots$$ Thirteen of the fourteen who previously said $0.\bar{9}&amp;lt;1$ now said $0.\bar{9}=1$. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and, later&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One [conflict factor] of great significance is that students often form a concept image of &quot;$s_n\to s$&quot; to imply $s_n$ approaches $s$, but never actually reaches there...  Thus [for these students] &quot;point nine recurring&quot; is not equal to one because the process of getting closer to one &lt;em&gt;goes on for ever&lt;/em&gt; without one ever being completed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now to some extent all of the above is unhelpful if the question is &quot;How to correct a wrong mental picture?&quot;  Nowhere in Tall &amp;amp; Vinner is an attempt made to &lt;em&gt;fix&lt;/em&gt; the problems caused by conflicting concept images; their goals were to describe the phenomenon, provide language for discussing it, theorize about the origins and nature of the problem, and document evidence for it existing.  But I would strongly suggest beginning there.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2018-03-08T21:44:09.933" LastActivityDate="2018-03-08T21:44:09.933" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="13777" PostTypeId="1" AcceptedAnswerId="13779" CreationDate="2018-03-09T11:25:28.053" Score="-4" ViewCount="248" Body="&lt;p&gt;Usually school children are taught fractions and decimal representations way before the notion of limit. So they must come to the idea that infinite decimal sequences like 0.999... are the same as an infinite sequence of partial sums 0.9 + 0.09 + 0.0009 + ... And when they are told that 0.999... = 1 they unavoidably get the wrong impression that the sequence has a fixed value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know from my own bad experience that many students are not aware of the fact that a strictly increasing sequence has no fixed numerical value since it increases without end. Usually mathematicians know that the limit has to be applied but from time to time I have even heard mathematicians claim: Every finite sequence fails but the infinite sequence is 1 - as if the infinite sequence of partial sums could contain an infinite partial sum or could somehow reach 1! Therefore it is very important to clear up these things early enough in order to prevent such mistakes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question: When is the appropriate time to correct this wrong picture and to tell them the truth about the meaning of the decimal representation?&lt;/p&gt;&#xA;" OwnerUserId="8557" LastEditorUserId="8557" LastEditDate="2018-03-10T08:06:09.070" LastActivityDate="2018-03-10T08:06:09.070" Title="When should the limit be introduced?" Tags="|limits|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13778" PostTypeId="1" AcceptedAnswerId="13783" CreationDate="2018-03-09T12:19:43.617" Score="11" ViewCount="411" Body="&lt;p&gt;Some integration techniques are just &amp;quot;tricks&amp;quot;, while some integrals are analytically significant in that they connect different fields of math or they embody higher level concepts.&lt;/p&gt;&#xA;&lt;p&gt;In the commonly stated form, &lt;a href=&quot;http://mathworld.wolfram.com/FrullanisIntegral.html&quot; rel=&quot;noreferrer&quot;&gt;the Frullani's integral&lt;/a&gt; is a definite integral that encompass a nontrivial family of integrand.&lt;/p&gt;&#xA;&lt;p&gt;One can &lt;a href=&quot;https://math.stackexchange.com/questions/62254&quot;&gt;apply it directly&lt;/a&gt; as a power-move, &lt;a href=&quot;https://mathoverflow.net/a/288178/100250&quot;&gt;out of necessity&lt;/a&gt;, or &lt;a href=&quot;https://mathoverflow.net/a/40543/100250&quot;&gt;more subtly&lt;/a&gt; in analysis.&lt;/p&gt;&#xA;&lt;p&gt;With some effort, it can be extended one way or the other. For example, see &lt;a href=&quot;https://math.stackexchange.com/a/1700507/356647&quot;&gt;the answer by fedja in this post&lt;/a&gt;, along with other valuable expositions on the derivation of Frullani integral. A certain generalization from the reals to complex &lt;a href=&quot;https://math.stackexchange.com/a/1808847/356647&quot;&gt;can also be made.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My Questions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How would you introduce the Frullani integral to freshmen calculus students?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How would you re-introduce Frullani integral to junior or senior math majors who presumably care about more than just getting the answers?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The issue with presenting it as a &amp;quot;trick&amp;quot; is that students might learn only the bare minimum (merely the mechanical maneuver). Students at various levels inevitably sometimes wonder if &amp;quot;tricks&amp;quot; are not worth learning at all. How does one carefully make the distinction between tricks and more substantial things (here specifically the Frullani integral).&lt;/p&gt;&#xA;" OwnerUserId="7268" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-03-13T04:58:31.827" Title="How would you introduce Frullani integral to students?" Tags="|undergraduate-education|mathematical-pedagogy|mathematical-analysis|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13779" PostTypeId="2" ParentId="13777" CreationDate="2018-03-09T13:19:18.597" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;When is the appropriate time to correct this wrong picture and to tell them the truth about the meaning of the decimal representation?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am not entirely clear yet on what the wrong impression here is, but a reasonable answer is: The appropriate time to discuss the precise meaning of decimal representation is when limits of sequences are discussed, and the level of rigour can be the same. Once infinite sums are also introduced, one can take a second look at the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A rigorous approach to limits and sequences seems to happen at around the first or second year of university mathematics studies in the universities with which I am familiar.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2018-03-09T13:19:18.597" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13780" PostTypeId="2" ParentId="13778" CreationDate="2018-03-09T14:46:18.143" Score="2" Body="&lt;p&gt;Just show them it as one more trick and emphasize that it has large scope of application.  I think that is enough.  Also, not clear what else you all are supposed to be working on (what course) and if spending extra time on this integral is worth it.  And if so, how worth it...like please don't skip other &quot;tricks&quot; that have large scope of application (e.g. partial fractions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whole question seems to read like &quot;I love this integral and want to push it into the curriculum&quot; not really &quot;how to teach&quot;.  But in terms of the literal question how to teach, you have your answer already.  Show it as one more trick, drill it a bit, test it.  And mention it's large extent of applications.  Is there really a teaching methodology conundrum?  Or a desire to justify the content to us or to colleagues?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if your students aren't in real analysis yet, than all you can really do is say something like &quot;it's useful in this higher math course see, where we do theory and such to rigorously prove things&quot;.  But I wouldn't bother showing them WHERE it is used in RA if they are not in RA at all yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I would only emphasize it in a regular calc class if they are a bunch of MIT, CalTech types who are taking some super enriched course anyways.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-09T14:46:18.143" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13781" PostTypeId="2" ParentId="13778" CreationDate="2018-03-10T02:53:42.383" Score="8" Body="&lt;p&gt;I almost certainly would not introduce that integral to freshmen. I probably wouldn't introduce it to sophomores or juniors either; not without some particular application in mind. (I don't know of one, but certainly it must have come up somewhere. Do you have an application in mind?) Still, playing along with a counterfactual world where I would do it, I'd do it using differentiation under the integral and elide questions of uniform convergence until after I'm done presenting the basic idea. Let&#xA;$$ \phi(a,b) = \int_0^\infty \frac{ f(ax)-f(bx) } {x} dx $$&#xA;Then&#xA;$$ \begin{align}&#xA;\frac{\partial}{\partial a} \phi(a,b) &#xA;&amp;amp;= \int_0^\infty \frac{\partial}{\partial a} \frac{ f(ax)-f(bx) } {x} dx \\&#xA;&amp;amp;= \int_0^\infty f'(ax) dx \\&#xA;&amp;amp;= \int_0^\infty \frac{1}{a} f'(u) du \\&#xA;&amp;amp;= -\frac{1}{a} \left( f(0)-f(\infty) \right) \\&#xA;\end{align} $$&#xA;Similarly, one can find&#xA;$$ \frac{\partial}{\partial b} \phi(a,b) = \frac{1}{b} \left( f(0)-f(\infty) \right) $$&#xA;Now you can use standard methods to solve those differential equations and get the result,&#xA;$$ \phi(a,b) = \left(f(0)-f(\infty)\right)\ln\left(\frac{b}{a}\right) $$&#xA;This is a technique found in &lt;em&gt;Advanced Calculus&lt;/em&gt; by Woods, which I found out about by reading people who were writing about Feynmann's book &lt;em&gt;Surely you're joking Mr. Feynmann&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-03-10T02:53:42.383" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13782" PostTypeId="2" ParentId="13712" CreationDate="2018-03-10T10:18:05.083" Score="0" Body="&lt;p&gt;According to my experience concrete examples like the following are best suited to support the understanding. Starting from the axes $y = 0$ and $x = 0$ I have always shown the following, one after another of course. &#xA;&lt;a href=&quot;https://i.stack.imgur.com/Wvzrh.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Wvzrh.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8557" LastActivityDate="2018-03-10T10:18:05.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13783" PostTypeId="2" ParentId="13778" CreationDate="2018-03-10T18:42:46.967" Score="2" Body="&lt;p&gt;I think it is hard to motivate this sort of issue at a typical U.S.-undergrad level. In my own recollection, I did not see such an integral/issue occur at all except in the discussion of the Gamma function in Whittaker-and-Watson, as a deus-ex-machina.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A saner context might be in discussion of principal-value integrals as distributions or tempered distributions. (The latter is more accessible, since the topology is easier to describe: Frechet rather than &quot;LF&quot;...) E.g., it is a reasonable exercise to prove that $f \to \lim_{\varepsilon\to 0^+} \int_{|x|\ge \varepsilon} {f(x)\over x}\;dx$ is a tempered distribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that context, considering $f\to \int {f(ax)-f(bx)\over x}\;dx$ as a tempered distribution, explicitly noting which seminorms enter in proving its continuity, would have some value, I think.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-03-10T18:42:46.967" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13784" PostTypeId="2" ParentId="10697" CreationDate="2018-03-10T19:03:33.633" Score="4" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I was introduced by the unit circle.  This seems simpler than the ratio of sides since hypotenuse is 1, although the ratio of sides is more general and physically meaningful.  But it is just simpler to grok initially.  To this day, I mentally picture the unit circle and picture a radar scope with the line moving counterclockwise.  Might be &quot;bad&quot; but I did fine in trig and further classes.  [It is probably an easier picture also when thinking about &quot;negative angles&quot; or angles greater than 180 or 360.  See the &quot;sweep&quot; returning in your mind.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think having physics class soon after trig class is VERY helpful since trajectories and such make such great use of trig in terms of vectors.  It gives good practice, motivation, and mental picture of the angles.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-10T19:03:33.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13785" PostTypeId="2" ParentId="380" CreationDate="2018-03-10T19:34:15.627" Score="3" Body="&lt;p&gt;&quot;For many topics, my students think they know the subject already, but all they actually know are a bunch of short-cuts or special cases. The problem is that when I try to illustrate the new ideas with simple problems, they fall back on their short-cuts rather than using what we've covered. Then when they get to the harder problems, they're completely lost, because the short-cuts don't work, but they haven't gotten any practice with the new material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can tell them &quot;you have to solve this problem using this method&quot;, but that feeds into the lesson that math is an arbitrary pile of methods which we choose based on the professor's whim, and it causes them to resent the new method.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Talk to them and explain why you are doing what you are doing!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the beginning of the course, say many of you have had calculus before but didn't do well enough to place out of it.  This can help you at times, but at other times can hurt you.  If you already know some of the material, that is great, makes it easier to study.  But it can lead you into false confidence and into goofing off and not studying material hard enough (both old and new).  Note, I think this is the bigger danger than using isolated shortcuts!  Be up front and tell them that you don't want them to take calculus three times!  This is the chance to learn it right, so let's get 'er done!    [Also mention that sometimes they may only know a shortcut or one method and not pay enough attention to rest of the course for problems that don't work with that shortcut or particular method.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of learning different methods, I would tell them that YES, you do want them to learn and drill particular methods using THAT method at the time.  Tell them you will allow them in tests to use whatever method works (if there are multiple approaches).  But it is important to have many tricks in the bag, many tools in the box.  So when learning a method, go ahead and use that method in the homework drill (and recitation).  Only way to learn new things.  Otherwise you rely too much on old methods and don't ever get the benefit of new.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compare sports for instance.  In wrestling practice, you will learn different takedowns (single leg, double leg, hip throw, etc.)  When you are learning that trick, you need to practice that trick.  In a match, you can use whatever seems best at the time.  And you may have some tricks you are better at and lean more towards.  But there will be situations where only one trick works or it works easier.  The more you can get good at different tricks, the better a wrestler you will be.  You won't be as good if you only have one takedown that you are good at. &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-10T19:34:15.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13786" PostTypeId="2" ParentId="4284" CreationDate="2018-03-11T15:25:45.220" Score="-1" Body="&lt;p&gt;We are so careful because $dy/dx$ is the &lt;em&gt;limit&lt;/em&gt; of a sequence of fractions. This limit is &lt;em&gt;not&lt;/em&gt; &quot;the final term&quot; of the sequence and does not entail that all terms of the sequence exist. We should be even more explicit when teaching this matter since there are even some contemporary logicians confessing in public to believe in the idea that the terms of a sequence could be exhausted step by step until the index $\omega$ is reached - in contradiction with the successor axiom.&lt;/p&gt;&#xA;" OwnerUserId="8557" LastEditorUserId="8557" LastEditDate="2018-03-12T08:03:43.187" LastActivityDate="2018-03-12T08:03:43.187" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13787" PostTypeId="2" ParentId="4333" CreationDate="2018-03-11T19:16:33.307" Score="-1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I suspect the issue is more one for the teacher than the students.  Having a proof (or at a lighter level, a derivation) is nice motivation, but the key thing to getting confidence is just drilling the method itself.  I suspect teacher is implicitly unhappy with teaching non-proved topics and that the student questions come from some implicit presentation (alluding to the proof gap by teacher).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If I wanted to show a little derivation for motivation, I would do something like this (showing the backwards derivation):&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;a. Show them a specific quadratic reciprocal integral (with numbers, not general).  Make a joke about how scary and evil it looks--we don't know what the heck to do with it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;b. hey, y'all...remember in grade school, when you combined fractions?  Well let's do one for practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[work through $\frac23 + \frac45 = \frac{22}{15}$]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c.  OK, then.  Well now we aren't in grade school any more, we need to do stuff with variables and all!  Remember in Algebra 2, when we learned about partial fractions.  &quot;Oh....don't you groan now...because it is finally going to have a use!&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Work through combination of $\frac{A}{x-c} + \frac{B}{x-d}$.  In other words, GO BACKWARDS.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;d.  Do the algebraic derivation forwards. (from the quadratic on the bottom to the two fractions.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;e.  Solve the specific &quot;scary looking&quot; integral using partial fractions.  Make a joke about how we kicked the butt of that scary looking integral using grade school fractions and high school algebra.  Not so scary any more!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;f.  Have the kids do one in class (simple, simple version).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;g.  &quot;Ok, guys, this is just a bunch of algebra.  It's not as hard as that darned integration by parts or those evil related rate word problems.  But you gotta practice it.  If you don't, you will mess up the grade school fractions and high school algebra.  So, I want each of you to do 10 of these problems tonight.  And we will have a one question quiz tomorrow.  And next week on the test, I guarantee you at least one problem will be a partial fraction integral.  And remember, you need to kick the integral's butt.  Not let it kick yours!&quot;&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="11" LastEditDate="2018-03-12T06:49:24.297" LastActivityDate="2018-03-12T06:49:24.297" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13788" PostTypeId="2" ParentId="4333" CreationDate="2018-03-12T02:21:54.470" Score="1" Body="&lt;p&gt;As long as your goal is not proof, but just helping people get a general feel for it, there are several ways you can go.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first is to say that, even if there were no proof, it is a worthwhile thing to try.  That is, we can imagine not having the proof that it &lt;em&gt;will&lt;/em&gt; work, but still attempting to do it anyway, just like we do with a lot of other methods.  We can see &lt;em&gt;that&lt;/em&gt; adding some numerator over $(x - c)$ and some numerator over $(x - d)$ will give us a sum that has a denominator of $(x - c)(x - d)$.  We also know that it would be a heck of a lot easier to do the integral if we could find some isolated number to put in the numerator.  So, why not see &lt;em&gt;if&lt;/em&gt; there is one there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The procedure of partial fractions is fairly straightforward if we just see &lt;em&gt;if&lt;/em&gt; something will turn up.  If we proceed just on the hunch that maybe we'll find something appropriate, then we can move forward.  If we succeed we have found something we can integrate, and if we fail then, just like any other attempt, we'll have to find some other way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact (I don't know if this is the case here), sometimes in math this is the way it works.  You have a method that you know can work sometimes, and then, you just kind of notice that it works always.  You don't know if it is &quot;always always&quot; or just &quot;always for my problems&quot;, so &lt;em&gt;then&lt;/em&gt; you search to see if you can prove it one way or the other.  In any case, seeing that it &lt;em&gt;could&lt;/em&gt; work is the first step to seeing &lt;em&gt;how&lt;/em&gt; it works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, given that it &lt;em&gt;could&lt;/em&gt; work, it isn't too hard to see why it should always work.  To see this, imagine what would happen if $(x - c)$ or $(x - d)$ had an $x$ in the numerator.  When cross-multiplying, that would bring the degree up to $x^2$!  Since we don't have any $x^2$ to work with, we can say for certain that either neither of them or both of them have $x$ (they &lt;em&gt;could&lt;/em&gt; both have $x$ if one were a negative of the other to cancel them out).  However, if they &lt;em&gt;both&lt;/em&gt; had an $x$ in the numerator, then they could both be simplified through polynomial long division, which would get rid of the $x$ and leave a $1$ and a $-1$ by themselves, and a fraction &lt;em&gt;without&lt;/em&gt; the $x$ in the numerator.  When the $1$ and the $-1$ cancel out, that leaves just the fractions without the $x$ in the numerator!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, we identified two possible cases - either both have an $x$ or neither do, and realized that the only way that both of them had an $x$ is if it were in a way that they canceled out and reduced to the case where neither one had an $x$.  Thus, we can always assume that neither one has an $x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's kind of convoluted, and not entirely a proof, but I think it goes a long way to showing why this method can work, and why they should trust that a proof exists.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2018-03-12T02:21:54.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13789" PostTypeId="2" ParentId="4477" CreationDate="2018-03-12T13:32:08.273" Score="2" Body="&lt;p&gt;Late answer for you, but for searchers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1)  I like how you consider the switching cost and would not underestimate it.  Don't let perfect be the enemy of better.  If teachers and department are used to one text, they can do a good job with it.  [Analogy: The weapon the DC sniper used was not ideal for his task as a theoretical choice of rifle, but because he was FAMILIAR with it, it was a great choice.] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Low cost is a reasonable rationale to switch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) If there is a way to use a text that doesn't change to new editions (e.g. a Dover edition or Schaum's). I think this is advantageous as it fits in with points 2 and 3.  But I don't have a specific suggestion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) I prefer texts to have the answers to the homework problems.  It is more conducive to drill.  (It seems strangely like this is an area where modern day practice is LESS liberal than older practice.)  This is a controversial point and your colleagues/department may not agree, but then if you can at least select something with large amount of problems and 50% answers would be good.  Or one where the students can buy the full solution manual (and not the evil &quot;student solution manual&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(5) I like Thomas and Finney back in the day but I used one of the last editions where T was part of T&amp;amp;F. Now it is a brand like &quot;Gray's Anatomy&quot;.  Probably any middle of the road text is fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(6)  I would avoid reform texts (see Amazon for very negative student reactions to the DHH text).  In addition to some experiments not working pedagogically, Harvard book has the feel of committee writing (too sterile). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(7) I would avoid ebooks.  Yes, this conflicts with (2). The physical object is easier to learn from.  Also, my impression is that a lot of ebooks have poor quality control (poorly edited).  The topic is confusing enough for new students--if the book has a lot of errata, that is bad.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(8) Similar to (7) and (1), I would avoid new editions.  Let someone else be the bleeding edge.  New programs have &quot;bugs&quot;.  Work with &quot;tested code&quot;. &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-12T13:32:08.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13790" PostTypeId="2" ParentId="4409" CreationDate="2018-03-12T14:17:48.283" Score="0" Body="&lt;p&gt;I think the stories here are about rather hard or minute issues that would distract the class. Quaternions?  Really?  For kids learning sqrt(-1)?  I would actually avoid these stories unless you have kids that are just crushing it on the topic and have time to waste.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one story that seemed simple and friendly was the bug on the ceiling for Descartes. A rectangular ceiling has a very Cartesian look to it.  Things like that are good.  Minutia of Leibniz versus Newton, or L'Hopital is wrong name, no.  That's confusion.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is useful instead is to emphasize how the courses have had value in the life of the instructor.  And you don't need to do detailed applied problems.  Just let them know that knowing trig and calc have been personally helpful to you in physics classes and in the military and oil exploration (or whatever from your personal life).  Just that knowing the stuff helped you earn some bread.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-12T14:17:48.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13791" PostTypeId="2" ParentId="13749" CreationDate="2018-03-12T17:03:51.907" Score="5" Body="&lt;p&gt;Seriously? Probability? There's a lot!&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Probability Puzzles:&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_puzzle#Probability&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Mathematical_puzzle#Probability&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Probability Paradoxes:&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Category:Probability_theory_paradoxes&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Category:Probability_theory_paradoxes&lt;/a&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_paradoxes#Probability&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_paradoxes#Probability&lt;/a&gt;&#xA;(statistics) &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_paradoxes#Statistics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_paradoxes#Statistics&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simulating stock prices using excel:&#xA;&lt;a href=&quot;https://www.investopedia.com/articles/investing/102715/simulating-stock-prices-using-excel.asp&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.investopedia.com/articles/investing/102715/simulating-stock-prices-using-excel.asp&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Computing the Historical Volatility in Excel&#xA;&lt;a href=&quot;https://www.investopedia.com/articles/investing/102715/simulating-stock-prices-using-excel.asp&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.investopedia.com/articles/investing/102715/simulating-stock-prices-using-excel.asp&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Book: Probability Problems:&#xA;Fifty Challenging Problems in Probability with Solutions&#xA;By: Frederick Mosteller&#xA;&lt;a href=&quot;http://store.doverpublications.com/0486653552.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://store.doverpublications.com/0486653552.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Book: Chapter 5 of A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality 1st Edition by Clifford A. Pickover (Author)&#xA;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0471690988&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Passion-Mathematics-Numbers-Puzzles-Religion/dp/0471690988&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Qjm7g.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Qjm7g.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h1&gt;alice in wonderland/Through the Looking-Glass example&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ZoTc0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZoTc0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Probabilistic arguments&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Category:Probabilistic_arguments&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Category:Probabilistic_arguments&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Probabilistic proofs&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_probabilistic_proofs_of_non-probabilistic_theorems&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_probabilistic_proofs_of_non-probabilistic_theorems&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Probabilistic proofs eg Independence for Euler:&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/427920/140308&quot;&gt;https://math.stackexchange.com/a/427920/140308&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Probabilistic proofs eg More on Euler:&#xA;&lt;a href=&quot;https://pdfs.semanticscholar.org/1377/8c9f711aa529ec7ce40ca664ac00b13a7ed6.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://pdfs.semanticscholar.org/1377/8c9f711aa529ec7ce40ca664ac00b13a7ed6.pdf&lt;/a&gt;&#xA;&lt;a href=&quot;https://projecteuclid.org/euclid.jap/1389370108&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://projecteuclid.org/euclid.jap/1389370108&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bayes (Bayes' Theorem, Bayesian vs Frequentist probability interpretations or statistics)&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Probability_interpretations&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Probability_interpretations&lt;/a&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/An_Essay_towards_solving_a_Problem_in_the_Doctrine_of_Chances&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/An_Essay_towards_solving_a_Problem_in_the_Doctrine_of_Chances&lt;/a&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lindley%27s_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Lindley%27s_paradox&lt;/a&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/R_v_Adams#Use_of_Bayesian_analysis_in_the_court&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/R_v_Adams#Use_of_Bayesian_analysis_in_the_court&lt;/a&gt;&#xA;&lt;a href=&quot;http://www.stat.rutgers.edu/home/hcrane/Teaching/582/lectures/chapter5-Bayes.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.stat.rutgers.edu/home/hcrane/Teaching/582/lectures/chapter5-Bayes.pdf&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4934658/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4934658/&lt;/a&gt;&#xA;&lt;a href=&quot;https://stats.stackexchange.com/questions/22/bayesian-and-frequentist-reasoning-in-plain-english&quot;&gt;https://stats.stackexchange.com/questions/22/bayesian-and-frequentist-reasoning-in-plain-english&lt;/a&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Base_rate_fallacy&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Base_rate_fallacy&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Geometric Probability&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Geometric_probability&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Geometric_probability&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Negative Probabilities:&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Negative_probability&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Negative_probability&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Negative Probabilities in Option Pricing:&#xA;&lt;a href=&quot;https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1773077&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1773077&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Risk-neutral probability (not just option pricing):&#xA;&lt;a href=&quot;https://quant.stackexchange.com/questions/103/how-does-the-risk-neutral-pricing-framework-work&quot;&gt;https://quant.stackexchange.com/questions/103/how-does-the-risk-neutral-pricing-framework-work&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Math with Bad Drawings' Series: Stories and Lessons in Probability&#xA;&lt;a href=&quot;https://mathwithbaddrawings.com/2013/09/30/the-bear-in-the-moonlight/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathwithbaddrawings.com/2013/09/30/the-bear-in-the-moonlight/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;From Quora: 'What are the most interesting or popular probability puzzles in which the intuition is contrary to the solution?'&#xA;&lt;a href=&quot;https://www.quora.com/What-are-the-most-interesting-or-popular-probability-puzzles-in-which-the-intuition-is-contrary-to-the-solution&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.quora.com/What-are-the-most-interesting-or-popular-probability-puzzles-in-which-the-intuition-is-contrary-to-the-solution&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Book: Probability with Martingales by David Williams&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h1&gt;Star Trek&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YUVa3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YUVa3.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wPDql.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wPDql.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1cEwu.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1cEwu.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Markov chains&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lJJAx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lJJAx.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/K2nkM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/K2nkM.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Laplace transform&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MaPLs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MaPLs.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Abracadabra&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gDcyD.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gDcyD.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2022-02-28T07:16:30.237" LastActivityDate="2022-02-28T07:16:30.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13792" PostTypeId="2" ParentId="13733" CreationDate="2018-03-12T17:12:27.003" Score="0" Body="&lt;p&gt;Why don't you:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Give a hint&lt;/li&gt;&#xA;&lt;li&gt;Go through the full solution if they don't get the hint&lt;/li&gt;&#xA;&lt;li&gt;Let them do another similar problem in front of you to confirm their understanding&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;?&lt;/p&gt;&#xA;" OwnerUserId="5987" LastActivityDate="2018-03-12T17:12:27.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13793" PostTypeId="2" ParentId="13778" CreationDate="2018-03-13T04:58:31.827" Score="4" Body="&lt;p&gt;The question asks &quot;How would you introduce Frullani integral to students?&quot;  To answer this, it's necessary to first ask a different question:  &quot;Why do you &lt;em&gt;want&lt;/em&gt; to introduce Frullani integral to students?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't mean this just as a rhetorical question.  If you want to introduce any new topic, the first thing you have to be clear on is:  Why is this worth teaching?  The universe of things you &lt;em&gt;could&lt;/em&gt; be teaching is always larger than the number of opportunities you'll have to teach something, so every topic that gets included carries with it opportunity costs.  Presumably if this is something you want to teach, it's because &lt;em&gt;you&lt;/em&gt; think it's worth knowing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So start there: What makes it worth knowing? What problems does this technique solve &lt;em&gt;for you&lt;/em&gt;?  What questions does it answer?  If it's not useful, is it at least interesting?  (Related:  See &lt;a href=&quot;https://matheducators.stackexchange.com/a/11410/29&quot;&gt;https://matheducators.stackexchange.com/a/11410/29&lt;/a&gt;.)  What makes it so?  Is it interesting because it seems surprising?  If so, why -- what expectation does it contradict?  Is it interesting because it reveals some underlying connection between other ideas?  If so, what are they?  Once you answer those questions, you will know how to introduce this topic to your students.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-03-13T04:58:31.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13794" PostTypeId="1" CreationDate="2018-03-13T09:56:00.007" Score="8" ViewCount="583" Body="&lt;p&gt;As well as mathematics I teach IB &lt;a href=&quot;http://www.ibo.org/programmes/diploma-programme/curriculum/theory-of-knowledge/&quot; rel=&quot;noreferrer&quot;&gt;Theory of Knowledge&lt;/a&gt;, which includes mathematics as an area of knowledge. The class are mainly not students with a maths focus, although they all study at least some for the IB, and all studied mathematics at GCSE level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the key concepts that I have struggled to get across to them is mathematics as a process of making logical deductions from a set of axioms, which has implications for the relationship between mathematics and reality. For example, I have talked to them about Euclidean versus non-Euclidean geometry, and how for a long time non-Euclidean geometry was considered an abstract curiosity but eventually turned out to apply to reality as a consequence of the non-Euclidean axioms applying to reality, but the example is fairly far from their frame of reference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any suggestions for a simple illustration, which ideally the students could try for themselves, of how starting from different axiom sets and following the same deductive rules leads to different consequences, that lends itself to a discussion of how reality could turn out to make one or the other set of axioms true?&lt;/p&gt;&#xA;" OwnerUserId="1295" LastActivityDate="2018-03-14T01:11:39.577" Title="Introducing an axiomatic method to high-school students" Tags="|secondary-education|axioms-foundations|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="13795" PostTypeId="2" ParentId="12165" CreationDate="2018-03-13T16:40:51.830" Score="1" Body="&lt;p&gt;I found another one I like. It's the example in this &lt;a href=&quot;https://math.stackexchange.com/q/2647300/167197&quot;&gt;question on MathSE&lt;/a&gt;. There are lots of other good ones in this post too.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Three friends Albert, Betty, and Chadwick ask the Game-Master &#xA;      to play a game with them. The Game-Master agrees, &#xA;      and proceeds to paint two colored dots on each of their foreheads. &#xA;      The dots are either blue or yellow,&#xA;      and among all the dots on their foreheads &#xA;      no color is used more than four times. &#xA;      They can each see the colors on the others' forehead, &#xA;      but not the colors on their own. &#xA;      The Game-Master asks the friends in the order &#xA;      Albert, Betty, Chadwick, Albert, Betty, Chadwick, ...&#xA;      whether or not they know the colors of their dots. &#xA;      If one says &quot;&lt;em&gt;no&lt;/em&gt;&quot;, the Game-Master asks the next friend. &#xA;      If one says &quot;&lt;em&gt;yes&lt;/em&gt;&quot;, that person has to state their guess, &#xA;      and if they are correct, the friends win the game, &#xA;      but if they are wrong, then they lose. &#xA;      The friends were not given any time to make up a strategy.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;They start to play the game, and their answers are&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;No, No, No, No, Yes!&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;and they win the game! What are the colors of the dots on Betty's head? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2018-03-13T16:40:51.830" CommentCount="0" CommunityOwnedDate="2018-03-13T16:40:51.830" ContentLicense="CC BY-SA 3.0" />
  <row Id="13796" PostTypeId="2" ParentId="13794" CreationDate="2018-03-13T21:11:58.917" Score="2" Body="&lt;p&gt;I think (based on some experience with math-for-broader-purposes courses at honors freshman university level, for example) that it may be suboptimal to try to introduce &quot;the axiomatic method&quot; to people who have little substantive mathematical or scientific (or philosophical) experience. Namely, in my observation, people very often over-interpret the apparently legitimate whimsicality or randomness of varying axiomatic systems. As though mathematicians apparently pick random variations of axiom systems, make whatever deductions they can, and occasionally &quot;get lucky&quot; in that they've done something &quot;useful&quot; (or descriptive of some aspect of &quot;reality&quot;?). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not think this at all faithfully represents professional mathematical practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more reasonable, and perhaps more intelligible, portrayal of genuine practice might be that we want &quot;economical&quot; or &quot;compressed&quot; &lt;em&gt;descriptions&lt;/em&gt; of &quot;real&quot; situations, sufficient to &lt;em&gt;deduce&lt;/em&gt; further genuine facts/truths _in_advance_ of further observations. Or, since people generally do not quite understand what &quot;deduce&quot; could possibly mean, just say &quot;predictive power&quot;...?!?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, while some people are fond of trying to distinguish &quot;pure&quot; from &quot;applied&quot; math, it seems to me that the real point is a &lt;em&gt;predictive&lt;/em&gt; capacity about things-of-interest, whether those things be abstract or very tangible, &quot;creating jobs&quot; or not, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, a very simple thing that struck me forcefully as a pretty small kid was the possibility of measuring how tall trees and houses and flagpoles are by using &quot;similar triangles&quot;. Sure, implicit in that use is the idea that Euclidean geometry approximates small-scale physical geometry quite well, and/but the idea that we can more-or-less &lt;em&gt;deduce&lt;/em&gt; heights by means other than raw measurements completely amazed me at the time...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-03-13T21:11:58.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13797" PostTypeId="2" ParentId="12165" CreationDate="2018-03-13T21:21:37.487" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://twitter.com/CmonMattTHINK&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Matt Enlow&lt;/strong&gt;&lt;/a&gt; recently posted a wonderful collection of problems on twitter in his tweet &lt;a href=&quot;https://twitter.com/CmonMattTHINK/status/972455526834589697&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He links to the &quot;More Questions than Answers&quot; pdf on dropbox &lt;a href=&quot;https://www.dropbox.com/s/jxwmd0s6or4yy8z/Enlow%20-%20MQTA%20-%20V1.pdf?dl=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;; it is a compilation of 100 math problems, which run the gamut in difficulty and range (for me) from well-known to never-before-seen.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2018-03-23T15:30:37.043" LastActivityDate="2018-03-23T15:30:37.043" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13798" PostTypeId="2" ParentId="13794" CreationDate="2018-03-13T23:22:20.827" Score="2" Body="&lt;p&gt;I suggested back in &lt;a href=&quot;https://matheducators.stackexchange.com/a/11840&quot;&gt;&lt;strong&gt;MESE 11836&lt;/strong&gt;&lt;/a&gt; the following example from RL Wilder:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9Dxhg.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9Dxhg.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regard to the question here:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does anyone have any suggestions for a simple illustration, which ideally the students could try for themselves, of how starting from different axiom sets and following the same deductive rules leads to different consequences, that lends itself to a discussion of how reality could turn out to make one or the other set of axioms true?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You could have students begin with the axioms above, but then add an additional axiom about the number of points. Assuming that the number of points is $n &amp;lt; \infty$, it is possible to prove that $n$ must be a perfect square. So, you can create various models that satisfy these axioms with different numbers of points. (I gave an even earlier answer in &lt;a href=&quot;https://matheducators.stackexchange.com/a/2530&quot;&gt;&lt;strong&gt;MESE 2528&lt;/strong&gt;&lt;/a&gt; showing how to satisfy these axioms with a nine point model consisting of cards from the game SET; note that, indeed, $9 = 3^2$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you dig deeply enough into writings about philosophy [of mathematics], then you will find there are differences in whether individuals believe things to be &quot;infinitely divisible&quot; (e.g., &lt;a href=&quot;https://en.wikipedia.org/wiki/Infinite_divisibility&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;wikipage&lt;/strong&gt;&lt;/a&gt;). So, the question of how many points there are &lt;em&gt;in reality&lt;/em&gt; is somewhat reasonable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Returning to cases in which the number of points is finite, you could have students tack on an additional axiom about the number of points being some perfect square, and then ask whether they can construct a model satisfying these axioms - and, if so, follow up with questions like, &quot;How many total lines are there?&quot;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-03-13T23:22:20.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13799" PostTypeId="2" ParentId="13794" CreationDate="2018-03-14T01:11:39.577" Score="2" Body="&lt;p&gt;I think (counter to some others) that this is indeed a useful thing to introduce and talk about. My personal opinion is that it's something of a shame/outrage for anyone to go to college and not ever hear about what a &quot;theorem&quot; is. Currently I have a 2-slide talk on the axiomatic method on the first day of any of my college courses, including (especially) remedial algebra and prealgebra. My take-away line here is: &quot;The goal is to reduce the facts that must be taken on faith, or from authority, to the &lt;em&gt;smallest number possible&lt;/em&gt;.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't try to model toy axiomatic systems or get students at this level to prove abstract relationships. But I do expect them to be able read and understand explanations from first principles. I work the terminology into what may seem like nigh-trivial results, e.g.:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Definition:&lt;/strong&gt; $\frac{a}{b} = \frac{c}{d}$ means $ad = bc$.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Theorem:&lt;/strong&gt; $\frac{ac}{bc} = \frac{a}{b}$.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Proof:&lt;/strong&gt; $abc = abc$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I actually get some traction out of this. Without getting too elaborate or side-tracked, the students can actually have a reasonable conversation about the definition/theorem/proof process in mathematics, not be intimidated by it, predict what's coming next, and frequently thank me for it. Personally I think that's a big win.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-03-14T01:11:39.577" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13800" PostTypeId="2" ParentId="10697" CreationDate="2018-03-14T08:42:12.633" Score="2" Body="&lt;p&gt;I'm really surprised by the way that some people here start defining trigonometric functions: as all those functions can have positive as well as negative values, it makes (in my humble opinion) no sense to define them as result of the division of two positive numbers. The way I learnt them started with the basis on the unity circle, in which an angle was drawn, and from that, the basic functions were defined (sin(x) as a the coordinate on the Y-axis and cos(x) as the coordinate on the X-axis).&lt;br/&gt;&#xA;I must admit that one thing was teached to me in the complete wrong way: the definition of the tangent of an angle (it was teached as being the division of sin(x) and cos(x)). I only realised how wrong this was, once I got faced with a student I was teaching: I had told him that sin(x) is the coordinate on the Y-axis, cos(x) is the coordinate on the X-axis, and tangent(x) was the division of both, to which he answered&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Sir, I don't believe you! If you say that both sin(x) and cos(x) are&#xA;  coordinates, then also tangent(x) must be a coordinate, or I don't&#xA;  believe what you say!&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although the student was known for his bad behaviour, I must admit he was right: tangent(x) is the coordinate of the angle, as cut off using the tangent line going through the point (1,0).&lt;br/&gt;&#xA;I do understand that it might be useful knowing the link between trigonometric functions and lengths of sizes of triangles, but this might not replace the pure definitions of the trigonometric functions.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastEditorUserId="8190" LastEditDate="2018-03-15T11:25:42.313" LastActivityDate="2018-03-15T11:25:42.313" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13801" PostTypeId="2" ParentId="8419" CreationDate="2018-03-14T22:38:35.980" Score="0" Body="&lt;p&gt;I recommend examining the site &lt;a href=&quot;http://students.brown.edu/seeing-theory/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Seeing Theory&lt;/strong&gt;&lt;/a&gt; to see whether it suits your needs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a corresponding text draft (&lt;a href=&quot;http://seeingtheory.civicomment.org/seeing-theory-notes&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;) on which the authors are actively seeking feedback.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a free, basic resource that covers some of the topics that you list, and perhaps can be useful in other ways, too. From my casual perusing of the site, it seems quite well-constructed/organized.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-03-14T22:38:35.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13802" PostTypeId="2" ParentId="8419" CreationDate="2018-03-15T06:51:40.553" Score="0" Body="&lt;p&gt;It sounds like a very basic, short survey of conventional prob/stat (&quot;probes and shafts&quot;).  But it's not clear to me that &quot;discreteness&quot; should be a limiter rather than a simple overall prob/stat unit...but your call on that. You want the basics so that students can do school projects using the concepts of standard deviation, etc.  Also, stats is more and more a thing in the every day information worker business world, so helps them with coops, etc.  And of course, stay the heck away from crazy hard measure theory grad student math stuff.  The kids need to know how to use some basic stats tools, not to prove how they were derived. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Couple ideas: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Take a look at the single chapter in Lindeberg's EIT/FE reference manual (your library should have it).  It covers the key topics (and does so very efficiently) within 11 pages (at least in the eighth edition).  You could just use it as some guidance and write some ppt slides to summarize the information.  Or you could pl...  ;-)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Came up with this from a quick Google search, looking for Six Sigma prob/stat review:  &lt;a href=&quot;http://www.qualitiamo.com/documenti/prob-stats-review.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.qualitiamo.com/documenti/prob-stats-review.pdf&lt;/a&gt; [It is pretty short and action oriented.  But it is basically all stats, lacking the permutations and combinations.  Personally I like perm/combinations, but distributions are more relevant to job type tasks.  You would need to supplement it a little or just bail on the games of chance objective.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-15T06:51:40.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13803" PostTypeId="1" AcceptedAnswerId="13812" CreationDate="2018-03-15T19:07:56.673" Score="11" ViewCount="542" Body="&lt;p&gt;What are some problems or theorems that motivate the distinction between continuity and uniform continuity? In particular, I would like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) A useful, appealing theorem that applies to uniformly continuous functions but not arbitrary continuous functions, and there is an easy example of a continuous (but not uniformly) function for which the conclusion of the theorem fails,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) A natural, fun problem that can be solved using a uniformly continuous function and can't be solved using a function that is merely continuous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest thing I currently have to what I'm looking for is a version of the Riemann-Lebesgue lemma: $\int_a^b \sin(Nx)f(x)dx \to 0$ as $N\to \infty$, if $f$ is uniformly continuous on $(a,b)$, but not if it's merely continuous. But my students do not yet know Fourier analysis, so they won't know why this is a useful or natural question to answer.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="2083" LastEditDate="2018-03-17T06:55:55.683" LastActivityDate="2018-03-17T06:55:55.683" Title="Motivation for uniform continuity" Tags="|undergraduate-education|concept-motivation|mathematical-analysis|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13804" PostTypeId="1" CreationDate="2018-03-16T00:50:38.583" Score="11" ViewCount="361" Body="&lt;p&gt;As a math teacher I often experiment new things (for instance non obvious ways to use technology in the classroom, or new ways to teach something), and I think it would be useful if I could share my results without having to write my own blog that few people would read. What are the most popular websites to do this? &lt;/p&gt;&#xA;" OwnerUserId="9197" LastActivityDate="2018-03-19T02:13:35.410" Title="What are the best places online where math educators can discuss their experience?" Tags="|teaching|teacher-development|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13805" PostTypeId="1" CreationDate="2018-03-16T02:41:12.263" Score="2" ViewCount="1147" Body="&lt;p&gt;What are some books or other resources for teaching primary school children logic?&lt;/p&gt;&#xA;" OwnerUserId="7971" LastActivityDate="2018-03-21T02:41:03.863" Title="Resources for Teaching Logic to Primary School Children?" Tags="|primary-education|logic|resource-request|resources|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13806" PostTypeId="2" ParentId="13805" CreationDate="2018-03-16T03:18:57.970" Score="0" Body="&lt;p&gt;A Google search for three terms logic, elementary, and school came up with the following relevant hits on the first page:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Page with links to many other resources on this topic:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://cathyduffyreviews.com/homeschool-extras/critical-thinking-logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cathyduffyreviews.com/homeschool-extras/critical-thinking-logic&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Another page listing several logic resources for young children (emphasis is on age appropriate puzzles).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://ourjourneywestward.com/my-favorite-logic-resources/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ourjourneywestward.com/my-favorite-logic-resources/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;1965 journal article on logic teaching for elementary school:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/41186878?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/41186878?seq=1#page_scan_tab_contents&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Hacker thread discussing teaching logic in elementary school (a university teacher argues no in well done post):&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://news.ycombinator.com/item?id=1867769&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://news.ycombinator.com/item?id=1867769&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;Reddit thread with similar debate (but many more posts) on the value proposition for this topic at this age:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.reddit.com/r/changemyview/comments/1wsalo/i_think_that_we_should_teach_formal_logic_to_very/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.reddit.com/r/changemyview/comments/1wsalo/i_think_that_we_should_teach_formal_logic_to_very/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  If you do your own Google Search, about 30% of the hits will be relevant (some are on the rapper Logic or on school discipline!), but you can quickly weed through that based on the preview descriptions.  So maybe worth while going to page 2 or 3 of the Google search if you want more info.  But I would start with the subordinate hits from items (1) and (2) above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.  I am skeptical of teaching real rigorous logic in this setting.  Although I was part of New Math and got the Venn diagrams and Boolian stuff in elementary school.  Didn't do me any harm I guess.  One of the problems with teaching real logical methods is that students may not have enough other references.  For example, if you teach truth tables than students may not have much experience with tabular data in general.  Probably better off to just work on that instead.  But idonno!&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-03-16T19:12:41.667" LastActivityDate="2018-03-16T19:12:41.667" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13807" PostTypeId="2" ParentId="13804" CreationDate="2018-03-16T03:36:27.627" Score="2" Body="&lt;p&gt;The &lt;a href=&quot;https://mathematicsteachingcommunity.math.uga.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics Teaching Community&lt;/a&gt; would seem to be a good place, but unfortunately it seems to be unavailable.  (It has been unavailable for a few years now, I think.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the &lt;a href=&quot;http://www.math.uga.edu/mathematics-education&quot; rel=&quot;nofollow noreferrer&quot;&gt;University of Georgia&lt;/a&gt; website:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The &lt;em&gt;Mathematics Teaching Community&lt;/em&gt; is an online community for those of us who want mathematics teaching to be a vigorous, vibrant profession. It's a place where we can learn with and from each other and build a repository of knowledge about mathematics teaching. Everyone who teaches (or taught) mathematics at any level from PreK through college is invited. Use the tags to search for topics of interest. Post submissions, which can be anything for or about mathematics teaching, such as activities, questions, or links to useful resources. Vote for postings that you find helpful or interesting.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An article in the &lt;a href=&quot;http://www.ams.org/notices/201406/rnoti-p621.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Notices of the American Mathematical Society&lt;/a&gt; provides more information.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-03-16T03:36:27.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13808" PostTypeId="2" ParentId="13804" CreationDate="2018-03-16T04:26:14.723" Score="3" Body="&lt;p&gt;For an absolute free-for-all with no restrictions on extended discussions (unlike here) or anything else for that matter, you could try sci.math at Google Groups. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;WARNING: It is currently infested with cranks and trolls, but you can just ignore them. They can't block your postings. I wouldn't recommend it for students. The trolls there seem to delight in misinforming naive readers. Best not to engage them at all. Three of them are really quite insane. Don't give out any personal info, e.g. full name, place of work, school board, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, it has come to this, but if you can filter out all the noise, serious discussions are still possible there. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: You may also get some input from professional mathematicians at sci.math. I have found them very helpful over the years. I couldn't have developed my proof software without them.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2018-03-18T21:45:52.623" LastActivityDate="2018-03-18T21:45:52.623" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13809" PostTypeId="1" CreationDate="2018-03-16T10:44:41.607" Score="4" ViewCount="283" Body="&lt;p&gt;How can I explain the fundamental concepts of limits and continuity to a student with a non-mathematical background?&lt;/p&gt;&#xA;&lt;p&gt;I am a PhD student in Mathematics working in Differential Geometry.&#xA;As a part of my teaching assignment I need to explain the concepts of &lt;strong&gt;Limits and Continuity&lt;/strong&gt; to some undergraduates majoring in Physics.&lt;/p&gt;&#xA;&lt;p&gt;I don't know how to explain these concepts easily to them. Will anyone at all listen to me?&lt;/p&gt;&#xA;&lt;p&gt;Should I make a slide presentation to them?&#xA;Is it possible to explain mathematical concepts using PowerPoint/LaTeX?&lt;/p&gt;&#xA;&lt;p&gt;Are there any sites or online resources which can help me to make them understand Mathematics in an easier way?&lt;/p&gt;&#xA;" OwnerDisplayName="user9721" LastEditorUserId="19777" LastEditDate="2022-08-31T16:35:38.753" LastActivityDate="2022-10-01T23:23:59.647" Title="How to explain the concepts of limits and continuity to non-mathematical students" Tags="|undergraduate-education|calculus|limits|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13810" PostTypeId="2" ParentId="13804" CreationDate="2018-03-16T20:17:11.120" Score="3" Body="&lt;p&gt;Many teachers use Twitter to share and discuss teaching. One large community includes teachers from early childhood to university level, mostly from North America but also from other countries. It calls itself the &quot;Math(s) Twitter-Blog-o-Sphere&quot; and uses the hashtag #MTBoS to identify tweets where they want to share or discuss their teaching ideas. You can see tweets with this hashtag without signing in to Twitter by searching for the hashtag. This link goes direct to the search ordered by most recent. &lt;a href=&quot;https://twitter.com/search?f=tweets&amp;amp;vertical=default&amp;amp;q=%23MTBoS&amp;amp;src=savs&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://twitter.com/search?f=tweets&amp;amp;vertical=default&amp;amp;q=%23MTBoS&amp;amp;src=savs&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2018-03-16T20:17:11.120" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13811" PostTypeId="2" ParentId="13804" CreationDate="2018-03-16T21:05:35.870" Score="2" Body="&lt;p&gt;There are a couple of Facebook groups that might be useful:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If you are interested in discussing &lt;em&gt;research&lt;/em&gt; in mathematics education (including &quot;action research&quot; in your own classroom, which sounds like what you are interested in), try the &lt;a href=&quot;https://www.facebook.com/groups/mathedresearchers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Education Researchers&lt;/a&gt; group (of which I am a moderator).&lt;/li&gt;&#xA;&lt;li&gt;If you want to discuss curriculum and best practices with other teachers, try the &lt;a href=&quot;https://www.facebook.com/groups/113674342553157/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Teachers Professional Learning Network&lt;/a&gt; group.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Both of these are closed groups, meaning you must ask for permission to join.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-03-16T21:05:35.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13812" PostTypeId="2" ParentId="13803" CreationDate="2018-03-16T21:15:20.563" Score="3" Body="&lt;p&gt;First, I think this is a great question for this site because it made me question how I teach uniform continuity in my own classes and sent me on an exploration for better ideas. So, thanks!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, to address the question. In the past, to motivate uniform continuity (after introducing it), I would use a challenging exercise about Cauchy sequences and an extension of a function's domain. I use Abbott's &lt;em&gt;Understanding Analysis (2nd Ed.)&lt;/em&gt;, and the exercise is 4.4.13 (Continuous Extension Theorem):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a) Show that a uniformly continuous function preserves Cauchy sequences; that is, if $f:A\to\mathbb{R}$ is uniformly continuous and $(x_n)\subseteq A$ is a Cauchy sequence, then show $f(x_n)$ is a Cauchy sequence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) Let $g$ be a continuous function on the open interval $(a,b)$. Prove that $g$ is uniformly continuous on $(a,b)$ if and only if it is possible to define values of $g(a)$ and $g(b)$ at the endpoints so that the extended function $g$ is continuous on $[a,b]$. (In the forward direction, first produce candidates for $g(a)$ and $g(b)$, and then show the extended $g$ is continuous.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is fine and should certainly appeal to us mathematicians. However, for the students, I'm sure this feels quite distant from what else they've seen about uniform continuity, when it's applied to specific functions (e.g. $x^2$ is uniformly continuous on $[0,1]$ but not $[0,\infty)$.) Moreover, when I teach Real Analysis, a significant course objective is to teach students how to think about (and, indeed, even &lt;em&gt;care about&lt;/em&gt;) abstraction and generality. So, I worry about heavy cognitive load in this section: why would a student care about this new concept (uniform continuity) when the reason to do so is merely to prove this rather general theorem about arbitrary sequences and continuous functions, especially if the student has difficulty even understanding what the theorem states?! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I've developed an idea that I am suggesting to you and myself. (Yes, I haven't tried this yet, but I believe it will work well.) &lt;strong&gt;Use this as an occasion to look back to some results from precalculus and calculus (and intro to proofs), as well as what has been discussed in class about the relationship between $\mathbb{Q}$ and $\mathbb{R}$. Specifically, prove in several steps that the exponential function $f(x)=2^x$ can be defined on the domain $\mathbb{R}$ and is continuous on that domain.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The list below could be completed as an investigative project by the students (perhaps in teams) or together with you in class, or some combination thereof.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Use the &quot;repeated multiplication&quot; concept to explain why $2^x$ exists for $x\in \mathbb{N}$.&lt;/li&gt;&#xA;&lt;li&gt;Use the idea that $2^{-x}=\frac{1}{2^x}$ to explain why $2^x$ exists for $x\in\mathbb{Z}$.&lt;/li&gt;&#xA;&lt;li&gt;Use the Intermediate Value Theorem to explain why $2^{1/n}$ exists for $n\in\mathbb{N}$. (If $2^{1/n}=y$ then $y^n=2$, so consider the function $f(y)=y^n$.)&lt;/li&gt;&#xA;&lt;li&gt;Explain why $2^x$ exists for $x\in\mathbb{Q}$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So far, this has been pretty straightforward, and the IVT is the only necessary result from analysis. Here comes the new &amp;amp; fun stuff, which was partly inspired by the discussion in &quot;Relations to the extension problem&quot; on the Wikipedia page for &quot;Uniform continuity&quot;: &lt;a href=&quot;https://en.wikipedia.org/wiki/Uniform_continuity#Relations_with_the_extension_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Uniform_continuity#Relations_with_the_extension_problem&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;Show that $f(x)=2^x$ is not uniformly continuous on the domain $\mathbb{Q}$. (Remember that we don't know yet whether $2^x$ exists for $x\notin\mathbb{Q}$.)&lt;/li&gt;&#xA;&lt;li&gt;Let $I$ be a bounded interval. Show that $f$ is uniformly continuous on the domain $I\cap \mathbb{Q}$. (You can do this for this specific function, or prove a more general result.)&lt;/li&gt;&#xA;&lt;li&gt;Show that $f$, restricted to $I$, preserves Cauchy sequences, &lt;em&gt;because it is uniformly continuous on that domain&lt;/em&gt;. (Again, do this specifically or generally, depending on the audience.)&lt;/li&gt;&#xA;&lt;li&gt;Explain why this means we can extend $f$ to the domain $I$ (not just $I\cap\mathbb{Q}$) in such a way that $f$ is continuous on $I\subseteq\mathbb{R}$. This will necessitate a discussion with students about the completeness of $\mathbb{R}$ and Cauchy sequences of rationals converging to irrationals, as well as continuity and limits of sequences. In other words, this should tie together many concepts from earlier in the course!&lt;/li&gt;&#xA;&lt;li&gt;Explain why, since this can be done for any bounded interval $I$, there must be a unique way to extend $f$ to all of $\mathbb{R}$ and make it continuous on that domain.&lt;/li&gt;&#xA;&lt;li&gt;If desired, share more general versions of some of these results. For example: If $X$ is a complete metric space and $f:S\to X$ is uniformly continuous for every $S\subseteq X$ that is bounded, then $f$ may be extended to a continuous function on the domain $X$.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Edit: I forgot to mention that step #7 is the one that truly addresses your question, specifically goal (a). A continuous function $f$ need not necessarily preserve Cauchy-ness of a sequence, but a uniformly continuous function does. The Math.SE link in the comments expands on this idea, but rather than just talk about that fact to the students, my answer here suggests using that idea within a specific (and, I hope, more compelling for them) example.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2018-03-16T21:15:20.563" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13813" PostTypeId="1" CreationDate="2018-03-16T22:29:26.113" Score="8" ViewCount="164" Body="&lt;p&gt;(This is only peripherally related to math education, but it seems to be slightly more on-topic here than in MSE; if people disagree, I will gladly remove it and post it there.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have recently started tutoring a college student who, I've noticed, uses a slightly nonstandard (?) way of speaking mathematics aloud.  Given an expression like&#xA;$$\frac{3\cdot n}{k}$$&#xA;she will say &quot;3 onto $n$, over $k$&quot;.  It's the use of the word &quot;onto&quot; that I am interested in here.  To my (Michigan-raised) ears, it's an unusual way of speaking this mathematics; I would expect to hear &quot;3 times $n$ over $k$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This particular student is from the southern US state of Georgia, which leads me to suspect that this is a regionalism.  So now I am wondering:  where in the English-speaking world is this way of speaking taught?&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-03-23T03:41:52.417" Title="Regions in which &quot;$a \times b$&quot; is read aloud as &quot;$a$ onto $b$&quot; rather than &quot;$a$ times $b$&quot;" Tags="|language-use|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13814" PostTypeId="1" CreationDate="2018-03-17T00:58:58.400" Score="10" ViewCount="336" Body="&lt;p&gt;One thing I notice frequently is that students don't have 'best practices' for doing algebra. Let me given an example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If students are trying to differentiate, say, $f(x) = (x^2 + x)^2$, they will often write: $2 (x^2 + x) [2x + 1]$, where the brackets are parenthesis that they don't write, so their final answer is $2 (x^2 + x) 2x + 1$. Observing them, I notice that this is because they tend to write $2 (x^2 + x)$ first, then they start to write $2x + 1$ next. But they forget to open a parenthesis between the two expressions!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always write the $[$ first, but I didn't notice this until I started to think about why students make this mistake. I'm sure there are lots of other similar best practices - I think it took me a lot of making mistakes to learn those best practice techniques. They are now mostly subconscious, so it can be hard even to notice that I'm doing them and students are not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes I try to tell students about specific mistakes they make, or give them tips for organizing algebra, but it's hard to catch all of them. So I'm wondering what other best practices people have noticed. I imagine that there is probably some research into this. (With a 'proof' that these best practices always work.)These are the kind of responses I'm looking for:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Suggestions of some similar best practices to point out to students (hopefully with helpful, battle tested exercises to help students learn these best practices)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;References on this (something systematic)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8029" LastEditorUserId="5220" LastEditDate="2018-03-17T15:11:11.720" LastActivityDate="2018-03-18T21:07:32.170" Title="Algebra best practices for students" Tags="|algebra|students-mistakes|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13815" PostTypeId="2" ParentId="13700" CreationDate="2018-03-17T15:09:38.543" Score="5" Body="&lt;p&gt;I just came across this discussion&#xA;&lt;a href=&quot;https://news.ycombinator.com/item?id=16605831&quot; rel=&quot;noreferrer&quot;&gt;https://news.ycombinator.com/item?id=16605831&lt;/a&gt;&#xA;which reminded me of your question. The perspective isn't exactly math and isn't exactly not-math but rather, a question in object-oriented programming. Several of the comments (search for rectangle and square to get the relevant comments) there are worth reading on their own, but let me provide some context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In object oriented programming, you define define types/classes which have properties and things you can do with them (methods). There is also a notion of &quot;inheritance&quot; where you take a class of type A and &lt;em&gt;extend it&lt;/em&gt; to a class of type B. Since B is an extension of type A, it has all of the methods and properties of A. Hence any instance of type B can also play the role of an instance of type A.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the classic examples which is given in programming text books is that of squares and rectangles. Squares are a type of rectangle, so if we were to create classes of type Rectangle and Square, it makes a certain amount of sense that Square should extend Rectangle. Similarly, Rectangle should be an extension of ConvexPolygon which extends Polygon, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, squares are formed by &lt;em&gt;constraining&lt;/em&gt; rectangles. They have a &lt;em&gt;single&lt;/em&gt; side length. Rectangles have &lt;em&gt;pairs&lt;/em&gt; of side lengths. The natural methods to implement for squares and rectangles should return different types: a single number vs. a pair. So perhaps having Square be an extension of Rectangle is not actually the best idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, none of this would prevent a Rectangle (type) from being a square (math) incidentally.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-03-17T15:09:38.543" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13816" PostTypeId="2" ParentId="13814" CreationDate="2018-03-17T15:37:11.483" Score="4" Body="&lt;p&gt;One thing I'll sometimes tell students is &quot;if you write a minus sign, and then some '&lt;em&gt;large&lt;/em&gt;' expression is going to follow the minus sign, trap that large expression in parenthesis just to be safe.&quot; I tell them this (with more of less specific phrasing depending on the class) after having seen many typos in exercises like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Use the definition to calculate the derivative of $f(x) = 3x^2 +1$. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Students will often write something like this &lt;em&gt;without the red parenthesis&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} \frac{3(x+h)^2 +1-\color{red}{\boldsymbol{(}}3x^2 +1\color{red}{\boldsymbol{)}}}{h}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now after seeing your example (and having seen that same mistake you're pointing out many times myself recently), I think it might be a good idea to generalize this line of advice and instead say something like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Anytime you're going to stick a new expression into an expression you're working with, like substituting for $f(x)$ or like using the chain rule or like changing variables (in integral calculus), be sure to trap that new expression in parenthesis just to be safe.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So to reinforce this, whenever I show students the chain rule now I'll be sure to write it with parenthesis:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\frac{\mathrm{d}}{\mathrm{d}x}\left(f(\text{stuff})\right) &#xA;=&#xA;f'(\text{stuff}) \color{blue}{\bigg(}\frac{\mathrm{d}}{\mathrm{d}x}\text{stuff}\color{blue}{\bigg)} &#xA;$$&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="8571" LastEditDate="2018-03-18T20:35:54.590" LastActivityDate="2018-03-18T20:35:54.590" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13817" PostTypeId="2" ParentId="13814" CreationDate="2018-03-17T16:01:53.860" Score="4" Body="&lt;p&gt;&lt;strong&gt;Correct the mistake and then drill&lt;/strong&gt; them with similar problems.  Vary the methods:  Lots of problems in class.  Lots of problems at home.  Kids to the blackboard, calling on students in class, maybe even chanting a problem together.  &lt;strong&gt;But the key is probably just volume of problems at home.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Use a problem source that has the answers so they can check their own work.  This gives them INSTANT feedback to check their own errors (&quot;Oh duh, I made that parentheses mistake again.&quot;)  If you are worried about them doing enough than have them hand in worked problems sets (checked for answers by them) and then just grade based on completion.  Even very small percentage participation grades can drive behavior.  If you are worried about honesty, then have a verification signature block at the top.  (If a few lie on that, then don't sweat it, they will get killed on the tests and it is not that much grade %.  Plus it will help the non-liars.)  Using a source that has the answers also takes off problem examination drudgery from you.  But most important feature is to accelerate the feedback time to students and allow them to be more engaged by correcting own mistakes.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This sort of mistake is typical of students that have not drilled enough. Teachers should not assume that poor explanation (finding the exact killer explanation to make a lightbulb go on) is the problem.  It is often lack of care by the students.  They get the concept.  They just aren't careful enough.  So you need to get them to do problems step by step (writing every step, not skipping some...you should do this as well until the whole class has proved efficacy...then you can skip some steps.)  And then they need to practice that a lot.  And on problems of easy to medium complexity (lots of one star and two star problems.  (Math teachers want to emphasize 3 star problems since they are more interesting, but this is like working on complicated football plays for players who need to concentrate on basic skills...a misallocation of time.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. I would not go back and do a review.   Kids will rightfully balk at that.  So keep teaching them calculus.  But even when they have LEARNED the calculus concepts and apply that part OK, if they are still making more algebra mistakes, you can work on that with lots of calculus drill.  In other words, you're sneakily improving their algebra while making them do calculus problems.  (Just like you get trig practice in physics or algebra practice in stoichiometry.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.  If you have several mistakes happening, think to hand out a written list for their reference.  At the beginning of each class.  But keep it action oriented.  NOT &quot;distributive law&quot; but instead &quot;algebra mistakes that will bite you in the butt in calculus class&quot;.  Do you capisce the difference in approach?  Former is an abstraction requiring a conversion from the idea to the practice.  The other is more of a &quot;tips&quot; type of coaching.  More immediately understandable and more &quot;in the trenches with the troops&quot;.  [Could be a fun pedagogical project for you to develop this little handout.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.s.  I also think the framing of the question is a little lacking.  It's a GREAT job of explaining the concept of the mistake.  But you don't give us enough info on the students or their training.  How good are they?  How much homework are they doing?  How long is the class?  What approach have you been using to teach them?  Etc.  I'm not sure if my answer would differ, but I think you have to see this as a multivariable problem and limitations of the student population (smarts, time) and class setting (year length, change of instructors by semester, etc.) are variables.  That said, sometimes using something that works in one setting maybe should even be used in another (who cares if it &quot;seems like high school&quot; if it gets the training done best.)  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-03-17T16:07:56.873" LastActivityDate="2018-03-17T16:07:56.873" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13818" PostTypeId="2" ParentId="13814" CreationDate="2018-03-18T01:08:33.273" Score="2" Body="&lt;p&gt;Here is an idea that I just came up with and haven't tried. (So take it with the appropriate amount of salt.) Try a game of &quot;telephone&quot;. That is, form one or more chains of students. Give the first person in the chain some multi-step problem and instruct them to do one step, write the result on a &lt;em&gt;new&lt;/em&gt; piece of paper, and pass it to the next person in the chain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason that I'm suggesting this in particular is the following: I think that it may be possible that when students write something like $2(x^2+x)2x+1$, they may be mentally grouping the $2x+1$. The failure to write the parens is simply (over?)confidence in their own ability to remember that those terms were grouped and a too-strong desire for brevity. In fact, they may be completely able to carry the grouping in their minds, without writing it correctly, in an environment where they are done with the problem quickly and their attention never breaks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, break that mentally carried grouping by activities which return them to their own written work after some time has passed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not think that more drilling, as others have suggested, is worth anything; at least with old material. If they are at the level of calculus, they have had plenty of drilling in that stuff already and it didn't help.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-03-18T01:08:33.273" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13819" PostTypeId="2" ParentId="13814" CreationDate="2018-03-18T21:07:32.170" Score="1" Body="&lt;p&gt;I try to model the idea of creating blanks to fill in when I do my work on the board.  The general idea is that there are certain procedures that we know are going to lead to terms that need to be grouped, so we should probably make sure that we establish those groups &lt;em&gt;first&lt;/em&gt;, then fill in the blanks.  Two examples:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Determine $\cos(15^\circ)$ using the angle addition formula for cosine.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In my own board work, would start the the problem by writing something like&#xA;\begin{align}&#xA;\cos(15^\circ)&#xA;= \cos(60^\circ - 45^\circ)&#xA;&amp;amp;= \cos(60^{\circ})\cos(45^{\circ}) - \sin(60^{\circ})\sin(45^{\circ}) \\&#xA;&amp;amp;= (\hspace1.3cm)(\hspace1.3cm) - (\hspace1.3cm)(\hspace1.3cm). \end{align}&#xA;The idea is to set up the form of the next step of the computation such a way that we can just &quot;fill in the blanks&quot;.  Once that is done we might, for example, determine that $\color{red}{\cos(60^{\circ}) = \frac{1}{2}}$, so the work written on the board becomes&#xA;\begin{align}&#xA;\cos(15^\circ)&#xA;= \cos(60^\circ - 45^\circ)&#xA;&amp;amp;= \cos(60^{\circ})\cos(45^{\circ}) - \sin(60^{\circ})\sin(45^{\circ}) \\&#xA;&amp;amp;= (\hspace0.45cm\color{red}{\frac{1}{2}}\hspace0.45cm)(\hspace1.3cm) - (\hspace1.3cm)(\hspace1.3cm). \end{align}&#xA;Next, $\color{blue}{\sin(60^{\circ}) = \frac{\sqrt{3}}{2}}$, giving&#xA;\begin{align}&#xA;\cos(15^\circ)&#xA;= \cos(60^\circ - 45^\circ)&#xA;&amp;amp;= \cos(60^{\circ})\cos(45^{\circ}) - \sin(60^{\circ})\sin(45^{\circ}) \\&#xA;&amp;amp;= (\hspace0.45cm\color{red}{\frac{1}{2}}\hspace0.45cm)(\hspace1.3cm) - (\hspace0.45cm\color{blue}{\frac{\sqrt{3}}{2}}\hspace0.45cm)(\hspace1.3cm), \end{align}&#xA;and so on.  Again, the big, important idea is to write down the form of the answer or solution before doing anything else.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Determine $f'(x)$ where $f(x) = (x^2+x)^2$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This can be addressed in a similar manner:  if $f(x) = (x^2+x)^2$, we want to use chain rule.  If I am feeling like writing lots of letters, I might let $g(x) = x^2$ and $h(x) = x^2+x$ so that&#xA;$$ f(x) = (g\circ f)(x), $$&#xA;or I might simply refer to the inner function and the outer function.  In either case, I would write something like&#xA;$$&#xA;f'(x)&#xA;= g'(h(x)) \cdot h(x)&#xA;= ( \underbrace{\qquad\qquad}_{\text{$g'$ at $h(x)$}} ) \cdot ( \underbrace{\qquad\qquad}_{h'(x)} ) $$&#xA;or&#xA;$$&#xA;f'(x)&#xA;= ( \underbrace{\qquad\qquad}_{\text{outer$'$ at inner}} ) \cdot ( \underbrace{\qquad\qquad}_{\text{inner$'$}} ).&#xA;$$&#xA;The &quot;outer&quot; function is the $x^2$ function, which has derivative $2x$, be we want to evaluate this function at $x^2+x$ rather than $x$ (or, if I want to be even more clear, I'll use a dummy variable, and say&#xA;$$ g(t) = t^2 \implies g'(t) = 2t, \qquad \text{evaluate at $t = x^2+x$}, $$&#xA;or something like that).  The implies that the derivative of the outer function, evaluated at the inner function, is $\color{red}{2(x^2+x)}$, so on the board we now have&#xA;$$&#xA;f'(x)&#xA;= ( \underbrace{\color{red}{2(x^2+x)}}_{\text{outer$'$ at inner}} ) \cdot ( \underbrace{\qquad\qquad}_{\text{inner$'$}} ).&#xA;$$&#xA;The inner function is $h(x) = x^2 + x$, which has derivative $\color{blue}{2x+1}$, so we finish the first step of the problem by plugging this into the second blank and getting&#xA;$$&#xA;f'(x)&#xA;= ( \underbrace{\color{red}{2(x^2+x)}}_{\text{outer$'$ at inner}} ) \cdot ( \underbrace{\color{blue}{2x+1}}_{\text{inner$'$}} ).&#xA;$$&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2018-03-18T21:07:32.170" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13821" PostTypeId="1" CreationDate="2018-03-19T08:05:35.683" Score="2" ViewCount="69" Body="&lt;p&gt;I am searching for ideas to construct Project Based Learning type that involves Modular arithmetics with eventually geometry and could fit in High School level &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What could a nice research question for a project that could explored which requires the use of modular arithmetics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any suggestion is welcome.&lt;/p&gt;&#xA;" OwnerUserId="9111" LastActivityDate="2018-03-20T04:04:43.010" Title="Project Based Learning or Applied Math involving modular arithmetics?" Tags="|problem-design|inquiry-based-learning|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13822" PostTypeId="1" AcceptedAnswerId="13826" CreationDate="2018-03-19T10:32:33.757" Score="8" ViewCount="1563" Body="&lt;p&gt;I'm teaching a course at a community college called &quot;Business Pre-Calculus&quot;, which is basically some varied rudimentary algebraic concepts integrated with some topics from basic economics and finance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I give in-class exams for which calculators are provided and the use of a phone is prohibited. On the last exam, a student who had conceptual difficulty adding fractions, on the same test was able to solve a 3x3 linear system, a complicated rational equation, and an even more complicated rational inequality. His work in the space provided is mimnimal and makes absolutely no sense (I've already considered the &quot;savant&quot; possibility), and I cannot fathom how he could have come up with the correct solutions to each with any of these ridiculous &quot;calculations&quot;. They resemble what you might see on a chalkboard in a cartoon: nonsense made to look like math to an ignorant observer. In all honesty, I feel like he gets the answers somehow, and then scribbles some &quot;work&quot; to avoid suspicion. But his answers are still correct, and seemingly out of nowhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've watched him take the test, but didn't see anything suspect. What am I missing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt; I've removed the picture. &lt;/p&gt;&#xA;" OwnerUserId="8529" LastEditorUserId="8529" LastEditDate="2018-03-21T05:44:34.633" LastActivityDate="2018-03-21T05:44:34.633" Title="A student is cheating and I don't know how" Tags="|exams|cheating|" AnswerCount="3" CommentCount="21" ContentLicense="CC BY-SA 3.0" />
  <row Id="13824" PostTypeId="1" CreationDate="2018-03-19T16:55:54.657" Score="4" ViewCount="125" Body="&lt;p&gt;We're using Pearson's Geometry in my class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As terms are defined there, Parallelograms include Rhombi (congruent sides), Rectangles (right angles), Squares (congruent sides and right angles, i.e. Rhombi that are Rectangles), and of course Parallelograms that are neither Rhombi nor Rectangles. Non-Parallelograms include  Kites (consecutive sides congruent but opposite sides never congruent) and Trapezoids (exactly one pair of parallel sides), and again, Non-Parallelograms that are neither Kites nor Trapezoids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder. What's the point of defining the Non-Parallelograms that way? No Trapezoid property depends on the fact that some opposite sides are not parallel, and no Kite property depends on the fact that opposite sides are not congruent. It seems to me it would be more interesting and more natural to drop those requirements and to say that Trapezoids are quadrilaterals with at least one pair of opposite sides parallel (so that all Parallelograms are Trapezoids) and Kites are quadrilaterals in which each side is congruent to at least one consecutive side (so that all Rhombi are Kites).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's the point in the lopsided way of defining terms, so that some categories intersect and some are disjoint?&lt;/p&gt;&#xA;" OwnerUserId="8258" LastActivityDate="2018-03-19T16:55:54.657" Title="Why define the names of quadrilaterals so that some categories (rhombus and rectangle) intersect and some (kite and trapezoid) are disjoint?" Tags="|geometry|definitions|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13825" PostTypeId="2" ParentId="13822" CreationDate="2018-03-19T18:07:10.113" Score="11" Body="&lt;p&gt;Do you give full credit for answers without clear work?? If you don't then I suggest you give partial credit and meet with the student in the guise of helping him write his work more clearly so that he can get full credit next time and possibly even a little more credit.  Put the responsibility on him to explain how he got his answer and help him write what he did clearly.  If he can't explain what he did, then explain that this is why he can't have any additional credit. IF he is interested, go over the method with him.  Find out if he has trouble writing down his thinking in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my years as a teacher, I have seen students who can do all sorts of things in their heads and have no clue how to write it down.  These students didn't cheat, just learned and worked differently. All their thinking was done in their heads and trying to write things down interfered with their thinking. If you required that they wrote something down it rarely matched up with what they did in their head.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2018-03-19T18:07:10.113" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13826" PostTypeId="2" ParentId="13822" CreationDate="2018-03-19T20:10:46.793" Score="13" Body="&lt;p&gt;First off, I challenge the framing of the question.  You seem to be seeking answers for the question&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How did the student get the correct answer from this work?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, I don't think that there is any possible way to answer that question without being the student.  Maybe they cheated.  Maybe they got lucky.  Maybe they did some work in their heads but were unable to communicate that work in writing.  Maybe they saw that exact problem in a homework assignment or study guide somewhere and somehow managed to memorize a correct answer.  Without being in the head of the student, it is entirely impossible to know how they got that answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, I think that there is a related question which can be answered:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If a student writes gibberish followed by a correct answer, what should I do?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the classes that I teach, I strongly emphasize that most of mathematics is not about &lt;em&gt;obtaining&lt;/em&gt; the correct answer, but about communicating to others &lt;em&gt;how&lt;/em&gt; that answer was obtained.  If you cannot communicate how you arrived at an answer, then you haven't done the basic work necessary for conducting mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an exam setting, I make it very clear (both on the syllabus and in the instructions for the exam) that correct answers without supporting work will receive no credit.  In this case, I don't see any work that I would be willing to give credit for (though I might be a harsh grader; Dave L Renfro sees some merit in a bit of what is there, and he isn't wrong; I do see an attempt to combine the terms with a common denominator&amp;mdash;that is a reasonable start; that being said, I still probably wouldn't give it any credit).  I don't need to know &lt;em&gt;how&lt;/em&gt; the student got the correct answer in order to know that they haven't &lt;em&gt;explained&lt;/em&gt; that answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, if the student came to my office to challenge the marking, I would be willing to hear them out.  If they could explain to me how to solve a similar problem that present to them on the fly, I would be willing to give them a handful of points on the exam (say 2-4 out of 10), with the admonition that, in the future, I won't be so generous (it is quite difficult to offer the option to revise a final, since grades are due pretty quickly after the final is graded).  I might even encourage this student to come and talk to me; not even under the &quot;guise&quot; of helping them&amp;mdash;they clearly need help, whether it be help with their mathematical skills, or help with their cheating technique. ;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will also point out that I am very carefully avoiding the issue of cheating entirely.  Yes, it is very possible that the student cheated.  But if you don't have any evidence of that (other than some suspicious answers on an exam), accusing the student of cheating does absolutely nothing to resolve the situation.  It is probably better to assume good faith and let the student prove in office hours that they have no idea what they are doing.  Whether they are cheating or not, they aren't going to get credit for something that they don't understand.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2018-03-19T22:47:27.557" LastActivityDate="2018-03-19T22:47:27.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13827" PostTypeId="2" ParentId="13821" CreationDate="2018-03-19T21:01:45.703" Score="1" Body="&lt;p&gt;Rotations of regular polygons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with an equilateral triangle. Label the vertices one, two, three. Define a rotation to be 60° counterclockwise, for example (or you could go clockwise). Note that the locations of the vertices have shifted. Consider this a new “state“ or “configuration”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, what “configuration” does one end up with after &lt;em&gt;n&lt;/em&gt; rotations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, extend this to any n-sided polygon. A real life application would be modeling the movement of molecules. Gallian’s abstract algebra book shows a cross-section of a 28-sided HIV virus, for example. Or, how about rendering graphics in video games? Modeling movements of satellites and space shuttles in outer space? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more accessible example would be a clock, or days of the week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, you could then use mod 12 or mod 7 to discuss notes on musical scales.&lt;/p&gt;&#xA;" OwnerUserId="5337" LastActivityDate="2018-03-19T21:01:45.703" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13828" PostTypeId="2" ParentId="13821" CreationDate="2018-03-20T04:04:43.010" Score="1" Body="&lt;p&gt;I think the idea of check digits is pretty compelling.  A bunch of examples of where check digits are used can be found &lt;a href=&quot;https://en.wikipedia.org/wiki/Check_digit&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, including things like government ID numbers in various countries, UPC and ISBN codes, credit cards, etc.  The basic idea is that you're given a number, say an ISBN number on a book.  You'd like to know if the number you've been given is a valid ISBN number.  What you do is calculate a certain (weighted) sum of all the numbers in the ISBN except the check digit (the last digit in the ISBN number, it takes on 11 values from 0 to X) and reduce the total mod 11.  If the result is not equal to the check digit, then you were not given a valid ISBN number.  If they're equal it might be inconclusive.  What's more, in many cases, the procedure can also identify what kind of error occurred and where.  &lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2018-03-20T04:04:43.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13829" PostTypeId="2" ParentId="13822" CreationDate="2018-03-20T09:29:38.737" Score="5" Body="&lt;p&gt;From the picture you gave, it seems like he wasn't cheating. The whole solution, including how he got there, is there, even though in a bad style and wrong in some places: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He started by getting the left hand side on one fraction by multiplying the first one by &#xA;$$\frac{x+2}{x+2}$$ and the second one by $$\frac{x-1}{x-1}.$$&#xA;This is the first line of the calculations block on the right.&#xA;In the next line, we get this, namely&#xA;$$\frac{4x+8}{(x-1)(x+2)} - \frac{2x-2}{(x-1)(x+2)}.$$&#xA;Now we have an arrow pointing to the left in his solution, towards&#xA;$$\frac{4x+8}{2x+2}.$$&#xA;I don't see how he got that, maybe he did a mistake substracting the fractions, maybe it has something to do with the unreadable word he put there.&lt;br&gt;&#xA;Now, he tries to test where this expression is positive or negative to get to the solution. I also don't see how he got $-1$ and $3$ here, but he surely had a reason...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All in all, I think he knew what he was doing. Of course there are some errors and the solution is almost unreadable, but still, I would not think that this student was cheating. I would think it is either that he thinks himself to clever to write such trivialities down, or he doesn't know how to properly document his work. Thus, I would suggest to talk to him, find out what is the case and then explain to him that he needs to properly write down things; both for you to understand it, and for him to avoid doing mistakes.&lt;br&gt;&#xA;If possible you should teach him resp. your whole class that writing things down properly, even in seemingly easy cases, helps a lot with avoiding and tracking down calculation errors. And yes, such errors happen, to everyone.&lt;br&gt;&#xA;Best example I can think of right now is the Gaussian algorithm to solve systems of linear equations. It might seem boring and students always want to be clever and do a really intelligent trick to solve the system faster, but in the end, just following the algorithm strongly reduces the number of errors due to slip of the pen.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2018-03-20T09:29:38.737" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13830" PostTypeId="1" CreationDate="2018-03-20T16:37:27.177" Score="10" ViewCount="274" Body="&lt;p&gt;Our (United States, undergraduate) math program is considering the idea of putting more mathematical modeling and computation into all levels of our curriculum. One of the hang-ups is that we can't decide on a specific mathematical software that could be uniformly adopted. Here are some of the criteria:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Friendly enough that Calculus I students (including a variety of STEM majors) could use it to graph functions and (symbolically) compute derivatives and anti-derivatives/integrals. Most likely at this level the students would work with faculty-created template files to allow for a gentle introduction to computing.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Advanced enough that students in mid- to upper-level courses (e.g. Multi-variable Calculus, Linear Algebra, Differential Equations, Mathematical Methods for Physics, Complex Analysis, etc.) can create their own files to use it to solve real-world modeling problems and/or do larger computational projects.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Be something that would likely be available to them after graduation in either graduate school or, more likely, government and industry.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Be able to create files/output that would be easily shown off during their post-graduation job searches (e.g. sharable on sites like GitHub).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Possible Example: Mathematica/Maple/MatLab. These seem to satisfy 1 and 2, but fail on 3 (except perhaps for grad schools) and I'm not sure about 4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possible Example: CoCalc (formerly called SageMathCloud). This seems to be a stretch for 1 and 2, but certainly satisfies 3 and 4.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does anyone have any advice/experience with implementing any of these,&#xA;  or some other option, across their whole mathematics curriculum? What&#xA;  did you discover to be the hardest/easiest parts of doing so? How do&#xA;  you think your choice fared in the above criteria? Are there other&#xA;  criteria that we should consider?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note: Our statistics faculty has already decided on using R in the statistics courses. So this question is only concerning mathematics courses.&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2018-03-20T18:20:41.087" LastActivityDate="2018-03-21T18:29:43.917" Title="Computational Software for the whole curriculum and beyond" Tags="|undergraduate-education|software|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13831" PostTypeId="1" CreationDate="2018-03-20T18:17:01.123" Score="4" ViewCount="538" Body="&lt;p&gt;Question 1: What are the literature, status, debates, references, etc regarding this matter please?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently, some (woohoo weasel words!) consider that squares are rectangles too advanced a topic for kindergartners so either&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The topic is avoided&lt;/li&gt;&#xA;&lt;li&gt;Kindergartners are allowed to think squares are not rectangles while the teacher does not confirm or deny such.&lt;/li&gt;&#xA;&lt;li&gt;Kindergartners are taught that squares are not rectangles.&lt;/li&gt;&#xA;&lt;li&gt;Same as #2 or #3 but the same kindergartners would be taught that squares are rectangles later on. In the case of #3, this is an outright contradiction.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Question 2: Which of the above 4 cases are acceptable? Which are not? Why/why not?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://korean.stackexchange.com/questions/3394/in-korea-are-squares-considered-rectangles&quot;&gt;In Korea, are squares considered rectangles?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/13831/&quot;&gt;Are kindergartners supposed to be steered from squares being rectangles?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/13700/in-what-curricula-are-rectangles-defined-so-as-to-exclude-squares&quot;&gt;In what curricula are “rectangles” defined so as to exclude squares?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/2249741/why-do-we-have-circles-for-ellipses-squares-for-rectangles-but-nothing-for-tria&quot;&gt;Why do we have circles for ellipses, squares for rectangles but nothing for triangles?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://physics.stackexchange.com/questions/394471&quot;&gt;What are/should kids (be) taught about the colour of the sun?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2018-03-21T19:01:30.730" LastActivityDate="2022-05-27T05:43:43.713" Title="Are kindergartners supposed to be steered from squares being rectangles?" Tags="|reference-request|geometry|curriculum|preschool-education|culture|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13832" PostTypeId="2" ParentId="13831" CreationDate="2018-03-20T18:36:58.537" Score="21" Body="&lt;p&gt;Kindergartners are generally at an early stage of geometric development, in which shapes are recognized by how well they resemble prototypical images, rather than by whether or not they conform to a definition.  Thus, for example, the shape on the left below is likely to be recognized as a &quot;triangle&quot; (despite the fact that it has four sides), the shape in the middle is is &lt;em&gt;unlikely&lt;/em&gt; to be recognized as a &quot;triangle&quot; (despite the fact that it has three sides), and the shape on the right is likely to be recognized as a &quot;diamond&quot; (and &lt;em&gt;not&lt;/em&gt; as a square, despite the fact that it is a regular quadrilateral).  In the &lt;a href=&quot;https://en.wikipedia.org/wiki/Van_Hiele_model&quot; rel=&quot;noreferrer&quot;&gt;Van Hiele model&lt;/a&gt;, children at this level would be described as being at the &quot;visualization&quot; level, or Level 0.  In order to recognize that a square is a rectangle they would need to be at the &quot;abstraction&quot; level, or Level 2, at which hierarchical relationships can be understood.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/uQWjU.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uQWjU.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say that one of the goals of early childhood education (roughly K-2) is to move kids from Level 0 to Level 1, with the transition to Level 2 taking place in later elementary (say grades 3-5 or 6).  While certainly &lt;em&gt;some&lt;/em&gt; kindergartners are developmentally ready for understanding &lt;em&gt;some&lt;/em&gt; hierarchical relationships (e.g. &quot;poodles are dogs, and dogs are mammals&quot;), I think expecting this to be a goal of kindergarten (something they are &quot;supposed to learn&quot;) is unrealistic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parenthetically, it's probably worth mentioning that the Van Hiele model is widely regarded by math ed researchers as outdated, overly simplistic, and extremely reductionist.  However,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;much as Euclidean geometry maintains a stable position in the&#xA;  secondary curriculum despite the efforts of some reformers of the&#xA;  early and mid-twentieth century to jettison it in favor of more modern&#xA;  approaches to the  field, so too does the van Hiele theory continue to&#xA;  play a dominant role in the discourse of students’ thinking about&#xA;  geometry, notwithstanding several decades of critique regarding its&#xA;  value and validity as an empirically-grounded theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Source:  Herbst, Fujita, Halverscheid and Weiss, &lt;a href=&quot;https://www.routledge.com/The-Learning-and-Teaching-of-Geometry-in-Secondary-Schools-A-Modeling/Herbst-Fujita-Halverscheid-Weiss/p/book/9780415856911&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Learning and Teaching of Geometry in Secondary Schools&lt;/em&gt;&lt;/a&gt;, pp. 92).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are decades and decades worth of research on these matter, but an accessible point of entry might be:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hannibal, M. A. (1999). Young children's developing understanding of&#xA;  geometric shapes. Teaching Children Mathematics, 5(6), 353+. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In fact, the entire issue of &lt;em&gt;Teaching Children Mathematics&lt;/em&gt; in which that article was published may also be of use, as it was a themed issue on geometry.  &lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2018-03-21T01:36:30.917" LastActivityDate="2018-03-21T01:36:30.917" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="13833" PostTypeId="2" ParentId="13830" CreationDate="2018-03-20T20:00:04.453" Score="7" Body="&lt;p&gt;(Disclaimer: I founded CoCalc in 2013 and SageMath in 2004.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does anyone have any advice/experience with implementing any of these, or some other option, across their whole mathematics curriculum? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've used Cocalc and SageMath, and interacting a lot with &lt;a href=&quot;https://github.com/sagemathinc/cocalc/wiki/Teaching&quot; rel=&quot;noreferrer&quot;&gt;courses using CoCalc&lt;/a&gt;.   Mike Croucher helped the entire math and physics departments switch to CoCalc at Univ of Sheffield, and &lt;a href=&quot;https://tutorial.cocalc.com/&quot; rel=&quot;noreferrer&quot;&gt;wrote about that here&lt;/a&gt;.  The Sheffield stats department use of CoCalc very significantly influenced the implementation of our web-based Jupyter notebooks; in particular, to better handle very large output, what packages are available, etc.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What did you discover to be the hardest/easiest parts of doing so? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For the users of CoCalc, stability &lt;em&gt;was&lt;/em&gt; an issue. The site has been very stable for months now, so the new main issue is obtaining or creating content for students (e.g., Jupyter notebooks).   We make installing software, distributing and collecting homework and handouts, etc., easy.  Instructors also have too much friction related to grading (both manual and automatic), and addressing this head on is our current top development priority.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're doing undergrad math using Sage, the book &lt;a href=&quot;http://www.gregorybard.com/Sage.html&quot; rel=&quot;noreferrer&quot;&gt;Sage for undergrads&lt;/a&gt; is very helpful with addressing point 1.  UCLA has also done an amazing job with developing Sage worksheets for their into undergrad LS30 course (they are one of the biggest CoCalc users).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding point 2, CoCalc provides a very complete Python/Jupyter/Anaconda stack, R, Octave (MATLAB clone), Julia, etc., so provides reasonable preparation for general industry work compared to Mathematica/Maple.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there other criteria that we should consider?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Price is also a relevant factor for some places.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definitely feel free to email me at wstein@sagemath.com if you have any questions you don't want to discuss here...&lt;/p&gt;&#xA;" OwnerUserId="1141" LastActivityDate="2018-03-20T20:00:04.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13834" PostTypeId="2" ParentId="13805" CreationDate="2018-03-21T02:41:03.863" Score="1" Body="&lt;p&gt;Here is a book that should fit the bill: &lt;a href=&quot;https://naturalmath.com/camplogic/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Camp Logic&lt;/a&gt;, by Mark Saul and Sian Zelbo.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Disclaimer: This comes from my publisher. I am guessing that it's good.]&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-03-21T02:41:03.863" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13835" PostTypeId="1" CreationDate="2018-03-21T16:53:44.930" Score="7" ViewCount="253" Body="&lt;p&gt;I teach a Grade 11 Functions class, but this relates to basically all grade levels.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Khan Academy (discussed in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10510/is-khan-academy-really-good&quot;&gt;Is Khan academy really good?&lt;/a&gt;) is a resource often being used by students as extra help, catchup for missed work, etc. etc.  &lt;strong&gt;Would it be a good policy for me to post links to each topic on Khan Academy on my class webpage&lt;/strong&gt;, so that students will use it to help them review and catchup, or will this encourage students to pay less attention in class and/or miss classes and extra help, since they know that they have a teacher-sanctioned backup?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Please help me with the tagging/formatting, as this is my first ME.SE question.)&lt;/p&gt;&#xA;" OwnerUserId="9250" LastEditorUserId="376" LastEditDate="2018-04-25T05:22:46.900" LastActivityDate="2018-07-01T16:38:40.980" Title="Would it be a good idea to provide students with Khan Academy links?" Tags="|online-instruction|review|resources|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13836" PostTypeId="2" ParentId="13830" CreationDate="2018-03-21T18:29:43.917" Score="5" Body="&lt;p&gt;I use Jupyter notebooks and depending on the class use Python or R.  Feel free to checkout some of my calc stuff (&lt;a href=&quot;http://calculus-notes.readthedocs.io/en/latest/&quot; rel=&quot;noreferrer&quot;&gt;http://calculus-notes.readthedocs.io/en/latest/&lt;/a&gt;).  They're a a bit rough but I use the Jupyter notebooks under the assumption that my students have laptops, and install Anaconda on their machines locally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We go back and forth between paper and pencil and computer, but I believe that Python syntax is incredibly simple and makes sense as a choice language due to its freeness, ubiquity, and versatility.  (Sometimes, I'll use R if the goal is more about basic work with data and introductory visualizations because I think matplotlib is more complex than ggplot2.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For students in precalculus or calculus, one of my main concerns is that students can quickly move between representations of functions and sequences -- tables, graphs, symbols, words.  In Python this is really quite simple.  For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;&#xA;def f(x): return x**2&#xA;x = np.linspace(-5, 5, 100)&#xA;plt.plot(x, f(x))&#xA;&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;defines a function, determines a domain, and generates a plot.  Ideas like finite differences and partial sums are easily accomplished using either built-in functions in Numpy or with plain python and basic for loops.  For example, derivatives can be examined by defining a derivative function per standard Calc definition, and examining the plot for $f, df$ together:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;def df(x, h): return (f(x+h) - f(x))/h&#xA;plt.plot(x, f(x))&#xA;plt.plot(x, df(x, 0.1))&#xA;&lt;/code&gt;&#xA;As a sequence with finite differences with Numpy's &lt;code&gt;diff&lt;/code&gt; method, this would be:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;an = [f(i, 0.0001) for i in np.linspace(-2,2,100)]&#xA;d_an = np.diff(an)&#xA;plt.plot(an, 'o')&#xA;plt.plot(d_an, 'o')&#xA;&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With basic lists, functions, and &lt;code&gt;matplotlib&lt;/code&gt; you can get a lot done on the cheap.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't believe there is a solid solution to the platform problem.  If you know somebody who has a little understanding of deploying software on servers, you might consider trying to setup a JupyterHub (&lt;a href=&quot;https://zero-to-jupyterhub.readthedocs.io/en/latest/index.html&quot; rel=&quot;noreferrer&quot;&gt;https://zero-to-jupyterhub.readthedocs.io/en/latest/index.html&lt;/a&gt;).  This would allow students to login to the materials in a browser and store all the work here.  You can integrate assignments and automatic grading with some work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, we would first support students getting a halfway decent computer, internet access, and a consistent safe and quiet space to work on the machine.  &lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2018-03-21T18:29:43.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13838" PostTypeId="2" ParentId="5730" CreationDate="2018-03-21T19:44:14.047" Score="-2" Body="&lt;p&gt;Well, why does he think his answer is right?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;'Don't criticize what you can't understand.'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If he gives an answer involving fractions, then perhaps there is a simplification error (Unlikely to have a simplification error so it's unlikely the answer will involve fractions)&lt;/p&gt;&#xA;" OwnerUserId="5987" LastActivityDate="2018-03-21T19:44:14.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13840" PostTypeId="2" ParentId="13340" CreationDate="2018-03-22T04:52:07.063" Score="0" Body="&lt;p&gt;I started to answer this question by pointing you to the International Correspondence Schools, which was founded over 125 years ago in Pennsylvania to educate miners, with the aim of improving worker safety. I know about them because I have one of their math textbooks from the 1890s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in researching ICS, I learned that it has expanded and split and splintered, and there are now several organizations that are ancestors of the original ICS:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.icslearn.co.uk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;ICS Learn&lt;/a&gt; offers A Levels including Maths (which doesn't appear to include linear algebra or discrete mathematics, but might include Calc II), plus career courses&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.icslearn.ca/&quot; rel=&quot;nofollow noreferrer&quot;&gt;ICS Canada&lt;/a&gt; offers mostly high school and career&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.pennfoster.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Penn Foster&lt;/a&gt; offers high school, career, and college programs&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Along the way, I learned about the &lt;a href=&quot;https://www.deac.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Distance Education Accrediting Commission&lt;/a&gt;, which you can consult to find an institution that offers what you're looking for. I found six listings for the subject &quot;mathematics&quot; under &quot;Search Institutions&quot;.&lt;/p&gt;&#xA;" OwnerUserId="3593" LastActivityDate="2018-03-22T04:52:07.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13842" PostTypeId="2" ParentId="8384" CreationDate="2018-03-22T05:36:56.810" Score="4" Body="&lt;p&gt;Let's take Kreyszig as the reference case, since I'm familiar with it.  In terms of next harder books, I would say:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Arfken and Weber:  unfortunately I HATE this book as do many physicists.  (It's best use is as a door stop or ganglion basher.)  It shows the signs of being written by someone(s) who was not a good math teacher or writer (not even talking about rigor, just basic math textook writing as for example Thomas Calculus or any regular math book).  It is sort of a grabbag for referring to and not that good to actually study from.  But it does have some harder topics that Kreyszig eschews.  But where there is overlap to K, stick with K.  That is a real book that is designed to actually teach math topics, not be a glorified, poorly organized reference.  So, while I hate it, would be remiss not the mention it as many people see it as next hardest book.  And it is on my shelf.  But you have been warned about its flaws (see the rips on Amazon also.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Books on asymptotics and perturbation methods.  The one by Bender is best known but there are a few other good ones that are similar content.  [This is probably the closest to a real &quot;next book&quot; in spirit.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Full texts on some of the topics in Kreyszig (PDEs, complex analysis, numberical analysis, linear algebra, ODEs)  Note, I think K actually does a rather good job of covering these topics.  (High bang for the buck.)  But there are standalone books that are more encyclopedic in terms of methods.  This is even without getting into books that are super rigorous or that go deeply theoretical.  Just more computational content.  Some are quite good, with lots of problems and answers.  In addition to PDEs and complex analysis, there are whole long good books on numeric analysis (including by Hamming).  Some of these even have programs or discs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A good primer on Calculus of Variations: there are a few that come up as most popular on Amazon.  Stick to the easier ones.  K doesn't cover it.  Arfken (as usual) does a bad job of covering it.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A primer on integral equations.  (Some of the advanced math methods books similar to Bender already have this.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Handbooks on special functions or on differential equations or integrals.  Some of these are really just like CRC tables.  Others have some good discussion of usage or derivation or manipulations.  Don't know any that have drill problems.  Some names:  Abromowitz and Stegun, Bates manuscript project, NIST website, Zwillinger.  There are others.  [As with all of these texts, check out the description on Amazon along with a few reviews and read inside to the preface description.  Once on Amazon, you will usually get a list of competing similar books.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Whitacker and Watson second half has a decent set of chapters on special functions.  Some content above what you get in K.  Also has a section on Mathieu functions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Monograph by Watson on Bessel functions is  whole book just on that.  Also, can look at Byerly treatise on Fourier series and spherical harmonics.  Both of these lack homework problems though.  But they are pretty straight forward and have insights used in Jackson E&amp;amp;M problems or the like.  There is one well known text on Mathieu finctions by McLachlan (not sure if it has problems.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some specialized texts on diffy Qs: Ince ODE and Forsyth Treatise of DE and Theory of DE (6 volumes).  These have some theoretical aspects but are really mostly in the realm of engineers who want to solve equations.  Not Fraktur letter craziness and Sobolov spaces.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Joseph Edwards 1000 page (!) text on Integral Calculus.  Full of Cambridge math problems and tricky integrals.  Sort of like having a handbook with problems.  More tricks than in a normal course like Thomas or Granville or the like.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I would also say that there are some topics that are not really in the main line of Adv Eng Math (so not logically &quot;next) but worth considering as forks for an engineer or scientist who just wants more math tools:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Probability and statistics is a whole wide field from intro courses to all kinds of applied statistics and design of experiments.  The book by Box, Hunter, Hunter is a great one for DOE.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Operations research (search problems, queuing, etc.)  Emphasize a text that is practical and from an industrial engineer perspective or military operations.  Not super theoretical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Group theory.  Some math guy will say entire topic of abstract algebra but that is way overkill for a technical guy and maybe even unpleasant.  stick to a course or text on group theory for physics or chemistry.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-03-22T05:49:46.893" LastActivityDate="2018-03-22T05:49:46.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13843" PostTypeId="2" ParentId="13745" CreationDate="2018-03-22T12:03:44.840" Score="0" Body="&lt;p&gt;The OP claims the limit symbol is left out. But the limit symbol, thus taking the limit, is not left out by convention. Its still there. Its part of the definition of $\sum_{n=1}^\infty$ which is defined as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\sum_{k=1}^\infty a_k = \lim_{n\rightarrow\infty} \sum_{k=1}^n a_n = a$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The infinite sum is defined as the limit of its partial sums. That the partial sums are different from $a$ is to expect when some summands $a_k$ are non zero. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The symbol $\infty$ in the infinite sum is compound to the symbol that is defined and doesn't refer to some point. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Series_%28mathematics%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;standard&lt;/a&gt; is that the limit operator is invoked for an infinite sum.&lt;/p&gt;&#xA;" OwnerDisplayName="user7145" LastEditorDisplayName="user7145" LastEditDate="2018-03-22T22:39:35.553" LastActivityDate="2018-03-22T22:39:35.553" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13845" PostTypeId="2" ParentId="13813" CreationDate="2018-03-22T21:27:07.430" Score="6" Body="&lt;p&gt;It's possible that she said, or meant to say, &quot;into&quot;, which denoted multiplication in the late 19th and early 20th centuries.  Perhaps she was taught using rather antiquated terminology. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The sign $\times$ is called the sign of multiplication, and $a \times b$ is read thus &quot;$a$ into $b$.&quot; &quot; Algebra for Beginners, I. Todhunter, 1872.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Multiplication, $\times$, multiplied by, or into; $A \times B$, $A$ into $B$.&quot; , First Lessons on Geometry, Alpheus Crosby, 1847&lt;/p&gt;&#xA;" OwnerUserId="5005" LastEditorUserId="8571" LastEditDate="2018-03-23T03:41:52.417" LastActivityDate="2018-03-23T03:41:52.417" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13846" PostTypeId="1" CreationDate="2018-03-22T21:36:39.577" Score="8" ViewCount="364" Body="&lt;p&gt;Among other courses, I'm teaching a (basic) &lt;strong&gt;matrix algebra course&lt;/strong&gt; for 1st year university students (they are studying Economics, and the cursus leads them to management, finance, or econometrics in 3rd univ year).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As it's quite common in France, these notions are taught in a rather abstract way, we teach them: matrix operations, determinants, rank, matrix equivalence, row echelon form, application to linear systems, &lt;a href=&quot;https://en.wikipedia.org/wiki/Hessian_matrix&quot; rel=&quot;nofollow noreferrer&quot;&gt;optimum of functions&lt;/a&gt;, etc. (but not eigenvalues/vectors, this is done the next year).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a ~ 40-hour course in a semester, I have ~ 40 students per group, and I'm doing nearly 1/3 theory, 2/3 training exercises during these 40 hours.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The problem:&lt;/strong&gt; even if 1) it works quite well after a few years teaching this course 2) noone complains 3) these notions are important for them for their following years (econometrics, they will also learn R, etc.), &lt;strong&gt;I feel the course is a bit &quot;too abstract&quot;&lt;/strong&gt; / not enough connected to applications and not &lt;strong&gt;pro-active enough&lt;/strong&gt; for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Idea:&lt;/strong&gt; I would like to test something like that: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Do the usual 40 hours in ~ 34 hours (I can reduce the number of exercices of 20% more or less)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In the 6 remaining hours, I'd like to test something more &lt;strong&gt;active&lt;/strong&gt; for them, more project based, example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;First 1.5hr sequence:&lt;/em&gt; I present the project briefly, and give &quot;topics&quot;, they start working in ~8 groups of 5 students&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Homework:&lt;/em&gt; they do some research on their &quot;project&quot;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Second 1.5hr sequence:&lt;/em&gt; they continue their research, I help them / answer questions, give them directions if needed, they bring books from university library + their laptop to find additional resources&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Third 1.5hr sequence:&lt;/em&gt; half of the groups do a presentation of their &quot;project&quot;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Fourth 1.5hr sequence:&lt;/em&gt; the other groups do the same&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Have you done such &quot;project based&quot; courses for university students? Any feedback?&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The biggest question: &lt;strong&gt;what kind of math project could they work on?&lt;/strong&gt; (using some matrix/linear algebra). I have a few ideas (see &lt;a href=&quot;https://math.stackexchange.com/questions/1520832/real-life-examples-for-eigenvalues-eigenvectors&quot;&gt;here for example&lt;/a&gt;, I used some of them for 3rd year students), but nearly all of these applications are too complex for students studying matrices for the first year.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What kind of setup would you use? Provided they are econ students, I was thinking about taking business-related examples.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7558" LastEditorUserId="376" LastEditDate="2020-04-18T08:00:37.730" LastActivityDate="2021-02-12T14:12:23.837" Title="Make a matrix algebra course (1st university year) more &quot;project-based&quot;" Tags="|undergraduate-education|course-design|student-motivation|linear-algebra|applied-mathematics|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13847" PostTypeId="2" ParentId="13846" CreationDate="2018-03-23T03:35:49.953" Score="3" Body="&lt;p&gt;Could take a look at some texts like this:  &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/047176941X&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Matrix-Algebra-Business-Economics-Shayle/dp/047176941X&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also:  &lt;a href=&quot;http://www.vmi.edu/media/content-assets/documents/academics/appliedmath/Fundamentals-of-Matrix-Algebra-3rd-Edition.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.vmi.edu/media/content-assets/documents/academics/appliedmath/Fundamentals-of-Matrix-Algebra-3rd-Edition.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like the idea of a a special project business case: optimizing a factory, refinery, catalog business, or insurance agency.  Because I think economists are weak on basic behavior of firms.  But I worry, you don't really have time for all that.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the second link has some very simple business problems (not even projects, just normal word problems).  Could consider to intersperse some of these into the course rather than an end project.  Also, the level of difficulty seems similar to what you are doing.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-23T03:35:49.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13849" PostTypeId="1" AcceptedAnswerId="13859" CreationDate="2018-03-23T12:31:21.013" Score="4" ViewCount="254" Body="&lt;p&gt;A Calculus class for 1st year students may have two subclasses (with &lt;em&gt;two different lecturers&lt;/em&gt;) : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The main class (which covers the theories and concept), &lt;/li&gt;&#xA;&lt;li&gt;and the 'response' class (which provides and explain how to solve problems in exercises)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;For the latter, what is the ideal way to teach?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some logical ideas that have been think of are :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Give them problems (homework) and call some students forward to work on the board in the next meeting&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give &lt;strong&gt;open book&lt;/strong&gt; quizzes (open book so that the students are also studying at the same time)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give problems and explanation as many as possible? (sometimes does this makes students lazy or not..?)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some facts : several students that rarely focus on the lecture (skip classes, or even working on other things inside the class) have higher grades. I presume that online resources (such as &lt;em&gt;Khan Academy&lt;/em&gt;, &lt;em&gt;Youtube&lt;/em&gt;, ... works better for them?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use slides or the board?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="9439" LastActivityDate="2018-03-28T19:37:37.513" Title="What is the ideal teaching style for Calculus exercises only?" Tags="|calculus|teaching|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13850" PostTypeId="1" CreationDate="2018-03-23T16:21:25.620" Score="2" ViewCount="198" Body="&lt;p&gt;Many math classes benefit from occasionally  being held in a computer lab. My question is about the pros and cons of different layouts and mechanics of a lab and &quot;solutions&quot; you have found to be effective and the ones you thought did not work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some of issues and concerns:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1- What practical setup fosters collaboration in a lab environment?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2- How to engage the students and reduce distractions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3- Are large TVs a good substitute for projection systems?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding collaborative lab layouts: If students are to huddle and help each other and discuss problems then the traditional layout by rows is not that convenient. However using circular tables prevents students from seeing the instructor at the board. Have you come across a design for a lab layout that you found particularly effective?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have you used the the &quot;plus/cross&quot; table design? (I have seen such designs where 4 tables are arranged in the shape of a plus sign, with six chairs per table. The instructor station is at the intersection/mid point of tables. There are four projection screens.)&#xA;What are the issues you faced when using these new &quot;cross-shaped&quot; designs?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you use movable tables how do solve the problem of having power to the table without creating the danger of students tripping on the power cables? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On reducing distractions: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To explain, I find that as soon as students are in front a computer they feel compelled to check their email or view a random video. How do you solve the resulting distraction problem? One solution that comes to mind is to have a common power button for all monitors that at the discretion of instructor will turn all monitors off. Another solution is to have a software that allows instructor to see and project student's screen on the projection screen. What works and what does not work?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another solution is to use desks that allow monitors to be hidden until needed (monitor is flipped into the desk or lowered below table top). This style appears to give the advantage of having a dual-use class. When computers are needed they come out and when they are not needed they are out of view. Have you used such furniture? What are the pros and cons (other than the expense)? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have you used a large TV in class in place of a projector? How do you compare the two? Have you used a set of 2 or 4 TVs to duplicate the size of a typical projection screen? What software did you use and how do you evaluate it?&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2018-03-30T15:33:24.577" LastActivityDate="2018-03-30T16:09:55.097" Title="Effective computer lab layouts for a university math class" Tags="|undergraduate-education|technology-in-education|graduate-education|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13851" PostTypeId="2" ParentId="269" CreationDate="2018-03-23T17:39:56.513" Score="-3" Body="&lt;p&gt;I think there is no specific field to use in our daily life. However such type of course explored the logical thinking capacity of readers.&lt;/p&gt;&#xA;" OwnerUserId="9742" LastActivityDate="2018-03-23T17:39:56.513" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13852" PostTypeId="1" CreationDate="2018-03-23T22:37:17.653" Score="7" ViewCount="437" Body="&lt;p&gt;I'm teaching algebra to lower ability grade 11 students. I've tried to give them fair grounding in algebraic manipulation. I'm trying to explain how to solve a linear equation like $2 x +1 =3$ (or $- 2.3 x + 1.2 =-3.7 $ but I'll treat the simpler one). The first step is clear &quot;do the opposite of plus $1$&quot; ie &quot;subtract $1$ from both sides&quot;:&#xA;$$&#xA;2x+1=3 \\&#xA;2x+1-1=3-1 \\&#xA;2x+0=2 \\&#xA;2x=2&#xA;$$&#xA;The middle steps are quite intuitive and my students seem happy skip them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next it comes to &quot;do the opposite of multiply by $2$&quot; ie &quot;divide by 2&quot;&#xA;$$&#xA;\frac{2x}{2}=\frac22 \\&#xA;x=1.&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How do I explain this step?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideas: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$\frac{x+x}2=x$&lt;/li&gt;&#xA;&lt;li&gt;$\frac{2x}{2}=\frac22x=1x=x$&lt;/li&gt;&#xA;&lt;li&gt;$\frac{2x}{2}=x\frac22=x1=x$&lt;/li&gt;&#xA;&lt;li&gt;other explanations&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I'll appraise these explanations. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;seems intuitive but won't easily handle the complex example above. &lt;/li&gt;&#xA;&lt;li&gt;The obvious choice, but why is it obvious and why is it true? Would I say that $\frac{a \times b}c = \frac{a}{c} \times b $? This seems confusing.&lt;/li&gt;&#xA;&lt;li&gt;exchanges &quot;$2$ lots of $x$&quot; with &quot;$x$ lots of $2$&quot;, which is valid but seems confusing. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="998" LastEditorUserId="11" LastEditDate="2018-03-24T01:15:08.547" LastActivityDate="2022-08-30T15:02:35.247" Title="How to solve $a x = b$?" Tags="|algebra|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="13853" PostTypeId="2" ParentId="9907" CreationDate="2018-03-24T02:49:38.093" Score="0" Body="&lt;p&gt;As a student, I was taught all three major line formulas:  point-slope, slope-intercept, and two-point (and even exposed to the intercept, intercept oddity).  I always devolved to just using point slope as it feels smaller and more intuitive.  Still to this day is how I think of lines.  I think there is a physical intuition to it as well:  initial condition and rate of change.  But I was forced to be exposed to all of the equations.  And I agree that, provided you remember them, the other equations work faster for problems suited to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the teacher, of course, you should let him do things how he wants when you are the assistant and the class belongs to him.  When you get your own dojo, you can teach karate how you like it.  &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-24T02:49:38.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13854" PostTypeId="2" ParentId="13852" CreationDate="2018-03-24T13:37:30.650" Score="4" Body="&lt;p&gt;I recommend that you carefully read the first part of the algebra book you use for your course. Usually there is a section on &quot;Properties of Real Numbers&quot;, which effectively presents the axioms for the field structure of real numbers, and hence your course. Usually these are: (1) commutativity (add &amp;amp; multiply), (2) association (add &amp;amp; multiply), (3) distribution (multiply over add), (4) identities (add &amp;amp; multiply), (5) inverses (add &amp;amp; multiply). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also there will be a formal definition of subtraction: $a - b = a + (-b)$, and division: $\frac{a}{b} = a \cdot \frac{1}{b}$. (Without saying this would be my top choice for axiomatic fundamentals, it is very standard.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First note that properties of equivalent equations can be proven by basic properties of how equality is defined. E.g., For multiplication: If $a = b$, then $ac = bc$. Proof: $ac = ac$ by reflexivity; so $ac = bc$ by substitution ($a = b$ on right). Likewise for other operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if desired, we can step through solving $2x = 2$ by the fundamental axioms. $\frac{2x}{2} = \frac{2}{2}$ by the equivalent-equation principle (prior paragraph, for division). This is equivalent to saying $2x \cdot \frac{1}{2} = 2 \cdot \frac{1}{2}$, by the definition of division. The left hand side is equivalent to $x \cdot 2 \cdot \frac{1}{2}$ (commutativity of multiplication), which is equal to $x \cdot 1$ (inverse property of multiplication), which is just $x$ (identity property of multiplication). The right hand side is immediately equal to $1$ (inverse property of multiplication). And so we have our $x = 1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, how much of these details you walk through with your students is up to you. Somewhat against popular opinion, I do actually find value and traction in my courses to walk through the axiomatic steps at least on one day, quizzing the class as a whole for which fundamental property was used at each step (as long as one or more students can give the answers in this presentation, I think it at least gives a model for everyone what real math looks like). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, of course, your students start generating an intuition that division to both sides of an equation cancels any multiplication there, as they did for the addition problem. The main problem I think we encounter is if they've previously been trained for addition problems to &lt;a href=&quot;https://math.stackexchange.com/questions/1505354/can-i-think-of-algebra-like-this&quot;&gt;&quot;move a term across the sides and change the sign&quot;&lt;/a&gt;, then that puts them on an incorrect path for any other operations. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-03-26T00:35:29.617" LastActivityDate="2018-03-26T00:35:29.617" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13855" PostTypeId="2" ParentId="11542" CreationDate="2018-03-25T00:28:30.500" Score="1" Body="&lt;p&gt;I would say just read the book in a careful engaged manner (much more engaged than a narrative history book).  And especially work all the example problems (along with the text, on paper).  Also, any derivations, work the derivation on paper along with the text.  &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-25T00:28:30.500" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13856" PostTypeId="1" AcceptedAnswerId="16582" CreationDate="2018-03-25T02:01:07.077" Score="9" ViewCount="508" Body="&lt;p&gt;I remember we were taught in high school (Eastern Europe) the difference&#xA;between implication ($\Rightarrow$) and equivalence ($\Leftrightarrow$)&#xA;and were instructed, when solving equations to be mindful to&#xA;always proceed with equivalances, not implications so that we may&#xA;trace our way backwards (since otherwise any&#xA;solution which we arrived at might not satisfy the original equation we set&#xA;out to solve).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we had to write things like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$x^2=y^2 \Leftrightarrow |x|=|y|$ (which makes perfect sense)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and (IIRC):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$x=y \Leftrightarrow x^2=y^2 \land x\cdot y\ge0$ (which also makes&#xA;sense, but only if someone explains it to you)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fast-forward twenty years I am revisiting old subjects to tutor my&#xA;son. To my astonishment, I notice that people (e.g. Wikipedia or&#xA;Wolfram MathWorld) don't bother using any kind of notation when they&#xA;move from one equation to the next.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.g. I see things like the following (while deriving the equation of&#xA;an ellipsis):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\sqrt{(x+c)^2+y^2} = 2a - \sqrt{(x-c)^2+y^2}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;immediately followed by:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(x+c)^2+y^2 = 4a^2 - 4a\sqrt{(x-c)^2+y^2} + (x-c)^2+y^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which seems sloppy in my view as I would expect it to be followed by:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(x+c)^2+y^2 = 4a^2 - 4a\sqrt{(x-c)^2+y^2} + (x-c)^2+y^2 \land 2a \ge&#xA;\sqrt{(x-c)^2+y^2}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has there been a trend in recent years to de-emphasize the distinction between&#xA;implication and equivalence when solving education or maybe this kind of&#xA;notation was never really used in the West at the secondary education&#xA;level? Or maybe it's not a good idea after all to insist on these&#xA;nuances?&lt;/p&gt;&#xA;" OwnerUserId="9744" LastActivityDate="2019-05-04T05:33:24.767" Title="implication vs equivalence when solving equations" Tags="|secondary-education|algebra|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13857" PostTypeId="2" ParentId="13852" CreationDate="2018-03-25T10:12:32.220" Score="0" Body="&lt;p&gt;@Pdmclean, &lt;strong&gt;My answer is a perspective :&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Keeping it simple while comprehensive would work, but it needs time. Sometimes, trying really hard to explain a concept will end up confusing the students.&lt;/p&gt;&#xA;&lt;p&gt;Keeping it simple, compact, and firm may let the students understand it and to them the concept is actually simple, logical, and easier to grasp.&lt;/p&gt;&#xA;&lt;p&gt;Build through habits and repetition, I remember a quote from a mathematician, which i think that it is quite true, it is something like this :&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;In mathematics, you don't understand things. You just get used to them.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is quite makes sense, because mathematics is a tool that &lt;em&gt;we&lt;/em&gt; have created, a logical way of using numbers and symbols.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example of explanation&lt;/strong&gt; :&lt;/p&gt;&#xA;&lt;p&gt;Solve &lt;span class=&quot;math-container&quot;&gt;$$2x + 1 = 4$$&lt;/span&gt;&#xA;The equation states that the value in left-hand side is equal to the value in right-hand side. If we change the value of either side, of course the equality will not apply anymore. So what we can do in order to get &lt;span class=&quot;math-container&quot;&gt;$$ x = .... (\text{something}) ? $$&lt;/span&gt;&#xA;To do this, manipulate each side the same way, adding or subtract by a number, divide or multiply by a number :&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 2x + 1 = 4 \implies 2x = 3 .....(\text{subtract by} -1)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now what can we do to get the equation &lt;span class=&quot;math-container&quot;&gt;$x = \text{a number}$&lt;/span&gt;? divide both side by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 2x = 3 \implies x = \frac{3}{2} .....(\text{divide by} 2) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;*NOTICE that you can also do it by &lt;strong&gt;multiply by &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;After this, repeat with different examples, or same one with different manipulation.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Try to solve only one problem on the board, with the students watching&lt;/strong&gt;. Then repeat this steps several times, to 5-6 problems. (like watching a video, but the difference is they can ask directly to you)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;After this, students must try to solve problems themselves, give them time 10-15 minutes&lt;/strong&gt;. If they &lt;strong&gt;cannot&lt;/strong&gt;, do the &lt;strong&gt;previous step for this particular problem only&lt;/strong&gt;. This way, after the students have tried working on it, they now know in which part they made mistakes, and therefore know how to fix.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The key is that make the students get used to it, then the logic will eventually come to them.&lt;/p&gt;&#xA;" OwnerUserId="9439" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-03-25T18:25:55.393" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13859" PostTypeId="2" ParentId="13849" CreationDate="2018-03-26T02:59:10.587" Score="1" Body="&lt;p&gt;My approach would likely be as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Reserve time for questions. In my experience this will not actually take much of the time. Don't be afraid to not answer certain questions. For example, &quot;what do you think is on the next test&quot; is usually not wise to entertain. Unless you are writing the test and have a specific sense of how to use that question to leverage studying. For example, I will sometimes ask in return, &quot;well, tell me something you're not done studying&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prepare a few examples to give lecture style. Perhaps just 1/3 of the time for this, but it assures there is real content delivered. This could be at the beginning, middle or end, in fact all three of these can be rearranged as you see fit for your audience and setting.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Put all the students names on 3x5 cards. Write about 5-10 problems on the board and randomly select names to work those problems. Give them 10 or so minutes and emphasize that everybody should be trying to work through the problems to follow along. When the time is up, go through the problems critiquing both answers and presentation. Mark the cards with dates and short description for your record. Next time, go to new students, or for fun repeat to keep them on their toes. Eventually everybody comes up front to work problems in this fairly low-pressure setting. This works best if you have some course points to assign. If you have no influence over their grade then sadly I have not much hope. I mean, try talking to people anywhere about math for 2 hours when you have no control over their grade. You'll find yourself alone in a room long before the end. (statistically, people are rarely mathematicians, so the sentence before is said in that manner of thinking)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The benefit of 3. is it gives them incentive to work on the class regularly and it gives you a chance to warn against common mistakes and/or to show better ways to solve the given problems. Finally, it is probably useful for the students to see that everybody (for the most part) struggles with the material. Too often students refuse to ask questions because they think they are alone in their confusion. In fact, the confusion is the rule. Ideally, this process helps some of the students to start asking good questions. We probably need to teach them what is a &quot;good question&quot;, but I'll leave that for another post.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-03-26T02:59:10.587" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="13860" PostTypeId="1" CreationDate="2018-03-26T15:31:38.067" Score="3" ViewCount="440" Body="&lt;p&gt;I am looking for ideas for an activity for high school students, which involves plane geometry and another field, such as algebra, series, etc...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in junior high there is a nice activity in which students are investigating the idea of pi, by taking rounded shapes, such as plates, measuring the perimeter and radius of each one of them, then setting an XY Cartesian axes, putting points from the measures and calculating the slope, which is pi (if we take 2r on the x axis).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for similar activity, has to be geometry, but slightly harder, for high school students. It has to involve algebra or any other field (perhaps apart from Trigonometry because that makes the geometry too easy).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any ideas will be most appreciated !&lt;/p&gt;&#xA;" OwnerUserId="9752" LastActivityDate="2018-04-19T08:33:37.720" Title="Multidisciplinary problem" Tags="|algebra|geometry|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13861" PostTypeId="2" ParentId="13860" CreationDate="2018-03-26T17:25:55.577" Score="5" Body="&lt;p&gt;Pick's Theorem is a great link between area of polygons and the algebraic skill of writing expressions. Nrich maths has a ready to go investigation of Pick's theorem: &lt;a href=&quot;https://nrich.maths.org/pickstheorem&quot; rel=&quot;noreferrer&quot;&gt;https://nrich.maths.org/pickstheorem&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Coordinate geometry and transformations are another area with a huge amount of overlap between geometry and algebra.&lt;/p&gt;&#xA;" OwnerUserId="5603" LastActivityDate="2018-03-26T17:25:55.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13862" PostTypeId="2" ParentId="13846" CreationDate="2018-03-26T17:40:36.007" Score="4" Body="&lt;p&gt;I've tried something of the same form, engineering students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Line of projects were both applied and theoretical:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Simple linear programming problems&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simple population models &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Discovering the link between determinant and volume&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Optimization problems presented in game form (which could be solved by setting a problem and solving linear systems)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Problems on Graphs&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My experience is that time runs out very fast. I underestimated completely the time required for my students to understand that some problems could be formalized with linear algebra. Mostly because they did not know how to model a problem altogether. So that they were drawn towards &quot;finding solution&quot; more than &quot;building a good model&quot;. They needed a lot of directions on how to build a model and understanding even the basic issues in the model.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have many groups and you're alone in interacting, you will have to jump  from one to the other. Asking questions of 8 groups? No one of the groups understand much about questions of the other therefore gets distracted and may become noisy. Organize this session VERY carefully.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the positive my students had a lot of fun and they participated in this very actively. It was a good experience, it took me a lot of time to build up decent models, maybe they haven't learned as much as I would have hoped out of it, but overall going back I would do it again (considering also what I gained from it in terms of teaching experience).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Project example&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Car Rental Company.&#xA;Rents car in three cities A, B, and C. Each person can return the car in the same or in a different city. We have tables on statistical habits of people: 70% of renters in city A returns in city A, 20% in B, 10% in C and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We start with 100 cars and a certain distribution of them along A, B, C. What will be the distribution of cars after 1 month, 6 months, 1 year, in the long run?&#xA;Is there an initial distribution of cars granting equilibrium?&#xA;What initial distribution of cars should I have to have them equally distributed after 3 months?&#xA;How do I take into account cars being stolen at a different rate in the three cities?&#xA;What about new cars being bought?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(and so on)&lt;/p&gt;&#xA;" OwnerUserId="465" LastEditorUserId="376" LastEditDate="2020-04-18T07:54:28.640" LastActivityDate="2020-04-18T07:54:28.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13866" PostTypeId="2" ParentId="9907" CreationDate="2018-03-26T21:05:53.847" Score="10" Body="&lt;p&gt;Point slope form teaches a &lt;em&gt;very&lt;/em&gt; important concept in a &lt;em&gt;very&lt;/em&gt; simple way - that of translation.  When you get into quadratics or any other equation, you can move the equation around by taking a standard equation and bumping the origin around.  Point slope is the simplest way to do this.  It takes a line of a particular slope which passes through the origin and translates it on the graph by moving the origin point to (a, b).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This technique (replacing $y$ with $(y - b)$ and replacing $x$ with $(x - a)$ to get a translation of the graph) is useful to every other form of graph translation and applies to every other equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a parabola shifted over so the vertex is at $(2, 3)$:&lt;br&gt;&#xA;$$(y - 3) = (x - 2)^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a circle of radius $5$ whose center is at $(-2, 7)$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ (x - -2)^2 + (y - 7)^2 = 5^2 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also argue that the most important thing you can learn in a math class is how to connect different ideas, and see the same ideas in different ways.  Solving problems is an important but &lt;em&gt;secondary&lt;/em&gt; consideration in a math class.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2018-03-26T21:05:53.847" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13867" PostTypeId="2" ParentId="8313" CreationDate="2018-03-27T02:59:44.110" Score="-1" Body="&lt;p&gt;You should give full marks for an &quot;easy&quot; answer, if it is mathematically valid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the classic example was when future mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carl Friedrich Gauss&lt;/a&gt; was a grade school boy, and the teacher asked the students to add the numbers between 1 and 100. Gauss solved the problem in about 30 seconds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He noticed that you could add 100+1, 99+2, 98+3, etc., and that there were 50 such pairings adding up to 101. So 101 X 50 =5050.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2018-03-27T02:59:44.110" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13868" PostTypeId="1" CreationDate="2018-03-27T07:09:49.903" Score="-2" ViewCount="130" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/13700&quot;&gt;Squares are special types of rectangles.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Are circles special types of ellipses/ovals?&lt;/p&gt;&#xA;&lt;p&gt;Are cones special types of pyramids? I guess the answer is no because of the 2D basis: circles are not special types of however you categorise the bases of pyramids. However, cubes are cuboids/(right?) rectangular prisms because&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Why/why not to all of the above? How do we know if a certain thing is a special type of another thing?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Who decides? Or is there a way to determine logically? If someone decides, should this be on some official list on the state where this someone has authority?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Maths is supposed to be about logic. If someone decides that squares are rectangles (that is defines the words to be that way) rather than that there is some logical way to deduce such, then these all seem so arbitrary. I think it's similar to how English says 'The plants are tall' not 'The plants is tall' whereas in French or Spanish, I think, the linking verb may contain an 's' if the noun would contain an 's'.&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Why isn't there a term for equilateral triangle but there is a term for equilateral rectangle?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why does the word rectangle exist? Why don't we just use oblong? Is this an English thing? Or a maths thing? Or what?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2021-10-10T18:22:25.803" LastActivityDate="2021-10-10T18:22:25.803" Title="When a geometrical figure a special case of another" Tags="|geometry|curriculum|terminology|definitions|language-use|" AnswerCount="1" CommentCount="4" ClosedDate="2018-03-27T12:43:29.960" ContentLicense="CC BY-SA 4.0" />
  <row Id="13869" PostTypeId="2" ParentId="13868" CreationDate="2018-03-27T07:53:09.707" Score="3" Body="&lt;p&gt;The question, as you pose it, is a bit too generic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What does it mean to be &quot;A special case&quot;? It means there is a definition of a class of math objects such that this object belongs it. A square is a special case of rectangle (a rectangle with equal sides) and a special case of quadrilateral, and a special case of closed simple curve in the plane and so on...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are cones special cases of pyramids? It depends on your definition of pyramid. If you intend pyramid as a cone over a closed simple plane curve than yes, while if you insist that the base is a polygon then no. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an authority accepting formally definitions? No. It is the community of mathematicians that decides what should be an accepted definition of a concept: so it has to do with &quot;sociological issues&quot; as well as to logical ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For years the definition of parallel lines was &quot;with no common points&quot;. Now it is usually said that two lines are parallel if there exists a non trivial vector to which they are both parallel. This is because we want to stress the fact that being parallel is an equivalence relation and therefore we want to say that any line is parallel to itself. Two centuries ago maybe equivalence relation and equivalence classes were not used as ubiquitously as now. This may explain the difference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course it is a matter of conventions: but such conventions are not certified by authorities but by the community of mathematicians as a whole. Historical reasons enter the scene, here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a concept has a constant widespread use it will sooner or later find its name...&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2018-03-27T07:53:09.707" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13870" PostTypeId="2" ParentId="13860" CreationDate="2018-03-27T14:59:26.277" Score="2" Body="&lt;p&gt;This was a problem set in a Spanish newspaper in the summer of 2011. I've stripped away some fluff about t-shirt designs: you can add your own &quot;real world&quot; motivation if desired.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Take two straight lines meeting at a point. Let's call that point the origin $O$, so the lines are $O + sA$ and $O + tB$. Now draw a zigzag starting at $O$. The first segment runs from $O$ to $O + A$, the second segment runs from $O + A$ to $O + t_2 B$ for some $t_2$; the third segment runs from $O + t_2 B$ to $O + s_3 A$ for some $s_3$; etc. We add two more constraints: each segment has the same length, and the 20th segment runs perpendicular to $A$. What is the angle between the two lines which form the boundary?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There's an extremely elegant solution using only basic geometry. But an alternative approach goes via the matrix power $$\left[ \begin{matrix}-1 &amp;amp; 1 + \cos\frac{\pi}{2n} \\ -2 &amp;amp; 1 + 2\cos\frac{\pi}{2n}\end{matrix} \right] ^n$$ (where $n=10$ for the stated problem), so the basic geometry solution ends up giving a nice identity for the matrix power.&lt;/p&gt;&#xA;" OwnerUserId="8005" LastActivityDate="2018-03-27T14:59:26.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13871" PostTypeId="2" ParentId="9907" CreationDate="2018-03-27T17:51:08.787" Score="8" Body="&lt;p&gt;This is a follow-up and add-on to &lt;a href=&quot;https://matheducators.stackexchange.com/a/13866/29&quot;&gt;johnnyb's excellent and important answer.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;One of the most important themes in Algebra 2 and higher-level classes is that of &lt;em&gt;function transformation&lt;/em&gt;.  This theme runs throughout virtually every topic in those courses. Consider the following function types:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Linear functions:  &lt;span class=&quot;math-container&quot;&gt;$y = A(x-x_0) + y_0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Quadratic functions:  &lt;span class=&quot;math-container&quot;&gt;$y = A(x-x_0)^2 + y_0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Exponential functions:  &lt;span class=&quot;math-container&quot;&gt;$y = Ab^{x-x_0} + y_0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Sinusoidal functions:  &lt;span class=&quot;math-container&quot;&gt;$y = A\sin(x-x_0) + y_0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Logarithmic functions:  &lt;span class=&quot;math-container&quot;&gt;$y = A\log_b (x-x_0) + y_0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;All of these functions have the same basic form:  take a &amp;quot;toolkit&amp;quot; function (&lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=b^x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=\sin x$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$y=\log_b x$&lt;/span&gt;) and perform (in sequence) the following transformations:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;shift it horizontally&lt;/li&gt;&#xA;&lt;li&gt;rescale it vertically&lt;/li&gt;&#xA;&lt;li&gt;shift it vertically&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Now you might have noticed that &amp;quot;rescale it horizontally&amp;quot; is missing from the list above.  For linear and quadratic functions, there's a good reason for that:  if &lt;span class=&quot;math-container&quot;&gt;$f(x)=x$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$f(k\cdot x) = k\cdot f(x)$&lt;/span&gt;, which shows that a horizontal &amp;quot;compression&amp;quot; by a factor of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is indistinguishable from a vertical &amp;quot;stretch&amp;quot; by the same factor; similarly if &lt;span class=&quot;math-container&quot;&gt;$g(x)=x^2$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$g(k\cdot x) = k^2\cdot g(x)$&lt;/span&gt;, so a horizontal &amp;quot;compression&amp;quot; by a factor of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is the same as a vertical &amp;quot;stretch&amp;quot; by a factor of &lt;span class=&quot;math-container&quot;&gt;$k^2$&lt;/span&gt;.  (Incidentally, this is one of the reasons why when students are learning about quadratic functions in vertex form they  often mis-recognize &amp;quot;make it taller&amp;quot; as &amp;quot;make it skinnier&amp;quot;.)  Even for exponential functions, a horizontal rescaling can be absorbed into a change of the base, so you don't (strictly speaking) &lt;em&gt;need&lt;/em&gt; them.  It's really only for trigonometric functions that the horizontal scale factor plays a role that can't be accounted for by one of the other parameters.&lt;/p&gt;&#xA;&lt;p&gt;In high school, transformations are usually taught as a discrete topic, confined to an early chapter of Algebra 2 and only occasionally mentioned again, but I really think it is one of the most important throughlines of mathematics at the secondary and early postsecondary level.&lt;/p&gt;&#xA;&lt;p&gt;In teaching university-level Calculus, for example, I will often begin the semester with a set of questions like this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Find a formula for a linear function that passes through &lt;span class=&quot;math-container&quot;&gt;$(2,5)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(9,15)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Find a formula for an exponential function that passes through &lt;span class=&quot;math-container&quot;&gt;$(2,5)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(9,15)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Find a formula for a quadratic function that has its vertex at &lt;span class=&quot;math-container&quot;&gt;$(2,5)$&lt;/span&gt; and passes through &lt;span class=&quot;math-container&quot;&gt;$(9,15)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Find a formula for a trigonometric function that has a minimum at &lt;span class=&quot;math-container&quot;&gt;$(2,5)$&lt;/span&gt; and whose first maximum after that minimum is at &lt;span class=&quot;math-container&quot;&gt;$(9,15)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These are meant to be review problems, although students always struggle with questions 2-4.  Needless to say, every student can answer question #1. Most will use slope-intercept; a few will use point-slope form, and then rewrite their answer in slope-intercept form.&lt;/p&gt;&#xA;&lt;p&gt;However, when we move on to #2, most students go through elaborate gyrations to find the values of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$y=A\cdot b^x$&lt;/span&gt; passes through the given points.  Then I show them that the function can be described as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt; set &lt;span class=&quot;math-container&quot;&gt;$y=5$&lt;/span&gt;, and multiply &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; every time &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; increases by &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and that this translates directly into the formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y=5\cdot 3^{(x-2)/7}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Students are usually astonished that it can be done that easily, at which point I remark that this is really the difference between slope-intercept form (which requires you to describe the line in terms of its &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept, regardless of whether that point is actually important for you) and point-slope form (which allows you to directly use the points you are given).  We then spend the next couple of days approaching the rest of the problems using similar methods.  This not only works well as a review of prerequisite material but also sets the stage for finding the equations of tangent lines, and (much later) Taylor series.&lt;/p&gt;&#xA;&lt;p&gt;The point-slope form is (or has the potential to be) students' first encounter with transformational thinking and lays the groundwork for these more advanced topics.  I think it is &lt;em&gt;essential&lt;/em&gt; material for high school algebra.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2021-07-25T23:38:39.350" LastActivityDate="2021-07-25T23:38:39.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13872" PostTypeId="1" AcceptedAnswerId="13884" CreationDate="2018-03-28T09:10:34.397" Score="4" ViewCount="551" Body="&lt;p&gt;This exact quote from my standard American Algebra 1 textbook states when  first introducing rational square roots:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{49} = 7$&lt;/span&gt;  is read &amp;quot;The &lt;em&gt;positive&lt;/em&gt; square root of &lt;span class=&quot;math-container&quot;&gt;$49$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$-\sqrt{49} = -7$&lt;/span&gt; is read &amp;quot;The &lt;em&gt;negative&lt;/em&gt; square root of &lt;span class=&quot;math-container&quot;&gt;$49$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$-7$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the next lesson, the expression &lt;span class=&quot;math-container&quot;&gt;$5\sqrt{3}$&lt;/span&gt; is used in an example of simplification, but no instructions are given on how to &amp;quot;read&amp;quot; it?&lt;/p&gt;&#xA;&lt;p&gt;I think these are some possible ways:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$5\sqrt{3}$&lt;/span&gt; is read &amp;quot;5 the positive square root of 3.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;5 square root 3.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;5 root 3.&amp;quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Following the textbook's instructions from the previous lesson, the first way seems most appropriate; but the third way has the least words? Is  there a &amp;quot;correct&amp;quot; way to read &lt;span class=&quot;math-container&quot;&gt;$5\sqrt3$&lt;/span&gt; and are there any other possibilities?&lt;/p&gt;&#xA;" OwnerUserId="9757" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-05-30T17:00:08.470" Title="How to Teach Middle School Students to Read Square Roots?" Tags="|secondary-education|algebra|terminology|middle-school|" AnswerCount="2" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="13873" PostTypeId="2" ParentId="8313" CreationDate="2018-03-28T16:50:14.790" Score="7" Body="&lt;p&gt;I think it's important here that you think about the purpose of an assessment.  I tell my first year college students (often to their shock) that the purpose of a test question isn't to see if a student can find the correct answer.  The purpose is to see if a student understands the concepts discussed in class.  From that perspective, I think you're justified in not giving credit for answers using methods that didn't demonstrate the skills you were trying to evaluate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the other side of that coin is that it's up to the instructor to write questions that can't be answered using other methods.  Sometimes this is as simple as specifying a method.  For example, if I want to see if students can use the elimination method to solve a system of linear equations, I'll start the question with, &quot;Solve the following systems of equations using the elimination method . . .&quot;&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2018-03-28T16:50:14.790" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13874" PostTypeId="2" ParentId="353" CreationDate="2018-03-28T17:20:02.050" Score="0" Body="&lt;p&gt;I use a specific example:  If I'm in the rain then I'm wet.  I also emphasize that  we're talking about whether it's possible for the statement to be true.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now think about the four cases:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;true/true:&lt;/strong&gt;  If you are in the rain then it's possible for you to be wet so this is a true result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;true/false:&lt;/strong&gt;  If you are in the rain then it &lt;strong&gt;isn't&lt;/strong&gt; possible for you not to be wet so this is a false result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;false/true (which I think is the most confusing):&lt;/strong&gt;  If you aren't in the rain, is it possible that you're wet?  The answer to that is yes because you may have gotten wet by other means.  That makes this a true result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;false/false:&lt;/strong&gt;  If you aren't in the rain it's possible that you aren't wet so this is also a true result.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2018-03-28T17:20:02.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13875" PostTypeId="2" ParentId="13849" CreationDate="2018-03-28T19:37:37.513" Score="-1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;To the extent the class does not cover problem solving, you should cover a a few in lecture style as examples.  I don't totally believe (maybe don't want to believe) that the regular class is all derivations and theory and no example problems.  But if so, definitely showing some examples would be helpful.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Emphasis even in case of 1, should be on drill still though.  Pop quizzes, work together assignments, kids to the board, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When explaining example problems, you will need to be efficient.  Don't have time to cover all, so cover things that get messed up most often or are most likely to be covered on tests.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would keep the tone in your section more collaborative and informal and friendly than the normal class.  More &quot;buddy telling you how to get it done&quot; and less &quot;herr doktor professor&quot;.  Students love feeling  like they are getting the inside scoop and cutting through the pretensions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use the board rather than slides.  If you want super extra credit A+ points, than compose your remarks ahead of time.  But still do the exposition on the board, rather than slides.  Slides are a crutch.  And they are so much different in connection than writing as you go.  When you show slides, it's like showing the book.  When you write on the board, you are a fellow warrior fighting in the trenches.  Maybe a little bit better of one.  But still &quot;in the fight&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Talk to other TAs and get their experience.  Coming here was a nice step. But do it IRL.  And get to the nitty gritty (time management, discipline, loudness, handwriting, etc.)  Not just high level stuff.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You avoided the question about country and student ability.  Realize that some issues of pedagogy are the same from CalTech to RN JuCo.  But in other cases, it makes a difference what the student limitations are in brains, desire, time, prep, etc.  Think about these variables.  Pedagogy is not a linear y(x) function but nonlinear, multifunction and full of stochastic noise (and student ability, etc. are confounding variables to some sort of same for everyone solutuon)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-28T19:37:37.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13876" PostTypeId="2" ParentId="6163" CreationDate="2018-03-28T20:30:10.353" Score="1" Body="&lt;p&gt;We can certainly do a lot to enliven discussions about triangles (and more generally polygons) in both K-12 geometry classes and college geometry classes. Looking at different ways to measure distance in the &quot;plane&quot; consisting of ordered pairs of real numbers is one approach to doing this. For example, after discussing taxicab distance one talk about whether in the taxicab plane triangles can be equilateral even when they are not equiangular. In axiomatic terms the taxicab plane does not obey the congruence axiom usually used in the Euclidean approach so there are many aspects of triangles in the taxicab plane that students have to get used to.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2018-03-28T20:30:10.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13877" PostTypeId="2" ParentId="353" CreationDate="2018-03-28T22:45:54.397" Score="1" Body="&lt;p&gt;I think the intuition for implication is aided when you consider it in the context of universal quantification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A claim of the form $(\forall x)(A(x) \Rightarrow B(x))$ can be translated roughly as every $x$ that has property $A$ must also have property $B$. Most of the use of conditional claims used in practice have this form implicitly assumed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, using Brendan Sullivan's example which I agree is good for explanation, when someone says, &quot;If it rains, I use an umbrella,&quot; the implicit reading is that this is a universal principle with the logical structure of $(\forall t)(\text{it rains at time } t \Rightarrow \text{I use an umbrella at time } t)$. Intuitively, we grasp that whatever happens when it is not raining cannot affect the truth value of the universal implication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The challenge in teaching this, of course, is that universal quantification is often not introduced until well after logical implication, but maybe that should be done differently.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2018-03-28T22:45:54.397" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13878" PostTypeId="2" ParentId="13860" CreationDate="2018-03-28T23:28:24.913" Score="0" Body="&lt;p&gt;I don't have a specific project in mind but you might give some thought to Diophantine equations, i.e. equations whose solutions have to be integers.  This connects coordinate geometry and number theory.  (For example, number theory concepts are used to show when the equation (a/b)x + (c/d)y = 1 has integer solutions and when it doesn't.)&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2018-03-28T23:28:24.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13880" PostTypeId="2" ParentId="12731" CreationDate="2018-03-29T03:36:06.690" Score="0" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f(x) := \sqrt{x+3}$.  What are the the domain and range $f$?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You are right to be concerned about such questions. Technically, you cannot define a function without specifying its domain and codomain. And if you haven't specified the domain, you cannot determine the range. The domain in this case could be $[-3,+\infty)$. Or it could be $[0,1]$. Such questions should probably be banned from the curriculum. There must be other ways to test students' understanding of the terminology.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2018-03-29T05:48:31.467" LastActivityDate="2018-03-29T05:48:31.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13881" PostTypeId="2" ParentId="13860" CreationDate="2018-03-29T08:03:41.247" Score="0" Body="&lt;p&gt;If you want something a little lighter, easier, consider the derivation of pi with experimental probability.  Kind of fun to draw the grid and do the experiment.  And it doesn't involve math, they don't know yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=statistical+derivtion+of+pi+needle&amp;amp;ie=utf-8&amp;amp;oe=utf-8&amp;amp;client=firefox-b-1-ab&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=statistical+derivtion+of+pi+needle&amp;amp;ie=utf-8&amp;amp;oe=utf-8&amp;amp;client=firefox-b-1-ab&lt;/a&gt; (various good links after)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can get more or less into the statistics of if if you need more math content:  &lt;a href=&quot;https://www.statslife.org.uk/the-statistics-dictionary/1976-statistics-bootcamp-estimating-pi-with-r-and-buffon-s-needle&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.statslife.org.uk/the-statistics-dictionary/1976-statistics-bootcamp-estimating-pi-with-r-and-buffon-s-needle&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or you can just make it an experimental physical activity with a 9th grade algebra equation as the math content.  &lt;a href=&quot;https://www.sciencefriday.com/articles/estimate-pi-by-dropping-sticks/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.sciencefriday.com/articles/estimate-pi-by-dropping-sticks/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wouldn't use the computer simulator as I feel like, we are physical monkeys and get more from dropping sticks than looking at a screen.  But it is an option if you want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia has a decent article on the concept (Buffon's Needle), but they go off into a bunch of minutia about using needles that are shorter or longer than the grid spacing.  I think if you use an equal spacing, the thing is more intuitive and the pi value pops out clearly.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-29T08:03:41.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13882" PostTypeId="2" ParentId="6163" CreationDate="2018-03-29T08:15:02.543" Score="0" Body="&lt;p&gt;Classically, in the US, high school geometry has had a heavy proof emphasis (basically from Euclid).  And most of that is about triangles. [Except make sure you hit the vertical angle theory hard.  SAT test LOVES that concept and it is useful in a lot of math contest problems.]  In this manner it is really sort of different from algebra 1 and 2, trig, analytic geometry, precalc, calculus, where we are really learning a lot of tools applicable to solving real world word problems.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, there is a practical mensuration aspect of geometry also (formulas for areas and volumes and the like).  And those end up being important for AP calculus (darned related rate word problems with geometry tricks like in the Stand and Deliver cheating problem!)  But really, it feels like we have this horse that wants to run down the calculus track and we hank it off into proof land.  And then getting back into mensuration formulas is short shrifted because that is yanking it back onto the type of thinking and algebraic problems solving we yanked away from!  Don't know the answer as there is good argument that the emphasis on proofs in plane geometry is a good way to get some proof solving and reading experience that is missing or more difficult in the other subjects.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-29T08:15:02.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13883" PostTypeId="2" ParentId="12731" CreationDate="2018-03-29T11:55:50.290" Score="3" Body="&lt;p&gt;This is bit too long for a good comment so I'll add it as an answer. When I teach this material to calculus students, many of whom are not math majors, I take the following approach:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If we are given $f(x)$ as some formula in $x$ then we define the $\text{dom}(f)$ to be the largest set of real numbers for which the evaluation of $f$ returns a single value. Or, in other words, make the domain of $f$ as greedy as possible in real terms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For example, $f(x) = \ln(x)$ has $\text{dom}(f) = (0, \infty)$ or $f(x) =1/ \sqrt{1-x}$ has $\text{dom}(f) = (-\infty, 1)$. Then I warn them that we have the freedom to make the domain smaller than this greedy choice. We can always make it smaller. How? Well, just state it directly. This gives me a chance to remind them that proper mathematics is about more than mere formulas. We need sentences and logic to communicate the structure we wish to analyze. Hopefully this discussion then naturally feeds into our discussion of inverse functions where the idea of local inverse demands some discussion of restriction. For example, we start with $f(x) = \sin(x)$ (so, the greedy domain is $\mathbb{R}$ as $\sin(x)$ makes sense for any real $x$). But, $\sin(x)$ is not one-to-one (it fails the horizontal line test) if our domain includes more than one cycle of sine. So, to obtain an inverse we restrict sine to $[-\pi/2, \pi/2]$... I think Hurkyl is right, we do think about partial functions in such courses without admitting it.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-03-29T11:55:50.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13884" PostTypeId="2" ParentId="13872" CreationDate="2018-03-29T14:36:12.883" Score="7" Body="&lt;p&gt;I would say that the word &lt;strong&gt;of&lt;/strong&gt; is pretty important here, and as long as you get that correct, the rest is less important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the word &lt;strong&gt;of&lt;/strong&gt; consistently when a function is being applied to an input, and talking about the fact that you are doing this, will help ward off a lot of the following kinds of mistakes that I see in my (much later) classes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Question: Solve for $x$. $\sin x = z$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Student solution: $x = \frac{1}{\sin}z = \csc z$  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The mistake here is that it is unclear to the student that there are different kinds of mathematical objects: some are numbers, but some are functions/operations. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So my answer is that saying &quot;3 root 5&quot; for $3\sqrt5$ is a dangerous abbreviation, because it begins to confuse this issue earlier than it needs to be confusing (What kind of object is &quot;root&quot; anyway?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a result I would prefer at least &quot;3 &lt;strong&gt;square root of&lt;/strong&gt; 5.&quot; &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Similarly &quot;sine &lt;strong&gt;of&lt;/strong&gt; x&quot; for $\sin x$ and &quot;ell en &lt;strong&gt;of&lt;/strong&gt; x&quot; for $\ln x$.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-03-29T14:36:12.883" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13885" PostTypeId="2" ParentId="13872" CreationDate="2018-03-29T16:27:47.350" Score="3" Body="&lt;p&gt;I'm not aware of any &quot;official&quot; version, the text that you quoted not withstanding, so any response is going to be mostly just personal preference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that said, I wouldn't read either expression the way your textbook wrote them.  &quot;the positive square root&quot; is redundant.  The square root of a number, in this context, is always positive so I would just read the first version as &quot;the square root of 49&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For your second expression, I wouldn't refer to &quot;the negative square root&quot; for the same reason, i.e. there's no such thing.  This phrasing implies that there's a positive version of the square root and a negative version which isn't the case.  I would read it as &quot;minus the square root&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With respect to your last example, I agree that (1) seems to be the most consistent with what you posted from the textbook.  However, it's also excessively wordy and the lack of a verb after the five makes it sound awkward.  I would read it as, &quot;three times the square root of five&quot;.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2018-03-29T16:27:47.350" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13886" PostTypeId="2" ParentId="13860" CreationDate="2018-03-30T00:25:07.013" Score="0" Body="&lt;p&gt;I have some calculus students who are working on a mini &quot;research project&quot; with me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea is the following:  when we define a tangent line, we take two points $(a,f(a))$ and $(a+h,f(a+h))$.  We find the unique line passing through these points, then we take a limit as $h \to 0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are investigating what happens if you take the three points $P= (a,f(a))$, $Q= (a+h,f(a+h))$, $R=(a-h,f(a-h))$, find the unique circle passing through all three points, and then take a limit as $h \to 0$.  It took most of the semester but we have it!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even the task of finding the equation of the circle probably the hardest algebra and geometry problem these students have ever done:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Juggling terms like $f(a+h)-f(a-h)$ and interpreting these things geometrically really stretches their understanding of graphs of functions.&lt;/li&gt;&#xA;&lt;li&gt;Figuring out the geometry of the situation:  I can find the center of the circle by finding the intersection of the perpendicular bisectors of the two segments $PQ$ and $PR$.&lt;/li&gt;&#xA;&lt;li&gt;Translating that geometric statement into an algebra problem.  Understanding how similar triangles show that the perpendicular bisectors have opposite reciprocal slope, finding the midpoint, and using point slope form.&lt;/li&gt;&#xA;&lt;li&gt;Solving the resulting system of equations (which has a lot of variables!) for the center of the circle.&lt;/li&gt;&#xA;&lt;li&gt;Finally using the center and one of the three points to get the equation of the circle.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This would be a fine stopping point for your students!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To find the limit of the resulting expressions as $h\to 0$ you need to do some careful work with Taylor expansions.  For instance, you need to expand the terms $f(a+h) \approx f(a)+f'(a)h+\frac{1}{2} f''(a) h^2$.  In the end everything works out, and you get a &quot;tangent circle&quot;!  Of course, this is already well known as the &quot;osculating circle&quot;, but this is a pretty novel and fun way to go about generating it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are two desmos links, one to the 3 point approximation, and one to the actual osculating circle that you get from taking a limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/eehupqcbtg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.desmos.com/calculator/eehupqcbtg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/dhzsnvcifz&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.desmos.com/calculator/dhzsnvcifz&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-03-30T00:25:07.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13887" PostTypeId="2" ParentId="13850" CreationDate="2018-03-30T16:09:55.097" Score="2" Body="&lt;p&gt;I think a fancy lab like you are talking about is kind of a dinosaur of the past.  It requires a person to maintain it (maybe even an attendant) and a lot of effort on scheduling...unless you make the whole school a lab which is cost prohibitive and a waste.  I really only see labs like this for dedicated classes and then the needs of the specific class drive it (not a use by several classes).  Even then, I generally just see a set of networked PCs at conference tables (e.g. RBN school of energy) and generally very high fees for the class for something like that make it a wash...works in corporate setting but not good fit for a school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to have students work on the computer, ask them to bring a laptop.  If there are any who lack one, than they can work with a classmate.  This is actually much closer to what almost all of them will do in a work setting (work on a laptop).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think there is anything wrong with telling them not to use the Internet or email during the class though.  Would you let a kid read the newspaper in your class?  I would kick him out.  (Don't underestimate the effect on other students...we are social monkeys, not AynRandian robots.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, the more you can work with common programs the better.  Excel actually has huge amounts of functionality.  And is so, so, sooo common in work environments.  R is free.  Etc.  If it doesn't do what you need, issue a specific formula or program.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't underestimate the time spent and confusion factor generated with incorporating technology into math classes.  I have seen this come a cropper since at least the 80s.  It sounds so easy and important...but then the kids doing ODE with computers hate it and learn less than the kids doing regular ODEs. Just watch out.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-30T16:09:55.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13889" PostTypeId="1" AcceptedAnswerId="13891" CreationDate="2018-03-31T18:31:09.130" Score="4" ViewCount="150" Body="&lt;p&gt;In my country the education system is very expensive, so I’m considering starting an organization to teach lower-income students for free. In particular, to motivate them into the science fields. So, I’m asking for general recommendations on good mathematics books. Books that are useful for both children and adults, and at the same time are helpful and powerful for them to improve their skills in mathematics. Any suggestions?&#xA; Thank you very much.&lt;/p&gt;&#xA;" OwnerUserId="9766" LastActivityDate="2018-04-01T06:31:15.483" Title="Suggestion and advices" Tags="|applied-mathematics|math-puzzle|" AnswerCount="3" CommentCount="7" ClosedDate="2018-04-03T20:07:15.853" ContentLicense="CC BY-SA 3.0" />
  <row Id="13890" PostTypeId="2" ParentId="13889" CreationDate="2018-03-31T21:13:13.860" Score="1" Body="&lt;p&gt;&lt;strong&gt;Overall:  I recommend two tracks:&lt;/strong&gt;  (1) books that are motivational and (2) books that are helpful.  For all of them, I would emphasize books that are more common/popular as it will be more likely to get a native language translation.  Also, for the charitable bang for the buck, emphasize books that have cheap editions (more likely in the popular ones as there will be current paperback prints).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;You don't mention it, but:&lt;/strong&gt;  I think bigger challenges than the content will be (1) funding and (2) outreach.  I feel like your question is already super broad (but lacking in detail on your situation, like WHAT country or age target or ability target, which would allow us to help you better), so won't address them.  But these are probably bigger challenges than the books.  I do see something like a national (perhaps participative) exam with prizes as a useful idea.  I'm not sure how to restrict it to low income awardees though.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Motivational books:&lt;/strong&gt;  Feynman &quot;what do you care what other people think&quot; and &quot;surely you are joking&quot;.  Microbe Hunters.  Men of Mathematics.  All of the Heinlein juveniles, especially Starman Jones, Have Spaceship Will Travel, and Citizen of the Galaxy (as they emphasize education and a bit of Horatio Alger). All of these should be available in Arabic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Useful books:&lt;/strong&gt;  Schaum's Outlines, especially the older ones for First Year College Math, Calculus, etc.  Practical man's Guide to Trig, Calc, etc.  Also Calculus made Easy.  These may all be English only.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-03-31T21:13:13.860" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13891" PostTypeId="2" ParentId="13889" CreationDate="2018-04-01T00:15:46.427" Score="3" Body="&lt;p&gt;I'd put my money on:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://www.goodreads.com/book/show/849866.Mathematics_and_the_Imagination&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics and the Imagination&lt;/a&gt;&lt;/em&gt; by Edward Kasner and James Newman&#xA;The four-volume set &lt;em&gt;&lt;a href=&quot;https://www.goodreads.com/book/show/705566.The_World_of_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;The World of Mathematics&lt;/a&gt;&lt;/em&gt; that Newman edited.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These books appeal to people in all subject areas. I myself taught a bit on math through literature at the secondary level, and I gleaned three short stories from the corresponding section in &lt;em&gt;World&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For inspiration, &lt;em&gt;&lt;a href=&quot;https://www.goodreads.com/book/show/186749.Innumeracy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Innumeracy&lt;/a&gt;&lt;/em&gt; by John Allen Paulos, which shows why people are so ready to say they're bad at math. He's a mathematics professor, so he's seen it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://www.goodreads.com/book/show/2112067.Mathematical_Snapshots&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Snapshots&lt;/a&gt;&lt;/em&gt; by Hugo Steinhaus is festooned with examples of connections within and outside of mathematics. I could easily get 25 conference presentations from this book alone. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess it's clear that I am not one for the traditional curriculum, preferring books with a student-oriented (rather than math-oriented) focus. I &lt;em&gt;loved&lt;/em&gt; &lt;a href=&quot;https://www.goodreads.com/book/show/1936376.Calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus: A Liberal Art&lt;/em&gt;&lt;/a&gt; by W. M. Priestley, and liberally used it when teaching AB and BC Calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're looking for specific textbooks, I can vouch for &lt;em&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/1570397759&quot; rel=&quot;nofollow noreferrer&quot;&gt;Contemporary Pre-Calculus through Applications&lt;/a&gt;&lt;/em&gt; and &lt;em&gt;Contemporary Calculus through Applications&lt;/em&gt;, both developed by the North Carolina School of Science and Mathematics faculty. I've taught using both of them and they are phenomenal, although the Calculus one might be hard to find.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My favorite series of high school books is from the University of Chicago School Mathematics Project. I was only able to use two of them: &lt;em&gt;Functions, Statistics, and Trigonometry&lt;/em&gt;, and &lt;em&gt;Precalculus and Discrete Mathematics&lt;/em&gt;, found in their &lt;a href=&quot;http://ucsmp.uchicago.edu/secondary/curriculum/&quot; rel=&quot;nofollow noreferrer&quot;&gt;entire series for grades 6–12&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ever since they came up with the acronym, I've contended that although the STE parts belong together, the M isn't like STE. M should exist between the humanities and the sciences, and the closer you get to the humanities the better. Have a look at &lt;a href=&quot;https://www.goodreads.com/book/show/6232657-a-mathematician-s-lament&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form&lt;/em&gt;&lt;/a&gt; to see why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of these books exist in inexpensive Dover editions. They're worth investigating. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm only allowed 8 links, and have probably written more than I know already. &lt;/p&gt;&#xA;" OwnerUserId="9241" LastActivityDate="2018-04-01T00:15:46.427" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13892" PostTypeId="2" ParentId="13889" CreationDate="2018-04-01T06:31:15.483" Score="3" Body="&lt;p&gt;It seems that your main concern is cost.  I therefore suggest you look for sources of free textbooks.  If you're looking for books for undergraduate students, then you might want to take a look at the books from &lt;a href=&quot;https://openstax.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From its &lt;a href=&quot;https://en.wikipedia.org/wiki/OpenStax&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia entry&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;OpenStax&lt;/strong&gt; (formerly OpenStax College) is a nonprofit ed-tech initiative based at Rice University. Since 2012, OpenStax has created peer-reviewed, openly licensed textbooks, which are available in free digital formats and for a low cost in print.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They have free textbooks on Math, Science, Social Sciences, Humanities, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Advanced_Placement&quot; rel=&quot;nofollow noreferrer&quot;&gt;AP&lt;/a&gt;.  For math, they currently have textbooks on Prealgebra, Elementary Algebra, Intermediate Algebra, College Algebra, Algebra and Trigonometry, Precalculus, Calculus (Volumes 1, 2, and 3), Introductory Statistics, and Introductory Business Statistics.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-04-01T06:31:15.483" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13893" PostTypeId="1" CreationDate="2018-04-01T23:11:59.997" Score="2" ViewCount="1747" Body="&lt;p&gt;I am exploring ideas to design a secondary-level-project-based-10-lessons-unit-learning-plan which can end with a creation from the students involving a tensegrity structure. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;such as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kltlU.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kltlU.jpg&quot; alt=&quot;cube&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Dxk8d.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Dxk8d.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My general questions are related to designing lessons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Question 1: what could be essential pre-lesson that leads to such creation? &lt;/li&gt;&#xA;&lt;li&gt;Question 2: what learning can be derived in each lesson? (... that is generally aligned with secondary math knowledge and skills) &lt;/li&gt;&#xA;&lt;li&gt;Question 3: Challenging: How could you involve in a &quot;simplified manner&quot; (secondary level) the use of modular arithmetics, and other element of abstract algebra such rotations, permutation, groups...&lt;/li&gt;&#xA;&lt;li&gt;Question 4: what general problem can be be asked that can lead to such creation? essential questions?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So far :&#xA;From readings, students will have to understand the concept of &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Set of Vertices configure distance constraints that define figure or solid &lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;struts&lt;/em&gt; and &lt;em&gt;cables&lt;/em&gt; (terminology used in tensegrity) holding vertices apart and keeping close together respectively.&lt;/li&gt;&#xA;&lt;li&gt;A structure is stable if it respect specific distance constraints. the literature use the concept of super stable.&lt;/li&gt;&#xA;&lt;li&gt;Not only distances between &lt;em&gt;Edges&lt;/em&gt; but also distances between &lt;em&gt;Diagonals&lt;/em&gt; have to follow specific constraints to shape a specific structure.&lt;/li&gt;&#xA;&lt;li&gt;Geometrical versus physical rigidity&lt;br&gt;&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/490242/what-is-the-precise-definition-of-a-rigid-shape&quot;&gt;https://math.stackexchange.com/questions/490242/what-is-the-precise-definition-of-a-rigid-shape&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So far my responses to those questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1st group of lessons: triangle, its property, and its physical rigidity &lt;/li&gt;&#xA;&lt;li&gt;2nd group of Lessons: properties of diagonals in quadrilaterals and its physical rigidity&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Students will understand (1) the importance of respecting the diagonals distances constraint to shape 2 dimensional figures by building simple figures  (using for  of popsicle sticks and fishing lines) using the least amount of struts (2) We can permute struts and cable for the edge and diagonals&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/s6NKp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/s6NKp.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe this lesson can be fruitful as students have little sense of seeing the need of understanding the properties associated to diagonal of quadrilaterals&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;3rd group of lessons: geometrical rigidity  of polygon and transformation (1)enlargement (2) Rotation (3) reflexion (4) translation &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Basically here I would use the idea of Abbas Jeffary &quot;Start with an equilateral triangle. Label the vertices one, two, three. Define a rotation to be 60° counterclockwise, for example (or you could go clockwise). Note that the locations of the vertices have shifted. Consider this a new “state“ or “configuration”. Now, what “configuration” does one end up with after n rotations? Then, extend this to any n-sided polygon.&quot;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Last lesson: the project build the highest tower using water pipes and metal strings which can support a weight at the top. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;That's an initial brainstorm that I want to refine. Any suggestion regarding this project? What specific other pre-lesson could be taught? I find this final project a bit hazardous as to see what exactly could be assessed in term of school mathematical knowledge. &lt;/p&gt;&#xA;" OwnerUserId="9111" LastEditorUserId="9111" LastEditDate="2018-04-02T03:51:07.450" LastActivityDate="2018-04-02T03:51:07.450" Title="Using tensegrity structure to teach high school math?" Tags="|course-design|problem-design|inquiry-based-learning|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13894" PostTypeId="2" ParentId="13893" CreationDate="2018-04-01T23:55:30.047" Score="1" Body="&lt;p&gt;Maybe just do a general unit on solid geometry--it is a bit undercovered in schools.  Just have the last class or two be model building and discussion.  With the self standing nature of the models being a little extra fun.  Can discuss the mensuration formulas or the like a little, but just have fun too.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think that the theoretical mechancis issues are appropriate for the grade level or easily taught.  So just make it more of a fun activity and maybe have a little student discussion where they speculate about how they stand up.  Maybe show some pictures of suspension bridges or the like so they can see some connection to real world engineering.  But don't teach them statics.  At the end of the day, some parts of early education may be more about evoking interest and having fun rather than proving things.  Gives the motivation for maybe proving it, learning it later.  And even the kids who never learn the math behind the equilibrium structures get the idea that such things can happen.  Which is maybe even more important than knowing the details.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also skip discussion of point symmetry operations as that is better done in a chemistry class, because of the direct connect to an application.  It is also more easily done with rigid models that don't fall apart when you rotate them.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-04-02T00:03:00.180" LastActivityDate="2018-04-02T00:03:00.180" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13895" PostTypeId="2" ParentId="1134" CreationDate="2018-04-02T02:20:33.437" Score="2" Body="&lt;p&gt;Disadvantages include cost and efficiency, gaps in explanations, differences in notation, differences in quality of pedagogy, &quot;switching cost&quot; of being used to one way of doing things in a text versus another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the answers poo-poo the student dislike of this jumping around as irrational.  But I would not be quick to condemn the students' hesitation.  Students may sense that perfect is the enemy of better.  Picking one thing and immersing and doing it well may be better than a smorgasbord that is not well learned.  Also, I suspect the teacher desire for this sort of jumping around is strongly motivated by a desire to have perfect rigor or very particular coverage rather than good learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I basically agree with the students. The hard part of math class is not finding the perfect proof or ideal coverage but doing the hard slog and learning something.  Why not set yourself up for success and pick one best text and use it?  There will ALWAYS be alternate choices that could be made, enrichments, or follow-on classes that go deeper.  But leave that for later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, would never do this jumping around for any service class (diffyQ, calc, LA, prob/stat, and probably not for any required math major course).  I just honestly think the hurdles to learning are MORE around doing the homework and learning complicated material, rather than bestest ever set of ODE chapters from 4 different books.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We evolved to kill things and make love, not to do math.  Think more about what is involved in training well rather than training the most perfect things.    &lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-04-02T16:22:13.073" LastActivityDate="2018-04-02T16:22:13.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13897" PostTypeId="1" CreationDate="2018-04-02T18:03:44.763" Score="-6" ViewCount="185" Body="&lt;p&gt;&lt;a href=&quot;https://academia.stackexchange.com/questions/95860/mathematics-stack-exchange-as-an-indicator-for-academic-career#comment280835_95897&quot;&gt;Similar question as in Academia StackExchange&lt;/a&gt; except education instead of academia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;High reputation on a technical Stack Exchange site might indicate that you're a good teacher of that subject, since well-explained, easy-to-understand answers tend to get higher reputation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Paraphrasing (or whatever is the term to change words of another to suit an analogous situation)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm due to start my PhD in mathematics education next month, fully funded by a scholarship, at an average university, I'll be specialising in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13700&quot;&gt;how language affects mathematical definitions&lt;/a&gt;, and I have a strong First in Mathematics from a Russell group university. My goal is to become a kindergarten/primary/secondary maths teacher and an academic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is my Mathematics Stack Exchange (MSE) profile. I've been an active user since 201X, visiting pretty much daily, but my reputation is currently just [4 digits] and I haven't used MathOverflow at all (as the questions there every time I visit, which isn't that often, are beyond me). This seems quite low. Should this worry me?&lt;/p&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2018-04-02T18:43:49.213" LastActivityDate="2018-04-02T21:37:06.570" Title="To what extent is MSE reputation an indication of potential in education?" Tags="|interactive-teaching|online-instruction|teacher-preparation|" AnswerCount="2" CommentCount="0" ClosedDate="2018-04-03T00:26:53.780" ContentLicense="CC BY-SA 3.0" />
  <row Id="13899" PostTypeId="2" ParentId="13897" CreationDate="2018-04-02T18:33:09.597" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It's an imperfect correlation.  The best athletes are not always the best coaches.  Sometimes those who struggled to perform can teach better than the star athletes that did it all naturally.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The how language affects definitions and triangle square nitpicking scares me.  And the &quot;Russell group&quot;.  And the academic eduction versus practical education slant.  I think this is not as useful as being a salty and empathetic (but not soft) teacher.  Go coach a sport.  Lot of really good things you can learn from that that translate to being a good teacher.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-04-02T18:33:09.597" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13900" PostTypeId="2" ParentId="13897" CreationDate="2018-04-02T21:31:09.140" Score="3" Body="&lt;p&gt;I'm not a tenured faculty member so take this with a grain of salt: No one is going to care, your reputation in the field and job prospects will almost certainly be based on your PhD work. Many people don't use these forums and even if they did academic hiring is based on publication output and the opinion of your recommenders. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know of one person who bothered to put their stack exchange reputation on their cv and it's because at the time they had something like the third highest reputation. But they put it on as a joke, and the jobs they got were certainly more due to the highly influential papers they published than their reputation on a website. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you start answering a lot of questions thoughtfully it might be good for your name recognition, but no where near as applicable to success as writing a single well thought out paper. Furthermore, you could potentially do yourself some harm if your postings are shoddy, low quality, or just incorrect. I would say don't worry about trying to inflate your score, use stack exchange as a resource to understand what people in the field think and to practice carefully writing and answering questions. Pay close attention to the feedback people give you and try to improve your writing accordingly. &lt;/p&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="8571" LastEditDate="2018-04-02T21:37:06.570" LastActivityDate="2018-04-02T21:37:06.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13901" PostTypeId="1" CreationDate="2018-04-02T23:20:53.627" Score="-2" ViewCount="369" Body="&lt;p&gt;Why high school teachers do not emphasize knowing the proofs of properties and theorems in math. In my 40 years of teaching prospective high school teachers, I rarely found students who can derive formulas they learned in school. For example properties of logarithms, changing from one base to another, Law of Sines and Law of Cosines proving major Geometry theorems. These &quot;formulas&quot; and theorems can be presented as problems to solve so with help, students can discover them on their own. Then they would &quot;own&quot; the theorems.&lt;/p&gt;&#xA;" OwnerUserId="9777" LastActivityDate="2018-04-04T03:59:19.653" Title="Proving basic Theorems and properties in high school" Tags="|proofs|curriculum|" AnswerCount="1" CommentCount="4" ClosedDate="2018-04-04T05:08:28.330" ContentLicense="CC BY-SA 3.0" />
  <row Id="13903" PostTypeId="1" AcceptedAnswerId="13905" CreationDate="2018-04-03T17:45:01.623" Score="6" ViewCount="908" Body="&lt;p&gt;I'm wondering if anyone knows of calculus books or other work towards teaching calculus in a &lt;em&gt;less&lt;/em&gt; mathematically rigorous way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm thinking mostly of American-style college level calculus courses introducing the derivative and integral.  For whatever reason, it seems to be general practice that these courses should have a significantly higher level of mathematical rigor than the courses that proceed them.  The textbooks I've seen are concerned with proofs and about rigorously justifying facts.  (They usually don't prove everything, but they're typically concerned about this - for instance, noting when a claim has to wait for a more advanced course for a proof - in a way that pre-calculus textbooks aren't.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an example of what I mean, there's been some discussion (including here) about whether to use the conventional epsilon-delta approach to limits, or an approach using some kind of rigorous infinitesimals.  But has anyone tried going all the way to working with informal infinitesimals?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that's just an example - I'm interested in other ways people might have tried dialing down the rigor in calculus.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2018-04-04T13:42:58.297" LastActivityDate="2018-04-13T13:31:15.967" Title="Teaching Calculus Less Formally" Tags="|calculus|textbooks|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13905" PostTypeId="2" ParentId="13903" CreationDate="2018-04-03T18:56:38.843" Score="9" Body="&lt;p&gt;Below are some books I know about. FYI, I read approximately the first half of a public library copy of Ferrar's book back in January or February of 1974 while enrolled in a 9th grade Algebra 1 course (however, in my independent reading/study of math I had completed the Algebra 2 material by this time), and I found it to be something I could understand and follow, which was definitely NOT the case with some college calculus texts I had seen at public libraries. I mention this because the book seems to be rarely mentioned on the internet, and thus will likely be overlooked by people looking for something like this. The other books, except for Brewster's book, are either well known or (because they're old enough to be in public domain) are in the process of becoming well known. Sawyer's book is not really a textbook, but I'm including it because I suspect it introduced a lot of &quot;mathy types&quot; in the 1960s and 1970s to calculus, as it was in the &quot;New Mathematical Library&quot; series of books that high school libraries back then often had several volumes from.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Calculus_Made_Easy&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Calculus Made Easy&lt;/strong&gt;&lt;/a&gt; by Silvanus P. Thompson (1910, 1914)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0486404536&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Calculus: An Intuitive and Physical Approach&lt;/strong&gt;&lt;/a&gt; by Morris Kline (1967, 1977)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0198531338&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Calculus for Beginners&lt;/strong&gt;&lt;/a&gt; by W. L. Ferrar (1967)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/W.W.SawyerWhatIsCalculusAbout&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;What is Calculus About?&lt;/strong&gt;&lt;/a&gt; by W. W. Sawyer (1962).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/differentialcal01knoxgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Differential Calculus for Beginners&lt;/strong&gt;&lt;/a&gt; by Alexander Knox (1884)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/elementaryillust00demouoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Elementary Illustrations of the Differential and Integral Calculus&lt;/strong&gt;&lt;/a&gt; by Augustus De Morgan (1832, 1899)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.worldcat.org/title/commonsense-of-the-calculus/oclc/1671551&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Commonsense of the Calculus&lt;/strong&gt;&lt;/a&gt; by George William Brewster (1923)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;(ADDED NEXT DAY)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't believe I overlooked &lt;strong&gt;Calculus in Context&lt;/strong&gt; below. I don't remember how I learned about it, but I've had a copy of the 1995 hardback edition since 1995 or 1996. I've used some of the material in classes (for example, Section 12.1: &lt;em&gt;Stirling's Formula&lt;/em&gt; on pp. 690-699 and &lt;em&gt;Bessel’s Equation&lt;/em&gt; on pp. 564-573 in classes &lt;a href=&quot;https://www.lsmsa.edu/page&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; during the late 1990s), and I often used to flip through this book for ideas about how to introduce certain topics in class (such as Section 3.6: &lt;em&gt;The Chain Rule&lt;/em&gt; on pp. 138-147). There’s a bit of computer program stuff in the book that’s probably out of date now, and there's some material on discrete dynamical systems topics that was avant garde in the late 1980s to late 1990s, but I was never much interested in this material and my interest in the book was for its exposition and its inclusion of certain topics that you rarely see in standard calculus texts. I’m almost certain I’ve written about this book on the internet, but for some reason I can’t find anything nontrivial now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even better, this book is now (legally) freely available on the internet, although if you find you really like it, then you'll probably want to get a print copy. The book (1995 edition; I haven't seen a print copy of the 2008 edition) opens nicely and the font size is medium to large (I'd guess either 12 point or 13 point), which makes it very easy to use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://math.smith.edu/~callahan/intromine.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Calculus in Context. The Five College Calculus Project&lt;/strong&gt;&lt;/a&gt; by James Callahan, et al. (1995, 2008)&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2018-04-04T19:54:28.833" LastActivityDate="2018-04-04T19:54:28.833" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13906" PostTypeId="1" CreationDate="2018-04-04T01:33:33.930" Score="1" ViewCount="217" Body="&lt;p&gt;I am teaching Differential Equations this semester. The pre-req is Calculus II, not even Multivariable Calculus. I made my peace with that, can't fight with the big-shots of the department. So, I have been teaching stuff from Multivariable calculus on a need basis (partial derivatives, chain rule, etc.) I also spent one lecture to teach basic matrix algebra, determinants. &#xA;Then, evidently, I need a little bit of complex algebra as well, so I did that too. Now, I am teaching higher order differential equations, so kids need a bit further explanations for finding complex roots. Well, these kids will eventually take linear algebra, and multivariable. And it is the department's fault not to require these as pre-req. But what about complex algebra/variables? There is no calculus course, single or multivariable, where we introduce complex numbers and complex algebra. Why? What is the fix for this? I mean I am checking calculus books, syllabus from different universities and their calculus courses, no complex algebra whatsoever. How can this be? &lt;/p&gt;&#xA;" OwnerUserId="7253" LastActivityDate="2018-04-04T10:33:00.607" Title="when should we teach basic complex algebra?" Tags="|mathematical-pedagogy|technology-in-education|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13907" PostTypeId="2" ParentId="13906" CreationDate="2018-04-04T01:58:37.883" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Complex roots of the characteristic equation are no big deal.  Kids have seen that plenty with the quadratic formula in high school algebra.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I worry from your question that you are dragging the course down too much by feeling the need for prereqs and then diverting to quasi teaching them (matrices, partial derivatives, complex algebra, etc.)  The vast bulk of ODEs can be taught without getting into that stuff. And the course is so full that you really have a hard time covering everything in a semester anyhow.  Even when some tiny aspect of one of these things, say a determinant, is used, and it is not necessary to even cover that in a 15 week ODE course, you should just do the minimum, rather than seeing a need/opportunity to divert the class.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-04-04T01:58:37.883" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13908" PostTypeId="2" ParentId="13906" CreationDate="2018-04-04T03:45:12.857" Score="4" Body="&lt;p&gt;In the United States, high school students typically encounter complex arithmetic and algebra for the first time in the context of an Algebra 2 course, and then revisit the topics the following year in Precalc.  (For most students this would be either 10th and 11th grade, or 11th and 12th grade.)  Complex numbers also appear on the SAT and ACT.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem you are having is not that students haven't seen complex numbers before -- it's that they haven't seen them or used them &lt;em&gt;recently&lt;/em&gt;.  If you are teaching Diff Eq, most of your students are in their second year of college (if not later), which means that it has been roughly two years since they've had to work with or use complex numbers.  As you point out, they do not play a role in single-variable Calculus -- but why should they?  Calc 1 and Calc 2 are the study of functions of a single real variable; it's hard for me to even think of places where I could make use of complex numbers in those courses, even if I wanted to.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-04-04T03:45:12.857" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13909" PostTypeId="2" ParentId="13901" CreationDate="2018-04-04T03:59:19.653" Score="5" Body="&lt;p&gt;I'm going to propose two reasons:  the &lt;em&gt;problem of assessment&lt;/em&gt;, and the &lt;em&gt;design of curricula&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The problem of assessment&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One reason high school teachers do not emphasize proof and reasoning (outside of Geometry) is that it is difficult to assess.  Take, for example, the proof that $\log(ab) = \log(a) + \log(b)$.  A teacher can show the proof to students; a teacher might even be able to guide the students to discover the proof themselves.  But how can the teacher assess whether the students understand the proof?  The only real options are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ask them to replicate the proof on a test -- but this may degenerate into a test of &lt;em&gt;memorization&lt;/em&gt;, rather than comprehension; or&lt;/li&gt;&#xA;&lt;li&gt;Ask them to prove something different, but similar -- but the scope of available things to prove at the high school level is limited, and to a certain extent each one is &lt;em&gt;sui generis&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Instruction tends to default to that which can be assessed.  One of the reasons the Geometry course has developed the highly artificial, stylized kind of proof that is common in that context is precisely because it is easy to generate many, many problems of a similar level of difficulty.  The two-column proof form in this sense &quot;solves a problem of teaching&quot;, making it viable for teachers to hold students accountable for learning something.  (See &lt;a href=&quot;https://link.springer.com/article/10.1023/A:1020264906740&quot; rel=&quot;noreferrer&quot;&gt;Herbst, 2002&lt;/a&gt; for more on this.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The design of curricula&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A second reason teachers do not hold their students accountable for deriving, justifying, and proving formulas in high school is because for the most part the curriculum materials they follow provide little support for such instruction.  &lt;a href=&quot;http://www.jstor.org/stable/10.5951/jresematheduc.43.3.0253?origin=JSTOR-pdf&amp;amp;seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;Thompson, Senk &amp;amp; Johnson (2012)&lt;/a&gt; studied 20 contemporary high school mathematics textbooks and found that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Overall, about 50% of the identified properties in the 3 topic areas&#xA;  were justified, with about 30% of the addressed properties justified&#xA;  with a general argument and about 20% justified with an argument about&#xA;  a specific case. However, less than 6% of the exercises in the&#xA;  homework sets involved proof-related reasoning, with developing an&#xA;  argument and investigating a conjecture as the most frequently&#xA;  occurring types of proof-related reasoning.  (p. 253)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The last of the statistics quoted above -- that fewer than 6% of the exercises in the homework sets involved proof-related reasoning -- is obviously related to the problem of assessment I mentioned above.  &lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-04-04T03:59:19.653" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13910" PostTypeId="2" ParentId="13906" CreationDate="2018-04-04T10:33:00.607" Score="5" Body="&lt;p&gt;Let me give you a slightly different perspective. I also come from a European background and have had the chance to TA and teach undergrad courses in the US. As a caveat I have not taught DE, but have taught many calc classes including multivariate and have graded for some DE.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First I understand your utter frustration if you come from Europe and are used to the European approach to teaching and the European undergrad students the US is a very very rude awakening. I remember being at one of my first calc I lectures and seeing the professor add infinite sums by terms without as much as a nod to convergence and present it as an ok thing to do and it was actually physically painful as I was clenching my fists in frustration that this is taught to someone. So believe me I understand that you want to teach the right stuff in the right way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second Xander is right. Noone is asking you to teach what you intend/try/are teaching. The expectation of the department and of the students and maybe most importantly of the follow up courses is that you will teach the students bags of tricks. The courses in the US do not generally correspond to similarly named courses in Europe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everything in the calculus sequence including DE is mostly about tricks and learning to handle the very basic formalism. Calculus students may be taught epsilon-delta proofs but 90% of them do not understand them and they are not really tested on whether they do. In the same way they are not taught real derivatives, no one cares about boundary conditions and pathologies and they are essentially never asked to think for themselves in the problems. Not really think anyhow I'm convinced that all the finals I've seen could be solved by a an easy program with minimal notational standardization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are reasons for this though (not necessarily good ones mind you). The students are generally not math majors and aren't going to continue on in math. Many of them have a fairly shaky background in math alltogether, I remember in calcII classes still consistently having students make elementary level errors such as sum of squares is square of sum and even splitting fractions in wrong ways. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All together if you try to teach complex algebra, (and other stuff you mentioned) along with DE you will be doing a disservice to your students. They will be expected to know the things that the department actually wants you to teach them and due to the extra stuff they will no have properly learned the bag of tricks that teachers further along will expect of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complex algebra will be taught later on to those that continue down to math majors or physics majors (and possibly not even those. In some colleges/Universities many math majors require no advanced complex algebra course at all).&lt;/p&gt;&#xA;" OwnerUserId="5896" LastActivityDate="2018-04-04T10:33:00.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13912" PostTypeId="1" AcceptedAnswerId="13922" CreationDate="2018-04-04T15:00:33.633" Score="5" ViewCount="620" Body="&lt;p&gt;&lt;strong&gt;How should I teach Comparison Tests in Calculus II, and why?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I will cover comparison tests in some way, and students will be expected to justify their answers to questions about series convergence by using tests, but they will not be expected to reproduce or even understand the proofs of the tests. Within these constraints, what do you think, or what do you do, and why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below are some options for &quot;how&quot; to get the discussion started, but I'm most interested in your &quot;why&quot; in any case. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Teach the Limit Comparison Test only, and avoid any problems that&#xA;require Direct Comparison.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even some oddballs like $\sum \frac{\ln n}{n}$ yield to the integral test, and we can avoid asking students about anything that would require a direct comparison, like $\sum \frac{5 + \sin n}{n}$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Teach both the Limit Comparison Test and the Direct Comparison&#xA;Test, and strategies for choosing.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most things are handled with limit comparison, and we can handle oddballs like $\sum \frac{5 + \sin n}{n}$ with direct comparison. We teach both methods and emphasize that the students should use the &quot;Easy&quot; method in general and the &quot;Hard&quot; method for harder problems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Teach the Direct Comparison Test only; there is no reason to&#xA;introduce Limit Comparison.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everywhere limit comparison is useful, direct comparison would have worked too, so we can simplify our course by omitting the &quot;easy&quot; test and focusing on getting students to be more expert at the estimates needed for Direct Comparison.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I've used textbooks that use approaches (2) and (3) and I also have had colleagues who would prefer (1), and there are probably other options, so any insight you all have would be nice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks! See also &lt;a href=&quot;https://matheducators.stackexchange.com/q/2428/11&quot;&gt;how to make Calculus II seem motivated, interesting, and useful.&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2018-04-04T20:36:03.947" LastActivityDate="2018-04-12T17:10:33.330" Title="Comparison Tests in Calculus" Tags="|calculus|series|" AnswerCount="8" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13913" PostTypeId="2" ParentId="13912" CreationDate="2018-04-04T16:12:09.000" Score="2" Body="&lt;p&gt;I think we should at least &lt;em&gt;mention&lt;/em&gt; direct comparison, and show a brief example like $\sum \frac{1}{x^2+1} &amp;lt; \sum \frac{1}{x^2}$, because that seems like the most straightforward way for students to develop some feeling for why any comparison-type results for series should be true in the first place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I also think it's fine to underemphasize direct comparison after it first comes up, and point out that limit comparison is more &quot;robust&quot; because you can't have the phenomenon of an unlucky inequality symbol that points in the &quot;wrong&quot; (i.e. unhelpful) direction.&lt;/p&gt;&#xA;" OwnerUserId="9249" LastEditorUserId="11" LastEditDate="2018-04-04T20:37:17.720" LastActivityDate="2018-04-04T20:37:17.720" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13915" PostTypeId="2" ParentId="13912" CreationDate="2018-04-04T22:06:39.260" Score="6" Body="&lt;p&gt;I believe option #2 (teach both and methods for choosing) is best. I'll try to illustrate with examples.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$\displaystyle{\sum_{n=1}^\infty \frac{n}{n^2+1}}$  The intuition (which most students see quickly) is that this series is &quot;pretty much&quot; the same as $\sum \frac{1}{n}$ because the terms being added approach $\frac{1}{n}$ &quot;in the long run&quot; (as $n\to\infty$). However, &lt;em&gt;direct comparison does not work here&lt;/em&gt; (at least not easily), because the inequality $\frac{n}{n^2+1}&amp;lt;\frac{n}{n^2}=\frac{1}{n}$, which means the series in question is &quot;less than $+\infty$&quot; which tells us nothing meaningful. &lt;em&gt;This is precisely where the Limit Comparison Test is helpful: when our intuition says, &quot;This series is pretty much like another series&quot;, but the Direct Comparison is not fruitful.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;$\displaystyle{\sum_{n=1}^\infty \frac{n}{n^3+1}}$ Again, the intuition is that this series is &quot;pretty much&quot; $\sum\frac{1}{n^2}$. But in this case, the Direct Comparison test works: $\frac{n}{n^3+1}&amp;lt;\frac{n}{n^3}=\frac{1}{n^2}$. Thus, the series in question is &quot;less than a finite value&quot; which means it is also finite, and therefore convergent. Certainly, the Limit Comparison Test works here, as well, but I believe students will agree that the Direct Comparison is far more (ahem) direct and efficient. Moreover, it relies on (and reminds students of) the distinction between convergent (&quot;adds to a finite value&quot;) and divergent.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My example #1 is, I believe, a reason to teach Limit Comparison. There are, indeed, examples (like this one) where inequalities do not work out the way we wish and Limit Comparison can be used to confirm our intuition. Furthermore, I think this particular intuition (&quot;the terms are pretty much $1/n$&quot;) is obvious to almost all students fairly quickly. Showing them that there is a test to confirm that intuition will (a) appeal to students who intend on studying further math, especially analysis, and want to know &lt;em&gt;why&lt;/em&gt; intuitions are valid, and (b) at least help the other students (who just want to know &quot;what the answer is&quot;) solidify their intuition and give them practice finding an answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My example #2 is, I believe, a reason to teach Direct Comparison. There are, indeed, examples (like this one) where inequalities do work out the way we wish and a Direct Comparison is clear, succinct, and effective. This should also help students solidify their understanding of concepts regarding con/divergence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, it should be especially helpful with students who intend to pursue further math, especially analysis. Just the other day, I was working with students in Real Analysis and I appealed to precisely this example! They sought to prove $\lim_{n\to\infty} \frac{n^2}{n^4+1}=0$ using the $\varepsilon$ definition of a sequential limit. They were struggling with the algebraic steps in the inequalities. I asked them, &quot;Could we establish an &lt;em&gt;intermediary goal&lt;/em&gt;, one that is (a) easier to prove and (b) would imply what we need?&quot; Using this exact example, and appealing to their intuition from Calculus II, I was able to guide them into realizing that $\frac{n^2}{n^4+1}&amp;lt; \frac{1}{n^2}$ and that it is much easier to prove $\frac{1}{n^2}\to 0$ using the formal definition.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2018-04-04T22:06:39.260" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13916" PostTypeId="1" CreationDate="2018-04-04T23:20:52.980" Score="4" ViewCount="105" Body="&lt;p&gt;I imagine it varies by publisher, but I was just working with a student who uses the textbook &lt;em&gt;Big Ideas Math Accelerated: A Common Core Curriculum&lt;/em&gt; by Larson and Boswell, and I found an error within minutes of looking at the book.  What do I do with this information?  (It's entirely possible the correct answer is:  just live with it.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-04-05T01:15:17.103" Title="Is there a way to report errors in K-12 textbooks?" Tags="|textbooks|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13917" PostTypeId="2" ParentId="13916" CreationDate="2018-04-05T01:15:17.103" Score="5" Body="&lt;p&gt;Treat it like any other book you find a mistake in:  send a letter or email to the author by way of the publisher.  I would wait until the end of the term and send all the feedback in at one time, then.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-04-05T01:15:17.103" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13918" PostTypeId="2" ParentId="8428" CreationDate="2018-04-05T04:30:10.757" Score="4" Body="&lt;p&gt;What makes a textbook engaging:  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The absence of &quot;it can be shown that&quot; or &quot;it is obvious&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Clear writing.  (Just general English quality of any writing.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Friendly &quot;person to person&quot; style of discussion.  The actual definitions can be precise in terms.  But the explication should be more the way you would talk when tutoring. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Absence of typos or other errors like that (especially in formulas, derivations, problems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Answers to the exercises&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Programmed instruction.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A preface aimed at the reader (student), not the teacher/specifier.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mentioning common errors (&quot;watch out&quot;).  Note to convey this requires the writer know STUDENTS as well as he knows the content.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="62" LastEditDate="2018-04-05T16:01:23.217" LastActivityDate="2018-04-05T16:01:23.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13919" PostTypeId="2" ParentId="13912" CreationDate="2018-04-05T16:24:20.343" Score="6" Body="&lt;p&gt;You should do option (2). As you already mentioned, there are series that cannot be done through limit comparison and can only be done with direct comparison. Why would you purposely leave those out? (OK, I can think of a few reasons, but they all lead me down the rabbit hole that is &quot;What's the point of Calculus II?&quot;) Anyway, from a mathematical point of view, this is sufficient to prove to me that both methods are necessary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a pedagogical point of view, having the two options (direct and limit) forces the student to actually think. If you only teach Limit Comparison, I often feel like they're turning a crank on a machine they really don't understand and/or appreciate. If you only teach Direct Comparison, there's a lot of wasted time trying to find just the right clever manipulation to make it all work. When you have both, the student is challenged to think: Can I do this easily with one method vs. the other? What does &quot;easy&quot; mean to me?&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2018-04-05T16:24:20.343" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13920" PostTypeId="2" ParentId="7199" CreationDate="2018-04-05T16:27:10.247" Score="1" Body="&lt;p&gt;Why is this assumption that you should do proofs at all, even in simple fashion for the activities?   There are activities that are fun and ability appropriate that have nothing to do with proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some ideas (on a super slow net connection, so no links):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Google Eleanor Dickey on Youtube.  Her main interest is education in Roman times (what they actually did).  While a lot of her talk is about language instruction or cultural learnings (slavery accepted, cleanliness emphasized), she does have a video showing a little example Roman school with current little English girls taking part.  Shows them pushing beads around on the little strange segmented rectangle counting boards.  Teachers would like that as it is in some ways similar (at least in appearance) to the new fangled boxes and stuff for math...at least it shows them something to think about.  I would even let the troops dress up if they want...might be surprised at the creativity and enjoyment they get out of it...even if they aren't MSE Rudin studs, can be surprised by how some people like amateur arty activities.  And it increase the sugar to medicine ratio.  More sugar, more sugar, more sugar...help that medicine go down!&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finding pi statistically using needles dropped on a board.  (There are some math lessons about stats and angles and such...but don't underestimate the fun and the traction from being able to doing something real and physical.)  There are some good 'net articles on it and after the exercise, you can teach the history.  But I would keep it simple and design the test using a line spacing equal to needle (or toothpick or matchstick)--the Wiki article needlessly complicates the exercise with making spacing not equal to the needle.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a nice Six Sigma exercise using a catapult...gives learning about angles and statistics and such.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Guessing the jelly beans in a jar (average, range, etc.).  Leads into a general discussion of Bayesian guessing--please don't get all wrapped up with the conditional math, but talk a little about how this is an important concept nowadays and helps people make money (efficient markets, election and sports betting, etc.)  These are topics that excite people since they have impact on their lives ($$).  [Always try to have some &quot;motivation&quot; for science or math subjects...usually this will be economic, but it might be social or military.  And by motivation, I don't mean a long complicated word problem derivation of a diffyQ...I mean mentioning that it helps people make money or save lives.]  Let 'em eat the beans afterwards.  Kids love that sort of stuff.  Adults are big kids.  Leaves them with a good feeling about the experience also.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Less formed idea, but you could do something with shooting stars (or sun or moon).  Don't teach them celestial nav (right ascension of Ares, blabla) or all the details of every measurement.  But just a couple measurements so they get to &quot;learn with fingers&quot; and just play a little.  Then a discussion of how development of trig, navigation, time-keeping, etc happened.  This was math with both economic and military consequences and very historical.  Furthermore in the (probable) event that your students are never able to take all the general science that would benefit them as a high school teacher, at least you have given a little exposure here.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Less formed idea, but some simple card game or gambling exercise and the stats insight from it.  Don't go super in depth on the stats, but do a little afterwards so they get that it wasn't just a game.  Can also discuss some of the history of probability and games of chance (this is fun and excites interest).  Also, there is an important cross-cultural component in that anthropologists have seen evidence across many cultures (rain forest to Eskimo, etc.) that gambling is a concept that people understand.  [Need to research to find some good articles on this, but I have seen it talked about at times.] &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even less formed idea but some simple prob/stats game with pretend farms.  Much of probability and stats and even genetics comes out of agriculture.  It is also an area that intuitively resonates with people (getting food to eat) and some of the first words and concepts we learn as a child are related to food and animals.  Many business frameworks use barnyard terminology since it resonates well with people.  [You probably don't need to design this thing from scratch...I bet someone has had same idea and designed an exercise, you could research.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-04-05T16:27:10.247" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13921" PostTypeId="2" ParentId="7199" CreationDate="2018-04-05T19:55:55.793" Score="3" Body="&lt;p&gt;It might be difficult to find paper copies these days, but the first edition of &lt;em&gt;A History of Mathematics&lt;/em&gt; by Carl Boyer had great exercises at the end of each section. For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ia800706.us.archive.org/3/items/AHistoryOfMathematics/Boyer-AHistoryOfMathematics.pdf#page=265&quot; rel=&quot;nofollow noreferrer&quot;&gt;Exercises from Chapter 12&lt;/a&gt; on China and India&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ia800706.us.archive.org/3/items/AHistoryOfMathematics/Boyer-AHistoryOfMathematics.pdf#page=41&quot; rel=&quot;nofollow noreferrer&quot;&gt;Exercises from Chapter 2&lt;/a&gt; on Egypt&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ia800706.us.archive.org/3/items/AHistoryOfMathematics/Boyer-AHistoryOfMathematics.pdf#page=664&quot; rel=&quot;nofollow noreferrer&quot;&gt;Exercises from Chapter 26&lt;/a&gt; on the Rise of Abstract Algebra&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Later editions of the book dropped the exercise sets, unfortunately.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-04-05T19:55:55.793" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13922" PostTypeId="2" ParentId="13912" CreationDate="2018-04-06T02:07:38.650" Score="5" Body="&lt;p&gt;Another big advantage of direct comparison (if you have the time) is that it can be used to establish error estimates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, if I want to estimate&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\sum_1^\infty \frac{n+1}{n^4+n+7}&#xA; $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;to within $\frac{1}{100}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I can make the estimates&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\frac{n+1}{n^4+n+7} \leq \frac{2n}{n^4} = \frac{2}{n^3}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, by drawing a picture we can see that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\begin{align*}&#xA;\sum_N^\infty \frac{2}{n^3} &amp;amp;&amp;lt; \int_{N-1}^\infty \frac{2}{x^3} \textrm{ d} x\\&#xA;&amp;amp;=\lim_{b \to \infty} \left. \frac{-1}{x^2} \right\vert_{N-1}^{b}\\&#xA;&amp;amp;=\frac{1}{(N-1)^2}&#xA;\end{align*}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we can ensure that the tail of our series is less than $\frac{1}{100}$ if we can make &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\frac{1}{(N-1)^2} &amp;lt; \frac{1}{100}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This first occurs when $N=12$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since almost every series usually covered in Calc 2 can be analyzed by comparison with a geometric series or a p-series, one can always either use the explicit sum of a geometric series, or a p-integral, to estimate the tail of the series.  So you can always decide how many terms you need to obtain a given level of accuracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The limit comparison test gives no such explicit quantitative information:  only the bare fact of convergence or divergence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These kinds of questions force students to really think through what the tests are saying.  They also give students a reason to care about them:  in an application if we obtain a series as a solution to a problem, we do not only need to know if it converges or diverges, we have to estimate the sum to a desired level of accuracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had good success getting students to think through this during a summer calculus session.  Several students told me that the experience changed their perspective on mathematics!&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-04-06T02:07:38.650" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13923" PostTypeId="2" ParentId="13912" CreationDate="2018-04-06T16:58:01.653" Score="1" Body="&lt;p&gt;I would lean towards 1.  You need to prioritize.  Your students don't have unlimited time or ability.  A very sound case could be made for not including this content at all.  [Note, I like the content, I did the content when I took the course, blabla.  I am just saying priorities are a factor!]  Given your likely situation based on school, student population, needs of the student for future course, this stuff is a bell/whistle.  Do the minimum.  I'm sure there is somewhere else some attention could be used!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So choice 1 over 2 as &quot;less stuff&quot;.  [You can either give some time back to the &quot;normal calculus&quot; or just do a less rushed coverage of the limit method if you keep time constant.]  Choice 1 over choice 3 as the more important method and what people expect the kids to have covered when you say you are expected to hit the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you assign homework, obviously make sure that none of the problems need the method you don't teach and you might just mention that you are only teaching one method (without even describing the other method...just so they know they can't use the alternate problems for drill or so they know &quot;don't need to know it for the test&quot;.)  Any kids who are motivated, can obviously learn the direct method and drill it from the text.  But be clear that you are limiting scope for the course.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-04-06T16:58:01.653" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13924" PostTypeId="1" AcceptedAnswerId="13937" CreationDate="2018-04-07T07:01:05.573" Score="7" ViewCount="609" Body="&lt;p&gt;In single variable calculus, integral calculus is taught &quot;back to back&quot; with differential calculus. This is generally true, although to a lesser extent, with multivariable calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet I don't see the teaching of &lt;a href=&quot;https://en.wikipedia.org/wiki/Integral_equation&quot; rel=&quot;noreferrer&quot;&gt;integral equations&lt;/a&gt; accompanying that of (ordinary) differential equations; instead, differential equations are often paired with linear algebra or partial differential equations. For instance, on this site, there was a tag for differential equations but not for integral equations (until I created one).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is that?&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2018-10-27T08:19:32.200" LastActivityDate="2018-10-27T08:19:32.200" Title="Why aren't integral equations often taught &quot;back to back&quot; with differential equations?" Tags="|differential-equations|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="13926" PostTypeId="2" ParentId="13912" CreationDate="2018-04-07T18:00:27.023" Score="3" Body="&lt;p&gt;I tend to teach option (2.) for many of the reasons already listed in the existing answers. If we just teach (1.) then the problem solving becomes formulaic which means the whole endeavor is essentially pointless. Why do we study convergence and divergence in calculus II? It's not really about learning whether these series converge or diverge. The real reason is to expose students to problems which require analysis. In particular, the use of theorems, attention to estimates, and perhaps most importantly the idea that $\Rightarrow$ is not the same as $\Leftrightarrow$. For example, a student who succeeds is one who realizes that showing $a_n \rightarrow 0$ does not imply $\sum a_n $ converges despite the fact that $a_n \nrightarrow 0$ does implie $\sum a_n$ diverges. Or for that matter, I view it as a success if the student has at least understood that the convergence of $a_n$ and $\sum a_n$ are related, but distinct, questions. So, again, the point of studying convergence and divergence is to introduce them to mathematical reasoning. Admittedly, they've had doses of it here and there, but in the study of convergence and divergence we have something like one-dozen theorems to apply. It is way harder logically then their previous work. I often argue it doesn't really belong in the standard calculus sequence because it is really at a completely different level if we are to do it well. Again, I fear not doing it well makes the whole discussion worthless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pragmatically, what students really need to succeed in engineering and physics is a mastery of power series calculation. Excellence in applying the dozen or so convergence and divergence tests does not help me a bit when asked to find the series expansion of $f(x) = \frac{1}{1+x^2}$ or the first three terms in $\sin(e^x)$ or $\tan(2x+1)$ or you name it... Sure, convergence logically matters, but the binomial series is far more important than the limit comparison test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, again, if I am to teach convergence and divergence it really should be with the goal of getting students to be better at mathematical analysis. With that goal in mind the choice of (2.) is clear since it requires the most sincere analysis from students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, there is another tool worth exploring which I don't think anyone has yet to cover. Substitution: consider,&#xA;$$ \sum_{n=1}^{\infty} \frac{1}{n^2+6n+9} = \sum_{n=1}^{\infty} \frac{1}{(n+3)^2} $$&#xA;Set $k=n+3$ then $n=1$ provides $k=4$ and thus,&#xA;$$ \sum_{n=1}^{\infty} \frac{1}{n^2+6n+9} = \sum_{k=4}^{\infty} \frac{1}{k^2}. $$&#xA;Since this is the &lt;strong&gt;tail&lt;/strong&gt; of the $p=2$ series it converges (since $p=2$ converges and the tail of any convergent series likewise converges). But, if we know the $p=2$ series converges to $\pi^2/6$ then we can calculate the tail by some simple arithmetic.&#xA;$$ \sum_{n=1}^{\infty} \frac{1}{n^2+6n+9} = \sum_{k=4}^{\infty} \frac{1}{k^2} = \sum_{k=1}^{\infty} \frac{1}{k^2} - 1 - \frac{1}{4}-\frac{1}{9} = \frac{\pi^2}{6} - 1 - \frac{1}{4}-\frac{1}{9}. $$&#xA;Combining these ideas with the direct comparison test leads to further options for analysis. But, sometimes, the Limit Comparison Test is just a better option if the problem is just about convergence. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-04-07T18:00:27.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13927" PostTypeId="2" ParentId="1598" CreationDate="2018-04-08T00:09:02.443" Score="0" Body="&lt;p&gt;Bad idea.  It's not such a radical idea that people can make mistakes that the students need to be exposed to it.  Except for the next Fields medalists as a child, all of your students, even the strong ones, will have plenty of experience with making mistakes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, giving some helpful tips on places where it is easy to go wrong is very helpful (like the Dummies book with the watch out symbols).  But here, note that it is not just important to be strong at MATH, you have to be strong at KNOWING common student mistakes.  And this is something that mostly comes from experience, from being open to experience and tabulating it, and perhaps discussion with other experienced teachers (who are real teachers, not just mathematicians).  You can do cute things like coming up with nicknames for common errors (forgetting the constant, say).  Or use red chalk for the watch out tips.  But on purpose derailing the lecture (or even worse showing up unprepared and derailing by accident) is not a good move.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Understanding &quot;how research mathematicians research&quot; is crushingly irrelevant to most students through junior year of math classes (i.e. ALL service classes in uni and all HS math classes, and much of the ug math classes).  Don't think this is some wonderful gift you are sharing.  They will probably learn common issues of problem solving and research in the workplace more from their engineering co-op jobs.  For the math undergrad, they can have a course on math research methods and do a senior project.  Good enough.  &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-04-08T00:09:02.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13928" PostTypeId="2" ParentId="12133" CreationDate="2018-04-08T15:47:14.170" Score="0" Body="&lt;p&gt;Not sure the level of knowledge your students and what ODE they have had to date.  I am going to assume they have had some exposure (the baby DE you get towards end of a calculus class, but not a regular survey course).'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that, I would do predator prey.  (1) Likely to be new to them.  (2) visually interesting, the graphs and all.  (3) intuitively interesting problem (the eaters and the eaten), (4) is an ODE and connected rate of change problem. (5) easy example in terms of the actual manipulation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An alternate one to consider is xenon transients in nuclear reactors.  It's a real system of ODEs that affects navy nuclear cores (older worse, higher recent power worse).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://roko.eng.hokudai.ac.jp/studentadm/chiba_data/others/nre_g/20161031%20(2).pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://roko.eng.hokudai.ac.jp/studentadm/chiba_data/others/nre_g/20161031%20(2).pdf&lt;/a&gt; (see page 6 especially)&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-04-08T15:47:14.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13929" PostTypeId="2" ParentId="8577" CreationDate="2018-04-08T16:41:46.510" Score="1" Body="&lt;p&gt;I think Joel gave you the standard list.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Caveat one:  I am used to control systems courses (well OK...the single college survey course I took) being essentially DE reviews. 2nd order ODE is huge but also some LaPlace transforms, etc.  Maybe one different thing is doing some modeling and learning to draw those little block diagrams.  But still very mathematical and not even much real modeling/applications.  But perhaps text I had was bad this way (and if so, it was bad on purpose...they felt covering the math was more global and helpful in the future...I kinda disagreed since it seemed like such a repeat of DEs!)  In contrast, taking a practical mechE engineer/contractor course from Landys and Gyr, it was much more about listing numbers of control points, types of controllers (analog/digital, PID, etc.)  More about cataloguing I/O points so you can buy something and wire it.  There probably should be (probably is) something between those two extremes...but I don't know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Caveat two:  I don't think your 'has to be non complicated' and 'fine if it is nonlinear' are self consistent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer:  Some standard application areas (sorry I don't have all the details...would think they are in common texts):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Building HVAC or chemical process controls.  Your students will deal with in workplace, mechE/chemE.  These are huge real work areas so anything you can do here, even with simplified examples will resonate. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Military systems:  the dampened spring shock absorber of aircraft landing gear (very hard/important problem for example for heavy jet landing on an aircraft carrier).  Also swinging a gun from position A to position B (you want it to happen fast, but you also have to limit the final oscillations).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Everyday life:  A final interesting example is how hydraulic door closing mechanisms work--had a great chat with a smart janitor about how these work--there is a main spring, main damper and then even a little adjustment that gives it the &quot;final push&quot;.  You might even have one on the classroom door and be able to do some demonstration or messing with it...kids love stuff like that.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-04-08T16:41:46.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13932" PostTypeId="2" ParentId="13912" CreationDate="2018-04-09T12:30:31.093" Score="4" Body="&lt;p&gt;While the harmonic series shows us that $a_n$ tending to $0$ is not sufficient to guarantee convergence, the comparison and limit comparison tests are strong &quot;almost replacements&quot;: they justify the important idea that a series of positive terms will converge if its terms tend to $0$ quickly enough. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the typical examples of series appearing in a calculus course, the limit comparison test settles so many convergence questions so quickly (provided you already know the way geometric series and $p$-series work) that it would be a severe disservice to leave that topic out, almost like forcing students to use the limit definition of a derivative by withholding rules of differentiation from them. (Well, not that bad, but it would still be pretty bad.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The limit comparison test lets you discuss the effect of &lt;em&gt;growth rates&lt;/em&gt; on series convergence in a way the comparison test does not. Its purpose is to let you efficiently &lt;em&gt;simplify&lt;/em&gt; the terms under consideration: lower-order additive parts of the terms in a (positive) series can be dropped. That is the message of this test. Only having the comparison test available forces the student to deal with rather tricky inequalities (unless you deliberately water things down), suggesting to the student that the key ideas involved here are much more complicated than they need to be. For the engineering and physics students in your class, practice with the idea of neglecting all lower-order terms is worthwhile.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The comparison test itself should not be ignored, because the limit comparison test alone lets you drop lower-order additive pieces of the terms in a series, but how would you deal with $\sum 1/(n2^n)$? You can't use the limit comparison test here, and trying the comparison with both $1/n$ and $1/2^n$ is a nice, well, comparison to discuss. Something else to keep in mind, even if it is beyond the scope of the course, is that the proof of the limit comparison test &lt;em&gt;uses&lt;/em&gt; the comparison test.  Also, the proof of the ratio test &lt;em&gt;uses&lt;/em&gt; the comparison test with a geometric series; I point that out in class to indicate why the ratio test criteria sort of look like the geometric series convergence criteria. (This is also why it would be circular to use the ratio test to explain convergence of geometric series, like the circularity of using L'Hospital's rule to find the limit of $(\sin x)/x$ as $x \rightarrow 0$.) Concerning my example above of $\sum 1/(n2^n)$, it of course can be handled &quot;without thinking&quot; by using the ratio test, and doing this example first by the comparison test and later by the ratio test allows a sense of progress to better (or at least a wider range of) tools to be displayed to the class.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2018-04-09T12:40:50.423" LastActivityDate="2018-04-09T12:40:50.423" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13933" PostTypeId="2" ParentId="13903" CreationDate="2018-04-10T09:15:23.710" Score="4" Body="&lt;p&gt;i would add the &lt;a href=&quot;https://www.math.upenn.edu/~ghrist/calculus.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;FLCT (funny little calculus text)&lt;/a&gt; by robert ghrist. lovely done, focusing on ideas and concepts and still rigorous.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastEditorUserId="9004" LastEditDate="2018-04-13T13:31:15.967" LastActivityDate="2018-04-13T13:31:15.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13934" PostTypeId="2" ParentId="13712" CreationDate="2018-04-10T10:49:03.003" Score="2" Body="&lt;p&gt;Imagine the simple equation of a straight line: $ y=2x$.&lt;br&gt;&#xA;You want to move this up one point (in the $Y$-direction), and you get: $y=2x+1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to move this up one point in the $X$-direction, then you first need to write this with $x$ at one side: $x=y/2$.&lt;br&gt;&#xA;So you get : $x=y/2+1$.&lt;br&gt;&#xA;Writing this back to the $y=f(x)$ notation becomes: $y=2(x-1)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the appearance of the minus is a simple consequence of the fact that we write our functions in the notation $y=f(x)$.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2018-04-10T10:49:03.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13935" PostTypeId="1" AcceptedAnswerId="13936" CreationDate="2018-04-10T15:22:41.867" Score="2" ViewCount="1005" Body="&lt;p&gt;I am familiar with the tombstone symbol, &quot;$\blacksquare$&quot;, that is used to signify the end of a proof.  However, it is my understanding that an example isn't technically a proof. For instance, one can't just find an example of a proposition being true and then claim the proposition to be true. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I was wondering if there is a symbol that should be used to signify the end of a counterexample?&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2018-04-10T16:09:26.457" Title="What is the correct symbol to use for ending a counterexample?" Tags="|proofs|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13936" PostTypeId="2" ParentId="13935" CreationDate="2018-04-10T16:09:26.457" Score="10" Body="&lt;p&gt;Two answers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First: These symbols are not standardized. You will find all kinds of &quot;end of ...&quot; symbols, say, black squares, white squares, black triangles, or any symbol that's contained in a dingbat font. Some authors distinguish between the symbols for &quot;end of proof&quot;, &quot;end of example&quot;, and &quot;end of definition&quot;, some authors use the same symbol everywhere, and some authors mark only the end of proofs, but not the end of examples or definitions. In other words: Do what you like, or whatever the publisher's style guide requires.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second: Technically, a &lt;strong&gt;counter&lt;/strong&gt;example &lt;em&gt;is&lt;/em&gt; a proof of the negation of the assumption. A counterexample for &quot;for every x and y there is a z such that xz = y&quot; is a proof for &quot;not for every x and y there is a z such that xz = y&quot;, or, if you push the negation inside, &quot;there exist x and y such that for every z we have xz ≠ y&quot;.&lt;/p&gt;&#xA;" OwnerUserId="8472" LastActivityDate="2018-04-10T16:09:26.457" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13937" PostTypeId="2" ParentId="13924" CreationDate="2018-04-10T17:01:18.863" Score="9" Body="&lt;p&gt;While there is a logical symmetry between differential equations and integral equations, it seems that (as they say) &quot;laws of nature&quot; are written in differential equations, not integral equations. As far as I can tell, rigorous study of integral equations arose in the late 1800's in part because of the relative difficulty of rigorous study of (especially partial) differential equations. That is, bounded operators (e.g., inverses of differential operators!), especially compact operators, were far easier to prove theorems about than unbounded operators (differential operators).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, people &lt;em&gt;using&lt;/em&gt; differential equations throughout the 19th and 20th centuries often were not slowed-down by worries about mathematical niceties (and very often reached correct conclusions with or without &quot;proofs&quot; whose legitimacy could be understood in the context of the times). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, integral equations seem mostly to be of interest either as a (temporary?) substitute for differential equations, or as a technical warm-up for rigorous treatment of differential equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The relatively more-recent Schwartz' Kernel Theorem, Grothendieck's &quot;nuclear spaces&quot; (and pseudo-differential operators, singular integral operators, etc.) can certainly be viewed as extrapolations of ideas about integral equations, but it is my impression that people &lt;em&gt;applying&lt;/em&gt; such ideas are also less finicky about distinguishing &quot;generalized functions&quot; (as in Schwartz' distributions) from classical functions, so may consider Schwartz' Kernel theorem as &quot;obvious&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, perhaps, we don't teach introductory integral equations along with introductory differential equations because the differential equations have the more direct applications, and because the greater ease of rigorizing integral equations is irrelevant.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-04-10T17:01:18.863" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13938" PostTypeId="2" ParentId="13903" CreationDate="2018-04-10T17:55:09.497" Score="0" Body="&lt;p&gt;I'm currently working on a Calculus book and solution guide that is much less formal, but no less deep.  The working title is &quot;Calculus from the Ground Up&quot;.  In fact, in some ways, it goes deeper than other calculus books, but without straining to do so.  It works using infinitesimals informally.  It goes onto a more formal explanation of hyperreals and hyperreal arithmetic, but only towards the end, after the student has gotten used to the idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anyone is interested, feel free to email me at jonathan@bplearning.net and I'll send you a PDF of the current version.  I would love any feedback on it!&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2018-04-10T17:55:09.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13939" PostTypeId="2" ParentId="7199" CreationDate="2018-04-11T00:47:06.417" Score="4" Body="&lt;p&gt;The MAA (Math Assoc Amer) maintains a collection of resources at the&#xA;website for&#xA;&lt;a href=&quot;https://www.maa.org/press/periodicals/convergence&quot; rel=&quot;nofollow noreferrer&quot;&gt;Convergence&lt;/a&gt;&#xA;to support teaching math via its history.&#xA;They have a link to a &quot;Treasures&quot; list which includes facsimiles&#xA;of many, many historical documents.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/vwdaP.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vwdaP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;The &quot;Witch of Agnesi&quot;: Maria Gaetana Agnesi's cubic curve.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;The publication includes articles that could each form a classroom module,&#xA;e.g.,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Nicholas A. Scoville, &quot;Connecting Connectedness: A Mini-Primary Source Project for Topology Students.&quot; 2016.&#xA;  (&lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/connecting-connectedness-a-mini-primary-source-project-for-topology-students&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-04-11T00:52:41.510" LastActivityDate="2018-04-11T00:52:41.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13941" PostTypeId="2" ParentId="2376" CreationDate="2018-04-11T05:28:07.027" Score="7" Body="&lt;p&gt;When I taught high school, I offered each students a choice between doing two different worksheets, making it clear that one worksheet had many problems at a low level for lots of practice and only a few challenge problems, and that the other worksheet had fewer repetitive problems and many challenge problems. My students generally self-selected the worksheet that was appropriate for their level. This had the added benefit of having my gifted students delve &quot;deeper&quot; in to the material we were currently learning, instead of zooming ahead with strong intuition but possibly weaker procedural skills. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To avoid having them call on you constantly during this time, ask those students who are doing the challenging worksheet to work together, and only call on you when everyone in their group of 3-4 students is stuck. This promotes collaboration and ensures that you spend the majority of your time on helping the students who need more directed guidance.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2018-04-11T05:28:07.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13944" PostTypeId="2" ParentId="1245" CreationDate="2018-04-11T20:41:55.433" Score="0" Body="&lt;p&gt;In terms of does a sum converge, I motivate by comparing growth rates of functions.  So powers of n grow slower than exponential functions of n, which in turn grow slower than factorials (as n goes to infinity). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \Sigma \frac{2^n}{n!}$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;will converge simply because the denominator has the faster growth rate.  Using this idea, my students are able to determine if a series will converge just by looking at it.  They then use a convergence test to confirm their intuition.  They also know what to compare t, since they know to ignore slowly changing terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a bonus they also learn about growth rates of functions which has a lot of applications.&lt;/p&gt;&#xA;" OwnerUserId="9815" LastActivityDate="2018-04-11T20:41:55.433" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13945" PostTypeId="2" ParentId="13912" CreationDate="2018-04-12T14:52:13.733" Score="1" Body="&lt;p&gt;I would like to give the argument for option (3). Even though I am not necessarily a strong supporter of it, my calculus book does it and I think it's fine to do so: to &lt;strong&gt;present the Direct Comparison Test and skip the Limit Comparison Test&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The most common argument against this is that students need to be too clever to get the right estimates, but it does not require too much cleverness if presented carefully:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First try to guess what the answer is by looking at &amp;quot;important terms.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Use this guess to choose either &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\geq$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;For each &amp;quot;unimportant&amp;quot; term, either downgrade the term or upgrade it to be important.&lt;/li&gt;&#xA;&lt;li&gt;At the end, make sure that the comparison test gives a result.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For this question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Find whether &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=2}^\infty \frac{4n^2 - 5}{7n^4 - 5}$&lt;/span&gt; converges or diverges using the comparison test.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A student's thought process would then be:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Guess: &lt;span class=&quot;math-container&quot;&gt;$\sum \frac{n^2}{n^4}$&lt;/span&gt; converges because it is a p-series &lt;span class=&quot;math-container&quot;&gt;$p=2&amp;gt;1$&lt;/span&gt;, so we try to show it converges.&lt;/p&gt;&#xA;&lt;p&gt;So start with &lt;span class=&quot;math-container&quot;&gt;$\frac{4n^2 - 5}{7n^4 - 5} \leq$&lt;/span&gt; .&lt;/p&gt;&#xA;&lt;p&gt;Can I drop the 5 on top? &lt;span class=&quot;math-container&quot;&gt;$\frac{4n^2 - 5}{7n^4 - 5} \leq \frac{4n^2}{7n^4-5}$&lt;/span&gt; Yes, that's fine.&lt;/p&gt;&#xA;&lt;p&gt;Can I drop the other 5? &lt;span class=&quot;math-container&quot;&gt;$\frac{4n^2 - 5}{7n^4 - 5} \leq \frac{4n^2}{7n^4-5}  \leq \frac{4n^2}{7n^4}$&lt;/span&gt;  No, that's not true.&lt;/p&gt;&#xA;&lt;p&gt;So I'll upgrade it instead: &lt;span class=&quot;math-container&quot;&gt;$\frac{4n^2 - 5}{7n^4 - 5} \leq \frac{4n^2}{7n^4-5}  \leq \frac{4n^2}{7n^4 - 5n^4}$&lt;/span&gt; which is true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then the student's solution is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{4n^2 - 5}{7n^4 - 5} \leq \frac{4n^2}{7n^4-5}  \leq \frac{4n^2}{7n^4 - 5n^4} = \frac{4n^2}{2n^4} = 2\frac{1}{n^2}$&lt;/span&gt;. The series 2 &lt;span class=&quot;math-container&quot;&gt;$\sum\frac{1}{n^2}$&lt;/span&gt; converges because it is twice a p-series with &lt;span class=&quot;math-container&quot;&gt;$p&amp;gt;1$&lt;/span&gt;, and the original series is smaller, so it converges by the comparison test.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My experience is that this kind of solution is very reachable by Calc 2 students, as it has been reduced to a &amp;quot;process&amp;quot; that can be digested. It also still requires students to use their intuition to guess at the answer (to decide whether to start with &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\geq$&lt;/span&gt;) but then makes it clear that &lt;strong&gt;your intuition is not the entire solution&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Separating out that the student's intuition is different from the student's solution is likely the key thing we are getting out of teaching series tests at all. So this approach seems somewhat elegant to me.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-04-12T17:10:33.330" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13946" PostTypeId="2" ParentId="1245" CreationDate="2018-04-12T17:25:58.213" Score="5" Body="&lt;p&gt;I start on the other end of the spectrum.  I start off by &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Noting that computers can't compute sine, cosine, or anything like it.  Computers can basically only compute polynomials.&lt;/li&gt;&#xA;&lt;li&gt;Sine and cosine can't be written in terms of polynomials&lt;/li&gt;&#xA;&lt;li&gt;However, we can get around #2 if we use an &lt;em&gt;infinite&lt;/em&gt; number of terms.  I point out how the degree of polynomial tells you how many humps you have, and so, to get something with infinite humps (like sine/cosine) you certainly need polynomials.&lt;/li&gt;&#xA;&lt;li&gt;Then, we go over simple series for basic functions with MacLauren and Taylor series.&lt;/li&gt;&#xA;&lt;li&gt;Then, I show that we can compute impossible integrals using infinite series (like integral of cos(x^2))&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So, at this point, they have a basic set of reasoning skills about infinite series.  They know why they are important (it allows you to perform calculations using computers, and it allows you to do previously-impossible integrals).  So, now they know the importance of series.  Once they know the importance of series, they can learn their limitations.  This is the point where you can teach about convergence and divergence.  For instance, you can use the Taylor series for natural log about 1, and show, using information about the rules for series, why this series only succeeds for a very small domain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the notion of &lt;em&gt;starting&lt;/em&gt; with a bunch of abstract concepts about series is a severe turn-off.  If you start with why we care about series, and start using series, you can then back up and talk about conditions required for the series to converge to an actual value.  This keeps students attention much better than the standard approach.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2018-04-12T17:25:58.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13949" PostTypeId="1" AcceptedAnswerId="13950" CreationDate="2018-04-14T13:41:15.187" Score="5" ViewCount="451" Body="&lt;p&gt;A close friend of mine is investigating Associate Degrees in Mathematics with the goal of assessing the plausibility of offering an online A.A.S. at a US institution. I'm curious if anyone here has any experience with such a degree and/or ideas about how to best set-up such a degree. My knee-jerk reaction to the question is that such a 2-year degree would mainly be intended as a transitional degree with the aim of entering a 4-year program in:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;engineering&lt;/li&gt;&#xA;&lt;li&gt;actuarial &lt;/li&gt;&#xA;&lt;li&gt;mathematics&lt;/li&gt;&#xA;&lt;li&gt;finance&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;With these goals in mind I would expect Calculus I, II and possibly III should be required as well as an entry-level linear algebra course. Probably Calculus-based statistics and discrete math would be wise as well. I would also want differential equations. It might be the A.A.S. should allow several varieties depending on the targeted transition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, thoughts on how to set-up an A.A.S in Math or experience you have in dealing with such a program would be quite helpful to my friend. Thanks!&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-04-14T16:25:33.647" Title="Associate Degree in Mathematics" Tags="|curriculum|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13950" PostTypeId="2" ParentId="13949" CreationDate="2018-04-14T16:25:33.647" Score="7" Body="&lt;p&gt;Yes, the A.S. in Mathematics at my community college is pretty much exactly as you expect (I'm the head of assessment for this particular academic degree; CUNY/Kingsborough in NYC, USA). &lt;a href=&quot;http://www.kbcc.cuny.edu/academicdepartments/math/Pages/Requirement.aspx&quot; rel=&quot;noreferrer&quot;&gt;See requirements here&lt;/a&gt; (must click through to PDF; apologies for it not being HTML). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One corrected observation: The number of students taking such a degree is miniscule (maybe the smallest single major at our college?). This means that we can't support any course with only math majors as the clientele; every course must be a support course with mostly engineering and computer science majors, etc. There isn't any flexibility for offering different varieties of the major (which would instantly resolve as one-per-individual-student). There is no calculus-based statistics at our community college, or others with which I'm familiar. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-04-14T16:25:33.647" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13951" PostTypeId="1" CreationDate="2018-04-14T21:31:13.167" Score="6" ViewCount="303" Body="&lt;p&gt;let $f$ and $g $ be two real valued function , I have asked many students what is the derivative of $(fg)'$ they answered me :it is $f' \cdot g'$, then I seek why most people (students) guess that ?&lt;/p&gt;&#xA;" OwnerUserId="7583" LastEditorUserId="4791" LastEditDate="2018-04-15T00:59:44.637" LastActivityDate="2018-04-15T11:05:53.660" Title="Why most people think that :$(fg)'=f' \cdot g'$?" Tags="|undergraduate-education|derivative|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="13952" PostTypeId="2" ParentId="13951" CreationDate="2018-04-14T21:55:10.650" Score="-2" Body="&lt;p&gt;A. Maybe because the dash is in the exponent place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B.  It's not a bad guess, if you had little info and had to make a quick guess.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C.  Humans are not computers and reason by analogy, first.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;On the practical side, who cares?  Will knowing why people guess that make the problem go away that you need to train them on the power rule?  That there is no quick fix to this sort of insight (some quick key that unlocks a puzzle and means kids don't need to do examples and homework and analogies to other functions (e.g. log or sin) where linearity does not hold)?&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-04-14T21:55:10.650" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="13953" PostTypeId="2" ParentId="13951" CreationDate="2018-04-15T11:05:53.660" Score="8" Body="&lt;p&gt;As mentioned, a probable cause is an implicit reasoning as if every operation were a homomorphism (similar to the implicit reasoning by linearity). Similar errors include $\ln(x+y) = \ln(x)+\ln(y)$, $e^{xy}=e^x e^y$, $\int f(x)g(x) dx = \int f(x) dx \int g(x) dx$, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such reasoning by (very loose) analogy can be caused by not understanding that whenever some similar rules do hold, they do for a reason. In particular, it is important to actually explain why $e^{x+y}=e^x e^y$ (at least when $x,y\in\mathbb{N}$), why $(f+g)'=f'+g'$, etc. So the answer to give to the error you mention implies discussing &lt;em&gt;other&lt;/em&gt; formulas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that one can also do is to show how erroneous formulas lead to obvious wrongs. For example, if it where true that $(fg)'=f'g'$, then taking $f=1$ would lead to $g'=0$ whatever $g$ actually is.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2018-04-15T11:05:53.660" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13954" PostTypeId="1" AcceptedAnswerId="13957" CreationDate="2018-04-15T11:40:30.897" Score="2" ViewCount="166" Body="&lt;p&gt;Throughout my undergrad, I dreaded probability. I hated it, I was horrible in it, I just never got it, and felt stupid when the professors used &quot;summation/marginalization&quot; equations out of the blue to get rid of variables here and there. I just didn't understand it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I started my Masters degree, I was dreading that one Statistics/Prob course that I have to take, but then I was introduced to our lord and savior Bayes, and to be specific, Bayesian Networks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It suddenly all made sense, all these equations that I suffered with, they were mostly variable eliminations on special Bayes Networks. Everything started clicking so much and by the end of the course I was dying to take more prob/stats.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my questions boils down to, why isn't Bayesian Networks and Variable elimination introduced in undergraduate courses? I've looked at the vast majority of undergrad stat/probability courses in the top 10 us universities and none of them come close to explaining it, even though I feel it makes probability a whole lot easier and understandable!&lt;/p&gt;&#xA;" OwnerUserId="7538" LastActivityDate="2018-04-16T20:12:27.943" Title="Why aren't Bayesian Networks and Variable Elimination introduced earlier?" Tags="|undergraduate-education|statistics|probability|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13955" PostTypeId="1" AcceptedAnswerId="13958" CreationDate="2018-04-16T12:26:59.587" Score="3" ViewCount="609" Body="&lt;p&gt;This is a reduced quote from &lt;a href=&quot;https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;There’s more to mathematics than rigour and proofs&lt;/a&gt; of Terrence Tao (emphasis mine):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. The emphasis is more on computation than on theory. &lt;strong&gt;This stage generally lasts until the early undergraduate years.&lt;/strong&gt;&lt;/li&gt;&#xA;  &lt;li&gt;The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner. The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. &lt;strong&gt;This stage usually occupies the later undergraduate and early graduate years.&lt;/strong&gt;&lt;/li&gt;&#xA;  &lt;li&gt;The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. The emphasis is now on applications, intuition, and the “big picture”. &lt;strong&gt;This stage usually occupies the late graduate years and beyond.&lt;/strong&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I ask a person on Reddit, and &lt;a href=&quot;https://www.reddit.com/r/math/comments/8bpai4/taking_notes_is_the_art_of_making_poetry_in/dxc49f2/?st=jg3edtew&amp;amp;sh=aabe1d77&quot; rel=&quot;nofollow noreferrer&quot;&gt;they say that&lt;/a&gt; (emphasis theirs):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;The rigorous stage, which undergrads &lt;em&gt;never&lt;/em&gt; get past&lt;/li&gt;&#xA;  &lt;li&gt;Post-rigorous only happens to the rare person who succeeds in grad school and begins doing actual high-level &lt;em&gt;unguided&lt;/em&gt; research.&lt;/li&gt;&#xA;  &lt;li&gt;There is no post-rigorous stage of grade 12 math, it's simply pre-rigorous end of story. You may very well completely understand it, but that has nothing to do with whether or not you've reached a level of rigor in your thinking.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Why do such stages have specific timestamps but not relative? And is this math-only? If an undergraduate can sketch out a proof of an grade 12 student, why doesn't this mean they are in post-rigorous stage of the grade 12? I think what Tao means simply about the requirements to &lt;em&gt;contribute new knowledge&lt;/em&gt;. But if one solves enough problems on Pythagoras triangle and is ready to move to the next topic, can we say that they have reached the post-rigorous stage on that &quot;field&quot;? A vehicle technician may have never been to college, but they can very well know what's wrong with an engine by just passingly hearing its sound.&lt;/p&gt;&#xA;" OwnerUserId="8875" LastEditorUserId="8875" LastEditDate="2018-04-17T08:28:56.600" LastActivityDate="2018-04-17T08:28:56.600" Title="Why do the stages of rigorousness have specific timestamps?" Tags="|undergraduate-education|graduate-education|intuition|undergraduate-research|problem-solving|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="13956" PostTypeId="1" AcceptedAnswerId="13966" CreationDate="2018-04-16T19:46:22.217" Score="4" ViewCount="967" Body="&lt;p&gt;I've just started teaching congruent triangles to a class of 14/15 year olds in the UK. All that they are required to know for the purpose of national exams here is that two triangles are congruent if they have the same size and shape (obviously not a precise definition) and how to write out SSS / SAS / ASA / RHS proofs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read in a few sources that congruence is often badly taught in schools, and that most students do not have a very strong grasp of what congruence is. For example, most students have no idea why SSS / SAS / ASA etc are sufficient for two triangles to be congruent. Maybe they can convince themselves by drawing pictures but that doesn't demonstrate they fully understand. This bothers me - what is the point of learning how to prove two triangles are congruent if one does not understand why the proof works? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Am I doing my students a disservice if I omit any discussion of why the proofs work? That's what most teachers here do. Is there a nice way to explain this without introducing lots of other precise definitions and proving the relevant theorems? &lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2018-04-20T04:48:39.607" Title="Teaching congruent triangles non-rigorously" Tags="|secondary-education|geometry|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13957" PostTypeId="2" ParentId="13954" CreationDate="2018-04-16T20:12:27.943" Score="3" Body="&lt;p&gt;(Expanded from my comment.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The short answer is: because monads aren't burritos.  That's a reference to &lt;a href=&quot;https://byorgey.wordpress.com/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;, which I highly recommend.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We've all had the experience of getting one new perspective that makes a previously mysterious subject suddenly click into place, and it's tempting to credit the understanding to that idea and to be frustrated that no one explained it so clearly before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, often it was really all the work and thinking that went into trying to understand the subject before that prepared us to understand the topic; when that's the case, getting the great explanation earlier wouldn't have led to a breakthrough - it would have been another seemingly unhelpful explanation (which actually secretly made progress towards understanding) - and some later idea would have led to the breakthrough.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-04-16T20:12:27.943" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13958" PostTypeId="2" ParentId="13955" CreationDate="2018-04-17T03:50:50.970" Score="3" Body="&lt;p&gt;I don't see any specific timestamps in the piece quoted.  There are some vague time frames given, all marked by phrases like &quot;generally&quot; and &quot;usually&quot;, which indicates that they're far from universal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, I read those time frames as being intended as clarification - they're there to help readers understand the stages by pointing out when they typically happen.  For instance, it helps to understand what Tao means by stage 1 to compare it to our own observations of typical high school and early undergraduate math.  But it only helps to compare it to what's &lt;em&gt;typical&lt;/em&gt; at those levels - there's no claim that it's a universal description of every mathematical activity by every person at those levels.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-04-17T03:50:50.970" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="13959" PostTypeId="2" ParentId="13956" CreationDate="2018-04-17T04:04:39.310" Score="1" Body="&lt;p&gt;My experience with trying to focus on rigor with slightly older students (18-19 years old) is that they don't get much out of it when 1) they don't see what they get out of rigor that plain old &quot;memorizing what the teacher says didn't get them&quot;, 2) they don't have the skills to distinguish between rigorous and non-rigorous argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The later, in particular, causes a lot of problems when trying to introduce rigorous content, especially if it's done piecemeal by dropping a few rigorous proofs into a largely formulaic course.  Since students can't tell the difference between rigorous reasoning, loosely correct but non-rigorous reasoning, and plainly incorrect reasoning, the whole project becomes a game of &quot;find the thing the teacher agrees is rigorous enough&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue that it's more important, at that stage, to be setting the groundwork for appreciating the value of proofs and informal reasoning.  For example, that's a great topic to begin thinking about counterexamples - students are likely to consider the idea of an SSA rule, which can be refuted by an example.  That can then raise the question of how one knows that the SAS rule doesn't have a counterexample.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This isn't based on concrete experience teaching geometry, but based on what I've seen with other subjects, I would expect students to get more out of trying and failing to find counterexamples to the rules that work, and perhaps finding informal ways to express why they can't seem to find a counterexample, than they would out of seeing a formal proof.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-04-17T04:04:39.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13960" PostTypeId="1" AcceptedAnswerId="13961" CreationDate="2018-04-17T09:53:33.107" Score="4" ViewCount="303" Body="&lt;p&gt;In order to explain Euler's notation of the triangle, it is desirable to use a triangle that does not look equilateral or rectangular or obtuse. There are well known ways to draw a parabola, but I personally have never heard whether tricks have come down from famous teachers to draw a &quot;typical&quot; triangle freehand on the blackboard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &quot;theoretical&quot; treatment of the general triangle is given in Friedrich Wille's book (in German) Humor in der Mathematik.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://mathoverflow.net/questions/128716/what-is-the-best-general-triangle&quot;&gt;https://mathoverflow.net/questions/128716/what-is-the-best-general-triangle&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8557" LastEditorUserId="8557" LastEditDate="2018-04-17T12:01:21.507" LastActivityDate="2018-04-17T12:01:21.507" Title="Tricks to draw a typical triangle?" Tags="|presentation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13961" PostTypeId="2" ParentId="13960" CreationDate="2018-04-17T11:15:25.647" Score="3" Body="&lt;p&gt;How about drawing a straight line, then fixing a point above that line that is not directly above the middle, and take this as the third vertex? This way, the two upper sides will not have the same length; and the chance that one of them has the same length as the straight line you drew at random is very low.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;edit: Looking at Thales theorem might help to avoid angles right angles.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2018-04-17T11:15:25.647" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13962" PostTypeId="1" CreationDate="2018-04-17T13:44:36.110" Score="3" ViewCount="361" Body="&lt;p&gt;Many universities in Germany have special math courses for engineers. These courses are often called &quot;Höhere Mathematik&quot; (higher mathematics) or &quot;Ingenieursmathematik&quot; (engineering mathematics). The content and depth varies a little, but usually they contain&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;one variable calculus (limits, series, differentiation, integration), quite rigorous&lt;/li&gt;&#xA;&lt;li&gt;linear algebra (general vector spaces, linear system, eigenvalues, Gauß elimination)&lt;/li&gt;&#xA;&lt;li&gt;ODEs (including elementary methods, Picard iteration, linear systems)&lt;/li&gt;&#xA;&lt;li&gt;vector calculus (including the divergence theorem and Stokes theorem, Lagrange multipliers)&lt;/li&gt;&#xA;&lt;li&gt;complex analysis, Fourier and Laplace transforms&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some example of books that cover the range are &lt;a href=&quot;https://www.amazon.de/Mathematik-Ingenieure-Naturwissenschaftler-Arbeitsbuch-Grundstudium/dp/3528942363&quot; rel=&quot;nofollow noreferrer&quot;&gt;Papula's&lt;/a&gt; or &lt;a href=&quot;https://www.amazon.de/H%C3%B6here-Mathematik-Rezepten-zahlreiche-Lerneinheiten/dp/3662548089/ref=sr_1_1?s=books&amp;amp;ie=UTF8&amp;amp;qid=1523972314&amp;amp;sr=1-1&amp;amp;keywords=karpfinger&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karpfinger's&lt;/a&gt; books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of my students are international and have started to learn German. They would appreciate an English textbook, but I am not sure what is appropriate. I found the two volumes by Tom Apostol quite good, but still look for more recommendations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So:&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;What are good books for international engineering students in a math course in Germany?&lt;/h3&gt;&#xA;" OwnerUserId="3398" LastEditorUserId="376" LastEditDate="2018-04-17T19:02:26.580" LastActivityDate="2018-04-18T07:46:37.463" Title="English book for math for (electrical) engineering similar to German &quot;Higher Mathematics&quot;" Tags="|textbooks|engineering-mathematics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13963" PostTypeId="2" ParentId="13962" CreationDate="2018-04-17T14:36:00.760" Score="1" Body="&lt;p&gt;Maybe E. Kreyszig, &lt;em&gt;Advanced Engineering Mathematics&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2018-04-17T14:36:00.760" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13964" PostTypeId="2" ParentId="13962" CreationDate="2018-04-17T16:04:13.703" Score="1" Body="&lt;p&gt;One option is &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0130268585&quot; rel=&quot;nofollow noreferrer&quot;&gt;Engineering Mathematics: A Foundation for Electronic, Electrical, Communications, and Systems Engineers&lt;/a&gt; by Croft et al. I have taught using this for several years. It is replete with electrical engineering examples and applications, and includes material on such subjects as the Z-transform and Fourier transform, in addition to Fourier series and the Laplace transform. That said, though, it does not usually include proofs and the coverage of some topics may be too light for your purposes. For instance, complex numbers get a chapter, but there's no complex analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note that link above is to the third edition, with some reviews and a detailed table of contents available. The fifth edition was recently published and has a briefer table of contents &lt;a href=&quot;http://catalogue.pearsoned.co.uk/educator/product/Engineering-Mathematics/9781292146652.page&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. In both the fourth and fifth editions, the main change I've noticed is that engineering applications have been added or made more prominent. A couple of sections have also been added - for instance, one on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Sinc_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;sinc&lt;/a&gt; function and one introducing Bessel functions.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2018-04-18T07:46:37.463" LastActivityDate="2018-04-18T07:46:37.463" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13965" PostTypeId="2" ParentId="10488" CreationDate="2018-04-17T16:16:47.900" Score="1" Body="&lt;p&gt;Other answers have concentrated on the base, but the problem can also be ascribed to the exponents. $(x^m)^n=x^{mn}$ is true for any base, as long as m and n are integers. This follows simply from the definition of exponentiation as repeated multiplication.  $x^m$ is m copies of x multiplied together. $(x^m)^n$ is n copies of (m copies of x multiplied together) multiplied together, and so is mn copies of x multiplied together. This works for any base, integer, rational, irrational, positive, negative, complex. In fact, it works in any ring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem comes when we try to extend the concept of exponents to non-integers. If we want to define $y = x^{\frac{1}{n}}$, we can take $y^n$ and extend the rule to say $y^n = (x^{\frac{1}{n}})^n =x^{\frac{1}{n}n}=x$. With this extension, y is a number such that $y^n$ = x. This is a description of y, but it's not a &lt;em&gt;definition&lt;/em&gt; of y, as it not does unique specify y. Thus, when we take $(x^2)^{\frac{1}{2}}$, we are finding a number that when squared is equal to $x^2$. And while it's tempting to look at the phrase &quot;a number that when squared is equal to $x^2$&quot; and say &quot;Well, that's a rather complicated way of saying 'x' &quot;, x isn't the only number that satisfies this description. Thus, we have to make qualifications such as &quot;$x^{\frac{1}{n}}$ is the &lt;em&gt;principal&lt;/em&gt; nth root of x&quot;, and its those qualifications that cause the rule to fail with negative numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, while we can save this rule by saying &quot;x has to be a positive number&quot;, we can also save it by saying &quot;m and have to be integers&quot;. Or combine these qualifications and say &quot;This works for non-integers only when we have positive numbers&quot;.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2018-04-17T16:16:47.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13966" PostTypeId="2" ParentId="13956" CreationDate="2018-04-17T16:28:40.267" Score="4" Body="&lt;p&gt;The main points are that knowing a length restricts you to a circle, and knowing an angle restricts you to a ray. So for SSS, you can draw one side, then for each endpoint draw a circle for the adjacent side length. Those two circles should intersect at two points, given two different, but congruent, triangles. For SAS, you can draw an side, then draw a ray from one of the endpoints, and a circle centered around the same endpoint. Obviously the circle and ray will intersect at only one point. For ASA, you can draw a side, and then rays at both endpoints. Obviously those rays intersect at one point. For SSA, you can draw a side, then draw a circle centered at one endpoint, and a ray from another. The circle and ray will intersect at two points, given two non-congruent triangles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is about as rigorous as you can expect your students to follow, and it's a sufficient level of understanding. Using this, they should be able to re-derive the rules, rather than just rote memorizing which ones work and which don't.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2018-04-17T16:28:40.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13967" PostTypeId="2" ParentId="13962" CreationDate="2018-04-17T21:36:49.717" Score="5" Body="&lt;p&gt;(edited)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have used Kreyszig, Advanced Engineering Mathematics, fifth edition, as an undergrad, a practicing engineer, and in grad school.  I love it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(1)&lt;/strong&gt; It is in English, composed in English, good English, but &lt;strong&gt;written by a German native speaker&lt;/strong&gt; mathematician.  Not sure that this matters but might be positive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(2)&lt;/strong&gt; &lt;strong&gt;One of high repute&lt;/strong&gt;, check out his books on differential geometry or functional analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(3)&lt;/strong&gt; It is written by a mathematician and better than that, &lt;strong&gt;a math TEACHER.&lt;/strong&gt;  So while he is (was, as he is dead) capable of hard core math like above, he has a good feel for EE students, for writing enough text but not too much, for good problems (not just a few hard ones, but lots of different hardness, and plenty that just allow trying the new concepts to get familiarity).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(4)&lt;/strong&gt; &lt;strong&gt;Real mathematician as opposed to practitioner.&lt;/strong&gt;  This can be advantageous over math books written by physicists or engineers.  (They tend to be more confusing...and it is easier to cover some of the math within physics from that physics slant, but then to get it from a math slant in math class...makes things click to see it from both sides: vector calc in Wangness E&amp;amp;M versus 3rd semester calc class, kinematics in college physics versus derivates/integrals in calc 1/2 classes.)  The one example of a practioner book that I own is Arfken Weber for physics:  has some overlap, but a little more advanced...but very spotty and poorly written...read the negative reviews on Amazon.     &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(5)&lt;/strong&gt;  While written by a mathematician, it is a mathematician with both an &lt;strong&gt;familiarity of engineering/science and a sympathy to engineers and scientists.&lt;/strong&gt;  This can be in contrast to many mathematicians and math teachers who lack strong general engineering/science education or sympathy.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(6) coverage&lt;/strong&gt; includes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;a. ODE:&lt;/strong&gt; basic, efficient, coverage in five chapters, 250 pages (excepting systems covered in LA part and numeric methods covered in NM part (see below).  All the topics in longer books (first and second order ODEs, series method, Bessel functions, Sturm Louisville, phase plane, Laplace Transform, etc.), but more efficiently (IMO).  Most reviewers are high on this part of the book as just well done.  [Note however, that in the US, it is common for engineering students to cover this with a specialized text and dedicated ODE course.  But it is in there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;b. Linear algebra:&lt;/strong&gt; two chapters, 100 pages.  Efficient, engineer's coverage of matrices and key operations.   In my experience, many engineers and physicists get LA only within their &quot;math methods&quot; or &quot;engine math&quot; course.  But some of course have dedicated courses (as with all the topics).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;c.  Vector calculus:&lt;/strong&gt;  Two chapters, 100 pages.  Good, efficient coverage.  Note that in the US, this is normally covered in 3rd semester calculus, usually with a book that covers calc 1/2/3 (Thomas, Swokowski, etc.).  However, K has it as a topic on it so it matches your need. But I had covered ODE and VC previous to this book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;d.  Fourier series and PDEs.&lt;/strong&gt;  A chapter on each, 100 pages total.  K covers F series first, which I quite like as there are a lot of applications BEFORE you get to PDEs.  And it just seems a little easier to cover it this way straight up versus as an evolution of harder PDE problems.  I think this is nice for EEs also who use FS a lot (as the en passent coverage in typical PDE book may not be optimal.)  The PDE chapter covers all the major topics (heat equation, circular membrane, blabla).  However it is just a single chapter with 13 sections.  I think it is good for an engineers treatment and as part of a math methods course.  But as with any of these topics, you can say maybe you should get a book on each topic instead.  That said, I find the PDE section very straight forward.  Not too applied (PDEs can be confusing because of the strong emphasis on applications...and kids will see that slant in their thermo classes regardless.)  But also not too rigorous as to be annoying.  Bessel functions are revisited within PDE chapter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;e.  Complex analysis:&lt;/strong&gt;  Five chapters, 200 pages:  mathematical but an emphasis on topics of interest to engineers (mapping, contour, residue blabla, etc....and derivations as motivation but not proof as the main emphasis.)  Series are covered strongly in a chapter here as well.  There is some &quot;real analysis&quot; (like the series) but more useful to engineers &quot;advanced calculus&quot;. Not annoying topology proof stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;f.  Probability and statics:&lt;/strong&gt; One chapter, 100 pages.  I didn't use this section, had a separate class for it.  I think a lot of math methods courses would not use this section.  But it's in there.  Looks straightforward and clean and fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;g. Numerical methods and optimization:&lt;/strong&gt; Two chapters 60 pages. Good section on different methods and error analysis.  LP and simplex method in optimization.  Looks fine.  Never used it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Coverage note:&lt;/strong&gt;  First and second semester calculus were NOT covered AND considered a prereq.  Epsilon-delta was not covered either.  In the US, epsilon-delta is typically covered (quickly) within normal calc course...so you get an exposure...but then you move to various manipulations and applications (partial fractions integration, min/max problems etc.)  I would also add that I never felt my lack of MORE epsilon-delta awareness stopped me from doing hard calculus problems through EE, physics (even some grad courses).  I bet Feynman felt the same way.  (I bet Euler did too!)  Now Bessel functions is a different story...those darned Yo and Jo things are everywhere.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also normal in the US for the first year calc course to have (towards course end) a short ODE section and a short sequence/series coverage.  [This is helpful for students who stop here with math.  However, engineers will normally get a full ODE course as a requirement later.  Deeper knowledge of series divergence and the like may not be a required course.]  But K doesn't require any carryover of this &quot;baby ODE&quot; or &quot;baby series&quot; knowledge (nor do other books in general in the US...that just ends up being review for the kids.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book does not cover calculus of variations or group theory.  (Arfken and Weber do, poorly.)  FWIW, I don't think these are high priority topics for typical EE undergrad, but FYI.  Of course some specialized topics like signal processing or Boolean algebra are also not covered.  I find most EEs do fine with how that is covered in the EE texts themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Length note:&lt;/strong&gt;  this book is about 4-6 semesters depending on student strength.  So picking and choosing becomes a need for shorter courses.  My one semester class covered LA, small part of F series and large part of PDEs.  A two semester class for physicists would add the complex analysis chapters.  It is a good reference though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Modeling:&lt;/strong&gt;  Preface talks a bunch about emphasis on modeling and interpretation.  I am really not in favor of so much of that in a math book (you get it in thermo and controls class regardless).  But it didn't end up bugging me.  So either he did it in a non-annoying manner or he just wrote about it a bunch in the preface to keep engineer professors happy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Layout note:&lt;/strong&gt; I found the book to be very well printed, few mistakes (I only know of one typo), and well laid out...text appropriate amount/tone (not too formal, not too chatty).  Enough examples for engineers.  Nice layout with learning objectives and chapter summaries and nice font usage.  No Fraktur (darned Altschrift)...which is good...I hate that.  The Greek letters &quot;squiggle&quot; and &quot;other squiggle&quot; may have turned up a teensy amount but not too much and not ostentatiously (like when a gamma would work fine and be normal engineering usage).  There were also very short historical footnotes on major mathematicians (nationality and birth/death).  I thought it was positive as there are a lot of new names (Legendre and the like) that come in and some other reference for them is helpful.  But it was just tiny notes and didn't get in the way of the math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Size:&lt;/strong&gt;  It has a nice heft to it and can be used to bash ganglions down.  In this it is similar to Arfken and Weber or Brealey and Myers.  ;-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Answers:&lt;/strong&gt;  5th edition had half the answers.  I hate books that don't have the answers.  So it is batting 50% on that, with me.  But it is better than some books like A/W that had minimal answers.  You should check more recent vintage though.  There may be some accompanying answer manuals out there but you would have to check out the coverage (is an annoying commercial practice of US publishers of having &quot;student solution manuals&quot; that only have half the answers...this is driven by commercial aspect of professors preferring it this way...is an annoying way that we have become LESS LIBERAL than c. 1900 math books.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Availability:&lt;/strong&gt;  It is probably the &quot;Coke&quot; of US Engine Math books so easily available.  I would go for an earlier edition (used) if price is an issue.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-04-17T22:47:42.200" LastActivityDate="2018-04-17T22:47:42.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13968" PostTypeId="2" ParentId="12788" CreationDate="2018-04-18T04:50:43.303" Score="1" Body="&lt;p&gt;Some students are easy to remember the numbers, some students are easy to remember the narrations such as history, some other students are easy to learn languages, etc. &lt;br&gt;&#xA;My 8 year son also hard to remember time tables, and we are still struggle to help him. The more we drill, the more he stress. So I think it's a bad idea. &lt;br&gt;&#xA;After tried so many methods, finally what we are doing is to make him understand how the numbers works. How to do addition, subtraction, multiplication, and division. We tell him &lt;b&gt;the basics&lt;/b&gt; to count, and use the &lt;b&gt;stories&lt;/b&gt; to make some example. Still he hard to remember the times table, but at least he can count correctly because he know the logic (even not yet fast enough).&lt;/p&gt;&#xA;" OwnerUserId="9830" LastActivityDate="2018-04-18T04:50:43.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13969" PostTypeId="2" ParentId="13315" CreationDate="2018-04-18T10:11:56.613" Score="3" Body="&lt;p&gt;&lt;strong&gt;A1/&lt;/strong&gt; I'm not sure why you would need a companion to Euclid. Euclid is already a textbook and you can progress through it as leisurely as you like... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I prefer Sir Thomas Heath's version. But the two most popular works back in the 19th century (when Euclid teaching was at its height in the UK) were Todhunter's &lt;em&gt;The Elements of Euclid for Schools and Colleges&lt;/em&gt; and Potts' &lt;em&gt;Euclid's Elements of Geometry&lt;/em&gt; which are more adapted for teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back in the day the sequence usually followed was BOOK I, BOOK III (excl. 35-37), BOOK II (plus III.35-37). Then a selection from BOOK IV as problems in geometrical drawing. BOOK V on theory of proportions was usually substituted with a treatment in algebra. Then BOOK VI and finally a limited selection of propositions from BOOK XI and XII.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're only doing the first two books then &lt;em&gt;Euclid Books I,II&lt;/em&gt; by Charles Dodgson (aka Lewis Carroll) is great.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A2/&lt;/strong&gt; There are several old collections of geometry problems or &quot;riders&quot; linked to progress through Euclid. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Todhunter includes exercises in his work and he separately published the answers in &lt;em&gt;Key to Exercises in Euclid&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might like to consult Smith, J Hamblin, &lt;em&gt;Riders in Euclud (I, II, III, IV &amp;amp; VI)&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are quite a few other books dedicated solely to exercises in BOOKS I and II, eg Deakin, Rupert &lt;em&gt;Rider Papers on Euclid (BOOKS I &amp;amp; II)&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="9832" LastActivityDate="2018-04-18T10:11:56.613" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13970" PostTypeId="2" ParentId="10674" CreationDate="2018-04-18T18:48:33.437" Score="2" Body="&lt;p&gt;I think you will find this paper very interesting. It addresses your first question about any formal connection between geometric figures and geometric proofs and argues the two are inseparable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper explores whether Euclidean geometric proofs are axiomatic-deductive as usually believed or diagrammatic as in the sense &quot;one reasons &lt;em&gt;in&lt;/em&gt; the diagram.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The author concludes that &quot;[a] Euclidean demonstration is not, then, diagram-based, its inferential steps licensed by various features of the diagram. It is properly diagrammatic. One reasons in the diagram, in Euclid, that is, through lines, &lt;em&gt;dia grammon&lt;/em&gt;, just as the ancient Greeks claimed.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.researchgate.net/profile/Danielle_Macbeth/publication/265361973_Diagrammatic_reasoning_in_Euclid%27s_Elements/links/55981ca908ae99aa62ca20f8/Diagrammatic-reasoning-in-Euclids-Elements.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Diagrammatic Reasoning in Euclid's Elements by Danielle Macbeth&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9832" LastActivityDate="2018-04-18T18:48:33.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13971" PostTypeId="2" ParentId="10674" CreationDate="2018-04-19T06:59:14.177" Score="0" Body="&lt;p&gt;For mathematical monologue, dialogue, discourse, in short for communication including proofs (&quot;A proof is any completely convincing argument&quot; &#xA;[Errett Bishop: &quot;Schizophrenia in contemporary mathematics&quot;, Amer. Math. Soc. Colloquium Lecture, Seventy-eighth summer meeting, University of Montana, Missoula, Montana (1973)]), every language is suitable. This includes abstract logic-based languages as well as ordinary spoken language like German, English, Latin, geometrical drawings and even gestures. The snooty Bourbaki-attitude admitting only a very narrow spectrum of languages may be welcomed by those who have problems with other languages. But with respect to the failure of this approach to recognize the fatal mistakes of modern set theory (see for instance &lt;a href=&quot;https://philosophy.stackexchange.com/questions/51038/will-mcduck-go-bankrupt&quot;&gt;https://philosophy.stackexchange.com/questions/51038/will-mcduck-go-bankrupt&lt;/a&gt;) this language cannot be recommended, and &quot;machine checked proof&quot; is not a mark of quality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Pure mathematics and science are finally being reunited and, mercifully, the Bourbaki plague is dying out.&quot; [Murray Gell-Mann: &quot;Nature conformable to herself&quot;, Bulletin of the Santa Fe Institute 7 (1992) p. 7]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course a rigorous geometrical proof is as correct as any other. For people with corresponding sensual perception it is even better.&lt;/p&gt;&#xA;" OwnerUserId="8557" LastActivityDate="2018-04-19T06:59:14.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13972" PostTypeId="2" ParentId="13956" CreationDate="2018-04-19T07:44:58.143" Score="2" Body="&lt;p&gt;I teach Euclid to groups of young students... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 3 tests for congruence of triangles all appear in Elements Book I: SAS (I.4), SSS (I.8) and ASA/AAS (I.26).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SAS is proved by superposition, ie one triangle placed over the other starting at one point and is rather unsatisfying. The proof for SSS is a little more interesting as superposition is only used to align the two bases; the remainder is proved from the fact triangles with two given sides form a unique rigid polygon on one side of the same base. Lastly, ASA is proved using &lt;em&gt;reductio ad absurdum&lt;/em&gt; to prove another side is equal and then SAS is employed to complete the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compared with other propositions in Book I the above do not stand out as being either interesting or satisfying to do. They are certainly far from deductive geometry's finest moment! I only include the proofs as part of the overall work to maintain consistency and because the results are so important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I had to teach the three in isolation I would be very tempted to approach them using practical geometry - ie the circle and ray method already alluded to in earlier replies - starting from just one given side or angle and building up from there, part by part, until students discover sufficiency for congruence.&lt;/p&gt;&#xA;" OwnerUserId="9832" LastActivityDate="2018-04-19T07:44:58.143" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13973" PostTypeId="2" ParentId="13860" CreationDate="2018-04-19T08:33:37.720" Score="0" Body="&lt;p&gt;I have a couple of suggestions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) The natural extension to your junior high exercise is to use areas instead of circumference to estimate pi. This can be done the same way Archimedes is said to have experimented with it - by making circles with various diameters out of the same material and weighing them. It is a great exercise in proportion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Euclid's Book II of the Elements is mostly geometric proofs that have direct algebraic equivalents, eg a(a+b); a(b+c+d); (a+b)^2; (a+b)(a-b); (a-b)^2 etc. It is both fun and challenging to explore algebraic expansions visually through geometry.  &lt;/p&gt;&#xA;" OwnerUserId="9832" LastActivityDate="2018-04-19T08:33:37.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13974" PostTypeId="1" CreationDate="2018-04-19T15:23:18.200" Score="7" ViewCount="1236" Body="&lt;p&gt;How would you explain to students the difference between &lt;em&gt;drawing&lt;/em&gt; and &lt;em&gt;constructing&lt;/em&gt;? &quot;Accuracy&quot; seems to be a go to word, but that's not really what the difference is. I want to say more, but I also don't want to lead answers, so I'll stop here. =)&lt;/p&gt;&#xA;" OwnerUserId="6675" LastActivityDate="2018-04-20T19:00:23.260" Title="Drawing vs Constructing" Tags="|geometry|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13975" PostTypeId="1" CreationDate="2018-04-19T15:25:39.233" Score="12" ViewCount="1576" Body="&lt;p&gt;If you look for word problems about exponential growth or decay, in text books or on the internet, certain settings arise again and again:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;bank accounts,&lt;/li&gt;&#xA;&lt;li&gt;populations of animals, people or bacteria, and&lt;/li&gt;&#xA;&lt;li&gt;temperature changes, as when something is taken from an oven or a refrigerator and allowed to come to room temperature.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I'd like to find more varied settings for exponential functions, including (it is hoped) some that are concave up or down, and for each of those some that are increasing and some that are decreasing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9F7Xn.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9F7Xn.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8258" LastEditorUserId="8258" LastEditDate="2018-04-19T17:31:46.900" LastActivityDate="2018-04-26T20:07:59.973" Title="Interesting settings for exponential growth or decay" Tags="|word-problems|exponential|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="13976" PostTypeId="1" CreationDate="2018-04-19T15:38:36.080" Score="2" ViewCount="116" Body="&lt;p&gt;At present, I teach Guldin's rules for surface and volume of rotation, and give an example task from the textbook. The textbook uses procedure 1 (below) for calculation (below), but I advocate that you should be able to use procedure 2 as well. &#xA;Where is the flaw in my argument for equivalence between the two procedures (or in my calculation of the surface area)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Guldin's first rule is also known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Pappus%27s_centroid_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pappus first centroid theorem&lt;/a&gt;, &#xA;and states that a plane figure creates an areal when rotated that is the product of the distance the centroid is moved and the length of the figure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should therefore get the same resulting area when I rotate a triangle, as when I rotate the three segments making up the sides of the triangle.&#xA;Thus, I must do something wrong when I do not get that. Here are my calculations, provided triangle $ABC$, where $A=(1,1)$, $B=(1,4)$ og $C=(4,3)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In procedure 1, I shall calculate the contribution from each segment, then add.&#xA;In procedure 2, I shal find the centroid and the perimeter and &quot;directly&quot; calculate the surface area of the volume of rotation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Procedure 1&lt;/strong&gt;: &#xA;$r_{AB}=1,L_{AB}=3,A_{AB}=2\cdot\pi\cdot r_{AB}\cdot L_{AB}=6\pi$.&#xA;$r_{BC}=\frac{5}{2},L_{BC}=\sqrt{(4-1)^2+(3-4)^2}=\sqrt{10},A_{BC}=2\cdot\pi\cdot r_{BC}\cdot L_{BC}=5\pi\sqrt{10}$.&#xA;$r_{AC}=\frac{5}{2},L_{AC}=\sqrt{(4-1)^2+(3-1)^2}=\sqrt{13},A_{AC}=2\cdot\pi\cdot r_{AC}\cdot L_{AC}=5\pi\sqrt{13}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consequently, the total surface area amounts to $A_1=A_{AB}+A_{BC}+A_{AC}=\pi\left(6+5\sqrt{10}+5\sqrt{13}\right)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Procedure 2:&lt;/strong&gt;&#xA;Triangle $ABC$ perimeter $L=L_{AB}+L_{BC}+L_{AC}=3+\sqrt{10}+\sqrt{13}$.&#xA;Radius of rotation for triangle $ABC$: $r=2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Total surface area: $A_2=2\pi r L=4\pi\left(3+\sqrt{10}+\sqrt{13}\right)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I expected $A_1=A_2$, but I get $A_1=A_2+\pi\left(-6+\sqrt{10}+\sqrt{13}\right)$.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/CyruK.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CyruK.png&quot; alt=&quot;Calculation of surface area of triangle rotated around the y-axis&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3502" LastEditorUserId="3502" LastEditDate="2018-04-22T12:37:00.693" LastActivityDate="2018-04-22T12:37:00.693" Title="Why does result depend on procedure in my calculation of surface area using Guldin?" Tags="|secondary-education|geometry|students-mistakes|" AnswerCount="0" CommentCount="5" ClosedDate="2018-04-23T06:56:32.607" ContentLicense="CC BY-SA 3.0" />
  <row Id="13977" PostTypeId="2" ParentId="13975" CreationDate="2018-04-19T15:54:55.827" Score="12" Body="&lt;p&gt;(edited)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a.  For all of these, I would think to &lt;strong&gt;expand the category into subcategories&lt;/strong&gt;.  So not just &quot;bank account&quot; but time value of money and NPV, bond details, etc.  You learn something when doing this finer description.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b.  Maybe it fits under your (2) but &lt;strong&gt;radioactive decay&lt;/strong&gt; is a huge one you did not mention.  Applications include carbon (and other) dating, radioisotopes for health use, after the A bomb, or a leak/accident from a power station.  If you look at the source, it decays toward zero, if you look at a (stable) product, it approaches a set amount.  It is possible to complicate the problems slightly with some algebra by looking at concentration versus activity or different fission products or decay pathways (meaning it is not a 1:1 factor) or with uranium enrichment/depletion  (some proportion needed at beginning, end of problem).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c.  Analogous to thermal gradient equalization, there are also &lt;strong&gt;chemical gradient equalizations&lt;/strong&gt; or &lt;strong&gt;pressure equalizations&lt;/strong&gt; (through an orifice).  [These are not academic problems, a relevant one that I know is buildup of chlorides in a heat exchanger with a pinhole leak to saltwater side.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;d.  &lt;strong&gt;Capacitor discharge (or charge)&lt;/strong&gt; in a DC circuit.  The former goes to zero (assuming this is the power source), the latter to applied voltage (usually the battery in the circuit).  Look up &quot;RC time constant&quot;.  (There is also &quot;RL time constant&quot; for an AC circuit with an inductor, but high school kids usually grok DC better than AC.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;e.  &lt;strong&gt;Some chemical reactions&lt;/strong&gt; (depending on rate order).  Usually this is way too fast to be a meaningful problem for tabletop or industrial reactions but if you look at &lt;strong&gt;geochemistry&lt;/strong&gt; or the like it can be meaningful.  Note that this is a kinetics problem approaching an equilibrium value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;f.  (From googling) &lt;strong&gt;compaction of sediments&lt;/strong&gt; and &lt;strong&gt;buildup to terminal velocity&lt;/strong&gt; (falling in air resistance).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;g.  &lt;strong&gt;Attenuation of a flux through a media&lt;/strong&gt; (light, X-rays, neutrons).  [Note the dimension is no longer time but thickness.  So it is a nice difference.  but math is still an exponential.  Instead of looking at &quot;half lifes&quot; (5.5 years for cobalt 60), you normally talk about &quot;tenth thicknesses&quot; (inch of lead for gamma rays).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;h.  &lt;strong&gt;Decline of an oil well.&lt;/strong&gt;  (Note there are several different models, but the simplest is exponentional, which corresponds to a bounded permeable reservoir.)  Look up Arps decline for reference (lots of good explanatory pages on the web so you don't need to, but may, read the iconic journal article itself.)  Note also, while it might be too much for an &quot;Algebra 2&quot; student, that the other models (hyperbolic, harmonic, etc.) are quite nice mathematically and you can see how they change into each other via &quot;b factor&quot; (b=1 is harmonic, b=0 is exponential).  Might be OK for an honors class or pre-calc class that is doing a lot of function graphing.  I had to come up to speed in this field and (having no geology background) found that strong high school functions knowledge helped me to nuke it out.  So all that high school stuff can make a difference!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;i. &lt;strong&gt;Product/market introduction.&lt;/strong&gt;  See &quot;product life cycle&quot; or the work of Christensen from Harvard Business School.  The model is concave up exponential at beginning, roughly linear in middle, then concave down to a plateau (in other words a sigmoid curve, with an inflection point).  [If the product ends up being displaced, you will have a have a bell curve effect of going down as substitute products come in, and then concave up with the tail of usage.  (This is also what graph of the market leader that is losing share looks like.) For a given product, may approximate a Gaussian if the plateau is short in duration or two sigmoids with an approximate level plateau in between.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;j.  Another curve that approximates a Guassian is the famous &lt;strong&gt;Hubbert &quot;peak oil&quot; bell curve&lt;/strong&gt;.  I would be leery of saying &quot;this is how it works&quot; since of course you can have deviations from shape (Hubbert admitted this, it is a generalization) and even radical departures, like US recent oil production.  But if you look at areas like North Sea or Cantarell, model has good fit.  And part of life in engineering/science is just to try models and they make work for a while (even a very good fit and theoretical sound fit like buildup of toxins in fish in a stream might change if the hydrology changes for instance.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;k.  Of course &lt;strong&gt;any normal distribution&lt;/strong&gt; (arrows hitting a target, observers estimating a value by eye, IQ, etc.) will have a Gaussian distribution.  I would probably skip this as it is harder to introduce the more abstract concept of a distribution than to discuss time variance.  [But including for completeness.]&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Bonus:  Not directly responding to your question,&lt;/em&gt; but I find that showing people &lt;strong&gt;graphing on log-log or semilog scales&lt;/strong&gt; is helpful in coming to grips and seeing application.  You can actually go to the engineering store and get that paper with the tiny green lines and such...you can just feel the science/engineering application come through by being a little physical.  (Mathematically it makes no difference, but we are psychological creatures, not computers...while you are there check out the other cool paper like the triangular phase diagrams!)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lot of common effects are logorithmic. [You can also have the kids graph some functions from above a-h list in linear and semilog paper.] &lt;strong&gt;Sonar/sound&lt;/strong&gt; is one I know well, but maybe little bit of a tangent to bring in decibels.  But &lt;strong&gt;earthquakes and Richter scale&lt;/strong&gt; is one that people know of by word and useful to learn it is a log scale.  So a 7 earthquake is not marginally worse than 6 quake but ten times worse.  And a 3.5 is some wimpy stuff!&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-04-19T17:26:31.117" LastActivityDate="2018-04-19T17:26:31.117" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="13978" PostTypeId="2" ParentId="13974" CreationDate="2018-04-19T16:58:29.557" Score="0" Body="&lt;p&gt;I would say that:&lt;br&gt;&#xA;construction yields an exactly &lt;em&gt;reproducible&lt;/em&gt; and unambiguous result, of which all properties can be &lt;em&gt;measured&lt;/em&gt; as expected (within the accuracy of the instruments used).&lt;/p&gt;&#xA;" OwnerUserId="9837" LastEditorUserId="1147" LastEditDate="2018-04-20T19:00:23.260" LastActivityDate="2018-04-20T19:00:23.260" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13979" PostTypeId="2" ParentId="13974" CreationDate="2018-04-19T18:15:17.737" Score="7" Body="&lt;p&gt;I would emphasize the &lt;strong&gt;algorithmic nature of a construction&lt;/strong&gt; over the ad hoc nature of a sketch/drawing. Tell your students that a construction must be accompanied by a specific sequence of steps (a &quot;recipe&quot;, if you will) so that, in the future, anyone else could come along and follow that recipe to accomplish the relevant task, as well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the perpendicular bisector of a line segment can be constructed by following a particular recipe. (&lt;a href=&quot;https://www.mathopenref.com/constbisectline.html&quot; rel=&quot;noreferrer&quot;&gt;See this link for the construction and its proof.&lt;/a&gt;) Starting from &lt;em&gt;any line segment&lt;/em&gt;, following that recipe will create the perpendicular bisector of that segment. However, any particular sketch/drawing of a perpendicular bisector of a line segment, without any accompanying information, is merely that: a picture of one example. &lt;strong&gt;The recipe is the key component of the construction, not any one example of carrying out the construction.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2018-04-19T18:15:17.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13980" PostTypeId="2" ParentId="13315" CreationDate="2018-04-19T18:18:12.273" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://kasperpeulen.github.io/&quot; rel=&quot;noreferrer&quot;&gt;Euclid the game&lt;/a&gt; is a set of challenges built in Geogebra.  It allows the student to progress through the elements gaining additional tools as they go or solve each using only straight edge and compasses. It is accessible and enjoyable. I have used it with students from the age of 12.  I don't know whether you were looking for something more formal.&lt;/p&gt;&#xA;" OwnerUserId="9838" LastActivityDate="2018-04-19T18:18:12.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13981" PostTypeId="2" ParentId="1200" CreationDate="2018-04-19T20:02:19.243" Score="0" Body="&lt;p&gt;The answer to the question depends on the definition of number. If we define that a number is the set of its common representations, then the number 2 is the set {2, II, two, deux, 4/2, 6/3, ...}. Then 4/2 and 6/3 should be called different representations of the number 2. In doubt the representations or names should be put in quotation marks while the standard representation without quotation marks is used as &lt;em&gt;the number&lt;/em&gt;: &quot;4/2&quot; and &quot;6/3&quot; are different representations of the number 2. The standard representation is also used in ordinary language as &lt;em&gt;the number&lt;/em&gt; when we read that the number of citizens has increased from 22478 to 22675 in 2017. Here &quot;number&quot; is in fact a variable and 2017 bis the standard representation of the fixed number identifying the year. But an exact application of all these details would make the text difficult to read and certainly less informative. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the standard representation is only a representation too because in our texts no real things appear. That is the same as in texts about parts of reality like &quot;Jonny married June&quot; or about ideas like &quot;Tarzan married Jane&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When dealing with fractions we should distinguish between the &lt;em&gt;rational number&lt;/em&gt; and the different &lt;em&gt;fractions&lt;/em&gt; representing it. Here usually the reduced fraction is used as the standard representation or simply as &quot;the number&quot;. An example is Cantor's enumeration of the rational numbers where he eliminated all fractions that were not cancelled down - although the enumeration of all fractions would have yielded the same countable result.&lt;/p&gt;&#xA;" OwnerUserId="8557" LastActivityDate="2018-04-19T20:02:19.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13982" PostTypeId="2" ParentId="1200" CreationDate="2018-04-19T20:24:33.360" Score="5" Body="&lt;p&gt;You should say the fraction is the number.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the rationale for why to do this, see Feynman:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.fs.blog/2016/07/richard-feynman-teaching-math-kids/&quot; rel=&quot;noreferrer&quot;&gt;https://www.fs.blog/2016/07/richard-feynman-teaching-math-kids/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The point became philosophical.  It was crucial, he argued, to distinguish clear language from precise language. The textbooks placed a new emphasis on precise language: distinguishing number from numeral, for example, and separating the symbol from the real object in the modern critical fashion— pupil for schoolchildren, it seemed to Feynman. He objected to a book that tried to teach a distinction between a ball and a picture of a ball— the book insisting on such language as color the picture of the ball red.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;'I doubt that any child would make an error in this particular direction,' Feynman said, adding:  'As a matter of fact, it is impossible to be precise … whereas before there was no difficulty. The picture of a ball includes a circle and includes a background. Should we color the entire square area in which the ball image appears all red? … Precision has only been pedantically increased in one particular corner when there was originally no doubt and no difficulty in the idea.  In the real world absolute precision can never be reached and the search for degrees of precision that are not possible (but are desirable) causes a lot of folly.'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feynman has his own ideas for teaching children mathematics.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  The more precise the math language does NOT equate to the more math taught.  In fact, it could actually lead to less math being learned if the emphasis is to an extent of spending too much time on fussy definitions and not enough on basic work (both the student time and the teacher attention).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.  I think 1/2, 3/6, 0.5 and 50% are all numbers and the same thing.  If I'm wrong, it has not hurt me in a lifetime of work in science, engineering, and business.  &lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-04-19T20:30:59.790" LastActivityDate="2018-04-19T20:30:59.790" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13983" PostTypeId="2" ParentId="13975" CreationDate="2018-04-19T23:23:26.717" Score="3" Body="&lt;p&gt;&lt;strong&gt;Catenary curves&lt;/strong&gt; are the sum of two exponentials -- One concave upward but decreasing, and the other concave upward and increasing.  They are a good model for the curve of suspension bridge cables.  (The key assumptions are that the weight of the cables is small compared to the weight of the bridge deck, and the weight of the bridge deck is constant along the length of the bridge.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Amplitudes of bouncing motions&lt;/strong&gt; (and &lt;strong&gt;pendulums&lt;/strong&gt;) tend to have exponential decay.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The shape of absurdly tall buildings&lt;/strong&gt; or &lt;strong&gt;space elevators&lt;/strong&gt;.  If you tried to build a building many miles high, how would the area of the building have to change as you went up, just to support its own weight?  If you haven't gone up far enough for the acceleration of gravity to change much, the curve will be an exponential.  If you try to make an elevator cable extending down from geosynchronous orbit, the cross-section would also vary exponentially while the cable was still &quot;short&quot;.  But if you try to reach down to the planetary surface, the change in the acceleration of gravity (adjusted for centrifugal acceleration) would distort the exponential into something like a Gaussian curve.  A longer overview (and a short bibliography) are in Hans Moravec's article &quot;Cable Cars in the Sky&quot;, in &lt;em&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0441206646&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Endless Frontier&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2018-04-20T18:23:20.157" LastActivityDate="2018-04-20T18:23:20.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13984" PostTypeId="2" ParentId="13956" CreationDate="2018-04-20T02:58:59.130" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You might consider defining a pair of triangles to be congruent&#xA;based on 3 pairs of equal sides. That would take care of SSS.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Make some puzzle worksheets asking students to pick out the&#xA;congruent triangles based on this definition. Give the length of&#xA;each line segment. Do not mark the angles.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain how SAS works. A &quot;shortcut&quot; -- you only have to know that&#xA;the length of 2 pairs of sides are equal, but you have will need a third&#xA;piece of information: The angles between pairs of sides must be&#xA;equal. (OK, not much of &quot;shortcut,&quot; but it sounds good!)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give students another puzzle sheet for SAS only.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give students another puzzle sheet mixing up SSS and SAS.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain how ASA works.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give students another puzzle sheet for ASA only. A &quot;shortcut&quot; -- you only need one pair of equal sides. But you need the angles at either end to match.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give students another puzzle sheet mixing SSS, SAS and ASA.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Forget Euclid's &quot;Elements.&quot; Avoid words like &quot;proofs,&quot; &quot;theorems&quot; and &quot;postulates.&quot; Think of them as &quot;rules&quot; and &quot;shortcuts.&quot; You are developing students' spatial sense. Rigor should come later.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2018-04-20T04:48:39.607" LastActivityDate="2018-04-20T04:48:39.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13988" PostTypeId="1" CreationDate="2018-04-20T23:42:26.280" Score="6" ViewCount="187" Body="&lt;p&gt;The Wason Selection Task (described by Pete Clark &lt;a href=&quot;https://matheducators.stackexchange.com/a/412/117&quot;&gt;here&lt;/a&gt;) is a great problem for getting students to grapple with all of the intricacies of logical implication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will be teaching a discrete mathematics course for the first time in about a month, and I would love to have a sizable repository of similar questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have a link to a big list of similar questions?  If not, I am happy to hear your ideas about similar problems which might stimulate a similar amount of thinking about the other logical operators.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-04-20T23:42:26.280" Title="Questions similar to Wason Selection Task" Tags="|proofs|exercises|discrete-math|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="13989" PostTypeId="2" ParentId="1028" CreationDate="2018-04-21T15:39:31.443" Score="10" Body="&lt;p&gt;Sorry for the necromancy but I had a slant to add and too long for comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Viewpoint:&lt;/strong&gt; It's not reasonable for the students to eschew previous techniques.  But it's ALSO not reasonable for you to be so super surprised that they lack perfect recall of previous material.  Instead of this all-or-nothing attitude, why not a more sympathetic approach:  &quot;yeah, I know partial fractions integration was a while ago and it was at the end of a long course...but we actually need that thing now...don't worry, we won't rehash every tough integration technique but this snake needs to come out of its hole again...no sweat, let's walk through the problem slowly and show the steps we use.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Course design (you asked):&lt;/strong&gt;  I'm not sure that proofs is the key thing to working on with a course where you are trying to give &quot;exposure&quot;.  For someone who uses a technique a proof is really more like a motivation for why you can use the technique.  But the technique is the important thing, not the proof.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Course structure:&lt;/strong&gt;  I would also suggest designing the course to follow a simple text, not as an ad hoc set of subjects or something that requires following lecture notes or stitched together handouts.  This gives the students some supporting scaffolding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pedagogy:&lt;/strong&gt;  The surprise at imperfect students and the attention to proofs makes me suspect someone who is good at understanding MATH but weak on understanding STUDENTS.  The math is half the problem.  But understanding students and how to improve them is PART of the problem.  Yes, absolutely, it is!  You can't ignore it, just as you can't ignore other independent variables in multivariable calculus, just as you can't ignore interactions and higher terms in stats problems that are nonlinear.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Course analysis:&lt;/strong&gt;  Take a look at your course and take a look at the previous calc course (or even algebra).  Figure out the key concepts that usually trip students up and make a little cheat sheet or refresher formula list or even list of key sections in the actual text used most commonly in your uni (or in nearby high school).  [This would be given to the students on day one.] But note, the point here is to be thoughtful and analytical and find the MINIMUM that they need to know.  Force yourself to list what is needed and not needed.  And then within the needed, list the things that typically confound students, not the ones that are usually no problem.  So, instead of saying &quot;you should know that whole calc book&quot;....well, sure they SHOULD.  But we are dealing with imperfect beings.  Not angels.  Not robots.  So if they need integration by parts (and it usually is an issue) but they don't need partial fractions, list parts integration and don't list partial fractions.  Figuring this out is part of your JOB.  And note that it is a relative thing (very different from Euclean proofs).  This is about max gain for min pain (AND with imperfect information).  At the end of the day, teaching math is more like science, engineering, business (practical activities using preponderance of evidence) than it is like math itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Coverage of old material:&lt;/strong&gt;  I would take a middle road when using topics that students are &quot;supposed to know&quot; but in reality often don't remember.  Don't just run remedial sessions and reteach the topic in general form.  Don't just expect students to have ready facility.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, something like (class before, end of class):  &quot;OK, for the next class...we are going to need to use partial fractions...I know that was a frustrating topic for many people in calc class and you may not have remembered it, so please take a look back at it tonight, because we will have to use it next class.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then when you do whatever derivation you are doing and the partial fractions come in...&quot;here are those darned partial fractions coming back.  I know it has been a while, so let's walk through the algebra carefully here.&quot;  [Then do show all the steps...and for the problem of interest, not a general partial fractions review.]&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-04-21T15:39:31.443" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13990" PostTypeId="2" ParentId="13975" CreationDate="2018-04-22T03:58:54.607" Score="4" Body="&lt;p&gt;One basic model of atmospheric pressure has it decaying exponentially as a function of height above sea level. Two places to look for this are &#xA;&lt;a href=&quot;https://people.clas.ufl.edu/kees/files/AtmosphericPressure.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://people.clas.ufl.edu/kees/files/AtmosphericPressure.pdf&lt;/a&gt; and &#xA;&lt;a href=&quot;http://nova.stanford.edu/projects/mod-x/ad-expatm.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://nova.stanford.edu/projects/mod-x/ad-expatm.html&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2018-04-22T03:58:54.607" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13991" PostTypeId="2" ParentId="13975" CreationDate="2018-04-22T22:31:04.823" Score="2" Body="&lt;p&gt;Some examples of exponential growth that don't seem to have been mentioned yet:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Moore's law&lt;/li&gt;&#xA;&lt;li&gt;the expansion of the universe (when dominated by dark energy, as it essentially is right now)&lt;/li&gt;&#xA;&lt;li&gt;time required for monkeys banging on typewriters to produce $n$ words of Shakespeare, or for a brute-force search to crack a password of length $n$&lt;/li&gt;&#xA;&lt;li&gt;motion of a pencil balanced on its tip (until the amplitude gets large)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;An example that might more naturally be described using the logarithm:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;time required to look up a word in a dictionary of a given length&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-04-22T22:31:04.823" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13993" PostTypeId="2" ParentId="13835" CreationDate="2018-04-24T19:16:16.010" Score="1" Body="&lt;p&gt;In my experience, the videos can be quite shallow, but then the textbook I use for A-level (16-18 year olds) has a very prescriptive approach. I can assist with the understanding, making-links aspect of their learning.  I think providing links is fine and probably a good idea.  I would expect students to soon realise that attending classes, and particularly extra help, is a much more effective use of their time than sitting through the videos, albeit ones that form part of a great resource.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do have students that would enjoy learning from Khan Acadamy but these are the ones who spend time watching Vsauce and similar. As they are so motivated, they would be unlikely to replace classes should they be given the choice whether to attend.&lt;/p&gt;&#xA;" OwnerUserId="9838" LastEditorUserId="376" LastEditDate="2018-04-25T05:22:07.507" LastActivityDate="2018-04-25T05:22:07.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13994" PostTypeId="2" ParentId="13712" CreationDate="2018-04-24T19:43:52.593" Score="1" Body="&lt;p&gt;Sorry, I don’t know of any literature but I trialed this approach with a class recently.  I felt strongly that applying different approaches to transformations in the x and y directions was less than helpful when they would subsequently meet, for example, circles where the centre is other than the origin and which can be considered as transformations.  It didn’t work for the majority; the idea of adding a constant to the right hand side to increment y by that amount was too intuitive for them to relinquish.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My next approach will be to keep the changes to the right side of the equals sign with translations of a in the x direction resulting in $f(x-a)$ and translations of b in the y direction resulting in $f(x)+b$.  I will go beck to making the link and showing consistency in the two directions when I later introduce circles.&lt;/p&gt;&#xA;" OwnerUserId="9838" LastActivityDate="2018-04-24T19:43:52.593" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13995" PostTypeId="2" ParentId="13712" CreationDate="2018-04-24T23:05:58.523" Score="1" Body="&lt;p&gt;The only &quot;literature&quot; I can think of is how &lt;em&gt;nearly every&lt;/em&gt; algebra textbook I've seen presents the &quot;point-slope&quot; form of a linear function: $$y - y_1 = m(x - x_1)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, the translation of axes is applied the same to both variables.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-04-24T23:05:58.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="13996" PostTypeId="2" ParentId="377" CreationDate="2018-04-25T00:55:31.407" Score="1" Body="&lt;p&gt;They failed to deliver the punchline: the uniqueness of the reals, as a [complete] ordered field.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Once numbers were under­stood as forming an ordered field, and the&#xA;  positive integers among them as a certain inductive subset, and once&#xA;  the language of sets became standard, so that statements with&#xA;  quantifiers made sense, then and only then would students see... &#xA;  (&lt;a href=&quot;http://web.math.rochester.edu/people/faculty/rarm/igno.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://web.math.rochester.edu/people/faculty/rarm/igno.html&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Most of the alterations to the curriculum narrated by Kline can be seen from this perspective. One introduces congruences to show examples of other fields that fail to meet some of the axioms of the reals; one introduces basis different of ten in order to show other ways to express the reals that actually are the same, isomorphic if you wish, field (Note particularly how binary notation was introduced but not related to set theory operations, contrary to its use in modern computer applications). In some presentations even equations were subordinated to the reinforcement of the idea of expanding, or completing, a numerical set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Discussions such as &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2763/is-it-advisable-to-avoid-teaching-multiplication-as-repeated-addition&quot;&gt;Is it advisable to avoid teaching &amp;quot;multiplication as repeated addition&amp;quot;?&lt;/a&gt; , asking if multiplication and addition should be considered independent, are a remmant of this perception; what is discussed at the end is how fast can students be presented to the concept of field, and then how to argue that the combination of multiplication and sum in the reals is the most natural or unique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surely New Math had other intermediate goals, but the real line was its final act of fireworks. Very much as the final fireworks bouquet of Euclid &quot;classical curriculum&quot; is the existence of five and only five regular solids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: The unpublished Mathématique moderne 4 of Papy, with notes on the real field, is avalaible online at &lt;a href=&quot;http://www.rkennes.be/Articles%20de%20Papy/ListesArticles.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.rkennes.be/Articles%20de%20Papy/ListesArticles.htm&lt;/a&gt; (for free, but lot of popups)&lt;/p&gt;&#xA;" OwnerUserId="9854" LastEditorUserId="9854" LastEditDate="2018-05-15T20:52:09.953" LastActivityDate="2018-05-15T20:52:09.953" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13998" PostTypeId="2" ParentId="13975" CreationDate="2018-04-26T20:07:59.973" Score="1" Body="&lt;p&gt;Here's the sequence I often use.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;The Smallpox Case&lt;/h1&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Imagine terrorists bioengineer a new version of smallpox that infects 3 people.&lt;/li&gt;&#xA;&lt;li&gt;Each day, each infected person infects one more person.&lt;/li&gt;&#xA;&lt;li&gt;Make a table of values relating the number of days that have passed to the number of people infected for two weeks. Begin with the day 0 corresponding to 3 infections.&lt;/li&gt;&#xA;&lt;li&gt;Predict the number of infections in 5 weeks.&lt;/li&gt;&#xA;&lt;li&gt;Predict how many days it will be until every human being on earth is infected?&lt;/li&gt;&#xA;&lt;li&gt;Create a model (equation) relating the number of days that have passed to the number of people infected. Prove that it is a correct model and explain how it works.&lt;/li&gt;&#xA;&lt;li&gt;Go back to the model in #6. How can you use it to help with the previous two questions?&lt;/li&gt;&#xA;&lt;li&gt;Graph the data that you have. Is it linear? Why or why not and what does that mean?&lt;/li&gt;&#xA;&lt;li&gt;Generalize your model with respect to both the number of initial infections and the number of people each sick person infects each day. Prove that this is a correct generalization and explain what will happen to your graph as your parameters change in value.&lt;/li&gt;&#xA;&lt;li&gt;List at least three reasons how and why the model will &lt;strong&gt;not&lt;/strong&gt; be perfectly accurate at predicting the spread of disease over time. Example: &quot;If the infected people die, then they will stop infecting other people. This will slow the rate of new infections meaning the model will overpredict the number of infections.&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-04-26T20:07:59.973" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="13999" PostTypeId="1" CreationDate="2018-04-27T10:37:49.040" Score="15" ViewCount="2823" Body="&lt;p&gt;I am looking for various mnemonics which help students to remember some important properties or theorems. Very often students confuse signs or relations such as $\leq$ and $\geq$ in some expressions. I wonder if there are some mnemonics that can be used to help the students not to confuse some particular expressions, theorems and the like. One example of a particular mnemonic which is concerned with the Min-Max-Inequality, is given in my answer below so that you can see what I am after. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion such mnemonics can help students a lot and make learning much easier.&lt;/p&gt;&#xA;" OwnerUserId="7608" LastEditorUserId="7608" LastEditDate="2018-04-28T07:33:05.500" LastActivityDate="2018-05-03T19:38:36.273" Title="Mnemonics for some properties in mathematics" Tags="|mathematical-pedagogy|student-motivation|examples|students-mistakes|recreational-learning|" AnswerCount="13" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="14000" PostTypeId="2" ParentId="13999" CreationDate="2018-04-27T10:37:49.040" Score="11" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Minimax_theorem&quot; rel=&quot;noreferrer&quot;&gt;minimax theorem&lt;/a&gt; states the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $X\subset   \mathbb{R}^{n}$ and $Y\subset \mathbb {R} ^{m}$ be&#xA;  compact convex sets. If $ f:X\times Y\rightarrow \mathbb {R} $ is a&#xA;  continuous function that is convex-concave, i.e.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$f(\cdot ,y):X\rightarrow \mathbb {R}  \text{ is convex for fixed }&#xA; y, \text{and}$$ $$   f(x,\cdot ):Y\rightarrow \mathbb {R} \text{ is concave for fixed } x.$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Then we have that&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$ \min _{x\in X}\max _{y\in Y}f(x,y)=\max _{y\in Y}\min _{x\in&#xA;&amp;gt; X}f(x,y).$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For arbitrary functions $f$ the equality does not hold in general. However,  the &lt;a href=&quot;https://en.wikipedia.org/wiki/Max%E2%80%93min_inequality&quot; rel=&quot;noreferrer&quot;&gt;Max-Min-Inequality&lt;/a&gt; is always satisfied.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For any function $f: Z \times W \to \mathbb{R}$ we have&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$  \inf _{w\in W}\sup _{z\in Z}f(z,w) \geq  \sup _{z\in Z}\inf_{w\in W}f(z,w) .$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Since this property always holds for arbitrary functions $f$, it is well worth to be kept in mind by students. Naturally, many students will tend to confuse the order of the max and min operations as well as the direction of the inequality. In a lecture on nonlinear optimization our professor told us the following mnemonic to remember the property, which, as I think, makes it really easy to remember:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;The shortest giant is at least as tall as the tallest dwarf.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here, the shortest giant refers to the inf sup on the left hand side, which is at least as tall (i.e. $\geq$) as the tallest dwarf, which corresponds to sup inf. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't forgotten the Max-Min-Inequality since I learned the above mnemonic which is why I posted the question whether you are aware of other such neat mnemonics which make students' learning easier. &lt;/p&gt;&#xA;" OwnerUserId="7608" LastEditorUserId="7608" LastEditDate="2018-04-27T13:56:55.747" LastActivityDate="2018-04-27T13:56:55.747" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="14001" PostTypeId="2" ParentId="13999" CreationDate="2018-04-27T10:53:30.153" Score="10" Body="&lt;p&gt;Fatou's Lemma states: for nonnegative measurable functions $f_n$,&#xA;$$&#xA;\int_E \liminf_{n\to\infty} f_n\;d\mu&#xA;\le&#xA;\liminf_{n \to \infty}\int_E f_n\;d\mu&#xA;$$&#xA;The mnemonic is &#xA;$$&#xA;\text{ILLLLLI},&#xA;$$&#xA;meaning &quot;the Integral of the Lower Limit is Less than the Lower Limit of the Integral&quot;.&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2018-04-27T11:01:05.943" LastActivityDate="2018-04-27T11:01:05.943" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="14002" PostTypeId="2" ParentId="13999" CreationDate="2018-04-27T11:25:09.810" Score="4" Body="&lt;p&gt;One I recently learned -- for the order of the signs in factoring a sum or difference of cubes, remember &lt;strong&gt;SOAP&lt;/strong&gt;: Same sign, Opposite sign, Always a Plus.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sum of Cubes: &lt;span class=&quot;math-container&quot;&gt;$x^3 + a^3 = (x + a)(x^2 - ax + a^2)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Difference of Cubes: &lt;span class=&quot;math-container&quot;&gt;$x^3 - a^3 = (x - a)(x^2 + ax + a^2)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Credit: &lt;a href=&quot;https://openstax.org/details/books/college-algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax College Algebra&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-04-29T17:00:24.597" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="14003" PostTypeId="2" ParentId="13999" CreationDate="2018-04-27T13:20:55.080" Score="3" Body="&lt;p&gt;Here is my way of memorizing the three main trigonometric functions.  An angle $\theta$ is in standard position locating a point $(x,y)$ on a circle with a radius $r$ centered at the origin.  There is a convertible (a car with the roof removed) being driven down a road during the daytime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sun (sine) is above (vertical $y$) the road ($r$): $\sin\theta=y/r$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The car (cosine*) is moving horizontally ($x$) over the road ($r$): $\cos\theta=x/r$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the sun is above ($y$) the car ($x$), the driver gets a tan: $\tan\theta=y/x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*In Filipino (my language), &quot;car&quot; is &quot;kotse&quot; which is pretty close to &quot;cos.&quot;&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-04-27T13:20:55.080" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="14004" PostTypeId="2" ParentId="13999" CreationDate="2018-04-27T15:02:14.357" Score="15" Body="&lt;p&gt;Recently, a student in my beginning algebra course offered the following to the class, regarding signed number multiplication:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming positivity is like &lt;em&gt;love&lt;/em&gt;, and negativity is like &lt;em&gt;hate&lt;/em&gt;, then...&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;If you love love, that's love.&quot; $\Rightarrow$ &lt;em&gt;positive&lt;/em&gt; $\times$ &lt;em&gt;positive&lt;/em&gt; = &lt;em&gt;positive&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&quot;If you love hate, that's hate.&quot; $\Rightarrow$ &lt;em&gt;positive&lt;/em&gt; $\times$ &lt;em&gt;negative&lt;/em&gt; = &lt;em&gt;negative&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&quot;If you hate love, that's hate.&quot; $\Rightarrow$ &lt;em&gt;negative&lt;/em&gt; $\times$ &lt;em&gt;positive&lt;/em&gt; = &lt;em&gt;negative&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&quot;But if you hate hate, that's love.&quot; $\Rightarrow$ &lt;em&gt;negative&lt;/em&gt; $\times$ &lt;em&gt;negative&lt;/em&gt; = &lt;em&gt;positive&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;[Read this by treating the first instance of &lt;em&gt;love&lt;/em&gt; or &lt;em&gt;hate&lt;/em&gt; as a verb.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-04-27T15:02:14.357" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="14005" PostTypeId="2" ParentId="13999" CreationDate="2018-04-27T15:34:42.080" Score="11" Body="&lt;p&gt;I tell students to visualize $&amp;lt;$ and $&amp;gt;$ as mouths.  They always want to eat the bigger number.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-04-27T15:34:42.080" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="14006" PostTypeId="2" ParentId="13999" CreationDate="2018-04-27T16:37:01.690" Score="1" Body="&lt;p&gt;For the 4 quadrants of a Cartesian graph I say &quot;All Students Take Calculus&quot; counterclockwise (in order) to remember which trig fxns are positive in which quadrants.&lt;/p&gt;&#xA;" OwnerUserId="9865" LastActivityDate="2018-04-27T16:37:01.690" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="14007" PostTypeId="1" CreationDate="2018-04-27T19:27:19.837" Score="6" ViewCount="267" Body="&lt;p&gt;A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen without a lot of resources available in terms of additional graders.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question essentially boils down to: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are tips/tricks/techniques for creating quiz and exam questions that both&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;test students at various levels of Bloom's hierarchy and&lt;/li&gt;&#xA;  &lt;li&gt;minimize the amount of work for the grader&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Again, the subject is particularly (pre)calculus, but more general tips/tricks/techniques are great too. I only mention the subjects so that techniques that work particularly well for topics in (pre)calculus are mentioned.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have some ideas:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Formatting can get rid of a lot of the time spent &lt;em&gt;looking&lt;/em&gt; for various components of an answer. E.g. for a question about convergence/divergence of a series, you can label spaces for saying whether the series converges or diverges, for what test they are using, and what work needs to be done in order to use that test.&lt;/li&gt;&#xA;&lt;li&gt;True/False questions in which the student must give a counterexample or an explanation in case of False (and perhaps some explanation in case of True as well).&lt;/li&gt;&#xA;&lt;li&gt;Matching problems for topics in which this makes sense, such as polar/parametric graphs or conics (given Cartesian or polar equations).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm curious to hear what other things people have used.&lt;/p&gt;&#xA;" OwnerUserId="449" LastActivityDate="2018-04-29T16:31:05.607" Title="Effective Assessment that's Easy to Grade" Tags="|undergraduate-education|mathematical-pedagogy|grading|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="14008" PostTypeId="2" ParentId="13999" CreationDate="2018-04-27T20:10:49.333" Score="1" Body="&lt;p&gt;SOH-CAH-TOA &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to remember the ratios for the 3 main trig functions. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-04-27T20:10:49.333" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="14009" PostTypeId="2" ParentId="14007" CreationDate="2018-04-27T20:10:56.743" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://webwork.maa.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WeBWorK&lt;/a&gt; is an open-source online homework system for math courses. You could supplement written homework with an online component, thereby cutting down on the amount of grading while still providing your students with enough feedback (both written feedback on the written components, as well as WeBWorK's &quot;wrong, try again&quot; feedback online.)&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2018-04-27T20:10:56.743" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="14010" PostTypeId="2" ParentId="13999" CreationDate="2018-04-27T21:26:42.143" Score="1" Body="&lt;p&gt;Last year I heard of &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{Lo De Hi Mi Hi De Lo}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{(sing: &quot;Low Dee High my High Dee Low!&quot;)}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;as a mnemonic for the numerator in the quotient rule:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\left(\frac{f}{g}\right)' = \frac{g\cdot Df - f \cdot Dg}{g^2}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(of course Lo(w) = denominator, De = derivative, Hi(gh) = numerator, Mi = minus). Make sure you emphasize you have to divide the whole thing&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\text{over LoLo}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;though.&lt;/p&gt;&#xA;" OwnerUserId="8931" LastActivityDate="2018-04-27T21:26:42.143" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="14011" PostTypeId="2" ParentId="14007" CreationDate="2018-04-28T02:36:06.130" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Grade tests, not homework.  (Homework is drill for tests, is practice problems.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Make a key before GIVING (not grading) the test.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Construct tests by using unassigned drill problems from the text.  (Easy and relevant.  If a kid does the extra drill and has it easier on the test, that's a feature, not a bug--encourages drill.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think a couple sections of 35 people is arduous grading.  Look what HS teachers do.  Just buckle up and do it.  I don't think the size is sufficient to justify the work in making some system, nor the drop in question quality from going to multiple choice.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let the pre-calc person do the pre-calc grading.  Have him take the test and then construct a key that you review.  After that, leave him be.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;[Edit addition]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comment about HS grading was not meant to be harsh.  More calculational.  Like if you told me the wife/husband had the family car and the job is 2 kilometers away, what to do (the answer is walk, walk faster, wear running shoes and keep dress shoes to change in office, etc.).  But if it is across town, that is another story and we can brainstorm larger systems (bike, beater bike, carpool, bus, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Figure 16 week semester, MWF single periods.  If you do a test every second week, that is 8 (7 if you skip last test), plus a final that is twice as long.  Figure 5 essay questions per test, that is 35*5*2=350 questions per test.  Figure 30 seconds per question, that is 120 questions per hour.  (When you are flying, 30 seconds is achievable.)  So three hours grading.  Call it four, with adding totals and such.  Double that for the final.  So 7*4+8=36 hours of grading.  Not great, but not end of the world.  You could reduce it significantly  by doing 3 interim tests plus a final (3*4+8=20 hours).  It will be less effective coaching for the trainees, but will be more research time.  They won't complain about less tests either...so that is a moral choice, but an easy expedience.  [Of course, it also takes time to construct a test and a key.  However, if you are using drill problems (or very close approximates), the test design is not too bad...key is a little more.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make the tests physically easy to grade and easy to take.  No blue books.  Each test is 5 &quot;essay&quot; questions with each question worth 10 points (easy percentages and addition!)  You can do multiparter questions though.  Use 1 (ONE) page per essay question.  Make it single-sided (yes, kill the trees...paper is free,  to you.)  This will be easier for you to grade.  And it gives kids the back if they are prolix.  Use a standard format with a box in upper right for the partial score (and for front page the total score).  Tell the kids to box their final answers (you can still grade on work done for partial credit/deductions but at least finding their final answer within the chicken-scratch is easy).  Note your key should already say how much you expect for each subpart of the problem.  The AP system is a good guide for how to do this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would grade the exams in order (question 1 for all, then question 2 for all).  Think it will give you a little better speed and definitely better accuracy for the students.  I would probably not separate the pages since you only have one grader (you) and since it becomes a nightmare to reassemble them (even with a numeric indicator).  Keep a stapler to hand to deal with the inevitable exams that start separating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use your judgment on curving or not.  Personally, I think if you can shoot for tests of reasonable difficulty level so that the standard 90-80-70-60 divisions work, that is better.  But if you have some calamity, you might consider curving.  (With multiple evaluation points, I would not sweat it too much as the students have a chance to recover.)&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-04-29T16:31:05.607" LastActivityDate="2018-04-29T16:31:05.607" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="14012" PostTypeId="2" ParentId="13999" CreationDate="2018-04-28T08:54:48.930" Score="3" Body="&lt;p&gt;First heard it from a former classmate of mine, might be her own invention:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When the second derivative is positive, the function is happy (i.e., its graph looks like a smile). When the second derivative is negative, the function is sad (i.e., its graph looks like a frown).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;added&lt;/strong&gt; (I hope Federico doesn't mind ... Gerald Edgar)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pictorially,&lt;br&gt;&#xA;second derivative positve, second derivative negative:&lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/lWeOu.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lWeOu.jpg&quot; alt=&quot;second&quot;&gt;&lt;/a&gt;&lt;br&gt;&#xA;&lt;strong&gt;Two&lt;/strong&gt; plus signs signifies &lt;strong&gt;second&lt;/strong&gt; derivative &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First derivative may be added, if you somehow remember they guy faces to our left):  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First derivative positive, first derivative negative : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eGcvA.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eGcvA.jpg&quot; alt=&quot;first&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4930" LastEditorUserId="127" LastEditDate="2018-04-28T13:03:48.853" LastActivityDate="2018-04-28T13:03:48.853" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="14013" PostTypeId="2" ParentId="13999" CreationDate="2018-04-28T21:16:09.573" Score="1" Body="&lt;p&gt;To remember concave Up (vs concave down), I remember that the U shape is concave up. Similarly, in conVex, the V is convex. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(If you like, v is the only letter in the word which is the graph of a function. Sadly, concave also has a v, but this mneumonic seems to work anyway. You just have to remember which word (convex) you've assigned the mneumonic to, which seems easier than remembering which word means which.)&lt;/p&gt;&#xA;" OwnerUserId="8029" LastActivityDate="2018-04-28T21:16:09.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="14014" PostTypeId="2" ParentId="13999" CreationDate="2018-04-30T09:30:37.297" Score="-2" Body="&lt;p&gt;In the 11th grade our math teacher taught us following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;was the maiden brave&lt;/p&gt;&#xA;&lt;p&gt;the belly stays concave&lt;/p&gt;&#xA;&lt;p&gt;but unprotected sex&lt;/p&gt;&#xA;&lt;p&gt;makes the belly convex&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9876" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-04-30T11:08:25.377" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="14015" PostTypeId="1" CreationDate="2018-04-30T09:32:03.483" Score="9" ViewCount="989" Body="&lt;p&gt;I came across this method to perform subtraction using addition and not using the &quot;borrow&quot; concept, apparently because it is harder to learn it that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Video - &lt;a href=&quot;https://www.youtube.com/watch?v=PKOd6S4-iXk&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=PKOd6S4-iXk&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method is referred to as &quot;Austrian method&quot; in &lt;a href=&quot;https://en.wikipedia.org/wiki/Subtraction#The_teaching_of_subtraction_in_schools&quot; rel=&quot;noreferrer&quot;&gt;wikipedia article&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is it really hard for students to learn subtraction using &quot;borrow method&quot;&lt;/li&gt;&#xA;&lt;li&gt;Is the &quot;austrian method&quot;, better than the borrow method? &lt;/li&gt;&#xA;&lt;li&gt;Are there any studies into which methods of teaching subtraction are more/less helpful to students?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I found reference to this method &lt;a href=&quot;https://archive.org/stream/teachingarithme00klapgoog#page/n194/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;in the book &quot;The Teaching of Arithmetic; a Manual for Teachers&quot; by Paul Klapper, published in 1921&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rPhtH.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rPhtH.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has this &quot;controversy&quot; been settled by now?&lt;/p&gt;&#xA;" OwnerUserId="1945" LastEditorUserId="1945" LastEditDate="2018-04-30T10:32:01.197" LastActivityDate="2019-09-17T18:23:19.130" Title="Subtraction using Addition (Austrian Method), is it useful to learn this method instead of the usual &quot;borrow&quot; method?" Tags="|primary-education|arithmetic-operations|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="14016" PostTypeId="2" ParentId="14015" CreationDate="2018-04-30T13:07:53.463" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Subtraction&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; says the Austrian method is now used in some countries in Europe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I looked up what the Austrian method is.  I recognized it from the patter in one of Tom Lehrer's comic songs, &quot;New Math&quot;.  Here it is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;and&#xA;  if you're under 35 or went to a private school you say seven from three is six, but if you're over 35 and went to a public school you say eight from four is six;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So perhaps the Austrian method was still used in America up to 25 years before that song, whenver that was...&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2018-04-30T13:07:53.463" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="14017" PostTypeId="2" ParentId="12234" CreationDate="2018-05-01T09:58:50.990" Score="1" Body="&lt;p&gt;In the actual teaching:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definitely, independence of more than 2 events or random variables. Even teachers or professors make this mistake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Independence of 3 events means pairwise independence and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$P(\bigcap_{i=1}^3 A_i) = \prod_{i=1}^{3}P(A_i).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Few students think it's only pairwise. Many will think it's only the latter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Independence_(probability_theory)#cite_note-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Independence_(probability_theory)#cite_note-4&lt;/a&gt;&#xA;&lt;a href=&quot;http://www.engr.mun.ca/~ggeorge/MathGaz04.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.engr.mun.ca/~ggeorge/MathGaz04.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the applications of probability or statistics in real life:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nassim Nicholas Taleb aka &lt;a href=&quot;https://math.stackexchange.com/users/88078/nero&quot;&gt;Nero&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5987" LastActivityDate="2018-05-01T09:58:50.990" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="14020" PostTypeId="2" ParentId="12234" CreationDate="2018-05-01T17:23:29.493" Score="1" Body="&lt;p&gt;A very common example (but not mentioned yet) is the birthday problem.  I think this is a nice teaching example since it is so counter-intuitive.  Also because it is easy to demonstrate (right in class) if you have a decent sized section.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See:  &lt;a href=&quot;http://www.math.cornell.edu/~mec/2008-2009/TianyiZheng/Birthday.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.cornell.edu/~mec/2008-2009/TianyiZheng/Birthday.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I don't have my textbook any more (darned book borrower/thieves) but I would think your assigned textbook or alternate texts would contain several examples.  (Not meant to deflect the question, but just that we don't have to solve this entirely from scratch.  And/or that it might be useful to know in what ways the written examples are inadequate to guide further ideation.) &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-05-01T17:23:29.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="14021" PostTypeId="1" AcceptedAnswerId="14024" CreationDate="2018-05-02T01:07:17.293" Score="1" ViewCount="136" Body="&lt;p&gt;I was taught under the New Math, so I should know this, but I am afraid I was tricked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the cardinal, it is easy to define a multiplication, as the cardinal of the cartesian product is the product of cardinals, and a substraction, as the cardinal of a subset is the cardinal of the total set minus the cardinal of the complementary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the obvious use of the union of sets fails to build an addition of cardinals. You need to add also the cardinal of the intersection, and then you need a definition of sum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there some way to get out of this loop, or an alternate construction? The SMSG textbook for the teacher simply suggests to use nonintersecting sets and that it is right to do not tell the pupils about the issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps is there some canonical family of sets so that all of them are not intersecting? Say, in the way that to define max and min we can build the family of ordinal sets.&lt;/p&gt;&#xA;" OwnerUserId="9854" LastActivityDate="2018-05-02T10:13:48.647" Title="How to build addition with sets?" Tags="|primary-education|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14022" PostTypeId="1" CreationDate="2018-05-02T03:10:44.370" Score="3" ViewCount="377" Body="&lt;p&gt;I am working on a unit plan dealing with tranformation geometry level secondary education. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students do enjoy creating those visual transformation but&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to figure out what could be a nice project with transformation geometry... that can answer a dilemma existing in the real life and that can be appealing to a teenager.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;NEW EDIT:&#xA;I have been reading/prospecting the very interesting link below. the central point is symmetry and transformation. The math of it seems to be focused on the creating groups (i.e. group symmetry, the &quot;17-wallpapers&quot;), how symmetry is fundamental in physics, defining or constructing a transformation or a composition of transformation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far, I envision as final project where students create an expository gallery of (let's say) 6 tessellations. Starting with some tessellations involving simple plane-figures and simple transformation up to having a final  tessellation involving a composition of figure (maybe a pentagon as mandatory to create a constraint as it does not &quot;tessellate by itself&quot;). For each tessellation, 2 parts: The 1st part the image of the construction phases done with geogebra + the description of those of the transformation fitting with mathematics phrasing, 2nd part the creative result and a paragraph explaining the &quot;beauty&quot; of it. The &quot;beauty of it&quot; would essentially be about describing the symmetries and (eventually, if it is not too complex for students) the operation under this group. Students will then assemble those in a gallery or in an electronic portfolio.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far this is what I heading to ... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Symmetry (and tessllation) is a first time encounter for me. What do you think about what I am proposing? What could the drawbacks be? &lt;/p&gt;&#xA;" OwnerUserId="9111" LastEditorUserId="9111" LastEditDate="2018-05-06T03:47:15.733" LastActivityDate="2018-05-06T03:47:15.733" Title="Project with transformation geometry" Tags="|student-motivation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14023" PostTypeId="2" ParentId="561" CreationDate="2018-05-02T04:23:47.980" Score="11" Body="&lt;p&gt;The way a calculus course is usually built, before integrals you do Riemann sums and before that you do sums. I try to stress from the beginning in the sums that the running index is a &quot;dummy variable&quot; that disappears when you evaluate the sum. So&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\displaystyle \sum_{i=1}^n a_i = \text{some formula in } n, \text{but not in } i$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for example&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\displaystyle\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} \text{ (-- the } i \text{ has vanished, what matters is the upper bound $n$)}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And when one is at the integrals, completely similarly the $t$ under the integral is a dummy variable, so&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\displaystyle \int_{0}^{x} f(t) dt = \text{some formula in } x, \text{but not in } t$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for example &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\displaystyle \int_{0}^{x} t^2 dt = \frac{1}{3}x^3 + C \text{ (-- the } t \text{ has vanished, what matters is the upper integral limit $x$)}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So regarding your specific example, I would say writing $\int_0^x x^2dx$ is like writing $\sum_{i=1}^n n^2$. (I know in a specific sense it is not, actually; but I think this perspective can help students see the issue.)&lt;/p&gt;&#xA;" OwnerUserId="8931" LastActivityDate="2018-05-02T04:23:47.980" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14024" PostTypeId="2" ParentId="14021" CreationDate="2018-05-02T10:13:48.647" Score="2" Body="&lt;p&gt;You can force the union to be of disjoint sets by pairing: instead of the cardinal of $A \cup B$ you take the cardinal of $(A \times \{0\}) \cup (B \times \{1\})$.&lt;/p&gt;&#xA;" OwnerUserId="8005" LastActivityDate="2018-05-02T10:13:48.647" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14025" PostTypeId="2" ParentId="561" CreationDate="2018-05-02T12:15:52.270" Score="4" Body="&lt;p&gt;I believe this problem is caused by this needless desire to abbreviate everything into confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's face it : everybody writes an integral as: $\displaystyle \int_{0}^{1} f(x) dx$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this is completely wrong: in fact it should be: $\displaystyle \int_{x=0}^{x=1} f(x) dx$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the second case it is clear that the boundaries mean that those are the values, taken by $x$ (you start by $x$ being 0, and you let $x$ go to 1). Only when this is very clear to the students, you might get go further and introduce the short notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So let's get to the question, what about: $\displaystyle \int_{0}^{x} f(x) dx$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can simply get back to the definition: $\displaystyle \int_{x=0}^{x=x} f(x) dx$ : do you mean that you want $x$ to go from 0 to $x$? (Here the student will need to admit that the whole question makes no sense) :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2018-05-02T12:15:52.270" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14026" PostTypeId="2" ParentId="561" CreationDate="2018-05-02T15:24:04.190" Score="1" Body="&lt;p&gt;For an example, try the following, which must be nearly the simplest possible example. Ask them to calculate directly the area of the triangle with vertices $(0, 0)$, $(x, 0)$, and $(x, x)$ as a function of $x$. Then do the calculation using integrals (so calculate $\int_{0}^{x}y\,dy$). Compare the result with the also sensible integral $\int_{0}^{x}x\,dy$, and investigate with them how this second integral calculates something else and what it is that it calculates (the area of the unique rectangle with the aforementioned vertices). This should help with explaining that the formally similar expression $\int_{0}^{x}x\,dx$ is meaningless. It might be helpful to do all of the following with some other letter, e.g. $b$, in place of $x$. I suspect many students will find that less confusing - because the variable name $x$ has for them magical significance as the universal argument of generic functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see various issues underlying student confusion in this context. Although I find it hard to articulate them well, I'll give it a try.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Students generally have a shaky understanding of the function concept. They see $f(x)$ and $f(y)$ as different objects, so can't see how the integral of $f(x)$ could be written as $\int_{0}^{x}f(y)\,dy$. To the student, $f(x)$ and $f(y)$ are different functions rather than a single function taking differently labeled arguments. This confusion is sometimes reinforced by the common practice of writing $f(s)$ as an abbreviation of the composition $f(r(s))$ (or something similar). This is somehow the same difficulty students have in understanding the chain rule, or even something apparently simpler such as that the variable of the sine functions is measured in radians not degrees.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students don't think of the integral notation as a symbolic shorthand indicating what needs to be done. In general, they like to translate notation literally into operations, and the integral notation has to be interpreted before it is made operational. Although they tend to think formally, they don't think operationally. This difficulty is apparent when one writes something like $\int_{\Gamma}E\cdot dr$ for the line integral of a vector field $E$ along a curve $\Gamma$. To actually calculate such an integral one has to parameterize the curve as $\gamma(t)$ for some $t \in [a,b]$ and compute the one-dimensional integral $\int_{a}^{b}E(\gamma(t))\cdot \dot{\gamma}(t)\,dt$ (and that the result does not depend on the choice of parameterization). The notation $\int_{\Gamma}E\cdot dr$ is a symbolic shorthand for this recipe and not something that can be put into some sort of 1-1 correspondence with the operations involved. With ordinary integrals the same interpretative issues arise, but are somehow harder to see because of the apparent simplicity of the context. The student doesn't read $\int \dots \,dy$ as notation indicating an operation with two arguments, one functional and one geometric, and the student doesn't interpret $\int_{a}^{b}$ as the integral over the interval $[a, b]$, so much as indicating a certain formal substitution procedure. From this point of view, $b$ will always be substituted by a number, and allowing it to vary, and treating it as a variable, and calling it $x$ rather than $b$, are all unnatural acts. In the context of ordinary integrals, the difficulty in interpreting the notation is compounded by the circumstance that many students incorrectly regard the so-called fundamental theorem of calculus as a tautology. For them the integral &lt;em&gt;is&lt;/em&gt; the antiderivative operator formally inverse to differentiation, that sends a function to the equivalence class of its primitives, rather than the limit of some complicated (to them) sums (they really don't see why the professor fusses so much about these sums since they apparently never get used to calculate anything ...) and so the putative theorem &lt;em&gt;is&lt;/em&gt; a tautology, showing that something incomprehensible can be evaluated in the usual way. The limits of the integral are then just placeholders indicating what to do with the primitive so obtained. They are indeterminate numbers (operationally always given fixed values) rather than variables.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;To my mind the solution to these difficulties requires a variety of approaches.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;One has to communicate that the integral is a limit of sums, motivated by the calculation of areas, and give examples/exercises that require it to be approximated.&lt;/li&gt;&#xA;&lt;li&gt;One has to communicate that actually calculating primitives is in general impossible except via integration. That is, one doesn't generally know an explicitly given function that is the primitive of a given function. Too many calculus courses give the impression that we &lt;em&gt;know&lt;/em&gt; the primitives of most functions.&lt;/li&gt;&#xA;&lt;li&gt;One has to train them well in the function concept, usually before teaching integration. It makes an early problematic appearance with the chain rule for calculating the derivative of a change of variables. In general, students who don't understand composition of functions well will not deal well with integrals. &lt;/li&gt;&#xA;&lt;li&gt;The confusion mentioned in the question is related to the difficulties students have with change of variables in integration. Change of variables is often taught as a formal procedure with no reference to composition of functions or change of domain. &lt;/li&gt;&#xA;&lt;li&gt;An integral has two arguments, not one. One is functional (the integrand), the other is geometric (the interval of integration), and the integral is a pairing assigning a number to a function and an interval. This point of view makes it as natural to vary the interval as to vary the function. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-05-02T15:24:04.190" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14028" PostTypeId="2" ParentId="14022" CreationDate="2018-05-02T22:18:05.987" Score="0" Body="&lt;p&gt;Do teenagers still play minigolf or pool? Both of those activities may be analyzed using transformation geometry. Not exactly a single project, but an area to explore perhaps&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2018-05-02T22:18:05.987" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14029" PostTypeId="1" AcceptedAnswerId="14030" CreationDate="2018-05-02T22:30:51.687" Score="13" ViewCount="564" Body="&lt;p&gt;Is the current state of math teaching in undergrad college courses struggling with the availability of easy cheap access to Wolfram Alpha?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The homework problem below, one of 40 assigned from one community college lecture, took my son and me each several hours to solve. (He struggled because he's new at this; I struggled because I was rusty at it.) He is enrolled in a second-quarter calculus course which he's already failed once after a lifetime of not struggling in school. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find the arclength between 2 and 5 of the function: &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$f(x)$ $=$ $x^5 \over 10$ $+$ $1 \over 6x^3$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The answer is $618639 \over 2000$, taken through the arclength integral &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\int_{a}^{b} \sqrt{1+(\frac{dy}{dx})^2} dx$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That is: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\int_{2}^{5} \sqrt{1+\frac{(-1 + x^8)^2}{4x^8}} = {618639 \over 2000}$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;His question is: Why bother with the tedious symbol manipulation given WolframAlpha as a cheap tool? The answer I'm looking for is a description for why current college maths teaching still sets problem sets like this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can order him to buckle down for a good result and already have; no need for those &quot;it's good for you to struggle&quot; answers. One analogy that justifies him is the fact that teachers no longer require students to learn how to cipher a square root. &lt;strong&gt;So, why would we require them to go through algebra contortions like this if the tool to do the &quot;monkey work&quot; exists and he can spend his time reviewing its output to make sure he set the problem up right?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I disagreed: This problem exposed me to insights about the fundamental theorem of calculus and the shortcuts possible with definite integral problems. The tedious problems teach you the shortcuts Wolfram Alpha is using between steps. You're using these kinds of problems to hone your thinking skills. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there anything salient I could add to that argument? Is this a controversy in maths teaching, or is my son just plain wrong and needs to drill for other reasons? &lt;/p&gt;&#xA;" OwnerUserId="9891" LastEditorUserId="9891" LastEditDate="2018-05-03T01:36:33.720" LastActivityDate="2018-05-03T13:10:21.117" Title="Is higher-math pedagogy responding properly to Wolfram Alpha's existence?" Tags="|mathematical-pedagogy|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14030" PostTypeId="2" ParentId="14029" CreationDate="2018-05-02T23:24:51.197" Score="5" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Why have it at all:  You need to know it to solve physics E&amp;amp;M problems.  (Not even junior level E&amp;amp;M, just calculus-based E&amp;amp;M.)  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In terms of why ever learn to do it by scratch, it's really part of a much bigger question (could ask the same about a lot of math content).  My personal take is that if you are going to do engineering or a physical science, that this is reasonable to learn how to do, explicitly.  Sure, if you have a computer and you are solving these problems all the time, you automate it.  But if you are reading papers, following derivations on the blackboard, etc., you should be able to do this sort of problem.  If you only have a black box knowledge (plug it into WA), you'll struggle in standard physics and engineering classes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think it is really that much symbol manipulation.  Lot less than a series solution to a diffyQ or the like.  Just memorize the formula wih the dy/dx squared or how to derive it (think it is a Pythagorean relation).  Or both.  The only real hard part with arc length problems is remembering the formula.  The work itself is not that intricate (not some double integral washers thingy.)  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-05-02T23:24:51.197" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14031" PostTypeId="2" ParentId="14022" CreationDate="2018-05-03T07:38:11.013" Score="2" Body="&lt;p&gt;Transformation geometry is a beautiful subject with a lot of possibilities for projects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It gives a dynamic view on geometry - as opposed to a static view, e.g. on the notion of congruence. It is - in my opinion - a good starting point to experience typical mathematical ideas on the road to abstraction.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;optimization problems: Understanding the &lt;a href=&quot;https://www.eduhk.hk/apfslt/v5_issue3/manyk/manyk2.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;law of reflection&lt;/a&gt;, &lt;a href=&quot;https://www.researchgate.net/publication/273403825_Fermat%27s_point_from_five_perspectives&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fermat point&lt;/a&gt;, &lt;a href=&quot;https://www.cut-the-knot.org/proofs/napoleon.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;Napoleon's theorem&lt;/a&gt; and alike...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;structuring congruence transformations (isometries of the Euclidean plane): see how every rotation can be written as combination of two reflections; to see which transformations are orientation preserving and which ones are orientation reversing or understanding glide reflections reveals a base to understanding the Euclidean group of isometries eventually. this provides a nice entry point to group theory. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;understanding symmetry as a property of transformations leaving a figure invariant e.g. to understand &lt;a href=&quot;https://en.wikipedia.org/wiki/Frieze_group&quot; rel=&quot;nofollow noreferrer&quot;&gt;frieze&lt;/a&gt; or &lt;a href=&quot;http://www2.clarku.edu/~djoyce/wallpaper/seventeen.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;wallpaper patterns&lt;/a&gt; as well as &lt;a href=&quot;http://www.tessellations.com/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;tessellations&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;exploring subgroups as a strategy to structuring relations among figures, as e.g. in the &lt;a href=&quot;http://subs.emis.de/journals/ZDM/zdm053a8.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;house of quadrilaterals&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Possibly, if you don't want to get too abstract (e.g. studying symmetry groups), comparing frieze and wallpaper patterns from your surrounding is a nice way to see the world around through &quot;math glasses&quot;. I usually let my students take pictures from patterns to analyze, or we look at patterns from different cultures, or they are asked to create their own patterns with some properties given. If you have dynamical geometry software such as &lt;a href=&quot;https://www.geogebra.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;geogebra&lt;/a&gt; available, that naturally gives a nice tool for possible projects. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far so good, I hope you find this answer helpful. I'm sure there are other uses, ideas for projects, maybe other people are willing to share their ideas within further answers or comments.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2018-05-03T07:38:11.013" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14032" PostTypeId="2" ParentId="14029" CreationDate="2018-05-03T13:10:21.117" Score="9" Body="&lt;p&gt;I think I agree with the sentiment of the OP which is why in my classes (which do not allow technology on exams) I only ask that they set up the integral correctly (including correct limits, an integrand where derivatives have been performed, and the correct differential). The hard part about this section of the content is remembering the right formula when you start to have lots of properties that are being computed: arc length, area below, area to left, and surface area and volume when being revolved. (And this is most easily achieved by remembering the geometrical reason behind it, e.g. Pythagorean Theorem.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The great thing about this approach is that you’re free to give the students more realistic and challenging scenarios; you don’t have to worry whether the algebra in the anti-derivative will be doable. (Heck, you don’t even have to worry if the function is algebraically integrable.) And, yes, on homework and projects (or in their science classes) I expect that, once they have the correct integral, they will use Wolfram Alpha to find what they need from there.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2018-05-03T13:10:21.117" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14033" PostTypeId="2" ParentId="13999" CreationDate="2018-05-03T19:38:36.273" Score="4" Body="&lt;p&gt;This only makes sense in Spanish but it's pretty fun. For integration by parts,&#xA;$$&#xA;\int u dv = u v - \int v du&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;S&lt;/strong&gt;i &lt;strong&gt;u&lt;/strong&gt;n &lt;strong&gt;d&lt;/strong&gt;ía &lt;strong&gt;v&lt;/strong&gt;i &lt;strong&gt;u&lt;/strong&gt;na &lt;strong&gt;v&lt;/strong&gt;aca &lt;strong&gt;menos&lt;/strong&gt; &lt;strong&gt;s&lt;/strong&gt;exy &lt;strong&gt;v&lt;/strong&gt;estida &lt;strong&gt;d&lt;/strong&gt;e &lt;strong&gt;u&lt;/strong&gt;niforme&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which translates roughly to:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I saw one day a not-so-sexy cow wearing a uniform. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This mnemonics has generated &lt;a href=&quot;http://img.desmotivaciones.es/201105/vacauniforme2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;awesome memes&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="5922" LastActivityDate="2018-05-03T19:38:36.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14034" PostTypeId="1" CreationDate="2018-05-03T23:04:49.680" Score="42" ViewCount="3031" Body="&lt;p&gt;Alice (not real name) is a student in one of my Math 100 (calculus) classes. It's a course offered by my college as a dual credit course at a high school, so the whole class is about 17/18 years old, and I'm aware that there's sometimes a maturity issue. Still, my high school teacher colleagues agree that Alice is very special.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alice is really into math -- in her own way. Before or after class she comes to me and shows me something she figured out. She plays around with functions on her graphical program and notices cool things about them when she changes the parameters. She looks up which polynomials approximate the exp function and graphs that and finds it cool. She reads about complex numbers and how trigonometric identities find a natural explanation there. After we covered Newton's method, she'll try to find every zero of every function with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And that's where the problem starts: Every zero of every function. In an optimisation problem, where the derivative is an easy parabola, she will not use the quadratic formula to find a critical number, but insists on applying Newton's method. In another optimisation problem she introduces a second variable and wants to find a more general solution, trying to develop multi-variable calculus on her own in the middle of a test, and fails. In a curve sketching question, she does not get beyond the first derivative because she first wants to give a proof of the quotient rule from scratch. In class, she continues to ask about generalisations of the material we cover to the complex setting, even after I have tried to make clear to her that complex numbers are not part of the curriculum and it's important to first get our material straight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, she's very enthusiastic about some mathematics, but only that which catches her attention, and she seems to neglect a lot of the actual material of the course for that. To the extent that she's lost many points on assignments and tests and is at risk to fail the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously I've tried to make the issue clear to her, but I see no success so far. I admit it's hard because often she's enthusiastic about things I'm enthusiastic about myself, and I actually like to chat with her about the Riemann sphere and stuff after class. Should I rigorously cut down such conversations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, part of me &lt;em&gt;likes&lt;/em&gt; that she thinks outside the box. Where most students' minds are too compartmentalised, or they can solve problems only with a memorised standard method -- Alice tries to use her own approaches, or methods from different sections of the course. Problem is she often makes mistakes then, and it takes so much time that she cannot work on other questions. To a lesser degree, I've had students like that before, and I've always tried to reward original or uncommon approaches, even if they don't entirely work out. But with Alice it's on a new level: She just refuses to use standard approaches even if they are obviously the shortest, most practical etc.; but her own approaches, although never stupid, basically always fail to work out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What can I do to make Alice pass the course, ideally without crushing her enthusiasm for mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Maybe I should clarify the following: While I do think that Alice has talents that are underappreciated, as far as I can tell she is &lt;strong&gt;not&lt;/strong&gt;  a hidden genius or savant. When I say &quot;we chat about the Riemann sphere&quot;, it's not like she has a deep understanding of complex geometry, rather like she read about it on a Wikipedia level, understands the basic projection idea, is all fascinated about having a way of putting &quot;infinity&quot; into the numbers. (Not in a crackpot &quot;I can divide by zero&quot; way, but not in a profoundly insightful way either.) I can tell she misunderstands certain concepts, then I try to talk her out of it, which sometimes works and sometimes doesn't. Her approaches are always original, but also often flawed, sometimes for reasons that are obvious to anyone with a formal math education. She still makes basic mistakes. The point I admire about her is the genuine fascination with, and enthusiasm for, mathematics, as well as her potential for original thinking; but her thoughts show a lack of precision and rigour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is not how to save a new Ramanujan from a hostile conformist education system. It's how to help a student getting their priorities straight without crushing their curiosity and motivation for self-learning.&lt;/p&gt;&#xA;" OwnerUserId="8931" LastEditorUserId="8931" LastEditDate="2018-05-04T23:33:58.140" LastActivityDate="2024-02-23T09:14:45.023" Title="How can I help a student who has a &quot;wrong&quot; kind of enthusiasm?" Tags="|undergraduate-education|secondary-education|calculus|student-motivation|" AnswerCount="5" CommentCount="23" ContentLicense="CC BY-SA 4.0" />
  <row Id="14035" PostTypeId="2" ParentId="14034" CreationDate="2018-05-04T00:12:22.883" Score="-11" Body="&lt;p&gt;&quot;She seems to neglect a lot of the actual material of the course...to the extent that she's lost many points on assignments and tests and is at risk to fail the class.  Obviously I've tried to make the issue clear to her, but I see no success so far. I admit it's hard because often she's enthusiastic about things I'm enthusiastic about myself, and I actually like to chat with her about the Riemann sphere and stuff after class. Should I rigorously cut down such conversations?&quot;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Stop encouraging her behavior.  If she were getting done what she needs to get done, then fine, no problem.  But her interest in estimation is interfering with mastering basic learning objectives for analytical solution.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Discourage the behavior.   &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Get the parents involved.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Slam her on the tests.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;[If it still doesn't work, oh well.  You can't save them all.  And it is her life.  People fail classes for a lot of reasons (ability, laziness, disinterest, and misprioritization).]  &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-05-04T00:12:22.883" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14036" PostTypeId="2" ParentId="14034" CreationDate="2018-05-04T11:58:16.690" Score="4" Body="&lt;p&gt;I envy you: having such a student is very, very interesting and pleasant, but obviously, if she fails due to this immature behaviour, she (and you) will have a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first thing which comes in my mind: how smart is she?&lt;br&gt;&#xA;Let me explain: you say that, for solving a simple parabola, she uses Newton instead of simple quadratic solving formula.&lt;br&gt;&#xA;So why not use both? She can use the Newton way, use the quadratic formula, and be amazed that both are converging to the same result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This approach (using her enthusiasm for leading her to the actual course content) is only applicable if she is smart enough to cover both in a reasonable timeframe. Is this the case? (and can you even know that? Are you sure she's not neglecting her other courses because of her interest in mathematics?)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2018-05-04T11:58:16.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14037" PostTypeId="2" ParentId="14034" CreationDate="2018-05-04T12:46:32.107" Score="17" Body="&lt;p&gt;My answer is maybe a little bit off. Still, I have had some luck in the past with two separate similar students to yours by communicating roughly the following concept using the outline below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;An expert knows both the theorem and the proof.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask this student what is $3 \cdot 5$. Then ask the student how they know for sure that this is the right value of $3 \cdot 5$. An expert multiplier will &lt;em&gt;both&lt;/em&gt; know the answer is 15 and that the reason this works is because you can make three rows of five dots and count them up. Probably your student is an expert multiplier (If it turns out she is not, move back to addition and use $3 + 4$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Establish with the student that it would be possible for a student to make it through elementary school in either of the following two unfortunate situations:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The student has memorized that $3 \cdot 5 = 15$, but does not know why and would not be able to convince anyone who had memorized it incorrectly.&lt;/li&gt;&#xA;&lt;li&gt;The student has not memorized that $3 \cdot 5 = 15$, and when presented with this problem, always draws all the little dots and counts them up.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Both of these situations would be bad. Try to get her to explain why they would be bad.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explain to the student that neither one of these hypothetical situations spells permanent doom. A student who refuses to learn reasons can eventually be convinced to shift focus toward reasons. A student who refuses to buckle down and learn facts can eventually be convinced to shift focus toward facts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually we end up needing this idea to tell students that they should seek understanding beyond mere shortcuts and cookbook methods that they are interested in learning. Tell the student this. It will be useful to engage the student where she is, which is that she thinks learning facts and efficient methods is &lt;strong&gt;boring&lt;/strong&gt;. She knows she is exceptional and it is useful and good to acknowledge this.  You could even go so far as to say that of the two unfortunate situations we talked about, the second one is &lt;strong&gt;better&lt;/strong&gt; because it has a higher likelihood to lead to success in the end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, at the end you need to tell her that she is in danger. That she is too focused on one part of math -- the part of math that most students find uninteresting, which is really good and awesome -- but that she could fall through the cracks and needs to refocus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That in the end, it's cool that she is interested in the proofs, but can't spend her whole life multiplying numbers using arrays of dots. She has to learn the theorems too.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-05-04T12:46:32.107" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14038" PostTypeId="1" AcceptedAnswerId="14040" CreationDate="2018-05-05T21:14:26.003" Score="2" ViewCount="408" Body="&lt;p&gt;Do you think that 10 options for multiple choice questions is too much? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am talking about Calculus Courses, if it makes difference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks &lt;/p&gt;&#xA;" OwnerUserId="9255" LastActivityDate="2019-10-17T11:31:38.043" Title="Ten options for multiple choices questions" Tags="|student-question|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14039" PostTypeId="2" ParentId="14038" CreationDate="2018-05-05T23:45:47.510" Score="10" Body="&lt;p&gt;Presumably, you are trying to reduce the chance that someone will get a correct answer by guessing. In that case, I think that 10 options are more than you need. Suppose that we have an $N$ question multiple choice exam, each of which has $M$ options. Then a randomly guessing student gets $Q$ questions correct, where $Q$ is sampled from the binomial distribution $B(N,1/M)$. If we define 60% as passing, what's the probability of passing for some values of $N$ and $M$? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\begin{array}{c|c|c|c|}&#xA;    &amp;amp; N=5    &amp;amp; N=6    &amp;amp; N=7    \\ \hline&#xA;M=3 &amp;amp; 0.2098 &amp;amp; 0.1001 &amp;amp; 0.0453 \\ \hline&#xA;M=4 &amp;amp; 0.1035 &amp;amp; 0.0376 &amp;amp; 0.0129 \\ \hline&#xA;M=5 &amp;amp; 0.0579 &amp;amp; 0.0170 &amp;amp; 0.0047 \\ \hline&#xA;M=6 &amp;amp; 0.0355 &amp;amp; 0.0087 &amp;amp; 0.0020 \\ \hline&#xA;M=7 &amp;amp; 0.0233 &amp;amp; 0.0049 &amp;amp; 0.0010 \\ \hline&#xA;\end{array}&#xA;$$&#xA;So there is maybe a case for moving from 4 to 5, if that's what your default was before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the randomly guessing student model is essentially assuming that the student knows &lt;em&gt;nothing&lt;/em&gt;, which is hopefully not true. They will be able to eliminate some. I don't think that I, or anyone else, would be very good at generating 9 incorrect, but plausible, options. It would end up something like&lt;br&gt;&#xA;&quot;What is $\int 2x dx$?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) $x^2+C$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) $x^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) $2x^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) $2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) $2dx$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6) 42&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7) aardvark&lt;/p&gt;&#xA;&#xA;&lt;p&gt;8) hat&lt;/p&gt;&#xA;&#xA;&lt;p&gt;9)  ¯\ _ (ツ)_ /¯&lt;/p&gt;&#xA;&#xA;&lt;p&gt;10) don't pick this one, it is not the answer&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So there are only 4 or 5 possibilities that are not &lt;em&gt;obviously&lt;/em&gt; wrong.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-05-05T23:45:47.510" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14040" PostTypeId="2" ParentId="14038" CreationDate="2018-05-06T00:45:00.567" Score="4" Body="&lt;p&gt;I would say it completely depends on what is being asked. For example, something simple like &quot;What is the derivative of &lt;span class=&quot;math-container&quot;&gt;$\cos(x)$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;?&quot; &lt;em&gt;could&lt;/em&gt; have as many as 10 options, because no real work or computations are needed -- either you know this one or you don't. [Assuming, of course, that you don't include both &lt;span class=&quot;math-container&quot;&gt;$-\sin(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-\frac{\pi}{180}\sin(x)$&lt;/span&gt; as options.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because answers in calculus can often be simplified, it would be tedious for a student to have to verify if their answer was among 10 different options, just in a different form. Therefore, I am fine with a large number of possible options if they are fairly simple but obviously distinct.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="238" LastEditDate="2019-10-11T19:53:34.390" LastActivityDate="2019-10-11T19:53:34.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14041" PostTypeId="2" ParentId="14034" CreationDate="2018-05-06T03:07:53.783" Score="9" Body="&lt;p&gt;First, a response to a particular excerpt:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I admit it's hard because often she's enthusiastic about things I'm enthusiastic about myself, and I actually like to chat with her about the Riemann sphere and stuff after class. Should I rigorously cut down such conversations?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Stepping back for a moment: This is a situation in which a student is enthusiastic about mathematics, you are enthusiastic about mathematics, and you are asking whether to &lt;em&gt;stop&lt;/em&gt; conversing enthusiastically with her about mathematics. To this I say: &lt;strong&gt;No&lt;/strong&gt;! How wonderful it is that you have an opportunity to engage meaningfully with a student whose interests in mathematics are both burgeoning and variegated. Keep the conversations going.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Back to the remaining issues: Alan Schoenfeld has a nice book called &quot;Mathematical Problem Solving&quot; (1985) and another called &quot;How We Think&quot; (2010) that I've mentioned elsewhere on MESE in the past. The gist of his ideas, for the purpose of this question, boil down as follows: problem solving (in mathematics) involves &lt;strong&gt;beliefs&lt;/strong&gt; (e.g., how long does a typical problem take to solve?), &lt;strong&gt;strategies&lt;/strong&gt; or &lt;strong&gt;heuristics&lt;/strong&gt; (e.g., quadratic formula, Newton's method), and &lt;strong&gt;metacognition&lt;/strong&gt; (thinking about thinking in the context of problem solving - e.g., how should I deploy my strategic resources in order to make headway on this problem? or: I've been trying a particular strategy for a while; should I adjust?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student that you are describing seems to be doing well with &lt;strong&gt;beliefs&lt;/strong&gt; (e.g., she is enthusiastic about mathematics, wants to discuss mathematics, believes herself capable of engaging with mathematics) and seems to be interested in various &lt;strong&gt;strategies&lt;/strong&gt; (even delving into material that is not formally part of your course). But, I think that the &lt;strong&gt;metacognitive&lt;/strong&gt; component is where some issues are arising: An important part of mathematical thinking is matching problems with problem solving methods, and (in my interpretation) that means picking an approach as a function of the problem. (So, not picking the approach - e.g., Newton's method - beforehand, and then trying to apply it willy-nilly irrespective of the problem at hand.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Here is my concrete suggestion&lt;/strong&gt; (and maybe I will reedit this later if you update your question, or leave a comment, etc): Since the student seems not short on energy/enthusiasm, see if you can get her to read the article &quot;What's All the Fuss About Metacognition?&quot; mentioned in my MESE answer &lt;a href=&quot;https://matheducators.stackexchange.com/a/2399&quot;&gt;here&lt;/a&gt;. If you cannot find a full copy, then let me know and I'll upload one. Then, see if you can engage with the student about mathematics not being &lt;em&gt;just&lt;/em&gt; strategies/resources (which she seems to be accruing impressively!) but also thinking about how they are used when faced with a particular mathematical question for which the method of solution is unknown at the outset (i.e., attending to metacognition: fitting a method to a problem rather than the other way around).&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-05-06T03:07:53.783" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14042" PostTypeId="2" ParentId="14038" CreationDate="2018-05-06T23:02:33.407" Score="5" Body="&lt;p&gt;I do think that 10 options for a multiple-choice question is excessive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few things to consider: This will be outside the range that standard automation tools can handle (TestGen application, Scantron sheets, premade testbanks, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I started teaching, I thought that using multiple-choice questions would be a time saver for my classes. But what I found was that writing (and organizing, scrambling) the distractor options took more time than I was saving. Especially if the problems should change each semester, the only way that's feasible is with an automated tool (see above). That was just with 4 options each; the 10-question proposal would be infeasible × 2.5 in my experience. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would already recommend that at the level of college credit-bearing courses, that the questions be short-answer anyway. Feel free to grade the problems on a simple 3- or 5-point rubric each. I think you'll find that to be actually less work than trying to write 10 distractor options per question. And you'll also get to see where the actual problems and strengths of the students lie, as opposed to being masked through the multiple-choice filter.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-05-07T21:59:57.200" LastActivityDate="2018-05-07T21:59:57.200" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14043" PostTypeId="1" AcceptedAnswerId="14044" CreationDate="2018-05-07T02:14:01.107" Score="4" ViewCount="236" Body="&lt;p&gt;In order to teach students what &quot;statistical modeling&quot; is, it would be helpful to show them some very concrete, specific (and maybe classic) examples of statistical models. Something you can point to and say, &lt;em&gt;this&lt;/em&gt; is a statistical model. Do you recommend any examples of statistical models to use for this purpose?&lt;/p&gt;&#xA;" OwnerUserId="9633" LastActivityDate="2018-05-07T04:48:01.060" Title="What are some very concrete examples of statistical models to present to students?" Tags="|statistics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14044" PostTypeId="2" ParentId="14043" CreationDate="2018-05-07T03:06:11.020" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Oil decline curves and EUR calculation.  Lot of good articles on the web and Youtube videos.  [I would link, but seriously there are so many explanations...if one is too technical or long, skip around and find another.]  Very accessible (to the teacher, I would hope..little translation perhaps helpful for the students.  You have a time series that is imperfect (even for a single instance) and then you have multiple instances (natural resources have variability even within a specific area).  Typical regression is an exponential decline but some wells (especially shale) have hyperbolic decline (so the regression is not only getting an initial decay constant but the &quot;b&quot; exponent of the hyperbolic function).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Election predictions (see 538 blogsite or Andrew Gelman's blog).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Many chemical or other manufacturing processes.  Sorry, I don't have an exact one to share but again basic Google or Youtube search should come up with many real data examples.  This is sort of a classic type of model since you can regress temp, pressure, reactants, flow, etc. and there can be some complexity (nonlinearity, multiplicative factors).  Typically rate of production of one product is the key output variable, to optimize, but it can be more complex...optimizing profit for instance, which will depend both on output of product and cost of the inducing conditions.  Often the situation is so complex that there is no theoretical equation, even the machinery dimensions can change things, thus you gather data and make a model statistically rather than using a function.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Of course crop yields (this is where statistics started).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Safety models (nuclear reactors, aircraft, spaceships, etc.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Lots of business problems ($ output as a function of salesperson skill, customer classification, etc.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hiring efficacy and performance prediction (Journal of Applied Psychology is the best journal for thoughtful HR statistics).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Biostatistics:  both disease detection and causation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The whole field of sports statistics.  An example I like is outcome of NFL players by draft position.  An analog output variable might be years in the league or a digital one is specific outcomes (made roster, started, made Pro Bowl, All Pro, etc.)  The cool thing about building this model is that you actually have a lot more input gradation than just &quot;round&quot; (a common sports misconception) if you use numerical position.  End of the first is more like beginning of the second than beginning of the first.  Also, NFL has implemented compensatory picks so the picks per round is variable.  A confounding variable to watch out for is size of the league (if you go back far enough).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Most of the field of climatology AND meteorology.  A nice specific one is annual tree ring size as a function of temp and moisture.  Lots of variability for uncontrolled factors (insects, fire, spacing, etc.)  Even tree ring size on a specific tree (at a specific height) has variability as the rings are not perfect circles (usually 3 cores are taken and averaged). Cutting a section gives better control of course (and more accurate measurement, less human factor confounding...but it kills the tree!)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Student performance (ed research) as output based on specific training methods used (and population controls: SES, IQ, age, sex, yada yada).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Advertising reaction (clickthrough, purchase decision, etc.) as a function of type and frequency of touch as well as demographic customer factors: age, income, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Running rats through mazes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(Probably a gazillion others I'm not thinking of)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Just saw that you want a specific recommendation to use:  I would use the chemical process.  It is very intuitive and people have kitchen cooking experience to know that temp, time, etc. will affect output.  This is also the first model that Box, Hunter, Hunter use as a classic explanation.  Think this is an example, where you don't really need to get too much into discussing physics or industry specifics (reactants A, B, C, etc.).  Think it is easier for the first example to be rather broad and hypothetical rather than using real data or a real business problem.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An alternate would be biostatistics (say cancer as a function of variables like age, smoking, etc.).  This is one that is also very intuitive to students.  If you use that I would keep it simple in terminology at the beginning of a course (avoid having to teach specific biology).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I don't mean this aggressively but am puzzled how a teacher would not already know some of these (or many others).  Any textbook will be full of them.  Box, Hunter, Hunter Statistics for Experimenters is an excellent reference.  Very easy to read and full of examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.  Stats rocks.  Remember what Hamming said (&quot;I won't fly in an aircraft that depends on the definition of the Lebesgue Integral.&quot;)  But he sure flew in some that depended on statistics!&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-05-07T04:48:01.060" LastActivityDate="2018-05-07T04:48:01.060" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14045" PostTypeId="2" ParentId="7976" CreationDate="2018-05-07T06:10:00.377" Score="0" Body="&lt;p&gt;I think the thing that makes it hard is the restriction to less than 35 elements.  There are many sources of vast and intricate real world data but not so many with so few elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very simple one (to do means, SD, histograms, etc.) is number of regular season wins by NFL team.  There are only 32 teams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some simple exercises are mean for the league (forced to be 8, if you call a tie half a win..I would)  You can not tell them this and just see how many note it, after they calculate it.  Also, then SD (this is not forced).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can do some comparisons (AFC versus NFC mean).  Is the difference statistically relevant? (p value).  How about division versus division?  (Soon you get into a Bonferroni situation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also do something a little more complicated by taking 2016 results and 2017 results and see how much 2016 performance predicts 2017 results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a couple sites with tables of the data:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.espn.com/nfl/standings&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.espn.com/nfl/standings&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nfl.com/standings/division/2017/REG&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nfl.com/standings/division/2017/REG&lt;/a&gt; (this one has a little menu where you can grab 2016, etc. also)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can cut and paste the data into excel. Have to clean up the repeated headers a little and maybe cut some columns if you want. But it should be pretty easy.  Maybe add a column for division and conference.  Really playing with something like this on excel (including using the &quot;filters&quot;) is an easy exercise and good thing for people to know how to do in the work world.  We haven't gotten to the point where admin assistants are proficient in playing around with filtering data in excel (but of course students are new and some will need to learn...so good). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the size of the data is small enough so that it is very intuitive but complex enough to be interesting (32 rows for the 32 teams and then somewhere between 4 and 12 columns depending on how complicated you want to get). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it bothers you that it is sports, you could turn it into some sort of artificial business example...but then you're not real any more.  I would just use it as is.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-05-07T06:10:00.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14046" PostTypeId="2" ParentId="7976" CreationDate="2018-05-07T06:33:06.270" Score="4" Body="&lt;p&gt;An alternate suggestion:  EIA.gov has WONDERFUL statistics and easily downloaded.  The site has a gazillion pages so you may want to navigate around.  But almost all of them that have a table or graph also have a &quot;download data&quot; button (usually in upper right).  It usually comes over in a nice formatted excel spreadsheet.  See here for the site overall:  &lt;a href=&quot;https://www.eia.gov/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.eia.gov/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A lot of what they have is time series and that won't be so useful to you.  But you can just take a snapshot in time.  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;For instance, if you look at 2017 oil production (annual M bpd), you get a nice Pareto pattern (80% production from 20% states).  Here is the link for 2017 oil production:  &lt;a href=&quot;https://www.eia.gov/dnav/pet/pet_crd_crpdn_adc_mbblpd_a.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.eia.gov/dnav/pet/pet_crd_crpdn_adc_mbblpd_a.htm&lt;/a&gt; (download data is in upper left).  Yes the units is thousands of barrels per day...but at least it is not acre feet!  And the states should be recognizable!  (well actually there are a few subtleties like the GOM offshore and PAC offshore &quot;states&quot; and then the states that have zero production, not in database.)  You may want to clean it up a little to eliminate the PADDs (or leave them in if you find that interesting).  This is close enough to a real &quot;business problem&quot; (production of an economic good) as to be interesting.  I think it just barely meets the 35 element limit (if not, combine the two Alaska series and/or the two offshore series).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Here is natural gas by state:  &lt;a href=&quot;https://www.eia.gov/dnav/ng/ng_prod_sum_a_EPG0_VGM_mmcf_a.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.eia.gov/dnav/ng/ng_prod_sum_a_EPG0_VGM_mmcf_a.htm&lt;/a&gt;  (Probably clean it up a little by combining some of the state offshore/onshore).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a lot of others stuff on the site (pricing, oil by API gravity, overall energy, other forms of energy: coal, renewables, per capita consumption, etc.)  Again a lot of it is time series, but you can pick a year.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-05-07T06:33:06.270" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14047" PostTypeId="2" ParentId="7976" CreationDate="2018-05-07T15:29:14.470" Score="2" Body="&lt;p&gt;... since it's not yet listed: &#xA;&lt;a href=&quot;https://www.gapminder.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gapminder&lt;/a&gt; is not only a beautiful tool for data visualization, it also allows to &lt;a href=&quot;https://www.gapminder.org/data/&quot; rel=&quot;nofollow noreferrer&quot;&gt;download the data&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2018-05-07T15:29:14.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14048" PostTypeId="2" ParentId="14038" CreationDate="2018-05-08T03:46:26.313" Score="6" Body="&lt;p&gt;There's a fair amount of research about this, mainly from psychology and medical education as well es test development research for high stakes tests.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practical development, I usually resort to the &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/13058/how-to-design-multiple-choice-questions/13063#13063&quot;&gt;gold standard&lt;/a&gt;&quot; and either collect sample answers analyzing the frequency and types of errors of students. Or I consult research articles on common misconceptions and typical mistakes to guide the formulation of distractors. In this sense:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Done right: 3 is enough.&quot; (see e.g. &lt;a href=&quot;https://sites.educ.ualberta.ca/staff/ldelia/M%27Choice%27Tests=Haladyna_Guidelines_AME_2002.pdf&quot; rel=&quot;noreferrer&quot;&gt;Haladyna et al (2002)&lt;/a&gt; and &lt;a href=&quot;http://edmeasurement.net/8268/Rodriguez-2005-Random-effects-meta-analysis-three-options.pdf&quot; rel=&quot;noreferrer&quot;&gt;Rodriguez (2005)&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I usually go with 4 Options on the most frequent mistakes (or anticipated from them). sometimes typical &quot;slips of the pen&quot;-mistakes make valid distractors as well (such as a change of &quot;+&quot; and &quot;-&quot; sign in answer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Upshot: In the answer given by &lt;a href=&quot;https://matheducators.stackexchange.com/a/14039/9004&quot;&gt;Adam&lt;/a&gt;, take the 4 or 5 options that are not &lt;em&gt;obviously&lt;/em&gt; wrong and leave out the rest.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2018-05-08T03:46:26.313" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14051" PostTypeId="1" CreationDate="2018-05-08T18:58:27.077" Score="5" ViewCount="466" Body="&lt;p&gt;Venn diagrams have got to infiltrate the modern curricula as a way to explain probability theory, discussing sets of events. Not surprising, and we can say that they never left us really in topology and other usages of the lattice of union and intersection of sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I have recently noticed that they are being used also for a different task, the analysis of factors of integer numbers and polynomials, and the subsequent calculation of the greatest common divisor and least common multiple.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact there are two different usages here: simply to represent the set of divisors of two numbers A and B and its intersection, $A \cap  B$ or directly represent the aggregates (?) of prime factors of A, B and its gcd, $A \wedge B$. The first way is very in the spirit of New Math, but in order to be able to expose the LCM it needs, I believe, to use $AB=LCM \times GCD$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second way is more explicit, one gets the LCM, as $A \vee B$, by multiplying all the factors in the diagram and the GCD by restricting to only the &quot;wedge&quot; part of it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4QF6n.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4QF6n.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But we can not see the factors as elements of a set, because of course we can have repeated factors. The problem appears when we remember that the number of subsets of a set is $2^N$, but the number of divisors of 12 = [2,2,3] needs of course to account for the repetition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still it looks, this second way, an useful representation and I am tempted to use it. The caveats are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Could it cause to the student some contradiction with other usages of the diagrams, as e.g. in probability theory?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is it actually a good tool to understand the concept? I mean, compared with the usual recipe of writing both numbers as a prime factorization and choosing the upper / lower exponent. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9854" LastEditorUserId="9854" LastEditDate="2018-05-10T11:30:56.427" LastActivityDate="2018-05-10T14:54:40.027" Title="Experiences with Venn diagrams as didactic tool for factors, GCD, LCM?" Tags="|secondary-education|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14053" PostTypeId="2" ParentId="13672" CreationDate="2018-05-08T19:57:43.907" Score="6" Body="&lt;p&gt;One way to avoid (explicit) division is to make a change of change variables $\, X = x-a\,$ which reduces it to the following simpler special case    $$ X\mid P(X) \iff \color{#c00}{P(0)} = 0$$&#xA;Explicitly we have $$\qquad\qquad\begin{eqnarray}  &#xA;&amp;amp; x-a\,&amp;amp;\mid\,\ p(x) \\[.21em]&#xA;\iff\ &amp;amp; X\!\!\!\!\!\!\!\! &amp;amp;\mid\ \ p(X+a)\\[.2em]&#xA;\iff\ &amp;amp; &amp;amp;\  \color{#c00}{\bigl[p(X+a)\bigr]_{X=0}} = 0\quad\text{by above special case}\\[.2em]&#xA;\iff\  &amp;amp; &amp;amp;\ \ p(a) = 0&#xA;\end{eqnarray}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark&lt;/strong&gt; $  $ This approach implicitly uses the fact that the above shift map is a &lt;em&gt;ring automorphism&lt;/em&gt; - something that is implicitly known since high school but not usually proven &lt;em&gt;rigorously&lt;/em&gt; until a course in abstract algebra. See &lt;a href=&quot;https://math.stackexchange.com/a/94729&quot;&gt;this MSE answer&lt;/a&gt; for further discussion on such pedagogical subtleties.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2018-05-08T20:17:08.167" LastActivityDate="2018-05-08T20:17:08.167" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14054" PostTypeId="2" ParentId="14051" CreationDate="2018-05-09T00:03:09.873" Score="6" Body="&lt;p&gt;I have a worked example on MSE &lt;a href=&quot;https://math.stackexchange.com/a/1923222&quot;&gt;here&lt;/a&gt; in which I cite (RE: your comment on pointers to articles):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Feldman, Z. (2014). Rethinking factors. &lt;em&gt;Mathematics Teaching in the Middle School, 20&lt;/em&gt;(4), 230-236. &lt;a href=&quot;http://www.jstor.org/stable/10.5951/mathteacmiddscho.20.4.0230&quot; rel=&quot;noreferrer&quot;&gt;JSTOR&lt;/a&gt; (paywall).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Check out that paper (if you can gain access) and its citations; specifically, you may wish to investigate work by Zazkis and Campbell (&lt;a href=&quot;https://scholar.google.com/scholar?hl=en&amp;amp;as_sdt=0%2C33&amp;amp;q=Zazkis+Campbell&quot; rel=&quot;noreferrer&quot;&gt;google scholar&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not see any issues with students conflating the use in probability with the use for factors: it is simply a way of representing what is unique to individual numbers (with regard to their factors, or in some approaches, their prime factors) as well as what is shared by the numbers (specifically, their greatest common factor).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that Venn Diagrams, like all tools, are not intrinsically good, but rather only as good as they are used in practice. I happen to like them as tools for working out the LCM or GCF in cases where a non-routine problem has been posed; you can find an example in a paper I co-wrote with B Sriraman:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Sriraman, B., &amp;amp; Dickman, B. (2017). Mathematical pathologies as pathways into creativity. &lt;em&gt;ZDM, 49&lt;/em&gt;(1), 137-145. &lt;a href=&quot;https://www.researchgate.net/publication/303920550&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt; (no paywall).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Section 4 in the aforecited/aforelinked discusses exactly how one can broach non-routine problems around LCMs, and includes (for example) the observation that one can draw a 3-Venn Diagram to represent the LCM and GCF of three whole numbers.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I close by wordily indicating how I think about LCMs in (at least) the k12 context: When I am looking at a list of whole numbers and trying to compute their LCM - and this can arise, e.g., when adding up fractions with different denominators - I think about each whole number as telling me what prime numbers I need to include. Specifically, I go through each number under consideration and look at their prime factors to see whether I can form the requisite numbers. Here is a fully worked example for how I would complete the task of finding the LCM of the first ten positive integers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1$: All whole numbers are divisible by one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$2$: I need a $2$. Right now I have: $2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$3$: I need a $3$. Right now I have: $2 \cdot 3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$4$: Since its prime factorization is $2^2$, and I only have one $2$ now, I need a second one. Right now I have: $2^2 \cdot 3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$5$: I need a $5$. Right now I have: $2^2 \cdot 3 \cdot 5$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$6$: Since its prime factorization is $2 \cdot 3$, all of which I already have, there is no change to my candidate for LCM.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$7$: I need a $7$. Right now I have: $2^2 \cdot 3 \cdot 5 \cdot 7$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$8$: Since its prime factorization is $2^3$, and I only have two $2$s now, I need a third one. Right now I have: $2^3 \cdot 3 \cdot 5 \cdot 7$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$9$: Since its prime factorization is $3^2$, and I only have one $3$ now, I need a second one. Right now I have: $2^3 \cdot 3^2 \cdot 5 \cdot 7$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$10$: Since its prime factorization is $2 \cdot 5$, all of which I already have, there is no change to my candidate for LCM.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So:&lt;/strong&gt; LCM$(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) = 2^3 \cdot 3^2 \cdot 5 \cdot 7$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This approach clarifies in my own thinking how the LCM is a multiple of each number (it contains all the prime numbers to their appropriate powers necessary to make each number) and it is the least such number with this property (it contains nothing superfluous: a common error in the above worked example would be to take the product of all ten numbers).&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-05-09T00:03:09.873" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14056" PostTypeId="2" ParentId="10981" CreationDate="2018-05-09T14:43:05.480" Score="0" Body="&lt;p&gt;It is convention, but is it arbitrary?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us consider just multiplication and addition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we are in a shop and buy 4 tins of beans, 3 loaves of bread, and 2kg of cheese. The beans are £0.50/tin, the bread is £1/loaf, and the cheese £1.50/kg.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So that is &lt;span class=&quot;math-container&quot;&gt;$4\times0.50+3\times1+2\times1.5$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or if addition has precedence, or we evaluate left to right, then we need brackets. &lt;span class=&quot;math-container&quot;&gt;$(4\times0.50)+(3\times1)+(2\times1.5)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus on a simple calculator that evaluates left to right, and has no brackets, or memory feature (as attached to the trolleys at my local supermarket), it is &lt;strong&gt;impossible&lt;/strong&gt; to calculate the sum of you shop (unless you do multiplication as repeated addition, this is ok if you only by a few of any item, and not good for imprecisely cut cheese.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally I do not believe that there is a use case, where left to right evaluation does better.&lt;/p&gt;&#xA;" OwnerUserId="5452" LastEditorUserId="5452" LastEditDate="2019-03-06T08:15:50.973" LastActivityDate="2019-03-06T08:15:50.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14057" PostTypeId="2" ParentId="13672" CreationDate="2018-05-09T17:52:04.030" Score="0" Body="&lt;p&gt;I'm not sure if there's a great way of explaining it without division (I feel like any good explanation will strongly hint at division) but maybe you could &quot;disguise&quot; the division by talking about repeated subtraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can subtract any multiple of $x-17$ from a polynomial. That includes multiples like $10x^{82}(x-17)=10x^{83}-170x^{82}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, given &lt;em&gt;any&lt;/em&gt; polynomial of degree $\ge1$, say its leading term is $ax^n$, you can subtract $ax^{n-1}(x-17)$ and get a polynomial of strictly smaller degree. And as long as you keep getting polynomials of degree $\ge1$, this process can be repeated. But it can't continue forever. So eventually we get a polynomial of degree $&amp;lt;1$, i.e., a constant. Call that constant $c$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, if the original polynomial was $p(x)$, we get $p(x)-(\mbox{some multiple of ($x-17$)})=c$. Then evaluate at $x=17$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that I look at what I've typed, it's not very different from the standard argument. But it's possible that phrasing it in terms of repeated subtraction, rather than polynomial division, may be helpful to some of your students.&lt;/p&gt;&#xA;" OwnerUserId="9249" LastActivityDate="2018-05-09T17:52:04.030" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14058" PostTypeId="2" ParentId="13672" CreationDate="2018-05-09T19:34:18.427" Score="0" Body="&lt;p&gt;To &lt;em&gt;prove&lt;/em&gt; the assertion, you need polynomial division. But to &lt;em&gt;demonstrate&lt;/em&gt; with enough examples that they develop an intuition (you're probably going to help them discover it on their own, right?) all they need is the ability to &lt;em&gt;factor&lt;/em&gt; polynomials. I'd think you could start simple with something like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f(x) = (x+1)(x-5)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is $f(-1)$? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the class calculates $f(-1)$ and finds it is zero. (Will you need to discuss what a root means in terms of a polynomial?) Do this three or four times with different functions, asking for a single root. Then ask for all roots, not just one. Ask the class what pattern they are seeing and if they think that pattern holds for all polynomials. See if they can guess the other root of $f(x)$, above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may not get the class to be able to construct a proof that goes both ways, but it should give them an understanding of how to apply the proof -- which is really the aim (assuming the class is not composed exclusively of math majors).&lt;/p&gt;&#xA;" OwnerUserId="9883" LastActivityDate="2018-05-09T19:34:18.427" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14059" PostTypeId="2" ParentId="353" CreationDate="2018-05-09T21:44:17.777" Score="1" Body="&lt;p&gt;&lt;em&gt;Here is an example I sometimes use to motivate the truth table for implication. The example taps in to our ability to recognize cheaters.&lt;/em&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you go to the vending machine. The price of a soda is one dollar. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose you put a dollar in the vending machine and receive a soda. Do you feel cheated? &lt;strong&gt;No!&lt;/strong&gt; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose you put a dollar in the vending machine and do not receive a soda. Do you feel cheated? &lt;strong&gt;Yes!&lt;/strong&gt; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose you do not put a dollar in the vending machine but receive a soda anyways. Do you feel cheated? &lt;strong&gt;No!&lt;/strong&gt; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose you do not put a dollar in the vending machine and do not receive a soda. Do you feel cheated? &lt;strong&gt;No!&lt;/strong&gt; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4665" LastActivityDate="2018-05-09T21:44:17.777" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14060" PostTypeId="2" ParentId="14051" CreationDate="2018-05-10T14:54:40.027" Score="3" Body="&lt;p&gt;Perhaps worth emphasis is that the use of prime factorizations (either directly or implicitly via (multiset) Venn diagrams or exponent vectors) may actually &lt;em&gt;obfusctate&lt;/em&gt; the essence of the matter. For example, lets solve the problem linked in Benjamin's answer without using prime factorizations but instead using gcd &amp;amp; lcm distributive laws to pull out and cancel a common factor $12$ from $A,B$. Namely using $\gcd(A,B) = \gcd(12a,12b) = 12 \gcd(a,b),\,$ and similarly for $\rm lcm,$ yields&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align} &#xA;\gcd(A,B) &amp;amp;= 12,\ \  360 = {\rm lcm}(A,B)\\[.2em]&#xA;\iff\ 12\gcd(a,b) &amp;amp;= 12,\ \ 360 = 12\,{\rm lcm}(a,b)\\[.2em]&#xA;\iff\quad\ \gcd(a, b) &amp;amp;= 1,\ \ \quad 30 = {\rm lcm}(a,b)\ [= ab]&#xA;\end{align}\qquad$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus we have reduced it to factoring $\,30 = ab\,$ into coprime factors $a,b,\,$ which is easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key simplification arises from applying the&lt;a href=&quot;https://math.stackexchange.com/a/113045/242&quot;&gt; $\,\rm gcd$ &lt;strong&gt;distributive law&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$  \gcd(ca,cb) = c \gcd(a,b)$$&#xA;and similarly for $\rm lcm$. Solutions using prime factorizations or Venn diagrams are essentially doing the same thing, but the key role played by distributivity is highly obfuscated from that viewpoint. This is pedagogically detrimental because such distributive laws are fundamental, so this innate algebraic structure should be highlighted - not obfuscated. The solution of many number theory problems works the same way as above - exploit distributivity to cancel out a common factor so reducing to the simpler case when $a,b$ are coprime (so their lcm = product). One will be at a great disadvantage if one is forced to rediscover this distributive structure every time in terms of Venn diagrams vs. simply invoking well-known fundamental gcd laws.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus if you use Venn diagrams or similar devices, strive to also teach the more algebraic approaches using basic gcd laws so that students will obtain a better grasp of the algebraic essence of the matter.&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2018-05-10T14:54:40.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14061" PostTypeId="2" ParentId="5730" CreationDate="2018-05-11T22:05:16.447" Score="0" Body="&lt;p&gt;This answer is assuming that the question is talking about the probability of drawing &lt;strong&gt;EXACTLY&lt;/strong&gt; 1 marble out of the bag and determining the likelihood of it being a black marble or being white marble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, something that many of the school teachers who teach math, (no offense to anyone) including the one who &quot;taught&quot; me probability is that they always leave out the idea that math is a formal system that is always based on fundamental assumptions which mathematicians call Axioms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First off though, let's start with defining the word probability: one of the dictionary definitions is &quot;Probability is the extent to which an event is likely to occur, measured by the &lt;strong&gt;ratio of favored cases to the total number of possible cases&lt;/strong&gt;. Or using mathematical notation, given event E:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P(E)=\frac{favorable\:outcomes}{total\:possible\:outcomes}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea behind this is that we are making the assumption that each individual outcome is equally likely. That means that we are making the assumption that there exists a probability for picking a single white marble such that the probability of doing so is exactly equal to the probability of picking a single black marble. Or, expressed in mathematical notation, P be predicate for &quot;Probability of picking&quot;, let w be predicate for &quot;white marble&quot;, let b be predicate for &quot;black marble&quot;: &#xA;$\exists \:P\left(w\right):\:P\left(w\right)=P\left(b\right)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, we represent the probability of an event that will &lt;strong&gt;NEVER&lt;/strong&gt; happen, as a 0% chance of happening meaning that for every 100 times the event could happen theoretically, it happens 0 times out of 100 and the way we represent this is: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P\left(E\right)=\frac{0}{100}=0$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, in a similar fashion, if the probability of an event is &lt;strong&gt;CERTAIN&lt;/strong&gt;, then it will happen 100 times for every 100 times it can happen i.e. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P\left(E\right)=\frac{100}{100}=1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we use the definition of certainty being 100%, i.e. 1, and we assume out of all the possible outcomes, at least one of the individual outcomes must occur (in this case a person has to either draw a white marble or black marble, can't draw zero marbles or have a random grey marble magically pop into existence),&#xA;Then the sum of all the individual outcomes must equal to 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you let A,B,C ... be the individual outcomes then mathematically:&#xA;$P\left(A\right)+P\left(B\right)+P\left(C\right)+...\:=1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus if you combine the assumption that each of the individual outcomes are equally likely and the definition that all individual outcomes must sum up to 1, then if you number all the white marbles (from 1-10) and all the black marbles from (1-20) then you get: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P\left(w_1\right)=P\left(w_2\right)=...=P\left(w_{10}\right)= P\left(b_1\right)=P\left(b_2\right)=...\:P\left(b_{20}\right)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and you also get: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P\left(w_1\right)+P\left(w_2\right)+...+P\left(w_{10}\right)+P\left(b_1\right)+P\left(b_2\right)+...\:P\left(b_{20}\right)=1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solving this gives you: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P\left(b_1\right)=\frac{1}{30}$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;and since there are 20 black marbles, if you let E rep. the event of selecting a black marble you get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$P(E) = \frac{20}{30} = \frac{2}{3}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and, you would use the same assumptions and reasoning for the other bag with just 2 black marbles and 1 white marble to get the same probability. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the &lt;strong&gt;KEY IDEA&lt;/strong&gt; here is that we made the assumption that each individual outcome is equally likely. This is not necessarily true in real life. For example, if you toss a coin (and actually depending on how you toss the coin) the probability of heads is actually slightly higher than the probability of tails (51% and also there's a very small probability of landing exactly in the middle). Another example would be if you had a loaded die, then the probability of each number wouldn't necessarily be $\frac{1}{6}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The simulations that run in theory will not necessarily have the same results as actually conducting the experiment in practice because real life is usually not as ideal and beautiful as it is in mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, I believe the best answer to the question &quot;which bag is more likely to yield a black marble?&quot; is: &#xA;&quot;It depends on how you draw the marble out of the bag, depends on the composition of each marble (maybe like some marbles are heavier than others) etc. But, if you assume the likelihood of drawing each individual marble is equally likely then, you should have an equal chance of drawing a black marble from bag 1 as you do in bag 2.&quot; &lt;/p&gt;&#xA;" OwnerUserId="9909" LastActivityDate="2018-05-11T22:05:16.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14062" PostTypeId="2" ParentId="12226" CreationDate="2018-05-12T02:10:02.593" Score="5" Body="&lt;p&gt;Look at patterns in decimal expansions: what is the period of the repeating decimal of $1/n$? From numerical data, the period is at most $n-1$, and you only get equality when $n = p$ is prime (but not conversely: $1/11$ has decimal period $2$, not $10$). If you look at the period of the decimal of $1/p$ for primes $p$ besides $2$ and $5$, you find that although the period is not always $p-1$ it &lt;em&gt;does&lt;/em&gt; always appear to divide $p-1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That divisibility (sharper than an inequality) is basically Fermat's little theorem. The period of the decimal for $1/n$ is the order of $10 \bmod n$ (for $n$ relatively prime to $10$), so the fact that the period of the decimal for $1/p$ when $p$ is a prime (besides $2$ or $5$) is a factor of  $p-1$ is equivalent to saying $10^{p-1} \equiv 1 \bmod p$. This is possibly how Gauss was led to Fermat's little theorem.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2018-05-12T12:30:59.570" LastActivityDate="2018-05-12T12:30:59.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14063" PostTypeId="1" CreationDate="2018-05-12T14:36:41.010" Score="1" ViewCount="428" Body="&lt;p&gt;I am a high school senior student.&#xA;Soon, I am giving a presentation about the Fibonacci sequence and I am searching for a creative way to start my speech. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering whether someone in this community can help me a little bit?&lt;/p&gt;&#xA;" OwnerUserId="9912" LastEditorUserId="2083" LastEditDate="2018-05-12T15:59:02.200" LastActivityDate="2018-05-13T10:45:52.657" Title="How can I introduce a speech about the Fibonacci sequence creativiely?" Tags="|secondary-education|presentation|sequences|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14064" PostTypeId="2" ParentId="14063" CreationDate="2018-05-12T19:14:51.443" Score="3" Body="&lt;p&gt;&lt;strong&gt;ADVICE:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;When you have a question about a speech, you should give us more information to help you.  Especially &lt;strong&gt;how long is the speech&lt;/strong&gt; (it affects how much intro you do).  But also the audience.  And even how important this is to you and how long you will spend preparing it.  It helps us help you and may help you strategize also.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;ANSWER:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I would start with a biographical &quot;hook&quot;, using Fibonacci, the man.  People are interested in people.  We evolved with incredible brains at facial recognition, but not analytical computer minds.  So if there is some personality hook.  It can draw people in and make the &quot;medicine&quot; (abstract math) go down easier.  Not a full biography, since that is not your subject.  But enough to get us &quot;hooked&quot; before you start having the chalkboard covered with numbers added together.&lt;/em&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;This is not purely motivation for a slick intro, but also serves a pedagogical purpose.  Our human minds tend to form patterns and associations to aid memory.  If people have a little mental image of the Fibonacci guy as a man, it may aid their remembering the topic itself as a topic (even if they forget the math).&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;OUTLINE:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Intro:  Something about how he learned from the Arabs about numbers and wanted to show everyone in Europe that Arabic numerals (really from India) were way easier than Roman numerals for doing math with.  He made a great contribution here and wrote a famous textbook, teaching people to use the numbers we all know now instead of the strange Roman numeral movies that we see in movies mostly.  But this wasn't his most important contribution.  His most important contribution was just a little example in his book that has made people marvel for centuries and showed a beautiful pattern hiding in numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Tell the rabbit story.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do some math and show numbers on the chalkboard.  &lt;em&gt;Gotta have some of that!&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give some applications or examples of the pattern in nature.  &lt;em&gt;Use visual aids, here....slide with flower petals or pine cone.  Girl flipping her hair in the spiral.  Etc.&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Conclude...circle back with some &quot;moral&quot; of the talk, in addition to the recap.  &quot;In the Fibonacci sequence, we have learned how to create the numbers of the sequence by adding the previous two numbers to make a new one.  Then we saw that this sequence is found in many places in nature (the spiral, flowers, pine cones, etc.)  We've also seen that what can seem like a small example in an arithmetic book, can end up being marveled at by people for centuries in the future.  Sometimes great things can come out of what seem like silly math puzzles.&quot;  [&lt;em&gt;Or whatever,  you think a &quot;moral&quot; might be...being  open to other cultures, becoming involved in work as a teenager, whatever you think is interesting from a human perspective.  Doesn't matter per se, but have some narrative message or moral at the end, like an Aesop's Fable (just find one).  A computer could care less, but pattern forming humans need some meaning.&lt;/em&gt;]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="143" LastEditDate="2018-05-13T10:45:52.657" LastActivityDate="2018-05-13T10:45:52.657" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14065" PostTypeId="2" ParentId="1447" CreationDate="2018-05-13T03:46:01.360" Score="3" Body="&lt;p&gt;How about antiderivatives? Evaluate a simple definite integral, like $\int_0^2 x \, dx$. Do it using the standard representative $\frac{x^2}2$, and then using a different one, like $\frac{x^2}2+7$. When you do the subtraction, it becomes immediately obvious that it doesn’t matter which representive you use, and the students get insight onto why we always put $+C$ ...&lt;/p&gt;&#xA;" OwnerUserId="9914" LastActivityDate="2018-05-13T03:46:01.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14066" PostTypeId="2" ParentId="377" CreationDate="2018-05-13T18:04:00.597" Score="6" Body="&lt;p&gt;There is an article by Phillips:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://doi.org/10.1086/678170&quot; rel=&quot;nofollow noreferrer&quot;&gt;Phillips, C. &quot;In accordance with a `more majestic order': the new math and the nature of mathematics at midcentury.&quot; Isis 105 (2014), no. 3, 540--563&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;that presents a thoughtful analysis of this curious episode from the history of math education. The way I understand his argument, the sequence of events was as follows.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Feeding the dissatisfaction with what was felt to be an inadequate situation with math education was the fever characterizing the Sputnik era mentality, when it was felt (at least in the US) that we must catch up with the Russians at any cost.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Certain philosophical assumptions that had originally developed in the community of (mainly pure) mathematicians trickled into the thinking of education decision makers who looked up to the professors.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A significant role here was played by a Piaget--Dieudonne interaction and equivocation on &lt;em&gt;mathematical structures&lt;/em&gt; where child psychology was mixed up with mathematical foundations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A set of measures was proposed that may have been somewhat effective at the highschool level, but at the elementary and middle school level was doomed from the start.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Vast federal moneys were fed into the effort led by the National Science Foundation which was ill-equipped to decide on issues of elementary education.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The proposal was implemented rapidly without proper testing via pilot programs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;By the time people started realizing something was terribly wrong, vast numbers of schools had already been affected, as well an entire generation of schoolers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Phillips also published a follow-up book&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Phillips, Christopher J. The new math. A political history. University of Chicago Press, Chicago, IL, 2015.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I personally found the book a big disappointment. In his 2015 book, he toned down many of the arguments in the 2014 paper and even reversed himself on some issues. One gets the impression that in order to get the book published he had to toe a politically correct line so as to make sure not to be perceived as pouring oil on the conservatives' agenda. In short, I recommend the article but not the book.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2018-05-14T08:50:45.193" LastActivityDate="2018-05-14T08:50:45.193" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14067" PostTypeId="1" AcceptedAnswerId="14093" CreationDate="2018-05-14T11:09:10.007" Score="5" ViewCount="336" Body="&lt;p&gt;Computer science is becoming more and more important for mathematicians nowadays. Terms like &lt;em&gt;big data&lt;/em&gt;, &lt;em&gt;algorithm&lt;/em&gt;, &lt;em&gt;artificial intelligence&lt;/em&gt; and others are frequently on the news. Many mathematical problems are tackled via computer scientific methods because many problems cannot be (efficiently) solved by a human being in practice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many such problems, like the &lt;em&gt;knapsack problem&lt;/em&gt; can be nicely illustrated and are easily accessible for students. However, there are problems (such as the &lt;em&gt;knapsack problem&lt;/em&gt; again) which even modern computers struggle to solve in acceptable time for large inputs. When talking about a problem's complexity we will quickly arrive at terms like &lt;em&gt;NP-hard&lt;/em&gt; and &lt;em&gt;NP-complete&lt;/em&gt;. As these are somewhat abstract concepts I was wondering how these could be explained to students such that they become familiar with the terminology and begin to understand the difference in the complexity classes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Admittedly, students will find it easy to believe that computers will take longer to solve a given problem if the input size grows. However, this is also true for problems which can be solved in polynomial time on a deterministic machine and completely misses the focus on NP-hardness and NP-completeness.&lt;/p&gt;&#xA;" OwnerUserId="7608" LastEditorUserId="7608" LastEditDate="2018-05-14T12:46:22.297" LastActivityDate="2018-05-21T10:23:25.513" Title="How to explain NP-hardness and NP-completeness to students" Tags="|mathematical-pedagogy|technology-in-education|teaching|terminology|computer-science|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14068" PostTypeId="1" CreationDate="2018-05-14T11:47:06.420" Score="21" ViewCount="785" Body="&lt;p&gt;While teaching students mathematics, I have noticed that some seem sloppy in the way that they write down their work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, a student is given a question: What is the area of the rectangle? &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;   4     3&#xA;┌──────┬─────┐&#xA;│      │     │ 2&#xA;└──────┴─────┘&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;(sorry for the terrible word picture, the original was a picture.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The student writes&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$3+4=7 \times 2 = 14$&lt;/span&gt; &#xA;This is clearly wrong, though they got the correct answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can also see their thought process. They first added 3 and 4. Then multiplied by 2. However what they wrote is wrong, since &lt;span class=&quot;math-container&quot;&gt;$3+4 ≠ 7 \times 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone cite any research to show what effect this has on the progress of the student? Or share personal experience?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: as this is not a question about units, I have not included any.&lt;/p&gt;&#xA;" OwnerUserId="5452" LastEditorUserId="238" LastEditDate="2019-10-11T19:40:15.207" LastActivityDate="2019-10-11T19:40:15.207" Title="How students write their work, and learning outcomes" Tags="|students-mistakes|notation|" AnswerCount="5" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14069" PostTypeId="2" ParentId="2157" CreationDate="2018-05-14T15:34:01.253" Score="2" Body="&lt;p&gt;I think the &lt;em&gt;Squeeze Theorem&lt;/em&gt; can provide a good opportunity for this phenomenon. The example I describe below arose in a Real Analysis course for upper-level mathematics majors that I taught this semester, but I can envision similar examples arising in calculus courses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Exercise:&lt;/em&gt; Use the formal definition of convergence of a sequence (&quot;For every $\epsilon&amp;gt;0$ there exists an $N&amp;gt;0$ such that ...&quot;) to prove that the $\lim_{n\to\infty} \frac{n^2}{n^4+1}=0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The algebra of the desired inequality $-\epsilon&amp;lt;\frac{n^2}{n^4+1}&amp;lt;\epsilon$ can get a little messy. So, when students came to my office hours for assistance with this problem, I used questioning to guide them into realizing that $0&amp;lt;\frac{n^2}{n^4+1}&amp;lt;\frac{n^2}{n^4}=\frac{1}{n^2}$, and then realizing that it is quite easy to use the formal definition to show $\lim_{n\to\infty}\frac{1}{n^2}=0$ and apply the Squeeze Theorem (discussed in class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admit this is not quite &quot;prove something &lt;em&gt;stronger&lt;/em&gt;&quot;, in the sense that the necessary observation is almost trivial (and not really &quot;proving&quot; something), and that observation is arguably not &quot;stronger&quot; than the claimed result to be proven. However, for students who are first learning how to figure out and communicate proofs of this nature ... they certainly need this kind of advice for a problem like this.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2018-05-14T15:34:01.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14070" PostTypeId="2" ParentId="14068" CreationDate="2018-05-14T17:09:05.057" Score="17" Body="&lt;p&gt;This example strikes me as resembling a phenomenon I've seen where a problem is of the form&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given A, find C&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;where the main approach to actually solving the problem is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Using A, obtain B&lt;/li&gt;&#xA;  &lt;li&gt;Using B, obtain C&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;however, the student seems entirely incapable of even &lt;em&gt;conceiving&lt;/em&gt; of doing these as separate steps. Any attempt to help them obtain B just results in them trying to arrange the advice into yet another guess as to a formula formula for computing C. Even if you &lt;em&gt;explicitly&lt;/em&gt; ask them what B is!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find it quite plausible that this difficulty could be the product of the very sort of sloppiness you describe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sort of formula you write is a sort of run-on sentence starting with &quot;find the width&quot; and running into &quot;find the area&quot;. Imagine, now, that their teachers correct this problem by showing how to combine the two run-on sentences into one compound sentence: $a = (3+4) \times 2 = 14$,  but never teach the student to separate things into two separate sentences; e.g. $w = 3+4 = 7$ followed by $a = w \times 2 = 14$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So now, you have a student whose &lt;em&gt;only&lt;/em&gt; means of expressing a complex mathematical idea is by doing so all at once &amp;mdash; they simply don't have any experience in expressing the individual ideas separately.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2018-05-14T17:09:05.057" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14071" PostTypeId="1" CreationDate="2018-05-14T19:37:43.023" Score="5" ViewCount="155" Body="&lt;p&gt;Asking for methods to produce the sum of natural numbers from the disjoint union of sets, &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14021/how-to-build-addition-with-sets&quot;&gt;it seems&lt;/a&gt; that the obvious way is to use the general definition, as coproduct, of the sum of sets. The accepted answer in that question suggested to use pairing with a &quot;label&quot;, but the coproduct just needs a pair of &quot;disjoint&quot; injections, for example here in &quot;Sets for Mathematicians&quot;, chapter 2:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AJ5gs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AJ5gs.png&quot; alt=&quot;Lawvere Sets&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Amusingly I have found a textbook in Spain, around 1974, that show the required injections.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6XOZe.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6XOZe.jpg&quot; alt=&quot;InjectionCoproduct&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I found another one &lt;a href=&quot;https://i.stack.imgur.com/j0QmF.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://i.stack.imgur.com/j0QmF.jpg&lt;/a&gt; EDIT but in this case the author confirms me that the intention was just to show the idea of &quot;increasing by one&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see, it is very like the examples from Lawvere category of sets. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And now I am puzzled if the Spanish textbooks could have originated from some foreign, probably USA, material for primary school. Is this kind of representation, or drawing, used in New Math textbooks? Do you know any?&lt;/p&gt;&#xA;" OwnerUserId="9854" LastEditorUserId="9854" LastEditDate="2018-05-16T00:23:12.070" LastActivityDate="2018-05-16T00:23:12.070" Title="Textbooks explicitly showing the injections for the sum of sets" Tags="|reference-request|primary-education|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14072" PostTypeId="2" ParentId="1550" CreationDate="2018-05-16T19:48:20.587" Score="1" Body="&lt;p&gt;Simple. Put a tin of beer on the table and ask the students if the packaging is the optimal solution. i.e. a cylinder has a volume of 500cm^3, what is the minimum possible surface area. Then compare to the actual surface area. I'm doing this in class tomorrow :)&lt;/p&gt;&#xA;" OwnerUserId="9921" LastActivityDate="2018-05-16T19:48:20.587" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14074" PostTypeId="2" ParentId="1550" CreationDate="2018-05-17T07:13:37.667" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;Profit maximization is a normal problem in the business world. Not sure that calculus is used that often as opposed to graphing though.  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://phx.corporate-ir.net/External.File?item=UGFyZW50SUQ9NjkzMTgyfENoaWxkSUQ9NDA0NDAyfFR5cGU9MQ==&amp;amp;t=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://phx.corporate-ir.net/External.File?item=UGFyZW50SUQ9NjkzMTgyfENoaWxkSUQ9NDA0NDAyfFR5cGU9MQ==&amp;amp;t=1&lt;/a&gt;  (see slide 15 upper right and 17 lower right.  The problem is to optimize financial return (PV, present value) versus number of wells drilled.  Too many and you leave too much oil behind.  Too many and you are only getting a little extra oil for the work of the new wells.)&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Many manufacturing problems (e.g. optimal batch size).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-05-17T07:13:37.667" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14075" PostTypeId="1" CreationDate="2018-05-17T16:20:56.140" Score="15" ViewCount="536" Body="&lt;p&gt;I am teaching a calculus class for business this summer (6 students) and all of them do not have the math background needed for the class. We are supposed to cover derivatives and integrals, but they don't even know how to evaluate functions. For example, I gave them $f(x)=x^2+x$ and the following happened:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;4 of them had no idea what $f(2+h)$ was supposed to be. They tried $f(2+h)=2^2+h$, or $2^2+x$ or similar incorrect expressions.&lt;/li&gt;&#xA;&lt;li&gt;2 of them had no idea what $f(2)$ was supposed to be. They tried $f(2)=(x^2+x)\cdot 2$ (yes, $x^2+x$ times 2), or $2x^2+2$ or $f(2)=fx^2+x2$, or similar incorrect expressions. After spending nearly the whole class explaining how evaluation works, they understood how to compute $f(2)$ and had some idea of $f(2+h)$. Then, I changed to another function $g(x)=-x^2+1$ and they were stuck again.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;NOTES&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The example above is just an example. Please do not give suggestions about how to explain evaluation to a student. &lt;/li&gt;&#xA;&lt;li&gt;I have taught the course before, but this problem has been getting worse and worse over the years&lt;/li&gt;&#xA;&lt;li&gt;I have no idea how they passed previous classes&lt;/li&gt;&#xA;&lt;li&gt;This is the last math class the students will take&lt;/li&gt;&#xA;&lt;li&gt;My chair and dean verbally told me students are getting worse (math-background-wise) and that nobody cares what I do with this particular class. They won't say that in writing, of course.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How to deal with the problem of ALL students having a very poor background?&lt;/li&gt;&#xA;&lt;li&gt;Do I just teach the required topics and likely fail all students?&lt;/li&gt;&#xA;&lt;li&gt;Do I spend time making sure some students understand even if I only cover 50% of the material?&lt;/li&gt;&#xA;&lt;li&gt;Do I just curve the grades even if they do not understand the material?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9923" LastEditorUserId="9924" LastEditDate="2018-05-17T22:46:30.163" LastActivityDate="2018-05-20T19:19:46.590" Title="What to do if all students lack prerequisites?" Tags="|undergraduate-education|mathematical-pedagogy|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14076" PostTypeId="2" ParentId="10923" CreationDate="2018-05-17T18:36:55.990" Score="1" Body="&lt;p&gt;I use a beautiful little story narrated by Helmut Kracke in order to demonstrate the size of the largest three-digit-number 9^9^9.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wotan, the God of the Teutons, had a very precious ring called Draupnir. Every ninth night nine rings of the same shape dropped from it. He could use them for paying maintenance to his several women. But these rings did not drop further rings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now assume, since in modern times everything goes faster, we have a ring that drops nine rings in every second, and every dropped ring will also do so. Then after 10 seconds we have 10 billion rings, after 20 seconds we have as many as there are combinations of the Rubik cube, after 82 seconds as many as there are atoms in the universe. How long do we have to wait for 9^9^9? More than 11 years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more detail see &lt;a href=&quot;https://www.hs-augsburg.de/~mueckenh/HI/HI01.PPT&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.hs-augsburg.de/~mueckenh/HI/HI01.PPT&lt;/a&gt;, slides 32-45.&lt;/p&gt;&#xA;" OwnerUserId="8557" LastActivityDate="2018-05-17T18:36:55.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14077" PostTypeId="1" CreationDate="2018-05-17T22:16:01.030" Score="10" ViewCount="256" Body="&lt;p&gt;Is there a resource or textbook that presents the basics of vector calculus, specifically the gradient, directional derivatives, curves and surfaces, and extrema, from a more algebraic geometry flavored perspective? I suppose I mean algebraic geometry flavored as a opposed to differential geometry flavored, which I am less familiar with.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an example of what I mean by the two flavors, you can present some surfaces in $\mathbf{R}^3$ as the graph of a function $f(x,y)$. The tangent plane to this surface at a point $(x_0,y_0)$ is given by &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ &#xA;z = f(x_0,y_0) &#xA;  + \left(\frac{\partial f}{\partial x}(x_0,y_0)\right)(x-x_0) &#xA;  + \left(\frac{\partial f}{\partial y}(x_0,y_0)\right)(y-y_0)&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is how the textbook I'm using now presents it initially, but it's strikingly asymmetric because we're thinking of the surface as a graph. Instead you could think of your surface as the vanishing set of the function $p(x,y,z) = f(x,y) - z$ and the equation of our tangent plane at $(x_0,y_0,z_0)$ is much more symmetric and easy to write down in terms of the gradient:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ &#xA;0 = \nabla p(x_0,y_0,z_0) &#xA;    \cdot &#xA;    \langle x-x_0, y-y_0, z-z_0 \rangle&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm looking for such a resource because it feels like the textbook I'm teaching from talks about things from a mix a both perspectives, and I'm suspicious that doing this leads to unnecessary confusion. I want to know if there is a complete treatment from a single perspective (preferably algebraic geometry), and to see if restricting to that one perspective leads to any problems within the scope of an undergraduate course is vector calculus.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-05-18T13:57:46.480" LastActivityDate="2021-01-31T12:07:30.383" Title="Recommend a vector calculus textbook/resource with an algebraic geometry flavor" Tags="|reference-request|textbooks|reference-books|vector-calculus|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14078" PostTypeId="2" ParentId="14075" CreationDate="2018-05-17T23:52:34.673" Score="10" Body="&lt;p&gt;I'm going to take the crummy-feeling position of upholding the legitimacy of the course (possibly) at the expense of the students' education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I presume there is a department-approved syllabus on what topics should be covered in this University course, right? Those topics are the topics that students who pass the course are expected to have understood by the end of it. Assuming I accurately understand your position in this, unfortunately you are more than just an educator in this situation, but you are also a gatekeeper as you have to grade them and decide if they pass. If it shows on their transcript that a student passed this course, it means that you've signed off on the fact that the student has a sufficient understanding of the material listed in the University's (or Math Department's) description of the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So teach the material that's usually taught, and examine them on that material. You can still teach it to them &lt;em&gt;softly&lt;/em&gt; with the understanding that they're lacking some basic prerequisite knowledge (go slowly, write out more details that you would normally think are necessary, etc). But at the end of the day, it's got to be a calculus course. Heck, maybe they'll surprise you at how much calculus they've learned by the end of the course. :) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this of course is pending any further word from the dean and chair. &lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-05-18T01:56:50.220" LastActivityDate="2018-05-18T01:56:50.220" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14079" PostTypeId="2" ParentId="14075" CreationDate="2018-05-18T05:18:03.097" Score="-2" Body="&lt;p&gt;I'm going to take a contrarian stance and say you should avoid the drama of failing them and just get by.  This institution is not Harvard.  And these kids are not little Feynmans.  Everyone gets that they are not real calculus students.  Just deal with it.  It's not your place to challenge it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do some basic evaluation, but don't kill yourself (like going back to teach them trig).  If you can cover basics of compound interest (exponentials or the like), this would be helpful for business.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Avoid teaching theory of differentiation and integration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Emphasize simple, graphical understanding of the topic and mechanical problem solving (like differentiation and antidifferentiation of polynomials).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use frequent business examples (e.g. rate of production growth)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Curve them and pass them if they are trying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  Learning how to use Excel shortcuts would do them more good than calculus anyways.  It's not like you have to worry about them taking physics or diffyQs.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-05-18T05:18:03.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14080" PostTypeId="1" AcceptedAnswerId="14082" CreationDate="2018-05-18T12:11:36.003" Score="3" ViewCount="636" Body="&lt;p&gt;It seems to me that Euler's method for solving ODEs is a topic that nicely illustrates the main strategy of calculus, which is to approximate a nonlinear function locally by a linear (or affine) function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose that a function $y:\mathbb R \to \mathbb R$ is differentiable and satisfies&#xA;$$&#xA;y'(t) = f(y(t)) \,\text{ for all } t \in \mathbb R, \quad y(t_0) = y_0.&#xA;$$&#xA;Here $f:\mathbb R \to \mathbb R$ is a continuous function.&#xA;If $t_1$ is close to $t_0$, then from the definition of the derivative we have&#xA;$$&#xA;y(t_1) \approx y(t_0) + y'(t_0)(t_1 - t_0).&#xA;$$&#xA;Indeed, this approximation is the key idea of calculus. It follows that&#xA;$$&#xA;y(t_1) \approx y(t_0) + f(y(t_0))(t_1 - t_0).&#xA;$$&#xA;This gives us an estimate of the value of $y$ at $t_1$. We can now repeat this procedure to obtain estimates of the values of $y$ at points $t_2,t_3,\ldots,t_n$, where $t_{i+1} = t_i + \Delta t$ for all $i$ and $\Delta t$ is some small positive number. That's Euler's method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is accessible to calculus students, and is something they could implement with a computer program to obtain a cool visual result (a graph of $y$). Do you think this is a good topic to include in a first-semester calculus course?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Maybe the answer is &quot;obviously yes, just like we teach Newton's method.&quot; But I'm curious to see if anyone thinks this is a bad idea. Perhaps it would confuse students more than I realize.)&lt;/p&gt;&#xA;" OwnerUserId="9633" LastActivityDate="2018-05-18T18:07:46.800" Title="Should Euler's method be taught in calculus 1 courses?" Tags="|calculus|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14081" PostTypeId="1" AcceptedAnswerId="14131" CreationDate="2018-05-18T12:30:10.193" Score="6" ViewCount="553" Body="&lt;p&gt;I don't know if this is the correct site to ask this question, but I felt it was off-topic on the Mathematics forum. I really like Number Theory and would like study some on my own. Which books should I read to get a good grasp on the subject?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a 20 year old Software Engineering student. I've taken courses on introduction to Discrete Math, Linear Algebra, and Calculus I. I'm taking Calculus II and Graph Theory now. My Calculus lacks analysis of series/indefinite integrals though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for ability I wouldn't know how to answer. I've aced all those math subjects but they were fairly basic, they're designed for an engineering course after all. They're mostly introductions to the subjects&lt;/p&gt;&#xA;" OwnerUserId="9925" LastEditorUserId="376" LastEditDate="2020-09-10T13:02:49.207" LastActivityDate="2021-04-13T16:08:38.437" Title="Source material to study number theory?" Tags="|self-learning|number-theory|resource-request|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14082" PostTypeId="2" ParentId="14080" CreationDate="2018-05-18T14:40:51.970" Score="3" Body="&lt;p&gt;I think that it may be reasonable to spend some time on this, if you have the time for it.  The content is, after all, essentially the definition of the tangent line and elementary ODEs are sometimes taught at the Calc 1/2 boundary.  However, in a 1st semester calculus class, you should expect to heavily support the students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the explanation should include a visual component. The explanation you give above will probably not lead to much understanding. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unless you know for certain that these students have CS skills, writing a program from scratch is probably too much to ask. Write it for them and explain in broad detail what it does. Visualization of the output is important here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good set of 1st order ODEs, where they can adjust the parameters, would be useful. (Ex: Exponential, Logistic, $y'=y^3+cy$ for positive and negative values of $c$) A comparison to the exact solutions would also be helpful. Make the step size adjustable so that they can see how convergence works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Done right, it could be a good place for more qualitative questions and explorations, but for sure don't make them do it by paper and pencil on an exam.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-05-18T14:40:51.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14083" PostTypeId="2" ParentId="14080" CreationDate="2018-05-18T18:07:46.800" Score="4" Body="&lt;p&gt;Personally, if I were not constrained by the heavy chains of course coordination, I would make Euler's method a major theme of my Calculus 1 course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, I would do Euler's method before defining the definite integral.  Then applying Euler's method to $\frac{dy}{dx} = f(x)$ on the interval $[a,b]$ yields a Riemann sum.   Defining the limit of this Riemann sum to be the &quot;definite integral&quot; of function $f$, the fundamental theorem of calculus is immediately obvious (assuming they already believe in the convergence of Euler's method).  You can reinterpret the definite integral as a signed area later. &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-05-18T18:07:46.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14084" PostTypeId="1" CreationDate="2018-05-18T18:31:56.613" Score="7" ViewCount="1218" Body="&lt;ol&gt;&#xA;&lt;li&gt;'Marginally generalized theorems' refers to &lt;a href=&quot;http://pirate.shu.edu/~kahlnath/Top100.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;theorems&lt;/a&gt; that were generalized so marginally that they can still be taught, and understood by the students, in the same course. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;h3&gt;Examples of theorems NOT marginally generalized:&lt;/h3&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;A skilled high-schooler can understand &lt;a href=&quot;https://en.wikipedia.org/wiki/Puiseux_series#Generalization&quot; rel=&quot;nofollow noreferrer&quot;&gt;Puiseux series&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions#Generalizations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dirichlet's theorem on arithmetic progressions&lt;/a&gt;, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem#Generalizations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange's four-square theorem&lt;/a&gt;. But their generalizations are too cavernous.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;h3&gt;Examples of theorems that can be marginally generalized:&lt;/h3&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Spivak proves $\sqrt{2}$'s irrationality fully, but banishes &lt;a href=&quot;https://math.stackexchange.com/q/4467/53259&quot;&gt;$a^{\frac 1b}$&lt;/a&gt;'s to the exercises. Isn't proving (only) the latter more efficient?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why don't multivariate calculus textbooks  introduce &lt;a href=&quot;https://math.stackexchange.com/a/326774/53259&quot;&gt;Stokes's Theorem&lt;/a&gt; (with a double integral, not differential forms) first, and then state Green's as a corollary? &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0495011665&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus: Early Transcendentals&lt;/em&gt; (6 edn 2007)&lt;/a&gt; introduces the latter as §16.4, and former as §16.8. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why not prove first &lt;a href=&quot;https://en.wikipedia.org/wiki/Wilson%27s_theorem#Gauss&amp;#39;s_generalization&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss's generalization&lt;/a&gt; of Wilson's Theorem:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;$\forall \; p \; \text{odd prime}, \alpha \in \mathbb{N} : \quad \prod_{k = 1 \atop \gcd(k,m)=1}^{m} \!\!k \ \equiv&#xA;\begin{cases}&#xA;-1      \pmod{m}  &amp;amp; \text{if } m=4,\;p^\alpha,\;2p^\alpha \\&#xA;\;\;\,1 \pmod{m}  &amp;amp; \text{otherwise} &#xA;\end{cases}?$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not referring to &lt;a href=&quot;https://math.stackexchange.com/questions/2784679/why-dont-textbooks-foreground-marginally-generalized-theorems?noredirect=1#comment5742546_2784679&quot;&gt;the further generalization of Wilson's Theorem requiring group theory&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user155" LastEditDate="2018-05-19T07:59:53.130" LastActivityDate="2018-05-20T09:22:46.067" Title="Why don't textbooks foreground marginally generalized theorems?" Tags="|textbooks|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14085" PostTypeId="2" ParentId="14084" CreationDate="2018-05-18T19:41:31.410" Score="25" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Spivak proves $\sqrt{2}$'s irrationality fully, but banishes $a^{1/b}$'s to the exercises. Isn't proving (only) the latter more efficient?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes, it's certainly more efficient.  So what?  Efficiency is not a primary goal of a textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generality often makes concepts harder to understand.  That's true for professional mathematicians, and it's even more true for students, who have less experience doing the task of concretizing ideas on their own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proof that $\sqrt{2}$ is irrational adequately conveys the idea of the proof - someone who has truly understood it will be able to generalize it to $a^{1/b}$ without much difficulty.  But many students will find the idea easier to understand the first time they see it by looking at a specific number like $2$ rather than an abstract number $a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side benefit, it gives students the opportunity to see whether they understand the argument by doing the generalization themselves.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Most multivariate calculus textbooks devote 1 chapter to Green's Theorem. Why not introduce Stokes's first, and then state Green's as a corollary?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Because Stokes' Theorem is too abstract for many students at that level to understand in a single go.  Green's Theorem is a warm-up, getting students more used to the idea, before they encounter the more general theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Abstraction imposes cognitive work on the learner, to think through the scope of the abstraction and to produce concrete examples themselves.  A good textbook has to balance that work against other important cognitive work by the learner, like thinking about the intuition and proof of a theorem.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2018-05-20T02:11:04.900" LastActivityDate="2018-05-20T02:11:04.900" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14086" PostTypeId="2" ParentId="14084" CreationDate="2018-05-18T20:10:21.887" Score="11" Body="&lt;p&gt;I'll answer the question with respect to the Green's/Stokes' theorem special case. The summary is that, from a pedagogical point of view, little is gained by passing directly to the general case, and much is potentially lost (students for example). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If by Stokes' theorem is meant the version relating surface integrals and line integrals, this theorem is substantially harder to teach than its planar special case usually known as Green's theorem. Green's theorem relates an ordinary double integral over a region with a line integrals over the region's boundary. The version for surfaces requires defining surface integrals, and this introduces substantial new difficulties. In particular students have a lot of trouble with parameterizing surfaces, calculating normal vectors, and understanding orientations of surfaces. With Green's theorem only the orientation issue is present as such, and it is more easily visualized in the planar context, where it already is difficult for many students. In any case, objects in three dimensions are almost always harder to understand than their counterparts in two dimensions, if only because it is easier to draw pictures of planar objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If by Stokes' theorem one means the full blown theorem in terms of differential forms, then the difficulty, both technically and conceptually, is substantially greater. Moreover, the theorem is usually formulated in terms of differential forms (why stop there? one could go on to currents and just teach out of Federer ...). Understanding differential forms requires understanding duality of vector spaces, a topic often omitted from introductory linear algebra courses, often omitted completely from instruction given to physicists and engineers, and itself requiring some effort to assimilate properly. Many professional engineers never learn that the curl of a vector field is the vector dual to the exterior differential of the one-form dual to the vector field; physicists mostly can interpret such a phrase, but even applied physicists (as opposed to high energy theoreticians or general relativists) don't usually think that way, and don't write Maxwell's equations using forms (obviously theoreticians do). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While in fact the general theorem is not much more difficult to prove rigorously than are its various special cases (although to do so in full generality one has to deal with triangulations, or partitions of unity, or something along those lines) doing so obscures the essential ideas, which are just as well understood in proving special cases such as Green's or Gauss's (the divergence) theorem. In fact, once one understands the proof of Green's theorem, one basically understands what is needed to prove the general theorem; all the essential issues are present already. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Very few students learn top down (only rarely does one have a future Grothendieck as a first-year student). Most learn best proceeding from the special to the general. One pares away as many inessential aspects as one can, to focus the student's attention on the core issues. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-05-18T20:10:21.887" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14087" PostTypeId="1" CreationDate="2018-05-19T00:47:15.057" Score="3" ViewCount="490" Body="&lt;p&gt;A high-school teacher in the US whom I know, is&#xA;teaching &lt;a href=&quot;https://en.wikipedia.org/wiki/AP_Calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;AP AB Calculus&lt;/a&gt; for the first time.&#xA;He would like to use a &lt;a href=&quot;https://web.archive.org/web/20171217164642/http://mathforum.org/mathed/constructivism.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;constructivist&lt;/em&gt; approach&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;students explore mathematical problems and ideas and then, through whole-class discussions, students present their thinking and we as a class generalize the ideas and codify their findings.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He is seeking materials in line with this approach, to address the following&#xA;topics:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Limits and their properties&lt;/li&gt;&#xA;&lt;li&gt;Differentiation&lt;/li&gt;&#xA;&lt;li&gt;Applications of Differentiation&lt;/li&gt;&#xA;&lt;li&gt;Integration&lt;/li&gt;&#xA;&lt;li&gt;Logarithmic, Exponential, and other Transcendental Functions&lt;/li&gt;&#xA;&lt;li&gt;Applications of Integration&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I can make a suggestion for the functions item,&lt;sup&gt;*&lt;/sup&gt;&#xA;but not so much the other topics.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;*&lt;/sup&gt;&#xA;&amp;quot;From Pop-Up Cards to Coffee-Cup Caustics: The Knight's Visor.&amp;quot;&#xA;(&lt;a href=&quot;https://arxiv.org/abs/1206.1312&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv abs&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="15884" LastEditDate="2022-09-20T18:20:56.350" LastActivityDate="2022-09-20T18:20:56.350" Title="Calculus via a constructivist approach" Tags="|secondary-education|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14088" PostTypeId="2" ParentId="14087" CreationDate="2018-05-19T01:05:28.880" Score="2" Body="&lt;p&gt;I'd suggest that he check out the POGIL approach to teaching.  They have a &lt;a href=&quot;https://pogil.org/educators/become-a-pogil-practitioner/curricular-materials/materials-science-engineering-math-computer-science/calculus-i-a-guided-inquiry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus&lt;/a&gt; book that he might find useful.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2018-05-19T01:05:28.880" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14089" PostTypeId="1" CreationDate="2018-05-19T02:08:10.253" Score="7" ViewCount="227" Body="&lt;p&gt;I teach at a primarily undergraduate 4-year college in the US and we don't cover the proof of Cayley-Hamilton theorem in our linear algebra courses. I did however see both the computational and the abstract proof of the theorem when I studied from Hoffman-Kunze in my undergraduate (long time ago). &#xA;I am wondering if anyone can share at what point in their institution's linear algebra courses the theorem is usually taught and at what level. Is it after eigenvalues/eigenvectors? Is it right after determinants are introduced? Is it taught in its most generality, such as &quot;T is a linear operator over a vector field V with scalars k&quot; or is it taught in a specialized case with linear transformations over C^n?&#xA;I'm mostly interested to hear from people teaching at 4-year institutions in the US, but if the topic is taught at a 2-year college linear algebra course in the US, I would like to hear that too.&#xA;Thank you for any information.&lt;/p&gt;&#xA;" OwnerUserId="9926" LastActivityDate="2020-10-23T15:37:49.010" Title="Timing of when Cayley-Hamilton theorem is taught in Linear Algebra" Tags="|linear-algebra|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14090" PostTypeId="2" ParentId="1042" CreationDate="2018-05-19T08:10:23.963" Score="0" Body="&lt;p&gt;Maybe something with 3D point groups (not all the details of the group theory, but describing the types of them (mirror plane, rotation, etc.) and classifying some common objects.  (E.g. it is interesting that octahedron and cube have same symmetry.  also, 2-D tiling patterns (but not 3d space groups...too complicated).  &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-05-19T08:10:23.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14091" PostTypeId="2" ParentId="14089" CreationDate="2018-05-19T08:57:10.247" Score="5" Body="&lt;p&gt;I see no compelling reason to teach Cayley-Hamilton in a first course directed at students who are not studying math as such. For math majors at a typical institution it probably makes more sense to teach Cayley-Hamilton in a first course on commutative algebra, rather than in a first-course on linear algebra. The principal use of Cayley-Hamilton in such courses is probably in the context of relating the minimal and characteristic polynomials and the deduction of canonical forms other than the Jordan form, and none of that material is typically included in the standard introductory linear algebra course. The theorem is not easy to state carefully (so that it does not appear to be a tautology) and the abstraction required to prove it is above the level of such a course. If one feels the need to indicate it, one can state it, and give exercises/examples checking it for particular linear transformations/matrices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth, I teach in Spain, in an engineering school. Our linear algebra class is taught in the first-year and has a higher level (more formal, more rigor) than what is standard in the sort of of US institution you ask about, but we do not teach Cayley-Hamilton. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-05-19T08:57:10.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14092" PostTypeId="2" ParentId="14084" CreationDate="2018-05-19T09:53:12.040" Score="8" Body="&lt;p&gt;I would like to add that progressing from the special to the general is a good way to teach how research actually works. Frequently you discover some relation by accident and only after working with that relation for some time you realize that there could be some generalization for it. By using intuition and experience you are probably able to extract what is common to all the special cases you have explored so far and finally derive the general theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The textbooks already make things more 'efficient' in that they reveal to the reader that the general theorem indeed exists which is usually already a great leap/shortcut for the learner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I understand very well the desire to express things more compactly than the textbooks do. I have always done this for myself, just to discover in the end, that my 'summaries' can only be understood by myself and because I had read the original textbook before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although reading a textbook takes a lot more time, I find it also a lot more enjoyable than reading a formulary for example, just because a (good) textbook gives me the feeling as if I were taking part in the original discovery of the facts it presents. And joy is certainly the number one success factor for learning.&lt;/p&gt;&#xA;" OwnerUserId="9927" LastEditorUserId="9927" LastEditDate="2018-05-20T09:22:46.067" LastActivityDate="2018-05-20T09:22:46.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14093" PostTypeId="2" ParentId="14067" CreationDate="2018-05-20T01:02:00.030" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Admittedly, students will find it easy to believe that computers will take longer to solve a given problem if the input size grows. However, this is also true for problems which can be solved in polynomial time on a deterministic machine and completely misses the focus on NP-hardness and NP-completeness.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is an important point which is often neglected in more &amp;quot;accessible&amp;quot; accounts of NP completeness.  It gets neglected because (charitably) the author cannot assume familiarity with basic complexity theory and particularly big O notation or (realistically) the author is repeating something that they were told by someone else.  So we hear about &amp;quot;problems which can be solved/verified &lt;em&gt;quickly&lt;/em&gt;,&amp;quot; which is a flat lie by any reasonable definition of &amp;quot;quickly&amp;quot; (a stopwatch does not neglect constant factors).&lt;/p&gt;&#xA;&lt;p&gt;The educator is in a better position here.  We presumably have multiple lectures of time to work with, and complexity theory is an important subject anyway.  So you can start with something simple, like comparing linear search to binary search.  Place strong emphasis on scalability rather than speed.  In this example, you could suggest that the processor is more performant with linear access than random access due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Locality_of_reference&quot; rel=&quot;nofollow noreferrer&quot;&gt;locality of reference&lt;/a&gt; (to provide concrete differences in constant overhead of the algorithms).  Emphasize that we don't care so much about what the hardware is doing, or about how long it takes to compare two elements, etc.  All we care about is the overall growth rate.  Depending on the level of the class, it is probably a good idea to give the formal definitions of big O and big Theta notation at this point.  Regardless, the students should see &lt;span class=&quot;math-container&quot;&gt;$\Theta(n)$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$\Theta(\log n)$&lt;/span&gt; and a derivation of those expressions (even if only informal).&lt;/p&gt;&#xA;&lt;p&gt;You can then progress to something more challenging, like insertion sort vs. quicksort.  It is tempting to throw in some examples of recurrence relations and dynamic programming, but this is not essential and can be deferred to a later course.  This trains the students on how the big O and big Theta work and how to manipulate them.&lt;/p&gt;&#xA;&lt;p&gt;Once the students are at an acceptable level of comfort with complexity analysis in general, and big O in particular, you can move on to complexity classes.  At this point, you can &lt;em&gt;mostly&lt;/em&gt; get away with giving the algebraic definition (i.e. &amp;quot;the term inside the O(...) is a polynomial&amp;quot;).  However, one sticking point is defining the &amp;quot;verification&amp;quot; of NP in a way that is both intuitive and correct.  Strictly, a decision algorithm is only supposed to output true or false, but verification requires a certificate describing the solution in greater detail.  It may be helpful to use deterministic and nondeterministic finite automata as a simpler point of comparison.  You might even show the &lt;a href=&quot;https://en.wikipedia.org/wiki/Powerset_construction&quot; rel=&quot;nofollow noreferrer&quot;&gt;powerset construction&lt;/a&gt; to demonstrate that NFAs are potentially a lot &amp;quot;simpler&amp;quot; than the corresponding DFAs, depending on the available class time.  For an introductory course, you may be able to just gloss over this definition.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The other issue comes when working with younger students, for example in a high school computer science class.  Here, you may not have the time or syllabus to provide a full account of complexity theory, but still want students to walk away with some kind of understanding.  In this case, it may be best to start with one of the &amp;quot;classic&amp;quot; NP-complete problems and explain, in simple terms, why it seems to be so difficult to solve.  The knapsack problem is one example, but I don't like it for this purpose because it is &lt;a href=&quot;https://en.wikipedia.org/wiki/Pseudo-polynomial_time&quot; rel=&quot;nofollow noreferrer&quot;&gt;pseudo-polynomial&lt;/a&gt;.  Explaining the difference between pseudo-polynomial time and polynomial time is difficult, and can make it look like the knapsack problem is only NP-complete because of a definition (which is not the case).  Instead, I would use the traveling salesman problem, for these reasons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Its input size is straightforward to define and does not require talking about the &amp;quot;number of bits&amp;quot; of anything.&lt;/li&gt;&#xA;&lt;li&gt;It is easy to illustrate with graphs.&lt;/li&gt;&#xA;&lt;li&gt;The exponential algorithm is intuitively simple.&lt;/li&gt;&#xA;&lt;li&gt;The exponential algorithm is obviously non-scalable.&lt;/li&gt;&#xA;&lt;li&gt;The exponential algorithm exhibits noticeable growth even for very small inputs.&lt;/li&gt;&#xA;&lt;li&gt;It is a &amp;quot;classic&amp;quot; problem that people may have heard of elsewhere.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Explain that there is a simple exponential algorithm and what that means.  After providing a basic TSP definition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;obvious&amp;quot; way to solve TSP is to check every possible route and output the best one.  But this scales very poorly.  When I add one more city, I have to consider every possible placement of that city in every route that I was looking at previously, and that adds up fast!  &lt;strong&gt;The number of additional routes is proportional to the previous number of routes,&lt;/strong&gt; so the overall number of routes we need to check goes through the roof almost immediately.&lt;/p&gt;&#xA;&lt;p&gt;[Show graphical examples for small N, if possible, or alternatively show a graph of an exponential function.  You could also describe rabbits or other small animals reproducing exponentially, or any number of other accessible examples.  Adapt this to your audience.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The bolded phrase is a complete (if very brief) definition of exponential growth, but it may be too brief for students to understand.  An example or two certainly won't hurt.&lt;/p&gt;&#xA;&lt;p&gt;(You'll notice that I cleverly avoided talking about superpolynomial subexponential complexity without explicitly denying its existence.  That's because it is a fairly narrow category which serves primarily to distract rather than to inform.  If directly asked, you could acknowledge its existence or &lt;a href=&quot;https://en.wikipedia.org/wiki/Lie-to-children&quot; rel=&quot;nofollow noreferrer&quot;&gt;lie&lt;/a&gt;, but don't get side-tracked.)&lt;/p&gt;&#xA;&lt;p&gt;You can then finish up by explaining that, while the exponential solution is suboptimal, we're still a long way away from P:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mathematicians and computer scientists have been studying the TSP and other problems like it for some time, and they've found a few tricks to avoid having to try every route.  But those techniques still don't scale very well for larger inputs.  &lt;strong&gt;We would like the &amp;quot;growth factor&amp;quot; to be proportional to a power of the number of cities.&lt;/strong&gt;  Most mathematicians and computer scientists think that's actually impossible, but nobody has managed to prove it yet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This gives a succinct definition of polynomial growth and an extremely abbreviated statement of &lt;span class=&quot;math-container&quot;&gt;$P \ne NP$&lt;/span&gt;.  If you also want to cover the broader notion of NP and NP-complete, that's harder, but doable:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If we want to check whether a given route has at most a given length, that's easy: we just add up the distances along each of the roads.  If there is one more city, we have to add one more number, which is pretty fast.  But if we want to &lt;a href=&quot;https://en.wikipedia.org/wiki/Decision_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;decide&lt;/a&gt; whether there is a route of at most a certain length, that seems to be really difficult.  All of the known algorithms scale very poorly for larger inputs, like we saw earlier.&lt;/p&gt;&#xA;&lt;p&gt;It turns out that there are a lot of problems that are easy to check, but only some of them seem easy to solve.  We call these problems &lt;span class=&quot;math-container&quot;&gt;$NP$&lt;/span&gt;, and if they have scalable solutions we also call them &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;.  Certain problems, like the traveling salesman problem, are very general and powerful, and are called &lt;span class=&quot;math-container&quot;&gt;$NP$&lt;/span&gt;-complete.  If you have a scalable solution to one of these special &lt;span class=&quot;math-container&quot;&gt;$NP$&lt;/span&gt;-complete problems, then you can generalize it to every problem in &lt;span class=&quot;math-container&quot;&gt;$NP$&lt;/span&gt;.  That would give you a scalable solution to everything in &lt;span class=&quot;math-container&quot;&gt;$NP$&lt;/span&gt;, which would mean that &lt;span class=&quot;math-container&quot;&gt;$P = NP$&lt;/span&gt;.  But most mathematicians and computer scientists think that none of the &lt;span class=&quot;math-container&quot;&gt;$NP$&lt;/span&gt;-complete problems have any scalable solutions, so that &lt;span class=&quot;math-container&quot;&gt;$P \ne NP$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8474" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-05-20T01:15:07.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14094" PostTypeId="2" ParentId="14075" CreationDate="2018-05-20T19:19:46.590" Score="1" Body="&lt;p&gt;You said the chair and dean don't care what you do with this class, so I wouldn't worry too much about upholding historical standards for rigor or what you think a calculus class should look like. Also, with six students you can and should customize the course for the students you have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In ten years, what impact do you hope this course will have on them? It will probably be useful for them to understand standard mathematics notation, and reason about core concepts from calculus, like slopes and areas. Actually doing symbolic manipulation and simple calculation is (I would guess) going to be irrelevant. I wouldn't cut that material though, but use it only as a means to an end. They don't have to be good at taking derivatives and integrals, they just need enough exposure for some of the big ideas to sink in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For concrete advice, I would start from the basics, recognizing where they are now, moving along very quickly but also giving them frequent reminders for basic material. I would focus only on very simple computational problems, and/or make traditional problems strictly homework or open-book assessment. Go heavy on conceptual problems, even challenging ones. Maybe try some discussions problems in class, ask for short essays, or do applied projects. I would pass them if they get some core concept and might have a hint of a benefit in ten years. I would assign grades A-D in line with recent standards, adjusted for changes to the course.&lt;/p&gt;&#xA;" OwnerUserId="7707" LastActivityDate="2018-05-20T19:19:46.590" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14095" PostTypeId="2" ParentId="14067" CreationDate="2018-05-20T23:58:01.490" Score="2" Body="&lt;p&gt;I've had some success focusing on a single problem that is easy to grasp.&#xA;Given a &lt;em&gt;carpenter's rule&lt;/em&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/fSomq.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fSomq.jpg&quot; alt=&quot;carprule&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://www.dreamstime.com/stock-photo-antique-carpenter-s-boxwood-folding-rule-th-century-white-background-image48577005&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;with specific given (integer) lengths for its $n$ ruler sections, and an integer $L$, can it be folded so that its total length is $\le L$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is NP-complete by reduction from &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Partition_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;number partitioning&lt;/a&gt;.&#xA;The reduction is not difficult. Playing around with example rulers quickly leads&#xA;to the intuition that one must try all possible folding sequences, which is&#xA;easily felt to be exponential, as the only choice at&#xA;each joint is: either leave two adjacent ruler&#xA;sections open end-to-end at $0^\circ$, or folded at $180^\circ$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, these might be the lengths of the ruler sections:&#xA;$$\{84, 42, 4, 5, 2, 5, 6, 6, 8, 3, 1, 2, 42, 84\} ,$$&#xA;and the challenge is to decide if the ruler can be folded into a length&#xA;of at most $84$. It can be if and only if&#xA;$$\{4, 5, 2, 5, 6, 6, 8, 3, 1, 2\}$$&#xA;can be partitioned into two subsets that sum to $21$ each.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In comparison, computing the length of a given ruler when every joint&#xA;is folded $180^\circ$ is a linear-time calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is discussed in &lt;em&gt;&lt;a href=&quot;http://www.gfalop.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometric Folding Algorithms&lt;/a&gt;&lt;/em&gt; Section 2.2.2, but there are many other sources.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-05-21T10:23:25.513" LastActivityDate="2018-05-21T10:23:25.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14098" PostTypeId="1" AcceptedAnswerId="14102" CreationDate="2018-05-23T17:27:25.730" Score="5" ViewCount="188" Body="&lt;p&gt;I teach at an American research 1 university. I am planning a course on combinatorial topology for undergraduates whose background is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;multivariable calculus&lt;/li&gt;&#xA;&lt;li&gt;linear algebra&lt;/li&gt;&#xA;&lt;li&gt;at least one proof course&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In particular, I can't assume they have taken a class on analysis. I would like the course to &lt;em&gt;carefully&lt;/em&gt; prove the classification of piecewise-linear surfaces with boundary. Can anyone recommend a textbook that accomplishes this? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: &quot;Topology Now!&quot; is close to what I am looking for, but I would prefer something that builds surfaces out of triangles, in part to avoid having to talk about what it means to be locally euclidean. &lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="376" LastEditDate="2018-05-24T20:01:25.170" LastActivityDate="2018-05-24T20:01:25.170" Title="Reference request: undergraduate combinatorial topology" Tags="|undergraduate-education|reference-books|topology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14099" PostTypeId="2" ParentId="14068" CreationDate="2018-05-23T18:15:11.040" Score="4" Body="&lt;p&gt;My experience with this type of written work is because students use &quot;=&quot; in a different way than we mathematicians use it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes it is used to mean “the result of this calculation is”, like the above student did, and then they continue with the next calculation, just like a calculator would do. And students' understanding of equality or the equal sign affects their success in mathematics in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, without a true understanding of equality, students struggle with solving equations in a general way, such as in solving a differential equation or solving an equation with the variables being polynomials or vectors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't remember the first paper I read about this but a quick Google search came up with the following news article about related research which might help:&#xA;&lt;a href=&quot;https://www.sciencedaily.com/releases/2010/08/100810122200.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.sciencedaily.com/releases/2010/08/100810122200.htm&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="9926" LastEditorUserId="5452" LastEditDate="2018-05-25T12:57:36.137" LastActivityDate="2018-05-25T12:57:36.137" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14100" PostTypeId="2" ParentId="14068" CreationDate="2018-05-23T22:07:49.043" Score="4" Body="&lt;p&gt;The problem stems from the proliferation of worksheets, so entrenched into the American school culture. Get rid of the worksheets and teach students how to record a problem and a solution on a clean sheet of paper, preferably in a notebook, so that the record stays for at least the duration of a school year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, &lt;strong&gt;use GFSA (Given-Find-Solution-Answer) or GFAS (Given-Find-Assumptions-Solution) process&lt;/strong&gt;, which is used in technical colleges, but is completely absent from schools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Given:&lt;/strong&gt; write down what is given without repeating the problem verbatim; instead introduce constants and variables that later will be used in the solution; draw a picture if needed, use introduced symbols in the picture. The problem must be completely recognizable by reading the Given section without referring to the original problem in a textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Find:&lt;/strong&gt; clearly state what should be found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Assumptions:&lt;/strong&gt; this may be part of a Given part, for example, when solving geometrical problems involving vehicles or walking we may assume that the Earth's surface is flat, or when solving an equation we may assume that the roots are real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; a step-by-step process, which usually involve some textual explanations, but not too much. There should be enough text to understand what numbers mean. Solve symbolically first, then plug in the numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Answer:&lt;/strong&gt; clearly state the answer, some professors require to underline it for clarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above problem can be solved as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given: rectangle (draw a picture) with sides (mark sides with letters, say &quot;a&quot; for the longer side, and &quot;b&quot;). On the picture mark segments of the longer side as a1 and a2, for example. Then write a1 = 4m; a2 = 3m; b = 2m&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Find: Area of the rectangle (introduce a letter, say &quot;s&quot;), write s - ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solution: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Generic formula: s = a * b = (a1 + a2) * b;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Plug in numbers: &#xA;        s = (3 + 4) * 2 = 7 * 2 = 14;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;alternatively,&#xA;        s = (3 + 4) * 2 = 3 * 2 + 4 * 2 = 6 + 8 = 14;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each separate calculation is terminated with semicolon. It must be beaten into the subconcious that equal sign means that stuff on the left equals to the stuff on the right, so 3+4=7×2=14 is incorrect solution, and the problem is marked as not done, plain and simple. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students must be taught to split task into clearly marked steps, like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;s = s1 + s2, where s1 = a1 * b, s2 = a2 * b;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then students would write:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Area of the left part: s1 = 3 * 2 = 6 (m&lt;sup&gt;2&lt;/sup&gt;)&#xA;Area of the right part: s2 = 4 * 2 = 8 (m&lt;sup&gt;2&lt;/sup&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice (m&lt;sup&gt;2&lt;/sup&gt;) in parentheses. There are two approaches to dimensions: one is to write dimensions everywhere throughout the expression, like: s = 4m * 2m = 8m&lt;sup&gt;2&lt;/sup&gt; (no parentheses), this is useful for dimensional analysis, like meters times meters equals meters squared, but can be clunky for school math. Another option is to specify dimension of the resulting value only, in this case put it in parentheses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;s = 6 + 8 = 14 (m&lt;sup&gt;2&lt;/sup&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer: the area of the rectangle is 14 m&lt;sup&gt;2&lt;/sup&gt;. (underline &quot;14 m&lt;sup&gt;2&lt;/sup&gt;&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, this particular problem does not require such an elaborate solution, but the students must be taught the principle, the scaffolding of the generic solution. The above problem can be solved simply as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) s = a * b;&#xA;2) s = (3 + 4) * 2 = 7 * 2 = 14;&#xA;Answer: 14 (m&lt;sup&gt;2&lt;/sup&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Introduce symbolic solution as early as possible, this improves conceptual understanding, and will pay back when they start algebra.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2018-05-26T00:25:03.187" LastActivityDate="2018-05-26T00:25:03.187" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14101" PostTypeId="2" ParentId="13340" CreationDate="2018-05-23T23:10:37.990" Score="0" Body="&lt;p&gt;Does your community college allow gaining credit by passing the final exam for a course?  Or does it allow you to skip a course if you either pass the final for the course, or pass an interview about the course?  (These options are often called &quot;challenging&quot; the course.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If so, you could take the online course to learn the material, even if you cannot find a proctor for the exam(s).  Then when you get back to your community college, &quot;challenge&quot; the course.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2018-05-23T23:10:37.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14102" PostTypeId="2" ParentId="14098" CreationDate="2018-05-24T14:17:47.307" Score="3" Body="&lt;p&gt;Check out Kinsey's book &lt;a href=&quot;http://a.co/gYOuEf0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Topology of Surfaces&lt;/em&gt;&lt;/a&gt;. Kinsey proves the classification theorem of surfaces by considering triangulations of surfaces and then going through the algorithm of cutting-and-pasting to classify them. I'm pretty sure he doesn't talk about surfaces having to be locally Euclidean before this, or at least I don't remember it featuring in his proof. It's all just cutting-pasting-triangles on polygons where the edges are associated in some manner. The classification comes in Chapter 4, after only discussing some point-set topology in the previous chapters.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-05-24T15:15:17.580" LastActivityDate="2018-05-24T15:15:17.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14103" PostTypeId="1" CreationDate="2018-05-24T19:45:18.067" Score="1" ViewCount="74" Body="&lt;p&gt;I'm teaching a precalculus course and also taking a class on how to teach mathematics constructing a specific cognitive model for different topics. So, I have this assignment to build a cognitive model where I have to introduce my students to Trigonometric Ratios but in a constructive approach. One of the condition is create different kind of exploration activities where the student should be able to conclude properties and ratios till the only thing that’s missing are the names of the relations (sine, cosine, etc.), now I’m kinda lost here because I always being teaching this in the traditional way, Pythagorean theorem, special triangles, sohcahtoa, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I have now is the idea to make them build several equilateral triangles. First make them realize the relation between the measures on the sides when we cut in half, then using triangle similarities try to make them realize on how they always get the same proportion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyone has implemented something like this in your classroom? Where I can find information on the matter? No luck so far and any help is very much appreciated.&lt;/p&gt;&#xA;" OwnerUserId="8853" LastActivityDate="2018-05-24T19:45:18.067" Title="How to introduce trigonometric ratios (HS) through a cognitive model?" Tags="|secondary-education|precalculus|trigonometry|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14104" PostTypeId="2" ParentId="14068" CreationDate="2018-05-24T19:59:22.447" Score="1" Body="&lt;p&gt;In my experience use notebooks where you can easy check their work and give instructions on how to correct it. To my students I always say that writing mathematics is like writing a small meaningful phrase. It has to be aesthetically appealing, must be correct, good grammar etc. So they have to write it as beautiful as they can and respecting rules and meanings. A lot of them complain that I call mathematics, or any expression beautiful. However with this they (after 2 or 3 months) start to get it, and apply it constantly. At the beginning of this school year I had easily 35 out of 40 students doing this kind of abuse to the notation, right now is around 15/40. And the other are slowly getting it. I want to point out that a lot of this mistakes are imprinted by private tutors, peers or sometimes a teacher that allows this kind of notation use.&lt;/p&gt;&#xA;" OwnerUserId="8853" LastEditorUserId="5452" LastEditDate="2018-05-27T00:33:27.817" LastActivityDate="2018-05-27T00:33:27.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14105" PostTypeId="2" ParentId="14068" CreationDate="2018-05-25T03:18:09.457" Score="6" Body="&lt;p&gt;I've read that this issue goes all the way back to elementary school where a teacher will write on the board, &quot;3 x 4 =   &quot; and ask students to fill in the blank.  This creates a subtle impression that an = sign is a procedural thing rather than a statement of equivalence, i.e. that it means something like &quot;the next step is&quot;.  This leads to statements like yours where each of the steps is separated by an = sign.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2018-05-25T03:18:09.457" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14106" PostTypeId="2" ParentId="13340" CreationDate="2018-05-25T03:22:36.367" Score="1" Body="&lt;p&gt;Ottawa University is a fully accredited university in Kansas that offers an &lt;a href=&quot;https://www.ottawa.edu/adult-and-graduate/academics/arts-and-sciences/degree-programs/bachelor-of-arts-in-mathematics-(1)/&quot; rel=&quot;nofollow noreferrer&quot;&gt;online math degree&lt;/a&gt; including all of the classes that you listed.  I used to teach for them and none of my classes ever required a proctored exam.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2018-05-25T03:22:36.367" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14108" PostTypeId="1" CreationDate="2018-05-25T19:08:54.340" Score="4" ViewCount="578" Body="&lt;p&gt;Is there any research that substantiates the helpfulness of using colour in presenting math?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.meta.stackexchange.com/a/23192/53259&quot;&gt;Colour has been used effectively&lt;/a&gt; by user &lt;a href=&quot;https://math.meta.stackexchange.com/users/242/bill-dubuque&quot;&gt;Bill Dubuque&lt;/a&gt; on Math SE,  &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0495011665&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus: Early Transcendentals&lt;/em&gt; (6 edn 2007)&lt;/a&gt; and David Lay's &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0321385179&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Linear Algebra&lt;/em&gt;&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;p&gt;I'm not asking about practical drawbacks of using color, like increased printing costs or &lt;a href=&quot;https://math.meta.stackexchange.com/q/4195/53259&quot;&gt;harm to the visually impaired&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="262" LastEditDate="2020-09-25T04:00:23.610" LastActivityDate="2020-10-01T12:33:38.383" Title="Ought color be used more in math?" Tags="|education-research|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14109" PostTypeId="1" CreationDate="2018-05-26T00:53:11.093" Score="10" ViewCount="351" Body="&lt;p&gt;This semester I taught the third semester of my college's freshman physics sequence. Nearly all the students are engineering majors. Compared to previous semesters when I've taught this course, I went into a little more detail on quantum mechanics, including introducing the very handy &lt;a href=&quot;https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation&quot; rel=&quot;noreferrer&quot;&gt;bra-ket notation&lt;/a&gt;. My students were first exposed to the description of a wavefunction $\Psi$ as a complex-valued function of position, including expressions such as $\int \Psi^*\Psi dx$ for probabilities. All of them had been exposed to complex numbers before. Later we did wavefunctions that were not expressed in position space, and they had to do things like taking the ket $i|\uparrow\rangle$ for an electron in a spin-up state and constructing its bra version $-i\langle\uparrow|$. In mathematicians' language, these notations describe a vector and its dual.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this stage I felt that there was some awkwardness, as expressed by student questions such as &quot;What's the difference between $|\uparrow\rangle$ and $\langle\uparrow|$?&quot; Most but not all of them had had a semester of linear algebra. With those students, I tried making the analogy to row and column vectors, and that sort of worked. However, it didn't seem like they had been exposed to the idea of taking the complex conjugate when doing a Hermitian adjoint. I don't know how much of their lin alg course was done over the reals only, but I get the impression that most of that course consists of mind-numbing concrete computations, with page after page of homework problems involving matrices with integer elements. It's also a little awkward to say that bras and kets are row and column vectors, since many of the vector spaces we deal with are infinite-dimensional.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could have just told them that a bra is a linear operator on ket space, but I think that would have been much too abstract for their level of mathematical development. And really it seems unaesthetic to me to break the symmetry between the two mutually dual vector spaces by describing one as more fundamental and the other as derived from it. The way they occur in quantum mechanics, and the way in which they relate to experimental observables, is completely symmetric. (In this respect they are different from the covariant and contravariant vectors of general relativity, where the convention is that infinitesimal displacements, i.e., elements of the tangent space, are represented by contravariant vectors, and we do have quantities like frequency that are then more naturally represented as covariant vectors.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With students who hadn't had any linear algebra at all, I seemed to have some luck when I tried explaining the general idea of duality. The best example I could think of using math they would know was that you could have a function and its derivative, with each one basically expressing the same information, two different representations of the same thing. But this isn't a completely accurate example of duality, because we have constants of integration, so a round-trip transformation loses information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone suggest other analogies or examples to use for the idea of duality, ones that would be accessible to people whose most advanced training is a year of calculus? Maybe it would be helpful to introduce complex conjugation itself as an example of duality.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2018-05-26T01:03:42.607" LastActivityDate="2018-07-27T13:00:44.347" Title="Analogies or explanations for duality, at the college sophomore level" Tags="|linear-algebra|physical-sciences|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14110" PostTypeId="2" ParentId="14108" CreationDate="2018-05-26T18:52:01.373" Score="1" Body="&lt;p&gt;Not research but some thinking, too long for comment:&lt;/p&gt;&#xA;&lt;p&gt;I have seen the use of &amp;quot;4 color&amp;quot; graphs (it's 2 colors with shading, and one of the colors is orange, thus really one extra printing color over black ink) in 1980s Thomas Finney for graphs and especially solid structure pictures (e.g. rotations of solids).  They touted it as beneficial, but I have also seen the practice criticized for cost.  (but it's not like color photo plates).  In general, there is much less need for it in math compared to other subjects where it really does help (e.g. biology, chemistry, history, or especially art history).  I suspect most places it is not needed, but it will help marginally in showing certain graphs.&lt;/p&gt;&#xA;&lt;p&gt;Also, in statistics (or really data analysis), there is a benefit to colors used in bar charts or line charts or the like with multiple data series.&lt;/p&gt;&#xA;&lt;p&gt;I don't see much benefit to the use in equations from the other question.  Probably get clearer emphasis with curly brackets underneath or something like that.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="14696" LastEditDate="2020-10-01T12:33:38.383" LastActivityDate="2020-10-01T12:33:38.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14111" PostTypeId="1" CreationDate="2018-05-27T11:10:55.560" Score="14" ViewCount="480" Body="&lt;p&gt;I am TA-ing a introductory course on ODEs and PDEs this year. At my university most introductory math courses can be taken at &quot;basic&quot; and &quot;extended&quot; levels. This one is the extended one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My students seem to be not engaged in the subject. I don't blame them. They have already seen some abstract mathematical theories -- general topology, measure theory, abstract algebra. What I am teaching them is a lot of methods for solving specific ODEs. There are of course parts of general theory (existence theorems and some qualitative theory) and exercises to prove something, but mostly its calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They long for some abstraction, general framework. Unfortunately, i don't know how to provide it to them without functional analysis (which they haven't had).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please, give me any of your thoughts and advice on this problem.&lt;/p&gt;&#xA;" OwnerUserId="9950" LastActivityDate="2018-05-28T16:57:16.670" Title="How to teach ordinary differential equations to good students?" Tags="|undergraduate-education|differential-equations|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14112" PostTypeId="2" ParentId="14111" CreationDate="2018-05-28T06:00:36.923" Score="3" Body="&lt;p&gt;Here's a few things I think are worthwhile:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;deeper substitutions, solve $y''+y=0$ by explicit substitution and integration. It's a nice think to touch base with as you go on in the course.&lt;/li&gt;&#xA;&lt;li&gt;theory of n-th order linear systems. How to capture linear independence. Discuss the distinction between LI for solution sets vs. arbitrary sets of functions. Abel's formula holding for the n-th order problem is a bit shocking really. Spend some time on the Wronskian and get them thinking about how cool it is we can use finite dimensional linear algebra in function space and get away with it.&lt;/li&gt;&#xA;&lt;li&gt;hand in hand with last bullet point, operators theory. Discuss and develop the theory of smooth operators. Noncommutativity is easy to exhibit, yet we largely solve problems of the form $L[y]=f$ where $L \in \mathbb{R}[D]$ and $D=d/dt$. Why can we factor such $L$. What about when $L = L_1L_2\cdots L_k$ and $L_1,\dots , L_k$ are not mere polynomials in the differentiation operator? How to solve such a problem? It's a lost art, but it's easy.&lt;/li&gt;&#xA;&lt;li&gt;again hand in hand in hand with the bullets above, the annihilator method. Rather than just forcing them to memorize how to construct the particular solution, this gives them a way to derive it. Also, gives us a good reason to care about larger than order two problems since it is entirely possible for a spring mass problem to beg us solving a $4$-th order or higher ODE.&lt;/li&gt;&#xA;&lt;li&gt;Variation of parameters is general whereas undetermined coefficients, obviously not. Notice from the operator approach you can even frame Cauchy-Euler problems as polynomials in $td/dt$ and much of the discussion can be mirrored. Or, you could be boring and just plug in $x^r$ and $e^{rt}$ and do the less inspired calculus.&lt;/li&gt;&#xA;&lt;li&gt;So much to see in systems! Notice the fact that $y''+2y'+y=0$ has solution $y = c_1e^{_t}+c_2te^{-t}$ already indicates the corresponding system obtained by reduction of order is not diagonalizable. Furthermore, much riches can be found simply by dwelling on the question of that mysterious $t$ in the solution here.&lt;/li&gt;&#xA;&lt;li&gt;matrix exponential. Need I say more?&lt;/li&gt;&#xA;&lt;li&gt;PDEs are quite challenging, but, for better students perhaps this is an opportunity to remind them where much of our soul searching stems from in analysis. Fourier series beg questions about interchange of limits and illustrate why details matter. Ironically, details of analysis matter less to the problems you're likely to cover in this course. If you can work in the solution via Laplace transform material I think it's a win here. I sometimes succeed.&lt;/li&gt;&#xA;&lt;li&gt;complex analysis. Comment on connections when you get a chance. &lt;/li&gt;&#xA;&lt;li&gt;last but certainly not least, the differentiate the parameter trick. Why does it work? Where does it work? Specifically, $\frac{d}{dr}e^{tr} = te^{tr}$ and $\frac{d}{dr}(x^r) = \ln(x)x^r$...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you're really interested and happen to have students who have had abstract algebra before your class, I have additional suggestions, but they're way outside the box.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; ok, so what's outside the box? How about this, to solve &lt;strong&gt;any&lt;/strong&gt; constant coefficient differential equation over an algebra we can simply select the component functions of the exponential over the characteristic extension algebra. For example, to solve $y''+y=0$ over $\mathbb{R}$ we work on $\mathcal{A} = \mathbb{R}[k]/(k^2+1)$ where $k^2=-1$. In this context,&#xA;$$ e^{kt} = 1+kt+\frac{1}{2}k^2t^2+ \cdots = \cos(t)+k \sin t $$&#xA;thus the solution is formed by the special functions $\cos t$ and $\sin t$ of the characteristic extension algebra $\mathcal{A}$. But, lest you think this is just complexification. Notice this technique works for any problem. For example, $y''-2y'+y=0$ suggests $\mathcal{A} = \mathbb{R}[k]/(k^2-2k+1)$ which is to say $(k-1)^2 = 0$. Once more consider, the map $t \mapsto e^{kt}$ and calculate:&#xA;$$ e^{kt} = e^{(k-1)t+t} =e^te^{(k-1)t} = e^t(1+(k-1)t) = (1-t)e^t+kte^t $$&#xA;thus $(1-t)e^t, te^t$ form a fundamental solution set. I have proof that the component functions of the exponential always form an $\mathcal{A}$-LI solution set even when the solution set is a module (as is the case when the base algebra is not a field, in both of my examples the base algebra was just $\mathbb{R}$). If you understand what I am saying here, you'll realize I just put forth a method which solves &lt;strong&gt;all&lt;/strong&gt; n-th order ODEs via the same technique. Studying differential equations over algebras which are not fields brings new challenges if you wish to continue down this path. For example, we can show the differential equation $(D-1)(D-j)[y]=0$ where $D = d/dz$ and $j^2=1$ in the hyperbolic numbers $\mathcal{H} = \mathbb{R}\oplus j \mathbb{R}$ has solutions $e^z$ and $e^{jz}$ which are &lt;strong&gt;linearly dependent&lt;/strong&gt;. It can be shown that:&#xA;$$ w = c_2e^{jz}+c_1e^{jz}\left( z+&#xA;\sum_{n=1}^{\infty} \frac{1-j}{(n-1)!(n+1)} z^{n+1}\right). $$&#xA;is the general solution. You see, the distinct characteristic values $\lambda_1 = 1$ and $\lambda_2 = j$ behave as if they are repeated... almost...The reason for this bizarre behavior is simply that $(1-j)(1+j)=0$ so the difference of the characteristic values is a zero divisor of the algebra. My experience with these things is that zero divisors bring rather different structure than that we've grown accustomed to see over the fields $\mathbb{R}$ or $\mathbb{C}$. I have more to say in my &lt;a href=&quot;https://arxiv.org/abs/1708.04137&quot; rel=&quot;nofollow noreferrer&quot;&gt;joint paper with Nathan BeDell&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The abstract algebra used here is mostly the extension of Kronecker's Theorem to polynomials which need not be irreducible. The key construction is just that $\mathcal{A}[k]/(p(k))$ for an $n$-order monic polynomial $p$ is in fact a free $\mathcal{A}$-module with basis $1,k,k^2, \dots , k^{n-1}$. In the case $\mathcal{A}=\mathbb{R}$ this is just an $n$-dimensional vector space. Beyond that, I had to develop a bit of calculus over an algebra. Details are found in my papers on these topics.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2018-05-28T16:57:16.670" LastActivityDate="2018-05-28T16:57:16.670" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14113" PostTypeId="2" ParentId="14109" CreationDate="2018-05-28T11:49:52.517" Score="1" Body="&lt;p&gt;The symmetry between vectors and co-vectors in your example (i.e. the possibility to identify a vector with a co-vector) exists only since you have a scalar/hermitian product on the vector space $V.$ The general notion of duality in linear algebra doesn't have that symmetry, and I would rather break the symmetry if I wanted them to understand this more general duality. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If my aim were for them to &quot;only&quot; understand the difference between $|v\rangle$ and $\langle v|$, I'd tell them that the hermitian product on $V$ induces two &lt;em&gt;different&lt;/em&gt; maps of type $V \to (V \to \mathbb{C})$. The first one is $v\mapsto |v \rangle$ and the second one is $v\mapsto \langle v|$. I'd then tell them a little bit about the funny notation and why it is useful.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2018-05-28T11:49:52.517" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14114" PostTypeId="1" AcceptedAnswerId="24544" CreationDate="2018-05-28T16:03:20.553" Score="10" ViewCount="783" Body="&lt;p&gt;Here's the foundational thing that irritates me the most when teaching college algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Up through the secondary level, I think that instructors and students are trained to understand subtraction and division in terms of the inverse operation. Focusing on division here, if one asked &quot;Why is $6/2 = 3$?&quot;, then one would most likely say it's because:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$3 \times 2 = 6$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in every college-level algebra book I've seen, a different definition is given (and this goes for any texts in remedial elementary algebra, intermediate algebra, college algebra, etc.). Specifically, such books begin with the accepted &quot;properties of real numbers&quot;, which are basically a restatement of the axioms for a field. In particular, one of the basic axioms is the existence of inverses: e.g., for multiplication, for any $a \ne 0$, there exists a value $1/a$ such that $a \times 1/a = 1$. (This is already problematic because students at this level are not yet familiar with statements involving existential quantifiers.) Thereafter, division is defined this way: $a/b$ means $a \times 1/b$. Of course, that's exactly what we see for a definition in most abstract algebra texts. But then technically this commits us to justifying &quot;Why is $6/2 = 3$?&quot; with something like the following chain of reasoning from the axioms:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$6/2 = 6 \times 1/2 = (3 \times 2) \times 1/2 = 3 \times (2 \times 1/2) = 3 \times 1 = 3$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which I'm pretty sure no one actually ever does. Rather, they continue to use the secondary-school justification, even though this is technically out-of-synch (although, obviously, provably consistent with) our starting textbook axiom-properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore: When radicals are defined in the college algebra text, then the definition will once again look like the understanding of inverses from secondary-school subtraction and division (so it is additionally irritating to have these definitions and justifications out-of-synch with each other). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary: Advantages of the secondary-school definition: (1) it's what students are familiar with, (2) it provides shorter justifications, (3) it better lays the groundwork for the definition of radicals. Advantages of the standard college-algebra definition: (1) it complies with any standard textbook, and (2) it synchronizes with standard abstract algebra definitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I go back-and-forth about this &lt;a href=&quot;https://www.quora.com/What-phrase-replaces-tall-poppy-syndrome-and-the-proud-nail-gets-hammered-down-when-the-new-generation-has-never-seen-a-poppy-or-knows-what-a-proud-nail-is&quot; rel=&quot;noreferrer&quot;&gt;proud nail&lt;/a&gt; every semester. It seems like there would be more advantages to redefining subtraction and division as per the customary secondary-school rules, and thus smooth the way for student entry and understanding of the course; but the labor of going off-book and rewriting everything always deters me. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the best resolution to this problem? &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-01-05T15:28:10.277" Title="How to resolve the new definition of subtraction and division seen in college algebra?" Tags="|undergraduate-education|algebra|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14115" PostTypeId="2" ParentId="14114" CreationDate="2018-05-28T16:39:26.077" Score="9" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;But then technically this commits us to justifying &quot;Why is $6/2 = 3$?&quot;&#xA;  with something like the following chain of reasoning from the axioms:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$6/2 = 6 \times 1/2 = (3 \times 2) \times 1/2 = 3 \times (2 \times 1/2) = 3 \times 1 = 3$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Which I'm pretty sure no one actually ever does.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am reminded of &lt;em&gt;Principia Mathematica&lt;/em&gt; famously taking hundreds of pages to prove that $1+1=2$. I'm pretty sure no-one actually ever does that either. (Actually, I doubt that anyone at secondary-school level, much level college level, bothers to justify $6/2=3$ at all).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, since this troubles your conscience, the best resolution to an apparent mis-match between an old, familiar, system and a new system is surely to prove their equivalence and then to work in whichever system is most convenient for the task at hand. So if you prove the lemma $a / b = c \iff a = b \times c$ you can then apply your secondary-school justification without any twinges of conscience. And if you explicitly &lt;em&gt;teach the principle&lt;/em&gt; of proving equivalence and then working in the more convenient system you're doing your students a big favour. I think it can be argued that that principle is as foundational to mathematical thinking as axiomatisation, and probably more so.&lt;/p&gt;&#xA;" OwnerUserId="8005" LastActivityDate="2018-05-28T16:39:26.077" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14116" PostTypeId="1" CreationDate="2018-05-28T23:33:06.980" Score="5" ViewCount="172" Body="&lt;p&gt;I have struggled to offer an intuitive explanation &#xA;(to U.S. college students)&#xA;why the number of regular polytopes in dimension $d$ is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$d=2$, number: $\infty$.&lt;/li&gt;&#xA;&lt;li&gt;$d=3$, number: $5$, the five Platonic solids.&lt;/li&gt;&#xA;&lt;li&gt;$d=4$, number: $6$, with the $24$-cell the polytope with no clear $\mathbb{R}^3$ analog.&lt;/li&gt;&#xA;&lt;li&gt;$d \ge 5$, number: $3$, the simplex, hypercube, and its dual the cross-polytope (or orthoplex).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The derivations are convincing without providing clear intuition.&#xA;Is there some intuitive explanation, perhaps connected to the maximum volume&#xA;of a unit-ball achieved in dimension $5$?&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/Rcv8u.png&quot; width=&quot;400&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Plot: &lt;a href=&quot;https://divisbyzero.com/2010/05/09/volumes-of-n-dimensional-balls/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dave Richeson's blog&lt;/a&gt;, 2010.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;(But see &lt;a href=&quot;https://mathoverflow.net/a/53130/6094&quot;&gt;Bill Thurston's remarks on the unit-ball volume&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-06-02T09:43:33.143" LastActivityDate="2018-06-02T09:43:33.143" Title="Intuition: 5 regular polyhedra, 6 regular 4-polytopes, and then 3 regular d-polytopes" Tags="|undergraduate-education|geometry|intuition|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14117" PostTypeId="2" ParentId="14108" CreationDate="2018-05-30T08:51:14.970" Score="4" Body="&lt;p&gt;There is a fairly old &lt;a href=&quot;http://www.jstor.org/stable/41190497?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; by William A. Ewbank (from 1978) in which the author argues that more colors should be used when teaching mathematics. Moreover, the references therein might also be interesting to you.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ewbank, William A. &quot;The Use of Color for Teaching Mathematics.&quot; The Arithmetic Teacher 26.1 (1978): 53-57.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7608" LastEditorUserId="7608" LastEditDate="2018-06-01T06:59:08.077" LastActivityDate="2018-06-01T06:59:08.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14118" PostTypeId="1" AcceptedAnswerId="14120" CreationDate="2018-05-31T14:17:12.373" Score="9" ViewCount="318" Body="&lt;p&gt;The following is an expectation for our &quot;course-based research initiative&quot;. I'll include the complete wording so you can best understand my question.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Designing a Research Proposal/Project&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;p&gt;Embark on a research proposal/project by being able to:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-select an area of interest&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-develop a topic&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-identify a problem&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-and/or develop a research question&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Clarify by conducting a literature or artifact review or developing a main argument&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Find needed information/data by:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-collecting relevant sources/evidence (including, but not limited to opposing or supporting arguments, frameworks, methodologies, and/or&#xA;  theories)&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Evaluate the information by identifying:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-the credibility of sources&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-feedback from peers and professionals in the field to improve the proposal/project&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Organize the presented content and/or project timeline in a manageable, usable, and discipline appropriate format that can result&#xA;  in implementation of the research proposal or project.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Critically analyze the strengths and weakness of the information collected, including, but not limited to:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-supporting and opposing arguments&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-methodologies&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-the proposed process&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Synthesize by constructing or summarizing all information/data into a research proposal/project that includes rigorous, researchable&#xA;  questions based on new understandings.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My school apparently wants all of this integrated into an existing course in the Math major. My question is this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do you know any school where all undergraduate Math majors in a regular course are expected to &lt;em&gt;do research&lt;/em&gt; in the sense outlined above ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What is the best way to explain the distinction between Math and other disciplines where the above &quot;research&quot; might be plausible. Am I correct to say this is not a reasonable request &lt;strong&gt;if&lt;/strong&gt; we accept the term research has its usual meaning in Mathematics ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I appreciate an advice and/or experience you might have to share. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Extra Note: this is subtle, but &lt;a href=&quot;https://matheducators.stackexchange.com/questions/11326/course-based-undergraduate-research-experiences-in-math&quot;&gt;this question&lt;/a&gt; is less specific than my own. My question concerns modifying an existing course, not creating a new one with extra resources and/or a special subcategory of students.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-06-03T13:51:58.260" Title="Can or should students do research in standard major math courses" Tags="|undergraduate-education|undergraduate-research|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14119" PostTypeId="2" ParentId="14118" CreationDate="2018-05-31T14:27:59.360" Score="3" Body="&lt;p&gt;Do some applied projects:  statistics or linear programming optimization or diffy Qs.  Not pure math research.  Doesn't need to be publication type finding a new law either.  Could be analyzing a physical problem or a company or an industry or social issue.  Would do some good to have that outreach anyways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that said, I think there is enough going on in a regular math course (majors or even pre-major) so that incorporating research is not a good idea. &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-05-31T14:27:59.360" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14120" PostTypeId="2" ParentId="14118" CreationDate="2018-05-31T17:33:40.233" Score="5" Body="&lt;p&gt;I think the term research causes a lot of confusion here.  When mathematicians think about research, we immediately think about proving new theorems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that doesn't seem to be what's asked here.  As I read the outline, the end product is supposed to be a research &lt;em&gt;proposal&lt;/em&gt;, not research itself.  That is, students are asked to propose a suitable question for research, including investigating prior work to determine whether the question is an appropriate and interesting one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't see anything unreasonable about including that that in a suitable upper level undergraduate math course.  Taken literally, the outline seems to call for students to skim some recent papers on a related subject in enough detail to identify what is and isn't known and make a conjecture supported by results in the literature.  That doesn't require students to prove anything new, or even understand the proofs in those papers in any detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not aware of a course with an assignment framed that way, but the main steps are similar to assignments I've seen, and given.  I've had my discrete math class do a writing project where they had to mostly-independently read a few outside sources and synthesize enough information to present the material to the class.  That's a similar level of &quot;research&quot; to what this outline asks for - the point is to start learning to parse information and context from the academic literature (as opposed to things pre-digested in textbooks).&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-05-31T17:33:40.233" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14125" PostTypeId="2" ParentId="14114" CreationDate="2018-06-02T11:58:15.167" Score="2" Body="&lt;p&gt;I'm very much with you concerning the problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Striving for conceptual understanding and not merely exercising procedure following, I actually do bother with the question to justify why 6/2=3.&#xA;But maybe one has to clarify more clearly, what 'justify' in school context actually means. To me, it doesn't mean to give a mathematically valid proof as in the &lt;em&gt;Principia Mathematica&lt;/em&gt;, but merely an explanation that allows students to connect to other concepts and to model meaning; loosely speaking...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case you mention, I see the problem mainly in a change of concept:&#xA;In middle school - students develop an understanding of inverse &lt;em&gt;operation&lt;/em&gt;: They learn to see division as inverse operation of multiplication.&#xA;In high school or later in college, students then should develop an understanding of inverse &lt;em&gt;element&lt;/em&gt;: they learn that there is a neutral element with respect to an operation and the meaning of an inverse element is that its action on (i.e. operation with) the element itself results in the neutral element. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the way that I read your line of reasoning. The first equal sign reads as interpretation what 'division' means: multiplication with the inverse element. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, students often have a lot of trouble with this abstraction: To see as object what they used to know as an operation. Frankly I don't know what the &lt;em&gt;best&lt;/em&gt; way is to approach this problem. That might much depend on the local context you encounter. But I do have some strategies as an approach. Said in advance, it anyway just takes time, and patience, and care...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I explicitly discuss different models for the objects and the operations. For example, taking the number line as model for the numbers (objects), what are models for the operations? addition and subtraction might be straight forward, but if you model muliplication as concatenation of &quot;arrows&quot;, that lets you explain what $5 \times \frac{2}{3}$ is, but does not work for a calculation like $\frac{8}{7}\times \frac{2}{3}$. That works if you model multiplication as scaling (e.g. I use the following &lt;a href=&quot;https://ggbm.at/Dcpczszz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geogebra-Applet&lt;/a&gt; as part of the many ways to explain why the product of two negative numbers yields a positive number as a result). You can model division in a similar way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using explicit models I'm usually pretty successful in talking about the distinction of &lt;em&gt;inverse operation&lt;/em&gt; and &lt;em&gt;inverse element&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure there are other ideas and better ways to address this problem with students, I would love to hear more about how what people in this community think about this question. &lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2018-06-02T11:58:15.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14126" PostTypeId="2" ParentId="14114" CreationDate="2018-06-02T15:05:09.447" Score="1" Body="&lt;p&gt;The problem stems from effectively defining the division operator twice. You should begin by defining division on &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; as usual:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For all &lt;span class=&quot;math-container&quot;&gt;$x, y, z \in R$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$y\neq 0$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$x/y=z$&lt;/span&gt; iff&#xA;&lt;span class=&quot;math-container&quot;&gt;$x=y\cdot z$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, &lt;span class=&quot;math-container&quot;&gt;$6/2=3$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$6 = 2\times 3$&lt;/span&gt; by the definition of division on &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From the field axioms, we know that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For all &lt;span class=&quot;math-container&quot;&gt;$x\in R$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$x\neq 0$&lt;/span&gt;, there exists &lt;span class=&quot;math-container&quot;&gt;$y\in R$&lt;/span&gt; such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$x\cdot y=1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note that I do not make use of the division operator here.&lt;/p&gt;&#xA;&lt;p&gt;Now, we can prove:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For all &lt;span class=&quot;math-container&quot;&gt;$a\in R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a\neq 0$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$1/a\in R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a\cdot (1/a)=1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$a\in R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a\neq 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then, from the field axioms, there must exist &lt;span class=&quot;math-container&quot;&gt;$b\in R$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$a\cdot b =1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Applying the definition of division, we have &lt;span class=&quot;math-container&quot;&gt;$b=1/a$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Substituting, we have &lt;span class=&quot;math-container&quot;&gt;$a\cdot (1/a)=1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We conclude as required that:&lt;/p&gt;&#xA;&lt;p&gt;For all &lt;span class=&quot;math-container&quot;&gt;$a\in R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a\neq 0$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$1/a\in R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a\cdot (1/a)=1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Follow-up some years later...&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;When does a theorem become an axiom? You might point out that the real numbers themselves can be constructed (i.e. proven to exist) using only the axioms of set theory. Likewise the addition, subtraction, multiplication and division functions. My &amp;quot;definition&amp;quot; of division will correspond to a theorem about the the multiplication and division functions. Likewise, each of the field &amp;quot;axioms&amp;quot; will correspond to a theorem about each of these functions. Now, these proofs consist of thousands of lines of formal proof and are way beyond the scope of any high school and almost every university math course  To get on with real analysis, the field axioms are usually taken as a given even in advanced courses.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-01-05T15:28:10.277" LastActivityDate="2023-01-05T15:28:10.277" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14127" PostTypeId="2" ParentId="14118" CreationDate="2018-06-02T15:22:49.780" Score="1" Body="&lt;p&gt;I have seen other schools and extracurricular programs aspire to include math &quot;research&quot; at younger levels without really understanding what that entails. Many programs don't understand that this usually doesn't happen until the end of a doctorate.&#xA;Also (as you brought up in the comments below), focusing on the part where one just sets up the research question is not a good way to simplify the task because that part is typically done by the senior researcher or a doctoral student's advisor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that a modified version of this might be a valuable activity for very advanced and motivated students. As written, I agree that it would be excessive to require every student to do the program you described. I also think it's unrealistic to expect every teacher of being qualified to carry it out. An alternative I would suggest is to first have the teacher find a handful of math papers that seem accessible (I would look only at expository journals, not research journals), then have a student pick one of those papers, and the project would simply be to read it (very carefully, over a long time), and write a review of it explaining the idea and what they think about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You also asked about explaining the distinction between math and other disciplines when it comes to research. I agree that is probably the source of the problem here. I'm including some comments on that referencing the items from your list.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;2&quot;&gt;&#xA;  &lt;li&gt;&quot;Clarify by conducting a literature or artifact review or developing a main argument.&quot;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Math research literature is incredibly specialized. Not only does one usually need an advanced degree to be able to read it, but one needs to have specialized in a similar area as the author. And if we are talking about literature pertaining to open problems, it would be very difficult for a student (probably even the teacher) to do this review.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;4&quot;&gt;&#xA;  &lt;li&gt;&lt;p&gt;Evaluate the information by identifying:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-the credibility of sources&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-feedback from peers and professionals in the field to improve the proposal/project&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In math, the credibility of an article is provided by whether or not it has been published in a peer reviewed journal. The math research community has a rigorous revision and refereeing process for articles before publication. So I would interpret this part as: restricting the student's sources to peer reviewed journals, as opposed to Wikipedia, blogs, or other information. But I think it would be more appropriate for students to be looking at those &quot;less credible&quot; sources because they are friendlier and potentially accessible.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;5&quot;&gt;&#xA;  &lt;li&gt;Organize the presented content and/or project timeline in a manageable, usable, and discipline appropriate format that can result&#xA;  in implementation of the research proposal or project.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Nobody, not even seasoned research mathematicians, can foresee the twists and turns that happen along the process of making a new mathematical discovery. For example, sometimes some trivial thing you overlooked at first turns out to be significant enough that it gets its own paper. It's not uncommon for a doctoral student to think he's a year away from graduating, then run into a theoretical problem that pushes that back a couple of years (or even indefinitely). The path is very unpredictable by nature.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;6&quot;&gt;&#xA;  &lt;li&gt;&lt;p&gt;Critically analyze the strengths and weakness of the information collected, including, but not limited to:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-supporting and opposing arguments&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-methodologies&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-the proposed process&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are no supporting or opposing arguments among proven math theorems. It's not like the social sciences where we are relying on evidence. We are relying on proofs of abstract logical statements. (There are some cases where mistakes end up published, but that's not what we're talking about here.) A known, checked math result has the consensus of the mathematical community, making this step basically irrelevant.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;7&quot;&gt;&#xA;  &lt;li&gt;Synthesize by constructing or summarizing all information/data into a research proposal/project that includes rigorous, researchable&#xA;  questions based on new understandings.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That is essentially what one does as a doctoral student after completing all their qualifying exams, and even then it is done under the guidance of a professor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, I would be happy that the school is expressing interest in something like this, but I would try to (in a friendly way) gear them toward something more realistic.&lt;/p&gt;&#xA;" OwnerUserId="9091" LastEditorUserId="9091" LastEditDate="2018-06-03T13:51:58.260" LastActivityDate="2018-06-03T13:51:58.260" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14128" PostTypeId="2" ParentId="14081" CreationDate="2018-06-02T15:57:47.820" Score="3" Body="&lt;p&gt;As a software engineer, the book &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Concrete_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;concrete mathematics&lt;/a&gt;&quot; by Graham, Knuth and Patashnik is a must. Chapter 4 is on number theory.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2018-06-02T15:57:47.820" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14129" PostTypeId="2" ParentId="779" CreationDate="2018-06-02T17:31:43.177" Score="2" Body="&lt;p&gt;One certain source can be found by looking through winners of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Morgan_Prize&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Morgan Prize&lt;/strong&gt;&lt;/a&gt;, which is awarded for mathematical research by undergraduate students. Besides the wikipage linked above, there is also an &lt;a href=&quot;http://www.ams.org/profession/prizes-awards/ams-prizes/morgan-prize&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;AMS&lt;/strong&gt;&lt;/a&gt; page and &lt;a href=&quot;https://www.maa.org/programs-and-communities/member-communities/maa-awards/research-awards/morgan-prize&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;MAA&lt;/strong&gt;&lt;/a&gt; page.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although you can find some very &quot;strong&quot; mathematics and mathematicians among award recipients, I feel it is worth noting which institutions show up repeatedly (listed alongside winners and runners-up on the wikipage). If the goal is &quot;motivating young students&quot; (per the OP) then it might be worth stepping back to wonder why the same names of certain universities keep showing up; I imagine it could be de-motivating for students at other schools to see this list and perhaps, mistakenly, believe that quality undergraduate research is outside of their own reach by virtue of the university that they attend.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-06-02T17:31:43.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14130" PostTypeId="2" ParentId="779" CreationDate="2018-06-02T17:59:54.780" Score="0" Body="&lt;p&gt;This story is very motivating.  It is not direct answer to your specification of UG research (was a grad student).  But it meets the spirit.  And serves the motivation rationale:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.snopes.com/fact-check/the-unsolvable-math-problem/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.snopes.com/fact-check/the-unsolvable-math-problem/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-02T17:59:54.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14131" PostTypeId="2" ParentId="14081" CreationDate="2018-06-02T20:53:47.450" Score="8" Body="&lt;p&gt;I've heard very good reviews of the 2017 book, &amp;quot;An Illustrated Theory of Numbers&amp;quot; by MH Weissman.&lt;/p&gt;&#xA;&lt;p&gt;The book's main site is &lt;a href=&quot;http://illustratedtheoryofnumbers.com/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;; a write-up, along with some reviews, by the American Mathematical Society can be found &lt;a href=&quot;https://bookstore.ams.org/mbk-105/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. To quote from the latter (emphasis added by me):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;An Illustrated Theory of Numbers&lt;/em&gt; gives &lt;strong&gt;a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises.&lt;/strong&gt; Its exposition reflects the most recent scholarship in mathematics and its history.&lt;/p&gt;&#xA;&lt;p&gt;Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship&lt;/strong&gt; to trace the development of number theory within and outside the Western tradition.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory.&lt;/strong&gt; It is also suitable for mathematicians seeking a fresh perspective on an ancient subject.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-06-02T20:53:47.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14132" PostTypeId="1" CreationDate="2018-06-03T10:15:29.857" Score="4" ViewCount="309" Body="&lt;p&gt;I would like to create some visual aids for illustrating principles in statistics, similar to the kind of graphic found here: &lt;a href=&quot;https://en.wikipedia.org/wiki/Linear_regression#/media/File:Anscombe%27s_quartet_3.svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Linear_regression#/media/File:Anscombe%27s_quartet_3.svg&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One choice is manually drawing such graphics in google slides or other slide/presentation software.  Are there better, easier, more efficient tools for doing this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is another example of the kind of illustration wanted:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/hxz1u.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hxz1u.jpg&quot; alt=&quot;Infographic that shows permutations and combinations of dice&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9964" LastEditorUserId="9964" LastEditDate="2018-06-04T11:22:47.840" LastActivityDate="2018-06-04T11:22:47.840" Title="What tools are available for creating visual aids?" Tags="|course-design|examples|graphing|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14133" PostTypeId="2" ParentId="14132" CreationDate="2018-06-03T12:29:55.583" Score="0" Body="&lt;p&gt;MS Excel is one way... eh... just for things like box plots and (non linear) regressions ... doesn't look nice - personal opinion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess the graphics you linked to are made using &lt;a href=&quot;https://www.tinkerplots.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;tinkerplots&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But any good stat's program will allow you to create graphics. Usually I teach with &lt;a href=&quot;https://www.r-project.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;R&lt;/a&gt; so that's the tool of my choice.&#xA;In combination with &lt;a href=&quot;https://yihui.name/knitr/&quot; rel=&quot;nofollow noreferrer&quot;&gt;knitr&lt;/a&gt; - i find it very useful. &lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2018-06-03T12:29:55.583" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14134" PostTypeId="2" ParentId="14132" CreationDate="2018-06-03T14:31:53.367" Score="0" Body="&lt;p&gt;Use MSFT  Excel.  It is the defacto standard (even for people with fancier programs).  Download the Analysis addin, if it is not active already (free, but often not enabled because of saving disk space).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The presentation may be a little ugly but you can pretty it up (adjust the default colors, eliminate gridlines, etc.)  There are other dedicated stats programs with a little bit prettier default output (but again, you can pretty up Excel charts to look better).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, for people very serious about chart appearance (e.g. book publishers, analyst reports), you have a graphics artist redo the chart in a drawing program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But really for classroom use, working research, etc., Excel is the default.    &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-03T14:31:53.367" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14135" PostTypeId="1" AcceptedAnswerId="14136" CreationDate="2018-06-03T23:30:09.710" Score="8" ViewCount="4333" Body="&lt;p&gt;I'm looking for a good way to explain how multiplication and division by fractions applies in the real-world the mechanics are receiving reasonably straight forward.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can $2$ divided by $1/2$ being $4$ be explained in terms of a real-world example? And the same for $2$ multiplied by $1/2$ being $1$?&lt;/p&gt;&#xA;" OwnerUserId="8764" LastEditorUserId="5220" LastEditDate="2018-06-13T13:47:23.333" LastActivityDate="2018-06-25T08:34:11.313" Title="How to explain multiplying and dividing by fractions with real-world examples" Tags="|intuition|fractions|" AnswerCount="6" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14136" PostTypeId="2" ParentId="14135" CreationDate="2018-06-04T00:11:44.730" Score="8" Body="&lt;p&gt;We have two cookies. We divide them into pieces of 1/2 cookie each and end up with four pieces. Thus 2 divided by 1/2 equals 4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have two cookies. We take 1/2 of the collection which is one cookie. Thus 2 multiplied by 1/2 equals 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each of those examples can be criticized. In the first example, one could claim that it shows that 2 divided by 4 equals 1/2, not what I claimed. But a/b=c is equivalent to a/c=b for non-zero b and c. Likewise, the second could be claimed to show 1/2 multiplied by 2 is 1. In the second example, remember that the word &quot;of&quot; usually means multiplication.  And a·b=c is equivalent to b·a=c. I can think of other criticisms also, but these examples do seem to work well.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2018-06-04T13:05:03.283" LastActivityDate="2018-06-04T13:05:03.283" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14137" PostTypeId="2" ParentId="14132" CreationDate="2018-06-04T00:15:36.910" Score="2" Body="&lt;p&gt;The style of your example graphics is very similar to &lt;a href=&quot;https://matplotlib.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;matplotlib&lt;/a&gt;, an extremely popular graphics package for the Python computer language. The package was designed to produce &quot;publication quality figures&quot; and version 2 made this even more so than version 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main disadvantage of matplotlib is that it requires programming in Python. There is no standalone, GUI package of it that I know of. On the other hand, that makes it extremely configurable.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastEditorUserId="1905" LastEditDate="2018-06-04T08:47:53.037" LastActivityDate="2018-06-04T08:47:53.037" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14138" PostTypeId="1" AcceptedAnswerId="14141" CreationDate="2018-06-04T18:13:08.270" Score="1" ViewCount="238" Body="&lt;p&gt;I have never seen this notation, but I think that it follows in a similar vein for function notation.  So if $y=f(x)$, then $y''=f''(x)$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then by that, can we say that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f^{(n)}(x)=y^{(n)}$$&lt;/p&gt;&#xA;" OwnerUserId="4728" LastActivityDate="2018-06-05T05:14:04.217" Title="Can $y^{(n)}$ be used as a way of representing higher order derivatives?" Tags="|calculus|notation|derivative|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14141" PostTypeId="2" ParentId="14138" CreationDate="2018-06-05T05:14:04.217" Score="2" Body="&lt;p&gt;It can be used, but it's a bad piece notation, just as $y'$ is. I have two reasons for saying that.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$y'$ doesn't indicate with respect to which variable you differentiate. So for example, if $y=t^2$ and $t=e^x$, what should $y'$ denote? Is it $dy/dt$ or $dy/dx$? I see students getting confused by this when they try to derive with the chain rule using the prime notation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even if you object and say: in my context $y'$ will always denote derivative with respect to $x$, I still consider it a bad notation since it's the same notation we use for $f'$. Hence it suggest to students that $y$ and $f$ are the same type of objects.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The only justification I see for using it is laziness and tradition.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2018-06-05T05:14:04.217" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14142" PostTypeId="1" CreationDate="2018-06-05T10:23:05.587" Score="11" ViewCount="1108" Body="&lt;p&gt;All of the middle and high school math educators I´ve encountered choose not to use LaTeX in preparing their documents or presentations.  I would assume that most of them had to use the program in the past for their Bachelors or Masters programs.  What are some reasons why it is not used before the undergraduate level?&lt;/p&gt;&#xA;" OwnerUserId="9968" LastEditorUserId="2083" LastEditDate="2022-11-25T12:29:42.173" LastActivityDate="2022-11-26T22:59:35.967" Title="Middle / High School Math Teachers and LaTeX" Tags="|secondary-education|software|presentation|latex|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14143" PostTypeId="2" ParentId="12200" CreationDate="2018-06-05T16:21:48.020" Score="5" Body="&lt;p&gt;Input-output machine is the most intuitive thing to first give, especially to a weaker student.  Ideally, you eventually want to have multiple frames of reference for a function, but to start with...go with input/output machine.  For God's sake, don't start with ordered pairs.  That is so aphysical and theoretical.  It is probably the most powerful and intuitive TO A MATH SOPHISTICATE.  But it is a bad intro for a new student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Schaum's Outline (Frank Ayres, 1958) has a good, simple explanation for new students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Intuitively, I still just think of functions as every x only give one y and relations as being able to give more than 1.  And a picture of a parabola versus a circle in my mind.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-05T16:21:48.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14144" PostTypeId="2" ParentId="12945" CreationDate="2018-06-06T01:41:21.260" Score="1" Body="&lt;p&gt;With a little effort you could include a generalization of the directional derivative towards the (std) covariant derivative:&#xA;For two vector fields $F,G:\Bbb R^3\to\Bbb R^3$ the construction&#xA;$$D_FG=[JG]F$$ &#xA;captures how the components of $G$ vary in the $F$ direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that you could ask for example: How the normal field of a surface varies in the tangential direction giving you the Weingarten's relations (or the Gauss' map, which is the derivative of the Shape operator). From the Weingarten's relations you get a $2\times2$ matrix whose determinant defines the Gaussian Curvature of the surface. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The algebraic, differential and geometrical elementary concepts combine on a clever fashion giving a wide range of cases to be amused at the exercising activities.&lt;/p&gt;&#xA;" OwnerUserId="5582" LastActivityDate="2018-06-06T01:41:21.260" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14145" PostTypeId="2" ParentId="14142" CreationDate="2018-06-07T02:45:07.437" Score="4" Body="&lt;p&gt;When I taught high school, I used &lt;a href=&quot;http://www.dessci.com/en/products/mathtype/&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathtype&lt;/a&gt; to type up all exams.  There was a trial version and a discount for teachers. I learned about mathtype at an NCTM conference.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At other times I used equation editor in Microsoft word and math tools for the smartboard (which meant I could type up things on my computer at home as long as I used the smartboard software.  I learned how to use the smartboard at various workshops I went to as well as an online course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I never used Latex for three reasons: &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;1. the tools I had were adequate&#xA;2.  I never heard of Latex until I joined this forum &#xA;    (NCTM doesn't mention it anywhere) and never had any training.&#xA;3. The tools I had were considerably more intuitive than Latex&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Finally, in response to one of the comments, I don't consider it sad that I never used Latex.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2018-06-07T02:45:07.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14146" PostTypeId="1" AcceptedAnswerId="14307" CreationDate="2018-06-07T03:32:11.473" Score="2" ViewCount="555" Body="&lt;p&gt;I hope never to study physics. Can I still ace undergraduate and PhD math?&#xA;Or ought I major in something else?&lt;/p&gt;&#xA;&lt;h3&gt;Background&lt;/h3&gt;&#xA;&lt;p&gt;I finish high school in 2019, and I'm mulling majoring in math/statistics. I'm mostly interested in probability and statistics (not number theory, notwithstanding my user name).&lt;/p&gt;&#xA;&lt;p&gt;Please presuppose my scorn for physics that can't be changed. I've never been interested in physics, because I'm interested in using probability and statistics to help the poor. I can't grok how physics can instantly help penniless African orphans, as it feels too ivory-tower. I know that many pure math courses are taught with physics applications (like PDEs).&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-07-11T07:27:20.287" Title="Can I succeed in university math, if I loathe physics?" Tags="|self-learning|" AnswerCount="5" CommentCount="22" ContentLicense="CC BY-SA 4.0" />
  <row Id="14147" PostTypeId="1" CreationDate="2018-06-07T04:51:10.853" Score="3" ViewCount="213" Body="&lt;p&gt;I have a math degree and have been hired to teach a proof class at a summer program. Our goal is to help the students learn the material they need for school (they take an algebra class separately) while also helping them improve their problem-solving skills more generally and find joy in doing math. They are mostly 16-17 years old and test into Algebra I or II in the program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've observed and helped with this course in previous years, but the curriculum is not well-defined and there aren't many past resources for me to use. So, my question is: what are your ideas for discussing proof, logical reasoning, etc. with a group of students who may struggle with math and don't necessarily like it very much? I will be focusing on the development of mathematical self-confidence and new ways of thinking, more so than specific content knowledge.&lt;/p&gt;&#xA;" OwnerUserId="9971" LastEditorUserId="9971" LastEditDate="2018-06-07T05:11:04.417" LastActivityDate="2018-06-07T12:08:53.023" Title="Ideas for high-school proof class?" Tags="|mathematical-pedagogy|proofs|teaching|struggling-students|summer|" AnswerCount="2" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14148" PostTypeId="2" ParentId="14147" CreationDate="2018-06-07T06:38:55.000" Score="5" Body="&lt;p&gt;Thanks for the description of the kids level (below average).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I actually think their time would be better spent on either doing remediation or advancement in their core topics or in just some useful use of the summer (archery, building stuff, whatever).  This is NOT to discourage you, though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My advice would be to keep things light and fun.  NOT rigor city.  Brain teasers, properties of numbers.  Polya (and maybe even easier than his stuff).  Simple number theory and even properties of numbers (divisible by 3, etc.).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of direct applicability to algebra, the one key proof topic is mathematical induction and the classic problems in any general algebra 2 text that are under that chapter.  If you want to do something useful, covering this, even if it is pre-covering, might help them.  I worry though that this is a topic that is slightly hard even for the average student within their algebra 2 class.  But perhaps you could really water it down or do it gently or the like.  And not make it a hill to die on but just more of giving some exposure.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Actually as I write this (sorry for stream of consciousness), I think more and more that proof ability is extremely unimportant for these kids (that probably lack computational skills).  Proof classes are something that the superstars of math contest kids take in HS.  Not your guys.  However, you do have an alternate mission which is to awaken some interest in math.  So I would let them call it a proof class but concentrate on fun activities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One fun area is symmetry and point groups.  Don't take a group theory approach to it, but instead just list the type of symmetry elements (rotations, mirror planes, etc.) and then have them catalog them on objects.  If possible, have physical objects to pass around.  (If you don't have enough, have the kids do teams...this is the sort of thing where teams actually makes sense, unlike computational problem solving.)  You can do 3d and 2d.  For 2D, you might do something with wallpaper patterns (space group) but don't do anything with 3d space groups (too hard).  And by do something, I don't mean mastering it, but coloring in the birds on some Escher print or listing the rotations and planes and such that exist in some prints.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also there was another thread on here (can't recall title) about experimental (i.e. physical) math projects for high school.  Stuff like dropping the needle and finding pi.  I would do some things where the kids can use their hands and such.  It is summer after all.  Plus really, we are physical creatures (all of us, even Andrew Wiles).  And these kids are obviously not abstract thinking superstars.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bottom line:  lots of &quot;sugar&quot; and not much &quot;medicine&quot;.  Keep the emphasis on fun rather than accomplishment and you will end up with more accomplishment (than if you torture them with proof strategies). &lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="9519" LastEditDate="2018-06-07T07:24:09.507" LastActivityDate="2018-06-07T07:24:09.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14150" PostTypeId="2" ParentId="14147" CreationDate="2018-06-07T12:08:53.023" Score="3" Body="&lt;p&gt;Dana Ernst has a fantastic collection of problems which he bases a course around at Northern Arizona University.  Here is a link to the course:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://danaernst.com/teaching/mat220s18/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://danaernst.com/teaching/mat220s18/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a collection of puzzles which do not require any background knowledge.  So the focus can be purely on working on problem solving skills, communicating mathematics effectively, employing a variety of different modes of reasoning, etc.  I am using this for ``half of'' a discrete mathematics course right now and it is very fun.  I am not sure if the level of the questions would be appropriate, but I think it is at least worth a shot.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-06-07T12:08:53.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14151" PostTypeId="2" ParentId="14146" CreationDate="2018-06-07T13:44:09.657" Score="2" Body="&lt;p&gt;Some math courses will have: simple applications in physics, simple applications in economics, simple applications in biology, simple applications in engineering, and so on.  Beginning calculus, or differential equations, for example.  So you perhaps will not &lt;strong&gt;ace&lt;/strong&gt; those courses, but still you could do OK.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now in most cases, when you work on a Ph.D. in mathematics, you will want to (or even be required to) work as a teaching assistant on low-level undergraduate courses.  And it may not be possible to avoid beginning calculus courses.  So in the end you will, indeed, have to know a small amount of physics.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back when I was a student taking basic calculus, I had a TA who explained that she could get by with just a few basic notions from physics, and that she never intended to learn any more physics than that.&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2018-06-08T12:13:33.583" LastActivityDate="2018-06-08T12:13:33.583" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14152" PostTypeId="2" ParentId="5948" CreationDate="2018-06-07T14:27:52.277" Score="7" Body="&lt;p&gt;Flatland. W. Abbott, 1884&#xA;Here's the Project Gutenberg link to read or download in your preferred e-version: &#xA;&lt;a href=&quot;https://www.gutenberg.org/ebooks/97&quot; rel=&quot;noreferrer&quot;&gt;https://www.gutenberg.org/ebooks/97&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an advanced student in elem school, my engineer single-parent mother gave me a copy while I was surviving a two year nothing-to-learn transition into public school from lifelong private. I believe I was 11 when I read it, and it entirely sorted dimensionality for me. In about 1962. &lt;/p&gt;&#xA;" OwnerUserId="9972" LastActivityDate="2018-06-07T14:27:52.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14154" PostTypeId="1" CreationDate="2018-06-07T20:23:41.607" Score="20" ViewCount="3775" Body="&lt;p&gt;I am a teaching assistant for a sophomore engineering laboratory. We use standard deviation a lot during the semester. It is an incredibly useful concept that can be used in a lot of engineering applications. We also teach students how to calculate variance, which is simply the square of standard deviation. I have never used variance for anything and I struggle to imagine how it could be applied. Is there any way that variance can actually be applied? Is there a reason to spend time on it at all?&lt;/p&gt;&#xA;" OwnerUserId="9976" LastActivityDate="2020-08-27T09:21:30.710" Title="What is the point of teaching variance?" Tags="|undergraduate-education|statistics|" AnswerCount="8" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14155" PostTypeId="2" ParentId="14154" CreationDate="2018-06-07T21:10:05.737" Score="19" Body="&lt;p&gt;An analogy:  speed is to standard deviation, as kinetic energy is to variance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Energies can be added usefully; speeds can only be added in very limited circumstances.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, variances can be added across data points, or across alleged sources of the total variance; standard deviations cannot be directly added.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When analysts say things like, &quot;This factor accounts for 20% of the total effect&quot;, they are usually performing a calculation involving either a ratio of variances, or the differences between means.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2018-06-09T21:17:41.537" LastActivityDate="2018-06-09T21:17:41.537" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14156" PostTypeId="2" ParentId="14142" CreationDate="2018-06-07T22:11:51.963" Score="2" Body="&lt;p&gt;&lt;em&gt;(what follows was assembled from several comments, now deleted)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;At least in the U.S., once you get away from academic research environments that are strongly connected to math, and you become immersed in environments such as academic education departments, academic business departments, education administration, pretty much any business or financial setting, even the preparation of math questions for standardized math tests, you'll find essentially zero official use of LaTeX. Even the journal &lt;em&gt;The Mathematics Teacher&lt;/em&gt;, which by a wide margin is the best known and most influential journal in the U.S. for middle and high school math teachers, strongly desires manuscripts written in MS Word with MathType. Thus, to answer your question, the middle and high school math teachers you’ve encountered probably have had very little contact with anyone using LaTeX.&lt;/p&gt;&#xA;&lt;p&gt;That said, there is at least one alternative that does not involve as steep a learning curve as directly coding LaTeX, which incidentally I did when writing my dissertation (1993, 200+ pages with many complex math expressions), something I did by making heavy use of Leslie Lamport's 1986 spiral bound manual &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B000GQDG0O&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;LaTeX. A Document Preparation System&lt;/strong&gt;&lt;/a&gt;. Because I did not have access to a computer or any software the 18 or so months after Summer 1993, I still hand-wrote tests and handouts and such (as I'd always done, since the first class I taught, Summer 1983). However, a colleague had &lt;a href=&quot;https://www.mackichan.com/index.html?products/sw.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scientific Word&lt;/a&gt; (Version 2.0, I believe), which is a &lt;a href=&quot;https://en.wikipedia.org/wiki/WYSIWYG&quot; rel=&quot;nofollow noreferrer&quot;&gt;wysiwyg format&lt;/a&gt; for LaTeX, and I &amp;quot;borrowed it&amp;quot; (installed it on a &lt;a href=&quot;https://en.wikipedia.org/wiki/Intel_80386&quot; rel=&quot;nofollow noreferrer&quot;&gt;386 PC computer&lt;/a&gt; that I had just gotten use of from the math dept., around Jan. or Feb. of 1995 I think) to write some math papers and other documents (but not yet for any classes I taught). Beginning in Fall 1996 I was teaching somewhere else, where at that time faculty did not have computers (this changed a year later), so around October 1996 I purchased my first computer and Scientific Workplace (SWP) for it (Version 2.5; had a computer algebra system attached, my first ever use of one), and at that time I began writing what eventually became several thousand teaching documents (quizzes, tests, handouts, etc. --- I posted &lt;a href=&quot;https://www.google.com/search?q=%22Dave%20L.%20Renfro%22%20%22servlet/JiveServlet/download%22%20site:mathforum.org%20%22pdf%22&amp;amp;filter=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;a few of them&lt;/a&gt; before this option at Math Forum became unavailable) that I had written by 2005, when I left teaching.&lt;/p&gt;&#xA;&lt;p&gt;I upgraded my 1996 Scientific Workplace program to Version 5.5 in Summer 2013, and I probably spend at least two hours with it each day (often more) working on various documents and such. However, I DON'T use LaTeX the way others here probably do, because the symbols are all pop-up and easy to use --- much easier than MathType (and looks much nicer also), and trust me, this is not because I don't have much experience with MathType (for the past 12 years my full-time job involved using it extensively; or see &lt;a href=&quot;https://web.archive.org/web/20160927012226/http://pballew.net/Constructable_17gon.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;). The main downside is the cost, which is about 800 dollars (386 dollars for students). I paid around 400 dollars in 1996, although at the time I believe the actual price was 600 dollars, but in my dealings with the company I think they were hoping --- indeed, they basically told me this --- that my use of SWP would result in the school buying a site license, so they were willing to knock off a couple hundred dollars.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="6053" LastEditDate="2022-11-24T21:09:43.020" LastActivityDate="2022-11-24T21:09:43.020" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14157" PostTypeId="2" ParentId="12618" CreationDate="2018-06-07T23:56:38.713" Score="0" Body="&lt;p&gt;With the advent of the era of Wolfram Alpha.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A great deal is to calculate the curvature of a planar curve. The formula is &#xA;$$\frac{x'y''-x''y'}{\sqrt{(x'^2+y'^2)^3}}$$&#xA;and a crash problem is calculate that for the projective coordinates of a circle of radius $R$ which is &#xA;$$x=\frac{R(1-t^2)}{t^2+1}$$&#xA;$$y=\frac{2R}{t^2+1}.$$&#xA;Only very well concentrated minds can do the calculation without software aid, but with Wolfram Alpha every body can see how this calculation should be performed to be elegant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The WA command for an specific, say $R=12$, is: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;x=(12)(1-t^2)/(t^2+1); y=(24)t/(t^2+1); (D[x,t]*D[D[y,t],t]-D[y,t]*D[D[x,t],t])/(D[x,t]^2+D[y,t]^2)^(3/2)&lt;/p&gt;&#xA;" OwnerUserId="5582" LastActivityDate="2018-06-07T23:56:38.713" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14158" PostTypeId="2" ParentId="14154" CreationDate="2018-06-08T02:38:45.850" Score="52" Body="&lt;p&gt;Actually, your definitions are backwards: the standard deviation is the square root of the variance. In other words, one defines variance first --- it has a simpler formula, and it has simpler mathematical properties. Then one defines the standard deviation as the square root of the variance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for why to study it, there are many mathematical formulas which are simpler when expressed in terms of variance rather than in terms of standard deviation. One key example is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The variance of a sum of independent random variables is the sum of the variances. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can, of course, translate this into standard deviations, but its more complicated when you do that: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The standard deviation of a sum of independent random variables is the square root of the sum of the squares of the standard deviations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7258" LastEditorUserId="1147" LastEditDate="2018-06-08T02:46:44.497" LastActivityDate="2018-06-08T02:46:44.497" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14159" PostTypeId="1" CreationDate="2018-06-08T03:35:33.857" Score="8" ViewCount="457" Body="&lt;p&gt;I was helping my niece (7th grade) with homework and one of the topics was the absolute mean deviation.  It's basically the same thing as standard deviation except instead of squaring the difference between the data points and the mean, you take the absolute value of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the one hand I can almost understand this because if I told her she had to square all those numbers instead of just dropping negative signs, she wouldn't have liked it.  That being said, she was using a calculator so squaring the numbers and dealing with big numbers should have been trivial anyway.  The absolute mean deviation isn't something that is used by even statisticians (I don't think) but standard deviation is pretty common in general business (at the very least) so why teach 90% of a real world topic instead of just teaching the real thing?&lt;/p&gt;&#xA;" OwnerUserId="1652" LastEditorUserId="376" LastEditDate="2018-06-08T06:52:36.337" LastActivityDate="2018-06-09T07:51:24.847" Title="Why teach absolute mean deviation?" Tags="|secondary-education|statistics|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14160" PostTypeId="2" ParentId="14159" CreationDate="2018-06-08T05:46:40.863" Score="6" Body="&lt;p&gt;The foundations of statistical inference are very hard to teach at any level, and almost certainly, at the 7th grade level, little or no serious motivation is given for the rules presentd. Probably at the 7th grade level what is being taught are rote rules for calculation of defined, but generally poorly if at all motivated, quantities. The presence or absence of a square is hardly of much importance if that is the situation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In quantifying error, the power used refects some subjective judgment about the importance attached to large values or outliers. One can use the absolute value, the square, or any $p$th power (or the $\infty$-norm). Different choices are affected differently by large values - the mean error calculated using a higher power is more affected by large values than is the error calculated using a lower power. The widespread use of the square has sound justifications in scientific practice, but the ordinary mean also can make sense. When working with large data sets on a computer, computational considerations can dictate using the square. Fundamentally the preference for the square is because the squared mean is given by an inner product. For example, the regression line defined using mean squared error is more easily calculated on a computer than that defined using mean absolute error because the &quot;normal equations&quot; can be solved efficiently and robustly using the singular value decomposition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if one refers to computation by hand, then the calculation without squares requires fewer operations and so is simpler. At the $7$th grade level little or no effort is made to interpret, and to the extent that any is made, the difference between ordinary mean and mean squared is not of great importance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is nothing any more &quot;real world&quot; about any of this stuff than there is about the standard exercises given to calculus students. One teaches mathematics to all $7$th graders not because of its real world utility, but because the habits of thought learned by learning mathematics are themselves useful quite generally. One teaches elementary statistical ideas in order to develop the ability to formulate qualitative interpretations of masses of data. What one can hope to teach students is that there is some sort of second order criterion that can distinguish distributions with the same mean. One hopes that a student can distinguish (visually/intuitively) between data uniformly distributed over a range and data all repeated on the midpoint of that same range, and then see that the measure of deviation (with or without squares) differentiates between these two data points. This is in the same spirit as teaching the student that distributions with the same mean can have different medians, but it is different, because mean and median are both first order statistics, whereas deviation is somehow second order, measuring spread around something first order. There are different ways to measure spread, but it becomes difficult to justify, on principled grounds understandable at the middle school level, the choice of one measure of spread over another. If I were teaching this material, I would probably use the mean squared error as the default, but I have little experience teaching $7$th graders per se, and am prepared to imagine the leaving out the square facilitates the realization of computations with little conceptual loss of a nature relevant at that level.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-06-08T05:46:40.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14162" PostTypeId="2" ParentId="14154" CreationDate="2018-06-08T07:51:51.153" Score="6" Body="&lt;p&gt;In statistics, everything is focused about variance as the second central moment of a distribution.  For example, you may be using the &quot;sample standard deviation&quot; formula which differs from the standard deviation of a complete distribution because of a correction that makes the &lt;em&gt;square&lt;/em&gt; of the sample standard deviation an unbiased estimator of the &lt;em&gt;variance&lt;/em&gt; of the distribution.  Without (re-)squaring that formula, it is an unbiased estimator of nothing useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, that bias can be expressed as VX=EVX + VEX, namely the actual variance of a distribution is the expected variance of a sampling plus the variance of the sample means over the samplings.  For equally weighted samples this leads to the well-known ${n\over n-1}$ correction factor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The principal advantage of the standard deviation is that it is in the same unit/scale as the sampled variable.  Which is nice for engineering purposes, namely looking up and relating material constants rather than doing actual statistical calculations.&lt;/p&gt;&#xA;" OwnerDisplayName="user9979" LastActivityDate="2018-06-08T07:51:51.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14163" PostTypeId="2" ParentId="14154" CreationDate="2018-06-08T09:50:44.267" Score="3" Body="&lt;p&gt;You are talking of &quot;use&quot; and &quot;application&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But when it comes to &quot;finding&quot; the standard deviation then the route in probability theory is always: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1) find the variance.&lt;/li&gt;&#xA;&lt;li&gt;2) find the standard deviation by taking the square root. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In short: in practice there is no way to find the standard deviation without first finding the variance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exceptions on this might exist, but are very very exceptional.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This justifies the focus on variance in probability theory.&lt;/p&gt;&#xA;" OwnerUserId="5662" LastActivityDate="2018-06-08T09:50:44.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14164" PostTypeId="2" ParentId="14159" CreationDate="2018-06-08T13:49:38.230" Score="1" Body="&lt;p&gt;Consider, when constructing Bayes estimators:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Using the squared error loss gives the basis for the Bayes estimator of the mean. &lt;/li&gt;&#xA;&lt;li&gt;Using the absolute error loss gives the basis for the Bayes estimator of the median.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;From: Casella/Berger, &lt;em&gt;Statistical Inference&lt;/em&gt;, Chapter 10, Theorem 10.3.2.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-06-08T13:49:38.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14165" PostTypeId="2" ParentId="14159" CreationDate="2018-06-08T14:48:19.500" Score="5" Body="&lt;p&gt;You're in the U.S. (according to your profile), and in the U.S. square roots are not generally introduced until the 8th grade (for example, &lt;a href=&quot;http://www.corestandards.org/Math/Content/8/EE/A/2/&quot; rel=&quot;noreferrer&quot;&gt;8.EE.A.2&lt;/a&gt; in the Common Core). I believe the topic &quot;absolute mean deviation&quot; (also called &quot;mean absolute deviation&quot;) is a relatively recent addition to middle grade level mathematics, probably within the last 10 years, that allows teachers to give a more nuanced measure of data variability than &quot;range&quot; and to give a cognitive stepping stone to &quot;standard deviation&quot; (taught in high school), before students have been introduced to square roots. In the Common Core, absolute mean deviation is taught in the 6th and 7th grades (&lt;a href=&quot;http://www.corestandards.org/Math/Content/6/SP/B/5/c/&quot; rel=&quot;noreferrer&quot;&gt;6.SP.B.5.c&lt;/a&gt; and &lt;a href=&quot;http://www.corestandards.org/Math/Content/7/SP/#CCSS.Math.Content.7.SP.B.3&quot; rel=&quot;noreferrer&quot;&gt;7.SP.B.3&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2018-06-08T14:48:19.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14167" PostTypeId="1" CreationDate="2018-06-08T16:34:36.120" Score="6" ViewCount="86" Body="&lt;p&gt;I'm designing some lessons on Game Theory, various versions to adapt to grades 1 through 9. In one of them I use the game of Hex to talk about games where we can prove some stuff about it even though its game tree is ridiculously huge. (For instance, that somebody always wins, that there is a winning strategy for player one even though we don't know what it is.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering what board size would be appropriate so that a game can be played in about 15 minutes. The usual 11x11 is definitely too big.&lt;/p&gt;&#xA;" OwnerUserId="9091" LastEditorUserId="9091" LastEditDate="2018-06-08T16:46:44.720" LastActivityDate="2018-06-09T01:31:24.697" Title="What is a good hex board size for kids to learn on, playing short games?" Tags="|games|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14168" PostTypeId="2" ParentId="14154" CreationDate="2018-06-08T17:25:41.923" Score="3" Body="&lt;p&gt;I'll work through a toy problem with both SD and Variance, and examine which is simpler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you roll 100 fair six sided dice.  Is it surprising if they sum up to 360?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, E(d6) = 3.5, E(100 d6) is 350.  Var(d6)=(7*5)/(6^2)=35/36, Var(100 d6)=3500/36.  SD(100 d6)=~sqrt(Var(100 d6))=9.72&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(E(100 d6)-350)/SD(100 d6) =~ 1.03, or 1.03 standard deviations.  Events 1.01 standard deviations away from the mean are not surprising.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without variance, we are stuck with SD(d6)=sqrt(7*5/6^2)=~0.972.  SD(100 d6)=sqrt( SD(d6)^2 )=sqrt( 100 * 0.972 ) = 9.72.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rest follows the same.  But instead of the nice, linear Var(100 d6) is 100 Var(d6), we had to do sqrt( 100 SD(d6)^2 ).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That step -- taking a value, squaring it, scaling it, then taking its square root -- is a bunch of steps that can introduce error into a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Things get even worse when we introduce covariance.  First, Variance is the Covariance of a random variable with itself, a nice property.  Second, once Covariance is measured or calculated, you get Var(A+B)=Var(A)+Var(B)+2Cov(A,B) (this generalizes).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then look at: SD(A+B) = sqrt( SD(A)^2 + SD(B)^2 + Cov(A,B) )&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same kinds of statement with standard deviations is painful.  You're squaring and taking the square root of everything all over the place, and usually the way you'd calculate the things you square would have been taking the square root in the previous step!&lt;/p&gt;&#xA;" OwnerUserId="9985" LastEditorUserId="9985" LastEditDate="2018-06-12T09:43:44.930" LastActivityDate="2018-06-12T09:43:44.930" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14169" PostTypeId="2" ParentId="14159" CreationDate="2018-06-08T18:45:35.083" Score="4" Body="&lt;p&gt;When I introduce measures of dispersion, the usual question from students is why do we use standard deviation and variance instead of absolute deviation, which is a lot simpler to interpret and compute. That is, they ask the opposite as the OP.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since we know more statistics (and calculus), we know that absolute deviation is not very useful. However, when introducing measures of dispersion, it's useful because it's one of the most intuitive measures and understanding it helps introducing and motivating variance and standard deviation.&lt;/p&gt;&#xA;" OwnerUserId="7178" LastEditorUserId="7178" LastEditDate="2018-06-09T07:51:24.847" LastActivityDate="2018-06-09T07:51:24.847" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14170" PostTypeId="2" ParentId="14154" CreationDate="2018-06-08T22:42:02.597" Score="8" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The variance is calculated directly, while the SD is calculated in terms of the variance.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The variance is additive for independent variables.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The effect of sample size is a lot easier to explain using both variance and SD: taking n samples multiplies the variance by n, and then when we divide by n to get the mean, we divide the &lt;em&gt;SD&lt;/em&gt; by n. It is this difference that results in larger sample sizes having less variation in the sample mean.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's a lot easier to talk about the accuracy of linear regression in terms of squared error: you have the total squared error (variance of the data points), the squared error of the residual (variance of residuals), and the explained variance (total SSE minus residual SSE). You can then take the ratio between explained variance and total SSE to get $r^2$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2018-06-08T22:42:02.597" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14171" PostTypeId="2" ParentId="14159" CreationDate="2018-06-08T23:24:55.970" Score="10" Body="&lt;p&gt;The question we pose to students is:  &lt;em&gt;How far away, on average, are these values from their mean?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;natural&quot; way to answer that question is to compute the deviations of the individual data points and compute the average.  When students do this they are usually surprised to find out that the answer comes out to be precisely $0$ — that the positive deviations and the negative deviations always cancel exactly.  Pointing out that this is actually a signal that we did everything right -- that the mean value is &lt;em&gt;supposed&lt;/em&gt; to be in the middle of the data, with just as much deviation above it as below it -- often triggers an &quot;aha&quot; moment for many students, for whom the average was just a formula disconnected from any intuitive meaning (see &lt;a href=&quot;https://matheducators.stackexchange.com/q/7663/29&quot;&gt;How to Teach Averages (Arithmetic Mean) to a Teenager?&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/q/922028/124095&quot;&gt;https://math.stackexchange.com/q/922028/124095&lt;/a&gt; for examples of this).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, once we have established that &quot;average deviation&quot; is not a useful thing to compute, the obvious next question is:  how could we get rid of those negative signs?  The idea of just dropping them all is both simple and intuitive, and &quot;average of the absolute values of the deviations&quot; is a literal interpretation of the idea of &quot;average distance from the mean&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, as others have pointed out, it's the &lt;em&gt;standard deviation&lt;/em&gt; that seems unnatural and unnecessarily complicated.  We square the deviations, which makes them all positive, and then we average them, but that comes out too big because we squared them, so then we take the square root...  Why do all of that?  I usually tell my students that the function $y=x^2$ has nicer mathematical properties than the function $y=|x|$, and in particular that the sharp corner in the absolute value function makes it hard to work with.  I'm not sure how intellectually honest that response is, but it at least makes a plausible-seeming connection with Calculus (for the students who have seen it) or lays some groundwork for it (for the students who have not yet but may someday).&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-06-08T23:24:55.970" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14172" PostTypeId="2" ParentId="14167" CreationDate="2018-06-09T01:31:24.697" Score="4" Body="&lt;p&gt;I've found 7x7 to be a pretty good size for my non-math major college students to play a few games in a reasonable time frame. My own (kindergarten and 2nd grade) children find it a good size as well. Here's my rendering of the board: &#xA;&lt;a href=&quot;https://www.instanton.org/hex-board.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.instanton.org/hex-board.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-06-09T01:31:24.697" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14173" PostTypeId="1" AcceptedAnswerId="14175" CreationDate="2018-06-10T15:28:33.440" Score="6" ViewCount="423" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose that $f(x,y,z) = x + 2xy^2 - yz$, and that $\gamma(u,v) = \langle uv, u\sin(v), u\cos(v)\rangle$. &lt;strong&gt;Use the chain rule&lt;/strong&gt; to calculate $\partial(f \circ \gamma)/\partial u$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is an exercise that typically shows up in a multi-variable calculus course. Sometimes the bit explicitly asking students to use the chain rule isn't there. The chain rule here is the fact that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\operatorname{D}(f \circ \gamma) = \operatorname{D}\!f(\gamma) \operatorname{D}\!\gamma\,,&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;so properly responding to this exercise entails calculating the vectors $\operatorname{D}\!f(\gamma)$ and $\operatorname{D}\!\gamma\,,$ multiplying them, and picking out the coordinate corresponding to $\partial u$ (or some semblance of that). But many students tell me that this is dumb: why deal with all those derivative matrices when you can just write the the composite function and take the partial derivative of that? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\frac{\partial(f \circ \gamma)}{\partial u}&#xA;= \frac{\partial}{\partial u} \big( uv + 2u^3\sin^2(v) - u^2\cos(v)\sin(v) \big)&#xA;= \dotsb &#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can you respond to these students? How can you explain the utility of using the multi-variable chain rule over just writing out the composite function? And is there a better way to ask this sort of exercise that makes any advantages of using the chain rule clear? &lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-06-10T19:23:02.600" LastActivityDate="2018-06-13T11:16:33.043" Title="Justifying the multi-variable chain rule to students" Tags="|undergraduate-education|calculus|concept-motivation|vector-calculus|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14174" PostTypeId="2" ParentId="14173" CreationDate="2018-06-10T16:31:12.827" Score="3" Body="&lt;p&gt;I think that the thing is to introduce meaning to the intermediate variables. Then each of $f,\gamma$ in $f\circ \gamma$ have meaning themselves. A physics example might have $f$ being the energy of a system and $\gamma$ be the time-evolution of the system. Now the intermediate variables mean things like position and momentum and we care about $Df$ and $D\gamma$ for their own sake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, you can now have these functions &quot;hot-swappable&quot;. That is, what if you vary $\gamma$ or the potential energy part of $f$? What happens to the composite?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Otherwise, the students are basically correct; it &lt;em&gt;is&lt;/em&gt; easier to not add in a bunch of extra machinery.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-06-10T16:31:12.827" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14175" PostTypeId="2" ParentId="14173" CreationDate="2018-06-10T19:50:30.793" Score="6" Body="&lt;p&gt;You can tell them the values of some of the partial derivatives without giving them the entire function. This prevents the students from creating the composite function, since they don't even know one of the pieces of it. For example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You know from your chemistry class that for a fixed amount of gas, the pressure, volume, and temperature are related by $PV = nRT$ where $n$ and $R$ are constants. Suppose that right now, $P = 10$ kPa, $V = 8$ L, and $T = 300^\circ$. If the temperature is increasing at a rate of $10^\circ$ per second and the pressure is decreasing at $2$ kPa per second, how fast is the volume changing?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You could also mix it up a little bit, and tell them $\frac{dP}{dt} = -2$ kPa per second but then give them a function $T = 10 + 3t - 4t^2$ and tell them that $t = 2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if you want to go further, you can make $T$ and $P$ depend on more than just time; toss in another parameter, again sometimes giving them the values of the partial derivatives and sometimes giving them functions.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-06-10T19:50:30.793" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14176" PostTypeId="1" AcceptedAnswerId="14177" CreationDate="2018-06-10T20:52:19.917" Score="19" ViewCount="11513" Body="&lt;p&gt;Sometimes, when preparing some calculus exams I try to have a &quot;funny question&quot; such as:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;T/F I love mathematics   &lt;/li&gt;&#xA;&lt;li&gt;T/F Calculus 2 is easier than Calculus 1, ...&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So my questions are:  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Do you think it is a good idea to have such questions in an exam?  &lt;/li&gt;&#xA;&lt;li&gt;Can you suggest more questions like that?  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9255" LastEditorUserId="2083" LastEditDate="2018-06-14T13:18:22.560" LastActivityDate="2018-06-16T00:39:44.043" Title="Adding irrelevant humorous questions to a quiz exam" Tags="|exams|" AnswerCount="11" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="14177" PostTypeId="2" ParentId="14176" CreationDate="2018-06-10T21:05:48.607" Score="36" Body="&lt;p&gt;I think there's room for differences of opinion on this, and the answer might depend on the ages of the students or who the specific students are.  When I was a student I enjoyed questions like this, but I'll explain why I don't include &quot;funny&quot; questions now that I'm the one making exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It comes down to cost and benefit.  I think the actual benefits are between small and non-existent.  I could imagine some theoretical benefits---maybe lighthearted questions help stressed students relax and do better on the exam; maybe they make the course more fun for some students, which might motivate them to work more or to continue to take math courses.  But I don't think either of these things really happen in practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, I think there are real costs.  Whenever I think about doing something funny on the exam, I start to worry about students who are really confused, or taking the exam in a second language, who might have trouble realizing that the question isn't intended seriously, and lose time trying to figure out what they're supposed to do with it.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-06-10T21:05:48.607" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14178" PostTypeId="2" ParentId="14176" CreationDate="2018-06-10T22:06:57.320" Score="5" Body="&lt;p&gt;This is a really interesting question, and I've been on both sides of this. Here's where I'm at with it now.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;On Funny Questions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think they're lighthearted great ways to relieve stress.  I also find it fun to add an interesting (bonus) question at the end.  For example, there was an exam on March 15, so a question on the Ides of March was asked.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The Other Point&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I would strive to make sure that the bonus / funny questions have &lt;strong&gt;NO IMPACT&lt;/strong&gt; on performance.  The test / the problems should be weighted that the grade comes from the math, not the extraneous.  It should not be possible to go from an A to B or vice versa from these types of questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sure, a 94 to a 95 is fine, but you should not go from a 69 to a 70 or whatever the cut off is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The Bottom Line&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They're great and fun, imho, so long as they don't severely impact the grade.&lt;/p&gt;&#xA;" OwnerUserId="9968" LastActivityDate="2018-06-10T22:06:57.320" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14179" PostTypeId="2" ParentId="14176" CreationDate="2018-06-10T23:01:30.773" Score="9" Body="&lt;p&gt;I think it's fine in moderation, but keep in mind that what you think is funny might not be funny to one or two of your students, so you'll want to tread very carefully when doing this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Occasionally I've written tests in which, after several attempts at making reasonable point-value assignments to the problems, I couldn't quite get the total to be $100$ points --- I'd get it to add to something like $98$ or $99$ points, but not $100.$ For example, there might be one $25$ point problem and three $15$ point problems and two $10$ point problems and three $3$ point problems, and I didn't want to increase any of these by $1$ point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way I handled this was to include an extra problem at the end, something like $1+1= {}_{---}$ that was worth $1$ or $2$ points to get the total to be $100.$ Yes, I know I could just compute a percent by calculating their point total divided by $98$ (or by $99),$ but when you have $50$ or more tests, it's well worth a little extra up-front time getting the total to be $100$ so that you don't have to grab your calculator and write something like $72/98 = 73$ (circling the $73)$ for each of the $50$ or more tests. Of course, whenever I did this, I told them as I was handing out the tests to not overlook the last problem if they're short on time at the end, and the instructions for the last problem would say something like &quot;Freebie problem to get points to add to $100$&quot;.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2018-06-10T23:01:30.773" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14180" PostTypeId="2" ParentId="14176" CreationDate="2018-06-11T01:45:41.387" Score="4" Body="&lt;p&gt;No.  Let them concentrate on the exam.  Humor should be relegated to the victory party afterwards.  Or the consoling sorrows party.  Funny questions is too cute and distracting while in exam.  It's not even that it will disturb their exam.  But that the humor will fall dead.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-11T01:45:41.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14181" PostTypeId="2" ParentId="14176" CreationDate="2018-06-11T04:23:16.143" Score="28" Body="&lt;p&gt;No, I think this is a really silly idea. Let's assume that students are trying to get a 100% score. What is the correct answer to &quot;I love mathematics - true/false&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is the &quot;I&quot; there the teacher? The student? If it is the student what is demonstrably the correct answer? How can the teacher marking the exam possibly know if the student loves mathematics or not? (The student might hate it, but be doing it because it is a course prerequisite).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since there is no answer that can be shown to be correct, then the student has to guess &quot;what am I supposed to say here?&quot;. Whatever answer they guess, if they are wrong, they won't get a top mark for the paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you might say &quot;I'll accept any answer and mark it correct&quot;. But the student will have &lt;strong&gt;wasted time&lt;/strong&gt; on the question, trying to work out what it means, and therefore having less time to spend on genuine questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some students might try to read the question as a riddle or puzzle. For example, &quot;I&quot; = 9, &quot;L&quot; = 12, &quot;O&quot; = 15 and so on, and then try to work out what that means mathematically.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As an example of why this can fail, see &lt;a href=&quot;https://www.smh.com.au/education/vce-authorities-apologise-for-phantom-robot-20121119-29lbl.html&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt; about a VCE (year 12) exam held in Australia in 2012. A history exam paper inadvertently had a &lt;strong&gt;giant robot&lt;/strong&gt; on the artwork &lt;em&gt;Storming the Winter Palace&lt;/em&gt; by Nikolai Kochergin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/p6Yxe.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/p6Yxe.jpg&quot; alt=&quot;Giant Robot&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students complained that they wasted time trying to work out the significance of the robot, for example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I went through a thought process something like this, 'What's that thing? It's definitely a robot. But it's on my history exam, so it's not a robot. But it can't be anything else. LOOK AT THOSE GUNS!' I stared at the image long and hard. It was extremely off-putting because you just don't expect there to be a mistake on your exam, especially one like that.'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Don't do it. It might be a fun game to you, but for the students this is an off-putting, time-wasting exercise that could well cause them to do more badly on the exam than they might otherwise have, and then subsequently complain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of the Australian exam the assessment authority had to apologise afterwards for any impact that the image might have had on students completing the exam. And in their case it wasn't even intentional!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, the authority had to &lt;a href=&quot;https://www.theage.com.au/national/victoria/vce-scores-changed-over-battle-tech-marauder-confusion-20130208-2e2qn.html&quot; rel=&quot;noreferrer&quot;&gt;adjust the scores&lt;/a&gt; for 130 students over the confusion about the robot.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Dave L Renfro  said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I sometimes had students answer my 1+1=−−− question (see my answer) with things like 10(in base 2),&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes, I didn't want to bring that up, but see &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Quotations/Russell.html&quot; rel=&quot;noreferrer&quot;&gt;Quotations by Bertrand Russell&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;But,&quot; you might say, &quot;none of this shakes my belief that 2 and 2 are 4.&quot; You are quite right, except in marginal cases -- and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a meter. Two must be two of something, and the proposition &quot;2 and 2 are 4&quot; is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. &quot;Well, at any rate there are four animals,&quot; you may say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. &quot;Well, then living organisms,&quot; you say. But there are things of which it is doubtful whether they are living organisms or not. You will be driven into saying: &quot;Two entities and two entities are four entities.&quot; When you have told me what you mean by &quot;entity,&quot; we will resume the argument.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, even a &quot;simple&quot; question about what is &quot;1+1&quot; could have arguably difficult answers.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Another issue is, by making joke questions you are treating the exam as a joke. That is, it is trivializing the experience. Some students may have studied long and hard, and have expectations from their parents that they do well. To have jokes introduced into the exam appears to say that the examination is being trivialized.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This question is out of the &lt;em&gt;domain&lt;/em&gt; of the subject. Asking a question about a state of mind (&quot;do you like something?&quot;) is more appropriate in, say, psychology. It isn't a calculus question and is therefore inappropriate. For example, if you asked &quot;How do you boil a hard-boiled egg?&quot; people may well object that this is not a relevant question in a calculus exam, even if they knew how to boil eggs.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Adding in questions that (assuming the reader can tell are a joke) alters the effective score in the exam. Let's say your joke questions are worth 5%. That means that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Someone can get 5% in Calculus even knowing nothing about the subject; and&lt;/li&gt;&#xA;&lt;li&gt;The other answers are scaled in value. For example a question which claims to be worth 10% of the score is really 10% of 95 and not 10% of 100, altering what the question is really worth.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Having joke questions, which are intended for everyone to get right unconditionally, could penalize a student who, because of time constraints, does not reach that question. They may say afterwards &quot;if only I had known there was a 'free' 5% from answering questions 19 and 20 I would have answered them first&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might respond &quot;but I'll give everyone full marks for my joke questions &lt;em&gt;even if they don't answer them&lt;/em&gt;&quot; now makes the questions seem even sillier. An exam question that is always marked correct? Even if you don't answer it? What is the point of having it?&lt;/p&gt;&#xA;" OwnerUserId="10002" LastEditorUserId="10002" LastEditDate="2018-06-12T23:33:07.027" LastActivityDate="2018-06-12T23:33:07.027" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14182" PostTypeId="2" ParentId="14176" CreationDate="2018-06-11T06:41:05.203" Score="2" Body="&lt;p&gt;Personally, I don't think that funny questions are such a good idea for the reasons already mentioned (such as: confusing students, skewing marks, etc.). Now, if your goal is to simply integrate some elements of humor into your math exam, one possibility could be to put funny pictures/&quot;memes&quot; at the end of the paper for students to enjoy once they have finished. One of my chemistry teachers always used to put picture jokes at the end of the exam relating to chemistry (obviously not the content), and there were never any problems. Another one of my teachers liked to put quotes at the end of the exam instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both were examples which didn't adversely affect the student's performance by giving them another question to think about, but instead provided something that might serve to &quot;lighten up&quot; the student's mood.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, so to eliminate any confusion, they would always put these things after an &quot;end of exam&quot; text line.&lt;/p&gt;&#xA;" OwnerUserId="10005" LastActivityDate="2018-06-11T06:41:05.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14183" PostTypeId="2" ParentId="14176" CreationDate="2018-06-11T08:21:26.227" Score="25" Body="&lt;p&gt;Not a good idea in my opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am on the Autism spectrum and as such I am a very very literal person and I am often derailed by trying to understand a poorly-defined question. Neurotypical people can look at a poorly defined question and naturally understand that some of the interpretations are absurd and most will naturally understand the meaning. I have to analyse each interpretation in turn before I can discount some.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's look at your two questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;T/F I love mathematics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know who 'I' refers to in this context. Does it mean me or does it mean the question setter? Trying to work this out could cause me some stress and effect the rest of the paper. I think that this refers to me, the question answerer but because I cannot be sure I will keep coming back to this question just to be sure. Most people will automatically understand to whom 'I' refers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;T/F Calculus 2 is easier than Calculus 1, ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is highly subjective, the definition of easy is unclear and even to me the term 'easy' is not clear. easy lectures, easy tutorials, simulation work easy. How to I aggregate the different parts and scale the different levels of 'easy into a single result. If I found lectures in one easier to follow and the tutorial work in the other easier I could be stuck trying to decide which was easier overall.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While it may be fun to put in jokes please be aware that it can cause a lot of stress and worry to some, often the quiet people who would not indicate that they are having problems.&lt;/p&gt;&#xA;" OwnerUserId="10006" LastActivityDate="2018-06-11T08:21:26.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14184" PostTypeId="2" ParentId="14176" CreationDate="2018-06-12T04:43:19.730" Score="3" Body="&lt;p&gt;I've read the answers and realize it is inappropriate to put a question such as this on an exam, because some students may not &quot;get it&quot; and may stress about it instead of focusing on the exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, as an elementary school teacher, I sometimes made individualized makeup tests or additional work as needed.  Since these were individualized and I knew my students well, I think there were times when it might have been  appropriate.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bottom line - this idea is not for a group, but if you make something up for an individual student who will appreciate it, definitely put it in.  If in doubt though leave it out!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do put in a question like this, I suggest that you put above it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;you can skip this question without penalty since it is just for fun&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can also put the above on a general test if you explain this direction the day before and remind the students of it during the test.  That way kids who are disinterested and/or confused by it will skip it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some examples of cute questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Add the number of ears that Mickey Mouse has to the number of dwarfs in the Snow White movie.  Divide by the number of letters in the 5th month.  &lt;/li&gt;&#xA;&lt;li&gt;True or False - This question is funny&lt;/li&gt;&#xA;&lt;li&gt;Why did the chicken cross the diameter of the circle?&lt;/li&gt;&#xA;&lt;li&gt;How many mathematicians does it take to screw in a lightbulb?&lt;/li&gt;&#xA;&lt;li&gt;True or False - Math is the best subject ever.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2018-06-12T04:43:19.730" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14187" PostTypeId="1" CreationDate="2018-06-12T12:07:05.687" Score="6" ViewCount="221" Body="&lt;p&gt;I am looking for an extensive source (often called &quot;table of integrals&quot;) listing primitives of various classes of functions including the &quot;elementary&quot; ones (rational functions, functions involving radicals, ...), &lt;strong&gt;indicating also a/the convenient method to obtain the result&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, one entry could be similar to &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int x^2e^x\mathrm{d}\!x=e^x(x^2-2x+2)+\mathrm{Constant}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;with the indication: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\textrm{By parts}\; (x^2,e^x); \textrm{by parts}\;(x,e^x)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or &quot;integrate by parts two times&quot; or just &quot;by parts&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there is a substitution to perform, it should be explicitely indicated, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A free and user friendly computer program would also do. But for example Wolfram Alpha doesn't seem to &quot;do the steps&quot; or to give indications on the procedure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit.&lt;/strong&gt; Please note that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) the example integral that I wrote above is, indeed, merely an example, not an indication of the intended complexity (or not) of the integrals! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) I am &lt;em&gt;not&lt;/em&gt; looking for textbooks that explain integration &lt;em&gt;techniques&lt;/em&gt; (though I realize the distinction may not be sharp). I am merely looking for a list of primitives, endowed with indications about how each result has been obtained. Hypothetically, to explain how to obtain the primitive in my (easy) example above just literally one or two words may be enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) I do not compute integrals all day long. I may very well forget which is the right substitution to perform in a certain instance. I am just looking for an easy way to get: &quot;this is the result ok, and -aha- this is the method. So if I wanted to do the integral by myself I would know, without trying a bunch of useless substitutions/methods&quot;. &lt;/p&gt;&#xA;" OwnerUserId="10018" LastEditorUserId="10018" LastEditDate="2018-06-12T16:54:34.460" LastActivityDate="2018-06-12T16:54:34.460" Title="Tables of primitives with indication of solution method" Tags="|calculus|reference-request|integration|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14188" PostTypeId="2" ParentId="14154" CreationDate="2018-06-12T12:10:28.627" Score="2" Body="&lt;p&gt;Consider: &lt;a href=&quot;https://en.wikipedia.org/wiki/Analysis_of_variance&quot; rel=&quot;nofollow noreferrer&quot;&gt;Analysis of Variance&lt;/a&gt; (ANOVA), a basic technique included as a chapter in most elementary statistics textbooks. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-06-12T12:10:28.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14189" PostTypeId="2" ParentId="14187" CreationDate="2018-06-12T14:35:49.680" Score="5" Body="&lt;p&gt;I don't think there exists a published reference book that goes into as much detail as you want with the kind of very elementary integral you gave as an example (a simple exponential times a polynomial, which is the protype for the &lt;a href=&quot;https://www.google.com/search?q=integration+tabular+method&quot; rel=&quot;noreferrer&quot;&gt;tabular method&lt;/a&gt;), since the example you gave would be well within the assumed background for a reader and would be too time consuming and too cluttering to be worth repeatedly mentioning (in a publication). Indeed, for the google search in my last sentence, I got nearly 9 million hits for the integration of parts tabular method. However, what you’re asking for DOES seem reasonable for software, even at the present time, and if something like this doesn't yet exist, I bet that it will within 10 years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, a once well-known book, Peirce &lt;strong&gt;[7]&lt;/strong&gt; (FYI, I have a hardback copy of the 1957 edition, revised by Ronald Martin Foster) gives brief descriptions of many integration methods. The descriptions appear at the beginning of many of the various chapters and sections. In fact, the reason I got a copy of this book (see &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/B00005WUAI&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; if you’re interested in getting a copy) was because I kept seeing it cited over and over again in older literature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are literally hundreds of freely available calculus books from the 1800s and early 1900s that provide extensive details on various integration methods, especially methods that have been edited out of calculus texts in the last 30 to 40 years. A few of the more comprehensive treatments I know about are &lt;strong&gt;[1]&lt;/strong&gt;, &lt;strong&gt;[2]&lt;/strong&gt;, &lt;strong&gt;[5]&lt;/strong&gt;, &lt;strong&gt;[8]&lt;/strong&gt;, &lt;strong&gt;[9]&lt;/strong&gt;, &lt;strong&gt;[10]&lt;/strong&gt; below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for books that are primarily devoted to methods of finite form integration, the best such books I know about (that are not designed for researchers in present day computer methods of finite form integration) are Fichtenholz &lt;strong&gt;[3]&lt;/strong&gt; (the only item below that, as far as I can tell, is not freely available on the internet), Hardy &lt;strong&gt;[4]&lt;/strong&gt;, and MacNeish &lt;strong&gt;[6]&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the items below, the dates and links are to the latest edition I could find freely available on the internet. I’m using the latest editions to minimize typos and errors and such that an early edition might have. In some cases (especially Todhunter’s book), the date is much later than when the book first appeared.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Joseph Edwards, &lt;a href=&quot;https://archive.org/details/treatiseonintegr01edwauoft&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;A Treatise on the Integral Calculus, Volume 1&lt;/strong&gt;&lt;/a&gt; (1921)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Joseph Edwards, &lt;a href=&quot;https://archive.org/details/treatiseonintegr02edwauoft&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;A Treatise on the Integral Calculus, Volume 2&lt;/strong&gt;&lt;/a&gt; (1922)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Grigorii Mikhailovich Fichtenholz, &lt;a href=&quot;http://www.worldcat.org/title/indefinite-integral/oclc/36960758&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Indefinite Integral&lt;/strong&gt;&lt;/a&gt; (1971)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Godfrey Harold Hardy, &lt;a href=&quot;https://archive.org/details/cu31924001539570&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Integration of Functions of a Single Variable&lt;/strong&gt;&lt;/a&gt; (1916)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[5]&lt;/strong&gt; Horace Lamb, &lt;a href=&quot;https://archive.org/details/AnElementaryCourseOfInfinitesimalCalculus&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;An Elementary Course of Infinitesimal Calculus&lt;/strong&gt;&lt;/a&gt; (1942)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[6]&lt;/strong&gt; Harris Franklin MacNeish, &lt;a href=&quot;https://catalog.hathitrust.org/Record/000576986&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Algebraic Technique of Integration&lt;/strong&gt;&lt;/a&gt; (1952)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[7]&lt;/strong&gt; Benjamin Osgood Peirce, &lt;a href=&quot;https://archive.org/details/cu31924073894754&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;A Short Table of Integrals&lt;/strong&gt;&lt;/a&gt; (1914)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[8]&lt;/strong&gt; William Benjamin Smith, &lt;a href=&quot;https://archive.org/details/infinitesimalan00smitgoog&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Infinitesimal Analysis&lt;/strong&gt;&lt;/a&gt; (1898)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[9]&lt;/strong&gt; Isaac Todhunter &lt;a href=&quot;https://archive.org/details/treatiseonintegr00todhrich&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;A Treatise on the Integral Calculus&lt;/strong&gt;&lt;/a&gt; (1906)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[10]&lt;/strong&gt; Benjamin Williamson, &lt;a href=&quot;https://archive.org/details/cu31924031264769&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;An Elementary Treatise on the Integral Calculus&lt;/strong&gt;&lt;/a&gt; (1888)&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2018-06-12T14:41:02.353" LastActivityDate="2018-06-12T14:41:02.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14192" PostTypeId="2" ParentId="14176" CreationDate="2018-06-12T18:30:33.297" Score="7" Body="&lt;p&gt;I strongly recommend you never use these kinds of questions, for the same reasons others have brought up. I once asked &quot;what was your favorite topic from this class?&quot; on a final exam, and gave full points for any reasonable answer. Unfortunately many students gave non-answers like &quot;thanks this class was fun&quot; and &quot;have a good summer&quot;, and other students used significant exam time writing thoughtful answers. This was a bad question, and the ones you suggest strike me as more confusing and less useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're looking to add more cheer to an exam, here are some ideas.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Include interesting quotes, comics, or jokes related to the material (make sure it is obvious that they are not part of a problem).&lt;/li&gt;&#xA;&lt;li&gt;Take a class picture and put it in the exam.&lt;/li&gt;&#xA;&lt;li&gt;Use lots of colors and figures.&lt;/li&gt;&#xA;&lt;li&gt;During the exam, have snacks or fidget toys available to students.&lt;/li&gt;&#xA;&lt;li&gt;Find some relevant puzzle-like problems to do in class/homework, and if students seem to enjoy them, put a particularly easy one on the exam.&lt;/li&gt;&#xA;&lt;li&gt;If and only if you make it abundantly clear that they won't be graded, add optional thought-provoking questions at the end for students who finish early. I like math brainteasers, math philosophy questions, and the Turing test (&quot;in the space below do your best to convince me you are a human being and not an artificial intelligence&quot;).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For all of these, be aware of how your individual students may respond to them in a high-stress situation. I would not recommend these ideas for every group of students. Err on the side of caution, and avoid potential distractions.&lt;/p&gt;&#xA;" OwnerUserId="7707" LastEditorUserId="7707" LastEditDate="2018-06-12T18:38:16.353" LastActivityDate="2018-06-12T18:38:16.353" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14193" PostTypeId="1" AcceptedAnswerId="14195" CreationDate="2018-06-13T00:30:20.393" Score="5" ViewCount="548" Body="&lt;p&gt;Days ago a professor told me to ask in class when I don't understand something and that he could keep explaining until it's clear for me. Being honest I'm a little slow to understand mathematics in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thing is I really need to understand what he teach otherwise I'm going to fail the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I said before, I'm not very smart so if I do what my professor suggest I might end up upsetting him because I'm going to keep telling him 'again please I still don't get it' at least 4 or more times..&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question to you, as experienced mathematics educators,  is &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Should I  ask only one or two times (to not upset him) if I don't understand something and to search by myself?  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Should I keep asking (many ... many times) until I understand something?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If I do 2. Does that can have negative consequences to me as student?&lt;/p&gt;&#xA;" OwnerDisplayName="user10026" LastActivityDate="2018-06-22T20:29:57.150" Title="Limit of questions that a student should ask in class without upsetting professor?" Tags="|undergraduate-education|teaching|teachers|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14194" PostTypeId="2" ParentId="14176" CreationDate="2018-06-13T01:12:52.233" Score="1" Body="&lt;p&gt;I'm coming at this question from a slightly different angle.  It's possible you've specified &quot;exam&quot; but have meant something more akin to an in class &quot;quiz&quot; or &quot;pop quiz&quot; in common vernacular.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word exam has the formal connotation in that the result has real bearing on a grade or GPA.  If this is not the case, and the situation is less formal such as an in class quiz, then using humour in a quiz would be more forgivable, perhaps even bolstering your reputation as a teacher and educator with regards to your relationship with your students.  I've been a student of a teacher with this mindset and it did allow the students to feel closer to the teacher, considering him as the &quot;cool teacher&quot; if you will.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As others have stated, if this is a formal exam or has any bearing on grade, humourous questions should be avoided.&lt;/p&gt;&#xA;" OwnerUserId="10027" LastActivityDate="2018-06-13T01:12:52.233" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14195" PostTypeId="2" ParentId="14193" CreationDate="2018-06-13T05:42:26.840" Score="4" Body="&lt;p&gt;How many times one can ask a given professor a question depends on the professor. There are professors who do not like questions (&quot;this is not a good question&quot;). There are professors who will forget about the rest of the class to attend to a question, however, misguided, spending five minutes answering whatever nonsensical question, boring the rest of the class in the process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The size of the class matters. It can be difficult and even problematic to handle numerous questions in large lecture halls. Once there are more than 80-100 students in the room the dynamic of answering questions changes. The student can be physically far away from the professor. Attending to individual needs does not always help in maintaining the concentration of a large group of people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nonetheless, my sentiment, I think shared by many professors, is that most students do not ask enough questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Questions can be valuable for the professor because sometimes they signal or highlight particular misunderstandings and confusions, and generally what is asked by one student could/should be asked by many. The professor is gleefully talking about computing the flux of an electric field across some surface and the student asks what that $\Omega$ thing below the integral sign is ... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, there are occasional students who really do ask too many questions. Most semesters I don't have one of these students in class, but last semester I did, and it requires some work to handle it well. The potential problem is that the constant questions interrupt the flow of class and distract other students. These problematic students tend to act like class is a private tutoring session, and what characterizes them is the indiscriminate nature of the questions they ask - they ask about notation, about the meaning of the logarithm, about philosophically why we need to parameterize surfaces, about famous French engineers, etc. Usually this is a sign of immaturity, of someone who has not yet learned to censor himself or lives in a mental bubble, blissfully unaware of those around him. Also it can indicate a poorly developed ability to distinguish between the trivial (the professor's handwritten $u$ and $v$ look too similar, but from context it is obvious, upon reflection, which is which) and the meaningful (the problem turns on whether that $u$ is really a $v$). Some very strong students ask a lot of questions too, but these are usually distinguishable from the problematic ones because of the quality of the questions they ask. I don't mean that they ask sophisticated high level questions - I mean that they ask questions that address a basic and key issue, even if elementary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a student, how to tell if you are asking too many questions? If you are asking yourself that question, it is highly likely that you are not asking too many questions in class. The self-awareness to pose the question suggests a sense of limits that suggests it is unlikely that who asks it is passing those limits.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-06-13T05:42:26.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14196" PostTypeId="2" ParentId="14193" CreationDate="2018-06-13T05:43:35.590" Score="4" Body="&lt;p&gt;I think this depends a lot on the type of questions (or rather prompts);&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you keep asking the professor &quot;again, please&quot;, you shouldn't do it perhaps more than once. Most likely, (s)he will repeat what was just said,  but that's useful only if you had problems understanding it due to noise or being distracted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you could understand the words clearly but not the meaning, you &lt;em&gt;should think hard about the exact point where the professor lost you and ask about this point in the most specific way possible, showing your thought process and your current obstacle&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The latter kind of question is way more difficult than a simple &quot;again, please&quot;,  but also generally useful, also for other students.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2018-06-13T05:43:35.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14197" PostTypeId="2" ParentId="14173" CreationDate="2018-06-13T11:16:33.043" Score="0" Body="&lt;p&gt;The underlying issue can be phrased in your terms. The problem is not to compute $\mathrm{D}(f \circ \gamma)$; the problem is to compute&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\mathrm{D}(f \circ \gamma) \; \widehat{e}_1 = \mathrm{D}f(\gamma)\; \mathrm{D}\gamma \;\widehat{e}_1  $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's going on is that you are insisting computing this in an inefficient manner. You insist that students compute the covector-matrix product $\mathrm{D}f(\gamma)\; \mathrm{D}\gamma $ first and then right multiply by $\widehat{e}_1$, when its much more efficient to first compute $\mathrm{D} \gamma \; \widehat{e}_1$ product and then left multiply by $\mathrm{D}f(\gamma)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is further exacerbated by the fact that the product $\mathrm{D} \gamma \; \widehat{e}_1$ can itself be computed directly; you can obtain its first column without bothering with the rest of the matrix. You shouldn't be computing the entire matrix $\mathrm{D} \gamma$ when solving this problem.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;If you cut out the waste, applying the multi-variable chain rule here is the same thing as computing (in the traditional notation that I hate)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \frac{\partial(f \circ \gamma)}{\partial u}&#xA;= \frac{\partial f}{\partial x}\frac{\partial x}{\partial u} &#xA;+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial u} &#xA;+ \frac{\partial f}{\partial z}\frac{\partial z}{\partial u} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which is much more likely to be viewed as an improvement over computing the composite, and students can be encouraged to see the product $\mathrm{D}f(\gamma) \; \mathrm{D} \gamma$ as showing part of the recipe for how to determine what calculation is the correct one.&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2018-06-13T11:16:33.043" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14198" PostTypeId="1" CreationDate="2018-06-13T13:13:45.887" Score="4" ViewCount="624" Body="&lt;p&gt;I have seen two different methods of showing algebraic work when solving equations. I show both of them below for the same simple math problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;\begin{alignat}{8}&#xA;x+3 &amp;amp;\;=&amp;amp;\; 5 \qquad&amp;amp;&amp;amp;\qquad x+3 &amp;amp;= 5&#xA;\\&#xA;-3 &amp;amp;\;&amp;amp;\;-3 \qquad&amp;amp;\text{or}&amp;amp;\qquad x+3-3 &amp;amp;= 5-3  &#xA;\\&#xA;x &amp;amp;\;=&amp;amp;\; 2 \qquad&amp;amp;&amp;amp;\qquad x &amp;amp;= 2  &#xA;\end{alignat}&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which method do you show when teaching and require of your students when they show work? Why do you feel one method has merit over the other?&lt;/p&gt;&#xA;" OwnerUserId="6675" LastEditorUserId="5220" LastEditDate="2018-06-13T15:34:45.917" LastActivityDate="2018-06-13T23:39:46.070" Title="Method of Showing Algebraic Work" Tags="|algebra|primary-education|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14199" PostTypeId="2" ParentId="14198" CreationDate="2018-06-13T16:45:34.287" Score="2" Body="&lt;p&gt;I like the left-hand example, because it highlights the step, which is something being done to both sides. This is how I add or subtract. Multiplying and dividing look different, of course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you were doing the right-hand version on a whiteboard, you could do it in a different color. But that isn't something the students could easily do in the same way.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-06-13T16:45:34.287" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14200" PostTypeId="2" ParentId="14198" CreationDate="2018-06-13T19:27:57.503" Score="3" Body="&lt;p&gt;At my institution, your 2$^{nd}$ option is the standard in the algebra sequence. This was chosen for three reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;[Most convincing for me] Immediately before solving equations, students practice simplifying an expression by combining like-terms. They are already accustomed to simplifying things like $$3x+5+(-3x),$$ so adding a term to both sides of an equation, in-line with everything else, reduces to just simplifying two expressions -- something they should be comfortable doing.&lt;/li&gt;&#xA;&lt;li&gt;Writing $+(-3x)$ under each side of an equation may not appear to some students as being the same as what they've been practicing, because we emphasize horizontal expression-writing $$13x + 25x$$ over stacked notation: $$\begin{array}{c}&#xA;\phantom{\99}13x\\&#xA;\underline{+\phantom{9}25x}\\&#xA;\phantom{\times9}\\&#xA;\end{array}$$&#xA;Better to keep consistent with the prevailing notation.&lt;/li&gt;&#xA;&lt;li&gt;It preserves notation. While many people add or subtract terms underneath expressions, students often write these changes too low, not low enough or sometimes up above at an angle. These variations often leave students (and instructor) wondering if a value is really a term being added, a subscript on a term or an exponent.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;All of that said, many of my students have writing underneath ingrained in their math habits. I will only make them change their habit if they are consistently making mistakes and need a new approach. For students who don't have any algebra habits developed, I suggest it as the standard in the class, and recommend that they learn this approach. In later courses, I am less pedantic about how one shows their algebra, provided they show something complete and mathematically correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I demonstrate that method with colored chalk/pens, and many students follow suit by bringing their own colored pens.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-06-13T19:27:57.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14201" PostTypeId="2" ParentId="14198" CreationDate="2018-06-13T21:02:06.037" Score="1" Body="&lt;p&gt;I do not mandate either method over the other, only that the students are able to demonstrate they understand how/why each step in their solution is done. I usually make my students go to the whiteboard, show all steps, and be prepared to present/explain each step.  I like the geometry way of using a &quot;t-gram&quot; with each step on one side and justification (usually in the form of mathematical property) on the the other. Another thing, in order to make sure mere memorization is not at hand, the students know they may be challenged or asked for evidential answers to challenging questions either by classmates or myself. A bit slow at first, but soon enough, speed is comparable to using only verbal explanations; and a writing element (as well as critical thinking) is now included.&lt;/p&gt;&#xA;" OwnerUserId="10034" LastActivityDate="2018-06-13T21:02:06.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14202" PostTypeId="2" ParentId="13136" CreationDate="2018-06-13T22:47:34.167" Score="2" Body="&lt;p&gt;Reference angles connect triangles (especially the 2 special triangles, 30-60-90 and 45-45-90) to the unit circle. If I want to think about the sine of 120 degrees, knowing that its reference angle is 60 degrees allows me to figure out the sine from the 30-60-90 triangle. &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-06-13T22:47:34.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14203" PostTypeId="2" ParentId="14198" CreationDate="2018-06-13T23:39:46.070" Score="2" Body="&lt;p&gt;I'm not sure that I actually like either presentation all that much.  If I were explaining the work on the board, I would probably start by writing&#xA;$$ x + 3 \phantom{{}-3} = 5 \phantom{{}-3} $$&#xA;leaving a little bit of space where I know that I am going to have to fill something in a later.  Then, in a different color, fill in the blank to get&#xA;$$ x + 3 \color{red}{-3} = 5 \color{red}{-3}. $$&#xA;Finally, the last statement needs to be connected to what is already written.  In this case, an implication suffices&#xA;\begin{align}&#xA;x + 3 \color{red}{-3} &amp;amp;= 5 \color{red}{-3} \\&#xA;\implies x &amp;amp;= 2.&#xA;\end{align}&#xA;This may be pedantic and nit-picky, but I generally insist that statements not be written one atop another without some kind of logical connection.  This becomes particularly important when, for example, working with equations that can have extraneous solutions.  For example&#xA;\begin{align}&#xA;&amp;amp;\frac{1}{x-2} = \frac{3}{x+2} - \frac{6x}{x^2 - 4} \\&#xA;&amp;amp;\qquad\implies x\ne 2,\quad x\ne -2,\quad \text{and}\quad x + 2 \color{red}{{}+3x-2} = -3x-6 \color{red}{{}+3x-2} \\&#xA;&amp;amp;\qquad\implies x\ne 2,\quad x\ne -2,\quad \text{and}\quad 4x = -8 \\&#xA;&amp;amp;\qquad\implies x\ne 2,\quad x\ne -2,\quad \text{and}\quad x=-2. \\&#xA;\end{align}&#xA;Since we arrive at a contradiction, we know that the original statement must have been false, i.e. the equation has no solutions.  By carefully keeping track of the argument and writing down the details (including the way in which statements are connected to each other, as well as the full set of implications of a particular identity), there is no need to come back and check for extraneous solutions.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The situation is slightly different for the work that student turn in to me (since they don't know, ahead of time, how much space they might need to add or multiply).  For the original problem, I might expect to see something like&#xA;\begin{align}&#xA;x+3 &amp;amp;= \  5 \\&#xA;{\small -3} &amp;amp;\ \ \ \ {\small{-}3} \\&#xA;\implies x \phantom{{}+3} &amp;amp;= \  2.&#xA;\end{align}&#xA;My preference would be that they write out the extra line (primarily for the third point that Nick C makes in his answer), but I recognize that students often lack the time to write more clearly (say, in an exam setting), and I think that it is more important for them to be comfortable with the big picture idea of keeping equations &quot;balanced&quot; than it is to nit-pick their notation too much.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Multiplication and division, as well as exponentiation and taking logarithms, follow a similar pattern.  On the board, I use color to emphasize what is changing from step-to-step, whereas in student work, I rather expect to see everything done in-line. For example, on the board I might write&#xA;$$ \color{red}{\tfrac{1}{3}(} 3x \color{red}{)} = \color{red}{(} 15 \color{red}{)\tfrac{1}{3}} = 5&#xA;\implies x = 5&#xA;$$&#xA;or&#xA;$$ \color{red}{\frac{\color{black}{3x}}{3}} \color{white}{\frac{\color{black}{=}}{-}} \color{red}{\frac{\color{black}{15}}{3}} \color{white}{\frac{\color{black}{=}}{-}} \color{white}{\frac{\color{black}{5.}}{-}}&#xA;$$&#xA;I would expect students to write basically the same thing in their work, except without the use of color.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2018-06-13T23:39:46.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14204" PostTypeId="2" ParentId="14135" CreationDate="2018-06-14T04:31:42.333" Score="1" Body="&lt;p&gt;Take anything that can easily be divided into fourths.  For example a cracker that has four sections or index cards.  Make sure the fourths are clearly marked.    Take 6 of them for your example.  Tell the students that someone uses 3/4 of this everyday and you want to know how many days it will last.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then count or physically separate each group of 3/4.  There should be 8 of them.  This is a great real world example of 6 divided 3/4 is 8.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2018-06-20T17:21:41.127" LastActivityDate="2018-06-20T17:21:41.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14205" PostTypeId="1" CreationDate="2018-06-14T08:36:13.533" Score="2" ViewCount="187" Body="&lt;p&gt;In Germany there exists a concept whose name might be translated as &quot;basic ideas.&quot; The core of the concept is to offer the students an idea of what they are learning, e.g. in the context of derivatives of a function, to interpret the derivative as instantaneous rate of change and so on. Is there a comparable concept in the US?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Greetings, Rico&lt;/p&gt;&#xA;" OwnerUserId="10036" LastEditorUserId="11" LastEditDate="2018-06-14T14:30:59.870" LastActivityDate="2018-06-15T00:29:37.980" Title="&quot;Basic ideas&quot; as a concept in teaching maths?" Tags="|teaching|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14206" PostTypeId="2" ParentId="14135" CreationDate="2018-06-14T12:13:32.840" Score="1" Body="&lt;p&gt;If some object moves at the rate of 5 kilometers per minute, how long does it take to go 100 kilometers?  Divide: 100 / 5 = 20 minutes&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If some object moves at the rate of $\frac{1}{10}$ kilometer per minute, how long does it take to go $\frac{1}{2}$ kilometer?  Again: divide $\frac{1}{2}$ / $\frac{1}{10}$ = 5 minutes&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2018-06-14T12:13:32.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14207" PostTypeId="1" AcceptedAnswerId="14216" CreationDate="2018-06-14T20:18:36.647" Score="4" ViewCount="142" Body="&lt;p&gt;As part of a free summer enrichment program for highly-motivated high school students, I need to plan eight hour-long lessons for mini-courses titled as &quot;Algebra 2&quot; and &quot;Calculus&quot; separately. Despite the titles of these mini-courses, they do not give students any actual school credits and they should primarily be seen as somewhat informal (I say &quot;somewhat&quot; because I will have to evaluate their progress) small-group meetings where I help students get ready for their real credit-bearing Algebra 2 and Calculus classes that they will take in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, these classes would &lt;strong&gt;not&lt;/strong&gt; be designed with the idea that students should achieve mastery. The summer program also requires to be in other kinds of mini-humanities, science, and art courses, so the program and the students are not all exclusively focused on math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem I am having is deciding how to map the course and plan instruction for both classes given that I would want to make sure that I engage every kind of student regardless of prior ability or interest. This objective is important to me because I am not actually teaching a &quot;real-deal&quot; Algebra 2 or Calculus class and so I want to help the students feel confident when they enter high school again in the fall.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What kinds of topics do you think I should cover for our short time together? (If you can't answer this, how do you suggest I go about paring down the list of topics before I even get to see all my students?) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;My own Google searches always seem to turn up pacing for standard high school courses, but I won't be able to follow these. If I did, it's possible I wouldn't cover both differentiation and integration in Calculus. Would covering only integration be a good call?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't know where to begin in designing my Algebra 2 course. I don't think I would want to risk boring these highly-motivated students with possibly unnecessary Algebra 1 review. Would you suggest giving a diagnostic assessment on our very first class meeting?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user8277" LastActivityDate="2018-06-18T07:35:03.800" Title="Exposure to Algebra 2/Calculus Under Time Constraints" Tags="|mathematical-pedagogy|course-design|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14208" PostTypeId="2" ParentId="14207" CreationDate="2018-06-14T23:06:44.623" Score="1" Body="&lt;p&gt;Start with:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What is student background (grade level, etc.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is student ability (very smart?  Below average)?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Total hours of classroom?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;REASONABLE estimate on out of class time available.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Based on 3 mostly and 4 partially, figure how much you can cover and want to cover.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Keep it light and motivational.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;USE CLASS TIME.  Assume they do NOTHING outside of class (bullet 4).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do something fun and non indoor class related.  Class outdoors. A demonstration.  A physical experiment.  Doesn't matter.  Just at least one &quot;thing&quot; that is more summer camp and less teacher in a room.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-14T23:06:44.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14209" PostTypeId="2" ParentId="14205" CreationDate="2018-06-15T00:29:37.980" Score="2" Body="&lt;p&gt;Most U.S. middle-school or high-school textbooks have a box labelled &quot;Objectives&quot; that lay out the basic ideas in the coming chapter or section. Many textbooks also begin a section with a thought question that introduces one or more basic idea of the section. The question is open-ended and thought-provoking and cannot be answered until some basic ideas are mastered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are so many examples--look at just about any recent U.S. textbook for an example.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2018-06-15T00:29:37.980" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14210" PostTypeId="2" ParentId="14176" CreationDate="2018-06-15T08:59:06.603" Score="1" Body="&lt;p&gt;First, I see that you wrote &quot;funny question&quot; like that, in quotes. You really mean a question that's unrelated to the exam problems. (If only for the fact that the questions were not humorous.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've seen teachers use this to help relate to their students, to know them better. &#xA;Sometimes on a personal level e.g. &quot;What are your plans for weekend / what do you do in your spare time? or as it relates to the class, e.g. &quot;What part of this past chapter did you struggle with or find confusing?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the question, answered truthfully, will help you teach the class or help build your relationship, you'll find value in doing so. Make it the last question, and clear that if left blank, no points lost. You don't want such a question to add any anxiety to the testing process. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-06-15T08:59:06.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14213" PostTypeId="1" CreationDate="2018-06-17T17:03:02.550" Score="4" ViewCount="417" Body="&lt;p&gt;There is a Mathematics Stack Exchange question on &lt;a href=&quot;https://math.stackexchange.com/questions/143501/a-good-book-on-inverse-problems-for-engineers&quot;&gt;a good book on inverse problems for engineers&lt;/a&gt;. Here, I would like to ask for suggestions on how to approach &lt;em&gt;teaching&lt;/em&gt; undergraduate upper-division science and engineering students about &lt;a href=&quot;https://en.wikipedia.org/wiki/Inverse_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;inverse problems&lt;/a&gt;, without requiring relatively heavy mathematical prerequisites such as functional analysis. What can be assumed is some exposure to linear algebra (including least squares, but not necessarily the singular value decomposition), calculus and introductory ordinary differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The students I teach have also had introductory courses on statistics, control systems, signal processing and image processing, but I would like to keep the question broadly applicable and not assume a very specific background.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2018-06-18T07:25:10.287" LastActivityDate="2018-06-20T14:54:38.580" Title="Teaching science and engineering students the field of inverse problems" Tags="|undergraduate-education|course-design|applied-mathematics|engineering-mathematics|inverse-problems|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14214" PostTypeId="5" CreationDate="2018-06-17T17:09:51.597" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2018-06-17T17:09:51.597" LastActivityDate="2018-06-17T17:09:51.597" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14215" PostTypeId="4" CreationDate="2018-06-17T17:09:51.597" Score="0" Body="For questions about teaching inverse problems. Inverse problems involve for example reconstruction of an object based on physical measurements and finding a best model/parameters out of a family given observed data. Typically the corresponding &quot;forward&quot; problems are well-posed and can be solved straightforwardly, while the inverse problems are often ill-posed. Not to be confused with the (inverse) tag." OwnerUserId="376" LastEditorUserId="376" LastEditDate="2018-06-18T15:05:54.553" LastActivityDate="2018-06-18T15:05:54.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14216" PostTypeId="2" ParentId="14207" CreationDate="2018-06-18T07:35:03.800" Score="1" Body="&lt;p&gt;I do not know exactly what the content of courses like Algebra 1 or Algebra 2 is supposed to be. Having said so one possibility is to carefully plan a modelization of some &quot;real life&quot; situation in which problems can be solved by algebraic techniques. An example of an activity I once performed with high school students is the following. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Divide them in groups and make them compete in a sort of city-buliding &amp;amp; war game like the many one you can found in Internet (e.g. Ikariam, ClashofClans, Grepolis...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually this leads to solve some linear optimization problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say: you can build two type of war units, each with its offensive and defensive value and each with maintenance cost. Which is he combination &#xA;that gives you the best value for money in offence? and in defence?... This sort of problems lead through:&#xA;- recognizing how to model this situation in a math framework. This is where they'll have more difficulty and you'll have to help them a bit in this.&#xA;- solve linear equations at first, then system of linear equations. &#xA;- if you think they can handle this you may enter also some linear inequalities and guide them through geometrical solutions of easy linear programming models.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The approach could be that of having them discover rules and solutions, with you playing the role of &quot;tutor&quot;. Competition can be used to engage them more actively (you can consider a prize for the winner group at the last lecture).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So rather than &quot;lecturing&quot; organize some sort of game in which they'll win thorugh understanding mathematically the kind of problems you pose them.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2018-06-18T07:35:03.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14217" PostTypeId="1" CreationDate="2018-06-18T09:25:27.827" Score="2" ViewCount="91" Body="&lt;p&gt;I am working with AI methods and thinking about how my daughter (also other kids) can learn mathematics with AI help easier (like showing where in learning path was biggest mistake and etc.)? I know it's a broad question, but any thoughts will be welcome.&lt;/p&gt;&#xA;" OwnerUserId="10049" LastActivityDate="2018-06-18T14:39:26.197" Title="How AI can help learn mathematics" Tags="|education-research|" AnswerCount="1" CommentCount="4" ClosedDate="2018-06-18T15:10:02.973" ContentLicense="CC BY-SA 4.0" />
  <row Id="14218" PostTypeId="1" CreationDate="2018-06-18T09:40:28.560" Score="8" ViewCount="467" Body="&lt;p&gt;I want to set the context for me asking this question before stating it properly.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I teach at college/university level. This question deals with first-year students, fresh from school. So think calculus, but with some vectors and proof by induction etc. to keep them on their toes. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I provide my students with almost-complete notes, which they can find online (on the course VLE). The notes have no proofs or examples in them. Instead, they have big, blank boxes where the students are expected to write down the proofs and the solutions to examples. This is meant to be a compromise between giving them the complete notes and helping them to pay attention in lectures.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I try my best at the start of term to make sure the students understand their responsibility in lectures is to copy down proofs and examples (and, of course, understand them etc. etc.).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Recently, I had the following conversation with a student.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Student&lt;/strong&gt;: How do I do question X?&#xA;  &lt;br&gt;&lt;strong&gt;Me&lt;/strong&gt;: Look at example Y in the notes. It is basically identical.&#xA;  &lt;br&gt;&lt;strong&gt;Student&lt;/strong&gt;: Okay....its blank.&#xA;  &lt;br&gt;&lt;strong&gt;Me&lt;/strong&gt;: I know. Did you take notes?&#xA;  &lt;br&gt;&lt;strong&gt;Student&lt;/strong&gt;: No.&#xA;  &lt;br&gt;&lt;strong&gt;Me&lt;/strong&gt;: ......&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So I didn't know how to respond (thus the &quot;.....&quot;), and upon reflection still don't know how to respond.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Giving them the notes will undermine the concept of bullet point (2), above - it will undermine the concept of the lecture! It will also give them unrealistic expectation of their later years at university.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Not giving them the notes will ensure they fail the course. (The &quot;almost-complete notes&quot; might suffice for a good student, but a poor student would likely need more.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Telling them how to do this one question will not help - the issue runs much deeper. (We can assume that they have not been taking notes since week 1.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Telling them to &quot;find a friend&quot; is just slightly cheap...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How to deal with poor students who don't take notes? What should I have said in the above conversation, when I discovered they had no notes?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="169" LastEditorUserId="5220" LastEditDate="2018-06-18T20:59:54.193" LastActivityDate="2018-06-18T20:59:54.193" Title="How to deal with poor students who don't take notes?" Tags="|undergraduate-education|mathematical-pedagogy|class-participation|lecture-notes|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14220" PostTypeId="2" ParentId="14217" CreationDate="2018-06-18T14:39:26.197" Score="4" Body="&lt;p&gt;Audrey Watters does some good writing on the (long) history of education technology, which I would recommend you read. At a minimum, I think it would be useful to be aware of the history. For example, this spring, &lt;a href=&quot;http://hackeducation.com/2018/04/26/cuny-gc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching Machines, or How the Automation of Education Became 'Personalized Learning'&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“Personalized learning,” depending on how you define it, dates back to&#xA;  Rousseau. Or it dates back further still – to Alexander the Great’s&#xA;  tutor, some guy named Aristotle. It dates to the nineteenth century.&#xA;  Or to the twentieth century. It dates to the rise of progressive&#xA;  education theorists and practitioners. To John Dewey. Or to Maria&#xA;  Montessori (often the only woman ever mentioned in any of these&#xA;  tales). Or it dates to the rise of educational psychology. To B. F.&#xA;  Skinner. To Benjamin Bloom. It dates to special education-related&#xA;  legislation passed in the 1970s or to the laws passed the 1990s. Or it&#xA;  dates to computer scientist Alan Kay’s 1972 essay “A Personal Computer&#xA;  for Children of All Ages.” Or it dates to the Gates Foundation’s&#xA;  funding grants and political advocacy in the early 2000s. Take your&#xA;  pick.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also, not mentioned above, you should be aware of &lt;a href=&quot;https://en.wikipedia.org/wiki/PLATO_%28computer_system%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;PLATO&lt;/a&gt; (computer-assisted instruction system from U. Illinois, 1960-2006). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-06-18T14:39:26.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14221" PostTypeId="2" ParentId="14218" CreationDate="2018-06-18T15:53:05.480" Score="10" Body="&lt;p&gt;You said that most of your students are first year college students.  Such students often lack basic academic skills.  Perhaps they were smart enough in high school to get away with not taking notes or studying, perhaps they had teachers that provided too much scaffolding (and never took it away), or perhaps the standards were just lower.  There are a lot of reasons that successful high school students might struggle in college, many of which have nothing to do with the students themselves being &quot;poor&quot;.  Thus the first step, I think, it to change your mindset.  These aren't poor students, they are good students (they are in college, after all) who have received poor training.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From that point of view, you might consider that part of your job is helping your students to learn the study skills that will enable them to be successful in the future.  I think that what you describe in your question is an excellent start&amp;mdash;providing blank notes for the students to fill in emphasizes that the students are at least partially responsible for their own success.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition to providing notes, you might consider continually explaining to the students that they should be taking notes.  Particularly early in the term, it is not unreasonable to stop with some frequency and suggest &quot;You really should be writing this down in your notes.  Do you need a little more time?&quot;  Then give them a good 15&amp;ndash;30 seconds to write.  That 15&amp;ndash;30 seconds will probably feel like an eternity to you, but I guarantee that you'll have several students who are struggling to keep up, and will need all of that time.  A suggestion:  bring a bottle of water or cup of coffee to class.  Leave it on the far side of the room.  When you ask them to write, put down your writing implements, walk over to your water or coffee, slowly take a swig, then go back to your lecture.  This slows things down a little, but feels less awkward than standing silently at the front of the room.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another suggestion:  for the first week, print out the blank notes for you students.  Make it very, very clear that this is a &quot;one time&quot; deal, and that they need to start printing out notes ahead of time.  This will hopefully help your students to understand that the notes are important.  You could even make up some song and dance about how you would like to continue printing out notes for them all term, but the department lacks funding for that, so they are going to have to be responsible for their own notes (this probably isn't even a lie in most departments).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet another suggestion:  quiz them on their notes.  On a (perhaps) weekly basis, give them a quiz with a problem taken directly from the notes.  Give them only a little bit of time ($&amp;lt;5$ minutes), and allow them to use their notes for these quizzes.  Try to keep such quizzes fairly low stakes, but make it clear that they are &quot;free points&quot; toward their final grade.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2018-06-18T15:53:05.480" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14222" PostTypeId="1" CreationDate="2018-06-18T17:04:44.707" Score="5" ViewCount="216" Body="&lt;p&gt;I am interested in creating a curriculum that helps cultivate students abilities to teach one another. Specifically good one-on-one instruction includes elements like: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples, &#xA;Pictures, &#xA;Humor, &#xA;Socratic Questioning,&#xA;Challenges appropriate to the students abilities. One could even evaluate an instructor (and a potentially a student-instructor) on their ability incorporate these elements into their teaching. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We might ask students to teach one another certain skills like how to add fractions with different denominators and most teachers have a sense of who would be good at teaching and who might be a tad patronizing or dismissive but I haven't ever seen a curriculum which directly addresses and develops instruction abilities in students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Does such a curriculum already exist? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) What research is out there on having students teach one another in one-on-one contexts? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) How does this community feel about students teaching one another directly? &lt;/p&gt;&#xA;" OwnerUserId="9841" LastActivityDate="2018-06-19T00:22:27.980" Title="Should students teach other students?" Tags="|curriculum|teaching|tutoring|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14223" PostTypeId="2" ParentId="14213" CreationDate="2018-06-18T18:25:31.340" Score="2" Body="&lt;p&gt;First, let me qualify this answer:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I am not familiar with the book mentioned in the original post.&lt;/li&gt;&#xA;&lt;li&gt;I have not taught any courses.&lt;/li&gt;&#xA;&lt;li&gt;My engineering courses had weekly problem sets.&lt;/li&gt;&#xA;&lt;li&gt;I am not familiar with very complicated inverse problems, such as X-ray crystallography, CAT scanning, seismography, or echo-based petroleum exploration.&lt;/li&gt;&#xA;&lt;li&gt;The course developer will need to identify categories of inverse problem that are likely to be relevant to the students in the future, tools that are available to the students for solving these problems, and examples of typical problems and results.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am a mechanical engineer who works as a computer programmer, so I have the students' background described in the original post.  As a student, I would have found the proposed course interesting.  In my professional career, I would have applied what I learned in such a course.  Indeed, I have found myself independently learning parts of such a course over the course of my career.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some of the things I would hope to see in such a course.  This list can be adapted to a unit that is anywhere from a week long to most of a course.  A week-long unit might only have some of the introductory problem categories; a course might cover several complicated problem categories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A one- or two- lecture overview of the topic, with:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The overall concept of inverse problems.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Given &quot;what happened&quot;, be able to figure out &quot;how&quot; and &quot;why&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A summary of the course's learning objectives:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Be able to recognize categories of inverse problems that can be solved.&lt;/li&gt;&#xA;&lt;li&gt;Know how to use some typical tools to solve such problems.&lt;/li&gt;&#xA;&lt;li&gt;Be able to recognize categories of inverse problems that cannot be (practically) solved (if such categories exist).&lt;/li&gt;&#xA;&lt;li&gt;Be able to estimate how precise an answer is reasonable, given the nature of the inputs to a problem.&lt;/li&gt;&#xA;&lt;li&gt;Be able to identify what extra information would be needed to get a more precise answer.&lt;/li&gt;&#xA;&lt;li&gt;Be able to check your answers.&lt;/li&gt;&#xA;&lt;li&gt;Be able to explain your answers.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A slideshow (with simple graphics) listing inverse problems that the students are already thoroughly familiar with, and a few that they might not be familiar with:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Addition/Subtraction&lt;/li&gt;&#xA;&lt;li&gt;Multiplication/Division&lt;/li&gt;&#xA;&lt;li&gt;Exponents/Logarithms&lt;/li&gt;&#xA;&lt;li&gt;Trig functions/Inverse trig functions&lt;/li&gt;&#xA;&lt;li&gt;Differentiation/Integration&lt;/li&gt;&#xA;&lt;li&gt;Inverting matrices&lt;/li&gt;&#xA;&lt;li&gt;Solving differential equations&lt;/li&gt;&#xA;&lt;li&gt;Conversion between time domain and frequency domain&lt;/li&gt;&#xA;&lt;li&gt;Inferring statistical distributions from data sets&lt;/li&gt;&#xA;&lt;li&gt;Photoshop's blurring / sharpening filters&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A slideshow (also with simple graphics) showing a few kinds of inverse problems that will be covered in the course.  For each kind of problem, show:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A field where the problem occurs.&lt;/li&gt;&#xA;&lt;li&gt;What a typical problem looks like.&lt;/li&gt;&#xA;&lt;li&gt;What a typical solution looks like.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What tools are available to the students to solve these problems during the course, and what fancier tools they might be able to obtain in their future work.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Then teach how to solve relevant categories of problems.  Start with easy ones, and move on to harder ones.  The first problem set or two could cover these topics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Find the formula of a linear equation.&lt;/li&gt;&#xA;&lt;li&gt;Estimate the formula of a linear equation given noisy data.  In the process, mention that the statistical formulas are based on an objective function, and the objective function minimizes the squared error of the data from the estimated equation.&lt;/li&gt;&#xA;&lt;li&gt;Use linear interpolation to find options along a Production Possibility Frontier.  (This is an economics concept, but very useful in Manufacturing Engineering.)  The PPF does not have to be linear.  The linear approximation can be piece-wise, and just has to be &quot;good enough&quot;.  Explain that the results from inverse problems are often unstable, so they can provide ranges of answers.&lt;/li&gt;&#xA;&lt;li&gt;Use the quadratic formula to find the roots of a parabola.&lt;/li&gt;&#xA;&lt;li&gt;Use Newton's method (or a similar method) to find the roots of a parabola.  Explicitly discuss how the objective function is set up and used.&lt;/li&gt;&#xA;&lt;li&gt;Use Newton's method (or a similar method) to find the roots of an arbitrarily complicated mathematical function of one variable.&lt;/li&gt;&#xA;&lt;li&gt;Simulate convolving two functions.  (Many real-world situations can be described by such a process.)  For example, apply a blurring algorithm to a function.  Teach how to estimate parameters of two (hidden) functions whose convolution can be seen.&lt;/li&gt;&#xA;&lt;li&gt;Show more-or-less inverse convolutions.  For example, blurring and sharpening filters.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Later problem sets could cover multi-variable inverse problems.  Some fields that use such problems include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Finance: estimating sets of parameters that would allow a system to have observed behavior.  For example, a rapidly growing field of business.  What are plausible combinations of fixed costs, variable costs, profit margins, productivity improvement rates that would allow a business to grow at a certain rate sustainably using internally generated cash flows?  If the business is growing too fast for this, or the business does not yet have that cost structure, how much external investment is required to achieve the observed growth rate?  How quickly must the external investment grow?  How much external investment is required before the cost structure can reach the necessary parameters?  How big will the business need to be at that point?  Side lessons that can be learned:  The importance of break-even analysis, and some of the risks of relying on outside financing.&lt;/li&gt;&#xA;&lt;li&gt;Use of the Laplace Transform and/or Fourier Transform (and their inverses) to convert between the time domain and the frequency domain.&lt;/li&gt;&#xA;&lt;li&gt;X-ray crystallography&lt;/li&gt;&#xA;&lt;li&gt;CAT scanning&lt;/li&gt;&#xA;&lt;li&gt;seismography&lt;/li&gt;&#xA;&lt;li&gt;echo-based petroleum exploration&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Try to design each problem set so that the student has to:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Identify why an inverse-problem needs to be solved.&lt;/li&gt;&#xA;&lt;li&gt;Identify available parameters in the observed data / images.&lt;/li&gt;&#xA;&lt;li&gt;Identify the category of inverse problem.&lt;/li&gt;&#xA;&lt;li&gt;Identify a tool that can estimate a solution.&lt;/li&gt;&#xA;&lt;li&gt;Determine the parameters of a solution.&lt;/li&gt;&#xA;&lt;li&gt;Verify that the solution can generate the observed data / images.&lt;/li&gt;&#xA;&lt;li&gt;Identify the limitations of the solution (such as instability / range of possible solutions, imprecision of parameters, &lt;em&gt;et cetera&lt;/em&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Notice whether the solution is significantly more accurate than a much simpler solution.&lt;/li&gt;&#xA;&lt;li&gt;Make a picture of the problem and solution.&lt;/li&gt;&#xA;&lt;li&gt;Be able to explain the solution to someone who has never taken calculus.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Early on in the course, some of the &quot;identify&quot; steps may be provided for the students.  But by the end of the course, the students should be able to answer these questions themselves.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2018-06-19T04:32:29.817" LastActivityDate="2018-06-19T04:32:29.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14224" PostTypeId="2" ParentId="14222" CreationDate="2018-06-19T00:22:27.980" Score="1" Body="&lt;p&gt;There are two related concepts which go by similar names:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) &lt;em&gt;Peer Teaching&lt;/em&gt;: &quot;a method by which one student instructs another student in material on which the first is an expert and the second is a novice.&quot;&#xA;See &lt;a href=&quot;https://www.opencolleges.edu.au/informed/features/peer-teaching/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this opencolleges link&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) &lt;a href=&quot;https://en.wikipedia.org/wiki/Peer_instruction&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Peer Instruction&lt;/em&gt;&lt;/a&gt;: Eric Mazur's flipped classroom technique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Peer teaching is one model for using students to teach students and&#xA;circumvent TheChef's problem that &quot;Students know very little.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This source may help:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://www.teachthought.com/pedagogy/the-definition-of-peer-teaching-a-summary-of-existing-research/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Definition Of Peer Teaching: A Sampling Of Existing Research&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-06-19T00:22:27.980" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14225" PostTypeId="2" ParentId="14213" CreationDate="2018-06-19T09:21:49.923" Score="0" Body="&lt;p&gt;Some more recent books on inverse problems that also stress applications are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Statistical and Computational Inverse Problems&lt;/strong&gt;, Jari Kaipio and Erkki Somersalo, &lt;a href=&quot;https://www.springer.com/de/book/9780387220734&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.springer.com/de/book/9780387220734&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Linear and Nonlinear Inverse Problems with Practical Applications&lt;/strong&gt;, Jennifer L. Mueller and Samuli Siltanen, &lt;a href=&quot;https://epubs.siam.org/doi/book/10.1137/1.9781611972344&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://epubs.siam.org/doi/book/10.1137/1.9781611972344&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A light introduction (short, concise, a little more focused on math, but still with good applications) is&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;A Taste of Inverse Problems: Basic Theory and Examples&lt;/strong&gt;, Martin Hanke, &lt;a href=&quot;https://epubs.siam.org/doi/book/10.1137/1.9781611974942&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://epubs.siam.org/doi/book/10.1137/1.9781611974942&lt;/a&gt; &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In a similar spirit, but older and a little longer, but also very well written is&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Inverse Problems in the Mathematical Sciences&lt;/strong&gt;, Charles W. Groetsch, &lt;a href=&quot;https://www.springer.com/la/book/9783322992048&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.springer.com/la/book/9783322992048&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2018-06-19T09:21:49.923" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14226" PostTypeId="1" AcceptedAnswerId="14244" CreationDate="2018-06-19T16:43:26.377" Score="8" ViewCount="257" Body="&lt;p&gt;I am a graduate math student and I believe that a nice way to raise the mathematical skills of people(especially students!) is to familiarize them with games and encourage them to use their minds and think on their own ideas. &#xA;I have studied too many puzzle games and I have found too many excellent puzzle games.&#xA;But I believe that board games and card games are more effective and I want to find a board game or a card game which have almost all of the following properties:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The game can be played by 8-9 players; almost all of them are 9-12 years old. (preferably 8 players!) &lt;/li&gt;&#xA;&lt;li&gt;I Prefer that the game can be played physically, without the use of any computer! &lt;/li&gt;&#xA;&lt;li&gt;The time of the workshops is Limited to 60-90 minutes. Let's assume the time is 90 minutes! (I must learn them the rules in at most 10-15 minutes, and I want a game such that we can do it 2 or 3 times in one workshop.)&lt;/li&gt;&#xA;&lt;li&gt;Also, after the first round that the game was played, I want to give them a low-level analysis of the game. &lt;/li&gt;&#xA;&lt;li&gt;Also, I Prefer the games, which had a possibility to make some low-level riddles and challenges to involve their minds. (For example, Yahtzee gives me the opportunity to give them some simple probabilistic problems. Also by a problem, I do not mean a mathematical problem, every exciting challenge is welcome.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I know so many exciting board games but most of them cannot be played by 8 players, and the others are not fit in the limited time of the workshop, because they have too many rules, I prefer a game with simple rules!&#xA;For example, &lt;a href=&quot;https://en.wikipedia.org/wiki/Kriegspiel_(chess)&quot; rel=&quot;noreferrer&quot;&gt;Kriegspiel&lt;/a&gt; is a very nice board game, but it is played by only two players and it is very hard to handle it in workshops! If the number of players was 2, then I could reduce chess to a very very simple board game, and then they can play it like Kriegspiel!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I have a trouble to find a suitable tag!&lt;/p&gt;&#xA;" OwnerUserId="8883" LastEditorUserId="8883" LastEditDate="2019-05-27T17:19:59.350" LastActivityDate="2019-05-27T17:19:59.350" Title="Targeted group game for 8 or 9 players" Tags="|mathematical-pedagogy|games|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14227" PostTypeId="2" ParentId="14135" CreationDate="2018-06-19T18:58:01.893" Score="1" Body="&lt;p&gt;I think students can generally grasp multiplication by fractions, both in why the answers are reasonable and in applying real-world examples. Once students understand that multiplying by a number less than 1 gives a product that is smaller than the original number, remind students that division is the inverse of multiplication, and as such, dividing by a number less than 1 (but greater than zero) gives an answer that is larger than the dividend.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Real-world examples using length and time are easy to work with in a story problem. An example: I have 2 feet of ribbon and cut it into lengths of 1/2 foot each. How many pieces of ribbon do I have? Answer: 4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example: Judy can make a friendship bracelet in 1/3 hour. She spends 3 hours making bracelets. How many bracelets does she make? Answer: 9.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This last example may have several ways to solve. Some would argue, therefore, it is not a good example. However, in the US, the CCSS allows for and encourages the development of multiple strategies in solving a problem. So, even though one student might think, &quot;Judy makes three bracelets an hour&quot; and multiply 3 times 3, this is mathematically equivalent to 3 divided by 1/3 -- we even teach kids to multiply by the reciprocal! I would suggest, then, that a problem like this be used during lecture time, allowing students to discuss the way they approached the problem. &lt;/p&gt;&#xA;" OwnerUserId="9883" LastActivityDate="2018-06-19T18:58:01.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14228" PostTypeId="2" ParentId="7528" CreationDate="2018-06-20T02:36:01.533" Score="4" Body="&lt;p&gt;Curves of constant width, the simplest of which is the Reuleaux Triangle, occur in a variety of applications. As a shape it consists of pieces of three circles. To construct a Reuleaux Triangle start with an equilateral triangle of side length h and with a compass from each vertex draw a circular arc with radius h between the other two vertices. The resulting set, like a circle has constant width h. Read more about the Reuleaux Triangle and its interesting properties here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Reuleaux_triangle&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Reuleaux_triangle&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/9u3cs.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9u3cs.jpg&quot; alt=&quot;Coin&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://de.ucoin.net/coin/bermuda-1-dollar-1997/?tid=50025&quot; rel=&quot;nofollow noreferrer&quot;&gt;de.ucoin.net&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="1865" LastEditorUserId="511" LastEditDate="2018-06-20T10:01:51.480" LastActivityDate="2018-06-20T10:01:51.480" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14229" PostTypeId="2" ParentId="14226" CreationDate="2018-06-20T04:46:42.540" Score="2" Body="&lt;p&gt;I taught elementary school for many years and had games every Friday for large groups of kids in grades 3-6 at lunchtime.  Here are some of the favorites.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://boardgamegeek.com/boardgame/444/escape-hidden-castle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Midnight Party&lt;/a&gt; from Discovery Toys (now called Escape from the&#xA;Hidden Castle by Ravensberger.  It teaches negative numbers and is&#xA;good for 2-8 kids.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://boardgamegeek.com/geeksearch.php?action=search&amp;amp;objecttype=boardgame&amp;amp;q=bazaar&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bazaar&lt;/a&gt;  It teaches algebraci thinking and is for 2-4 kids BUT I usually had 8 kids and they played in pairs.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/B00000IRW1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Presto-Change-o&lt;/a&gt;  It teaches making change and is for 2-4 kids but again you can play in pairs.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.24game.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;24&lt;/a&gt; My all time favorite'&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you like these and want ones similar to these, let me know. If these aren't what you are looking for, let me know more specifics&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2018-06-20T04:46:42.540" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14230" PostTypeId="2" ParentId="14135" CreationDate="2018-06-20T08:10:36.667" Score="0" Body="&lt;p&gt;Multiplication and division are the inverse of each other.I try to give a simple example. When we multiply a number say 10 by 2, we get 10x2=20 and when we divide the same number 10 by 1/2, we again get 20. It shows that when 2 and 1/2 are the inverse of each other, then multiplication and division are also the inverse of each other.So whenever, fraction comes in multiplication or division, then firstly remove the fraction by reversing it and then simplify.&lt;/p&gt;&#xA;" OwnerUserId="10055" LastActivityDate="2018-06-20T08:10:36.667" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14231" PostTypeId="2" ParentId="7528" CreationDate="2018-06-20T13:31:25.457" Score="4" Body="&lt;p&gt;A square antiprism at &lt;a href=&quot;https://en.wikipedia.org/wiki/One_World_Trade_Center&quot; rel=&quot;nofollow noreferrer&quot;&gt;One World Trade Center&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/thumb/a/ac/OneWorldTradeCenter.jpg/600px-OneWorldTradeCenter.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/thumb/a/ac/OneWorldTradeCenter.jpg/600px-OneWorldTradeCenter.jpg&quot; alt=&quot;Antiprism&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7258" LastActivityDate="2018-06-20T13:31:25.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14232" PostTypeId="2" ParentId="14213" CreationDate="2018-06-20T14:54:38.580" Score="3" Body="&lt;p&gt;You should start by figuring out what you want students to understand and be able to do with the material that you will teach them.  This includes issues of course content but you should also consider the kinds of exercises and projects that you will assign to students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some questions that you should consider include:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What should the balance between theory and computational practice be?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To what extent do you want to follow a statistical approach?  Classical or Bayesian or both?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How much programming do you want students to do?  With what languages and libraries?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What kinds of inverse problems do you want to deal with?  Inverse scattering?  Inverse heat conduction?  Travel time and attenuation tomography?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What mathematical prerequisites can you assume?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the course that I teach to first year graduate students in science, engineering, and math, the answers to these questions were very clear and determined the course syllabus and the content of the book that’s wrote.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, we assumed that students had background in calculus, ordinary differential equations, and probability and statistics at the undergraduate level.  The students needed a fairly applied course with a strong computational focus because they would mostly be doing those kinds of computations in their own research projects.  We expected them to treat inverse problems as statistical parameter estimation problems and wanted them to see both classical and Bayesian approaches.  Because students might not have background in partial differential equations and almost certainly had no background in functional analysis, we did not deal with inverse scattering and limited our attention to discretized inverse problems.  &lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2018-06-20T14:54:38.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14233" PostTypeId="1" AcceptedAnswerId="14237" CreationDate="2018-06-21T00:44:52.973" Score="6" ViewCount="273" Body="&lt;p&gt;Nowadays, when I look at mathematics programs of study, &quot;algebra&quot; (at the abstract level) and &quot;analysis&quot; are treated as equally important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm &quot;dating&quot; myself, but this did not appear to be true in the 1970s, when I studied math. This was just past the &quot;space&quot; age, and at that time, &quot;analysis&quot; (and its connections with calculus) was all the rage, with &quot;algebra&quot; (abstract algebra, algebraic topology, etc.) being relegated to the &quot;bench.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it less than 40 years that algebra has become the equal of analysis importance? If so, why might that be; perhaps the greater use of &quot;strings&quot; and &quot;arrays&quot; by computers?&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2018-06-23T05:31:44.097" Title="What made (abstract) algebra grow in relative importance?" Tags="|mathematical-analysis|abstract-algebra|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14234" PostTypeId="1" CreationDate="2018-06-21T09:40:58.540" Score="64" ViewCount="9268" Body="&lt;p&gt;This question is primarily discussing maths education for adult learners, on courses teaching engineering mathematics at an undergraduate level. These students generally never set out specifically to learn mathematics, but need to in order to obtain necessary qualifications for their career.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I frequently find that students arrive in my classes with unfortunate preconceptions about mathematics. In particular &quot;I was never good at maths&quot;, &quot;Maths is too complex for me&quot; or &quot;I just don't understand maths&quot;. This notion that mathematics is beyond the ken of mere mortals seems to be a widely accepted cultural phenomenon (How would a movie show the audience that this scientist is a genius? By showing them working on a blackboard filled with impenetrable equations). I've had onlookers walk past my office, and remark how complex my whiteboard looks, when it's normally some simple calculus revision that I was going over with a student. I politely say that it's only complex to them because they haven't seen it before, but I always just get the &quot;Well I was never good at maths&quot; answer. I've even over-heard colleagues teaching more qualitative modules say these exact words to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This preconception is often so pernicious that it is very damaging to their future prospects. Whenever such students are taught some particular method for dealing with a problem, (e.g. transforming an integrand into a different coordinate system) they see it as &quot;this was a technique I never could have seen myself, and therefore this is too hard for me to learn&quot;, rather than &quot;this is the technique that I need to familiarise myself with and learn when to apply&quot;. When they struggle with a problem, as everyone does from time to time, they see it as reinforcing that &quot;they cannot do maths&quot;, rather than seeing it as something that can be overcome through practice and effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although many academic subjects have this problem, I feel that maths suffers from this &quot;mystique&quot; issue more than most. Are there any teaching strategies that are good at trying to break down the &quot;I just can't do maths&quot; barrier?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;: Some comments are raising the issue of &quot;what if the student actually &lt;em&gt;can't&lt;/em&gt; do the maths&quot;? This is a complex secondary issue with a lot of history behind it, that I believe is beyond the scope of this question. I think it is manifestly true that there exist students who have the ability to learn the material, but mistakenly believe they cannot. This question is purely concerned with the best way of helping &lt;em&gt;those&lt;/em&gt; students, and not on adjacent issues.&lt;/p&gt;&#xA;" OwnerUserId="10057" LastEditorUserId="10057" LastEditDate="2018-06-21T20:55:29.423" LastActivityDate="2022-09-28T15:43:54.563" Title="How to get past the &quot;mystique&quot; of Maths" Tags="|mathematical-pedagogy|student-motivation|" AnswerCount="13" CommentCount="20" ContentLicense="CC BY-SA 4.0" />
  <row Id="14235" PostTypeId="2" ParentId="14135" CreationDate="2018-06-21T10:07:10.520" Score="0" Body="&lt;p&gt;Every fraction has two parts, numerator and denomenator. The upper part is the numerator and the lower part is the denomenator where denomenator should never be equal to zero. Every decimal fraction should be converted into numerator and denomenator fraction before applying multiplication or division. The basic rule of multiplication is multiply numerator with numerator and denomenator with denomenator e.g.a/b multiplied with c/d will give you ac/bd. In division, the second number should be inversed. e.g. on dividing a\b by c\d, we  will inverse the second number c\d to d\c and then multiply. So on dividing a\b by c\d, we will multiply a\b with d\c and the result will be ad\bc. Similarly, we can do multiplications and divisions for multiple fractions.&lt;/p&gt;&#xA;" OwnerUserId="10055" LastEditorUserId="10055" LastEditDate="2018-06-25T08:34:11.313" LastActivityDate="2018-06-25T08:34:11.313" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14236" PostTypeId="2" ParentId="14234" CreationDate="2018-06-21T11:51:07.550" Score="34" Body="&lt;p&gt;This is indeed a challenge, especially for adults. Three suggestions, none&#xA;of which is a panacea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Emphasize a &lt;a href=&quot;https://www.mindsetworks.com/science/&quot; rel=&quot;noreferrer&quot;&gt;growth mindset&lt;/a&gt;.&#xA;Make it clear to them that learning math is a skill accessible to everyone,&#xA;with effort. It is not only accessible to those with a&#xA;mythical &quot;math gene.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Compare understanding the abstractions of math and its notations as&#xA;learning a language, or learning to read music.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) Wherever possible, concretize the math, avoiding equations.&#xA;Here is one example I use: &lt;a href=&quot;https://math.stackexchange.com/a/1338716/237&quot;&gt;Cutting out a triangle&lt;/a&gt;.&#xA;See also &lt;a href=&quot;https://matheducators.stackexchange.com/a/10724/511&quot;&gt;Physical vs. Virtual manipulatives in the high school classroom&lt;/a&gt;.&#xA;Exploit the many &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_without_words&quot; rel=&quot;noreferrer&quot;&gt;Proofs without words&lt;/a&gt;&quot; collections.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Added&lt;/em&gt;&lt;/strong&gt;. (3') To quote user @Mars, &quot;then bring the ideas back to the equations.&quot; In other words, use the concretization to help the students&#xA;to grasp the concept, and then use that to walk toward the abstraction&#xA;that generalizes the concrete, usually via symbolic equations. This&#xA;responds to user @Džuris' concern.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-06-25T00:27:29.983" LastActivityDate="2018-06-25T00:27:29.983" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14237" PostTypeId="2" ParentId="14233" CreationDate="2018-06-21T12:57:03.233" Score="2" Body="&lt;p&gt;One guess might be the importance of algebraic logic for, among other things, proving that computer programs behave as desired. Algebraic logic is also quite close to topology --e.g., via category theory-- however this is close to analysis and so the two have a bit of a bridging between them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another reason, also from computing, is the rising trend in functional programming which is essentially executable mathematics and thus supports equational --read algebraic-- reasoning. This form of programming, among others, makes abstract algebra not only concrete but also accessible to a wide range of people with no heavy mathematical training.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the algebraic and analysis notions of monoid, topology, and derivative arise quite naturally as, respectively, systems of composition, notions of computability, and as the one-hole contexts of regular data types. For more on these particular examples, see&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Monoids and their uses in the functional language Haskell: &#xA;&lt;a href=&quot;http://blog.sigfpe.com/2009/01/haskell-monoids-and-their-uses.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://blog.sigfpe.com/2009/01/haskell-monoids-and-their-uses.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Synthetic topology of data types and classical spaces: &lt;a href=&quot;http://www.cs.bham.ac.uk/~mhe/papers/barbados.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.cs.bham.ac.uk/~mhe/papers/barbados.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;The Derivative of a Regular Type is its Type of One-Hole Contexts: &lt;a href=&quot;http://strictlypositive.org/diff.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://strictlypositive.org/diff.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7139" LastActivityDate="2018-06-21T12:57:03.233" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14238" PostTypeId="2" ParentId="14234" CreationDate="2018-06-21T15:43:16.263" Score="17" Body="&lt;p&gt;I think it's something you have to tackle head on on the first day of class.  Some materials I pull from are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ed.stanford.edu/news/myths-about-mathematics&quot; rel=&quot;noreferrer&quot;&gt;Jo Boaler's work on mathematical myths&lt;/a&gt;, such as: Some people are math people and some are not; struggle is a sign of weakness; math is about methods and procedures; you are not good at math unless you are quick at math; math is best learned alone.  &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Something else I learned from Jo Boaler: if you put people in an MRI and show them math problems, the same part of their brains lights up as if you had shown them pictures of snakes and spiders.  So the myths aren't real, but the anxiety is.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://danaernst.com/setting-the-stage/&quot; rel=&quot;noreferrer&quot;&gt;Dana Ernst's “Setting the Stage”&lt;/a&gt; lesson plan, in which he has students discuss in groups what they think it means to learn, and what they want from their education.  It might be different with nontraditional students, but I'm sure there are relevant themes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="446" LastActivityDate="2018-06-21T15:43:16.263" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14239" PostTypeId="2" ParentId="14234" CreationDate="2018-06-21T18:54:15.190" Score="12" Body="&lt;p&gt;As a child, I was subjected to aptitude testing in various settings partly due to my very poor grades in mathematics.  What these tests showed was had high aptitude with regard to understanding math concepts but that my skill level with computation was very poor.  I think it was something like 30th percentile.  I did manage to pass a fair number of college level mathematics courses but I struggled greatly.  I hope that I can provide some insight that will assist you in your worthy goal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your mention of the whiteboard hits home for me.  To this day, when I am faced with any mathematical formula of moderate complexity, I get a anxious feeling in my stomach.  I can labor through these if I am familiar with the symbols but it takes a long time and I find it really unpleasant.  However, a description of the same formula in text is generally quite easy for me to understand.  The difference between the teachers whose classes I enjoyed and the ones I hated was largely related to whether they 'spoke' in symbols.  Often a teacher would fill up an entire board with formulas while I was still trying to make sense of the first one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I find interesting is that in software development, there's a general agreement that one-letter variables or terse names are a bad practice.  But in math a single letter or symbol is used to convey what are often extremely complex concepts or algorithms and practitioners seem fine with that.  I think I understand why that is: time and space concerns.  So one suggestion would be that when teachers are writing formulas they should describe what they mean out loud as they are written.  just writing and saying things like 'and that gives us', 'and that leads to' is extremely frustrating and disheartening to someone who cannot easily read mathematical symbols.  I'm not suggesting that you are doing this but I have had many teachers that do this, especially in higher education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other big thing that helps me a is using pictures.  I took two statistics courses in undergrad and I don't recall being shown a histogram or having anyone even describe what one was.  Just lots of nasty-looking formulas.  I decided I hated statistics.  Then I took a graduate course in statistics and everything was pictures and demonstrations.  I loved it and continue to learn more on my own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my experiences, it seems that there is a general belief that skill in mathematics means the ability to execute calculations flawlessly and the ability to read and understand obscure symbols.  If someone is not good at those two things, they will probably say something like: &quot;I'm not good at math(s).&quot;  You are right in questioning this and I applaud you but you need to accept that what may seem like a simple matter of effort to you may be a life-long struggle to others.  Make the mystical hieroglyphic symbology of mathematics secondary and the conceptual understanding primary and you may get better results.&lt;/p&gt;&#xA;" OwnerUserId="10064" LastEditorUserId="10064" LastEditDate="2018-06-21T19:49:09.563" LastActivityDate="2018-06-21T19:49:09.563" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14240" PostTypeId="2" ParentId="14234" CreationDate="2018-06-22T00:56:15.547" Score="3" Body="&lt;p&gt;Not an educator but couldn't resist replying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always struggled (and still struggle) with math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of years ago, randomly browsing YouTube, I came across &lt;a href=&quot;https://youtu.be/HgNJwg2GISs?list=PLpH1IDQEoE8QWWTnWG5cK4ePCqg9W2608&quot; rel=&quot;nofollow noreferrer&quot;&gt;this home made video&lt;/a&gt; asking how they figured out the distance to the moon before modern technology. The host starts out small scale showing he can calculate the distance to things in his back yard using trigonometry and then scales it up to the moon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My mind was blown, because no one ever told me that. It was simple, anyone could understand it. When I was in school, all I was told was to memorize abstract formulae like calculating the length of sides of triangles based on angles and known length of one side. It was never contextualized to any actual, let alone interesting or fascinating, applications.&lt;/p&gt;&#xA;" OwnerUserId="10067" LastEditorUserId="10078" LastEditDate="2018-06-23T00:08:52.837" LastActivityDate="2018-06-23T00:08:52.837" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14242" PostTypeId="2" ParentId="14234" CreationDate="2018-06-22T06:56:53.523" Score="8" Body="&lt;p&gt;I think that MadScientist described perfectly well the &amp;quot;&lt;em&gt;I just can't do the math&lt;/em&gt;&amp;quot; barrier. It is something that most of us recognized in (a subset of) our students sooner or later.&lt;/p&gt;&#xA;&lt;p&gt;The question is: what kind of teaching is effective in breaking this barrier? That's tough.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Why&lt;/strong&gt;, first of all, the barrier is there? Because &lt;strong&gt;doing math is frustrating&lt;/strong&gt;. In the end your exercise is right or wrong, nothing in between. And if it's wrong, it's frustrating. And at times you do not even know how to start. Nothing of this happens if you're studying history or writing an essay on literature or discussing biology. This frustration often happens very early in your life. It is a common experience that many of the &amp;quot;I am not a math guy&amp;quot; will start you talking about bad experiences in primary school. And, unfortunately, many math teachers at the primary school level are content with landing on you the label &amp;quot;not math guy&amp;quot;. So that many (young) adults who are math blind have a long record of bad feelings associated to math: &lt;strong&gt;frustration&lt;/strong&gt;, &lt;strong&gt;low self esteem&lt;/strong&gt;. No one wants to face a situation in which they feel like they are not &amp;quot;fit to&amp;quot;. Funnily enough, often research mathematicians have more ease in recognizing these feelings than a high school teacher, since in their research work they have to deal on a daily basis with this kind of frustration...&lt;/p&gt;&#xA;&lt;p&gt;How can one reverse this? On a very trivial level by associating math also with positive feelings, i.e. building a basis of successful experiences in math. Building situations in which you will feel proud by having accomplished a task (which is, by the way, how research mathematicians are able to cope with periods of frustration: balancing them with successful moments).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Which&lt;/strong&gt; is the first &lt;strong&gt;obstacle&lt;/strong&gt; about this? While in sociology or history or literature there is clearly a relation between the effort you put in studying and the results you're obtaining (I'd say almost a linear relation), this is not the case with math. Often our students have &lt;strong&gt;no idea about how you study math&lt;/strong&gt;. They'd like to make an effort but how can they do it? They sit in front of the exercise book: they fail the first, they fail the second; if they're brave enough they fail the third, and at this point their level of frustration is high enough for the afternoon. Maybe they'd go for an ice-cream at this point :D.&lt;/p&gt;&#xA;&lt;p&gt;I often compare studying math to learning to ski. I feel that really a small percentage of people is not capable of learning skiing at a decent level (say a full red slope without falling). But imagine this situation. Learning how to ski is compulsory and everyone has to go to a class where a teacher teaches you the principles of good skiing and then you're left alone with your skis on a slope. After you fell 10 times and hurt yourself a bit you start feeling frustrated. After the 100th fall you decide you're not the skiing kind of guy.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;We should tell students that frustration is something they will experience in learning math, just like they will feel goofy on their first vacation on snow. There is nothing bad/wrong about this.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We should tell students that mistakes are not something to blame but something to learn from: if most of your mistakes are because of sloppiness, you have to work on your concentration skills, while if you make big conceptual ones they will signal to you that there is something you've understood incorrectly.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Practice is important. You would not go skiing without some training.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We should tell them how you do study a definition, a theorem, a whole theory. We all know what it means to give a definition. It is not so clear to our students. Definition = set of rules that identifies uniquely all objects that deserve that name. Have your students try to define what is a chair, or what is a car or even what is a student. Use this to show them how tricky a definition can be. Teach your students how you study the proof of a theorem, like identify at all points in a proof which hypothesis is used, check that all hypotheses are used at least once, signal possible &lt;em&gt;hidden hypotheses&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give them a rule on what to do when they feel completely stuck. Like a behavioural psychologist would do with someone in distress.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teach them to work in groups in an effective manner. Learning from peers some times can put less pressure. It may be easier to admit you do not understand something to a friend than to a professor.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Build confidence. They are math guys. They can learn this. It is not their fault if they've not been able to learn math correctly up to now, it is just that they've not found an effective way of working on it. They will do and succeed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;It is difficult&lt;/strong&gt;. The obstacle is mainly psychological. As such each individual has to walk through their very specific path to overcome the barrier. If you teach someone on a 1-1 basis, it is easier. If you teach a class of 20-30, you can do something. If you teach a class of 60, you'd be happy if you succeed in helping one of them.&lt;/p&gt;&#xA;" OwnerUserId="465" LastEditorUserId="60" LastEditDate="2022-09-28T15:43:54.563" LastActivityDate="2022-09-28T15:43:54.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14243" PostTypeId="2" ParentId="14226" CreationDate="2018-06-22T07:58:46.083" Score="2" Body="&lt;p&gt;Almost any board/card game can be converted to play with 8 players by simply &lt;strong&gt;making the children/students play in pairs&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if playing a board game, each pair has one pawn and together they decide where to move on each turn. If playing a card game, one hand of cards between two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Playing in a pair means that students have to discuss their strategies with their partner, making the thinking and problem-solving more explicit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Games that could be useful are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Go Fish&lt;/li&gt;&#xA;&lt;li&gt;Prime Climb&lt;/li&gt;&#xA;&lt;li&gt;Cluedo&lt;/li&gt;&#xA;&lt;li&gt;Any basic board game like Snakes and Ladders but with two dice and the rule that you choose one of the two options to go forward or backward.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2018-06-22T07:58:46.083" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14244" PostTypeId="2" ParentId="14226" CreationDate="2018-06-22T08:40:13.810" Score="1" Body="&lt;p&gt;A very combinatorial game is &lt;a href=&quot;https://en.wikipedia.org/wiki/Ricochet_Robot&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Ricochet Robot&lt;/strong&gt;&lt;/a&gt;, and any number of player can play (from $1$ to $\aleph_0$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting game involving probabilities is &lt;a href=&quot;https://boardgamegeek.com/boardgame/41/cant-stop&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Can't Stop&lt;/strong&gt;&lt;/a&gt;; the number of player is more limited in the commercial edition, but it is not difficult to add more players (although the waiting might be problem). One could make teams, and possibly split the roles (e.g one throws and groups the dice, the other chooses to launch again or not, with or without communication between them).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both have simple rules, suitable for kids, but are interesting for adults too. &lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2018-06-22T08:40:13.810" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14246" PostTypeId="2" ParentId="14234" CreationDate="2018-06-22T18:04:23.987" Score="3" Body="&lt;p&gt;As a mature student in engineering I definitely had a lot of math anxiety at the beginning of my studies. My professor mentioned to us on the first class, and regularly throughout the semester that anyone in the class can do well and that he is there to help us all get the marks we want, be that the 50% required to pass or 100% or anything in between. He also mentioned that when he was a student, he really struggled in his first semester math class and only scored in the 50s. He kept putting the effort in anyway and his marks increased each semester after that. It was very reassuring to have someone be so candid about earlier struggles and to know that you can come out on the other side. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Through a lot of hard work on my part and support from him I managed to get over 100% in both of my math classes last year and have been hired as a tutor by the school for the first year math class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My recommendations for you, based on my professors attitude and methods as well as everything I did in class:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Be open about your math journey. Was a topic difficult for you? How did you get past it?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Share some of your more &quot;unique&quot; study strategies. For example, I am a tactile learner. I find it really helpful to write out the steps to a math problem on flashcards and then tape them to the floor. I then actually walk myself (think hopscotch) through each of the steps to solving the problem. This helps me focus on what I'm doing at that part of the problem and helps prevent me from skipping steps.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Create a positive space in the classroom, where students can ask and answer questions without worrying about being made fun of or looked down on. My professor was really great at this, and would explain questions 6 or more times if necessary if several different people were getting the answer wrong.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Encourage students to work together in groups and to access any tutoring services available on campus. My school has 3 hours of free (!) peer tutoring that each student is eligible for weekly for every semester they are enrolled. There are also special tutorials for first year math classes. The students who took advantage of these options do an average of 14% better in math classes than the students who don't, according to the school's own data.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10080" LastActivityDate="2018-06-22T18:04:23.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14247" PostTypeId="2" ParentId="14193" CreationDate="2018-06-22T20:29:57.150" Score="2" Body="&lt;p&gt;As a math professor myself, I want to point out one of the biggest things students in your position tend to forget: &lt;em&gt;you are not alone&lt;/em&gt;. If you're confused about something, odds are that lots of other students are just as confused. I can't tell you how many times I've taught a class how to do a problem, and then discovered when they turned in the next assignment that eighty percent of the class had missed something crucial. If any one of them had asked for clarification, they would have all gotten the question right. So &lt;em&gt;ask&lt;/em&gt;. Many professors, myself included, will redirect you (&quot;that's a great question; how about we talk about it in my office hours?&quot;) if you start to ask too many questions or questions that the professor judges the other students don't need to hear about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're concerned, though, here's a good rule of thumb: try not to ask the &lt;em&gt;same&lt;/em&gt; question more than twice. &quot;Can you repeat that?&quot; is fine, and &quot;Sorry, I still didn't get it, could you go through it one more time?&quot; is okay, but after that you should probably make a note to yourself about what lost you and then go see the professor in office hours. This is partly because at a certain point class time needs to be spent on the next thing, but there's a more important concern: generally, if a student needs repetition of something more than twice, it's because there's an &lt;em&gt;earlier&lt;/em&gt; thing they aren't understanding. One-on-one, the professor can help you figure out what that was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to be as specific as you can. &quot;Can you repeat that?&quot; is okay if you think you just missed something, but &quot;Can you explain how you got from step 5 to step 6?&quot; or even &quot;where did that $x^2$ in step 6 come from?&quot; is much better. The more specific you can be, the better the professor can address the issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, your instructor might have his or her own opinions about what is and is not a &quot;good question&quot;. But the professor has specifically told you that you should ask questions if you're confused; that means they think, based on their interactions with you outside of class, that your questions are good ones. So go ahead and ask. The worst thing that's likely to happen is that you ask a few annoying questions one day and the professor asks you to back off a little!&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2018-06-22T20:29:57.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14248" PostTypeId="2" ParentId="14233" CreationDate="2018-06-22T23:55:14.987" Score="-1" Body="&lt;p&gt;I think a big area has been computers and crypto and such.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But even before that, I would not underestimate chemistry/physics (molecules with point symmetry affecting IR modes and crystals with space groups and such in x ray crystallography).  Is a very practical application.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sad thing is the math field has so lost track of what benefits are for practical types that it ends up being better to take a group theory for chemists class (even the miserable Cotton book) than deal with normal abstract algebra.  [Same thing is of danger in happening to statistics!]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of course, the people who first founded the field would be at home with practical types more than math theory types.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Whole comment above is a guess, a Bayesian assessment on Friday with a beer.  But still...trust your heart.]&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-22T23:55:14.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14249" PostTypeId="2" ParentId="14234" CreationDate="2018-06-23T00:03:50.973" Score="0" Body="&lt;p&gt;Math is a practical, physical skill.  Like tackling a football player, doing a backflip, tying a tie, riding a bike, etc.  Teach it as such.  Practical...repetitive, etc.  End, end, end of story. These are NOT natural processes like running, talking, loving or recognizing faces.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Humans are not computers.  They are better at certain things than other things. They have a lot of flexibility...so they can perform abstract tasks.  But it is not natural.  So approach it with that in mind.  And DON'T assume that all the kids are budding Ph.D. students destined to teach community college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  Sure there is more to math than that.  But 95%+ of students will benefit from that.  For the 5% who need more or suffer from that, they will claw your arm off for something different.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-23T00:03:50.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14250" PostTypeId="2" ParentId="14234" CreationDate="2018-06-23T01:01:34.557" Score="4" Body="&lt;p&gt;One of the things I find most helpful for teaching adults to math is math history.  Math is known for is precision.  Everything fits together with the precision of a fine Rolls Royce carburetor. (which, if I might say, &lt;em&gt;are&lt;/em&gt; black magic and should be burned at the stake for their heresy against the laws of physics!)  Frankly, it's terrifying to grapple with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like math history for showing how it was hard for others too.  Math did not spring into existence, cleaved from its father's head with a gleaming axe.  Real men and women struggled to get it there.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Have a student that's having trouble with negative numbers?  So did the ancient Greeks.  The Greek mathematician Diophantus even argued that an equation equivalent to &quot;4x+20 = 4&quot; was absurd.  The Chinese were the first to grapple with the concept, and they had the advantage that Chinese philosophy was amenable to the concept.  So yes, it's okay to struggle with them! (taken from &lt;a href=&quot;https://en.wikipedia.org/wiki/Negative_number#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=jfQ9E0u4pLAC&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;A student has trouble with irrational numbers?  Remember &lt;a href=&quot;https://en.wikipedia.org/wiki/Hippasus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hippasus&lt;/a&gt;.  As the story goes, the Greeks were so certain that every number was rational that when he proved that irrational numbers must exist, he was literally thrown overboard.  Okay, so maybe if you follow my link, the connections between Hippasus and drowning is a little less extreme.  But it does demonstrate that these numbers are infuriating enough that such a drowning story could grow and persist!&lt;/li&gt;&#xA;&lt;li&gt;Having trouble with limits?  Zeno had trouble with them too.  His famous paradox argued that we could not possibly move anywhere, despite the obvious observational evidence that we do move.  His argument dealt with infinite sums.  This was not satisfactorily resolved until the invention of limits, which provided a concrete set of tools to manipulate infinite sums in ways which aligned with our observed reality.&lt;/li&gt;&#xA;&lt;li&gt;Admit that we sometimes just make things up, and see if it sticks.  This is a very coarse way of phrasing what Michael Stevens does in his VSauce video: &lt;a href=&quot;https://www.youtube.com/watch?v=SrU9YDoXE88&quot; rel=&quot;nofollow noreferrer&quot;&gt;How To Count Past Infinity&lt;/a&gt; at around the 12 minute mark.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can also look at the attempts made to grapple with these problems, and see where they fell apart.  It's not that math is only for brilliant people, it's that it's really good at letting a thousand people try, and moving forward on the best path any one of them came up with.  I, for one, find it fascinating to look at the attempts of Whitehead and Russel to create a self-consistent proof system, only to have Godel demonstrate that the specific task they were trying to undertake could only possibly fail, with the twist of the knife in the form that the system they created would provide the proof that it must fail. (from my understanding, Whitehead later left mathematics behind and got himself well known for his philosophy)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now about those carburetors...&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2018-06-23T01:01:34.557" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14251" PostTypeId="2" ParentId="14233" CreationDate="2018-06-23T05:31:44.097" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Theoretical Physics: since the 50ies the geometrization of theoretical physics has been costantly increasing. While in 50ies-80ies this meant mostly differential geometry/algebraic topology nowadays this means mostly algebraic geometry/category theory. And for that you need a lot of algebra.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Numerical linear algebra. This field has been progressing enormously. To understand its usage you should be familiar with advanced linear algebra and quite a lot of matrix theory. Applications are countless.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Graph theory. Ubiquitous in math and in applications as a very powerful tool to summarize complex informations. 30 years ago you would probably study it &quot;by hands&quot; while nowadays you have all sort of algebraic tools to apply to it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Nevertheless I tend to agree with the essence of Lee Mosher's comment. All this is very much dependent on where you are and what you look at. While there certainly was an increase in interest on algebra outside math, from the inside it is very difficult to see such a defined path. Personally, for example, I'd say that the biggest success story in last 20 years of math is geometric PDE. But I may be completely wrong.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2018-06-23T05:31:44.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14252" PostTypeId="2" ParentId="14234" CreationDate="2018-06-23T13:32:05.873" Score="5" Body="&lt;p&gt;From my experience in professional training (teaching adults things like accounting, economics and statistics, and so unavoidably using some mathematics), it is not math, it is abstract thinking in general. This is the activity with which many people have a problem.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I strongly support the advice given in another answer: &lt;strong&gt;suggest to the adult students to treat mathematics as a foreign language, and one moreover written in a foreign alphabet. Add to this warning the complexity that this foreign alphabet includes also some familiar symbols, but with a different meaning and functionality that they have in the language they speak.&lt;/strong&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This psychologically mitigates the shock of looking at &quot;unintelligible scribble&quot; on the whiteboard &lt;em&gt;because they anticipate it&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And remind them constantly about this &quot;frame of mind&quot; by saying things like &quot;let's now &lt;em&gt;translate&lt;/em&gt; in the language of mathematics the following real-world situation&quot;, etc. By being able to show them that two different specific situations that they fully understand, have the exact same translation in mathematics, is a concrete step in helping them understand the essence of abstraction.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not everybody is good at learning a foreign language, but those who are, they have a good chance of understanding that there is nothing special about learning to understand mathematics, and go further. In fact one could argue that mathematics is an easier to learn language compared to human languages, because the rules of grammar and syntax of the former is much more rigid and without exceptions compared to the latter.&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2018-06-23T17:44:28.987" LastActivityDate="2018-06-23T17:44:28.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14253" PostTypeId="2" ParentId="14234" CreationDate="2018-06-23T17:03:26.517" Score="1" Body="&lt;p&gt;My experience tutoring algebra 1 students in college led me to believe the best way to get past this issue is to do 1-on-1 sessions with them. I'm a programmer, so diagnosing how a system is malfunctioning is one of my strengths. I would look at the questions they asked, and diagnose what was actually going on in their minds to confuse them, then address those issues on a case-by-case basis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Emphasize how important it is to write every step down, so that you can help them learn to diagnose their own mistakes once they gain more confidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using this method I was able to get several second- and third-timers pass the class. They expressed how amazed they were that it wasn't nearly as hard as they initially thought.&lt;/p&gt;&#xA;" OwnerUserId="1727" LastActivityDate="2018-06-23T17:03:26.517" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14254" PostTypeId="2" ParentId="14234" CreationDate="2018-06-23T19:31:37.970" Score="2" Body="&lt;p&gt;I have experienced this problem, from both sides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Words won't break the barrier.  Action and examples might.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd suggest approaching the problem by answering two questions for yourself, then devising a way to approach the issue that implicitly answers those questions: &quot;Why are you teaching this?&quot; &quot;When would I use this?&quot; &quot;Only mathematicians do that stuff, and I'm not a mathematician.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a child first learning functions, I remember asking my teachers to explain what they were and why. The best answers I ever got were &quot;just because&quot; and &quot;it's a black box that does whatever 'f' is to 'x' to produce 'y'&quot;. I was taught to stop asking those questions; math is just to be trusted and memorized, not understood (I doubt I'm alone). But there are good answers to both what and why. An example: the Sum() function in Excel. It takes in a bunch of numbers, and returns a new number. f(x)=Sum(x)=y.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe not everyone appreciates parallels to programming as much as I do, but the point is that there is a tangible reason why a topic is being taught, but so often educators focus on the mechanics absent the motivation, (or the educators don't really understand what they're teaching well enough to actually explain it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cringe when people suggest that &quot;I can't do math,&quot; just like I cringe when people say &quot;I just can't write well.&quot; Neither are things beyond the means of people who want to learn those things, but those people carry baggage of past failures, and people are afraid of failing, so they don't try. They also think that they've been fine thus far without those skills, so why bother now. I think a practical approach can help here. Math isn't magical, and so much of math is practical, especially for the types of courses I understand you to be describing (eg, an engineering student learning boundary value problems vs a physics student studying abstract algebra).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside, if people are scared off by the Greek, then translate the Greek. If capital sigma is scary, then write it as Sum(X for 23 steps). If delta is scary in a derivative, then remove the shorthand: the change in x for each unit of time. In math, we use symbols for clarity and expressiveness and convention. If the &quot;shorthand&quot; impedes learning, then remove it, and ease it back in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, math is hard to learn because it's harder to teach, not because it's hard.&lt;/p&gt;&#xA;" OwnerUserId="10087" LastActivityDate="2018-06-23T19:31:37.970" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14255" PostTypeId="1" CreationDate="2018-06-24T08:14:40.837" Score="6" ViewCount="163" Body="&lt;p&gt;I would like to make math videos in the manner of &lt;a href=&quot;https://www.youtube.com/watch?v=sVFWrMCfje8&amp;amp;list=PLCgncMh0TrCmrkvT7zW0x32flMJLuYMT5&quot; rel=&quot;nofollow noreferrer&quot;&gt;this YouTube video&lt;/a&gt;. The specific features which I like in this video are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the interactive freestyle drawing (see the beginning of the video)&lt;/li&gt;&#xA;&lt;li&gt;the way different colours are selected smoothly (00:28 into the video)&lt;/li&gt;&#xA;&lt;li&gt;the way pre-written text can appear (01:27)&lt;/li&gt;&#xA;&lt;li&gt;the way existing drawings can be replaced by written text smoothly  (05:37)&lt;/li&gt;&#xA;&lt;li&gt;the support for maths equations&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have tried Microsoft OneNote and Microsoft Powerpoint, which have been mentioned in similar questions here, but:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I understand how to make pre-written text appear in Microsoft PowerPoint, but I cannot see how to allow freehand drawing, or the interactive nature of the presentation&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Conversely, Microsoft OneNote supports freehand drawing, but I am not sure about the text effects I mentioned.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10090" LastEditorUserId="5220" LastEditDate="2018-06-24T17:36:04.427" LastActivityDate="2018-06-24T17:36:04.427" Title="How was this maths educational video made?" Tags="|technology-in-education|software|presentation|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14256" PostTypeId="2" ParentId="14146" CreationDate="2018-06-24T17:41:02.827" Score="4" Body="&lt;p&gt;I don't see why you would need any physics knowledge to get a math degree.  I have a masters and took one physics class that I took purely out of personal interest.  There was no physics involved in any of my mathematics classes.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2018-06-24T17:41:02.827" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14257" PostTypeId="2" ParentId="14234" CreationDate="2018-06-24T21:44:19.480" Score="2" Body="&lt;p&gt;While I'm not a maths teacher, I tutored some GCSE students in afterschool sessions a few years ago and I found the easiest way was to show them some other math concept they understood like long division and explain that while they understood it and knew how simple it was, somebody who had never seen that concept before would find it &quot;too complex&quot; the same way they do about things that are ultimately quite simple.&lt;/p&gt;&#xA;" OwnerUserId="10092" LastActivityDate="2018-06-24T21:44:19.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14258" PostTypeId="1" CreationDate="2018-06-25T02:32:40.060" Score="2" ViewCount="312" Body="&lt;p&gt;I'm looking for a simple tool (preferably online or at least that works on Linux) to draw &lt;a href=&quot;https://en.wikipedia.org/wiki/Stencil_%28numerical_analysis%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;stencils&lt;/a&gt;. When teaching and creating notes for students, I often feel that it would make understanding in some topic much easier if I could draw stencils like the one below. But doing stencils like that by hand (with tools like Gimp of by typing code) can be very cumbersome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any tool specifically designed to make things like this easy?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gfMbo.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gfMbo.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10093" LastEditorUserId="376" LastEditDate="2018-06-25T10:38:40.250" LastActivityDate="2018-08-10T22:04:51.943" Title="Easy tool to draw stencils" Tags="|differential-equations|discrete-math|graphing|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14259" PostTypeId="2" ParentId="7312" CreationDate="2018-06-25T21:24:18.230" Score="2" Body="&lt;p&gt;See the information at the following link: &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/problems-for-journey-through-genius-the-great-theorems-of-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/periodicals/convergence/problems-for-journey-through-genius-the-great-theorems-of-mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Description of link: On the 25th anniversary of the publication of Journey Through Genius, Convergence is pleased to make available to our readers William Dunham’s own problem sets to accompany the book. The 162 problems are divided into 12 problem sets, one for each chapter of the text. &lt;/p&gt;&#xA;" OwnerUserId="10100" LastEditorUserId="401" LastEditDate="2018-06-26T23:05:56.277" LastActivityDate="2018-06-26T23:05:56.277" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14260" PostTypeId="2" ParentId="14234" CreationDate="2018-06-26T07:14:30.997" Score="1" Body="&lt;p&gt;Just to add something to the already very good existing answers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you solve an exercise-like problem in course in front of the students, start with a very intuitive, but &lt;strong&gt;wrong&lt;/strong&gt; approach. Explain your thoughts and why you’re doing what you’re doing, then when it can be seen that the approach fails, explain what went wrong and what to take from it. Then come up with a better approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think doing this could be fruitful, because very often I think that this unbelievably common conception comes from the fact that in class, the professor seems to solve everything instantly and everything is immediately clear to him, and so when they go about solving a problem, they expect the same to happen for them, and when it doesn’t, they are disappointed and give up. They don’t even try. Showing that it is normal, even essential, to just start, and when something is wrong to just restart with a more adapted approach, could led to the students trying more.&lt;/p&gt;&#xA;" OwnerUserId="10102" LastActivityDate="2018-06-26T07:14:30.997" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14261" PostTypeId="1" CreationDate="2018-06-26T13:57:37.060" Score="5" ViewCount="264" Body="&lt;p&gt;I am looking for academic references related to this questions. In fact, I need to write a paper which shows the importance of problem-solving approach for everyone. Certainly, this should include mathematical problem solving, but not only that. In other words, why education needs to focus on the problem-solving approach nowadays.&lt;/p&gt;&#xA;&lt;p&gt;I have been reading some books (see list below) for this purpose.&lt;/p&gt;&#xA;&lt;p&gt;I really appreciate your help in advance.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Robertson, S. I. (2016). &lt;em&gt;Problem solving: perspectives from cognition and neuroscience&lt;/em&gt;. Psychology Press.&lt;/p&gt;&#xA;&lt;p&gt;Sternberg, R. J. (Ed.). (1994). &lt;em&gt;Thinking and problem solving&lt;/em&gt;. San Diego, CA: Academic Press.&lt;/p&gt;&#xA;&lt;p&gt;Schoenfeld, A. H. (1985). Mathematical Problem Solving, Academic Press, Orlando, FL.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1789" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-10-13T02:11:25.343" Title="Why is problem solving important for all of us?" Tags="|education-research|problem-solving|" AnswerCount="1" CommentCount="5" ClosedDate="2019-10-13T11:16:56.470" ContentLicense="CC BY-SA 4.0" />
  <row Id="14262" PostTypeId="2" ParentId="11583" CreationDate="2018-06-27T02:47:08.280" Score="0" Body="&lt;p&gt;One idea not mentioned already is to have students write a computer program to start checking every even number and see how quickly the code runs.  This also requires that the program be able to generate prime numbers, so it's not completely obvious to write.  Are there any tricks that can make the algorithm run faster?  And how does the program slow down over time?  If you graph n vs time to find n = p1 + p2, do you get a logarithmic curve?  Lots to explore here.&lt;/p&gt;&#xA;" OwnerUserId="10105" LastActivityDate="2018-06-27T02:47:08.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14263" PostTypeId="1" CreationDate="2018-06-27T07:05:27.547" Score="13" ViewCount="292" Body="&lt;p&gt;I'm teaching a graduate course in mathematics next semester. I'm planning to have a midterm and a final exam. But I'm thinking about having weekly (or once-every-two-weeks) in-class quizzes instead of homework assignments. My plan (for each week) is to post five problems for the students and one week later randomly select one of the problems for the students to solve during a 10-minute closed-book quiz in class.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've done this type of thing before in first-year calculus and linear algebra classes without issues. Some students reported that they liked the quizzes because it forced them to keep up and internalize the material more than homework does. I agree with them on that point. I also like quizzes over homework because it is more difficult to cheat and that there is less grading to do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is doing quizzes instead of homework a good idea or a bad idea in a graduate course? What are the pros and cons? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: It's a standard first-year lecture-style graduate course. Think complex analysis or abstract algebra or measure theory. I expect 10-20 students, mostly graduate students, but a few senior undergraduates. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: A valid answer could be neither. I'm open to hearing the case for that. &lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2018-06-27T19:29:29.037" LastActivityDate="2018-07-04T15:15:36.873" Title="Quizzes (with questions known in advance) instead of homework in a graduate mathematics class. Good Idea or Bad Idea? Pros and Cons?" Tags="|mathematical-pedagogy|graduate-education|homework|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14264" PostTypeId="2" ParentId="6107" CreationDate="2018-06-27T14:07:20.680" Score="0" Body="&lt;p&gt;If you use $*$ as multiplication modulo 6 you get very strange behavior. The only  non-negative &quot;prime&quot; is 5. All other numbers are composite. Even zero is $2 * 3$. But with only one prime, unique decomposition into primes has no real meaning. Also $3 * 3 = 3$ and $3 * 5$ is also $3$. That isn't a prime decomposition, of course. Some numbers in this situation have no prime decomposition at all. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;You can either interpret this as multiplication over [0..5], or as over the integers with $=$ meaning &quot;equivalent to&quot;. i.e. $15 = 5$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This isn't exactly what you asked for, but can motivate the fundamental theorem in a parallel way. &lt;/p&gt;&#xA;" OwnerUserId="8756" LastActivityDate="2018-06-27T14:07:20.680" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14265" PostTypeId="1" AcceptedAnswerId="14266" CreationDate="2018-06-27T17:11:02.207" Score="4" ViewCount="575" Body="&lt;p&gt;Given a question like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find the coordinates of the minimum point on the curve $y=3x^2+2x+9$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;students are often taught to solve this by completing the square. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The class I am currently teaching this to find completing the square difficult, and I think they would find it much easier to use the quadratic formula to find the roots and then average them to find the $x$ coordinate of the minimum point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is finding minima/maxima of quadratic functions not usually taught like this? What are the advantages of solving these problems by completing the square?&lt;/p&gt;&#xA;" OwnerUserId="8900" LastEditorUserId="8900" LastEditDate="2018-06-27T17:19:33.927" LastActivityDate="2018-06-27T21:36:46.577" Title="Why bother completing the square to find the minimum/maximum of a quadratic function?" Tags="|secondary-education|mathematical-pedagogy|algebra|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14266" PostTypeId="2" ParentId="14265" CreationDate="2018-06-27T17:21:17.997" Score="8" Body="&lt;p&gt;&lt;code&gt;it much easier to use the quadratic formula to find the roots and then average them to find the x coordinate of the minimum point&lt;/code&gt;&#xA;&lt;br/&gt;&#xA;1. It doesn't seem easier at all.&lt;br/&gt;&#xA;2. How did you derive the quadratic formula or is it black magic&lt;br/&gt;&#xA;3. Why does the image of the average of the roots give the extremum?&lt;br/&gt;&#xA;4. Completing the square are usually presented way before the quadratic formula (depends on the country)&lt;br/&gt;&#xA;5. It'll help later when studying circles and conic sections in analytic geometry.&lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2018-06-27T17:21:17.997" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14267" PostTypeId="2" ParentId="14263" CreationDate="2018-06-27T19:42:21.760" Score="4" Body="&lt;p&gt;This is rather rare, but I have had two or three graduate classes that did this (out of an unusually lengthy graduate career spanning several universities). The best use of this for me was in the first semester of a two-semester graduate algebra course that used &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0387905189&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hungerford's book&lt;/a&gt;. I thought it worked extremely well in that case because Hungerford's book (especially Chapter I and Chapter III) contains a lot of fairly simple and short techniques in the text proofs and exercises that one should master before they pile up too much and one gets overwhelmed by all the important and unimportant details. This was a MWF class, and there were at least 2 quizzes each week, except maybe the week of an hour test (there were probably 2 hour tests, and 1 final exam), when at most one quiz was given, and probably none if the hour test was on a Wednesday (since a Monday quiz wouldn’t get handed back to us until Wednesday, too late to help in studying for the hour test, and there would be no new material to base a Friday quiz on).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, this reminds me of when I taught an elementary differential equations course (4 times), when so many methods for solving ODE’s pop up at the beginning --- separable equations, exact equations, homogeneous equations, first order linear equations, integrating factors, Bernoulli equations, etc. I gave a short quiz nearly each class meeting in which they knew which method I would be asking them to use. The more difficult task in which the students had to determine which method (or methods, as often more than one method could be used to solve an equation) to use on a particular differential equation, without being told, was dealt with in the first major test (and the review for that test).&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2018-06-27T19:42:21.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14268" PostTypeId="2" ParentId="14265" CreationDate="2018-06-27T21:36:46.577" Score="1" Body="&lt;p&gt;I don't think that I ever even &lt;em&gt;heard&lt;/em&gt; of completing the square until graduate school when I was forced to teach it along with a number of other semi-archaic things like &lt;a href=&quot;https://en.wikipedia.org/wiki/Synthetic_division&quot; rel=&quot;nofollow noreferrer&quot;&gt;synthetic division&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that you can make a case for completing the square as a technique for deriving the quadratic formula and possibly for answering the question &quot;why do all quadratics have parabola graphs&quot;. Now, once you actually &lt;em&gt;have&lt;/em&gt; those, there may not be much of a case for having them do it by scratch again and again. The quadratic formula really is quite effective; why not use the tools we are given?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question of &quot;why parabolas&quot; is really given by one of the formulas you get along the way:&#xA;$$ ax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4a}$$&#xA;and observing how it tells you that the graph must be a shift and scale of the standard $x^2$ graph. That's worthwhile and you can push the idea along to cubics! There, you have the notion of writing a general cubic in the form of a &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubic_function#Reduction_to_a_depressed_cubic&quot; rel=&quot;nofollow noreferrer&quot;&gt;depressed cubic&lt;/a&gt; by a similar stretch and shift. The new, depressed cubic now looks like&#xA;$$ t^3 +pt+q. $$&#xA;Notice the lack of a quadratic term. You can now figure out what any kind of cubic graph looks like by playing around with $p$ and $q$.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-06-27T21:36:46.577" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14269" PostTypeId="1" AcceptedAnswerId="14270" CreationDate="2018-06-28T05:08:26.753" Score="4" ViewCount="89" Body="&lt;p&gt;This is my first question here, so please correct me if I'm wrong in my way of asking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is suggested directly by the title. In my college institution, for each subject, half of the semester is handled by one lecture, and the next half by another one. The reasoning is that each lecturer needs to have near equal teaching responsibility. If one lecturer only is applied, there might be some lecturers who have lower or higher workloads than others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another reasoning is that this way we can minimize the bias that can happen with just one lecturer. If a lecturer has bias over some students but the other does not, it will somehow even out, rather than just this first lecturer handles the whole semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some negative effects are there of course, such as content bias and assessment related problem, but how do you think the negative or positive effects are affecting the learning and teaching experience? My heart says the negatives are just too much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eventhough this has been the tradition, my math program, if possible, would like to evaluate this and possibly make changes.&lt;/p&gt;&#xA;" OwnerUserId="10108" LastEditorUserId="10108" LastEditDate="2018-06-28T14:02:41.577" LastActivityDate="2018-06-28T14:02:41.577" Title="One course/subject, one class, two lecturers" Tags="|teaching|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14270" PostTypeId="2" ParentId="14269" CreationDate="2018-06-28T12:29:22.313" Score="2" Body="&lt;p&gt;(1)  Definitely sounds like a system optimized for the teacher, not the student. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2)  Somehow the rest of the free world manages to deal with the opposite arrangement and teaching workload varies from semester to semester for individual teachers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Therefore, 3) I suggest you are right.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) To strengthen the argument and help you versus the opposition...come up with (a) examples (e.g. student critiques) to show the damage to student and (b) come up with examples of other schools which just manage with teachers having variability...maybe sample 10 randomly and see how many do what you do (I bet 1 or less).&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-28T12:29:22.313" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14271" PostTypeId="2" ParentId="11583" CreationDate="2018-06-28T12:44:46.070" Score="1" Body="&lt;p&gt;&lt;strong&gt;Too top down and too proofy/hard in places.  Do something more experiential and experimental.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Explain the conjecture, provide some simple examples to begin&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Nope, run an exercise and have students find if they can figure out the sums (small numbers).  Even let group pick a couple small numbers (maybe two digit ones) and let group find sum that works.&lt;/p&gt;&#xA;&lt;p&gt;AT THAT POINT, state the conjecture (not as a famous conjecture, but just the concept) and just let class have a little debate on their own &amp;quot;if we had more time, do we think we could find a counterexample&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;AFTER class debate...have them vote as a group (true/not true).&lt;/p&gt;&#xA;&lt;p&gt;Then wrap after the vote, by giving the name Goldbach, saying when he lived and how long the conjecture has been around...computers checked numbers up to a gazillion (fill in number) and no counterexample found.  BUT nobody can prove the concept (yet) either.  What to do, what to do?  (Class discussion, let them debate if conjectures like this should just be accepted or not...let them as a group debate the practical versus rigorous slant.)&lt;/p&gt;&#xA;&lt;p&gt;THEN tell the story of FLT and how world had given up for centuries of ever finding a proof...and some nut in an attic for 7 years figured it out.  [Play a little clip of Andrew crying or the like from the documentary...it's fair use to steal a snippet.]&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;(For secondary students, having not mentioned whether its proved or not), practice forming an induction argument. Why does it fail? (eg. this post.)&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;No.  Too hard.  Too boring.  EVen for those with good algebra 2 background and having had induction method.  Plus it doesn't get to a proof.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Talk a bit about the twin prime conjecture and how it's related.&amp;quot;  OK.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;(For any age) discuss the difference between showing the conjecture is true for many numbers, and a rigorous proof. (Eg. We know the pythagorean theorem is always true)&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;OK, but bring in earlier.  And don't TELL.  Have class debate the concept.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-06-28T12:44:46.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14272" PostTypeId="2" ParentId="12368" CreationDate="2018-06-28T13:19:33.127" Score="2" Body="&lt;p&gt;&quot;How I can use the two students' mistake constructively?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.teachingchannel.org/videos/class-warm-up-routine&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; warm up routine was shown to me a couple years ago as a way to show common math mistakes in a constructive manner and I used it some of the time (not everyday like the teacher in the video does). &lt;br&gt;&lt;br&gt;&#xA; 1. Post warm up problems that has common misconceptions or easy to fall into traps. &lt;br&gt;&#xA; 2. Have students complete the problems &lt;i&gt;anonymously&lt;/i&gt;&lt;br&gt;&#xA; 3. Collect the answers and display one or two of the wrong answers to show on a document cam &lt;br&gt;&#xA; 4. since the class knows the answer is wrong, have a discussion or Think-Pair-Share about what is wrong in the problem &lt;br&gt;&#xA; 5. Say something along the lines of &quot;Thank you to whoever wrote this, because otherwise we wouldn't have had a chance to have a great class discussion and now everyone who made the same mistake will learn from your submission.&quot;&lt;br&gt;&lt;br&gt;&#xA;Perhaps step 5 is the most important as it makes students feel helpful for making a mistake and there is no penalty for doing so, only helping other who had the same mistake. Additionally, this way there is no confrontation with the students who got it wrong (the teacher approaching them and asking to show their mistake to the class can feel intimidating perhaps).&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2018-06-28T13:19:33.127" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14273" PostTypeId="1" AcceptedAnswerId="14275" CreationDate="2018-06-28T13:23:15.923" Score="7" ViewCount="240" Body="&lt;p&gt;I am doing exercises with middle grade students and looking at their capacities to create arguments for simple number theory conjectures. I want three tasks, and I have two so far: 1. The sum of three consecutive integers is divisible by three. 2. The sum of two odd numbers is an even number. I need one more task that is similar in difficulty, but not too similar that the same warrants would arise. Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="7550" LastActivityDate="2018-06-29T11:22:39.100" Title="Simple Number Theory Task" Tags="|middle-school|" AnswerCount="6" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14274" PostTypeId="2" ParentId="14273" CreationDate="2018-06-28T13:42:14.210" Score="2" Body="&lt;p&gt;The &lt;a href=&quot;https://nrich.maths.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;NRICH&lt;/a&gt; website has several posters that I like to pick interesting problems like this from. Right now, these three are my favorite: &lt;br&gt;&#xA;1. &lt;a href=&quot;https://nrich.maths.org/content/id/6176/NRICH-poster_BigPowers.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;Big Powers&lt;/a&gt; &lt;br&gt;&#xA;2. &lt;a href=&quot;https://nrich.maths.org/content/id/7205/NRICH-poster_Make37.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;Make 37&lt;/a&gt;&lt;br&gt;&#xA;3. &lt;a href=&quot;https://nrich.maths.org/content/id/8013/NRICH-poster_MysticRose.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mystic Rose&lt;/a&gt;&lt;br&gt;&#xA;They have more poster problems for &lt;a href=&quot;https://nrich.maths.org/8114&quot; rel=&quot;nofollow noreferrer&quot;&gt;secondary&lt;/a&gt; and &lt;a href=&quot;https://nrich.maths.org/8113&quot; rel=&quot;nofollow noreferrer&quot;&gt;primary&lt;/a&gt; ages. The problems are generally simple to state and allow students to play around with the math a bit before making conjectures and proving them.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2018-06-28T13:42:14.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14275" PostTypeId="2" ParentId="14273" CreationDate="2018-06-28T13:44:32.687" Score="1" Body="&lt;p&gt;Maybe discuss the twin primes conjecture briefly and then ask them to investigate how many triplet primes of the form $p,p+2,p+4$ there are.  They should be able to prove (using the kinds of arguments you'd use in tasks 1. and 2.) that there is only one such triplet.  I think this both shows them that number theory is pretty deep and that conjecture is part of the natural cycle of doing mathematics.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2018-06-28T13:44:32.687" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14276" PostTypeId="2" ParentId="14273" CreationDate="2018-06-28T14:26:29.067" Score="3" Body="&lt;p&gt;A nice third problem arises by continuing from the second problem to show the power of parity arithmetic for solving problems in integer arithmetic. For example, we can use it to show that large classes of polynomials have no integer roots. Let's consider a simple example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$f(x) = x(x\!+\!1)+2n\!+\!1\,$ has no integer roots since $\,x(x\!+\!1)\,$ is even, so adding $2n\!+\!1$ yields an odd so  $\ne 0;\,$ equivalently $\,f(0)\equiv 1\equiv f(1),\,$ i.e. $f\,$ has no roots mod $\,2,\,$ so no integer roots.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark&lt;/strong&gt; $ $ This idea generalizes widely as follows&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Parity Root Test&lt;/strong&gt; $\ $ A polynomial $\,f(x)\,$ with integer coefficients &#xA;has no integer roots when its $\rm\,\color{#0a0}{constant\,\ coefficient}\,$ and $\,\rm\color{#c00}{coefficient\,\ sum}\,$ are both odd. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proof&lt;/strong&gt; $\ $ If so then $\  \color{#0a0}{f(0)} \equiv  1\equiv \color{#c00}{f(1)}\,\pmod 2,\ $ i.e. $\:f\:$ has no roots in $\,\Bbb Z/2 = $ integers mod $\,2,\,$ therefore $\,f\,$ has no integer roots. $\ $ &lt;strong&gt;QED&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the same way, we can often reduce problems to &quot;smaller&quot;, simpler problems in modular images. Because (ring) homs preserve the ambient algebraic structure, as above, we can often deduce information about the original problems from information gleaned in the simpler modular images. Such problem solving by modular reduction is an algebraic way of &quot;dividing and conquering&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As above, one can often introduce the germ of such ideas in very early grades through simple instances such as parity arithmetic.  When teaching at such early levels I strive to present examples that will provide good intuition for the generalizations that students will encounter in later studies. The Parity Root Test is a model example of such. You can find much further discussion of such in my many &lt;a href=&quot;https://math.stackexchange.com/search?tab=votes&amp;amp;q=parity%20root%20test&quot;&gt;MSE posts on the Parity Root Test.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2018-06-28T14:45:35.610" LastActivityDate="2018-06-28T14:45:35.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14277" PostTypeId="2" ParentId="14263" CreationDate="2018-06-28T17:10:41.200" Score="1" Body="&lt;p&gt;I think this is a great approach.  Just because you are in fancy schmancy grad school doesn't mean basic pedagogy or psychology has changed.  I don't know much about measure theory but for complex analysis or modern algebra, think it would be easy to write questions of appropriate time length that test something relevant (drill problems from the homework).  You go, girl!&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-28T17:10:41.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14278" PostTypeId="1" CreationDate="2018-06-28T18:52:21.203" Score="3" ViewCount="271" Body="&lt;p&gt;Context: I am an assistant professor at a small college in the US.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next semester I am teaching real analysis for the first time, and we are using Steven R. Lay's book.  (It also happens to be the same text that I used at my undergraduate institution.)  I like to write my own homework assignments, and I sometimes draw on other texts for inspiration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That brings me to my question: What are affordable and level-appropriate texts that I could use to supplement Lay's book?  I have my analysis texts from graduate school, but I am aiming for something softer than Baby Rudin.&lt;/p&gt;&#xA;" OwnerUserId="5988" LastEditorUserId="2083" LastEditDate="2018-06-29T06:43:01.217" LastActivityDate="2018-06-29T22:34:48.557" Title="Supplemental text for undergraduate real analysis" Tags="|undergraduate-education|textbooks|course-design|mathematical-analysis|homework|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14279" PostTypeId="2" ParentId="6107" CreationDate="2018-06-28T19:44:31.117" Score="1" Body="&lt;p&gt;In addition to the $\sqrt{-5}$ example, there is also the $R=\mathbb Z[\sqrt{-3}]$ example, which fails to be unique factorization for the &quot;secret&quot; reason that it is not integrally closed: we should really adjoin $\omega={-1+\sqrt{-3}\over 2}$, and then we have a Euclidean ring. A virtue of this example is that the details of non-unique factorization in $R$ can be studied &quot;externally&quot; by using the integral closure. (And that integral closure has finitely-many units, etc., so it is hardly more complicated than $\mathbb Z$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, the $\sqrt{-5}$ example has the virtue that there is no &quot;easy fix&quot; to the non-uniqueness (apart from switching over to factorization into ideals).&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-06-28T19:44:31.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14280" PostTypeId="1" CreationDate="2018-06-28T22:16:56.393" Score="2" ViewCount="73" Body="&lt;p&gt;Several years ago I have read about a study in a US state where standard test scores were used to identify teachers whose students consistently improved far above the average and then film and analyse their teaching for common traits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An important aspect was that these teachers were quite different in many respects. However, quite a few commonalities were identified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would really like a reference for this study if someone recognizes it.&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2018-06-28T22:16:56.393" Title="Reference for study about good teachers in a US state" Tags="|secondary-education|reference-request|usa|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14281" PostTypeId="2" ParentId="14273" CreationDate="2018-06-29T03:28:09.477" Score="0" Body="&lt;p&gt;The divisible by 3 thing (digits addition)?&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-29T03:28:09.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14282" PostTypeId="2" ParentId="12368" CreationDate="2018-06-29T03:33:21.977" Score="1" Body="&lt;p&gt;I love the Ruferd suggestion.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I think that instructors here think too much that students learn from explanation or correction (the way a computer would) versus drill and practice and example and mimicing.  I suggest the human animal is not not a completely rule based system.  Thus methods to correct back inferences should include some sort of feedback method for getting it wrong, versus getting it right, seeing other students get it right, etc.  And being forced to use the proposed general method.  Not just &quot;nice&quot; explanation of why they were wrong.&lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-29T03:33:21.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14283" PostTypeId="2" ParentId="14273" CreationDate="2018-06-29T03:49:20.777" Score="2" Body="&lt;p&gt;I like your question a lot :) and got some new ideas from other answers. So I like to contribute some of mine in the hope they provide useful to others as well. I usually distinguish if problems are  relating to place value system or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;1. Palindromic numbers ($ANNA$-numbers)&lt;/em&gt;. Using two different digits, you can build two different&#xA;$ANNA$-numbers. For example using $7$ and $3$, you can build $7447$&#xA;and $4774$. What can you say (and prove) about their difference?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;2. Mirrors&lt;/em&gt;. Sometimes, the product of a two digit numbers and their mirrors are equal. Like $23\times 64= 32\times 46$. Or $34\times 86 = 43\times 68.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you figure (and prove) a special property of numbers where this is true? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;3. Sum of consecutive numbers&lt;/em&gt;. Actually, your first problem generalizes a lot and allows to equal or more challenging problems, like : What numbers can &lt;em&gt;not&lt;/em&gt; be written as a sum of consecutive numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark-1: Actually, this problem is also on &lt;a href=&quot;https://nrich.maths.org/summingconsecutive/solution&quot; rel=&quot;nofollow noreferrer&quot;&gt;nrich&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;4. Gaps in Prime numbers&lt;/em&gt;. For each pos. integer $n$ there is a sequence of $n$ numbers, none of which is prime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark-1: this is nice to show that the general solution (i.e. following a rule like take $(n+1)!$, $(n+1)!+1$, $(n+1)!+2$, $\ldots$, $(n+1)!+(n+1)$ does not always produce the easiest example possible, as you can see e.g. setting $n=8$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark-1: If your students know Euclids proof that there are infinitely many primes, you can also do the following problem: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prove that for $n\geq 3$, there is at least one prime number between $n$ and $n!$. &lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2018-06-29T03:49:20.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14284" PostTypeId="2" ParentId="14273" CreationDate="2018-06-29T11:22:39.100" Score="1" Body="&lt;p&gt;The product of an even number and an odd number is an even number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The product of two odd numbers is again an odd number.&lt;/p&gt;&#xA;" OwnerUserId="212" LastActivityDate="2018-06-29T11:22:39.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14286" PostTypeId="2" ParentId="14278" CreationDate="2018-06-29T17:42:42.603" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Bartle and Sherbert, Introduction to Real analysis, Wiley&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A. Kumar and S. Kumaresan, A Basic course in Real analysis, CRC Press&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Both these books are at introductory level for Real analysis course. You will find exercises in the second book particularly useful in developing your own exercises.&lt;/p&gt;&#xA;" OwnerUserId="10114" LastActivityDate="2018-06-29T17:42:42.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14287" PostTypeId="2" ParentId="14278" CreationDate="2018-06-29T22:34:48.557" Score="6" Body="&lt;p&gt;&lt;em&gt;Understanding Analysis&lt;/em&gt; by Stephen Abbott is a very gentle and solid introduction to undergraduate analysis. Good exercises, too.&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2018-06-29T22:34:48.557" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14288" PostTypeId="1" AcceptedAnswerId="14293" CreationDate="2018-06-30T04:09:05.453" Score="7" ViewCount="844" Body="&lt;p&gt;&lt;strong&gt;The context&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my country, when the concept of function needs to be introduced in math classes, most teachers will simply talk about $f(x)=c$, $f(x)=ax+b$ and $f(x)=1/x$ (constant, linear and inverse functions). (This is enough for the first year; the following $2$ years deal with second degree polynomials, exponential and logarithmic functions, trigonometric functions, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic properties of functions (domain, range, maximum, etc.) as well as the concept of reciprocal functions ($f^{-1}(x)$) are also briefly discussed.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;However, it is rarely explained what functions truly are. Most students don't understand the usefulness of the &quot;$f(x)$&quot; notation versus &quot;$y$&quot; (for example, $f(x)=ax+b$ versus $y=ax+b$). When asked: &quot;explain what a function is&quot;, most will say &quot;it's a line&quot;, &quot;it's a curve&quot;, &quot;it's a graph&quot; or &quot;it's a formula&quot; (these are not too far from the truth, but not quite right), or worse, will not be able to say &lt;em&gt;anything&lt;/em&gt; about what functions are.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is even a scary amount of students who have studied functions for more than two years and still struggle to explain the concept clearly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Source of the problem&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I believe part of the problem stems from the fact that it is never quite &lt;em&gt;explained&lt;/em&gt; clearly what functions are. For example, it is never mentioned that functions are a rule that associates elements from one set to another. Incidentally, students never develop a real intuition of what functions do; they think &quot;formula&quot; and not &quot;input&quot; $\Rightarrow$ &quot;output&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note: I might be making generalizations here, but it is my perception as the same problem was there years ago when I was in school and is still present today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example of one way to introduce functions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example that I've thought of but I'm sure is not optimal is to present a set:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\mathbb{L}=\{a,b,c,d,...,x,y,z\}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and another set:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\mathbb{E}=\{\text{vowel},\text{consonant}\}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and invent a function $f$ whose job, whose &quot;function&quot;, is to assign each letter to the right category (vowel and consonant).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I can explain many things like why $f(x)=y$ works (because for example, $f(a)=\text{vowel}$ but $f(b)=\text{consonant}$). I can also explain that $\mathbb{L}$ represents the domain and $\mathbb{E}$ the range of the function. However, the example would be bad to explain what $f^{-1}(x)$ is and there is ambiguity if I try to say that $f(x)=y$ because really $f(x)=\text{consonant}$; so it is definitely not a perfect example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do you know of any &lt;strong&gt;simple&lt;/strong&gt; (think &quot;short&quot;), &lt;strong&gt;elegant&lt;/strong&gt;, &lt;strong&gt;unforgettable&lt;/strong&gt; and ideally &quot;&lt;strong&gt;all-encompassing&lt;/strong&gt;&quot; ways to explain the concept of functions before actually starting to teach about real functions, formulas and graphs?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thank you for your time and propositions.&lt;/p&gt;&#xA;" OwnerUserId="9341" LastActivityDate="2018-07-02T19:37:45.057" Title="Simple, elegant ways to teach the idea of what functions are for the first time" Tags="|secondary-education|examples|functions|" AnswerCount="6" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14289" PostTypeId="2" ParentId="14288" CreationDate="2018-06-30T12:17:12.420" Score="0" Body="&lt;p&gt;I would be careful not to generalize from your own situation (good or bad).  Many courses teach the concept of the input/output machine and teach kids multiple ways of thinking of functions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An input/output machine&#xA;An equation of y and x&#xA;An equation of f(x) and x&#xA;A graph&#xA;A graph that has no more than one y for a given x, unlike relationships like circle graphs (more precise)&#xA;Ordered pairs&#xA;Probably some more&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My recommendation is to introduce the concept with the input/output machine.  It is the most intuitive and most related to the word function.  Ordered pairs is probably the least intuitive.  All that said, eventually students should be able to think of a function in several ways.  But pick your battles and the timing of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a lot of math to teach (and sometimes definitions and concepts are clearer in retrospect) so don't think that killing yourself over the definition of the function is the most important thing.  Get the concept across and then start teaching manipulation, graphing, calculus, etc.  &lt;/p&gt;&#xA;" OwnerUserId="9519" LastActivityDate="2018-06-30T12:17:12.420" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14291" PostTypeId="2" ParentId="14288" CreationDate="2018-06-30T17:39:11.013" Score="2" Body="&lt;p&gt;An expectation that there is a simple and elegant way to teach the concept of function seems unrealistic. There is much literature about this.  The APOS (Action-Process-Object-Schema) theory of Dubinsky et al. holds that the student’s concept image of function must evolve and undergo revision before “functions” can be “objects.” There are many references, one of which is &lt;em&gt;APOS Theory: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research&lt;/em&gt;,  by  Ed Dubinsky, Michael A. Mcdonald.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2018-06-30T17:39:11.013" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14292" PostTypeId="1" AcceptedAnswerId="14306" CreationDate="2018-06-30T18:19:13.627" Score="1" ViewCount="125" Body="&lt;p&gt;At my University, we have the optional feature to write a &lt;em&gt;project&lt;/em&gt; like a Bachelor Thesis. This semester have finished and I would like to work in the summer in project like this. So, I'm searching for a topic and I would appriciate your opinion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More specifically, I would like to work on a topic relative with &lt;strong&gt;Number Theory&lt;/strong&gt; (like Diophantine Equations etc) or &lt;em&gt;secondarily&lt;/em&gt; with &lt;strong&gt;Abstract Algebra&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; Could you suggest me some topics accompanied by&#xA;  the corresponding bibliography?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;PS 1: I should inform you that I have been taught Elementary Number Theory, Abstract Algebra (Group &amp;amp; Ring Theory) and some Galois Theory (all in an undergraduate level, please let me know if you want more information about this).&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;PS 2: I hope this post is not off topic. Otherwise inform me and I will delete it.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Thank you in advance.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="10117" LastActivityDate="2018-07-03T12:31:25.250" Title="Subject advice in Number Theory" Tags="|undergraduate-education|undergraduate-research|number-theory|advising|" AnswerCount="2" CommentCount="8" ClosedDate="2018-07-03T13:37:41.140" ContentLicense="CC BY-SA 4.0" />
  <row Id="14293" PostTypeId="2" ParentId="14288" CreationDate="2018-06-30T19:16:04.333" Score="7" Body="&lt;p&gt;I think that the phrase &quot;input-output &lt;em&gt;machine&lt;/em&gt;&quot; strongly suggests that the function must correspond to a formula. If there weren't a formula, how would you describe how the &lt;em&gt;machine&lt;/em&gt; works? It is a machine after all, so when students hear that word they think of the internals of the machine as having to be mechanical and formulaic. So for the sake on not further ingraining the impression that a function has to involve a formula, and so must involve numerical inputs and outputs, I would avoid this concept. Instead, I draw this picture to show students what a function is, and say&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/d/df/Function_color_example_3.svg&quot; alt=&quot;A set theoretic illustration of a function, with arrows pointing from the inputs in one set to the outputs in another set&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A function is something that takes &lt;strong&gt;any&lt;/strong&gt; object from your set of &lt;em&gt;inputs&lt;/em&gt; (point to the oval labelled with X) and will tell you the &lt;em&gt;output&lt;/em&gt; (follow the arrows with your finger to the oval labelled Y) that corresponds to it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And then I would draw this picture for many different examples! The above example inputs a (colored) shape and returns its color. You can draw this picture with your example of the function $\mathbb{L} \to \mathbb{E}$ that classifies a letter as a vowel or consonant. Of course you &lt;em&gt;have to&lt;/em&gt; use the example where your function is going from $\mathbb{R} \to \mathbb{R}$; pick a specific formula for the function, and pick some sample inputs and draw the arrow going from that input to its output. I think that drawing a picture like this, and relating this picture to the more specific case of functions $\mathbb{R}\to\mathbb{R}$ that they typically see is the key to getting students to understand the distinction between a function and &lt;em&gt;the graph of&lt;/em&gt; a function. There are plenty of other wacky examples of functions you can invent just to hammer home the fact they don't have to be based on numbers and formulas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While giving the accompanying explanations to any of this, I would avoid all mathematical jargon. Don't say &lt;em&gt;ordered pair,&lt;/em&gt; or &lt;em&gt;domain&lt;/em&gt;, or &lt;em&gt;codomain,&lt;/em&gt; or &lt;em&gt;range&lt;/em&gt;. Just get the students comfortable with this idea of a function. Saying &lt;em&gt;set&lt;/em&gt; is fine because that word will have non-mathematical meaning to the students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when teaching functions for this first time, I would personally spend maybe only day on this, since usually the curriculum demands you talk about real-valued functions rather quickly. But then at each new concept I teach about functions, like &lt;em&gt;inverse functions&lt;/em&gt;, and the &lt;em&gt;composition of functions&lt;/em&gt;, and &lt;em&gt;one-to-one&lt;/em&gt; and &lt;em&gt;onto&lt;/em&gt; functions if you get to those ideas, I would pull this picture out again and draw what these concepts mean in terms of this picture. There are very nice illustrations of these ideas in terms of this picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few key points/pitfalls regarding showing students this picture:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;That picture has only finitely many things in each oval (set), and when you draw this picture you can only draw finitely many things in your domain and codomain. But since many functions have an infinite domain and codomain, like functions $\mathbb{R} \to \mathbb{R}$, it's probably a good idea to put some ellipses in the domain and codomain ovals and to say explicitly in these examples that although you're drawing only finitely many sample inputs and outputs, there are many many more there.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The bold word &quot;&lt;strong&gt;any&lt;/strong&gt;&quot; in my explanation above is important. A function has to tell you the output of &lt;em&gt;any&lt;/em&gt; possible input, otherwise it's not technically a function. The rigorous way to avoid this is to define a function to be &lt;em&gt;a domain&lt;/em&gt; equipped with a rule for assigning outputs (the domain is included in the definition). But most elementary math textbooks ignore this, like when they present classic exercises such as, &quot;&lt;em&gt;What is the domain of the function $\log(x)$?&lt;/em&gt;&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-06-30T19:21:08.367" LastActivityDate="2018-06-30T19:21:08.367" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14294" PostTypeId="2" ParentId="14292" CreationDate="2018-06-30T22:34:17.823" Score="3" Body="&lt;p&gt;I think it would be a good Bachelor topic to explain, at a high-level,&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich.27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;the MRDP theorem&lt;/a&gt;&#xA;(Matiyasevich–Robinson–Davis–Putnam),&#xA;which settled Hilbert's 10th problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A set of integers is &lt;a href=&quot;https://en.wikipedia.org/wiki/Diophantine_set&quot; rel=&quot;nofollow noreferrer&quot;&gt;Diophantine&lt;/a&gt;&#xA;  if and only if it is computably enumerable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And its many consequences and refinements. E.g., just $11$ integer variables are&#xA;enough to lead to &#xA;undecidable Diophantine equations.&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And there are many open problems&lt;sup&gt;2&lt;/sup&gt; to which you could refer. For example,&#xA;find all integer solutions to&#xA;$$ \binom{x}{2} = \binom{y}{5} \;.$$&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/XF38M.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XF38M.jpg&quot; alt=&quot;JuliaRobinson&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://www.ams.org/notices/200805/&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMS Notices&lt;/a&gt;: Julia Robinson and Hilbert’s Tenth Problem.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; James Jones.&#xA;&quot;Undecidable diophantine equations.&quot;&#xA;&lt;em&gt;Bull. Amer. Math. Soc.&lt;/em&gt; (N.S.).&#xA;Volume 3, Number 2 (1980), 859-862.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt; &lt;a href=&quot;https://www.math.leidenuniv.nl/~evertse/07-workshop-problems.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Some open problems: PDF download&lt;/a&gt;.&#xA;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-06-30T23:57:59.343" LastActivityDate="2018-06-30T23:57:59.343" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14295" PostTypeId="2" ParentId="10017" CreationDate="2018-06-30T23:56:07.160" Score="6" Body="&lt;p&gt;Despite the reputation that mathematics has for &quot;logical order&quot; and such, that assertion is misleading about how to study it, and how to understand it, I think. For one thing, it is &lt;em&gt;not&lt;/em&gt; the case that there is a &quot;true natural logical order&quot;, but only that things are order-&lt;em&gt;able&lt;/em&gt;, in various ways, depending on tastes/prejudices. And, then, still, I doubt that any one of these logical/linear orderings really serves anyone well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A fundamental symptom of a failure of linearizing to accurately reflect the sense of mathematics is that examples, counter-examples, motivations, and applications (within mathematics) cannot (in my experience) be made to naturally fit into any such scheme. Artificially, sure, we can refuse to mention motivations and examples, but that's terrible, not helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, possibly contrary to various myths, studying &quot;logic&quot; is not a prerequisite for much of anything. For that matter, studying from a book/course about &quot;how to do proofs&quot; is a very peculiar thing, often making artificial mountains out of intuitive molehills... leaving the reader with no sense of &lt;em&gt;why&lt;/em&gt; we need &quot;proof&quot; of trivial things, nor how to prove &lt;em&gt;non-trivial&lt;/em&gt; things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, altogether, don't just read one thing at a time. Go back and forth between. And don't feel obliged to do all the exercises, or finish each chapter in every detail before moving on. Keep in mind that textbook ordering is invariably a choice, and is inescapably artificial. Nothing sacred, and not even necessarily/reliably helpful to the reader.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sure, the topics you've mentioned are good... don't think of them as restricting you!&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-06-30T23:56:07.160" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14296" PostTypeId="1" CreationDate="2018-07-01T00:48:43.053" Score="-2" ViewCount="121" Body="&lt;p&gt;Can anybody say me what is differentiation and integration and what is the use of these ? &lt;/p&gt;&#xA;" OwnerUserId="10119" LastActivityDate="2018-07-01T02:17:29.143" Title="What are the uses of calculus in every day life?" Tags="|calculus|" AnswerCount="2" CommentCount="1" ClosedDate="2018-07-01T08:13:28.077" ContentLicense="CC BY-SA 4.0" />
  <row Id="14297" PostTypeId="2" ParentId="14296" CreationDate="2018-07-01T01:06:51.047" Score="1" Body="&lt;p&gt;If you've traveled in an automobile, you've experienced both the integral and derivative&#xA;viscerally. Stepping on the gas pedal accelerates your speed and pushes you&#xA;back into the seat cushion. This acceleration&#xA;is the derivative of your velocity. When you brake, your speed decreases, the derivative is negative, and you are pulled forward in your seat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Traveling 60 km/hr for an hour moves the car 60 km of course. This distance &#xA;traveled is the integral of your velocity. The same works when your speed varies&#xA;over time: the distance covered is the integral of the time-varying velocity &#xA;with respect to time.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-07-01T01:06:51.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14298" PostTypeId="2" ParentId="14296" CreationDate="2018-07-01T02:17:29.143" Score="1" Body="&lt;p&gt;Differentiation has to do with &lt;em&gt;rates&lt;/em&gt;. (What your car's value will be in a few years, how much money you'll have in a bank account with interest rates, how long will it take for your garden's carrots to be ripe, determining hunting quotas (in order to let preys reproduce enough), etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Integration has to do with &lt;em&gt;adding multiple small pieces to know more about the whole&lt;/em&gt;. For example, you can figure out volumes and areas of almost anything you want using integration. It has many applications in physics for example (as shown by Jospeh O'Rourke's example).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue that integration, unlike differentiation, doesn't really have many &lt;em&gt;direct&lt;/em&gt; ways in which it can be used in &lt;em&gt;every day life&lt;/em&gt;. That being said, there are many things you &lt;em&gt;do&lt;/em&gt; use in every day life that couldn't exist if integration (and calculus in general) was not developed (cellphones, computers, engines, etc.).&lt;/p&gt;&#xA;" OwnerUserId="9341" LastActivityDate="2018-07-01T02:17:29.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14299" PostTypeId="2" ParentId="10017" CreationDate="2018-07-01T06:35:45.090" Score="3" Body="&lt;p&gt;Complementing Paul Garrett's advice in his &lt;a href=&quot;https://matheducators.stackexchange.com/a/14295/376&quot;&gt;answer&lt;/a&gt;, you might want to try a book with minimal prerequisites to get more of a feel for PDEs before going into more depth. One option could be Borthwick's &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/3319489348&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Partial Differential equations&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the publisher's &lt;a href=&quot;https://www.springer.com/us/book/9783319489346&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise. &#xA;  Within each section the author creates a narrative that answers the five questions: &lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;What is the scientific problem we are trying to understand?&lt;/li&gt;&#xA;  &lt;li&gt;How do we model that with PDE?&lt;/li&gt;&#xA;  &lt;li&gt;What techniques can we use to analyze the PDE?&lt;/li&gt;&#xA;  &lt;li&gt;How do these techniques apply to this equation?&lt;/li&gt;&#xA;  &lt;li&gt;What information or insight did we obtain by developing and analyzing the PDE?&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspiration for the development of methods.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you do wish to follow a learning path of sorts, then you might try something along the lines of the following: calculus (single variable and multivariable/vector), linear algebra, real analysis, complex analysis, general topology (or at least metric spaces), measure theory &amp;amp; integration, and functional analysis. However, no set-in-stone ordering is intended, as some subjects can be studied in parallel or you can jump back and forth in grasshopper fashion. Along the way, picking up a little knowledge of ODEs would make sense, but no need to make a big detour just to get started on PDEs.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2018-07-18T11:43:52.547" LastActivityDate="2018-07-18T11:43:52.547" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14300" PostTypeId="2" ParentId="14288" CreationDate="2018-07-01T14:30:49.393" Score="-2" Body="&lt;p&gt;Suggestion: Introduce calculator functions. Very concrete. Use function notation to record the results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Examples&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;sin(1) = 0.017 (rounded)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;sin(2) = 0.035&lt;/p&gt;&#xA;&#xA;&lt;p&gt;log(1) = 0&lt;/p&gt;&#xA;&#xA;&lt;p&gt;log(2) = 0.301&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then introduce simple functions like f(x) = 2x.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;f(1)=2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;f(2)=4&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try substitutions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;f(f(1)) = f(2) = 4&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generalize: For any real number x, f(x) is a &lt;em&gt;unique&lt;/em&gt; real number. (Logically, &quot;unique&quot; is redundant here. Added for emphasis.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If x=y, then f(x) = f(y), by substituting in f(x)=f(x).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Converse not necessary true, e.g. g(x) = x^2.  g(-1) = g(1)&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2018-07-01T17:47:51.627" LastActivityDate="2018-07-01T17:47:51.627" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14301" PostTypeId="2" ParentId="14288" CreationDate="2018-07-01T20:05:07.130" Score="2" Body="&lt;p&gt;I'll be provocative and claim that the modern notion of function is not inherently more difficult to learn than other concepts in mathematics. What makes it &lt;em&gt;really&lt;/em&gt; difficult is an additional man-made problem: we don't practice what we preach in front of students! This has historical reasons, and is hard to fight against.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've written about this elsewhere, but I'll summarise: for more than 200 years (since Bernoulli until ~1930), the word function was &lt;strong&gt;&lt;em&gt;not&lt;/em&gt;&lt;/strong&gt; defined as a rule that transforms inputs to outputs. Instead, the official definition involved a &lt;em&gt;relation between two things&lt;/em&gt; $a$ and $b$ and went roughly like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$a$ is a function of $b$ iff the value of $a$ is uniquely determined as soon as the value of $b$ is given.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is still the predominant use of the word function among physicists, engineers and other scientists (including some mathematicians). Think of expressions like &quot;the area of a circle is a function of the radius&quot;, &quot;the temperature of the gas is a function of the pressure&quot; etc. Moreover there is a long tradition (at least since Euler) of dropping the &quot;of ...&quot; when it's clear from the context, and just calling the temperature a function etc. But notice that the area or the temperature are &lt;em&gt;definitely not&lt;/em&gt; rules that transform inputs to outputs. This follows from the fact that the area of a circle is not only a function of the radius but also of the diameter or of the circumference etc. So whatever the $a$ in the original definition is, it's not a function in the modern sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Add to this the very popular notation $y=y(x)$ which goes back to at least 1830 (and was meant to express that $y$ is a function of $x$, not that $y$ is a rule that maps input to outputs) and I think the confusion for students must be quite big. How for example should they interpret something like $v=v(t)=v(h)$? Such confusions are are further supported by notations like $\frac{\partial f}{\partial x_1}$ for the partial derivative of a &quot;modern function&quot; $f:\mathbb{R}^n\to \mathbb{R}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been trying to address these points when I teach calculus to engineers, but I've found that students already arrive with all these contradicting uses of the word function and the notation f(x) in their head. They also constantly see these contradicting uses in their other courses. So it would take me significant more time to get things straight in their heads. Even if I'd manage, I suspect they still would not fully understand what they see in other courses. So I'm rather pessimistic about the effectiveness of the well meant suggestions you have received here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, let me mention that Euler wrote his whole &lt;a href=&quot;https://en.wikipedia.org/wiki/Institutiones_calculi_differentialis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Institutionum calculi differentialis&lt;/a&gt; &lt;em&gt;without&lt;/em&gt; using the $f(x)$ notation (as far as I can tell). So I'll make another provocative claim: the $f(x)$ notation is not more useful than the $y$ notation. At least not for those parts of mathematics which most engineers or natural scientist need.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2018-07-01T20:05:07.130" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14302" PostTypeId="1" AcceptedAnswerId="14312" CreationDate="2018-07-02T19:21:34.747" Score="3" ViewCount="271" Body="&lt;p&gt;I am looking for a nice app that would enable me to create &quot;automatically&quot; nice urns filled with balls of two different colors, following the illustration below (actual colors do not matter, being able to write something on the balls is a plus):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;An urn is filled with 10 balls, 5 blue and 5 red. If you draw a red one, you earn 2 000€, if you draw a blue one, you earn 1 000€&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Jni3d.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Jni3d.png&quot; alt=&quot;An urn filled with blue and red balls&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for an automatic way to do it as I need to create a 100 of those urns, with different probabilities (number of balls of each color) and amounts. Have you ever heard of some kind of app which would do that? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you!!&lt;/p&gt;&#xA;" OwnerUserId="10124" LastActivityDate="2018-07-04T16:34:18.600" Title="Urn (containing colored balls) generator?" Tags="|technology-in-education|probability|online-instruction|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14303" PostTypeId="2" ParentId="14288" CreationDate="2018-07-02T19:37:45.057" Score="0" Body="&lt;p&gt;I use the example of a vending machine to describe the concept of a function, v(b), where &quot;b&quot; is the value of the button that is pressed and v(b) outputs the type of food you receive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A vending machine has clear inputs: Buttons that can be pressed. It has clear outputs: The food that it dispenses. The rule can be summarized by a table: Press this button, get this soda. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can talk about domain and range: Is there a button labeled &quot;X&quot; on this machine? No? So &quot;X&quot; is not in the domain at all, and v(&quot;X&quot;) is undefined. Perhaps the vending machine itself tells you that there is no valid output for button &quot;C&quot;. You can then say that v(&quot;C&quot;) is undefined, even though you could think about trying to press &quot;C&quot;. Can you get coca-cola from this vending machine? Perhaps it's a pepsi machine, so coca-cola is not in the image or range of this function. Is it an invertible function? Perhaps there's actually three different buttons that will give you Pepsi, so it isn't.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2018-07-02T19:37:45.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14304" PostTypeId="2" ParentId="14302" CreationDate="2018-07-03T02:05:30.357" Score="2" Body="&lt;p&gt;This is not that hard to do manually.  I bet it takes you longer to research a program or build one than to just do it by hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make a PowerPoint (or other drawing program that is the urn.  Big enough to hold two stacks of at least 9 balls.  You will have the blue ones always on the left and red ones always on the right.  In separate stacks (not mingled as you showed here).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I assume the urns don't themselves need to follow some distribution (like stat mech microcanonical ensembles).  So write out the urns in terms of number of blue and number of red:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;blue: 0-9&lt;/p&gt;&#xA;&#xA;&lt;p&gt;red: 0-9&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All the combinations of that give 100 choices (you do have one empty set, but if that is a problem just add the two 10-0 combinations and have 101 choices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make PPT drawing objects of 0-9 blue balls (stack them into 3-3).  Use the group command (not critical for blue) and save a slide with each choice on it (in the urn).  Now duplicate each slide 10 times (in slide sorter view with cut and paste command).  You will then have a 100 slide presentation.  When you do the copy do it 10 slides at a time (iow sort of &quot;collated&quot;...not 0 balls 10 times and then 1 ball 10 times, but ten times of 0, 1, 2, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Create the ten different red ball objects (just make ten more slides at end of your presentation or make a separate presentation for the red ball objects.  Here you do need to use the group command for the drawing object, because you will be cutting and pasting the objects into the 100 urns.  Go one at a time and paste each of the ten objects into ten slides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably take you a couple hours max.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boom.  Done.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastEditorUserId="10126" LastEditDate="2018-07-03T02:12:10.837" LastActivityDate="2018-07-03T02:12:10.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14305" PostTypeId="2" ParentId="14146" CreationDate="2018-07-03T03:58:58.047" Score="3" Body="&lt;p&gt;You definitely can do well in mathematics, without knowing (almost) any physics. Some first year mathematics subject, most notably calculus, will have some physical application (spring/circuit) when teaching  differential equations for example, but even here not having the 'real' understanding of the pysical concepts will not hinder your progress in any significant way. &#xA;These are two very different fields, in the sense that they employ different types of thinking.&#xA;A lot of theory from mathematics is applied or is useful in physics, some was developed for that purpose. But again, in a maths department, it is mathematics that is taught. &#xA;Some mathematicians will have a very good grasp of the field and may be working with physicists or on developing theory for physicists. You obvioysly don't need to be working with these: There are many branches of mathematics, and plenty with no obvious link to real world (at least for now).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: I am not aware of any pure mathematical course being taught with applications in physics. It is mathematics that is taught that is likely 'useful' in physics and that is all you are likely to here (rather than have physics taught - unlikely to happen  even in an applied maths class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than believe the above, check it yourself. Universities will have subject lists with brief descriptions of topics covered, together with prerequisits and corequisites. &lt;/p&gt;&#xA;" OwnerUserId="10125" LastEditorUserId="10125" LastEditDate="2018-07-03T04:04:36.417" LastActivityDate="2018-07-03T04:04:36.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14306" PostTypeId="2" ParentId="14292" CreationDate="2018-07-03T12:31:25.250" Score="2" Body="&lt;p&gt;As mentioned in the comments, I used the topic of Continued Fractions when I had a similar paper to write. The applications include: rational approximations to irrational numbers, Diophantine Equations, Pell's Equation, Factoring Algorithms, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of my sources included:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;William Stein: &quot;Elementary Number Theory: Primes, Congruences, and Secrets&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A. Rockett and P. Szusz: &quot;Continued Fractions&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hugh L. Montgomery, Ivan Niven, and Herbert S. Zuckerman: &quot;An&#xA;Introduction to the Theory of Numbers&quot;&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2018-07-03T12:31:25.250" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14307" PostTypeId="2" ParentId="14146" CreationDate="2018-07-03T15:14:02.927" Score="8" Body="&lt;p&gt;I left a comment a while back, but I have been mulling over this (off and on) since then, and feel that I should expand on it.  As a first-term master's student, I made some off-hand comment about not liking applied mathematics.  My advisor cracked down on me pretty hard at the time, and essentially explained that I lacked the mathematical maturity to have that opinion.  You can't dismiss a field until you adequately understand what that field is all about.  I could either take some applied mathematics classes and then make up my mind, or I could shut up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I took a couple of applied classes and learned some nifty stuff.  I still don't really dig on applied mathematics, but I have a better appreciation for applied math than I did before.  I'm glad that I took those classes, and I now feel pretty confident in my opinion that applied mathematics isn't right for me.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Another anecdote:  I have taken a grand total of two physics classes in my life.  I took one semester of a combined precalc / physics class in high school, and I took one quarter of quantum mechanics a little over a year ago (I'm a pure math Ph.D. candidate at the moment, so I still take the occasional class).  By some measures, I have been relatively successful as a mathematician without a strong background in physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there are a lot of places where I can see that a physical intuition would be really, really helpful to have.  My master's advisor had an uncanny ability to know when my computations were wrong after just a second or two.  It turns out that he was doing a lot of quick dimensional analysis in his head, and noticing that the units came out wrong.  This struck me as very odd, since we were working in pure fractal geometry, and there were no units anywhere in any of my computations.  However, he had a physical intuition about what all of those quantities represented, so it was easy for him to check units.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I teach multivariable calculus, I can happily explain Stoke's theorem to my students, and even give a pretty convincing proof.  What I cannot do is adequately explain to my students how to interpret that theorem physically---I have no intuition for what a flux or curl are (I can parrot back what I've been told, but I don't have a good internal model for these ideas).  I don't think that this makes me much less effective as a mathematician (I still work in fractal geometry, after all), but I do think that it makes me somewhat less effective as an instructor in that class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a third example, I am currently working with a colleague to study autocorrelation and diffraction measures associated to certain kinds of sets.  The project is really my colleague's idea---he is studying quasicrystals---but I am a reasonably good analyst, and there are some tricky questions of functional analysis that come up (how &lt;em&gt;do you&lt;/em&gt; take the Fourier transform of the convolution of two measures which may not be finite?).  My colleague has a much stronger background in optics, and has an intuitive grasp for what the answers should be based on his knowledge and experience with experimental data.  Again, my lack of physics knowledge slows me down a bit (though I am currently learning all about optics and diffraction---yay!).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Long story short:  if you have not even completed a bachelor's degree, you are too young and academically immature to have already decided that you abhor physics and don't want to learn it.  You simply have not been exposed to enough physics to hold that opinion.  It would be good for you to take a few classes, and will make your life easier later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, it is entirely &lt;em&gt;possible&lt;/em&gt; to be successful as a mathematician without a strong background in physics.  As I tell my students, I'm a mathematician, not a physician... er... physicalist... uh... &lt;em&gt;physicist&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2018-07-03T15:14:02.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14308" PostTypeId="1" CreationDate="2018-07-03T19:55:45.650" Score="4" ViewCount="605" Body="&lt;p&gt;I'm a high school senior. What can I do to be really interested in mathematics ? I know some of my batch-mates (from meeting them at several math Olympiads) are like insanely good at math (know lots of things from college/higer math, also are really good at solving extremely hard math olympiad problems, for example can solve IMO P3/P6 routinely)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I don't believe that your mathematical skills are genetically predetermined, those guys I know are insanely interested in math. I mean every time I see them they're working on some math problems or exploring on some thing on math they find interesting or checking whether somethings are true or not (i.e always thinking about math from pure interest and not because of fame or glory in competitions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do I become as intersted in mathematics like them ? I am to some extend interested, but not so much insanely interested in math. There are some topics I find interesting (eg: Graph theory), but I don't feel the &quot;spark&quot; or the &quot;urge to check and find out if something is true or not&quot; or the &quot;urge to explore&quot; even while reading books I find interesting (for example, Diestel Graph theory). I also feel my interest is somewhat (not too much, but to some extend) hampered because of obsessing over my scores on various math Olympiads.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks everyone for your advice in advance :)&lt;/p&gt;&#xA;" OwnerUserId="10129" LastActivityDate="2018-08-25T00:40:12.650" Title="How to become really interested in math" Tags="|self-learning|" AnswerCount="1" CommentCount="6" ClosedDate="2018-07-04T03:33:40.817" ContentLicense="CC BY-SA 4.0" />
  <row Id="14309" PostTypeId="2" ParentId="14308" CreationDate="2018-07-04T00:16:51.347" Score="3" Body="&lt;p&gt;I suggest you put aside the Olympiad training problem-collections,&#xA;and put aside textbooks, and read something like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dunham, William. &lt;em&gt;Journey Through Genius: The Great Theorems of Mathematics&lt;/em&gt;. Wiley, 1990.&#xA;  (&lt;a href=&quot;https://www.maa.org/press/maa-reviews/journey-through-genius-the-great-theorems-of-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA review&lt;/a&gt;.)&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/q9r7W.jpg&quot; width=&quot;150&quot; /&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;If you are not captivated by this (or a similar book),&#xA;then you should rethink your priorities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added&lt;/strong&gt; (&lt;em&gt;4Jul2918&lt;/em&gt;). Responding to the OP's request: At a higher level:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Villani, Cédric. &lt;em&gt;Birth of a Theorem: A Mathematical Adventure&lt;/em&gt;. Farrar, Straus and Giroux, 2015.&#xA;  (&lt;a href=&quot;https://www.theguardian.com/books/2015/feb/25/birth-of-a-theorem-mathematical-adventure-cedric-villani-review&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guardian review&lt;/a&gt;.)&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;  &lt;a href=&quot;https://i.stack.imgur.com/wQOlG.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wQOlG.jpg&quot; alt=&quot;Villani&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Added&lt;/em&gt;&lt;/strong&gt; (17Aug2018) in response to @usr0192's apropos comment.&#xA;I've read it and it&#xA;is fantastically well-written, destined (I predict) to become a classic of mathematics exposition.&#xA;It covers the same ground three times, each time assuming a stronger background:&#xA;no calculus, calculus, complex analysis. &#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/SvPyE.jpg&quot; width=&quot;180&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-08-25T00:40:12.650" LastActivityDate="2018-08-25T00:40:12.650" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14310" PostTypeId="1" AcceptedAnswerId="14313" CreationDate="2018-07-04T13:31:18.190" Score="13" ViewCount="642" Body="&lt;p&gt;I am going to be teaching a discrete math class in the fall.  One of the major goals of the course is a solid understanding of the basics of logic: the precise meanings of &quot;and&quot;, &quot;or&quot;, &quot;not&quot;, &quot;implies&quot;, &quot;if and only if&quot;, &quot;there exists&quot;, and &quot;for each&quot;  and how to logically manipulate statements involving these.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often Discrete Math tries to illustrate all of these concepts in the context of set theory, graph theory, combinatorics, elementary number theory, etc.  This has the disadvantage that the student is learning new content at the same time they are trying to master the basic logical ideas.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Logic is ubiquitous in mathematics, so I figure that I should be able to illustrate all of these concepts in a &lt;strong&gt;familiar&lt;/strong&gt; mathematical setting: namely &quot;high school algebra&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example: does the first line imply the second line, does the second line imply the first line, or both ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;x^2 &amp;amp;= 2x \tag 1\\&#xA;x &amp;amp;= 2\tag 2&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am sure that I could come up with &quot;algebra problems&quot; which illustrate all of the concepts I am trying to convey.  For instance, partial fraction decomposition is a fairly logically complex idea (you want to find the only values of some variables (&lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; , etc) which make an equation true &lt;strong&gt;for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/strong&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Does anyone have any textbooks which would fit well with what I am trying to do?  This could either be a book on logic or discrete math which already does what I am proposing, or a book on high school algebra which pays careful attention to logic.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Has this idea been tried before?  In particular, is there any relevant research on this approach?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastEditorUserId="238" LastEditDate="2019-11-20T16:58:17.030" LastActivityDate="2019-11-20T16:58:17.030" Title="Teaching logic through &quot;high school algebra&quot;?" Tags="|algebra|proofs|logic|discrete-math|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14311" PostTypeId="2" ParentId="14263" CreationDate="2018-07-04T15:15:36.873" Score="1" Body="&lt;p&gt;I had a graduate abstract algebra course (part of my minor to a PhD in Computer Science), in which weekly (or each two weeks, it was long ago) we got homework. Usually four quite tough questions (not &quot;exam style&quot;!), and only one (chosen at random each time) was graded. If some question gave too much grief, it was discussed in class after handing homework in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One particular point that stuck was that often it was asked to prove something, and your first reaction was &quot;That can't possibly be true!&quot;. I enjoyed the course immensely.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2018-07-04T15:15:36.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14312" PostTypeId="2" ParentId="14302" CreationDate="2018-07-04T16:34:18.600" Score="2" Body="&lt;p&gt;I whipped up this: &lt;a href=&quot;https://ol-gym-sal.de/tools/urn_creator.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;urn creator&lt;/a&gt;. Maybe it's useful. Feel free to suggest improvements.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2018-07-04T16:34:18.600" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14313" PostTypeId="2" ParentId="14310" CreationDate="2018-07-04T16:53:07.537" Score="7" Body="&lt;p&gt;Obviously, one place to look is in the huge amount of “new math” curriculum material that was written during the late 1950s to early 1970s, but I’ll leave that for you or someone else to search through.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the papers listed below, I looked through a lot of papers that I've collected over the years on elementary logic issues (far more than what’s below) and I’ve singled out some papers that might be worth looking at. I’ve divided these papers into two categories: &lt;strong&gt;(A)&lt;/strong&gt; $7$ papers in which I saw some explicit connection with high school algebra topics being made (identified by italic bold numbers --- &lt;strong&gt;&lt;em&gt;[9]&lt;/em&gt;&lt;/strong&gt;, &lt;strong&gt;&lt;em&gt;[16]&lt;/em&gt;&lt;/strong&gt;, &lt;strong&gt;&lt;em&gt;[18]&lt;/em&gt;&lt;/strong&gt;, &lt;strong&gt;&lt;em&gt;[19]&lt;/em&gt;&lt;/strong&gt;, &lt;strong&gt;&lt;em&gt;[21]&lt;/em&gt;&lt;/strong&gt;, &lt;strong&gt;&lt;em&gt;[23]&lt;/em&gt;&lt;/strong&gt;, &lt;strong&gt;&lt;em&gt;[29]&lt;/em&gt;&lt;/strong&gt;), and &lt;strong&gt;(B)&lt;/strong&gt; the remaining $22$ papers, which seem to have something of possible pedagogical interest for what you’re dealing with, even though they do not appear to make any explicit connection with high school algebra topics. I googled the titles to find the digital versions I’ve given links to, but nearly all of them are behind paywalls. There might be some papers that I’ve given a paywall link to that can be found elsewhere not behind a paywall, but I did not put any effort into looking for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Oliver D. Anderson, &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/0020739880190208?journalCode=tmes20&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;How many Venn diagrams are there?&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;International Journal of Mathematical Education in Science and Technology&lt;/strong&gt; 19 #2 (1988), 299-305.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Oliver D. Anderson, &lt;em&gt;A universal Venn diagram&lt;/em&gt;, &lt;strong&gt;Mathematics and Computer Education&lt;/strong&gt; 22 #2 (Spring 1988), 78-80.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Keith Austin, &lt;em&gt;Set algebra: word proofs and Venn diagrams&lt;/em&gt;, &lt;strong&gt;International Journal of Mathematical Education in Science and Technology&lt;/strong&gt; 18 #3 (1987), 482-484.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Christopher Baltus, &lt;a href=&quot;http://sheilamorr.weebly.com/uploads/5/6/8/0/5680346/frederixon_cs1_articlefortask.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;A truth table on the island of truthtellers and liars&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematics Teacher&lt;/strong&gt; 94 #9 (December 2001), 730-732.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[5]&lt;/strong&gt; Margaret E. Baron, &lt;a href=&quot;https://www.jstor.org/stable/3614533&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;A note on the historical development of logic diagrams: Leibniz, Euler and Venn&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 53 #383 (May 1969), 113-125.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[6]&lt;/strong&gt; John D. Baum, &lt;a href=&quot;https://www.jstor.org/stable/3615252&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;An arithmetic method in symbolic logic&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 56 #396 (May 1972), 91-95.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[7]&lt;/strong&gt; Javad Behboodian, &lt;em&gt;Set identities and set equations&lt;/em&gt;, &lt;strong&gt;International Journal of Mathematical Education in Science and Technology&lt;/strong&gt; 22 #1 (1991), 123-126.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[8]&lt;/strong&gt; Max Black, &lt;a href=&quot;https://www.jstor.org/stable/3607475&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The relevance of mathematical philosophy to the teaching of mathematics&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 22 #249 (May 1938), 149-163.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;[9]&lt;/em&gt;&lt;/strong&gt; Brother T. Brendan, &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1949-8594.1959.tb08254.x&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;A popular fallacy&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;School Science and Mathematics&lt;/strong&gt; 59 #7 (October 1959), 509-513.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[10]&lt;/strong&gt; Arthur H. Copeland, &lt;a href=&quot;https://www.jstor.org/stable/24345108&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The algebra of logic&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Pi Mu Epsilon Journal&lt;/strong&gt; 2 #7 (Fall 1957), 317-323.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[11]&lt;/strong&gt; Carmine DeSanto, &lt;em&gt;A classroom note on Venn diagrams for four and five sets&lt;/em&gt;, &lt;strong&gt;Mathematics and Computer Education&lt;/strong&gt; 18 #2 (Spring 1984), 107-110.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[12]&lt;/strong&gt; Melvin Fitting, &lt;em&gt;Propositional logic using elementary algebra&lt;/em&gt;, &lt;strong&gt;Mathematics and Computer Education&lt;/strong&gt; 16 #3 (Fall 1982), 204-207. [related to Goodstein’s paper]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[13]&lt;/strong&gt; Reuben Louis Goodstein, &lt;em&gt;Solving equations in the algebra of classes&lt;/em&gt;, &lt;strong&gt;Mathematical Spectrum&lt;/strong&gt; 2 #1 (1969-1970), 25-28. [related to Fitting’s paper]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[14]&lt;/strong&gt; David W. Henderson, &lt;a href=&quot;https://www.jstor.org/stable/2311865&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Venn diagrams for more than four classes&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;American Mathematical Monthly&lt;/strong&gt; 70 #4 (April 1963), 424-426.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[15]&lt;/strong&gt; Clive Kelly, &lt;em&gt;Set algebra via Boolean functions&lt;/em&gt;, &lt;strong&gt;International Journal of Mathematical Education in Science and Technology&lt;/strong&gt; 20 #3 (1989), 478-480.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;[16]&lt;/em&gt;&lt;/strong&gt; Ernest Albert Kuehls, &lt;a href=&quot;https://www.jstor.org/stable/2688742&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The truth-value of&lt;/em&gt; $\{\forall, \; \exists, \; P(x,y)\}:$ &lt;em&gt;a graphical approach&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematics Magazine&lt;/strong&gt; 43 #5 (November 1970), 260-261.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[17]&lt;/strong&gt; B. Meltzer, &lt;a href=&quot;https://www.jstor.org/stable/3613605&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mathematics, logic and undecidability&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematics Gazette&lt;/strong&gt; 51 #375 (February 1967), 16-25. [See &lt;a href=&quot;https://www.jstor.org/stable/3614388&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Corrigendum&lt;/em&gt;&lt;/a&gt; in &lt;strong&gt;MG&lt;/strong&gt; 51 #376, May 1967, p. 134.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;[18]&lt;/em&gt;&lt;/strong&gt; Karl Menger, &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1949-8594.1939.tb03989.x&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;On necessary and on sufficient conditions in elementary mathematics&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;School Science and Mathematics&lt;/strong&gt; 39 #7 (October 1939), 631-642.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;[19]&lt;/em&gt;&lt;/strong&gt; Kenneth A. Retzer, &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1949-8594.1984.tb09539.x&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Logic, tradition, truth in algebra, and inferences for geometry&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;School Science and Mathematics&lt;/strong&gt; 84 #3 (March 1984), 181-188.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[20]&lt;/strong&gt; Kenneth A. Retzer, &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1949-8594.1984.tb09564.x&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Proofs with visible inference schemes&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;School Science and Mathematics&lt;/strong&gt; 84 #5 (May-June 1984), 367-376.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;[21]&lt;/em&gt;&lt;/strong&gt; Joseph V. Roberti, &lt;a href=&quot;https://www.jstor.org/stable/27965222&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The indirect method&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematics Teacher&lt;/strong&gt; 80 #1 (January 1987), 41-43.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[22]&lt;/strong&gt; Mariano Rodrigues, &lt;em&gt;Coding Venn diagrams&lt;/em&gt;, &lt;strong&gt;Mathematics and Computer Education&lt;/strong&gt; 15 #1 (Winter 1981), 46-49.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;[23]&lt;/em&gt;&lt;/strong&gt; Myron Frederick Rosskopf and Robert M. Exner, &lt;a href=&quot;https://www.jstor.org/stable/27954897&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Some concepts of logic and their application in elementary mathematics&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematics Teacher&lt;/strong&gt; 48 #5 (May 1955), 290-298.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[24]&lt;/strong&gt; M. Stephanie, &lt;a href=&quot;https://www.jstor.org/stable/27956754&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Venn diagrams&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematics Teacher&lt;/strong&gt; 56 #2 (February 1963), 98-101.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[25]&lt;/strong&gt; Ian Stewart, &lt;a href=&quot;https://www.jstor.org/stable/3615644&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The truth about Venn diagrams&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 60 #411 (March 1976), 47-54.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[26]&lt;/strong&gt; Charles W. Tryon, &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1949-8594.1973.tb08961.x&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The logic of implication&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;School Science and Mathematics&lt;/strong&gt; 73 #1 (January 1973), 56-60.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[27]&lt;/strong&gt; C. H. Wild, &lt;a href=&quot;https://www.jstor.org/stable/3613800&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The &quot;characteristic function&quot; and set theory&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 54 #389 (October 1970), 296-297.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[28]&lt;/strong&gt; Margaret Wiscamb, &lt;a href=&quot;https://www.jstor.org/stable/27958215&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Graphing true-false statements&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematics Teacher&lt;/strong&gt; 62 #7 (November 1969), 553-556.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;[29]&lt;/em&gt;&lt;/strong&gt; John D. Wiseman, &lt;a href=&quot;https://www.jstor.org/stable/27968038&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Complex contrapositives&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematics Teacher&lt;/strong&gt; 58 #4 (April 1965), 323-326.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2018-07-04T16:53:07.537" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14314" PostTypeId="2" ParentId="9940" CreationDate="2018-07-04T17:01:19.243" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;The most well-known organization that teaches algebra (along with physics and data science) using programming is &lt;a href=&quot;http://www.bootstrapworld.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bootstrap&lt;/a&gt;. Its team even involves the university professors that have a number of &lt;a href=&quot;http://www.bootstrapworld.org/impact/&quot; rel=&quot;nofollow noreferrer&quot;&gt;publications&lt;/a&gt; about the Bootstrap teaching approach. It's not implemented in the curriculum at the official level but they organize a lot of professional development workshops for teachers across the United States.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7m2cJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7m2cJ.png&quot; alt=&quot;Bootstrap&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Another popular initiative for teaching math using coding is &lt;a href=&quot;https://www.computerbasedmath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Computer-Based Maths&lt;/a&gt; (CBM). Its founder is Conrad Wolfram, a brother of Stephen Wolfram who created the famous computer algebra system Mathematica. The CBM course in statistics has been run as a pilot study in Estonian schools by the University of Tartu (see the results &lt;a href=&quot;https://www.etis.ee:2345/File/DownloadPublic/416b4efa-e23f-4a5c-a27c-bef0ae1c8600?name=Teachers%20Feedback%20to%20Secondary-Level%20Statistics%20Course%20Innovation%20Computer-Based%20Education%20Pilot%20in%20Estonia.pdf&amp;amp;type=application%2Fpdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). It hasn't become, however, the part of an official curriculum yet.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/niMiW.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/niMiW.png&quot; alt=&quot;CBM&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Finally I wouldn't be able to end this post without mentioning legendary &lt;a href=&quot;https://en.wikipedia.org/wiki/Seymour_Papert&quot; rel=&quot;nofollow noreferrer&quot;&gt;Seymour Papert&lt;/a&gt; and his influential &lt;a href=&quot;http://el.media.mit.edu/logo-foundation/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;LOGO project&lt;/a&gt;. This research project was supported by NSF and its report can be found &lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED038034.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. There is no official curriculum that uses LOGO principles but they have been widely applied by many teachers. Since then the LOGO turtle has become a standard Python module and can even be accessed online using &lt;a href=&quot;https://trinket.io/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trinket&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lMs8t.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lMs8t.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. There is much more action in the universities and, for example, the Brock University has the complete undergraduate program called &lt;a href=&quot;https://brocku.ca/mathematics-science/mathematics/mica-mathematics-integrated-with-computers-and-applications/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics Integrated with Computers and Applications&lt;/a&gt;. It is also supported by their own research studies that can be accessed &lt;a href=&quot;https://link.springer.com/article/10.1007/s40751-015-0009-x&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6306" LastEditorUserId="6306" LastEditDate="2018-07-04T17:11:58.227" LastActivityDate="2018-07-04T17:11:58.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14316" PostTypeId="2" ParentId="9940" CreationDate="2018-07-04T17:15:22.003" Score="-1" Body="&lt;p&gt;In a paradoxical way, I think coding and beginning coding of math (for neophytes) has gone backwards recently.   Read the article &quot;Johnny can't code&quot; by David Brin for reference:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.salon.com/2006/09/14/basic_2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.salon.com/2006/09/14/basic_2/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The calculator push and graphing calculator push has ended up being more show than push (and you could see the signs of it going back to early 80s with NCTM and NSF).  Gotta have some of them thar technology. So we throw gazillions at ipads in the classroom.  And the kids who really get it have no problem picking up technology.  And the ones who don't have same issues.  And in some cases have them cloaked by calculator crutch (check out college confidential posts for example).&lt;/p&gt;&#xA;" OwnerUserId="10132" LastActivityDate="2018-07-04T17:15:22.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14318" PostTypeId="2" ParentId="14310" CreationDate="2018-07-04T17:43:17.813" Score="-1" Body="&lt;p&gt;My advice is to teach basic Boolean stuff, with an EE slant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;FWIW, I have never had a course in this stuff, but at least have the very very basics from Ne Math and from EE survey course (for general engineers).  I was working at McKinsey, where in theory we do &quot;issue trees&quot; but rarely do.  I was told to sketch out a problem once in an issue tree before a partner discussion, did it with and and or gates (abnormal for McK and often causing issues in the though process).  Partner liked it.  So maybe there is use in places other than strict logic or EE or the like.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-07-04T17:43:17.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14319" PostTypeId="2" ParentId="14154" CreationDate="2018-07-05T18:35:42.267" Score="0" Body="&lt;p&gt;The intuitive concept of variability/dispersion around a central tendency, if any, is the (mean) absolute deviation, the distance. Historically, it did not become the primary concept because the absolute value is a difficult operator to manipulate.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, the roundabout way: to eliminate the direction of the deviation, use the square and then sum the squares and average (variance). Then, take the square root of the variance and obtain the standard deviation, in order to go back to units of measurement that are compatible with the real-world phenomenon under study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the variance is the parent of the standard deviation, the latter exists because the former exists. &quot;Teaching the variance&quot; is required for a better comprehension as to what is the standard deviation, but apart from that, it is required if one will face &lt;em&gt;calculating theoretically/empirically the standard deviation&lt;/em&gt; -because inherently, one will then have to unavoidably calculate the variance first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, a lot of times it is the variance that is given as datum for the dispersion, with the (silent) understanding that when one wants to obtain actual measures comparable to reality they will take the square root of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also the estimators that estimate directly the variance have usually better statistical properties than estimators that attempt to estimate directly the standard deviation (this is because of the non-linearity arising from the square root). Estimating the variance of an i.i.d. sample,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\hat \sigma^2 = \frac 1{n-1} \sum_{i=1}^n (x_i-\bar x)^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is an unbiased estimator. Estimating the standard deviation as simply as that is not&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\hat \sigma = \sqrt{\frac 1{n-1} \sum_{i=1}^n (x_i-\bar x)^2}$$&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2018-07-05T18:40:52.577" LastActivityDate="2018-07-05T18:40:52.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14320" PostTypeId="1" CreationDate="2018-07-05T21:39:17.067" Score="9" ViewCount="346" Body="&lt;p&gt;Something my students* often struggle with is how to react on being confronted by 'new things', including functions, notation or definitions for which they are given sufficient definition but with which they are unfamiliar. I would like to get them used to the process a mature mathematician goes through on encountering new things (e.g. 'read the definition carefully, try to find concrete examples/counterexamples'), but of course to do this I need a stockpile of new things to present them with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples of the sort of things I am describing include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Floor and ceiling functions (e.g. '$\lfloor x \rfloor$ is the largest integer $n$ such that $n\leq x$. Sketch the graph of $y=\lfloor x \rfloor$.')&lt;/li&gt;&#xA;&lt;li&gt;Properties of relations (e.g. 'A relation $R$ is &lt;em&gt;transitive&lt;/em&gt; if whenever $aRb$ and $bRc$ it is also true that $aRc$. If $aRb$ whenever $a-b$ is even, show that $R$ is transitive.')&lt;/li&gt;&#xA;&lt;li&gt;New definitions (e.g, 'A value $x\in \mathbb{R}$ is a &lt;em&gt;limit point&lt;/em&gt; of a set $A\subseteq \mathbb{R}$ if for any positive $\varepsilon$ there is an element (distinct from $x$) of $A$ less than $\varepsilon$ away from $x$. What are the limit points of the set $(0,1)$? What about $\mathbb R$, $\mathbb N$, or $\mathbb Q$?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Further suggestions would be very welcome!&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;*My personal context here is helping UK sixth-formers (16–18-year-olds) prepare for university admissions tests/interviews for mathematics, but the question should be more generally applicable.&lt;/p&gt;&#xA;" OwnerUserId="1295" LastEditorUserId="1295" LastEditDate="2018-07-06T11:36:22.460" LastActivityDate="2018-07-24T01:01:35.270" Title="Small 'new things' to confront talented high-schoolers with" Tags="|secondary-education|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14321" PostTypeId="2" ParentId="14320" CreationDate="2018-07-06T02:51:48.537" Score="2" Body="&lt;p&gt;I submit the function $f$ with rule:&#xA;$$f(x) = \begin{cases} 1 &amp;amp; \text{if $x$ is rational} \\&#xA;0 &amp;amp; \text{if $x$ is not rational} \end{cases}. $$&#xA;Graph $y=f(x)$. Does there exist a point at which $f$ is continuous? &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-07-06T02:51:48.537" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14322" PostTypeId="2" ParentId="14320" CreationDate="2018-07-06T04:27:15.690" Score="3" Body="&lt;p&gt;You specified high school students, so I am pretty sure that the following recommendations aren't going to be all that helpful, but I am going to give them anyway.  First, a few books (in what I think is a meaningful order):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sergiy Klymchuk and Susan Staples, &lt;a href=&quot;https://www.maa.org/press/books/paradoxes-and-sophisms-in-calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Paradoxes and Sophisms in Calculus&lt;/em&gt;&lt;/a&gt;.  MAA Press, 2013.  This book is full of surprising results that are true, as well as diabolical results which look true but aren't (my favorite example being DOOM KITTY, a poor kitten/kinetic impactor at the top of a ladder which is sliding down a wall).  This would be good to have in an AP classroom, I think.&lt;/li&gt;&#xA;&lt;li&gt;Lynn Arthur Steen, J. Arthur Seebach, Jr., &lt;a href=&quot;http://store.doverpublications.com/048668735x.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Counterexamples in Topology&lt;/em&gt;&lt;/a&gt;.  Dover, 1978.  The basic content of this book is probably a bit too high level for secondary students, but there are some really gnarly examples which, I think, could be interesting in a high school setting.  It would take work to make it fit in the spirit of the question, though, so it is (as I prefaced above) not a great answer to the question.&lt;/li&gt;&#xA;&lt;li&gt;Bernard R. Gelbaum, John M. H. Olmsted, &lt;a href=&quot;http://store.doverpublications.com/0486428753.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Counterexamples in Analysis&lt;/em&gt;&lt;/a&gt;. Dover, 1962.  This is probably way too high level for a high school audience, but is good for undergrads.  Same caveats as above.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The Cantor set is another example that I think is rich for mining.  Showing that it is non-empty requires some careful thinking, and showing that it is uncountable can be done by a careful reading of the usual diagonalization argument.  One could also introduce a notion of measure and ask how &quot;long&quot; the Cantor set is.  Or even just give students the two functions&#xA;$$\varphi_1(x) = \frac{1}{3}x&#xA;\qquad\text{and}\qquad&#xA;\varphi_2(x) = \frac{1}{3}x + \frac{2}{3}$$&#xA;and ask your students to find $\varphi_j([0,1])$ (while introducing the image of a &lt;em&gt;set&lt;/em&gt; rather than a point).  Then iterate.  And do it again.  Ask them what they get if they do this infinitely many times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another possible source of material might be an introductory undergradate algebra text.  Addition modulo $n$ is interesting; even more interesting is the difference between prime and composite values of $n$.  Small finite groups could also be interesting to play with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I really like to introduce students to things like sawtooth, square, and triangular waves before getting into trigonometry.  These provide interesting examples of periodic functions that are &lt;em&gt;not&lt;/em&gt; trig functions.  They are really good for parsing both the definition of periodicity, and how periods are changed (or not) by elementary transformations (e.g. if $f$ is $T$ periodic, what is the period of $S_a \circ f$ or $f\circ S_a$, where $S_a(x) = ax$?).&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2018-07-06T04:27:15.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14323" PostTypeId="1" AcceptedAnswerId="14333" CreationDate="2018-07-06T04:45:25.960" Score="10" ViewCount="2521" Body="&lt;p&gt;Last semester I had a teacher who let us use any type of information in the exam, for example the course notes, books, solved exercises, etc. The only thing he did not let us use was something electronic like smartphone. Of all the teachers that have given me classes (like 30) this is the first one to evaluate in this way and I do not think I will have it again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why do not all teachers evaluate in this way?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, it served me enough to have these freedoms to be able to pass the course.&lt;/p&gt;&#xA;" OwnerDisplayName="user10026" LastEditorUserId="376" LastEditDate="2018-08-06T10:28:20.570" LastActivityDate="2018-08-06T10:28:20.570" Title="Why don’t all professors let students use notes, books, etc. on exams?" Tags="|undergraduate-education|exams|student-question|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14324" PostTypeId="2" ParentId="14323" CreationDate="2018-07-06T13:35:40.587" Score="4" Body="&lt;p&gt;When you have to solve a problem later, in real life, you will also be allowed to use all available sources of information (even including internet and your coworkers). Therefore it is a good idea to let students use their notes, scripts, books (including dictionaries if their native language is not that of the exam).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, to answer your question, there are some disadvantages. Some students may not afford to buy all helpful material (if, for instance, there are only few copies of a useful book in the library) and the situation easily becomes unpracticable and confusing when students hide behind mountains of material. Too much material may even be a handicap. Further it is obvious that no material must be allowed when there are questions to check pure knowledge and memory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in case of exams where problem solving is the major topic, like it is often the case in mathematics or physics, it is certainly a matter of fairness, when a script or formulary is allowed. I used to allow students to write into their formularies whatever they wanted (if they owned personal copies, not taken from the library) and I allowed them to write one sheet of paper for use in the exam. But it must be handwritten by themselves! I always told this during the very first lesson. The idea behind was that they sit down at home immediately after the lessons and write down what they find important. Of course after some lessons the space was filled, and they had to sum up and to rewrite their sheet. So they wrote many sheets and learnt the important formulas - and often did not need at all the sheets in the exam.&lt;/p&gt;&#xA;" OwnerUserId="8557" LastEditorUserId="8557" LastEditDate="2018-07-06T21:02:38.093" LastActivityDate="2018-07-06T21:02:38.093" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14325" PostTypeId="2" ParentId="14323" CreationDate="2018-07-06T14:54:24.040" Score="13" Body="&lt;p&gt;I think that allowing students to use resources on exams can work well in certain situations, such as when the knowledge they can pull from those resources is peripheral (not of core importance to the course) or terminal (not going to be built upon in the future). However, I think it is very rare for all of the material in a math class to fall into one of these categories. For peripheral knowledge you'd want the instructor to select specific items and include something like a formula sheet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Terminal knowledge is quite rare in mathematics. Math curriculum is generally cumulative. A math course builds on previous courses and sets a foundation for future courses. Suppose for example that I ask you &quot;what is three times five?&quot; and you have to first look up how multiplication works. This will slow you down, but perhaps it won't cripple you completely and you'll be able to answer in a few minutes. But if we take it a step further, suppose you have to look up what the quantities &quot;three&quot; and &quot;five&quot; refer to, how to add quantities, and (eventually) how to multiply quantities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might correctly argue that this example is too extreme, and of course students will internalize these concepts along the way. This is true, but exams say to students &quot;you should have this concept internalized &lt;em&gt;now&lt;/em&gt;&quot;. If you teach material one semester, but let students wait until a future semester to actually internalize it, in practice I think they will forget it and have to learn it all over again.&lt;/p&gt;&#xA;" OwnerUserId="7707" LastActivityDate="2018-07-06T14:54:24.040" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14326" PostTypeId="2" ParentId="14320" CreationDate="2018-07-06T16:59:50.017" Score="2" Body="&lt;p&gt;&lt;strong&gt;Telescopy&lt;/strong&gt; (or &lt;strong&gt;telescopic induction&lt;/strong&gt;) is a nice example. It is a simple, prototypical example of induction that will provide strong intuition for later studies. Telescopic cancellation is usually clearer to visualize  in products (vs. sums), so I start teaching the multiplicative form first, &lt;a href=&quot;https://math.stackexchange.com/a/2042532/242&quot;&gt;e.g. below&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad\qquad\, \displaystyle (x-1)(x+1)(x^{\large 2}\!+1)(x^{\large 4}\!+1)\qquad\! \cdots\qquad (x^{\large 2^{\rm N}}\!+\,1)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad\ \ \ = \ \displaystyle  \frac{\color{#0a0}{x-1}}{\color{#90f}1} \frac{\color{brown}{x^{\large 2}-1}}{\color{#0a0}{x-1}}\frac{\color{royalblue}{x^{\large 4}-1}}{\color{brown}{x^{\large 2}-1}}\frac{\phantom{f(3)}}{\color{royalblue}{x^{\large 4}-1}}\, \cdots\,  \frac{\color{#c00}{\large x^{\large 2^{\rm N}}\!-1}}{\phantom{f(b)}}\frac{x^{\large 2^{\large \rm N+1}}\!-1}{\color{#c00}{x^{\large \rm 2^N}\!-1}} \,=\,  \frac{x^{\large 2^{\rm N+1}}-1}{\color{#90f}1} $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a nice example of additive telescopy, in &lt;a href=&quot;https://math.stackexchange.com/a/1803901/242&quot;&gt;this answer&lt;/a&gt; I show how the &quot;proof by picture&quot; below can be &lt;em&gt;mechanically discovered&lt;/em&gt; and rigorized by applying a  2-dimensional form of telescopy - which allows us to view a sequence of rectangles as being built-up layer-by-layer from successive differences of prior rectangles - as if they were built by a 2-D printer in FlatLand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \large  \color{PaleVioletRed}1 + \color{DarkViolet}5 + \color{DodgerBlue}9 + \dots + \color{LightCoral}{(4n-3)}\, =\, n(2n-1) = 2n^2-n$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/aR18c.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aR18c.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many more examples - both simple and complex - are in my &lt;a href=&quot;https://math.stackexchange.com/search?page=1&amp;amp;tab=Newest&amp;amp;q=user%3a242%20telescopy&quot;&gt;diverse MSE posts on telescopy.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2018-07-06T17:05:24.383" LastActivityDate="2018-07-06T17:05:24.383" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14327" PostTypeId="2" ParentId="14323" CreationDate="2018-07-06T21:26:19.930" Score="12" Body="&lt;p&gt;I think it is usually counterproductive.  Instead of building skills, it rewards students who are good at information retrieval.  And I say that as one of those advantaged.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still remember a P-chem course where prof said he would do this AND he also curved the course.  I ended up with highest grade, but did not &quot;master&quot; the course as I had calculus, freshman chem, etc.  Would have gotten much more benefit from the opposite policy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And...I am 50+ and routinely work with things where I used the Internet, books, etc.  So this is not something I don't grok.  But to really master a subject, the memory and recall in a test situation is much more powerful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just a contrary view to others, but hope it has some additive insights.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-07-06T21:26:19.930" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14328" PostTypeId="2" ParentId="14323" CreationDate="2018-07-07T10:27:05.767" Score="11" Body="&lt;p&gt;I allow students to have one index card of notes during their exams.  I believe this allows me to test for understanding and skills that are more important than the ability to memorize.  I also believe that it reduces student anxiety and creates a level playing field (instead of having only a few students using &quot;cheat sheets,&quot; all of them are given this advantage).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think, however, that allowing students to use a lot more resources (such as books and pages of notes) during exams is counterproductive.  The first time I taught, that was what I did, and I quickly learned that this discouraged students from studying.  They believed that they did not need to study before the exam because they could study during the exam.  A lot of students failed that exam, but, more importantly, a lot of students didn't learn.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-07-07T10:27:05.767" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14329" PostTypeId="2" ParentId="5612" CreationDate="2018-07-07T18:14:21.753" Score="0" Body="&lt;p&gt;I like method B, but go ahead and call people to the blackboard versus taking volunteers.  Of course if they say &quot;not prepared&quot;, just go to the next person.  But they should at least feel some mild social pressure from being put on the spot.  Sometimes you will have kids like me that try to brave it out and figure it out on the fly and make a hash out of it...but not enough to derail the class.  And it can be fun to watch them flail.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-07-07T18:14:21.753" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14330" PostTypeId="2" ParentId="7907" CreationDate="2018-07-07T19:11:31.613" Score="0" Body="&lt;p&gt;I don't think this is really needed/optimal for the following rationale.  In general, you are covering a few select topics and student competency varies on those topics.  The bell curve will sort itself out and the general theme of someone getting 70% of the easy right and 30% of the hard right will be more common than the opposite: students getting all difficulty at easy, zero at harder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead I would just think of the exam construction this way.  Normal books for classes from grade school through diffyQs tend to have problems at the I/II/III levels.  I is complete plug and chug.  II is slightly harder, some manipulation.  III can be challenging (at different levels of challenging but still not as easy).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would just look at all the topics you covered (say 5 lessons from a MWF class, 2 weeks with test on last F).  Design a 50 minute exam to cover the 5 topics.  It is probable 5 II problems (one for each section) as well as a smattering of I problems, perhaps to hit subtopics in the lessons, to the extent time allows.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to assign one III problem, do it at the end of the exam (on any of the topics).  The purpose of that is mostly student differentiation (B versus A or +/- shades).  It can be required or extra credit as you think best (if timing tight or students struggling to reach reasonable percentage of 80%/90% cuttofs, make it extra credit).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think it is reasonable to give different points for the II and I problems.  Just do it based on rough time required.  As a ballpark, 2:1, but use your judgment and keep it simple.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The III problem can probably be 3 in 3:2:1. (15, 10, 5 is easy point math for students under the gun.) Given how this drives the % cuttoffs, it is another argument for making that problem EC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[When I say I/II/III, I mean probably very close analogs of the problems, but you can even consider to include unassigned HW problems that are right out of the book, especially if the drill materials are lengthy.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Avoid the temptation to get cute and assign problems different in form from the homework or requiring a lot of synthesis of various skills.  Test what was in the lessons!  That is hard enough, usually and the students have accomplished a lot if they master that. If you can't resist the temptation to get cute, creative, and show how smart you are, then do it with the III problem at the end (and probably DO make it extra credit).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of A/B/C differentiation, it will handle itself fine accross the questions.  Getting 90% right of a bunch of II problems with a few Is mixed in is non trivial in a tested 50 minute environment, with no reference materials.  Kid probably deserves an A for that.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastEditorUserId="10126" LastEditDate="2018-07-07T19:20:14.350" LastActivityDate="2018-07-07T19:20:14.350" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14332" PostTypeId="1" CreationDate="2018-07-08T15:00:21.713" Score="7" ViewCount="561" Body="&lt;p&gt;At least in my country, the explanation of the basic operations over rational numbers is done very near to the concept of prime numbers, prime factorization, and the calculation of the least common multiple. In fact, usually l.c.m is explained &lt;strong&gt;previous&lt;/strong&gt; to fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a real and typical example of text:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Addition and subtraction of fractions with same denominator: add or subtract the numerators and keep the denominators [... some examples ...]. To add or subtract fractions of different denominator: convert all fractions to their common denominator, by obtaining the l.c.m. of all denominators [...]&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can see in the previous text how the concept of common denominator is based on the concept of l.c.m., that easily one of them replaces the other. The drawback of this approach is that students lose the basic concept of the operations in a mix of concepts and a nightmare of l.c.m algorithm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is it done in this way in most other countries ? &lt;/li&gt;&#xA;&lt;li&gt;What are the advantages and disadvantages of discarding the concept and mechanics of l.c.m.  until more advanced learning stages?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7441" LastEditorUserId="238" LastEditDate="2019-10-09T19:54:22.097" LastActivityDate="2019-10-09T23:40:27.450" Title="On fractions and the least common multiple" Tags="|mathematical-pedagogy|fractions|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="14333" PostTypeId="2" ParentId="14323" CreationDate="2018-07-08T15:38:10.587" Score="4" Body="&lt;p&gt;An exam is a totally artificial scenario. In real life you will never be in a scenario with all these fact at same time: a) you can not consult external resources or colleagues; b) you have limited time; c) you do not known the activity before to start it. As conclusion, exams should be deprecated in all forms, with or without consultation of external resources. They exists only because they are a fast and practical way to obtain a numerical evaluation for a big (massive) group of students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Starting from this point and returning to the question about usage of external sources of information, we can analyze some different scenarios:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) if a concept is basic, could be it is already acquired before the exam (not the same than memorized) or student will fail the test, because he/she has no time to acquire the concept and use it only during the time allowed for the exam. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) thus, to have or not to have access to external sources of information is only a difference in the case of specific items of data information, like some particular formulas, taxonomies or algorithms. In these cases, allow notes or books has only advantages. In fact, if something can be successfully retrieved during the exam it means &lt;strong&gt;it is not knowledge but data&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) allow notes minimizes some of the usual drawbacks of the (artificial method) exams: when anxiety causes a fail in the memorization of a data term; a question that fails into a paragraph that has been ignored as unimportant; it made useless hidden notes that benefits only cheaters; ... .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;d) if the student has not acquired confidence in its book, course notes and/or write own abstracts, the presence of them during the exams will not help. Thus, allow these resources encourages the necessity of write abstracts, organize sources, ... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One classical example is the periodic table in chemicals. It is the perfect example of a resource to allow, not to memorize, and instead use this time to most conceptual activities (in my country, students loss months memorizing it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In conclusion: skip usage of exams; if exams are a must, allow books, notes, even intranet.&lt;/p&gt;&#xA;" OwnerUserId="7441" LastEditorUserId="7441" LastEditDate="2018-08-04T09:58:37.207" LastActivityDate="2018-08-04T09:58:37.207" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14334" PostTypeId="2" ParentId="14323" CreationDate="2018-07-09T08:49:43.197" Score="2" Body="&lt;p&gt;I believe the choice is based on the expectations: what does the teacher expect the student to be able to do with the course content?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If the student needs to show that (s)he is able to use the course content to do something which is not stated in the course, then let the student use all the course material.&lt;/li&gt;&#xA;&lt;li&gt;If the student needs to show that (s)he knows the course material by heart, then it obviously makes no sense to allow the course material during the exam.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As most of the course are of the second kind, in most cases course material is not to be allowed during an exam.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2018-07-09T08:49:43.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14335" PostTypeId="1" AcceptedAnswerId="14341" CreationDate="2018-07-09T11:07:55.030" Score="8" ViewCount="466" Body="&lt;p&gt;One of the most important ideas of calculus is&#xA;$$&#xA;f(x) \approx f(x_0) + f'(x_0)(x - x_0).&#xA;$$&#xA;The approximation is good when $x$ is close to $x_0$. This approximation is very useful because the complicated, nonlinear function $f$ is approximated well by the very simple function on the right (at least when $x$ is near $x_0$), and we can use this approximation to simplify many calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When teaching freshman calculus, I would really like to call the function $L(x) = f(x_0) + f'(x_0)(x - x_0)$ something like &quot;the linear approximation to $f$ near $x_0$&quot;. I am afraid to use the word &quot;affine&quot; at this stage, because psychologically the word &quot;affine&quot; may sound unfamiliar and intimidating, and does not convey the fact that the graph of $L$ is a straight line and that $L$ is a very simple, easy, friendly and non-intimidating function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am considering using the term &quot;linear&quot; in this sense when teaching calculus, but emphasizing that I am using the &quot;high school&quot; definition of a linear function, and that in more advanced math (such as linear algebra) students will find that the word &quot;linear&quot; means something slightly different, and that in a more advanced class $L$ would be called &quot;affine&quot; rather than &quot;linear&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you think is the best way to handle this issue? Do you use a term such as &quot;local linear approximation&quot; to $f$ when teaching calculus?&lt;/p&gt;&#xA;" OwnerUserId="9633" LastActivityDate="2018-11-16T19:18:37.243" Title="Should I describe the function $x \mapsto f(x_0) + f'(x_0)(x - x_0)$ as &quot;linear&quot; in a freshman calculus class?" Tags="|calculus|terminology|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14336" PostTypeId="2" ParentId="14332" CreationDate="2018-07-09T13:07:16.170" Score="6" Body="&lt;p&gt;Hung-Hsi Wu seems to agree with you in &lt;a href=&quot;https://math.berkeley.edu/~wu/fractions2.pdf&quot; rel=&quot;noreferrer&quot;&gt;Some remarks on the teaching of fractions in elementary school&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The worst case is the rule of adding two fractions. In book after book (with very few exceptions, such as Lang (1988)), $\frac{a}{b}$ + $\frac{c}{d}$ is defined as $(pa + cq)/m$, where $m = \mathrm{lc}m\{b, d\}$ and $m = bp = dq$. Now at least two things are wrong with this definition. First, it turns off many students because they cannot differentiate between lcm and gcd. This definition therefore sets up an entirely unnecessary roadblock in students’ path of learning. Second, from a mathematical point of view, this definition is seriously flawed because it tacitly implies that without the concept of the lcm of two integers, fractions cannot be added.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Typos present in the PDF from Wu's site have been corrected in the above quote.)&lt;/p&gt;&#xA;" OwnerDisplayName="user5114" LastEditorDisplayName="user5114" LastEditDate="2018-07-09T18:00:13.243" LastActivityDate="2018-07-09T18:00:13.243" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14337" PostTypeId="2" ParentId="14323" CreationDate="2018-07-09T17:01:57.190" Score="15" Body="&lt;p&gt;Making a course easier to pass is not the same as making a good course. There is a certain corpus of knowledge with which practitioners are expected to have instant recall (a.k.a: &quot;automaticity&quot;). If one were to be looking up all of these basic facts constantly, one would not be able to accomplish anything practical. These basic topics are the ones mostly likely tested by standard college exams. Cognitive scientist Daniel Willingham writes (&lt;a href=&quot;https://www.aft.org/periodical/american-educator/spring-2004/ask-cognitive-scientist&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Automaticity is vital in education because it allows us to become more&#xA;  skillful in mental tasks. An effective writer knows the rules of&#xA;  grammar and usage to the point of automaticity—and knows automatically&#xA;  to begin a paragraph with a topic sentence, include relevant detail,&#xA;  etc. The effective mathematician invokes important math facts and&#xA;  procedures automatically. Readers who are able to visualize a map of&#xA;  the world will find various books and assignments easier to read (and&#xA;  learn more from them). In each field, certain procedures are used&#xA;  again and again. Those procedures must be learned to the point of&#xA;  automaticity so that they no longer consume working memory space. Only&#xA;  then will the student be able to bypass the bottleneck imposed by&#xA;  working memory and move on to higher levels of competence.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Secondarily (as stated in a comment), &quot;all of my classmates approved this course, so it was a good idea to do this&quot; is not a valid justification. There is evidence that given the choice between different pedagogical strategies, students tend to choose the strategy from which they learn the least. See Clark, &quot;Antagonism between Achievement and Enjoyment in ATI Studies&quot;, Educational Psychologist 17, no. 2 (1982)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-08-06T10:17:20.703" LastActivityDate="2018-08-06T10:17:20.703" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14338" PostTypeId="1" CreationDate="2018-07-10T00:00:19.173" Score="4" ViewCount="135" Body="&lt;p&gt;From: Philip Johnson-Laird &lt;a href=&quot;https://dof.princeton.edu/about/clerk-faculty/emeritus/philip-nicholas-johnson-laird&quot; rel=&quot;nofollow noreferrer&quot;&gt;BA PhD Psychology (UCL)&lt;/a&gt;, Stuart Professor of Psychology Emeritus at Princeton.    (Author isn't  a logician.)    &lt;a href=&quot;https://www.amazon.ca/How-We-Reason-Philip-Johnson-Laird/dp/0199551332&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How We Reason  (1st edn 2008)&lt;/em&gt;&lt;/a&gt;. p. 112.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;  You might suppose that logic does at least underlie the mental abilities of&#xA;  mathematicians and logicians. Some of them have thought so too, but the&#xA;  disciplines demand more than just deduction. The Indian genius Srinivasa&#xA;  Ramanujan, who died in 1920, left behind a body of work that continues to&#xA;  occupy mathematicians, but proof was not crucial to his many conjectures. As&#xA;  another great mathematician, G.H. Hardy wrote about him, “[he] combined&#xA;  a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his&#xA;  own peculiar field, without a rival in his day.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is this  psychologist correct that mathematical discoveries demand more than deduction? If yes:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What's the term for this creativity?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How can undergraduate students learn it? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2018-07-11T00:32:53.193" Title="How can students self-learn creativity in math that &quot;demand more than just deduction&quot;?" Tags="|self-learning|" AnswerCount="1" CommentCount="6" ClosedDate="2018-07-12T19:53:48.433" ContentLicense="CC BY-SA 4.0" />
  <row Id="14339" PostTypeId="2" ParentId="14335" CreationDate="2018-07-10T00:37:15.357" Score="0" Body="&lt;p&gt;The original post's formula specifies a &quot;tangent line&quot;.  In other words, the line that is tangent to the function f at the point (x, f(x)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students are familiar with the term &quot;tangent&quot; from their geometry classes:  in particular, lines that are tangent to circles.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2018-07-10T00:37:15.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14341" PostTypeId="2" ParentId="14335" CreationDate="2018-07-10T10:45:30.000" Score="14" Body="&lt;p&gt;In calculus one calls $x \to ax + b$ a &lt;em&gt;linear function&lt;/em&gt;. In linear algebra one calls $x \to ax + b$ an &lt;em&gt;affine transformation&lt;/em&gt;, and says that it is &lt;em&gt;linear&lt;/em&gt; only if $b = 0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first order Taylor approximation to a differentiable map is reasonably called the &lt;em&gt;linear approximation&lt;/em&gt; of the map. The terminology is reasonable because one is doing function theory, not linear algebra, and the meaning of the qualifier &lt;em&gt;linear&lt;/em&gt; is context dependent. A formal way of giving content to the word &quot;context&quot; is to speak in terms of categories. Said more formally, what &lt;em&gt;linear&lt;/em&gt; means when qualifying a morphism depends on the category to which the morphism pertains. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When working in the category of vector spaces (that is, doing linear algebra), a morphism is &lt;em&gt;linear&lt;/em&gt; if it preserves the vector space structure, so $x \to ax + b$ is not linear, viewed as a map between one-dimensional vector spaces (it can be called affine if one works in the affine category).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When working in the category of smooth (to whatever degree) manifolds (that is, doing calculus), there is no notion of &lt;em&gt;linear&lt;/em&gt; without some additional structure. A mapping ought to be called &lt;em&gt;linear&lt;/em&gt; if it maps lines to lines, but for this to make sense there has to be a notion of &lt;em&gt;line&lt;/em&gt;. The structure necessary to speak of lines is an affine connection - the &lt;em&gt;lines&lt;/em&gt; are the geodesics of the connection. Properly speaking these are really &lt;em&gt;lines&lt;/em&gt; only if the affine connection is torsion-free and flat (in this case there exist localy coordinates in which a geodesic really is a line). So, the additional structure that is used implicitly in speaking of a &lt;em&gt;linear function&lt;/em&gt; is a flat torsion-free affine connection (on $\mathbb{R}$ it will be the usual derivative operator). A smooth mapping $f:M \to N$ between connected differentiable manifolds equipped with flat torsion-free affine connections can be defined to be &lt;em&gt;linear&lt;/em&gt; if the covariant derivative of its differential vanishes (to make sense of this the differential $Df$ has to be viewed as a section of the bundle $T^{\ast}\otimes f^{\ast}TN$ equipped with the induced product connection; the condition is $\nabla Df = 0$). This is akin to defining a smooth map between connected differentiable manifolds to be constant if its differential vanishes, but is different in that it requires the extra structure of the flat connection (in fact the flatness is mostly irrelevant - dropping it one obtains the definition of a totally geodesic map - the flatness, which in the one-dimensional case is in any case automatic, is just there to justify using the word &lt;em&gt;linear&lt;/em&gt;, because without flatness, there's no reason to think of geodesics as &lt;em&gt;linear&lt;/em&gt;). If $M = N = \mathbb{R}$ with the usual flat connection, then this just means that $f^{\prime\prime} = 0$, so that $f$ has the form $f(x) = ax + b$.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2018-07-11T05:05:51.837" LastActivityDate="2018-07-11T05:05:51.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14342" PostTypeId="2" ParentId="14258" CreationDate="2018-07-10T18:30:58.567" Score="1" Body="&lt;p&gt;I'd whip up something in LaTeX using TikZ (see for example &lt;a href=&quot;https://cremeronline.com/LaTeX/minimaltikz.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cremer's short tutorial&lt;/a&gt;, the full documentation is a real bear; or snoop around in TeX.stackexchange.com).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For standalone, I'd probably use something like &lt;a href=&quot;http://mcj.sourceforge.net&quot; rel=&quot;nofollow noreferrer&quot;&gt;xfig&lt;/a&gt; (for hand drawing) or &lt;a href=&quot;http://asymptote.sourceforge.net&quot; rel=&quot;nofollow noreferrer&quot;&gt;asymptote&lt;/a&gt; (for &quot;programming&quot; a more complex figure). Under Linux, there probably are Mac (and even Windows) versions available.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2018-07-10T18:30:58.567" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14343" PostTypeId="2" ParentId="14338" CreationDate="2018-07-11T00:32:53.193" Score="1" Body="&lt;p&gt;Polya's classic was already mentioned by @vonbrand.&#xA;It is full of insights.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Polya, George. &lt;em&gt;How to solve it: A new aspect of mathematical method&lt;/em&gt;. No. 246. Princeton university press, 2004.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/N3cBY.jpg&quot; width=&quot;150&quot; /&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;I hesitate to recommend the following as I have not read it, but it is in the&#xA;same vein, albeit more algorithmically oriented:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Michalewicz, Zbigniew, and David B. Fogel. &lt;em&gt;How to solve it: modern heuristics&lt;/em&gt;. Springer Science &amp;amp; Business Media, 2013. (&lt;a href=&quot;https://www.springer.com/us/book/9783540224945&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer link&lt;/a&gt;.)&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/YGZde.jpg&quot; width=&quot;150&quot; /&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-07-11T00:32:53.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14345" PostTypeId="2" ParentId="14332" CreationDate="2018-07-11T09:14:02.717" Score="4" Body="&lt;p&gt;In Italy it is standard to teach addition of fractions using l.c.m.. An example from a widely used textbook: &quot;&lt;em&gt;Per addizionare due frazioni aventi denominatori diversi dapprima si riducono le frazioni al minimo comun denominatore e poi si addizionano i rispettivi numeratori&lt;/em&gt;&quot; (to sum two fractions having different denominators you first reduce them both to the l.c. denominator and then you had the two respective numerators).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the reason is somewhat historical. At some period, a huge stress was placed on all fractions being reduced to minimal terms (do not know if this is the correct terminology: numerator and denominator with no common divisors). I remember as a kid (some 40 years ago) endless exercises on reducing fractions to minimal terms. There no discussion about &quot;equivalence of fractions&quot; defining a single rational number but rather it was implicit that a non-reduced fraction was not a &quot;real&quot; fraction but only something not well defined. If the result of an exercise was a fraction, giving the result in non-reduced form was considered an error. Fractions corresponding to integer numbers are still called &lt;em&gt;apparent fractions&lt;/em&gt; (with a rather misleading terminology in my opinion).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not see any special advantage in this approach which I find rather clumsy, and I agree with what Wu writes on the topic.&lt;/p&gt;&#xA;" OwnerUserId="465" LastEditorUserId="511" LastEditDate="2018-07-12T11:11:44.727" LastActivityDate="2018-07-12T11:11:44.727" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14348" PostTypeId="1" CreationDate="2018-07-12T00:05:51.613" Score="2" ViewCount="116" Body="&lt;p&gt;From: Philip Johnson-Laird &lt;a href=&quot;https://dof.princeton.edu/about/clerk-faculty/emeritus/philip-nicholas-johnson-laird&quot; rel=&quot;nofollow noreferrer&quot;&gt;BA PhD Psychology (UCL)&lt;/a&gt;, Stuart Professor of Psychology Emeritus at Princeton.    (Author isn't  a logician.)    &lt;a href=&quot;https://www.amazon.ca/How-We-Reason-Philip-Johnson-Laird/dp/0199551332&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How We Reason&lt;/em&gt;  (1st edn 2008)&lt;/a&gt;. p. 457.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;172&quot;&gt;&#xA;  &lt;li&gt;For an account of Ramanjuan and his ignorance of proofs, see Kanigel (1991, p. 92, and&#xA;  see p. 207 for the comment quoted from the great Cambridge mathematician, G.H.&#xA;  Hardy). With a finite number of finite models, we can always in principle determine&#xA;  whether an inference is valid. Mathematicians, however, have to think about the positive integers, 1, 2, 3, . . . , and there are infinitely many of them. &lt;strong&gt;How they reason about&#xA;  infinite sets is a mystery&lt;/strong&gt;, which psychologists have yet to elucidate.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is the emboldened phrase true? If yes, how and why? Mathematicians have been doing this for millenia?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If it isn't, please recommend books or papers? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2018-07-12T00:05:51.613" Title="Are there any books on how mathematicians reason about infinite sets?" Tags="|infinity|" AnswerCount="0" CommentCount="11" ClosedDate="2018-07-12T03:16:51.957" ContentLicense="CC BY-SA 4.0" />
  <row Id="14350" PostTypeId="1" CreationDate="2018-07-13T00:36:21.337" Score="9" ViewCount="13922" Body="&lt;p&gt;I am a graduate student teaching college algebra at a larger state school, and currently I'm covering transformations of graphs of function, i.e.: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given the graph of a function &lt;span class=&quot;math-container&quot;&gt;$y =f(x)$&lt;/span&gt;, what do the following graphs look like, compared to the graph of $y = f(x)?&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = f(x) \pm C$&lt;/span&gt;, &lt;/li&gt;&#xA;  &lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = f(x \pm C)$&lt;/span&gt;, &lt;/li&gt;&#xA;  &lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = Cf(x)$&lt;/span&gt;, &lt;/li&gt;&#xA;  &lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = f(Cx)$&lt;/span&gt;, &lt;/li&gt;&#xA;  &lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = -f(x)$&lt;/span&gt;, &lt;/li&gt;&#xA;  &lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = f(-x)$&lt;/span&gt;. &lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is a topic that I always struggle to come up with concrete “real world” applications of. My audience is typically freshman who are looking to get the class out of the way, so they can move on to business calculus or so they can be done with math altogether. Consequently, teaching material through a real scenario is a good way to hold my students attention. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However with this topic, I have a hard time coming up with good examples. One such example I gave was the following: I presented them with the graph &lt;span class=&quot;math-container&quot;&gt;$y = f(x)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is the function that takes in the number of miles/hour over the speed limit a driver is going and returns the value of the corresponding fine for speeding in a particular city. I then asked them how the graph would change as well as how they could modify &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; using these elementary transformations if (a) the city decides to crack down on speeding and raise all fines amount by \$50 ( &lt;span class=&quot;math-container&quot;&gt;$y = f(x) + 50$&lt;/span&gt;) and (b) the city decides to be more lenient and give drivers a 5 mph “buffer” (&lt;span class=&quot;math-container&quot;&gt;$y = f(x-5)$&lt;/span&gt;) – meaning that a driver would not receive a fine unless they exceeded the speed limit by more than 5 mph.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone else have better examples that are rooted in a real scenario?&lt;/p&gt;&#xA;" OwnerUserId="9884" LastEditorUserId="238" LastEditDate="2019-10-11T19:16:54.490" LastActivityDate="2019-10-11T19:16:54.490" Title="Good real-life examples of transformations of function graphs" Tags="|algebra|examples|graphing|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14351" PostTypeId="2" ParentId="14350" CreationDate="2018-07-13T01:46:22.953" Score="7" Body="&lt;p&gt;My favorite example of a graphical transformation is waves. Consider,&#xA;$$ y = A \sin( x-vt) $$&#xA;Here $A$ gives the &lt;strong&gt;amplitude&lt;/strong&gt; of the wave and it tells us from a graphical perspective how much the unit-sine wave is vertically stretched. If we fix $t$ then the term $-vt$ is just some phase-shift and we can see the graph is just a horizontal shift of the sine wave. In particular we shift the $y = A \sin(x)$ graph $vt$-units to right to form $y = A \sin(x-vt)$. This is a &lt;strong&gt;right-moving wave&lt;/strong&gt; if we assume $v&amp;gt;0$. To see it, I think setting $0 \leq x-vt \leq 2\pi$ might be helpful, then $vt \leq x \leq 2\pi+vt$ and if we animate $t$ we can actually see the wave travel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other more complicated wave graphs could be studied. I happen to have &lt;a href=&quot;https://www.desmos.com/calculator/bk4pfkdcz5&quot; rel=&quot;nofollow noreferrer&quot;&gt;this graph of a solution to the wave equation&lt;/a&gt; sitting around. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, adding graphs has interesting interpretations in terms of constructive and destructive interference. Or, if we add time I think you can get beats. There is much to explore here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; as Dave Renfro pointed out, I am using forbidden trigonometry. To, give a less exciting, but very much applied example consider:&#xA;$$ y = y_o- \frac{g}{2}t^2 $$&#xA;If we start with $y=t^2$ then multiplying b $-g/2$ stretches the graph and flips it over. The significance is that $g$ is the acceleration due to gravity. Then, adding $y_o$ shifts the graph $y=-gt^2/2$ up by $y_o$ and in this discussion $y_o$ has the significance of being the initial condition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more ambitious formula to explain would be $y = y_o+v_ot-\frac{g}{2}t^2$. Completing the square gives,&#xA;$$ y = y_o+\frac{v_o^2}{2g}-\frac{g}{2}\left( t-\frac{v_o}{g} \right)^2 $$ &#xA;We can view this as the graph $y=-\frac{g}{2}t^2$ which is shifted vertically by $y_o+\frac{v_o^2}{2g}$ and translated horizontally by $\frac{v_o}{g}$. If $v_o&amp;gt;0$ then $\frac{v_o}{g}&amp;gt;0$ so it would signify a right-shift of the graph. The vertex of the parabola would be at $\frac{v_o}{g}$ which is where the maximum height of $y_o+\frac{v_o^2}{2g}$ is attained. In contrast, $y=-\frac{g}{2}t^2$ has maximum height $y=0$ at $t=0$. To simplify this discussion, you could say it is all taking place on the Dwarf Moon of Convenience where $g/2=1$.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2018-07-13T16:00:04.680" LastActivityDate="2018-07-13T16:00:04.680" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14352" PostTypeId="2" ParentId="12618" CreationDate="2018-07-13T19:01:46.907" Score="0" Body="&lt;p&gt;There is a great YouTube series on this topic from the Berkley graduate William Flannery called &lt;a href=&quot;https://www.youtube.com/watch?v=Ns4BCDH2keI&amp;amp;list=PLn2BtAO3VAR4f3fn3FhIcVdcV1KOwuxMc&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Mathematical Modeling and Computational Calculus I&quot;&lt;/a&gt; and &lt;a href=&quot;https://www.youtube.com/watch?v=xyY8Nkw-YYY&amp;amp;list=PLn2BtAO3VAR5ew6A9nc_ZHbE3ZtPlKcGf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Mathematical Modeling and Computational Calculus II&quot;&lt;/a&gt;. He uses MATLAB to simulate different physics problems that involve calculus. The multivariable calculus comes more in the 2nd series when he starts working with the partial differential equations. He has also two books that support these courses and can be purchased on Amazon (&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0976413868&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vol. 1&lt;/a&gt; and &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0976413876&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vol. 2&lt;/a&gt;). Finally, he has a website called &lt;a href=&quot;http://berkeleyscience.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Berkeley Science&lt;/a&gt; where you can find his other materials.&lt;/p&gt;&#xA;" OwnerUserId="6306" LastActivityDate="2018-07-13T19:01:46.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14353" PostTypeId="2" ParentId="14350" CreationDate="2018-07-13T19:26:45.993" Score="7" Body="&lt;p&gt;Unit conversions often provide a natural application of these transformations. Since temperature unit conversions often include both scaling and shifting, they are particularly useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose you have a function $H(c)$ which gives the rate of heat stroke death in a region, per thousand occupants, as a function of the temperature $c$, in Celsius. You wish to change units to have the input of the temperature be in Fahrenheit, represented by $f$, and to instead output the rate of heat stroke death per hundred thousand people. Your new function would be $100 H(\frac{5}{9}(f-32))$. What would the graph of the new function look like? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose you have a function $C(d)$ giving the temperature $C$, in Celsius, on day $d$ of the year. You wish to instead input the approximate month, m, of the year, considering each month to be 30 days, and having m=0 represent the first day of January. You wish to have your function output the temperature in degrees Fahrenheit below 90 degrees (since, perhaps, this value is what determines the heating costs of your 90-degree greenhouse). Your new function will be of the form $ 90- (\frac{9}{5}C(30m)+32)$. What will the graph of your new function look like?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of course, these are intentionally challenging examples that are intended to pack as much possible into one problem. Some other specific ways to incorporate each individual transformation could include: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$f(x\pm C)$: Measuring traffic density in &quot;hours after 5am&quot; instead of &quot;hours after midnight&quot;. Tax deductions (taxable and non-taxable income). &lt;/li&gt;&#xA;&lt;li&gt;$f(x)\pm C$: Surcharges and flat fees. Measuring altitude vs. height above a fixed altitude, e.g. &quot;The tree line is ____, and located 10,000 feet in altitude. If $A(t)$ is the altitude of a climber at time t, write a function describing their height above the tree line at time t.&quot; &lt;/li&gt;&#xA;&lt;li&gt;$f(-x)$: Instead of measuring time &quot;after&quot; an event measure time &quot;before&quot; an event. &lt;/li&gt;&#xA;&lt;li&gt;$f(Cx)$: Convert the units of the input of the function by a scalar, e.g. feet to inches&lt;/li&gt;&#xA;&lt;li&gt;$Cf(x)$: Convert the units of the output of the function by a scalar&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="724" LastActivityDate="2018-07-13T19:26:45.993" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14354" PostTypeId="1" CreationDate="2018-07-14T01:36:22.233" Score="5" ViewCount="366" Body="&lt;p&gt;I would be quite interested in learning of trends (in the U.S. or internationally) of student interest at the undergraduate level in pure math vs. applied math,&#xA;say, measured by majors at graduation. Even just math-major trends &#xA;could be useful, especially since there is often no distinction between pure &amp;amp; applied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course this leads to the question of what constitutes &quot;applied math,&quot; and&#xA;in particular where statistics fits along this spectrum. This connects to&#xA;the surgence of data science and its relation to traditional statistics. This is why &#xA;I am having difficulty grasping where student interest is trending.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask this as an administrator, responsible for allocating faculty lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/PjF2z.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PjF2z.jpg&quot; alt=&quot;Chart&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://www.educationnext.org/solving-america%E2%80%99s-math-problem/&quot; rel=&quot;nofollow noreferrer&quot;&gt;EducationNext&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added&lt;/strong&gt; (&lt;em&gt;14Aug2018&lt;/em&gt;). I found some data from IPEDS: Integrated Postsecondary Education Data System. This shows the percentage of all U.S. undergraduate degrees in mathematics,&#xA;from 1984 to 2018. (Apologies for the low resolution.)&#xA;There has been a gradual rise in the last $17$ years or so.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/mi8H8.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mi8H8.jpg&quot; alt=&quot;MathNational&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Data from IPEDS. Percentage of all undergraduate degrees in Math.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-08-14T23:35:30.797" LastActivityDate="2018-08-14T23:35:30.797" Title="Trends in math education: Majors? Applied math? Statistics?" Tags="|undergraduate-education|statistics|applied-mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14356" PostTypeId="2" ParentId="7907" CreationDate="2018-07-15T04:41:51.850" Score="3" Body="&lt;p&gt;I once took a Topology course at the University of Michigan in which the exam questions were constructed in roughly the form of a menu:  some questions were worth 10 points, some were worth 20, etc.  There were roughly 160 points worth of questions available, and students were instructed to answer any combination of questions that added up to a maximum of 100 points.  So, for example, a student who wanted to mainly stick to routine exercises could answer eight 10-point problems and one 20-point problem; a student who wanted to really challenge him or herself could answer two 40-point problems and one 20-point problem.  (These are just examples; I don't recall the exact configuration of options.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many years later, when I taught a Math for Elementary Teachers course, I adapted the scheme, but I found that virtually without exception every single student chose to answer only low-value questions, so after a couple of years I abandoned the model, as I was putting a lot of effort into designing the exams and not getting much out of it.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-07-15T04:41:51.850" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14357" PostTypeId="2" ParentId="7907" CreationDate="2018-07-15T16:10:23.643" Score="2" Body="&lt;p&gt;The exam construction strategy which I have been using recently, and which seems to work pretty well, is as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make the numbers easy, suppose that I want to write a 10 question exam, where a student who gets 90% or more of the possible points gets an A, 80-90% a B, 60-80% a C, and anything less is D/F territory  (obviously, this can be scaled up to make longer exams).  This exam will consist of four categories:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Passing level questions:  on the exam described above, 6 of the 10 questions will be in this section.  These questions represent the bare minimum of material that needs to be mastered in order to pass the class at the lowest possible level.  These questions typically test recall and very simple computations---students should not be required to show any work here, and should be able answer these questions in less than 30 seconds---it really &lt;em&gt;should&lt;/em&gt; be just a matter of writing down the answer.  On a 10 question exam, the whole section should take less than 5 minutes.  In a recent business calculus exam, such questions included&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Determine $\frac{\mathrm{d}}{\mathrm{d}x} \pi^2$.&lt;/li&gt;&#xA;&lt;li&gt;Complete the statement of the product rule:  if $f$ and $g$ are differentiable functions, then $(fg)'(x) = $___.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;C level questions:  the next 2 questions are in this section.  These questions are typically very computational in nature, and relatively straight-forward and mechanical, but might require a bit more work and should depend on more than &lt;em&gt;just&lt;/em&gt; recall.  These questions can also be taken word-for-word from the homework assignments (maybe fudging some numbers here and there).  I expect a typical student to be able to answer one of these questions in about 5 minutes (or, hopefully, less).  These are applications of ideas in relatively simple settings.  For example&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Find $\frac{\mathrm{d}}{\mathrm{d}x} 3\sqrt{9-x^2}.$&lt;/li&gt;&#xA;&lt;li&gt;Alice paints and sells her own paintings. Each painting that she produces requires \$150 in supplies, and can be sold for \$300. If she has fixed costs of \$1,350 per month, how many paintings must she sell each month in order to break even?&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;B level questions: on this 10 question exam, there is only one B level question.  The goal of a B level question is, in general, to go a little bit beyond just the mechanics and see whether or not a student can do more than one step at a time.  These questions ask students to link several ideas to solve a single problem, without any extra scaffolding given.  For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Use the limit definition of the derivative to compute $\frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{\sqrt{x}}$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I expect a B level question to take about 10 minutes to complete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They might know this from the power rule, but would not have seen this particular derivative computed &quot;by hand&quot; in class.  The question requires them to know a definition, to be able to apply that definition, and to actually compute a limit.  Another example (from a precalc class):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Find an equation for a line tangent to the graph of $2x^3 + 2$ which passes through the point $(1,4)$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is actually an appropriate B level question in the precalc class (it would be a C level question in a calc class), as they have been given an algebraic notion of tangency, but no derivative function.  Hence they have to remember that definition, apply it to the function, and work through the computations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A level questions: on a 10 question exam, there would be one of these.  An A level question is meant to push the students to use what they know in a novel situation.  I don't actually expect many students to give a complete answer to an A level question, and I expect that the ones that can answer these questions will spend 15-20 minutes on each of them.  The goal is to get the students to show how they think about hard problems.  A level questions might require knowing multiple definitions and applying them to a novel &lt;em&gt;type&lt;/em&gt; of problem.  There may be some computation in such a question, but the real impediment to answering should be conceptual rather than computational.  For example&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Suppose that $$f(x) = \begin{cases} 5x + 4 &amp;amp; \text{if $x \le 0$, and} \\ ax^2 + bx + 7 &amp;amp; \text{if $x &amp;gt; 0$.} \end{cases}$$  Are there values of $a$ and $b$ such that $f$ is differentiable at zero?  If so, find them.  If not, explain why not.&lt;/li&gt;&#xA;&lt;li&gt;(For a precalc class:) Suppose that $x$ is a real number, and let $\lceil x \rceil$ denote the ceiling of $x$.  That is, $\lceil x \rceil$ is the smallest integer that is greater than $x$, or what we get when we &quot;round $x$ up&quot; to the next integer.  For example, $\lceil -1.5 \rceil = -1$ and $\lceil \pi \rceil = 4$.  Note that integers &quot;round up&quot; to themselves, e.g. $\lceil -2 \rceil = -2$.  Let $f:\mathbb R\to\mathbb R$ be defined by the formula&#xA;\begin{equation*}&#xA;		f(x) = \lceil x \rceil - x.&#xA;	\end{equation*}&#xA;For what values of $p$ is $f$ $p$-periodic?  Does $f$ have a fundamental period?  If so, what is it?&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Every question is worth exactly the same quantity of points.  So the A level question which requires 20 minutes to answer is just as &quot;valuable&quot; as the passing level question which can be dashed off in 10 seconds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students &lt;em&gt;seem&lt;/em&gt; responsive to this scheme.  They really like to know exactly how much effort they have to put into something in order to get the result that they want, and they really like to know where they stand.  I've tried other schema where, for example, students choose some number of problems to complete; or where there are hard cutoffs for the number of questions at a given level which must be answered in order to earn a particular grade; or where questions at different levels are weighted differently, but these seem to confuse students.  Students also &lt;em&gt;really&lt;/em&gt; like percentage-based grading scales---every time I have ever tried to move away from such scales, I've been creamed in evaluations (not that one should base their teaching on getting good numerical evaluations, but this doesn't seem like a good hill to die on).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the instructor side of things, I don't really think that these exams are &quot;game-able&quot; to any significant degree.  You can't get the A without making significant progress on the A level question(s).  They are also not hard to grade, since there is no choice involved on the part of the students, and one doesn't have to go back and forth between pages of the exam to count the number of questions answered in each section.  You also don't lose time explaining the structure of the exam to the students---it is an exam like any other that they have ever seen (just with sections labeled &quot;passing&quot;, &quot;C&quot;, &quot;B&quot;, and &quot;A&quot;).&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2018-07-15T16:10:23.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14358" PostTypeId="1" CreationDate="2018-07-16T08:56:48.840" Score="6" ViewCount="191" Body="&lt;p&gt;What is meant by learning math historically (NOT learning math history only, but learning math with a historical development perspective)? I've seen some sources that to learn a math topic X, you need to look at the historical development of the topic X and go over the famous questions by yourself to develop a good understanding of the subject. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also find this method (learning in a historical context) nicer because more often that not traditional books (say a book on Group theory) starts like here are the axioms (eg the group axioms), memorize it and look at the theorems and corollaries which follows from the axioms. Without the historical context it keeps me wondering what was the point of the axioms in the first place (i.e what motivated that).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But on the other hand, regarding learning things historically, how I am supposed to &quot;go over the famous questions by myself&quot; when the problems took tens of years to solve ? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a concrete example, I'm learning Ring theory now. What should I do to learn it in a historical way ? Am I supposed to work on problems like $x^3+y^3 = z^3$ or UFD over cyclotomic integers and &quot;rediscover&quot; Ring theory ? But how I am supposed to sensibly work on the problems without knowing the theory when they took really long time to solved in the first place ? Or &quot;learning math historically&quot; is done just like reading math from textbook, you just &quot;passively&quot; read the history of how some ideas were developed instead of &quot;actively&quot; working on some historically important questions which took really long time to solve ?&lt;/p&gt;&#xA;" OwnerUserId="10161" LastEditorUserId="10161" LastEditDate="2018-07-17T16:10:20.587" LastActivityDate="2018-07-17T16:10:20.587" Title="Learning math historically" Tags="|undergraduate-education|self-learning|history|inquiry-based-learning|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14359" PostTypeId="1" CreationDate="2018-07-17T11:32:12.897" Score="6" ViewCount="261" Body="&lt;p&gt;When I teach complex numbers to undergraduate engineering students, I invariably start, as appears to be customary, with $a + bi$ (or $a + bj$ for electrical engineers) and then follow up with the polar form $r\angle\theta$ or $r(\cos\theta+i\sin\theta)$ and the exponential form $re^{i\theta}$. Given how often the polar and exponential forms arise in later material and applications, including phasor diagrams for AC circuits, various integral transforms and solving differential equations, it makes me wonder whether it is more effective to introduce the polar form first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the pros and cons of introducing the polar and exponential forms of complex numbers before the Cartesian form? I am especially interested in the context of teaching engineering students, although I am also curious how a similar approach would work for mathematics majors.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2018-07-17T15:30:51.967" Title="Polar form before Cartesian form when introducing complex numbers" Tags="|undergraduate-education|engineering-mathematics|complex-numbers|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14360" PostTypeId="2" ParentId="14359" CreationDate="2018-07-17T15:30:51.967" Score="5" Body="&lt;p&gt;Let me suggest the following story: if we consider $z = x+iy$ as a point in the complex plane then we have $z = (x,y)$ in the usual Cartesian coordinate notation. Since &#xA;$$ z=x+iy = x(1)+y(i) = x(1,0)+y(0,1)$$ &#xA;we have $1=(1,0)$ and $i=(0,1)$ in this approach. Viewing $z=(x,y)$ as a vector based at the origin we have:&#xA;$$ \boxed{z = r\hat{r}} $$&#xA;where $\hat{r}$ is a unit-vector in the direction of increasing distance from the origin $r$. Again, we are viewing $z$ as a directed line-segment based at the origin. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Where is the polar form?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Let $\theta$ be the standard angle measured CCW from the positive $x$-axis. From trigonometry, if $z = r\hat{r} = x+iy$ then we can see that $x=r\cos \theta$ and $y= r \sin \theta$ hence apparently we must have $\hat{r} = (\cos \theta,\sin \theta)$  hence in our complex notation $\hat{r} = \cos \theta + i \sin \theta$.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What does $i^2=-1$ have to do with anything?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Let us suppose $i^2=-1$ and we multiply complex numbers as you were (hopefully) taught in an earlier part of your education,&#xA;$$ (a+ib)(c+id) = ac-bd+i(ad+bc) $$&#xA;Consider what happens when we multiply the direction vectors of two complex numbers with standard angle $\theta_1, \theta_2$ respective: &#xA;\begin{align} \hat{r}_1\hat{r}_2 &amp;amp;=(\cos \theta_1+i\sin \theta_2)(\cos \theta_2+i\sin \theta_2) \\&#xA;&amp;amp;= \cos \theta_1\cos \theta_2-\sin \theta_1\sin \theta_2+i \left[&#xA;\sin \theta_1\cos \theta_2+\sin \theta_2\cos \theta_1 \right] \\&#xA;&amp;amp;= \cos( \theta_1+\theta_2)+i \sin(\theta_1+\theta_2)&#xA;\end{align}&#xA;where we used the adding angles formulas in the last step (which you can have them derive from the law of cosines and the appropriate triangle). We find $i^2=-1$ paired with the usual arithmetic implies the product of unit-vectors in the complex plane gives us a new unit-vector in the direction which has standard angle $\theta_1+\theta_2$. If we use the notation $\hat{r}_1=e^{i\theta_1}$ and $\hat{r}_2=e^{i\theta_2}$ then we have shown that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$ e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1+\theta_2)} $$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course, you might have assumed this anyway, but we have &lt;strong&gt;proved&lt;/strong&gt; this law as a consequence of trigonometry you already know and the rule $i^2=-1$. In a nutshell, what makes the complex plane with points $z=x+iy$ the genuine complex plane is precisely the rule $i^2=-1$. Most everything else is calculus and algebra you already &lt;em&gt;know&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a different perspective, if you accept $e^{i\theta} = \cos \theta + i \sin \theta$ by faith (which is not without reason given the preceding paragraphs) this is a terribly useful way to recall trig. identities. It is easy to derive&#xA;$$ \cos \theta = \frac{1}{2}\left( e^{i\theta}+e^{-i\theta} \right)&#xA;\qquad \&amp;amp; \qquad \sin \theta =\frac{1}{2i}\left( e^{i\theta}+e^{-i\theta} \right) $$&#xA;Hence,&#xA;\begin{align} &#xA;\cos A \cos B &amp;amp;= \frac{1}{2}\left( e^{iA}+e^{-iA} \right)\frac{1}{2}\left( e^{iB}+e^{-iB} \right) \\&#xA; &amp;amp;= \frac{1}{4}\left( e^{i(A+B)}+e^{-i(A+B)} +e^{i(A-B)}+e^{-i(A-B)} \right)\\&#xA; &amp;amp;= \frac{1}{2}\left( \cos(A+B)+\cos(A-B)\right).&#xA;\end{align} &#xA;Could you derive the identity above before? Perhaps you were taught how to derive it via trigonometry, notice we just derived is from a logically linear argument using mere algebra. In fact, with Euler's Identity $e^{i\theta}= \cos \theta+i\sin \theta$ in hand we can derive nearly any trigonometric identity in real notation you desire. This is the basis for the success of the phasor methor, we are able to put into place some subtle trigonometric relations via relatively simple algebraic calculations with these so-called &lt;em&gt;imaginary exponentials&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try to introduce these ideas as soon as possible if I'm teaching a course where serious complex arithmetic is utilized. Notice, I fail to even distinguish between polar and exponential form in this answer. I'm sorry about that, they seem to be the same thing. Much like the abuse $x+iy=(x,y)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, it's fun to think some about $j$, well, fine, $k$ such that $k^2=1$. In that plane with algebra $(a+bk)(c+dk)=ac+bd+k(ad+bc)$ we would find a very different story. It happens $e^{k\phi}= \cosh \phi + k \sinh \phi$, but I don't motivate that geometrically. Rather, analytical arguments force those hyperbolic sines and cosine to arise from the exponential series. Moreover, there are numbers with no multiplicative inverses along the numbers $z=r(1\pm k)$ for $r \in \mathbb R$. The analog of exponential representation breaks into four distinct cases in the regions separated by $y= \pm x$. In fact, there is some connection with Special Relativity which can be made here with these so-called &lt;strong&gt;hyperbolic numbers&lt;/strong&gt;. For example, there is a paper (&lt;em&gt;Hyperbolic Calculus&lt;/em&gt; by Motter and Rosa, Advances in Applied Clifford Algebras  8 No. 1, 109-128 (1998)) which solves the wave equation on the uniformly accelerated rigid rod by a clever application of such numbers. Personally, knowing what happens when we study non-complex numbers has given me much greater appreciation for $i^2=-1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this said, the fundamental theorem of algebra is reason enough to study $\mathbb C$.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-07-17T15:30:51.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14361" PostTypeId="1" AcceptedAnswerId="14362" CreationDate="2018-07-18T06:22:37.820" Score="3" ViewCount="266" Body="&lt;p&gt;I'm just starting graduate school (master in mathematics) and there we have absolute freedom in choosing the number of courses that we can attend each term. In consequence, I'm wondering what is the best strategy when taking courses:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To take three courses and broaden fast my knowledge of mathematics but doing less problems on every course, or&lt;/li&gt;&#xA;&lt;li&gt;To take two courses and slow my rate of knowledge of mathematics but make a lot of more problems on every course.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I should say that I'm considering the first strategy because I want to get as soon as possible to research activities (mainly because I have seen that all graduate students are getting fast to research activities and end up their masters degree with at least one article and it is known that this is important if you want a place in the best doctoral programs in the world).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in terms of education and pedagogy and so on, what is the best strategy?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="5402" LastActivityDate="2018-07-18T15:30:24.833" Title="Better strategy on math graduate school" Tags="|graduate-education|" AnswerCount="1" CommentCount="3" ClosedDate="2019-10-10T02:02:25.603" ContentLicense="CC BY-SA 4.0" />
  <row Id="14362" PostTypeId="2" ParentId="14361" CreationDate="2018-07-18T15:30:24.833" Score="2" Body="&lt;p&gt;Academia in general, and Mathematics in particular, tends to reward specialization over generalization. Not in every case, but doctoral studies are typically very deep and narrow. If that is your goal, then focus on the parts of math that you enjoy and have strength in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand it is good to have a lot of mathematical interests at your current state so that you have a lot of ways you can jump into research. The further you move along, the more will be the pressure to specialize. Once you hold a doctorate you are pretty free to make your own path. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other consideration is about how hard you want to work right now, and how much risk you want to accept of getting in over your head with a heavy work load and subjects that are different enough from what you already know that you have a non-trivial learning curve. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no reason why you need to make a decision now and stick to it for all time. If you are taking really new courses, fewer is likely better (safer). If you are already somewhat comfortable with the new material and want to work really hard (all the time), then plunge in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But better that you should consider your strengths (and any weaknesses) and find a way to excel. &lt;/p&gt;&#xA;" OwnerUserId="8756" LastActivityDate="2018-07-18T15:30:24.833" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14363" PostTypeId="1" AcceptedAnswerId="14365" CreationDate="2018-07-18T20:59:07.057" Score="0" ViewCount="136" Body="&lt;p&gt;In the following multiplication example,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align}&#xA;34\;&amp;amp;&#xA;\\\underline{\times\;\; 7\;}&amp;amp;&#xA;\end{align}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;first one would multiply the units digits, producing the partial product $28$ as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align}&#xA;\color{maroon}{2}\;\;\;\!\&amp;gt;\!\\&#xA;34\;&amp;amp;&#xA;\\\underline{\times\;\; 7\;}&amp;amp;&#xA;\\\color{maroon}{8}\;&amp;amp;&#xA;\end{align}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How would this partial product operation best be described?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;( 4 \times 7 ) &#xA;\qquad\text{or}\qquad&#xA;( 7 \times 4 )&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, how would the next partial product operation, which adds the carry, best be described?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;	( 3 \times 7 ) + 2&#xA;\quad\text{or}\quad&#xA;	( 7 \times 3 ) + 2&#xA;\quad\text{or}\quad&#xA;	2 + ( 3 \times 7 )&#xA;\quad\text{or}\quad&#xA;	2 + ( 7 \times 3 )&#xA;$$&lt;/p&gt;&#xA;" OwnerUserId="10170" LastEditorUserId="5220" LastEditDate="2018-07-19T04:35:06.177" LastActivityDate="2018-07-19T05:06:33.443" Title="How to present the order of factors and summands for the usual multiplication procedure" Tags="|primary-education|arithmetic-operations|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14364" PostTypeId="2" ParentId="14335" CreationDate="2018-07-19T00:26:56.210" Score="1" Body="&lt;p&gt;I think both &quot;the linear approximation of $y=f(x)$ at $x=a$&quot; and &quot;local linear approximation&quot; are fine. Now, it should be made clear that the linear approximation is a function. Next, what kind of function? A linear function! It has the form $y=mx+b$ after all. But what about $b$? I think you are right that it would be confusing to bring (the correct term) &quot;affine&quot; in this context. I have been trying to use &quot;linear operator&quot; for $y=mx$ in calculus 3 and linear algebra to distinguish the two.&lt;/p&gt;&#xA;" OwnerUserId="10168" LastActivityDate="2018-07-19T00:26:56.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14365" PostTypeId="2" ParentId="14363" CreationDate="2018-07-19T04:41:01.500" Score="1" Body="&lt;p&gt;This answer is very culture-dependent. If the primary language used in your classroom is English, then since in English we read left-to-right and top-to-bottom I would pick the way where these two directions agree. If the language you primarily use in the classroom differs from English in this way, then change the directions accordingly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Following your example, in the first step, reading downward the order is $4$ then $7$, so write $(4 \times 7)$. In the second step, the order is $2$ then $3$ then $7$, so write $2+(3 \times 7).$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Really though, pending a study into this that I imagine is &lt;em&gt;way&lt;/em&gt; too specific to ever get funded, I don't think it matters too much to a students' understanding what order you choose. &lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2018-07-19T05:06:33.443" LastActivityDate="2018-07-19T05:06:33.443" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14366" PostTypeId="1" CreationDate="2018-07-19T15:57:56.437" Score="0" ViewCount="157" Body="&lt;p&gt;The function $P$ that takes an event $A$ as input and returns the probability $P(A)$ as output is called a &quot;probability measure&quot; when we are developing probability using measure theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have also seen some authors use the term &quot;probability law&quot;. (I think Bertsekas uses this term.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it is fairly common to refer to $P$ as a &quot;probability distribution&quot;, but there is an issue here that I want to be careful about: I have seen that some authors use the term &quot;distribution&quot; only when referring to the distribution of a random variable $X$. For example, I believe that Folland makes this distinction. Should I make this distinction also in an undergraduate probability course (that does not use measure theory)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the best or most standard term to use instead of &quot;probability measure&quot; in an introduction to probability course?&lt;/p&gt;&#xA;" OwnerUserId="9633" LastActivityDate="2018-07-19T15:57:56.437" Title="What is the best term for &quot;probability measure&quot; in an undergrad introduction to probability course?" Tags="|terminology|probability|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14367" PostTypeId="1" AcceptedAnswerId="14369" CreationDate="2018-07-19T16:38:44.823" Score="11" ViewCount="1026" Body="&lt;p&gt;It seems like the internet has made it very difficult to prevent cheating -- simply looking up the solutions to textbook problems -- on undergraduate math homework. So there is a danger that grading math homework assignments could be unfair to the students who do not cheat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the best way to handle this? One system is to check homework for completion only and have a weekly ten minute quiz in class that consists of one or two problems taken directly from homework. Is that a good way to go?&lt;/p&gt;&#xA;" OwnerUserId="9633" LastActivityDate="2018-07-29T16:33:41.893" Title="Should homework be graded in an undergraduate math course?" Tags="|grading|homework|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14368" PostTypeId="2" ParentId="14367" CreationDate="2018-07-20T00:24:27.640" Score="0" Body="&lt;p&gt;Nope.  Nor should HS math stuff.  Just do kids to the board or grade each other's papers for no recorded grade.  and do frequent quiz and testing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Graded homework is a waste of instructor time.  And then in college (especially higher end ug or grad school), hw tends to become too &quot;project-y&quot;.  Loses the key pedagogy of drill.)&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-07-20T00:24:27.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14369" PostTypeId="2" ParentId="14367" CreationDate="2018-07-20T01:02:12.673" Score="8" Body="&lt;p&gt;I should preface this by saying that I am working in the US, at a large public university that serves a population that largely consists of first-in-family college attendees, low income students, and many students who speak a language other than English at home.  I have previously taught in low-income public high schools with similar demographic profiles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Homework gives students an opportunity to practice performing the kinds of computations that they are expected to learn in a course, or to build familiarity with the definitions and theorems that they are expected to understand by the end of a course.  I find it incredibly difficult to believe that a student can master material (at any level) without the kind of practice that is provided by homework assignments.  Since the goal of homework is to give students an opportunity to &lt;em&gt;practice&lt;/em&gt; the skills that they will ultimately be tested on, I am incredibly sympathetic to the argument that homework should not be graded, as the students that actually complete the homework will master the material and do well on exams, while students who don't complete the homework will probably fail those exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, students (especially undergraduates, and even more especially high school students) are highly motivated by &quot;points&quot; and often don't value anything that isn't worth any &quot;points&quot; in a class.  For example, one quarter I had two precalc classes, one right after the other.  In one section, I took attendance every day and factored that into a &quot;participation&quot; grade.  In the other section, I took attendance, but did not factor that into the grade at all.  The classes were otherwise as identical as they could be.  In the class where I gave credit for attendance, I had an average of about 90% daily attendance, while the other class averaged out to about 60%.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As such, I think that it is necessary to give students some kind of grade for completing their homework.  Moreover, I think that such a grade should be directly tied to the homework (rather than to quizzes based on homework problems; I think that students often fail to see the connection).  That being said, I think that the key work here is &lt;em&gt;completion&lt;/em&gt;.  I typically don't knock myself out grading every problem, and will often grade homework in a completely binary manner (1 point for turning in a substantially completed assignment, 0 points for not turning in anything or turning in something that is incomplete).  Alternatively, if homework can be done online (my institution uses either WebWork or WebAssign for most lower division classes), I tend to take the score given by the computer, and &quot;round up&quot; to the next letter grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal is to give students an incentive, in the form of &quot;points&quot;, to complete their homework.  I am trying to exploit my students' innumeracy (they'll work quite hard on their homework for a tiny fraction of their overall grade) and the &lt;a href=&quot;https://www.simplypsychology.org/operant-conditioning.html&quot; rel=&quot;noreferrer&quot;&gt;Skinner box-like&lt;/a&gt; reinforcement that comes from giving students &quot;points&quot; in order to get them to do something that is good for them (i.e. finish the homework).  At the end of the day, the homework grade shouldn't have that much effect on their overall grade (exams or projects will be much more important).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, &lt;strong&gt;yes&lt;/strong&gt; I grade homework, but (1) I focus on completion over accuracy and (2) I don't make the homework grade a huge component of the overall course grade.  In answer to the secondary question (&quot;Should I give homework quizzes?&quot;), I think that is a fine strategy, but I would employ it &lt;em&gt;in addition to&lt;/em&gt; assigning some kind of homework grade (with, perhaps, more weight given to the quizzes) and I would be careful not to spend too much lecture time on quizzes.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2018-07-20T12:42:28.757" LastActivityDate="2018-07-20T12:42:28.757" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14370" PostTypeId="1" CreationDate="2018-07-20T01:55:20.687" Score="5" ViewCount="201" Body="&lt;p&gt;Background: I’m not exactly a Math educator, but I’m currently a TA of an elementary algebra course aiming at students of age 14-15.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found that a lot of people have misconception about probability theory. Usually, when the (undergraduate) course director from my department encourages us to take the introductory probability course (a 3000-Level one), not so many students follow his advice. While he emphasized that learning probability theory is useful (for example, as a preparation for advanced courses like Financial Maths and Stochatic Processes), students don’t take the advice seriously. Instead,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;students who struggle in both analysis and algebra find it hard to fulfil the graduation requirement, and take the probability course because they think it is easier;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;students who are good at analysis or algebra think that probability theory is too applied (and in fact it isn’t...the course starts at the Kolmogorov axioms and ends at an elementary proof of the Strong Law of Large Number, and exams are all about proofs);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;students who love applied stuff (in particular Statistics) have usually chosen Statistics as their major, and they take their STAT courses instead of MATH courses.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This misconception doesn’t only appear in the department. During the tutorial I talked to some (young) students. Unfortunately, their understandings on probability theory only concern elementary combinatorics (such as finding out the number of ways to take out r objects out of n objects with certain constraints, which, I admit, is not quite Math-like compared to other subjects such as solving equations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did take the probability course as a major elective one and I find it interesting. However, I’m not sure whether I should rectify others’ misconceptions concerning probability theory, and if I should, how I should do so (perhaps as a piece of advice given to students/close friends at year 1-2).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any help is appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit:&#xA;Essentially I’m asking:&#xA;A lot of students think that probability theory is too applied, and this is certainly a misconception. How to clarify what probability theory actually involves (as an incentive for them to take the course), perhaps only using a few words? (e.g. during a short discussion concerning course selection)&#xA;This aims at persuading students who come from either pure (who think that probability theory is less theoretical) or applied (who treat the probability course as a very easy one) background.&lt;/p&gt;&#xA;" OwnerUserId="10174" LastEditorUserId="10174" LastEditDate="2018-07-21T05:31:56.400" LastActivityDate="2018-07-26T01:32:08.263" Title="Encouraging students to learn probability" Tags="|student-motivation|probability|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14371" PostTypeId="2" ParentId="14354" CreationDate="2018-07-20T10:52:03.813" Score="2" Body="&lt;p&gt;Thanks to @user541 for pointing to the CMBS survey:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Conference Board of Mathematical Sciences (CMBS) 2015 Survey Report.&#xA;  (&lt;a href=&quot;http://www.ams.org/profession/data/cbms-survey/cbms2015&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMS link&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It contains lots of data. The following is interesting, although it&#xA;would be more useful if normalized to the total enrollments at the&#xA;institutions. Without that, an increase in math degrees could be due&#xA;to an increase in all degrees. &#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/S6xx8.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/S6xx8.jpg&quot; alt=&quot;CMBS_fig&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;CMBS, Chapter 3, p.97.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/VZJtB.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VZJtB.jpg&quot; alt=&quot;CMBS_txt&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;CMBS, Chapter 3, p.85.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-07-20T10:52:03.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14372" PostTypeId="2" ParentId="14367" CreationDate="2018-07-20T21:53:41.017" Score="3" Body="&lt;p&gt;Great answer is given here &lt;a href=&quot;https://rryshke.blog/2011/09/04/whats-the-value-of-homework-and-should-it-be-graded/&quot; rel=&quot;nofollow noreferrer&quot;&gt;What's the value of homework and should it be graded&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Homework should be given only if the teacher is prepared to give&#xA;  students feedback on their performance.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Additionally, even cheaters can be forced to understand the solution they have found in internet and to adapt it on the problems they were given.&lt;/p&gt;&#xA;" OwnerUserId="1520" LastActivityDate="2018-07-20T21:53:41.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14374" PostTypeId="2" ParentId="3952" CreationDate="2018-07-23T22:29:56.100" Score="5" Body="&lt;p&gt;To explain the answer, I found this question while thinking about asking a similar question.  In teaching both pre-calculus and calculus, this is an issue that seems to come up over and over again in my professional life&amp;mdash;students need to be given some notion of limits so that the rest of calculus can be made to work, but the full rigor of $\varepsilon$-$\delta$ arguments are not appropriate (here, I am thinking of a US-centric curriculum, with a student population that consists mostly of non-math majors who will never take analysis).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the last year, I have been experimenting more with a sequential-first approach to teaching limits.  In addition to your list of pros, I would suggest the following as motivation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pros:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I think that sequential limits are more intuitive.  In fact, something that I see students do a &lt;em&gt;lot&lt;/em&gt; on exams is try to compute limits sequentially.  For example, they might try to compute&#xA;$$ \lim_{x\to 5} \frac{x^2-25}{x-5} $$&#xA;by evaluating the rational expression at several values of $x$ that are close to 5.  They are clearly &lt;em&gt;thinking&lt;/em&gt; of some kind of sequential definition of a limit, hence I think that it is reasonable to harness that intuition and make it rigorous.&lt;/li&gt;&#xA;&lt;li&gt;In some sense, sequential limits are &quot;simpler&quot;.  When taking the limit of a function at a point, there are &lt;em&gt;two&lt;/em&gt; quantities to keep track of:  the limit point in the domain, $\color{red}x$, and the values in the codomain, $\color{blue}{f(x)}$:&#xA;$$ \lim_{\color{red}{x\to a}} \color{blue}{f(x)}. $$&#xA;In contrast, all sequential limits are taken at infinity.  We don't have to worry about the points in the domain, and can focus on the behaviour of the function (i.e., the sequence) itself.  I am almost tempted to introduction notation like&#xA;$$ \operatorname{Lim} a_n := \lim_{n\to \infty} a_n \qquad\text{or}\qquad \operatorname{Lim}\, \left(a_n\right) := \lim_{n\to \infty} a_n$$&#xA;to denote a sequential limit, thereby removing some extraneous notation from the first introduction to a limit (I've not tried this yet, and am still not convinced that it is a good idea, but the overall point is that we don't have to worry about notation for $n$ going to infinity&amp;mdash;at worst, the notation $\lim_{n\to\infty}$ has some extra decoration which we can explain later).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There is, unfortunately, a fairly important downside to this approach that I think has been missed, which is going to take some explanation to get to.  If we want to be really rigorous, we would probably start with a good definition of a sequential limit (your students &lt;em&gt;might&lt;/em&gt; not see this, but it should be in your notes somewhere, I think):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Definition:&lt;/strong&gt;  Let $(a_n)_{n\in\mathbb{N}}$ be sequence and let $L \in \mathbb{R}$.  We say that&#xA;  $$ \lim_{n\to\infty} a_n = L $$&#xA;  if for any $\varepsilon &amp;gt; 0$ there exists some $N$ sufficiently large such that&#xA;  $$ |a_n - L| &amp;lt; \varepsilon $$&#xA;  whenever $n \ge N$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Because the Greek letter $\varepsilon$ is often kind of scary, I typically write $\text{error} &amp;gt; 0$ and $|a_n - L| &amp;lt; \text{error}$ when I first introduce the idea, then transition to Greek a bit later.  I also try to spend a good amount of time explaining the basic idea of this definition, i.e. we set an error tolerance, then seek to find a value of $N$ that is so large that every term of the sequence is closer to the limit $L$ than the error tolerance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The alternative idea is to pull out a calculator and work a bunch of examples.  For example, we might consider&#xA;$$ \lim_{n\to \infty} \frac{ 5n^2 - 8}{3(n+1)(n+2)}, $$&#xA;and try to get some intuition for the limit by evaluating this expression at various large values of $n$ (say, $n=1$, $10$, $100$, $1000$, $10^6$, and so on).  It becomes apparent fairly quickly that this is converging.  In an introductory class (i.e. &lt;em&gt;not&lt;/em&gt; real analysis, &lt;em&gt;not&lt;/em&gt; an honors calculus class; but maybe a regular calculus and definitely a pre-calculus class), the examples are generally enough to drive home the point, and a rigorous definition can be elided.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once an intuition about sequential limits is developed, the next step is to work with continuous or functional limits.  The &quot;right&quot; definition in this case is something like the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Definition:&lt;/strong&gt; Let $f : X \to \mathbb{R}$ be a function defined on some set $X \subseteq \mathbb{R}$, let $L \in \mathbb{R}$, and assume that $I$ is an open interval such that $I\setminus\{a\} \subseteq \mathscr{D}(f)$.  We say that&#xA;  $$ \lim_{x\to a} f(x) = L $$&#xA;  if for any sequence $(x_n)$ such that $\lim_{n\to\infty} x_n = a$ we have $\lim_{n\to\infty} f(x_n) = L$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A lot of the technicalities here can be elided&amp;mdash;the real issue here is the use of quantifiers.  In order for the limit of $f$ to exist at $a$, we require that &lt;em&gt;for every&lt;/em&gt; sequence $(x_n)$ such that $x_n \to a$, we have $f(x_n) \to L$.  You can't just check a few examples&amp;mdash;you have to check every possible sequence, which is hard.  Counterexmples are easier to deal with, since it is necessary only to find one sequence which fails to converge (e.g. consider $\lim_{x\to 0} \sin(\frac{1}{x})$), but actually showing that anything converges is much harder.  In short:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cons:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Students at an introductory level have great difficulty with quantifiers.  The fact that $\lim_{n\to \infty} f(x_n)$ must be $L$ for all possible sequences $(x_n)$ which tend to $a$ is a really, really difficult idea for many students.&lt;/li&gt;&#xA;&lt;li&gt;Beyond just coming to terms with quantifiers, showing that something is true for &lt;em&gt;every&lt;/em&gt; sequence $x_n \to a$ is a pretty big leap for most students.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In an introductory class, we can elide many of these these issues, and claim that it is &quot;good enough&quot; to check several sequences in a couple of key examples, then move on to continuity (and use properties of continuity to sort of &quot;backfill&quot; the skipped technicalities).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a class for math majors, this becomes much more delicate, and I think that there is no way of avoiding the eventual introduction of the $\varepsilon$-$\delta$ definition (though I don't think that the original question is seeking to avoid such definitions; only to put them off a bit).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A reference:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding references, I don't really have any that are &lt;em&gt;really&lt;/em&gt; appropriate for an introductory class.  However, a text that approaches real analysis from a chronological or historical approach (rather than an axiomatic approach) may give some good ideas.  For example, I recently picked up a cheap used copy of Saul Stahl's &lt;em&gt;Real Analysis: A Historical Approach&lt;/em&gt;, which seems (so far) to be a pretty solid text.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2018-07-24T14:46:09.513" LastActivityDate="2018-07-24T14:46:09.513" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14375" PostTypeId="2" ParentId="14320" CreationDate="2018-07-23T22:47:52.310" Score="5" Body="&lt;p&gt;As is often the case on this site, I slightly disagree with the specific premises of the question... and claim that tweaking those premises does help to solve the problem at hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Namely, although the logically linearized version of mathematics often does present definitions first, this is mildly perverse, since the people who conceived those definitions rarely wrote them down and then tried to look for examples, or tried to think what they could prove about the just-defined things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather, a &quot;definition&quot; is nearly an end-product of usually very substantial inquiries, including many, many examples whose nature needs to be considered &lt;em&gt;before&lt;/em&gt; the &quot;definition&quot; is made, since the definition needs to be inclusive enough, but not accidentally too inclusive to inadvertently lose the key motivations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, seriously, I myself only like &quot;definitions&quot; as sort of &quot;wrap-up&quot; of discussions. When grad students giving talks or writing papers want to tell me their &quot;definition&quot; first, I politely interrupt to ask for the examples that led to their definitions...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This does &lt;em&gt;approximately&lt;/em&gt; lead back to the operational point of coming up with illustrative examples! Indeed! But/and to more genuinely mimic or simulate or emulate genuine mathematical thinking I'd claim that the examples should be given first. &lt;em&gt;And&lt;/em&gt; then ask the students to formalize/definitionalize those examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This also has the psychological benefit of giving students/people more validation, since they're not having definitions dictated to them by some authority-entity, but get to make their own. :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to let the other shoe drop... Groups of symmetries of geometric objects. E.g., regular polygons... first in 2D, and then considering reflections. Then, regular polyhedra... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, Galois theory &quot;for citizens&quot;, talking about the &lt;em&gt;algebraic&lt;/em&gt;  indistinguishability of the various roots...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Irrationality of various numbers. A trope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complex numbers as a two-dimensional incarnation of &quot;number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, quaternions...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, then, ask people what was the goal there... ? :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2018-07-24T01:01:35.270" LastActivityDate="2018-07-24T01:01:35.270" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14376" PostTypeId="2" ParentId="3952" CreationDate="2018-07-24T03:35:38.257" Score="-3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;you don't need epsilon delta to cover limits of functions.  Just derivatives.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(As someone else said) you lose the visual aspect of an asymptote.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You have more messy algebra with the terms of the series.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Other aspects of series/sequences taught later in the course (e.g. Maclaurin and Taylor).  And if you move it all to earlier, it just delays learning some calculus (min/max problems, rates, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Net, net:  There is a good pedagical reasons for the current order.  This is another suggestion from someone who thinks in terms of already knowing the material and thinks in terms of logical exposition versus the most incremental steps for a new learner.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-07-24T03:35:38.257" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14377" PostTypeId="1" AcceptedAnswerId="14378" CreationDate="2018-07-24T17:05:54.413" Score="5" ViewCount="253" Body="&lt;p&gt;I've been out of a math program for about four years now. My wife is starting a CS degree, and finished her first calculus course last semester. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tutored calculus throughout my entire undergrad, and have read quite a few calculus and real analysis textbooks. When &lt;em&gt;I'm&lt;/em&gt; referring to a candidate for an $x$-value at which a point of extremum is located, I've always called it a &lt;strong&gt;critical point&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But my wife's professor taught three different terms:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Critical number&lt;/strong&gt;: the $x$-value at which the possible extremum is located&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Critical value&lt;/strong&gt;: the corresponding $y$-value of the critical number&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Critical point&lt;/strong&gt;: a pair $(\text{critical number}, \text{critical value})$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I don't understand the pedagogical value in creating these three terms.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;This doesn't look like standard terminology. I was confused at first when I was trying to help my wife learn about critical points.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Critical values&quot; are rarely used in what I specialize in: statistics. I care more about &lt;em&gt;where&lt;/em&gt; the extrema are possibly located, rather than what the actual $y$-values are at these possible candidates of extrema. And I like dropping constants whenever possible when it comes to having to find points of extrema, so the $y$-value is useless.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Can someone provide some rationale for why one would want to distinguish between these three bolded terms above?&lt;/p&gt;&#xA;" OwnerUserId="1907" LastEditorUserId="1907" LastEditDate="2018-07-25T02:18:20.980" LastActivityDate="2018-07-27T01:43:52.877" Title="What is the value in creating distinguishing terminology between the $x$, $y$, and $(x, y)$ values of a possible point of extremum?" Tags="|calculus|terminology|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14378" PostTypeId="2" ParentId="14377" CreationDate="2018-07-24T18:37:05.297" Score="3" Body="&lt;p&gt;One rationale for introducing terms for these three objects is to encourage students to be clear about whether they are describing the input, output or pair $(x, y)$ for a function, and giving them language to do so efficiently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the question &quot;&lt;em&gt;Where is the maximum of the function $f$ on the given domain&lt;/em&gt;?&quot; I've seen this type of wording trip students up, particularly if the back of the book just says &quot;$5$&quot;. It's totally unclear (to me) whether the directive is to find the maximum &lt;em&gt;output&lt;/em&gt; value, the &lt;em&gt;input&lt;/em&gt; corresponding to that maximum output, or possibly the ordered pair. Giving names to these three things avoids a wordier rephrasing such as &quot;&lt;em&gt;At what x-value is $f(x)$ maximized on the given domain&lt;/em&gt;?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, while my immediate reaction to introducing non-standard terminology is not positive, I do respect an attempt to get students to exercise care in how they communicate their ideas -- and one good way is by giving them language to do so. However, I think even more care should be exercised when introducing any subject-specific terminology, and new or non-standard terminology should almost never be given formally to a class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Note: On my shelf, Hughes-Hallett, Taalman, Anton and Faires all use &quot;critical point&quot; for the $x$-value, while two of them mention &quot;critical value&quot; and &quot;stationary point&quot;. Sherwood and Taylor (c 1942) use &quot;critical point&quot; for the $(x,y)$ pair and &quot;critical &lt;em&gt;value&lt;/em&gt;&quot; for the $x$-coordinate.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-07-24T18:37:05.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14379" PostTypeId="1" CreationDate="2018-07-25T03:45:32.637" Score="8" ViewCount="142" Body="&lt;p&gt;I am looking for an online quiz that can generate multiple versions of a multiple choice question. What I want is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The same question can have a different text for each student&lt;/li&gt;&#xA;&lt;li&gt;Alternatives are randomized and picked from a larger pool (the student sees 5 alternatives, 1 correct, but I provided 10, 2 correct, and the software chose)&lt;/li&gt;&#xA;&lt;li&gt;Cloud based (so that I do not have to maintain a system running)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Does such a thing exist?&lt;/p&gt;&#xA;" OwnerUserId="1662" LastEditorUserId="1147" LastEditDate="2018-07-26T02:14:23.417" LastActivityDate="2018-07-26T02:14:23.417" Title="Online quiz with slightly different questions for each student" Tags="|online-instruction|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14380" PostTypeId="2" ParentId="14377" CreationDate="2018-07-25T07:47:25.183" Score="5" Body="&lt;p&gt;The standard terminology in use among mathematicians is to call a &lt;em&gt;critical point&lt;/em&gt; a point $p$ where the differential of a differentiable function $f:M \to \mathbb{R}$ vanishes ($M$ is a smooth manifold, for example an open subset of $\mathbb{R}$), and to call a &lt;em&gt;critical value&lt;/em&gt; the value $r$ it assumes at such a point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proposed terminology, calling $p$ a &lt;em&gt;critical number&lt;/em&gt; only works in the limited context where $M$ is an open subset of $\mathbb{R}$ or $\mathbb{C}$, so can properly be called a &lt;em&gt;number&lt;/em&gt;. It is confusing to call the pair $(p, r)$ a critical point for another reason. The point $p$ and the value $r$ &quot;live&quot; in different spaces. The pair $(p, r)$ lives in $M \times \mathbb{R}$, while $p$ lives in $M$ and $r$ lives in $\mathbb{R}$. In the special case where $M$ is an open subset of $\mathbb{R}$, then $(p, r)$ can be viewed as a point in $\mathbb{R}^{2}$, but this seems to me to generate unnecessary confusion, because the roles of the two coordinates are not symmetric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, one certainly wants to distinguish between the point at which a function is critical and the value it assumes there. This is just formalizing the distinction between the day of the year on which the coldest temperature occurs and what that temperature is, or the distinction between the point on the earth where the elevation above sea level is highest, and what that elevation is. One needs different names for different things. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-07-25T07:47:25.183" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14382" PostTypeId="2" ParentId="14379" CreationDate="2018-07-25T09:56:33.547" Score="4" Body="&lt;p&gt;My institute is using this system: &lt;a href=&quot;http://www.math.rwth-aachen.de/~OKUSON/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.rwth-aachen.de/~OKUSON/&lt;/a&gt; While it takes some time to understand how to use it  - and it is sometimes a little annoying, as it was written by Mathematicians who needed something for their exercises, not by Computer Scientists aiming to sell a perfect product - it can do what you want and it is free and open source.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2018-07-25T09:56:33.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14383" PostTypeId="2" ParentId="14377" CreationDate="2018-07-25T10:12:27.380" Score="1" Body="&lt;p&gt;In primary, secondary, and some basic undergraduate courses, a point is an $n$-tuple (an ordered pair or an ordered triple), that is, a point is not a number.  For a function of a single variable $y=f(x)$, we refer to $(x,y)$ as the point with abscissa $x$ and ordinate $y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I can imagine that it would be confusing for these students if, given $(x_0,f(x_0))$, one refers to $x_0$ as a point.  It would make more sense for them to call $(x_0,f(x_0))$ a point.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-07-25T10:12:27.380" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14384" PostTypeId="2" ParentId="14379" CreationDate="2018-07-25T12:03:48.550" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://webwork.maa.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;WeBWorK&lt;/a&gt; of course&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;WeBWorK is an open-source online homework system for math and sciences courses. WeBWorK is supported by the MAA and the NSF and comes with a National Problem Library (NPL) of over 20,000 homework problems. Problems in the NPL target most lower division undergraduate math courses and some advanced courses. Supported courses include college algebra, discrete mathematics, probability and statistics, single and multivariable calculus, differential equations, linear algebra and complex analysis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="127" LastActivityDate="2018-07-25T12:03:48.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14385" PostTypeId="1" CreationDate="2018-07-25T13:07:04.523" Score="7" ViewCount="2750" Body="&lt;p&gt;I have always taught my students that the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept of a line is the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-coordinate of the point of intersection of a line with the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis, that is, for the line given by the equation &lt;span class=&quot;math-container&quot;&gt;$y=mx+y_0$&lt;/span&gt;, the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept is &lt;span class=&quot;math-container&quot;&gt;$y_0$&lt;/span&gt;.  I emphasize that that the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept is the &lt;em&gt;number&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$y_0$&lt;/span&gt; and not the &lt;em&gt;point&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$(0,y_0)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But I was quite surprised when I recently looked at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Intercept&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; and &lt;a href=&quot;http://mathworld.wolfram.com/x-Intercept.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wolfram&lt;/a&gt; &lt;a href=&quot;http://mathworld.wolfram.com/y-Intercept.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathWorld&lt;/a&gt; entries for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept because these define the intercept as a point and not as a number (&amp;quot;the point where a line crosses the y-axis&amp;quot; and &amp;quot;The point at which a curve or function crosses the y-axis&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Further investigation yielded inconsistencies:  the Wikipedia entry for &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Line_(geometry)#On_the_Cartesian_plane&quot; rel=&quot;nofollow noreferrer&quot;&gt;Line (geometry)&lt;/a&gt;&amp;quot; states that in the equation &lt;span class=&quot;math-container&quot;&gt;$y=mx+b$&lt;/span&gt;, &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is the y-intercept of the line&amp;quot;; the Wolfram MathWorld entry for &amp;quot;&lt;a href=&quot;http://mathworld.wolfram.com/Line.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Line&lt;/a&gt;&amp;quot; states that &amp;quot;The line with &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and slope &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is given by the slope-intercept form &lt;span class=&quot;math-container&quot;&gt;$y=mx+b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;Edit made on February 21, 2021&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;According to the &lt;em&gt;Dictionary of Analysis, Calculus, and Differential Equations&lt;/em&gt; (edited by Douglas N. Clark, published by CRC Press in 2000),&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;intercept&lt;/strong&gt; The point(s) where a curve or graph of a function in &lt;span class=&quot;math-container&quot;&gt;$\mathbf R^n$&lt;/span&gt; crosses one of the&#xA;axes. For the graph of &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbf R^2$&lt;/span&gt;, the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-&lt;em&gt;intercept&lt;/em&gt; is the point &lt;span class=&quot;math-container&quot;&gt;$(0,f(0))$&lt;/span&gt; and the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-&lt;em&gt;intercepts&lt;/em&gt; are the points &lt;span class=&quot;math-container&quot;&gt;$(p,f(p))$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f(p)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately, the book does not consistently use that definition.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;slope-intercept equation of line&lt;/strong&gt; An equation of the form &lt;span class=&quot;math-container&quot;&gt;$y=mx+b$&lt;/span&gt;, for a straight line in &lt;span class=&quot;math-container&quot;&gt;$\mathbf R^2$&lt;/span&gt;. Here &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is the slope of the line and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept; that is, &lt;span class=&quot;math-container&quot;&gt;$y=b$&lt;/span&gt;, when &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus, even though the book defines an intercept as a point, it uses the term to denote a number.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Is there a trusted source targeted at mathematics educators (from, say, a government agency, an educational institution, or an organization) that defines &amp;quot;intercept&amp;quot; and consistently uses that definition?&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="376" LastEditDate="2021-02-21T15:27:26.300" LastActivityDate="2021-02-21T15:27:26.300" Title="What is an intercept?" Tags="|reference-request|terminology|definitions|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14386" PostTypeId="2" ParentId="14385" CreationDate="2018-07-25T15:29:05.137" Score="18" Body="&lt;p&gt;This is a case where you might be looking for a distinction that's pretty subtle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By definition, the y-intercept occurs at x=0. In one notation, it's literally f(0), where the x is explicitly offered. I'd be ok with a student's answer to &quot;What is the y-intercept?&quot; to be simply the y value, or the $(0,y_0)$ point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a teacher prefers one, you can ask&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is the y value of the y-intercept?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;or &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Give the point (coordinate) of the y-intercept.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;When I was in high school, one math teacher was fussy about 'negative' vs 'minus'. He insisted that an answer, &quot;-4&quot; should never be pronounced &quot;minus four&quot;. He declared &quot;negative is an adjective, minus is a verb.&quot; While I suppose this is true, I never found value in correcting a student who is otherwise doing the math correctly. This case may be similar, if they are getting the concept, don't focus on a point (pun intended) that may be a matter of preference. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-07-25T15:29:05.137" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14387" PostTypeId="2" ParentId="14385" CreationDate="2018-07-25T15:44:23.020" Score="8" Body="&lt;p&gt;Some time spent with the Google does not seem to turn up very much.  The best answer that I can find seems to be from the Common Core Standards, and is ambiguous, at best.  On page 69 of the &lt;a href=&quot;https://www.engageny.org/resource/new-york-state-p-12-common-core-learning-standards-for-mathematics&quot; rel=&quot;noreferrer&quot;&gt;Core Learning Standards&lt;/a&gt; for mathematics (as provided by the State of New York; the same line appears in &lt;a href=&quot;https://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf&quot; rel=&quot;noreferrer&quot;&gt;California's document&lt;/a&gt; on page 94), it states that students should be able to&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[i]nterpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are other references to intercepts, but they are ambiguous, at best.  This seems to indicate that the Common Core regards intercepts as values, not points.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;A quick perusal of the books on my (and my office-mate's) shelves gives an equally ambiguous answer.  Most (though not all) of the precalculus texts say that the line given by $y = mx + b$ has the value $b$ as it's $y$-intercept, while most of my calculus texts define the $y$-intercept to be the point $(0,b)$.  A notable exception (notable as it is quite widely used) is a copy of the 6th edition of Stewart's &lt;em&gt;Calculus&lt;/em&gt;, which gives the $y$-intercept as a value, not a point.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2018-07-25T15:44:23.020" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14388" PostTypeId="1" CreationDate="2018-07-25T18:15:00.350" Score="9" ViewCount="428" Body="&lt;p&gt;This is a soft question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What is the purpose of teaching techniques of integration at the college level?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;More specifically, in the sense of putting integration into practice, what value does teaching these techniques serve in a world which increasingly depends on approximation and computer algebra systems?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If the use of these techniques in practice is limited, then what &lt;em&gt;should&lt;/em&gt; integral calculus be teaching in order to make the content imparted by the class more actionable?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am assuming that, in general, the students of integral calculus will not go on to be mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="10192" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-07-30T23:11:59.017" Title="Integral calculus from the modern viewpoint" Tags="|teaching|interactive-teaching|integration|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14389" PostTypeId="2" ParentId="14388" CreationDate="2018-07-25T21:39:12.767" Score="7" Body="&lt;p&gt;1: Up until the collegiate level, most math students memorize the 'right' way to do certain types of problems, and then repeat that hundreds of times on homework, so that they can repeat it on a test.&lt;/p&gt;&#xA;&lt;p&gt;However, with integration there are too many possible forms to recognize and memorize each one. Instead the student is forced to try to figure out their own way of solving the problem from a common set of rules. Even though this process can often be automated, there are similar reasoning tasks, such as proof-writing, which are impractical to automate at this time. There are reasoning tasks in other fields such as programming and writing which also require the same kind of thinking: manipulation by predefined rules.&lt;/p&gt;&#xA;&lt;p&gt;This does not always have its intended effect, as many students are stuck in the memorization-mindset, and it is easier for professors to say what to do without explaining why or what it means. There is (unfortunately) still much memorization involved, but it asks for more original thinking than prior math questions have.&lt;/p&gt;&#xA;&lt;p&gt;2: For approximations, people try not to approximate things that could be computed directly to get better accuracy; approximations are usually for things that &lt;em&gt;can't&lt;/em&gt; be integrated analytically.&lt;/p&gt;&#xA;&lt;p&gt;Knowing what an integral &lt;em&gt;really means&lt;/em&gt; (as opposed to knowing how to operate a CAS) is both useful and important even if students do not become mathematicians. The concepts of integrals and other infinitesimals lay the foundation for differential equations and stats. These are heavily utilized by all STEM fields, business, and the social sciences.&lt;/p&gt;&#xA;&lt;p&gt;Rather, they should be viewed as two complementary skills that are valuable in different ways: learning how to use a CAS, and learning the theory (what it really means, why it is that particular way).&lt;/p&gt;&#xA;&lt;p&gt;3: Maybe &amp;quot;integrate this function&amp;quot; problems gets boring and distant. It feels really useless if you could just type it into a CAS. Good professors will ask questions whose difficulty does not come from bookkeeping symbols but from figuring out what to do next.&lt;/p&gt;&#xA;&lt;p&gt;Example: explain why the area of a circle is &lt;span class=&quot;math-container&quot;&gt;$\pi r^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Try to answer this question on your own.&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; If you imagine the circle like an onion slice with concentric rings of width &lt;span class=&quot;math-container&quot;&gt;$dr$&lt;/span&gt; (just a weird looking name for a small number), then you could 'unroll' each ring into a rectangle whose length would be the circumference of that ring &lt;span class=&quot;math-container&quot;&gt;$2 \pi r$&lt;/span&gt; and width would be &lt;span class=&quot;math-container&quot;&gt;$dr$&lt;/span&gt;. Adding up all of the rings, and &lt;em&gt;voilà&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\int 2 \pi r\ dr = \pi r^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Or maybe don't teach integral calculus at all&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have answered thus far as a 'Devil's advocate', and I am sympathetic to your viewpoint. I think there is some use to learning higher-level math so that students become more logical thinkers, but maybe not integral calculus. Integration involves more memorization than I would like, and ends up being tedious rather than rewarding.&lt;/p&gt;&#xA;&lt;p&gt;I would be happy to see a number theory course be offered in place of calculus. Number theory asks questions like &amp;quot;how many prime numbers are there?” These questions sometimes have counterintuitive and interesting answers.&lt;/p&gt;&#xA;" OwnerUserId="10193" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-07-30T15:12:57.937" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14390" PostTypeId="2" ParentId="14370" CreationDate="2018-07-26T01:32:08.263" Score="2" Body="&lt;p&gt;Perhaps you could give a sample of the important central subjects of study. I would say something like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In this course, we will utilize basic axioms to prove many results, culminating in the Law of Large Numbers: in the limit that the number trials goes to infinity, the observed average converges to its theoretically expected value. Calculus and proof-reasoning are heavily utilized.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It signals the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;this is a 'real' math course with proofs, not just combinations/permutations.&lt;/li&gt;&#xA;&lt;li&gt;the depth of the course; interested students know that the course culminates in the Law of Large Numbers, so they have a better idea of how applied it is.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10193" LastActivityDate="2018-07-26T01:32:08.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14391" PostTypeId="1" AcceptedAnswerId="14418" CreationDate="2018-07-26T01:48:15.493" Score="8" ViewCount="507" Body="&lt;p&gt;I am currently tutoring a student who is really lacking in math.  At first I thought she was just resistant; perhaps she thought that her teacher had no choice but to pass her.  What truly stunned me is that she is going to tenth-grade and has very little common sense with numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once asked her what is 3 minus 1/2, and she could not answer it.  So, I drew three circles in front of her and shaded out one half.  Now I asked how many circles are left, and she still could not answer it ....  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An episode like this is not a rarity with her.  Moreover, too often she would give responses that even a third-grader would know were nonsense.  (What's half-way between 19.5 and 20?  She said 9.)  According to her mom, she is fine with her other classes.  That's why I am wondering if a person can have issues with learning math specifically.  &lt;/p&gt;&#xA;" OwnerUserId="8586" LastEditorUserId="2083" LastEditDate="2018-07-30T10:42:55.910" LastActivityDate="2019-02-28T11:49:34.447" Title="Is There Such Thing as Learning Disability in Math?" Tags="|tutoring|struggling-students|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14392" PostTypeId="2" ParentId="12451" CreationDate="2018-07-26T02:28:52.690" Score="1" Body="&lt;p&gt;The accuracy limitations of a straight-edge and compass are more obvious than the accuracy limitations of a drawing program.  And drawing programs do have accuracy limitations, which are sometimes worse than those of a straight-edge and compass.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Floating point numbers are quantized, and not especially consistently.&lt;/li&gt;&#xA;&lt;li&gt;Many drawing programs only pretend to draw arcs and circles.  They actually draw regular polygons instead.&lt;/li&gt;&#xA;&lt;li&gt;Many drawing programs only pretend to draw ellipses.  Instead they draw &quot;4-circle ellipses&quot; that are actually 4 pieces of regular polygons.&lt;/li&gt;&#xA;&lt;li&gt;Rasterized screenshots and printouts are either quantized enough to be less accurate than a straight-edge, or are fuzzy enough to be less accurate than a straight-edge.&lt;/li&gt;&#xA;&lt;li&gt;Paper sizes can vary by a fraction of a percent from day-to-day.  Most drawing software does not have any feature for calibrating print outs.&lt;/li&gt;&#xA;&lt;li&gt;Some drawing programs are not designed to simply shade the area of a shape; they insist on outlining the shape in black as well.  Ellipses and circles are especially likely to have this problem.&lt;/li&gt;&#xA;&lt;li&gt;Many drawing programs do not make clear where the mid-line of the outline is with respect to the theoretical shape being drawn.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2018-07-26T02:28:52.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14393" PostTypeId="2" ParentId="14385" CreationDate="2018-07-26T03:21:45.943" Score="2" Body="&lt;p&gt;It's worth noting that a real number is a point on the real number line (according to, for instance, Stewart's &lt;em&gt;Calculus&lt;/em&gt;, Appendix A).  The $y$-axis is a copy of the real number line.  So, one could take the defensible position that a $y$-intercept is just as much a point $(0,b)$ as it is a point $b$ along the $y$-axis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a little pedantic gloss: linear equations and lines in the plane are different concepts, yet they both have a $y$-intercept.  Authors disagree on what the $y$-intercept of a linear equation $y=mx+b$ should be: is it $b$ itself? or is it the point where the graph of the equation intersects the $y$-axis?  In any case, an equation is not literally a line, so we have to recognize that it's the same terminology for two different things.  In the end, the whole $y$-intercept concept is about capturing a connection between geometry (lines) and algebra (linear equations), and since the connection is so strong it might not be worth forcing a distinction, unless it's to make a pedagogical point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I did some digging into geometry, and there are the concepts of intercepted line segments and intercepted arcs.  The $y$-intercept is related to the intercepted line segment on the $y$-axis, bounded by the $x$-axis and the given line.  Strangely, I couldn't find any mention of calling the actual point where two intersecting lines meet the intercept, except for the $x$- and $y$-intercepts.)&lt;/p&gt;&#xA;" OwnerUserId="8981" LastEditorUserId="8981" LastEditDate="2018-07-26T03:38:15.070" LastActivityDate="2018-07-26T03:38:15.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14394" PostTypeId="2" ParentId="14385" CreationDate="2018-07-26T06:24:24.893" Score="7" Body="&lt;p&gt;This questions reflects the dangers in over formalization of the language used to discuss simple things. A pedantic speaker might distinguish between the &lt;em&gt;y-intercept b&lt;/em&gt; and the &lt;em&gt;intercept (0, b)&lt;/em&gt; (although &lt;em&gt;intersection point (0, b)&lt;/em&gt; might be a better name for the latter), but very little is gained by fussing about such a distinction, particularly at the elementary level. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example is very different from that some commenters have referenced that treats the distinction between critical point and critical value. The point on the earth where the temperature is lowest and what that temperature is are two very different things in qualitative terms, and so one needs terminology to distinguish them. In the present case, there is very little difference qualitatively, the fussing is about whether one choose to refer to the point or its ordinate as the &lt;em&gt;y-intercept&lt;/em&gt;. Both choices are reasonable and neither is consequential. The difficulty arises mostly because of overly rigid teachers who insist that one is right and the other is wrong because &lt;em&gt;I said so&lt;/em&gt; or for some similar reason similarly lacking in reason.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice I think it's best to adopt a consistent usage, simply because that is easier for students, but not to fuss when individual students, for having read a book on their own, gone to a tutor, or simply not speaking with the same pedantic care as the teacher, use the other terminology, as long as it is clear that they have answered correctly whatever question requires its use.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-07-26T06:24:24.893" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14395" PostTypeId="2" ParentId="14385" CreationDate="2018-07-26T10:00:23.870" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/14387/6275&quot;&gt;JoeTaxpayer's answer&lt;/a&gt; says the distinction is subtle. To me, the distinction is non-existent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't see any benefit in discerning two concepts that are &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Closely related&lt;/li&gt;&#xA;&lt;li&gt;Have a trivial bijection between the two concepts&lt;/li&gt;&#xA;&lt;li&gt;Do not appear separately in any other context&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In this case, the intercept value $y_0$ is trivially bijected to the intercept point $(0, y_0)$, and the large part of mathematical education is teaching students to recognize equivalent concepts and group them together.&lt;/p&gt;&#xA;" OwnerUserId="6275" LastActivityDate="2018-07-26T10:00:23.870" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14396" PostTypeId="2" ParentId="14385" CreationDate="2018-07-26T14:47:23.820" Score="0" Body="&lt;p&gt;It comes down to convention, ultimately, since specifying a unique point $y_0$ associated to $x_0$ is equivalent to specifying the ordered pair $(x_0,y_0)$ on the graph of $f(x)=y.$ That is, since a function is well defined by definition, then there is no ambiguity if one requires that the intercept &quot;$y_0$&quot; be associated to a given input &quot;$x_0$&quot;.&lt;/p&gt;&#xA;" OwnerUserId="10192" LastActivityDate="2018-07-26T14:47:23.820" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14397" PostTypeId="2" ParentId="12451" CreationDate="2018-07-26T18:31:55.617" Score="1" Body="&lt;p&gt;One simple answer is that they are such basic tools to make and use. Obviously, there are historical reasons for using a compass and straight edge, but they are also very practical in carpentry.&lt;/p&gt;&#xA;&lt;p&gt;For example, just last weekend I was helping a friend build a deck, and we had to miter-cut some boards to fit some arbitrary (unmeasured) angle. We needed the miter-cut to bisect the inside obtuse angle so the cut edges would be the same length, and hence fit well together.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gGIpU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gGIpU.png&quot; alt=&quot;Angle Cut&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Not being adept with the tools at hand (speed square) for this task, I made marks on the two boards (equidistant from the vertex point) and &amp;quot;made&amp;quot; a beam compass (i.e. held a straight stick) to bisect the angle. It turned out fine, as expected.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zGFcb.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zGFcb.png&quot; alt=&quot;Angle Bisection&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, I am thankful that I not only knew the theory of bisecting an angle with a compass and straight edge, but that I had also actually used such devices in various forms. I think exclusive use of a drawing program that allows one to simply &amp;quot;bisect an angle&amp;quot; with a button click would not have given me the requisite understanding to perform this task, but if the program makes you create circles and straight lines individually, that's another story.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-07-26T18:31:55.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14398" PostTypeId="2" ParentId="14385" CreationDate="2018-07-26T20:13:57.277" Score="-1" Body="&lt;p&gt;$y$ is a value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difference between the two would essentially boil down to &lt;em&gt;dimensions&lt;/em&gt; of $y$, with y-as-value being uni-dimensional and y-as-point being n-dimensional. Now, even with multiple $x$'s, all information about the intercept (think about difference in the intercepts of two models) is captured in y-as-value.&#xA;The other dimensions never come into play. &lt;/p&gt;&#xA;" OwnerUserId="10200" LastActivityDate="2018-07-26T20:13:57.277" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14400" PostTypeId="2" ParentId="14388" CreationDate="2018-07-27T01:18:23.050" Score="7" Body="&lt;p&gt;There are many techniques of integration. Some of them, like integration by parts, are important theoretically. Integration by parts shows up in the derivation of the Euler-Lagrange equations in classical mechanics and in other situations where we study functionals on a domain with boundary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other techniques for integrals are there to complete the table of integrals of basic functions. It sounds like box-checking, but stats and physics can come up with some pretty wacky functions. That being said, I wish I had a non-artificial example of when $\int \sin^m x \cos^n x\ dx$ would come up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I think that it would be nice to have Applied Calculus classes actually be about using computers for these more approximate algorithms (e.g. Monte Carlo integration), rather than being slightly easier classes with more examples. (Hughes-Hallett does appox. integration by Riemann sums, but it is done by hand and with pitifully small datasets. Plus, no estimates on error are made.) However, I don't think that there is a market for that right now.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-07-27T01:18:23.050" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14401" PostTypeId="2" ParentId="14377" CreationDate="2018-07-27T01:43:52.877" Score="1" Body="&lt;p&gt;I think all three terms have value because they allow us to make important distinctions which would be muddled if we simply used the same term for all three. I'll focus this answer for a function $f$ on $\mathbb R$; that is, a real-valued function of a real-variable.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;critical number&lt;/strong&gt; a number $x_o$ is critical if either $f'(x_o)$ does not exist or $f'(x_o)=0$. Notice, I do not assume $x_o$ is even in the domain of $f$. This becomes important if we analyze functions which have vertical asymptotes. It is possible for a function to change from increasing to decreasing at a point outside the domain of $f$. To give a complete account of the increase and decrease, concavity etc. it is necessary to consider critical numbers both inside and outside the domain (of course, for concavity we also must consider numbers for which the second derivative exists and is zero).&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;critical value&lt;/strong&gt; a number $y_o$ is called a critical value if it is the output of $f$ at a critical number in the domain of $f$; $y_o = f(x_o)$. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;critical point&lt;/strong&gt; if $x_o$ is a critical number in the domain of $f$ then $(x_o,f(x_o))$ is a critical point.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;All of this said, I don't usually penalize students for confusing the terminology. Honestly, I'm just happy if they get any of the above via correct calculus and analysis.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-07-27T01:43:52.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14402" PostTypeId="1" AcceptedAnswerId="14405" CreationDate="2018-07-28T03:35:48.513" Score="3" ViewCount="618" Body="&lt;p&gt;Let's take an example in Wu-Ki Tung, &lt;em&gt;Group theory in physics&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem 3.4:&lt;/strong&gt; Irreducible representations of any abelian group must be of dimension one.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof:&lt;/strong&gt; Let &lt;span class=&quot;math-container&quot;&gt;$U(G)$&lt;/span&gt; be an irreducible representation of the abelian group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;. Denote by &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; a ﬁxed element of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;. Due to the abelian nature of the group, we have &lt;span class=&quot;math-container&quot;&gt;$U(p)U(g) = U(g)U(p)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$g ∈ G$&lt;/span&gt;. According to Schur’s Lemma, &lt;span class=&quot;math-container&quot;&gt;$U(p) = λ_p E$&lt;/span&gt;. This applies to all &lt;span class=&quot;math-container&quot;&gt;$p ∈ G$&lt;/span&gt;. Hence, the representation &lt;span class=&quot;math-container&quot;&gt;$U(G)$&lt;/span&gt; is equivalent to the one dimensional representation &lt;span class=&quot;math-container&quot;&gt;$p → λ_p ∈ C$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$p ∈ G$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would rewrite it as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Because in an abelian group every irrep commutes with another irrep, then according to the Schur’s Lemma, they are just multiples of the identity operator, &lt;strong&gt;forcing them to be one dimension only&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In my opinion, the rewrite is much better. There is no need to prove the theorem anymore, but they are blended into a flowery but self-contained text with no single symbol, and readily to connect to the next idea. 7 sentences in 6 lines reduce to 1 sentence in 3 lines, but the true intuition is still preserved, and I don’t see how this is vague or makes the formalism lost at all. In other words, we have the best of both worlds.&lt;/p&gt;&#xA;&lt;p&gt;But in the discussion &lt;a href=&quot;https://www.reddit.com/r/math/comments/92fb8w/making_intuitive_formalism_and_concrete_flowery/?st=jk4cphfr&amp;amp;sh=3c9f4d63&quot; rel=&quot;nofollow noreferrer&quot;&gt;Making intuitive formalism and concrete flowery text&lt;/a&gt;, it is said to be awful or incomprehensible, even to those who understand the topic. Do you know why is that?&lt;/p&gt;&#xA;" OwnerUserId="8875" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-07-28T21:32:03.677" Title="Why are proofs written in flowery language incomprehensible?" Tags="|undergraduate-education|proofs|self-learning|intuition|writing|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14403" PostTypeId="2" ParentId="12087" CreationDate="2018-07-28T13:08:53.077" Score="5" Body="&lt;p&gt;You can read on PE's website that questions are either&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Classic questions&lt;/li&gt;&#xA;&lt;li&gt;Questions that are derived from some theorem in some way&lt;/li&gt;&#xA;&lt;li&gt;Other questions are believed to be original&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are now over 600 questions and they have admitted in the past that they are starting to run out of questions (since they've already covered a lot). At some point, they started releasing questions every two weeks instead of every week; not sure what the frequency is right now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They rely on the community to submit question ideas, so you are not too far off in claiming that there is some &lt;em&gt;army&lt;/em&gt; of people working on the project.&lt;/p&gt;&#xA;" OwnerUserId="9341" LastActivityDate="2018-07-28T13:08:53.077" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14404" PostTypeId="2" ParentId="14402" CreationDate="2018-07-28T13:11:30.593" Score="5" Body="&lt;p&gt;Honestly, both seem pretty bad and don't prove the statement in an obvious way. For example, there is confusion betwen irreps and the representations of individual elements. You can make the $G$-equivariant maps in Schur's lemma into intertwiners because $G$ is abelian, but it really deserves to be said that this is what you are doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that it is useful to reduce notation if possible, but your rewrite isn't even grammatical. i.e. It isn't &quot;&lt;em&gt;the&lt;/em&gt; Schur's lemma&quot;, but simply &quot;Schur's lemma&quot;. That lets alone the question of what it means for an irrep to commute with another irrep. What you are using is the commutation on the individual elements. There &lt;em&gt;is&lt;/em&gt; such a thing as a representation ring (&lt;a href=&quot;https://en.wikipedia.org/wiki/Representation_ring&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Representation_ring&lt;/a&gt;) where you could talk about the irreps commuting or not, but this is logically further down the line from the basic result you are proving.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2018-07-28T15:14:32.073" LastActivityDate="2018-07-28T15:14:32.073" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14405" PostTypeId="2" ParentId="14402" CreationDate="2018-07-28T21:17:31.363" Score="16" Body="&lt;p&gt;To answer the question in the title, I would say that one problem with no-symbols reasonings is that one need to use a lot of pronouns. Problem is, pronouns usually leave too much ambiguity. At some point, mathematical sentences where written without symbols; the introduction of letters to denote mathematical object improved quite a lot the depth and intricacy of reasoning available and communicable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While it is sometimes possible to write mathematical statements and proofs with few to no mathematical symbols (and I do prefer statement using fewer math symbols), this is not always reasonably possible, and always requires to be very precise (notably with grammar). Your example is telling in this regard: you confound commutation of two representations (which does not make much sense in the context) and commutation of images of two elements under one given representation. This makes your proposition quite problematic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even written with great precision, a proof without math symbols can be difficult to decipher, because the reader also needs to be hyper precise in reading it. The slightest slip in the interpretation of which pronoun refers to which object, and the meaning is lost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, the habit of using symbols can sometime impose itself too heavily, and it is often possible to improve a proof by making it lighter, changing a sentence, limiting the introduction of notation. &lt;strong&gt;But the question is not math symbols versus flowery language, it is about how to best convey the proof to your intended audience. This is a craft that can be improved without limit.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, note that separating the statement from its proof is very important, as they play very different roles. You want the statement to be prominent, because it will usually be referred to several times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last, let me give a try at the proof you give in example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proof (most natural version to me).&lt;/strong&gt; Let $U$ be an irreducible representation of an Abelian group $G$ and consider any $p\in G$. Since $G$ is Abelian, for all $g\in G$ we have &#xA;$$U(p)U(g) = U(pg) = U(gp) = U(g)U(p)$$&#xA;and the operator $U(p)$ commutes with $U$. By Schur's lemma, this implies that $U(p)$ is a scalar operator. Since this is true for all $p$, $U$ is a scalar representation and, being irreducible, it must have dimension $1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proof (attempt without symbols).&lt;/strong&gt; Given an irreducible representation of an Abelian group, the operator associated with any element of the group must commute with the representation. By Schur's lemma, this implies that this operator is scalar. Since this holds for all elements of the group, the representation is scalar, and being irreducible it must have dimension one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not think much is gained in replacing $G$ by &quot;the group&quot; and $p$ by &quot;element of the group&quot;, and in the first proof it was possible to give more details (only suitable for beginners, but possibly crucial for them). Expressing the displayed equations without symbols would be clumsy and certainly unclear. Moreover, this is a very easy proof; try to only &lt;em&gt;state&lt;/em&gt; the &lt;a href=&quot;https://en.wikipedia.org/wiki/Five_lemma&quot; rel=&quot;noreferrer&quot;&gt;Five Lemma&lt;/a&gt; without maths symbols!&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2018-07-28T21:32:03.677" LastActivityDate="2018-07-28T21:32:03.677" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14406" PostTypeId="2" ParentId="14367" CreationDate="2018-07-29T16:33:41.893" Score="3" Body="&lt;p&gt;You complain about access on the internet to solutions to textbook problems. Here is a way around that: write &lt;em&gt;your own&lt;/em&gt; homework problems. &lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2018-07-29T16:33:41.893" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14407" PostTypeId="2" ParentId="14391" CreationDate="2018-07-30T13:17:37.463" Score="8" Body="&lt;p&gt;I do not specialize in learning disabilities, but it's possible that your student has &lt;em&gt;dyscalculia&lt;/em&gt;.  From &lt;a href=&quot;https://en.wikipedia.org/wiki/Dyscalculia&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dyscalculia is difficulty in learning or comprehending arithmetic, such as difficulty in understanding numbers, learning how to manipulate numbers, and learning facts in mathematics. It is generally seen as the mathematical equivalent to dyslexia.&lt;/p&gt;&#xA;&lt;p&gt;It can occur in people from across the whole IQ range – often higher than average – along with difficulties with time, measurement, and spatial reasoning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The Wikipedia page gives some examples of signs and symptoms.  For example,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Surprisingly, students with dyscalculia often do exceptionally in writing, reading, and speaking.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-07-30T13:17:37.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14408" PostTypeId="2" ParentId="14388" CreationDate="2018-07-30T23:11:59.017" Score="2" Body="&lt;p&gt;Reposting a comment as an answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since all integration techniques come down to substitution or by parts, I view them as a chance for students to really master basic algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Corollaries to this line of thought: Don’t spend gobs of time on the techniques. (I take about 2 - 3 weeks total between substitution in first semester Calc and the rest in second semester.) And in later sections (surface area, volume, work, hydrostatic pressure, etc.) I focus on setting up the integral and assume/encourage them to use technology to actually get an answer when needed.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2018-07-30T23:11:59.017" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14409" PostTypeId="1" AcceptedAnswerId="14410" CreationDate="2018-07-31T08:51:24.193" Score="8" ViewCount="934" Body="&lt;p&gt;In a quiz, there was a question asking students to show something doesn’t exist. A lot of them gave proofs by contradiction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Initially, I designed the marking scheme so that an incomplete proof by contradiction would lead to 0 marks, because I thought that exhausting all cases is very important in a proof by contradiction. Otherwise, there may not be any contradiction at all (another case is valid). Some students may impose an (perhaps, obviously contradictory) assumption and claim that their proofs are complete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I found that a lot of students missed some cases, or imposed extra assumptions which were neither given nor proved. Some of them were certainly with loss of generality. I gave them 0 marks for that question. I’m not sure whether marking in this way is too harsh.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should students be given partial marks if they gave an incomplete proof by contradiction?&lt;/p&gt;&#xA;" OwnerUserId="10174" LastActivityDate="2018-09-19T18:40:20.000" Title="Should students be given partial scores when they gave an incomplete proof by contradiction?" Tags="|proofs|exams|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14410" PostTypeId="2" ParentId="14409" CreationDate="2018-07-31T12:27:57.403" Score="20" Body="&lt;p&gt;There's no abstract reason that an imperfect proof by contradiction should categorically fail to get credit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A proof should generally get partial credit based on how much knowledge of the relevant material it demonstrates.  An incomplete proof by contradiction could correctly get the main idea but omit some of the technical material needed to make the argument work; I would give that close to full credit.  An incomplete proof by contradiction might lead to an argument that makes real use of material from the course, but is ultimately a dead end; I would give that some credit for demonstrating knowledge of the material.  An incomplete proof by contradiction could also end up accidentally trivializing the problem by imposing a contradictory assumption, missing the point entirely; I wouldn't give credit for that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd recommending avoiding the question &quot;how far is this from a correct proof&quot;, which I don't think is a useful way to grade.  I prefer &quot;how well does this demonstrate mastery of the material&quot;.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-07-31T12:27:57.403" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14411" PostTypeId="1" CreationDate="2018-08-01T20:47:34.260" Score="3" ViewCount="334" Body="&lt;p&gt;I'll be teaching graph theory this fall for the first time.  The only undergraduate graph theory book I am familiar with is Doug West's book, which I like.  But I'd like to consult some other undergraduate books as I prepare.  Any suggestions?&lt;/p&gt;&#xA;" OwnerUserId="10222" LastEditorUserId="376" LastEditDate="2018-08-02T08:39:35.763" LastActivityDate="2018-08-04T16:09:32.710" Title="What are standard (or good) textbooks for undergraduate graph theory?" Tags="|undergraduate-education|textbooks|graph-theory|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14412" PostTypeId="2" ParentId="14409" CreationDate="2018-08-01T21:38:18.513" Score="14" Body="&lt;p&gt;I agree completely with Henry, but let me try to mention some specific, practical advice that you or others may find helpful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I strongly encourage you to adopt a more wholistic manner of marking/grading, based on the student's mastery of the material. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use the following method of scoring student work in mathematics, where all problem solutions are graded on a five-point scale. It has worked well for me for about twenty years, and I would recommend it to anyone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5 = solution is &lt;strong&gt;basically perfect&lt;/strong&gt;, completely acceptable&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4 = solution is marred by a few &lt;strong&gt;minor errors&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3 = the solution has displayed basically the &lt;strong&gt;right idea&lt;/strong&gt; or approach to the problem&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 = the solution has displayed some understanding of &lt;strong&gt;relevant ideas&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 = the solution has displayed some understanding of &lt;strong&gt;some ideas&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0 = the solution has &lt;strong&gt;no merit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find it easy to use this rubric when evaluating student work. I just keep the phrases in mind, and assign the corresponding score. Thus, the work is evaluated as a whole, on the basis of whether the student has demonstrated mastery over the topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My quizzes usually have just one problem, graded out of five. The final exam will have ten or fiften problems, each graded out of five. I often give extra credit for participation on math.stackexchange or whatever, and these are also worth up to five points.&lt;/p&gt;&#xA;" OwnerUserId="90" LastActivityDate="2018-08-01T21:38:18.513" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14413" PostTypeId="2" ParentId="14411" CreationDate="2018-08-01T23:19:37.383" Score="5" Body="&lt;p&gt;I use &quot;&lt;a href=&quot;http://store.doverpublications.com/0486247759.html&quot; rel=&quot;noreferrer&quot;&gt;Introductory Graph Theory&lt;/a&gt;&quot; by Gary Chartrand (Dover, 1985) because it is heavy on applications and introduces some of the concepts only after presenting examples that would need them.  (A warning though, some of its definitions are not the same as those used by other authors.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a newer book, &quot;&lt;a href=&quot;https://press.princeton.edu/titles/10314.html&quot; rel=&quot;noreferrer&quot;&gt;The Fascinating World of Graph Theory&lt;/a&gt;&quot; by Arthur Benjamin, Gary Chartrand, and Ping Zhang (Princeton University Press, 2015) but I haven't tried using it in class yet.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-08-01T23:19:37.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14414" PostTypeId="2" ParentId="14411" CreationDate="2018-08-02T05:00:36.153" Score="0" Body="&lt;p&gt;The book that I used was &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0470458380&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Applied Combinatorics&lt;/strong&gt;&lt;/a&gt; by Alan Tucker.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a preview available for free on google books, &lt;a href=&quot;https://books.google.com/books?id=hdgbAAAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;. Here is part of the table of contents:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5Qk9V.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5Qk9V.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[FWIW: I mentioned this same book back in MESE 12221: &lt;a href=&quot;https://matheducators.stackexchange.com/a/12223&quot;&gt;&lt;strong&gt;What should I learn after calculus?&lt;/strong&gt;&lt;/a&gt;]&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-08-02T05:00:36.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14416" PostTypeId="5" CreationDate="2018-08-02T08:43:18.727" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2018-08-02T08:43:18.727" LastActivityDate="2018-08-02T08:43:18.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14417" PostTypeId="4" CreationDate="2018-08-02T08:43:18.727" Score="0" Body="Use this tag for questions about teaching graph theory. Here a graph is a collection of vertices and connecting edges. Use (graphing) instead if your question is about teaching graphing or plotting functions." OwnerUserId="376" LastEditorUserId="376" LastEditDate="2018-08-02T11:33:51.680" LastActivityDate="2018-08-02T11:33:51.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14418" PostTypeId="2" ParentId="14391" CreationDate="2018-08-02T11:07:07.660" Score="12" Body="&lt;p&gt;The inabilities described in the question don't strike me as either particularly unusual nor as clear signs of a neurological learning disability. They strike me as more likely the passivity and mental shutdown that one frequently encounters in students who, whether consciously or unconsciously, are hiding a sense of incapacity, inability, confusion, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students learn arithmetic operationally, as a series of rules for obtaining an answer, without internalizing the significance of the rules or their outcomes. Such a student neither has the habit of associating some mental model (e.g. the circles) with an arithmetic problem nor in many cases is able to relate such a model, posed by someone else, with the problem at hand. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often students are not asked to consider whether their answers are reasonable (in some qualitative sense). Often they are rewarded (in grading) for performing certain operations correctly, whether or not the outcome is reasonable (with respect to extra-mathematical considerations). So a student who does not think much about what halfway means might easily obtain an answer that is less than either of the averaged numbers. That's not a sign of a learning disability, it's a sign of someone who hasn't learned how to connect arithmetic operations with common sense (in any case, common sense is mostly learned).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an innumerate student progresses through the educational system, the psychological need to hide inadequacies becomes stronger and stronger, and the possibilities to correct basic misunderstandings become rarer and more remote. The algebra teacher often won't have time, energy, or inclination to teach arithmetic to a student who supposedly has passed several courses that require knowing arithmetic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's an extreme case, but once I taught a fourth year university student who was functionally illiterate (how could such a situation occur? This was in the US, and he was the starting tailback on the football team) and largely innumerate. He was adept at coaxing hints from the teacher, and at concealing his ignorance (of which he was fully and quite self-consciously aware). It took considerable care and effort to develop with him the confidence necessary to deal with the problem openly and honestly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own suspicion is that far more students have difficulties like this than most teachers care to admit. However, to detect such problems, one needs to interact with such a student personally, individually, and with the confidence sufficient to break down the barriers that prevent honest realization of the psychological obstacles (and the institutional context often makes that difficult or impossible). At the university one experiences that most students who struggle with calculus do so because they don't know basic algebra, even basic arithmetic. (A typical example is that students know that the logarithm and exponential satisfy functional identities, but can't remember if it is $f(xy) = f(x) + f(y)$ or $f(xy) = f(x)f(y)$ of $f(x + y) = f(x)f(y)$ and so guess.) Surely for students struggling in high school mathematics the common difficulty is this sort of inability to relate basic arithmetic to something other than a chain of operations aimed at getting points on an exam.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-08-02T11:07:07.660" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14419" PostTypeId="1" CreationDate="2018-08-02T19:49:56.767" Score="3" ViewCount="171" Body="&lt;p&gt;Note - this question &lt;a href=&quot;https://stackoverflow.com/questions/50381724/existing-tools-for-simple-math-expression-equivalence-logic&quot;&gt;was posted here&lt;/a&gt; and garnered some decent replies before it was closed as off-topic in Stack Overflow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Online systems, such as ALEKS, Cengage's WebAssign, and even Khan Academy employ some kind of logical matching for polynomial expressions and numerical values (ie, fractions). What unpaid tools (libraries, command line programs, scripts, etc) exist that can provide expression/numerical matching? For example, a student enters the expression&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 2p^2(p+5)-8, $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but the following expression is equivalent and would also be acceptable:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 2(p^2+4p-4)(p+1). $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question about how to do this mathematically has an excellent answer in &lt;a href=&quot;https://math.stackexchange.com/questions/584230/how-do-you-validate-that-two-math-expressions-are-equal&quot;&gt;this post&lt;/a&gt;, and a question addressing one particular way to implement this has a partial answer in &lt;a href=&quot;https://stackoverflow.com/questions/8583301/check-if-two-math-answers-are-equivalent&quot;&gt;this post&lt;/a&gt;. &lt;a href=&quot;http://www.sympy.org/en/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sympy&lt;/a&gt; looks promising, but the command line Maxima could work, and so could the WolframAlpha API, Maple, MatLab, and any number of symbolic computer algebra systems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's fine to talk about things that &quot;could work&quot;, but what tools are already being used? How has this already been implemented? Can anyone speak from experience about what online math learning programs are using on the backend? Give examples or direct to existing projects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify the question, I'm talking about logically comparing simple expressions(middle/high school math), minimally complicated, with canonical forms typically easy to obtain. The implementation will be online (html+nifty_tool) and input will most likely be captured as a string unless someone can suggest a better input method for math learners - a LaTeX front-end perhaps?&lt;/p&gt;&#xA;" OwnerUserId="10227" LastEditorUserId="8571" LastEditDate="2018-09-28T23:38:41.390" LastActivityDate="2018-09-28T23:38:41.390" Title="Existing Tools for Math Expression Equivalence Logic" Tags="|mathematical-pedagogy|online-instruction|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14420" PostTypeId="1" AcceptedAnswerId="14422" CreationDate="2018-08-02T23:22:26.603" Score="6" ViewCount="135" Body="&lt;p&gt;I've noticed that some undergraduate introductory probability textbooks and courses emphasize or seem to prefer the moment generating function $m(t) = \mathbf E(e^{tX})$ of a random variable $X$ rather than the characteristic function $f(t) = \mathbf E(e^{i t X})$. However, the characteristic function seems to have nicer properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a reason for an undergraduate introduction to probability course to emphasize the moment generating function instead of the characteristic function? Or is the characteristic function simply better. I wonder if the characteristic function is avoided only because some students feel uncomfortable with complex numbers.&lt;/p&gt;&#xA;" OwnerUserId="9633" LastEditorUserId="376" LastEditDate="2018-08-03T12:26:28.973" LastActivityDate="2018-08-03T13:18:59.927" Title="Why emphasize moment generating function over characteristic function in a probability course?" Tags="|undergraduate-education|probability|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14421" PostTypeId="2" ParentId="14419" CreationDate="2018-08-02T23:25:18.303" Score="7" Body="&lt;p&gt;Almost 20 years ago, I faced a similar issue in the course of designing some on-line calculus exercises... For me, it turned out to be simplest to use a Monte-Carlo idea, namely, to evaluate both expressions at a &quot;random&quot; set of locations, and test for agreement within some not-too-tight bound. Yes, some necessity of tolerating invalid inputs and such, but such &quot;errors&quot; can be &quot;caught&quot; and dealt with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Getting the details reasonably correct was much easier than implementing a genuine symbol-manipulation version... which would inevitably have some open-ended issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(In particular, this was implemented in Java, and &quot;expressions&quot; were converted to tree structures, and so on. Pretty typical.)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-08-02T23:25:18.303" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14422" PostTypeId="2" ParentId="14420" CreationDate="2018-08-03T13:18:59.927" Score="2" Body="&lt;p&gt;At least around here, complex calculus just isn't part of the regular engineering curricula. To introduce generating functions already makes most brains explode...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2018-08-03T13:18:59.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14424" PostTypeId="2" ParentId="14411" CreationDate="2018-08-03T18:59:35.047" Score="0" Body="&lt;p&gt;I've had success with &lt;a href=&quot;https://global.oup.com/academic/product/graphs-colourings-and-the-four-colour-theorem-9780198510628?cc=us&amp;amp;lang=en&amp;amp;&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graphs, Colourings and the Four Colour Theorem (R.A. Wilson)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have a predetermined list of graph theory topics to cover it might not work for you, but it does a good job of stretching away from just 4CT-related material into wider topics.   &lt;/p&gt;&#xA;" OwnerUserId="10159" LastActivityDate="2018-08-03T18:59:35.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14425" PostTypeId="1" CreationDate="2018-08-04T02:02:03.763" Score="7" ViewCount="4278" Body="&lt;p&gt;Someone has made &lt;a href=&quot;https://hackernoon.com/i-wanted-to-learn-computer-science-so-i-created-my-own-degree-heres-my-curriculum-bad56f28c278&quot; rel=&quot;noreferrer&quot;&gt;this template for a self taught CS degree&lt;/a&gt;. I am a CS grad and found it quite good. As I don't have much idea about a proper pathway to learn degree level mathematics in my own. Please guide me for a similar template for mathematics. The name of the papers would be enough, as I will try to search for MOOCs, however if someone provides that too, it will be great help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Initially, I &lt;a href=&quot;https://mathoverflow.net/questions/307492/curriculum-for-self-taught-maths-degree&quot;&gt;posted this question in mathoverflow&lt;/a&gt;. Someone suggested to post it here.&lt;/p&gt;&#xA;" OwnerUserId="10236" LastEditorUserId="5220" LastEditDate="2018-08-04T23:13:13.290" LastActivityDate="2018-08-05T18:49:09.683" Title="Curriculum for self taught Maths degree" Tags="|self-learning|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14426" PostTypeId="2" ParentId="14425" CreationDate="2018-08-04T03:53:27.343" Score="6" Body="&lt;p&gt;That's an interesting question and it's a little difficult to answer without knowing what you're starting point is and what your interests are.  If you don't have a solid, high school level background then you should start with algebra (often called College Algebra) and some kind of pre-calculus class that includes trigonometry.  Once you've got those down, you're ready to get serious with the college level material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester One&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Calculus I&lt;/strong&gt; - calculus of two variables, primiarly focused on differentiation with a basic introduction to integration near the end&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester Two&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Calculus II&lt;/strong&gt; - more calculus of two variables primarily focused on integration techniques and convergence of series&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Transition to Higher Math&lt;/strong&gt; - High school gives students the very mistaken impression that math is about solving equations and answering word problems.  It would be more accurate to say that it's about proving if and when a solution exists.  Actually finding the things is a question for engineers.  In other words, math is about developing formal proofs about mathematical structures.  This is a &lt;strong&gt;huge&lt;/strong&gt; transition for a lot of students and some degree programs offer a specific class intended to help students make it.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester Three&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Calculus III&lt;/strong&gt; - multivariate calculus&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Differential Equations&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester Four&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Abstract Algebra 1 or Real Analysis 1&lt;/strong&gt; - Abstract algebra is the grown up, way more mature cousin of the algebra you learned in high school.  Real Analysis is the 20th century version of calculus.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Set Theory and/or Logic&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Semester Five&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Abstract Algebra 2 or Real Analysis 2&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Linear Algebra&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;That covers what an aspiring mathematician really needs to know although you could arguably replace the two algebra classes with classes in real analysis.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From here, it really becomes a question of choosing electives.  If you already have an interest in computer science, you might consider classes in &lt;strong&gt;combinatorics&lt;/strong&gt; and &lt;strong&gt;graph theory&lt;/strong&gt;.  If your interests are more abstract, you could go with &lt;strong&gt;number theory&lt;/strong&gt; a semester or two of &lt;strong&gt;topology&lt;/strong&gt; and add &lt;strong&gt;real analysis&lt;/strong&gt; or &lt;strong&gt;abstract algebra&lt;/strong&gt; if you haven't done both already.  If engineering interests you, you could do &lt;strong&gt;vector analysis&lt;/strong&gt;, more &lt;strong&gt;differential equations&lt;/strong&gt; and &lt;strong&gt;complex analysis&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastEditorUserId="4845" LastEditDate="2018-08-05T18:49:09.683" LastActivityDate="2018-08-05T18:49:09.683" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14427" PostTypeId="2" ParentId="14411" CreationDate="2018-08-04T16:09:32.710" Score="1" Body="&lt;p&gt;For completeness, I do recommend several &quot;Moore method&quot;/inquiry-based sets of notes found at the Journal of Inquiry-Based Learning in Mathematics website.  I have successfully used a pastiche of these resources for a first course, needing only basic sets and induction as background.  Graph theory is very ideally suited to such presentation.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For a fairly applied one, &lt;a href=&quot;http://jiblm.org//downloads/dlitem.php?id=54&amp;amp;category=jiblmjournal&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Clark's notes at JIBLM&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://jiblm.org//downloads/dlitem.php?id=22&amp;amp;category=mathnerdscollection&quot; rel=&quot;nofollow noreferrer&quot;&gt;An unrefereed one with interesting info on blocks&lt;/a&gt; by James Benedict&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://jiblm.org//downloads/dlitem.php?id=23&amp;amp;category=guilfordjournal&quot; rel=&quot;nofollow noreferrer&quot;&gt;From the now-defunct Guilford Journal&lt;/a&gt;, Martin Lewinter's has quite interesting challenge sections in the second half&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2018-08-04T16:09:32.710" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14428" PostTypeId="1" AcceptedAnswerId="14750" CreationDate="2018-08-06T10:48:56.387" Score="1" ViewCount="229" Body="&lt;h1&gt;OMR in exit tickets&lt;/h1&gt;&#xA;&lt;p&gt;I plan to use an&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Optical_mark_recognition&quot; rel=&quot;nofollow noreferrer&quot;&gt;OMR, Optical Mark Recognition systems&lt;/a&gt; at the end of (some of) my classes.&#xA;I want to use the same OMR system for exit tickets scattered over the academic year (not necessarily on exiting &lt;strong&gt;every&lt;/strong&gt; class).&lt;/p&gt;&#xA;&lt;p&gt;At present I am looking into &lt;a href=&quot;https://sdaps.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;SDAPS&lt;/a&gt; for building the forms in LaTEX and for registerering the printed, hand-filled-in forms. There are other OMS systems, though, supporting automated registering of marked paper forms.&lt;/p&gt;&#xA;&lt;h2&gt;Teacher making the marks - student answering open questions&lt;/h2&gt;&#xA;&lt;p&gt;My plan is to provide open questions and white space in order for the&#xA;&lt;strong&gt;student to answer in writing (and drawing)&lt;/strong&gt;,&#xA;while the fields preprinted for marking are reserved for the teacher's&#xA;feedback.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/YZ3OK.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YZ3OK.png&quot; alt=&quot;Draft form&quot; /&gt;&lt;/a&gt;&#xA;Here, of course the white space for answer may need to be enlarged.&#xA;Cosider the above screendump as a draft.&lt;/p&gt;&#xA;&lt;h2&gt;Why?&lt;/h2&gt;&#xA;&lt;p&gt;By training handwriting for assessment, I want to take the&#xA;tool (Maple, TI-NSpire, ...)&#xA;out of the solution&#xA;and inviting the student to focus on the theory,&#xA;the mathematical principles.&lt;/p&gt;&#xA;&lt;p&gt;By posing the question and collecting responses,&#xA;I want to tell the class as a group of students&#xA;why focus is where it is in the next session.&lt;/p&gt;&#xA;&lt;p&gt;By registering the marks,&#xA;I want the student to feel recognized.&lt;/p&gt;&#xA;&lt;h2&gt;- But how?&lt;/h2&gt;&#xA;&lt;p&gt;Would OMR be the way for me to go towards my objective of&#xA;giving fast and visible feedback to students&#xA;and registering individual levels of skills for&#xA;guidance (and assessment)?&#xA;Or would some other way be far more obvious?&lt;/p&gt;&#xA;&lt;p&gt;Having experience with using LaTEX packages&#xA;such as &lt;code&gt;tasks&lt;/code&gt; and &lt;code&gt;exsheets&lt;/code&gt;,&#xA;what would be a good choice for me implementing an OMR?&#xA;&lt;strong&gt;Which OMR&lt;/strong&gt; would you recommend, why?&lt;/p&gt;&#xA;&lt;p&gt;What is considered good practice regarding&#xA;such questionnaires?&#xA;I mean regarding&#xA;the &lt;strong&gt;language&lt;/strong&gt; (including use of mathematical notation and diagrams)&#xA;in questions,&#xA;and in the &lt;strong&gt;requested answers&lt;/strong&gt; indicated by the form layout?&lt;/p&gt;&#xA;" OwnerUserId="3502" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-11-07T18:12:53.650" Title="Evaluation and feedback using Optical Mark Recognition systems in secondary school" Tags="|secondary-education|course-design|technology-in-education|assessment|interactive-teaching|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14429" PostTypeId="2" ParentId="12728" CreationDate="2018-08-07T15:52:40.237" Score="3" Body="&lt;p&gt;The debate seems to be where it was 20 years ago. The reason is that most universities have a &quot;generic&quot; mathematics curriculum, so if the engineers need calculus as it is now, everyone will also get it. That said, the quotation rings true to me: &quot;The world has moved from analog to digital, and it is time for our mathematics curriculum to change from analog to digital—from the more classical continuous mathematics to the more modern discrete mathematics.&quot; I just think that the latter can't and shouldn't exclude calculus. What might make this possible is &lt;em&gt;discrete calculus&lt;/em&gt;. Discrete calculus is nothing but the familiar calculus before we take that fateful limit. This limit makes the algebra simpler and easier to handle manually but when the problem falls outside the familiar scope, one must use approximations, i.e., circle back to discrete calculus. Maybe we can emphasize the discrete at the expense of the continuous?&#xA;$$\lim_{\Delta x\to 0}\left( \begin{array}{cc}\text{ discrete }\\ \text{ calculus }\end{array} \right)= \text{ calculus }$$&#xA;An attempt to develop the latter and the former in parallel is in a draft of a book of mine, &lt;a href=&quot;http://inperc.com/wiki/index.php?title=Calculus_Illustrated&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus Illustrated&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="10168" LastActivityDate="2018-08-07T15:52:40.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14430" PostTypeId="1" CreationDate="2018-08-07T18:09:55.237" Score="3" ViewCount="156" Body="&lt;p&gt;I've been extremely excited by my reading of the book &lt;em&gt;Elementary and Middle School Mathematics: Teaching Developmentally&lt;/em&gt; by Johan A. Van de Walle et al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of similar books (or other references) for high school mathematics (grade 9-12), by which I mean either of the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;books that explain in detail how to set up a problem-based classroom in grade 9-12,&lt;/li&gt;&#xA;&lt;li&gt;books that provide a perspective of the mathematical content taught at grade 9-12 together with how teenagers best learn that content.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;PS: This question is a more specific version of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13115/a-good-book-for-juniors-senior-high-school-similar-to-vandewalle-middle-school&quot;&gt;A good book for Juniors/senior/high school similar to VanDeWalle middle school?&lt;/a&gt; which has not been answered.&lt;/p&gt;&#xA;" OwnerUserId="10248" LastActivityDate="2018-10-11T21:00:03.370" Title="Books similar to &quot;Teaching Developmentally&quot;, but for high school math" Tags="|secondary-education|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14431" PostTypeId="1" CreationDate="2018-08-08T19:33:24.837" Score="2" ViewCount="529" Body="&lt;p&gt;I am looking for multiple choice question (MCQ) based tests on some Mathematics' topics (details below), which could be downloaded in most preferably tex (LaTex) format or doc/docx format. Kindly please share any info on this if you have. Please do mention services or websites with payment as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I particularly need Linear Algebra and Multivariable Calculus (undergrad level) and basic Group Theory (undergrad level).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any help would be gratefully appreciated.&lt;/p&gt;&#xA;" OwnerUserId="10253" LastEditorUserId="8571" LastEditDate="2018-08-08T20:39:26.403" LastActivityDate="2018-08-22T06:06:30.763" Title="Downloadable MCQs on Mathematics" Tags="|linear-algebra|vector-calculus|test-design|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14433" PostTypeId="2" ParentId="14258" CreationDate="2018-08-09T19:12:43.903" Score="1" Body="&lt;p&gt;Excel comes to mind. This took just a few seconds to make:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tGcyB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tGcyB.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10168" LastEditorUserId="10168" LastEditDate="2018-08-10T13:06:54.777" LastActivityDate="2018-08-10T13:06:54.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14434" PostTypeId="2" ParentId="14258" CreationDate="2018-08-09T23:50:39.423" Score="1" Body="&lt;p&gt;I've made some fairly nice graphics using &lt;a href=&quot;http://sagemath.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sage&lt;/a&gt;. I think that it uses matplotlib, at least partly, as a backend. It can render to a variety of image formats and supports LaTeX in labels. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edited to add what the code referenced in the comment produces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AuUeE.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AuUeE.png&quot; alt=&quot;stencil&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2018-08-10T22:04:51.943" LastActivityDate="2018-08-10T22:04:51.943" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14436" PostTypeId="2" ParentId="2763" CreationDate="2018-08-10T10:26:45.650" Score="0" Body="&lt;p&gt;There are numbers and there are magnitudes (3 vs. 3cm). Multiplication of numbers is repeated addition (i.e. 3x5 = 5+5+5), multiplication of magnitudes is not (i. e. 3N x 5cm is not an addition).&#xA;The problem is that standard mathematical education does not explain what are magnitudes and how to calculate with them (although it is obvious, at least to physicists). Math education pretends that there is no calculation with magnitudes and instead of mathematically rigorous (400 m / 50 sec) / 10 sec = 8 m/sec*2 use mathematically incorrect (400 / 50 ) / 10  = 8 m/sec*2 or non-explanatory (400 / 50 ) / 10  = 8. This is just ignorance. C.f. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.fsb.unizg.hr/matematika/download/ZS/clanci/what_are_magnitudes_and_how_to_calculate_with_them.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.fsb.unizg.hr/matematika/download/ZS/clanci/what_are_magnitudes_and_how_to_calculate_with_them.pdf&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.dropbox.com/home/zs-mm-share/web_dostupni_seminari_i_predavanja/fizika?preview=Magnitudes+dbk.docx&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.dropbox.com/home/zs-mm-share/web_dostupni_seminari_i_predavanja/fizika?preview=Magnitudes+dbk.docx&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, we should appreciate number/magnitude distinction and put magnitudes and calculations with them back into math curricula (where they belong).  &lt;/p&gt;&#xA;" OwnerUserId="10261" LastActivityDate="2018-08-10T10:26:45.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14437" PostTypeId="2" ParentId="10004" CreationDate="2018-08-11T15:00:12.973" Score="1" Body="&lt;p&gt;If I had to explain common usage to a layperson then I might use an analogy with travelling to beautiful places, as follows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A theorem is like a beautiful place that you would highlight to anyone exploring the area. Lemmas and propositions are signposts and guides that help one to efficiently reach the beautiful places and better understand their importance. Corollaries are other notewothy things that are easily visible once one reaches the beautiful destination.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unlike some other analogies, this has the benefit of explicitly highlighting the key role that beauty plays in the judgement of such matters. Of course the judgement of beauty, importance, etc is highly subjective and not everyone may agree on just which results should be elevated to theorems. For example, the eminent algebraist Irving Kaplansky equates them all in his beautiful influential textbook &lt;em&gt;Commutive Rings&lt;/em&gt;, with the following explanation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the style of Landau, or Hardy and Wright, I have presented the material as an unbroken series of theorems. I prefer this to the n-place decimal system favored by some authors, and I have also grown tired of seeing a barrage of lemmas, propositions, corollaries, and scholia (whatever they are). I admit that this way the lowliest lemma gets elevated to the same eminence as the most awesome theorem. Also, the number of theorems becomes impressive, so impressive that I felt the need to add an index of theorems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="163" LastEditorUserId="1147" LastEditDate="2018-08-11T15:59:44.157" LastActivityDate="2018-08-11T15:59:44.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14438" PostTypeId="2" ParentId="14430" CreationDate="2018-08-12T19:29:08.423" Score="1" Body="&lt;p&gt;This is not exactly what you're asking for, but it might come close: &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0807734810&quot; rel=&quot;nofollow noreferrer&quot;&gt;What's Happening in Math Class?&lt;/a&gt;, edited by Deborah Schifter. This is volume 1, and there is a volume 2.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-08-12T19:29:08.423" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14439" PostTypeId="1" AcceptedAnswerId="14468" CreationDate="2018-08-13T15:32:05.213" Score="5" ViewCount="198" Body="&lt;p&gt;I and my colleague will teach an elementary linear algebra next few weeks, but the way our course is planned mostly is by turn teaching. By that I mean, my partner will teach the first 8 weeks of the semester, and I will teach the last 8 weeks. That's how it works here. Tradition you may say. However, we both agree that this should not be the only way. The course itself is scheduled at 2 hours on each Monday and 2 hours on each Friday.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My Q is: Are there other possible suggestions about better teaching method that we can use? The aim is that we want to distribute our teaching load a bit more even, not like this which eventhough seems even in total, but separated in distribution. We also want better result for student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we think can be the solution:&#xA;1. We follow the planned responsibility but with a small difference. My partner teaches first 8 Mondays, and I serves as tutor in first 8 Fridays. This means I help student focus more on problem solving, applications, etc each Friday related to the material/basic concept given last Monday by my partner, and then we swap roles last 8 weeks. We think about this option often but wonder a lot abour whether all materials will be covered.&#xA;2. We simply teach the planned course but my partner teaches on Monday and I teach on Friday. This way, it seems pretty confusing to be honest for the student.&#xA;3. ...?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we think CANNOT be the solution:&#xA;1. CoTeaching in class. This actually just means that my partner does not really like being observed by someone else who is not the student. I wonder why but we will talk about this too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eventhough we do turn teaching a lot here, we still communicate very often between lecturers. Even some pairs of lecturers do co-teaching already, including me with another partner, and help breaking the tradition for the sake of our students.&lt;/p&gt;&#xA;" OwnerUserId="10108" LastActivityDate="2018-08-24T02:43:10.823" Title="CoTeaching Elementary Linear Algebra" Tags="|teaching|linear-algebra|teacher-preparation|teachers|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14440" PostTypeId="2" ParentId="12618" CreationDate="2018-08-13T15:40:55.380" Score="2" Body="&lt;p&gt;There are already several &quot;yes&quot; answers so I'll give you a qualified &quot;no&quot;: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you do in class is what your students learn, so it depends on what you want to teach them. They should of course be aware the tools exist because it's hard to start learning how to visualize equations in 3d, but the more you as a teacher lean on them the less time they will have to see how to visualize equations without the help of the computer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why teach them visualize equations at all? In my opinion, the geometry skills we're building in Calc 3 are&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Looking at 2 to 4 variable equations and understanding their rough geometric structure and analytic structure (do they max extrema, self intersections, etc).&lt;/li&gt;&#xA;&lt;li&gt;Looking at low dimensional shapes and understanding the equations that could model them.&lt;/li&gt;&#xA;&lt;li&gt;Understanding &lt;em&gt;in general&lt;/em&gt; how the features of an equation give rise to geometric and analytic structures that we may want to analyze. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I think that computer systems are decent for (1) since students can explore different equations and see how the geometry of their solutions change. I think they're much less helpful for (2) and virtually worthless for (3). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For (2) and (3), you have to understand the theory of why, say, both $z = x^2 - y^2$ and $z = xy$ give rise to saddles points. If you can manipulate the equation, draw the level curves and trace curves, compute the Jacobean and really understand how all those structures explicitly give rise to the behavior of the solutions to the equation, then you'll be able to start forming fundamental connections between equations and geometry. Computer visualizations are &lt;em&gt;useful&lt;/em&gt; to check your work and to get an idea of the visual vocabulary, but at the end of the day the connections made rely on memorization, much like the quadratic zoo most text books have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the hard, but important, skill is doing geometry without a computer. You can spend class time showing students the easy crutch that frankly they can (and will) all figure out on their own, or you can spend your class time helping them through the actual hard work. Computer visualization systems and solvers are useful, and should be discussed in class, but the more you rely on them, the students will come to understand that background skill is worthless, when in fact it's just difficult. &lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2018-08-13T15:40:55.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14441" PostTypeId="1" AcceptedAnswerId="14444" CreationDate="2018-08-14T03:26:41.437" Score="-1" ViewCount="862" Body="&lt;p&gt;High school student here...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This coming school year I'm scheduled to start AP Calculus AB and then my school is looking into taking Multivariable Calculus at a local university. The school's calculus teacher thinks it isn't necessary for me to take BC since AB and BC are so similar. Over the summer, I did some studying and learned limits, derivatives, and am currently working on integrals. Looking over common curriculums for AB, there doesn't seem to be much beyond that and I'm worried the class may be a little boring. Do teachers really think AB is necessary or would I be better off asking to transfer to BC? &lt;/p&gt;&#xA;" OwnerUserId="10273" LastActivityDate="2018-08-14T21:37:24.863" Title="Is AP Calculus AB really necessary?" Tags="|secondary-education|calculus|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14442" PostTypeId="2" ParentId="14354" CreationDate="2018-08-14T14:34:17.853" Score="2" Body="&lt;p&gt;This is a rather broad ranging response to your post. Some time ago the Dean at York approached me about suggestions for what would be a fair distribution of faculty lines among the different departments at York that would take into account student interests as reflected in number of credits taken in various departments, number of majors in different departments and the fact that faculty &quot;work&quot; for teaching, say, a 3 credit course varies a lot. Computer science courses which teach a programming language and English courses which teach writing often have higher work loads than other courses with the same number of credits (or contact hours, which sometimes are not equal numbers). I was surprised that I could not find much of a &quot;literature&quot; on this issue, and I am not sure I was of much use to the Dean!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the CBMS reports, and the AMS has not only the most recent report available but earlier reports, too, are a very valuable resource in understanding statistical patterns for mathematics related educational issues. However, I want to point out that the CBMS report is generated using a stratified sample, and that over time the &quot;definition&quot; of what numbers go into the counts has changed. Some colleges have statistics degree programs within the mathematics department and some colleges have a separate statistics department. Some schools had one mathematics and CS department and now they have separate departments for mathematics and CS. So, I think one has to be somewhat careful with regard to time series data. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The graphic in your post is vague about where the data comes from (and whether two year college data is included) but in particular I want to call attention to the fact that when people refer to STEM, some include computer science in STEM and some, since there is no C (or CS) in the acronym, don't. And when data is collected about computer science one has to be careful what is included, since for many colleges there are information science degrees that are not computer science. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Until relatively recently (50 years ago), oversimplifying, people with mathematical talent majored in mathematics but now many such people have the choice of mathematics or computer science degrees, and many mathematically talented people also major in economics, physics, or engineering. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trying to understand the trends is not easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regards,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Joe&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2018-08-14T14:34:17.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14443" PostTypeId="1" AcceptedAnswerId="14446" CreationDate="2018-08-14T17:26:06.270" Score="24" ViewCount="837" Body="&lt;p&gt;In the 1930's, &lt;a href=&quot;https://en.wikipedia.org/wiki/Louis_P._B%C3%A9n%C3%A9zet&quot; rel=&quot;noreferrer&quot;&gt;Louis Bénézet&lt;/a&gt;, a superintendent of several schools in New Hampshire made the interesting experiment of teaching no formal arithmetic until grade 6:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite - my new Three R's. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language. I picked out five rooms - three third grades, one combining the third and fourth grades, and one fifth grade.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He wrote about it in &lt;em&gt;&lt;a href=&quot;http://www.inference.org.uk/sanjoy/benezet/&quot; rel=&quot;noreferrer&quot;&gt;Teaching of Arithmetic I, II, III: The Story of an Experiment&lt;/a&gt;&lt;/em&gt;, and the results seem very impressive. On the other hand, the findings are described in a style that seems more anecdotal than objective at times, and I would like to know more details.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Did Bénézet or someone else publish more details on the experiment (like the curriculum used, or how teachers were trained for it)? Did someone follow up on what happened with the children that were thought with this method?&lt;/li&gt;&#xA;&lt;li&gt;Has the experiment been reproduced since then? Was further research done in this direction?  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="590" LastEditorUserId="590" LastEditDate="2018-08-14T17:33:57.520" LastActivityDate="2018-08-15T18:44:05.797" Title="Has Benezet's teaching experiment ever been reproduced?" Tags="|primary-education|arithmetic-operations|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14444" PostTypeId="2" ParentId="14441" CreationDate="2018-08-14T18:28:18.357" Score="0" Body="&lt;p&gt;Taking the AB course, and then trying to fill in the gaps independently for a BC course is risky.  I do not recommend it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the county I grew up in, there was a student who took the first semester of a one-year high school AP calculus course, and then tried to take second semester calculus at the local junior college.  He was woefully underprepared, and did not succeed.  This was despite his being very bright and a very good student.  (He had the second-highest Academic Decathlon scores in the county.)  This was also despite having excellent instructors for both his high school calculus course, and for his junior college calculus course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A solid understanding of calculus through the Reynolds Transport Theorem is key to doing well in many science and engineering subjects.  If you have holes in your understanding of calculus -- especially holes that you do not realize you have -- you will suddenly find yourself in a difficult position in later classes.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2018-08-14T18:40:02.507" LastActivityDate="2018-08-14T18:40:02.507" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14445" PostTypeId="2" ParentId="14441" CreationDate="2018-08-14T21:37:24.863" Score="3" Body="&lt;p&gt;Stronger students should take BC, not AB.  AB is basically about 0.5-0.7 of what is in the BC course.  BC pretty closely matches the standard first two semesters of college calc.  AB is a little bit more than a semester of college calc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general (i.e. in theory), BC starts at the same point as AB but just goes faster.  &lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-08-14T21:37:24.863" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14446" PostTypeId="2" ParentId="14443" CreationDate="2018-08-15T18:44:05.797" Score="10" Body="&lt;p&gt;You can find some further studies that &lt;em&gt;cited&lt;/em&gt; the original writing on google scholar: &lt;a href=&quot;https://scholar.google.com/scholar?cites=11335995320848008936&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mahajan (&lt;a href=&quot;https://people.rit.edu/svfsps/perc2001/Mahajan.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;PDF&lt;/strong&gt;&lt;/a&gt;) remarks that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Etta  Berman,&#xA;  one  of  the  teachers  in  the  program,  studied&#xA;  it  for  a  masters  degree  (1935);  she  gave  a&#xA;  battery  of  quantitative  tests  to  106  control&#xA;  and  82  experimental  students  in  the  sixth&#xA;  grade. understatement:  ‘The  results  of  this  study&#xA;  cast  doubt  upon  whether  we  are  justified  in&#xA;  devoting five years to the drilling of formal&#xA;  arithmetic’ (Berman 1935, p. 40).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Despite  or  because  of  these  results,  in&#xA;  1936  the  Manchester  school  board  ended&#xA;  the experiment. Two years later Benezet was&#xA;  forced out of his job, and today few know of&#xA;  his  experiment. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The citation for this paper is given as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Berman, E. (1935). Result of deferring systematic teaching of arithmetic to grade six as disclosed by the deferred formal arithmetic plan at Manchester, New Hampshire (Doctoral dissertation, Boston University).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is clearly incorrect, since it is &lt;em&gt;not&lt;/em&gt; a doctoral dissertation: It is mentioned above as being for a &quot;masters degree,&quot; and Boston University lists the thesis &lt;a href=&quot;https://open.bu.edu/handle/2144/20451&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; with the correct classification of Ed.M. (i.e., Master's in Education). The paper's 223 page PDF is available in full through archive: &lt;a href=&quot;https://archive.org/details/resultofdeferrin00berm&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An excerpt from p. 2:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7TXYg.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7TXYg.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-08-15T18:44:05.797" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14447" PostTypeId="2" ParentId="14439" CreationDate="2018-08-16T04:35:18.007" Score="1" Body="&lt;p&gt;This is a student perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Moscow (during my third year) I took an algebraic topology course which was split between class lecture and a problem-solving session. These roles were not strictly adhered to, and indeed some easier proofs were deferred to the problem solving session and likewise instrumental examples were solved by the class in lecture (things like homotopy invariance for pairs via the five lemma.)&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Pros&lt;/h2&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Deferring some proofs had the advantage of not interrupting the flow of lecture with technical points, and gave students the feeling of &quot;doing&quot; some algebraic topology in the problem session.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A diversity of opinion on what points were central in the lecture. The second teacher supplied the problems for the class and also assigned homework.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;h2&gt;Cons&lt;/h2&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;There were days where the problem-solving session assumed knowledge not covered in lecture. This sometimes led to unreasonable homework assignment, but this may have had something to do with an attitude of &quot;just go learn it.&quot;&lt;/li&gt;&#xA;&lt;li&gt;The difference in approach between two professors was sometimes baffling. One would hand-wave away certain issues, and the other could be pedantic about similar problems and vice versa.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Altogether, I think I liked the approach, and I think doing something like this would require some collaboration on the problem set (especially if one wants to cover some central/personally important material in the second half.) I also think the tutoring approach requires some extra attention to what is going on in class/ where confusion is, so it seems like a lot of communication is needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry this is not new advice  (not that I am equipped to give it) but it was too long for a comment. Maybe also what I've said is obvious.&lt;/p&gt;&#xA;" OwnerUserId="6276" LastActivityDate="2018-08-16T04:35:18.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14448" PostTypeId="1" AcceptedAnswerId="14449" CreationDate="2018-08-16T07:37:20.610" Score="3" ViewCount="393" Body="&lt;p&gt;Here's the most common way that I've seen letter grades assigned in undergrad math courses. At the end of the semester, the professor: 1) computes the raw score (based on homework, quizzes, and tests) for each student; 2) writes down all the raw scores in order; 3) somewhat arbitrarily clusters the scores into groups corresponding to grades of A, B, C, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a few problems with this approach though. One might object that the assignment of letter grades is too arbitrary. Students also might object that they don't know in advance what grade they are likely to get in the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, it's difficult to assign letter grades in a less arbitrary manner. For example, if we declare in advance that a score of 80-90 on an exam corresponds to a B, we might find that the exam was too difficult and that scores on the exam were lower than expected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What do you think is the best approach to assigning letter grades in an undergraduate math course?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="9633" LastEditorUserId="376" LastEditDate="2018-08-16T10:22:04.423" LastActivityDate="2018-08-17T02:32:51.067" Title="What is the best way to assign letter grades in a math class?" Tags="|undergraduate-education|grading|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14449" PostTypeId="2" ParentId="14448" CreationDate="2018-08-16T17:43:27.057" Score="4" Body="&lt;p&gt;What I do is a blend of the two methods that you describe. At the beginning of the course I give students the strictest grading scale that I might possibly use (a standard 90-80-70 scale). At the end of the term, I reserve the right to lower the scale as needed. In this way, any deviations from the stated scale are in the student's favor but I can account for overall difficulty of the class. &lt;/p&gt;&#xA;" OwnerUserId="6949" LastActivityDate="2018-08-16T17:43:27.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14450" PostTypeId="1" CreationDate="2018-08-16T21:19:31.013" Score="15" ViewCount="693" Body="&lt;p&gt;This term I am teaching a very quiet precalculus class. I am having trouble getting students to answer questions or respond to verbal prompts, even questions I &lt;em&gt;know&lt;/em&gt; they can answer. What I initially chalked up to first-day shyness has persisted into the 3rd week of class, so I'm reaching out. This is unusual for me, and all of my usual tricks are failing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Symptoms:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students will not answer basic questions from me.&lt;/li&gt;&#xA;&lt;li&gt;Students will not work with or talk to each other (they sit at tables of four), even when I give very clear instructions on that point.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Why This Matters to Me:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Creating dialogue is, at least, an educational habit I've long had. When I show an example, I am always asking questions of the class so they can take part in the work, laboring to understand why certain steps are taken and how to avoid common pitfalls. At their most basic level, these questions can involve recall of basic, level-appropriate facts (&quot;what is the period of sin(x)?&quot; or &quot;which side of the triangle is the hypotenuse?&quot;). On the other end of the spectrum, I will ask concept or open-ended questions, or try to play devil's advocate. I have now nearly ceased asking questions as I can anticipate a full 15 seconds without an answer from my class (I generally will not wait longer than this before giving the answer to a basic-fact question or just asking a different question). If my question was too hard, I will ask a simpler one to the same effect. I have relied on this verbal feedback from students so that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I can tell that someone is following the argument I'm making.&lt;/li&gt;&#xA;&lt;li&gt;Hearing fellow students answer basic questions sets an expectation for the class about what things they should know or be learning.&lt;/li&gt;&#xA;&lt;li&gt;I can get the students' consent to make an aside comment or touch on tangential information.&lt;/li&gt;&#xA;&lt;li&gt;I can give students the opportunity to give their own ideas or pose their own questions about a problem.&lt;/li&gt;&#xA;&lt;li&gt;(maybe more important than I'd like to admit) I don't feel like I'm just engaging in mathematical performance art in front of an audience.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I want my students to learn to talk to each other, as appropriate, when they solve a problem. I also want them to take a basic direction from their instructor. I think my students (these are mostly engineering or science majors) should be discussing the work with each other, able to ask and/or answer questions about a solving process. When I pose an involved problem for them to solve, often my first direction (after I read the problem statement or ask them to read it for themselves) is for them to just talk together for 1-2 minutes about ideas for how to approach the problem. I want everyone on the same page so that nobody just sits there without any idea of what to do. I'll be very specific about what I want them to do. Example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Don't take out your calculator yet. First just try to correctly draw the situation from the problem, and label what what numbers you know. I will give you 1 minute to draw. Then share your drawing with your group and see if you can improve it in any way before we jump in and do any algebra.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After a minute, I'll say:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;OK -- that was a full minute. Now everyone take a minute to share your drawing with the group. See if you have all the important details or if you forgot something. Ready...go.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Absolute silence. No sharing of papers. Maybe one person glances at the person's paper next to them without saying anything.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What I want:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a math class where you receive almost no verbal feedback from your students, how do you encourage it? I want to address this as pertains to a math classroom, so I'm looking for your experience -- that is why I'm asking this here (not at the general education stack exchange). I want to help foster communication in my class while still teaching the topic at hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have any math-specific activities/strategies that encourage student/instructor and student/student communication? Have they helped you with a particularly quiet class?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, this class is generally doing fine on individual assessments. Am I just asking or expecting too much from this group?&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2018-08-17T14:00:39.433" LastActivityDate="2019-12-21T12:26:10.367" Title="How To Help a Quiet Class" Tags="|classroom-management|class-participation|community-colleges|classroom-interaction|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14451" PostTypeId="2" ParentId="14448" CreationDate="2018-08-17T00:08:20.453" Score="1" Body="&lt;p&gt;As @shoover hints at in their comment, the question assumes that you've already answered the question of why you are assigning letter grades at all.  Assuming the answer to that is something along the lines of &quot;because I have no choice&quot; or &quot;that's just how it's done&quot;, here's how I approach it.  (Perhaps useful info: my college has a brief description of what letter grades should mean across disciplines.  For example, a B is an indicator that the student is ready---and encouraged---to pursue further studies in the discipline.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't use numbers at all any more, and grade assignments like (I assume) my colleagues in the humanities do.  Does the work demonstrate mastery of the material?  Put an A on it.  Is it littered with errors, both small and large, but with evidence of a reasonable amount of understanding of the basics?  C.   Of course, there's written feedback on individual errors/questions and the assignment/exam as a whole as well.  This written feedback should, and usually does, correlate with the grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Course grades are then compiled from these individual grades in some somewhat arbitrary but pre-defined and known way.  (Best 6 assignments from 8 count 60% and the final exam counts 40%, or whatever.)  I also reserve the right to move a student's grade up or down by upto one letter based on promptness of submission and class participation and whatever other sundries pop up during the semester.  (This is much more often a boon for doing something not explicitly assigned credit rather than a dock.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, this bypasses a lot of the concerns that seem to have driven the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Bonus anecdote; related but not strictly part of an answer: I studied and then taught a little in the UK in places where 70ish earned an A and a passing mark was around 40.  I moved to the US and started teaching without knowing about the usual US system.  In my first Calc 1 assignment my students did well with some tough (in context) questions and most scored between 60 and 80 (so, I thought, at least a B- and some really strong As).  They were not happy.  This is part of what prompted the switch to my current system, but I like it regardless of its origin.]  &lt;/p&gt;&#xA;" OwnerUserId="10159" LastActivityDate="2018-08-17T00:08:20.453" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14452" PostTypeId="2" ParentId="14450" CreationDate="2018-08-17T00:37:33.663" Score="12" Body="&lt;p&gt;I've had a remarkable (remarkable to me) success in a different environment on a different topic, so I am not sure this will translate. But I'll mention in anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With one change, the same quiet &quot;labs&quot; turned into such lively conversations that I now have to raise my voice above the din to interrupt them with instructions. The change was from allowing students to work in pairs of their choice, to working with randomly assigned partners. At the start of each class, I run a program (projected for all to see) that randomizes the pairings, then they get up, move around the room, introduce themselves, and start working on the &quot;lab&quot; exercises. This with about $30 \pm 5$ students. Somehow the random mixing and the weekly surprise partners totally changes the atmosphere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course I'm not addressing the content of the exercises, reporting out, creating a non-threatening environment, etc., etc. I recognize those considerations may dominate.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-08-17T00:37:33.663" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14454" PostTypeId="2" ParentId="14448" CreationDate="2018-08-17T02:32:51.067" Score="1" Body="&lt;p&gt;60 D&#xA;70 C&#xA;80 B&#xA;90 A&lt;/p&gt;&#xA;&#xA;&lt;p&gt;IF you are familiar with the course, you should be able to set appropriate difficulty tests.  If you are less familiar, than sneak in easy test (or hard one) if you think you need to correct a little for scores being out of whack halfway through.  but even then try to have some personal guess on if the issue is your previous testing being off or just student performance being different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nothing is perfect.  So don't sweat it so much.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-08-17T02:32:51.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14455" PostTypeId="2" ParentId="14439" CreationDate="2018-08-17T02:35:24.503" Score="0" Body="&lt;p&gt;split it half way as the traditional solution.  Better for the students and better for the teachers. Allows focus.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-08-17T02:35:24.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14456" PostTypeId="2" ParentId="14450" CreationDate="2018-08-17T14:52:27.223" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Am I just asking or expecting too much from this group?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I empathize with your goals completely, and they are honorable. Bear in mind that sometimes, due to a conflation of factors, a particular class climate may be such that this just never gels no matter what you do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my master teaching folder I have a half-dozen post-it notes for (hopefully very rare) emergency or worst-case scenario events. One of them reads as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If class is uniformly dumb (all blank stares to questions), just fire&#xA;  out answers to verbal questions! (else class gets slowed down &amp;amp; runs&#xA;  out of time)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I haven't had to implement that for about the last year, I think. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, in smaller computing labs recently (where I didn't have particular content I needed to get through in the hour), I've found myself resorting to the exact opposite tactic -- Asking a question and then remaining scrupulously silent, intentionally letting the silence get really awkward and uncomfortable, until someone says &lt;em&gt;something&lt;/em&gt;. I think I've had to go as long as 5 minutes doing basically nothing in silence (maybe looking out the window, or taking a drink of water) but at some point with this group someone says something on-topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully you find some better solution in your current case, but do keep these in your toolbox as a last-gasp response. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2019-11-30T16:27:11.063" LastActivityDate="2019-11-30T16:27:11.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14457" PostTypeId="2" ParentId="14450" CreationDate="2018-08-18T00:27:15.217" Score="9" Body="&lt;p&gt;Start calling on students individually without waiting for any one to raise their hand. For a long time I never did this since I am conflict avoidant and since I didn’t want to put students on the spot, but then I witnessed another instructor masterfully call on every student throughout the course of one class period. The key point to add is that when a student you call on doesn’t know the answer or gives an incorrect answer or gives no suggestion, start a dialogue with them - asking easier and easier questions until they get something right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit/addendum: I’ve quit doing this, and I think OOESCoupling’s comment below is good. I also just wanted to start covering more material, and students don’t like being put on the spot.&lt;/p&gt;&#xA;" OwnerUserId="10289" LastEditorUserId="10289" LastEditDate="2019-12-21T12:26:10.367" LastActivityDate="2019-12-21T12:26:10.367" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14459" PostTypeId="2" ParentId="14450" CreationDate="2018-08-18T20:38:05.930" Score="6" Body="&lt;p&gt;Here are some ideas:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Try using shared vertical surfaces for doing problems. For example use whiteboards/windows. According to Peter Liljedahl &lt;a href=&quot;http://www.peterliljedahl.com/publications/building-thinking-classrooms&quot; rel=&quot;noreferrer&quot;&gt;http://www.peterliljedahl.com/publications/building-thinking-classrooms&lt;/a&gt;, having to share the writing space can encourage discussion. He suggests vertical non-permanent surfaces, but any shared space is better than individual spaces. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Perhaps put the discussion on paper: students write comment/question and pass it between the table group. Alternatively, they all write down the idea they found most interesting/useful and you collect them all. I would shy away from asking them what they struggled with here. I don't have a reference for this being a good idea.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use the principles outlined in &quot;Five Practices for Orchestrating Productive Mathematical Discussions&quot; (&lt;a href=&quot;https://www.nctm.org/Store/Products/5-Practices-for-Orchestrating-Productive-Mathematics-Discussion,-2nd-Edition/&quot; rel=&quot;noreferrer&quot;&gt;NCTM in the USA link&lt;/a&gt;). The idea is to look at student work as they are working and ask particular students if you can share their work. Then you present their work/ask them to present it, and you lead class discussion about different approaches to the work.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2018-08-18T20:38:05.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14460" PostTypeId="2" ParentId="14431" CreationDate="2018-08-22T06:06:30.763" Score="2" Body="&lt;p&gt;As no-one answered yet... &#xA;This doesn't fit the subjects you ask for, as it covers single variable calculus; but I really like the project.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway: Cornell has a &quot;&lt;a href=&quot;http://pi.math.cornell.edu/~GoodQuestions/materials.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;good questions&lt;/a&gt;&quot; project. It is related to Stewarts calculus, but it is a good example of how one can get a good setting with useful questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;so maybe someone knows similar projects for all kinds of classical undergrad math themes? Actually I guess there are many math departments at higher ed institutions maintaining their databases with MCQs for their service lectures. Understandable or not, they are reluctant to make them publicly available. &lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2018-08-22T06:06:30.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14461" PostTypeId="1" CreationDate="2018-08-22T17:35:17.870" Score="3" ViewCount="311" Body="&lt;p&gt;For the purposes of this question, there are two kinds of differential equations: linear, and non-linear, which is to say that there could be two ways to teach the subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way is to separate linear from non-linear differential equations, and combine the former with other courses; e.g. by adding simple linear differential equations to the end of a calculus course or by combining their study with &lt;em&gt;linear&lt;/em&gt; algebra. A more advanced course would then cover topics such as series solutions, or phase plane methods, laPlace transforms, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other way is to teach the fairly simple solution methods for (many) linear differential equations in the same course as the more complicated topics. That's the way I was taught (decades ago. But it seemed incongruent to me to put the simpler methods for solving linear equations in the same course as using the more advanced methods to solve non-linear equations,even though they fell under the same heading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is differential equations still (mostly) taught this way today, or have many college programs reorganized the teaching of differential equations along the lines suggested in the second paragraph instead?&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2021-02-09T22:20:07.063" Title="What is the preferred method of teaching &quot;linear&quot; differential equations today?" Tags="|differential-equations|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14462" PostTypeId="1" CreationDate="2018-08-23T15:04:27.360" Score="1" ViewCount="511" Body="&lt;p&gt;Til today I've been using geogebra to sketch functions for my students quizzes or homework. Sometimes I use the ones that I found searching in google, but this takes a lot of time specially because I have multiple courses and I need to diversify the tests, homework's, etc.  Recently I found this image:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tU0Wh.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tU0Wh.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's there a way to make this type of graph quickly without having to define five different functions and so on?&lt;/p&gt;&#xA;" OwnerUserId="8853" LastActivityDate="2018-08-23T16:45:00.303" Title="How can I make &quot;complex&quot; graphs that combine multiple functions with a software?" Tags="|calculus|software|functions|precalculus|graphing|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14463" PostTypeId="2" ParentId="14462" CreationDate="2018-08-23T16:45:00.303" Score="2" Body="&lt;p&gt;Have you used &lt;a href=&quot;https://www.padowan.dk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graph&lt;/a&gt;? There, if you want piecewise functions, a quick way to do so is to select:  &lt;strong&gt;Function&lt;/strong&gt; $\rightarrow$ &lt;strong&gt;Insert Point Series&lt;/strong&gt;, and type the coordinates of a point. Exit that environment, and while holding &lt;strong&gt;Select&lt;/strong&gt;, just click other points on your graph. You can choose how to connect them (linear, half-cosine, cubic spline) and later change this interpolation style with a click.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the holes in the graph, you'd still have to write a formula (&lt;strong&gt;Function&lt;/strong&gt; $\rightarrow$ &lt;strong&gt;Insert Function&lt;/strong&gt;), specifying whether it &quot;ends&quot; at a point or a hole.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: This is a Windows application, though you can get to run on Mac with &lt;a href=&quot;https://www.winehq.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wine&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-08-23T16:45:00.303" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14464" PostTypeId="1" CreationDate="2018-08-23T18:51:08.807" Score="3" ViewCount="186" Body="&lt;p&gt;For teaching geometry, it could be useful to have a website where one can enter the names of vertices of a polygon, specify which diagonals should be depicted, and specify the measures of certain angles, and have the website generate a credible figure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some likely websites for this purpose, and what are their strengths or weaknesses for use in the classroom or mathematics lab?&lt;/p&gt;&#xA;" OwnerUserId="8258" LastEditorUserId="1608" LastEditDate="2019-10-10T01:46:24.367" LastActivityDate="2019-10-11T11:47:25.873" Title="Recommend a website for creating geometric figures" Tags="|geometry|software|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14465" PostTypeId="1" CreationDate="2018-08-23T23:42:04.293" Score="4" ViewCount="142" Body="&lt;p&gt;For a course that I am teaching I have produced pdf format &quot;skeleton notes&quot;, i.e. partially completed class notes which students can print and bring to class. In the lecture I then aim to go though these notes, filling in examples, missing defintions, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am hoping to be able to do this by projecting the pdf notes and then using an iPad which I have borrowed from my university to write directly on the notes during lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to find an app suited to this purpose. One comment is that the app needs to allow me to zoom in my view, to get fine control over the symbols etc, without zooming in the projected view, i.e. the pdf that is being projected should not change except for the writing appearing on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any software suggestions which might fit my needs? I have tried &lt;em&gt;many&lt;/em&gt; &lt;em&gt;many&lt;/em&gt; apps but not yet found a suitable one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, the app is free, or at least has a free trial, since I am reluctant to purchase a whole bunch of apps without being totally sure that they will work for what I have in mind.&lt;/p&gt;&#xA;" OwnerUserId="10303" LastActivityDate="2018-08-24T02:14:51.840" Title="Using an iPad to write on pdf slides" Tags="|technology-in-education|software|lecture-notes|ipads|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14467" PostTypeId="2" ParentId="14465" CreationDate="2018-08-24T02:14:51.840" Score="4" Body="&lt;p&gt;I’ve been using GoodNotes 4 for six years now, and I’m pretty happy with it. Others I’ve heard about but do not have extensive experience with are Notability and Nebo.&#xA;Notability has a audio recording feature which upon playback can sync the audio to your writing (but again, I have not used this feature).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good Notes 4 does have the zoom in feature you are asking about - you can zoom in on a pane that shows on your iPad but it won’t show the zoomed in part on the projection.&lt;/p&gt;&#xA;" OwnerUserId="10289" LastActivityDate="2018-08-24T02:14:51.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14468" PostTypeId="2" ParentId="14439" CreationDate="2018-08-24T02:43:10.823" Score="0" Body="&lt;p&gt;This doesn't address co-teaching per se, but I strongly suggest you check out:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://bentilly.blogspot.com/2009/09/teaching-linear-algebra.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://bentilly.blogspot.com/2009/09/teaching-linear-algebra.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This link covers how you can incorporate effective learning methods into homework policy and Q&amp;amp;A. It seems to have worked incredibly well in a very important course that many students struggle with.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-08-24T02:43:10.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14469" PostTypeId="1" CreationDate="2018-08-25T02:01:02.657" Score="5" ViewCount="296" Body="&lt;p&gt;I love Category Theory and I also love programming so I learn both of them in my spare time. However I find that programming takes up so much of my spare time. I reason that the less thing I can do now, the better I would do them so maybe I would try to give up on programming. But I'm still very indecisive. If I'm going on the road of Category Theory research, there would be no use for improving and keeping my programming skill?&lt;/p&gt;&#xA;" OwnerUserId="7540" LastActivityDate="2018-08-30T17:39:15.703" Title="If I want to pursue theoretical mathematics, especially Category theory, do I ever need programming skill?" Tags="|computer-programs|category-theory|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14470" PostTypeId="1" CreationDate="2018-08-25T06:12:53.510" Score="-2" ViewCount="225" Body="&lt;p&gt;I have a kid, 8 years old. The boy has phenomenal math skills. When he was 6 he was solving fractions with roots, writing Fibonacci series and much more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now he developed a new calculation algorithm that shocked his math teacher. Teacher has consulted with university professors and none of them knew this algorithm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What could be done either to claim copyright, or that he gets get any kind of credit or recognition for it?&lt;/p&gt;&#xA;" OwnerUserId="10312" LastEditorUserId="2083" LastEditDate="2018-08-25T08:57:27.437" LastActivityDate="2018-08-25T09:15:53.203" Title="Kid, 8 years old, developed a new calculation algorithm" Tags="|gifted-students|" AnswerCount="1" CommentCount="9" ClosedDate="2018-08-26T13:35:43.823" ContentLicense="CC BY-SA 4.0" />
  <row Id="14471" PostTypeId="2" ParentId="14470" CreationDate="2018-08-25T09:15:53.203" Score="6" Body="&lt;p&gt;It is highly unlikely that your child has developed a new algorithm for solving an interesting problem, but it is possible. It is a very good sign that they developed something that was new for themselves at that age.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I recommend not trying to publish the work. If the child turns into a phenomenal mathematical genius, then their future accomplishments will almost certainly dwarf the present achievement. I do not believe that trying to publish the result would be a good way of nurturing the mathematical talents. Playing with numbers and formulae and finding like-minded people would be much more useful.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You (plural) should still document the work as well as you can, maybe using mathematics teachers as an aide. The child can carry the algorithm with them and introduce it to future teachers, or just keep it in mind.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you do really want to find out if the algorithm is novel, then your goal is to introduce it to as highly educated mathematician as you can find. I do not know the educational system in your country, or your country for that matter, but a doctoral student (PhD student) should certainly be able to check the novelty quite fast. The problem is getting in touch with such a person.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If there is a mathematics education program at a nearby university, then you might try to contact them and ask for advice on how to best encourage your child to grow their mathematical talents. This would also offer a way of discussing the algorithm, but I do not recommend leading with that.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2018-08-25T09:15:53.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14472" PostTypeId="2" ParentId="11436" CreationDate="2018-08-25T12:18:34.807" Score="1" Body="&lt;p&gt;When I read the original question, I focused mostly on why in algebra 2 do we teach complex numbers. And, as I am preparing to teach algebra 2 after several years of having taught other courses, I also wonder if it is the right course to introduce that topic. Of course, I do not disparage the importance and the beauty of the complex set, and when I teach it at the pre-calculus level, I do spend some time (not enough!) to talk about the geometry of the complex plane where I find the beauty of the complex numbers particularly clear. (I also present the complex plane simply as the real plane with a special algebra structure that allows you to add, subtract, multiply, and divide POINTS!) I also love to introduce complex numbers with a simple question that only involves natural numbers but whose solution is so quick if you play with sqrt(-1). I ask my students to express the product of the sum of two square numbers as the sum of two square numbers (e.g. find a and b such that: (12^2+7^2)*(9^2+17^2) = a^2+b^2.) I give them time to think about it while on a corner of the board I quickly solve it. I then scratch my work and simply give them my numbers. I find to be a fun historical analogy with the way 15th century Italien mathematicians were &quot;secretly&quot; using sqrt(-1) to solve cubic equations.&#xA;But why in algebra 2? Last time I taught that course, I did cover it, but now, as I try to have a coherent and logical understanding of the algebra 2 topics, and especially how the curriculum should flow, I perceive complex numbers as a marginal topic that may hinder this logical flow I'm seeking. In my view, better left for the next course, pre-calculus. (With a greater emphasis than is usually given to that fundamental topic.)&lt;/p&gt;&#xA;" OwnerUserId="10313" LastActivityDate="2018-08-25T12:18:34.807" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14473" PostTypeId="1" AcceptedAnswerId="14475" CreationDate="2018-08-25T19:23:15.607" Score="8" ViewCount="395" Body="&lt;p&gt;Let's say I'm teaching an undergraduate Calculus 1 course using Stewart's Calculus textbook. Should I spend a fair amount of time in lecture doing examples that are taken straight from the textbook chapters?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been doing this, because often the examples are quite good. This book is highly polished, and user-tested over many years, and so I'm unlikely to be able to routinely think of better examples to teach the material. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, my lectures might then sometimes feel a bit redundant. A student who has already read the relevant textbook sections might think, &quot;I've seen these examples already, so why do I need to be here?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is your opinion about this? My current viewpoint is that it is useful to go over textbook examples in class, because I can't assume the students will have read the book in advance. (And some students find it easier to understand things when explained in a lecture than by purely reading.) I'd like to see what other people think about this issue.&lt;/p&gt;&#xA;" OwnerUserId="9633" LastActivityDate="2018-08-27T19:21:51.067" Title="Should I go over examples straight from the textbook in Calculus lectures?" Tags="|textbooks|" AnswerCount="4" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14474" PostTypeId="2" ParentId="14473" CreationDate="2018-08-25T21:43:19.213" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;My advice is to follow the book very closely in topics, methods, notation, and explication.*  But I would not literally use the same examples unless you feel time pressed in preparation. Do an analogous problem but slightly different numbers or formula.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You could also look at the examples and the homework problems and at least do a sanity check.  Is there a &quot;missing example&quot; for doing the problems?  Or if there are too many examples to work in class (along with time for drill), put your brain to work on selecting those you think more important.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;*The desire by instructors to be creative ends up doing more harm than benefit.  I believe it comes from a sort of romantic view of the university lecturer rather than a sound pedagogical view of how humans learn (at all levels and ages).&lt;/p&gt;&#xA;" OwnerUserId="10126" LastEditorUserId="10126" LastEditDate="2018-08-26T01:56:01.613" LastActivityDate="2018-08-26T01:56:01.613" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14475" PostTypeId="2" ParentId="14473" CreationDate="2018-08-26T01:21:15.047" Score="5" Body="&lt;p&gt;Sometimes Stewart picks exactly the example I would have picked, even if I hadn't been prepping from his book. It is usually when the natural example is relatively simple. I wouldn't worry too much if you reuse a few of them -- as you say, they probably didn't read the book and if they did, who knows if they understood it -- but it is good to diversify. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can do things like covering homework questions which you aren't going to assign, picking a second calculus text for your reference, mining the textbooks of other STEM fields, and keeping a log of good examples you find incidentally to keep things from being verbatim Stewart's explanation.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-08-26T01:21:15.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14476" PostTypeId="1" CreationDate="2018-08-26T06:10:21.600" Score="1" ViewCount="205" Body="&lt;p&gt;By &quot;tutoring&quot;, I mean when parents hire someone to help their child one-on-one. I am referring to secondary education. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This &quot;help&quot; can take many forms, some of which are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Re-teaching concepts that were not understood in class&lt;/li&gt;&#xA;&lt;li&gt;Pre-teaching concepts that will be taught in the near future&lt;/li&gt;&#xA;&lt;li&gt;Homework help (&quot;help&quot; or &quot;guide&quot;, not do it for them of course)&lt;/li&gt;&#xA;&lt;li&gt;Review concepts before exams&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The students could:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Have learning disabilities and really need outside help&lt;/li&gt;&#xA;&lt;li&gt;Have some difficulty and require some help&lt;/li&gt;&#xA;&lt;li&gt;Have no difficulty but still wish to have help (to validate their understanding or be reassured in some way)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I know there are many cases and each student is different, but I simply wanted to know if:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Tutoring is viewed as a good thing by teachers&lt;/li&gt;&#xA;&lt;li&gt;If so, &lt;strong&gt;what would they expect the tutors to do/not to do with the student?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;If not, why is it not a good thing?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt;: I'm not asking if tutoring is good or bad, as that might be controversial. I'm asking what is the point of view of teachers specifically (your point of view if you are one, or that of your colleagues) and what you would expect tutors to be doing. Anything you have to share is a candidate to 'answer' this question.&lt;/p&gt;&#xA;" OwnerUserId="9341" LastActivityDate="2018-08-26T06:10:21.600" Title="What are teachers' expectations on private tutoring in secondary education?" Tags="|secondary-education|tutoring|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14478" PostTypeId="2" ParentId="14469" CreationDate="2018-08-26T14:23:29.493" Score="4" Body="&lt;p&gt;Your profile suggests that you are still a teen ager. Therefore my advice is, for now, to explore a lot of things and don't try to specialize too much in any one thing. It is a big world with a lot of possibilities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not suggesting that you give up your study of Category Theory, but there is a lot of other mathematics that is interesting and which supports that. Likewise, if programming is fun, even if hard, it will &lt;em&gt;stretch your brain&lt;/em&gt; in a slightly different way than mathematics. That is very valuable in itself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There will be plenty of time for specialization - even extreme specialization if you reach the stage of doctoral level education. &lt;/p&gt;&#xA;" OwnerUserId="8756" LastActivityDate="2018-08-26T14:23:29.493" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14479" PostTypeId="2" ParentId="14469" CreationDate="2018-08-26T16:42:39.503" Score="1" Body="&lt;p&gt;My advice would be to be open to areas of math that allow for programming skill.  You can still take some theoretical classes and such.  But there is a lot of good stuff in optimization theory, data processing, cryptography, etc. that uses both math and programming (and even sometimes other fields like OR, business, mechE for finite element analysis, etc.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if you get a broad science/math background, you may find yourself doing something even different from the other two choices (e.g. physics or crystallography) which still involves a lot of the skills and interests you have.  And you will be a better teacher and maybe even researcher if you have some general breadth.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-08-26T16:42:39.503" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14480" PostTypeId="2" ParentId="14473" CreationDate="2018-08-27T17:10:24.133" Score="8" Body="&lt;p&gt;An approach I sometimes use is to call students' attention to an example from the book without presenting it in detail, and then present a variation on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if I were teaching optimization with constraints, I might say something like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In your textbook, which as always I assume you have read before class today, there is an example of minimizing the surface area of a cylindrical can with a given volume.  You should make sure you read and understand that example -- it's Example 3 on p. 426.  I'm going to do a variation on that problem:  let's take a can with a given &lt;em&gt;surface area&lt;/em&gt; and try to &lt;em&gt;maximize the volume&lt;/em&gt;.  Essentially I'm going to exchange the roles of the constraint function and the objective function.  Let's see how it looks:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let's take a second look at Example 3 in the textbook, which (as always) I assume you have read before class.  The text solves this problem by using the constraint equation to solve for one variable in terms of the other; when you plug that in to the equation for the surface area you end up with an equation of one variable.  But you can also solve this problem by using implicit differentiation.  Let's see how that method looks:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This serves multiple purposes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It reminds students that they are expected to read the textbook and that you will hold them accountable for what they read; at the same time,&lt;/li&gt;&#xA;&lt;li&gt;It makes &lt;em&gt;connections&lt;/em&gt; between what is covered in class and what is covered in the textbook (most important, in my opinion) without devoting class time to something that should be done outside of class.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One way to think about this is that the examples done in class should be &lt;em&gt;homomorphic&lt;/em&gt; to the ones in the textbook, and sometimes even &lt;em&gt;isomorphic&lt;/em&gt;, but they should very rarely be &lt;em&gt;identical&lt;/em&gt; in all respects.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-08-27T17:10:24.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14481" PostTypeId="1" CreationDate="2018-08-27T17:36:05.947" Score="8" ViewCount="202" Body="&lt;p&gt;I'm looking to create animations for a linear algebra course. I need things like writing and changing equations, including matrices, plotting of 2- and 3-dimensional axes with points, vectors, lines and planes, and so on. The intention is to incorporate the animations into a separate video, to demonstrate and visualize what's being said.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I go about creating such animations? I'm ideally looking for a free solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm aware of 3Blue1Brown's youtube channel and vPython package, but I'm looking for other options. Mostly because I don't like working with vPython.&lt;/p&gt;&#xA;" OwnerUserId="10328" LastActivityDate="2019-03-11T18:08:47.317" Title="How to create educational linear algebra animations?" Tags="|undergraduate-education|linear-algebra|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14482" PostTypeId="2" ParentId="14473" CreationDate="2018-08-27T19:21:51.067" Score="2" Body="&lt;p&gt;When I choose to closely follow the book, I use &lt;em&gt;exercises as examples&lt;/em&gt;. For me, it's more work but also more freedom. Meanwhile, the students are exposed to both a polished presentation in the book and an informal discussion during the lecture.&lt;/p&gt;&#xA;" OwnerUserId="10168" LastActivityDate="2018-08-27T19:21:51.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14483" PostTypeId="2" ParentId="12024" CreationDate="2018-08-28T04:07:37.693" Score="-2" Body="&lt;p&gt;Instructors are looking for M &amp;amp; M : memorization and manipulation. We're training brains, not button pushers. Thus, no calculator on math tests. For engineering/applied courses, a scientific is allowed. Your graphing calculator required in high school? Sell it on eBay. When high schools stop teaching button pushing, education will take a turn for the better.&lt;/p&gt;&#xA;" OwnerUserId="10333" LastActivityDate="2018-08-28T04:07:37.693" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14484" PostTypeId="1" CreationDate="2018-08-28T13:32:33.443" Score="1" ViewCount="127" Body="&lt;p&gt;What is the most motivating way to introduct Pythagorean triples to undergraduate students? I am looking for an approach that will have an impact. Good interesting or real life examples will help. Is there any resources for this?&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="376" LastEditDate="2018-08-29T10:37:25.760" LastActivityDate="2018-08-29T10:37:25.760" Title="Pythagorean triples" Tags="|undergraduate-education|teaching|number-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14485" PostTypeId="2" ParentId="14484" CreationDate="2018-08-28T17:04:35.847" Score="1" Body="&lt;p&gt;I saw &lt;a href=&quot;https://www.youtube.com/watch?v=3wVYeg7LXc8&amp;amp;list=PL613A31A706529585&amp;amp;index=18&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; video a long time ago, I thought it was interesting. The talk is aimed to be a talk about one of the Millenium Problems but aimed at a general audience (so undergrads should understand/follow it). It is  the first a in  series about connecting Congruent Number Problem to the BSD Conjecture, and the 2nd video in the series describes the idea of generating Pythagorean Triples to help find Congruent Numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=P28b43RcDgk&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; video Might be a bit elementary for Undergrad, but it demonstrates a way to program Excel to find Pythagorean Triples for you by using Brute Force. The &lt;a href=&quot;https://www.youtube.com/watch?v=voGeePhmjQw&amp;amp;t=1s&quot; rel=&quot;nofollow noreferrer&quot;&gt;follow up video&lt;/a&gt; shows the algebraic manipulations to generate all the primitive triples.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2018-08-28T17:04:35.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14486" PostTypeId="1" AcceptedAnswerId="14491" CreationDate="2018-08-28T21:21:46.263" Score="1" ViewCount="1318" Body="&lt;p&gt;Can these be taught at a high school level?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;multivariable calculus&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;linear algebra&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;abstract algebra&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;real analysis&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;complex analysis&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(non-measure theoretic) probability theory&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Maybe even other more math courses.&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="-1" LastEditDate="2018-08-29T06:59:07.630" LastActivityDate="2018-08-30T11:28:34.253" Title="Can undergraduate math courses be taught at a high school level?" Tags="|secondary-education|" AnswerCount="6" CommentCount="8" ClosedDate="2018-08-30T14:07:02.360" ContentLicense="CC BY-SA 4.0" />
  <row Id="14487" PostTypeId="2" ParentId="14486" CreationDate="2018-08-28T22:50:26.917" Score="6" Body="&lt;p&gt;Why not? As long as the students have the appropriate prerequisites and aptitude, go for it. Our local private/boarding school regularly offers one of multivariate, linear algebra, or discrete math to the seniors. A lot of these kids take precalculus freshman year and are looking for more once they complete second semester calculus.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2018-08-28T22:50:26.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14488" PostTypeId="2" ParentId="14486" CreationDate="2018-08-28T23:20:16.677" Score="5" Body="&lt;p&gt;This paper discusses Linear Algebra as an elective (&quot;Mathematics III&quot;) for science&#xA;high-school students in South Korea.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Choe, Young Han. &quot;Teaching Linear Algebra to High School Students.&quot; &lt;em&gt;Research in Mathematical Education&lt;/em&gt; 8 (2004).&#xA;  &lt;a href=&quot;http://www.mathnet.or.kr/mathnet/kms_tex/981589.pdf&quot; rel=&quot;noreferrer&quot;&gt;PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So certainly Linear Algebra can be and has been taught at the high-school level.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-08-28T23:20:16.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14489" PostTypeId="2" ParentId="14486" CreationDate="2018-08-29T06:24:04.970" Score="5" Body="&lt;p&gt;Well, in India some of the topics you mentioned above are already being taught at high school level, infact they are necessary for students who want to become future engineers. You can check out the JEE-MAINS syllabus India(earlier known as IIT-JEE).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit:&#xA;&lt;a href=&quot;https://jeemain.nic.in/WebInfo/Handler/FileHandler.ashx?i=File&amp;amp;ii=86&amp;amp;iii=Y&quot; rel=&quot;nofollow noreferrer&quot;&gt;link to syllabus&lt;/a&gt; - look at page 23 for required topics for mathematics&lt;/p&gt;&#xA;" OwnerUserId="10336" LastEditorUserId="10338" LastEditDate="2018-08-29T17:53:27.743" LastActivityDate="2018-08-29T17:53:27.743" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14490" PostTypeId="2" ParentId="14486" CreationDate="2018-08-29T11:50:05.300" Score="3" Body="&lt;p&gt;In the UK, some students take A-Level Further Mathematics, which covers bits of linear algebra (determinants, solving and interpreting systems of 3 linear equations, possibly also eigenvalues and eigenvectors and diagonalizing matrices) and some simple first and second order ODEs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also gives students the option to study some elementary group theory (the definition of a group, basic examples, and Lagrange's Theorem) and elementary number theory (Euclidean algorithm, modular arithmetic, solving congruence equations).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One syllabus can be found &lt;a href=&quot;https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/a-level-l3-further-mathematics-specification.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2018-08-29T11:50:05.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14491" PostTypeId="2" ParentId="14486" CreationDate="2018-08-29T12:14:42.073" Score="1" Body="&lt;p&gt;In Spain, some linear algebra is taught in high school and is examined on the exams required to enter the university. Topics taught include matrices, determinants, solvability of and solving linear systems of equations, and Euclidean geometry of lines and planes in the plane and three space. Basic probability is also taught, focusing on the binomial and normal distributions. One variable calculus is taught in more detail, and notions such as continuity and differentiability are included. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One might ask how much students really learn. The answer is that most of this material has to be retaught, albeit rapidly, in the first few weeks of the university. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the US, with the great flexibility of its otherwise generally undemanding school system, advanced students often find a way to learn multivariable calculus, linear algebra, and more. The limiting factor is often the availability of an instructor competent to teach these things.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-08-29T12:14:42.073" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14493" PostTypeId="1" AcceptedAnswerId="14498" CreationDate="2018-08-29T18:15:35.277" Score="5" ViewCount="453" Body="&lt;p&gt;preface: I took AB and not BC but I feel both are similar in nature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By bad course, I mean that I feel as though &lt;a href=&quot;https://en.wikipedia.org/wiki/AP_Calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus AB/BC&lt;/a&gt; sets students up for 'failure,'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for starters, how successful are BC students in the future multivariable course?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get a 5 in BC, one needs to pass with almost a 60% of questions correct on the exam. The exam is also all computation (which to be fair is much of Calculus II anyway), but because a lot of the exam is computations, proofs are magically handwaved or ignored in favor of computation (which gives students a bad impression of what math really is). For example, delta-epsilon is never touched on and so when I first encountered delta-epsilon, I was very confused.&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="376" LastEditDate="2018-08-31T08:00:44.550" LastActivityDate="2018-08-31T08:00:44.550" Title="Is Calculus AB/BC a 'bad course?'" Tags="|calculus|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14494" PostTypeId="1" AcceptedAnswerId="14502" CreationDate="2018-08-29T19:27:43.593" Score="4" ViewCount="292" Body="&lt;p&gt;For example, there is an integration bee and lots of youtube videos tackling hard or impossible integrals with just tools of Calculus I &amp;amp; II. However, a lot of these integrals found on youtube need assistance from Multivariable (Gaussian integral) or Complex Analysis (use residue theorem to solve a real integral) or some other tricks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/2249352/integrals-that-cant-be-solved-by-real-methods&quot;&gt;Examples of these integrals&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.mit.edu/~same/pdf/qualifying_round_2018_test.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;More Examples of Integrals&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="10335" LastEditDate="2018-08-31T00:52:22.540" LastActivityDate="2018-08-31T12:50:38.807" Title="Would it be constructive to teach a whole course on how to evaluate certain hard integrals?" Tags="|integration|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14495" PostTypeId="2" ParentId="14494" CreationDate="2018-08-30T01:51:50.553" Score="3" Body="&lt;p&gt;I think it would be hard to get the demand for it, especially as a formal course.  Even for those students who were interested, there might be more &quot;bang for the buck&quot; to just take a diffyQ course or the like (more advanced material).&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-08-30T01:51:50.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14496" PostTypeId="2" ParentId="14493" CreationDate="2018-08-30T02:25:43.373" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think there is traditionally much need for epsilon-delta in third semester calculus.  After all, it is not a real analysis course. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When I took calc in the 80s, there was a short section on epsilon-delta.  But who cares.  It's not like you need it to do hard calc problems.  Euler didn't.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The 60% is more troubling, but all that said, I bet ETS has plenty of statistics to back up the idea that a 5 is like an A for a typical (GA Tech, not Cal Tech) calc course.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-08-30T02:25:43.373" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="14497" PostTypeId="2" ParentId="14486" CreationDate="2018-08-30T11:28:34.253" Score="3" Body="&lt;p&gt;If you want to teach more advanced courses and have sufficient students, I would advise 3rd semester calc, or ODEs.  The reason is these are all &quot;no regrets&quot; moves from the student point of view.  Pretty much any hard science, math or engineering will require them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Abstract algebra is only required for math students which is a tiny tiny percentage of your class population (with undeclared major high school students).  Just because they are good or advanced in math do not assume they will become math majors.  Common sense and personal experience and statistic should show that most will go into engineering or chemistry or physics or the like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Linear algebra is not a bad choice either, especially if it is one with emphasis on matrices and calculation (not proofs).  For one thing, it is an easier course than 3rd sem calc or ODEs.  (Perhaps important for kids who already feel they are essentially doing an elective.)  It won't be as central for engineers or physicists (most don't have a semester requirement, just pick up a little in midst of engine math class).  But it will be more central and required for comp sci students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe do the LA because of the easiness and serving the future comp sci students.  It won't hurt the engineers either.  Plus it will appeal to you more.  At least it is algebra even if not abstract.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-08-30T11:28:34.253" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14498" PostTypeId="2" ParentId="14493" CreationDate="2018-08-30T13:19:52.523" Score="8" Body="&lt;p&gt;Yes the AP exam is bad and not an adequate substitute for a rigorous college class at, say, a state flagship university.  This is for two main reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The exam allows calculators and the calculators are really good at Calculus, so you can answer many of the questions just by learning how to get the calculator to do calculus.&lt;/li&gt;&#xA;&lt;li&gt;The grade range is too small.  As you point out a 5 is 60%, which at most strong colleges would be a failing grade.  Of course the vast majority of students go to schools that are weaker than state flagships so maybe a 5 is an A at most schools, but the cutoff for credit at my university should be something more like a 6 rather than a 4.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="345" LastActivityDate="2018-08-30T13:19:52.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14499" PostTypeId="1" CreationDate="2018-08-30T13:57:42.483" Score="2" ViewCount="3502" Body="&lt;p&gt;I am looking for good motivational examples of how quadratic equations can naturally arise in real life for someone starting high school. The high school book my child is using just jumps into factoring and solving them without any clear purpose. The few motivations seem to be contrived, such as the flight of a ball. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A related question I found is the following: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/11310/everyday-example-problems-for-solving-linear-and-quadratic-equations&quot;&gt;Everyday Example Problems for Solving Linear and Quadratic Equations&lt;/a&gt;. My question is different because I am looking for motivations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible starter example:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;A farmer has 2400 ft. of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? [1]&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only concern I have here is that the quadratic function does come up but the solutions to the obvious quadratic equation are not immediately intuitive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally an example of how they arose in antiquity would be great.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://studentsuccess.asu.edu/sites/default/files/optimization.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://studentsuccess.asu.edu/sites/default/files/optimization.pdf&lt;/a&gt; &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="555" LastActivityDate="2018-09-06T03:01:11.540" Title="Good (natural) motivational examples for quadratic equations" Tags="|algebra|precalculus|" AnswerCount="3" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14500" PostTypeId="2" ParentId="12024" CreationDate="2018-08-30T17:19:01.557" Score="2" Body="&lt;p&gt;In a perfect world, after students learn some new technique they would program &lt;em&gt;themselves&lt;/em&gt; how to do it into a computer and then they would be allowed to that computer program to solve problems.  The problem isn't using a calculator, it's using a calculator that was programmed by someone else.  That's not testing the students' knowledge, that's testing the knowledge of the person who programmed the calculator.&lt;/p&gt;&#xA;" OwnerUserId="345" LastActivityDate="2018-08-30T17:19:01.557" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14501" PostTypeId="2" ParentId="14469" CreationDate="2018-08-30T17:39:15.703" Score="1" Body="&lt;p&gt;Learning how to code and code well is an incredibly useful skill in math research in all fields.  (I can code, but not code well, but I've been very lucky to have collaborators who are much better than I am.  Around half of my papers involve serious computer calculations)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But perhaps more importantly, category theory is one of the closest fields in math to computer science!  For some examples, category theory is used in &lt;a href=&quot;http://math.mit.edu/~dspivak/informatics/talks/CTDBIntroductoryTalk&quot; rel=&quot;nofollow noreferrer&quot;&gt;studying databases&lt;/a&gt;, &lt;a href=&quot;https://wiki.haskell.org/All_About_Monads&quot; rel=&quot;nofollow noreferrer&quot;&gt;monads&lt;/a&gt; are an important notion in category theory and also in Haskell, and Homotopy Type Theory is a computer science-inspired approach to &lt;a href=&quot;https://homotopytypetheory.org/book/&quot; rel=&quot;nofollow noreferrer&quot;&gt;homotopy theory&lt;/a&gt; and &lt;a href=&quot;https://arxiv.org/abs/1705.07442&quot; rel=&quot;nofollow noreferrer&quot;&gt;higher category theory&lt;/a&gt;.  Some &lt;a href=&quot;https://www.cs.ox.ac.uk/people/jamie.vicary/&quot; rel=&quot;nofollow noreferrer&quot;&gt;category theorists&lt;/a&gt; actually work in CS departments.  I'd say that if you want to do category theory that's of broad interest, it's almost essential to have a solid background in computer science.&lt;/p&gt;&#xA;" OwnerUserId="345" LastActivityDate="2018-08-30T17:39:15.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14502" PostTypeId="2" ParentId="14494" CreationDate="2018-08-31T03:12:56.447" Score="4" Body="&lt;p&gt;Much of what follows is a matter of opinion, and it is kind of stream-of-conciousness.  &lt;em&gt;Caveat lector.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think that courses (particularly &lt;em&gt;informal&lt;/em&gt; courses) on &quot;hard integrals&quot; already exist (at least in the United States).  These courses are an outgrowth of the graduate school admissions process here.  Specifically, many PhD programs in the US require applicants to take (and do well on) the &lt;a href=&quot;https://www.ets.org/gre/subject/about/content/mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Subject GRE&lt;/a&gt;.  While this examination ostensibly covers the breadth of undergraduate mathematics, it seems to emphasize the calculus sequence quite a bit.  As such, GRE prep classes often spend a great deal of their time on techniques of integration, and focus on &quot;tricky&quot; integrals.  While such courses are about more than &lt;em&gt;just&lt;/em&gt; integration, they are likely the closest thing to a course on &quot;hard integrals&quot; that you will find.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think that a course in &quot;Computational Techniques&quot; would be an interesting and valuable course to have running around in the curriculum, but I think that it fills a niche that doesn't really exist at the moment.  I have argued to colleagues that the undergraduate calculus sequence focuses too much on computational techniques and not enough on theory, and that one way of mitigating this might be to rejigger the curriculum a bit to change this.  Instead of teaching three semesters of calculus, then a semester of linear algebra and another (or two) of differential equations, I think it might be worthwhile to condense things a bit, focus on theory, then offer a semester long course on computational techniques.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such a course would likely cover a lot of ground---I might defer most of the trigonometric integrals to such a course, play around with &quot;integration bee&quot; style problems, and hit the Cartesian-to-spherical integration problems pretty hard.  The course should also spend some time on transformational techniques for solving DEs (e.g. the Laplace and Fourier transforms), some of the more clever tricks in linear algebra, and perhaps some of the techniques from complex analysis (though the target audience that I have in mind probably wouldn't have the background for complex, so maybe not).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My feeling is that such a course would appeal primarily to non-math majors (particularly physics majors), but that it could also find an audience among the kind of students that currently enroll in more informal GRE prep classes.  However, I can't see there being a great deal of demand in the current environment, as much of the material already covered, albeit spread out over a lot of other undergraduate courses.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In short, my answer to the headline question is a &lt;em&gt;very&lt;/em&gt; qualified &quot;Yes?&quot;&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2018-08-31T12:50:38.807" LastActivityDate="2018-08-31T12:50:38.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14503" PostTypeId="2" ParentId="14499" CreationDate="2018-08-31T08:15:30.930" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The few motivations seem to be contrived, such as the flight of a&#xA;  ball.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Ideally an example of how they arose in antiquity would be great.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The latter question might be better addressed on &lt;a href=&quot;//hsm.stackexchange.com&quot;&gt;History of Science and Mathematics&lt;/a&gt;, but it seems quite likely that they arose in antiquity largely to tackle such issues as... the flight of a ball. War is a powerful driver of technology, and understanding the trajectory of missiles launched from a catapult would improve the effectiveness of your artillery.&lt;/p&gt;&#xA;" OwnerUserId="8005" LastActivityDate="2018-08-31T08:15:30.930" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14504" PostTypeId="1" CreationDate="2018-08-31T21:34:54.930" Score="6" ViewCount="259" Body="&lt;p&gt;This semester I am a TA for a Calc 2 course. At my first meeting with my instructor, he mentioned in passing that &quot;Homework is always easier than an exam, because homework questions come from the textbook, so they always know which section the problems are from.&quot;. It's just anecdotal, but that does align with my own experience taking calculus courses: each section covers a very specific technique, and the problems in that section are designed to be solved using that technique, and often very little else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It occurred to me that, since we regularly have 2-3 sections due for a single homework assignment, this 'bug' could be worked around. The 'manual' way to do this would be to (1) type up every homework problem into LaTeX, (2) randomize their order into a document, and (3) hand them the worksheet as a homework assignment&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[ I would probably say explicitly that they are taken from the textbook. The goal here isn't to hide the fact that they're textbook problems, merely to make the section numbers less visible so that the student has to pause to think about the tools they need to start a problem. ]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course step (1) is fairly labor-intensive, even if there isn't any TikZ work involved. If I am using a very common book, like Stewart, is there an online tool to, e.g. create worksheets by typing in a list of problem numbers?&lt;/p&gt;&#xA;" OwnerUserId="10344" LastActivityDate="2018-09-08T08:42:50.780" Title="Automatically creating homework worksheets from textbook problems" Tags="|undergraduate-education|homework|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14505" PostTypeId="2" ParentId="14504" CreationDate="2018-08-31T21:47:45.103" Score="2" Body="&lt;p&gt;If you have access to an online version of the textbook, you could take screenshots of each problem you want to use and compile them in a TeX document very quickly. It's also very easy to rearrange the order of the problems.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2018-08-31T21:47:45.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14506" PostTypeId="2" ParentId="14504" CreationDate="2018-08-31T22:01:21.733" Score="4" Body="&lt;p&gt;Easier than the proposed solution may be to use another book, or online worksheets, and look for one of the following features:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Mixed practice&quot; (a.k.a. &lt;a href=&quot;https://en.wikipedia.org/wiki/Varied_practice&quot; rel=&quot;nofollow noreferrer&quot;&gt;varied practice&lt;/a&gt;) exercise sets, which are intended to solve precisely this problem. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;End-of-chapter review exercises, which may generally have the same&#xA;composition.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A test-generator program with premade testbank problems (e.g., Stewart links to &quot;Cengage Learning Testing Powered by Cognero&quot; see bottom &lt;a href=&quot;https://stewartcalculus.com/media/16_inside_supplements.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;; my preferred choice is Pearson Testgen with some Pearson text's testbank).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-09-01T06:07:30.417" LastActivityDate="2018-09-01T06:07:30.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14508" PostTypeId="2" ParentId="14504" CreationDate="2018-09-01T01:26:19.667" Score="4" Body="&lt;p&gt;There are lots of ways around this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;As Daniel R. Collins has suggested, many textbooks feature end-of-chapter reviews or &quot;mixed practice&quot; problems.  Indeed, my recollection is that Stewart's calculus features such review sections quite prominently at the end of each chapter.  It might not be a bad idea to assign exercises from those sections, if only as review before an exam.&lt;/li&gt;&#xA;&lt;li&gt;If you use an online homework system, such as MyLab or WebAssign, you can create homework assignments which mix problems from various sections of a single book, or which include problems from multiple books.  Alternatively, you can &lt;em&gt;create&lt;/em&gt; assignments using these tools, then print them out and give them to the students as handouts.  This may actually be the solution that you are looking for---I believe that ebook version of Stewart's calculus comes bundled with some online homework system which you could potentially use to construct assignments.&lt;/li&gt;&#xA;&lt;li&gt;Frankly, it isn't that difficult to $\TeX$ up a bunch of problems for homework---it just takes some practice.  If you want to include images, that takes a little longer, but if you use &lt;code&gt;\includegraphics&lt;/code&gt; or some such package to include .eps or .png images, you don't have to muck about with TikZ or pstricks.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;That being said, I don't think that it is necessarily a bad thing to tell students which tools they are expected to use, at least not when they are first learning to use those tools.  At the very least, I think that there should be a very clear distinction made between &quot;These are the problems that you are doing in order to get a handle on this new tool introduced in section x.y,&quot; vs &quot;These are the questions which could potentially use any tool you have ever learned, up to an including those introduced in section x.y.&quot;&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2018-09-04T12:54:44.787" LastActivityDate="2018-09-04T12:54:44.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14509" PostTypeId="1" CreationDate="2018-09-01T13:39:31.047" Score="9" ViewCount="1311" Body="&lt;p&gt;In the U.S. students in grades $\{9,10,11\}$ often learn long division of two polynomials, e.g.:&#xA;$$&#xA;\frac{x^4 + 6x^2 + 2}{x^2 + 5} = x^2 + 1 - \frac{3}{x^2 + 5} \;.&#xA;$$&#xA;I believe it is fair to say that almost never is any motivation provided:&#xA;Why would anyone ever want to divide two polynomials?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. How can polynomial division be motivated at that grade level?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The dilemma seems to be that, where polynomial division is really needed&amp;mdash;for example, &#xA;in error-correcting codes (&lt;a href=&quot;https://en.wikipedia.org/wiki/Cyclic_redundancy_check&quot; rel=&quot;noreferrer&quot;&gt;cyclic redundancy checking&lt;/a&gt;)&amp;mdash;requires&#xA;more advanced math to understand.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-09-04T02:32:03.617" Title="Motivation for polynomial long division" Tags="|secondary-education|algebra|solving-polynomials|" AnswerCount="5" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="14510" PostTypeId="2" ParentId="14499" CreationDate="2018-09-01T17:37:59.907" Score="0" Body="&lt;p&gt;Note mathematics are models of the real world. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What makes quadratic equations nice is they either have a maximum or minimum. So for many problems we are trying to solve; our systems have maxima and minima. a quadratic representation lets us model such a system's behavior close to the maximum or minimum value. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In trying to help my students to understand the motivation for various models, I will sometimes show curve fitting (done via software) to real data. This illustrates real data can be noisy and that our models may not be exact but can be good enough to predict from the data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside MAA has a nice textbook - Functions, Data, and Models An Applied Approach to College Algebra, that explains from this perspective, motivating the algebra from how we have to apply it to real world data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when you look back at the flight of the ball the quadratic form happens naturally, a ball will rise to a maximum point fighting gravity until gravity stops its upward motion and then fall towards the ground with gravity. Since gravity works in the up and down direction (y or h) in the equation this motion will be symmetric around that point. This language is also how we would describe a graph of a quadratic equation. If you progress to calculus you can derive the quadratic nature of this system by knowing that the acceleration due to gravity is constant. But that is another discussion for a later time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some resources about the thought process of choosing the model. &#xA;Once a model is selected you use the tools of algebra to bring out all the predictive power of that model.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ck12.org/book/CK-12-Basic-Algebra-Concepts/section/10.10/&quot; rel=&quot;nofollow noreferrer&quot;&gt;CK-12  Linear, Exponential, and Quadratic Models&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U10_L2_T1_text_container.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Monterey Institute Algebra Course&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.mathsisfun.com/algebra/quadratic-equation-real-world.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math is Fun Examples&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note many of the resources you find will start with the flight of a projectile because this is a real life experience that is common to many people so it is used to bridge conceptually from the formula to application.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To Specifically answer your questions about the historical motivation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Quadratic_etc_equations.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;University of Virginia Math History&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.mathnasium.com/the-history-behind-the-quadratic-formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathnasium a nice summary timeline&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8952" LastActivityDate="2018-09-01T17:37:59.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14511" PostTypeId="2" ParentId="14509" CreationDate="2018-09-02T15:23:52.553" Score="0" Body="&lt;p&gt;Needed for partial fractions (calculus topic) and for LaPlace transforms (normal part of ODE course).  In terms of applications, common need is in controls problems (very simple level systems engineering classes that most engineers take).  EE will have more frequent needs.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-09-02T15:23:52.553" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14512" PostTypeId="2" ParentId="14509" CreationDate="2018-09-02T18:10:47.157" Score="4" Body="&lt;p&gt;There are many example of when this may be useful, I would point to integral calculus as a place where right hand side of your example is easier to evaluate with introductory tools than the left hand side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though I want to look past examples in application and speak to the idea of putting expressions into a particular form to &quot;see&quot; patterns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you look at the right hand side of the equation the form generated lets us talk about the end behavior of x visually.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The leading coefficient is positive. The highest degree term is even.&#xA;So as x nears negative infinity this expression approaches positive infinity. As x nears positive infinity this expression approaches positive infinity. As x reaches zero this expression is at $\frac{2}{5}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can be done visually and with minimal computation. This parallel structure in form helps us with most of the expressions we evaluate and to compare expressions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So by doing this we can compare rational expressions to the polynomials and notice the similarities and the differences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also expressions can have asymptotic behavior to a line and polynomial division can show this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:&#xA;$f(x)=\frac{4x^3+x^2+3}{x^2-x+1}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long division gives us:&#xA;$f(x)=4x+5+\frac{x-2}{x^2-x+1}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The large x behavior of x has the fraction reaching 0 when x goes to either infinity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this function approaches $4x+5$  is has an asymptote to this line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means if we are talking generally about data if we are not in close to where this has curved behavior the linear model could work for us simplifying our calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Graphing it we can see this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NqVC9.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NqVC9.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8952" LastActivityDate="2018-09-02T18:10:47.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14513" PostTypeId="2" ParentId="14509" CreationDate="2018-09-02T20:14:29.233" Score="1" Body="&lt;p&gt;How about the mother of all elephants in the room... It forms the basis for our positional number system...in the sense that carrying out polynomial division and the substituting x=10 will give just the normal quotient and remainder....&lt;/p&gt;&#xA;" OwnerUserId="998" LastActivityDate="2018-09-02T20:14:29.233" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14514" PostTypeId="1" CreationDate="2018-09-02T22:40:55.793" Score="2" ViewCount="652" Body="&lt;p&gt;** Edit&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks everyone for some great suggestions.  I should have been more clear though.  I am actually looking for a college level proof that pertains to algebra or leads to algebra in some form.  My professor has already shot down any proof that would be easy enough to teach to a high school algebra I or II class.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am currently in my final year studying mathematics education.  I am doing a 1 credit math seminar project.  I will be writing a paper and presenting on this topic.  I have chose integrating mathematical proofs into high school math.  However, I need a detailed proof for this topic.  Anyone have any suggestions?  I am looking for a proof that I can expand on and talk about for nearly 30 minutes.&lt;/p&gt;&#xA;" OwnerUserId="10348" LastEditorUserId="10348" LastEditDate="2018-09-04T13:53:15.853" LastActivityDate="2018-09-05T21:31:30.503" Title="Undergraduate Math Seminar topic" Tags="|proofs|seminar|" AnswerCount="6" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14515" PostTypeId="2" ParentId="14509" CreationDate="2018-09-02T22:55:29.647" Score="5" Body="&lt;p&gt;One reason is that it's essential to determining the slant asymptote of a rational expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another reason is that it's a useful step in factoring large polynomials.  For example, say you're trying to factor $x^3 + 4x^2 -4x-1$.  You can determine, using the Rational Root Theorem, that x - 1 is a factor.  Since x + 1 isn't, my next step would be to divide $x^3 + 4x^2 -4x-1$ by x - 1.  That would tell me that the factorization is $(x - 1)(x^2 +5x + 1)$ and I could get the final result using the Quadratic Formula.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2018-09-02T22:55:29.647" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14516" PostTypeId="2" ParentId="14514" CreationDate="2018-09-03T03:12:37.330" Score="3" Body="&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt; - Following the edit made to the OP, I must admit it is a bit unclear what you are looking for exactly. Here is what I understand. The proof must:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Pertain or lead to algebra&lt;/li&gt;&#xA;&lt;li&gt;Algebra must be harder than algebra I and II&lt;/li&gt;&#xA;&lt;li&gt;Be college level&lt;/li&gt;&#xA;&lt;li&gt;Be &lt;em&gt;hard&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Not be 'related to any subject'&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;First, I would ask your professor for an example of something she'd find acceptable, because there are contradictions here. No college level proof is &lt;em&gt;hard&lt;/em&gt;, no proof is &lt;em&gt;unrelated to any subject&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She is basically asking you to find a proof hard enough that it is unteachable at algebra level, but at the same time, you are somehow expected to find a way to make it teachable. This sounds as absurd as asking you to find a way to teach colors to blind people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All I can think of is the usual calculus/analysis proofs:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Rolle's theorem&lt;/li&gt;&#xA;&lt;li&gt;Intermediate value theorem&lt;/li&gt;&#xA;&lt;li&gt;Unicity of zero theorem&lt;/li&gt;&#xA;&lt;li&gt;Lagrange theorem&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;They 'pertain to algebra' because they use the concept of functions. They would be difficult to teach to algebra students, who think of $f(x)$ as a formula, and not some abstract object. They are college level.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I suggest you read section 1.1 of Stephen Abbott's &lt;em&gt;Understanding Analysis&lt;/em&gt;. In it, Abbott talks about G.H. Hardy and two of the proofs he used as a 'defense' of mathematics (more precisely, about how mathematics is an artistic endeavour).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first is the proof that $\sqrt{2}$ is irrational.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second is the proof that there are infinitely many prime numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both proofs are quite short, relatively easy to understand and although they are very old, they are still &quot;as fresh and significant as when [they] where discovered&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you could talk about one or both of these proofs in the first half of your presentation and then explain why they are still relevant and how easily they could be included in the high school curriculum in the second half.&lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="9341" LastEditDate="2018-09-05T21:31:30.503" LastActivityDate="2018-09-05T21:31:30.503" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14517" PostTypeId="2" ParentId="14514" CreationDate="2018-09-03T07:01:08.863" Score="1" Body="&lt;p&gt;In high school students usually see the statement that if a continuous function changes sign in a closed interval then it has zero in the interval. This is usually given some sort of informal proof in the form of a drawing, but it can be given a constructive proof that is interesting in its own right. Namely, what is called the method of bisection for root finding yields a proof of the theorem. (Recall that the bisection method consists simply in choosing a half of the given interval in which the given function changes sign, and iterating; an explicit error bound is not hard to obtain.) This approach has the virtue that the proof is not hand-wavy (not some informal appeal to geometric intuition the student might not yet have) and yields an algorithm for actually calculating the root (given a concrete function) that the student can actually do by hand (or on a computer if one has such facilities available). Presenting the proof in this way also gives an introduction to numerical methods, and there is a lot that can be said there, e.g. $x^{2} - 2$ has a root at $\sqrt{2}$, but how does one obtain a decimal approximation of $\sqrt{2}$? (In practice, bisection is not the way to do it, but that's another story, that comes later.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A virtue of this example is that students are made to spend a lot of time factoring polynomials and finding their roots. The context of the problem is not unfamiliar to them. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-09-03T07:01:08.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14518" PostTypeId="2" ParentId="14514" CreationDate="2018-09-03T08:36:56.687" Score="2" Body="&lt;p&gt;You could take a look at basic set theoretic proofs, like for example showing that&#xA;$$A \cap (B \cup C) = (A \cap B) \cup (A \cup C).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can draw a diagram here and you can show and emphasize formal proofs. This property is not hard to prove, but it teaches the students how to do formal proofs, step by step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another such sort of proofs would be the classical proofs by induction, like for example&#xA;$$\sum_{i=1}^n i = \frac{n(n+1)}{2}.$$&#xA;Again, the proof is not the hard part here, you can use it to show and discuss formal procedures and rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, check out your lecture notes from the first few weeks at university. Most courses start with a formal introduction to proofs and give proofs and concepts that are both easy to grasp and important to learn for beginning students, so they would fit well with high school students who just start to learn about formality and proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compared to the other answers you don't have the nice effect of showing off a great result, but you are teaching your (future) students valuable skills and are starting with the basics (of course one can argue if a fancy example to motivate them first would be good).&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2018-09-03T08:36:56.687" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14519" PostTypeId="2" ParentId="14514" CreationDate="2018-09-03T12:29:54.170" Score="1" Body="&lt;p&gt;Might I suggest:  Every polygon triangulation can be $3$-colored.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/DbNcf.png&quot; alt=&quot;3-coloring&quot;&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;sup&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Art_gallery_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia image&lt;/a&gt;.&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;See &lt;a href=&quot;https://matheducators.stackexchange.com/a/2454/511&quot;&gt;this earlier ME post&lt;/a&gt;&#xA;for surrounding details and links.&#xA;This can be motivated by and leads to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Art_gallery_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;art-gallery theorem&lt;/a&gt;.&#xA;Art-gallery-like theorems continue to be an active area of research today.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-09-03T12:29:54.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14520" PostTypeId="1" CreationDate="2018-09-03T13:16:24.030" Score="1" ViewCount="124" Body="&lt;p&gt;I was recently at &lt;a href=&quot;https://dronedays.epfl.ch/fr/index-fr-html/&quot; rel=&quot;nofollow noreferrer&quot;&gt;EPFL drone days&lt;/a&gt; and enjoyed a demo of a robot that could follow a black line like in the sketch (I can improve the sketch on demand): &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/qLMQR.jpg&quot; width=&quot;200&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I remembered my good all times at the qualitative theory of differential equations where we learnt about the conditions for a particle not to go out of a certain region under the influence of a scalar field. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I came up with the following idea, why not do design a robot that would behave according to a scalar field in the plane and that could be used for teaching differential equations, specially the qualitative theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would supersede the old idea of just &quot;visualizing&quot; the vector field generated by the differential equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps this has been done before?&lt;/p&gt;&#xA;" OwnerUserId="6363" LastActivityDate="2018-09-08T15:14:32.837" Title="A robot to simulate differential equations for undergraduate students." Tags="|undergraduate-education|technology-in-education|differential-equations|" AnswerCount="1" CommentCount="2" ClosedDate="2018-09-13T16:31:26.097" ContentLicense="CC BY-SA 4.0" />
  <row Id="14521" PostTypeId="2" ParentId="14514" CreationDate="2018-09-03T14:45:01.807" Score="2" Body="&lt;p&gt;The binomial theorem is a great way to introduce proofs, since it provides a &quot;shortcut&quot; for high school algebra problems using Pascal's triangle (which students enjoy drawing). You can ask why it keeps working and lead them to proof by induction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also pivot to a question about choosing from sets, and let students realize they're the same numbers. There's a ton of fun stuff you can follow up with in future sessions.&lt;/p&gt;&#xA;" OwnerUserId="7707" LastActivityDate="2018-09-03T14:45:01.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14524" PostTypeId="2" ParentId="14514" CreationDate="2018-09-03T18:01:03.403" Score="1" Body="&lt;p&gt;I recommend choosing an elementary proof that includes both elements of surprise and beauty - one that might spark students to further study mathematics. Along these lines I've had much success employing the striking geometric generation of the tree of primitive Pythagorean triples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a brief sketch. Given such a triple, i.e. coprime naturals $\,(x,y,z)\,$with $\,x^2 + y^2 = z^2$ we can divide by $z^2$ to obtain $\,(x/z)^2\!+(y/z)^2 = 1,\,$ so each triple corresponds to a rational point $(x/z,\,y/z)$ on the unit circle. Aubry showed that we can generate all such triples by a very simple geometrical process. Start with the trivial point $(0,-1)$. Draw a line to the point $\,P = (1,1).\,$ It intersects the circle in the &lt;em&gt;rational&lt;/em&gt; point $\,A = (4/5,3/5)\,$ yielding the triple $\,(3,4,5).\,$ Next reflect the point $\,A\,$ into the other quadrants by taking all possible signs of each component, i.e. $\,(\pm4/5,\pm3/5),\,$ yielding the inscribed rectangle below. As before, the line through $\,A_B = (-4/5,-3/5)\,$ and $P$ intersects the circle in $\,B = (12/13, 5/13),\,$ yielding the triple $\,(12,5,13).\,$ Similarly the points $\,A_C,\, A_D\,$ yield the triples $\,(20,21,29)\,$ and $\,(8,15,17).$ &#xA;&lt;a href=&quot;https://i.stack.imgur.com/9xudY.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9xudY.jpg&quot; alt=&quot;Generating triples on the unit circle&quot;&gt;&lt;/a&gt;&#xA;We can continue this process with the new points $\,B,C,D\,$ doing the same we did for $A,\,$ obtaining further triples. Iterating this process generates the primitive triples as a ternary tree&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\qquad\qquad$   &lt;a href=&quot;https://i.stack.imgur.com/sUg9E.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sUg9E.jpg&quot; alt=&quot;Ternary tree of primitive Pythagorean triples&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Descent in the tree is given by the formula (whose &lt;em&gt;reflective&lt;/em&gt; geometric genesis is &lt;a href=&quot;https://math.stackexchange.com/a/1127199&quot;&gt;given here&lt;/a&gt; )&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{eqnarray} (x,y,z)\,\mapsto &amp;amp;&amp;amp;(x,y,z)-2(x\!+\!y\!-\!z)\,(1,1,1)\\ = &amp;amp;&amp;amp;(-x-2y+2z,\,-2x-y+2z,\,-2x-2y+3z)\end{eqnarray}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;e.g. $\ (12,5,13)\mapsto (12,5,13)-8(1,1,1) = (-3,4,5),\ $ yielding $\,(4/5,3/5)\,$ when reflected into the first quadrant.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ascent in the tree by inverting this map, combined with trivial sign-changing reflections:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\quad\quad (-3,+4,5) \,\mapsto\, (-3,+4,5) - 2 \; (-3+4-5) \; (1,1,1) = ( 5,12,13)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\quad\quad (-3,-4,5) \,\mapsto\, (-3,-4,5) - 2 \; (-3-4-5) \; (1,1,1) = (21,20,29)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\quad\quad (+3,-4,5) \,\mapsto\, (+3,-4,5) - 2 \; (+3-4-5) \; (1,1,1) = (15,8,17)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Continuing in this manner one may reflectively generate the entire tree of primitive Pythagorean triples, e.g. the topmost edge of the triples tree corresponds to the ascending $C$-inscribed zigzag line &#xA;$(-1,0), (3/5,4/5), (-3/5,4/5), (5/12,12/13), (-5/12,12/13), (7/25,24/25), \ldots$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The linked elegant presentation using geometry of quadratic spaces is a bit beyond high-school level (though elementary and accessible to bright high-school students). Instead we can derive the formulas directly though simple algebraic calculations, e.g.  below is a sketch John Conway gave in email (1998/3/9),  slightly edited. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take the typical rational point  $(x,y) = (p/r, q/r )$&#xA;on the circle, and join it to the nearest corner of the square.&#xA;The typical point on this join is $(x',y') = (1-c(1-x),1-c(1-y))$ &#xA;which lies on the circle provided that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align} 1  \,=\quad\   &amp;amp;1 - c(2 - 2x) + c^2 (1-x)^2 \\&#xA;         \ +\ &amp;amp;1 - c(2 - 2y) + c^2(1-y)^2&#xA;\end{align}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;hence  $\,0  =   1-c-c(3-2x-2y)+ c^2(3-2x-2y)\, $ after using $  x^2+y^2 = 1  $    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;giving  $\ c\, =\, (1\ {\rm  or})\ 1/(3-2x-2y) = r/(3r-2p-2q) $     &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore  $\,x' = 1-c(1-p/r)  = (2r-p-2q)/(3r-2p-2q)$   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;and its denominator is strictly between $0$ and $r$ since  $p+q$  is between  $r$ and  $\,r\sqrt 2 &amp;lt; 3r/2 $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This shows that if you join to the nearest corner the denominator&#xA;decreases, but also conversely that if you join to any other corner,&#xA;it increases.  By induction, we see that there &lt;em&gt;is&lt;/em&gt; a path to any point,&#xA;and then that this path is unique, because if an increasing move is&#xA;followed by a decreasing one, they must cancel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remark&lt;/strong&gt; The diagram is excerpted from the terse presentation on p. 172 of Conway and Guy's delightful &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/038797993X&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Book of Numbers&lt;/a&gt; - which is a good source to browse for other possible elementary topics worthy of presentation at this level. John Conway informed me that the entry was contributed by Richard Guy, and Richard informed me that it is based on a talk given by Roger Vogeler at 15:40 on Fri 89-04-07 at the Intermountain Section meeting of the MAA at Brigham Young University.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example provides an elementary glimpse of some beautiful and deep connections between number theory and geometry (reflective lattices). See the &lt;a href=&quot;https://math.stackexchange.com/a/1127199&quot;&gt;linked MSE post&lt;/a&gt; for further discussion.&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2018-09-03T18:01:03.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14526" PostTypeId="2" ParentId="14509" CreationDate="2018-09-04T02:32:03.617" Score="5" Body="&lt;p&gt;There is no sane motivation for introducing this topic to kids in grade 9, 10, or 11. Not all these kids are even bound for college. Of those who are, only some will take calculus. This kind of manipulation of polynomials can be applied to certain things like curve sketching, but that's peripheral in a calculus course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Topics like this just accumulate in math curricula like useless crap in someone's garage.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-09-04T02:32:03.617" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14527" PostTypeId="1" AcceptedAnswerId="14529" CreationDate="2018-09-04T15:06:44.620" Score="6" ViewCount="341" Body="&lt;p&gt;I have a 13-year-old nephew that liked learning with me, but I've recently moved 400km away. I wonder if it is possible to still learn with him long distance. The subjects I care about are maths and chemistry. We were usually doing additional assignments and puzzles on paper, and I was showing him often, how to write things better (e.g. keep some things vertically aligned) or how to draw an auxiliary drawing for an exercise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It gets extremally hard when we don't share a piece of paper, so I wanted to invest in a graphical tablet for both of us. I have a few questions.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Have you tried such solutions? Do they work?&lt;/li&gt;&#xA;&lt;li&gt;Do we need something with a touchscreen or does a graphical tablet connected to computer suffice? A pad with a stylus is usually significantly cheaper, but I am not sure how hard it is to point to the right thing while looking at a computer screen and writing on a pad. On the other hand, touchscreen solutions are less precise and sometimes annoying for writing.&lt;/li&gt;&#xA;&lt;li&gt;Do you know a &quot;collaborative whiteboard&quot; app or website that we could use as a shared piece of paper?&lt;/li&gt;&#xA;&lt;li&gt;Do you recommend any concrete devices?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10354" LastActivityDate="2020-07-02T12:06:22.087" Title="Teaching math long distance" Tags="|interactive-teaching|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14528" PostTypeId="2" ParentId="14527" CreationDate="2018-09-05T06:53:54.103" Score="1" Body="&lt;p&gt;I occasionaly use Zoom for working web conferences with colleagues. It allows to share screen and to share a whiteboard. You can give it a try.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2018-09-05T06:53:54.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14529" PostTypeId="2" ParentId="14527" CreationDate="2018-09-05T06:56:58.573" Score="3" Body="&lt;p&gt;I have used (and still use) a &lt;em&gt;Huion H420&lt;/em&gt; graphic tablet (it costs around 40$). It has pressure recognition which really helps the software understand and interpret your writing and give a nice output. (Some software does not take pressure recognition into acount and you don't get the full benefit of your tablet in that case).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After the first session (say one hour), you should both be pretty much at ease with the way writing works with that tablet. It's like normal writing, except you have to do it in a constrained space, but in my experience, it quickly becomes second-nature so I wouldn't worry about that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The software I liked the most was &lt;em&gt;Microsoft OneNote 2010&lt;/em&gt; (but it also works in &lt;em&gt;Microsoft Word 2010&lt;/em&gt; and I assume more recent versions). I &lt;em&gt;think&lt;/em&gt; you can have access to the free online version of these softwares by making a Microsoft account, if you want to try it out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These software recognized the handwritting pretty well. The curves are not converted into rough lines, even if you write really fast, so you don't have to spend 60 seconds writing one sentence or one equation in order for it to be readable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then it's a matter of finding software to &quot;share-screen&quot;. I have used &lt;em&gt;Skype&lt;/em&gt; in the past but a downside is that only one screen could be shared at a time (I do not know if this is still the case and if this has been improved). So for example, I could share my screen with someone else, give explanations, but they couldn't show me their work or write on top of what I wrote without me ending my screen sharing and them opening theirs (so it can get tedious if you need to switch often).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Things to consider are the &lt;strong&gt;synch&lt;/strong&gt;. Ideally, audio is in synch with video (so that your nephew hears the explanations and sees them written out simultaneously) and your tablet is in synch with &lt;em&gt;your&lt;/em&gt; video (for obvious reasons). If your sessions are to last an hour or more, then even a 2 second delay might become annoying. One thing that's happened to me in the past is that I always cut the other person as they were asking questions; all of this due to bad sync (although that was not with Skype). Turning off webcam &lt;em&gt;could&lt;/em&gt; help in this situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last I checked &lt;em&gt;Skype&lt;/em&gt; is also free software so you could try that out today with him and see how your connection is and get a feel for the screen sharing. For example, try to draw things in Paint and ask your nephew if things appear smoothly enough for his taste on his end (does the image refresh fast enough or is it too sloppy).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe there are some &quot;online whiteboards&quot; that are free can be shared in real-time, but all the ones I found where nowhere near as precise as OneNote was. For example, writing at a fast pace would be perfectly readable in OneNote but complete gibberish using other software. (A $4$ might look like a $9$ because every curve you draw is turned into a rough &quot;line&quot;.) I believe there might be two possible causes to this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The software isn't made to recognize as much precision as my graphic tablet is capable of (and it's a really cheap one)&lt;/li&gt;&#xA;&lt;li&gt;There are issues with the way data is transferred due to poor Internet connection&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As for the &quot;relational aspect&quot;, sure it's different than having him work next to you, but there is no &quot;stranger barrier&quot;; you already know each other and have worked together in the past and if that worked out, I don't believe that sessions being online will make things that different. You say he liked working with you, so it's not as though you had to discipline him into not going to Facebook while you both work. It's definitely do-able.&lt;/p&gt;&#xA;" OwnerUserId="9341" LastActivityDate="2018-09-05T06:56:58.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14531" PostTypeId="1" AcceptedAnswerId="14535" CreationDate="2018-09-06T00:52:19.747" Score="2" ViewCount="637" Body="&lt;p&gt;Students often misuse the equals sign to indicate &quot;I've done this operation&quot; rather than the proper use indicating numerical equivalence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eg. Tax is paid using the rule: \$3 572 plus 32.5c per \$1 over $37 000.&#xA;How much tax is paid on \$78 540?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer 1: 78 540-37 000 = 41 540 x 32.5%  = 13 500.50 + 3572 = $17 072.50&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answer 2: 3 572+(78 540-37 000)x 32.5% = 3 572+41 540 x 32.5% = 3 572+13 500.50 = $17 072.50&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both methods produce the correct answer. But Answer 1 misuses &quot;=&quot; to mean &quot;now we do this&quot;, perhaps it also shows a lack of understanding of order of operations and  algebra, but fundamentally it is a good method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How could this method be written to not misuse the equals sign?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How could this method be used to introduce the correct use of the equals sign?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observation: What they are doing is writing the expression (a+(b-c) x d) using a form of postfix notation: bc-dxa+ which is quite a rational thing to do.&lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="998" LastEditDate="2018-09-06T02:22:28.360" LastActivityDate="2018-09-07T00:34:01.927" Title="Harnessing misuse of equals sign" Tags="|algebra|equal-sign|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14532" PostTypeId="2" ParentId="14531" CreationDate="2018-09-06T01:15:28.600" Score="1" Body="&lt;p&gt;Update. It appears that the intent of this answer was not clear to some readers, so I will elaborate. The most common way to fix the student's computations is simply to break out the subexpressions onto separate lines (as in the other answers). There are however other noteworthy approaches that offer some advantages. Below we briefly present one such approach. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observe that the student uses the equal sign as it functions on an (infix) calculator, i.e. to evaluate the preceding expression. Using variables (and disambiguating parentheses) the student's expression is essentially the first one displayed below, where the equal sign has been (ab)used to annotate (label) subexpressions with their values $\,\color{#0a0}n,\color{#c00}m.\,$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align} ((a\!-\!b =\color{#0a0}n) \times c = \color{#c00}m) + d &amp;amp;\\[.2em]&#xA; \underbrace{\overbrace{(a\ \ -\ \ b)}^{\Large\color{#0a0} n}\ \ \times\ \ c}_{\Large\color{#c00} m}\ \ +\ \ d\ \ &amp;amp;&#xA;\end{align}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can greatly improve this by using better ways to annotate the values  $\,\color{#0a0}n,\color{#c00}m.\,$ e.g. as above using under/overbraces. Notice how this helps clarify the &lt;em&gt;algebraic structure&lt;/em&gt; of the expression, e.g. now we can see with a single glance  that it depends &lt;em&gt;linearly&lt;/em&gt; on $c$ (interest rate) so we can make inferences about how its value changes with changes in the interest rate. But this  innate algebraic structure is greatly obfusctated if instead we dissect the expression into many subexpressions and break them out onto separate lines (into a &quot;straight line computation&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my point is that we should teach students &lt;em&gt;various&lt;/em&gt; ways to present such computations - not only the common straight-line methods. When one encounters more complex expressions the structure-preserving approaches can make a huge difference in reducing complexity. For a less trivial example, below is an excerpt from one of my &lt;a href=&quot;https://math.stackexchange.com/a/134925&quot;&gt;MSE posts on telescopic induction.&lt;/a&gt;. Notice how the over/underline annotations below allow one to comprehend in a single glance the effect of the telescopic cancellations - which would be greatly obfuscated if the intermediate annotated expressions were broken out onto separate lines.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hint&lt;/strong&gt; $\: $ First trivially inductively prove the &lt;strong&gt;Fundamental Theorem of Difference Calculus&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\rm\ F(n)\ =\ \sum_{i\: =\: 1}^n\:\ f(i)\ \  \iff\ \ \ \color{#c00}{F(1)=f(1)},\,\ \ \color{#0a0}{F(n) - F(n\!-\!1)\ =\ f(n)}\ \ {\rm for}\ \ n&amp;gt; 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;whose proof is simply a rigorous inductive proof of the following telescopic cancellation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\rm \underbrace{\overbrace{\color{#c00}{F(1)}}^{\large \color{#c00}{f(1)}}\phantom{-\color{#c00}{F(1)}}}_{\large =\  0}\!\!\!\!\!\!\!\!\!\!\!\!\!\overbrace{\color{#0a0}{-\,F(1)\!+}\!\phantom{F(2)}}^{\large\color{#0a0}{ f(2)}}\!\!\!\!\!\!\!\!\! \underbrace{\color{#0a0}{F(2)} -F(2)}_{\large =\ 0}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\overbrace{\phantom{-F(2)}+ F(3)}^{\large f(3)}\!\!\!\!\!\!\!\!\!\!\underbrace{\phantom{F(3)}-F(3)}_{\large =\ 0}+\,\underbrace{\cdot\ \cdot\ }_{\large =\,0\,}\overbrace{\cdot\ +F(n)}^{\large f(n)}\ =\ F(n) $$&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2018-09-07T00:34:01.927" LastActivityDate="2018-09-07T00:34:01.927" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14533" PostTypeId="2" ParentId="14531" CreationDate="2018-09-06T01:16:40.437" Score="1" Body="&lt;p&gt;Answer 1,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$78 540-37 000 = 41 540 \times 32.5\% = 13 500.50 + 3572 = $17 072.50$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;could be rewritten as:&#xA;\begin{align*}&#xA;78540-37000 &amp;amp;=41540\\&#xA;41540\times 32.5\% &amp;amp; =13500.50\\&#xA;13500.50+3572 &amp;amp; =17072.50&#xA;\end{align*}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The use of $\leadsto$ has been suggested in the comments below to indicate each next step in the reasoning. At the level where a student makes this kind of mistake, they might not be ready for the introduction of further notation (if they get confused with $=$, they might get even more confused with some new symbol). As &quot;guest&quot; has suggested below, simply writing each step on a different line should be sufficient.&lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="9341" LastEditDate="2018-09-06T04:41:19.153" LastActivityDate="2018-09-06T04:41:19.153" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14534" PostTypeId="2" ParentId="14499" CreationDate="2018-09-06T03:01:11.540" Score="0" Body="&lt;p&gt;I think something a little different that what's normally done in most high school classes is finding equilibria in second order difference equations (but it certainly can be done if one discusses recursive functions as models).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me give an example:  suppose you have a population of fish that grow logistically (this can be explained to a high school student without too much effort) with a carrying capacity of 20000 and a growth rate of .02.  Then the population $P(t)$ at time $t$ years can be given by &#xA;$$P(t) = P(t-1) +(.02)\cdot(1- P(t-1)/20000)\cdot P(t-1).$$&#xA;Find the positive populations $s$ at which the population doesn't change (i.e., find the equilibria).  What if we allow fisherpeople to fish 80 fish a year from the population?  How do the equilibria change?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From there you could talk about stable vs unstable equilibria, draw a phase diagram, modify the model and see what happens, etc.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2018-09-06T03:01:11.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14535" PostTypeId="2" ParentId="14531" CreationDate="2018-09-06T08:21:14.497" Score="4" Body="&lt;p&gt;I would write this as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Earnings over $\$37,000$: $\$78,540-\$37,000 = \$41,540$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Base tax: $\$3,572$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additional tax: $\$41,540\times 0.325 = \$13,500.50$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Total tax: $\$3,572+\$13,500.50 = \$17 072.50$&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2018-09-06T08:21:14.497" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14536" PostTypeId="2" ParentId="14504" CreationDate="2018-09-08T08:42:50.780" Score="0" Body="&lt;p&gt;In university, you should be able to work out a deal with the prof and/or other TAs to say this to the students.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Your homework assignment is to make a compilation of the 10 most&#xA;likely types of questions you expect on your next test. You&#xA;must:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Give each question a descriptive title.&lt;/li&gt;&#xA;&lt;li&gt;Include step-by-step solutions, with explanations, cues (how you would know what to do), diagrams, checks, comments/comparisons/contrasts, etc. as necessary.&lt;/li&gt;&#xA;&lt;li&gt;Rate each question as easier, as hard as, or harder than you expect on the test. At least one question must be conceptually harder than what you expect on the test.&lt;/li&gt;&#xA;&lt;li&gt;Compare your set of questions and solutions to those of 2 of your peers. Keep editing until everyone approves of everyone's work.&lt;/li&gt;&#xA;&lt;li&gt;Hand in your 10 questions at least 2 weeks before the test.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;You will receive feedback from your TA 1 week before the final exam, mostly to tell you if you are or are not on the right track. Needless to say, you should have mastered everything that you hand in.&lt;/p&gt;&#xA;&lt;p&gt;This will count for 2% of your final grade.&lt;/p&gt;&#xA;&lt;p&gt;I will take what the TA considers the best questions and minor variations on them will constitute 25% of your test.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This will force them and incentivize them to think through all the different concepts and techniques they've learned and how to distinguish among them. It will also give you an idea of how clueless or on-the-ball they really are, well in advance of the test.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-09-08T08:42:50.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14537" PostTypeId="1" CreationDate="2018-09-08T13:13:24.813" Score="5" ViewCount="575" Body="&lt;p&gt;I am no professional educator; I am a student myself! But apparently I come up with useful tricks that help my younger brother do better in maths. I just want to hear your feedback, is all.&lt;/p&gt;&#xA;&lt;p&gt;My younger brother is in Year &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; (that is, seventh grade or class in Australia) and was learning the following: &lt;span class=&quot;math-container&quot;&gt;$$a=\frac bc\implies b=ac\;\mathrm{and}\;c=\frac ba.$$&lt;/span&gt; He wanted me to help him remember, and I came up with a trick (although I told him that he should know &lt;em&gt;why&lt;/em&gt; certain rules are as is, and not find some nifty tricks to help him remember, but he doesn't listen). So here was my trick:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Imagine the vinculum (fraction bar) as the horizon, such that &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; are positions of the sun, sunrise and sundown respectively. Then &lt;strong&gt;sunrise multiplies&lt;/strong&gt; and &lt;strong&gt;sundown swaps around&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So if &lt;span class=&quot;math-container&quot;&gt;$a=b\div c$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; (sunrise) is equal to &lt;span class=&quot;math-container&quot;&gt;$ac$&lt;/span&gt; (multiplies) and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is equal to &lt;span class=&quot;math-container&quot;&gt;$b\div a$&lt;/span&gt; (swaps).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What is your feedback? I think the word &lt;em&gt;sunset&lt;/em&gt; is more common than &lt;em&gt;sundown&lt;/em&gt; so I am just making sure if this &amp;quot;trick&amp;quot; is actually doing good for my brother. Any thoughts for improvements, perhaps?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;To anybody who wants to &lt;span class=&quot;math-container&quot;&gt;$100\%$&lt;/span&gt; mentally grasp the idea of fractions, I absolutely recommend seeing @&lt;a href=&quot;https://matheducators.stackexchange.com/users/163&quot;&gt;Number&lt;/a&gt;'s (Bill Dubuque's) comment below @&lt;a href=&quot;https://matheducators.stackexchange.com/users/501/in%C3%A9quation&quot;&gt;inéquation&lt;/a&gt;'s answer! Please give that an upvote!&lt;/p&gt;&#xA;" OwnerUserId="10366" LastEditorUserId="163" LastEditDate="2023-01-26T22:36:45.890" LastActivityDate="2023-01-26T22:36:45.890" Title="Is This Trick Helpful?" Tags="|secondary-education|algebra|division|" AnswerCount="4" CommentCount="28" ContentLicense="CC BY-SA 4.0" />
  <row Id="14538" PostTypeId="2" ParentId="14537" CreationDate="2018-09-08T13:41:21.680" Score="-2" Body="&lt;p&gt;The trick is more difficult than the proposition. You can look at the fraction $\dfrac{ a}{b}=\dfrac{c}{d}$ from the right, from the left or from the bottom. For example looking from the right gives $\begin{array}{r} \rightarrow\\\rightarrow\end{array} \dfrac{ a}{b}=\dfrac{c}{d} \Longleftrightarrow \dfrac{a}{c}=\dfrac{b}{d}$. Looking at $a=\dfrac{b}{c}$ from the left gives $c=\dfrac{b}{a}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way is permuting the diagonals. For example permuting $a$ and $d$ in $\dfrac{ a}{b}=\dfrac{c}{d}$ gives $\dfrac{d}{b}=\dfrac{c}{a}$.&lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2018-09-08T13:41:21.680" CommentCount="27" ContentLicense="CC BY-SA 4.0" />
  <row Id="14539" PostTypeId="2" ParentId="14520" CreationDate="2018-09-08T15:14:32.837" Score="1" Body="&lt;p&gt;I think projects like this are useful for the first undergrad class in control systems (required majors course for engineers).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think it would be that useful for a standard undergrad diffyQ course, which is rather rushed and has a hard time spending lots of time on applications, estimation methods etc.  (course is just the proverbial ten pounds in five pound bag...it is).  If you spend time doing this lab experiment, you are likely slighting time that should be spent on the standard methods.  Note, I LOVE applications.  But I also believe in prioritization and efficiency.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think it is a very cool lab experiment though.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-09-08T15:14:32.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14540" PostTypeId="2" ParentId="14537" CreationDate="2018-09-08T15:34:27.767" Score="17" Body="&lt;p&gt;I do not think that such tricks are helpful:  in fact I believe they are deeply damaging.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These types of basic relationships should not be memorized:  they should be derived on the fly from an understanding of the meaning of the operations and the meaning of equality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Namely, you should understand that $a = \frac{b}{c}$ is asserting that the number $a$ &lt;strong&gt;is identical&lt;/strong&gt; to the number $\frac{b}{c}$.  Since these numbers are equal, the result of multiplying each by $c$ will also be equal.  Thus $ac = \frac{b}{c} c$.  Now, if you understand that $\frac{b}{c}$ mean &quot;take $b$ and split it into $c$ equal pieces&quot;, and you understand that multiplication by $c$ means &quot;take $c$ of those pieces&quot;, then you should instantly recognize that $\frac{b}{c} c = b$.  Thus $ac=b$.  This reasoning might take a considerable amount of thought the first few times, but it eventually becomes automatic. &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-09-08T15:34:27.767" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14541" PostTypeId="2" ParentId="10597" CreationDate="2018-09-08T17:47:47.540" Score="1" Body="&lt;p&gt;Property (3), i.e., the $\varepsilon-\delta$ definition of continuity, has numerous motivations/interpretations. For example, continuity can be interpreted as &lt;em&gt;accuracy&lt;/em&gt;. Suppose we are shooting a cannon located at the top of a hill. Even when the mathematics is perfect, our limited knowledge of the many parameters that affect the accuracy of our shot will make us under- or overestimate where the cannonball hits the ground. Let's take just one: the accuracy of our measurement of the height of the hill. If the height, $H$, varies, then so will the placement, $D$, of the shot. Of course, the smaller error in the former will lead to a smaller error in the latter and, furthermore, we can achieve any required degree of accuracy, $\varepsilon$, of our shot if we can ensure a sufficient accuracy, $\delta$, of the measurement of the height of the hill. In other words, the dependence of $D$ on $H$ is continuous. But if we replace this target shooting with a game of tennis, we will face discontinuity.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ETCEf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ETCEf.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10168" LastActivityDate="2018-09-08T17:47:47.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14542" PostTypeId="2" ParentId="14537" CreationDate="2018-09-09T11:54:04.190" Score="3" Body="&lt;p&gt;$$a=\frac bc\implies \;c=\frac ba.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you propose above is one of the if/thens that students should pick up after seeing it in practice a couple times. The 'long' way has them multiplying both sides by c and dividing by a. It only takes seeing this, with variables or integers a few times before they should be able to do that swap (of a and c) intuitively. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd note that you mentioned grade 7. This would be a student who is a bit advanced, as in the US this material is 8th or 9th grade depending on the school. I'd save the 'tricks', the mnemonics, etc, for the things that will really help, such as SOHCAHTOA for the trig identities. While the trick may be cute, it takes more time to think about it than to understand why that variable swap is allowed and just practice it, and embrace it. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-09-09T11:54:04.190" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14543" PostTypeId="2" ParentId="14537" CreationDate="2018-09-10T11:24:58.300" Score="3" Body="&lt;p&gt;I teach Year 7 and I’m with Steven on this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sort of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a “trick” reminds you of a correct understanding that you’ve already established, then it is a procedural memory aid and shortcut. Nothing tricky about that. If a trick is memorised and applied without an understanding of how it was derived and why it works, then it may prevent getting to that essential fuller understanding later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This sunrise/sunset metaphor you’ve developed with your brother isn’t necessarily bad. But he does need to understand balancing equations first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\require{cancel}&#xA;\begin{align}&#xA;a&amp;amp;=\frac{b}{c}\\&#xA;\Rightarrow a\times c&amp;amp;=\frac{b}{c}\times c\\&#xA;\Rightarrow a\times c&amp;amp;=\frac{b}{\cancel{c}}\times\cancel{c}\\&#xA;ac&amp;amp;=b&#xA;\end{align}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach this explicitly, make sure all my students can do it, and make sure they can explain it (not just describe it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But once this is understood, I start talking informally about &lt;em&gt;moving $c$ to the top on the left&lt;/em&gt;. It is on the bottom (denominator) on the right (RHS), so we can move it to the top (numerator) on the left (LHS). You can figure out what I might say about moving $c$ back, or for $+$/$-$ balancing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This informal talk describes the end result of the algebraic manipulations. I think your sunrise/sunset talk is figuratively equivalent. But maybe more loosely so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Steven says these kinds of basic results should be derived on the fly. I would modify that slightly to say that you should be &lt;em&gt;able&lt;/em&gt; to derive them on the fly, but that applying a correct memorised result is an appropriate shortcut. Mathematicians do this all the time. For example, identities. I could re-derive $\sin^2 x+\cos^2 x=1$ pretty quickly, but that might distract me from thinking about why I’m selecting this identity when I’m trying to do a difficult integration by parts (senior secondary maths).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Identities are a totally different thing from balancing equations, but the truth is equivalent: mathematics is made easier by applying rules for previously established results—but only in the context of appreciating how those results were obtained.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, encourage your brother to take the time to understand balancing equations, and then to think of a better metaphor/trick which more closely links to the ways terms are allowed to move around in a balanced equation.&lt;/p&gt;&#xA;" OwnerUserId="7046" LastActivityDate="2018-09-10T11:24:58.300" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14545" PostTypeId="1" CreationDate="2018-09-11T16:43:07.410" Score="2" ViewCount="65" Body="&lt;p&gt;I am teaching intro to Geometry using Moise and Downs textbook.  It is an excellent text but somewhat old.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know if there are sample tests that are available for use with this textbook? &lt;/p&gt;&#xA;" OwnerUserId="10373" LastActivityDate="2018-09-11T16:43:07.410" Title="Geometry sample tests" Tags="|geometry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14547" PostTypeId="1" AcceptedAnswerId="14548" CreationDate="2018-09-12T01:35:15.277" Score="11" ViewCount="2853" Body="&lt;p&gt;Third grade grandchild had this for homework. Can someone explain the intent here?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/j7APi.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/j7APi.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10376" LastEditorUserId="238" LastEditDate="2019-10-25T16:18:52.177" LastActivityDate="2023-10-04T13:07:28.427" Title="Third Grade Question -- This makes no sense to me" Tags="|primary-education|arithmetic-operations|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14548" PostTypeId="2" ParentId="14547" CreationDate="2018-09-12T02:42:56.667" Score="6" Body="&lt;p&gt;I am not too familiar with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Common_Core_State_Standards_Initiative&quot; rel=&quot;noreferrer&quot;&gt;Common Core State Standards Initiative&lt;/a&gt; (whose standards I assume the question above is intended to follow), but according to &lt;a href=&quot;http://www.corestandards.org/Math/Content/3/introduction/&quot; rel=&quot;noreferrer&quot;&gt;this introduction to the standards for Grade 3&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To me, it seems that the emphasis is on &quot;equal-sized groups,&quot; &quot;same-size units of area,&quot; &quot;identical rows,&quot; and &quot;identical columns.&quot;  The child's teacher could have emphasized this in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If so, then perhaps one &quot;valid&quot; answer to this higher-order thinking question is &quot;You can add objects together if they belong to the same group.  You can multiply groups of objects if they are of the same size.&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For example, say that students are riding in $3$ buses:  one bus has $30$ students, another has $30$, and another has $32$.  How many students are there in total?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is not $30+30+32+3$, that is, the number of buses is not added because buses are not students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is not $30\times 3$, because not all the buses have exactly $30$ students.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2018-09-13T01:05:45.983" LastActivityDate="2018-09-13T01:05:45.983" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14549" PostTypeId="1" AcceptedAnswerId="14551" CreationDate="2018-09-12T10:31:26.590" Score="3" ViewCount="413" Body="&lt;p&gt;I often see a situation when professors use words &quot;logic&quot;, &quot;mathematical proof&quot; and even prove logically while actually knowing that students are not even familiar with logic itself, i.e. no formal understanding of equivalence, implications, inference rules, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How are students supposed to understand such &quot;proofs&quot; because they can actually accept as a proof any &quot;intuitively&quot; explained reason why a theorem is true and never even suspect they were deceived unless they know the exact definition of an argument and true (sound) argument? Is knowing logic always taken as granted like a prerequisite? Even so, shouldn't the lecturer at least designate some time for explaining basic logic?&lt;/p&gt;&#xA;" OwnerUserId="9168" LastEditorUserId="9168" LastEditDate="2018-09-13T07:47:03.413" LastActivityDate="2018-09-13T20:39:26.277" Title="How is it correct for a lecturer to prove and &quot;explain&quot; a proof while explicitly knowing students are not familiar with logic itself?" Tags="|undergraduate-education|proofs|" AnswerCount="2" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="14550" PostTypeId="2" ParentId="14549" CreationDate="2018-09-12T17:23:17.560" Score="2" Body="&lt;p&gt;A mathematical proof has (among others) the purpose to convince someone of some fact, given some already established facts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether or not a proof is valid does not depend on who presents it. That is one of the key features of math - it does not matter at all if the &quot;professor knows what he does&quot; or if she sounds clever. If I can't help but agree that the arguments clearly show that the new fact follows from known facts, I have to agree to the new fact as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are however several levels of difficulty, for example:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Direct proof: showing $A \Rightarrow B$ by showing $A \Rightarrow C_1 \Rightarrow C_2 \Rightarrow \dots \Rightarrow B$&lt;/li&gt;&#xA;&lt;li&gt;Proof by contradiction.&lt;/li&gt;&#xA;&lt;li&gt;Proof by (complete) induction.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I agree that point 3 is somewhat hard to employ without any prerequisite in logic, but the first two points are doable. I will give examples for each.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a &lt;em&gt;huge&lt;/em&gt; difference between explaining (or understanding) a proof and finding and writing it down. Since the question is asking about a lecturer explaining a proof, I'll not bother with the latter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the $a \text{ even} \Rightarrow a^2 \text{ even}$ example:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The lecturer can explain the bigger picture of the proof: &quot;We take any even number and show that its square must also be even&quot;.&lt;/li&gt;&#xA;&lt;li&gt;The lecturer can remind the students that for even $a$, there exists $k \in \mathbb N$ so that $a = 2k$. Give examples if neccessary. This should be working knowledge already. If not, this has to be considered a gap in mathematical basics and not in logic.&lt;/li&gt;&#xA;&lt;li&gt;The manipulation to express $a^2$ in terms of $k$ can be executed by either students or lecturer, leading to $a^2 = 4k^2$ if done correctly -- regardless of whoever did it.&lt;/li&gt;&#xA;&lt;li&gt;The last step, showing that the number $4k^2$ is also even because $4k^2 = 2 \cdot k'$ with $k' := 2k^2$ can also be done by the lecturer. This decomposition always works and again does not depend on who does it.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;None of these steps requires &quot;deep&quot; understanding&quot; of logic except &quot;simple&quot; implications. But in the end, the students should be able to understand that for any even $a$, also $a^2$ must be even.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is indeed a completely different story for the students to come up with a similar proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding proofs by contradiction: it is important for the students to understand that apart from the cleverly chosen starting fact, each conclusion is right and again does not depend on who presents the reasoning. If one then arrives at a contradiction, the only remaining option is that the premise was wrong. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's look at Euler's proof of the fact that there are infinite prime numbers. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Assume that there are finitely many prime numbers. We don't know yet if this is true or false, but we can assume either.&lt;/li&gt;&#xA;&lt;li&gt;If the students have mathematical background about divisors, they must agree that the product of all primes, increased by one, is prime or has a prime divisor that is not in the list we started with. This argument is somewhat complex, but uses only known facts about divisibility.&lt;/li&gt;&#xA;&lt;li&gt;The reasoning in point 2 is sound. The only way to fix the error that we did not start with &quot;all primes&quot; is to assume that there is no such thing.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;To summarize: Understanding proofs does not require formal knowledge of logic. Common sense is enough, together with the attitude that arguments are not valued by authority.&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2018-09-13T20:39:26.277" LastActivityDate="2018-09-13T20:39:26.277" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14551" PostTypeId="2" ParentId="14549" CreationDate="2018-09-13T05:56:31.603" Score="4" Body="&lt;p&gt;In practice, one does have to explain (even to university engineering students) that although A implies B, it need not be the case that B implies A. However, there is a confounding factor, which is that mathematical logic is not what is used in scientific reasoning. As is explained by Polya in one of his books (the one on plausible reasoning) and by V. I. Arnold in some of his essays (he tells a story about his paper being rejected), in scientific contexts, when it is known that A implies B, the truth of B provides evidence for the plausibility of A (this is precisely what Polya calls plausible reasoning). It is therefore incumbent on the teacher of mathematics to distinguish mathematical logic from ordinary reasoning (dare I say common sense reasoning?) and it is generally helpful to do this explicitly rather than implicitly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, one does not need to teach logic formally, but the basic consequences of negations of implications should be explained if they are used, because they are not obvious to many, and they are less so still in contexts in which reasoning is not strictly mathematical (an error often made by mathematicians is to assume that there are no such contexts). This is particularly so if one proves a statement by proving its contrapositive, or by contradiction. The validity of either line of argument is not clear to a novice, and the strategy underlying such an argument needs to be explained to such a student. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-09-13T05:56:31.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14552" PostTypeId="1" CreationDate="2018-09-13T15:54:27.797" Score="10" ViewCount="1119" Body="&lt;p&gt;What are good examples of proofs by induction that are relatively low on algebra? Examples might include simple results about graphs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My aim is to help students get a sense of the logical form of an induction proof (in particular proving a statement of the form 'if $P(k)$ then $P(k+1)$'), independent of the way one might show that in a proof about series formulae specifically.&lt;/p&gt;&#xA;" OwnerUserId="1295" LastActivityDate="2021-10-22T12:10:31.360" Title="'Low-algebra' examples of induction" Tags="|secondary-education|induction|" AnswerCount="6" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14553" PostTypeId="2" ParentId="14552" CreationDate="2018-09-13T17:39:26.787" Score="6" Body="&lt;p&gt;How about: A tree with $n\ge 1$ vertices has $n-1$ edges.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2018-09-13T17:39:26.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14554" PostTypeId="2" ParentId="14552" CreationDate="2018-09-13T17:39:55.527" Score="4" Body="&lt;p&gt;A couple of simple examples come to mind:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Prove that there are $2^n$ subsets of an $n$-element set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Prove the power rule of derivatives for non-negative integer powers using the product rule.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastActivityDate="2018-09-13T17:39:55.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14555" PostTypeId="2" ParentId="14552" CreationDate="2018-09-13T17:41:04.930" Score="4" Body="&lt;p&gt;I think tiling problems are good for this kind of thing.  See, for example, &lt;a href=&quot;https://www.math.hmc.edu/funfacts/ffiles/20002.4.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;.  There they describe how to prove the statement &quot;if you have a $2^n\times 2^n$ chessboard with one square missing, then you can tile it with L-shaped trominoes.&quot;  There are other tiling questions, as well, such as the ones &lt;a href=&quot;http://www.mathcircles.org/wp-content/uploads/2017/10/tiling.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; that deal with triangular chessboards and trominoes.  Yet others ask students to count the number of ways to tile something (e.g., &lt;a href=&quot;http://www.algorithmist.com/index.php/UVa_10918&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) via linear recurrences (which usually can easily be proved inductively).&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2018-09-13T17:41:04.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14556" PostTypeId="2" ParentId="14552" CreationDate="2018-09-13T18:19:48.000" Score="9" Body="&lt;p&gt;Tiling problems might meet your constraints. A nice simple example is Golomb's Theorem that a chessboard of side $2^n$ with &lt;em&gt;any&lt;/em&gt; square omitted can be tiled by trominoes (&quot;L&quot; shapes of 3 squares).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact we can modify it to give an example of how &lt;em&gt;strengthening the induction hypothesis&lt;/em&gt; is often needed: simply replace &quot;any square omitted&quot; by &quot;central square omitted&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The induction step is easy and vivid: divide the board into four smaller $2^{n-1}$ boards. By induction we can tile the board with the missing (pink) square, and we can tile the other three omitting their (purple) corner squares (in the center of the big square), leaving 3 central squares that form an &quot;L&quot;, which we tile with one final tromino.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XDHYP.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XDHYP.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2018-09-13T18:19:48.000" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14557" PostTypeId="1" CreationDate="2018-09-13T19:38:19.370" Score="2" ViewCount="413" Body="&lt;p&gt;I'm covering section 2.5 of Stewart (on continuity) and stewarts treatment seems needlessly complicated.  It seems like the following theorem would streamline a lot of it: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If $f(x)$ and $g(x)$ are functions which are continuous at every point in their domains, then each of the following functions is continuous at every point in its domain:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$1.~~~~~~~ f(x)+g(x)$$&#xA;$$2.~~~~~~ f(x)-g(x)$$&#xA;$$3.~~~~~~ f(x)g(x)$$&#xA;$$4.~~~~~~ c\cdot f(x)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$5. ~~~~~~ f(x)/g(x)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$6.~~~~~~f(g(x)).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there something pedagogically wrong with giving this theorem? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems like it would simplify examples like the ones below because it reduces them to just finding the domain of the function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GRutW.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GRutW.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YiMq1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YiMq1.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5969" LastEditorUserId="5969" LastEditDate="2018-09-13T20:44:58.713" LastActivityDate="2018-09-13T20:44:58.713" Title="a theorem to simplify continuity in Stewart's calculus: early transcendentals" Tags="|calculus|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14558" PostTypeId="2" ParentId="14552" CreationDate="2018-09-13T20:36:49.227" Score="2" Body="&lt;p&gt;How about the &lt;strong&gt;Tower of Hanoi&lt;/strong&gt; puzzle and finding the optimal number of moves? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This link describes the recursive solution procedure and a proof of optimality using induction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://proofwiki.org/wiki/Tower_of_Hanoi&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://proofwiki.org/wiki/Tower_of_Hanoi&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2018-09-13T20:36:49.227" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14559" PostTypeId="1" AcceptedAnswerId="14560" CreationDate="2018-09-14T00:11:39.607" Score="3" ViewCount="252" Body="&lt;p&gt;I have been thinking, is it possible, or a good thing to do when learning from a textbook without ever writing anything down. But have to verbally give out the solution from beginning to the end with explaination of how the solution works and just records the solution in audio files. Is this a good way to do math? Because writing down is so slow I think this could be an alternative. Should I try this? I aware that this could be harder to find mistakes in solutions after it is done though. But when doing the recording, I think when you get stuck you will easily points out where you are not good enough.&lt;/p&gt;&#xA;" OwnerUserId="7540" LastEditorUserId="64" LastEditDate="2018-09-15T02:18:17.680" LastActivityDate="2018-09-15T02:18:17.680" Title="Can we solve math just by speaking and not using any other extra devices?" Tags="|problem-solving|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14560" PostTypeId="2" ParentId="14559" CreationDate="2018-09-14T02:13:38.710" Score="5" Body="&lt;p&gt;Mathematics has a highly optimized writing style.  It is like shorthand, except that there are widely agreed standards for how to read and write it.  This means that hand-writing numbers, formulas, and equations can be faster than saying them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Written mathematics is much easier to check than verbal mathematics.  Most mathematical transformations have symmetries and near-symmetries.  With a few exceptions, it is easier to notice mistakes when the transformations are written out, rather than when they are spoken aloud.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7243#7254&quot;&gt;exceptions&lt;/a&gt; are things like negative signs, and the difference between exponentiation and multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to convince someone of a mathematical argument, it helps if they (think they) can check it.  Since it is much easier to check a mathematical argument that is written in standard notation, most people are suspicious of anyone who is not willing write out &quot;the steps&quot; of any mathematical argument that requires more than a couple of simple arithmetic steps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is possible to perform mathematics verbally, or &quot;in one's head&quot;.  This is useful for practicing math problems, and for becoming faster at doing math.  But it has the same risk as any other form of practice -- if you practice the wrong thing, you will reinforce bad habits.  It is also useful for impressing people, if you can come up with the right answer &quot;in your head&quot; faster than they can compute it using tools.  But there are better ways to &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/B001O2HNSA&quot; rel=&quot;noreferrer&quot;&gt;win friends and influence people&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because it hard to keep track of many details &quot;in one's head&quot;, performing mathematics verbally is best done with relatively simple problems.  For example, first-order approximations and second-order approximations.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2018-09-14T02:20:23.747" LastActivityDate="2018-09-14T02:20:23.747" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14561" PostTypeId="1" AcceptedAnswerId="14562" CreationDate="2018-09-14T09:36:24.667" Score="6" ViewCount="1849" Body="&lt;p&gt;If $a = \frac{x}{b}$ and $a = \frac{c}{b}$, and I solve for $x$ I get $x = c$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$b$ has been removed because it appeared in the numerator and the denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is it called in English what happened to $b$?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$b$ ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;In German I would say &quot;$b$ kürzt sich weg.&quot;&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="10385" LastEditorUserId="1680" LastEditDate="2018-09-16T14:43:33.680" LastActivityDate="2018-09-16T14:43:33.680" Title="What is it called when terms disappear when reducing fractions?" Tags="|language-use|" AnswerCount="4" CommentCount="21" ContentLicense="CC BY-SA 4.0" />
  <row Id="14562" PostTypeId="2" ParentId="14561" CreationDate="2018-09-14T09:57:20.490" Score="10" Body="&lt;p&gt;If you continue the operations until&#xA;$$\require{cancel}\frac{x}{b}=\frac{c}{b},\qquad\left(\frac{x}{b}\right)b=\left(\frac{c}{b}\right)b,\qquad x\left(\frac{b}{b}\right)=c\left(\frac{b}{b}\right),\qquad x\left(\frac{\cancel b}{\cancel b}\right)=c\left(\frac{\cancel b}{\cancel b}\right)$$&#xA;then $x=c$, I would say that you &lt;em&gt;cancelled the common factor $b$&lt;/em&gt; in the fraction.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-09-14T09:57:20.490" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14563" PostTypeId="2" ParentId="14561" CreationDate="2018-09-14T12:29:26.570" Score="13" Body="&lt;p&gt;So German &quot;$b$ kürzt sich weg&quot; becomes in English &quot;$b$ cancels out&quot;.  We may also say &quot;$b$ is eliminated&quot;.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2018-09-14T12:29:26.570" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="14564" PostTypeId="2" ParentId="14561" CreationDate="2018-09-14T16:13:41.940" Score="8" Body="&lt;p&gt;In general, as others have noted, if you have an equation such as&#xA;$$\frac{x}{b}=\frac{c}b$$&#xA;The step to get from there to&#xA;$$x=c$$&#xA;is typically referred to as &lt;strong&gt;cancelling the denominator&lt;/strong&gt;. More generally, if you can just remove some piece of the equation, you can use the verb &quot;cancel&quot; both with that piece as the object and the subject. For instance:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;We cancel the denominators.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The denominators cancel.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can also use the phrasal verb &quot;cancel out&quot; as in &quot;The denominators cancel out.&quot;&lt;/p&gt;&#xA;" OwnerUserId="3229" LastActivityDate="2018-09-14T16:13:41.940" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14565" PostTypeId="1" CreationDate="2018-09-14T17:43:44.017" Score="4" ViewCount="102" Body="&lt;p&gt;I am teaching a student basic equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For now, I want him to drill:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Multiplication&lt;/li&gt;&#xA;&lt;li&gt;Exponentiation (by exponents 2, 3, 4 and 5)&lt;/li&gt;&#xA;&lt;li&gt;Substituting values in first degree equations&#xA;(what is &lt;span class=&quot;math-container&quot;&gt;$(x-2)(5x+3)+3x+5$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x=0,1,2,3$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Simplifying first degree equations&#xA;(take the same expression and convert it to the shortest 2 degree polynomial available)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For now this is my strategy for the student. This question is &lt;strong&gt;not about whether this a good idea or not&lt;/strong&gt;, though I intend to ask that good or bad question in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: where can I find auto-grading drills for those skills, that I can pass to my student to do? I want the computer to say right away if he got it right or wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I looked at Khan Academy, but they seem to mix too many different concepts in a basket and limit the amount of exercises. I want my student to be able to do series of 20 exercises or so, every week (until we get it right)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;A small word of justification: My student gets the concepts involved. When I explain to him that you cannot make &lt;span class=&quot;math-container&quot;&gt;$3x^2+4x=7x$&lt;/span&gt;, he gets it. The problem is a lack of familiarity with the operations, so that some times he short-circuits the operations. That is why drills seem to be in order&lt;/p&gt;&#xA;" OwnerUserId="1662" LastEditorUserId="6053" LastEditDate="2019-04-04T15:49:35.750" LastActivityDate="2019-04-04T16:31:01.367" Title="Automated drilling sites with some specific problems" Tags="|online-instruction|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14566" PostTypeId="2" ParentId="14561" CreationDate="2018-09-14T17:58:43.090" Score="3" Body="&lt;p&gt;Coming at this from a slightly more abstract point of view than the other answers, this is an application of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cancellation_property&quot; rel=&quot;nofollow noreferrer&quot;&gt;multiplicative cancellation law&lt;/a&gt; (over the rationals if $a$, $b$, and $c$ are all integers; or over the reals if $a$, $b$ and $c$ are real numbers; or over the complex numbers; or whatever...).  Specifically, in this context, the cancellation law says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $q$, $r$, and $s$ be rational numbers (or real numbers, or complex numbers) with $q\ne 0$.  Then $q\cdot r = q\cdot s$ if and only if $r=s$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that it is quite important here that $q \ne 0$.  If $q = 0$, then both $r$ and $s$ may be chosen freely, and no cancellation is possible. &#xA; Taking $q = \frac{1}{b}$ (assuming that $b\ne 0$, $q$ is well-defined and we automatically have $q\ne 0$), $r = x$, and $s = c$, the multiplicative cancellation law gives us&#xA;$$&#xA;\frac{x}{b} = \frac{c}{b}&#xA;\iff \frac{1}{b} \cdot x = \frac{1}{b} \cdot c&#xA;\iff x = c. $$&#xA;Because we are using the multiplicative cancellation law, the process is called &lt;em&gt;cancelling&lt;/em&gt; or &lt;em&gt;cancelling out&lt;/em&gt; the common factor (in this case, we are cancelling a factor of $\frac{1}{b}$).  Indeed, I think that a properly rigorous reading of this step would be &quot;We cancel out a common factor of $\frac{1}{b}$.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might also be reasonable to say that &quot;We cancel the common factors from the denominators,&quot; or more simply &quot;We cancel the denominators.&quot;  That said, because I am kind of pedantic, I would be a little hesitant to say that anything is being done to $b$.  We aren't really cancelling a factor of $b$, but rather a factor of $\frac{1}{b}$.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2018-09-14T17:58:43.090" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14567" PostTypeId="2" ParentId="14565" CreationDate="2018-09-14T21:33:55.063" Score="4" Body="&lt;p&gt;If the goal of the first bullet point is to practice multiplying single-digit integers, then I have a site for that specific skill:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.automatic-algebra.org/timestables.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.automatic-algebra.org/timestables.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That site also has basic-level drills for operations on negatives, order-of-operations, etc., but not your other bullet points. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-09-15T18:07:41.947" LastActivityDate="2018-09-15T18:07:41.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14569" PostTypeId="1" CreationDate="2018-09-15T19:55:43.043" Score="12" ViewCount="741" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;: What are good examples of functions &lt;span class=&quot;math-container&quot;&gt;$f: \Bbb R \rightarrow \Bbb R$&lt;/span&gt; (or &lt;span class=&quot;math-container&quot;&gt;$f: D \rightarrow \Bbb R$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$D \subseteq \Bbb R$&lt;/span&gt;) which are not just given by &quot;a formula&quot; (or finitely many formulae on explicitly given intervals), but make sense to a beginner / good high school student?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I know that for a wide enough definition of &quot;formula&quot;, that's exactly what a function is. But here, by a formula I mean a finite algebraic expression of real powers of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as well as (inverse) trigonometric functions, exp, logs and absolute values; more or less what one would call elementary functions, and what many students falsely believe to be &lt;em&gt;all&lt;/em&gt; functions, which is exactly what I want to counter with such examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt; Teaching a first-year Calculus course (in North America), the first thing I need to make sure is that the students know what a function is. I am aware of the effort pre-calculus teachers put into explaining functions as &quot;input-output-machines&quot;, cf. &lt;a href=&quot;https://matheducators.stackexchange.com/q/14288/8931&quot;&gt;this question&lt;/a&gt;. And when talking of functions whose domain and/or range are arbitrary sets, letters, geometric shapes, colours, animals, cars etc., as in those discussions, students know that those functions are not given by an &quot;algebraic formula&quot;. However, as soon as we start talking about &quot;real&quot; functions (meaning strictly: those whose domain and range are subsets of &lt;span class=&quot;math-container&quot;&gt;$\Bbb R$&lt;/span&gt;) I see that, subconsciously, almost every student falls back to the belief that &quot;a function is a formula&quot;, in the above narrow sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first week of such a course one puts in a little (p)review of what functions can be, typically using piecewise defined functions (including ones with some jumps or removable discontinuities). However, these surely feel very artificial to students. Also, even if they quickly take in that information, I fear they now basically believe that a function is a collection of possibly several &quot;elementary formulae&quot;, just one at a time / for each &quot;piece&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously I cannot use any of the &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_special_functions_and_eponyms&quot; rel=&quot;nofollow noreferrer&quot;&gt;many non-elementary functions which are given as integrals, power series, or solutions to differential equations&lt;/a&gt;, because these are accessible at the &lt;em&gt;end&lt;/em&gt; of a calculus lecture (at best).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;strong&gt;Edit&lt;/strong&gt;: And, cf. discussion in the comments, I should have said that these non-elementary but highly applicable functions are what I eventually want to get at here, and I think every calculus course -- even if it's not aimed at mathematicians -- &lt;em&gt;should&lt;/em&gt; care about them. The FT of Calculus (and later, the theories of differential equations and power series) gives us tons of differentiable functions which cannot be expressed by elementary formulae. I firmly believe that many functions encountered in higher engineering and physics are actually of this kind. And students have a very hard time understanding this. Teaching the course before, near the end of the course I have assigned them questions like &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are the zeroes of the function &lt;span class=&quot;math-container&quot;&gt;$F(x) = \int_2^x \frac{dt}{\log(t)}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;With the usual curve sketching routine, sketch the graph of the function &lt;span class=&quot;math-container&quot;&gt;$G(x) = \int_0^x \frac{\sin(t)}{t} dt$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which should be easy resp. doable for anyone who understands the Fundamental Theorem and the basics of calculus (for the first one, I even give them as hint: there's an obvious one, and then think about whether the function increases). But students struggled a lot with these questions, and I think one of the biggest reasons for that is that they don't &quot;get&quot; what the actual function is; many try to answer the questions for the only functions-given-as-formula they see in there, &lt;span class=&quot;math-container&quot;&gt;$1/\log(x)$&lt;/span&gt; resp. &lt;span class=&quot;math-container&quot;&gt;$\sin(x)/x$&lt;/span&gt;. &lt;strong&gt;End Edit&lt;/strong&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best examples I have come up with, which have sort of worked in the classroom, are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the prime-counting function &lt;span class=&quot;math-container&quot;&gt;$\pi(x)=$&lt;/span&gt; number of primes &lt;span class=&quot;math-container&quot;&gt;$\le x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;the Dirichlet function &lt;span class=&quot;math-container&quot;&gt;$Dir(x) = \mathbf 1_\Bbb Q = \begin{cases} 1 \text{ if } x \in \Bbb Q \\ 0 \text{ if } x \notin \Bbb Q. \end{cases}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have also played around with ideas like&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x) =$&lt;/span&gt; the first decimal place in the standard decimal expansion of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; where a 7 appears.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What would you suggest?&lt;/p&gt;&#xA;" OwnerUserId="8931" LastEditorUserId="8931" LastEditDate="2018-09-22T05:09:31.350" LastActivityDate="2018-09-22T05:09:31.350" Title="Examples (for beginners) of real functions which are not given by elementary formulae" Tags="|calculus|examples|functions|precalculus|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14570" PostTypeId="2" ParentId="14569" CreationDate="2018-09-16T03:55:02.620" Score="1" Body="&lt;p&gt;I think it is enough to discuss $f(x) = x^2$ and toy around with what happens when different domains are given for $f$. Or, better yet, to discuss the idea of restriction and extension. It is important for them to know the default custom that we take the domain to be as large as the formula allows. For $f(x)=x^2$ that is naturally the whole set of reals $\mathbb{R}$. So, what if $x$ represents something that cannot be negative ? Then we state $\text{dom}(f) = [0, \infty)$ for physical reasons. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good physical example would be projectile motion where we hit a ball at height $y_o$ above the ground at some angle $\theta$ with speed $v_o$ then&#xA;$$ y = y_o+tv_o\sin \theta-\frac{g}{2}t^2 $$&#xA;what is the domain for $y$ as a function of time ? If we just go by the formula above then the answer would appear to be $\mathbb{R}$. However, the formula fails to be applicable in the time before the ball is thrown or hit. In fact, for time a bit smaller than zero, probably $y = y_o$ is constant. Where the ball was before that is a truly complicated question which no reasonable person would try to find a formula to describe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point is simply that formulas for functions are often sensible mathematically far outside their common sense application. Just because the math says something, that doesn't make it so. However, usually the math means something at least hypothetical. In my projectile example, if we solve $y=0$ that gives a quadratic in $t$ with a positive and negative time solution. The meaning of the positive time solution is simply the time it hits the $y=0$ ground. On the other hand, the negative time solution corresponds to the time when a person would have thrown a ball from ground level so that it appears as if you threw it with the given speed and angle at $y_o$. Is that interesting? Perhaps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure I've helped with your quest. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a purely mathematical one. Use the ceiling or floor function followed by keeping the remainder upon division by $n$ of your choosing. Guess you can't write a formula for that without a lot of trouble, but we should understand the rule.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-09-16T03:55:02.620" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14571" PostTypeId="2" ParentId="14569" CreationDate="2018-09-16T08:27:35.597" Score="4" Body="&lt;p&gt;Consider also very simple piecewise functions, such as the sign function and Heaviside function. Though they are trivially given by formulae, the formulae are so simple (being only constants) that the students are unlikely to think of them as formulae. They can be composed with other functions; of the composite functions of sign and sine, for example, one is more interesting than the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea here is to try to crate intuition from the direction of very easy examples, which might not fit the intuition the students have of functions. The sign function, for example, has a very nice interpretation and can still be written by using the typical notation for piecewise defined functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, both the sign function and Heaviside function have applications and might be encountered later by the students. In particular the (surprisingly tricky) absolute value function can be written with the help of a sign function, if one wishes to.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2018-09-16T08:27:35.597" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14572" PostTypeId="1" AcceptedAnswerId="14575" CreationDate="2018-09-16T14:30:45.607" Score="4" ViewCount="886" Body="&lt;p&gt;This happens a lot when I try to explain the commutative property, mostly in elementary grade levels. I say&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;2 + 3 = ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and then the student usually replies with 5. Albeit they're not wrong, it's not the idea of commutative property. I feel that students so conditioned early on to see that 2 + 3 is necessarily 5 as opposed to understanding that 2 + 3 can just be that: 2 + 3. I had this problem myself growing up.&lt;/p&gt;&#xA;&lt;p&gt;How can I address this issue?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;2 + 3 = 3 + 2&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;more examples,&lt;/p&gt;&#xA;&lt;p&gt;I was explaining cross-cancellation and this is the visualization&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{2}{7} \cdot \frac{7}{3} = \frac{2 \cdot7}{7 \cdot 3} = \frac{2 \cdot \require{cancel} \cancel{7}}{\require{cancel} \cancel{7} \cdot3}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;instead, they visualize the following&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{2}{7} \cdot \frac{7}{3} = \frac{14}{21}$&lt;/span&gt; {we are stuck here since we cannot see the cancellation working out}&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;so what I mean is that students cannot comprehend that it isn't necessary to always work out the arithmetic right away but can keep them as expressions to observe patterns. I hope this is a bit more clear.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Addition of a story:&#xA;A famous example, the story of Gauss. Once upon a time, there was a teacher who was bored and asked the class, included Gauss, who asked students to sum up from 1 to 100. While the rest of the class struggled to solve it, Gauss got it with ease. This is the idea I was trying to convey.&lt;/p&gt;&#xA;&lt;p&gt;Instead of sitting down and computing hard, compute smarter kind of idea.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \sum_{i=0}^n i = \dfrac{n\times(n+1)}2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \sum_{i=0}^{100} i = \dfrac{100 \cdot 101}2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-09-19T17:23:56.890" Title="How to correct visualization of mathematical expressions?" Tags="|primary-education|arithmetic-operations|" AnswerCount="4" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="14573" PostTypeId="2" ParentId="14569" CreationDate="2018-09-16T14:36:42.367" Score="12" Body="&lt;p&gt;How about $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = y$ where $y$ is the unique solution to $y^5+x^2y+5=0$?  This does not have an elementary formula, but students can understand that for any value of $x$ the function $g(y) = y^5+x^2y+5$ is increasing (and the limits at $\pm \infty$ are $\pm \infty$), so it must have a unique root for each $x$.  You could contrast this with $y^5+xy+5$, which does not always have unique solutions, and so does not define a function.  Desmos can graph these easily as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be better to use this example as part of your introduction to implicit differentiation though.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-09-16T14:36:42.367" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14574" PostTypeId="2" ParentId="14572" CreationDate="2018-09-16T15:18:31.693" Score="11" Body="&lt;p&gt;One way to see if the student understands the commutative property of addition is to have &quot;fill-in-the-blank&quot; questions such as&#xA;$$2+3=3+\_\_$$&#xA;$$2+3=\_\_+2$$&#xA;$$2+\_\_=3+2$$&#xA;$$\_\_+3=3+2$$&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2018-09-16T22:52:07.650" LastActivityDate="2018-09-16T22:52:07.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14575" PostTypeId="2" ParentId="14572" CreationDate="2018-09-16T16:18:41.440" Score="9" Body="&lt;p&gt;The problem is due to &lt;em&gt;imprecise&lt;/em&gt; specification of the intended result. Here's a more precise way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\text{Recall that the }{\bf  commutative\ law}\ \color{#c00}X+ \color{#0a0}Y = Y + X\ \text{ is true for all reals } X,Y$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\text{Use the above law to $ $  simplify }\ 2\, +\, \color{#c00}{\pi}\, +\, \color{#0a0}3\ \text{ to the form }\, n + \pi\,\text{ for some integer }n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt; $ $ In case it wasn't obvious, the idea is to choose a sum where it is clear that performing the commutation simplifies the addition. If you can't use $\,\color{#c00}{\pi}\,$ (or $x)$ then it is clear how to tweak it to use &quot;simpler&quot; numbers, e.g. $\ 9 + 1/123 - 9\ $ or $\ 1/11+123+10/11,\,$ etc. But these forms have the disadvantage that they don't &lt;em&gt;prohibit&lt;/em&gt; the student from diving head-first into computation, i.e they might try computing $9 + 1/123$ before commuting - something they can't do with $\,2+\pi\,$ or $\,2+x$. Hence using a  transcendental forcefully guides the student along the &lt;em&gt;correct&lt;/em&gt; solution path.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt; $ $ You added a new problem exhibiting the utility of &lt;em&gt;lazy&lt;/em&gt; (vs. eager) evaluation (of arithmetic) in order to help apply simplifying transformations. This idea should be &lt;em&gt;explicitly&lt;/em&gt; taught using multiple types of examples to better lend intuition, e.g. besides your example of delaying multiplication in order to exploit cancellations, it would help to give other examples where one can exploit innate arithmetical structure before diving head-first into brute-force arithmetical computation. Below are some further complementary examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Special Polynomial structure&lt;/strong&gt; $ $  If we notice that the arithmetical expression has the form of a well-known polynomial formula then applying that first may simplify the arithmetic. For a simple example consider &lt;em&gt;differences of squares&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$123^2 - 122^2 = (123-122)(123+122) = 245$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is simpler than brute-force arithmetic computation, i.e. squaring both $123$ and $122$ then taking their difference. It will be &lt;em&gt;much&lt;/em&gt; simpler for analogous examples with much larger numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Reflection symmetry&lt;/strong&gt; $\ $ Sums symmmetric about a midpoint can be rearranged as follows&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align}&#xA;1 + 2 + 3\\&#xA;+\ 6 + 5 + 4\\&#xA;\hline&#xA;= 7 + 7 + 7&#xA;\end{align}$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;which yields an easy proof that the sum is divisible by $7$. If we view the above sum $\!\bmod 7\,$ then it is an &lt;em&gt;additive&lt;/em&gt; form of the cancellations in your fractions since $\ 6\equiv -1,\  5\equiv -2,\ 4\equiv -3$ &#xA;so the sum is $\equiv 1 + 2 + 3 -1 -2 - 3$. So here the reflection is &lt;em&gt;negation&lt;/em&gt; and the &lt;em&gt;key idea&lt;/em&gt; is to preprocess the sum by pairing each summand with its negation in order to simplify the arithmetic (this is the &lt;a href=&quot;https://math.stackexchange.com/a/44964/242&quot;&gt;key idea behind one proof Wilson's Theorem&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally, before diving head-first into brute-force solution methods, it is wise to first perform some &quot;meta level&quot; preprocessing - searching for interesting innate structure that may help simplify it or shed further intuition on the heart of the matter. With that idea in mind I am sure you can come up with many interesting examples appropriate to the level of your class.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2018-09-19T17:23:56.890" LastActivityDate="2018-09-19T17:23:56.890" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14576" PostTypeId="2" ParentId="14572" CreationDate="2018-09-16T23:56:25.740" Score="4" Body="&lt;p&gt;One way to address this &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;students cannot comprehend that it isn't necessary to always work out the &#xA;  arithmetic right away but can keep them as expressions to observe&#xA;  patterns.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is to do (or have the students do) a problem in as many different ways as possible, then take the time to compare the solutions (to see that they agree) and the methods. Some will be faster, some more conceptual.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often considering a few problems in depth is more useful than a list of many where you just check for correctness.&lt;/p&gt;&#xA;" OwnerUserId="4984" LastActivityDate="2018-09-16T23:56:25.740" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14577" PostTypeId="2" ParentId="14572" CreationDate="2018-09-17T12:44:29.450" Score="0" Body="&lt;p&gt;You have not stated the grade level here, but one way to approach this is by giving problems where commuting actually makes the computation easier.  For example $97+26+13$.  You could have students do &quot;number talks&quot; on things like this.  This also gives them an opportunity to showcase their understanding of place value, which is another major goal at this age.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some resources on &quot;number talks&quot; I suggest watching the following video by Sherry Parrish.  She also has some books on number talks which might be helpful to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=twGipANcIqg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=twGipANcIqg&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2018-09-19T16:39:45.710" LastActivityDate="2018-09-19T16:39:45.710" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14578" PostTypeId="2" ParentId="7392" CreationDate="2018-09-17T14:12:31.107" Score="1" Body="&lt;p&gt;A complete cut of a strip (= tape jointed to itself) is a cut which forms a loop in the strip interacting one, two, three… times with each ‘parallel’ of the strip. Namely a cut with one, two, three,... turns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A complete cut of a strip individuates a geometrical object adding, at each turn, the number of edges of the initial strip. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, for instance &lt;/p&gt;&#xA;&#xA;&lt;p&gt;•   a complete cut with a turn of a strip with two edges generates an object with 2+2 edges, namely two strips with two edges,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;•   a complete cut with two turns of a strip with one edge (such as the Möbius strip) generates an object with 1+1+1 edges, namely a strip with two edges and a strip with one edge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since a ‘one third’ cut of a Möbius strip, to be complete, must turn twice around the strip, dividing the strip in two parts, this cut necessarily individuates a strip with two edges and a Möbius strip.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the two external parts individuated by the cut must be connected and must form a strip with two edges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, this cut separates the central part of the original Möbius strip from its external part, which is given by the conjunction of two strips as long as the central part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, since the edges of these strips are interlaced, the two resulting  strips are linked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;NOTE. It is interesting that when the internal part disappears, the cut becomes a cut with a turn which individuates only the external part (with two edges).&lt;/p&gt;&#xA;" OwnerUserId="10403" LastEditorUserId="10403" LastEditDate="2018-09-18T01:23:00.810" LastActivityDate="2018-09-18T01:23:00.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14580" PostTypeId="2" ParentId="7837" CreationDate="2018-09-17T17:53:14.783" Score="3" Body="&lt;p&gt;Below are a various methods of presenting fraction inversion based on various innate &lt;strong&gt;symmetries.&lt;/strong&gt; Unlike some other methods, these ideas are more to the heart of the matter, so they generalize more nicely. The exposition is at the level of the teacher - who can then decide how to best scale the ideas to the appropriate level - be it primary, secondary or higher level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1. Inversion as a special case of fraction scaling symmetry&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can invert fractions using the fact that they are &lt;em&gt;preserved under equal scalings&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$s\neq 0$&lt;/span&gt; of the numerator and denominator, i.e &lt;span class=&quot;math-container&quot;&gt;$\,\dfrac{a}b = \dfrac{as}{bs}.\,$&lt;/span&gt; So &lt;span class=&quot;math-container&quot;&gt;$\,\dfrac{1}{2/3} = \dfrac{3}2\,$&lt;/span&gt; follows by scaling by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; to (remove the denominator &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; in the denominator). Viewed geometrically, in terms of similar triangles we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\qquad\qquad\qquad\qquad\quad$&lt;/span&gt;&lt;a href=&quot;https://i.stack.imgur.com/rnJx9.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rnJx9.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;by scaling the smaller triangle by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; to cause its magnified base &lt;span class=&quot;math-container&quot;&gt;$\color{#c00}{2/3}$&lt;/span&gt; to have &lt;em&gt;integer&lt;/em&gt; length &lt;span class=&quot;math-container&quot;&gt;$2.$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the same scaling method works for your example &lt;span class=&quot;math-container&quot;&gt;$\,\dfrac{4}{2/7}\,$&lt;/span&gt; by scaling by &lt;span class=&quot;math-container&quot;&gt;$s = 7$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So once students know the fundamental scaling symmetry of fractions, the &quot;inversion by flipping&quot; reflection follows as a special case. If the class already knows some algebra then it is well worth emphasizing that the scaling symmetry of fractions is just a reformulation in &quot;root language&quot; of the scaling symmetry of the associated linear equation of which they are roots, i.e.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x= \dfrac{a}b\iff b\,x = a\iff bs\, x = as\iff x = \dfrac{as}{bs},\quad {\rm f&#xA;or}\ \ s\neq 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a special (linear) case of the obvious fact that scaling a polynomial by a scalar &lt;span class=&quot;math-container&quot;&gt;$s\neq 0$&lt;/span&gt; does not alter its roots &lt;span class=&quot;math-container&quot;&gt;$\,f(r)=0\iff s f(r) =0$&lt;/span&gt; . For another concrete example let's consider the &lt;em&gt;quadratic&lt;/em&gt; case. Here the &quot;root language&quot; analog of the above linear case &lt;span class=&quot;math-container&quot;&gt;$\,a/b = as/(bs)\,$&lt;/span&gt; is that  replacing &lt;span class=&quot;math-container&quot;&gt;$\,a,b,c\,$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\,as,bs,cs\,$&lt;/span&gt; in the quadratic formula does not alter the set of roots. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2. Inverse of algebraic number by reverse-reflecting a polynomial of which it is a root&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given a linear polynomial &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; having a nonzero fraction as a root, we can scale &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$x^{-1}$&lt;/span&gt; to get a linear polynomial having &lt;span class=&quot;math-container&quot;&gt;$x^{-1}$&lt;/span&gt; as a root, namely &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x= \dfrac{a}b\iff b\,x = a\iff  ax^{-1} = b\iff x^{-1} = \dfrac{b}a$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This generalizes to higher degrees polynomials too:  &lt;span class=&quot;math-container&quot;&gt;$x\neq 0$&lt;/span&gt; is a root of a polynomial &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;  &lt;span class=&quot;math-container&quot;&gt;$\iff x^{-1}$&lt;/span&gt; is a root of &lt;span class=&quot;math-container&quot;&gt;$\,x^{-n} f(x)\, =: \bar f(x^{-1}),\,$&lt;/span&gt; e.g. for quadratics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ ax^2 + bx+c = 0\iff  c x^{-2} + b x^{-1} + a = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that the map &lt;span class=&quot;math-container&quot;&gt;$\,f\mapsto \bar f$&lt;/span&gt; simply &lt;strong&gt;reverses&lt;/strong&gt; the coefficient sequence&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ a,b,c\ \mapsto\ c,b,a\quad$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus the flipping in fraction inversion is just a special case of the above general reversal reflection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\bar f(x)$&lt;/span&gt; is called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Reciprocal_polynomial&quot; rel=&quot;nofollow noreferrer&quot;&gt;reciprocal polynomial&lt;/a&gt; (or reflected or reversed). It arises &lt;a href=&quot;https://math.stackexchange.com/search?q=reciprocal+user%3A242&quot;&gt;arises frequently in algebra&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3. Symmetries of equal fractions via symmetries of a square&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Inversion can also be be seen as a special case of the symmetries of equal fractions, which can be viewed as symmetries of the square &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{A}B = \dfrac{a}b\iff \dfrac{B}A = \dfrac{b}a\iff \dfrac{B}b = \dfrac{A}a$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which hold true because these reflections and rotations preserve the diagonal cross-products, hence preserve fraction equality (&lt;a href=&quot;https://matheducators.stackexchange.com/questions/14537/is-this-trick-helpful#comment36613_14538&quot;&gt;see here&lt;/a&gt; for further discussion). The special case &lt;span class=&quot;math-container&quot;&gt;$B=1$&lt;/span&gt; above yields inversion of &lt;span class=&quot;math-container&quot;&gt;$\,A = a/b\,$&lt;/span&gt; by flipping (reflection around the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis).&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2018-09-21T14:55:34.540" LastActivityDate="2018-09-21T14:55:34.540" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14581" PostTypeId="2" ParentId="7392" CreationDate="2018-09-18T16:03:38.763" Score="6" Body="&lt;p&gt;Alternatively, a pictorial explanation can be used (perhaps more effectively)&lt;a href=&quot;https://i.stack.imgur.com/o6dzk.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/o6dzk.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10403" LastActivityDate="2018-09-18T16:03:38.763" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14582" PostTypeId="2" ParentId="12185" CreationDate="2018-09-18T19:20:28.230" Score="7" Body="&lt;p&gt;I was never a professional mathematician, and I am no longer a professional educator.  I hope this answer isn't too far off the mark.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the problem is the expression &quot;solve the equation.&quot;  I simply would never use that expression in a basic algebra class.  I didn't come across it myself until using differential equations in college.  (Before that I would have physics professors as us to &quot;solve the equation&quot; in class, and that expression left the class scratching their heads. Solve it &lt;em&gt;in what way?&lt;/em&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At least from my experience, such a request was always to &quot;solve the equation &lt;strong&gt;for &lt;em&gt;x&lt;/em&gt;&lt;/strong&gt;.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said that, if your colleagues or you wish to use the expression &quot;solve the equation,&quot; then the trick is to clearly and unambiguously define what that means. That will almost certainly require a compare and contrast to the more explicit &quot;solve for x&quot; request.  &lt;strong&gt;As long as you define your terms&lt;/strong&gt;, and don't expect the students to just intuit the distinction, then the rest is a matter of how pedantic you want to be about &quot;standard form.&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For students that are still learning basic algebra of linear equations, I have a feeling that the distinction between &quot;a solution set with one member&quot; and &quot;an equivalent equation&quot; may be stretching their ability to take on too much abstraction at once.  A better time might be after they have mastered quadratic equations.  Then the distinction between a solution set, and a particular solution, might make more sense to them.&lt;/p&gt;&#xA;" OwnerUserId="5465" LastEditorUserId="5465" LastEditDate="2018-09-19T21:57:08.420" LastActivityDate="2018-09-19T21:57:08.420" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14583" PostTypeId="2" ParentId="13032" CreationDate="2018-09-18T22:16:20.590" Score="4" Body="&lt;p&gt;Often students wrongly believe that integer factorization is unique because there is a &lt;em&gt;deterministic&lt;/em&gt; algorithm for factorization: simply continue to pull out the least prime factor (found by trial division). To help them debunk this belief it suffices to show them another ring where a similar method fails. An elementary  example is factoring (monic) quadratics with coefficients being integers $\!\bmod m.\,$ Here - as above - there is a simple deterministic factorization algorithm: simply test in order $r\equiv 0,1,\cdots,m-1$ till we find a root $\,r\,$ hence a factor  $\,x-r\,$ (by the Factor Test).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But here factorization need not be unique, e.g. $ (x-1)(x+1) \equiv (x-3)(x+3)\,\pmod{\! 8}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applying the above algorithm would find only the first factorization. This shows clearly and simply that the existence of deterministic algorithm for solution does not imply that the solution is unique.&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2018-09-18T22:16:20.590" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14584" PostTypeId="1" CreationDate="2018-09-19T01:05:17.923" Score="3" ViewCount="152" Body="&lt;p&gt;A course that my kids are doing in conic sections insists that all positions are represented as standard 2D cartesian coordinates $(x,y)$ (i.e. row vectors), and all translations are written as 2D column vectors $\begin{pmatrix}x\\y\end{pmatrix}$. Any deviation from this is marked incorrect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iCVLV.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iCVLV.png&quot; alt=&quot;textbook image showing positions as row vectors and translations as column vectors&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe the reasoning is that positions are not vectors and so are written in row form, while translations are vectors and therefore must be written in column form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I can recall, I have seen positions and translations in either orientations according to whatever works in context, but usually both written as row vectors or both written as column vectors. If position and translation vectors are written with different orientations, translations are unable to be obtained from subtracting position vectors, and unable to be applied to positions by addition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any basis in theory or tradition for a strict separation of positions as row vectors and translations as column vectors?&lt;/p&gt;&#xA;" OwnerUserId="3493" LastActivityDate="2018-09-19T01:05:17.923" Title="When to use row or column vectors in points and translations" Tags="|secondary-education|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14585" PostTypeId="1" CreationDate="2018-09-19T05:30:55.860" Score="4" ViewCount="1226" Body="&lt;p&gt;I am teaching a very small intro to proofs class with Dana Ernst &lt;a href=&quot;http://danaernst.com/IBL-IntroToProof/&quot; rel=&quot;nofollow noreferrer&quot;&gt;IBL book&lt;/a&gt;, and came to theorem 2.56.  The section is about proof by contradiction but I felt that the solution I came up with is unsatisfying or not at all what is expected.  The students have not gotten here yet, I am preparing for when they do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quick Note - IBL is inquiry based learning and the students are meant to come up with the proof themselves.  In this case, to learn how to use proof by contradiction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The instructions are: Prove the following theorem via contradiction. Afterward, consider the difficulties one might encounter when trying to prove the result more directly. (Also, here $\mathbb{N} = \{1,2,3,\ldots\}$).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Assume that $x,y \in  \mathbb{N}$. If $x$ divides $y$, then $x \leq y$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My proof goes something like this.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Assume to the contrary that $x&amp;gt;y$.  Since $x$ divides $y$, there is some integer, in this case positive, $k$, such that $xk = y$.  Note that $xk&amp;gt;x$, and hence $y&amp;gt;x$, a contradiction.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This feels unsatisfying because if we can say that $xk&amp;gt;x$,  then this is hardly worth proving. AND it is unnecessary to prove using proof by contradiction, which makes me wonder about the instructions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What am I missing here?  I assume that there must be some other proof which makes more sense.  Any ideas are appreciated.&lt;/p&gt;&#xA;" OwnerUserId="514" LastActivityDate="2019-10-08T18:26:38.957" Title="Intro to Proof: if $x$ divides $y$, then $ x \leq y$" Tags="|inquiry-based-learning|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="14586" PostTypeId="1" CreationDate="2018-09-19T13:46:51.540" Score="8" ViewCount="613" Body="&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have a nice list of &quot;no effort&quot; activities that parents can employ to promote numeracy? I am primarily interested in K-8 activities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Exposition&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often parents ask me about what 'math activities' they can be doing at home. I could supply many in-depth math lessons that will never be taught but I would much rather give them a short list of things that they can incorporate in to activities that they are already doing. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Every time you go out to eat at a restaurant: Your child should be calculating tip. 10%, 20% are easy computations 18.5% requires some more thinking...  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When you are driving in the car: Your child could be looking for equations in license plates. Something like &lt;a href=&quot;http://www.sheppardsoftware.com/conlic0.htm&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;. Or factoring! License plates are a good way to cultivate our factoring skills. Please keep your eyes on the road when checking your child's computations... &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When you go to the grocery store you can ask your child to estimate the number items and the cost. This is somehow a skill that every adult has managed to develop but children rarely know the cost. After showing the child the receipt a few times and asking him/her to figure out how much the family spends on ice cream... We should see an improvement in their ability to crunch numbers and estimate. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When you play monopoly (or any game that requires some math), your child should be doing at least some of the banking (or whatever the part of the game requires the math...).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am looking for more activities that shouldn't really feel like a 'math activity' and should feel more like daily life: your kid is talking way too much on a long road trip? Why aren't they factoring large numbers in their head? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The activities I am looking for &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Should promote numeracy&lt;/li&gt;&#xA;&lt;li&gt;Should require no manipulatives or planning (This is what I mean by &quot;no effort&quot;)&lt;/li&gt;&#xA;&lt;li&gt;Should be an activity that a middle class family may do once a month: Go out to eat, Drive in a car, collect groceries, etc. (This is also what I mean by &quot;no effort&quot;)&lt;/li&gt;&#xA;&lt;li&gt;Have a type of built in differentiation: we should be able to format the activity for a 3rd grader (What is 10% of the cost of the meal?) or an 8th grader (How much can we spend on desert and still have the meal be under $100 after applying an 18% tip?). &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9841" LastEditorUserId="5321" LastEditDate="2018-09-24T17:58:14.937" LastActivityDate="2018-09-25T22:50:57.703" Title="Math Everywhere Activities" Tags="|primary-education|mathematics-in-daily-life|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14587" PostTypeId="2" ParentId="14586" CreationDate="2018-09-19T14:08:06.707" Score="9" Body="&lt;p&gt;I would recommend activities that can be done at least once a day (instead of once a month).&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When setting the dinner table (especially when guests are present), request the child to help get the dishes and cutlery, asking &quot;How many dishes will we need?  How many forks will we need?&quot;  If each guest is to be given, say, three pieces of dessert, then how many pieces would need to be brought out?&lt;/li&gt;&#xA;&lt;li&gt;When performing a time-dependent activity (for example, when food in a pot on a stove will be ready to eat in, say, 15 minutes), ask the child to read the current time on a clock, then ask the child to predict the time the activity will finish.  (Digital clocks for the younger children; analog clocks for the older ones.)&lt;/li&gt;&#xA;&lt;li&gt;When an event is to happen in a few days, ask the child to read the current date on a calendar, then ask questions such as &quot;How many days is that event from today?  If that event was postponed for a week, what date would the new schedule be?&quot;&lt;/li&gt;&#xA;&lt;li&gt;If a child is reading a book, then ask what page he or she is currently reading, then how many pages there are in total in the book, then how many more pages are to be read to finish the book.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-09-19T14:08:06.707" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14588" PostTypeId="1" CreationDate="2018-09-19T14:14:58.980" Score="4" ViewCount="269" Body="&lt;p&gt;I have found very little online that compares &amp;amp; critiques the MVP vs traditional&#xA;curricula. Any suggestions &amp;amp; pointers would be welcomed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://www.mathematicsvisionproject.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MVP&lt;/a&gt; is an implementation of Common Core Standards in the USA aimed at&#xA;the K-12 grade levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have one child who followed fairly standard curriculum which was challenging from a mathematics perspective and provided a solid basis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have another child, two years younger who has been following the MVP. Frankly,&#xA;I have nothing good to say about it. I am surprised to find so little about it online. I fear that my second child's interest in mathematics has been irreparably damaged and the skills taught insufficient in comparison to their&#xA;older sibling.&lt;/p&gt;&#xA;" OwnerUserId="1836" LastEditorUserId="1836" LastEditDate="2018-09-19T15:04:10.497" LastActivityDate="2018-09-19T15:04:10.497" Title="Are there any studies evaluating the impact of the Mathematics Vision Project?" Tags="|curriculum|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14589" PostTypeId="2" ParentId="14586" CreationDate="2018-09-19T17:58:40.880" Score="4" Body="&lt;p&gt;Grade appropriate elementary number theory:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Counting even and odd license plates to see if parity is equally&#xA;distributed.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Same for the residue mod 3. using the sum of digits algorithm.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Factoring numbers you stumble on. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Play buzz: count &quot;i 2 3 4 5 6 buzz 8 ...&quot; with &quot;buzz&quot; for multiples of 7 and numbers containing 7. Then do it with numbers other than 7, or with two at a time with a different word for each. I recall &quot;skinimerink&quot; for 3, so 21 was buzz-skinimerink.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Invent more for yourself.&lt;/p&gt;&#xA;" OwnerUserId="4984" LastEditorUserId="4984" LastEditDate="2018-09-19T22:41:13.687" LastActivityDate="2018-09-19T22:41:13.687" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14590" PostTypeId="2" ParentId="12516" CreationDate="2018-09-19T18:38:54.583" Score="3" Body="&lt;p&gt;Joel David Hamkins recently gave a sufficiently excellent argument for a &lt;a href=&quot;https://matheducators.stackexchange.com/a/14412/5563&quot;&gt;5-point grading system&lt;/a&gt; that I'd like to quote it here for the record. (By implication I think this would argue for something like a 4/5 or 80% credit on the short algebra question above.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I use the following method of scoring student work in mathematics,&#xA;  where all problem solutions are graded on a five-point scale. It has&#xA;  worked well for me for about twenty years, and I would recommend it to&#xA;  anyone.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;5 = solution is &lt;strong&gt;basically perfect&lt;/strong&gt;, completely acceptable&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;4 = solution is marred by a few &lt;strong&gt;minor errors&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;3 = the solution has displayed basically the &lt;strong&gt;right idea&lt;/strong&gt; or approach to&#xA;  the problem&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;2 = the solution has displayed some understanding of &lt;strong&gt;relevant ideas&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1 = the solution has displayed some understanding of &lt;strong&gt;some ideas&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;0 = the solution has &lt;strong&gt;no merit&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I find it easy to use this rubric when evaluating student work. I just&#xA;  keep the phrases in mind, and assign the corresponding score. Thus,&#xA;  the work is evaluated as a whole, on the basis of whether the student&#xA;  has demonstrated mastery over the topic.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;My quizzes usually have just one problem, graded out of five. The&#xA;  final exam will have ten or fiften problems, each graded out of five.&#xA;  I often give extra credit for participation on math.stackexchange or&#xA;  whatever, and these are also worth up to five points.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-09-19T18:38:54.583" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14592" PostTypeId="1" CreationDate="2018-09-20T21:58:52.690" Score="3" ViewCount="127" Body="&lt;p&gt;I would like suggestions for a good text on concentration inequalities (examples here &lt;a href=&quot;https://en.wikipedia.org/wiki/Concentration_inequality&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Concentration_inequality&lt;/a&gt;). I am looking for sources (texts) that can give strong intuitions in applying these bounds for practical problems. Sources (not limited to textbooks) which have practice problems applied to CS theory or applied math style is ideal for my purpose. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer: I am not &quot;yet&quot; an educator, but a researcher who wants to get a good introduction to it before I apply them to my research.&lt;/p&gt;&#xA;" OwnerUserId="10411" LastEditorUserId="10411" LastEditDate="2018-09-21T00:32:44.390" LastActivityDate="2018-09-21T00:32:44.390" Title="On concentration inequalities" Tags="|textbooks|graduate-education|" AnswerCount="0" CommentCount="5" ClosedDate="2018-09-23T02:29:04.900" ContentLicense="CC BY-SA 4.0" />
  <row Id="14593" PostTypeId="1" CreationDate="2018-09-22T04:33:26.053" Score="4" ViewCount="449" Body="&lt;p&gt;Just now I'm thinking a crazy reversed idea: Is it good to teach math to elementary students without starting from its real world motivation or making it intuitive first, but rather by realizing that it is a game with some rules which revealed later to have real uses?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am thinking that if we start with the intuitive ideas of some concepts, then we are required to make almost every concept we introduce intuitive..which is not quite obvious for a lot of students. Thus, sometimes we have to skip introducing some concepts intuitively, but we still introduce only some easily intuitive concepts.&#xA;Doesn't this affect students' motivation, making them ask: &quot;I can't grasp this. What's the real world example?&quot; when there is not an easy connection..&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mean, since math is abstract, although many ideas in it are intuitive, (many are not)..if we start by introducing a concept as a game with rules and students have to master this game, how would this impact the learning process?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: When teacher teach about fraction, they usually start with dividing cake or pizza or something, and then put it in the mathematical notation of fraction that we know well. I used to (and probably still) think this is a reasonably good way to teach math. However, as implied in my question, can the teacher not start with this real world motivation, but instead start immediately by introducing the notation for fraction, and give some of the rules of summing, multiplying, etc, and the challenge for the students is to apply this rule..which sounds pretty much like playing a game of fraction?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, I don't mean that we use games when teaching math, but I mean math itself is made as collection of games. Then, at some point, the teacher relates that fraction to real world uses, if any,..such as dividing pizza, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't assume this idea is new. In fact, I ask whether some have considered this method or not. I just think that this way, students won't rely too much on real world uses of math before studying it, hence not being overly dependent on that..since some math concepts such as negative times negative, or even the whole integers, or square root, or even some geometrical objects, are quite abstract in the minds of elementary students, right?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am asking this only relating to elementary math, not higher math, since I know the higher we go, we have motivation before abstraction..and at some point math students will learn that..but can we start by giving &quot;the rules of the game&quot; and &quot;play&quot; for elementary students..&lt;/p&gt;&#xA;" OwnerUserId="10108" LastEditorUserId="10108" LastEditDate="2018-09-23T03:55:28.693" LastActivityDate="2018-10-24T02:44:52.277" Title="Teaching math by serving it as games with rules first, not intuition?" Tags="|teaching|teachers|games|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14596" PostTypeId="1" CreationDate="2018-09-23T21:35:35.807" Score="5" ViewCount="222" Body="&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Would it be feasible to teach undergraduate math students a &quot;map&quot;-centric view early on? If so, how early on?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that I'm preparing for a phd program, I'm also reflecting on my undergraduate education and wishing that I had adopted a so-called &quot;map&quot;-centric point of view earlier on. Here are toy-examples:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A point &lt;span class=&quot;math-container&quot;&gt;$x \in X$&lt;/span&gt; is a map &lt;span class=&quot;math-container&quot;&gt;$f:\{\bullet\} \hookrightarrow X$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or maybe the product as the cartesian product &lt;span class=&quot;math-container&quot;&gt;$A \times B$&lt;/span&gt; equipped with canonical projections vs. &lt;span class=&quot;math-container&quot;&gt;$A \sqcup B$&lt;/span&gt; (and injections.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Either way, these are things of a categorical &quot;flavor&quot; that probably wouldn't have really profoundly affected the math at an early stage, but &lt;em&gt;did&lt;/em&gt; clarified some things for me later on (paths in &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; are continuous maps &lt;span class=&quot;math-container&quot;&gt;$[0,1] \to X$&lt;/span&gt;, resp. the difference between direct product/sum in algebra.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question I guess is not really &lt;strong&gt;should this be done?&lt;/strong&gt; But rather &lt;strong&gt;can this be done successfully?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An answer to my question would be either someone who has personally taken up this point of view (un)successfully or arguments for why it can(not) be done.&lt;/p&gt;&#xA;" OwnerUserId="6276" LastEditorUserId="6276" LastEditDate="2018-09-24T01:40:21.863" LastActivityDate="2018-09-24T12:23:50.397" Title="Is it feasible to expose undergraduates to a &quot;map&quot;-centric point of view early on?" Tags="|undergraduate-education|mathematical-pedagogy|course-design|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14597" PostTypeId="2" ParentId="562" CreationDate="2018-09-23T23:31:26.253" Score="0" Body="&lt;p&gt;I think some reasonable fluency is needed.  But I would mostly stick with dy/dx and y'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also avoid using the most obscure (least used) notation when you are in the middle of introducing a new derivation or something complicated (e.g. integration by parts).  Try to stay with something familiar in those cases (probably dy/dx).&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-09-23T23:31:26.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14598" PostTypeId="2" ParentId="14596" CreationDate="2018-09-24T12:23:50.397" Score="3" Body="&lt;p&gt;Yes!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the text &quot;Conceptual Mathematics&quot;, which aims to teach set theory notions, among other things, from a categorical perspective to complete novices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's an excellent text!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.amazon.ca/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.ca/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7139" LastActivityDate="2018-09-24T12:23:50.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14600" PostTypeId="2" ParentId="1691" CreationDate="2018-09-25T08:37:45.493" Score="-1" Body="&lt;p&gt;The motivation for learning the different names of the categories is just that students may encounter these names in the future in papers or books or the like.  Knowing what they mean is then helpful.  Also, in general, human beings remember things and interpret them easier if they can categorize them.   So, when having to deal with something hard and new and difficult, it is just nice to have some framework to rest on.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is why equation classification is often at the front of a PDE text.  It's just easy and gentle and gets you into dealing with the topic and just getting some hands on looking at the formulas.  Rather than the first thing being some tough problem full of Bessel functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The whole thing is really a very minor part of a PDE course so I really wouldn't make any fuss about it or feel the need for some motivation.  Just teach the concept.  I'm not even sure that it is needed to test them on ability to classify PDEs.  It's pretty easy and not that important compared to actually working the classic problems themselves.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-09-25T08:37:45.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14601" PostTypeId="1" CreationDate="2018-09-25T13:55:21.613" Score="3" ViewCount="307" Body="&lt;p&gt;What is the name for a situation where the obvious thing turns out to be true, but the reasoning is more complicated?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In abstract algebra we can say the rational numbers - the fractions, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt; form a &lt;em&gt;field&lt;/em&gt;.  Then we can adjoin the number &lt;span class=&quot;math-container&quot;&gt;$x = \sqrt{-1}$&lt;/span&gt; which solves &lt;span class=&quot;math-container&quot;&gt;$x^2 + 1 = 0$&lt;/span&gt;, and we can have the field &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}(\sqrt{-1}) = \mathbb{Q}[x]/(x^2 + 1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doesn't it make sense that the ring of integers should be &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}[i]$&lt;/span&gt;.  We are going to take the ordinary integers &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt; and adjoin the number &lt;span class=&quot;math-container&quot;&gt;$i = \sqrt{-1}$&lt;/span&gt;.   Unfortunately, this conflicts with the definition of ring of integers.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Definition&lt;/strong&gt; The ring of integers &lt;span class=&quot;math-container&quot;&gt;$\mathcal{O}_K$&lt;/span&gt; is the set of solutions &lt;span class=&quot;math-container&quot;&gt;$x \in K$&lt;/span&gt; of monic irreducible polynomials &lt;span class=&quot;math-container&quot;&gt;$x^n + a_0 x^{n-1} + \dots + a_{n-1}$&lt;/span&gt; with coefficients in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And then it's not at all obvious that &lt;span class=&quot;math-container&quot;&gt;$\mathcal{O}_{\mathbb{Q}(i)} = \mathbb{Z}[i]$&lt;/span&gt;.  Would it be safe to skip this definition entirely and just say here's an interesting mathematical object:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \mathbb{Z}[i] = \big\{ a + bi : a , b \in \mathbb{Z} \big\} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and discuss whatever topic of interest, e.g. unique factorization.&lt;/p&gt;&#xA;" OwnerUserId="1654" LastEditorUserId="63" LastEditDate="2018-09-26T22:14:19.133" LastActivityDate="2021-06-20T18:10:30.780" Title="Succinct description of situations where naively obvious is correct, but for far more complicated reasons?" Tags="|undergraduate-education|graduate-education|abstract-algebra|number-theory|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="14602" PostTypeId="2" ParentId="3989" CreationDate="2018-09-25T20:03:32.353" Score="0" Body="&lt;p&gt;(1) If you add more time, what do you take away?  At least start to address this issue.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) I agree that there is a huge amount of material covered in multivariable calc, but I fear that your solution of two semesters (adding lots of extra content!) is not even addressing the issue in terms of developing the topics themselves with more practice and familiarity.  In other words, you are adding more time, but also more content.  How can I tell if this is better/worse?  (2 unknowns with one equation.)  Why not do something simple like adding more time for the same topics?  At least then I know the directionality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3)  The two quarter idea is interesting in that it allows performing the idea in (2) [deeper practice with unfamiliar and less intuitive topics like line integrals and div/grad/curl and all that, but not adding a bunch of extra content in.]  The question then becomes what do you do with the rest of the calc sequence?  ODE really could use 2 quarters for the same rationale.  If you have advanced students (above average, exposed to strong pre-calc), you could do some quarter sequence like (1) differential calc (2) integral calc (3) calc 3 part 1 (4) calc 3 part 2 (5) ODE part 1 (6) ODE part 2.  The extra time for calc 3 and ODE is nice.  Disadvantages would be students that really don't know regular calc that well (after all they did not place out of it!)  Also, you stretch multivariable calc over a summer.  (that said, there are disparate topics.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4)  I actually think the fundamental topics of single variable calc (and to a lesser extent diffy Qs) are more important to STEM students.  Perhaps covering multivariable calc in some less satisfactory manner is sufficient.  Sort of how LaPlace transform is covered in a short manner...so that students have at least seen it...but will need to learn deeper within the context of whatever EE or control systems course needs it later.  In other words, not really developing competence (as we do with integration methods), but familiarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4.5) The main need for  multivariable calc is in E&amp;amp;M.  However, the standard physics 1 E&amp;amp;M basically uses line integrals and not that much else (yes, you may vaguely touch a little of it when hitting Maxwell's equations, but line integrals are the main thing for problem solving).  It's not ideal, but probably sufficient for the majority of STEM majors (who all agree that first semester physics is like high school warmed over and then second semester is a pain in the butt...but you get through it and move on to other things.)  Physics majors really do have a need to do more applied vector calculus, but this is covered very hard (get lots of review AND practice) in the context of junior year E&amp;amp;M, ITSELF.  For instance, look at the first chapter of Wangsness (it's one ~80 page vector calc lesson with almost no physics).&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-09-25T20:03:32.353" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14603" PostTypeId="2" ParentId="3989" CreationDate="2018-09-25T20:57:41.057" Score="1" Body="&lt;p&gt;At Columbia University, &quot;calculus&quot; goes from Calculus I to Calculus IV, along the lines I've described. At Carnegie Mellon University, there is a course between Calculus II and Calculus III called &quot;Integration and Approximation,&quot; that covers advanced integration techniques (e.g partial fractions), improper integration, Taylor and Power Series, Simpson's Rule, introductory differential equations, and Euler's method. Call it &quot;Calculus 2.5,&quot; but it effectively lengthens the calculus sequence to four courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At some of the state schools, e.g. Georgia Tech and the University of Pittsburgh, Calculus I, II, and III are (reasonably) four- rather than three- credit courses. In this case, it makes sense to break them up into four three credit courses instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Carnegie Mellon does have a 17 course (51 credit) math major, so it comes at the expense of humanities and social science courses. Their engineering program allows for lot of &quot;technical&quot; electives, so the greater emphasis on math comes at the expense of &quot;traditional&quot; engineering for these students.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2018-09-25T20:57:41.057" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14604" PostTypeId="2" ParentId="14586" CreationDate="2018-09-25T22:50:57.703" Score="2" Body="&lt;p&gt;Play good games as a family. I have &lt;a href=&quot;http://mathmamawrites.blogspot.com/p/games.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;a list&lt;/a&gt; (more shape and logic - oriented, than numeracy).&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-09-25T22:50:57.703" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14605" PostTypeId="2" ParentId="562" CreationDate="2018-09-26T09:54:32.810" Score="-1" Body="&lt;p&gt;If one ponders seriously the question &quot;&lt;a href=&quot;https://mathoverflow.net/q/115416/745&quot;&gt;If &lt;span class=&quot;math-container&quot;&gt;$d/dx$&lt;/span&gt; is an operator, on what does it operate?&lt;/a&gt;&quot;, one comes to the conclusion that it doesn't operate on &lt;em&gt;functions&lt;/em&gt; in the modern sense of the word (i.e. objects that map elements from a subset of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; to elements of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;). Instead, it operates on objects of a different kind, for which the &lt;a href=&quot;https://hsm.stackexchange.com/q/6104/3462&quot;&gt;historically correct&lt;/a&gt; name is &lt;em&gt;functions of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Warning: although we use these other objects all the time, it seems that they were &lt;a href=&quot;https://mathoverflow.net/q/307947/745&quot;&gt;never formalized&lt;/a&gt; in modern mainstream mathematics. So we are in the awkward situation of teaching something which we can't really make precise. But I'll just pretend that we all know what a functions of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we compare the prime and the &lt;span class=&quot;math-container&quot;&gt;$d/dx$&lt;/span&gt; notations from this perspective, we see that they are not two different notations for the same thing, but operators on two different spaces. &lt;a href=&quot;https://mathoverflow.net/q/115416/745&quot;&gt;Eulers prime&lt;/a&gt; operates on (modern) functions, while Leibniz' &lt;span class=&quot;math-container&quot;&gt;$d/dx$&lt;/span&gt; operates on functions of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the convention I currently use when I teach. So for example I would never write &lt;span class=&quot;math-container&quot;&gt;$(x^2+1)'$&lt;/span&gt;, since &lt;span class=&quot;math-container&quot;&gt;$x^2+1$&lt;/span&gt; is not a function (in the modern sense), but a function &lt;em&gt;of&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Instead I write &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}(x^2+1)$&lt;/span&gt;. (There are other reasons for avoiding the prime notation when differentiating functions of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, like the impossibility to do the chain rule with it.).  Similarly, if &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a function like &lt;span class=&quot;math-container&quot;&gt;$x\mapsto (x^2+1)$&lt;/span&gt; I would never write &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dx}$&lt;/span&gt; since &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Instead I write &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt;. On the other hand, if &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is a function depending on a parameter, like &lt;span class=&quot;math-container&quot;&gt;$x\mapsto x^2+c$&lt;/span&gt;, then it does make sense to write &lt;span class=&quot;math-container&quot;&gt;$\frac{dg}{dc}$&lt;/span&gt;, but this is not the same as &lt;span class=&quot;math-container&quot;&gt;$g'$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2018-09-26T09:54:32.810" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14606" PostTypeId="1" CreationDate="2018-09-26T22:36:15.300" Score="1" ViewCount="112" Body="&lt;p&gt;For simple expressions with easily derived canonical forms (eg polynomials and simple rational expressions), is there a way to leverage existing tools to verify that two expressions are equal when those expressions are expressed in programming form (ie, 2x^2+3x+1 and (4x^2+6x+2)/2) in two separate cells of a spreadsheet?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm drawn toward Google Sheets and Gapps script because I could leverage java libraries, but this might not be a programming question if there is a simpler approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make the connection to math education more clear, the purpose of this question is to allow a teacher to get math responses in a free response quiz (eg, via google forms), then grade the quiz automatically so much as is possible, largely cutting down on the time required to grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is possible via multiple choice responses, but this format is less accurate for assessing student understanding. This question is in the educator's forum because it's possible that someone has accomplished this task already or by using a different approach. Otherwise, this is a programming question (which is asked in the appropriate forum as well). &lt;/p&gt;&#xA;" OwnerUserId="10227" LastEditorUserId="2083" LastEditDate="2018-09-27T19:55:04.273" LastActivityDate="2018-09-28T03:22:49.860" Title="Verifying Simple Expression Equivalence in a Spreadsheet" Tags="|mathematical-pedagogy|test-design|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14608" PostTypeId="2" ParentId="14606" CreationDate="2018-09-28T03:22:49.860" Score="1" Body="&lt;p&gt;Generally speaking, this issue is exactly why the global direction to &quot;simplify your answer&quot; is given. Properly defined for different structures, it provides a way of generating a canonical single answer that can be more easily checked (among two students, student-and-book, student-and-instructor, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I don't think that's merely a pedagogical or ease-of-grading issue. In practice and industry, there has to be some language in which two people communicate and express answers and check whether they agree or not. That is precisely the point of the simplification process (again, defined for a variety of different objects). I would recommend you use that; include the direction &quot;simplify your answer&quot; (and define it in class!) and then you'll just have one form of each answer to check. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-09-28T03:22:49.860" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14611" PostTypeId="1" CreationDate="2018-09-28T20:02:24.877" Score="3" ViewCount="255" Body="&lt;p&gt;I am teaching a mid-level calculus course, and I see my students making the &lt;a href=&quot;https://en.wikipedia.org/wiki/Freshman%27s_dream&quot; rel=&quot;nofollow noreferrer&quot;&gt;freshman's dream&lt;/a&gt; mistake of thinking that every function is a homomorphism. In particular, they think that exponents can be distributed, so &lt;span class=&quot;math-container&quot;&gt;$\sqrt{a + b} = \sqrt{a} + \sqrt{b}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(a + b)^2 = a^2 + b^2$&lt;/span&gt; in their world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have discussed this in class, and I've demonstrated how this will lead to contradictions like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2} = 2$&lt;/span&gt;. When I see this mistake, I struggle with grading it because I have the opinion that properties of exponents should have been learned long ago. I came here to ask how I can gently get my students to unlearn this mistake as soon as possible. Am I being unreasonable in thinking that this is a big problem at this stage of learning mathematics?&lt;/p&gt;&#xA;" OwnerUserId="1903" LastActivityDate="2018-09-28T20:02:24.877" Title="How to remedy the &quot;freshman's dream&quot;?" Tags="|students-mistakes|" AnswerCount="0" CommentCount="2" ClosedDate="2018-09-30T15:49:12.120" ContentLicense="CC BY-SA 4.0" />
  <row Id="14612" PostTypeId="1" CreationDate="2018-09-29T09:01:55.647" Score="6" ViewCount="1381" Body="&lt;p&gt;For finding counter examples. That does not sound convincing enough, at least not always. Why as a object in its own right the study of Cantor Set has merit ? &lt;/p&gt;&#xA;" OwnerUserId="2131" LastActivityDate="2018-10-07T21:26:11.853" Title="Why do we study Cantor Set?" Tags="|mathematical-analysis|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14613" PostTypeId="2" ParentId="4379" CreationDate="2018-09-29T13:38:34.867" Score="3" Body="&lt;p&gt;I recommend &lt;a href=&quot;https://ziteboard.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ziteboard, the realtime online whiteboard&lt;/a&gt; for math tutors. It has really useful features for math teaching, like the &lt;a href=&quot;https://ziteboard.com/online-whiteboard-features/&quot; rel=&quot;nofollow noreferrer&quot;&gt;graphing calculator synced&lt;/a&gt; in real time: &lt;a href=&quot;https://i.stack.imgur.com/Zvtxa.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Zvtxa.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10430" LastActivityDate="2018-09-29T13:38:34.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14614" PostTypeId="2" ParentId="14612" CreationDate="2018-09-29T14:40:36.030" Score="11" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;in beginning real analysis: to counter the naive notion that a &quot;closed set&quot; is a union of closed intervals, plus a few single points.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In beginning Lebesgue measure: the easiest example of an uncountable set of measure zero&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In general topology: sets homeomorphic to the Cantor set are useful in proofs&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Fractal geometry: many fractals are homeomorphic to the Cantor set.  Mandelbrot calls such a thing a &lt;em&gt;Cantor dust&lt;/em&gt; to suggest its appearance.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="127" LastActivityDate="2018-09-29T14:40:36.030" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14615" PostTypeId="2" ParentId="14612" CreationDate="2018-09-29T16:16:28.960" Score="5" Body="&lt;p&gt;I think that you may be selling short the value of a counterexample! They are quite useful for making sure that you have not &lt;em&gt;proven&lt;/em&gt; &lt;em&gt;too&lt;/em&gt; &lt;em&gt;much&lt;/em&gt;. i.e. When you have a plausible but only semi-formal argument, how do you tell if it is worth the effort in making it rigorous? Checking against counterexamples is often a useful step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, there are relations to unbounded paths in an infinite binary tree without leaves. That is, suppose that you start at the root and write &lt;span class=&quot;math-container&quot;&gt;$0.$&lt;/span&gt;, thought of as a ternary number. As we travel to the left or right, write a &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, respectively, for the following digit. Continuing on we get a ternary expansion defining a real number in the Cantor set. Correspondingly, the ternary expansion of an element of the Cantor set gives rise to a path from the root in the tree.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-09-29T16:16:28.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14616" PostTypeId="1" CreationDate="2018-09-29T18:17:52.363" Score="11" ViewCount="377" Body="&lt;p&gt;I was teaching elementary school kids (aged 10) about the mean. The intuition I gave them is roughly as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are trying to find a value such that the sum of  all the distances from the mean for the values above the mean is the same as the sum of all the distances from  the mean for the values below the mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They were happy with this and we supplemented it by plotting sets of points and drawing the line representing the mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then I told them how to compute the mean. That is add up all the values and divide by the number of values.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The method of calculation seemed completely disconnected from the intuition for what the mean is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is a good way to give intuition for the standard method of computing the mean?&lt;/p&gt;&#xA;" OwnerUserId="10432" LastEditorUserId="10432" LastEditDate="2018-09-29T19:58:05.483" LastActivityDate="2019-10-20T20:00:31.367" Title="Intuition for the mean for elementary school kids" Tags="|primary-education|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14617" PostTypeId="2" ParentId="14616" CreationDate="2018-09-29T22:46:17.277" Score="9" Body="&lt;p&gt;Take 3 piles of toothpicks (different amounts in each).  Shove them together and split them into 3 equal piles.  Make the kids do it. Do it with marbles.  Do it with money.  Do it a few times: &quot;lather, rinse, repeat&quot;.  ;)&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-09-29T22:46:17.277" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14618" PostTypeId="1" CreationDate="2018-09-29T23:18:43.413" Score="7" ViewCount="324" Body="&lt;p&gt;Cubic polynomials are crucially important in computer graphics: for example, cubic &lt;a href=&quot;https://en.wikipedia.org/wiki/B%C3%A9zier_curve&quot; rel=&quot;noreferrer&quot;&gt;Bézier curves/surfaces&lt;/a&gt;,&#xA;and&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Spline_(mathematics)#General_expression_for_a_C2_interpolating_cubic_spline&quot; rel=&quot;noreferrer&quot;&gt;cubic splines&lt;/a&gt;,&#xA;which have many practical applications. Essentially visual continuity constraints force the polynomials to be cubic;&#xA;quadratic would not suffice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. Are there practical applications that lead&#xA;  to a need to study polynomials of degree 4 or higher?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am seeking justification for exploring polynomials of higher&#xA;degree, justification that is external to pure mathematics (as is graphics).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/ZPIn0.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZPIn0.gif&quot; alt=&quot;Bezier2/3&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://helpx.adobe.com/flash/kb/bezier-pen-tool-flash-professional.html&quot; rel=&quot;noreferrer&quot;&gt;Adobe bezier-pen-tool&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-09-30T11:28:56.180" LastActivityDate="2018-10-01T17:34:37.230" Title="Beyond cubic polynomials: Applications?" Tags="|undergraduate-education|solving-polynomials|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14619" PostTypeId="2" ParentId="14618" CreationDate="2018-09-29T23:55:23.657" Score="2" Body="&lt;p&gt;One important example is the &lt;a href=&quot;http://www.chebfun.org/about/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chebfun software package&lt;/a&gt;, which uses higher order polynomial interpolants on carefully chosen points to accurately represent a variety of functions.  &lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2018-09-29T23:55:23.657" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14620" PostTypeId="2" ParentId="14616" CreationDate="2018-09-30T00:16:48.360" Score="3" Body="&lt;p&gt;[Note: I interpreted the question as how to explain the relationship between the two definitions]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's first consider the underlying algebra before turning to real-world models.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, suppose we have four reals &lt;span class=&quot;math-container&quot;&gt;$a_1 &amp;lt; a_2 &amp;lt; a_3 &amp;lt;a_4$&lt;/span&gt; with mean &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; between &lt;span class=&quot;math-container&quot;&gt;$a_2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a_3.\,$&lt;/span&gt; Here the equivalence connecting the two views of the mean is as follows&lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\overbrace{a+a+a+a = a_1 + a_2 + a_3 + a_4}^{\large\text{definition of mean } a}\iff \overbrace{a\!-a_1\, +\, a\!-a_2\, =\, a_3 - a\, +\, a_4 -a}^{\large{ a_i\ \text{are balanced around the mean } a}} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The direction &lt;span class=&quot;math-container&quot;&gt;$(\Rightarrow)$&lt;/span&gt; arises as follows: in the definition, we can cancel a LHS &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; from each &lt;span class=&quot;math-container&quot;&gt;$a_i \ge a$&lt;/span&gt; on the RHS, and we can cancel each RHS &lt;span class=&quot;math-container&quot;&gt;$a_i &amp;lt; a$&lt;/span&gt; from an &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; on the LHS, yielding the equivalent balanced form on the right. This method can be used more generally to check an equality of sums by rewriting it more simply, e.g.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\qquad\qquad\qquad\begin{align}&#xA;   &amp;amp;\color{#c00}{222}+\color{#0a0}{200}+1000+4119\\&#xA;=\ &amp;amp;\color{#c00}{100}+\color{#0a0}{328}+2113+3000\end{align}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\ \iff\ \begin{align}  &#xA;   &amp;amp;\color{#c00}{122} + 1119\\ &#xA;=\ &amp;amp;\color{#0a0}{128} + 1113&#xA;\end{align}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\iff \begin{align}&#xA; &amp;amp;6\\&#xA;=\ &amp;amp;6&#xA;\end{align}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Above we cancelled &lt;span class=&quot;math-container&quot;&gt;$\color{#c00}{100}$&lt;/span&gt; from both sides, leaving the summand &lt;span class=&quot;math-container&quot;&gt;$\color{#c00}{122}$&lt;/span&gt; on the new LHS.&lt;br&gt;&#xA;Next, &lt;span class=&quot;math-container&quot;&gt;$\, $&lt;/span&gt; we cancelled &lt;span class=&quot;math-container&quot;&gt;$\color{#0a0}{200}$&lt;/span&gt; from both sides, leaving the summand  &lt;span class=&quot;math-container&quot;&gt;$\color{#0a0}{128}$&lt;/span&gt; on the new RHS, etc. The OP is just the special case when all summands are equal on one side, say &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. This replaces each &lt;span class=&quot;math-container&quot;&gt;$a_i$&lt;/span&gt; by its distance from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, doing it on the equation side that keeps all the summands nonnegative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conversely, direction &lt;span class=&quot;math-container&quot;&gt;$(\Leftarrow)$&lt;/span&gt; follows by inverting the prior, i.e. by adding terms to both sides of the equation in order to move all negated terms to the opposite side of the equation, i.e. by eliminating all subtractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A simple real-world model is an old-fashioned balance scale. Then the prior additions and subtractions to both sides amount to adding or removing equal weights from both sides of the scale - which preserves balance. One can illustrate the above equivalence by explicitly performing the steps in the above sketched proof using weight manipulations. Of course one should use much smaller numbers than I did above (I chose those larger numbers only to highlight that the method can yield &lt;em&gt;nonntrivial&lt;/em&gt; simplifications).&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2018-09-30T01:32:29.260" LastActivityDate="2018-09-30T01:32:29.260" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14621" PostTypeId="2" ParentId="14616" CreationDate="2018-09-30T01:01:24.600" Score="3" Body="&lt;p&gt;For basic intuition with kids this age, I would simply draw a number line and locate 2 and 4 as dots. Ask them what number is half-way in between. They say 3. Then demonstrate that &lt;span class=&quot;math-container&quot;&gt;$(2+4)/2=3$&lt;/span&gt;. Of course this isn't a proof, but the goal is to build intuition, not to prove anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To amplify on this simple example, describe a situation where Alice has 2 cookies and Betty has 4, but they want to be fair and share equally. How many should each have? Describe the fact that they gain and lose equal amounts (what Betty loses equals what Alice gains), and also the fact that each one gets half the cookies.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-09-30T01:01:24.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14622" PostTypeId="2" ParentId="14618" CreationDate="2018-09-30T07:48:35.170" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Shamir%27s_Secret_Sharing&quot; rel=&quot;noreferrer&quot;&gt;Shamir's secret sharing&lt;/a&gt; is a cryptographic application of polynomial interpolation where the order of the polynomial depends on the number of shares which you want to be required to access the secret. E.g. if you want 12 stars worth of generals to be required to launch your missiles, you take a random polynomial of degree 11 which passes through &lt;span class=&quot;math-container&quot;&gt;$(0, \text{launch code})$&lt;/span&gt; and give four points on the polynomial to each 4-star general, three to each 3-star general, etc.&lt;/p&gt;&#xA;" OwnerUserId="8005" LastActivityDate="2018-09-30T07:48:35.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14623" PostTypeId="2" ParentId="14618" CreationDate="2018-09-30T15:18:45.650" Score="4" Body="&lt;p&gt;Quartics and higher degree polynomials frequently arise when intersecting lower degree curves, e,g, the intersection of two conics, or a line and a torus. Such curve intersections often occur in physical problems. For example, &lt;a href=&quot;https://en.wikipedia.org/wiki/Quartic_function#Applications&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; lists the following examples.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;&lt;p&gt;In computer-aided manufacturing, the torus is a shape that is&#xA;     commonly associated with the endmill cutter. To calculate its&#xA;     location relative to a triangulated surface, the position of a&#xA;     horizontal torus on the z-axis must be found where it is tangent to&#xA;     a fixed line, and this requires the solution of a general quartic&#xA;     equation to be calculated.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;A quartic equation arises also in the process of solving the &lt;a href=&quot;https://en.wikipedia.org/wiki/Crossed_ladders_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;crossed&#xA;     ladders problem,&lt;/a&gt; in which the lengths of two crossed ladders, each&#xA;     based against one wall and leaning against another, are given along&#xA;     with the height at which they cross, and the distance between the&#xA;     walls is to be found.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;In optics, &lt;a href=&quot;https://en.wikipedia.org/wiki/Alhazen%27s_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alhazen's problem&lt;/a&gt; is &quot;Given a light source and a&#xA;     spherical mirror, find the point on the mirror where the light will&#xA;     be reflected to the eye of an observer.&quot; This leads to a quartic&#xA;     equation.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Similarly for surfaces and higher dimensions, e.g. intersections of such are ubiquitous in geometric modelling etc, e.g. see the excerpt below from a &lt;a href=&quot;https://doi.org/10.1016/j.comgeo.2005.10.004&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper on quadric intersection algorithms&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The simplest of all the curved surfaces, quadrics (i.e., algebraic surfaces of degree two), are fundamental geometric&#xA;  objects, arising in such diverse contexts as geometric modeling, statistical classification, pattern recognition, and&#xA;  computational geometry. In geometric modeling, for instance, they play an important role in the design of mechanical&#xA;  parts; patches of natural quadrics (planes, cones, spheres and cylinders) and tori make up to 95% of all mechanical&#xA;  pieces according to Requicha and Voelcker [19].&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Computing the intersection of two general quadrics is a fundamental problem and an explicit (i.e., parametric)&#xA;  representation of the intersection is desirable for most applications. Indeed, computing intersections is at the basis of&#xA;  many more complex geometric operations such as computing convex hulls of quadric patches [12], arrangements of&#xA;  sets of quadrics [2,10,18,22,29], and boundary representations of quadric-based solid models [13,21].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2018-09-30T15:40:18.663" LastActivityDate="2018-09-30T15:40:18.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14624" PostTypeId="2" ParentId="14616" CreationDate="2018-09-30T22:39:33.090" Score="1" Body="&lt;p&gt;You are trying to find a number &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt;, the &quot;balance point,&quot; with the property (let’s take &lt;span class=&quot;math-container&quot;&gt;$n=4$&lt;/span&gt; to avoid summation notation)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(x_1-\mu)+(x_2-\mu)+(x_3-\mu)+(x_4-\mu)=0.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can solve for &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$\mu=\frac{x_1+x_2+x_3+x_4}{4}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For elementary school kids, you will have to explain why at least one term &lt;span class=&quot;math-container&quot;&gt;$(x_j-\mu)$&lt;/span&gt; will need to be negative, and another positive. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2018-09-30T22:39:33.090" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14625" PostTypeId="1" CreationDate="2018-10-01T11:28:19.207" Score="0" ViewCount="477" Body="&lt;p&gt;Looking for the necessary hardware to share, in real time streaming, the writings and drawings done in a paper or notebook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It looked like something easy to find but after two hours goggling, nothing found. Obviously, I can use a webcam with a support focusing to the white paper. But it seems a solution far from optimal.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A small interactive white-board seems another solution, but no idea if they can do real time streaming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notebooks with touch screen are a another alternative, but my experience with electronic/digital pens/stylos is disappoint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any hint ? thanks in advance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/4379/real-time-interactive-whiteboard-for-tutoring-math&quot;&gt;this question&lt;/a&gt; is related, but focus more in the software and server than in the local method to capture/record.&lt;/p&gt;&#xA;" OwnerUserId="7441" LastEditorUserId="7441" LastEditDate="2018-10-01T17:26:32.357" LastActivityDate="2018-10-01T23:22:03.363" Title="Hardware for real time streaming of writings and drawings" Tags="|tutoring|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14626" PostTypeId="1" CreationDate="2018-10-01T15:23:53.980" Score="8" ViewCount="129" Body="&lt;p&gt;I am in charge in my institution of a class where students prepare math interviews to enter university. Part of the time is devoted to the review of the topics of PreCalculus and Calculus the students studied the previous. This time is mostly filled with problem solving: standard practice problems and examination problems, together with problems tailored for interviews.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To help students review, I prepared many summaries (called &quot;cheat sheet&quot;) of various topics. I feel unsatisfied with these cheat sheets. First of all, because it is time-consuming. Secondly, because I believe they should be made by students. My rationale is that &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;actively making cheatsheets and summaries help students to memorize them. &lt;/li&gt;&#xA;&lt;li&gt;The other reason is that, by making a summary, students show they understand the bigger picture of a topic, how things are related and what are the key points on a topic. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I will have soon have some time that I planned to have students make their own cheatsheet on a given topic. Since it will be my first time doing it, I am concerned about &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How should I organize such an activity?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in people's experience on the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the sake of the question, you can consider it will be three times one hour spread in a week. The students are senior high-school students (12th grade, so about 18 years old). I would qualify their average organizational skills are low (with a high standard deviation): the best of them will use highlighters to... highlight important information in their textbook, together with additional comments in their margin. Seldom have I seen them writing summaries (while it is customary of senior high school students in my own country, France). This happens in Asia (but I don't think it really matters).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt;: I am open to answers that challenge the premise of the question and explain why cheatsheets/summaries made by students are a bad idea.&lt;/p&gt;&#xA;" OwnerUserId="1538" LastActivityDate="2018-10-01T23:35:50.323" Title="How to organize a &quot;cheatsheet-making session&quot; with students" Tags="|classroom-management|classroom-interaction|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14627" PostTypeId="2" ParentId="14618" CreationDate="2018-10-01T17:34:37.230" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/BCH_code&quot; rel=&quot;nofollow noreferrer&quot;&gt;BCH&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reed-Solomon&lt;/a&gt; codes are classes of error correcting codes constructed from polynomials over finite fields. The polynomials occurring in such constructions have higher degrees. These examples also provide motivation for studying vector spaces over finite fields.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-10-01T17:34:37.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14628" PostTypeId="2" ParentId="14625" CreationDate="2018-10-01T18:49:31.133" Score="2" Body="&lt;h3&gt;Tablet Solutions&lt;/h3&gt;&#xA;&lt;p&gt;For online office hours, I have access to an iMac with an attached &lt;a href=&quot;https://us-store.wacom.com/Catalog/pen-displays/cintiq-pro/wacom-cintiq-pro-24#/undefined1&quot; rel=&quot;nofollow noreferrer&quot;&gt;24&amp;quot; Cintiq Pro display&lt;/a&gt; from Wacom.  Also connected to this computer is a relatively high-end webcam and a good microphone (you really don't want to be using the in-built microphone).  This little studio is also equipped with good set of LED lights, which seems like overkill, but makes a big difference in the ability of your students to see you.&lt;/p&gt;&#xA;&lt;p&gt;For video conferencing, I use &lt;a href=&quot;https://zoom.us/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zoom&lt;/a&gt;.  I am sure that there is other video conferencing out there (Skype, for example, or perhaps Apple's FaceTime), but Zoom seems pretty solid to me.  Zoom can be used for free, but in a somewhat limited capacity (for example, I think that you are limited to 45 minutes of streaming before you have to restart your conference). In addition to letting students see your face (or whatever else a camera might be pointed at), Zoom makes it relatively easy to share whatever is on your screen.  My understanding is that meetings can be recorded via Zoom, but I have never used that capability.&lt;/p&gt;&#xA;&lt;p&gt;I conduct most of my office hours by setting up a desktop with &lt;a href=&quot;http://gingerlabs.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Notability&lt;/a&gt;.  Notability is designed for note-taking, and does a pretty good job of it.  The only real downside that I have encountered with Notability is that it produces very large .pdf files.&lt;/p&gt;&#xA;&lt;p&gt;While I was not a part of writing the grant which paid for this studio, my understanding is that it took about \$10,000 to set up, and the subscription to Zoom probably costs the institution something as well, so such a system probably requires institutional support and/or a healthy grant, and is likely not useful for all use cases.&lt;/p&gt;&#xA;&lt;p&gt;If the above system is out of your price range, it can be made cheaper.  Get a cheaper webcam and microphone, use a cheaper computer, and get a smaller tablet display.  You could also use a tablet with a stylus as a writing pad, routing the video output through a computer to the internets (this is a strategy that I have seen used in large lecture halls).  However, at some point you are going to start running into disappointing stylus experiences, poor sound quality, etc.  Indeed, you can broadcast directly from a tablet using Zoom, if that is what you want to do---I assume that other video conferencing software behaves similarly.&lt;/p&gt;&#xA;&lt;p&gt;Alternatively, skip the tablet entirely, and get a couple of high-quality webcams.  You will want one camera that shows your face, and a second set up on a tripod which can be used as a document camera.  Get some good lights to illuminate what you are writing, and stream that.  There are a couple of downsides to this solution:  for example, it is an extra step if you want to upload written notes, and while you are writing your hand is going to be in the way.  However, the biggest expense here is likely to be the computer, so it could be done this way much more cheaply.&lt;/p&gt;&#xA;&lt;h3&gt;Learning Glass&lt;/h3&gt;&#xA;&lt;p&gt;Another possibility is to use a &lt;a href=&quot;https://www.learning.glass/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Learning Glass&lt;/a&gt; setup.  The Learning Glass is essentially a large piece of glass with lights embedded around the edges which shine into the glass and illuminate anything that is written on it with neon whiteboard markers.  You stand on one side of the glass, and a camera is set up on the other side (the image is flipped horizontally so that you can write naturally on your side of the glass, but your students will see everything with the correct orientation---there is only a problem if you are overly attached to your students believing that you are right handed).  Our department uses the Learning Glass to produce &amp;quot;&lt;a href=&quot;http://microtutorials.ucr.edu/videos.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;micro-tutorials&lt;/a&gt;&amp;quot;, which you can have a look at if you are interested.&lt;/p&gt;&#xA;&lt;p&gt;I am not a huge fan of the Learning Glass.  It takes up a lot of space (you need to setup a large sheet of glass, and have space both for lecturers and cameras), and it is quite expensive (on the order of \$12,000-15,000 for a studio setup, though they do sell a desktop version for about \$5000).  You have to be careful about where you stand, as you can stand right behind something you have written, which will make it hard to read.  Also, make sure that you have a black backdrop, and get used to wearing black (ideally).  There is also an issue of resolution---you set up the camera, then have to perform for that camera.  In contrast to what is done in Notability (or other similar software), you cannot zoom in on an important part of the screen in real time.  Finally, and this is maybe a bit nitpicky, I find the neon markers hard to erase.&lt;/p&gt;&#xA;&lt;p&gt;More cheaply, you could set up a camera in front of a whiteboard, then simply lecture as usual.&lt;/p&gt;&#xA;&lt;p&gt;For streaming, something like Zoom is still a viable option.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-10-01T18:49:31.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14629" PostTypeId="2" ParentId="14625" CreationDate="2018-10-01T23:22:03.363" Score="1" Body="&lt;p&gt;One solution that looks promising is the combination of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;~$100 &lt;a href=&quot;http://www.microcenter.com/product/472959/boogie-board-sync-97-lcd-ewriter&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kent Displays Boogie Board Sync&lt;/a&gt; for the tablet and software to capture your writing and drawing&lt;/li&gt;&#xA;&lt;li&gt;free &lt;a href=&quot;https://myboogieboard.com/support/apps-software&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sync Virtual Desktop Companion&lt;/a&gt; software to add live-streaming capability&lt;/li&gt;&#xA;&lt;li&gt;Skype or GoToMeeting or Zoom or similar for sharing your screen&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The Sync is a 9.7-inch pressure-sensitive LCD eWriter that captures what you're writing and drawing and allows you to store it or send it to a computer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The VDC software has Windows and Mac versions, and extends the batching communications capabilities to add live-streaming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the &lt;a href=&quot;https://myboogieboard.com/uploads/boogie-board-sync-9-user-manualvdc-mac.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;VDC Mac manual&lt;/a&gt; and &lt;a href=&quot;https://myboogieboard.com/uploads/boogie-board-sync-9-user-manualvdc-windows.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;VDC Windows manual&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Button #1.1: Live Drawing&lt;/strong&gt;&lt;br/&gt;&#xA;  Opens Live Draw window. This option allows you to draw on your &#xA;  Boogie Board Sync eWriter&#xA;   with the Sync Stylus and &#xA;  have your drawing appear in real time on your computer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, it looks as though you can hook up the BB Sync to your computer, launch the VDC software on your computer, start a Skype/GTM/Zoom meeting on your computer and start sharing your screen. Then when you draw and write on the tablet, what you're drawing or writing should be visible to your Skype/GTM/Zoom audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer: I have no affiliation with any of the above products, and in fact have not even tried any of them.&lt;/p&gt;&#xA;" OwnerUserId="3593" LastActivityDate="2018-10-01T23:22:03.363" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14630" PostTypeId="2" ParentId="14626" CreationDate="2018-10-01T23:35:50.323" Score="2" Body="&lt;p&gt;I think it is a great idea.  Given their age and organizational skills, I don't think you will get total victory.  At the same time, I have learned from weight training that the often the beginners make the biggest gains (easier to improve from weakness).  So you should feel good about anything that introduces the method and that helps them some.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have 3 sessions, I recommend to try the following exercise [just BSing, never done it.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A.  Session 1: individual work.  Have each student prepare a 1 page &quot;cheat sheets&quot;.  Should be limited to cover material they already know.  Limited to in class time.  To be graded based on appearance, completeness, selectivity (yes, these two conflict), prioritization, innovations, etc. [have a 5 minute discussion ahead of time, for the value of cheat sheet summaries.]  Let the students know they will be graded. (low point count, but something, and on a curve, maybe gentle like 8/9/10, don't tell them you will be gentle ahead of time though...and if you have someone completely blow it off...slam them of course, but any honest attempt should get 8+).  Also give the student with the best one a big bar of chocolate (let them know they all compete for a small prize, but keep the prize a secret for the fun aspect).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;B.  Session 2: First hand out the graded results of the first session, and the chocolate.  Avoid much class discussion and move to the main work...paired work.  Same thing, but pair the students (you assign them) and have them make a best version together.  [Use your judgment on how much to debrief the previous work assignment.  My gut is not to debrief too much, so they have some uncertainty and freedom to innovate.  Also so as not to derail the purpose of session 3.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C.  Session 3:  debrief and discussion.    Give grades out again for the paired submissions.  Go over (on the overhead, or better yet, hand out copies) of the &quot;best submission.  Do another bar of chocolate for the winner.  Have class discussion (what was missing, what was too much, what were things they disagree with).  Let them all out 10-30 minutes early as a group reward assuming they played along the past two sessions (or give them free time in class if wandering halls not allowed is not allowed).  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I think that is plenty.  If you wanted to do less, you could probably just do two sessions (second paired one, not graded, and shorter, with in class discussion last half hour).  Anyhow, this will totally give them an exposure to making their own cheat sheets.  As well as some effort to think about how to do the best ones.  After that, it is up to them to progress and use them.  But you have exposed them enough to &quot;fishing&quot; that they have some potential to feed themselves.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-10-01T23:35:50.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14631" PostTypeId="1" AcceptedAnswerId="14634" CreationDate="2018-10-03T12:56:57.620" Score="15" ViewCount="670" Body="&lt;p&gt;One easy integration of &lt;span class=&quot;math-container&quot;&gt;$\sec x$&lt;/span&gt; substitutes &lt;span class=&quot;math-container&quot;&gt;$u=\sin x$&lt;/span&gt;, viz.&lt;span class=&quot;math-container&quot;&gt;$$\int\frac{\cos x}{1-\sin^2 x}\,\mathrm{d}x=\frac{1}{2}\ln\left|\frac{1+\sin x}{1-\sin x}\right|+C.$$&lt;/span&gt;Multiplying top and bottom by &lt;span class=&quot;math-container&quot;&gt;$1+\sin x$&lt;/span&gt; and writing &lt;span class=&quot;math-container&quot;&gt;$1-\sin^2 x=\cos^2 x$&lt;/span&gt; gives the more familiar &lt;span class=&quot;math-container&quot;&gt;$$\ln\left|\sec x+\tan x\right|+C.$$&lt;/span&gt;A proof written so as to get to that result first requires more elaborate tricks, such as comparing the derivatives of &lt;span class=&quot;math-container&quot;&gt;$\sec x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tan x$&lt;/span&gt; to spot a solution to &lt;span class=&quot;math-container&quot;&gt;$y'=y\sec x$&lt;/span&gt;, or to use the fact that &lt;span class=&quot;math-container&quot;&gt;$f:=\sec x,\,g:=\tan x$&lt;/span&gt; satisfy&lt;span class=&quot;math-container&quot;&gt;$$f^2=g^2+1\implies\frac{f'}{g}=\frac{g'}{f}\implies\frac{d}{dx}\ln\left|f+g\right|=\frac{f'}{g}=\sec x.$$&lt;/span&gt;So why is the sec/tan formula for an antiderivative universally taught, instead of the sine-based one?&lt;/p&gt;&#xA;" OwnerUserId="5794" LastEditorUserId="5794" LastEditDate="2021-02-06T09:42:28.393" LastActivityDate="2021-02-06T09:42:28.393" Title="Why do we state the antiderivative of $\sec x$ as $\ln |\sec x +\tan x|+C$?" Tags="|trigonometry|integration|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14632" PostTypeId="2" ParentId="14631" CreationDate="2018-10-03T14:53:48.223" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/1285741552&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stewart's text&lt;/a&gt; (&lt;a href=&quot;https://books.google.com/books?id=AcQJAAAAQBAJ&amp;amp;pg=PA320&amp;amp;dq=%22We%20could%20verify%20Formula%201%20by%20differentiating%20the%20right%20side,%20or%20as%20follows%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;) derives the result via:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int \sec x\ dx = \int \sec x \frac{\sec x +\tan x}{\sec x +\tan x}\ dx = \int \frac{\sec^2 x +\sec x \tan x}{\sec x +\tan x}\ dx$$&lt;/span&gt;&#xA;and then &lt;span class=&quot;math-container&quot;&gt;$u=\sec x +\tan x$&lt;/span&gt;. I don't think this is any worse than rewriting &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\cos x} = \frac{\cos x}{1-\sin^2x}$&lt;/span&gt;, using a &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-sub, and then partial fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if you take &lt;span class=&quot;math-container&quot;&gt;$\sec$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt; to be functions in their own right, then &lt;span class=&quot;math-container&quot;&gt;$\ln|\sec x+\tan x|$&lt;/span&gt; requires fewer computational steps than &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}\ln\left|\frac{1+\sin x}{1-\sin x}\right|$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorDisplayName="user378" LastEditDate="2018-11-10T12:49:35.060" LastActivityDate="2018-11-10T12:49:35.060" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14633" PostTypeId="2" ParentId="10021" CreationDate="2018-10-03T17:21:29.430" Score="6" Body="&lt;p&gt;I found this post as a student trying to figure out why induction proofs are so difficult to understand. There are good answers here, but I think many are not specific enough to why induction proofs (rather than proofs in general) are so challenging.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The metaphor and principle of induction is straightforward. For me, the real issues arise in following along with what's happening in an actual induction proof, and being able to replicate it myself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1: Most of the proofs involve &quot;algebraic cleverness.&quot;&lt;/strong&gt; Every induction proof I've seen so far involves some unusual algebra trick&#xA;    that I have never had a reason to use outside of the context of&#xA;    induction. Examples: removing an element from a factorial, or&#xA;    adding/subtracting some element so that something else can be&#xA;    substituted in. From the perspective of someone who has only used algebra in a conventional, solution-oriented way, this is difficult to wrap my head around. As a result, as I'm trying to read or watch an induction proof, most of my time  is spent just figuring out why the algebraic manipulations make sense, and convincing myself that they are correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2: The notation.&lt;/strong&gt; Up to this point in my proofs course, we have generally dealt with notation that is easy to distinguish in terms of &quot;what it is.&quot; As an example, it's very clear that the notations &lt;span class=&quot;math-container&quot;&gt;$p/q$&lt;/span&gt; for rational numbers, &lt;span class=&quot;math-container&quot;&gt;$2k$&lt;/span&gt; for even numbers, etc. are integers, and generally these are the same letters that are always used. But because induction proofs often deal with sums, products, and factorials, suddenly we're dealing with &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;'s, &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;'s, &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;'s, &lt;span class=&quot;math-container&quot;&gt;$n + 1$&lt;/span&gt;'s, and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;'s, depending on the statement. This is frustrating because the inconsistent notation from problem to problem distracts from the logic and makes the student expend energy just keeping track of what represents what.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Two things I think would help are:&lt;/strong&gt; First, giving some preceding exercises that will help students improve their skills at being clever with algebra. Many of the &quot;tricks&quot; used in induction proofs probably feel mundane to an experienced professor, but to a student, they are utterly flabbergasting the first time you see them. Second: Be very explicit about what each piece of notation is. Write it on the board during the first few problems, so that students are not trying to keep track in their head while also following along with the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also agree that it's hard to see the motivation behind a proof by induction. All but the most number theory-oriented students will probably not care that the sum of some amount of numbers is equal to some algebraic expression. If time permits, providing some examples of where this is actually useful (like in applied math or statistics) may help struggling students replace their frustration with curiosity.&lt;/p&gt;&#xA;" OwnerUserId="10437" LastActivityDate="2018-10-03T17:21:29.430" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14634" PostTypeId="2" ParentId="14631" CreationDate="2018-10-03T18:09:40.003" Score="10" Body="&lt;p&gt;First idea: Maybe just because it's easier to memorize the derivatives and antiderivatives of the trig functions this way: &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; always pairs with &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt; always pairs with &lt;span class=&quot;math-container&quot;&gt;$\sec$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\cot$&lt;/span&gt; always pairs with &lt;span class=&quot;math-container&quot;&gt;$\csc$&lt;/span&gt;. These are the same pairs you already know from the sum/difference of squares formulas too, if your precalculus trig studies included all&lt;sup&gt;*&lt;/sup&gt; 6 functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may put the dividing line between &quot;stuff to memorize&quot; and &quot;stuff to re-derive when needed&quot; in a different place, but at least some students have been expected to memorize those 6 derivatives and 6 antiderivatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second idea: if you arrived at &lt;span class=&quot;math-container&quot;&gt;$\int \sec x\ dx$&lt;/span&gt; through a trig substitution, then an answer involving &lt;span class=&quot;math-container&quot;&gt;$\sec x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tan x$&lt;/span&gt; will be easier to translate back to the original variable than one with a pair of &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;, since you already figured out what &lt;span class=&quot;math-container&quot;&gt;$\sec x$&lt;/span&gt; is, and again, the relationship between &lt;span class=&quot;math-container&quot;&gt;$\sec x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tan x$&lt;/span&gt; is one of those easy difference-of-squares-equals-1 deals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;*&lt;/sup&gt;Sorry, exsecant and haversine.&lt;/p&gt;&#xA;" OwnerDisplayName="user5114" LastActivityDate="2018-10-03T18:09:40.003" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14635" PostTypeId="1" AcceptedAnswerId="14640" CreationDate="2018-10-04T19:53:19.463" Score="6" ViewCount="314" Body="&lt;p&gt;I'm going to be giving a five minute &quot;elevator pitch&quot; presentation to undergraduate math majors in order to advertise an introduction to statistics class that I'll be teaching in the spring. It's an upper division course and the students will have already taken an upper division probability course. Here are a few objectives for the elevator pitch:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Briefly explain what statistics is. I'd like to give a quick, big picture summary of what statistics is that captures the essence of the subject.&lt;/li&gt;&#xA;&lt;li&gt;Explain, in a compelling way, why statistics is so useful.&lt;/li&gt;&#xA;&lt;li&gt;Explain what makes statistics beautiful or exciting. I'd like to mention some &quot;gems&quot; of statistics that could be covered or touched on in the course. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The presentation should also just give an overview of the specific topics that will be covered in the course, ideally in a way that sounds interesting. I have some ideas about what to say but I'd like to get your suggestions for how to make a statistics class sound exciting and interesting in a five minute presentation.&lt;/p&gt;&#xA;" OwnerUserId="9633" LastActivityDate="2018-10-06T13:55:51.853" Title="What to include in an &quot;elevator pitch&quot; for an undergraduate statistics class" Tags="|statistics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14636" PostTypeId="1" AcceptedAnswerId="14637" CreationDate="2018-10-04T20:24:01.920" Score="7" ViewCount="3131" Body="&lt;p&gt;I am teaching a math class where the students, most of them, tell me that they can understand the materials given by me during the course. I test them during the course too and they seem to get it. However, when it is time to face quiz or exam problems, they don't know what to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At first, I thought that my exam is too hard. However, I usually, definitely, give problems that need the understanding of materials I give in the class to solve, but add some &quot;spices&quot; to them. By spices, I mean I always assume that in an exam, there should be a little bit of challenge, not just repeating exercises. It is a test after all, right? I will not repeat problems I gave previously during the course. Sometimes, even when I intentionally give the same problems, they still don't know what to do, most likely because they do not repeat until they understand after class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example of the problem, instead of just asking how many ways we can arrange a group of 4 letters, I ask how many ways we can arrange a group of n letters. The point is that I won't repeat and will give a small added challenge during exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I thought I should give them homeworks and also tutorial sessions, but the math program is new (just two batches now), so we have no tutor..nor grader. I do it myself almost always. We have no money for tutors either. Thus, I can't do anything else (like finishing paper, doing administrative task, etc) other than helping my students. Indeed, it's gonna be so tiring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I genuinely want them to succeed and this is just my second year of formal lecturing. So happy knowing they understand me when I teach, but no idea yet how to make them exercising more often, other than reducing the covered materials maybe? Doesn't seem wise. Or should I make the exam..a repetition of problems, where only the information about the numbers given is different? It will kinda help but I am just a little bit afraid they will only be able to do specific problems with no small added restriction, resulting in less flexibility of problem solving.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ADDED: It could also be due to my experience as an undergrad student in one of the leading universities in my country. I am still so fresh to remember my days in college where I was given problems in exams (most of them) that was served more like challenges rather than competency tests. Hence my mindset was like &quot;Exam is supposed to challenge you with extra work&quot;. Now I am teaching in a not so well-known university where I should've known the standard set would be much lower, which is in the form of fewer materials given. Hence, I am still in the mindset that while university is a place to learn, it is also a place for those who can flexibly apply what they learn, now matter how high or low the standards are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any suggestion on how do I demonstrate flexibility to students other than by having them learn the basics and giving nonrepeated problems (in exam) to test them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of my students do fine with the extra work exams, but most of them, predicted from quizzes, still need more work and guidance before midterm.&lt;/p&gt;&#xA;" OwnerUserId="10108" LastEditorUserId="10108" LastEditDate="2018-10-05T13:21:56.237" LastActivityDate="2018-10-13T00:04:49.983" Title="Students understand during course but can't solve exam" Tags="|exams|teaching|tutoring|problem-solving|exercises|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14637" PostTypeId="2" ParentId="14636" CreationDate="2018-10-04T21:13:00.007" Score="27" Body="&lt;p&gt;Do &lt;strong&gt;NOT&lt;/strong&gt; give exam questions that are intentionally &lt;em&gt;more&lt;/em&gt; challenging than homework or in-class problems. I would recommend precisely the opposite. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point of the exam is really a spot-check that students know the basics and aren't just faking their way all through the class. If there is a time-limit, then that is already a lot more pressure/high-stakes than open-ended homework problems. Clearly the hardest problems confronted should be in the time-free context, with available open resources and study partners, etc. -- not with the simultaneous additional overhead of a time limitation; that's geometrically increasing the difficulty, and arguably really is unfair. Use the exam situation as a spot-verification that students have the expected individual &quot;automaticity&quot; with basic skills. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think I can sympathize in that I made the similar mistake of writing &quot;interesting&quot; (i.e., with-&quot;spices&quot;) test problems in my first year of teaching. That simply never works. My rule of thumb now is that each test question needs to be completely transparent to me on first sight. Writing the answer sheet should take at most one-quarter of the time available to students. If I look at a problem and think that it's &quot;interesting&quot;, then that's a red flag that it needs to be cut from the test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're new to teaching (or just unsure of what direction to go in), then I recommend reading Steven G. Krantz, &lt;em&gt;How to Teach Mathematics&lt;/em&gt;, published by the American Mathematical Society. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-10-05T14:22:59.357" LastActivityDate="2018-10-05T14:22:59.357" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14638" PostTypeId="2" ParentId="14636" CreationDate="2018-10-04T21:39:46.243" Score="9" Body="&lt;p&gt;I always make homework (from the textbook and online in WeBWorK) and written assignments MORE difficult than exam questions. I tell my students this, with the reason being “if you can run 10 miles in training running 5 miles on race day is easy”. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keep the course learning goals in mind. Your exam questions should be a chance for your students to demonstrate competency. &lt;/p&gt;&#xA;" OwnerUserId="8559" LastEditorUserId="8559" LastEditDate="2018-10-04T21:50:35.880" LastActivityDate="2018-10-04T21:50:35.880" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14639" PostTypeId="2" ParentId="651" CreationDate="2018-10-04T22:46:28.937" Score="-4" Body="&lt;p&gt;Groupwork sucks. In any field groupwork is used because it is necessary - you need to row in sync to make the boat go faster, or you need four people to carry a coffin - not because people actually enjoy it. Do not introduce groupwork where it is not needed. Comparison is good. Competition is good. As long as you don't grade on a curve your students should be all right (because grading on a curve sucks).&lt;/p&gt;&#xA;" OwnerUserId="7930" LastActivityDate="2018-10-04T22:46:28.937" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14640" PostTypeId="2" ParentId="14635" CreationDate="2018-10-04T23:21:34.940" Score="5" Body="&lt;p&gt;I suggest you tie the &quot;undergraduate statistics class&quot; to &lt;em&gt;data science&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) For example,&#xA;Target’s pregnancy prediction algorithm:&#xA;&lt;a href=&quot;https://www.quora.com/How-do-you-explain-data-science-to-non-computer-science-people&quot; rel=&quot;nofollow noreferrer&quot;&gt;How do you explain data science to non computer science people?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Or &lt;a href=&quot;https://www.cio.com/article/3221621/analytics/6-data-analytics-success-stories-an-inside-look.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;data analytics success stories&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) &lt;a href=&quot;https://www.svds.com/machine-learning-vs-statistics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Machine learning vs. Statistics&lt;/a&gt;: &quot;Machine learning may emphasize prediction, and statistics may focus more on estimation and inference, but both focus on using mathematical techniques to answer questions.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-10-06T00:13:35.223" LastActivityDate="2018-10-06T00:13:35.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14641" PostTypeId="2" ParentId="14636" CreationDate="2018-10-05T08:57:20.217" Score="3" Body="&lt;p&gt;If you ask me in how many ways I can arrange 4 letters or digits, this is what I will do:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;1234&#xA;1243&#xA;1324&#xA;1342&#xA;1423&#xA;1432&#xA;2134&#xA;2143&#xA;2314&#xA;2341&#xA;2413&#xA;2431&#xA;3124&#xA;3142&#xA;3214&#xA;3241&#xA;3412&#xA;3421&#xA;4123&#xA;4132&#xA;4213&#xA;4231&#xA;4312&#xA;4321&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Then, I'll count them and answer your question: &quot;Sir, I know, it's 24!!!&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you ask me how many there are for a general &lt;code&gt;n&lt;/code&gt; letters, I don't have the slightest idea, and honestly, when you think about it, it's indeed extremely difficult.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2018-10-05T08:57:20.217" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14642" PostTypeId="2" ParentId="14635" CreationDate="2018-10-05T22:21:46.580" Score="3" Body="&lt;p&gt;Here is the course description I wrote for a (high school) class on Statistics and Probability. I think you could modify it ever so slightly if you are covering Statistics, specifically, and that it could serve as a real elevator pitch.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What do the words &lt;strong&gt;probability&lt;/strong&gt; and &lt;strong&gt;statistics&lt;/strong&gt; mean? How are &lt;strong&gt;probability&lt;/strong&gt; and &lt;strong&gt;statistics&lt;/strong&gt; used or not used, correctly or incorrectly, in research journals, popular media (newspapers, television), and social media (blogs, Twitter, Facebook)?&lt;/p&gt;&#xA;&lt;p&gt;How is it possible that the same areas of mathematics can be applied to meteorology (e.g., forecasting), sports (e.g., oddsmaking), and elections (e.g., polling)?&lt;/p&gt;&#xA;&lt;p&gt;In this class, we will investigate topics of contemporary interest, and position ourselves better to be analytical and skeptical readers by using statistical and probabilistic tools to inform our critical consumption of information and data.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think that the second paragraph above is something that can be especially exciting, as many know this to be true about statistics, but (for whatever reason) may not have paused to marvel at the wondrous nature of analytical tools that can be applied to such disparate areas of the world.&lt;/p&gt;&#xA;&lt;p&gt;I also like to give one real world example about causation versus correlation, which (as far as I know/as best I remember) is hypothetical - but may have appeared somewhere else. (I'd appreciate a source if you know one!)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Sensible Gum Laws&lt;/strong&gt;: Consider the question of whether chewing gum leads to cancer. Specifically, whether frequently chewing gum makes one more likely to suffer from mouth cancers later in life. Looking through the data, one may observe that, indeed, those who are chewing gum with a greater frequency are also being diagnosed with mouth cancers later on at a greater frequency. From here, one can argue that chewing gum is a bad idea, or look to pass laws about the ingredients in gum, and so on and so forth. &lt;strong&gt;But&lt;/strong&gt;, a deeper look at the data can reveal that those who frequently chew gum are also much more likely to be cigarette smokers, who are looking to mask the smell of smoke throughout the day. And smoking cigarettes &lt;em&gt;definitely&lt;/em&gt; causes mouth cancers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am deeply concerned that this gum example (with its pithy title!) is but one of many common conflations of causation and correlation. Considering the frequency with which articles are retracted after faulty statistical methods are discovered, and some questionable practices within certain disciplines around the use of statistics (I am thinking of &lt;a href=&quot;https://projects.fivethirtyeight.com/p-hacking/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;p-hacking&lt;/strong&gt;&lt;/a&gt;, for example) it is very important that we work towards a level of quantitative literacy that will enable us to be better consumers of information.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-10-05T22:21:46.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14643" PostTypeId="2" ParentId="14635" CreationDate="2018-10-06T13:55:51.853" Score="3" Body="&lt;p&gt;You can give the example of election polling. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many popular explanations of election polling give a false impression of what statisticians can tell us about an upcoming election. For example, the so-called &quot;probability clocks&quot; on the front pages of major newspapers during the 2016 election gave the false impression that what statisticians do is to estimate probability (leading to false confidence of newpaper readers). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, of course, what statisticians do is to compute the confidence of a prediction, based on polling data. You can explain this very clearly in a statistics course.&lt;/p&gt;&#xA;" OwnerUserId="7258" LastActivityDate="2018-10-06T13:55:51.853" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14644" PostTypeId="2" ParentId="14612" CreationDate="2018-10-06T14:09:54.883" Score="5" Body="&lt;p&gt;Spaces that are homeomorpic to the Cantor set arise naturally in many mathematical settings, particularly in dynamical systems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For one dynamical example, the Cantor set is homeomorphic to the phase space of any infinite &lt;a href=&quot;https://en.wikipedia.org/wiki/Bernoulli_process&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernoulli process&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For another, the &quot;nonescaping set&quot; of many simple dynamical systems in the real line (or the complex plane) is homeomorphic to the Cantor set (this is a &quot;Cantor dust&quot; example as in the answer of @GeraldEdgar). Consider for example the dynamical system &#xA;&lt;span class=&quot;math-container&quot;&gt;$$z_n = (z_{n-1})^2 + 10&#xA;$$&lt;/span&gt;&#xA;(You can replace &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; by any real or complex number of magnitude &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;2$&lt;/span&gt;). One can prove that there is a subset &lt;span class=&quot;math-container&quot;&gt;$C \subset \mathbb C$&lt;/span&gt; homeomorphic to the Cantor set such that if &lt;span class=&quot;math-container&quot;&gt;$z_0 \in C$&lt;/span&gt; then the sequence &lt;span class=&quot;math-container&quot;&gt;$(z_n)$&lt;/span&gt; is bounded (in fact it stays in &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;), whereas if &lt;span class=&quot;math-container&quot;&gt;$z_0 \not\in C$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\lim_{n \to \infty} |z_n| =\infty$&lt;/span&gt;. In short, points not in &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; escape to infinity, points in &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; do not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, there are important theoretical descriptions/properties of the Cantor set, for example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A topological space is homeomorphic to the Cantor set if and only if it is compact, metrizable, has no isolated points, and every component is a point. &lt;/li&gt;&#xA;&lt;li&gt;Any compact zero-dimensional metrizable topological space is homeomorphic to a subspace of the Cantor set.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Cantor sets even occur naturally in number theory! The &lt;a href=&quot;https://en.wikipedia.org/wiki/P-adic_number#Topology&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic integers &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z_p$&lt;/span&gt; are homeomorphic to the Cantor set&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7258" LastEditorUserId="7258" LastEditDate="2018-10-07T21:26:11.853" LastActivityDate="2018-10-07T21:26:11.853" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14645" PostTypeId="2" ParentId="14612" CreationDate="2018-10-07T00:20:51.997" Score="7" Body="&lt;p&gt;The Cantor set is quite useful in its own right. My preferred way to think of the Cantor set is as &quot;the most general compact metrizable space.&quot; That it is the most general such space means that it is often good for counterexamples because it lacks the particulars. At the same time, one can construct things by specialization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The formal expression of the Cantor set being the most general compact metrizable space is that every compact metrizable space is the image of the Cantor set under a continuous function. One can use this as a method of proof. For example, to show that a continuous surjection from &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; onto &lt;span class=&quot;math-container&quot;&gt;$[0,1]^2$&lt;/span&gt; exists, one can use the fact that &lt;span class=&quot;math-container&quot;&gt;$[0,1]^2$&lt;/span&gt; is the image of the Cantor set under a continuous function and then extend this continuous function by linear interpolation (the complement of the Cantor set consists of open intervals) to all of &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A lot more can be proven this way, including a number of very surprising results. A wonderful source for such arguments is the following paper, which (deservedly) won an award for mathematical exposition: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Benyamini, Yoav. &quot;&lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Benyamini832-839.pdf&quot; rel=&quot;noreferrer&quot;&gt;Applications of the universal surjectivity of the&#xA;  Cantor set.&lt;/a&gt;&quot; The American mathematical monthly 105.9 (1998): 832-839.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Among the results proven there is the entirely classical result of Banach and Mazur that every separable Banach space is linearly isometric to a subspace of &lt;span class=&quot;math-container&quot;&gt;$C[0,1]$&lt;/span&gt;; a result as far from being a counterexample as can be.&lt;/p&gt;&#xA;" OwnerUserId="8799" LastActivityDate="2018-10-07T00:20:51.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14646" PostTypeId="2" ParentId="135" CreationDate="2018-10-07T00:47:20.150" Score="4" Body="&lt;p&gt;I think every really convincing argument must make use of the fact that one can take integrals with respect to arbitrary measures. If all one ever does is integrating functions with respect to Lebesgue measure, one is really doing absolute &lt;a href=&quot;https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henstock-Kurzweil integration&lt;/a&gt; which can be introduced much like the Riemann integral and actually integrates more functions than the Lebesgue integral (the Henstock-Kurzweil integral is a non-absolute integral). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that, the Lebesgue integral can be easily motivated by probability theory. A  function on a probability space is measurable (a random variable) exactly when one can meaningfully calculate the probability that the value lies within some interval. If one then wants to calculate the expectation of a bounded random variable, the Lebesgue integral is the obvious correct way to do it. Indeed, many people learned the Lebesgue integral first in the context of probability theory (I'm one of them).&lt;/p&gt;&#xA;" OwnerUserId="8799" LastActivityDate="2018-10-07T00:47:20.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14647" PostTypeId="2" ParentId="135" CreationDate="2018-10-07T07:08:38.123" Score="0" Body="&lt;p&gt;I would not let the desire for motivation lead you to pushing something in a manner that gives the student the wrong insights.  The bottom line is that there is almost no important scientific or commercial applications of the L integral (consider that none of the answers to date here on this site, full of very smart people, has listed one).  Also consider that most chemists, physicists, engineers go their entire  training without bothering to learn it or having it as a required tool. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just be clear with the students that this is a logical oddity necessary for a more complete course in analysis.  I think it will actually all go over better if you are just sort of &quot;man to man&quot; that it is a sort of curiosity of math versus contorting yourself to motivate it as useful (it is not).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.&quot; -Hamming&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hamming's point is more than cute.  A modern airplane is a complicated device that requires several sciences and engineering discplines to make all its devices (metallurgy, lift, electronics, heat cycles, radar, petroleum refining, etc.).  You could probably write an article like the famous &quot;I, Pencil&quot; about everything that goes into a modern airplane.  In doing so, many fields of math would be encountered as needed tools: differential equations and the calculus, complex numbers [key part of EE], algebra, statistics, trigonometry and spherical trigonometry.  But where would you find Lebesgue integration?  You honestly would just...not.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-10-07T07:08:38.123" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14649" PostTypeId="1" AcceptedAnswerId="16974" CreationDate="2018-10-07T21:22:31.463" Score="3" ViewCount="765" Body="&lt;p&gt;I don't understand why common core: High School Algebra II (NYC, NY, USA) removed so many topics. I think these ideas are pretty important and useful (especially since a handful of these topics come up in SAT). These also help with critical thinking, thinking abstractly and mathematical reasoning (proofs and problem solving/algorithms).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Rationalizing binomial denominators&lt;/li&gt;&#xA;  &lt;li&gt;Dividing complex numbers (incl. rationalizing binomial denominators, ex: 10/(3 – i)&lt;/li&gt;&#xA;  &lt;li&gt;Solving absolute value equations algebraically&lt;/li&gt;&#xA;  &lt;li&gt;Solving absolute value inequalities&lt;/li&gt;&#xA;  &lt;li&gt;Given a real-life scenario, write an absolute value inequality that models it&lt;/li&gt;&#xA;  &lt;li&gt;Solve problems involving direct and inverse variation&lt;/li&gt;&#xA;  &lt;li&gt;Simplifying complex fractions&lt;/li&gt;&#xA;  &lt;li&gt;Using L. of Sines and L. of Cosines to solve triangles&lt;/li&gt;&#xA;  &lt;li&gt;Finding area of a triangle using (1/2)abSinC&lt;/li&gt;&#xA;  &lt;li&gt;Ambiguous case (SSA, using L. of Sines)&lt;/li&gt;&#xA;  &lt;li&gt;Binomial Probabilities/Bernoulli experiments&lt;/li&gt;&#xA;  &lt;li&gt;Finding probabilities based on comparing areas&lt;/li&gt;&#xA;  &lt;li&gt;Finding probabilities using permutations and combinations&lt;/li&gt;&#xA;  &lt;li&gt;Composition of functions: writing an algebraic rule for f(g(x)) given f(x) and g(x) and finding the domain of f(g(x)) given f(x) and&#xA;  g(x)&lt;/li&gt;&#xA;  &lt;li&gt;Co-functions (applying the idea that cos(A) = sin(90 – A) in various ways)&lt;/li&gt;&#xA;  &lt;li&gt;Angle Sum, Angle Difference and Double Angle identities&lt;/li&gt;&#xA;  &lt;li&gt;Solving Trig equations (linear, quadratic, equations requiring use of the identities above)&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="2083" LastEditDate="2019-08-16T06:51:43.620" LastActivityDate="2019-09-01T01:06:48.787" Title="why did common core remove so many topics from Algebra II?" Tags="|common-core|" AnswerCount="2" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="14650" PostTypeId="1" AcceptedAnswerId="14651" CreationDate="2018-10-09T10:02:36.553" Score="4" ViewCount="275" Body="&lt;p&gt;I am now pretending to be a newbie student. I write the following sample space for the Monty Hall problem (It is a famous brain teaser, I assume you know it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;S=\{&#xA;(C,G1),(C,G2), (G1,G2), (G2,G1)&#xA;\}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where the first tuple represents the first choice taken by the guest and the second tuple represents the optional switch  offered by the host.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a teacher I have difficulty to explain that &lt;span class=&quot;math-container&quot;&gt;$\{(C,G1),(C,G2)\}$&lt;/span&gt; must be simplified to just one &lt;span class=&quot;math-container&quot;&gt;$\{(C,G)\}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n(S)=3$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is the car, &lt;span class=&quot;math-container&quot;&gt;$G1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$G2$&lt;/span&gt;  are the goats. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have any idea to explain it?&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Edit&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;More information:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;for &lt;span class=&quot;math-container&quot;&gt;$(C,G1)$&lt;/span&gt;, if you switch you will lose&lt;/li&gt;&#xA;&lt;li&gt;for &lt;span class=&quot;math-container&quot;&gt;$(C,G2)$&lt;/span&gt;, if you switch you will lose&lt;/li&gt;&#xA;&lt;li&gt;for &lt;span class=&quot;math-container&quot;&gt;$(G1,G2)$&lt;/span&gt;, if you switch you will win&lt;/li&gt;&#xA;&lt;li&gt;for &lt;span class=&quot;math-container&quot;&gt;$(G2,G1)$&lt;/span&gt;, if you switch you will win&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5696" LastEditorUserId="5696" LastEditDate="2018-10-09T12:07:21.680" LastActivityDate="2018-10-10T02:44:33.613" Title="How to explain the sample space of Monty Hall problem?" Tags="|probability|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14651" PostTypeId="2" ParentId="14650" CreationDate="2018-10-10T02:44:33.613" Score="6" Body="&lt;p&gt;I think the key here is that &lt;span class=&quot;math-container&quot;&gt;$(C, G_1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(C, G_2)$&lt;/span&gt; are each only half as likely as each of the other two cases - and the standard &quot;counting&quot; approach to probability only works if all the cases are equally likely. To fix it, you need a third event - think of it as Monty flipping a coin. If it comes up heads, he picks the leftmost goat that's still hidden; if tails, the rightmost. If the player picks a goat, the coin flip doesn't matter, but it happens anyway. This gives us six outcomes: &lt;span class=&quot;math-container&quot;&gt;$(C, H, G_1), (C, T, G_2), (G_1, H, G_2), (G_1, T, G_2), (G_2, H, G_1), (G_2, T, G_1)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2018-10-10T02:44:33.613" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14653" PostTypeId="1" CreationDate="2018-10-11T23:06:22.240" Score="15" ViewCount="1541" Body="&lt;p&gt;Is there some single, persuading visualization that can be used to convince students of the intuitive truth of the fundamental&#xA;theorem of calculus, in the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_a^b  f(t) \, dt = F(b) - F(a) \;?&#xA;$$&lt;/span&gt;&#xA;I'm not seeking a &quot;proof-without-words,&quot; but rather an&#xA;image that, when supplemented by an appropriate verbal explanation,&#xA;is quite convincing.&#xA;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Corollary&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia image&lt;/a&gt; doesn't quite do it for me.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-12-29T10:36:16.480" Title="Iconic image to explain the fundamental theorem of calculus?" Tags="|calculus|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14654" PostTypeId="2" ParentId="14653" CreationDate="2018-10-12T00:34:51.183" Score="9" Body="&lt;p&gt;I'd make this a comment, but I'm not certain I know how. Maybe this is the picture to lead students to &lt;strong&gt;see&lt;/strong&gt; FTC I (I think of your theorem as FTC II, but I know books vary on this terminology)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YfVKz.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YfVKz.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully this leads students to see &lt;span class=&quot;math-container&quot;&gt;$\frac{dA}{dx} = f(x)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$A(x) = \int_a^x f(t) \, dt$&lt;/span&gt;. I'm not sure what the picture is for FTC II.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: I thought about FTC II a bit and here is my best stab at it currently:&#xA;If &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is an antiderivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\frac{dF}{dx} = f$&lt;/span&gt;. Apply the Mean Value Theorem on &lt;span class=&quot;math-container&quot;&gt;$[x_{i-1}, x_i]$&lt;/span&gt; to select &lt;span class=&quot;math-container&quot;&gt;$x_i^* \in [x_{i-1}, x_i]$&lt;/span&gt; for which &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{F(x_i)-F(x_{i-1})}{x_i-x_{i-1}} = f(x_i^*) $$&lt;/span&gt;&#xA;Then if we set &lt;span class=&quot;math-container&quot;&gt;$x_i-x_{i-1} = \Delta x$&lt;/span&gt; we find&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ F(x_i)-F(x_{i-1}) = f(x_i^*) \Delta x  $$&lt;/span&gt; &#xA;Therefore, the signed-area &lt;span class=&quot;math-container&quot;&gt;$A_i$&lt;/span&gt; bounded by &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$[x_{i-1},x_i]$&lt;/span&gt; is well approximated by:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ A_i = f(x_i^*) \Delta x =  F(x_i)-F(x_{i-1}) $$&lt;/span&gt;&#xA;Notice the signed area under &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt; we defined to be &lt;span class=&quot;math-container&quot;&gt;$\int_a^b f(x)dx$&lt;/span&gt; and it is well approximated by &lt;span class=&quot;math-container&quot;&gt;$A_1+A_2+ \cdots + A_n$&lt;/span&gt;. Thus,&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align} &#xA;\int_a^b f(x)dx &amp;amp;= A_1+A_2+ \cdots + A_n \\&#xA;&amp;amp;= F(x_1)-F(x_o)+F(x_2)-F(x_1)+ \cdots + F(x_{n})-F(x_{n-1})&#xA;\end{align}&lt;/span&gt;&#xA;But, we set-up the partition of &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x_o=a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_n=b$&lt;/span&gt; hence,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int_a^b f(x)dx = F(x_n)-F(x_o) = F(b)-F(a). $$&lt;/span&gt;&#xA;My picture for the argument above at the moment is:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/jOihK.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jOihK.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess a proof by picture alone must somehow picture both &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; and somehow connect the area under &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; to the secant line for &lt;span class=&quot;math-container&quot;&gt;$y=F(x)$&lt;/span&gt;. I'm not sure I have a good generic picture for that. Certainly I can do it for simple graphs such as &lt;span class=&quot;math-container&quot;&gt;$y = constant$&lt;/span&gt; or even &lt;span class=&quot;math-container&quot;&gt;$y = x$&lt;/span&gt;, but I'm not sure that is what we're after here. See pages 231-232 of &lt;a href=&quot;http://www.supermath.info/OldschoolCalculusII.pdf&quot; rel=&quot;noreferrer&quot;&gt;my calculus notes&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="8571" LastEditDate="2018-10-13T16:08:10.210" LastActivityDate="2018-10-13T16:08:10.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14656" PostTypeId="1" CreationDate="2018-10-12T16:36:45.310" Score="8" ViewCount="193" Body="&lt;p&gt;My institution is now in the process of &quot;standardizing&quot; our calculus classes. One issue we have is the variation among instructors in grading problems. I am interested if there are ways to objectively study such variation and so my question is whether anyone knows of a standard set of questions where there is an agreed upon rubric as to how to grade the problem and also how to deduct points for cases that don't fall nicely into the rubric. I know these types of rubrics do exist for AP Calculus tests for example, so I was wondering if anyone in math ed has ever constructed a bank of such questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So as a brief example: consider the problem of finding the derivative of ln(sin(x) + tan(x)) that is worth, say 10 points. The rubric would have a breakdown of what points are awarded to each part (maybe +4 points for knowing to use chain rule, +2 for the derivative of each trig function, etc) but would also consider all the possible common student errors (such as ln(x+y) = ln(x) + ln(y)) and how to deduct points for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure that such a set of problems whose grading had been standardized would actually answer my question, but I am thinking it might be a first step in understanding how much variation we have among the instructors and how to interpret it.&lt;/p&gt;&#xA;" OwnerUserId="1877" LastEditorUserId="64" LastEditDate="2018-10-14T16:17:54.097" LastActivityDate="2018-10-14T16:17:54.097" Title="Are there standard questions for testing how an instructor grades calculus?" Tags="|calculus|assessment|testing|" AnswerCount="0" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="14658" PostTypeId="2" ParentId="14636" CreationDate="2018-10-13T00:04:49.983" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;At first, I thought that my exam is too hard. However, I usually, definitely, give problems that need the understanding of materials I give in the class to solve, but add some &quot;spices&quot; to them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Part of the problem is the &quot;extra spice&quot; you're giving them during the test. Why are you surprising them with extra spice &lt;em&gt;for the first time&lt;/em&gt; during the test? That's literally the worst possible time to introduce new material!&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I am teaching a math class where the students, most of them, tell me that they can understand the materials given by me during the course. I test them during the course too and they seem to get it. However, when it is time to face quiz or exam problems, they don't know what to do.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This suggests a standard blocking vs interleaving illusion. Many textbooks are organized by concept and procedures rather than by difficulty, so students:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Get &lt;strong&gt;all&lt;/strong&gt; the fraction addition and subtraction questions right in the adding and subtraction fractions section of the textbook.&lt;/li&gt;&#xA;&lt;li&gt;Get &lt;strong&gt;all&lt;/strong&gt; the  multiplying fractions questions right in the multiplying fractions section of the textbook,&lt;/li&gt;&#xA;&lt;li&gt;Then they get &lt;strong&gt;all&lt;/strong&gt; the dividing fractions questions right in the dividing fractions section of the textbook,&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Students and teachers conclude &quot;Holy cow, look at all these right answers, this is amazing, we're &lt;em&gt;really&lt;/em&gt; getting it!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then students arrive at the test and face:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;'There are 5 and 1/4 equal pieces of cake and 6 people to share them. How much should each person get?'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And various students respond:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5\frac{1}{4}\times6=31\frac{1}{2}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5\frac{1}{4}\times6=\frac{21}{4}\times6=\frac{126}{4}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5\frac{1}{4}+6=11\frac{1}{4}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5\frac{1}{4}-6=?$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$6\div5\frac{1}{4} = 6 \times 5\frac{4}{1}$&lt;/span&gt; ... &quot;When we divide, the big number goes first, then you flip and multiply...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;i.e. The students used section titles of textbook to select the operation correctly instead of the meaning of the word problems. This is an illusion of understanding. When those section titles are not there on the exam, those non-meaningful habits continue and students are forced to guess somewhat randomly, which, of course, is terrible in a math class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try interleaving! It drastically reduces speed and accuracy &lt;em&gt;in the short run&lt;/em&gt; but forces meaningful thinking and retention in the longer run, and effect sizes are huge. [Google it.]&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-10-13T00:04:49.983" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14659" PostTypeId="2" ParentId="14593" CreationDate="2018-10-13T00:27:51.757" Score="8" Body="&lt;p&gt;You may be overestimating how much students learn from symbolic rules. Most textbooks already teach symbolic rules first then put word problems last. This is backasswards.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most students succeed at exercises such as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Compute the exact missing value.  &lt;span class=&quot;math-container&quot;&gt;$\frac{7}{9}\times39=\_\_\_$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But, if you do not show them that prior question and instead begin with this, most students bomb horribly:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Without calculating, determine if the missing value is greater than 40, less than 40, equal to 40, or if it is not possible to tell without calculating exactly. Explain your answer, then rate your confidence as certain, fairly high, medium, or low.  &lt;span class=&quot;math-container&quot;&gt;$\frac{7}{9}\times39.884536=\_\_\_$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Not only do many students not know the meaning of multiplication with fractions, they will &lt;em&gt;fight you&lt;/em&gt; when you try to tell them they should be able to do the 2nd exercise trivially. They will &lt;em&gt;fight you&lt;/em&gt; when you tell them they don't understand what it means but need to. They will &lt;em&gt;fight you&lt;/em&gt; if you do anything other than tell them how to compute the exact missing value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parents will generally take the student's side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[If you don't believe me, start asking your students my 2nd question and ask gentle probing questions to elicit their thinking aloud.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because the first type of question - which can be answered by symbolic rules - is always on the test. The second type of question - which requires meaning - is almost never on the test. Testing symbolic rules - i.e. rote learning - has already convinced them that math is just rote rules &lt;em&gt;and nothing more&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These kinds of issues are especially prevalent with fractions because many students &lt;em&gt;never&lt;/em&gt; learn that a fraction is a number, so it never occurs to them to compare &lt;span class=&quot;math-container&quot;&gt;$\frac{7}{9}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; whole.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, no, do not teach math as a bunch of symbolic rules. Make math meaningful.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2018-10-23T20:16:16.653" LastActivityDate="2018-10-23T20:16:16.653" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14660" PostTypeId="2" ParentId="14653" CreationDate="2018-10-13T16:49:37.883" Score="1" Body="&lt;p&gt;Try the &lt;a href=&quot;https://www.youtube.com/playlist?list=PLmm7E-Sos_YWjcjWIBDdtck4OnXeSNs4R&quot; rel=&quot;nofollow noreferrer&quot;&gt;Derek Owens YouTube playlist for the FTC&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He starts with a few worked examples, then goes into the intuition for why the FTC is true. As far as visuals go, this is quite good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is &lt;em&gt;not&lt;/em&gt; a single, iconic image, but it is still very good, nonetheless, at getting across the basic idea.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-10-13T16:49:37.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14661" PostTypeId="2" ParentId="14653" CreationDate="2018-10-14T00:34:52.663" Score="1" Body="&lt;p&gt;This may not be the iconic image for the introduction of the fundamental theorem of calculus, but I think it is the iconic image of Calculus Reform.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem associated to the image is to use FTC to determine the sign of &lt;span class=&quot;math-container&quot;&gt;$\displaystyle{\int_0^2 f''(x)\, dx}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\displaystyle{\int_0^2 f'(x)\, dx}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\displaystyle{\int_0^2 f&#xA;(x)\, dx}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RT0pZ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RT0pZ.png&quot; alt=&quot;NewCenturyIconiImage&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2018-10-14T00:34:52.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14662" PostTypeId="1" CreationDate="2018-10-14T10:57:57.133" Score="5" ViewCount="197" Body="&lt;p&gt;I'm a private math tutor. I'm fairly new at this, and this semester is the first time I've been tutoring for a statistics class at a community college. I enjoy experimenting and learning about ways to explain math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to get some feedback on explaining sampling and distributions to students who struggle, especially those who don't have a lot of cognitive power.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any math topic, to help them understand math concepts I relate those to everyday experiences. Here is my idea for sampling and distributions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I say imagine that you are a hairdresser (or substitute other profession they can relate to). I say, imagine it's the end of your work day and you're reflecting back on what happened. You think about the number of clients you had (an example of a discrete random variable), the amount of time spent cutting (a continuous random variable), or anything you like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I say, imagine explaining your workday to a friend. I get them to imagine this hypothetical day and the language they would use. For instance, would they say to their friend, it was an unusual day with a lot of clients? Or a typical day with an average number, or typical number of minutes spent cutting? What are some ways in which days can be unusual?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully they start to form this mental picture of both the experiences of a single day (a sample) and how days vary over time, including what patterns are more frequent (a distribution).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We talk about some of the factors that might affect the distribution of clients. Are holidays very busy, while Fridays slow? We think about the population of all potential clients in this city and parameters of this population, and how they would affect what the hairdresser sees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't done this yet, but even the Central Limit Theorem could be explained by asking them to differentiate between day to day variation, and the variation of weekly averages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, just wondering if the statisticians here have similar or better ideas.&lt;/p&gt;&#xA;" OwnerUserId="10466" LastActivityDate="2018-10-15T15:02:57.670" Title="What are some suggestions fo teaching statistics concepts to struggling college students?" Tags="|mathematical-pedagogy|statistics|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14663" PostTypeId="1" CreationDate="2018-10-14T20:56:07.653" Score="6" ViewCount="145" Body="&lt;p&gt;For concreteness lets keep our discussion to calculus courses where there is a balance of proof and computations (computing limits but also doing epsilon-delta proofs)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can understand that in more computational based courses one would like to use flow diagrams such as this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-7/a/limit-strategies-flow-chart&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-7/a/limit-strategies-flow-chart&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where a student just wants to obtain a bag of tools. In more proof based courses the argument against using them is that it encourages the bad habit of rote memorization and so when a slightly different situation arises, the student will be stuck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In discussions with other instructors, they told me there is research showing that using such summarizing diagrams is harmful for learning. So I would appreciate if you can also include some references in your answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some arguments I can think of for either side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Good for learning&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Let me first take the perspective of a good student who actively does problems. A good student could use such a diagram to obtain a bird eye view of the techniques. Then as new techniques and refinements arrive, he/she will update their mental model.&lt;/li&gt;&#xA;&lt;li&gt;These diagrams are not good or bad per se, but only as good as the instructor using them. So ,for example, the instructor can accompany these diagrams with homeworks that refine them and challenges the students to modify the existing techniques.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bad for learning&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;From the POV of a bad student this diagram is a god send. So it might discourage them from doing problem solving to help them figure out the skeleton of possible techniques and their refinements. They will rely on these diagrams and give up on a problem that doesn't quite fit them. In other words, they will take the summarizing strategies as absolutes. &lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students will try to save time and focus on learning the strategies and not the material. Therefore, we are depriving them of learning and discovering the techniques for themselves. So they will soon forget that material because they were not challenged with it and didn't form those precious neural connections. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In more proof based courses, the emphasis should be on deriving and thinking, so any such diagram will deviate them from that. The instructor should guide the students to form their own mental models, rather than imposing external ones. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;: I now realize that this goes under the debate of IBL (inquiry based learning) versus procedural based learning. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Q2: Are there any ways we can use strategy diagrams in an IBL setting? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In IBL we want the students to discover many of the techniques themselves. So we avoid giving these diagrams at the start, that way students will be forced to do a lot of problems themselves.  &lt;/p&gt;&#xA;" OwnerUserId="8714" LastEditorUserId="8714" LastEditDate="2018-10-16T15:08:44.040" LastActivityDate="2018-10-16T21:32:26.420" Title="Flow diagrams and summarizing strategies in proof-computation courses: good or bad for learning? Unsuitable for Inquiry-based learning?" Tags="|mathematical-pedagogy|reference-request|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14664" PostTypeId="2" ParentId="14662" CreationDate="2018-10-14T22:56:41.573" Score="3" Body="&lt;p&gt;Another way to make this idea concrete is to physically enact it.  It requires some additional preparation, but you could do something like this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Print out many thousands of slips of paper, each with a single number printed on it.  These numbers can be generated randomly using a spreadsheet (e.g. Excel or Google Sheets), and if you are clever you can get the numbers to be drawn from a normal distribution with a mean and standard deviation of your choice.&lt;/li&gt;&#xA;&lt;li&gt;Put all of the slips of paper in a box and ask the student:  How could you figure out the mean of this data?  Obviously actually adding them all up and dividing by the number of slips is impractical -- you need some way to &lt;em&gt;estimate&lt;/em&gt; it, since finding it exactly is impossible.&lt;/li&gt;&#xA;&lt;li&gt;So you take a sample of, say, 40 slips and find the mean of the sample.  That gives you an estimate -- but how reliable is it?&lt;/li&gt;&#xA;&lt;li&gt;If you take a different sample, you'd get a different sample mean.  What if you took 50 different samples?  You'd get 50 different sample means.  How could you use those to estimate the population mean?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This kind of activity really clarifies the difference between the population mean (which is unknown, and in a sense unknowable) and a sample mean.  Follow this out and you get to the idea of the distribution of sample means, and how it relates to the (unknown) background distribution, and then to the CLT.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-10-14T22:56:41.573" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14666" PostTypeId="2" ParentId="14653" CreationDate="2018-10-15T11:54:27.197" Score="3" Body="&lt;p&gt;Here’s a picture that illustrates the comment of littleO (with enough time staring at it). It uses the graph of &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, while the pictures in the answer of James S. Cook use the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/d2TQT.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/d2TQT.jpg&quot; alt=&quot;FTCII&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I’m using FTC II in the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_{x=a}^b \frac{dy}{dx}dx =y|_{x=b} - y|_{x=a}&#xA;$$&lt;/span&gt;&#xA;with &lt;span class=&quot;math-container&quot;&gt;$y=F(x)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}=f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(It becomes even more obvious if one writes &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}dx$&lt;/span&gt; in the integral. This is mathematically correct, but it seems that many people feel uncomfortable doing this.)&lt;/p&gt;&#xA;" OwnerUserId="590" LastEditorUserId="590" LastEditDate="2021-12-29T10:36:16.480" LastActivityDate="2021-12-29T10:36:16.480" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14667" PostTypeId="2" ParentId="14662" CreationDate="2018-10-15T15:02:57.670" Score="1" Body="&lt;p&gt;I agree with @mweiss that you need to physically enact this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Luckily, your students have random number generators on them at all times. Statistics classes these days are set up with some kind of technology, either a TI-83 or Excel or StatCrunch or something similar. You can do the following experiment:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain that &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is the random variable of random numbers generated&#xA;by (on the TI-83 or 84) &lt;span class=&quot;math-container&quot;&gt;$MATH \to PRB \to rand$&lt;/span&gt;. If you have covered&#xA;uniform distributions in any detail, great -- you can ask them to&#xA;remember that &lt;span class=&quot;math-container&quot;&gt;$X \sim U(0,1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mu_X = 0.5$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sigma_X =&#xA;   \sqrt{\frac{1}{12}} \approx 0.289$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sample from &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and get a picture of your data. On the calculator&#xA;this is easy; press enter like 30 times. The data will be all over&#xA;the place. Each time you do this, and this is crucial, make sure the&#xA;student knows that these are values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Decide on a value of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;; 10 is fine. Explain that after you have&#xA;chosen &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\overline{X}$&lt;/span&gt; is the random variable of means of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s.&#xA;If you have covered the Central Limit Theorem, you can write down the&#xA;distribution &lt;strong&gt;but don't do that&lt;/strong&gt;!!! If you do that, you will rob&#xA;them of the surprise.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sample from &lt;span class=&quot;math-container&quot;&gt;$\overline{X}$&lt;/span&gt; and get a picture of your data. On the TI&#xA;calculator you can actually type&#xA;&lt;span class=&quot;math-container&quot;&gt;$(rand+rand+rand+rand+rand+rand+rand+rand+rand+rand)/10$&lt;/span&gt; so that you&#xA;can just press enter over and over. If you have a big class of&#xA;students, I would instead recommend that you have each student actually&#xA;get 10 values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and average them to get their own &lt;span class=&quot;math-container&quot;&gt;$\overline{x}$&lt;/span&gt; and &#xA;write it on the board. Either way, the point is to &#xA;make sure the student realizes these new numbers are values of &lt;span class=&quot;math-container&quot;&gt;$\overline{x}$&lt;/span&gt; and&#xA;not &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. The point here is to tell the difference between those&#xA;things. Of course the data will not be all over the place.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Then you can ask: Cool; this looks like a normal distribution. I wonder which normal distribution it is?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that by making the whole thing a concrete exercise, you can get them to tell the difference between the letters &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\overline{X}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\overline{x}$&lt;/span&gt;, which is the real hurdle of this topic.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-10-15T15:02:57.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14669" PostTypeId="1" CreationDate="2018-10-15T18:01:38.707" Score="4" ViewCount="133" Body="&lt;p&gt;In Germany, one usual example to start teaching about integrals is to look at a simple (piecewise constant or with constant slope) functions that make up a water flow vs. time diagram and ask about the total flow during a given time span. This leads to the idea that the area under the curve is of interest and represents exactly the  desired quantity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking about a discretisation of this example because my dog requires pain medication and I plotted a daily dose vs. day graph. This is basically the same as the water flow diagram, but already with discrete steps, probably leading to the &quot;rectangular stripes&quot; method. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any research or experience about this approach? &lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2018-10-15T18:59:38.193" Title="Using discrete examples in the beginning of integration" Tags="|examples|integration|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14670" PostTypeId="2" ParentId="14653" CreationDate="2018-10-15T18:10:24.097" Score="16" Body="&lt;p&gt;This is about the Riemann sums but the logic is there. &lt;a href=&quot;https://i.stack.imgur.com/VzcSp.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VzcSp.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10168" LastActivityDate="2018-10-15T18:10:24.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14671" PostTypeId="2" ParentId="14669" CreationDate="2018-10-15T18:59:38.193" Score="2" Body="&lt;p&gt;Some of my students have a real a-ha moment when they first graph the integral of a constant function (using the area-so-far concept), seeing a new line of non-zero slope emerge. Then having them graph piecewise constant functions reinforces the &lt;em&gt;height&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\rightarrow$&lt;/span&gt; &lt;em&gt;slope&lt;/em&gt; connection they will depend on when they learn the short-cut rules for antiderivatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No research to speak of here, but I think beginning with something simple and concrete like this is an excellent idea.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-10-15T18:59:38.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14672" PostTypeId="1" CreationDate="2018-10-16T08:31:14.290" Score="9" ViewCount="1522" Body="&lt;p&gt;I'm a fairly new private math tutor, and I'm good at math (I have a BS from Caltech with lots of graduate level math), but becoming good at &lt;em&gt;teaching&lt;/em&gt; math is something else, which I strive to improve at every day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've starting working with my youngest student so far, a 6th grader. I'm dismayed by the problems in the way her class is taught. I'd like to know if this really is a problem or if I'm missing something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I'm new to tutoring, I've been teaching non-math topics to adults for years. From that experience, I believe people learn best by starting in a secure place and moving by small steps. Of course they need to enter into new and potentially confusing situations, but the teacher can help them stay oriented to something solid along the way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first session with my 6th grader, she was confused by word problems that tested her ability to know whether to use LCM or GCD. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word problems seemed designed to be confusing. They read as if some teacher somewhere knew that kids get these two concepts confused in real-life models or word problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I would do about this is introduce them as separate topics, probably spaced far in time. Let them develop a secure &lt;em&gt;understanding&lt;/em&gt; of one &lt;em&gt;before&lt;/em&gt; introducing the other. Only then mix them up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this textbook introduced them at the &lt;em&gt;same&lt;/em&gt; time and then gave them challenging problems in which it's hard to tell them apart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That seems like &lt;em&gt;exactly&lt;/em&gt; the wrong approach. I was able to help my student, however, by teaching her to differentiate between problems that involve &quot;cutting up things&quot; (GCD) and &quot;extending things&quot; or &quot;laying things end to end&quot; (LCM). Fortunately this was enough to solve her problem, in part because the language used by the word problems was consistent, so she wasn't being tricked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next two weeks, she was taught about multiplying and dividing by fractions and given word problems on those. The word problems asked her questions like &quot;if I divide 2/3 of an acre of land into plots each 1/6 of an acre, how many plots do I get?&quot; So that's division by a fraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the problems also asked her to &lt;em&gt;multiply&lt;/em&gt; by fractions, such as starting by referring to &quot;1/6 of a garden&quot; (of unspecified size) and then later mentioning the garden was 90 square yards, and asking what &quot;1/6&quot; of that is. So she had to &lt;em&gt;multiply&lt;/em&gt; 1/6 by 90. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having just been introduced to dividing by fractions, she wanted to divide 90 by 1/6.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I decided to teach her the difference between counting the number of pieces or sections, and finding the size of each piece. I ran through a bunch of simple examples, then started to model why division or multiplication would be appropriate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the next problem referred to a class of students, and asked &quot;what is the number of students in the class?&quot; Since I had just told her about counting the &quot;number of&quot; pieces, she naturally thought this was asking about counting sections or pieces. But it was really about the size of the whole class! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is bad question-writing. If the sadists who designed these word problems were determined to confuse students, the least they could do is be consistent in their language. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This session was 90 minutes, because she needed so much comprehension built up from scratch, that's as far as we got. I skimmed the rest and there was no consistency. Each problem was a variation, without simple patterns that she could &quot;hook&quot; into to decide whether to add, subtract, multiply or divide.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can only guess that the textbook writers think it's a good idea to help students differentiate concepts by mixing them together and making them as hard to discern as possible, but as I previously stated, I think that's bad teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at the biggest picture, most students don't grow up into jobs that require much math, so teaching hard math (and abstract math like algebra) is a kind of &quot;rite of passage&quot; that in my opinion, despite being employed as a math tutor, serves little purpose. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But given that our schools teach math (for possibly justifiable reasons), shouldn't they at least refrain from making it more confusing than necessary?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm fairly new to this, so maybe there's something I'm missing.&lt;/p&gt;&#xA;" OwnerUserId="10466" LastActivityDate="2022-08-21T00:40:24.130" Title="Don't these word problems seem designed to be confusing?" Tags="|fractions|word-problems|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14673" PostTypeId="2" ParentId="14672" CreationDate="2018-10-16T11:13:59.830" Score="2" Body="&lt;p&gt;Even though your question seems like a rant about textbooks, I'm assuming that you are looking for advice on how to deal with this situation. So let me give my opinion on the topic:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From what you are telling, it sounds like this textbook wants to be very clever and do two things at once. When teaching these things like, for example, GCD and LCM, there are two things to do:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Understand how to compute them.&lt;/li&gt;&#xA;&lt;li&gt;Understand when to use which one.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you make such challenging tasks, you can do both with the same exercise. The students first have to figure out which one to use, and then they have to do a (likely difficult) computation to get the result.&lt;br&gt;&#xA;I think your approach to split both points up to teach them to a struggling student is good. I would suggest to take it even further and, after the student understood the algorithms and concepts, start really slow with the second point, so instead of taking &lt;span class=&quot;math-container&quot;&gt;$1/6$&lt;/span&gt; of a &lt;span class=&quot;math-container&quot;&gt;$90$&lt;/span&gt; square yards, give her half a cake (real cake or figurative cake, you decide^^) and then let her eat half of it, so that she is left with &lt;span class=&quot;math-container&quot;&gt;$1/2*1/2 = 1/4$&lt;/span&gt; of a cake. Make her understand the differences and develop a feeling for the concepts with such trivial examples, and teach her how to retract to such cases, e.g. &quot;replace every integer by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and every fraction by &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; to figure out if you have to multiply or divide. Once you know that, do it with the actual numbers given in the problem.&quot;&lt;br&gt;&#xA;Furthermore, it can also help to teach students how to check their solutions and develop an intuition for such things. Many students, especially struggling ones, are relieved when they finally have a solution to write down, so they don't even realize that &quot;&lt;span class=&quot;math-container&quot;&gt;$1/6$&lt;/span&gt; of the garden of size &lt;span class=&quot;math-container&quot;&gt;$90$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$720$&lt;/span&gt;&quot; sounds a little strange.&lt;br&gt;&#xA;There are many different ways to develop that intuition, you might have her draw images, use the number line, etc. Just make sure that it is not you telling her &quot;this is wrong, because...&quot; but, whenever possible, ask her to verify her result. Of course also ask about verification of true results, don't let her associate verification with wrong work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;TL;DR version: Teach your student to verify her own results and help her to develop an intuition. Whenever possible, only give hints and let her do all the work. Do not teach algorithms blindly, always aim for an understanding of the underlaying ideas and concepts.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2018-10-16T11:13:59.830" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14674" PostTypeId="2" ParentId="14672" CreationDate="2018-10-16T13:33:59.250" Score="25" Body="&lt;p&gt;I think there's a countervailing issue that the book you're describing is trying to deal with.  I teach college students, so I don't know what the particular approach it's taking is age appropriate for a 6th grader, but it is trying to solve a real problem.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What I would do about this is introduce them as separate topics, probably spaced far in time. Let them develop a secure understanding of one before introducing the other. Only then mix them up.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The problem with this approach - and what I assume is exactly what the book is trying to avoid - is that you &lt;em&gt;can't&lt;/em&gt; develop an understanding of (for instance) LCM in isolation.  You can do a bunch of calculations, but if you try to ask problems which get at understanding LCM, students will just autopilot through them, taking the LCM of whichever two numbers they find.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But the problems also asked her to multiply by fractions, such as starting by referring to &quot;1/6 of a garden&quot; (of unspecified size) and then later mentioning the garden was 90 square yards, and asking what &quot;1/6&quot; of that is. So she had to multiply 1/6 by 90.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is pretty clearly a pre-algebra problem.  (Which I'm pretty sure &lt;em&gt;is&lt;/em&gt; age appropriate; I think it's pretty common to start algebra in 7th grade, which makes 6th grade a good time to lay groundwork like this.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your student wants to just take numbers in the problem and combine them while ignoring the words, which is a pretty normal thing for students to want to do, but not the point of a word problem.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I think this is bad question-writing. If the sadists who designed these word problems were determined to confuse students, the least they could do is be consistent in their language.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If they used &quot;consistent&quot; language, students could shortcut through the problems by using those linguistic clues, rather than by understanding the situation.  It sounds like that's exactly what your student is trying to do, so in that sense word problems of &lt;em&gt;some&lt;/em&gt; kind are probably exactly on target.  (Obviously I can't assess whether these specific ones are appropriate, and I imagine that's a hard thing to do, since it depends on how good the students' language skills are as well as their calculation skills.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I'd suggest is that it sounds like you're thinking of, for instance, LCM and GCD as the main topic here.  I'm not sure the textbook agrees - it sounds to me like her textbook thinks the topic is word problems (that is, the process of interpreting situations mathematically), and things like LCM/GCD are incidental bits of content that provide topics to practice word problems on.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-10-16T13:33:59.250" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="14675" PostTypeId="2" ParentId="12451" CreationDate="2018-10-16T21:30:54.143" Score="1" Body="&lt;p&gt;As previous answers have explained, geometric construction with just those two tools has a long and rich history. There is much to learn from playing with ideas this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, personally, geometric construction is much more pleasurable, and more informative, when done with electronic tools. I love &lt;a href=&quot;https://sciencevsmagic.net/geo/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this site&lt;/a&gt; (which gamifies geometric construction) and I sometimes enjoy &lt;a href=&quot;https://www.euclidea.xyz/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this site&lt;/a&gt;. (I get frustrated when I get stuck, because you have to finish each step to go to the next.) Once I've gotten engaged in a problem, I've also used geogebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always felt clumsy with a compass. I would press down too hard, and it would change size. Now I don't have to worry about that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have learned so much in the past few years by playing with geometric construction this way.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-10-16T21:30:54.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14676" PostTypeId="2" ParentId="14672" CreationDate="2018-10-17T05:30:05.660" Score="12" Body="&lt;p&gt;I feel like I always post the same thing in these threads, but this again sounds like an issue of blocking vs interleaving. In this case, the textbook may have started interleaving different problems a little bit too early, but generally it should be introduced earlier than feels intuitively right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's because the algorithms for &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{6}\times 90$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$90\div\frac{1}{6}$&lt;/span&gt; are not that tough to execute, but &lt;em&gt;knowing what they mean and when to do them is the hard part.&lt;/em&gt; [If you don't believe me, just go to a community college and ask a bunch of students if the prior 2 expressions are greater than 90 or less than 90 or if it's not possible to tell and how confident they are.] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a partial solution to your current textbook problem. Tell her:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To practice algorithms on her own or with another book that has an answer key.&lt;/li&gt;&#xA;&lt;li&gt;Use your time with her to practice sketching and/or parsing the word problems. &quot;Instead of calculating answers, today we're going to just read the word problems and you're going to tell me if LCM is relevant to each word problem or not and how you know.&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This way, she does the easy stuff on her own and the much harder, more abstract stuff with you.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-10-17T05:30:05.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14677" PostTypeId="1" AcceptedAnswerId="14678" CreationDate="2018-10-17T12:56:15.383" Score="2" ViewCount="390" Body="&lt;p&gt;I had a question from a student which I'm unable to answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We were practicing the rule &lt;span class=&quot;math-container&quot;&gt;$\int \frac{f'(x)}{f(x)} \, dx=\ln(f(x))$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student noticed that if applied naively it gives the following two results &lt;span class=&quot;math-container&quot;&gt;$$\int \frac{2}{2x} \, dx = \ln(2x) +c$$&lt;/span&gt;&#xA;and &lt;span class=&quot;math-container&quot;&gt;$$\int \frac{2}{2x} \, dx = \int \frac{1}{x} \, dx = \ln(x) +c$$&lt;/span&gt;&#xA;which appear contradictory.&lt;/p&gt;&#xA;" OwnerUserId="5237" LastEditorUserId="5237" LastEditDate="2018-10-30T18:16:20.287" LastActivityDate="2018-10-30T18:16:20.287" Title="Integrating derivatives over functions problem" Tags="|secondary-education|calculus|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14678" PostTypeId="2" ParentId="14677" CreationDate="2018-10-17T13:39:29.233" Score="10" Body="&lt;p&gt;As guest mentioned in a comment, the two expressions are equivalent. Suppose &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 0$&lt;/span&gt; so the logarithm is defined without any problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For logarithms, &lt;span class=&quot;math-container&quot;&gt;$\log(2x) = \log (2) + \log (x)$&lt;/span&gt;. Consider graphing &lt;span class=&quot;math-container&quot;&gt;$x \mapsto \log (x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x \mapsto \log (2x)$&lt;/span&gt;, and maybe their difference, to illustrate. A good window would be &lt;span class=&quot;math-container&quot;&gt;$x=1\ldots 10$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=0\ldots 5$&lt;/span&gt;, as recommended by user52817.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, the first integral equals &lt;span class=&quot;math-container&quot;&gt;$\log (x) + \log (2)  + C_1$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$C_1$&lt;/span&gt; is the constant of integration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second integral equals &lt;span class=&quot;math-container&quot;&gt;$\log (x) + C_2$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$C_2$&lt;/span&gt; is the constant of integration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see that these are the same thing, choose &lt;span class=&quot;math-container&quot;&gt;$C_1 = C_2 - \log(2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, as &lt;span class=&quot;math-container&quot;&gt;$C_1$&lt;/span&gt; goes through all the real numbers, and as &lt;span class=&quot;math-container&quot;&gt;$C_2$&lt;/span&gt; does so, you get precisely the same collections of functions from both integrals.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This would be a nice moment to discuss the meaning of the constant of integration, and of the indefinite integral in general. You might want to see this questions and answers for motivation: &lt;a href=&quot;https://matheducators.stackexchange.com/q/2338/2083&quot;&gt;Should we avoid indefinite integrals?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This might also be useful: &lt;a href=&quot;https://matheducators.stackexchange.com/q/5901/2083&quot;&gt;Explaining the symbols in definite and indefinite integrals&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2018-10-18T06:09:19.917" LastActivityDate="2018-10-18T06:09:19.917" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14679" PostTypeId="2" ParentId="1348" CreationDate="2018-10-18T16:39:51.640" Score="5" Body="&lt;p&gt;Not sure if you're still teaching, but the really right answer to this question in terms of fraction &lt;em&gt;content&lt;/em&gt; is to read &lt;a href=&quot;https://math.berkeley.edu/~wu/&quot; rel=&quot;noreferrer&quot;&gt;the work of Hung-Hsi Wu&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a brief summary of some practical implications.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Fractions are not food, nor are they shapes that you color or shade. Fractions are &lt;em&gt;numbers&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;To teach the fact that fractions are numbers, use a number line from 0 to 3 with the whole numbers marked and perhaps thirds marked as well. Students can then practice counting by whole numbers (0,1,2,3), thirds (1 third, 2 thirds, 3 thirds, 4 thirds, 5 thirds), and mixed numbers (2 thirds, 1, 1 and 1 third, 1 and 2 thirds, etc.). [This may seem trivial, but it is an epiphany for many kids.] Have them notice that &lt;span class=&quot;math-container&quot;&gt;$\frac{5}{3}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1\frac{2}{3}$&lt;/span&gt; share a point on the number line, which is why they are equal.&lt;/li&gt;&#xA;&lt;li&gt;If you restrict yourself to number lines from 0 to 1 or shapes where the implied whole numbers range from 0 to 1, then students can get 100% correct answer while completely ignoring what 0 and 1 are in that context, implying that fractions are ~meaningless in their mind. That's why the number line must go from 0 to 3.&lt;/li&gt;&#xA;&lt;li&gt;Explain that mathematicians need fractions to name quantities between consecutive whole numbers.&lt;/li&gt;&#xA;&lt;li&gt;Continue to use the number line to show equivalent fractions, fraction and whole number comparisons, addition, subtraction, multiplication, division, etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-10-18T16:39:51.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14680" PostTypeId="2" ParentId="14677" CreationDate="2018-10-18T21:04:14.417" Score="5" Body="&lt;p&gt;Show your students this integral as well (maybe later): Find &lt;span class=&quot;math-container&quot;&gt;$\int \sec^2x\tan x\,dx$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Method 1: &lt;span class=&quot;math-container&quot;&gt;$u = \sec x, \,du = \sec x \tan x \,dx$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int \sec^2x\tan x\,dx = \int u\,du = \frac{u^2}{2} + C = \frac{\sec^2 x}{2} + C$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Method 2: &lt;span class=&quot;math-container&quot;&gt;$u = \tan x, \,du = \sec^2 x \,dx$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int \sec^2x\tan x\,dx = \int u\,du = \frac{u^2}{2} + C = \frac{\tan^2 x}{2} + C$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How can this be?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, those two answers are the same up to a constant. &lt;span class=&quot;math-container&quot;&gt;$\sec^2 x = \tan^2 x + 1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-10-18T21:04:14.417" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14682" PostTypeId="2" ParentId="12080" CreationDate="2018-10-22T05:27:18.393" Score="3" Body="&lt;p&gt;&quot;Order of Operations&quot; as commonly taught and tested is just a mess.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a picture from a real standardized test in New York. It was quoted in one of &lt;a href=&quot;https://math.berkeley.edu/~wu/order5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hung-Hsi Wu's essays&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ejRIS.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ejRIS.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The order of operations as generally taught says you &lt;em&gt;must&lt;/em&gt; evaluate &lt;span class=&quot;math-container&quot;&gt;$4^2$&lt;/span&gt; before evaluating &lt;span class=&quot;math-container&quot;&gt;$\frac{6}{2}$&lt;/span&gt;. Huh? Is that how ANYONE who knows mathematics thinks about it? Is the universe going to explode if you divide 6 by 2 first?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the actual question from the test, the next line of work should look like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3-16+3$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if you did the exponent and division simultaneously and the universe did not explode - you rebellious daredevil! -  we can then move on to trying to figure out who the Hell cares if the rest of the evaluation is &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3-16+3=-13+3=-10$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3-16+3=3-13=-10$&lt;/span&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Either addition or subtraction could be the final operation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So... should the real mnemonic be &quot;PEMDAS - EWIDTMAA&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[PEMDAS - Except When It Doesn't Matter At All]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps it's OK to use the mnemonic to teach initial calculations, but the real goal of the PEMDAS mnemonic ought to be to show students what can or should be considered a single entity, whether there are brackets to emphasize it or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, many students consider both of the following to be factored expressions: &lt;span class=&quot;math-container&quot;&gt;$(x-7)(x+1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x-7(x+1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students should be able to look at the second one and be able to consider &lt;span class=&quot;math-container&quot;&gt;$7(x+1)$&lt;/span&gt; &lt;em&gt;as a single number&lt;/em&gt;. This will allow them to see that the second expression does not &quot;end up&quot; with two quantities being multiplied like the first one does, so it is not a factored expression. Students should be able to interpret that second expression as the &lt;em&gt;difference&lt;/em&gt; between &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$7(x+1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of this may seem obvious to those who are steeped in math, but interpreting that second expression as a difference is surprisingly rare. Many students doing well in high school algebra don't see a difference until I have them evaluate the expression for &lt;span class=&quot;math-container&quot;&gt;$x=5$&lt;/span&gt; in: &lt;span class=&quot;math-container&quot;&gt;$x-[7(x+1)]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x-7(x+1)$&lt;/span&gt; then ask them to compare/contrast. Sometimes this turn the light bulb on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Algebra tiles can also help with this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have other or better ways to do it, I am all ears!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Sorry for the somewhat off-topic rant. Returning to the question, I believe that PEMDAS was basically invented to ensure that students learned at least one correct sequence of calculations that would always be correct.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2018-10-23T04:14:40.813" LastActivityDate="2018-10-23T04:14:40.813" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14683" PostTypeId="2" ParentId="2338" CreationDate="2018-10-22T07:17:31.960" Score="4" Body="&lt;p&gt;Teach your students that &lt;span class=&quot;math-container&quot;&gt;$$ \int \sin x \,\mathrm{d}x = -\cos x + k$$&lt;/span&gt; is simply  (very convenient) shorthand for this precise but long-winded statement:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is defined on an interval and has an antiderivative.&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is defined by &lt;span class=&quot;math-container&quot;&gt;$f(x)=\sin x$&lt;/span&gt;, then its antiderivatives are &lt;em&gt;exactly&lt;/em&gt; those functions &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$g(x)=\cos x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The above statement is not &amp;quot;utterly difficult&amp;quot;, is fairly precise, and can be understood by even average high school/introductory calculus students.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Repeat the above statement (in speech, examples, exercises, test questions) as often as is judged necessary.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user9798" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-02-19T06:02:43.617" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14684" PostTypeId="1" CreationDate="2018-10-22T07:33:15.140" Score="2" ViewCount="184" Body="&lt;p&gt;I am teaching on probability. I found a question that seems to be ambiguous as follows.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Four students are randomly chosen from a place. Assuming the birthdays of people are equally likely to occur in any month, find the probability that 4 selected students are not born in the same months.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The answer key of this question is &lt;span class=&quot;math-container&quot;&gt;$$1-\frac{12}{12^4}$$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I agree with this key using negation principle then I think the question must be rewritten as &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find the probability that at least one student of the selected students is not born in the same month.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Because the negation of &quot;all students are born in the same month&quot; is &quot;at least one of the selected students is not born in the same month&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus for me, the original question &quot;find the probability that 4 selected students are not born in the same months&quot; is badly written.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Question&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;What do you think? Is it also ambiguous for you? Is there any better expression for the original question to get rid of its ambiguity?&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Edit&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Other examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;A = 4 balls are red.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A' = at least one ball is not red (rather than 4 balls are not red)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A = 4 cars are sold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A' = at least one car is not sold (rather than 4 cars are not sold)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5696" LastEditorUserId="5696" LastEditDate="2018-10-22T17:35:48.230" LastActivityDate="2018-10-22T22:19:39.177" Title="What does &quot;Four selected students are not born in the same months&quot; mean?" Tags="|logic|probability|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14685" PostTypeId="2" ParentId="14684" CreationDate="2018-10-22T09:01:15.447" Score="3" Body="&lt;p&gt;The original question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Four people are randomly chosen from a place. Assuming the birthdays of people are equally likely to occur in any month, find the probability that 4 selected students are not born in the same months.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I understand that the intended event includes cases such as [Jan, Feb, Mar, Apr] and [Jan, Feb, Feb, Feb] and excludes cases such as [Feb, Feb, Feb, Feb].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your proposed revision is to edit the last clause to:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;find the probability that at least one student of the selected students is not born in the same month.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In my opinion, both the original question and your proposed revision could be slightly edited to improve clarity:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Four students are randomly selected.  Assuming that the students are equally likely to be born in any month, find the probability that the four selected students are not all born on the same month.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Four students are randomly selected.  Assuming that the students are equally likely to be born in any month, find the probability that at least one of the four selected students is not born on the same month as another selected student.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2018-10-22T22:19:39.177" LastActivityDate="2018-10-22T22:19:39.177" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14686" PostTypeId="2" ParentId="14593" CreationDate="2018-10-23T20:53:20.613" Score="0" Body="&lt;p&gt;Here is a more visceral reason why we have to ensure math is meaningful and that it makes sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=kibaFBgaPx4&quot; rel=&quot;nofollow noreferrer&quot;&gt;This video&lt;/a&gt; shows student responses to the following exercise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cS18M.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cS18M.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And reactions like this happen &lt;a href=&quot;https://mystudentvoices.com/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff&quot; rel=&quot;nofollow noreferrer&quot;&gt;quite often&lt;/a&gt;. Students aren't making mathematical sense of problems, they're just overwhelmingly brainwashed into thinking they need to jump into calculation procedures.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-10-23T20:53:20.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14688" PostTypeId="2" ParentId="14593" CreationDate="2018-10-24T02:44:52.277" Score="2" Body="&lt;p&gt;The approach you envision aligns with that of the Bourbaki group. They espoused a formal, self-contained, and abstract approach to mathematics. Their thinking influenced the School Mathematics Study Group of the early 1960s, which produced the infamous “New Math” curriculum reform in the US. This reform was for both primary and secondary education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So yes, your idea has been considered and even implemented at the elementary school level. Examples at this level included teaching of set theory and numeral systems other than base-ten. The reform movement was criticized by many, including influential thinkers such as Richard Feynman. In his 1965 article “New Textbooks for the ‘New’ Mathematics,” he criticized this approach and emphasized that &quot;we do not want to just teach words,” and “subjects should not be introduced without explaining the purpose or reason.” &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2018-10-24T02:44:52.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14689" PostTypeId="2" ParentId="1164" CreationDate="2018-10-24T15:29:27.230" Score="1" Body="&lt;p&gt;Perhaps consult the appropriate parts of &lt;a href=&quot;https://mathoverflow.net/questions/2556/real-world-applications-of-mathematics-by-arxiv-subject-area&quot;&gt;Real World Applications of Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also try &lt;a href=&quot;https://www.comap.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;COMAP&lt;/a&gt;.  Their materials include free materials and materials available to members.  Many (at least the members' material) are classified according to the subject area of the application and the math topic to be applied.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2018-10-24T15:29:27.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14690" PostTypeId="2" ParentId="12080" CreationDate="2018-10-25T22:01:44.673" Score="6" Body="&lt;p&gt;Based on published works, it appears that both PEMDAS (the US version of the acronym) and BODMAS (the version taught in the UK) began to appear in print only in the 1980s, and really began to spike in 1990.  See the &lt;a href=&quot;https://books.google.com/ngrams/graph?content=PEMDAS%2C%20BODMAS&amp;amp;year_start=1800&amp;amp;year_end=2000&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t1%3B%2CPEMDAS%3B%2Cc0%3B.t1%3B%2CBODMAS%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;Google Ngram&lt;/a&gt; below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Hkal6.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Hkal6.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, searching for PEMDAS within Google Books in the range 1960-2009 suggests that the acronym first started appearing in print in a large-scale way in &lt;em&gt;test preparation books&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/uItb0.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uItb0.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This doesn't mean, of course, that PEMDAS and BODMAS were not part of the &lt;em&gt;oral tradition&lt;/em&gt; of teaching mathematics for a long time before they began to appear in print.  In fact if you dig around older hits for BODMAS you find lots of people referring (in print) to it as something they were taught in school.  But the acronyms don't seem to have themselves made it into print until the test prep industry normalized them to a certain extent.  I suspect something similar is true about other acronyms like SOHCAHTOA.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(By the way, in response to those who commented under the OP that mnemonics like this do more harm than good:  I mostly agree with that sentiment, but the example of SOHCAHTOA is a good reminder that the &lt;em&gt;names of things&lt;/em&gt; are arbitrary matters of convention.  In any right triangle there are indeed six possible side ratios to consider, but there is no way to &quot;figure out&quot; which one is called &quot;sine&quot; and which one is called &quot;cosine&quot;; at some point you just have to &lt;em&gt;memorize&lt;/em&gt; it.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-10-25T22:01:44.673" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14691" PostTypeId="1" AcceptedAnswerId="14697" CreationDate="2018-10-26T14:03:18.537" Score="5" ViewCount="109" Body="&lt;p&gt;My son has just finished his first half-term at secondary school (year 7). I have just received an email saying that the school has set him in a maths class based on current and previous results, and that he will be covering level 5-8 topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It does not mention whether he's in a top, middle or bottom set and I have no frame of reference for the levels he is studying. Where does level 5-8 stand in the grand scheme of things?&lt;/p&gt;&#xA;" OwnerUserId="10499" LastActivityDate="2018-10-28T17:23:40.567" Title="UK Secondary School - how advanced (or not) is level 5-8?" Tags="|secondary-education|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14692" PostTypeId="1" CreationDate="2018-10-26T20:44:00.360" Score="7" ViewCount="2312" Body="&lt;p&gt;I am a student who would like to become a teacher, so I am currently following courses in education. One of the things I learned, is that students like authentic, realistic problems. However, much of the extremum problems (in one variable) appear very 'fabricated'. I only know a few 'realistic extremum problems':&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;minimizing some cost function&lt;/li&gt;&#xA;&lt;li&gt;maximizing some return function&lt;/li&gt;&#xA;&lt;li&gt;maximizing the volume of a box. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A realistic example of the third problem can be found &lt;a href=&quot;https://www.maa.org/node/117464&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, where the volume of a box is maximized. This box is the kind of box you receive a package in: it is closed, with overlapping flaps. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With respect to multivariable functions, I can think of one extra example: determining a regression line. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this list seems rather short. Can anyone give extra examples of real-life situations where we wish to maximize/minimize some sort of function?&#xA;(For my class, single variable functions are the most interesting, but I equally like multivariable examples.)&lt;/p&gt;&#xA;" OwnerUserId="7788" LastEditorUserId="1147" LastEditDate="2018-10-27T04:25:30.027" LastActivityDate="2022-11-28T12:51:57.847" Title="List of realistic extremum problems" Tags="|secondary-education|calculus|student-motivation|big-list|" AnswerCount="8" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14693" PostTypeId="2" ParentId="14692" CreationDate="2018-10-26T23:00:00.017" Score="6" Body="&lt;p&gt;There are many volume-of-a-box questions. I like this one, simpler than&#xA;what the OP cites:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given a rectangle,&#xA;cut out squares from the corners so you can fold it up to a box, without a top,&#xA;of maximal volume.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The rectangle might be specialized to a square, as below.&#xA;See also &lt;a href=&quot;https://web.archive.org/web/20171220175135/http://mathforum.org/library/drmath/view/53560.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Math Forum&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;[![BoxVol][1]][1]&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from [patrickJMT](https://www.youtube.com/watch?v=_oS_LjKse38) YouTube video.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;In response to [@RoryDaulton](https://matheducators.stackexchange.com/questions/14692/list-of-realistic-extremum-problems#comment37319_14692),&#xA;here is the box problem to which the&#xA;OP @Student points:&#xA;[![BoxFlaps][2]][2]&#xA;" OwnerUserId="511" LastEditorUserId="6053" LastEditDate="2022-11-27T21:24:59.937" LastActivityDate="2022-11-27T21:24:59.937" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14694" PostTypeId="2" ParentId="14692" CreationDate="2018-10-27T04:24:01.867" Score="9" Body="&lt;p&gt;Here are some optimization problems that were harder than a simple homework problem:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Given the pavillion angle of a diamond, &lt;a href=&quot;http://www.folds.net/diamond/index.html&quot; rel=&quot;noreferrer&quot;&gt;what crown angle produces optimal light return&lt;/a&gt;?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What percentage of the lignin should be removed from wood, if one wishes to squeeze the wood into a cheap structure similar to carbon fiber with &lt;a href=&quot;https://www.researchgate.net/publication/322991664_Processing_bulk_natural_wood_into_a_high-performance_structural_material&quot; rel=&quot;noreferrer&quot;&gt;an optimum strength-to-weight ratio&lt;/a&gt;?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://pubs.usgs.gov/pp/1395/report.pdf&quot; rel=&quot;noreferrer&quot;&gt;When choosing the parameters for a map projection&lt;/a&gt;, which parameters produce the least worst-case distortion of the needed area of the map?  For example, which two latitudes should have undistorted lengths in an Albers Equal-Area map of the 48 conterminous United States?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Notice that in these examples, more effort went into designing the functions to be optimized, and in choosing the parameter to be varied, than went into performing the calculus.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2018-10-27T04:34:57.597" LastActivityDate="2018-10-27T04:34:57.597" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14695" PostTypeId="2" ParentId="14692" CreationDate="2018-10-27T14:44:41.990" Score="11" Body="&lt;p&gt;Some example types:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Minimizing potential energy of any realistic physical system. Examples:&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).&lt;/li&gt;&#xA;&lt;li&gt;1D: The curve described by a hanging chain/flexible rope (in equilibrium).  &lt;/li&gt;&#xA;&lt;li&gt;2D: The surface of a soap film (in equilibrium).&lt;/li&gt;&#xA;&lt;li&gt;Generally, anything that doesn't move or change: look at the world and wonder.  [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Maximizing a utility function subject to a budget constraint. &lt;/li&gt;&#xA;&lt;li&gt;Fermat's principle: Light takes the path of least time.&lt;/li&gt;&#xA;&lt;li&gt;Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)&lt;/li&gt;&#xA;&lt;li&gt;In a network of resistors connecting voltage &lt;span class=&quot;math-container&quot;&gt;$V_A$&lt;/span&gt; to voltage &lt;span class=&quot;math-container&quot;&gt;$V_B$&lt;/span&gt;, the voltages at the nodes of the network minimize power dissipation.&lt;/li&gt;&#xA;&lt;li&gt;Space trajectories that minimize fuel use (&lt;a href=&quot;http://www.cds.caltech.edu/~marsden/books/Mission_Design.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mission Design&lt;/em&gt;&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;li&gt;When using the central difference formula &lt;span class=&quot;math-container&quot;&gt;$[f(x+h)-f(x-h)]/(2h)$&lt;/span&gt; to approximate the derivative &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; using floating-point numbers with a &lt;a href=&quot;https://en.wikipedia.org/wiki/Machine_epsilon&quot; rel=&quot;nofollow noreferrer&quot;&gt;machine epsilon&lt;/a&gt; of &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;,&#xA;what choice of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; minimizes the relative error of the approximation? [Note: This is a somewhat ill-defined problem, since it may depend on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.]&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2018-10-28T16:00:17.423" LastActivityDate="2018-10-28T16:00:17.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14696" PostTypeId="2" ParentId="14692" CreationDate="2018-10-27T20:35:49.710" Score="6" Body="&lt;p&gt;Some physics examples: --&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that the range of a projectile is &lt;span class=&quot;math-container&quot;&gt;$R=(v^2/2g)\sin\theta\cos\theta$&lt;/span&gt;, prove that the maximum range is achieved for &lt;span class=&quot;math-container&quot;&gt;$\theta=45$&lt;/span&gt; degrees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An electrical transmission line of resistance &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is in series with a load of resistance &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. For a fixed voltage &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, the useful power dissipated in the load is &lt;span class=&quot;math-container&quot;&gt;$P= V^2y(x+y)^{-2}$&lt;/span&gt;. Show that this is maximized, for fixed &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, when &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two atoms will interact through electrical forces between their protons and electrons. One fairly good approximation to the total electrical energy &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; is the Lennard-Jones formula, &lt;span class=&quot;math-container&quot;&gt;$U(r) = k\left[\left(\frac{a}{r}\right)^{12}-2\left(\frac{a}{r}\right)^{6}\right]$&lt;/span&gt;. Infer the units of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. Find the equilibrium value of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, which occurs at the minimum of &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt;, and show that your answer has units that make sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes doors are built with mechanisms that automatically close them after they have been opened. The designer can set both the strength of the spring and the amount of friction. If there is too much friction in relation to the strength of the spring, the door takes too long to close, but if there is too little, the door will oscillate.&#xA;For an optimal design, we get motion of the form &lt;span class=&quot;math-container&quot;&gt;$x=ct e^{-bt}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the position of some point on the door, and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are positive constants. (Similar systems are used for other mechanical devices, such as stereo speakers and the recoil mechanisms of guns.) In this example, the door moves in the positive direction up until a certain time, then stops and settles back in the negative direction, eventually approaching &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;. This would be the type of motion we would get if someone flung a door open and the door closer then brought it back closed again. (a) Infer the units of the constants &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.  (b) Find the door's maximum speed (i.e., the greatest absolute value of its velocity) as it  comes back to the closed position. (c) Show that your answer has units that make sense.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-10-27T20:35:49.710" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14697" PostTypeId="2" ParentId="14691" CreationDate="2018-10-28T17:23:40.567" Score="1" Body="&lt;p&gt;Although the numbers don't correspond exactly to lettered grades, a 5 is an old C (or C+), an 8 is an A or A+.  There's no way to tell if he's in a top set, as there might be a set doing level 7 to 9, or there might not be.&lt;/p&gt;&#xA;" OwnerUserId="5237" LastActivityDate="2018-10-28T17:23:40.567" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14700" PostTypeId="1" AcceptedAnswerId="14704" CreationDate="2018-10-28T21:38:33.150" Score="5" ViewCount="224" Body="&lt;p&gt;I've been thinking that in an ideal world math classes would be self-paced, with a student being required to demonstrate mastery in a collection of topics and techniques in order to complete the course. Some students might complete the course much faster than others, and the slower students would not be forced to try to learn more advanced material when they are still struggling with the basics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difficulty with implementing this idea is: how do you check whether the student has mastered a given topic? They could take an exam, but if they don't do well on the exam what then? I suppose then they would take a different exam which covers the same topics. The student might end up taking 5 exams or more on the same topics before they demonstrate mastery. This poses a practical problem, as it would be difficult to create so many exams and also difficult to grade so many exams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the Khan Academy learning system (at least when I tried it out several years ago) students demonstrated mastery by getting a sufficiently long streak of questions correct. For example, if they get ten chain rule questions correct, then they have demonstrated mastery. This wasn't a perfect system, but it was at least an interesting step towards practical self-paced math courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;: Has there been much discussion or research on how students can demonstrate mastery so that self-paced math courses can be implemented in a practical way? What are some of the ideas people have come up with? Have there been any efforts to create large banks of exams that can be used for self-paced math courses? &lt;/p&gt;&#xA;" OwnerUserId="9633" LastActivityDate="2018-10-29T04:23:11.117" Title="Practical ways to check if a student has mastered a particular topic" Tags="|exams|assessment|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14702" PostTypeId="2" ParentId="14692" CreationDate="2018-10-28T21:58:05.690" Score="1" Body="&lt;p&gt;Many important machine learning algorithms, such as training a neural network, require solving a large optimization problem (essentially tuning the neural network weights to minimize the classification error on a training dataset). The whole deep learning revolution is powered by optimization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In radiation therapy (for cancer treatment), an optimization problem is solved to find optimal beam intensities that target the tumor most effectively while sparing healthy tissue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Magnetic resonance imaging (MRI) works by solving an optimization problem that finds an image of the human body which is most consistent with the measurements obtained by the MRI machine (and also consistent with prior knowledge we have of what images of the human body should look like). Other medical imaging techniques also use optimization for the image reconstruction step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Statistics, many important optimization problems arise as special cases of Maximum Likelihood Estimation. Linear regression is a classic example with a ton of applications. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book Convex Optimization by Boyd and Vandenberghe discusses many applications of convex optimization.&lt;/p&gt;&#xA;" OwnerUserId="2049" LastActivityDate="2018-10-28T21:58:05.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14703" PostTypeId="2" ParentId="11566" CreationDate="2018-10-29T00:30:45.973" Score="3" Body="&lt;p&gt;There is a perfectly good reason to write &lt;span class=&quot;math-container&quot;&gt;$(5)(2)$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$5\cdot2$&lt;/span&gt;, which is that the dot can easily be confused with a decimal point. I believe Russians use a comma as a decimal point, so it makes sense that Russians would not see any danger of error in writing &lt;span class=&quot;math-container&quot;&gt;$5\cdot2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, scientists and engineers typically think of numbers as having units (unlike most mathematicians, who seem to think of units as part of the definition of the number), and therefore consider it scientifically illiterate to write numbers without units. So suppose we're going to multiply 5 m/s by 2 s. Writing it as &lt;span class=&quot;math-container&quot;&gt;$5\ \text{m}/\text{s}\cdot2\ \text{s}$&lt;/span&gt; is ugly and hard to read, so we write it as &lt;span class=&quot;math-container&quot;&gt;$(5\ \text{m}/\text{s})(2\ \text{s})$&lt;/span&gt;. I find, e.g., &lt;span class=&quot;math-container&quot;&gt;$2\pi r=2\pi(1.2\ \text{cm})$&lt;/span&gt;, much more readable than &lt;span class=&quot;math-container&quot;&gt;$2\pi r=2\pi\cdot1.2\ \text{cm}$&lt;/span&gt;. Even for unitless numbers, an expression like &lt;span class=&quot;math-container&quot;&gt;$2\cdot7.8\cdot0.134$&lt;/span&gt; looks like a hot mess to me compared with &lt;span class=&quot;math-container&quot;&gt;$2(7.8)(0.134)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The disease that I've been seeing a lot within the last 5 years or so is the use of parentheses in totally unnecessary ways like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(2)(x)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$((2)(x))/((y)(z))$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect this is actually because of the increasing use of software on computers, phones, and calculators that has fancier symbolic math capabilities than most calculators used to have. Math and science students are typically entering their homework answers into computer software systems such as Mastering Physics or MyMathLab.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many such systems require parentheses for function evaluation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Computers don't have common sense or understanding of context or meaning, so they can't interpret expressions like &lt;span class=&quot;math-container&quot;&gt;$\sin\omega t$&lt;/span&gt; the way any engineer would (as &lt;span class=&quot;math-container&quot;&gt;$\sin(\omega t)$&lt;/span&gt;, not &lt;span class=&quot;math-container&quot;&gt;$(\sin\omega)t$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, students will have had the experience of having the software interpret their input according to the normal rules of precedence of operations, just as a human would, but the result isn't what the student expected because the student doesn't understand the order of operations. For example, the student intends &lt;span class=&quot;math-container&quot;&gt;$a/(b+c)$&lt;/span&gt; but types &lt;span class=&quot;math-container&quot;&gt;$a/b+c$&lt;/span&gt;. Instead of learning from this experience that they need to get more fluent with the order of operations, they just decide that it's safer to use lots and lots of parentheses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing that's going on now with systems like Mastering Physics or MyMathLab is that many lazy teachers take them as excuses to avoid ever reading students' work and writing comments on it with a red pen. Therefore when students make stylistic mistakes such as &lt;span class=&quot;math-container&quot;&gt;$(2)(x)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt;, they never get any feedback from a human telling them that their style is wrong.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2018-10-29T00:38:10.413" LastActivityDate="2018-10-29T00:38:10.413" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14704" PostTypeId="2" ParentId="14700" CreationDate="2018-10-29T04:23:11.117" Score="6" Body="&lt;p&gt;I presume you are fishing for practical ways in an online (digital) environment. First off, there's no objective definition of mastery. At the most rudimentary level you have standardised assessments that rely on overall accuracy to indicate mastery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you said Khan Academy uses an enhanced model that factors in for the success streak and not just accuracy. Duolingo (although unrelated to math) uses a similar model with another sophisticated layer added to it - which is to account for forgetting what you've learned with time (which happens naturally without sustained practice). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;ALEKS uses something called Knowledge Space Theory which uses stochastic methods to efficiently assess student proficiency in a given set of topics. It's able to do this by asking the least number of questions. More importantly, the result of ALEKS is not a numerical score but a detail of what the student has mastered, and what he is ready to learn next.&#xA;Lastly, Cognitive Tutor goes even further to consider step by step attempt of student, along with his hint seeking behaviour and other parameters to navigate a student through a course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There must be many other burgeoning ideas in this field that I haven't mentioned above. Research on this is thriving mostly from the perspective of developing Intelligent Tutoring Systems (ITS) with the aid of big data and machine learning. The idea is to capture as much metadata about the student as possible and use it to strengthen the metric defining mastery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, regarding your question on availability of question banks, do appreciate that with the help of programming, making a question bank is not such a big issue. This is particularly true for maths and even more so for primary and secondary level education. For instance, ALEKS question bank has dynamic questions based on the problem type. Every time a problem type is invoked, a randomly generated problem (with the numbers being randomised) is thrown to the student. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answer would be incomplete if I don't mention that defining mastery will be ultimately specific to the subject, the level of education and the assessors' objectives.&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2018-10-29T04:23:11.117" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14705" PostTypeId="2" ParentId="1816" CreationDate="2018-10-29T20:15:56.213" Score="8" Body="&lt;p&gt;There is a website called &lt;a href=&quot;https://openstax.org/&quot; rel=&quot;noreferrer&quot;&gt;OpenStax&lt;/a&gt; that has pdf versions of textbooks (for various high school level classes including math) that you can download for free; you can also pay like $45 each to get hardback print copies. There are teacher resources as well that you can get access to after creating an account &amp;amp; verifying that you are really an instructor.&lt;/p&gt;&#xA;" OwnerUserId="10507" LastEditorUserId="8571" LastEditDate="2018-10-30T02:25:37.830" LastActivityDate="2018-10-30T02:25:37.830" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14706" PostTypeId="1" CreationDate="2018-10-29T21:41:51.853" Score="6" ViewCount="137" Body="&lt;p&gt;I'm a private math tutor--been doing it a couple of years, so I'm not terribly experienced yet. I have junior high through college students in most math topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It goes the best when the parents hire me to see their student at least twice per week. Then I have a continuous feel for what they are struggling with, and we have time not only to finish their homework, but to review old topics or work ahead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;But many of my families only want one session per week.&lt;/strong&gt; I sometimes feel we can barely make progress. They come to me with different issues every week and there's no time to do anything but rush through the latest homework. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my questions are: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;how can I work with this situation, infrequent tutoring?&lt;/strong&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Can I refer them to outside websites or books&lt;/strong&gt; they can use to study in-between sessions with me? (note: we probably aren't allowed on this stack exchange site to make recommendations for specific sites or products, but if you could just speak generally about the types of sites or books - like a &quot;video lecture site&quot; or &quot;YouTube videos&quot; or &quot;a practice math problem site&quot;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In particular I'm asking about how to &lt;strong&gt;make their math experience more continuous,&lt;/strong&gt; so the student has a clear idea of the topic and moves through it at a good pace, with a chance to clear up any confusion, &lt;strong&gt;in spite of obstacles presented by the school, their teacher, their textbook, or infrequent tutoring.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's a situation with a discontinuous/fractured experience for a different reason -- the student is using a textbook which tries to show connections between different areas of math--by jumping all over the place! So my student can't tell me what topic to prepare him for, and the book obscures some of the typical ways that topics are sequenced.&lt;/p&gt;&#xA;" OwnerUserId="10466" LastActivityDate="2018-11-01T01:33:11.163" Title="How to help my tutoring students study between sessions" Tags="|tutoring|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14707" PostTypeId="2" ParentId="14706" CreationDate="2018-10-30T04:13:50.443" Score="4" Body="&lt;p&gt;Here are a few pointers and questions.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Do the families have a realistic picture of how much help the students actually need? It is common in my work as a tutor to find 14-year-old kids who are getting a B+ or better in math, but cannot calculate &lt;span class=&quot;math-container&quot;&gt;$2000-1997$&lt;/span&gt; mentally. (They have to use the vertical stacking algorithm to get an answer and are unaware of any other methods.) Having students try to do simple &lt;em&gt;conceptual&lt;/em&gt; questions - not calculation questions - while thinking aloud while the parent watches, often causes the horrifying epiphany that parents need. I'm happy to share some of those with you if you request them by private message.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are you sure they are willing to do homework or have a second session in the first place? A lot of students already have so much on their plate that it is very hard to prioritize math. This is especially true for those who perceive math as a credentialing and hoop-jumping exercise with no intrinsic value. They just want to do the minimal effort to get the minimally required grade for passing or not going to summer school or admission into some other program, etc. How many of your students does this describe? Do they even want you to try to change their attitudes?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Related to #2, are the students: (a) rote cramming for tests and homework completion or (b) really going for a deep understanding and lasting mastery?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Instead of a 2nd weekly session, would the students be willing to do homework that you assigned and send it to you by email (or Schoology, EdModo, moodle, etc.) the night before the sessions?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2018-11-01T01:33:11.163" LastActivityDate="2018-11-01T01:33:11.163" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14708" PostTypeId="1" CreationDate="2018-10-30T07:16:58.897" Score="3" ViewCount="75" Body="&lt;p&gt;I want to introduce formal linear combinations in an upper-level undergraduate combinatorics class. By this I mean expressions like &lt;span class=&quot;math-container&quot;&gt;$7 \operatorname{cat} + 5 \operatorname{dog} - \sqrt{2} \operatorname{molerat}$&lt;/span&gt; (and infinite versions thereof), as well as (situation-dependent) rules of multiplication on such expressions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Formally speaking, this means that I have a commutative ring &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; and define the free &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;-module &lt;span class=&quot;math-container&quot;&gt;$R^{(I)}$&lt;/span&gt; for a set &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; (usually not &lt;span class=&quot;math-container&quot;&gt;$\left\{1,2,\ldots,n\right\}$&lt;/span&gt;) and its &quot;completion&quot; &lt;span class=&quot;math-container&quot;&gt;$R^I$&lt;/span&gt;, and define some &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;-algebra structures on these modules. I will briefly introduce commutative rings first, but I cannot expect anyone to have prior experience with them, and so I would like to avoid direct sum, direct products, universal properties, etc.. The two examples I really want to get to are the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The polynomial ring &lt;span class=&quot;math-container&quot;&gt;$R\left[X\right]$&lt;/span&gt; and the ring &lt;span class=&quot;math-container&quot;&gt;$R\left[\left[X\right]\right]$&lt;/span&gt; of formal power series. (A major pain point with US student audiences is having to unteach the misconceptions that polynomials are polynomial functions and that power series have to converge.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The group ring &lt;span class=&quot;math-container&quot;&gt;$R\left[S_n\right]$&lt;/span&gt; of the symmetric group. (The students have seen a lot of permutations; my plan is to let them have some fun with sums of permutations -- e.g., Young-Jucys-Murphy elements, Tsetlin libraries, etc.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But I am interested in other, &lt;strong&gt;motivating&lt;/strong&gt; examples -- ideally baby versions of these objects I can have them play around with that show the structure and give some hint of its usefulness (which means, in particular, that there should be a multiplication -- linear combinations per se are boring). I know of only one for now:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Hardy%E2%80%93Weinberg_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hardy-Weinberg principle&lt;/a&gt; can be proven using a tailor-made non-associative ring: Consider the free module with basis &lt;span class=&quot;math-container&quot;&gt;$\left(a,A\right)$&lt;/span&gt;, and define a (non-associative) multiplication on it by &lt;span class=&quot;math-container&quot;&gt;$a^2 = a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A^2 = A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$aA = Aa = \dfrac{1}{2}\left(a+A\right)$&lt;/span&gt;. This multiplication is literal multiplication, as in &quot;be fruitful and multiply&quot;, as &lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt; is the (probability distribution of) the allele of the offspring of an individual with allele &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with an individual with allele &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. Now, the Hardy-Weinberg principle boils down to the identity &lt;span class=&quot;math-container&quot;&gt;$\left(pa+qA\right)^2 = \left(p+q\right)\left(pa+qA\right)$&lt;/span&gt; in this ring, which is easily checked.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I suspect this is just an example for more general probability-type implications -- is there such a thing as a Markov chain with several transition matrices?&lt;/p&gt;&#xA;" OwnerUserId="5423" LastActivityDate="2018-10-30T07:16:58.897" Title="Formal linear combinations: motivating examples" Tags="|undergraduate-education|abstract-algebra|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14710" PostTypeId="1" AcceptedAnswerId="14734" CreationDate="2018-10-30T12:20:36.123" Score="3" ViewCount="604" Body="&lt;p&gt;Probably it's just another advice-seeking CS student about math.. Well. I have math anxiety. Im a CS student, back then in undergrad I deal with code everyday, as long as my program work smoothly, i'm fine. I graduated with cumlaude honor in my undergrad school. I can say I'm pretty good with my coding skill. Now that I get into grad school (master), everything needs fundamental motivation, reasoning.. and a lot of math.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm always afraid of math, and never really good at numbers, can't remember formulas, brains so slow at processing equation. Especially when someone's watching me, I kind of revert back to kindergarten stage. I even failed a simple matrix multiplication when my prof was watching me did it on the white board, I was so devastated, it was just matrix multiplication ! I felt like a failure to the point that I cried when I got back home. Whenever I read papers and it comes to method section with a lot of equations, I got very nervous, I'm not even exaggerating. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to talk to people I trust about this problem, and turn out this is a phobia, they call it math anxiety. I wish I had known sooner so I could do something about it, I thought I was just stupid. Also, I came from a country when math is &quot;God&quot;, you're a good kid if you're smart at math. So I was called stupid for a while back then.. which I think makes my anxiety even worse. &#xA;Good point, I continued college, and even grad school so no one call me stupid again. I was a teaching assistant in several coding courses in undergrad, even my juniors told me they understand a lot from my course explanation, I'm good at presentation, demo program, anything .. i'm just so bad, at math.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I want to change. I want to learn math again. I read article about &quot;How to Science&quot; and how math is in everyday life. I really want to continue PhD, but the future idea that I will deal with a lot of math stuff always fright me out. I also read &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10679/what-methods-successfully-identify-and-eliminate-severe-math-anxiety&quot;&gt;What methods successfully identify and eliminate severe math anxiety?&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/6026/how-to-overcome-mathematics-related-anxiety&quot;&gt;How to overcome mathematics-related anxiety?&lt;/a&gt; and I got several ideas and book suggestion from the answer&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I've been doing :&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Study from Khan Academy (this will be to basic for grad school level, but I feel that I need to start all over).&lt;/li&gt;&#xA;&lt;li&gt;Study from Daniel Shiffman channel on youtube. He's a professor and programmer. And sometimes in his coding he's using math, and he's very good at explaining stuff, even I can understand his math better than what my college professor taught me.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I will need to deal with computational logic, and artificial intelligence that requires a lot of math. I need some advice for my future PhD plan : either is there anyway we can actually continue PhD with minimum math ability ? or, I'll try to be real here, I read &lt;a href=&quot;https://www.quora.com/How-do-I-learn-mathematics-from-level-zero-I-am-a-computer-science-graduate-and-fairly-decent-at-my-programming-role-I-miss-the-context-of-using-maths-in-life-I-cant-visualize-it&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, that's it's too late to learn math when you're already in Grad School. Do you think I should just go, and continue and improve on aspects that I'm already good at (like, coding) ? or do you think no matter what I still need the fundamental understanding about math ? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="10511" LastActivityDate="2018-11-06T19:02:18.017" Title="Math Anxiety get in the way of my Graduate School, I want to continue PhD" Tags="|student-motivation|anxiety|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14711" PostTypeId="2" ParentId="14710" CreationDate="2018-10-30T12:42:37.870" Score="2" Body="&lt;p&gt;It seems like you are on a good way already; recognizing and facing your problems are often the hardest parts. Keep at it, get all the help you can, and you should be good. I would not try to find a PhD with less math, you might damn yourself to a bad topic or thesis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might help if you have a certain problem (in CS, maybe something NP complete?) that you want to understand or solve and then research all the math needed for it. That might be a better motivation than people telling you to do this or that without a reason.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2018-10-30T12:42:37.870" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14712" PostTypeId="1" AcceptedAnswerId="14715" CreationDate="2018-10-30T18:18:23.973" Score="3" ViewCount="212" Body="&lt;p&gt;I work as a math tutor mostly for talented high-school students that are passionate about mathematics and want to learn more of it beyond school programs. They are very smart kids, but I noticed that the following happens sometimes, so I am trying to understand how to deal with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose I am explaining a solution to a problem by dividing it in many step. After each step I ask the student if it understands it. If not, I divide it in more sub-steps, and repeat the procedure. Otherwise, I go to the next step. I do this until the student tells me that he understands every single step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, according to this algorithm, I should always be sure that the student understands the solution. However, it happens sometimes that the student says that he/she understands every step, but still have problems with the solution as a whole.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering. Should I try and add additional intermediate steps (even though, sometimes I really cannot figure out how to slit the problem more) or maybe test the student to check if he/she actually understand the single steps or just believe he/she does? What would be more helpful and likely to solve the problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. I apologize for the question being a bit general. Any suggestion or consideration is appreciated, thank you!&lt;/p&gt;&#xA;" OwnerUserId="10218" LastActivityDate="2020-01-31T02:50:07.653" Title="More intermediate steps or check well-understanding" Tags="|teaching|tutoring|problem-solving|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14713" PostTypeId="2" ParentId="1816" CreationDate="2018-10-30T23:26:39.110" Score="2" Body="&lt;p&gt;The &lt;a href=&quot;https://www.floridashines.org/orange-grove&quot; rel=&quot;nofollow noreferrer&quot;&gt;Orange Grove&lt;/a&gt; is Florida's &quot;Open Educational Resource Repository&quot;.  It has resources both for K-12 and higher education.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2018-10-30T23:26:39.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14714" PostTypeId="1" CreationDate="2018-10-31T01:42:02.783" Score="0" ViewCount="282" Body="&lt;p&gt;Say you have college A, B and C that are each ranked in the top 100 in the US News or other similar rankings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Admissions (SAT scores, GPA, et cetera) standards, quantitatively, rank A&gt;B&gt;C as the hardest and the easiest of the three to get into.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's clear that college A has the 'best' students according to the metrics the entrance standards measure. It could measure IQ (SAT), effort and communication skills and critical thinking skills (GPA). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would you say, in mathematics and elsewhere, that the content of the material is markedly different? For instance, could the undergraduate textbooks used be different?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, why do students from prestigious colleges have such a leg up in terms of finding jobs and getting into graduate schools of similar prestige, if the material taught is similar, in terms of the amount of material taught and the difficulty of the exams?&lt;/p&gt;&#xA;" OwnerUserId="10514" LastEditorUserId="901" LastEditDate="2018-11-12T12:17:43.363" LastActivityDate="2018-11-12T12:17:43.363" Title="Is there a difference in the content taught at each university?" Tags="|exams|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14715" PostTypeId="2" ParentId="14712" CreationDate="2018-10-31T05:35:25.873" Score="9" Body="&lt;h1&gt;Part 1: Do they really understand?&lt;/h1&gt;&#xA;&lt;p&gt;My first thought is that you are running into the limits of working memory. As students try hard to understand step 5, they are pushing previous thoughts about step 1 and 2 out of their working memory, before having really processed that information. That ~guarantees they will forget steps 1 and 2 almost immediately, whether or not they understood it in the first place.&lt;/p&gt;&#xA;&lt;p&gt;My second thought is false positives. If the student understands, they will likely say they understand. But maybe they incorrectly &lt;em&gt;believe&lt;/em&gt; they understand and state that they understand. Or, maybe they don't understand but don't want to admit it. Or perhaps their understanding is very inflexible (i.e. cannot withstand contextually superficial changes), but they don't realize it.&lt;/p&gt;&#xA;&lt;p&gt;The evidence from cognitive science, by the way, is overwhelming that virtually everyone is a poor judge of when real learning is happening. [See book &amp;quot;Make It Stick&amp;quot; or YouTube &amp;quot;How We Learn vs How We Think We Learn&amp;quot; or &amp;quot;Reject self-report&amp;quot; from book &amp;quot;Teach Like a Champion&amp;quot;.] So, false positives are likely a big problem.&lt;/p&gt;&#xA;&lt;p&gt;Instead of asking &amp;quot;Do you understand?&amp;quot;, you need to develop more probing questions. For example, many teachers will write &lt;span class=&quot;math-container&quot;&gt;$(x^m)(x^n)=x^{m+n}$&lt;/span&gt; and some examples of it on the board, then attempt to check understanding by saying, &amp;quot;Any questions?&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Student responses and non-responses to that are untrustworthy.&lt;/p&gt;&#xA;&lt;p&gt;A better check would be to write, say, a variety of 20 different exponential expressions on a piece of paper and ask if that exponent property could apply and why or why not. If you mix in expressions such as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(x^n)(y^n)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$9^m9^3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$m^4m^7m^8$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$m^4m^7+m^8$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(x-3)^7(x-3)^{10}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(x-3)^7(3-x)^{10}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;etc.&lt;/p&gt;&#xA;&lt;p&gt;then, you can &lt;em&gt;really&lt;/em&gt; check for understanding.&lt;/p&gt;&#xA;&lt;p&gt;Asking if something is an example or a non-example or a partial example of a concept you're discussing is one way to see if they &lt;em&gt;really&lt;/em&gt; understand.&lt;/p&gt;&#xA;&lt;p&gt;It is, unfortunately, difficult and time-consuming to develop really great probing questions like this. If you find a great source of such questions, let me know. :)&lt;/p&gt;&#xA;&lt;h1&gt;Part 2: What does &amp;quot;one step at a time&amp;quot; mean?&lt;/h1&gt;&#xA;&lt;p&gt;Say students are learning to solve the systems such as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x+y=12$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\hskip 0.5cm$&lt;/span&gt; (1)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$y-2x=6$&lt;/span&gt;  &lt;span class=&quot;math-container&quot;&gt;$\hskip 0.5cm$&lt;/span&gt; (2)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The Traditional Sequence&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Isolate one unknown, say &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, in equation (1).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Substitute into equation (2).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Solve (2) for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Substitute that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-value into (1).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Plug the values into (1) and (2) to check if they really are solutions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This may &lt;em&gt;seem&lt;/em&gt; like &amp;quot;one step at a time&amp;quot;, but to a learner, it feels like five steps at a time.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A different approach to &amp;quot;one step at a time&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;First, provide students with five different linear systems and bunch of ordered pairs. Have students verify which of those pairs are solutions to which equations.&lt;/p&gt;&#xA;&lt;p&gt;Start by learning to check, not to solve.&lt;/p&gt;&#xA;&lt;p&gt;Then, once they know how to check their own work, have students solve systems, but ramp up difficulty slowly.&lt;/p&gt;&#xA;&lt;p&gt;Have a bunch of VERY EASY systems such as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x=7$&lt;/span&gt; [obviously sub this into the next equation]&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x+y=-10$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then ramp them up to a bunch of EASY systems such as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2x-1=7$&lt;/span&gt; [solve for x, then sub into the next equation]&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x+y=-10$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Ramp up to a bunch of MEDIUM systems such as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2x-1=7x+16$&lt;/span&gt; [solve for x, then sub into next equation]&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x+y=-10$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then, MEDIUM-HARD&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2x-1=y$&lt;/span&gt; [sub immediately into next equation]&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x+y=-10$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;HARD&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2x-1=y+13$&lt;/span&gt; [isolate an unknown first, then sub into the next equation]&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x+y=-10$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I call this &amp;quot;backwards horizontal&amp;quot; teaching. Students first master the step of checking. Once they can do that, they master the last step before checking, then they can check their answers. Then they move on to doing the last two steps before checking, then actually checking. Repeat until they can do the entire thing from the start.&lt;/p&gt;&#xA;&lt;p&gt;The purpose of the steps then is crystal clear the entire way and the students &lt;em&gt;really are&lt;/em&gt; learning &lt;em&gt;one&lt;/em&gt; step a time.&lt;/p&gt;&#xA;&lt;p&gt;After doing this, I'd ask the &lt;em&gt;students&lt;/em&gt; to name all the steps required to solve and check a system of equations. If there were a bunch of students, I'd have them try to do it alone, then check their work with a peer.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-10-31T06:15:24.370" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14716" PostTypeId="2" ParentId="14714" CreationDate="2018-10-31T10:09:12.960" Score="3" Body="&lt;p&gt;Different universities teach different courses and different content in them, though typically there are similarities in the basic courses. They are likely to use different books. Custom lecture notes are not uncommon, either. (They might be uncommon in USA, but the question does not, at the moment, specify a country).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The specific lecturer often has a fair amount of freedom in deciding the specific content of a course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not know whether the difficulty or learning outcomes of courses are correlated with rankings or other measures of quality of a university. At least the US News rankings also include social factors like sports teams and fraternities, the quantity of alumni donations and access to facilities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a plausible hypothesis that universities with higher rankings teach more or better, but it is also a plausible hypothesis that the quality of incoming students determines the quality of outgoing students. It would be interesting to read studies about this, but I am not an expert on the field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One should also remember that the university rankings tend to be controversial, and even if such rankings make sense, the quality of mathematics departments (or any other specific departments) need not follow the general rankings.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2018-11-01T09:45:14.013" LastActivityDate="2018-11-01T09:45:14.013" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14718" PostTypeId="1" CreationDate="2018-10-31T22:05:22.993" Score="10" ViewCount="5182" Body="&lt;p&gt;I'm a baby boomer who was taught that the constant term of a polynomial is a coefficient, being the constant factor for the x^0 term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's not what's taught today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Current text books are vague on their definition of coefficient, but, when they ask for a student to list all coefficient in a given polynomial, they do not include the constant term in their answer keys.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Kahn Academy video on the topic does include the constant term as a coefficient, as does the related Wikipedia article.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, all modern text books written to the Common Core that I have seen imply that the constant term is not a coefficient. A lot of web sites created by today's teachers who are teaching to the Common Core explicitly state that the constant term is not a coefficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have a polynomial of one variable, aren't the coefficients supposed to uniquely determine that polynomial? Aren't they all a computer program needs to know to work with polynomials? Without the constant term, you get the related polynomial that represents a vertical displacement of the original polynomial by the opposite of the constant term. But, you don't get the original polynomial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, if we are talking about natural number constants, isn't each digit, including the ones' place, a coefficient of a power of 10? We don't segregate that digit by calling it something else; it's a digit similar to all the other digits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Isn't mathematics supposed to be non-arbitrary and consistent?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to see this taught as: after combining like terms, all constant factors, including the constant term, are coefficients. Two of the coefficients are special. The coefficient of the term with the largest exponent is the leading coefficient. The coefficient of the term with the smallest exponent (aka zero) is the constant term. That shouldn't be too much for eighth graders who were taught exponent properties, including the zero exponent property, in seventh grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this going to be yet another contributor to generational divide?&lt;/p&gt;&#xA;" OwnerUserId="10517" LastEditorUserId="2083" LastEditDate="2018-11-01T10:00:15.303" LastActivityDate="2018-11-02T13:39:18.770" Title="Is the constant term a coefficient?" Tags="|secondary-education|terminology|common-core|middle-school|polynomials|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14719" PostTypeId="2" ParentId="14718" CreationDate="2018-10-31T22:50:29.057" Score="8" Body="&lt;p&gt;Your question is kind of two parts: one about a convention &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is the constant term a &quot;coefficient&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and one about a philosophy, which I perhaps find to be a more important question to answer.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Isn't mathematics supposed to be non-arbitrary and consistent?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Different fields of mathematics have different conventions; this can lead to some mathematicians using &quot;sigma&quot; for &quot;standard deviation&quot; in statistics and other mathematicians using &quot;sigma&quot; for a map defining a simplex in algebraic topology, and others using it to represent a group element in a permutation group. Even more egregiously, some people consider &quot;Natural numbers&quot; to include zero, and others consider &quot;natural numbers&quot; to be only positive integers. Even in higher mathematics, the definition of a word, or the statement of a theorem, may differ slightly between two textbooks. What is important is that 1) their definitions are clearly stated when they are introduced, and 2) that when a single mathematician uses the word in one body of work, that they are consistent. We are capable of handling the fact that different texts have slightly different definitions of what, exactly, a &quot;coefficient&quot; is, because we know how to interpret mathematical definitions and classify objects according to whether or not they fit that definition. That skill -- determining whether an object satisfies certain properties, or classifying mathematical objects based on definition, and recognizing properties of objects satisfying a particular definition -- is often an unstated goal in teaching mathematical vocabulary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that sense, I do not think that the definition of &quot;coefficient&quot; needs a universal definition for schoolchildren. When we teach students the meaning of mathematical words, the intent is often to build in them the ability to think about different &quot;parts&quot; of what they are learning and clearly articulate their thinking mathematically. In this sense, teachers and students being consistent with themselves and their textbook matters. However, being consistent across different states, or even different school districts, seems unimportant. Ultimately, they may grow up and realize that what their school called a coefficient differs slightly from what their college called a coefficient. This realization may help them understand the importance of an &quot;operational definition&quot; in many contexts! &lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2018-10-31T22:50:29.057" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14720" PostTypeId="2" ParentId="14718" CreationDate="2018-11-01T00:54:44.053" Score="4" Body="&lt;p&gt;It sometimes happens that slightly different definitions of the same word each have advantages and disadvantages. In such cases, I wouldn't be surprised to see some people supporting one definition and other people supporting a different definition.  In the case at hand, though, I can't think of any advantages for defining &quot;coefficient&quot; to exclude the constant term. It seems to me that, if one adopts this definition, one will repeatedly have to say &quot;coefficients and constant term&quot;. In addition to your example, &quot;a polyomial is determined by its coefficients and its constant term&quot;, we'd have &quot;to multiply a polynomial by &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;, multiply all its coefficients and its constant term by &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;&quot; and &quot;to add two polynomials, add the corresponding coefficients and add the constant terms.&quot; All of these become simpler and clearer if we just include the constant terms among the coefficients instead of treating them separately. (And I refuse to even think about multiplying two polynomials if I'm expected to think about constant terms separately from the other coefficients.)&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2018-11-01T00:54:44.053" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14721" PostTypeId="2" ParentId="14718" CreationDate="2018-11-01T14:22:35.590" Score="2" Body="&lt;p&gt;Let's keep in mind that &quot;mathematics&quot; and &quot;mathematics education&quot; are different subjects. This question brings forth this distinction. At one end of the &quot;Piaget&quot; spectrum of mathematical stages of development, explaining that the constant term &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the same as &lt;span class=&quot;math-container&quot;&gt;$cx^0$&lt;/span&gt; might be too much cognitive overload. However, towards the other end of the spectrum, say in calculus, the construct &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=0}^N c_nx^n$&lt;/span&gt; needs to be a primary object. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2018-11-01T14:22:35.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14722" PostTypeId="2" ParentId="14714" CreationDate="2018-11-01T16:01:03.927" Score="2" Body="&lt;p&gt;The short answer to your question is &quot;of course there is&quot;.  I think that other responses will probably adequately take care of the many exceptions to this in terms of how questionable university rankings are.  But there is absolutely no question that even in a course like calculus which (in the United States) has a fairly well-defined set of topics (see the &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA calculus report&lt;/a&gt;) there is considerable difference in how rigorously things are done, how difficult of applications are considered, etc.   (An example might be whether one barely talks about implicit differentiation, whether it's connected to parametric curves, whether one then does logarithmic differentiation, or whether one then introduces the implicit function theorem.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I want to focus on a different problem with this, which is that at many universities (most?) there may be different versions of the &quot;same&quot; course taught.  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In graduate school we had three different varieties of calculus to teach, for instance, ranging from fairly standard to using &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=1&amp;amp;ved=2ahUKEwjVruObyLPeAhWCmOAKHT6pCZIQFjAAegQIEBAB&amp;amp;url=https%3A%2F%2Fwww.amazon.com%2FCalculus-4th-Michael-Spivak%2Fdp%2F0914098918&amp;amp;usg=AOvVaw0wwHTZaG-4XgUBys6rhOxH&quot; rel=&quot;nofollow noreferrer&quot;&gt;Spivak&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Some institutions will have separate linear algebra (often not in a course by that name) for engineers, for math majors, for computer scientists, and for others - all depending on the application.&lt;/li&gt;&#xA;&lt;li&gt;Even (US) upper-division courses like number theory or geometry might come in several varieties; for instance, particularly if there is a large teacher preparation program, you could have two courses covering non-Euclidean geometry, one focusing on the axiomatic viewpoint and one on the differentiable viewpoint.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In all of these cases the content might be extremely different, certainly with different texts, but in general the &quot;hard&quot; version at University C probably would fall higher under your hierarchy than the &quot;easiest&quot; version at University A.  If there were just one calculus or linear algebra (or whatever) at each university it might be easier to make such comparisons, but there isn't.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2018-11-03T01:00:29.397" LastActivityDate="2018-11-03T01:00:29.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14723" PostTypeId="2" ParentId="8218" CreationDate="2018-11-01T19:47:10.703" Score="1" Body="&lt;p&gt;I can not count the times that I used or read expressions as &lt;span class=&quot;math-container&quot;&gt;$\epsilon+\epsilon^2=\epsilon$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(1+dx)(1+dy)-1=dx+dy$&lt;/span&gt;. It is usual infinitesimal  reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example, any time that in the evaluation of a limit we use a Taylor series ended as dots we are implicitly doing an intermediate limit evaluation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{x \to 0}~ \sqrt{\frac{1+2x}{x^2}}-\frac{1}{x} = \lim_{x \to 0}~{\frac{1}{x}+1-\frac{x}{2}+\dots-\frac{1}{x}}=1&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, state rules ( &quot;never replace parts of an expression by their limits when taking a limit&quot;, &quot;not to do it. ... take all limits in a single step&quot;) is not the way (never it is, rules are ok only for computers) and, when done, it is full of exceptions. Instead it is better help students to practice, learn the dangers, advantages, problems and concepts under some approach. In this way, students experience and acquire the concept of infinitesimal, asymptotic behavior, ... .  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From this point of view, a student who writes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{n \to \infty}~ n^2-n = \lim_{n \to \infty}~ n^2 &#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{n \to 0}~ (n^2+n)/n = \lim_{n \to 0}~ n/n = 1&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is showing he/she understands the relation between infinity/infinitesimal and powers, the concept of asymptotic behavior, .... &lt;/p&gt;&#xA;" OwnerUserId="7441" LastEditorUserId="7441" LastEditDate="2018-11-02T12:24:15.007" LastActivityDate="2018-11-02T12:24:15.007" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14724" PostTypeId="1" AcceptedAnswerId="14906" CreationDate="2018-11-01T21:52:53.957" Score="3" ViewCount="224" Body="&lt;p&gt;I am 37 years old and I have a mild learning difficulty. I never went to a special school, but yes, I got some help and I never got held back. I left my grades on borderline. I always loved to learn how things function and work in the world and my strongest point was computing and tech.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I want to continue my life work. I have always liked maths and physics but never grasped them until now. I love to solve things, and now all I want to do is learn enough maths for physics and engineering and to help me get a good reputation on maths and physics stack exchange sites.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use Khan Academy and I was going to go from level to level, but I find that boring. I love to learn new things. I have a maths capability of a 4th or 5th grader, and understand maths better now. My interests are electronics. For example, I love to learn robotics and how the parts work, and chemistry and bonds, and particle and quantum physics and particle field physics just like the new age physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thing is, I accidentally deleted Khan Academy 1 month ago when I got down to AP Calculus level and had to restart because I was stuck. Now I am wondering whether I should continue from geometry, trigonometry and algebra 1 and 2, or continue from where I left off before I deleted the old account? I am a bit rusty on, but familiar with, the old studies. Or should I just start from functions, conic sections, polynomial and quadratics, then the trig pre-calc then back onto AP Calculus and onwards?&lt;/p&gt;&#xA;" OwnerUserId="10519" LastEditorUserId="11" LastEditDate="2019-10-10T00:25:38.683" LastActivityDate="2019-10-10T00:25:38.683" Title="I want to learn how things function and continue my life work of learning" Tags="|self-learning|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14725" PostTypeId="2" ParentId="12285" CreationDate="2018-11-01T22:42:23.390" Score="4" Body="&lt;p&gt;I apologize for not leaving this as a comment, but being new to this StackExchange I will reply in the way in which I am able.&lt;/p&gt;&#xA;&lt;p&gt;I do not have the hard data you are seeking, but I'll share my personal experience.&lt;/p&gt;&#xA;&lt;p&gt;I am a math PhD student, and each semester I teach a 100 level math course (in our department these are (primarily) retreads of things the students should have learned in high school).  Examples include Algebra, trigonometry and Pre-Calculus.  Incoming students are placed in these courses based on a placement test taken during orientation (provided they do not have credit for AP Calculus, or transfer credits from a community college, etc.)&lt;/p&gt;&#xA;&lt;p&gt;However, many students still struggle with basic things like exponent rules, fraction addition, and knowing the 'legal' algebraic moves they may make to solve an equation.  So in this sense I'm not sure how useful our placement exams are.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I address this issue in the following way.  If I'm about to teach something, for example an introduction to exponential  functions, I will do something like the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Motivate with some discrete examples&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give the students problems to try that require a good understanding of exponent rules&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;circulate and give differentiated help&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;work thorough the problem while explaining my reasoning&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;tell the students that if they struggled with exponent rules that that is something they are assumed to know, so it is beyond the scope of the lecture&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;direct them to resources for additional practice and offer to help in office hours&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;give a general algorithm for solving similar problems&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;give word problems to try and help students to develop basic number sense&lt;/p&gt;&#xA;&lt;p&gt;I do this for two reasons.  First, is that these courses are prerequisites for others in the department, and the instructors of those courses will assume a certain set of material has been covered.  Second, I want my students to take ownership of the learning process.  I will do everything reasonable to help them to succeed, but I also make them earn their success.  To use a trite saying: I will lead them to water, but I will not force said water down their throats.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Having taught high school before university, I find that many students in high school, and indeed in the first couple years of college, have been conditioned to think that it is the instructors job to make them pass.  I do my best to break them of this notion.&lt;/p&gt;&#xA;&lt;p&gt;While it is true that I occasionally get bad reviews from individual students, and indeed more students fail my course (roughly 30-40% are likely to fail this semester) than might be ideal to pass them on when they haven't demonstrated some level of mastery does no one favors.  This is how we have students in multivariate calculus that still cannot correctly do basic algebra.&lt;/p&gt;&#xA;" OwnerUserId="10521" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-11-01T22:42:23.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14726" PostTypeId="1" CreationDate="2018-11-01T20:22:16.770" Score="2" ViewCount="223" Body="&lt;p&gt;I am a tutor for a student and I work with him 7 days a week, for about 2-3 hours a day. The student severely struggles with math, although I am a tutor for every subject (he is in high school). His first chemistry exam he got a 77, and I brought it up to a 96. I studied with him for history and he received 100 on that test. However, his first math test was a 73. I spoke with the parents and told them that the homework he is receiving is very different (in terms of difficulty than the exam). The homework is much easier. So I decided that despite how easy his homework is, I will come up with challenging problems that are similar to his exam problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He had the test today and said he really messed up on it. I spent 4 hours yesterday and 4 hours the day before preparing with him. I gave him a mock exam, I came up with questions that are similar to his review sheet. I did everything I thought I could to help him. What should I do?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I noticed that during the mock exam, he could barely answer any questions without turning to me to ask for clarification and then he would stop midway and though his logic was sometimes correct, he would make silly errors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there anything I can do in order to fix this situation? I feel responsible for his bad grade. What more can I offer to help him, or what more can I do in order to make him succeed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like I failed him, and that it's my fault. During the sessions I am very attentive and any small thing he doesn't understand, I make sure he gets it. So I don't know what to do. I've never experienced this before. &lt;/p&gt;&#xA;" OwnerDisplayName="user8290579" LastActivityDate="2018-11-02T04:20:28.950" Title="What are tutoring strategies for students struggling in math?" Tags="|tutoring|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14727" PostTypeId="2" ParentId="14726" CreationDate="2018-11-02T03:25:19.597" Score="1" Body="&lt;p&gt;A raw score on an exam doesn't mean anything in and of themselves, so we have no context for evaluating what a 73% means. If this is a class where the grading scale is 90%=A, 80%=B, 70%=C, then this student passed his math exam, which doesn't seem like a bad outcome for someone who really struggles in math. Grade inflation notwithstanding, not everyone is going to get an A in every subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I spent 4 hours yesterday and 4 hours the day before preparing with him.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You don't say whether this is college or high school. If it's college, then the typical expectation would be that for a 4-unit math class, the student would spend about 8 hours a week outside of class on the work. So 8 hours of work (ordinarily independent work) would be a typical week. And 8 hours a week isn't a maximum, it's more like a minimum. For a student who is particularly bad at a certain subject, and who wants to get a good grade like a B, probably more like 10, 15, or 20 hours a week will be needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math is a subject that requires constant, steady practice. Mastering most skills (violin, chess, ...) requires about 10 years of steady, intense work. What has this student's work been like for the last 10 years?&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-11-02T03:25:19.597" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14728" PostTypeId="2" ParentId="14726" CreationDate="2018-11-02T04:20:28.950" Score="2" Body="&lt;p&gt;This is very hard to answer without details such as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is this student generally a sense-making student or one who just seeks answers? How do you know?&lt;/li&gt;&#xA;&lt;li&gt;Is the student lacking prior knowledge? Many students get high grades for years in math because they rote memorize procedures while understand ~nothing. For example, they can do long division to calculate &lt;span class=&quot;math-container&quot;&gt;$249\div15.3$&lt;/span&gt; but cannot estimate an answer nor can they generate a word problem that might go with it or distinguish it from &lt;span class=&quot;math-container&quot;&gt;$15.3\div249$&lt;/span&gt;. What have you done or are you planning to do to assess for this? Weak prior knowledge is almost always the biggest problems with students who struggle with math but succeed in all other courses.&lt;/li&gt;&#xA;&lt;li&gt;You said &quot;... any small thing he doesn't understand, I make sure he gets it.&quot; How are you making sure of this? How do you know he is only appearing to understand? Inferring understanding of mathematical concepts from typical school math is not like push ups. If you do 10 push ups, you can obviously do 10 push ups. If you do 10 division calculations correctly, you might have &lt;em&gt;zero&lt;/em&gt; understanding of division.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I can give this advice, though. Generally, it is not a good idea to help students part way through a problem. Make them try their hardest first, then share all their thoughts, and then you both have a clear idea of where they're at. Now, you have the information you need to provide appropriate feedback and hints. After they succeed, then they should try a &lt;em&gt;mix&lt;/em&gt; of different types of problems, at least one of which is like the on you just helped them with. [If it isn't a mix, they might just robotically apply the same algorithm without thinking.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assistance before their best effort means that anything that makes it to the page is a &lt;em&gt;joint&lt;/em&gt; effort. Furthermore, students typically ask for help with the hardest part of the question, e.g. deciding if it's permutations, combinations, or neither. If you say  &quot;It's permutations&quot;, they might then spend 5 minutes hand-calculating an answer, getting it right, then thinking they've mastered it... when really they had no conceptual idea of what was happening at all and would have gotten a zero on the test.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-11-02T04:20:28.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14729" PostTypeId="1" AcceptedAnswerId="14744" CreationDate="2018-11-02T05:08:01.650" Score="17" ViewCount="2036" Body="&lt;p&gt;Has anyone performed a study on the differences between student interpretations of these words? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Background: When I taught high school geometry and later undergraduate precalculus, I noticed that even when explicitly taught how to read and interpret the words &quot;true&quot; and &quot;false,&quot; that students have a hard time circling &quot;false&quot; when a statement is &quot;sometimes true.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in my experience, students struggle with &quot;True/False: Dividing two polynomials results in another polynomial,&quot; but have less difficulty with the question &quot;True/False: Dividing two polynomials always results in another polynomial,&quot; even though these two statements have equivalent truth value and even when students are explicitly taught to interpret the word &quot;true&quot; as meaning &quot;ALWAYS true&quot; and the word &quot;false&quot; as meaning &quot;SOMETIMES false&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With this in mind, I hypothesize that before students have been expected to master mathematical logic, that they will perform better on &quot;Always True /Sometimes but Not Always True/ Never True&quot; questions, and we would get a more accurate picture of their understanding than if we ask true/false questions. Is there any research supporting (or not!) the use of always/sometimes/never &lt;em&gt;instead&lt;/em&gt; of true/false questions, before the more formal mathematical understanding of the meaning of &quot;true&quot; and &quot;false&quot; are expected to be grasped?  &lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2018-11-05T01:03:16.070" LastActivityDate="2018-11-05T20:58:26.553" Title="&quot;Always/Sometimes/Never&quot; vs. &quot;True/False&quot; questions for mathematical reasoning" Tags="|reference-request|students-mistakes|logic|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14730" PostTypeId="2" ParentId="14714" CreationDate="2018-11-02T07:02:08.030" Score="0" Body="&lt;p&gt;&lt;strong&gt;Yes.&lt;/strong&gt;  MIT and Caltech are notorious for having freshman math and physics classes that go 30 percent faster than corresponding courses at other highly ranked institutions.  Sadly, this means that 20% of MIT freshmen do poorly in their freshman math classes, despite being some of the best-prepared college students in the country.  MIT's standard freshman physics courses used to have similar rates of Ds and Fs, prior to a major pedagogy change during the first decade of this century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;And no.&lt;/strong&gt;  In California, almost anyone can attend a community college.  And either after passing enough remedial classes, or after demonstrating that the remedial classes are not needed, you can take essentially the same calculus classes as are offered by the University of California at Berkeley.  If you manage to get in to MIT, you can use those the non-remedial classes to get out of MIT's classes.  But because MIT's classes go 30% faster, you will need to have taken 30% more semesters-worth of non-remedial classes at the community college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many colleges offer multiple versions of some courses, depending on students' interests and planned majors.  These courses can vary widely in difficulty, content, and teaching approach.  Some &quot;honors&quot; versions of classes are only offered to students who meet special admissions standards.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2018-11-03T00:27:55.497" LastActivityDate="2018-11-03T00:27:55.497" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14731" PostTypeId="1" CreationDate="2018-11-02T12:48:47.070" Score="4" ViewCount="458" Body="&lt;p&gt;In the introduction of the limits chapter of a scholar book I can read (translated and abstract) the following examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&quot;the price of a product has a limit value that, from this price, the&#xA;number of sales start to decrease, reducing the benefits&quot; &lt;/li&gt;&#xA;&lt;li&gt;&quot;the light speed has a limit&quot; &lt;/li&gt;&#xA;&lt;li&gt;&quot;the population of animals is limited due to the amount of food&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For me, these examples are not correct examples of a limit in the sense used in calculus: 1 and 3 are a function maximum, example 1 is a maximum of the function f:price-&gt;benefit and example 2 is a empirical maximum, base of the special relativity theory (I do not known any formula that gives light speed as root, maximum or limit of an expression ).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true that in a lot of situations in normal life, limit and maximum are synonyms. But not in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, I wonder if in the classroom we must strengthen the usage of words as &quot;tendency&quot;, &quot;trend&quot;, ... even when in the written math expression still using &lt;span class=&quot;math-container&quot;&gt;$\lim$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7441" LastEditorUserId="7441" LastEditDate="2018-11-02T14:59:10.920" LastActivityDate="2022-08-09T16:05:10.093" Title="About the word &quot;limit&quot; used in calculus" Tags="|limits|vocabulary|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14733" PostTypeId="2" ParentId="14718" CreationDate="2018-11-02T13:39:18.770" Score="4" Body="&lt;p&gt;I have to admit I was skeptical of the OP's claim that contemporary textbooks do not identify the constant term as a coefficient, so I checked the first book that I had handy -- and indeed it does seem to be the case, in at least my sample of 1.  Here is some evidence:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/srHRa.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/srHRa.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/xv5hs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xv5hs.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Source:  McDougal Littell &lt;em&gt;Algebra 2&lt;/em&gt;, 2004.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note however that 2004 &lt;em&gt;precedes the Common Core&lt;/em&gt; by about 6 years, so this cannot be laid at the feet of the CCSS. &lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-11-02T13:39:18.770" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14734" PostTypeId="2" ParentId="14710" CreationDate="2018-11-02T14:59:11.373" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In the immortal words of Dirty Harry, a man's gotta know his limitations.  I recommend not to bang your head against the wall on the most theoretical and math intensive parts of CS, when you are weak at math and good at coding.  And there are lots of needs for good coders without much math.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's strongly likely that your issue is not nervousness (knowing the stuff and failing in tests) but general weakness on the material.  Nevertheless, whether your issue is test anxiety or skill, the way to remedy that is with drill (&quot;do homework until you puke, then do some more&quot;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Kahn Academy videos are time inefficient.  I would work through the Frank Ayers Schaum's first year college math book (geometry, algebra, trig, analytical geometry) instead. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In addition, you should use the regular course problems you work on as an opportunity to increase your skill level in the lower level skills by practice.  For instance if you are doing some linear algebra and you get the matrices right but mess up some arithmetic, DO THE PROBLEM OVER.  Don't say, &quot;oh, I got the higher level concept&quot;.  Instead look on it as an opportunity to perfect the fundamental skills.  (Like doing pushups for punishment...you get good at pushups and stop messing up the football plays also...it's just how the human animal works...we are not computers...we have to be trained like dogs.  We are protein, not silicon.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would also talk to an instructor; devise a plan for remediation/improvement and then check in with him as you work through it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;All that said (2-5), GOTO (1).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-11-02T14:59:11.373" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14736" PostTypeId="2" ParentId="14731" CreationDate="2018-11-02T17:00:30.573" Score="7" Body="&lt;p&gt;One of the difficult aspect of learning and teaching mathematics is that it is in part akin to learning or teaching a new language. Words are used with precise technical meanings that are often divorced from their colloquial, everday uses. A simple, commonly problematic example is the word &quot;sphere&quot;, that a mathematician uses to mean the surface of a perfectly round object, but which is used by many as a synonym for the solid that the mathematician calls a &quot;ball&quot;. An example that vexes students of linear algebra is the use of &quot;vector&quot; to refer to an element of an abstract vector space, for example a space of polynomials or matrices. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My sense is that this issue is best addressed openly and explicitly, by explaining to students that the mathematical use of whatever terminology is in play, although perhaps partly motivated by some (perhaps long ago) colloquial use, is different from the colloquial use, and has a particular meaning specific to the context of its mathematical use. So, in a mathematical context, a mathematician would speak of a maximum admissible speed, or an absolute bound on the speed of a moving object, reserving the word &quot;limit&quot; for reference to a particular mathematical construct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At early stages of learning this problem is sometimes even more pronounced because the distinction between mathematical speech and colloquial speech is emphasized less, or sometimes intentionally set aside. When an algebraic geometer speaks of &quot;perverse sheaves&quot; there is not much danger of thinking degenerate thoughts about agricultural products, but when a function is defined to be convex if its graph lies below its secant lines and the student is also learning about convex and concave lenses in physics, there is a lot of potential for confusion, particularly if no one points out to the student that the words convex and concave as used technically by mathematicians and physicists are not exact synonyms, at least not without some additional clarification. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-11-02T17:00:30.573" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14737" PostTypeId="2" ParentId="14729" CreationDate="2018-11-02T17:08:43.337" Score="6" Body="&lt;p&gt;You say that the questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Dividing two polynomials results in another polynomial,&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Dividing two polynomials always results in another polynomial ?&quot;&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;are equivalent. In addition, you say that &quot;false&quot; must be understood as &quot;sometime false&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in front of a question &quot;always &amp;lt;...&gt; ?&quot; answer &quot;sometimes false&quot; is not a valid construction. Thus, it seems students are receiving contradictory instructions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About the other possible set of answers &quot;always/sometimes/never&quot;. Is &quot;sometimes&quot; exclusive of &quot;always&quot; ? In mathematics we use &quot;for all&quot; and &quot;exists&quot;. If &quot;sometimes&quot; is equivalent to &quot;exists&quot; (and assuming &quot;always&quot; is equivalent to &quot;for all&quot;), then &quot;sometimes&quot; is not exclusive to &quot;always&quot;, it should be clarified as &quot;sometimes but not always&quot;.&lt;/p&gt;&#xA;" OwnerUserId="7441" LastActivityDate="2018-11-02T17:08:43.337" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14738" PostTypeId="1" CreationDate="2018-11-03T20:25:39.533" Score="5" ViewCount="495" Body="&lt;p&gt;I have to do a teaching demo on the following:&lt;/p&gt;&#xA;&lt;p&gt;Please treat the committee as students in your Calculus II class. Please take 12-15 minutes to introduce your lesson on the Taylor Series and its applications. Take the final two minutes to indicate how you would finish your lesson for the day&lt;/p&gt;&#xA;&lt;p&gt;Here's my plan:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Begin by stating that a Taylor Series is an expansion of a function into an infinite sum of terms.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Present the Taylor expansion for &lt;span class=&quot;math-container&quot;&gt;$\cos x=1-x^2/2!+x^4/4!+...$&lt;/span&gt; as an example&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use Desmos (a computer will be provided) to show how the more sequence of terms that we have, the better the approximation of the &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt; graph. Example:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qc1yH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qc1yH.png&quot; alt=&quot;Approximation of cos x&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The red line is the cos x graph and the blue line is the approximation.&lt;/p&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Talk quickly about why the formula works (getting &lt;span class=&quot;math-container&quot;&gt;$c_0, c_1, c_2$&lt;/span&gt;) using &lt;span class=&quot;math-container&quot;&gt;$f(x) = c_0 + c_1(x-a) + c_2(x-a)^2 + c_3(x-a)^3+... $&lt;/span&gt; to obtain &lt;span class=&quot;math-container&quot;&gt;$f(x) = f(a) +  f'(a)/1! (x-a) +  f''(a)/2!(x-a)^2 +  f'''(a)/3!(x-a)^3 + ...$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use this formula to obtain the Taylor series expansion of &lt;span class=&quot;math-container&quot;&gt;$cos x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have a student centered formative assessment last 2-3 minutes and find the Taylor expansion for sin x&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;My only concern is: Is this too much for a 12-15 minute demo?&lt;/p&gt;&#xA;" OwnerUserId="2198" LastEditorUserId="6053" LastEditDate="2021-03-03T03:52:44.540" LastActivityDate="2021-03-25T21:54:51.123" Title="Advice on Full Time Community College Math Instructor Position" Tags="|calculus|community-colleges|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14739" PostTypeId="1" CreationDate="2018-11-03T21:40:35.507" Score="2" ViewCount="121" Body="&lt;p&gt;Is it possible and is it a good idea to introduce the additive group and metric space &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}/2i\pi \mathbb{Z}$&lt;/span&gt; &lt;strong&gt;very soon&lt;/strong&gt;, at the same time as the complex logarithm  &lt;span class=&quot;math-container&quot;&gt;$\log(r e^{i \theta}) = \ln(r)+i \theta, \theta \in (-\pi,\pi], r &amp;gt; 0$&lt;/span&gt; ? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With &lt;span class=&quot;math-container&quot;&gt;$\log : \mathbb{C}^* \to \mathbb{C}/2i\pi \mathbb{Z}$&lt;/span&gt; we can say &lt;span class=&quot;math-container&quot;&gt;$\log(ab) = \log(a)+\log(b), \log(a^n) = n \log(a)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\prod_n (1+a_n)$&lt;/span&gt; converges iff &lt;span class=&quot;math-container&quot;&gt;$\sum_n \log(1+a_n)$&lt;/span&gt; converges. &lt;/p&gt;&#xA;" OwnerUserId="10524" LastActivityDate="2018-11-03T21:40:35.507" Title="Complex logarithm and $\mathbb{C}/2i\pi \mathbb{Z}$" Tags="|complex-numbers|logarithm|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14740" PostTypeId="2" ParentId="14738" CreationDate="2018-11-03T23:41:51.567" Score="4" Body="&lt;p&gt;Caveat:  I'm just a random Internet poster, not an instructor, so take this with a grain of salt.  And get some advice from blooded veterans.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A.  It's too much.  1 and 2 are plenty for 12 minutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B.  Write out the entire 50 minute lecture, practice it at once, and write up a synthesis of the last 35 minutes.  I suggest something like topic, time, key points, perhaps purpose (you can draw a 4 column table and fill it out in the 3 minute).  Make it simple enough so that you can actually explain it in 2 minutes.  Test yourself to see that you can (it is simple enough summary).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C.  Your beginning should have some intro that is a little more motivational, less dry (1 is too theoretical).  Maybe something like &quot;sins and cosines are complicated and polynomials are easy so we like to change them into something easier to work with&quot;  Or &quot;you will need this in physics&quot;  or whatever the practical rationale is for why this is in the curriculum (not too real analysis-y, please).  [I do quite like 2, though.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;D.  Remember that your target audience at a juco is not math superstars.  They are a lower skill set than you or even than an AP calculus class.  Good people who want to get through this to support their chem/physics class or nursing degree or what have you.  They are looking to progress and get jobs or to transfer to lower level state schools (and then get jobs).  Have some sympathy for this and for them (without being obvious or patronizing about it either). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.  I suspect the interview committee wants to see that you can manage time, organize your thoughts, are practiced, command the room, have some energy, etc.  And can get through the topic without getting too tied up into every nuance.  I think you have plenty of math chops and that will not be their main worry (that you know the topic well).  Sure review the standard lesson and be absolutely up to speed on it (especially if they probe...but if they don't, don't feel a need to flaunt.). But the objective is probably 20% math skill (and mostly about being above a skill threshold rather than how high above it).  80% is instructional ability...which is very, very strongly correlated to planning the lecture and practicing it at least 3 times.  [Don't bother with that level of time investment when doing the job, except for first lecture, but definitely for the interview.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;F.  Do a little reconnaissance and figure out what text they use.  See how it addresses this topic and work the homework problems in that text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;G.  Really this whole topic is a little bit of a pain for the student and not the most important material.  Maybe even why they picked it.  Show you can get the kids through the damned thing.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-11-03T23:41:51.567" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14741" PostTypeId="2" ParentId="14738" CreationDate="2018-11-04T00:47:32.830" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Please take 12-15 minutes to &lt;em&gt;introduce&lt;/em&gt; your lesson&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Your intuition is correct that this is far too much for a 12-15 minute lesson. The committee knows that there is no way you could deliver a whole lesson, even an introductory one, on this topic.&lt;/p&gt;&#xA;&lt;p&gt;Since you've only got 15 minutes, plan to use it to spark interest in the topic, knowing that you're only showing the committee the very beginning.&lt;/p&gt;&#xA;&lt;p&gt;For example, why not show students the cosine function on a small interval around &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;, and ask them what polynomial it reminds them of? &amp;quot;Does it look like some kind of &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x^3$&lt;/span&gt;?&amp;quot; Let them decide which one, and then graph their guess on top of &lt;span class=&quot;math-container&quot;&gt;$y=\cos(x)$&lt;/span&gt;. You will have the opportunity to discuss the vertical intercept and the width of the graph – that it can be adjusted (by multiplying a term by a constant). In this way, your lesson could start as a game about building the cosine function from simpler functions, and students will be involved in the guessing process. I see this occupying up to 5 minutes. Then do your steps 1 and 2, and you're probably out of time.&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, using your lesson time to draw on your students' previous experience with trig and power functions helps them see that they are already capable of doing much of the intuitive work leading to Taylor polynomials, and it validates that this &lt;em&gt;new&lt;/em&gt; thing is something that doesn't have to seem like just applying another magic math formula.&lt;/p&gt;&#xA;&lt;p&gt;You should definitely plan to tell the committee what else the lesson would include (if you had the time), but I also would tell them what they should already know. [e.g. &amp;quot;Hey, class...let's remember back when we studied graphs of power functions. Do you all remember how to draw the graph of &lt;span class=&quot;math-container&quot;&gt;$y=-x^2$&lt;/span&gt;?&amp;quot;]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Have a student centered formative assessment last 2-3 minutes and find the Taylor expansion for sin x&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You could considerably simplify this assessment (for an introduction) by asking them what &lt;em&gt;just the first term&lt;/em&gt; of the series expansion should be. If you took the time to remind them of the &amp;quot;long-run behavior&amp;quot; of a polynomial when you discussed the cosine function, you could let them use their calculators to determine what the 2nd term of the series expansion of &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt; is.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="6053" LastEditDate="2021-03-25T21:54:51.123" LastActivityDate="2021-03-25T21:54:51.123" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14742" PostTypeId="2" ParentId="14738" CreationDate="2018-11-04T07:49:59.587" Score="0" Body="&lt;p&gt;I'd take an entirely different approach, though the faculty that is assessing you may not view it favorably.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematicians tend to view the logic of math as intrinsically beautiful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The vast majority of people do not. The vast majority considers math to be hoop jumping and weirdly arbitrary rules that must be followed to get check marks. They have no idea why anyone would care about formulae.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simply telling students &quot;Taylor Series are important!&quot; does not motivate them. Glossing over it or lecturing them for a few minutes on it will also fail to motivate them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To begin addressing this, I would use intellectual need to motivate the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in lower levels of math, one might teach fractions as part of a whole or something like that, or sections of circles, or naming amount of food. Or, you could make them feel intellectual need by having them how they'd share 5 pieces of licorice among 3 kids and name the amount each kid got.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could teach exponents as repeated multiplication then do a bunch of notational drills. Or, you could make students feel the need for exponents by teaching them about the spread of infectious diseases. &quot;If 3 people are infected initially and each sick person infects one more person each week, what will happen in 6 months?&quot; Then, you make a table, and let them struggle for a while to figure out that in week, say, 10, they calculate the number of infections by going &lt;span class=&quot;math-container&quot;&gt;$3\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2$&lt;/span&gt;, which is just a pain to read and write.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the eyes of most mathematicians, the average person, including college students, have &lt;em&gt;shockingly&lt;/em&gt; little interest in the logic of math because they just want to get answers and check marks. So, what will you do to make them feel the &lt;em&gt;need&lt;/em&gt; for a Taylor Series?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Check out the work on &lt;a href=&quot;https://www.math.ucsd.edu/~jrabin/publications/ProblemFreeActivity.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Intellectual Need&quot; in math education&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The risk: The &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Curse_of_knowledge&quot; rel=&quot;nofollow noreferrer&quot;&gt;Curse of Knowledge&lt;/a&gt;&quot; unfortunately, implies a very high probability that the people assessing you assume that students will intuitively understand that Taylor Series are interesting and important, and thus just need to be told so. Experts, very frequently, are shocked to find out that basic concepts, such as why anyone cares about Taylor Series, completely fly over the head of students who ace quizzes and tests on Taylor Series.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2018-11-04T07:56:57.160" LastActivityDate="2018-11-04T07:56:57.160" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14744" PostTypeId="2" ParentId="14729" CreationDate="2018-11-05T20:58:26.553" Score="7" Body="&lt;p&gt;With extensive anecdotal experience, by now I scrupulously avoid such questions (and analogous ones that exactly hit at the incompatibilities between &quot;ordinary&quot; language and mathematical language), at all levels, exactly because of the ambiguities you mention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the problem is that &quot;order of quantifiers&quot; even in very careful ordinary English need not match the order of occurrence in a sentence. E.g., `property &lt;span class=&quot;math-container&quot;&gt;$P(x)$&lt;/span&gt; holds for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;' has the for-all after the assertion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use of articles (a, an, the) is highly charged in mathematics, and fairly charged in ordinary English, but in different ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And the classic &quot;sequence&quot; versus &quot;series&quot;, which are synonyms in English, but (traditionally?) we make a big deal about in math?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Altogether, it seems to me perverse to exactly &quot;test&quot; students (at all levels) on the incompatibility of ordinary language versus mathematical usage, when there are more direct and substantive questions available. And when it is easily possible to spend a few more words to clarify what the &lt;em&gt;mathematical&lt;/em&gt; (as opposed to linguistic) question really is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've seen this on long-answer calculus problems, long-answer upper-division stuff, graduate qualifying exams, and so on, for 40+ years now... and, yes, some colleagues do assert that &quot;this is part of what math is about&quot;.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2018-11-05T20:58:26.553" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14745" PostTypeId="2" ParentId="14710" CreationDate="2018-11-06T19:02:18.017" Score="1" Body="&lt;p&gt;I have had much of the same history and ended up as a math major with good grades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I very strongly recommend &lt;strong&gt;Schaum's outline series&lt;/strong&gt; in any subject but especially mathematics. They have all the problems you will ever see. In time you will find any new problem will be a different version of one you have seen elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you must do is keep going into the area you fear, whether the dark forest or the other side of the tracks, until the fear is worn down to nothing and can be managed. Doing problems at home or other safe zone until you cannot be surprised or intimidated by them is the way to do this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your past is similar to mine but is not your future if you work at it. You were not born as a programmer but enjoyed the reinforcements of success you had when you studied it. It will take time to find that reinforcement while doing math but it is there. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Working while being watched is also a common problem. It splits your focus and allows you to be distracted. A tough problem. I had a friend sing (loudly) at me while I worked sets of new problems. It was exhausting but fun in the end. Such things cannot &quot;cure&quot; but do take the edge off such problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry about sounding trite but; your goals are worth the trouble you will need to go through to achieve them. If you give these up you will still find trouble in any other direction you choose. You may as well pursue one you care about.&lt;/p&gt;&#xA;" OwnerUserId="10531" LastActivityDate="2018-11-06T19:02:18.017" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14746" PostTypeId="1" CreationDate="2018-11-06T20:36:59.040" Score="2" ViewCount="442" Body="&lt;p&gt;Is there a case to be made that the topic of line integrals should only involve vector fields?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My colleagues and our textbook take the position that line integrals should only be taught from a vector field perspective (specifically for calculating &quot;work&quot;). [In fact, our textbook &lt;em&gt;defines&lt;/em&gt; a line integral as &lt;span class=&quot;math-container&quot;&gt;$\int _C \mathbf{F} \cdot d\mathbf{r}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\mathbf{F}$&lt;/span&gt; is a vector field and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is some parametrized curve.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it makes more pedagogical sense to introduce line integrals as a way to generalize what students should have just done in their integral calculus class: integrate a function along some 1-dimensional direction. Now, that &quot;direction&quot; can be a path in 2-space or 3-space, so we can see an area as the result of the line integral. Then, after introducing vector fields, we can consider other, meaningful things to integrate along a path, such as a dot product of the field with the path.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My motivation here is that I would like new calculus topics to be easily connected to old topics, if possible. If we jump right into calculating work without any tie back to the &quot;calculate the area&quot; problem students are used to, I fear they may come away thinking that line integrals are just these weird things with their own rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short: Is there a prevailing setting for introducing line integrals? If so, has there been a movement to minimize the teaching of line integrals over scalar fields, focusing primarily on work calculations?&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-11-07T11:05:27.010" Title="Line Integral Motivation" Tags="|calculus|community-colleges|vectors|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14747" PostTypeId="2" ParentId="14746" CreationDate="2018-11-06T22:43:38.287" Score="2" Body="&lt;p&gt;The problem is that there isn't a direct connection with area.  Sure, you can do a &quot;&lt;span class=&quot;math-container&quot;&gt;$\textrm{d}s$&lt;/span&gt;&quot; line integral, but that isn't quite the same thing is it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can we motivate &lt;span class=&quot;math-container&quot;&gt;$\int_\gamma P(x,y)\textrm{d}x + Q(x,y)\textrm{d}y$&lt;/span&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that it is very hard to do.  Some physical problems do require this, as you mention, but that seems like a much more narrow motivation than our motivation for the single variable integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only &quot;general&quot; motivation I have found is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In calculus, we use integrals to anti-differentiate. The analogue of the derivative in Calc 3 is the gradient. So it is natural to ask &quot;If you tell me that &lt;span class=&quot;math-container&quot;&gt;$\nabla f = \langle P(x,y) , Q(x,y) \rangle$&lt;/span&gt;, then what is &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; &quot;?  This is &quot;anti-gradienting&quot;.  In other words we want to be able to find a function &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{R}^2 \to \mathbb{R}$&lt;/span&gt; which satisfies&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{cases}&#xA;\frac{\partial f}{\partial x} = P(x,y)\\&#xA;\frac{\partial f}{\partial y} = Q(x,y)&#xA;\end{cases}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We cannot expect to be able to solve this system of PDEs in general, because we have equality of mixed partial derivatives.  So this equation cannot have a solution if &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial P}{\partial y} \neq \frac{\partial Q}{\partial x}$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say we &lt;strong&gt;know&lt;/strong&gt; that there is such a primitive function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; though.  How can I approximate it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say we know &lt;span class=&quot;math-container&quot;&gt;$f(1,1) = 5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\nabla f = \langle P(x,y), Q(x,y)\rangle$&lt;/span&gt;, and I want to approximate &lt;span class=&quot;math-container&quot;&gt;$f(2,3)$&lt;/span&gt;.  The gradient's defining feature is that it allows you to approximate small changes in the function value:  &lt;span class=&quot;math-container&quot;&gt;$f(\vec{p}+\vec{h}) \approx f(p)+\nabla f  \cdot \vec{h}$&lt;/span&gt;.  So if we know the gradient &lt;strong&gt;everywhere&lt;/strong&gt;, we can approximate &lt;span class=&quot;math-container&quot;&gt;$f(2,3)$&lt;/span&gt; by picking a path from &lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(2,3)$&lt;/span&gt;, chopping it up into a zillion tiny vector changes, and summing those small changes dotted with &lt;span class=&quot;math-container&quot;&gt;$\nabla f$&lt;/span&gt;.  You will end up recovering the notion of a line integral of &lt;span class=&quot;math-container&quot;&gt;$\nabla f$&lt;/span&gt; along this curve. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As noted though, not every vector field is conservative (because of Clairaut's theorem).  But we can still &quot;do a line integral&quot; to them exactly the same way we did line integrals of gradient vector fields. So I cannot interpret that line integral of a general vector field by thinking of it as the change in values of some potential function, but it is still useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One reason this is useful is because it gives us a test for whether a given vector field is conservative.  If &lt;span class=&quot;math-container&quot;&gt;$\langle P(x,y), Q(x,y)\rangle$&lt;/span&gt; were conservative, then the line integral would have to be independent of path.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I still do not have really &quot;down to earth&quot; examples of where I would want to use a line integral of a non-conservative vector field outside of physics or pure mathematics though.  I would love to find economic examples (perhaps connected to arbitrage?), engineering examples, biological examples, etc.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-11-06T22:43:38.287" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14748" PostTypeId="2" ParentId="14746" CreationDate="2018-11-07T11:05:27.010" Score="2" Body="&lt;p&gt;The integral of a scalar function on a curve is easily motivated as follows, and doing so is helpful for later motivating the definition of the integral of a vector field along an oriented curve. First, an interval on the real axis can be viewed as a very special sort of curve in the plane. Its length is the integral of the constant function &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; along this curve. The length of a polygonal curve is the sum of the lengths of its constituent segments. Any reasonable curve in the plane or space can be approximated by polygonal curves (parameterize the curve; partition the parameter domain and take the images of the points of the parition of the parameter domain as the vertices of the polygonal curve) (that a curve be &quot;reasonable&quot; will be defined a posteriori to mean that the approximation works). Write down the sum expressing the length of the approximating polygonal curve and observe that it can be interpreted as a Riemann sum approximating a certain integral. Take this integral as the definition of the length of the parameterized curve and note that its value does not depend on the choice of parameterization, so depends only on the image curve. Next observe that the preceding can be modified to calculate the mass of a infinitesimally thin wire. The mass density is interpreted as a scalar field along the curve. Assuming it to be approximately constant on each segment of an approximating polygonal curve one obtains a Riemann sum approximating an integral, that can be taken as the integral of the mass density along the curve. Again, observe that the result does not depend on the choice of parameterization of the curve. Finally, observe that everything continues to make sense dropping the positivity of the mass density (one can think of charge density). All this works for plane curves or space curves equally well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The preceding is helpful in motivating the definition of the integral of a vector field along an oriented curve. The line integral of a vector field formalizes the physical notion of the work done in moving a particle subject to a field of force represented by the vector field. This simple physical motivation requires that, unlike the integral of a scalar field, the integral of a vector field should be defined so as to depend on the orientation of the curve traversed, simply because the work done in moving against the stream is the negative of the work done in moving with the stream. Again, one can approximate the curve by an oriented polygonal curve, and suppose the vector field approximately constant on each segment of the approximating polygonal curve. The work done along each segment of the approximating polygonal curve is given by the projection of the vector field along this segment. Summing up the result gives a sum that can be viewed as a Riemann sum for an integral that is now taken as the definition of the integral of the vector field along the curve. The definition depends on the choice of a parameterization of the curve, and one checks that the number it yields does not depend on the parameterization, as along as the orientation is unchanged, and changes sign if the orientation is reversed. Directly this defines the integral of the vector field &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; along the oriented curve &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\int_{a}^{b}F(\gamma(t))\cdot \dot{\gamma}(t)\,dt$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\gamma:[a, b] \to C$&lt;/span&gt; is a parameterization of &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; consistent with the orientation of &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;. The &lt;em&gt;notation&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\int_{C}F\cdot dr$&lt;/span&gt; is shorthand that indicates the pairing of the functional argument (the vector field &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;) and the geometric argument (the oriented curve &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;); the &lt;span class=&quot;math-container&quot;&gt;$\cdot dr$&lt;/span&gt; has no &lt;em&gt;meaning&lt;/em&gt;, rather it should be viewed as part of the notation &lt;span class=&quot;math-container&quot;&gt;$\int_{\,\,} \,\,\cdot dr$&lt;/span&gt; that could equally well be written &lt;span class=&quot;math-container&quot;&gt;$\langle\,\,,\,\,\rangle$&lt;/span&gt; for the pairing of the functional and geometric argument (in fact, in more sophisticated contexts, one often sees notations such as &lt;span class=&quot;math-container&quot;&gt;$\langle F, C\rangle$&lt;/span&gt;). This notation is meant to suggest the operations that need to be performed in order to evaluate the pairing without indicating the choice that needs to be made to effect such an evaluation (and on which the result does not depend).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The preceding approaches adapt straightforwardly to the integration of scalars and vector fields over surfaces. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea of eliminating the integration of scalar fields along curves and surfaces from the curriculum seems to me wrongheaded. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The calculation of things as simple as the mass or center of mass of a plate, calculations all engineering students have to make, requires the notion of the integral of a scalar quantity. From a pedagogical point of view it is better to take a steppingstone approach - first extend the familiar integration of scalar quantities over intervals to the integration of scalar quantities over curves - then extend the latter to the integration of vector fields. An integral has two arguments, one functional and one geometric. First generalize the geometric argument (intervals to curves), then generalize the functional argument (scalars to vectors) - this is better than generalizing both at once simply because it compartamentalizes ideas. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-11-07T11:05:27.010" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14749" PostTypeId="1" CreationDate="2018-11-07T16:08:49.267" Score="4" ViewCount="245" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is a point of the circle of diameter &lt;span class=&quot;math-container&quot;&gt;$JK$&lt;/span&gt;. The perpendicular bisector of &lt;span class=&quot;math-container&quot;&gt;$JK$&lt;/span&gt; cut the semi-circle not containing &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; be the orthogonal projections  of &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$IJ$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$IP$&lt;/span&gt;. Prove that &lt;span class=&quot;math-container&quot;&gt;$\angle JMN=\angle PMK$&lt;/span&gt;. The next question asks to prove that &lt;span class=&quot;math-container&quot;&gt;$JMN$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$KMP$&lt;/span&gt; are congruent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CVEsS.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CVEsS.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How I did it:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\angle JMN=90-\angle MJN=90-\left(\angle MJK+\angle KJI\right)=90-(45+\angle KJI)=90-\angle KJI-45=\angle IKJ -45=180-45-\angle MKP -45=90-\angle MKP=\angle PMK$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe there's a shorter way in this case but how can I simplify angle chasing for students?&lt;/p&gt;&#xA;" OwnerUserId="10535" LastActivityDate="2019-11-25T12:18:47.827" Title="How to explain angle hunting to students" Tags="|secondary-education|geometry|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14750" PostTypeId="2" ParentId="14428" CreationDate="2018-11-07T18:12:53.650" Score="2" Body="&lt;p&gt;You might wish to take a look at some online grading applications:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;GradeCam    -   &lt;a href=&quot;https://gradecam.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gradecam.com/&lt;/a&gt;   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;AKINDI      --      &lt;a href=&quot;https://akindi.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://akindi.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exam Reader --  &lt;a href=&quot;https://bebyaz.com/ExamReader&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://bebyaz.com/ExamReader&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All In Learning  -  &lt;a href=&quot;http://www.allinlearning.com/scanning/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.allinlearning.com/scanning/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;FlashGrade  -    &lt;a href=&quot;https://www.flashgrade.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.flashgrade.com/&lt;/a&gt;    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gradescope  -   &lt;a href=&quot;https://gradescope.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gradescope.com/&lt;/a&gt;     &lt;/p&gt;&#xA;&#xA;&lt;p&gt;QuickKey    -   &lt;a href=&quot;https://get.quickkeyapp.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://get.quickkeyapp.com/&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;ZipGrade    -   &lt;a href=&quot;https://www.zipgrade.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.zipgrade.com/&lt;/a&gt;   &lt;/p&gt;&#xA;" OwnerUserId="10536" LastActivityDate="2018-11-07T18:12:53.650" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14751" PostTypeId="1" AcceptedAnswerId="14754" CreationDate="2018-11-07T20:10:59.153" Score="13" ViewCount="6987" Body="&lt;p&gt;Is there a mathematical reason (like a proof) of why this happens? You can do it with examples and it is 'intuitive.' But the proof of why this happens is never shown in pedagogy, we just warn students to remember to flip the inequality when&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;multiply or divide by a negative number both sides&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$-2&amp;gt;-3 \implies 2 &amp;lt; 3$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;take reciprocals of same sign fractions both sides&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{3}{4} &amp;gt; \frac{1}{2} \implies \frac{4}{3} &amp;lt; 2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-11-09T18:02:20.047" Title="Why do inequalities flip signs?" Tags="|proofs|" AnswerCount="9" CommentCount="4" ClosedDate="2018-11-10T03:43:27.377" ContentLicense="CC BY-SA 4.0" />
  <row Id="14752" PostTypeId="2" ParentId="14751" CreationDate="2018-11-07T21:30:36.413" Score="12" Body="&lt;p&gt;Depending on the context and the previous curriculum, the following might work:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&quot;less than&quot; means &quot;to the left of&quot; on the number line. &lt;/li&gt;&#xA;&lt;li&gt;Multiplying by a negative number flips numbers around 0.&lt;/li&gt;&#xA;&lt;li&gt;Thus, &quot;left of&quot; becomes &quot;right of&quot;, or &quot;greater than&quot;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="667" LastActivityDate="2018-11-07T21:30:36.413" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14753" PostTypeId="2" ParentId="14751" CreationDate="2018-11-07T23:41:34.257" Score="5" Body="&lt;p&gt;For multiplying or dividing by -1...&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align}&#xA;a&amp;amp;&amp;gt;b\\&#xA;a-b&amp;amp;&amp;gt;0\\&#xA;\\-(a-b)&amp;amp;&amp;lt;0\\&#xA;-a&amp;amp;&amp;lt;-b&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;(You can then extended to arbitrary negative numbers by multiplying or dividing by the [positive] magnitude.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For taking reciprocals... assuming &lt;span class=&quot;math-container&quot;&gt;$ab&amp;gt;0$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align}&#xA;a&amp;amp;&amp;gt;b\\&#xA;\left(\frac{1}{ab}\right)a&amp;amp;&amp;gt;\left(\frac{1}{ab}\right)b\\&#xA;\frac{1}{b}&amp;amp;&amp;gt;\frac{1}{a}\\&#xA;\frac{1}{a}&amp;amp;&amp;lt;\frac{1}{b}\\&#xA;\end{align}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="10537" LastActivityDate="2018-11-07T23:41:34.257" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14754" PostTypeId="2" ParentId="14751" CreationDate="2018-11-07T23:43:31.083" Score="39" Body="&lt;p&gt;I'm slightly concerned that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a mathematical reason (like a proof) of why this happens?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is a purely mathematical question, but since you write &quot;we just warn students&quot; I will assume that this question is purposefully asked here on Math Educators StackExchange.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to a proof:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;b$&lt;/span&gt;, subtract &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; from both sides: &lt;span class=&quot;math-container&quot;&gt;$0 &amp;gt; b-a$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, subtract &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; from both sides: &lt;span class=&quot;math-container&quot;&gt;$-b &amp;gt; -a$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this final inequality is equivalent to &lt;span class=&quot;math-container&quot;&gt;$-a &amp;lt; -b$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And so we have proved: If &lt;span class=&quot;math-container&quot;&gt;$a &amp;gt; b$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$-a &amp;lt; -b$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-11-07T23:43:31.083" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14756" PostTypeId="2" ParentId="14751" CreationDate="2018-11-08T10:22:39.390" Score="6" Body="&lt;p&gt;Perhaps one way to see (and explain intuitively to children) the &quot;multiply by &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; part&quot; is the following. Imagine your two numbers, &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, lying on the numberline. Multiplying by &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; is like 'rotating the numberline through 180°': imagine it's a straight metal pole, lying flat on the ground; pick it up by the middle, and rotate it 'long ways' (ie not 'barrel roll') 180°. (One only needs to consider the line segment &lt;span class=&quot;math-container&quot;&gt;$[-\max\{a,b\},\max\{a,b\}]$&lt;/span&gt; for this, in case you get a smart-arse saying you can't move something of infinite mass!)&#xA;It is hopefully clear to most people why the point that was to the right is now to the left.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is only a visual intuition -- one of course needs to make this a rigorous proof, and the other answers do this -- however such intuition can often be valuable to early-learners.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar, but slightly less clean, statement can be made regarding inversion: take just the positive real axis &lt;span class=&quot;math-container&quot;&gt;$(0,\infty)$&lt;/span&gt;; rotate it around the number &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. (Since, again, we can look at &lt;span class=&quot;math-container&quot;&gt;$(0,M)$&lt;/span&gt; for say &lt;span class=&quot;math-container&quot;&gt;$M = a + b$&lt;/span&gt; (with &lt;span class=&quot;math-container&quot;&gt;$a,b &amp;gt; 0$&lt;/span&gt;), one could imagine some sort of ellipse.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would draw graphics to illustrate... if only I were better at doing so!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. In complex analysis, multiplying by &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; really &lt;em&gt;is&lt;/em&gt; rotating by 180°!&lt;/p&gt;&#xA;" OwnerUserId="10542" LastEditorUserId="10542" LastEditDate="2018-11-08T16:37:15.333" LastActivityDate="2018-11-08T16:37:15.333" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14757" PostTypeId="1" CreationDate="2018-11-08T14:17:09.327" Score="11" ViewCount="799" Body="&lt;p&gt;I teach math in university, in France. This semester I have first-year bachelor students. I am becoming increasingly annoyed that they cannot remember the simple fact that &lt;span class=&quot;math-container&quot;&gt;$x^2 = a$&lt;/span&gt; has &lt;strong&gt;two&lt;/strong&gt; solutions for positive &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x = \sqrt{a}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x = -\sqrt{a}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know what to do. I've told them several time during lectures and exercise sessions. We had a whole class on determining square roots of complex numbers. And yet, I'm grading partial exams right now, and about half of them still wrote &lt;span class=&quot;math-container&quot;&gt;$x^2 = 3/2 \iff x = \sqrt{3/2}$&lt;/span&gt; when trying to find out the domain of definition of a rational function (assuming they wrote the &quot;iff&quot; sign at all, but it's a different question). And I have the nagging suspicion that if the equation was written as &lt;span class=&quot;math-container&quot;&gt;$x^2 - 3/2 = 0$&lt;/span&gt;, they would realize it's a quadratic equation that has two solutions (some of them do solve the question this way and painstakingly find &lt;span class=&quot;math-container&quot;&gt;$x = \pm \sqrt{6}/2$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What can I do?&lt;/p&gt;&#xA;" OwnerDisplayName="user1362" LastEditorUserId="64" LastEditDate="2019-04-04T00:50:47.977" LastActivityDate="2019-04-04T00:50:47.977" Title="How to help students understand/remember that $x^2 = a$ has two solutions?" Tags="|undergraduate-education|roots|" AnswerCount="9" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14758" PostTypeId="2" ParentId="14751" CreationDate="2018-11-08T14:50:27.560" Score="2" Body="&lt;p&gt;In a more general case this happens if you deal with a strictly monotonic decreasing function like f(x) = -x or f(x) = 1/x (in the positive or negative numbers).&lt;/p&gt;&#xA;" OwnerUserId="10545" LastEditorUserId="10545" LastEditDate="2018-11-08T17:33:00.613" LastActivityDate="2018-11-08T17:33:00.613" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14759" PostTypeId="2" ParentId="14751" CreationDate="2018-11-08T16:24:28.617" Score="1" Body="&lt;p&gt;Which would you rather I give you: 10 cookies or 8 cookies? Okay, now which would you rather have me take away: 10 cookies, or 8 cookies? Adding 10 gives you a larger number than adding 8, but subtracting 10 means leaves you with a smaller number than subtracting 8 does. Similarly, multiplying by 3 gives you more than multiplying by 2, but dividing by 3 gives you less than dividing by 2.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2018-11-08T16:24:28.617" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14760" PostTypeId="2" ParentId="14757" CreationDate="2018-11-08T16:38:24.837" Score="0" Body="&lt;p&gt;The square root symbol &lt;span class=&quot;math-container&quot;&gt;$\sqrt {\cdot}$&lt;/span&gt; means &quot;the principal square root&quot;. For positive numbers, it's understood that it means the positive square root. But your students clearly haven't advanced enough for this to be implicit. Insist that if they want to represent the positive square root of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, they have to write &lt;span class=&quot;math-container&quot;&gt;$\sqrt[+] x$&lt;/span&gt;. Note that I didn't write &lt;span class=&quot;math-container&quot;&gt;$+\sqrt x$&lt;/span&gt;. I'm treating positive sign as &lt;em&gt;part&lt;/em&gt; of the square root symbol. When you take the square root, write it as &lt;span class=&quot;math-container&quot;&gt;$\sqrt[\pm] x$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$\pm \sqrt x$&lt;/span&gt; to emphasize that you're  not taking &quot;the&quot; square root and then taking plus and minus that number; the square root operation itself gives two outputs. You should eliminate as much as possible the concept of &quot;the&quot; square root from your students' minds.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2018-11-08T16:38:24.837" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14761" PostTypeId="2" ParentId="14757" CreationDate="2018-11-08T16:49:32.927" Score="7" Body="&lt;p&gt;Telling and explaining is really not enough. Working exclusively in &lt;em&gt;square&lt;/em&gt; roots of numbers all but guarantees blocking vs interleaving problems and encourages shallow robotic thinking and generalizing involving roots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few ideas:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Error analysis. Have them look at a &lt;em&gt;variety&lt;/em&gt; of equations that boil down to something of the form &lt;span class=&quot;math-container&quot;&gt;$x^n=m$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is an even or odd whole number and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is a real number. Show them various (fictitious) students' work of varying qualities. Have them check and assess the work, then discuss. Why do some have two solutions and others have only one? Why do some have more?&lt;/li&gt;&#xA;&lt;li&gt;Multiple representations. Have them graph &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt; and have them use the graph to show the solution(s), if there are any, to &lt;span class=&quot;math-container&quot;&gt;$x^2=9$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^2=0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^2=-9$&lt;/span&gt;, etc. Then use the graph of &lt;span class=&quot;math-container&quot;&gt;$y=x^3$&lt;/span&gt; to solve &lt;span class=&quot;math-container&quot;&gt;$x^3=1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^3=8$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^3=-8$&lt;/span&gt;, etc. Compare and contrast. Encourage them to generalize as to how many solutions the equations will have depending on the degree of the polynomial. Relate this to the error analysis above.&lt;/li&gt;&#xA;&lt;li&gt;Positive incentivizes. Make a list of the 5 or 10 common mistakes that you see in class. Create a bonus quiz where students, on a blank piece of paper, recall all 10 mistakes from memory, explain and/or prove the wrongness in each, and demonstrate and check the right way to think about it. Another positive incentive: Tell them that regardless of the topic the class is on, at least 5% of every assessment will involve one of those 10 mistakes, so they have to keep studying them. Distributed practice makes a huge difference.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Negative incentives. Given the list of 5 or 10 most common mistakes has been released, tell them that any mistakes of that sort on the test will result in a zero (or a max score of 50% or something like that) for that portion of the test regardless of all other work.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A gentler version of those negative incentives. Only give feedback and grades on work that does not contain those common mistakes. e.g. Write something like this on their quiz. &quot;On question 8, you made one of the common mistakes. Search for it and fix it, then I will grade it. If you choose not to fix it, you get zero for question 8.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Good luck with this!&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2018-11-15T19:17:14.137" LastActivityDate="2018-11-15T19:17:14.137" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14762" PostTypeId="2" ParentId="14751" CreationDate="2018-11-08T19:34:50.487" Score="1" Body="&lt;p&gt;I see it as a combination of the following, when we assume a field with a positive cone, and trichotomy. We want to show that if &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c&amp;lt;0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$cb&amp;lt;ac$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It is natural to expect the product of positive numbers to be positive. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We denote the additive inverse of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$-a$&lt;/span&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We note, from &lt;span class=&quot;math-container&quot;&gt;$a+(-a)=0$&lt;/span&gt;, that &lt;span class=&quot;math-container&quot;&gt;$-(-a)=a$&lt;/span&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We note that &lt;span class=&quot;math-container&quot;&gt;$(-a)b=-(ab)$&lt;/span&gt;. Proof: &lt;span class=&quot;math-container&quot;&gt;$(-a)b+ab=(-a+a)b=0\times b=0$&lt;/span&gt; (one can earlier deduce that multiplication by &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; from distributivity). &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Now &lt;span class=&quot;math-container&quot;&gt;$(-a)(-b)=-(a(-b))=-((-b)a)=-(-ab)=ab$&lt;/span&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;From the above we conclude that the product of two negative numbers is positive, and that a positive number times a negative is negative. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Now if &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c&amp;lt;0$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$b-a&amp;gt;0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-c&amp;gt;0$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$(-c)(b-a)&amp;gt;0$&lt;/span&gt;. That is, &lt;span class=&quot;math-container&quot;&gt;$-cb+ac&amp;gt;0$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$cb&amp;lt;ac$&lt;/span&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="475" LastActivityDate="2018-11-08T19:34:50.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14763" PostTypeId="2" ParentId="1" CreationDate="2018-11-08T20:44:19.790" Score="2" Body="&lt;p&gt;&lt;strong&gt;Establish that participation is compatible with being &quot;cool&quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think one aspect that has not been metioned yet, is that there is often social pressure to not participiate too much in the class in order to not to be seen as a &quot;uncool math geek&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This might sound strange to many math educators, because you are surrounded by people who are fascinated by math and you are one yourself. You enjoy passionate discussions about it. You can freely share that passion with your peers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;general population&quot; often still has a stereotypical, negative, view of math professors teachers or in simply people who are good at it. If you really like math and talk to your &quot;normal&quot; friends about how facinating certain math concepts are, often you will get blank stares and people will change the subject, especially during the school years. Thus many of us learned to surpess this passion and not showing it publicly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When they reach college, many will avoid participating too much, to avoid being seen like that. They don't want their &quot;cool&quot; friends (who are sitting next to them) to think: &quot;Oh look, he answers again. I bet, all he does in his free time is read math books. He probalby has no friends and is a virgin.&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when the teacher poses a question, they will not say anything although they could answer it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What imho can alleviate this, is saying something like &quot;Come on people, I know you are all tired from last night's partying, but let's see if we can solve this together.&quot; or &quot;Come on people, I know this is hard but let's crack this one together&quot;. [There are probably better examples]. This gives students an excuse and kind of implies that they are one of those tired partying people and therefore not a &quot;nerd&quot; but someone with a &quot;social life&quot;. As soon as some of the &quot;cool kids&quot; start participating it becomes &quot;socially acceptable&quot; and the general participation will be  much higher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please don't think: &quot;Well, if they think this way, I don't care if they learn something or not.&quot; Imho thats how social groups and social pressure works and educators should be aware of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, don't think students are only afraid to be embarassed by being wrong, they are also afraid of being seen as the &quot;uncool math nerd that has no friends&quot;. Change that!&lt;/p&gt;&#xA;" OwnerUserId="10550" LastActivityDate="2018-11-08T20:44:19.790" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14764" PostTypeId="2" ParentId="1" CreationDate="2018-11-09T01:28:22.177" Score="2" Body="&lt;p&gt;Consider using the advice given by Catherine Attard in her 'Framework for Engagement with Mathematics' &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Engagement with Mathematics: What Does It Mean and What Does It Look Like?&#xA;  Attard, Catherine&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Australian Primary Mathematics Classroom, v17 n1 p9-13 2012&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;When discussing issues surrounding mathematics education, the topic of student engagement (or lack of) often dominates conversations. The low levels of engagement with mathematics experienced by students during the middle years have been of some concern to Australian mathematics educators and stakeholders in recent decades. Lowered engagement with mathematics has the potential to affect communities beyond the need to fill occupations that require the use of high level mathematics. It can also limit one's capacity to understand life experiences through a mathematical perspective. This article explores the concept of engagement against the backdrop of a recent longitudinal study into the influences on student engagement during the middle years of schooling, provides some insight into students' perceptions of engaging mathematics lessons and introduces a &quot;framework for engagement with mathematics&quot; that could be used to inform planning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Full-text at &lt;a href=&quot;https://eric.ed.gov/?id=EJ978128&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://eric.ed.gov/?id=EJ978128&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a related article is at the following link: &lt;a href=&quot;http://archive.uuworld.org/2003/03/interpretation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://archive.uuworld.org/2003/03/interpretation.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might also be worth taking note of Steve Sigur’s remark that he preferred teaching high school students because they were still inquisitive, whereas college students have to be continually dragged along. Here is a link to the Youtube interview in which he makes that remark: &lt;a href=&quot;https://www.youtube.com/watch?v=h256d1sfPwk&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=h256d1sfPwk&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user10552" LastEditorDisplayName="user10552" LastEditDate="2018-11-10T16:52:16.430" LastActivityDate="2018-11-10T16:52:16.430" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14765" PostTypeId="1" AcceptedAnswerId="18971" CreationDate="2018-11-09T03:02:11.637" Score="8" ViewCount="511" Body="&lt;p&gt;I like to make the &quot;dominoes&quot; analogy when I teach my students induction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently came across the following video:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=-BTWiZ7CYoI&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=-BTWiZ7CYoI&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this video, a sequence of concrete block wall caps are set up like dominoes on the top of a wall.  The first wall cap is knocked down, setting off the domino effect.  The blocks are spaced so that they are resting on each other when they fall, but just barely. So rather than resting flat each block is supported slightly by its successor.  When the last block falls, however, it falls flat (having no subsequent block to rest on).  This causes the block behind it to slip off, and lay flat, which causes the brick behind it to slip off and lie flat, until all the blocks are lying flat perfectly end to end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any instance of a similar phenomena occurring in mathematics?  I am thinking of a situation in which you want to prove both &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q(n)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n = 1, 2, 3, \dots, 100$&lt;/span&gt; (say).  If you are able to prove: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$P(1)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall k \in \{1,2,3, \dots, 99\} P(k) \implies P(k+1)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$P(100) \implies Q(100)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall k \in \{ 100, 99, 98, \dots, 3,2\},  Q(k) \implies Q(k-1)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then it will follow that both &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q(n)$&lt;/span&gt; are true for &lt;span class=&quot;math-container&quot;&gt;$n = 1, 2, 3, \dots, 100$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If an example is found, it could be a great example for teaching because it would force students to think through the logic of why induction works rather than blindly following a certain form of &quot;an induction proof&quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2018-11-09T17:43:13.090" LastActivityDate="2020-11-28T17:29:36.070" Title="Is there any example of a &quot;forwards/backwards&quot; induction?" Tags="|proofs|discrete-math|induction|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14766" PostTypeId="2" ParentId="14757" CreationDate="2018-11-09T05:50:55.993" Score="0" Body="&lt;p&gt;I have similar issues, and handle it in the following way:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;On the first day of class I do a review of basic algebra, things like:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{a + b} \neq \sqrt{a} + \sqrt{b} $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{ab} = \sqrt{a}\sqrt{b}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$a(b + c) = ab + ac$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Give a list of what I call 'fatal errors', things like what you mention, give demonstrations of why these things are the way they are, e.g.:&#xA;if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; even:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^n = b \hspace{.5in} (b &amp;gt; 0)$$&lt;/span&gt;&#xA;has 2 solutions, namely &lt;span class=&quot;math-container&quot;&gt;$$\pm\sqrt[n]{b} $$&lt;/span&gt;&#xA;if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; odd:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^n = b \hspace{.5in} (b &amp;gt;0)$$&lt;/span&gt;&#xA;has 1 solution, namely &lt;span class=&quot;math-container&quot;&gt;$$\sqrt[n]{b}$$&lt;/span&gt;&#xA;and use some numbers that work out 'nicely' as examples, e.g.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(-3)^2 = (-3)(-3) = 9 = (3)(3) = 3^2 $$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Tell them I will stop grading their solution the moment I see one of these errors.&lt;/li&gt;&#xA;&lt;li&gt;Give them lots of examples to work, circulate to help individual students&lt;/li&gt;&#xA;&lt;li&gt;Work out solutions while being sure to explain my reasoning&lt;/li&gt;&#xA;&lt;li&gt;Make sure to write a specific comment on their exams about having stopped grading because they made one of these fatal errors&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10521" LastEditorUserId="10521" LastEditDate="2018-11-13T04:05:39.703" LastActivityDate="2018-11-13T04:05:39.703" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14767" PostTypeId="2" ParentId="14751" CreationDate="2018-11-09T06:51:36.487" Score="0" Body="&lt;p&gt;Other answers address multiplication by a negative number on both sides better than I can. For taking reciprocals of both sides, it may be helpful to examine why we know we can do that and maintain equality. This is one simple demonstration:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Starting with the equation &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}=\frac{c}{d}$&lt;/span&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;we can multiply both sides by &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{c}$&lt;/span&gt;, giving &lt;span class=&quot;math-container&quot;&gt;$\frac{ad}{bc}=\frac{cd}{dc}$&lt;/span&gt;, which simplifies to &lt;span class=&quot;math-container&quot;&gt;$\frac{ac}{bd}=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can then do the same with &lt;span class=&quot;math-container&quot;&gt;$\frac{b}{a}$&lt;/span&gt;, giving &lt;span class=&quot;math-container&quot;&gt;$\frac{bad}{abc}=\frac{b}{a}\cdot 1$&lt;/span&gt;, which simplifies to &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{c}=\frac{b}{a}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This shows that if two numbers are equal, then their reciprocals are equal. However, notice that we didn't just turn each number into its reciprocal. We also flipped their positions: the &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; on the left became &lt;span class=&quot;math-container&quot;&gt;$\frac{b}{a}$&lt;/span&gt;, but on the right. When we're talking about equality, this is inconsequential, since equality is symmetric. However, if you take the same example with &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; replaced with &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt;, the fact that the values &quot;switched places&quot; does matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This demonstration also shows why the sign only flips when both sides have the same sign. If they have different signs, then their reciprocals also have different signs, and exactly one of the multiplications by a reciprocal causes the inequality to flip. So you could argue that it does flip, but then it flips again, making it seem like it hasn't flipped. Similarly, if both sides are negative, it flips a total of three times, which has the same result as flipping once.&lt;/p&gt;&#xA;" OwnerUserId="10553" LastActivityDate="2018-11-09T06:51:36.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14768" PostTypeId="2" ParentId="14751" CreationDate="2018-11-09T10:22:56.380" Score="2" Body="&lt;p&gt;I can't help but feel this question highlights the self defeating nature of teaching maths via &quot;cheat tricks&quot;. In my opinion it is probably best not to teach students ever to just &quot;switch the sign&quot; of the inequality in an arbitrary manner, which can be confusing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As elegantly illustrated by @Benjamin_Dickman's proof, the alternative notation &#xA;&lt;span class=&quot;math-container&quot;&gt;$$a&amp;gt;b \quad \text{implies} \quad -b&amp;gt;-a$$&lt;/span&gt;&#xA;is much clearer, and invites the student to treat the inequality as a fixed equivalence in the same way they would treat an equality. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel it might be less confusing for students were this convention adopted, and the inequality was never reversed but for as a last resort. That would mean one less &quot;rule&quot; for students to &quot;remember&quot;.&lt;/p&gt;&#xA;" OwnerUserId="10555" LastActivityDate="2018-11-09T10:22:56.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14769" PostTypeId="1" CreationDate="2018-11-09T17:26:43.553" Score="5" ViewCount="246" Body="&lt;p&gt;Building off my own experience and the responses to &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/10478/rings-before-groups-in-abstract-algebra&quot;&gt;Rings before groups in abstract algebra?&lt;/a&gt;&quot; I've decided to teach Abstract Algebra using a rings-first approach. However the various texts mentioned in that post are not available for free to students. Our department tries to use open texts as much as possible.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does anyone know of and/or recommend an open text for Abstract Algebra that uses a rings-first approach?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Note: In the past, when taking a groups-first approach, I was very happy with &lt;a href=&quot;http://abstract.ups.edu/aata/aata.html&quot; rel=&quot;noreferrer&quot;&gt;Judson's text&lt;/a&gt;. However, the path to rings is a bit long in that text.)&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2018-11-19T00:49:43.487" Title="Open Rings-First Abstract Algebra Text" Tags="|textbooks|abstract-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14772" PostTypeId="1" AcceptedAnswerId="14773" CreationDate="2018-11-10T19:37:57.340" Score="5" ViewCount="308" Body="&lt;p&gt;I'm teaching an after school workshop for a few 7th graders. I was having them try to predict the next item in a complicated sequence. After some failed attempts, one of the kids started analyzing the differences between the numbers, then the differences between those differences, and used that to predict the next few items on the list. (This worked because the list was generated by a quadratic function but they don't know this part; they can predict the next list items but cannot write a formula yet).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm trying to assess the students and am not sure whether they just discovered that technique or had learned it in school already.  In what grade are sequences and common differences usually covered? I'm talking about in the US, specifically NYC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also interested in pushing the idea further with them, but ideally in a different direction form what will be done in school. How often does the following concept get covered in MS-HS, and when?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given a sequence &lt;span class=&quot;math-container&quot;&gt;$S=(s_1,s_2,s_3,\dots)$&lt;/span&gt;,&#xA;its sequence of common differences is&#xA;&lt;span class=&quot;math-container&quot;&gt;$D(S)=(s_2-s_1,s_3-s_2,s_4-s_3,\dots).$&lt;/span&gt;&#xA;Now take a polynomial&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(n)=c_kn^k+c_{k-1}n^{k-1}+\dots+c_0,$$&lt;/span&gt;&#xA;and form the sequence&#xA;&lt;span class=&quot;math-container&quot;&gt;$F=\big(f(0), f(1), f(2), \dots\big)$&lt;/span&gt;.&#xA;Then &lt;span class=&quot;math-container&quot;&gt;$D^k(F)$&lt;/span&gt;&#xA;is a constant string of the number &lt;span class=&quot;math-container&quot;&gt;$c_kk!$&lt;/span&gt;,&#xA;and using this one can determine the polynomial completely.&lt;/p&gt;&#xA;" OwnerUserId="9091" LastActivityDate="2018-11-11T23:29:59.083" Title="In what grade do kids (New York, US) learn common differences?" Tags="|curriculum|interactive-teaching|inquiry-based-learning|sequences|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14773" PostTypeId="2" ParentId="14772" CreationDate="2018-11-11T04:47:50.863" Score="5" Body="&lt;p&gt;The linear function component is covered early(ish) in Algebra 1, and quadratic functions are covered towards the end of Algebra 1; so, the former by 7th/8th grade and the latter - if at all - by 8th grade. But, even for students in New York (where I teach) taking Geometry in 9th grade, more general polynomials may not be covered until Algebra 2 in 10th grade. Even then, using (nth order) constant differences to solve for the corresponding (unique, degree n) polynomial is unlikely to be covered: It requires solving n equations in n variables, which is one reason that DL Renfro remarks that it may be saved for a first course in (post-secondary) Linear Algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I teach it in a Problem Solving &amp;amp; Problem Posing course to 12th grade students who have already taken, or are currently taking, differential Calculus. This is not the norm in the state/city, and/but I do not think that the students would otherwise see such an idea during (middle school or) high school beyond the linear and quadratic cases.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-11-11T04:47:50.863" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14774" PostTypeId="2" ParentId="14757" CreationDate="2018-11-11T18:56:29.337" Score="2" Body="&lt;p&gt;We are all dealing with this, of course. In a theoretical/philosophical sense, you can't &quot;make&quot; students learn anything (&lt;a href=&quot;https://www.phrases.org.uk/meanings/you-can-lead-a-horse-to-water.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;You can lead a horse to water...&quot;&lt;/a&gt;). In a practical/punitive sense, this is what course grades are for. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, consider one or more of the following options:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Simply grade exams rigorously on this concept. &lt;/li&gt;&#xA;&lt;li&gt;Give a lead-in multiple-choice or short-answer question on the Fundamental Theorem of Algebra. &lt;/li&gt;&#xA;&lt;li&gt;Award only half the maximum score for the question if only one of two solutions is given.&lt;/li&gt;&#xA;&lt;li&gt;Require in applications where only a positive solution is practically useful (e.g., solving a right triangle) that the mathematical two solutions be expressed (for extra practice), and then a separate natural-language statement using only the positive result.&lt;/li&gt;&#xA;&lt;li&gt;For the most radical approach, consider mastery grading; if this particular item is absolutely critical, then do not pass any student from the course until they can successfully answer such a question. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2018-11-11T18:56:29.337" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14775" PostTypeId="2" ParentId="14772" CreationDate="2018-11-11T19:13:46.787" Score="3" Body="&lt;p&gt;This is a topic I could imagine not being adequately covered in all U.S. schools, although (as Dave L. Renfro pointed out in a comment), it is listed in the Common Core Mathematics standards under high school functions (&lt;a href=&quot;http://www.corestandards.org/Math/Content/HSF/BF/A/2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;HSF.BF.A.2&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could imagine it being presented successfully anywhere from the 3rd grade up to an advanced college class for mathematics educators. As one example, the NYC community college where I teach probably has at least 5 different math courses where this gets treated, including our lowest-level liberal-arts quantitative-reasoning course; as well as (briefly) in our highest-level terminal course for mathematics and computing majors. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2018-11-11T23:29:59.083" LastActivityDate="2018-11-11T23:29:59.083" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14776" PostTypeId="2" ParentId="14757" CreationDate="2018-11-12T02:33:57.740" Score="8" Body="&lt;p&gt;There are possibly two different issues here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Issue 1 is that some students are under the mistaken impression that the symbol &lt;span class=&quot;math-container&quot;&gt;$\sqrt{5}$&lt;/span&gt; actually designates &lt;em&gt;two different numbers&lt;/em&gt;, one positive and one negative.  I have even heard high school teachers (in the United States) tell students this; it seems to be a very pervasive belief and you can find many examples of it on MSE.  If this is the case, then the problem is not that they don't know that there are two solutions, it's that they don't know the correct way to denote it.  Notation, of course, is just a convention, so this would be more of a failure to adapt to the common culture of mathematical writing than it would be a failure of comprehension.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to identify if this is, in fact, what's happening would be to give an example in which the square root is a simple whole number.  If you give the equation &lt;span class=&quot;math-container&quot;&gt;$x^2=9$&lt;/span&gt;, would they say &lt;span class=&quot;math-container&quot;&gt;$x=3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x=\pm 3$&lt;/span&gt;?  If the issue is purely notation -- that is, if the problem is that they think the &lt;span class=&quot;math-container&quot;&gt;$\pm$&lt;/span&gt; symbol is already &quot;built in&quot; to the radical sign -- then they will include it explicitly when the square root can be evaluated exactly, and drop it when the square root cannot be completely evaluated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second issue is this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I've told them several time during lectures and exercise sessions. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Maybe the problem is that you're &lt;em&gt;telling&lt;/em&gt; them during exercise sessions.  I don't know what the format of your exercise sessions is -- in particular I don't know whether &lt;em&gt;you're&lt;/em&gt; presenting the solutions, or having &lt;em&gt;students&lt;/em&gt; present solutions -- but if the students are the ones showing their work, and such a mistake occurs, you can stop and &lt;em&gt;ask&lt;/em&gt; &quot;Are you sure that's right?&quot;  In my experience this is even more effective when the answer is extremely simple.  I have had Calculus students go from &lt;span class=&quot;math-container&quot;&gt;$x^2=25$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x=5$&lt;/span&gt;; asking &quot;Are you sure that's right?&quot; typically catches them off-guard, precisely because &lt;em&gt;of course&lt;/em&gt; they're sure that &lt;span class=&quot;math-container&quot;&gt;$5^2=25$&lt;/span&gt;.  If that doesn't give them enough of a hint, prod them &quot;Is &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; the &lt;em&gt;only&lt;/em&gt; number whose square is &lt;span class=&quot;math-container&quot;&gt;$25$&lt;/span&gt;?&quot;  However you get them there, the goal should be for &lt;em&gt;them&lt;/em&gt; to tell &lt;em&gt;you&lt;/em&gt; that there's a second, negative solution.  If they figure it out themselves once, they will be much more likely to remember it than if they are told it fifty times.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-11-12T02:33:57.740" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14777" PostTypeId="1" CreationDate="2018-11-12T09:10:40.503" Score="5" ViewCount="276" Body="&lt;p&gt;To begin with I am working in a high school classroom where the students are working on the applications of vectors. The beginning of the lesson is about calculating direction and magnitude of vectors and follows with an example question that states: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Two ropes are tied to a wagon. a child pulls one with a force of 20 pounds, while another child pulls the other with a force of 30 pounds. If the angle between the two ropes is 28 degrees, how much force must be exerted by a third child, standing behind the wagon, to keep the wagon from moving? Hint: assume the wagon is at the origin and one rope runs along the positive x-axis. Find the resultant forces on the wagon from the ropes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now I know that students can solve this by thinking about it in terms of a parallelogram and applying the law of cosines and quadratic equation.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They can also use the magnitude formula &lt;span class=&quot;math-container&quot;&gt;$V = \sqrt {x^2+y^2}$&lt;/span&gt; where child 3 is approximately &lt;span class=&quot;math-container&quot;&gt;$(47.67, 9.39)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally it has been a while since I have worked with vectors and can't really think of other ways to approach this although I know there probably are. Is there other methods that would be appropriate to show high school students? This is an advanced math course where they have just finished law of sines and cosines, and end goal of vectors is mostly just airplane problems. Students are not expected to have any background knowledge in physics.&lt;/p&gt;&#xA;" OwnerUserId="10567" LastActivityDate="2018-11-13T21:46:31.310" Title="Presenting ways to find a resultant force" Tags="|secondary-education|vectors|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14778" PostTypeId="1" CreationDate="2018-11-12T10:47:30.133" Score="4" ViewCount="152" Body="&lt;p&gt;This is a pattern even school kids could discover (when gently pointed to). I never did conciously, and cannot remember to have been pointed to explicitly, neither at school nor later:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\color{red}{\mathbf{2}}\cdot 9 = 1\color{red}{\mathbf{8}}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\color{red}{\mathbf{8}}\cdot 9 = 7\color{red}{\mathbf{2}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\color{blue}{\mathbf{3}}\cdot 9 = 2\color{blue}{\mathbf{7}}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\color{blue}{\mathbf{7}}\cdot 9 = 6\color{blue}{\mathbf{3}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\color{green}{\mathbf{4}}\cdot 9 = 3\color{green}{\mathbf{6}}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\color{green}{\mathbf{6}}\cdot 9 = 5\color{green}{\mathbf{4}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which may come as kind of a miracle when first discovering it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematical terms&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\boxed{a\cdot (10-1) \equiv b \mod 10\ \ \ \ \Leftrightarrow\ \ \ \ \ b\cdot (10-1) \equiv a \mod 10 \\&#xA;a\cdot (10-1) \equiv b \mod 10\ \ \ \ \Leftrightarrow\ \ \ \ \ a + b = 10 \equiv 0 \mod 10}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This holds not only for &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; but for every &lt;span class=&quot;math-container&quot;&gt;$p \in \mathbb{N}$&lt;/span&gt;, i.e. in every &quot;number system&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\boxed{a\cdot (p-1) \equiv b \mod p\ \ \ \ \Leftrightarrow\ \ \ \ \ b\cdot (p-1) \equiv a \mod p \\&#xA;a\cdot (p-1) \equiv b \mod p\ \ \ \ \Leftrightarrow\ \ \ \ \ a + b = p \equiv 0 \mod p}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and is responsible for the fact that the &lt;a href=&quot;https://math.stackexchange.com/questions/2988910/graphical-multiplication-tables-for-mathbbz-p-mathbbz-and-mathbbz&quot;&gt;graphical multiplication tables of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}/p\mathbb{Z}$&lt;/span&gt;&lt;/a&gt; always looks the same for &lt;span class=&quot;math-container&quot;&gt;$p-1$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TSar2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TSar2.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder if there are attempts (in educational research and literature) to make use of the simple observability of the pattern above to explain to (clever) school kids that the observed regularity is not by pure coincidence, why it is so, and what it does &quot;mean&quot;.&lt;/p&gt;&#xA;" OwnerUserId="10568" LastEditorUserId="10568" LastEditDate="2018-11-12T12:38:55.670" LastActivityDate="2018-11-12T16:07:37.823" Title="Making modular arithmetic interesting for school kids" Tags="|secondary-education|student-motivation|primary-education|concept-motivation|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14780" PostTypeId="2" ParentId="14778" CreationDate="2018-11-12T16:07:37.823" Score="6" Body="&lt;p&gt;I would say &lt;strong&gt;probably not&lt;/strong&gt;; the &lt;strong&gt;work&lt;/strong&gt; (algebra mod n) and the &lt;strong&gt;payoff&lt;/strong&gt; (those digits come out in a neat pattern) are not in a good ratio.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your goal is to get students to do some of the &lt;strong&gt;work&lt;/strong&gt; (modular arithmetic), you should choose a different &lt;strong&gt;payoff&lt;/strong&gt;. I recommend &lt;a href=&quot;https://www.maa.org/press/periodicals/loci/joma/patterns-in-pascals-triangle-with-a-twist-first-twist-what-is-it&quot; rel=&quot;noreferrer&quot;&gt;Pascal's Triangle mod n&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Outline:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Introduce modular arithmetic as a &quot;what if:&quot; What if 4 = 0?&lt;/li&gt;&#xA;&lt;li&gt;Practice adding and multiplying numbers under the assumption that 4 = 0.&lt;/li&gt;&#xA;&lt;li&gt;Make a multiplication table where 4 = 0.&lt;/li&gt;&#xA;&lt;li&gt;Show them how Pascal's Triangle is generated with regular old integers.&lt;/li&gt;&#xA;&lt;li&gt;Give them &lt;a href=&quot;https://www.printablepaper.net/category/hexagon_graph&quot; rel=&quot;noreferrer&quot;&gt;hexagonal graph paper&lt;/a&gt; and have them use it to generate Pascal's Triangle where 4 = 0.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Km2Rb.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Km2Rb.png&quot; alt=&quot;From the linked MAA page&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a better graphical effect, tell them not to write any of the 0's (leave them blank). Then when you look at the picture from farther away, you will see all the cool patterns. If your students are super-diligent, you could demand different colors for different values, but I really recommend at least leaving the 0's blank; negative space is a great artistic technique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then let the students pick a different n and make their own. Bring multiple sizes of hexagonal graph paper so that students who are fired up can try mod 10 or mod 8 or mod 11 or whatever while some others will just make a little bit of mod 4. They will find mod 2 to be hilarious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify why I think this answers the question: I think the &lt;strong&gt;payoff&lt;/strong&gt; here is pretty great -- the students get to play in a weird field of math, they get the idea that math is huge and mysterious and cool, and they walk away with a pretty picture. The &lt;strong&gt;work&lt;/strong&gt; here doesn't even look like work to them. But if they want to later do the application you suggested, they will be more able to do so after this introduction anyway.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-11-12T16:07:37.823" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14781" PostTypeId="1" CreationDate="2018-11-12T16:09:42.517" Score="29" ViewCount="18784" Body="&lt;p&gt;I am curious why students who take calculus in high school often do so poorly in college calculus. I am an instructor at an engineering college and I've noticed a decent number of students who have calculus 1 on their high school transcript and yet fail abysmally at the college level. Is there such variance among what constitutes &quot;calculus&quot; in high school curriculum or that big a difference between high school and college calculus? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give you a bit more context about my situation, my institution is a moderately competitive engineering school that offers a 3 sequence calculus class that is mandatory for our engineers. With the one qualifier that &lt;strong&gt;&lt;em&gt;we do not allow our students to use advanced calculators that can do graphing or symbolic manipulation on our exams&lt;/em&gt;&lt;/strong&gt;, I would characterize our calculus 1 class as a pretty standard typical one where students focus on 5 main areas: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;They learn to use the traditional &quot;plug and chug&quot; formulas for differentiation and integration.&lt;/li&gt;&#xA;&lt;li&gt;They learn to &quot;translate&quot; word problems into mathematical problems they can solve with the tools of calculus (esp problems involving physics and engineering applications).&lt;/li&gt;&#xA;&lt;li&gt;They learn how to take limits (but we do not do delta-epsilon proofs).&lt;/li&gt;&#xA;&lt;li&gt;They develop an understanding of what an integral and derivative mean mathematically by applying the limiting processes along within a geometric setting to a few simple power functions.&lt;/li&gt;&#xA;&lt;li&gt;They learn and apply some of the standard theorems and definitions (continuity, IVT, Extreme Value Theorem, MVT, FTC). &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;One observation I have made is that the students who take calculus in high school are very reliant on formulas: I think those that struggle at the college level find it, for example, hard to apply the definition of the limit to actually derive a derivative. Which leads me to believe that at least some high school curricula teach very little of the underlying math and teach calculus as a bunch of formulas. But at the same time, I doubt this can account for the huge disparity in performance between I would expect from a student who took high school calculus and what I see.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any insight would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="1877" LastEditorUserId="1147" LastEditDate="2018-11-13T01:15:34.587" LastActivityDate="2018-11-13T04:58:08.360" Title="Difference between high school and college calculus courses" Tags="|secondary-education|calculus|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="14782" PostTypeId="2" ParentId="14781" CreationDate="2018-11-12T18:17:55.697" Score="33" Body="&lt;p&gt;Assuming we're talking about mostly US students, most American high schools teach calculus in a way that's very focused on the AP test.  The pressure to get students through that with an adequate success rate seems to have lead to some streamlining and cutting corners in ways that mean that what gets taught doesn't really stick, and also doesn't set students up for success when they retake calculus in college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've noticed a number of ways that my students' high school courses seem to cause them trouble in college:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Reliance on formulas: As you noticed, a lot of high schools teach calculus in way that's overly reliant on formulas.  That has some direct effects - students struggle with figuring out how to apply formulas which are more abstract (like the definition of the derivative), and aren't prepared for problems which call for a non-algorithmic decision about which formula to use.  I think it also has an indirect effect: students have learned that formulas are what math &lt;em&gt;is&lt;/em&gt;, and some of them have learned to tune out anything other than formulas.  I've seen students who rely on their notes leave out whole sections of class because we spent half an hour on an idea rather than a formula, and they didn't see anything in it worth writing down.&lt;/li&gt;&#xA;&lt;li&gt;Pre-calc preparation: High schools are under a lot of pressure to push students through calculus, so they often skimp on algebra and trigonometry.  Calculus courses, on the other hand, often work with problems that assume students can reliably do algebra.  Even when students can eventually do the algebra, it's slow and difficult for them; for instance, I notice that some of my students have trouble learning calculus because they get distracted by the algebra - it takes them forever and they have to spend a lot of time thinking about it, so they can't focus as much on the new material.&lt;/li&gt;&#xA;&lt;li&gt;Streamlined problems: To accommodate the previous item, high school courses seem to primarily give problems where the algebra is simple.  I notice that my students are sometimes uncomfortable with answers that aren't very &quot;clean&quot;.  For example, my students can actually factor quadratics on sight very well (better than I can), but aren't very comfortable with the quadratic formula (to the point that some of them don't remember having seen it).  That presumably reflects that in high school, problems were always rigged so that the quadratic would factor.  &lt;/li&gt;&#xA;&lt;li&gt;Straightforward problems: My students expect problems to always be short, with only one idea in play.  I see a lot of students completely flummoxed by problems which need them to apply two or three steps in a row - even when they've mastered each step.  Again, this is a way that the high school and college versions can say they cover the same material, and yet mean something very different by it.  (A student once told me that she wasn't prepared to start problems unless she already knew what all the steps were going to be to finish it.  I think it reveals a lot about what sort of problems she was being asked to solve in high school that she'd made it to college with that approach.)&lt;/li&gt;&#xA;&lt;li&gt;Overconfidence: Finally, as already noted in the comments, because students don't realize that a college course has different expectations, some of them are overconfident and don't put in the necessary work to relearning things they think they know.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-11-12T18:17:55.697" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14783" PostTypeId="2" ParentId="14777" CreationDate="2018-11-12T18:23:40.050" Score="4" Body="&lt;p&gt;Physically, the problem is to find a third force &lt;span class=&quot;math-container&quot;&gt;$\vec F_3$&lt;/span&gt; that balances the two given forces &lt;span class=&quot;math-container&quot;&gt;$\vec F_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec F_2$&lt;/span&gt;. (This is Newton's Second Law &lt;span class=&quot;math-container&quot;&gt;$\vec F_{net}=m\vec a$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\vec F_{net}=\vec F_1+\vec F_2+\vec F_3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec a=\vec 0$&lt;/span&gt; for equilibrium.) So, &lt;span class=&quot;math-container&quot;&gt;$\vec F_3= -(\vec F_1+\vec F_2)$&lt;/span&gt;.&lt;BR&gt;&#xA;In Physics classes, this is an opportunity to practice vector addition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your post, you describe using the parallelogram rule [tails-together] to find &lt;span class=&quot;math-container&quot;&gt;$(\vec F_1+\vec F_2)$&lt;/span&gt; along the diagonal of the parallelogram. Then &lt;span class=&quot;math-container&quot;&gt;$\vec F_3$&lt;/span&gt; is the vector opposite to &lt;span class=&quot;math-container&quot;&gt;$(\vec F_1+\vec F_2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a geometry problem, &lt;span class=&quot;math-container&quot;&gt;$\vec F_1+\vec F_2+\vec F_3=\vec 0$&lt;/span&gt; means that the vectors added head to tail form a closed polygon (here, a triangle). You are given have two legs and an exterior angle. Find the missing leg (magnitude and direction)... using trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Physics classes, we usually don't use the closed polygon method.&lt;BR&gt;&#xA;Instead, we usually draw a Free Body Diagram with the force tails&#xA;at the origin, then use components:&lt;BR&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$F_{1,x}+F_{2,x}+F_{3,x}=0$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$F_{1,y}+F_{2,y}+F_{3,y}=0$$&lt;/span&gt;&#xA;and try to find unknowns &lt;span class=&quot;math-container&quot;&gt;$F_{3,x}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F_{3,y}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I get &lt;span class=&quot;math-container&quot;&gt;$\vec F_3=(( -46.488)\hat x+(-14.084)\hat y) \rm{\ lbs}$&lt;/span&gt;,&#xA;then, if needed, find magnitude and direction.)&lt;/p&gt;&#xA;" OwnerUserId="10537" LastActivityDate="2018-11-12T18:23:40.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14784" PostTypeId="2" ParentId="14781" CreationDate="2018-11-12T21:11:32.443" Score="11" Body="&lt;p&gt;This is not really an independent answer, but more of a supplement to Henry Towsner's answer, proposing some more possible reasons.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;7&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;AP classes in the US have become such an entitlement that there is often little pretense of actually preparing students for the AP exam. Particularly in school districts that draw from low levels of socioeconomic status, it's not uncommon to see that zero students pass the AP exam in a particular course. In my area (near Los Angeles), this has prompted some high schools to get the community college to teach classes at the high school campus, for college credit. That way the students don't have to pass the AP exam.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students' skills decay, and this decay is especially rapid among students who memorized rules and algorithms without understanding why they worked. The less you understand, the more you have to rely on memorization.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You don't say whether you're at a state university, but if so, then that may explain a lot. There has been a huge, long-term trend of grade inflation, but the one place where this has been least pronounced is in STEM courses at state universities, especially state universities with low admissions standards. While these schools have continued to apply relatively consistent academic standards over the decades, public high schools, private universities, and the humanities and social sciences have dropped their standards gradually but, in the end, dramatically. Therefore the students who show up in your class may be experiencing real grading standards for the first time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Since many old-timers like to groan about grade inflation, and there may be some skepticism about my claims above, here is some info:--&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stuart Rojstaczer is retired Duke professor who has made a project out of researching this sort of thing. His web site is &lt;a href=&quot;http://www.gradeinflation.com/&quot; rel=&quot;noreferrer&quot;&gt;gradeinflation.com&lt;/a&gt;. &lt;a href=&quot;https://www.washingtonpost.com/opinions/the-rise-of-the-gentlemans-a-and-the-gpa-arms-race/2016/03/28/05c9e966-f522-11e5-9804-537defcc3cf6_story.html&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt; is a newspaper op-ed piece summarizing some of his research. An interesting book-length treatment, with original research and a prescription for reform, is Valen Johnson, Grade Inflation: A Crisis in College Education, 2003. A big theme of Johnson's is that grade inflation is driven by teaching evaluations, and he offers clever experimental evidence demonstrating that there really is a cause-and-effect relationship between grades and evaluations. He also gives data demonstrating differences between standards in STEM and other fields, including pretty convincing comparisons of grades students earn within their major and outside it.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2018-11-12T21:17:10.377" LastActivityDate="2018-11-12T21:17:10.377" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14785" PostTypeId="2" ParentId="14781" CreationDate="2018-11-13T04:58:08.360" Score="4" Body="&lt;p&gt;This is a follow on to Ben Crowell's follow on answer.  First, I would like to agree with much of what he says, and say that my personal experience agrees with it.  I hold my students to a high standard (and sometimes get poor reviews for it) but am in a fortunate position to have the support of my department (at a public engineering university) in doing so.  By way of illustration, my grade distribution in the current semester is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2/63 - A&lt;/p&gt;&#xA;&#xA;&lt;p&gt;13/63 - B&lt;/p&gt;&#xA;&#xA;&lt;p&gt;17/63 - C&lt;/p&gt;&#xA;&#xA;&lt;p&gt;19/63 - D&lt;/p&gt;&#xA;&#xA;&lt;p&gt;12/63 - F&lt;/p&gt;&#xA;&#xA;&lt;p&gt;along with 18 drops since the start of the semester.  I place a heavy emphasis on basic number sense and word problems.  I heavily penalize errors in basic algebra (as a graduate student I teach primarily college algebra, trigonometry, and pre-calculus) and do not spend class time on things they should already know, though I offer to help in office hours.  I also categorically refuse to give students a &quot;I showed up&quot; C.  I have sometimes looked at a students Grade in my course vs their overall GPA and can say with confidence that there is grade inflation in other disciplines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anecdotally, after trying out multiple majors, the harsh grading is part of what drew me to mathematics.  It was the first subject I found both interesting and challenging enough to hold my interest, without being told to memorize reams of information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having taught high school before, I would say that there is how I wanted to teach my courses and how I was told to teach my courses.  That said, there are school districts with excellent calculus classes.  The issue is it is very hit or miss.  It requires both a good teacher and a district willing to care about more than just the AP exam pass rate.  I was fortunate to have such a course in my own high school career. &lt;/p&gt;&#xA;" OwnerUserId="10521" LastActivityDate="2018-11-13T04:58:08.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14787" PostTypeId="2" ParentId="14757" CreationDate="2018-11-13T19:35:47.857" Score="2" Body="&lt;p&gt;Not a perfect solution, but I would insist on the meaning of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\cdot}$&lt;/span&gt;: I have observed that this symbol quickly becomes a meaningless mantra to students. I regularly ask (notably in 1st year of master for future high school teacher, also in France) what &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; means. First time students stare at me, and I go with them through the definition: &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is the unique positive real number whose square is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;. Then I question the definition: does it make sense? Why is there one such number? Why only one? Is there a non-positive one? (intermediate value theorem / monotony / look at the graph of the square function and draw a horizontal line).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It will not replace drill, but it complements it. Often, this kind of mistake come from the loss of meaning. Bring meaning back any way you can.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2018-11-13T19:35:47.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14788" PostTypeId="2" ParentId="14777" CreationDate="2018-11-13T21:46:31.310" Score="1" Body="&lt;p&gt;I hope this approach isn't too elementary for your question.  I taught myself trig during the summer between my junior and senior year in high school, and developed the following approach.  For the &quot;airplane problems&quot; you described, I started with a 4-column table:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;    r  |   angle  |   x = r cos(theta)  |   y = r sin(theta)&#xA;=============================================================&#xA;  3 m  |   45 deg |                     |&#xA;2.5 m  |  127 deg |                     |  &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;For the vectors that were given, I worked left-to-right to decompose them into x and y projections:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;   r  |   angle  |   x = r cos(theta)  |   y = r sin(theta)&#xA;=============================================================&#xA;   3m |  45 deg  |  2.12 m             |  2.12 m&#xA;2.5 m |  127 deg |  -1.5 m             |  1.99 m&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;This worked well for any number of vectors (not just two).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then in the last row I worked backwards from right-to-left:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sum the x and y components separately&lt;/li&gt;&#xA;&lt;li&gt;Let Pythagoras calculate the distance&lt;/li&gt;&#xA;&lt;li&gt;Use the inverse tangent formula to determine the angle&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;pre&gt;&#xA;           r  |   angle  |   x = r cos(theta)  |   y = r sin(theta)&#xA;     =========|==========|=====================|===================&#xA;           3m |  45 deg  |  2.12 m             |  2.12 m&#xA;        2.5 m |  127 deg |  -1.5 m             |  1.99 m&#xA;     =========|==========|=====================|===================&#xA;        4.16 m| 81.4 deg |   0.62 m            |  4.11 m&#xA;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;At first, I paid attention to the quadrant that a vector was in so that I could determine the signs of the x and y projections.  But then I realized with delight that my calculator would do that for me.  Neither did I need to worry whether I was adding or subtracting vectors.  The angle took care of the correct signs (so that some components were negative), so I could just add the columns of positive and negative numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The WWII-era trig book I read would use different axes as the 0 degree reference line.  It took examples from navigation, surveying, artillery, and several others I no longer remember.  So I couldn't always use the exact same formulas to decompose a vector into its x and y components.  That's when I started to write the formulas in the header row of the table: I thought through it (and maybe had to diagram it) just once.  But then I could rapidly apply the same formula to all of the rows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was all very cut-and-dried. And systematic enough that I still remember the approach today, more than 30 years later.  That might be it's greatest strength.&lt;/p&gt;&#xA;" OwnerUserId="5465" LastActivityDate="2018-11-13T21:46:31.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14790" PostTypeId="1" AcceptedAnswerId="14793" CreationDate="2018-11-15T06:36:52.287" Score="2" ViewCount="156" Body="&lt;p&gt;In game theory, the Nash Indifference Theorem states that if a mixed strategy &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is a best response to a mixed strategy &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, then every pure strategy in the support of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is also a best response to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; (with the same expected payoff).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for a written proof of this theorem that could be read by an undergraduate with a high school but not a college level math background. Are you aware of such a proof?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.ucla.edu/~tom/Game_Theory/mat.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;This document&lt;/a&gt; contains a proof (section 3, on pp. 18-19) but the sigma notation, double subscripts, and terse proof style will all be very hard for the students I have in mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not familiar with many standard texts in Game Theory; I'm hoping you may be aware of some that prove this theorem in a wordier and less notationally intense way.&lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2018-11-16T14:15:06.230" Title="Accessible written proof of the Nash Indifference Theorem (game theory)?" Tags="|undergraduate-education|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14793" PostTypeId="2" ParentId="14790" CreationDate="2018-11-16T14:15:06.230" Score="1" Body="&lt;p&gt;In Saul Stahl's &lt;a href=&quot;https://bookstore.ams.org/mawrld-13&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Gentle Introduction to Game Theory&lt;/a&gt; there is a simple proof of the following statement on page 30:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Theorem [3.]5.  In any &lt;span class=&quot;math-container&quot;&gt;$2\times 2$&lt;/span&gt; zero-sum game, if one players&#xA;  employs a fixed [possibly mixed] strategy, then the opponent has an&#xA;  optimal counterstrategy that is pure.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The extension to the non zero-sum case is an exercise later in the book.  Naturally, this is weaker than what you are looking for, but maybe this will be enough.  I don't really see a way to rigorously prove such statements without sigma notation in the &lt;span class=&quot;math-container&quot;&gt;$n\times m$&lt;/span&gt; case, though probably there are nice graphical intuitions.  Stahl proves the &lt;span class=&quot;math-container&quot;&gt;$2\times 2$&lt;/span&gt; case of the existence of a (mixed) Nash equilibrium at the end of the book in that way.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would think that perhaps you could find a way to generalize these to the case you need for your student to the level of rigor you desire?  Good luck.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2018-11-16T14:15:06.230" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14794" PostTypeId="2" ParentId="14335" CreationDate="2018-11-16T19:18:37.243" Score="1" Body="&lt;p&gt;There are two unambiguous terms that you can use in any context: ‘homogeneous linear’ and ‘affine linear’; ‘linear’ by itself is ambiguous, and its meaning depends on the circumstances.  It's not that one of these meanings is right and the other is wrong, so you should feel free to use whichever best fits your class (perhaps following the textbook).  That said, you could certainly point out the ambiguity to your students (especially if they might read a different text later).&lt;/p&gt;&#xA;" OwnerUserId="408" LastActivityDate="2018-11-16T19:18:37.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14795" PostTypeId="1" AcceptedAnswerId="14796" CreationDate="2018-11-16T22:21:44.263" Score="8" ViewCount="250" Body="&lt;p&gt;I'm training to be a teacher and I am doing a maths lesson later next week. The topic is geometry, the students are 12-year-olds. &#xA;More concretely, I've been given a selection of exercises that I may use if I wish so. For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7gt9c.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7gt9c.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exercise text: This has been built according to the plan given. Draw all 4 side views.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cR2O5.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cR2O5.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exercise text: For a given cube building you find here two side views and the view from above. Which of the following plans fits all three views?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My guess is that the goal of this lesson is to train mental rotation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have come up with an exercise where first each student builds one of these cube houses, then draws the side views and then I pair up the students. One gives the views to the other who has to try and build it, then they switch roles. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any better more fun ideas for a lesson?&lt;/p&gt;&#xA;" OwnerUserId="10589" LastActivityDate="2018-11-17T03:05:54.623" Title="Fun classroom exercise for mental rotation" Tags="|geometry|primary-education|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14796" PostTypeId="2" ParentId="14795" CreationDate="2018-11-17T03:00:35.053" Score="4" Body="&lt;p&gt;I think these are great exercises as you outline them, especially the object-building aspect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe it also would be useful to ask if these two views show the same object (your 1st; they do), or not?&#xA;One could delete one or more cubes so that the answer is &lt;em&gt;No&lt;/em&gt;.&#xA;Just for 3D-training.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/W3TY4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/W3TY4.jpg&quot; alt=&quot;Rot01&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Two views of the same object.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;I'm thinking that going beyond side orthogonal views may be useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, one could arrange two &lt;a href=&quot;https://en.wikipedia.org/wiki/Polycube&quot; rel=&quot;nofollow noreferrer&quot;&gt;polycube objects&lt;/a&gt;&#xA;such that no two viewpoints could distinguish them, because one has a cavity and the other does not.&#xA;The cavity would be invisible externally if the cubes are opaque.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This could lead into research questions exploring the shadows of polycube&#xA;objects, including the famous cover of &lt;em&gt;Gödel, Escher, Bach&lt;/em&gt;.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/W3TY4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uRWuE.jpg&quot; alt=&quot;JOE&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;MSE question: &lt;a href=&quot;https://math.stackexchange.com/a/2390355/237&quot;&gt;Cover of “Gödel, Escher, Bach”&lt;/a&gt;&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;This is a fascinating topic that could serve to show &#xA;these students that there is more beyond what is known, &#xA;and even they can (dimly) glimpse the research frontier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;MO question: &lt;a href=&quot;https://mathoverflow.net/q/73449/6094&quot;&gt;Regular polygon shadows of convex&#xA;polyhedra&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Lagarias, Jeffrey C., Yusheng Luo, and Arnau Padrol. &quot;Moser's Shadow&#xA;Problem.&quot;  &lt;a href=&quot;https://arxiv.org/abs/1310.4345&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv:1310.4345&lt;/a&gt; (2013).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-11-17T03:05:54.623" LastActivityDate="2018-11-17T03:05:54.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14798" PostTypeId="2" ParentId="14692" CreationDate="2018-11-17T19:52:17.660" Score="1" Body="&lt;p&gt;Here is a real life example with millions of dollars (maybe billions) depending on the answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://phx.corporate-ir.net/phoenix.zhtml?c=197380&amp;amp;p=irol-presentations&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://phx.corporate-ir.net/phoenix.zhtml?c=197380&amp;amp;p=irol-presentations&lt;/a&gt; (MAY18, IR presentation, page 20).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key insight is to maximize unit net present value (middle row), not per well performance.  You have a tradeoff when adding extra wells to a section.  Each well gives you more total oil from the land, BUT the per well performance goes down (because there is a mix of cannibalization with old wells versus accessing new rock by the extra fracturing) and as the spacing gets tighter, it becomes more and more cannibalization.  And each well has a capital cost (there is a small capital economy of scale from running on the same pad [and even smaller OPEX scale savings from pad operations] but most of the capital cost remains.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is almost the exact same problem as adding salaried business development (sales) resources to a territory.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm NOT saying to present this problem to the kids with all the modeling involved (taking business insights and converting into algebra and Excel).  I would actually opt more for giving them idealizing problems with formula shown where they just do the derivatives and check for extrema.  But maybe when they are encountering these issues in the work world (and tradeoffs like this exist in almost every business), they have some &quot;feel&quot; for the issue from having done it in calculus class and it just resonates off a remote memory.  And I don't even care that this problem is discrete (can't have half a well) versus continuous.  The concepts convey.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And note, this is a &quot;return&quot; problem.  So it doesn't meet your restrictions of some other category.  But I just want you to see how rich, rich, rich that area is.  I'm not even saying it for the kids benefits but for yours.  So that you feel like the concept is worthwhile.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-11-17T19:52:17.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14799" PostTypeId="1" CreationDate="2018-11-18T20:17:44.620" Score="20" ViewCount="2109" Body="&lt;p&gt;Over the years I have occasionally encountered a number of Algebra 2 textbooks that make an incorrect (or at very least extremely misleading) claim along the lines that &quot;all solutions of a polynomial equation can be found using a combination of the Rational Roots Theorem, Synthetic Division, and the Quadratic Formula.&quot;  For example, the following image is from a McGraw-Hill &lt;em&gt;Algebra 2&lt;/em&gt; ebook:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NcFVH.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NcFVH.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found a similar claim in the 2007 edition of Holt &lt;em&gt;Algbra 2&lt;/em&gt; (section 6-5, p. 441).  While I haven't done an exhaustive curriculum search, I suspect this is fairly widespread.  Even more to the point, I expect that many Algebra 2 teachers &lt;em&gt;tell&lt;/em&gt; their students this, even if it is not written in the textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is of course true that the problems that &lt;em&gt;students encounter in the course&lt;/em&gt; can all be handled by these methods, but that is because they have been carefully curated for that purpose.  More generally, while the Fundamental Theorem of Algebra guarantees the &lt;em&gt;existence&lt;/em&gt; of solutions, there are certainly polynomials of degree 3 and 4 that are irreducible over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;, and finding zeros for such polynomials requires methods far beyond what is normally taught in high school.  For degree 5 and higher, the Abel-Ruffini Theorem shows that it is in general not possible &lt;em&gt;at all&lt;/em&gt; to find the irrational roots algebraically, and no &quot;exact form&quot; even exists, although the roots can be approximated to arbitrary precision using numerical methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even when a text or teacher does not explicitly make the false claim that all polynomials can be solved using just a few methods, failing to state that in fact the opposite is true (while only presenting examples in which those methods work) seems to me to be at least a sin of omission, and I imagine most students come away believing that any polynomial equation can be solved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quite apart from the fact that we shouldn't be teaching students things that are actually false, the fact that some polynomial equations &lt;em&gt;just can't be solved, no matter how clever you are&lt;/em&gt;, and that mathematics is capable of proving the impossibility of something, seems like an important piece of meta-knowledge to me.  I put it in the same category as &quot;you can't trisect an angle using only a compass and unmarked straightedge&quot; -- we don't expect students to understand the proof, but the result seems worth knowing, if for no other reason than that it makes the constructions you &lt;em&gt;can&lt;/em&gt; do seem more worthwhile.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions, after all this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How pervasive is this error in textbooks?&lt;/li&gt;&#xA;&lt;li&gt;Are there &lt;em&gt;any&lt;/em&gt; high school textbooks that explicitly acknowledge that the methods included in the text are not adequate to solve all 3rd and 4th degree polynomial equations, and that in higher degrees that are no general methods at all?&lt;/li&gt;&#xA;&lt;li&gt;Has anybody ever studied this as an issue of teacher knowledge?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2018-11-18T20:28:18.310" LastActivityDate="2018-11-19T17:50:03.883" Title="Algebra 2 textbooks that incorrectly claim that all solutions of polynomial equations can be found" Tags="|algebra|textbooks|teachers|solving-polynomials|" AnswerCount="2" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="14800" PostTypeId="1" CreationDate="2018-11-18T20:27:37.050" Score="4" ViewCount="902" Body="&lt;p&gt;Should a more general concept of the &quot;form&quot; of an equation or expression, be taught to math students as young as elementary school? I'm a fairly new tutor--do more experienced teachers think this would be useful (or perhaps have done it already)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Some people are answering as if I was asking about teaching specific form, but my question is really about teaching general form for multiple purposes. Perhaps it is unfortunate I gave all my examples as specific forms. I'm editing all of the following text to make my pedagogical suggestions clearer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Specific&lt;/strong&gt; forms appears in modern teaching in several ways. In Algebra I, students are taught the &quot;point-slope form&quot; of a line, the &quot;slope-intercept&quot; form, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Algebra 2 they are taught the standard and vertex forms of parabolas, the standard form of an exponential equation, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet, my typical tutoring student doesn't understand form &lt;strong&gt;as a general concept&lt;/strong&gt; well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I give an Algebra 2 student a question like this: give them vertex form of a parabola, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ y = a(x-h)^2 +k $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;then a specific parabola such as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ y = -(x+1)^2 -1 $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and ask what has been substituted for &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; (here, -1, -1, and -1) they usually are confused about what I'm asking. Then, if they have some idea, they get the sign wrong on &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; or have no idea that &lt;span class=&quot;math-container&quot;&gt;$a=1$&lt;/span&gt; in a parabola like &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they understand &lt;strong&gt;in a general way&lt;/strong&gt; what it means to express the &lt;strong&gt;form&lt;/strong&gt; of an equation as &lt;span class=&quot;math-container&quot;&gt;$y= a(x-h)^2 +k$&lt;/span&gt;, that is if they saw each letter and each mathematical structure as having a higher meaning to them, then working with a new specific form would be trivial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then what I do is take them to remedial discussion of form. This could actually be done at the pre-algebra level, such as giving a student an expression like &lt;span class=&quot;math-container&quot;&gt;$$\frac{AB}{C}$$&lt;/span&gt; then asking which of the following expressions are in that form?&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{1}{2 \cdot 3}, \frac{1 \cdot 2} {3} , \frac{1 \cdot 2 \cdot 3}{1}$$&lt;/span&gt;&#xA;My 7th grade students usually are confused by what I mean for a second, but quickly get it. They answer &quot;the second one.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I ask &quot;in the second expression with numbers, what number is in the same place as A? As B? As C?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again they are confused for a moment, but if I use that language &quot;in the same place as...&quot; they see.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From there is takes off rapidly. I can give them laws of manipulating expressions like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{AB}{C} = (A)\frac{B}{C} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and have them do three things: (1) write a similar expression by substituting specific numbers for A, B, and C, (2) identify matching patterns and name A, B, and C, (3) put the rule to use by manipulating expressions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are several goals here. Partly I'm working toward an understanding of specific forms. But more generally I'm training them to see the visual structure of a form and training them to conceive of &quot;manipulating&quot; the expression as &quot;changing the form.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also working up to giving them the rules of form manipulation in a way they can understand. A student who has been trained in general form from an early age will know right away what to make of a new factoring rules such as &lt;span class=&quot;math-container&quot;&gt;$$ a^2 - b^2 = (a-b)(a+b) $$&lt;/span&gt;. They'll know to ask if this rule is constrained by the independent variable (i.e. whether any &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is untouchable or not part of the form parameters) or whether this form identity is true for any substitution into &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They can learn to recognize ways of manipulate an equation and learn how to express them as &quot;form diagrams&quot;: for instance &lt;span class=&quot;math-container&quot;&gt;$$ r(x-a) = b \implies  x-a = \frac{b}{r} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They can practice writing their own &quot;form diagrams&quot; to communicate their thoughts back to the teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students will &quot;get&quot; much more deeply what it means to say that &quot;the vertex form is&quot; &lt;span class=&quot;math-container&quot;&gt;$ y = a(x-h)^2+k $&lt;/span&gt;, the main goal is far beyond learning specific forms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what I'm wondering is whether this approach (teaching general form) is used in any schools. My students, who are in both private and public schools in the Southern California area and range from 6th grade to college, pretty much universally have not been introduced to form as a general concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would it help all math teaching to introduce form in this way?&lt;/p&gt;&#xA;" OwnerUserId="10466" LastEditorUserId="10466" LastEditDate="2018-11-22T08:47:38.683" LastActivityDate="2018-11-22T17:34:04.310" Title="Should young math students be taught an abstract concept of form?" Tags="|mathematical-pedagogy|teaching|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="14801" PostTypeId="2" ParentId="14799" CreationDate="2018-11-19T00:17:13.697" Score="12" Body="&lt;p&gt;At the moment, I can answer bullet point two:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there &lt;em&gt;any&lt;/em&gt; high school textbooks that explicitly acknowledge that the methods included in the text are not adequate to solve all 3rd and 4th degree polynomial equations, and that in higher degrees that are no general methods at all?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Yes&lt;/strong&gt;, you can find this on p. 267 of CME Project's (2009) Algebra 2 text. See the final paragraph below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xmVqw.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xmVqw.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Separately, Cardano and Tartaglia are mentioned on p. 205.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case you have a copy that may be lying around, here is a photo of the cover of the textbook:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/V1pua.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/V1pua.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Side-note: In the second paragraph pictured above, it is claimed that &quot;the first rigorous proof of the [fundamental] theorem [of algebra] was given by Gauss in his doctoral thesis in 1799.&quot; This assertion sometimes comes under scrutiny; for more on a gap and its potential fix, cf:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Basu, S., &amp;amp; Velleman, D. J. (2017). On Gauss's first proof of the fundamental theorem of algebra. &lt;em&gt;The American Mathematical Monthly, 124&lt;/em&gt;(8), 688-694. &lt;a href=&quot;https://arxiv.org/abs/1704.06585v1&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt; (arXiv).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-11-19T00:17:13.697" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14802" PostTypeId="2" ParentId="14769" CreationDate="2018-11-19T00:49:43.487" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://www.csun.edu/~asethura/GIAAFILES/GIAAV1.0/GIAAV1.0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.csun.edu/~asethura/GIAAFILES/GIAAV1.0/GIAAV1.0.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;A Gentle Introduction to Abstract Algebra&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;this text is pretty good (I used it in my undergraduate algebra course), starts with rings, and uses a distribution license similar to that of linux&lt;/p&gt;&#xA;" OwnerUserId="10521" LastActivityDate="2018-11-19T00:49:43.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14803" PostTypeId="2" ParentId="14799" CreationDate="2018-11-19T06:36:50.943" Score="2" Body="&lt;p&gt;I think every textbook I've used for pre-calculus (at community college level) has the Fundamental Theorem of Algebra in it, so I add  the context that, &quot;Yeah, we know there &lt;em&gt;are&lt;/em&gt; n solutions to an nth degree polynomial, but we also know that for degree 5 and higher, there will be cases where we can't find those solutions. Math is weird sometimes.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This post made me realize that we could probably quantify that, and say &lt;em&gt;most&lt;/em&gt; equations of degree 5 and higher won't have algebraic solutions.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="1147" LastEditDate="2018-11-19T17:50:03.883" LastActivityDate="2018-11-19T17:50:03.883" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14804" PostTypeId="2" ParentId="2612" CreationDate="2018-11-19T12:56:01.023" Score="1" Body="&lt;p&gt;This is an old question now, but I would &lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=J3FgDgAAQBAJ&amp;amp;oi=fnd&amp;amp;pg=PA97&amp;amp;dq=info:MIl6OduydEwJ:scholar.google.com&amp;amp;ots=VRw8HEtggD&amp;amp;sig=26T_cVuAccogD4qTRvVST1gc3NI#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;toot a horn&lt;/a&gt; and suggest that perhaps basic finite group representation theory would fit in well, given that there are applications to &lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=I5lNDwAAQBAJ&amp;amp;oi=fnd&amp;amp;pg=PR4&amp;amp;dq=MATHEMATICS%20MOTIVATED%20BY%20THE%20SOCIAL%20AND%20BEHAVIORAL%20SCIENCES&amp;amp;ots=Phnv-JaPPA&amp;amp;sig=uWgVoe40l5zilB4WPkcVyYr1Y1s#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;noncooperative&lt;/a&gt; and cooperative game theory as well as social choice.  There are many wonderful areas of algebra that have good connections to things LSE cares about.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2018-11-19T12:56:01.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14805" PostTypeId="1" CreationDate="2018-11-19T21:55:26.647" Score="9" ViewCount="303" Body="&lt;p&gt;I am going to teach a flipped real analysis class next term, using &lt;a href=&quot;https://www.springer.com/us/book/9781493927111&quot; rel=&quot;noreferrer&quot;&gt;Abbott's book&lt;/a&gt;. Does anyone know of resources for such a class? I have found the article:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.tandfonline.com/doi/pdf/10.1080/10511970.2016.1162889&quot; rel=&quot;noreferrer&quot;&gt;&quot;Flipping the Analysis Classroom&quot; by Christine Ann Shannon&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but would welcome student-friendly notes or videos, exercises, etc. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For context: I teach at research 1 public university in the USA. Students taking this course have passed an intro to proof class.&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2018-12-03T16:23:58.943" Title="Flipped introductory real analysis resources?" Tags="|undergraduate-education|mathematical-analysis|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14806" PostTypeId="2" ParentId="14800" CreationDate="2018-11-19T22:02:39.653" Score="3" Body="&lt;p&gt;I have not done this, but now that I consider it I think it might be helpful.  I have certainly had students who, when applying the quadratic formula, will look at an equation like &lt;span class=&quot;math-container&quot;&gt;$3x^2+5x-10$&lt;/span&gt; and write &lt;span class=&quot;math-container&quot;&gt;$A=3x^2, B=5x, C=-10$&lt;/span&gt;.  Then they plug those monomials into the quadratic formula and stare helplessly as they try to figure out what to do with all of those &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Really, though, I think the best way to teach things like the point-slope form of a line, or the vertex form of a parabola, is to &lt;em&gt;first&lt;/em&gt; do many specific examples, and &lt;em&gt;then&lt;/em&gt; observe that they all have the same pattern to them.  Take the point &lt;span class=&quot;math-container&quot;&gt;$(2,3)$&lt;/span&gt; and a slope of &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;, and &lt;em&gt;show&lt;/em&gt; them (don't just tell them the end result) why any other point &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; on that line should satisfy the equation &lt;span class=&quot;math-container&quot;&gt;$y-3=6(x-2)$&lt;/span&gt;.  Then repeat, using a different point and a different slope.  Do this 3 or 4 times, and then ask:  You notice how these all have the same basic pattern to them?  What do they all have in common?  Then write the general form, following the students' responses.  You can point out that the letters you use (&lt;span class=&quot;math-container&quot;&gt;$h, k, m$&lt;/span&gt;) really don't matter -- it's just as fine to write&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$y - \textrm{y-coord of point} = \left(\textrm{slope}\right)(x - \textrm{x-coord of point})$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$y-y_0 = m(x-x_0)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$y-k=a(x-h)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A new form should never be taught before examples; rather, the form should be a generalization or summary of a number of examples that have just been encountered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They should, of course, be taught what the standard conventions are, and you should make a habit of using them yourself so that when they are in other contexts they speak the same language as is used in other classrooms.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2018-11-19T22:26:17.800" LastActivityDate="2018-11-19T22:26:17.800" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14807" PostTypeId="2" ParentId="1834" CreationDate="2018-11-20T22:18:29.350" Score="7" Body="&lt;p&gt;Some students that I've encountered, especially those who have struggled with solving linear equations in the past, have taken well to what is sometimes called &quot;backtracking&quot; [in CME Project: Algebra 1] or &quot;the arrow method&quot; [in LINCT Transition Course curriculum (note: this is the program I work for)]. The illustration below may be enough for a master algebraist, but if you seek more description, I recommend searching for either of the two curricula mentioned above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eZSCr.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eZSCr.jpg&quot; alt=&quot;Arrow Method&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First the &lt;strong&gt;black&lt;/strong&gt; text and arrows is written in, from left to right. Generally students are able to follow the order of operations that occur to the variable and can generate this step with relative ease. Next the &lt;strong&gt;green&lt;/strong&gt; text is handled, where students just need to focus on the inverse operation from the black arrows. Last, the &lt;strong&gt;red&lt;/strong&gt; text allows students to follow along with the operations from the green arrows. I like this method particularly because it also opens the door for fruitful conversations around inverse operations and their relationship to solving algebra equations of this type.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, I have also seen instances where students and/or teachers prefer to leave out the black text below the arrows, and just leave empty circles (or &quot;bubbles&quot;) to be filled in where the red text is. If the only goal is to solve a particular equation, then I think this is fine; however, I think there are many more benefits to seeing the expression grow and its equivalent values with the complete picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are limitations with this method, notably when there is more than one variable in the equation, but your initial question doesn't seem to ask about this. I think this is a viable option for students to see who struggle with solving basic linear equations that provides a visual representation for inverses that is otherwise obscured in many other methods. It does not work universally, but is a good place to start and can develop some very strong conversations as well.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2018-11-20T22:18:29.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14808" PostTypeId="1" CreationDate="2018-11-21T01:57:56.510" Score="8" ViewCount="783" Body="&lt;p&gt;I was thinking to make classroom illustrations of some 3D mathematical objects, such as graphs of 2 variable functions, minimal surfaces, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, what would be a good way to go about it? I thought about 3D printers, but I have no experience with it whatsoever. I would really appreciate if someone with an experience with these things point me to a right direction.&lt;/p&gt;&#xA;" OwnerUserId="2205" LastActivityDate="2018-11-26T00:05:45.203" Title="Making physical 3D models" Tags="|calculus|geometry|classroom-interaction|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14809" PostTypeId="2" ParentId="1834" CreationDate="2018-11-21T09:31:35.610" Score="2" Body="&lt;p&gt;For students a problem such as &lt;span class=&quot;math-container&quot;&gt;$3x + 4 = 16$&lt;/span&gt; presents several conceptual difficulties even before the question of &lt;em&gt;how&lt;/em&gt; to solve it is addressed.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The notion of a variable/indeterminate is somehow quite abstrct. The idea that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a placeholder for a putative solution (which in general may or may not solve the equation once &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is replaced by the putative solution) requires a substantial mental leap when compared with operating only with concrete numbers, e.g. &lt;span class=&quot;math-container&quot;&gt;$3(5) + 4 = 19$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The notion of equality of formal expressions is also quite abstract. In the equation &lt;span class=&quot;math-container&quot;&gt;$3x + 4 = 16$&lt;/span&gt;, the left-hand side is a formula involving a formal variable, while the right-hand side is a particular number. These are objects that do not occupy the same mental space. The difficulty in some sense is that the particular number, in this case &lt;span class=&quot;math-container&quot;&gt;$16$&lt;/span&gt;, has to be promoted to be viewed on an equal footing as the formal expressioN &lt;span class=&quot;math-container&quot;&gt;$3x + 4$&lt;/span&gt;. This is not a straightforward step.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The identification of equations whose solutions are the same presents the fundamental difficulty, encountered at many stages of learning mathematics, of the difference between equality an identification. The equations &lt;span class=&quot;math-container&quot;&gt;$3x + 4 = 16$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$3x = 12$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$x = 4$&lt;/span&gt; are all &lt;em&gt;identified&lt;/em&gt; in the sense that their sets of solutions are the same, but they are (obviously!) not &lt;em&gt;equal&lt;/em&gt; as formal expressions (the symbols appearing in them are different!). Whatever algorithmic procedure is used for solving the equation requires identifying the &lt;em&gt;different&lt;/em&gt; equations obtained at each step of the procedure. Further compounding the difficulty is that the final equation, &lt;span class=&quot;math-container&quot;&gt;$x = 4$&lt;/span&gt;, is usually conflated with its set of solutions (consisting of the number &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;), and this is done without any comment. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The issue of understanding an abstract variable is the most easily addressed. Repetition of problems of a similar form, e.g. &lt;span class=&quot;math-container&quot;&gt;$3x + 4 = 16$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$3x + 4 = 19$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$3x + 4 = 22$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2x + 4 = 16$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2x + 4 = 18$&lt;/span&gt;, etc. (minor variations) helps develop a sense of the formal meaning of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Later these same problems can be formulated in words - I have &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; widgets and I buy some boxes each containing &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; widgets, how many boxes need I buy to have &lt;span class=&quot;math-container&quot;&gt;$16$&lt;/span&gt; widgets in total? - and the exercise of identifying the relation of the formal equation &lt;span class=&quot;math-container&quot;&gt;$3x + 4 = 16$&lt;/span&gt; with the word problem will help understand what &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is (the student can be asked to formulate a word problem that corresponds to the equation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue of the identification of equations whose solution sets are the same seems to me much harder to teach and explain. Even at the university level, with students who are quite practiced at solving single variable equations, the issue reoccurs when one tries to teach them methods for solving systems of equations (e.g. Gaussian elimination). Part of the problem is that instructors rarely give enough credit to the notion that &lt;span class=&quot;math-container&quot;&gt;$3x + 4 = 16$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3x = 12$&lt;/span&gt; are entirely different equations. On the one hand this is so obvious that it seems to require no comment at all, on the other hand, we treat them (operationally) as the same when they occur in the series of steps leading to &lt;span class=&quot;math-container&quot;&gt;$x = 4$&lt;/span&gt;. Probably one needs to be quite explicit that, although different, these equations have the same solution (their interpretations might be quite different, if, for example, the coefficients have physical or economic meaning - if I have no widgets and I buy &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; boxes looking to have &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; the situation is not the same as that described in the word problem above). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That a smallish child can readily assimilate the notion of a variable as a placeholder is an indicator of certain talent for the sort of abstraction that makes for a good mathematics student. Many, maybe most, children don't have this talent, or at least require that it be developed and directed, usually via a lot of drill with minor variations of a simple theme. Even with university engineering students (generally excellent students in high school!) one frequently encounters a lack of understanding relating to formal substitution of variables and its role vis-a-vis the function concept; this is part of why so many have trouble with things like the chain rule and changes of variables in integrals. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-11-21T09:31:35.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14810" PostTypeId="2" ParentId="14808" CreationDate="2018-11-21T13:07:16.387" Score="11" Body="&lt;p&gt;Here are some examples of physical 3D-models of mathematical surfaces.&lt;BR&gt;&#xA;They might give you some ideas.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://raisingcalculus.winona.edu/&quot; rel=&quot;noreferrer&quot;&gt;https://raisingcalculus.winona.edu/&lt;/a&gt; (Raising Calculus to the Surface - Aaron Wangberg - Winona State)&lt;BR&gt;&#xA;uses a plastic mold of a surface that students can study (and markup with a dry-erase marker).&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7L8pn.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7L8pn.jpg&quot; alt=&quot;RaisingCalculus-Wangberg&quot;&gt;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.3dprintmath.com/&quot; rel=&quot;noreferrer&quot;&gt;http://www.3dprintmath.com/&lt;/a&gt; (Visualizing Mathematics with 3D Printing - Henry Segerman - Oklahoma State)&lt;BR&gt;&#xA;uses a 3D-printer to make various objects&lt;BR&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/I5ee3.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/I5ee3.jpg&quot; alt=&quot;VisualizingMathematicsWith3DPrinter-Segerman&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://math.okstate.edu/people/segerman/talks/design_of_3d_printed_math_art.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://math.okstate.edu/people/segerman/talks/design_of_3d_printed_math_art.pdf&lt;/a&gt;&lt;BR&gt;&#xA;&lt;a href=&quot;https://youtu.be/JIM-IWh_-n0&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/JIM-IWh_-n0&lt;/a&gt; (video)&lt;BR&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://tangible.media.mit.edu/project/inform/&quot; rel=&quot;noreferrer&quot;&gt;https://tangible.media.mit.edu/project/inform/&lt;/a&gt; ( inFORM - MIT Media Lab -Tangible Media Group)&lt;BR&gt;&#xA;is a Dynamic Shape Display that can render 3D content physically.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/WJOq5.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WJOq5.jpg&quot; alt=&quot;inFORM-MITMediaLab-TangibleMediaGroup&quot;&gt;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://pi.math.cornell.edu/~dtaimina/&quot; rel=&quot;noreferrer&quot;&gt;http://pi.math.cornell.edu/~dtaimina/&lt;/a&gt; (Daina Taimina - Cornell)&lt;BR&gt;&#xA;creates surfaces for hyperbolic space by crocheting&lt;BR&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Ag0I2.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ag0I2.png&quot; alt=&quot;DainaTaimiņa-CrochetingHyperbolicPlanes&quot;&gt;&lt;/a&gt;&lt;BR&gt;&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=w1TBZhd-sN0&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=w1TBZhd-sN0&lt;/a&gt; (video)&#xA;&lt;a href=&quot;http://pi.math.cornell.edu/~dwh/papers/crochet/crochet.html&quot; rel=&quot;noreferrer&quot;&gt;http://pi.math.cornell.edu/~dwh/papers/crochet/crochet.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.andrewlipson.com/mathlego.htm&quot; rel=&quot;noreferrer&quot;&gt;http://www.andrewlipson.com/mathlego.htm&lt;/a&gt; (Andrew Lipson's Mathematical LEGO® Sculptures)&lt;BR&gt;&#xA;creates mathematical surfaces with LEGO.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/EnTa1.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EnTa1.jpg&quot; alt=&quot;www.andrewlipson.com-images-mobiusmen.jpg&quot;&gt;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Of course, it's probably easy to find paper models of various solids (e.g. Paper Icosahedron, etc...) For example, &lt;a href=&quot;https://www.korthalsaltes.com/&quot; rel=&quot;noreferrer&quot;&gt;https://www.korthalsaltes.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;UPDATED:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/3Dprinting&quot; rel=&quot;noreferrer&quot;&gt;https://www.geogebra.org/3Dprinting&lt;/a&gt;&lt;BR&gt;&#xA;3D Printing with GeoGebra&#xA;by Diego Lieban&lt;BR&gt;&#xA;&quot;Create your own 3D shapes with GeoGebra and a 3D printer. In this tutorial you will learn how easy it is to export 3D constructions from our GeoGebra 3D Graphing app to the STL file format which can be printed directly on any 3D printer.&quot;&#xA;&lt;BR&gt;&#xA;&lt;BR&gt;&#xA;It seems that it uses &lt;a href=&quot;https://www.geogebra.org/3d&quot; rel=&quot;noreferrer&quot;&gt;https://www.geogebra.org/3d&lt;/a&gt; and the STL output from the &quot;Download as&quot; menu item.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10537" LastEditorUserId="10537" LastEditDate="2018-11-21T18:20:19.340" LastActivityDate="2018-11-21T18:20:19.340" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14811" PostTypeId="2" ParentId="14808" CreationDate="2018-11-21T14:10:04.983" Score="4" Body="&lt;p&gt;This is not what you are looking for, but I have had success explaining&#xA;the &lt;a href=&quot;https://en.wikipedia.org/wiki/Voronoi_diagram&quot; rel=&quot;nofollow noreferrer&quot;&gt;Voronoi diagram&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Medial_axis&quot; rel=&quot;nofollow noreferrer&quot;&gt;medial axis&lt;/a&gt; by pouring dry sand.&#xA;Here's an example of the medial axis. &#xA;In (b) of the figure, one can discern a parabolic arc caused by the reflex vertex.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/A4x3B.png&quot; width=&quot;500&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Figure from &lt;a href=&quot;http://cs.smith.edu/~jorourke/DCG/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discrete and Computational Geometry&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-11-21T14:33:25.267" LastActivityDate="2018-11-21T14:33:25.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14812" PostTypeId="2" ParentId="13460" CreationDate="2018-11-21T15:07:04.703" Score="5" Body="&lt;p&gt;I teach mostly physics, but have taught calculus a couple of times. From the physics end, I see things in almost the opposite way that you do. Here is a typical way that this plays out in my class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have a homework problem where a cable is stretched between two buildings, with a streetlight hanging from the middle, so that the cable makes the shape of a &quot;V&quot;. The students calculate a formula for the tension &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; in the table as a function of the height &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; by which it sags in the middle. The problem asks them to do some interpretation of the result, including checking its units, the trend of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; as a function of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, and the special case of &lt;span class=&quot;math-container&quot;&gt;$h=0$&lt;/span&gt;. (Note that we automatically have &lt;span class=&quot;math-container&quot;&gt;$h\ge 0$&lt;/span&gt; in this problem.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student comes to my office hours for help with the problem, and we get to the point where we're checking the special case &lt;span class=&quot;math-container&quot;&gt;$h=0$&lt;/span&gt;. They say, &quot;Oh, it's undefined, because zero is in the denominator.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point they think they're done, and they need to be disabused of this notion that has been inculcated in them. They are missing out on the insight that is to be gained by realizing that the result is infinite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you like, you can dress this up in various language such as the language of limits, or saying that the result is undefined &lt;em&gt;as a real number&lt;/em&gt;, but only because the result is infinite, and we don't have infinite real numbers. But these are issues of mathematical formalism that are way above the level of this type of student. To see the actual level that they're operating at, it may be useful to consider the following fragment of dialog, which I have had many times with many students:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student: &quot;Oh, it's undefined.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Me: &quot;That's great that you've had such good mathematical training, and your math teachers have told you that dividing by zero is undefined. But suppose we really did divide by zero. What do you think you would really get?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student: &quot;Zero?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Me: &quot;Well, hang on, what kind of result would you get if you divided by a really small number, like 1/0.001?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student: &quot;It would come out really small?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Me: &quot;Let's try that on your calculator.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student: &quot;Oh, it's big. I see.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, they're not at the level where they understand that the expression diverges but they just don't know how to formalize that statement. They're at the level where they lack the idea that it blows up at all. They need a first exposure to the idea that it blows up, and then on some later pass maybe they can learn about calculus, limits, the extended reals, and so on. The first exposure needs to happen in grade school, when they first learn about division. If they learn division at age 9, and calculus (possibly) at age 18, then IMO the right answer for them to give from age 9 to age 18 is that &lt;span class=&quot;math-container&quot;&gt;$1/0=\infty$&lt;/span&gt; (or possibly &lt;span class=&quot;math-container&quot;&gt;$\pm\infty$&lt;/span&gt;, depending on context).&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-11-21T15:07:04.703" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14813" PostTypeId="2" ParentId="1834" CreationDate="2018-11-21T17:38:12.167" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;For instance, in the example, &lt;span class=&quot;math-container&quot;&gt;$3x+5=14$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First, decide what to do to both sides (subtract 5)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Funny that no one mentioned &lt;strong&gt;balance scale&lt;/strong&gt;, because equation is exactly this: balance scale being in balance. &amp;quot;Do the same thing to both sides&amp;quot; meaning add the same number (like the same weight) to both sides, remove the same number (weight), divide or multiplie by the same number (multiplying an apple by three means replacing the apple with three times as many) to &lt;strong&gt;keep the scale in balance&lt;/strong&gt;. For anyone who have seen balance scale daily by going into a corner store to buy groceries or by going to a farmer's market, this concept is extremely easy to grasp.&lt;/p&gt;&#xA;&lt;p&gt;From there you proceed with what you want to do: to isolate x on one side of the scale. You do this by collecting like terms first, then moving all constants to another side (that is, adding or subtracting to both sides to isolate a monomial), then getting rid of the coefficient by dividing both sides by it. The process is highly algorithmical, at least for simple equations studied in middle school.&lt;/p&gt;&#xA;&lt;p&gt;I would suggest introducing the notions of monomial, polynomial, and like terms before introducing solving equations. On another hand, kids as young as four-grades can easily solve systems of linear equations without knowing they are doing it if you phrase it somewhat like &amp;quot;a bar of soap and two tubes of toothpaste cost 2.50, while one tube of toothpaste costs 0.80, how much the bar of soap costs?&amp;quot; and you can go from here. The idea or &lt;strong&gt;substitution&lt;/strong&gt; seems to be not the most obvious one for many kids, and one has to get used to the idea that things can be substituted with equal things. After it, solving equations becomes much easier.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-11-21T17:38:12.167" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14814" PostTypeId="2" ParentId="14808" CreationDate="2018-11-21T21:07:34.093" Score="2" Body="&lt;p&gt;It would help a little if you gave us some discussion of your limits in cost or time.  The high cost, high quality approach is to buy models from vendors.  If you can expense it go for it.  The high quality (and you learn something), but also high effort and time, is to go into the machine shop (find one at your college or high school) and just mill something out of plastic or wood (metal not needed).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A happy medium would be to use sculpting clay and fire it.  It will still be a fair amount of work, especially if you want it to look good.  &lt;/p&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-11-21T21:07:34.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14815" PostTypeId="2" ParentId="14800" CreationDate="2018-11-21T21:08:40.187" Score="1" Body="&lt;p&gt;I think an &quot;abstract concept of form&quot; is not helpful in mathematical education up to university level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To quote YiFans comment, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I honestly think all this hype about &quot;point slope form&quot;, &quot;vertex&#xA;  form&quot;, etc. is really unnecessary&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and I'd like to expand on this. One of the core concepts of algebra (or math in general) is the equation. Students have enough trouble grasping the concept of variables and if you add &quot;form of a term&quot; as another concept, this will lead to one extra thing to learn for the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there is no need to add this layer of abstraction if the students really understand the meaning of variables and terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My guess is that students that are &quot;fluent&quot; in algebra can easily understand an abstract concept of form, but those who struggle to interpret (let alone manipulate) terms will only be confused by this new point of view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quoting OP:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;They can learn to recognize ways of manipulate an equation [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Again, this boils down to &quot;applying the same operation to both sides of the equation&quot;. This is what matters most and there is no need for the extra step of introducing the abstract concept of form.&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2018-11-22T17:34:04.310" LastActivityDate="2018-11-22T17:34:04.310" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14816" PostTypeId="2" ParentId="7354" CreationDate="2018-11-21T22:52:47.910" Score="1" Body="&lt;p&gt;Like another poster, I also see two separate questions here, but different ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Should we explicitly write &lt;strong&gt;identity value&lt;/strong&gt;? 1 is identity for multiplication and division, while 0 is identity for addition and subtraction. Adding 0 does not change the result, likewise dividing by 1 does not change the result. Which is why they are omitted for brevity, but if one wants to use it, it will not change the result, although when I was in school I would get half a grade off for extra constant or parentheses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Should one discern subtraction of a positive value from addition of a opposite negative one? There was a custom to use inline + and - for addition and subtraction, and to specify the sign of a term above the term. But I guess because mathematicians are a lazy bunch, they use the property of &quot;adding a number is the same as subtracting an opposite number,&quot; so two operations - &lt;strong&gt;binary subtraction and unary sign&lt;/strong&gt; - have been condensed into one. This is, in fact, quite a hard idea for many students. I would not mind if some wrote -(5+x) as (-1) * (5+x) or something like this, if it helps.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastActivityDate="2018-11-21T22:52:47.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14817" PostTypeId="2" ParentId="13460" CreationDate="2018-11-22T07:12:33.847" Score="0" Body="&lt;p&gt;If the students have sufficient algebraic skills, point out that division is defined in terms of multiplication as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For any real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;, we define &lt;span class=&quot;math-container&quot;&gt;$x/y = z$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$y\neq 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=zy$&lt;/span&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Why the condition &lt;span class=&quot;math-container&quot;&gt;$y\neq 0$&lt;/span&gt;? When &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$x=z\times 0$&lt;/span&gt; (the equation on the RHS) would have no unique solution for &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 1:&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt;, then the equation on the RHS becomes &lt;span class=&quot;math-container&quot;&gt;$0=z\times 0$&lt;/span&gt;. Then any value would work for &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example 2:&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt;, then the equation on the RHS becomes &lt;span class=&quot;math-container&quot;&gt;$1=z\times 0$&lt;/span&gt;. Then no value would work for &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;. &lt;/p&gt;&#xA;" OwnerUserId="6103" LastActivityDate="2018-11-22T07:12:33.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14818" PostTypeId="1" CreationDate="2018-11-22T18:34:12.103" Score="2" ViewCount="489" Body="&lt;p&gt;My 12-year-old cousin thinks &lt;a href=&quot;https://math.stackexchange.com/a/133348/53259&quot;&gt;this explanation&lt;/a&gt; is the most comprehensible, but she still can't relate the analogy with wealth inequality&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If I say &lt;code&gt;mymoney = yourmoney + 1&lt;/code&gt; are you richer than me? No, you're poorer by a dollar! That's easy to see because when the variables have these names, of course you instantly and intuitively put yourself into the &lt;code&gt;yourmoney&lt;/code&gt; frame of reference, and easily see that although 1 is being added to &lt;code&gt;yourmoney&lt;/code&gt;, the result of that formula expresses my money and not yours! &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;to &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; system must remain our frame of reference if we are to visualize the motion,&#xA;  and so what we do is rework the equations to isolate x and y:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \begin{align} s = x + 1 &amp;amp; \iff   x = s - 1 \\ t = y + 3 &amp;amp; \iff y = t - 3  \end{align} $$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Now you can see that it's a move left and down and this is consistent with the minus signs.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The &quot;backwards&quot; issue arises when you put yourself into the frame of reference of the transformation, but continue using &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; to think about that frame. You have to think about the backwards mapping: how do you recover &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; from that other frame of reference?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;When you see &lt;span class=&quot;math-container&quot;&gt;$x + 1$&lt;/span&gt;, you must not imagine that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is moving. (This may be &quot;brain damage&quot; from working in imperative computer programming languages, especially ones like C and BASIC where the &lt;code&gt;=&lt;/code&gt; sign is used for assignment: &lt;code&gt;x = x + 1&lt;/code&gt; moves coordinate &lt;code&gt;x&lt;/code&gt; to the right.) Rather, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is used as the input to a calculation that produces some other value. And so &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is that other value, minus 1.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;To wit, she knows that &lt;span class=&quot;math-container&quot;&gt;$s = x + 1 \implies$&lt;/span&gt; s is richer than x. But how does this imply  a rightward shift? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similarly, how does (&lt;span class=&quot;math-container&quot;&gt;$t = y + 3 \implies$&lt;/span&gt; t is richer than y) &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; downward shift?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user155" LastEditDate="2018-11-24T18:01:08.803" LastActivityDate="2019-07-05T16:27:19.763" Title="How can I explain horizontal shifts to a 12-year-old by analogizing with $\text{your money} = \text{my money} + 1?$" Tags="|functions|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14819" PostTypeId="2" ParentId="14818" CreationDate="2018-11-23T10:09:40.320" Score="4" Body="&lt;p&gt;To be honest, it also took me some time reading the answers you linked to see what you are talking about, so I can fully understand how it might not be intuitive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you want to discuss the shift of the origin in a coordinate system, it might be best to draw said system and do lots and lots of examples. The simplest one I can think of right now would be something like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y = x + a$$&lt;/span&gt;&#xA;and then figure out where the graph meets the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; axis, e.g. by drawing it. As it turns out, this is the case for &lt;span class=&quot;math-container&quot;&gt;$x = -a$&lt;/span&gt;, so kind of the same thing happens here as in what you are trying to discuss.&lt;br&gt;&#xA;Once that is clear and well understood, you can explain how this is the same as moving the origin along the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; axis and then go into the topic you want to discuss.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, I have to agree with the comments that this is rather advanced for a 12 (or 10?) year old to fully understand, beyond just learning the formula without thinking.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2018-11-23T10:09:40.320" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14820" PostTypeId="2" ParentId="376" CreationDate="2018-11-23T19:06:48.347" Score="6" Body="&lt;p&gt;As far as I know, vector calculus is applied by financial analysts in exotic derivatives pricing. The Black-Scholes Model is actually a special form of Schrödinger equation. Thus, if you want to establish high precision models to price exotic derivatives, you will have the chance to apply vector calculus. However, these kinds of applications presumes students have solid background in economics and finance.&lt;/p&gt;&#xA;" OwnerUserId="10611" LastEditorUserId="10611" LastEditDate="2018-11-23T19:11:49.497" LastActivityDate="2018-11-23T19:11:49.497" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14821" PostTypeId="1" CreationDate="2018-11-24T15:37:04.390" Score="5" ViewCount="196" Body="&lt;p&gt;I'm starting my job as a mathematics teacher in an intermediate school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to follow some new way of assessment and I'm really curious about take-home exams; are they suitable for any subjects? if not, for which subjects are they suitable?, for which grade levels?, what kind of questions does they include? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I read a lot but can't find accurate answers for my questions, hope I'll find them here.&lt;/p&gt;&#xA;" OwnerUserId="10618" OwnerDisplayName="Nour" LastActivityDate="2018-11-24T16:43:56.780" Title="For what subjects and grade levels is the take-home exam suitable?" Tags="|teaching|exams|assessment|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14822" PostTypeId="2" ParentId="14821" CreationDate="2018-11-24T16:38:16.033" Score="4" Body="&lt;p&gt;This depends on how you teach. If you stress creativity rather than memorization and regurgitation then you should be fine with take home. Of course it can be difficult to come up with problems that require creativity and whose solutions can't just found on the internet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just don't put forth rules that can't be enforced and are likely to be ignored. Assume (and make it ok) that students will use all available resources including the internet. Hence the need for creativity. Don't grade on right-wrong but on the basis of intelligent query and response. It may be hard, but possible. Note that even making collaboration improper may be difficult to enforce. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use peer evaluation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have to assume that students will collaborate no matter what you say, then make it possible to produce joint work. They are a bit young for this, of course, but peer evaluation stresses the positive contributions of the partner, not the &quot;quality&quot;. &quot;What was your partner's main contribution.&quot; No young person is likely to &quot;rat out&quot; their partner and you don't really want to encourage it. But asking for positive statements avoids this problem. Students doing peer evaluation can/should also do self evaluation: &quot;What was your own chief contribution?&quot; &lt;/p&gt;&#xA;" OwnerUserId="8756" LastEditorUserId="8756" LastEditDate="2018-11-24T16:43:56.780" LastActivityDate="2018-11-24T16:43:56.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14825" PostTypeId="2" ParentId="376" CreationDate="2018-11-26T00:02:12.923" Score="0" Body="&lt;p&gt;I'm not totally sure the scope of your course but I think there is a wealth of applications from constrained optimization problems, gradient descent, dynamical systems, etc.  Check Economic Dynamics: Methods and Models by Giancarlo Gandolfo as a starting place.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2018-11-26T00:02:12.923" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14826" PostTypeId="2" ParentId="14808" CreationDate="2018-11-26T00:05:45.203" Score="2" Body="&lt;p&gt;If you have access to Rhino, you can develop 3D models and feed the model to a 3D printer.  You can generate models using basic Python scripts here.  I've done this with limited success.  &lt;a href=&quot;https://wiki.mcneel.com/rhino/3dprinting&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://wiki.mcneel.com/rhino/3dprinting&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of basic 3D model rendering, I've also used VPython with Calculus students to make solids with known cross sections and solids of revolution.  &lt;a href=&quot;http://vpython.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://vpython.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2018-11-26T00:05:45.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14827" PostTypeId="2" ParentId="14692" CreationDate="2018-11-26T00:23:29.507" Score="0" Body="&lt;p&gt;I also like the example of regression and least squares as a nice application of optimization a differentiation with respect to something other than &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my class, I also discuss &lt;em&gt;partial derivatives&lt;/em&gt; and give some examples of envelope problems.  Suppose a 13 foot ladder slides down a wall, what is the curve that it traces out?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Describe the ladder as an equation determined by its &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; an &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; intercepts and some parameter &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, and consider a specific value &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, say &lt;span class=&quot;math-container&quot;&gt;$x = 4$&lt;/span&gt;.  Then, we have &lt;span class=&quot;math-container&quot;&gt;$y = 17 -a -52/a$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we look at what happens as we run through different values of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; we find that the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;'s increase then decrease, moving first towards and then away from the envelope, as we see below with the little red dots.  Thus, we have the curve at the maximum value when we differentiate our expressions with respect to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EK2uC.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EK2uC.png&quot; alt=&quot;Envelope&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, more generally, we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$y = \frac{a - 13}{a}(x - a) = 13 + x - a - \frac{13x}{a}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\partial y}{\partial a} = -1 + 13x/a^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a = \sqrt{13x}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and we can substitute this value for &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; back in our original equation, yielding the description for the curve.  This example is from &lt;em&gt;Analysis by its History&lt;/em&gt; by Hairer and Wanner.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2018-11-26T00:23:29.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14828" PostTypeId="1" AcceptedAnswerId="14829" CreationDate="2018-11-26T07:37:36.020" Score="5" ViewCount="168" Body="&lt;p&gt;I am teaching probability on mutually exclusiveness and dependency of events. Let me take a simple example as follows.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A box contains 2 red balls and 3 purple balls. They are identical except for their color. A ball is randomly taken, the color is noted, and it is replaced back to the box. We repeat this action twice.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; be the event that a red ball is taken in the first draw.&lt;/li&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; be the event that a purple ball is taken in the second draw.&lt;/li&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; be the event that there are 2 red balls in the 2 draws.&lt;/li&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; be the event that there are 2 purple balls in the 2 draws.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;First Fallacy&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A student answers that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are mutually exclusive because there is no intersection between &lt;span class=&quot;math-container&quot;&gt;$A=\{R_1,R_2\}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B=\{P_1,P_2,P_3\}$&lt;/span&gt;. He then calculates the probability that the first ball is red or the second ball is purple is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;p(A\cup B) = 2/5 +3/5 =1&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Second Fallacy&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A student answers that &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; are independent because &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; don't interfere each other. He then calculates&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;p(C\cap D) = 4/25 \times 9/25 = 36/625&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Question&lt;/h1&gt;&#xA;&lt;p&gt;How do we correct him with easy-to-digest explanation?&lt;/p&gt;&#xA;" OwnerUserId="5696" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-12-02T04:24:53.360" Title="Better ways to explain mutually exclusiveness and dependency of events" Tags="|probability|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14829" PostTypeId="2" ParentId="14828" CreationDate="2018-11-26T14:00:15.860" Score="4" Body="&lt;p&gt;Emphasize to the student that&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Every probability has an associated experiment.&lt;/li&gt;&#xA;&lt;li&gt;Every experiment has an associated sample space.&lt;/li&gt;&#xA;&lt;li&gt;Every event is a subset of the sample space.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In this case, the associated experiment is &quot;randomly take one ball from a box, note the color, then return the ball to the box.  Repeat.&quot;  The associated sample space is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\{P_1P_1,P_1P_2,P_1P_3,P_1R_1,P_1R_2,\\&#xA;P_2P_1,P_2P_2,P_2P_3,P_2R_1,P_2R_2,\\&#xA;P_3P_1,P_3P_2,P_3P_3,P_3R_1,P_3R_2,\\&#xA;R_1P_1,R_1P_2,R_1P_3,R_1R_1,R_1R_2,\\&#xA;R_2P_1,R_2P_2,R_2P_3,R_2R_1,R_2R_2\}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The event that a red ball is taken in the first draw is &lt;em&gt;not&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\{R_1,R_2\}$&lt;/span&gt; because it is not a subset of the sample space.  Similarly, the event that a purple ball is taken in the second draw is &lt;em&gt;not&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\{P_1,P_2,P_3\}$&lt;/span&gt;.  It should be easy to see that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$A=\{R_1P_1,R_1P_2,R_1P_3,R_1R_1,R_1R_2,R_2P_1,R_2P_2,R_2P_3,R_2R_1,R_2R_2\}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$B=\{P_1P_1,P_1P_2,P_1P_3,P_2P_1,P_2P_2,P_2P_3,P_3P_1,P_3P_2,P_3P_3,R_1P_1,R_1P_2,R_1P_3,R_2P_1,R_2P_2,R_2P_3\}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;are not mutually exclusive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your other concerns are addressed in a similar manner.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-11-26T14:00:15.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14830" PostTypeId="1" CreationDate="2018-11-26T15:39:14.637" Score="8" ViewCount="309" Body="&lt;p&gt;I understand that at high school level or below, teachers usually spend extra effort helping those students who are struggling. However, how about at college/university level? Here are two philosophies:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A college teacher should be well prepared for the lectures and explain concepts as clearly as possible. He/she may have office hours when students can seek assistance if needed. But a teacher is not responsible if a student is failing the course and does not take the initiative seeking help. If a small percentage of students do poorly in the class, do not bother to find out the reason; just given them whatever grade they deserve.&lt;/li&gt;&#xA;&lt;li&gt;Teaching well is not enough; a college teacher should also be a nice guy helping whoever are struggling in the course, even though the student never take the initiative seeking assistance. Spend extra time inviting them for a talk and find out the problems; help them as much as possible.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It seems that the second philosophy is big-hearted and will likely benefit most, if not all, struggling students. But it also means significant time investment and extra effort. Am wondering what is generally expected from a college teacher? Is philosophy 1 good enough?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2018-11-26T19:44:00.070" LastActivityDate="2018-12-08T05:22:05.533" Title="Is it the college teacher's responsibility to help the struggling students?" Tags="|undergraduate-education|struggling-students|" AnswerCount="0" CommentCount="8" ClosedDate="2018-12-09T21:24:03.693" ContentLicense="CC BY-SA 4.0" />
  <row Id="14831" PostTypeId="1" CreationDate="2018-11-26T23:24:44.390" Score="7" ViewCount="2420" Body="&lt;p&gt;I am finding it difficult to convince my kid that 2/4 and 1/2 are same. As per the kid, 2/4 is more than 1/2 since in first case the boy gets 2 candies out of 4 and in second case he gets 1 candy out of the 2. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any ideas or any books I can buy which can help me to explain this?&lt;/p&gt;&#xA;" OwnerUserId="10619" LastEditorUserId="667" LastEditDate="2018-11-29T00:22:30.307" LastActivityDate="2020-03-30T16:37:52.973" Title="How to explain fractions to 7 year old kid" Tags="|primary-education|fractions|" AnswerCount="10" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14832" PostTypeId="2" ParentId="14831" CreationDate="2018-11-27T08:46:02.353" Score="11" Body="&lt;p&gt;Comparing fractions only works when the whole is the same size.&#xA;Here's a few examples to get help the 7 year old understand what happens when things aren't the same size:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;1/2 will be greater than 2/4 when you compare 1/2 of a watermelon&#xA;and 2/4 of an apple.&lt;/li&gt;&#xA;&lt;li&gt;1/2 will be less than 2/4 when you compare 2/4 of a watermelon and&#xA;1/2 of an apple.&lt;/li&gt;&#xA;&lt;li&gt;If you take 1/2 of 4 pieces of candy, you will take 1 from the first&#xA;2 and 1 from the second 2 which is the same 2 out of 4 or 2/4.&lt;/li&gt;&#xA;&lt;li&gt;Furthermore 1/2 of 6 pieces of candy is 3 candies and 2/4 of 4&#xA;candies is 2 candies.  In this case 1/2 is greater than 2/4.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;By showing the examples the 7 year old will start to understand that comparing fractions is can only work if the wholes that you take the fraction of are identical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My suggestion is to show all these examples, and then explain that since 1/2 is sometimes greater than 2/4 (see 1 and 4), sometimes equal to (see 3), and sometimes less than  2/4 (see 2)- the mathematicians decided we would only compare 1/2 and 2/4 (or any fractions) when we are taking 1/2 and 2/4 of the same thing.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2018-11-27T08:46:02.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14833" PostTypeId="2" ParentId="14831" CreationDate="2018-11-27T16:59:23.940" Score="4" Body="&lt;p&gt;7? I work in the math dept at a local high school, and students 14-18 struggle with fractions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found 2 things that help. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Money Math&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Somehow, the same student struggling to take 1/2 of 2-1/2 can easily tell me that half of \$2.50 is \$1.25. For your son, 1/2 of a dollar is 50 cents. No need to stack the pennies, he knows (right?) that 50 cents is 2 quarters. But wait, we just said two quarters, 2/4. Half of a dollar is literally 2 quarters, assuming, of course, that your currency works this way. I am in the US. After this discussion, ask him what half of \$4 is, and then what 2 fourths of that same \$4 is. Further on, you can show how this all changes if the second item, the thing we're multiplying, changes. Of course, he'd prefer to have 1/4 of \$10 than half of \$1. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pizza/food Math&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm a fan of the circular pizza cut into 8 equal sectors. It makes discussing fractions easy for any student, as the fractions 1/2, 1/4, 1/8 are right there. Similar to the money discussion, you can start with what 1/2 means, and how 1/2 of 1/2 is 1/4, and then go into the above explanation of how the whole is our 8 slice pizza. When we ask &quot;how much pizza would you like?&quot; it's essential to know how much is there to begin with. 1/4 of one pizza is not the same as 1/4 of the 10 pizzas I bought to treat the class at year end. As far as 1/2 vs 2/4 is concerned, the answer is right there, they are both the same 4 slices of our standard 8 slice pizza. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In both cases, money and food lend themselves to divisibility. And having them right there avoids the issue with things being too abstract. Even in high school, I find that numbers and variables create a struggle for some students, and the faster I can move the conversation to solid objects or real life examples, the easier they get it. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-11-27T16:59:23.940" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14835" PostTypeId="2" ParentId="14672" CreationDate="2018-11-27T22:16:54.740" Score="8" Body="&lt;p&gt;I would say that this book is trying to get students to &lt;em&gt;think about what the calculations mean&lt;/em&gt;, rather than simply execute an algorithm, and I strongly believe that is something that should be done right from the beginning, not saved for later or for more advanced students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot tell you how many students I have worked with as a private tutor who, if you ask the question &quot;How many pencils do you need if you want to put 7 pencils on every table and there are 12 tables?&quot; will assiduously count by sevens until they reach 84...  but if you ask &lt;em&gt;those same students&lt;/em&gt; just &lt;em&gt;five minutes later&lt;/em&gt; &quot;What is &lt;span class=&quot;math-container&quot;&gt;$7\times12$&lt;/span&gt;?&quot; they will immediately write down the standard algorithm for multiplication and find that the answer is 84.  And then they look at those two questions, and their identical answers, and &lt;em&gt;it never occurs to them that they are connected&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For too many students -- I am even tempted to say most! -- operations like multiplication, division, finding the LCM or GCD, etc., are &quot;things you do with numbers&quot; that are entirely disconnected from any meaning or sense.  &lt;em&gt;Everything&lt;/em&gt; we do as teachers or as tutors should be about making math reasonable, not just easy to learn.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-11-27T22:16:54.740" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14841" PostTypeId="1" CreationDate="2018-11-29T16:56:31.437" Score="3" ViewCount="665" Body="&lt;p&gt;May I know how can I teach a proof-based Multivariable Calculus and linear algebra as a single course? While there are quite a few known books in the field such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Vector Calculus, Linear Algebra and Differential forms by Hubbard &amp;amp; Hubbard&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Multivariable Mathematics by Theodore Shifrin, etc&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which are proof based, I wonder if students can easily catch the proof part since most students have not taken Real Analysis. I believe I will need to spend sometime to explain about proof and in doing so, I wonder if I will be able to cover the whole general syllabus for Multivariable Calculus and Linear Algebra of the US education system? If I exclude the proof part and focus on the application and computation part, then there will be no issue at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, has anyone tried doing a proof based course and can share experience on how to handle such course?&lt;/p&gt;&#xA;" OwnerUserId="10633" LastActivityDate="2018-11-29T16:56:31.437" Title="A proof based Multivariable Calculus and Linear Algebra" Tags="|mathematical-analysis|linear-algebra|vector-calculus|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14843" PostTypeId="1" CreationDate="2018-11-30T00:53:51.400" Score="10" ViewCount="311" Body="&lt;p&gt;In the comments under &lt;a href=&quot;https://matheducators.stackexchange.com/q/14830/29&quot;&gt;another question&lt;/a&gt;, a couple of people expressed interest in how Calculus is taught at the University of Michigan.  I'm not convinced a question that narrow is appropriate for this site, so I am opening the topic up to a broader discussion.  I suggest that this question be made Community Wiki to encourage a broad range of different responses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What format does Calculus instruction take at different post-secondary institutions, both in the United States and elsewhere in the world?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;Format&quot;, here, includes such issues as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Whether classes meet in a large lecture, and if so how many students are typically enrolled in each lecture&lt;/li&gt;&#xA;&lt;li&gt;Whether classes meet in smaller sections, and if so how large those sections are, and what they are called (&quot;discussions&quot;, &quot;practica&quot;, &quot;problem sessions&quot;, &quot;labs&quot;...)&lt;/li&gt;&#xA;&lt;li&gt;Whether students meet in one-on-one tutorials, or small group tutorials, and if so who leads those tutorials (graduate students, lecturers, advanced undergraduates...)&lt;/li&gt;&#xA;&lt;li&gt;Whether classes are held in a combination of the above (e.g. &quot;large 300-student lecture that meets twice per week, supplemented by optional 20-student problem sessions twice per week&quot;)&lt;/li&gt;&#xA;&lt;li&gt;How many times per week, how long each lecture or small session lasts, how many weeks per semester (or &quot;term&quot;, or equivalent)&lt;/li&gt;&#xA;&lt;li&gt;Whether instructors have broad autonomy to teach however they want, or whether instruction is coordinated across sections to ensure some kind of consistency&lt;/li&gt;&#xA;&lt;li&gt;and anything else that seems relevant for a specific institution.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am hoping to gather a wide range of responses, including from small liberal arts colleges, large research institutions, community colleges, and others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't particularly think &quot;textbook&quot; is one of the most important variables here, but feel free to include that information if you think it is salient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feel free to include information about how things vary from, say, &quot;Precalculus&quot; to &quot;Calculus 1&quot; to &quot;Calculus 2&quot; to &quot;Calculus 3&quot; (or whatever they are called at your institution).&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-12-02T14:15:02.727" Title="Formats for Calculus instruction at different colleges and universities" Tags="|undergraduate-education|calculus|community-colleges|large-lecture|" AnswerCount="3" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14844" PostTypeId="2" ParentId="14843" CreationDate="2018-11-30T01:18:09.517" Score="4" Body="&lt;p&gt;At the University of Michigan, all instruction at the level of Precalculus, Calculus 1, and Calculus 2 is conducted in small &quot;Recitation&quot; sections -- no lectures!.  Traditionally (since the mid-1990s), these sections were capped at a maximum enrollment of 30 students; since 2016, the University has invested in lowering the class sizes to a ceiling of 20 students, which has resulted in many more sections and instructors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a typical Fall semester we have upwards of 1600 students in Calculus 1, distributed across 80-90 sections, taught by 60+ instructors; these numbers are roughly halved in the Winter semester. A tremendous amount of effort goes into trying to ensure that the instruction is inquiry-based and centered around group work, and that we deliver a consistent experience to all students in every section.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A recent article (&lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/10511970.2017.1315474&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tandfonline.com/doi/abs/10.1080/10511970.2017.1315474&lt;/a&gt;) summarized the systems that Michigan has put in place to manage a system with this many moving parts, and some of the challenges that remain in consistently delivering quality instruction.  Among the highlights:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;All sections are taught by an instructor who has full responsibility&#xA;  for running each class, and most of the instructors are graduate&#xA;  students or postdoctoral assistant professors. All of the sections&#xA;  exist in an extensive structural framework: the day-by-day course&#xA;  schedule, all deadlines, homework assignments, and exams are set by a&#xA;  course coordinator, who also runs weekly meetings with all instructors&#xA;  to address conceptual and logistical questions. Individual instructors&#xA;  teach three days a week, with 80 minutes per class-session, and are&#xA;  expected to have an engaged classroom in which students are working,&#xA;  in groups, on meaningful problems for about half of each class period.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Student assessment in the program has five primary components: two&#xA;  midterm exams and a final; individual, on-line web homework; team&#xA;  homework worked on by groups of four students, with each team&#xA;  submitting a solution paper that requires extensive written work;&#xA;  “gateway” (skills) tests, as noted below; and quizzes written and&#xA;  graded by each instructor. Exams are written by the course&#xA;  coordinator, are highly conceptual, and require significant problem-&#xA;  solving skills. They are written with the expectation that students&#xA;  will be using (graphing) calculators, and they have few “standard”&#xA;  problems that reappear semester to semester (an example exam is shown&#xA;  in Appendix A). Basic skills are developed and assessed with “gateway&#xA;  tests,” which are run in a proctored lab environment in which&#xA;  calculators are not allowed...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Ideally, a Michigan calculus classroom is noisy, with students sitting&#xA;  at tables of four happily engaged in inquiry-focused problem solving.&#xA;  While the students collaborate on solving the problem(s) at-hand, the&#xA;  instructor moves from table to table, ensuring students stay on track&#xA;  and gently guides them when necessary. In particular, instructors are&#xA;  encouraged to spend less than half of each class meeting lecturing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although the article quoted above describes Calc 1 (Math 115), much the same description applies to our Precalculus course, Math 105 (which is not actually called &quot;Precalculus&quot;; we call it &quot;Data, Functions, and Graphs&quot;, and it is the lowest-level mathematics course offered at Michigan), and to Calc 2 (Math 116).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first course taught in a lecture format is Calc 3 (Math 215, Multivariable Calculus), and which is usually capped at 110 students per section.  Concurrent with their Calc 3 lectures, students also take a computer-based lab section, run by a graduate student, in which they use various software packages (currently MatLab; in the past, both Mathematica and Maple have been used) to visualize and explore surfaces, contour plots, and other representations of multivariable functions that are difficult to produce and modify by hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also worth mentioning that, completely separate from all of the above, Michigan also has multiple independent &quot;honors&quot; options.  These are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A three semester &lt;em&gt;applied honors sequence&lt;/em&gt;, Math 156-285-286.  From the &lt;a href=&quot;https://lsa.umich.edu/math/undergraduates/undergraduate-math-courses/courses-for-freshmen/honors-sequences.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt;:  &lt;em&gt;Applied Honors Calculus II (MATH 156) is designed for engineering and science majors who received a score of 4 or 5 on the AP exam (AB or BC). MATH 156 is an alternative to MATH 116 with more emphasis on science applications and theory. The sequence continues with courses in multivariable calculus and differential equations (MATH 285-286), which are alternatives to MATH 215-216.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;A four semester &lt;em&gt;honors seminar mathematics sequence&lt;/em&gt;, Math 175-176-285-286.  From the &lt;a href=&quot;https://lsa.umich.edu/math/undergraduates/undergraduate-math-courses/courses-for-freshmen/honors-sequences.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt;:  &lt;em&gt;MATH 175 (Introduction to Cryptography) and MATH 176 (Explorations in Topology and Analysis) are taught in the Inquiry-Based Learning (IBL) style. The IBL method emphasizes discovery, analysis, and investigation to deepen understanding. These courses assume a knowledge of calculus roughly equivalent to MATH 115; they cover a substantial amount of basic number theory (MATH 175) and provide a good, high level understanding of calculus (MATH 176). The sequence concludes with Honors versions of multivariable calculus (MATH 285) and differential equations (MATH 286).&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;A four semester &lt;em&gt;honors calculus sequence&lt;/em&gt;, Math 185-186-285-286.  From the &lt;a href=&quot;https://lsa.umich.edu/math/undergraduates/undergraduate-math-courses/courses-for-freshmen/honors-sequences.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt;:  &lt;em&gt;Honors Calculus I and II (MATH 185 and 186) rigorously develop the concepts of calculus. These courses are intended for students who desire a complete understanding of the theoretical underpinnings of calculus, and they lay a solid foundation for future Mathematics courses. The sequence concludes with Honors versions of multivariable calculus (MATH 285) and differential equations (MATH 286). Most students who take MATH 185 have taken a high school calculus course, but it is not required.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;And a four-semester &lt;em&gt;honors mathematics sequence&lt;/em&gt;, Math 295-296-395-396.  From the &lt;a href=&quot;https://lsa.umich.edu/math/undergraduates/undergraduate-math-courses/courses-for-freshmen/honors-sequences.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt;:  &lt;em&gt;The Honors Mathematics Sequence provides a rigorous introduction to theoretical mathematics. These courses require an extremely high level of interest and commitment and provide excellent preparation for mathematics at the advanced undergraduate and beginning graduate level. Most students electing MATH 295 have completed a thorough high school calculus course.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The honors courses are also taught in small &quot;Recitation&quot; sections, typically 20 students or less, and meet either four days per week for 50 minute sessions, or three days per week for 80 minute sessions, depending on the course.  Because there are only a few sections of each honors course, my sense is that instructors typically have much more latitude in how they decide to run things.  Also, as both the description and the course numbers suggest, the &lt;span class=&quot;math-container&quot;&gt;$x9x$&lt;/span&gt; sequence is taught at a &lt;em&gt;much&lt;/em&gt; higher level of abstraction than the other honors courses; in many ways it really is the equivalent of an upper-division, proof-based analysis course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of these courses -- 105 (Data, Functions &amp;amp; Graphs), 115 (Calc 1), 116 (Calc 2), 215 (Calc 3), and the various honors alternatives -- are 4 credit courses.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2018-12-02T00:29:57.280" LastActivityDate="2018-12-02T00:29:57.280" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14845" PostTypeId="2" ParentId="14843" CreationDate="2018-11-30T06:53:05.100" Score="6" Body="&lt;p&gt;I teach in the Universidad Politécnica de Madrid, which is a fairly large public engineering school with research objectives. The students are comparable to those I have taught in engineering degrees at places like Georgia Tech or the University of Washington, although they enter the univesity with better preparation. The teaching of calculus in the UPM varies somewhat from school to school (the UPM is divided into many different schools) but is in broad strokes similar to what is standard in Spanish universities, which are more uniform in their practices than are their counterparts in the US (although of course there are lots of variations in details). It is strongly conditioned by the structure of Spanish degree programs and by the comparativaley poor financing of Spanish universities (when compared with other countries having a comparable ecomonic level professors and administrative staff are few and poorly paid).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Classes typically meet in large lectures, somewhere in the 50-150 students range. There is plenty of willingness to teach in smaller groups, but there are usually not enough professors available to do so. A typical class meets 4 or 5 hours a week, perhaps 1 or 2 of which are formally or nominally dedicated to working problems (this depends on local practices), although this would often be done in the usual classroom, by the usual instructor. For example, in my school we have 4 hours of instruction in a large lecture hall, and 1 hour in a room with movable tables so students can work problems in small groups or in the computer lab to use Matlab, R, or whatnot (with the one professor circulating to help them - I have only about 70 students so this is viable). Sometimes these problem sessions, computer lab session are broken into smaller groups but not for educaitonal reasons, simply because they don't fit in the rooms used (this requires assigning additional professors). Semesters are 15 weeks by law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are no teaching assistants or graders in the US sense, so it is generally not viable to assign homework in calculus classes. Often one gives some sort of quiz or weekly in class assignment or computer exercise, that might be evaluated automatically via some cell phone app, or something of this sort, or might be done in groups, so as to make grading viable, but this is often left up to individual professors, and is not universal. The main components of grading typically are one or two tests and a final exam. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Whether instructors have broad autonomy to teach however they want, or&#xA;  whether instruction is coordinated across sections to ensure some kind&#xA;  of consistency and anything else that seems relevant for a specific institution.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Such autonomy is basically inconceivable in the Spanish system. Each degree program has an official plan of study approved by a national accreditation agency. Modifying the study plan requires substantial bureaucratic movement at the level of a dean's office, so principally occurs in response to flagrant problems, or at regular 5 year intervals when these plans are obligatorialy revised. The study plan stipulates course contens, and for something like calculus it sometimes essentially establishes the curriculum. In engineering degrees the contents of even calculus courses are in any case basically obligated by law and professional accreditation requirements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every year the department responsible for teaching a course approves a &quot;study guide&quot; for the course (later approved by the large faculty - this all occurs many months before the semester starts). This is written by the official course coordinator in conjunction with the other professors assigned to teach the class. It is a document of some pages that stipulates course objectives and learning goals, and details the syllabus (in my school to the point of indicating what contents will be covered in what lecture). It describes the evaluation scheme to be used (and the description is supposed to be obligatory for the professor). It contains recommended books and other information about learning resources. The professor is expected to adhere to it more or less rigidly. Of course there are professors that don't, and not much can be done about this, but the general spirit is to do what has been stipulated in advance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This sort of planning guarantees considerable uniformity, at least where such uniformity is desired. I teach one class where the study guide intentionally leaves 20% of the grading, as well as the structuring of problem sessions, at the discretion of the individual professor (there are 9 professors). In other classes all grading is done in common. Certainly there is no grading on a curve, or a posteriori changing of grading schemes, etc. Such practices, common in the US, are generally viewed as unprofessional, unfair to the student, and potentially unethical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally speaking books are only recommended, not assigned. It is economically impossible to do anything else. Many families can't afford to shell out euros for books and no one thinks it reasonable to force them to do so. In any case everything can be downloaded from the web, and everyone knows this. Books are formally recommended, but it is essentially unheard of to assign problems from a book. Typically professors prepare weekly problem sheets (voluntary) and provide lecture notes that constitute unedited books. When several professors teach together this effort is often coordinated. There might be an official course problem list accompanied by official lecture notes, often distributed through the university's internal publication service.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the university level there is no precalculus as this is not considered (essentially in a legal sense) a university level subject and could not count towards an official university degree. The high school courses required to enter the university in a degree program that involves calculus include calculus and linear algebra. Typical first year calculus courses assume knowledge of basis differentiation and integration, and typical first year linear algebra courses assume ability to analyze and solve systems of linear equations. A typical first year program in mathematics includes calculus, one variable and multivariable, linear algebra, and often gets to more advanced topics such as Laplace and Fourier transforms, ODEs, a bit of numerical methods. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the old days (a decade ago, before the Bologna treaty), pass rates were in the 10-30 percent range, at least in engineering programs. Now they are typically more in the 30-70 percent range. (I'm guessing a bit at the numbers, but the upwards shift has been undoubtedly pronounced, a response to changing administrative exigencies). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Professors are obligated to hold office hours, in my institution 6 hours a week, and in my experience this obligation is taken more seriously than it is in the US, although students come to office hours just as infrequently as they do in the US.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bureaucratic overhead associated with teaching a calculus course is substantial in Spain, in my experience more onerous than in the US and with substantially less administrative support and fewer instructors per student. This involves things like the annual preparation of the aforementioned study guide, various offical reports on class outcomes, etc., in addition to the day to day handling of matters related to enrolment and the like. Teaching a course of 150 in the US I would have 4 TAs and several graders at my disposition, also some departamental administrative personel to attend to purely administrative matters. In Spain there is one administrative person serving three departments with a total of 50 professors, and there are no TAs or graders. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Teaching style is generally a professor's prerogative. When teaching calculus I mainly use a blackboard (when I teach programming this is done entirely in a computer lab). Some professors teach entirely using a computer/projector. Most instruction is classic lecture format, but there are people trying other approaches and this is encouraged administratively.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-11-30T06:53:05.100" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14846" PostTypeId="1" CreationDate="2018-11-30T13:30:02.320" Score="9" ViewCount="593" Body="&lt;p&gt;I am currently teaching a high-school student, 1st grade Social Science. He is weak in mathematics. My initial strategy was to explain basic concept but with high repetitions, so that he can have a strong foundation. Initially, I gave  &quot;Solving 1 Variable Linear Equations&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has been 4 months (2 hours a week in 1 day), and we have discussed up to &quot;Solving Inequalities with Square Roots: &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2x-3} = x + 1$&lt;/span&gt;... etc.&quot; Until recently, I give him an easy problem of solving 1 variable lin.eqn, which we have discussed many times before, and he still has not got strong understanding of the concept. He keeps asking easy stuffs like : &quot;&lt;span class=&quot;math-container&quot;&gt;$-2 + -4 = -6$&lt;/span&gt;.. right?&quot; or &quot;&lt;span class=&quot;math-container&quot;&gt;$2 + x = 3(\frac{x}{2} + 3) \implies  x = 3(\frac{x}{2} + 3 - 2)$&lt;/span&gt;...right?&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to solve this problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I have tried:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Using markers with different colors to indicate different terms so that the written solution looks clearer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Told him to become independent with respect to me as tutor. To get used to mathematics and read the book.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Repeat exercises of basic concept (1 var. linear equation) many times. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I also often give extra hours.. (up to 3 at most)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Impact:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;He is &lt;strong&gt;improving&lt;/strong&gt;, but not enough to get good marks (or even average). If I continue the method, there could be two possibilities : either he will be good in the &lt;em&gt;long term&lt;/em&gt;, or... not.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But still does not show good understanding of the concept. Very stiff, it seems that he thinks mathematics as &lt;em&gt;instructions that have to be memorized&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Particular Questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Should I go back to the very basics, teaching arithmetics, understand brackets, etc..? What book or article is good for this..?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;From my experience, I understand mathematics not through tutors but by reading math books. So, is it better to teach the student: how to read math books? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="10637" LastEditorUserId="10637" LastEditDate="2018-12-02T11:43:58.327" LastActivityDate="2020-09-14T20:08:19.003" Title="Teaching methods to make a weak student good at math? (particularly student from social science background)" Tags="|secondary-education|algebra|tutoring|private-lessons|" AnswerCount="3" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="14847" PostTypeId="2" ParentId="14831" CreationDate="2018-12-01T09:16:59.390" Score="6" Body="&lt;p&gt;Typically, it is better to introduce mathematical ideas after the need for that new idea is obvious. For example:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How should 2 children share 8 pieces of licorice?&lt;/li&gt;&#xA;&lt;li&gt;How should 4 children share 8 pieces of licorice?&lt;/li&gt;&#xA;&lt;li&gt;How should 3 children share 15 pieces of licorice?&lt;/li&gt;&#xA;&lt;li&gt;How should 3 children share 2 pieces of licorice? And why is this different from the prior cases?&lt;/li&gt;&#xA;&lt;li&gt;How should 5 children share 4 pieces of licorice?&lt;/li&gt;&#xA;&lt;li&gt;How should 4 children share 5 pieces of licorice?&lt;/li&gt;&#xA;&lt;li&gt;etc.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Challenge them to show and assess all of their work according to three main criteria:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Everyone must have the same amount.&lt;/li&gt;&#xA;&lt;li&gt;There should be no leftovers.&lt;/li&gt;&#xA;&lt;li&gt;They can name the amount everyone has. [This is where formal fractional language comes in handy.]&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Ensure you compare and contrast whole numbers vs fractional numbers. That is really the heart of the matter - that fractions are numbers, most of which are used to name quantities between consecutive whole numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bonus criteria and tasks can include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How can you do the sharing with as few cuts as possible?&lt;/li&gt;&#xA;&lt;li&gt;Is there some other way you can name the amount everyone received?&lt;/li&gt;&#xA;&lt;li&gt;Is there a systematic way to solve all of these problems?&lt;/li&gt;&#xA;&lt;li&gt;Generate some sharing examples that do require fractions, then generate some sharing examples that do not require fractions. How do you know when fractions are necessary?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Eventually you can guide them toward the situations of 2 kids sharing one unit of food vs 4 kids sharing 2 units of food. When they have to name the amounts, you can show them the amount of food is exactly the same and then show them that even the cuts can be the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also show them counting by fractions on a number line to show that 1/2 and 2/4 occupy the same point on a number line.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-12-01T09:16:59.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14848" PostTypeId="2" ParentId="14831" CreationDate="2018-12-02T00:19:57.440" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;OP (@NotAgain):&#xA;  &quot;Any ideas or any books I can buy which can help me to explain this?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;May I suggest games (rather than books)? For example, &#xA;BrainPop's &lt;a href=&quot;https://www.brainpop.com/games/refraction/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Refraction&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/G6dQE.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/G6dQE.jpg&quot; alt=&quot;Refraction&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Or the &lt;a href=&quot;https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Fraction-Game/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fraction Game&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Bi2Ta.png&quot; width=&quot;400&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Or &lt;a href=&quot;https://www.mathgames.com/fractions&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathGamesFractions&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/ZHYhK.png&quot; width=&quot;400&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;or &lt;a href=&quot;https://www.weareteachers.com/fun-with-fractions-7-tactile-and-kinesthetic-games/&quot; rel=&quot;nofollow noreferrer&quot;&gt;7 Fun Fraction Games for Kids&lt;/a&gt;,&#xA;or &lt;a href=&quot;https://www.weareteachers.com/fun-with-fractions-7-tactile-and-kinesthetic-games/&quot; rel=&quot;nofollow noreferrer&quot;&gt;33 Fraction Games&lt;/a&gt;, ... or many other online fraction games. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is the current locus of creative ideas, not books.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-12-02T01:01:47.380" LastActivityDate="2018-12-02T01:01:47.380" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14849" PostTypeId="2" ParentId="14828" CreationDate="2018-12-02T04:24:53.360" Score="2" Body="&lt;p&gt;Its easy for even &quot;seasoned&quot; mathematicians to make mistakes in such straightforward probability questions. With that in mind I suggest the following - &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Insist on solving the problems &lt;strong&gt;systematically&lt;/strong&gt; (especially for beginners). This means - writing out the sample space (if not in its entirety, at least the pattern), rereading what the event is, choosing the outcomes according to the event only &lt;strong&gt;from&lt;/strong&gt; the sample space and so on.&lt;/li&gt;&#xA;&lt;li&gt;Make it clear from the beginning that the two formulas &lt;span class=&quot;math-container&quot;&gt;$p(A\cup B) = p(A)+p(B)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p(A\cap B)=p(A) \times p(B)$&lt;/span&gt; are both &lt;strong&gt;conditional&lt;/strong&gt;, not universal. The former is conditional on the events being mutually exclusive and the latter on independency of the two events. So before applying each of the formulas, the conditions &lt;strong&gt;must&lt;/strong&gt; be checked.&lt;/li&gt;&#xA;&lt;li&gt;In order to check the mutual exclusiveness of two events, students may be encouraged to think &lt;strong&gt;inversely&lt;/strong&gt; i.e., &quot;Do any of the outcomes that are in favour of event &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; are also favourable to event &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;?&quot;&lt;/li&gt;&#xA;&lt;li&gt;Take the aid of &lt;strong&gt;Venn diagrams&lt;/strong&gt; to visualise the events. This will help see both mutual exclusion and independence. This approach is particularly useful to check whether events are truly independent.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;You can use an easy-to-digest explanation for a few problems initially, but if students do not practice such problems systematically until they get an intuition for it, they are bound to struggle with the concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, do encourage the student(s) for their efforts and specifically point out that it can be a hard concept to learn in the beginning but becomes easier with practice.&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2018-12-02T04:24:53.360" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14850" PostTypeId="2" ParentId="14843" CreationDate="2018-12-02T14:15:02.727" Score="4" Body="&lt;p&gt;I'm studying physics at Eötvös Loránd University, Hungary, so my answer will not be complete, but I will try my best.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first semester, we can choose between &quot;advanced&quot; and &quot;normal&quot; level calculus (there aren't many proofs on calculus for us). Both of them last for a &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; semester, with &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;hr lecture + &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;hr practice/week. &quot;In my times&quot;, there were &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; lecture for both levels and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; practice group for the advanced class (with about &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; students), and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; practice groups for the normal class (with about &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; students/group). In the remaining part of the semester, we have a &quot;differential equations in physics 1&quot; class, with only &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; difficulty level, but &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; practice group + &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; lecture group (and the same &lt;span class=&quot;math-container&quot;&gt;$3+3$&lt;/span&gt; hr/week). Also, we are having a &quot;vector calculus&quot; class, &quot;advanced&quot; and &quot;normal&quot; level, &lt;span class=&quot;math-container&quot;&gt;$4+4$&lt;/span&gt; (lecture + practice) hr/week, and every occasion was &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;hr long. About &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; part of this class is about linear algebra, and &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; is about differentiating and integrating in higher dimensions (It's a rather practical class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the second semester, there's the &quot;differential equations in physics 2&quot; class, with &lt;span class=&quot;math-container&quot;&gt;$1+1$&lt;/span&gt; hr/week, with &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; lecture and &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; practice groups. There's also a &quot;mathematical methods in physics&quot; class, with only &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; group and &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; hr/week, all at once. It's a practice and lecture at the same time. This class contains &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; parts: Fourier analysis, Complex analysis, Calculus of variations and Curvilinear coordinates. It's also a practical class. This class will be reformed this year, there'll be &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; levels here as well, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also &quot;Analysis 1&quot; and &quot;Analysis 2&quot; classes on the theoretical physics specialization, with only lectures (there were an &quot;Analysis 3&quot; before as well, but it's on the MSc now). I don't know much about them, because Analysis 1 is on the 4th semester, but I know that they are differentiating in normed spaces. On the Analysis 2, there's some complex analysis and measure theory, and on Analysis 3, there's some functional analysis.&lt;br&gt;&#xA;&lt;br&gt;&lt;br&gt;&#xA;I'm taking the &quot;Analysis 1&quot; class at the moment (which is a first semester subject for mathematicians), so I can talk about it a little bit as well. Basically, they can choose between &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; paths: They can either take the &quot;Analysis 1&quot; and &quot;Analysis 2&quot;, or the &quot;Calculus 1&quot;, &quot;Calculus 2&quot; and the &quot;Introduction to analysis&quot; classes. I don't know much about the calculus path, but I was told that it's much easier, so I think they are going slower, and proving fewer things. Back to the analysis, there is only &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; group for the lecture, and they take &lt;span class=&quot;math-container&quot;&gt;$2+1$&lt;/span&gt; hr/week. But the practices are a bit different: There is an intensive group (this is the hardest) with &lt;span class=&quot;math-container&quot;&gt;$24$&lt;/span&gt; student, and &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; advanced groups, with  &lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt; student/group. And all of the groups are having &lt;span class=&quot;math-container&quot;&gt;$2+2$&lt;/span&gt; hr/week of practices.&lt;br&gt;&#xA;&lt;br&gt;&lt;br&gt;&#xA;So usually the lectures are collective, while there are smaller groups on the practices, with different difficulty levels.&lt;br&gt;&#xA;&lt;br&gt;&lt;br&gt;&#xA;Remark 1: This kind of separation is quite common for us, but there are some classes without it, like the mathematical methods in physics was.&lt;br&gt;&#xA;Remark 2: In our university, the &quot;Analysis&quot; and &quot;Calculus&quot; are different, as you could see. But I didn't see this kind of distinction between them elsewhere in Hungary. For example the engineers at different universities are having only analysis, which is, of course, not as deep as the analysis here. (Or at least this is how I saw it).&lt;/p&gt;&#xA;" OwnerUserId="10644" LastActivityDate="2018-12-02T14:15:02.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14851" PostTypeId="2" ParentId="14805" CreationDate="2018-12-03T16:23:58.943" Score="3" Body="&lt;p&gt;I've found that a lot of success for introductory advanced math hinges on getting students to feel comfortable talking about their math ideas and actually having a &lt;em&gt;discussion&lt;/em&gt; wherein they can start to develop an awareness of how to precisely express what they do and don't know.For example, in an RA course I once took, we were put into study groups and had a (twice) weekly GoogleDoc discussion of the problem set.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So maybe one thing to do would be to model some problem set discussions early on in the semester with a &lt;a href=&quot;https://www.cultofpedagogy.com/speaking-listening-techniques/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fishbowl discussion strategy&lt;/a&gt; (maybe use some of the stronger students or call in a couple of grad students).  Then, monitor how the groups are doing on their independent GoogleDocs and pull out exemplary segments to analyze in detail in class.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for other resources, I've found the early portions of &lt;a href=&quot;https://bookstore.ams.org/real&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frank Morgan's &quot;Real Analysis&quot;&lt;/a&gt; have a simple conversational tone that's pretty good for developing baseline intuition about how unexpectedly weird the Real Numbers can be once you delve into their structure past what's necessary for basic arithmetic.&lt;/p&gt;&#xA;" OwnerUserId="10572" LastActivityDate="2018-12-03T16:23:58.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14852" PostTypeId="1" AcceptedAnswerId="14865" CreationDate="2018-12-03T18:28:37.967" Score="6" ViewCount="323" Body="&lt;p&gt;I'm working one-on-one with a student who is part of a sponsored refugee family. He's bright and a good learner, but has had a lot of interruptions to his education. No indication of any learning disability, and language is an issue, but not a challenging one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The family is hoping to get him working at grade level (grade 9, from about a grade 5 level) as soon as possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is this: is it a better strategy to:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;go full-tilt on one topic (for example, fractions) from start to finish, then tackle the next topic, and the next, and so on&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;OR&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;follow standard curricula - cover all of grade 5, then 6, etc.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I realize this is a bit open-ended and opinion based, but I'm looking for suggestions based on people's experience and best practice recommendations.&lt;/p&gt;&#xA;" OwnerUserId="10647" LastActivityDate="2018-12-05T16:37:53.620" Title="Mainstreaming math student" Tags="|secondary-education|mathematical-pedagogy|curriculum|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14853" PostTypeId="1" AcceptedAnswerId="14860" CreationDate="2018-12-03T20:35:50.313" Score="4" ViewCount="426" Body="&lt;p&gt;I am teaching entry calculus to a bunch of students outside class (more like complementary to their math classes, without making much connections) and I can teach on a much more individual level than in a class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There seems to be a tendency to overlook mistakes such as a minus sign instead of a plus sign. If the answer is -0.954 and I point out that it should be 0.954, some of them are taking it more lightly (&quot;oh, it was just the sign&quot;), while if the answer doesn't look similar, they spend more time and think more about why is it wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This goes also when analysing deeper concepts. For example, in doing some rather practical applications of the material (such as applications of the limit in modeling/estimating some phenomena), they are taking mistakes in modelling much lighter than mistakes in the computation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to convince them that those are actually real mistakes, and not to be taken more lightly than other mistakes?&lt;/p&gt;&#xA;" OwnerUserId="8147" LastEditorUserId="10647" LastEditDate="2018-12-14T22:30:44.937" LastActivityDate="2018-12-14T22:30:44.937" Title="How to make a student not overlook easy mistakes such as the wrong sign" Tags="|mathematical-pedagogy|" AnswerCount="6" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14854" PostTypeId="2" ParentId="10122" CreationDate="2018-12-03T22:02:11.977" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think you understand what the students need.  You seem like a math type, not a scientist, not a math teacher serving scientists.  Not meant adversarially, honest.  Consider this a blind spot.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would be hesitant about iconoclastic courses (not a laundry list, different from traditional) given (1).  You may make it worse.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would go with reversing calculus among your choices.  Just looking at a diffyQ, you see a bunch of derivatives.  It is similar to integral calc, which students have had.  Note, that this basically means a set of techniques, similarly to integral methods in calc 2.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would avoid modeling since kids who need it will get it in required classes.  Word problems are hard.  Plus ther is limited time in ODEs to hit various forms, just as math.  Let them learn the basic second order constant coefficient nonhomo as an algebraic form.  Then when they encounter it in a myriad of places (e.g. controls), they will at least feel they know the math, while dealing with the physical concepts (how are inductors and capacitors different; dashpots versus springs; etc).  Also, I don't think you have time to do justice to word problems as well as normal ones.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;No on we can't solve a lot.  They will learn that soon enough.  And they need analytical solutions for lots of derivations (e.g harmonic oscillator).  There is a difference between designing a specific engineering model and derivation of broad equations in fluids, thermo, physics, chemistry, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;No on change of basis.  Too hard, too end of course to be a theme for an intro class.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10126" LastActivityDate="2018-12-03T22:02:11.977" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14855" PostTypeId="1" CreationDate="2018-12-03T22:46:18.187" Score="11" ViewCount="1451" Body="&lt;p&gt;I'm currently teaching a couple of courses that have a calculus prerequisite, and within the last week I've had two students make notational mistakes that amount to writing &lt;span class=&quot;math-container&quot;&gt;$y\frac{d}{dx}$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}y$&lt;/span&gt; (although in terms of different variables than &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;). E.g., they might write something like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x^2 \frac{d}{dx}=2x.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At least one of them actually seemed relatively fluent with calculus, in the sense of knowing facts like &lt;span class=&quot;math-container&quot;&gt;$\int x^n dx=x^{n+1}/(n+1)$&lt;/span&gt;. I pointed out his mistake and made the analogy with the ungrammatical expression &lt;span class=&quot;math-container&quot;&gt;$4\sqrt{}$&lt;/span&gt;. I mentioned that I had just had another student make a similar mistake recently, and I asked him if he could explain more about what he was thinking when he wrote it. His response was that that was just the way he had gotten used to seeing it in his calculus book! (I assume this is not actually true.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone provide any insight into why this would be a common enough mistake that I would see it this frequently? I wonder if there is some confusion because the &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; notation involves a postfix operator, or because in an integral we usually write the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; at the end...?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would seem obvious to me that their notation wouldn't make sense if you think of the derivative as the ratio of two infinitesimals, but presumably many of them haven't been exposed to that way of thinking. Part of their confusion may also be because they are used to  everything being &lt;span class=&quot;math-container&quot;&gt;$y(x)$&lt;/span&gt;, never &lt;span class=&quot;math-container&quot;&gt;$\Psi(x)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$v(t)$&lt;/span&gt;. They also never seem to have been asked to think about the meaning of notation and why it makes sense, and don't seem to understand attempts to elicit discussion of this kind of thing. E.g., if I ask them whether they've been taught that &lt;span class=&quot;math-container&quot;&gt;$d/dx$&lt;/span&gt; is an operator, like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\sin{}$&lt;/span&gt;, they generally look at me blankly, as though this is not the kind of thing that was ever discussed in their calculus class.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="3453" LastEditDate="2021-03-27T14:26:25.067" LastActivityDate="2021-03-27T14:26:25.067" Title="Multiple students writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ -- why?" Tags="|calculus|students-mistakes|notation|community-colleges|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14856" PostTypeId="2" ParentId="14853" CreationDate="2018-12-04T12:39:43.707" Score="1" Body="&lt;p&gt;Let me elaborate Rusty's last comment: you go to one of the students, and you tell him/her you'll give him/her -20 euros (or dollars, whatever the currency in your country), and you just open your hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student will be waiting for you for the money but then you explain that, giving -20 euros means, (s)he has to give you 20 euros, and then you say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;That's the importance of a sign!&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2018-12-04T12:39:43.707" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14857" PostTypeId="2" ParentId="14855" CreationDate="2018-12-04T16:09:14.027" Score="21" Body="&lt;p&gt;When a student writes incorrect notation, &lt;strong&gt;ask them to read it out loud&lt;/strong&gt;. I would say something like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Something here doesn't look right, but we can fix it. Could you read this work out loud? I think probably you are not super familiar with this topic, and that's okay, but this can help us fix it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I had success with this when dealing with a student who wrote &quot;&lt;span class=&quot;math-container&quot;&gt;$\sqrt 4 = \sqrt 2$&lt;/span&gt;&quot;. After some prodding and reassurance, they read it to me: the student said &quot;The question is the square root of 4. The answer is the square root is 2.&quot; The key here was that the student didn't understand what the &quot;&lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;&quot; meant (which is a &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7964/issues-with-equals-where-does-this-come-from-and-how-do-i-combat-it&quot;&gt;whole topic of discussion on its own&lt;/a&gt;) nor did the student really understand what the &quot;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{}$&lt;/span&gt;&quot; meant (see also &lt;a href=&quot;https://matheducators.stackexchange.com/q/1040/11&quot;&gt;this question&lt;/a&gt; which is the source of my understanding).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in this case you may be surprised by something similar. Maybe the student will say that &quot;&lt;span class=&quot;math-container&quot;&gt;$x^2 \frac{d}{dx} = 2x$&lt;/span&gt;&quot; says &quot;&lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; prime is &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;&quot;? Or slightly better &quot;&lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;'s derivative is &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;&quot;. If the student reads &quot;d d x,&quot; don't accept that: get them to translate what it means to them inside the sentence. Maybe they think &quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}=$&lt;/span&gt;&quot; is a group of symbols that separates a question and an answer, like my student above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No matter what they say, the correction is relatively easy: you can explain that &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}$&lt;/span&gt; actually precisely means &quot;The derivative of.&quot; Then you can read their work back to them for effect to help them see why it looks strange: &quot;&lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; the derivative of equals &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;.&quot; It's important that you emphasize this is not &lt;em&gt;your opinion&lt;/em&gt;, but that these symbols have an accepted precise meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2018-12-04T16:09:14.027" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14858" PostTypeId="2" ParentId="14853" CreationDate="2018-12-04T18:20:19.543" Score="1" Body="&lt;p&gt;I don't know &quot;how deep&quot; is your &quot;course&quot; and their knowledge, but a good demonstration of the importance of the sign might be an example with multiple parts, when you need to rely on your previous answers. Based on my failures, the complex analysis is a good example. There are contour integrals, where you need to break them apart into smaller ones. And once or twice I did miss the sign of an integral. And it did really mess up the final answer, for example, it could make the &lt;span class=&quot;math-container&quot;&gt;$2\pi + 2\pi =4 \pi$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$2\pi -2\pi=0$&lt;/span&gt;, which is not just a sign error.&lt;/p&gt;&#xA;" OwnerUserId="10644" LastActivityDate="2018-12-04T18:20:19.543" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14859" PostTypeId="2" ParentId="14853" CreationDate="2018-12-04T22:17:35.880" Score="3" Body="&lt;p&gt;It might help to give costly examples of such mistakes.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.worldsciencefestival.com/2014/11/six-tiny-scientific-mistakes-created-huge-disasters/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.worldsciencefestival.com/2014/11/six-tiny-scientific-mistakes-created-huge-disasters/&lt;/a&gt;&lt;BR&gt;&#xA;&lt;a href=&quot;https://www.nature.com/news/2004/041018/full/041018-1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nature.com/news/2004/041018/full/041018-1.html&lt;/a&gt;&lt;BR&gt;&#xA;Reports on the crash of a Genesis return capsule because some sensors were installed backwards (i.e. with the wrong sign).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some other examples from the first URL describes errors due to unit-conversion (some used feet while others used meters).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://boscoh.com/protein/a-sign-a-flipped-structure-and-a-scientific-flameout-of-epic-proportions.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://boscoh.com/protein/a-sign-a-flipped-structure-and-a-scientific-flameout-of-epic-proportions.html&lt;/a&gt;&lt;BR&gt;&#xA;describes a sign-error in a computer program that misled researchers in reporting their results, leading to some likely embarrassing retractions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You could &quot;accidentally&quot; make a sign error in tallying their score on something ( an exam or a game )...&lt;BR&gt; and use their &quot;it was just a sign&quot; excuse.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10537" LastActivityDate="2018-12-04T22:17:35.880" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14860" PostTypeId="2" ParentId="14853" CreationDate="2018-12-04T22:42:24.547" Score="6" Body="&lt;p&gt;This is an interesting puzzle, for sure! A few thoughts come to my mind.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;The Joy of Peer Review&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I'm a huge fan of having students review each other's work. All the research shows that if done properly, it is one of the most effective ways to reinforce existing knowledge and deepen learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since you aren't working with these students in a group setting, this becomes more challenging. However, I still think it is possible. Make copies of their work as they bring it to you or take pictures of what they write on a whiteboard. Then present one student's work to another and say &quot;so and so was working on this problem and came up with this solution. Do you agree with it? Why?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since our educational system places a high emphasis on original, independent work, I would recommend telling your students why you are doing this beforehand. My favorite talking points include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Most &quot;real&quot; mathematics is done in groups. Adult mathematicians who discover interesting, cool things don't work by themselves.&lt;/li&gt;&#xA;&lt;li&gt;If you are researching a new topic and find a solution online, you need to be able to understand if it works and point out any flaws that it has. This helps everyone do better math.&lt;/li&gt;&#xA;&lt;li&gt;Thinking about work from this angle provides different exercise for their brains. Athletes don't only do one kind of work out and expect to get stronger. Flexible thinking will make them see different things.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Present them with a wide variety of other work. Make sure to give them correct solutions as well as incomplete or incorrect ones. You can also give them a correct solution but where work was not clearly shown. They can learn from all of these examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When they spend 30 minutes trying to understand why someone else's function doesn't graph the way that it should only to find that the sign was flipped, they will be more diligent with their own work.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Let Them Find Their Own Mistake&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;In addition to letting them find each other's mistakes, you can let them find their own just like they would if they were editing a paper for an English class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When they come to show you work, don't instantly tell them whether their answer is right or wrong. Instead, ask them to prove its correctness. Again, you can cite the idea that mathematicians who do interesting work aren't working on problems where the answers are known already.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask them how they can prove it to you/themselves. If they aren't sure, you can offer suggestions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Graph it!&lt;/li&gt;&#xA;&lt;li&gt;Explain how they reached the answer&lt;/li&gt;&#xA;&lt;li&gt;Solve it &quot;backwards&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you have a chance to gather several students together, you can have them &quot;present&quot; on problems and encourage them to ask hard questions, point out problems in each other's solutions, or express confusion with the way something was explained or proven.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They will get into a habit of checking their own work, which will make them even more independent as learners and will eliminate this sort of problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck! It sounds like you have an awesome group of students!&lt;/p&gt;&#xA;" OwnerUserId="10626" LastActivityDate="2018-12-04T22:42:24.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14862" PostTypeId="2" ParentId="1655" CreationDate="2018-12-05T02:30:27.003" Score="9" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I stated that the cotangent function is decreasing where it's defined, and gave some explanation based on the sine and cosine functions. She insisted cotangent was increasing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How about this: &quot;Hmm...I'm pretty sure my argument was right, but maybe I'm just having a brain fade this morning, so let's check. Those of you who have a calculator handy, could you pull it out and calculate cotangent of 0.1 and cotangent of 0.2, so we can check?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Advantages of this approach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) It demonstrates the correct real-world technique for checking your own work, which is to find some &lt;em&gt;independent&lt;/em&gt; way of checking it, rather than just going over your own steps and looking for a mistake. Recapping your own work usually doesn't work because you're mentally locked in to your own steps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) If the class sees two different ways of establishing the same fact, it should erase any doubt in their mind as to whether you were right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) It models civil behavior toward other people when you are disagreeing over an objective fact. You are humble enough to admit that you could be wrong, and you don't dismiss the other person's objection based on status or bluster. You approach it as a situation where the fact is what is to be discussed, and if there is a mistake, what should be criticized is the mistake, not the person.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) Admitting the possibility of your own error has the paradoxical effect of increasing your authority as an expert; denying that you could have made an error is behavior that people will easily recognize as a sign of insecurity. The expert is not in doubt about their knowledge or self-worth, so they approach the situation in a relaxed way.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2018-12-05T02:39:16.977" LastActivityDate="2018-12-05T02:39:16.977" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14863" PostTypeId="2" ParentId="14852" CreationDate="2018-12-05T03:27:13.087" Score="2" Body="&lt;p&gt;Do you have a favorite curriculum? If you don't, and he likes math, he might enjoy the &lt;a href=&quot;https://beastacademy.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beast Academy&lt;/a&gt; curriculum. Be aware that their level 3 feels like it's for older than 3rd grade (etc). They have books for grades 2 through 5, which I think is more like grades 3 through 8. Definitely all that's needed before algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I were working with him, I'd try to structure the time in a way that encourages interest. Which parts does he find most interesting? What motivates him in math? Beast has 12 chapters per grade level, and I don't think you have to do them in order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're not using something that goes by grade level, just keep it connected from one session to the next, and focus on what he can figure out. (I tutored a very mathematically inclined boy for a few years. He did multi-digit multiplication totally backwards from the standard algorithm. It worked for us.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-12-05T03:27:13.087" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14864" PostTypeId="2" ParentId="14853" CreationDate="2018-12-05T13:15:19.367" Score="3" Body="&lt;p&gt;Perhaps one approach would be to emphasize to them some general number sense and estimation. That way, they can get in the habit of stopping and thinking &quot;Hmmm....does this answer seem reasonable?&quot; before they circle it and move on. For example, an exponential growth problem could go wrong by this &quot;simple mistake:&quot; the student taking the rate of 4% and adding it to 1 to get 5 and then multiply the starting amount by 5 and moved on. So a population of 250 that grew by 4% ended up at &lt;span class=&quot;math-container&quot;&gt;$250(1+4\%)=1250$&lt;/span&gt;. Hopefully with some number sense, they will pause and re-evaluate their steps (&quot;That seems like a really big jump for just a small 4%...it shouldn't be that big...Oh woops, I forgot to change 4% into 0.04 before adding the 1, it should really be &lt;span class=&quot;math-container&quot;&gt;$250(1+0.04)=260$&lt;/span&gt;, that looks better.&quot;)&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2018-12-05T13:15:19.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14865" PostTypeId="2" ParentId="14852" CreationDate="2018-12-05T16:37:53.620" Score="1" Body="&lt;p&gt;If you have the luxury of ignoring what's going on in math class in school, do so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want a student to make big progress, you need to focus on common bottlenecks that span multiple grades, not the entire content of each individual grade. Students rarely cry or give up on their intellect because they don't understand volume. Many students will cry or give up on their intellect because they don't understand fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, focus exclusively on:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Whole number sense. That's place value, the operations, relationships between the operations, estimates, math facts (e.g. times table such as &lt;span class=&quot;math-container&quot;&gt;$9&#xA;\times 8=72 $&lt;/span&gt;), and extended math facts (&lt;span class=&quot;math-container&quot;&gt;$900\times80 = 72000)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Extended whole number sense to fractional sense. Ensure the student can place any set of any rational number on a number line. Make sure they can intelligently compare and contrast whole number arithmetic and fractional arithmetic.&lt;/li&gt;&#xA;&lt;li&gt;Introductory algebra. That means solving and checking simple linear equations (e.g. &lt;span class=&quot;math-container&quot;&gt;$5x+2=3x+28$&lt;/span&gt;), then using that knowledge to, say, determine, check, and use formulae for &lt;a href=&quot;http://visualpatterns.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Visual Patterns&lt;/a&gt;, or other simple word problems. Representing relationships in graphs, tables, equations, words, drawings, etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2018-12-05T16:37:53.620" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14867" PostTypeId="2" ParentId="14831" CreationDate="2018-12-06T15:36:02.310" Score="2" Body="&lt;p&gt;&lt;strong&gt;The numbers in a fraction are different parts of speech.  Denominators are nouns.  Numerators are adjectives. Denominators say 'what'  Numerators say 'how many'&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This becomes somewhat more apparent in words.  I have three quarters of a pie.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But you have to take it a step further:  A fraction needs a prepostional phrase:  It's always a fraction of &lt;em&gt;something&lt;/em&gt; When we go abstract, the something can be thought of as being just 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So a half of 4 candies is 2 candies.&#xA;Two quarters of 4 candies is 2 candies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A half of 12 m&amp;amp;m's is 6 m&amp;amp;m's&#xA;A third of 12 m&amp;amp;m's is 4 m&amp;amp;m's&#xA;A third of 6 m&amp;amp;m's is 2 m&amp;amp;m's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you are doing these, you can lay them out in rows and arrays.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An array of 3x4 m&amp;amp;m's with one row being yellow:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What fraction of m&amp;amp;m's are yellow.  Four twelfths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Divide into columns each with 2 browns and a yellow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are the columns the same?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What fraction of the column is yellow?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One third&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Play with this for a while.  At the end let him eat the manipulatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are on a no-sugar regime, coins, cards and poker chips also work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this prepositional phrase requirement is what gets people screwed up with percents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The incidence of measles fatalities has risen 70% due to anti-vaxers&quot;  Sounds like an epidemic.  But when you find out that it went from 10 to 17 out of 60 million kids it's not as alarming.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can illustrate this with a puzzle:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I'm sorry Mike, I have to give you a 50% pay cut.  Instead of &lt;span class=&quot;math-container&quot;&gt;$12/hour I can only pay you $&lt;/span&gt;6 and hour.  Maybe next month...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A month goes by.  &quot;Mike, things are looking up.  I'm giving you a 50% pay raise to $9/hour&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;WHERE's my other 3 bucks!&lt;/p&gt;&#xA;" OwnerUserId="2114" LastActivityDate="2018-12-06T15:36:02.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14869" PostTypeId="1" CreationDate="2018-12-06T19:08:30.647" Score="14" ViewCount="846" Body="&lt;p&gt;I'm teaching introductory real analysis at a large public university in the US. A common question from students is of the form &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Why &lt;strong&gt;can't&lt;/strong&gt; I just do it like this?&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;i.e. Often a student has come up with a not fully rigorous but quick or easy version of a correct analysis proof and then demand to know &lt;em&gt;exactly&lt;/em&gt; &quot;&lt;strong&gt;where&lt;/strong&gt;&quot; the mistake is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A concrete example would be e.g. showing that the series &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^{\infty}(a_n + b_n)$&lt;/span&gt; diverges to &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt; say by using&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sum_{n=1}^{\infty}(a_n + b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n&#xA;$$&lt;/span&gt;&#xA;and then saying that &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^{\infty}a_n$&lt;/span&gt; converges and &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^{\infty}b_n$&lt;/span&gt; diverges to &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the difficulties of the teacher is that the student happens &lt;em&gt;not&lt;/em&gt; to have made a fatal mistake... they have &quot;got the right answer&quot; and now you are telling them that something is still wrong. The issue of course this isn't a real equation; it's a nonsense equation like &quot;&lt;span class=&quot;math-container&quot;&gt;$+\infty = 1+ \infty&quot;$&lt;/span&gt; and really they need to argue with the partial sums. i.e. The real mistake is usually just... &quot;you've used this like its a general fact but it isn't a general fact&quot;... and that usually seems like a weak answer to the student (in my experience). I think this is partly because until you are more experienced and have done analysis in an unknown setting, you are not used to deliberately considering all of the things that could go wrong, so they are focussed on the fact that in this example nothing went wrong so what's the big deal?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clearly I can't be expected to come up with the exact explicit example that highlights where their thinking would have let them down, but sometimes it feels like I'm trapped into doing that (and coming up with an example live at the blackboard is tricky!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there are any good tricks or tips to deal with this sort of issue when teaching analysis?&lt;/p&gt;&#xA;" OwnerUserId="10001" LastEditorUserId="10001" LastEditDate="2018-12-06T19:49:51.490" LastActivityDate="2019-01-14T16:11:21.733" Title="How to deal with &quot;Why can't I just do ......&quot; in real analysis?" Tags="|mathematical-analysis|students-mistakes|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14870" PostTypeId="2" ParentId="14869" CreationDate="2018-12-07T08:04:57.450" Score="6" Body="&lt;p&gt;The issue here is that the student is still trying to learn a fundamental property of mathematics as a field of study - that the truth of any claim can, in principle, be reduced back to a relatively simple set of axioms and relatively simple rules of inference. Furthermore, this is something one should be capable of doing to any claim one wants to make use of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the rigorous stage of learning mathematics, according to the classification of Terence Tao: &lt;a href=&quot;https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;noreferrer&quot;&gt;https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this stage, the appropriate response to an uncertain claim is to provide a proof. If something is obviously true, then it should be proven from first principles or reduced to known facts proven from them. This is something you have to tell to the student, and also to demonstrate, until they understand the idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the specific example, the reasoning is correct on a moral level: A converging series is essentially just a constant (up to an epsilon of error when the index is big enough), which will not affect the convergence of the summed series, so the diverging part dominates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proof could go by reducing the problem to sequences and showing that the sum of a converging and a diverging sequence diverges (maybe only in the case the divergence is towards plus infinity). You want the sum to be eventually bigger than a fixed &lt;span class=&quot;math-container&quot;&gt;$M \in \mathbb{R}$&lt;/span&gt;, and the converging sequence is almost &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; (up to some small epsilon that you can choose to be less than one) when the index is big enough, so consider indices so large that the diverging sequence is bigger than &lt;span class=&quot;math-container&quot;&gt;$M-A+1$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some arguments that might help the student see this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What if you have a sum of two converging sequences, or two diverging ones, or one diverging to infinity and the other oscillating within some bounded interval, or one converging to plus infinity and the other to minus infinity, etc. Can they answer what happens in each of those cases?&lt;/li&gt;&#xA;&lt;li&gt;There is a lot to remember unless they learn to prove things. It is a compression technique - large amounts of information are compressed to a much smaller number of proof techniques. This is especially true, since there will be more and more mathematics to master in further courses and studies.&lt;/li&gt;&#xA;&lt;li&gt;Someone needs to know and verify which computational techniques and shortcuts are viable. You are training to become that one.&lt;/li&gt;&#xA;&lt;li&gt;It increases one's confidence when doing mathematics when one can simply prove things whenever doubt arises. As a working mathematician I do this often for quite simple claims, even though I have a good idea what the correct answer is, because this helps to avoid simple mistakes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One should remember that not everyone can learn this, or at least it is terribly difficult to learn for some people (non-mathematicians I have talked to); maybe they could learn the techniques, but the philosophy (of being able to prove anything, in principle) is genuinely difficult for some.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2018-12-07T08:04:57.450" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14871" PostTypeId="2" ParentId="14831" CreationDate="2018-12-07T09:28:25.753" Score="1" Body="&lt;p&gt;7 years old seems quite young, so my answer is not specifically thought for this age (it is much inspired by a recent &lt;a href=&quot;https://twitter.com/mtiques/status/1066783105229443073&quot; rel=&quot;nofollow noreferrer&quot;&gt;twitter thread&lt;/a&gt; about these questions for 12 yo), but I hope it can still help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example you give shows precisely how using &quot;real world&quot; interpretations can be misinterpreted, therefore I do not think that coming up with more real-world interpretation helps (such interpretation &lt;em&gt;are&lt;/em&gt; needed in the course of learning fraction, but they are in my opinion not the solution to your particular problem).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What seems is missing is the very meaning of the symbols &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac24$&lt;/span&gt;. First and foremost : &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt; is a &lt;strong&gt;number&lt;/strong&gt;, not an &lt;strong&gt;operation&lt;/strong&gt;. This is a common belief that needs to be dispelled, but is very understandable given the way we write. In your example, &lt;span class=&quot;math-container&quot;&gt;$\frac24$&lt;/span&gt; is interpreted by the pupil as &quot;2 out of 4&quot;, which &lt;strong&gt;it is not&lt;/strong&gt;. I think that it might actually be easier to stick to the maths here in order not to confuse these things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;em&gt;what is&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac24$&lt;/span&gt; ? By definition, it is the unique number that, when multiplied by &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, gives &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;. Nothing less, nothing more. You can lean on the relative integers to explain this: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Do you remember that we could not perform the operation &lt;span class=&quot;math-container&quot;&gt;$3-5$&lt;/span&gt; at some point, because we would have needed a number that, when added &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;, gives &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;? And we only had non-negative number, none of which would do? Then we introduced new numbers, e.g. &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;, that yields solutions to such problems. Here it is the same: with only integers, we cannot find any number which, when multiplied by &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, would give &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;. So we just added numbers to that effect, and denoted them with this somewhat weird notation that, you'll see, will prove very convenient in computation.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note 1.&lt;/strong&gt; If you have not yet dealt with negative numbers, maybe it would be easier to do so first. You can use level buttons in an elevator for concreteness,  find this much better then the usual money/debt considerations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note 2.&lt;/strong&gt; This is a fantastic opportunity to recall that the division &lt;em&gt;operation&lt;/em&gt; is performed by actually solving an &lt;em&gt;equation&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, how can one see that &lt;span class=&quot;math-container&quot;&gt;$\frac12=\frac24$&lt;/span&gt;? We should use the definition, and e.g. look at what happens to &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt; when we multiply it by &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;. Its own definition only says what happens when we multiply it by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, so let us use that &lt;span class=&quot;math-container&quot;&gt;$4=2\times 2$&lt;/span&gt; to observe that multiplying &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; is the same as multiplying it by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, and then multiplying the result by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;. By definition &lt;span class=&quot;math-container&quot;&gt;$\frac12\times 2 = 1$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$\frac12 \times 4 = \frac12\times 2\times 2 = 1\times 2 = 2$&lt;/span&gt;. Now by definition, the number that, when multiplied by &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, gives &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, is &lt;span class=&quot;math-container&quot;&gt;$\frac24$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$\frac12 = \frac24$&lt;/span&gt;. That seems long and hard only because it is very detailed; this motivates the learning of computation techniques that will simplify these kind of reasoning and &quot;factor&quot; them quite a lot. Because &lt;em&gt;we&lt;/em&gt; are used to these computations, we easily forget where they come from, but they need to be grounded in the definition in order to avoid them being merely magic formulas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other erroneous conceptions that will probably need to be dispelled along the way:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;&quot;two different sets of symbols must represent two different numbers&quot;&lt;/em&gt;: here it can be a problem to even imagine that we can use both &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac24$&lt;/span&gt; to denote the very same thing; this is unavoidable from our definitions: in our words, we want the number we added to be a solution to &lt;span class=&quot;math-container&quot;&gt;$4x=2$&lt;/span&gt; to be unique, but it may well have been added as solution of another equation (here, &lt;span class=&quot;math-container&quot;&gt;$2x=1$&lt;/span&gt;),&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;&quot;equality is like the = button of my pocket calculator: it is an oriented symbols that tells what happens when I perform the operation mentioned in the left term&quot;&lt;/em&gt;: very common conception, even among undergrads. Often &lt;span class=&quot;math-container&quot;&gt;$\frac12 =\frac24$&lt;/span&gt; will puzzle students much more than &lt;span class=&quot;math-container&quot;&gt;$\frac24 = \frac12$&lt;/span&gt;, but in any case to understand what we mean by this they need to be told that '=' merely expresses that the terms on both sides are different guises for the very same object.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="187" LastActivityDate="2018-12-07T09:28:25.753" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14872" PostTypeId="2" ParentId="14869" CreationDate="2018-12-07T09:49:03.020" Score="1" Body="&lt;p&gt;(I know this is not an answer, but for readibility reasons, I'm putting this comment as an answer)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Am I reading your question correctly?&lt;br&gt;&#xA;So, the student is saying:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^{\infty}a_n$&lt;/span&gt; converges&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^{\infty}b_n$&lt;/span&gt; diverges to &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, the conclusion of the student is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^{\infty}(a_n + b_n)$&lt;/span&gt; diverges to &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And your reaction is: this is wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, this is right. Can you give me an example of why this is wrong?&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2018-12-07T09:49:03.020" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14873" PostTypeId="2" ParentId="14869" CreationDate="2018-12-07T16:21:45.290" Score="1" Body="&lt;p&gt;It sounds like the students may be treating sums with &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; in them as just as easy to manipulate as the finite algebraic versions - presumably not just in this particular example, right?  Which is very unsurprising, since a lot of times we &lt;em&gt;do&lt;/em&gt; treat them this way and because they &quot;look&quot; like things you can treat wholly algebraically.  (To really confuse them, spend a week on generating functions!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More seriously, here is a possible thing to try if this happens in class:  Can they write the (in general wrong) equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sum_{n=1}^{\infty}(a_n + b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in terms of the definition of the infinite sum/series?  This is a good in-class &quot;ask not a specific student, but everyone&quot; question - restate this step in terms of its definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you do that (and if nobody can, then it's time to review it slowly), things look messier and presumably less &quot;obvious&quot; to the student(s).  Now we are saying &quot;limit = limit + limit&quot;.  &quot;Okay, can anyone think of a case back in calculus where the limit of a sum doesn't equal the sum of the limits?&quot;  Hopefully someone comes up with a situation like &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}+\frac{-1}{x}=0$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you can say &quot;in this case, we have to be careful too&quot;, and hopefully this analogy from something they should recall from freshman calculus can get you to convince them of it.  And note that we are not asking them to remember things about series, which presumably they are still gnawing on, but function limits.  (Hopefully someone will think of connecting the calculus example to the series with &lt;span class=&quot;math-container&quot;&gt;$\sum \frac{1}{n}+\sum \frac{-1}{n}\neq \sum 0$&lt;/span&gt;, but that will depend on the class.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a bonus, presumably you have a nice theorem about when the equation is true (such as, if both sums converge or whatever) which, when applied, &lt;em&gt;would&lt;/em&gt; make their proof complete.  Then on an exam you can give one where they are allowed to use that theorem, and one where they are not (do it &quot;from the definitions&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admit this is just one possible idea, but for some students it might snap them out of the blind algebraic manipulation, which I believe lies at the heart of many challenges in teaching real analysis.  Analysis is really topology and the Archimedean axiom applied to the real line, but it still looks as algebraic as &lt;span class=&quot;math-container&quot;&gt;$(x^2)'=2x$&lt;/span&gt; in many texts because of the notation coming (historically) before the proofs.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2018-12-07T16:21:45.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14874" PostTypeId="2" ParentId="14855" CreationDate="2018-12-07T17:03:09.960" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I didn't like f(x) — that looked to me like f times x. I also didn't like &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; — you have a tendency to cancel the d's. So I made a different sign, something like an &amp;amp; sign. For logarithms it was a big L extended to the right, with the thing you take the log of inside, and so on.&lt;/p&gt;&#xA;&lt;p&gt;I thought my symbols were just as good, if not better, than the regular symbols — it doesn't make any difference what symbols you use — but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, &amp;quot;What the hell are those?&amp;quot; I realized then that if I'm going to talk to anybody else, I'll have to use the standard symbols, so I eventually gave up my own symbols.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;~ Richard Feynman&lt;/p&gt;&#xA;&lt;p&gt;So, explain to your students that this is an accepted notation for differentiation, nothing more. Explain, that it should be taken as a single operator, not a fraction; that d's cannot be canceled. Show them other notations and allow using them even if your course and the textbooks stick to Leibniz's notation.&lt;/p&gt;&#xA;&lt;p&gt;I think that Lagrange's notation is much cleaner for single-variable derivative than Leibniz's:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ (x^2)'=2x$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-12-07T17:03:09.960" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14875" PostTypeId="2" ParentId="10828" CreationDate="2018-12-07T18:46:39.847" Score="2" Body="&lt;p&gt;For your first question, I don't have a general answer for you, but there is this for New York State.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;On the 2015 Algebra I exam, which was supposed to align with the new Common Core curriculum, the percentage of students passing fell to 63 percent, down nine points from the old exam last year. &lt;a href=&quot;https://www.nytimes.com/2015/12/01/nyregion/algebra-scores-prompt-second-look-at-revamped-regents-exams.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;New York Times, Nov. 2015&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would imagine there is more recent data to be found on this. My first hunt at answers might be here: &lt;a href=&quot;http://www.p12.nysed.gov/irs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;NYS Information and Reporting Services&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2018-12-07T18:46:39.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14876" PostTypeId="1" CreationDate="2018-12-07T20:22:58.320" Score="8" ViewCount="762" Body="&lt;p&gt;Math textbooks for undergraduate and graduate students are almost always structured in the same way. Each chapter/section/etc. has it's definitions, theorems, propositions, etc. with proofs following the statements directly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we now take a look at, for example, a physics text book, we will probably find a completely different approach in the structure. Instead of building blocks (theorems, etc.) we will find a continuous text, where the problem is usually introduced in the beginning, some calculations and reasoning done in the middle part with the result at the end of the chapter/section/etc. Instead of physics we could have taken almost any other field, I at least don't recall seeing such a strict way of dividing a book in building block as it is done in mathematics in any other field, and found the same result. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course there are some exceptions, at least in physics one can find books that are almost math-like in structure (often called &lt;em&gt;Some-field-in-physics for mathematicians&lt;/em&gt;), but generally speaking this separation in theorems/etc., or more generally in such building blocks, seems to be something almost exclusive to mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;And now to the question:&lt;/strong&gt; Why do we have such a strict separation in math textbooks? What are advantages and what are disadvantages? Do you know some examples of textbooks outside of mathematics that have attempted a similar structure? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Honestly, I always preferred the approach that math textbooks had over that of other disciplines (mostly physics, not really familiar with the rest). There are obviously some problems when it comes to the math-approach, like f.e. steps that are not motivated enough, etc. but the fact that I am able to see the thing that we are going to prove on the next few pages in form of a concrete statement (theorem, etc.) helped me to not lose sight of what we are actually trying to do. On the other hand in physics you often start with something and have no clue where exactly you want to go/what you want to show (well, the lecturer/author/etc. knows but the student/reader has usually no idea)... So I was wondering is there a reason why such an approach is usually avoided in physics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Notes:&lt;/strong&gt; This question was originally asked on PhysicsSE (&lt;a href=&quot;https://physics.stackexchange.com/questions/445670/structure-of-physics-textbooks&quot;&gt;see here&lt;/a&gt;) but put on-hold for not being on topic. I therefore re-framed it a bit in the hope to make it on-topic for this site. I hope the structure of math textbooks is also part of math education... If not, please feel free to suggest a better site to ask this question.&lt;/p&gt;&#xA;" OwnerUserId="10658" LastActivityDate="2019-01-07T07:06:01.533" Title="Structure of math textbooks" Tags="|textbooks|physical-sciences|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14877" PostTypeId="2" ParentId="10828" CreationDate="2018-12-07T22:28:15.023" Score="-1" Body="&lt;p&gt;Here is a study on it:  &lt;a href=&quot;https://digitalcommons.fiu.edu/dissertations/AAI3249717/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://digitalcommons.fiu.edu/dissertations/AAI3249717/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Algebra I and other predictors of high school dropout&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A Google search on the two terms algebra and dropout reveals several other articles:  &lt;a href=&quot;https://www.google.com/search?q=algebra+dropout&amp;amp;ie=utf-8&amp;amp;oe=utf-8&amp;amp;client=firefox-b-1-ab&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=algebra+dropout&amp;amp;ie=utf-8&amp;amp;oe=utf-8&amp;amp;client=firefox-b-1-ab&lt;/a&gt; (I lack the status to post several links, but I clicked through to most of the articles on first page of the search and they seemed decent).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your question statement discusses several other ideas than just wanting that number, but I think some issue analysis is needed to grapple with that (causation versus correlation, purpose of a high school degree, passing local grade versus passing a national test evaluation, failure first year versus later years,  etc.).  So I don't think just getting that number will resolve all that you are wondering about.&lt;/p&gt;&#xA;" OwnerUserId="10659" LastActivityDate="2018-12-07T22:28:15.023" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14878" PostTypeId="2" ParentId="14876" CreationDate="2018-12-07T23:01:41.370" Score="2" Body="&lt;p&gt;(Comment)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not really sure that your predicate is as you've described.  I took down a copy of my Calc book (Thomas Finney 1980) and my college chemistry book (MSS fifth edition) and both are really rather similar in explication, having examples, derivations, chapters, subsections, etc.  Yes, there are some differences:  subsections in TF a little more standalone than in MSS; also TF has problems for each subsection, each lesson while MSS has end of chapter problems. But I still don't think it is radically different and TF definitely doesn't come off as a theorem-proof abstraction (despite doing a decent job at derivations and understanding of why methods work).  Looking at my bookshelf and about two feet of stuff and none of it is theorem-proof monstrosities.  OK, maybe one or two tiny little books I won in science fairs and never got any use out of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly the mass of textbooks from algebra through engineering math (post ODE), were not in the proof mode.  Yes, there is Euclid's Elements and Rudin, but I really think that this is a minority of &quot;math books&quot;.  As with many Q&amp;amp;As here, I think there is a blind spot, not realizing the (patently obvious) fact that math majors are a tiny minority of UG math teaching/courses. And zero in high school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So really before answering the question, which may be based on a false predicate, it would be worthwhile to do some development of the predicate itself.  Heck, even if it is correct, I think this exercise helps you to answer some of your question (by exposure to the materials and looking in detail into how they really are organized).&lt;/p&gt;&#xA;" OwnerUserId="10659" LastActivityDate="2018-12-07T23:01:41.370" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14879" PostTypeId="2" ParentId="14855" CreationDate="2018-12-08T01:29:21.067" Score="2" Body="&lt;p&gt;A good way to start is setting clear that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{d}{dx}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;as a block, is an &lt;strong&gt;operator&lt;/strong&gt;, and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Operators are things that act on their right side.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx} y$&lt;/span&gt; means &quot;derivative of what is at the right&quot;, i.e. derivative of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest you to highlight that sentence in yellow. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$y \ \frac{d}{dx}$&lt;/span&gt; just makes no sense because it is &quot;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; times derivative of... nothing&quot;, as there's nothing at the right side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also like starting with &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}$&lt;/span&gt; and not &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt;, so that it is clearer that it is not a fraction, but a whole-block operator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And finally, I'd suggest to read it as &quot;derivative &lt;strong&gt;of&lt;/strong&gt;&quot;, so that the students are tempted to complete the sentence in their minds. It's more serious than it seems haha. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helped.&lt;/p&gt;&#xA;" OwnerUserId="10660" LastActivityDate="2018-12-08T01:29:21.067" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14882" PostTypeId="1" CreationDate="2018-12-09T15:42:26.110" Score="3" ViewCount="327" Body="&lt;p&gt;The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector &lt;span class=&quot;math-container&quot;&gt;$(x,y,z)$&lt;/span&gt; is in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;. I am tempted to say that it has &quot;length &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;&quot;. But then some student interprets this as saying that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x^2+y^2+z^2} = 3$&lt;/span&gt;. Is there some word other than &quot;length&quot; I could introduce and use to consistently refer to the number of coordinates in a vector? (To point out that I am not nuts here, the commands to get this number in &lt;a href=&quot;https://reference.wolfram.com/language/ref/Length.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematica&lt;/a&gt; and &lt;a href=&quot;http://matlab.izmiran.ru/help/techdoc/ref/length.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MATLAB&lt;/a&gt; are &lt;code&gt;Length[]&lt;/code&gt; and &lt;code&gt;length()&lt;/code&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2018-12-09T16:38:30.957" LastActivityDate="2018-12-09T21:59:48.890" Title="Word for the dimension of the vector space in which a vector lives?" Tags="|linear-algebra|vocabulary|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14883" PostTypeId="2" ParentId="14882" CreationDate="2018-12-09T16:34:03.300" Score="5" Body="&lt;p&gt;I say &quot;This is a 3D vector&quot; or &quot;This is a 7D vector&quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-12-09T16:34:03.300" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14884" PostTypeId="2" ParentId="14882" CreationDate="2018-12-09T17:45:55.237" Score="5" Body="&lt;p&gt;If all your spaces are &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; then you can say n-dimensional (three-dimensional for &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;). However the space of matrices &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \left(\begin{array}{cc}a&amp;amp;b\\c&amp;amp;d\end{array}\right)$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$a,b,c,d\in\mathbb{C}$&lt;/span&gt;) is four-dimensional over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt;, eight-dimensional over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; and infinite-dimensional over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;. So be clear what the base field is. Alternatively you could just write &lt;span class=&quot;math-container&quot;&gt;$(a,b,c)\in\left(\mathbb{R}^3,+,\cdot\right)$&lt;/span&gt; or more briefly &lt;span class=&quot;math-container&quot;&gt;$(a,b,c)\in\mathbb{R}^3$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2018-12-09T17:45:55.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14885" PostTypeId="2" ParentId="14831" CreationDate="2018-12-09T17:54:58.890" Score="3" Body="&lt;p&gt;With regard to this question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Any ... books I can buy which can help me to explain this?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is a specific book recommendation, which is a source that I have personally found helpful:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;National Council of Teachers of Mathematics. (2010). Developing Essential Understanding of Rational Numbers for Teaching Mathematics in Grades 3-5. &lt;a href=&quot;https://eric.ed.gov/?id=ED512630&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;ERIC&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is the table of contents:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MeiiG.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MeiiG.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Wrz9j.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Wrz9j.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2018-12-09T17:54:58.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14886" PostTypeId="2" ParentId="14882" CreationDate="2018-12-09T21:59:48.890" Score="6" Body="&lt;p&gt;Along with &quot;dimension&quot;, you could also use &quot;component&quot;.&#xA;A vector in three dimensions has three components.&lt;/p&gt;&#xA;" OwnerUserId="10537" LastActivityDate="2018-12-09T21:59:48.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14888" PostTypeId="1" AcceptedAnswerId="14890" CreationDate="2018-12-10T03:43:59.873" Score="6" ViewCount="521" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How to introduce Group Theory to a general audience in 15 minutes?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I know that it will be quite tough to introduce Groups to a general audience in such a short time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what will be a good way to introduce Groups using Chalk-Duster approach?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the topics in Group Theory which can be taught in 15 mins&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any help. &lt;/p&gt;&#xA;" OwnerDisplayName="user10666" LastEditorDisplayName="user507" LastEditDate="2021-07-01T16:37:42.567" LastActivityDate="2021-07-01T16:37:42.567" Title="How to introduce group theory to a general audience in 15 minutes?" Tags="|undergraduate-education|teaching|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14890" PostTypeId="2" ParentId="14888" CreationDate="2018-12-10T04:52:09.977" Score="10" Body="&lt;p&gt;One way to introduce group theory to a general audience is to talk about how to flip a mattress so that it is in &quot;a different position each time, so as to pound down the lumps and fill in the sags on all the various surfaces.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Brian Hayes has a good discussion of this in his &lt;a href=&quot;http://bit-player.org/wp-content/extras/bph-publications/AmSci-2005-09-Hayes-mattress.pdf&quot; rel=&quot;noreferrer&quot;&gt;September-October 2005 Computing Science&lt;/a&gt; column in &lt;em&gt;American Scientist&lt;/em&gt;.  You'll probably need to make a drawing (on a blackboard) like the one below (which I took from his article).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rD6Vc.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rD6Vc.png&quot; alt=&quot;Klein 4-group&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2018-12-11T04:54:54.357" LastActivityDate="2018-12-11T04:54:54.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14892" PostTypeId="1" CreationDate="2018-12-10T23:09:09.150" Score="13" ViewCount="1737" Body="&lt;p&gt;I am wondering if it is a bad idea to use an old textbook, such as&#xA;&lt;a href=&quot;https://archive.org/details/cu31924015990108/page/n3&quot; rel=&quot;noreferrer&quot;&gt;Differential and integral calculus, with examples and applications&lt;/a&gt; by George A. Osborne. This book was published in 1906 and there are no known copy right restrictions, which means students may use a free e-version if they would like to save money. On the other hand, hard copies are still available for sale. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, this book is very well written and contains all the basic materials that need to be covered in a traditional calculus course. Furthermore, it also contains a large number of examples, which is very helpful to the students. On the other hand, I am wondering if there is any issue with using an old textbook like this. For example,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Are there any terminologies and notations that are considered outdated?&lt;/li&gt;&#xA;&lt;li&gt;Are there any new discoveries in the past 110 years or so that need to be included into the calculus course which were not found in an old book?&lt;/li&gt;&#xA;&lt;li&gt;What will my students and peers think about the idea of using an old textbook?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I personally do not know any teacher who uses such an old book as the textbook; but is it really a bad idea to do so?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2018-12-11T20:52:03.983" LastActivityDate="2018-12-11T20:52:03.983" Title="Is it a bad idea to use an old textbook such as Differential and integral calculus, with examples and applications for calculus course?" Tags="|undergraduate-education|calculus|textbooks|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14893" PostTypeId="2" ParentId="14892" CreationDate="2018-12-11T00:27:06.153" Score="3" Body="&lt;p&gt;I'm all for using old editions and/or free e-texts. But, this is a bit too outdated in my opinion. I could have missed it, but I did not spot a clear section on:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;related rates&lt;/li&gt;&#xA;&lt;li&gt;mean value theorem&lt;/li&gt;&#xA;&lt;li&gt;L'hopital's Rule&lt;/li&gt;&#xA;&lt;li&gt;surface integration&lt;/li&gt;&#xA;&lt;li&gt;Green's, or Divergence or Stokes' Theorems&lt;/li&gt;&#xA;&lt;li&gt;modern vector notation&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Yes, there is a preponderance of examples on basic calculational techniques, quite impressive. But, I do think the application of calculus to curve sketching and applications to say circuits, biology, finance are missing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More to the point, the organization is very nonstandard when framed against the usual USA-based sequence. In summary, calculus II and III are mixed together in a rather strange way. Also, missing as far as I remember:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;introduction to differential equations&lt;/li&gt;&#xA;&lt;li&gt;second order constant coefficient ODEs&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;We can agree or disagree about whether or not these belong, but some schools need these covered early to help engineering keep their students up to speed with engineering curriculum which needs this basic ODE stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the worst thing, the lack of nice diagrams and organizing boxes. All Calculus texts for about the last 5 decades have pretty nice pictures and a lot of organizational aids for studying. I think some of these are worth it. Of course, you could use this book as a backdrop for adding all that nice stuff if you want to work on it, but it seems like a lot of work when you could just as reasonably say use the 4th edition of Thomas or some such thing which is widely available for 10's of dollars.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-12-11T00:27:06.153" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14895" PostTypeId="2" ParentId="14888" CreationDate="2018-12-11T04:06:58.473" Score="3" Body="&lt;p&gt;Well, it depends a lot on what your real goal is.  If it's just the wow factor, you could introduce the &lt;a href=&quot;https://ruwix.com/the-rubiks-cube/mathematics-of-the-rubiks-cube-permutation-group/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rubik's Cube group&lt;/a&gt; or do &lt;a href=&quot;http://doc.sagemath.org/html/en/reference/groups/sage/groups/perm_gps/cubegroup.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;computations&lt;/a&gt; about it.  If they know what a matrix is, then doing matrix computations would be a very nice way (I mean &lt;span class=&quot;math-container&quot;&gt;$GL_n(\mathbb{R})$&lt;/span&gt; here, not additive matrices).  Or symmetries in art/frieze groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, my personal favorite (even though the groups are commutative) is simply introducing clock arithmetic or one's digit arithmetic (which you may know as &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}_{12}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}_{10}$&lt;/span&gt;, or as &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}/12\mathbb{Z}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}/10\mathbb{Z}$&lt;/span&gt; if you are a notational purist).  Just a little messing around with these can be quite powerful if you have people truly new to the concepts.  And the huge benefit is that everyone is very familiar with them, even if they didn't know it.  Naturally this is for a less mathematically inclined audience than some others might be.  But definitely can do everything in 15 minutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Also: Want multiplication but there are zero-divisors?  No problem, just ignore them and now you have another, &quot;multiplicative&quot;, group of units.  Want a non-commutative variant?  Finish up by talking about clock arithmetic as a rotation of the clock face - and ask how flips would work.)&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2018-12-11T04:06:58.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14896" PostTypeId="2" ParentId="14892" CreationDate="2018-12-11T04:16:59.427" Score="3" Body="&lt;p&gt;The text is fine (even good), but I would opt for Granville instead.  Granville: Very clear by using simple vocabulary (low grade level English).  Brief but not in a Rudin manner...more in a Schaum's manner.  Excellent exercises.  Most answers provided.  Granville was the standard text from ~1910-1960.  Also uses American English, which will serve you better unless you are in the Commonwealth. (minor point).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the Granville Sage example is clearly a labor of love and well done, I would avoid that and just use the actual Granville text.  Learning calculus is hard enough.  When you add in some programming, it makes it harder.  (Even easy stuff...it just does.)  Also, the drill exercises are much more numerous in Granville and more helpful in terms of drill (which beginners need) as opposed to &quot;cool project&quot; type problems that professors like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a link:  &lt;a href=&quot;https://archive.org/details/elementsdiffere02goog/page/n18&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/elementsdiffere02goog/page/n18&lt;/a&gt;  (may need a slightly earlier edition for copyright, but I don't know...1941 could have lapsed also.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has a very nice set of chapters on series.  Also the diffyQ survey is good.  There are a few chapters at the end that verge towards calc 3, but don't cover it all.  You can skip those.  Also a few special topics (e.g. Mercator projection) that may be of less interest in this day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think the absence of EE problems is a flaw since many calc students are still weak on behavior of inductors, capacitors, etc.  Would leave that for physics or perhaps ODE course applications.  Problems with motion or with fluids are more easy to visualize.&lt;/p&gt;&#xA;" OwnerUserId="10659" LastEditorUserId="10659" LastEditDate="2018-12-11T04:31:43.530" LastActivityDate="2018-12-11T04:31:43.530" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="14897" PostTypeId="1" CreationDate="2018-12-11T04:33:30.867" Score="4" ViewCount="4230" Body="&lt;p&gt;I need to appear for an interview for the post of Assistant Professor in Mathematics in an undergraduate college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My Backgorund&lt;/strong&gt; : I have studied topics like Algebra comprising of Group Theory,Ring Theory and Field Theory.&#xA;I have also studied Linear Algebra and Matrices,Complex Analysis,Functional Analysis.&#xA;The above topics are of interest to me and I study these topics in my PhD.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Background of College&lt;/strong&gt;:The college also teaches undergraduate students in the courses mentioned above and also teaches some applied mathematics as a part of Bachelors in Science Course.The syllabus of the college in the website at least says this .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I received a call letter from the college that I need to give a demonstration about  some topic of interest of mine for 15 minutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is I have never given such a demonstration before and I really dont know how to handle this.In fact I have only given talks about my PhD topics for my PhD and I am pretty sure that I cant say those things for a 15 minute demonstration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that there are numerous topics on which such a demonstration can be made but I am not sure what will fit into a 15 minutes demonstration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone please help me on how to select some topics and how to manage them in 15 minutes?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;NOTE&lt;/strong&gt;:There is no powerpoint presentation involved.It should be done in Blackboard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any help will be appreciated.&lt;/p&gt;&#xA;" OwnerDisplayName="user9721" LastEditorDisplayName="user9721" LastEditDate="2018-12-11T05:11:07.453" LastActivityDate="2018-12-11T19:00:58.537" Title="Topics in Mathematics for a 15 minute demonstration" Tags="|undergraduate-education|teaching|classroom-management|interactive-teaching|classroom-interaction|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14898" PostTypeId="1" CreationDate="2018-12-11T08:28:07.057" Score="10" ViewCount="428" Body="&lt;p&gt;In the early 1900s Felix Klein lay out his vision for secondary mathematics curriculum. He wanted schools to teach calculus, so that universities would not be burdened by it. And at the core of the curriculum was to be the notion of function, which was conceptually necessary for the foundations of calculus. (I know I make a very messy story short here, but I think the basic points are more or less correct.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this was over 100 years ago, and to this day we teach according to Klein's vision. My question has to do with whether now, given the current state of the art, we would chose to design the mathematics curriculum in the same way. Specifically:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Do you think functions should be the backbone of the curriculum.  (Is it too abstract for students? Do the students leave secondary school understanding why functions is central?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Do you think calculus is the appropriate &quot;end&quot; of the high school curriculum. If not, what would you suggest in its place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you in advance for your responses!  &lt;/p&gt;&#xA;" OwnerUserId="5658" LastActivityDate="2018-12-28T18:42:04.110" Title="The royal road to calculus" Tags="|secondary-education|calculus|curriculum|precalculus|history|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14901" PostTypeId="2" ParentId="14892" CreationDate="2018-12-11T09:30:00.543" Score="2" Body="&lt;p&gt;While the style of writing is quite readable, many theorems are given without a proof.  That is a severe deficiency.  I find this text by E. Goursat (1902) much superior:&#xA;&lt;a href=&quot;https://archive.org/details/mathematicalanal021323mbp/page/n1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/mathematicalanal021323mbp/page/n1&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10673" LastActivityDate="2018-12-11T09:30:00.543" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14903" PostTypeId="1" AcceptedAnswerId="14912" CreationDate="2018-12-11T13:14:56.563" Score="19" ViewCount="4093" Body="&lt;p&gt;It seems that the following functions are not only excluded from a course in trigonometry, they are almost never taught in any course:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Chord_(geometry)&quot; rel=&quot;noreferrer&quot;&gt;Chord&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Versine&quot; rel=&quot;noreferrer&quot;&gt;Versine&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Versine#cvs&quot; rel=&quot;noreferrer&quot;&gt;Coversine&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Versine#Haversine&quot; rel=&quot;noreferrer&quot;&gt;Haversine&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Exsecant&quot; rel=&quot;noreferrer&quot;&gt;Exsecant&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Exsecant#excsc&quot; rel=&quot;noreferrer&quot;&gt;Excosecant&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I could have asked the same question with the title &quot;why have these functions lost their popularity&quot; at math.stackexchange, but I fear that they will consider this question as &quot;opinion-based&quot;.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2018-12-11T18:47:50.297" LastActivityDate="2018-12-13T00:30:52.627" Title="Why we don't normally teach chord, versine, coversine, haversine, exsecant, excosecant any more?" Tags="|trigonometry|" AnswerCount="4" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="14904" PostTypeId="2" ParentId="14897" CreationDate="2018-12-11T13:24:55.380" Score="1" Body="&lt;p&gt;Maybe just pick a topic/area you're interested in and search &lt;a href=&quot;https://math.stackexchange.com/&quot;&gt;Mathematics Stack Exchange&lt;/a&gt; for questions/answers on this topic/area until you find something that works for you. What I often did in the 1990s and early 2000s for job talks that I gave (roughly 5 to 8 during each of five different search years) was to go to a university library and browse through books (and sometimes look through the tables of contents in bound volumes of MAA journals and Math. Gazette) until I found something that seemed appropriate. Of course, I also sometimes just used things based on my own interests (such as large numbers &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=380011&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;, &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=380002&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;, &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=379996&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;) or something based on (or inspired by) a supplementary handout I'd written for students (done this several times, one example being &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=303245&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; with 9-year later comments &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=7191964&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, the following comments extracted from &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=303245&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; might be useful in how to think about what's appropriate, although it's important to note that I had 60 minutes for this particular talk that I gave in March 2001.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I decided to change a job interview (a tenure-track math position) presentation the evening before I was to leave town because I decided that what I had originally planned to talk about might not be sufficiently focused to pull off very easily. [Square/cube law in biology and engineering, applications of scaling principles in mathematics (the Pythagorean theorem can be proved in this manner, for instance), how the Cantor set and other &quot;fractals&quot; scale, and using big 'Oh' and little 'Oh' to describe growth rates (those already discussed, along with linear approximation errors, trapezoid and Simpson rule errors, etc.).] The audience would consist of upper level mathematics students and &#xA;  I could assume a background through sophomore level multivariable calculus. The topic needed to be something relating to undergraduate level real analysis. Although I could assume a bit more mathematical maturity than the typical student who had just finished multivariable calculus (the students were, after all, upper level math majors), I shouldn't assume that they would have already taken a real analysis or advanced calculus course. I decided to discuss the rate at which the harmonic series diverges. By grouping the terms in groups of &lt;span class=&quot;math-container&quot;&gt;$2,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$4,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$8,\; \ldots,$&lt;/span&gt; in two different ways, you can obtain a divergence rate that is within a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; of log-base-&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; of the number of terms added. By looking at appropriate upper and lower Riemann sums for &lt;span class=&quot;math-container&quot;&gt;$y = 1/x,$&lt;/span&gt; you can show that the divergence is asymptotic to log-base-&lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; of the number of terms added. Indeed, you're able to show that the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; terms lies between &lt;span class=&quot;math-container&quot;&gt;$\ln(n) + 1/n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\ln(n) + 1,$&lt;/span&gt; a much stronger result. By considering appropriate triangular regions that get ignored during the lower Riemann sum calculation, you can improve this by showing that the sum lies between &lt;span class=&quot;math-container&quot;&gt;$\ln(n) + 1/2 + 1/(2n)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\ln(n) + 1.$&lt;/span&gt; At this point I would show (a bounded monotone sequence argument) that the difference between the sum and &lt;span class=&quot;math-container&quot;&gt;$\ln n$&lt;/span&gt; actually approaches a limit---Euler's constant &lt;span class=&quot;math-container&quot;&gt;$\gamma,$&lt;/span&gt; and I would then show some of the many ways Euler's constant &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; comes up in math.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Incidentally, this was before Julian Havil's well known 2003 book &lt;strong&gt;Gamma: Exploring Euler's Constant&lt;/strong&gt; was published, so the stuff I talked about was not as well-trodden for a talk such as this as it probably would be now.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2018-12-11T13:24:55.380" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14905" PostTypeId="2" ParentId="14903" CreationDate="2018-12-11T13:29:22.713" Score="22" Body="&lt;p&gt;More of a comment than an answer but:  They are all composites of more basic functions.  In fact, all of the trig functions could can expressed in terms of sine, linear changes in coordinates, and rational functions.  For instance:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\tan(\theta) = \frac{\sin(\theta)}{\sin( \frac{\pi}{2}-\theta)}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We certainly don't want children to have to memorize 20 different trig functions.  That seems a bit silly.  Do we even really &lt;em&gt;need&lt;/em&gt; to teach all six ``standard'' trig function?  Personally I tend to avoid &lt;span class=&quot;math-container&quot;&gt;$\csc$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cot$&lt;/span&gt; in my work...&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2018-12-11T13:29:22.713" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14906" PostTypeId="2" ParentId="14724" CreationDate="2018-12-11T17:58:19.400" Score="1" Body="&lt;p&gt;My suggestion is to follow your comfort level. Start in wherever you feel comfortable - if it's a struggle, go back to earlier material. If it's smooth sailing, see that as an opportunity to strengthen your foundations. It's surprising, sometimes what new connections or understanding you gain from revisiting topics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One huge factor in your favour is the desire to learn and a curiosity about how things work. IMO, that's half the battle with learning - having it matter and be meaningful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may also help if there's someone you can go to with questions for when you get stuck. Sometimes a different explanation can do wonders at removing roadblocks.&lt;/p&gt;&#xA;" OwnerUserId="10647" LastActivityDate="2018-12-11T17:58:19.400" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14907" PostTypeId="2" ParentId="14888" CreationDate="2018-12-11T18:16:14.070" Score="1" Body="&lt;p&gt;I like the mattress example.  I was going to say 3d symmetry point groups (since it is visual and useful plus huge impact in chemistry and physics).  But...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Reyes answer does what I want and makes it even simpler. I would really just cover that example, maybe even make some 3x5 cards with a lump and let audience play with them.  Covering that one example and the table of results is fine.  And the insight that two motions can equal a third one is COOL INSIGHT.  That will get them a good feeling.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On top of that just give a generic discussion of group theory to answer basic questions:  what course is it taught in (what year) for math majors, when chemists/physicists cover it.  and that it has uses in chem/physics (spectroscopy, molecule ID, etc.) as well as &quot;proving you can't solve 5th order polynomials the way you do with quadratic). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;DO NOT show anything like wierd Z letter thing from kccrisman answer.  NO, NO, NO. NO matrices, group names.  NONE OF THAT. You don't have time to cover wierd notation in 15 minutes.  Just do the one mattress example and then give them orienting information about the topic AS A TOPIC (not proving anything).&lt;/p&gt;&#xA;" OwnerUserId="10659" LastActivityDate="2018-12-11T18:16:14.070" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14908" PostTypeId="2" ParentId="14897" CreationDate="2018-12-11T18:40:26.060" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think you should tell us what the audience is.  Just the selection committee or more general?  Is the intent to be more of a research talk or a teaching demonstration or general interest talk?  How prestigious is the school?  HPYS?  UCal?  CalState?  Juco?  Is this a teaching position or a tenure track research position? (I doubt it since you are not doing an hour talk, but let us know.)  I'm not being nosy but my point is that these are &quot;variables&quot; that affect the &quot;solution&quot;.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Length:  you should be able to do a talk in 5 minutes, covering the world or all day covering a small part of your Ph.D.  Note:  this, of course, is easier said than done.  But just do the sweat and figure it out. It is a life skill.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;2.5.  And after you write your talk, then practice it 3 times.  And then cut it more.  It ALWAYS TAKES LONGER in real life.  And practice it several times!  Skill in public speaking for neophytes is highly correlated (rsq &gt; 0.9) to practice amount.  Yes, it is &quot;cheating&quot; in that this is not your extemperaneous skill. Just double down and do the cheat.  It will be worth it regardless if you get the job and of all variables (to include topic selection and writing, I would say it is the single most important variable...highest partial derivative.)&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;My sneaky advice to getting hired is find something mildly topical (elections, fracking, Facebook social graph, global warming, TV viewing, etc.) and make some talk that plays off of it but shows something of interest of yours.  Show your basic topic and how it affects something.  People love that stuff.  You don't have to go overboard, but get a hook. Even mathematicians earn money, make love, read the newspaper, fight with their kids, etc.  Human interest sells.  Used to see great articles in The American Scientist. Hint, hint.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(Feel free to convert to comment.  -cookiemouse)&lt;/p&gt;&#xA;" OwnerUserId="10659" LastEditorUserId="10659" LastEditDate="2018-12-11T19:00:58.537" LastActivityDate="2018-12-11T19:00:58.537" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14909" PostTypeId="2" ParentId="14903" CreationDate="2018-12-11T20:41:50.357" Score="4" Body="&lt;p&gt;They are almost never needed for applications problems in physics or engineering.  Really sine, cosine and tangent are mostly what you need.  Not even the reciprocal functions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm talking with respect to how formulas and problems are normally written in those courses.  Ask yourself, did you encounter those functions in any of your science course and see a need for math to cover them as a service?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the versine and such are useful in navigation.  But even for celestial nav as of post ww2, it was mostly done with tables and worksheets that don't require you to use these functions (or really know any of the math in what you do).  I believe there was a time when there was more need for them before celestial nav became so work sheet oriented.  And now celestial nav itself is a dying art because GPS is so common.  Ask the average QM to use a sextant and see how he does...&lt;/p&gt;&#xA;" OwnerUserId="10659" LastEditorUserId="10659" LastEditDate="2018-12-13T00:30:52.627" LastActivityDate="2018-12-13T00:30:52.627" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14910" PostTypeId="2" ParentId="14903" CreationDate="2018-12-11T20:47:59.133" Score="6" Body="&lt;p&gt;Ask yourself if you would miss anything useful if you didn't know the functions you mentioned. I doubt it! I'm teaching high school math happily without having heard of them up to this point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To the contrary, there are several downsides to teaching them:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/14903/why-we-dont-normally-teach-chord-versine-coversine-haversine-exsecant-exco#comment38091_14905&quot;&gt;Quoting Steven Gubkin's comment&lt;/a&gt;: &quot;Note that teaching students haversine, versine, etc will actually make it &lt;strong&gt;harder&lt;/strong&gt; for them to communicate with others, because most other people have not learned these things.&quot;&lt;/li&gt;&#xA;&lt;li&gt;Cognitive overload. &lt;a href=&quot;https://xkcd.com/809/&quot; rel=&quot;noreferrer&quot;&gt;Obligatory xkcd&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Personal opinion: Math should be more about (creative) problem solving than about applying memorized formulas. By not teaching above functions, students can solve more problems by creative usage of &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt; and Pythagorean Theorem.&lt;/li&gt;&#xA;&lt;li&gt;Calculators. If I express a solution using one of the fancy functions, I'm stuck with a calculator and can't come up with an approximate solution. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2018-12-12T05:39:18.750" LastActivityDate="2018-12-12T05:39:18.750" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14911" PostTypeId="2" ParentId="14853" CreationDate="2018-12-12T01:32:32.620" Score="7" Body="&lt;p&gt;Can I answer as a learner? Three observations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(i) When I was at school, I was impatient to get on with learning the next bit. So I rushed through a maths exercise partly because of enthusiasm. Presumably the enthusiasm is something to be encouraged, but also steered in the right direction. (Can it be harnessed into enthusiasm for finding mistakes?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(ii) I honestly thought that by doing as many steps as I could at once in my head, I was being good at maths. What I was actually doing was making sign errors and the like inevitable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(iii) Finding misplaced minus signs and suchlike was &lt;strong&gt;hard.&lt;/strong&gt; Most likely made worse by (ii). That was part of my reluctance to do it. I &lt;em&gt;didn't actually believe&lt;/em&gt; that I could avoid the mistakes, however careful I was. So why try?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, based on this historical sample of one person, what might work? What I needed to be taught was:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Practice thinking clearly, not thinking fast.   &lt;/li&gt;&#xA;&lt;li&gt;Don't do lots of steps at once. Do small enough steps for each one to be easy, and for there to be no doubt whether it's right or not. You'll also have more idea what you did.  &lt;/li&gt;&#xA;&lt;li&gt;If you do make the steps simple ones, it's much easier to check them afterwards and since you &lt;em&gt;can&lt;/em&gt; find the errors, it's worth trying to after all.  &lt;/li&gt;&#xA;&lt;li&gt;If you go slowly enough to avoid the errors, maths is much more fun because you spend more time finding things out and less time wondering why it's not working.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10663" LastEditorUserId="10663" LastEditDate="2018-12-12T12:01:08.707" LastActivityDate="2018-12-12T12:01:08.707" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14912" PostTypeId="2" ParentId="14903" CreationDate="2018-12-12T06:34:55.627" Score="13" Body="&lt;p&gt;As several respondents have indicated, one could choose many different trigonometric functions to serve as the basic elements in terms of which other trigonometric functions are expressed and in (ancient) history other choices were made than those standard now. What we know now that was not known to the ancients that leads us to use cosine and sine is the central place in mathematics of linear ordinary differential equations and exponential functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The functions &lt;span class=&quot;math-container&quot;&gt;$\cos{t}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin{t}$&lt;/span&gt; are a basis for the space of solutions of &lt;span class=&quot;math-container&quot;&gt;$\ddot{x} + x = 0$&lt;/span&gt;, which is probably the most important differential equation in mathematics, because it is probably the most fundamental differential equation in physics. An alternative basis is &lt;span class=&quot;math-container&quot;&gt;$e^{i t}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e^{-i t}$&lt;/span&gt;, so, said another way, the choice of &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; is made because these are the real and imaginary parts of the complex exponential &lt;span class=&quot;math-container&quot;&gt;$e^{i t} = \cos{t} + i \sin {t}$&lt;/span&gt;. The exponential is clearly fundamental, and it is convenient to express as much as one can in terms of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The alternative trigonometric functions do not have such properties and this means that to make extensive use of them it turns out to be more convenient to express them in terms of cosine and sine, or, equivalently, exponentials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, this point of view makes apparent the parallelism with real exponentials, or rather the hyperbolic cosine and sine, corresponding to the differential equation &lt;span class=&quot;math-container&quot;&gt;$\ddot{x} - x = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; admit simply geometric interpretations and sets of functions such as &lt;span class=&quot;math-container&quot;&gt;$\{\cos{nt}, \sin{nt}: n \in \mathbb{Z}, n &amp;gt; 0\}$&lt;/span&gt; exhibit useful orthogonality properties (with respect to integration) that make them well adapted for use in contexts like Fourier series. There are of course many systems of orthogonal functions. What the most useful ones have in common is an origin as the solution of a homogeneous second order linear ODEs with some parameter (in this case &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;) in its coefficients. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What one teaches to children is in part dictated by what is needed later by those who will study more, and in part dictated by what has proved most tractable in diverse contexts. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-12-12T06:34:55.627" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14913" PostTypeId="1" AcceptedAnswerId="14914" CreationDate="2018-12-12T23:18:43.563" Score="-1" ViewCount="217" Body="&lt;p&gt;It is widely &lt;a href=&quot;https://web.archive.org/web/20130121100313/http://io9.com/5977095/why-we-should-switch-to-a-base+12-counting-system&quot; rel=&quot;nofollow noreferrer&quot;&gt;recognised&lt;/a&gt; that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Duodecimal&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duodecimal&lt;/a&gt; number system is superior to the decimal system. However, it is plainly obvious that trying to introduce such a system would be difficult, especially in a classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, could some advantages of such a system be gained by using 12 (and its multiples and divisors) more in the classroom, while still using the decimal system for writing down numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples I could think of are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I High school: Graph paper with major lines at every 12th minor line, rather than the usual 10th line&lt;/li&gt;&#xA;&lt;li&gt;In Primary school: &quot;Coins&quot; of denominations 1,2,3,4,6,12 &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What are the pros and cons of such an initiative?&lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="11" LastEditDate="2019-10-05T00:47:10.487" LastActivityDate="2019-10-05T00:47:10.487" Title="Duodecimal by Stealth" Tags="|notation|history|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14914" PostTypeId="2" ParentId="14913" CreationDate="2018-12-13T05:23:31.897" Score="5" Body="&lt;p&gt;A. Cons:  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Distraction from normal topics (which many kids need to work on, are not meeting state standards).  Just in that it is &quot;extra material&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Potentially confusing for kids struggling to master intricacies of the normal decimal system. (after the decimal point, scientific notation, repeating decimals, etc.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;B. Pros:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Tees up the idea of alternate bases (this has been a GT &quot;new math&quot; topic for decades).  However, I think it is better to use base 2 or 16.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I guess some forms of mental arithmetic might be easier (but this only if you do serious mastering of the new base so that it is automatic...which means a lot of time (Con A.1) as well as serious confusion unless they really master &quot;two math languages&quot; (and many kids struggle to master one), thus serious issues with Con A.2.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;C. Analysis:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think in effect it would just be a short special topic.  So you would have a mild amount of Con A.1 but not Con A.2.  Thus you don't get Pro B.2 either.  You do cover Pro B.1 but at the expense of a more difficult to learn (versus 2) and less useful base than 2 or 16 (either).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(As with many questions on this site) the intro statement itself is in debate (that 12 is better).  I have gotten by fine without 12.  And I don't feel like debating/researching the topic (would not be appropriate for this board anyway since it is a math topic, not math ed).  So bottom line, I don't &quot;know&quot; that the statement of it being better is correct (as I know 1 plus 1 equals 2 or even like smoking makes me at more risk for cancer).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The idea of trying to stealth change the world like this is sort of crankish.  Worry about teaching the kids and getting them to learn fractions and stuff that they have a hard time on.  Not stealth changing the world (since you never will anyway.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10659" LastEditorUserId="10659" LastEditDate="2018-12-13T21:16:10.433" LastActivityDate="2018-12-13T21:16:10.433" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14915" PostTypeId="1" CreationDate="2018-12-14T11:04:29.797" Score="-4" ViewCount="200" Body="&lt;p&gt;Who are in the current top 10-20 applied mathematics educators and in what fields? (Answers will be varied)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applied mathematics educators that teaches applied math topics such as applied statistics, differential equations, etc and are also have good influence in industry or organization or research.  Terrence Tao is not one example, because he is pure mathematician. Gilbert Strang is also not an example in this context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="10637" LastEditorUserId="10637" LastEditDate="2018-12-17T04:09:00.600" LastActivityDate="2018-12-17T04:09:00.600" Title="Who are in the current top 10-20 applied mathematics educators and in what fields?" Tags="|applied-mathematics|teachers|" AnswerCount="1" CommentCount="7" ClosedDate="2018-12-14T18:33:23.600" ContentLicense="CC BY-SA 4.0" />
  <row Id="14916" PostTypeId="2" ParentId="14915" CreationDate="2018-12-14T12:28:40.077" Score="2" Body="&lt;p&gt;Maybe look at recipients of major prizes in applied mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Norbert Wiener Prize, William Benter Prize, John von Neumann Lecture, ...&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2018-12-14T12:28:40.077" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14919" PostTypeId="1" CreationDate="2018-12-15T14:39:12.883" Score="8" ViewCount="1479" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/ISO_80000-2&quot; rel=&quot;noreferrer&quot;&gt;ISO 80000&lt;/a&gt;-2:2009 is a standard describing mathematical signs and symbols developed by the &lt;a href=&quot;https://en.wikipedia.org/wiki/International_Organization_for_Standardization&quot; rel=&quot;noreferrer&quot;&gt;International Organization for Standardization&lt;/a&gt; (ISO). In teaching mathematics, should one always follow this standard? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, there is no universal agreement on &lt;a href=&quot;https://math.stackexchange.com/questions/283/is-0-a-natural-number&quot;&gt;whether zero should be considered as a natural number or not&lt;/a&gt;. But since there is an international standard which considers zero as a natural number (though the teacher may not like this convention), should one teach base on this standard only in order to avoid any confusion?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is, people use &lt;span class=&quot;math-container&quot;&gt;$\log x$&lt;/span&gt; for both natural and common logarithm. In my teaching I usually spend 10 minutes explaining the difference and the fact that in most calculators, &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; means common logarithm and in &lt;a href=&quot;https://www.wolframalpha.com/input/?i=log(x)&quot; rel=&quot;noreferrer&quot;&gt;WolframAlpha&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; means natural logarithm. My personal rule of thumb is, &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; means natural logarithm starting at pre-calculus and means common logarithm before pre-calculus. Based on ISO 80000-2, one should use &lt;span class=&quot;math-container&quot;&gt;$\lg$&lt;/span&gt; for common logarithm and &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; for natural logarithm. Under this convention, ambiguity ceased to exist. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: The Chinese goverment published a &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=3&amp;amp;ved=2ahUKEwj96emrt6PfAhXrlOAKHYLrBHMQFjACegQIBBAC&amp;amp;url=http%3A%2F%2Flxbwk.njournal.sdu.edu.cn%2FCN%2Fitem%2FdownloadFile.jsp%3Ffiledisplay%3D20170223092530.pdf&amp;amp;usg=AOvVaw2UBIegpOqbr391oAYjugsx&quot; rel=&quot;noreferrer&quot;&gt;standard&lt;/a&gt; in 1993, requiring all institutions in China teach that &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is a natural number. So at least in China, the dispute is forcefully solved.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2018-12-16T03:48:05.943" LastActivityDate="2021-09-21T16:09:35.847" Title="In teaching mathematics, should one always follow some international standards such as ISO 80000-2?" Tags="|notation|terminology|logarithm|" AnswerCount="5" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="14920" PostTypeId="2" ParentId="14919" CreationDate="2018-12-15T18:53:32.970" Score="14" Body="&lt;h1&gt;No.&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;This standard may be useful for professionals in international settings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most teaching happens in smaller, localized settings and things will differ from country to country (e.g. how large numbers are packed in bunches of three or how decimal places are separated from the integral part).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Focusing on a single standard will make it harder for students to adapt to different nomenclature that they will encounter at some time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, in the ISO standard only &lt;span class=&quot;math-container&quot;&gt;$\varnothing$&lt;/span&gt; is listed as &quot;empty set&quot;. Not everyone agrees on this and I guess that &lt;span class=&quot;math-container&quot;&gt;$\{\}$&lt;/span&gt; is just as common. Your students will have a hard time to deal with this different notation if they are taught that there is the one way™ to write things down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consistency is important, but don't overdo it.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2018-12-15T18:53:32.970" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14921" PostTypeId="2" ParentId="14919" CreationDate="2018-12-16T08:44:42.110" Score="13" Body="&lt;p&gt;The standard that you link to (ISO 80000-2:2009) seems to be not available for free.  That is, in order for me to follow the standard, I have to be able to read it, and in order for me to read it, I have to pay for it.  (It currently costs 158 Swiss francs or approximately 158 US dollars.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To require mathematics educators to follow the standard implies requiring each to pay a relatively large amount of money.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that if you want to require mathematics educators to follow the standard then you should make the standard freely available.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2018-12-16T08:44:42.110" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14922" PostTypeId="2" ParentId="14898" CreationDate="2018-12-16T18:16:10.480" Score="4" Body="&lt;p&gt;this is an excellent question.  I would prefer to offer a different interpretation however.  My dissertation was in mathematics education, and I was interested in the history of the function concept in schooling.  Klein was not the only one who espoused an interest in utilizing the function concept as the backbone of the school curriculum, and it has been claimed as such for more than a century now.  If you look at the current New York State high school mathematics curriculum, it is called &lt;em&gt;A Story of Functions&lt;/em&gt;, CCSS identify functions as backbone, etc. etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is important however, is to pay closer attention to what calculus means for Klein in comparison to the nature of the discipline as it has subsequently developed in school and curricula.  Klein was German, and valued a Leibnizian view of calculus.  There is a difference here, and approximation methods play a more pronounced role in this vision.  If you notice, Klein later demands a more prominent role for &quot;The Finite Calculus&quot;, this includes things like finite differences and interpolation; ideas which never made their way into the standard curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think Klein's demands for a more humanistic and approximate style to the calculus resonates louder today with the rise of computing and statistics in culture, society, and mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While you can find something like a recursive sequence as a function in the CCSS, curricula and teaching still have a long way to go to valuing the finite and approximate.  Too often we see these as specialized topics for later study rather than ways to fill out existing concepts like linear functions that are such mainstays in the curriculum.  Some examples do exist however, maybe you'd be interested in the COMAP &lt;em&gt;Mathematical Modeling Our World&lt;/em&gt; materials, or some older textbooks made for North Carolina high schools before the CCSS when they were teaching vectors, matrices, and discrete mathematics topics in classes like Algebra I.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, functions should be the backbone -- but a function is not a function is not a function.  Functions have multiple interpretations through history, and we should aim for a less rigid notion where we study relationships in a variety of contexts as opposed to questions around function theory like &quot;is this a function&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, calculus should be the end game -- but calculus can be much more than what is presented in the AP Curriculum or MIT's first course.  People arguing for statistics to replace calculus ignore the fact that statistics is just an application of calculus, particularly when you value topics like finite calculus and the probability calculus in an introductory calc setting.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2018-12-16T18:16:10.480" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14923" PostTypeId="1" CreationDate="2018-12-16T20:02:35.997" Score="14" ViewCount="3490" Body="&lt;p&gt;I am guessing solving general cubic and quartic equations should be taught in a course somewhere between precalculus and Galois theory, though personally I do not recall learning this topic ever in any course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When should we teach students (mainly ones majoring in mathematics) how to solve general cubic and quartic equations? Or solving general cubic and quartic equations is so unimportant that it does not even deserve to be taught to even undergraduates majoring in mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I understand that it is impractical to solve such equations by hand. On the other hand, are those algorithms at least of some theoretical importance that an undergraduate student majoring in mathematics should know the algorithm before his graduation?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2018-12-18T12:42:36.577" LastActivityDate="2018-12-18T12:42:36.577" Title="In which course should we teach solving general cubic and quartic equations?" Tags="|undergraduate-education|solving-polynomials|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14924" PostTypeId="2" ParentId="14923" CreationDate="2018-12-16T21:21:40.813" Score="22" Body="&lt;p&gt;The place I’ve seen this is usually in a History of Math class. This makes the most sense to me since&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;solving polynomial equations (and the methods of reducing one type to another) plays an important role throughout the ages, but especially in 16th century mathematics.&lt;/li&gt;&#xA;&lt;li&gt;students of the sciences certainly don’t need this info for practical reasons, so there’s little motivation to include it in Precalculus or the Calculus sequence.&lt;/li&gt;&#xA;&lt;li&gt;that there is a general solution is enough to know for most upper level math classes, the actual methods and solutions being less important.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="401" LastActivityDate="2018-12-16T21:21:40.813" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14925" PostTypeId="2" ParentId="14898" CreationDate="2018-12-16T23:03:32.587" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I suspect there is an implicit thinking about what the curriculum for a nascent math Ph.D. should be.  But you need to realize that less than 1% of high school students will do this.  And you don't know who they are ahead of time.  So thinking in this mode is really not the right way to optimize the curriculum.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I got a functions module in pre-calc (domain/range emphasis).  Think it was fine to have it at that time.  Did not hurt me from learning the quadratic formula or trig or logs to not have it earlier.  Be careful about the impulse to want to move everything at the beginning.  You can't have everything at the beginning.  Also be careful about learning categorical frames of reference before important content (logs and antilogs, sin and arcsin, etc.).  Just because it makes sense in retrospect does not mean it makes sense when you are fresh to topics.  Also learning too much abstract theory before getting some math muscles can be distracting.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For a very bright or accellerated student, there is no reason why they can't go beyond calc 1/2 within high school.  The logical next step is calc 3 and ODEs because it is most common next step in a general STEM career.  But conceivably if school is big enough or student has an interest they could look at other topics (e.g. linear algebra).  I understand stats is in vogue nowadays in high schools also.  In addition, if a kid has tapped out the math classes at his school, he could look at a comp sci course (always useful).  But the ability to go on to next classes after calc for individual students is crushingly obvious.  But they are outliers and can go to a juco or self study or the like.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This question about &quot;should calc be the end&quot; shows the same mental gap as in (1).  Calc isn't even the end of MOST high school students.  I will admit that it is the end step for serious college prep students.  But don't act like the other bodies in the school are irrelevant.  They are citizens also.  Get out of the Yale or jail mindset.  Lots of great people never go to college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even for the college prep kids, I would say it is only the upper half (maybe even upper third) that really ace calculus and place out of it in college (say a 4 or 5 on the AP).  A vast amount of kids who are not in that upper tranche are going to repeat calculus anyways in college. Even for the kids that pass the AP, most do it as seniors (and were accelerated a year to even do that).  So worrying about accelerating further (or having extra classes for the outliers that already accelerated so much) is penny wise, pound foolish.  &lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;I'm sure the real analysis lovers will be here to say that calc course should be made more theoretical, etc.  But they have a tendency to miss points above.  They are super great at finding the hidden counterexamples and tracking epsilons.  But not so good at comparative analysis (points 1-3).  And education is a human activity with costs, benefits, limits, probabilistic outcomes, etc.  Any strategy for how education should be designed needs to think about it this way.  It's not about the best individual outcome possible, but about maximizing overall benefit for given limits in cost, IQ, time, etc.  [Also the vast majority of kids taking calculus for physics, engineering, chemistry and life sciences, not to mention nursing, business, etc. will never need to work their way through Rudin.]&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10659" LastActivityDate="2018-12-16T23:03:32.587" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14926" PostTypeId="2" ParentId="14923" CreationDate="2018-12-16T23:21:48.863" Score="5" Body="&lt;p&gt;I'm used to seeing them &quot;starred&quot; in an algebra 2 or &quot;college algebra&quot; course.  Starred means the extra topics that very advanced classes could cover.  My experience is even for those classes nobody bothers.  I have taken a huge amount of engineering and science courses and never really felt the loss of these techniques.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only time I solved cubics was in some of the harder AP chemistry stoichiometry problems where a cubic resulted in equilibrium problems.  and then I just got x by approximation (guessing and revising guesses after feeding the number in, mechanically without a program or algorithm).  And this worked fine for those chem problems (for example to get pH to single character after the decimal).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you do see a hyooooge amount of the time is the quadratic.  bsq-4ac needs to become iconic.  It is in a gazzillion chem rate and equilibrium problems and it is all over physics and engineering.  Either directly or in solving the characteristic equation of the 2nd order constant coefficient homo diffy Q (EE, controls/feedback, harmonic oscillator, quantum mechanics, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I can tell you that I have &quot;needed&quot; the quadratic a metric gazillion amount of times.  And the cubic almost never (and worked around by approximating).  And really never solved a quartic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I know you were asking about kids in math, but I just wanted to discuss why we don't bother with it in algebra 2 (where it sort of belongs by history and by topic).  Because of course, the needs of math undergrads are irrelevent to a general algebra 2 class because these kids are a tiny fraction of kids in a high school class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now.  The question becomes if you need to cover that stuff in other math classes.  Personally I doubt it.  Do you really encounter the need to analytically solve cubics or quartics in upper div math classes?  No.  And do you need it if you get a job that typical math grads do?  (Actuary?)  Probably not.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And in the tiny tiny fraction of the time that you do need it, you can just look it up in the &quot;starred section&quot; of your old book.  Provided that you have strong manipulational muscles, you can just apply the formula for your specific problem then, without any qualms.  But you really don't need to learn/remember it like you do the quadratic (not just knowing it, but using it readily).&lt;/p&gt;&#xA;" OwnerUserId="10659" LastActivityDate="2018-12-16T23:21:48.863" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14927" PostTypeId="2" ParentId="14923" CreationDate="2018-12-16T23:55:11.287" Score="9" Body="&lt;p&gt;I think it is very reasonable in grad abstract algebra to show that these formulas are &lt;em&gt;corollaries&lt;/em&gt; of very natural manipulations of &lt;em&gt;Lagrange&lt;/em&gt; &lt;em&gt;resolvents&lt;/em&gt;. These natural manipulations also illustrate some other points, about &quot;averaging&quot; and Vandermonde determinants, and so on. All these pre-date Galois by decades, and were the context in which Galois, Abel, and Ruffini worked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is slightly strange that Lagrange resolvents are not visible in most &quot;modern&quot; treatments of Galois theory. At least one edition of van der Waarden's algebra books &lt;em&gt;does&lt;/em&gt; treat them (following E. Noether?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own algebra notes (on-line at &lt;a href=&quot;http://www-users.math.umn.edu/~garrett/m/algebra/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www-users.math.umn.edu/~garrett/m/algebra/&lt;/a&gt; in chapter 23, section 3, as well as existing in a physical book) do in fact give &lt;em&gt;more&lt;/em&gt; emphasis to Lagrange resolvents (which I think illustrate many interesting things) than to the fact that extensions are expressible in radicals if and only if the Galois group is solvable. This is certainly not the main point of field theory and Galois theory, in any case, but, rather, an interesting aside, in my opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use of Lagrange resolvents (and a little algebraic number theory) to express roots of unity in radicals is at &lt;a href=&quot;http://www.math.umn.edu/~garrett/m/v/kummer_eis.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.umn.edu/~garrett/m/v/kummer_eis.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2018-12-17T19:32:35.973" LastActivityDate="2018-12-17T19:32:35.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14928" PostTypeId="1" CreationDate="2018-12-17T03:10:09.070" Score="4" ViewCount="193" Body="&lt;p&gt;Students in the basic statistics courses I teach often learn a little bit of probability and then learn hypothesis testing. The core concept that ties the course together is the p-value, but most students never adequately connect the p-value to the earlier topics in probability.&lt;/p&gt;&#xA;&lt;p&gt;I would like my students to have an &amp;quot;aha moment&amp;quot; about the p-value by completing an assignment that has the following outline:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Question 1: Assume [Situation X]. Calculate the probability of [unlikely data Y] occurring.&lt;/p&gt;&#xA;&lt;p&gt;Question 2: Complete a hypothesis test of the claim &amp;quot;We are in the situation [Situation X],&amp;quot; given the data [unlikely data Y].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In Question 1, the students would compute a probability. Then in Question 2, the students would find that the p-value is exactly the same thing as their answer to Question 1. This would illustrate what the p-value means precisely.&lt;/p&gt;&#xA;&lt;p&gt;Here are some almost-answers to my question that do not satisfy me:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Situation X is &amp;quot;You have a fair coin.&amp;quot; When students do the first&#xA;question to compute a probability assuming a fair coin, they use the binomial distribution.&#xA;But then inside a hypothesis test, students use the normal&#xA;approximation to the binomial distribution, so the probabilities are&#xA;not equal and the exercise does not work.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Situation X is &amp;quot;You have a normally distributed random variable with&#xA;a known population mean and standard deviation.&amp;quot; The problem here is&#xA;that when you go to the second question, you have to awkwardly&#xA;declare that you are absolutely sure about the distribution and the&#xA;population standard deviation, and that the hypothesis test is only&#xA;the test of a claim about the population mean. This seems really&#xA;unrealistic to me.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Thanks for your insight!&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2018-12-18T07:00:52.953" Title="What is a realistic situation that illustrates precisely what a p-value is?" Tags="|examples|statistics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14929" PostTypeId="2" ParentId="14923" CreationDate="2018-12-17T05:06:23.853" Score="-1" Body="&lt;p&gt;It should be taught in high school after students learn solving quadratic equations. The difficulties with having to work with roots of negative numbers should not be an objection here, rather such problems should be used as an extra motivation to later learn about complex numbers. And later at university it can be revisited again to introduce Galois theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Invoking the complexity of the general solution is not a good argument either. Solving third and fourth degree equations when the equations are specified using numeric values can be done quite fast and efficiently using the methods that lead to the general solution, while using the general solution directly is quite awkward. This happens for the same reason why solving a system of linear equations using Gaussian elimination is quite efficient, while the general solution is massively complex even for systems as small as 4 equations with 4 unknowns. The complexity of the solution increases exponentially with the number of coefficients, but if the coefficients are just numbers, you can always add up many different terms to get to short expressions.&lt;/p&gt;&#xA;" OwnerUserId="1856" LastActivityDate="2018-12-17T05:06:23.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14930" PostTypeId="1" CreationDate="2018-12-17T15:01:26.643" Score="4" ViewCount="207" Body="&lt;p&gt;I want to make a presentation of Burnside's lemma outside of group theory, and more as the stand-alone combinatorial tool that it can also be. My plan right now is to make it into a 15-20 minute video, a la 3blue1brown, or Numberphile, or any of a number of other great educational math videos out there. I have no idea whether I will ever get that far, but that's the plan anyways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My target audience would be more or less the same as those. Say &quot;math interested high schoolers&quot; to be concrete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The plan is to establish an example set with some group action, and then derive Burnside's lemma for that particular example. I think I have the theoretical parts more or less figured out, but I'm missing a hook. A great example. Something &quot;regular people&quot; can actually encounter and want to reflect upon outside of a math puzzle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it would be best to have an example using the dihedral group with 8 elements, or the full symmetry group on 3 elements. Either of those two groups would be a decent mix of non-trivial and still small enough to actually be able to work with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have thought about making necklaces / bracelets to appeal to the arts-and-craftsy people out there. I even got hold of a couple of A and B tags to actually make the necklaces. But it feels a bit too artificial, especially any restrictions on the number of tags to use (why exactly 3 letters, or exactly 4, when I clearly have at least 4 of each letter readily available, for instance).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've thought about arranging people around a table for a game of Uno, where it matters who you sit next to, but not so much who's on the right and who's on the left, as play swaps direction all the time. But then the orbits on the set of possible table seatings are all max size anyways because no two people are the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could to an entirely abstract example, with actual squares or triangles, with painting or marking the corners or sides, but I don't think that would be suitable as a motivational example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe I'm being too picky, and should just go with the bracelets. But I thought I should at least hear if anyone here had a great idea before I lock myself onto that track.&lt;/p&gt;&#xA;" OwnerUserId="7624" LastActivityDate="2018-12-17T23:24:16.123" Title="A good example to show group actions and Burnside's lemma" Tags="|examples|abstract-algebra|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14931" PostTypeId="2" ParentId="14930" CreationDate="2018-12-17T16:38:43.163" Score="4" Body="&lt;p&gt;Well, certainly nearly all the usual texts (&lt;a href=&quot;http://www.rellek.net/book/ch_polya.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here is an open source one&lt;/a&gt;!) do this via necklace/bracelet examples, and you can definitely get the classes this way.  The treatment in &lt;a href=&quot;https://www.pearson.com/us/higher-education/program/Anderson-Discrete-Mathematics-with-Combinatorics-2nd-Edition/PGM308903.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anderson's Discrete Math&lt;/a&gt; would be typical - coloring an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-sided polygon with &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; colors.  Even with two colors and four sides becomes nontrivial and leads you to stabilizers etc.  An interesting variant on this (which I don't recall where I first heard it) is to first talk about necklaces, then to talk about candles on birthday cakes, which of course can't be reflected so easily without a big mess on your table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want something a little more interesting, though, I'd recommend trying music theory, such as &lt;a href=&quot;http://www.mtosmt.org/issues/mto.07.13.4/mto.07.13.4.hook.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Why Are There Twenty-Nine Tetrachords&quot;&lt;/a&gt;, a nice article by Julian Hook.  Here &lt;span class=&quot;math-container&quot;&gt;$m=2$&lt;/span&gt; - either you have a note in a chord or you don't - and the rotation and reflection have very immediate rationales for certain genres of 'modern' music.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or if you want something more advanced (perhaps using Pólya enumeration) there is a nice exercise in Anderson: &quot;In how many ways can the squares on the top of a &lt;span class=&quot;math-container&quot;&gt;$3\times 3$&lt;/span&gt; chessboard by colored?&quot; That's a little more interesting than the square, but has the same symmetry.  There are many helpful &lt;a href=&quot;http://www.rellek.net/book/s_polya_exercises.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;exercises in Keller/Trotter&lt;/a&gt; as well, such as &quot;Xylene is an aromatic hydrocarbon having two methyl groups (and four hydrogen atoms) attached to the hexagonal carbon ring. How many isomers are there of xylene?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With all these and more options, I figure you'll find &lt;em&gt;something&lt;/em&gt; you find interesting enough to use as your base example, good luck!&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2018-12-17T16:38:43.163" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14932" PostTypeId="1" CreationDate="2018-12-17T20:55:31.987" Score="7" ViewCount="397" Body="&lt;p&gt;In the recent past, I've come across a pedagogical strategy for teaching/learning algebra that is sometimes called &quot;Indicated Arithmetic&quot; or &quot;Delayed Evaluation&quot;. However, I've been unable to find any literature on it, and only know it by name from a former colleague. I've searched resources and have come up empty. &lt;strong&gt;Is this a term anyone is familiar with, and if so, do you have any source materials?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In the interest of clarity, here is an example where Indicated Arithmetic (or Delayed Evaluation) is used.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Rommel brought $15 to a school fundraiser and spent it all on hot dogs and game tickets. Game tickets were 25 cents each, and hot dogs were 3 dollars each.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A number of standard algebraic questions can be asked following this set up, but they all generally get to the desire for an algebraic equation which describes the situation. A valuable tool for students who struggle with finding this equation is Indicated Arithmetic. The key attribute in IA is writing out all operations to be performed, but not performing any binary operations. Combined with the ideas of trying some explicit values, keeping track of your work with a table of values, and organizing your work, you might be able to create a visual like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UoBGr.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UoBGr.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The right column of the table builds towards the algebraic expression, and it is easy to see in this organized way, what different roles the different numbers play in the scenario and the expression. It is much easier for struggling students to see pattern that the &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; variable follows, rather than trying to find a pattern in the sequence of values 0, 3, 6, 9, 12, 15.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit: 1/29/19&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my searching I found a footnote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The term “indicated arithmetic” was shared by Nicholas Branca in 2004.&#xA;  It is used to be explicit about making used work visual so that&#xA;  emerging patterns can be captured.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Branca was teaching at San Diego State University until his passing in 2008. Still, this doesn't quite satisfy my needs. If anyone has more information, I'd be forever indebted.&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="267" LastEditDate="2019-01-29T16:00:21.333" LastActivityDate="2020-08-05T18:04:40.480" Title="&quot;Indicated Arithmetic&quot; or &quot;Delayed Evaluation&quot;" Tags="|reference-request|algebra|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14933" PostTypeId="2" ParentId="14930" CreationDate="2018-12-17T23:24:16.123" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I could to an entirely abstract example, with actual squares or triangles, with painting or marking the corners or sides, but I don't think that would be suitable as a motivational example.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think this &lt;em&gt;could&lt;/em&gt; be a motivational example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;True Story: My daughter and I tried to make a little card game once (when we were tired of the ones we owned), and discovered we needed Burnside's Lemma when it came to making the cards. No doubt this idea already exists somewhere and has a name, but we &quot;invented&quot; a game where players take turns laying square tiles (with colored edges) down on a board, where edges of neighboring tiles must match in color. When we allowed only two possible colors per edge, it was easy to make the cards and the deck was too small to make for much of a fun game. It was when we considered increasing the number of possible edge-colors and &quot;possible&quot; shapes for the tiles that we quickly lost our ability to even determine how many tiles there should be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was such a fun and natural way to stumble upon a need for a theorem, that I presented it to a group of undergrads (&lt;a href=&quot;https://drive.google.com/file/d/0B3TnktvmnquPWkdtSEJIZzVZSVNoOW1HUVVYMWJPZUt0alMw/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;PowerPoint presentation&lt;/a&gt;), having them build and play the game before diving into the counting problem.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2018-12-17T23:24:16.123" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14934" PostTypeId="2" ParentId="14923" CreationDate="2018-12-18T01:19:22.853" Score="-1" Body="&lt;p&gt;I think it should be taught in first year Calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it is taught in high school, there is a good chance many of the students will never use it again (a high proportion do not go on to college), so that information will be wasted. Teaching it to students who are already in college or university increases the odds that this information will actually be used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In first year Calculus, there is usually a brief review of some high school topics, trigonometry, algebra, etc, before charging ahead with the new material. Even if a derivation is not provided, or students are not required to work through a derivation, they should at least know these solutions exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my own background (engineering), we started solving third degree polynomials in second term of Physical Chemistry, when we had to work with Van der Waals Equation, a third-degree polynomial. Today, most university students have access to powerful software products that solve such equations numerically in the blink of an eye, but I did not. Working through the solutions for a third or fourth degree polynomial may have been a little tedious with a hand calculator, but it was doable. Instead of hitting a brick wall, we at least had tools at hand to proceed to a solution. &lt;/p&gt;&#xA;" OwnerUserId="11694" LastActivityDate="2018-12-18T01:19:22.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14935" PostTypeId="2" ParentId="14923" CreationDate="2018-12-18T01:57:13.113" Score="2" Body="&lt;p&gt;Abstract Algebra, under field extensions/Galois theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unlike quadratics, there are relatively few applications (hard-applied or applications to other theory) where exact solutions to cubic or quartic matter, and no reason you wouldn't just ask a computer algebra system or consult a reference book if you need them, rather than trying to memorize or work things out by hand.&lt;/p&gt;&#xA;" OwnerUserId="1146" LastActivityDate="2018-12-18T01:57:13.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14937" PostTypeId="2" ParentId="14923" CreationDate="2018-12-18T06:12:35.913" Score="1" Body="&lt;p&gt;It seems a mistake to include such formulas in middle school or high school education except perhaps in supplementary material or in classes directed at motivated and talented students. For the most part the interest of such formulas comes from number theoretic and algebraic considerations that are not of such broad interest or applicability. Their complexity is such that including them simply reinforces a problematic memorize formulas mentality already strong in students who can find the roots of a quadratic polynomial but have no idea how to complete the square. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems reasonable (although not necessary) to discuss such formulas in a calculus class as part of a broader discussion of root finding. It should be standard to include in calculus classes exercises directed at the localization of and determination of the number of roots of a cubic polynomial. Qualitative arguments such as this polynomial with leading coefficient &lt;span class=&quot;math-container&quot;&gt;$x^{3}$&lt;/span&gt; has a unique local minimum at which it takes a negative value so has three real roots, one of which is greater than number at which the local minimum occurs are well within the reach of a basic calculus class. Bisection is generally taught in calculus classes as well (often under some name such as Bolzano's theorem, with a nonconstructive proof ...), and it would be a good thing to teach Newton's method too. It is instructive to combine such methods to find (by hand) approximate roots of a cubic polynomial and compare with what the solutions in radicals give. Such a comparison highlights that an expression in radicals may not be terribly useful in a scientific/numerical context, because there remains the need to approximate the radicals! &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-12-18T06:12:35.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14938" PostTypeId="2" ParentId="14928" CreationDate="2018-12-18T07:00:52.953" Score="1" Body="&lt;p&gt;Without knowing about p-value, students could discuss their answers to:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Willy Wonka is putting a small number of golden tickets in his&#xA;  chocolate bars again. His claim is that the probability of winning is&#xA;  1/10. How many chocolate bars would you have to buy, with no winning&#xA;  ticket inside, to be reasonably skeptical about his claim?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then, the idea of p &amp;lt; 0.05, or 0.01, etc. could be introduced, and students could subsequently answer the question with these tolerances in mind.&lt;/p&gt;&#xA;" OwnerUserId="6845" LastActivityDate="2018-12-18T07:00:52.953" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14941" PostTypeId="1" AcceptedAnswerId="14942" CreationDate="2018-12-19T00:37:00.830" Score="3" ViewCount="125" Body="&lt;p&gt;I've noticed that some math majors seem to have arrived at their junior or senior years with some pretty severe gaps in their knowledge of the basics. Ideally, students would be constantly looking back and filling in gaps in their knowledge as they progress through their major. (Probably this is something that the top students do automatically, which is part of the reason for their success.) However, many students seem not to spend enough time or energy on improving their understanding of the basics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How could the structure of a degree program be modified in order to encourage or require students to keep honing their knowledge of the fundamentals?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are a few ideas (not necessarily good ideas, just ideas):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;All math majors could be required to take a cumulative exam at the end of every semester or every year. The cumulative exam would cover everything the students have learned in their math classes up to that point (but focusing on the fundamentals).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students could be required to work through the Khan Academy knowledge map and demonstrate proficiency in relevant subjects (as measured by the Khan Academy learning system).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students could be enrolled every semester in a 2 unit review class which has homework assignments which review the fundamentals of courses they've already taken.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'd be interested in hearing any non-traditional ideas for how to structure a degree program along these lines.&lt;/p&gt;&#xA;" OwnerUserId="9633" LastActivityDate="2018-12-19T01:02:32.273" Title="A system for students to fill in gaps throughout a math degree" Tags="|review|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14942" PostTypeId="2" ParentId="14941" CreationDate="2018-12-19T00:53:56.307" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think some sort of examination program like qualifier, but for undergrads, would be the simplest way to drive for what you want.  Perhaps like the Tripos, albiet not as hard.  And with motivation being scoring on it (but hopefully not just in ranking, but in absolute sense).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think you have to consider the disadvantage of spending more time on basics meaning less time on non basics (or even non majors work).  Now it may or may not be worthwhile to do so.  But you have to analyze trade-offs.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You also need to consider that there is a competitive market for students.  Kids can go to other schools or to other majors if you make them do too much that is distasteful.  So design a solution that seems fun or worthwhile (e.g. for job offers or grad school placement).  Or at least that appears or actually does minimize the unpleasant aspects of additional requirements.  I guess you could also try to attract by the nature of the demands (like how USMC or SEAL program play on the rigors).  But just consider this issue of &quot;selling&quot; the program.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's not clear to me what gaps you are talking about.  For instance, not just getting A+ mastery of existing courses?  (Do you expect every kid to have mastery?) Curricula that don't include some basics?  Is it computational and manipulational mastery that they are missing or something else?  More in a particular area (e.g. probability)?  Also, some clarity on extent of the gaps (what percent have what gaps, even if just by descriptive estimate of observations but ideally with an assessment).  The reason to say this is the more you know about &quot;what the problem is&quot;, the more you can address it.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think it's also worthwhile discussing what sort of students and school you are dealing with (Harvard vs UVA vs Cal State vs NW LA State).  The nature of the problem/solution may differ a lot.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="11700" LastEditorUserId="11700" LastEditDate="2018-12-19T01:02:32.273" LastActivityDate="2018-12-19T01:02:32.273" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14943" PostTypeId="1" CreationDate="2018-12-19T04:21:30.880" Score="6" ViewCount="290" Body="&lt;p&gt;I remember my linear algebra teacher mentioning tensor products as an advanced topic that would be covered in upper level algebra coursework. During undergraduate abstract algebra, tensor products were not mentioned. Later, my graduate algebra professor told me that the tensor product was required background knowledge that I should have learned in linear algebra. Ultimately, I ended up learning the tensor product in a special topics course in the physics department.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Several of my classmates who had attended different undergraduate institutions reported similar experiences. From my limited polling, this seems to imply that the tensor product has proven difficult to position within a conventional mathematics curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At what point during a mathematics degree should the tensor product be introduced?&lt;/p&gt;&#xA;" OwnerUserId="263" LastEditorUserId="263" LastEditDate="2018-12-20T02:17:25.367" LastActivityDate="2018-12-22T13:11:29.960" Title="At what point in the curriculum should the tensor product be introduced?" Tags="|curriculum|abstract-algebra|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14944" PostTypeId="2" ParentId="14943" CreationDate="2018-12-19T08:20:45.670" Score="9" Body="&lt;p&gt;&lt;em&gt;(This pertains to U.S. universities.)&lt;/em&gt; Times may have changed, but when I was in graduate school (several places, and yes I know this is unusual, but I mention it because I'm talking about more than one data point) this was a standard topic that showed up somewhere in the standard first year (2-semester) &lt;strong&gt;graduate algebra sequence&lt;/strong&gt; (e.g. &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/038795385X&quot; rel=&quot;noreferrer&quot;&gt;Lang&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0387905189&quot; rel=&quot;noreferrer&quot;&gt;Hungerford&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/048647187X&quot; rel=&quot;noreferrer&quot;&gt;Jacobson&lt;/a&gt;, etc.), and the only way an undergraduate would see this in a formal class would be when taking such a class as an undergraduate. Of course, someone also taking upper level physics classes might see physics/engineering tensors in classical mechanics, continuum mechanics, fluid dynamics, electrodynamics, etc. (in my case it was in the form of &lt;a href=&quot;https://en.wikipedia.org/wiki/Dyadics&quot; rel=&quot;noreferrer&quot;&gt;dyads&lt;/a&gt; that appeared in the second semester of a 2-semester sequence using &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0201073927&quot; rel=&quot;noreferrer&quot;&gt;Symon&lt;/a&gt;), and someone taking an honors level advanced calculus course at one of the few universities that offered such courses (think &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/9814583936&quot; rel=&quot;noreferrer&quot;&gt;Loomis/Sternberg&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0486480909&quot; rel=&quot;noreferrer&quot;&gt;Nickerson/Spencer/Steenrod&lt;/a&gt;, or &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0387902066&quot; rel=&quot;noreferrer&quot;&gt;Fleming&lt;/a&gt;) would see tensor products, but in these cases (and probably others I could imagine) we're either not talking about formal tensor products in which notions such as &quot;universal mapping property&quot; arise or we're not talking about standard undergraduate material. I checked my undergraduate linear algebra text, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0135367972&quot; rel=&quot;noreferrer&quot;&gt;Hoffman/Kunze&lt;/a&gt;, which was the text used for a fairly stiff undergraduate course in linear algebra that I took in 1978, and I see some discussion of tensor products in Sections 5.6: &lt;em&gt;Multilinear Functions&lt;/em&gt; and 5.7: &lt;em&gt;The Grassman Ring&lt;/em&gt; (pp. 166-180), but in looking over this now I definitely remember that we skipped these sections (further evidence is that I also have margin notes and homework problems marked in Section 5.4, then none until Chapters 6 and 7 and the beginning of Chapter 8).&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2018-12-19T08:20:45.670" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14945" PostTypeId="2" ParentId="14855" CreationDate="2018-12-20T06:39:27.977" Score="11" Body="&lt;p&gt;I see the same mistake in my Calculus students' work a lot. And from my observations, I think there's a very simple reason for that &amp;mdash; for typical students (here in the U.S.) this is multiplication, and in multiplication order doesn't matter. In fact, for typical students (&amp;hellip;) almost everything is multiplication, including function notation; and they simply extend their writing habits to this new symbol without giving it much thought. And I'm going to demonstrate below that I mentioned function notation for a reason.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclaimer: described below are my personal observations and opinions; I can't provide any supporting literature. But then, I have years' worth of these observations, which are consistent across hundreds of students I've taught and tutored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine a middle school student needs to solve an equation like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{x}{5}=3.$$&lt;/span&gt;&#xA;Of course, the approach is to multiply both sides by &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;. And the typical way students are going to show their work on paper is by writing the fact that they are multiplying by &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; by adding some markings in the same line, and then the result in the next line:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align} \color{blue}{5\times}\frac{x}{5}&amp;amp;=3\color{blue}{\times5} \\ x&amp;amp;=15 \end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As part of the reason for students developing this habit, I also blame the tradition (habit? expectation?) of showing what they are going to do &lt;em&gt;in the same line&lt;/em&gt;. There's no room between the equals sign and the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; to fit &quot;&lt;span class=&quot;math-container&quot;&gt;$\color{blue}{5\times}$&lt;/span&gt;&quot; in there, so writing this multiplication from the left on the left and from the right on the right is basically the only way to fit it. And at this point it's fine. Even though pedagogically it would be much more preferable to indicate multiplication from the same side on both sides of the equation, in the spirit of &lt;strong&gt;&lt;em&gt;doing the same thing to both sides of an equation&lt;/em&gt;&lt;/strong&gt;, arithmetically this works and leads to the correct answer. (Of course, then we really have to deal with this problem in our abstract algebra classes when solving or manipulating equations in non-abelian groups.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then we move on to more advanced classes and encounter equations such as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$e^x=10.$$&lt;/span&gt;&#xA;Don't forget that for most U.S. students (deep sigh&amp;hellip;) &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; means &quot;&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot;, &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt; means &quot;sine times &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot;, etc. Yeah, function notation is another sore topic&amp;hellip; So, by following the same recipe &amp;mdash; and because there's no room between the equals sign and the &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; anyway! &amp;mdash; students &quot;multiply&quot; both sides by &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align} \color{blue}{\ln}e^x&amp;amp;=10\color{blue}{\ln} \\ x&amp;amp;=\ln10 \end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then this pattern continues. One of the first instances when students have to write &lt;span class=&quot;math-container&quot;&gt;$\dfrac{d}{dx}$&lt;/span&gt; themselves (as opposed to a given problem with this notation) is in the topic of implicit differentiation. It's the same situation again: given an equation, perform some steps to get an answer, and so that's what they do. For example, starting with something like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^2+y^2=10$$&lt;/span&gt;&#xA;they &quot;multiply&quot; both sides by &lt;span class=&quot;math-container&quot;&gt;$\dfrac{d}{dx}$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align} \color{blue}{\frac{d}{dx}}(x^2+y^2)&amp;amp;=10\color{blue}{\frac{d}{dx}} \\ 2x+\cdots&amp;amp;=\cdots \end{align}$$&lt;/span&gt;&#xA;(where lots of different things may appear in place of those dots), and then continue to the subsequent steps.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Of course, as others have already said, a deeper reason is not understanding the differential operator &lt;span class=&quot;math-container&quot;&gt;$\dfrac{d}{dx}$&lt;/span&gt; as an &lt;strong&gt;&lt;em&gt;operator&lt;/em&gt;&lt;/strong&gt;, not treating mathematical notation properly in general, etc. But I wanted to address the specific question of where this particular mis-notation comes from, in my opinion.&lt;/p&gt;&#xA;" OwnerUserId="6762" LastEditorUserId="6762" LastEditDate="2019-12-25T05:56:47.477" LastActivityDate="2019-12-25T05:56:47.477" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="14946" PostTypeId="1" AcceptedAnswerId="14947" CreationDate="2018-12-20T10:36:18.570" Score="0" ViewCount="432" Body="&lt;p&gt;I am studying ap calculus now, before I move onto differential equations etc., but the thing I am unsure of is, should I learn set theory before continuing on my ap calculus sections?&lt;/p&gt;&#xA;" OwnerUserId="10519" LastEditorUserId="2083" LastEditDate="2018-12-20T12:21:06.007" LastActivityDate="2018-12-20T12:57:40.593" Title="When self teaching, should I learn set theory before continuing ap calculus?" Tags="|undergraduate-education|calculus|self-learning|set-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14947" PostTypeId="2" ParentId="14946" CreationDate="2018-12-20T12:57:40.593" Score="5" Body="&lt;p&gt;Short answer: No.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you really want to know a little bit about sets, the first section of &lt;a href=&quot;http://www.people.vcu.edu/~rhammack/BookOfProof/&quot; rel=&quot;noreferrer&quot;&gt;Book of Proof&lt;/a&gt; or &lt;a href=&quot;http://faculty.uml.edu/klevasseur/ads/&quot; rel=&quot;noreferrer&quot;&gt;Applied Discrete Structures&lt;/a&gt; would more than suffice for the purposes of most collegiate level courses.  (You'll want more of them for courses involving proof, of course, but no more set theory.)  Especially the first is quite suitable for self study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But while there are some basic calculus arguments (especially once you start looking at vectors and/or parametric things) which can involve sets a bit, if you are studying on your own, no need to complicate matters by studying them separately.  Whatever calculus text/resource you are using should have enough background to make progress - and if it doesn't, find another one.  &lt;a href=&quot;https://aimath.org/textbooks/approved-textbooks/&quot; rel=&quot;noreferrer&quot;&gt;This website&lt;/a&gt; has several English-language open access texts, one of which should suffice for your needs.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2018-12-20T12:57:40.593" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14948" PostTypeId="1" CreationDate="2018-12-20T22:10:14.513" Score="4" ViewCount="101" Body="&lt;p&gt;I have a large probability class and would like to do some memorable demonstrations of Markov chains for them.  Does anyone have any recommendations for a &quot;low-tech&quot; demo that doesn't involve computers (or at least, one that minimizes this requirement)?  I have access to all kinds of physical props like dice, poker chips, coin tosses, and so forth.&lt;/p&gt;&#xA;" OwnerUserId="11705" LastActivityDate="2018-12-21T10:47:49.210" Title="A Markov chain demonstration that doesn't require computers" Tags="|undergraduate-education|mathematical-pedagogy|examples|interactive-teaching|probability|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14949" PostTypeId="2" ParentId="14948" CreationDate="2018-12-20T22:45:47.303" Score="2" Body="&lt;p&gt;Per Wiki:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Markov_chain#Gambling&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Markov_chain#Gambling&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose that you start with \$10, and you wager \$1 on an unending, fair, coin toss indefinitely, or until you lose all of your money. If &lt;span class=&quot;math-container&quot;&gt;$X_n$&lt;/span&gt; represents the number of dollars you have after &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; tosses, with &lt;span class=&quot;math-container&quot;&gt;$X_0 = 10$&lt;/span&gt;, then the sequence &lt;span class=&quot;math-container&quot;&gt;$\{X_n: n\in \mathbb{N}\}$&lt;/span&gt; is a Markov process. If I know that you have \$12 now, then it would be expected that with even odds, you will either have \$11 or \$13 after the next toss. This guess is not improved by the added knowledge that you started with \$10, then went up to \$11, down to \$10, up to \$11, and then to \$12.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I recommend to make the following modifications though:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Use a smaller starting amount (3 or 5 or the like) so you can run a few examples.  Think it will take a while to exhaust \$10 bankroll.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Flip the coin, cover it, and then call on a student in class to guess it. Pick different students at random.  Pick a student (with good handwriting) or use a TA, to graph or tally the experiment as it goes, on the chalkboard.  (Makes it more interactive...human factor.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="11700" LastEditorUserId="8005" LastEditDate="2018-12-21T10:47:49.210" LastActivityDate="2018-12-21T10:47:49.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14951" PostTypeId="2" ParentId="14948" CreationDate="2018-12-21T08:28:24.340" Score="0" Body="&lt;p&gt;The simpliest example I know is the ball game with at least three people: it starts with one person, having the ball, and every second (s)he throws it to another person randomly. Question: what's the probability that the ball gets to person &lt;code&gt;X&lt;/code&gt; after &lt;code&gt;n&lt;/code&gt; seconds?&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2018-12-21T08:28:24.340" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14952" PostTypeId="1" AcceptedAnswerId="14961" CreationDate="2018-12-21T18:55:23.710" Score="4" ViewCount="294" Body="&lt;p&gt;My students found an exercise about cyclotomic polynomials in the AOPS precalculus text. They asked me why this construction exists in the first place and what it's good for... I am looking to give them a more satisfying answer than: &quot;We use it in the study of Galois theory.&quot; They are familiar with the symmetry group &lt;span class=&quot;math-container&quot;&gt;$S_3$&lt;/span&gt; and are very talented &lt;span class=&quot;math-container&quot;&gt;$7/8$&lt;/span&gt;th graders. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's a good answer to give them?  &lt;/p&gt;&#xA;" OwnerUserId="9841" LastActivityDate="2018-12-22T12:50:06.853" Title="Why are we even studying cyclotomic polynomials?" Tags="|concept-motivation|precalculus|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14953" PostTypeId="2" ParentId="14831" CreationDate="2018-12-21T23:36:12.307" Score="0" Body="&lt;p&gt;7 years old is not too young to draw fractions and think about them. I am slightly biased as I work specifically with the &quot;gifted and talented&quot; population (whatever that means... ). The best method I have found is to draw circles and chop them up. Denominator means &quot;De-number of equal parts&quot; and also Denom=DeName of the fraction. &quot;nom&quot; is name in latin. Also, I have found that Denomnomnom is particularly appealing to 7 year olds. Especially good for getting giggles if you are working with fractions that look like pacman or if you are using pizza as an example. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cS6aD.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cS6aD.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is a nice picture of &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; and we can draw another line to make&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/V1R91.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/V1R91.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2/4$&lt;/span&gt; but because the shaded section hasn't changed: we haven't changed the amount. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keep chopping up the fraction to discover that &lt;span class=&quot;math-container&quot;&gt;$4/8$&lt;/span&gt;ths is equal. We just need to bare in mind that the parts need to be equal and the denonominator represents the number of equal parts and the numerator is just the number of shaded parts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bonus points if you can include pizza in your explanation. I haven't met any 7 year olds that don't like pizza (this could be an observational bias on my part...).&lt;/p&gt;&#xA;" OwnerUserId="9841" LastActivityDate="2018-12-21T23:36:12.307" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14955" PostTypeId="2" ParentId="14952" CreationDate="2018-12-22T04:23:05.333" Score="3" Body="&lt;p&gt;On the topic of &lt;a href=&quot;http://mathworld.wolfram.com/SichermanDice.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sicherman dice&lt;/a&gt;: &quot;Cyclotomic polynomials play an important role in analyzing dice labelings of six-sided dice as well as other sizes of dice.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://buzzard.ups.edu/courses/2010spring/projects/jenkins-sicherman-dice-ups-434-2010.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://buzzard.ups.edu/courses/2010spring/projects/jenkins-sicherman-dice-ups-434-2010.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6845" LastEditorUserId="9841" LastEditDate="2018-12-22T04:43:11.613" LastActivityDate="2018-12-22T04:43:11.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14956" PostTypeId="1" AcceptedAnswerId="14958" CreationDate="2018-12-22T07:43:06.437" Score="7" ViewCount="1002" Body="&lt;p&gt;Is it appropriate to mention weird/exciting results in math (or use as cautionary tales why one cannot apply mathematics naively) in say high school level?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples of these results include the sphere eversion which turned into a meme (yes, I'm aware it's a true result). Banach-Tarski paradox, Vitali Sets, &quot;1+2+3+4 = &lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{12}$&lt;/span&gt;&quot; and Hilbert's Hotel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, I feel as though&#xA;&lt;span class=&quot;math-container&quot;&gt;$e^{i\pi} = -1$&lt;/span&gt; demonstrates that math despite seemingly random ties up at this beautiful equation.&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="10335" LastEditDate="2018-12-22T08:03:58.920" LastActivityDate="2018-12-22T22:55:44.717" Title="is it appropriate or beneficial to mention weird results in math?" Tags="|proofs|examples|general-pedagogy|" AnswerCount="6" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14958" PostTypeId="2" ParentId="14956" CreationDate="2018-12-22T11:13:10.610" Score="13" Body="&lt;p&gt;I would be careful with the type of result for which one needs a lot of new math to digest the explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I would avoid talking about &lt;span class=&quot;math-container&quot;&gt;$ 1 + 2+3+.. = -1/12$&lt;/span&gt; because there is basically no way to explain to highschoolers in what sense this could be true. First, one would need a good understanding of the limit and the value of a series (i.e., that this value is not the result of repeated addition) and then one can appreciate the weirdness even more (but the paradox is not removed). I guess zeta function regularization is out of reach for most high schoolers. Many of them will just remember this as one more data point of &quot;math is weird and I can't make sense of it&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could, however talk about results like the Goldbach conjecture - extremely simple to state and only partly solved recently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hilbert's hotel is nice but you would have to explain the notion &quot;infinity&quot; in that context. By the way, there is an upcoming book&quot;Life on the infinite farm&quot; which tells a story about infinity which is more for elementary school. One part is about a cow with an infinite number of legs who wears a show on each foot and gets a new pair of shoes - what should she do? I think that this story about infinity would go very well with kids.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2018-12-22T11:13:10.610" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14961" PostTypeId="2" ParentId="14952" CreationDate="2018-12-22T12:50:06.853" Score="5" Body="&lt;p&gt;Can we grant that the students think roots of unity are worthwhile? If so, point out that one of the ways to understand roots of polynomials (like roots of &lt;span class=&quot;math-container&quot;&gt;$x^n-1$&lt;/span&gt;) is to understand the &lt;em&gt;lowest degree&lt;/em&gt; polynomial with those roots. The &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; roots of unity do not all behave in the same way algebraically, but the primitive &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th roots of unity (in &lt;span class=&quot;math-container&quot;&gt;$\mathbf C$&lt;/span&gt;) do. That is what irreducibility of the cyclotomic polynomials tells us.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are two applications of these polynomials.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;They have a role in an elementary proof that for each &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;1$&lt;/span&gt; there are infinitely many primes &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$p\equiv 1 \bmod n$&lt;/span&gt; (Euclid's proof that there are infinitely many primes uses the 2nd cyclotomic polynomial.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They have a role in Witt's proof of Wedderburn's theorem that all finite division rings are commutative.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The cyclotomic polynomials are a nice example of a pattern that lasts a long time but eventually stops: the first 100 of them have coefficients in {0,1,-1}, but farther out this is not true (first instance is the 105th cyclotomic polynomial).&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2018-12-22T12:50:06.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14962" PostTypeId="2" ParentId="14943" CreationDate="2018-12-22T13:11:29.960" Score="2" Body="&lt;p&gt;In a math department (not physics department) I'd introduce it &lt;em&gt;after&lt;/em&gt; some more familiar constructions (quotient groups, cyclic groups with a choice of generator,...) have already been described a second time by a universal mapping property. Tensor products will be tough the first time no matter what, so it would help if the idea of a construction based on a universal mapping property is already known to the class in some other contexts.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2018-12-22T13:11:29.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14963" PostTypeId="2" ParentId="14956" CreationDate="2018-12-22T16:47:25.503" Score="2" Body="&lt;p&gt;At any level, I'd ask myself the question &quot;how many new things does the student need to know in order to understand this even at a surface level?&quot;. If that number gets beyond 2, I would think that it is a definite no. (So anything involving measure theory is right out.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^{\infty} n = -\frac{1}{12}$&lt;/span&gt; result is going to require zeta functions, analytic continuation, and something about convergent/divergent sums at a minimum. Even if you do this, they are likely to get some nonsense in their head about &quot;wrapping around infinity&quot;. I speak from experience here; I was teaching a &quot;Great Ideas in Mathematics&quot; class when a Numberphile video about this was making the rounds on the internet and it seemed timely to talk about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, &lt;span class=&quot;math-container&quot;&gt;$e^{\pi i} = -1$&lt;/span&gt; could maybe work for Calc AB students once they know the chain rule and the derivatives of &lt;span class=&quot;math-container&quot;&gt;$\ln,\sin,$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt;. There, you just need that you should treat &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; like any other constant and that  &lt;span class=&quot;math-container&quot;&gt;$e^{i\theta}=\cos\theta+i\sin\theta$&lt;/span&gt;, which you can demonstrate by taking a derivative. That is, &#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{eqnarray*}&#xA;\frac{d}{d\theta}\ln\left( \cos\theta + i \sin\theta \right)&#xA;&amp;amp; = &amp;amp; \frac{\frac{d}{d\theta}\left(\cos\theta + i \sin\theta\right)}{\cos\theta + i \sin\theta} \\&#xA;&amp;amp; = &amp;amp; \frac{-\sin\theta + i \cos\theta}{\cos\theta + i \sin\theta} \\&#xA;&amp;amp; = &amp;amp; \frac{i\left(\cos\theta + i \sin\theta\right)}{\cos\theta + i \sin\theta} \\&#xA;&amp;amp; = &amp;amp; i&#xA;\end{eqnarray*}$$&lt;/span&gt;&#xA;So, for some constant &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \ln\left( \cos\theta + i \sin\theta \right) = i\theta + C $$&lt;/span&gt;&#xA;The rest follows by pre-calc methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that you didn't mention, that might be worthwhile talking about, is the Weierstrass function: &lt;a href=&quot;https://en.wikipedia.org/wiki/Weierstrass_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Weierstrass_function&lt;/a&gt;&#xA;It was designed to combat the misconception that all functions are differentiable except at isolated points and plays an important role in the history of calculus.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2018-12-22T16:47:25.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14964" PostTypeId="2" ParentId="14956" CreationDate="2018-12-22T17:27:53.187" Score="11" Body="&lt;p&gt;If it's the right amount of &quot;weird&quot;, then &lt;em&gt;YES&lt;/em&gt; (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Zone_of_proximal_development&quot; rel=&quot;noreferrer&quot;&gt;Zone of Proximal Development&lt;/a&gt;). For example, I often try to show students how &lt;span class=&quot;math-container&quot;&gt;$0.\bar{3} = \frac{1}{3}$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$0.\bar{9} = 1$&lt;/span&gt;. This example alone has so much to unpack, that I often come back to it again and again while teaching rational numbers, series, algorithms, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could see some mathematical results being so far beyond the grasp of a student that it would at best have no impact, or at worst, be completely de-motivating. It truly is an art to read your audience and pick something that is &lt;em&gt;just&lt;/em&gt; beyond their ability, but &lt;em&gt;within&lt;/em&gt; their potential.&lt;/p&gt;&#xA;" OwnerUserId="6845" LastActivityDate="2018-12-22T17:27:53.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14965" PostTypeId="2" ParentId="14956" CreationDate="2018-12-22T20:29:27.387" Score="0" Body="&lt;p&gt;I would bring up some of these truly weird results to bring the humanity into the process.  We often teach mathematics as if it has been as it is forever, never changing.  These weird results are typically surrounded by a great deal of insight wrought by humans.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;span class=&quot;math-container&quot;&gt;$e^{\pi i} = -1$&lt;/span&gt; was not &lt;em&gt;always&lt;/em&gt; true.  Indeed, its validity stems around the work of Euler.  It cemented the meaning of logarithms.  Some people at the time were running on the belief that &lt;span class=&quot;math-container&quot;&gt;$log(-x) = log(x)$&lt;/span&gt;, which was a natural way of thinking if you didn't want to have complex numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, tying his work into Analytic Continuation would be a way to show the kind of way mathematicians think.  There's countless ways one could have gone about extending logarithms, but this was the one way which satisfied the expectations of mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Likewise I find there's no point in teaching the Banach-Tarski paradox unless you want to get into the fact that there is not a universal agreement about how to axiomize set theory surrounding the axiom of choice.  Few outside of the true set theorists will &lt;em&gt;ever&lt;/em&gt; care about whether the axiom of choice is true or not.  But the fact that mathematicians are divided on it is meaningful.  It could also be used to show how some theories may start out being proven in ZFC, and only later do proof in ZF surface.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, without that, the only conclusion you can really take from Banach-Tarski is that mathematicians are idiots, because only an idiot would fail to see that you can't make one sphere into two of the same size just by cutting rotating and gluing!  You have to show enough to demonstrate why it really bothers mathematicians that they can't just blindly accept the axiom of choice, as intuitive as it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Show that &lt;em&gt;humans&lt;/em&gt; are still figuring out how mathematics should work.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2018-12-22T20:29:27.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14966" PostTypeId="2" ParentId="14956" CreationDate="2018-12-22T21:27:40.713" Score="1" Body="&lt;p&gt;As a learner: weird results were one of the things that made me want to learn more. But they mostly had to be weird results of &lt;em&gt;what I already knew&lt;/em&gt;, or of something well enough explained for me to follow it. The &quot;birthday paradox&quot; was one example, as I remember, when first learning about probability. I think I may have got that from a Martin Gardner book though, not from what I was learning at school. (I think there were some much better examples, but I'm struggling to remember them at the moment.)&lt;/p&gt;&#xA;" OwnerUserId="10663" LastActivityDate="2018-12-22T21:27:40.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14967" PostTypeId="2" ParentId="14956" CreationDate="2018-12-22T22:55:44.717" Score="1" Body="&lt;p&gt;There's no point in showing people something weird to illustrate how their knowledge is valid and yet their expectations are not, if they don't have the knowledge in the first place. Anything that requires university-level topics should immediately be discarded.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A straightforward example that can be approached in a number of ways is the Monty Hall question, should you change your decision once the goat is revealed? Along the same lines, the birthday paradox and its explanation are both easy to communicate and to understand, and the resulting confusion much more easy to work around.&lt;/p&gt;&#xA;" OwnerUserId="6724" LastActivityDate="2018-12-22T22:55:44.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14968" PostTypeId="1" CreationDate="2018-12-23T03:30:32.967" Score="3" ViewCount="213" Body="&lt;p&gt;I'm teaching a college algebra course and I'm trying to design a few projects that involve modeling with quadratic functions. So far I have two ideas that involve downward-facing parabolas (projectile motion and the profit function given a linear demand curve).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to design a third project that involves some real-world application of an upward-facing parabola, preferably something where the minimum has some interesting physical interpretation.&lt;/p&gt;&#xA;" OwnerUserId="21" LastEditorUserId="2083" LastEditDate="2018-12-31T08:10:25.830" LastActivityDate="2018-12-31T08:10:25.830" Title="Quadratic modeling project with upward-facing parabola" Tags="|undergraduate-education|functions|projects|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14969" PostTypeId="2" ParentId="14968" CreationDate="2018-12-23T04:42:29.960" Score="2" Body="&lt;p&gt;One example that comes to mind is modelling the position of a diver (or of a diver's head).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; be the time elapsed since jump.&#xA;Let &lt;span class=&quot;math-container&quot;&gt;$d(t)$&lt;/span&gt; be the diver's distance from the water and define &lt;span class=&quot;math-container&quot;&gt;$d(t)&amp;gt;0$&lt;/span&gt; to be &quot;above water&quot; and &lt;span class=&quot;math-container&quot;&gt;$d(t)&amp;lt;0$&lt;/span&gt; to be below water. (It should be obvious why &lt;span class=&quot;math-container&quot;&gt;$d(t)=0$&lt;/span&gt; represents being &quot;at the surface&quot;). If we use a quadratic model, then:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$d(t)=a(t-h)^2+k$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is a positive parameter (to get an upward parabola), &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is the time at which the diver reaches the lowest point and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; indicates the furthest distance reached below the surface.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An even better model would be to have 2 parabolas (and make a function defined by parts), one that is downside for the initial jump and one that is upside for the rest of the movement (falling, going through the water and resurfacing).&lt;/p&gt;&#xA;" OwnerUserId="9341" LastActivityDate="2018-12-23T04:42:29.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14970" PostTypeId="1" CreationDate="2018-12-23T05:48:28.683" Score="2" ViewCount="193" Body="&lt;p&gt;Generally, in a Geometry test, you'd need to test proofs (Prove that triangle XYZ and ABC are congruent). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, proofs depends depends on theorems which depend on postulates, which are annoying (and ultimately useless) to memorize. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, when I learned Geometry, I had to memorize that postulate 13 is called the &quot;Parallel Postulate&quot; (If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, I didn't remember any of that, I had to Google it. That's why I feel that learning Geometry like this is useless. On the other hand, you do need to somehow document your steps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do others do it?&lt;/p&gt;&#xA;" OwnerUserId="11713" LastActivityDate="2018-12-25T06:32:06.793" Title="How to give a good Geometry test?" Tags="|geometry|testing|" AnswerCount="3" CommentCount="10" ClosedDate="2019-01-01T21:27:38.433" ContentLicense="CC BY-SA 4.0" />
  <row Id="14971" PostTypeId="2" ParentId="14968" CreationDate="2018-12-23T14:49:14.223" Score="0" Body="&lt;p&gt;Answer:  Air friction often modeled as a square (comes from compression of air in front of you).  I think there are some hyper complicated formulas that add even higher terms, but square is really common. If you want to get fancy, can add a linear term for sliding and/or rolling friction.  You can look at terminal velocity for a falling object or at automobile travel as common real world examples.  I guess it depends on your point of view with downward facing, but to me, the force is increasing (e.g. if in a car, takes more gas pedal (&quot;accelerator pedal&quot;) to match the force.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long comment:  I worry when teachers push &quot;projects&quot;.  Think it appeals to them as interesting content but is not necessarily best approach pedagogically.  &quot;Drill&quot; (much reviled) is often more effective pedagogically.  When I say interesting, I mean interesting TO THE INSTRUCTOR, not the students.  Recall that for the students the material is new anyways (or if repeating course, is challenging).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular if you are teaching college algebra in college, you are likely dealing with weaker kids (by selection process). I would think of experimenting in pedagogy or just make your own internal challenge to increase class performance versus last year on some semi-objective reference to give yourself motivation.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Watch the movie Stand and Deliver (note the different drills as well as the progressive improvement in results by year) and read the Jay Matthews biography.  Read the Escalante article in JNE:  &lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED345942.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://files.eric.ed.gov/fulltext/ED345942.pdf&lt;/a&gt; And it doesn't just need to be him (and he evolved and changed and improved over time, so not a static reference).  But look at other people, flipped classroom ideas, old salts, whatever.  You could even run some little study or publish something.  Do an A/B trial across two sections for instance.  (Sample size may not be statistically great, but still it really gets you thinking.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[This is a general concern I have with the content uber pedagogy attitude.  But, in your case, I actually found a thread on MSE where you specifically said you were bored with teaching.]  &lt;/p&gt;&#xA;" OwnerUserId="11700" LastActivityDate="2018-12-23T14:49:14.223" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14972" PostTypeId="2" ParentId="10895" CreationDate="2018-12-23T15:38:52.243" Score="3" Body="&lt;p&gt;It's impossible (like the joke about the Maine directions...you can't get there from here).  What I mean is you can't really communicate this idea kindly to your colleagues.  Furthermore, however nicely you get it across to your colleagues is not going to have the effect you want (to change their behavior).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, there is a little bit of a muddle here.  Do you need to &quot;tell your colleagues&quot; or talk to the students?  Really the students are your objective (and you control communications in your class).  So instead of &quot;how do I tell my colleagues&quot;, the question is how do I tell my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of your students, the easy way to communicate the difference in terms is by analogy to football players.  Teachers of math are like coaches of sports. (Even if some coaches/teachers play, the activity of coaching/teaching is different than playing/researching.)  You can even continue the analogy to pro football players (and note the small numbers of pro players versus school players...similar to small number of pro mathematicians.)    Bottom line is teaching versus doing.  In addition to research of course there are some &quot;doing&quot; aspects of math like calculating things, cracking codes, statistical modeling.  I think if you just differentiate coaching versus playing, teaching versus doing (to your students), it makes the point.  And make the conceptual point, but don't dwell on the &quot;this is the only way to use the word 'mathematician' and your other teachers were wrongity wrong-wrong&quot; picayune terminology aspect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other thing is to maybe give them some aspect/glimpse of what advanced math is about (and not terms like Sobolov spaces and manifolds and Lie group crap, but simpler concepts like &quot;these are the courses after calculus in the engineering track&quot;, &quot;these are the ones in the math track&quot;, &quot;these are the major schools of research math (analysis, algebra)&quot;, &quot;these are the major areas in applied math (stats, numerical methods, etc.)&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Realistically VERY FEW of your students will go on to get a BS in math, let alone a Ph.D. and then research activities.  So don't spend a bunch of time on trying to inspire the few.  Drop a few pearls, maybe, but limit the time/effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as some kid saying something wrong, don't let it get to you.  There are even people wrong on the Internet.  And sometimes I try to pin them down on the figurative high school wrestling mat of forum debates, but they never admit they were wrong.  And they're probably not worth the effort either.  ;-)&lt;/p&gt;&#xA;" OwnerUserId="11700" LastActivityDate="2018-12-23T15:38:52.243" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14973" PostTypeId="2" ParentId="10895" CreationDate="2018-12-23T16:15:58.337" Score="10" Body="&lt;p&gt;My favourite definition of a mathematician is the one given by Ian Stewart in his book &lt;em&gt;Letters to a young mathematician&lt;/em&gt; (on page 32).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is Mathematics? In despair, some have proposed the definition &amp;quot;Mathematics is what mathematicians do.&amp;quot; And what are mathematicians? &amp;quot;People who do mathematics.&amp;quot; This argument is almost Platonic in its circularity. But let me ask a similar question. What is a businessman? Someone who does business? Not quite. It is someone who &lt;em&gt;sees opportunities&lt;/em&gt; for doing business when others might miss them.&lt;/p&gt;&#xA;&lt;p&gt;A mathematician is someone who sees opportunities for doing mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I like this definition because when I stop to consider what makes a mathematician in this light, then I am forced to ask what it means to &lt;em&gt;see an opportunity&lt;/em&gt; for doing mathematics. I am sure it will be a revelation to many students that this is a valid way of looking at mathematics and mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;I am also confident that the businessman example will resonate with them, because it immediately clicked for me. I have absolutely no business sense in me, and I can clearly recognise it in those people around me who possess it. The idea that one can recognise a mathematician with the same (or similar) sense is an exciting idea when you are first introduced to it.&lt;/p&gt;&#xA;&lt;p&gt;You could also try making an analogy with sportspersons if your students are athletically inclined. For instance, even if a person goes through the motions properly during training, if they do not possess a keen insight on the playing field, then they will come up short against one who has that insight.&lt;/p&gt;&#xA;&lt;p&gt;The final thing that I like about this definition is that by accepting it you ensure that you're not taking the easy way out. The student can no longer say that Mr. X says he's a mathematician, so he is. The student must apply their own judgement based on their own understanding of what &lt;em&gt;seeing an opportunity&lt;/em&gt; for doing mathematics means.&lt;/p&gt;&#xA;&lt;p&gt;And that is the best one can hope for, in my opinion. The students with sufficient mathematical intuition will be able to pick the true ones out, and they are the ones who are ideally placed to take up mathematics later on, anyway. So, rather than hoping to convince everyone that &lt;em&gt;real&lt;/em&gt; mathematics is interesting and not like what your colleagues do, it would be better to place an effective tool in your students' hands that they can apply by themselves to decide what mathematics is and who mathematicians are.&lt;/p&gt;&#xA;" OwnerDisplayName="user7990" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-03-20T10:59:38.117" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14974" PostTypeId="2" ParentId="14970" CreationDate="2018-12-23T16:47:28.900" Score="5" Body="&lt;p&gt;If you don't want your students to have to memorize facts for your tests, then give them open-notes or open-book tests.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want them to memorize some things but not others, then hand out a sheet of notes that you wrote (and which they have seen in advance).&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-12-23T16:47:28.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14975" PostTypeId="2" ParentId="14970" CreationDate="2018-12-23T19:03:40.433" Score="2" Body="&lt;p&gt;You could allow any answer that unambiguously refers to the postulate or theorem even if wording is inexact or the name of the postulate or theorem isn't given.&lt;/p&gt;&#xA;" OwnerUserId="10665" LastEditorUserId="5321" LastEditDate="2018-12-25T06:32:06.793" LastActivityDate="2018-12-25T06:32:06.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14976" PostTypeId="1" CreationDate="2018-12-23T20:46:03.970" Score="19" ViewCount="1954" Body="&lt;p&gt;I know a math teacher who is going to teach a calculus class that will include the shell, disk, and washer methods for calculating volumes. My question is, is there some 3D kit she could use to demonstrate these methods physically? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally for example, the washer method could be demonstrated with a cylinder that has an inner cylinder that could be removed.&lt;/p&gt;&#xA;" OwnerUserId="11715" LastActivityDate="2022-12-29T06:27:25.827" Title="Is there a place to buy physical models to demonstrate the Calculus shell, disk, and washer methods?" Tags="|calculus|" AnswerCount="8" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14977" PostTypeId="2" ParentId="14976" CreationDate="2018-12-23T21:10:31.880" Score="11" Body="&lt;p&gt;It is probably easier to make 3D computer images of such methods, using even something simple like geogebra (or using CAD software if you know how to use it), that can be projected during class, than to obtain functional physical models. The latter can be machined or made using a 3D printer, but my impression is that computer visualizations are more adaptable and adjustable, and it's not clear to me that much is gained by using an actual physical model. In most any university physics or engineering department there is someone who knows how to make such physical models, but probably it is a better use of one's time to learn to use software designed for creating visualizations. (It is also cheaper - 3D printers and 3D printing is expensive - there is free software that can be used). Moreover, computer models can be interactive in ways that physical models simply cannot. The student can see a surface of revolution as it is generated, and so forth, and can tune parameters. Physical models will be made for a (usually fairly small) finite set of surfaces of revolution; with a computer model one can visualize the surface generated by &lt;em&gt;any&lt;/em&gt; admissible curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, some students probably benefit from touching an object. The intersection of two cylinders is good example of a surface for which a computer vizualization is probably less communicative than a physical model, simply because the surface is somehow quite hard to see.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One recommendation would be to look into using something like geogebra. The sorts of models you describe (appropriate for a multivariable calculus class) are not hard to realize in such software and there are already many examples available online. Making quality, usable physical models requires skills and  access to facilities not all of us have, and in this sense computer visualizations are more easily prepared/generated.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2018-12-24T17:13:36.323" LastActivityDate="2018-12-24T17:13:36.323" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14978" PostTypeId="2" ParentId="14976" CreationDate="2018-12-24T02:43:44.043" Score="21" Body="&lt;p&gt;3D printing is an attractive avenue.&#xA;I think there &lt;em&gt;is&lt;/em&gt; something to be gained by actual physical models.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/Ynj3Z.jpg&quot; width=&quot;400&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Image from &lt;a href=&quot;http://home.wlu.edu/~dennee/math_vis.html&quot; rel=&quot;noreferrer&quot;&gt;Elizabeth Denne's webpages&lt;/a&gt;.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;See also Rebecka Peterson's &lt;a href=&quot;http://www.epsilon-delta.org/2015/02/more-volumes-in-calculus-student-edition.html&quot; rel=&quot;noreferrer&quot;&gt;Epsilon-Delta blog&lt;/a&gt; for low-tech alternatives:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/AuUwD.jpg&quot; width=&quot;400&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Student volume models based on cross-sections.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-12-25T02:04:14.093" LastActivityDate="2018-12-25T02:04:14.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14979" PostTypeId="2" ParentId="14976" CreationDate="2018-12-24T03:04:24.280" Score="2" Body="&lt;p&gt;I suppose, if you wish to illustrate the washer method then perhaps slicing a pear (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Pear&quot; rel=&quot;nofollow noreferrer&quot;&gt;here for a picture&lt;/a&gt; into 10 or so horizontal slices would give you bunch of near washers with which you could estimate the volume of a pear. Then, you could even estimate the error in the calculation by the water displaced by the uncut pear. Ideally, you'd like a near circular pear, but you could estimate a mean radius if it was lumpy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For cylindrical shells, I have a less fruity idea involving foam sheets held together by large rubber bands around some center cylinder. But, I haven't had a chance to try it out myself yet. Onions are almost good, but the layers are too curved. If you could find some cylindrical onions. See, someone needs to grow cylindrical onions. Perhaps there is some other easily obtained food which illustrates the shell method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Actually, the foam sheets I had in mind were way to slim to make this plausible. Perhaps Bologna would work for the right shape. It has a regular thickness and it easy to cut. I'm not sure how to arrange it to stay together. &lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2018-12-24T23:58:47.197" LastActivityDate="2018-12-24T23:58:47.197" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14980" PostTypeId="2" ParentId="14970" CreationDate="2018-12-24T06:02:16.737" Score="5" Body="&lt;p&gt;Some ideas:&#xA;a) Consider giving extra questions and the student can pick eg 5 out of 6 or 5 out of 7. The advantage is that if the students forgets a postulate or theorem, they can pick a different question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) Don't insist on using postulate names, but allow either postulate names or the wording of the theorem and be flexible about the exact wording. (as in one the other answers)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) Speak to your colleagues or past geometry teachers in your school. They will have the best sense of what works in your population.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;d) I don't believe it's useless to memorize theorems just because you forget them when you no longer use them.  If students are doing classwork and homework, they should have the postulates and theorems at their fingertips without actually having to sit and memorize them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;e) Consider the idea (from the previous answer) of giving students a list of theorems to use during the test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck and please edit the question to tell us what level you teach and in what country.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2018-12-24T06:02:16.737" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14981" PostTypeId="2" ParentId="14976" CreationDate="2018-12-24T13:02:39.730" Score="3" Body="&lt;p&gt;Another substitute available nowadays: computer-animated videos.  They should be just as convincing as physical models.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Search &quot;shell integration animated&quot; on YouTube to start.  You may have to search many of the results to find one that you want to use.&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2018-12-24T13:14:56.410" LastActivityDate="2018-12-24T13:14:56.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14983" PostTypeId="2" ParentId="689" CreationDate="2018-12-25T04:44:59.887" Score="1" Body="&lt;p&gt;It's actually a very hard topic to teach. We don't want to teach them the Naive set theory definition of a cardinal number because Naive set theory is contradictory. In Zermelo-Fraenkel set theory, it's easy to define the property of there existing a bijection from one set to another set but the axiom of choice is not provable so it's very hard to define an object for each set such that two sets are assigned the same object when there's a bijection from one set to the other although that can be done using Scott's trick. Alternatively, you could create a provably consistent extension of ZF where you just declare that the cardinality of each set is a set such that two sets have the same cardinality when there is a bijection from one to the other. With either way of introducing them, you can show that there is in fact no set of all cardinal numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ordinal numbers are even easier to teach. First we have all the finite ordinals. Next comes &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;. Now starting from it, you can apply the successor function as many times as you want. The next larger ordinal is &lt;span class=&quot;math-container&quot;&gt;$\omega \times 2$&lt;/span&gt;. The supremum of all the ordinal numbers that can be expressed as &lt;span class=&quot;math-container&quot;&gt;$\omega \times$&lt;/span&gt; a finite ordinal is &lt;span class=&quot;math-container&quot;&gt;$\omega \times \omega$&lt;/span&gt;. If you then again apply the successor function as many times as you want, the next ordinal after all of them is &lt;span class=&quot;math-container&quot;&gt;$(\omega \times \omega) + \omega$&lt;/span&gt;. We also have &lt;span class=&quot;math-container&quot;&gt;$(\omega \times \omega) + (\omega \times 2)$&lt;/span&gt; We can keep going and get &lt;span class=&quot;math-container&quot;&gt;$(\omega \times \omega) + (\omega \times \omega)$&lt;/span&gt; which can also be written as &lt;span class=&quot;math-container&quot;&gt;$(\omega \times \omega) \times 2$&lt;/span&gt;. We can also add to that any ordinal number less than &lt;span class=&quot;math-container&quot;&gt;$(\omega \times \omega) \times 2$&lt;/span&gt;. The supremum of all ordinal numbers that can be gotten that way is  &lt;span class=&quot;math-container&quot;&gt;$(\omega \times \omega) \times 3$&lt;/span&gt;. The supremum of all ordinal numbers that can be expressed as &lt;span class=&quot;math-container&quot;&gt;$(\omega \times \omega) \times$&lt;/span&gt; a natural number is &lt;span class=&quot;math-container&quot;&gt;$(\omega \times \omega) \times \omega$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is really a totally different concept of an ordinal number than the ZF definition and probably easier to teach than ZF. It includes only those ordinal numbers that are small enough to use to create a proof system of pure number theory. By Godel's incompleteness theorem, you can always think of proof systems corresponding to larger ordinal numbers than you could have before. Yet, those ordinal numbers will always be smaller than the Church-Kleene ordinal. How can that be when there's no end to constructing ordinal numbers useable to create a proof system? Because the Church-Kleene ordinal is unfathomably large. Once you create a system of ordinal numbers useable in a proof system, you still don't have all of the ones smaller than the Church-Kleene ordinal so it doesn't necessarily mean that when you make a major extension to that system, you will get one large enough to include the Church-Kleene ordinal. This last paragraph can probably only be understood by adults and probably is not worth teaching to elementry school kids. I wrote it just to explain how the concept of ordinal numbers they could be taught really is a completely different concept from the ZF definition and that concept is the one that should be taught.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="11718" LastEditDate="2019-01-22T02:01:54.713" LastActivityDate="2019-01-22T02:01:54.713" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14984" PostTypeId="1" AcceptedAnswerId="14986" CreationDate="2018-12-25T06:33:09.223" Score="-4" ViewCount="460" Body="&lt;p&gt;Some students really struggle to learn fractions. Not only that but also, once they've mastered an understanding of real numbers, they can learn about fractions so much faster and more efficiently later. Maybe teaching real numbers renders teaching rational numbers totally unnecessary. In addition to that, teaching rational numbers too early might cause some people to form the misconception that rational numbers are the only numbers. I think so because I read on the internet about people who don't understand how irrational numbers exist. &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7718/effects-of-early-study-of-advanced-books&quot;&gt;This question&lt;/a&gt; and &lt;a href=&quot;https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page&lt;/a&gt; seem to support my theory. Also when they omit unnecessary material, they can fit in more other material while still moving slowly enough that all the students can actually keep up with learning what school is trying to teach them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may seem wierd but it might be better to use the base 2 notation for the fractional part of a real number and the base 10 notation for the integer part because of they way they construct the real numbers. Real numbers could be taught as follows. First we can define a natural number as a finite ordinal number. Next, we invent the negative numbers and then redefine +, &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; on them. Next, since each odd number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is not a solution to &lt;span class=&quot;math-container&quot;&gt;$2 \times y = x$&lt;/span&gt; in the integers, we invent a solution to that equation in &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;. I know that's how I say it but for them, it's better not to introduce a variable and just say that none of them get you that number when you multiply 2 by it. Let's call each invented solution a half integer. Each half integer &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is still not a solution to &lt;span class=&quot;math-container&quot;&gt;$2 \times z = y$&lt;/span&gt; so we can again invent a solution to each of them. Now it's easier to define +, &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; on this system than it is to teach fractions and how to multiply and divide them and determine which of two is greater. Some people may quickly figure out that not all numbers can be gotten by multiplying a number by 3 in this system and get confused but the teacher might just have to explain that that's how the system was defined and that they will later teach them a different system where there is a solution to &lt;span class=&quot;math-container&quot;&gt;$3 \times x = 1$&lt;/span&gt;. They can later be taught the concept of my definition of a Dedekind cut of the dyadic rationals which is not the real definition and my definition is a subset of them that has the property that it is not empty and its complement is not empty and for any dyadic rational in the subset, all smaller dyadic rationals are in the subset. They might start to notice that for some cuts, that cut has a maximal element and for some cuts, its complement has a minimal element and for some cuts, it has no maximal element nor does its complement have a minimal element. Now there's on obvious one-to-one correspondence from the cuts between 0 and 1 to all the functions from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt; to {0, 1}. However, we want to invent a new number for the cut only when there isn't already a maximal element of the cut or a minimal element of its complement. Now this gives an obvious binary notation for the fractional part of each real number but that notation forbids a string of trailing 1's just like some authors forbid a string of trailing 9's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can then redefine +, &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; on this system. - and &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; in this system are just defined in terms of + and &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; in this system. We can show that multiplication can be defined in that way in that system and that in that system, multiplication by any nonzero number is bijective and squaring restricted to the nonnegative numbers is also bijective on the nonnegative numbers. Now that they already know a lot of the laws of real numbers used as some of the defining criteria for a complete ordered field, all we need to do is teach them how to divide any real number by any nonzero real number and then say a real number is defined to be a rational number if and only if for some integer &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and nonzero integer &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;, it is &lt;span class=&quot;math-container&quot;&gt;$p \div q$&lt;/span&gt;. Then they might quickly figure out so many properties of rational numbers that some students are struggling to learn.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="11718" LastEditDate="2019-01-15T04:42:05.027" LastActivityDate="2019-01-15T04:42:05.027" Title="Could schools jump straight into teaching real numbers first then teaching fractions later?" Tags="|primary-education|fractions|" AnswerCount="2" CommentCount="29" ContentLicense="CC BY-SA 4.0" />
  <row Id="14985" PostTypeId="2" ParentId="14984" CreationDate="2018-12-25T17:27:55.187" Score="3" Body="&lt;p&gt;I'll give a brief defense of the somewhat conventional view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I understand history correctly, part of what made Newton's advance in calculus possible was the (then) recent introduction of the decimal expansion of a real number. Calculation to arbitrary precision was something that everyone could both do and easily communicate with the introduction of decimal expansion. Lexographic ordering of decimal expansions is simple to see. In contrast, comparing rational numbers is far more complicated. Even if you limit the scope to a particular class of rationals. The manifest ordering of decimal expansions cannot be beat for kids. In addition, the ease of multiplication and division by 10 and the whole of scientific computation. One can learn all this and still fail to understand the construction of negative numbers, much less the construction of the real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, the idea of a decimal representation is quite natural if the student has an intuition for length and/or direction. Surely many have such intuition. Therefore, I would argue, formulating real numbers as their decimal representations is a quick way to get very far. Furthermore, while questions of non-unique representation and convergence lurk under the surface, the problem of calculation in decimals is worthy of every student's attention. Rationals, negatives, the whole line is there. You just have to grapple with arithmetic of decimal expansions. Of course, the problem of justifying decimal expansions requires significant effort. But, that is the beauty. All that effort is wrapped up into something kids should and can learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in summary, I would say your idea is intriguing, but I would save it for an abstract algebra class. For children, the usual material is already quite good if it is taught well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, we come to another significant problem for your proposal. You'd have to find a way to convince elementary math educators to completely up-end their teaching and understanding of real numbers. Many of these teachers are quite poor at abstract algebra, as such getting them to undertake a reformulation which is abstract-algebraic at its core is a really hard sell.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2018-12-25T17:27:55.187" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14986" PostTypeId="2" ParentId="14984" CreationDate="2018-12-26T05:46:06.627" Score="7" Body="&lt;p&gt;I appreciate you raising the question of improving the teaching of fractions which is certainly needed.  I don't feel optimistic about your suggestion.  I taught gifted elementary math students for over 25 years.  I have a degree in math from MIT.  I have no sense of how I would present your suggestions to my very gifted students.  The elementary classroom teachers I worked with have problems with each change in the math curriculum and very little math background.  I don't see how your ideas could work in the regular classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Children bake with fractions, measure with fractions, and make sense of decimal arithmetic using their understanding of fractions. When I introduce decimal multiplication and division, they understand it only because of the work we've done with fractions.  My feeling is teaching fraction first has a great value. However, I agree with you that students struggle with many aspects of fractions and it would be great if they could be taught better by everyone.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2018-12-26T05:46:06.627" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14988" PostTypeId="1" CreationDate="2018-12-26T13:56:04.780" Score="4" ViewCount="1382" Body="&lt;p&gt;I am seeking a list of topics that students taking calculus should memorise. Some topics from Calculus I might include:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\varepsilon-\delta$&lt;/span&gt; definition of limit;&lt;/li&gt;&#xA;&lt;li&gt;Definition of the derivative of a function;&lt;/li&gt;&#xA;&lt;li&gt;Product rule, quotient rule and chain rule for derivative;&lt;/li&gt;&#xA;&lt;li&gt;Fundamental theorem of calculus;&lt;/li&gt;&#xA;&lt;li&gt;Definition of Riemann sum.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This list is far from being complete. Here I am asking for a more comprehensive list that covers topics in Calculus I, II and III. Thank you!&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorDisplayName="user507" LastEditDate="2018-12-27T01:35:48.143" LastActivityDate="2019-03-13T06:29:50.847" Title="Recommended list of things calculus students should be required to memorise?" Tags="|undergraduate-education|calculus|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14989" PostTypeId="2" ParentId="36" CreationDate="2018-12-26T14:17:00.207" Score="4" Body="&lt;p&gt;I made the following website with the aim of producing a Desmos-like experience in 3D for my multivariable calculus students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math3d.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;math3d.org&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can use &lt;a href=&quot;https://www.math3d.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;math3d.org&lt;/a&gt; to create simple surface plots or complex, animated visualizations. Some features:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Create and animate points, lines, vectors, curves, surfaces (explicit &amp;amp; implicit), and vecotr fields&lt;/li&gt;&#xA;&lt;li&gt;Intuitive math input (powered by same library, MathQuill, as Desmos)&lt;/li&gt;&#xA;&lt;li&gt;Save and share your scenes&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Here are three scenes that I particularly like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.math3d.org/sliders_intro&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simple Surface Plot&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.math3d.org/motion&quot; rel=&quot;nofollow noreferrer&quot;&gt;Parametric Curves, Velocity and Acceleration&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.math3d.org/horizontal_revolution_washer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes of Revolution, Shell Method&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.math3d.org/ruled_hyperboloid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hyperboloids as a Ruled Surface&lt;/a&gt; (+screenshot) &lt;a href=&quot;https://i.stack.imgur.com/IpG1b.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IpG1b.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This project is &lt;a href=&quot;https://github.com/ChristopherChudzicki/math3d-react&quot; rel=&quot;nofollow noreferrer&quot;&gt;on Github&lt;/a&gt;. If you find bugs or have ideas for improvements, please &lt;a href=&quot;https://github.com/ChristopherChudzicki/math3d-react/issues&quot; rel=&quot;nofollow noreferrer&quot;&gt;open an issue&lt;/a&gt;!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: This answer is mostly a re-post of my answer to a similar question &lt;a href=&quot;https://math.stackexchange.com/questions/690632/calculator-similar-to-desmos-but-for-3d/2235881#2235881&quot;&gt;Calculator Similar to Desmos but for 3D&lt;/a&gt; on Math StackExchange.&lt;/p&gt;&#xA;" OwnerUserId="1341" LastEditorUserId="1341" LastEditDate="2018-12-26T14:36:44.083" LastActivityDate="2018-12-26T14:36:44.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14990" PostTypeId="2" ParentId="14988" CreationDate="2018-12-26T15:40:11.693" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Derivatives of common functions (sin, cos, polynomial, exponential, log, etc.)  These should be at your fingertip versus needing to rederive them or refer to a table.  Since you will use them in harder problems (e.g. with chain rule).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Most of the common function antiderivatives follow from (1), but there may be a few that is worth memorizing or have used so often they are in working memory.  Sorry, can't recall which.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Formula for integration by parts.  (Yes, you could derive it but should have it in working memory like you do quadratic formula versus completing the square.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Formula for arc length.  Personally, I struggled to derive it under duress.  But you should probably force yourself to derive it occasionally also so it is not just a magical formula.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The concept of using tan theta over two substitution.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is no formula to remember with partial fractions, per se, (I think it is intuitive) but just having done enough problems to be confident to rock out the algebra.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Formula for integration of volume by discs.  Personally, this formula is rather intuititive so should not be hard to keep in memory.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[During a course and shortly after it)] formulas for common quadratic and radical forms.  Don't think these need to be retained for the rest of your engineering career though.  Can refer to tables.  But worth having had them in working memory once.  Makes using tables easier in the future and they do crop up.  But at least you can say &quot;oh I remember that now&quot; having once memorized them.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Definitions of div, grad, curl (and all that)  [Sorry can't remember much more from calc 3.  Liked calc 2 and ODE more.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="11700" LastActivityDate="2018-12-26T15:40:11.693" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="14991" PostTypeId="1" CreationDate="2018-12-26T16:12:56.053" Score="7" ViewCount="301" Body="&lt;p&gt;I am analyzing some data on ALEKS for my home institution (ALEKS, an acronym that stands for &quot;&lt;em&gt;Assessment and Learning in Knowledge Spaces&lt;/em&gt;&quot;, is an online tutoring and assessment program that includes course material in mathematics and many of the sciences). ALEKS has an adaptive assessment program that is used by many institutions of higher learning for placement purposes. I am currently in the process of analyzing the assessment scores from ALEKS for placement into our Calculus 1 class. The way ALEKS was administered was that the students were allowed to take the placement test multiple times and given the chance to improve their background (i.e. remediate) using ALEKS in between test attempts. My data set consists of the ALEKS placement test scores and the total time spent in remediation with ALEKS (for each student).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to find a ballpark estimate for the expected rate of improvement (in terms of number of new topics learned or % increase in score) a student should see from remediation. I would like to have some benchmark with which to compare the rate of improvement from my data set. In other words, are my students doing better, worse , or about the same as expected using the remediation available via ALEKS?&lt;/p&gt;&#xA;" OwnerUserId="1877" LastEditorUserId="2083" LastEditDate="2018-12-30T14:27:22.410" LastActivityDate="2018-12-30T14:27:22.410" Title="Measure of Improvement in Math Skills from Remediation with ALEKS" Tags="|undergraduate-education|assessment|online-instruction|community-colleges|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14992" PostTypeId="2" ParentId="9907" CreationDate="2018-12-26T22:02:46.180" Score="2" Body="&lt;p&gt;I see many good points here, addressing issues going forward. But I have a lower-level reason. I teach point slope form in preference to y=mx+b because it has better flow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When they use y=mx+b, I have seen students plug in a point to get b, and then stop. They forget to give the equation of the line, which is what they were trying to find. When they use point slope form, it flows better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try to help students see why they would want to use each form, and how one form can transform into another.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2018-12-26T22:02:46.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14993" PostTypeId="2" ParentId="14991" CreationDate="2018-12-27T02:32:03.240" Score="3" Body="&lt;p&gt;I don't think you are going to find  data yielding an &quot;expected rate of improvement.&quot; Much depends on the circumstances of how ALEKS is deployed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, one can find academic papers on the effectiveness of ALEKS, even if they&#xA;do not answer your question directly. For example, (1) says that &quot;ALEKS had a significant effect on students‘ mathematics achievement&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(1) Nwaogu, Eze. &quot;The effect of ALEKS on students' mathematics achievement in an online learning environment and the cognitive complexity of the initial and final assessments.&quot; Dissertation, Georgia State University, (2012). &#xA;  &lt;a href=&quot;https://scholarworks.gsu.edu/cgi/viewcontent.cgi?article=1098&amp;amp;context=msit_diss&quot; rel=&quot;nofollow noreferrer&quot;&gt;GSU PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Another study (2) says that&#xA;&quot;students randomly assigned to the ALEKS condition significantly out-performed students assigned to the teacher condition on a state assessment test (TCAP). However, this was only if the students received sufficient exposure to the program&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(2) Craig, Scotty D., Celia Anderson, Anna Bargagloitti, Arthur C. Graesser, Theresa Okwumabua, Allan Sterbinsky, and Xiangen Hu. &quot;Learning with ALEKS: the impact of students’ attendance in a mathematics after-school program.&quot; In &lt;em&gt;International Conference on Artificial Intelligence in Education&lt;/em&gt;, pp. 435-437. Springer, Berlin, Heidelberg, 2011.&#xA;  &lt;a href=&quot;https://www.researchgate.net/publication/221297524_Learning_with_ALEKS_The_Impact_of_Students&amp;#39;_Attendance_in_a_Mathematics_After-School_Program&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conference link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We have used ALEKS at my institution, with as-yet inconclusive results.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2018-12-27T12:22:03.750" LastActivityDate="2018-12-27T12:22:03.750" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14995" PostTypeId="1" CreationDate="2018-12-27T07:42:43.910" Score="4" ViewCount="682" Body="&lt;p&gt;I am tutoring a student who is in an honors Algebra II class. The class is definitely advanced and the student hasn't been exposed to this kind of material. The teacher is going beyond algebra II and into calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The major issue I have with the student is how incredibly lazy he is. He takes way too long to solve a problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The teacher normally posts a review sheet which I complete with the student. I additionally come up with several mock exams and give it to the student to complete. During the session I have trouble because as soon as he doesn't understand something, he asks me. I have tried to coach him in the right direction without giving answers, I've also flat out refused to give him answers. But in either case, if he doesn't understand it, he doesn't &lt;em&gt;try&lt;/em&gt; anything. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For his first couple exams, he scored in the 70's. The third exam he took, he knew everything on the exam, it was very easy (by his own admission) however when he took the test, he just froze. This happens constantly. I thought this was more of a psychological thing than anything so I started making mock exams to create a similar environment for him to work in but he is always getting distracted. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the exams, he constantly runs out of time. However on his 4th exam, he scored 28/30 which is excellent and he only got two points off for a silly error. I thought this was great progress and I was so happy. His most recent exam(the fifth one), I prepared him for about 10-15 hours total. I created multiple exams, I went through the review sheet. He got back his exam and he scored a 19/30. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm devastated, I don't know what more I should be doing as a tutor. It's really difficult for me to even understand what is going on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have also noticed that when he knows how to do a problem, he is very focused (as you would expect). However even if he understands a concept but hasn't been exposed to the specific wording or presentation of a problem, he just shuts down. I try my best to come up with questions that are similar to the review sheet however he still needs to know how to think. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know he's in high school but I give small and simple proofs for certain things because I want him to know why and not have it be a matter of memorization. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please help with any suggestions on what I can do moving forward and how I can best utilize my time and efforts working with my student. Thank you, any feed back is greatly appreciated!&lt;/p&gt;&#xA;" OwnerUserId="11725" LastEditorDisplayName="user507" LastEditDate="2018-12-28T02:08:57.717" LastActivityDate="2018-12-28T18:37:13.917" Title="How can I help a student who is constantly struggling in an honors Algebra II (high school) class" Tags="|secondary-education|exams|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14996" PostTypeId="2" ParentId="13233" CreationDate="2018-12-27T18:45:44.830" Score="2" Body="&lt;p&gt;Here's how I did mental multiplication. When I was in my grade 9 math class, my teacher was asking single digit multiplications and I keep getting the answer really quickly and I think I kept being the first to answer the question because I had already memorized the single digit multiplication table. I'm not sure whether I made the conscious effort to memorize it. Later in 2003 while I was at Magic Camp located at Camp White Pine, I was getting asked multiplication questions. I think some of them were 3 digits by 3 digits and I mentally calculated the answer by the following method. I applied the property that &lt;span class=&quot;math-container&quot;&gt;$\forall a \in \mathbb{R}\forall b \in \mathbb{R}\forall c \in \mathbb{R}\forall d \in \mathbb{R}\forall e \in \mathbb{R}\forall f\in \mathbb{R} (100a + 10b + c) \times (100d + 10e + f) = 10,000ad + 1,000(ae + bd) + 100(af + be + cd) + 10(bf + ce) + cf$&lt;/span&gt; I wrote that as a 5 term sum. I treated the digits like those variables and added the last term first then the second last term and retained what I got so far and kept going. I think I got the right answer to so many multiplication questions.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastActivityDate="2018-12-27T18:45:44.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14997" PostTypeId="2" ParentId="13233" CreationDate="2018-12-28T02:49:28.630" Score="2" Body="&lt;p&gt;As was already mentioned by @SCS, Art Benjamin is an exemplar:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://blog.ted.com/turn-arithmetic-into-mathemagic-arthur-benjamin-at-tedglobal-2013/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Turn arithmetic into mathemagic&lt;/a&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here you cannot but marvel at: &lt;span class=&quot;math-container&quot;&gt;$388 \times 388 = 150544$&lt;/span&gt;,&#xA;which he answers instantaneously, even under the pressure of &#xA;Colbert on national TV:&#xA;&lt;a href=&quot;http://www.cc.com/video-clips/suw63r/the-colbert-report-arthur-benjamin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Colbert&lt;/a&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Colbert checks him, and verifies the answer. But I think Art answered&#xA;&lt;span class=&quot;math-container&quot;&gt;$388 \times 3\textbf{8}8$&lt;/span&gt; while the posed question was &lt;span class=&quot;math-container&quot;&gt;$388 \times 3\textbf{9}8$&lt;/span&gt; instead. Easy to confuse in that environment.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-12-28T02:49:28.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14998" PostTypeId="2" ParentId="14995" CreationDate="2018-12-28T06:01:15.567" Score="2" Body="&lt;p&gt;It sounds like you are frustrated by his inconsistent performance and your inability to address what he needs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Inconsistent performance is often a sign of a learning disability.  Can he be evaluated by the Dept of Education to see if he needs an IEP (individualized education plan)? In the US (not sure where you are from), they would identify his learning issues and suggest modifications including more time.  You would have a better idea of what was going on and know how to address the issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would also be worthwhile to discuss his inconsistent performance with his teacher and perhaps his previous math teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also suggest you talk to him about his shutting down when he is not in the middle of a problem and has already shut down.  He will be more open to listening when he is not struggling with the problem.  Tell him the best way to learn is to continue even when he has no idea how.  Ask him to help you find strategies so you help him work through the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2018-12-28T06:01:15.567" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14999" PostTypeId="1" AcceptedAnswerId="15004" CreationDate="2018-12-28T10:28:52.970" Score="10" ViewCount="618" Body="&lt;p&gt;I want to ask a question that causes confusion. In the trigonometry, we use some units of measure of angle: degree and radian. Which is/are correct?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 180^{\circ} = \pi $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 180^{\circ} = \pi \ \  \ \text{radians}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the other words: Is &lt;span class=&quot;math-container&quot;&gt;$180^\circ$&lt;/span&gt; a real number?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My opinion &lt;span class=&quot;math-container&quot;&gt;$180^\circ$&lt;/span&gt; is real number like &lt;span class=&quot;math-container&quot;&gt;$3.14$&lt;/span&gt;. Thanks for your comments/answers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Some my friends high school mathematics teachers present in the form of a real number of &lt;span class=&quot;math-container&quot;&gt;$180^\circ$&lt;/span&gt; in the class, my other friends comment: &quot;No, this is not a number. Because they have units&quot;. I said that '' We can calculate &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt; function at the point &lt;span class=&quot;math-container&quot;&gt;$x = \pi/4$&lt;/span&gt;. Hence this argument tells us that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a real number. Because, the sine function is defined in real numbers''. Also, there are a third group. They says ''Don't worry about these! It doesn't matter what this concept is, we can make a calculation and go on our way''.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We could not agree on how to present this concept to students correctly.&lt;/p&gt;&#xA;" OwnerUserId="6372" LastEditorUserId="6053" LastEditDate="2019-07-14T14:25:41.763" LastActivityDate="2019-07-17T08:52:41.487" Title="Is $180^\circ = \pi$?" Tags="|trigonometry|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15000" PostTypeId="2" ParentId="14999" CreationDate="2018-12-28T16:23:19.007" Score="5" Body="&lt;p&gt;Both &lt;span class=&quot;math-container&quot;&gt;$180$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\pi/4$&lt;/span&gt; are real numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Units are an extramathematical notion related to measurement and indicating how to interpret a real number in some given (for example physical) context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can speak of &lt;span class=&quot;math-container&quot;&gt;$\pi/4$&lt;/span&gt; degrees and &lt;span class=&quot;math-container&quot;&gt;$180$&lt;/span&gt; radians just as one speaks of &lt;span class=&quot;math-container&quot;&gt;$\pi/4$&lt;/span&gt; radians and &lt;span class=&quot;math-container&quot;&gt;$180$&lt;/span&gt; degrees. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One definition of the sine function is that it is the unique solution of the initial value problem, &lt;span class=&quot;math-container&quot;&gt;$\tfrac{d^{2}x}{dt^{2}}= -x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x(0) = 0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\tfrac{dx}{dt}(0) = 1$&lt;/span&gt; (cosine is defined similarly, with different initial conditions). What might be called the natural units of this solution, &lt;span class=&quot;math-container&quot;&gt;$x(t) = \sin{t}$&lt;/span&gt;, are radians and not degrees. The functions &lt;span class=&quot;math-container&quot;&gt;$s(t) = \sin(\tfrac{2\pi}{360}t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c(t) = \cos(\tfrac{2\pi}{360}t)$&lt;/span&gt; are what what might be called the &quot;sine in degrees&quot; and &quot;cosine in degrees&quot;. However, note that their derivatives no longer satisfy the usual identities. For example, &lt;span class=&quot;math-container&quot;&gt;$s^{\prime}(t)$&lt;/span&gt; is &lt;em&gt;not&lt;/em&gt; the cosine in degrees function, rather it is a multiple of it, namely &lt;span class=&quot;math-container&quot;&gt;$s^{\prime}(t) = \tfrac{2\pi}{360}\cos(\tfrac{2\pi}{360}t) = \tfrac{2\pi}{360}c(t)$&lt;/span&gt;. That is &lt;span class=&quot;math-container&quot;&gt;$c^{\prime}(t) \neq -s(t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s^{\prime}(t) \neq c(t)$&lt;/span&gt;. The function &lt;span class=&quot;math-container&quot;&gt;$s(t)$&lt;/span&gt; solves the initial value problem &lt;span class=&quot;math-container&quot;&gt;$\tfrac{d^{2}x}{dt^{2}}= -\left(\tfrac{2\pi}{360}\right)^{2}x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x(0) = 0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\tfrac{dx}{dt}(0) = \tfrac{2\pi}{360}$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Put another way, although &lt;span class=&quot;math-container&quot;&gt;$(\sin^{\prime}(t), \sin(t))$&lt;/span&gt; parameterizes the unit circle, it is not true that &lt;span class=&quot;math-container&quot;&gt;$(s^{\prime}(t), s(t))$&lt;/span&gt; parameterizes the unit circle. If we work in degrees rather than radian identities such as the Pythagorean identity &lt;span class=&quot;math-container&quot;&gt;$\sin^{2}(t) + \cos^{2}(t) =1$&lt;/span&gt; fail in the sense that it is not true that &lt;span class=&quot;math-container&quot;&gt;$s^{2}(t) + (s^{\prime}(t))^{2}$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, the functions &lt;span class=&quot;math-container&quot;&gt;$(\cos(t), \sin(t))$&lt;/span&gt; in radians yield a unit speed parameterization of the unit circle, whereas the functions &lt;span class=&quot;math-container&quot;&gt;$(c(t), s(t))$&lt;/span&gt; in degrees only yield a constant speed parameterization of the unit circle. The choice of units corresponds to the choice of normalization of the speed of the parameterization. The non-unit speed parameterization is less aesthetically pleasing because it breaks the symmetry of certain standard identities, and practically a bit unwieldy because of the factors of &lt;span class=&quot;math-container&quot;&gt;$\tfrac{2\pi}{360}$&lt;/span&gt; it generates.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2019-07-17T08:52:41.487" LastActivityDate="2019-07-17T08:52:41.487" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15001" PostTypeId="2" ParentId="14731" CreationDate="2018-12-28T18:19:54.047" Score="2" Body="&lt;p&gt;This is an answer about terminology in general. (As always, I'm answering as a learner who has thought about the learning process rather than as a tescher, but I hope it will be relevant.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that in order to learn the true meaning of a term, one must first understand the concept that it refers to. &lt;strong&gt;Ideally in that order:&lt;/strong&gt; if possible, encounter the concept first and &lt;em&gt;then&lt;/em&gt; learn the name for it. (This is the same as when learning a foreign language.) Then when you do learn the name—probably immediately after learning the concept—you're not misled by the common meaning of the word. You already &lt;em&gt;know&lt;/em&gt; the meaning: what you didn't know was the word for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I think I was taught about limits not long after getting used to the idea of asymptotes in the context of curve sketching. First we were shown some examples of sequences and sums like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$S_n=1+\frac12+\frac14+. . .+\frac{1}{2^n}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We spent a bit of time playing with these and learning how to express the sums in shorthand using the &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; sign&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the above example, it was easy to see that increasing &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; made &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt; get closer and closer to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; without ever quite reaching it. And once we'd understood that, we were ready to be told something like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is called the &lt;strong&gt;limit&lt;/strong&gt; of the sequence as  &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; &lt;strong&gt;tends to&lt;/strong&gt; infinity and we write it as &lt;span class=&quot;math-container&quot;&gt;$$\lim_{n\rightarrow \infty }S_n =2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;At this point there was nothing confusing about the word &lt;em&gt;limit&lt;/em&gt; because we were already familiar with what it was describing. &lt;em&gt;Tends to&lt;/em&gt; seemed a slightly odd phrase, but was just an instance of mathematics sometimes using funny language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same went for the notation: we weren't introduced to a strange new symbol followed by a definition to grapple with and misunderstand—we were introduced to something needing the symbol, then to the symbol itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt; I don't think terms which differ from their ordinary usage have to be confusing, provided they're not the first thing we encounter. After you've seen what the limit of a series is, the most obvious English word for it is &lt;em&gt;limit&lt;/em&gt;, so it's not a problem being asked to use that word. The problem arises if we try to get the concept from its name, rather than the other way round.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastEditorUserId="10663" LastEditDate="2018-12-28T18:29:12.520" LastActivityDate="2018-12-28T18:29:12.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15002" PostTypeId="2" ParentId="14995" CreationDate="2018-12-28T18:37:13.917" Score="6" Body="&lt;p&gt;There doesn't actually seem to be any problem here. Your student sounds like a typical intelligent but unmotivated high school student who is getting along fine in school. His test grades sound like they were maybe 75%, 75%, 93%, and 63%. This averages out to about 76%. Raw scores don't mean much, but if you're intending us to assume a 90/80/70 scale, then your student is passing despite not trying very hard. There is nothing abnormal about this. High school is generally pretty easy, so it's common for smart kids to be able to pass without doing much work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there's a problem, perhaps the problem is that his parents are spending a lot of money for many hours of tutoring, when the only issue is that the kid simply doesn't care much.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-12-28T18:37:13.917" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15003" PostTypeId="2" ParentId="14898" CreationDate="2018-12-28T18:42:04.110" Score="0" Body="&lt;p&gt;There's certainly a tradition of dissent from the &quot;ladder&quot; approach to mathematics education. As examples I would recommend &lt;a href=&quot;https://www.maa.org/external_archive/devlin/LockhartsLament.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Mathematician’s Lament [PDF]&lt;/a&gt; and &lt;a href=&quot;http://poibella.org/papers/indefenseofmath.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;In Defense of Mathematics and its Place in Anarchist Education [PDF]&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10423" LastActivityDate="2018-12-28T18:42:04.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15004" PostTypeId="2" ParentId="14999" CreationDate="2018-12-28T18:48:11.273" Score="15" Body="&lt;p&gt;It is possible to treat degrees and radians as units, just as we treat inches and centimeters as units.  Just as we can convert inches to centimeters (1 in = 2.54 cm), we can convert degrees to radians (1 degree = &lt;span class=&quot;math-container&quot;&gt;$\pi/180$&lt;/span&gt; radians).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this point of view overlooks an important fact about radians: The radian measure of an angle is the ratio of two lengths (arc length divided by radius), so as a &quot;unit&quot; it is a pure number (cm/cm, or in/in, or furlong/furlong) independent of any arbitrary choice of units for measuring lengths (or masses or anything else). Because of this, I would prefer to identify &quot;radian&quot; with the pure number &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.  As a consequence, I would identify &quot;degree&quot; with &lt;span class=&quot;math-container&quot;&gt;$\pi/180$&lt;/span&gt;.  And I would therefore answer the title question affirmatively: &lt;span class=&quot;math-container&quot;&gt;$180^\circ=\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2018-12-28T18:48:11.273" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15005" PostTypeId="1" CreationDate="2018-12-28T19:12:28.467" Score="3" ViewCount="354" Body="&lt;p&gt;I am interested in collecting/creating a compendium of lesson plans that are essentially just this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lesson Plan: Appropriate for a precalc class and an algebra 2 course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Show the following graph: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EXmlx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EXmlx.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Talk to the students. &#xA;See what's going on in the eye with the triangle? I want you to make a graph in Desmos that does that. And I want the point to be able to move in time like &lt;a href=&quot;https://www.desmos.com/calculator/qs7stqbuxj&quot; rel=&quot;nofollow noreferrer&quot;&gt;this.&lt;/a&gt;. After you have completed this unit's work you can add all the embellishments you'd like. Give it sinusoidal eyebrows. You could turn it into a &lt;a href=&quot;https://www.desmos.com/calculator/ieebaw29ez&quot; rel=&quot;nofollow noreferrer&quot;&gt;bicycle&lt;/a&gt;. Whatever. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hand out Rubric: &#xA;Points for silliness. Points for creativity. Points for mathematical correctness. Blah blah whatever else goes on a rubric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tell the students to get laptops and get started. After the students are launched walk the classroom and engage with the students working on it and help the students who may be struggling. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have &lt;a href=&quot;https://www.desmos.com/calculator/nge0kpus3a&quot; rel=&quot;nofollow noreferrer&quot;&gt;many&lt;/a&gt; &lt;a href=&quot;https://www.desmos.com/calculator/gjvfjmmlbv&quot; rel=&quot;nofollow noreferrer&quot;&gt;many&lt;/a&gt; &lt;a href=&quot;https://www.desmos.com/calculator/buhcyqsnue&quot; rel=&quot;nofollow noreferrer&quot;&gt;many&lt;/a&gt; &lt;a href=&quot;https://www.desmos.com/calculator/ijx71vgsop&quot; rel=&quot;nofollow noreferrer&quot;&gt;many&lt;/a&gt; of these. I would like to know if others would like to help to contribute to these. I know Desmos already has a list of activities but these aren't quite what I am looking for. If others would like to contribute I would love to see a big list here. Does anybody else do this? And would they be willing to share their lessons? If you don't do this maybe you would enjoy making one? And then you can add to the list. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An ideal answer would specify the class it's appropriate to and the content that we should be focusing on as the heart of the lesson. Like in the example I gave above: The eye with the triangle is the point of the lesson: It is the skill you want them to master. An answer with a graph so we don't have to click any links is also good. &lt;/p&gt;&#xA;" OwnerUserId="9841" LastEditorUserId="9841" LastEditDate="2018-12-28T22:07:01.320" LastActivityDate="2019-06-26T19:45:26.920" Title="Desmos Assignment: Make a graph that does this" Tags="|homework|graphing|big-list|projects|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15006" PostTypeId="2" ParentId="15005" CreationDate="2018-12-28T19:38:34.083" Score="4" Body="&lt;p&gt;&lt;strong&gt;Edit (June 2019)&lt;/strong&gt;: I used this final project with minor tweaks again this year. You can find links to some of the output for Spring 2019 - both students' graphs and write-ups - &lt;a href=&quot;https://twitter.com/benjamindickman/status/1142674034729148417&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; and &lt;a href=&quot;https://twitter.com/benjamindickman/status/1135986735593205761&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I used a Desmos Make-A-Graph prompt for an Algebra 2 class' final project last year. The results were quite good; so, I expect to incorporate at least one similar project this year. Rather than specify graphs to produce, I asked students to pick one or more ideas from the course and engage in the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Make a graph that has a playable parameter;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Present on their graph and its mathematics;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hand in a formal write-up of the project.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I have made a copy of the assignment prompt publicly available &lt;a href=&quot;https://docs.google.com/document/d/1lRXJa-wAzh0j9uuHKIba2Uhs4G2KuVCM3e5kZdZxbdQ/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. It contains four sample graphs that I provided for the students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For examples of some student-pairs' graphs, see &lt;a href=&quot;https://www.desmos.com/calculator/dfvtdx7nk9&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; and &lt;a href=&quot;https://www.desmos.com/calculator/m4iawzsllv&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; and &lt;a href=&quot;https://www.desmos.com/calculator/ufzmeumsje&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2019-06-26T19:45:26.920" LastActivityDate="2019-06-26T19:45:26.920" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15007" PostTypeId="2" ParentId="294" CreationDate="2018-12-29T04:09:16.657" Score="2" Body="&lt;p&gt;The World of Mathematics (1950s).  Good popular essays to describe various fields of math (e.g. operations research).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Martin Gardner column collections (several).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Brian Hayes columns from The American Scientist (read in the mag, not aware if collected).&lt;/p&gt;&#xA;" OwnerUserId="11700" LastActivityDate="2018-12-29T04:09:16.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15008" PostTypeId="2" ParentId="14988" CreationDate="2018-12-29T15:47:14.417" Score="3" Body="&lt;p&gt;I'm thinking certain mental tricks or rules of thumb mght be worth memorising too: for example &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;to work out what trig substitution to use, draw a triangle and check which side is in the integral. &lt;/li&gt;&#xA;&lt;li&gt;when struggling to remember or follow an &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;–&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proof or definition, try mentally inserting &quot;however small&quot; after &quot;for any &lt;span class=&quot;math-container&quot;&gt;$\epsilon &amp;gt;0$&lt;/span&gt;&quot;, &quot;small enough&quot; in &quot;a number &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;&quot;, and &quot; large enough&quot; in &quot;a number &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;&quot;. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These aren't theorems or strict mathematical procedures, but they can help a lot.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastActivityDate="2018-12-29T15:47:14.417" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15009" PostTypeId="1" CreationDate="2018-12-29T17:15:15.633" Score="7" ViewCount="414" Body="&lt;p&gt;I sometimes encounter students who ask questions like 'Why are we learning this if it won't be on the exam?' If there is time to spare I like to teach interesting applications or natural extensions of the material we have covered. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do you respond to such students? &lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2019-01-09T01:06:52.203" Title="How to deal with students who object to me teaching material that won't be in the exam?" Tags="|undergraduate-education|secondary-education|student-motivation|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15011" PostTypeId="2" ParentId="15009" CreationDate="2018-12-29T23:04:58.840" Score="1" Body="&lt;p&gt;If you're going to teach it, than include it, or a credible threat of it (meaning sometimes on the exam).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, if the class is not nailing the regular topics (high percentage of strong grades) than perhaps you shouldn't do extra topics but drill more on the ones that aren't strong yet.  Remember drill is more effective than lecture. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not about how much you can &quot;get through&quot; but about how well you teach the majority of the students the topics they need to know.  It's like sports.  Fundamentals rule.&lt;/p&gt;&#xA;" OwnerUserId="11700" LastActivityDate="2018-12-29T23:04:58.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15013" PostTypeId="2" ParentId="14988" CreationDate="2018-12-30T01:06:28.117" Score="7" Body="&lt;p&gt;An argument can be made that one should memorize almost nothing&amp;mdash;or at least as little as possible&amp;mdash;and&#xA;instead derive the results needed on the fly.&#xA;I do this myself with &lt;a href=&quot;https://en.wikipedia.org/wiki/Rotation_matrix&quot; rel=&quot;noreferrer&quot;&gt;rotation matrices&lt;/a&gt;&#xA;(although I have to admit this is hardly necessary, as what needs to be memorized&#xA;is so little).&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/wHJhd.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wHJhd.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=qW6Ua50fTw8&quot; rel=&quot;noreferrer&quot;&gt;Brian McLogan YouTube&lt;/a&gt;:&#xA;Why you should never memorize the unit circle.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;For example, many trig identities &#xA;can be derived easily &#xA;from a few core relationships: See&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/1779226/237&quot;&gt;MSE @BrL&lt;/a&gt;&#xA;or&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/1354792/237&quot;&gt;MSE @zahbaz&lt;/a&gt;.&#xA;Similar arguments could be made for calculus equations.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2018-12-30T01:06:28.117" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15014" PostTypeId="2" ParentId="15009" CreationDate="2018-12-30T01:46:00.513" Score="2" Body="&lt;p&gt;Well, for me it would be a big deception if that happened to me, provided that the students didn't actually have reasons to complain. If so, avoid teaching this &quot;extra material&quot;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If the students do not have the required previous knowledge. Make sure they all have it (or at least the big majority). Having been taught before doesn't mean they remember it either.&lt;/li&gt;&#xA;&lt;li&gt;If the students have so much pressure. Nobody appreciates learning when they've got more urgent things.&lt;/li&gt;&#xA;&lt;li&gt;If those contents are quite unrelated to the main topic, while other issues are more important. I remember things like &quot;why are you teaching us group theory when we are needing Fourier transforms and nobody's helpping us?&quot;. That's very dissapointing.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So, if it is not any of those cases, or similar, then yes, the situation must change. I assume we're talking about lazy / unconcious students who just want to pass, and they don't really care about learning or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here my advice. In short, it is: &quot;okay, you won't need it for my exam, but next year you'll be using this. The teacher won't have time to explain this, and (s)he will go very fast through this, so you better understand it now so that you can follow he class&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it is important that you must &lt;strong&gt;transmit&lt;/strong&gt; how much one suffers when one gets lost in class. The suffering of being in your seat and ask yourself &quot;what am I doing here?&quot; And you can't leave. That's stressing and agonizing. And not only you have a bad time in class. Then you don't understand anything. Since you don't udnerstand it, lessons go by and you keep at the start point, so you can lose a whole year. Make them imagine and &quot;live&quot; the situation. That's what I'd do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: I didn't have much time and I left the answer like this. Now, re-reading it looks quite terrible . Of course, you should also do the opposite: transmit how satisfactoy is to learn something new, which helps people make sense of everything. Connecting concepts is always good. Those times when &quot;suddenly everything makes sense&quot; are enormously satisfying. You must transmit all this as well.&lt;/p&gt;&#xA;" OwnerUserId="10660" LastEditorUserId="10660" LastEditDate="2018-12-30T10:52:52.143" LastActivityDate="2018-12-30T10:52:52.143" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15015" PostTypeId="1" AcceptedAnswerId="15039" CreationDate="2018-12-30T04:29:53.690" Score="1" ViewCount="186" Body="&lt;p&gt;How to analyze the level of difficulty of mathematical computation of a problem on a standard mathematical test designed for high school students? I mean how to choose some indices that can reflect the difficulties of mathematical computation without giving a test in advance?&lt;/p&gt;&#xA;" OwnerUserId="10464" LastEditorUserId="2083" LastEditDate="2019-01-03T07:13:42.370" LastActivityDate="2019-01-03T15:20:50.297" Title="How can I measure the mathematical computation skills of high school students through a test?" Tags="|secondary-education|mathematical-pedagogy|test-design|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15016" PostTypeId="2" ParentId="15009" CreationDate="2018-12-30T16:30:07.813" Score="8" Body="&lt;p&gt;This is really two different questions: (1) applications of the standard material, and (2) additional material that extends the standard material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For applications, my answer, addressed to my students, would be something like this. We have just learned math topic X, and I've presented a couple of its applications, A and B. These applications will help you to understand X better, will help you to understand why we would care about X in the first place, and may come up in later coursework in your major. You ask whether A and B will be on the exam. Applications on the exam could include A, B and C, where C is some application you've never seen. If you actually understand what the material means, and have seen a couple of applications, then you should also be able to figure out new applications for yourself. If you can't do this, then you don't understand the material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For additional material, it really depends on what your motivation was. For example, if you're teaching first-year calculus but you want to introduce some vector calculus, the answer is probably that you just shouldn't. If the motivation is that it's super fun, then present it that way. Say, literally, that that's why. In general, make sure you can give a clear answer to the question and then preemptively answer it when you introduce the material.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-12-30T16:30:07.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15017" PostTypeId="1" CreationDate="2018-12-30T20:35:41.023" Score="0" ViewCount="196" Body="&lt;p&gt;I think that the standard practice in the first grades when addition (or other operation) is taught as a &quot;process&quot; may be not so good.&#xA;I always wondered why so many children lose interest in math immediately after arithmetics. And i think this may be one of the reasons.&#xA;1+2=3 viewed as: this is a process performed on two numbers to get a result.&#xA;I think that, with this approach, we affect their future capability to think abstractly in mathematics and their understanding of algebra. Sometimes for good!&#xA;I think instead we should view 1+2=3 as a relation between two numbers, on one side and another number on the other side.&#xA;Relation, and not process.&#xA;With this process idea they remain with the idea that &quot;1+2&quot; is not a true number, but only an &quot;exercise to be performed &quot; and &quot;=&quot; means just &quot;compute&quot;. Which would be disastruous in algebra, where &quot;=&quot; can't have such meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you think about it?&#xA;Could this be the biggest reason why there is this big gap, children who are intelligent and bright in arithmetics suddenly get stuck in algebra??&#xA;Who in the world invented this &quot;operation=process&quot; idea???&lt;/p&gt;&#xA;" OwnerUserId="5972" LastEditorUserId="8190" LastEditDate="2019-01-02T12:47:27.670" LastActivityDate="2019-01-02T12:47:27.670" Title="Viewing arithmetical operations as processes-possibly wrong and detrimental to long term math performance of the students" Tags="|arithmetic-operations|" AnswerCount="1" CommentCount="16" ClosedDate="2019-01-01T21:27:12.227" ContentLicense="CC BY-SA 4.0" />
  <row Id="15019" PostTypeId="1" CreationDate="2018-12-30T22:53:22.180" Score="6" ViewCount="87" Body="&lt;p&gt;When recently teaching Calculus II to college students, I instructed my students to read and be ready to work through the first 8 or so questions of James Walsh's climate modeling differential equations packet: &lt;a href=&quot;http://www2.oberlin.edu/math/faculty/walsh/UMAPclimate.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www2.oberlin.edu/math/faculty/walsh/UMAPclimate.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I enjoy this text, because: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) these problems are well within the reach of students in any Calc II class that addresses examples of the simplest kinds of ODEs (separable, linear first-order, etc.),&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) the questions attack the ODE from different angles, allowing the students to understand the ODE from different perspectives and to integrate new and old mathematical skills, and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) the familiarity that the long-form nature of this richly contextual problem engenders in the students allows them to translate their mathematical observations and deductions into scientific interpretations. Most of my students are science majors, so this is a win-win.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I capped off this unit in my Calc II class with a speculative question on a take-home quiz that had the students qualitatively examining the the energy-balance ODE for the atmosphere of Mars and making a predictive scientific interpretation for the dust storm phenomena observed by the Mariner 9 probe in 1971. Admittedly, I did perhaps prepare them a little heavy-handedly for answering that question on the quiz, but otherwise, I was fairly satisfied with how well this unit went over with the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to utilize more long-form, multi-step, skills-integrating applied mathematics problem packets like this in my calculus classes, and was wondering what you all had uncovered on the internet or had written yourselves that is in the vein of the Walsh packet? I am especially interested in Calc III (multi-variable calculus) applications, but all calculus topics are fine.&lt;/p&gt;&#xA;" OwnerUserId="6318" LastActivityDate="2018-12-30T22:53:22.180" Title="Long-form, multi-step, skills-integrating applied mathematics problems in calculus I, II, III" Tags="|undergraduate-education|mathematical-pedagogy|vector-calculus|applied-mathematics|critical-thinking|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15020" PostTypeId="1" CreationDate="2018-12-31T01:47:23.570" Score="11" ViewCount="529" Body="&lt;p&gt;Some of my students solve equations not by applying the same operations on the left and right sides of an equation, but by applying the operation to the whole equality. For example, they may write something like this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{x-1} = 3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\left(\sqrt{x-1} = 3 \right)^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x-1 = 9$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x=10$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;instead of the usual second line &lt;span class=&quot;math-container&quot;&gt;$(\sqrt{x-1})^2=3^2$&lt;/span&gt;. I have never seen such notation (I have been educated in France mostly). So my first reaction was to consider it incorrect and ask the student to write it the &amp;quot;right&amp;quot; way.&lt;/p&gt;&#xA;&lt;p&gt;On second thought, it seems that the notation is sound if we define&#xA;&lt;span class=&quot;math-container&quot;&gt;$f( LHS = RHS )$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$ f(LHS) = f(RHS) $&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Still, I have a bad impression any time I see a student using this notation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Should I prohibit my students to use this kind of notations?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Or, to make it less opinion-based, are there any objective (in the mathematical, logical, and/or pedagogical sense) reason I should ban the notation in my class?&lt;/p&gt;&#xA;" OwnerUserId="1538" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-01-02T07:09:54.960" Title="Are there any negative consequences in applying operations/functions to a whole equality?" Tags="|notation|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15021" PostTypeId="2" ParentId="8268" CreationDate="2018-12-31T03:51:01.623" Score="0" Body="&lt;p&gt;Few extra ideas:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Operations research wrt sub hunting.  Classic essay in World of Mathematics.  Very readable.  And topic will interest some students.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some simple naval architecture.  There are a few simple equations that your students would have the math chops to understand fine (e.g. how max speed is connected to length of ships for non hydroplaning hulls).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Economics of fracking (hydraulic fracturing) or water disposal.  For each just give them the formulas and let them play with scale.  /For the latter there are some simple things to do with percentages or ratios (oil/water/gas).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Fluid flow in pipes.  Often this is shown with lots of calculus, but the essential relations are algebraic.  This is how most of it is taught in the Navy.  for instance the power laws of velocity, head loss, and pump power (linear, quadratic, cubic): doubling flow requires 8 times the power.  Working engineers often don't even use any algebra but add up head loss by arithmetic and size pumps from the Ingersol Rand catalog.  But learning to use a nomograph is a skill itself.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Open channel flow.  Provide the equation and have students work a few problems doing inputs.  (Make realistic if possible, using numbers from codes, real sewer projects or even rivers.)  See Michael Lindberg texts for EIT or mechanical PE.  (simple formulas, just don't have to hand.)    &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="11700" LastEditorUserId="11700" LastEditDate="2019-01-01T15:30:07.823" LastActivityDate="2019-01-01T15:30:07.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15022" PostTypeId="2" ParentId="15020" CreationDate="2018-12-31T04:24:17.700" Score="11" Body="&lt;p&gt;The notation your students are using isn't &lt;em&gt;wrong&lt;/em&gt;.  It's perfectly clear - anyone who knows algebra instantly knows what they mean - it's not particularly confusing, and it isn't going to quickly lead to a dead end where it's incompatible with upcoming material.  Indeed, there's at least one good argument for it &lt;em&gt;over&lt;/em&gt; the conventional notation: it identifies what the actual step is, rather than requiring the reader to look at both sides of the equation and compare them to realize what's happening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's just not the conventional notation.  There might be a good reason for that (I haven't thought through the implications of using your students' notation very far), but if there's a reason not to use it, I don't think it's quickly accessible to your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you can't really explain to your students &lt;em&gt;why&lt;/em&gt; they shouldn't use that notation other than that it's not how other people write.  That creates some problems:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;student often have trouble reliably telling the difference between conceptual mistakes and notational mistakes, so focusing too much on notation can be very distracting,&lt;/li&gt;&#xA;&lt;li&gt;relatedly, having to write things a certain way imposes a real cognitive cost on students, when they're already trying to learn the material, and&lt;/li&gt;&#xA;&lt;li&gt;students get very frustrated if they feel like they're not getting credit for understanding material just because it's not expressed in the right notation; this can feed into a sense that math is just a list of arbitrary rules to be memorized, which discourages them from trying to understand it.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It might still be worth it to enforce the use of conventional notation.  Your students certainly need to be able to &lt;em&gt;read&lt;/em&gt; conventional notation, and at some point, if not now, they'll need to be able to write it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(It's worth comparing to the similar tradeoffs when children misspell words in their writing - on the one hand, children eventually need to learn to spell correctly, while on the other hand, when children are first learning to write, obsessing about spelling everything correctly bogs children down so much that they just learn to hate writing.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But you should consider whether it's worth the cost, and how you can enforce prohibiting it in a way that doesn't conflate notation errors with math errors or discourage students.  (For instance, if your situation makes it feasible, you might allow rewrites of notation errors.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-12-31T04:24:17.700" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15024" PostTypeId="2" ParentId="15020" CreationDate="2018-12-31T13:08:01.073" Score="8" Body="&lt;p&gt;This seems to me an interesting question, with variants in many contexts, whose answer is not self-evident. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The notation &lt;span class=&quot;math-container&quot;&gt;$(A = B)^2$&lt;/span&gt; should be considered problematic because it is &lt;em&gt;undefined&lt;/em&gt; (another way of saying &quot;nonstandard&quot;) in the same sense as &lt;em&gt;undefined behavior&lt;/em&gt; in a programming language is undefined - it might compile, it might be interpreted properly, but there is no guarantee that it will be understood. Mathematical syntax functions as a shorthand for indicating to who reads it what operations were performed or should be performed to pass from one statement to the next. It works best when the translation is straightforward (when correctly done). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A square amounts to multiplying something by itself and generally for this to make sense one has to operate within some mathematical framework (algebraic structure) in which multiplications are defined - one needs some sort of binary operation on the set of things that can be multiplied. What is the square of an equality? Usually one considers an equality as having a truth value - &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; or something nonzero - and in fact it makes sense to square a truth value - in a programming language such as C code of the form &lt;span class=&quot;math-container&quot;&gt;$(A == B)*(A == B)$&lt;/span&gt; is syntactically correct and would have a value equal to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are unequal and something nonzero (usually &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, but in principle dependent on the compiler) in the case &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are equal. However, note that, although coherent, this way of interpreting &lt;span class=&quot;math-container&quot;&gt;$(A = B)^2$&lt;/span&gt; is not what the student has in mind in this case! The problem is that the correspondence between the notation &lt;span class=&quot;math-container&quot;&gt;$\,^2$&lt;/span&gt; and the operations to be performed is not the usual one indicated by an exponent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student has in mind to indicate notationally the following reasoning: Since &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A^{2}$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$B^{2}$&lt;/span&gt;. It captures far better the sense of the reasoning to write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ A = B \implies A^{2}= B^{2}$$&lt;/span&gt;&#xA;than to write &lt;span class=&quot;math-container&quot;&gt;$(A = B)^{2}$&lt;/span&gt;. Often students are taught to write simply&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A = B\\&#xA;A^{2}= B^{2}$$&lt;/span&gt;&#xA;with no symbol indicating logical implication, such implication being indicated by passing from one line to the next. (Since the implication is left implicit many students do not realize it is there. Such a student thinks of a pattern of operations, rather than a series of implications.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summary: what is problematic about writing &lt;span class=&quot;math-container&quot;&gt;$(A = B)^2$&lt;/span&gt; is that the exponent does not correspond directly to squaring something in the sense of multiplying something by itself. Rather it is something that has to be unpacked and interpreted (like the notation for a line integral) if it is to be evaluated - and, moreover, there already exists a more easily and more directly interpretable notation for indicating the same operations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One encounters things like this in many contexts. In linear algebra one sees often identification of &lt;span class=&quot;math-container&quot;&gt;$1 \times n$&lt;/span&gt; row vectors, &lt;span class=&quot;math-container&quot;&gt;$n \times 1$&lt;/span&gt; column vectors, and elements of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^{n}$&lt;/span&gt;. Many a successful engineer/physicist makes such identifications regularly, yet making such identifications usually indicates that someone does not distinguish well between equality and isomorphism. When teaching a class where it matters to understand this distinction, I generally start out by advising students about the incorrectness of identifying two things that are not equal, marking it but not taking off points, but later in the semester, when the error is repeated, I take points off (it is never a substantial deduction, usually only a marginal one). The experience is that students who don't eventually manage to understand the distinction between equating and identifying rows and columns have trouble with many other aspects of the course, so that insisting on such a trivial point is not as formal as it may appear to some.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another similar example from linear algebra: one calculates ranks and determinants by applying elementary row (and, for determinants, column) operations. In the case of rank computations the matrices produced are (tautologically) equivalent via row operations but not equal, while in the case of determinants there results a string of equalities. Many students write &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; in both cases, and this usually signals conceptual deficits that are fairly serious in the context of learning linear algebra. Such students are generally memorizing sequences of operations and have very little understanding of what those operations are doing (e.g. that row operations correspond to operations on equations), or why they are being performed (e.g. eliminating redundancies/dependencies). One understands what such a student is doing, such a student might very well obtain the correct answer to the particular formal exercise being solved, but such a student almost always does not understand what he/she is doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Writing &lt;span class=&quot;math-container&quot;&gt;$(A = B)^2$&lt;/span&gt; in place of &lt;span class=&quot;math-container&quot;&gt;$A^2 = B^2$&lt;/span&gt; seems to me somewhere in between these two examples and how to react to it depends on the context in which it appears. Objecting to it is less fussy than objecting to identifying row and column vectors, but the possible conceptual problem underlying it is not so severe as in the case of writing &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; between matrices related by row operations. My approach would be to mark it in red and explain in detail what should be written, but to take off no points (since in this case it seems the student did compute correctly, just used nonstandard notation to communicate the computations). If the nonstandard notation is used repeatedly, I would probably progress at some point to taking off points. I say &quot;probably&quot; because exactly how to react surely depends on the context (the student, the class, the goals of the class, etc.). (If a university student writes this it is an indicator of low probability of success in university level math classes, but when written by a 10 year old it could even be an indicator of an autonomous and active mind - inventing reasonable notation is not so trivial).&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2018-12-31T13:08:01.073" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15025" PostTypeId="2" ParentId="18" CreationDate="2018-12-31T14:21:15.343" Score="0" Body="&lt;p&gt;For most courses, I feel like you learn the material better if it is no notes on the tests.  You've really internalized the material.  It sticks with you better years down the road (so even if you need to look at a book then, it comes back fast since you really learned it down pat before).  And then in near term, it gives you more mastery so the tools can be used very conveniently and quickly in physics, chemistry, engineering, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, I don't think most math courses really require crib sheets (it's not such a mass of tedium).  What math courses require is doing lots of practice problems.  When you do lots of practice problems, you're doing much more than memorizing the procedure...getting practice in using it and what mistakes to watch out for.  So remember the formula itself is least of the worries.  If you don't remember the formulas than you have a weak level of knowledge indeed.&lt;/p&gt;&#xA;" OwnerUserId="11700" LastActivityDate="2018-12-31T14:21:15.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15026" PostTypeId="2" ParentId="18" CreationDate="2018-12-31T16:45:28.273" Score="2" Body="&lt;p&gt;In many cases, there is a choice between memorizing and understanding. E.g., a calc student can memorize that &lt;span class=&quot;math-container&quot;&gt;$\cos'=-\sin$&lt;/span&gt;, or they can understand that it's a phase shift of 90 degrees and figure out the direction of the phase shift by visualization. Understanding is better, but it requires three things: (1) a teacher who has this type of understanding, (2) a teacher who emphasizes and aids in this kind of understanding, (3) a student who is intellectually capable of it, and (4) a student who is willing to do it. An advantage of memory-oriented teaching may be that it works in cases where &lt;span class=&quot;math-container&quot;&gt;$\neg$&lt;/span&gt;(1 and 2 and 3 and 4).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Especially in K-12 and community college education, there are many cases where 1 is false. When you get teachers who lack this level of understanding, they don't believe that it's useful or can substitute for a certain amount of memorization. For example, when I was a kid I did &lt;span class=&quot;math-container&quot;&gt;$6\times8$&lt;/span&gt; by remembering &lt;span class=&quot;math-container&quot;&gt;$3\times8=24$&lt;/span&gt; and then doubling that. But I have spoken to a lot of kids who were specifically told never to do this kind of thing, because it would be too slow. If they scribbled this kind of thing in the margin of their times-table test, they were penalized. This is what you get from teachers who don't themselves have the habit of understanding. They have never had the experience of doing this kind of memorization-avoidance strategy successfully, so they have never had the experience of seeing how it becomes faster and faster, until eventually it's very much like memorization.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2018-12-31T16:45:28.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15027" PostTypeId="2" ParentId="15017" CreationDate="2018-12-31T23:59:00.903" Score="2" Body="&lt;p&gt;@Henry Townsner:&#xA;Ok, so let's be specific: what exactly are the difficulties students experience when transitioning to algebra and why? Specific, not general  statements as &quot;more abstract understanding of numbers&quot; or &quot;building their own model of how math works&quot;, that is too general. &#xA;I just wonder WHY CLEVER students begin to hate math when they encounter algebra. In my opinion it HAS something to do with teaching. We cannot just lay there and just say &quot;algebra is difficult, so it is normal to struggle &quot; and watch them struyggle and stryggle and hating math more and more, while they once loved it. We should try to find what exactly are the difficulties, why they are happening and what we can do to make students overcome those difficulties.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And speaking about &quot;building their own way of how math works&quot;, even if this is a too general, non specific idea, i think the problem is just here, we teach arithmetical operations in the &quot;process-result&quot; manner , a manner which is unsuitable for understanding algebra. &#xA;And this is not an original idea of mine, there is a ton of research on the subject, see for example articles by Anna Sfard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's why they are so confused when starting to learn algebra, because the old process -result view of operations is no more suitable and they are just confused ...and it is quite normal that they start to hate math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don;t &quot;Act&quot; like algebra is simple or difficult, but let's be realistic, what's the main difference between arithmetics and algebra? It is simply the presence of variables and unknowns, aka the &quot;letters&quot;.&#xA;And i honestly don't think that the letters denoting variables and unknowns are the problem. And i say that because studnets solve equations in arithmetic too. It is just that instead of the &quot;intimidating&quot; letters we have blank spaces or question marks or small circles, etc...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arithmetic is full of equations like 2*?+3=17 so students perform excellently at equations of the form ax+b=c.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ones of the type ax+b=cx+d are the problem. Because in ax+b=c they still use the old arithmetic interpretation of &quot;=&quot; as meaning &quot;do the computation&quot; but in ax+b=cx+d &quot;=&quot; cannot have the same meanning. So here they encounter a major confusion! Produced exactly by the &quot;process-result&quot; way we teach them arithmetics!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What i want to say is that we should make the best effort from the start to avoid them seeing &quot;=&quot; as &quot;compute&quot; and &quot;a+b&quot; like a process with the result c. And &quot;=&quot; as compute! &#xA;We should from the start make them see &quot;2+3&quot; as a DESCRITPTION OF A NUMBER, and not a process, and &quot;=&quot; in 2+3=5 to mean &quot;is the same number(with different &quot;descriptions&quot;) and NOT &quot;compute&quot; or &quot;evaluate&quot; or other crap. &#xA;even in the practical way we teach them counting with real objects, the second you put 2 coins next to three coins you there have an exact number of 5 coins, there is no further &quot;process&quot;  to perform and no &quot;result&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what i think is that someone invented this ridiculous &quot;process-result&quot; interpretation of addition just to make arithmetic simpler to teach. But with the price that when advancing to algebra students will be confused and have more difficulties.&#xA;So we make arithmetic &quot;simpler&quot; (even if that might not be necessary) but with the price of compromising the tyransition to algebra for most students... How silly is that??&#xA;We just made them struggle with algebra, and know we just lay there and say &quot;algebra is difficult, iot is normal to struggle bla bla, tacher can't do anything, just tell them math is hard and they should struggle bla bla&quot;. How wrong is that? &#xA;No, IT IS NOT NORMAL that kids have to struggle, in a well designed teaching system! They should not struggle! If they struggle it means the teaching is not optimal!&lt;/p&gt;&#xA;" OwnerUserId="5972" LastEditorUserId="5972" LastEditDate="2019-01-01T00:21:53.867" LastActivityDate="2019-01-01T00:21:53.867" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15029" PostTypeId="2" ParentId="2623" CreationDate="2019-01-01T16:05:07.467" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I hate to be the one, but...a Google search gives several articles on the topic.  Didn't see one killer one.  But also did not see them all a waste.  Think skimming a few would help (and more content than in QA here).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Quora has several responses also.  Being even more the one.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One interesting insight in some articles is that it appears there is &quot;cannibalization&quot; of the visual cortex by blind people to use it for math!  Seems to be just with people blind from birth (presumably others don't since their bodies hope to get the sight back).  Also I don't mean to say they have an advantage.  Fer sure, still many hurdles for a blind man.  But still...fascinating that this accommodation is made.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some other articles comment on the obvious issues with following lectures (helps if teacher speaks out the equations).  Q&amp;amp;A helpful to blind person but there can be concern on slowing lecture down.  Office hours needed more, used more to get dedicated instruction.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nemeth Braille is one system for showing math characters:  &lt;a href=&quot;https://en.wikipedia.org/wiki/Nemeth_Braille&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Nemeth_Braille&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For some reason, blind R&amp;amp;D mathematicians tend to be more often geometers!  Not sure why this is (or even the proof through statistics, but several people mention it).  Maybe it is just perplexed combatitiveness?  Or maybe it is that the relationships of advance geometry are a strain for anyone to visualize?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="11700" LastActivityDate="2019-01-01T16:05:07.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15030" PostTypeId="2" ParentId="15020" CreationDate="2019-01-02T07:09:54.960" Score="1" Body="&lt;p&gt;My experience is that students resent having to write out all the steps in algebra when they can do so much of it in their head. Therefore shorthand is a kind of compromise that allows them to show their work without writing everything out. These students like to text and abbreviate as much as possible.  I would allow it while emphasizing that it is personal shorthand that wouldn't be used when writing mathematics correctly.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that you want them to learn the correct notation. Perhaps you could allow it but regularly give exercises where you give examples of shorthand that the students have used and have the students write the notation correctly. Shorthand quizzes (translating shorthand to notation) can send the message that this is important even if you don't insist on it all the time.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2019-01-02T07:09:54.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15031" PostTypeId="2" ParentId="7964" CreationDate="2019-01-02T17:43:06.243" Score="0" Body="&lt;p&gt;To summarize previous answers and comments, and adding some of my own, the reasons of students treating the equals sign as operation are: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the meaning of equal sign is not clearly explained in elementary school;&lt;/li&gt;&#xA;&lt;li&gt;the difference between chained expressions and equations not clearly explained;&lt;/li&gt;&#xA;&lt;li&gt;worksheets with pre-printed placeholders do not build the culture of writing chained expressions;&lt;/li&gt;&#xA;&lt;li&gt;early use of calculators builds wrong perception of equal sign as an operation.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Considering that such an abuse is widespread, this tells much about the lack of systematic approach in the elementary and middle school education.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastActivityDate="2019-01-02T17:43:06.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15033" PostTypeId="2" ParentId="4367" CreationDate="2019-01-03T00:43:12.403" Score="0" Body="&lt;p&gt;For each positive integer, you can determine the quotient and remainder on division by 2. I think the best way is to demonstrate an example. Here's one possible way to  teach what the binary notation is through an example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with the number 43. When you divide it by 2, you get a quotient of 21 and a remainder of 1. Now divide 21 by 2 and you get a quotient of 10 and a remainder of 1. Now divide 10 by 2 and you get a quotient of 5 and a remainder of 0. Now divide 5 by 2 and you get a quotient of 2 and a remainder of 1. Now divide 2 by 2 and you get a quotient of 1 and a remainder of 0. Now divide 1 by 2 and you get a quotient of 0 and a remainder of 1. Now that the quotient is 0, we stop here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you write down the remainders you got from each of those steps in reverse order so the binary notation of 43 is 101011. It may be worth pointing out that when ever you're writing the binary notation of a number, the last digit is the remainder on division of that number by 2 and the string of all digits but the last digit is the binary notation of the quotient on division of that number by 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then as a result, some students might figure out by themselves that we can recursively define the binary notation of every positive integer and once you get up to that integer, since you already defined the binary notation for its quotient on division by 2, you can define the binary notation of that number itself. However, it's not worth telling the students that. It may confuse them and they may struggle to understand what the teacher is saying. I think students can generally do better job of understanding things they notice all by themselves. Not only that but if the teacher teaches them something confusing, the students might take a long time struggling to understand what they're saying and devote less attention to another thing the teacher is trying to teach them and learn even less. According to the article &lt;a href=&quot;https://www.huffpost.com/entry/teaching-discovery-learning_b_856463&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching Is not Learning — The Guided Discovery Approach for Learning&lt;/a&gt;, teaching is not learning. I think the education system is not learning to accept what they cannot do and because schools are teaching so much difficult to learn material, the students are actually learning less. A lot of students are failing some courses in high school. That's probably because the education system won't take no for an answer and teach less in a more understandable way.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="11718" LastEditDate="2020-03-12T18:55:01.933" LastActivityDate="2020-03-12T18:55:01.933" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15034" PostTypeId="1" CreationDate="2019-01-03T05:30:54.813" Score="-1" ViewCount="641" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html&lt;/a&gt;, when students get taught a concept when they're so young, they're less likely to be able to learn it properly. For example, some students really struggle to learn what fractions are and some people also can't understand how it's possible that irrational numbers exist after they learned how to represent all rational numbers and make calculations on them. Maybe if they avoid teaching that so early, they can use the time to teach the students stuff they actually can understand such as number theory and go quite far with it and then later, they can teach about real numbers so much more easily as a result of the knowledge about number theory they learned earlier. For example, they could completely avoid teaching them about the concept of cardinality of a finite set and teach only the definition of a natural number as an ordinal number because some of them might not have yet thought of their own proof that distinct finite ordinal numbers always correspond to different cardinal numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe they could teach them how to write a formal proof in a certain weak system of number theory after giving them the resources from which they can form an intuition that everything it proves is true where that system only talks about natural numbers. They might later test them on their ability to write a formal proof of the statement they're asked to write a formal proof of. If the student writes an intuitive reason why a certain statement is obviously true, the teacher will not progress with the material until after the student learns that the task was to write a formal proof the way they were asked to to show that they're able to figure out how to write a formal proof of it and masters the skill of figuring how how to write formal proofs of different statements when they're asked to write formal proofs of them. By waiting until all the students in the class master the skill of figuring how how to write a formal proof of a new statement when they're asked on a test to write a formal proof of that statement before moving on, they might gain the ability to think and figure stuff out really well and understand other material they're getting taught later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without being told ahead that these are some of the statements they're going to be asked to write a formal proof of so that they won't collaborate with another student to figure out the formal proofs of those statements ahead of time, here are some of the statements they could be asked to write a formal proof of except that those statements in their current form aren't actually describable in the system so they will be given strings of characters that represent a different but similar statement. They could be told on the test that by definition&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x \in \mathbb{N} x + 0 = x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x \in \mathbb{N}\forall y \in \mathbb{N} x + S(y) = S(x + y)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x \in \mathbb{N} x \times 0 = 0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x \in \mathbb{N}\forall y \in \mathbb{N} x \times S(y) = (x \times y) + x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;without any prior being introduced to the concept of addition and multiplication and then asked to write a formal proof of the following statements.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} (x + y) + z = x + (y + z)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x \in \mathbb{N}\forall y \in \mathbb{N} x + y = y + x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} (x \times y) \times z = x \times (y \times z)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x \in \mathbb{N}\forall y \in \mathbb{N} x \times y = y \times x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} x \times (y + z) = (x \times y) + (x \times z)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It could be even later that they're taught the inductive definition of the decimal notation for each natural number which goes as follows&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get to the successor follow the following rules&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If the last digit is not 9, increase that digit by 1&lt;/li&gt;&#xA;&lt;li&gt;If the sting of characters is 9, change it to 10&lt;/li&gt;&#xA;&lt;li&gt;If the last digit is 9 but is not the only digit, change that digit to a 0 and change the string of all digits before it to the string that represents the successor of that string&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;They can also be left to figure out on their own how to multiply numbers in decimal notation mentally and to know how to divide two numbers when ever the division problem is defined as well as learning how to check whether it actually is defined and write undefined when it's not defined instead of only knowing how to form a division strategy that always works when the given problem is defined and being the type of person who would write 8 as the answer to 14 &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They might not get a feel that real numbers other than natural numbers actually exist but later, they could be taught how to become a savant at number theory who can discover results without reliance on a calculator, pencil and paper, a computer program, or the internet. They might later work in stronger systems of number theory dealing with more complex properties of the natural numbers the stronger system describes and when they read a book that uses the phrase real number, they might not understand what it's saying but will be so good at noticing patterns that they notice a certain connection between the text in the book and what they discovered about number theory and decide that when they talk about the real number &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;, they're really informally describing a certain property of the natural numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One down side of that is that not all real numbers correspond to a property of the natural numbers the system describes so the person might say that the real numbers it does describe are the only real numbers. They might later create their own version of type theory which is weaker than the already existing type theory to only extend number theory and not be at all an extension of ZF where Godel's first incompleteness theorem on proof systems of number theory is a meaningful statement and they don't assume unrestricted comprehension on all properties of the natural numbers anymore than normal type theory assumes unrestricted comprehension on all sets and will treat &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; like a proper class and not get along with mathematicans who work in Zermelo-Frankel set theory which does assert that there is a set of all subsets of the set of all finite ordinals and write formal proofs in it. Also, we can informally define ordinal numbers in terms of proof systems of pure number theory where stronger systems can define larger ordinal numbers. However, in that version of type theory, if we include the Church-Turing theris in the type theory, we could probably also derive that the ordinal numbers at and above the Church-Kleene ordinal don't exist at all. See &lt;a href=&quot;https://mathoverflow.net/questions/133597/what-would-remain-of-current-mathematics-without-axiom-of-power-set/133598#133598&quot;&gt;https://mathoverflow.net/questions/133597/what-would-remain-of-current-mathematics-without-axiom-of-power-set/133598#133598&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="11718" LastEditDate="2019-01-03T05:42:38.903" LastActivityDate="2022-08-28T18:26:06.420" Title="Could students learn a lot more from school if they're only taught number theory until way later?" Tags="|mathematical-pedagogy|number-theory|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15035" PostTypeId="2" ParentId="15015" CreationDate="2019-01-03T07:12:22.817" Score="1" Body="&lt;p&gt;An answer originally by &lt;a href=&quot;https://matheducators.stackexchange.com/users/672/dan-fox&quot;&gt;Dan Fox&lt;/a&gt; in a comment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One way to judge computational difficulty is by performing the computations oneself. With exams at the university level, my rule of thumb is that if I set a two hour exam, I should be able to physically write completely correct solutions in no more than 15-20 minutes (otherwise the exam is too hard). At lower levels, I would suppose the teacher should be able to complete (doing and writing all the required computations) whatever is tested in an hour test in a few minutes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="143" LastEditDate="2019-01-03T14:09:48.827" LastActivityDate="2019-01-03T14:09:48.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15036" PostTypeId="2" ParentId="15015" CreationDate="2019-01-03T07:12:56.070" Score="1" Body="&lt;p&gt;Answer by &lt;a href=&quot;https://matheducators.stackexchange.com/users/11700/guest&quot;&gt;guest&lt;/a&gt; in a comment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Give problems modeled on the homework drill problems. Typically they will be divided into 2-3 sets of increasing difficulty by number. (a) simple plug and chug (b) longer multistep problems (c) challenging problems requiring combining techniques, proofs, or deriving something new not taught yet. Tests should avoid much or any (c) problems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="143" LastEditDate="2019-01-03T14:09:12.290" LastActivityDate="2019-01-03T14:09:12.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15037" PostTypeId="2" ParentId="15034" CreationDate="2019-01-03T07:28:29.223" Score="10" Body="&lt;p&gt;The way I understand the question is: If students are not taught fractions, but instead formal deductive proofs of properties of natural numbers, would they learn mathematics better?&lt;/p&gt;&#xA;&lt;p&gt;I find it unlikely.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Students struggle with fractions in primary school. Students usually struggle with proofs in gymnasium (the level of school between middle school and university which prepares and qualifies for university studies) or university. It seems unlikely that primary school students would master proofs (a very abstract concept), when even the older students with more experience struggle with the matter. Furthermore, everyone attends primary school (around here, at least), while around half the population attends gymnasium (around here) and even fewer take university mathematics classes with proofs. Hence, there is some amount of filtering on who currently learns proofs; not everyone faces it, at the moment, and around here.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Fractions are useful outside formal mathematics and even engineering. Deductive proofs of properties of natural numbers are much less so. This argues for teaching fractions to everyone, rather than deductive proofs of properties of natural numbers. As far as I understand, evidence for the transfer effect (of studying mathematics helping with reasoning in general) is weak, so it is a good idea to teach content that is useful in and of itself, especially in primary school.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Many teachers would have an awfully hard time understanding deductive proofs of properties of natural numbers; in my teacher education, many future middle school teachers had serious problems with proofs and advanced mathematics, and the effect would likely be even more pronounced for elementary school teachers. They already teach mathematics that not all of them understand very well, which might be one cause of mathematics anxiety among the students. Asking them to teach more abstract mathematics would likely make the problem worse.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2022-08-28T18:26:06.420" LastActivityDate="2022-08-28T18:26:06.420" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15038" PostTypeId="2" ParentId="15034" CreationDate="2019-01-03T13:27:17.393" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I understand the question in the same way Tommi Brander did but I would like to give my own answer. Maybe I misunderstood the question or the age group you meant, but I assumed it to be children who are about 12-13 or under and who are beginners at learning fractions (for instance).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;First off, I’m not sure where you get your premise that “learning something young means that that thing is more likely not to be learned properly”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article you linked mostly cites studies about Denmark kids. In particular, it says at the end that “In Denmark […] there is universal access to decent pre-kindergarten. Relatively few American cities and towns offer the same thing.” Put that way, of course you would expect kids who are put through better preparation would do better later in life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article cites one study about OECD countries, including data from Canada and the United States which says that kids who started school later are more likely to attend University.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can’t conclude that “not attending University” means that I probably learned fractions wrong. So I’ll remain skeptical with the premise.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Second, even if we were to assume that the studies are right, the thing to do would be to postpone the age at which we start learning (at school), not replace the current content by other content. If kids are truly “getting fractions wrong” right now because they learn too soon, then they would learn number theory wrong instead…&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Third, you seem to think that learning fractions is harder than doing number theory proofs. Imagine for one minute that you are a &lt;em&gt;kid&lt;/em&gt; and you see this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\forall x\in\mathbb{N}x+S(y)=S(x+y)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2-\frac{5}{6}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which do you think is easier to understand, or at least would take less time to understand properly?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You also suggest that kids who give “an intuitive reason” to “a certain statement that is obviously true” would be told to start again until “a formal proof” is written and that they can’t “collaborate with another student”. The whole group stays &quot;locked&quot; in that spot until everybody gets it “right”. This is so disgustingly behaviorist and so far off from modern socioconstructivism that I don’t think this would be applicable in the near future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify, there is currently a shift in paradigm in teaching (well, it depends on the country but I think you'll understand). Essentially, it isn't about &quot;how should we teach this&quot;, it's about &quot;how should they learn this&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll try to explain it (grossly) here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before, the prerogative was that students were &quot;entirely ignorant&quot; and &quot;wrong&quot; until they got the same answer as the teacher. In other words, if a student does something that doesn't meet the teacher's standards, it's rubbish. So the teacher's job was simply to &quot;punish&quot; a student if he had different answers and &quot;reward&quot; if he had the same answers as the teacher. (By &quot;punish&quot;, I mean things like saying &quot;that is wrong&quot;, or marking 0, etc.) There is basically no regard to what goes on in the students mind, it's all about changing their behavior until they &quot;learn&quot; to do it right (like the teacher). This is, essentially, &lt;em&gt;behaviorism&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays, this kind of approach is less and less popular. One of the reasons is that we know that kids are capable of being creative and telling them to meet very specific standards essentially inhibits that creativity. (Instead of learning the content, they learn how to do what teachers' want). So if you try to teach a 12 year-old how to do proofs and you tell him to start over until he's doing it just like you want (which, if I understood you correctly, sounds like the work of a much more advanced student), he'll end up completely lost, discouraged and unmotivated. At beast, he'd be able to copy a proof that you wrote on the board, but without understanding very much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But an even better reason not to use this approach (behaviorism) is because there are better ones out there now, that implement what we now know about metacognition (knowing how we learn), with the advent of research in psychology, mainly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the picture:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students arrive in class with their own conception of reality (or &quot;mathematical truth&quot; in our case). They learn new things (what the teacher tells them, what they read in books, what they hear from other students). From this new knowledge, they modify and improve their original conception of reality; they &lt;em&gt;construct&lt;/em&gt; a new version of what they think reality is. It might not be perfect yet, but the idea is that this new version is (in general) better than the old one. In another course, or later in life, that same knowledge might be improved further as the student experiences more things. That's &lt;em&gt;constructivism&lt;/em&gt;. So this is a pedagogical approach where knowledge is not fixed in place, but rather, is something that is constructed throughout one's life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Socioconstructivism is essentially constructivism made through interactions with other people (peers, the teacher). Activities like group work, debates and such are characteristic of socioconstructivism. For example, you ask students a question. They answer it on their own. Then they trade their work with a neighbour. If they disagree, they have to argue with them to try and convince them that their answer is better. If they agree, they try to see if the neighbour's work had any interesting differences. By doing this, the students might glean some insight on the content that they wouldn't have had if they had only listened to the teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case you are worried that &quot;everyone constructs their own reality&quot; is a dangerous view in mathematics, that is where the teacher comes in. His/her role is (among other things) to make sure that what is assimilated (through constructivist activities) is not erroneous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if you have a 12 year old who &quot;proves&quot; in a very naive or intuitive way some statement, he's basically using all of his previous knowledge to try and come up with something convincing enough. To expect anything more abstract is a bit too much, since he/she doesn't have any background knowledge on how to do a proper proof. You can start doing proofs when you have some background knowledge about math. But when you first learn math, your background knowledge is just a bunch of intuitions and ideas that are probably littered with falsehoods; so that background knowledge ought to be fleshed out a bit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, when you first learn that &lt;span class=&quot;math-container&quot;&gt;$1+1=2$&lt;/span&gt;, do you have to know that you are using a binary operation that is closed in some field? No. But if you want to make proofs involving fields, you ought to know how &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; works in some intuitive way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to conclude with the &quot;paradigm shift&quot; I was talking about, instead of focusing on the content we want to pass on, we should focus on &lt;em&gt;how&lt;/em&gt; to pass on the content first and socioconstructivist strategies are much richer than behaviorist ones, for the learners. I'm not entirely against teaching proofs, but the way you presented it goes against the new paradigm, so I'm hoping this answers your question. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Children are not full-fledged mathematicians&lt;/strong&gt;. Doing proofs requires some mathematical maturity that kids don’t have. It sounds like you want them to be able to do math the way a mathematician does, before they even start learning anything about math.&lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="9341" LastEditDate="2019-01-03T16:00:45.113" LastActivityDate="2019-01-03T16:00:45.113" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15039" PostTypeId="2" ParentId="15015" CreationDate="2019-01-03T15:20:50.297" Score="1" Body="&lt;p&gt;The difficulty of a problem depends on a large number of factors.  Different students are going to find different problems difficult (for example, some students struggle with computation but do well on conceptual questions), and the material that you actually cover in lecture and homework is going to influence how easy or hard some problems are (to give a higher level example, half-angle formulae for the sine and cosine functions are relatively easy to prove if you already know Euler's formula; they are a little harder to prove if you can only use properties of the unit circle).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, there are a number of things that you can do in order to get a sense of how difficult an exam problem might be:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Use homework exercises from a text as a model for exam questions (see &lt;a href=&quot;https://matheducators.stackexchange.com/a/15036/8571&quot;&gt;this answer&lt;/a&gt;).  In addition to being organized from easiest to hardest (typically), these problems have probably been vetted by more than one person, edited to death, and checked carefully for alignment with the text.  This means that the level of difficulty should be fairly easy to assess (as per guest's answer) and that problems from the text (assuming that you are using that text) should align well with the material being taught and be &quot;fair&quot; questions to ask.&lt;/li&gt;&#xA;&lt;li&gt;Work the problems yourself (see &lt;a href=&quot;https://matheducators.stackexchange.com/a/15035/8571&quot;&gt;this answer&lt;/a&gt;).  This can't really be used to check if problems are &quot;too easy&quot; (since you might have inadvertently created problems that don't align well with what you taught), but it is certainly a good way to check if your problems are too hard.&lt;/li&gt;&#xA;&lt;li&gt;Exploit your colleagues.  Unless you are in a &lt;em&gt;very&lt;/em&gt; small school out in the middle of rural America, I would guess that you have colleagues that you can talk to (you say that you are designing a test for high school students, so talk to the other people in your math department).  Have them look over your exams and give an opinion about the difficulty.  If they have a lot more experience than you, they will likely be able to guide you quite a bit.  Even if they don't have that much more experience, they can still point out flaws or problems that are potentially too difficult.&lt;/li&gt;&#xA;&lt;li&gt;Consider student performance.  This isn't really an &lt;em&gt;a priori&lt;/em&gt; measure of difficulty, but it can be used to design better exams in the future.  Before you give an exam, determine how hard you think each question is.  After the exam, look at student performance on each question.  Are there questions that nearly every student got right or wrong?  Are there questions where students clearly burned through a lot of scratch paper or made common errors?  By understanding where students struggle on one exam, you gain information about how to write future exams.&lt;/li&gt;&#xA;&lt;li&gt;Finally, as hinted at by the previous two answers, let experience guide you.  If you are a young or new teacher, you are going to write some terrible exams---most of us do that.  Learn from your mistakes, consult your colleagues, and you'll do better the next time.  In particular, as you give more exams, you will have a better idea of which problems are easy or hard for your students.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2019-01-03T15:20:50.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15040" PostTypeId="1" CreationDate="2019-01-03T23:22:46.613" Score="15" ViewCount="699" Body="&lt;p&gt;My impression to students majoring in mathematics is, whenever we teach them a theorem, a proof should be given in the class, or at least as a reading assignment. However, how about students not majoring in mathematics?&#xA;One extreme is, proving everything, treating them as students majoring in mathematics. The other extreme is, not teaching any proof at all, only introducing the conclusions and their applications. Of course one can always teach some proofs while omitting others. Now the question is, what proofs should we teach and what proofs should we omit? What are the factors to consider?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2019-01-04T13:40:56.600" LastActivityDate="2019-01-22T14:00:09.573" Title="Should theorems be proved to students who are not majoring in mathematics?" Tags="|undergraduate-education|proofs|" AnswerCount="7" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="15041" PostTypeId="2" ParentId="15040" CreationDate="2019-01-04T00:00:10.187" Score="5" Body="&lt;p&gt;For the sake of making the question more definite, let's assume we are looking at calculus students.  Let's assume we are looking at a generalist class.  One with advanced strong high school students or decent college students.  Not calculus for business students).  But a standard course that meets the needs of physics and engineering students.  But also likely has future math majors in it (many will not even be declared by then).  But also chemists, oceanographers, and maybe even business students (for schools that don't do a weakened version).  But not Caltech or MIT.  Something close to the AP curriculum. Standard course of Thomas, Stweart, Granville, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say do the proofs you can (don't do ones that use concepts kids don't have...leave that for a later theoretical course).  Some, more intricate proofs may be put in an appendix.  I would treat the proofs as motivation, NOT objective.  The proofs give the kids a feeling that what they are doing is true.  And in some cases, they may help solidify knowledge and make people remember something.  But they are NOT the objective.  Doing problems is the objective.  (Like if we were in a war, shooting bad guys would be the objective, not explaining how the rifle works.)  I wouldn't even test on proofs much.  It's more a form of motivation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want an actual put your finger on it answer my advice is to go with Thomas level of proofs.  Not &quot;real analysis in sheep's clothing&quot; Spivak.  But a little more rigor than classical Granville or current Stewart.  Based on what I have gleaned about your class, Zuriel, between the lines, I think that would work well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But again...the objective is to teach, drill, and test problem solving.  So don't overthink this proofs issue.  &lt;/p&gt;&#xA;" OwnerUserId="11700" LastActivityDate="2019-01-04T00:00:10.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15042" PostTypeId="2" ParentId="15040" CreationDate="2019-01-04T08:52:36.503" Score="12" Body="&lt;p&gt;I would like to offer a different way of thinking about the question. The idea that you have to do the proofs for maths students tends to come from people who have a fixed idea of what it means to 'do maths' and what a maths degree is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, I believe the place to start is asking 'what skills do we want the students to learn?' (followed by 'why do we want them to learn &lt;em&gt;these&lt;/em&gt; things?'). I wrote down a list (not to hand) of about 20 different possible reasons for including a proof in a lecture. The value of each will vary according to the makeup of the class (whether maths students or otherwise). You might, for example, want to include a proof to give students the idea of how mathematicians prove things (and how this differs from experimental science), or because seeing the proof makes it easier to remember/use correctly the result etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another part of the question is 'why is including this proof more valuable than the alternative uses of the time?' Maths courses tend to be heavy on content, without time for students to really understand what they are being told. Everything you put &lt;em&gt;in&lt;/em&gt; comes at the cost of something you take &lt;em&gt;out&lt;/em&gt;. &lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-01-04T08:52:36.503" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15043" PostTypeId="2" ParentId="15040" CreationDate="2019-01-04T09:20:19.060" Score="10" Body="&lt;p&gt;It seems to me pointless and perhaps even irresponsible to teach results/algorithms/methods without explaining them, but between rigorous proof and adequate expanation there is a lot of terrain. Explanation and proof are somehow two extremes of the same thing, the difference being one of level of detail and formalization. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this question there is nothing special about calculus. This question is applicable to nearly everything one teaches in mathematics, at every level. (Generally speaking the least useful research talks are those that (try to) give all the details of the proofs ...) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The worst case scenario is the way my daughter was taught to take square roots in middle school. The students were shown a step by step algorithm for obtaining square roots and learned to perform it. No explanation of why it worked was given. The students learn something, albeit very little, by learning to correctly perform the operations and obtain the correct answer, but I am not sure there is not done more harm than good - the same students who can correctly calculate &lt;span class=&quot;math-container&quot;&gt;$\sqrt{34}$&lt;/span&gt; to two decimal places cannot say - beforehand - whether &lt;span class=&quot;math-container&quot;&gt;$\sqrt{34}$&lt;/span&gt; is bigger or smaller than &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;. Also the students learn this way that mathematics is a formal game with opaque rules, the objective of which is to get the correct answer. The essence of such a results only approach is authoritarian and anti-educational.&#xA;(After dissecting the algorithm in question, it automates writing the putative square root in its decimal expansion and solving iteratively for the unknown digits - something that certainly could be explained to middle school students, and that could probably profitable serve as a way to improve understanding of decimal representations of numbers. A separate question is whether the finding of square roots should be taught in middle school, but this is traditional and universal in Spain, and I think also many other European countries.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In calculus the situation is the same. One does not want to simply assert that a function with a positive derivative is nondecreasing. A student who has simply memorized this learns nothing. At a practical level such a student is likely to run into problems understanding critical points. At a deeper level, the student learns to treat mathematics as a purely deductive exercise of purely formal steps. However, the same statement can be explained reasonably well without a formal proof involving the mean value theorem if one has developed intuition relating the derivative to the slope of the tangent line, and it can be explained in such terms to a physicist's level of rigor. For many students, particularly those in engineering, such an approach may be more productive than dwelling on the formal details. On the other hand, as anyone who has tried to do this knows, this is more difficult to do than it sounds - precisely, giving a careful, clear intuitive explanation of the significance of the sign of the derivative without entering into formalities is a difficult task. In fact, I am not completely convinced by any attempt I have seen or made, and so, when I teach this to engineering students, I give both such intuitive explanations and something close to a formal proof. With proper motivation the formal proof is largely accessible and introduces tools useful in other parts of the course. This of course requires having developed previously the language of continuity and diferentiability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With students from social sciences, computing, or biological sciences the situation is more difficult (generally speaking such students have less interest in mathematics than do engineering students). I think the solution is not to omit explanations so much as to omit content. Better to explain well a few things than to cover a lot of ground without explaining anything. Very little of what is taught will be instrumentally useful for more than a handful of students, and so the goal is to teach a student the skills and background necessary to learn later, on her/his own, whatever aspects of the material might be needed later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, the classroom is not really the place to give all the details of most arguments. The goal in class is to communicate the main ideas and the motivations. The student who cares about the details is expected to learn them on her/his own, using a book, in problem sessions, or in office hours, etc. (In high school the pace is different and more is done in class, but the situation is not fundamentally different.) A good explanation is a &quot;proof&quot; if it contains the ideas/motivations, and a good &quot;proof&quot; really amounts to making concrete the steps necessary to formalize the ideas and motivations - in how much detail to develop each of those steps is really the hard decision for the teacher - and the answer will always depend on the context of who the students are - but identifying them and pointing them out seems essential if anything is to be really gained from the exercise. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-01-04T09:20:19.060" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15044" PostTypeId="2" ParentId="14999" CreationDate="2019-01-04T15:16:34.837" Score="4" Body="&lt;p&gt;Radians are &quot;pure numbers&quot;, inasmuch as they are ratios of the length along the arc subtending the angle to the radius of the circle.  As such, they don't require units, and you could (and should) write &lt;span class=&quot;math-container&quot;&gt;$180^{\circ} = \pi$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, there's a good reason to specifically write radians.  As Dan Fox points out that there is another natural unit for angle measure:  the turn.  In fact, the turn is arguably the most natural measure of an angle, since everyone understands what it is (though our language confuses the issue:  &quot;Turn around&quot; and &quot;Turn halfway around&quot; are used interchangeably).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is also a ratio (it's the fraction of a full rotation) and as such has no units.  So you could just as legitimately write &lt;span class=&quot;math-container&quot;&gt;$180^{\circ} = 1$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, this leads to writing &lt;span class=&quot;math-container&quot;&gt;$1 = \pi$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So while you could write &lt;span class=&quot;math-container&quot;&gt;$180^{\circ} = \pi$&lt;/span&gt; and not raise an eyebrow, it's probably better to write &lt;span class=&quot;math-container&quot;&gt;$180^{\circ} = \pi$&lt;/span&gt; radians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/_p0aF4si5Rs?list=PLKXdxQAT3tCuJku9nTlRZgx_RjGZ7djMc&quot; rel=&quot;nofollow noreferrer&quot;&gt;This is how I introduce angles to my trigonometry students&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="11747" LastActivityDate="2019-01-04T15:16:34.837" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15045" PostTypeId="2" ParentId="15040" CreationDate="2019-01-04T17:42:44.673" Score="3" Body="&lt;p&gt;I think there's also an issue of &lt;strong&gt;how&lt;/strong&gt; you present the proofs. As an engineering student—knowing that I didn't need to remember proofs, just results—I often sat in class happily following each step of a derivation, and I could see that because all the steps were right the result must be true, but most of the time I didn't have a clear idea of what the proof &lt;em&gt;outline&lt;/em&gt; was—the proof or derivation was almost a magical sequence of steps leading to the result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I would definitely have benefitted from understanding the proofs.It would have made the results easier to remember and understand, and knowing how the proofs worked would have given me a means of checking for myself whether I was remembering a particular formula or equation correctly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think: where possible, focus on giving a clear understanding of the main concepts and arguments of the proofs—but for people who will be using results rather than proving them, avoid getting bogged down in too much detail of the actual manipulations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though of course there might be students whose experience is the opposite: ones who see the outline straight away and have trouble following details so need to see every step written down. For me, being given all the details was actually an obstacle to understanding.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastActivityDate="2019-01-04T17:42:44.673" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15046" PostTypeId="1" CreationDate="2019-01-04T23:32:54.387" Score="2" ViewCount="115" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Julie Barnes, Jessica M. Libertini. &lt;em&gt;Tactile Learning Activities in Mathematics: A Recipe Book for the Undergraduate Classroom&lt;/em&gt;.&#xA;  2018. MAA Press.&#xA;  &lt;a href=&quot;https://bookstore.ams.org/clrm-54&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMS Bookstore link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Can anyone comment / review this collection? Perhaps too new for published reviews...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/GF7x4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GF7x4.jpg&quot; alt=&quot;Cover&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-01-04T23:32:54.387" Title="Tactile Learning Activities in Mathematics" Tags="|undergraduate-education|recreational-learning|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15047" PostTypeId="2" ParentId="15040" CreationDate="2019-01-05T02:34:14.563" Score="1" Body="&lt;p&gt;I like this comment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;@DanFox: &quot;between rigorous proof and adequate explanation there is a lot of terrain&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(1)&lt;/strong&gt;&#xA;I would try to sprinkle in a few very clear proofs of non-obvious theorems. So I would avoid the intermediate value theorem, because it seems so obvious. But proving that the geometric mean is at most the arithmetic mean,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sqrt[n]{ a_1 a_2 \ldots a_n} \le \frac{a_1 + a_2 + \cdots + a_n}&#xA;{n} \;'&#xA;$$&lt;/span&gt;&#xA;for positive reals &lt;span class=&quot;math-container&quot;&gt;$a_i$&lt;/span&gt;, is certainly not obvious. And usually requires induction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(2)&lt;/strong&gt;&#xA;Another example is described in this&#xA;response to an MESE question, &lt;a href=&quot;https://matheducators.stackexchange.com/a/2454/511&quot;&gt;Good, simple examples of induction?&lt;/a&gt;.&#xA;The line of logic is clear once explained, but certainly not obvious &lt;em&gt;de novo&lt;/em&gt;.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/DbNcf.png&quot; alt=&quot;3-coloring&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-01-05T02:34:14.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15048" PostTypeId="2" ParentId="15009" CreationDate="2019-01-06T19:09:29.973" Score="0" Body="&lt;p&gt;First, any time students ask &quot;Is this on the test?&quot;, it makes me reach for a vomit bag. But my reaction is irrelevant compared to the student experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most students [consider themselves] busy and [in their minds] you've already decided that the material is &lt;strong&gt;not&lt;/strong&gt; important enough to assess for grading. This incentive of grading is, for many students, the &lt;strong&gt;one&lt;/strong&gt; true signal of what is important enough work on. Once students know something isn't graded, they will turn their attention to friends, family, health, other classes, jobs, money, relationships, housing, etc. and other areas of life that they actually perceive as important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analogous situations:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;You call customer service and are told by an automated recording for 2 straight hours, that your call is very important. How much do you care about such claims of importance relative to the fact that the company doesn't hire (incentivize) enough people to work in customer service?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Your boss says: &quot;From now on, everyone in the company spends 10am-12pm every Monday, Wednesday, and Friday working on project X, under my direction. You're &lt;em&gt;not&lt;/em&gt; going to get paid for those hours but I expect you to show up cheerfully and work as hard as you can anyway. Why? Because we don't just work for pay! We work for many other reasons, such as fun, learning, and natural interest. Come on, guys, be intrinsically motivated! Like me! Project X is cool! As a bonus, some of you might find the experience with project X useful someday. [Edit: Besides, I'm not sure how to measure your productivity at all for X, so there's another reason for me not to pay you for your labor.] Anyone who doesn't want to put in those hours is just lazy and instrumentally motivated and that's a big problem.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2019-01-09T01:06:52.203" LastActivityDate="2019-01-09T01:06:52.203" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15049" PostTypeId="2" ParentId="15034" CreationDate="2019-01-06T20:40:23.273" Score="2" Body="&lt;p&gt;It does seem to me that this is the wrong way round. I remember, in primary school, being shown via a rectangular arrangement of &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; dots that &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; rows of &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; was the same as &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; columns of &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$5×4=4×5$&lt;/span&gt;. That wasn't a formal proof—it was an &lt;em&gt;explanation&lt;/em&gt;, in visible form, of why the order didn't matter. Neither was it presented so we'd learn what a proof was—it was there to help us understand how multiplication worked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proofs are abstract things because of needing to show that something is &lt;em&gt;always&lt;/em&gt; true, not just true of the concrete example in front of you. So we were shown the example with a simple explanation of how it worked, not given a proof involving all &lt;span class=&quot;math-container&quot;&gt;$m, n \in \mathbb N$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The natural progression is surely&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;simple explanations of day-to-day concepts and procedures&lt;/li&gt;&#xA;&lt;li&gt;more detailed explanations of less simple ideas&lt;/li&gt;&#xA;&lt;li&gt;explanations that are general enough to amount to informal proifs&lt;/li&gt;&#xA;&lt;li&gt;formal proofs with all the non-obvious holes plugged.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the early stages the purpose of the explanations is understanding of whatever concepts are essential to that stage of learning—understanding how explanations work and how they can be made rigorous comes later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly there's a progression in the ability to make arguments:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;understanding simple arguments (like the &lt;span class=&quot;math-container&quot;&gt;$5×4$&lt;/span&gt; grid) but not coming up with them&lt;/li&gt;&#xA;&lt;li&gt;arguing in order to solve a simple problem (if &lt;span class=&quot;math-container&quot;&gt;$3x=15$&lt;/span&gt;, dividing both sides by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; keeps them equal to each other and gives us &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;li&gt;arguing about more advanced or more general problems&lt;/li&gt;&#xA;&lt;li&gt;putting arguments together into proofs rather than just routes to solutions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But a the early stages, the aim isn't to produce future mathematicians—it's to produce people who can understand the maths that's needed in daily life.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastActivityDate="2019-01-06T20:40:23.273" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15050" PostTypeId="2" ParentId="14876" CreationDate="2019-01-07T07:06:01.533" Score="1" Body="&lt;p&gt;I think the short answer, particularly for graduate texts but to some extent for undergraduate texts in the style you describe, is that the textbooks are written that way because that is the way research papers are written. Part of the point of university, certainly at graduate level, is to learn how the academic research discipline operates.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-01-07T07:06:01.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15052" PostTypeId="1" CreationDate="2019-01-07T16:08:26.037" Score="3" ViewCount="256" Body="&lt;p&gt;I am a future teacher and am interested in incorporating a mathematics research unit where my students do their own research on an unsolved problem. I have a couple ideas on problems, but am difficulty finding scholarly results/benefits/examples of doing a research unit within the secondary setting. Would someone be able to point me to some scholarly articles and papers that could be used as evidence to convince administrators? Thank you!&lt;/p&gt;&#xA;" OwnerUserId="11755" LastActivityDate="2019-01-08T18:09:23.537" Title="Mathematics Research at Secondary Level (High school Level)" Tags="|secondary-education|education-research|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15053" PostTypeId="2" ParentId="15052" CreationDate="2019-01-07T17:56:28.767" Score="3" Body="&lt;p&gt;I have taught high school for two years, and am now a Ph.D. student in mathematics. I continue to teach undergraduate courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some high schoolers do participate in mathematics research. For example, the AMS put out a notice regarding the &lt;a href=&quot;https://www.ams.org/notices/201508/rnoti-p910.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PRIMES program for advanced high schoolers&lt;/a&gt; -- who, first of all, are passionate about mathematics (such a thing is necessary to do mathematics research), and second of all, have at least a calculus background. The article then details the problems which students tackle, and the difficulty of devising problems to which a high schooler with adequate mathematical maturity can be built up to over a few months. Unless you are going to be teaching at a highly specialized school, I strongly advise against attempting to get high school students to do mathematical research. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if you intend for them to learn &quot;research skills&quot; in generality, I believe that a mathematical history project that aligns with your course goals would be an education-research-supported unit. I refer you to &lt;a href=&quot;https://www.jstor.org/stable/40248010&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Using History in Mathematics Education&quot; (Fauvel, 1991)&lt;/a&gt;, which discusses the notion of using history in mathematics education and summarizes some methods of doing so. A more recent discussion of this topic is found in &lt;a href=&quot;https://hal.archives-ouvertes.fr/hal-01349230/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;History of Mathematics in Mathematics Education&quot; (2016)&lt;/a&gt; which discusses the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;• Which history is suitable, pertinent, and relevant to Mathematics Education (ME)?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;• Which role can History of Mathematics (HM) play in ME?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;• To what extent has HM been integrated in ME (curricula, textbooks, educational&#xA;  aids/resource material, teacher education)?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;• How can this role be evaluated and assessed and to what extent does it contribute to the teaching and learning of mathematics?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In my personal and professional opinion, tackling historical problems is much more accessible to high school students, and can give them the experience of devising their own question and seeking an answer in a real-world context. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another possible way you could get high school students having a research experience without needing advanced knowledge is is via a multidisciplinary unit together with a science teacher. Mathematics, in particular statistics, is useful in answering many science problems. If your students have a statistics unit, you could have them be the &quot;statistics team&quot; and a science class be a &quot;science team&quot;, and work together to produce a statistically and scientifically meaningful result for an end-of-year project. I did this together with an advanced science elective when I taught AP Statistics and it was very successful. I may add details later if requested.  &lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2019-01-08T18:09:23.537" LastActivityDate="2019-01-08T18:09:23.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15054" PostTypeId="2" ParentId="5794" CreationDate="2019-01-07T21:50:18.820" Score="0" Body="&lt;p&gt;I reformat the most upvoted answer (also my favorite)  with MathJax beneath, from &lt;a href=&quot;https://redd.it/8ni40x&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reddit&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;&lt;a href=&quot;https://www.reddit.com/r/explainlikeimfive/comments/8ni40x/eli5_why_is_1_x_1_1/dzvv55d/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zerotan 42.2k points 7 months ago&lt;/a&gt;&lt;/h3&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Repost from 2 years back:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I give you three \$20 notes: &lt;span class=&quot;math-container&quot;&gt;$+3 × +20 =$&lt;/span&gt; you gain $60&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I give you three \$20 debts: &lt;span class=&quot;math-container&quot;&gt;$+3 × -20 =$&lt;/span&gt; you lose $60&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I take three \$20 notes from you: &lt;span class=&quot;math-container&quot;&gt;$-3 × +20 =$&lt;/span&gt; you lose $60&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I take three \$20 debts from you: &lt;span class=&quot;math-container&quot;&gt;$-3 × -20 =$&lt;/span&gt; you gain $60&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user155" LastEditDate="2019-01-08T00:14:22.880" LastActivityDate="2019-01-08T00:14:22.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15055" PostTypeId="2" ParentId="14919" CreationDate="2019-01-07T23:00:15.697" Score="8" Body="&lt;p&gt;Another point. It seems to me that someone who's trained to think that there's only one way to write everything might well be trained to think less in general. If you know that &lt;span class=&quot;math-container&quot;&gt;$\mathbb N$&lt;/span&gt; may or may not include &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, you're forced to check which definition is meant, and the awareness that it's important to define the things &lt;em&gt;you&lt;/em&gt; use is reinforced. Later on, awareness of multiple conventions might lead to a greater willingness to define your own notation in situations where that's necessary or helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Someone who thinks the whole of mathematics has been handed down &quot;as is&quot;, including the set of symbols and operations and functions, might be less able to think flexibly and maybe, less able to make the contribution they might have been capable of.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastEditorUserId="10663" LastEditDate="2019-01-08T14:40:18.860" LastActivityDate="2019-01-08T14:40:18.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15056" PostTypeId="2" ParentId="14869" CreationDate="2019-01-07T23:31:59.663" Score="3" Body="&lt;p&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt; I'm only really at the beginning of teaching myself analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;BUT I have a copy of Gelbaum &amp;amp; Olmsted, &lt;em&gt;Counterexamples in Analysis&lt;/em&gt; which contains such examples as &quot;Two uniformly continuous functions whose product is not uniformly continuous&quot; (&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;), &quot;A convergent series with a divergent rearrangement&quot; (any conditionally convergent series, with a demonstration of how to get any predetermined limit as well), &quot;A power series convergent at only one point&quot;, etc. (There is a whole chapter of power series examples.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe there are other books in the series, covering different areas of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe books like this could be a useful resource? I didn't see a counterexample specific to your example, but I wouldn't be surprised if some of them might come in handy. And they're nicely collected together by subject, which might help towards being pre-armed with examples of what can go wrong.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastEditorUserId="10663" LastEditDate="2019-01-14T16:11:21.733" LastActivityDate="2019-01-14T16:11:21.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15057" PostTypeId="2" ParentId="5794" CreationDate="2019-01-08T00:14:49.190" Score="2" Body="&lt;p&gt;I reformat the 2nd most upvoted answer  with MathJax beneath, from &lt;a href=&quot;https://redd.it/8ni40x&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reddit&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://www.reddit.com/r/explainlikeimfive/comments/8ni40x/eli5_why_is_1_x_1_1/dzvp0s3/&quot; rel=&quot;nofollow noreferrer&quot;&gt;sjets3 14.1k points 7 months ago&lt;/a&gt;&lt;/h3&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Imagine you are watching a movie. The first number is how the person in the movie is moving. The second number is how you are watching the film (normal or in reverse).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1 \times 1$&lt;/span&gt; is a person walking forward, you watch it normal. Answer is you&#xA;see a person walking forward, which is 1.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1 \times -1$&lt;/span&gt; is a person walking forward, you watch it in reverse. You see a&#xA;person walking backwards. -1&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$-1 \times 1$&lt;/span&gt; is a person walking backward, you watch it normal. You see a person walking backwards. -1&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$-1 \times -1$&lt;/span&gt; is a person walking backwards, but you watch it in reverse. What you will see is a person that looks like they are walking&#xA;forward. 1&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As a funny example of the last para. overhead, compare &lt;a href=&quot;https://gfycat.com/PopularFrighteningCormorant&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gfycat.com/PopularFrighteningCormorant&lt;/a&gt; (original video) with &lt;a href=&quot;https://i.imgur.com/ZCw2C81.gifv&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://i.imgur.com/ZCw2C81.gifv&lt;/a&gt; (&lt;a href=&quot;https://www.reddit.com/r/explainlikeimfive/comments/8ni40x/eli5_why_is_1_x_1_1/dzvwe4e/&quot; rel=&quot;nofollow noreferrer&quot;&gt;film person walking backward then play backward&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-01-08T00:14:49.190" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15058" PostTypeId="2" ParentId="15052" CreationDate="2019-01-08T15:16:03.037" Score="4" Body="&lt;p&gt;This kind of echos Opal E's answer but with some specific problems I used for one of my &quot;research projects&quot; when I was teaching a class of standard level seniors (they were on the algebra II / Precal level). I also agree that modern day unsolved problems are extremely difficult to understand and even the easy sounding ones (Goldbach Conjecture, Collatz Conjecture, etc) are much deeper than they let on, making students believe they know much more about the problem than they really do. (EDIT: for a specific example of this, I showed a very bright student the &quot; &lt;a href=&quot;https://www.youtube.com/watch?v=Y30VF3cSIYQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;legendary question #6&lt;/a&gt;&quot; and he was convinced he had found a simple solution after only 30 minutes. Most times, students think a simple question means a simple solution) Although, if you really do want to used unsolved problems, then this &lt;a href=&quot;http://mathpickle.com/aboutus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt; could help get you started. They have unsolved problems for students to explore from every grade K-12. I haven't used their resources, but they do look like very interesting/engaging problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I wanted to do a &quot;research project,&quot; I wanted them to research the important landmark theorems of the past, the ones that marked major changes in the history or development of math. The students had to research what the problem meant why it was important in the history/development of math. I don't remember all of the problems I used but here's a short list: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Solving the Cubic Equation (eventually led to Galois Theory and Group Theory)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) 4 Color Theorem (first rigorous proof completed via computer)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Bridges of Konisberg (Led to Graph Theory)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) How easily can people factor a large number with the aid of a computer? (Leads to RSA encryption) Additionally, how can we tell if a large number is prime?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) Is it true that for a given line and a point not on the line, there is only one line through the point parallel to the given line? (leads to the debate about the parallel postulate and the discovery of non-euclidean geometry).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6) How did the ancient mathematicians figure out the decimal representation of transcendental numbers? How do we know &lt;span class=&quot;math-container&quot;&gt;$\pi=3.141592...$&lt;/span&gt;? How do  we find rational approximations, for example &lt;span class=&quot;math-container&quot;&gt;$\pi \approx \frac{22}{7} \approx \frac{333}{106} \approx \frac{355}{113}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7) Which set has more numbers in it: a) The set of positive integers b) The set of all integers c) The set of numbers between 0 and 1? (Leads to the ideas of multiple infinities and how to show two sets have equal cardinality).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;8) I think I also had a questions about what an irrational number is, how do we know &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is irrational? Why was Pythagoras so upset at this. Are most numbers irrational or rational?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't remember if I had other problems for them too look up, those are the ones I can remember. Some groups did better than others, but the point wasn't for them to understand the proof of the Four Color Theorem, but just know how that one theorem represented a big change in rigorous mathematics.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastEditorUserId="8527" LastEditDate="2019-01-08T17:51:15.853" LastActivityDate="2019-01-08T17:51:15.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15059" PostTypeId="1" CreationDate="2019-01-08T16:52:45.853" Score="1" ViewCount="2082" Body="&lt;p&gt;The title is self-explanatory, but what is the appropriate response to &quot;motivate your answer&quot;? Should you prove your answer? Should you just explain it, or give the intuition for it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, suppose students are given a true/false question, and &quot;motivation for the answer&quot; is required. A student could provide a formal, rigorous proof of the answer, which is obviously sufficient, but a student could also give, say, provide a sketch of a graph with an explanation that  demonstrates the validity of the answer. Although the latter is not a formal proof, it does point out -albeit, less rigorously - why the answer is correct. Thoughts?&lt;/p&gt;&#xA;" OwnerUserId="11759" LastEditorUserId="2083" LastEditDate="2019-01-09T07:10:49.327" LastActivityDate="2019-01-09T15:32:17.953" Title="Motivate your answer vs. Prove your answer" Tags="|exams|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15060" PostTypeId="2" ParentId="15040" CreationDate="2019-01-08T19:04:18.960" Score="2" Body="&lt;p&gt;Some lovely answers are here already. I would like to look at an example below the level of calculus. In beginning and intermediate algebra (at a community college), I love working through the proof of the quadratic formula with my students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's why:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They are highly likely to think of it as a complete black box. &lt;/li&gt;&#xA;&lt;li&gt;It is the nastiest looking formula they are likely to ever use if they are not going on in math. (Well, standard deviation looks worse, but statistics is sadly full of black boxes for them.)  &lt;/li&gt;&#xA;&lt;li&gt;They have seen it before. &lt;/li&gt;&#xA;&lt;li&gt;It is one of the reasons we work on completing the square, which has few uses for those not going on, but is a powerful tool worth learning, if put in a good context.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I use James Tanton's method (using an area model) which avoids fractions. (And fractions raise the anxiety level for many students.) We refer to the area model frequently as we go. His youtube videos on this are &lt;a href=&quot;https://www.youtube.com/watch?v=OZNHYZXbLY8&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://www.youtube.com/watch?v=bjH1HphOZ1Y&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, and &lt;a href=&quot;https://www.youtube.com/watch?v=eDwi96DCdTc&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. [Note: His method does not always work as described, you may need to multiply by 4 a second time. Watch the videos and see if you can find examples of this glitch. I am trying to figure out which types of problems cause it, and do not have an adequate description yet.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think proving this can help them see how math works. Our process allows them to see this proof / derivation as a sensible explanation, and as something they might be able to do themselves, if they cared enough. As we are working on it, I talk about us building this thing ourselves. I do it one day, and the next day, I have them tell me each step. My hope is that this gives them a better understanding of why we prove things.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2019-01-08T19:10:53.877" LastActivityDate="2019-01-08T19:10:53.877" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15061" PostTypeId="2" ParentId="10832" CreationDate="2019-01-08T20:57:15.663" Score="1" Body="&lt;p&gt;There are terms I have never liked because their meaning conflicts with what one would expect. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one that comes most readily to mind is '&lt;strong&gt;whole numbers&lt;/strong&gt;'. It sounds to me like it should include the negative whole numbers (the ones that don't have fractions), but of course it doesn't. Integers means what I want whole numbers to mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With fractions, I often say top and bottom instead of &lt;strong&gt;numerator&lt;/strong&gt; and &lt;strong&gt;denominator&lt;/strong&gt;, though perhaps I shouldn't.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-01-08T20:57:15.663" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15062" PostTypeId="2" ParentId="10832" CreationDate="2019-01-09T05:56:01.277" Score="3" Body="&lt;p&gt;The word &quot;multiply&quot; in English almost always means &quot;increase&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many people believe that &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{7}\times9$&lt;/span&gt; must be bigger than &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; is getting multiplied.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2019-01-09T05:56:01.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15063" PostTypeId="2" ParentId="5605" CreationDate="2019-01-09T08:23:52.267" Score="2" Body="&lt;p&gt;Stephen Boyd and Lieven Vandenberghe (also authors of a well-known &lt;a href=&quot;http://web.stanford.edu/~boyd/cvxbook/&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt; on convex optimization) have recently written an introductory linear algebra book, &lt;a href=&quot;http://vmls-book.stanford.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Applied Linear Algebra: Vectors, Matrices and Least Squares&lt;/a&gt; that may suit your needs. From the preface:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This book is meant to provide an introduction to vectors, matrices, and least&#xA;  squares methods, basic topics in applied linear algebra. Our goal is to give the&#xA;  beginning student, with little or no prior exposure to linear algebra, a good grounding in the basic ideas, as well as an appreciation for how they are used in many&#xA;  applications, including data fitting, machine learning and artificial intelligence, tomography, navigation, image processing, finance, and automatic control systems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also from the preface:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The book covers less mathematics than a typical text on applied linear algebra.&#xA;  We use only one theoretical concept from linear algebra, linear independence, and&#xA;  only one computational tool, the QR factorization; our approach to most applications relies on only one method, least squares (or some extension). In this sense&#xA;  we aim for intellectual economy: With just a few basic mathematical ideas, concepts, and methods, we cover many applications. The mathematics we do present,&#xA;  however, is complete, in that we carefully justify every mathematical statement.&#xA;  In contrast to most introductory linear algebra texts, however, we describe many&#xA;  applications, including some that are typically considered advanced topics, like&#xA;  document classification, control, state estimation, and portfolio optimization.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The book is indeed replete with applications, many of which come from machine learning (or data mining) and from statistics in the guise of data fitting. It's very much an introduction though, as you'll need another source for learning topics such as vector spaces, subspaces, LU decomposition, eigenvalues, eigenvectors, the singular value decomposition, and various other topics. It's very much at the other end of the spectrum from Axler's book, but they do have something in common - little or no emphasis on determinants. In fact, the word &quot;determinant&quot; does not even appear in Boyd &amp;amp; Vandenberghe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On balance, I think it's worth a look as a relatively gentle introduction with a wealth of applications.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2019-01-09T19:19:15.377" LastActivityDate="2019-01-09T19:19:15.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15064" PostTypeId="2" ParentId="10832" CreationDate="2019-01-09T13:29:55.693" Score="1" Body="&lt;p&gt;Two things come in my mind: continuity and circles:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Continuity :&lt;/strong&gt;&lt;br&gt;&#xA;In normal English, a curve is continuous when you can draw it without lifting your pencil.&lt;br&gt;&#xA;In mathematics, there's a whole definition with environments, basic environments, or epsilon-delta formulas to describe continuity, but no mathematics teacher has ever shown me the relation between those definitions and the fact that you don't need to lift your pencil while drawing the function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Circles :&lt;/strong&gt;&lt;br&gt;&#xA;In normal English, a circle is the typical round thing, and yes, in mathematics, a circle, being defined as all the points, located at the same distance of one point (the centre) also reveals the typical round thing.&lt;br&gt;&#xA;But you can also use different definitions of &quot;distance&quot;, e.g. the so-called 1-distance (or Manhattan distance) :&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;D(point_1,point_2) = abs(X2-X1) + abs(Y2-Y1)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;When drawing a circle, based on that distance, you end up with a square, which means that a square is a circle :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2019-01-09T13:29:55.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15065" PostTypeId="2" ParentId="15059" CreationDate="2019-01-09T15:32:17.953" Score="1" Body="&lt;p&gt;The word &quot;motivate&quot; isn't sufficiently specific to make clear what's meant by such an instruction without some further context.  (It might suffice a short hand on an exam, but only if students had seen examples in class and in practice problems.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would argue that the &lt;em&gt;same&lt;/em&gt; is true of the word &quot;prove&quot; - more seriously, since mathematicians often underestimate how much variability there is in what arguments count as proofs.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-01-09T15:32:17.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15066" PostTypeId="2" ParentId="14999" CreationDate="2019-01-09T15:42:38.197" Score="-1" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$180^{\circ} = \pi$&lt;/span&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly not : the surface of the unity circle is &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. If anyone tells me the surface of anything can be expressed in degrees, I'll tell this person (s)he's wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$180^{\circ} = \pi$&lt;/span&gt; radians?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That I can agree to.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2019-01-09T15:42:38.197" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15068" PostTypeId="2" ParentId="5605" CreationDate="2019-01-09T23:46:47.540" Score="2" Body="&lt;p&gt;Gilbert Strang has a new book (published January 2, 2019!) that focuses on linear algebra's connection to machine learning.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;G. Strang. &lt;a href=&quot;http://math.mit.edu/~gs/learningfromdata/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Linear Algebra and Learning from Data&lt;/a&gt; SIAM, 2019.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I obviously haven't had time to review this book yet, but the TOC looks like it might suit your needs.&lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2019-01-09T23:46:47.540" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15069" PostTypeId="1" AcceptedAnswerId="15072" CreationDate="2019-01-11T17:04:09.860" Score="5" ViewCount="127" Body="&lt;p&gt;I teach mostly physics and a little math at a community college in California. When I teach special relativity, my preferred pedagogy is to describe the Lorentz transformation using an almost purely geometrical and visual approach, with very little algebraic manipulation. My presentation uses what physicists would probably refer to as a passive transformation, in which we think of the points as staying fixed while the coordinate grid changes. Below is an example that shows the sort of presentation I use. (I actually use a slightly different set of graphical conventions so that it's intelligible in black and white.) In this example, the blue dot has coordinates (3,2) in the red coordinate system and (3,0) in the black coordinates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BrSfZ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BrSfZ.png&quot; alt=&quot;passive coordinate transformation using two grids overlaid&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I usually spend quite a bit of time introducing this type of graphical presentation, and I have a lesson worked out in which I initially use the Galilean transformation, which should be more familar, or at least less counterintuitive, before introducing the Lorentz transformation. But I'm wondering if my presentation could be improved or streamlined by presenting some totally different and more familiar mathematical example of this kind of change of variables. This seems like the kind of thing that people might have done more commonly back in the era of nomograms and slide rules. Basically I would like a pair of variables &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; with some familiar, easy real-world interpretation, where there is some interesting notion of transforming to some new variables &lt;span class=&quot;math-container&quot;&gt;$(x',y')$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best examples I've been able to come up with are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(\rho,v)$&lt;/span&gt;=density and velocity of water in a river; transformation is flying downstream or upstream in a plane&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(\text{lat},\text{lon})$&lt;/span&gt;=latitude and longitude, transformation is changing to some other set of coordinates&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm not really happy with either example. The &lt;span class=&quot;math-container&quot;&gt;$(\rho,v)$&lt;/span&gt; example is not necessarily easy for students with weak physical intuition, and it's not very exciting because &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt; doesn't change. The &lt;span class=&quot;math-container&quot;&gt;$(\text{lat},\text{lon})$&lt;/span&gt; requires projection to the plane of the page for purposes of drawing it, which is an extra complication, and there is not another set of coordinates that is necessarily familiar to most people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A rotation could be a good example, but it fails to illustrate that angles are not necessarily preserved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be happy with a good qualitative example. E.g., maybe something from economics or psychology, like &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;=happiness and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;=wealth.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2019-01-12T16:49:19.973" Title="Easy examples of visualizing passive transformations in the plane?" Tags="|graphing|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15071" PostTypeId="2" ParentId="14976" CreationDate="2019-01-12T14:25:16.777" Score="3" Body="&lt;p&gt;Links break. As a mod on another Stack, and a blogger, I'm painfully aware of this. &#xA;Please consider this a complement to Gerald's answer. i.e. He suggested a search instead of a hard link.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This particular very short video &lt;a href=&quot;https://youtu.be/BPK_4Ntq9Qo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Integration and Volume - Shell Method Animation&lt;/a&gt; was ok, as a nice example. But, as it was created with Desmos, I found I was able to 'steal' the graph equations to my own account. The Desmos link is &lt;a href=&quot;https://www.desmos.com/calculator/y46kazzniu&quot; rel=&quot;nofollow noreferrer&quot;&gt;Shells&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of the graph &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Otvda.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Otvda.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-01-12T14:25:16.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15072" PostTypeId="2" ParentId="15069" CreationDate="2019-01-12T16:49:19.973" Score="6" Body="&lt;p&gt;1 hotdog = 100 calories and 20 grams of protein. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 hamburger = 150 calories and 15 grams of protein. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any meal can be plotted in hamburger/hotdog space or calorie/protein space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might also be interested in my answer here:  &lt;a href=&quot;https://matheducators.stackexchange.com/a/5788/117&quot;&gt;https://matheducators.stackexchange.com/a/5788/117&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2019-01-12T16:49:19.973" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15073" PostTypeId="2" ParentId="5783" CreationDate="2019-01-13T17:20:34.057" Score="1" Body="&lt;p&gt;Surely you teach complex numbers in the curriculum. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing to do with matrices is to use them to create other mathematical objects, or at least a &lt;strong&gt;model&lt;/strong&gt; of the object. In particular, we can use &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; real matrices of the form &lt;span class=&quot;math-container&quot;&gt;$\left[ \begin{array}{cc} x &amp;amp; -y \\ y &amp;amp; x \end{array}\right]$&lt;/span&gt; to model the complex number &lt;span class=&quot;math-container&quot;&gt;$x+iy$&lt;/span&gt;. Matrix addition and multiplication then correspond naturally to the addition and multiplication of complex numbers and the complex number &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; corresponds naturally to the matrix &lt;span class=&quot;math-container&quot;&gt;$\left[ \begin{array}{cc} 1 &amp;amp; 0 \\ 0 &amp;amp; 1 \end{array}\right]$&lt;/span&gt; etc. In addition, in this discussion you can either look at the formula &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{z} = \frac{\bar{z}}{z\bar{z}}$&lt;/span&gt; as a method to find the multiplicative inverse of a complex number, or you can use the &lt;span class=&quot;math-container&quot;&gt;$2\times 2$&lt;/span&gt; inverse formula (if that is in the scope of the discussion). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably Steven Gubkin's answer is more in-line with your goals. But, I think highschool is the right place to start talking about this circle of ideas.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2019-01-13T17:20:34.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15075" PostTypeId="2" ParentId="14976" CreationDate="2019-01-14T01:12:27.883" Score="4" Body="&lt;p&gt;The most inexpensive and engaging material that you can easily slice is &lt;strong&gt;cake&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GJIUl.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GJIUl.jpg&quot; alt=&quot;Bundt Cake&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you purchase a Bundt cake pan, you can easily create something slice-able that is (roughly) a solid of revolution. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When you slice it in one direction (parallel to the table), you get washers. They are fairly compelling washers, too. You will be able to hold them up.&lt;/li&gt;&#xA;&lt;li&gt;When you slice it another way (awkwardly around the outside), you get cylindrical shells -- and, if your cake is moist enough, you will be able to unroll the shell to see that it roughly forms a rectangular box of volume &lt;span class=&quot;math-container&quot;&gt;$2\pi r h \Delta x$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Don't slice it the way you would normally slice this cake -- that doesn't form slices that are useful for calculus.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;When the students &lt;strong&gt;actually eat&lt;/strong&gt; part of a shell or part of a washer, they are more likely to remember the different shapes and what they mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember to bring something like a filet knife if you can; cutting the shells can be a little tricky. But you can do it on the document camera, so it's pretty compelling.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2019-01-14T01:12:27.883" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15076" PostTypeId="1" CreationDate="2019-01-14T16:10:25.887" Score="4" ViewCount="157" Body="&lt;p&gt;When I was in school, pupils were given numerical grades, or the equivalent of numerical grades but disguised as words, on their performance in various school subjects and also behaviour. A key feature of these is that it allows the pupils to compare their performance with their peers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a newspaper story, not recent anymore, a child was asked if they would be happy with receiving only information about their strengths and weakness, with no numerical grade. Their response was that it is okay, as long as they still get to know how good they are when compared to others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One might imagine that good students might get complacent or poor students might give up when such comparisons are possible, but one might also imagine good students getting self-confidence and poor students seeing that something is wrong and maybe deciding to fix the matter. One might assume that the negative effects outweight the positive ones; an answer to a question about grading assignments at a university certainly points at such a conclusion: &lt;a href=&quot;https://matheducators.stackexchange.com/a/2267/2083&quot;&gt;https://matheducators.stackexchange.com/a/2267/2083&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in any research concerning what effect quantitative, or quantifiable, feedback has on learning, particular when it comes to mathematics and to younger children. This would be compared to feedback which tells about the strengths and weaknesses of the pupil, but does not give an explicit means of comparing oneself to others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as this is quite specific, I am open to research where one or both of the conditions are loosened; the subject might not be mathematics and the students need not be young. I am particularly interested in the effect of evaluation that happens at the end or midpoint of school year, rather than the results of every exam or smaller test.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Some related studies&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Giving final grades is a common form of summative assessment. Finding studies about its effects on student learning seems tricky.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in &lt;a href=&quot;https://helda.helsinki.fi/handle/10138/41026&quot; rel=&quot;nofollow noreferrer&quot;&gt;Toimivatko päättöarvioinnin kriteerit? : Oppilaiden saamat arvosanat ja Opetushallituksen oppimistulosten seuranta-arviointi koulujen välisten osaamiserojen mittareina&lt;/a&gt; the author simply declares final grades as summative. This is reasonable for the article, which aims to investigate if the given grades are comparable in different schools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the article &lt;a href=&quot;https://tampub.uta.fi/bitstream/handle/10024/100405/978-952-03-0347-1.pdf?sequence=1&amp;amp;isAllowed=y#page=35&quot; rel=&quot;nofollow noreferrer&quot;&gt;Osaaminen ja hyvinvointi yläkoulusta toiselle asteelle&#xA;- Tutkimus metropolialueen nuorista&lt;/a&gt;, in second chapter, it is taken for granted that finishing grades of comprehensive school have an effect on future studies (in gymnasium or professional schools), but few references are given. There is nothing on comparison of giving or not giving such grades and the focus is on the transition from comprehensive school to the next stage in one's life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The developmental project &lt;a href=&quot;https://www.theseus.fi/bitstream/handle/10024/8044/J%C3%83%3Frvi.Hanna_Koski.Jari_Paananen.Heikki_Ranta.Mikko_Uimonen.Minna.pdf?sequence=2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arvioinnin vaikutus merkitykselliseen oppimiseen &#xA;opiskelijoiden ja opettajien näkökulmasta&lt;/a&gt; gets as close as I can find. They asked teachers and students in professional high school (university of applied sciences) and university about different ways of assessing in courses. There, both students and teachers found numerical assessment to be quite good and motivating. However, the study did not actually compare the effects on learning on different assessment methods.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-01-20T20:06:17.463" LastActivityDate="2019-01-20T20:06:17.463" Title="What effect does giving numerical or written grades have on learning?" Tags="|reference-request|primary-education|education-research|grading|general-pedagogy|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15078" PostTypeId="2" ParentId="7718" CreationDate="2019-01-15T00:01:52.227" Score="2" Body="&lt;p&gt;I was introduced to the concept of infinity early so I formed the intuition of the hyperreal number system. I think I also remember being taught in School that 0.999... = 1 and not understanding why that was true. I think that might have been after the time I was introduced to the concept of infinity and that might have been the reason. I was also told that &lt;span class=&quot;math-container&quot;&gt;$1 \div \infty = 0$&lt;/span&gt; outside of school but my intuition told be that &lt;span class=&quot;math-container&quot;&gt;$1 \div \infty \neq 0$&lt;/span&gt; which is how the hyperreal system really works. I think I may have also thought that only the notations that eventually terminate whether at finite or infinite position represent a number similar to the method of thinking shown at &lt;a href=&quot;https://www.youtube.com/watch?v=wsOXvQn3JuE&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=wsOXvQn3JuE&lt;/a&gt; as a result of being introduced to infinity early and from that assumption deduced that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{6} = \frac{1}{3} \div 2 = 0.333...333 \div 2 = 0.1666...6665$&lt;/span&gt; because I had already figured out on my own how to divide by 2 in decimal. Without realizing it, my assumptions as a whole were contradictory so it can in fact be shown that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{6} = 0.1666...6665 \neq \frac{1}{6}$&lt;/span&gt;. It may have come from the Burali-Forti paradox of Naive set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much later, I thought I finally figured out a way to construct the real numbers properly from the dyadic rationals and understood that the real number system is a different system where there are no infinite or infinitesimal numbers. I later discoved Cantor's paradox by myself. Even later, I discovered by myself that I cannot prove the axiom of choice. It turns out that without realizing it, I didn't really figure out how to construct the real numbers because I had also earlier thought of my own definition of a natural number independently which turned out to be the same as the definition on page 25 of the book &lt;em&gt;A first course in real analysis&lt;/em&gt; which said an inductive set is a subset of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; such that 0 is in the subset and for every real number in the subset, that number plus 1 is also in the subset and a natural number is a real number that every inductive set contains. Now I realize that you can construct the natural numbers as finite ordinal numbers then construct the integers then construct the dyadic rationals then construct the real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was 21 at the time I discovered on my own that I can't figure out how to prove the axiom of choice and I feel like after the time I discovered that, I got helped even more by discovering it on my own at that age than I would have by being told at a young age why it wasn't provable. I think by brain was muture enough that it could handle the change and I acutally liked breaking by old habits and learning how to adapt to think in a totally different way, unlike the Pythagoreans who hated the discovery that irrational numbers exist. I'm glad I didn't get told that at a younger age because then otherwise, my brain might have missed out on breaking its old habits at an older age and becoming even better as a result. &lt;a href=&quot;https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html&lt;/a&gt; seems to be a sign that my brain may have actually benefitted from discovering the unprovability of the axiom of choice at the age of 21.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="11718" LastEditDate="2019-01-15T00:17:58.250" LastActivityDate="2019-01-15T00:17:58.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15082" PostTypeId="2" ParentId="4018" CreationDate="2019-01-15T15:56:52.610" Score="2" Body="&lt;p&gt;Rob Beezer at the University of Puget Sound has an open &quot;&lt;a href=&quot;http://linear.ups.edu/scla/html/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Second Course in Linear Algebra&lt;/a&gt;&quot; which, though I believe still in draft form, he has used for an advanced linear algebra course a few different places, and contains many of the topics you just mentioned.  (Well, actually the vector spaces are in the &lt;a href=&quot;http://linear.ups.edu/fcla/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;first course&lt;/a&gt;.)  I'm not familiar with the other texts you mention, though, so I can't compare - this is just a data point, if a very late one.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-01-15T15:56:52.610" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15083" PostTypeId="2" ParentId="1113" CreationDate="2019-01-16T14:44:01.007" Score="4" Body="&lt;p&gt;Coming extremely late to this question.  I certainly agree with historical approaches of various kinds as being appropriate in this context (presumably also in a &quot;non-spiritually-uplifting context&quot; if that were requested as well) and infinity is always fun.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I notice that two issues of application have not been addressed by previous answers.  References abound on the internet so I will only put in a couple likely-to-die links.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a lot of discussion of the &quot;Unreasonable Effectiveness of Mathematics&quot; article of Wigner's.  That article itself has been endlessly discussed, including whether the effectiveness is reasonable, whether there is a reasonable ineffectiveness, and so forth, but at any rate this shows up a lot in discussions of this type, e.g. Chapter 8 of &quot;&lt;a href=&quot;https://www.harpercollins.com/9780062024473/mathematics-through-the-eyes-of-faith/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics Through the Eyes of Faith&lt;/a&gt;&quot;, and I think some discussion of it can be &quot;uplifting&quot; to some.  Obviously calculus has a lot to do with some of these modeling issues in the physical sciences.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Secondly (and this could cut even into &quot;morally uplifting&quot;, it's quite broad) there is the issue of which applications to pursue, and how.  Particularly in linear algebra, which underlies so much of modern data analysis/science/AI/you name it, there are real questions of how much to just let PCA give you answers and how much you are asking the right questions.  &quot;&lt;a href=&quot;https://weaponsofmathdestructionbook.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weapons of Math Destruction&lt;/a&gt;&quot; is a fairly strident take on this, and one might think it's not &quot;uplifting&quot; to dwell on such things - but considering that one has the opportunity to use mathematics to &lt;em&gt;discover&lt;/em&gt; places where &quot;the algorithm&quot; has helped perpetuate bias, and combat that, I think this would fit the bill as well.  In fact, many humanitarian NGOs now are using tools like linear algebra and OR to help inform their resource allocation to respond even better to crises in the future.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-01-16T14:44:01.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15084" PostTypeId="2" ParentId="7837" CreationDate="2019-01-17T16:11:12.100" Score="7" Body="&lt;p&gt;I was surprised to see such an old question with so many answers, all of them algebraic. Here is a proof (almost) without words that &lt;span class=&quot;math-container&quot;&gt;$1/\frac{a}{b} = \frac{b}{a}$&lt;/span&gt;, which I believe is the crux of the question. It uses the definition of multiplication as the area of a rectangle with the given side lengths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JPiTh.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JPiTh.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3381" LastEditorUserId="3381" LastEditDate="2019-01-17T17:15:44.593" LastActivityDate="2019-01-17T17:15:44.593" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15085" PostTypeId="2" ParentId="7837" CreationDate="2019-01-17T21:08:13.920" Score="4" Body="&lt;p&gt;I'll mention a very simple idea which, it seems, hasn't come yet: get some intuition from addition and subtraction, to which multiplication and division are analogous. I'll run through it with the divide-by-&lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; example asked about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can intuitively see that subtracting &lt;span class=&quot;math-container&quot;&gt;$2-3$&lt;/span&gt; is the same as adding &lt;span class=&quot;math-container&quot;&gt;$3-2$&lt;/span&gt;, because subtraction reverses addition. One needn't go into negative numbers with such an explanation as, &quot;to subtract an amount means to add its additive inverse, which for a difference of two numbers is that difference backwards&quot;. Instead one can get by with noting, &quot;the answer must be &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; more than if I'd subtracted &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, so  may as well add &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; before the subtraction, which means I'm adding &lt;span class=&quot;math-container&quot;&gt;$3-2$&lt;/span&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But however you explain it, the result is clear, exchanging the numbers we're subtracting (reverse-adding, if you'll forgive the Newspeak) has the effect of adding the post-exchange difference. Well, division reverses multiplication, so by the same logic dividing by &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; should mean multiplying by &lt;span class=&quot;math-container&quot;&gt;$3/2$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5794" LastEditorUserId="5794" LastEditDate="2020-02-10T19:58:11.383" LastActivityDate="2020-02-10T19:58:11.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15086" PostTypeId="1" CreationDate="2019-01-18T16:25:56.700" Score="7" ViewCount="301" Body="&lt;p&gt;I'm looking around for a text that covers vector calculus and multivariable calculus, and that is also &quot;free as in speech,&quot; not just &quot;free as in beer.&quot; In other words, I'm looking for texts that are not just available for downloading free of charge but that are also under a license such as GFDL or Wikipedia's CC-BY-SA, so that their contents can be reused, modified, remixed, etc., even for commercial purposes. Does anyone know of any good books that fit these criteria? The only one I know of is Hoffman, &lt;a href=&quot;http://scidiv.bellevuecollege.edu/dh/Calculus_all/Calculus_all.html&quot; rel=&quot;noreferrer&quot;&gt;Contemporary Calculus&lt;/a&gt;, which covers two years of calculus.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-03-07T13:36:10.340" Title="Vector calculus texts that are free-as-in-speech?" Tags="|undergraduate-education|textbooks|vector-calculus|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15088" PostTypeId="2" ParentId="15086" CreationDate="2019-01-19T00:09:05.217" Score="1" Body="&lt;p&gt;I believe &lt;a href=&quot;https://www.whitman.edu/mathematics/multivariable/&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Community Calculus&quot;&gt;Community Calculus&lt;/a&gt; is exactly what you're looking for. It is under a CC BY-NC-SA 3.0 license.&lt;/p&gt;&#xA;" OwnerDisplayName="user8624" LastEditorUserId="64" LastEditDate="2020-03-07T13:36:10.340" LastActivityDate="2020-03-07T13:36:10.340" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15089" PostTypeId="2" ParentId="13745" CreationDate="2019-01-19T00:42:52.320" Score="0" Body="&lt;p&gt;It seems to me that with infinite series there are several distinct concepts and would be good for the students (and everyone) to be able to distinguish between them:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a finite sum of infinitely many numbers (cf. Zeno)&lt;/li&gt;&#xA;&lt;li&gt;calculating what that sum must be, as an equivalent limit&lt;/li&gt;&#xA;&lt;li&gt;the meaning of the notation &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=0}^\infty x_n$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;how to safely manipulate infinite sums (by manipulating limits instead).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My issue is that if we say the infinite sum of all the &lt;span class=&quot;math-container&quot;&gt;$x_n$&lt;/span&gt; doesn't actually exist, only the limit, we're perilously close to saying (or maybe &lt;em&gt;actually&lt;/em&gt; saying) that the infinite set &lt;span class=&quot;math-container&quot;&gt;$\{x_n:n\in \mathbb N\}$&lt;/span&gt;  doesn't exist either. Or for that matter, any infinite set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd say the accurate picture is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;infinite sums exist, but not every infinite series has one.&lt;/li&gt;&#xA;&lt;li&gt;they can usually only be calculated via limits. (With exceptions, such as geometric series.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\sum_{n=0}^\infty$&lt;/span&gt; represents the result of taking the limit if one exists, not an instruction to &quot;start at &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and stop at &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;—and &quot;stop at &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;&quot; is the self-contradictorty concept we need to avoid acquiring.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastEditorUserId="10663" LastEditDate="2019-01-19T01:10:05.227" LastActivityDate="2019-01-19T01:10:05.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15090" PostTypeId="2" ParentId="15086" CreationDate="2019-01-19T00:45:10.350" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://openstax.org/subjects/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax&lt;/a&gt; textbooks are freely available under a Creative Commons license. They have a series of three calculus books, including multivariable calculus. I haven't used that book, but I have taught using their Calculus 2 book, which I found to be reasonably good in most respects (and without any really critical flaws, though there are some aspects that could use improvement).&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2019-01-19T00:45:10.350" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15091" PostTypeId="2" ParentId="7528" CreationDate="2019-01-20T23:58:04.790" Score="3" Body="&lt;p&gt;From an MathOverflow question,&#xA;&lt;a href=&quot;https://mathoverflow.net/q/321348/6094&quot;&gt;Six yolks in a bowl: Why not optimal circle packing?&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/pyDgT.jpg&quot; width=&quot;200&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Six yolks in a bowl.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-01-20T23:58:04.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15092" PostTypeId="2" ParentId="7528" CreationDate="2019-01-21T05:25:16.263" Score="0" Body="&lt;p&gt;Many molecules have striking shapes.  For example cubane:  &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubane&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Cubane&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11788" LastActivityDate="2019-01-21T05:25:16.263" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15093" PostTypeId="1" CreationDate="2019-01-21T18:57:40.720" Score="9" ViewCount="2556" Body="&lt;p&gt;For example, I am confident that very few students majoring in pure mathematics can write a complete proof to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem&quot; rel=&quot;noreferrer&quot;&gt;Abel–Ruffini theorem&lt;/a&gt; (there is no algebraic solution to general polynomial equations of degree five or higher with arbitrary coefficients) by the time of their graduation. I suspect many students with a Master's degree or Doctorate in pure mathematics could not prove this theorem either. They may know the conclusion, but may not be able to sketch an idea of the proof, let alone give a complete proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: should we educate pure mathematics major students in such a way that they should know how to prove most of the classical results in mathematics such as the Abel–Ruffini theorem and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra&quot; rel=&quot;noreferrer&quot;&gt;Fundamental Theorem of Algebra&lt;/a&gt; before getting their bachelor's degree, or at least their master's Degrees?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="-1" LastEditDate="2019-01-22T22:05:18.197" LastActivityDate="2019-02-07T21:06:10.147" Title="Are students majoring in pure mathematics expected to know classical results in mathematics very well by their graduation?" Tags="|undergraduate-education|algebra|graduate-education|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15094" PostTypeId="2" ParentId="5783" CreationDate="2019-01-21T20:02:41.523" Score="0" Body="&lt;p&gt;Try to handle and solve for the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s conceptually this system,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CcPQ6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CcPQ6.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or this one&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$A\mathbf x = \mathbf b$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is a &lt;span class=&quot;math-container&quot;&gt;$K\times K$&lt;/span&gt; matrix, and &lt;span class=&quot;math-container&quot;&gt;$\mathbf x, \mathbf b $&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$K \times 1$&lt;/span&gt; vectors, &lt;span class=&quot;math-container&quot;&gt;$K=10^6$&lt;/span&gt;. Hmm, why not set &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; equal to one trillion? Or a hundred thousand trillion.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And when the Matrix Algebra interrelations start to become more and more intricate and impressive, beyond the basic level of solving a linear system of equations, and still the dimensions can be as large as we want, this is how I fell for Linear Algebra and matrices: once I realized that this dance remains elegant irrespective of the size of the entities we dance with (while also being able to visualize somehow the enormity of them -in most other mathematics, the object dimensions are a bit too abstract for most people).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's like Circus Human Acrobatics perfectly performed by elephants. Or galaxies.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2019-01-21T20:02:41.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15095" PostTypeId="2" ParentId="15093" CreationDate="2019-01-21T20:19:20.593" Score="16" Body="&lt;p&gt;For the project, you would need to define what are &quot;classical results in mathematics&quot;.&#xA;I suspect that different people would disagree on the classicalness of various results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, it is doubtful if everyone should learn the same classical results - mathematics is a big field. You can, of course, cut it down by a sufficiently strong definition of &quot;pure mathematics&quot;, in which case, maybe everyone should know the same results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it would be more fruitful to ask what results a particular university or study programme should consider as key results, concepts and tools. This nicely sidesteps the logistical issue of coordinating mathematics study programmes worldwide and the potentially ugly definitional issue of who qualifies as a pure mathematician. Furthermore, since it retains the current situation of different mathematicians knowing different tools, it widens the scope of overall knowledge. Collaboration with and research periods at foreign universities, or at least universities with different priorities, is useful partially due to such differences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is the further issue of how much one can realistically ask of bachelor (or even master) students. As Steve Gubkin mentioned in a comment, many bachelor students have problems with even simple proofs. Depending on how wide one wishes to cast the net of &quot;classical results&quot;, it might be unfeasible to teach them to everyone, without making &quot;everyone&quot; a much smaller set then nowadays.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2019-01-21T20:19:20.593" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15096" PostTypeId="2" ParentId="15093" CreationDate="2019-01-21T23:21:46.503" Score="14" Body="&lt;p&gt;This is an interesting question, but, understandably, confounds at least two different things. E.g., is it really the case that to &quot;know&quot; a true mathematical fact is to be able to produce its proof on command? I think not. Another diagnostic question: must we understand thermodynamics and the Carnot cycle to drive a car usefully? Must we be able to prove the stability of the proton before setting our coffee cup on the table? Yes, of course, I'm exaggerating... but my exaggeration is in the direction I think is relevant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Namely, &lt;em&gt;awareness&lt;/em&gt; is the key point (and assimilation of the facts into one's world-view... to the extent that they might have some impact and affect one's own decisions). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My opinion on this is in the same vein as my objection to people being told to do every exercise before moving forward: not only are many of those exercises either make-work or pranks, but many are also incomprehensible without understanding what happens in the sequel... which one will not see until after? A bit perverse. Sure, some such pranks are &quot;fun&quot; in Math Olympiads and Putnam and such, but...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem that I see is that undergrads are too often conditioned to be paranoid that there's some unfathomable flaw in what they've written... that can only be adjudicated by the oracular professor. One of the worst corollaries of this is that kids are very inhibited about broadening their scope, because they're already worried about defending themselves with regard to a tiny, trivial &quot;plot of land&quot;, and are taught to give no credence to their own critical faculties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, I think this question raises some good issues, but is literally a bit mis-aimed in its assumptions.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2019-01-21T23:21:46.503" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15097" PostTypeId="2" ParentId="737" CreationDate="2019-01-22T00:06:28.163" Score="1" Body="&lt;p&gt;I agree that it is difficult to communicate the issue of well-definedness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, obviously, in real life there is nothing like this. We rarely accidentally make a reference to a thing that is non-existent (well, ok, let's ignore politics for a moment).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, really, people have absolutely no experience with this potential issue. And, in fact, I would claim that (by this year) a mathematical discussion that includes a requirement of proving something is &quot;well-defined&quot; has prior epistemological and ontological problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, this is a bit of a &quot;peeve&quot;. I claim that the issue should be about &lt;em&gt;existence&lt;/em&gt; of a (often &quot;universal&quot; in some way...) object... as opposed to writing formulas that may or may not make sense. Yes, I understand that a certain philosophical viewpoint retreats (!?) to only claiming that well-formed formulas are all we should contemplate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, I'd claim that students should rarely be asked to &quot;prove that X is well-defined&quot;. They have no idea, and discovering that fact proves nothing. The more genuine question, in my opinion, is whether the alleged X is sensible or not. Or is it random and pranky? For math people, I think we really should recover from the random-pranky lifestyle... supposedly (but dubiously) teaching students about logical precision.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quite seriously, what I try to communicate to students is that mere formulaic stuff is not what we want. Writing strings of symbols guarantees nothing. There should be a meaning behind the narrative. In that context, what the heck could &quot;well-definedness&quot; possibly be? Seriously! It is a bit amusing that mathematics has arrived in a situation where &quot;well-definedness&quot; should be a real issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By this year, I try to insist that the math grad students respond in a different way to such questions... In particular, that the notion of &quot;well-definedness&quot; is generally deficient, if lacking a larger context. Yes, I do claim that the notion itself is intrinsically problemmatical. Not that it won't arise in certain formalized situations, but that it doesn't capture anything truly useful.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2019-01-22T00:06:28.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15098" PostTypeId="2" ParentId="2679" CreationDate="2019-01-22T05:55:42.247" Score="2" Body="&lt;p&gt;Please warn me if my rewrite of &lt;a href=&quot;https://ma.reddit.com/r/explainlikeimfive/comments/58cdw3/eli5_the_monty_hall_problem/d8za610/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this /r/eli5 comment&lt;/a&gt; doesn't answer your bolded question.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;It helps to get rid of the standard trappings of the problem. The problem beneath is mathematically the same, but makes it clearer why you ought switch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are a superhero patrolling a crowded train station. A male stranger comes up to you, and asks you to pick a person at random, out of a crowd of thousands. We'll call your pick person A.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This stranger then tells you that he's part of a math-themed supervillain militant movement. He explains that one of the people in the crowd is their agent, who has a bomb that will blow up the city.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Seeing the worry in your eyes---and a total lack of mathematical thinking given the crisis—he says that they'll even up the odds a bit: he'll narrow all but two of the people in the crowd who may be carrying the bomb to person: A (the person you picked at random without even knowing what you were doing), and B (their agent). He guarantees that A or B has the bomb, which'll detonate in a few seconds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, who would you think has a better chance of carrying the bomb: your random pick (A) or his pick (B)? If you only had time to disarm one of them, would you go for A or B?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This ought clarify why you “switch” rather than just, say, flip a coin. The odds that the bomb is on A, are a truly random chance from the original cast of thousands. The odds that B has the bomb are obviously higher, since B MUST have the bomb if your original choice was wrong, and your original choice only had a &lt;span class=&quot;math-container&quot;&gt;$\dfrac{1}{thousands}$&lt;/span&gt; chance of being correct.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2019-01-22T05:55:42.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15100" PostTypeId="2" ParentId="15040" CreationDate="2019-01-22T14:00:09.573" Score="1" Body="&lt;p&gt;That’s not a yes or no question. The proper way to teach mathematics is to give all (or all main) derivations, sometimes even different ways of getting those derivations and formulas. The students must understand why the formulas are the way they are. They must have a feel of the logic behind. They should be able to prove and derive formulas with confidence. At the same time some books may be either excessively rigorous or not rigorous enough. Some books may contain too many proofs while others may cut down on proofs way too much. A good balance is needed. Much more serious problem is the lack of sound knowledge of of precalculus. Teaching calculus theorems and their proofs or theorems from linear algebra is hardly acceptable if a student can’t derive the &lt;span class=&quot;math-container&quot;&gt;$\sin(a+b)$&lt;/span&gt; formula or prove the law of cosines. Some can’t even derive the formula for the quadratic equation – totally unacceptable but it is common! So, basic stuff and most glaring gaps should be addressed right away, otherwise we are going to erect a building without any foundation that will tumble down in a light breeze.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a serious problem in teaching math. It can only be solved by filling out all the fundamental gaps and by filling them properly. It basically means that students should optimally take College Algebra (precalc) and should study it properly, i.e. with understanding all the basic proofs and derivations. Students should also be confident when they reason and make derivations. They should have a feel and a solid grasp of what they are doing. They should be able to debunk false proofs or false steps if those are intentionally thrown at them for confusion. It’s difficult when standard curriculum cuts down on proofs, derivations and explanations. Yet, just following algorithms or doing Schaums problems is not enough in itself. It’s not the right approach to understand math. Just plugging numbers in or following algorithms without real understanding won’t cut it. The balance between the amount of rigor and derivations should boil down to the course books chosen. I never use books that severely lack derivations and explanations. On the other hand a text like Calculus by Apostol might be overkill for students in engineering. It depends on students and the college. We have to choose middle ground. I gave extensive recommendations on books but that post is not available. Good books will have the right amount of rigor and derivations for engineering students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, I'm not going to leave this answer in the dark and I'm going to list a few books. For engineering students a good &quot;dip stick&quot; of how many proofs, derivations and how much rigor is a good balance will be the books on Calculus by Larson or Calculus by Marsden; Linear algebra by Strang or Linear Algebra by (this one is probably overkill) Kolman; &lt;strong&gt;Algebra and Trigonometry by Larson or the same title by Stewart&lt;/strong&gt; (this course is usu. &lt;strong&gt;called College Algebra and is a foundation without which no calculus or linear algebra can be studied because without it everything is just rote-learning in the worst sense of the word&lt;/strong&gt;). That's the frame of reference in terms of derivations and proofs. It' s the right balance for engineering students. &lt;strong&gt;What is important here is not to omit derivations, proofs and explanations in these books&lt;/strong&gt;. Said books already contain derivations/proofs in a very digestible form and in the minimal amount. Just doing template problems is a huge malpractice. Derivations and explanations are also part and parcel of studying math just like doing standard problems. &lt;strong&gt;Derivations should never be skipped or turned a blind eye to even if they are in the book's addendum! Your students are better off by skipping some chapters rather than explanations and derivations/proofs.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All the necessary proofs/derivations are in respected books I gave. Okay, you can cut down a bit more on them but I'd rather not. It's already the bare minimum. I suggest cutting down on chapters and material, not on explanations/derivations/proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Important note. All students say they know precalc. but they never do! All the know is a bit of selective rote-learning.&lt;/strong&gt; And then they plunge into calculus, linear algebra, etc. &lt;/p&gt;&#xA;" OwnerUserId="10689" LastActivityDate="2019-01-22T14:00:09.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15103" PostTypeId="1" CreationDate="2019-01-23T03:39:53.813" Score="3" ViewCount="128" Body="&lt;p&gt;This more specific &lt;a href=&quot;https://matheducators.stackexchange.com/questions/15093/are-students-majoring-in-pure-mathematics-expected-to-know-classical-results-in&quot;&gt;question&lt;/a&gt; relates to a more general question of what is a maths degree aiming for.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do any universities define a high level goal for pure mathematics degrees at all?&lt;/li&gt;&#xA;&lt;li&gt;If so, are there specific differences in the goals of higher degrees?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also interesting:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Whether related institutions of bodies have something along this line&lt;/li&gt;&#xA;&lt;li&gt;Whether the concept is seen as too narrowing and consciously avoided&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;@james-s-cook 's comments below draw out that any answer is likely country specific. An honours degree from the UK is much more specialized than typical for most of the US. Perhaps equivalent to a research specialization.&lt;/p&gt;&#xA;" OwnerUserId="3587" LastEditorUserId="3587" LastEditDate="2019-01-23T07:20:46.260" LastActivityDate="2019-01-23T14:11:01.217" Title="Is there any alignment on what a maths grad should know?" Tags="|undergraduate-education|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15104" PostTypeId="2" ParentId="15103" CreationDate="2019-01-23T06:55:22.607" Score="4" Body="&lt;p&gt;(Addressing the second half of the question)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The UK QAA has a subject benchmark for what it means to have an undegraduate degree in maths/stats/OR (and other ones for other subjects). In terms of content (explicitly or implicitly listed), it specifies:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;some linear algebra (ie vectors/matrices)&lt;/li&gt;&#xA;&lt;li&gt;some calculus (ie differentiation/integration)&lt;/li&gt;&#xA;&lt;li&gt;some modelling&lt;/li&gt;&#xA;&lt;li&gt;some estimation&lt;/li&gt;&#xA;&lt;li&gt;a mathematical computing package&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-01-23T06:55:22.607" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15105" PostTypeId="2" ParentId="15103" CreationDate="2019-01-23T14:11:01.217" Score="-1" Body="&lt;p&gt;This may be largely a matter of opinion, but:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal of a Ph.D. program is to train researchers.  The expectation is that, as a Ph.D. student, you will be guided by a mentor to produce some original, publishable work, called a thesis (or dissertation).  The expectation is that by the time you finish your Ph.D., you should be able to continue to work and write papers on your own, and perhaps to mentor students of your own (either at the doctoral level, or at the undergraduate level).  A Ph.D. program is, in essence, the end of an apprenticeship program in academia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, the job of a bachelor's degree program was to prepare younger students for advanced degrees&amp;mdash;the bachelor's program was, more or less, the first part of an apprenticeship in academia.  The goal of such a program was (and ideally, still is) to give students the background necessary to start their own research.  Depending on how specialized the program is, this might mean specializing in mathematics very early (this seems to be the European model), or acquiring a broad background for a couple of years, then specializing (this is a more American model).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking more specifically of the United States, someone with a bachelor's degree in mathematics should, ideally, have the background required in order to start a Ph.D. program (since the underlying goal of a bachelor's degree, in my opinion, is to provide training to nascent researchers).  In as much as there is a &quot;high level&quot; goal, I would say that this is it.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As you also seem to be interested in &lt;em&gt;curriculum&lt;/em&gt;, an undergraduate degree should give a student enough background to start a Ph.D. program.  The usual expectation is that students entering a Ph.D. program (in &quot;pure&quot; mathematics, rather than applied mathematics, statistics, or mathematical education; in the US) should know at least:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;computational techniques, which are typically taught in lower division calculus, linear algebra, and differential equations courses;&lt;/li&gt;&#xA;&lt;li&gt;some basic mathematical logic&amp;mdash;not necessarily anything foundational, but enough to argue mathematically using (for example) contraposition, induction, exhaustion, and so on;&lt;/li&gt;&#xA;&lt;li&gt;some basic real analysis (the theory of differentiation and Riemann integration over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^d$&lt;/span&gt;);&lt;/li&gt;&#xA;&lt;li&gt;some basic modern algebra (the study of groups, rings, and fields; maybe some Galois theory);&lt;/li&gt;&#xA;&lt;li&gt;point-set topology; and&lt;/li&gt;&#xA;&lt;li&gt;something a little more specialized.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Basically, a student should be able to perform computations, write a coherent proof, have familiarity with the essential pillars of modern mathematics (analysis and algebra, IMO), and have been exposed to something a bit more specialized which they might&amp;mdash;or might not&amp;mdash;follow up on in a Ph.D. program (complex analysis, knot theory, fractal geometry, graph theory, mathematical modeling, number theory, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are interested in specifics of what Ph.D. programs expect students to know, you might look at the &lt;a href=&quot;https://www.ets.org/gre/subject/about/content/mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;GRE Mathematics Subject Test&lt;/a&gt;.  Many (most?) US Ph.D. programs in mathematics require GRE math subject scores for admission.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2019-01-23T14:11:01.217" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15107" PostTypeId="2" ParentId="557" CreationDate="2019-01-23T22:17:22.793" Score="0" Body="&lt;p&gt;Ban technology.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you let kids in football practice or weightlifting session text on their phones?  Running the obstacle course? No.  Eff no.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note:  you are not &quot;living in Y2K&quot; or walling them off from tech or preventing them from playing on their phones at a bazillion other times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  And if someone opens a damned newspaper in class...grr.  Throw their ass out.  A little balls goes a long ways.  Look how the general came onto base in 12 O'clock High.  (People are emotional animals, not computers...learn from history and literature.  Pedagogy is a matter of practical psychology NOT some Rudin real analysis nitanoid gotcha.)&lt;/p&gt;&#xA;" OwnerUserId="11803" LastActivityDate="2019-01-23T22:17:22.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15108" PostTypeId="1" CreationDate="2019-01-23T22:25:04.633" Score="3" ViewCount="74" Body="&lt;p&gt;Can you please explain or provide resources that explain this issue (esp. to a lay audience):&#xA;How to understand the significance of a co-relation number that studies (esp. meta-analyses) produce. The difficulty is from moving from the number to the practical implication of the number (what does it mean for the reader?)  &lt;/p&gt;&#xA;&#xA;&lt;h2&gt;The contrast is between an implications for a lay individual vs group/academic research purposes.&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Please note that while comments about the following example are appreciated, the main purpose of the post is to be able to deal with correlations in general not the specifics of the example &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:, the correlation between general mental ability (GMA) and job performance is .5.&#xA;See the difficulty of interpreting that number here&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.facebook.com/groups/473795076132698/permalink/1072863099559223/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.facebook.com/groups/473795076132698/permalink/1072863099559223/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When a lay person sees the number .5, how can they better understand it? Is that high, low, moderate etc?? For me as an individual, is it worth considerable time and effort to raise my GMA? If I'm an employer, do I  eliminate a candidate because of a low GMA ? Do I specify from the beginning that my applicant pool will be from a certain GMA etc..?  How much important is GMA (.5) in comparison to a correlation of .4 or .2? If a correlation is .1, should I ignore it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;===&#xA;Answers/resources up to now:&#xA;1-  Against Individual IQ worriers (Scott Alexander)&#xA;&lt;a href=&quot;https://slatestarcodex.com/2017/09/27/against-individual-iq-worries/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://slatestarcodex.com/2017/09/27/against-individual-iq-worries/&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2-  A video explaining the concept &quot;Co-relation does not equal causation&quot;&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=gxSUqr3ouYA&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=gxSUqr3ouYA&lt;/a&gt;&#xA;I explain more about what I mean by &quot;resources&quot; and &quot;demonstrations&quot; here&#xA;&lt;a href=&quot;https://www.researchgate.net/post/List_of_Concept-Demonstrating_Resources&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.researchgate.net/post/List_of_Concept-Demonstrating_Resources&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11804" LastEditorUserId="11804" LastEditDate="2019-01-24T19:46:37.597" LastActivityDate="2019-01-25T08:06:14.553" Title="Practical Implication for Correlation (Lay individual vs Scholarly Group)" Tags="|education-research|general-pedagogy|statistics|applied-mathematics|applications|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15109" PostTypeId="1" AcceptedAnswerId="15136" CreationDate="2019-01-24T00:15:27.153" Score="11" ViewCount="4426" Body="&lt;p&gt;There are so many datasets available that it seems difficult to sort through them all and identify the best ones to use when teaching an introduction to statistics class. What are your favorite datasets to use when teaching statistics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want the students to have some interesting and fun datasets to work with when experimenting with data visualization, constructing confidence intervals, hypothesis testing, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's possible this question will be closed as being a duplicate of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7976/where-can-i-find-realistic-data-for-college-level-elementary-statistics-problems&quot;&gt;this question&lt;/a&gt; or for being opinion-based, but I'd like to know the specific datasets that you love to use when teaching. For example, DASL has been mentioned, but DASL is large -- which dataset within DASL do you like? &lt;/p&gt;&#xA;" OwnerUserId="9633" LastEditorUserId="238" LastEditDate="2019-10-03T13:12:51.150" LastActivityDate="2019-10-03T13:12:51.150" Title="Favorite datasets to use when teaching statistics" Tags="|statistics|big-list|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15110" PostTypeId="2" ParentId="15108" CreationDate="2019-01-24T06:45:15.863" Score="2" Body="&lt;p&gt;Correlation tells you how much one quantity can be relied upon as a measure of the other quantity. The usefulness of that is when you want to measure one but can't, and the other is easier to measure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, you can't easily measure how much people like your webpage. But you can measure what proportion click through to a second page, or send a link to a friend...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the key thing a lay person needs to know is that there is no simple interpretation of whether the value is high/low/important. In particular, they should learn not to take media reports at face value. Those who can should consider reading what they can of the original paper, as some of the time the interpretation/caveats are available and intelligible. Otherwise, if the idea seems like it might have an impact on their lives, they should file the information under 'find out more before making a decision'.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;More detail:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A high correlation says that if property A increases, it is very likely that property B increases too. If the correlation is negative, it means A increases is linked to B decreasing. But the correlation doesn't say anything about the size of the increase/decrease. For that you need to look at other information from the data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the context of the question: a correlation of 1 would say that smarter people always do better at their job. But that doesn't have anything at all to say about how much better they do. If the total variation is, say, worth 10 seconds a day, it's not going to be worth considering. But if the range is worth 5 days per month, that might make it worth the effort of considering. The correlation does not give that information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Deciding whether a particular correlation is high/low also requires additional information. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there are two data points, there will (almost) always be a correlation of 1 or -1, because the second reading was either higher or lower than the first (if your measure is discrete, it could also be the same, giving a correlation of 0). Similarly, with three data points, there are usually only two really-different options: the graph keeps going in the same direction, or it changes between up and down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In these cases, what appears to be a very strong correlation is more likely just the result of insufficient data. As the number of data points increases, you are less likely to get a high correlation coefficient by pure chance, and so getting a high number becomes more significant. The majority of researchers will check whether the number is 'high enough' by checking a table, rather than any direct understanding/calculation of their own, and any respectable researcher will report the result of this look-up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice, if your news source is reputable, it is reasonably safe to assume that if the result is being reported, the correlation coefficient has been found to be high enough. But, as mentioned above, that does not tell you how you should respond. There are several factors that still need to be considered:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the quality of the study: poorly-done research may not give true results. In extreme cases, there have been researchers who made up their data to get funding/meet job requirements.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the size of the effect: (as described above). One example here is 'learning styles'. Research shows that people do respond differently to different methods of teaching. However, it also says that this effect is swamped by other factors (such as the teaching style that works best for the content being taught).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the applicability of the study to you: each study occurs in a particular context. How much the result applies to you personally depends on how well that context matches your circumstances. For example, a study of women's health should only be applied to men with caution.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;other data: a single study, however large, is not guaranteed to give the truth. Other results might give conflicting information. You may or may not have access to these. Drug trials are like this - the drug company has to show that the drug does it's job and doesn't cause (too much) harm. However, they are not required to tell you how many times they tried to get evidence and failed.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;correlation does not imply causation: eating ice-cream is correlated with heat-stroke. This does not mean that eating less ice-cream will protect you from heat stroke (indeed, I would guess the reverse is true); it is simply that the two occur in the same conditions. Knowing there is a correlation does not tell you how you should act.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;cost/benefit ratio: being alive increases the chance of getting cancer. Deciding to stop being alive to avoid getting cancer would be ... ill-advised. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="4746" LastEditDate="2019-01-25T08:06:14.553" LastActivityDate="2019-01-25T08:06:14.553" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15111" PostTypeId="2" ParentId="15093" CreationDate="2019-01-24T08:48:20.093" Score="6" Body="&lt;p&gt;It just depend on what do you mean by &quot;knowing very well&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me &quot;knowing very well&quot; means: if you give me 1 hour I can tell you the basic ingredients that go in the proof (and the reason why the result is relevant) and if you give me 1 full day I can sketch a reasonably detailed proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I know how to fit the result in an area of math;&lt;/li&gt;&#xA;&lt;li&gt;I know on which books I'd rather look for it (which is very personal, depends on your own background);&lt;/li&gt;&#xA;&lt;li&gt;I am capable of looking back at the relevant prerequisites and recover them quite fastly.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Then yes, that's what we hope from our students. And it is already a quite high level of demand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(as for the example, I got a PhD in math without ever being exposed to a proof of Abel-Ruffini, so &quot;classical&quot; means different things to different people. I think to grasp the proof of it I'll probably need a week) &lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2019-01-24T08:48:20.093" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15112" PostTypeId="1" CreationDate="2019-01-24T10:43:12.740" Score="4" ViewCount="165" Body="&lt;p&gt;I am, at the moment, teaching calculus to students whose majors are, for example, biology, biochemistry, chemistry and geology. The course book is Claudia Neuhauser's &quot;Calculus for biology and medicine&quot;, third edition. The course focuses on mathematical modelling and there are also calculation, but very few proofs. My role is that of an øvingslærer; I make exercises and give roughly one fifth of the lectures, in addition to bureaucracy and communication with student assistants (teaching assistants) and students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I gave a lecture on equilibria and their stability of differential equations; essentially, for an an equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\text{d}y}{\text{d}t} = g(y)&#xA;$$&lt;/span&gt;&#xA;we have that &lt;span class=&quot;math-container&quot;&gt;$\hat y$&lt;/span&gt; is an equilibrium point if and only if &lt;span class=&quot;math-container&quot;&gt;$g(\hat y) = 0$&lt;/span&gt;, and such a point is stable if &lt;span class=&quot;math-container&quot;&gt;$g'(\hat y) &amp;lt; 0$&lt;/span&gt; and unstable if &lt;span class=&quot;math-container&quot;&gt;$g'(\hat y) &amp;gt; 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book justifies the stability by linearizing the equation. I tried to explain this during the lecture, but I am far from convinced the explanation was satisfactory. In addition to a small calculation I sketched a graph and a tangent line and explained how this relates to the linearization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How to best explain linearization in this case and use it to explain the stability conditions?&lt;/li&gt;&#xA;&lt;li&gt;A very tempting alternative is to bypass the justification and use the time to consider a nice example and do some modelling. How to decide between the mathematical justification and an example, in the context of this type of course?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2019-02-05T10:54:40.387" Title="How to explain linear approximation to an equation to calculus students?" Tags="|undergraduate-education|calculus|differential-equations|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15113" PostTypeId="1" AcceptedAnswerId="15124" CreationDate="2019-01-24T11:05:14.350" Score="4" ViewCount="292" Body="&lt;p&gt;I am giving some lectures on a calculus course in Norwegian. My Norwegian (or, rather, Scandinavic) is good enough to do so mostly without resorting to English, but I would, of course, like to improve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular I have problems in understanding questions from the audience; big reason being, I suppose, that I can not anticipate what they are asking and so understanding the question is much harder. But sometimes spontanous explanations I want to give during the lecture also stumble and I have to use non-native expressions or resort to English. Sometimes, notation also causes problems, as discussed here: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7978/resources-for-reading-mathematics-out-loud-in-different-languages&quot;&gt;Resources for reading mathematics out loud in different languages&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are best practices for preparing a (fairly traditional) lecture in a language one is not yet fluent in, given a large undergraduate calculus course as the specific context?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As per the usual Stack exchange best practices on subjective questions, please base your answer on personal experience or sources you cite. See &lt;a href=&quot;https://matheducators.meta.stackexchange.com/a/41/2083&quot;&gt;https://matheducators.meta.stackexchange.com/a/41/2083&lt;/a&gt; and especially the linked blog post.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-01-25T08:06:43.667" LastActivityDate="2019-02-07T20:43:01.617" Title="How to prepare for lecturing in a non-fluent foreign language?" Tags="|calculus|language-use|large-lecture|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15114" PostTypeId="2" ParentId="15113" CreationDate="2019-01-24T13:57:13.723" Score="0" Body="&lt;p&gt;Make a list of all relevant keywords in that particular language, and do two things with it:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Learn it by heart, and try to find out how those words are pronounced.&lt;/li&gt;&#xA;&lt;li&gt;Make a copy of this, which you put on the desk before you while you are giving the lecture. It must be readable while you're standing up (use bold font and the size should be large enough (take a font size so that your page is just filled completely)).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;At the beginning of the lecture, tell your audience that this is not your mother's tongue, you are trying your best but you might make mistakes while pronouncing local words (generally people are very understanding about this, especially when they belong to smaller language communities, like Scandinavian ones).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you are giving the lecture, when somebody asks you a question about such a word, always be sure to repeat the question. While you do that, take a peek at your list in order to be sure that you understand the word correctly. (This also gives you some time to think about the answer)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2019-01-24T13:57:13.723" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15115" PostTypeId="2" ParentId="15112" CreationDate="2019-01-24T13:59:54.660" Score="2" Body="&lt;p&gt;Linearization is a technique which is useful for systems of equations, but there are easier (and more powerful) ways of approaching stability in the single variable case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The major thing that you are getting out of &lt;span class=&quot;math-container&quot;&gt;$g(\hat{y})=0$&lt;/span&gt; and the sign of &lt;span class=&quot;math-container&quot;&gt;$g'$&lt;/span&gt; is that &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is changing sign around &lt;span class=&quot;math-container&quot;&gt;$\hat{y}$&lt;/span&gt;, and&lt;/li&gt;&#xA;&lt;li&gt;whether is it going from &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The direction of the sign change determines if the fixed point is stable or not. For example, if &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; goes from &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;, the fixed point is stable as the ODE moves points below &lt;span class=&quot;math-container&quot;&gt;$\hat{y}$&lt;/span&gt; up and points above &lt;span class=&quot;math-container&quot;&gt;$\hat{y}$&lt;/span&gt; down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I say that analyzing the change in sign is more powerful than looking at the derivative because of examples such as &lt;span class=&quot;math-container&quot;&gt;$g(y)=-y^3$&lt;/span&gt;. Here you have a stable fixed point at &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt; despite &lt;span class=&quot;math-container&quot;&gt;$g'(0)=0$&lt;/span&gt;. Further, students can graph &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; and get an immediate idea about stability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if you must use linearization for single variable 1st order ODEs, I suggest explaining it as a way to determine the type of sign change of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Linearization gives more benefits for the higher dimensional case since there are other questions, such as &quot;does the system oscillate?&quot;, which can be answered by computing eigenvalues.)&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2019-01-24T13:59:54.660" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15118" PostTypeId="1" AcceptedAnswerId="15119" CreationDate="2019-01-24T15:32:43.870" Score="5" ViewCount="464" Body="&lt;p&gt;I noticed that in my undergraduate class a few students understand things quite fast and some times see the proof before I even explain things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But some of them also have trouble understanding quite basic ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did a survey among students about the speed of lecture. Half of students thinks it's a bit slow, the other half think it's good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So do you have any suggestion about how to balance the need of students of very different levels?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2019-01-24T17:22:39.527" Title="How to balance the difficulty level and speed of lectures for students of very different levels?" Tags="|undergraduate-education|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15119" PostTypeId="2" ParentId="15118" CreationDate="2019-01-24T15:46:43.723" Score="5" Body="&lt;p&gt;For students who quickly solve the basic problems, what about having a list of &quot;next level&quot; problems for them to solve? Maybe if you want students to solve (or follow your argument for) five problems, try to have problems 6 and 7 be challenging enough for them, but accessible enough that they are possible without knowing other, special information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've tried this, and my stronger students (almost!) always go for it and try to solve the challenging problems. My goal, of course, is to teach what is in the course description, which means that everyone (regardless of ability) has been exposed to the basics. If a student has strong skills, I think they usually appreciate the opportunity to test that strength.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-01-24T15:46:43.723" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15121" PostTypeId="2" ParentId="15112" CreationDate="2019-01-24T17:43:42.047" Score="1" Body="&lt;p&gt;Tommi, you say&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;strong&gt;A very tempting alternative is to bypass the justification and use the time to consider a nice example and do some modelling.&lt;/strong&gt; How to decide between the mathematical justification and an example, in the context of this type of course?&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I agree with the temptation to eschew more proof(ish) activity.  Instead work on problem solving.  Here is the rationale for &quot;&lt;strong&gt;how to decide&lt;/strong&gt;&quot;.  (1) Nothing in your entire question (not just the quote but all of it) says that the students were unhappy with the rationale/justification/proof.  Instead you mention your OWN dissatisfaction.  Furthermore, you also mention (2) that proofs are not a big part of the course and (3) the audience is softer natural scientists (biologists, etc.)  So definitely not mathematicians, but not even engineers and physicists.  ALL of these factors (1-3) argue against pursuing a better rationale.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note to pre-empt deletion:  I am answering a solid HALF of the question.  This IS AN ANSWER.  Furthermore, you could even say someone who only gives a linear approximation lesson is ignoring THIS HALF of the question.  And if anything, you could say such an answer luxuriates in the mathematics itself rather than the teaching of it.&lt;/p&gt;&#xA;" OwnerUserId="11812" LastActivityDate="2019-01-24T17:43:42.047" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15122" PostTypeId="2" ParentId="15113" CreationDate="2019-01-24T17:53:01.547" Score="-1" Body="&lt;p&gt;KISS:  Keep it simple, s...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use small simple words versus trying for exact translations of the most complicated long words.  Follow the text closely.  Work example problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think if you follow this course, you will avoid opening up communication challenges that would occur if you emphasize abstractions or the most fancy technical terms (not just in the immediate lecture.  It's not just you spewing stuff out (but even there, I would worry that a translation may be imperfect) but in your inability to engage in discussion about most tricky abstract things during discussion because of adding language barrier to the other issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take an approach of &quot;three quarters of a loaf is better than none&quot; and &quot;perfect is the enemy of better&quot;.  Go in and give them a solid problem solving hour that helps them with their homework.  This will be time well spent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I hope these are not the same students who you are also trying to push more proofy non-proofs onto (separate question).  If so, you are really borrowing trouble.&lt;/p&gt;&#xA;" OwnerUserId="11813" LastActivityDate="2019-01-24T17:53:01.547" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15123" PostTypeId="2" ParentId="15113" CreationDate="2019-01-24T20:54:19.250" Score="4" Body="&lt;p&gt;Since you seem interested in experience in a similar situation, I will share my experiences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I moved overseas two years ago.  The biggest help to thinking about math in a foreign language (in my case Hebrew) was to watch videos of professors teaching math in that language.  I was lucky enough to find a coursera course given in Hebrew which really helped me to start to think about math in Hebrew.  You might be interested in this &lt;a href=&quot;https://www.independent.co.uk/life-style/bilingual-maths-language-process-differently-brain-a7953241.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;article from the Independent&lt;/a&gt; which discusses how bilingual people process math differently in different languages.  It will emphasize the importance of learning to think about math in foreign language.  If you can't find a Coursera course or open university course, there is always youtube. The bottom line is that you don't want to think about math in English and translate to Norwegian - you want to start thinking about math in Norwegian. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have also spent time reading math books which helps with the thinking, but created a different problem - that I often mispronounced words. This is very common.  As a high school student I took a class at a local college with a non-native professor.  He mispronounced words and none of us were comfortable asking, so we all got lost. I suggest that you tell the students that this is not your native language and you might mispronounce a word and ask them to correct you and/or ask if your pronunciation isn't clear. This will make them more sensitive to your predicament and perhaps more helpful&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2019-01-24T20:54:19.250" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15124" PostTypeId="2" ParentId="15113" CreationDate="2019-01-25T10:49:10.083" Score="5" Body="&lt;p&gt;When I first started teaching in Spanish I wrote my lectures out in full before every class, something I would never do in English (my native language). This required consulting textbooks written in Spanish and speaking to other teachers to find out what is standard usage and mathematical terminology. Such consultation is essential. Since mathematical terminology mostly comes from English, French, and German there sometimes are not native Spanish expressiosn for technical terms. Dictionaries rarely are useful for finding out what is correct mathematical usage. Much mathematical terminology in Spanish originates in French usage rather than English usage. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When actually giving class, it is important to write everything that one says on the board. If students struggle with one's accent, at least they can read what one writes. For this it is of course important that what one writes be more or less correct grammatically and orthographically. For this one sometimes has to write lectures in far more detail than one would in one's native language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something I have encountered frequently is that theorems are known by different names in different cultures. The (some) theorem expressing the dimension of the space of solutions of a system of linear equations in terms of its rank is known as the Rouché-Frobenius theorem in Spain. It is called differently in other countries (in the US it doesn't usually get given a name). One needs to learn these local customary names and terminology. It helps for communicating with students. Otherwise what happens is that students know some theorem as Bolzano's theorem and don't realize the professor is talking about Bolzano's theorem because he calls it something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one teaches at the university level, for example calculus or linear algebra, one needs to familiarize oneself with what is taught at the high school level. Terminology used in primary education is often peculiar, highly regional, and quite different from &quot;professional&quot; use. For example, in Spain what most (?) mathematicians would call &quot;convex&quot; is called &quot;concave up&quot; in high school, where students are taught to speak of &quot;concave up&quot; and &quot;concave down&quot;. Teaching without awareness of such issues one can generate a lot of confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found that explaining math in a second language obliged me to use a simpler, more colloquial language than I would use in my native language. My experience was that this often was positive from the pedagogical point of view. For example, I found myself explaining Cavalieri's principle in terms of presliced loaves of bread. It's a helpful metaphor I continue to use now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When giving exams, or any graded exercise, it is absolutely essential to have a mathematically competent native speaker proofread them carefully. I once (unknowingly) wrote a problem whose meaning changed materially depending on whether &lt;em&gt;como/cómo&lt;/em&gt; had or lacked an accent mark.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the difficulty of teaching in a second language will not last long. There is probably no better way to improve at speaking and writing a language than by giving classes in that language.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-01-25T10:49:10.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15125" PostTypeId="1" CreationDate="2019-01-25T23:01:30.123" Score="7" ViewCount="181" Body="&lt;p&gt;I am looking for studies and experiments in the literature that I can share with undergraduate students in an intro statistics course. I do not expect students to understand everything, and I plan to only have them read the abstract or other short excerpts and then discuss. They must be reasonably readable papers and apply concepts from introductory statistics (experimental design principles, confidence intervals, p-values, linear regression, etc).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to collect some examples where the conclusions are strong and others that are flawed. I would also prefer subject matter that is important and useful.&lt;/p&gt;&#xA;" OwnerUserId="7707" LastActivityDate="2019-01-26T07:13:06.060" Title="Published papers for Intro Stat students to read" Tags="|undergraduate-education|reference-request|examples|statistics|big-list|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15126" PostTypeId="1" CreationDate="2019-01-25T23:27:59.307" Score="4" ViewCount="318" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Nathan_Jacobson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nathan Jacobson&lt;/a&gt;'s Basic Algebra &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486471896&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;I&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/048647187X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;II&lt;/a&gt; covers many topics in Algebra that is probably even beyond many pure mathematics full professor's scope of knowledge, unless the professor is specialised in algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, if the following topics are included in Basic Algebra:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Modules over a PID;&lt;/li&gt;&#xA;&lt;li&gt;Galois Theory;&lt;/li&gt;&#xA;&lt;li&gt;Algebras over a Field;&lt;/li&gt;&#xA;&lt;li&gt;Lattices and Boolean Algebras.&lt;/li&gt;&#xA;&lt;li&gt;Classical Representation Theory;&lt;/li&gt;&#xA;&lt;li&gt;Homological Algebra;&lt;/li&gt;&#xA;&lt;li&gt;Commutative Ideal Theory;&lt;/li&gt;&#xA;&lt;li&gt;Field Theory;&lt;/li&gt;&#xA;&lt;li&gt;Valuation Theory;&lt;/li&gt;&#xA;&lt;li&gt;Dedekind Domains.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Then what is considered (according to N. Jacobson's standard) as &quot;Intermediate Algebra&quot; and what is considered &quot;Advanced Algebra&quot;? Any list of topics and recommended textbooks on Intermediate Algebra and Advanced Algebra?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2019-01-25T23:27:59.307" Title="According to Nathan Jacobson, what is Intermediate Algebra and Advanced Algebra?" Tags="|algebra|textbooks|self-learning|reference-books|abstract-algebra|" AnswerCount="0" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15127" PostTypeId="2" ParentId="15125" CreationDate="2019-01-26T00:42:16.013" Score="4" Body="&lt;p&gt;I did something very similar as a project when I taught AP statistics a few years ago. It was relatively effective and my students left that class with that project as one of the ones they felt was the most impactful, since it showed them how statistics was actually used in real life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that the most recent topics for this project were linear regression and experimental design, and I generally asked my students to summarize the methods and results. My goal was to have papers that did not take a large amount of technical knowledge to understand what, to some extent, the study was measuring and how the experiments or observations were carried out. I wasn't fully satisfied with the papers that I found, but some that served the purposes I wanted them to. It should be noted that I taught at a &quot;medical field focus high school.&quot; The papers I ended up using included: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/19526369&quot; rel=&quot;nofollow noreferrer&quot;&gt;Is grip strength a predictor for total muscle strength in healthy children, adolescents, and young adults?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://journals.sagepub.com/doi/abs/10.1177/00220345950740080201&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Prevalence of Dental Anxiety in Children from Low-income Families and its Relationship to Personality Traits&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/15925952&quot; rel=&quot;nofollow noreferrer&quot;&gt;Validity of self-reported height and weight in a Japanese workplace population.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I'm not sure if you'd consider this subject matter &quot;important and useful.&quot; One other possible paper that you could use snippets of that is well-known, relatively accessible, and has straightforward statistics : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.viriya.net/jabref/cognitive_and_attentional_mechanisms_in_delay_of_gratification.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cognitive and Attentional Mechanisms in Delay of Gratification&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which I don't think is &quot;the&quot; Stanford marshmallow experiment of lore, but could be potentially more interesting to students interested in psychology. In general, older psychological research seems to be the most accessible to students without domain specific knowledge.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2019-01-26T00:42:16.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15128" PostTypeId="2" ParentId="15125" CreationDate="2019-01-26T07:13:06.060" Score="5" Body="&lt;p&gt;I customarily use: Gillison, Maura L., et al. &quot;Prevalence of oral HPV infection in the United States, 2009-2010.&quot; Jama 307.7 (2012): 693-703. This has a nice mix of CI's, P-values, and statements about distribution shapes (bimodal) in the abstract.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At one point I used this article: Peck, Peggy, &quot;Long Work Hours Increase Hypertension Risk&quot;, MedPage Today. This had the intimate connection that my mother (school nurse) asked me to help interpret this for her as part of her continuing education credits, and many of my students tend to be nursing students or related fields (physical therapy, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other stuff from JAMA I've used in passing or to make exam questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Bobrow, &quot;Chest Compression–Only CPR by Lay Rescuers and Survival From Out-of-Hospital Cardiac Arrest&quot;&lt;/li&gt;&#xA;&lt;li&gt;Joynt, &quot;Quality of Care and Patient Outcomes in Critical Access Rural Hospitals&quot;&lt;/li&gt;&#xA;&lt;li&gt;Kayne, &quot;Progression of Left Ventricular Diastolic Dysfunction and Risk of Heart Failure&quot;&lt;/li&gt;&#xA;&lt;li&gt;Snitz, &quot;Ginkgo biloba for Preventing Cognitive Decline in Older Adults&quot;&lt;/li&gt;&#xA;&lt;li&gt;Sulheim, &quot;Helmet Use and Risk of Head Injuries in Alpine Skiers and Snowboarders&quot;&lt;/li&gt;&#xA;&lt;li&gt;Benson, &quot;Head and Neck Injuries Among Ice Hockey Players Wearing Full Face Shields vs Half Face Shields&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-01-26T07:13:06.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15129" PostTypeId="2" ParentId="15109" CreationDate="2019-01-26T23:21:52.517" Score="4" Body="&lt;p&gt;I always enjoy bringing in Bortkiewicz’s data on the annual deaths by horse kicks in the Prussian Army from 1875-1894.This is what inspired the discovery of the Poisson distribution. Sometimes, students enjoy thinking about how to collect data about rare events: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How often does the anaesthetist has fallen asleep during surgery? And other medical “Never Events.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The number of mutations in a given length of DNA.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The number of goals in a soccer game.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-01-26T23:21:52.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15130" PostTypeId="1" AcceptedAnswerId="15131" CreationDate="2019-01-28T07:44:06.737" Score="9" ViewCount="6061" Body="&lt;p&gt;I have asked in &lt;a href=&quot;https://math.stackexchange.com/questions/3090247/is-linear-algebra-done-right-too-much-for-a-beginner&quot;&gt;Mathematics stackexchange&lt;/a&gt;, but I think asking here is more appropriate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am self studying the book Linear Algebra Done Right by Axler. That's how I started using the great Stack Exchange in Mathematics!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not have a background in mathematics and just finished my undergraduate and want to learn Maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try to read every page and finish all exercises (now on Chapter 3: Linear Maps). And on average, it takes me an hour to finish a page (exercise included). I can finish 1/3, if not 1/4 of the questions on my own. For the rest, I need to look at the solutions to get some hints. Sometimes, I don't even understand the solutions...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just want to know if I am too slow on average? Are the questions hard for beginners on average (on  average, so not for IQ above 160 type)? (I will keep going and finish the book even if you tell me the questions are super easy.)&lt;/p&gt;&#xA;" OwnerUserId="11818" LastEditorDisplayName="user507" LastEditDate="2019-03-02T01:30:06.327" LastActivityDate="2019-06-15T02:53:59.670" Title="Is Linear Algebra Done Right too much for a beginner?" Tags="|self-learning|linear-algebra|book|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15131" PostTypeId="2" ParentId="15130" CreationDate="2019-01-28T11:37:42.293" Score="20" Body="&lt;p&gt;Unguided self-study of mathematics is difficult, and harder for someone with little experience at it. It is normal to take time to advance. One should think in terms of months not hours. A typical one semester class in linear algebra does not cover every detail in the typical linear algebra textbook, nor require a student to work every exercise in that book. When studying by oneself it is important to do exercises carefully and to be sure one is doing them right, and to really make sure that one is understanding B before progressing to studying C (for example, in general one should understand well Gaussian elimination and its consequences for the analysis of systems of linear equations before studying abstract linear maps). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are struggling, find someone knowledgeable to answer your questions and give you some guidance. There are many paths that lead to understanding linear algebra, but when one is lost it can be hard to find any of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book by Axler has an unfortunately tendentious title and a strange (and in some respects questionable) fixation on determinants. I like Sergei Treil's freely available &lt;a href=&quot;https://www.math.brown.edu/~treil/papers/LADW/LADW.html&quot; rel=&quot;noreferrer&quot;&gt;Linear algebra done wrong&lt;/a&gt; better. There are many books, and it is useful to look at several. What one person likes and can learn from is terrible for another person. You might learn well from Axler's book. To some extent the election depends on taste, to some extent on one's goals. Gilbert Strang's introductory textbook is one that seems to me that can work well for self study (although, maybe since my own inclinations are those of an algebraist, I find it too wordy and talkative for me to have learned from it). Strang's book works from the concrete to the abstract, showing a good pedagogical sense and emphasizing techniques are useful in practice. It is written for someone without much experience with more abstract mathematics (e.g. an engineering student) and such an approach has virtues for someone learning alone. It might also be of interest to look at some numerical linear algebra textbooks - the ones by Trefethen/Bau and Demmel are my favorites - as these communicate a completely different point of view than the standard textbooks.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2019-01-28T19:07:30.397" LastActivityDate="2019-01-28T19:07:30.397" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15132" PostTypeId="2" ParentId="15130" CreationDate="2019-01-28T14:29:10.693" Score="5" Body="&lt;p&gt;I know your struggle well. I enrolled at a distance learning university and the Mathematics is just very painful and slow to grab. I also struggle with every proof and have to force myself to think it through. What helps is to have someone talking me through or discussing certain proofs or concepts, unfortunately I only have a mailing list which makes questions tedious. As DanFox mentioned Gilbert Strang is a good teacher and I really like his lectures (they are on youtube: MIT open courseware Lectures in linear algebra) those really helped me to get the concepts and understand my lecture much better. His look on the concepts was so different than what I had read. In this way I at least had someone explaining the things written in the book with a few more hints. Maybe that would also be of help for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in general I think if you seem to understand whats written on a page your progress is fine, I would not measure it by the time you take. There are just people who get it faster than others. Speeding up for the sake of being better or comparable to others just is unnecessary struggle. But you will be getting faster as you progress. Its like learning a language. The first progress is very slow...you need vocabulary and grammar and need to put it together and after you know the basic sentence structures you get faster.&lt;/p&gt;&#xA;" OwnerUserId="11823" LastEditorUserId="11823" LastEditDate="2019-02-27T10:06:25.770" LastActivityDate="2019-02-27T10:06:25.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15133" PostTypeId="2" ParentId="15130" CreationDate="2019-01-28T16:44:40.050" Score="2" Body="&lt;p&gt;Get an easier book and one that emphasizes calculations more than proofs.  One with lots of &quot;drill&quot; exercises and the answers to them.  Then do the drill.  All of it.  Struggling with too hard a book is not time efficient self study.  If you can't do the exercises there is a problem.  (Do a linear programming optimization on that!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, you should look for &quot;programmed instruction&quot; texts when self studying.  If you can't find that look for the closest to it.  Don't listen to the Internet's Rudin lovers pushing you to difficult, even pedagogically flawed, texts.  If you learn an easy text 4.0, you will be way better off than learning a hard text 3.0 (or 2.0 as sounds like you are doing now).&lt;/p&gt;&#xA;" OwnerUserId="11824" LastActivityDate="2019-01-28T16:44:40.050" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15134" PostTypeId="1" CreationDate="2019-01-28T17:12:40.997" Score="4" ViewCount="413" Body="&lt;p&gt;I am teaching Calculus for non-maths major students. As far as I know, when we teach about derivatives, we should mention &quot;the rate of change&quot;. There are some practical examples to motivate this concept. For example, the velocity of a car moving on the street.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I would like to find an example that may fascinate our students a bit more. Could anyone help me with some suggestions?&lt;/p&gt;&#xA;" OwnerUserId="11825" LastEditorUserId="2083" LastEditDate="2023-04-27T07:47:14.667" LastActivityDate="2023-04-27T07:47:14.667" Title="Which examples should we mention when teaching the concept of derivatives?" Tags="|undergraduate-education|calculus|examples|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15135" PostTypeId="2" ParentId="15130" CreationDate="2019-01-28T23:27:04.403" Score="4" Body="&lt;p&gt;&lt;em&gt;Linear Algebra Done Right&lt;/em&gt; is intended as a second encounter (US curriculum) with linear algebra (it says so in the introduction), and some of the exercises are a bit tricky. If you don't have a background in math, then it's perfectly normal to take what feels like a very long time for a single page. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems like you have already progressed a fair bit into the book. If you like its style, then I would recommend to just keep going until you get stuck, and then look at other material. You also don't need to solve every single exercise. You can also try to find exercises in other books or online. Many lecturers upload problem sheets for their courses online, and you can download them freely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also recommend trying to do &quot;reality checks&quot; every now and then. Try to come up with some simple, maybe even silly examples. Compute some stuff. Doodle a bit. It will give a better feeling for what you have understood from a section of a book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides the books already mentioned in the other answers, I would like to recommend taking a look at the following books:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Linear Algebra Problem Book&lt;/em&gt; by Paul Halmos: this is written in an accessible, conversational style. It teaches linear algebra through short explanations and lots of exercises. All exercises have hints &lt;em&gt;and&lt;/em&gt; solutions at the end of the book. This book is great for self-studying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Linear Algebra&lt;/em&gt; by Klaus Jänich.&#xA;This is a solid introductory textbook and covers the content of a typical first semester course in linear algebra at a European university. It has regular exercises and a few quick tests interspersed within the text. There are also short sections introducing topics from multiple point of view. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck on your journey! &lt;/p&gt;&#xA;" OwnerUserId="9285" LastActivityDate="2019-01-28T23:27:04.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15136" PostTypeId="2" ParentId="15109" CreationDate="2019-01-29T00:05:21.403" Score="6" Body="&lt;p&gt;19 public data sets, from &lt;a href=&quot;https://www.springboard.com/blog/free-public-data-sets-data-science-project/&quot; rel=&quot;noreferrer&quot;&gt;Springborg blog&lt;/a&gt;, curated by T.J. DeGroat.&#xA;Summaries and links for each in DeGroat's page.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;United States Census Data&lt;/li&gt;&#xA;&lt;li&gt;FBI Crime Data&lt;/li&gt;&#xA;&lt;li&gt;CDC Cause of Death&lt;/li&gt;&#xA;&lt;li&gt;Medicare Hospital Quality&lt;/li&gt;&#xA;&lt;li&gt;SEER Cancer Incidence&lt;/li&gt;&#xA;&lt;li&gt;Bureau of Labor Statistics&lt;/li&gt;&#xA;&lt;li&gt;Bureau of Economic Analysis&lt;/li&gt;&#xA;&lt;li&gt;IMF Economic Data&lt;/li&gt;&#xA;&lt;li&gt;Dow Jones Weekly Returns &lt;/li&gt;&#xA;&lt;li&gt;Data.gov.uk&lt;/li&gt;&#xA;&lt;li&gt;Enron Emails &lt;/li&gt;&#xA;&lt;li&gt;Google Books Ngrams &lt;/li&gt;&#xA;&lt;li&gt;UNICEF &lt;/li&gt;&#xA;&lt;li&gt;Reddit Comments &lt;/li&gt;&#xA;&lt;li&gt;Wikipedia &lt;/li&gt;&#xA;&lt;li&gt;Lending Club &lt;/li&gt;&#xA;&lt;li&gt;Airbnb&lt;/li&gt;&#xA;&lt;li&gt;Walmart &lt;/li&gt;&#xA;&lt;li&gt;Yelp&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-01-29T10:50:41.083" LastActivityDate="2019-01-29T10:50:41.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15137" PostTypeId="2" ParentId="15134" CreationDate="2019-01-29T03:39:18.947" Score="4" Body="&lt;p&gt;The typical example of a rate of change is one that changes with respect to time.  I would strongly suggest introducing at least one example where the independent variable is not a quantity of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One relatively-easy-to-visualize example is to find the rate of change of a shape's area with respect to one of its lengths.  For example, a square's area &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is the square of the length of its side &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$\frac{\mathrm{d}A}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}x^2=2x$&lt;/span&gt;.  Note that it is a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  That is, the rate of change of a square's area with respect to its side length depends on what the side length currently is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This example easily leads to the discussion on the chain rule.  If a square's side length is changing at a time rate of, say, &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; meters per second, then its area is changing at a time rate of &lt;span class=&quot;math-container&quot;&gt;$\frac{\mathrm{d}A}{\mathrm{d}t}=\frac{\mathrm{d}A}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}t}=2x(2$&lt;/span&gt; m/s&lt;span class=&quot;math-container&quot;&gt;$)$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a quantity having meters as the unit of measurement, so &lt;span class=&quot;math-container&quot;&gt;$\frac{\mathrm{d}A}{\mathrm{d}t}$&lt;/span&gt; is in m&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;/s).  As the square gets bigger, its area is changing faster with respect to time.  (That is, the time rate of change of the side length is constant, but the time rate of change of the area is not.)&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2019-01-30T01:59:49.290" LastActivityDate="2019-01-30T01:59:49.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15138" PostTypeId="1" AcceptedAnswerId="15176" CreationDate="2019-01-29T13:00:29.663" Score="2" ViewCount="213" Body="&lt;p&gt;At my university, traditionally a few lectures of a course should be tutorial sessions. The idea is that instead of covering new materials, the teacher should go over many exercises so that student learn how to solve problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My colleagues do these sessions differently. Some go over some examples and let student work in group themselves for the rest of the time. Some would ask students to come to blackboard to solve problems in the front of the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you organize such tutorial sessions? What format do you use? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Update, I have arranged office hours, but so far no student has ever come. I suspect very few of them will eventually come to office hours.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also the lectures are of 105 minutes each time. I usually take 5 minutes break in between.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I am thinking of telling students this next tutorial&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You vote to decide which category of problems we will talk about.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A problem from the chosen category will be given, with some hint.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You work by yourself or in group on the problem for 5 minutes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A random student is chosen to present his/her solution to the class&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Name another student to represent you if you don't want to talk.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user11702" LastEditorDisplayName="user11702" LastEditDate="2019-02-07T10:47:36.763" LastActivityDate="2019-02-07T17:21:51.937" Title="What is a good format of tutorial sessions?" Tags="|undergraduate-education|tutoring|lecturing|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15139" PostTypeId="2" ParentId="15134" CreationDate="2019-01-29T18:21:59.917" Score="1" Body="&lt;p&gt;To emphasize the importance of rate-of-change vs. actual value a graph of the world population can be useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The world population now is more than 100 years ago, but more importantly the derivative now is much much greater than 100 years ago.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2019-01-29T18:21:59.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15140" PostTypeId="2" ParentId="15134" CreationDate="2019-01-29T20:45:10.710" Score="9" Body="&lt;p&gt;I dont know whether this is more or less &lt;em&gt;fascinating&lt;/em&gt; than the position/velocity/acceleration examples of derivatives with respect to time, but for a practical example of a derivative students are all used to, I ask them to put one hand on their wooden(?) desktop and grab the metal leg of the desk with the other hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Which is colder?&quot; I ask.&#xA;&quot;The table leg&quot;, they all reply.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But these have been in the room together for hours (years, probably), so surely they should be the same temperature. Thus, students are really encountering the difference in thermal conductivity of two materials, and they are experiencing &lt;em&gt;heat transfer&lt;/em&gt; at different rates (from their hand to the table leg or the desk). The common notion of &quot;cold&quot; here is a derivative that we all have built-in. Most students find this example pretty interesting. We then try to come up with other examples of things we experience and quantify that are actually rates.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-01-31T04:26:43.403" LastActivityDate="2019-01-31T04:26:43.403" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15141" PostTypeId="2" ParentId="15134" CreationDate="2019-01-30T00:02:12.487" Score="1" Body="&lt;p&gt;Suggestion from a non-teacher. When you get on to second and third derivatives, how about a simplistic model of global warming? Assuming things haven't had time to reach equilibrium, and making various possibly wild assumptions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the rate of warming depends on the difference between incoming and outgoing heat radiation&lt;/li&gt;&#xA;&lt;li&gt;that depends on the amount of CO&lt;span class=&quot;math-container&quot;&gt;$_2$&lt;/span&gt; in the atmosphere, so more CO&lt;span class=&quot;math-container&quot;&gt;$_2\Rightarrow$&lt;/span&gt; faster warming, not just higher temperature (unless equilibrium is reached)&lt;/li&gt;&#xA;&lt;li&gt;the rate of increase of CO&lt;span class=&quot;math-container&quot;&gt;$_2$&lt;/span&gt; depends on the rate at which we produce it (strictly, how much that exceeds the rate at which natural processes can absorb it)&lt;/li&gt;&#xA;&lt;li&gt;now relate the behaviour of the temperature to how fast the rate of emission is increasing (for example).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I think that can get you up to about the fourth derivative, with meaningful interpretations all along, and there should be a few ways to play around with it. (Unfortunately the real equations are probably hideously complicated.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually I've just realised you can get yet another derivative out of it if you assume the rate of ice loss is proportional to the excess temperature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or maybe save all this as an integration example for later on, or something to think about qualitatively just before introducing integration properly.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastEditorUserId="10663" LastEditDate="2019-01-30T03:21:39.007" LastActivityDate="2019-01-30T03:21:39.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15142" PostTypeId="1" CreationDate="2019-01-30T15:54:50.917" Score="6" ViewCount="699" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/inq3u.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/inq3u.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to a recent experiment conducted by user &lt;a href=&quot;https://matheducators.stackexchange.com/users/117/steven-gubkin&quot;&gt;Steven Gubkin&lt;/a&gt;, nearly one half of his students in a senior level Real Analysis course do not have any idea how to prove the quadratic formula. Is this a problem in our education of students majoring in mathematics? Or are we alright with students obtaining bachlor's degree in mathematics without knowing the proof of the quadratic formula? &lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2019-02-08T14:39:24.933" Title="Is it a problem if a senior student majoring in mathematics could not prove the quadratic formula?" Tags="|undergraduate-education|proofs|solving-polynomials|" AnswerCount="5" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="15144" PostTypeId="1" CreationDate="2019-01-31T04:23:12.447" Score="2" ViewCount="1204" Body="&lt;p&gt;Most problems require the student to simplify the final answer and in many context the meaning is obvious: For example, &lt;span class=&quot;math-container&quot;&gt;$3/9$&lt;/span&gt; should be simplified to &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{16}$&lt;/span&gt; should be simplified to &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, how about the answer &lt;span class=&quot;math-container&quot;&gt;$$6\left(\frac{2}{3}\right)^{1/5}?$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This same answer can be equivalently written as &lt;span class=&quot;math-container&quot;&gt;$\sqrt[5]{5184}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$2^{6/5}\cdot 3^{4/5}$&lt;/span&gt;, etc. Which one(s), if any, is considered as &quot;simplified as much as possible&quot;?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2019-01-31T17:13:20.777" Title="What is the standard for &quot;simplifying your answer&quot;?" Tags="|fractions|roots|powers|" AnswerCount="4" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="15145" PostTypeId="2" ParentId="15144" CreationDate="2019-01-31T04:56:10.723" Score="-3" Body="&lt;p&gt;I suggest to do a Google search and at least read a couple pages that give the classical rules for simplifying radical expressions.  You may still have some critique but will at least know what you are criticizing.  Question as stated shows lack of familiarity with basics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. &lt;span class=&quot;math-container&quot;&gt;$2  \sqrt[5]{162}$&lt;/span&gt; is the classical answer.  &lt;/p&gt;&#xA;" OwnerUserId="11833" LastEditorUserId="10663" LastEditDate="2019-01-31T15:17:26.590" LastActivityDate="2019-01-31T15:17:26.590" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15146" PostTypeId="2" ParentId="15144" CreationDate="2019-01-31T07:16:05.843" Score="3" Body="&lt;p&gt;I would say the canonical answer for what constitutes 'simplified as much as possible' is &lt;em&gt;whatever the exam board says it is&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;'Simplify' isn't a mathematical function. It is a pedagogical instruction trying to require students to make use of a selection of mathematical equivalences that they are expected to know. However, it is too vague a term to make clear exactly which such equivalences are expected in a particular case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{(x-1)(x-2)^2}{(x-2)(x+2)}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student might reasonably realise they should cancel the factor on the top with that on the bottom, when it's written like this. But if it was presented as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{x^3-5x^2+8x-4}{x^2-4}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;it is far less clear that cancelling can reasonably be expected. Also, whether the final answer should be factorised or expanded is dependent on context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would say that the instruction to 'simplify as much as possible' should only be used in situations where it is clear to the students what that means. Otherwise, more explicit instructions should be used (or more than one answer judged as correct).&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-01-31T07:16:05.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15147" PostTypeId="2" ParentId="15144" CreationDate="2019-01-31T08:55:28.097" Score="9" Body="&lt;p&gt;Rigid criteria for simplification seem to me largely a bad idea if they are not motivated by contextual considerations. The idea that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}/2$&lt;/span&gt; should be preferred to &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt{2}$&lt;/span&gt; struck me as unmotivated when I was a student, and now seems to me problematic to motivate. The situation is different with respect to writing rational numbers or rational functions. In both cases there is a canonical simplification, that in which the numerator and denominator are relatively prime. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a particular context it might make more sense to write &lt;span class=&quot;math-container&quot;&gt;$6(2/3)^{1/5}$&lt;/span&gt; than to write &lt;span class=&quot;math-container&quot;&gt;$2^{6/5}3^{-4/5}$&lt;/span&gt;. It might be that the &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; in front and the &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; have different extra-mathematical origins, e.g. the &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; counts something and the &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; reflects a ratio of measurements, and it's just an accident that both are expressible in terms of powers of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. In a purely mathematical context there are no such considerations, but then still one might prefer one form or the other (I find &lt;span class=&quot;math-container&quot;&gt;$6(2/3)^{1/5}$&lt;/span&gt; easier to interpret at a glance than &lt;span class=&quot;math-container&quot;&gt;$2^{6/5}3^{-4/5}$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Returning to the &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}/2$&lt;/span&gt; versus &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt{2}$&lt;/span&gt; example, imagine we are working in the field &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}[\sqrt{2}]$&lt;/span&gt;. Then, in fact, we can always write an element of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}[\sqrt{2}]$&lt;/span&gt; in the form &lt;span class=&quot;math-container&quot;&gt;$a + b\sqrt{2}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; rational. In this context there is a normalized expression, and it would be &lt;span class=&quot;math-container&quot;&gt;$(1/2)\sqrt{2} = \sqrt{2}/2$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt{2}$&lt;/span&gt; (working in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}[\sqrt{2}]$&lt;/span&gt; we have to prove (!) that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is invertible before we can give sense to &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt{2}$&lt;/span&gt; - as a shorthand for its inverse in this field extension of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;). On the other hand, if we are working in the field &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt{2}$&lt;/span&gt; is directly the expression for the inverse of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; &lt;em&gt;qua&lt;/em&gt; element of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. It's that thing that gives &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; when we multiply it by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; which we know exists from when we constructed the real field (when did we do that?). The conceptual apparatus necessary to make such distinctions/motivations comprehensible (in this case amounting to distinguishing between the inverse of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; in the field &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; and its inverse in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}[\sqrt{2}])$&lt;/span&gt;) is simply not available to (all but a very few highly exceptional) students in primary education where these things are taught. Insisting on the use of a &quot;canonical&quot; form without providing motivation for what makes it canonical (&quot;canonical&quot; is always with respect to some background context) has the feel of authoritarian dogma. This can alienate students, particularly those with a more creative or curious mindset. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Absent some context that motivates preferring one or another simplified form, demanding one or the other such form seems to me typical of the sort of teaching that turns students off from mathematics. What is arbitrary should generally be avoided.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-01-31T08:55:28.097" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15150" PostTypeId="2" ParentId="15144" CreationDate="2019-01-31T14:47:17.953" Score="2" Body="&lt;p&gt;There is a theorem which says that it is impossible to decide the equivalence of two elementary functions syntactically.  So there is not, and cannot, be a uniquely defined &quot;simplest form&quot; for a given expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://inst.cs.berkeley.edu/~cs282/sp02/readings/caviness.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://inst.cs.berkeley.edu/~cs282/sp02/readings/caviness.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2019-01-31T17:13:20.777" LastActivityDate="2019-01-31T17:13:20.777" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15151" PostTypeId="2" ParentId="15109" CreationDate="2019-01-31T18:21:18.903" Score="2" Body="&lt;p&gt;I think it depends on what you're trying to teach, the following are great for basic introductory stats:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Measures of Center: Income Data from US&lt;/li&gt;&#xA;&lt;li&gt;Linear Regression: Advertising Dataset from &lt;em&gt;Introduction to Statistical Learning&lt;/em&gt; &lt;a href=&quot;http://www-bcf.usc.edu/~gareth/ISL/data.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; or Ames Housing &lt;a href=&quot;https://www.kaggle.com/c/house-prices-advanced-regression-techniques&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Classification: Iris or Titanic Datasets, &lt;a href=&quot;https://en.wikipedia.org/wiki/Iris_flower_data_set&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://www.kaggle.com/wei10117/titanic-machine-learning-from-disaster&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Kaggle.com, UCI Machine Learning Repo, and NYC Open Data are three fun collections of datasets where you'll find plenty more.  These examples are pretty vanilla, but they're easy to find and don't require much preprocessing if any.  Otherwise, learn some webscraping and start making your own!&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2019-01-31T18:21:18.903" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15152" PostTypeId="2" ParentId="15134" CreationDate="2019-01-31T19:52:16.473" Score="7" Body="&lt;p&gt;One of my favorite examples is to explain why the derivative of the area of a circle is the circumference. And the derivative of the volume of a sphere is the surface area. If you try the same for the square and cube, it may not work at first, but try to not use the length of the side as the parameter, but half the length of the side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can argue in many ways. My favorite is to cut the ring and bend it out to get something that looks like rectangle with width equal to the circumference, and height equal to &lt;span class=&quot;math-container&quot;&gt;$\Delta r$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3566" LastEditorUserId="470" LastEditDate="2019-01-31T21:50:57.427" LastActivityDate="2019-01-31T21:50:57.427" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15153" PostTypeId="2" ParentId="15134" CreationDate="2019-01-31T21:59:23.150" Score="-3" Body="&lt;p&gt;I recommend to stick to kinematic motion as the initial, most common application. &quot;How fast the car goes.&quot; It is just the easiest to understand from daily life (even if you are a nurse, not an engineer/physicist).  Of course there are many other functions (thermal, financial, etc.)  But a lot of these are more abstract ESPECIALLY if time is NOT the independent variable.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember for weaker students, don't get too tricky at first.  While all the regulars here are good at math, you need to consider your audience is not them, but your actual students.  Practical pedagogy--not &quot;interesting&quot;.  In particular, an easy example to stay away from is the Joel's area example--too abstract for new, weak, students.  (Not to go after Joel but to make an example of what not to do.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, this question shows a lack of any initial effort.  I would say this to a student asking for homework help.  But a teacher?  Can't you at least scan a couple textbooks?  Do a 1 minute Google search?  We can help you better (and you help yourself more also) if you make some small initial effort yourself.&lt;/p&gt;&#xA;" OwnerUserId="11836" LastActivityDate="2019-01-31T21:59:23.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15155" PostTypeId="2" ParentId="15113" CreationDate="2019-02-04T02:41:04.343" Score="2" Body="&lt;p&gt;I have given lectures numerous times in Russian. The main hurdle at first was &lt;em&gt;notation&lt;/em&gt;, since books never discuss this. Ahead of time I thought of a lot of the notation I would need (how to pronounce all the Latin and Greek letters, integrals over a space with respect to a measure, a mapping from one set to another, a congruence mod m, a quotient space, a dual space, etc.), wrote it all down on a sheet or two of paper, and with a couple of native speakers (more than one at the same time is useful to check their agreement!) went through all of it. Later if new notation came up I checked with native speakers before the lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I prepared lectures thoroughly at first, and asked a native speaker to read them and correct errors. After doing this long enough, eventually I did not feel the need to have someone go over my notes in advance, but I might check with a native if there were some new technical phrases I was unsure about. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like you, I was concerned at first about not understanding questions from the audience, but this was not a problem since if I could catch half the words in a question then I often could tell what they wanted to ask, and if I did not understand the question after 1 or 2 tries they asked me the question in English! One student by default always asked his questions in English (even if I stuck to Russian when replying). In your case, don't all students in Norway already know English?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes, in a smaller class, I was corrected by a student (e.g., when using the Russian for &quot;remind&quot; instead of &quot;recall&quot;) or the students gave me tips about a simpler way to say something (analogous to &quot;m halves&quot; instead of &quot;m divided by 2&quot;), but there were never any problems with them understanding me language-wise. It is much easier for a native to understand their own language with some errors than a foreign language they are not used to (I did have students not yet used to understanding rapidly spoken English). As long as you have generally good knowledge of the grammar and can pronounce things mostly correctly, there shouldn't be a problem. But make sure to figure out how to say all the notation you need!!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The experience of lecturing definitely got me over the hang-up of being afraid to make mistakes when speaking in all settings, although I don't feel like it improved my comprehension of Russian in a comparable way since people on a train platform, say, tend not to be discussing unramified extensions of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic fields.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2019-02-04T02:49:39.153" LastActivityDate="2019-02-04T02:49:39.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15156" PostTypeId="2" ParentId="10832" CreationDate="2019-02-04T04:07:20.720" Score="2" Body="&lt;p&gt;The words &lt;strong&gt;rational&lt;/strong&gt; and &lt;strong&gt;irrational&lt;/strong&gt; mean very different things in math and in real life.&lt;/p&gt;&#xA;" OwnerUserId="11844" LastActivityDate="2019-02-04T04:07:20.720" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15157" PostTypeId="1" AcceptedAnswerId="15192" CreationDate="2019-02-04T13:16:34.853" Score="9" ViewCount="703" Body="&lt;p&gt;I will be teaching Euclidean geometry to future teachers, and I am feeling a bit lost (I know geometry, but I am not that familiar with mathematics education).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there some recent (as concise as possible and maybe classic) mathematics education book that focuses on geometry? I am reading Freudenthal's relevant chapter in &quot;Mathematics as an Educational Task&quot;, which is very interesting, but it does not contain any practical instructions on how to teach geometry (it is also a bit old).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, is there some short enough mathematics education book that provides some necessary background to the different schools of thought in mathematics education?&lt;/p&gt;&#xA;" OwnerUserId="11845" LastEditorUserId="376" LastEditDate="2019-02-04T14:31:43.477" LastActivityDate="2019-02-08T21:42:39.577" Title="Book recommendations on mathematics education focusing on geometry" Tags="|undergraduate-education|secondary-education|geometry|education-research|reference-books|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15160" PostTypeId="2" ParentId="15142" CreationDate="2019-02-05T03:26:31.803" Score="0" Body="&lt;p&gt;I didn't realize until coming across the earlier comment thread at &lt;a href=&quot;https://matheducators.stackexchange.com/questions/15093/are-students-majoring-in-pure-mathematics-expected-to-know-classical-results-in&quot;&gt;Are students majoring in pure mathematics expected to know classical results in mathematics very well by their graduation?&lt;/a&gt; that the students only had 10 minutes to do the quiz. This is a little bogus. How many students walked in late and barely had time to look at the question before their time was up? Even the full 10 minutes is not really a lot of time to set one's thoughts in order on something that is a total surprise on  some random topic. I'm not sure that 10 minutes would be long enough to tie my shoelaces if it was a morning class and I hadn't yet had my second cup of coffee.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was exposed to completing the square in 7th grade (roughly age 12 in the US system). This was the only time I ever saw the technique. I remember that it seemed kind of complicated and kind of like a trick. I didn't like learning tricks unless they seemed like they were generally useful. This one didn't seem generally useful, so I didn't bother studying it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For many students, remembering that this was how the quadratic formula was derived a long time ago would probably be a disadvantage, because they may have just seen Gubkin's test as a test to see whether they remembered this random procedure from a long time ago. If a student didn't remember that this was how the quadratic formula had been derived for them during their distant childhood, their brains would probably be freed up to attack the problem in the obvious way, which would be simply to take the known quadratic formula, plug in, and verify that it gives solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I've noticed about student psychology is that the idea of finding a solution and then &lt;em&gt;checking&lt;/em&gt; that it's a solution is not one that most of them will &lt;em&gt;ever&lt;/em&gt; do naturally unless the problem specifically says to. E.g., I could give them a system of 3 nonlinear equations in the unknowns &lt;span class=&quot;math-container&quot;&gt;$(x,y,z)$&lt;/span&gt; and ask them to solve the equations. They proceed to solve the equations by successively eliminating variables. At the end, &lt;em&gt;none&lt;/em&gt; of them will spontaneously plug back in to see if it's right. This may be partly because they're oriented toward carrying out algorithms and trying to do the required algorithm correctly, partly because they don't really care about whether their answers are right, and partly because they have never received any instruction in the general topic of debugging a calculation. (At many schools, you can get an A in many STEM courses without ever solving a single moderately complicated problem completely correctly.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be interesting to know whether this was a school with selective admissions, and whether the students who couldn't do this task were students who were doing well. If they're C students at a school with nonselective admissions, then this might be about the expected result. It's the kind of thing that I would expect from students who have received instruction in which they're told that the point is just to memorize problem-solving algorithms -- and &lt;em&gt;many&lt;/em&gt; math teachers at nonselective schools teach math &lt;em&gt;entirely&lt;/em&gt; with this philosophy. They produce people with math degrees who are competent enough to run a 7th grade math class reasonably well. This is not actually a terrible outcome when you consider that in many low-socioeconomic status school districts in the US, math is being taught by people who have no degree in math. The principal literally pulls the gym teacher off the field and puts them in a math classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course if this is Harvard, and these students have A averages and are applying to grad schools, then these results are pretty scandalous.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-02-08T05:09:59.823" LastActivityDate="2019-02-08T05:09:59.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15161" PostTypeId="1" CreationDate="2019-02-05T03:39:25.177" Score="1" ViewCount="309" Body="&lt;p&gt;In teaching students how to do iterated integrals, I would like to find some analogy using a finite task nested inside another finite task. It would be especially nice if it satisfied the following criteria:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;natural to visualize as a 2-dimensional grid&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;involves familiar objects&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;doesn't involve a sum (for initial simplicity), but can easily be altered so as to involve a sum&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;An example that isn't naturally grid-able would be: Go to the library and find all the books by your favorite author. For each book, open the book, read all the pages, and close the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another possibility would be something like building a wall out of legos. This seems almost optimal except that you would normally stagger the brick pattern.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2019-02-07T01:16:22.210" Title="Analogy for nested loops/integrals" Tags="|vector-calculus|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15163" PostTypeId="2" ParentId="15161" CreationDate="2019-02-05T04:09:37.953" Score="0" Body="&lt;p&gt;Developing a shale oil field?  There will be some overall grid of sections (usually 1 by 1 or 1 by 2 miles).  This grid has limits to it's size.  Tyically several parallel wells are needed to develop a unit.  However, the number of wells will vary (in general &quot;richer&quot; rock getting more wells per unit, because the economics drive more effort to be spent).  Total oil (EUR, estimated ultimate return) will be the sum of the oil from all the sections.  You can even make it 3D if you look at multibench formations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sales across territories for a company?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Population?  Voting?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sales across time for a company--4 quarters in the year?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rebuilding an engine cylinder, by cylinder (and valve by valve within that)? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just brainstorming... &lt;/p&gt;&#xA;" OwnerUserId="11849" LastActivityDate="2019-02-05T04:09:37.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15164" PostTypeId="2" ParentId="15161" CreationDate="2019-02-05T06:48:39.447" Score="1" Body="&lt;p&gt;Working in a factory packing boxes: fill a pallet with a pile of boxes, where for each box you need to fill it with packets of biscuits.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastEditorUserId="4746" LastEditDate="2019-02-05T19:26:25.377" LastActivityDate="2019-02-05T19:26:25.377" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15165" PostTypeId="2" ParentId="15138" CreationDate="2019-02-05T15:40:54.360" Score="0" Body="&lt;p&gt;I personally do a mix of what you mention, depending on what mathematical maturity is expected of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they are advanced students, I would work our various example problems for them (perhaps leaving some details to them). Most of the time it’s fairly beneficial to let the students work in small groups/pairs so they get the benefit of getting information they might not have gleaned earlier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in a differential equations class, I might ask the students to form groups of 4-5 and present problems (to the whole class) that cover the different subjects in the course: 1. A group on separable (or exact) 1st order DE, 2. A group working on a variation of parameters problem, 3. A group working on a similar problem instead having to use Laplace operators. By doing this the students will have practice with each type, and see how other students complete it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works even if it’s only the first part of a class - simply pick different problems that force students to look at different subtopics. &lt;/p&gt;&#xA;" OwnerUserId="10662" LastEditorUserId="10662" LastEditDate="2019-02-06T13:28:27.890" LastActivityDate="2019-02-06T13:28:27.890" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15166" PostTypeId="2" ParentId="15157" CreationDate="2019-02-05T19:31:14.880" Score="2" Body="&lt;p&gt;In addition to whatever book you choose, I recommend getting the students / future teachers to &lt;em&gt;play&lt;/em&gt; with geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two sites I love: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://sciencevsmagic.net/geo/&quot; rel=&quot;nofollow noreferrer&quot;&gt;sciencevsmagic.net&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.euclidea.xyz/&quot; rel=&quot;nofollow noreferrer&quot;&gt;euclidea.xyz&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And I have started finding great basic geometry puzzles which I am sharing with my college students in hopes of improving their visualization skills:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Geometry Snacks&lt;/em&gt;, by Ed Southall and Vincent Pantaloni  &lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;More Geometry Snacks&lt;/em&gt;, by Ed Southall and Vincent Pantaloni &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://drive.google.com/file/d/1hVP8tLURVDphmHsphz5BQLVzHCeTts29/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;Puzzles by Catriona Shearer&lt;/a&gt; (@catrionashearer)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-02-05T19:31:14.880" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15167" PostTypeId="2" ParentId="15142" CreationDate="2019-02-05T19:42:40.877" Score="0" Body="&lt;p&gt;The way things like this are usually presented is problematic, as is the pressure to move too fast. Many students memorize too much, and figure things out for themselves too little. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't have a phd, so I don't teach at that level (I'm a community college math prof), but my calculus course (1974, U of Mich) was an honors course, which made it close to real analysis. What I remember best about the course is the challenge problems Professor Piranian offered us, which required creative mathematical thinking. If this particular question were framed as a challenge problem, maybe more students would get it, knowing to give themselves plenty of time to explore.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The usual way involves division (by a) at the beginning. It is much easier when done the way James Tanton describes in his videos (though there may be an extra step for some problems, which he doesn't address). His method of completing the square does not feel like a trick to me. There are 3 videos: &lt;a href=&quot;https://www.youtube.com/watch?v=OZNHYZXbLY8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Completing the Square, Part 1&lt;/a&gt;, &lt;a href=&quot;https://www.youtube.com/watch?v=bjH1HphOZ1Y&quot; rel=&quot;nofollow noreferrer&quot;&gt;Completing the Square, Part 2&lt;/a&gt;, and &lt;a href=&quot;https://www.youtube.com/watch?v=eDwi96DCdTc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Deriving the Quadratic Formula&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-02-05T19:42:40.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15168" PostTypeId="1" AcceptedAnswerId="15169" CreationDate="2019-02-05T22:33:49.067" Score="6" ViewCount="176" Body="&lt;p&gt;I'm tutoring a first-semester calculus student, and we were looking over the slides the teacher has used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After teaching (or rather, repeating, for those who completed AP high school math) basic derivatives, there's a couple slides about &quot;linear approximations&quot; and how to find them. The student had a hard time grasping the idea until the end, when she realized &quot;oh, this is just the tangent to the function at that point&quot;, and realized she already knew this from high school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This, after going through the whole idea of &lt;span class=&quot;math-container&quot;&gt;$f(x) \approx f(a) + (x-a)f'(a)$&lt;/span&gt; for some constant &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In HS, she learned how to find a linear approximation by applying the point-slope formula, where the slope is the derivative at this point, and the point is the point at which the tangent touches the curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'm wondering is, will there come a time further down the line where the idea of a &quot;linear approximation&quot; is a better way of looking at it than as &quot;just&quot; a tangent?&lt;/p&gt;&#xA;" OwnerUserId="796" LastActivityDate="2021-03-25T13:35:17.513" Title="Is there a point at which it makes decidedly more sense to learn about a &quot;linear approximation&quot; to a function, rather than a &quot;tangent&quot;?" Tags="|calculus|derivative|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15169" PostTypeId="2" ParentId="15168" CreationDate="2019-02-05T23:21:44.687" Score="6" Body="&lt;p&gt;Thinking about tangent lines as linear approximations is a great way to foreshadow Taylor series. After linear approximations in calc 1, I usually give students a few prompts to get them thinking about quadratic approximations, and then I talk briefly about how it appears in calc 2.&lt;/p&gt;&#xA;" OwnerUserId="7707" LastActivityDate="2019-02-05T23:21:44.687" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15170" PostTypeId="2" ParentId="15142" CreationDate="2019-02-05T23:27:35.457" Score="2" Body="&lt;p&gt;We cannot reduce the assessment of mathematics education to knowledge of how to prove a specific theorem. To be sure, the inability of so many students to provide a proof for such a fairly simple mathematical proposition (in the context of asking the problem of senior math majors) is troubling, and reflects the fact that we don't give students enough time to truly understand many of the key underlying ideas of what they are learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, mathematics as a study serves a wide population of students. If a student intended to go to graduate school in mathematics and could not prove the quadratic formula, I'd be very skeptical of their chances at success in their chosen endeavor. But I would not necessarily feel the same away about a student planning to work in actuarial sciences or engineering. Indeed, one could design a bridge brilliantly without knowledge of a proof of the quadratic formula and conversely (as I can personally instantiate), one can know how to prove the quadratic formula and have not the slightest clue how to properly design a bridge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact that so few senior math majors can prove the quadratic formula brings up a lot of issues, but I don't think this single fact itself is a problem. Or certainly not the root problem. It really depends what the goals of mathematics education are and many students in the same mathematics class may have very different goals for what the course is supposed to help them achieve. If a student does not know how to prove the quadratic formula, depending on context, that could be a sign of a gross failure in their education or an indication that their studies were focused in a different direction more relevant to their future aspirations.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2019-02-05T23:27:35.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15171" PostTypeId="2" ParentId="15161" CreationDate="2019-02-06T03:30:32.527" Score="2" Body="&lt;p&gt;Eating all the chocolates in a multi-layered chocolate box. Each layer has a tray containing several rows of chocolates, and each one you eat is just a &lt;em&gt;tiny&lt;/em&gt; increment that only makes an infinitesimal contribution to your calorie intake . . .&lt;/p&gt;&#xA;" OwnerUserId="10663" LastActivityDate="2019-02-06T03:30:32.527" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15172" PostTypeId="2" ParentId="15142" CreationDate="2019-02-06T09:13:43.993" Score="0" Body="&lt;p&gt;I would not be able to prove the quadratic formula if you told me &quot;prove the quadratic formula&quot;, as I'm not entirely sure what it is (there are lots of formulas involving squares, after all).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadratic_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;googling&lt;/a&gt; it, I know that I never once saw a proof of it, simply because my math teacher back when I was around 14-15 said that it is easier to divide by &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; first and only memorize the &lt;span class=&quot;math-container&quot;&gt;$pq-$&lt;/span&gt;formula. If you give me the formula, I think I'll be able to prove it, but I can well understand if many students have forgotten the proof after such a long time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the first question I would ask Steven Gubkin is if he only said &quot;prove the quadratic formula&quot; or if he actually gave the formula to be proven. Next, it would be good to know how much time passed from them seeing the proof to him asking for it, how much time they were given to come up with a proof (e.g. was it a homework or did he ask in class for a complete proof on the spot). It would also be good to know how much proofs the students had to do up to that point, if they were used to writing formal proofs yet or not, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course we can discuss about students not learning the right things for ages, but such a short message without background is not really representative.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastEditorUserId="5563" LastEditDate="2019-02-08T14:39:24.933" LastActivityDate="2019-02-08T14:39:24.933" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15173" PostTypeId="2" ParentId="10832" CreationDate="2019-02-06T09:59:52.967" Score="5" Body="&lt;p&gt;I'm quite surprised no one has mentioned &lt;strong&gt;groups&lt;/strong&gt;, &lt;strong&gt;rings&lt;/strong&gt; and &lt;strong&gt;fields&lt;/strong&gt;, along with the other algebraic concepts like &lt;strong&gt;category&lt;/strong&gt; and &lt;strong&gt;algebra&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10206" LastActivityDate="2019-02-06T09:59:52.967" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15175" PostTypeId="2" ParentId="15161" CreationDate="2019-02-07T01:16:22.210" Score="1" Body="&lt;p&gt;I like stressing the connection between rates of change and integrals, and I carry this into multiple integrals as well.  As an example, you could talk about a row of apple trees producing (on average) &lt;span class=&quot;math-container&quot;&gt;$f(t,x)$&lt;/span&gt; apples per day per meter at time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; and location &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; along the row.  Probably &lt;span class=&quot;math-container&quot;&gt;$f(t,x)$&lt;/span&gt; is roughly periodic in &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; with a period of &lt;span class=&quot;math-container&quot;&gt;$365$&lt;/span&gt;, and has substantial variation along the row (different trees produce different amounts of fruit at different times).  You might be interested in the total number of apples produced in one year by the entire row. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to talk about how if you have rate data that looks like &lt;span class=&quot;math-container&quot;&gt;$\frac{\left(\frac{\textrm{apples}}{\textrm{day}}\right)}{\textrm{meter}}$&lt;/span&gt;, then you will need to do a double integral in &lt;span class=&quot;math-container&quot;&gt;$(\textrm{day},\textrm{meter})$&lt;/span&gt;-space to find total apples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if you have rate data like &lt;span class=&quot;math-container&quot;&gt;$\frac{\textrm{pressure change}}{\textrm{temperature change}}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{\textrm{pressure change}}{\textrm{volume change}}$&lt;/span&gt;, and these rates are variable depending on current temperature and volume, then you will need to do a line integral over the curve through &lt;span class=&quot;math-container&quot;&gt;$\textrm{temperature} - \textrm{volume}$&lt;/span&gt; space which your beaker is being exposed to to find the total change in pressure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everything makes sense when you pay attention to the units!&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2019-02-07T01:16:22.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15176" PostTypeId="2" ParentId="15138" CreationDate="2019-02-07T17:21:51.937" Score="2" Body="&lt;p&gt;&quot;Oh I just had my first tutorial today. First half I talked. The second half I told them they could stay to work on homework and I will answer questions. 3 students stayed. :D&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope you learned a lesson from that!  It's absolutely what I could have predicted.  You should have also.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally would mix it up some,  but emphasize drill/work throughout.  Have a handout, do a pop quiz (can be ungraded), do breakouts, do games (Family Feud or Jeapordy knockoffs).  Do recitation (student to board).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very simple way to get things moving is to have a handout that they work on first 25 minutes (make it reasonably short).  Then do in class recitation, calling on people, etc. for the solutions.  This kills first half of the period.  And it gets people using the math, in the mood, etc.  Allow self work or partnering/grouping as they choose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second half you could pivot to limited lecture time, followed by ad hoc student questions.  Keep it light &quot;tricks, hints&quot; during the lecture.  Do the horse sound and tap the floor like you are giving them best sooper secret scoop to pass tests.  Students love these corny games...life is short, have some fun.  Also, the interaction second period will be much better since people have had some recent experience with the material.  If they want to bail after the horse secrets during the ad hoc time that is OK.  They will have had a good experience for the majority of the time. &lt;/p&gt;&#xA;" OwnerUserId="11857" LastActivityDate="2019-02-07T17:21:51.937" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15177" PostTypeId="1" CreationDate="2019-02-07T17:53:11.710" Score="5" ViewCount="284" Body="&lt;p&gt;Context: I am an assistant professor of mathematics at a small institution in the US.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our department uses Stewart's &lt;em&gt;Essential Calculus&lt;/em&gt; for our calculus sequence, but I find that my students and I are definitely under-using the book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, I only use the text to assign &quot;suggested problems&quot;, which are simply practice to help my students prepare for quizzes/exams.  As such, Stewart's book is not of significant enough value for students to be willing to buy it, which means that many of them study less than they probably should.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This brings me to my question: Are there cheaper texts out there that are merely &quot;workbooks&quot;--just lists of practice problems with answers?  I'm hoping to find something that students could buy for less than $40 that would correspond (at least somewhat) with the material in Stewart's text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I still assign graded homework problems that I write myself, but I don't currently have the time to write practice problems too.)&lt;/p&gt;&#xA;" OwnerUserId="5988" LastActivityDate="2019-04-18T04:26:18.077" Title="Calculus workbook suggestions" Tags="|undergraduate-education|calculus|textbooks|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15178" PostTypeId="2" ParentId="15177" CreationDate="2019-02-07T18:36:12.410" Score="2" Body="&lt;p&gt;Of course there are many drill manuals out there.  Humongous, Schaum's, etc.  They are often superior to assigned texts in that they have the answers to all the drill (not just odds) and in some cases written solutions.  Usually they are less verbose (easier for kids to scan for formulae and examples).  And are often written with simpler English (less of a need to preen for text selection committees, more student oriented).&lt;/p&gt;&#xA;" OwnerUserId="11858" LastEditorUserId="143" LastEditDate="2019-02-09T00:35:47.557" LastActivityDate="2019-02-09T00:35:47.557" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15179" PostTypeId="2" ParentId="15177" CreationDate="2019-02-07T19:26:01.923" Score="4" Body="&lt;p&gt;You may consider using an Open Education Resource (OER) such as Matt Boelkins' text &lt;em&gt;Active Calculus&lt;/em&gt;. It is available on the web, utilizing WeBWorK for feedback on homework sets, and also has PDF and Print versions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this is much more than simply a &quot;workbook&quot;, but it would certainly serve the purpose of having a list of problems to solve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://activecalculus.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://activecalculus.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-02-07T19:26:01.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15181" PostTypeId="2" ParentId="15142" CreationDate="2019-02-07T20:07:21.117" Score="1" Body="&lt;p&gt;Prove it, or derive it? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had to derive it as my final task in 9th-grade Algebra I (1982). That is, I had to show&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$ax^2+bx+c =0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I memorized it, forgot it, then re-learned it years later as a high school math teacher just to say I could do it, which many of my colleagues couldn’t. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Algebra and Calculus are getting less and less paper-and-pencil manipulative, which I view as a good trend generally. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That probably means that techniques like completing the square aren’t taught as much, and they certainly aren’t reinforced very much, which lets them spill out of a 21-year-old brain pretty fast. I’m not sure I could derive it without that technique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While working at a major statistical software company, I polled our (hundreds of) statisticians, asking when was the last time they used certain things. The quadratic formula was one of those things, and it was universally unused after algebra class, with an occasional Calc I response. Quadratic firms? All day. Quadratic equation? Nope. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, I wouldn’t be too concerned about it. &lt;/p&gt;&#xA;" OwnerUserId="9241" LastActivityDate="2019-02-07T20:07:21.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15182" PostTypeId="2" ParentId="15157" CreationDate="2019-02-07T20:32:44.600" Score="1" Body="&lt;p&gt;Alfred Posamentir (granted, a generation ago) was big in Math Ed and big in Geometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I were teaching undergraduate Geometry to future teachers, I’d look for Ed Moise’s book &lt;em&gt;Elementary Geometry from an Advanced Standpoint&lt;/em&gt;. He wrote &lt;em&gt;the&lt;/em&gt; Geometry book back in the “New Math” days—perhaps more rigor than needed for high school Geometry—but it’s perfect for undergraduates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His book builds up Geometry as far as you can without the parallel postulate. I think he called that “Synthetic Geometry” (although it’s been since Fall ‘88 when I took the course). He then goes into the various cases where the parallel postulate gives us unique and tasty geometrical flavors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since then, I discovered Taxicab Geometry and wonder if it wouldn’t be a nice way to ease the undergoos into a non-Euclidean idea with surprising results (e.g., circles appear as squares in Taxicab Geometry).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing about Moise is that he holds out on Non-Euclidean Geometry until the very end, and doesn’t do a helluva lot with it. The title says exactly what the book is, and it also expresses what I think the course ought to be. If you’re looking for a serious Non-Euclidean Geometry book for undergraduates, I can make some recommendations, but you’ll have to tell me you at least glanced at Moise. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;NB: His high school Geometry text is available on the Internet. It started as a series of typed notes (search for SMSG, the cool kids way of saying “new math”) and you’ll find them. Also, if you open an account on the library section of archive.org, a relatively mature 1990s-era version of the book (co-authored by Floyd Downs) is available for checkout in Encrypted ePub and Encrypted PDF formats, accessible with Adobe Digital Editions software.  &lt;/p&gt;&#xA;" OwnerUserId="9241" LastActivityDate="2019-02-07T20:32:44.600" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15185" PostTypeId="2" ParentId="15157" CreationDate="2019-02-07T21:06:14.757" Score="1" Body="&lt;p&gt;I recommend to avoid the &quot;interesting&quot; suggestions of books that include high school geometry but with some advanced insights.  These suggestions likely come from the commenters here (many were research mathematicians at least in grad school) versus the needs and enjoyment of high school teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead assign a popular, straightforward example text actually used in schools.  For simplicity, I would pick one a bit more to the side of old fashioned.  This also gives you sort of a base case to compare different ed fads to.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition I recommend to assign a text on the different main methods of teaching geometry.  If you can't find one, then assign readings on each major &quot;school&quot;: articles, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not &lt;strong&gt;critical&lt;/strong&gt; to be encyclopedic.  Just give them some flavor of a few different approaches (Common Core, traditional, Saxon, etc.)  That way when they go teach, wherever they get a job they feel a bit confident that there are several ways of skinning a cat.  That there is not one correct school of geometry education (i.e. there is no Euclidean proof of how to teach Euclid).  Do NOT feel like you need to be an expert on every school of math ed (too harsh on yourself).  Just give them an honest exposure to at least three methods--you will learn something also.  [However I would NOT emphasize your own lack of knowledge--students find that a turnoff when the teacher prostrates himself like that...and it reduces confidence.]  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quick Amazon search found this for a math ed text:  &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0415856914&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Learning-Teaching-Geometry-Secondary-Schools/dp/0415856914#reader_0415856914&lt;/a&gt;  I actually like the modeling approach but would prefer it had a bit more of a review of different methods (has some in a historical context) versus just pushing their approach.  But take a look at it.  And do a Google/Amazon search yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, there is a very active blogosphere of math ed (secondary school edu refomers as well as practitioners).  It is NOT a well connected graph (haha) to this MESE community.  Below are a few links.  You can look through and find others as well.  But I think having the students read some blogs will give them excellent insights from real practitioners and also help them understand the whole multi cat skin idea.  Also I think they will enjoy that versus just articles and books (less dry).  Plus there is the possibility they just play the link game and wander onto other things that interest them on the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://talesofahighschoolmathteacher.blogspot.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://talesofahighschoolmathteacher.blogspot.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://crazymathteacherlady.wordpress.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://crazymathteacherlady.wordpress.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://thegeometryteacher.wordpress.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://thegeometryteacher.wordpress.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://blog.feedspot.com/high_school_math_blogs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://blog.feedspot.com/high_school_math_blogs/&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://educationrealist.wordpress.com/?s=geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://educationrealist.wordpress.com/?s=geometry&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://larrycuban.wordpress.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://larrycuban.wordpress.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://garyrubinstein.wordpress.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://garyrubinstein.wordpress.com/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11860" LastActivityDate="2019-02-07T21:06:14.757" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15186" PostTypeId="2" ParentId="689" CreationDate="2019-02-07T21:10:15.780" Score="2" Body="&lt;p&gt;Depending on their age, they may not have reached what Piaget called the “abstract operational” stage yet. If they’re still in the concrete operational stage, they will need physical or otherwise concrete explanations. This is one reason why we don’t attempt algebra at 5th grade—their minds haven’t gotten there yet. &lt;/p&gt;&#xA;" OwnerUserId="9241" LastActivityDate="2019-02-07T21:10:15.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15187" PostTypeId="2" ParentId="15157" CreationDate="2019-02-08T00:44:22.030" Score="0" Body="&lt;p&gt;Not what @Sumac seeks, but it would be good to keep this book&#xA;handy so you can add historical context at appropriate junctures.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first five chapters show how the ancient Greeks established geometry, together with its numerous practical applications, while more recent findings on Euclidean geometry are discussed as well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/5gsHs.jpg&quot; width=&quot;180&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ostermann, Alexander, and Gerhard Wanner. &lt;em&gt;Geometry by its history&lt;/em&gt;. Springer, 2012.&#xA;  &lt;a href=&quot;https://www.springer.com/us/book/9783642291623&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-02-08T00:44:22.030" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15188" PostTypeId="1" AcceptedAnswerId="15189" CreationDate="2019-02-08T10:53:28.293" Score="17" ViewCount="5420" Body="&lt;p&gt;I just received one assignment (by email) from a student. Out of 6 questions, &quot;I don't know&quot; is the answer to 4 of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also a comment at the end of the assignment which suggests &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;my lectures are not helpful for solving problems&lt;/li&gt;&#xA;&lt;li&gt;I am assuming students already know many of the things that I am talking about, so things are very confusing in class&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I replied that I will improve my lectures and go over basic things more clearly. And I also offered one-to-one tutoring to help him catch up. (Though I had a feeling that he will not take it up--he didn't ask any questions previously through email and he did not come to office hours)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now the problem is, I announced that assignments are graded by completion. His assignment certainly is far from complete. But I feel that if giving him 0 will further antagonize and discourage the student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, traditionally, assignments are not mandatory but only serve as bonus points in final evaluation. So less than half of the students will actually do the assignment. It seems that answering two questions are already better than most other students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How would you deal with such an assignment?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorDisplayName="user11702" LastEditDate="2019-02-08T12:48:49.430" LastActivityDate="2019-02-11T08:53:58.060" Title="How to deal with a &quot;protest&quot; assignment?" Tags="|undergraduate-education|assessment|" AnswerCount="6" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="15189" PostTypeId="2" ParentId="15188" CreationDate="2019-02-08T14:44:05.403" Score="45" Body="&lt;p&gt;You need to slam him on the grade.  That is what he earned.  Don't be so easily manipulated by his comments on your teaching.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also I would not have sent an email apology.  Just offered to meet with him to discuss his concerns.  But still slammed him on the assignment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you let these kids walk over you, you'll never survive.  You're in charge.  Doesn't mean your teaching is great (or not).  But don't be a wimp.  That doesn't help regardless.&lt;/p&gt;&#xA;" OwnerUserId="11864" LastActivityDate="2019-02-08T14:44:05.403" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15190" PostTypeId="2" ParentId="15188" CreationDate="2019-02-08T15:39:53.910" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Now the problem is, I announced that assignments are graded by completion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't see a problem if you were clear about how the assignments will influence the final grade, perhaps something like &quot;The grade on the assignments will be included in the final grade if and only if the grade on the assignments is better than the grade that would have been given without the assignments.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I would grade each assignment &quot;normally&quot; which would be a failing grade for at most 33% completion.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2019-02-08T15:39:53.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15191" PostTypeId="2" ParentId="15188" CreationDate="2019-02-08T18:31:49.463" Score="13" Body="&lt;p&gt;I am not a mathematics educator but I feel the need to chime in from the student side of this. I have taken numerous hard math classes during my BSc, and I have had my fair share of feeling hopeless, lost, and frustrated. You didn't indicate which math class you teach, and to be honest it is irrelevant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unlike the other answers I don't feel like this is necessarily an attempt to control you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Point (1) sounds like something I commonly heard from fellow students who struggled not because the professor didn't try to help, but because they are taught &quot;the solution is the goal&quot;. They get quickly discouraged, and at points frustrated enough to write smarmy marks and protest. Perhaps to address this point you could talk to the class about an anonymous assignment that said these things. Talk to them about how its more important to show how they arrived at their solution and open up the floor for a few minutes to let people tell you what they think. A quick way to find your flaws is to do this. Many complaints will be down right ignorable (this class is too hard, you're not good enough, etc) but occasionally a student will call out a character flaw worth addressing. Maybe you're not making something clear enough? Generally students don't protest unless they truly feel hopeless. A few classes will do this (mathematical statistics, proof-based math courses, etc) so it would be useful to figure out why at least. He could have came to your office to complain directly and start a dialog with you - so he gets a 0 if only to demonstrate snark wont be tolerated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Point (2) is sort of addressed inside of point (1). You're an undergraduate educator. The student has arrived at your class having either:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Taken an entrance exam demonstrating the minimum level of competence for the course&lt;/li&gt;&#xA;&lt;li&gt;Passed pre-requisites satisfactorily showing competence in the material&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;At any rate, people forget. There is a great book I'm sure you've heard of called &lt;em&gt;How To Solve It&lt;/em&gt; by Polya. In the book he talks about how if a student cannot grasp the harder stuff you need to prod them with questioning until you find out what they do know and build them up from there. If you're unwilling or unable to do this, then you need to look at your department. Students struggling this much generally are being passed by easier professors or lax examiners. A certain level of struggle is acceptable, but I have been in classes that felt hopeless for that exact reason - the professor was unwilling to help us learn what we didn't know quickly, and the department pre-requisites didn't prepare us for the class at all. Many of us nearly failed, and the professor smugly proclaimed we were all incompetent. It's not a good experience being that lost, and it's worth digging into the root cause so you can address it directly either at the department level, or assisting the class with their shortcomings. Either way, there are mature ways to deal with this at the student level so he gets a 0 regardless of what you choose to do.&lt;/p&gt;&#xA;" OwnerUserId="11868" LastActivityDate="2019-02-08T18:31:49.463" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15192" PostTypeId="2" ParentId="15157" CreationDate="2019-02-08T20:22:29.607" Score="4" Body="&lt;p&gt;Not to blow my own (and my co-authors') horn, but this may be exactly what you're looking for:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.routledge.com/The-Learning-and-Teaching-of-Geometry-in-Secondary-Schools-A-Modeling/Herbst-Fujita-Halverscheid-Weiss/p/book/9780415856911&quot; rel=&quot;nofollow noreferrer&quot;&gt;Herbst, A., Fujita, T., Halverscheid, S., and Weiss, M. (2017). &lt;em&gt;The Learning and Teaching of Geometry in Secondary Schools&lt;/em&gt; (IMPACT: Interweaving Mathematics Pedagogy and Content for Teaching), 1st Edition.  London:  Routledge.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0415856914&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Amazon link, if you prefer&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RbiWH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RbiWH.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the publisher's website:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;The Learning and Teaching of Geometry in Secondary Schools&lt;/em&gt; reviews&#xA;  past and present research on the teaching and learning of geometry in&#xA;  secondary schools and proposes an approach for design research on&#xA;  secondary geometry instruction.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Areas covered include:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;teaching and learning secondary geometry through history;&lt;/li&gt;&#xA;  &lt;li&gt;the representations of geometric figures;&lt;/li&gt;&#xA;  &lt;li&gt;students’ cognition in geometry;&lt;/li&gt;&#xA;  &lt;li&gt;teacher knowledge, practice and, beliefs;&lt;/li&gt;&#xA;  &lt;li&gt;teaching strategies, instructional improvement, and classroom interventions;&lt;/li&gt;&#xA;  &lt;li&gt;research designs and problems for secondary geometry.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;Drawing on a team of international authors, this new text will be&#xA;  essential reading for experienced teachers of mathematics, graduate&#xA;  students, curriculum developers, researchers, and all those interested&#xA;  in exploring students’ study of geometry in secondary schools.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To be clear, this is not a &quot;how-to&quot; handbook, but it does provide a pretty comprehensive summary and synthesis of what the mathematics education research community knows about the teaching and learning of Geometry, including a historical survey of how that has changed over the last 150 years.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2019-02-08T20:22:29.607" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15193" PostTypeId="2" ParentId="15157" CreationDate="2019-02-08T21:42:39.577" Score="2" Body="&lt;p&gt;This is likely not exactly what you have in mind, but a recent book by an author that is very critical of how geometry is traditionally taught and who offers a starting point for how it should be taught is &lt;em&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B00AFS6LSC&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Measurement&lt;/a&gt;&lt;/em&gt;, by Paul Lockhart. While it may not be the book you want to use as your reference, I think it is a very worthwhile read to understand how one particularly opinionated math educator thinks geometry should be taught. But that mathematician's viewpoint does represent what a lot of mathematicians think about geometry education (and math education more broadly), even if most of us aren't quite as passionate about it as Dr. Lockhart. (You can also read his famous &lt;em&gt;A Mathematician's Lament&lt;/em&gt; online &lt;a href=&quot;https://www.maa.org/external_archive/devlin/LockhartsLament.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; to get a sense of his criticisms of the traditional approach to teaching mathematics. He addresses geometry specifically in his critique.)&lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2019-02-08T21:42:39.577" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15194" PostTypeId="2" ParentId="15188" CreationDate="2019-02-08T23:27:40.930" Score="6" Body="&lt;p&gt;I'm not an educator, though I've been involved in the admin side of university-level  education and did a heavily mathematical degree at university. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I'm wondering about two things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Does he actually understand how to study maths?&lt;/strong&gt; I didn't when I was a student. I think I had the idea that I should be able to go to the lecture, understand the material, then go away able to do the assignment, and that a maths textbook could be read like an ordinary book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Do the prerequisites for the course, as taught, actually provide what's needed?&lt;/strong&gt; I say &quot;as taught&quot; because the way topics are listed in a course outline might not be a guide to how thoroughly each one is covered in the actual sessions. If something essential for your course isn't being adequately covered in one of the prerequisites, then you could probably do with knowing that—since other students might well be affected too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the grading, I think you jjust give the honest grade for what was handed in (one that would, say, convince an external examiner that you're applying identical criteria to everyone). It's perfectly consistent to both do that and try to find out what caused him to do so badly. Also being clear that it's done objectively might make it easier for the student not to take the grade personally, so he's more open to being helped.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastActivityDate="2019-02-08T23:27:40.930" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15195" PostTypeId="1" CreationDate="2019-02-09T00:44:32.500" Score="5" ViewCount="1052" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Non-standard_calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nonstandard calculus&lt;/a&gt; is a reformulation of calculus that is based on infinitesimals instead of epsilon-delta definitions. Of course, people had tried to use infinitesimals in calculus before; in fact, Calculus originally used infinitesimals. The problem was that it did not have a rigorous foundation, which is why mathematicians started using the epsilon-delta definitions instead, although they still used infinitesimals informally. Much later, nonstandard analysis came along, which did put infinitesimals on rigorous foundations, but by then it was not necessary since epsilon-delta definitions were available. Nonstandard calculus is calculus except based on nonstandard analysis instead of analysis. There are still people who think that nonstandard analysis should replace those definitions, however. I will put an informal introduction to how it works at the end of this post.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Would teaching students calculus using nonstandard calculus make it easier to learn?&lt;/strong&gt; Answers should also take into account future calculus learning, not just learning in the course itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would prefer to focus on &quot;normal&quot; students. I think for bright students, teaching a little bit of both would be beneficial, since I think comparing the approaches teaches the idea that there can be radically different ways to study math and arrive at the same results. However, for most students, I think mainly focusing one or the other would be better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some of my observations. They are kind of long, so feel free to skip/skin them. The most important is probably points 4 and 6, since they are about how standard and nonstandard calculus are different. The other points are about how they are the same.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Of course, the mathematical foundations of nonstandard analysis would be much too complicated to cover in such a course. The same is true of standard analysis though, so this is not a concern.&lt;/li&gt;&#xA;&lt;li&gt;Nonstandard calculus is compatible with calculus. By that I mean that anything you can prove in analysis can be proven in nonstandard analysis, and anything you can disproof is analysis can be proven in standard analysis, so there will never be a contradictory result. In terms of provable statements, the only difference is that nonstandard analysis proves some things about infinite and infinitesimal numbers that standard analysis is not concerned with (since it does not use such number), and even those can be translated into equivalent statements about real numbers in standard analysis. The main difference is how they got about proving them.&lt;/li&gt;&#xA;&lt;li&gt;Definitions in nonstandard calculus involve much fewer quantifiers than those in standard analysis. Limits, continuity, and uniform continuity all have a quantifier complexity of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, whereas in standard calculus they have 3 or 4. This however is balanced by the fact that it is much more difficult to do arithmetic with hyperreal numbers than the real numbers. You can not even build a hyperreal number calculator. All in all, proofs in both are about equally as long.&lt;/li&gt;&#xA;&lt;li&gt;Although the proofs are equally as long, they are much more intuitive in nonstandard calculus than in standard calculus. With standard calculus, you usually motivate theorems with ideas like infinity, or being infinitely close, or dividing infinitesimal numbers. Then you translate those intuitions into a series of epsilon delta definitions, with many inter-dependencies, never including your intuitions in your proof, just their translation. In nonstandard calculus, most of the intuitions are included literally in the proof. Infinity is no longer just an idea, its a number. They are also superior to informal proofs using infinitesimals (like scientists sometimes use) since it never proves an contradictions. You still have to do the work, but the work is not separate from the intuition.&lt;/li&gt;&#xA;&lt;li&gt;Both standard and nonstandard calculus generalize to other fields, such as topology, so that is not an issue.&lt;/li&gt;&#xA;&lt;li&gt;The main issue I see is when they take Calculus in the future. If the students never take another formal calculus course, this is not a concern, but if they do, all the proof methods they learned in nonstandard calculus will no longer be taught in light of new material. The grader may not know how to grade a proof using nonstandard calculus, as well (which is reasonable, since it invokes a whole host of concepts not used in standard calculus). One thing that helps though is that proofs from standard calculus still make sense in nonstandard calculus. For example, even though limits no longer have an epsilon-delta definition, they do have an epsilon-delta theorem. In fact, most limit proofs using the definition can be mechanically translated into a proof using the theorem. Since the epsilon-delta theorems are true by, well, definition in standard calculus, they would be compatible with future calculus courses. Additionally, but having the students carefully study the epsilon-delta theorems, they will likely be able to see the connection between the material in future calculus courses could be translated into nonstandard calculus. Unfortunately, this results an &quot;overhead&quot; cost of using nonstandard calculus over standard calculus. However, in my experience, I find that writing a proof first in nonstandard calculus and then translating it to standard calculus is easier mentally (unless I immediately see the standard calculus proof). This may be unique to myself however. However, for some courses (especially less proof based ones), learning nonstandard calculus will actually make the course easier. That's because many courses less concerned with rigor will just use infinitesimals anyways, ignoring foundational issues. Students who learned nonstandard calculus with have an advantage, since they know how to manipulate infinitesimals rigorously.&lt;/li&gt;&#xA;&lt;li&gt;Another potential issue is materials. For any introductory calculus course, this is just potential, since there is &lt;a href=&quot;http://www.math.wisc.edu/~keisler/calc.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;textbook&lt;/a&gt; that teaches introductory calculus, using nonstandard calculus proofs, that the author distributes for free. There are also some other ones of varying costs.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5030" LastEditorDisplayName="user507" LastEditDate="2019-02-10T17:10:45.897" LastActivityDate="2019-02-10T17:54:35.643" Title="Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?" Tags="|calculus|course-design|infinitesimals|axioms-foundations|infinity|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15196" PostTypeId="2" ParentId="15195" CreationDate="2019-02-09T01:48:08.830" Score="3" Body="&lt;p&gt;It's not really that relevant since the bulk of a normal calculus course (e.g. AP BC, Thomas Finney, Stewart) just does a small amount of epsilon-delta (so student is exposed to it) and then moves to &quot;x+h&quot;.  The bulk of the course is about learning derivatives, antiderivatives, methods of integration, classic applied problems, a bit if polar coordinates, bit of series, and small section on ODEs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are swinging at the wrong opponent with this obsession on the definition of a derivative (but not surprising given the theory bent of many math majors).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to make things easier, cut partial fractions, integration by parts, etc.  Do nurses and business students really need that?  (Of course this does mean tracking because science and engineering do.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I found your question hard to parse given the notation (upside down A).  This is ironic given you are planning to make things easier for knuckleheads worse than I.  Not showing it here...makes me leery of your proposals.  Can't you discuss pedagogy and coverage without such a segue into exploring the math theory like that? &lt;/p&gt;&#xA;" OwnerUserId="11869" LastActivityDate="2019-02-09T01:48:08.830" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15197" PostTypeId="2" ParentId="15195" CreationDate="2019-02-09T18:02:27.140" Score="5" Body="&lt;p&gt;Speaking as a former student, though an engineering one . . . It was hard enough learning to integrate tricky expressions and solve differential equations, without having to learn a new number system as well. And I don't remember ever needing the rigorous definition of a limit, standard or nonstandard. The most I needed even in complex calculus was an ability to manipulate limits, and an awareness that derivatives and antiderivatives were defined as limits and could be found as limits if necessary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You're not going to understand Maxwell's Equations or the Schrödinger Wave Equation or Fourier transforms without understanding integrals and derivatives, but you can perfectly well understand them without knowing the rigorous definition of a limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that an introductory calculus course isn't going to be an introduction to proving that calculus works, any more than an introductory algebra course is going to start with a rigorous definition of the real numbers—it's going to be an introduction to the concepts of differentiation and integration and how to use them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, &lt;em&gt;proving rigorously that calculus works&lt;/em&gt; does require rigorous definitions, and at that point something new has to be introduced. Maybe a strange new definition involving &lt;span class=&quot;math-container&quot;&gt;$ε$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt;, and maybe a strange new number system in which there are different sizes of zero which are defined not to equal each other or zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think awareness at that stage is likely to be:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; is defined as a limit, and so is &lt;span class=&quot;math-container&quot;&gt;$\int y dx$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;but we sometimes treat &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; like algebraic quantities&lt;/li&gt;&#xA;&lt;li&gt;this is a bit dodgy even though it works, since we're effectively either dividing &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; or multiplying &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;the way round this involves something to do with limits.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$ε–δ$&lt;/span&gt; is explained &lt;strong&gt;clearly&lt;/strong&gt; at that stage—ie in terms of making &lt;span class=&quot;math-container&quot;&gt;$ε$&lt;/span&gt; arbitrarily small by making &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt; small enough or &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; large enough, and &lt;strong&gt;not&lt;/strong&gt; as a dense expression full of quantifiers—then once someone has grasped it, they'll be convinced it's rigorous. But if they're introduced to hyperreal numbers instead, they'll have lingering doubts about whether arguments using them are actually valid. It means another layer of understanding is needed, namely the background theory of the new number system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think it's more useful to focus on understanding the principle of &lt;span class=&quot;math-container&quot;&gt;$ε–δ$&lt;/span&gt; definitions and techniques clearly than to introduce an unfamiliar number system. Apart from anything else, this will help them to understand other textbooks in the future, which must likely won't use hyperreals.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastActivityDate="2019-02-09T18:02:27.140" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="15198" PostTypeId="2" ParentId="15195" CreationDate="2019-02-09T19:02:09.233" Score="7" Body="&lt;p&gt;This has certainly been tried before.  See for example,  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.wisc.edu/~keisler/calc.html&quot; rel=&quot;noreferrer&quot;&gt;H. Jerome Keisler. Elementary Calculus: An Infinitesimal Approach.  On-line Edition.&lt;/a&gt;  This has also been published in print by Dover.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/2318657?origin=crossref&amp;amp;seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;Kathleen Sullivan.  The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach.&lt;/a&gt; The American Mathematical Monthly Vol. 83, No. 5 (May, 1976), pp. 370-375.&lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2019-02-09T19:02:09.233" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15199" PostTypeId="2" ParentId="15188" CreationDate="2019-02-10T02:05:56.210" Score="1" Body="&lt;p&gt;In contradistinction to many of the answers, I say the student may have actually done the assignment.  Are these reading questions about the material (e.g. &quot;explain what an integral is&quot;)?  If the questions are worded in such a way that &quot;I don't know&quot; is a grammatical answer to the question, then this one time you can throw him/her a bone and say &quot;Okay, you get the points.  Next time you'll know I mean for a mathematical answer.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the questions are of the form &quot;Compute 2+2&quot;, this doesn't obtain.  Even in that case, though, in a smallish class with someone who may really be completely unprepared and unable to recognize how to deal with it, you could again throw a bone and let the student get half points for submitting an actual answer a week late.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bottom line is to know your constituents and setting.  That is hard, or at least I find it hard, and it takes time.  But as an educator, it's your job (even if it's not what you are rewarded for).  It's possible this student is a brat; it's also possible (or, if you are in the United States, likely) the student has found an inappropriate way to respond to how overwhelming 'real math' with a lot less hand-holding is.  Give the student a chance to turn it around.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-02-10T02:05:56.210" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15200" PostTypeId="2" ParentId="15195" CreationDate="2019-02-10T17:36:51.873" Score="5" Body="&lt;p&gt;My understanding of this question is that it proposes the idea of a &quot;monolingual&quot; freshman calc course in which students mostly learn the language of NSA, and limits are largely or completely neglected. This makes it different from &lt;a href=&quot;https://matheducators.stackexchange.com/questions/12552/is-it-to-the-students-advantage-to-learn-the-language-of-infinitesimals&quot;&gt;this question&lt;/a&gt; by Mikhail Katz, which asks whether it's a good idea for students to be &quot;bilingual.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have some experience teaching some NSA-based material to first-semester calculus students at a community college. &lt;a href=&quot;http://www.lightandmatter.com/fund/&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt; is the book I wrote for that purpose. My approach is &quot;bilingual.&quot; The philosophy with which I approached this was not that students should learn NSA in excruciating detail -- that would be kind of silly IMO, and would detract from the main thrust of the course, which is basically to be able to do rule-based differentiation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, scientists and engineers still use Leibniz notation and manipulate infinitesimals using algebra. They've been doing it for 300 years, and they never stopped doing it just because there was a short gap between the invention of the limit and the invention of NSA. These students will see such manipulations in their physics courses, and they will see and be expected to use them in their careers. So we should give them some systematic idea of what techniques are appropriate when performing these manipulations. If this was 1850, we would teach them a body of techniques that we knew from experience gave correct results, e.g., throwing away higher orders of &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;. Today we have more secure knowledge that these procedures can be put on a sound logical footing, but that actually has little effect in reality on what body of techniques we use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal should not be to eliminate the &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; definition of limits. This is not possible because there is a clear consensus among mathematicians that &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; is something students should have already learned by the time they go on to their second-semester course, and we need to prepare these students properly. In fact,  &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt;-like ideas are fundamental to many of the common modes of reasoning about calculus. E.g., we can do a numerical simulation of the motion of the moon with a small time step, and say, &quot;If I make the step size small enough, I should be able to make the error as small as desired.&quot; Newton and Leibniz would have understood that idea, and so would every physicist of succeeding centuries, long before the definition of the limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you get right down to it, using NSA doesn't really turn out to produce any incredible simplification of freshman calculus. For example, it would be nice if we could just prove the chain rule and L’Hôpital’s rule by manipulating infinitesimals using algebra, but the nuts and bolts don't actually work out quite that trivially. In the case of the chain rule, you have the issue that the derivative is not the quotient of infinitesimals but the standard part of that quotient. In the case of L’Hôpital’s rule you also have all the various forms of the rule (repeated application, limits at infinity, &lt;span class=&quot;math-container&quot;&gt;$\infty/\infty$&lt;/span&gt;, etc.). These are complicated to prove (and many books don't prove them all), and the complication isn't really reduced very much by using NSA.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keep in mind also that at this level, our students never really get a full-blown introduction to the &lt;em&gt;real&lt;/em&gt; number system. No commercial textbook I've ever seen systematically introduces and applies anything beyond the first-order properties of the reals. (They may state the completeness property, but they never use it to prove things like the intermediate value theorem.) This is material that belongs in an upper-division analysis class. Since the hyperreals have the same first-order properties as the reals, there is actually very little that we can meaningfully say about the hyperreals to students at this level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do explicitly name the hyperreals and describe how their properties differ from those of the reals (this is mainly in section 2.9 of my book). However, I don't think it's a good idea to go into the kind of depth that Keisler does.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-02-10T17:54:35.643" LastActivityDate="2019-02-10T17:54:35.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15201" PostTypeId="1" CreationDate="2019-02-11T04:43:20.433" Score="1" ViewCount="339" Body="&lt;p&gt;I was sort of thrown into teaching calculus based physics to a bunch of non-physics majors, who have taken one semester of calculus, and are poor with that material. It is only a 50 minute per week class. So I've been stuck in between teaching elements they'd receive in the calc sequence (integration by parts, &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; vector notation, dot product, partial derivatives, etc), and somewhat random integral or derivative versions of stuff they've seen in their normal physics class running in parallel. I also don't like the calc based text I'm using in parallel to their regular non-calc class text. Since it's a 50 minute per week, 1 credit hour class with low expectations, can anyone recommend me some &quot;interesting&quot; activities, or else point me to a nice standard problem set, to do in there?&lt;/p&gt;&#xA;" OwnerUserId="11880" LastEditorUserId="8571" LastEditDate="2019-02-11T16:47:02.257" LastActivityDate="2019-02-11T16:47:02.257" Title="Activities for calc based physics" Tags="|calculus|physical-sciences|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15202" PostTypeId="2" ParentId="15201" CreationDate="2019-02-11T05:28:17.753" Score="5" Body="&lt;p&gt;I am going to assume that you are teaching a calculus &quot;helper&quot; versus the entire physics class.  Your initial statements don't match that.  But then all your content described is math, not physics.  And also 50 minutes per week sounds rather light for a whole class.  [If the converse is the case, I would spend your time on...physics.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that in mind, my advice is NOT to look for some interesting activities.  Instead use this as an opportunity for you to learn about the math required in stereotypical physics 101 classes.  And give the kids a brushup related to that.  Look at the homework and see what are the key math steps in them.  And give a little help on them.  Don't teach the physics of it (no word problems, they get that in real class).  But abstract it down just to the math itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would try to parallel whatever is being done in physics class.  If vectors are important this week, spend some time on vectors this week.  If integrals are important a different week, spend time on that.  Now, it may not match perfect match, and some remediation may be needed of issues in previous semester.  But let that be a general guide.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of format, I would try to do something interactive, using a handout each week. Some lecture and some practice, using the sheet.&lt;/p&gt;&#xA;" OwnerUserId="11881" LastActivityDate="2019-02-11T05:28:17.753" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15203" PostTypeId="2" ParentId="15188" CreationDate="2019-02-11T08:53:58.060" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Now the problem is, I announced that assignments are graded by completion. His assignment certainly is far from complete. But I feel that if giving him 0 will further antagonize and discourage the student.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems to me that his assignment was &lt;span class=&quot;math-container&quot;&gt;$\frac13$&lt;/span&gt; complete and he should get &lt;span class=&quot;math-container&quot;&gt;$\frac13$&lt;/span&gt; of the maximum grade for the assignment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recognise that this seems so simple that I'm probably overlooking something due to being from a different culture to you, but sometimes people overcomplicate things and overlook simple solutions.&lt;/p&gt;&#xA;" OwnerUserId="8005" LastActivityDate="2019-02-11T08:53:58.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15204" PostTypeId="2" ParentId="15201" CreationDate="2019-02-11T15:11:04.817" Score="0" Body="&lt;p&gt;I don't have any specific resources, but I suggest that you might find a little success by finding some physics applications where there really is a difference between the &quot;algebra-based&quot; and &quot;calc-based&quot; physics that use e.g. multivariate calc or integration by parts or something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, I think that there are some problems involving work (e.g. pumping water out of a conical tank?) that probably aren't exactly solvable without integrals, and which might require even trig substitution or worse.  If you have access to a solution manual for the &quot;real&quot; calc-based physics course, you can just browse it for problems where those specific techniques come up, and then just do the problem up to the point where calculus comes in.  Give them two minutes to talk to a neighbor to say &quot;now what do I do&quot;, and assuming they have no idea, now you review the relevant calc technique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is probably not really going to help them internalize new calculus they need, but then again a 1-credit hour class probably is going to have pretty minimal expectations so at least it will connect a bit.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-02-11T15:11:04.817" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15205" PostTypeId="1" CreationDate="2019-02-12T09:45:12.250" Score="6" ViewCount="285" Body="&lt;p&gt;I'm a young math student. And I live with the effort of always wanting to understand everything I study, in mathematics. This means that for every thing I face I must always understand every single demonstration, studying the basics every time if I don't remember them. And this makes it impossible for me to prepare the exams, because I can't go on, I fix myself on wanting to derive by myself a theorem and I lose days in it. And so I ask mathematicians if it is always necessary to be able to prove everything, or we must accept what the theorems say and give it for good. If possible I also ask you some advices to help me study, knowing my problem.&lt;/p&gt;&#xA;" OwnerUserId="11884" LastEditorUserId="8571" LastEditDate="2019-02-12T14:55:22.837" LastActivityDate="2019-02-12T19:16:23.900" Title="A question from a young student to mathematicians" Tags="|proofs|problem-solving|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15206" PostTypeId="2" ParentId="15205" CreationDate="2019-02-12T15:05:42.257" Score="2" Body="&lt;p&gt;Concentrate on doing the homework problems.  They are most important for developing your skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your approach is not time efficient.  You need to learn things at a surface level and then come back and learn it deeper.  The human brain is not a computer.  We are imperfect animals and learn by starting a groove and deepening it over time AND making connections over time to previous knowledge.  If you try to do it perfect immediately, you are fighting nature.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keep a notebook and use it for a parking lot.  Record questions, thoughts, ideas as you study something.  Often this will set your mind at ease as the concern is recorded at least.  Often later on, you will realize the answer or resolve the question.  But it won't have stopped you from advancing in the material.  Where it does not, at least you have an organized list of questions to discuss with your teacher.&lt;/p&gt;&#xA;" OwnerUserId="11885" LastActivityDate="2019-02-12T15:05:42.257" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15208" PostTypeId="2" ParentId="15205" CreationDate="2019-02-12T19:16:23.900" Score="2" Body="&lt;p&gt;You will never know everything in mathematics. One place you could read about this is Bill Thurston's 'On Proof and Progress in Mathematics'. He gives a (slightly fictitious) list of many, many ways of understanding the concept of derivative, some of which will almost certainly be beyond you at this point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It sounds like you need to recognise that your personality is making your studies difficult, and find ways of dealing with that. There is nothing wrong with wanting to understand everything: mathematicians would generally encourage that, and many wish their students took more of that attitude. But you should perhaps try to remember that you don't necessarily need to understand everything &lt;strong&gt;now&lt;/strong&gt;. If you have an exam coming up, you should change your purpose temporarily from 'understanding mathematics' to 'learning what you can to get enough marks in the exam'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also worth considering that the process of going through the motions of something, even if you don't understand it, may in fact help you to come to understand it. As you do examples, you may spot the pattern that you are not noticing from the general description.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-02-12T19:16:23.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15211" PostTypeId="2" ParentId="15177" CreationDate="2019-02-12T21:02:45.817" Score="2" Body="&lt;p&gt;Other Calculus OER resources include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://openstax.org/details/books/calculus-volume-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax Calculus I&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://scidiv.bellevuecollege.edu/dh/Calculus_all/Calculus_all.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Contemporary Calculus&lt;/a&gt;, by Dale Hoffman&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I like the problems in Hoffman.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-02-12T21:02:45.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15212" PostTypeId="1" AcceptedAnswerId="15231" CreationDate="2019-02-13T01:42:18.843" Score="48" ViewCount="19795" Body="&lt;p&gt;My son, who just turned 5, has been interested in the concept of infinity since long. He asks me a lot of questions regarding infinity. For example, not accepting my infinity + any number = infinity, he asked me how old I will be when he himself becomes infinity years old.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How should I explain to him this concept that resonates with his existing understanding of mathematics. If it matters, he knows how to add and subtract very large numbers, knows about negative numbers and has figured out tables of any number less than 100.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;UPDATE:&lt;/strong&gt; He derived one possible answer a few minutes after I typed the question on this forum. So I have written it as an answer below instead of a comment. Along with many other useful answers and comments here, this will be helpful to future questioners searching for the same to satisfy their children. Furthermore, it would be interesting for some to look at the questioning child's own thinking process.&lt;/p&gt;&#xA;" OwnerUserId="11887" LastEditorUserId="11887" LastEditDate="2019-02-18T23:38:46.380" LastActivityDate="2019-03-24T05:13:33.133" Title="The concept of infinity for a 5 year old" Tags="|preschool-education|infinity|" AnswerCount="21" CommentCount="25" ContentLicense="CC BY-SA 4.0" />
  <row Id="15214" PostTypeId="1" CreationDate="2019-02-13T03:01:36.767" Score="5" ViewCount="656" Body="&lt;p&gt;Throughout my geometry course, I was given many theorems and postulates, which I was were expected to memorize and apply. At the time, I sorta went along with it, but I couldn’t help but wonder where these came from. Probably 90% of these questions were answered when I took trigonometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take, for example, proving two triangles congruent. As we all know, one can prove them congruent by side-side-side, side-angle-side, angle-side-angle, angle-angle-side, and hypotenuse-leg. My teacher, I distinctly remember, made sure to emphasize to “stay away from the bad word, forward and backward” (angle-side-side). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But wait! I wondered at the time. Why doesn’t angle-side-side work, but hypotenuse-leg does? Isn’t hypotenuse-leg just a special case of angle-side-side, where the angle is a right angle? To this day I cannot come up with a rigorous proof that justifies these that doesn’t rely on the fact that &lt;span class=&quot;math-container&quot;&gt;$\sin\theta=\sin(\pi-\theta)$&lt;/span&gt;, meaning that, unless &lt;span class=&quot;math-container&quot;&gt;$\theta=\frac\pi2$&lt;/span&gt;, there are two possible side lengths corresponding to the same angle, and thus two triangles that contain the same angle-side-side but differing third sides. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I think, explaining the basic trigonometry functions and the unit circle as the very first thing in the course would make pretty much all of geometry make so much more sense. Granted, even the unit circle requires certain geometric knowledge (ex. definition of a circle, that all radii are congruent, etc.), but not too much that would seem to preclude some form of this working?&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is what I’ve described a typical geometry course worldwide, or at least US-wide, or did I happen to end up with a poor geometry course?&lt;/li&gt;&#xA;&lt;li&gt;If this is the normal way of teaching geometry, why? Why is the course focused more on memorizing theorems rather than understanding where they come from — something which, in my experience, seems to be nigh-impossible without the tools learned &lt;em&gt;after&lt;/em&gt; Geometry?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="11889" LastActivityDate="2023-05-15T23:36:38.013" Title="Why do standard geometry textbooks not start with trigonometry?" Tags="|secondary-education|course-design|geometry|curriculum|trigonometry|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15215" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T05:32:53.070" Score="54" Body="&lt;p&gt;This does not directly concern the &lt;span class=&quot;math-container&quot;&gt;$\infty+1=\infty$&lt;/span&gt; issue and I am not certain that I understand what you mean by &lt;em&gt;his previous understanding of mathematics&lt;/em&gt;, but I wanted to give the following suggestion:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Ask your child to name the biggest number he knows (besides &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;). (Let's say he answers &lt;span class=&quot;math-container&quot;&gt;$1000$&lt;/span&gt;);&lt;/li&gt;&#xA;&lt;li&gt;Tell him to add &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to it;&lt;/li&gt;&#xA;&lt;li&gt;Ask him &lt;em&gt;again&lt;/em&gt; what is the biggest number he knows. (It should be &lt;span class=&quot;math-container&quot;&gt;$1001$&lt;/span&gt;).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Repeat the process a few times and he should realize at some point that he can do this &lt;em&gt;indefinitely&lt;/em&gt;. He can just keep on adding &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; &lt;em&gt;for free&lt;/em&gt;. It doesn't matter if he can't &lt;em&gt;name&lt;/em&gt; the numbers eventually, as long as he understands that the next number is &lt;em&gt;one more than&lt;/em&gt; the previous one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While this does not necessarily show the various types of infinities that might exist, I think the idea that you can &quot;keep on going&quot; is a fair definition of infinity for a 5 year-old. It's not too hard to understand and it illustrates that infinity is not a number like &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, but rather an &lt;em&gt;idea&lt;/em&gt;: infinity is what you get if you keep on going forever. It is certainly better than the belief I had as a child that infinity was &lt;em&gt;the biggest&lt;/em&gt; number; there's no such thing as &lt;em&gt;the biggest&lt;/em&gt; if you can keep on adding &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. I hope this helps in some way!&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: As requested, I should mention that I haven't had the chance to test this with a 5 year-old, but it did work with teenagers (12-16) who had the same question (what is infinity?), as they seemed satisfied with the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also reiterate that this approach does not treat all types of infinity and should sound quite incomplete to a mathematician. However, there must be some limit to what we can and can't explain to a 5 year-old without sacrificing rigor and without &quot;burning&quot; them out. (Also, they'll have the opportunity to improve their understanding of the concept as they grow up). This particular approach seems to me to be &quot;viable&quot; for a 5 year-old (especially one who &quot;knows negative numbers&quot; and &quot;has figured out tables of any number less than 100&quot;, like OP's child).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, this approach involves the child in his own learning: he should be the one to realize on his own, inductively, what infinity is. This is much more convincing than &quot;being told&quot; what infinity is and should help with the &quot;resistance&quot; issue OP had.&#xA;&lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="-1" LastEditDate="2019-02-18T15:04:24.243" LastActivityDate="2019-02-18T15:04:24.243" CommentCount="23" ContentLicense="CC BY-SA 4.0" />
  <row Id="15216" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T08:46:01.800" Score="9" Body="&lt;p&gt;My son, also 6 yo, regularly talks about millions and billions and infinity. Obviously, large numbers have some attraction to children of this age.&lt;/p&gt;&#xA;&lt;p&gt;I try to explain that infinity is not a number. Instead, infinity is an &lt;em&gt;order of magnitude&lt;/em&gt; which has its own algebraic rules. Plus, minus, divison and multiplication do not work the way children learn in elementary school when applied to infinity.&lt;/p&gt;&#xA;&lt;p&gt;My first explanation is that this has also an impact on how we use the words &lt;em&gt;infinity&lt;/em&gt; and &lt;em&gt;infinite&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Three meters. (works)&lt;/p&gt;&#xA;&lt;p&gt;*Infinity meters. (completely wrong)&lt;/p&gt;&#xA;&lt;p&gt;*Infinite meters. (sounds wrong)&lt;/p&gt;&#xA;&lt;p&gt;Infinitely many meters. (works)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another approach is that the concept of numbers does not work. Numbers grow. For every number, there is another number that is larger. The mathematical notation to this concept is &lt;span class=&quot;math-container&quot;&gt;$\forall n \in \mathbb{N}: \exists m \in \mathbb{N}: m &amp;gt; n$&lt;/span&gt;. If infinity was a number, then this statement would be false, because let &lt;span class=&quot;math-container&quot;&gt;$n=\infty$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\infty + m &amp;gt; \infty$&lt;/span&gt; is false for all &lt;span class=&quot;math-container&quot;&gt;$m \in \mathbb{N}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$m &amp;gt; 0$&lt;/span&gt;. Surprisingly, children who already learnt addition upto, say, 100, understand this. They understand that 100 is not the largest of all numbers, neither is 1000, neither is a million, and so on. But the fact, that addition does not alter the &amp;quot;number&amp;quot;, makes them understand that infinity is not a number.&lt;/p&gt;&#xA;&lt;p&gt;In words that are better suited for children, you can also say: Adding any number to infinity does not change its size, because infinity expresses a magnitude, a size, rather than a number.&lt;/p&gt;&#xA;&lt;p&gt;Admittedly, my daughter, 8 yo, understands this point better, because my 6 yo son has not yet learnt addition of numbers larger than his 10 fingers provide.&lt;/p&gt;&#xA;" OwnerUserId="11891" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-02-16T09:54:05.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15217" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T10:28:28.530" Score="26" Body="&lt;p&gt;First of all, regardless of age, people need to understand that &quot;infinity&quot; is not a number, and not a placeholder for a number, but an &lt;em&gt;attribute&lt;/em&gt; of them (i.e. the fact that you can increase numbers without ever getting to an end).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For my children, the concept somehow came into their mind all alone due to the book &quot;Guess How Much I Love You&quot; by Sam McBratney. It plays with ever bigger numbers, and the kids can easily increase the distances used in the book on their own as soon as they learn that, for example, the sun is farther away than the moon, that stars are even farther, etc.. They &lt;em&gt;had&lt;/em&gt; to increas their number, because otherwise I would increase &lt;em&gt;mine&lt;/em&gt;, and &quot;win&quot; the contest laid out in the book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At some age (I cannot remember if it was 5 or older), the kids figured out that they can make numbers ever larger - even larger than in the book - by adding or multiplying (doubling as in &quot;there and back again&quot;), or whatever operation they learned in school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At that point, the concept of infinity seems to be represented by - literally - &quot;in-finite&quot;, or &quot;non-ending&quot;. I.e., they understand that there is no end to numbers, that you can go on and on adding ever more of them.&lt;/p&gt;&#xA;" OwnerUserId="8146" LastActivityDate="2019-02-13T10:28:28.530" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15218" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T11:13:26.803" Score="16" Body="&lt;p&gt;Speaking as someone who was that kid, you might be able to explain &lt;span class=&quot;math-container&quot;&gt;$\infty + 1 = \infty$&lt;/span&gt; via the Hilbert hotel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine a hotel that has an infinite number of rooms, one for every number. Imagine the hotel's full, and another guest shows up. You can make room for that guest by having the guest in room 1 move to room 2, the guest in room 2 move to room 3, and so on. So even though it's full, you can always fit more guests in, so you can always add one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He's also not totally wrong to insist that &lt;span class=&quot;math-container&quot;&gt;$\infty + 1 \neq \infty$&lt;/span&gt;. Infinity isn't really number, but an idea, that you can apply in different ways. There are different types of infinite numbers, with different rules, and in some of them, he'd be quite right that &lt;span class=&quot;math-container&quot;&gt;$\infty + 1 \neq \infty$&lt;/span&gt;. The hotel doesn't have a room &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;, but if it did, the room next to it would be room &lt;span class=&quot;math-container&quot;&gt;$\infty + 1$&lt;/span&gt;, and would be a different room.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Five might be a bit young to understand the idea that different rules lead to different maths, but he can probably grasp the idea the idea that there are different types of infinite numbers, and it's good to tell him that he's not wrong.&lt;/p&gt;&#xA;" OwnerUserId="11893" LastEditorUserId="11893" LastEditDate="2019-02-13T11:22:30.563" LastActivityDate="2019-02-13T11:22:30.563" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15219" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T12:00:09.833" Score="3" Body="&lt;p&gt;I suppose one problem is that your son looks at &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; the same way he looks at &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;. But infinity is not a natural or real number, even though it has a symbol and can be used in &quot;equations&quot; like &lt;span class=&quot;math-container&quot;&gt;$\infty + 1 = \infty$&lt;/span&gt;. These equations do not have the same meaning and do not follow the same rules as with &quot;normal&quot; numbers &amp;mdash; the reason being that at least one operand isn't one. Making clear that infinity is not a single number but is used to describe the unboundedness of number sequences should go a long way towards understanding and is totally within reach to a 5 year old.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me indulge in my computer science perspective. (I suppose it is correct in purely mathematical ways as well. Please correct me if that is not so.) How do we use &quot;infinity&quot; in math?  For example we say that the result of some sum is infinite: &lt;span class=&quot;math-container&quot;&gt;$\sum_{i=1}^\infty{f(i)} = \infty$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. Similarly for some integral, or simply the value of some function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches a certain value. The essence is that &lt;em&gt;we make statements of the outcome of procedures.&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; is not a static &quot;value&quot;; it is a statement of what happens to a value which is computed procedurally, provided we never stop. Specifically, it is the statement that we cannot name a limit that this value will not exceed. In other words, infinity can be considered &lt;strong&gt;the opposite of a static, fixed value.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Making this crucial distinction will go a long way explaining why &lt;span class=&quot;math-container&quot;&gt;$\infty + 1 = \infty$&lt;/span&gt; holds (even if I don't like writing it at all, as mentioned in a comment elsewhere in this thread): If I cannot put an upper threshold to a result, incrementing the result will still not yield an upper threshold, and that's all that &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; means.&lt;/p&gt;&#xA;" OwnerUserId="11894" LastEditorUserId="11894" LastEditDate="2019-02-14T11:56:15.293" LastActivityDate="2019-02-14T11:56:15.293" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15220" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T14:24:28.953" Score="3" Body="&lt;p&gt;I've no idea whether this would work, but would relating it to &lt;em&gt;forever&lt;/em&gt; hekp? Infinity is like &lt;em&gt;forever&lt;/em&gt; but for nunbers. Doing something for a week and then forever is the same as just doing it forever. When is &lt;em&gt;forever&lt;/em&gt;? That's when you'd be &quot;infinity years old&quot;, but there isn't a &lt;em&gt;when&lt;/em&gt; because &lt;em&gt;forever&lt;/em&gt; means &quot;never stop&quot; . . .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if you're travelling to infinity, which is travelling &lt;strong&gt;forever,&lt;/strong&gt; when do you get there? When you stop. &lt;strong&gt;When&lt;/strong&gt; do you stop? &lt;strong&gt;Never&lt;/strong&gt;! Because, &lt;em&gt;forever&lt;/em&gt; says &lt;strong&gt;never&lt;/strong&gt; stop. But maybe prepare for questions about whether infinity really exists . . .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think &lt;em&gt;forever&lt;/em&gt; a much more familiar concept to a five-year-old than infinity, so maybe you can start from that.&lt;/p&gt;&#xA;" OwnerUserId="10663" LastEditorUserId="10663" LastEditDate="2019-02-13T15:07:09.917" LastActivityDate="2019-02-13T15:07:09.917" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15221" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T14:30:06.940" Score="17" Body="&lt;p&gt;I'm not sure why the two basic things adults seem to say about infinity are &quot;infinity is not a number&quot; and &quot;∞+1=∞&quot;, both of which are at best misleading.  (Infinity doesn't name a number, but it does refer to a property some numbers can have.  ∞+1 is nonsense, &lt;span class=&quot;math-container&quot;&gt;$\aleph_0+1=\aleph_0$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\omega+1\neq\omega$&lt;/span&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem with talking about infinity with small children is that it's an imprecise term that covers multiple precise ideas that behave differently.  Children that age aren't ready for that, so I don't think there are any particular concepts about infinity that it's useful to convey.  I certainly don't think there's any reason to prioritize explaining the non-fact that ∞+1=∞ over other non-facts (say, that ∞+1&gt;∞).  Rather, I think the goal has to be to enjoy playing around with the concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your son's question - how old you'll be when he turns infinity - is a really good question, and points to the difficulty of dealing with the question: it's not at all clear what the right number system to discuss that in is.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-02-13T14:30:06.940" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15222" PostTypeId="2" ParentId="15214" CreationDate="2019-02-13T14:38:46.167" Score="5" Body="&lt;p&gt;My mental picture of why angle-side-side doesn't work as a congruence condition doesn't involve trigonometry and seems to me much simpler than trigonometry: Imagine that you're given a line segment &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; to serve as one of the two specified sides of the triangle and you're given the angle at &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, so you can draw a line &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; in the desired direction. The third vertex &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; of the desired triangle should be on &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; and should be at a specified distance from &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;. That gives you a circle &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; of possible locations for &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;,  centered at &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; must be at an intersection of &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;. In general, a circle will intersect a line twice; for the two choices to produce congruent triangles, you need that they're equidistant from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. In other words, the chord of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; joining the two intersection points should be bisected at &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; by the radius of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; that passes through &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. But a chord of a circle is bisected by a radius if and only if it's perpendicular to that radius, and that's why the angle-side-side condition works iff the angle is a right angle. (Strictly speaking, one should pay attention to the case where &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; meets &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; only once, i.e., where &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; is tangent to &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, but there again you have a right angle.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this example doesn't need (or, in my opinion, even benefit from) trigonometry. Offhand, I can't think of other examples either that would justify putting trigonometry earlier in the curriculum.  Trigonometry is a useful tool, but the essential ideas of geometry don't involve it.  &lt;/p&gt;&#xA;" OwnerUserId="326" LastEditorUserId="326" LastEditDate="2019-02-13T15:31:03.603" LastActivityDate="2019-02-13T15:31:03.603" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15223" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T15:02:11.533" Score="7" Body="&lt;p&gt;I am going to answer your question by suggesting a couple of books which might be fun to read with your son:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://www.penguinrandomhouse.com/books/89124/the-phantom-tollbooth-by-norton-juster-illustrated-by-jules-feiffer/9780394820378/&quot; rel=&quot;noreferrer&quot;&gt;The Phantom Tollbooth&lt;/a&gt;&lt;/em&gt; by Norton Juster.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book is a rather surreal adventure trip through a Wonderland-style setting populated by mad grammarians and mathemagical wizards (among others).  There is a section somewhere in the middle where the protagonist is sent on a quest to infinity, which is described in a couple of ways.  The concept of infinity discussed here is, if I recall correctly, more akin to the infinity that occurs when you compactify the the real numbers, so it isn't quite the same idea as &lt;span class=&quot;math-container&quot;&gt;$\infty + 1 = \infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any event, the book is quite a lot of fun.  Even if you aren't overly concerned with infinity, it is worth reading, and should be at about the level of a 5 year old (maybe a little advanced?  I seem to recall having it read to me when I was in first or second grade, so maybe just a little older?).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://www.goodreads.com/book/show/1275090.The_Cat_in_Numberland&quot; rel=&quot;noreferrer&quot;&gt;The Cat in Numberland&lt;/a&gt;&lt;/em&gt; by Ivar Ekeland.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This book might be a little advanced for a 5 year old, but maybe not–I think that it is intended for 3rd or 4th graders, but might be accessible with the help of a parent.  In any event, the book deals with infinity as a cardinal.  There is the basic &quot;a new number arrives, everyone moves up a room&quot; example, but my recollection is that the correspondence between rational numbers and the natural numbers is discussed, as well.  I had difficulty getting a copy of the book several years ago, but it appears that it might be back in print(?).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2019-02-13T15:02:11.533" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15224" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T15:55:24.860" Score="3" Body="&lt;p&gt;My children both learned about infinity at around four to five years old (now 5 and 7).  For both of them it was fairly straightforward; it came about with my eldest when he was talking to other kids at school about the biggest number.  We talked about trillion, quadrillion, etc.; as they were at a Montessori, it was easy to understand these.  Then we talked about googol, and googolplex.  Then we talked about a few other numbers, like Graham's number, and the idea of other extremely large numbers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, we talked about infinity.  Because we were talking in the context of very large numbers, the first thing we learned - before anything else - was that infinity is a &lt;em&gt;concept&lt;/em&gt;, not a number.  A way of thinking about extremely, impossibly large numbers, without actually naming one.  This brought a little confusion, until we went through the thought exercise of:  &quot;What's the largest number.  Okay, add one to it.&quot;  But ultimately they got the idea of 'concept' pretty easily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, at 5 (almost 6) and 7, they get some of the other ideas pretty easily - like 1/0 approaches infinity, infinty/n is infinity, but 0/0 = undefined.  Teaching infinity as a concept made it easy for them to understand these are basically just rules to follow, and that it's not the same as a number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I had a lot of sympathy that week for the poor preschool teacher who had to deal with the arguments 'infinty is the biggest number' 'no, it's a concept, not a number' between my children and the other children...&lt;/p&gt;&#xA;" OwnerUserId="1330" LastEditorUserId="1330" LastEditDate="2019-02-14T18:15:13.620" LastActivityDate="2019-02-14T18:15:13.620" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15225" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T16:07:34.747" Score="5" Body="&lt;p&gt;There is a well-known Christian hymn, &lt;em&gt;Amazing Grace&lt;/em&gt;, whose last lyric captures the idea of (countable) infinity quite well, and may be more effective to a five year old because it includes a context in which the notion of infinity can be applied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The lyrics goes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;When we’ve been there ten thousand years,&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Bright shining as the sun,&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;We’ve no less days to sing God’s praise&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Than when we'd first begun.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, this brings issues of religion into the matter. If you don't want a Christian hymn, you might rephrase the lyric in the framework of another religion or atheistically. If the child can grasp the notion of an infinite number of future years, then this lyric is &quot;explaining&quot; that &lt;span class=&quot;math-container&quot;&gt;$10,000 + \infty = \infty$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="1397" LastEditDate="2019-02-13T18:53:18.573" LastActivityDate="2019-02-13T18:53:18.573" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15226" PostTypeId="2" ParentId="15214" CreationDate="2019-02-13T16:26:26.113" Score="2" Body="&lt;p&gt;Anecdotal evidence from Germany: congruence is taught way before trigonometry (~7th grade vs ~9th grade for trig functions)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why is the course focused more on memorizing theorems rather than understanding where they come from.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The fact that two triangles are congruent if their sides are pairwise of equal length has nothing to do with trigonometry in the first place; one can give a constructive proof (that also shows why the triangle inequality must hold) without trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see the same fact using trigonometry one would have to apply the law of cosines twice which in my eyes is much harder. This law will also most likely be memorized without &quot;understanding where it comes from&quot; because the proof is again much harder than the constructive ones for congruence using compass, straightedge, and protractor &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Geodreieck&quot; rel=&quot;nofollow noreferrer&quot;&gt;(In Germany, we call this a Geodreieck)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2019-02-13T17:37:15.760" LastActivityDate="2019-02-13T17:37:15.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15227" PostTypeId="2" ParentId="15214" CreationDate="2019-02-13T16:28:38.170" Score="8" Body="&lt;p&gt;I cannot speak to curricula outside of the US, and I don't really buy your argument that geometry courses are &quot;focused more on memorizing theorems rather than understanding where they come from.&quot;  It is a poor geometry instructor indeed who admonishes students &quot;Don't be an ASS&quot; before first taking the time to discuss the failure of the angle-side-side congruence relation.  Because I don't find much in your question that I can directly respond to, let me start by questioning the framing of your question, and instead answer the question of &lt;em&gt;why&lt;/em&gt; geometry is taught the way that it is (which is, after all, the essence of the title of your question).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My impression is that geometry is usually taught (in the modern world, at least) because it is a relatively concrete and hands-on way of getting students to engage in mathematically rigorous argumentation.  Very few people actually care if two triangles are congruent or not, or how to construct a regular hexagon.  However, the process of &lt;em&gt;proving&lt;/em&gt; statements about triangles and hexagons introduces students to the idea of mathematical logic.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In the example you give, I would hope that the pedagogy follows something like the following outline (modulo some permutation of ideas, depending on the instructor, the author, or where the students lead you):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;We observe that a triangle has three sides and three angles.  To a mathematician, it is (I think) natural to ask how many of these data are actually necessary to uniquely specify a triangle, up to congruence.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;So, we get out our compass and straight-edge, and start doodling.  Experiment a little.  If we are clever, we should quickly discover that if two angles are specified, then the third angle is completely determined.  This gives us the &lt;strong&gt;Angle Sum Theorem&lt;/strong&gt;, which we can prove satisfactorily in the Euclidean setting.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;However, we might note that three (or really, two) angles are insufficient to uniquely specify a triangle.  Two triangles can have the same set of angle measures, but be of radically different sizes.  However, such triangles &lt;em&gt;will&lt;/em&gt; be similar (i.e. there is a constant ratio between the lengths of corresponding sides).  We can prove this result, and, perhaps, call it the &lt;strong&gt;AAA Similarity Theorem&lt;/strong&gt; (for angle-angle-angle).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At this point, we know that if we have determined two angles, we need at least one side length in order to uniquely determine the triangle.  The natural question here should be &quot;is this sufficient?&quot;  That is, if I know the measures of two angles and the length of one side, can I determine everything else about the triangle?  The answer is yes, giving us an &lt;strong&gt;AAS Congruence Theorem&lt;/strong&gt; (or, in your question SAA, for side-angle-angle).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Now we might want to know if we can lose even more information and still make progress.  For example, is there an AS theorem?  That is, if we know an angle and a side, is that enough to determine a triangle.  Very quickly, we should see that the answer is &quot;No&quot;–counterexamples are fairly easy to construct.  Fair enough.  If we only know one angle, knowing the length of only one side is insufficient.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Here the real fun starts:  if we are clever (or have good guidance), we should start to realize that if we know an angle and two sides, then the arrangement of those objects matters:  if the angle is &quot;between&quot; the two sides, then a triangle is uniquely determined, a result which we can prove as the &lt;strong&gt;SAS Congruence Theorem&lt;/strong&gt;.  On the other hand, if the angle is not &quot;between&quot; the two sides, we get into trouble.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The mathematically inclined might ask to classify the ways in which we can get into trouble.  After some investigation, we should determine that there are essentially three things which can go wrong (assume that &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; is a side with a specified angle at &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;; we wish to construct the side &lt;span class=&quot;math-container&quot;&gt;$BC$&lt;/span&gt; of specified length):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;We construct a circle of the specified radius centered at &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, but the radius is too small, and so we cannot construct any triangle with the specified angle and sides.&lt;/li&gt;&#xA;&lt;li&gt;We construct a circle of the specified radius centered at &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, but the circle intersects the the opposite side of the angle at &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; at two different points, leading to two possible triangles.&lt;/li&gt;&#xA;&lt;li&gt;We construct a circle of the specified radius centered at &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, and the circle intersects the opposite side of the angle at &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; at exactly one point, giving exactly one triangle.  There are actually a couple of ways that this can happen:  either the specified length is longer than &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; (in which case the circle will meet the line at only one point on the &quot;correct&quot; side of the the angle), or the circle will meet the line at a point of tangency (in which case, the side &lt;span class=&quot;math-container&quot;&gt;$BC$&lt;/span&gt; will be perpendicular to the side &lt;span class=&quot;math-container&quot;&gt;$AC$&lt;/span&gt;).&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Because there is ambiguity in the &quot;ASS&quot; case, we don't get a nice theorem, and we demand more information in such a case.  However, do note that the &quot;ASS&quot; case is not &lt;em&gt;always&lt;/em&gt; indeterminate:  for example, we get the &lt;strong&gt;Hypotenuse-Leg Congruence Theorem&lt;/strong&gt; as a special case.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This topic could be further expanded (it might be nice to carefully consider exactly when ASS fails and how and what extra data may be used to patch things up), but I've already hit the main points alluded to in the question, so I'll stop here.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Having laid out an outline for a possible series of lectures or activities for students, let me make the following points:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;There are (by my count) five theorems here which require proof.  Real mathematics is about rigorously proving statements, so there are a number of good exercises here for students to work on.  As an added bonus, each of the theorems to be proved should be more-or-less intuitively obviously true, and good constructions will help the students to see how to make the arguments.  In higher mathematics, it is sometimes hard to visualize an argument, and it is often difficult to tell &lt;em&gt;a priori&lt;/em&gt; whether or not a result is true (there are a lot of &quot;obvious&quot; algebraic statements that are a pain to prove in an abstract ring, and a lot of wicked estimates in analysis which are hard to visualize or otherwise understand intuitively), hence Euclidean geometry is a good sandbox for learning to pose and prove theorems.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There are a number of cases where things don't work.  This can be used to emphasize the power of a counter-example.  Once we show that there is an example in which ASS fails to produce a unique triangle, then we know that no ASS congruence theorem can possibly exist.  Counter-examples are &lt;em&gt;amazing&lt;/em&gt;!  Indeed, two of my favorite books are slim Dover volumes full of counter-examples in analysis and topology.  However, these require a lot more technical knowledge than a geometric construction.  Again, Euclidean geometry is a good sandbox for learning about counter-examples.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, to reiterate, no one really cares about most of the results in a high school geometry class.  By suggesting that we skip straight to trigonometry, which is its own nightmare of abstract notation and endless confusion for students, you are suggesting that the goal of such a class is to learn computational techniques for determining angles and sides (or whatever).  This completely misses the point, which is to explore a topic in a mathematically rigorous manner.  The &lt;em&gt;results&lt;/em&gt; are not at all important (for the most part); it is the &lt;em&gt;process&lt;/em&gt; by which those results are obtained that is important.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2019-02-13T16:28:38.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15228" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T16:37:38.600" Score="1" Body="&lt;p&gt;Before trying to explain &lt;span class=&quot;math-container&quot;&gt;$\infty + 1$&lt;/span&gt;, it'll help if he has an intuitive grasp of what infinity &lt;em&gt;means&lt;/em&gt;.  I recommend using the cardinals rather than the ordinals, because they can be constructed without needing to understand limits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below is a possible explanation that may help with that; I try to avoid using too much terminology, since he's 5, and in particular, when I say &quot;number&quot;, I mean &quot;non-negative integer&quot; and when I say &quot;infinity&quot;, I mean &lt;span class=&quot;math-container&quot;&gt;$\aleph_0$&lt;/span&gt;.  You might mention briefly that when you say &quot;numbers&quot; here, you mean numbers like &quot;3&quot;, but not numbers like &quot;3 and a half&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Start with a set with just one number in it: &lt;span class=&quot;math-container&quot;&gt;$\{1\}$&lt;/span&gt;.  We then add numbers to it one at a time, and we'll do it so that that the biggest number in the set is also &lt;em&gt;how many&lt;/em&gt; numbers are in the set.  So right now, there's &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; number in the set, and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is the biggest number in the set.  Next, we'll add the number that's one bigger than the biggest number in the set. &lt;span class=&quot;math-container&quot;&gt;$1+1=2$&lt;/span&gt;, so now our set is &lt;span class=&quot;math-container&quot;&gt;$\{1,2\}$&lt;/span&gt; and the biggest number in it is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.  We continue as such. &lt;span class=&quot;math-container&quot;&gt;$2+1=3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$3+1=4$&lt;/span&gt;, and so on.  So for &lt;em&gt;any&lt;/em&gt; number, we have the corresponding set containing every number from 1 to that one, and our number is how many numbers are in that set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No matter how many numbers we add, there's &lt;em&gt;always&lt;/em&gt; a biggest number in the set and that number is &lt;em&gt;always&lt;/em&gt; how many numbers there are in the set. But there's also always a number that's one bigger than that one. So we can keep adding numbers to the set &lt;em&gt;forever&lt;/em&gt; and never get all of them.  There &lt;em&gt;is&lt;/em&gt; no biggest number, and there are an unlimited number of numbers; no matter how many we have, we can always add one more.  So if we want to ask &quot;how many numbers are there altogether&quot;, we need a word for that that isn't a number.  That's what infinity is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$\infty + 1$&lt;/span&gt; doesn't mean anything, because &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; just means &quot;unlimited&quot;; it's specifically the thing we can't get to by adding 1 to numbers we already have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a few other kinds of infinity, but this is the easiest one to understand, so you want to make sure you &lt;em&gt;really&lt;/em&gt; understand this kind before moving on to the others.&lt;/p&gt;&#xA;" OwnerUserId="8080" LastActivityDate="2019-02-13T16:37:38.600" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15229" PostTypeId="2" ParentId="15214" CreationDate="2019-02-13T19:03:47.087" Score="12" Body="&lt;p&gt;If you really want to understand why the curriculum is structured the way it is, and how it got that way, you might want to read a brief history of the discourse around geometry education for the past 150 years or so; I recommend Chapter 1 of &lt;a href=&quot;https://www.routledge.com/The-Learning-and-Teaching-of-Geometry-in-Secondary-Schools-A-Modeling/Herbst-Fujita-Halverscheid-Weiss/p/book/9780415856911&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Learning and Teaching of Geometry in Secondary Schools: A Modeling Perspective&lt;/em&gt;&lt;/a&gt;, by Herbst, Halverscheid, Fujita, and me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I think the question actually relies on a (pardon the pun) circular argument:  Certainly one can introduce &lt;span class=&quot;math-container&quot;&gt;$\sin, \cos, \tan$&lt;/span&gt; etc as &quot;circular function&quot; defined via the coordinates of points on a unit circle.  But in order to make use of those functions in triangles, you have to know how to adapt them to the case when the hypotenuse is not &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.  How do you know that the circular functions are scale-invariant?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer, of course, is that you rely on &lt;em&gt;similarity&lt;/em&gt; properties:  specifically, if two angles of one triangle are congruent to two angles of a second triangle, then the ratios of corresponding sides are equal (&quot;Angle-Angle Similarity&quot;).  To solve any complicated problems involving triangles, you also need to know that it works in the other order:  if the legs of one right triangle are in the same ratio as the legs of a second right triangle, then the corresponding angles are congruent (a special case of &quot;Side-Angle-Side Similarity&quot;).  So if you really want to start with trigonometry, you have to preface it with a unit on similarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But similarity properties (and their proofs) depend in an essential way on the properties of parallels, so you can't really cover similarity without first covering parallels.  And at that point you have basically reproduced about 2/3 of the first few chapters of a traditional geometry course anyway, so why not do the whole thing?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2019-04-23T17:53:03.860" LastActivityDate="2019-04-23T17:53:03.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15230" PostTypeId="2" ParentId="15212" CreationDate="2019-02-13T23:43:32.580" Score="3" Body="&lt;p&gt;I would start by saying something along the following lines...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;You're asking some very grown-up questions for someone that's only 5.  Are you ready to do some really, really, grown-up thinking about the answers?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He will of course, answer, 'yes'.  I would respond...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;'Ok, but this is serious stuff.  You need to be ready to take this thinking very seriously.'  Only when you really have his attention, do all the 'infinity is not a number' stuff, and all the other things that people have suggested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A 5-year old, thinking about infinity, if he's really thinking about infinity, is a very smart kid indeed.  He is probably smart enough to confront the idea of types of thinking beyond what he's encountered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But he doesn't yet realise how he needs to change gear.  Teach him that, before you teach him about infinity, and 20 years from now, he'll thank you for all the other things that enabled him to do.&lt;/p&gt;&#xA;" OwnerUserId="980" LastActivityDate="2019-02-13T23:43:32.580" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15231" PostTypeId="2" ParentId="15212" CreationDate="2019-02-14T00:35:03.947" Score="19" Body="&lt;p&gt;On a piece of paper, he started with writing 10, then 100, then 1000, .... and he stopped after writing 40 zeros with 1. Then he came to me and said, &quot;I understand infinity now; infinity is a number with infinite zeros.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The main point is that as most of you suggested, he has now registered infinity in his brain as a concept rather than a number, which is why he used the expression 'a number with infinite zeros'.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ljdSN.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ljdSN.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11887" LastEditorUserId="11887" LastEditDate="2019-03-24T05:13:33.133" LastActivityDate="2019-03-24T05:13:33.133" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15236" PostTypeId="2" ParentId="15212" CreationDate="2019-02-14T18:34:59.583" Score="1" Body="&lt;p&gt;Have you considered geometry? Take two points and draw lines, one through each points.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Two lines meeting at an obtuse angle meet &quot;soon,&quot; that is a small number.&lt;/li&gt;&#xA;&lt;li&gt;Two lines meeting at an acute angle meet &quot;far away,&quot; that is a large number.&lt;/li&gt;&#xA;&lt;li&gt;Two parallel lines meet &quot;infinitely far away.&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In a way that gets you to limits, which may be too complicated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this is an &lt;a href=&quot;https://en.wikipedia.org/wiki/Alexandroff_extension&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alexandroff Extension&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann_sphere&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann Sphere&lt;/a&gt;, depending on how you look at it. With slightly older students, the Riemann sphere is a good way to &quot;visualize&quot; infinity.&lt;/p&gt;&#xA;" OwnerUserId="8201" LastEditorUserId="8201" LastEditDate="2019-02-18T17:15:15.110" LastActivityDate="2019-02-18T17:15:15.110" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15237" PostTypeId="1" CreationDate="2019-02-14T20:18:48.520" Score="15" ViewCount="824" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/176/knowing-mathematics-does-not-translate-to-knowing-to-teach-mathematics-why&quot;&gt;A question&lt;/a&gt; has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with one's teaching of lower level courses in mathematics. For example, I believe that a good student with bachelor's degree in mathematics should have sufficient knowledge to teach calculus. However, how does having a master's or doctorate degree help one's teaching in calculus, if any at all?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am teaching calculus now and I do not understand commutative algebra; I took a course on commutative algebra long time ago; I did poorly in the course and now I could hardly recall anything from this course. If I invest substantial among of time studying this subject well now, will it help me in my calculus course in any sense?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2022-11-07T14:26:35.283" Title="How does knowing more about mathematics help one's teaching of lower level course, such as calculus?" Tags="|undergraduate-education|calculus|" AnswerCount="8" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="15238" PostTypeId="2" ParentId="15237" CreationDate="2019-02-14T21:14:20.940" Score="0" Body="&lt;p&gt;My personal impression across many fields is that a certain amount of extra subject matter can help (especially with &quot;top high school&quot; classes like AP Calculus or AP Chemistry or AP Bio).  but that by and large, the issues in teaching and learning intro topics like first year college calc, chem, physics, bio, and through diffyQ and engine math are MUCH more about the basics AND about PEDAGOGY as opposed to being further advanced in your field.  I have seen in chem where the salty old (barely passed a Ph.D. and with the grace of his adisor) had way more empathy and ability to teach freshman chem (e.g. analogy of wheels to car for limiting reactant) versus the shiny research profs.  Plus he knew stoich and equilibrium problems inside out.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, net, net:  I don't think advanced education really helps you be a better teacher.  Often it can even be a negative (look at the peeps here wanting to &quot;do something interesting&quot; rather than horsing up the troops.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Really if you even have a math UNDERGRAD (as opposed to ed) , you are well, well equipped to teach calculus.  After that, really concentrate on being a salty, salty TEACHER.  Not a mathematician.&lt;/p&gt;&#xA;" OwnerUserId="11935" LastActivityDate="2019-02-14T21:14:20.940" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15239" PostTypeId="2" ParentId="15237" CreationDate="2019-02-14T21:28:36.827" Score="6" Body="&lt;p&gt;&lt;em&gt;I believe that a good student with bachelor's degree in mathematics should have sufficient knowledge to teach calculus.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree with this. I work with a few colleagues who came to teach at the community college after working in the high school, and their credential is a Master's of Science for Teachers (MST). They have never taken graduate (theory) courses in algebra, analysis, geometry, etc, but they are excellent and dedicated teachers/colleagues -- attributes I chalk up to their teaching experience. This experience speaks a lot to me about their potential, more so than a graduate degree.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;If I invest substantial among of time studying this subject well now, will it help me in my calculus course in any sense?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short -- no. While it is good to have a depth of knowledge in a topic so you can engage students when they ask something &quot;out of the box&quot; (like the &lt;em&gt;three&lt;/em&gt; students asking me about fractional derivatives in the past two weeks alone), its use has its limits, and the law of diminishing returns applies. It is a reminder that a teacher is not there to simply have knowledge to spew forth, but to help students take on new information. I believe a &quot;good teacher&quot; should be able to take something just out of reach of their own understanding and still help a student learn it. [Of course, they should be willing to learn it alongside that student.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;...how does having a master's or doctorate degree help one's teaching in calculus, if any at all&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simply put, a graduate degree is just a minimum qualification adopted by institutions of higher ed for their faculty. It (and a good curriculum vitae) will get you in the door for an interview, and then you must show your skills as an educator. If this is hard to hear, consider the perennial student complaint &quot;when am I &lt;em&gt;ever&lt;/em&gt; going to use this&quot;. The answer for them is the same as for the math PhD who wants to teach beginning algebra: &quot;Perhaps you won't.&quot;&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-02-14T21:28:36.827" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15240" PostTypeId="2" ParentId="15237" CreationDate="2019-02-14T22:10:23.960" Score="17" Body="&lt;p&gt;I will speak purely anecdotally.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;However, how does having a master's or doctorate degree help one's teaching in calculus, if any at all?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It isn't the degree per se that helps, but rather the process of having to learn, relearn, and reformulate gargantuan amounts of mathematics of all types that helps.  Do I use either my PhD research or my current research in teaching non-major freshman calculus?  Not at all.  (I do get to mention my current research in linear algebra, so it is possible, of course.)  But what I do use on a daily basis is the practice that comes from intense advanced study in seeing &lt;em&gt;every side of a mathematical problem&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give just one example, when teaching Riemann sums (often without using this phrase), one can just do left and right hand sums, or maybe trapezoid or midpoint if you are ambitious.  Fine.  What happens when a student decides to pick some left and some right hand endpoints?  What happens when a student wants to pick only integer points?  What about the student who insists that on the interval &lt;span class=&quot;math-container&quot;&gt;$[0,2]$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$n=2$&lt;/span&gt; you should use &lt;span class=&quot;math-container&quot;&gt;$f(0),f(1),f(2)$&lt;/span&gt; with equal weight?  Answering these questions beyond &quot;the book says that is wrong&quot; requires experience, and the ability to try to see what the many possibilities are from all sides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, naturally you can obtain this experience in a multitude of ways.  I have colleagues in other departments without a PhD whose deep understanding of the material they teach is unparalleled.  And getting a doctorate doesn't automatically mean you magically look at calculus (or lower-level courses - I find that precalc is actually conceptually a bigger target because it roams so far and wide) better.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, on the whole, I  find that significant advanced study in mathematics makes me a &lt;em&gt;much&lt;/em&gt; better calculus teacher than I would have been otherwise.  Don't worry about commutative algebra; do worry about finding ways to stretch yourself mathematically early and often - whether through formal means or informal.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Can't resist:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If I invest substantial among of time studying [commutative algebra] well now, will it help me in my calculus course in any sense?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you want to talk commutative algebra and calculus, take aside the two students who really want to do deep AI/machine learning/big data/whatever, and tell them about dual numbers, &lt;a href=&quot;https://en.wikipedia.org/wiki/Automatic_differentiation#Automatic_differentiation_using_dual_numbers&quot; rel=&quot;noreferrer&quot;&gt;automatic differentiation&lt;/a&gt;, and &lt;a href=&quot;https://math.stackexchange.com/questions/61763/dual-numbers-and-tangent-vectors&quot;&gt;tangents&lt;/a&gt;.  Hard to find parts of math that don't influence each other in some way - but of course you don't tell this story to 99% of students!  Anyway, the point isn't the specific math content, it's the process.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-02-14T22:10:23.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15241" PostTypeId="1" CreationDate="2019-02-15T00:13:23.650" Score="6" ViewCount="173" Body="&lt;p&gt;I’m not sure if this question belongs here, so I apologize if it doesn’t.&#xA;I work in a tutoring center at my university where we tutor every subject. Mathematics is in high demand, and occasionally my coworkers ask if I can tutor multiple students at the same time (commonly in different courses, but not always). I’ve always told them no, because I felt my style would be incompatible with that type of session.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is two fold: &#xA;1. Are there any studies on the effectiveness of group tutoring? What parameters have determined the effectiveness of the sessions?&#xA;2. How can I successfully host a session like this? It’d definitely help our overall numbers (to keep funding for our grant we need to reach certain goals), but I don’t want to reduce my effectiveness in helping the student. Perhaps I can adopt a different style that would be more effective in tutoring in this way?&lt;/p&gt;&#xA;" OwnerUserId="10662" LastActivityDate="2019-06-24T14:12:45.317" Title="Studies about group tutoring sessions" Tags="|undergraduate-education|reference-request|concept-motivation|education-research|tutoring|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15242" PostTypeId="1" CreationDate="2019-02-15T00:46:08.147" Score="-2" ViewCount="429" Body="&lt;p&gt;I will be teaching geometry for the first time ever this summer. I teach at a community college, and we only offer this course in the summer. (Mostly high school students take it, but it is a college course.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have just started to read the textbook we use: &lt;em&gt;Elementary Geometry for College Students&lt;/em&gt;, by Alexander and Koeberlein. So far I am not impressed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know of good textbooks others have used for such a course. Sadly, the only book I found that's OER is a photocopy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ETA:&#xA;Problems with the current textbook: It feels like a mish mash. I saw a definition that was incorrect. (&quot;Adjacent angles have one corresponding side.&quot; Well, that would be true if one angle were inside the other, and I don't believe we call those adjacent.) I don't care for the two-column proofs, and the way they try to teach them. I don't see what the goal is when I work through this book. There are 'theorems' with no proof in sight. How do you prove things if you don't know what your starting point (the axioms) is? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My criteria for a good textbook: It will work with me to strengthen the students' visual reasoning skills. It will do a good job with construction. (I can do that on the side, though.) It will do undefined terms to axioms / postulates to proofs well. It will have a good flow / progression. It will inspire me and the students (fun and meaningful). For example, I mostly like what I see in James Tanton's book.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="238" LastEditDate="2019-10-27T19:55:18.730" LastActivityDate="2019-10-28T16:32:59.103" Title="Good textbooks for a college Basic Geometry course?" Tags="|undergraduate-education|textbooks|geometry|community-colleges|" AnswerCount="4" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="15244" PostTypeId="2" ParentId="15241" CreationDate="2019-02-15T07:57:43.293" Score="2" Body="&lt;p&gt;I would think you could do this if you are giving the people work and just checking it occasionally.  I.e. not holding their hand constantly.  This would allow you to timeshare more than one student.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously one on one training is the best (for the student) but it is expensive, per capita.  This is why it is common that athletic training is done in groups.&lt;/p&gt;&#xA;" OwnerUserId="11942" LastEditorUserId="64" LastEditDate="2019-06-24T14:12:45.317" LastActivityDate="2019-06-24T14:12:45.317" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15245" PostTypeId="2" ParentId="15212" CreationDate="2019-02-15T11:51:56.107" Score="-3" Body="&lt;p&gt;Short answer: I would say, infinity is larger than the largest number anyone can count.&lt;/p&gt;&#xA;" OwnerUserId="11945" LastActivityDate="2019-02-15T11:51:56.107" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15246" PostTypeId="2" ParentId="15237" CreationDate="2019-02-15T21:00:43.293" Score="12" Body="&lt;p&gt;The title of the question asks how knowing &quot;more about math&quot; can help in teaching calculus, while the question itself asks specifically about the math learned towards a masters degree or a doctorate. I am going to focus on the &quot;more about math&quot; part, regardless of the stage at which it was acquired, because the math that one person learned in grad school could be learned as an undergraduate by someone else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A solid understanding of analysis of all kinds (such as real analysis, complex analysis, and Fourier analysis) gives an instructor in a calculus course a thorough mastery of the intricacies behind the often confusing topic of infinite series. It means the instructor understands what makes all those convergence tests work and really understands the math behind boundary behavior of power series (&quot;at the endpoints&quot;) or forming power series expansions of the same function around different points. This can help in the construction of examples for the class that emphasize a certain issue and it means the instructor should be up to the challenge of answering almost any non-routine question calculus students may ask about infinite series. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course a calculus instructor who has not taken a lot of analysis courses could teach himself/herself why all the convergence tests for series work, but I think the experience of using series in more advanced coursework gives you a perspective on the value of this topic and how it gets used later that someone whose analysis knowledge ends at calculus doesn't have. It's kind of like taking high school French from someone who just knows high school French very well compared to taking high school French from someone who is actually fluent in French (assuming that both instructors actually want to teach French!).  Maybe on most days the students won't notice a difference, but as soon as someone asks about something going even slightly beyond the regular curriculum, the instructor with more limited knowledge will be at a loss to answer the questions or point out connections that more mathematically experienced instructors know well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since you bring up commutative algebra in your question, let me give an example of how one topic from that could help an instructor understand calculus better: completions of commutative rings include as a special case the rings of &lt;em&gt;formal power series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\mathbf R[[x]]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf R[[x,y]]$&lt;/span&gt;. Some calculations with power series really work if you use just a formal manipulation of the terms, while others are genuinely numerical (convergence issues are essential for the result to even mean anything). When I was first learning calculus, I found the idea of division of power series (e.g., the ratio &lt;span class=&quot;math-container&quot;&gt;$\sin x/\cos x = (x - x^3/3! + \ldots)/(1 - x^2/2! + \ldots)$&lt;/span&gt; near &lt;span class=&quot;math-container&quot;&gt;$x = 0$&lt;/span&gt;) a bit puzzling. Knowing about both analytic functions in complex analysis and formal power series made me much more comfortable with division of power series. (For the record, my understanding of formal power series actually came from my reading about &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic analysis rather than commutative algebra.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Knowing some number theory and complex analysis gave me a much clearer idea of what is going on with integration of rational functions &lt;em&gt;in general&lt;/em&gt;, which if you look in a calculus textbook appears to be a large number of disconnected different cases, of which we may teach only 1 or 2 to avoid excessive hand calculation. If a student were to ask me for some &lt;em&gt;intuition&lt;/em&gt; behind decompositions like &lt;span class=&quot;math-container&quot;&gt;$1/(x^3+x) = 1/x - x/(x^2+1)$&lt;/span&gt;, I can point out a numerical analogy: &lt;span class=&quot;math-container&quot;&gt;$7/(3 \cdot 5)$&lt;/span&gt; can be broken up into some number of thirds and some number of fifths: &lt;span class=&quot;math-container&quot;&gt;$7/15 = 2/3 - 1/5$&lt;/span&gt;.  The principle is that when you have a denominator that is a product of &quot;relatively prime parts&quot;, you should expect to be able to break it up into a sum with those separate parts as their own denominator. It works for denominators of fractions just as much as it works for denominators of rational functions.  I don't know what other people would say if a student asked for intuition behind the partial fraction decomposition process: it just works?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Knowing some algebraic geometry lets me tell students why implicit differentiation has a surprising practical application that they use without even realizing it: the security in ATM cards and other smart cards is based on elliptic curve cryptography (ECC), and the math behind ECC involves being able to compute tangent lines to points on curves like &lt;span class=&quot;math-container&quot;&gt;$y^2 = x^3 - x$&lt;/span&gt;. (Technically, ECC uses elliptic curves over finite fields rather than over the real numbers, and the equations look more like &lt;span class=&quot;math-container&quot;&gt;$y^2 + xy = x^3 + x + 1$&lt;/span&gt; because of issues involving fields of characteristic &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, but the basic ideas for finding tangent lines are still largely the same as over the real numbers.) Algebraic geometry also lets me know the motivation behind the otherwise peculiar &lt;span class=&quot;math-container&quot;&gt;$\tan(t/2)$&lt;/span&gt;-substitution in calculus, which turns integrals of rational functions of &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt; into integrals of rational functions.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2019-02-16T10:43:45.780" LastActivityDate="2019-02-16T10:43:45.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15247" PostTypeId="2" ParentId="15212" CreationDate="2019-02-15T21:13:41.620" Score="0" Body="&lt;p&gt;The thing is, saying that infinity + any number equals infinity is a bit imprecice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two widely used concepts of infinity; they refer to cardinality and ordinality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In finite numbers, Cardinality is the concept of &quot;how many&quot; of something there are -- 1 sheep, 2 sheep, 3 sheep.  Ordinality is &quot;what order&quot; they come in -- 1st sheep, 2nd sheep, 3rd sheep.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With finite numbers they are highly tied to each other.  You can just &quot;count labels&quot; in a sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With infinite sets the two concepts diverge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We'll start with the natural numbers -- the set of all counting numbers.  0, 1, 2, 3 etc.  That'll be our &quot;first infinity&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you take the first infinity, and add another element to it, you get the &lt;em&gt;same cardinality&lt;/em&gt;.  This is what people talk about when they say &quot;infinity+1 equals infinity&quot;.  More than that, if you take the first infinity, and double it, you get ... the same cardinality.  You can even add an infinite number of infinities to it -- take &quot;first infinity times first infinity&quot; or &quot;first infinity squared&quot; -- and you get the same cardinality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It isn't until you reach (assuming continuum hypothesis) 2^&quot;first infinity&quot; that you reach a new cardinal.  This can either be described as the &quot;set of all the first infinity&quot; or &quot;the set of functions that go from the first infinity to yes/no&quot; (it shouldn't be hard to see they describe the same thing).  This is a &lt;em&gt;bigger&lt;/em&gt; cardinal than the first infinity, and is also the same cardinality as the &lt;em&gt;real numbers&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The cardinality game continues from there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So that is one branch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other is ordinals.  In ordinals, we talk about ordering things.  For any two things, you can say which is in front of the other in the order.  And for any collection of things ordered, we can find the &quot;least&quot; element (the one &quot;behind&quot; all the others), including the entire collection of ordered elements (we normally call this element 0).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;first infinity&quot; in ordinals is ordering everything by the natural numbers.  Everyone gets a tag that says &quot;1st&quot; or &quot;1 million and 7th&quot; or whatever.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, in ordinals, we can they have someone with the label &quot;1st in 2nd lineup&quot;, and we can state that the 2nd lineup goes after &lt;strong&gt;every&lt;/strong&gt; value in the first lineup.  This is &quot;infinity plus 1&quot; in ordinals, and it is a distinctly different way of ordering people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What more you can have 2 infinite lineups (where the 2nd goes after the first), or 3, or an infinite number of infinite lineups (where each lineup goes after the one before).  These are all distinct ordinals -- they describe fundamentally different ways of ordering things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And the game continues from there.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Now that I have disabused you of the notion that infinity+1 always equals infinity, how do we talk about it with a 5 year old?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could talk about that split.  Say &quot;up to infinity the idea of ordering and counting is the same.  At infinity they are different.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then talk about infinite ordering and lineups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; is a lineup that goes on forever.  There is someone in front.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$\omega + 1$&lt;/span&gt; is two lineups.  One that goes on forever, and one with a single person in it.  That single person goes &lt;em&gt;after&lt;/em&gt; the first lineup.  It will get very boring for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$\omega +2$&lt;/span&gt; has 2 people in the second lineup.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$\omega + \omega = 2 \omega$&lt;/span&gt; has two infinitely long lineups.  The second goes after the first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$k \omega$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; lineups, each infinitely long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$\omega \omega = \omega^2$&lt;/span&gt; has an infinite number of lineups, each infinitely long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$\omega^2 + 1$&lt;/span&gt; has an infinite number of lineups, each infinitely long, plus one person who gets to go after &lt;em&gt;everyone else&lt;/em&gt; is done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$\omega^2 + \omega$&lt;/span&gt; has an infinite number of lineups, each infinitely long, plus another lineup that goes after the previous infinite set of lineups are all done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$2\omega^2$&lt;/span&gt; has a two collections, each with an infinite number of lineups, each infinitely long, with one going after the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$\omega \omega^2 = \omega^3$&lt;/span&gt; has an infinite number of collections, each with an infinite number of lineups, each infinitely long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ordinal &lt;span class=&quot;math-container&quot;&gt;$\omega^\omega$&lt;/span&gt; has an infinitely long order of layers.  In each layer, there is an infinite number of the next layer, all ordered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You probably will break down before getting this far.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Also talk about cardinality.  Here the other answers cover things really well -- things like the Hilbert Hotel and the Cat in Numberland are great resources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A fun part of this is that every Cardinality has a whole bunch of Ordinalities associated with it.  You can look at the stories, like Hilbert Hotel, and talk about how the &lt;em&gt;Ordinality&lt;/em&gt; changed even when the &lt;em&gt;Cardinality&lt;/em&gt; didn't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And you can talk about how this doesn't work for &quot;normal&quot; numbers.  You cannot change the &lt;em&gt;fundamental&lt;/em&gt; ordinality without changing the cardinality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having two lines, one of 2 people, followed by a line of 3 people, is the ordinal 2+3, which is fundamentally the same as the ordinal 5.  You can &quot;just paste&quot; the 3 people onto the end of the first line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With infinite ordinalities, you cannot reach the &lt;em&gt;end of the first line&lt;/em&gt; to paste the second line on.  It is infinitely far away.&lt;/p&gt;&#xA;" OwnerUserId="9985" LastActivityDate="2019-02-15T21:13:41.620" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15248" PostTypeId="2" ParentId="15237" CreationDate="2019-02-15T21:19:07.167" Score="12" Body="&lt;p&gt;I'll give an answer by analogy. When you are a kid, you should be exposed to playing with other kids or participating in activities such as team sports. It's natural to ask, &quot;Does playing soccer at seven years old help a child become a better adult?&quot; And the answer seems to be that, in general, yes it does help. Not because of soccer per se, but because it is important for the child to learn how to interact with others, work with others in a cooperative environment, learn about the importance of practice and hard work, and learn how to listen to their coach to improve their skills. Of course, that's not what the child notices. Hopefully, they're just having fun and focusing on the immediate task in front of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And it doesn't mean that if a kid plays soccer, then they'll automatically be better at those bigger goals than a kid who doesn't play soccer, but on average playing soccer will be a positive influence. Those experiences tend to accumulate and help shape who we become as adults.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think studying advanced math and getting a Ph.D. work the same way. Most of the time is hopefully spent enjoying the opportunity to learn mathematics and focus on the problem in front of us. At the same time, we have an opportunity to learn a plethora of skills in the process. Among them are skills that help with our teaching - the ability to think about problems in greater depth, to question which assumptions are really needed, to better understand the essence of an argument or a calculation and to hopefully improve our ability to communicate mathematical insights to others. Again, not all Ph.D. recipients acquire all of these skills, nor is it the only possible route to acquire them, but on average, math Ph.D.'s are much more skilled in these areas than people whose math education stopped much earlier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Documenting exactly &lt;em&gt;how&lt;/em&gt; an advanced degree can make you a better teacher is non-trivial. It's not clear the extent to which the experienced process (which takes years to complete) can be summarized succinctly. But experience shows that those with a Ph.D. in mathematics (or related fields) are generally - but again not universally - much more prepared to teach the content required for a math course such as calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it's much more debatable if math Ph.D. recipients are better at the other aspects of teaching calculus (how they deliver the content) than others. And it's debatable how much relative importance good presentation of the material has compared to a good choice of content. But without a strong understanding of the content, no amount of brilliant presentation can help a teacher who either makes a poor choice of how to present specific content or, worse, presents that material incorrectly because they lack a full understanding of the relevant content and the proper perspective to view it in. &lt;/p&gt;&#xA;" OwnerUserId="1397" LastActivityDate="2019-02-15T21:19:07.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15249" PostTypeId="2" ParentId="15242" CreationDate="2019-02-16T18:41:59.923" Score="2" Body="&lt;p&gt;@BPP answered in a comment. Their answer suggested two Dover books: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle&lt;/em&gt;, by Nathan  Altshiller-Court&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Geometry: A Comprehensive Course&lt;/em&gt;, by Dan Pedoe&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dover does not do the common publisher scam of new editions every few years, and their books are inexpensive. Not OER, but way less burden on my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am also buying some high school-level books as resources. &lt;a href=&quot;https://www.mathed.page/proof/math-2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henri Picciotto recommends&lt;/a&gt; &lt;em&gt;Geometry: A Guided Inquiry&lt;/em&gt; by Chakerian, Crabill and Stein. (I use bookfinder.com to find the cheapest used copies of books. This title with Chakerian as author was very expensive, but the same title by Stein was inexpensive. Odd.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am still interested in hearing about other possibilities.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-02-16T18:41:59.923" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15250" PostTypeId="2" ParentId="15212" CreationDate="2019-02-16T20:36:39.703" Score="0" Body="&lt;p&gt;If you are 24 years older than him now, the answer can be &quot;I will be 24 years older than you.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Adjust the &quot;24&quot; to fit.&lt;/p&gt;&#xA;" OwnerUserId="11957" LastActivityDate="2019-02-16T20:36:39.703" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15252" PostTypeId="2" ParentId="15212" CreationDate="2019-02-17T16:59:49.897" Score="-4" Body="&lt;p&gt;Imagine a Steel ball the size of the earth. A fly lands on the surface once every hundred years. When the fly lands his feet wears away the surface of the steel ball by an amount that can only be described as next to nothing. When the whole of the Steel ball has been worn away to nothing, INFINITY has not even started.&lt;/p&gt;&#xA;" OwnerUserId="11962" LastActivityDate="2019-02-17T16:59:49.897" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15253" PostTypeId="1" AcceptedAnswerId="15258" CreationDate="2019-02-17T20:42:32.280" Score="6" ViewCount="170" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; be a finite dimensional vector space and let &lt;span class=&quot;math-container&quot;&gt;$B=\{v_1,v_2,\cdots,v_n\}$&lt;/span&gt; be a basis of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;. If a vector &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; can be written as &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$v=a_1v_1+a_2v_2+\cdots+a_nv_n,$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;we call &lt;span class=&quot;math-container&quot;&gt;$(a_1,a_2,\cdots,a_n)$&lt;/span&gt; the coordinates of &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; with respect to the basis &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I find teaching this definition contradicts our teaching that in a set, the order of elements does not matter. Here if we write &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\{v_2,\ldots,v_n,v_1\}$&lt;/span&gt;, it is still the same set; however, the coordinates of &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; will usually be different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in teaching the coordinates with respect to a basis, should we introduce the notion &quot;ordered basis&quot; (as an ordered &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-tuple of vectors) and define &quot;the coordinates with respect to an ordered basis&quot; instead? I find that in most Linear Algebra textbooks that I am aware of, this issue is never raised.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2019-02-18T02:37:49.563" Title="Difficulty in teaching the coordinates of a vector with respect to a basis $\{v_1,v_2,\ldots,v_n\}$" Tags="|undergraduate-education|linear-algebra|definitions|vectors|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15254" PostTypeId="2" ParentId="15212" CreationDate="2019-02-17T21:31:16.343" Score="0" Body="&lt;p&gt;Imagine you have a snack machine that contains infinite number of candies and gives one to you (for free) every time you press the button. So you have infinite number of candies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now your friend gives to you a candy of the same kind as a snack machine. Do you have more candies than you had before that? No as instead of your friend's candy you could just press a button and get it from there. The machine will continue to give them to you infinitly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So infinity plus one is still infinity - no profit in extending infinite resource by the same thing from a finite source.&lt;/p&gt;&#xA;" OwnerUserId="11965" LastActivityDate="2019-02-17T21:31:16.343" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15255" PostTypeId="1" AcceptedAnswerId="15259" CreationDate="2019-02-17T21:43:23.640" Score="3" ViewCount="276" Body="&lt;p&gt;I am aware of questions such as &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10086/teaching-arithmetic-operations-times-div-to-a-3-year-old&quot;&gt;this one&lt;/a&gt;. On the other hand, I still believe that teaching a bright three year old subtraction is possible. He counts from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; and backwards from &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$-10$&lt;/span&gt;, or some more negative integer. What I did was using a number line such as the following:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2MGdy.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2MGdy.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Say I want to teach &lt;span class=&quot;math-container&quot;&gt;$3-5$&lt;/span&gt;, I will ask the child to point his finger at &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, then move &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; units to the left, then ask the child where his finger is now. He will reply &amp;quot;negative two&amp;quot; and I will write down the equation &lt;span class=&quot;math-container&quot;&gt;$3-5=-2$&lt;/span&gt;. Here the minuend is not necessarily larger than or equal to the subtrahend.&lt;/p&gt;&#xA;&lt;p&gt;My question is, is the a right way to teach a three year old subtraction? I personally find that early exposure to negative numbers exciting but am not sure if it could be counter-productive or &amp;quot;haste makes waste&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me that most people teach by asking questions such as &amp;quot;&lt;em&gt;Here are five apples, if I take away three, how many are there left&lt;/em&gt;&amp;quot;? However, I find this method limited as the child may have the misconception that in &lt;span class=&quot;math-container&quot;&gt;$a-b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; cannot be larger than &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. Also, it seems to me that the usage of the number line reflects the spirit of more advanced mathematics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; When I asked this question about 10 months ago when the student is only a little over 3, she does not seem to get the number line idea well. Now as she is about to turn 4 (though technically she is still 3), she understands it and enjoys doing calculations such as &lt;span class=&quot;math-container&quot;&gt;$5-9$&lt;/span&gt; using the number line.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-01-03T20:44:45.557" Title="Teaching three-year-old subtraction using the number line" Tags="|gifted-students|arithmetic-operations|preschool-education|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15256" PostTypeId="2" ParentId="15255" CreationDate="2019-02-17T22:08:19.363" Score="0" Body="&lt;p&gt;Zuriel, START with the apples version.  Yes, it is not as good as the number line.  But it gets you STARTED.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are dealing with a 3 year old.  Build muscles gradually.  Would you teach a forward roll first or a front flip?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is this urge, urge, urge on this forum to want to jump to &quot;cleaner&quot; but more advanced formulations.  This shows a total misunderstanding of human psychology.&lt;/p&gt;&#xA;" OwnerUserId="11968" LastActivityDate="2019-02-17T22:08:19.363" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15257" PostTypeId="2" ParentId="15212" CreationDate="2019-02-18T00:30:06.583" Score="0" Body="&lt;p&gt;I don't think explaining the &lt;span class=&quot;math-container&quot;&gt;$\infty + 1 = \infty$&lt;/span&gt; thing directly is very productive. It's meaning is pretty fuzzy, it is very counter-intuitive, and it doesn't give any new perspective by itself. If I wanted to explain how equality works different on infinities, I would rather try to show an example of how 2 infinities can appear both equal and unequal, at the same time. This might be e.g.:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;(natural) numbers vs even (natural) numbers. There is obviously the same amount of both, as every even number is just &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. But equally obvious facts is, that there are twice as many natural numbers as even numbers, because there is &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; for each of them.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(natural) numbers vs &lt;span class=&quot;math-container&quot;&gt;$n+5$&lt;/span&gt; for each of them. Similar logic to the one above: there is obvious 1:1 relation between them, but also the first set has clearly 5 elements more.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This doesn't really prove anything like &lt;span class=&quot;math-container&quot;&gt;$\infty + 1 = \infty$&lt;/span&gt;, but it conveys a general fact of &quot;equality for infinities is really strange, and definitely different than for numbers&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is similar to the hotel example from a different answer, but I believe it is much better by having clearly separated sets. Instead of changing one set in some complicated way, we have two pretty simple things, lying next to each other, that we can compare in different ways.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure whether this is a useful answer for 5-year-old. I guess other answers are much better in this regard.  However, I myself struggled with these strange infinity eqalities, and examples like above were a key to get me to understand it, so I wanted to share. I believe they are the best way to explain where the &quot;&lt;span class=&quot;math-container&quot;&gt;$\infty + 1$&lt;/span&gt;&quot; paradox comes from.&lt;/p&gt;&#xA;" OwnerUserId="11969" LastEditorUserId="11969" LastEditDate="2019-02-18T21:25:49.190" LastActivityDate="2019-02-18T21:25:49.190" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15258" PostTypeId="2" ParentId="15253" CreationDate="2019-02-18T02:37:49.563" Score="4" Body="&lt;p&gt;At the level of standard linear algebra courses, where we consider bases only for finite-dimensional vector spaces, we should work with ordered bases, so that the coefficients of a vector come with an ordering (and thus constitute a vector in &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbb C^n$&lt;/span&gt;) and so that the matrices that represent linear transformations come with their rows ordered and their columns ordered (and might therefore be written on a page). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a more abstract level, we could use unordered bases. The coefficients of a vector &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; would then not be a sequence of scalars but rather a function from the basis into the field of scalars --- the function assigning to each basis element &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; the coefficient of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; in the expansion of &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;. Similarly, transformation matrices would have their rows and columns not ordered but rather  indexed by basis vectors. The point of this more abstract approach is that an ordering would be used only to match up which coefficient goes with which basis vector (or, in the case of matrices, which pair of basis vectors), and so we can forget the ordering and work directly with the matching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not tried to teach the more abstract approach in linear algebra classes, and I'm pretty sure I don't want to try. In classes I'll stick with ordered bases.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2019-02-18T02:37:49.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15259" PostTypeId="2" ParentId="15255" CreationDate="2019-02-18T04:23:09.603" Score="3" Body="&lt;p&gt;The following is anecdotal and rambling, but I think it goes a good way towards addressing the question (from a personal point of view).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My daughter and I play a variant of the card game &lt;a href=&quot;https://www.bicyclecards.com/how-to-play/war/&quot; rel=&quot;nofollow noreferrer&quot;&gt;War&lt;/a&gt; to work on arithmetic facts.  Originally, we played &quot;Addition War&quot;:  we would each play a card, and the first to sum the two numbers got to keep both cards.  When her first grade class started drilling subtraction facts, we started playing &quot;Subtraction War.&quot;  At one point, she turned up a 5 and I turned up a 3.  She got very angry with me when I said &quot;Oh, three minus five is negative two!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Papa!&quot; said, &quot;You can't minus five from three!  Five is too big!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point, I introduced her to a number line model of addition and subtraction.  The story that I gave her is that a frog is sitting on a lily pad in a pond.  Every lily pad has an address or number, counting up from 1, going from left to right.  When we add, the first number tells us where the frog starts, and the second number tells us how the frog hops.  For example, &lt;span class=&quot;math-container&quot;&gt;$3+4$&lt;/span&gt; means that the frog starts at the lily pad labeled &quot;3&quot;, then hops to the right four times, which takes it to the lily pad labeled &quot;7&quot;.  Therefore &lt;span class=&quot;math-container&quot;&gt;$3+4=7$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note that this was all done on a piece of paper with an actual toy frog to hop around, which makes life a little easier.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once she got the idea of addition with the frog, we started playing the same game with subtraction.  The only difference is that the frog now moves left instead of right.  For starters, we only worked &quot;proper&quot; subtraction problems, where the minuend is larger than the subtrahend.  I then asked about &lt;span class=&quot;math-container&quot;&gt;$4-4$&lt;/span&gt;.  Our lily pads started at 1, so this didn't seem to make sense (the frog splashed into the water).  However, since we are clever mathematicians, we can plant a lily seed and grow a new lily pad.  She had already seen zero before, so it made sense to call this new pad zero, and get on with life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, we can introduce negative numbers.  I got out a different color, and started &quot;planting&quot; more lily pads to the left of zero.  I labeled them going up from right to left in the new color (which happened to be purple).  She was really resistant to this kind of labeling at first&amp;mdash;she didn't like there being two different &quot;3s&quot;, for example, but we managed to talk about &quot;left three&quot; and &quot;right three&quot;, and she eventually bought the idea.  After a bit more play, she got pretty comfortable with the idea of problems like &quot;Five minus nine is 'purple four'&quot; (or 'left four').  A few months later, she is pretty comfortable with the idea of (small) negative numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now when we play subtraction war, black cards &lt;em&gt;always&lt;/em&gt; come first, so the &quot;correct&quot; difference if I play &lt;span class=&quot;math-container&quot;&gt;$5\spadesuit$&lt;/span&gt; and she plays &lt;span class=&quot;math-container&quot;&gt;$6\heartsuit$&lt;/span&gt; is &quot;Negative one!&quot;  If the cards have the same color, either positive or negative answers are fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any event, I think that this model of subtraction&amp;mdash;which really is exactly what you propose&amp;mdash;is not only appropriate for an introduction to subtraction, but is, perhaps, the &quot;right&quot; model for addition and subtraction in general.  It might &lt;em&gt;seem&lt;/em&gt; less concrete than &quot;three apples plus two more apples is five apples,&quot; but I think that it gives a &lt;em&gt;much&lt;/em&gt; more concrete meaning to negative numbers.  Frankly, I still have trouble with negative quantities (negative apples scare me), but I have no problem with negative coordinates or negative directions.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2019-02-18T04:23:09.603" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15260" PostTypeId="2" ParentId="15212" CreationDate="2019-02-18T06:03:10.700" Score="0" Body="&lt;p&gt;From what i know, infinity is a fun &quot;number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's say it takes one minute to climb on a rock.&#xA;If you have a tower made of 100 rocks, it takes you 100 minutes to climb on top of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's say you have a tower made of an infinite number of rocks. It's not just big, it goes higher that the sky, it goes higher than the sun, it never stops going higher than everything that isn't infinite too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you try to climb on top of it, you'll never achieve to, you'll keep climbing forever, and won't reach the top even after an infinite amount of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you possess a tower made of an infinity of rocks, and someone gives you another rock, you are now possessing an infinity of rocks from the tower, plus the rock you just have been given. You possess infinity +1 rocks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you manage to put this new acquiered rock in the tower, it still takes as much time as before to reach its top by climbing on it: forever. The tower still is made of an infinite amount of rocks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you possess two towers, each made of an infinite amount of rocks, you possess two infinities of rocks. But if you somehow manage to put the two towers one &quot;on top&quot; of the other, it takes the same amount of time to reach the top of it as it is taking to climb on the former tower. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's say you have an infinitely big hole, it's so big that you can put as many rock as you want in it, it will never get full. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if you put your infinitely big tower in it, there will still be room left for an infinite amount of rocks. Even if you put an infinite amount of infinite towers in it, it won't change that, there is still an infinite amount of room left for rocks. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you share your infinite amount of rocks with poeple around you, you can give them rocks forever, they all get an infinite amount of rocks. Even if you share it with an infinite amount of poeple, you'll never stop giving them rocks. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you build a square floor made of rocks, by putting an infinite straight line of rocks in front of you, and an infinite straight line of rocks aside every rock composing the first line, it's area will be infinity².&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can then take the rocks of your floor and build a tower with it, you will never stop building the tower, which means it would create an infinitely big tower. But the tower will take much more space than the floor, because it has &#xA; an infinite amount of floors which areas are also infinity². &lt;/p&gt;&#xA;" OwnerUserId="11972" LastEditorUserId="11972" LastEditDate="2019-02-18T06:28:00.897" LastActivityDate="2019-02-18T06:28:00.897" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15262" PostTypeId="1" CreationDate="2019-02-18T18:51:51.573" Score="4" ViewCount="401" Body="&lt;p&gt;I'm teaching a variety of undergraduate and graduate geometry classes (mostly for in-service teachers) which range from elementary axiomatic geometry to more advanced transformational geometry. I'm looking to bring various levels of abstract algebra into these classes that incorporate basic ring and field theory to bridge the gap between these two traditional approaches to geometry but without teaching a full algebraic geometry course. In particular, I'm looking for suggestions of textbooks that give a good treatment of:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Abstract affine spaces including finite and rational affine spaces,&lt;/li&gt;&#xA;&lt;li&gt;The relationship between affine spaces defined using abstract algebra and affine spaces defined by axioms (particularly affine planes and 3-dimensional affine spaces),&lt;/li&gt;&#xA;&lt;li&gt;Projective spaces using homogeneous coordinates from affine spaces.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Are there any books out there that cover this material in a way that doesn't get too heavy into advanced algebraic geometry?&lt;/p&gt;&#xA;" OwnerUserId="515" LastEditorUserId="376" LastEditDate="2019-02-20T12:37:07.720" LastActivityDate="2019-06-02T23:17:25.157" Title="Geometry textbook with an abstract algebra emphasis" Tags="|undergraduate-education|textbooks|geometry|graduate-education|abstract-algebra|" AnswerCount="3" CommentCount="8" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15263" PostTypeId="1" AcceptedAnswerId="15349" CreationDate="2019-02-18T23:20:52.947" Score="15" ViewCount="917" Body="&lt;p&gt;Until fairly recently, it was common for students in school to learn Euclidean geometry from a translation of Euclid. I get the impression that ca. 1700 this would have been in college and only for a tiny fraction of the population, whereas maybe ca. 1900 in the US it would have been at a younger age and perhaps in some cases the entire population.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm curious how this actually worked. In particular, what did these students do to practice their skills? Would students have mainly proved theorems, or would there have been more of mix? Were there editions of Euclid that had a set of theorems created for students to practice by proving them? Or maybe separate anthologies of problems? Can such materials still be found online?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were people using a straight translation of Euclid, like Heath in the English-speaking world, or modernizations? Were there any particularly good or widely used modernizations? The coolest one I'm aware of is &lt;a href=&quot;https://archive.org/details/firstsixbooksofe00eucl/page/n6&quot; rel=&quot;noreferrer&quot;&gt;Byrne&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-05-12T20:20:04.637" Title="When Euclid was used as a textbook, what exercises did students do?" Tags="|textbooks|geometry|homework|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15264" PostTypeId="1" AcceptedAnswerId="15265" CreationDate="2019-02-18T23:49:45.690" Score="5" ViewCount="286" Body="&lt;p&gt;I teach mostly physics and a little math at a California community college. I've never taught the multivariate calculus course, but I have taught the electricity and magnetism course for which the math course is largely a service course. I think our curriculum is fairly standard for the US: after taking a year of single-variable calculus, students take a 3rd-semester course covering multivariable calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a typical math course of this type, what kind of area integrals do students actually learn to do?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the E&amp;amp;M course, 90% of what we use area integrals for is just cases where we integrand is constant. We do a few simple examples that have some special symmetry, so that they can find the electric field of a given charge distribution. A typical case like this would be determining the field of a uniform line of charge, in which we pick the Gaussian surface to be a cylinder. Students mainly do minor variations on this, such as the field of a uniformly charged cylinder. There are maybe half a dozen simple examples of this type that you can make up, and that's about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the opposite extreme, it seems like the general case of evaluating such a surface integral would involve basically treating the surface as a manifold (even if we don't use the word) and picking coordinate charts on it (even if we don't call it that). Most surfaces can't be covered with a single chart, so in general we'd have to use multiple charts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are students actually taught to do something in between these extremes of complexity? For example, does the typical math text restrict attention to cases in which the surface can be projected onto a coordinatized plane in such a way that the mapping is either 1-to-1 or 2-to-1 (as for a sphere, but not for most non-convex surfaces)?&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="2083" LastEditDate="2019-02-19T06:50:37.683" LastActivityDate="2019-12-11T20:07:24.547" Title="In a typical 3rd-semester multivariate calculus course in the US, what kind of area integrals do students actually learn to do?" Tags="|undergraduate-education|vector-calculus|usa|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15265" PostTypeId="2" ParentId="15264" CreationDate="2019-02-19T00:31:44.040" Score="3" Body="&lt;p&gt;The following is based on my experience at as a TA and adjunct at a UC, a Cal State, and the University of Nevada, as well as observations of siblings and friends at other institutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The two texts which seem to be used most commonly are &lt;a href=&quot;https://books.google.com/books/about/Calculus_Early_Transcendentals.html?id=QyOYWR9RLI8C&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stewart's &lt;em&gt;Calculus&lt;/em&gt;&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books?id=yavIPgAACAAJ&amp;amp;dq=thomas+calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas' &lt;em&gt;Calculus&lt;/em&gt;&lt;/a&gt; (note that both Stewart and Thomas have been dead for a while, and neither has been an author on either text in a decade or two&amp;mdash;these are pretty well established franchises).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, both of these books give a somewhat cursory introduction to techniques of integration in multiple variables.  They give a basic overview of the theory (in terms of Riemann sums over partitions of higher dimensional domains of integration), then quickly introduce Fubini / Tonelli so that students can actually compute some integrals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next several sections deal with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Integration_by_substitution#Substitution_for_multiple_variables&quot; rel=&quot;nofollow noreferrer&quot;&gt;generalized change of variables formula&lt;/a&gt;, with an emphasis on polar and spherical coordinates.  The actual general change of variables formula is there, but is often elided in practice (as students often don't really have the necessary background in linear algebra to handle a more general change of coordinates).  Finally, both texts include sections on vector fields, line and surface integrals, and the big theorems of multivariable integration (Stokes, Green, Gauss, etc).  In my experience, these sections are typically taught very quickly near the end of the quarter / semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because linear algebra is not typically a prerequisite for multivariable calculus, many of the results that you are looking for are likely to be beyond the ken of the average student.  They likely won't know what a projection is, let alone a manifold or a cover by smooth charts.  I believe that the goal is to give students a very brief overview of the general shape of calculus in higher dimensions, but that specific computations get left for future classes in, for example, physics departments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long story short:  if it can't be done relatively simply in rectangular, polar, cylindrical, or spherical coordinates, or it cannot be done via an application of Stoke's theorem (and other related theorems) in a relatively quick way, then it is likely that the students haven't seen it.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2019-02-19T00:50:06.587" LastActivityDate="2019-02-19T00:50:06.587" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15266" PostTypeId="2" ParentId="15255" CreationDate="2019-02-19T02:44:51.573" Score="1" Body="&lt;p&gt;You might need to use both, the number line, and the apples. It is always interesting to have a visual idea of a concept, because raw numbers sometimes aren't really intuitive. &#xA;To get away from the limit where you can't take more apples than the number of apple you have, there is a simple trick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have to give me 5 apples, but you only have 3 apples. So, give me the first 3 apples, you have now zero apple. I need 2 more apples, but i'm a nice guy, so you can give me those two apples later.&#xA;You now have negative 2 apples, which means, if you find apples, the first two apples you find belong to me, not to you. &lt;/p&gt;&#xA;" OwnerUserId="11972" LastActivityDate="2019-02-19T02:44:51.573" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15267" PostTypeId="2" ParentId="9760" CreationDate="2019-02-19T03:06:18.633" Score="2" Body="&lt;p&gt;When I was a kid, (around 4 years old), I was playing super nintendo games, because my dad was playing these too. You easily find out that mario has 3 lives, and that once it gets to zero, it's game over. You find coins, and once you have 100 coins, you win one more life. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You aren't searching for maths, it comes to you directly. Then some games (like zelda), needs you to learn slightly more complex maths, because you find rupees (coins, basically), and you can use those to buy things. Do I have enough money to buy the stuffs I want ?&#xA;It also requires you to know what are the characters saying, as it's all text, you need to call your dad for him to read for you, but it encourages you to start learning how to read by yourself, and instead of just going to school and learn how to read there, you are the one asking poeple to teach you how to do so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was fun, because the game wasn't about learning how to read, or learning how to count to win, you had to rescue the princess, and learning how to read was just one of the requirements on how to do so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The side effect was to turn me into a geek, but entering school, I knew how to count and read.&lt;/p&gt;&#xA;" OwnerUserId="11972" LastEditorUserId="11972" LastEditDate="2019-02-19T18:15:24.660" LastActivityDate="2019-02-19T18:15:24.660" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15268" PostTypeId="2" ParentId="2462" CreationDate="2019-02-19T03:21:49.653" Score="0" Body="&lt;p&gt;Depending on the kind of education you want to give, &quot;Normal video games&quot; can work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By normal, i mean usual video games, in opposition to educationnal video games.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can play a RPG game with your child if you want to learn him how to read.&#xA;A game that has a money and shops for basic maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just make sure it isn't violent. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For non-video games, Yahtzee (a dice game) needs some calculations in it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the games that exist in the world will teach things to your child, as long as you're with him to help him and praise him when he wins. &lt;/p&gt;&#xA;" OwnerUserId="11972" LastActivityDate="2019-02-19T03:21:49.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15269" PostTypeId="2" ParentId="9760" CreationDate="2019-02-19T03:43:07.943" Score="2" Body="&lt;p&gt;Some of the math games on &lt;a href=&quot;http://www.abcya.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;abcya.com&lt;/a&gt; or &lt;a href=&quot;http://www.hoodamath.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;hoodamath&lt;/a&gt; could be good for this age level, especially if you find that logic puzzles fill your bill.  As an example, &lt;a href=&quot;https://www.hoodamath.com/games/tronix.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt; could be imagined to teach about planar graphs ... or just to be fun/maddening.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-02-19T03:43:07.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15270" PostTypeId="2" ParentId="15263" CreationDate="2019-02-19T05:01:56.867" Score="10" Body="&lt;p&gt;Euclid’s &lt;em&gt;Elements&lt;/em&gt; was used as a Geometry textbook in essentially the same way that Virgil’s &lt;em&gt;Æneid&lt;/em&gt; was used as a Latin textbook. Neither contained exercises anything else pedagogical, and both were used in the same way, in about the same time period.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their heyday was in the days before schooling as we know it existed. They would be used between a student and his Latin or math tutor, a person or people who worked one-on-one with the student, often at the well-to-do student’s house. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You “knew” the material once you plowed through it. Virgil (and Ovid and Horace) were simply translated by the student as a way of practicing Latin skills. The poems were translated when the tutor said they were. Similarly, the &lt;em&gt;Elements&lt;/em&gt; were read and demonstrated to the tutor’s satisfaction. Exercises? Why would you need exercises? That wouldn’t be the objective of the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Heath’s versions were mainly written for mathematicians, and were certainly not used in schools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our current course called Geometry doesn’t have much in common with reading the &lt;em&gt;Elements&lt;/em&gt;. For decades, Geometry was taught in two courses, Plane and Solid. Those were the days when math schooling only went to Algebra II or so, with Freshman Math at colleges being what we call Pre-calculus. My dad, high school Class of 1949 who entered Clemson Agricultural College that same year, went through mathematics like this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The set of postulates isn’t the same: many of the modern sets of postulates are due to contemporary writers. For example, the current formulation of Euclid’s 5th (“Given a line and a point not on that line, there is one and only one line through the point that is parallel to the given line”) is due to John Playfair, and is often referred to as Playfair’s axiom. In the full-on day’s of the 1950’s and 1960’s “New” Math, Edwin Moise’s &lt;em&gt;Geometry&lt;/em&gt; book turned the SAS triangle congruence theorem into a postulate: proving the others (ASA, SSS, etc.) given one of them is not too bad, but having to prove an initial one is pure hell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moise was later joined by Floyd Downs, and their Geometry book may still be in print. Alfred Posamentier and H.S.M. Coxeter soon came out with competing books, mostly sticking to the Moise postulate model with tweaks. Those tweaks included exercises, applications, word problems, career sections, computer-aided learning, and all sorts of applied things that would never pollute the pure reader of Euclid. Those that needed practical Geometry would learn it as an apprentice to a craftsman.&lt;/p&gt;&#xA;" OwnerUserId="9241" LastEditorUserId="9241" LastEditDate="2019-02-19T05:11:15.450" LastActivityDate="2019-02-19T05:11:15.450" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15271" PostTypeId="2" ParentId="9760" CreationDate="2019-02-19T07:13:02.137" Score="2" Body="&lt;p&gt;Ekapeli-matikka (available &lt;a href=&quot;http://www.lukimat.fi/matematiikka/materiaalit/Tietokoneohjelmat/ekapeli-matikka&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.lukimat.fi/matematiikka/materiaalit/Tietokoneohjelmat/ekapeli-matikka&lt;/a&gt; ) is intended for children who are not yet in school and who have problems learning mathematics. It teaches one-to-one correspondence, comparisons, order, number symbols and addition, at least, as per the quoted description below:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ekapeli-Matikka on tietokonepeli, joka harjoittaa yksi-yhteen vastaavuutta, vertailua, järjestämistä, lukusanan, -määrän ja numerosymbolin vastaavuutta sekä lukujono- ja yhteenlaskutaitoja. Ekapeli-Matikka on tarkoitettu niille esi- ja alkuopetusikäisille lapsille, joille matematiikan oppiminen on haasteellista.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have not used it, but I have used the sister program which teaches reading and letters. It seemed to be okay. There are some other mathematics games available: &lt;a href=&quot;http://www.lukimat.fi/matematiikka/materiaalit/Tietokoneohjelmat&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.lukimat.fi/matematiikka/materiaalit/Tietokoneohjelmat&lt;/a&gt; . According to the description, they are intended for children up to fourth grade, so around the age of ten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The games are free and produced by Niilo Mäki instituutti and University of Jyväskylä.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2019-02-19T07:13:02.137" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15272" PostTypeId="1" CreationDate="2019-02-19T16:01:35.713" Score="12" ViewCount="305" Body="&lt;p&gt;I'm going to start introducing my abstract algebra class to a variety of groups soon. Dihedral groups &lt;span class=&quot;math-container&quot;&gt;$D_n$&lt;/span&gt; arise out of symmetries on polygons. And the Symmetric group &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt; makes sense as the group of permutations. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a similarly naive or geometric way to give the Alternating group, &lt;span class=&quot;math-container&quot;&gt;$A_n$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I know the Alternating group is the subgroup of even parity elements in &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt;. But I thought it'd be nice for students to work with (an isomorphic version of) the group first and then later show it is the even parity elements. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm hoping someone out there has a simple (no pun intended) and independent construction of &lt;span class=&quot;math-container&quot;&gt;$A_n$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="6053" LastEditDate="2019-02-20T17:30:22.277" LastActivityDate="2019-05-06T08:06:27.417" Title="Alternating group without $S_n$" Tags="|undergraduate-education|abstract-algebra|" AnswerCount="2" CommentCount="6" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15273" PostTypeId="1" AcceptedAnswerId="15283" CreationDate="2019-02-20T10:38:50.787" Score="4" ViewCount="393" Body="&lt;p&gt;When giving a lecture, I feel that I write too fast on the blackboard. This causes me to have to clarify some letters and symbols after the fact, in order to make them more legible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that slowing down my speed of handwriting at least a little would help. I should mention that I have not received feedback about poor handwriting; rather, the process of writing a symbol in a sloppy way and then erasing or fixing it annoys me and is an issue I would like to fix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that I know this is the case, what are some practical tips or habits to make the change happen?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The present context is lecturing in a fairly big lecture hall.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As always, answers should draw from personal experience or credible sources in an explicit way.)&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="376" LastEditDate="2019-02-20T11:56:56.173" LastActivityDate="2019-02-22T23:58:05.660" Title="How to write slower on a blackboard?" Tags="|board-use|lecturing|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15274" PostTypeId="2" ParentId="15262" CreationDate="2019-02-20T11:12:56.407" Score="2" Body="&lt;p&gt;I think you are looking for a &quot;French style&quot; approach. So maybe &lt;a href=&quot;https://www.springer.com/gp/book/9783540434986&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometry by M. Audin&lt;/a&gt; is good for you. Note that a new edition in French is available. It covers pretty well points 2 and 3 but not 1. Also interesting is the book of P. Gabriel, &lt;a href=&quot;https://www.springer.com/gp/book/9783764353766&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matrizen,Geometrie, Lineare Algebra&lt;/a&gt;, unfortunatedly only available in German or a French translation. I hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="330" LastActivityDate="2019-02-20T11:12:56.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15275" PostTypeId="2" ParentId="15273" CreationDate="2019-02-20T15:33:45.367" Score="15" Body="&lt;p&gt;When I feel my blackboard presentation is a little sloppy, and remembering to &quot;just slow down&quot; doesn't help, I try writing &lt;em&gt;bigger&lt;/em&gt;. It has the benefit of actually slowing me down a bit, and students sitting far back can more clearly see what I've written.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-02-20T15:33:45.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15276" PostTypeId="2" ParentId="2196" CreationDate="2019-02-20T20:54:54.543" Score="2" Body="&lt;p&gt;I think my system solves your problem&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Assign homework due by date &lt;span class=&quot;math-container&quot;&gt;$X_1$&lt;/span&gt; and time &lt;span class=&quot;math-container&quot;&gt;$t_1$&lt;/span&gt;.  The submission is accepted electronically until that date and time.&lt;/li&gt;&#xA;&lt;li&gt;The solutions to this homework are automatically released on date &lt;span class=&quot;math-container&quot;&gt;$X_1$&lt;/span&gt; at time &lt;span class=&quot;math-container&quot;&gt;$t_1$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The students now have until date &lt;span class=&quot;math-container&quot;&gt;$X_2$&lt;/span&gt; to correct their homework using the supplied solutions.  I need to see evidence that each problem was compared to the given solution.  This might be a check-mark if it was entirely correct, all the way to completely redoing the problem if the original solution was not salvageable.&lt;/li&gt;&#xA;&lt;li&gt;I can very quickly &quot;grade&quot; their correction on essentially a binary scale: for each problem, have you shown me evidence of an initial attempt and a revision.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The first step is needed to prevent people from just tweaking the solutions and making it look like their own work.  If I suspect that is happening, I can look at the electronic submission which was accepted before the release of the solution.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2019-02-20T21:06:04.727" LastActivityDate="2019-02-20T21:06:04.727" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15277" PostTypeId="2" ParentId="15237" CreationDate="2019-02-20T21:25:57.283" Score="8" Body="&lt;p&gt;There's a general point I can't see explicitly in the other answers: knowing more maths (and generally having spent time knowing/thinking about maths) helps you have a bigger picture. A lot of maths starts to fit together better as you know more for longer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a (not very good) analogy, suppose someone left a sandwich on a bench, and an ant found it. The ant could teach all its ant friends how to reach the sandwich up one of the legs of the bench, and that is arguably enough. But it can't guess that there are three other legs they could climb up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having a sense of the bigger picture doesn't automatically make you better at teaching. But it does make it easier to be good at teaching. It means you have a better idea which bits are important, which aspects come up elsewhere in maths, how to reverse-engineer suitable exam questions...&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-02-20T21:25:57.283" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15278" PostTypeId="1" AcceptedAnswerId="15286" CreationDate="2019-02-21T04:21:20.037" Score="13" ViewCount="3728" Body="&lt;p&gt;I read on multiple web pages something that implies that that some students really struggle with fractions but I could never find a detailed explanation of why. This question is different from &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7826/are-fractions-hard-because-they-are-like-algebra&quot;&gt;Are fractions hard because they are like algebra?&lt;/a&gt;. I'm not sure whether this question is more suitable for Academia Stack Exchange or whether questions that would normally belong on Academia Stack Exchange but are about math belong here so I'm asking it here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also don't know in which way they struggle with fractions or what they mean exactly by it and probably cannot find the answer to that so that's why I'm not stating here in what way they struggle with fractions. I think there were researchers out there who researched how primary math education should be taught. If so, would it be possible for anyone to give me a lot of details about what those researchers discovered on the topic of teaching fractions to elementry school students and write them in a way that clearly explains in detail why some students are struggling so much with fractions and why certain methods of teaching are claimed to be the best way to teach given the current information researchers have about how students learn?&lt;/p&gt;&#xA;" OwnerUserId="11718" LastActivityDate="2022-12-28T18:17:42.923" Title="Why do some students struggle so much with fractions?" Tags="|primary-education|fractions|" AnswerCount="6" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15279" PostTypeId="2" ParentId="15263" CreationDate="2019-02-21T06:06:25.957" Score="6" Body="&lt;p&gt;As others have said, Euclid was often used as a reference book rather than what we now call a &quot;textbook&quot;. Students were often required to memorize and reproduce the contents. For example, at Oxford, until the mid-19th century, students had to memorize two books of Euclid (to the letter) and were not asked to solve any original problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Textbooks with practical exercises began to be introduced in the 18th century with Clairaut in France (1741) and Kästner in Germany (1758). These initial textbooks were not always taken seriously, but d'Alembert made a call for serious, practical textbooks and a new approach to geometry in 1784. The transition to a more pedagogical approach occurred slowly in the early 19th century. LeGendre wrote a geometry textbook in France which became very popular and a &lt;a href=&quot;https://archive.org/details/cu31924001166341/page/n6&quot; rel=&quot;nofollow noreferrer&quot;&gt;translated version&lt;/a&gt; eventually became the most popular geometry textbook in the U.S. People continued to use Euclid as a reference, but increasingly turned to other books for teaching and exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the preceding history can be found in the historical introduction to a 1912 report (Slaught, H. E., &amp;amp; others. (1912). Final report of the national committee of fifteen on geometry syllabus. The Mathematics Teacher, 5(2), 46-131.)&lt;/p&gt;&#xA;" OwnerUserId="6782" LastEditorDisplayName="user507" LastEditDate="2019-02-22T04:44:24.760" LastActivityDate="2019-02-22T04:44:24.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15280" PostTypeId="2" ParentId="15237" CreationDate="2019-02-21T10:46:41.837" Score="4" Body="&lt;p&gt;There are four ways an advanced knowledge of mathematics can help, in rough order of importance: Mastery, understanding student thinking, context and content and tricks. However, general teaching skill also matters, and I would guess that it does not reliably increase along with mathematics studies, or at least that the effect is small.&lt;/p&gt;&#xA;&lt;h1&gt;Mastery&lt;/h1&gt;&#xA;&lt;p&gt;All other things being equal, someone with more experience with difficult mathematics is likely to have mastered lots of easy mathematics. Even when the particular subject is new to them, they have learned how to easily learn a new mathematical idea or abstraction.&lt;/p&gt;&#xA;&lt;p&gt;Mastery gives confidence, speeds up preparation and makes it easier to find mistakes. It is all around helpful.&lt;/p&gt;&#xA;&lt;h1&gt;Understanding student thinking&lt;/h1&gt;&#xA;&lt;p&gt;You need to master several ways of thinking about something as simple as multiplication, not to speak of more advanced mathematics. It helps to have developed mathematical thinking to be able to understand these different ways of seeing (say) multiplication, and also thereby being able to understand non-standard solutions and solving methods of students. The same is true of common misunderstandings and explaining what is the problem, or maybe giving a spontaneous counterexample of exercise that illustrates the different ways of thinking.&lt;/p&gt;&#xA;&lt;p&gt;In particular, you need to do this on the fly, in front of a classroom. It is not easy!&lt;/p&gt;&#xA;&lt;h1&gt;Context&lt;/h1&gt;&#xA;&lt;p&gt;Having learned higher mathematics gives context for the basics. This helps in making the content meaningful and in finding examples and applications, which can be made into exercises or project work.&lt;/p&gt;&#xA;&lt;h1&gt;Tricks and content&lt;/h1&gt;&#xA;&lt;p&gt;Sometimes, though rarely, the specific techniques or content one has worked on perfectly line up with the teaching. I doubt this is very common, and the more advanced mathematics one is teaching, the more likely it is. Applied research and courses for students of relevant subjects might also increase the incidence.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2022-11-07T14:26:35.283" LastActivityDate="2022-11-07T14:26:35.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15282" PostTypeId="2" ParentId="5794" CreationDate="2019-02-22T13:52:19.527" Score="-2" Body="&lt;p&gt;The idea of using the concept of &lt;em&gt;direction&lt;/em&gt; and &lt;em&gt;flipping directions&lt;/em&gt; in order to understand the concept of negative numbers and their multiplication is helpful but maybe misleading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math is an expression/product of &lt;strong&gt;&lt;em&gt;Logic&lt;/em&gt;&lt;/strong&gt; instead, and therefore logical operators and analogies are better suited for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Think of the &lt;strong&gt;positive sign&lt;/strong&gt; as a symbol indicating &lt;strong&gt;something that is true&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In which case you can think of a &lt;strong&gt;positive number &lt;code&gt;n&lt;/code&gt;&lt;/strong&gt; as meaning it is &lt;strong&gt;true that something exists &lt;code&gt;n&lt;/code&gt; times&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, think of the &lt;strong&gt;negative sign&lt;/strong&gt; as a symbol indicating &lt;strong&gt;something that is NOT true&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In which case you can think of a &lt;strong&gt;negative number &lt;code&gt;n&lt;/code&gt;&lt;/strong&gt; as meaning it is &lt;strong&gt;NOT true that something exists &lt;code&gt;n&lt;/code&gt; times&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you &lt;strong&gt;multiply a positive X by a positive Y&lt;/strong&gt; you are saying that &lt;strong&gt;it is true Y times in the world that it is true that something exists X times&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;strong&gt;2 x 3&lt;/strong&gt; means &lt;strong&gt;it's true 3 times that (it is true that 2 unicorns exist)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, you are saying that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;it is true that 2 unicorns and it is true that 2 unicorns and it is true that 2 unicorns&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which can be written as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;it is true that (2 unicorns and 2 unicorns and 2 unicorns)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which can be written as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;it is true that 6 unicorns&lt;/strong&gt;,  &lt;em&gt;also written as &lt;code&gt;+6 unicorns&lt;/code&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you &lt;strong&gt;multiply a positive X by a negative Y&lt;/strong&gt; you are saying that &lt;strong&gt;it is NOT true Y times in the world that something is true X times&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;strong&gt;2 x -3&lt;/strong&gt; means &lt;strong&gt;it is NOT true 3 times that (it is true that 2 unicorns)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, you are saying that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;it is not true that 2 unicorns and it is not true that 2 unicorns and it is not true that 2 unicorns&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;which is &lt;strong&gt;logically equivalent to&lt;/strong&gt; saying:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;it is not true that (2 unicorns and 2 unicorns and 2 unicorns)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which can be written as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;it is not true that 6 unicorns&lt;/strong&gt;,  &lt;em&gt;also written as &lt;code&gt;-6 unicorns&lt;/code&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, when you &lt;strong&gt;multiply a negative X by a negative Y&lt;/strong&gt; you are saying that &lt;strong&gt;it is NOT true Y times in the world that it is NOT true that something exists X times&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;strong&gt;-2 x -3&lt;/strong&gt; means &lt;strong&gt;it is NOT true 3 times that (it is NOT true that 2 unicorns)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, you are saying that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;not true that (not true that 2 unicorns) and not true that (not true that 2 unicorns) and not true that (not true that 2 unicorns)&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if it's not true that something is not true then that something is actually true, so the above is &lt;strong&gt;logically equivalent to&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(true that 2 unicorns) and (true that 2 unicorns) and (true that 2 unicorns)&lt;/strong&gt; &#xA;which can be written as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;true that (2 unicorns and 2 unicorns and 2 unicorns)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;meaning: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;true that 6 unicorns&lt;/strong&gt;, &lt;em&gt;also written as +6 unicorns&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And that is why &lt;strong&gt;&lt;code&gt;-2 x -3 = +6&lt;/code&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="11993" LastEditorUserId="11993" LastEditDate="2019-02-22T16:01:09.790" LastActivityDate="2019-02-22T16:01:09.790" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15283" PostTypeId="2" ParentId="15273" CreationDate="2019-02-22T14:06:05.373" Score="4" Body="&lt;p&gt;You might benefit from some &lt;a href=&quot;https://en.wikipedia.org/wiki/Practice_(learning_method)#Deliberate_practice&quot; rel=&quot;nofollow noreferrer&quot;&gt;deliberate practice&lt;/a&gt;.  Take 15 or 20 minutes out of every day to give a &quot;lecture&quot; to an empty room (or a room with a colleague who can critique you).  During that &quot;lecture&quot;, pick some particular aspect of your board work to concentrate on (for example, writing larger, writing slower, keeping your lines straight, &amp;amp;c.&amp;mdash;you might even just practice writing particular mathematical symbols over and over again if your concern is having to erase; I once spent an afternoon practicing nothing but circles for a trigonometry class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the problem with trying to practice these skills while teaching is that you are &quot;on the spot.&quot;  If you make a mistake or write in a way that you dislike, it is easy to get flustered.  Once you mess up once, you are on edge, and far more likely to mess up again.  Personally, when I make a mistake, I often start moving a lot faster, too (this is a reaction to stress, I'm sure).  This makes it very hard to practice your pacing while in front of a live group of students.  Take the pressure off, and practice in some other setting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other bits and bobbles of general advice:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Write larger.&lt;/strong&gt; This not only forces you to slow down a little, but also makes your writing easier to see from the back of the room.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Write in block letters.&lt;/strong&gt; That is, pick a handwriting &quot;font&quot; that consists only of capital letters which are well separated from each other (though lowercase letters will necessarily show up for mathematical symbols).  These letters typically take a little more care to write, but are easier to read.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Write in narrower columns.&lt;/strong&gt; This might seem like a silly piece of advice, but I find that it helps.  When I watch other people lecture, they typically divide the whiteboard in half (roughly), and write a column of text on each half.  Because they can't quite reach the right-hand side of the column from the left-hand side, they end up moving back and forth a lot, or writing slanty-wise, or otherwise messing up their board work.  Divide the board into three columns.  Maybe even draw vertical lines on the board before you do anything else.  It will probably make your life easier.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Keep an eraser in your off hand.&lt;/strong&gt; I know that the question is about avoiding the use of an eraser, but... meh.  Typically, no matter how slowly you write, you will be ahead of your students.  Moreover, it is very difficult to write in a way that is visible to the students while you are writing.  If you write something that you don't like, it is very easy to quickly erase it and correct it before the students even notice.  Don't be afraid of making quick corrections.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Mentally separate notes from exposition.&lt;/strong&gt; You should not be trying to write down everything you say, and your students should not be copying down your words verbatim.  For example, if I am relating a theorem to students, I might write four words (i.e. state a hypothesis), then spend a minute or two explaining those four words.  If a theorem has a lot of hypotheses, it might take five or ten minutes just to write down a statement of the theorem.  The whole statement might take up only a couple of lines, but there will have been a lot of exposition between the lines.  At the end of the day, the students need a copy of the statement of the theorem in their notes, and they need to be cognizant of some motivation and explanation (but this doesn't &lt;em&gt;need&lt;/em&gt; to be written).  I have seen folk try to write down all of this exposition.  Don't do that.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2019-02-22T23:58:05.660" LastActivityDate="2019-02-22T23:58:05.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15284" PostTypeId="2" ParentId="15278" CreationDate="2019-02-22T18:40:35.340" Score="29" Body="&lt;p&gt;There are many reasons why fractions are so hard for students to learn. Mostly, they're taught gibberish and assessed according to such gibberish.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 1&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;You are a 12-year-old student who has learned that &amp;quot;a fraction is part of a whole, such as part of pizza&amp;quot;. So when you look at &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{3}\times\frac{7}{5}$&lt;/span&gt;, you now must multiply pizza slices by pizza slices. This nonsense is obviously impossible to understand. Whether or not your teacher says to just memorize the procedures for multiplying fractions, that's how you're assessed so that's what you do. You give up on sense-making. You never learn what it means to multiply by a fraction, so, even years later, you still think that &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{3}\times700$&lt;/span&gt; must be bigger than 700 because there's 700 and multiplication. You're not sure if you need common denominators, though, because all those other fraction rules you tried to memorize are getting confused. &amp;quot;I hate fractions!&amp;quot; you say.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 2&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Word problem: &amp;quot;A plane can go 600km/hour [fraction]. A retrofitted wing can cause it to go &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{10}$&lt;/span&gt; [another fraction] faster. How far can it go in half-an-hour [another fraction] with the new wing?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Student: &amp;quot;I see the fractions. Where are the pizzas? My teacher says to draw a picture if I don't understand a word problem, so I should draw a pizza cut into 600 slices? How do I slice the pizza into plane travel?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 3&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Teachers tell you over and over again that whatever you do to the top of a fraction, you must also do to the bottom of a fraction. But they never tell you &lt;em&gt;why&lt;/em&gt; or provide you with any sensory experiences or contexts to develop your intuition for this. They don't tell you &lt;em&gt;when&lt;/em&gt; this is appropriate or useful or even what it means, and you're definitely never tested or graded on how it compares/contrasts with all the other things you can/should do with fractions. You seem to get check marks when you cancel zeroes in &lt;span class=&quot;math-container&quot;&gt;$\frac{600}{7100}=\frac{6}{71}$&lt;/span&gt;, though, so, when you're doing first-year calculus at university, you think you can cancel the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s to simplify as follows:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{x+3}{x+4}=\frac{3}{4}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{sin(x)}{x}=sin$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And you're pretty sure that &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}\times2$&lt;/span&gt; is the same thing as &lt;span class=&quot;math-container&quot;&gt;$\frac{a\times2}{b\times2}$&lt;/span&gt;, right? Whatever you do to the top, you also do it to the bottom... right? That's usually what got you good enough grades in the past...&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 4&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Sometimes teachers say that when given &lt;span class=&quot;math-container&quot;&gt;$\frac{26}{6}$&lt;/span&gt;, you should type &lt;span class=&quot;math-container&quot;&gt;$26\div6$&lt;/span&gt; into the calculator. But I challenge you to find &lt;em&gt;any&lt;/em&gt; textbook in North America that explains &lt;em&gt;why&lt;/em&gt; this might be a reasonable thing to do and why the fraction bar and the &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; are &lt;em&gt;always&lt;/em&gt; interchangeable while the fraction bar and, say, &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; are not. Even harder: find &lt;em&gt;any&lt;/em&gt; textbook or test or teacher or tutor that asks students to compare and contrast &lt;span class=&quot;math-container&quot;&gt;$\frac{26}{6}$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$\frac{26\div6}{6\div6}$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$26\div6$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$\frac{26\div2}{6\div2}$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$\frac{26}{6}\div6$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$\frac{\frac{26}{6}}{6}$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$\frac{6}{\frac{26}{6}}$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$4.33333...$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$4\frac{1}{3}$&lt;/span&gt; etc.&lt;/p&gt;&#xA;&lt;p&gt;One reason that students are not taught or tested this way: they tend to bomb horrifically, immediately. If a student who types &lt;span class=&quot;math-container&quot;&gt;$26\div6$&lt;/span&gt; into his calculator when he sees &lt;span class=&quot;math-container&quot;&gt;$\frac{26}{6}$&lt;/span&gt; - only because he was told to yesterday - then a teacher can feel that student understanding is confirmed. The student gets a good report card, moves on to the next topic or grade-level, can apply to a prestigious educational program, etc.&lt;/p&gt;&#xA;&lt;p&gt;The bad news - that the student is horribly confused by all the superficially similar expressions above - could have been uncovered immediately, but it almost never is. It's uncovered, say, when the student is in grade 12 trying to learn calculus and intelligent interpretations of all of those expressions is just assumed &lt;em&gt;even though it has never been tested&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You may have been in comparable situations in the past; perhaps you're in some now. Are you actively trying to collect evidence that student understanding and retention are vastly weaker than they appear so that, if they are, grades will plummet immediately and cause hideous political fights for you, your students, your job, your admin, etc.? Or do you give into the pressure to give students sufficient grades to move them towards the next educational goal posts?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;There are at least two ways main ways to avoid all this gobbledygook. The first is to &lt;strong&gt;teach actual math instead of gobbledygook&lt;/strong&gt;. In the realm of fractions, that means teaching that &lt;strong&gt;fractions are numbers&lt;/strong&gt; and in every single lesson and worksheet that involves fractions, to compare and contrast them with whole number(s), and to not introduce &lt;em&gt;anything&lt;/em&gt; with two fractions until the student can do the same thing with one or two whole numbers and one fraction. Don't bother with &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{6}-\frac{1}{6}$&lt;/span&gt; until the student can draw a picture for &lt;span class=&quot;math-container&quot;&gt;$2-\frac{1}{6}$&lt;/span&gt;. Don't bother with common denominator fraction comparisons until the student can easily use number lines to compare &lt;span class=&quot;math-container&quot;&gt;$\frac{19}{4}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$19$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I affirm those who say to read the &lt;a href=&quot;https://math.berkeley.edu/%7Ewu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;work of Hung-Hsi Wu&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The second way to avoid this mental gobbledygook is meaningful interleaving.&lt;/p&gt;&#xA;&lt;p&gt;If students are given a blocked (monotonous) set of exercises, such as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Find equivalent fractions for &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{5}{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{10}{8}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{6}{7}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{2}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{8}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then the normative response will be to mindlessly multiply the top and bottom number by, say, 3, over and over and never thinking about meaning. The students will just rush to finish with correct answers.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, if students are given an interleaved set of exercises:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Create a word problem and draw a picture for each of the following: &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}\times2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{1\times2}{3\times2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}\div2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2\div\frac{1}{3}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}=1\div3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then the students &lt;em&gt;must&lt;/em&gt; think about meaning. They might need a caring teacher to gradually build them up to this kind of challenge, but there's really no comparison as to which set of exercises will make students think more meaningfully. In addition to being pedagogically difficult, though, this is also motivationally hard as students and parents - and most teachers - tend to care most about what's on the test, and these kinds of exercises are rarely on tests.&lt;/p&gt;&#xA;&lt;p&gt;The short-run politics of determining the truth about student understanding is a nightmare. It lowers grades, so it prevents students from making honor roll or getting into an enriched math class or private school, it might make them lose a scholarship, it will cause tension in families, it will harm student status, parents scream &amp;quot;This isn't how I was taught math!&amp;quot;, people will be convinced you're a bad teacher because student grades are low, students try to transfer out of your class, admin gets angry, etc.&lt;/p&gt;&#xA;&lt;p&gt;I haven't figured out how to deal with such politics yet. :P&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2022-12-28T18:17:42.923" LastActivityDate="2022-12-28T18:17:42.923" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="15286" PostTypeId="2" ParentId="15278" CreationDate="2019-02-24T23:23:04.727" Score="8" Body="&lt;p&gt;As for research on fractions education... there's a TON.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for research on fractions education that rigorously measures cause and effect through randomized controlled experiments and long-term follow up... there is, I think, &lt;a href=&quot;https://ies.ed.gov/ncee/wwc/PracticeGuide/15&quot; rel=&quot;noreferrer&quot;&gt;little to nothing as of 2010&lt;/a&gt;. And, since the field of education rarely uses that kind of research design, I doubt there is any need to update that report.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If someone out there is more familiar with the research on fractions pedagogy than I am, please know I would love to be contradicted on anything I've written. :)&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2019-02-24T23:23:04.727" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15287" PostTypeId="2" ParentId="15237" CreationDate="2019-02-25T11:20:37.257" Score="6" Body="&lt;p&gt;An aspect that does not show up much in other's answer is: exercise &amp;amp; test design. To construct an exercise, it is very often quite useful to have more advanced knowledge, either to be able to choose the values correctly, or to know what will echo later courses.&lt;/p&gt;&#xA;&lt;p&gt;Some example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;knowing finite fields helps understanding that the coincidence between equality of polynomials and equality of their associated polynomial functions is not obvious, and needs a proof (it does not make it so much easy to convey, though),&lt;/li&gt;&#xA;&lt;li&gt;having learned quite some differential geometry, I know that constructing a smooth function that is zero on &lt;span class=&quot;math-container&quot;&gt;$(-\infty,0]$&lt;/span&gt; and positive on &lt;span class=&quot;math-container&quot;&gt;$(0,+\infty)$&lt;/span&gt; is not just for practice, but turns out to be very useful (much) later on,&lt;/li&gt;&#xA;&lt;li&gt;when I give a family of vectors for students to decide whether it is free, and whether it is generating, I am happy to have a set of available tools (e.g. dimension theory) that helps me knowing some of the answers in advance, and checking theirs,&lt;/li&gt;&#xA;&lt;li&gt;Knowing the partial fraction decomposition of a rational function helps designing exercises on sums, integrals, and systems of linear equation that will put these subject closer together.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Additional examples on the already discussed fact that knowing more gives you better perspective (and warns you from some misconception, and help you answer student's questions):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;knowing Weierstrass continuous nowhere differentiable functions helps me understand better the difference between continuous and differentiable functions; so does knowing that generic continuous functions are nowhere differentiable,&lt;/li&gt;&#xA;&lt;li&gt;knowing Taylor series helps understanding derivatives, in particular proving the formula &lt;span class=&quot;math-container&quot;&gt;$(fg)'=f'g+fg'$&lt;/span&gt; is much more transparent¹ when one defines &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; as the number (when it exists) such that &lt;span class=&quot;math-container&quot;&gt;$f(x+h)=f(x)+f'(x)h+o_{h\to 0}(h)$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;knowing that the antiderivatives of &lt;span class=&quot;math-container&quot;&gt;$x\mapsto e^{x^2}$&lt;/span&gt; cannot be expressed with usual functions helps warning students that finding explicit anti-derivatives (or solutions of differential equations) is not always possible, and helps explaining why we define new function names, and what math is really about.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;¹ &lt;strong&gt;Added in edit.&lt;/strong&gt; I was asked by email to give more details on this point, as well doing it here. With the first-order Taylor series definition of the derivative, whenever &lt;span class=&quot;math-container&quot;&gt;$f,g$&lt;/span&gt; are differentiable at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; we get&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA; (fg)(x+h) &amp;amp;= \big(f(x) + f'(x) h +o(h)\big) \big(g(x)+g'(x)h+o(h)\big) \\&#xA;  &amp;amp;= f(x)g(x) + \big(f'(x)g(x)+f(x)g'(x)\big)h + f'(x)g'(x)h^2 + o(h) \\&#xA;  &amp;amp;= (fg)(x) + (f'g+fg')(x) h + o(h)&#xA;\end{align*}&lt;/span&gt;&#xA;so that &lt;span class=&quot;math-container&quot;&gt;$fg$&lt;/span&gt; is differentiable at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, with the derivative as we know it. This is much more transparent than the usual trick of computing the ratio by adding and subtracting some quantity in order to factorize, as the cross products &lt;span class=&quot;math-container&quot;&gt;$f'g$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$fg'$&lt;/span&gt; appear naturally when developping, and the product &lt;span class=&quot;math-container&quot;&gt;$f'g'$&lt;/span&gt; that student might think should be the answer appears clearly, but as a second-order term. (Of course, beware that the higher-order Taylor formulas do not entail high differentiability, a misconception that this approach could unfortunately enforce.)&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2022-11-07T10:27:05.793" LastActivityDate="2022-11-07T10:27:05.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15288" PostTypeId="1" AcceptedAnswerId="15292" CreationDate="2019-02-25T18:38:29.167" Score="8" ViewCount="757" Body="&lt;p&gt;I am a pure mathematics PhD student and graduate teaching assistant at a major state university. During the summers here, teaching assistants are typically appointed to teach an entire course, rather than simply leading a separate discussion section. I have been appointed to teach a Precalculus course and am of course a bit surprised.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though the fall and spring semesters are quite standardized, we have much more flexibility over the summer. As I could modify explanations to my liking, I was considering perhaps taking a slightly more rigorous approach to teaching Precalculus. In particular, I am hoping to give more detailed explanations of objects that they will be working with. As an example, many students in my discussion sections will express belief that a line is nothing but an equation of the form &lt;span class=&quot;math-container&quot;&gt;$y = mx+b$&lt;/span&gt;. In reality, for &lt;span class=&quot;math-container&quot;&gt;$m,b \in \mathbb{R}$&lt;/span&gt;, a line with slope &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is the set of points &lt;span class=&quot;math-container&quot;&gt;$ L =\{(x,y):y = mx+b\}$&lt;/span&gt;. I look through many different books and even the official lecture notes of my university and they don't mention this explicitly. The closest to this is one that said a line is &lt;strong&gt;determined&lt;/strong&gt; by such an equation (this of course isn't wrong at all, just not the most complete). This issue I think this causes is when they try to understand why, to solve where two lines intersect, we must set their equations equal to eachother. Even with pictures, some students wonder how that geometric picture translates to the equations being set equal to one another. I personally feel that it could be alleviated through a more set-theoretical explanation since it could be explained via the sets. So to find points of intersection of two lines &lt;span class=&quot;math-container&quot;&gt;$L_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L_2$&lt;/span&gt;, one could simply look at elements of their intersection &lt;span class=&quot;math-container&quot;&gt;$L_1 \cap L_2$&lt;/span&gt; and see that such a point must satisfy both of their defining features, which means that the second coordinate, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, must be equal to &lt;span class=&quot;math-container&quot;&gt;$m_1x + b_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m_2x + b_2$&lt;/span&gt; (this would of course hypothetically be explained slowly and more carefully than I presented here).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another issue some students in my discussion sections have had is understanding the difference of when two functions are different or the same. For example, if &lt;span class=&quot;math-container&quot;&gt;$f(x) = \frac{(x-1)(x-2)}{(x-2)}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x) = x-1$&lt;/span&gt;, the students are likely to be convinced that they are the exact same, despite having different domains (the first not defined at &lt;span class=&quot;math-container&quot;&gt;$x = 2$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book and lecture notes will simply say that &lt;span class=&quot;math-container&quot;&gt;$x \neq 2$&lt;/span&gt; on the left hand margin as some kind of warning against the equality of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, yet it doesn't seem sufficient for students and often seems ad hoc to them. It would seem that it wouldn't be much harder for students to try to view the functions differently via the notations &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{R} - \{2\} \rightarrow \mathbb{R}$&lt;/span&gt; versus &lt;span class=&quot;math-container&quot;&gt;$g:\mathbb{R} \rightarrow \mathbb{R}$&lt;/span&gt;, so they have different domains, but also understanding that &lt;span class=&quot;math-container&quot;&gt;$f(x) = g(x)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{R} - \{2\}$&lt;/span&gt; (or &lt;span class=&quot;math-container&quot;&gt;$(-\infty,2) \cup (2,\infty)$&lt;/span&gt; however is nicer to notate it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me it seems that main education programs are (understandably) afraid to use any kind of rigor, or even set-theoretic notation, whatsoever, with some kind of all-or-nothing approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone here had any experience with teaching precalculus in this manner? Surely one would not want to go extreme and push rigor too hard on students, especially when they have likely not encountered such ways of thinking before hand, but rather a nice balance between the two. Are there any textbooks or references that follow this kind of style? Thanks in advance for any advice.&lt;/p&gt;&#xA;" OwnerUserId="11999" LastEditorUserId="64" LastEditDate="2019-02-26T23:24:10.193" LastActivityDate="2019-03-16T02:04:24.903" Title="A more rigorous approach to Precalculus" Tags="|mathematical-pedagogy|precalculus|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15289" PostTypeId="1" AcceptedAnswerId="15290" CreationDate="2019-02-25T18:40:53.770" Score="6" ViewCount="576" Body="&lt;p&gt;Can someone point me to papers indicating whether or not a midterm is an important part of a course?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect I can find many 'experiential anecdotes' that midterms are good/bad/moot but I would really like some concrete evidence (papers?) that a midterm is a useful  assessment element in a mathematics/computer science course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, what would happen if I replaced midterms with many small quizzes? Do we get a change in student learning?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: If it helps, I am at a Canadian University and at our institution, a course grade break down routinely consists of 10% assignments, 30% for a 2 hour test assessment (a midterm in the middle of a course) and a 2.5 hour 60% final exam assessment after the course lecturing. These percentages fluctuate a bit from course to course but are relatively stable. I'm thinking about removing the midterm and redistributing the weight to more quizzes, assignments and other components like this.&lt;/p&gt;&#xA;" OwnerUserId="12000" LastEditorUserId="12000" LastEditDate="2019-02-26T21:32:57.040" LastActivityDate="2019-02-26T21:32:57.040" Title="Midterm in Mathematics Courses" Tags="|reference-request|course-design|assessment|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15290" PostTypeId="2" ParentId="15289" CreationDate="2019-02-25T19:23:06.393" Score="17" Body="&lt;p&gt;&quot;Cheating Lessons&quot; by James M. Lang argues (and has many references to back up) the claim that smaller, more frequent, lower stakes assessment both improves student learning outcomes and decreases the frequency of cheating.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2019-02-25T19:23:06.393" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15292" PostTypeId="2" ParentId="15288" CreationDate="2019-02-25T21:59:56.977" Score="18" Body="&lt;p&gt;My experience in teaching Precalculus is that you will be surprised about what you actually need to teach compared to what you thought you needed to teach.&lt;/p&gt;&#xA;&lt;p&gt;For example, if you ask students to compute and simplify the difference quotient for the function &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt;, even after several lessons and group work, you may be surprised that &lt;strong&gt;by far&lt;/strong&gt; the most difficult part of this is the idea that &lt;span class=&quot;math-container&quot;&gt;$f(x+h)$&lt;/span&gt; is an expression and that it is equal to another expression. You will get student work that is confusing to you at first. For example, the following was real work from a student:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x) = \frac{f(x+h) - f(x)}{h}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x^2 + h) - f(x^2)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{h}{h}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then you will either dismiss the student as hopeless or you will identify that the student has two critical skill deficiencies (the ones discussed in two of the all-time great MESE questions &lt;a href=&quot;https://matheducators.stackexchange.com/q/926/11&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/q/7964/11&quot;&gt;here&lt;/a&gt;) that can be overcome.&lt;/p&gt;&#xA;&lt;p&gt;As a result I would offer the advice to &lt;strong&gt;not try anything extraordinary on your first time through&lt;/strong&gt;. Instead, take all that extra time and effort and use it to think about why students make the errors they make and resolve to restructure the course to teach them the things they actually lack (for me, I would restructure the course to be basically a class on mathematical maturity, the equals sign, and the idea that students can tell whether things are true without asking the teacher).&lt;/p&gt;&#xA;&lt;p&gt;After you teach the class once, you may be inspired to try something new and radical and that would be great. However, postpone any unconventional solution to a problem until after you have seen the problem.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-02-25T21:59:56.977" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15293" PostTypeId="1" AcceptedAnswerId="15294" CreationDate="2019-02-26T00:35:08.520" Score="5" ViewCount="378" Body="&lt;p&gt;My kid, 5 years old, knows addition, subtraction and multiplication now, of course, in a basic level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also he understands that parentheses means &quot;whichever inside shall be computed first&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I try explain the rules of &quot;taking off the parentheses&quot;, he seems thinks &quot;so long as we can compute, why take it off?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;span class=&quot;math-container&quot;&gt;$10 - (5-3) = 10 - 2 = 8$&lt;/span&gt;, why shall it be done as &lt;span class=&quot;math-container&quot;&gt;$10 - (5-3) = 10 -5 +3 = 5 + 3 = 8$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="9059" LastEditorUserId="2083" LastEditDate="2019-02-26T06:54:28.230" LastActivityDate="2019-10-26T10:24:02.453" Title="How to explain the motivation of parentheses in addition, subtraction and multiplication?" Tags="|mathematical-pedagogy|primary-education|arithmetic|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15294" PostTypeId="2" ParentId="15293" CreationDate="2019-02-26T03:44:08.317" Score="10" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Why take off the parentheses?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Because sometimes &quot;taking off the parentheses&quot; results in an easier to calculate expression.  For example,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$123456789-(-9876543210+123456789)$$&lt;/span&gt;&#xA;is easier to evaluate by using &lt;span class=&quot;math-container&quot;&gt;$123456789+9876543210-123456789=9876543210$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2019-02-26T03:44:08.317" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15295" PostTypeId="2" ParentId="15288" CreationDate="2019-02-26T14:56:31.310" Score="1" Body="&lt;p&gt;Pre-calc that includes a lot of work on function domains is not so strange.  I took a class like that in 80s.  It was even called &quot;functions&quot;.  Covered a lot of domain, range stuff, and inequalities and a little derivatives and antiderivatives.  Really, if you just look at a few different courses, texts, you will see this--try a GT track at a high school--won't need to ask the web if this is something new to the world. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that you need to think about the qualities of your students, not just the coverage of the course.  It's not something new to the world to have a course with the content you mention.  Has really been the trend since the 60s or so.  However, the content that stronger kids take may not be as suitable for weaker kids.  You may need to do more patching up of bad algebra 2 and trig and a little prep for calc.  And less domain stuff (like really, how often is x = 2 exactly anyways!)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, that you didn't discuss the student quality shows a big whole in your thinking (worse than missing x = 2).  Teaching involves content, recipients, and methods.  You are ignoring a lot of &quot;z variables&quot; and only thinking of x versus y, if you say &quot;I'm a pure math guy, let's cover more content, more strictly&quot;.  Now kids at a major state uni aren't going to be bottom of the barrel.  At the same time, if they are taking pre-calc in college, that is remedial in this day and age.  Calculus is the expected freshman college starting course.  And better kids validate that, even.  But not even being in calc for frosh year?  Those are weaker kids.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to consider how strong the kids are versus if you can add more load on their backs.  An easy thing to look at would be how they do on the other content (e.g. in standard content courses).  If they are crushing it, sure add more plates on the bar.  If they are struggling, then absolutely don't add more load--are you crazy?  If they are kind of doing OK (not too much of a gut, not getting killed), than I would still lean to not toughing the course up, especially given your own low experience and status.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This doesn't even address what Dave said about summer section capabilities.  I'm not sure, might find they are stronger or weaker (need to check).  Plus I actually find students can often concentrate easier on a single course in the summer than during the year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on the likely weaknesses of your students (and yourself, as being a new relatively new teacher), I recommend not to change the content when given more ability in a summer section.  Concentrate on working on your pedagogy instead.  Teach the rock bottom standard course and see if you can horse the kids up on that, better.  You need to think about your own progression as a teacher and not change too much at once.  It sounds like you will have more responsibility in the summer. Learn to deal with that first, versus changing the content also.&lt;/p&gt;&#xA;" OwnerUserId="12001" LastActivityDate="2019-02-26T14:56:31.310" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15296" PostTypeId="2" ParentId="7964" CreationDate="2019-02-26T15:13:03.150" Score="2" Body="&lt;p&gt;A strong pre-algebra class can do a lot to deal with this problem.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kids need to laboriously &quot;do same thing to both sides of an equation&quot; over and over.  And writing down all the steps.  Not &quot;moving 2 to other side of equation and making it negative&quot;.  But writing original equation, then writing it with a -2 on each side (doing same thing to each side), then evaluating it by subtracting the 2 (to presumably eliminate it on left side).&lt;/p&gt;&#xA;" OwnerUserId="12001" LastActivityDate="2019-02-26T15:13:03.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15297" PostTypeId="1" CreationDate="2019-02-26T21:33:59.867" Score="8" ViewCount="180" Body="&lt;p&gt;This is fairly open ended, so I understand if people consider this to be off-topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested in creating math lessons where two groups each learn how to use a different simple math skill, and then the groups are combined and challenged with problem types that require both skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a simple example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Group A&lt;/strong&gt; would learn the definition of factorial and run some easy factorial problems.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Group B&lt;/strong&gt; would learn that if you have certain fractions (like &lt;span class=&quot;math-container&quot;&gt;$\frac{8 \cdot 15}{5}$&lt;/span&gt;) it's easier to cancel first and then multiply.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Combined Groups&lt;/strong&gt; form with a mix of people from both groups A and B.  They are given problems like evaluating &lt;span class=&quot;math-container&quot;&gt;$\frac{10!}{8!}$&lt;/span&gt; where having both skills makes running the problem pretty easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a specific name for this sort of collaborative learning? What are some fun/interesting problems to run in this way? &lt;/p&gt;&#xA;" OwnerUserId="8642" LastEditorUserId="143" LastEditDate="2019-03-11T10:16:57.467" LastActivityDate="2019-03-11T10:16:57.467" Title="Math Lessons with Two Parts and a Combination" Tags="|mathematical-pedagogy|terminology|classroom-management|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15298" PostTypeId="1" AcceptedAnswerId="15307" CreationDate="2019-02-26T23:22:20.633" Score="8" ViewCount="925" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The amplitude of a sinusoid is the distance from its axis to a high&#xA;  point or a low point.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When we read this, it follows that Tan and Cot don't have an amplitude. Nor do SEC or CSC. Now, I'm in an odd situation. (Note, I work in a high school, and function as an in house tutor and occasional sub). The trig classwork is using this term as in y=2tan(x) for example. The students would be asked to identify the 2 as the &quot;amplitude&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given my relationship to the teachers, I am there to support them, not criticize them or appear a threat. And I am at the point in my life where I don't need to be right or adhere to the past (Like what happened to the septagon? It worked well, and fit into the series sept, oct, non, dec.) The question is - &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Am I doing damage by using non-standard terms with my students? In the same manner that I talk about 1/0 not equalling infinity, but rather, as the angle approaches say 90 degrees, Tan approaches infinity, do I need to awkwardly say &quot;what your teacher calls 'amplitude'&quot; / &quot;The 'A' term in this equation&quot; and not feed into the misnomer? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can use the term intelligently with no sarcasm if needed -&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we look at TAN and COT, we see the parent functions have 2 points, tan(45)=1 and tan(-45)=-1. A translated function would see these points shift vertically, so the difference in Y value is consistent, twice the new amplitude. &#xA;Similarly, when graphing SEC/CSC, I am ok to identify the space between the graph segments as twice the amplitude. All in the spirit of not going against what the teachers are choosing use as a colloquialism. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit - an example, for context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KUEAC.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KUEAC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was from a worksheet. Which was from the section of chapter that specifically was beyond SIN &amp;amp; COS. To be clear, if it had all the trig functions, I'd put the amplitude for SIN and COS and N/A for the other 4. As of this moment, I don't know if the teachers created it, or if this was from a third party. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-02-27T12:33:38.410" LastActivityDate="2019-03-11T21:23:00.360" Title="&quot;Amplitude&quot; of Tan and Cot functions" Tags="|terminology|trigonometry|" AnswerCount="3" CommentCount="21" ContentLicense="CC BY-SA 4.0" />
  <row Id="15299" PostTypeId="2" ParentId="15297" CreationDate="2019-02-27T02:59:13.250" Score="7" Body="&lt;p&gt;This sounds something like a &quot;Jigsaw&quot; activity &lt;a href=&quot;http://www.readingrockets.org/strategies/jigsaw&quot; rel=&quot;nofollow noreferrer&quot;&gt;(1)&lt;/a&gt;, &lt;a href=&quot;https://www.teachhub.com/jigsaw-method-teaching-strategy&quot; rel=&quot;nofollow noreferrer&quot;&gt;(2)&lt;/a&gt;, &lt;a href=&quot;https://serc.carleton.edu/NAGTWorkshops/coursedesign/tutorial/jigsaw.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;(3)&lt;/a&gt;.  The basic idea is as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Divide your students into groups of equal size; say, four students in each group.  For specificity, lets label these groups &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, and so on.  Don't actually have them get into these groups yet.  In each group, assign each student to a different task, say tasks &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Have the students work individually to learn the task which they were assigned.  Give them some specific instructions and some relatively simple or straight-forward problems to complete in order to become familiar with their given task.&lt;/li&gt;&#xA;&lt;li&gt;Have all of the students who have been assigned to the same task collaborate with each other in an &quot;expert group&quot; to master that particular task.  That is, put all of the &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;'s together into a group, and have them learn the thing that they need to learn.  Same with the &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;'s, &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;'s, &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;'s, etc.  For example, you might have one group learn about Pascal's triangle, another group think about combinations, a third consider expressions of the type &lt;span class=&quot;math-container&quot;&gt;$(a+b)^n$&lt;/span&gt;, and so on.  Give the group more complicated tasks (within the area that they are meant to master) to complete together.&lt;/li&gt;&#xA;&lt;li&gt;After a reasonable amount of time has passed, have the students get into the original groups &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, etc.  Give these groups a task which requires the expert knowledge obtained by each member of the group.  For example, ask them to prove the binomial theorem.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There is a nice diagram on the page linked as (3), above:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://d32ogoqmya1dw8.cloudfront.net/images/NAGTWorkshops/coursedesign/tutorial/jigsaw_diagram.jpg&quot; alt=&quot;Jigsaw schematic&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The links I gave above give quite a bit more detail, but they seem to be more geared towards an elementary audience and reading comprehension-style exercises.  However, I am sure that a reasonably clever instructor could figure out how to make it more mathy. ;)&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2019-02-27T13:21:33.793" LastActivityDate="2019-02-27T13:21:33.793" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15300" PostTypeId="2" ParentId="15298" CreationDate="2019-02-27T14:32:29.263" Score="0" Body="&lt;p&gt;This looks to me like a balance between two concerns:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Tradition: Students should learn correct and standard terminology.&lt;/li&gt;&#xA;&lt;li&gt;Professionalism: Everyone involved should show a united front to the students.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Suppose the teacher, Mr. Flample, wants to call the coefficient of a cosecant function the &quot;&lt;strong&gt;flamplitude&lt;/strong&gt;.&quot; You have two choices of what to say to the student:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;Which number does your teacher call the 'flamplitude'?&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;Which number do we call the 'flamplitude'?&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This is not a subtle distinction, nor is it a difficult choice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Choosing the first option is actively giving the student a reason to disengage from the class. That's bad for the student. Don't help students develop excuses to not learn mathematics. &lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2019-02-27T14:32:29.263" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15301" PostTypeId="2" ParentId="13073" CreationDate="2019-02-27T20:33:36.097" Score="6" Body="&lt;p&gt;One reason for not dismissing the complaint that graduate math textbooks do too much handwaving is that some of the authors of graduate textbooks have the same complaint.  For example, in his introduction to &lt;strong&gt;Stochastic Integration Theory&lt;/strong&gt;, Peter Medvegyev writes:  &quot;[The] books concentrating on the general theory were, for me, . . . a bit sketchy.  I very often had quite serious problems trying to decode what the ideas of the authors were, and it took me a long time, sometimes days and weeks, to understand some basic ideas of the theory.  I was nearly always able to understand the main arguments but, looking back, I think some simple notes and hints could have made my suffering shorter.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematicians who would publish their &quot;notes and hints&quot; for &quot;textbooks written by masters&quot; as some sort of systematic commentary would be doing a great service to the profession.&lt;/p&gt;&#xA;" OwnerUserId="6625" LastActivityDate="2019-02-27T20:33:36.097" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15304" PostTypeId="2" ParentId="15297" CreationDate="2019-02-28T13:37:23.493" Score="2" Body="&lt;p&gt;I did something like a jigsaw activity in an undergraduate 300-level topics in combinatorics course. I split the class into two groups, and each was assigned a reading and several exercises related to Stirling numbers. One group worked with Stirling numbers of the first kind and the other group worked with Stirling numbers of the second kind. The next class, the two groups split in half and matched with half of the other group. Each of these mixed groups took turns presenting their material (within the mixed group), and then the two mixed groups were given another assignment including proofs of identities involving both kinds of Stirling numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also use similar strategies in an undergraduate exploration/discovery-based math course, but in a more informal way, as the course is completely student-driven. In that course, if I find a student struggling to prove something, instead of giving them a hint, I'll find another student not already busy, give them an idea (which is a hint for the first student), and then suggest they work together. In that class, I spent most of my in-class time looking for opportunities to facilitate things like this.&lt;/p&gt;&#xA;" OwnerUserId="7707" LastActivityDate="2019-02-28T13:37:23.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15306" PostTypeId="2" ParentId="12146" CreationDate="2019-03-01T21:24:12.053" Score="4" Body="&lt;p&gt;There isn't enough information to answer the question, but I'll speculate, based on what often happens in the world of math education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;People who enjoy &amp;amp; teach math are often shell-shocked at how many people consider math to be a meaningless, tear-filled, horrible subject &lt;em&gt;in which effort is pointless&lt;/em&gt;. My guess is that most students have given up on trying because 287 adults before you have told them to try harder, then given them rote or inappropriately difficult work... &lt;em&gt;in which trying doesn't make a difference&lt;/em&gt;. See my &lt;a href=&quot;https://matheducators.stackexchange.com/a/15284/7151&quot;&gt;recent post on fractions&lt;/a&gt; for examples of how hard work combines with rote gibberish to demoralize and confuse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect many of your students view you as the 288th adult who's giving them more rote gibberish and empty pep talks. Some likely signs of this: your lessons are very heavy on symbols and students (rightly?) don't believe you when you say they should pay attention and try harder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their response: The bare minimum effort to: (a) get you off their backs and (b) avoid summer school or repeating the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider doing gentle but probing &quot;math biography&quot; interviews with some of your students. Tell them you need their input to become a better teacher and want to begin by understanding when they stopped trying. Ask them when math became a painful subject for them. In kindergarten, math is for many children the favorite academic subject; among the general population, about 37% of people specifically use the word &quot;hate&quot; to describe how they feel about math. [Many others use other negative words to describe their feelings.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Expect a lot of stories such as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;I loved math until grade 4. Then my teacher didn't really teach us division or fractions and never tested us on anything. Sometimes we'd go weeks at a time without a math class, months without a test. I've been terribly confused in most math classes since then. I noticed that studying really hard didn't help, so I just don't try hard any more because it's not fun.&quot; Note that this is likely true: given terrible confusion regarding grade-4-level division, does it matter if you study really hard to master proportional reasoning and ratios? Of course not. Hard work didn't matter then even though adults falsely told them it would. Why are you different from those adults?&lt;/li&gt;&#xA;&lt;li&gt;&quot;My teacher made the class do all these timed exercises. I was the slowest one in the class no matter how hard I tried and that's when I realized I was stupid.&quot; Notice that maybe the student just needed to learn, say, how &lt;span class=&quot;math-container&quot;&gt;$6+6=\_\_\_$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$6+7=\_\_\_$&lt;/span&gt; are related, then they could have stopped counting by ones to do all calculations. But if you don't know such relationships, then, well, math = counting by ones, which is slow and makes people feel stupid. It's a hopelessly weak foundation for future math. Are you trying to build on their hopelessly weak foundations?&lt;/li&gt;&#xA;&lt;li&gt;&quot;The teacher would make me come to the front of the class and tell me to do math I couldn't do and that was humiliating. I hate math. So just tell me how to pass this class so I don't have to go to summer school.&quot; How often do your students feel &lt;em&gt;great&lt;/em&gt; about what they've learned and accomplished in your class?&lt;/li&gt;&#xA;&lt;li&gt;&quot;Math is just all these steps. I just memorize those steps. I'm the memorizing &lt;em&gt;type&lt;/em&gt;. I'm not interested in understanding because I don't think there is anything to understand. Just tell me the steps and I'll do them.&quot; Are you sure you're &lt;em&gt;really&lt;/em&gt; testing for understanding? Are you really convincing your students that math is understandable?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2019-12-03T04:33:02.260" LastActivityDate="2019-12-03T04:33:02.260" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15307" PostTypeId="2" ParentId="15298" CreationDate="2019-03-02T07:04:18.913" Score="5" Body="&lt;p&gt;I used to teach my PreCalc students an early unit on the Function Toolbox. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt;. Something happens to it when you add a term: &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2 + a$&lt;/span&gt;. Also if you change it to &lt;span class=&quot;math-container&quot;&gt;$f(x) = (x - b)^2$&lt;/span&gt;. And how ‘bout &lt;span class=&quot;math-container&quot;&gt;$f(x) = cx^2$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now change the function to &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^3$&lt;/span&gt;. And &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sqrt{x}$&lt;/span&gt;. And &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sin(x)$&lt;/span&gt;. And &lt;span class=&quot;math-container&quot;&gt;$f(x) = \frac1x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point is that the constant that is traditionally called amplitude in trig class actually does something more general, and to a ton of functions, even to those with asymptotes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think I fall on your side when calling it “amplitude,” but would love students to know what &lt;span class=&quot;math-container&quot;&gt;$2f(x)$&lt;/span&gt; does to &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, regardless of the function. &lt;/p&gt;&#xA;" OwnerUserId="9241" LastEditorUserId="11" LastEditDate="2019-03-11T21:23:00.360" LastActivityDate="2019-03-11T21:23:00.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15308" PostTypeId="2" ParentId="15298" CreationDate="2019-03-02T15:54:09.960" Score="0" Body="&lt;p&gt;The other teach is wrong.  But since you are in a support role, I would just ignore it and roll onward.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surely there are a bazillion other things to work on, to help the kids with.  It's not like 50% of your time is on this wrong amplitude.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it still bugs you the wrongness, you could say something like &quot;I don't know amplitude so well--have Mrs. WrongAmp help on that one.  Just circle it and move on, kid.  I'll help with next problem.&quot;  That way you don't endorse the wrongness or tie yourself in knots promulgating it.  But you still move along and do something useful.  And don't make students lose faith in WrongAmp lady.&lt;/p&gt;&#xA;" OwnerUserId="12015" LastActivityDate="2019-03-02T15:54:09.960" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15309" PostTypeId="1" CreationDate="2019-03-02T22:28:41.200" Score="3" ViewCount="207" Body="&lt;p&gt;I've got a very smart and capable student who needs to finish his college degree. He has severe limitations at this time. Because of a severe illness, he is unable to write and is now deaf and blind. However, he still retains enough sight that he can read on and use the computer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are trying to figure out how he can finish his degree requirements. We are looking for college-level software that can aid him in reading, writing, and calculating math. He will need to finish Business Math, Calculus, and Statistics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What software could aid us in making this accomplishment spossible for him&lt;/p&gt;&#xA;" OwnerUserId="12017" LastEditorDisplayName="user507" LastEditDate="2019-03-03T20:36:47.277" LastActivityDate="2019-07-31T22:46:22.013" Title="What software and technology could help this deaf and blind student do math?" Tags="|technology-in-education|software|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15310" PostTypeId="1" CreationDate="2019-03-02T22:55:39.497" Score="3" ViewCount="188" Body="&lt;p&gt;I've taught how to use &lt;span class=&quot;math-container&quot;&gt;$\epsilon, \delta$&lt;/span&gt; to prove that a function is continuous at a point, and I'm about to teach how to prove that a function is uniformly continuous over an open interval.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually, the examples I can think of that seem easy enough on the outside, require some algebraic trickery that might make it seem more daunting than it needs to be, and may inspire a &quot;damn, this is too difficult&quot; mentality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there some examples of functions that are almost painfully straightforward to give a soft introduction to these, that I may increase the difficulty more smoothly?&lt;/p&gt;&#xA;" OwnerUserId="796" LastEditorUserId="498" LastEditDate="2019-03-03T17:30:20.940" LastActivityDate="2019-03-03T17:30:20.940" Title="When teaching someone how to prove a function is uniformly continuous, using epsilon/delta, which example would be among the simplest?" Tags="|calculus|limits|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15311" PostTypeId="2" ParentId="15309" CreationDate="2019-03-03T00:24:02.067" Score="0" Body="&lt;p&gt;Both edX and Coursera have free, quality courses:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.edx.org/learn/calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;edX Calculus&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.coursera.org/courses?query=calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Coursera Calculus&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.edx.org/learn/statistics&quot; rel=&quot;nofollow noreferrer&quot;&gt;edX Statistics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.coursera.org/courses?query=statistics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Coursera Statistics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/Suqjj.jpg&quot; width=&quot;300&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Image from &lt;a href=&quot;https://www.coursera.org/learn/single-variable-calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Ghrist's Coursera course&lt;/a&gt;.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Concerning accessibility, they are both US-based companies and subject to government-mandated requirements (as @BenCrowell details). Here are their accessibility statements:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.edx.org/accessibility&quot; rel=&quot;nofollow noreferrer&quot;&gt;edX accessibility&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://learner.coursera.help/hc/en-us/articles/209818883-Coursera-s-accessibility-policy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Coursera accessibility&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-03-03T00:24:02.067" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15312" PostTypeId="2" ParentId="15310" CreationDate="2019-03-03T00:38:50.587" Score="6" Body="&lt;p&gt;I think this cannot be understood without a contrasting example where it fails.&#xA;So perhaps, in addition to a linear function as suggested by @paw88789, consider &lt;span class=&quot;math-container&quot;&gt;$f(x) = \frac{1}{x}$&lt;/span&gt; over the open interval &lt;span class=&quot;math-container&quot;&gt;$(0,1)$&lt;/span&gt;. &#xA;It is continuous over that interval, but not uniformly continuous.&#xA;Fix an &lt;span class=&quot;math-container&quot;&gt;$\epsilon &amp;gt; 0$&lt;/span&gt;; then for any &lt;span class=&quot;math-container&quot;&gt;$\delta &amp;gt; 0$&lt;/span&gt; one can&#xA;arrange the difference in &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;-values to exceed &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; by getting&#xA;close enough to &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-03-03T00:38:50.587" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15313" PostTypeId="1" CreationDate="2019-03-03T03:39:06.913" Score="2" ViewCount="138" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;If words of command are not clear and distinct, if orders are not thoroughly understood, the general is to blame. But if his orders are clear, and the soldiers nevertheless disobey, then it is the fault of their officers. - &lt;em&gt;Sun Tzu, The Art of War&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Situation:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am helping a calculus student, and it seems the professor of the class has &lt;a href=&quot;https://math.stackexchange.com/questions/3129816/surface-area-of-revolution-on-an-area-crossing-the-axis/3132328#3132328&quot;&gt;given a homework item&lt;/a&gt; on computing the volume and surface area of a solid revolution obtained by rotating a region about an axis that cuts the region's interior such that Pappus's centroid theorems don't apply. There are no prior examples in class discussions or class tutorials, and both the student and I are unable to find examples in &lt;em&gt;Calculus by James Stewart&lt;/em&gt; or the student's textbook, &lt;em&gt;Calculus by George B. Thomas&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student has been told that a classmate has asked the professor about this, and the professor responded, 'There is nothing wrong with that item.'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently, examples of these are very difficult to find, and according to the answer given in the question linked above,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;'There is not an unanimous interpretation of what is the rotation solid in this case.'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My questions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is this fair to give this to students without prior &lt;strong&gt;definitions&lt;/strong&gt; AND without prior &lt;strong&gt;instructions&lt;/strong&gt; that it is up to the students to research the definitions?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Where can I read about this both for maths and maths education?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;For elementary maths: What elementary textbooks give definitions for situations like these?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For advanced maths: What advanced textbooks discuss geometric, algebraic and physical interpretations for these?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For maths education: What does the theory of teaching solids of revolution to beginning calculus students have to say in this case?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;I should note&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This isn't the first time the student has encountered a homework item without a definition.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In the semester prior to this, students were expected to compute the complex exponential &lt;span class=&quot;math-container&quot;&gt;$e^z$&lt;/span&gt; in a homework item even though they had only the definition of &lt;span class=&quot;math-container&quot;&gt;$e^{iy}$&lt;/span&gt;, that is they were not told that &lt;span class=&quot;math-container&quot;&gt;$e^{x+iy} := e^x e^{iy}$&lt;/span&gt;. The student e-mailed the professor, who replied that the student was right that the students were not given this definition. This may seem like a small thing, but mistakes like this could lead to fallacies like because '&lt;span class=&quot;math-container&quot;&gt;$e^{iz} = \cos z + i \sin z$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$Im(e^{iz}) = \sin z$&lt;/span&gt;', in my opinion. (Obviously, complex analysis assumes calculus among others, so the class prior taught only complex variables, along with some probability and linear algebra, which had a little calculus but only up to what they knew in secondary school)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The class also has an &lt;a href=&quot;https://math.stackexchange.com/questions/3122373/a-rigorous-yet-intuitive-summary-of-inflection-and-critical-points-for-beginning&quot;&gt;unresolved issue about critical or inflection points&lt;/a&gt;, though the calculus student in question has yet to contact the professor, and they are on break for the upcoming weeek.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2019-03-03T11:50:10.097" LastActivityDate="2019-03-03T17:04:10.657" Title="Are questions on overlapping solids of revolutions without prior definitions and instructions fair given that there are divided interpretations?" Tags="|mathematical-pedagogy|calculus|reference-request|geometry|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15314" PostTypeId="2" ParentId="15313" CreationDate="2019-03-03T07:03:53.353" Score="7" Body="&lt;p&gt;I take it from the content that this student is at university level. Based on that, I'd say:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Yes, it is fair to give questions without having given examples before.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;University is &lt;em&gt;higher education&lt;/em&gt;. A key point of higher education, as opposed to further education, is learning to think for yourself. You don't learn to do that by never being given anything to think through for yourself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The student does not need the definition&#xA;&lt;span class=&quot;math-container&quot;&gt;$$e^{x+iy}=e^x e^{iy}$$&lt;/span&gt;&#xA;because they have already learned, years earlier, that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^{a+b}=x^a x^b.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the volume of revolution: there may not be one single definition in use in the world, but there will be one definition in use in the class. The student should work from the definition they have been given. If theorems don't apply, then work from the definition from first principles instead. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Employers are not looking for graduates who can only follow previously defined algorithms. They can get robots to do that. Or high school leavers. They want graduates who meet a situation they haven't been programmed to deal with and at least try to find a solution.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-03-03T07:03:53.353" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="15315" PostTypeId="1" CreationDate="2019-03-03T13:24:21.923" Score="4" ViewCount="240" Body="&lt;p&gt;I am not a native math teacher. I have a question related to a terminology when solving an algebraic equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume that we are solving some complicated equation like &lt;span class=&quot;math-container&quot;&gt;$x^{3}-\sqrt{1-x^{2}}-\frac{1}{x+1}=2$&lt;/span&gt;. After doing a bunch of algebraic manipulation, we come up with some finite possible solutions: &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt;,.... Now, because I see that &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; is not a solution (for example, by inserting into the equation to check). Then I might say that &quot; &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; does not satisfy the equation, so it is not a solution.&quot;. However  I don't like this  sentence. I would prefer to use &quot; &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; is excluded by not satisfying the equation&quot;. Is it ok with this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="12021" LastActivityDate="2019-03-16T00:28:27.243" Title="The terminology for an excluded solution" Tags="|terminology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15316" PostTypeId="2" ParentId="15315" CreationDate="2019-03-03T13:54:33.070" Score="5" Body="&lt;p&gt;As noted in a comment, the word for this is &quot;extraneous solution&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I explain it to high school students this way:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you are trying to remove the square root, i.e. by squaring both sides of an equation, you risk introducing extraneous solutions, e.g. &lt;span class=&quot;math-container&quot;&gt;$ 1 = -1$&lt;/span&gt; (not true), whereas after squaring, &lt;span class=&quot;math-container&quot;&gt;$1 = 1$&lt;/span&gt; is indeed true. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure it can arise from other manipulations, but this one seems most common. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YHZgI.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YHZgI.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll add: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find that both for students and myself, graphing almost always adds to an understanding of the answer involved. I can analyze your equation and see that for the fact that &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; is the highest &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; can go (higher, and you have the square root of a negative number, same for -1, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; cannot be lower). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That immediately tells me that the equation can never be true. For some, this might be obvious, but for the fraction of students who are more visual, the graph really makes the point. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-03-16T00:28:27.243" LastActivityDate="2019-03-16T00:28:27.243" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15317" PostTypeId="2" ParentId="15313" CreationDate="2019-03-03T17:04:10.657" Score="-1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Boy that's a leading question:  &quot;given that there are divided interpretations?&quot;  Seems like you are fishing for agreement or endorsement versus asking a question.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;My practical advice is to just help out versus second guessing the teach.  Even if the teach is actually wrong, then just say you can't help on that problem and move to a next one. But you're not helping anything with this &quot;I'm a TA who disagrees with the professor&quot; (and BOY, do we hear a lot of these types of questions).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For that matter, why not go talk to the prof yourself?  Why come running to the damned Internet?  Just engage in real life.  (Even if you are not an assigned TA, but a tutor...then you have to realize there will come times it is difficult to be a tutor, given specific teachers.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also, you don't give us enough info about the specific homework problem (what object, what rotation, etc.) to see if the question is ambiguous or just difficult.  Or even how difficult--maybe it's not as hard as you think.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The last example just seems like more of a rant about the professor.  If you are going to tutor people and not engage directly with the professors, you are going to have these issues.  You don't know everything that was said in class, for example.  Doesn't mean you can't tutor.  Just that you need to realize that it's not a perfect process.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12022" LastActivityDate="2019-03-03T17:04:10.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15319" PostTypeId="1" AcceptedAnswerId="15320" CreationDate="2019-03-05T08:54:54.310" Score="0" ViewCount="484" Body="&lt;p&gt;I have heard that Peter Lax's linear algebra book is the hardest you can find on the subject, but that it is the greatest linear algebra text in the world.  But I also hear great things about Gil Strang's book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do I go about choosing between the two?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's an idea in my mind:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Choose Strang if I'm headed towards engineering mathematics, else choose Lax if I'm headed towards more theoretical, say, functional analysis.&lt;/p&gt;&#xA;" OwnerUserId="12029" LastActivityDate="2019-03-05T16:10:52.423" Title="Gil Strang or Peter Lax? Which linear algebra book to use?" Tags="|textbooks|linear-algebra|" AnswerCount="1" CommentCount="8" ClosedDate="2019-03-05T18:54:10.213" ContentLicense="CC BY-SA 4.0" />
  <row Id="15320" PostTypeId="2" ParentId="15319" CreationDate="2019-03-05T16:10:52.423" Score="1" Body="&lt;p&gt;Use the easier text. You can go back if needed and do more advanced, trickier work.  But as a self studier (implied by your question), you need to make sure this is a hill you can climb.  Remember Feynman was a big fan of Calculus for the Practical Man...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually I would advise to see if there is a text with the answers.  Because for self studiers, you lack much feedback, instruction.  So having the answers is very helpful in checking your drill, motivating yourself.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even better is if you could find a text that is &quot;programmed instruction&quot;.  You get huge amounts of feedback and they are designed for self studiers.  I don't know of a specific one, but do a library search.  &lt;/p&gt;&#xA;" OwnerUserId="12030" LastActivityDate="2019-03-05T16:10:52.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15321" PostTypeId="2" ParentId="2196" CreationDate="2019-03-06T16:02:01.430" Score="1" Body="&lt;p&gt;My answer/suggestion is in the form of a question. I'm new here, so please forgive me if this is not with the zeitgeist:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm curious whether anyone out there has ever tried a version of the following idea:  have the students grade each other's homework, by hand and eye with a red pen, but penalize the grader if the work of the person he's grading does not correlate to his exam score on a proctored exam.  The exams would of course need to closely resemble the homework!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Novel here would be the new cost: in this case any significant gap between a student's performance on the homework and the grader's evaluation of that homework would be expensive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would place a premium on accurate grading, which is a great way to learn math!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the homework would be worth points and a poor performance would garner fewer points, and the exams would still be worth what the exams are worth.&lt;/p&gt;&#xA;" OwnerUserId="12034" LastActivityDate="2019-03-06T16:02:01.430" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15322" PostTypeId="1" AcceptedAnswerId="15327" CreationDate="2019-03-06T18:56:34.707" Score="2" ViewCount="267" Body="&lt;p&gt;I have browsed the website a lot and I encountered many similar questions but not a question that asks the same question as I intend to. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In introductory undergraduate classes in Analysis, usually, Rudin is assigned which has a pretty straightforward way of introducing topics. The author provides the definitions, then concludable theorems along with their proofs and at the end exercises.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My main concern is with the proof for the theorems. When I am going through the proof for the theorems, what should my aim be? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should I be making sure that I understand each step, that's to say, that I will be able to understand and explain the proof &lt;strong&gt;provided&lt;/strong&gt; that I am given the proof?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;OR&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should I try to read the proof again and again so as to be able to rebuild the proof from memory and intuition alone (and of course understand the proof in the process)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Possible Constraint I would like the reader to keep in mind:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware the optimal strategy is to try to approach the proof myself first, along with coming up with counterexamples for the conditions in the theorems, however, given the fast pace of these courses and the fact that an undergraduate is not just taking the analysis course, really minimizes the time a person can devote to such sort of analysis.&lt;/p&gt;&#xA;" OwnerUserId="12036" LastActivityDate="2019-03-10T14:35:33.093" Title="Learning proofs in introductory analysis courses" Tags="|undergraduate-education|proofs|mathematical-analysis|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15323" PostTypeId="2" ParentId="15322" CreationDate="2019-03-06T19:16:28.120" Score="1" Body="&lt;p&gt;The latter of your two choices.  You don't know it if you can't recreate it.  Just reading stuff and saying &quot;yeah I get it&quot; is the lazy way and you don't really learn from it.  This was true in technique based math courses as well or chem or physics or languages.  (Bio/history/lit/psych/etc. can be read a bit more passively with large gains of knowledge.  However, even here some &quot;drill&quot; will make them better.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it's OK not to bang your head against the wall too long to independently derive every proof while reading (not needing the text at all).  However, at least a short attempt at first deriving the proof will make you more invested in the text once you do read it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After seeing the proof, practice rederiving it.  It will approximate the learning from first deriving.  Hopefully the exercises as well will build some skills into your brain.  This is especially because it is near impossible to just regurgitate everything as a memorized script.  Thus some of it you will internalize in terms of building the ability to rederive it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you are practicing, try to read the proof and then put the paper away and see if you can redo it yourself with a blank paper.  If needed, look at the book, but only for the smallest quickest hint and the go back and finish the proof.  If you had to look at the text (at all), force yourself to do the problem right over with no peeking.  This is really a similar study skill to doing technique based problems as well.  You need to do this in order to grind some grooves into your head, to internalize the knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are struggling with Rudin, try other easier texts like Cummings or Abbot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1724510126&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Real-Analysis-Long-Form-Mathematics-Textbook/dp/1724510126/ref=sr_1_3?keywords=real+analysis&amp;amp;qid=1551898925&amp;amp;s=books&amp;amp;sr=1-3&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1493927116&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116/ref=sr_1_4?keywords=real+analysis&amp;amp;qid=1551898925&amp;amp;s=books&amp;amp;sr=1-4&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel you on having other courses and needing to be time efficient.  But some level of moderate struggle is needed to drive learning.  An hour of reading a text like a novel (or for &quot;recognition&quot;) is far inferior to an hour of working problems.  So it's actually time inefficient to be too weak and passive in the learning process.  As said, this was the same in your technique courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s. I have never taken a real analysis course and I'm not a teacher.  (Caveats.)&lt;/p&gt;&#xA;" OwnerUserId="12037" LastEditorUserId="12037" LastEditDate="2019-03-06T19:22:23.363" LastActivityDate="2019-03-06T19:22:23.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15324" PostTypeId="2" ParentId="1817" CreationDate="2019-03-07T06:18:21.007" Score="10" Body="&lt;p&gt;Thanks to Google Doodle, today I learned about &lt;a href=&quot;https://en.wikipedia.org/wiki/Olga_Ladyzhenskaya&quot; rel=&quot;nofollow noreferrer&quot;&gt;Olga Ladyzhenskaya&lt;/a&gt; and her fantastic life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qKsHw.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qKsHw.jpg&quot; alt=&quot;screenshot of Google Doodle for Olga Ladyzhenskaya&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;From Wikipedia&lt;/strong&gt;. Ladyzhenskaya was born and grew up in Kologriv. She was the daughter of a mathematics teacher who is credited with her early inspiration and love of mathematics. In October 1939 her father was arrested by the NKVD and soon killed. Young Olga was able to finish high school but, because her father was an &quot;enemy of the people&quot;, she was forbidden to enter the Leningrad University.&#xA;After the death of Joseph Stalin in 1953, Ladyzhenskaya presented her doctoral thesis and was given the degree she had long before earned. She went on to teach at the university in Leningrad and at the Steklov Institute, staying in Russia even after the collapse of the Soviet Union and the rapid salary deflation for professors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ladyzhenskaya was on the shortlist for potential recipients for the 1958 Fields Medal, ultimately awarded to Klaus Roth and René Thom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Her Mathematical Work.&lt;/strong&gt; She was known for her work on partial differential equations (especially Hilbert's nineteenth problem) and fluid dynamics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think some aspect of the fluid dynamics might be introduced in high school. There is an attempt &lt;a href=&quot;https://peer.asee.org/introducing-talented-high-school-students-to-engineering-via-fluid-mechanics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. In particular, looking at the male-dominated history of the subject, I feel she should find her place in the photo used in the aforementioned attempt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4DR2G.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4DR2G.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="8336" LastEditDate="2020-05-08T14:18:48.613" LastActivityDate="2020-05-08T14:18:48.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15327" PostTypeId="2" ParentId="15322" CreationDate="2019-03-09T15:09:50.717" Score="2" Body="&lt;p&gt;There are two separate things you want to take away from a proof, and they should be the focus of your study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first important takeaway from a proof is the driving idea (or ideas) that make the proof work. You should generally be able to express these ideas using as much ordinary language as possible. As an example, I will consider the theorem that the uniform limit &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; of a sequence of continuous functions &lt;span class=&quot;math-container&quot;&gt;$$f_n : D \subseteq \mathbb{R} \rightarrow \mathbb{R}$$&lt;/span&gt; is continuous. (This can actually be considered as a special case of a more general theorem, but that's no matter here.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The driving idea behind the proof is to use the triangle inequality to bound &lt;span class=&quot;math-container&quot;&gt;$$|f(x) - f(x_0)|\, \text{ by } \,|f(x) - f_N(x)| + |f_N(x) - f_N(x_0)| + |f_N(x_0) - f(x_0)|$$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in a suitable disk of radius &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; centered at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; using a suitably large choice of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The three terms in the latter summand can be made less than &lt;span class=&quot;math-container&quot;&gt;$\epsilon/3$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$f_n$&lt;/span&gt; converges to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; uniformly, &lt;span class=&quot;math-container&quot;&gt;$f_N$&lt;/span&gt; is continuous, and &lt;span class=&quot;math-container&quot;&gt;$f_N(x_0)$&lt;/span&gt; converges point-wise to &lt;span class=&quot;math-container&quot;&gt;$f(x_0)$&lt;/span&gt;, respectively. Note that this part of understanding a proof is creative and requires understanding the subject in a greater context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second important takeaway from a proof is how to turn the driving idea of the proof into a rigorous argument. In the example above, this is where you manipulate &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;'s and &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;'s and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;'s appropriately, carefully considering the order of quantifiers and what depends on what. Note that this part of understanding a proof is technical and can be improved upon with constant practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you are first learning proofs, the second part can be very challenging. The more experience you get, the more you will come to realize that the first part is the generally harder part for discovering a new theorem. The second part can largely be automated, but the first part requires a genuine insight. (Constructing new proofs is actually a bit more complex than that. Generally, one comes with lots of potential strategies for a proof, then analyzes whether they can be turned into rigorous proofs. Often, there is a gap in the argument, and then one must go back to the creative stage and figure out a way to close the gap or a new approach to the proof.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of your two options, the second one is the one that will help you to learn how to approach proofs successfully. Focus on understanding the big picture and then on how to turn those ideas into rigorous arguments. Don't memorize proofs line-by-line. It's important that your goal should be to &lt;em&gt;understand&lt;/em&gt; the proof, not to merely be able to recite it.&lt;/p&gt;&#xA;" OwnerUserId="1397" LastEditorUserId="238" LastEditDate="2019-03-10T14:35:33.093" LastActivityDate="2019-03-10T14:35:33.093" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15328" PostTypeId="1" CreationDate="2019-03-10T08:12:26.367" Score="10" ViewCount="2164" Body="&lt;p&gt;I am a math teacher from China, teaching a course in English. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some students of mine are really good at finding answers for math problems designed in a quiz, however they are unable to write down  full answers with details or explanations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do I decribe this situation in English? I am about to prepare an exam in which for those who do not give detailed answers would receive low scores. Is the following sentence ok: &quot;Do not write only your answers (numbers), give me your detailed solutions.&quot;&lt;/p&gt;&#xA;" OwnerUserId="12043" LastEditorUserId="143" LastEditDate="2019-03-10T14:37:00.117" LastActivityDate="2019-08-13T13:22:51.123" Title="Vocabulary for giving just numbers, not a full answer" Tags="|terminology|" AnswerCount="10" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15329" PostTypeId="2" ParentId="15328" CreationDate="2019-03-10T09:55:44.147" Score="14" Body="&lt;p&gt;The problem with the word &quot;solution&quot; is that it could mean the final answer or it could mean how the final answer was obtained, so I suggest that you don't use it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How about the following?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Clearly show how you got your final answer if you want to get full credit for it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If the phrase &quot;full credit&quot; is not familiar to them, you may use &quot;all the points&quot; or &quot;the complete score.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also suggest having the student make explicit what the final answer is.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Put your final answer in a box.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastActivityDate="2019-03-10T09:55:44.147" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15330" PostTypeId="2" ParentId="15328" CreationDate="2019-03-10T13:32:51.150" Score="13" Body="&lt;p&gt;(From a comment)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On American math exams, I see the phrase &quot;justify your answers&quot;.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2019-03-10T13:32:51.150" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15331" PostTypeId="2" ParentId="15328" CreationDate="2019-03-10T15:21:34.507" Score="9" Body="&lt;p&gt;There's no phrase that's going to reliably convey what you want - if it's not what they've been taught to do, they're not going to know what you want.  Whatever phrase you use, you're going to have to back it up with a discussion about what it means and what the expectations are.  &lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-03-10T15:21:34.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15333" PostTypeId="2" ParentId="15328" CreationDate="2019-03-10T18:25:48.617" Score="5" Body="&lt;p&gt;When I see a student write down an answer with absolutely no work I will point out a problem to them. If I don’t know them personally, I do not know if they are a gifted student who is able to skip many or all steps, or if they simply sat near a smart student and copied just her circled answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I tell that student that a teacher is looking to see all the work, step-by-step, how they went from the question to the final answer. I remind them that if they have an incorrect answer, the teacher has no ability to offer partial credit, say three points out of the four points this question might be worth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might also help for you to provide two good examples of the process. Whatever level you are teaching, you should be able to find two different types of problems this would apply to and how a good answer is composed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;TLDR - please show all steps/work that shows how you calculated the answer. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-03-10T18:25:48.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15335" PostTypeId="2" ParentId="15328" CreationDate="2019-03-10T19:19:33.967" Score="9" Body="&lt;p&gt;The typical British English phrase would be &quot;show your working&quot;, but of course if the students are not familiar with the idea, you need to explain what that means in practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the instructions printed on a &lt;a href=&quot;https://revisionmaths.com/level-maths/level-maths-past-papers/edexcel-level-maths-past-papers&quot; rel=&quot;noreferrer&quot;&gt;UK national exam paper in mathematics&lt;/a&gt; (at the level where a high enough grade in the exam would be a requirement for university admission) is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You should show sufficient working to make your methods clear. Answers&#xA;  without working may not gain full credit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7677" LastEditorUserId="7677" LastEditDate="2019-03-10T19:25:06.567" LastActivityDate="2019-03-10T19:25:06.567" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15336" PostTypeId="2" ParentId="15328" CreationDate="2019-03-10T21:02:18.497" Score="6" Body="&lt;p&gt;Let me frame challenge the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of focusing on the terminology how to &quot;make the students provide details/explanations&quot; you could also provide the numeric result up front and simply ask how they would solve this using technique &quot;xyz&quot;. By providing the result up front it should be clear to anyone that just providing the solution will earn them no points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you would like to see specific intermediate steps you could also divide the problem into multiple parts (optionally provide the numeric answer for each part). Each part could also have a score attached to it so there's also no question about the grading. This has the benefit (although that may be subjective) that if a student gets stuck at the beginning they don't have to skip the complete question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realize that depending on the questions this might not be possible.&lt;/p&gt;&#xA;" OwnerUserId="12048" LastActivityDate="2019-03-10T21:02:18.497" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15337" PostTypeId="2" ParentId="15328" CreationDate="2019-03-10T22:29:59.820" Score="5" Body="&lt;p&gt;I always state the number of points possible on exam problems. If you have this practice, then take it a step further. Suppose a 10-point problem says:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Find the volume of the rectangular prism shown below.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask yourself what you want to see a student do. In my class, I would want (based on class demonstrations, homework problems, etc.) them to write an expression that involves a basic formula (no numbers), followed by that expression where the particular lengths have been substituted into that formula (using the correct units), followed by the answer they obtain from their calculator, concluding with the correctly rounded answer, according to the significant digits given (again, with the correct units).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of writing &lt;strong&gt;[10 points]&lt;/strong&gt;, I will often write something like &lt;strong&gt;[Correct numerical answer = 7 points, Correct units throughout = 1 point, Showed the formula = 1 point, Correctly rounded = 1 point]&lt;/strong&gt;. This has the benefit of giving me a rubric to follow and publishing that to the student. If you're demonstrating all of these steps in class and awarding/withholding points for them consistently on homework/quizzes, then seeing this breakdown given on the exam should make sense to your class.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-03-10T22:35:20.377" LastActivityDate="2019-03-10T22:35:20.377" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15338" PostTypeId="2" ParentId="14481" CreationDate="2019-03-11T14:08:33.733" Score="2" Body="&lt;p&gt;As an example (I encourage the commenters to make their comments answers), one could use the technology used in &lt;a href=&quot;http://merganser.math.gvsu.edu/david/linear.algebra/ula/ula/sec-eigen-intro.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Activity 4.1.2 of Austin's linear algebra&lt;/a&gt;, which makes some reasonable extensive use of javascript.  You can see a bunch of examples together &lt;a href=&quot;https://pretextbook.org/examples/sample-article/html/section-interactive-authored.html#interactive-html5-canvas&quot; rel=&quot;nofollow noreferrer&quot;&gt;in the PreTeXt sample article&lt;/a&gt; and how to make them in &lt;a href=&quot;https://github.com/rbeezer/mathbook/tree/dev/examples/sample-article/code/austin&quot; rel=&quot;nofollow noreferrer&quot;&gt;the code that goes with it&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-03-11T14:08:33.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15339" PostTypeId="2" ParentId="14481" CreationDate="2019-03-11T18:08:47.317" Score="0" Body="&lt;p&gt;You may find some useful things already exist at&#xA;&lt;a href=&quot;https://demonstrations.wolfram.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://demonstrations.wolfram.com/&lt;/a&gt;&#xA;but it's not at all clear to me what the copyright situation is for using them for videos.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-03-11T18:08:47.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15340" PostTypeId="2" ParentId="15328" CreationDate="2019-03-12T03:04:26.037" Score="4" Body="&lt;h1&gt;&amp;quot;Show your work&amp;quot; . . .&lt;/h1&gt;&#xA;&lt;p&gt;. . . is the phrase my instructors always used to declare that answers, absent the written steps involved in arriving at such an answer, are worth only partial credit.  This was at the elementary, high-school and university levels.&lt;/p&gt;&#xA;&lt;p&gt;For context, I attended school in the USA in the midwest in the 1970s through the 1980s.&lt;/p&gt;&#xA;" OwnerUserId="12059" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-03-12T03:09:32.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15341" PostTypeId="2" ParentId="15328" CreationDate="2019-03-12T07:22:16.333" Score="2" Body="&lt;p&gt;We use the word 'demonstrate' (or prove).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Demonstrate/prove that the answer is correct&quot;. That way, the student must go through the deductive procedure when resolving math issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a note, I only partially agree as a teacher to making everyone follow a manual-like demonstration. That is because sometimes students can find alternate unique undocumented ways to solve a math problem. So I split them into 3 categories: the ones finding a new way to solve the math problem (best of the students), the normal ones that reach the result by the manual and a special category that can directly put in the result without following an existing or new demonstrated procedure. This last category can be more complex to deal with - some can cheat and copy the result from someone else, in which case I do not take their answer into account but some are of very high skills and can put the correct result no matter what without copying it from someone. I found such skilled students that could state the correct result of any logarithmic and exponential-based problems. No reason not to grant them good scores.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on what I said above, I developed the scoring system.&lt;/p&gt;&#xA;" OwnerUserId="12060" LastActivityDate="2019-03-12T07:22:16.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15342" PostTypeId="1" CreationDate="2019-03-12T23:06:03.503" Score="4" ViewCount="236" Body="&lt;p&gt;Sometimes I want to explain to laymen/new students/laywomen how addition modulo N works. There are some instructive analogies: Addition on the clock (12), Addition on weekdays (7). They illustrate the idea &quot;N=0&quot; well.&#xA;However, what I struggle with is to give an analogy for multiplication mod N. Something like &quot;Thursday times Tuesday is Monday&quot; - but it should have some interpretation &quot;in real life&quot;.&#xA;What kind of examples can I use?&lt;/p&gt;&#xA;" OwnerUserId="12061" LastActivityDate="2019-03-16T23:27:34.920" Title="Analogy for multiplying modulo N" Tags="|undergraduate-education|number-theory|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15343" PostTypeId="2" ParentId="15342" CreationDate="2019-03-13T02:14:17.877" Score="9" Body="&lt;p&gt;Maybe the issue is that if the values you're multiplying have units, then the result of multiplying will have the product of those units. Therefore, your result can't really be equivalent to a value in the base set, because the units are different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, if applying this to the real world, I suggest considering the repeated-addition interpretation (scalar multiplication) so you can treat the result as equivalent to something in the base set.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-03-13T02:14:17.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15344" PostTypeId="2" ParentId="14988" CreationDate="2019-03-13T06:29:50.847" Score="0" Body="&lt;p&gt;They should know by heart how to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5586#5595&quot;&gt;set up, solve, and check problems&lt;/a&gt;.  I discuss this in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/8195#9613&quot;&gt;my answers&lt;/a&gt; to &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/8195&quot;&gt;What are non-math majors supposed to get out of an undergraduate calculus class?&lt;/a&gt;&quot; and &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/5586&quot;&gt;How can you be perfect at maths (highschool)?&lt;/a&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The list in the original post is a good start for calculus techniques that the students should know.  I would add to it the Reynolds Transport Theorem.  In particular, they should be able to convert &quot;what goes in, either stays in, or comes out&quot; into a mathematical description.  To do this, they need to know how to set up:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;An area integral&lt;/li&gt;&#xA;&lt;li&gt;A volume integral&lt;/li&gt;&#xA;&lt;li&gt;A dot product (to calculate the component of a vector that is perpendicular to a surface in the outward direction)&lt;/li&gt;&#xA;&lt;li&gt;A surface integral&lt;/li&gt;&#xA;&lt;li&gt;The time derivative of a surface integral (a.k.a. flux)&lt;/li&gt;&#xA;&lt;li&gt;The time derivative of a volume integral&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2019-03-13T06:29:50.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15346" PostTypeId="2" ParentId="15342" CreationDate="2019-03-14T07:07:30.150" Score="4" Body="&lt;p&gt;Perhaps the easiest thing would just be to use mod 10 multiplication. While only caring about the last digit isn't something we particularly do in real life, it might be simple enough for people to picture anyway.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-03-14T07:07:30.150" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15347" PostTypeId="1" AcceptedAnswerId="15348" CreationDate="2019-03-14T17:43:52.853" Score="9" ViewCount="836" Body="&lt;p&gt;Today in my calculus class we were going over L'Hopital's Rule and were dealing with limits of the following form&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$h(x)=f(x)^{g(x)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Three examples we considered are as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(1)\; \displaystyle \lim_{x\rightarrow \infty}\left(1+\frac{1}{x}\right)^x,$&lt;/span&gt; and its analog,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(2)\;\displaystyle \lim_{x\rightarrow 0}\left(1+x\right)^{1/x},$&lt;/span&gt; and then&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(3)\; \displaystyle \lim_{x\rightarrow 0}\left(\sin{x}\right)^x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am able to explain the applications involving the first two (exponential continuous growth), but when a student asked me what the function &lt;span class=&quot;math-container&quot;&gt;$\sin{x}$&lt;/span&gt; raised to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; would be used for, I couldn't answer it.  What would be some of the applications or phenomenon where that function or that limit would show up?  I tried checking the &lt;a href=&quot;https://dlmf.nist.gov/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Digital Library of Mathematical Functions&lt;/a&gt; but was not finding anything useful. &lt;/p&gt;&#xA;" OwnerUserId="4728" LastEditorUserId="238" LastEditDate="2019-03-16T06:06:31.133" LastActivityDate="2019-03-16T06:06:31.133" Title="Real World use of the Function $(\sin{x})^x$" Tags="|calculus|applications|limits|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15348" PostTypeId="2" ParentId="15347" CreationDate="2019-03-15T15:27:20.717" Score="6" Body="&lt;p&gt;I suspect that if &lt;span class=&quot;math-container&quot;&gt;$(\sin x)^x$&lt;/span&gt; shows up in any physical situation, it will be highly specific and not really a natural or worthwhile thing for a calculus student to spend their time on. Perhaps students should think of it as &quot;cross training&quot;, where you do exercises for a somewhat different sport to get better at the main one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, as &lt;span class=&quot;math-container&quot;&gt;$\sin x \approx x$&lt;/span&gt; for small &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, the function reminds me of &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt;. That is an example of &lt;a href=&quot;https://en.wikipedia.org/wiki/Tetration&quot; rel=&quot;noreferrer&quot;&gt;Tetration&lt;/a&gt; which shows up in the theory of computability.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2019-03-15T15:27:20.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15349" PostTypeId="2" ParentId="15263" CreationDate="2019-03-15T18:29:27.417" Score="7" Body="&lt;p&gt;For a history of how Geometry textbooks (and the way they were used) in the United States evolved from the mid-19th century into the 20th century, see:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Herbst, P. (2002) Establishing a custom of proving in American school&#xA;  geometry: Evolution of the two-column proof in the early twentieth&#xA;  century. &lt;em&gt;Educational Studies in Mathematics&lt;/em&gt; 49(3), 283-312.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Herbst distinguishes among three eras in Geometry instruction in US high schools:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The Era of Text, beginning in the 1840s&lt;/li&gt;&#xA;&lt;li&gt;The Era of Originals, beginning with the texts of Greenleaf (1858) and Chauvenet (1870)&lt;/li&gt;&#xA;&lt;li&gt;The Era of Exercise, resulting from the recommendations of the Committee of Ten&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Regarding the Era of Text, Herbst writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;American high schools started to offer geometry courses in the 1840’s&#xA;  as universities started to make it a requisite for admission (Quast,&#xA;  1968, p. 36). During this period, which I call the Era of Text, the&#xA;  study of geometry entailed mastering the Euclidean body of knowledge&#xA;  as developed by a text. Texts that served that purpose included Robert&#xA;  Simson’s (1756) &lt;em&gt;Elements of Euclid&lt;/em&gt;, John Playfair’s (1795/1860)&#xA;  &lt;em&gt;Elements of Geometry&lt;/em&gt;, and Adrien-Marie Legendre’s &lt;em&gt;Elements of Geometry&lt;/em&gt;&#xA;  (translated and edited by John Farrar in 1819). There are notable&#xA;  differences in how the three authors – Simson, Playfair, and&#xA;  Legendre – develop the Euclidean body of knowledge (Jones, 1944), but&#xA;  as far as what proving could have meant for students who studied those&#xA;  texts, the three texts are similar...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The purpose of studying geometry in school was to grasp the necessary&#xA;  character of the relationships between geometrical objects. It was&#xA;  essential to that acquisition to study the proofs of those&#xA;  relationships. The word &lt;em&gt;demonstration&lt;/em&gt; was used in the text, but it was&#xA;  neither used as the name of an object to be studied nor used as the&#xA;  name of a singular skill to be trained on...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The study of geometry was done through reading and reproducing a text;&#xA;  such work would train the reasoning faculties of students. &lt;strong&gt;But, the&#xA;  texts do not hint at the existence of official mechanisms to verify or&#xA;  steer the evolution of students’ reasoning&lt;/strong&gt;. To know geometry and to be&#xA;  able to prove the theorems of geometry were indistinguishable. And &lt;strong&gt;the&#xA;  difference between knowing geometry and remembering the text was&#xA;  immaterial in school&lt;/strong&gt; (see Quast, 1968, p. 40). As the geometry course&#xA;  became more common in high schools, geometry instruction began to move&#xA;  away from simply reproducing a geometry text.  (Emphasis added)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As has been remarked by others, during this era geometry texts (including Euclid's) did not contain problems for students to solve, and it was not expected that students write any proofs other than the ones that were already in the text, which they memorized and had to reproduce.  The first textbooks to include &quot;originals&quot; (that is, original problems for students to solve) were those of Greenleaf (1858) and Chauvenet (1870), regarding which Herbst writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The texts by Greenleaf (1858) and later by William Chauvenet (1870)&#xA;  differed from those of the Era of Text in that they included exercises&#xA;  at the end. As these exercises were spoken of as opportunities for&#xA;  students to do original work, I call this transitional period in the&#xA;  evolution of proving the Era of Originals. These originals were&#xA;  corollaries of propositions proved in the main text and additional&#xA;  theorems that might not have deserved a place in the main text. Thus,&#xA;  they afforded opportunities for students to use their reasoning to&#xA;  further their geometric knowledge. In his preface to a later edition&#xA;  of Chauvenet’s (1887) text, William Byerly said the purpose of&#xA;  originals was to “compel the student to think and to reason for&#xA;  himself” beyond just learning “to understand and demonstrate a few set&#xA;  propositions” (p. 5). Changes in the way later texts would give&#xA;  original exercises seem to respond to that purpose of training&#xA;  students’ “power of grasping and proving a simple geometrical truth,”&#xA;  a power that “can never be gained by memorizing demonstrations”&#xA;  (Chauvenet, 1887, p. 5). In that spirit, George Wentworth (1878) would&#xA;  give long lists of original exercises at the end of each chapter, and&#xA;  Wentworth (1888) would also interject some of those original exercises&#xA;  in between the propositions of a given chapter.  &lt;strong&gt;The presence of&#xA;  originals presumed that students would learn to reason by reasoning&lt;/strong&gt;.&#xA;  (Emphasis added)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is much more in Herbst's paper than I have mentioned in this brief summary; in fact as the title suggests most of the paper is devoted to the evolution of the &quot;two-column proof&quot; as both an instructional form and as a norm specific to the genre of high school Geometry textbooks.  But the background material on how things were done &lt;em&gt;before&lt;/em&gt; the Era of Exercise is very relevant for this question.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2019-03-15T18:29:27.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15350" PostTypeId="2" ParentId="15288" CreationDate="2019-03-15T18:42:24.257" Score="6" Body="&lt;p&gt;The notion that more formalism and precision will help beginning students learn is a common one among mathematicians, especially ones who are near the beginning of their teaching experience.  It is rooted, I believe, in a conflation between two closely-related notions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Some ideas come &lt;em&gt;logically&lt;/em&gt; after others;&lt;/li&gt;&#xA;&lt;li&gt;Some ideas are &lt;em&gt;taught sequentially&lt;/em&gt; after others.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Since you are a mathematician, you can think of the curriculum as a partially-ordered set, in which some topics come before others, but with &lt;em&gt;two different order relations&lt;/em&gt; on the same underlying set.  (This paragraph is, in fact, a self-conscious an example of using unnecessarily formal language to explain a simple idea.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are certainly not the first person to ask a question on this site that is based on the premise that teaching &quot;the basics&quot; would be more effective if we were really careful about the &quot;foundations&quot;.  But Hilbert's &lt;em&gt;Grundlagen&lt;/em&gt; is not a book for beginners!  Regarding a related question (&lt;a href=&quot;https://matheducators.stackexchange.com/questions/10620/why-would-you-teach-calculus-before-teaching-real-analysis&quot;&gt;Why would you teach Calculus before teaching Real Analysis?&lt;/a&gt;), I wrote the following answer:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You may as well ask: Why teach elementary school children how to perform whole-number arithmetic without teaching them the Peano axioms&#xA;  first? Why teach high-school Algebra without starting with the basics&#xA;  of groups, rings and fields? Why, for that matter, teach children to&#xA;  read first, instead of starting with the fundamentals of grammar and&#xA;  linguistics?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Everything we know about how learning takes place tells us that&#xA;  complicated ideas are formed and absorbed by building them on a&#xA;  foundation of simpler ones. &lt;strong&gt;It is important to realize that &quot;simpler&quot;&#xA;  does not mean here &quot;more fundamental&quot;, i.e. logically more primary;&#xA;  rather &quot;simpler&quot; is here used in the sense of &quot;easier&quot;.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A human being is not a formal system, and our minds do not develop&#xA;  according to deductive principles. Almost nobody understands anything&#xA;  complicated the first time they encounter it; we master things by&#xA;  revisiting them, over and over again, gaining deeper insight into it&#xA;  each time. We teach people Calculus first because if we taught them&#xA;  Analysis first they would have no experiential or conceptual substrate&#xA;  on which to build.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This applies &lt;em&gt;even more so&lt;/em&gt; to Precalculus students, who are even closer to the beginning of their understandings of functions, and therefore have even less of a foundation on which to build more sophisticated understandings.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2019-03-15T18:42:24.257" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15351" PostTypeId="1" CreationDate="2019-03-16T01:21:44.867" Score="6" ViewCount="380" Body="&lt;p&gt;Many students think&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{a} \sqrt{b}=\sqrt{a\ b}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{a^2}=a$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;but none of the above are true when (a) and (b) are negative.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;To avoid such problems, students need to learn the properties below are true for any (a, b, c) that  meet the conditions specified.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\ a^b a^c = a^{b+c}$&lt;/span&gt; when both sides are defined.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a^c b^c=(a\ b)^c$&lt;/span&gt; when c is an integer and both sides are defined.&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$a^c b^c=(a\ b)^c$&lt;/span&gt; when (0 &amp;lt; a &amp;lt; &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;) or (0 &amp;lt; b &amp;lt; &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\left(a^b\right)^c=a^{b\ c}$&lt;/span&gt; when c is an integer and both sides are defined.&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\left(a^b\right)^c=a^{b\ c}$&lt;/span&gt; when (0 &amp;lt; a &amp;lt; &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;).&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\left(a^b\right)^c=a^{b\ c}$&lt;/span&gt; when (-1 &amp;lt; b &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; 1).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a^c b^{-c}=\frac{a^c}{b^c}$&lt;/span&gt; for all&lt;span class=&quot;math-container&quot;&gt;$\ $&lt;/span&gt; a, b, c.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a^c b^{-c}=\frac{a^c}{b^c}=a^c \left(\frac{1}{b}\right)^c=\left(\frac{a}{b}\right)^c$&lt;/span&gt;  when &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is an integer.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a^c b^{-c}=\frac{a^c}{b^c}=a^c \left(\frac{1}{b}\right)^c=\left(\frac{a}{b}\right)^c$&lt;/span&gt;  when (0 &amp;lt; b &amp;lt; &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The caveat &amp;quot;when both sides are defined&amp;quot;, excludes couter examples such as&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\ 0^5\ 0^{-2} \neq 0^{5-2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\left(0^{-2}\right)^{-3}\neq0^6.$&lt;/span&gt;&lt;br /&gt;&#xA;What books include most of the above properties?&lt;/p&gt;&#xA;&lt;p&gt;-----&lt;strong&gt;Edit&lt;/strong&gt;------&lt;br /&gt;&#xA;Actually, I only expect people to learn how some of the above identities extend to complex numbers when they are in certain fields of study. The above properties actually apply when the variables are complex numbers. An electrical engineering student should learn most of that before they study Discrete Fourier Transforms, or Z-Transforms. Sadly, some books actually tell you the fallacies above are true. One example is the following from: Ron Larson et al, Algebra II, Maryland Edition, McDougal Little, 2008.&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/4yBPq.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4yBPq.png&quot; alt=&quot;Ron Larson, et al&quot; /&gt;&lt;/a&gt;&#xA;Consider the next example from Daniel Zwillinger, CRC Standard Mathematical Tables and Formulae, CRC Press 1996.&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/xQb4E.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xQb4E.png&quot; alt=&quot;CRC Press&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;The second example is a math reference for engineers and scientists. It covers elliptical integrals and partial differential equations, but nothing is mentioned about the properties of &lt;span class=&quot;math-container&quot;&gt;$a^b$&lt;/span&gt; that leads to complex numbers. Many books only mention how the properties work for integer and rational exponents. What do the authors think people will do when working with approximate exponents having 16 digits of precision? Do they think people will convert the approximate exponents to rational numbers and work the problem from there? Hopefully by the time students take Pre-Calculus they lean that that the properties work in the case of a positive number to a real power.&lt;/p&gt;&#xA;&lt;p&gt;For those in math intensive fields, the subject should be revisited by the third year at a university. I don't remember what my professors mentioned about the properties of &lt;span class=&quot;math-container&quot;&gt;$a^b$&lt;/span&gt;, but I can't find a book that covers what I am looking for. I once took a course on functions of complex variables and the text book said nothing about the properties of &lt;span class=&quot;math-container&quot;&gt;$a^b$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7001" LastEditorUserId="7001" LastEditDate="2020-09-22T20:26:21.033" LastActivityDate="2020-09-22T20:26:21.033" Title="What books properly address the properties of $a^b$?" Tags="|reference-request|algebra|exponent|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15352" PostTypeId="2" ParentId="15288" CreationDate="2019-03-16T02:04:24.903" Score="1" Body="&lt;p&gt;One of the important first steps in course design is to know your audience.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, do your students need that level of understanding?  It sounds like you're wanting to approach this like a math class but you need to keep in mind that many undergraduate math classes (including pre-calculus) are so-called &quot;courtesy classes&quot; that are taught by the math department for other departments like physics and engineering.  Your average engineer doesn't care that &quot;a line with slope m and y-intercept b is the set of points L={(x,y):y=mx+b}.&quot;  They just need to know how to find the equation.  That's the kind of distinction that would be important to a math major but you probably won't find a lot of them in a college pre-calc class.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2019-03-16T02:04:24.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15353" PostTypeId="2" ParentId="10716" CreationDate="2019-03-16T04:20:59.873" Score="0" Body="&lt;p&gt;I would think the material that the American printing house for the blind has material that raised lines can be created&lt;/p&gt;&#xA;" OwnerUserId="12069" LastActivityDate="2019-03-16T04:20:59.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15354" PostTypeId="1" AcceptedAnswerId="15356" CreationDate="2019-03-16T10:30:31.077" Score="6" ViewCount="295" Body="&lt;p&gt;I'm introducing my kids to the concepts of group theory. To make abstract things tangible, I'm trying the geometry way, adopting Arnold's in &quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/9048166098&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abel's Theorem&lt;/a&gt;&quot;, so far I've explained, by using symmetry group of triangle, square, and tetrahedron&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;group (closure, identity and invertibility, I omitted the associativity part)&lt;/li&gt;&#xA;&lt;li&gt;abelian &lt;/li&gt;&#xA;&lt;li&gt;subgroup&lt;/li&gt;&#xA;&lt;li&gt;cyclic group&lt;/li&gt;&#xA;&lt;li&gt;isomorphic&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm more or less confident as explaining what is a normal subgroup, by interpreting internal automorphisms as &quot;relabeling&quot; -- by &lt;em&gt;algebra&lt;/em&gt; style definition internal automorphism is &lt;span class=&quot;math-container&quot;&gt;$gag^{-1}$&lt;/span&gt;, but one can interpret in a &lt;em&gt;geometry&lt;/em&gt; way that &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is &quot;relabeling&quot; then &lt;span class=&quot;math-container&quot;&gt;$gag^{-1}$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; under that action.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then I hesitate on how to move on, for example, how to introduce such concepts in a &lt;em&gt;geometry&lt;/em&gt; way, not relying on &lt;em&gt;algebra&lt;/em&gt; formula?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;direct product&lt;/li&gt;&#xA;&lt;li&gt;coset&lt;/li&gt;&#xA;&lt;li&gt;quotient groups&lt;/li&gt;&#xA;&lt;li&gt;homomorphism&lt;/li&gt;&#xA;&lt;li&gt;commutator &amp;amp; commutant (the commutator subgroup)&lt;/li&gt;&#xA;&lt;li&gt;solvable group&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;How do I introduce such concepts in a geometry way?&lt;/p&gt;&#xA;" OwnerUserId="9059" LastEditorUserId="851" LastEditDate="2019-03-17T00:20:01.193" LastActivityDate="2019-03-24T00:03:04.457" Title="Group theory by geometry" Tags="|geometry|abstract-algebra|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15355" PostTypeId="2" ParentId="15354" CreationDate="2019-03-16T16:29:42.660" Score="3" Body="&lt;p&gt;Interesting question, here are some ideas just off the top of my head:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Direct product groups act on Cartesian products, so the symmetry group of two two squares rotating independently (you could then also talk about how this is a subgroup of the full symmetry group where you're also allowed to swap the squares if they're identical.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cosets: This isn't great, but the cosets of the dihedral group wrt the subgroup of flips are the rotations and the cosets of the rotations wrt are the flips&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Quotient Groups: Same as above, but say &quot;We can ignore the rotations rigorously by forming a new group where we've set the elements of the subgroup of rotations to 1. What will be left in this group?&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Group homomorphism: Show that the rotation group for the triangle maps into the group for the hexagon (by threefold symmetry), then show that when you try to form a mapping back, you have to explain what happens to the generator &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. You could demonstrate rotating them together, where you rotate a (marked) hexagon 6 times while rotating a (marked) triangle 6 times and show that rotations respect the groups structure.  Then, try it with the pentagon and see that when you've rotated the pentagon 5 times you're not back to the origin for the triangle. A contradiction! You could also use the cube to try this with nonableian groups. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Look at the commutator subgroup of the dihedral group. First challenge the students to tell you how many transformations of the triangle commute with every other transformation (none). What about the square (one)? Then derive the result. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Someone else is going to have to jump in with solveable groups, it's been too long since I've taken group theory.  &lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2019-03-16T16:29:42.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15356" PostTypeId="2" ParentId="15354" CreationDate="2019-03-16T16:49:55.957" Score="5" Body="&lt;p&gt;Since you are doing this geometrically, I would suggest a different tact on your treatment of normal subgroups: You can think of them as stabilizers of certain extra decorations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in a dihedral group we could label the various configurations with a right or left handed orientation. Then you obtain the (normal) rotation subgroup as the stabilizer of these orientations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the dihedral group is a &lt;span class=&quot;math-container&quot;&gt;$D_{2n}$&lt;/span&gt;, then you can give two different 2-colorings the vertices and look at the (normal) subgroup which preserves these colorings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you look at the group generated by the reflections on the lines here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/P40tx.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/P40tx.png&quot; alt=&quot;Tetrakis square tiling&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Tetrakis_square_tiling&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Tetrakis_square_tiling&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can give 2-colorings to the triangles. The symmetries are transitive on the triangles, but they don't all preserve colorings. The stabilizer of a coloring is again a normal subgroup.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also ask for preservation of the four rotation configurations of the triangles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are more complicated things as well:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XClF2.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XClF2.png&quot; alt=&quot;Kisrhombille tiling&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Truncated_trihexagonal_tiling#Kisrhombille_tiling&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Truncated_trihexagonal_tiling#Kisrhombille_tiling&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Wallpaper group&quot; would be a good thing to google.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quotients by these normal subgroups can be seen as acting on the decorations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Non-normal subgroups can also be analyzed by what they preserve --- but now they will not be uniform on the undecorated space.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2019-03-16T16:49:55.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15357" PostTypeId="2" ParentId="15342" CreationDate="2019-03-16T23:27:34.920" Score="2" Body="&lt;p&gt;How about this: I made &lt;span class=&quot;math-container&quot;&gt;$\$12$&lt;/span&gt; per hour for &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; hours. I then bought as many books as I could for &lt;span class=&quot;math-container&quot;&gt;$\$$&lt;/span&gt;15 each. How much money did I have left?  This would illustrate &lt;span class=&quot;math-container&quot;&gt;$7\cdot 12\equiv ? \pmod {15}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2019-03-16T23:27:34.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15358" PostTypeId="2" ParentId="15354" CreationDate="2019-03-17T00:02:22.237" Score="4" Body="&lt;p&gt;Permit me to amplify @XanderHenderson's comment pointing&#xA;to Nathan Carter's &#xA;&lt;a href=&quot;http://web.bentley.edu/empl/c/ncarter/vgt/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Visualizing Group Theory&lt;/em&gt;&lt;/a&gt;,&#xA;which I think is the perfect vehicle for your intentions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The MESE question,&#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/a/10245/511&quot;&gt;What makes cosets hard to understand?&lt;/a&gt;&#xA;pointed to his&#xA;illustration of cosets in &lt;span class=&quot;math-container&quot;&gt;$A_4$&lt;/span&gt; over several slides, of which this is one&#xA;(&lt;a href=&quot;http://groupexplorer.sourceforge.net/TalkWithoutWords.ppt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Download .ppt&lt;/a&gt;):&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ID6WW.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ID6WW.jpg&quot; alt=&quot;Cosets&quot;&gt;&lt;/a&gt;&#xA;&lt;hr /&gt;&#xA;Here is more on&#xA;&lt;a href=&quot;https://tomcircle.wordpress.com/2014/05/20/visualizing-group-theory/&quot; rel=&quot;nofollow noreferrer&quot;&gt;automorphisms&lt;/a&gt;, as &quot;rewirings.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a Cayley diagram,&#xA;showing &quot;interactions between group generators&quot;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/UGhhe.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UGhhe.jpg&quot; alt=&quot;CayleyDiagram&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Image from &lt;a href=&quot;https://sourceforge.net/projects/groupexplorer/#screenshots&quot; rel=&quot;nofollow noreferrer&quot;&gt;GroupExplorer&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-03-17T13:00:46.743" LastActivityDate="2019-03-17T13:00:46.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15359" PostTypeId="2" ParentId="4351" CreationDate="2019-03-17T01:17:17.870" Score="1" Body="&lt;p&gt;Here's a great application of the little-mentioned &lt;a href=&quot;https://en.wikipedia.org/wiki/Abel%27s_theorem#Theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abel's theorem&lt;/a&gt;:  If &lt;span class=&quot;math-container&quot;&gt;$\sum_{i=0}^\infty a_i,\sum_{i=0}^\infty b_i$&lt;/span&gt; and their convolution product &lt;span class=&quot;math-container&quot;&gt;$\sum_{i=0}^\infty \left(\sum_{j=0}^i a_ib_{j-i}\right)$&lt;/span&gt; all converge in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; (absolutely or not!) to sums &lt;span class=&quot;math-container&quot;&gt;$A,B,C$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$AB=C$&lt;/span&gt;.  In other words, the product series can't converge to something other than my first guess.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's gotten as a corollary by powers-seriesifying (generating-functionifying?) the given series in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and using absolute convergence on the interior where &lt;span class=&quot;math-container&quot;&gt;$x \in (-1,+1)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4679" LastActivityDate="2019-03-17T01:17:17.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15363" PostTypeId="2" ParentId="15272" CreationDate="2019-03-19T17:15:55.763" Score="1" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/15_puzzle&quot; rel=&quot;nofollow noreferrer&quot;&gt;15 puzzle&lt;/a&gt; is a nice example of an alternating group. The classic puzzle gives a concrete model for &lt;span class=&quot;math-container&quot;&gt;$A_{15}$&lt;/span&gt;. Generalizations extend it to &lt;span class=&quot;math-container&quot;&gt;$A_{n-1}$&lt;/span&gt; for any composite &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students could work out the way in which it is a group, and then parity could be introduced as an answer to the question of which configurations of the puzzle board are reachable from the start state.&lt;/p&gt;&#xA;" OwnerUserId="6891" LastEditorUserId="6891" LastEditDate="2019-05-04T17:28:22.500" LastActivityDate="2019-05-04T17:28:22.500" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15364" PostTypeId="1" CreationDate="2019-03-20T07:37:10.530" Score="5" ViewCount="280" Body="&lt;p&gt;I teach mathematics at MSc and PhD levels. My preferred method of teaching is old-fashioned: talking and writing on the blackboard at the same time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why? Because it has many advantages:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;em&gt;Handwriting:&lt;/em&gt; imposes few restrictions on notation and illustration. (Complicated figures I could project from my laptop, but I have no need for this in my courses.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Flexibility:&lt;/em&gt; whenever this is useful, it is easy to 'deviate from the script'.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Natural speed:&lt;/em&gt; it imposes a natural speed on the speaker. Preparing slides using LaTeX or PowerPoint and just clicking through them, I find myself proceeding way too fast.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Parallel displays:&lt;/em&gt; having several boards available for writing makes it easy to keep some text/examples on display on one board, while writing on another.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Dynamics:&lt;/em&gt; referring to information on the different boards allows me to move through the room, adding a more dynamic aspect to the lecture.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Ease:&lt;/em&gt; it is a low-tech way of achieving all these things simultaneously with easily available means.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The main disadvantage of this method is that I spend a significant amount of time of each lecture with my back to the audience. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What would you recommend as a means of communication that combines the six features above (most importantly, the handwriting and parallel displays), but facing the audience? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, a low-budget solution would be appreciated, but my institute is usually pretty generous in investing in technology that improves teaching, so don't let that restrict you!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What I tried:&lt;/strong&gt; Many things, including writing by hand on tablets (iPads, Digital Paper, reMarkable, etc) and projecting this in the classroom. Perhaps I haven't found the optimal device for this yet, but it often comes out pixly, delayed, and less readable than my usual handwriting on paper or the blackboard. Using a document camera to project my handwriting on paper. Works well, but can project only about half an A4 paper at a time to keep it readable for people in the back of the room and, like other approaches, has the disadvantage of not having parallel displays: it's hugely important to be able to keep definitions, examples, theorems from earlier on for easy reference.&lt;/p&gt;&#xA;" OwnerUserId="12079" LastActivityDate="2019-03-20T07:37:10.530" Title="Mimic lecturing on blackboard, but facing audience" Tags="|mathematical-pedagogy|technology-in-education|lecturing|" AnswerCount="0" CommentCount="5" ClosedDate="2019-03-20T15:51:29.710" ContentLicense="CC BY-SA 4.0" />
  <row Id="15365" PostTypeId="1" CreationDate="2019-03-20T08:20:07.043" Score="5" ViewCount="318" Body="&lt;p&gt;My child, now four (soon five) years old, is interested in time, knowing what time it is and how long until something happens. We do not, at the moment, have analog clocks and a digital clock is not very convenient, especially since it shows time in the usual 24 hour manner, whereas we often say that &quot;clock is seven&quot; (word-for word translation) when the time is 7:00 or 19:00.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other problem is the lack of concreteness; two-digit numbers and the symbol &quot;:&quot; as a separator between hours and minutes, and the idea that once the minutes reach 60 they reset while the hours keep on going; there is much to learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems a pity to not use the child's interest to teach a little bit about the subject matter.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is getting an analog clock the way to go, or are there alternatives?&lt;/li&gt;&#xA;&lt;li&gt;How to proceed, given the analog clock or other educational tools, and what are reasonable learning goals at this age?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(As always, please answer based on experience or reliable sources.)&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-03-24T07:51:27.497" LastActivityDate="2019-03-24T07:51:27.497" Title="Teaching a child time and hours in a digital world" Tags="|preschool-education|private-lessons|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15366" PostTypeId="1" AcceptedAnswerId="15374" CreationDate="2019-03-21T16:25:17.390" Score="13" ViewCount="3898" Body="&lt;p&gt;Yesterday a student in my calculus class attempted something like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Problem statement&lt;/strong&gt;: Find the derivative of &lt;span class=&quot;math-container&quot;&gt;$3^{(5x+1)}$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proposed solution&lt;/strong&gt;: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Let the inner function be given by &lt;span class=&quot;math-container&quot;&gt;$g(x)=3,$&lt;/span&gt; and the outer by &lt;span class=&quot;math-container&quot;&gt;$f(z)=z^{(5x+1)}$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$$f(g(x))=3^{(5x+1)}.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f'(z)=(5x+1)\cdot z^{(5x)}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g'(x)=0$&lt;/span&gt;, so by the chain rule, &lt;span class=&quot;math-container&quot;&gt;$$\frac{d\left(3^{(5x+1)}\right)}{dx} = f'(g(x))g'(x)=0.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I had difficulties explaining what's wrong with this, and basically just told the student &quot;the&quot; right way to do it. Although I now have a rough idea of what's wrong, I'd like to hear from others:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Have you seen similar attempts?&lt;/li&gt;&#xA;&lt;li&gt;How would you explain to a beginning calculus student what's wrong with this specific solution?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="590" LastEditorUserId="238" LastEditDate="2019-03-23T23:02:52.940" LastActivityDate="2024-01-09T00:49:29.003" Title="How to explain what's wrong with this application of the chain rule?" Tags="|calculus|students-mistakes|derivative|" AnswerCount="11" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15367" PostTypeId="2" ParentId="15366" CreationDate="2019-03-21T16:30:08.697" Score="13" Body="&lt;p&gt;f is not a function of (only) z - f here is a function of x as well as z.  I think this explanation is intelligible to a calc 1 student, and gets at the heart of the matter.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-03-21T16:30:08.697" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15368" PostTypeId="2" ParentId="15366" CreationDate="2019-03-21T19:37:21.800" Score="6" Body="&lt;p&gt;This is a VERY VERY typical problem.  In fact, it's a problem even for &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}3^x$&lt;/span&gt;, much less your example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way I try to deal with this is one of two ways.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What has to happen first?  To evaluate &lt;span class=&quot;math-container&quot;&gt;$3^{5x+1}$&lt;/span&gt;, you have to evaluate &lt;span class=&quot;math-container&quot;&gt;$5x+1$&lt;/span&gt; first.  So that is the inside function in the chain rule, just like in &lt;span class=&quot;math-container&quot;&gt;$\sin(x^2)$&lt;/span&gt; you have &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; to evaluate first, so it is the inside function.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You could rethink how we notate or talk about exponential functions.  In particular, Excel has &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; written as &lt;code&gt;exp(x)&lt;/code&gt; (I think as an option).  So one can ask what the &quot;input&quot; is here.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;However, on the plus side the student &lt;em&gt;does&lt;/em&gt; seem to have the chain rule down; it's just the exponential notation that is causing trouble.  So there is definitely hope here.  And again, you should not be surprised at encountering this, so it is worth your time to come up with several possible responses for it in the long run.  Good luck!&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2019-03-23T01:09:12.460" LastActivityDate="2019-03-23T01:09:12.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15369" PostTypeId="2" ParentId="15365" CreationDate="2019-03-21T20:20:42.947" Score="5" Body="&lt;p&gt;My experience: In the early 2000's, I had my first child (&lt;span class=&quot;math-container&quot;&gt;$\approx$&lt;/span&gt; 5 yrs) help me &quot;build&quot; an analog clock (it was just a paper model, no gears). We would occasionally set it to the correct time (given on my phone), perform some task, and then come back to the clock to reset the time. I think this gave him the sense of how fast/slow the minute hand took to move, and it reinforced the meaning of the tick marks for minutes. This made an easy transition to other analog clocks. I never did this with my 2nd child, and there was a definite delay in analog clock reading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not know what the benchmarks were for this age back then, but the current Common Core puts &quot;reading time&quot; in &lt;a href=&quot;http://www.corestandards.org/Math/Content/1/MD/&quot; rel=&quot;nofollow noreferrer&quot;&gt;1st grade&lt;/a&gt; (ages 6-7), but restricts it to whole and half hours. Then, in 2nd grade (ages 7-8), students must tell and write time &lt;a href=&quot;http://www.corestandards.org/Math/Content/2/MD/C/7/&quot; rel=&quot;nofollow noreferrer&quot;&gt;to the nearest 5 minutes&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-03-22T19:58:30.803" LastActivityDate="2019-03-22T19:58:30.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15370" PostTypeId="2" ParentId="15366" CreationDate="2019-03-21T21:17:59.263" Score="30" Body="&lt;p&gt;The root of the difficulty is that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; appears free in &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt;, but we are trying to &quot;capture&quot; it with &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;, which is illegal. When we substitute &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$f(g(x))$&lt;/span&gt;, we have a variable clash:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(g(\color{red} x)) = 3^{5\color{blue}x + 1}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The red (first) &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a &lt;em&gt;different variable&lt;/em&gt; from the blue (second) &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. This is clearer if we rename the bound variable:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(g(\color{red} y)) = 3^{5\color{blue}x + 1}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original expression had &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; bound to the &lt;span class=&quot;math-container&quot;&gt;$\mathrm d x$&lt;/span&gt;, so by unbinding it, we have changed the meaning of the expression:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\mathrm d}{\mathrm d \color{blue} x} f(g(\color{red}y)) \ne \frac{\mathrm d}{\mathrm d \color{red}y} f(g(\color{red}y)) &#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Incidentally, this is one reason I dislike the notation &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;, because it hides the variable of differentiation. Students must still be taught it, unfortunately, because Leibniz's notation is verbose in some contexts, but it should only be used as shorthand. Students should understand that it is a shorthand, and that there is still a variable of differentiation, even if it is not shown.)&lt;/p&gt;&#xA;" OwnerUserId="8474" LastEditorUserId="8474" LastEditDate="2019-03-21T21:22:59.823" LastActivityDate="2019-03-21T21:22:59.823" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15371" PostTypeId="1" CreationDate="2019-03-22T03:52:30.603" Score="11" ViewCount="3377" Body="&lt;p&gt;As far as I can tell, it's only a slight exaggeration to say that every text has a different notation for a change of basis matrix from (say) &lt;span class=&quot;math-container&quot;&gt;$\mathcal{B}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathcal{C}$&lt;/span&gt;.  That's not even to talk about the &quot;standard&quot; matrix for representation, e.g., changing from &lt;span class=&quot;math-container&quot;&gt;$\mathcal{B}$&lt;/span&gt; to the standard basis &lt;span class=&quot;math-container&quot;&gt;$\mathcal{E}$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is whether anyone has any ideas for why any particular notation is superior &lt;em&gt;in the classroom&lt;/em&gt; (or other educational outlet) to other such notations.  I'm not asking for a list of actual notations currently used - that would be more appropriate for math.se anyway.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to know the specific notation that you feel, in your experience, helps students understand this notably thorny concept.  (Yes, if you wrote a book, feel free to explain why you chose that notation!)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Aside:&lt;/em&gt; Seriously, at this point, I need a dictionary just to translate from one text to another.  A web search just now showed three or four notations I'd never even seen before, and I teach this course regularly!&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="238" LastEditDate="2019-03-23T01:04:24.683" LastActivityDate="2021-12-10T12:09:27.070" Title="Notation for change of basis matrix" Tags="|linear-algebra|notation|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15372" PostTypeId="2" ParentId="15371" CreationDate="2019-03-22T07:22:20.283" Score="8" Body="&lt;p&gt;I like to use&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ {}_{\mathcal{C}}A_{\mathcal{B}}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for the change from &lt;span class=&quot;math-container&quot;&gt;$\mathcal{B}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathcal{C}$&lt;/span&gt; because then the subscripts match up when you try to compose the matrices (in the usual convention) - you put a vector in &lt;span class=&quot;math-container&quot;&gt;$\mathcal{B}$&lt;/span&gt; coordinates into the right hand side, and get out one in &lt;span class=&quot;math-container&quot;&gt;$\mathcal{C}$&lt;/span&gt; coordinates. If you combine multiple matrices (which is the point), you need the subscripts to match up in the same way as the matrix sizes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ {}_{\mathcal{B}}D_{\mathcal{C}}\,{}_{\mathcal{C}}M_{\mathcal{C}}\,{}_{\mathcal{C}}A_{\mathcal{B}}\, v_{\mathcal{B}} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here I can see what I'm doing: changing into &lt;span class=&quot;math-container&quot;&gt;$\mathcal{B}$&lt;/span&gt; coordinates, doing the linear map using &lt;span class=&quot;math-container&quot;&gt;$\mathcal{C}$&lt;/span&gt; coordinates, and then convert back to &lt;span class=&quot;math-container&quot;&gt;$\mathcal{B}$&lt;/span&gt; coordinates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The use of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is not important in this notation.)&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-03-22T07:22:20.283" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15373" PostTypeId="2" ParentId="15366" CreationDate="2019-03-22T10:52:59.980" Score="3" Body="&lt;p&gt;I had a similar problem with a student last week and could not succinctly explain why she could not select &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; as the 'inner' function here: &lt;span class=&quot;math-container&quot;&gt;$$f(x) = e^{8x +4}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best explanation I have seen thus far, &lt;a href=&quot;http://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul's Notes&lt;/a&gt;, explains it in this way:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Recall that the 'outside' function is the last operation that we would perform in an evaluation. In this case if we were to evaluate this function the last operation would be the exponential. Therefore, the outside function is the exponential function and the inside function is its exponent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12086" LastActivityDate="2019-03-22T10:52:59.980" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15374" PostTypeId="2" ParentId="15366" CreationDate="2019-03-22T11:14:05.367" Score="8" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{d (3^{5x+1})}{dx} = f'(g(x))g'(x)= \frac{d \left(3^{5x+1}\right)}{d(3)} \times \frac{d (3)}{dx}.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However &lt;span class=&quot;math-container&quot;&gt;$\dfrac{d (3^{5x+1})}{d(3)}$&lt;/span&gt; is undefined.&lt;/p&gt;&#xA;" OwnerUserId="4780" LastEditorUserId="238" LastEditDate="2019-03-23T00:34:59.550" LastActivityDate="2019-03-23T00:34:59.550" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="15375" PostTypeId="2" ParentId="15366" CreationDate="2019-03-22T12:34:39.537" Score="4" Body="&lt;p&gt;This idea is fine, and you can use the multivariable chain rule to do it this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say we want to differentiate &lt;span class=&quot;math-container&quot;&gt;$h(x) = f(x)^{g(x)}$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Notice that we can write &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; as the composite of &lt;span class=&quot;math-container&quot;&gt;$p: \mathbb{R} \to \mathbb{R}^2$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$p(t) = (f(t),g(t))$&lt;/span&gt; with the function &lt;span class=&quot;math-container&quot;&gt;$E: \mathbb{R}^2 \to \mathbb{R}$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$E(u,v) = u^v$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the multivariable chain rule, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;Dh\big|_{x} &amp;amp;=  DE\big|_{p(x)} \circ Dp\big|_{x}\\&#xA;&amp;amp;= \left.\begin{bmatrix} \frac{\partial E}{\partial u} &amp;amp; \frac{\partial E}{\partial v} \end{bmatrix} \right|_{(u,v) = (f(x),g(x))} \circ  \left.\begin{bmatrix} \frac{\partial f}{\partial t} \\ \frac{\partial g}{\partial t}\end{bmatrix}\right|_{t = x}\\&#xA;&amp;amp;= \left.\begin{bmatrix} vu^{v-1} &amp;amp; \ln(u) u^v \end{bmatrix} \right|_{(u,v) = (f(x),g(x))} \circ  \left.\begin{bmatrix} f'(t) \\ g'(t)\end{bmatrix}\right|_{t = x}\\&#xA;&amp;amp;= \begin{bmatrix} g(x)(f(x))^{g(x)-1} &amp;amp; \ln(f(x)) (f(x))^{g(x)} \end{bmatrix} \circ  \begin{bmatrix} f'(x) \\ g'(x)\end{bmatrix}\\&#xA;&amp;amp;=f'(x)g(x)(f(x))^{g(x)-1}+g'(x)\ln(x)f(x)^{g(x)}&#xA;\end{align}&#xA;$$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applying this to the problem in question, we see that &lt;span class=&quot;math-container&quot;&gt;$f'(x) =0$&lt;/span&gt;, so the first term disappears.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in a sense, the student was trying to apply the multivariable chain rule (using the two variables &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;), but didn't know how to do that yet.  So you could tell them it is a good approach, but they will learn how to properly execute that approach in calc 3. &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2019-03-22T12:34:39.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15378" PostTypeId="2" ParentId="248" CreationDate="2019-03-23T22:44:35.547" Score="1" Body="&lt;p&gt;Not being an algebraist, I am not annoyed with cheating a bit &#xA;(well, we can use fictional history, as Tim Gowers defends so well...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with a fundamental, natural question: given a polynomial &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; &#xA;with integer coefficients, find a numerical set of sorts (i.e., a small field) &#xA;with a root of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We all know the exquisite answer: consider 'arbitrary' polynomials and operate &lt;span class=&quot;math-container&quot;&gt;$\mod p$&lt;/span&gt;.&#xA; Which in turn gets codified by passing a quotient by... an ideal &#xA;(and irreducibility comes up...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A normal subgroup is an ideal's smaller brother, in a rather tight context, &#xA;when only one well behaved operation is being considered.&lt;/p&gt;&#xA;" OwnerUserId="12091" LastEditorUserId="8571" LastEditDate="2019-03-24T00:04:16.223" LastActivityDate="2019-03-24T00:04:16.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15379" PostTypeId="2" ParentId="15354" CreationDate="2019-03-24T00:03:04.457" Score="2" Body="&lt;p&gt;There are some references to using the geometric art of M.C. Escher get across group theory (and other) ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example this article of Marjorie Senechal:&#xA;&lt;a href=&quot;https://upcommons.upc.edu/bitstream/handle/2099/1134/st15-07-a3-ocr.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://upcommons.upc.edu/bitstream/handle/2099/1134/st15-07-a3-ocr.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED583835.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://files.eric.ed.gov/fulltext/ED583835.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally you can look for papers related to using the wallpaper groups to teach group theory ideas.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2019-03-24T00:03:04.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15380" PostTypeId="2" ParentId="15365" CreationDate="2019-03-24T06:23:03.660" Score="3" Body="&lt;p&gt;Have you considered buying an analog watch for your child?  My parents got me one for my sixth birthday and I was able to use it to tell time.  I would think the child could also use it to learn to tell time.  The advantages are that your child will always have it handy and be able to compare the watch to the digital time. Whenever you want to talk about time in or out of the house, your child will have it ready.  Because it will belong to your child, he will know that you are supporting him in his quest to learn analog time.  Just make sure to get a watch with numbers that are easy to read.   &lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2019-03-24T06:23:03.660" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15381" PostTypeId="2" ParentId="15366" CreationDate="2019-03-24T08:06:56.390" Score="5" Body="&lt;p&gt;The other answers have completely missed the mistake.&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\def\rr{\mathbb{R}}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Your student's error has nothing to do with exponentiatiation. Consider the following based on exactly the same error:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{red}{\text{Let (???)}}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$f(y) = x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x) = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$1 = \frac{dx}{dx} = (f∘g)'(x) = f'(g(x))·g'(x) = f'(1)·0 = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The error lies in the &lt;strong&gt;very first line&lt;/strong&gt;! It is extremely obvious once you actually attempt to make it rigorous. Recall that to define a &lt;strong&gt;function&lt;/strong&gt; you must provide a domain as well as a rule that specifies the output for each input in the domain. And of course the rule has to be meaningful &lt;strong&gt;in the context&lt;/strong&gt; where you want to define the function. So see what you get:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{red}{\text{Let (???)}}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$f : \rr→\rr$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f(y) = x$&lt;/span&gt; for each &lt;span class=&quot;math-container&quot;&gt;$y∈\rr$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$g : \rr→\rr$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$g(x) = 1$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$x∈\rr$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The definition of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is fine. The definition of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is &lt;strong&gt;not&lt;/strong&gt; fine! What on earth is &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;? The rule has to specify the output for each input &lt;span class=&quot;math-container&quot;&gt;$y∈\rr$&lt;/span&gt;, so where did &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; pop up from?&lt;/p&gt;&#xA;&lt;p&gt;As explained above, the error has &lt;strong&gt;nothing to do with differentiation&lt;/strong&gt;. Rather, it is in the &lt;strong&gt;illegal definition of the function&lt;/strong&gt;!&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-03-24T08:06:56.390" CommentCount="31" ContentLicense="CC BY-SA 4.0" />
  <row Id="15383" PostTypeId="1" CreationDate="2019-03-25T12:31:32.500" Score="21" ViewCount="2108" Body="&lt;p&gt;I encountered the following concern when teaching indefinite integrals. I believe that many of us may overlook this. May I be wrong?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's consider the following example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Find the indefinite integral &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;I=\int\dfrac{dx}{x\sqrt{x^{2}-1}}.&#xA;$$&lt;/span&gt;&#xA;Some of my students gave the following answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$t=1/x$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$dx=-1/t^{2}dt$&lt;/span&gt;, so we get &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;I=\int\dfrac{-1/t^{2}dt}{\frac{1}{t}\sqrt{\frac{1}{t^{2}}-1}}=\int\dfrac{-dt}{\sqrt{1-t^{2}}}=-\arcsin\left(t\right)+C=-\arcsin\left(\frac{1}{x}\right)+C.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes, I accept this answer since it gives a quick general antiderivative.&#xA;However, the problem here is that we should write &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int\dfrac{-1/t^{2}dt}{\frac{1}{t}\sqrt{\frac{1}{t^{2}}-1}}=\int\dfrac{-\left|t\right|dt}{t\sqrt{1-t^{2}}}.&#xA;$$&lt;/span&gt;&#xA;Then we end up with the answer &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int\dfrac{dx}{x\sqrt{x^{2}-1}}=\begin{cases}&#xA;-\arcsin\left(\dfrac{1}{x}\right)+C &amp;amp; \text{for }x&amp;gt;1,\\&#xA;\arcsin\left(\dfrac{1}{x}\right)+C &amp;amp; \text{for }x&amp;lt;-1.&#xA;\end{cases}&#xA;$$&lt;/span&gt;&#xA;In your teaching practice, how would you usually proceed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. We may encounter the same issue in many other problems. For example, find &lt;span class=&quot;math-container&quot;&gt;$\int\sqrt{1-x^{2}}dx$&lt;/span&gt;. Then if we let &lt;span class=&quot;math-container&quot;&gt;$x=\sin\left(t\right)$&lt;/span&gt; then&#xA;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{1-\sin^{2}\left(t\right)}$&lt;/span&gt; should be &lt;span class=&quot;math-container&quot;&gt;$\left|\cos\left(t\right)\right|$&lt;/span&gt;. So now we need to explain a bit here to our naive students. Of course, avoiding these kinds of problems is the quickest way to make our teaching job easier. However, we need to prepare a good way of explanining or handing these types of problems. That's what I want to know.&lt;/p&gt;&#xA;" OwnerUserId="12043" LastEditorUserId="12043" LastEditDate="2019-03-25T22:05:59.827" LastActivityDate="2019-04-13T09:20:32.253" Title="Teaching indefinite integrals that require special-casing" Tags="|teaching|teacher-preparation|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15384" PostTypeId="2" ParentId="15383" CreationDate="2019-03-25T13:27:39.477" Score="20" Body="&lt;p&gt;I'd avoid giving problems like that to students first learning indefinite integrals (either by not asking it at all, or specifying the range x&gt;1 in the question).  It's a subtle algebraic trap, and if the goal is to teach students the mechanics of integration, it's going to be distracting rather than helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be an interesting question in a more advanced class, or as a question which is marked as difficult where students are expected (or told) to investigate their answer more carefully.  (For instance, graphing the functions will quickly reveal that there's a problem with the first solution, and looking at the graphs is probably enough to figure out what the fixed solution should look like, though figuring out why might take students a while.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-03-25T13:27:39.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15385" PostTypeId="2" ParentId="15383" CreationDate="2019-03-25T13:49:59.347" Score="12" Body="&lt;p&gt;This is a hard question, because students are so used to manipulation of this kind.  I have found you are right that absolute values can cause the worst of these examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an example I ran into recently, which I hope will help your thinking.  Observe that there are two different limits here:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to\pm\infty} \frac{x}{\sqrt{x^2+1}} = \pm 1$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;usual&quot; way to proceed with these (informally, in many texts nowadays) is to divide numerator and denominator by the highest power, so:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to\pm\infty} \frac{x\cdot 1/x}{\sqrt{(x^2+1)\cdot 1/x^2}} = \lim_{x\to\pm\infty} \frac{1}{\sqrt{1+1/x^2}}=1$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course bringing the &lt;span class=&quot;math-container&quot;&gt;$1/x$&lt;/span&gt; inside the root like that is the same invalid manipulation you are mentioning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, we actually talked through it at an even more naive level, not more rigorous!  Namely, as &lt;span class=&quot;math-container&quot;&gt;$x\to -\infty$&lt;/span&gt;, the numerator is negative and the denominator is positive.  So the overall answer &lt;em&gt;must&lt;/em&gt; be negative, no matter what the manipulation says.  (You can graph it for them too.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in your case, I would go more naive as well.  Do a very rough sketch of &lt;span class=&quot;math-container&quot;&gt;$\arcsin(1/x)$&lt;/span&gt; (you can basically do this by drawing &lt;span class=&quot;math-container&quot;&gt;$-\arcsin(x)$&lt;/span&gt; and then &quot;flipping over &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; to infinity&quot;), and then ask them whether this function is increasing or decreasing. When &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;-1$&lt;/span&gt; it should be increasing (in fact, it should be increasing on the whole domain), so its derivative should be positive (by whichever numbering of the fundamental theorems of calculus you like).  But &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x\sqrt{x^2-1}}$&lt;/span&gt; is definitely negative there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/uzKZr.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uzKZr.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you can explain &lt;em&gt;why&lt;/em&gt; you are picky, instead of just being picky because of some &quot;dumb&quot; &lt;span class=&quot;math-container&quot;&gt;$|x|$&lt;/span&gt; thing students may find to be a little too abstract.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Another answer brings up the question of whether this is a good question at all.  But I think it is reasonable.  What you may want to do, though, is find a way to discuss this &quot;naturally&quot;, i.e. using the disconnect between what people write and then if they see what seems to be a &quot;wrong&quot; answer in the back of a book or something.  Taking it as a first example in class probably will not register with them unless they are quite good at the concepts of calculus (not just mechanics), whereas pointing out why it is wrong/right should be better.  (On another note, presumably there are branch cut issues here as well but presumably your class isn't ready for that!)&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-03-25T13:49:59.347" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15386" PostTypeId="2" ParentId="15366" CreationDate="2019-03-25T15:41:00.960" Score="2" Body="&lt;p&gt;Some users have expressed doubt at the validity of the accepted answer, so let me make it rigorous. To do that, we first need to make sense of the notation &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt;. (Which is &lt;a href=&quot;https://math.stackexchange.com/questions/1258923&quot;&gt;not&lt;/a&gt; a trivial &lt;a href=&quot;https://mathoverflow.net/questions/115416&quot;&gt;task&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We start by changing perspective:  instead of thinking of variables &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as numbers, we think of them as smooth &lt;em&gt;real valued functions&lt;/em&gt; on some manifold &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$x\colon M\to \mathbb{R}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y\colon M\to \mathbb{R}$&lt;/span&gt;. You might rightly ask: why would we do that and which manifold &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; are you talking about? The answer to the first question is: because I'd like to talk about the differentials &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; and say everyday stuff like &quot;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is a constant&quot; or &quot;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot;. All of that is impossible inside first order logic + ZFC if we simply interpret &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as elements of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. Concerning the second question: think of &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; as the physical state space underlying the problem we are trying to model and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as observables. If that sounds too unfamiliar: it's similar to how people in probability theory assume an underlying space of outcomes &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;, in order to talk about random variables. (Which are the things we really care about and historically came before &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;, just like &lt;span class=&quot;math-container&quot;&gt;$dx,dy$&lt;/span&gt; historically came before manifolds, but I'm drifting of.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, having fixed the background manifold &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, whenever you hear me say &lt;em&gt;variable&lt;/em&gt;, what I mean is a thing of type &lt;span class=&quot;math-container&quot;&gt;$M\to \mathbb{R}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Definition.&lt;/strong&gt; Given two variables &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, call &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; a &lt;strong&gt;function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/strong&gt; if &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; is not zero almost everywhere and there exists a variable &lt;span class=&quot;math-container&quot;&gt;$q\colon M \to \mathbb{R}$&lt;/span&gt; such that&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;dy = q \cdot dx.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Intuitively, the equation &lt;span class=&quot;math-container&quot;&gt;$&#xA;dy = q \cdot dx&#xA;$&lt;/span&gt; says that the change of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is determined by the change of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, i.e. that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; depends on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not hard to show that &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is uniquely determined by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, hence we decide to denote it with &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; and call it the &lt;em&gt;derivative of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; wrt. &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/em&gt;. It was originally called the &lt;em&gt;differential coefficient&lt;/em&gt;, cause that's what it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to this definition, &lt;span class=&quot;math-container&quot;&gt;$3^{5x+1}$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (assuming &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a &lt;em&gt;true variabel&lt;/em&gt;, i.e. &lt;span class=&quot;math-container&quot;&gt;$dx\neq 0$&lt;/span&gt;), but it's also a function of &lt;span class=&quot;math-container&quot;&gt;$5x+1$&lt;/span&gt;. On the other hand, &lt;span class=&quot;math-container&quot;&gt;$3^{5x+1}$&lt;/span&gt; is &lt;em&gt;not&lt;/em&gt; a function of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; since &lt;span class=&quot;math-container&quot;&gt;$d3=0$&lt;/span&gt; (what we call a constant). In particular &lt;span class=&quot;math-container&quot;&gt;$\frac{d3^{5x+1}}{d3}$&lt;/span&gt; is undefined, as user21820 has been pointing out emphatically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can now state the chain rule in Leibniz form&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; is also a function &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and their differential coefficients satisfy&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{dz}{dx}=\frac{dz}{dy}\cdot \frac{dy}{dx}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The proof of this is trivial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From this perspective, what the student in my question was trying to do, was to let &lt;span class=&quot;math-container&quot;&gt;$z=3^{5x+1}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=3$&lt;/span&gt;. But the theorem does not apply, since &lt;span class=&quot;math-container&quot;&gt;$3^{5x+1}$&lt;/span&gt; is not a function &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. The same point of view can be used in the example discussed &lt;a href=&quot;https://math.stackexchange.com/questions/401122&quot;&gt;here&lt;/a&gt;, where a student attempted to differentiate &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt; by taking &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as inner function. Although that's allowed it just leads to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{dx^x}{dx}=\frac{dx^x}{dx}\cdot \frac{dx}{dx}&#xA;$$&lt;/span&gt;&#xA;which is of not much use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not to say that I don't appreciate the other answers (also users 21820). Taemyr's is just one of the three perspectives that haven been proposed. It might seem like it needs a lot of background to make it rigorous. But consider that mathematicians understood this stuff for at least 200 years without requiring manifolds to formalize it. And consider that the other approaches also require quite some background to make them rigorous (like quantifiers, variable bindings, the idea of dummy/bound variables etc. or derivatives of functions of multiple variables). Each has it advantages and disadvantages and none seems more right than the others, methinks.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2019-03-25T15:41:00.960" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15387" PostTypeId="2" ParentId="15383" CreationDate="2019-03-25T18:47:56.133" Score="11" Body="&lt;p&gt;You are right to be concerned that the students are &quot;missing something&quot;, but IMO the real problem here is that the question is completely artificial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any &lt;em&gt;application&lt;/em&gt; of this type of integral, most likely &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; will be known to be either positive or negative, but not both, and only one part of the &quot;either-or&quot; answer would apply. And there had better be a good reason why the rest of the problem needs &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to be negative, when it could have been replaced by &lt;span class=&quot;math-container&quot;&gt;$-x$&lt;/span&gt; right from the start!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same is true for the more common case of the indefinite integral of &lt;span class=&quot;math-container&quot;&gt;$1/x$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x &amp;lt; 0$&lt;/span&gt;, of course.&lt;/p&gt;&#xA;" OwnerUserId="7677" LastActivityDate="2019-03-25T18:47:56.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15388" PostTypeId="2" ParentId="15383" CreationDate="2019-03-25T22:10:37.497" Score="8" Body="&lt;p&gt;To me, the key point here is that the integral runs over a singularity.  If you naively calculates a definite form that runs over the singularity you get the wrong answer.  This is something I have done enough so that I have taught myself to be careful in this case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am more a physicist than a mathematician, so what I care about is the connection to a practical situation, rather than the formal manipulation of symbols.  If you or the students are of a similar inclination, then the presence of the singularity is what tips you off.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you know, different students respond well to different approaches, so I mention this one so that you might add it to your arsenal.&lt;/p&gt;&#xA;" OwnerUserId="12098" LastActivityDate="2019-03-25T22:10:37.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15389" PostTypeId="1" CreationDate="2019-03-26T14:39:08.903" Score="3" ViewCount="743" Body="&lt;p&gt;I will give a 2-hour long revision lecture on a Mathematics area. The aim of this lecture is to help the students prepare for the exam test. The course was rather long, covering 5 chapters and a number of topics. I have been thinking of giving the students a refresh of key definitions and results, or to present instead some particular exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, what are efficient structures for a revision lecture? Are there any guidelines about how should the material be presented in order to make the learning more efficient?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, students would like to see the same exercises that will appear in the exam. I am trying to avoid serving them the exam test in a silver tray but at the same time helping them learn the material in the course and prepare for the exam.&lt;/p&gt;&#xA;" OwnerUserId="12106" LastActivityDate="2019-03-26T15:56:03.417" Title="Revision Lecture, what are the aims?" Tags="|exams|teaching|testing|review|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15390" PostTypeId="2" ParentId="15389" CreationDate="2019-03-26T15:56:03.417" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;...what are efficient structures for a revision lecture?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Things I've actually tried, sorted by increasing student engagement:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Merely provide list of topics for students to study on their own time.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This was simple and fast, and it made it clear that they should expect questions from the entire course's material. It also put the review entirely on them to perform -- no &quot;silver tray&quot; of problems to memorize. However, calling this &lt;em&gt;review&lt;/em&gt; was really false advertising, and it seemed quite ineffective. Students were not engaged and basically had to figure out what problems to study on their own.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Provide list of topics (see (1) above) and spend the session answering student questions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Without fail, the questions students asked were &quot;Can you work a problem from topic &lt;em&gt;X&lt;/em&gt;&quot;? Then I would have to make one up or find one in the text and solve it (or &lt;em&gt;we&lt;/em&gt; would solve it as a class). This was a better use of time, but it worked slowly and wasn't tuned to the needs of everyone in class. For example, a few students dominated the questions and the session could have become a de facto tutoring session for them only. Mostly, students sat as observers -- not at all what their upcoming exam would be like.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Provide list of topics (see (1) above) and prepare a list of problems to demonstrate/solve in lecture. Pause between each problem to answer questions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This was basically a problem-based lecture, and it had the benefit of exposing students to problems &lt;em&gt;I&lt;/em&gt; felt were important for them to review. By varying the problem type and depth (all the way back to first-day or even prerequisite material), I could emphasize how cumulative the exam would be, and we could revisit the &lt;em&gt;older&lt;/em&gt; or more basic material.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Provide list of topics (see (1) above) and break class session into two parts. In the first 30 minutes, break your class into groups of 3 or 4 and randomly&lt;span class=&quot;math-container&quot;&gt;$^*$&lt;/span&gt; assign to each group two or three topics. Have them create a problem for each topic and write them out on paper. Collect all problem sheets. During break time (or have a helper) make copies of all problems for the class. After break, display the problems with a document camera and quickly read over them with the class. Identify  which problems are &lt;em&gt;easy&lt;/em&gt;, &lt;em&gt;medium&lt;/em&gt; or &lt;em&gt;difficult&lt;/em&gt;. Use the second half to have students work on those problems in groups.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(*) Knowing your students' abilities pretty well by the end of the term, you may &quot;stack the deck&quot; by giving certain topics to students who you know have a good grasp of the material -- enough to create a problem that isn't impossible, for instance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was much more engaging for students, and they actually seemed to enjoy the creative aspect. Some struggled to write clear questions, so I was checking in on them as they wrote, and we (as a class) had to &quot;clean up&quot; some of the language in their problems after copies were made. I now had a good idea of what students thought of when presented with a list of topics, though they did not know what &lt;em&gt;I&lt;/em&gt; thought about those topics.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;Provide list of topics (see (1) above) and a large list of problems, sorted by topic. Problems should involve some &quot;easy&quot; things (definitions, basic calculation, etc.) but should be more difficult than the exam, overall. Break class into groups of 3 or 4, have them look over the problems, and then give the entire period for them to work the problems in an order of their choosing. Walk around and answer questions as they come up. If a question is particularly insightful or generally helpful, address the whole class in response.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This has been the most effective method for me. By this point in the term, students have already seen me solve dozens or hundreds of problems, and they struggle with different things, student-by-student. The flexibility of being able to work with others and still focus their time on problems they struggle with, all while having me to answer questions, means there's no &quot;down time&quot;. Students often thank me for these class sessions. Of course, this means you must create a bunch of problems for this review, but over time your list will grow. [One of my former professors turned his problem sets into &lt;a href=&quot;http://web.pdx.edu/%7Eerdman/LINALG/Linalg_licensepage.html&quot; rel=&quot;noreferrer&quot;&gt;books&lt;/a&gt;.]&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;Do (5) on one day, and then (4) on the next day.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This has only worked out twice, as I don't usually end the term with an extra day to spare. On the first day (item 5), students finished the review problems for homework and we were able to discuss a few of them at the start of the last day. Then I was able to have them create problems (item 4) with the understanding that they could &lt;em&gt;not&lt;/em&gt; merely duplicate something from my review problem set -- it could resemble an old problem, but it had to be something &lt;em&gt;new&lt;/em&gt;. Many students thanked me, saying it &quot;put everything together&quot; for them, and other things you want to believe as your class goes into final exams.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-03-26T15:56:03.417" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15391" PostTypeId="2" ParentId="15383" CreationDate="2019-03-26T16:50:53.963" Score="3" Body="&lt;p&gt;It is interesting to note that if we instead write the antiderivative as &lt;span class=&quot;math-container&quot;&gt;$I=-\arctan\left(\frac{1}{\sqrt{x^2-1}}\right)$&lt;/span&gt;, then this form is valid for both &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;1$&lt;/span&gt;. In other words, using this arctan representation, we avoid a need for a piecewise representation for the antiderivative. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This problem is well suited for &lt;em&gt;formative&lt;/em&gt; assessment. I would give full credit for an answer of the form &lt;span class=&quot;math-container&quot;&gt;$I=-\arcsin\left(\frac{1}{x}\right)$&lt;/span&gt;, and then use this as a launching point for a discussion of these more involved issues that you have uncovered, depending on the mathematical maturity of the student. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-03-26T16:50:53.963" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15392" PostTypeId="2" ParentId="15366" CreationDate="2019-03-26T22:31:44.717" Score="1" Body="&lt;p&gt;Applying the naive approach of a non-mathematician, to me the expression &lt;span class=&quot;math-container&quot;&gt;$z^{(5x+1)}$&lt;/span&gt; points to a bivariate function,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(z,x) = z^{(5x+1)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(because &quot;I see two variables in here&quot;), and with &lt;span class=&quot;math-container&quot;&gt;$g(x) = 3$&lt;/span&gt; we have defined&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(g(x), x) = 3^{(5x+1)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac {df(g(x),x)}{dx} = \frac {\partial f(g(x),x)}{\partial g(x)}\cdot \frac {dg(x)}{dx} + \frac {\partial f(g(x),x)}{\partial x}\cdot \frac {dx}{dx}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$=\frac {\partial f(g(x),x)}{\partial g(x)}\cdot 0 + \frac {\partial f(g(x),x)}{\partial x}\cdot 1 = \frac {\partial f(g(x),x)}{\partial x} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$=\frac {\partial}{\partial x} \left(3^{(5x+1)}\right) $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This appears to be correct, although not useful, since we ended up back in the beginning. Am I doing something wrong here?&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2019-03-26T22:31:44.717" CommentCount="20" ContentLicense="CC BY-SA 4.0" />
  <row Id="15393" PostTypeId="1" CreationDate="2019-03-27T00:54:10.070" Score="9" ViewCount="232" Body="&lt;p&gt;I saw this &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=12&amp;amp;ved=2ahUKEwivouSTiaHhAhVTo54KHarsC90QFjALegQIBxAC&amp;amp;url=https%3A%2F%2Fpdfs.semanticscholar.org%2F90c5%2Fe9d6d94280b81e6a66973e3635aa9d163f98.pdf&amp;amp;usg=AOvVaw2p65O1EXIuj_IFdik0K0qF&quot; rel=&quot;noreferrer&quot;&gt;partial derivative machine&lt;/a&gt; yesterday, and it got me excited about other physical devices for exploring calculus concepts in a &quot;lab&quot; setting (e.g. &lt;em&gt;make a prediction, collect data, etc.&lt;/em&gt;) Do you use any devices like this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are certainly some nice &lt;a href=&quot;https://raisingcalculus.winona.edu/&quot; rel=&quot;noreferrer&quot;&gt;objects for visualizing graphs&lt;/a&gt;, and &lt;a href=&quot;https://www.monroecc.edu/faculty/paulseeburger/calcnsf/CalcPlot3D/&quot; rel=&quot;noreferrer&quot;&gt;great computer apps&lt;/a&gt;, but I am looking for physical devices that can be used to explore calculus (or precalculus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/8444/interesting-physical-implements-for-the-classroom&quot;&gt;Here is a question&lt;/a&gt; about interesting physical implements for the classroom, but it was specifically about wall-mountable, display items. I want great ideas for things to build and bring to class that assignments/labs can be built around.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-03-28T04:12:09.387" Title="Physical devices for exploring calculus or pre-calculus" Tags="|calculus|precalculus|classroom-interaction|manipulatives|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15394" PostTypeId="2" ParentId="15393" CreationDate="2019-03-27T08:14:32.890" Score="1" Body="&lt;p&gt;In chemistry, it is still common to integrate some test results by cutting the curve out and weighing the paper.&lt;/p&gt;&#xA;" OwnerUserId="12110" LastActivityDate="2019-03-27T08:14:32.890" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15395" PostTypeId="2" ParentId="15393" CreationDate="2019-03-27T14:38:20.530" Score="2" Body="&lt;p&gt;I have a &lt;a href=&quot;http://www.lightandmatter.com/velocity/&quot; rel=&quot;nofollow noreferrer&quot;&gt;browser-based app&lt;/a&gt; I use where students wiggle the mouse and see graphs of position, velocity, and acceleration. These are the tasks I use with this.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;task 1&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Your goal is to produce an x-t graph that looks like a staircase going down and to the right. Discuss with your group (a) how you would need to move the mouse in order to accomplish this, and (b) how the v-t graph would look.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pass out blank graphs, one per group, and do laps around room.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As each group agrees on a prediction, tell them to type in the URL they see on the demo screen and try it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;task 2&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Imagine -- but don't do it yet -- that you repeatedly wiggle the mouse up and down, doing it fast but smoothly. (You will actually find it physically smoother to move the mouse rapidly in a circle; the horizontal part of the motion will be ignored.)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;As a group, predict what the x-t, v-t, and a-t graphs would look like.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;It takes some physical practice to get good results, e.g., for task 2 they often do it too slowly at first. The quality of the results is limited by the poor resolution of the mouse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also sonar sensors that can be used for this. They're expensive and have their own difficulties and limitations.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-03-27T16:07:40.310" LastActivityDate="2019-03-27T16:07:40.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15396" PostTypeId="1" CreationDate="2019-03-27T21:03:27.140" Score="3" ViewCount="130" Body="&lt;p&gt;I want help phrasing the instructions in a math question. The issue is the correct way to express mixed units.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if an answer is “25 inches,” I don’t want to accept “25 inches” or “1 foot 13 inches,” but only “2 feet 1 inch.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess you might say that I want the total number of units to be small as possible. But I don’t think anyone would understand what I meant by that. Similarly, I don't think it would do to say that I want the answer &quot;simplified.&quot;&lt;/p&gt;&#xA;" OwnerUserId="8258" LastActivityDate="2019-04-04T18:03:01.597" Title="How to verbalize the correct statement of mixed units?" Tags="|terminology|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15397" PostTypeId="2" ParentId="15396" CreationDate="2019-03-27T21:18:23.057" Score="1" Body="&lt;p&gt;There's no conventional phrase that clearly expresses that.  The only way to convey that is to give an example.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-03-27T21:18:23.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15398" PostTypeId="1" AcceptedAnswerId="15408" CreationDate="2019-03-27T22:03:02.933" Score="2" ViewCount="174" Body="&lt;p&gt;With  the  image below I try to explain in which way substituting (x-a) &#xA;( with a&gt; 0) for x in the expression defining a function results in a shift to the right, although &quot; intuition&quot; tells us it should result in a shift to the left. To do this I use the auxiliary idea of &quot;shifting&quot; as &quot;copying&quot; or &quot;imitating&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be interested in knowing whether this explanation could be efficient in the classroom at the high school level. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we have &lt;span class=&quot;math-container&quot;&gt;$y=f(x)=x^2$&lt;/span&gt; and want to graph &lt;span class=&quot;math-container&quot;&gt;$y=g(x)=f(x-1).$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; maps each &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to the image of &lt;span class=&quot;math-container&quot;&gt;$(x-1)$&lt;/span&gt; under the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, each  &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; value has an &quot;&lt;span class=&quot;math-container&quot;&gt;$ x-1$&lt;/span&gt; &quot; ( his own  &quot;&lt;span class=&quot;math-container&quot;&gt;$x-1$&lt;/span&gt;&quot; ) , and&#xA;copies his (x-1) 's image under the &quot;old&quot; mapping &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; . &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since each &quot;imitator&quot; ( each &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; value) is to the RIGHT of it's &quot;model&quot;, that is it's &quot; &lt;span class=&quot;math-container&quot;&gt;$x-1 $&lt;/span&gt;&quot; , the change from the function  &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt;  to the function  &lt;span class=&quot;math-container&quot;&gt;$ g(x)=f(x-1)$&lt;/span&gt;  results , for each &quot;old&quot; point of the graph of f, in a translation of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; unit to the RIGHT.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ktmcI.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ktmcI.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. - I first used the idea of &quot; stealing&quot; , for which I substitute the idea of &quot; copying&quot; or &quot;imitating&quot;. I've just seen a post of Hyperpallium using the idea of &quot; sampling&quot; , even better.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Link to Hyperpallium's post : &lt;a href=&quot;https://math.stackexchange.com/questions/2813164/seeing-why-horizontal-shifts-are-reversed&quot;&gt;https://math.stackexchange.com/questions/2813164/seeing-why-horizontal-shifts-are-reversed&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user12116" LastActivityDate="2019-03-28T18:16:55.303" Title="Could this visual explanation of horizontal shift be helpful ? ...( if not beautiful...)" Tags="|algebra|graphing|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15399" PostTypeId="2" ParentId="15396" CreationDate="2019-03-27T22:22:25.950" Score="1" Body="&lt;p&gt;Assuming this is about some kind of exam: If there's still time in class to clear this up, use it. Explain why one might consider 1 foot 13 inches as bad and why 2 foot 1 inch might be better than 25 inches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there's no time, the most simple and precise way is to use an example to show the students which answer you like and which you don't.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2019-03-27T22:22:25.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15400" PostTypeId="2" ParentId="14757" CreationDate="2019-03-27T22:50:11.453" Score="0" Body="&lt;p&gt;Maybe the first question to ask is : WHY are they doing that mistake. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As says a famous 17th century philosopher, falsehood is nothing in itself, but just incomplete truth. So to understand the mistake, it can be usefull to look for the incomplete truth that lies in that mistake. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the &quot; incomplete truth&quot; here is the &quot;rule&quot;: &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;       IF a²=b then  sqrt(a²) = sqrt (b)   and then    a = sqrt(b). &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Yes, this rule is true, but with that complement : IF AND ONLY IF it is first given that the number a is positive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The complete truth is that, when one does not know whether the number a is positive or not, one has to write : &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;                              sqrt (a²) = |a|. &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Applying what precedes to the specific question, one has : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) a²= b &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) sqrt (a²) = sqrt (b) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) |a|  = sqrt (b) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) a = sqrt(b) OR -a = sqrt (b) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(5) a = sqrt (b) OR a=  - sqrt (b) &lt;/p&gt;&#xA;" OwnerDisplayName="user12116" LastEditorDisplayName="user12116" LastEditDate="2019-03-28T18:37:50.483" LastActivityDate="2019-03-28T18:37:50.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15401" PostTypeId="2" ParentId="14757" CreationDate="2019-03-27T23:49:26.913" Score="2" Body="&lt;p&gt;Use repetition and carrot/stick (e.g. weekly period-long exams, daily one question pop quizzes, in class games).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question shows an unconscious assumption that clear explanation is the key criteria.  (&quot;I told them several times.&quot;)  It's not.  We are not computers that get fixed forever with a line of code.  We are physical, imperfect beings.  Our brains adapted to make sense of our universe but are subject to various fallacies (e.g. optical illusions).  We learn to avoid various errors by &quot;imitation and practice&quot; (cf. Aristotle).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of taking the attitude &quot;how can these students make this mistake&quot; have a position that is more sympathetic and tough at the same time.  &quot;I get why you made the error but I will keep hammering you until you get it right.&quot; Drill sergeant.  Or even the &quot;yeah, this stuff is tough...let me tell you a sneaky trick&quot; (like you are a fellow criminal, fellow imperfect human).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make little silly names for the errors.  A computer would not need them.  It would grok the iff symbol.  But humans are humans.  If you call it &quot;offsides&quot; or &quot;penalty kick&quot; or whatever, it will make a weird association that helps them.&lt;/p&gt;&#xA;" OwnerUserId="12117" LastActivityDate="2019-03-27T23:49:26.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15402" PostTypeId="2" ParentId="15393" CreationDate="2019-03-28T04:12:09.387" Score="2" Body="&lt;p&gt;A &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Planimeter&quot; rel=&quot;nofollow noreferrer&quot;&gt;planimeter&lt;/a&gt; is a wonderful little device that measures the area surrounded by a simple closed curve by tracing it. It is based in the version of Green’s Theorem that computes area by integrating vertical/horizontal displacements along the curve.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2019-03-28T04:12:09.387" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15403" PostTypeId="1" AcceptedAnswerId="15406" CreationDate="2019-03-28T06:22:18.377" Score="7" ViewCount="248" Body="&lt;p&gt;I have been encouraging my classmates to connect with me on Google Docs to work collaboratively on taking notes. Still, no takers though. I imagine that if I were a professor, I would attempt to get them on board with collaboratively taking notes and improving on old notes as I go through the material. A major feature would be that I could see their comments if they ask a question, even anonymously. They would also be able to help each other if I can't get to their question sooner. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I'm not very clear on what research exists that would support or debunk such methods as best practices in pedagogy. Can anyone offer research sources and/or personal experiences in attempting different forms of collaborative note taking of this sort?&lt;/p&gt;&#xA;" OwnerUserId="12118" LastActivityDate="2019-04-06T21:34:26.090" Title="Collaborative note taking" Tags="|mathematical-pedagogy|technology-in-education|general-pedagogy|lecture-notes|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15405" PostTypeId="2" ParentId="14757" CreationDate="2019-03-28T13:35:53.007" Score="0" Body="&lt;p&gt;When you just know the basics of complex number theory, then you know that &lt;span class=&quot;math-container&quot;&gt;$x^n = a$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; solutions, so for &lt;span class=&quot;math-container&quot;&gt;$n=2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^n = a$&lt;/span&gt; has two solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's the big deal?&lt;br&gt;&#xA;For &lt;span class=&quot;math-container&quot;&gt;$n=2$&lt;/span&gt;, all roots (in case &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;0$&lt;/span&gt;) are real, which means you have 2 real roots.&lt;br&gt;&#xA;For &lt;span class=&quot;math-container&quot;&gt;$n=3$&lt;/span&gt;, there is one real root and two purely complex ones.&lt;br&gt;&#xA;For &lt;span class=&quot;math-container&quot;&gt;$n=4$&lt;/span&gt;, (in case &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;0$&lt;/span&gt;) two roots are real, and two are purely complex.&lt;br&gt;&#xA;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can easily show this in a nice drawing:&lt;br&gt;&#xA;For &lt;span class=&quot;math-container&quot;&gt;$n=5$&lt;/span&gt;, the roots are on a regular pentagon.&lt;br&gt;&#xA;For &lt;span class=&quot;math-container&quot;&gt;$n=4$&lt;/span&gt;, the roots are on a regular quadrilateral.&lt;br&gt;&#xA;For &lt;span class=&quot;math-container&quot;&gt;$n=3$&lt;/span&gt;, the roots are on a regular triangle.&lt;br&gt;&#xA;For &lt;span class=&quot;math-container&quot;&gt;$n=2$&lt;/span&gt;, the roots are on a line segment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I truely believe, once you've shown this in a graphical manner, they won't forget it anymore.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2019-03-28T13:35:53.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15406" PostTypeId="2" ParentId="15403" CreationDate="2019-03-28T15:46:40.373" Score="5" Body="&lt;p&gt;Personal experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an intro to probability course I had the chance to have a good friend (who used to be a lab partner in another course) and we decided to use &lt;a href=&quot;https://www.sharelatex.com/&quot; rel=&quot;noreferrer&quot;&gt;ShareLatex&lt;/a&gt; to build our own course notes (a &quot;digested&quot; version of what was shown in class). (It is better than Google Docs for the typesetting of formulas, but it is only free for up to two people that work at the same time on a document).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the note taking was made &quot;by hand&quot; in class and we would update the &lt;em&gt;Latex&lt;/em&gt; document a few times per week with new notes, formulas, etc. We also had a separate document where we solved some of the exercices. We also built a 'cheat sheet' of all the important stuff with the rule that it had to fit in one page (for the final exam).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, it was a great experience because we had to check what the other had written and make sure that there was no mistake and that it made sense (so that made us think about the content of the course, it was &lt;em&gt;not&lt;/em&gt; just about copying notes). Also, if I made mistakes or if something was unclear, I had feedback from my partner and we would talk about it in order to reach a consensus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the main takeaways for me are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It worked well because we were &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and we worked well together (with more people or people I didn't like or trust, there tends to be more disagreements and time wasted, so results might have been different). Also, we were serious about this 'project' and it was from our own initiative; if a teacher were to force this on students, some students might not like that at all!&lt;/li&gt;&#xA;&lt;li&gt;Because the note taking was collaborative, I felt more motivation to work on these notes (we both expected the other to do some work) it helped reduce procrastination&lt;/li&gt;&#xA;&lt;li&gt;The fact that we forced ourselves to digest the 'in-class' notes actually helped us learn because it forced us to think about the content, not just mindlessly copying formulas&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If I had to do it again, I would do some sort of collaborative &lt;em&gt;portfolio&lt;/em&gt;. So instead of just digesting notes and solving problems, I would add a new dimension where each participant can ask (and try to answer) questions such as &quot;&lt;em&gt;when could this formula be used?&lt;/em&gt;&quot;, &quot;&lt;em&gt;what content from other courses should be reviewed before studying this section?&lt;/em&gt;&quot; or &quot;&lt;em&gt;is there a real life example where this technique was useful?&lt;/em&gt;&quot;. This would have helped putting the content into perspective and learn how to &lt;em&gt;use&lt;/em&gt; the stuff we learned in other areas (transferring of knowledge).&lt;/p&gt;&#xA;" OwnerUserId="9341" LastActivityDate="2019-03-28T15:46:40.373" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15407" PostTypeId="1" CreationDate="2019-03-28T17:45:43.857" Score="3" ViewCount="225" Body="&lt;p&gt;I'm searching for exercises for practising operations with terms. They should involve &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;working with decimal numbers and fractions (ideally one should convert decimal numbers to simple fractions like halves, fourths or fifths to simplify calculations and be able to cancel out factors)&lt;/li&gt;&#xA;&lt;li&gt;expanding products and getting rid of parenthesis, in particular deal with plus and minus signs and possibly use binomial formulas&lt;/li&gt;&#xA;&lt;li&gt;using power laws&lt;/li&gt;&#xA;&lt;li&gt;combining terms&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and ideally give a possibly simple result. An example of what I have in mind would be something along the lines of&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$1.25(6a-4b)^2(0.4a^2b)^2 - 2.2(5a-4b)(4a-5b)(0.5ab^2)(2.5a^3b)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(however, this example doesn't simplify well at all and gives many terms with ugly coefficients).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hints on how to design such problems myself or even automatize the generation of such problems with a little program are also welcome!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for any input!&lt;/p&gt;&#xA;" OwnerUserId="3542" LastEditorUserId="11" LastEditDate="2019-10-05T00:53:56.450" LastActivityDate="2019-10-05T00:53:56.450" Title="Where to find good exercises for term operations?" Tags="|problem-design|fractions|exercises|powers|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15408" PostTypeId="2" ParentId="15398" CreationDate="2019-03-28T18:16:55.303" Score="0" Body="&lt;p&gt;I tried two other ways along the graphical representation you are proposing.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A lookup table:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In the first line note some values for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;: -2, -1, 0, 1, 2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the second line note the corresponding values of &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt;: 4, 1, 0, 1, 4&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the third line note the corresponding values of &lt;span class=&quot;math-container&quot;&gt;$g(x)=f(x-1)=(x-1)^2$&lt;/span&gt;: 9, 4, 1, 0, 1&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fill them out together with the pupils and let them observe that all the values of &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; are shifted by one to the right. Let them think about why this is the case and how they can get the values of the third line from those of the second line.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;A function with some particular &quot;feature&quot; like the extremum of a parabola or the first extremum or zero of a sine. Say, in the parabola case, to get the extremum you need to plug in &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt;. So which value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; needs to be plugged in into the shifted parabola &lt;span class=&quot;math-container&quot;&gt;$g(x)=(x-1)^2$&lt;/span&gt;? Obviously, a &lt;strong&gt;larger&lt;/strong&gt; one to &lt;strong&gt;compensate&lt;/strong&gt; for the 1 which is being &lt;strong&gt;subtracted&lt;/strong&gt; before the squaring comes. Therefore, we will get the extremum at &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt;, shifted to the &lt;strong&gt;right&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Both methods can (and probably should) be used together with the graphical representation as in the original question.&lt;/p&gt;&#xA;" OwnerUserId="3542" LastActivityDate="2019-03-28T18:16:55.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15409" PostTypeId="2" ParentId="5614" CreationDate="2019-03-28T20:23:55.170" Score="3" Body="&lt;p&gt;I once saw a Calculus book written in the form of a dialogue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me try something like that about function transformation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; If the function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is such that &lt;span class=&quot;math-container&quot;&gt;$g(x)= f(x-1)$&lt;/span&gt;, should not the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; be at every point, 1 unit to the left of the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; Why do you think it should be the case?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; Well I see that in the equation &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; (on the left) becomes &lt;span class=&quot;math-container&quot;&gt;$f(x-1)$&lt;/span&gt; (on the right). Apparently, the function symbol &quot;&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;&quot; is substituted for the function symbol &quot;&lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;&quot; and the variable &quot;&lt;span class=&quot;math-container&quot;&gt;$x-1$&lt;/span&gt;&quot; is substituted for the variable &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot;. So the X-coordinate of each &quot;new point&quot; decreases by 1 unit, and the whole graph is moved 1 unit to the left.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; Let me ask you a question: which variable represents, according to you, the input of the function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; Normally it should be the variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, since the left side of the equation has &quot;&lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;&quot;, but the whole equation says that instead of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, the input becomes &lt;span class=&quot;math-container&quot;&gt;$(x-1)$&lt;/span&gt;. In the expression &quot;&lt;span class=&quot;math-container&quot;&gt;$f(x-1)$&lt;/span&gt;&quot;, does not the number &lt;span class=&quot;math-container&quot;&gt;$(x-1)$&lt;/span&gt; play the role of the input?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; I think your mistake as to the direction of the transformation comes precisely from not understanding that the inputs that g takes are represented by the letter &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot;. The number &lt;span class=&quot;math-container&quot;&gt;$(x-1)$&lt;/span&gt; does not play the role of input (for &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;); it is simply &quot;used&quot; by the function-machine &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; (so to say) to calculate the image of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; under &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; Could you show an example?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; Let me take a general example. Suppose that a point &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; belongs to the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;. Knowing that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$g(x) = f(x-1)$&lt;/span&gt;, how could you rewrite the couple defining this point?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; Probably &lt;span class=&quot;math-container&quot;&gt;$(x-1, f(x-1))$&lt;/span&gt;, no?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; Why did you change the X-coordinate of this point? You've &quot;moved&quot; that point to the left without any reason! The only new information I gave you is about &quot;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&quot; , not about &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; You're right! Using the information you gave me, I'll now say that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(x, y) = (x, (f(x-1))$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; Right! Now, tell me, which point belonging to the graph of f corresponds to our &quot;&lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;-point&quot;: &lt;span class=&quot;math-container&quot;&gt;$(x, f(x-1))$&lt;/span&gt;? Obviously, the Y-coordinate of that point is &lt;span class=&quot;math-container&quot;&gt;$f(x-1)$&lt;/span&gt;, but what is its X-coordinate?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; The answer is in the question itself, isn't it? It's obviously the point&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(x-1, f(x-1))$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; I agree! But do you realize what it means? You've just told me that for any arbitrary point &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; belonging to the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, the corresponding point belonging to the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has the same Y-coordinate, but an X-coordinate that is smaller by 1 unit! That means that any arbitrary &quot;&lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;-point&quot; (so to say) is one unit to the RIGHT of it's corresponding point &quot;&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;-point&quot;!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; If I now understand what I said myself, I've just claimed, contrary to what I contended at the beginning, that the transformation is a translation of the whole graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; 1 unit to the RIGHT!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; That's what you said, actually!&lt;/p&gt;&#xA;" OwnerDisplayName="user12116" LastEditorUserId="6053" LastEditDate="2019-04-01T09:24:15.530" LastActivityDate="2019-04-01T09:24:15.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15410" PostTypeId="1" CreationDate="2019-03-28T20:57:26.503" Score="1" ViewCount="152" Body="&lt;p&gt;I am a Missouri High School Geometry teacher.  WE are adopting textbooks this year.  I would like opinions on which books are most closely aligned with the Missouri Learning Standards because at the end of the year my students must take an EOC (End of Course Exam)&lt;/p&gt;&#xA;" OwnerUserId="9889" LastEditorUserId="1608" LastEditDate="2019-04-07T00:10:29.260" LastActivityDate="2019-04-07T00:27:57.517" Title="Missouri EOC and the best Geometry book" Tags="|textbooks|geometry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15411" PostTypeId="1" CreationDate="2019-03-28T23:44:31.080" Score="5" ViewCount="168" Body="&lt;p&gt;I know &lt;a href=&quot;https://www.moe.gov.sg/docs/default-source/document/education/syllabuses/sciences/files/primary_mathematics_syllabus_pri1_to_pri5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Singapore&lt;/a&gt; and &lt;a href=&quot;http://movimentopelabase.org.br/wp-content/uploads/2018/03/BNCC_Competencias_Progressao.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brazil&lt;/a&gt; explicitly adopt &lt;a href=&quot;https://en.wikipedia.org/wiki/Metacognition&quot; rel=&quot;nofollow noreferrer&quot;&gt;metacognition&lt;/a&gt; as one of their maths curricular pillars. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The relevance of metacognition is recognized by OECD that has written a &lt;a href=&quot;http://www.oecd.org/publications/critical-maths-for-innovative-societies-9789264223561-en.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;state-of-the-art report&lt;/a&gt; about how to use the theory and its impact in math performance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given this importance, I would like to know if there are other countries that also adopt metacognition in their official math curricula.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="8828" LastEditDate="2019-04-03T20:27:20.627" LastActivityDate="2019-04-03T20:27:20.627" Title="Which countries adopt metacognition in their official math curricula?" Tags="|curriculum|meta-cognition|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15412" PostTypeId="2" ParentId="5614" CreationDate="2019-03-29T01:20:05.850" Score="-1" Body="&lt;p&gt;My advice is a little different/contrarian.  IN ADDITION TO BUT BEFORE explaining the stuff like &quot;signal delayed&quot; or &quot;making &lt;span class=&quot;math-container&quot;&gt;$2x = y$&lt;/span&gt;&quot; or whatever:  Just be honest and tell the kids:  &quot;This is a sneaky, tricky thing to watch out for.  People miss it all the time and get points taken off.&quot;  Then give them the blabla explanation.  But the honest expression before will concentrate their minds much more than just the explanation itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that, drill and carrot/stick them.  Some things make more sense after you just have spent a lot of time doing them.&lt;/p&gt;&#xA;" OwnerUserId="12120" LastEditorUserId="8571" LastEditDate="2019-03-29T17:08:43.663" LastActivityDate="2019-03-29T17:08:43.663" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15414" PostTypeId="1" AcceptedAnswerId="15423" CreationDate="2019-03-30T11:51:25.377" Score="5" ViewCount="4351" Body="&lt;p&gt;When I was a math undergraduate 30 years ago in India, we were taught what was then called &quot;classical algebra&quot; (as opposed to abstract algebra), and we were taught among other things solving polynomial equations using techniques like Cardan's method, synthetic division, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't kept in touch with math since leaving university, but now when I look at any standard undergraduate course, I don't see any of these things. All I see is abstract algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do students nowadays not learn how to solve polynomial equations? If not why not?&lt;/p&gt;&#xA;" OwnerDisplayName="user11871" LastActivityDate="2019-04-01T23:59:10.530" Title="Where do students learn to solve polynomial equations these days?" Tags="|algebra|abstract-algebra|solving-polynomials|polynomials|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15415" PostTypeId="2" ParentId="15414" CreationDate="2019-03-30T12:39:52.497" Score="10" Body="&lt;p&gt;Students learn linear and quadratic equations in high school algebra.  And then, if they have forgotten it, re-learn it in college, in courses called &quot;pre-calculus&quot; or something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unless they specialize in mathematics at the college level, they do not learn any more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why not?  Because we have computers now, so most people do not need to solve polynomial equations by hand any more.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2019-03-30T12:39:52.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15416" PostTypeId="2" ParentId="15414" CreationDate="2019-03-30T12:45:56.827" Score="7" Body="&lt;p&gt;In the United States, solving linear and quadratic equations is a standard part of Algebra 1, which most students take in 8th or 9th grade. Students will return to polynomials and see long division and synthetic division in Algebra 2. This is also when students will learn the remainder theorem, Descartes’ Rule, and how to identify the only possible rational roots for a polynomial.&lt;/p&gt;&#xA;" OwnerUserId="8670" LastActivityDate="2019-03-30T12:45:56.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15417" PostTypeId="1" AcceptedAnswerId="15421" CreationDate="2019-03-30T16:19:07.127" Score="3" ViewCount="122" Body="&lt;p&gt;If I'm not in error, old style algebra books ( before 1945) concentrated on equation solving, and modern ones concentrate more on functions and their graphs ( as a preparation to calculus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there classical references, popular amongst math teachers  - not too old references ( I'd better like not before mid 40's, for fear terminology could be out of date ) - concentrating specially on equation solving ( at the intermediate level).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once saw a post dealing with absolute value expressions as exponents with an example taken from Dorofeev's Elementary mathematics. That kind of reference ( with that kind of exercise) would be of great interest to me. ( Unfortunately it is rather difficult to find a copy of Dorofeev's book).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. When I say &quot; not too old ones&quot;, don't understand I have prejudices against old math books. To the contrary I highly appreciate math books written during the period 1945-1970 first for typography ,  second for clarity and elegance in explanations, third for a certain &quot; free thinking air&quot; ( so to say)  one can breathe in these books. As an example, I'd take Moses Richardon's books. &lt;/p&gt;&#xA;" OwnerDisplayName="user12116" LastEditorDisplayName="user12116" LastEditDate="2019-03-30T16:43:09.927" LastActivityDate="2019-03-31T00:06:07.963" Title="Classical references on equation solving" Tags="|algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15419" PostTypeId="2" ParentId="15414" CreationDate="2019-03-30T20:08:15.423" Score="3" Body="&lt;p&gt;Most math majors never learn cardano's methods. As for synthetic division, it is part of most high school curricula and linear algebra courses. Cubic and quartic equations aren't part of any curriculum I'm aware of, not because of computers, but because students should learn &quot;more important topics&quot; and because the solutions of these equations are very long so teachers can't really test the students.&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="501" LastEditDate="2019-03-30T20:46:12.117" LastActivityDate="2019-03-30T20:46:12.117" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15420" PostTypeId="2" ParentId="15414" CreationDate="2019-03-30T23:32:15.803" Score="4" Body="&lt;p&gt;Synthetic division is a standard part of the stereotypical &quot;algebra 2&quot; course in the US (~grade 11) and is normally covered including drill problems and examination.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, cubics and quartics are in the algebra 2 book (I just checked and they are in mine) but in sections with an asterisk. Typically the asterisk sections are not covered because they are considered more difficult diversions and the time, especially for non-superstar students, is better spent solidifying the unasterisked sections.  But most teachers will at least say something verbally like&quot; &quot;There are complicated formulas, sort of like the quadratic solution, that can be used to solve general cubics or quartics.  We won't cover them because of time, but they are in the book and you should know that methods exist.  However, there is no general algebraic solution of 5th or higher order polynomials.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that it is standard also to have teaching, drill, and examination on simple (pencil and paper) numerical estimation methods for finding roots to polynomials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth, I think the quadratic really is more fundamental/important in terms of applied problems.  Some people say the reason more applied problems use the quadratic is because via circular logic, the students know the equation and this is probably true in some instances.  However, we also routinely get people asking questions for fundamental formulas using cubic or higher equations and there really aren't many.  Yes, you can construct problems that have enough complication to get to that, but I would not say they are fundamental in the way that the harmonic oscillator is for instance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth, I do think there is a small benefit to covering cubic and quartic methods but would only bother with very high end classes with time to burn.  The benefit is from manipulational skill, not from the methods themselves.  But being trained, confident in more elaborate manipulations is helpful when you get into physics problems and the ability juggle Taylor series solutions, Bessel functions, Fourier, or change of coordinate systems (without messing up algebraically) is helpful.  [Despite what the math and compsci and Maple lovers say, &quot;19th century mathematics is alive and well in physics departments&quot;.]&lt;/p&gt;&#xA;" OwnerUserId="12127" LastEditorUserId="12127" LastEditDate="2019-03-30T23:41:53.047" LastActivityDate="2019-03-30T23:41:53.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15421" PostTypeId="2" ParentId="15417" CreationDate="2019-03-30T23:57:40.467" Score="1" Body="&lt;p&gt;I know what you mean about terminology getting too ornate or different in some older books.  Despite my love of them, I struggle with Forsyth for instance just based on the writing.  In contrast, I find Granville extremely direct and readable, even to a modern audience (probably one reason why it had such a long run from ~1905-1965 as the &quot;standard text&quot;).  So it is possible to find some older books that are readable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below is a classic algebra 2 text that I love.  Note that it has all the answers.  Is written very directly and economically (enough info but not florid or a doorstop).  Consider that the audience was US military in WW2, deployed and away from classrooms.  (My father literally carried his copy onto islands in the Pacific during WW2.)  So there is an intention of being accessible and usable by the student, versus a book written for teachers or selection committees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B00AM3HS5A&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/College-Algebra-Department-Education-Manual/dp/B00AM3HS5A&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. FWIW, I did not find much difference between the Hart book and my 1980s algebra 2 HS text, so you may be exaggerating the differences versus current texts.  But I don't know.  1980s was before the TI time fetish (although the storm clouds were gathering even then).&lt;/p&gt;&#xA;" OwnerUserId="12127" LastEditorUserId="12127" LastEditDate="2019-03-31T00:06:07.963" LastActivityDate="2019-03-31T00:06:07.963" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15423" PostTypeId="2" ParentId="15414" CreationDate="2019-03-31T09:11:11.360" Score="9" Body="&lt;p&gt;The rational root theorem, synthetic division, the remainder theorem, Descartes rule of signs, and similar lower level topics were fairly widely taught in U.S. high school algebra-2 courses before and during the 1980s, but they've slowly been de-emphasized as graphing calculators (allows for numerical equation solving) came onto the scene at the end of the 1980s. Some schools still cover this material as much as ever, and others mostly avoid it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding more advanced topics such as cubic equations, Vitae's formulas and their uses, etc., these were a standard part of the U.S. undergraduate curriculum when &quot;theory of equations&quot; courses were offered, which was one of the more common upper level math courses a mathematics major would take. These courses were mostly phased out during the mid 1950s to early 1960s as undergraduate abstract algebra courses began taking hold. Older college algebra texts (before 1960s, say) often had some of this material, but I doubt it was actually covered much, if at all, in the standard first year college algebra classes. [Typical college math schedule in U.S. for someone not especially advanced in math, but not necessarily behind either: three 1-semester courses consisting of college algebra (sometimes this was 2 semesters) and trigonometry (sometimes included a bit of spherical trigonometry) and analytic geometry (often included some solid analytic geometry), after which one began studying calculus (often as a 2nd year student, with better students maybe the 2nd semester of their first year).] Since the 1960s, these more advanced topics tend to arise only in abstract algebra courses, typically when Galois theory is covered, but the amount of coverage varies from almost none to a brief treatment, depending on how interested the instructor is in it and how much time is available for what amounts to a diversion on a supplementary topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, I suspect students are now more aware of many of these topics than they were in the 1990s (but not more than they were in, say, the 1940s) because of the increasing number of (and increased access to information about) mathematics competitions and the rise of the internet, both of which have greatly helped students to become exposed to these topics.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2019-03-31T17:50:51.177" LastActivityDate="2019-03-31T17:50:51.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15425" PostTypeId="2" ParentId="15407" CreationDate="2019-03-31T19:31:04.990" Score="3" Body="&lt;p&gt;It costs $5/month (for educators) to use Wolfram Alpha in its practice worksheets model. It will generate a lot of problems for you, but I'm not 100% sure it gives you the granularity you want. I really like it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.wolframalpha.com/pro-for-educators/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.wolframalpha.com/pro-for-educators/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, Math.com has a worksheet generator, which allows some specification of fraction use and difficulty of problems, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.com/students/worksheet/algebra_sp.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.com/students/worksheet/algebra_sp.htm&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9241" LastActivityDate="2019-03-31T19:31:04.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15426" PostTypeId="2" ParentId="15242" CreationDate="2019-03-31T19:59:29.897" Score="1" Body="&lt;p&gt;This sounds like a great opportunity to teach the course a few times, keeping track of what you do, then writing up your own book. I've done that, and even have written a blog about each day's topics, which gets pre-publication feedback.&lt;/p&gt;&#xA;" OwnerUserId="9241" LastActivityDate="2019-03-31T19:59:29.897" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15428" PostTypeId="2" ParentId="15242" CreationDate="2019-03-31T22:08:16.577" Score="1" Body="&lt;p&gt;I really fell &quot;in love&quot; with &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0201508672&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Elementary Geometry from an Advanced Standpoint&quot; by Edwin E. Moise&lt;/a&gt; when I have taught undergraduate geometry.  Many of the students, but not all, have been secondary math education majors.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book is rigorous, as it has challenged the brightest of the undergraduates.&lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2019-03-31T22:08:16.577" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15429" PostTypeId="2" ParentId="12986" CreationDate="2019-04-01T00:07:10.823" Score="3" Body="&lt;p&gt;Many answers are opining on the 2dp notation, which is not the subject of the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;FWIW, no the two are not equivalent.  Approximately 3.14 does not mean +/- .005.  It could be a higher (or slightly lower error).  Consider science papers listing an approximate value.  &lt;/p&gt;&#xA;" OwnerUserId="12132" LastEditorUserId="64" LastEditDate="2019-07-21T14:08:31.497" LastActivityDate="2019-07-21T14:08:31.497" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15431" PostTypeId="1" AcceptedAnswerId="15437" CreationDate="2019-04-01T16:03:59.023" Score="3" ViewCount="235" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How to explain to pupils that &quot;&lt;span class=&quot;math-container&quot;&gt;$\frac n{100}$&lt;/span&gt; OF &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&quot; is equivalent to &quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{n}{100}\times a$&lt;/span&gt;&quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is some difficulty in explaining that the first sentence, containing &quot;OF&quot; (which could suggest something like &quot;taking a part of&quot; and therefore something like division) has to be mathematically translated by the second sentence (with its &quot;TIMES&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the most efficient method to show that the &quot;TIMES&quot; translation is correct? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematically speaking, is the intuition that &quot;of&quot; suggests a division totally wrong? &lt;/p&gt;&#xA;" OwnerDisplayName="user12116" LastEditorUserId="238" LastEditDate="2019-04-04T21:56:10.563" LastActivityDate="2019-04-04T21:56:10.563" Title="How to explain to pupils that &quot;$\frac n{100}$ OF $a$&quot; is equivalent to &quot;$a$ TIMES $\frac{n}{100}$&quot;?" Tags="|algebra|fractions|middle-school|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15432" PostTypeId="2" ParentId="15431" CreationDate="2019-04-01T16:52:41.430" Score="4" Body="&lt;p&gt;The earliest mathematical insight I remember from childhood is that the word &amp;quot;of&amp;quot; almost always means &amp;quot;times.&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Half of a dozen &lt;span class=&quot;math-container&quot;&gt;$= \frac12 \cdot 12 = 6$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Three-fourths of a mile &lt;span class=&quot;math-container&quot;&gt;$= \frac34 \cdot 5280$&lt;/span&gt; feet &lt;span class=&quot;math-container&quot;&gt;$= 3960$&lt;/span&gt; feet.&lt;/p&gt;&#xA;&lt;p&gt;I'll take 6 of those thousand-count boxes &lt;span class=&quot;math-container&quot;&gt;$= 6 \cdot 1000 = 6000$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I remember feeling like I had secret knowledge that no one else had.&lt;/p&gt;&#xA;&lt;p&gt;Showing children this &amp;quot;secret&amp;quot; might give some of them the same moment I still remember to this day.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a one-hundredth of six &lt;span class=&quot;math-container&quot;&gt;$= \frac{1}{100} \cdot 6 = 0.06$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The fact that &amp;quot;a one-hundredth of six&amp;quot; is the same thing as &amp;quot;six hundredths&amp;quot; is not a trivial fact.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-04-01T16:52:41.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15433" PostTypeId="2" ParentId="15431" CreationDate="2019-04-01T17:19:11.490" Score="3" Body="&lt;p&gt;The word &quot;of&quot; means &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;. For example &quot;half of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&quot; means &lt;span class=&quot;math-container&quot;&gt;$\frac12\times a$&lt;/span&gt;. Note that the product of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; by a number isn't necessarily less than &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; ; it'll be less than &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; iff the number is less than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; ; in that case we can write the number as &lt;span class=&quot;math-container&quot;&gt;$\frac{m}{n}$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$m&amp;lt;n$&lt;/span&gt;) so that &lt;span class=&quot;math-container&quot;&gt;$\frac{m}{n}\times a$&lt;/span&gt; means to divide &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; equal parts and take &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; parts of them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we take a rectangle of dimensions &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, its area is &lt;span class=&quot;math-container&quot;&gt;$na$&lt;/span&gt; (you could partition it into &lt;span class=&quot;math-container&quot;&gt;$na$&lt;/span&gt; unit squares). If we divide &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; we obtain &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; smaller identical rectangles so the area of each one is &lt;span class=&quot;math-container&quot;&gt;$n\times \frac{a}{10}$&lt;/span&gt; which is also the tenth of the initial rectangle &lt;span class=&quot;math-container&quot;&gt;$\frac{na}{10}$&lt;/span&gt;. If we divide &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; we get &lt;span class=&quot;math-container&quot;&gt;$\frac{n}{10}\times a=\frac{na}{10}$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$\frac{n}{10}\times a=n\times\frac{a}{10}$&lt;/span&gt;. Of course we can &quot;prove&quot; more generally that &lt;span class=&quot;math-container&quot;&gt;$ab=ba$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; : &lt;span class=&quot;math-container&quot;&gt;$ba$&lt;/span&gt; is the area of the rectangle obtained by rotating the rectangle &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt; by a quarter revolution.&lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2019-04-01T17:19:11.490" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15434" PostTypeId="1" CreationDate="2019-04-02T12:47:04.713" Score="9" ViewCount="350" Body="&lt;p&gt;I am finding it difficult to motivate students on why they should how to prove mathematical results. They learn them just to pass examinations but show no real interest or enthusiasm for this.&#xA;How can I inspire them to love essential kind of mathematics? They love doing mathematical techniques. Any resources or any answers would really help me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additional information: 1st year undergraduates who dislike proofs. The university mostly thinks the maths courses are there for supplying other subjects. The students don't like proof based courses.&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="2083" LastEditDate="2019-04-02T15:58:21.740" LastActivityDate="2019-04-04T22:14:37.117" Title="How to motivate students to do proofs?" Tags="|undergraduate-education|proofs|student-motivation|teaching|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15437" PostTypeId="2" ParentId="15431" CreationDate="2019-04-02T18:29:38.153" Score="2" Body="&lt;p&gt;One can only &lt;em&gt;see&lt;/em&gt; that this is true just as one saw what the symbols &lt;span class=&quot;math-container&quot;&gt;$1,2,3,\cdots$&lt;/span&gt; meant. What's happening in this case is that we are seeing a ratio. Suppose we want &lt;span class=&quot;math-container&quot;&gt;$n/100$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$Q.$&lt;/span&gt; Then you may explain this to them as &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Divide &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; equal parts, and take &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; &lt;strong&gt;of&lt;/strong&gt; such parts of &lt;span class=&quot;math-container&quot;&gt;$100.$&lt;/span&gt; That is, we're taking &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; of something. But this simply means we add &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; such things together, which means we are multiplying such things by &lt;span class=&quot;math-container&quot;&gt;$n.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;NB. The explanation above will be clearer if you've previously introduced positive integer multiplication as a short notation for iterated addition, and will make sense therefore only if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a positive integer. Using this convention more generally is just a matter of agreement, as with most other operations outside &lt;span class=&quot;math-container&quot;&gt;$\mathbf N.$&lt;/span&gt; However, the explanation for &lt;span class=&quot;math-container&quot;&gt;$n\in\mathbf N$&lt;/span&gt; should be a good motivation as to why this convention of thinking of 'of' as some type of multiplication (later when they learn about functions and the notation &lt;span class=&quot;math-container&quot;&gt;$f(x),$&lt;/span&gt; they're going to have a nod of déjà vu!) is most useful, convenient, and agrees with the &lt;em&gt;fact&lt;/em&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbf N.$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="9204" LastActivityDate="2019-04-02T18:29:38.153" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15439" PostTypeId="2" ParentId="15407" CreationDate="2019-04-02T22:17:28.253" Score="1" Body="&lt;p&gt;Something like this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a. \left({a+b \over a-b} + {a-b \over a+b}\right) \div \left({a^2 \over a^2-b^2} + {1 \over {a^2 \over b^2}-1}\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$b. \left({x^2y - xy^2 \over x-y} + xy\right) \times \left({y \over x} + {x \over y}\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$c. \left({n \over m-n} + {m \over {m + n}} \right) \times \left({m^2 \over n^2} + {n^2 \over m^2} - 2\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$d. \left({x + 1 \over x - 1} - {x - 1 \over {x + 1}} + 4x \right) \times \left(x - {1 \over x} \right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$e. \left(1 + {a \over x} + {a^2 \over {x^2}} \right) \left(1 - {a \over x} \right) \times {x^3 \over {a^3 - x^3}} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f. {4x - 3 \over 3 - 2x} - {4 + 5x \over {3 + 2x}} - {3 + x - 10x^2 \over {4x^2 - 9}} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There should be plenty of exercises like these in any decent textbook, I pulled these from a &lt;a href=&quot;https://drive.google.com/file/d/1L-RfYwRHy2qTjeWYABxs-WY5CB_wpGOW/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;7th grade Russian algebra textbook&lt;/a&gt;. Start from Chapter 4, page 59 for monomials, and scroll forward for polynomials, factoring, completing the square, etc. Some exercises have answers in the back of the book.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2019-09-06T20:16:05.473" LastActivityDate="2019-09-06T20:16:05.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15440" PostTypeId="2" ParentId="15434" CreationDate="2019-04-03T13:54:03.707" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Surprise them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Especially in a (mathematical) culture where &quot;getting the right answers&quot; is prized and excellence on such examinations is valued in general, it is reasonable that students will value this above other expressions of mathematics.  (I have a little teaching experience in that context, though most of my experience is in the &quot;just get through it&quot; American culture of math.)  So one way to get students interested in proof is to express something where &lt;em&gt;the mathematical techniques are irrelevant and do not help in solving an interesting problem&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, you would be able to have students in a course like number theory or graph theory - both of which can be quite applied - and then give them a problem which simply doesn't have a (purely) computational answer.  Like &quot;is there an integer point on this elliptic curve&quot;, or &quot;can I embed this graph on a computer chip with no crossings of circuits&quot;, or something.  Maybe those are too advanced, but I hope you take my meaning.  The only way to know for sure is to prove.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difficulty is that most course content in this situation is more in the precalculus-through-differential equations trajectory, which doesn't always lend itself well to this.  You can prove things like the intermediate value theorem or existence of solutions to equations, but students will not care about these.  Because they seem obvious, like the fact there are no integers between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; - you can prove it from some axioms, but students will not see the point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But (perhaps ironically) in numerical methods in these areas, proofs are perhaps much more useful.  How accurate is Simpson's rule?  How much computer power do you need to get within &lt;span class=&quot;math-container&quot;&gt;$0.0001$&lt;/span&gt; for your integral?  Proofs are key to this - especially the somewhat (to students) bizarre requirements about absolute values of various derivatives.  Show an example that shouldn't work, but it does.   Now we need a proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another idea is to talk about integration techniques.  Give them a really hard integration by partial fractions or nasty multiple parts/substitution, then another.  When does it work, when not?  You can prove it - or that it isn't, Liouville's Theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't guarantee even a little success; it's hard to find resources at times (my examples are probably half-baked), and you have a lot of headwinds, as is evident from your post.  But in my experience, showing people what mathematics &lt;em&gt;cannot&lt;/em&gt; do is a first direction that will motivate them to love math for its own sake, not just for the satisfaction of a good successful computation.  We need both.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-04-03T13:54:03.707" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15441" PostTypeId="2" ParentId="15434" CreationDate="2019-04-03T20:24:34.803" Score="2" Body="&lt;p&gt;As usual, my advice is contrarian, but you need to consider alternate viewpoints.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you would be better off not pushing so hard.  It's like pestering a girl who doesn't like you. Not a good approach.  Slightly pathetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Make proofs the spice or garnish or relish or what have you to the main course.  Don't try to convince people they are better than the meat and potatoes.  They are entitled to their point of view of enjoying technique more.  (For that matter they are CORRECT in saying calculus and diffyQs, for most students, are taught as service courses for engineering and science.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also communicate some sympathy and exceptionalism. &quot;We've spared most of the proofs, but this is a really important one, so pay attention.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are forcing them to learn/test more proofs than are in the text (or drilling them more than the homework percentage of proofs), stop.  Just stop.  The students are seeing you pushing MORE peas at them than even in the balanced diet.&lt;/p&gt;&#xA;" OwnerUserId="12138" LastActivityDate="2019-04-03T20:24:34.803" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15442" PostTypeId="2" ParentId="1266" CreationDate="2019-04-03T20:36:30.600" Score="1" Body="&lt;p&gt;I don't think the other answers answer what you have appeared to be asking so I will write my own answer. It looks like your question was whether pointing out the mistake is useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it depends on what you mean by pointing it out. Do you mean just stating that that's not how it's supposed to be written or do you mean giving a reason why that's not how it's supposed to be written? I think for some students, just pointing out that that's not how it's done might not actually work. I think some students when they're taught to do things a certain way will keep insisting on understanding why it's supposed to be done that way and of those students, the ones who wrote 1/3x to mean (1/3)x, they might need to have it explained that until other people are told how to do things, to them, 1/3x could either mean 1/(3x) or (1/3)x. I don't know if it's true or not but if it is true that by convention, 1/3x means 1/(3x), then they could be given that reason. It could also be explained to them that when they write the other notation for 1/3, others may take it as writing that notation and be confused because they haven't been taught to use the other notation to indicate that they mean (1/3)x and not 1/(3x).&lt;/p&gt;&#xA;" OwnerUserId="11718" LastActivityDate="2019-04-03T20:36:30.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15443" PostTypeId="2" ParentId="15434" CreationDate="2019-04-03T23:41:07.350" Score="5" Body="&lt;p&gt;Maybe you can motivate the value of proofs by showing seemingly true&#xA;claims which are in fact false, justifying the need for a proof&#xA;of even an &quot;obvious&quot; claim.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example,&#xA;this appears to be a dissection of an &lt;span class=&quot;math-container&quot;&gt;$8 \times 8$&lt;/span&gt; square&#xA;to a &lt;span class=&quot;math-container&quot;&gt;$13 \times 5$&lt;/span&gt; rectangle: &lt;span class=&quot;math-container&quot;&gt;$64$&lt;/span&gt; vs. &lt;span class=&quot;math-container&quot;&gt;$65$&lt;/span&gt; unit squares.&#xA;It takes some effort to uncover the flaw.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/5SQOk.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5SQOk.gif&quot; alt=&quot;fake-dissection&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://www.ics.uci.edu/~eppstein/junkyard/dissect.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Eppstein&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Perhaps better is this MSE question,&#xA;&lt;a href=&quot;https://math.stackexchange.com/q/820686/237&quot;&gt;'Obvious' theorems that are actually false&lt;/a&gt;.&#xA;For example, &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/822137/237&quot;&gt;There are more rational numbers than there are integers&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/820703/237&quot;&gt;One can arbitrarily rearrange the terms in a convergent series without changing its value.&lt;/a&gt;. This one's quite complicated, as the comments indicate, which is great for discussions.&lt;/li&gt;&#xA;&lt;li&gt;False induction proofs: &lt;a href=&quot;https://math.stackexchange.com/q/627969/237&quot;&gt;Induction without a base case&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-04-04T22:14:37.117" LastActivityDate="2019-04-04T22:14:37.117" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15444" PostTypeId="1" CreationDate="2019-04-03T23:43:29.637" Score="0" ViewCount="756" Body="&lt;p&gt;One of the selling points of re-hashed American 1990s high school math programs is that they are &quot;integrated&quot;, that is, combine algebra, geometry, statistics, trigonometry just like the European programs do, and the Europeans know better (obligatory nod to Finland follows).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering that SE is visited from all over the world, I am interested in validity of this statement. I wonder which countries use integrated math course for HIGH school (not so much interested in middle school, but factoids like starting geometry in 7th grade are appreciated). Also, which countries use courses like algebra and geometry simultaneously in the same grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, in Russia algebra starts at 7th grade (elementary and secondary school altogether is 11 years). Geometry also starts at 7th grade simultaneously with algebra. Two subjects are taught in parallel, but the textbooks are separate for each subjects. Physics also starts at 7th grade (there is no &quot;science&quot; subject in Russian schools, there are physics, chemistry, biology, geography, astronomy). All three subjects mentioned above go for five years until the end of school. The last two years of algebra deal with trigonometry and pre-calculus (basics of differential and integral calculus), all mandatory. The last two years of geometry deal with stereometry (3D geometry). Physics do not use any calculus in it, only some trigonometry is used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I especially would appreciate links to specific programs and textbooks. Searching online is not that easy, considering that I don't know French, German or, say, Malay (meaning that I welcome info from all over the world, not only from Europe).&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2020-01-14T20:02:05.377" LastActivityDate="2020-01-14T20:02:05.377" Title="Integrated math curriculum in different countries" Tags="|secondary-education|algebra|geometry|curriculum|physical-sciences|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15445" PostTypeId="1" CreationDate="2019-04-04T03:26:25.997" Score="7" ViewCount="441" Body="&lt;p&gt;I read on the internet that Finland has the best education system in the world and that in Finland, students are taught to love mistakes and that's how they learn and become smarter. I could not find the details of how math is taught and why that method of teaching works. I would like a detailed explanation of how they do it and why it works, or a link to a detailed description if it is available. Maybe a scientific journal or a book explaining not how to teach students, but a researched property of how it is getting taught and why satisfying that property works. Although I might not bother with that book for now, I might bother with it sometime because it's a particularly useful book because it summarizes the education system in Finland and why it works so well.&lt;/p&gt;&#xA;&lt;p&gt;For example, for the students who insist that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; is not the same thing as &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{6}$&lt;/span&gt;, do they listen to how the student thinks the number system works and discuss the topic with them in the form of research instead of telling them they are crazy and wrong and do something like construct the multiples of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;, the multiples of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; and the multiples of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{6}$&lt;/span&gt; separately and explain that although &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{6}$&lt;/span&gt; belongs to a completely different system than &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; and so it's an entirely different entity, if we work in the system of multiples of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{6}$&lt;/span&gt;, we find that &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{6} \times 3 = 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{6} \times 2 = 1$&lt;/span&gt; so we can redefine &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; to have a different meaning than we previously defined it to have like you can redefine the meaning of a symbol in Python and we now redefine &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{6}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{6}$&lt;/span&gt; so we can calculate that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="7365" LastEditDate="2022-11-10T15:47:06.310" LastActivityDate="2022-11-10T15:47:06.310" Title="How is math taught in elementary school in Finland?" Tags="|mathematical-pedagogy|primary-education|finland|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15446" PostTypeId="2" ParentId="11813" CreationDate="2019-04-04T05:00:25.283" Score="-3" Body="&lt;p&gt;When I learned the rule for differentiating a power, I actually understood why it worked. I'm not sure if it's the case for everybody. If it's not the case for everybody, I have a possible idea but it's only my theory. It's to change the entire education system to teach a whole lot less and have people discover things on their own based on what they were taught. That way, students will become smarter and know how to check things themselves with their smartness in an engineering job and not just put blind trust into rules they were taught always working. They may later encounter new and different formula that either are actually wrong or are liable to be misunderstood so I think it's good to get them into the habit of checking things on their own. Also, some people if they do just get taught how to do it without understanding and then later get an engineering job, even if the boss of that job knows that those formula are mathematically correct, that person might still refuse to put blind trust into using formulae they don't know how to prove are mathematically correct because they don't know that other people aren't making a mistake in thinking they're correct and will have to be respected for refusing to blindly apply formulae they don't know how to prove in the job.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before that change is made, it will have to be really well researched exactly how the whole system and job market should change where a lot of knowledge moves to work place specific training. Finland has a really good education system so the Finish National Agency for Education might be able to help them with that research. Here's my possible idea which may need researching whether or not to do. I think a lot of students when they get taught that much end up learning less because the curriculum is too fast for them to follow so once we remove the need to be taught so much, they might end up learning more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some basics of how Polish notation works. To denote a sum of two expressions, you write + followed by each of those expressions but with spaces in between. Polish notation also expresses natural numbers in standard decimal notation with no spaces between the digits but spaces on both sides of the decimal notation. We could use a modified Polish notation that I invented. We could use ∅ to denote 0, S to denote the successor operation, and + to denote addition where once you've constructed a notation for a natural number, you can stick an S onto the beginning to denote its successor but without a space in between. Also, once you've constructed two notations for a natural number, you can use + followed by one of those expressions and then the other expression with no spaces in between to denote their sum. After learning just a few basics like the number 0 denoted ∅, the successor operation denoted S, and the addition operation denoted + and write it in Polish notation, the students can be left to do almost all play with very little teaching. Then they might discover what they're interested in discovering based on those basics and become really smart, but they will also have the right idea of how natural numbers work because there's no dispute about how they work. Also by using Polish notation, the notation is already unambiguous and already has no brackets so the're no need to learn PEDMAS and the time can instead be used to teach something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, we could switch to senary notation but wait until way later to teach it when they can actually learn it more efficiently. They could be taught that the senary notation of a number is the string of characters that describes the method of getting to that number by starting from 0 and then applying a series of operations each of which is of the form of left multiplying by 6 then left adding a number from 0 to 5. So the senary notation for 95 would be +SSSSS∅×SSSSSS∅+SSS∅×SSSSSS∅+SS∅×SSSSSS∅∅. I think senary is the base that people should use because it makes it easy for students to teach themselves how to find the quotient and remainder of any division problem because it has such a simple single digit multiplication table because every natural number from 1 to half of 6 is a factor of 6. They probably should not even be given the single digit multiplication table and instead be left to completely teach themselves how to find the quotient and remainder in senary, and they will realize that some of the steps are to mentally compute the product of two single digit numbers all on their own without ever having seen a multiplication table. Here, the problem will be considered entirely a problem about pure number theory because they have not yet been taught how to compute a division problem between natural numbers and give the answer as a non whole number in mixed fraction notation. Maybe way later after they've got losts of experience performing calculations in senary notation, then they could be taught that&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the character 0 is short hand for +∅×SSSSSS∅&lt;/li&gt;&#xA;&lt;li&gt;1 is short hand for +S∅×SSSSSS∅&lt;/li&gt;&#xA;&lt;li&gt;2 is short hand for +SS∅×SSSSSS∅&lt;/li&gt;&#xA;&lt;li&gt;3 is short hand for +SSS∅×SSSSSS∅&lt;/li&gt;&#xA;&lt;li&gt;4 is short hand for +SSSS∅×SSSSSS∅ and&lt;/li&gt;&#xA;&lt;li&gt;5 is short hand for +SSSSS∅×SSSSSS∅&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Then the compact senary notation for 95 would be 532∅. I know the digits appear in backwards order from conventional senary notation but I think it's worth it for the new school kids in order to define a digit as an operation and have it proceed the operand. If they're introduced to it too early, they might forget how they're really performing calculations in senary and adopt an autopilot method of performing them that they find so intuitive and can't figure out how to break down further. Also by convention, those digits should strictly be used to denote an operation of multiplying by 6 then adding a number and not to denote an operation of multiplying by 10 and then adding a number because it might be confusing having multiple meanings. The convention should be that you use decimal when writing numbers the normal way and senary when using digits to denote operations in modified Polish notation. Then the switch to senary will occur at the same time as the switch to modified Polish notation. If somebody wants to use another base like quinary, they should write it as a long string of characters which really is a valid notation for that number instead of using a digit to represent a different operation than it really means. So it's correct to denote 95 as +∅×SSSSS∅+SSSS∅×SSSSS∅+SSS∅×SSSSS∅∅ but it's not correct to denote 95 as 043∅.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later when they're a lot smarter and can learn it better, maybe they could be introduced to the negative numbers and even later to all the real numbers, but probably should not be left to discover properties all on their own because they might get the wrong idea of how real numbers work. Instead, the teacher could construct the dyadic rational numbers, the numbers that have a terminating notation in binary except that they don't actually introduce the binary decimal expansion, and then the rest of the real numbers from those Dedekind cuts of the dyadic rational numbers where the lower part has no maximal element nor does the higher part have a minimal element. According to &lt;a href=&quot;https://nrich.maths.org/2550&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://nrich.maths.org/2550&lt;/a&gt;, students work well with halves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the time they start taking calculus and already know how the real numbers work, I think they can be taught to teach themselves Calculus. It might actually be better for them to teach themselves because they might become smarter in the long run as a result. They should probably just about once a month be given some homework problems and be given a whole month to complete them so that they will have more than enough time to teach themselves Calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They could be told just a few basic laws such as &lt;span class=&quot;math-container&quot;&gt;$\forall a \in \mathbb{R}\forall b \in \mathbb{R}\forall c \in \mathbb{R}$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is positive, then &lt;span class=&quot;math-container&quot;&gt;$a^{b + c} = a^b \times a^c$&lt;/span&gt; and exponentation to a natural number exponent is iterated multiplication where 1 is the empty product and exponentation is continuous. The criterion that exponentation is continuous allows you to evaluate a positive number raised to an irrational power. They could also be taught that a logarithm is the inverse of a left exponentation and a root is the inverse of a right exponentation such as taking the third power. They could be taught using a student centered approach until they understand the basics and then guided to figure out other properties all on their own without being told what they are such as &lt;span class=&quot;math-container&quot;&gt;$\forall a \in \mathbb{R}\forall b \in \mathbb{R}\forall c \in \mathbb{R}$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is positive, then &lt;span class=&quot;math-container&quot;&gt;$(a^b)^c = a^{b \times c}$&lt;/span&gt;. After they get taught what a derivative is, then they could be told that &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is defined to be the number such that &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}e^x = e^x$&lt;/span&gt;. Then they could be left to teach themselves how to differentiate any function that can be gotten from the functions they were already introduced to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe schools could get students to teach themselves how to differentiate any function that can be gotten from the functions they know how to differentiate by giving them one homework problem for every 3 school days. For example, their first homework problem could be to differentiate &lt;span class=&quot;math-container&quot;&gt;$e^{e^x}$&lt;/span&gt;. Then they'll teach themselves the rule for differentiating a composition. Next, they could be asked to differentiate &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt;, then they will teach themselves the rule for differentiating an inverse. Next, they could be given the problem of differentiating &lt;span class=&quot;math-container&quot;&gt;$x^{2.5}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2^x$&lt;/span&gt; and told that it will not be for 6 school days that they are given more homework problems. Then they'll figure out that they can do that by expressing &lt;span class=&quot;math-container&quot;&gt;$x^{2.5}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$e^{\ln x \times 2.5}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2^x$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$e^{\ln 2 \times x}$&lt;/span&gt;. 6 days later, they could be asked to differentiate &lt;span class=&quot;math-container&quot;&gt;$(x^2 + 1)^{x^3}$&lt;/span&gt;. Then they'll teach themselves that for any binary operation, if you know how to differentiate any expression of the form of that binary operation on two seperate operands where one operand of that binary operation is constant and the other operand is a function you already know how to differentiate, then you use the rule for differentiating the expression treating one operand as constant and then the rule for differenting while treating the other operand as constant and then add the two results to get the derivative of the whole expression. 3 days later, they could be asked to differentiate a product and they'll see that they can do it by using the general rule for differentiating a binary operation. Next, they one possible idea of what they could be asked is to differentiate &lt;span class=&quot;math-container&quot;&gt;$\log_x2$&lt;/span&gt;. Then they'll figure out to use the general binary operation rule in reverse. Then they'll figure out that &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}x^{\log_x2} = \frac{d}{dx}2 = 0$&lt;/span&gt; but also &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}x^{\log_x2} = x^{\log_x2}\ln x(\frac{d}{dx}\log_{x}2) + \log_{x}2(x^{\log_{x}2 - 1})$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}x^{\log_x2} = x^{\log_x2}\ln x(\frac{d}{dx}\log_{x}2) + \log_{x}2(x^{\log_{x}2 - 1}) = 0$&lt;/span&gt;. Then they can use that equation to solve for &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}\log_x2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, that doesn't force them to use the binary operation rule in reverse because that's not the only way because &lt;span class=&quot;math-container&quot;&gt;$\log_x2$&lt;/span&gt; can also be expressed as &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\log_2x}$&lt;/span&gt;. I haven't researched enough but one possible idea is to introduce tetration to real heights. According to the Complex heights section of the Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Tetration&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tetration&lt;/a&gt;, one such way was proposed. Then when they're asked to differentiate, the teacher will have to be clear what forms students are allowed to write the derivative of a function as because once tetration is introduced, the derivative of any right tetration or left tetration operation probably still cannot be expressed in terms of functions the students were introduced to before. The super root of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to the base &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is the number &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; tetrated to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. Maybe they could specifically introduce two other binary operations. One assigns to each pair of numbers the derivative of left tetration by the first part of the pair applied to the second part. The other assigns to each pair the derivative of right tetration by the second part of the pair applied to the first part. Then they could be asked to express the derivative of the super root of 2 to the base &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in terms of all the functions they were already introduced to and their inverses. In that case, using the binary operation rule in reverse is probably the only way. I'm not sure if it's worth introducing tetration so that students can be given a differentiation problem where using the binary operation rule in reverse is the only way and then they'll keep thinking until they come up with the idea of using the binary operation rule in reverse all on their own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article &lt;a href=&quot;https://www.huffpost.com/entry/teaching-discovery-learning_b_856463&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching Is not Learning — The Guided Discovery Approach for Learning&lt;/a&gt; seems to support my idea that discovery learning could work so well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Update: My opinion has changed. Modified Polish notation should use octal, not senary, because 8 is a power of 2. I don't like hexadecimal as much because its single digit multiplication table is so large. I want the system of all notations that can be expressed using the characters ∅, S, and +, to be one of the first things the students get taught and after that, I want the associative law of natural number addition to be demonstrated in an engaging way. After that, I want them to be left to do almost almost all play for a really long time and nudged to do very general thinking about number theory. Study of prime numbers will probably naturally be extremely little on their mind so there's no need to use senary for the purpose of prime numbers. The idea of computing the binary notation of a natural number or something like that will probably come naturally, and some study of properties of the binary notation will probably also come naturally, and I don't want to divert attention away from that study by using senary instead of octal. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schönhage–Strassen algorithm&lt;/a&gt; is a really fast multiplication algorithm. As a result of that study, they may eventually come up with something like the Schönhage–Strassen algorithm for octal all on their own. However, the Schönhage–Strassen algorithm should not be introduced to them in school. When they're given homework problems, they should be infrequent so that they can explore each question in great depth, and they should be left to figure out the answer all on their own. Then they will come up with their own method. Introducing them to the Schönhage–Strassen algorithm will get them blindly following the rules they're taught instead of getting them to think for themselves how to figure out the answer to a problem.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="11718" LastEditDate="2020-02-24T04:35:31.647" LastActivityDate="2020-02-24T04:35:31.647" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15447" PostTypeId="2" ParentId="15444" CreationDate="2019-04-04T06:35:23.047" Score="1" Body="&lt;p&gt;If by high school you mean lukio/gymnasium, the Finnish info is available in Finnish and Sápmi &lt;a href=&quot;https://www.oph.fi/saadokset_ja_ohjeet/opetussuunnitelmien_ja_tutkintojen_perusteet/lukiokoulutus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.oph.fi/saadokset_ja_ohjeet/opetussuunnitelmien_ja_tutkintojen_perusteet/lukiokoulutus&lt;/a&gt; and in Swedish at &lt;a href=&quot;https://www.oph.fi/lagar_och_anvisningar/laroplans-_och_examensgrunder/gymnasiet&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.oph.fi/lagar_och_anvisningar/laroplans-_och_examensgrunder/gymnasiet&lt;/a&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The English version does not seem to be free, but does seem to be available, supposing by &quot;high school&quot; you mean upper secondary education: &lt;a href=&quot;https://www.oph.fi/english/curricula_and_qualifications/general_upper_secondary_education&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.oph.fi/english/curricula_and_qualifications/general_upper_secondary_education&lt;/a&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case of link rot, one should search the webpages of Opetushallinto/Utbildningsstyrelsen, the Finnish national agency of education, for more topical links.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one wishes to read the official document via machine translation, I would recommend starting with the Swedish document as the baseline. Swedish is a larger language and an Indo-European one, so machine translation to other Indo-European languages is likely to be of higher quality, and the document by itself is also easier to understand for someone who knows Indo-European languages. (Use Finnish if machine translating to e.g. Estonian and maybe Hungarian.)&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Structure and role of lukio/gymnasium studies&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;After compulsory education, a child may go to lukio for general studies or to ammattiopisto/yrkeskola to qualify for a profession. Both lukio and ammattiopisto qualify one for higher vocational studies (bachelor level) and lukio also prepares one for university studies (typical default is master level).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In lukio one has to take a certain amount of courses, in addition to which there is the national matriculation examination. Some courses are compulsory. One can take either &quot;short&quot; or &quot;long&quot; mathematics studies, with different compulsory courses. It is possible to change from one to the other during studies.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Books in mathematics for lukio/gymnasium, in Finnish&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;There are Finnish books in natural science available here: &lt;a href=&quot;https://www.akateeminen.com/kirjavalikoima/oppikirjat/lukiokirjat/matematiikka&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.akateeminen.com/kirjavalikoima/oppikirjat/lukiokirjat/matematiikka&lt;/a&gt; . Matematiikka is mathematics, pitkä means long, lyhyt means short.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Biologia is biology, fysiikka is physics, kemia is chemistry.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Books in short mathematics: &lt;a href=&quot;https://tuotteet.sanomapro.fi/tuotteet/lukio/lyhyt-matematiikka.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://tuotteet.sanomapro.fi/tuotteet/lukio/lyhyt-matematiikka.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Books in long mathematics: &lt;a href=&quot;https://tuotteet.sanomapro.fi/tuotteet/lukio/pitka-matematiikka.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://tuotteet.sanomapro.fi/tuotteet/lukio/pitka-matematiikka.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Books in both mathematics: &lt;a href=&quot;https://otava.kauppakv.fi/sivu/lukio-matematiikka/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://otava.kauppakv.fi/sivu/lukio-matematiikka/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mostly books in lukio mathematics: &lt;a href=&quot;https://www.huuto.net/osasto/kirjat-ja-lehdet/lukion-oppikirjat/matematiikka/285&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.huuto.net/osasto/kirjat-ja-lehdet/lukion-oppikirjat/matematiikka/285&lt;/a&gt; (this is an Ebay-like site.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of open textbooks of unknown quality: &lt;a href=&quot;https://avoinoppikirja.fi/mat-lukio&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://avoinoppikirja.fi/mat-lukio&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-04-13T08:23:37.907" LastActivityDate="2019-04-13T08:23:37.907" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15449" PostTypeId="1" CreationDate="2019-04-04T14:19:20.093" Score="0" ViewCount="300" Body="&lt;p&gt;My sibling is done with high school and has always scored A in Maths and am not in position to advise her on the future in line of her niche.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She's not yet in university and she's in her vacation but I hope that by societies, she can get inspiration from like-minds through engagement and interaction to achieve greater (she's a big dreamer.) she currently has little career guidance other than her love for numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any societies to join or courses to take that would be good for them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The location is Uganda, Kampala.&lt;/p&gt;&#xA;" OwnerUserId="12140" LastEditorUserId="11" LastEditDate="2019-04-09T13:04:43.717" LastActivityDate="2020-09-16T04:23:26.373" Title="Future in mathematics" Tags="|undergraduate-education|student-motivation|africa|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15450" PostTypeId="2" ParentId="15445" CreationDate="2019-04-04T16:02:31.117" Score="6" Body="&lt;p&gt;You can find the official document describing the Finnish education system at Opetushallinto's website; in Finnish and Sápmi at &lt;a href=&quot;https://www.oph.fi/saadokset_ja_ohjeet/opetussuunnitelmien_ja_tutkintojen_perusteet/perusopetus&quot; rel=&quot;noreferrer&quot;&gt;https://www.oph.fi/saadokset_ja_ohjeet/opetussuunnitelmien_ja_tutkintojen_perusteet/perusopetus&lt;/a&gt; and in Swedish at &lt;a href=&quot;https://www.oph.fi/lagar_och_anvisningar/laroplans-_och_examensgrunder/grundlaggande_utbildningen&quot; rel=&quot;noreferrer&quot;&gt;https://www.oph.fi/lagar_och_anvisningar/laroplans-_och_examensgrunder/grundlaggande_utbildningen&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is some information in English at &lt;a href=&quot;https://www.oph.fi/english/curricula_and_qualifications/basic_education&quot; rel=&quot;noreferrer&quot;&gt;https://www.oph.fi/english/curricula_and_qualifications/basic_education&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case the links decay, the new links should be located somewhere in the webpages of Opetushallinto/Utbildningsstyrelsen, the national agency for education. Their current webpages are &lt;a href=&quot;https://www.oph.fi/english&quot; rel=&quot;noreferrer&quot;&gt;https://www.oph.fi/english&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-04-05T05:37:23.387" LastActivityDate="2019-04-05T05:37:23.387" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15451" PostTypeId="2" ParentId="14565" CreationDate="2019-04-04T16:31:01.367" Score="0" Body="&lt;p&gt;You didn't indicate whether you cared about paying for the website, but &lt;a href=&quot;https://www.ixl.com/math/algebra-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;IXL.com&lt;/a&gt; provides the granularity that you're looking for. From the link provided, I'd guess you'd want to drill your student on skills B.1-B.3 (operations),  I.1-I.4 (variable expressions and equations), and perhaps V.1-V.11 (exponent properties). The website only provides 10 free problems per day, however, and it takes about 8 correct problems in a row to reach the most difficult questions in each section. Paying for access to all math questions is around $10/month.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2019-04-04T16:31:01.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15453" PostTypeId="2" ParentId="15396" CreationDate="2019-04-04T18:03:01.597" Score="3" Body="&lt;p&gt;The best way to teach this concept would be using money. For example if you were in the US, you can say that you want the least coins and bills for $5.80 cents. Only a five dollar bill, 3 quarters, and a nickel would satisfy that criteria.   Once the class understands the idea of the &quot;least coins and bills&quot;, ask them how you would apply a similar concept to feet and inches.  They can then help you give a name to the idea, maybe least feet and inches?&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2019-04-04T18:03:01.597" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15454" PostTypeId="1" CreationDate="2019-04-04T21:29:25.273" Score="5" ViewCount="465" Body="&lt;p&gt;I work in a University Tutor Lab that covers content up to Calculus II. However, when a student in a Calculus III or Differential Equations class comes in, some other tutors and I will still tutor these students if they ask for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, some students in an Introduction to Proofs class and a intro level abstract algebra course have come in for help as well. But some tutors are more unwilling to help these students based on a cut-off point they believe in when it comes to tutoring mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a conversation that often comes up often amongst the tutors. Some believe there should be a cut-off right after Calculus II, while others feel it comes after Differential Equations. Even some believe there really isn't a cut-off point at all. (We only help these students in our free time, but are not required to do so.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there is such a cut-off when it comes to tutoring and helping students out with mathematics, where is it or should it be?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I am not discussing the lab's policy, but as educators should we have a cutoff point where we expect students to learn and do mathematics on their own? Or should they be able to seek tutoring at any level?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="12141" LastEditorUserId="64" LastEditDate="2019-04-07T21:14:51.670" LastActivityDate="2019-04-07T21:14:51.670" Title="When should students stop receiving tutoring for mathematics?" Tags="|tutoring|" AnswerCount="4" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="15455" PostTypeId="2" ParentId="15454" CreationDate="2019-04-04T21:47:13.980" Score="4" Body="&lt;p&gt;This is completely up to you to whoever is in charge of this center. There is no apparent reason that tutoring should stop after Calculus II, III, IV or V or something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, what is important is that all of you follow the same policy (and it would be the best to announce this policy publicily). (If you have one a cut-off point, you could fix one hour per week where you answer questions beyond your cut-off point.) If there is no coherent policy, students are likely to complain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When discussing which policy to use, keep in mind the following questions:&#xA;- In which courses students struggle most?&#xA;- Do you have enough time for low-level courses and their questions if you allow questions to, say, abstract algebra?&#xA;- For which courses are the tutors qualified?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I don't quite understand: why is &quot;Introduction to proofs&quot; only after some other courses in your university? Do people not do proofs the first few semesters?&lt;/p&gt;&#xA;" OwnerUserId="12144" LastActivityDate="2019-04-04T21:47:13.980" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15456" PostTypeId="2" ParentId="15454" CreationDate="2019-04-05T05:51:41.227" Score="11" Body="&lt;p&gt;At University of Jyväskylä there is Ratkomo twice a week for some hours. It means at least one faculty member and typically some PhD students wander a corridor with tables and chairs and help everyone with whatever questions they ask. From my limited experience with Ratkomo, must of the students were from mathematics courses intended for the first two years, but some were asking for help with more advanced material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I doubt there is any level in mathematics where discussing the problem with someone else stops being useful - this is one reason why there is cooperation among research mathematicians, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, since the students could ask about any and all courses, it was by no means certain that the person helping them was familiar with the material, or even if they were, there were no guarantees that they could just spit out a solution. Hence, especially for students asking about more advanced topics, it often became a process of figuring out the solution together. Engaging in this process with a peer or a more mature mathematician certainly remains useful far into one's career.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe there is no cut-off as such, but at some point explaining one's problems becomes sufficiently difficult that anyone who understands them and provides helpful insight is likely to become a co-author.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-04-05T14:48:05.797" LastActivityDate="2019-04-05T14:48:05.797" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15458" PostTypeId="2" ParentId="11548" CreationDate="2019-04-06T06:51:09.223" Score="1" Body="&lt;p&gt;Teacher College should have full sets of Unified Modern Mathematics, 1-6 course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the first pages of the textbook (FYI - Columbia is in New York City, US)-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3kR9G.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3kR9G.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RYyJi.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RYyJi.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7935" LastEditorUserId="64" LastEditDate="2019-04-06T15:22:47.110" LastActivityDate="2019-04-06T15:22:47.110" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15459" PostTypeId="2" ParentId="15403" CreationDate="2019-04-06T20:58:25.830" Score="1" Body="&lt;p&gt;There have been many positive reports on the use of collaborative note taking in the STEM disciplines, in part because they help students learn not only the &lt;em&gt;content&lt;/em&gt; of a lecture, but they also help students to model and replicate independently &lt;em&gt;how to take good notes&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is consistent with research which demonstrates that the best mode of instruction comes not only from lecturing content, but also modeling how to learn and understand it best.  This would include &lt;em&gt;how to take good notes&lt;/em&gt; when encountering delivered content, whether in the form of a lecture, or text, (or powerpoint, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See, e.g., &lt;a href=&quot;https://www.cultofpedagogy.com/note-taking&quot; rel=&quot;nofollow noreferrer&quot;&gt;Note-taking: A research round-up&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should also note that such efforts (collaborative note-taking, etc.) and research on the effectiveness of those efforts has focused primarily on secondary education students (high-school), and undergraduates in their first year or two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also note that collaborative note-taking doesn't &lt;em&gt;require&lt;/em&gt; a computer, if socio-economic concerns are of issue.  Guidance can be given in class, on transparencies, or at periodic intervals during classroom reviews (on transparencies of instructor's notes) during which students can compare their own notes to those of the instructor (prof/TA/teacher).  Also an option is occasional group work or study groups in which students can pair off into two or more team members to compare their notes with other students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps setting up &lt;em&gt;study-groups&lt;/em&gt; where you and other students in your class can meet in person to compare notes is also an idea worth considering.  Don't hesitate to ask your professor or TA whether such a study-group/study session can be announced in class, and same wrt using Google Docs for collaborative notetaking.  If you're at a university/college, there are often class websites on the university/college's server, and often these have discussion boards/chats, over which you can suggest your ideas. If you're looking to help get the word out, your prof or TA or teacher will likely be amenable to your efforts. &lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-04-06T21:34:26.090" LastActivityDate="2019-04-06T21:34:26.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15461" PostTypeId="2" ParentId="15454" CreationDate="2019-04-07T00:59:39.003" Score="3" Body="&lt;p&gt;There's no cutoff point after which people, students or not, should be expected to learn and do mathematics exclusively on their own.  I'm a mathematics professor, and I have no qualms about &quot;receiving tutoring&quot; - that is, asking other mathematicians about when I'm struggling to learn something new.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your center might, of course, have a cutoff for any of a number of practical reasons, but there's no pedagogical reason that there's something wrong with helping students after a certain point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The flip side is that there's no level at which students should be working entirely with tutors: students at all levels should &lt;em&gt;also&lt;/em&gt; be working to understand material independently.  Students in Intro to Proofs shouldn't be &quot;handheld&quot;, but students in Calculus I shouldn't be either.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-04-07T00:59:39.003" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15463" PostTypeId="2" ParentId="15449" CreationDate="2019-04-07T06:14:31.050" Score="3" Body="&lt;p&gt;(Sticking my lion claw in.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;User18858:  That is great about your sister.  I just want to give you a slight caution.  The vast majority of people who &quot;like math&quot; or &quot;do well at math&quot; go on to work in fields OTHER than straight mathematics.  There are many opportunities to use skill in math in engineering, business, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you listen to advice here, you are apt to get a blind spot to this since the vast majority of people here are teachers of mathematics (and even while teaching service classes), tend to ignore the statistics of their students.  It is not meant unkindly, but it is still a blind spot that many have.&lt;/p&gt;&#xA;" OwnerUserId="12153" LastActivityDate="2019-04-07T06:14:31.050" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15465" PostTypeId="2" ParentId="15454" CreationDate="2019-04-07T16:23:41.493" Score="3" Body="&lt;p&gt;As others have said, for a broad definition of tutoring, there really isn't a point at which anyone at any level should be completely alone. However, in the more narrow sense of a university tutoring lab, there is some point at which this is just not the right venue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At some point, students need to get used to talking to professors directly, and not just about grades. (Also, their peers.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they are taking upper level courses, they are likely majors or minors in math or a related field. They are going to need letters of recommendation and to develop personal relationships. They aren't going to get that in the relatively anonymous tutoring lab.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, you also have the practical question of whether or not they can get good answers for advanced questions from the staff of the lab. I remember being one of the only people both willing and able to answer these questions when I was in grad school. A fair number of the tutors answering the factoring questions were only a class or two beyond that material themselves.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2019-04-07T16:23:41.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15466" PostTypeId="1" CreationDate="2019-04-07T23:58:18.977" Score="6" ViewCount="632" Body="&lt;p&gt;I work in the University with students in a situation of disability, specifically, teaching them math and related things. I have a few students that are very visually impaired; they work with JAWS or NVDA, usually.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we are working in Calculus or some course which depends a bit in visual things to be understood, it goes slow. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like that the dynamics of the teaching were more fluid. So, I'm searching for some way to teach &lt;a href=&quot;https://www.geogebra.org/?lang=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geogebra&lt;/a&gt;, so they can apply these things for their study. Sadly, I think that Geogebra isn't too accessible for the visually impaired &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder if anyone have some good ideas to do a good geometric work with some accessible software, or if you know somehow to work with Geogebra for this purpose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance and sorry for my bad English :)&lt;/p&gt;&#xA;" OwnerUserId="12160" LastEditorUserId="238" LastEditDate="2019-04-08T20:57:48.700" LastActivityDate="2019-04-11T05:01:38.213" Title="Geogebra for Blind People" Tags="|geometry|technology-in-education|special-needs-education|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15469" PostTypeId="1" CreationDate="2019-04-09T03:11:23.840" Score="-2" ViewCount="170" Body="&lt;p&gt;Link to article cited in title, &lt;a href=&quot;https://en.wikipedia.org/wiki/Rational_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia rational number&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14831/how-to-explain-fractions-to-7-year-old-kid/14833#14833&quot;&gt;this answer&lt;/a&gt;, some students 14-18 are still struggling to understand fractions. Maybe some students know how to perform the calculations on rational numbers given in fraction notation but don't understand why it works. If they're taught the method described in that section, they might actually understand why it works so well. At first, the teacher could teach the student how to do creative thinking and then ask the student to explain why addition and multiplication can be defined that way and then guide them through to come up with an answer showing that addition and multiplication of ordered pairs in the same class always gives you a result in the same class so two ordered pairs can be defined to represent the same rational number when they're in the same class. Later, they could say that by definition, subtraction is addition of the additive inverse and division is multiplication by the multiplicative inverse. Later so that the student will become smarter, the student could be left with the task to teach themself how to compute a division problem on rational numbers given in fraction notation to get a result expressed in fraction notation. That way, they not only will know how to perform the calculations but will have also been guided to show that they have a real understanding of what the calculations actually mean. Even later to make them even smarter, they could be guided to come up with a proof that &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt; is a field.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="64" LastEditDate="2019-05-01T13:30:43.653" LastActivityDate="2019-05-01T13:30:43.653" Title="Could students be taught the concept of rational numbers the same way as in the Formal construction section of the Wikipedia article rational number?" Tags="|self-learning|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15470" PostTypeId="2" ParentId="15469" CreationDate="2019-04-09T03:37:32.620" Score="4" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Bad idea.  You are going in exactly the opposite direction.  Weak students do not need more formalism.  For instance, your Q field comment at end  (WTF?).  Or basically any Wikipedia article--which all assume formal proof or definitions are explanations--they are not.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If your math explanations are anything like your communication here, you will make things worse, not better with students.  The whole question itself is very hard to parse.  No paragraph breaks.  And one incredible run-on stream of consciousness sentence in the middle:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;At first, the teacher could teach the student how to do creative thinking and then ask the student to explain why addition and multiplication can be defined that way and then guide them through to come up with an answer showing that addition and multiplication of ordered pairs in the same class always gives you a result in the same class so two ordered pairs can be defined to represent the same rational number when they're in the same class.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Oh...I love the &quot;first teach creative thinking&quot; comment.  As if this was a well understood spice to just pull out of the rack and add to the soup.  No issue with how/what (and with weak students nonetheless).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12163" LastActivityDate="2019-04-09T03:37:32.620" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15475" PostTypeId="1" CreationDate="2019-04-10T05:48:23.817" Score="6" ViewCount="399" Body="&lt;p&gt;Inspired by this question: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/15474/what-makes-education-in-finland-so-good&quot;&gt;What makes education in Finland so good?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finland has marketed itself as a top country in education. Indeed, at some time, the Pisa results in Finland were quite good. However, the Pisa results started going down when a significant number of immigrants arrived (this is based on recollection from studying to become a teacher in Finland; might be wrong, and please correct if it is), and the gap between their results and the results of the native population used to be big. This suggests that the high Pisa results might have been due to a homogenous population of native Finnish speakers, as much as due to a successful curriculum. But maybe not, and maybe Finland has a very good school system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a research-based reason to believe that the educational system in Finland is particularly excellent, or just average, or worse than average? Is there a consensus among researchers about this?&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-04-15T13:48:08.000" LastActivityDate="2019-04-18T08:59:36.183" Title="Is the education system in Finland particularly good?" Tags="|reference-request|curriculum|education-research|finland|pisa|" AnswerCount="4" CommentCount="23" ContentLicense="CC BY-SA 4.0" />
  <row Id="15477" PostTypeId="1" CreationDate="2019-04-10T19:17:20.370" Score="8" ViewCount="666" Body="&lt;p&gt;I am teaching introductory real analysis this term and realize that my students have problem coming up with sequence in some arguments in real analysis. Let's take this example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Theorem: Given a function &lt;span class=&quot;math-container&quot;&gt;$ f: [a,b] \to \mathbb R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_0\in [a,b]$&lt;/span&gt;. If for all sequence &lt;span class=&quot;math-container&quot;&gt;$\{a_n\}_{n=1}^\infty$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$[a,b]\setminus \{x_0\}$&lt;/span&gt; which converges to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\{f(a_n)\}_{n=1}^\infty$&lt;/span&gt; converges to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has a limit &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof&lt;/strong&gt; Assume the contrary that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; does not have limit &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;. Then there is &lt;span class=&quot;math-container&quot;&gt;$\epsilon_0 &amp;gt;0$&lt;/span&gt; such that for all &lt;span class=&quot;math-container&quot;&gt;$\delta&amp;gt;0$&lt;/span&gt;, there is &lt;span class=&quot;math-container&quot;&gt;$x\in [a,b]\setminus\{ x_0\}$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$|x-x_0|&amp;lt;\delta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|f(x) - L|\ge \epsilon_0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then the next step is to choose (e.g.) &lt;span class=&quot;math-container&quot;&gt;$\delta = 1/n$&lt;/span&gt; and come up with a sequence &lt;span class=&quot;math-container&quot;&gt;$\{x_n\}_{n=1}^\infty$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$|x_n - x_0|&amp;lt;1/n$&lt;/span&gt;....&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This step involves the (countable) Axiom of choice. Every time I perform a similar argument in class, they seem to understand it. But they are failing in the HW/midterm. It seems that their complaint is that they cannot see how to choose the sequence.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me that their confusion is legit, since this is the major reason why the Axiom of choice got some criticisms.&lt;/p&gt;&#xA;&lt;p&gt;I would just throw &amp;quot;Hey! This is Axiom of Choice!&amp;quot; to them, but (1) this is not how we study real analysis here, where they don't have a solid background in set theory, and (2) that does not seem to help them understand the concept.&lt;/p&gt;&#xA;&lt;p&gt;So my question is, how do we in general motivate the (implicit) use of AC in real analysis?&lt;/p&gt;&#xA;" OwnerUserId="9793" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-04-12T06:32:20.987" Title="How to make students comfortable with the use of axiom of choice in analysis" Tags="|mathematical-analysis|" AnswerCount="3" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="15480" PostTypeId="2" ParentId="15477" CreationDate="2019-04-11T01:17:13.273" Score="5" Body="&lt;p&gt;I agree that the AOC is a red herring; that is not what the students find challenging here.  My suggestion (and it is only a suggestion) is to consider taking a certain portion of your course and making it more &quot;inquiry-based&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a bigger topic than can be adequately addressed in this space, obviously, but I have found that even quite weak students can really &quot;get&quot; at least some piece of truly difficult arguments, whereas when I've taught more traditional real analysis they seem to only partly get everything.  For instance, you might have the expert in the Dirichlet function and where it is useful, or the expert in showing things are continuous, etc.  (The best students will be expert in everything.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For possible resources you may wish to peruse the following (disclosure; I've been affiliated with some of these by publishing or editing):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.jiblm.org&quot; rel=&quot;noreferrer&quot;&gt;Journal of Inquiry-Based Learning in Math&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.tandfonline.com/loi/upri20&quot; rel=&quot;noreferrer&quot;&gt;PRIMUS&lt;/a&gt; has many articles on analysis, IBL, and their intersection&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.maa.org/press/books/the-moore-method-a-pathway-to-learner-centered-instruction&quot; rel=&quot;noreferrer&quot;&gt;Coppin et al. on the Moore method&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www2.kenyon.edu/Depts/Math/schumacherc/public_html/Professional/Research/closer/Closer.htm&quot; rel=&quot;noreferrer&quot;&gt;Carol Schumacher's book on the subject&lt;/a&gt; could be used in many types of classes&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Real analysis is one of the more popular topics to teach this way.  Naturally, you aren't going to &quot;get as far&quot; and it's not some kind of panacea that makes students magically get these arguments.  Your mileage may vary.  But I have found that, properly done, it can help some students who would otherwise always be lost understand at least one type of analytic argument fully, and sometimes helps the best students really know what is going on topologically and not just know how to parrot proofs.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-04-11T01:17:13.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15483" PostTypeId="2" ParentId="15466" CreationDate="2019-04-11T05:01:38.213" Score="6" Body="&lt;p&gt;Desmos Geometry is not as fully featured as GeoGebra. However, Desmos does have specific features for the visually impaired such a screen reader support, braille support, and audio trace. These features may not currently work for the &lt;a href=&quot;http://www.desmos.com/geometry&quot; rel=&quot;noreferrer&quot;&gt;www.desmos.com/geometry&lt;/a&gt; section, but perhaps the &lt;a href=&quot;http://www.desmos.com/calculator&quot; rel=&quot;noreferrer&quot;&gt;www.desmos.com/calculator&lt;/a&gt; fits some of your needs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See more details and instructions here: &lt;a href=&quot;https://www.desmos.com/accessibility&quot; rel=&quot;noreferrer&quot;&gt;https://www.desmos.com/accessibility&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4962" LastActivityDate="2019-04-11T05:01:38.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15484" PostTypeId="1" AcceptedAnswerId="15485" CreationDate="2019-04-11T05:11:14.260" Score="1" ViewCount="149" Body="&lt;p&gt;I am studying the basics of Quantum Physics (involving the characteristics of Schrodinger's Wave equation without actually analyzing it rigorously mathematically) this semester. I was wondering, however, if I did study it in the context of a mathematical background, it would be a great help. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wish to know what the prerequisites for studying PDEs and Probability Distributions are. I have already studied the following topics: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Single-Variable Calculus&lt;/li&gt;&#xA;&lt;li&gt;Partial Differentiation&lt;/li&gt;&#xA;&lt;li&gt;Vectors&lt;/li&gt;&#xA;&lt;li&gt;ODEs (just the basics)&lt;/li&gt;&#xA;&lt;li&gt;Combinatorics&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I would be grateful to learn what roadmap I should follow for learning to analyze and solve PDEs and also Probability Distributions. Also, kindly list a few textbooks to follow for the same. Thanks &lt;/p&gt;&#xA;" OwnerUserId="12170" LastEditorUserId="1608" LastEditDate="2019-04-11T12:46:21.730" LastActivityDate="2019-04-11T12:46:21.730" Title="Roadmap to studying PDEs for analyzing Quantum Physics better" Tags="|reference-request|textbooks|curriculum|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15485" PostTypeId="2" ParentId="15484" CreationDate="2019-04-11T06:26:03.503" Score="5" Body="&lt;p&gt;I recommend do more ODEs, first step.  You mention &quot;just the basics&quot;.  This is the first step to upgrade.  Take a normal ODE text (not the snippet of ODEs within a calculus book).   Make sure you are familiar with all of it, including the later chapters on series solutions, etc.  Do some of the harder homework problems with Legendre polynomials (shows up in solution of hydrogen atom).  Get exposed to Bessel functions and Hermite polynomials as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, you actually won't need deep understanding of several different PDEs for quantum course, since you just work with one or two of them (and turn them into ODEs when solving them).  Also, you don't need really probability theory or even a computational probability course.  As long as you understand the concept of a distribution (intuitively, not formally) that is plenty.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that said, PDEs and probes/shafts (probability and statistics) are great courses themselves.  But for much, much more than just support of your quantum course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Realistically, you don't need deep extra math course prep for your quantum course.  Just to stay up to speed on what you are doing in that course.&lt;/p&gt;&#xA;" OwnerUserId="12171" LastActivityDate="2019-04-11T06:26:03.503" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15486" PostTypeId="2" ParentId="10297" CreationDate="2019-04-11T20:02:34.540" Score="3" Body="&lt;p&gt;You might be interested in Brian C. Hall's book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/146147115X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantum Theory for Mathematicians&lt;/a&gt;. The author writes on his &lt;a href=&quot;https://www3.nd.edu/~bhall/book/quantum.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;webpage&lt;/a&gt; about the book:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This book aspires to be a self-contained and reasonably comprehensive treatment of quantum mechanics (excluding quantum field theory) from a mathematical perspective. No prior knowledge of physics is required, but only the basics of Hilbert spaces and real analysis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://www3.nd.edu/~bhall/book/quantum.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www3.nd.edu/~bhall/book/quantum.html&lt;/a&gt; for more information. There's also a helpful, detailed &lt;a href=&quot;https://www.amazon.com/gp/customer-reviews/R3Q509T6Z13W5G/ref=cm_cr_dp_d_rvw_ttl?ie=UTF8&amp;amp;ASIN=146147115X&quot; rel=&quot;nofollow noreferrer&quot;&gt;review&lt;/a&gt; on Amazon.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2019-04-13T07:34:55.333" LastActivityDate="2019-04-13T07:34:55.333" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15487" PostTypeId="2" ParentId="15477" CreationDate="2019-04-11T20:07:48.790" Score="10" Body="&lt;p&gt;As others who answered have pointed out, this issue is not the countable choice involved in defining the sequence that makes this challenging for learners. Rather, the difficulty is the semantic complexity of the negation of the statement to be proved. When I get close to this theorem or similar ones when teaching analysis, I like to give a &quot;fun&quot; assignment a couple of days in advance:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your friend asserts that for every &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;lephant, there is a &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;ay such that if the day is rainy, then the elephant forgets to bathe. How would you prove your friend incorrect?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note the logical structure of the elephant sentence is very similar to the definition of a limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am always surprised at the variation in incorrect answers, the rarity of correct answers, and the cognitive challenge this poses. It is revealing to listen to students discuss this challenge amongst themselves. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-04-11T20:07:48.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15488" PostTypeId="2" ParentId="15477" CreationDate="2019-04-12T06:32:20.987" Score="3" Body="&lt;p&gt;Like others, I don't think AC is the real issue. I don't think most students mean 'how can we do this infinitely many times?', but rather 'I don't know how to work out what to do'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I would use the idea of 'what information do we have available to us?' I (in the role of a student) don't have any ideas for creating a sequence, but instead of giving up I should just play around with anything I can do, and see if that gives me new ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I have at my disposal is one definition I know to be true (the non-existence of the limit). Depending on where your students are up to (it sounds like they must be very strong students), that could take a few steps. If dealing with the abstract statement is too hard to comprehend, try it for specific values. &lt;span class=&quot;math-container&quot;&gt;$\epsilon_0$&lt;/span&gt; is outside of my control (we could get away with pretending it is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;), but I get to choose a small &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;. Choosing &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; is a reasonable first step. I get handed back an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Then try some other small values (some students might pick the sequence &lt;span class=&quot;math-container&quot;&gt;$1/n$&lt;/span&gt;, others &lt;span class=&quot;math-container&quot;&gt;$2^{-n}$&lt;/span&gt;). Each try hands me an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Now making a sequence out of these seems less strange.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-04-12T06:32:20.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15489" PostTypeId="2" ParentId="10297" CreationDate="2019-04-12T07:48:50.427" Score="2" Body="&lt;p&gt;From a comment by Michael Joyce:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The notes are by Igor Dolgachev and can be accessed here: &lt;a href=&quot;http://math.lsa.umich.edu/~idolga/lecturenotes.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://math.lsa.umich.edu/~idolga/lecturenotes.html&lt;/a&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2019-04-12T07:48:50.427" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15490" PostTypeId="2" ParentId="10297" CreationDate="2019-04-12T18:01:20.163" Score="2" Body="&lt;p&gt;See Michael Spivak's &quot;&lt;a href=&quot;http://mathpop.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Physics for Mathematicians&lt;/a&gt;&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also the &lt;a href=&quot;http://alpha.math.uga.edu/~shifrin/Spivak_physics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;notes that the book came from&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2019-04-12T18:01:20.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15492" PostTypeId="2" ParentId="15383" CreationDate="2019-04-13T09:20:32.253" Score="4" Body="&lt;p&gt;This sort of example is not artificial and needs more careful treatment than it often receives. The inherent difficulty of the example is compounded by the tendency to discuss primitives without discussing their domains.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The badly named &quot;indefinite integral&quot; of a function is really an incompletely defined primitive of a function. By &quot;incompletely defined&quot; I mean that when specifying a primitive of a function one should also specify its domain. When this is done, the issue described in the question becomes clearer. Frequently the domain is not mentioned, and the indefinite integral, which is in any case an equivalence class of objects (also confusing for students), is treated as a single well-defined entity whose precise meaning is determined by the context in which it is used. This is like the common practice of physicists, and works well when done well (it usefully avoids too much fussing about what for beginning students are perhaps marginal issues), but generates great confusion when not handled with a certain care (attention to the confusion it possibly generates).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue arises when solving ordinary differential equations. For example, consider the ODE whose solutions include the examples given in the question. Which case is relevant depends on the initial conditions. One approach when teaching this is to always state initial value problems rather than asking for the &quot;general solution&quot;. At any rate, it is probably not helpful to students to ask for the &quot;general solution&quot; when there is an ambiguity like that in the question, unless of course one wants to focus attention on this ambiguity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once taught calculus with a professor who insisted on never writing an integral without limits. This practice eliminates the ambiguity described in the question. It worked quite well. While I'm not as inflexible about it as he was, I try to minimize writing indefinite integrals in contexts where confusion could arise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One also runs into essentially the same issue when teaching multivariable calculus, where it is related to real physical phenomena. Whether a vector field is conservative or not depends on the domain in which it is considered. Classical examples are vector fields such as &lt;span class=&quot;math-container&quot;&gt;$\left(\tfrac{-y}{x^{2} + y^{2}}, \frac{x}{x^{2}+ y^{2}}\right)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^{2}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\tfrac{x}{|x|^{3}}$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^{3}$&lt;/span&gt;, which are singular at the origin, and admit a potential in a ball not containing the origin, but do not in a ball containing the origin, a fact related to the Gauss-Coulomb law in electrostatics. Failure to condition the definition of &quot;conservative&quot; or &quot;exact&quot; on the domain leads to confusion for students who are trying to understand how it can be that an irrotational vector field is not a potential.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-04-13T09:20:32.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15494" PostTypeId="2" ParentId="8428" CreationDate="2019-04-14T17:48:55.167" Score="1" Body="&lt;p&gt;I read Henri Roudier's &lt;a href=&quot;https://www.amazon.fr/Alg%C3%A8bre-lin%C3%A9aire-cours-exercices-agr%C3%A9gation/dp/2711724859&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algèbre linéaire&lt;/a&gt; (in french) after reading Serge Lang's book. They are polar opposite in term of exposition, clarity, engagement and efficiency. I was forcing my self to read Lang's book back then to learn linear algebra but it doesn't seem Lang put any effort in this book; maybe he finished it in a year or two while copying the first chapter from his other books. It feels more like a handbook of theorems than a textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand I read Roudier's book without much difficulty. While reading the book you feel like the author is human, like he's your professor in class or your more experienced colleague. He explains the concepts and motivate them, he proves the theorems and gives many remarks about the utility of these results, he warns you about common mistakes or abuse of notation, etc. The book is better than most of the famous linear algebra books. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For abstract algebra there's &lt;a href=&quot;https://math.stackexchange.com/a/940148/81595&quot;&gt;Godement&lt;/a&gt;'s book. Not as engaging as Roudier's but still very clear.&lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2019-04-14T17:48:55.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15495" PostTypeId="2" ParentId="15475" CreationDate="2019-04-15T00:33:12.220" Score="3" Body="&lt;p&gt;A few quotes from this 2016 article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Krzywacki, Heidi, Leila Pehkonen, and Anu Laine. &quot;Promoting mathematical thinking in Finnish mathematics education.&quot; In &lt;em&gt;Miracle of Education&lt;/em&gt;, pp. 109-123. SensePublishers, Rotterdam, 2016.&#xA;  &lt;a href=&quot;https://sites.nationalacademies.org/cs/groups/pgasite/documents/webpage/pga_173315.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;The outcomes of Finnish mathematics education have proved to be excellent according to PISA testing.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;We elaborate on Finnish mathematics education especially from the perspective&#xA;of the teachers, who can be seen as autonomous professionals, meaning that they are responsible&#xA;for the planning, implementation, and assessment of teaching and learning mathematics. As a&#xA;result of the autonomous role of the teachers, the nature of teaching mathematics in Finnish&#xA;classrooms is highly dependent on individual teachers.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Finnish pupils seem to like mathematics especially at the primary school level based on studies&#xA;that have found pupils’ attitudes towards mathematics to be quite positive.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Using teacher-conducted assessments instead of national tests and&#xA;examinations especially gives teachers enough scope to independently plan and teach&#xA;mathematics.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;Another paper (which I have not yet read), but now a decade old:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Kupari, Pekka. &quot;Mathematics education in Finnish comprehensive school: Characteristics contributing to student success.&quot; In &lt;em&gt;Proceedings of the XI International Congress in Mathematics Education&lt;/em&gt;. 2008.&#xA;  &lt;a href=&quot;https://www.mathunion.org/fileadmin/ICMI/files/About_ICMI/Publications_about_ICMI/ICME_11/Kupari.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-04-15T13:15:56.530" LastActivityDate="2019-04-15T13:15:56.530" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15496" PostTypeId="2" ParentId="15475" CreationDate="2019-04-15T02:26:36.440" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;REMO MOREIRA BRITO BASTOS Instituto Brasileiro de Geografia e&#xA;  Estatística, Fortaleza, CE, Brazil &quot;&lt;a href=&quot;https://docsbay.net/the-surprising-success-of-the-finnish-educational-system-in-a-global-scenario-of-commodified-education&quot; rel=&quot;nofollow noreferrer&quot;&gt;The surprising success of the&#xA;  Finnish educational system in a global scenario of commodified&#xA;  education&lt;/a&gt;&quot; in Revista Brasileira de Educação v. 22 n. 70&#xA;  jul.-set.2017 &lt;a href=&quot;https://docsbay.net/the-surprising-success-of-the-finnish-educational-system-in-a-global-scenario-of-commodified-education&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF Download&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;“This paper, supported by bibliographic qualitative research, makes use of state of the art sources in studies of the educational system of Finland, as well as official government and multilateral institutions’ documents that investigate and seek to influence national decisions in the area of education. Additionally, it discusses the emergence, in 2001, of the international recognition of the the success of the country’s educational model. In view of the astonishing results obtained by students in the first Programme for International Student Assessment, which was conducted in 2000, we address the factors that contribute to the consistency and the success of Finland’s educational paradigm. Among the achieved results, emerges the conclusive understanding that there are successful alternative educational systems that are deeply opposed to the global corporate standard of education, and which can serve as educational models for other nations.”&lt;/p&gt;&#xA;" OwnerUserId="12178" LastEditorUserId="12178" LastEditDate="2019-04-15T02:39:37.753" LastActivityDate="2019-04-15T02:39:37.753" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15497" PostTypeId="2" ParentId="15475" CreationDate="2019-04-15T10:35:49.537" Score="0" Body="&lt;p&gt;I suspect the pre-immigration differences were more based on &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Better pupils (smarter kids).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Good, controlled culture. (Low crime, good behavior, etc.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;People routinely want to blame/praise educational methods when the differences are more fundamental.  For example for factor (1), are the teachers really better at Harvard in teaching entry level calculus or are the students just smarter than at some junior college?  I would argue often the instruction is superior at the lower ranked school since they have more of a teaching versus research mission.  [In any case, even if I'm wrong or haven't Euclidean proved my point, ignoring this factor is a logical mistake...like taking a function of x and y into z and assuming you can tell exactly how x affects z, ignoring changes in y that also affected the z.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two is less of a factor than one, but I suspect it matters, especially with very at risk urban schools in the US.  You can look at commercial businesses and see how the cost of security in &quot;war zone&quot; (literal or figurative) increases the cost of groceries.  It's not that people don't know how to run a supermarket.  But I don't think the issue is substantial at the majority of US schools (even the poorer ones), just at very high risk urban settings.  [Have read some articles to this effect, but not finding one immediately.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  &quot;Is there a research-based reason...&quot; from your question would be improved if you showed the results of your own initial Google Scholar search.  &quot;My question is X.  I did a quick search and found 1, 2, 3, etc.  A few of them gave insights blablabla, but I was unable to find key issue umptifratz.&quot;  Otherwise, you're asking that ONLY answers reflecting a deep scholarly examination be allowed and have done nothing yourself.  And you are a bright guy, capable of starting the discussion off with a little more.&lt;/p&gt;&#xA;" OwnerUserId="12180" LastEditorUserId="12180" LastEditDate="2019-04-15T10:41:07.313" LastActivityDate="2019-04-15T10:41:07.313" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15498" PostTypeId="2" ParentId="15371" CreationDate="2019-04-15T11:25:41.913" Score="6" Body="&lt;p&gt;I think it is best to use no notation for change of basis! Let me explain what I mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One defines the matrix of a linear map &lt;span class=&quot;math-container&quot;&gt;$T:V \to W$&lt;/span&gt; with respect to (ordered) bases &lt;span class=&quot;math-container&quot;&gt;$\{v_{1}, \dots, v_{n}\}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\{w_{1}, \dots, w_{m}\}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; to be the &lt;span class=&quot;math-container&quot;&gt;$m \times n$&lt;/span&gt; matrix whose &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;th column comprises the coordinates of &lt;span class=&quot;math-container&quot;&gt;$T(v_{i})$&lt;/span&gt; with respect to the (ordered) basis &lt;span class=&quot;math-container&quot;&gt;$\{w_{1}, \dots, w_{m}\}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When &lt;span class=&quot;math-container&quot;&gt;$V = W$&lt;/span&gt;, the &quot;change of basis matrix&quot; is then simply the matrix of the identity transformation &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Id}_{V}:V \to V$&lt;/span&gt; with respect to the basis &lt;span class=&quot;math-container&quot;&gt;$\{v_{1}, \dots, v_{n}\}$&lt;/span&gt; in the domain and &lt;span class=&quot;math-container&quot;&gt;$\{w_{1}, \dots, w_{n}\}$&lt;/span&gt; in the codomain. More prosaically, the &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;th column of this matrix are the coordinates of &lt;span class=&quot;math-container&quot;&gt;$v_{i}$&lt;/span&gt; in the (ordered) basis &lt;span class=&quot;math-container&quot;&gt;$\{w_{1}, \dots, w_{n}\}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I have been fussily writing &quot;ordered&quot; in parentheses. This is because these notions do not depend simply on the basis (which is a set). They depend on the ordering of this basis implicit in the choice of indices. Reordering a given basis leads to a nontrivial change of coordinates! (By a permutation matrix). So when one speaks of the &quot;change of basis&quot; matrix one should really speak of the &quot;change of ordered basis matrix&quot;. Of course no one does this, but this sometimes causes confusion for students, particularly if one writes something like &lt;span class=&quot;math-container&quot;&gt;${}_{B}M_{C}$&lt;/span&gt; and speaks of bases &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;. Are &lt;span class=&quot;math-container&quot;&gt;$\{(1, 0, 0), (0, 1, 0), (0, 0, 1\}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\{(1, 0, 0), (0, 0, 1), (0, 1, 0\}$&lt;/span&gt; different bases? Not if one's definition of a basis is a linearly independent set that spans. But they are different ordered bases, and the usual conventions for change of basis matrices assign to them the permutation matrix&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation*}&#xA;\begin{pmatrix} 1 &amp;amp; 0 &amp;amp; 0\\ 0 &amp;amp; 0 &amp;amp; 1 \\ 0 &amp;amp; 1 &amp;amp; 0 \end{pmatrix}.&#xA;\end{equation*}&lt;/span&gt;&#xA;Notation like &lt;span class=&quot;math-container&quot;&gt;${}_{B}M_{C}$&lt;/span&gt; tends to obscure this subtle point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More problematically, notation like &lt;span class=&quot;math-container&quot;&gt;${}_{B}M_{C}$&lt;/span&gt; is really quite sophisticated. Without going into details, this is essentially a transition function for the frame bundle (this explains why it satisfies cocycle identities like &lt;span class=&quot;math-container&quot;&gt;${}_{B}M_{C} = ({}_{C}M_{B})^{-1}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;${}_{B}M_{C}{}_{C}M_{D} = {}_{B}M_{D}$&lt;/span&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, most of us can't remember whether &lt;span class=&quot;math-container&quot;&gt;${}_{B}M_{C}$&lt;/span&gt; means what it means or &lt;span class=&quot;math-container&quot;&gt;$({}_{B}M_{C})^{-1}$&lt;/span&gt; (that is, on which side do &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; go? Equivalently, when do I write &lt;span class=&quot;math-container&quot;&gt;$A = P^{-1}BP$&lt;/span&gt; and when &lt;span class=&quot;math-container&quot;&gt;$A = PBP^{-1}$&lt;/span&gt;?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My solution is to use commutative diagrams. This sounds horrible, but my experience is that it works with first year engineering students. I draw diagrams whose vertices are vector spaces and edges are arrows between them labeled to indicate linear transformations. Choices of bases are indicated by writing them at each vector space and writing the label of the matrix determined by the linear transformation and the choices of bases below the corresponding edge. (I would show here what I mean but I can't figure out how to get xypic to work here.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The relation expressing the change of the matrix of a linear transformation takes a form such as &lt;span class=&quot;math-container&quot;&gt;$PA = BP$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$A = P^{-1}BP$&lt;/span&gt; (or &lt;span class=&quot;math-container&quot;&gt;$B = PAP^{-1}$&lt;/span&gt;), where &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is the matrix of the identity transformation. There is no need to memorize a convention for placing inverses (equivalently the labels &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;) because the correct choices are forced by the directions of the arrows.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-04-15T11:25:41.913" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15500" PostTypeId="2" ParentId="15371" CreationDate="2019-04-15T14:50:05.740" Score="2" Body="&lt;p&gt;We shouldn't need a new notation for this, there's a long established one: partial derivatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me explain: When people write something like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left(\frac{\partial T}{\partial p}\right)_V \quad \text{ or } \quad \left.\frac{\partial T}{\partial p}\right|_V&#xA;$$&lt;/span&gt;&#xA;it means &quot;the coefficient in front of &lt;span class=&quot;math-container&quot;&gt;$\text{d} p$&lt;/span&gt; in the expansion of &lt;span class=&quot;math-container&quot;&gt;$\text{d}T$&lt;/span&gt; using the basis &lt;span class=&quot;math-container&quot;&gt;$\text{d} p,\text{d}V $&lt;/span&gt;&quot;. This might sound awkward, but that's what it means. (In fact, people used to talk of &quot;differential coefficients&quot; before the derivative terminology became standard. And they used to write &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{\text{d} T}{\text{d} p}\right)$&lt;/span&gt; for partial derivatives, before someone thought we'd gain something by changing &lt;span class=&quot;math-container&quot;&gt;$\text{d}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we're extracting a coefficient from the expansion of a vector wrt. to a basis. That's something meaningful in any vector space, not just the space of differentials and it's useful in any linear algebra class. So let's generalize the notation to arbitrary vector spaces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is a finite dimensional vector space with a basis &lt;span class=&quot;math-container&quot;&gt;$B=\{b_1,\ldots,b_n\}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v\in V$&lt;/span&gt;, denote with&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left(\frac{v}{b_i}\right)_B \quad \text{ or } \quad \left.\frac{v}{b_i}\right|_B&#xA;$$&lt;/span&gt;&#xA;the coefficient in front of &lt;span class=&quot;math-container&quot;&gt;$b_i$&lt;/span&gt; in the expansion of &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; wrt. to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;. In other words&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;v = \sum_{i=1}^n \left.\frac{v}{b_i}\right|_B  \cdot b_i.&#xA;$$&lt;/span&gt;&#xA;Using Einstein's summation convention we could just write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;v = \left.\frac{v}{b_i}\right|_B  b_i.&#xA;$$&lt;/span&gt;&#xA;and if we are lazy/sloppy (as we are with partial derivatives) we could also drop the &lt;span class=&quot;math-container&quot;&gt;$|_B$&lt;/span&gt; and claim it's clear from the context which basis &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; we're using. So we arrive at &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;v = \frac{v}{b_i}  b_i.&#xA;$$&lt;/span&gt;&#xA;Using this notation, the matrix of a linear map &lt;span class=&quot;math-container&quot;&gt;$A\colon V \to W$&lt;/span&gt; with respect to bases &lt;span class=&quot;math-container&quot;&gt;$B=\{b_1,\ldots,b_n\}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C=\{c_1,\ldots,c_m\}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; is written as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left( \left.\frac{A\, b_i}{c_j}\right|_C \right)_{i,j} &#xA;$$&lt;/span&gt;&#xA;or in the sloppy variant&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left( \frac{A\, b_i}{c_j} \right)_{i,j} &#xA;$$&lt;/span&gt;&#xA;The nice thing about this, is that change of basis becomes as natural as the chain rule in multi variabe calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have to admit that I haven't tested this in class. Mainly because I teach parallel classes with other instructors, who object to using notation that was not used when they were students. But I think it's the natural consequence if you take partial derivative notation seriously and want to make the link between linear algebra and multi variable calculus more explicit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also think it is good to have an explicit notation for a concept that's often used, like &quot;coefficients of a vector with respect to a basis&quot;.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2019-04-15T14:50:05.740" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15501" PostTypeId="1" AcceptedAnswerId="15502" CreationDate="2019-04-15T14:50:20.923" Score="-2" ViewCount="210" Body="&lt;p&gt;The students are aware of mathematical logic and proof but have not come across any of the notions of a set. What is the most natural and motivating way to introduce set theory?&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="238" LastEditDate="2019-04-15T18:21:45.477" LastActivityDate="2019-04-15T18:21:45.477" Title="What is the best way of introducing set theory?" Tags="|discrete-math|set-theory|" AnswerCount="1" CommentCount="10" ClosedDate="2019-04-17T11:37:46.560" ContentLicense="CC BY-SA 4.0" />
  <row Id="15502" PostTypeId="2" ParentId="15501" CreationDate="2019-04-15T15:14:07.190" Score="-2" Body="&lt;p&gt;The student background in proofs and logic is not necessary for initial explanation/motivation of the concepts.  And actually it worries me that you mention that when wanting to know how to do an initial introduction!  I remember learning the basics of sets (and union, intersection, etc.) as an elementary schoolkid in the New Math.  You do not need a huge background to start looking at the topic.  This is very different from calculus where you really do need some background in algebra, functions, analytic geometry, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The easiest way to first explain the concept is as a collection of things (e.g. stamps, books, etc.) see pages 5-6 of Tenenbaum Ordinary Differential Equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course you should move to generalize to infinite sets and more difficult concepts.  And the students have background to handle that.  But to START, just do exercises with  simple sets of integers.  E.g.  (1, 3, 5, 7, 9) versus (2, 4, 6, 8, 10) versus (3, 7, 11, 214) versus (3, 5, 7, 1, 9).  I recommend using integer collections as the starting point as it is easy to look at them and discuss union/intersection.  Of course, you can move to more fancy complicated things.  But start simple.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can illustrate with Venn diagrams also.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, given the level of the students you can move to some formal proofs as well.  AFTER they have been introduced to the topic.  but I would not START any topic with a definition, theorem, proof, example style of teaching.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I also suggest looking at some books (hard and easy) for their approaches.  This should not just be something to be solved de novo by a question here.  [Or with an approach of constructing a course from scratch.  Anyone who DOES construct a course from scratch should have some deep perspective before doing so...and I don't mean deep math perspective.  I mean in addition to that having seen different pedagogical approaches and having some formed ideas on what works/doesn't.]&lt;/p&gt;&#xA;" OwnerUserId="12181" LastEditorUserId="12181" LastEditDate="2019-04-15T15:41:20.800" LastActivityDate="2019-04-15T15:41:20.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15503" PostTypeId="1" AcceptedAnswerId="15504" CreationDate="2019-04-15T16:36:27.370" Score="2" ViewCount="208" Body="&lt;p&gt;I'm looking for recommendations for a good textbook to use for a hypothetical lower-division course in complex analysis, at a level of sophistication comparable to a second or third semester course in calculus.  (In particular, the audience are not necessarily math majors, and the course would not be proof-based.)  I'm particularly interested in books that have a good collection of exercises at this level: hopefully a good number of basic computational exercises, plus some more conceptual but not overly difficult (and in particular not proof-based) ones.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2019-04-15T17:51:49.033" Title="Lower-division complex analysis textbook" Tags="|undergraduate-education|textbooks|complex-numbers|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15504" PostTypeId="2" ParentId="15503" CreationDate="2019-04-15T16:45:26.480" Score="3" Body="&lt;p&gt;I really liked using &lt;a href=&quot;https://www.maa.org/press/maa-reviews/a-first-course-in-complex-analysis-with-applications&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zill and Shanahan &quot;A First Course in Complex Analysis with Applications&quot;&lt;/a&gt; when I taught this course, certainly at a low level of sophistication.  Loads of pictures, nice wide margins, careful with algebra saved &quot;for the reader&quot; in other texts.  I had a version from 2003 in softcover, not sure if that is available now.  I also find it prepared those with better uptake for what they might encounter next.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;On a note I cannot vouch for from personal use, there is a text (same publisher!) by &lt;a href=&quot;http://mathfaculty.fullerton.edu/mathews/complex.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Howell and Mathews&lt;/a&gt; which is closely connected to several pedagogical initiatives surrounding complex analysis, for further details on which see for instance the contents of &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/10511970.2017.1312652&quot; rel=&quot;nofollow noreferrer&quot;&gt;this special issue of PRIMUS&lt;/a&gt;.  Not that you have to read or use those, but just for information as to where other people are no doubt teaching this same cohort and trying to share best practices.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-04-15T16:45:26.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15505" PostTypeId="2" ParentId="15503" CreationDate="2019-04-15T17:46:30.693" Score="-1" Body="&lt;p&gt;Take a look at the &lt;strong&gt;Schaum's Outline of Complex Variables Theory and Practice&lt;/strong&gt;.  I find it friendly in being very problem based.  Warning, it does have SOME proofs but not a huge amount.  Or &quot;proofs&quot; that are smaller, easier little derivations.  And it has a gazillion (technical term) calculational problems.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you look at it, you can see if you like or dislike the review text style.    There's less text than normal in a doorstopper book but at the same time it's not &quot;hard&quot; like some very terse short that is more monograph style (e.g. Adrian Albert Higher Algebra).  More short and practical like a language review grammar is economical (which I prefer to standard language texts).  I personally like that but see how you feel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The level is pretty friendly in terms of not expecting students to have had real analysis or &quot;advanced calculus&quot; first.  Just starts with the standard review of complex alebra, doesn't get into a bunch of series convergence assumptions (or expect students to know them for homework problems) at least not in the beginning or without developing needed tools.  For comparison, it's at an easier level than Carrier Krook Pearson where first chapter had HW problems that blithely ask for proofs in the well known theory of real variables (not well known to kids with basic calculational calculus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Caveat: I have the first edition.  But I suspect the second edition is fine.  But just letting you know I haven't looked at it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Price:  $14 new, for second edition, paperback.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Link:  &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B002A6RRYO&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Schaums-Outline-Complex-Variables-Outlines-ebook/dp/B002A6RRYO#customerReviews&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12181" LastEditorUserId="12181" LastEditDate="2019-04-15T17:51:49.033" LastActivityDate="2019-04-15T17:51:49.033" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15506" PostTypeId="1" CreationDate="2019-04-15T22:27:28.963" Score="12" ViewCount="499" Body="&lt;p&gt;This question is about references in current best practices in &lt;em&gt;teaching math to future elementary teachers&lt;/em&gt; at a university level. I am asking it because I do not see any question so far on this site regarding this specific type of course (other than a textbook request, &lt;a href=&quot;https://matheducators.stackexchange.com/questions/8131/reference-request-textbooks-in-mathematics-for-future-kindergarten-teachers&quot;&gt;here&lt;/a&gt;), and I think it is worth discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These courses are generally focused on helping future elementary teachers understand CCSS (Common Core) language (edit: or other equivalent standards-basedf language) and rephrase it into child-friendly language, helping them understand arithmetic through arithmetic in other bases and understand and be able to explain visual models of arithmetic, including understanding and identifying mistakes that people can make when using these models. The general focus tends to be on mastery of addition, subtraction, multiplication, division, divisibility rules, fractions, negative numbers, geometry, proportional reasoning, and equations, among other things in the same vein.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arithmetic in alternate bases is a common part of these courses. While I strongly believe that this helps future elementary teachers conceptualize the learning of mathematics, I have noted (while tutoring students taking similar courses) that this message does not get through very clearly. So, my question is truly about &quot;best practices&quot; in this kind of course, but for clarity about what I mean by &quot;best practices&quot; I pose the following more specific questions, all of which are within my scope of asking about &quot;best practices&quot;: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;What are some references regarding good pedagogical approaches in this type of course? For example, I would lean to, but do not currently have evidence for:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;De-emphasis of lecturing in favor of activity-based, constructivist approach: Students already &quot;know&quot; the material (though I recognize that they often do not quite &quot;know&quot; it, but feel like they do), and are asked to make sense of it in a deeper way, and so should rarely need to be 'told' material (there are arguments against ever lecturing in any math class, but that is a different question entirely)&lt;/li&gt;&#xA;&lt;li&gt;Use of different bases to help students understand the underlying structure of their number systems instead of rote &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What are additional types of exercises which help elementary teachers process this information and be prepared to use it when they teach mathematics to children?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;What do elementary teachers really need to know, vs. simply be able to do? What evidence can I provide &lt;em&gt;them&lt;/em&gt; that they need to know it rather than simply be able to do it?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Clarification: This is NOT about teaching a &quot;methods&quot; course: That is for experienced elementary educators to teach. This is instead about teaching a math course TO elementary teachers, where the elementary teachers are expected to have a firm grasp on the basics before they take their &quot;methods&quot; course on how to &lt;em&gt;teach&lt;/em&gt; the basics.&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2019-04-16T18:34:23.683" LastActivityDate="2019-07-01T17:05:35.320" Title="Best practices in teaching math to future elementary teachers" Tags="|undergraduate-education|reference-request|primary-education|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="15509" PostTypeId="2" ParentId="10571" CreationDate="2019-04-16T10:44:27.817" Score="0" Body="&lt;p&gt;Use printed log tables instead.  You see the patterns more clearly than with a calculator or a slide rule.  We learned how to use them in chemistry class--they were still provided on the AP chem exam until the late 90s.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, instead of needing to remember, prove or the like the addition of logs, you are DIRECTLY USING it, repeatedly.  Nothing motivates belief and memory like actually having dependence on the property.  Have a choice between manually adding logs from a table or doing the multiplication of the original numbers and it is a no-brainer which you go to.  You also get very used to characteristic and mantissa.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I never used the log tables much after the 70s (calculators became cheap).  But just that initial use in high school chem was very powerful for me.  Much less cryptic than a button on a box or moving a ruler stick side by side.  (It also set me up to use trig and steam tables and later Bessel function tables, probability tables, financial tables, etc.  Just developing a facility with tables is good thing to have.  You still need it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=VRzH4xB0GdM&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=VRzH4xB0GdM&lt;/a&gt; (more motivating and slick but not as quick to just show the tables)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=4ebnr_h4m7E&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=4ebnr_h4m7E&lt;/a&gt; (very hands on and practical, but heavily accented English)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To print tables for your class just zox the few pages out of your CRC.  (Of course you have a CRC.  No math/science person would not.  Plus it is good for ganglion bashing.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. Please start with common logs, not Napierian.  As a veteran scientist or mathematician, it is fine to use ln instead of log because we already know that stuff.  But for kids just learning the thing for first time, base 10 is much easier.  You see the exponents of 10, 100, 1000 etc. very clearly. &lt;/p&gt;&#xA;" OwnerUserId="12187" LastActivityDate="2019-04-16T10:44:27.817" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15513" PostTypeId="1" CreationDate="2019-04-17T05:25:23.760" Score="7" ViewCount="339" Body="&lt;p&gt;In remedial algebra, we learn that the graph of &lt;span class=&quot;math-container&quot;&gt;$y=(\sqrt x)^2$&lt;/span&gt; is only in the first quadrant. We know this is the correct graph for the equation. This is because we know &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x \ge 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, a student in a higher level algebra class got the full line where &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is all real numbers. Her logic was that if you take &lt;span class=&quot;math-container&quot;&gt;$x=-4$&lt;/span&gt;, you'd take the square root and get &lt;span class=&quot;math-container&quot;&gt;$2i$&lt;/span&gt;, and then you'd square that to get &lt;span class=&quot;math-container&quot;&gt;$-4$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$i^2 = -1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2^2 = 4$&lt;/span&gt;. This would give you a line that goes into the negatives, which is different from the correct answer we are told, which is a line only in the first quadrant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This student did not thrive in class, however she made a very interesting argument. She did not see the point of what she came up with and just wanted to pass the class. How would I, as a teacher, try to celebrate this students solution? Although we don’t know if it’s really right or wrong, how could a teacher try and justify her answer and what could a teacher say to encourage her thinking and show a connection to how complex numbers were made?&lt;/p&gt;&#xA;" OwnerUserId="12191" LastEditorUserId="2083" LastEditDate="2019-04-17T07:41:51.787" LastActivityDate="2019-05-12T19:57:35.570" Title="Complex numbers and encourage justification" Tags="|secondary-education|algebra|student-motivation|concept-motivation|complex-numbers|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15514" PostTypeId="1" CreationDate="2019-04-17T09:52:43.720" Score="4" ViewCount="2628" Body="&lt;p&gt;How can I convince students that &quot;P implies Q&quot; is true when P is false, independent of what truth value Q takes? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any real life or a convincing argument for this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have given the analogy of if being elected then you will abolish the death penalty which seems ok but rather crass. Are there any other examples?&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="4791" LastEditDate="2019-04-17T11:36:48.307" LastActivityDate="2019-05-14T14:09:19.773" Title="How to convince students of the implication truth values?" Tags="|teaching|" AnswerCount="6" CommentCount="10" ClosedDate="2019-04-25T00:46:28.510" ContentLicense="CC BY-SA 4.0" />
  <row Id="15515" PostTypeId="2" ParentId="15514" CreationDate="2019-04-17T10:45:06.037" Score="24" Body="&lt;p&gt;I make the statement &quot;If it is raining, then I have an umbrella.&quot;  Did I lie?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it is raining and I do not have an umbrella, then I lied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it is raining and I do have an umbrella, then I didn't lie.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it is not raining, then it doesn't matter whether or not I have an umbrella; I still did not lie.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2019-04-17T10:45:06.037" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15516" PostTypeId="2" ParentId="15514" CreationDate="2019-04-17T13:12:39.503" Score="0" Body="&lt;p&gt;You can introduce implication and equivalence side by side to make the difference clear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Implication: If A, then also B. (But if not A, this statement does not tell us anything. See the umbrella example in Joel's answer.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Equivalence: If &lt;strong&gt;and only if&lt;/strong&gt; A, then B. This includes that not A implies not B. &lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2019-04-17T14:35:20.197" LastActivityDate="2019-04-17T14:35:20.197" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15517" PostTypeId="2" ParentId="15513" CreationDate="2019-04-17T14:52:13.647" Score="0" Body="&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$z \neq 0$&lt;/span&gt;. Define &lt;span class=&quot;math-container&quot;&gt;$z^{1/2} = \{ \sqrt{z}, - \sqrt{z}\}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\sqrt{z}$&lt;/span&gt; is defined by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{z} = \sqrt{|z|}e^{i\text{Arg}(z)/2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\text{Arg(z)}$&lt;/span&gt; is the standard angle of &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$(-\pi, \pi]$&lt;/span&gt;. We can study &lt;span class=&quot;math-container&quot;&gt;$w \in (z^{1/2})^2$&lt;/span&gt; in the sense that &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; is in the set formed by squaring the the set &lt;span class=&quot;math-container&quot;&gt;$z^{1/2}$&lt;/span&gt; element-wise. In particular,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (z^{1/2})^2 = \{ (\sqrt{z})^2, (-\sqrt{z})^2 \} = \{ (\sqrt{z})^2 \}$$&lt;/span&gt;&#xA;but, the square of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{z}$&lt;/span&gt; can be calculated directly:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (\sqrt{z})^2 = \left(\sqrt{|z|}e^{i\text{Arg}(z)/2} \right)^2  = (\sqrt{|z|})^2 \left(e^{i\text{Arg}(z)/2} \right)^2 = |z|e^{i\text{Arg}(z)} = z.$$&lt;/span&gt;&#xA;So, if &lt;span class=&quot;math-container&quot;&gt;$w \in (z^{1/2})^2$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$w \in \{ z \}$&lt;/span&gt; so we might as well say &lt;span class=&quot;math-container&quot;&gt;$w=z$&lt;/span&gt;. I'm sure someone else can find a better way to communicate this to the level of student you intend.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2019-04-17T14:52:13.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15518" PostTypeId="2" ParentId="10571" CreationDate="2019-04-17T15:20:07.587" Score="8" Body="&lt;p&gt;As Daniel's answer states, the most likely reason why slide rules are rarely used to teach logarithms boils down to them being hard to come by. As they have been replaced by electronic calculators, it is common for many educators now to have never seen one, or to only be peripherally aware of them. Many don't even think of including them in lesson plans.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this doesn't mean you shouldn't. In fact, if you have the time available, and you don't have a slide rule to hand, you can do something even better - have your class make them!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This doesn't take long (&amp;lt;20mins), and I personally find it a demonstration of laws of logarithms that is so obvious and intuitive that it seems almost &quot;too obvious&quot;. You will need the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Standard ruler&lt;/li&gt;&#xA;&lt;li&gt;Pen/pencil&lt;/li&gt;&#xA;&lt;li&gt;2 strips of card/paper (just cut a standard piece length-ways down the middle)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/H9opU.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/H9opU.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now align both pieces, and make a mark along the edge every 2 cm (other graduations are available). Write a simple linear scale on the top and bottom piece such that each gradation is 1 added to the value of the previous gradation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/neeDl.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/neeDl.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, use your simple scale to perform an addition. If you want perform &lt;span class=&quot;math-container&quot;&gt;$2 + 3$&lt;/span&gt;, offset the top scale so the zero lines up with the &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; on the bottom scale. Now find the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; on the top scale, and look at the number below it on the bottom scale. Tada! 5!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DkbF0.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DkbF0.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(annotations added in for emphasis)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ok, at this point, who cares. We are only introducing the idea that two parallel sliding scales can add two values together. Now lets make a real slide rule. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Flip your pieces of card over, and make a new set of graduated scale lines. However, instead of incrementing by one each time, multiply by some base. I use two. It is worth pointing out that this is a logarithmic scale and each entry is just the linear scale value, raised as a power of two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it helps to make a larger mark for the &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; line, and to write &lt;span class=&quot;math-container&quot;&gt;$2^n$&lt;/span&gt; above each graduation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vajb8.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vajb8.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now perform the same operation as before, only as a multiplication. We will find &lt;span class=&quot;math-container&quot;&gt;$4 \times 8$&lt;/span&gt;. Line up the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; of the top scale with the &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; on the bottom. Find the &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; on the top scale, and then look at what it lines up with on the bottom. &lt;span class=&quot;math-container&quot;&gt;$4 \times 8=32$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jzQOI.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jzQOI.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Congratulations, you have now made a simple slide rule, and used it to demonstrate the first law of logarithms. The action that we performed is the same &quot;addition&quot; action we performed with the linear scales, but instead of adding, we ended up with a multiplication, because our scales are logarithmic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have more time, you can use this to explore further concepts:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;2nd and 3rd laws of logarithms&lt;/li&gt;&#xA;&lt;li&gt;Different bases (use a 10 times multiplier instead of 2)&lt;/li&gt;&#xA;&lt;li&gt;Increase the precision. Use a calculator to find the logarithmic values of numbers in-between the ones already present, and mark them in the correct place on the scale. This will allow you to perform multiplication with more values, and helps familiarise students with logarithmic scales.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10057" LastEditorUserId="10057" LastEditDate="2019-04-17T15:30:04.257" LastActivityDate="2019-04-17T15:30:04.257" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15519" PostTypeId="2" ParentId="15514" CreationDate="2019-04-17T17:31:06.193" Score="3" Body="&lt;p&gt;You could say it means &quot;Whenever P is true, Q is true&quot;. So &quot;If it rains, I will bring an umbrella&quot; means &quot;Every time it rains, I bring an umbrella&quot;. It's not possible to disprove this statement by looking at what happens when it doesn't rain.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2019-04-17T17:31:06.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15520" PostTypeId="2" ParentId="15514" CreationDate="2019-04-17T20:18:58.657" Score="0" Body="&lt;p&gt;It's obvious if you view a proposition as a subset of all values in the proposition domain for which the proposition is true. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, a proposition &quot;this umbrella is red&quot; selects only red umbrellas form the set of all umbrellas that exist in the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A proposition &quot;I own only red umbrellas&quot; can be viewed as implication - &lt;code&gt;P&lt;/code&gt; is &quot;I own this umbrella&quot;, &lt;code&gt;Q&lt;/code&gt; is &quot;this umbrella is red&quot;, and it can be interpreted as &quot;the set of umbrellas that I own is a subset of red umbrellas&quot;, that is, implication corresponds to &quot;being a subset&quot; relation in the set interpretation (similarly, conjunction corresponds to set intersection and disjunction corresponds to set union).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;False proposition corresponds to an empty set - there are no values in the proposition domain for which it's true. From set theory axioms we know that empty set must be a subset of any set. This immediately gives you &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;P implies Q&quot; is true when P is false&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It's essentially a formality, but a convenient one - for example if you don't own any umbrellas at all you can safely say &quot;I own only umbrellas hand-made by unicorns from rainbows&quot; and it's assumed to be true because it can't be disproved.&lt;/p&gt;&#xA;" OwnerUserId="12195" LastActivityDate="2019-04-17T20:18:58.657" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15521" PostTypeId="2" ParentId="15514" CreationDate="2019-04-17T22:18:41.500" Score="0" Body="&lt;p&gt;My approach in the past has always been to work via negation. However, I've recently read one of Tim Gowers' 'discussions' that showed me a new side to the problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically he points out that in everyday usage (and in proofs too) we are looking for &lt;em&gt;causal&lt;/em&gt; relationships. We want to say that 'if A then B' is true when we can understand A as leading to B. But when dealing with more formal logic, we are actually only concerned with &lt;em&gt;evidence&lt;/em&gt;. We say 'if A then B' is true when 'in every case where A is true, B is also true', regardless of whether this is something you would expect or just plain chance.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-04-17T22:18:41.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15522" PostTypeId="1" AcceptedAnswerId="15530" CreationDate="2019-04-17T23:38:16.837" Score="7" ViewCount="275" Body="&lt;p&gt;I have noticed working with bright undergraduates that&#xA;it is not uncommon for them to have difficulty&#xA;easily converting between a point&amp;mdash;say, a point &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; on a surface &lt;span class=&quot;math-container&quot;&gt;$S \subset \mathbb{R}^3$&lt;/span&gt;&amp;mdash;and a vector &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; from the origin &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt; of the coordinate system in which &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is embedded: &lt;span class=&quot;math-container&quot;&gt;$v=p-o$&lt;/span&gt;.&#xA;For example, a normal vector &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;&#xA;might be computed as based on &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt;, and then needs&#xA;to be moved out to &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; for display on the surface of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$n + v$&lt;/span&gt;.&#xA;To me the transition between points and vectors is&#xA;natural and easy, but not so to my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having learned from &#xA;@AlexandreEremenko's&#xA;answer to my question,&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/q/8488/606&quot;&gt;When were vectors invented?&lt;/a&gt;,&#xA;I now see that however natural vectors are to us now,&#xA;they were not natural in the past.&#xA;And perhaps this is a hint that there is an intellectual&#xA;hurdle here I am not seeing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd appreciate comments on this issue.&#xA;Wherein lies the difficulty converting between points and vectors out to those points?&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-04-19T07:50:48.087" Title="Why is it difficult to freely change between points and vectors?" Tags="|undergraduate-education|geometry|vectors|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15523" PostTypeId="2" ParentId="15522" CreationDate="2019-04-18T01:27:28.523" Score="6" Body="&lt;p&gt;My first thought (which is just barely this side of a comment) is that when students are introduced to both vectors and points, they are always with respect to a specific origin.  Even if lip service is paid to vectors as a magnitude and direction in some coordinate-free way, realistically the examples have the origin as &quot;the origin&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my conclusion is that at least part of this is that vectors are introduced this way, so people get used to it.  This is an unscientific answer, though, so I welcome others - doubtless it is indeed more challenging to think of vectors in this more abstract way.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-04-18T01:27:28.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15525" PostTypeId="2" ParentId="15514" CreationDate="2019-04-18T08:56:32.340" Score="1" Body="&lt;p&gt;&lt;em&gt;If I do the work, I will get paid.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If I do the work and get paid, all is good. &lt;/li&gt;&#xA;&lt;li&gt;If I do the work and not get paid, it's bad!  &lt;/li&gt;&#xA;&lt;li&gt;If I don't do the work and don't get paid, I cannot complain - it's all good.&lt;/li&gt;&#xA;&lt;li&gt;If I don't do the work but still get paid - now this is good!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I know it's not exactly how one should interpret implication, but it worked for me. :)&lt;/p&gt;&#xA;" OwnerUserId="12198" LastEditorUserId="12255" LastEditDate="2019-05-14T14:09:19.773" LastActivityDate="2019-05-14T14:09:19.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15526" PostTypeId="2" ParentId="15475" CreationDate="2019-04-18T08:59:36.183" Score="6" Body="&lt;p&gt;The following research paper, by a Finnish researcher, is expressing more concern than enthusiasm about the effectiveness of the &quot;Finnish way&quot; to education:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://elib.mi.sanu.ac.rs/files/journals/tm/23/tm1221.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://elib.mi.sanu.ac.rs/files/journals/tm/23/tm1221.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 2015PISA performance again does not necessarily backup a stunning success of Finnish educational system&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://data.oecd.org/pisa/mathematics-performance-pisa.htm&quot; rel=&quot;noreferrer&quot;&gt;https://data.oecd.org/pisa/mathematics-performance-pisa.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;what data are registering is that Finland is registering a significant difference (in positive) for the performance of &lt;em&gt;female&lt;/em&gt; students both in scientific literacy and in math, and that the overall good result is strongly dependent on this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me add that when you look a bit closer into data you have that in many european countries there are subregions where scores are significantly above average. So, for example, Italy NorthEast area scores in math better than Finland (with 3-4 times the number of people living in an area that comprises both urban and rural areas). Finland's results are on the contrary much more uniform (you have to rely on my words for this: I read a research report in Italian where data are just scattered through and not readable easily). This of course question much what PISA is really measuring: efficiency of the educational system or context-dependent hidden variables (like those depending on economic welfare).&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2019-04-18T08:59:36.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15527" PostTypeId="2" ParentId="15522" CreationDate="2019-04-18T16:18:27.383" Score="5" Body="&lt;p&gt;You might want to look at how vectors are introduced to your students in previous coursework. At my school (a community college in California), it would be something like this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The remedial trig course includes vectors in the course outline, but not all students take that course, and if they do, the instructor will often skip the topic.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A first semester physics course spends a lot of time on vectors. Probably some but not all of your students have taken such a course.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The sophomore multivariate calculus course introduces vectors from scratch on the assumption that students have never seen them before.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The linear algebra course spends most of the semester teaching students algorithms for manipulating matrices and row and column vectors. I doubt that students connect these manipulations in any way to any possible geometric interpretation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you ask your students to define what a vector is, you will probably get a variety of answers, depending on which of these experiences they've had. My guess it that some will say it's a thing that has both a magnitude and a direction (a common definition in physics textbooks), while others will say that it's an n-tuple of numbers (basically the only representation they ever seem to work with in the math courses). Neither of these is quite the same as the definition that would apply best to your example, which is that a vector is an equivalence class of displacements between points in space. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I teach my students about vectors in freshman physics, I find that they need a lot of encouragement to think about vectors as &quot;portable.&quot; For example, they may draw a free-body diagram in which two force vectors act on an object, and the force vectors are drawn tail to tail. Then if they want to sketch graphical addition of the vectors, they have to slide them around into tip-to-tail position. This is not natural to them, nor is it natural to them to understand that the vectors have to be slid around without rotation (parallel transported, in fancy terminology).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my office hours, I spend a lot of time demonstrating this with pens laid on the tabletop. I place two pens on the tabletop, tail to tail, and ask the student if they can slide them around so they're tip to tail. Usually they don't understand how to do this. Then I usually take one pen, move it around some path and bring it back to the start again, to illustrate the idea of sliding without twisting. I may also explicitly demonstrate twisting, in order to show that it's not allowed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, students usually do not get much practice with the conceptual aspects of vectors. For example, I have them do this think-pair-share exercise:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;In example 4 [computing the components of a vector given in magnitude-direction form], we dealt with components that were negative. Does it make sense to classify vectors as positive and negative?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may seem extremely elementary to a mathematician, but for many students it's very difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the particular example you give, of converting between points and vectors, note that for students who have learned about vectors in a physics class, their stock of mental examples of vectors probably has a couple of primary examples in it: force vectors and displacement vectors. For force vectors, it isn't even true that you can convert back and forth between a point form and a vector form, if the point is to be a point in the ordinary Euclidean space that we swim around in. It could of course be a point in a more abstract &quot;space&quot; of force vectors, but this is not a level of abstraction that would occur to students at this level.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-04-18T17:15:08.960" LastActivityDate="2019-04-18T17:15:08.960" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15529" PostTypeId="2" ParentId="13116" CreationDate="2019-04-19T07:24:53.263" Score="2" Body="&lt;p&gt;Your premise is not entirely true. We do distinguish between an equation and an identity in English &lt;em&gt;sometimes&lt;/em&gt;, but most of the time we use 'equation' for both. 'Identity' is only really used on 'special occasions'. In particular, students are only likely to come across the term in the context of 'trig identities' such as &lt;span class=&quot;math-container&quot;&gt;$\sin^2(x)+\cos^2(x)=1$&lt;/span&gt;. There are no notable instances of 'identities with inequality signs' that students come across, so there is no reason to use different terms to distinguish them. &lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-04-19T07:24:53.263" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15530" PostTypeId="2" ParentId="15522" CreationDate="2019-04-19T07:50:48.087" Score="7" Body="&lt;p&gt;I strongly dislike the “magnitude and direction” description. Yes, it comes from physics, and is helpful sometimes, but it hides the true nature of vectors, and makes later understanding more difficult (what is the direction of the zero vector?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I prefer to focus from the very beginning on the core property of vectors: they can be &lt;strong&gt;added&lt;/strong&gt; and &lt;strong&gt;rescaled&lt;/strong&gt;; and more importantly, these two operations behave in a friendly manner with each other: scalar factors can be &lt;strong&gt;distributed&lt;/strong&gt; among two added vectors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In class I start with something along the line of “you cannot add your address plus mine, but you can add motion instructions. The address is akin to a point coordinate, the instruction can be applied anywhere that you are standing.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I stand in the middle of the room, and ask the entire class to point to my nose: every arm points ‘differently’. After this, ask them to point North: every arm points ‘the same’.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually, these three approaches work together to give students a good grasp. To clinch it, I am meticulous about writing point coordinates like so (2,3), and vector coordinates like so &amp;lt;2,3&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2019-04-19T07:50:48.087" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15531" PostTypeId="2" ParentId="1455" CreationDate="2019-04-19T10:24:35.273" Score="2" Body="&lt;p&gt;There are two aspects to this question. One is how to handle mistakes with respect to the class in which they are made. The other is how to handle them in terms of one's preparation for giving class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When doing a long computation on the board I sometimes make some arithmetical or sign mistake. I anticipate this, warn students about, and use it as a vehicle for teaching them strategies to catch their own mistakes. Students hardly ever complete a computation without making mistakes themselves, and need to learn first, to expect to make mistakes, and second, to realize and organize computations and thinking in such a way as to facilitate detection, localization, and correction of the errors that are inevitably made. (People who teach computer programming all understand this and call it &quot;debugging&quot;.) For example, when solving an initial value problem for an ODE or calculating an integral one can always subsequently check the putative answer by differentiating it; this is a good practice for students, and I usually do it in class also. One wants students to see that developing a sense of when something is wrong is also part of learning how to do mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is who views making an error in computations as evidence of lack of preparation. This can be true, as sometimes such errors are evidence that the instructor did not think through ahead of time all the details of the computation and all the difficulties that it might entail, or evidence of a pedagogically poor choice of example (e.g. one that leads to omplications in the computations that are not relevant to the principal point being taught). This sort of error should be avoided precisely because one should be prepared in this sense. If one find oneself making errors too frequently, one should take it as an indicator that one needs to prepare more carefully. On the other hand, with routine calculations (say solving a standard integration), one should not know already all the difficulties involved and should be able to do the computations on the fly, so to speak, doing so is often sound pedagogically because it permits the student to see one's thought process (also overpreparation can lead to a presentation that is too fast and too slick) and one's consistency checks and the like, and one can save preparation time (often in short supply) by not needlessly preparing in detail computations one can do in one's sleep - provided one anticipates that this approach inevitably leads to occasional errors on the board. What I mean in terms of an example - whether I pose the problem of finding the general solution of &lt;span class=&quot;math-container&quot;&gt;$\ddot{x} + 2\dot{x} + 3x = \sin{t}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\ddot{x} + 5\dot{x} + 7 x = e^{t}$&lt;/span&gt; is all sort of the same as long as I can calculate the discriminant of the characteristic equation in my head on the fly, to recognize that formally speaking these in the essential points the same problem. The real parts of the general solutions of the associated homogeneous equations are respectively &lt;span class=&quot;math-container&quot;&gt;$e^{-t}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e^{-5t/2}$&lt;/span&gt; and I might write &lt;span class=&quot;math-container&quot;&gt;$e^{t}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$e^{5t/2}$&lt;/span&gt; on the board by accident; the key then is not to declare the exercise finished until having checked the results. (Moreover, in my case, I'm just as likely to make such a mistake in preprepared notes as I am to make it on the fly.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A different sort of error is a conceptual error. This should never happen. One should know what one is talking about and adequate preparation definitely entails thinking through all the subtle points (whether or not one mentions them to students - sometimes precisely so as to decide which points to avoid or hide). If it does happen, the only thing to do is to be open and honest about it and to correct it clearly and insistently. Students will lose faith completely in a teacher who hides having made an error, but most everyone pardons once or twice an error corrected in good faith. On the other hand, making repeated errors of a conceptual sort will cause students to lose faith in the instructor, as it should.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I notice that I make minor errors more frequently when presenting material that for me is routine, boring, or lacks interesting contextualization (e.g. the ODE examples cited above would be less boring were they to arise in a physical context - and such a context also would provide intuition that would decide the sign of the real part of the solution). It can be a sign of going too fast, too blithely. Often one has to present such material, and as the years go by more and more material feels this way, but making errors doing it can be a sign that one lacks the engagement and enthusiasm necessary to concentrate adequately. If that's the case, one needs to do something to shake up one's own approach, simply to restore vitality to one's teaching.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-04-19T10:24:35.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15532" PostTypeId="2" ParentId="1455" CreationDate="2019-04-19T13:55:02.733" Score="3" Body="&lt;p&gt;I bring them doughnuts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This happens to all of us from time to time, sometimes in little ways sometimes in bigger ones.  I want to encourage my students to question things so every time they catch me making a math mistake, the class gets a point.  When we get to 15 points (they are 13 week semester courses), I bring in doughnuts the next day.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2019-04-19T13:55:02.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15535" PostTypeId="2" ParentId="1455" CreationDate="2019-04-19T19:05:26.637" Score="3" Body="&lt;p&gt;You don’t mention the level, but the issue is applicable at all levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I work in a High School.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One skill I think has faded over time is the ability/desire to check one’s work. I’m a fan of teaching students to check to see that their answer is correct. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I am at the board, lecturing, I hope to engage the class by asking them to verify my work as I move along. When a student finds I’ve made a mistake, I thank them, and tell the class that mistakes and the process of correcting them are part of the process. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do my best to engage with the students in a way that encourages them to not be intimidated by what they don’t know, but rather to use what they know to get a valid result. It’s important that they understand no one is infallible and errors along the way are ok, as long as we learn from them. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-04-20T13:00:31.053" LastActivityDate="2019-04-20T13:00:31.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15536" PostTypeId="1" CreationDate="2019-04-20T14:25:38.640" Score="6" ViewCount="1318" Body="&lt;p&gt;I am in charge of some practice lesson for Calculus II.&#xA;I have to show how to apply the theory for unconstrained optimization (mainly Hessian analysis) and constrained optimization (Lagrange multipliers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to use functions and boundaries taken from the real world. These examples must be easy and solvable and they should appear 'not-too-boring' to students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have any suggestion?&lt;/p&gt;&#xA;" OwnerUserId="12209" LastEditorUserId="64" LastEditDate="2019-04-21T20:40:29.420" LastActivityDate="2019-09-25T15:28:01.650" Title="Unconstrained/Constrained optimization real life example" Tags="|calculus|exercises|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15540" PostTypeId="2" ParentId="1455" CreationDate="2019-04-22T15:10:46.520" Score="1" Body="&lt;p&gt;In some classes, for nontrivial errors:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I pay them - &lt;em&gt;immediately&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Several other people have mentioned rewards.  I like the individual reward of chocolate in &lt;a href=&quot;https://matheducators.stackexchange.com/a/1460/1608&quot;&gt;this answer&lt;/a&gt; and the class reward in &lt;a href=&quot;https://matheducators.stackexchange.com/a/15532/1608&quot;&gt;this answer&lt;/a&gt;, but one fails to be public, and the other fails to reward the individual initiative that pointing out such an error requires. (At least in my post-secondary experience, it takes some guts to do so.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't pay much; a quarter, usually, or occasionally a random coin from another country I have in my pocket - and sometimes I don't have any money on me.  If you have your own lecture notes you make public in some fashion and they find errors, you should definitely pay them - you'd pay an editor to find embarrassing mistakes, right?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't forget this might be a bad choice of strategy in many contexts, depending on the socioeconomic stratum you deal with (you may be either much poorer or much richer than your students, for example).  Also, sometimes you do not want to encourage the loudest to always be loud, in which case the donuts might be a better idea.  But, properly handled, students never fail to be both baffled and more keen-eyed after doing it a few times - and eventually you don't have to have the carrot any more.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Related anecdote: I first heard of this idea after I'd been teaching for some time.  A professor at a college I shall not name routinely took the first day of calculus and asked all kinds of questions, giving students a dollar per correct answer.  Then anybody who had gotten money was abruptly told they were required to register for the next higher level of calculus!  Good way to weed out the 'easy-A' seekers.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-04-22T15:10:46.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15541" PostTypeId="2" ParentId="13116" CreationDate="2019-04-22T16:47:19.770" Score="14" Body="&lt;p&gt;You ask &quot;Why isn't the term &lt;em&gt;inequation&lt;/em&gt; widely used in English?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer, however tautological, is that the term isn't used in English because it isn't used in English.  Perhaps more usefully, remember that languages change and evolve as speakers of the language (for example) start using new words, stop using old words, alter pronunciation, and play with grammatical structure.  When a language possesses a word which is used infrequently, the answer to the question of &lt;em&gt;why&lt;/em&gt; that word is used infrequently is typically &quot;Because speakers of that language do not frequently require that word in order to communicate ideas to each other.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One important facet of language evolution is that you will almost never find two words which have exactly the same meaning.  If a language possesses two synonymous words, one of two things typically happens:  either the meanings of the two words diverge (for example, big, large, and huge all mean basically the same thing in English, but have different connotations), or one of the two words will fall out of favor (for example, the word &quot;empathic&quot; in English seems to be in the process of being replaced by &quot;empathetic&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In English usage, the words &quot;inequation&quot; and &quot;inequality&quot; have nearly identical meanings.  Moreover, it appears that there are very few instances where a native speaker of English might want to draw a distinction between these two concepts.  As such, one of the two words is used quite a lot, while the other is used rarely.  We &lt;em&gt;can&lt;/em&gt; draw distinctions between the two words (e.g. via the analogy &quot;equality : equation :: inequality : inequation&quot;) and assert that the words have quite different meaning, but the fact of the matter is that most English speakers &lt;em&gt;don't&lt;/em&gt; feel that distinction, and therefore do not require the word &quot;inequation.&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;You also, secondarily, ask why the use of &quot;inequation&quot; went into decline in the mid-20th century.  I dispute that interpretation of events.  If you expand the time scale to cover the last 200 years, you get a very different picture:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0quzu.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0quzu.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First off, if you look at the vertical scale, it indicates that this word has &lt;em&gt;never&lt;/em&gt; been all that common, even in the late 50s and early 60s when it hit it's peak.  More importantly, it appears to me that the word was hardly used &lt;em&gt;until&lt;/em&gt; the late 50s, when usage spiked.  The decline after that is simply a return to normalcy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that the more interesting question is &quot;Why did the use of the term &lt;em&gt;inequation&lt;/em&gt; appear to spike in the late 50s?&quot;  A couple of hypotheses occur off the top of my head:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The term has never been very common.  That spike could easily be the result of the word being used frequently by a very small number of authors.  The spike, then might represent the output of one or two people's careers:  a lot of publications early on, with fewer publications as they take on other duties.&lt;/li&gt;&#xA;&lt;li&gt;The use of the term might be related to the adoption of the &quot;new math&quot; curriculum.  It starts trending up a little bit early, but it is possible that the term was brought into vogue by new math advocates, and that the decline coincides with the decline of new math.&lt;/li&gt;&#xA;&lt;li&gt;(This is they hypothesis which I favor.) After WWII, tensions between the US and the USSR increased, and a great deal of emphasis was placed on mathematics and engineering education.  US institutions recruited a large number of European mathematicians in that time, many of whom were not native English speakers.  The spike may be due to their writing in English but with some of their own native vocabulary.  The decline could then be explained by acculturation, as well as the deaths and retirements of these European scholars.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Finally, because this website is about mathematics education and not (for example) English language usage, there is a mild pedagogical argument against using the term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every time a new term is introduced, you create extra cognitive work for your students.  The goal should be to introduce as little jargon and technical terminology as is necessary to understand the underlying concepts.  Certainly, there are times when we really need distinct words to describe things.  For example, the word &lt;em&gt;continuous&lt;/em&gt; is very useful, as it hides a couple of distinct (though equivalent) definitions (&lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;? left- and right-limits agree with the function? etc.).  Still, it is a bit of vocabulary that students have to learn, and they struggle with it.  If some bit of jargon does not make future discourse easier, then it should probably be skipped.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my own teaching, I typically try to avoid using the word &quot;identity,&quot; and will typically use it interchangeably with &quot;equality&quot; and &quot;equation&quot; if I do happen to use it.  Moreover, I can typically avoid using any jargon by simply asking students to do what I want them to do.  For example, instead of asking them to prove an identity, I can simply give them the statement and ask them to prove it.  For example, I would write&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Show that &lt;span class=&quot;math-container&quot;&gt;$\sin(t)^2 + \cos(t)^2 = 1$&lt;/span&gt; for any real number &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;instead of&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Prove the identity &lt;span class=&quot;math-container&quot;&gt;$\sin(t)^2 + \cos(t)^2 = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I &lt;em&gt;might&lt;/em&gt; (depending on the class) finish this by stating &quot;This formula (or equation or identity or statement) is called the 'Pythagorean identity',&quot; but I can do that without ever explicitly defining the word &quot;identity.&quot;&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2019-10-05T19:08:45.953" LastActivityDate="2019-10-05T19:08:45.953" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15542" PostTypeId="6" CreationDate="2019-04-23T22:56:11.040" Score="0" Body="&lt;p&gt;I have been a member of MESE for 5+ years. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've also served as a mod on &lt;a href=&quot;https://money.stackexchange.com/&quot;&gt;Money.SE&lt;/a&gt; for over 5 years and feel I can bring what I've learned from that experience to MESE. As I am retired, just out part time, I can offer the attention to members here when called to handle urgent matters, or simply the fast response of a moderator when needed. I'd look forward to being on the team to help it reach &quot;graduation&quot; and lose the &quot;Beta&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that my reputation on Money.SE is one of patience, inclusiveness and team building. In real life, I offer students my respect and patience, both critical to the learning process. When one explanation fails, I'll explain a topic a second or third way until I hear the magic words &quot;Now I get it.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most recent best things I've heard? &quot;I got an A in math for the first time with your help&quot; &amp;amp; &quot;I got into Cornell! It was because you helped me through math these 3 years that I took BC calc, and I know that's what did it.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I reserve the right to edit, add/delete, etc.)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-04-24T11:16:23.870" LastActivityDate="2019-04-24T11:16:23.870" CommentCount="13" CommunityOwnedDate="2019-04-23T22:56:11.040" ContentLicense="CC BY-SA 4.0" />
  <row Id="15543" PostTypeId="1" AcceptedAnswerId="16543" CreationDate="2019-04-23T22:59:45.470" Score="4" ViewCount="763" Body="&lt;p&gt;I am self learning the book &lt;em&gt;Linear Algebra Done Right&lt;/em&gt;. I tried to complete all exercises in each chapter. I am currently at Chapter 3 and found that it is not feasible to complete all of the exercises. (I want to finish the book this year and I started in January.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, I don't know whether I should complete all of the exercises. Many of the text's questions are difficult for me. Most of the time, I have try a few times. So, I cannot decide whether to skip a question or not. If I can know the approach to the question by just looking at it, I can decide to skip. But often this is not the case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I am wondering, if anyone has taken a course that use the book (3rd edition), and can list some problems for each chapter (maybe from problem sets)?&lt;/p&gt;&#xA;" OwnerUserId="11818" LastEditorUserId="238" LastEditDate="2019-04-27T12:12:57.653" LastActivityDate="2019-04-27T12:12:57.653" Title="Selected Exercise for Linear Algebra Done Right Edition 3" Tags="|self-learning|homework|linear-algebra|" AnswerCount="2" CommentCount="1" ClosedDate="2019-04-25T17:26:14.710" ContentLicense="CC BY-SA 4.0" />
  <row Id="16542" PostTypeId="1" CreationDate="2019-04-24T03:18:12.160" Score="2" ViewCount="217" Body="&lt;p&gt;(this is in the US and at a high school level)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;why don't we dedicate a day of the week each week to do a lab for math for exploration? I mean we already do that for Earth Science, Physics, Chemistry and Biology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples of mathematics labs would be in applied math or math itself&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Discovering numbers: constructing the real numbers (probably not by doing Dedekind cuts)&lt;/li&gt;&#xA;&lt;li&gt;Pan balance problem = algebra&lt;/li&gt;&#xA;&lt;li&gt;Proportional relationships (exploring a physics constant/application eg Hooke's Law)&lt;/li&gt;&#xA;&lt;li&gt;Finding pi by measuring circular objects&lt;/li&gt;&#xA;&lt;li&gt;Proof workshop&lt;/li&gt;&#xA;&lt;li&gt;Working on a very very hard math problem or proof (e.g., volume of frustum or motion problems, &lt;span class=&quot;math-container&quot;&gt;$d=rt$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(The examples below are of beyond high school level, but I wanted to include them to show examples of ideas of what I thought would be good labs adjusted to the algebra level despite being of calculus and beyond level):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In calculus II, it would be something like: show that &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \int_{-\infty}^{\infty} e^{(-x^2)} dx= \sqrt{\pi}.$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\lim_{x\to 0} \frac{\sin x}{x} = 1$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://math.berkeley.edu/~giventh/lim.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Exploring l'hopital's rule.&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="238" LastEditDate="2019-04-24T14:28:27.447" LastActivityDate="2019-04-24T14:28:27.447" Title="why don't we do labs in/for math?" Tags="|secondary-education|mathematical-pedagogy|usa|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16543" PostTypeId="2" ParentId="15543" CreationDate="2019-04-24T12:49:59.857" Score="5" Body="&lt;p&gt;See &lt;a href=&quot;https://www.google.com/search?client=firefox-b-1-d&amp;amp;ei=21XAXOrwM8K4tQWCgpzgAw&amp;amp;q=syllabus+for+linear+algebra+using+Linear+Algebra+Done+Right&amp;amp;oq=syllabus+for+linear+algebra+using+Linear+Algebra+Done+Right&amp;amp;gs_l=psy-ab.12...15311.25049..27150...0.0..2.643.3906.0j19j1j5-2......0....1..gws-wiz.......0i71j33i10.LOC2eJsjkjs&quot; rel=&quot;noreferrer&quot;&gt;Syllabus for linear algebra using Axler's &quot;Linear Algebra Done Right&quot;&lt;/a&gt; for current/past courses being taught with the text as the primary text in a linear algebra class.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, see &lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/18-700-linear-algebra-fall-2013/syllabus/&quot; rel=&quot;noreferrer&quot;&gt;MIT's Open Courseware (OCW) model for Linear Algebra using Axler's text&lt;/a&gt;.  You can download the course materials for the MIT course, to supplement your study of the text &lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/18-700-linear-algebra-fall-2013/download-course-materials/&quot; rel=&quot;noreferrer&quot;&gt;here, including assigned problem sets&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also see the &lt;a href=&quot;https://math.berkeley.edu/~ogus/Math_110--008/index.html&quot; rel=&quot;noreferrer&quot;&gt;on-line syllabus from Berkeley for &quot;Linear Algebra&quot;&lt;/a&gt; using the text.  On the linked course outline, you'll find links to homework assignments, supplementary study material, and some solutions to assigned homework.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-04-24T13:00:22.480" LastActivityDate="2019-04-24T13:00:22.480" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16544" PostTypeId="6" CreationDate="2019-04-24T13:29:01.920" Score="0" Body="&lt;p&gt;My strengths:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I am one of the earliest members of the site. I love this place. I see its potential. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In review queues, I work to help new users whose questions are very nearly on-topic. Often these questions become passable or even good.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am happy to create value from the mundane tasks of moderation: helping with edits, deleting obvious spam.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My traits that might be important to some:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I am (currently) unconnected to Twitter; a lot of the math education wisdom which could appear on this site is instead being disseminated on Twitter. This is unfortunate, because Twitter is not designed to be a repository of knowledge the way that StackExchange sites are. This site needs a Twitter-connected leader to help archive some of that wisdom here.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I have zero patience for &lt;a href=&quot;https://matheducators.meta.stackexchange.com/questions/555/answers-and-comments-on-a-question-attacking-an-entire-subset-of-our-profession&quot;&gt;gatekeeping&lt;/a&gt; or &lt;a href=&quot;https://matheducators.meta.stackexchange.com/questions/564/handling-edgy-jokes-in-comments&quot;&gt;political trolling&lt;/a&gt;; my stances on these are not &quot;moderate.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I would be honored to become a moderator of this lovely place.  &lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2019-04-24T13:29:01.920" LastActivityDate="2019-04-24T13:29:01.920" CommentCount="17" CommunityOwnedDate="2019-04-24T13:29:01.920" ContentLicense="CC BY-SA 4.0" />
  <row Id="16545" PostTypeId="2" ParentId="15543" CreationDate="2019-04-24T19:42:14.273" Score="0" Body="&lt;p&gt;One book I read over the last few years, in the preface, mentions that some problem sets contain problems logically covered in a later chapter. The author says these problems were set as challenges requiring a little creative insight, beyond the explicit content of the current chapter but within the scope of the book. I think it was this book.&lt;/p&gt;&#xA;" OwnerUserId="6202" LastActivityDate="2019-04-24T19:42:14.273" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16548" PostTypeId="1" CreationDate="2019-04-25T21:01:42.743" Score="1" ViewCount="185" Body="&lt;p&gt;I need to give an option talk (a 10 min talk given to students who are selecting their options for sophomore mathematics) about an elementary number theory module. The students will have completed a course in calculus, linear algebra and discrete mathematics so are familiar with mathematical proofs. I will discuss how it is the study of positive integers particularly the primes and give some cryptography applications. What is a good hook to stipulate in this talk regarding an introduction to elementary number theory?&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="376" LastEditDate="2021-02-03T19:40:55.293" LastActivityDate="2021-02-03T19:40:55.293" Title="What should I say about elementary number theory?" Tags="|undergraduate-education|number-theory|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16549" PostTypeId="1" CreationDate="2019-04-26T03:56:10.227" Score="6" ViewCount="240" Body="&lt;p&gt;When I say &quot;framing,&quot; I mean things like choosing zoom, x-axis/y-axis step, horizontal/vertical shift from the origin, choosing how/when to number steps, labeling axes, as well as, purely aesthetic details like using a grid (or not), labeling the axes &quot;x&quot;/&quot;y&quot;, colouring/texturing the graphed functions, thickness of the lines, resolution, pixel dimensions, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondary education mathematics like algebra, geometry, trigonometry, precalculus, calculus. So lots of polynomials, trigonometric functions, logarithmic and exponential functions, rational functions, conics, functions in polar coordinates, parametric equations/curves, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example I can already think of is: if you're graphing a conic, it's a good idea to frame the graph in such a way that the student can clearly read the center/foci/vertices/etc because identifying them will likely be a necessary or helpful task in understanding the material.&lt;/p&gt;&#xA;" OwnerUserId="12228" LastEditorUserId="64" LastEditDate="2019-04-26T10:48:52.583" LastActivityDate="2019-04-27T18:24:39.423" Title="What are some tips for framing a graph in the most readable, relevant, and aesthetic way, for secondary education mathematics?" Tags="|secondary-education|graphing|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16550" PostTypeId="2" ParentId="16549" CreationDate="2019-04-26T11:44:31.657" Score="6" Body="&lt;p&gt;I think your last paragraph was most of a good answer. i.e. a graph should contain the points of interest. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Linear Equation - X and Y intercepts, and depending on the lesson topic, highlight points that were part of the problem as stated. (e.g. the problem statement may have been &quot;given the 2 points, solve for the equation of the line and produce a graph. In this case, you can ask they highlight those 2 points on the line.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parabola - Typical to ask for &quot;five points&quot; - Vertex, 2 X-intercepts, Y-intercept, and symmetric point. [Note, this is the generic approach, of course if the parabola doesn't intersect X-axis, those 2 points don't exist.] Focus/directrix for the higher level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cubic - The X-intercepts, and Y-intercept if appropriate. Let me offer a graph of a potential problematic situation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$y=\left(x-5\right)\left(x+1\right)\left(x+8\right)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jonRc.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jonRc.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I say &quot;problematic&quot; for one common issue I see. A teacher might give a worksheet/exam where the blank graph has a scale already printed. That scale can be 10X10 or 20X20 and it would cover the 3 X-intercepts, but not let the student draw the local extrema. The request sometimes feels ambiguous to the student. A 'sketch' might mean general shape of the curve and identify X-intercepts, along with Y (with a note on the graph, indicating &quot;-40&quot;). For advanced students, usually pre-calc, the exact extrema values are important as well. And note that to show them in a pleasant graph, the scales are different. I'd prefer graphs with unlabeled axis for the more advanced students' use. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the other conics, circle/ellipse, hyperbola, a similar approach. These have their own points of interest, foci, vertex, directrix, asymptotes, and it's important to have problems be clear, i.e. if they are given the graph, those points should be labeled or identified, if they are given the equation, their graph should should these points and the drawing itself be neat. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trig graphs - There are a number of directions problem come from. &quot;Graph this equation&quot;, &quot;Look at this graph and write the appropriate equation&quot;, &quot;Here is a word problem. Graph the sinusoidal function, and give the equation.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In each case, I'd suggest (a) Identify the minimum and (b) maximum values, as well as the (c) sinusoidal axis / midline. Next, Identify the (d) period. From there, the appropriate starting point (the horizontal shift) is determined and the graph drawn. For some problems, one period is enough, others, multiple periods. I'd note that if min/max if a high enough number, the graph need not always show the X-axis, only the scale for x values. e.g. a value ranges from 100 to 110 over time. I've seen frustrated students produce a scale that has a blank area from 0-100, and a tiny graph above that. Better just to show them that not all graphs require starting at zero at either axis.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-04-27T18:24:39.423" LastActivityDate="2019-04-27T18:24:39.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16551" PostTypeId="6" CreationDate="2019-04-26T15:08:33.553" Score="0" Body="&lt;p&gt;As suggested by Chris Cunningham...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have some experience with a number of different SE sites, though no mod experience. I try to answer and help where I can - answer, edit, vote, flag. I can handle even heated discussions and hostile online interactions calmly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possibly controversial opinions. Vote accordingly:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Comments are strictly for improving questions and answers. No partial answers.&lt;/li&gt;&#xA;&lt;li&gt;Research-based answers are the best, explicit personal experience is fine too, while speculation and opinions are not valid answers.&lt;/li&gt;&#xA;&lt;li&gt;Old questions, answers and comments are as deserving of attention as new ones. The website is also a repository of knowledge.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I would not start rigidly enforcing policies about these (without consensus via meta discussions), but would doubtless shift the moderation in the direction of my preferences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Background: Mathematics postdoc, some lecturing and plenty of small group teaching experience, 60 ECTS credits of pedagogical studies, but not a researcher in education.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-04-26T15:08:33.553" LastActivityDate="2019-04-26T15:08:33.553" CommentCount="6" CommunityOwnedDate="2019-04-26T15:08:33.553" ContentLicense="CC BY-SA 4.0" />
  <row Id="16553" PostTypeId="2" ParentId="16548" CreationDate="2019-04-27T02:46:22.977" Score="5" Body="&lt;p&gt;I have so many ideas for this ... but will select one.  I promise.  But all my ideas about this satisfy this criterion:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Try talking about something they can start computing while you are talking, but that can quickly be connected to something interesting and mysterious (if not to say unsolved).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Today's suggestion is the &quot;number of divisors&quot; or &quot;sum of divisors&quot; functions, often called &lt;span class=&quot;math-container&quot;&gt;$\tau$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; (or &lt;span class=&quot;math-container&quot;&gt;$\sigma_0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sigma_1$&lt;/span&gt;).  It is easy to get people computing these for small positive integers, and perhaps to very quickly see patterns (e.g. &lt;span class=&quot;math-container&quot;&gt;$\tau(p)=2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sigma(p)=p+1$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; prime).  And prime numbers/factorization obviously are useful in computing them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But one can quickly ask related questions.  Such as, &quot;What is the &lt;em&gt;average&lt;/em&gt; number of divisors of a number?&quot;  Of course, that requires definitions, but for this kind of talk you can sweep that under the rug and say &quot;here is a sample definition&quot; - like &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{n}\sum_{k=1}^n \tau(k)$&lt;/span&gt;.  Turns out &lt;a href=&quot;https://sagecell.sagemath.org/?z=eJxVkNFqwzAMRd8D-QdRApaJuyUPeykEtvf-QROK1yipwbGD42zt308mC2MCg-R7dXxxTwOc0clTngGXXaCBS7cNfo08Vdsw-AAGjIOg3UhYK1fW-9ZuLhtYzDhpNKqSf5JdXvQ8k-uRBfa9GvmrBoprcGzIszybntY44hfPWFdcMl2-Gxcp6FvMs56jXnFKAVlXb3ySLzVbV3V7oOXuvzHRcINeTlOnLI2c4Wr1J9lGfHwxdiTwAxRtG_XKn1AICSXM1ke0fsSHVPhQtZqkgpu3PjQiUC9UIi7xaakRx6NQ8A98KNq0y7CDlD_eXF7X&amp;amp;lang=sage&amp;amp;interacts=eJyLjgUAARUAuQ==&quot; rel=&quot;noreferrer&quot;&gt;this is asymptotically&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$\log(n)+2\gamma-1$&lt;/span&gt; - yes, &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;that&lt;/em&gt; gamma-the-number&lt;/a&gt;.  On the other hand, I don't think a hard &lt;em&gt;error&lt;/em&gt; has been given - you may have to peruse Apostol, Stopple, or Hardy/Wright to see, I think it is only known to be between &lt;span class=&quot;math-container&quot;&gt;$O(1/\sqrt[4]{n})$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$O(1/\sqrt[3]{n})$&lt;/span&gt; but my info may be out of date.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternately, to show how quickly new questions can arise, one can make up functions like &lt;span class=&quot;math-container&quot;&gt;$\tau_{o}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tau_{e}$&lt;/span&gt;, the number of odd and even divisors, respectively - are they equally good?  What is &quot;good&quot; here? They are easy to compute small values of, but different - can we ask/answer the same questions about formulas and so forth?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or if the students are quite erudite you can talk about its Dirichlet series (&lt;a href=&quot;http://mathworld.wolfram.com/DivisorFunction.html&quot; rel=&quot;noreferrer&quot;&gt;number 19 here&lt;/a&gt;) but that probably is beyond what you can do ...  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In any case, you don't have to take this example, but I very much recommend finding some topic that is very easy to investigate that leads really fast to unknown/hard questions.  Whether you look for one related to cryptography, algebra, or geometry, you will find one!  I hope they all pick number theory.  And that many other people answer this question so it becomes a repository for ideas for talks advertising it!&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-04-27T02:46:22.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16554" PostTypeId="2" ParentId="16549" CreationDate="2019-04-27T03:18:48.987" Score="5" Body="&lt;p&gt;Assuming you are teaching these topics, I think this is a perfect opportunity to show students why they need to know how to actually solve these things - because there is no &quot;one answer&quot; for what zoom/min/max is best.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some more complex functions there will be no one frame that will even show all significant features, as some may only be visible close by and others only zoomed way out - or if you have an aspect ratio very different from 1.  Consider the &lt;a href=&quot;https://sagecell.sagemath.org/?z=eJxzyMwrSS1KTC7h5UpJTVOI18izjTbUMdUxNNAxNQCSIMogVtOKl0sBCIoz8ss1CnLySzQq4kx087Qq4ox1NCp0dPN08rQNNTU1AUrhFL8=&amp;amp;lang=sage&amp;amp;interacts=eJyLjgUAARUAuQ==&quot; rel=&quot;noreferrer&quot;&gt;family of curves&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$x^4-nx^3$&lt;/span&gt; for various positive &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and you will see what I mean, I hope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that doesn't exactly answer the question, but I think it answers a possible question &lt;em&gt;behind&lt;/em&gt; it about how to remove distraction from instruction surrounding graphing.  And this is one possible answer to the &quot;why are we doing this&quot; surrounding careful plotting - it won't satisfy pure utilitarians, but it might be a nice way to say &quot;the calculator did it&quot; won't always work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To bring it back to your actual question, I might recommend asking students to vote on different zooms, or to have a real-time zoom/frame choice if your computation method supports that.  This may align closer to your pedagogical goals than just picking &quot;the perfect one&quot;, at least after one's initial examples which should be simple enough not to have to worry about the choice.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;On a very practical note unrelated to (and possibly undermining?) my main comment, if you can get your plotter to plot &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt; and friends with &lt;a href=&quot;https://sagecell.sagemath.org/?z=eJwryMkv0SjOzNOo0NTRqNAx0DHSKsjU1CnJTM4uti3I1DcGM-PT8otyE0tKUouAYpoAxcIRuA==&amp;amp;lang=sage&amp;amp;interacts=eJyLjgUAARUAuQ==&quot; rel=&quot;noreferrer&quot;&gt;horizontal axis tick marks in (rational) multiples of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;&lt;/a&gt;, that is a big win!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hnZT7.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hnZT7.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-04-27T03:18:48.987" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16555" PostTypeId="2" ParentId="7528" CreationDate="2019-04-27T12:36:49.297" Score="4" Body="&lt;p&gt;Hexagonal basalt columns at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Giant%27s_Causeway&quot; rel=&quot;nofollow noreferrer&quot;&gt;Giant's Causeway&lt;/a&gt;&#xA;in Northern Ireland:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/hBOnD.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hBOnD.jpg&quot; alt=&quot;HexCols&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Columnar_jointing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/gyxO1.jpg&quot; width=&quot;400&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://www.rob-tomlinson.com/places/giants-causeway&quot; rel=&quot;nofollow noreferrer&quot;&gt;RTomlinson&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-04-27T12:36:49.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16556" PostTypeId="1" CreationDate="2019-04-27T16:13:57.917" Score="6" ViewCount="180" Body="&lt;p&gt;I'd like to learn some things about undergraduate mathematical  education in Syria &lt;strong&gt;EDIT: In particular I'm interested in students 15-16 years old.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What are the main differences from the European countries?&lt;/li&gt;&#xA;&lt;li&gt;Are there any interesting facts about undergraduate mathematical education in Syria?&lt;/li&gt;&#xA;&lt;li&gt;Where could I find (books, articles) information about undergraduate mathematical education in Syria?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'd like the material to be in English. I know almost nothing about undergraduate education in Syria. What I found most about was about education in general whereas I'd like information about mathematics in specific.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;SECOND EDIT : Any answer regarding countries at war that have immigrants is also acceptable&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="12234" LastEditorUserId="12234" LastEditDate="2019-05-01T15:53:42.740" LastActivityDate="2019-05-01T15:53:42.740" Title="Mathematical undergraduate education in Syria" Tags="|undergraduate-education|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16557" PostTypeId="1" CreationDate="2019-04-30T07:40:45.190" Score="5" ViewCount="303" Body="&lt;p&gt;I will be teaching a calculus-type course in Norwegian. Our textbook is unfortunately in English (the curse of a small language), but any custom exercises should be and all exams have to be in Norwegian; exams in both bokmål and nynorsk, actually.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the notations for (polynomial) long division seem to vary from country to country, I would like to know what is the notation used for long division of polynoms in Norway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia does not seem to have the relevant articles in Norwegian or even Danish, for that matter. Of culturally related Wikipedia articles I found:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Swedish: &lt;a href=&quot;https://sv.wikipedia.org/wiki/Polynomdivision&quot; rel=&quot;noreferrer&quot;&gt;https://sv.wikipedia.org/wiki/Polynomdivision&lt;/a&gt; with no notation, but there is notation for the long division of numbers: &lt;a href=&quot;https://sv.wikipedia.org/wiki/Liggande_stolen&quot; rel=&quot;noreferrer&quot;&gt;https://sv.wikipedia.org/wiki/Liggande_stolen&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Finnish: &lt;a href=&quot;https://fi.wikipedia.org/wiki/Polynomin_jakokulma&quot; rel=&quot;noreferrer&quot;&gt;https://fi.wikipedia.org/wiki/Polynomin_jakokulma&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;German: &lt;a href=&quot;https://de.wikipedia.org/wiki/Polynomdivision&quot; rel=&quot;noreferrer&quot;&gt;https://de.wikipedia.org/wiki/Polynomdivision&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What is the relevant notation in Norway?&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2019-05-02T15:54:45.487" Title="What is the notation for polynomial long division in Norway?" Tags="|undergraduate-education|notation|solving-polynomials|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16558" PostTypeId="2" ParentId="16557" CreationDate="2019-04-30T07:40:45.190" Score="3" Body="&lt;p&gt;The fruitful search terms turn out to be &quot;divisjon&quot; and &quot;polynomdivisjon&quot;. Both give several results on Youtube. This example includes both: &lt;a href=&quot;https://www.youtube.com/watch?v=dTfgvuuALBE&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=dTfgvuuALBE&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;dividering&quot; seems to be another useful term for finding relevant content.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The notation seems similar to the notation in the German Wikipedia article.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-05-02T15:54:45.487" LastActivityDate="2019-05-02T15:54:45.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16559" PostTypeId="1" CreationDate="2019-04-30T19:21:12.887" Score="5" ViewCount="397" Body="&lt;p&gt;Next year I volunteered to lead the math class for a group of 6th graders (ages 11 - 12). Here are the pertinent details:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;About 5 - 8 (U.S.) students, for about 45 minutes, 3 days a week (they'll meet with the regular teacher the other 2).&lt;/li&gt;&#xA;&lt;li&gt;These are the &quot;advanced&quot; students. This year, as 5th graders, they are in the 6th grade math class. The school (small, private) only goes up to 6th grade, so taking math with the 7th graders isn't really an option.&lt;/li&gt;&#xA;&lt;li&gt;All (or almost all) of them have completed &lt;a href=&quot;https://www.khanacademy.org/math/cc-sixth-grade-math&quot; rel=&quot;nofollow noreferrer&quot;&gt;Khan Academy's 6th grade math curriculum&lt;/a&gt;, and most have already made significant progress through &lt;a href=&quot;https://www.khanacademy.org/math/cc-seventh-grade-math&quot; rel=&quot;nofollow noreferrer&quot;&gt;Khan Academy's 7th grade math curriculum&lt;/a&gt;. (This is probably what they'll continue to work on during the two &quot;off&quot; days.)&lt;/li&gt;&#xA;&lt;li&gt;In addition to Khan (mainly as homework), their current teacher uses a lot of inquiry-based and group-work approaches in the classroom, so they should already be relatively comfortable with loosely structured lessons that involve exploration. In general the school uses the Singapore/Eureka Math curriculum.&lt;/li&gt;&#xA;&lt;li&gt;The minimal goal would be to prepare these students to enter Algebra 1 in their 7th grade. But since a lot of them are close already, it seems like an opportunity to do more.&lt;/li&gt;&#xA;&lt;li&gt;I am a college math professor, but I don't have any Math-Ed experience/training at this level.&lt;/li&gt;&#xA;&lt;li&gt;Yes, my son is one of the kids. And if my daughter continues her trajectory, I'll be doing this again in 2 years.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So here are the main questions: &lt;strong&gt;What would you do?&lt;/strong&gt; What topics would you cover? Is there a collection/book of interesting problems/topics we could work through? Any other advice?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A couple of notes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;While one or two ideas of what to do for a day or week are appreciated, I'm more concerned about having/creating enough material to last 3 days a week for the whole year. I'm looking for a larger resource or coherent theme that can generate a wealth of math topics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One idea that's been suggested by a colleague is prepping for and doing various math competitions for this age group. Most of them do compete on the school's math team. But, prep specifically in this direction is usually considered an after-school activity and probably wouldn't be appropriate. Perhaps, though, the topics for such competitions might be a good starting place for ideas.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="401" LastEditorUserId="238" LastEditDate="2019-05-01T11:36:52.930" LastActivityDate="2019-05-07T19:16:44.610" Title="Curriculum for Advanced 6th Graders" Tags="|secondary-education|primary-education|gifted-students|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16560" PostTypeId="2" ParentId="16559" CreationDate="2019-04-30T21:39:09.823" Score="-2" Body="&lt;p&gt;&lt;strong&gt;Basic content:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you expect them to get into Algebra 1 in 7th grade, you need to cover pre-algebra,now.  This is essentially one equation with one unknown.  I would make sure you cover this very solidly as there become a lot of problems later on when kids lack the ability to manipulate sides of an equality.  It is important that they get some solid practice with mechanically &quot;doing the same thing to both sides of the equation&quot; E.g. adding to both sides, not &quot;throwing it over from one side to another and then reversing the sign&quot;.  You need to take the time and give them lots of drill.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Assuming too much brains/knowledge:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, you need to be cautious about assuming your level of knowledge or even brains in kids with much less experience and whose brains are not physically mature.  I see this problem all the time (expert blindness).  I'm not saying not to progress the kids.  Just to be sensitive to your likely blind spot and to watch for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Learning what you will teach:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also good that you recognize your level of ignorance of the content.  Math prof is fine, but you need to get yourself up to speed on the content for this course.  A question on SE is not adequate to educate yourself. IOW, you need more detail than my first para here.  But at least you recognize your gap. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Go get some competing textbooks and read them (maybe outline them in topics covered, not saying read word for word) so that you have some basic feel for typical pre-algebra content and how much it may vary.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition to learning what topics are covered, you should practice a homework problem or few; work through some of the examples, etc.  Don't just assume you are smartypants math prof.  Make sure you can do all the manipulations.  And also, it puts you in some &quot;walking in their moccasins&quot; so that you get the feel of what the students must learn.  After all, your own experience will be in dim past.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, glance over the content in the year before pre-algebra (I can't even recall what that was) and of algebra itself.  This is so that you have sufficient battlespace situational awareness that you don't give the kids a gap.  For the year before and after, I think just looking at their past year book and likely next year book is sufficient (don't need to look at competing texts).  You don't need to work any problems here either.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you have done all that (maybe 2-3 long evenings), you will be able to at least converse about the content with some baseline.  At that point, I recommend interviewing a couple teachers (you will have generated some questions from the task I gave you).  Go over content coverage, likely hard/easy parts, and ask for a few tricks.  Do this with real teachers face to face.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This site is mostly junior college pre/calculus teachers.  It has weak coverage of K-12.  There are also blogs, forums etc. in the greater web (not SE) where you have middle school teachers.  But I would try for some face to face.  But at least an electronic exchange with them will be better than at MESE.  In addition, do your &quot;task&quot; first, rather than reaching out before you have educated yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Level&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, that I have a little confusion as to their level.  In the dark ages, Algebra 1 was stereotypically 9th grade.  Pre-algebra was 8th.  Kids who were one year advanced covered Algebra 1 in 8th.  Of course these kids may be a couple years ahead.  And there has been some movement to push 8th grade algebra for all (a failure).  But certainly the most advanced kids may be able to handle Algebra in 7th.  The point I am making is just to make sure that you know their real level (this is a key thing to assess) and don't miss out on some grade 5-8 content that they need to be getting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Offspring&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Treat your boy similarly to the other students (no familiarity or references to family/home life) and ask him to call you Mr. Umptifratz.  Don't be harsh with him.  But just have a normal teacher/student mode of engagement.  No favoritism or the converse.  Talk to him ahead of time, so he understands the need for this and is actively supporting this policy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Class time&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use a book/workbook.  Do not just invent what you want to talk about (same applies for college courses...just pick a book as a scaffold...it may not be perfect but nothing is, including your mishmash).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to keep it active, with some games, drill, solve and pass papers, etc.  But do not make it math contest focused because then when do they learn basic content.  Teach a standard pre-algebra course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mesh with other 2 days&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is actually kind of a puzzle to me.  How do they learn all the stuff they need for her class, with less time.  how does your stuff work, when they don't have full year of her stuff.  Etc.  But at least we know it's a puzzle.  Baseline, I assume that she will be covering stuff from standard 6th or 7th (whatever that is, advanced arithmetic?) and you will cover pre-algebra, which is stereotypically 8th?  But I am not sure if this is correct.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should at a minimum review what content will be covered in their regular course, so you know it.  I guess if the class is structured so that they just spend a lot of time drilling arithmetic, perhaps your smart kids can just do with less drill (there) and simultaneously learn pre-algebra on 3 days per week.  Possible maybe.  But keep your finger on the pulse and see how things are going.  Ideally the other teach covers major principles on the 2 days she has the kids and uses the other 3 for more practice. (I imagine she has a bigger class and you are selecting out the top kids for a few days/week?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She also may know a lot about basic middle school math, so you should consult her as one of your face to face interviews.  (Not just how you mesh together, but she may have advice for you on your own 3 days/week.  She is an SME.)  After doing the &quot;task&quot; so you are educated enough to ask better questions and learn more.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This whole thing sounds tricky in and of itself, so it is important not to have any rivalry with the other teach (one more straw on the donkey).  Be down to earth and talk to her as a colleague and win her over.&lt;/p&gt;&#xA;" OwnerUserId="12241" LastEditorUserId="12241" LastEditDate="2019-04-30T21:54:11.863" LastActivityDate="2019-04-30T21:54:11.863" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16561" PostTypeId="1" CreationDate="2019-05-01T06:31:22.307" Score="2" ViewCount="708" Body="&lt;p&gt;That was the most boring title I could choose but in all honesty, it is what the question is. Here is a reminder of the FOIL method that is used for multiplying two binomials. For example, to multiply &lt;span class=&quot;math-container&quot;&gt;$$(x+2)(x-3),$$&lt;/span&gt; the &quot;FOIL&quot; algorithm has students sum the following four products:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;First terms: &lt;span class=&quot;math-container&quot;&gt;$x.x$&lt;/span&gt;, &lt;/li&gt;&#xA;&lt;li&gt;Outer terms: &lt;span class=&quot;math-container&quot;&gt;$x.(-3)$&lt;/span&gt;, &lt;/li&gt;&#xA;&lt;li&gt;Inner terms: &lt;span class=&quot;math-container&quot;&gt;$2.x$&lt;/span&gt;, &lt;/li&gt;&#xA;&lt;li&gt;Last terms: &lt;span class=&quot;math-container&quot;&gt;$2.(-3)$&lt;/span&gt;. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Personally, I never used it as a teaching tool. But, I was writing something on the use and misuse of signs, that I came to the FOIL method. The &quot;common&quot; method of doing it took me by surprise, as I personally never take the terms with their signs. For me, it is going like this: &lt;span class=&quot;math-container&quot;&gt;$x.x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x.3$&lt;/span&gt; (then looking at the sign of the second binomial) I write &lt;span class=&quot;math-container&quot;&gt;$x^2-3x$&lt;/span&gt;, and so on. Basically, I multiply the terms and then decide about the signs.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is whether there is any pedagogical advantage or disadvantage to either of these approaches. Obviously, I am biased towards one of them :)  Having confessed that, let me put you in the shoes of a student who knows how to do something like &lt;span class=&quot;math-container&quot;&gt;$$(100+2)(100-3)$$&lt;/span&gt; but still doesn't know negative numbers. Which approach would be more &quot;natural&quot; for the student when doing &lt;span class=&quot;math-container&quot;&gt;$(x+2)(x-3)$&lt;/span&gt;? &lt;/p&gt;&#xA;" OwnerUserId="1217" LastEditorUserId="238" LastEditDate="2019-05-01T11:37:56.287" LastActivityDate="2024-03-12T20:50:20.463" Title="Why in the FOIL Method the terms are taken with their signs?" Tags="|algebra|arithmetic-operations|negative-numbers|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="16562" PostTypeId="1" CreationDate="2019-05-01T09:35:01.323" Score="6" ViewCount="275" Body="&lt;p&gt;I understand that this website is designated particularly for teaching-related topics. However as many of us do agree that being a lecturer, we are supposed to run both teaching and research duties at the same time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My problem is that I am having quite a heavy teaching load, roughly 40 hours/week. In order to maximize the profit, the university where I am working tries their best to put more pressure on the lecturers, such as they increase the number of students in a classroom but always keep a very basic salary. Doing research and producing publications are also required for lecturers. At the end of each academic year, you need to have at least one paper in some peer-reviewed journals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing research, learning new knowledge is also my interest. It is a motivation for me to begin a new day. The problem here is that I cannot get focused on only one thing recently. I often try to spend all the night for doing research, however, my mind then continuously keeps thinking to things that I would have for the class tomorrow. Whereas, during the time when I am teaching, I constantly keep thinking on a problem in my research.     &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you give me some suggestions or recommendations (even small ones) to help me to run both duties more effectively and properly? I really appreciate your help. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry for my English.&lt;/p&gt;&#xA;" OwnerUserId="12043" LastEditorUserId="12255" LastEditDate="2019-05-07T10:03:58.537" LastActivityDate="2019-05-07T10:03:58.537" Title="How to overcome the frustration in teaching and doing research at the same time" Tags="|education-research|teaching|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="16564" PostTypeId="2" ParentId="16561" CreationDate="2019-05-01T15:18:56.727" Score="12" Body="&lt;p&gt;I'm going to answer with something of a polemical frame challenge:  FOIL is evil, and probably shouldn't even be taught.  Okay... that's a bit extreme.  How's this:  FOIL is a mnemonic that is, in my opinion, not all that useful, and should not be taught.  In my own experience teaching college algebra and precalculus courses, students come to rely on FOIL far too much without understanding what they are really doing.  Then they have to multiply trinomials, and it all falls apart.&lt;/p&gt;&#xA;&lt;p&gt;Instead, I think that students should spend more time actually working with the distributive property of multiplication over addition.  FOIL then becomes a special case.  That is,&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;(a+b)(c+d)&#xA;&amp;amp;= (a+b)c + (a+b)d &amp;amp;&amp;amp; \text{(distribute $(a+b)$)} \\&#xA;&amp;amp;= ac + bc + (a+b)d &amp;amp;&amp;amp; \text{(distribute $c$)} \\&#xA;&amp;amp;= ac + bc + ad + bd. &amp;amp;&amp;amp; \text{(distribute $d$)}&#xA;\end{align}&lt;/span&gt;&#xA;While I think that your concern about working with negative numbers is overblown (students who are working at this level of symbol manipulation should already be familiar with negative numbers), but even so, it can be handled fairly naturally.  In your example&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;(x+2)(x-3)&#xA;&amp;amp;= (x+2)x - (x+2)(3) &amp;amp;&amp;amp; \text{(distribution over subtraction)} \\&#xA;&amp;amp;= x^2 + 2x - (3x + 2(3)) \\&#xA;&amp;amp;= x^2 + 2x - 3x - 6 &amp;amp;&amp;amp; \text{(distribute the negative)} \\&#xA;&amp;amp;= x^2 - x - 6.&#xA;\end{align}&lt;/span&gt;&#xA;Personally, I find this a little awkward, as one has to remember to distribute the negative (I would have preferred to keep the negation and the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; together, i.e. write &lt;span class=&quot;math-container&quot;&gt;$(x+2)(x-3) = (x+2)x + (x+2)(-3)$&lt;/span&gt;, but this is a minor point).&lt;/p&gt;&#xA;&lt;p&gt;I prefer to put the emphasis on the distributive property of multiplication over addition (and subtraction, which is really just addition in the other direction) because it is far more generalizable.  For example, if I want to multiply polynomials of degree two and three, I would write&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp;\color{red}{(ax^2 + bx + c)}(dx^3 + ex^2 + fx + g) \\&#xA;&amp;amp;= \color{red}{(ax^2 + bx + c)}dx^3 + \color{red}{(ax^2 + bx + c)}ex^2 + \color{red}{(ax^2 + bx + c)}fx + \color{red}{(ax^2 + bx + c)}g \\&#xA;&amp;amp;= (ax^2 + bx + c)\color{blue}{dx^3} + (ax^2 + bx + c)\color{green}{ex^2} + (ax^2 + bx + c)\color{orange}{fx} + (ax^2 + bx + c)\color{purple}{g} \\&#xA;&amp;amp;= ax^2\color{blue}{dx^3} + b\color{blue}{dx^3} + c\color{blue}{dx^3} + ax^2\color{green}{ex^2} + bx\color{green}{ex^2} + c\color{green}{ex^2} + ax^2\color{orange}{fx} + bx\color{orange}{fx} + c\color{orange}{fx} + ax^2\color{purple}{g} + bx\color{purple}{g} + c\color{purple}{g} \\&#xA;&amp;amp;= adx^5 + bdx^4 + aex^4 + cdx^3 + bex^3 + afx^3 + cex^2 + bfx^2 + agx^2 + cfx + bgx + cg \\&#xA;&amp;amp;= adx^5 + (bd + ae)x^4 + (cd + be + af)x^3 + (ce + bf + ag)x^2 + (cf + bg)x + cg.&#xA;\end{align}&lt;/span&gt;&#xA;This is horribly tedious and rather difficult to convey in text (this is what would end up on the board after five minutes of lecture; the story really needs to unfold in real time).  Even so, I typically very quickly get to the idea of distributing terms one at a time (i.e. first distribute the &lt;span class=&quot;math-container&quot;&gt;$ax^2$&lt;/span&gt; to each term of the cubic, then distribute &lt;span class=&quot;math-container&quot;&gt;$bx$&lt;/span&gt; to each term of the cubic, etc).  I also tend to suggest that students line up like terms as they multiply, which makes it easier to add coefficients in the end:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp;(ax^2 + bx + c)(dx^3 + ex^2 + fx + g) \\&#xA;&amp;amp;\qquad \begin{matrix}&#xA;=&amp;amp; adx^5 &amp;amp;+&amp;amp; aex^4 &amp;amp;+&amp;amp; afx^3 &amp;amp;+&amp;amp; agx^2 &amp;amp;&amp;amp;&amp;amp;&amp;amp;&amp;amp; \text{(distribute $ax^2$)} \\&#xA;      &amp;amp;&amp;amp;+&amp;amp; bdx^4 &amp;amp;+&amp;amp; bex^3 &amp;amp;+&amp;amp; bfx^2 &amp;amp;+&amp;amp; bgx &amp;amp;&amp;amp;&amp;amp; \text{(distribute $bx$)} \\&#xA;      &amp;amp;&amp;amp;&amp;amp;        &amp;amp;+&amp;amp; cdx^3 &amp;amp;+&amp;amp; cex^2 &amp;amp;+&amp;amp; cfx &amp;amp;+&amp;amp; cg. &amp;amp; \text{(distribute $c$)} \\&#xA;\end{matrix} \\&#xA;\end{align}&lt;/span&gt;&#xA;The columns may be added to get the same thing as above.&lt;/p&gt;&#xA;&lt;p&gt;Note that FOIL completely fails here.  If we were to only multiply the first, outer, inner, and last terms, we would get&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align} (ax^2 + bx + c)(dx^3 + ex^2 + fx + g)&#xA;&amp;amp;= adx^5 + agx^2 + cdx^3 + cg. &#xA;&amp;amp;&amp;amp; \text{$\leftarrow$ this is wrong!}&#xA;\end{align}&lt;/span&gt;&#xA;I hear the objection:  &amp;quot;But Xander, we don't teach students to FOIL like that!&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;You are correct, we don't teach students to do that.  And yet, they show up in my class and do it anyway, because they don't understand the deeper concept, which is distribution.  The FOIL mnemonic is not all that general, hides the reality of what is going on, and seems to confuse students.  Don't teach it.&lt;/p&gt;&#xA;&lt;p&gt;(Please note:  I have intentionally written this to convey a strongly held opinion in an intentionally confrontational manner.  However, this really is just my opinion—albeit one backed by my own experience.  Do with this opinion as you like.)&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2024-03-12T20:50:20.463" LastActivityDate="2024-03-12T20:50:20.463" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16565" PostTypeId="2" ParentId="16561" CreationDate="2019-05-01T19:53:48.863" Score="4" Body="&lt;p&gt;Simple answer: Including the signs/negative numbers is algorithmic. Working out the sign afterwards requires thinking.&#xA;Imagine writing computer code to multiply out the brackets. You would do it by keeping the minus signs with the relevant terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Pedagogically: keeping the signs with the magnitude will extend to bigger brackets in a more natural way than working out the sign afterwards.)&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-05-01T19:53:48.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16566" PostTypeId="1" CreationDate="2019-05-01T22:02:01.960" Score="2" ViewCount="163" Body="&lt;p&gt;When factoring quadratic expressions &lt;span class=&quot;math-container&quot;&gt;$ax^2+bx+c$&lt;/span&gt; it is common to the guess and check factors (AKA the cross method).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would involve factoring &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and considering particular combinations of these factors that add to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; taking into account the signs of &lt;span class=&quot;math-container&quot;&gt;$a,b,c$&lt;/span&gt;. This is typically presented using a cross and just the coefficients. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am worried about this in that there seems to be no way to record what guesses have been tried, and no way to record the &quot;checksums&quot;. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadratic_equation#Factoring_by_inspection&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quadratic Equations&lt;/a&gt; on wikipedia. (Surely this search is justified by &lt;a href=&quot;https://en.wikipedia.org/wiki/Gauss%27s_lemma_(polynomial)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss's Lemma&lt;/a&gt; and won't work if the quadratic is irreducible.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;I would like a way to explicitly and concisely enumerate all possible combinations. From this, one could demonstrate heuristics to simplify the process.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's take the monic &lt;span class=&quot;math-container&quot;&gt;$x^2+5x+6$&lt;/span&gt; with all positive coefficients.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way would be to systematically list of all factors which give the correct leading coefficient and constant and chose one which gives the correct cross term. &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(x+1)(x+6)\\&#xA;(x+2)(x+3)\\&#xA;(x+3)(x+2)\\&#xA;(x+6)(x+1)&#xA;$$&lt;/span&gt;&#xA;From this we see the middle two give the right answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have tried something like: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ohywv.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ohywv.jpg&quot; alt=&quot;trial&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where the rows and columns are labelled by the factors of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and the cells contain the sums. I'm unsure how this extends to negative coefficients and non-monic quadratics.&lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="11" LastEditDate="2019-05-02T13:10:50.970" LastActivityDate="2019-05-02T13:10:50.970" Title="Explicit Cross Method" Tags="|secondary-education|algebra|" AnswerCount="1" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16567" PostTypeId="2" ParentId="16566" CreationDate="2019-05-01T22:47:09.000" Score="4" Body="&lt;p&gt;For factoring a quadratic, when &quot;A&quot; = 1, it's relatively easy. Answer the question, &quot;What 2 numbers multiply to C but add to B?&quot; This process may be guess and check, but when A is 1, there's little need to spend much time on this. When one factor has a negative integer, the rule doesn't change, so much as the student needs to be mindful of where the minus sign lands. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's when A is non-zero we need to spend more time. The A-C Method, aka &quot;Split the middle term&quot; is useful in this case. This is my notes for that -&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JGd0m.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JGd0m.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Above, to solve for the 2 numbers AC=60, B=19, I have students start listing the pairs of number that multiply to 60, along with their sum -&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1 + 60 = 61&lt;/li&gt;&#xA;&lt;li&gt;2 + 30 = 32&lt;/li&gt;&#xA;&lt;li&gt;3 + 20 = 23&lt;/li&gt;&#xA;&lt;li&gt;4 + 15 = 19 (Done)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And an example when there's negative numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AyBt6.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AyBt6.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we needed 2 numbers, adding to -1, but multiplying to -72. We can use the same method by looking for the pairs that multiply to 72 but have a difference of 1.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;36 - 2 = 34&lt;/li&gt;&#xA;&lt;li&gt;24 - 3 = 21&lt;/li&gt;&#xA;&lt;li&gt;18 - 4 = 14&lt;/li&gt;&#xA;&lt;li&gt;12 - 6 = 6&lt;/li&gt;&#xA;&lt;li&gt;9 - 8 = 1 &gt;&gt; 9 &amp;amp; 8 have a difference of 1, but since we want the sum to be -1, we use -9 and 8. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note: The tutorial above assumes the student is already proficient at factoring when A = 1, so these extra details aren't on these sheets, but would appear on the lesson for the other cases.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-02T09:25:08.060" LastActivityDate="2019-05-02T09:25:08.060" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16568" PostTypeId="1" CreationDate="2019-05-02T18:49:12.633" Score="6" ViewCount="186" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/16561/why-in-the-foil-method-the-terms-are-taken-with-their-signs&quot;&gt;This question about FOIL&lt;/a&gt;, comments and answers made me think about the two roles of &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;: as a sign and as an operator. This struck me because the title &lt;/p&gt;&#xA;&#xA;&lt;h2&gt;&quot;Why in the FOIL Method the terms are taken with their signs?&quot;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;seems odd to me, because in the example given, &lt;span class=&quot;math-container&quot;&gt;$$(x+2)(x-3),$$&lt;/span&gt; there are no signs, just one subtraction operator. To get a sign appear, one first would have to convert to &lt;span class=&quot;math-container&quot;&gt;$$(x+2)(x+(\color{red}{-}3))$$&lt;/span&gt; to get the red minus sign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Germany, we have &quot;Vorzeichen&quot; (sign, literally &quot;pre sign&quot;) and &quot;Rechenzeichen&quot; (operator, literally &quot;calculation sign&quot;) and I think a similar distiction can be made in english with sign and operator, but I get the impression that the use is rather sloppy.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When first teaching negative numbers, is a clear distinction between the two roles of &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt; made?&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2019-05-12T16:15:48.983" Title="Is a clear distinction made between signs and operators?" Tags="|arithmetic-operations|negative-numbers|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16569" PostTypeId="1" CreationDate="2019-05-02T23:09:20.597" Score="10" ViewCount="688" Body="&lt;p&gt;I greatly admire Paul Lockhart's&#xA;&lt;em&gt;Measurement&lt;/em&gt; &#xA;(&lt;a href=&quot;http://www.hup.harvard.edu/catalog.php?isbn=9780674284388&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harvard Press&lt;/a&gt;).&#xA;Many of you know him through&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/A_Mathematician%27s_Lament&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Mathematician's Lament&lt;/a&gt;.&#xA;One review of &lt;em&gt;Measurement&lt;/em&gt; said,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“Here Lockhart offers the positive side of the math education story by showing us how math should be done. &lt;em&gt;Measurement&lt;/em&gt; offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But I do not see a way to integrate this into a conventional&#xA;mathematics course (say, for undergraduates).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. Can anyone see a way to use Lockhart's &lt;em&gt;Measurement&lt;/em&gt;&#xA;  in mathematics education, either in a conventional course,&#xA;  or as a supplement, or for guided independent study?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;&lt;em&gt;Measurement&lt;/em&gt;: Table of Contents:&#xA;&lt;/sup&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/c1als.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/c1als.jpg&quot; alt=&quot;MeasContents&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Added&lt;/em&gt;. One could imagine Part One supplementing a first course&#xA;in Geometry, and Part Two supplementing Calculus I. But the style&#xA;is so unique that I can't decide if it would be better &#xA;for these sections to proceed or to follow geometry/calculus. But really the true thrust of the book&#xA;is to understand what it means to pursue mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like this quote (where &quot;we&quot; = &quot;we mathematicians&quot;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We're always working at the edge of the unknown,&#xA;  and we're always stuck.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;A few SE citations of &lt;em&gt;Measurement&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;MESE: &lt;a href=&quot;https://matheducators.stackexchange.com/a/4199/511&quot;&gt;What are some great books for exploring mathematics? (...)&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;MSE: &lt;a href=&quot;https://math.stackexchange.com/a/284446/237&quot;&gt;Explaining what real math is to a high school student&#xA;&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;HSM: &lt;a href=&quot;https://hsm.stackexchange.com/q/9567/606&quot;&gt;When was 4D space “conceived of”?&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-05-07T23:10:11.400" LastActivityDate="2019-05-07T23:10:11.400" Title="Use of Lockhart's *Measurement* in a course?" Tags="|undergraduate-education|geometry|undergraduate-research|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16570" PostTypeId="2" ParentId="15109" CreationDate="2019-05-03T03:38:48.890" Score="1" Body="&lt;p&gt;I'm amazed that neither &lt;a href=&quot;https://www.gapminder.org/data/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gapminder&lt;/a&gt; nor the &lt;a href=&quot;https://data.worldbank.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;World Bank&lt;/a&gt; data sets have come up; they have tons of interesting development indicators, plus Gapminder comes with entertaining videos!&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-05-03T03:38:48.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16571" PostTypeId="2" ParentId="16562" CreationDate="2019-05-03T05:49:18.393" Score="3" Body="&lt;p&gt;You state that you just started your teaching job. I would assume that you have a lot of work that needs to be done once, like writing a script or getting familiar with a textbook, designing interesting slides for the lectures, etc. On top of that, you need to get experience as a teacher.&lt;br&gt;&#xA;Much of this can be recycled once you give the same class again a few semesters later, so I would strongly suggest to properly store it and already think about how to best use it again.&lt;br&gt;&#xA;Furthermore, it might be possible to come to an agreement with your dean/supervisor on that if you are able to show that you want to not do mediocre teaching and mediocre research but set up really good classes now and then do really good research in a few semesters. If that is possible in your special case I don't know of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the long run, teaching and research can be combined by supervising theses. If you don't write a solo paper yourself but instead publish one or two per year together with your Master or PhD students, that should also be accepted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For now, you have a few options and need to decide for yourself which one to pick (the list is not ordered in any way):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Suck it up and work a lot.&lt;/li&gt;&#xA;&lt;li&gt;Talk to people and come to an agreement.&lt;/li&gt;&#xA;&lt;li&gt;Quit.&lt;/li&gt;&#xA;&lt;li&gt;Do mediocre work and somehow get along.&lt;/li&gt;&#xA;&lt;li&gt;Be a genius and produce an earth shaking paper in a few days.&lt;/li&gt;&#xA;&lt;li&gt;Copy others (in teaching, maybe not in writing the paper...).&lt;/li&gt;&#xA;&lt;li&gt;Analyze your workflow and improve it.&lt;/li&gt;&#xA;&lt;li&gt;...&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2019-05-03T05:49:18.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16572" PostTypeId="2" ParentId="13856" CreationDate="2019-05-03T13:40:02.483" Score="2" Body="&lt;p&gt;In the U.S. we forgot the spirit of mathematics as we embraced the race toward solution-finding in the mid-1900's (think NASA and military-supported efforts based on those branches' interests).  In the business of finding a solution, you are allowed to be sloppy with these implications a bit more and then the idea is to sharpen their understanding of the logic later on at university.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth I totally disagree with this way of approaching mathematics, but I can't help but think this is exactly what has happened.  Once it becomes normal, it takes over, and becomes ubiquitous.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You're right, the lack of distinction is terrible.  But it's fairly clear why there is no distinction if you think about the way a culture values mathematics.&lt;/p&gt;&#xA;" OwnerUserId="6589" LastActivityDate="2019-05-03T13:40:02.483" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16573" PostTypeId="1" AcceptedAnswerId="16584" CreationDate="2019-05-03T14:23:58.213" Score="8" ViewCount="1103" Body="&lt;p&gt;What are some common ways students get confused about finding an inverse of a function?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One I can think of is conflating multiplicative inverses of rational numbers with functional inverses. e.g. thinking &quot;the inverse of the function &lt;span class=&quot;math-container&quot;&gt;$f(x)=x$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$f^{^{-1}}(x)=\frac{1}{x}$&lt;/span&gt;.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm inexperienced and I'm trying to anticipate the ways which someone would incorrectly find an inverse and what thinking motivates it.&lt;/p&gt;&#xA;" OwnerUserId="12228" LastEditorUserId="238" LastEditDate="2019-05-03T14:31:31.117" LastActivityDate="2019-05-04T17:22:56.363" Title="What are some common ways students get confused about finding an inverse of a function?" Tags="|students-mistakes|functions|inverse|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16574" PostTypeId="2" ParentId="16573" CreationDate="2019-05-03T15:42:09.490" Score="-2" Body="&lt;p&gt;The inverse (or reciprocal) of a bijection &lt;span class=&quot;math-container&quot;&gt;$f~:~E\longrightarrow F$&lt;/span&gt; (denoted &lt;span class=&quot;math-container&quot;&gt;$\stackrel{-1}{f}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\stackrel{①}{f}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt;; the last notation is the worst and the number inside the circle should be &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;) satisfy &lt;span class=&quot;math-container&quot;&gt;$\stackrel{-1}{f}\circ f=f\circ \stackrel{-1}{f}=\mathrm{id}_E=1_E$&lt;/span&gt;.&lt;br/&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is not a function; it is an element of &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; and its inverse (if it exists) is &lt;span class=&quot;math-container&quot;&gt;$\left(f(x)\right)^{-1}=\frac{1}{f(x)}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have the same word means two different things: the &lt;em&gt;inverse&lt;/em&gt; of a function and the &lt;em&gt;inverse&lt;/em&gt; of a real number. The same &quot;problem&quot; occurs in other words: compare &lt;code&gt;a root of the polynomial P&lt;/code&gt;, &lt;code&gt;a root of the equation...&lt;/code&gt; and &lt;code&gt;a root of 4&lt;/code&gt;. To avoid the confusion, just state the question clearly.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Find the inverse of the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; defined over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^3$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$\color{green}{Good}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Find the inverse of the function &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^3$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$\color{red}{Bad}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Find the inverse of &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^3$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$\color{red}{Very~Bad}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Find the inverse of &lt;span class=&quot;math-container&quot;&gt;$x^3$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$\color{red}{Horrible}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="501" LastActivityDate="2019-05-03T15:42:09.490" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16575" PostTypeId="2" ParentId="16562" CreationDate="2019-05-03T16:40:46.950" Score="5" Body="&lt;p&gt;I think there's a lot of variability in publishing rates between different STEM fields and this can lead to administrators imposing unrealistic demands on Mathematics faculty. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anecdotally I've heard publishing in Biology once a year is pretty straight-forward: just collect a bunch of new data from a new location and publish the results. Maybe even get a few undergrads to do the dirty work and call it field experience. (My apologies to any bio people if I'm trivializing your work.) Mathematics is different in that procuring new mathematical truths or methods takes a lot of time; working alone and/or with other professionals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To that end, the &lt;a href=&quot;https://www.maa.org/programs-and-communities/member-communities/committee-on-departmental-review/guidelines-programs-departments&quot; rel=&quot;noreferrer&quot;&gt;MAA Guidelines&lt;/a&gt;, Article C.5.1.2, states:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Faculty for whom personnel decisions are based upon assessment of&#xA;  contributions in teaching, scholarship, and service should have&#xA;  teaching assignments that reflect these multiple expectations and&#xA;  allow for attention to non-classroom responsibilities. Teaching&#xA;  assignments above three courses [nine credit hours] per semester, when&#xA;  combined with other faculty responsibilities, do not allow the time&#xA;  needed to develop and maintain a program of sustained scholarship with&#xA;  the result that tenure and promotion might be effectively&#xA;  unattainable. For such faculty, teaching assignments above the level&#xA;  of three courses [nine credit hours] per semester must be avoided.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(It goes on to address faculty at institutions that do require more research and what are acceptable levels of teaching assignments for them.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found it helpful to communicate to administration that this is what my main professional organization states and that it's not just me making excuses for my lack of research when teaching 12 credits a semester. Hopefully the guidelines can also help you assess your situation and communicate your needs. Good luck.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2019-05-03T16:40:46.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16576" PostTypeId="2" ParentId="16573" CreationDate="2019-05-03T17:28:53.540" Score="3" Body="&lt;p&gt;From a comment by the OP:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I'm trying to come up with &quot;plausible&quot; wrong answers for a multiple choice question about finding inverses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Per an answer given to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13058/how-to-design-multiple-choice-questions&quot;&gt;this question&lt;/a&gt;, you might be able to collect data on your students' possible answers by giving them a fill-in-the-blank quiz on inverse functions. Then, keep track of the most-common wrong answers by frequency.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, I've asked questions like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Suppose the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is invertible, and &lt;span class=&quot;math-container&quot;&gt;$f(3) = -4$&lt;/span&gt;. Find the value of &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(-4)$&lt;/span&gt;.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some common wrong answers (in no particular order) are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{3}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I do not know for sure what motivates these answers, but if I had to guess, I would say the answer of &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt; suggests the student correctly remembered something about &lt;em&gt;switching input and output&lt;/em&gt;, but applied the &quot;exponent&quot; as a factor. The answer of &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; suggests the student didn't remember the input-output switch, but used the &quot;exponent&quot; as a factor. The answer of &lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{3}$&lt;/span&gt; suggests the student knows the action of raising a number to the &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; exponent, but misinterpreted the notation for an inverse function.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-05-03T18:22:57.797" LastActivityDate="2019-05-03T18:22:57.797" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16577" PostTypeId="1" AcceptedAnswerId="16581" CreationDate="2019-05-03T18:47:16.693" Score="1" ViewCount="256" Body="&lt;p&gt;Reading and commenting on &lt;a href=&quot;https://matheducators.stackexchange.com/questions/16573/what-are-some-common-ways-students-get-confused-about-finding-an-inverse-of-a-fu&quot;&gt;What are some common ways students get confused about finding an inverse of a function?&lt;/a&gt; I was kindly set straight that the use of &lt;span class=&quot;math-container&quot;&gt;$\sin^{^{-1}}(x)$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\arcsin(x)$&lt;/span&gt; has been rendered obsolete. &lt;a href=&quot;https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;International Standard ISO 80000-2&lt;/a&gt; offers the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rlfwU.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rlfwU.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This notation is preferable to me, with one point of confusion. The text we use, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0201232537&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Algebra and Trigonometry&lt;/em&gt;&lt;/a&gt; by Paul Foerster, uses &lt;span class=&quot;math-container&quot;&gt;$\textrm{Arcsin}(x)$&lt;/span&gt; to indicate the restricted domain, and the lower case as above for all real domain. This notation leaves me with my question - &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do we ask for the arcfunction with no domain restriction, i.e we actually want the multiple answer as in &quot;solve for the first 6 values when &lt;span class=&quot;math-container&quot;&gt;$y=2$&lt;/span&gt; for the following sinusoidal function&quot; ? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Disclosure - This question is in reaction to my discovering that a notation that I never cared for has been eliminated by an international standard for nearly 10 years, but still used at my high school. I am trying to understand where to go with this as people in general aren't so open to change. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-03T20:59:17.090" LastActivityDate="2019-05-03T20:59:17.090" Title="How to formulate this type of arcsin problem?" Tags="|secondary-education|trigonometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16578" PostTypeId="2" ParentId="13116" CreationDate="2019-05-03T18:49:06.640" Score="-1" Body="&lt;p&gt;I have to disagree with the answers claiming that the word &quot;inequation&quot; isn't necessary in english or that it isn't widely used because &quot;people&quot; chose not to. The fact is that most english speakers don't know it exists and &lt;a href=&quot;http://mathworld.wolfram.com/Inequation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathworld&lt;/a&gt; does a horrible job explaining it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I taught mathematics in four languages. Of these four english was the most different in its terminologies and notations. For example we denote an open interval &lt;span class=&quot;math-container&quot;&gt;$]a,b[$&lt;/span&gt; which is similar to the closed one &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt;; in english they use &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; for the open interval. That doesn't only cause unnecessary confusion with the pair (unordered couple) &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; but also isn't compatible with the notations of geometry (&lt;span class=&quot;math-container&quot;&gt;$[AB]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$]AB[$&lt;/span&gt; for segments and &lt;span class=&quot;math-container&quot;&gt;$(AB)$&lt;/span&gt; for the line).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same goes for many other notations and terminologies like a skew field not being a field, 5/8/2000 means &quot;8 may 2000&quot; instead of the international &quot;5 august 2000&quot;. And of course the term &quot;inequation&quot; was one of them. The books published in US or UK never use this term; they use &quot;inequality&quot; to mean two different things. The books translated to english in other countries usually use such terms and terminologies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that this term hasn't entered school books yet. If it didn't in french there wouldn't be many people using the term &quot;inéquation&quot;; that wouldn't mean that this word isn't necessary. So the necessity or importance of the word isn't correlated with the frequency of its use. If a curriculum reform introduced this term in schools, its use will grow exponentially at all levels. See &lt;a href=&quot;https://math.stackexchange.com/questions/1688205/terminology-how-to-call-this-relation-inequality-inequation&quot;&gt;this question&lt;/a&gt; on another site and &lt;a href=&quot;https://math.stackexchange.com/questions/144959/how-to-solve-this-inequation?rq=1&quot;&gt;this question&lt;/a&gt; where he had to use the inequality tag because there's no inequation tag until today [3 May 2019].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The image below is from the book &lt;em&gt;The A to Z of Mathematics&lt;/em&gt; by Thomas Sidebotham.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CYZsr.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CYZsr.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally comparing the english wikipedia article with the french one, the deutsch or the català articles, you see much less explanations in english. So a student who consults wikipedia won't learn much. Other websites explain it better with examples like &lt;a href=&quot;https://www.sangakoo.com/en/unit/first-degree-inequations&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt; or &lt;a href=&quot;https://www.bbc.com/bitesize/guides/zwgdb82/revision/3&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12246" LastActivityDate="2019-05-03T18:49:06.640" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="16579" PostTypeId="5" CreationDate="2019-05-03T19:06:23.057" Score="0" Body="&lt;p&gt;An inequality is a question, in the form of an inequality between two algebraic quantities. This inequality contains unknowns. To solve an inequality is to find the values of these unknowns that make inequality true. For example &lt;span class=&quot;math-container&quot;&gt;$3x-2&amp;gt;0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x^2-5x+6)(x^3-1)\leq 0$&lt;/span&gt; are inequations to be solved.&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="501" LastEditDate="2019-05-06T12:55:44.840" LastActivityDate="2019-05-06T12:55:44.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16580" PostTypeId="4" CreationDate="2019-05-03T19:06:23.057" Score="0" Body="For questions about solving an inequation or a system of inequations." OwnerUserId="501" LastEditorUserId="501" LastEditDate="2019-05-05T17:46:45.573" LastActivityDate="2019-05-05T17:46:45.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16581" PostTypeId="2" ParentId="16577" CreationDate="2019-05-03T20:19:00.810" Score="6" Body="&lt;ul&gt;&#xA;&lt;li&gt;Solve the equation &lt;span class=&quot;math-container&quot;&gt;$\sin x=0.5$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$[0,7\pi]$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The equation &lt;span class=&quot;math-container&quot;&gt;$y=-10\cos(3t)$&lt;/span&gt; represent the motion of a weight hanging on a spring after it has been pulled &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; cm below its equilibrium point and released. The output &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; gives the position of the weight in cm above (positive &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;) or below (negative &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;) the equilibrium point after &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; seconds. Find the first four times when the weight is &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; cm above its equilibrium point. &lt;/li&gt;&#xA;&lt;li&gt;A ferris wheel with a diameter of &lt;span class=&quot;math-container&quot;&gt;$36$&lt;/span&gt; m rotates counterclockwise at &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; revolutions per minute. At its lowest point, each seat is &lt;span class=&quot;math-container&quot;&gt;$0.9$&lt;/span&gt; m from the ground. For a rider starting at the lowest point, his height (in meters) above the ground after &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; minutes is given by &lt;span class=&quot;math-container&quot;&gt;$y=18.9-18\cos(10\pi t)$&lt;/span&gt;. Find all the times in the first half minute when the rider is &lt;span class=&quot;math-container&quot;&gt;$24$&lt;/span&gt; m above the ground. Round to the nearest tenth of a second.&lt;/li&gt;&#xA;&lt;li&gt;An alternating current generator produces the current (in amperes) &lt;span class=&quot;math-container&quot;&gt;$i=30\sin(120\pi t)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is the time in seconds. Find the least positive &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$i=25$&lt;/span&gt; A. &lt;/li&gt;&#xA;&lt;li&gt;A polarizing filter for a camera contains two parallel plates of polarizing glass, one fixed and the other able to rotate. If &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is the angle of rotation from the position of maximum light transmission &lt;a href=&quot;https://en.wikipedia.org/wiki/Polarizer#Malus&amp;#39;s_law_and_other_properties&quot; rel=&quot;noreferrer&quot;&gt;then the intensity of light leaving the filter&lt;/a&gt; is &lt;span class=&quot;math-container&quot;&gt;$\cos^2 \theta$&lt;/span&gt; times the intensity entering the filter. Find the least positive &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; so that the intensity of light leaving is &lt;span class=&quot;math-container&quot;&gt;$70$&lt;/span&gt; % of that entering.&lt;/li&gt;&#xA;&lt;li&gt;The planet Mercury travels around the sun in &lt;a href=&quot;https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#First_law_of_Kepler&quot; rel=&quot;noreferrer&quot;&gt;an elliptical orbit given by&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$\displaystyle r=\frac{3.44\times 10^7}{1-0.206\cos\theta}$&lt;/span&gt;. Find the least positive &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; for which mercury is &lt;span class=&quot;math-container&quot;&gt;$3.78\times 10^7$&lt;/span&gt; miles from the sun.&lt;/li&gt;&#xA;&lt;li&gt;Given the equation &lt;span class=&quot;math-container&quot;&gt;$xy=-2$&lt;/span&gt; replace &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x=u\cos\alpha-v\sin\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=u\sin\alpha+v\cos\alpha$&lt;/span&gt;. Find the least positive &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; so that coefficient of the &lt;span class=&quot;math-container&quot;&gt;$uv$&lt;/span&gt; term will be &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;What are the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-intercepts of the graph of the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$f(x)=4\sin^2x-3$&lt;/span&gt; on the interval &lt;span class=&quot;math-container&quot;&gt;$[0,2\pi]$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;li&gt;Blood pressure is a way of measuring the amount of force exerted on the walls of blood vessels. It is measured using two numbers: systolic (as the heart beats) blood pressure and diastolic (as the heart rests) blood pressure. Blood pressures vary substantially from person to person, but a typical blood pressure is 120 80, which means the systolic blood pressure is 120 mmHg and the diastolic blood pressure is 80 mmHg. Assuming that a person’s heart beats 70 times per minute, the blood pressure P of an individual after t seconds can be modeled by the function &lt;span class=&quot;math-container&quot;&gt;$P(t)=100+20\sin\left(\frac{7\pi t}{3}\right)$&lt;/span&gt;. In the interval &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;, determine the times at which the blood pressure is between 100 and 105 mmHg.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="501" LastEditorUserId="501" LastEditDate="2019-05-03T20:33:48.233" LastActivityDate="2019-05-03T20:33:48.233" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16582" PostTypeId="2" ParentId="13856" CreationDate="2019-05-03T21:31:52.370" Score="0" Body="&lt;p&gt;Not that I mind the slight rant in the other answer, but&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... maybe this kind of notation was never really used in the West at the secondary education level&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;seems closer to the truth. Those arrows are relatively &lt;a href=&quot;http://jeff560.tripod.com/set.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;modern&lt;/a&gt; inventions, from around 1920 onwards. I'm not aware of any widely used alternative notation in the mathematics and physics literature pre 1920. Even today I don't see people using &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\iff$&lt;/span&gt;  that often, neither in research literature nor undergraduate nor graduate books. One either uses words like &quot;hence&quot; &quot;therefore&quot; &quot;equivalently&quot; etc. or one doesn't write anything, assuming the reader will fill in the details. So I doubt this was ever standard in secondary education. (But I'd be interested in being proven wrong.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning your second question, I would personally rather emphasise the difference between &quot;implies&quot; and &quot;is equivalent&quot;, but I'm not sure at what level and to which extent.&lt;/p&gt;&#xA;" OwnerUserId="590" LastEditorUserId="590" LastEditDate="2019-05-04T05:33:24.767" LastActivityDate="2019-05-04T05:33:24.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16584" PostTypeId="2" ParentId="16573" CreationDate="2019-05-04T17:22:56.363" Score="6" Body="&lt;p&gt;I've noticed a few issues when students solve problems of the form, &quot;Find the inverse of this function&quot;, and &lt;em&gt;not all&lt;/em&gt; of the issues are necessarily because of the students' misunderstanding of what an inverse function means!&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Misunderstanding/forgetting the &quot;one input &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; one output&quot; defining feature of a function.&lt;/strong&gt; This issue arises because implicitly-defined curves do have &quot;inverses&quot;, they just may not be the graph of a &lt;em&gt;function&lt;/em&gt;. For example, you can use desmos to plot something like &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^3-4x$&lt;/span&gt;, and then if you type &quot;&lt;span class=&quot;math-container&quot;&gt;$x=f(y)$&lt;/span&gt;&quot;, it will plot the inverse relation for you. The student has to know that this is not the graph of a function (because it fails the &quot;vertical line test&quot;). &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Algebraic troubles when solving for the inverse.&lt;/strong&gt; A good example of this is something like &lt;span class=&quot;math-container&quot;&gt;$f(x)=\frac{x+1}{2x-3}$&lt;/span&gt;. Even if students are following the instructions &quot;set &lt;span class=&quot;math-container&quot;&gt;$f(x)=y$&lt;/span&gt;, swap all instances &lt;span class=&quot;math-container&quot;&gt;$x\leftrightarrow y$&lt;/span&gt;, they often will correctly set up &lt;span class=&quot;math-container&quot;&gt;$x=\frac{y+1}{2y-3}$&lt;/span&gt; and then, unfortunately, have no idea how to start solving for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. (I believe it's the multiple instances of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; that seem daunting. But I've seen this happen with even simpler examples like &lt;span class=&quot;math-container&quot;&gt;$y=\frac{1}{\sqrt{2x-1}}$&lt;/span&gt;.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;&quot;Left-inverse&quot; vs. &quot;right-inverse&quot; and colloquial understanding of &quot;inverse&quot;.&lt;/strong&gt; The best example of this is &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)=\sqrt{x}$&lt;/span&gt;. Many students intuit that these functions are &quot;inverses&quot; of each other because that is how they have been used in practice to solve equations. However, this glosses over the fact that &lt;span class=&quot;math-container&quot;&gt;$f(g(x))=x$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and yet &lt;span class=&quot;math-container&quot;&gt;$g(f(x))=|x|$&lt;/span&gt; which is not necessarily &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Inverse vs. reciprocal.&lt;/strong&gt; This was mentioned in the comments but it's worth reiterating. The issue goes beyond merely notation, e.g. &lt;span class=&quot;math-container&quot;&gt;$\arcsin(x)=\sin^{-1}(x)\neq \frac{1}{\sin(x)}$&lt;/span&gt;. I've even seen this notion appear when students think of the inverse of &lt;span class=&quot;math-container&quot;&gt;$\ln(x)$&lt;/span&gt; means to &quot;divide by ln&quot;! (Yes, I have genuinely seen students circle an answer of &lt;span class=&quot;math-container&quot;&gt;$x=5/\ln$&lt;/span&gt;, blissfully unaware that it makes no sense.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Inability to see the relationship between the graphs of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt;.&lt;/strong&gt; I don't know how to alleviate this problem, but I genuinely find it challenging to explain (and get students to understand and subsequently remember) that their graphs are mirror images across &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;. We look at multiple examples, we use a table of data points, we see that &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; becomes &lt;span class=&quot;math-container&quot;&gt;$(b,a)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt; ... I think it's just a challenging concept.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For all of these reasons, I would recommend against having multiple choice questions that test (or &quot;trap&quot;) students on these issues. These concepts are subtle and tricky, and it would be better to be able to see student work on a problem about these topics. (If you are &lt;em&gt;required&lt;/em&gt; to use multiple choice questions by your institution, I would try to break the question down as much as possible so that you are asking about a specific step in the process of solving for an inverse function, instead of essentially giving them &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and asking them for &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt; overall.)&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2019-05-04T17:22:56.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16586" PostTypeId="1" AcceptedAnswerId="16590" CreationDate="2019-05-04T22:31:30.873" Score="10" ViewCount="357" Body="&lt;p&gt;I am developing an assessment piece where the content is the same but the particular numbers are different for each student. It involves finding Triangle Centers given points using coordinate geometry. The particular skills it is assessing (for/as/of) are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;calculating distances, gradients and midpoints&lt;/li&gt;&#xA;&lt;li&gt;forming equations of lines&lt;/li&gt;&#xA;&lt;li&gt;perpendicular gradients&lt;/li&gt;&#xA;&lt;li&gt;simultaneous equations&lt;/li&gt;&#xA;&lt;li&gt;graphing points and lines&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am giving each student different randomly generated vertices and asking them to work out 4 triangles centres graph the results. As I've written it is quite guided with the main tasks being calculations between suitably chosen points and graphing. I'll be generating individualised answers as well. See below for a draft.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I am using Mathematica to generate the initial points and answers including graphs. I export these to Excel and mailmerge them in Word. )&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/r29Ui.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/r29Ui.png&quot; alt=&quot;triangle centres draft&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are the pedagogical issues at play? References welcomed!&lt;/strong&gt; My own ideas are shown below. I am not so interested in the technical issues involved (Google &quot;latex random test&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pros&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To encourage students to share skills not answers&lt;/li&gt;&#xA;&lt;li&gt;To give students a sense of ownership&lt;/li&gt;&#xA;&lt;li&gt;To combat cheating (not a concern in my case, it is more assessment &lt;em&gt;for&lt;/em&gt; learning)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Cons&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Difficult to create individualised questions, needs a thorough understanding of the problem and it's algebraic solution &lt;/li&gt;&#xA;&lt;li&gt;Unintentional differenation (eg one set of numbers may require a much easier solution than another)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have particular objective in mind for this project at this stage (paper based, an activity not a test, papers should be of the same difficulty, I like this triangle centres task), but there are some obvious extensions (other maths content, differentiation, automated marking).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt; Having run this in  highschool class the main feedback I got was that it was too challenging for most. The most important pedagogical thing which emerged was the &lt;strong&gt;need to differentiate.&lt;/strong&gt; To modify the task to make it easier one could:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;present a partial solution&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;algorithmically, select points which have more simple intermediate steps.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Both of which require more work for me! Hence, &lt;em&gt;it is &quot;difficult to create individualised questions, needs a thorough understanding of the problem and its algebraic solution.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="64" LastEditDate="2019-06-08T10:36:40.543" LastActivityDate="2019-06-08T10:36:40.543" Title="Pros and cons of randomised question generation" Tags="|mathematical-pedagogy|reference-request|technology-in-education|general-pedagogy|assessment|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16587" PostTypeId="1" CreationDate="2019-05-05T02:56:14.733" Score="4" ViewCount="316" Body="&lt;p&gt;I need to give a short presentation on introducing a class of engineering students to the concept and definition of the derivative. I'm to assume that the students are currently at the appropriate place in first quarter calculus to support the delivery. In particular, they would have been introduced to the limit, be able to compute limits for many interesting examples, had a treatment of continuity and understand what a removable discontinuity is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the general question here is, just take a time-tested standard treatment from a textbook, or try to do something fancier or outside the box? More particularly, introduce it as a textbook would (derive the definition and then compute examples), or jump right into computation using the definition of derivative, and then &quot;motivate&quot; or derive this definition?&lt;/p&gt;&#xA;" OwnerUserId="11880" LastActivityDate="2021-03-24T00:29:27.833" Title="Introducing derivative concept and definition" Tags="|calculus|engineering-mathematics|derivative|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16588" PostTypeId="2" ParentId="16561" CreationDate="2019-05-05T03:46:08.027" Score="6" Body="&lt;p&gt;The issue isn't unique to factoring or the FOIL process. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The roots of &lt;span class=&quot;math-container&quot;&gt;$(ax^2 + bx + c)$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;can be found via the quadratic equation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, this prompts the issue; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(2x^2 - 3x - 4)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the value for a,b,c that we will plug into the quadratic equation? Isn't c = -4 ? And we need to accept that the minus operator transmogrifies into a negative sign when the term is isolated. Even though you see 2 'subtractions', how do you respond to the request to identify the 2nd or 3rd term?  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-05T13:03:25.047" LastActivityDate="2019-05-05T13:03:25.047" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16589" PostTypeId="2" ParentId="16586" CreationDate="2019-05-05T13:07:20.593" Score="5" Body="&lt;p&gt;Here are some problems I have run into:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If problems are randomly generated with unlimited attempts on a computer, then pattern matching is a real problem.  Maybe the question is to calculate the radius of convergence of the power series &lt;span class=&quot;math-container&quot;&gt;$\sum_1^\infty k(\frac{x-a}{c})^n$&lt;/span&gt; for various &lt;span class=&quot;math-container&quot;&gt;$k, a, c$&lt;/span&gt;.  The students will just guess randomly one of the constants in the expression or its reciprocal until they get the right answer.  Then they will memorize which guess is correct, and use it to &quot;solve&quot; many more such problems.  This exercise accomplishes nothing.  Students will always find the easiest way to play the game, which will often circumvent the concept you thought you were trying to get them to engage with.&lt;/li&gt;&#xA;&lt;li&gt;It is quite difficult to fully debug randomly generated problems.  You will often end up with some cases which are unsolvable or which have physically nonsensical solutions (people who weigh less than a mouse, etc).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2019-05-05T15:11:47.353" LastActivityDate="2019-05-05T15:11:47.353" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16590" PostTypeId="2" ParentId="16586" CreationDate="2019-05-05T14:31:44.253" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;To encourage students to share skills not answers&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is great in principle, but the question in my mind, based on your sample, is what would be the intellectual content presented by one student to the other. Because the problem is stylized and you're generating a lot of them, the danger might be that your students would infer that the important thing here is to learn to execute an algorithm, without any conceptual understanding. So maybe student A succeeds in getting the right answers to their problem, based on a model you did on the board. Then student A shows their scratch work to student B, who works through the same algorithm with their numbers.&lt;/p&gt;&#xA;&lt;p&gt;Now you could say this is fine, and they've successfully shared a skill. A has helped B to learn. But it may be that they don't understand the concepts, and therefore their mastery is basically fragile and useless. All they know how to do is fill out a worksheet with exactly this stylized form.&lt;/p&gt;&#xA;&lt;p&gt;I definitely see this happening with physics students at my school, where some of the teachers use a commercial answer-checking system provided by the book publisher. Many students pay for a monthly membership with a sleazy web site called Chegg, which provides them with worked solutions to all the problems. The numbers are different, but the students just work through the steps with their numbers. They don't understand what they're doing, and their skill doesn't generalize to other problems.&lt;/p&gt;&#xA;&lt;p&gt;What has worked better for me is to take the time to write problems that are actually qualitatively different, and then randomly assign different problems to different students. Here are two problems I assign that basically use the same skills and are at comparable levels of difficulty, but differ from each other in qualitative ways:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Aircraft carriers originated in World War I, and the first landing on a carrier was performed by E.H. Dunning in a Sopwith Pup biplane, landing on HMS Furious. (Dunning was killed the second time he attempted the feat.) In such a landing, the pilot slows down to just above the plane's stall speed, which is the minimum speed at which the plane can fly without stalling. The plane then lands and is caught by cables and decelerated as it travels the length of the flight deck. Comparing a modern US F-14 fighter jet landing on an Enterprise-class carrier to Dunning's original exploit, the stall speed is greater by a factor of 4.8, and to accomodate this, the length of the flight deck is greater by a factor of 1.9. Which deceleration is greater, and by what factor?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In college-level women's softball in the U.S., typically a pitcher is expected to be at least 1.75 m tall, but Virginia Tech pitcher Jasmin Harrell is 1.62 m. Although a pitcher actually throws by stepping forward and swinging her arm in a circle, let's make a simplified physical model to estimate how much of a disadvantage Harrell has had to overcome due to her height. We'll pretend that the pitcher gives the ball a constant acceleration in a straight line, and that the length of this line is proportional to the pitcher's height. Compare the acceleration Harrell would have to supply with the acceleration that would suffice for a pitcher of the nominal minimum height, if both were to throw a pitch at the same speed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This particular pair of problems is actually from a set of 5. So certainly a couple of my students can get together at Starbucks and work on their homework together, but it's unlikely that they were both assigned the same problem. Therefore if they help each other, they are more likely to be giving help at the conceptual level, not the algorithmic level. One student can't just walk through the other student's algorithm.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-05-05T14:38:55.030" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16591" PostTypeId="2" ParentId="16587" CreationDate="2019-05-05T19:56:37.933" Score="3" Body="&lt;p&gt;I think you should take whatever motivation you are most excited to show them based on (a) their interests and background (b) your passion and go that way.  Everyone will have a different answer to this and so it's important to, with careful consideration, do whatever you, the educator, feels is best.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, as someone with a geometric bent, I would ask how we could define that tangent line we all know how to draw, using only our techniques for finding the line between two points.  Motivating the definition via pictures usually feels best given my inclinations and my students' preparations.  But you know your students better than us.&lt;/p&gt;&#xA;" OwnerUserId="6589" LastActivityDate="2019-05-05T19:56:37.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16592" PostTypeId="1" CreationDate="2019-05-05T23:55:17.800" Score="3" ViewCount="172" Body="&lt;p&gt;When choosing names for arbitrary constants either during a lesson or while working with a single student, should one use&lt;span class=&quot;math-container&quot;&gt;$\{n_1,n_2,n_3,\dotsc\}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\{n, n', n'', \dotsc\}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\{a,b,c,\dotsc\}$&lt;/span&gt;? Is there any evidence that one of these is easier for students to work with than others? Or is less of a barrier to understanding concepts than others?&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2019-05-06T05:29:53.623" Title="Naming arbitrary constants: subscripts, primes, or just more letters?" Tags="|undergraduate-education|mathematical-pedagogy|notation|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16593" PostTypeId="2" ParentId="16592" CreationDate="2019-05-06T04:57:45.690" Score="1" Body="&lt;p&gt;If the context is integral calculus and/or differential equations, I would definitely use &lt;span class=&quot;math-container&quot;&gt;$C_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$C_2$&lt;/span&gt;, etc.  After all the student is used to &quot;&lt;span class=&quot;math-container&quot;&gt;${}+C$&lt;/span&gt;&quot;.  So keep it as close to what they are used to seeing and thinking about.&lt;/p&gt;&#xA;" OwnerUserId="12251" LastEditorUserId="2074" LastEditDate="2019-05-06T05:29:53.623" LastActivityDate="2019-05-06T05:29:53.623" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16594" PostTypeId="2" ParentId="15272" CreationDate="2019-05-06T08:06:27.417" Score="4" Body="&lt;p&gt;For all &lt;span class=&quot;math-container&quot;&gt;$n \geq 3$&lt;/span&gt; the alternating group &lt;span class=&quot;math-container&quot;&gt;$A_{n}$&lt;/span&gt; can be realized as the orientation-preserving symmetries of the regular &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-simplex (triangle, tetrahedron, etc.). In this case the full symmetry group is &lt;span class=&quot;math-container&quot;&gt;$S_{n}$&lt;/span&gt;, and the relation between &lt;span class=&quot;math-container&quot;&gt;$A_{n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$S_{n}$&lt;/span&gt; can be seen geometrically. For instance, when &lt;span class=&quot;math-container&quot;&gt;$n = 3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A_{3}$&lt;/span&gt; comprises the rotations through an angle of &lt;span class=&quot;math-container&quot;&gt;$2\pi/3$&lt;/span&gt; of an equilateral triangle about its centroid, while &lt;span class=&quot;math-container&quot;&gt;$S_{3}$&lt;/span&gt; includes additionally the reflections through the angle bisectors. One can relate these geometric descriptions to the usual descriptions in terms of permutations by looking at what the geometric transformations do to the triangle's vertices. The symmetries of the regular tetrahedron can be examined in a similarly explicit fashion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The cardinality &lt;span class=&quot;math-container&quot;&gt;$60$&lt;/span&gt; alternating group &lt;span class=&quot;math-container&quot;&gt;$A_{5}$&lt;/span&gt; is also the group of rotational (orientation-preserving) symmetries of a regular icosahedron. (This example has the virtue of being visualizable in three-dimensional space. It has the defect of being peculiar to &lt;span class=&quot;math-container&quot;&gt;$n = 5$&lt;/span&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although somewhat complicated for beginning students, this can be written down explicitly in the following way that uses only introductory linear algebra. Take as the &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; vertices of the isosahedron the cyclic permutations of &lt;span class=&quot;math-container&quot;&gt;$(\pm 1, \pm w, 0)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$w = (1 + \sqrt{5})/2$&lt;/span&gt;. The icosahedron is the convex hull of these &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; points. &lt;span class=&quot;math-container&quot;&gt;$A_{5}$&lt;/span&gt; is generated by the order &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; map sending &lt;span class=&quot;math-container&quot;&gt;$x = (x_{1}, x_{2}, x_{3}) \to (-x_{1}, -x_{2}, x_{3})$&lt;/span&gt;, the order &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; map sending &lt;span class=&quot;math-container&quot;&gt;$x = (x_{1}, x_{2}, x_{3}) \to (x_{2}, x_{3}, x_{1})$&lt;/span&gt; (which is rotation by &lt;span class=&quot;math-container&quot;&gt;$2\pi/3$&lt;/span&gt; about the axis &lt;span class=&quot;math-container&quot;&gt;$(1, 1, 1)$&lt;/span&gt;), and the order &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; map given by rotating by &lt;span class=&quot;math-container&quot;&gt;$2\pi/5$&lt;/span&gt; around the axis spanned by any one of the vertices. (Adding &lt;span class=&quot;math-container&quot;&gt;$x \to -x$&lt;/span&gt; yields the cardinality &lt;span class=&quot;math-container&quot;&gt;$120$&lt;/span&gt; full symmetry group of the icosahedron; note, however, that this is &lt;span class=&quot;math-container&quot;&gt;$A_{5}\times \mathbb{Z}_2$&lt;/span&gt; and not the symmetric group &lt;span class=&quot;math-container&quot;&gt;$S_{5}$&lt;/span&gt;.) It is instructive for students to think about what is permuted by this action of &lt;span class=&quot;math-container&quot;&gt;$A_{5}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comparing the two different geometric descriptions of &lt;span class=&quot;math-container&quot;&gt;$A_{5}$&lt;/span&gt;, as the rotational symmetries of an icosahedron and the rotational symmetries of a regular simplex, would be instructive (they are associated with different irreducible representations of &lt;span class=&quot;math-container&quot;&gt;$A_{5}$&lt;/span&gt;, of dimensions &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-05-06T08:06:27.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16595" PostTypeId="2" ParentId="16587" CreationDate="2019-05-06T11:36:30.440" Score="7" Body="&lt;p&gt;It occurs to me that maybe you can approach this from the viewpoint of approximations. Briefly, I’m thinking of the kinds of approximations they likely use in practice, and which are often given in an appendix at the back of texts, such as the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$x \approx 0$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{1-x} \approx 1+x$&lt;/span&gt; (multiply numerator and denominator of left side by &lt;span class=&quot;math-container&quot;&gt;$1+x$&lt;/span&gt; and then drop the &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; term).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$\theta \approx 0$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$\sin \theta \approx \theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tan \theta \approx \theta$&lt;/span&gt; (use a geometric argument).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$\delta \approx 0$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$\sqrt{a^2 + \delta} \; \approx \; a + \frac{\delta}{2a}$&lt;/span&gt; (see &lt;a href=&quot;https://math.stackexchange.com/a/1837196/13130&quot;&gt;here&lt;/a&gt; for an algebraic derivation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both the product rule for derivatives (expand &lt;span class=&quot;math-container&quot;&gt;$(u + \Delta u)(v + \Delta v)$&lt;/span&gt; and drop the &lt;span class=&quot;math-container&quot;&gt;$“\text{small} \times \text{small}”$&lt;/span&gt; term) and the quotient rule for derivatives (multiply numerator and denominator of &lt;span class=&quot;math-container&quot;&gt;$\frac{u + \Delta u}{v + \Delta v}$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$v - \Delta v$&lt;/span&gt; and drop the two &lt;span class=&quot;math-container&quot;&gt;$“\text{small} \times \text{small}”$&lt;/span&gt; terms) can be heuristically obtained by methods that they will be (or at least should be) familiar with. The idea is that you want the simplest non-constant approximation, which winds up being a linear (first degree) approximation, and this arises as the linear part when you are able to expand the quantity as a (or some simple algebraic combination of) possibly infinite sum(s) of non-negative integer powers of “small things”. Then maybe briefly discuss aspects of &lt;strong&gt;(a)&lt;/strong&gt; why we can drop the higher order small terms (this &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=4487158&quot; rel=&quot;noreferrer&quot;&gt;graphical exploration idea&lt;/a&gt; is worth considering) and &lt;strong&gt;(b)&lt;/strong&gt; the value of a linear approximation (easy to integrate linear functions, can apply proportionality when linear, simple to numerically calculate or estimate when linear, etc. --- note that I’m trying to avoid simply saying “linear” is good or “graph locally looks like a line” is good, because this leaves unanswered why linear/lines are good/useful in the first place).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the books I list in my answer to &lt;a href=&quot;https://matheducators.stackexchange.com/a/13905/745&quot;&gt;Teaching Calculus Less Formally&lt;/a&gt; (especially &lt;strong&gt;Calculus Made Easy&lt;/strong&gt; by Silvanus P. Thompson and &lt;strong&gt;Calculus in Context. The Five College Calculus Project&lt;/strong&gt; by James Callahan, et al.) might be worth looking through for ideas in general.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2019-05-06T11:41:50.683" LastActivityDate="2019-05-06T11:41:50.683" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16596" PostTypeId="2" ParentId="16562" CreationDate="2019-05-07T03:00:22.717" Score="1" Body="&lt;p&gt;As you said &quot;even small [ideas/suggestions]&quot;, here's what came to mind first when I read your question.  Richard Feynman once said of the Institute for Advanced Studies that it never really produced much, or a least far below its potential, and that he believed the reason for this is that, I paraphrase: &quot;If you aren't teaching, you don't have to think about how to answer the students' questions.&quot;  Meaning that it shut off an avenue of deep thought that led to strong research questions.  This may be somewhat different for math than for physics, but it still might be something that helps you with your situation psychologically.  It was a dream to go to IAS because &quot;you didn't have to deal with students, grading, etc&quot;--but it didn't (in Feynman's opinion, anyway) turn out to be that great for turning out good research.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your case, it sounds like they've given you too much, but it can help to recognize that while it's too much, it's at least too much of a good thing, and that can possibly make you feel more positive as you work to figure out how to handle the load.&lt;/p&gt;&#xA;" OwnerUserId="371" LastActivityDate="2019-05-07T03:00:22.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16597" PostTypeId="2" ParentId="2376" CreationDate="2019-05-07T03:38:58.010" Score="0" Body="&lt;p&gt;I agree to those who would motivate their &quot;fast&quot; learners to become peer tutor to their classmates because at a certain point they could use their means of letting their classmates understand the topic accordingly.&lt;/p&gt;&#xA;" OwnerUserId="12252" LastActivityDate="2019-05-07T03:38:58.010" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16598" PostTypeId="2" ParentId="16568" CreationDate="2019-05-07T04:13:19.207" Score="5" Body="&lt;p&gt;It can definitely be confusing, and a clear distinction should be made when teaching. It's not always made, and students can in fact remain lost for a long time. I've seen students who thought &lt;span class=&quot;math-container&quot;&gt;$1 \cdot (-1) = 0$&lt;/span&gt;, unsure of the role of &lt;span class=&quot;math-container&quot;&gt;$&quot;\cdot&quot;$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$&quot;-&quot;$&lt;/span&gt;, and parentheses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as symbol, it's not too confusing if when teaching you use parentheses, e.g. &quot;&lt;span class=&quot;math-container&quot;&gt;$(-5)$&lt;/span&gt;&quot;, and call it &quot;negative five&quot; (rather than &quot;minus five&quot;). When students are ready, explain that it's OK to omit parentheses when there is no confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's how &lt;a href=&quot;https://beastacademy.com/books&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beast Academy&lt;/a&gt; approaches it (I am not involved in it, but like the books). First they introduce negative numbers, and it's only after some practice that they get to adding a negative number, and later yet to subtracting:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Explain negative numbers on the number line, explaining that &lt;span class=&quot;math-container&quot;&gt;$-4$&lt;/span&gt; is read &quot;negative four&quot;, etc.&lt;/li&gt;&#xA;&lt;li&gt;Comparing negative numbers, e.g. &lt;span class=&quot;math-container&quot;&gt;$-15 &amp;lt; -7$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Adding on the number line, e.g. &quot;To add 3, we move 3 units to the right&quot;, with example of &lt;span class=&quot;math-container&quot;&gt;$5+3$&lt;/span&gt; and then &lt;span class=&quot;math-container&quot;&gt;$-5+3$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;But if &lt;span class=&quot;math-container&quot;&gt;$-5 + 3$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;, then it nust be that &quot;3 plus -5 also equals -2&quot;, but how does that work on the number line?&lt;/li&gt;&#xA;&lt;li&gt;To add a negative number, we move left, and write it as &lt;span class=&quot;math-container&quot;&gt;$3 + (-5) = -2$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;There is a whole separate section to remind students of subtraction: on the number line, we go left to subtract a number, e.g. with &lt;span class=&quot;math-container&quot;&gt;$9 - 5$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;We can use the same method to subtract e.g. &lt;span class=&quot;math-container&quot;&gt;$5 - 9$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Another way to solve &lt;span class=&quot;math-container&quot;&gt;$5-9$&lt;/span&gt;, we can find the number to add to 9 to get 5. Since &lt;span class=&quot;math-container&quot;&gt;$(-4) + 9 = 5$&lt;/span&gt;, it must be that &lt;span class=&quot;math-container&quot;&gt;$5-9=-4$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So that's how you can explain why &lt;span class=&quot;math-container&quot;&gt;$3 + (-5)$&lt;/span&gt; is the same as &lt;span class=&quot;math-container&quot;&gt;$3 - 5$&lt;/span&gt;. (Here you can draw the students attention to the choice of the symbol for &quot;subtraction&quot; and for &quot;negatives&quot; -- it's the same symbol, because the two concepts are in fact closely connected.) Then there is a whole other section about subtracting a negative.&lt;/p&gt;&#xA;" OwnerUserId="12253" LastEditorUserId="12253" LastEditDate="2019-05-12T16:15:48.983" LastActivityDate="2019-05-12T16:15:48.983" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16599" PostTypeId="2" ParentId="12964" CreationDate="2019-05-07T04:37:23.963" Score="4" Body="&lt;p&gt;Extracurricular math, or math enrichment programs often do introduce proofs in the realm of number theory. At the earliest level, it can include direct proofs such as &quot;even + even is always even&quot;, &quot;odd + odd is always even&quot;, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The geometry proofs are usually taught as very structured -- a build-up from definitions and earlier theorems to the result, justifying each step, usually formatting the steps on the page in a particular way. There are many examples of interesting statements to prove, at a similar level of difficulty. Plus there is a benefit to visualizing things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the answer to your question is that beyond very simple statements, number theory is not as simple an environment as it might seem:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Some statements in number theory are reasonably easy to prove (e.g. that &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$m^3 - m$&lt;/span&gt;), but it's not so easy to structure these arguments as a sequence of simple steps.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For many statements, to prove them rigorously, you need a proof by induction. But that's certainly a higher level of difficulty.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I think that's why number theory proofs are more commonly seen in advanced math programs, rather than in standard curriculum.&lt;/p&gt;&#xA;" OwnerUserId="12253" LastActivityDate="2019-05-07T04:37:23.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16601" PostTypeId="2" ParentId="16559" CreationDate="2019-05-07T19:16:44.610" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://beastacademy.com/&quot; rel=&quot;noreferrer&quot;&gt;Beast Academy&lt;/a&gt; is a new(ish) curriculum, meant for math-loving students, from the Art of Problem Solving folks. It has 4 levels for each 'grade' (from 2nd to 5th), and each level has a Guidebook and a Practice book. But don't believe the grade levels, they can be challenging for much older students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although your students are advanced 6th graders, I believe the level 5 will still be challenging, and Beast Academy is a lot of fun. You can use the books or do it online. If you do it online, they can do lower levels too.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-05-07T19:16:44.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16602" PostTypeId="1" CreationDate="2019-05-09T07:50:10.577" Score="3" ViewCount="280" Body="&lt;p&gt;I have to choose a topic to give a presentation. The topic should be high-school level or at most Linear Algebra 1 and Calculus 1. Conics and polygons in the Euclidean geometry are some fine topics but have already been worked out by the professor so they are not available. I am required to bring up a question about the topic and answer it, using Mathematica or Geogebra as a visual tool, and then show mathematical theory related to the problem. My question is, what would be a nice topic fitting this requirements?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The presentation is between 30 minutes and an hour long.&lt;/p&gt;&#xA;" OwnerUserId="12260" LastEditorUserId="11" LastEditDate="2019-05-09T19:25:42.740" LastActivityDate="2019-05-27T03:19:51.280" Title="What is an interesting high-school level topic to discuss using Mathematica or Geogebra?" Tags="|undergraduate-education|secondary-education|technology-in-education|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16603" PostTypeId="2" ParentId="16602" CreationDate="2019-05-09T13:56:59.483" Score="4" Body="&lt;p&gt;If it doesn't have to be related to Calculus and/or Linear Algebra, why not give the story of the seven bridges of Königsberg and show how Euler developed graph theory and solved that problem at the same time?  No prior mathematics required :)&lt;/p&gt;&#xA;" OwnerUserId="6589" LastActivityDate="2019-05-09T13:56:59.483" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16604" PostTypeId="2" ParentId="16602" CreationDate="2019-05-09T19:57:47.443" Score="1" Body="&lt;p&gt;Having linear algebra as an option opens up a whole lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related to @cheyne's answer, you could introduce the adjacency matrix of a graph and how it and its matrix powers can be used to count paths. The rules for matrix multiplication are essentially the same as those for path composition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing would be to write the recurrence relation for the Fibonacci numbers as a two by two matrix. Then the asymptotic formula can be expressed in terms of eigenvalues.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2019-05-09T19:57:47.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16605" PostTypeId="1" CreationDate="2019-05-10T00:45:45.950" Score="4" ViewCount="176" Body="&lt;p&gt;For proof-based math courses, the gist of the learning happens in problem sets and so it is essential to design them well. We would appreciate responses containing references (eg. from active learning) and personal experiences on designing problem sets to give us ideas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some criteria and ideas we use in crafting our assignments in proof-based calculus and senior years analysis (such as epsilon-delta arguments, constructing the reals, proving properties about families of functions):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Have them prove a statement similar to one in class so that they at least get the main ideas.&lt;/li&gt;&#xA;&lt;li&gt;Include many true (prove why) or false (prove why not) questions. This is the quintessential skill of modern mathematical research, where one has to build intuition to figure out which directions to go first. &lt;/li&gt;&#xA;&lt;li&gt;Have them work through toy problems first before attacking a general statement.&lt;/li&gt;&#xA;&lt;li&gt;The more all the questions tie together, the better, e.g., via the use of themes. For example, we can devote a problem set to studying a convenient application that relates to many of the class's results.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8714" LastEditorUserId="8714" LastEditDate="2019-05-10T17:39:20.120" LastActivityDate="2019-05-10T17:39:20.120" Title="The art of designing of problem sets" Tags="|homework|problem-design|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16607" PostTypeId="1" CreationDate="2019-05-10T12:53:31.243" Score="8" ViewCount="330" Body="&lt;p&gt;I am teaching the exercise sessions for a 3rd year algebra course (intro to field theory, Galois theory and Algebraic geometry). The format of the course is as follows: for every 2 hour lecture by the prof, I teach a 2 hour exercise session where the students are supposed to actively work on exercises and I answer their questions. In practice this often means I give quite a few solutions on the blackboard, but not all of them. There is no homework for this course, just an exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Preparing for this course, I made latex notes containing the solutions of all the exercises. And so my question is: should I give the pdf of this file to the students? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the one hand, having detailed, rigorous solutions for the exercises will help prepare the students for similar questions that might be on the exam. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, on the other hand, I am worried that if the students have all the the solutions on hand, they will not learn the perseverance needed to struggle with a new problem (as might be presented on the exam).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have some experience with this? Do students tend to do better or worse if they have access to all the solutions?&lt;/p&gt;&#xA;" OwnerUserId="12264" LastEditorUserId="238" LastEditDate="2019-05-10T13:37:22.957" LastActivityDate="2019-05-13T13:36:00.620" Title="Solutions to exercises" Tags="|mathematical-pedagogy|exercises|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16609" PostTypeId="2" ParentId="16607" CreationDate="2019-05-10T17:06:54.543" Score="0" Body="&lt;p&gt;They probably can't finish all exercises in class. At the end of each chapter give then the solutions of the remaining chapters. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having detailed solutions for the exercises will help the interested and hard working students progress and check their solutions with yours. As for the other students they will probably just read the solutions. Is this bad? Of course, but if they don't have the solutions they will probably just skip the remaining exercises or, at best, they will take the solutions of other students and read them. Needless to say these solution could have many mistakes. So it's better to give them the solutions at the end of each chapter, it'll help the weak and the hardworking students. &lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2019-05-10T17:06:54.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16610" PostTypeId="2" ParentId="16607" CreationDate="2019-05-10T19:23:12.293" Score="-1" Body="&lt;p&gt;There's differences of opinions on this.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Liberal view:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm of the opinion that drill problems are for the benefit of the student and am used to using them on my own.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have never had an issue with making use of them and/or deciding how long to struggle with something.  I would note that even if you don't &quot;get&quot; the magic aha on your own, you can get some of the brain groove training effect by mechanically working the problem EVEN THOUGH you had to look at part/all of the answer/solution.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feel like not providing answers removes a feedback mechanism. Also I disagree with providing the solutions days later.  We are not computers.  We are animals that learn from near term feedback.  Behavioralism.  We're closer to puppies than to Univac.  More dog than God. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One other thing since you are NOT making this graded homework it is one more reason to be liberal on the solution availability. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Orthodox view:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many paid teachers feel different.  I have seen prefaces where authors said they removed some/all answers because buying committees preferred that.  I'm sure that those espousing this view claim/think it is best.  But I wonder if part of it is preservation of control for the teachers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since I disagree the conservative view, I probably don't explain or defend it thoroughly.  But given board opinions you will get a lot of people backing up the orthodox view.  More than the liberal.  Want you to at least consider heresy.&lt;/p&gt;&#xA;" OwnerUserId="12265" LastActivityDate="2019-05-10T19:23:12.293" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16611" PostTypeId="2" ParentId="16607" CreationDate="2019-05-10T20:45:57.600" Score="5" Body="&lt;p&gt;If these are exercises from a published textbook, then it's probably self-deluding to imagine that the students don't have access to them already. Chegg et al. probably already have the solutions available to anyone willing to pay  the monthly membership fee.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your lead instructor has already chosen a philosophy and a set of rules. They've made the homework not count in the students' grades. That implicitly means that they're expecting the students to act like adults and do what is best educationally. These are upper-division math majors, after all. Even if you have doubts about their maturity and willingness to apply themselves and resist the temptation of looking at the solutions, your opinion about their maturity has already been preempted by the instructor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since there are only 15 students in the class, and presumably only some fraction of them will show up to the exercise sessions, it seems like you have the ability to interact with them one on one while they work on the exercises. This is an optimal situation, and one in which they can't really pretend not to be doing the work, if they actually aren't.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-05-13T13:36:00.620" LastActivityDate="2019-05-13T13:36:00.620" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16612" PostTypeId="2" ParentId="15513" CreationDate="2019-05-11T14:03:15.163" Score="1" Body="&lt;p&gt;After squaring she just added some extra roots to the equation &lt;span class=&quot;math-container&quot;&gt;$y=-4$&lt;/span&gt; :&#xA;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{y}=2i=2\textrm{e}^{in\pi/2}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Squaring again we get &lt;span class=&quot;math-container&quot;&gt;$y=4\mathrm{e}^{in\pi}$&lt;/span&gt; in which &lt;span class=&quot;math-container&quot;&gt;$-4$&lt;/span&gt; is one of the solution.&lt;/p&gt;&#xA;" OwnerUserId="12267" LastEditorUserId="501" LastEditDate="2019-05-12T19:57:35.570" LastActivityDate="2019-05-12T19:57:35.570" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16613" PostTypeId="2" ParentId="15513" CreationDate="2019-05-11T16:14:10.263" Score="5" Body="&lt;p&gt;I don't see any good way to talk about this without explicitly addressing the issue of domains and the fact that the same square root symbol is used for two different functions depending on context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first context, the square root symbol refers to the single-valued, nonnegative real square root. In the second context, the square root symbol refers to the multi-valued, complex square root. &lt;em&gt;These are different functions.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I think the answer is to clearly distinguish between the two functions. You can do so by specifying which one you're using, and giving a precise definition of &lt;em&gt;square root&lt;/em&gt; in both contexts:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Given a real number &lt;span class=&quot;math-container&quot;&gt;$x \geq 0$&lt;/span&gt;, the &lt;em&gt;nonnegative real square root&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the unique real number &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x} \geq 0$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$(\sqrt{x})^2 = x$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Given a complex number &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;, the &lt;em&gt;complex square root&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; is the set of complex numbers &lt;span class=&quot;math-container&quot;&gt;$\{\sqrt{z}, -\sqrt{z}\}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\sqrt{z}$&lt;/span&gt; is &lt;em&gt;any&lt;/em&gt; complex number such that &lt;span class=&quot;math-container&quot;&gt;$(\sqrt{z})^2 = z$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Once you've provided these definitions, you can give a simple answer to a student who gives an answer for the complex square root: That's the correct answer for the complex square root, but not for the nonnegative real square root (because the domain is different).&lt;/p&gt;&#xA;" OwnerUserId="3505" LastActivityDate="2019-05-11T16:14:10.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16614" PostTypeId="1" CreationDate="2019-05-12T12:43:03.960" Score="7" ViewCount="227" Body="&lt;p&gt;In most of the science textbooks I read, I observed that most of them contain the terms, definitions and &lt;strong&gt;etymology&lt;/strong&gt; too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But nowadays, the mathematics textbooks are becoming more formal and contain only the relevant terms and their corresponding definitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the following scenario:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The probability mass function is a function from the powerset of a sample space to [0,1]&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The probability density function is a positive function that integrates to 1 on totality.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both are formal definitions. Needs to be remembered. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is some intuitive explanation regarding why we call &lt;strong&gt;mass&lt;/strong&gt; function and &lt;strong&gt;density&lt;/strong&gt; function. First, one tells about the value directly and hence called mass and the second one gives information about probability or area under it as probability and hence it is called a density function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do the textbooks covering all these aspects in historical etymology view exist for major fields of mathematics? &lt;/p&gt;&#xA;" OwnerUserId="8052" LastEditorUserId="238" LastEditDate="2019-05-18T04:50:06.343" LastActivityDate="2019-05-18T04:50:06.343" Title="Are there textbooks that cover most etymological aspects of mathematics?" Tags="|terminology|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16615" PostTypeId="2" ParentId="16614" CreationDate="2019-05-12T14:23:49.733" Score="9" Body="&lt;p&gt;There are already some books dedicated to etymological aspects of mathematics. Here are two of them:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Origins of Mathematical Words: A Comprehensive Dictionary of Latin, Greek, and Arabic Roots by Anthony Lo Bello (&lt;a href=&quot;https://www.maa.org/press/maa-reviews/origins-of-mathematical-words&quot; rel=&quot;noreferrer&quot;&gt;https://www.maa.org/press/maa-reviews/origins-of-mathematical-words&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English by Steven Schwartzman (&lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00029890.1995.12004620&quot; rel=&quot;noreferrer&quot;&gt;https://www.tandfonline.com/doi/abs/10.1080/00029890.1995.12004620&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For mass:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Bello: mass The Latin noun massa means a lump.&lt;/li&gt;&#xA;&lt;li&gt;Schwartzman: mass (noun): from Greek maza &quot;kneaded dough.&quot;  The Indo-European root is mag - &quot;to knead, fashion,  fit,&quot; as seen in native English make. A related borrowing from Greek is magma, a fiery mass of molten  rock. The nontechnical sense of mass is &quot;stuff, substance.&quot; In physics, mass is defined as a measure of a  body's tendency to resist changes in velocity.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For density:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Bello: From the Latin adjective densus meaning thick, close was formed the noun densitas by the addition of the suffix -itas to the stem of the adjective. This noun entered French as densité and from there came into English as density.&lt;/li&gt;&#xA;&lt;li&gt;Schwartzman: dense (adjective), density (noun): from Latin densus &quot;having its particles close together, crowded,  thick.&quot; The word is used with its meaning virtually  unchanged in English. In topology a set S is said to  be dense in a space P if every point in P is a point or  a limit point of S. In physical applications, density is defined as mass of matter per unit of volume. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2019-05-12T14:23:49.733" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16618" PostTypeId="2" ParentId="16602" CreationDate="2019-05-14T23:53:59.877" Score="1" Body="&lt;p&gt;How Archimedes figured out the value for pi is a pretty cool construction. The lower bound is found through inscribed hexagon, then 12-gon then ..., there is lots of algebra with the Pythagorean theorem, along with constructing angle bisectors (which is also the perpendicular bisector of the chord, and you could go into proving that).&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-05-14T23:53:59.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16619" PostTypeId="2" ParentId="16602" CreationDate="2019-05-15T00:20:31.733" Score="1" Body="&lt;p&gt;The geometry of the cycloid could make a nice demonstration.&#xA;You could derive (for a unit-radius circle):&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{eqnarray}&#xA;x(t) &amp;amp; = &amp;amp; t - \sin t \\&#xA;y(t) &amp;amp; = &amp;amp; 1 - \cos t \\&#xA;t &amp;amp; \in &amp;amp; [0,2 \pi]&#xA;\end{eqnarray}&lt;/span&gt;&#xA;Then you could derive the length of the curve: &#xA;&lt;span class=&quot;math-container&quot;&gt;$L(t) = 4 - 4 \cos t/2$&lt;/span&gt;,&#xA;which then shows that &lt;span class=&quot;math-container&quot;&gt;$L=8$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$t=2\pi$&lt;/span&gt;,&#xA;i.e., the length of one arch is four diameters of the circle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/AQ9zu.gif&quot; width=&quot;200&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Cycloid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Either Mathematica or Geogebra could be used to make the&#xA;animation above.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-05-15T00:26:30.437" LastActivityDate="2019-05-15T00:26:30.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16621" PostTypeId="1" CreationDate="2019-05-16T11:22:53.083" Score="4" ViewCount="194" Body="&lt;p&gt;I am looking for textbooks pertaining to drawing intersecting planes, intersecting point (among others) as follows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/fOSIE.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fOSIE.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure this topic is taught in high school around the globe. That is why I am a bit difficult to find the book in bookstores. &lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Edit&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;As a reference:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In some secondary mathematics curricula, there is a topic called&#xA;  Stereometry that deals with investigating the position and finding the&#xA;  intersection, angle, and distance of lines and planes defined within a&#xA;  prism or pyramid. Coordinate system is not used. The metric tasks are&#xA;  solved using Pythagoras’ theorem, trigonometric functions, and sine&#xA;  and cosine rules. The basic problem is to find the section of the&#xA;  figure by a plane that is defined by three points related to the&#xA;  figure. In this article, a formula is derived that gives the positions&#xA;  of the intersection points of such a plane and the figure edges, that&#xA;  is, the vertices of the section polygon. Spreadsheet implementations&#xA;  of the formula for cuboid and right rectangular pyramids are&#xA;  presented. The user can check his/her graphical solution, or proceed&#xA;  if he/she is not able to complete the section.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is quoted from &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/0020739X.2011.618553?journalCode=tmes20&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5696" LastEditorUserId="5696" LastEditDate="2019-05-16T13:53:40.167" LastActivityDate="2019-05-17T03:15:05.437" Title="What books are good for drawing an intersecting plane?" Tags="|textbooks|reference-books|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16622" PostTypeId="1" CreationDate="2019-05-16T14:17:16.900" Score="9" ViewCount="2848" Body="&lt;p&gt;What are some real-life exceptions to the PEMDAS rule?&#xA;I am looking for examples from &quot;real&quot; mathematical language --- conventions that are held in modern mathematics practice, such as those appearing in papers, books, and class notes, that are not covered (or contradicted) by PEMDAS. I am not interested in contrived examples, such as those constructed &lt;a href=&quot;https://slate.com/technology/2013/03/facebook-math-problem-why-pemdas-doesnt-always-give-a-clear-answer.html&quot; rel=&quot;noreferrer&quot;&gt;to confuse people on Facebook&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The PEMDAS convention says that order of operations are evaluated as:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Parentheses&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Exponents&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Multiplication and Division, from left to right&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Addition and Subtraction, from left to right&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So far I know of one exception: sometimes the placement of text indicates &quot;implicit&quot; parentheses: text that is a superscript, subscript, or above/below a fraction.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2^{3 + 5}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, &lt;span class=&quot;math-container&quot;&gt;$3 + 5$&lt;/span&gt; is evaluated before the exponentiation&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{3+5}{2}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, &lt;span class=&quot;math-container&quot;&gt;$3 + 5$&lt;/span&gt; is evaluated before the division&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Are there other exceptions?&lt;/p&gt;&#xA;" OwnerUserId="200" LastEditorUserId="200" LastEditDate="2019-05-16T21:27:39.043" LastActivityDate="2023-07-11T22:50:32.887" Title="Real-life exceptions to PEMDAS?" Tags="|algebra|primary-education|examples|" AnswerCount="5" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="16623" PostTypeId="2" ParentId="16622" CreationDate="2019-05-16T14:50:08.870" Score="16" Body="&lt;p&gt;The examples you give aren't exceptions. The parentheses aren't needed because there is no other way to interpret the expressions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In applications in engineering and the physical sciences, variables have units associated with them, and the units often disambiguate the expression without the need for parentheses. For example, if &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; has units of radians per second, and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; has units of seconds, then &lt;span class=&quot;math-container&quot;&gt;$\sin\omega t$&lt;/span&gt; has to be interpreted as &lt;span class=&quot;math-container&quot;&gt;$\sin(\omega t)$&lt;/span&gt;, not as &lt;span class=&quot;math-container&quot;&gt;$(\sin\omega) t$&lt;/span&gt;, because the sine function requires a unitless input. Omitting the parentheses is preferable in examples like these because it simplifies the appearance of expressions and makes them easier to parse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, if a force &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; acts on an object of mass &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; for a time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, then, starting from rest, the object is displaced by &lt;span class=&quot;math-container&quot;&gt;$Ft^2/2m$&lt;/span&gt;. This can only be interpreted as &lt;span class=&quot;math-container&quot;&gt;$(Ft^2)/(2m)$&lt;/span&gt;, because otherwise the units wouldn't come out to be units of distance. Examples like this aren't just typewriter usages. Built-up fractions that occur inline in a paragraph of text are awkward typographically: &lt;span class=&quot;math-container&quot;&gt;$\frac{Ft^2}{2m}$&lt;/span&gt;. They either have to be written very small, which makes them hard to read, or they force the spacing of the lines to be uneven, which looks ugly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are examples involving physics and units, but more generally, when people read mathematics, they apply their understanding of the content, which could be economics or number theory. People are not computers. Mathematical notation is a human language written for humans to read. Human languages are ambiguous, and that's a good thing. When we don't require all ambiguities to be explicitly resolved, it helps with concision and expressiveness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides the meaning, another factor that resolves ambiguities and allows concise and readable notation is that we have certain conventions that we follow, such as the convention that tells us to write &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$x2$&lt;/span&gt;. If someone means &lt;span class=&quot;math-container&quot;&gt;$(\sin x)(y)$&lt;/span&gt;, then they'll write it as &lt;span class=&quot;math-container&quot;&gt;$y\sin x$&lt;/span&gt;, not &lt;span class=&quot;math-container&quot;&gt;$\sin xy$&lt;/span&gt;. There is also a convention that if we have a long chain of multiplications and divisions, such as &lt;span class=&quot;math-container&quot;&gt;$abcdefgh/ijklm$&lt;/span&gt;, we put all the multiplicative factors to the left of the slash, and then it's understood that we mean &lt;span class=&quot;math-container&quot;&gt;$abcdefgh/(ijklm)$&lt;/span&gt;. We wouldn't write this as &lt;span class=&quot;math-container&quot;&gt;$a/ibc/j/kdef/lgh/m$&lt;/span&gt;, which would be extremely difficult to read. We wouldn't interpret &lt;span class=&quot;math-container&quot;&gt;$abcdefgh/ijklm$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$(abcdefgh/i)jklm$&lt;/span&gt;, because if that had been the intended meaning, it would have been written like &lt;span class=&quot;math-container&quot;&gt;$abcdefghjklm/i$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-05-17T18:18:39.920" LastActivityDate="2019-05-17T18:18:39.920" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16624" PostTypeId="2" ParentId="16622" CreationDate="2019-05-16T18:11:02.080" Score="8" Body="&lt;p&gt;That's a good question. In addition to PEMDAS there's grouped symbols; they occur under a radical (&lt;span class=&quot;math-container&quot;&gt;$\sqrt{~}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sqrt[n]{~}$&lt;/span&gt;), in the numerator and denominator of a fraction and in exponents. These grouped symbols are treated as if they are between parenthesis. That means that the expressions &lt;span class=&quot;math-container&quot;&gt;$a^{\text{expression}}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sqrt[n]{\text{expression}}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\text{expression}_1}{\text{expression}_2}$&lt;/span&gt; mean &lt;span class=&quot;math-container&quot;&gt;$a^{(\text{expression})}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sqrt[n]{(\text{expression})}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\dfrac{(\text{expression}_1)}{(\text{expression}_2)}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The horizontal fractional line and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Vinculum_(symbol)&quot; rel=&quot;nofollow noreferrer&quot;&gt;vinculum&lt;/a&gt; of the radical act as symbols of grouping. That's why we write &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\text{expression}}$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{~}(\text{expression})$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\text{expression}_1}{\text{expression}_2}$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$(\text{expression}_1)/(\text{expression}_2)$&lt;/span&gt;. The exponent, written in superscript, is also a grouped symbol, so we write &lt;span class=&quot;math-container&quot;&gt;$a^{\text{expression}}$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$a\wedge(\text{expression})$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other functions use parentheses around the input to avoid ambiguity. The parentheses are sometimes omitted if the input is a monomial. That doesn't have anything to do with units. Thus &lt;span class=&quot;math-container&quot;&gt;$\sin 2x$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$\sin (2x)$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$\sin a+b$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$\sin (a)+b$&lt;/span&gt;. Personally I prefer to write the parenthesis even around a monomial like &lt;span class=&quot;math-container&quot;&gt;$\ln (5x)$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What does &lt;span class=&quot;math-container&quot;&gt;$a/bc$&lt;/span&gt; means? Is it &lt;span class=&quot;math-container&quot;&gt;$\dfrac{a}{bc}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\dfrac{a}{b}c$&lt;/span&gt; ? We could argue which convention we like but the matter of fact is that different people have different convention and different softwares have different conventions. For example &lt;span class=&quot;math-container&quot;&gt;$18/2\times 3$&lt;/span&gt; gives &lt;a href=&quot;https://www.wolframalpha.com/input/?i=18%2F2*3&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$27$&lt;/span&gt;&lt;/a&gt; on wolframalpha but &lt;span class=&quot;math-container&quot;&gt;$a/bc$&lt;/span&gt; gives &lt;a href=&quot;https://www.wolframalpha.com/input/?i=a%2Fbc+where+a%3D18+and+b%3D2+and+c%3D3&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;&lt;/a&gt; when &lt;span class=&quot;math-container&quot;&gt;$a=18$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b=2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c=3$&lt;/span&gt;. The same software has two conventions for the same expression! That's why I always prefer to write &lt;span class=&quot;math-container&quot;&gt;$\dfrac{a}{b}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{a}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a^b$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{}a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a\wedge b$&lt;/span&gt;. They don't cause any ambiguity. The same issue is present with &lt;span class=&quot;math-container&quot;&gt;$\sin ab$&lt;/span&gt; which WA interprets as &lt;a href=&quot;https://www.wolframalpha.com/input/?i=sin2*3&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$b\sin a$&lt;/span&gt;&lt;/a&gt; or &lt;a href=&quot;https://www.wolframalpha.com/input/?i=sin+ab+where+a%3D2+and+b%3D3&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin (ab)$&lt;/span&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="501" LastEditDate="2019-05-18T00:00:42.933" LastActivityDate="2019-05-18T00:00:42.933" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16625" PostTypeId="2" ParentId="16621" CreationDate="2019-05-16T18:40:05.017" Score="0" Body="&lt;p&gt;This is not an answer and will not help.&#xA;But it may be useful to know there are webpages&#xA;that need nothing more than a browser to interactively explore&#xA;cross-sections: &lt;a href=&quot;https://www.geogebra.org/m/XCZwsytr&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here is one based on Geogebra&lt;/a&gt;.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/SpEuF.png&quot; width=&quot;200&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Image from &lt;a href=&quot;https://www.geogebra.org/m/XCZwsytr#material/NywUsyXp&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;The underlying calculations, however, are not evident.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-05-16T18:40:05.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16626" PostTypeId="2" ParentId="16621" CreationDate="2019-05-16T20:46:37.343" Score="2" Body="&lt;p&gt;A quick Google search or two (various keyword combinations) shows some old pedagological articles on teaching solid geometry and the importance of drawing.  You will need to create a free JSTOR profile to read them.  Also mechanical drafting textbooks (more often the ones called intermediate) treat solid geometry drawing as a typical test problem (and show how to do it using construction lines).  There's a YouTube video of a drafting teacher showing how to make a correctly dimensioned drawing of a square pyramid truncated by an angled plane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I didn't find a quick canned answer to your question.  But there are several decent resources that if you parse/combine will help you if you take the effort.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example:  &lt;a href=&quot;https://www.youtube.com/watch?v=Cghf62w6f_k&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=Cghf62w6f_k&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another:  &lt;a href=&quot;https://www.youtube.com/watch?v=kohKsPo59Ts&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=kohKsPo59Ts&lt;/a&gt; (at second 14, he mentions a textbook that perhaps you can track down)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't have enough of an account to assemble all the links...but really I think you benefit from doing the simple Google research/reading yourself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the ones from a drafting coverage are better and more detailed than those from a geometry slant.  There is a little assumption that you will have learned, be interested in overall drafting though.&lt;/p&gt;&#xA;" OwnerUserId="12276" LastActivityDate="2019-05-16T20:46:37.343" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16628" PostTypeId="2" ParentId="16621" CreationDate="2019-05-17T03:15:05.437" Score="0" Body="&lt;p&gt;Can't help myself.  Second answer.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One more cool video that even almost has calculus insights.  Or anyhow a bunch of math-y insights.  Instructor works more on the drawing, doesn't discuss the math insights...but y'all can see them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=6ITbWNTXemA&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=6ITbWNTXemA&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He's drawing a helix.  To start with there is an issue of projecting a cylinder (which is a circle in reality (extended up) onto a page (2-D surface).  So he draws the construction lines.  The key insight is that when you stare at a cylinder (beer can, e.g), you see half of it and you see the &quot;middle&quot; straight on.  But as you move closer to the edges, there is more and more of a ratio between apparent length and length on the cylinder.  For example, the lettering is easy to read in the middle, but more and more visually distorted because of foreshortening as you move to can edges.  At the edge (or tangent of your sightline if you are close enough for noticeable parallax), this foreshortening becomes infinite.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So anyhow, these construction lines he draws, help to give the varying distances in flat view, for the curved surface.  Thus when he draws the helix, it has to be curved, not a straight line to connect the construction lines.  He could draw segments, but knows that he is approximating a curve of steepening slope as he nears the &quot;edge&quot;.  Of course in reality, the helix has a constant vertical slope.  But as viewed or projected, it becomes infinite (vertical), right at the edge of the cylinder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  You might just track the YT video guy down and send an email and ask for book advice.  He's not just a shop teacher.  Has other videos showing pentagon construction with compass.  So I think he will be sympathetic to your geometry aims, not just machine design drawing skills.&lt;/p&gt;&#xA;" OwnerUserId="12278" LastActivityDate="2019-05-17T03:15:05.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16629" PostTypeId="2" ParentId="11054" CreationDate="2019-05-17T13:30:05.607" Score="0" Body="&lt;p&gt;As Dave notes, the zeroes are key. Factoring is one method to arrive at that. I deal with students that benefit from visual examples where appropriate. To that end, I encourage these students to sketch a quick graph. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9hLWC.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9hLWC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Presumably, when they are given this particular question, the zeroes are enough for them to sketch the graph in a way that makes the inequality solution clear. By this, I mean that they won't use these X-Axis intercepts but produce a flipped graph. And 'sketch' means that the local maxima/minima don't really matter, as far as accuracy is concerned. If the graph shows (-1,2) as the local max, no harm. No impact to the question as asked.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-05-17T13:30:05.607" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16630" PostTypeId="1" CreationDate="2019-05-18T00:40:29.593" Score="5" ViewCount="206" Body="&lt;p&gt;What are the benefits of an expertly curated learning pathway? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like that provided by a major publisher's textbook - &lt;a href=&quot;https://cpm.org/&quot; rel=&quot;noreferrer&quot;&gt;CPM&lt;/a&gt;, a school district's mandated curriculum - &lt;a href=&quot;https://im.openupresources.org/&quot; rel=&quot;noreferrer&quot;&gt;IM's Open Up Resources&lt;/a&gt; or a learning game - &lt;a href=&quot;https://www.stmath.com/&quot; rel=&quot;noreferrer&quot;&gt;ST Math&lt;/a&gt;, &lt;a href=&quot;https://dragonbox.com/&quot; rel=&quot;noreferrer&quot;&gt;Dragonbox&lt;/a&gt;, or &lt;a href=&quot;https://www.amplify.com/programs/amplify-fractions/&quot; rel=&quot;noreferrer&quot;&gt;Amplify Fractions&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask this question with no particular age-level in mind, but I could see how the younger a learner is, the more useful a defined learning pathway would be - partly due to the inexperience of the learner&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Research has been done on the idea of learning maps as a way of assisting learners along a learning pathway. See &lt;a href=&quot;https://intentionalfutures.com/insights/portfolio/individual-learning-maps/&quot; rel=&quot;noreferrer&quot;&gt;Gates Learning Maps&lt;/a&gt; projects. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One led by Jere Confrey's team has had some interesting &lt;a href=&quot;http://www.oecd.org/education/2030/A-Synthesis-of-Research-on-Learning-Trajectories-Progressions-in-Mathematics.pdf&quot; rel=&quot;noreferrer&quot;&gt;research results&lt;/a&gt;  and a &lt;a href=&quot;https://sudds.ced.ncsu.edu/&quot; rel=&quot;noreferrer&quot;&gt;design concept&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I still find it difficult to believe that the future of learning will involve standardized linear learning pathways... true personalization will bend and weave the trajectory of each learner, right? As &lt;a href=&quot;http://www.lmi.ub.edu/cursos/s21/REPOSITORIO/documents/knewton-adaptive-learning-whitepaper.pdf&quot; rel=&quot;noreferrer&quot;&gt;Knewton described&lt;/a&gt; or InBloom &lt;a href=&quot;https://datasociety.net/pubs/ecl/InBloom_feb_2017.pdf&quot; rel=&quot;noreferrer&quot;&gt;once visaged&lt;/a&gt;... Or as a math prof recently argued to me, &quot;Whose canon?&quot; when discussing canonical examples to teach undergraduate intro stats.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an age of brokering experiences, I offer my &lt;a href=&quot;http://bit.ly/futureEducation&quot; rel=&quot;noreferrer&quot;&gt;non-expert postgraduate data science pathway&lt;/a&gt; and ask the question another way, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why pay for an expert's subjective structuring of content when you can&#xA;  structure your own learning based on your own personal interest and&#xA;  passion projects for free?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1693" LastEditorUserId="1693" LastEditDate="2019-05-20T01:11:56.247" LastActivityDate="2019-05-21T00:12:26.607" Title="What are the benefits of an expertly curated learning pathway?" Tags="|textbooks|course-design|curriculum|education-research|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16631" PostTypeId="1" AcceptedAnswerId="16635" CreationDate="2019-05-19T07:17:32.637" Score="4" ViewCount="313" Body="&lt;p&gt;Nowadays, a student should be able to learn the course material at home through reading the textbook or follow one of the many free online courses. Some universities record video or audio of lectures so students can watch them at a time convenient for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So as mathematics educators, how should we justify that students should still come to classes? Is there any research about the correlation between attending classes and learning outcomes?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2019-05-20T18:55:41.673" Title="How to justify that students should come to class?" Tags="|undergraduate-education|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16632" PostTypeId="2" ParentId="16631" CreationDate="2019-05-19T14:18:55.963" Score="6" Body="&lt;p&gt;&lt;em&gt;&quot;Nowadays, a student should be able to learn the course material at home through reading the textbook or follow one of the many free online courses. Some universities record video or audio of lectures so students can watch them at a time convenient for them.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have to disagree with your premise - especially in the case of technical education.  It quite often isn't enough just to &quot;read the textbook&quot; or &quot;watch the video&quot;.  In fact, many undergraduates aren't at the level where they can do that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have you actually tried this with a class?  I can see how it would work with graduate students or even upper division undergrads but, other than that, I'm having a hard time imagining a scenario where this ends well.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2019-05-19T14:18:55.963" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16633" PostTypeId="2" ParentId="16630" CreationDate="2019-05-19T14:22:49.553" Score="0" Body="&lt;p&gt;Who is going to create your &quot;non-linear learning pathway&quot;?  I agree that there's a lot of promise to the idea of customized learning paths although I don't think we're as far along in that direction as some.  However, I don't see this as being any less curated than the traditional linear path that you would get from a textbook.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2019-05-19T14:22:49.553" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16635" PostTypeId="2" ParentId="16631" CreationDate="2019-05-20T14:50:52.060" Score="10" Body="&lt;p&gt;A student-attended class does not have to be a one way flow of information. Even if there are 200 in the lecture theatre, the lecturer can set a problem, break off from teaching, ask the students in pairs to come up with a solution (hopefully productive chaos ensues for a few minutes) , then the lecturer rings a loud bell bringing the class to order, takes a vote between various solutions (some of which contain pernicious errors), then presents a solution or leaves the solution unfinished  for an assignment. This encourages students to engage with the material and debate it with someone else.&lt;/p&gt;&#xA;" OwnerUserId="7954" LastActivityDate="2019-05-20T14:50:52.060" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16636" PostTypeId="1" AcceptedAnswerId="16640" CreationDate="2019-05-20T18:42:54.760" Score="18" ViewCount="7503" Body="&lt;p&gt;Suppose that I want to show a student that empirical evidence in mathematics is not enough and we do need proofs, what kind of examples can I use?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By empirical evidence, I mean that (most of the time) you cannot simply check the statement &lt;span class=&quot;math-container&quot;&gt;$S(n)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n \in \{1,\dots, 10^9\}$&lt;/span&gt; and conclude it's true for all &lt;span class=&quot;math-container&quot;&gt;$n \in \mathbb N$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorUserId="7337" LastEditDate="2019-05-22T01:21:29.167" LastActivityDate="2022-05-04T05:41:28.627" Title="Is there a simple example that empirical evidence is misleading?" Tags="|undergraduate-education|" AnswerCount="13" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16638" PostTypeId="2" ParentId="16630" CreationDate="2019-05-20T19:02:27.130" Score="3" Body="&lt;p&gt;What are the benefits of having experts create curriculum ? I gather the alternative you entertain is some sort of user-driven machine learning path which is custom fit to the student. Ok, so, if that automation is initiated and curated by experts in math then I don't see the distinction. In some sense it would be a return to the old apprentice system that predates the mass production of university education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, &lt;strong&gt;if experts in math instruction do not play a significant role&lt;/strong&gt; in the delivery of this new automated custom-built learning experience then what will guide the experience and content ?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;strong&gt;students&lt;/strong&gt; guide the content then they will likely choose candy over meat. Sometimes the pay-off of learning deep skills and/or ideas is part of building the larger mathematical whole. Inculcation into the society and customs of Mathematics is surely going to be short-changed.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;strong&gt;companies&lt;/strong&gt; guide the content then again their will be specialization which favors the needs of the company over the needs of the student. That which is beautiful, but not seemingly practical will likely suffer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So, I see a lot of problems with control of education resting in the wrong hands. To my knowledge, PhD's in Mathematics care the most about Mathematics. Of course there are exceptions, but those who have invested so deeply want to see the subject thrive, prosper, grow and attain proper appreciation by the wider society. Math education is a major vehicle for those goals. We know directions the student might pursue that they simply cannot know as they don't know what they don't know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These guided learning path ways could misappropriate interest of students in the same way that TV commericals manipulate us to buy things we really neither want nor need. Of course, I think we all took courses in the standard curriculum we feel the same way about. The difference then is ultimately accountability. New products are almost by default unaccountable and self-policing doesn't always serve well. Take a look at online education at the moment. Where is the quality control ?&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2019-05-21T00:12:26.607" LastActivityDate="2019-05-21T00:12:26.607" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16639" PostTypeId="2" ParentId="16636" CreationDate="2019-05-20T19:36:53.127" Score="21" Body="&lt;p&gt;Strangely, just this morning I asked &lt;a href=&quot;https://www.wolframalpha.com/input/?i=sum%20for%20n%3D1%20to%20infinity%20of%201%2F(n*sin(n))&quot; rel=&quot;noreferrer&quot;&gt;Wolfram|Alpha&lt;/a&gt; to compute the sum &lt;span class=&quot;math-container&quot;&gt;$$\sum_{n=1}^{\infty}{\frac{1}{n\sin(n)}}$$&lt;/span&gt; and it returned the approximate value of &lt;span class=&quot;math-container&quot;&gt;$-0.863507$&lt;/span&gt;. I asked it to &quot;show more digits&quot;, and it returned a new approximation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$-94.377284731050845020943145217217734512865979242824685504875914407196948018$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was trying to illustrate a series for which convergence (or divergence) is difficult to determine and was treated to some very different approximations. Note that Wolfram did not tell whether the series converges.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-05-22T19:06:26.353" LastActivityDate="2019-05-22T19:06:26.353" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16640" PostTypeId="2" ParentId="16636" CreationDate="2019-05-20T22:31:30.473" Score="31" Body="&lt;p&gt;There are some collections of such examples at sister sites:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples&quot;&gt;Conjectures that have been disproved with extremely large counterexamples?&#xA;&lt;/a&gt; at Mathematics Stack Exchange.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/q/15444/36173&quot;&gt;Examples of eventual counterexamples&lt;/a&gt; at MathOverflow.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;One rather simple example that can be checked with a calculator is the conjecture by Fermat, that all numbers of the form &lt;span class=&quot;math-container&quot;&gt;$$2^{2^n}+1, \qquad n \in \mathbb N_0$$&lt;/span&gt; are prime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2^{2^0} +1 = 3$&lt;/span&gt; is prime&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2^{2^1} +1 = 5$&lt;/span&gt; is prime&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2^{2^2} +1 = 17$&lt;/span&gt; is prime&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2^{2^3} +1 = 257$&lt;/span&gt; is prime&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2^{2^4} +1 = 65537$&lt;/span&gt; is prime&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2^{2^5} +1 = 4294967297$&lt;/span&gt; is &lt;strong&gt;not&lt;/strong&gt; prime: &lt;span class=&quot;math-container&quot;&gt;$4294967297 = 641 \cdot 6700417$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So the original conjecture is clearly false, but it took nearly 100 years to find the counterexample. All following Fermat numbers appear to be composite, but this is an open problem.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2019-05-20T22:31:30.473" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="16641" PostTypeId="2" ParentId="16636" CreationDate="2019-05-21T05:42:32.450" Score="5" Body="&lt;p&gt;There is an old joke describing how various scientific disciplines &quot;prove&quot; that all odd numbers are prime. The part for mathematicians goes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1 is a special case, 3 is prime, 5 is prime, 7 is prime, the proof by induction is left to the reader. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8201" LastActivityDate="2019-05-21T05:42:32.450" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="16642" PostTypeId="2" ParentId="16636" CreationDate="2019-05-21T06:11:06.943" Score="3" Body="&lt;p&gt;The Pell equation &lt;span class=&quot;math-container&quot;&gt;$x^2-Dy^2=1$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are positive integers) admits solutions if the natural number &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; is not a square number.  But for &lt;span class=&quot;math-container&quot;&gt;$D=61$&lt;/span&gt; the smallest solution is astonishingly great: you can derive an example from this fact. &lt;a href=&quot;https://proofwiki.org/wiki/Pell%27s_Equation/Examples/61&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://proofwiki.org/wiki/Pell%27s_Equation/Examples/61&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12290" LastEditorUserId="12255" LastEditDate="2019-05-21T20:30:23.750" LastActivityDate="2019-05-21T20:30:23.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16643" PostTypeId="2" ParentId="16636" CreationDate="2019-05-21T15:02:04.760" Score="2" Body="&lt;p&gt;If you are open to other fields that strongly rely on mathematics, you can use the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ultraviolet_catastrophe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rayleigh-Jeans catastrophe&lt;/a&gt; as an example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The term &quot;ultraviolet catastrophe&quot; was first used in 1911 by Paul Ehrenfest, but the concept originated with the 1900 statistical derivation of the Rayleigh–Jeans law. The phrase refers to the fact that the Rayleigh–Jeans law accurately predicts experimental results at radiative frequencies below 105 GHz, but begins to diverge with empirical observations as these frequencies reach the ultraviolet region of the electromagnetic spectrum.&lt;a href=&quot;https://en.wikipedia.org/wiki/Ultraviolet_catastrophe&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; Since the first appearance of the term, it has also been used for other predictions of a similar nature, as in quantum electrodynamics and such cases as ultraviolet divergence.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The beginning times of quantum physics is full of such examples. This particular one was solved by Max Planck and physics has never been the same since then.&lt;/p&gt;&#xA;" OwnerUserId="8099" LastEditorUserId="7337" LastEditDate="2019-05-22T01:21:48.533" LastActivityDate="2019-05-22T01:21:48.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16644" PostTypeId="2" ParentId="16636" CreationDate="2019-05-21T18:21:58.163" Score="5" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test&quot; rel=&quot;noreferrer&quot;&gt;Miller-Rabin&lt;/a&gt; test may be a good example.  It identifies &quot;industrial strength primes,&quot; numbers which are strongly indicated to be prime through testing, but have not actually been proven to be prime.  It is based off of the proof that &lt;span class=&quot;math-container&quot;&gt;$a^d \equiv 1 (mod\;p)$&lt;/span&gt; for all prime numbers.  The test flips that around into its contrapositive, and explores the idea that if we look for an &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; which does not satisfy that equation, then we have found a &quot;witness&quot; which shows that &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is not prime.  It can be shown that all composite numbers have at least one witness, but a simple algorithm for determining it has never been found.  Instead, we test many &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and, if we find no witnesses, then we say &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is an &quot;industrual strength prime&quot; because we took a good swing at it and failed to disprove its primality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As it turns out, testing small values of a is &lt;em&gt;enormously&lt;/em&gt; successful.  So successful that you might make the assumption that testing just &lt;span class=&quot;math-container&quot;&gt;$a=2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a=3$&lt;/span&gt; is sufficient to prove a number is prime.  And you'd be right, so long as the number is smaller than 1,373,653, which is the first point where that test suggests a composite number is prime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maple's &lt;code&gt;isprime()&lt;/code&gt; function relies on this test.  It tests &lt;span class=&quot;math-container&quot;&gt;$a=2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a=3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a=5$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a=7$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$a=11$&lt;/span&gt;.  This turns out to be a very fast and enormously effective test.  But there are numbers which fail this.  In fact, one mathematician identified a particular 397-bit number which passes all tests from &lt;span class=&quot;math-container&quot;&gt;$a=2$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$a=307$&lt;/span&gt;, and yet is composite.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2019-05-21T18:21:58.163" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16645" PostTypeId="1" AcceptedAnswerId="16648" CreationDate="2019-05-21T19:47:02.277" Score="5" ViewCount="463" Body="&lt;p&gt;A friend of mine that teaches math has made many times the following experiment : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;drawing two circles on the blackboard representing two sets A and B such that A and B are disjoint &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;writing on the blackboard  &lt;span class=&quot;math-container&quot;&gt;$A \cap B$&lt;/span&gt; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;writing below : &quot; true, or false?&quot; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I let you guess what is the most frequent answer ( this answer being not worse, in fact, that the other, for they are both inappropriate). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The interesting thing, according to me, is not which answer is given but the fact that very few students say: &quot; but, thtis is neither true nor false! this denotes a &lt;em&gt;set&lt;/em&gt; ! &quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this experiment is interesting inasmuch as it puts into light the fact that not grasping the &lt;strong&gt;&lt;em&gt;category a symbol belongs to&lt;/em&gt;&lt;/strong&gt; might be one of the major causes of misundertanding in mathematics.Here the confusion is between a symbol denoting an object and a symbol expressing a sentence.  I express this as a hypothesis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you give examples of analogous misunderstandings? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you imagine analogous experiences that would enable a math teacher to clear up students' confusions? &lt;/p&gt;&#xA;" OwnerDisplayName="user12295" LastEditorDisplayName="user12295" LastEditDate="2019-05-21T20:23:29.247" LastActivityDate="2019-06-08T13:20:12.043" Title="Category mistakes regarding symbols and their impact on math (mis) understanding. ( Object symbol/ sentence symbol confusion)" Tags="|mathematical-pedagogy|teaching|students-mistakes|set-theory|mathematical-discourse|" AnswerCount="7" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16646" PostTypeId="2" ParentId="16602" CreationDate="2019-05-21T23:51:30.110" Score="4" Body="&lt;p&gt;I've used Geogebra to show the Taylor polynomial of a function. Adding terms one by one shows how the series converge to the function and how approximation improves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is my Taylor polynomial of exponential &lt;a href=&quot;https://ggbm.at/tszpraxe&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ggbm.at/tszpraxe&lt;/a&gt; . However, I think sine function could be more effective.&lt;/p&gt;&#xA;" OwnerUserId="7178" LastActivityDate="2019-05-21T23:51:30.110" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16647" PostTypeId="2" ParentId="16645" CreationDate="2019-05-22T00:36:57.163" Score="2" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$A\cap B$&lt;/span&gt; is not true or false; but it is true that the intersection of two disjoint sets is the empty set, that is, that &lt;span class=&quot;math-container&quot;&gt;$A\cap B = \varnothing$&lt;/span&gt; is true. This comes down to a question about whether A\cap B is a boolean statement (a proposition): that which can be answered with &quot;true or false&quot;. Asking if &lt;span class=&quot;math-container&quot;&gt;$A\cap B$&lt;/span&gt; is true or false is  no different than asking if 2+3 is true or false.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think your presentation to the students invites such confusion. Why not draw two sets, &lt;span class=&quot;math-container&quot;&gt;$A, B,$&lt;/span&gt; on the blackboard, disjoint, and ask &lt;span class=&quot;math-container&quot;&gt;$A\cap B = ?.$&lt;/span&gt; By your own description, drawing two disjoint sets, A, B, and then writing &lt;span class=&quot;math-container&quot;&gt;$A\cap B$&lt;/span&gt;, and then asking &quot;true or false&quot; (without the option &quot;neither true nor false&quot;) sets the students up for &quot;failure&quot;, as you measure it. –&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among sets, we are able to know decisively, when given sets A, B the following boolean statements are such that they are always either true or false:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$|A|&amp;gt;|B|, |A|&amp;lt; |B|,$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$|A| = |B|$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$A\subset B, B\subset A, \text{ or } A = B$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In arithmetic we get boolean statements when we compare of equate numbers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$5+8 = 13$&lt;/span&gt;  (true)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$8&amp;lt;5$&lt;/span&gt;       (false)&lt;/li&gt;&#xA;&lt;li&gt;The set of real numbers is a subset of the set of rational numbers.  (false).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;On the contrary, &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$5+8$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$3x^2-4$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx} (8x-4 - x^2)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Are not in the category of sentences that be be deemed true or false.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$A=\{1, 3, 5, 7, 9\}; B=\{2, 4, 6, 8, 10\}$&lt;/span&gt;.  Then &lt;span class=&quot;math-container&quot;&gt;$A\cap B = \varnothing$&lt;/span&gt;, and is a true statement, whereas &lt;span class=&quot;math-container&quot;&gt;$A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$&lt;/span&gt;, which is also a true statement.  But merely writing &lt;span class=&quot;math-container&quot;&gt;$A\cap B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A\cup B$&lt;/span&gt; yields no sentence with a truth-value, and hence to ask if either is true or false misleads.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think what is most misleading about the set of your friend is what appears to be an abuse of notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's partition the set of positive integers, which I refer to as &lt;span class=&quot;math-container&quot;&gt;$\mathbb N,$&lt;/span&gt; into two sets:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$A=\{2n-1\mid n \in \mathbb N\}$&lt;/span&gt;;&lt;/li&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$B= \{2n\mid n \in \mathbb N\}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Suppose your friend then writes below these sets: &lt;span class=&quot;math-container&quot;&gt;$$A\cap B$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then asks &quot;True or False ?&quot;.  The only way such a question has meaning is if the teacher is abusing the notation &lt;span class=&quot;math-container&quot;&gt;$A\cap B$&lt;/span&gt; (which denotes the intersection between sets A and B), to mean &quot;Is the intersection between A and B nonempty&quot;?  That's not what &lt;span class=&quot;math-container&quot;&gt;$A\cap B$&lt;/span&gt; in fact means.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-05-22T12:30:26.987" LastActivityDate="2019-05-22T12:30:26.987" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="16648" PostTypeId="2" ParentId="16645" CreationDate="2019-05-22T10:40:28.757" Score="6" Body="&lt;p&gt;I like the idea here, but I agree that it misleads students, and might have the opposite of intended effect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why not hand out a paragraph to the students, and ask them to critique it. Say that the paragraph is a fake student response to an exam question. One sentence in the paragraph could be something like ``Since &lt;span class=&quot;math-container&quot;&gt;$A \cap B$&lt;/span&gt;, there must be an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$x \in A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x \in B$&lt;/span&gt;''.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to pack in other common errors you notice into this paragraph.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The resulting conversation could be really valuable, and will address the issue you were trying to target in a more natural way.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2019-05-22T10:40:28.757" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16650" PostTypeId="2" ParentId="16645" CreationDate="2019-05-22T17:01:16.447" Score="1" Body="&lt;p&gt;I'll go with the common &lt;strong&gt;&quot; '&lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;' is a key on the calculator&quot;&lt;/strong&gt; misconception.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students don't see a problem with and write down things like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$3 \cdot 4  = 12 - 5 = 7$$&lt;/span&gt;&#xA;when asked to calculate &lt;span class=&quot;math-container&quot;&gt;$3\cdot 4 - 5$&lt;/span&gt;. &#xA;This error is caused (or at least reinforced) by the fact that &quot;&lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;&quot; can in almost all cases be read as &quot;calculate the left hand side and write the result on the right hand side&quot; when doing exercises where only numbers are involved. So there is a lot of time in a student's live to memorize this misconception about &quot;&lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It of course breaks down when you are dealing with terms and equations which depending on your curriculum is probably around 7th grade. Perhaps one huge obstacle for the students is not only the increasing level of abstraction but also the need to &quot;re-learn&quot; what &quot;equals&quot; actually means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Things I've tried to deal with this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Emphasize that &quot;&lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;&quot; means &quot;is equal to&quot; in my speech and encourage students to spell it out completely: not &quot;three times four &lt;strong&gt;is&lt;/strong&gt; twelve&quot; but &quot;three times four &lt;strong&gt;is equal to&lt;/strong&gt; twelve&quot;. If I feel very pedantic, I'd also elaborate that &quot;three times four&quot; is a multiplication while &quot;twelve&quot; is not, so one thing can't possibly be the other.&lt;/li&gt;&#xA;&lt;li&gt;Emphasize transitivity of &quot;&lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;&quot; and look at &lt;span class=&quot;math-container&quot;&gt;$3 \cdot 4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Have the students actually look at their calculator. TI uses &quot;Enter&quot;, Casio uses &quot;EXE&quot; on their advanced models. There are sadly many calculators that support the misconception because they have the dreaded &quot;&lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;&quot; key.&lt;/li&gt;&#xA;&lt;li&gt;Forbid multiple equals signs in a single row altogether.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2019-05-22T18:43:23.900" LastActivityDate="2019-05-22T18:43:23.900" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16651" PostTypeId="2" ParentId="16645" CreationDate="2019-05-22T17:42:57.577" Score="0" Body="&lt;ul&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are two disjoint sets, writing &lt;span class=&quot;math-container&quot;&gt;$A\cap B=0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$\ln x=2$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$x=\dfrac{2}{\ln}$&lt;/span&gt;. &lt;/li&gt;&#xA;&lt;li&gt;How many corners a circle have? &lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$\sin x&amp;lt;\sin\frac{\pi}{4}$&lt;/span&gt; than, after simplifying the sines, we get &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;\frac{\pi}{4}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}{6}&amp;lt;x&amp;lt;\frac{\pi}{3}$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\cos\frac{\pi}{6}&amp;lt;\cos x&amp;lt;\cos\frac{\pi}{3}$&lt;/span&gt;. What's wrong? I just added a cosine !&lt;/li&gt;&#xA;&lt;li&gt;The area of a circle is given by &lt;span class=&quot;math-container&quot;&gt;$\pi r^2$&lt;/span&gt;. Since when does a circle have an area?&lt;/li&gt;&#xA;&lt;li&gt;This is a square not a rectangle ! This is an equilateral triangle not isosceles !&lt;/li&gt;&#xA;&lt;li&gt;Using L'Hôpital's rule to find the limit of a sequence : &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim \frac{2n^2-5}{n^2+1}=\lim\frac{4n}{2n}=2$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is increasing. The function is &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; not &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="501" LastActivityDate="2019-05-22T17:42:57.577" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16652" PostTypeId="2" ParentId="16645" CreationDate="2019-05-22T19:44:03.193" Score="-1" Body="&lt;p&gt;To a computer programmer familiar with the Lisp family of languages, it is the original poster who has the misconception.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lisp is one of the most ancient computer languages.  It is about as old as Fortran and Algol; it is older than Basic, C, Pascal, and all of the object-oriented languages.  Lisp was designed by a mathematician to illustrate how set theory could be used as pseudo-code.  It was simple enough that his students promptly turned it into an actual programming language.  The Logo turtle graphics language is a dialect of Lisp.  Lisp is still used as an extension language for the emacs text editor and the AutoCAD drafting system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Lisp, everything is an ordered set, a number, or a string.  In particular, the concept of false is implemented as the empty set, a.k.a. &quot;nil&quot;.  So if A and B are disjoint, the intersection of A and B is empty, which evaluates to false.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2019-05-23T02:21:41.723" LastActivityDate="2019-05-23T02:21:41.723" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16653" PostTypeId="2" ParentId="16622" CreationDate="2019-05-23T04:21:22.283" Score="5" Body="&lt;p&gt;I would add the following to Paracosmiste's reply:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Numerical negation (e.g. the '&lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;' sign in in the expression &lt;span class=&quot;math-container&quot;&gt;$-x^2$&lt;/span&gt;) usually has an order of precedence that is less than that of exponentiation and greater than that of multiplication, division, addition and subtraction in order from right to left. So we have &lt;span class=&quot;math-container&quot;&gt;$-x^2=-(x^2)$&lt;/span&gt;. And &lt;span class=&quot;math-container&quot;&gt;$--x = -(-x))$&lt;/span&gt;. And &lt;span class=&quot;math-container&quot;&gt;$x+-y= x+(-y)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When exponentiation is printed using different levels of typeface (e.g. &lt;span class=&quot;math-container&quot;&gt;$x^{y^z}$&lt;/span&gt;) then they are done in order from right to left. So we have &lt;span class=&quot;math-container&quot;&gt;$x^{y^z}=x^{(y^z)}$&lt;/span&gt;  If, on the other hand, it is printed using the '^' symbol (or '&lt;span class=&quot;math-container&quot;&gt;$**$&lt;/span&gt;') as in many programming languages (e.g. &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;^&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;^&lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; or x&lt;span class=&quot;math-container&quot;&gt;$**$&lt;/span&gt;y&lt;span class=&quot;math-container&quot;&gt;$**$&lt;/span&gt;z), they are usually done in order from left to right. So we have  &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;^&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;^&lt;span class=&quot;math-container&quot;&gt;$z = ((x$&lt;/span&gt;^&lt;span class=&quot;math-container&quot;&gt;$y)$&lt;/span&gt;^&lt;span class=&quot;math-container&quot;&gt;$z)$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2019-05-27T14:56:34.913" LastActivityDate="2019-05-27T14:56:34.913" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16654" PostTypeId="2" ParentId="16636" CreationDate="2019-05-23T07:43:03.097" Score="2" Body="&lt;p&gt;Try representability by the sum of three cubes,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$  x^3 + y^3 + z^3 = k  \text{.}  $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Early computer search paper, that turned up no new representable &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;s.: Gardiner, V.L., R.B. Lazarus, and P.R. Stein, &lt;a href=&quot;https://www.ams.org/journals/mcom/1964-18-087/S0025-5718-1964-0175843-9/S0025-5718-1964-0175843-9.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solutions of the Diophantine Equation &lt;span class=&quot;math-container&quot;&gt;$x^3 + y^3 = z^3 - d$&lt;/span&gt;&lt;/a&gt;, 1964.  In this paper, no representations are found for &lt;span class=&quot;math-container&quot;&gt;$d = k = 30$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d = k = 33$&lt;/span&gt; (and several more &lt;span class=&quot;math-container&quot;&gt;$d = k &amp;lt; 1000$&lt;/span&gt;).  Contains the lines &quot;Nevertheless, it is in our opinion rather unlikely that all the missing &lt;span class=&quot;math-container&quot;&gt;$|d|$&lt;/span&gt;'s will turn out to be expressible as sums of three cubes.  It would be of interest to attempt a proof that, say, 30 cannot be so represented.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recent paper:  Huisman, S., &lt;a href=&quot;https://arxiv.org/abs/1604.07746&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newer sums of three cubes&lt;/a&gt;, 2016.  This paper finds that &lt;span class=&quot;math-container&quot;&gt;$33$&lt;/span&gt; is the least &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; with no known representation, so a representation for &lt;span class=&quot;math-container&quot;&gt;$k = 30$&lt;/span&gt; was found in the interim.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recent video: Haran, B. and A. Booker, &lt;a href=&quot;https://www.youtube.com/watch?v=ASoz_NuIvP0&quot; rel=&quot;nofollow noreferrer&quot;&gt;42 is the new 33 - Numberphile&lt;/a&gt;, 12 March 2019.  A much larger empirical search found a representation for &lt;span class=&quot;math-container&quot;&gt;$k = 33$&lt;/span&gt;, leaving &lt;span class=&quot;math-container&quot;&gt;$k = 42$&lt;/span&gt; as the least &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; without a known representation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moral: empirical searches keep being inadequate.&lt;/p&gt;&#xA;" OwnerUserId="5800" LastActivityDate="2019-05-23T07:43:03.097" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16655" PostTypeId="2" ParentId="16636" CreationDate="2019-05-23T15:24:24.797" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://xkcd.com/1047/&quot; rel=&quot;nofollow noreferrer&quot;&gt;XKCD compiled a list of approximations&lt;/a&gt; several years back:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many are just funny trivia, but you could use most of them as examples of the limits of empirical evidence, because empirical estimations using these formulas would seem to be strong evidence that they are correct, when mathematically they are complete nonsense.&lt;/p&gt;&#xA;" OwnerUserId="12309" LastActivityDate="2019-05-23T15:24:24.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16656" PostTypeId="2" ParentId="16636" CreationDate="2019-05-23T15:39:58.130" Score="5" Body="&lt;p&gt;There is a classic undergrad example (apparently going back to Euler) still missing in this list:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$n^2+n+41 \text{ is prime}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which is true for all &lt;span class=&quot;math-container&quot;&gt;$n&amp;lt;40$&lt;/span&gt; but fails at &lt;span class=&quot;math-container&quot;&gt;$n=40$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8988" LastActivityDate="2019-05-23T15:39:58.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16657" PostTypeId="2" ParentId="16636" CreationDate="2019-05-23T16:08:17.403" Score="4" Body="&lt;p&gt;Here is a straightforward example that knowing the first &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; terms (no matter how big &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is) of a sequence &lt;span class=&quot;math-container&quot;&gt;$(x_n)$&lt;/span&gt; tells you very little about the term &lt;span class=&quot;math-container&quot;&gt;$x_{k+1}$&lt;/span&gt;: I can generate a sequence whose first &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; terms are &lt;span class=&quot;math-container&quot;&gt;$1,2,\dots,k$&lt;/span&gt; but whose &lt;span class=&quot;math-container&quot;&gt;$(k+1)^{{\rm st}}$&lt;/span&gt; term is an arbitrary &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; of my choosing.  For the given &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, let &lt;span class=&quot;math-container&quot;&gt;$x_n = n + \frac {m-n}{k!}(n-1)(n-2)\dots(n-k)$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, for any &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; up to &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, a factor in the product is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, forcing the entire product to be &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, leaving &lt;span class=&quot;math-container&quot;&gt;$x_n=n$&lt;/span&gt;.  Meanwhile, at &lt;span class=&quot;math-container&quot;&gt;$n=k+1$&lt;/span&gt;, the product &lt;span class=&quot;math-container&quot;&gt;$(k+1-1)\dots(k+1-k)=k!$&lt;/span&gt;, and straightforward arithmetic shows &lt;span class=&quot;math-container&quot;&gt;$x_{k+1}=m$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much more in fact is true.  Linear algebra can be used to show, given completely arbitrary first terms &lt;span class=&quot;math-container&quot;&gt;$a_1,\dots,a_{k+1}\in\mathbb R$&lt;/span&gt;, there is a sequence &lt;span class=&quot;math-container&quot;&gt;$(x_n)$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$x_1=a_1,\dots,x_{k+1}=a_{k+1}$&lt;/span&gt;, and the generic term &lt;span class=&quot;math-container&quot;&gt;$x_n$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$(x_n)$&lt;/span&gt; is given as a polynomial in &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; with degree at most &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;. (Indeed, the terms don't even need to be the ``first'' terms, we just need to know &lt;em&gt;which&lt;/em&gt; terms they are.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without knowing something more about a sequence of real numbers than just ``these are some of the terms'', nothing of substance can be concluded about the sequence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit (30 May 2019):&lt;/strong&gt;  Many standardized tests have questions along the lines of ``If a sequence starts with the numbers &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;, what is the next term in the sequence?&quot; then proceeds to have a multiple choice answer, providing four numbers/answers.  One main point behind this example is that &lt;em&gt;all four answers are correct&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12310" LastEditorUserId="12310" LastEditDate="2019-05-30T13:57:13.083" LastActivityDate="2019-05-30T13:57:13.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16658" PostTypeId="2" ParentId="16636" CreationDate="2019-05-23T18:58:57.127" Score="6" Body="&lt;p&gt;There's a great paper out there called &quot;The Strong Law of Small Numbers&quot;, by Richard K Guy, which provides numerous examples of misleading patterns. A PDF copy can be found here: &lt;a href=&quot;https://www.ime.usp.br/~rbrito/docs/2322249.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.ime.usp.br/~rbrito/docs/2322249.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The YouTube channel &quot;Numberphile&quot; (if you aren't already a subscriber, which you should be) has a video by the same name that's also worth a look. It's where I first learned about the paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a personal note, I remember stumbling across Guy's Example 8 myself in a different form (very simply: construct a list starting with 0, ending with 1, and containing all unique fractions between the two where the denominator is &lt;span class=&quot;math-container&quot;&gt;$\leq n$&lt;/span&gt;. Now count the members of the list. You'll find that &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt; produces &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$n=2\rightarrow3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$n=3\rightarrow5$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$n=4\rightarrow7...$&lt;/span&gt;) The sense of being on to something weird and exciting and possibly new, then suddenly seeing it all fall apart was something I'll never forget. Finding out it had been written up in a paper twenty-plus years ago was the cherry on the sundae, for sure.&lt;/p&gt;&#xA;" OwnerUserId="12313" LastActivityDate="2019-05-23T18:58:57.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16659" PostTypeId="1" AcceptedAnswerId="16661" CreationDate="2019-05-24T10:18:17.443" Score="2" ViewCount="362" Body="&lt;p&gt;I am reading the paper &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/full/10.1111/jcal.12347&quot; rel=&quot;nofollow noreferrer&quot;&gt;Effects of game‐based learning on students' mathematics achievement: A meta‐analysis&lt;/a&gt; and can't find a definition for the term &quot;PreK‐12th‐grade students&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I know that &quot;K-12&quot; is used to refer to education from kindergarten to 12th grade I am not sure about &quot;pre&quot;. Does it mean that the study refers exclusively to children before kindergarten aged 0-3? Or to all students aged 0-18+ years?&lt;/p&gt;&#xA;" OwnerUserId="12316" LastEditorUserId="2083" LastEditDate="2019-05-24T10:59:41.817" LastActivityDate="2019-05-24T14:14:13.963" Title="What are &quot;PreK‐12th‐grade students&quot;?" Tags="|education-research|terminology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16660" PostTypeId="1" CreationDate="2019-05-24T10:51:43.377" Score="3" ViewCount="134" Body="&lt;p&gt;When talking to kids before they are taught complex numbers, I would really like to give some examples of why it will be exciting to learn them.  I am comfortable explaining the intellectual excitement and mystery but it would be great to combine that with real practical examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any really simple examples of tasks you can do, problems you can solve, using imaginary numbers but which you couldn't otherwise?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By way of comparison, you can motivate negative numbers (to even younger children) simply by talking about going backwards, or paying back money etc.&lt;/p&gt;&#xA;" OwnerUserId="10432" LastEditorUserId="2083" LastEditDate="2019-05-24T11:00:58.403" LastActivityDate="2019-05-24T11:00:58.403" Title="A compelling example of what complex numbers are for, before teaching them" Tags="|mathematical-pedagogy|applications|complex-numbers|" AnswerCount="0" CommentCount="6" ClosedDate="2019-05-24T19:29:05.447" ContentLicense="CC BY-SA 4.0" />
  <row Id="16661" PostTypeId="2" ParentId="16659" CreationDate="2019-05-24T10:53:25.710" Score="9" Body="&lt;p&gt;From the article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Grade level was a categorical factor consisting of three levels: &lt;strong&gt;preschool–kindergarten&lt;/strong&gt; students (or its equivalent if outside of the United States), &lt;strong&gt;1st‐ to 6th‐grade&lt;/strong&gt; students, and &lt;strong&gt;7th grade and above&lt;/strong&gt;. This variable evaluated whether the effectiveness of mathematics game‐based learning varied across different grade levels as to accommodate for a continued increase in difficulty of mathematics skills and decrease in student motivation to learn from &lt;strong&gt;preschool‐elementary&lt;/strong&gt; to &lt;strong&gt;middle‐high school&lt;/strong&gt; settings &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So &quot;PreK‐12th‐grade students&quot; means any student in school, preschool or kindergarten.&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="501" LastEditDate="2019-05-24T14:14:13.963" LastActivityDate="2019-05-24T14:14:13.963" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16662" PostTypeId="2" ParentId="16622" CreationDate="2019-05-26T15:08:05.840" Score="11" Body="&lt;p&gt;Briefly: The thing the OP seems to be orbiting around is that some mathematical symbols, in addition to indicating a given operation, also have the secondary function of serving as &amp;quot;grouping&amp;quot; symbols. These symbols chunk sub-expressions together, in the same way that parentheses do, without extra notation. That is: parentheses are just one type of grouping symbol. None of these are exceptions to the standard precedence syntax. Examples include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Fraction bars&lt;/li&gt;&#xA;&lt;li&gt;Radical overbars&lt;/li&gt;&#xA;&lt;li&gt;Absolute value bars&lt;/li&gt;&#xA;&lt;li&gt;Superscripts and subscripts&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-07-11T22:50:32.887" LastActivityDate="2023-07-11T22:50:32.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16663" PostTypeId="2" ParentId="16645" CreationDate="2019-05-26T15:21:11.137" Score="0" Body="&lt;p&gt;Among my top frustrated wishes are that there should be much more emphasis around the first year of algebra on directly connecting natural-language grammar to algebraic grammar. My students are extremely hazy on what counts as a statement or non-statement (expression/fragment) throughout the curriculum; including second-year math and computing majors at my community college. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, I wish there was solid drilling on the difference between an &quot;operation&quot;, which has the grammatical status of an adjective phrase (e.g.: arithmetic or set operations like intersection), and &quot;relations&quot;, which have the grammatical status of verbs. Without the latter we have a fragment, not a propositional statement that can be assessed as true or false. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-05-26T15:21:11.137" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16664" PostTypeId="2" ParentId="16602" CreationDate="2019-05-26T16:15:27.163" Score="2" Body="&lt;p&gt;Related to Joseph O'Rourke's cycloid idea, I've used the following to discuss the motion of a train wheel, part of which is always moving backwards when the train is moving forwards (which now has &lt;a href=&quot;https://math.stackexchange.com/questions/721935/vector-valued-function-what-part-of-the-train-is-always-moving-backward&quot;&gt;an explanation on MSE&lt;/a&gt;.  The length of the code is due to the ornamentation of the demonstration. The color-coded horizontal components of the velocity (which are constant at a given level above or below the rail) are plotted in the background; they are also plotted below for some of the actual positions of the point on the wheel.  There are some spokes and points on the wheel to help visualize the rolling motion when the &lt;code&gt;t&lt;/code&gt; slider is moved.  The distance &lt;code&gt;R&lt;/code&gt; of the traced point from the center is adjustable, too.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Manipulate[&#xA; With[{&#xA;   cyc0 =   (* full cycloid path, computed once for all t *)&#xA;     Table[{θ}~Join~cycloid[R, θ], {θ, 0, 3 π, π/18}],&#xA;   bg = {   (* background: vertical grid, horizontal arrows *)&#xA;     {Thin, LightGray,&#xA;      Line[&#xA;       Table[{{t0, -2.2}, {t0, 2.2}}, {t0, π/3, 3 π, π/3}]],&#xA;      Gray,&#xA;      Line[{{{0, -2.2}, {0, 2.2}}, {{2 π, -2.2}, {2 π, 2.2}},&#xA;        {{-2.1, 0}, {3 π + 2.1, 0}}}]},&#xA;     {Arrowheads[Small],&#xA;      Table[{Hue[(y + 1)/5, 0.7, 1],&#xA;        Arrow[&#xA;         Partition[&#xA;          Table[{t0, y}, {t0, &#xA;            If[y &amp;gt; -1, 0 - 5 π, 3 π + π], &#xA;            If[y &amp;lt; -1, 0 - 5 π, 3 π + π], (y + 1) π/6}],&#xA;          2, 1]]&#xA;        }, {y, {-2, -15/10, -5/10, 0, 5/10, 1, 15/10, 2}}]},&#xA;     Thickness[Medium], Line[{{{-2.1, -1}, {3 π + 2.1, -1}}}]}},&#xA;  Dynamic@Show[&#xA;    With[{   (* cycloid path 0 ≤ θ ≤ t *)&#xA;      cyc = &#xA;       Append[Cases[cyc0, x_?(#[[1]] &amp;lt;= t &amp;amp;) :&amp;gt; Rest[x]], cycloid[R, t]]},&#xA;     Graphics[{&#xA;       bg,&#xA;       Line[{{{-2.1, -1}, {3 π + 2.1, -1}}}],  (* rail *)&#xA;       With[{L = Max[R, 1], l = Min[R, 1]},    (* spokes *)&#xA;        {Lighter@Gray, &#xA;         Line[Table[{circle[{t, 0}, L, θ + t], &#xA;            circle[{t, 0}, -L, θ + t]}, {θ, 0, 2 π/3, π/3}]]}],&#xA;       {   (* rest of wheel *)&#xA;        Thick, Circle[{t, 0}], &#xA;        Thickness[Medium], Circle[{ t, 0}, R], PointSize[Medium], &#xA;        Point[Table[&#xA;          circle[{t, 0}, 1, θ + t], {θ, 0, 5 π/3, π/3}]],&#xA;        Line[{{ t, 0}, cycloid[R, t]}],&#xA;        Disk[{t, 0}, 1, {-t + π, -t + 5 π/6}], &#xA;        Disk[{t, 0}, 1, {-t, -t - π/6}]},&#xA;       {   (* Velocity vector plots below plot *)&#xA;        Arrowheads[Small],&#xA;        Table[{Hue[(R Sin[-π/2 - t0] + 1)/5, 0.7, 1],&#xA;          Arrow[With[{y0 = &#xA;                 If[#[[1, 1]] &amp;gt; #[[2, 1]], -2.4, &#xA;                  If[t0 &amp;gt; 2 π, -2.8, -2.6]]},&#xA;            {{#[[1, 1]], y0}, {#[[2, 1]], y0}}]&#xA;           ] &amp;amp;[{cycloid[R, t0 - π/12], cycloid[R, t0 + π/12]}]&#xA;          }, {t0, 0, t, π/6}]},&#xA;       {        (* cycloid *)&#xA;        {Thickness[0.008], Gray, Line[cyc]}, (* frames the cycloid *)&#xA;        {       (* leading point of the trace of the cycloid *)&#xA;         Hue[(R Sin[-π/2 - t] + 1)/5, 1, 1], EdgeForm[Black], &#xA;         Disk[cycloid[R, t], 0.075]},&#xA;        Thick,  (* the cycloid &amp;amp; mesh points *)&#xA;        Line[cyc, VertexColors -&amp;gt; (Hue[(#[[2]] + 1)/5] &amp;amp; /@ cyc)], &#xA;        Black, Point[cycloid[R, #] &amp;amp; /@ Range[0, t, π/6]], Gray, &#xA;        Point[cycloid[R, #] &amp;amp; /@ Range[π/12, t, π/6]]}&#xA;       }]&#xA;     ],&#xA;    ImageSize -&amp;gt; 600, &#xA;    PlotRange -&amp;gt; {{-2.1 , 3 π + 2.1}, {-2.9, 2.2}}&#xA;    ]&#xA;  ],&#xA; {t, 0., 3. π}, {{R, 2}, 0., 2.},&#xA; Initialization :&amp;gt; (&#xA;   cycloid[rad_, θ_] := {θ, 0} + &#xA;     rad {Cos[-π/2 - θ], Sin[-π/2 - θ]};&#xA;   circle[ctr_, rad_, θ_] := &#xA;    ctr + rad {Cos[-π/2 - θ], Sin[-π/2 - θ]};)&#xA; ]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/nTSgV.png&quot; alt=&quot;Mathematica graphics&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2019-05-27T03:19:51.280" LastActivityDate="2019-05-27T03:19:51.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16665" PostTypeId="1" AcceptedAnswerId="16670" CreationDate="2019-05-27T14:28:07.360" Score="6" ViewCount="266" Body="&lt;p&gt;A three by three Leslie matrix looks like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{bmatrix}&#xA;f_0 &amp;amp; f_1 &amp;amp; f_2 \\&#xA;s_0 &amp;amp; 0   &amp;amp; 0   \\&#xA;0   &amp;amp; s_1 &amp;amp; 0&#xA;\end{bmatrix},&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$f_0 \ge 0$&lt;/span&gt; and everything else is strictly positive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The application is in population dynamics; &lt;span class=&quot;math-container&quot;&gt;$f_0$&lt;/span&gt; tells how many children a zero-year old creature has on average, &lt;span class=&quot;math-container&quot;&gt;$f_1$&lt;/span&gt; how many a first year old has, and &lt;span class=&quot;math-container&quot;&gt;$f_2$&lt;/span&gt; how many a second year old has. &lt;span class=&quot;math-container&quot;&gt;$s_0$&lt;/span&gt; tells the proportion of zero year olds that survive to become one year old, &lt;span class=&quot;math-container&quot;&gt;$s_1$&lt;/span&gt; that of the one year olds that survive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to come up with nice and easy numbers for &lt;span class=&quot;math-container&quot;&gt;$f_j$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s_k$&lt;/span&gt; so that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f_0 = 0$&lt;/span&gt; or a small fraction&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f_1 &amp;gt; 1$&lt;/span&gt;, maybe around 5, and an integer or a friendly fraction.&lt;/li&gt;&#xA;&lt;li&gt;Ideally &lt;span class=&quot;math-container&quot;&gt;$1 \le f_2 &amp;lt; f_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f_2$&lt;/span&gt; is also an integer or a friendly fraction.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; s_0, s_1 &amp;lt; 1$&lt;/span&gt; and they should also be friendly fractions. They should not be microscopically small, and probably not very close to 1, either.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;By a &quot;friendly fraction&quot; I mean a fraction which looks friendly to a student, rather being a horrible mess or intimidating. The other restrictions and wishes are to make the model correspond to the application.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How do I choose the parameters so that the eigenvalues are nice and friendly, or at least so that the highest eigenvalue is friendly?&lt;/strong&gt; Preferably, the highest eigenvalue would be greater than one, as the population is roughly &lt;span class=&quot;math-container&quot;&gt;$\lambda_3^t$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\lambda_3$&lt;/span&gt; is the greatest eigenvalue and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is the time. Given the model, this should be achievable with the parameters at roughly the desired values, though whether the values can be friendly is a different question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The eigenvalues are, modulo miscalculations, the zeroes of the polynomial&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;-\lambda^3 +f_0 \lambda^2 +f_1 s_0 \lambda + f_2 s_0 s_1.&#xA;$$&lt;/span&gt;&#xA;The second order term vanishes if one takes &lt;span class=&quot;math-container&quot;&gt;$f_0 = 0$&lt;/span&gt;, which one can do if one wishes to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There exists a closed formula for the solutions, but I do not immediately see how to proceed from there to choosing nice parameters. See the Cardano formula at Wikipedia: &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubic_function#Cardano_formula&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Cubic_function#Cardano_formula&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some intuition on the model, suppose &lt;span class=&quot;math-container&quot;&gt;$f_0 = f_2 = s_1 = 0$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$s_0$&lt;/span&gt; tells the proportion of animals that survive the winter, while &lt;span class=&quot;math-container&quot;&gt;$f_1$&lt;/span&gt; tells how much each procreates (on average); then all of them die. Hence, the yearly growth should be multiplication by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{f_1 s_0}$&lt;/span&gt; and the eigenvalues tell the same story.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-05-29T07:31:02.480" LastActivityDate="2019-05-30T16:22:42.623" Title="How to come up with a Leslie matrix with convenient eigenvalues?" Tags="|linear-algebra|applied-mathematics|solving-polynomials|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16667" PostTypeId="1" AcceptedAnswerId="16690" CreationDate="2019-05-28T00:28:51.943" Score="5" ViewCount="180" Body="&lt;p&gt;Are there any mobile phone apps that would allow for students to anonymously &quot;buzz&quot; me in real time? What I have visualized is a free app students can download that features a button they can press that will signal me by making a light blink on a website. A noise might be distracting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would be especially suitable during the days where I teach more difficult topics. On certain days I anticipate more, maybe smaller confusions. Only some students feel comfortable asking questions, and those that do often will limit themselves to one or two. &lt;/p&gt;&#xA;" OwnerUserId="12329" LastActivityDate="2019-06-03T17:10:58.120" Title="Mobile phone apps that would allow for students to anonymously &quot;buzz&quot; me in real time?" Tags="|technology-in-education|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16668" PostTypeId="1" CreationDate="2019-05-28T05:22:44.340" Score="7" ViewCount="1554" Body="&lt;p&gt;I'm asking about definite integrals that can effortlessly be found numerically by high schoolers using software. For example,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/q/562694/53259&quot;&gt;This link&lt;/a&gt; shows the numerical integration done by software.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/&quot; rel=&quot;noreferrer&quot;&gt;This Reddit comment&lt;/a&gt; substantiates that the exact solution is knotty, but not why:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least &lt;em&gt;be understandable&lt;/em&gt; had you taken Calc I-III and Complex Analysis. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How can I explain to someone who is just getting started in calculus why it's so hard to find the exact solution by symbolic integration, when it's so trivial to get a numerical answer using software?&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user507" LastEditDate="2019-05-30T13:13:53.263" LastActivityDate="2020-01-17T04:04:53.700" Title="How can I explain why numerical integration is easy, but symbolic integration is hard?" Tags="|integration|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16669" PostTypeId="2" ParentId="16668" CreationDate="2019-05-28T20:56:23.387" Score="1" Body="&lt;p&gt;This reminds me of &lt;a href=&quot;http://eulerarchive.maa.org/pages/E695.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;one of Euler's papers&lt;/a&gt;, where he evaluates (not the definite integral, but indefinite integral) &lt;span class=&quot;math-container&quot;&gt;$$\int\frac{dz}{(3\pm z^2)\sqrt[3]{1\pm 3z^2}}$$&lt;/span&gt; using an amazing variety of substitutions and other tricks, among other things the formula &lt;span class=&quot;math-container&quot;&gt;$\arctan(x)-\arctan(y) = \arctan\left(\frac{x-y}{1+xy}\right)$&lt;/span&gt;.  (In fact, he evaluates it two different ways!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you can feel perfectly free to say that one the greatest mathematicians of all time struggled with various unusual integrals, even of the type which are actually doable with elementary antiderivatives, and then published papers about them.  Maybe we don't do that as much today.  And that apparently he also thought it wasn't evident which ones are easy and which ones aren't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But far from being a discouragement, this should be instead freedom for a student.  Use creativity, and see what you get!  Use all the tricks you have access to now, acquire more later, and keep trying.  And these are good thoughts for life; know what you've got, have a growth learning mindset to get more, and persevere.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-05-28T20:56:23.387" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16670" PostTypeId="2" ParentId="16665" CreationDate="2019-05-28T23:22:43.713" Score="8" Body="&lt;p&gt;If I use your simplification that &lt;span class=&quot;math-container&quot;&gt;$f_0 = 0$&lt;/span&gt;, then I suggest just choosing a real eigenvalue &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; and writing out the relation for the other parameters:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$-\lambda^3+f_1s_0\lambda + f_2s_0s_1 = 0$$&lt;/span&gt;&#xA;Now isolate a parameter, say &lt;span class=&quot;math-container&quot;&gt;$s_1$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$s_1 = \frac{\lambda^3-\lambda f_1 s_0}{f_2 s_0}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then just find values that satisfy your requirements. It should be clear that choosing nice values of these parameters (especially &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;) will help to make &lt;span class=&quot;math-container&quot;&gt;$s_1$&lt;/span&gt; &quot;friendly&quot;. Also, because of real-world constraints, you'll need &lt;span class=&quot;math-container&quot;&gt;$\lambda^3 &amp;gt; \lambda f_1 s_0$&lt;/span&gt; (so &lt;span class=&quot;math-container&quot;&gt;$s_1$&lt;/span&gt; will be positive) and for &lt;span class=&quot;math-container&quot;&gt;$f_2 s_0 &amp;gt; \lambda^3 - \lambda f_1 s_0$&lt;/span&gt; (so &lt;span class=&quot;math-container&quot;&gt;$s_1$&lt;/span&gt; will be a proper fraction). Since you want &lt;span class=&quot;math-container&quot;&gt;$f_1 &amp;gt; f_2$&lt;/span&gt;, we will need the numerator to be fairly small.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Messing around, here are a few examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Choose &lt;span class=&quot;math-container&quot;&gt;$\lambda = 2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f_1 = 5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s_0 = \frac{3}{4}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This makes the numerator &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;. Now, &lt;span class=&quot;math-container&quot;&gt;$f_2$&lt;/span&gt; can be any integer greater than zero, so I choose &lt;span class=&quot;math-container&quot;&gt;$f_2 = 2$&lt;/span&gt;, and we get &lt;span class=&quot;math-container&quot;&gt;$s_1 = \frac{1}{3}$&lt;/span&gt;, and our matrix becomes:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left[\begin{matrix}&#xA;0 &amp;amp; 5 &amp;amp; 2\\&#xA;\frac{3}{4} &amp;amp; 0 &amp;amp; 0\\&#xA;0 &amp;amp; \frac{1}{3} &amp;amp; 0&#xA;\end{matrix}\right]$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And we can confirm that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left[\begin{matrix}&#xA;16\\&#xA;6\\&#xA;1&#xA;\end{matrix}\right]$$&lt;/span&gt;&#xA;is the eigenvector for &lt;span class=&quot;math-container&quot;&gt;$\lambda = 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Just because I tried this one out and it didn't quite work:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\lambda = 3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f_1 = 5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s_0 = \frac{6}{10}$&lt;/span&gt;. This makes the numerator &lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$f_2$&lt;/span&gt; must be larger than &lt;span class=&quot;math-container&quot;&gt;$30$&lt;/span&gt; in order to have a positive value of &lt;span class=&quot;math-container&quot;&gt;$s_1$&lt;/span&gt;. Therefore, you'll have to be thoughtful concerning the sizes of the parameters you choose.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-05-29T01:07:18.147" LastActivityDate="2019-05-29T01:07:18.147" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16671" PostTypeId="2" ParentId="16668" CreationDate="2019-05-29T01:21:29.860" Score="9" Body="&lt;p&gt;It might be fun to have your students pretend that the only functions they know are sums of monomials &lt;span class=&quot;math-container&quot;&gt;$cx^n$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$n\in{\bf Z}$&lt;/span&gt;, and in particular, play like they know nothing about the function &lt;span class=&quot;math-container&quot;&gt;$f(x)=\ln(x)$&lt;/span&gt;. You see where this is going: we can easily graph &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^n$&lt;/span&gt; between &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, and find an antiderivative with the one annoying exception of &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^{-1}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what is done in some developments of calculus? We &lt;em&gt;define&lt;/em&gt; a new function &lt;span class=&quot;math-container&quot;&gt;$g(x)=\int_{1}^x \frac{1}{t}\, dt$&lt;/span&gt; precisely so that we can have an antiderivative for the situation where we were previously stuck, and give &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; the special name &lt;span class=&quot;math-container&quot;&gt;$\ln(x).$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that integration, unlike differentiation, takes one outside the function type of the integrand, e.g., the derivatives of rational functions are rational functions, but their antiderivatives might not be. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wrap up the lesson with the xkcd comic found here &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://xkcd.com/2117/&quot; rel=&quot;noreferrer&quot;&gt;https://xkcd.com/2117/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-05-29T01:21:29.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16673" PostTypeId="1" CreationDate="2019-05-29T08:49:29.203" Score="4" ViewCount="604" Body="&lt;p&gt;I was very unsatisfied with how I taught Calc III a couple years ago, and this summer I have to do it again. Are there any general resources for teaching this course? It seems like there should be, since it's a very standard thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular I'm interested in tools to show students visual graphics so they can really see what's happening in spaces we work in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I'm also interested in general insight on how to reach students at this level. Any tips or tricks are welcome. Anything you can point me to.&lt;/p&gt;&#xA;" OwnerUserId="12332" LastEditorUserId="2083" LastEditDate="2019-05-29T09:57:41.307" LastActivityDate="2019-05-31T06:04:07.997" Title="Resources for teaching calc III" Tags="|calculus|resource-request|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="16674" PostTypeId="2" ParentId="16668" CreationDate="2019-05-30T06:38:37.413" Score="1" Body="&lt;p&gt;One way of thinking about it (although I'm not sure whether it would be helpful for the student of interest to you):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are you doing when you 'graph' the function? At each point on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis, you calculate the value of the function, and plot that on the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis. That is something that can be done at each value. In practice, we start by knowing this is a 'reasonably nice' function (ie mostly continuous), so you can get away with finding the value at finitely many points and either stopping (on a computer screen) or joining the dots (in your head).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, what are you doing when you 'integrate' the function? Finding the area under the graph. Well, sort of. Finding the area under the graph can be done. An integral does exist (ignoring issues about measure etc). That's a point students don't always appreciate. There &lt;em&gt;is&lt;/em&gt; an integral, and it is &lt;em&gt;some&lt;/em&gt; function. The problem is how to write down what that function is. When a student asks 'can I integrate this?' what they really mean is 'can I write the integral of this function in a reasonably nice formula using functions I am familiar with?' &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think phrased that way, it becomes clear why integration is 'harder' than graphing, which should be sufficient explanation for most 17-year-olds. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Now I've written it out, I can see that this is similar to the point made by user52817 .)&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-05-30T06:38:37.413" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16676" PostTypeId="2" ParentId="16668" CreationDate="2019-05-30T13:06:56.903" Score="3" Body="&lt;p&gt;When students learn integration, it's usually the first time they've ever encountered a type of mathematical calculation for which there are techniques, but no general algorithm that always succeeds. A good way of explaining why differentiation is rule-based, but integration is not, is to make the analogy with discrete, finite sums.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to evaluate the sum &lt;span class=&quot;math-container&quot;&gt;$f(n)=1+2+3+\ldots+n$&lt;/span&gt;, you can easily find the result for any given &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, but it's harder to find a general formula for &lt;span class=&quot;math-container&quot;&gt;$f(n)$&lt;/span&gt;. OK, so next show the student how to visualize triangles of dots and use geometrical reasoning to find that the answer is &lt;span class=&quot;math-container&quot;&gt;$n(n+1)/2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But you can now demonstrate that going the opposite way is just a straightforward calculation of &lt;span class=&quot;math-container&quot;&gt;$f(n)-f(n-1)$&lt;/span&gt;, with a simplification to show that the result is &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. This rule-based calculation is like a derivative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in your example of using software to numerically evaluate a definite integral, what the software is doing is essentially like finding that &lt;span class=&quot;math-container&quot;&gt;$1+2+3=6$&lt;/span&gt;. That's straightforward and rule-based. But that doesn't mean that we can find the general formula &lt;span class=&quot;math-container&quot;&gt;$n(n+1)/2$&lt;/span&gt; in the same rule-based way.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-05-30T13:16:40.477" LastActivityDate="2019-05-30T13:16:40.477" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16677" PostTypeId="2" ParentId="16665" CreationDate="2019-05-30T16:22:42.623" Score="4" Body="&lt;ol&gt;&#xA;&lt;li&gt;Choose three numbers &lt;span class=&quot;math-container&quot;&gt;$\lambda_1,\lambda_2,\lambda_3$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Compute the coefficients &lt;span class=&quot;math-container&quot;&gt;$a,b,c$&lt;/span&gt; of the polynomial &lt;span class=&quot;math-container&quot;&gt;$(x-\lambda_1)(x-\lambda_2)(x-\lambda_3) = x^3-ax^2-bx-c$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Choose &lt;span class=&quot;math-container&quot;&gt;$s_i$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f_i$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f_0=a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f_1s_0=b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f_2s_0s_1=c$&lt;/span&gt;. It shouldn't be hard to play around with the coefficients so that those inequalities are satisfied, since you have so many degrees of freedom.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you want to have &lt;span class=&quot;math-container&quot;&gt;$f_0=0$&lt;/span&gt;, you'll need to choose the numbers so that &lt;span class=&quot;math-container&quot;&gt;$\lambda_1+\lambda_2+\lambda_3=0$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4930" LastActivityDate="2019-05-30T16:22:42.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16678" PostTypeId="2" ParentId="16673" CreationDate="2019-05-30T23:05:23.467" Score="10" Body="&lt;p&gt;Since no one's posted an answer I will get things started with some general advice. Calculus 3 is my favorite course to teach, but it can be a bear to wrestle with the first few times. Some more specific information about your situation would be helpful (are you at Community College? Ivy League? Are the students Engineers? Math Majors? Is the class 20 students? 200?) but here is some general advice I can give you about the lecture portion of the course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;General Advice:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This is a course about the interplay between geometry and analysis, you need to focus on both. &lt;/li&gt;&#xA;&lt;li&gt;Assign and grade homework. There is so much content students &lt;em&gt;must&lt;/em&gt; do it progressively to keep up. &lt;/li&gt;&#xA;&lt;li&gt;There are some good open source books out right now (APEX for example) but if you’re struggling to teach the course just use Stewart, it has good examples and proofs throughout and by far the most interesting selection of problems. &lt;/li&gt;&#xA;&lt;li&gt;You need to do examples, you need to do them carefully, but you won't have time to do more than 1 or 2 per class, pick them wisely and don’t be afraid to use the examples in the book. &lt;/li&gt;&#xA;&lt;li&gt;You can teach a semester long course without getting to Stokes Theorem, but if you need to get to Stokes Theorem, cut at the end, not the beginning. Your students have to know what a vector space is and how to perform differentiation and integration.&lt;/li&gt;&#xA;&lt;li&gt;Lagrange multipliers are going to confuse the hell out of them because they don’t know multivariable algebra. Do a lot of problems connecting the math to the geometry, but also remind them of the pitfalls (IE, if you divide an equation by &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt; is also a solution). Give them a road map as much as possible.&lt;/li&gt;&#xA;&lt;li&gt;Display your reasoning, don’t be afraid to make mistakes and fix them live. The challenge in Calc III is understanding a problem well enough to know what kind of math solves it, not plugging numbers into a formula.&lt;/li&gt;&#xA;&lt;li&gt;Always leave time for questions. If you’re not getting them in Calc III you’re (almost certainly) going too fast.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Visualization:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Computer: Use Geogebra, MATLAB, Mathematica or whatever you're comfortable with. &lt;/li&gt;&#xA;&lt;li&gt;Board: You need to be able to draw a minimum, a maximum, one general space, a saddle and a sphere. Practice those and you’ll be fine, they’ll be your work horses. Always add your axes last when you’re freehanding. &lt;/li&gt;&#xA;&lt;li&gt;Add animations if you can, and some visualizations that show the interplay between the numbers, formulas and the geometry. If you're not a strong coder check out &lt;a href=&quot;https://savkar.math.uconn.edu/calculus-3-visuals/&quot; rel=&quot;noreferrer&quot;&gt;https://savkar.math.uconn.edu/calculus-3-visuals/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Other Resources:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Pauls Notes &lt;a href=&quot;http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx&quot; rel=&quot;noreferrer&quot;&gt;http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx&lt;/a&gt; is great, cross reference with it and let your class know about it. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;MIT’s Open Courseware is always worth watching &lt;a href=&quot;https://www.youtube.com/watch?v=PxCxlsl_YwY&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=PxCxlsl_YwY&lt;/a&gt;. I’ve often found it gives me ideas for explaining content different than my own intuitions. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="376" LastEditDate="2019-05-31T06:04:07.997" LastActivityDate="2019-05-31T06:04:07.997" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16679" PostTypeId="2" ParentId="16636" CreationDate="2019-06-01T14:21:03.063" Score="2" Body="&lt;p&gt;If you keep a running tally of primes of the form &lt;span class=&quot;math-container&quot;&gt;$4k+1$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$4k+3$&lt;/span&gt;, it seems like the latter are always at least as numerous. But it was eventually shown each type takes the lead infinitely many times; it's just you have to get into huge numbers before the first lead for &lt;span class=&quot;math-container&quot;&gt;$4k+1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5794" LastActivityDate="2019-06-01T14:21:03.063" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16680" PostTypeId="2" ParentId="16668" CreationDate="2019-06-01T14:29:42.500" Score="8" Body="&lt;p&gt;You should expect numerical integration to be &quot;easier&quot; [1] than symbolic integration because it is answering a fundamentally weaker question. That is, symbolic integration, if you can do it, gives you an infinitely precise answer. Numerical integration gives you an answer which is approximate, ideally with some bound on the amount of error.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is like the difference between &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1.414 \pm 0.0003$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1]: You might not find numerical integration quite as easy if you were to do it by hand. If you level the playing field and allow the computer for symbolic integration, it can often be pretty easy too. &lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2019-06-01T14:29:42.500" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16681" PostTypeId="1" CreationDate="2019-06-02T03:54:48.373" Score="7" ViewCount="516" Body="&lt;p&gt;I was working on &lt;a href=&quot;https://math.stackexchange.com/questions/3248162/i-need-help-with-proving-vector-space-given-the-operations-confused-by-complex/3248165?noredirect=1#comment6679998_3248165&quot;&gt;this question&lt;/a&gt; on math, where (among other things), the OP was asked to prove that &lt;span class=&quot;math-container&quot;&gt;$$x \oplus y=\sqrt[3]{x^3+y^3}$$&lt;/span&gt; is associative. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After some prompting, the offered proof was &lt;span class=&quot;math-container&quot;&gt;$$\begin{align} x \oplus (y \oplus z) &amp;amp;= (x \oplus y) \oplus z\\ \\ &amp;amp; \implies x \oplus\sqrt[3]{y^3 + z^3}\\ \\ &amp;amp; \implies \sqrt[3]{x^3 + (\sqrt[3]{y^3+z^3})^3}\\ \\ &amp;amp; \implies \sqrt[3 ]{x^3 + y^3 + z^3} \\ \\ &amp;amp;\implies  \sqrt[3]{x^3 + y^3} \oplus z \\ \\ &amp;amp;\implies(x \oplus y) \oplus z\end{align}$$&lt;/span&gt;&lt;br&gt;&#xA;How big of a sin is this?  The algebra is right, but the implications start with what is to be proved.  If we just deleted the conclusion at the start and changed the implications to equalities, I would be happy.  As it is we start with what is to be proved and have implications between expressions, not between sentences.  I have seen this kind of write-up a number of times and would like to know how to advise the poster.&lt;/p&gt;&#xA;" OwnerUserId="12340" LastEditorUserId="238" LastEditDate="2019-06-03T17:16:28.477" LastActivityDate="2019-06-07T21:23:55.343" Title="Writing up a proof that assumes what is to be proven?" Tags="|proofs|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16682" PostTypeId="2" ParentId="15262" CreationDate="2019-06-02T07:51:20.100" Score="-1" Body="&lt;p&gt;Audin's book is great but may require a little effort I suggest &lt;strong&gt;FOUR PILLERS OF GEOMETRY - JOHN STILLWELL&lt;/strong&gt;&#xA;apart from transformation group it will give broad aspect of non-euclidean geometry. &lt;/p&gt;&#xA;" OwnerUserId="12341" LastActivityDate="2019-06-02T07:51:20.100" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16683" PostTypeId="2" ParentId="16681" CreationDate="2019-06-02T07:56:08.463" Score="4" Body="&lt;p&gt;I think there are two problems that have to be addressed individually:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Wrong usage of implications. Implications just don't describe a relation between terms of the kind that appear here. They can be used between things that have a truth value such as equations or boolean expressions. I'd say there is no way other than fixing this to use only the correct symbols. &lt;a href=&quot;https://matheducators.stackexchange.com/questions/16645/category-mistakes-regarding-symbols-and-their-impact-on-math-mis-understanding&quot;&gt;This Question about symbols and category errors seems closely related&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Starting with what is to prove. This might be OK depending on the target audience and if the relations linking the individual steps are symmetric (i.e. equivalence or equality), so the order could be arranged so that the goal of the proof appears at the end. The Question at Mathexchange sounds as if OP there is in some sort of beginner algebra course. I hope that proper proof etiquette is taught there, so for this specific case, I'd also say that &quot;starting with the goal&quot; is also wrong.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Edit: I now realize that once the first point is taken care of, the second one becomes a non-issue because the proof then will be a chain of equations. I'd still mention the second point as a general advice. &lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2019-06-02T08:56:00.507" LastActivityDate="2019-06-02T08:56:00.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16684" PostTypeId="1" CreationDate="2019-06-02T10:32:52.523" Score="4" ViewCount="682" Body="&lt;ol&gt;&#xA;&lt;li&gt;How many &lt;strong&gt;hours&lt;/strong&gt; does it take for the average child to memorize the &lt;span class=&quot;math-container&quot;&gt;$10\times 10$&lt;/span&gt; &lt;strong&gt;addition&lt;/strong&gt; table?&lt;/li&gt;&#xA;&lt;li&gt;How many &lt;strong&gt;school years&lt;/strong&gt; does it take for the average child to memorize the &lt;span class=&quot;math-container&quot;&gt;$10 \times 10$&lt;/span&gt; &lt;strong&gt;addition&lt;/strong&gt; table?&lt;/li&gt;&#xA;&lt;li&gt;How many &lt;strong&gt;hours&lt;/strong&gt; does it take for the average child to memorize the &lt;span class=&quot;math-container&quot;&gt;$10\times 10$&lt;/span&gt; &lt;strong&gt;multiplication&lt;/strong&gt; table?&lt;/li&gt;&#xA;&lt;li&gt;How many &lt;strong&gt;school years&lt;/strong&gt; does it take for the average child to memorize the &lt;span class=&quot;math-container&quot;&gt;$10 \times 10$&lt;/span&gt; &lt;strong&gt;multiplication&lt;/strong&gt; table?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12342" LastEditorUserId="238" LastEditDate="2019-06-03T13:14:17.923" LastActivityDate="2019-06-03T20:48:56.777" Title="How many hours / school years does it take for the average child to memorize the $10\times 10$ addition and multiplication tables?" Tags="|primary-education|multiplication-table|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16685" PostTypeId="2" ParentId="16684" CreationDate="2019-06-02T14:41:48.607" Score="4" Body="&lt;p&gt;Assuming that the German &quot;Kerncurriculum&quot; (actually, there are 16 of them, one for each federal state. I'm referring to the one from Lower Saxony) for primary school is at least somewhat tailored for the &quot;average person&quot;: at most two years, because at the end of second grade, pupils are supposed to&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;state the addition table (up to 10) and confidently derive the inverse operations &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;state the multiplication table (up to 10), confidently derive the inverse operations  and the results of further problems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;While I have no idea what the last part (&quot;further problems&quot;) is supposed to mean, the additions &quot;derive inverse operations&quot; is a hint that the tables should be thoroughly memorized by the end of second grade. Because I think that the memorization does not start straightaway from day one, less than two years is a reasonable estimate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sources for the translated expectations:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Die Schülerinnen und Schüler geben die Zahlensätze des kleinen 1 + 1 automatisiert wieder und leiten deren Umkehrungen sicher ab. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Die Schülerinnen und Schüler geben die Kernaufgaben des kleinen 1 x 1 automatisiert wieder und leiten deren Umkehrungen und die Ergebnisse weiterer Aufgaben ab. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Kerncurriculum für die Grundschule, Mathematik, Niedersachsen, 2006.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2019-06-02T14:41:48.607" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16686" PostTypeId="2" ParentId="16681" CreationDate="2019-06-02T18:18:34.613" Score="11" Body="&lt;p&gt;I always provide the following example whenever a student assumes what they want to prove:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Suppose 0=1.&lt;/li&gt;&#xA;&lt;li&gt;Then 1=0 must be true.&lt;/li&gt;&#xA;&lt;li&gt;Then we can add both equations to deduce that 1=1.&lt;/li&gt;&#xA;&lt;li&gt;This is a true statement (1=1) and therefore our original assumption was valid, so 0=1.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In our &quot;Intro to Proofs&quot; course for math majors, I show this on the first day of class. Any time later in the semester, either during class or on a homework submission, when someone attempts to assume the desired conclusion, I start to write this proof on the board. Eventually, it gets to the point where I write &quot;0=1&quot; and a student says, &quot;Okay, okay, yeah yeah, I get it.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it's beneficial to have a &quot;standard incorrect example&quot; like this. I get the feeling that students learn to catch this mistake themselves by thinking about this very example.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I don't usually get into it further, but I'm looking at @Jasper's answer where they mention &quot;wrong usage of implications&quot; and thinking that this example could spark further discussion. You can point out that this is a &quot;valid proof&quot; in the sense that &lt;em&gt;if it really were true that 0=1&lt;/em&gt; then we could deduce that 1=1. However, this is separate from whether or not 1=1 is true on its own. Moreover, you can discuss whether any of these &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; implications are reversible. Specifically, in my example as outlined above, (1) and (2) imply (4) (while (3) is just stating what is being done in that step). However, one cannot take (4) and deduce that both (1) and (2) are true. That is the implication whose converse is false, and this example can be used to motivate and explain all of these concepts and terminology.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2019-06-02T18:18:34.613" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16687" PostTypeId="2" ParentId="15262" CreationDate="2019-06-02T23:17:25.157" Score="3" Body="&lt;p&gt;Permit me to endorse @halfbloodprince's recommendation.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/EJKUp.jpg&quot; width=&quot;180&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;Stillwell's four pillars are:&#xA;Euclidian straight-edge/compass constructions,&#xA;linear algebra &amp;amp; coordinates, projective geometry,&#xA;transformational groups &amp;amp; non-Euclidean geometry.&#xA;The most important aspect is that he views geometry from&#xA;several different viewpoints, which to my mind is much more effective than  focussing solely on one viewpoint.&#xA;This allows him to show how algebra emerges from projective geometry,&#xA;and how projective geometry leads to the hyperbolic plane.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-06-02T23:17:25.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16688" PostTypeId="2" ParentId="16681" CreationDate="2019-06-03T02:01:36.597" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How big a sin is this?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It's flat-out wrong and nonsensical for multiple reasons (as you point out here). The person writing those things is in need of correction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that similarly, the person in question also has serious problems writing standard-grammar English sentences. E.g.: Inconsistent use of a capital for the personal pronoun &quot;I&quot;, inconsistent capitals to start sentences, inconsistent use of punctuation to end a sentence, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They seem to be aware that they're a beginner and have written comments such as, &quot;It was a typo, i'm pretty tired and determined to understand this&quot;. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-06-03T02:01:36.597" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16690" PostTypeId="2" ParentId="16667" CreationDate="2019-06-03T17:10:58.120" Score="3" Body="&lt;p&gt;You might just want to use a game buzzer app. There are tons of them for jeopardy like games/pub quizes that you can find online. Students have to sign up but they don’t have to use their real names. Looking around a bit I found the following free version&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://buzzin.live/play&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://buzzin.live/play&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which you can use in a browser. It’s very simple but I played around with a bit and it does everything you need it to do. There’s probably something more robust out there but I haven’t used any other myself and so don’t want to recommend. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’ve thought about using such a system for students to tell me to speed up or slow down, but for the life of me I cant find a free multiple choice version. &lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2019-06-03T17:10:58.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16691" PostTypeId="2" ParentId="16684" CreationDate="2019-06-03T20:48:56.777" Score="5" Body="&lt;p&gt;When/where I grew up in California, ordinary second-graders (7 - 8 year-olds) were expected to learn the times tables well enough to answer randomly chosen problems without writing anything down or looking anything up; but 100 percent accuracy was more of a hope than an expectation.  Flashcards, mental shortcuts, and other memorization aids were used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the end of fourth grade (10 year-olds), above-average students &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7283#7292&quot;&gt;were expected&lt;/a&gt; to be able to see, recognize, and write down the correct answer to randomly chosen times table problems in 2 - 3 seconds with 97 - 100 percent accuracy.  The problems were randomly chosen between 0 x 0 and 12 x 12.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2019-06-03T20:48:56.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16693" PostTypeId="1" CreationDate="2019-06-04T11:38:38.147" Score="2" ViewCount="279" Body="&lt;p&gt;I am struggling with that book which I find to be more of second-guessing type than a book for self-study: it has cryptically written sections, no examples (and those given, and rarely, are even more cryptic than the text they are supposed to illustrate), with chapter VIII being a good example of abysmal indigestion...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good introductions are found in the online series: Socratica's and Macauley's. The Harvard online series uses another book...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question: Do you know of any online series on abstract algebra that uses MacLane's &lt;em&gt;Algebra&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerUserId="12346" LastEditorUserId="8571" LastEditDate="2019-06-04T16:36:38.250" LastActivityDate="2019-06-04T16:36:38.250" Title="Online open-course-ware that uses Maclane's book &quot;Algebra&quot;" Tags="|graduate-education|abstract-algebra|online-instruction|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16694" PostTypeId="1" AcceptedAnswerId="16695" CreationDate="2019-06-04T20:30:09.803" Score="8" ViewCount="232" Body="&lt;p&gt;When we encounter &quot;real world problems&quot; in math, one can chose different levels of detail with regard to units: from leaving them out completely up to using them everywhere. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd argue that both extremes have their merits and inconsistent usage of units (using them in some places but not always) is wrong and leads to confusion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, the most important aspect is that units are great in telling you if a calculation makes sense - and more often than not, a calculation is right if and only if the units are right. For example, due to the units alone it is evident that distance has to be divided by time to get speed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, units can get hard, for example if you're solving for time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there scientific research or results that is pro/contra using units in math? &lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2019-06-05T17:49:50.093" LastActivityDate="2019-06-05T17:49:50.093" Title="Scientific results on the usefulness of physical units in secondary education?" Tags="|secondary-education|physical-sciences|meta-cognition|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16695" PostTypeId="2" ParentId="16694" CreationDate="2019-06-04T23:10:57.720" Score="10" Body="&lt;p&gt;This is not exactly what you seek, but at least you can find educational research articles&#xA;under the key phrase &lt;em&gt;dimensional analysis&lt;/em&gt;. Below [1] says it's useful, [2] questions that usefulness. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[1] Hrayr Ohanyan. &quot;The Application of the Method of Dimensional Analysis When Solving Problems.&quot;&#xA;  &lt;em&gt;American Journal of Educational Research&lt;/em&gt;.&#xA;  Vol. 4, No. 1, 2016, pp 5-7.&#xA;  &lt;a href=&quot;http://pubs.sciepub.com/education/4/1/2/index.html&quot; rel=&quot;noreferrer&quot;&gt;Journal link&lt;/a&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[2] Canagaratna, Sebastian G. &quot;Is dimensional analysis the best we have to offer?&quot; &lt;em&gt;Journal of Chemical Education&lt;/em&gt; 70, no. 1 (1993): 40.&#xA;  &lt;a href=&quot;https://pubs.acs.org/doi/10.1021/ed070p40&quot; rel=&quot;noreferrer&quot;&gt;Journal link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And this is interesting: &lt;a href=&quot;https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem&quot; rel=&quot;noreferrer&quot;&gt;Buckingham's &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; theorem&lt;/a&gt;.&#xA;It states that, if there are &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; physical variables in a &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-dimensional equation,&#xA;then the equation can be rewritten in &lt;span class=&quot;math-container&quot;&gt;$n-k$&lt;/span&gt; dimensionless parameters.&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem#Speed&quot; rel=&quot;noreferrer&quot;&gt;For example&lt;/a&gt;,&#xA;if we have an equation in the &lt;span class=&quot;math-container&quot;&gt;$n=3$&lt;/span&gt; variables&#xA;&lt;span class=&quot;math-container&quot;&gt;$\{$&lt;/span&gt;distance &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, velocity &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, and time &lt;span class=&quot;math-container&quot;&gt;$t$$\}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$k=2$&lt;/span&gt;&#xA;(&lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; &amp;amp; &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;), and the &lt;span class=&quot;math-container&quot;&gt;$n-k=1$&lt;/span&gt; dimensionless parameter is &lt;span class=&quot;math-container&quot;&gt;$\pi_1 = t v / d$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-06-05T11:37:22.220" LastActivityDate="2019-06-05T11:37:22.220" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16696" PostTypeId="2" ParentId="16681" CreationDate="2019-06-06T22:32:17.253" Score="4" Body="&lt;p&gt;This sort of writing is quite common, in my experience, among young mathematicians (in their early years of college) who have had little feedback or training in formal mathematical writing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first line is simply a statement of the problem, boiled down to its algebraic bones, so to speak.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the arrow, which in mathematical writing is read as an implication (or &quot;approaches&quot; in the case of a single arrow), students usually mean something like &quot;the next step is....&quot;  The arrow is a nice, natural indication of the order in which they thought up the steps and you should read them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no real sentence structure here, and probably the student is not accustomed to writing out their thinking.  Instead they've written down only the expressions they've arrived at by means of their thinking.  This is how they interpret the instruction &quot;show all steps&quot; from their high school and perhaps early college days.  I gather it is accepted in many high schools by how often my first-year students write in this way.  I don't have a good feel for how often it is accepted in college.  I endeavor to get my students to write correctly, but some of these habits are so ingrained that they are difficult for them to kick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reading the proof as a relatively easy puzzle given as a sequence of hints, one might find it a innocuous pastime.  The main problem is when there is an error in their thinking and one cannot easily fill in the gaps, but that is not an issue here.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that the proof came from a comment, I might have written, if I wanted to address their writing, &quot;I think this reads better: &lt;span class=&quot;math-container&quot;&gt;$x \oplus (y \oplus z) = x \oplus\sqrt[3]{y^3 + z^3} =$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{x^3 + (\sqrt[3]{y^3+z^3})^3} =$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3 ]{x^3 + y^3 + z^3} =$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{x^3 + y^3} \oplus z =$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$(x \oplus y) \oplus z$&lt;/span&gt;,&quot; and leave it to them the appreciate the difference.  Perhaps add, &quot;The proper use of the equal sign makes clearer the connection between the steps.&quot;  I guess at some point I'd wonder how much effort would be effective trying to help the user write mathematics better via StackExchange comments.  The user might have been writing the comment less formally than on an exam, say.  In class, in which I see the students for many weeks, I take a different approach.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2019-06-06T22:32:17.253" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16697" PostTypeId="1" CreationDate="2019-06-07T15:39:10.967" Score="6" ViewCount="710" Body="&lt;p&gt;In terms of helping students to understand propositional and predicate logic, with quantifiers, is there any research regarding when it is most advantageous for students studying mathematics, to first be &lt;em&gt;explicitly&lt;/em&gt; taught the basics in logic?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In part, my question is motivated now, in relation to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/16681/writing-up-a-proof-that-assumes-what-is-to-be-proven&quot;&gt;this question revealing a student's lack of understanding of basic propositional logic, and specifically, their lack of understanding of implication&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that in undergraduate mathematics education, propositional and predicate logic with quantifiers may be covered over four weeks of a discrete math class, or first formally introduced in the early portion of an &quot;introduction to proof&quot; class offered in a math department, or first introduced after Calc I- III, DE, perhaps in a proof-based linear algebra course. But there are many undergraduate math programs that don't explicitly require majors to complete a full semester (or two) in logic, or math logic.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am concerned that perhaps students are not taught early enough about logical operators like implication, or that &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$\lt$&lt;/span&gt; OR &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;, so that any expression &lt;span class=&quot;math-container&quot;&gt;$a\leq b$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$a \lt b$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt;, and if either is true, then &lt;span class=&quot;math-container&quot;&gt;$a\leq b$&lt;/span&gt; is true.  Not to mention, there are students struggling with &lt;span class=&quot;math-container&quot;&gt;$0\lt x \lt 12$&lt;/span&gt;, which relies on, implicitly, an understanding of the logical operation of &quot;AND&quot;:  If &lt;span class=&quot;math-container&quot;&gt;$0\lt x \lt 12$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$(x&amp;gt; 0$&lt;/span&gt; AND &lt;span class=&quot;math-container&quot;&gt;$x\lt 12)$&lt;/span&gt;, that is, &lt;span class=&quot;math-container&quot;&gt;$0\lt x \lt 12$&lt;/span&gt; is only true if BOTH &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0$&lt;/span&gt; AND &lt;span class=&quot;math-container&quot;&gt;$x\lt 12$&lt;/span&gt; are true.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(But notice in all these scenarios, it is assumed that students already understand the definition of material implication: &quot;if p, then q&quot;.  And that isn't always the case for many students.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've just listed only a few important concepts that students seem to struggle with in understanding, even in high school.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anyone has significant experience with respect to teaching logic, or is aware of programs that introduce basic logic explicitly, to students prior to college, or is aware of research in math education relating to when it is most advantageous to introduce logic to students studying math, I'd love to hear your input.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-06-17T11:26:49.700" LastActivityDate="2019-06-17T11:26:49.700" Title="Propositional and predicate logic, with quantifiers: Is there any research when it is ideal to explicitly teach in mathematics education?" Tags="|mathematical-pedagogy|education-research|students-mistakes|logic|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16699" PostTypeId="2" ParentId="16645" CreationDate="2019-06-08T13:20:12.043" Score="2" Body="&lt;p&gt;Another example of this sort of issue:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have often seen students write things like &lt;span class=&quot;math-container&quot;&gt;$\frac84$&lt;/span&gt; when they mean &lt;span class=&quot;math-container&quot;&gt;$4\mid 8$&lt;/span&gt; (i.e., &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It takes a while (for some) to see that the former has a numerical value, and the latter has a truth value.&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2019-06-08T13:20:12.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16702" PostTypeId="1" CreationDate="2019-06-09T12:14:35.657" Score="11" ViewCount="811" Body="&lt;p&gt;Students sometimes ask whether the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in the expression&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2x$$&lt;/span&gt; the same kind of thing as the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in the equation &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2x = 4.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the expression &lt;span class=&quot;math-container&quot;&gt;$2x, \;x$&lt;/span&gt; can be any real value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in the equation &lt;span class=&quot;math-container&quot;&gt;$2x = 4, \;x$&lt;/span&gt; can only be the value or values (from the some given set of real numbers) that makes the equation true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there some mathematical principles by which we can explain why and how the use of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is different? That is, is there a better explanation than simply &lt;strong&gt;telling students&lt;/strong&gt; that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; has a different meaning in these situations? For example, could/should we tell students to interpret the equals sign as a question that asks which value(s) of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; make the expressions on both sides of the equation equal?&lt;/p&gt;&#xA;" OwnerUserId="12356" LastEditorUserId="5321" LastEditDate="2020-10-26T13:27:30.220" LastActivityDate="2020-10-27T15:24:14.390" Title="Different Kinds of Variables" Tags="|algebra|definitions|" AnswerCount="8" CommentCount="27" ContentLicense="CC BY-SA 4.0" />
  <row Id="16703" PostTypeId="2" ParentId="16702" CreationDate="2019-06-09T12:38:08.063" Score="1" Body="&lt;p&gt;For the function &lt;span class=&quot;math-container&quot;&gt;$f(x)=2x$&lt;/span&gt;, the letter &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a variable; it can have any value in the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. For the polynomial &lt;span class=&quot;math-container&quot;&gt;$p(x)=2x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is an &lt;a href=&quot;https://en.wikipedia.org/wiki/Indeterminate_(variable)&quot; rel=&quot;nofollow noreferrer&quot;&gt;indeterminate&lt;/a&gt; that is just a placeholder. We can write the polynomial &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; without it : &lt;span class=&quot;math-container&quot;&gt;$p=(0,2,0,0,\ldots)$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt; is an (algebraic) equation and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the unknown to be found.&lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2019-06-09T12:38:08.063" CommentCount="22" ContentLicense="CC BY-SA 4.0" />
  <row Id="16704" PostTypeId="2" ParentId="16702" CreationDate="2019-06-10T16:27:06.640" Score="5" Body="&lt;p&gt;Your question is based on a false premise.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;However, in the equation &lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; can only be the value or values (from the some given set of real numbers) that makes the equation true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;No! In the equation &quot;&lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt;&quot;, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &lt;strong&gt;merely a variable&lt;/strong&gt;, and the equation is &lt;strong&gt;meaningless&lt;/strong&gt; without any further context. You can ask many &lt;strong&gt;different&lt;/strong&gt; questions about that equation, such as what &lt;strong&gt;real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/strong&gt; &lt;strong&gt;satisfies&lt;/strong&gt; it, or what is its &lt;strong&gt;graph&lt;/strong&gt; in the (cartesian) &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;-plane (it is a vertical line), or what are the &lt;strong&gt;free variables&lt;/strong&gt; in it (just &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;), and so on. Furthermore, that equation may be just one part of a larger expression such as &quot;&lt;span class=&quot;math-container&quot;&gt;$∀x∈\mathbb{R}\ ( 2x = 4 ∨ ∃y∈\mathbb{R}\ ( (x-2)·y = 1 ) )$&lt;/span&gt;&quot;, in which certainly &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is not only &quot;the value or values that makes the equation true&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt;&quot; is an equation, but equations do not imply anything about the symbols in them. &quot;What real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; satisfies the equation &lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt;?&quot; is a question with a well-defined answer. If you do not specify &quot;real&quot;, then it is not only &lt;strong&gt;not&lt;/strong&gt; a well-defined question, but also a common example of imprecision. Namely, &quot;Solve &lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt;.&quot; is not a precise question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An equation is a (mathematical) &lt;strong&gt;statement&lt;/strong&gt; of the form &quot;&lt;span class=&quot;math-container&quot;&gt;$A = B$&lt;/span&gt;&quot; where &lt;span class=&quot;math-container&quot;&gt;$A,B$&lt;/span&gt; are &lt;strong&gt;terms&lt;/strong&gt; (also called expressions). A statement has a &lt;strong&gt;truth-value&lt;/strong&gt; once all its free variables have been &lt;strong&gt;bound&lt;/strong&gt; (i.e. their values are specified). Namely, &quot;&lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt;&quot; has a truth-value once you specify what &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is. Hence it is meaningful to ask what real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &lt;strong&gt;satisfies&lt;/strong&gt; that equation (i.e. what value[s] for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; makes that equation true).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;&quot; and &quot;&lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;&quot; are simply terms, and &quot;&lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;&quot; is a compound term built from the terms &quot;&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;&quot; and &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot;, where multiplication has been represented by &lt;strong&gt;juxtaposition&lt;/strong&gt;. Yes, every variable is also a term, and as with statements, terms are meaningless until you specify the values of all its free variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &quot;&lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;&quot; and &quot;&lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt;&quot; are actually &lt;strong&gt;exactly the same&lt;/strong&gt; kind of variable, and mean exactly the same thing. Just as you can ask a &lt;strong&gt;question&lt;/strong&gt; about the equation &quot;&lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt;&quot; (e.g. what real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; satisfies it?), you can likewise ask a question about the term &quot;&lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;&quot; (e.g. what real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; makes it equal to &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;?). No difference.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="1550" LastEditDate="2019-06-10T16:39:13.840" LastActivityDate="2019-06-10T16:39:13.840" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16705" PostTypeId="2" ParentId="16697" CreationDate="2019-06-10T20:00:15.043" Score="6" Body="&lt;p&gt;I'm afraid I don't know of any research that has been done to my satisfaction on this pedagogical issue of teaching logic, because all of those that I found so far (over many years) investigate methods that I consider very pedagogically unsuitable. Based on my experience in teaching mathematics, there are a few important aspects if one wishes to teach logic to every student (not just those who can figure it out on their own) so that they are capable of utilizing it in all their subsequent mathematical work:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;There are many levels of logic (including both syntax and reasoning), and different levels are suitable for different kinds of people/students. At the &lt;strong&gt;basic&lt;/strong&gt; level, one must be able to use full classical logic (including quantifiers, equality and predicate/function-symbols), which requires complete grasp of the intended meaning of the syntax (including the meaning of variables and scoping and contexts) as well as the deductive rules and why they are sound. This basic level should be taught early, and must be taught before calculus! I call this &quot;using logic&quot; (in contrast with &quot;studying logic&quot;). It would be foolish to try to teach students about logic (namely invoking mathematical tools to define and analyze first-order logic) before they even know how to use it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There are many different possible variants of syntax and deductive systems, and choosing the wrong ones will spell pedagogical failure for almost all except students who already know how to reason logically (i.e. use classical logic, even though they don't know the name). I think that it is necessary to use a &lt;strong&gt;Fitch-style system&lt;/strong&gt; because we want students to be able to use logic for more than trivial toy exercises.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But the exact &lt;strong&gt;syntax&lt;/strong&gt; need not be the standard one. After all, it is true that the strange symbols &quot;&lt;span class=&quot;math-container&quot;&gt;$∀∃¬∧∨⇒$&lt;/span&gt;&quot; can be intimidating because it results in a significant compression of expressed meaning and requires decoding, which is not actually a crucial part of logic. For young students, even before learning any mathematical symbols, we can use standard English words. Of course, propositional logic should be taught first, and the only non-English we need to explain is brackets for indicating which parts of the sentence is evaluated first in the intended interpretation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For students with programming background, it is especially easy to show how propositional logic can be expressed in terms of programs that comprise solely &lt;code&gt;assert&lt;/code&gt;ions and &lt;code&gt;if&lt;/code&gt;-structures, corresponding to statements and if-subcontexts in the Fitch-style system respectively. I have personally used this to ensure that students have a 100% grasp of the deductive rules as well as have no problem with proof-by-contradiction. Even for students without programming background, one can do almost as well by showing them how to validate a Fitch-style proof given any truth-values of the atomic statements, in the same way the program would, and the goal is to ensure that every statement validated is in fact true. (See below for a fuller example.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To move on to classical logic, one must teach &lt;strong&gt;variables&lt;/strong&gt; properly as placeholders for objects, making sure to emphasize that different variables may refer to the same object. Again, there is a choice of variants in syntax and semantics. I think the pedagogically correct choice is to enforce that &lt;strong&gt;every variable must be declared&lt;/strong&gt; (i.e. bound), whether in a universal-subcontext header such as &quot;Given apple x:&quot; or an existential instantiation such as &quot;Let c be an apple such that ...&quot;. One concrete deductive system with all these features can be found &lt;a href=&quot;http://math.stackexchange.com/a/1684204/21820&quot;&gt;here&lt;/a&gt;, which includes axiomatizations of PA and set theory.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Related to quantifiers, I do not think it is wise to use unrestricted quantifiers. Not only does natural language use restricted quantifiers all the time, unrestricted quantifiers are unwieldy and result in asymmetric representations of restricted quantification. When I want to affirm &quot;Every fruit comes from a flower.&quot;, I do not want to have to say &quot;For every object x, if x is a fruit then x comes from a flower.&quot;, if I want to deny it I want to simply say &quot;Some fruit does not come from a flower.&quot;. Also, we cannot easily put together various axiomatizations of common structures if we do not use restricted quantifiers. Equivalently, we should work within &lt;strong&gt;multi-sorted&lt;/strong&gt; first-order logic, even though one-sorted first-order logic is in theory sufficient.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think we should teach &lt;strong&gt;game semantics&lt;/strong&gt;, because it would solidify the meaning of quantifiers and the effect of their order, and because it works perfectly even for an infinite domain. This ties in nicely with the Fitch-style semantics as well, since a universal-subcontext is of the same nature; &quot;Given apple x:&quot; literally introduces a subcontext in which you are given an arbitrary apple x, and all you know about x is that it is an apple. Also, students would not make the common logical error of invalid quantifier switch. This clearly can be taught as early as students understand 2-player games.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Certainly all &lt;strong&gt;syntactic sugar&lt;/strong&gt; must be explained in full at some point. For example, precedence rules for boolean operations, condensed quantifiers over the same type (e.g. &quot;∀x,y∈S ( ... )&quot;), chained relations (e.g. &quot;A &amp;lt; B ≤ C = D&quot;), short-forms (e.g. &quot;A ≤ B&quot; for &quot;A &amp;lt; B ∨ A = B&quot; and &quot;A ≠ B&quot; for &quot;¬ A = B&quot;). Also, chained relations may be written over multiple lines, and students must be made aware of how exactly to extract the desired conclusion (e.g. from &quot;A &amp;lt; B ≤ C = D&quot; to &quot;A &amp;lt; D&quot;) that is often omitted from a proof.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I believe &lt;strong&gt;induction&lt;/strong&gt; should actually be considered a core part of logic, rather than just being part of some mathematical theories. The litmus test of whether a student has complete grasp of basic logic is whether they can perform induction over predicates with quantifiers. Naturally, induction is best taught only after the students have a complete understanding of full classical logic and can prove simple theorems of PA−.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, and most importantly, the teacher must strive to be &lt;strong&gt;100% precise&lt;/strong&gt;. The age at which the teacher should expect 100% precision in the Fitch-style proofs by the student is the same age at which the student can do simple programming (including &lt;code&gt;for&lt;/code&gt;-loops). There is a sad reason why after one semester, almost all undergraduate CS students can write programs that the compiler accepts, but almost no undergraduate mathematics student can write perfect proofs in an actual deductive system. Incidentally, I often say that students who can easily use a deductive system no longer need to do so on paper, because they can do it mentally. Unfortunately, it is invariably people who cannot actually perform rigorous reasoning that fail to see the benefit of a deductive system, and often do not want to learn to use any (if it is not going to be tested in the exam).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think 4 weeks is enough for a proper teaching of logic. I think an entire semester is just nice, because one should cover everything in my above points. Also, ideally all introductory mathematics courses should be done within the &lt;strong&gt;same deductive framework&lt;/strong&gt; (such that the teacher skips steps but can furnish any omitted steps if asked). Without this, there is always a tendency for students to compartmentalize logic as &quot;just another topic to be learned and later forgotten&quot;. Of course, if one really covers everything up to PA, and a little bit of real analysis, students may get a vague idea of really how everything in mathematics can be expressed in logical form. But if one fails to convey that all-encompassing nature of logic, one cannot expect students to really learn to use it. (It is like teaching them Latin for one semester but teaching all other subjects in English.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Based on all the above points, as well as my personal experience in learning and teaching, I estimate that an optimal pedagogy would enable us to teach propositional logic at age 9-10, classical logic at age 11-12, along with programming, and then full PA at age 13-14. Note that basic logic should be applied across all subjects whenever applicable, and not just confined to &quot;mathematics&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But since we will probably never get what we want, the next best is to have a whole semester-length course (≈50 hours over 4 months) dedicated to it, right at the beginning of undergraduate studies. I do not think it should be restricted to mathematics majors either, since basic logic is a critical component in any serious pursuit of knowledge. In my opinion, mathematics majors ought to be taught a fair bit more than basic logic, probably up to the completeness and compactness theorems for countable first-order theories, spanning two courses in total, but that is for another discussion. =)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As an example of proof-by-contradiction (i.e. using ⊥-elim), consider the following Fitch-style proof:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;If P:&#xA;  If not P:&#xA;    P.&#xA;    not P.&#xA;    false.&#xA;  not not P.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;By definition of subcontext, the first subcontext under &quot;If P:&quot; represents the subcontext (sub-scenario) in which (P) is true. The second subcontext under &quot;If not P:&quot; is nested inside the first subcontext, so it represents the further subcontext in which (not P) is true. Well, can this inner subcontext be possible? No, that scenario simply cannot occur, and we can see that from the proof itself since both (P) and (not P) are true within it. Thus we can conclude that the negation of the innermost subcontext header (not P) is true in its context. Note that we cannot claim &quot;not not P&quot; in the global context (i.e. under no conditions). The outer subcontext may occur, and it is just the inner subcontext that is impossible, so &lt;strong&gt;within&lt;/strong&gt; the scenario where (P) is true, we can conclude that (not P) is false.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this kind of explanation must be done slowly and repeated until it clicks. People who meet logical reasoning for the first time need time to warm up their brains.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="1550" LastEditDate="2019-06-16T07:01:18.233" LastActivityDate="2019-06-16T07:01:18.233" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16706" PostTypeId="2" ParentId="16702" CreationDate="2019-06-11T19:42:13.593" Score="16" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; is an expression, a phrase. Compare it to &quot;two ducks&quot;. This is neither true nor false. It doesn't have a 'truth value'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2x = 4$&lt;/span&gt; is an equation, a statement. Compare it to &quot;two ducks have four legs&quot;. This is true (edit: for the ducks, but not necessarily for the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The meaning of the word &quot;ducks&quot; has not changed. The grammar of what is with that word has.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you replace &quot;ducks&quot; with &quot;dogs&quot;, the &quot;two dogs&quot; phrase makes us think of something else, but doesn't feel much different grammatically than &quot;two ducks&quot;. But &quot;two dogs have four legs&quot; would be false. Different values for the variable give different 'truth values'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This comparison isn't perfect, because the &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; turned into &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; legs some magical way. But I hope it helps you to see that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; isn't changing, just the grammar around it.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2019-06-23T21:28:19.000" LastActivityDate="2019-06-23T21:28:19.000" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16707" PostTypeId="2" ParentId="1042" CreationDate="2019-06-11T21:47:22.620" Score="2" Body="&lt;p&gt;COMAP (The Consortium for Mathematics and Its Applications) publishes a newsletter for its members called Consortium. The current issue of Consortium can be downloaded for free from the web page below (which also includes other free COMAP materials) until the next issue appears, at which time the &quot;current&quot; issue goes behind a firewall.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.comap.com/Free/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.comap.com/Free/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consortium has many articles that can inspire student research (middle school to graduate school) research, In particular, the last couple of issues of Consortium have featured a &quot;new column&quot; that I edit, that discusses research problems specifically designed for (but not limited to) pre-college students. There problems appear under the title: &quot;Student Research Corner.&quot; If you use the COMAP search feature you can find the titles of these research problems which are quick starting but all are geometrical questions nearly all involving graph theory and polyhedra.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2019-06-11T21:47:22.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16709" PostTypeId="1" AcceptedAnswerId="16719" CreationDate="2019-06-12T07:57:00.573" Score="0" ViewCount="646" Body="&lt;p&gt;I posted this &lt;a href=&quot;https://math.stackexchange.com/questions/3189449/practical-applications-of-integration-by-substitution-where-integrand-is-unknown&quot;&gt;question&lt;/a&gt; on the Mathematics Stack Exchange a while ago, and got no responses, so I thought I would ask it here. I'm looking for any real-life applications of integration by substitution where the integrand is unknown as a way of motivating the topic, as opposed to giving repetitive questions which can be answered by typing the integral into WolframAlpha.&lt;/p&gt;&#xA;" OwnerUserId="8383" LastActivityDate="2020-02-04T16:33:16.763" Title="Practical applications of integration by substitution where integrand is unknown" Tags="|calculus|concept-motivation|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16710" PostTypeId="2" ParentId="8335" CreationDate="2019-06-12T09:14:14.550" Score="0" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I also wonder, What should math student do with proofs? [...] just keeping the idea of the proof and reconstruct the details herself when needed?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes, and not just a mathematics student. In my opinion, to learn mathematics properly we must not only make sure we fully understand the proofs of the theorems covered, but also ensure that we can reconstruct it on our own, and not from memory!&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To be concrete, in the last few weeks, I was studying Boolean algebra, ordinals and Godel first incompleteness theorem, Is it natural that I forget most of the proofs after sometime only keeping the main ideas and the main results from those topics?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is natural, and in fact I would say it is a good thing if you pinpoint the core bits. For example, I would say that the core behind all the basic facts about ordinals are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;We can perform transfinite induction/recursion along any well-ordering.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Any two well-orderings are comparable by embedding, and embedding implies isomorphism to an initial segment (or the whole thing), and any set of well-orderings have a minimal element under embedding.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given any set &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;, the well-orderings on subsets of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; modulo isomorphism forms a well-ordering &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; into which &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; injects, and hence &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is well-orderable. This is the crucial theorem that enables transfinite iteration through any set. Choosing the minimum initial segment of &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; that bijects with &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; permits iterations relying on the 'number of steps' so far always being 'less than' (no surjection onto) the 'total number of steps'.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In ZFC, we can construct a canonical representative for any well-ordering &lt;span class=&quot;math-container&quot;&gt;$(W,&amp;lt;)$&lt;/span&gt; by transfinite recursion along &lt;span class=&quot;math-container&quot;&gt;$(W,&amp;lt;)$&lt;/span&gt;, namely by &lt;span class=&quot;math-container&quot;&gt;$f(x) := \{ f(y) : y∈W ∧ y&amp;lt;x \}$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$x∈W$&lt;/span&gt;, and then &lt;span class=&quot;math-container&quot;&gt;$ord(W,&amp;lt;) := \{ f(x) : x∈W \}$&lt;/span&gt;. Note that both the transfinite recursion and the final step need the replacement schema. These canonical representatives are also called von Neumann ordinals, and we can define cardinals as ordinals that do not biject to any smaller ordinal.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Similarly for the first incompleteness theorem:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Godel's version requires ω-consistency and the β-lemma for encoding into PA, and translating the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fixed-point_combinator#Fixed_point_combinators_in_lambda_calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Y combinator&lt;/a&gt; into PA yields the fixed-point lemma, which can be applied to &lt;span class=&quot;math-container&quot;&gt;$( A ↦ ¬⬜A )$&lt;/span&gt; to give the incompleteness theorem. Along the way you can check that the Godel sentence is (equivalent to) a &lt;span class=&quot;math-container&quot;&gt;$Σ_1$&lt;/span&gt;-sentence because &lt;span class=&quot;math-container&quot;&gt;$⬜A$&lt;/span&gt; is a &lt;span class=&quot;math-container&quot;&gt;$Π_1$&lt;/span&gt;-sentence.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Rosser's version requires only consistency and can be obtained either by using &quot;has a proof and no shorter proof of its negation&quot; in place of &quot;provable&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Arithmetic with addition and multiplication is in fact a red herring to the incompleteness theorem, and any computable system that can do basic reasoning about finite binary strings is already essentially incomplete. With ω-consistency we can reduce to the halting problem. With mere consistency we can reduce to a weaker problem which I call zero-guessing &lt;a href=&quot;https://math.stackexchange.com/a/2486349/21820&quot;&gt;here&lt;/a&gt;. You can see that all I need to remember to recover the full proof of the generalized incompleteness theorem is the zero-guessing problem, as all the rest is quite easy to figure out.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Reasoning about provability within a theory that interprets arithmetic is cleanly represented in &lt;a href=&quot;https://plato.stanford.edu/entries/logic-provability&quot; rel=&quot;nofollow noreferrer&quot;&gt;provability logic&lt;/a&gt; as the Hilbert-Bernays' provability conditions plus the modal fixed-point theorem. This representation makes it easier to grasp, and also makes it easy to figure out how to prove &lt;a href=&quot;https://math.stackexchange.com/a/1766631/21820&quot;&gt;Lob's theorems (both internal and external forms)&lt;/a&gt; by simply 'translating' Curry's paradox, namely applying the modal fixed-point theorem to obtain &lt;span class=&quot;math-container&quot;&gt;$B ⇔ (⬜B ⇒ A)$&lt;/span&gt; and then working from there.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Basically, since I know I can obtain all the basic facts from these core ideas, there is not much I need to keep around in my head for that purpose. Of course, the more we learn, the more core ideas we need to remember, but it should be much less than the amount we can reconstruct from them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The hard part is in identifying these core ideas, since many textbooks do not do so. Sometimes, a textbook may also choose a longer route because it yields stronger results, but it may be better to understand the key mathematical structures before going deeper. For instance, arithmetization of syntax (via Godel's β-lemma) is essentially needed only to establish that PA can do basic reasoning about finite strings, so in my opinion it is not necessary to understand that at all if we simply want to understand the incompleteness phenomenon. Yet we should know it and isolate its relevance in the manner I just did.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2019-06-12T09:14:14.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16712" PostTypeId="2" ParentId="8335" CreationDate="2019-06-13T02:46:17.220" Score="1" Body="&lt;p&gt;As an undergraduate math major, there's a radical change in structure around late-sophomore to early-junior year (U.S. experience here). Prior to that point, most of the work is to calculate things with given formulas or algorithms. After that point, it switches to mostly proof-writing exercises. The big picture here is that the junior-senior years of the program are meant to introduce and build skills for possible graduate study, and someday, hopefully, discovering and proving novel theorems. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What should math student do with proofs? What is she expected to do&#xA;  with it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To a large degree one is likely expected to use the tricks/techniques in those proofs to prove closely-related exercises, and expand those skills as time goes on. (I'd say secondarily you'll be expected to sketch and explain those proofs to hypothetical students or skeptical interlocutors, but that will be more the work of a teaching assistant than classwork.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a very simple example: Say you're shown the proof that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is irrational. The next exercise is probably for you to prove that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3}$&lt;/span&gt; is irrational (which is almost the exact same piece of work, with a few numbers slightly modified). A test question might ask for a proof that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{5}$&lt;/span&gt; is irrational. A somewhat more advanced exercise would be to generalize the proof for the square root of any non-perfect-square integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, or other roots above &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; (which again are very similar, but veer off a bit more from the original each time). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you proceed, you'll be expected to handle more sophisticated new proofs, with larger amounts of creative thinking involved, and a deeper toolkit of tricks/techniques from which to draw on. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2019-06-13T14:25:09.793" LastActivityDate="2019-06-13T14:25:09.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16713" PostTypeId="1" CreationDate="2019-06-13T09:03:18.607" Score="5" ViewCount="6093" Body="&lt;p&gt;The companies that run these expensive abacus programs for children claim it has all kinds of benefits for their mathematics abilities and speed. Apparently it starts with a child learning the mechanics of using an abacus, giving them practice in doing so, and eventually, the child is thought to visualize an abacus in their head, as a &quot;mental&quot; abacus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don’t doubt that some kids doing Soroban classes &lt;em&gt;do&lt;/em&gt; manage to do very fast arithmetic tasks when strings of numbers are thrown at them. But I don’t know if it’s worth doing, other than for speed in calculations.&lt;/p&gt;&#xA;" OwnerUserId="12360" LastEditorUserId="238" LastEditDate="2019-06-17T02:42:53.893" LastActivityDate="2019-10-29T14:19:13.083" Title="Are soroban (Japanese abacus) classes worth doing?" Tags="|primary-education|arithmetic|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16714" PostTypeId="2" ParentId="16713" CreationDate="2019-06-13T11:54:20.933" Score="9" Body="&lt;p&gt;The article [1], according to the abstract, [claims various benefits.](&lt;a href=&quot;https://eric.ed.gov/?id=EJ1105219&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://eric.ed.gov/?id=EJ1105219&lt;/a&gt; , &lt;a href=&quot;https://scholar.google.no/scholar?cluster=12532307503119935328&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://scholar.google.no/scholar?cluster=12532307503119935328&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it is not widely cited at all. I do not know the landscape of education journals well enough to know if the journal is of high quality. I checked the Finnish publication forum, which has rated the journal as scientific but not exceptionally good since 2017.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The method used was asking teachers if the students taking abacus classes are doing better than others. This method can not distinguish between mathematically gifted students taking abacus classes and between students taking abacus classes becoming better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, one would expect any engagement with mathematics to lead to improvement, so a pertinent question is if this method leads to more improvement than other activities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The conclusions section of the article refers to other studies that might be of interest in answering the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] Altiparmak, K. (2016). The  teachers'  views on  Soroban  abacus  training. International Journal of Research in Education and Science (IJRES), 2(1), 172-178.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="238" LastEditDate="2019-06-18T15:14:35.340" LastActivityDate="2019-06-18T15:14:35.340" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16715" PostTypeId="1" CreationDate="2019-06-13T14:42:06.047" Score="10" ViewCount="356" Body="&lt;p&gt;During my studies I observed that while most lecturers try to explain theorems and their proofs, only very few of them try to explain definitions. However, in my opinion, definitions are the base of maths education. In principle, it is possible to take them as given and just work with them, but establishing a link between the content of a definition and the name of the object or property we define seems very important to me. However, while some people asked questions concerning theorems and proofs, I never saw anybody asking about the plausibilty of definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me give you an example: The definition of linear (in)dependency. Quoting Wikipedia: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The vectors in a subset &lt;span class=&quot;math-container&quot;&gt;$S=\{\vec v_1,\vec v_2,\dots,\vec v_k\}$&lt;/span&gt; of a&#xA;  vector space &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; are said to be linearly dependent, if there exist&#xA;  scalars &lt;span class=&quot;math-container&quot;&gt;$a_1,a_2,\dots,a_k$&lt;/span&gt;, not all zero, such that: &#xA;  &lt;span class=&quot;math-container&quot;&gt;$$a_1\vec v_1+a_2\vec v_2+\cdots+a_k\vec v_k= \vec 0,$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\vec 0$&lt;/span&gt; denotes&#xA;  the zero vector.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now, imagine a student asking you: Why do we actually call this property &quot;linear dependency&quot;? Please take a minute or two if you can't come up with an answer immediately before looking into the spoiler below. Hint: The answer &quot;It's a definition and there is no point in asking why a definition is the way it is!&quot; is not the one I had in mind.&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; A variable &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is called dependent on another variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; if there is a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt;. This also works for several dependencies: A variable &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is called dependent on other variables &lt;span class=&quot;math-container&quot;&gt;$x_1, ..., x_n$&lt;/span&gt;, if there is a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$y=f(x_1, ..., x_n)$&lt;/span&gt;. This also works for a vector valued function which takes vectors as arguments: A vector variable &lt;span class=&quot;math-container&quot;&gt;$\vec y$&lt;/span&gt; is dependent on other vector variables &lt;span class=&quot;math-container&quot;&gt;$\vec x_1, ..., \vec x_n$&lt;/span&gt; if there is a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\vec y=f(\vec x_1, ..., \vec x_n)$&lt;/span&gt;. If the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is linear, it can be written as &lt;span class=&quot;math-container&quot;&gt;$f(\vec x_1, ..., \vec x_n)=c_1\vec x_1 + ... + c_n \vec x_n$&lt;/span&gt; for some coefficients &lt;span class=&quot;math-container&quot;&gt;$c_1, ..., c_n$&lt;/span&gt;. Then it is natural to say that &lt;span class=&quot;math-container&quot;&gt;$\vec y$&lt;/span&gt; is linearly dependent on the vector variables &lt;span class=&quot;math-container&quot;&gt;$\vec x_1, ..., \vec x_n$&lt;/span&gt;, if there are coefficients &lt;span class=&quot;math-container&quot;&gt;$c_1, ..., c_n$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\vec y=c_1\vec x_1 + ... + c_n \vec x_n$&lt;/span&gt;.&#xA;&#xA; Now, how does this relate to the definition quoted above? If there is a non-trivial solution of &lt;span class=&quot;math-container&quot;&gt;$a_1\vec v_1+a_2\vec v_2+\cdots+a_k\vec v_k= \vec 0$&lt;/span&gt;, there is an &lt;span class=&quot;math-container&quot;&gt;$a_i\neq 0$&lt;/span&gt;, so we can solve for the corresponding &lt;span class=&quot;math-container&quot;&gt;$\vec v_i$&lt;/span&gt; and express it through the other &lt;span class=&quot;math-container&quot;&gt;$\vec v_j$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$j\neq i$&lt;/span&gt; and get that &lt;span class=&quot;math-container&quot;&gt;$\vec v_i$&lt;/span&gt; is linearly dependent on the other &lt;span class=&quot;math-container&quot;&gt;$\vec v_j$&lt;/span&gt; in the sense as described above. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now that I have written it down, the explanation looks trivial and lengthy to me, but I recall the time when I was wondering what the relation between linear dependency of vectors and dependency in the sense of a functional dependency of two variables is and had a tough time to really pinpoint it. Also, there are definitions (for example from algebra, say, the different kinds of field extensions) where I am still missing the link between the name and the content (hints to good literature for this topic are highly appreciated).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I am wondering:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is it a common phenomenon that giving intuition for definitions, in particular, relating name and content is treated as an orphan, or is it just my subjective perception?&lt;/li&gt;&#xA;&lt;li&gt;Would it make sense to spend more time on giving intuition for definitions?&lt;/li&gt;&#xA;&lt;li&gt;What are possible reasons for the phenomenon?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3542" LastEditorUserId="3542" LastEditDate="2019-06-13T14:50:17.437" LastActivityDate="2019-06-23T15:21:26.293" Title="How important is making definitions plausible?" Tags="|intuition|definitions|" AnswerCount="3" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="16716" PostTypeId="2" ParentId="16713" CreationDate="2019-06-13T15:17:55.577" Score="4" Body="&lt;p&gt;&quot;The companies that run these expensive programs&quot; just want to make money. Some sell \$1B worth of iPads, other sell abaci.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do your school budget a favor and get an abacus at Ikea for \$10. It has horizontal wires with ten beads on each, which is similar to Russian-style abacus. The wires are not curved, and the colors are just for fun not for easier grouping, but it is good enough to teach decimal system and place value. Ten beads on a wire may seem excessive — you need only nine — but come handy when you want to break a higher-place value into lower one, like breaking one thousand into ten hundreds. This abacus is much closer representation of decimal system than soroban — one wire is one decimal place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Adding and subtracting is simple. Multiplication is more involved. Division requires some mental math, which itself can be a good reason to refresh multiplication table and division shortcuts. I think if you just explain how addition and subtraction works, that would be good enough return for your \$10.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ibiblio.org/eldritch/ac/tutor.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;This is a fun short story&lt;/a&gt; by Chekhov about a high-school tutor failing to solve a middle-school problem that the merchant father of the middle-schooler solves in seconds using abacus (called counting board in this translation). The problem can be solved arithmetically and algebraically, and the merchant solves it arithmetically, cannot do a system of equations with abacus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9sTbd.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9sTbd.jpg&quot; alt=&quot;MULA abacus from Ikea&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2019-06-13T15:33:52.127" LastActivityDate="2019-06-13T15:33:52.127" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16717" PostTypeId="2" ParentId="16715" CreationDate="2019-06-13T16:03:00.020" Score="7" Body="&lt;p&gt;(1). It's going to depend on level to a huge degree. In my experience, up into advanced undergraduate you could expect definitions to be named sensibly and an instructor to explain how a sensible name like &lt;em&gt;linear dependence&lt;/em&gt; corresponds to the definitions content. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3). Here would be my guess: the more advanced you get, definitions become more utilitarian and axiomatic, with the purpose of picking out formal objects that we don't have ordinary language to describe. For example, if I want to talk about a &lt;em&gt;normal subgroup of a connected semisimple Lie group&lt;/em&gt;, I'm going for precision and brevity in my naming conventions over more English language explanatory power. We start giving definitions that we don't have short pithy English language referents for. So in higher math your question largely doesn't apply, since the naming conventions are trying to do something different. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bringing it back, I wouldn't be surprised if mathematicians lecturing at lower levels can forget the importance of the expository power of a sensibly named definition. The attitude is much more &quot;A definition separates out classes of objects that we would like to talk about. These classes must be justified by theoretical results, or a practical intuitive definition. Oh, an it's called a principle component for historical reasons seem less important than it's use.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2). Do I think we could spend more time on it? Yes, absolutely, especially if the naming conventions are explanatory. Your example of linear dependence is a good one: It's a great place to teach/reinforce mathematical vocabulary like &lt;em&gt;linear&lt;/em&gt; and &lt;em&gt;dependence&lt;/em&gt;. Such an explanation will hopefully help students make connections between the content and the name, but also between the content of the given definition and other parts of math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;(Edited to get the number order correct)&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="9019" LastEditDate="2019-06-14T17:01:14.143" LastActivityDate="2019-06-14T17:01:14.143" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16718" PostTypeId="2" ParentId="16713" CreationDate="2019-06-13T22:48:41.670" Score="4" Body="&lt;p&gt;I took abacus classes when I lived in Japan for several years as a child.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was interesting because one learned a mechanical way of solving arithmetic problems, but I didn't learn &quot;number sense&quot; which was much more useful later on in my academic career.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think my experience meshed with Richard Feynman's conclusions in his excellent story &lt;em&gt;&lt;a href=&quot;https://www.ee.ryerson.ca/~elf/abacus/feynman.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Feynman vs. the Abacus&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How did the customer beat the abacus?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The number was 1729.03. I happened to know that a cubic foot contains&#xA;  1728 cubic inches, so the answer is a tiny bit more than 12. The&#xA;  excess, 1.03 is only one part in nearly 2000, and I had learned in&#xA;  calculus that for small fractions, the cube root's excess is one-third&#xA;  of the number's excess. So all I had to do is find the fraction&#xA;  1/1728, and multiply by 4 (divide by 3 and multiply by 12). So I was&#xA;  able to pull out a whole lot of digits that way.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A few weeks later, the man came into the cocktail lounge of the hotel&#xA;  I was staying at. He recognized me and came over. &quot;Tell me,&quot; he said,&#xA;  &quot;how were you able to do that cube-root problem so fast?&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I started to explain that it was an approximate method, and had to do&#xA;  with the percentage of error. &quot;Suppose you had given me 28. Now the&#xA;  cube root of 27 is 3 ...&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;He picks up his abacus: zzzzzzzzzzzzzzz— &quot;Oh yes,&quot; he says.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I realized something: he doesn't know numbers. With the abacus, you&#xA;  don't have to memorize a lot of arithmetic combinations; all you have&#xA;  to do is to learn to push the little beads up and down. You don't have&#xA;  to memorize 9+7=16; you just know that when you add 9, you push a&#xA;  ten's bead up and pull a one's bead down. So we're slower at basic&#xA;  arithmetic, but we know numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;All that being said, I'm glad I learned it, but it's useful now mainly as a parlor trick.  I also demonstrate how to multiply with a slide rule!  And it was interesting when I lived there to watch shopkeepers sum up checks.&lt;/p&gt;&#xA;" OwnerUserId="12362" LastEditorUserId="12362" LastEditDate="2019-06-17T17:13:23.260" LastActivityDate="2019-06-17T17:13:23.260" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16719" PostTypeId="2" ParentId="16709" CreationDate="2019-06-14T19:11:15.763" Score="4" Body="&lt;p&gt;This is probably too abstract for students just learning about integration by substitution. But &lt;em&gt;convolution&lt;/em&gt; is a &quot;real-world application of integration,&quot; finding applications in such things as image processing. One defines&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\, d\tau$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and then to show &lt;span class=&quot;math-container&quot;&gt;$f*g=g*f$&lt;/span&gt; you use integration by substitution. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2019-06-14T23:59:12.237" LastActivityDate="2019-06-14T23:59:12.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16720" PostTypeId="2" ParentId="16715" CreationDate="2019-06-14T20:57:05.120" Score="4" Body="&lt;p&gt;I agree with Nate Bade about the helpfulness of making connections between the names of properties and how those properties are defined, particular when there are direct connections that can be articulated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, there are terms in mathematics, with specific definitions, for which the choice of specific terms may best be explained by understanding the &lt;a href=&quot;https://www.google.com/search?client=firefox-b-1-d&amp;amp;q=etymology+of+mathematical+terms&quot; rel=&quot;nofollow noreferrer&quot;&gt;etymology of those terms&lt;/a&gt; (when those terms were first coined, and by whom.)  I.e., there are historical factors that figure in to the choice of some mathematical terms/properties their definitions, that make sense w.r.t the time period at which the term/property was defined, but may not be so clear &lt;em&gt;at this point in time&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd argue that explaining the etymology of such terms and properties, when introducing them to students, and their definitions, is also useful in helping students develop an understanding/intuition connecting terms/properties with their definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I empathize with and share Daniel R. Collins' reality-check on the idea of calling on teachers and profs, etc, to spend time when teaching to help students make connections between terms, theorems, etc.. Even teaching math at the secondary school level, responsible for covering a curriculum state-dictated curriculum, on which students will ultimately be tested, makes teaching math a sort of &quot;triage&quot; based, sometimes remediation, leaving teachers very limited opportunities for them to wonder off into things like the origin of terms, etc.  When it can be done, great.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, why presume it ought to be, from grade school through undergraduate years, that educators &lt;em&gt;ought to give&lt;/em&gt; students such connections. Better yet, we need to encourage independent thinking, critical reading, and independent research skills of students, starting from the primary grades, through middle school.  So, e.g., when a student wonders &quot;I know I'm taking &lt;em&gt;Algebra&lt;/em&gt;, and my textbook tells me so. But where did the name &quot;Algebra&quot; come from?&quot; they take initiative to first find out, if they can. &quot;Hmm, let me plug &lt;em&gt;Algebra&lt;/em&gt; into a Google search bar ... Wow, lots of hits including Wikipedia which discusses the history of algebra, and here we go, I see 'The origins of Algebra'! ...ahhh, oh cool, I get it!&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The capacity to think independently, work and research independently and within groups, read critically, etc., are skills students need to develop from their first day in first grade, and all the way through the pre-college education. Hopefully, a student will then be able to better engage in their own education, and realize that their education is not merely defined by what teachers/profs lecture on, or the texts they read from, but also, perhaps more importantly, it's determined also by what the student &lt;em&gt;does with what information&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This applies to much more than math education. Students, for too long, have been described as &quot;consumers&quot; of educational &quot;products&quot;, which seems to imply the student/consumer is passive:  &quot;Give me (the student) the educational product you promote, and I'll consume it. And I can't be bothered to hunt or gather or otherwise find what I might also need to consume to supplement the delivered product.&quot;   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We would all be better off if students thought of themselves as partners in their education.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-06-23T11:53:12.227" LastActivityDate="2019-06-23T11:53:12.227" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16721" PostTypeId="2" ParentId="15130" CreationDate="2019-06-15T02:53:59.670" Score="2" Body="&lt;p&gt;This book is better appreciated after a first exposure to the subject. However, one can also start reading this book. It is beautifully written. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as determinants are concerned, there are varying opinions on determinants: Consider this, if we have a linear transformation, then how do we define its determinant? We have to take help of a basis, fix coordinates to get a matrix representation and then define determinant of linear transformation using its matrix representation (after proving that it is well-defined!). For understanding his view point, one must read his article &quot;Down with determinants&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This book presents results separately depending on whether vector space is over Real numbers or Complex numbers. I liked this aspect of the book. Most of the other books assume that vector space is over field of Complex numbers, and we are in dark as far as real vector spaces are concerned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The new edition (3rd) also has exercises placed at the end of each section.&lt;/p&gt;&#xA;" OwnerUserId="12365" LastActivityDate="2019-06-15T02:53:59.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16724" PostTypeId="2" ParentId="16715" CreationDate="2019-06-17T06:16:26.497" Score="1" Body="&lt;p&gt;I sympathize very much with this. Re: &quot;What are possible reasons for the phenomenon?&quot;, I think the primary issue is simply time limitations. As a math/computing lecturer at a 2-year school, I'm wildly desperate for time in class, and already have to slash out half of any standard book section's content to try and get my students up and running on the core required skills. On the other hand, I also know of some faculty who do prefer a purely mechanical class presentation, and are simply disinterested in drawing connections to (even slightly) larger social or historical meaning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've made the point numerous times in the past that math/computing courses inherently run at a higher conceptual level than other courses. E.g.: Other faculty reference Bloom's taxonomy and bemoan why they're operating at a low level of base knowledge (testing on definitions of terms), and wishing they can escalate to higher levels (like applications and analysis). But I don't know any math faculty who test on definitions -- we don't have time, and all of the work is jumping instantly to applications all the time already. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's a lot of times when I get a bit frustrated with students not picking up the &quot;obvious&quot; meaning of a certain name for a particular definition. E.g.: In a statistics course I speak of a distribution's &quot;tail&quot;, and then on tests some students write this as &quot;tale&quot;, having heard what I verbalized but not associating any meaning to it. Sometimes in response I wind up drawing a distribution with a cat's face on one end to emphasize the meaning. Is that really a good use of time? If I did that in a lecture, what percentage of students would be bored by the pedantry, and what other topic would be getting pushed aside by it? (Other examples: Say I'm in a discrete mathematics course presenting basic logic identities; in one table we have the Identity laws, Domination laws, Commutative laws... and about 20 others. Are these names all obvious? Should I spend time drawing out the English meaning of all these names?) That's the dilemma for me, and I suspect for many others. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If those naming backgrounds were in sidebar in the text, and my students could dependably read them (which they can't), then I would consider that ideal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, it might make for an interesting research question to survey students and see what proportion can identify/reason out the motivation for different naming traditions on their own, versus those that need explicit instruction (and so help to prioritize use of in-class time). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2019-06-23T15:21:26.293" LastActivityDate="2019-06-23T15:21:26.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16726" PostTypeId="2" ParentId="16702" CreationDate="2019-06-17T21:49:37.113" Score="-2" Body="&lt;p&gt;With older students (university age, or maybe late school) I'd be inclined to try and show them that they probably don't really know what they are asking, rather than actually trying to give a definitive answer (which is much harder to do properly).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Write &lt;span class=&quot;math-container&quot;&gt;$y-2x=3$&lt;/span&gt; next to a pair of axes.&#xA;Write it again separately with &lt;span class=&quot;math-container&quot;&gt;$y+x=1$&lt;/span&gt; underneath.&#xA;Now ask if the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;s are the same.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would expect most students to think of the first instance as a line to plot, while the second is some equations to solve. While these are in some sense the same, I think a student who asks the opening question probably means these to be considered as different. Using the same equation in both cases helps to show that the issue isn't so much adding &lt;span class=&quot;math-container&quot;&gt;$=4$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; as how you choose to think about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Telling the students to interpret the equals sign as asking a question is great, so long as they understand in which contexts they should think of it that way and when they should think of it a different way. (That students generally don't know this is, I think, probably one of the big reasons they find maths confusing.)&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-06-17T21:49:37.113" CommentCount="21" ContentLicense="CC BY-SA 4.0" />
  <row Id="16727" PostTypeId="1" CreationDate="2019-06-18T13:18:58.490" Score="8" ViewCount="1702" Body="&lt;p&gt;(&lt;a href=&quot;https://math.stackexchange.com/questions/3239037/maclane-birkhoffs-algebra-vs-jacobsons-basic-algebra-i-ii-vs-langs-algeb&quot;&gt;Cross-posted&lt;/a&gt; at Math.Stackexchange)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I'm searching for an apt textbook(s) on Abstract Algebra for a very advanced undergraduate/graduate level course in Algebra, and would be grateful for any help. I've thought of the aforementioned texts, but additional suggestions would be welcome too.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some points about my background: I’ve taken a course in Linear Algebra where I read Roman’s &lt;em&gt;Advanced Linear Algebra&lt;/em&gt; in addition to Halmos’ &lt;em&gt;Finite Dimensional Vector Spaces&lt;/em&gt; and Axler’s &lt;em&gt;Linear Algebra Done Right&lt;/em&gt; as primary texts. I’ve already had a course of Abstract Algebra from Artin’s &lt;em&gt;Algebra&lt;/em&gt;. Additionally, I have finished the first 7 chapters of Baby Rudin, and plan to try Loomis and Sternberg’s &lt;em&gt;Advanced Calculus&lt;/em&gt; next.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These upcoming 3 semesters I have courses in Algebra (the syllabus for which is attached at the end). The prescribed texts are Jacobson’s &lt;em&gt;Basic Algebra I, II&lt;/em&gt;, and Lang’s &lt;em&gt;Algebra&lt;/em&gt;. Some of the material is familiar, so I’m looking to self-study beyond the syllabus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've looked through Lang's &lt;em&gt;Algebra&lt;/em&gt;, MacLane and Birkhoff’s &lt;em&gt;Algebra&lt;/em&gt;, and Jacobson’s &lt;em&gt;Basic Algebra I,II&lt;/em&gt;. So far, &lt;em&gt;Basic Algebra I&lt;/em&gt; seems much easier and more ‘leisurely’ than the other two. Understanding the exposition was not an issue for any of the books (I used G. Bergman's &lt;em&gt;Companion to Lang&lt;/em&gt; for some assistance with Lang's &lt;em&gt;Algebra&lt;/em&gt;). Unfortunately &lt;em&gt;Basic Algebra I&lt;/em&gt; usually gives explicit constructions as opposed to using categories or universal properties. I would prefer to learn Abstract Algebra using Category Theory and Universal Properties openly; to do this from Jacobson’s book, I would have to use both volumes together. I’m not sure how to do this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I referred to the &lt;a href=&quot;https://www.ocf.berkeley.edu/~abhishek/chicmath.htm&quot; rel=&quot;noreferrer&quot;&gt;Chicago Undergraduate Mathematics Bibliography&lt;/a&gt;, which suggested that a few portions from Basic Algebra II (such as Group Representation Theory) were best done from elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could somebody please compare using &lt;em&gt;Basic Algebra I, II&lt;/em&gt; , MacLane and Birkhoff’s &lt;em&gt;Algebra&lt;/em&gt; , and Lang’s &lt;em&gt;Algebra&lt;/em&gt;. In particular, please explain their relative merits/demerits, levels of difficulty, ‘modernness’ of the treatment, and quality of the exercises? I enjoy struggling through the texts that are terse, and leave significant gaps (such as in proofs) for the reader to fill in, something like Rudin or Lang's books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, is it better to do any one of these books from cover to cover? Or is it better to do individual sections from each book, or perhaps one book followed by another? If it is the latter two, then could the relevant chapters/order please be pointed out to me?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All comments and answers are greatly appreciated. Thank you so much for your time!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Syllabus&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Rings, ideals, homomorphisms, quotient rings, fraction fields, maximal ideals, factorization, UFD, PID, Gauss Lemma, fields, field extensions, finite fields, function fields, algebraically closed fields.&lt;/li&gt;&#xA;&lt;li&gt;Galois theory: separable and normal extensions, purely inseparable extensions, fundamental theorem of Galois theory. &lt;/li&gt;&#xA;&lt;li&gt;Module theory, structure theorem for modules over PIDs. &lt;/li&gt;&#xA;&lt;li&gt;Multilinear algebra: tensor, symmetric and exterior products, tensor product of algebras. &lt;/li&gt;&#xA;&lt;li&gt;Categories and functors, some notions of homological algebra.&lt;/li&gt;&#xA;&lt;li&gt;Non-commutative rings, semisimplicity, Jacobson theory, Artin-Wedderburn theorem, group-rings, matrix groups, introduction to representations.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user12374" LastEditorUserId="238" LastEditDate="2019-06-18T14:51:39.997" LastActivityDate="2019-07-01T19:17:17.667" Title="MacLane-Birkhoff's &quot;Algebra&quot; vs Jacobson's &quot;Basic Algebra I,II&quot; vs Lang's &quot;Algebra&quot;" Tags="|undergraduate-education|textbooks|graduate-education|abstract-algebra|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16728" PostTypeId="2" ParentId="16727" CreationDate="2019-06-18T16:42:07.947" Score="3" Body="&lt;p&gt;Search [mathematics.se] using the (reference-request) tag, and the (abstract-algebra) tag.  There have been a number of posts, from students at various levels of study in abstract algebra, seeking text recommendations.  One such post, mentions &lt;a href=&quot;https://math.stackexchange.com/questions/239734/a-book-for-abstract-algebra&quot;&gt;Lang's &lt;em&gt;Algebra&lt;/em&gt;&lt;/a&gt;.  It's a very thorough text (scan the &lt;a href=&quot;https://math24.files.wordpress.com/2013/02/algebra-serge-lang.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Table of Contents here&lt;/a&gt;, but doesn't cover category theory explicitly.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another [mathematics.se] post of possible interest to you is &lt;a href=&quot;https://math.stackexchange.com/questions/354221/is-serge-langs-algebra-still-worth-reading?noredirect=1&amp;amp;lq=1&quot;&gt;Is Serge Lang's &lt;em&gt;Algebra&lt;/em&gt; still worth reading?&lt;/a&gt;.  Jacobson's texts are both also recommended, as is Lang's text, in that post, as are other texts you haven't mentioned.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, see a question on [mathematics.se] which addresses &lt;a href=&quot;https://matheducators.stackexchange.com/questions/12812/is-maclane-and-birkoffs-algebra-suitable-today-as-either-an-undergraduate-or&quot;&gt;MacLane and Birkoffs' &lt;em&gt;Algebra&lt;/em&gt;&lt;/a&gt; and it's suitability as a text for upper-level undergrads and/or graduate students. (There are additional posts on [mathematics.se] about this text.  This text seems to include more of the items listed in your syllabus, than the other options. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think all three choices you list are fine choices.  I'd suggest browsing through each, to discern which style, approach(es) best suit(s) your needs.  My decisions, when it comes to self-study, inevitably involves the use of two highly recommended texts.  Inevitably, one of them emerges as by &quot;basis text&quot;, but when struggling with a particular topic, the second text is very useful to have.  I just would not recommend juggling too many texts at one time.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of the level at which you want to self-study Algebra, I came across a rather highly regarded but classic text in pdf format: &lt;a href=&quot;https://bookstore.ams.org/gsm-165/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Advanced Modern Algebra: Part 1&lt;/em&gt; by Joseph Rotman&lt;/a&gt;. (There's an associated text &quot;Part 0&quot; in the series, to be read first, which is Martin's &lt;em&gt;Algebra&lt;/em&gt; text. There's also the sequel (3rd edition) &lt;em&gt;Advanced Modern Algebra: Part 2&lt;/em&gt;.  Combining Parts I and II, the topics covered are shown in the &lt;a href=&quot;https://bookstore.ams.org/gsm-165-180/~~FreeAttachments/gsm-165-180-toc.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;TOC&lt;/a&gt;. In a library, you may only find Rotman's 2nd Edition &lt;em&gt;Advanced Modern Algebra&lt;/em&gt;, which covers much of Part I and II, in one volume.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All I see missing from these texts is an explicit chapter/coverage on Non-commutative Rings. If you can master the other topics, and have time to &quot;taste&quot;/sample non-commutative Algebra: Review &lt;a href=&quot;https://pages.uoregon.edu/anderson/rings/COMPLETENOTES.PDF&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;, &lt;a href=&quot;http://www-math.mit.edu/~etingof/artinnotes.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;, or &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=8&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwi32aWQvPPiAhUI7qwKHctEBgIQFjAHegQIARAC&amp;amp;url=http%3A%2F%2Fmath.uga.edu%2F~pete%2Fnoncommutativealgebra.pdf&amp;amp;usg=AOvVaw3zZ0noShLH8r6KjOH1pR2U&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've used all of the texts I discuss here, (and Artin's Notes on non-commutative algebra), and the choice sometimes comes down to a text that students can use over two semesters studying Algebra, and/or the level of the course and the caliber at which students are at, etc.  &lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-06-18T18:10:02.733" LastActivityDate="2019-06-18T18:10:02.733" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="16729" PostTypeId="1" AcceptedAnswerId="16730" CreationDate="2019-06-18T22:51:34.230" Score="-6" ViewCount="253" Body="&lt;p&gt;Want to ask if someone knows a official site where all kind of rules like &lt;span class=&quot;math-container&quot;&gt;$\infty-\infty$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\infty^0$&lt;/span&gt; are classified. Ment an paper rule collector for that kind of definitions which has an certified character. ok wikipedia is also fine, but thats not that what i ment. in the IT sector you have RFC's similar to papers but an really uniqe form, because the RFC's are strict reglementated by an CA ietf.org. Exists in mathematic also an CA with that kind of strict reglementations? If yes which is it and where can i find all kind of rules which exists in mathematics in one place? And i dont ment wikipedia :D it's cool but still in progress to became an regular CA. thx&lt;/p&gt;&#xA;" OwnerUserId="12353" LastActivityDate="2019-06-19T00:46:53.693" Title="math norms for all kind of ruls" Tags="|resource-request|" AnswerCount="1" CommentCount="9" ClosedDate="2019-06-19T01:39:46.470" ContentLicense="CC BY-SA 4.0" />
  <row Id="16730" PostTypeId="2" ParentId="16729" CreationDate="2019-06-19T00:46:53.693" Score="1" Body="&lt;p&gt;The field of mathematics does not have a central regulatory board, like many computing platforms do (e.g., the IETF RFC's). Definitions of terms become standardized only through common use and consensus -- it is very common for different textbooks, papers, etc. to define the same term differently. There is no site for &quot;official&quot; definitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example of an academic paper reflecting on this issue:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gómez, Bernardo. &quot;Historical conflicts and subtleties with the √ sign&#xA;in textbooks.&quot; &lt;em&gt;6th European Summer University on the History and&#xA;Epistemology in Mathematics Education&lt;/em&gt;. HPM: Vienna University of&#xA;Technology, Vienna, Austria (2010).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-06-19T00:46:53.693" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16731" PostTypeId="2" ParentId="16727" CreationDate="2019-06-20T01:55:02.213" Score="2" Body="&lt;p&gt;I taught myself the basic material from the first 6 chapters of the 2nd edition of Rotman's Advanced Modern Algebra, and I loved it and felt things were explained clearly. Earlier I had tried teaching myself out of Artin's Algebra but got stuck or spent too much time trying to understand with details of the wallpaper group (which in retrospect I wish I hadn't done) and my recollection is that it postpoines galois theory for quite a bit. Rotman on the other hand gets to basic Galois theory at least by chapter 6, possibly earlier. &#xA;I wish I had learned the subject out of Dummitt and Foote, as it is very comprehensive and has lots of examples. &#xA;Paul Cohn's Basic Algebra was also a book I found clearly and concisely written and I taught myself things from there. &#xA;I'm didn't try Lang or Jacobson, so I can't exactly answer your question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should also mention Prof. Anthony Knapp has made basic algebra analysis textbooks he wrote freely available, but I haven't read them but the table of contents look impressive&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.stonybrook.edu/~aknapp/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.stonybrook.edu/~aknapp/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10289" LastActivityDate="2019-06-20T01:55:02.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16733" PostTypeId="1" CreationDate="2019-06-20T16:37:28.950" Score="1" ViewCount="566" Body="&lt;p&gt;This is a variant on the question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14492/small-real-numbers&quot;&gt;small real numbers&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a disagreement with someone about the meaning of &quot;bigger&quot; real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say we have the real number &lt;span class=&quot;math-container&quot;&gt;$-1.$&lt;/span&gt; Is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; &quot;bigger&quot; or &quot;smaller&quot; than &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;? In other words, should we interpret &quot;bigger&quot; to be a synonym of &quot;greater than&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may be considered an &quot;opinion-based&quot; question, but is there a consensus about how to assign meaning to &quot;bigger&quot; or &quot;smaller&quot; (real) numbers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My interpretation is that &quot;bigger&quot; real numbers have larger magnitude, irrespective of sign. For example, I consider that &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; is &quot;bigger&quot; than &lt;span class=&quot;math-container&quot;&gt;$-1.$&lt;/span&gt; To avoid ambiguity (we are mathematicians!), I think it better to say/write &quot;&lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; has a larger magnitude/size than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, I try to avoid using &quot;bigger&quot; or &quot;smaller&quot; with respect to comparing numbers unless it is clear we are comparing their magnitude (e.g. vectors or complex numbers), where we are only comparing non-negative values. I would stick to &quot;greater / less than&quot; or &quot;closer to / further from zero&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regards,&lt;/p&gt;&#xA;" OwnerUserId="7954" LastEditorUserId="11" LastEditDate="2019-06-24T23:36:39.937" LastActivityDate="2019-06-24T23:36:39.937" Title="Interpretation of how to define &quot;bigger&quot; and &quot;smaller&quot; real numbers" Tags="|secondary-education|terminology|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16734" PostTypeId="2" ParentId="16733" CreationDate="2019-06-20T18:20:36.327" Score="3" Body="&lt;p&gt;Since &quot;bigger&quot; and &quot;smaller&quot; are ambiguous, it is best to avoid them, as you mention. The methods you mention seem reasonable, though I am not native English speaker.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Someone may be able to refer to a credible source or official standard, but regardless, you can only know how someone else understands the terms by asking them. So you might as well do that and dispel the confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A mathematician should know that definitions are matters of convenience and communication. As such, a mathematician can easily adapt to the terminology the other person prefers, for the purposes of a given discussion. Most people have less practice with working with explicit and sometimes arbitrary definitions than mathematicians (and philosophers) do.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="64" LastEditDate="2019-06-20T18:33:16.100" LastActivityDate="2019-06-20T18:33:16.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16735" PostTypeId="2" ParentId="16733" CreationDate="2019-06-20T18:39:17.240" Score="1" Body="&lt;p&gt;The words &quot;big&quot; and &quot;small&quot; are relative.  Asking a student: &quot;Is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; small?&quot; is confusing, and meaningless.  Likewise, &quot;bigger&quot; and &quot;smaller&quot; are relative. If you consider six numbers, &lt;span class=&quot;math-container&quot;&gt;$1, 2, 4, 5, 7, 8$&lt;/span&gt;, then each of &lt;span class=&quot;math-container&quot;&gt;$2, 4, 5, 7, 8$&lt;/span&gt; is bigger &lt;strong&gt;&lt;em&gt;than&lt;/em&gt;&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$1,$&lt;/span&gt; &lt;em&gt;and&lt;/em&gt; each of them is &quot;smaller&quot; &lt;em&gt;than&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What &lt;em&gt;you&lt;/em&gt; seem to want to convey, when comparing two real numbers and calling one &quot;bigger than&quot; the other, is that one is &quot;bigger in magnitude&quot; or one is &quot;smaller in magnitude&quot;, where &lt;em&gt;magnitude&lt;/em&gt; can be thought of as the &lt;em&gt;distance&lt;/em&gt; of a number from zero. So &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; is bigger (greater) in &lt;em&gt;magnitude&lt;/em&gt; than is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, where the magnitude of a real number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is given by &lt;span class=&quot;math-container&quot;&gt;$|x|$&lt;/span&gt;.  But without adding the qualifier &lt;em&gt;in magnitude,&lt;/em&gt; I'm afraid you will not be well understood.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, indeed, &lt;span class=&quot;math-container&quot;&gt;$|-2| &amp;gt; |1|$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The more common and well understood terms used to compare two real numbers, 'greater than' and 'less than',  (or, 'equal to'), are used to refer to the stricter interpretation in which the one further to the left on the real number line is less than the other. (And the number further to the right on the real number is greater than the other.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence we write, e.g., &lt;span class=&quot;math-container&quot;&gt;$-83\lt -3 \lt 0 \lt 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd suggest that the use of &quot;greater than&quot; or &quot;less than,&quot; when comparing magnitudes of two numbers, that, e.g., &quot;greater in magnitude&quot; (&quot;lesser in magnitude&quot;) is more widely used than is &quot;bigger than, smaller than.&quot;&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-06-24T16:54:34.517" LastActivityDate="2019-06-24T16:54:34.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16737" PostTypeId="1" AcceptedAnswerId="16762" CreationDate="2019-06-21T16:20:47.757" Score="6" ViewCount="565" Body="&lt;p&gt;I want to introduce to my class to the derivative, but I am still searching for a good, realistic context that isn't too hard to understand, without seeming to be contrived. Do you have an ideas for that?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I want to use a problem that needs to be solved by the use of the first derivative, so that the students come to a preliminary understanding of the derivative as representing the instantaneous &lt;em&gt;rate of change.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Further edit:&lt;/strong&gt; I am introducing students aged 17-18 years, to the first derivative of a function of one variable. So I would say that my question is about teaching high-school students. I am looking for an example to demonstrate the first derivative that has is roots in the everyday life of the students. [Consolidation of a comment from the author of this post.]&lt;/p&gt;&#xA;" OwnerUserId="10036" LastEditorUserId="238" LastEditDate="2019-06-23T02:28:21.023" LastActivityDate="2019-07-01T05:07:34.497" Title="Ideas for the introduction of the derivative?" Tags="|mathematical-pedagogy|calculus|examples|derivative|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="16741" PostTypeId="1" CreationDate="2019-06-23T16:39:48.203" Score="9" ViewCount="745" Body="&lt;p&gt;What is the point of exercises for which answers aren't provided?  (That is to say, what is the pedagogical justification for such exercises? - Edit by someone other than original poster.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Commentary behind the question by original poster:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most if not all courses I've taken, math or otherwise, come with books with exercises where answers aren't always provided. What are students supposed to learn from such exercises?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not necessarily talking about &quot;difficult&quot; exercises here. Indeed it's even worse when &quot;exercise 1&quot; doesn't come with an answer - the student has no way of confirming that he understood even that first paragraph of the entire book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't know how to solve something right away then you need the answer to guide you. The activity of trying and trying until you're completely exhausted and just have to give up, need not be fruitful, the approaches you tried may have been way off.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, providing answers may cause some students to not bother taking the time to try on their own, but that's their responsibility.&lt;/p&gt;&#xA;" OwnerUserId="12389" LastEditorUserId="1608" LastEditDate="2019-06-25T02:49:58.900" LastActivityDate="2019-06-25T02:49:58.900" Title="What's the point of exercises without answers?" Tags="|undergraduate-education|homework|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16743" PostTypeId="2" ParentId="16737" CreationDate="2019-06-23T18:08:21.693" Score="4" Body="&lt;p&gt;OP: &quot;something, that is rooted in the everyday life of the students,&quot;&#xA;&quot;aged 17-18 years.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are two ideas, using data rather than explicit functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1)&#xA;Charts of height vs. age (in the U.S.) show that girls at age &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; grow about &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; cm/yr,&#xA;whereas by age &lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt; the derivative of growth is nearly zero.&#xA;Boys are still growing about &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; cm/yr at age &lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt;.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/Onbv5.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Onbv5.png&quot; alt=&quot;CDC_Girls&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Image from U.S. &lt;a href=&quot;https://www.cdc.gov/growthcharts/clinical_charts.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Center for Disease Control (CDC)&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) Data on hours of sleep vs. age show a negative derivative&#xA;from about age &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; onward, from almost half of a day to &#xA;a third of a day (&lt;span class=&quot;math-container&quot;&gt;$8+$&lt;/span&gt; hours) by age &lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt;,&#xA;representing a slope of about &lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt; minutes less sleep per year. &#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/isDSn.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/isDSn.png&quot; alt=&quot;Sleep&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Image from &lt;a href=&quot;https://faculty.washington.edu/chudler/sleep.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;E.Chudler&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-06-23T20:44:49.523" LastActivityDate="2019-06-23T20:44:49.523" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16744" PostTypeId="2" ParentId="16741" CreationDate="2019-06-23T19:04:10.233" Score="8" Body="&lt;p&gt;To add to Namaste's answer above, two of the thing we're trying to teach in math is literacy and competency with the algorithms used to solve problems and the ability to solve problems for which you don't know the solution. The first is pretty well served by problems with solutions since you need to check your work. The second is the actual activity of doing mathematics, and is heavily informed by the first. And it's hard. But in order to do it, you need to practice doing it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, there's a whole list of skills that surround tackling an unknown problem: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Identifying the problem as being solvable with a certain kind of mathematics.&lt;/li&gt;&#xA;&lt;li&gt;Planning out a logical series of steps and computations that will allow you to solve the problem.&lt;/li&gt;&#xA;&lt;li&gt;Following those steps carefully and adjusting them as you run into issues.&lt;/li&gt;&#xA;&lt;li&gt;Finally, analyzing the solution from a couple of different angles to know if you got something reasonable (The area under the curve is 4) or unreasonable (The area under the curve is -3).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;All of these things take practice and confidence to do well. And they're &lt;em&gt;hard&lt;/em&gt;, but just like playing guitar you can learn all the music theory and tabs you want but eventually you have to actually play. The fact that the doing is different skills is something you want to learn at home, not the first time you get on stage. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematically, it is the difference between being able to do known analyses and being able to come up with a novel prediction mechanism, model, or formal description, use it to say something and be able to robustly describe your reasoning and back up the validity of your claims. To be clear, there's no reason to knock practical competency! A lot of engineering, computer modeling and practical statistics, uses known mathematics; the problems they deal with are interesting for different reasons and involve different kinds of novelty. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can follow your logic through an unknown exercise, check it three ways, and really understand your result, that is when you &lt;em&gt;know&lt;/em&gt; a mathematical concept. Once you are comfortable with these steps, someone can hand you a novel problem and you can begin to reason through it.&lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2019-06-23T19:04:10.233" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16745" PostTypeId="2" ParentId="1671" CreationDate="2019-06-23T20:11:54.377" Score="2" Body="&lt;p&gt;I've used racing cars as an example. In Formula One they have pitstops were the tires are changed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The nuts that attach the tires have to be within a lower and upper bound. If the nut is below the lower bound it'll be too small and not fit on the tire. If the nut is above the upper bound it'll fit too loosely and fly off the tire causing a crash.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when they check if the nuts have been manufactured correctly, they'll have a margin of error corresponding to the bounds.&lt;/p&gt;&#xA;" OwnerUserId="12391" LastActivityDate="2019-06-23T20:11:54.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16746" PostTypeId="2" ParentId="15241" CreationDate="2019-06-23T21:11:32.697" Score="1" Body="&lt;p&gt;If they are going over the same material, cross-help by students is valuable: it shores up confidence, and can lead to useful discussion on approaches that do (or not) work.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2019-06-23T21:11:32.697" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16747" PostTypeId="2" ParentId="16702" CreationDate="2019-06-23T23:57:57.567" Score="5" Body="&lt;p&gt;I have read much of what is here, so, I'd wager the answer I'm about to offer will not be welcomed by some. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I think it might be useful to students to give the following answer:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; we have variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; appearing in the &lt;strong&gt;expression&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;In &lt;span class=&quot;math-container&quot;&gt;$2x = 4$&lt;/span&gt; we have variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; being &lt;strong&gt;constrained&lt;/strong&gt; by the equation &lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt;. We could also call &lt;span class=&quot;math-container&quot;&gt;$2x=4$&lt;/span&gt; a &lt;strong&gt;condition&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Really, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the same thing in both cases. It is a label for a quantity we do not know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In both cases we assume &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; can be multiplied by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;. There is often a natural context for possible choices of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; given the source of the question. Especially if this is from an applied problem it may make sense to assume &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a real variable. But, I think (depending on the student's attention span) expanding the point of context may be unwise in the first pass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At least this is one reasonable interpretation before we find more abstract algebraic ways to think about &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Those abstract algebraic interpretations can wait for the next grade since this is just a question for the 10-year-old crowd.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2019-06-23T23:57:57.567" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16748" PostTypeId="1" CreationDate="2019-06-24T17:15:00.000" Score="5" ViewCount="280" Body="&lt;p&gt;Intuitively speaking, one space is homemorphic to another if one can be deformed continuously to another without tearing and gluing. It is more or less easy to convince the students that a square is homeomorphic to a triangle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How about a strip with two full twists and a standard annulus then? How can I illustrate to my students that it can be continuously deformed to a standard annulus? I failed to find an easy way to convince the students that we can &quot;untwist&quot; the two twists using continuous deformations without tearing and gluing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zMHHu.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zMHHu.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2019-07-09T16:19:35.347" Title="How to intuitively convince the students that a strip with two full twists is homeomorphic to the standard annulus?" Tags="|undergraduate-education|intuition|topology|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16749" PostTypeId="2" ParentId="16748" CreationDate="2019-06-24T17:58:40.777" Score="3" Body="&lt;p&gt;As suggested, I will turn my comment into an answer. Perhaps you could provide each student with a rubberband (or something similar) and provide different challenges for them by showing different shapes and asking which ones are possible and which ones aren't. You could even color the sides of the rubberbands different colors in case the different orientation matters (for something like a mobius strip).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This idea obviously doesn't create any rigorous proof, but it does provide the intuition/visual confirmation for what is/isn't going to be homeomorphic.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2019-06-24T17:58:40.777" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16750" PostTypeId="2" ParentId="16741" CreationDate="2019-06-24T18:06:59.930" Score="5" Body="&lt;p&gt;Texbooks can be used for:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Self-study. In this case they have a healthy dosage of theoretical&#xA;material and a bunch of exercises, which have answers.&lt;/li&gt;&#xA;&lt;li&gt;Guided study, when the teacher explains theoretical material, shows&#xA;how to solve exercises and then checks and grades homework and&#xA;classwork. Some of these textbooks come in two flavors: for students&#xA;and for teachers, in this case the teachers' version has answers and&#xA;some additional guidance.&lt;/li&gt;&#xA;&lt;li&gt;Calling up a student to a whiteboard to solve an exercise in front&#xA;of the class. Then the class is asked whether the exercise was&#xA;solved correctly, and a discussion ensues. If an answer is known&#xA;there will be no discussion.&lt;/li&gt;&#xA;&lt;li&gt;Tests. Split students into two or three or four groups and give each group an exercise that has no answer. Make sure that members of these groups are not able to talk to each other. Works best when the desks are arranged in rows.&lt;/li&gt;&#xA;&lt;li&gt;Search for older material. Most decent books build skills and&#xA;knowledge step by step, so if you missed a step, you need to flip&#xA;back several sections to find what you are missing. This implicitly&#xA;forces you to figure out what exactly you are searching for, an&#xA;important skill by itself.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In most cases the exercises that have no answers can be verified by the student himself: plug in the answer and verify that the equation holds true. If an exercise does not have a single best answer, which happens rarely in school math, then the student has to wait for the teacher to check and grade his homework. This waiting adds additional emotional tension, although most students don't care.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If you don't know how to solve something right away then you need the&#xA;  answer to guide you.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;No, if you don't know how to solve something, then you need to work with the textbook, flip back several pages and see what has been explained there. Search engines condition students to sloppy thinking, sloppy phrasing, and do not require memorizing anything. Stay away from search engines, use your textbook.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2019-06-24T18:42:08.347" LastActivityDate="2019-06-24T18:42:08.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16751" PostTypeId="1" AcceptedAnswerId="16756" CreationDate="2019-06-25T13:10:11.057" Score="6" ViewCount="252" Body="&lt;p&gt;My university is eliminating its developmental math courses, and moving to a system using &lt;a href=&quot;https://www.insidehighered.com/views/2018/07/17/data-already-tell-us-how-effective-co-requisite-education-opinion&quot; rel=&quot;noreferrer&quot;&gt;corequisite remediation&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to develop a coreq for the first course in our &quot;Mathematics for Elementary Educators&quot; series.  We use Beckmann's book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am struggling a bit because we really do build everything &quot;from the ground up&quot; in the regular course, so I do not really see that the course has &quot;pre-requisite&quot; material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is whether anyone is aware of other universities which have developed successful coreqs for these courses.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorDisplayName="user507" LastEditDate="2019-06-25T17:06:30.703" LastActivityDate="2019-07-01T14:02:18.230" Title="Corequisite remediation for &quot;Mathematics for Future Elementary Teachers&quot;" Tags="|primary-education|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16752" PostTypeId="2" ParentId="16737" CreationDate="2019-06-25T15:49:42.817" Score="1" Body="&lt;p&gt;Don't call the slope of a line slope, rather call it rate of change of a function and explain to them the concept again but in term of numbers, not geometrically, so that they get the gist of it, since geometry won't really help them when they'll see derivatives in higher dimensions. Then from the fact that &lt;span class=&quot;math-container&quot;&gt;$a=k\frac{a}{k}$&lt;/span&gt;, tell them that for function which are not linear, we use this to see how the values change locally &lt;span class=&quot;math-container&quot;&gt;$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12395" LastActivityDate="2019-06-25T15:49:42.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16753" PostTypeId="1" CreationDate="2019-06-25T17:31:01.727" Score="2" ViewCount="134" Body="&lt;p&gt;I’m working on a section of a course covering Taylor expansions, and have found that, although there is great notation for simplifying the formula for the coefficients of a general infinite binomial series (i.e. not limited to positive, integer, powers), the computation of the coefficients for any actual series is rather long and messy and (most importantly) unenlightning (i.e. the computations won’t help students gain a better understanding of the expansion). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On Math.SE, I found &lt;a href=&quot;https://math.stackexchange.com/a/767316/179404&quot;&gt;this answer&lt;/a&gt; to a question about the expansion of finite binomial expansions, which is handy, and it made me wonder if anyone had some tips / ideas of how to present the computation of &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The general term, and&lt;/li&gt;&#xA;&lt;li&gt;The first 4-5 nonzero terms&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;of these expansions, quickly and efficiently? For clarity, I’m speaking of expansions of functions of the form &lt;span class=&quot;math-container&quot;&gt;$(a+x)^k = a^k\big(1+\frac{x}{a}\big)^k$&lt;/span&gt;, for &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$k \in \mathrm{I\!R}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if an alternative method emphasized some sort of mathematical reasoning more than the brute-force computation, that would be preferable. E.g. solving the coefficients of &lt;span class=&quot;math-container&quot;&gt;$f(x) = \frac{1}{(1+x)^2}$&lt;/span&gt; by differentiating the series expansion of &lt;span class=&quot;math-container&quot;&gt;$F(x) = \frac{1}{1+x}$&lt;/span&gt;; the later method underscores one of the advantages of working with series, and involves related skills, where as Taylor’s formula for the coefficients is just basic derivatives and pattern recognition... &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;[Edit: addition of specific example, for clarification]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any type of binomial expansion that will have infinitely many terms, i.e. &lt;span class=&quot;math-container&quot;&gt;$(a+x)^k$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is not a positive integer.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;E.g. 1: Find the Maclaurin series for &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sqrt{2-x}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;E.g. 2: Find the Taylor series for &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sqrt[3]{x}$&lt;/span&gt; about &lt;span class=&quot;math-container&quot;&gt;$x = 8$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;E.g. 3: Find the Maclaurin series for &lt;span class=&quot;math-container&quot;&gt;$f(x) = (1-x)^{3 /4}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Currently, I have the following method&lt;/strong&gt; when the goal is to provide both a general term, as well as the first 3-4 nonzero terms. I’ll use the first example to illustrate:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;1. Compute the first few derivatives &lt;span class=&quot;math-container&quot;&gt;$\mathbf{f^{(i)}(x)}$&lt;/span&gt; to determine the general pattern for &lt;span class=&quot;math-container&quot;&gt;$\mathbf{f^{(n)}(x)}$&lt;/span&gt;&lt;/strong&gt;&#xA;  On my own, usually the first 3 suffice to identify the general pattern; when illustrating for students, I’ll sometimes include the 4th and 5th derivatives. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;N.B. : if the desired series can be obtained by integration or differentiation of a simpler series, I will prioritize that method; see the example in the last paragraph, before the edit, for an example&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{array}{ll}&#xA;f(x) = \sqrt{2-x}&#xA; \\ \hline&#xA;%&#xA;f’(x) = \tfrac12(2-x)^{-1 /2} (-1) &#xA;       &amp;amp;  = \quad  -\tfrac12(2-x)^{-1 /2} \\ \hline&#xA;%&#xA;f’’(x) = (-\tfrac12) (\tfrac 12)(2-x)^{-3/2} (-1)^2 &#xA;       &amp;amp;  =\quad  -(\tfrac12) (\tfrac 12)(2-x)^{-3/2} \\ \hline&#xA;%&#xA;f’’’(x) = (-\tfrac32)(-\tfrac12) (\tfrac 12)(2-x)^{-5/2} (-1)^3 &#xA;        &amp;amp; =\quad  -(\tfrac32)(\tfrac12) (\tfrac 12)(2-x)^{-5/2} \\ \hline&#xA;\end{array}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;From this we see that, for &lt;span class=&quot;math-container&quot;&gt;$n \geq 3$&lt;/span&gt;:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f^{(n)}(x) = -(\tfrac12)(\tfrac12)(\tfrac32)(\tfrac52)\cdots(\tfrac{2n-3}{2})(2-x)^{-(2n-1)/2},$$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;which can be simplified as:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f^{(n)}(x) = - \frac{1\cdot 3 \cdot 5 \; \cdots \; (2n-3)}{2^n}(2-x)^{-(2n-1)/2}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;2. Evaluate &lt;span class=&quot;math-container&quot;&gt;$\mathbf{f^{(n)}(x)}$&lt;/span&gt; at the centre to find the general expression for &lt;span class=&quot;math-container&quot;&gt;$\mathbf{f^{(n)}(c )}$&lt;/span&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Plugging &lt;span class=&quot;math-container&quot;&gt;$c = 0$&lt;/span&gt; into the expression for the &lt;span class=&quot;math-container&quot;&gt;$n^{th}$&lt;/span&gt; derivative yields: &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f^{(n)}(0) = - \frac{1\cdot 3 \cdot 5 \; \cdots \; (2n-3)}{2^n}(2)^{-(2n-1)/2},&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;where we can combine the exponents on the powers of 2 to obtain: &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f^{(n)}(0) = - \frac{1\cdot 3 \cdot 5 \; \cdots \; (2n-3)}{2^{(4n-1)/2}},&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;3. Plug the result for &lt;span class=&quot;math-container&quot;&gt;$\mathbf{f^{(n)}(c)}$&lt;/span&gt; into Taylor’s formula to obtain the general term.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Recalling that the coefficients of the Taylor series expansion are giving by &lt;span class=&quot;math-container&quot;&gt;$a_n = \frac{1}{n!} f^{(n)}(c)$&lt;/span&gt;, we have:&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x) = \sqrt{2-x} &#xA;%&#xA;      = 2 - \frac{1}{2\sqrt{2}}x - \frac{1}{16\sqrt{2}}x^2 - \sum_{n=3}^{\infty}  \frac{1\cdot 3 \cdot 5 \; \cdots \; (2n-3)}{n! \; 2^{(4n-1)/2}} x^n,&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What I dislike about the above is that there are so many simple pitfalls that are only overcome either by extensive practice (in which nothing is learnt other than how to find the expansion of these specific series, which will inevitably fade with time and has little-to-no transfer value), or by writing out many intermediate steps, which makes solving them even longer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.g. a common error is to forget to include the factorial terms when writing out the series expansion. An extra line, in which every element of the general term is individually accounted for (e.g. writing &lt;span class=&quot;math-container&quot;&gt;$a_3$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$- \frac{1}{3!} \cdot \frac{1\cdot 3}{2^{11/2}}$&lt;/span&gt; and then computing the final value), avoids these errors, but is quite long to write out for multiple terms (using up time which could be better spent on actually understanding series expansions). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Does anyone know of a more efficient way to get the general term of the sum and/or the first 4-5 nonzero terms of the series?&lt;/strong&gt; The general formulas that use combinatorial notation are fine for summarizing the results, but don’t seem very practical for generating specific series. &lt;/p&gt;&#xA;" OwnerUserId="5797" LastEditorUserId="5797" LastEditDate="2019-06-26T23:35:29.893" LastActivityDate="2019-07-01T05:20:59.887" Title="Student-friendly / efficient approach to computing Taylor coefficients of infinite binomial series expansions?" Tags="|calculus|problem-solving|presentation|calculations|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16754" PostTypeId="2" ParentId="10209" CreationDate="2019-06-25T19:34:06.093" Score="2" Body="&lt;p&gt;I think the existing answers, at this time, all raise important points, and I have nothing to add to the ones already made. However, I feel it is worth emphasizing that, when teaching, it is important to bear in mind the ultimate purpose of the education, and not “make the material fit our teaching” but “make our teaching fit the material.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is simply no question that, &lt;strong&gt;if you are teaching students who are studying&lt;/strong&gt; (or will go on to study) &lt;strong&gt;physics, engineering, math or any other subject that uses applied mathematics, they would be quite literally “uneducated” if they did not understand the difference&lt;/strong&gt; (or lack thereof) &lt;strong&gt;between “Power Series,” “Taylor Series” and “Maclaurin Series.”&lt;/strong&gt; It is simply part of the language and culture of these fields, even if confusing when first learning, and/or inconvenient when teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, if the students will not go on to use these tools and the content is covered in a course that marks the end of their studies, then it is surely less important (for the reasons covered above). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A related issue is the language used to express the centre of convergence: “centred at &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt;” and “in powers of &lt;span class=&quot;math-container&quot;&gt;$(x-a)$&lt;/span&gt; are both equivalent expressions that, for better or worse, students who will use these ideas in the future need to know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;In short, the language really does matter for some students; however, the distinctions are easily introduced once a firm understanding of the ideas exists&lt;/strong&gt; (much as the difference between different breeds of dogs is easily introduced once one has a firm grasp of “dogs” as a group, first!); &lt;strong&gt;in fact, confusion in the terminology can be used to diagnose a lack of understanding of the basic ideas and can, in that way, be turned into a pedagogical advantage.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;On the practical considerations associated with teaching these topics:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found the most success (in avoiding confusion surrounding the language used) by introducing the idea of a power series, developing the related idea of power-series representations, and then simply mention, as an historical / cultural note, that “Taylor/Maclaurin series” is commonly used instead of “power series representation,” once students are comfortable with the ideas. Similarly, I have found that students are better equipped to understand the connection with “tangent lines” and “linear approximation,” after they have grown comfortable with the definitions and computations involved in power series (representations).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’m not sure if the phenomenon is widespread, but students in my neck of the woods generally seem to have a very poor understanding of power series and power series representations (reflecting an equally poor understanding of tangent lines and linear approximation), in general, so focusing on clearing up these points usually makes the later clarification of terminology a non-issue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As suggested by others in the comments and other answers, I have found that emphasizing power-series expansions as an extension of the tangent line is fundamental to helping students understand (again, from an applied-maths perspective, I would want all my engineering and science students to know that “first-order Taylor series expansion,” “tangent line” and “(best) linear approximation” are all equivalent expressions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, a strong emphasis of graphical presentation of the material always seems to be essential for most students. &lt;/p&gt;&#xA;" OwnerUserId="5797" LastActivityDate="2019-06-25T19:34:06.093" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16755" PostTypeId="2" ParentId="10209" CreationDate="2019-06-25T20:08:51.050" Score="4" Body="&lt;p&gt;Even though &quot;Taylor series&quot; and &quot;power series&quot; are the same thing, the names suggest different viewpoints. When I hear &quot;Taylor series&quot;, I think that there was first a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; under consideration and then a series was obtained by evaluating &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and the derivatives &lt;span class=&quot;math-container&quot;&gt;$f',f'',\dots$&lt;/span&gt; at some point, etc. When I hear &quot;power series&quot;, I get no such preconception of where the series came from. It might have been obtained as the Taylor series of some function, but it might also have arisen as the generating function for some combinatorially defined sequence of numbers, or it might have arisen in some totally different way.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2019-06-25T20:08:51.050" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16756" PostTypeId="2" ParentId="16751" CreationDate="2019-06-26T02:35:31.683" Score="5" Body="&lt;p&gt;I would advocate for arguing for no (de facto, even if not de jure) corequisites for such a course.  We don't have any, nor prerequisites, and I think that is the case for a number of institutions in Massachusetts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This course often is a challenge for some pre-service teachers, to be sure.  But I can't think of any course that would bear college credit that would resolve that issue - honestly, not even some of those that cannot bear college credit would do so for the type of content usual in these courses.  Properly done, this course often starts to fill in what no prerequisite course can - that is to say, a sense of understanding why mathematics is not just a set of arbitrary rules to memorize.  You do need a high-quality and compassionate instructor for this - we have been very fortunate to have someone in that role for many years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What might sweeten the pot for your administrators is if you were to have a &quot;lab component&quot; or &quot;tutorial&quot; for students who would not meet whatever prerequisite had formerly been in place. (As SvH points out in the comments, this can be formally labeled as a corequisite.) Especially for ones who still have trouble with times tables or kinds of triangles or percents (and I wouldn't be surprised if you had some), that might be a mechanism that satisfies all constituencies while preserving your pedagogical insight that a corequisite as such would be a hindrance.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2019-07-01T14:02:18.230" LastActivityDate="2019-07-01T14:02:18.230" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16757" PostTypeId="1" AcceptedAnswerId="16764" CreationDate="2019-06-26T14:19:54.680" Score="7" ViewCount="430" Body="&lt;p&gt;I am looking for examples of how polynomial division is presented to secondary students outside of the US. Wikipedia has a nice presentation of this for integer division &lt;a href=&quot;https://en.wikipedia.org/wiki/Long_division#Notation_in_non-English-speaking_countries&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. Do you know of anything similar for polynomial division?&lt;/p&gt;&#xA;" OwnerUserId="1346" LastActivityDate="2019-07-02T19:50:38.530" Title="Non-US polynomial division notation" Tags="|polynomials|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16758" PostTypeId="1" CreationDate="2019-06-26T14:42:44.157" Score="4" ViewCount="424" Body="&lt;p&gt;I am tutoring a 16-year-old student from my home country (in Asia) in, roughly speaking, precalculus. I would like to give him a feeling of procedural justice, so to speak, in modern mathematics, which I wish at some point in my youth a math teacher would somewhat stress. Since my home country's culture possibly overly stresses that sort of engineering mathematics all through the math classes prior to college education, I especially would like to show him a sense of pure mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it seems that perhaps &lt;em&gt;I need to find a way to make this process more smoother&lt;/em&gt;; I found that I am losing this student little by little gradually. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can understand that many people naturally won't appreciate or even care about how one arrives at a result, let alone the discernment of logical rigor. I have no intention to convert any student to become an automatic reasoner; but some experience suggests that many, without a proper sense of procedural justice, would hold instead a business mindset that only learning or seeing important results matters and how a result, important or not, is obtained does not matter. This concerns me in the sense that such a thinking always looks anti-scientific to me; I mean I cannot think of any esteemed field of study that does not stress a sense of procedural justice. Besides, a sense of procedural justice is natural in a sense, since I can hardly find someone who is not curious about what really happened after a magic trick was done for them. On the contrary, many seem surprisingly impatient when being asked to prove &lt;span class=&quot;math-container&quot;&gt;$1+1 = 4/2$&lt;/span&gt;, whose proof (with properly delimited deepness) involves nothing beyond and possibly well below most people's working knowledge. &lt;/p&gt;&#xA;" OwnerUserId="1952" LastEditorUserId="238" LastEditDate="2019-06-26T16:09:31.410" LastActivityDate="2019-07-01T06:23:20.317" Title="Enlighten younger students about the concept of &quot;procedural justice&quot; in mathematics?" Tags="|mathematical-pedagogy|teaching|precalculus|advising|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16759" PostTypeId="2" ParentId="16758" CreationDate="2019-06-26T15:30:59.040" Score="13" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;On the contrary, many seem surprisingly impatient when being asked to prove 1+1=4/2, whose proof (with properly delimited deepness) involves nothing beyond and possibly well below most people's working knowledge.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The practice of starting students out with trivial arithmetic proofs like proving 1+1=4/2 seems to be pretty common, but I'm very skeptical of it.  In fact, I think the problem is precisely that the proof is so trivial.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It doesn't look to students like procedural justice, it looks like a new arbitrary imposition from their teacher.  Suddenly their teacher has decided that some facts are now called axioms, and those count as obvious, while other equally obvious facts for some reason don't count, and they now have to do this new pointless, tedious exercise involving some technical rules about how these axioms work.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I have yet to see anyone successfully produce a list of rules at this level which is intelligible to students, and also has no hidden assumptions.  Getting the technical details about things like equality rules and substitution of terms correct is pretty subtle.  So, in practice, students are usually given a list of rules and are told &quot;you can only use these, and no other facts, even if they seem obvious&quot;, right up until they run against one of the gaps in the system they've been given, at which point they get told something like &quot;of course you can substitute equivalent terms (or whatever the gap is); that's obvious&quot;.  Which just highlights for them how utterly arbitrary the entire activity is.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, for all that work, the students don't see the value added.  The students see no &quot;procedural justice&quot; in the activity, because these new formal arguments are &lt;em&gt;less persuasive&lt;/em&gt; to them than their previous reasoning (&quot;both sides are obviously equal to 2!&quot;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I think you're right that some curiosity about where these things come from is natural, but I don't think proofs of trivial facts like &lt;span class=&quot;math-container&quot;&gt;$1+1=4/2$&lt;/span&gt; actually triggers that curiosity.  I think you need to start with things where students can see the value added, i.e., in places where the answers aren't obvious, and where students can make mistakes, and then see how a more rigorous argument can avoid mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once students have internalized the idea that something can seem obvious but be subtly flawed, &lt;em&gt;then&lt;/em&gt; you can hope to go back to things they thought were obvious and convince them it's worthwhile to be more rigorous.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2019-06-29T13:51:09.033" LastActivityDate="2019-06-29T13:51:09.033" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="16761" PostTypeId="1" AcceptedAnswerId="16765" CreationDate="2019-06-27T00:50:20.720" Score="10" ViewCount="483" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Owing to an injury, I need an alternative solution for delivering a semester-long mathematics course that I was originally going to teach using chalk and blackboard. It seems that a good option might be to use a tablet and stylus to simulate this classic delivery format. Indeed, most lecture rooms here are equipped with a projector and VGA/HDMI connectors. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After browsing around, it &lt;em&gt;seems&lt;/em&gt; like a Microsoft Surface product combined with software such as Drawboard PDF or OneNote &lt;em&gt;could&lt;/em&gt; be what I'm looking for. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Desired attributes:&lt;/strong&gt;&#xA;I essentially want a virtual chalkboard that'll be cast via projector. Added perks might include&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I also tend to touch hand-to-surface when writing, so the monitor should solely react to the stylus in lecture-mode&lt;/li&gt;&#xA;&lt;li&gt;Control over fore/background colours and other nice visual/drawing features&lt;/li&gt;&#xA;&lt;li&gt;The ability to display 'multiple boards' on the projector at once, even while I'm just primed on a specific 'board' on the tablet (to simulate the true lecture theatre experience)&lt;/li&gt;&#xA;&lt;li&gt;Being able to flip back to 'erased' boards&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Could anyone offer any further advice, recommendations or guidance? It'd be great to have some more specific, up-to-date hardware/software recommendations with setup advice. Does anyone have any experience doing this?&lt;/p&gt;&#xA;" OwnerUserId="12398" LastEditorUserId="12398" LastEditDate="2019-06-27T05:22:20.263" LastActivityDate="2020-03-17T21:35:46.263" Title="Delivering mathematics lectures via tablet and projector" Tags="|undergraduate-education|technology-in-education|lecturing|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16762" PostTypeId="2" ParentId="16737" CreationDate="2019-06-27T09:00:23.497" Score="4" Body="&lt;p&gt;There are good examples in the other answers. You can use some of them as an introductory example to the concept of derivative, then propose the case is population growth (the population could be pretty much whatever you want), where &lt;span class=&quot;math-container&quot;&gt;$y'(x)$&lt;/span&gt; is proportional to &lt;span class=&quot;math-container&quot;&gt;$y(x)$&lt;/span&gt;. This will give them a hint of why &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; is so beloved by Math teachers&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can then jump into nuclear desintegration, where &lt;span class=&quot;math-container&quot;&gt;$y(x)$&lt;/span&gt; is the number of remaining atoms and radiation danger is related to &lt;span class=&quot;math-container&quot;&gt;$y'(x)$&lt;/span&gt;, which itself depends on &lt;span class=&quot;math-container&quot;&gt;$y(x)$&lt;/span&gt; (if they are taking this math course, they're probably interested in science)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that, rather than presenting one particular illustrative case they can relate too, I would refer to a wide variety of different applications, since:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Not every student will relate to/be interested by the same examples.&lt;/li&gt;&#xA;&lt;li&gt;If the same solution (calculus) applies to different problems, then they will understand that it's a solution worth mastering rather than looking for a different workaround for each particular case.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of course, everything comes at a cost, and presenting such a variety of examples may take some time that can no longer be dedicated to other things. In the end, it is a compromise, but I think it's much more important that they key concepts of what a derivaitve represents stick in their head, even if they don't get a great skill at computing derivatives (in the real world, when they need it, a computer will do that for them anyway. What the computer won't tell them is when they actually need derivaitves)&lt;/p&gt;&#xA;" OwnerUserId="12399" LastActivityDate="2019-06-27T09:00:23.497" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16763" PostTypeId="2" ParentId="16761" CreationDate="2019-06-27T11:14:15.090" Score="4" Body="&lt;p&gt;My colleagues have been doing this using a Surface and Acrobat. If you just want to write onto a plain page, OneNote will probably serve the purpose, and has the best writing experience. The advantage of Acrobat is you can write onto pre-prepared slides. The disadvantage of it is price. I'm going to try using OpenBoard, which is free, but still somewhat buggy (most noticably the button sizes are off on my screen).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These can generally cope with the hand/stylus issue, but can allow finger-writing as a back-up option. There is some choice of background. Returning to earlier pages is fine too. However, I've not come across anything that allows multiple boards to show at once (unless you spend a LOT more on tech). On the flip-side, you can save the file and upload it to your VLE, so it's not as bad if students don't have time to copy everything in the lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one situation we haven't found a solution to yet is: if you have slides that reveal in stages, and write on an earlier part of the slide, the writing 'disappears' when you bring up more of the slide.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-06-27T11:14:15.090" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16764" PostTypeId="2" ParentId="16757" CreationDate="2019-06-27T16:21:55.297" Score="9" Body="&lt;p&gt;Here is a Russian 7th grade algebra textbook (&lt;a href=&quot;https://catalog.prosv.ru/item/4872&quot; rel=&quot;noreferrer&quot;&gt;publisher's website&lt;/a&gt;). Attached is a complete section dedicated to polynomial division, it is marked as optional.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FDrEL.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FDrEL.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7930" LastActivityDate="2019-06-27T16:21:55.297" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16765" PostTypeId="2" ParentId="16761" CreationDate="2019-06-27T16:31:05.293" Score="6" Body="&lt;p&gt;Another idea: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Use a document camera with lots of paper and markers. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This gives you the advantage of drawing things before class without much planning and/or going back to previous &quot;slides&quot; by simply retrieving the relevant paper. I think you can route a document camera into an hdmi or vga connection. I like this as a solution since it is relatively low-tech and is more close to your usual style. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I think the electronic pens leave a lot to be desired and I've spent a few hours trying to create legible content for grading purposes on pdfs. I own a Bamboo tablet which attempts to recreate some of the texture we take for granted with paper/pen/pencil writing. It does help, but it is still rather ugly at my current skill level. Maybe after I spend a couple hundred hours it'll start feeling natural...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Time pressure teaching, markers on paper ftw.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, this is more or less how I've made some of my videos for advanced students' independent study. The videos linked below I use a camcorder fastened to a wood frame I built about 50cm above where I'm writing. A document camera is similar if you get the right one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=JCor1st0d2E&amp;amp;list=PLBY4G2o7DhF38OEvEImfR2heX7Szmq5Gs&quot; rel=&quot;noreferrer&quot;&gt;My Differential Geometry Playlist&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2019-06-27T16:31:05.293" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16767" PostTypeId="2" ParentId="16761" CreationDate="2019-06-30T00:54:44.443" Score="6" Body="&lt;p&gt;I am teaching at the high school level, not university, but this has been my standard set up for a couple of years now:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I connect my iPad to my notebook, and screen capture my iPad onto my notebook. I then display my notebook via a projector. I use the Notability app on iPad to write notes (using the Apple pencil), and this is projected for students to see. (This app is very flexible with different pens / highlighters / selection tools / etc, and the developers are regularly updating it with new functions) The purpose of also doing the screen capture is to (i) record the lessons, and (ii) allow me to quickly switch from the screen capture application to project other useful applications (ie. a computer algebra system; graphing software; etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main disadvantage of this method is the high cost of the Apple products.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This also does not allow me to display multiple &quot;boards&quot; at once, but it is very quick to scroll back through earlier pages, or select an earlier page from a list of thumbnails.&lt;/p&gt;&#xA;" OwnerUserId="9455" LastActivityDate="2019-06-30T00:54:44.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16768" PostTypeId="2" ParentId="16713" CreationDate="2019-06-30T11:57:56.450" Score="1" Body="&lt;p&gt;I'm learning abacus myself now and am really enjoying it. I actually find it relaxing, whilst I usually find arithmetic tiring and frustrating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like the physical feel of the beads on my fingers rather than looking at ink on paper. I'm able to understand arithmetic better by watching how the beads move and where they end up rather than just looking at a question then an answer. It's one thing to memorise &quot;8 + 7 = 15&quot;, which I never did before, and another to think &quot;what's the best way get those beads to move to get to the answer, as I see the beads as an answer rather than just two numerals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've bought a soroban for my child but I'm not paying a fortune for soroban classes. I'll teach him myself, and if he enjoys it we'll see where it leads to.&lt;/p&gt;&#xA;" OwnerUserId="12360" LastActivityDate="2019-06-30T11:57:56.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16769" PostTypeId="2" ParentId="16757" CreationDate="2019-06-30T12:15:38.017" Score="9" Body="&lt;p&gt;In Italy, polynomial long division is usually presented as in the following example taken from one of the most widely used textbooks for first year high school students (M. Bergamini, G. Barozzi, A. Trifone, &lt;em&gt;Matematica.blu 1&lt;/em&gt;, Zanichelli):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vxmmH.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vxmmH.png&quot; alt=&quot;Polynomial long division&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5531" LastActivityDate="2019-06-30T12:15:38.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16770" PostTypeId="2" ParentId="16761" CreationDate="2019-07-01T04:31:28.120" Score="2" Body="&lt;p&gt;You might consider to just use an overhead projector, with grease pencil and acetate.  You won't have the multiple boards, but you can go back to earlier pages if needed.  (Number as you go, to make this easier, on the fly.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to keep the same feel as blackboard talk, I recommend to write them as you do the discussion.  If you have a canned talk, it is a very different energy.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One other advantage of the transparencies is that they can easily be converted to notes for web (or even handouts) by running through a scanner (or copier).  Note, that there is a little practice in how to do this best in terms of putting your hands down when done writing (not occluding the screen) and which side you sit on.  One small advantage of the OHP is you are facing the audience and it is easier to glance at the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the downsides is probably the energy level, since people don't see you striding around (we are social animals, not computing machines).  Try to compensate for that, by being loud (not yelling, but projecting) and modulating your voice and intensity/interest.  Also, maybe add a little more interaction in terms of Q&amp;amp;A, exercises, etc.  Nothing over the top, but just a little bit more than normal to compensate for the lack of movement energy given by a chalk talk.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, consider to incorporate small amounts of recitation (student to the board), if not already doing this. [Figure out how to do this, if it is impromptu and in a large group, do the better students only...so you don't waste class time.  Other option is volunteers from after group/individual problem solving time.]  This will at least, get your boards filled sometimes.  Obviously, if you need to have the slates erased for next classes, you will need a student volunteer(s) to do this--discuss this with the class during first session.  I anticipate you will get some sympathy and not have issues with volunteers.  (If not, then just order the kids who wrote stuff up, to erase it at class end, as a general practice.  Again, set the expectations during first session when you describe how class will be conducted.) &lt;/p&gt;&#xA;" OwnerUserId="12405" LastActivityDate="2019-07-01T04:31:28.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16771" PostTypeId="2" ParentId="16758" CreationDate="2019-07-01T04:54:45.200" Score="3" Body="&lt;p&gt;&lt;em&gt;&quot;Since my home country's culture possibly overly stresses that sort of engineering mathematics all through the math classes prior to college education, I especially would like to show him a sense of pure mathematics.&#xA;However, it seems that perhaps I need to find a way to make this process more smoother; I found that I am losing this student little by little gradually.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This statement screams of &quot;I like proofy stuff and want to force people to do more of it, despite them not liking it like I do, and despite it not being the curriculum they are supposed to learn to support chem/physics/engineering classes.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My recommendation is to BACK OFF.  Get a win first on the core topic.  Then, and only then introduce the extra proofy stuff (that you love).  Note, if the kid is a weak student, than you basically never will get to this.  [You did not specify if the tutoring was remedial, normal, or enrichment.  But remember, the pedagical &quot;equation&quot; has a variable of student in it, not just of material and method.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even in the case that the kid is above average, consider using other &quot;lures&quot; to interest him in pure math.  It wasn't the picky details of rigorous proof that drew Andrew Wiles into algebraic number theory.  It was a cool problem.  Consider small amounts of brain teasers related to definitions.  For example, &quot;If the second derivative is zero, does that mean it's an inflection point?  If no, give me a simple smooth function to show this.&quot;  [Answer is y=x^4.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, even if the kid is strong (which is a pre-requisite for any push towards pure math), that many strong kids prefer to do physics or the like.  Look at Onsanger, Feynman, etc.  Even for people like this, you can probably still interest them in some details of pure math, but it is more likely to be around definitions, suitability of methods, etc.  More than procedural justice.  Again, I think even the strongest pure mathematicians discovering huge new breakthroughs, don't discover them via procedure.  Rather that is what they do to check their work and publish it.  Again, Wiles is great example.  But many of the writings of Polya* are also applicable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Speaking of which, you might also look at his books on the Stanford math problems (pre calculus, but tricky and interesting) or even his volume of analysis problems (but be very selective here...since many are post calculus). Again...for this even to be an option, it only applies if you have a stronger than average student and he is mastering the base materials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  And it's not like normal curriculum has no proofs.  That's the whole point of geometry--lot of proof writing.  More emphasis on that, than on the actual calculations.&lt;/p&gt;&#xA;" OwnerUserId="12405" LastEditorUserId="4746" LastEditDate="2019-07-01T06:23:20.317" LastActivityDate="2019-07-01T06:23:20.317" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16772" PostTypeId="2" ParentId="16737" CreationDate="2019-07-01T05:07:34.497" Score="1" Body="&lt;p&gt;Velocity is the classic and best example.  Not sure why you don't mention this.  Have you looked at any standard textbooks?  (It's as if you were asking how to introduce combinatorics and were unfamiliar with dice, cards, coin flips.)  If you're dissatisfied with the classics, fine, specify that in your question.  But to not even mention it as if this were a new situation in all pedagogy to be solved from first principles...seems strange.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your students don't sound terribly advanced or accelerated, so I think using the classical example is fine.  Remember that even if it's boring to you...it's new to the students.  They will get plenty of practice later on in, looking at other situations.  But start with the simple one to understand that is part of their lives, kinematics.  Avoid the more complicated ideas from other answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, that there's good crossover for any students in high school physics class as they see same concept in two classes--helps the learning.&lt;/p&gt;&#xA;" OwnerUserId="12405" LastActivityDate="2019-07-01T05:07:34.497" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16773" PostTypeId="2" ParentId="16753" CreationDate="2019-07-01T05:20:59.887" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think the bug can actually be a feature.  Kids need work on the &quot;muscles&quot; of computation.  It's not like this is the only setting where doing long calculations is needed (can be the norm in physics and engineering problems.)  Look at all the questions here about the frustration of dealing with kids that can't perform algebra.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It can be good to look at something new in a slowed down and more mechanical manner, to build familiarity with it.  Or even to look at something in differing amounts of assistance.  Calculating by hand, calculating by scientific calculator, use of spreadsheets, use of CAS.  Graphing by hand (yes on actual paper, with a pencil), graphing with Excel (shows you the data points), graphing with Desmos.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Of course, there is a happy medium.  Don't expect kids to calculate stuff out to 20 terms or the like, as Gauss or Euler might. But also don't be totally put off by making them do some grunt algebra.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12405" LastActivityDate="2019-07-01T05:20:59.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16774" PostTypeId="2" ParentId="11017" CreationDate="2019-07-01T06:54:15.183" Score="3" Body="&lt;p&gt;The one thing I would add--hopefully not considered a segue--is to avoid justifying or opening up a discussion of justifying.  The natural impetus of the students is to assume competence of the instructor.  Don't weaken that.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may feel an urge to confession (&quot;first time teaching this&quot; or &quot;never took this&quot;), but it is likely you are trying to gain sympathy or to set expectations lower.  This is a natural human thing to do, especially for bright people with inverse Dunning-Kreuger.  But don't do it.  Stay positive, stay in control, and just do your best to teach a great course...with a stiff upper lip about your lack of experience.  Nobody wants to hear your issues...they want to learn the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source: instructor training for first aid and skiing.  Absolutely do not tell your classes, you are a first timer.  And I believe the commonality here is psychology and pedagogy.  Not course content.  IOW, the &quot;educators&quot; part is dominant, not the &quot;mathematics&quot; part.&lt;/p&gt;&#xA;" OwnerUserId="12405" LastActivityDate="2019-07-01T06:54:15.183" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16775" PostTypeId="2" ParentId="15506" CreationDate="2019-07-01T16:55:37.650" Score="2" Body="&lt;p&gt;In terms of practical advice, I recommend to concentrate on in-class exercises rather than enrichment or discovery.  In class because students won't do out of class work.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Core exercises rather than discovery or enrichment as that is what they need work on.  I think they will also appreciate that it has high carryover to actual teaching demands (relevant, rather than rarity).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course do some research on the content and what areas are trickiest for students/teachers, within the curriculum.  This is on you to make an effort here.  Not asking for a deep study...but make an attempt.  This shows care in that you consider what is best use of the time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, areas of greater difficulty tend to be from the later grades so spend time on that.  For instance word problems (especially rate, time, distance), percentages, decimals, fraction operations, long division.  But not times tables or basic adding and subtracting.  But come up with your own view...doesn't have to be perfect, but make an effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make an illusion of it not being drill/kill, you can mention that point is to put selves in roles of the children and things they struggle on.  This will be perceived as progressive.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can probably also make some liberal charade about progressiveness in that you are not doing a lot of lecture, but practice instead.  It's actually the old becoming new again, but don't tell them that.  There is probably some newer word for practice...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. I think that asking for &quot;references for best practice&quot; for something that is not a settled solved question is not practical.  You would do better to ask for opinions, anecdotes, etc.  In addition, asking for references and not showing that you have done a lit search yourself (and what you found, didn't) is wrong.&lt;/p&gt;&#xA;" OwnerUserId="12407" LastEditorUserId="12407" LastEditDate="2019-07-01T17:05:35.320" LastActivityDate="2019-07-01T17:05:35.320" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16776" PostTypeId="2" ParentId="16727" CreationDate="2019-07-01T19:17:17.667" Score="-1" Body="&lt;p&gt;other possibilites&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Joseph Gillian - Contemporary Abstract Algebra &#xA;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1133599702&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/1133599702&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/rkLdL.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="1654" LastActivityDate="2019-07-01T19:17:17.667" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16778" PostTypeId="1" AcceptedAnswerId="16783" CreationDate="2019-07-02T01:04:34.467" Score="4" ViewCount="92" Body="&lt;p&gt;We are teaching a real analysis course each week for two hours one day and one hour another day. We use a blackboard and we upload most of the class notes to encourage students to participate instead of furiously taking notes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are trying to figure out a good way to gauge how much time a given proof or example will take to present in real class time (including time for student questions).  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;of course, one can just spend an additional three hours teaching the material to an empty classroom, but we are trying to find more time efficient ways.&lt;/li&gt;&#xA;&lt;li&gt;keep a list of how much proving some theorems took in actual class time. Then use that list to help decide how much time to devote for the current proof, if it is similar to any of the listed ones.&lt;/li&gt;&#xA;&lt;li&gt;Split the lecture into time blocks and add as much material as possible to each of these blocks. For example, we can split a two hour class into 30 minute blocks and then devote each block to some theme/topic/theorem. When we hit the 30 minute mark, we must stop and move to the next topic. The price of this is that you might have rush the proof, which is bad, so we might have to slip into the next block. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8714" LastActivityDate="2019-07-03T07:30:02.650" Title="Gauging correspondence between classtime and material rate" Tags="|classroom-interaction|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16779" PostTypeId="2" ParentId="27" CreationDate="2019-07-02T13:32:49.203" Score="4" Body="&lt;p&gt;Maybe you can do &lt;a href=&quot;https://www.khanacademy.org/math/basic-geo/basic-geometry-pythagorean-theorem/basic-geometry-pythagorean-proofs/v/garfield-s-proof-of-the-pythagorean-theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;President Garfield's proof&lt;/a&gt; and combine it with a history lesson.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;page 161 of the New-England Journal of Education, April 1, 1876 (image from Google Books)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/n1EJy.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/n1EJy.png&quot; alt=&quot;image&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note: M. C. = Member of Congress.&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-07-08T14:57:32.853" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16780" PostTypeId="2" ParentId="27" CreationDate="2019-07-02T17:27:31.110" Score="2" Body="&lt;p&gt;This was in my math folder, I don't know its source. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lqwT3.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lqwT3.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was shown as 'not needing any further explanation', but of course, for lower level students, I connect the 3 points on the circle's perimeter to form the larger right triangle and remind them of the ratios formed by dropping an altitude (line b). &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-07-02T17:27:31.110" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16781" PostTypeId="2" ParentId="16757" CreationDate="2019-07-02T19:50:38.530" Score="9" Body="&lt;p&gt;In Greece the students are taught polynomial division in the second class of upper high school (grade 11 at US educational system). It is the same algorithm as in Italy and Russia. Whole book in pdf form is available &lt;a href=&quot;http://ebooks.edu.gr/modules/document/document.php?course=DSGL-B133&amp;amp;download=/4e09fabdb67e/4e09fac5l2ig&quot; rel=&quot;noreferrer&quot;&gt;here.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BUHJp.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BUHJp.png&quot; alt=&quot;Polynomial division in Greece secondary education&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8643" LastActivityDate="2019-07-02T19:50:38.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16782" PostTypeId="2" ParentId="16778" CreationDate="2019-07-03T06:19:36.447" Score="2" Body="&lt;p&gt;As already suggested in the comment:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Type out everything in the notes. If you want to draw a figure, have it in the notes and know how long it takes to draw. If you want to explain something, have it in the notes and write it down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also one thing that I myself also only learned after multiple lectures: Try to not explain verbally if possible, write everything down. Yes, a verbal explanation might seem nicer at first, but a considerable number of students is writing down everything and only later starts to process and understand it. If you do verbal explanations that are neither on the board nor in the textbook, that might be nice for the good students that already fully understood the topic and listen carefully. But the average or below average students, that are still struggling, will have a lot of trouble reproducing it later at home if part of the explanation is missing.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2019-07-03T06:19:36.447" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16783" PostTypeId="2" ParentId="16778" CreationDate="2019-07-03T07:30:02.650" Score="3" Body="&lt;p&gt;I also tried to estimate this for my lectures in advance for some years but couldn't really figure it out. However, after a several years of teaching math I got pretty good at estimating. Here is what I got:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If I prepare lectures by hand on paper, I do roughly one page per 20 minutes for introductory courses (first year students) and up to one page per 15 minutes for advanced students. (I figured this out by putting time markers in my handouts during lectures.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If I prepare TeXed handouts, I take a little more minutes per page, but not much.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This needs adjustment: proof-heavy pages take a little longer, matrix-heavy pages take much less time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I always plan conservative, i.e. a little less material than I would expect to cover at first guess.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If I am very unsure, I plan even less material but bring some additional examples to fill up the lecture.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is possible to explain the same thing/proof/example in different time without actually just speeding up or slowing down, but by taking a different route. This is quite difficult to do on the fly at the blackboard, but it is a skill that can be practiced (with good planning).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A final note: It does not really matter, how much material you can cover, but how much the students learn. So do not only track how long it takes to cover the material, but do an effort to try to figure out if the students are still up to date with the stuff you are teaching.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2019-07-03T07:30:02.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16784" PostTypeId="1" CreationDate="2019-07-03T10:39:15.737" Score="5" ViewCount="669" Body="&lt;p&gt;I’ve been teaching my kids addition tables (1+3=4, 2+3=5, 3+3=6, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I only just found out about number bonds (1+4=5, 2+3=5, 4+1=5). This seems a better method because it’s mastering all the components of a single number, rather than mastering what a single number like 2 will do to all the other numbers. Am I right? Or should I teach rote memorisation of both?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My 5 year old seems pretty good at problem solving but I’d like him to improve his rote memory. He actually enjoys it so I’m not taking time away from focusing on understanding.&lt;/p&gt;&#xA;" OwnerUserId="12360" LastEditorUserId="238" LastEditDate="2019-07-04T01:39:53.780" LastActivityDate="2019-09-13T15:24:05.600" Title="What’s better: number bonds, or addition tables?" Tags="|primary-education|arithmetic|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16785" PostTypeId="2" ParentId="16784" CreationDate="2019-07-03T13:00:12.453" Score="2" Body="&lt;p&gt;From the Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Number_bond&quot; rel=&quot;nofollow noreferrer&quot;&gt;Number Bond&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In mathematics education at primary school level, a number bond&#xA;  (sometimes alternatively called an addition fact) is a simple addition&#xA;  sum which has become so familiar that a child can recognise it and&#xA;  complete it almost instantly, with recall as automatic as that of an&#xA;  entry from a multiplication table in multiplication.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When I try to paraphrase this, I get &quot;A &lt;em&gt;number bond&lt;/em&gt; is to simple addition what strong memorization is to the multiplication table.&quot; In effect, the phrase itself is not a distinction of two processes, rather something closer to a grade school teacher saying, &quot;last year, M. Smith did so well with their students, memorizing their addition facts, that all but 2 have really internalized it, forming a Number Bond, and not grabbing the calculator as much as they used to.&quot; In other words, it appears to be used in context as a level of understanding not a process in and of itself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: The article also states &quot;the term &lt;em&gt;number bond&lt;/em&gt; is sometimes derided as a piece of unnecessary new mathematical jargon.&quot; A &lt;a href=&quot;http://mental-arithmetic.co.uk/Number-Bonds-to-20-Worksheet-1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;worksheet&lt;/a&gt; asking students to find a number, which when added to the first, sums to 20, is linked as an example. These sheets have been around for 50+ years, and were just referred to as &quot;addition drills&quot;. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-07-03T13:05:30.283" LastActivityDate="2019-07-03T13:05:30.283" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16789" PostTypeId="2" ParentId="16784" CreationDate="2019-07-04T09:06:19.723" Score="4" Body="&lt;p&gt;My sense is that the formalism of &quot;number bonds&quot; is a bad idea. Here are some reasons.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;At this level there is no need for new jargon and little good is accomplished by introducing it. It only creates potential new confusion. &quot;Number bond&quot; is particularly terrible terminology, as children are very unlikely to be familiar with the word &quot;bond&quot;, whose usual meanings have, moreover, little to do with addition. Additionally, there is already well established terminology for these &quot;bonds&quot; - they are (ordered) partitions of the integer into two parts. It is never a good idea to invent new terminology for well established notions. &quot;Partition&quot; is better terminology because it reflects the (easily explained to a child) idea of representing the number by a linear series of dots and dividing the series somewhere.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;While it's quite reasonable to pose, as an exercise, the problem of finding all (ordered or unordered - two different problems) pairs (implicitly of positive integers) that sum to a given integer (e.g. &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;), using such an exercise as the basis for teaching arithmetic seems misguided, because on the one hand no one can answer it that has not understood how to sum natural numbers, and, on the other hand, there is no general reason to link pairs whose sums are equal (unless one is studying partitions of integers, a topic that is not generally broached at the elementary school level).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't see how &quot;number bonds&quot; differs operationally from usual rote memorization except in that it organizes the material to be memorized in what might (and possible should) appear to a child as a bizarre way.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-07-04T09:06:19.723" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16790" PostTypeId="1" CreationDate="2019-07-04T14:05:10.310" Score="-2" ViewCount="382" Body="&lt;p&gt;I see many teachers use slow methods to solve a given problem where there's another faster methods that doesn't demand much more effort. I'm not looking for mistakes like saying that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is the slope of &lt;span class=&quot;math-container&quot;&gt;$y=ax-x$&lt;/span&gt; or that &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;\sqrt{2}$&lt;/span&gt; is the solution of &lt;span class=&quot;math-container&quot;&gt;$x^2&amp;lt;2$&lt;/span&gt; (yes these are real examples), I'm looking for methods that are correct but slow and inefficient. Here are some examples.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Simplifying radicals&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Simplify &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2^6}$&lt;/span&gt;. I see most middle school teachers do it like this &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2^6}=\sqrt{2^2\times 2^2\times 2^2}=2\times 2\times 2=8$&lt;/span&gt; instead of teaching students to take half of the exponent.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Table of signs&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;To solve the inequation &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \dfrac{(x^2-9)(2x+1)}{(x-2)(x+1)(-x^2-2)}&amp;gt;0$&lt;/span&gt;, we study the sign of &lt;span class=&quot;math-container&quot;&gt;$\displaystyle f(x)=\dfrac{(x^2-9)(2x+1)}{(x-2)(x+1)(-x^2-2)}$&lt;/span&gt;. We can do it the long way studying the sign of each factor &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vGJNn.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vGJNn.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or the short way&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/aIVYz.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aIVYz.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We take the highest degree term of each factor: &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-x^2$&lt;/span&gt;. The product of their coefficients is negative so we start with a minus on the right and change the sign each time we encounter a simple root (or root of odd multiplicity) or pole. This way is more efficient not only because the table is much smaller but also because students don't have to memorize the table of signs of &lt;span class=&quot;math-container&quot;&gt;$ax+b$&lt;/span&gt; and of &lt;span class=&quot;math-container&quot;&gt;$ax^2+bx+c$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Integration by parts&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Writing &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$u'$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v'$&lt;/span&gt; three times to find &lt;span class=&quot;math-container&quot;&gt;$\int x^3 \mathrm{e}^x \mathrm{d}x$&lt;/span&gt; is tiresome and takes a whole page whereas with &lt;a href=&quot;https://en.wikipedia.org/wiki/Integration_by_parts#Tabular_integration_by_parts&quot; rel=&quot;nofollow noreferrer&quot;&gt;tabular integration&lt;/a&gt; it takes 2 lines.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Differentiation&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;To find the derivative of &lt;span class=&quot;math-container&quot;&gt;$\displaystyle f(x)=\dfrac{(x^2+x-1)^3 \sqrt{x^2+1}}{\left(\ln\left(\cos x\right)\right)^2}$&lt;/span&gt;, I prefer to do it this way :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f'(x)=\dfrac{\left((x^2+x-1)^3 \sqrt{x^2+1}\right)'\left(\ln\left(\cos x\right)\right)^2-\left((x^2+x-1)^3 \sqrt{x^2+1}\right)\left(\left(\ln\left(\cos x\right)\right)^2\right)'}{\left(\ln\left(\cos x\right)\right)^4}\\=\dfrac{\left(3(2x+1)(x^2+x-1)^2 \sqrt{x^2+1}+\frac{2x(x^2+x-1)^3}{\sqrt{x^2+1}}\right)\left(\ln\left(\cos x\right)\right)^2+2\tan x \ln(\cos x)\left((x^2+x-1)^3 \sqrt{x^2+1}\right)}{\left(\ln\left(\cos x\right)\right)^4}$$&lt;/span&gt;&#xA;Although it is long (~ half page), it's still better than naming &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; each time. The best method to do this is the logarithmic derivative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;These examples are from school but feel free to add examples from undergraduate mathematics. &lt;span class=&quot;math-container&quot;&gt;$\color{red}{\text{I'm not asking why do teachers still use them}}$&lt;/span&gt;; I'm asking for other examples of such methods.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="501" LastEditDate="2019-07-05T15:50:28.190" LastActivityDate="2019-07-05T15:50:28.190" Title="Inefficient methods" Tags="|undergraduate-education|secondary-education|teachers|middle-school|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="16791" PostTypeId="2" ParentId="16790" CreationDate="2019-07-04T15:07:51.647" Score="0" Body="&lt;p&gt;To solve the system &lt;span class=&quot;math-container&quot;&gt;$$\left\lbrace\begin{array}{l}\dfrac{1}{x+5}\geq\dfrac{1}{x-2}\\ \dfrac{(x-3)^2}{3}\leq (x-1)^2-8\\ -x(x+5)&amp;lt;0\end{array}\right.$$&lt;/span&gt;&#xA;many teachers draw a table of signs for each inequation and then find the intersection of the three partial solutions on an axis. It's faster to use one table to solve it.&lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2019-07-04T15:07:51.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16792" PostTypeId="2" ParentId="16784" CreationDate="2019-07-04T16:09:55.463" Score="5" Body="&lt;p&gt;I agree with the now-deleted answer from @Jedrek_Mansfield. It's not so much that one is better than the other. The approaches complement one another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have had plenty of experience with adults who still have trouble with basic addition and subtraction. Yesterday a student couldn't do 104 - 8 in her head. I showed her that I broke the 8 down into 4 and 4. Of course she knew 104 - 4, and it turned out she could do 100 - 4 also. I don't use the phrase 'number bonds' in a situation like that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once in a while I have a student who is willing to put some work into improving their basic arithmetic skills. And then I might mention the term to help them see the concept. I most often focus on the number pairs for 10. Many of my students don't see how to use those to their advantage: 16 + 7 = 16 + 4 (now I've 'made a ten') + 3 more = 23. Practicing breaking a problem down this way can be a tremendous help to students who didn't have strategies for this sort of thing before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a lovely picture book, &lt;a href=&quot;https://www.betterworldbooks.com/product/detail/Quack-and-Count-9780152928582&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quack and Count&lt;/a&gt;, by Keith Baker, that uses this concept well, showing all the combinations of ducks that add to 7. “Slipping, sliding, having fun, 7 ducklings, 6 plus 1.” I have long wanted to teach a math for elementary teachers course, so I could get students to make picture books like this for other sums. Years ago, my son was in a preschool and I watched the teacher play a game with them where the squirrel was hiding nuts. Here are 3, something is hidden, now there is 1. The kids &lt;em&gt;loved&lt;/em&gt; telling her that 2 were hidden.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're interested in how life-changing something like this can be, you might want to read the chapter The Math Haters Come Around, by Tiffany Bearup, in my book, &lt;em&gt;&lt;a href=&quot;https://naturalmath.com/playingwithmath/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers&lt;/a&gt;&lt;/em&gt;. (You can download a pdf of this CC-licensed book for free. But I believe it's worth the cover price, if you want to read more than that one chapter.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2019-09-13T15:24:05.600" LastActivityDate="2019-09-13T15:24:05.600" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16793" PostTypeId="2" ParentId="16790" CreationDate="2019-07-05T00:20:59.910" Score="0" Body="&lt;p&gt;Simplifying equations in one unknown.  I seem to remember some formulaic system of seven steps or so, including showing explicitly operations on each side (e.g. subtract 4 from each side, rather than toss over the equals and change sign).  Note, the justification is more understanding, as well as less errors.  Probably benefits to doing it long way at first, before learning faster method (similar to integration by parts).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this is not a segue, but one similar thing is teaching of Excel design, where the advice is to show each operation with a new column or row.  &quot;Rows are cheap.&quot;  Instead of some super long computer program like formula in the cell formula, use a new column or row for each operation.  Makes future debugging or modifying easier.&lt;/p&gt;&#xA;" OwnerUserId="12413" LastActivityDate="2019-07-05T00:20:59.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16794" PostTypeId="2" ParentId="16784" CreationDate="2019-07-05T02:39:24.140" Score="1" Body="&lt;p&gt;During this time I've been playing around with a Japanese abacus (soroban) which is a base-5 method, and I'm thinking it's more valuable for learning addition &amp;amp; subtraction quickly than rote memorisation and number bonds. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With a soroban, you have to mechanically operate the beads to get the correct answer. You start off moving them one by one and as you run out of beads on a rod you start a new rod. Sometimes you get stuck... eg. I need to add 4 beads in the units rod, but there's only 2 beads left. So I'll move 2 beads in the units rod, reset the rod to 0 and move 1 bead on the next adjactent 10's rod, then move one more bead in the units rod again. With some practice, it start to become second nature and you stop moving them one by one and just start forming the image of what you know the answer will look like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It gets even trickier when you have to add 10, minus 5 and add 1 just to add a single bead. But once you get the hang of that you can do the same bead operation for any size number. Hundreds, thousands, billions, they all look exactly the same on a soroban so they're not at all intimidating because it's an identical operation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if I want to do 17 + 36, what I'd do on a soroban is 1+3 in the 10's rod, 7+6=3 in a unit rod and add 1 to the 10's rod. I get to see the answer 53 without having to use any intuition. With practice I start to figure out how and why it's that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I might do away with all rote memorisation of tables and just focus on the abacus. So long as my child does it often (haven't been able to so far) he'll manage to learn number combinations AND traditional number+number calculations with a single method.&lt;/p&gt;&#xA;" OwnerUserId="12360" LastActivityDate="2019-07-05T02:39:24.140" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16795" PostTypeId="2" ParentId="16784" CreationDate="2019-07-05T03:48:03.420" Score="0" Body="&lt;p&gt;Another option is to just teach the addition tables, then tell him that subtraction uses the same numbers. So 3+2=5, therefore 5-2=3, and 5-3=2.&lt;/p&gt;&#xA;" OwnerUserId="12360" LastActivityDate="2019-07-05T03:48:03.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16796" PostTypeId="2" ParentId="16790" CreationDate="2019-07-05T12:41:04.303" Score="1" Body="&lt;p&gt;The three methods for solving for the zeros of a quadratic equation -&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Factoring&lt;/em&gt; (and perhaps realizing that when the discriminant isn't a perfect square, no integer solution)&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Completing the Square&lt;/em&gt; - and the process for this, which many students struggle with.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Quadratic Equation&lt;/em&gt; - Their 'goto' method, often despite a potential simple factored result.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Use Calculator Solver&lt;/em&gt; - Ironically not 100% as students can still have a typo entering the original numbers. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-07-05T12:41:04.303" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16797" PostTypeId="2" ParentId="16790" CreationDate="2019-07-05T15:19:09.417" Score="0" Body="&lt;p&gt;One explanation is that the teacher doesn't know any better.  I think this is going to be more common at the earlier the education levels, i.e. elementary school teachers are going to be less fluent than middle school teachers, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another reason is that the student may not yet know the concepts that you think are better.  Your radical simplification example stood out as an example of this.  I've never taught middle school math but I've had children at that age and I'm pretty sure rational exponents are still years down the road.  In this situation, the teacher is working with the tools available.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A third reason, that comes up more often in my college classes, derives from the nature of undergraduate math education.  The vast majority of my students, even into the calculus series, aren't going to become mathematicians.  Their interest in the material is almost certainly never going to extend beyond using it to solve non-mathematics problems.  In that sense, the real goal of the class is to give them the tools that they need to succeed in their future classes in physics, engineering, chemistry, computer science, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that goal in mind, my goal is to give them as wide a selection of tools as possible.  Some of them will be quick and easy to use, e.g. l'Hopital's Rule, where others may be less straightforward, e.g. evaluating a limit with radicals by rationalizing the numerator, but by putting both methods in their toolbox I'm maximizing the options that they have going forward.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2019-07-05T15:19:09.417" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16798" PostTypeId="2" ParentId="14818" CreationDate="2019-07-05T15:46:34.063" Score="0" Body="&lt;p&gt;I don't know if I understood your question completely. You're asking for an intuitive explanation why &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \left\lbrace\begin{array}{l}s=x+5\\t=y+3\end{array}\right.$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider an arbitrary point &lt;span class=&quot;math-container&quot;&gt;$M(x,y)$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are the distance from the projections of &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; on the axis to the origin. What are &lt;span class=&quot;math-container&quot;&gt;$(s,t)$&lt;/span&gt;? There's &lt;a href=&quot;https://en.wikipedia.org/wiki/Active_and_passive_transformation&quot; rel=&quot;nofollow noreferrer&quot;&gt;two ways&lt;/a&gt; to answer it.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Active transformation&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;In this case &lt;span class=&quot;math-container&quot;&gt;$(s,t)$&lt;/span&gt; are the coordinates of the point &lt;span class=&quot;math-container&quot;&gt;$M'$&lt;/span&gt; image of &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; by the translation of vector &lt;span class=&quot;math-container&quot;&gt;$(5,3)$&lt;/span&gt;. For example the image of &lt;span class=&quot;math-container&quot;&gt;$A(2,2)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$A'(7,5)$&lt;/span&gt;. Here we transform the point leaving the coordinate system intact.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Passive transformation&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;I think what you're looking for is a passive transformation. In this case we don't find the image of &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, we find a new coordinate system in which the coordinates of &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$(s,t)$&lt;/span&gt;. This is equivalent to a translation of the coordinate system by a vector opposite to the previous vector, that is &lt;span class=&quot;math-container&quot;&gt;$(-5,-3)$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way to see it is to translate &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$M'$&lt;/span&gt; using the vector &lt;span class=&quot;math-container&quot;&gt;$(5,3)$&lt;/span&gt; leaving the coordinate system intact (active transformation until now) then translate back &lt;span class=&quot;math-container&quot;&gt;$M'$&lt;/span&gt; (attached to the coordinate system) into &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, that is translate &lt;span class=&quot;math-container&quot;&gt;$M'$&lt;/span&gt; and the coordinate system by the vector &lt;span class=&quot;math-container&quot;&gt;$(-5,3)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometrically, &lt;span class=&quot;math-container&quot;&gt;$x+5$&lt;/span&gt; is the horizontal distance between &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; &amp;amp; &lt;span class=&quot;math-container&quot;&gt;$(-5,0)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y+3$&lt;/span&gt; is the vertical distance between &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(0,-3)$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; have coordinates &lt;span class=&quot;math-container&quot;&gt;$(x+5,y+3)$&lt;/span&gt; iff the origin is &lt;span class=&quot;math-container&quot;&gt;$(-5,-3)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JZHtZ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JZHtZ.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="501" LastActivityDate="2019-07-05T15:46:34.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16799" PostTypeId="2" ParentId="14818" CreationDate="2019-07-05T16:27:19.763" Score="2" Body="&lt;p&gt;Looking at the analogy in your question, suppose someone was confused about whether &lt;code&gt;mymoney = yourmoney + 1&lt;/code&gt; made me richer, or you richer, compared to &lt;code&gt;mymoney = yourmoney&lt;/code&gt;. How would you help that person understand?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is pretty clear: you would tell them to try some values. Ask them: If &lt;code&gt;mymoney = 5&lt;/code&gt;, what is &lt;code&gt;yourmoney&lt;/code&gt; in the two cases?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Now looking at your question, suppose someone is confused about whether &lt;code&gt;y = x+1&lt;/code&gt; moves a point to the left, or to the right, compared to &lt;code&gt;y = x&lt;/code&gt;. How should you help that person understand?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my answer to your question is that you should also tell them to &lt;strong&gt;try some values&lt;/strong&gt;. Ask them: if &lt;code&gt;y = 5&lt;/code&gt;, what is &lt;code&gt;x&lt;/code&gt; in the two cases?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2019-07-05T16:27:19.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16801" PostTypeId="1" CreationDate="2019-07-09T14:05:25.760" Score="6" ViewCount="514" Body="&lt;p&gt;I just had a look at the curriculum of a university's math faculty 100 years ago. Most of the topics there are the same as the topics taught today, including complex analysis, differential equations, etc, but there are some major differences:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Functional analysis had not developed fully at that time, so this was not in the curriculum. Real analysis was also not in it.&lt;/li&gt;&#xA;&lt;li&gt;There seemed to be a course about special functions such as &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt; and beta functions in the past. But now this seems to be out of fashion - students don't learn it until they have to use it.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I am just wondering which course is more old fashioned than which. For example, functional analysis and quantum mechanics are definitely more modern than gamma functions and beta functions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What other topics have been added/excluded in the past 100 years? For what reason were they added/excluded?&lt;/p&gt;&#xA;&lt;p&gt;(I would be interested if anyone can predict what topics will be added in the future as well, although this might be too opinion-based.)&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="376" LastEditDate="2022-11-24T12:47:26.593" LastActivityDate="2022-11-24T12:47:26.593" Title="&quot;Out of fashion&quot; topics in degree level math" Tags="|undergraduate-education|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16802" PostTypeId="2" ParentId="16748" CreationDate="2019-07-09T16:19:35.347" Score="7" Body="&lt;p&gt;First of all, let me echo all the comments -- the key point here is that these surfaces are homeomorphic, but this homeomorphism cannot be realized by an isotopy. This is an important distinction! It is probably worth taking some time to explain the issue. In particular, students should understand that topological spaces don't naturally come embedded in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; and that homeomorphism is a property of the abstract spaces. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think I would draw this picture on the black board and &lt;b&gt;ask the students&lt;/b&gt; to complete the bottom diagram.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/H0nBk.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/H0nBk.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't done this particular example in class, but I do use this sort of pictures for  cylinders, Mobius strips, tori, Klein bottles and projective planes all the time, so it would tie into that familiarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can imagine a fun activity here. Pass out the three types of strips to the students and have them triangulate them by drawing with pen. Have them carefully record which vertices are joined by edges and which three cycles are filled with triangles. Then pass that list of combinatorial data to another team and see if that team can determine which simplicial complex is which strip. If this works for you, let me know how it goes!&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2019-07-09T16:19:35.347" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16803" PostTypeId="2" ParentId="16801" CreationDate="2019-07-09T18:36:24.403" Score="9" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Spherical trig: huge one that was very much the norm, through WW2 (heavy navigation applications), now not a part of the standard course.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Solid geometry: still a part of the standard course, but emphasized more back in the day.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Projective geometry: (See bullet 2, but even more so.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As for &quot;real analysis&quot;, I think advanced analysis concepts (epsilon delta, continuity, etc.) were out there, if not emphasized the same way (and not a topological interpretation).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's probably also useful to note that the Tripos emphasized a very computational, puzzle-y, problem-solving type of approach to math.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;General editorial comments:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not all good or bad. It's probably still a mildly better course for people going into applied areas (physics/engineering) where &quot;the 19th century math is alive and well&quot; as opposed to the extreme rigor/proof emphasis that graduates students who can't solve AP Chemistry stoichiometry problems with two equations in two unknowns and a few conversion factors.  Also, note that Ramanujan got massive inspiration from an encyclopedic mechanistic prep book for Tripos students (of the most abhorrent kind to those who obsess over abstraction at the expense of all else).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All in all, I think the way to look at &quot;old school&quot; stuff is as a source of hidden/forgotten gems.  But not to rely on.  Perhaps not to be returned to completely (for one thing the writing can be too flowery), and also of course, there are huge overlaps with current practice or things you miss, or don't want.  But still dipping into the old stuff can be a great idea starter.  Edwards Integral Calculus Treatise (a supposedly starter calculus course!) has freaking polylogs in it, with references to 1700s papers and such.  And many problems can be found in old books that are good homework (for advanced students).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, older books (especially in English or from England) will click more with mathematical physicists or applied mathematicians and less with &quot;don't want to do algebra because it's too trivial for me to do the work on&quot; abstraction lovers and definition/caveat eagle-eye types.  But even for them, the writing can be a hurdle if too flowery (a fraction are not flowery...but many are.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, note that there are (usually) modern treatments of the less fashionable topics.  So you are not forced to use books that are sometimes hard to read.  All that said, I think occasionally dipping into old stuff is a good way to find ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I think it's very fair to find many areas where the course has not changed much.  If you had a great knowledge of algebra 1, 2, trig, geometry, and first year calculus, and ODEs/PDEs, you would be just as good as someone from nowadays at least in terms of engineering math.  Maybe within that course, the one thing lacking would be strength in vectors (not even vector analysis) as it is emphasized more now and earlier.&lt;/p&gt;&#xA;" OwnerUserId="12420" LastEditorUserId="11" LastEditDate="2019-07-13T15:05:22.367" LastActivityDate="2019-07-13T15:05:22.367" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16804" PostTypeId="2" ParentId="16801" CreationDate="2019-07-10T20:10:28.667" Score="10" Body="&lt;p&gt;I'm not a historian nor would I trust one unless they were also a mathematician, but it seems to me &lt;strong&gt;Abstract&lt;/strong&gt; or &lt;strong&gt;Modern Algebra&lt;/strong&gt; was probably not part of the undergraduate curriculum at non-elite schools. This is in part due to the fact that parts of what we teach now were not yet discovered. For example, Noether's treatment of ring theory brought out into the open some things which were probably only known to experts before her time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the &lt;strong&gt;Introduction to Proofs&lt;/strong&gt; course we find a lot of places is also a relatively modern invention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Differential Equations&lt;/strong&gt; as an introductory course has changed a lot in terms of rigor. If you look at older texts you find much more attention to analysis and/or computational generality. Now we have books like Blanchart and Devaney and Hall which go so far as to relegate power series solutions to an Appendix. Forget about any real attention to orthogonal polynomials or the Frobenius method. Of course this is done in the name of giving attention to &lt;em&gt;modelling&lt;/em&gt;, but I can't shake the feeling it is just a ploy to not face the music of student's inability to hack real calculus calculations. I suspect, the older DEqns course was more of a junior or senior level course where much more mathematical maturity was assumed. Or, it was a course that was taken by engineering students who were more elite having passed through a serious matriculation process. Anyway, my point is just that this course may have the same name now and in 1919, but it is a radically different course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, now that I think about your question a little while, it occurs to me the real answer is that &lt;strong&gt;none&lt;/strong&gt; of the courses are the same. We have introduced new notation, philosophy and technology to all subjects which change the very nature of each course. Furthermore, even in the most pure of topics, surely we learned something in the last century to improve how we teach the material. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2019-07-10T20:10:28.667" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16805" PostTypeId="1" CreationDate="2019-07-11T00:10:20.933" Score="7" ViewCount="304" Body="&lt;p&gt;I participated in some high-school-level math competitions, but I never invested much of my time in them at that time, something that I regret immensely. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that I am an engineering grad student, I want to do something about it and grab the opportunities to compete in college-level math competitions. But, I am having a hard time finding free materials to prepare for them, online. I found a lot of high-school-level resources, but haven't had much luck in finding college-level resources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know I'm already asking for a lot of help, but I'm poor in a country where the dollar is really expensive. So if possible (I know it might not be possible), please help me find only free resources for preparation.&lt;/p&gt;&#xA;" OwnerUserId="12423" LastEditorDisplayName="user7990" LastEditDate="2019-07-11T19:06:49.113" LastActivityDate="2019-07-16T18:09:33.030" Title="How to study for university level math olympiads and competitions?" Tags="|reference-request|self-learning|math-contests|" AnswerCount="1" CommentCount="12" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16806" PostTypeId="2" ParentId="14146" CreationDate="2019-07-11T07:27:20.287" Score="8" Body="&lt;p&gt;I know I'm late to the party, but I'd like to offer a different type of answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say that, as described, you are not in a good position to do a PhD, in mathematics or otherwise (and may find an undergraduate degree hard too), not because &lt;em&gt;physics&lt;/em&gt; is essential, but because &lt;em&gt;learning&lt;/em&gt; is. As a teenager you have said that you are not willing to consider changing your views. This sort of mindset will not serve you well in trying to discover new knowledge.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-07-11T07:27:20.287" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16807" PostTypeId="1" CreationDate="2019-07-11T19:26:15.557" Score="6" ViewCount="859" Body="&lt;p&gt;As a high school teacher, I sometimes wonder about the usefulness of certain topics.  Some topics seem to be in the textbook because they have always been there, not because they lead anywhere interesting.&lt;/p&gt;&#xA;&lt;p&gt;For instance, I am fairly sure that rhombuses and kites are pretty useless. In fact, once you get past alt-int angles, parallelograms are not horribly useful later.  I do not recall needing any of this in any math afterwards, at least up to and including calculus.&lt;/p&gt;&#xA;&lt;p&gt;Since I have been told to cut out some geometry to make way for statistics / probability, it seems to me that rhombuses, kites and a good lot of parallelograms are perfect candidates for the chopping block.&lt;/p&gt;&#xA;&lt;p&gt;Am I right? Or do rhombuses and kites turn out to be really useful in the conceivable future of any random student?  I am not denying their beauty etc.&lt;/p&gt;&#xA;&lt;p&gt;If proofs could be be put back into the state test, of course, rhombuses etc. would just be more practice in proofs.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A rhombus is a quadrilateral with four congruent sides (which must be a parallelogram), and a kite is a quadrilateral with two pairs of adjacent congruent sides. Useful facts about rhombuses are that their diagonals are perpendicular to each other and also that the diagonals bisect the angles of the rhombus.&lt;/p&gt;&#xA;" OwnerUserId="12426" LastEditorUserId="2083" LastEditDate="2021-01-04T10:21:38.820" LastActivityDate="2021-01-09T20:54:07.330" Title="Rhombuses, kites etc" Tags="|secondary-education|geometry|concept-motivation|" AnswerCount="8" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16808" PostTypeId="2" ParentId="16807" CreationDate="2019-07-11T19:52:46.840" Score="4" Body="&lt;p&gt;I am just now close to completing my first time teaching geometry (at a community college, a course that parallels the typical high school course) Using the fact that parallelograms (and hence rhombuses) have bisecting diagonals makes a short proof for why the line segment midpoint construction works. Other than that, I don't see rhombuses as a big part of geometry. I would keep some of parallelograms in. Have you looked at the common core standards to see which topics are meant to be kept?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Henri Picciotto, a former high-school teacher and (current) curriculum developer, has written a well-thought out pair of blog posts, &lt;a href=&quot;https://blog.mathed.page/2019/02/20/in-defense-of-geometry-part-i/&quot; rel=&quot;nofollow noreferrer&quot;&gt;In Defense of Geometry&lt;/a&gt;, which you might find useful.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-07-11T19:52:46.840" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16809" PostTypeId="2" ParentId="13628" CreationDate="2019-07-11T20:55:47.940" Score="-3" Body="&lt;p&gt;Students in my gen ed college math class often ask me &quot;what is math good for?&quot; What they mean is &quot;what information will this class provide that I can use in other classes, or at work, or at the casino, or managing my 401-K&quot; I have come to the conclusion that there is nothing that I can teach them that wouldn't be better served by giving them an appropriate app for their smart phone. I have put together a power point presentation that illustrates this that helps them to learn critical thinking skills they can use in the class as well as after they leave the class. Of course most of my colleagues want to burn me at a stake for disrespecting our discipline. I hope it doesn't upset you too much: &lt;a href=&quot;http://faculty.ccc.edu/jnadas/FDW-2019.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://faculty.ccc.edu/jnadas/FDW-2019.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12202" LastActivityDate="2019-07-11T20:55:47.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16810" PostTypeId="2" ParentId="16807" CreationDate="2019-07-12T00:06:00.920" Score="8" Body="&lt;p&gt;Parallelograms are useful for understanding:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Paths taken by light, especially through a layer of a medium with a different refractive coefficient&lt;/li&gt;&#xA;&lt;li&gt;Shear, and related deformations&lt;/li&gt;&#xA;&lt;li&gt;Area = height * width (but not necessarily the product of the sides' lengths)&lt;/li&gt;&#xA;&lt;li&gt;Dot products&lt;/li&gt;&#xA;&lt;li&gt;Surface integrals&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Paths taken by light are useful for understanding which routes people and other animals will take to minimize their effort.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2019-07-12T00:06:00.920" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16811" PostTypeId="2" ParentId="16801" CreationDate="2019-07-12T13:27:21.950" Score="4" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/q/15414/501&quot;&gt;The theory of equations&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Spherical astronomy/Spherical trigonometry/Spherical geometry. Also related The hedronometry of &lt;a href=&quot;https://math.stackexchange.com/users/409/blue&quot;&gt;@Blue&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com.lb/books?id=EfvZAQAAQBAJ&amp;amp;lpg=PP1&amp;amp;pg=PT8#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inequalities&lt;/a&gt; which are limited to high school competitions nowadays.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can read &lt;a href=&quot;http://www.ams.sunysb.edu/~tucker/MathHistory.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The History of the Undergraduate Program in Mathematics in the United States&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="501" LastEditorUserId="501" LastEditDate="2019-07-12T14:33:11.970" LastActivityDate="2019-07-12T14:33:11.970" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16812" PostTypeId="2" ParentId="16807" CreationDate="2019-07-12T17:05:26.703" Score="1" Body="&lt;p&gt;I would keep the parallelograms.  Too much application in SAT questions--they love the vertical angle theorem.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Agreed on ditching kites and rhombuses.  (In the interest of time.)  I have never used them in engineering, physics, other math classes that support engineering/science.  Actually, I don't even remember covering them much in 1980s geometry. If we did, it was super quick (a day at most).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also probably cut solid geometry some (even though this pains me to see this--just give them a handout with the mensuration formulas and don't even test it).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Really, even proofs could be cut.  (I'm not saying to cut it all...but if the point is exposure to proofs, do you really need so much for &quot;exposure&quot;.)  Geometry is such a diversion from the other courses in their headlong build towards calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyone saying to keep the topics, ought to suggest other places to cut.  Hours are zero sum.  And not doing the stats doesn't sound like an option.  But it would help if you told us how many days you are trying to free up (how long spending on stats).&lt;/p&gt;&#xA;" OwnerUserId="12433" LastActivityDate="2019-07-12T17:05:26.703" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16814" PostTypeId="1" CreationDate="2019-07-15T02:52:15.407" Score="3" ViewCount="130" Body="&lt;p&gt;If there is a Data Science class (for final year high school students, not necessarily from the same school) with the following syllabus:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Python programming (first 3 meetings)&lt;/li&gt;&#xA;&lt;li&gt;Data cleaning and Twitter Data visualization using Matplotlib (4 meetings)&lt;/li&gt;&#xA;&lt;li&gt;Probability &amp;amp; Statistics Theory, Normal Distribution, and the Central Limit Theorem (2 meetings)&lt;/li&gt;&#xA;&lt;li&gt;Bootstrap Algorithm (2 meetings)&lt;/li&gt;&#xA;&lt;li&gt;Some Advanced methods: Clustering, classification and regression trees (CART) (4 meetings)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My question is, how should the entry test be? (the test questions)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or better not give test? since we will teach from basic&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have initial idea that is to give test that includes basic calculus such as application of derivatives (like graphing a function etc) to test their mental ability to visualize abstract math concept.  I wonder better approaches. The test should not be restrictive.&lt;/p&gt;&#xA;" OwnerUserId="9439" LastEditorUserId="1147" LastEditDate="2019-07-16T00:34:51.033" LastActivityDate="2019-07-16T06:04:44.460" Title="Entry Test for Statistical/Data Science class" Tags="|secondary-education|statistics|applied-mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16815" PostTypeId="1" CreationDate="2019-07-15T07:31:20.423" Score="11" ViewCount="622" Body="&lt;p&gt;I've graduated high school and I am joining college soon. The problem with me is that I'm not finding less tangible math interesting at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people find abstract math to be very beautiful, and I'm exactly the polar opposite. I am turned off by seemingly pointless abstract mathematical structures with no use whatsoever. For example, some people find group theory to be very interesting after learning the definition of a nilpotent group; I couldn't care less as I never got why would someone make those weird definitions in the first place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found that I will find a topic to be (very) interesting iff that has mathematical &quot;real-life&quot;/concrete/tangible/physics related applications (e.g., proving isoperimetric inequality using Fourier Analysis or Prime number theorem with complex analysis). But to be able to appreciate how that theory relates to the more concrete applications (e.g., understanding the proof of PNT with complex analysis), I need to navigate through the material I find &quot;boring&quot;, which turns me off since if an exposition of the topic doesn't routinely provide examples of such concrete applications, I get bored very quickly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, currently I'm now working my way through Stein and Shakarchi, Complex Analysis and Stein and Shakarchi, Fourier Analysis. What really intrigued me at first is that you can prove really cool number theoretic result with complex + fourier analysis and motivate the Olympiad coloring proofs (see &lt;a href=&quot;https://artofproblemsolving.com/community/c6h90046p525984&quot; rel=&quot;noreferrer&quot;&gt;this link&lt;/a&gt; for the details of what I mean by this) with discrete fourier analysis. So when I started reading I was very interested and breezed through the first few chapters (Chapters 1,2,3 in both books). But now I'm on the section &quot;The action of Fourier transform on F&quot; and I find the chapter to be too much boring so I am now feeling disinterested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, math educators: What are some tricks/strategies to keep students like me motivated with the material even when it feels less tangible to me?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: It's not that I have problem with understanding the material. In the chapters I read in SS, I didn't face any difficulty with any of the exercises/unstarred problems (though I did face lots of difficulty with starred problems.)&lt;/p&gt;&#xA;" OwnerUserId="12438" LastEditorUserId="11" LastEditDate="2019-07-15T22:45:58.557" LastActivityDate="2023-07-26T19:42:43.093" Title="How to stay interested in less-tangible math" Tags="|student-motivation|self-learning|" AnswerCount="4" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="16817" PostTypeId="2" ParentId="16814" CreationDate="2019-07-16T02:55:39.860" Score="3" Body="&lt;p&gt;Consider the following papers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Johnson, Marianne, and Eric Kuennen. &quot;&lt;a href=&quot;https://tandfonline.com/doi/pdf/10.1080/10691898.2006.11910581&quot; rel=&quot;nofollow noreferrer&quot;&gt;Basic math skills and performance in an introductory statistics course.&lt;/a&gt;&quot; Journal of Statistics Education 14.2 (2006).&lt;/li&gt;&#xA;&lt;li&gt;Lunsford, M. Leigh, and Phillip Poplin. &quot;&lt;a href=&quot;https://tandfonline.com/doi/pdf/10.1080/10691898.2011.11889604&quot; rel=&quot;nofollow noreferrer&quot;&gt;From research to practice: Basic mathematics skills and success in introductory statistics.&lt;/a&gt;&quot; Journal of Statistics Education 19.1 (2011).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These feature a multiple-choice diagnostic test used at the start of a statistics course, with various data and conclusions drawn from the results. The test is given in its entirety in each paper. The Johnson paper uses a 15-question test, and the Lunsford paper builds on it with an additional 5 questions. I've used this as the basis for a first-day diagnostic in my community-college elementary statistics course for several years. I find that it has a small amount of predictive power with regard to final exam scores (R^2 = 0.13, N = 238). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="1147" LastEditDate="2019-07-16T06:04:44.460" LastActivityDate="2019-07-16T06:04:44.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16818" PostTypeId="2" ParentId="4165" CreationDate="2019-07-16T03:31:19.250" Score="2" Body="&lt;p&gt;It might bear pointing out that there is a rather wild kaleidoscope of different definitions for continuous functions seen in basic calculus and analysis texts (which might possibly inform different answers to this question). For example, this issue is highlighted in Steven Krantz's &lt;em&gt;How to Teach Mathematics&lt;/em&gt; (Sec. 2.12; cites &quot;some calculus books&quot; that conclude the reciprocal function is discontinuous at &lt;span class=&quot;math-container&quot;&gt;$x = 0$&lt;/span&gt;). And a short paper on the issue was written by J.F. Harper:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Harper, J. F. &quot;What Really is a Continuous Function?.&quot; (2007).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-07-16T03:31:19.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16820" PostTypeId="2" ParentId="16805" CreationDate="2019-07-16T14:41:21.207" Score="3" Body="&lt;p&gt;With acknowledgement to @RoryDaulton - there is an archive for the &lt;a href=&quot;https://kskedlaya.org/putnam-archive/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Putnam Math Competition&lt;/a&gt;. I believe this is exactly what you requested. Free. Let us know if you have trouble accessing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also download the entire book &lt;a href=&quot;http://www.cms.zju.edu.cn/UploadFiles/AttachFiles/20108232376816.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The William Lowell Putnam Math Competition 1985-2000&lt;/a&gt;. Note: the copywrite page offers “Reproduction. The work may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents.” I need to state this because of the abundance of online pirated books.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-07-16T18:09:33.030" LastActivityDate="2019-07-16T18:09:33.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16821" PostTypeId="2" ParentId="16815" CreationDate="2019-07-16T19:46:48.090" Score="7" Body="&lt;p&gt;Your remarks about group theory (&quot;I couldn't care less as I never got why would someone make those weird definitions in the first place&quot;) suggest that you need to see the ultimate goal to appreciate the path to that goal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) For nilpotent groups, a natural goal may be to understand which polynomial roots can be expressed in terms of radicals. Once you see that solvable groups are a key to this question, and that nilpotent groups are solvable, the pursuit makes more sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Continuing with group theory, a natural goal is understanding the symmetric wallpaper&#xA;patterns. Then studying the &lt;a href=&quot;https://en.wikipedia.org/wiki/Wallpaper_group&quot; rel=&quot;noreferrer&quot;&gt;crystallographic groups&lt;/a&gt;, ultimately showing there are &#xA;exactly &lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt;, is motivated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) And just for fun, &lt;em&gt;Group Theory in the Bedroom&lt;/em&gt; gives an interesting application&#xA;of groups to turning a mattress to balance usage.&#xA;In fact, this whole book is one example after another of interesting questions/goals that&#xA;lead to substantive mathematics. The math is motivated by the questions in a way &#xA;not always evident in textbooks.&#xA;(I see this is along one of the lines suggested by @DaveLRenfro.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hayes, Brian. &lt;em&gt;Group Theory in the Bedroom and other Mathematical Diversions&lt;/em&gt;. Hill and Wang, 2008. &lt;a href=&quot;https://www.ams.org/notices/200902/rtx090200237p.pdf&quot; rel=&quot;noreferrer&quot;&gt;AMS review (PDF)&lt;/a&gt;.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;  &lt;img src=&quot;https://i.stack.imgur.com/V7ihy.png&quot; height=&quot;200&quot; /&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-07-17T18:23:25.013" LastActivityDate="2019-07-17T18:23:25.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16822" PostTypeId="1" CreationDate="2019-07-16T23:31:50.110" Score="5" ViewCount="435" Body="&lt;p&gt;I’m teaching my five year old to read an analogue clock and realised how easy it is because (against teacher advice) I already taught him the 5x tables by rote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He knows other tables too but they’re meaningless numbers to him. I’d love to hear ideas of practical everyday use of as other times tables so he can practice and improve them on a daily basis. I don’t mean games and exercises, I mean something you use without even thinking it. The 10x tables are easily covered with using money.&lt;/p&gt;&#xA;" OwnerUserId="12360" LastActivityDate="2019-07-17T00:00:39.473" Title="What’s a practical daily application of each times table?" Tags="|mathematics-in-daily-life|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16823" PostTypeId="2" ParentId="16822" CreationDate="2019-07-17T00:00:39.473" Score="5" Body="&lt;p&gt;A few off the top of my head:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7: calculating how many days away an event is when you know how many weeks it is (or vice versa)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of them: estimating costs while grocery shopping&lt;/p&gt;&#xA;" OwnerUserId="7673" LastActivityDate="2019-07-17T00:00:39.473" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16824" PostTypeId="2" ParentId="2754" CreationDate="2019-07-17T14:01:53.347" Score="2" Body="&lt;p&gt;You should calculate with magnitudes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;m/s = (1/1000)km/(1/3600)h = 3.6 km/h&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My experience is that math teachers detest calculations with magnitudes. They left them to physicists. Their problem is that standard mathematical education does not explain what are magnitudes and how to calculate with them. In the paper linked below I define magnitudes and show that calculations with them are respectable mathematical calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.fsb.unizg.hr/matematika/download/ZS/clanci/what_are_magnitudes_and_how_to_calculate_with_them.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.fsb.unizg.hr/matematika/download/ZS/clanci/what_are_magnitudes_and_how_to_calculate_with_them.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10261" LastEditorUserId="10261" LastEditDate="2019-07-18T16:21:03.437" LastActivityDate="2019-07-18T16:21:03.437" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16825" PostTypeId="1" AcceptedAnswerId="16826" CreationDate="2019-07-19T10:56:03.147" Score="11" ViewCount="5095" Body="&lt;p&gt;I am currently trying to build a flow chart to visualize all tests there are to tell whether an ordinary differential equation is solvable and how to solve it. This is for tutoring purposes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The inspiration for this project comes from another flowchart summarizing all tests to tell whether an infinite series converges.&#xA;&lt;a href=&quot;http://www.math.hawaii.edu/~ralph/Classes/242/SeriesConvTests.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.math.hawaii.edu/~ralph/Classes/242/SeriesConvTests.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to make a document similar to this, but instead for ways to solve an ordinary differential equation (or determine that it is not solvable).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what I have so far:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8XSME.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8XSME.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to build this, I have written down every method that I know of to solve ODE's and have indicated the situation in which it can be used and the type of solution it gives, in a table. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the link to the chart:&#xA;&lt;a href=&quot;https://docs.google.com/document/d/1RYDoOI5Y3eQnEr9WV8b9tlwY4yFBqOIWE3ZIVSp3zCQ/edit?usp=sharing&quot; rel=&quot;noreferrer&quot;&gt;https://docs.google.com/document/d/1RYDoOI5Y3eQnEr9WV8b9tlwY4yFBqOIWE3ZIVSp3zCQ/edit?usp=sharing&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The trouble is that I am not sure when each of these methods can be used, and which one is preferable if there are multiple approaches that could be used. Additionally, if there are any other methods that you know of that I could add, or any resources that might be useful on this project, then feel free to mention it. It would be helpful if there was a list somewhere online of all known methods of solving an ODE, and especially if it was a more exhaustive list than that on Wikipedia (&lt;a href=&quot;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Summary_of_exact_solutions&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Summary_of_exact_solutions&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wanted to extend the chart's section &quot;Is Solvable&quot; to include more specific tests. Are there any tests I should include besides Picard's theorem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, &quot;Matrix methods&quot; is very general. I was wondering if there is a list somewhere online of matrix-based methods that can be used, and when they work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="12452" LastActivityDate="2019-07-19T11:20:46.553" Title="Diagram of Methods to Solve Differential Equations" Tags="|undergraduate-education|teaching|differential-equations|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16826" PostTypeId="2" ParentId="16825" CreationDate="2019-07-19T11:20:46.553" Score="13" Body="&lt;p&gt;I would just like to mention that other similar flowcharts have been developed, of varying degrees of generality, which you might consult.&#xA;Here is &lt;a href=&quot;http://web.uvic.ca/~monahana/ode_flowchart.pdf&quot; rel=&quot;noreferrer&quot;&gt;one (by Adam Monahan)&lt;/a&gt;. &#xA;And &lt;a href=&quot;https://www.pdx.edu/tutoring/sites/www.pdx.edu.tutoring/files/Differential%20Equations%20ODE%20Flow%20Chart.pdf&quot; rel=&quot;noreferrer&quot;&gt;another (by Jeremy Higgins)&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/6ELOw.jpg&quot; width=&quot;600&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;And &lt;a href=&quot;http://www.enriqueareyan.com/files/math/M343%20Introduction%20to%20Differential%20Equations%20I/Resources/DifferentialEquations-Flowchart.pdf&quot; rel=&quot;noreferrer&quot;&gt;another (by Enrique Areyan)&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Zf6wd.png&quot; width=&quot;400&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-07-19T11:20:46.553" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16827" PostTypeId="1" CreationDate="2019-07-20T23:17:21.290" Score="10" ViewCount="199" Body="&lt;p&gt;This is such a vague topic that I hesitate to post.&#xA;I constantly struggle between the time-tradeoff between&#xA;motivating a topic, and delving into the rigorous details necessary&#xA;to fully &lt;a href=&quot;https://en.wikipedia.org/wiki/Grok&quot; rel=&quot;noreferrer&quot;&gt;&quot;grok&quot;&lt;/a&gt; the topic.&#xA;For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The details of polynomial long-division &#xA;(in US middle schools) vs., e.g., &lt;a href=&quot;https://en.wikipedia.org/wiki/Cyclic_redundancy_check&quot; rel=&quot;noreferrer&quot;&gt;cyclic redundancy checks (CRC)&lt;/a&gt; for communication.&lt;/li&gt;&#xA;&lt;li&gt;The need to parse programming languages, such as Python, vs. the (formidable) details of &lt;a href=&quot;https://en.wikipedia.org/wiki/LR_parser&quot; rel=&quot;noreferrer&quot;&gt;LR-parsing&lt;/a&gt; (in US college curricula).&lt;/li&gt;&#xA;&lt;li&gt;Basic number theory (US colleges) vs. truly &lt;a href=&quot;https://doc.sagemath.org/html/en/thematic_tutorials/numtheory_rsa.html&quot; rel=&quot;noreferrer&quot;&gt;understanding the RSA algorithm&lt;/a&gt; &#xA;(graduate school?).&lt;/li&gt;&#xA;&lt;li&gt;Abstract group theory (US colleges) vs. applications, such as turning a &lt;a href=&quot;https://matheducators.stackexchange.com/a/16821/511&quot;&gt;mattress to minimize wear&lt;/a&gt;, or the&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Wallpaper_group&quot; rel=&quot;noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt; wallpaper groups&lt;/a&gt;&#xA;(in no curricula?).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What I try to avoid is the &quot;just do it&quot; attitude, e.g., divide one polynomial by another without any indication of why one might every need to or care to do this. But it is too easy to spend all class time on motivation&#xA;and leave not enough for the rigorous results needed to undergird the applications.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. How do you balance motivation vs. rigorous mathematical development?&#xA;  Have you any principles that might apply across several domains?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-07-21T09:52:32.900" Title="Motivation vs. Rigor" Tags="|undergraduate-education|proofs|student-motivation|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16828" PostTypeId="2" ParentId="16827" CreationDate="2019-07-21T08:42:42.613" Score="11" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think there are more like three (or even four) aspects of education here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A. Theoretical understanding (basis, range of applicability, exceptions, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B. Motivation (why care, what's it good for, is it fun, will it make me money, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C. The basic technique (what you do)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;D. Practice (we are not biological computes--we burn grooves in the brain with practice, not just with a program).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Obviously there's a balance, but I think the key is to look at what you're doing and how well it works. And to consider cost/benefit.  And the audience.  Emphasizing theoretical niceties or rare exceptions (1A) when students don't have 1C or 1D down is probably a waste of time and even frustrating.  For one thing, it's not even good on the cost/benefit side (perfect is the enemy of better).  For another, it's probably going to be a waste, since it is hard to process more difficult aspects of a topic before getting arms around the basic aspects of the topic.  Learning something once and imperfectly is better than nothing.  And it doesn't mean you never return to it and learn it better.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it's also true that too much emphasis on 1C and 1D can be a waste of time for some students or even turn them off.  How to know which is which?  HAVE THEY MASTERED THE BASIC TECHNIQUE.  If in general, they are struggling, then don't kill them with abstractions. If on the other hand, they are crushing it, than move on and enrich.  (This would seem to be brutally obvious pedagogy, but somehow we see questions and such where it eludes many commenters and questioners here, especially those who are basically good at math part of math ed but not the ed part of math ed.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Think about the audience.  How smart are they?  How advanced are they?  How motivated are they?  Pick your battles strategically.  And please, please realize they are different from you. And that &quot;what I like&quot; may not be what is best for the general class.  Or even what is best for the smartest of 30 kids may not be best for the bulk of the class.  Think of it as an optimization problem.  &lt;a href=&quot;https://en.wikipedia.org/wiki/Linear_programming&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Linear_programming&lt;/a&gt;  By which, I mean think of it that way intuitively, not as an actual calculation.  But at least you have some ideas of the factors, perhaps priorities, etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I have not really address motivation, 1B.  My advice is to do a very minimum of motivation and move rapidly to 1C.  In many cases, &quot;motivation&quot; ends up being some physical word problem. But guess what?  Word problems are harder than math problems!  Lots of tests have shown this.  You are compounding a need to understand some basic physics or econ (even if simple) with the math and also adding the need to &quot;translate&quot; word relations into math relations. So I would generally eschew specific application to set up the discussion.  (And I say this as a guy who is an application ADVOCATE.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the issues with why many PDE texts are hard for engineers is that they start out with engineering applications.  Rather than the way ODE texts start with just the algebraic technique...and then do applications of it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, there are some motivations like advancing to the next class, supporting science courses, etc.  But I don't think that you need to dwell on these.  Occasional en passant references are plenty. Really &quot;you have to do this because it will be tested&quot; is fine (yes I mean it) as an immediate motivator.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One useful &quot;motivator&quot; is just the ability to do something we haven't (in the course) been able to do before.  So, we learned a specific technique of integration or the like but it doesn't work for some type of problem. Now we give you another tool that will solve those.  This feels like a natural connection to earlier work and like we are getting something useful.  But still, it doesn't take huge time to say this.  OK to mention the connection and how the new tool will help with a harder but related problem.  And then move on and show them the darned integration by parts or characteristic equation or what have you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some general expressions of sympathy, humor, camaraderie, &quot;in it together&quot; and &quot;been in your shoes&quot; are useful as well.  By which I mean don't be a robot and have some warmth.  But also don't dwell on this stuff.  Lots of students tend to be very immediate in terms of wanting to get the show to move on.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think there is also something that is a blend of 1A/B/C, which is to have an intuitive concept of the new material.  Note this doesn't mean an exact understanding.  It's the opposite of that actually.  But having a quick mind picture or label to be familiar with a topic.  So for instance, in algebra, we solve algebraic equations for y and in ODE, we solve ODEs for y.  [Yes, there is more to each course than that. Yet the generalization is still a very useful first order description.]  Examples of analogies are things like &quot;darn doesn't the second order ODE with constant coefficients look a lot like a quadratic equation?  Wouldn't it be nice if we could use the quadratic to help solve it!  These may be rather far from 1A perfection or proof-based explication.  But I assert they are damned close to &quot;groking&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;P.s.  &quot;What I try to avoid is the &quot;just do it&quot; attitude, e.g., divide one polynomial by another without any indication of why one might every need to or care to do this.&quot;  I would encourage you to have some self questioning here.  Often questions on this site, espouse a position and look for sympathetic support.  And you may get some (likely will based on dynamics of the community).  And you might even be right.  But at least continue to consider if you are wrong.  An analagous example:  if you were teaching lock-picking, perhaps some technique-based description and even drill might be preferable to start, before the &quot;why&quot;.  (But it's a balance.  Sometimes the &quot;why&quot; makes the &quot;how&quot; easier to remember/absorb.  But not always...sometimes it makes it harder, for new students.  If the why is easy, by which I mean not complex, not counter-intuitive, than it's more likely to help, initially.)&lt;/p&gt;&#xA;" OwnerUserId="12462" LastEditorUserId="12462" LastEditDate="2019-07-21T09:52:32.900" LastActivityDate="2019-07-21T09:52:32.900" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16829" PostTypeId="2" ParentId="10021" CreationDate="2019-07-21T10:21:30.890" Score="0" Body="&lt;p&gt;I think a lot of the issue is the algebraic complexity.  Having to write a lot, carry a lot of relationships, etc.  It's just more work.  Note, humans are far from computers in our ability to carry a lot of relations at one time.  And many students are not as used to being drilled and working on writing a lot of expressions.  We even routinely see &quot;questions&quot; (more like arguments) to have everyone use a CAS versus doing manipulations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Add onto that, the limited utility (at least early) of the technique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I'm a little wondering why you talk about teaching it in calculus. It is normally a part of the algebra 2 course.  Arguable one of the harder, less pleasant parts of it.&lt;/p&gt;&#xA;" OwnerUserId="12462" LastActivityDate="2019-07-21T10:21:30.890" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16830" PostTypeId="2" ParentId="7837" CreationDate="2019-07-21T10:30:05.863" Score="2" Body="&lt;p&gt;As I said in my comments to the OP, I think this is just a bit of a tricky topic.  A little like quantum mechanics, where you sort of have to get used to it, versus understand it immediately like kinematics.  So hoping for some secret key to unleash their groking may not work (for this item, other topics there may be a nice mental key).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple practical ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a.  Have the student just use the technique, after learning it, on several simple problems.  Perhaps even simple enough so that they could guess the answer without any manipulations.  Seeing &quot;that it worked&quot; even if not sure why may be helpful motivation to accepting the technique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b.  Play it up a little as some secret sauce.  &quot;Here's a sneaky trick.&quot; Why are emotional, human creatures, not only logic processors.  Feeling like you got the magic password is motivating.&lt;/p&gt;&#xA;" OwnerUserId="12462" LastActivityDate="2019-07-21T10:30:05.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16831" PostTypeId="2" ParentId="4526" CreationDate="2019-07-21T10:45:39.917" Score="2" Body="&lt;p&gt;As a sharpshooter in the military (but nothing special, never used it) I would say the big difference is that you can just align things linearly:  your eye, the first sight (usually a post), the second one, usually a notch and the target.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm assuming a conventional shot here, but even shots requiring windage or Coriolis or drop, can be still be sighted in the normal manner.  You just use a vernier to adjust for them, using the sights, but the picture is still linear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The loopedy-loop thing, you don't have sights to align.  Instead, you need to use some equation.  &lt;em&gt;Plus, really, you need to know the distance (to know the part of the loop and then to compensate by off angle shot and/or adjustment of rotation phase).  But for a straight conventional shot, you just need angle, not distance.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[There is an interesting segue here to target motion analysis and passive sonar (bearing only) versus radar (bearing and distance) and how much harder sonar TMA is.  See. e.g. Ekelund ranging.]&lt;/p&gt;&#xA;" OwnerUserId="12462" LastActivityDate="2019-07-21T10:45:39.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16832" PostTypeId="2" ParentId="8150" CreationDate="2019-07-21T11:33:14.250" Score="1" Body="&lt;p&gt;If you are dealing with those who are naive, I would consider to just give them situational awareness.  One of the things that is frustrating about Wikipedia, is if I want to learn ABOUT a math topic, not learn THE math topic, I get very little out of the articles.  This is because math types devolve to full proof or rigorous definition (worst) or teaching the topic itself (better, but still too intricate).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead if you are dealing with naive people, just give them information to help them classify it within overall knowledge (not to learn the logarithms themselves...if you want to teach logarithms, than take algebra 2...and follow the pedagogical sequence therein.)  So some things ABOUT logarithms, sans learning them, sans defining them:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*what course taught in: algebra 2, typically 11th grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*what later courses require it: calculus, chemistry, physics.  Don't say bio (don't need it for freshman bio).  Don't say differential equations.  &quot;Calculus&quot; is enough...big hard course they have heard of but not taken.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*what related to:  &quot;like powers of stuff, you know squaring and cubings, but on steroids.  Not the same thing as powers, but you get to logarithms after mastering your powers.&quot;  [This assumes they have heard of the powers, even if not knowing exponent rules]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*useful for: helps to do complicated arithmetic.  Calculators use it to do their calculations of big arithmetic stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*involved with:  pH in chemistry (how acid stuff is), Richter scale for earthquakes, sound level (&quot;decibels&quot;).  [RESIST the impulse to lecture about nonlinearity...you basically have to explain what a logarithm is to do that...and for God's sake don't use the word &quot;nonlinear&quot; with the peeps.  It's enough to just give them some motivational touchpoints that the topic relates to, that they have heard about.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*funky looking graph paper:  hand each student a page of log-log and of semi-log graph paper.  Don't make it yourself.  Go to the specialty engineering store and buy a pad of each paper.  Nice and green and professional looking...you practically feel crew cutted 1950s engineers sending rockets into space from the paper itself.  Let the audience keep the pages you give them. Resist any urge to run an exercise or graph something (too hard, benefits from knowing logs first).  But you can say &quot;look at that funky scale with the 10, 100, 1000, etc. and missing the zero&quot;.  &quot;Once you learn what logs are, that weird scale will make sense.&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Consider to make a joke or pun or two.  Icebreaker and tension reliever.  Something that does NOT require understanding what a log is. Something like &quot;it's different from Lincoln's cabin type of log&quot;  (look it up, sure there is some site that has math puns on it...GIYF.)&lt;/p&gt;&#xA;" OwnerUserId="12462" LastActivityDate="2019-07-21T11:33:14.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16833" PostTypeId="2" ParentId="10339" CreationDate="2019-07-21T11:51:22.410" Score="0" Body="&lt;p&gt;I would consider how intricate your homework is.  For &quot;drill&quot;, more of the advantage comes from solo trials.  For &quot;projects&quot;, collaboration may have an advantage, or at least less of a disadvantage.* I would also consider if there are ways to get some social interaction without full joint responsibility (half a loaf, quarter of a loaf).  &quot;Trade papers and grade.&quot;  &quot;Do you agree with Johnny's answer?  Anything to add/change?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also like the suggestion of separate parts of the homework, because it allows some grading independence.  Perhaps you can do something like a part A and part B (separate) that merge into a joint part C.  Not really sure how to implement, just brainstorming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*I would add that professors tend to overvalue &quot;project type&quot; homeworks and undervalue &quot;drill&quot; because the former is closer to their research role...and just kind of more interesting to them, than duller problems.  But that's a different battle to fight.  (There's a reason why Gene Hackman makes the kids do drill rather than just scrimmage all practice long, in Hoosiers.  And gets static for it from the townspeople.)&lt;/p&gt;&#xA;" OwnerUserId="12462" LastActivityDate="2019-07-21T11:51:22.410" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16835" PostTypeId="2" ParentId="10737" CreationDate="2019-07-21T12:40:56.057" Score="0" Body="&lt;p&gt;Sure. Many, many examples.  For instance algebraic understanding of electricity and fluids flow is much more intuitive than the tensor calculus versions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In some cases, you need high powered pre-reqs to even do the proof.  Also, in many cases, students don't need a proof.  Do enlisted reactor operators need a tensor calculus and quantum mechanics and stat mech understanding of reactor dynamics?  I think a lot of mathematicians lose sight of the audience. and what they will use something for. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm glad stuff is correct and somebody proved it.  But if I should is a different question. And sure if we all had unlimited time, money, and IQ points we could all learn everything rigorously.  But life (and education is part of life, not of math) involves trade-offs and constraints.  If the trade-off means skipping the proof, so be it.  &lt;/p&gt;&#xA;" OwnerUserId="12462" LastActivityDate="2019-07-21T12:40:56.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16836" PostTypeId="2" ParentId="7528" CreationDate="2019-07-21T13:06:56.867" Score="5" Body="&lt;p&gt;Spiral = snail shell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/a7kgc.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/a7kgc.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Broccoli = fractal&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7iBUD.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7iBUD.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-or- broccoli = decision tree (but a tree can also be a decision tree).  Note that the slang term in the Navy for broccoli is &quot;trees&quot; (like sliders for burgers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vTIc3.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vTIc3.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wankel engine rotor has a similar curve-y triangle shape to the criticized coin above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FfRkH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FfRkH.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Saddle = saddle (3rd semester calc)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iGSjs.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iGSjs.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Drill chuck = truncated cone (also some of the internals of an automotive differential)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MVp1u.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MVp1u.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Stadiums&quot; for trapezoidal cylindrical shells (calculus volume of rotation problems)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VDqy0.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VDqy0.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lots of other cool gear-type shapes (prop screw for a ship, pump lobes, camshaft, chevronic separators in boilers, tricone rotary drill bit).  Not a 100% sure what they all correspond to math-name wise, but they definitely engage some wonder about shape to function.  &lt;/p&gt;&#xA;" OwnerUserId="12462" LastEditorUserId="12462" LastEditDate="2019-07-21T20:57:54.757" LastActivityDate="2019-07-21T20:57:54.757" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16837" PostTypeId="2" ParentId="2338" CreationDate="2019-07-21T13:10:02.900" Score="3" Body="&lt;p&gt;Indefinite integrals are easy to avoid, it only takes a will to do so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basic antiderivatives:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x^2 =\frac{d}{dx}\frac{x^3}{3},\quad&#xA;\sin x =\frac{d}{dx}(-\cos x),\quad&#xA;\frac{1}{x} =\frac{d}{dx}\ln|x|.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Antidifferentiation by change of variable:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;e^x\sin(e^x) =\sin(e^x)\frac{d}{dx}e^x =\frac{d}{dx}\sin(e^x),&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Antidifferentiation by parts:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x e^x = x \frac{d}{dx}e^x = x \frac{d}{dx}e^x + (\frac{d}{dx} x)e^x - (\frac{d}{dx} x)e^x\\&#xA;=\frac{d}{dx}(x e^x) - (\frac{d}{dx} x)e^x = \frac{d}{dx}(x e^x) - e^x\\&#xA;=\frac{d}{dx}(x e^x) -\frac{d}{dx}e^x = \frac{d}{dx}(x e^x - e^x).&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using &lt;a href=&quot;https://en.wikipedia.org/wiki/Differential_form&quot; rel=&quot;nofollow noreferrer&quot;&gt;differential forms&lt;/a&gt;, equivalent calculations can be carried out with simpler notation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x^2\,dx = d\frac{x^3}{3},\quad&#xA;\sin x\,dx = d(-\cos x),\quad&#xA;\frac{dx}{x} =d(\ln|x|),&#xA;$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;e^x\sin(e^x)\,dx =\sin(e^x)d e^x =d\sin(e^x),&#xA;$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x e^x\,dx = x d e^x = x d e^x + (dx)e^x - (dx)e^x\\&#xA;= d x e^x - (dx)e^x = d x e^x - e^x\,dx\\&#xA;= d x e^x - de^x = d(x e^x - e^x).&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I teach definite integrals (or try to teach indefinite integrals despite being fuzzy myself about their exact meaning), I prefer to carry out calculations with differential forms first and apply definite (or indefinite) integral at the end.&lt;/p&gt;&#xA;" OwnerUserId="6987" LastActivityDate="2019-07-21T13:10:02.900" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="16840" PostTypeId="2" ParentId="11813" CreationDate="2019-07-22T05:40:13.540" Score="4" Body="&lt;p&gt;There is evidence that both a computational and conceptual approach are needed: &lt;a href=&quot;https://www.jstor.org/stable/3482237&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/3482237&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper of Sfard linked to does seem to agree that the scale must tip first toward computation, in the beginning, but with a view toward conceptualizing.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper aims to uncover why mathematics is hard for many to learn, and makes the interesting claim that process and concept are prerequisites of one another...which is clearly problematic. Unsurprisingly, &quot;programming&quot; students with procedures and very few concepts won't work for the reasons we expect (students end up memorizing many things with few relations between them) and overemphasizing abstract concepts without sufficient work with procedures amounts to talking about things completely divorced from student experience...and this is no good, either.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2019-07-22T07:52:28.180" LastActivityDate="2019-07-22T07:52:28.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16841" PostTypeId="2" ParentId="16815" CreationDate="2019-07-22T12:48:16.863" Score="5" Body="&lt;p&gt;&lt;em&gt;(expansion of a comment made a week ago)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;When you get to college I recommend actively discussing math topics with undergraduate peers and graduate students (assuming a university with a graduate math program). This will help insert you into the mathematics community --- undergraduates will tell you about the Putnam Exam and which professors to approach (and which professors to actively avoid) and the like, while graduate students are useful for giving you an idea of what graduate school is like and advice on applying to graduate school and the hurdles for beginning research in math.&lt;/p&gt;&#xA;&lt;p&gt;Also, to maintain interest, it helps to read books that could serve as inspiration, such as &amp;quot;What is Mathematics?&amp;quot; by Courant/Robbins, &amp;quot;A Course of Pure Mathematics&amp;quot; by Hardy, the 3-volume &amp;quot;Feynman Lectures on Physics&amp;quot; by Feynman (yes, it's physics, but despite Feynman's sometimes disdain for aspects of pure mathematics, I find his writing to be very VERY strong with what Terrance Tao calls the &lt;a href=&quot;https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;post-rigorous&lt;/a&gt; stage), &amp;quot;Visual Complex Analysis&amp;quot; by Needham, &amp;quot;A Concrete Approach to Abstract Algebra&amp;quot; by Sawyer, &amp;quot;Adventures of a Mathematician&amp;quot; by Ulam, etc.&lt;/p&gt;&#xA;&lt;p&gt;Spend some time browsing university library shelves to get an idea of what exists in mathematics (besides what random googling provides) and to help regain interest in mathematics when you're too study-tired and jaded to do any formal work (working problems, studying texts, etc.). Look through back issues of expository and undergraduate level journals you find on library shelves for things that might catch your interest and which you otherwise would not have encountered. &lt;a href=&quot;https://groups.google.com/g/sci.math/c/gF6LNhFT-T8/m/8i8iEj2LvuoJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's a possibly dated list&lt;/a&gt; of such journals that I posted back in 2005. Probably the 3 MAA journals and The Mathematical Gazette are what I'd most strongly recommend browsing. Also, keep in mind that unlike the case with many other academic fields, math articles from 70 or 80 years ago can be just as useful as those written in the last 5 or 10 years.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2023-07-18T11:56:25.200" LastActivityDate="2023-07-18T11:56:25.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16842" PostTypeId="2" ParentId="7281" CreationDate="2019-07-23T17:34:02.063" Score="0" Body="&lt;p&gt;Relation between division and fraction  have to be understood to understand the relationship between fraction and ratio:&#xA;1. Division: finding how many times of divisor in the dividend, or the value of a part &#xA;  1/5 means 5 is 0.20 times in 1;  value of one part is 0.20.&#xA;2. Fraction: It is how many times the denominator in numerator or how many parts of denominator in numerator. It  means multiplying the value of one part with numerator:&#xA; 2/5 = 1/5 X 2 = 0.40&#xA;3 Ratio: Ratio is a division finding how many times one entity in to other. And it is a fraction for one in numerator.&#xA;4. All are written in fraction form with same value&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Division and Fractions have applications in daily life where as ratio being an another form of division and fraction does not find place in daily life applications. Therefore ratio is not an independent entity but a name to refer how many times in a division in context. Probably it is good idea not to teach ratio as an independent entity. &#xA;Therefore ratio is an application of fractions like percentage, rebate, loss and profit. They all use the properties of equivalent fractions. Proportion too is an equivalent fraction.&lt;/p&gt;&#xA;" OwnerUserId="12465" LastActivityDate="2019-07-23T17:34:02.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16843" PostTypeId="2" ParentId="7281" CreationDate="2019-07-24T06:40:15.430" Score="-1" Body="&lt;p&gt;Above while representing ratio in a  flow diagram above one branch is represented by part/part. There is no such concept as part/part in fraction form. The denominator is always a 'whole'. While comparing two items  one of them should be taken as whole and other as part of the whole. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The diagram in the present form violates the fundamental principle of fraction. &lt;/p&gt;&#xA;" OwnerUserId="12467" LastActivityDate="2019-07-24T06:40:15.430" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16844" PostTypeId="2" ParentId="7281" CreationDate="2019-07-24T08:19:40.143" Score="4" Body="&lt;p&gt;The ratio &lt;span class=&quot;math-container&quot;&gt;$a:b$&lt;/span&gt; and the fraction &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt; are generally not synonymous and should not be treated as such, at least without making clear their interpretations. In many contexts, &lt;span class=&quot;math-container&quot;&gt;$a:b$&lt;/span&gt; corresponds to the fraction &lt;span class=&quot;math-container&quot;&gt;$a/(a+b)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$b/(a+b)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if one cuts a pizza into &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; slices, one of which has anchovies, the fraction of slices with anchovies is &lt;span class=&quot;math-container&quot;&gt;$1/6$&lt;/span&gt;, while the ratio of slices with anchovies to slices without anchovies is &lt;span class=&quot;math-container&quot;&gt;$1:5$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One context, unfortunately familiar to many students, where &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a:b$&lt;/span&gt; are not synonyms is when speaking of odds in the context of bookmaking. That the odds be given by the ratio &lt;span class=&quot;math-container&quot;&gt;$3:2$&lt;/span&gt; means roughly that the bookmaker expects the bet upon side to win &lt;span class=&quot;math-container&quot;&gt;$2/5$&lt;/span&gt; times and the payout in the event of a win will be &lt;span class=&quot;math-container&quot;&gt;$3/5$&lt;/span&gt; of the sum of the payout and money staked. That is when speaking of odds, the ratio &lt;span class=&quot;math-container&quot;&gt;$a:b$&lt;/span&gt; corresponds to the fraction &lt;span class=&quot;math-container&quot;&gt;$a/(a+b)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$b/(a+b)$&lt;/span&gt; (which form is relevant depends on interpretation/use). So &lt;span class=&quot;math-container&quot;&gt;$9:1$&lt;/span&gt; odds against means that the horse is not expected to win, so the payout is big if it does, &lt;span class=&quot;math-container&quot;&gt;$900$&lt;/span&gt; would be paid on a &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; bet, while &lt;span class=&quot;math-container&quot;&gt;$9:1$&lt;/span&gt; odds in favor (usually quoted as &lt;span class=&quot;math-container&quot;&gt;$1:9$&lt;/span&gt;) means that the horse is expected to win, and the payout is small if it does, approximately &lt;span class=&quot;math-container&quot;&gt;$11 \simeq 100/9$&lt;/span&gt; would be paid on a &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; bet.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2019-07-25T14:00:14.767" LastActivityDate="2019-07-25T14:00:14.767" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16845" PostTypeId="1" CreationDate="2019-07-24T13:16:14.067" Score="12" ViewCount="256" Body="&lt;p&gt;Is there evidence in the education literature to support or refute the claim that students who are given problems without solutions to work out on their own do better on different problems in the future (assuming that they took the time to do the problems)? Does it matter if they get the problems correct/incorrect?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know there is evidence that worked examples help (Sweller and Cooper in 1985 comes to mind as the first reference point). However is there benefit after providing worked examples to provide non-worked examples? Has there been any research justifying this? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key here is has someone actually done the research supporting this. I do believe that not seeing answers to everything helps but I don't have proof of this.&lt;/p&gt;&#xA;" OwnerUserId="12000" LastEditorUserId="12000" LastEditDate="2019-07-29T11:30:29.227" LastActivityDate="2019-07-29T11:30:29.227" Title="Problems Without Posting Available Solutions" Tags="|problem-design|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16846" PostTypeId="2" ParentId="16845" CreationDate="2019-07-24T13:31:51.270" Score="1" Body="&lt;p&gt;I am assuming you mean that answers are not available in the back of the textbook for the exercises assigned. If no one does find research on this, it would be pretty hard to test. Students have access to &lt;a href=&quot;https://www.photomath.net/en/&quot; rel=&quot;nofollow noreferrer&quot;&gt;photomath&lt;/a&gt;, which can solve their problems for them.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-07-24T13:31:51.270" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16847" PostTypeId="1" AcceptedAnswerId="16848" CreationDate="2019-07-25T05:48:39.877" Score="4" ViewCount="348" Body="&lt;p&gt;In a book i'm writing, i want to introduce students to equations in slightly more general terms. Solving an equation with some unknown is just &#xA;one example of finding an object by fact that it is a part of &#xA;a certain equivalence class - specifically a class in the &quot;equals&quot;-relation. Same goes for inequalities too. In both scenarios we use various operations on both sides, make sure that none of these operations change the equivilance class of any member, and then isolate some desired expression. This is really intuitive and natural to work with for equality, but i think it could be a great exercise for the mind and maturity to try something very different. Does there exist any other nice similair problems and techniques for different relations? Maybe something using congruence or similarity of figures? We can obviously work with set equalities too, though that may be a little very abstract. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For information, my readers has not yet at this point in the book learned calculus. It's at just the point before. The target audience is math-interested high-schoolers. &lt;/p&gt;&#xA;" OwnerUserId="12400" LastEditorUserId="12400" LastEditDate="2019-07-25T10:06:12.173" LastActivityDate="2019-07-25T17:36:30.407" Title="Examples of solving for unknowns using equivalence relations that are not equality, inequality, or boolean truth?" Tags="|examples|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="16848" PostTypeId="2" ParentId="16847" CreationDate="2019-07-25T07:07:54.540" Score="2" Body="&lt;p&gt;Maybe proportions. Granted there's an equality in there but the emphasis is on proportions. You can even generalize the idea to SAT analogies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps conversions or dimensional analysis would fit well in the book. Another idea is the piano tuner business case, estimating methods ala Fermi. Note these are not strictly relations. But might fit well into what you are trying to do and would work with the audience. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe also some simple probability stuff with application to gambling and cards and dice and such.  Things like 4:1 odds means 20% win chance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A little bit of finance math nice also. Compound interest.  Time value of money to convert between future and present values.  &lt;/p&gt;&#xA;" OwnerUserId="12472" LastEditorUserId="11" LastEditDate="2019-07-25T12:30:43.530" LastActivityDate="2019-07-25T12:30:43.530" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16849" PostTypeId="2" ParentId="7281" CreationDate="2019-07-25T12:27:48.343" Score="12" Body="&lt;p&gt;There are several possible fractions one could associate with a given ratio.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say that a recipe for lemonade calls for &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; cups of lemon juice and &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; cups of water.  This could be expressed with the ratio &lt;span class=&quot;math-container&quot;&gt;$2:5$&lt;/span&gt;.  Here are some different associated fractions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2/7$&lt;/span&gt; of the lemonade is lemon juice.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$5/7$&lt;/span&gt; of the lemonade is water.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2/5$&lt;/span&gt; of a cup of lemon juice is the amount which needs to be added to 1 cup of water.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$5/2$&lt;/span&gt; of a cup of water is the amount which needs to be added to 1 cup of lemon juice.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I don't think any of these fractions has a right to be called &quot;the fraction&quot; associated with the ratio.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some additional evidence that ratio and fraction are distinct concepts:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ratios 0:5 and 5:0 both make sense.  For instance, you could have 0 parts lemon juice to 5 parts water.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can have ratios between more than 2 quantities.  The ratio 1:2:5, as in 1 part sugar, 2 parts lemon juice, and 5 parts water makes perfect sense.  This ratio would be equivalent to 2:4:10 (since the proportions are the same), but there is no single associated fraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a mathematical definition one could use the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An &lt;strong&gt;ratio of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; quantities&lt;/strong&gt; is an equivalence class of elements of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n-\{\mathbf{0}\}$&lt;/span&gt; under the equivalence relation &lt;span class=&quot;math-container&quot;&gt;$\mathbf{v} \sim \mathbf{w}$&lt;/span&gt; iff there exists a &lt;span class=&quot;math-container&quot;&gt;$c \in \mathbb{R}-\{0\}$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\mathbf{v} = c \mathbf{w}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, ratios are just elements of real projective spaces!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please don't tell this to your 5th graders.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2019-07-25T12:55:14.567" LastActivityDate="2019-07-25T12:55:14.567" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16850" PostTypeId="2" ParentId="16847" CreationDate="2019-07-25T12:48:57.597" Score="5" Body="&lt;p&gt;Consider the equivalence class of knot diagrams depicting the unknot.&#xA;Given a knot diagram, Reidemeister moves do not change the knot type.&#xA;So one can apply these moves to a knot diagram because &#xA;&quot;none of these operations change the equivalence class.&quot;&#xA;If you reach the unknot, then you know your original diagram was&#xA;just a different drawing of the unknot.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/J8JRp.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/J8JRp.png&quot; alt=&quot;Reidemeister&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;Fig: Dominic Goulding,&#xA;&quot;Knot Theory:&#xA;The Yang-Baxter Equation, Quantum Groups and&#xA;Computation of the Homfly Polynomial,&quot; 2010.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-07-25T12:48:57.597" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16851" PostTypeId="1" AcceptedAnswerId="16852" CreationDate="2019-07-25T15:37:13.577" Score="6" ViewCount="250" Body="&lt;p&gt;What kind of computer software do you recommend for math teacher? Like if student asks for some difficult integral or asks some probability that requires computer software. Do teachers know some software by heart (like Sagemath, Maple, Matlab) that he or she can answer it by short code and its output or is it enough to answer something like &quot;Well, a suitable computer program solves that in seconds. I don't know yet how to do that but I can write a short tutorial and demo to the next lesson.&quot;&lt;/p&gt;&#xA;" OwnerUserId="12474" LastEditorUserId="12474" LastEditDate="2019-07-25T15:45:38.653" LastActivityDate="2019-07-26T13:13:53.323" Title="Should a math teacher know some computer algebra system &quot;by heart&quot;?" Tags="|software|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16852" PostTypeId="2" ParentId="16851" CreationDate="2019-07-25T16:38:57.390" Score="7" Body="&lt;p&gt;I think you only need to have this knowledge if you are teaching a unit or significant part of the course, using a CAS.  This is probably a small minority of courses.*  In that case, what makes sense is whatever CAS you will be expecting the kids to use (I recommend picking one specific one). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one I know is Maple, but I'm sure the others are fine too.  Probably whatever is easiest, cheapest for the class.  I remember Maple having a lot of academic business and free/cheap license in the day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*In order to have a course like this, you need kids who are relatively advanced (so they have time to learn to code Maple), but didn't quite validate the frosh calc course either.  Even then, most kids end up hating the &quot;with computers&quot; courses (or other &quot;enriched&quot; courses) as being harder and prefer to just get through the standard track and advance with that.  I regularly advise kids in college to take the standard basic courses (first year physics, chem, calculus, diffyQs) or to validate them.  But avoid enriched stuff like the plague.  This is different from high school where enriched courses &quot;GT track&quot; are usually a good option.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  FWIW, I do NOT recommend telling the kids &quot;don't worry about harder (standard) problems because the computer does it for you&quot;.  There is a strong benefit to learning the classical calculus course and the manipulations required.  Need to build those muscles for physics and engineering derivations and homeworks.  As someone who uses tables occasionally, or just a basic textbook to look up an equation, I think it is easier to do so having mastered the stuff once versus never learning it and always having been reliant on an aid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.p.s.  I would also not emphasize problems that are so complex they require a CAS (for mortals). Most of the standard books are fine.  You don't need a CAS for them.  Stick to getting them up to speed on normal stuff first.  Walk before run...&lt;/p&gt;&#xA;" OwnerUserId="12475" LastEditorUserId="64" LastEditDate="2019-07-25T17:26:53.603" LastActivityDate="2019-07-25T17:26:53.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16853" PostTypeId="2" ParentId="16847" CreationDate="2019-07-25T17:27:33.830" Score="2" Body="&lt;p&gt;Also, let's not forget isomorphisms in group theory,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;nor congruence relations in geometry, and, e.g., similarity in geometry.  Further, in geometry, relations between lines might include &lt;span class=&quot;math-container&quot;&gt;$\overline{AB} \parallel\, \overline{CD}$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$\overline{EF} \perp \overline{GH}$&lt;/span&gt; (two lines being parallel, or two lines being perpendicular, respectively&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also also congruence equations,&lt;span class=&quot;math-container&quot;&gt;$\mod n: \;\; 3 \equiv 10 \pmod 7$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-07-25T17:36:30.407" LastActivityDate="2019-07-25T17:36:30.407" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16854" PostTypeId="2" ParentId="16851" CreationDate="2019-07-26T06:33:02.837" Score="7" Body="&lt;p&gt;Just one small thing to consider for you:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you like to do non-standardized exercises (i.e. the ones where the solution isn't available in a textbook), then a software can help a lot with that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example I like the most is the calculation of Eigenvalues. For an exercise, you usually want nice Eigenvalues (e.g. &lt;span class=&quot;math-container&quot;&gt;$1,2,3,4$&lt;/span&gt;), but how do you get an interesting matrix having these Eigenvalues?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Take the matrix &#xA;&lt;span class=&quot;math-container&quot;&gt;$$A = \begin{pmatrix}1 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 2 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 3 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 4\end{pmatrix}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Create a random &lt;span class=&quot;math-container&quot;&gt;$4 \times 4$&lt;/span&gt; matrix over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}_2$&lt;/span&gt; and check that its determinant over the integers (taking &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to be the elements of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}_2$&lt;/span&gt;) is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;. I just did that, and found the random matrix&#xA;&lt;span class=&quot;math-container&quot;&gt;$$M = \begin{pmatrix} 0 &amp;amp; 1 &amp;amp; 1 &amp;amp; 1 \\ 1 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 1 &amp;amp; 0 &amp;amp; 0 &amp;amp; 1 \\ 0 &amp;amp; 1 &amp;amp; 0 &amp;amp; 0\end{pmatrix}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Compute &lt;span class=&quot;math-container&quot;&gt;$B := MAM^{-1}$&lt;/span&gt;. In this case,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$B = \begin{pmatrix}3 &amp;amp; -1 &amp;amp; 1 &amp;amp; -1 \\ 0 &amp;amp; 1 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; -3 &amp;amp; 4 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 2\end{pmatrix}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; has the same Eigenvalues as &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, so we have a nice exercise.&#xA;One could argue that &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is too easy, you can directly see two Eigenvalues. To make it more complicated, replace &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; by the companion matrix of the polynomial &lt;span class=&quot;math-container&quot;&gt;$(x-1)(x-2)(x-3)(x-4)$&lt;/span&gt;, or look for a different &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;. You can also play around with the lift from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}_2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt;, e.g. take some entries as &lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt; and some as &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;, or replace some &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, take a different finite field to start, etc; but make sure that the determinant stays &lt;span class=&quot;math-container&quot;&gt;$\pm 1$&lt;/span&gt;. The reason for that is that doing the conjugation in step 3 will usually introduce a factor of &lt;span class=&quot;math-container&quot;&gt;$\det(M)^{-1}$&lt;/span&gt; into the result, and we don't want fractions in the matrix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This example is based on the fact that I already know some computer algebra systems. If it is worth to learn it just to be able to create nice, custom made exercises is up to you; of course also taking into account the quality of exercises your textbook provides.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2019-07-26T06:33:02.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16855" PostTypeId="2" ParentId="16851" CreationDate="2019-07-26T13:13:53.323" Score="2" Body="&lt;p&gt;I don't think that learning a CAS is in the category of &quot;must-do&quot;, but it certainly is useful. Personally I use &lt;a href=&quot;http://www.sagemath.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sage&lt;/a&gt; but Mathematica is also a good choice. Mathematica is highly polished and widely used while sage is open source, based on a general purpose programming language, and free of charge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't find myself fielding random integral questions very often and, honestly, &lt;em&gt;no program is all that great&lt;/em&gt; at giving simple closed forms. Nor is it really possible. Expect the answers to involve things a Calc student does not know about like the gamma or hypergeometric functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For my Calc students, the main thing that I use Sage for is generating graphs and animations. (Sagetex allows you to put sage code in latex.) I point them to Desmos when they want to do the same, unless I know they can program. Another use is for numerical results which are a bit tedious when done to high accuracy such as Newton's method or numerical integration.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2019-07-26T13:13:53.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16856" PostTypeId="2" ParentId="3839" CreationDate="2019-07-26T14:14:59.773" Score="6" Body="&lt;p&gt;I teach at community college. I often publish the homework problems at the beginning of the semester, listed by section.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have never had a student work ahead (that I know of). And I have had a few students who loved math, asked deep questions, and were interested in doing extra. I don't think that is likely to happen, except in very rare instances.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One benefit for me (and my students, really) has been that I don't have to tell them the homework. I used to forget sometimes to tell them what homework to do, and they would do none. That got us behind. Now I tell them, whatever sections of the textbook we work on in class, those sections of homework are due the next day (or at the end of the week).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My students know that I am completely willing to change my plans depending on what they bring to class (questions, misunderstandings, stuck points), so I know that no one will take my homework list as being inflexible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the problems are just numbers on pages, they do not help students to see the &quot;story arc&quot; of the course. I have considered giving them a practice final at the beginning for that purpose. I do talk about this arc in my first lecture, and repeatedly during the course.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-07-26T14:14:59.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16857" PostTypeId="2" ParentId="2162" CreationDate="2019-07-26T15:58:25.910" Score="12" Body="&lt;p&gt;The apparent conflict between points of view expressed in the OP is illusory. There is no real conflict. The mathematics education researcher quoted in the OP is arguing that students find the definition difficult to appreciate and master. The mathematicians quoted in the OP are arguing that, &lt;em&gt;once mastered&lt;/em&gt;, it provides clarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have questions about the representativeness of the quotes of Kleinfeld and of Bishop among mathematicians: see the postscript, for one contrasting data point (and the addenda for two others); also, both pieces were written in explicit reaction against directions in college-level math education that were being championed by other mathematicians (as the first addendum illustrates). But putting that aside, not even these quotes themselves are asserting that students &lt;em&gt;find the epsilon-delta proof easy&lt;/em&gt;. More specifically:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kleinfeld is asserting not that the definition is easy to master, but that once mastered, it provides clarity that is otherwise unavailable. &quot;Without the proper definition, things are difficult. With the definition, they are simple and clear.&quot; This sentence asserts that the definition, once it has been fully understood, clarifies and illuminates the matters that the notion of &quot;limit&quot; is intended to deal with. This does not imply that the definition itself is easy. Indeed, its difficulty is implicit in her celebration of it, quoting Russell, as &quot;the greatest achievement of the human intellect in 2000 years.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quote from Bishop acknowledges the definition as &quot;notorious&quot;. More importantly, it is taken out of context. Here it is with the 6 prior and 5 following words as well:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although it seems to be futile, I always tell my calculus students that mathematics is not esoteric: It is common sense. (Even the notorious epsilon, delta definition of limit is common sense, and moreover is central to the important practical problems of approximation and estimation.) They do not believe me. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;https://www.ams.org/journals/bull/1977-83-02/S0002-9904-1977-14264-X/S0002-9904-1977-14264-X.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no assertion that students are readily assimilating to Bishop's point of view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Postscript:&lt;/strong&gt; &lt;a href=&quot;https://flm-journal.org/Articles/7FC3DCB00614F668235473F38508F6.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; is David Tall, writing in 1981:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;During the last century the epsilon-delta method has led to fruitful advances in analysis and a clarification of the meaning of certain concepts for the professional mathematician, but it is too complex and intractable for the beginning student in calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1981, Tall was making a transition into mathematics education research. Still, he was by then established as a mathematician. I think the quote nicely illustrates how one can simultaneously appreciate the definition's clarificatory power and the difficulty it presents to students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addendum 8/12:&lt;/strong&gt; I just bumped into another mathematician (this time, one who is not also a math education researcher) whose point of view is decidedly in contrast to the quotes of Kleinfeld and Bishop given in the OP. &lt;a href=&quot;http://www.ams.org/notices/199705/comm-mumford.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; is David Mumford, writing in 1997 in defense of the Gleason-Hallett Calculus Consortium's textbook's less-formal definition of continuity (namely, &quot;the closer &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; gets to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, the closer &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; gets to &lt;span class=&quot;math-container&quot;&gt;$f(a)$&lt;/span&gt;&quot;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If one says instead, &quot;For any &lt;span class=&quot;math-container&quot;&gt;$\epsilon &amp;gt; 0$&lt;/span&gt;, there is a &lt;span class=&quot;math-container&quot;&gt;$\delta &amp;gt; 0$&lt;/span&gt; such  that  whenever  &lt;span class=&quot;math-container&quot;&gt;$|x-a|&amp;lt;\delta$&lt;/span&gt;,  then &lt;span class=&quot;math-container&quot;&gt;$|f(x)-f(a)|&amp;lt;\epsilon$&lt;/span&gt;,&quot; what happens  to  most  students? First off, since Greek letters and complex English syntax (&quot;for any ... there is ... such that whenever ... then ...&quot;) are used, the student is convinced that something very complicated must be going on. What is worse, even if you give the simple description of the meaning afterwards, the  student  will  be  sure  that  something  more complex is going on or else why did you put it in such an opaque way? I think it is impossible to explain to most students that we prefer the complex syntax of the &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; definition because we have crafted it precisely to squeeze out all the ambiguity of normal English. The important question is: do most people learn a new concept most efficiently by being exposed to elegant definitions of this sort? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addendum 8/21:&lt;/strong&gt; And &lt;a href=&quot;https://launchings.blogspot.com/2014/06/beyond-limit-i.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; is David Bressoud, writing in 2014:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The very first [pedagogical problem] is that any definition of limit that is mathematically correct makes little sense to most students. Starting with a highly abstract definition and then moving toward instances of its application is exactly the opposite of how we know people learn. This problem is compounded by the fact that first-year calculus does not really use the limit definitions of derivative or integral. Students develop many ways of understanding derivatives and integrals, but limits, especially as correctly defined, are almost never employed as a tool with which first-year calculus students tackle the problems they need to solve in either differential or integral calculus. The chapter on limits, with its attendant and rather idiosyncratic problems, is viewed as an isolated set of procedures to be mastered.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Post-postscript:&lt;/strong&gt; Tall's quote (addendum: as well as Mumford's and Bressoud's) includes something that goes beyond an assessment either of the definition's ease of mastery or its clarificatory value -- namely, an assessment regarding its appropriate place in the curriculum. Kleinfeld also makes such an assessment -- presumably an opposing one, although she doesn't specify whether she's talking about the &quot;beginning student&quot; of Tall's quote. Dawkins makes no such assessment (see &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0732312313000813&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) nor, as far as I can tell, does any of the research he cites on the difficulty of delta-epsilon proofs (see &lt;a href=&quot;http://u.cs.biu.ac.il/~katzmik/sgtdirectory/ely10.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://reader.elsevier.com/reader/sd/pii/S0732312396900152?token=A277E5207A7C01B95BCD202F1922004C72675D658F720257E2313BCC703E1752F950ED966AB194497AEFB3EBED42D16A&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://staff-old.najah.edu/sites/default/files/Concept%20Image%20and%20Concept%20Definition%20in%20mathematics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, or &lt;a href=&quot;https://www.jstor.org/stable/749075?seq=1&amp;amp;cid=pdf-reference#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="140" LastEditDate="2019-08-21T22:56:06.337" LastActivityDate="2019-08-21T22:56:06.337" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16858" PostTypeId="1" CreationDate="2019-07-27T07:06:19.137" Score="23" ViewCount="1575" Body="&lt;p&gt;I wonder why topics examined in high school math contests are so different from the maths learned by those who are seriously studying a math major at a university. Firstly, contests like IMO, ARML, AMC and most of the others seem to focus on a &lt;strong&gt;very small area&lt;/strong&gt; of math (graph theory, combinatorics, elementary number theory, planar geometry, etc).  Topics in analysis (which include several courses like complex analysis and functional analysis), topics in mathematical physics (quantum mechanics, electromagnetism), and more advanced algebra (rings, modules) are almost completely ignored. I cannot see any obvious reasons why those important topics should be omitted -- it is true that some mathematicians specialise in small fields like combinatorics, but they are definitely not the majority. Most mathematicians are specialists in more &quot;modern&quot; math, like the ones I listed above (analysis, physics, algebraic and analytic number theory, etc).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, contests tend to dig unnecessarily deep into something &quot;obsolete&quot;. Take the 6th question of this year's IMO as an example:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/RzQk8.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RzQk8.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see six or seven extra lines that must be added before one can answer the question. Such questions involve a lot of small tricks that one hardly ever needs in math research. In fact, nobody needs to know how to do this question unless they are a contest math teacher. Yes, one can really dig very deep into planar geometry, and there is beauty in it. But contest-style geometry is very &quot;ancient&quot;, and is no longer the focus of research nowadays. Today, computers can already do such geometric proofs in a much more rigorous way than people in the contest. It doesn't seem to be a good idea to play with ancient things too much -- this IMO question is just like making a horse run faster than an airplane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another &quot;ancient&quot; technique commonly seen in contests is to construct inequalities without using calculus. It used to be part of the math course at university 100 years ago, but now it is no longer a topic that must be studied -- students simply learn those inequalities when necessary. However, such inequalities are still a large part of the contest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, that leads to the question: &#xA;&lt;strong&gt;what it the reason why there are almost no high school math contests that are even close to the style of math at universities?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This might be too opinion based, but note that I am asking &lt;strong&gt;&quot;why&quot;&lt;/strong&gt;, not &quot;what should we do&quot;, so I believe one can write a very objective answer to this question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, don't tell me that is because degree level maths is too hard -- clearly that IMO question above is much harder than math at &lt;strong&gt;any&lt;/strong&gt; universities. In fact, many PhD students of leading math departments cannot do such questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: I know that students learn a little bit of degree level math while training for IMO or other competitions, but their knowledge about those more advanced math is likely to be fragmented and incomplete -- they are likely to end up with false impressions about math at universities.&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="376" LastEditDate="2019-07-27T12:50:05.450" LastActivityDate="2022-05-20T01:28:41.747" Title="Why are the contents of contest maths so different from contents of degree-level maths?" Tags="|undergraduate-education|math-contests|" AnswerCount="5" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16859" PostTypeId="2" ParentId="16858" CreationDate="2019-07-28T11:46:22.513" Score="12" Body="&lt;p&gt;I don’t think it is remotely reasonable to include the areas you mentioned, since (in the U.S., at least) all those that you mentioned are at least 3rd and 4th year undergraduate subjects. Indeed, functional analysis is usually a 2nd year graduate course (often not even required for a Ph.D. in math), and mathematical physics topics such as quantum mechanics and electromagnetism are studied by only a few university mathematics majors (almost certainly fewer than 10%, and probably fewer than 5%, will have taken upper level courses in both of these during their undergraduate years), to say nothing of high school students. Anyway, for these subjects there is the Putnam exam in the U.S. and Canada, the Schweitzer Competition in Hungary, and probably others in other countries. Even these exams avoid mathematical physics (that I am aware of), and the Putnam exam in particular focuses mostly on lower undergraduate level mathematics (with an emphasis on combinatorics and other similar topics). By the way, in the U.S. and Canada, high school students taking one or more college classes are eligible to take the Putnam exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given this, one might wonder if a reasonable argument could be made for including elementary single-variable calculus topics on high school math contests. I think part of the reason for not doing this is historical inertia, part of the reason for not doing this is lack of reasonable access to calculus by many students (especially at the many smaller high schools dotted throughout the U.S., even now few offer the equivalent of a second semester college course in which integration techniques, sequence and series convergence, and the like are covered), and part of the reason for not doing this is in determining what would be fair game for calculus-background problems. Regarding this last reason, would subtle issues such as the fact that a derivative can be discontinuous or derivatives satisfy the intermediate value property (even when discontinuous) be suitable topics? Would the use of topics such as the Leibniz rule for the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;’th derivative of a product and the more specialized series convergence tests one sees in an advanced calculus course be suitable topics? At least for the examples I’ve mentioned (last two sentences), it seems to me that we primarily would be testing for content knowledge, as someone who isn’t aware of these results would essentially have to discover them while taking the test (an extremely high bar that seems only useful for detecting the next Gauss), and anyone aware of the relevant results will have a MUCH easier task.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose one could counter with the argument that the same is true for the topics that actually appear in contest problems --- the subtle issues and the specialized topics/methods still apply to combinatorics, number theory, Euclidean geometry, etc. However, students taking the tests will have been familiar with these other subjects for a much longer time than with calculus, and thus it is reasonable, I think, to expect students to have explored the more subtle issues and specialized topics/methods in these subjects than is the case with calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the underlying assumption in all of the above is that these tests are primarily for measuring mathematical ability and potential, not for mathematical achievement/knowledge. We want the questions to be chosen mainly for their cognitive difficulty than for their background-knowledge difficulty. The fact that “many PhD students of leading math departments cannot do such questions” is simply a reflection of the fact that IMO problems are pitched at cognitive difficulty levels beyond that which is needed to obtain a Ph.D. in math. Just to take the U.S., for example, each year about 900 U.S. citizens receive a Ph.D. (in the U.S.) in a mathematical science field (about 940 in 2015-2015 --- see top of p. 354 [= .pdf file page 116] &lt;a href=&quot;https://www.ams.org/journals/notices/201803/201803FullIssue.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;), and each year 6 U.S. citizens participate in the IMO. Thus, even if every IMO participant eventually gets a Ph.D. in a mathematical science field, then fewer than 1% of those earning a Ph.D. in a mathematical science field will have participated in the IMO. The fact that some IMO participants take the IMO more than one time only reduces the percentage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given all this, I certainly sympathize with your view, because when I was in high school I wasn’t interested in learning “contest math topics”. One reason is that there was hardly anything available to me (this was around 1973-1977, in a rural location). Another reason is that I was more interested in learning calculus and other college level math than I was in solidifying my knowledge of advanced/specialized high school type topics. In the years since then, especially since the mid 1990s, I’ve managed to somewhat rectify this (solidifying my knowledge of …), but it was primarily a result of teaching such topics to very strong high school students, and a result of the rise of some internet math discussion groups in the mid 1990s where such topics tend to come up quite a bit more often than, say, the Lebesgue density theorem or the Cantor-Bendixson theorem.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2019-07-28T11:46:22.513" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16860" PostTypeId="1" CreationDate="2019-07-29T03:34:51.087" Score="6" ViewCount="746" Body="&lt;p&gt;How to make a mathematical dynamic graph?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen a lot of mathematics teaching videos containing a lot of mathematical dynamic graphs; I think it is very interesting, and it is very attractive!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such a video can help us have a good understanding of mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what tool or software should I use to make such a mathematical dynamic picture and insert it into the video? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example of what I mean, I feel that one of the excellent teaching science videos is &lt;a href=&quot;https://www.youtube.com/watch?v=sD0NjbwqlYw&amp;amp;list=PLoXwKge5yMmz_1reli_mqMhEfTSWfySgv%E2%80%9C%E2%86%92here%E2%80%9D&quot; rel=&quot;noreferrer&quot;&gt;here, from 3Blue1Brown&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12484" LastEditorUserId="11" LastEditDate="2019-07-29T19:19:32.933" LastActivityDate="2019-08-21T06:52:06.007" Title="How to make a mathematical dynamic graph?" Tags="|technology-in-education|graphing|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16861" PostTypeId="2" ParentId="16860" CreationDate="2019-07-29T07:11:24.733" Score="4" Body="&lt;p&gt;Somewhere in the comments somebody asked the same question and this link is the answer: &lt;a href=&quot;https://talkingphysics.wordpress.com/2018/06/11/learning-how-to-animate-videos-using-manim-series-a-journey/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://talkingphysics.wordpress.com/2018/06/11/learning-how-to-animate-videos-using-manim-series-a-journey/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A somewhat easier way but far less powerful would be the program geogebra where you can export animations as gif.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2019-07-29T07:11:24.733" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16862" PostTypeId="2" ParentId="16807" CreationDate="2019-07-30T14:22:01.550" Score="5" Body="&lt;p&gt;The following remarks do little more than amplify on a comment made above by Andreas Blass. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A basic use of parallelograms is to represent the sum of vectors in the plane as the diagonal of the parallelogram they determine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Visualizing the action of a linear transformation of the plane can be achieved by examining the parallelogram determined by the images of the standard basis vectors. This is helpful for understanding the qualitatively different sorts of linear transformations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another use of parallelograms is when one formalizes notions such as area and volume, as is typically done in university courses in linear algebra and calculus. The best way to define the determinant of a &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; matrix is as the signed area of the parallelogram generated by the rows (or columns) of the matrix. The basic properties of the determinant - its antisymmetry and linearity in the rows/columns of the matrix and its behavior with respect to elementary row operations can be interpreted in terms of orientations of parallelograms and cutting and pasting parallelograms and their subtriangles. All of this is useful for students who will study things such as the theorems of Green/Stokes, surface integrals, and so forth, and need to assimilate notions such as orientation and signed area. For example, motivating the definition of the integral of a function over a surface by constructing a Riemann sum approximating a parametric form of the integral requires interpreting the summands of the Riemann sum as areas of infinitesimal parallelograms in planes tangent to the surface.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-07-30T14:22:01.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16863" PostTypeId="2" ParentId="16860" CreationDate="2019-07-30T23:12:57.963" Score="4" Body="&lt;p&gt;The video to which you link is much more than a dynamic graph: It is an&#xA;entire lecture using many dynamic components, requiring considerable skill&#xA;with several technologies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To return to a single dynamic graph,&#xA;in response to this MO question, &lt;a href=&quot;https://mathoverflow.net/q/77175/6094&quot;&gt;Taking “Zooming in on a point of a graph” seriously&lt;/a&gt;,&#xA;I created this &lt;a href=&quot;https://mathoverflow.net/a/77227/6094&quot;&gt;little animation&lt;/a&gt;&#xA;for &lt;span class=&quot;math-container&quot;&gt;$y=x(x−1)(x+1)$&lt;/span&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/8mVd4.gif&quot; alt=&quot;Zoom&quot;&gt;&#xA;&lt;hr /&gt;&#xA;I created this animated &lt;code&gt;.gif&lt;/code&gt; in &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Wolfram_Mathematica&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematica&lt;/a&gt;.&#xA;Similar graph animations could be made in&#xA;&lt;a href=&quot;https://www.geogebra.org/?lang=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geogebra&lt;/a&gt; &#xA;or &lt;a href=&quot;https://www.cinderella.de/tiki-index.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cinderella&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-07-31T12:26:23.013" LastActivityDate="2019-07-31T12:26:23.013" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16864" PostTypeId="2" ParentId="16858" CreationDate="2019-07-31T09:04:03.987" Score="17" Body="&lt;p&gt;Mathematics constests are a kind of game or puzzle, like chess, poker, sudoku, etc. Not all mathematics adapts well to the context of a competition in a limited amount of time. While it's true that research mathematicians can be very competitive (in Yau's recent autobiography, he several times describes mathematics as a competitive activity - note though that Yau has advanced a &lt;a href=&quot;http://www.yau-awards.org/?iafgji=avmqi2&quot; rel=&quot;noreferrer&quot;&gt;model of mathematics competition&lt;/a&gt; very different from contest math), the time scale of their competition, when it occurs, is months or years (even if a paper is written in weeks, the preparation required to write occurs on the time scales of months or years), and it is in any case an implicit competition, not occuring in a regimented and programmed context, such as contest mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sort of mathematics amenable to contests necessarily cannot require deep, novel, creative thinking, as such just is not possible on the relevant time scales. Contests require mathematics that has been well assimilated into fairly general teaching (to give the illusion of accessibility) and that can be broken into small parts stated in terms that require relatively little formalism. This favors combinatorics, plane and space geometry of the most classical sort, graph theory, finite group theory, elementary number theory, and disfavors that which requires much analysis, any kind of modern algebra or representation theory, or physically related mathematics, just to name a few areas. On the other hand, often problem writers try expressly to avoid the mathematics that is taught in standard courses, so that the contest is not simply testing who is best trained, or something like that. See &lt;a href=&quot;http://www.math.uiuc.edu/~reznick/putnam.pdf&quot; rel=&quot;noreferrer&quot;&gt;this essay&lt;/a&gt; by Bruce Reznick about writing problems for the Putnam exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most fundamental difference between mathematics contests and doing mathematics is that in a contest one knows there is an answer that can be obtained in a few hours at most. In research sometimes a big part of the problem is deciding what an answer would look like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That there are people who are successful both at contest math and at doing deep research is in no way an argument against what I just wrote, but it is not the case that ability at either necessarily signals ability at the other (although it is surely an indicator) - for every deep thinking Tao or Perelman who was successful at contest math there is an equally deep thinking mathematician who was not so good at contest math or simply did not like it (I'm tempted to suggest Thurston and Grothendieck as examples, although I have no idea what either felt about or how either fared in mathematics contests).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me a mistake to think of contest mathematics as indicative of what mathematics as a discipline is. There is some part of mathematics contests that is remanent research mathematics, in the same way (although less exaggeratedly so) that solving a sudoku is reminiscent of mathematics (there are even some nontrivial mathematical problems about solvability of generic sudokus, enumeration of sudokus, etc.), but primarily mathematics contests are oriented at giving prizes for solving a certain kind of problem rapidly. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-07-31T09:04:03.987" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16865" PostTypeId="2" ParentId="7528" CreationDate="2019-07-31T13:28:31.657" Score="6" Body="&lt;p&gt;I've found that students are not very clear on the image that is being invoked when I call &lt;span class=&quot;math-container&quot;&gt;$z = x^2-y^2$&lt;/span&gt; by its traditional name of &quot;saddle point&quot;, but they are all very clear on what a Pringles potato chip looks like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cnsfY.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cnsfY.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2019-07-31T13:28:31.657" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16866" PostTypeId="2" ParentId="7528" CreationDate="2019-07-31T13:45:16.293" Score="5" Body="&lt;p&gt;A nice challenge for a calculus class with a little physics: If particles are thrown out from a common point in all directions at the same speed, then allowed to fall freely, the shape they will sweep out is a &lt;a href=&quot;https://en.wikipedia.org/wiki/Envelope_(mathematics)#Example_5&quot; rel=&quot;noreferrer&quot;&gt;parabola&lt;/a&gt;. (Of course, the trajectory of each particle is also a parabola, that's a simpler fact.) The Fourth of July might suggest some examples:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/G5Jq2.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/G5Jq2.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5fvQm.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5fvQm.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was in high school, I saw a cutting board lying on an angle in a sink with the water pouring from the faucet onto a point on it. The water splashed out to form a parabolic arc. I wonder if you could actually bring something like that into the classroom and trace the edge of the water?&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2019-07-31T13:45:16.293" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16867" PostTypeId="2" ParentId="16858" CreationDate="2019-07-31T18:15:06.287" Score="4" Body="&lt;p&gt;I figured that I can expand my comment into an answer, even though it is still more a comment than an answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You say that math contest questions &quot;involve a lot of small tricks that one hardly ever needs in math research&quot;, and because of this you find contest questions useless for future career as a mathematician. But I do not think that math contests are meant to be a preparatory step for university math, I think they are meant to teach how to attack problems with what you are calling &quot;tricks&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The contest problems are rarely about breakthroughs in mathematics, but rather about smart combination of known techniques. They foster the way of thinking more attributable to an engineer than to a &quot;pure&quot; mathematician: take existing building blocks — &quot;tricks&quot; — and use them to achieve your goal, not much different from chess, where good players know a lot of existing combinations and apply them accordingly. Invention strategies like &lt;a href=&quot;https://en.wikipedia.org/wiki/TRIZ&quot; rel=&quot;nofollow noreferrer&quot;&gt;TRIZ&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Advanced_Innovation_Design_Approach&quot; rel=&quot;nofollow noreferrer&quot;&gt;AIDA&lt;/a&gt; rely on this approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Contest problems do not require lots of base knowledge, hence are more approachable, and usually can be solved without &quot;tricks&quot;, but usage of a particular &quot;trick&quot; can produce a fast and simple solution, while not using it would result in a wall of calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a simple problem that can be solved by a middle-schooler:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Every day an engineer arrives to a station at 8 a.m. by train. Exactly&#xA;  at the same time a car, sent from a factory, drives up to the station,&#xA;  picks up the engineer and takes him to the factory. One day the&#xA;  engineer arrived at 7 a.m., decided not to wait for the car, and&#xA;  started walking towards the car. When the car met the engineer, it&#xA;  picked him up, turned back and arrived to the factory 20 minutes&#xA;  earlier than usual. For how long did the engineer walk? Consider the&#xA;  speeds of the engineer and the car constant.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It can be solved using a rather elaborate approach that produces a system of two linear equations. Alternatively, using a &quot;trick&quot; it can be solved with mental elementary-school math in less than a minute.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a way, such problems are on the opposite end of the spectrum compared to &quot;pure&quot; math. Whether this sort of mental gymnastics is valuable is a matter of opinion. But I think it is more democratic, and more applicable in real life than &quot;pure&quot; university-level math.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastActivityDate="2019-07-31T18:15:06.287" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16868" PostTypeId="2" ParentId="16858" CreationDate="2019-07-31T18:23:10.440" Score="7" Body="&lt;p&gt;I teach a lot of contest math and I have very mixed feelings. Contests reward repertoire, perspicacity and speed. These are all certainly useful at undergrad, but the emphasis given to them feels wrong. On the other hand, there is a decent overlap between students who can assail these problems and students who will develop the other qualities and attitudes to be good mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We cannot expand the syllabus because there is not time enough to teach students at this age about all of those things in any non-superficial way whilst providing a well-rounded education and preparing them for university entrance exams. It's also not worth it because they'd just do it again at undergrad anyway, and I get the sense universities don't want their undergrads turning up with inchoate conceptions of vector spaces and lebesgue integration - they would rather have a voracious and powerful tabula rasa. Also, for some exams, the syllabus is traditional and pretty well fixed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that, I think the answer to your question is quite optimistic: the questions become that complicated and arcane &lt;em&gt;because students are that good&lt;/em&gt;. The questions cannot go outside of the syllabus, but they need to really challenge the top students, so examples like the one you gave are the result. Students really are amazing in what they can achieve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, hard problems which get the balance between expounding interesting theory and results, conveying the spirit of university mathematics and providing a fair amount of mechanical difficulty can be found in Cambridge's STEP Papers. I prefer to use these problems over competition problems to challenge my students because they give me the same feeling I had when I was an undergraduate solving problems and putting things together, instead of picking through tedious geometry puzzles.&lt;/p&gt;&#xA;" OwnerUserId="901" LastEditorUserId="901" LastEditDate="2019-11-03T22:05:53.997" LastActivityDate="2019-11-03T22:05:53.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16869" PostTypeId="2" ParentId="15309" CreationDate="2019-07-31T22:46:22.013" Score="2" Body="&lt;p&gt;Perhaps what he needs (or needed) to finish his courses&#xA;is a screen-reader that can handle mathematics?&#xA;The answer to this question, &lt;a href=&quot;https://www.washington.edu/doit/are-there-screen-readers-can-read-math-equations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Are there screen readers that can read math equations?&lt;/a&gt;, is &lt;em&gt;Yes&lt;/em&gt;: There are screen-readers that can handle&#xA;MathML. They rely on &lt;a href=&quot;http://www.dessci.com/en/products/mathplayer/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathPlayer&lt;/a&gt;,&#xA;&quot;a universal math reader that now enables math to be spoken in assistive technology products.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-07-31T22:46:22.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16871" PostTypeId="2" ParentId="16860" CreationDate="2019-08-02T20:44:38.897" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator&quot; rel=&quot;nofollow noreferrer&quot;&gt;Desmos&lt;/a&gt; may be less powerful than geogebra, but is worth mentioning in this regard. Graphs on desmos are very easily animated (with sliders), and look great with little modification. You can find much better examples, but &lt;a href=&quot;https://www.desmos.com/calculator/4ajmb7srtg&quot; rel=&quot;nofollow noreferrer&quot;&gt;here's a very simple one I made&lt;/a&gt; for thinking about the tangent line to a curve.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-08-02T20:44:38.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16872" PostTypeId="1" CreationDate="2019-08-03T01:53:18.303" Score="4" ViewCount="1697" Body="&lt;p&gt;What are the best resources to learn trigometry? I recently decided to pursue a BS in mathematics at uni. I used to fail all my math classes with D's or F's until I started teaching myself, and so far I've done well enough teaching myself geometry, algebra, etc., to get consistent A's. However, within my first couple days of studying trigonometry I've noticed its already been a frustrating experience (I've tried about 3 books so far and still struggle understanding the first chapters). Of course, in order to do well in calculus and beyond I need to overcome this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally would prefer a textbook recommendation I can download or pick up that is [preferably] not old and does not make trigonometry intimidating to approach (especially one that emphasizes understanding proofs behind properties/theorems). I'm fine with being recommended online lectures or series of videos if you insist, but I just think I learn best relying on textbooks based on my experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also like a platform one recommends to test my trigonometry skills (I don't want to rely on constantly flipping pages to find the answers to odd or even problems on the back of textbooks in order to test myself), preferably one that is cheap or free. I currently use Khan Academy's website, but for a variety of reasons they aren't ideal for me.&lt;/p&gt;&#xA;" OwnerUserId="12501" LastActivityDate="2019-12-13T21:04:51.190" Title="Best Way to Learn Trigonometry" Tags="|self-learning|teaching|precalculus|trigonometry|resource-request|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16873" PostTypeId="2" ParentId="16872" CreationDate="2019-08-03T02:13:09.197" Score="4" Body="&lt;p&gt;Schaum's outlines are very practical in general and cheap.  Well suited to an older learner.  Often the answers are right after the problems versus at the end.  And you get all the answers, not the odd/even gyp.  Thus suited to self learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like this one, overall and own it:  &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0070026505&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/gp/product/0070026505/ref=dbs_a_def_rwt_hsch_vapi_taft_p1_i10&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is from 1960ish, so the language is not archaic, but it's not &quot;new&quot;.  Not sure what benefit other than language you want from newer versions, but if you want a newer one, they have a recent 4th edition College Math you can get instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, this is a general pre-calc book (and probably what you need).  But if you just want a trig primer, Schaum's has that as well.  Obviously more trig problems in the trig book than the precalc book (which has all normal high school courses covered).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  It would be easier to advise you if had told us what books failed you.  Like did I write a long answer in vain?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.  I'm not sure why trig is so much a hurdle to people.  But I do recommend to first think about the sin and cos and such in the context of the unit circle, not ratios of sides of triangles.  It's just a little simpler concept and without a ratio to keep track of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:trig/x2ec2f6f830c9fb89:unit-circle/v/unit-circle-definition-of-trig-functions-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:trig/x2ec2f6f830c9fb89:unit-circle/v/unit-circle-definition-of-trig-functions-1&lt;/a&gt;  (Kahn makes it a little more complex here by talking about ratios.  But when I learned it the big benefit was a very first introduction with no ratios...just x and y axes of the unit circle.&lt;/p&gt;&#xA;" OwnerUserId="12502" LastEditorUserId="12502" LastEditDate="2019-08-03T02:18:33.977" LastActivityDate="2019-08-03T02:18:33.977" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16874" PostTypeId="2" ParentId="2786" CreationDate="2019-08-03T12:45:12.837" Score="5" Body="&lt;p&gt;I would like to mention this recent opinion article in the &lt;em&gt;AMS Notices&lt;/em&gt; (vol.66, no.7; &lt;a href=&quot;https://www.ams.org/journals/notices/201907/rnoti-p1045.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;) by Colin Adams (author of &lt;a href=&quot;https://bookstore.ams.org/knot&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Knot Book&lt;/em&gt;&lt;/a&gt;). His main point is&#xA;that we should try &#xA;&quot;to impart a love of mathematics.&quot; &#xA;He ends with this anecdote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I remember seeing a lecture by a professor who had won&#xA;  a variety of teaching awards. As I always am, I was curious&#xA;  to see what it was he did that made him so successful in&#xA;  the classroom. He gave a relatively standard lecture, but&#xA;  definitely well organized and clear. Okay, that’s fine, but&#xA;  it’s not going to win you prizes. There was only one thing&#xA;  he did differently that really stood out. During the entire&#xA;  lecture, he was grinning from ear-to-ear. It was clear for all&#xA;  to see how much he enjoyed the mathematics. He was in&#xA;  his element and everyone in the room could see and feel&#xA;  that. Of course, this doesn’t mean that you should lecture&#xA;  with a grin plastered across your face. If it is not naturally&#xA;  you, students can tell, and you end up looking like a raving&#xA;  lunatic.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;The &quot;Early Career&quot; section of the &lt;em&gt;AMS Notices&lt;/em&gt; (in which&#xA;the above article appears) is in general &#xA;an excellent resource.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-08-03T22:35:46.240" LastActivityDate="2019-08-03T22:35:46.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16875" PostTypeId="2" ParentId="16872" CreationDate="2019-08-03T13:05:55.563" Score="4" Body="&lt;p&gt;Maybe a visual approach could supplement your study? There are many such resources available on the web, not in textbooks. E.g., &#xA;&lt;a href=&quot;https://betterexplained.com/articles/intuitive-trigonometry/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trig Intuitively&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/EZ32N.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EZ32N.jpg&quot; alt=&quot;TrigTriangles&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Note: the labels show where each item &quot;goes up to.&quot;&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Another: &lt;a href=&quot;https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Interactive Unit Circle&lt;/a&gt;.&#xA;Another: &lt;a href=&quot;https://www.intmath.com/analytic-trigonometry/7-inverse-trigo-functions.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inverse Trig Functions&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-08-06T12:58:01.333" LastActivityDate="2019-08-06T12:58:01.333" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16876" PostTypeId="1" CreationDate="2019-08-03T13:25:44.710" Score="13" ViewCount="2669" Body="&lt;p&gt;What are the pros and cons for students of taking introductory real analysis before or after introductory abstract algebra, assuming they are going to take both?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recognize that the overlap between the two courses is minimal, and therefore they are largely independent of one another. (A debatable minor exception could be discussing fields when introducing the real numbers axiomatically, when the notion of a field is discussed in more detail in an abstract algebra course.) That said, students might cope better with real analysis than abstract algebra because it initially covers familiar ground (calculus). On the other hand, they might find the challenges of epsilontics, including working with double quantifiers and inequalities particularly difficult in contrast to the early proofs in abstract algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(By introductory real analysis, I mean primarily a course covering the theory of single variable calculus and possibly also introducing metric spaces; a typical textbook could be Abbott's &lt;em&gt;Understanding Analysis&lt;/em&gt;; by introductory abstract algebra, I mean a first course on groups, rings and fields; a typical textbook might be Fraleigh's &lt;em&gt;A First Course in Abstract Algebra&lt;/em&gt; or Gallian's &lt;em&gt;Contemporary Abstract Algebra&lt;/em&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2019-08-05T06:01:22.003" LastActivityDate="2019-08-05T06:01:22.003" Title="Introductory real analysis before or after introductory abstract algebra?" Tags="|undergraduate-education|mathematical-analysis|abstract-algebra|course-sequence|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16877" PostTypeId="2" ParentId="2786" CreationDate="2019-08-03T20:53:31.953" Score="2" Body="&lt;p&gt;Study the evidence. People believe all kinds of things about teaching, but in many cases their beliefs are contradicted by evidence. A classic, important paper in physics is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hake, &quot;Interactive Engagement Versus Traditional Methods: a Six-Thousand Student Survey of Mechanics Test Data for Introductory Physics Courses,&quot; Am. J. of Phys, 66 (1997) 64&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These findings appear to hold as well in other fields, including math:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Freeman et al., &quot;Active learning increases student performance in science, engineering, and mathematics&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.pnas.org/content/early/2014/05/08/1319030111&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.pnas.org/content/early/2014/05/08/1319030111&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2019-08-03T20:53:31.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16878" PostTypeId="2" ParentId="16876" CreationDate="2019-08-03T23:04:09.987" Score="9" Body="&lt;p&gt;Despite the names of these fields, as a student I found real analysis more &#xA;abstract than abstract algebra: &lt;em&gt;real&lt;/em&gt; analysis was less real and more abstract to me than &lt;em&gt;abstract&lt;/em&gt; algebra. I don't think I can justify this, but let me give two examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)&quot; rel=&quot;noreferrer&quot;&gt;Lagrange's&#xA;theorem&lt;/a&gt;&#xA;in abstract algebra: The order of a subgroup &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; of a finite group&#xA;&lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; divides the order of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;. Sure, this is abstract, but it is&#xA;discrete and definite and understandable from a thorough grasp of cosets.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem&quot; rel=&quot;noreferrer&quot;&gt;Heine-Borel theorem&lt;/a&gt;&#xA;in real analysis:&#xA;Closed and bounded iff every open cover has a finite subcover.&#xA;Requires understanding limit points, accumulation points, triangle&#xA;inequality.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Certainly one can pluck out a theorem from abstract algebra that is decidedly &#xA;more abstract than a particular theorem in real analysis, to make&#xA;the opposite point. But to me&#xA;abstract algebra as a whole was (and still is) more concrete than real analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would argue: Abstract algebra before real analysis, just because &#xA;proof sophistication would improve in abstract algebra to help with the&#xA;more difficult (and abstract) proofs in real analysis.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-08-04T19:13:56.023" LastActivityDate="2019-08-04T19:13:56.023" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16880" PostTypeId="2" ParentId="16876" CreationDate="2019-08-05T03:44:34.833" Score="8" Body="&lt;p&gt;Is 'at the same time' an option?  I mean, by junior year, math majors should be taking at least two math classes per semester, right?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was an undergraduate at Penn State, these two courses were the only 300 level math courses, both designed to be taken first semester junior year.  The introduction to abstract algebra used &quot;Numbers, Groups, and Codes&quot;, and the introduction to real analysis used &quot;Elementary Analysis: The Theory of Calculus&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both courses were designed to teach how to write proofs, and I don't think having two courses focused on teaching this was a waste of time.&lt;/p&gt;&#xA;" OwnerUserId="12506" LastActivityDate="2019-08-05T03:44:34.833" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16881" PostTypeId="1" AcceptedAnswerId="16883" CreationDate="2019-08-06T20:31:39.763" Score="8" ViewCount="475" Body="&lt;p&gt;I am currently writing some basic introductory texts to &#xA;complex numbers for third-year high school students (Denmark). My main goal is to introduce complex numbers as a practical tool that both simplifies the overall algebraic structure of math (simplifying work with trigonometric functions&#xA;and polynomials), and can work in and of itself as a practical&#xA;tool for modelling certain geometric objects. Due to the nice interplay between&#xA;rotation, multiplication and exponentiation, numbers in the complex plane can on occasions be a better choice to work with. Two pretty &lt;em&gt;mathematical&lt;/em&gt; examples are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Finding the centroid or circumcenter of a triangle&lt;/li&gt;&#xA;&lt;li&gt;Working with rotated conics: Finding intersections, amount of intersections, transformations, ect. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Conics have lots of obvious applications, but circumscribed triangles is a bit too specific for me to find any good applications/modelling exercises. Rotation is &lt;em&gt;so&lt;/em&gt; much nicer with complex numbers, so surely there must be more geometrical applications not? &lt;/p&gt;&#xA;" OwnerUserId="12400" LastEditorUserId="376" LastEditDate="2019-08-07T05:30:49.557" LastActivityDate="2019-08-07T05:30:49.557" Title="Examples of application problems of coordinate geometry in the complex plane?" Tags="|examples|applications|complex-numbers|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16882" PostTypeId="1" CreationDate="2019-08-06T21:23:28.990" Score="45" ViewCount="7153" Body="&lt;p&gt;I am struggling to help some of my new precalculus students accept &lt;em&gt;function notation&lt;/em&gt; -- something new to them this term. I am looking for strategies to help them adopt this new notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their main complaint:  &quot;&lt;em&gt;Why do we need &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; when we can just talk about &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;?&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For these students, an equation already represents an &lt;em&gt;input/output&lt;/em&gt; relationship, so they see no need to introduce a new letter to refer to something they already have a name for. I would like them to feel some ownership over the notation, as though it is useful-enough to them that they actually &lt;em&gt;want&lt;/em&gt; to use it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you have students with these complaints, how do you help them? What strategies/exercises have worked?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this level of student, assume they are versed in linear and quadratic equations, and they are learning function notation for the first time, including composition. They are being asked questions related to functions with formulas, word problems, graphs and numerical data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Note: While I could list many particular things my students struggle with in this regard, the issue I want to address is helping them feel function notation may be useful and, therefore, something worth really adopting.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-08-12T00:01:29.457" LastActivityDate="2019-09-29T04:30:47.307" Title="How to help new students accept function notation" Tags="|notation|precalculus|functions|" AnswerCount="21" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16883" PostTypeId="2" ParentId="16881" CreationDate="2019-08-06T22:44:34.430" Score="6" Body="&lt;p&gt;Here is a possibility, taken from &#xA;Tristan Needham,&#xA;&lt;a href=&quot;http://usf.usfca.edu/vca//&quot; rel=&quot;noreferrer&quot;&gt;Visual Complex Analysis&lt;/a&gt; (Oxford Univ. Press).&#xA;&lt;br /&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/CXmuw.jpg&quot; alt=&quot;Needham cover&quot;&gt;&#xA;&lt;br /&gt;&#xA;The advantages of this theorem are: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;it is certainly not obvious,  &lt;/li&gt;&#xA;&lt;li&gt;&quot;it would require a great deal&#xA;of ingenuity&quot; to prove this &lt;em&gt;without&lt;/em&gt; complex numbers, &lt;/li&gt;&#xA;&lt;li&gt;it is&#xA;elementary planar geometry, and &lt;/li&gt;&#xA;&lt;li&gt;it is more engaging than &quot;finding&#xA;the centroid or circumcenter of a triangle.&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/SxLXx.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SxLXx.png&quot; alt=&quot;Fig12&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;T. Needham, Fig.[12], p.16.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;The proof uses rotations throughout.&#xA;For example, the point &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;&#xA;is obtained by moving &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; halfway along the &lt;span class=&quot;math-container&quot;&gt;$2a$&lt;/span&gt; edge of the quadrilateral,&#xA;and then turning &lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt; counterclockwise via &lt;span class=&quot;math-container&quot;&gt;$i a$&lt;/span&gt;. &#xA;So &lt;span class=&quot;math-container&quot;&gt;$p=a+i a = (1+i) a$&lt;/span&gt;.&#xA;(OP: &quot;Rotation is &lt;em&gt;so&lt;/em&gt; much nicer with complex numbers.&quot;) &#xA;Eventually the theorem is proved by showing that&#xA;&lt;span class=&quot;math-container&quot;&gt;$A + iB = 0$&lt;/span&gt;, &quot;the verification of which is a routine calculation.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://math.stackexchange.com/a/85244/237&quot;&gt;Visual research problems in geometry&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-08-07T01:30:16.203" LastActivityDate="2019-08-07T01:30:16.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16884" PostTypeId="2" ParentId="16881" CreationDate="2019-08-07T00:45:51.300" Score="1" Body="&lt;p&gt;(comment) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why the need and is there really a pedagogical benefit to a non-standard presentation of complex numbers?  This feels more like something that appeals to you, that thus you want to push on students.  But without considering if it really benefits them or why it wasn't done before.  Or even if the non-standard approach is detrimental.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other than that, useful to think about if this is for high capability students or average students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oh...and the most obvious applications of complex numbers are (real life) alternating current and (math) roots to the quadratic.  Oh...and those are boring and familiar to math shmarties. But for kids learning complex numbers for the first time, they are not boring.&lt;/p&gt;&#xA;" OwnerUserId="12502" LastActivityDate="2019-08-07T00:45:51.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16886" PostTypeId="1" CreationDate="2019-08-07T07:38:33.227" Score="6" ViewCount="198" Body="&lt;p&gt;I am very new to this platform so I may have misunderstood the intent of this site, or might seem a bit off, but please bear with me because I know what I want for certain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I always wanted to study mathematics back in high school and I was reading up on anything I could find. But due to certain problems, I ended up studying medicine for my family. I still did read a few magazines on math, but I found myself unable to remember a single thing last week when I wanted to go over my high school notes which, and that got me very frustrated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know, regardless what happens, that I want to be a mathematician. I may lack the knowledge, but I know I see the world from a mathematician's eyes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if any of you can spare some time for me, please suggest books or study patterns for me. I don't expect this to be a quick process, but I would love to establish the basics for my journey to begin. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you so much already.&lt;/p&gt;&#xA;" OwnerUserId="12510" LastEditorUserId="238" LastEditDate="2019-09-30T23:17:29.360" LastActivityDate="2021-09-12T05:56:20.813" Title="Forming a Study Model for a Self-Study Beginner" Tags="|textbooks|curriculum|self-learning|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16887" PostTypeId="2" ParentId="16882" CreationDate="2019-08-07T08:26:17.153" Score="24" Body="&lt;p&gt;Start by talking about functions in general, not only about functions that can be expressed by a simple formula in x and y. Examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The function that maps every non-empty list to its first element.&lt;/li&gt;&#xA;&lt;li&gt;The function that maps every finite set to its size.&lt;/li&gt;&#xA;&lt;li&gt;The function that maps color names to RGB triples.&lt;/li&gt;&#xA;&lt;li&gt;The function that maps days to sunrise times at a particular location.&lt;/li&gt;&#xA;&lt;li&gt;The fubction that maps locations to sunrise times on a particular day.&lt;/li&gt;&#xA;&lt;li&gt;The function that maps the age of a particular person to their height.&lt;/li&gt;&#xA;&lt;li&gt;The function that maps the length of a square to its area.&lt;/li&gt;&#xA;&lt;li&gt;The function that maps my age (in days) to your age (in days).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some of these can be expressed by a simple mathematical formula, some can be expressed by a complicated formula, or by program, or by a table, and for some (&quot;age -&gt; height&quot;) there is no formal presentation at all, even though one can still state certain properties (domain, range, monotonicity, ...) of the function.&lt;/p&gt;&#xA;" OwnerUserId="8472" LastActivityDate="2019-08-07T08:26:17.153" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16888" PostTypeId="2" ParentId="16882" CreationDate="2019-08-07T09:17:25.707" Score="22" Body="&lt;p&gt;You should tell them these two main benefits:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Function notation is concise! For example, instead of writing &quot;Find &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x=5$&lt;/span&gt;&quot; one can simply write &quot;Find &lt;span class=&quot;math-container&quot;&gt;$f(5)$&lt;/span&gt;&quot; This becomes very appreciable when dealing with long or complicated problems asking for a lot of information. We also shorten things like this all the time. For instance, instead of writing &lt;span class=&quot;math-container&quot;&gt;$\{x|-\infty&amp;lt;x&amp;lt;\infty \}$&lt;/span&gt; we may prefer to write &lt;span class=&quot;math-container&quot;&gt;$x\in \mathbb{R}$&lt;/span&gt;. So these nice conventions are things students will just have to get used to (I'm sure that won't be hard once they see there are very good reasons entire math communities agree on specific notation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) It clears a lot of confusion. If we use &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; for everything then we'd have to use subscripts every time we are dealing with two or more functions. If we were given two functions where one refers to say, the number of blue marbles and the other the number of red marbles, I'd much rather see &lt;span class=&quot;math-container&quot;&gt;$B(x), \ R(x) \ \text{than} \ y_1, y_2$&lt;/span&gt;. I don't want to think &quot;wait, what which one is which, again?&quot; in my head more than once. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same issue comes up in just about anything involving functions (function mapping, defining categories, etc.), but let's say we're talking about function composition. &lt;span class=&quot;math-container&quot;&gt;$(y_1 \circ y_2)(x)=y_1(y_2(x))$&lt;/span&gt; is far more confusing (especially when there is context from a word problem) than &lt;span class=&quot;math-container&quot;&gt;$(f \circ g)(x)=f(g(x))$&lt;/span&gt;. It's just not immediately clear nor pleasing to look at. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure after enough practice, function notation will come naturally to your students. &lt;/p&gt;&#xA;" OwnerUserId="12501" LastEditorUserId="12501" LastEditDate="2019-08-07T18:29:39.517" LastActivityDate="2019-08-07T18:29:39.517" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16889" PostTypeId="1" CreationDate="2019-08-07T10:48:56.893" Score="13" ViewCount="3040" Body="&lt;p&gt;&lt;strong&gt;Background/rant:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am in charge of teaching our single quarter course on vector calculus (don't ask me why the department head thinks the area can be covered in half a semester). The two biggest groups of students in that course are A) math/statistics majors, B) physics majors. There are smaller groups, but those two form the bulk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The course is designed to be taken roughly by the sophomores. During their respective freshman years the math/statistics majors have seen a variant of 1st analysis that concentrates more on concepts and proofs as opposed to learning methods of calculus. The physics majors have seen a variant of the same, but with emphasis on the methods, but also a touch of epsilons, upper and lower sums when defining integrals and such. Furthermore, the physics majors have taken a year long &quot;methods&quot; course. I am definitely concentrating on giving the students insight/intuition on how to apply the tools of calculus/elementary analysis in several dimensions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the past the students, math majors more than the others, have come forward and asked, whether they can use the fancy calculators they bought for their high school work in the finals. The general department policy is to only allow &quot;calculators from the 70s-80s&quot; (guess what decade the senior staff, yours truly included, did time in high school!): no graphical output, nothing symbolic... The thinking has been that the students need to learn to do the integrals by hand, just in case their survival on a desolate island depends on... Yeah, right. So to make allowances for all this I have had to dedicate a bit of time to extra exercises (outside of lecture hours) just to make sure that the math majors also know how to evaluate &quot;cheat sheet&quot; integrals like&#xA;&lt;span class=&quot;math-container&quot;&gt;$\int_0^{\pi/2}\cos^mx\sin^nx\,dx$&lt;/span&gt;. Otherwise they would be helpless when calculating integrals in spherical coordinates and such. &lt;em&gt;This is also time that would be better spent learning something more essential.&lt;/em&gt; Such as figuring out how to set up the boundaries of an iterated integral &amp;ndash; something that is essential when doing several variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The idea.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be happy to accommodate for this sorry state of affairs, and offer the students their choice of a final exam. Either their fancy symbolic/graphical calculator or the standard edition. Making the two variants and their grading absolutely comparable/fair is not easy, but I think I can manage.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it a bad idea to offer variants of a final exam based on the type of allowed calculators?&#xA;  Have you ever tried this? What pitfalls did you meet? Do you foresee any?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remarks, sporadic thoughts:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For a long time I have been rewarding the students for performing&#xA;&quot;reality checks&quot; on their answer. Say, if I ask them to calculate the&#xA;volume of some object that (judging from the plot I gave) is on top&#xA;of the unit disk, with height varying between &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, they&#xA;should expect an answer in the ball park of &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt;. I have been&#xA;gradually making such mental checks &lt;em&gt;obligatory&lt;/em&gt; parts of the answer,&#xA;and can use this to compensate.&lt;/li&gt;&#xA;&lt;li&gt;On the other hand, I do suffer from pangs of guilt. Am I just&#xA;sweeping things under the rug instead of making sure the students&#xA;know how to calculate basic integrals? Irrespective of what was&#xA;covered in their freshman courses.&lt;/li&gt;&#xA;&lt;li&gt;I asked the colleague in charge of organizing the exams, whether he can arrange the proctoring of such &quot;dual&quot; exams. He did not foresee any problems. We have several exams simultaneously in the same auditorium, and it is trivial to arrange the seating in such a way that the students sit on distinct rows according to the type of allowed aids, making rule enforcement feasible.&lt;/li&gt;&#xA;&lt;li&gt;I have no idea yet, whether the students would actually buy this idea :-)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="282" LastEditorUserId="143" LastEditDate="2019-08-11T17:36:31.853" LastActivityDate="2019-09-13T00:20:12.393" Title="Is it a bad idea to offer variants of a final exam based on the type of allowed calculators?" Tags="|undergraduate-education|exams|calculators|" AnswerCount="6" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="16890" PostTypeId="2" ParentId="16889" CreationDate="2019-08-07T12:02:03.337" Score="21" Body="&lt;p&gt;Let's start by saying that I strongly advice against such a dual-exam. Even if you and everyone involved in the planning think it is fair, students might think differently. In this way, you open up the floodgates for grade complaining. They might not succeed, but even the fact that some might try will cost you a lot of time and possibly reputation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, in the hope that you are still planning your class and aren't already two weeks before the exam, let's tackle the main problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the past the students, math majors more than the others, have come forward and asked, whether they can use the fancy calculators they bought for their high school work in the finals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course they have, and you can't blame them. They learned in high school that life is easy, you just enter the information into the calculator and it returns the desired result. Why should they do it the hard way when they don't have to? Thus, your lecture needs to show them the why. Include examples where the calculators fail or, even worse, make wrong assumptions on the parameters included and thus output wrong information.&lt;br&gt;&#xA;If you want to, you can also include the calculators. These are mostly stupid beasts, so it should be possible to have an integral that they can't solve, but after applying some transformations that the students learn in class, they suddenly can solve the simpler integrals constructed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, make sure to keep your numbers easy. If I have to compute&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_1^2 \frac{1}{x} + 5x^2 dx,$$&lt;/span&gt;&#xA;then I feel like this should be feasible by hand. If, on the other hand, the same problem comes with numbers like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_1^2 \frac{7.56893}{x} + 5x^2 - 1.7653\cdot 10^8 dx,$$&lt;/span&gt;&#xA;even though the steps to take are the exact same, I can fully understand if no one wants to do it by hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, you can ask for intermediate steps and explanations in the computation, something that a calculator can most likely not provide.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So to make allowances for all this I have had to dedicate a bit of time to extra exercises (outside of lecture hours) just to make sure that the math majors also know how to evaluate &quot;cheat sheet&quot; integrals ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And why was this outside of lecture hours? I'm sure everyone would profit from learning how to do it. Even if they already knew beforehand, repetition is always good.&#xA;If you can't do it in a single repetition lecture because the differences are way too big, you need to reevaluate the course outline and the prerequisites. See also the next point about that.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I am in charge of teaching our single quarter course on vector calculus (don't ask me why the department head thinks the area can be covered in half a semester).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You are in charge of teaching that class, so you should &lt;em&gt;definitely&lt;/em&gt; know how and why to do it in a single quarter. Push back by demanding more time or reducing the content you are teaching. As long as you don't do anything, people will be happy that the students learn that much in a single quarter and don't bother checking if that is even possible.&lt;br&gt;&#xA;It is your responsibility as a teacher to make sure that the students have the right knowledge to take the class and that the goals of the class can be achieved in one quarter. If you think that it not possible, something has to change. Not only is it stressful for you, but imagine how the students feel, taking extra lectures and maybe still failing because it is too much.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know how bad it is in your case, I wish that your only problem are lazy students that got too used to a machine thinking for them. Still, I think that meeting their demands is not the right way to handle things. Take time to rethink your lecture so that there is no need for two exams anymore.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastActivityDate="2019-08-07T12:02:03.337" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16891" PostTypeId="2" ParentId="16886" CreationDate="2019-08-07T12:50:41.470" Score="6" Body="&lt;p&gt;If you have forgotten things, it is because you either didn't learn them deeply the first time, or you've been so long away, they are buried. Start as low-level as you have to. Find things that are fun and challenging. You haven't yet shared what your level is, so it is hard to respond with specific resources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach at community college in the U.S. and am quite comfortable with most topics. I still had a lot to learn before teaching geometry for the first time this summer. (And loved learning it.) Are you completely comfortable with the tools that algebra and geometry provide? Are you stuck at calculus level? (Or previous to that, or beyond that?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you describe your level, many of us can answer with specific resources you might find helpful.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-08-07T12:50:41.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16892" PostTypeId="2" ParentId="16882" CreationDate="2019-08-07T16:27:23.177" Score="16" Body="&lt;p&gt;The crucial thing the students need to realise is that the (e.g.) &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that turns up in the function definition is a &lt;a href=&quot;https://en.wikipedia.org/wiki/Free_variables_and_bound_variables&quot; rel=&quot;noreferrer&quot;&gt;bound variable&lt;/a&gt;. That's what allows it to be freely renamed or indeed omitted without changing the semantics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, education tends to completely obscure this facet by &lt;strong&gt;a)&lt;/strong&gt; always using the same dumb variable names as if there were a particular meaning to &lt;code&gt;f&lt;/code&gt;, &lt;code&gt;x&lt;/code&gt; and &lt;code&gt;y&lt;/code&gt; (when in fact the power of these abstractions is that the naming is completely arbitrary) and &lt;strong&gt;b)&lt;/strong&gt; by themselves confusing functions with their results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, I so often hear people talking/writing about “the function &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt;”. That's wrong, &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt; is not a function, the function is &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt;. Why this is important only becomes properly apparent when using &lt;em&gt;higher-order functions&lt;/em&gt;, the simplest being indeed function composition. So that's the example I would focus on: give deliberately exercises where the notation would clash without the proper notion of &lt;em&gt;function&lt;/em&gt;. Like, just ask them to compose the functions &lt;span class=&quot;math-container&quot;&gt;$f : x\mapsto x^2+x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g : x\mapsto x-1$&lt;/span&gt;. This clearly doesn't work when just talking about “the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;s”.&lt;/p&gt;&#xA;" OwnerUserId="6902" LastActivityDate="2019-08-07T16:27:23.177" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16894" PostTypeId="2" ParentId="16882" CreationDate="2019-08-07T19:13:40.333" Score="14" Body="&lt;h1&gt;Because x and y are just variable names&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;It happens that sometimes &lt;strong&gt;&lt;em&gt;y&lt;/em&gt;=&lt;em&gt;f&lt;/em&gt;(&lt;em&gt;x&lt;/em&gt;)&lt;/strong&gt;, but other times &lt;strong&gt;&lt;em&gt;z&lt;/em&gt;=&lt;em&gt;f&lt;/em&gt;(&lt;em&gt;x&lt;/em&gt;,&lt;em&gt;y&lt;/em&gt;)&lt;/strong&gt;, &lt;strong&gt;&lt;em&gt;w&lt;/em&gt;=&lt;em&gt;f&lt;/em&gt;(&lt;em&gt;x&lt;/em&gt;,&lt;em&gt;y&lt;/em&gt;,&lt;em&gt;z&lt;/em&gt;)&lt;/strong&gt;, or &lt;strong&gt;&lt;em&gt;x&lt;/em&gt;=&lt;em&gt;f&lt;/em&gt;(&lt;em&gt;y&lt;/em&gt;)&lt;/strong&gt; for that matter.  All of these variable names are syntactically equivalent, and the mere existence of &quot;x&quot; and &quot;y&quot; in an equation does not necessarily connote that &quot;x&quot; is the independent variable and &quot;y&quot; the dependent.  Thinking of the function &lt;em&gt;as a function&lt;/em&gt;, making explicit each variable's role, is a new level of understanding that's missing from a mere equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The common practice of presenting &lt;strong&gt;&lt;em&gt;y&lt;/em&gt;&lt;/strong&gt; as a function of &lt;strong&gt;&lt;em&gt;x&lt;/em&gt;&lt;/strong&gt; builds in the minds of many students the notion that &lt;strong&gt;&lt;em&gt;y&lt;/em&gt;&lt;/strong&gt; &lt;em&gt;must always be&lt;/em&gt; a function of &lt;strong&gt;&lt;em&gt;x&lt;/em&gt;&lt;/strong&gt;.  Thus, when we introduce &lt;strong&gt;f(&lt;em&gt;x&lt;/em&gt;)&lt;/strong&gt; notation, it seems redundant.  Textbook authors should deliberately present other scenarios to avoid creating this notion.&lt;/p&gt;&#xA;" OwnerUserId="5117" LastEditorUserId="5117" LastEditDate="2019-08-12T17:17:56.077" LastActivityDate="2019-08-12T17:17:56.077" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16895" PostTypeId="2" ParentId="16882" CreationDate="2019-08-07T19:36:45.787" Score="3" Body="&lt;p&gt;Function notation is a next step in mathematical maturation. In the language of Dubinsky &lt;em&gt;et al.&lt;/em&gt;, your students are in the process of encapsulating functions as primary objects. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At one point in mathematical development, after learning to count, positive integers are &quot;encapsulated&quot; by children as primary objects. Later, while learning algebra, variables such as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; become encapsulated as primary objects. Many people never get past this stage. It sounds like your students have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then when learning calculus, functions like &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; might become encapsulated (abstracted) as primary objects. But it's a messy process. The very fact that your students are asking this question is evidence that they are in the process of encapsulating functions as primary mathematical objects. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An analogy I like is learning to tie one's shoe laces. At first, it seems very mysterious and difficult. It would just be easier to remove the laces, or go around barefoot. But eventually, the task becomes muscle memory, and it's impossible to recall the learning difficulty. And then it is on to the next struggle. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2019-08-07T22:28:44.080" LastActivityDate="2019-08-07T22:28:44.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16896" PostTypeId="2" ParentId="16889" CreationDate="2019-08-07T20:58:13.490" Score="9" Body="&lt;p&gt;I think this is a bad idea, because it invites students to try to game the system: those who own a fancy calculator suddenly have to decide if they’re better off using it and taking an exam they think will be harder, and those who don’t have to decide if it would offer enough of an advantage to be worth getting one and taking the other version.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Worse, you invite students to decide, retroactively, that they made the wrong choice. They won’t make that assessment based on full knowledge of the exams - they’ll hear “oh, I had such and such problem on my version” and decide that they would have been better off taking the other exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Issues like this come up a lot - where there are a couple reasonable ways to structure an exam (with or without a calculator, a few long problems or an larger number of shorter problems, etc), and it’s tempting to offload that decision to students in the hopes that they might make the best choice for themselves. I think this rarely works out; the weakest students are often the ones most prone to choosing badly for themselves, and it invites students to fixate on these discrete, superficially high stakes decisions rather than the core content of the course.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-08-07T20:58:13.490" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16897" PostTypeId="2" ParentId="16882" CreationDate="2019-08-07T22:08:38.393" Score="24" Body="&lt;p&gt;You might remind them that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is just a name for a number.  When they draw a plot, they draw a bunch of points: maybe &lt;span class=&quot;math-container&quot;&gt;$y=3$&lt;/span&gt; here, &lt;span class=&quot;math-container&quot;&gt;$y=5$&lt;/span&gt; there, and &lt;span class=&quot;math-container&quot;&gt;$y=-2$&lt;/span&gt; over there.  But at some point (no pun intended) we want to talk about the entire shape: we want to say that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is symmetric, that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is concave, that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has an asymptote.  We can't do that with &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;; saying &quot;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is invertible&quot; is as nonsensical as saying &quot;4 is invertible&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; lets us talk about a shape instead of a number.  Or, if you prefer, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; gives us a way to talk about the forest, whereas before we only had &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; for talking about trees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider &lt;span class=&quot;math-container&quot;&gt;$y=x^2+x+7$&lt;/span&gt;, an equation about &quot;trees&quot;.  We might have no idea what number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is, and no idea what number &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is, but do we know they're related quadratically. &#xA; Now we can write equations like &lt;span class=&quot;math-container&quot;&gt;$g(x)=f(x-1)+1$&lt;/span&gt;, an equation about &quot;forests&quot;.  Again, we might have no idea what shape &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is, and no idea what shape &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is, but we do know they're translations of each other.  There was no way to express this idea when we could only talk about trees.&lt;/p&gt;&#xA;" OwnerUserId="12515" LastEditorUserId="12515" LastEditDate="2019-08-10T14:48:50.847" LastActivityDate="2019-08-10T14:48:50.847" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16898" PostTypeId="2" ParentId="16882" CreationDate="2019-08-08T01:06:01.620" Score="13" Body="&lt;p&gt;&lt;code&gt;TL;DR:&lt;/code&gt; &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A function is a verb. It's an action.&lt;/li&gt;&#xA;&lt;li&gt;Variables are nouns, objects. &lt;/li&gt;&#xA;&lt;li&gt;Verbs (functions) connect nouns semantically, i.e. &lt;em&gt;how&lt;/em&gt; A (or x) relates to B (or y), how to get from &lt;em&gt;here&lt;/em&gt; to &lt;em&gt;there&lt;/em&gt;. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;Long version&lt;/code&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some context: I learned maths from my father who was a physics / engineering guy at heart, so &lt;strong&gt;&lt;em&gt;everything&lt;/em&gt;&lt;/strong&gt; had to be 'tangible' or 'observable' to him. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus the lesson would always begin with an &quot;ELI5&quot; overview of the essential concept (and indeed, I was between 6-18...), involving drawings and a lot of gestures, but more importantly was always &lt;strong&gt;based on a real-world tangible use-case/example&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Typically, simple physics. That's how I learned most math: as a tool to solve some problem I could 'feel' (often even solve mentally for simple/extreme cases, which is great to build initial understanding). To this day I think¹ of maths &lt;em&gt;mostly&lt;/em&gt; in visual terms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It helped me &quot;internalize&quot; or &quot;own&quot; mathematical objects &lt;strong&gt;&lt;em&gt;exactly&lt;/em&gt;&lt;/strong&gt; the way you seek for your students to understand. So that's my recommendation: make them 'feel' it through &lt;em&gt;real stuff&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below (in ““quotes””) is just one extremely simple example of how I'd do it (it's about going from a tone to the same tone 1 octave higher in music). [my remarks in sq. brackets]&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;““A musical tone is defined by its frequency in Hertz: for instance, 440 Hz is an &lt;code&gt;A&lt;/code&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you double the frequency, you obtain the same tone, just 1 octave higher: so 880 Hz is an &lt;code&gt;A&lt;/code&gt; too, higher in pitch. We usually number the tones to know which octave on a piano, 440 Hz is &lt;code&gt;A4&lt;/code&gt; and 880 Hz is &lt;code&gt;A5&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when we go from &lt;code&gt;A4&lt;/code&gt; to &lt;code&gt;A5&lt;/code&gt;, we apply a &lt;strong&gt;&lt;em&gt;function&lt;/em&gt;&lt;/strong&gt; which doubles the frequency. That's what going 1 octave higher &quot;does&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's write this in math:&lt;br&gt;&#xA;we define the &lt;strong&gt;function &lt;code&gt;Oct&lt;/code&gt;&lt;/strong&gt; of &lt;strong&gt;&quot;going one octave higher&quot;&lt;/strong&gt; as: &lt;strong&gt;&lt;code&gt;Oct(x) = 2x&lt;/code&gt;&lt;/strong&gt;  with &lt;code&gt;x&lt;/code&gt; the frequency of a music tone.&lt;br&gt;&#xA;[notice, &quot;going&quot; is a &lt;em&gt;verb&lt;/em&gt;, as it should: it expresses a relation between nouns, between 'things' aka variables; also we do away with &lt;code&gt;F&lt;/code&gt; and go more literal to open the mind to &lt;em&gt;meaningful&lt;/em&gt; abstractions — thereby laying some groundwork for an other important metaconcept, how to select/create/cookie-cut variables to solve problems] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we input &lt;code&gt;x=A4&lt;/code&gt; we get twice that in result: &lt;code&gt;Oct(A4)=A5&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's the equivalent of moving your finger 12 keys to the right on a piano, or some length down on a cello: that's what this function does &lt;em&gt;in those contexts&lt;/em&gt;.&lt;br&gt;&#xA;[bonus points if you can show them the movement of the arm, the three physical elements:&lt;br&gt;&#xA;key &lt;code&gt;A4&lt;/code&gt;, arm &lt;code&gt;movement&lt;/code&gt;, key &lt;code&gt;A5&lt;/code&gt;; in other common words &lt;code&gt;x&lt;/code&gt;, &lt;code&gt;F&lt;/code&gt; and &lt;code&gt;y&lt;/code&gt;.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that, we can take any tone, we get the same tone 1 octave higher: this particular function works for &lt;strong&gt;&lt;em&gt;all pairs&lt;/em&gt;&lt;/strong&gt; between/from the origin space (say x is bound to octaves 0-7) and/to the destination space (&lt;code&gt;Oct(x)&lt;/code&gt;, the 'y's (all of them), would thus be bound to octaves 1-8).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Forget about an &quot;equality&quot; between some terms of x and y, only consider the movement &lt;em&gt;from&lt;/em&gt; x &lt;em&gt;to&lt;/em&gt; y: a &lt;strong&gt;&lt;em&gt;relation&lt;/em&gt;&lt;/strong&gt; that takes x and transforms it into y. This relation, this transformation, this change is called a &lt;strong&gt;function&lt;/strong&gt;.””&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The important part here, imho, is to de-associate the function itself from variables. Have them see:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;a &quot;definition space&quot; (terms like 'origin' or 'from' work well because they prepare concepts like bijection, surjection etc).&lt;/li&gt;&#xA;&lt;li&gt;a &quot;result space&quot; (again, 'destination' or 'to' work well)&lt;/li&gt;&#xA;&lt;li&gt;a &quot;connection&quot; or &quot;relation&quot; between the two, a &lt;em&gt;way&lt;/em&gt; to get &lt;em&gt;from&lt;/em&gt; 1 &lt;em&gt;to&lt;/em&gt; 2, from the origin to the destination.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A function is a movement, a process, a change, a variation, a computation, a derivation, an integration, a representation, an abstraction. A fucntion is a verb, an action, something we &lt;strong&gt;&lt;em&gt;do&lt;/em&gt;&lt;/strong&gt;. It's not a &lt;em&gt;thing&lt;/em&gt;, it's what &lt;em&gt;changes&lt;/em&gt; things. It's a very-transparent-box that takes stuff &lt;strong&gt;&lt;em&gt;in&lt;/em&gt;&lt;/strong&gt; and chews some other stuff &lt;strong&gt;&lt;em&gt;out&lt;/em&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have them think like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;don't ask &quot;how much of x is equal to y?&quot; because that's entirely static, there is no function (verb, change) even if y=2x&lt;/li&gt;&#xA;&lt;li&gt;rather wonder &quot;what do I need to &lt;em&gt;&lt;strong&gt;do to x&lt;/strong&gt;&lt;/em&gt; to obtain y?&quot; and that's an action, a verb, you take x and change it into something else and call that thing y. &lt;/li&gt;&#xA;&lt;li&gt;Well, this change is a function.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Later on you can expand on the idea of &quot;for any X&quot; (domain/space of definition).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could elaborate further on the musical (or whatever) example you used in introduction:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;““Now you can also see how there are tons of possible functions in music to describe going from&lt;/em&gt; some tone &lt;em&gt;to&lt;/em&gt; some other, &lt;em&gt;and these &lt;strong&gt;variations&lt;/strong&gt; more than the tones themselves are what makes music: for proof we can totally change the tonality of a song (play it in &quot;D&quot; instead of playing it in &quot;C&quot;) and it's still the same song to everyone listening: because we're in fact listening to the functions, the variations).””&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd caution against using non-static variables (involving time, progressions), it may be confusing in this particular context (you want 'x' and 'y' as static as can be, like music tones don't ever change by themselves: the only 'moving part' in the picture should be the function).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know where you're going with this (I'm not American so I don't really know in which order you learn math topics), but there are interesting functions to visualize. I personally like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;code&gt;i&lt;/code&gt; as a new object defined by a function (&lt;code&gt;sqrt(-1)&lt;/code&gt; ). How cool is that.&lt;/li&gt;&#xA;&lt;li&gt;Complex functions, rotations in the complex plane etc.&lt;br&gt;&#xA;Plug with trig or not (I'd say as little as necessary in this context) or compare x,i with traditional x,y linear, show them how functions can appear under different forms yet may remain identical (or close enough for the &quot;feel&quot; like multiplication by i; and how it's also just a rotation of coordinates themselves if you'd rather fix variables in the plane).&lt;br&gt;&#xA;[note: also great electricity math to be done with these, and current works much like water for the most part (e.g. &quot;tension&quot; is a difference of potential between two points, much like gravity/altitude on Earth) so it's very easy to picture the effects of some well-crafted circuits/functions, mentally.]&lt;/li&gt;&#xA;&lt;li&gt;&quot;Divide by two&quot; is more complex but more fun that our &lt;code&gt;2x&lt;/code&gt; example above.&lt;br&gt;&#xA;Like halving 1 infinitely approximates to 0 (or 1/2+1/4+1/8... goes to 1).&lt;/li&gt;&#xA;&lt;li&gt;Likewise the &lt;code&gt;α=1+1/α&lt;/code&gt; equation (&lt;a href=&quot;https://math.stackexchange.com/a/315376&quot;&gt;https://math.stackexchange.com/a/315376&lt;/a&gt;). Generally these infinite expressions can be &lt;em&gt;represented&lt;/em&gt; as infinite compositions of the same function, infinite recursions, and these kinda 'fractal' &lt;em&gt;movements&lt;/em&gt; or &lt;em&gt;evolutions&lt;/em&gt; can not only be mentally pictured but via computers as well (or even simple pieces of paper, add one half of the remaining half until you can't and everyone gets the point). Think animation, step by step, whatever the medium.&lt;br&gt;&#xA;&lt;em&gt;Tell the story of &quot;this specific function&quot; in &quot;this specific context&quot;, then shift one (function or context), and again, and again, to further isolate the set of variables from the set of functions, in the student's mind.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm sorry I took so long but I wanted to arm you with all the terms, views and 'tricks' I could think of, so you can choose. I feel that's easily in the top 5 most important concepts to really grasp in math and any applied field.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;¹ Although this is slightly off-topic, I need to plug 3Blue1Brown's video on the Euler identity, it's a visual graal if there ever was one on a math relation.&lt;br&gt;&#xA;Link: &lt;a href=&quot;https://youtu.be/mvmuCPvRoWQ&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/mvmuCPvRoWQ&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edits: clarifications, typos&lt;/p&gt;&#xA;" OwnerUserId="12516" LastEditorUserId="12516" LastEditDate="2019-08-08T01:30:42.620" LastActivityDate="2019-08-08T01:30:42.620" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16899" PostTypeId="2" ParentId="16889" CreationDate="2019-08-08T01:35:04.610" Score="-1" Body="&lt;p&gt;Just make the kids do a test sans calculators.  Knowing how to manipulate the integrals helps when you are reading derivations or papers in physics and engineering.  Then you only need one version.  The no calculator version.&lt;/p&gt;&#xA;" OwnerUserId="12502" LastEditorUserId="143" LastEditDate="2019-08-11T17:34:17.037" LastActivityDate="2019-08-11T17:34:17.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16900" PostTypeId="2" ParentId="16889" CreationDate="2019-08-08T15:49:54.630" Score="3" Body="&lt;p&gt;Wanting your students to learn how to solve problems, instead of having them memorize some formulas and do basic arithmetic by for two hours, is a great goal imo.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first step towards that goal is making the formulas available to your students, just as they would be in the real world, but preventing them from accessing the actual solution of the problem (or something that calculates the solution). The simple answer to that is a cheat sheet with the formulas. You can create a curated cheat sheet with all necessary formulas (in your case it would be some of the most common integrals, if I understood correctly), distribute it to your students or somehow make it accessible to them, use it during the course, and allow it on the exam. Your students will be able to focus on the problem instead of the formula, and they will learn how to solve unknown problems by reducing them to known problems (the formulas themselves).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second step is to remove the calculators, to prevent any unfair advantage from having an extremely advanced and expensive calculator, or arguments about whether this or that model is allowed and if not why wasn't it made clear before the student wasted money buying it. You wanted to allow calculators in the first place because that would remove basic arithmetic from the grading (i.e. eliminate incorrect results that show correct reasoning but basic arithmetic errors). You can actually achieve that to a very acceptable degree by using constants that are easy to operate with, such as small integers. Dirk has already made a great point about that, so I won't elaborate further.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you find it impossible or unfeasible to write exam questions that don't need a calculator, having the department acquire a number of very basic calculators is an option, though a more difficult one. In the long run it could benefit many other courses and help standardize rules throughout the department's courses, so imho it's not a bad idea at all, but you're in a better position to judge that and the feasibility.&lt;/p&gt;&#xA;" OwnerUserId="12523" LastActivityDate="2019-08-08T15:49:54.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16901" PostTypeId="2" ParentId="16882" CreationDate="2019-08-08T16:16:03.010" Score="5" Body="&lt;p&gt;I have never worked with students of that skill-level, so take this with a grain of salt.&lt;/p&gt;&#xA;&lt;p&gt;I like to thinks of functions as values, just a different kind of value from numbers. This can help demystify stuff like &lt;span class=&quot;math-container&quot;&gt;$\circ$&lt;/span&gt; as it just like &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;, except it works on a different type of value. Once you get to vectors you also have a very nice parallel, since they are just yet another type of value.&lt;/p&gt;&#xA;&lt;p&gt;One thing that you cannot do if you only talk about &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is to talk about the operation/computation itself; you can only talk about the &lt;em&gt;result&lt;/em&gt; of the computation. When thinking of it this way, when I write &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, I am talking about some value of function type: it contains an operation that is not yet performed. When I write &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; I am talking about the number that I got from performing that operation on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Understanding &lt;span class=&quot;math-container&quot;&gt;$y=2x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)=2x$&lt;/span&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let's look at the difference between &lt;span class=&quot;math-container&quot;&gt;$y=2x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)=2x$&lt;/span&gt;. So it all starts with some number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Now, if &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is the same as two multiplied with that number, then &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; must also be a number. Numbers multiplied with numbers give numbers.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, when I write &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt;, by the same logic, I get that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is a number. And it is! It is the number you get when you apply the operation. Let's try translating &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt; into words.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is “the result of the computation/operation/function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; performed on the number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;”&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; is “is equal to”&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; is “two multiplied by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;”, but let's leave it as &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So what &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt; means is “the result of the computation/operation/function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; performed on the number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is equal to &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;”.&lt;/p&gt;&#xA;&lt;p&gt;So what we're saying is that whatever &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; does to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, the result will be &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I think it is important to see numbers inside the parentheses, e.g. notice how &lt;span class=&quot;math-container&quot;&gt;$f(5)$&lt;/span&gt; is the number &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;. This will emphasize why &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is a number.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What can you use the function for?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Well just like you can do math on numbers, you can do math on functions. For example, just like you can add numbers, you can combine functions. One example would be:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We have the functions &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x) = x + 3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$f \circ g$&lt;/span&gt;? Well &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; are functions, so &lt;span class=&quot;math-container&quot;&gt;$f \circ g$&lt;/span&gt; is another function. You can then define an &lt;span class=&quot;math-container&quot;&gt;$h(x) = 2x + 6$&lt;/span&gt; and explain why &lt;span class=&quot;math-container&quot;&gt;$h = f \circ g$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note that it can also make sense to use &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; with two functions, or a function and a number.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$f + 3$&lt;/span&gt;? Well it is a function that first computes &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, then adds three to the result.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, what might you want to do now that you can talk about a function? Well one thing is that you can talk about the relation between the results when you give it different values.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The function &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2 + 1$&lt;/span&gt; satisfies &lt;span class=&quot;math-container&quot;&gt;$f(x) = f(-x)$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The function &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt; is one-to-one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now you can introduce stuff like inverse functions. Just like the square root, not all functions have an inverse, but some do. One thing I like to be aware of is that all equation solving is just making use of inverse functions.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let's solve &lt;span class=&quot;math-container&quot;&gt;$x + 5 = 10$&lt;/span&gt;. Well the left side can be thought of &lt;span class=&quot;math-container&quot;&gt;$f(x) = 10$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$f(x) = x+5$&lt;/span&gt;. The inverse function of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is given by &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(x) = x - 5$&lt;/span&gt;, so we can apply it to both sides.&lt;/p&gt;&#xA;&lt;p&gt;We now have &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(f(x)) = f^{-1}(10)$&lt;/span&gt;, and the f's cancel, so we now have &lt;span class=&quot;math-container&quot;&gt;$x = 10-5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course this is much much more verbose than what we normally do, but if you point out how the inverse function is just the function mirrored across the &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt; line, then it helps to understand why equation solving corresponds to drawing a horizontal line from the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-coordinate and finding where it intersects.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Functions of functions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A function is a value right? So can you put it in a function‽&lt;/p&gt;&#xA;&lt;p&gt;I don't know if you want to talk about this yet, but when you get to calculus, it can be nice to look back and recognize that the derivative is just a function that takes a function and results in a function.&lt;/p&gt;&#xA;&lt;p&gt;If you do want to talk about them now, here are some examples of functions of functions.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Finding the inverse can be thought of a function. So you have &lt;span class=&quot;math-container&quot;&gt;$\text{Inv}(f) = f^{-1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The composition operator takes pairs of functions. So you have &lt;span class=&quot;math-container&quot;&gt;$\circ(f, g) = f \circ g$&lt;/span&gt;. Here you can also mention &lt;span class=&quot;math-container&quot;&gt;$+(1,2) = 3$&lt;/span&gt;, as a weird way of writing plus.&lt;/p&gt;&#xA;&lt;p&gt;Maximum/minimum value: You can have stuff like &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\min(f) = 0$&lt;/span&gt;. This is a function that takes a function and returns a number (or negative infinity, which is not a number) (or it doesn't work if it has no minimum; maybe this is just like square root, and doesn't work for all functions)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1989" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-08-08T16:16:03.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16902" PostTypeId="2" ParentId="16882" CreationDate="2019-08-08T20:35:48.360" Score="8" Body="&lt;p&gt;(First, I should mention that I've never taught this, so my approach does not come from experience.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you have students who think of something like &lt;span class=&quot;math-container&quot;&gt;$y = f(x) = x^2 + 3$&lt;/span&gt; as a relationship between two “specific” quantities &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. As intuitions go, that's not so bad: it serves physicists quite well. But it's incomplete, and you're looking for ways to illustrate that. I think a key point is to take &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at several points in the same context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has an interesting property: for any &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(-x) = f(x)$&lt;/span&gt;. How can this be expressed with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;? It's a bit awkward, isn't it? You have to say that when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is replaced by &lt;span class=&quot;math-container&quot;&gt;$-x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; doesn't change. Now take another example: &lt;span class=&quot;math-container&quot;&gt;$z = g(x) = 3 x$&lt;/span&gt;. An interesting property of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is that &lt;span class=&quot;math-container&quot;&gt;$g(w + x) = g(w) + g(x)$&lt;/span&gt;. What could &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; even be here? “Which” &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; are you talking about?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can build up to more complex examples involving multiple functions. For example, take &lt;span class=&quot;math-container&quot;&gt;$h(x) = v = x^3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$i(x) = u = x^4$&lt;/span&gt;. An interesting property of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; is that &lt;span class=&quot;math-container&quot;&gt;$h(i(x)) = i(h(x))$&lt;/span&gt;. Which in terms of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; means that, er, well, if you take &lt;span class=&quot;math-container&quot;&gt;$x = v$&lt;/span&gt; then you get &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; which is the same &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; as if you take &lt;span class=&quot;math-container&quot;&gt;$x = u$&lt;/span&gt;? If that makes any sense? Right, it doesn't. But functions let us express this unambiguously.&lt;/p&gt;&#xA;" OwnerUserId="170" LastActivityDate="2019-08-08T20:35:48.360" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16903" PostTypeId="1" AcceptedAnswerId="16907" CreationDate="2019-08-09T01:26:22.193" Score="2" ViewCount="9331" Body="&lt;p&gt;I see a lot of places where &quot;Calculus 1&quot; is referred to as &quot;Introduction to Calculus&quot;, or &quot;Single-variable Calculus.&quot; &quot;Calculus 3&quot; is referred to as &quot;Multiple-variable calculus.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an alternate name for &quot;Calculus 2&quot;? Am I mistaken in thinking single variable calculus is Calculus 1?&lt;/p&gt;&#xA;" OwnerUserId="12529" LastEditorUserId="143" LastEditDate="2019-08-10T22:57:44.820" LastActivityDate="2019-08-17T16:20:40.393" Title="Is there a more telling name for &quot;Calculus 2&quot;?" Tags="|calculus|curriculum|terminology|" AnswerCount="6" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16904" PostTypeId="2" ParentId="16872" CreationDate="2019-08-09T01:30:51.297" Score="0" Body="&lt;p&gt;Do Saylor Academy or edX have anything that will help you? They are both free platforms with math courses. Saylor Academy is almost exclusively uses a textbook - you can actually get credit through them. Modernstates.org also may help you - they have a self-guided course with videos to teach it. Rootmath may be a good resource too. Are you planning on getting credit for this course through the Clep?&lt;/p&gt;&#xA;" OwnerUserId="12529" LastEditorUserId="12529" LastEditDate="2019-08-09T14:50:39.600" LastActivityDate="2019-08-09T14:50:39.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16905" PostTypeId="2" ParentId="16882" CreationDate="2019-08-09T02:07:11.283" Score="0" Body="&lt;p&gt;I hear the problem, but seeing as they are just beginning to learn function notation, they probably are not up to higher level mathematics. I think you can just blame it on the math world being crazy and tell them that as they learn more math it will become necessary for them to use it, so they need to get used to it. &lt;/p&gt;&#xA;" OwnerUserId="12529" LastActivityDate="2019-08-09T02:07:11.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16906" PostTypeId="1" AcceptedAnswerId="16909" CreationDate="2019-08-09T02:07:49.217" Score="2" ViewCount="165" Body="&lt;p&gt;I just finished self study of Calculus One and I am looking to begin Calculus Two. Is there a free platform that can help me with this?&lt;/p&gt;&#xA;" OwnerUserId="12529" LastActivityDate="2019-08-09T11:53:05.143" Title="Self-Study of Calculus Two" Tags="|calculus|self-learning|" AnswerCount="3" CommentCount="2" ClosedDate="2019-10-10T21:50:26.620" ContentLicense="CC BY-SA 4.0" />
  <row Id="16907" PostTypeId="2" ParentId="16903" CreationDate="2019-08-09T02:10:17.947" Score="20" Body="&lt;p&gt;In my job, I evaluate university math courses for transfer equivalency on a regular basis. In the US, &quot;Calculus 1&quot; typically refers to single variable &lt;em&gt;differential&lt;/em&gt; calculus up to the fundamental theorem of calculus. So the course includes limits, the definition of the derivative, techniques and applications of the derivative including trigonometric and exponential functions, and an introduction to anti-differentiation. &quot;Calculus 2&quot; typically starts with FTC, and works through techniques of integration and typically includes sequences, series, and Taylor series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I see a course titled &quot;Introduction to Calculus,&quot; my first instinct is that it will be a course that rushes through limits, differentiation, and integration all in one semester with reduced attention to theory, and probably does not include trigonometric functions. Such a course would &lt;em&gt;not&lt;/em&gt; be part of the calculus sequence taken by students majoring in STEM fields. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course there is variation on these observations. One source of confusion in US universities can result from institutions with an academic calendar based on a quarter system, as opposed to a semester system. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-08-09T02:10:17.947" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16908" PostTypeId="2" ParentId="16906" CreationDate="2019-08-09T03:25:48.393" Score="2" Body="&lt;p&gt;I suspect there are several. I know edX has some calculus courses, but I’m not sure if any are “Calculus 2” per se.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Khan Academy definitely has Calc 2:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.khanacademy.org/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.khanacademy.org/math&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12506" LastActivityDate="2019-08-09T03:25:48.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16909" PostTypeId="2" ParentId="16906" CreationDate="2019-08-09T03:40:04.787" Score="2" Body="&lt;p&gt;To supplement @Joe's posting,&#xA;here are the edX's &lt;a href=&quot;https://www.edx.org/learn/calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;calculus courses&lt;/a&gt;,&#xA;the three below from MIT:&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/fmf3X.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fmf3X.jpg&quot; alt=&quot;edX_Calc&quot;&gt;&lt;/a&gt;&#xA;&#xA;(And there are others at that link.)&#xA;I am not sure where to draw the line between Calc 1 and Calc 2 ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And also check out &lt;a href=&quot;https://www.math.upenn.edu/~ghrist/calculus.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Ghrist's pages &amp;amp; videos on calculus&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/5SOv8.jpg&quot; width=&quot;300&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-08-09T03:45:42.083" LastActivityDate="2019-08-09T03:45:42.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16910" PostTypeId="2" ParentId="2268" CreationDate="2019-08-09T06:48:15.333" Score="4" Body="&lt;p&gt;You may wish to view the MAA's report &lt;em&gt;Transitions to Proof&lt;/em&gt; by Carol Schumacher, Susanna Epp, and Danny Solow:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/sites/default/files/Transitions%20to%20Proof.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/sites/default/files/Transitions%20to%20Proof.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has some suggestions for these kinds of courses, and gives references to some relevant books and articles. Ultimately, the course design is very open-ended, so it's hard to give definitive prescriptions. But still the suggestions should be helpful.&lt;/p&gt;&#xA;" OwnerUserId="12530" LastActivityDate="2019-08-09T06:48:15.333" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16911" PostTypeId="2" ParentId="16889" CreationDate="2019-08-09T11:34:30.053" Score="2" Body="&lt;p&gt;There are already answers suggesting you to make an exam that doesn't require a calculator at all, so I won't elaborate it here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you are concerned about students with different backgrounds and different expectations about what they know after it ends, I see no issue with splitting it along these lines. I have had a few classes in a similar setup in my years (including one which was for both grad and post-grad students at the same time). The post-grad group got some additional questions to raise their bar and it worked fine.&lt;/p&gt;&#xA;" OwnerUserId="12532" LastActivityDate="2019-08-09T11:34:30.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16912" PostTypeId="2" ParentId="16906" CreationDate="2019-08-09T11:43:12.557" Score="2" Body="&lt;p&gt;Please also see &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=2&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwjO_u6W2vXjAhUFaq0KHTvxB14QFjABegQIAxAB&amp;amp;url=http%3A%2F%2Ftutorial.math.lamar.edu%2FClasses%2FCalcI%2FCalcI.aspx&amp;amp;usg=AOvVaw1qMSE1q-6I2n9McFukwsE7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul's Notes; Calc I&lt;/a&gt;, and &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=1&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwjO_u6W2vXjAhUFaq0KHTvxB14QFjAAegQIAhAB&amp;amp;url=http%3A%2F%2Ftutorial.math.lamar.edu%2FClasses%2FCalcII%2FCalcII.aspx&amp;amp;usg=AOvVaw3SM7E3h9Go4DB1lhUMhBPx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calc II&lt;/a&gt;.  Calc III is offered too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please see the left-most column (Table of contents) &lt;a href=&quot;http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; to get an overview of what Calc I, II, III offer.  There are tutorials, worked examples, exercises, and solutions to problems in each sub-topic throughout the Calculus sequence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more information about Paul's notes, please see the website: &lt;a href=&quot;http://tutorial.math.lamar.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul's Online Math Notes&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also the available courses in calculus at &lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MIT using OpenCourseWare (OCW)&lt;/a&gt;: assigned text, lectures, assigned exercises, etc.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-08-09T11:53:05.143" LastActivityDate="2019-08-09T11:53:05.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16913" PostTypeId="2" ParentId="16882" CreationDate="2019-08-09T11:55:49.000" Score="4" Body="&lt;h2&gt;Different notation for different things&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The key thing that the students seem to be missing is the conceptual distinction between &lt;code&gt;f&lt;/code&gt; and &lt;code&gt;y&lt;/code&gt; (in this example), so this apparently needs to be explicitly explained to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;IMHO the way to go at this is to tell them that there are two &quot;things&quot; that we may want to talk about - the &lt;em&gt;transformation process&lt;/em&gt; (the function) and &lt;em&gt;it's result with a particular input&lt;/em&gt;; and so we need separate names and notation to be clear which we mean. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to spend a bit of time to demonstrate both concepts separately, compare and contrast them. Perhaps it's worth to use some &lt;em&gt;very&lt;/em&gt; trivial function such as '+2' as an illustrative example; perhaps if you demonstrate the idea of a function as a &quot;machine that does a thing to transform the input&quot;, and the need to talk about such &quot;machines&quot; and their properties then that analogy may work better for some students than the abstract math definitions proposed in other answers.&lt;/p&gt;&#xA;" OwnerUserId="567" LastActivityDate="2019-08-09T11:55:49.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16914" PostTypeId="2" ParentId="16882" CreationDate="2019-08-09T15:04:38.647" Score="2" Body="&lt;p&gt;From a computer science perspective understanding that functions are first class objects is also pretty difficult. There are cases where functions can be parameters to other functions, the classic example being &lt;code&gt;sort&lt;/code&gt; accepting a &lt;code&gt;compare&lt;/code&gt; function. This case would be impossible to explain using just the &lt;code&gt;y&lt;/code&gt; output. In the case of &lt;code&gt;sort&lt;/code&gt; you don't even need to know the &lt;code&gt;compare&lt;/code&gt; outputs for &lt;em&gt;all&lt;/em&gt; the pairs of inputs, though this is moving away from math a bit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be fair I think the students are not at the level of CS to really understand this reason, but maybe you can explain it as a nice teaser?&lt;/p&gt;&#xA;" OwnerUserId="8122" LastActivityDate="2019-08-09T15:04:38.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16915" PostTypeId="2" ParentId="16882" CreationDate="2019-08-10T06:08:25.620" Score="6" Body="&lt;p&gt;Prior to my final year of high school, I was sent to a maths tutor for a couple of sessions, to give me a headstart on calculus. It helped a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He introduced me to the concept of functions. He described it as a monster, living inside a box, that accepted a thing through one (!) tube, and pushed out a thing through another tube. The monster was consistent - every time it saw a particular thing, it would always push out the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Monsters that dealt with numbers were just a special case - there were some monsters that would always spit out a number twice as big as the number that went in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;name&lt;/em&gt; of a particular monster was &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; was a description of its behaviour&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It sounds silly, but this description stuck with me and helped a lot (including in my Math major at uni).&lt;/p&gt;&#xA;" OwnerUserId="12536" LastActivityDate="2019-08-10T06:08:25.620" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16916" PostTypeId="2" ParentId="16882" CreationDate="2019-08-10T08:35:16.287" Score="1" Body="&lt;p&gt;Some rudimentary programming exercises might make it obvious why it's useful to &lt;em&gt;encapsulate functionality.&lt;/em&gt; When you write &lt;code&gt;y = f(x)&lt;/code&gt; in Python, for example, it's clear that &lt;code&gt;y&lt;/code&gt; is just a static result, while &lt;code&gt;f&lt;/code&gt; is the thing that does the work. You can't reuse &lt;code&gt;y&lt;/code&gt; to change another variable &lt;code&gt;z&lt;/code&gt; in the same way - you have to refer to &lt;code&gt;f&lt;/code&gt; to do that.&lt;/p&gt;&#xA;" OwnerUserId="737" LastActivityDate="2019-08-10T08:35:16.287" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16917" PostTypeId="2" ParentId="16903" CreationDate="2019-08-10T21:30:17.030" Score="1" Body="&lt;p&gt;When I was an undergrad, I feel like our Calc 2 was “Multivariable Calculus”, and Calc 3 was “Vector Calculus”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Calc 1, we covered limits and the limit definitions for differentiation and integration, and the fundamental theorem of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Calc 2, we covered multivariable calculus, such as implicit differentiation, and integrals in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;, to include change of variables with the Jacobian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Calc 3, we learned about divergence, gradients, and curl, plus line/surface/volume integrals, and covered Stokes’ theorem and Green’s theorem.&lt;/p&gt;&#xA;" OwnerUserId="12506" LastActivityDate="2019-08-10T21:30:17.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16918" PostTypeId="2" ParentId="16903" CreationDate="2019-08-10T23:59:57.450" Score="1" Body="&lt;p&gt;If Calculus 2 means second-semester calculus to you, then call it second-semester calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If Calculus 2 means multivariable calculus to you, then call it multivariable calculus.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2019-08-10T23:59:57.450" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16919" PostTypeId="2" ParentId="16903" CreationDate="2019-08-11T04:17:56.010" Score="11" Body="&lt;p&gt;The course titles &quot;Calculus 1&quot;, &quot;Calculus 2&quot;, etc. are not meaningful terms outside of the specific institutions where there are courses with these titles.  These are names of classes, and not some internationally decided-upon list of topics or curriculum.  The actual content of a class called &quot;Calculus 1&quot; might vary quite a lot from one institution to another, thus the best way to decide on a better course title would be to read the course catalog for your institution or track down a syllabus and determine what is actually taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few examples from my own experience:&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;UNR&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;At the &lt;a href=&quot;https://catalog.unr.edu/content.php?filter%5B27%5D=MATH&amp;amp;filter%5B29%5D=&amp;amp;filter%5Bcourse_type%5D=-1&amp;amp;filter%5Bkeyword%5D=&amp;amp;filter%5B32%5D=1&amp;amp;filter%5Bcpage%5D=1&amp;amp;cur_cat_oid=25&amp;amp;expand=&amp;amp;navoid=11109&amp;amp;search_database=Filter#acalog_template_course_filter&quot; rel=&quot;noreferrer&quot;&gt;University of Nevada Reno&lt;/a&gt; (where I did my undergraduate and masters work), there is a three semester sequence of courses taught.  The courses are titled Calculus I, II, and III.  From the course catalog:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Calculus I (Math 181)&lt;/strong&gt; Fundamental concepts of analytic geometry and calculus; functions, graphs, limits, derivatives and integrals.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Calculus II (Math 182)&lt;/strong&gt; Methods of integration. Sequences and series, power series.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Calculus III (Math 283)&lt;/strong&gt; Continuation of MATH 182 ; infinite series, three-dimensional calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Given these course descriptions, reasonable titles for these might be:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Calculus I: Introduction to Calculus in One Variable&lt;/li&gt;&#xA;&lt;li&gt;Calculus II: Techniques of Integrations; Sequences and Series&lt;/li&gt;&#xA;&lt;li&gt;Calculus III: Three Dimensional Calculus&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h3&gt;AP Calculus&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;In the US, high school students are often given the opportunity to to AP (or &quot;Advanced Placement&quot;) courses, which prepare students for exams which may be in place of college courses.  There are two &lt;a href=&quot;https://en.wikipedia.org/wiki/AP_Calculus&quot; rel=&quot;noreferrer&quot;&gt;AP calculus exams&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Calculus AB&lt;/strong&gt; The material includes the study and application of differentiation and integration, and graphical analysis including limits, asymptotes, and continuity. An AP Calculus AB course is typically equivalent to one semester of college calculus.&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Analysis of graphs (predicting and explaining behavior)&lt;/li&gt;&#xA;  &lt;li&gt;Limits of functions (one and two sided)&lt;/li&gt;&#xA;  &lt;li&gt;Asymptotic and unbounded behavior&lt;/li&gt;&#xA;  &lt;li&gt;Continuity&lt;/li&gt;&#xA;  &lt;li&gt;Derivatives&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Concept&lt;/li&gt;&#xA;  &lt;li&gt;At a point&lt;/li&gt;&#xA;  &lt;li&gt;As a function&lt;/li&gt;&#xA;  &lt;li&gt;Applications&lt;/li&gt;&#xA;  &lt;li&gt;Higher Order derivatives&lt;/li&gt;&#xA;  &lt;li&gt;Techniques&lt;/li&gt;&#xA;  &lt;/ul&gt;&lt;/li&gt;&#xA;  &lt;li&gt;Integrals&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Interpretations&lt;/li&gt;&#xA;  &lt;li&gt;Properties&lt;/li&gt;&#xA;  &lt;li&gt;Applications&lt;/li&gt;&#xA;  &lt;li&gt;Techniques&lt;/li&gt;&#xA;  &lt;li&gt;Numerical approximations&lt;/li&gt;&#xA;  &lt;/ul&gt;&lt;/li&gt;&#xA;  &lt;li&gt;Fundamental theorem of calculus&lt;/li&gt;&#xA;  &lt;li&gt;Antidifferentiation&lt;/li&gt;&#xA;  &lt;li&gt;L'Hôpital's rule, starting in the 2016-17 school year&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Calculus BC&lt;/strong&gt; Calculus BC is a full-year course in the calculus of functions of a single variable. It includes all topics covered in Calculus AB plus additional topics...Students who take an AP Calculus course should do so with the intention of placing out of a comparable college calculus course.[5]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;AP Calculus BC includes all of the topics covered in AP Calculus AB, as well as the following:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Convergence tests for series&lt;/li&gt;&#xA;  &lt;li&gt;Taylor series&lt;/li&gt;&#xA;  &lt;li&gt;The use of parametric equations&lt;/li&gt;&#xA;  &lt;li&gt;Polar functions (including arc length in polar coordinates)&lt;/li&gt;&#xA;  &lt;li&gt;Calculating curve length in parametric and function equations&lt;/li&gt;&#xA;  &lt;li&gt;Integration by parts&lt;/li&gt;&#xA;  &lt;li&gt;Improper integrals&lt;/li&gt;&#xA;  &lt;li&gt;Differential equations for logistic growth&lt;/li&gt;&#xA;  &lt;li&gt;Using partial fractions to integrate rational functions&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Calculus AB is typically used in lieu of the first semester of college calculus in the US, i.e. it is equivalent to Calculus 1.  The curriculum here is basically the same as the first semester of calculus at UNR, though it is usually taught with any eye towards computation, rather than theory.  Calculus AB might reasonably be given the same title as UNR's Calculus I.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus BC covers all of the same material, and also includes techniques of integration.  The curriculum is approximately equivalent to about a year of UNR's calculus, so this course is, perhaps, the equivalent of Calculus I and II.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;UCR&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;At my current institution (&lt;a href=&quot;http://students673.ucr.edu/docsserver/registrar/UCR_Catalog_1819_web.pdf&quot; rel=&quot;noreferrer&quot;&gt;University of California Riverside&lt;/a&gt;), we are on a quarter system.  Calculus is taught over five quarters:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;First Year Calculus (Math 9A)&lt;/strong&gt;  Introduction&#xA;  to the differential calculus of functions of one&#xA;  variable.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;First Year Calculus (Math 9B)&lt;/strong&gt; Introduction to the integral calculus of functions&#xA;  of one variable.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;First Year Calculus (Math 9C)&lt;/strong&gt;  Further topics from integral calculus,&#xA;  improper integrals, infinite series, Taylor’s series,&#xA;  and Taylor’s theorem.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Calculus of Several Variables (Math 10A)&lt;/strong&gt; Topics include Euclidean geometry, matrices and&#xA;  linear functions, determinants, partial derivatives,&#xA;  directional derivatives, Jacobians, gradients,&#xA;  chain rule, and Taylor’s theorem for several&#xA;  variables. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Calculus of Several Variables (Math 10B)&lt;/strong&gt;  Covers&#xA;  vectors; differential calculus, including implicit&#xA;  differentiation and extreme values; multiple&#xA;  integration; line integrals; vector field theory; and&#xA;  theorems of Gauss, Green, and Stokes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here, there are really only two introductory calculus classes:  the first year course, and the multivariable course.  Hence Calculus 1 and 2 are, respectively, &quot;Single Variable Calculus&quot; and &quot;Multivariable Calculus&quot;.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2019-08-11T04:17:56.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16920" PostTypeId="2" ParentId="8487" CreationDate="2019-08-11T18:08:27.123" Score="5" Body="&lt;p&gt;You mention the desire for a book with content and exercises, but less fluff.  But also, seem not to have done a lot of comparison research (other commenters recommend this to you, also). Two specific texts to look at would be: Schaum's Outline and Granville, et al.  Both have a short, simple presentation.  And there is definitely a student market for books like this versus the doorstops that entice purchasing committees.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if you still feel the need for something different, better, I think what you do will be better if you have looked at other texts with the same objective as yours.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  Yes, I realize the question is old, but the intention of this Q&amp;amp;A site is to create archived Google-able answers for others with similar questions in the future.  For instance, the &quot;related questions&quot; list freely pushes up other questions and there is no time limit on responses.&lt;/p&gt;&#xA;" OwnerUserId="12543" LastActivityDate="2019-08-11T18:08:27.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16921" PostTypeId="2" ParentId="13003" CreationDate="2019-08-11T18:54:36.587" Score="0" Body="&lt;p&gt;This is a hard comment to answer given the caveat about without a constraint on time.  (If anything, one of the advantages of self study is more time efficiency versus lectures, as well as more flexibility to move faster.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But on the uncaveated question, I think yes, you can do as well or better with self study.  In particular, with lower level courses that have settled content and well groomed texts, full of exercises (and answers).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the downsides of lack of forcing discipline and social interaction are much more serious than the issue of not having the instructor to ask questions of.  Provided you pick a helpful text (ideally a programmed text, but certainly not a &quot;derivation left to the reader&quot; text), than you can get almost everything you need, especially if already a strong student.  (I would not recommend such self study for a weak student.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you take a disciplined approach of writing down questions in your notebook (as a sort of &quot;parking lot&quot; to allow you to proceed), you will often find that with the benefit of time (just later in the chapter or homeworks), you are able to answer your own question.  In the few cases, you can't, the questions are easy to take to a Q&amp;amp;A site (again, I recommend trying to parking lot and then ask versus stopping).  Note, this is also the best way to use your instructor!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it possible, that self study will leave gaps?  Sure.  But then again, how many instructed students are gap-less?  Let's be real about the comparable.  For instance, if I had to choose between reading the book and working extensive drill problems versus listening to lectures and doing minimal drill, it's a no brainer, which approach is better.  See also the following essay on how lecture is overrated:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.scotthyoung.com/blog/2012/11/13/why-lectures/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.scotthyoung.com/blog/2012/11/13/why-lectures/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I like watching lecture videos, and I’d watch them whenever they were available. Videos are nice, and watching a video is certainly less strenuous than reading a textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But after having finished the classes, I came to an unfortunate conclusion—the videos don’t matter too much. Having a video explanation of a concept is nice, but it’s rarely superior to the same explanation in text. Text even has the advantages of searchability and nonlinearity, features missing in video.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What mattered was having practice problems and projects. Under time constraints, speeding up lecture watching to spend more time on practice or insight-building tactics was almost always a good tradeoff. Knowing I’d need to pass the exam later, I’d always pick a course without lectures than one without practice problems.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Now of course, you can make the comment that you could have BOTH lectures and drill, offline.  But please consider how the conjoint either/or thought experiment reveals something about the value of instruction.  And especially if we consider time efficiency.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also quite like the comment about strength training as an analogy.  Consider that Lou Ferrigno worked out for most of the beginning of his career in a little home gym.  Diligent study is several times more important than having an instructor.  (Sorry, people-who-get-paid-to-teach.)&lt;/p&gt;&#xA;" OwnerUserId="12543" LastActivityDate="2019-08-11T18:54:36.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16923" PostTypeId="2" ParentId="16886" CreationDate="2019-08-11T19:28:52.357" Score="2" Body="&lt;p&gt;Buy this book and work through it.  Good review of HS math and an intro to calculus.  Very user friendly.  Concise but simple, not cryptic.  Answers to the problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0070026505&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/gp/product/0070026505/ref=dbs_a_def_rwt_hsch_vapi_taft_p1_i10&lt;/a&gt;  (shows $6.75 for price, plus whatever shipping cost.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After you have done every problem report back, here.  This will show your dedication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't understand something, then write it down (in a dedicated section of your notebook).  At the end of each study session, ask questions on the regular math site [mathematics.se].  In general, I suspect, by just writing the question, and continuing to work,  you will discover most answers to you questions yourself, during the course of a study session.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Changing to a career in math, when you don't remember it is not a good idea.  It's already a steep hill for those who are good at it and current.  You're not current AND have forgotten a lot.  Not signs of aptitude.  (Not meant to hurt you, just being analytical.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, I recommend to treat math as a bit of an avocation to start.  This will test both your aptitude and dedication and see if there is any point in doing more.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, I suggest to look for areas with some synergy.  For instance doing a bit of data analysis on some medical work.&lt;/p&gt;&#xA;" OwnerUserId="12543" LastEditorUserId="238" LastEditDate="2019-10-01T14:28:22.000" LastActivityDate="2019-10-01T14:28:22.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16924" PostTypeId="2" ParentId="10467" CreationDate="2019-08-11T20:58:48.350" Score="1" Body="&lt;p&gt;Be it undergraduate or professor, you need hands-on examples to &quot;see&quot; why some concept or technique is worthwhile. Sure, you can take it that more advanced people are better able to come up with their own examples and applications, or have a richer experience to which new material relates. So examples, applications, cross-connections are certainly needed for undergraduates. If you don't provide them, your audience will get lost. If you don't intersperse motivation for some more exotic proof technique, or don't lay out a overall plan for a proof, they will get bored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a Discrete Math class I've taught, I found out the hard way that it is important to give a general structure and format for e.g. proofs by induction, and stick to it. State each time &quot;We use proof by contradiction/counterpositive/if and only if by proving implication in both directions/...&quot;. Students get lost otherwise. Same for proofs that some problem is NP-complete: need to prove in NP and NP-hard. Do it every time, even if it's obvious/one line. &lt;strong&gt;Write it down.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2019-08-11T20:58:48.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16925" PostTypeId="2" ParentId="10467" CreationDate="2019-08-12T04:42:26.200" Score="1" Body="&lt;p&gt;As someone who probably took undergraduate courses more recently than most, I can attest to the fact that applications are necessary. If not given, students very often get lost - not understanding the big picture. They may be able to apply specific operations to specific problems, but it is the applications that allows them to see the forest instead of just the trees. I don't have a research paper to quote, but I talk from my own experience as a student - which I believe is your ideal case study. &lt;/p&gt;&#xA;" OwnerUserId="12529" LastActivityDate="2019-08-12T04:42:26.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16926" PostTypeId="2" ParentId="8487" CreationDate="2019-08-12T23:05:41.607" Score="6" Body="&lt;p&gt;Although I admire the enthusiasm of the OP,&#xA;I'd like to make a point similar to &lt;a href=&quot;https://matheducators.stackexchange.com/a/16920/511&quot;&gt;@guest's post&lt;/a&gt;:&#xA;There are quite a few calculus textbooks out there.&#xA;Each thumbnail below is the cover of a calc textbook.&#xA;I would be wary of investing too much time in such a crowded&#xA;environment without investigation of how it is already&#xA;populated.&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/2AQ2a.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2AQ2a.jpg&quot; alt=&quot;CalcCovers&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;&lt;a href=&quot;https://www.google.com/search?q=calculus+textbook&amp;amp;source=lnms&amp;amp;tbm=isch&amp;amp;sa=X&amp;amp;ved=0ahUKEwimnr3Qtv7jAhWwd98KHbJwBEoQ_AUIEygC&amp;amp;biw=1182&amp;amp;bih=681&quot; rel=&quot;noreferrer&quot;&gt;Google images link&lt;/a&gt;&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Also, there are free, high-quality online calculus courses available,&#xA;e.g., this &lt;a href=&quot;https://www.coursera.org/learn/single-variable-calculus&quot; rel=&quot;noreferrer&quot;&gt;Coursera course&lt;/a&gt;. Not quite as crowded an environment, but still not&#xA;easy to compete with the professionals.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-08-13T00:40:00.650" LastActivityDate="2019-08-13T00:40:00.650" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16927" PostTypeId="2" ParentId="15328" CreationDate="2019-08-13T13:22:51.123" Score="0" Body="&lt;p&gt;&quot;Your goal is to demonstrate that you understand this material.&quot;&lt;/p&gt;&#xA;" OwnerUserId="10168" LastActivityDate="2019-08-13T13:22:51.123" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16928" PostTypeId="1" AcceptedAnswerId="16929" CreationDate="2019-08-13T15:13:08.307" Score="3" ViewCount="205" Body="&lt;p&gt;To start with, I'm not sure whether this is the correct stackexchange site to ask this, but I can't find one for physics teachers and my problem could easily be extended to math teachers with the same problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm working for my university's physics department. They're looking for some software or tool that allows teachers to generate physics problems by plugging in values to predefined variables and generating different exercises based on that (as in, having an exercise archetype and generating different exercises by changing the values of these variables), then sending a different exercise to each student in the course, have them answer them, and then comparing those answers with the real values, taking into account errors (admitting a 1-2% error, for example)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does such a tool exist? Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="9925" LastEditorUserId="11" LastEditDate="2019-08-13T21:46:20.150" LastActivityDate="2019-08-13T21:46:20.150" Title="Software for validating answers from students" Tags="|software|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16929" PostTypeId="2" ParentId="16928" CreationDate="2019-08-13T15:51:15.987" Score="10" Body="&lt;p&gt;There are many such programs, but I highly recommend &lt;a href=&quot;http://webwork.maa.org&quot; rel=&quot;noreferrer&quot;&gt;WeBWorK&lt;/a&gt;.  The &lt;a href=&quot;https://www.ams.org/publications/journals/notices/201605/rnoti-p539.pdf&quot; rel=&quot;noreferrer&quot;&gt;founders received the American Mathematical Society award for impact on teaching math in 2016&lt;/a&gt;, it is used at hundreds of institutions (primarily in the United States, but I believe not exclusively at all) and is open source.  You can (I think still) pay for hosting or set it up locally, which I think is what most places do.  I believe you can try it out with a few sample logins at the MAA instance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the user interface is a bit retro, it is still actively developed and I have had only very rare complaints from students about it.  Currently, you can even &lt;a href=&quot;https://www.facebook.com/The-Green-Bar-You-Get-When-All-the-Answers-are-Correct-106246586078386/&quot; rel=&quot;noreferrer&quot;&gt;like the green bar you get when your answers are correct on Facebook&lt;/a&gt;.  You will want to keep the &lt;a href=&quot;https://github.com/openwebwork/webwork-open-problem-library/tree/master/OpenProblemLibrary&quot; rel=&quot;noreferrer&quot;&gt;problem library&lt;/a&gt; up to date.  There are even &lt;a href=&quot;https://github.com/openwebwork/webwork-open-problem-library/tree/master/Contrib/BrockPhysics&quot; rel=&quot;noreferrer&quot;&gt;contributed physics problems&lt;/a&gt; though I'm not sure how integrated these are in the problem library.  I am quite sure that the developers would welcome a strong set of physics problems, and of course there are many rote physics questions of the type you are talking about that are essentially algebra or calculus dressed up in physics garb (not that I am claiming that is what physics &quot;really is&quot;!).  Good luck!&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-08-13T15:51:15.987" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16931" PostTypeId="2" ParentId="16903" CreationDate="2019-08-14T12:03:44.590" Score="3" Body="&lt;ul&gt;&#xA;&lt;li&gt;Calculus 1 is Differential Calculus. You start off by learning how to find limits of Algebraic functions, then you learn how to derive every function you learned in High School Algebra.&lt;/li&gt;&#xA;&lt;li&gt;Calculus 2 is Integral Calculus. You learn how to find the area under a curve and between two curves, which are solved using integrals. You will also learn the various techniques to solving integrals. Calculus 2 also covers sequences and series, as well as polar coordinates.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://calcworkshop.com/calculus-3/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus 3&lt;/a&gt; is Multivariable Calculus. I never took this course, but I have heard that it involves vectors and three-dimensional space.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://calcworkshop.com/differential-equations/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus 4&lt;/a&gt; is usually known as Differential Equations. This part of Calculus is a broad field with many classes specialized in Differential Equations. It is common for Engineering and Applied Math majors to take these classes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12554" LastActivityDate="2019-08-14T12:03:44.590" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16932" PostTypeId="1" CreationDate="2019-08-14T12:53:10.977" Score="18" ViewCount="13512" Body="&lt;p&gt;I would say it is &lt;em&gt;not&lt;/em&gt; the Fundamental Theorem of Calculus,&#xA;but rather some notion connecting limits and continuity,&#xA;perhaps the &lt;span class=&quot;math-container&quot;&gt;$(\epsilon,\delta)$&lt;/span&gt;-definitions of limits and continuity.&#xA;But I would be interested to learn from experienced calculus &#xA;instructors, as it would help me in courses that have&#xA;Calculus as a prerequisite, to understand where to&#xA;anticipate weaknesses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;For the usual meaning of &quot;Calculus 1&quot; (or &quot;Intro to Calculus&quot;), see &#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/q/16903/511&quot;&gt;Is there a more telling name for “Calculus 2”?&lt;/a&gt;&#xA;See also Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_calculus_topics&quot; rel=&quot;nofollow noreferrer&quot;&gt;List of calculus topics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;em&gt;Summary by 16Aug2019&lt;/em&gt;:&#xA;I did not anticipate such lack of consensus.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;@vonbrand: &quot;the whole &lt;span class=&quot;math-container&quot;&gt;$\epsilon/\delta$&lt;/span&gt; dance.&quot;&lt;/li&gt;&#xA;&lt;li&gt;@SueVanHattum: implicit derivatives and optimization&lt;/li&gt;&#xA;&lt;li&gt;@guest2: &quot;multistep problem solving&quot;&lt;/li&gt;&#xA;&lt;li&gt;@Maesumi: &quot;the concept of a variable and that of a function&quot;&lt;/li&gt;&#xA;&lt;li&gt;@JamesSCook: &quot;the chain rule is most troublesome&quot;&lt;/li&gt;&#xA;&lt;li&gt;@user1527: &quot;the point of the mean value theorem&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-08-16T23:03:23.883" LastActivityDate="2019-09-02T23:15:47.703" Title="What is the most difficult concept to grasp in Calculus 1?" Tags="|undergraduate-education|calculus|limits|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="16933" PostTypeId="2" ParentId="16932" CreationDate="2019-08-14T14:11:11.333" Score="21" Body="&lt;p&gt;At my (community) college, I believe many teachers skip the (,)-definitions of limits. I give the main definition, and teach it with an example I call the cookie crispness index. But I don't assign problems in that section, and I don't test on it. I figure actually working with it is for an analysis course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of the topics that I do test on, implicit derivatives and related rates seem to be the hardest for students to understand. They also have trouble with optimization, because too many of them can't actually do anything with math other than follow the steps. (Hmm, I wonder if I can somehow have them optimize throughout the course. It seems like the most important topic that they do badly on.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-08-14T14:11:11.333" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16934" PostTypeId="2" ParentId="16932" CreationDate="2019-08-14T15:28:36.077" Score="6" Body="&lt;p&gt;Related rates problems show the most issues on AP grading. (No reference, just my own observation)  Topic involves word problems, geometry and some multistep problem solving.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think the concept of epsilon delta is so difficult as the tediousness of the algebra along with students openly or unconsciously questioning the value add for science and engineering.  Even for the calculus class itself there is an issue that you do it st the beginning and set it aside.  Not really a tool you are building with.  These create an engagement issue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit. Just noticed your question on prereqs.  I think for calc 2, 3, and diffy Qs, the most important issue is just having solid drilled ability to work problems.  And having solidified algebra.  Calculus tends to do that. For real analysis, the course itself tends to cover the dive into theory versus assuming it.  So even here main benefit of calculus is the muscle building.  Although of course having seen definitions and theorems before once has a benefit in terms of repetition when doing a theoretical calc course. &lt;/p&gt;&#xA;" OwnerUserId="12472" LastEditorUserId="64" LastEditDate="2019-08-15T00:39:17.393" LastActivityDate="2019-08-15T00:39:17.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16935" PostTypeId="2" ParentId="1857" CreationDate="2019-08-14T23:33:44.087" Score="4" Body="&lt;p&gt;I have really enjoyed using “Beast Academy” to homeschool my son.  I wouldn’t say that it resembles a college textbook at all, but I do think it matches some of your other criteria:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) “I don't necessarily want it to be readable by 8 year olds, that might be asking a lot.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is readable by 8 year olds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) “give them a more solid foundation and inspire more of an interest in math. Something well written, that wants to make the subject interesting and beautiful”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe it does give them a more solid foundation. I find that it explains things using the same mathematical intuition that I would use, as someone with a master’s in math who loves math and has taught math in college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The books (for U.S. common core grades 2-5) are written as comics, with characters that are ‘beasts’ attending “Beast Academy”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don’t know for sure that they PROVE many assertions, but I’d say that they motivate the material, and make many connections, to help students gain a more thorough understanding.  I do recall that they do have some exercises in which there are some informal proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The authors also believe in giving some challenging problems at every level, so that even 2nd graders can start to learn that math isn’t about just having equations memorized, but rather that problems can take some time and mental effort to solve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After grade 5, they do have books that may be more similar college textbooks, but I haven’t used them yet, so I cannot attest to their quality. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find them here:   &lt;a href=&quot;https://artofproblemsolving.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://artofproblemsolving.com&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12506" LastActivityDate="2019-08-14T23:33:44.087" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16936" PostTypeId="2" ParentId="16882" CreationDate="2019-08-15T08:25:55.473" Score="2" Body="&lt;p&gt;Well...&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\def\zz{\mathbb{Z}}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f : \zz → \zz$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$f(n) = n+1$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$n∈\zz$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$f(0) = 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(f(0)) = 2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(f(f(0))) = 3$&lt;/span&gt; and so on. It is now obvious why having functions as &lt;strong&gt;first-class objects&lt;/strong&gt; is useful, since we can &lt;strong&gt;repeatedly&lt;/strong&gt; apply them. Similarly, the Mandelbrot fractal is defined in terms of iterating an elegant function.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2019-08-15T08:25:55.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16937" PostTypeId="2" ParentId="16872" CreationDate="2019-08-15T12:38:54.660" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I personally would prefer a textbook recommendation I can download or pick up that is [preferably] not old and does not make trigonometry intimidating to approach (especially one that emphasizes understanding proofs behind properties/theorems).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't have textbooks to recommend, but I can recommend an approach to &lt;strong&gt;doing&lt;/strong&gt; trigonometry that facilitates mathematical understanding of it by crystallizing the &lt;strong&gt;logical&lt;/strong&gt; foundation of trigonometry and &lt;strong&gt;algebraic&lt;/strong&gt; structure of trigonometric expressions. There are two 'levels' to this, depending on whether you want to go straight to complex numbers or stay within real trigonometry. In either case, the focus is on identifying the &lt;strong&gt;intrinsic&lt;/strong&gt; core of trigonometry and reducing everything else to that.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;Real trigonometry&lt;/h3&gt;&#xA;&lt;p&gt;The key quantities are &lt;span class=&quot;math-container&quot;&gt;$\cos(t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin(t)$&lt;/span&gt;, which are the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; coordinates of the point &lt;span class=&quot;math-container&quot;&gt;$P_t$&lt;/span&gt; on the unit circle that subtends an arc of length &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; anti-clockwise from the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis, as depicted in the image from &lt;a href=&quot;https://en.wikipedia.org/wiki/Trigonometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ksepZ.png&quot; alt=&quot;cos(θ) and sin(θ) as coordinates of the point on the unit circle that subtends an angle of θ from the x-axis&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here arc length is measured along the unit circle, and &lt;span class=&quot;math-container&quot;&gt;$π$&lt;/span&gt; is &lt;strong&gt;defined&lt;/strong&gt; as the arc length of the semicircle, so &lt;span class=&quot;math-container&quot;&gt;$2π$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$360°$&lt;/span&gt;. (This way of measuring angles is often called measuring them in &amp;quot;radians&amp;quot;, but I personally think it is an unnecessary term.) Note that &lt;span class=&quot;math-container&quot;&gt;$P_t = P_{t+2πk}$&lt;/span&gt; for any integer &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, because &lt;span class=&quot;math-container&quot;&gt;$2πk$&lt;/span&gt; would be an integer multiple of full rounds. Also note that increasing &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; moves &lt;span class=&quot;math-container&quot;&gt;$P_t$&lt;/span&gt; anti-clockwise, while decreasing &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; moves &lt;span class=&quot;math-container&quot;&gt;$P_t$&lt;/span&gt; clockwise. Related to that, &lt;span class=&quot;math-container&quot;&gt;$P_{-t}$&lt;/span&gt; is the reflection of &lt;span class=&quot;math-container&quot;&gt;$P_t$&lt;/span&gt; across the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis.&lt;/p&gt;&#xA;&lt;p&gt;Note that the signs of &lt;span class=&quot;math-container&quot;&gt;$\cos(t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin(t)$&lt;/span&gt; match exactly the signs of the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; coordinates of the point on the circle. (Do not listen to people who tell you to memorize something to determine which of them is positive in which quadrant.)&lt;/p&gt;&#xA;&lt;p&gt;And just by definition, &lt;span class=&quot;math-container&quot;&gt;$\cos(t)^2+\sin(t)^2 = 1$&lt;/span&gt; for every real &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;. This is the &lt;strong&gt;first key algebraic fact&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Next, &lt;span class=&quot;math-container&quot;&gt;$\tan(t)$&lt;/span&gt; is &lt;strong&gt;defined&lt;/strong&gt; as &lt;span class=&quot;math-container&quot;&gt;$\sin(t)/\cos(t)$&lt;/span&gt;. (Historically, we also have defined &lt;span class=&quot;math-container&quot;&gt;$\sec(t) := 1/\cos(t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\csc(t) := 1/\sin(t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cot(t) := 1/\tan(t)$&lt;/span&gt;, but frankly there is little benefit to have so many when &lt;span class=&quot;math-container&quot;&gt;$\cos,\sin$&lt;/span&gt; alone suffice.) Whenever you wish to simplify any trigonometric expression involving &lt;span class=&quot;math-container&quot;&gt;$\cos,\sin,\tan,\sec,\csc,\cot$&lt;/span&gt;, you probably should perform the standard mathematical technique of &lt;strong&gt;rewriting in canonical form&lt;/strong&gt;, which in this case means rewriting in terms of &lt;span class=&quot;math-container&quot;&gt;$\cos,\sin$&lt;/span&gt; alone, while taking note of where the original expression is not defined (for instance, &lt;span class=&quot;math-container&quot;&gt;$1/\csc(t) = \sin(t)$&lt;/span&gt; for any real &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; only when &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is not a multiple of &lt;span class=&quot;math-container&quot;&gt;$π$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The &lt;strong&gt;other key algebraic facts&lt;/strong&gt; arise from considering rotation matrices applied to vectors. (If you are unfamiliar with matrices as operators on vectors, please read &lt;a href=&quot;https://math.stackexchange.com/a/1698898/21820&quot;&gt;this&lt;/a&gt; first. For an introduction to vectors in euclidean space, see &lt;a href=&quot;https://www.mathsisfun.com/algebra/vectors.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.) Let &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; be any rotation about the origin in the plane. Then &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; satisfies three properties:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$R(u+v) = R(u)+R(v)$&lt;/span&gt; for any vectors &lt;span class=&quot;math-container&quot;&gt;$u,v$&lt;/span&gt; (i.e. summing two vectors then rotating the result gives the same as rotating the two vectors first before summing them).&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$R,S$&lt;/span&gt; are anti-clockwise rotations of angles &lt;span class=&quot;math-container&quot;&gt;$t,u$&lt;/span&gt; respectively, then &lt;span class=&quot;math-container&quot;&gt;$R∘S$&lt;/span&gt; is an anti-clockwise rotation of angle &lt;span class=&quot;math-container&quot;&gt;$t+u$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is an anti-clockwise rotation of angle &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, then:&lt;br /&gt;&#xA;a. &lt;span class=&quot;math-container&quot;&gt;$R(⟨x,0⟩) = ⟨x·\cos(t),x·\sin(t)⟩$&lt;/span&gt; for any real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;br /&gt;&#xA;b. &lt;span class=&quot;math-container&quot;&gt;$R(⟨0,y⟩) = ⟨-y·\sin(t),y·\cos(t)⟩$&lt;/span&gt; for any real &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;We can take these properties as axioms (assumption) about rotations. After all, if &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; does not satisfy them then we would not call &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; a rotation to begin with. To see why, property (1) captures the intuition that rotating two connected rods will rotate both rods by the rotation angle while preserving where they connect. Property (2) is only needed in conjunction with property (3). Property (3a) follows from the definition of &lt;span class=&quot;math-container&quot;&gt;$\cos,\sin$&lt;/span&gt;, and property (3b) follows from the same definition rotated &lt;span class=&quot;math-container&quot;&gt;$90°$&lt;/span&gt; anti-clockwise.&lt;/p&gt;&#xA;&lt;p&gt;Properties (1) and (3) yield the matrix form of a 2d rotation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is an anti-clockwise rotation of angle &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$R = \small \pmatrix{ \cos(t) &amp;amp; -\sin(t) \\ \sin(t) &amp;amp; \cos(t) }$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And then using property (2) we get:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\small \pmatrix{ \cos(t+u) &amp;amp; -\sin(t+u) \\ \sin(t+u) &amp;amp; \cos(t+u) } = \pmatrix{ \cos(t) &amp;amp; -\sin(t) \\ \sin(t) &amp;amp; \cos(t) } \pmatrix{ \cos(u) &amp;amp; -\sin(u) \\ \sin(u) &amp;amp; \cos(u) }$&lt;/span&gt; for any reals &lt;span class=&quot;math-container&quot;&gt;$t,u$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Multiplying the matrix product on the right and comparing with the matrix on the left immediately gives the angle-sum identities:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(t+u) = \cos(t)·\cos(u) - \sin(t)·\sin(u)$&lt;/span&gt; for any reals &lt;span class=&quot;math-container&quot;&gt;$t,u$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(t+u) = \cos(t)·\sin(u) + \sin(t)·\cos(u)$&lt;/span&gt; for any reals &lt;span class=&quot;math-container&quot;&gt;$t,u$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Whenever you wish to simplify expressions involving trigonometric functions on sums of angles, you should consider using these identities to &lt;strong&gt;reduce&lt;/strong&gt; the expression to be in terms of &lt;span class=&quot;math-container&quot;&gt;$\cos,\sin$&lt;/span&gt; of as few angles as possible.&lt;/p&gt;&#xA;&lt;p&gt;In fact, all trigonometric identities involving only arithmetic operations and trigonometric functions can be proven using just the above definitions and key algebraic facts. A bit curiously, even the symmetry properties can be proven algebraically as follows.&lt;/p&gt;&#xA;&lt;p&gt;Given any real &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;  &lt;span class=&quot;math-container&quot;&gt;$1 = \cos(t+(-t)) = \cos(t)·\cos(-t) - \sin(t)·\sin(-t)$&lt;/span&gt;.   [angle-sum]&lt;/p&gt;&#xA;&lt;p&gt;  &lt;span class=&quot;math-container&quot;&gt;$0 = \sin(t+(-t)) = \cos(t)·\sin(-t) + \sin(t)·\cos(-t)$&lt;/span&gt;.   [angle-sum]&lt;/p&gt;&#xA;&lt;p&gt;  &lt;span class=&quot;math-container&quot;&gt;$\cos(t) = \cos(t)^2·\cos(-t) - (\cos(t)·\sin(-t))·\sin(t)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;    &lt;span class=&quot;math-container&quot;&gt;$ = \cos(t)^2·\cos(-t) + (\sin(t)·\cos(-t))·\sin(t)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;    &lt;span class=&quot;math-container&quot;&gt;$ = (\cos(t)^2+\sin(t)^2)·\cos(-t)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;    &lt;span class=&quot;math-container&quot;&gt;$ = \cos(-t)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;  &lt;span class=&quot;math-container&quot;&gt;$\sin(t) = (\sin(t)·\cos(-t))·\cos(t) - \sin(t)^2·\sin(-t)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;    &lt;span class=&quot;math-container&quot;&gt;$ = -(\cos(t)·\sin(-t))·\cos(t) - \sin(t)^2·\sin(-t)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;    &lt;span class=&quot;math-container&quot;&gt;$ = -\sin(-t)·(\cos(t)^2+\sin(t)^2)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;    &lt;span class=&quot;math-container&quot;&gt;$ = -\sin(-t)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Going on to real analysis, we would need the following facts, which can be taken as axioms for now (and justified separately later):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin' = \cos$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos' = -\sin$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;As before, everything can be reduced to these, so there is no real need to memorize anything more (even though it may be convenient to do so).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;Complex trigonometry&lt;/h3&gt;&#xA;&lt;p&gt;Personally, I think it is best to go straight to the complex-valued trigonometric functions, if one desires a complete and rigorous foundation for the mathematical field of &lt;strong&gt;analysis&lt;/strong&gt;. One simply defines:&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\def\rr{\mathbb{R}}&#xA;\def\cc{\mathbb{C}}&#xA;\def\lfrac#1#2{{\large\frac{#1}{#2}}}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\exp(z) := \sum_{k=0}^∞ \lfrac{z^k}{k!}$&lt;/span&gt; for every complex &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; (after proving that the sum converges).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(z) := \lfrac{\exp(iz)+\exp(-iz)}{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(z) := \lfrac{\exp(iz)-\exp(-iz)}{2i}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$π$&lt;/span&gt; is twice the first positive root of &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt; (after proving that it exists).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The motivation is that we want &lt;span class=&quot;math-container&quot;&gt;$\exp : \cc→\cc$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\exp' = \exp$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\exp(0) = 1$&lt;/span&gt;, to be able to solve general linear differential equations, and we want &lt;span class=&quot;math-container&quot;&gt;$\cos,\sin : \rr→\rr$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\cos'' = -\cos$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin'' = -\sin$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$⟨\cos(0),\cos'(0)⟩ = ⟨1,0⟩$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$⟨\sin(0),\sin'(0)⟩ = ⟨0,1⟩$&lt;/span&gt;, to be able to solve simple harmonic motion, and Taylor expansion brings us to the above definitions for &lt;span class=&quot;math-container&quot;&gt;$\exp,\cos,\sin$&lt;/span&gt;, which we can prove to converge on the entire complex plane. The above definition of &lt;span class=&quot;math-container&quot;&gt;$π$&lt;/span&gt; is the easiest one that I know of that does not depend on any geometry. (For more details on this motivation, see &lt;a href=&quot;https://math.stackexchange.com/a/2360276/21820&quot;&gt;this post&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;Suffice to say that with these definitions, we can prove by basic analysis that &lt;span class=&quot;math-container&quot;&gt;$\exp,\cos,\sin$&lt;/span&gt; satisfy the desired motivating properties as well as another &lt;strong&gt;key property&lt;/strong&gt; of &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\exp(z+w) = \exp(z)·\exp(w)$&lt;/span&gt; for any complex &lt;span class=&quot;math-container&quot;&gt;$z,w$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Using this property, we can prove all the trigonometric identities via algebraic manipulation alone (and they hold for complex variables and not just real variables).&lt;/p&gt;&#xA;&lt;p&gt;For instance, given any complex &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;  &lt;span class=&quot;math-container&quot;&gt;$\cos(z)^2+\sin(z)^2 = \lfrac{(\exp(iz)+\exp(-iz))^2}{4} - \lfrac{(\exp(iz)-\exp(-iz))^2}{4}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;    &lt;span class=&quot;math-container&quot;&gt;$ = \exp(iz)·\exp(-iz) = \exp(0) = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Nevertheless, it is often still easier to first prove the same key algebraic facts for &lt;span class=&quot;math-container&quot;&gt;$\cos,\sin$&lt;/span&gt; and then use them to prove other identities, than to reduce everything to &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-08-15T12:38:54.660" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16938" PostTypeId="2" ParentId="16932" CreationDate="2019-08-15T14:21:49.607" Score="7" Body="&lt;p&gt;I'd say the whole &lt;span class=&quot;math-container&quot;&gt;$\epsilon/\delta$&lt;/span&gt; dance. Just because it involves &lt;em&gt;inequalities&lt;/em&gt;, while students just have trained &lt;em&gt;equalities&lt;/em&gt; all their life. To just work with (sometimes very rough) bounds, when you were drilled to get exact answers goes against the grain. &lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2019-08-16T20:29:11.603" LastActivityDate="2019-08-16T20:29:11.603" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16940" PostTypeId="2" ParentId="16932" CreationDate="2019-08-15T15:05:18.487" Score="2" Body="&lt;p&gt;In terms of issues affecting most students I believe the concept of a variable and that of a function are still the most difficult concepts for calculus 1 students, even though the concepts are introduced in precalculus. Writing a full and correct mathematical sentence is a topic most students struggle with.&lt;/p&gt;&#xA;" OwnerUserId="1472" LastActivityDate="2019-08-15T15:05:18.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16941" PostTypeId="1" AcceptedAnswerId="16961" CreationDate="2019-08-15T18:21:51.783" Score="6" ViewCount="1685" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Suppose you bought four boxes of pencils having five pencils in each, how many pencils do you have altogether?&quot; — &quot;Nine.&quot; — &quot;How come?&quot; — &quot;Because 4 plus 5 is 9.&quot; — &quot;But you cannot add boxes to pencils!&quot; — &quot;Why?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Indeed, why? Why you can multiply boxes and pencils, but cannot add? This is sort of self-evident for most adults, but how do you explain it to an elementary-school student?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I came up with an approach calling a box a &quot;bunch of things&quot; (Common Core likes to use the word &quot;group&quot;), so a box by itself has no meaning, what does have meaning is that it groups, combines, ties together several things that we are actually interested in, say pencils. If each box combines the same amount of things, then we can define and use multiplication as quick addition of the same number of things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar approach can be used to explain why you cannot add two tens of flowers and five flowers as 2 + 5 = 7, because it is not clear seven of what we are getting. First, we &quot;unbunch&quot; two tens into one, two, three, ..., twenty flowers, then add five flowers to them, so we can count them, twenty five flowers. It just so happens that by having ten-based positional system we can simply write 5 to the right of 2 to get the correct number, it won't work if we had two dozen flowers instead of two tens.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another phrase commonly used is that you can add &quot;like things&quot;, things that are similar. All this kinda works, but does not feel perfect, does not seem rigorous, persuasive enough. Does anyone have a better idea, approach, script to explain to kids &lt;strong&gt;why adding apples to apples is ok, but adding apples to apple crates is not?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="7930" LastActivityDate="2019-08-28T09:19:32.147" Title="Adding things to bunches of things vs multiplication" Tags="|primary-education|arithmetic|" AnswerCount="7" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16943" PostTypeId="2" ParentId="16941" CreationDate="2019-08-15T20:15:10.957" Score="15" Body="&lt;p&gt;Here is where it helps to get more concrete instead of more general. Have the student draw a picture of the problem and similar problems. First, you demonstrate drawing one box of pencils (a square) with five pencils inside (perhaps five tally marks) and make sure they understand the picture. Then ask them to draw four boxes of pencils (four squares) each with five pencils inside. When you ask them how many pencils there are at this point (by saying the original question again, and tying it to their drawing), they should get a correct answer (perhaps encourage them to skip-count if they are not good at multiplying yet), and at that point you should be able to ask them why they think adding 4 pencils + 5 boxes doesn't work to answer the question. Their self-explanation will probably be much better at making sense to them than any way we try to do it, because they will have processed it in context of what they already know about adding. You can always help clarify their wording at this point, and help them refine their statement so that they get at the crux of the issue: &quot;pencils&quot; and &quot;boxes of pencils&quot; are different &quot;wholes.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To take it a step further if they still have a hard time explaining it, have them draw three pencils plus two boxes of pencils and repeat the process. How many pencils are there? Why didn't 3 + 2 work? Draw 3 pencils + 2 pencils. How is this drawing different than 3 pencils + 2 boxes of pencils?&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2019-08-15T20:15:10.957" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16944" PostTypeId="2" ParentId="16941" CreationDate="2019-08-15T20:46:43.133" Score="5" Body="&lt;p&gt;You can add, though it's more tedious.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Suppose you bought four boxes of pencils having five pencils in each, how many pencils do you have altogether?&quot;  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;First box has five pencils. Second box has five pencils. Third box has five pencils.  Fourth box has five pencils.  Altogether, then, we have &lt;span class=&quot;math-container&quot;&gt;$\underbrace{5 \text{ pencils}}_{\text{box} 1} + \underbrace{5  \text{ pencils}}_{&#xA;\text{box} 2}+\underbrace{5  \text{ pencils}}_{\text{box} 3} + \underbrace{5  \text{ pencils}}_{\text{box} 4} = 20$&lt;/span&gt; pencils. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or more conveniently, we can write &lt;span class=&quot;math-container&quot;&gt;$4 \text{ box } \times \dfrac{5 \text{ pencils }}{\text{box}} = 20$&lt;/span&gt; pencils.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;What you can't do is add one box of six apples, with 3 oranges, to get either 9 apples nor nine oranges, nor four apples, nor 4 oranges.    &lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-08-15T20:54:47.390" LastActivityDate="2019-08-15T20:54:47.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16945" PostTypeId="2" ParentId="16941" CreationDate="2019-08-16T03:30:37.150" Score="5" Body="&lt;p&gt;I wonder if it would help to have them make up the problems, instead of you? Clearly, these students are already discounting the meaning of math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is lots of research on the efficacy of well-led classroom discussions about math topics. (Deborah Ball has written eloquently about this.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One book I loved, set at this level, is &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0325004315&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Little Kids: Powerful Problem Solvers&lt;/a&gt;, by Angela Andrews.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;===&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see in one comment that you refer to 'him', so I now think you are working with one individual student. If he really doesn't see why you can only add like things, then he has already decided that math does not make sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I were working with this student, I might focus on food. 1. You have 3 oranges on the counter and 2 oranges in the fridge. How many oranges do you have? 2. You have 3 oranges on the counter and 2 apples in the fridge. Can we put them together? How would we talk about them? (5 fruit.) After focusing on sensible problems, maybe you want to play with absurdity so you can laugh at silly answers together. You have 3 cars on the counter (they must be toys, not real, huh?) and 2 blueberries in the fridge. Can you put them together? (Maybe. As objects.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once he is clear on why adding only makes sense with like objects, then you can work on the multiplication issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your student can make sense of this if you talk in language that works for him. You can find that through your connection with him better than we can find it through general principles.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2019-08-16T15:21:54.510" LastActivityDate="2019-08-16T15:21:54.510" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16946" PostTypeId="2" ParentId="16941" CreationDate="2019-08-16T04:15:16.847" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;When they are first learning multiplication, keep the numbers very small and allow them to do repeated addition.  2*4 and 4*2 are good ones to start with.  Do 3*5 before 4*5.  (Or a sequence of 1*5, 2*5, 3*5, 4*5, etc.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to keep things simpler.  Not boxes of pencils.  But groups of pencils.  Boxes of pencils is a bit of a word problem and a conversion problem.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teach them the multiplication table via memorization and drill.  Don't only approach multiplication in this manner...use concrete counting examples as well.  But take a belt and suspenders approach.  IOW, don't exclude learning of this kind.  Having learned the table, kids are more ready to use it.  Also, do not underestimate the joy in memorization and recall.  Look how kids are with state capitals.  Or how kids compete in games even simple drill.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Just persist and prevail.  Don't assume they are as smart as you or as experienced. Repeat, repeat, repeat.  That is the path to instruction more than &quot;killer explanation&quot;.  But of course, as in 3, use explanations AS WELL.  But don't expect concepts themselves to magically unlock a stuck door.  Some people even need to just learn by imitation, practice, and correction. (See coaching in sports and music.) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12568" LastActivityDate="2019-08-16T04:15:16.847" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16947" PostTypeId="2" ParentId="16941" CreationDate="2019-08-16T07:07:04.330" Score="11" Body="&lt;p&gt;The thing is, it &lt;em&gt;is&lt;/em&gt; possible to add 4 boxes to 5 pencils.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;4 boxes + 5 pencils = 9 things.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So start by showing the student what they can do with addition, and what its real-world application is.&#xA;But then remind them what you really want to know - how many pencils are in the boxes (which they can easily count to see it's not 9).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then you can show them how we can use multiplication to figure out how many pencils are actually in the boxes.&lt;/p&gt;&#xA;" OwnerUserId="12571" LastEditorUserId="60" LastEditDate="2019-08-16T20:00:31.503" LastActivityDate="2019-08-16T20:00:31.503" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16948" PostTypeId="1" CreationDate="2019-08-16T07:07:55.593" Score="20" ViewCount="3181" Body="&lt;p&gt;My understanding is that students find absolute value to be challenging to learn or understand. Off the top of my head, I can come up with two possible reasons for this.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Absolute value is a piecewise defined function. Piecewise defined functions are more difficult due to increased abstractness: they are not a simple formula, but include a conditional. (I do not know if this is true, but it sounds plausible enough.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Absolute value is difficult, because it combines algebra (changing sign) with a geometric interpretation as a distance of number. At least according to the thesis of Hähkiöniemi on derivative [1], it is challenging for students to change between perspectives in a fruitful way.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Is one of these the reason for the difficulties, or maybe it lies elsewhere? As always, answers using scientific literature are the most valuable and ones relying on explicit personal experiences are also fine.&lt;/p&gt;&#xA;&lt;p&gt;[1] Hähkiöniemi, Markus. The role of representations in learning the derivative. No. 104 in Reports of university of Jyväskylä, department of mathematics and statistics. University of Jyväskylä, 2006. &lt;a href=&quot;http://urn.fi/URN:ISBN:951-39-2639-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://urn.fi/URN:ISBN:951-39-2639-7&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2023-03-17T07:26:59.980" LastActivityDate="2023-03-17T17:40:06.767" Title="Why is absolute value difficult?" Tags="|undergraduate-education|secondary-education|absolute-value|" AnswerCount="11" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16949" PostTypeId="2" ParentId="16948" CreationDate="2019-08-16T11:07:28.660" Score="30" Body="&lt;p&gt;(My answer is just a guess and not based on any formal research.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect the absolute value function may be difficult to understand because it involves &quot;negative numbers that aren't negative.&quot;  One way to define the absolute value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$|x|=\left\{\begin{array}{rl}-x, &amp;amp; x&amp;lt;0\\x, &amp;amp; 0\le x\end{array}\right.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the &lt;span class=&quot;math-container&quot;&gt;$-x$&lt;/span&gt; confuses students, causing them to think that &quot;sometimes the output of the function is negative.&quot;&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2019-08-16T11:07:28.660" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16950" PostTypeId="2" ParentId="16882" CreationDate="2019-08-16T16:16:38.270" Score="0" Body="&lt;p&gt;It is generally easier to learn something when you can see why it's useful. One use mentioned already is that you can effectively describe even and odd functions using function notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I most want to help my precalculus students understand function notation so that they can deal with the &lt;em&gt;f(x+h)&lt;/em&gt; that they'll see repeatedly when working with derivatives.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-08-16T16:16:38.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16953" PostTypeId="1" CreationDate="2019-08-17T08:58:19.993" Score="8" ViewCount="291" Body="&lt;p&gt;In my locality, many schools have this tendency to partially include this and that chapter in the syllabus (for almost every subject). For example, (most of the chapters are subdivided in two or more parts), they will take chapter 2, chapter 3.1, chapter 5.1, .2, chapter 14 and make this as a syllabus for a term/semester (out of 15 chapters, say). They complete the full textbook in a whole year, by the way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the textbook is actually well organised, and a student would understand the topics better if the syllabus followed the order the book is written. The concept of syllabus selection committee is to 'mix hard and easy chapters (parts)' and 'to give different tastes'. In my opinion, the hard parts do not need more work once the prerequisite chapters are done; I mean if you go step by step through the text book, it is always the same amount of difficulty. And for the 'taste', if we keep tasting this and that food frequently, we do not get any taste at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is this normal in other places/countries too? If so, how does this practice benefit the students in learning (mathematics)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12549" LastEditorUserId="238" LastEditDate="2019-08-18T00:11:08.657" LastActivityDate="2019-08-26T21:04:29.873" Title="Should school syllabus include chapters partially?" Tags="|mathematical-pedagogy|curriculum|syllabus|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16954" PostTypeId="2" ParentId="16903" CreationDate="2019-08-17T16:20:40.393" Score="1" Body="&lt;p&gt;Here in Chile, at UTFSM in the '70 we had:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Cálculo I: Sequences, limits, derivatives (the whole &lt;span class=&quot;math-container&quot;&gt;$\epsilon - \delta$&lt;/span&gt; dance)&lt;/li&gt;&#xA;&lt;li&gt;Cálculo II: Integrals (Riemann), integration techniques. Taylor series.&lt;/li&gt;&#xA;&lt;li&gt;Cálculo III: Multiple variables. Taylor theorem, line integrals, Green's theorem, vector calculus (rudiments)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;We also had a first semester class called Álgebra, which was more some of analytic geometry and a smattering of trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today's curriculum is different (the topics stretch out over four courses now), but the first ones are more or less the same.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2019-08-17T16:20:40.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16955" PostTypeId="2" ParentId="16953" CreationDate="2019-08-18T00:58:37.003" Score="3" Body="&lt;p&gt;I tend to see this in upper level college courses a lot.  My impression is that the schools are trying to appear to have a solid course by using an iconic textbook (not often the best pedagogically).  Often the course is too short to really cover the content properly.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't like the practice.  Prefer to spend more time or more realistically, use a book tailored for the course length.  But I understand the psychological basis in having shorter courses using iconic, much too long texts. It's the normal human desire to look fancier than one is.&lt;/p&gt;&#xA;" OwnerUserId="12568" LastActivityDate="2019-08-18T00:58:37.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16956" PostTypeId="2" ParentId="16953" CreationDate="2019-08-18T02:52:36.023" Score="4" Body="&lt;p&gt;My sense is that often authors are encouraged (required?) by publishers to make books very 'complete'. This means the book has all the topics that any instructor would be likely to want for a subject.  However, it also means that books are often too big to entirely cover in a semester (or year).  Thus an instructor must pick and choose chapters/sections based on the goals/objectives of their particular course.  So depending on what is to be accomplished in the class, a certain amount of selection from the book is likely to occur, and may result in some topics in the book being skipped or being covered out of order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer one of your specific questions, I don't think this particularly benefits students, except for perhaps giving them material for further self-study if the entire book didn't get covered during the course.&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2019-08-18T02:52:36.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16957" PostTypeId="2" ParentId="16953" CreationDate="2019-08-18T05:45:38.930" Score="4" Body="&lt;p&gt;Yes they do so in many places, I think the point is that the students are forced to finish a certain curriculum (depends on the country ofcourse), for example when students finish the &lt;span class=&quot;math-container&quot;&gt;$9^{th}$&lt;/span&gt; grade they must know this and this and that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in the years before the &lt;span class=&quot;math-container&quot;&gt;$9^{th}$&lt;/span&gt; grade the teachers focus on what have to be finished and leave the other topics that are in the book either because they are not in the curriculum, or because they see that this topic is hard for students in this age so they explain it in the year after, and sadly there is another reason also, most teachers just ask the older teachers what they explain and in which order and do just the same just because the other teachers that used to teach before them do it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as a benefit for students, I think there isn't really any benfit, and that is why all around the world you see students and teachers, that wisely think about this issue, complaining about the curriculum and that it must be changed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not so sure about this, but I think the ministries of education put the curriculum right? So the question is, are there really qualified people in these ministries to specify what the Mathematics curriculum should and should not include? I don't think so myself, and I see that this is the problem.&lt;/p&gt;&#xA;" OwnerUserId="12486" LastActivityDate="2019-08-18T05:45:38.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16958" PostTypeId="2" ParentId="16882" CreationDate="2019-08-18T14:47:34.300" Score="0" Body="&lt;p&gt;My answer would be: &quot;What if there is no formula?&quot; Just as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; might stand for an unspecified number, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; might stand for an unspecified function.&lt;/p&gt;&#xA;" OwnerUserId="10168" LastActivityDate="2019-08-18T14:47:34.300" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16961" PostTypeId="2" ParentId="16941" CreationDate="2019-08-18T19:57:16.090" Score="8" Body="&lt;p&gt;Roughly speaking, adding apples to apples is &lt;em&gt;factoring&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can reasonably interpret &quot;four apples&quot; as the result of multiplying the number &quot;four&quot; with the unit &quot;apple&quot; &amp;mdash; in fact, I would even argue that you &lt;em&gt;should&lt;/em&gt; think in this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, when you add like things, what you are &lt;em&gt;actually&lt;/em&gt; doing is invoking the distributive law to factor the unit out:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 4 \cdot \text{apple} + 5 \cdot \text{apple} = (4 + 5) \cdot \text{apple} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But when you add unlike things, such as &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 4 \cdot \text{apple} + 5 \cdot \text{oranges} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;there's nothing to factor out: no further simplification is possible. Note, however, that if we're willing to forget information, and just say apples and oranges are both fruits, we can then make a substitution&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 4 \cdot \text{fruit} + 5 \cdot \text{fruit} = (4 + 5) \cdot \text{fruit}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and then simplify.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The examples you consider are similar. If we add&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 4 \cdot \text{apple} + 5 \cdot \text{&amp;lt;crate of ten apples&amp;gt;} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;we can't factor anything out. However, if we're willing to ignore the containers, we do have an &lt;em&gt;identity&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 10 \cdot \text{apple} = \text{&amp;lt;crate of ten apples&amp;gt;} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means we can make a substitution into the original formula&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align} 4 \cdot \text{apple} + 5 \cdot \text{&amp;lt;crate of ten apples&amp;gt;} &#xA;&amp;amp;= 4 \cdot \text{apple} + 5 \cdot (10 \cdot \text{apple})&#xA;\\&amp;amp;= 4 \cdot \text{apple} + (5 \cdot 10) \cdot \text{apple}&#xA;\\&amp;amp;= (4 + 5 \cdot 10) \cdot \text{apple}&#xA;\\&amp;amp;= 54 \cdot \text{apple}&#xA;\end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;so we also get the explanation of why you should multiply &lt;span class=&quot;math-container&quot;&gt;$5 \cdot 10$&lt;/span&gt;. (of course, you can actually compute the arithmetic anywhere in the calculation you like, rather than waiting until the end as I did)&lt;/p&gt;&#xA;" OwnerDisplayName="user797" LastActivityDate="2019-08-18T19:57:16.090" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16962" PostTypeId="1" CreationDate="2019-08-19T00:07:14.337" Score="8" ViewCount="472" Body="&lt;p&gt;In contemporary US secondary mathematics textbooks, geometric means occasionally make a brief appearance.  For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In Geometry, students learn that when an altitude is dropped to the hypotenuse of a right triangle, three different geometric relationships are created, each involving a geometric mean.  (See the two figures below.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/syAgz.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/syAgz.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/SFpwG.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SFpwG.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Also in Geometry, students learn that the length of a tangent segment to a circle is equal to the geometric mean of two secant segments.  See the figure below, which -- curiously -- does not use the phrase &quot;geometric mean&quot;, even though it had been previously taught two chapters earlier.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qSJnn.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qSJnn.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(All three of the above images are taken from Glencoe / McGraw-Hill &lt;em&gt;Geometry&lt;/em&gt;, 2008.)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In Algebra 2, students commonly learn that a &quot;geometric mean&quot; is a number between two given (nonconsecutive) terms of a geometric sequence, as in the image below.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/61sA5.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/61sA5.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Source:  Glencoe / McGraw-Hill &lt;em&gt;Algebra 2&lt;/em&gt;, 2008.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that all of these examples have in common is that students never find the geometric mean of &lt;em&gt;more than two numbers&lt;/em&gt;.  And yet there are plenty of instances in which it makes sense to compute the geometric mean of three or more numbers!  Compound interest is one such context: for example, if the value of an investment increases in three consecutive years by 10%, 15%, and 2%, the correct way to describe the average annual growth rate is not to compute the arithmetic mean &lt;span class=&quot;math-container&quot;&gt;$\frac{10+15+2}{3} = 9$&lt;/span&gt;, but rather the geometric mean &lt;span class=&quot;math-container&quot;&gt;$$\sqrt[3]{1.10\cdot 1.15 \cdot 1.02}\approx 1.0887$$&lt;/span&gt; (so the average annual growth rate is 8.87%, not 9%).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I can tell, the topic of geometric means used to play a larger role in the secondary curriculum.  The first &quot;old&quot; Algebra 2-type textbooks I looked at (&lt;em&gt;Higher Algebra&lt;/em&gt;, 1891, by Hall &amp;amp; Knight) included a discussion of general geometric means, as well as a statement of the AM-GM inequality (i.e. that the arithmetic mean is always greater than or equal to the geometric mean).  (See figure below).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Y3J96.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Y3J96.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So (assuming these isolated examples are representative), it seems that once upon a time the GM was part of the curriculum, and now it is not, except in the limited case of the GM of two numbers.  I was in high school in the late 1980s, and I seem to recall the general case was still being taught at the time -- but my memory may be wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question, then:  When did the geometric mean (of more than two numbers) disappear from the curriculum?  And does anybody have any thoughts as to why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just to be clear, I don't expect any one person to be able to give a definitive answer to this.  I would be more than happy with some additional data points, of the form &quot;In this 1985 textbook, I find...  but in this 1994 textbook from the same publisher, it's gone.&quot;&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="2083" LastEditDate="2019-08-19T06:26:15.327" LastActivityDate="2019-08-19T06:26:15.327" Title="When (and why) did geometric means of more than two numbers exit the secondary curriculum?" Tags="|secondary-education|algebra|curriculum|history|usa|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16963" PostTypeId="2" ParentId="16962" CreationDate="2019-08-19T02:15:44.720" Score="8" Body="&lt;p&gt;Your last para was very reasonable.  (I was going to give a mean sarcastic answer, but can't now.)  We can crowdsource this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Frank Ayres, First Year College Math&lt;/strong&gt; (algebra 1 to precalc; Schaum's Outline) 1958 but still in print:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only has geometric mean of 2 objects, but does spend quite a lot of time on geometric progressions.  And also discusses getting multiple geometric means between two numbers. Note also that it is covering a lot of territory (4 years of math in a 430 page text, with problems and answers).  Note the discussion is all algebraic, not geometric.  But there are word problems, not just calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;College Algebra, William Hart.&lt;/strong&gt; I have the 1938 War Department edition, but it was a popular text in different editions from the 20s to the 50s, I think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a similar length section on algebraic, geometric and harmonic progressions (even some infinite ones).  Geometric mean is discussed only as between two endpoints, but similar discussion of having several means between the first and last member.  Some very nice word problems (ancestors, money) that make you think.  This is all in a chapter called &quot;Progressions&quot;.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;100 pages LATER in the book, the geometric mean of multiple numbers is discussed with problems.&lt;/em&gt;  This is within a section on computation with logarithms.  &lt;em&gt;Presumably this is because the log concept and log tables make taking the cube and larger roots possible.&lt;/em&gt;  (Think logs were not yet taught in the Progressions chapter.)  Makes sense.  Nice problems again (e.g. with money, volumes, and just numbers).  But very short subsection (not even a section).  No geometric discussion per se.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus:  There is one problem within &lt;strong&gt;Thomas Finney Calculus and Analytical Geometry&lt;/strong&gt; (1980, the iconic blue textbook with kaleidoscope cover that you see in the movie Stand and Deliver) where the concept of geometric means, of several numbers, is used (and explained en passent).  It's not to teach geometric means per se but some sort of problem with a ratio of arithmetic and geometric means as a series varies, requiring calculus and stuff.  However &lt;strong&gt;Granville Calculus&lt;/strong&gt; (iconic 1900-1960s text) does not reference the concept, at least per the index.  &lt;strong&gt;Swokowski Calculus&lt;/strong&gt; (1975 and 1983 editions) do not refer to geometric means either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometry/trig:  Also none of my &lt;strong&gt;geometry, analytical geometry or trigonometry&lt;/strong&gt; texts (about 2 of each, and 1940-1980 ages) reference or use geometric means at all, at least per the index (and I looked under both m's and g's!)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Post calculus:  Nor does &lt;strong&gt;Kreyszig Advanced Engineering Mathematics&lt;/strong&gt; (1983), even within a problem), although geometric series are discussed wrt convergence.  Situation is similar wrt &lt;strong&gt;Arfken Webber&lt;/strong&gt; or the &lt;strong&gt;various ODE, PDE, and complex analysis texts&lt;/strong&gt; I have.  No mention/use of geometric mean.  Some discussion of geometric series or of mean convergence.  (Nor is it in my 1980s &lt;strong&gt;Giancoli General Physics&lt;/strong&gt;, even within a homework problem on something else.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Net/net:  It's really not that important a concept in daily life.  Even the regular geometric mean, let alone the multinumber one.  Can't recall ever using it in physics, engineering, chemistry or finance...although I'm sure you could torture up a homework problem with effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(EDIT) While the one text that covered multinumber geo mean (Hart) waited until after logs and covered it in the context of calculations with logs, this is not so necessary in a modern course.  Provided the properties of exponents and roots other than the square, have been already been covered, the student should be fine to find the 3rd, 4th, 5th root.  As in all likelihood, he will use a scientific calculator for the root.  (Or the problems can be made to be obviously extractable with numbers like 125, 64, etc.) Thus, it could be included within a slightly longer chapter on progressions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, it maybe is a good thing to build muscles with for stronger kids. And maybe it reverberates in memory long enough to have some touchpoint when they look at geometric series convergence (but I doubt it).  But it's probably not the end of the world to give it short shrift for the average kid.  In the extreme even that use of it or manipulation of it is needed by some strong STEM kid in later classes, they can probably look it up and use it and will have enough general manipulational skills from other topics that it won't throw them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I do very commonly solve for &quot;CAGR&quot;.  &lt;a href=&quot;https://investinganswers.com/dictionary/c/compound-annual-growth-rate-cagr&quot; rel=&quot;noreferrer&quot;&gt;https://investinganswers.com/dictionary/c/compound-annual-growth-rate-cagr&lt;/a&gt;  This may even be the same thing as geometric mean (not sure, but it's smelling that way now).  But I never think about it in terms of geometric mean.  Just think of CAGR.  Also, you really don't need a multinumber geometric mean (even with several different growth rates, over a several year period) since you can solve for the CAGR still using the last and first values of the revenue (with all the growth multiplied in).  Probably this shows some linkage of the concept of two value geo means and multinumber means.   &lt;/p&gt;&#xA;" OwnerUserId="12568" LastEditorUserId="12568" LastEditDate="2019-08-19T03:53:10.280" LastActivityDate="2019-08-19T03:53:10.280" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16964" PostTypeId="1" CreationDate="2019-08-20T07:16:56.273" Score="8" ViewCount="3843" Body="&lt;p&gt;Increasing numbers of students seem to be using the word &amp;quot;unique&amp;quot; incorrectly.  A common example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;.  Define the term &amp;quot;one-to-one&amp;quot; for a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Answer&lt;/strong&gt;.  Every &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; has a unique &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The answer is nothing new, and in the past it would have indicated that the student has the idea the wrong way around: &amp;quot;every &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; has a unique &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; would be fine.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me, however, that nowadays this answer often indicates not that the student has the wrong idea, but that they think &amp;quot;unique&amp;quot; means &amp;quot;different&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;different &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; values have different &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; values&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is a perfectly reasonable answer to the question.  Possibly the error has been &amp;quot;popularised&amp;quot; by web designers who like to proclaim &amp;quot;this site has had 1000000 unique visitors&amp;quot;, when they actually mean &amp;quot;1000000 different visitors&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Is there anything (apart from keeping on about it to students, and marking their assessments wrong) that we can do about this?  Or do we just have to accept that the English language has changed?&lt;/p&gt;&#xA;" OwnerUserId="12591" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-08-21T09:31:33.867" Title="Can we save the word &quot;unique&quot;?" Tags="|language-use|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="16965" PostTypeId="2" ParentId="16964" CreationDate="2019-08-20T11:48:40.183" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I haven't noticed casual everyday use of that word involving a misuse.  As for instance, the word &quot;literally&quot; is often misused by millenials.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even were the word being misused, I wouldn't abandon correct usage.  Instead teach and require correct usage.  But again, I don't notice the word unique shifting in normal day to day usage.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think the issue here is more one of fussy correctness in discussing functions. When the kids use the word unique, they mean (tacitly) unique to that x, i.e. single.  I would say single (where defined) to differentiate from relationships, like the graph of a circle, that have more than one y to an x. The problem with unique, in the context of a function, is that it implies monotonic increase or decrease (no two or more x's having the same y).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12568" LastActivityDate="2019-08-20T11:48:40.183" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16966" PostTypeId="2" ParentId="16882" CreationDate="2019-08-20T14:03:41.587" Score="1" Body="&lt;p&gt;When identifying &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; we implicitly consider a point &lt;span class=&quot;math-container&quot;&gt;$(x;y)$&lt;/span&gt; lying on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;'s graph. However, this doesn't have to be the case. Say, we take &lt;span class=&quot;math-container&quot;&gt;$f: x\mapsto x^2$&lt;/span&gt;, then the point &lt;span class=&quot;math-container&quot;&gt;$(x;y)=(2;f(2))=(2;4)$&lt;/span&gt; lies on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;'s graph but I can well consider the point &lt;span class=&quot;math-container&quot;&gt;$(x;y)=(2;3)$&lt;/span&gt; lying below the graph or the point &lt;span class=&quot;math-container&quot;&gt;$(x;y)=(2;5)$&lt;/span&gt; lying above the graph, whose &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; coordinates are not equal to &lt;span class=&quot;math-container&quot;&gt;$f(1)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3542" LastActivityDate="2019-08-20T14:03:41.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16967" PostTypeId="2" ParentId="16964" CreationDate="2019-08-20T15:03:34.987" Score="19" Body="&lt;p&gt;I don't see this as a major issue, nor do I believe that the word &amp;quot;unique&amp;quot; is in any particular need of saving.  There are a large number of terms in mathematics which correspond to vernacular English words, but which have distinct technical meaning in mathematics.  For example, an &amp;quot;odd&amp;quot; number is an integer which is not a multiple of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, rather than a number which is in some way strange or peculiar (though, oddly enough, the mathematical definition of &amp;quot;odd&amp;quot; predates the modern vernacular usage).  As another example, in mathematics &amp;quot;onto&amp;quot; is a synonym of &amp;quot;surjective&amp;quot;, while in vernacular English &amp;quot;onto&amp;quot; is a preposition which is of general utility.  Or, as a particularly perverse example, what does &amp;quot;normal&amp;quot; mean (hint:  even in mathematics, this little word has a lot of different meanings, depending on context)?&lt;/p&gt;&#xA;&lt;p&gt;Note, also, that this isn't an issue unique (heh heh) to mathematics:  in archaeology (and geology, maybe?), the word &amp;quot;flint&amp;quot; refers to a specific type of toolstone which is associate with limestone deposites; in vernacular English, many types of crypto- and microcrystalline silicates are referred to as &amp;quot;flint&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;One of the jobs of an educator is to introduce their students to the technical jargon of their field, and to help students to understand that words may have a precise definition which is different from the vernacular meaning, or different from the technical meaning in another field (indeed, &amp;quot;unique visitors&amp;quot; is a well-understood term and has its own technical meaning).&lt;/p&gt;&#xA;&lt;p&gt;In the example question and answer posed above, I would regard it as an opportunity to discuss this distinction between vernacular usage and mathematical usage.  For example, on an exam, I might write something like the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;  Define the term &amp;quot;one-to-one&amp;quot; for a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Answer:&lt;/strong&gt;  Every &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; has a unique &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Response:&lt;/strong&gt; I understand what you mean by this answer, but this is not the correct usage of &amp;quot;unique&amp;quot; in mathematics.  It would be better to say &amp;quot;Every &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; has a &lt;em&gt;different&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;,&amp;quot; or even better to say &amp;quot;Every &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is mapped to (or sent to) a distinct value by &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-08-20T19:06:46.263" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16968" PostTypeId="2" ParentId="15536" CreationDate="2019-08-20T15:31:18.023" Score="2" Body="&lt;p&gt;Typical constrained optimization (useful to &quot;get it&quot;) is asking for the rectangle of largest area that can be enclosed in a fence of given length. Sure, it can be reduced to one-dimensional, but leave that option out. Or ask for the largest volume box with given surface area.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2019-08-20T15:31:18.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16969" PostTypeId="2" ParentId="16860" CreationDate="2019-08-21T06:52:06.007" Score="2" Body="&lt;p&gt;You mentioned 3Blue1Brown's videos. If you're interested in making similar animations, Grant (the man behind 3B1B) actually released the entire python library he wrote for creating them: &lt;a href=&quot;https://github.com/3b1b/manim&quot; rel=&quot;nofollow noreferrer&quot;&gt;manim on github&lt;/a&gt;. Obviously, this is very open ended and requires a working knowledge of python to begin with. The other examples of software (Desmos, Mathematica, etc) will generally be much more friendly if you just need to whip something together for a lecture or slide. However, manim+python will open up loads of customization and options. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The subreddit &lt;a href=&quot;https://www.reddit.com/r/manim/&quot; rel=&quot;nofollow noreferrer&quot;&gt;/r/manim&lt;/a&gt; has additional examples and resources!&lt;/p&gt;&#xA;" OwnerUserId="7260" LastActivityDate="2019-08-21T06:52:06.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16970" PostTypeId="2" ParentId="16964" CreationDate="2019-08-21T09:31:33.867" Score="7" Body="&lt;p&gt;To my mind the defintion &quot;Every x has a unique f(x)&quot; of &lt;em&gt;one-to-one&lt;/em&gt; is problematic because &quot;has a unique&quot; is neither clear English nor precise. The definition is usually stated as &quot;&lt;span class=&quot;math-container&quot;&gt;$f(x) = f(y)$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$x = y$&lt;/span&gt;&quot;, which is unambiguous and avoids any potential interpretative problems related to the use of the word &quot;unique. Alternatively, one could define &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to be &lt;em&gt;one-to-one&lt;/em&gt; if &lt;span class=&quot;math-container&quot;&gt;$x \neq y$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$f(x) \neq f(y)$&lt;/span&gt;; the issue with this definition is that it only corresponds to the terminology &lt;em&gt;one-to-one&lt;/em&gt; after negating both sides. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a mistake to try to make mathematical definitions more accesible by framing them in colloquial language. Doing so tends to generate far more confusion than it saves. It is better to advise students (repeatedly if necessary) that mathematical usage and colloquial usage differ, and that part of mathematics involves developing a precise language, and to use this precise language carefully and correctly (even if one is more tolerant of student usage).&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-08-21T09:31:33.867" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16971" PostTypeId="1" CreationDate="2019-08-21T12:24:09.133" Score="8" ViewCount="280" Body="&lt;p&gt;Although the implementation of curricula and assessments that align to the Common Core has been a slow roll-out across many states, the standards themselves are close to ten years old now. I don't imagine that anyone ever thought that Common Core would be etched in stone, a perfect set of standards that would remain forever unchanged. But I have not heard of any attempt to revise them, or even any mechanism by which that would be possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the best way of coming to a good set of standards would be by some iterative process of revision over the course of many years. If Common Core is to be successful, at least from the standpoint of coherence of the standards, it only makes sense to revisit them after they've been in use for some time and their failings become apparent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Will we ever see CCSSM version 2.0 that takes into account lessons learned from actual implementation of the standards?&lt;/p&gt;&#xA;" OwnerUserId="5088" LastActivityDate="2019-08-21T15:22:50.937" Title="Will there ever be a revision of Common Core?" Tags="|common-core|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16972" PostTypeId="2" ParentId="16971" CreationDate="2019-08-21T15:22:50.937" Score="6" Body="&lt;p&gt;Not a completely satisfying answer, but there is this from &#xA;the FAQ at &lt;a href=&quot;http://www.corestandards.org/about-the-standards/frequently-asked-questions/#faq-2319&quot; rel=&quot;noreferrer&quot;&gt;corestandards.org&lt;/a&gt;,&#xA;emphasizing that the onus is on the states:&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Who will manage the Common Core State Standards in the future?&lt;/strong&gt;&#xA;  &lt;br /&gt;&#xA;  The Common Core State Standards are and will remain a state-led effort, and adoption of the standards and any potential revisions will continue to be a voluntary state decision. The National Governors Association Center for Best Practices and the Council of Chief State School Officers will continue to serve as the two leading organizations with ownership of the Common Core and will make decisions about the timing and substance of future revisions to the standards in consultation with the states. Federal funds have never and will never be used to support the development or governance of the Common Core or any future revisions of the standards. Any future revisions will be made based on research and evidence. Governance of the standards will be independent of governance of related assessments.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This may be a bit more informative (and recent), from &lt;a href=&quot;https://edtechtimes.com/2017/02/15/common-core-revisions-what-are-states-really-changing/&quot; rel=&quot;noreferrer&quot;&gt;edtechtimes&lt;/a&gt;,&#xA;February 15, 2017:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Mathematics changes&lt;/strong&gt;: Across all grade levels in mathematics, the nine states revised 26.5 percent of the standards, with 73.5 percent of the standards were kept the same. The number of math standards revised ranged from 17 changes in one state to 282 changes in another. Moreover, eight of the nine states added math standards, with a total of 51 new standards added.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Similar information is in an &lt;em&gt;Education Week&lt;/em&gt; article, &lt;a href=&quot;https://www.edweek.org/ew/articles/2017/11/13/even-when-states-revise-standards-the-core.html&quot; rel=&quot;noreferrer&quot;&gt;Even When States Revise Standards, the Core of the Common Core Remains&lt;/a&gt;, &#xA;Vol. 37, Issue 14, Pages 1, 14, November 13, 2017.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-08-21T15:22:50.937" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16973" PostTypeId="2" ParentId="16948" CreationDate="2019-08-22T13:11:59.620" Score="10" Body="&lt;p&gt;I am writing this based on pure observation (e.g., entering year four of teaching this topic to secondary school students, and having co-taught a minicourse for teachers on absolute value functions&lt;span class=&quot;math-container&quot;&gt;$^\star$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a lot of definitions/interpretations of absolute values:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the (abstract) axiomatic one; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the piecewise or &quot;case-based&quot; one (provided by &lt;strong&gt;Joel Reyes Noche&lt;/strong&gt;); &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the colloquial function one (erase any negative sign and return the result); &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the equivalent function one of &lt;span class=&quot;math-container&quot;&gt;$x \mapsto \sqrt{x^2}$&lt;/span&gt;; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the geometric one (where &lt;span class=&quot;math-container&quot;&gt;$|a-b|$&lt;/span&gt; is the distance on the number line between real numbers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;); &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the positive difference interpretation (subtract the lesser number from the greater one, i.e., &lt;span class=&quot;math-container&quot;&gt;$|a-b|=\max(a,b)-\min(a,b)$&lt;/span&gt; for real numbers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;); &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the graphical one communicated by a &lt;span class=&quot;math-container&quot;&gt;$\mathsf{V}$&lt;/span&gt;-shaped curve; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;etcetera.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A complication that has already arisen in the above collection of definitions/interpretations is its ambiguity around whether we are defining a function whose input is one number or two numbers; the one number version can, of course, be viewed as the two number version where (at least) one of the entries is zero. Nevertheless, I think that sometimes the topic is introduced with multiple approaches and without drawing this distinction; e.g., if you try to help a student get a grip on absolute value functions by asking the difference in age between them and a friend, then you are effectively asking about the two input interpretation, which could be confusing if you had just introduced it as, e.g., a piecewise function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There may also be an issue of timing: When I delve into absolute value functions, equations, and inequalities, it is in an Algebra 2 course; in our present course sequence, this means that students have done Algebra 1 already, and then spent a year on Geometry. In other cases, you may have this topic broached in an Algebra 1 course; so, students are in the throes of matching graphical representations, geometric representations, symbolic representations, and so forth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to covering absolute values pedagogically, I think there is a great value in viewing more general absolute value functions of the form &lt;span class=&quot;math-container&quot;&gt;$x \mapsto a|x-h|+k$&lt;/span&gt; for real parameters &lt;span class=&quot;math-container&quot;&gt;$a, h, k \in \mathbb{R}$&lt;/span&gt; in terms of transformations, and, in addition, connecting this to quadratic functions written in vertex form, &lt;span class=&quot;math-container&quot;&gt;$x \mapsto a(x-h)^2 + k$&lt;/span&gt;, to emphasize similarities and differences (and to reinforce terminology: intercepts, roots, vertex, concavity, end behavior, domain, range).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\star$&lt;/span&gt;: I would be remiss if I failed to link our (my colleague/department chair, Liz Brennan, and my) freely available materials from a minicourse taught through &lt;em&gt;Math for America&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://docs.google.com/document/d/1NLRybvRXQ1CkOgaFtl_0kDzWZ_VeF1NUfbKRP1yzqXY/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Absolute Values are Absolutely Valuable: Secondary School Problem Posing&lt;/strong&gt;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are a lot of materials in there, which range from a question I asked on MathOverflow (&lt;a href=&quot;https://mathoverflow.net/q/301514&quot;&gt;&lt;strong&gt;MO 301514&lt;/strong&gt;&lt;/a&gt;) to a misformulated problem (p. 9 #6 is impossible!) to problems that participants formulated during our final meeting (&lt;a href=&quot;https://www.desmos.com/calculator/jbykvt3bs7&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Desmos link&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The three-meeting course linked above is by no means exhaustive: In fact, I have been thinking more about absolute values and their role in defining equations as one endeavors to impose domain restrictions; for example, see the sequence of tweets that begins &lt;a href=&quot;https://twitter.com/benjamindickman/status/1160056746603421696&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2019-08-22T13:11:59.620" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16974" PostTypeId="2" ParentId="14649" CreationDate="2019-08-22T14:05:15.047" Score="4" Body="&lt;p&gt;The Common Core State Standards for Mathematics (CCSSM) provides a list of standards that &lt;em&gt;should&lt;/em&gt; be covered; the inference that a topic not mentioned in the algebra strand should, therefore, be omitted does not necessarily follow, although time constraints no doubt ensure that will be the case for many courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than going into the process for developing CCSSM, I believe it may be helpful to wonder about the '2' in the course title: What exactly is meant by Algebra 2?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It follows a course called Algebra 1&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It precedes a course called Algebra 3? (No, unless you mean Linear Algebra; followed by Abstract Algebra; then Abstract Algebra 2 = Galois Theory; then Algebraic Geometry or Algebraic Topology or Algebraic Number Theory or...)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It precedes pre-Calculus, but doesn't that make it - for all those who eventually take Calculus - fall under the umbrella of (literally) pre-Calc?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It &lt;em&gt;is&lt;/em&gt; pre-Calculus, and the next course is to be Calculus&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The last two points are especially important: Is this course leading in to a follow-up course that examines further function families (e.g., rational functions), trigonometry, and other topics? Or is it headed straight for secant lines that segue into tangent lines and eventually limits in a first course on Calculus? Alternatively still, is Algebra 2 supposed to position students to make a decision around whether they should continue on a path to Calculus &lt;em&gt;or&lt;/em&gt; Statistics/Probability? This could have an effect on topics for inclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, could Algebra 2 be expected as a terminal course? For some US colleges/universities, yes, this is the expectation; in fact, some do not require even Algebra 2. Consider &lt;a href=&quot;https://www.insidehighered.com/news/2016/07/06/michigan-state-drops-college-algebra-requirement&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;this&lt;/strong&gt;&lt;/a&gt; piece from &lt;em&gt;Inside Higher Ed&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Michigan State University has revised its general-education math requirement so that algebra is no longer required of all students. The revision reflects an increasing view on college campuses that there is no one-size-fits-all math curriculum -- and that math is often best studied in connection with everyday life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(I note with no irony or sarcasm -- only interest -- that MSU truly has one of the best programs in the country/world for mathematics education.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Returning to your list of topics, it would probably take me a full year to cover a proper subset of those topics in ways that are mathematically meaningful. If you will permit a mixed-metaphor, then I note that one can &quot;cookbook-blitz&quot; these topics by asserting facts and having students carry out various procedures (e.g., transforming trig identities into statements about just sine and cosine, and then using a very select few axioms to prove the desired consequence). Personally, if I were to teach about trig identities in a way that felt meaningful (in some ways) to me, then I might opt to use them as a way to introduce more formal proof writing (cf. my answer to &lt;a href=&quot;https://math.stackexchange.com/a/3310782&quot;&gt;&lt;strong&gt;MSE 2905643&lt;/strong&gt;&lt;/a&gt;) but this would diverge from certain expectations around &quot;mathematical maturity&quot; in an Algebra 2 course, and it would take far too long for me to get to all the &quot;omitted&quot; topics in your post - as well as those that remain!&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For a fun mathematical closer, here is one example of a cookbook-blitz approach to trigonometry: Accept the &lt;strong&gt;fact&lt;/strong&gt; (whatever it means) that &lt;span class=&quot;math-container&quot;&gt;$e^{ix} = \cos(x) + i\sin(x)$&lt;/span&gt;. Consider next two ways to change the LHS to &lt;span class=&quot;math-container&quot;&gt;$e^{i2x}$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Squaring both sides&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Replacing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Since each of these results in the same LHS, the corresponding RHS's must be equal. By accepting that &lt;strong&gt;fact&lt;/strong&gt;, we arrive, then, at the following identity:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$[\cos^2(x) - \sin^2(x)] + i[2\sin(x)\cos(x)] = \cos(2x) + i[\sin(2x)]$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observe that matching real and imaginary parts immediately yields the double-angle formulae for cosine and sine. Now, I happen to believe this is an interesting derivation (and you can use this to derive many other identities provided, e.g., you know a special value or two of sine and cosine from the unit circle) but I do not think that a one day lesson on this &lt;strong&gt;fact&lt;/strong&gt; is an ideal way to build meaningful mathematics with (most) students. In particular, I am concerned that the given is too strong, and that the techniques (observing two ways of getting to the same LHS; matching real parts and matching imaginary parts) are too novel to be sprung all at once. Imagine instead a course that developed these ideas in multiple ways, and in which student discovery was centered rather than speed or initial efficacy. Under such conditions, I would not, personally, be able to do justice to these ideas along with all the many other ones that could fall under as broad a name as Algebra 2.&lt;/p&gt;&#xA;" OwnerUserId="262" LastEditorUserId="262" LastEditDate="2019-08-22T19:17:47.690" LastActivityDate="2019-08-22T19:17:47.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16975" PostTypeId="2" ParentId="16948" CreationDate="2019-08-22T21:42:47.920" Score="5" Body="&lt;p&gt;At my school (a community college in California) the curriculum is set up so that students first take an algebra course in which essentially every function is a linear function, and only later do they do anything at all with nonlinear functions. Nonlinear functions are just more complicated, so they require more thought about the logic. Thinking about logic is harder than manipulating formulas according to recipes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, if I tell a student that &lt;span class=&quot;math-container&quot;&gt;$x^2=4$&lt;/span&gt; and ask them to solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, the formula-manipulation approach would be &quot;do the same thing to both sides,&quot; so they may get &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt;. It requires an extra logical step to realize that there are two roots. One way to write this is &lt;span class=&quot;math-container&quot;&gt;$|x|=2$&lt;/span&gt;, but I'll see students do things like &lt;span class=&quot;math-container&quot;&gt;$x=|2|$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$|x|=|2|$&lt;/span&gt; (which is correct but kind of silly). Understanding why &lt;span class=&quot;math-container&quot;&gt;$|x|=2$&lt;/span&gt; is the right choice requires some more logical thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example that comes up in physics is that students will memorize the fact that &lt;span class=&quot;math-container&quot;&gt;$a=-g$&lt;/span&gt; for free fall, despite instruction to the effect that this depends on the coordinate system, and that they need to pick a coordinate system first. Some textbooks even encourage this. Picking a coordinate system requires an extra logical step, which is hard if you aren't used to thinking about math logically. I can tell them that &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is defined as &lt;span class=&quot;math-container&quot;&gt;$|a|$&lt;/span&gt; so that we can put a value of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; in the book without reference to any coordinate system, but again, this requires some logical thinking about topics that they aren't used to thinking about, and have gotten the impression that they never have to think about.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2019-08-22T21:42:47.920" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16977" PostTypeId="1" CreationDate="2019-08-24T20:48:03.230" Score="4" ViewCount="342" Body="&lt;p&gt;I am seeking the most transparent exposition of Bayes' Theorem (for undergraduates). I would prefer to avoid mentioning &quot;prior&quot; and &quot;posterior,&quot; and instead focus on frequencies.&#xA;The &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bayes%27_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia entry&lt;/a&gt;&#xA;is quite good in some ways, but I disagree with this: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;The role of Bayes’ theorem is best visualized with tree diagrams.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The tree diagrams do little for me. But the&#xA;associated table is quite useful:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/tevpZ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tevpZ.png&quot; alt=&quot;Frequency&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bayes%27_theorem#Frequentist_interpretation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia image&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$P( A \&amp;amp; B ) = \frac{2}{20}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P( A | B ) = \frac{2}{8}$&lt;/span&gt;&#xA;and &#xA;&lt;span class=&quot;math-container&quot;&gt;$P(B) = \frac{8}{20}$&lt;/span&gt;.&#xA;So &lt;span class=&quot;math-container&quot;&gt;$$P( A \&amp;amp; B ) = P( A | B )\; P(B) = \tfrac{2}{8} \tfrac{8}{20} =  \tfrac{2}{20}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P( B | A ) = \frac{2}{5}$&lt;/span&gt;&#xA;and &#xA;&lt;span class=&quot;math-container&quot;&gt;$P(A) = \frac{5}{20}$&lt;/span&gt;.&#xA;So &lt;span class=&quot;math-container&quot;&gt;$$P( A \&amp;amp; B ) = P( B | A )\; P(A) = \tfrac{2}{5} \tfrac{5}{20} =  \tfrac{2}{20}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And this leads to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$P( A \&amp;amp; B ) = P( A | B )\; P(B) = P( B | A )\; P(A),$$&lt;/span&gt;&#xA;which is one form of Bayes' Theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here are my questions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Q1&lt;/strong&gt;. What is a compelling story one can associate with&#xA;  this specific table, identifying &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; with real-world&#xA;  events that make sense with that table's data?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Q2&lt;/strong&gt;. What is the clearest justification &#xA;  one could make for&#xA;  &lt;span class=&quot;math-container&quot;&gt;$P( A \&amp;amp; B ) = P( A | B )\; P(B)$&lt;/span&gt;,&#xA;  independent of any particular interpretation of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-08-24T20:56:31.510" LastActivityDate="2019-08-26T01:31:43.010" Title="The most transparent exposition of Bayes' Theorem" Tags="|undergraduate-education|probability|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16979" PostTypeId="2" ParentId="16977" CreationDate="2019-08-26T01:07:40.500" Score="4" Body="&lt;p&gt;Q1. A real-world story that I've heard many times over when Bayes' Theorem is explained concerns medical lab tests where one of the events in your table would be &quot;actually has the disease the test is designed to detect&quot; while another is &quot;experiences a positive test result when the lab test is administered&quot;; however, these scenarios tend to be confusing for people who don't understand positive and negative in the context of test results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q2. One way I think about this is to view the conditional probability as being defined by &lt;span class=&quot;math-container&quot;&gt;$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and  &lt;span class=&quot;math-container&quot;&gt;$P(A|B) := 0$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$P(B) = 0$&lt;/span&gt; rather than defining the probability of the intersection as the product of the other two probabilities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The exposition that goes along with it is &quot;Assume within our universe, event B has occurred (i.e., &lt;em&gt;given&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;). Then &lt;span class=&quot;math-container&quot;&gt;$P(A|B)$&lt;/span&gt; is the probability of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; in the new, restricted universe&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastEditorUserId="1230" LastEditDate="2019-08-26T01:31:43.010" LastActivityDate="2019-08-26T01:31:43.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16980" PostTypeId="2" ParentId="16953" CreationDate="2019-08-26T21:04:29.873" Score="2" Body="&lt;p&gt;Also, realize that many authors write &quot;complete&quot; (self-contained) texts, and by their own recommendations, may suggest various ways an instructor may want to approach the subject.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, an author of a text with 13 chapters, may suggest, in the event that an instructor is focusing on foo, to focus on chapters 1 - 3, chapter 4 (sections 1-3), and chapters 7, 9, 10. Occasionally, an instructor may designate two or three chapters and their sections to be addressed by students in presentations, or in a semester paper, based on the foundation of the text that's covered in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a common practice in classes, because some texts cannot be thoroughly covered  in one or two semesters.&lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2019-08-26T21:04:29.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16981" PostTypeId="2" ParentId="7281" CreationDate="2019-08-27T03:10:28.587" Score="-1" Body="&lt;p&gt;I'm not sure how you would translate the following into the language of 10-year-olds, but the teacher should first understand this much:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The variables x and y are respectively in the ratio a:b means only that x/y = a/b (for non-zero values of a, b, x and y).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, the variables x, y and z are respectively in the ratio of a:b:c means only that x/y = a/b and y/z =b/c (for non-zero values of a, b, c, x, y and z).&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2019-08-27T05:07:50.243" LastActivityDate="2019-08-27T05:07:50.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16982" PostTypeId="2" ParentId="2679" CreationDate="2019-08-27T18:27:59.393" Score="1" Body="&lt;p&gt;Judea Pearl &amp;amp; Dana Mackenzie, in their new book &#xA;&lt;a href=&quot;https://www.penguin.co.uk/books/289/289825/the-book-of-why/9780141982410.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Book of Why&lt;/em&gt;&lt;/a&gt; &#xA;(p.190ff), explain the paradox in a way I hadn't seen before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pearl imagines changing the rules to &quot;Let's Fake a Deal,&quot;&#xA;where &quot;Monty opens one of the two doors you didn't choose, but his choice is completely at random.&quot; Of course he could open the door&#xA;containing the car/prize, ruining the game. But his point is that, with this random choice, there is no longer any advantage to switching. If Monty opens&#xA;a door containing a goat, there is a &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-in-&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; probability of&#xA;winning either by sticking or by switching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That Monty's door-opening is not random, but rather&#xA;constrained by Monty's need to avoid revealing the car, now makes it more clear that significant information is conveyed by his goat door-opening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He assumes you open Door &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, and then:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/fkFWJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fkFWJ.png&quot; alt=&quot;Snippet&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Let's &lt;strong&gt;Make&lt;/strong&gt; a Deal, vs. Let's &lt;strong&gt;Fake&lt;/strong&gt; a Deal.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-08-27T23:34:22.430" LastActivityDate="2019-08-27T23:34:22.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16983" PostTypeId="2" ParentId="16941" CreationDate="2019-08-28T09:19:32.147" Score="2" Body="&lt;p&gt;I am right now teaching Numbers to kids and I think it is the toughest part because Numbers are the true abstract thing in math. So I first taught them what 1(One) means and they got it pretty quickly. So now I am approaching bigger Numbers and I am trying to make them understand that bigger numbers are &quot;COLLECTION&quot; of smaller numbers. So the problem with &quot;things&quot; and representing them as &quot;numbers&quot; is that there is a transition between this real world, &quot;things&quot; and Mathematical Universe where those &quot;things&quot; are represented as numbers and this transition may not be as automatic and intuitive as that of a grown up. Addition can be used as an aid to explaining collection. There are additions where constituents are not visible after addition(7+2=9, or milk in milk) and add then there are additions where constituents are visible after addition(2 pencils + 4 pencils), these additions are collections of things. &#xA;Also one of the things I try to make clear is there are various aspects of addition:&#xA;1. What is addition?&#xA;2. What is it used for?&#xA;3. What is the process of addition?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Difference between individual things, collections and parts of a thing are important and must be made clear to kids...&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2019-08-28T09:19:32.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16984" PostTypeId="1" AcceptedAnswerId="16986" CreationDate="2019-08-30T09:53:26.023" Score="7" ViewCount="1188" Body="&lt;p&gt;Does anyone have any good sources to teach upper primary so grade 7 students basics on graph theory? The only ones I can find are for like upper middle school and high school, and having trouble finding friendly sources for grade 7 students. I would appreciate the sources if any is provided. &lt;/p&gt;&#xA;" OwnerUserId="12614" LastActivityDate="2019-09-10T01:35:31.407" Title="Graph theory teaching materials for young students" Tags="|teaching|interactive-teaching|teacher-preparation|resources|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16985" PostTypeId="2" ParentId="16984" CreationDate="2019-08-30T14:44:05.053" Score="4" Body="&lt;p&gt;Here are GeoGebra materials: &lt;a href=&quot;https://www.geogebra.org/m/Pr3bvbx3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graph Theory for Kids&lt;/a&gt;,&#xA;inspired by Joel Hamkins' notes, to which @A.Goodier pointed.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/JNXj3.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JNXj3.jpg&quot; alt=&quot;FourColors&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;&lt;a href=&quot;https://www.geogebra.org/m/Pr3bvbx3#material/pjPgJdhV&quot; rel=&quot;nofollow noreferrer&quot;&gt;Four-color challenge&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-08-30T18:38:16.287" LastActivityDate="2019-08-30T18:38:16.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16986" PostTypeId="2" ParentId="16984" CreationDate="2019-08-30T19:56:05.473" Score="11" Body="&lt;p&gt;One of the charms of graph theory is that people of all ages often enjoy learning graph theory ideas and tools. One place one can read about these ideas is in the book called For All Practical Purposes (I am a co-author) which has gone through 10 editions. The book was designed for college liberal arts students who might have almost no proficiency with algebra. However, &quot;brave&quot; 5th graders and above can read the beginning chapters, which treat graph theory related ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, many treatments of basic graph theory don't use applications settings as a carrot to interest beginners. Thus, while the historical roots of graph theory are often tied to Euler's work on a recreational mathematics problem involving the traversal of bridges, a much more natural collection of settings for Euler's work relates to operations research—finding an efficient route for a pot-hole inspector, a snow plow, mail delivery, or distributing advertising flyers. The beautiful theorem is that a graph in one piece has a tour starting at any vertex and finishing at the same vertex which traverses each edge once and only once if and only if the graph has an even number of local line segments (edges) at a vertex. Few &quot;real world&quot; graphs have this property but one can modify the applied and theoretical questions so as to look for the tour which minimizes the number of repeated edges or minimizes the total weight of the tour (where each edge has some positive weight, time or cost attached to it). This approach leads to the beautiful &quot;model&quot; often called the &quot;Chinese postman problem.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other nifty applied graph theory problems lead to questions which relate to hamiltonian circuits in graphs, coloring the vertices, edges or faces of a graph, or finding matchings of different kinds in a graph.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The important point for me is that applications settings and appealing geometric diagrams quickly lead to nifty theory (the proof of the theorem mentioned above is very intuitively appealing) that can &quot;hook&quot; lower grade students (or adults) who might not find &quot;algebra&quot; based mathematics as having the &quot;charm&quot; that graph theory mathematics does for many.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2019-08-30T19:56:05.473" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16988" PostTypeId="2" ParentId="14649" CreationDate="2019-09-01T01:06:48.787" Score="2" Body="&lt;p&gt;With apologies to those who gave us the CCSS-M and who were well meaning, in my view the CCSS-M have been a &quot;disaster&quot; for mathematics, computer science, mathematics education, and most importantly American public school students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the years since World War II the world has been transformed by new technologies, in particular those that were driven by the development of the digital computer. The digital computer has also dramatically changed the practice of mathematics and the content of mathematics, as well as helping create the academic discipline of computer science. Mathematics curriculum in K-12 should reflect the tremendous growth in mathematics in the last 75 years as well as the growing number of domains where mathematics is proving itself useful in providing significant insight. There was a time when biology was viewed as being too complex to be amenable to mathematical investigation, but I don't think many feel this way anymore.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you read what I write below, be aware that without my writing a book length treatment many issues are over vastly simplified. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, remember some &quot;boundary conditions.&quot; In America, unlike most European democracies, education is not &quot;directly&quot; under the control of the United States government. Since education is not mentioned in the US Constitution, educational policy is left to the states, and more precisely, local school boards. Whatever benefits this arrangement may have, it does not include efficiency in the delivery of mathematics education to American students. Second, America was the lucky beneficiary of the NCTM's decision to develop what have come to be called the &quot;NCTM Standards,&quot; without external funding. Part of the idea was to develop a nation-wide way of looking at the mathematics curriculum and the teaching of mathematics that could serve as the basis for what individual states did with regard to mathematics education. In a general way one goal of the NCTM Standards was to get rid of tracking of students by race, gender, and income that &quot;encouraged&quot; some to pursue STEM careers (more generally, college careers) and dissuaded others. The development of the CCSS-M was not primarily a grass roots movement of those involved in K-12 teaching. In some ways it was the &quot;negative&quot; response of the mathematics &quot;research&quot; community about how influential the NCTM Standards had become. The issue of the high stakes testing that accompanied the CCSS-M is in part independent of the curriculum issues involved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were many differences between the NCTM Standards and CCSS-M. One can compare and contrast the NCTM Standards and the CCSS-M regarding  continuous mathematics over discrete mathematics, algebraically rooted mathematics over &quot;geometrically&quot; rooted mathematics, the importance of &quot;theory&quot; over the &quot;applicability&quot; of mathematics, and mathematical tools that would be useful for students planning STEM careers vs. other careers. The CCSS-M implicitly assumes that all public school students should be &quot;prepared&quot; to take Calculus even when data show that perhaps 1/3 of high school students who go on to college major in STEM fields or ones that are &quot;artificially&quot; required to take Calculus. (Lots of research in biology takes advantage of Calculus-based ideas but lots of biology research also benefits from knowing about mathematics used for phylogeny and genomics which is more or less Calculus-free. One commonly stated aim of the CCSS-M was to change K-12 mathematics curriculum from one that was a mile wide to a curriculum which was less broad and to give students a better &quot;conceptual&quot; understanding of mathematics by downplaying some procedural aspects of the prior curriculum. This, despite the fact that for many students procedural fluidity was a path to conceptual understanding, and, for me, emphasizing a conceptual understanding, say, of complex numbers for all students is misguided. For most students having more breadth in the mathematics they see (graph theory, recursion, mathematical modeling, social choice (voting, apportionment, game theory) ideas, etc.) is a wiser choice. Thus, that certain algebra topics treated in the past in K-12 are no longer present does not bother me. In fact, my feeling is that K-11 mathematics education should be designed to serve the needs of students independent of their career plans and that the 12th grade be used to treat more technical parts of mathematics for those who know in high school that they want to major in STEM fields when they get to college. Having students have their first exposure to Calculus in college has much to recommend it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopefully, NCTM will return to the curriculum leadership role it had when it developed the Standards, and America can return to having a K-12 curriculum that serves all American students better than the patchwork of modified CCSS-M that the states have turned to when the &quot;national&quot; version of the CCSS-M &quot;collapsed,&quot; in many cases not for the &quot;right&quot; reasons.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2019-09-01T01:06:48.787" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16989" PostTypeId="1" AcceptedAnswerId="16992" CreationDate="2019-09-01T05:10:17.300" Score="8" ViewCount="1020" Body="&lt;p&gt;I'm a high school teacher in New York State (US), starting in on my first year of teaching Geometry.  One of the things that really intrigues me is that the Regents exam (the state-mandated final exam) accepts a graphic organizer for proofs that is different than the traditional two column Statement-Reason style that I learned.  Here, for instance, is a fully correct model response in a recent exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/uhGBW.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uhGBW.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My colleagues and I are really intrigued with this style and the potential that it has to make the &quot;flow&quot; of a proof more visually intuitive for students.  However, I'm afraid that I don't even know what this style is named, much less research on whether it helps students master skills in proof-writing and strategies for teaching it in a class.  Does anyone have any leads that could point me in the right direction?&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="2083" LastEditDate="2019-09-01T14:43:40.597" LastActivityDate="2019-09-01T15:54:28.560" Title="An alternative to &quot;two column&quot; geometry proofs" Tags="|secondary-education|geometry|proofs|terminology|usa|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16990" PostTypeId="1" CreationDate="2019-09-01T06:00:06.717" Score="8" ViewCount="505" Body="&lt;p&gt;I teach at community college, and have often encountered others talking about helping students learn to take good notes. I have never felt that I took good notes as a student. I was too busy thinking about the ideas to carefully write down good notes. Until recently, I thought my lack of skill in note-taking was a deficit. But then I read &lt;a href=&quot;https://blog.mathed.page/2019/07/16/taking-notes-vs-doing-math/&quot; rel=&quot;noreferrer&quot;&gt;this article by Henri Picciotto&lt;/a&gt;. It resonated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I'd like to convince my colleagues that there are better options than trying to help students take good notes in math class. But I'd like to find some research if I can.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The better options, in my opinion, include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If we have a tutor in the class, they take the notes, and post them&#xA;for the class. &lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If not, the students take turns taking notes to post.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The teacher has notes available.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-09-01T14:35:30.000" Title="Is there research on the efficacy of taking good notes in math class?" Tags="|mathematical-pedagogy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16991" PostTypeId="2" ParentId="16990" CreationDate="2019-09-01T14:35:30.000" Score="8" Body="&lt;p&gt;You seem to be seeking research to show there are better options&#xA;than note-taking. I can't answer the question you posed,&#xA;but you will need to overcome the evidence that note-taking&#xA;&lt;em&gt;is&lt;/em&gt; effective. For example: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jennifer Gonzalez.&#xA;  &quot;Note-taking: A Research Roundup&quot;&#xA;  September 9, 2018.&#xA;  &lt;a href=&quot;https://www.cultofpedagogy.com/note-taking/&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;Whether it’s taking notes from lectures (Kiewra, 2002) or from reading (Rahmani &amp;amp; Sadeghi, 2011; Chang &amp;amp; Ku, 2014), note-taking has been shown to improve student learning. In other words, if we want our students to remember more of what they learn in our classes, it’s better to have them take notes than it is to not have them take notes.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-09-01T14:35:30.000" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16992" PostTypeId="2" ParentId="16989" CreationDate="2019-09-01T15:54:28.560" Score="12" Body="&lt;p&gt;This is an example of what is usually called a &lt;strong&gt;flowchart proof&lt;/strong&gt; (or sometimes a &lt;strong&gt;flow proof&lt;/strong&gt; for short).  A quick Google search for &quot;flowchart proof&quot; or &quot;flow proof&quot; shows many, many contemporary examples of the form, including a whole genre of YouTube videos teaching this style of presentation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This style of proof has been promoted at various points since the 1960s.  As it happens, way back in 2006 (when I was a grad student) I once prepared an annotated bibliography of articles from &lt;em&gt;The Mathematics Teacher&lt;/em&gt; on proof and proving; one section of that bibliography was about &quot;Articles proposing alternatives / modifications to the 2-Column Form&quot;, and a subsection of that was &quot;II.   Articles about 'Flow Proofs'&quot;.  I am reproducing the entirety of that section below (with a few minor edits and elisions for clarity).  (In what follows, MT is an abbreviation for &lt;em&gt;Mathematics Teacher&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It seems that the &quot;Flow Proof&quot; gets rediscovered every eight years or so:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Ness, H.  (1962).  &quot;A method of proof for high school geometry&quot;.  MT, 55, 567-569.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Thorsen, C.  (1963).  &quot;Structure  diagrams for geometry proofs.&quot;  MT, 56, 608-609.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Hallerberg, A.  (1971).  &quot;A form of proof.&quot; MT, 64, 203-214.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;McMurray, R. (1978).  &quot;Flow proofs in geometry.&quot;  MT, 71, 592-595.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Basinger, D. (1979).  &quot;More on flow proofs in geometry.&quot; MT, 72, 434-436.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Brandell, J.  (1994).  &quot;Helping students write paragraph proofs in geometry.&quot;  MT, 87, 498-502.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A few words about the articles above:     Ness (1962) introduces (is this&#xA;  the first reported case of it?) the flow proof.  He begins with a&#xA;  concise summary of the drawbacks of the 2CF, and then goes on to say:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;The School Mathematics Study Group, in its geometry course,&#xA;  introduces the paragraph type of proof presumably to eliminate some of&#xA;  the objections listed above.  However, in my opinion, the paragraph&#xA;  type of proof would be extremely difficult for sophomore geometry&#xA;  students, and these proofs are often vague and lack precision.  I&#xA;  would like to introduce a method of proof that we tried last year as&#xA;  an experiment... that I believe eliminates the difficulties of the&#xA;  two-column proof without  introducing the vagueness and lack of&#xA;  precision of the paragraph proof.&quot;    &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Ness then goes on to give an example of the same proof, done both in 2CF and as a flow proof (although&#xA;  he does not name his 'method').  Curiously, his flow proof consists&#xA;  solely of what are normally called &quot;statements&quot;; there is no space in&#xA;  his diagram for the &quot;reasons&quot;.  Curiously, he seems not to notice this&#xA;  – at least he makes no mention of it.     &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Thorsen (1963) is a response to&#xA;  Ness (1962).  Apparently Thorsen had also been  using an essentially&#xA;  identical format, the only difference being that Thorsen's statements&#xA;  are enclosed  in boxes (&quot;balloons&quot;).  Like Ness, Thorsen does not&#xA;  include reasons in his flow proofs, which he calls &quot;structure&#xA;  diagrams&quot;.  Thorsen cautions against using structure diagrams instead&#xA;  of a 2CF; rather, he uses them as a planning tool, to help understand&#xA;  the logic of an argument, which still needs to be written as a 2CF&#xA;  &quot;which includes 'authorities'&quot; (i.e. reasons).   The structure diagram&#xA;  thus serves as a scaffolding device only.     &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Hallerberg (1971)&#xA;  re-discovers the flow proof independently (?) eight years later.  His&#xA;  flow proofs are extremely formal in nature; nodes in the proof&#xA;  correspond to implications (of the form P -&gt; Q), with incoming edges&#xA;  representing known assertions (P) and outgoing edges representing&#xA;  deduced conclusions (Q).  Thus Hallerberg, like Berger, Schacht, and&#xA;  Shields, makes the law of &lt;em&gt;modus ponens&lt;/em&gt; the central feature of his&#xA;  format.   Hallerberg's flow proofs lack the readability of Thorsen's&#xA;  and Ness's, largely because his diagrams consist solely of code labels&#xA;  (&lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$G_k$&lt;/span&gt;, etc.) with a &quot;legend&quot; alongside the diagram decoding the&#xA;  symbols.  The result might please a logician but is manifestly&#xA;  classroom –unfriendly.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Two remarkable features of Hallerberg's&#xA;  article are that he makes no claim that this has worked well in the&#xA;  classroom (I believe his article is unique in not making this claim),&#xA;  and that he offers examples of non-geometric proofs as well.  He also&#xA;  addresses using the flow proof with indirect proofs.      &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Flow proofs&#xA;  are rediscovered once again in McMurray (1978).  It is here that the&#xA;  name &quot;flow proof&quot; is first (?) used.  McMurray's flow proofs consist&#xA;  of statements, joined by arrows; each arrow is labelled with a number,&#xA;  and under the flow proof the numbers are associated with &quot;reasons&quot;. &#xA;  As in Hallerberg, McMurray stresses that the law of modus ponens&#xA;  (which he calls the &quot;law of detachment&quot;) is made manifest through this&#xA;  representation.   &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Basinger (1979) is a response to McMurray (1978). &#xA;  Basinger reports that he has also used flow proofs, with a few key&#xA;  superficial differences in notation.  Interestingly, Basinger writes&#xA;  &quot;The flow proof format does seem to show graphically the 'flow of&#xA;  logic' in complicated proofs.  But overemphasis of this proof format&#xA;  may be just as bad as over-emphasis on the traditional formats...  It&#xA;  may be better to use several proof formats, with near-equal emphasis&#xA;  on each of them...  And certainly the paragraph proof should not be&#xA;  left out.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;My final paper in this section is Brandell (1994).  Despite its title (&quot;Helping &#xA;  students write paragraph proofs in&#xA;  geometry&quot;) the paper has much more to say about flow proofs than about&#xA;  paragraph proofs.  In fact, Brandell's approach to teaching paragraph&#xA;  proofs seems to be:  (1)  First, make a flow proof.  (2) Then, turn&#xA;  the flow proof into sentences and paragraphs.     This approach seems&#xA;  amazingly backwards to me.  The whole point of paragraph proof is that&#xA;  it's less rigorously formal, more like natural language.  (Isn't it?) &#xA;  To teach paragraph proof as just a highly-condensed form of flow proof&#xA;  is just weird.    Brandell's flow proofs have statements in boxes,&#xA;  linked by arrows, with the reasons labelling the arrows. &#xA;  Interestingly, he suggests as a learning tool that unnecessary&#xA;  statements could be included in the flow diagram; when the paragraph&#xA;  is finally written, those extraneous statements are jettisoned.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="29" LastActivityDate="2019-09-01T15:54:28.560" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16995" PostTypeId="1" CreationDate="2019-09-02T15:35:10.617" Score="4" ViewCount="367" Body="&lt;p&gt;To give an example of what I mean, I'll answer a similarly worded question: “is there a pre-calculus introduction to the derivative?” I would say yes, since there already are the ideas of a slopes of a line, the tangent and secant lines. The major difference in calculus is that it includes the limit into the equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, what about limits, and specifically the formal definition of a limit? What would prepare students for it? I suppose the absolute value would need to be included since we’re talking about being within values of epsilons and deltas, but what would be a good starting off point?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you had to teach the formal definition of a limit, how would you end the phrase, “As you know, from precalculus…”&lt;/p&gt;&#xA;" OwnerUserId="12626" LastActivityDate="2019-09-09T18:14:29.697" Title="Is there a pre-calculus introduction to the formal definition of a limit?" Tags="|calculus|precalculus|limits|absolute-value|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16996" PostTypeId="2" ParentId="16995" CreationDate="2019-09-02T17:56:49.603" Score="1" Body="&lt;p&gt;Perhaps what would best prepare a student for the formal &lt;span class=&quot;math-container&quot;&gt;$\epsilon{-}\delta$&lt;/span&gt; definition of a limit is an animation which shows both &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;&#xA;approaching &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\epsilon \neq \delta$&lt;/span&gt;, something like this&#xA;(apologies for the too-high speed):&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/Wmx21.gif&quot; width=&quot;300&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://a4509-958957.cluster90.canvas-user-content.com/courses/4509~19985/files/4509~958957/course%20files/lvosbury/CalculusI_Folder/CalcIexamplesExam1.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;(This would be more effective with a non-linear curve, but I couldn't quickly find such an animation.)&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-09-02T18:15:13.283" LastActivityDate="2019-09-02T18:15:13.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16997" PostTypeId="1" CreationDate="2019-09-02T20:18:09.033" Score="4" ViewCount="173" Body="&lt;p&gt;Why it is said that Finland has a particularly good education system, but Finland's performance on international mathematics competitions is quite often at relatively intermediate level? &lt;/p&gt;&#xA;" OwnerUserId="12629" LastEditorUserId="11" LastEditDate="2019-09-03T15:20:03.107" LastActivityDate="2019-09-03T15:44:48.777" Title="Finland's performance on international competitions" Tags="|mathematical-pedagogy|math-contests|finland|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16998" PostTypeId="1" AcceptedAnswerId="17001" CreationDate="2019-09-02T20:38:47.563" Score="8" ViewCount="111" Body="&lt;p&gt;I'm a new adjunct faculty member. I haven't taught math yet (I'm teaching a related subject for now), but I've been thinking about my approach to teaching Calculus I should I be asked to teach it in the future. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One idea that has come to my mind is the possibility of fewer, more difficult homework problems that represent &quot;real-world&quot; practice (e.g., maximum likelihood estimation) over doing hundreds of computational problems (like what one would do through, say, WebAssign or MyMathLab for homework problems). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have there been any studies that attempt something similar to what I've said here?&lt;/p&gt;&#xA;" OwnerUserId="1907" LastActivityDate="2019-09-02T23:42:01.707" Title="Studies on more simple problems, or fewer difficult problems?" Tags="|calculus|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16999" PostTypeId="1" CreationDate="2019-09-02T21:01:19.237" Score="8" ViewCount="544" Body="&lt;p&gt;I'm teaching freshman electricity and magnetism this semester, and as usual in this type of course, I will need to teach my students a lot of vector calculus before they see it in a math course. The usual way this works in most curricula in my country (US) is that students only learn the integral form of Maxwell's equations. This semester, I'm attempting to do more of a balanced presentation, so that students will have some clue as to the local form of these equations, and of the correspondence between the differential and integral forms. This is meant to be mostly at a pretty simple level, without a lot of emphasis on coordinate-based calculations. They will see Gauss's law next week (the second week of the semester), but will not see Stokes's theorem until the very last week of the semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this context, I'm looking for simple, easy examples of this kind of interplay of global and local perspectives in math. The simplest example I've come up with, to get across the idea that there can be local statements and global ones, is a linguistic one:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Xf7RU.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Xf7RU.jpg&quot; alt=&quot;linguistic example, MY KATS EATS RATS, with R backwards&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are three errors in the sentence, only one of which can be detected if you're looking through a keyhole and can only see one letter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A second example would be that if one of the kitchen staff at a fancy restaurant is stealing the gold-plated silverware, you can detect it either by seeing them put it in their pocket, or by seeing that the count of pieces at the end of the day is short. This is the discrete analog of the fundamental theorem of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a third, visual example very specifically applied to Gauss's law, I have this picture:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NJtdT.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NJtdT.jpg&quot; alt=&quot;field lines&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here a vector field in two dimensions is represented by field lines. If this is a vacuum, then Gauss's law is being violated. Such a violation at the global level (line escaping the large square) can always also be detected at the local level as well (line escaping the small square).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like there ought to be a nice example from some branch of mathematics that would interpolate between examples 2 and 3 in terms of obviousness or surprisingness. Basically I would like an example that is sort of interesting in its own right, and easy to verify once you see it, but that isn't like the silverware example in being kind of trivial and uninteresting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems like Euler's polyhedral formula is exactly right in spirit, and maintains some simplicity because it's entirely discrete. However, I feel like I would probably need too big a chunk of a class session to do justice to this example. Is there some simpler example from geometry or topology that demonstrates the ideas without getting into calculus?&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-09-02T23:16:30.910" LastActivityDate="2019-09-09T15:06:28.707" Title="Easy examples of correspondence between global and local, as preparation for Gauss's theorem and Stokes's theorem" Tags="|examples|vector-calculus|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17000" PostTypeId="2" ParentId="16932" CreationDate="2019-09-02T23:03:53.277" Score="3" Body="&lt;p&gt;I'd say the concept of zero, infinity, and the infinitesimal. Often they're thought of like numbers and this leads to calculations which don't make sense where, even if they work numerically, it's not actually correct since the process is wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With calculus, I think this would be the idea of the limit and everything that revolves around it, which is basically everything.&lt;/p&gt;&#xA;" OwnerUserId="12626" LastEditorUserId="12626" LastEditDate="2019-09-02T23:15:47.703" LastActivityDate="2019-09-02T23:15:47.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17001" PostTypeId="2" ParentId="16998" CreationDate="2019-09-02T23:29:42.100" Score="6" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If you want to find academic studies, do an academic literature search.  Didn't you learn to do this during your studies?  Even if the results show little, if you give us the parameters that you searched with, it helps others to respond/add. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also recommend you not to expect too much from formal education research.  Teaching is a complex affair, full of confounding factors.  Many academic studies are poorly controlled, biased (both statistical and colloquial terms), and have low population counts.  Sure, look for studies--why not?  But realize the state of the subject.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The impulse to do fewer, more difficult, problems is a frequent desire by skilled academics moving from research to teaching.  It tends to mimic their research.  And is more intellectually interesting (to them).  However, it ignores several possible traits of pedagogy that relate to more/simpler problems, such as progression and repetition.     (Would you learn judo by fighting a match first thing?)  Also realize that the typical student is weaker than his teacher (except maybe at Cal Tech) in terms of both aptitude and experience.  Don't be blind to this difference. So my advice is not to replace drill for &quot;project&quot; style questions.  Especially given your lack of either research or deep experience to support changing the traditional approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not to say there is no value (motivational or conceptional) in more complicated problems.  But make it the paprika in the goulash, not the meat and onions.  Also realize the benefit even of the harder problems towarsd the end of a homework section.  Even a &quot;simple&quot; (to you) word problem is complex for someone new to the techniques.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also realize humans are not computers.  We have to do things several times to learn/master them.  A computer can execute a program instantly and repeatedly, including one with a complex number of steps.  Humans need repetition.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Get some experience teaching the traditional approach before deciding to try something different.  Don't make your first rodeo an alternate rodeo.  Even if you still want to change, you'll know more about why/what to change.  But don't confound being a new instructor with teaching an alternate approach.  If it bombs, you won't even know if the fault was you or was the approach (one equation, two unknowns).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12630" LastEditorUserId="12630" LastEditDate="2019-09-02T23:42:01.707" LastActivityDate="2019-09-02T23:42:01.707" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17002" PostTypeId="2" ParentId="16999" CreationDate="2019-09-02T23:38:20.283" Score="5" Body="&lt;p&gt;Here are two similar ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) The shortest geodesic between two points on the surface of a polyhedron (generally) bends as it crosses edges, when viewed&#xA;globally in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;, but from the local point of view of an ant walking along the path, it is straight, i.e., straight when each crossed edge is unfolded flat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is commonly illustrated on a cube.&#xA;Here is a more complex example.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/VNLyU.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VNLyU.gif&quot; alt=&quot;ShortestPaths&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Shortest paths from upper left corner to each vertex. Image from &lt;a href=&quot;https://cs.smith.edu/~jorourke/ShortestPaths/Welcome.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;(2) Imagine an ant walking around the apex vertex in the left image&#xA;below. Locally it has no way of knowing whether it is exploring&#xA;a protrusion, or a dent (right image). The curvature is the same&#xA;at that central vertex, but the embeddings in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; are different.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/dgX7L.png&quot; width=&quot;360&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Image from &lt;a href=&quot;http://cs.smith.edu/~jorourke/DCG/&quot; rel=&quot;noreferrer&quot;&gt;Discrete &amp;amp; Computational Geometry&lt;/a&gt;, p.177.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;A bit of a stretch, but you could connect this to the difficulty&#xA;we have understanding the geometric shape of the universe.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-09-02T23:38:20.283" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17004" PostTypeId="1" AcceptedAnswerId="17015" CreationDate="2019-09-03T00:21:26.533" Score="14" ViewCount="2293" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\rightarrow a} f(x)=L$$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which way should students best get in the habit of?&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The limit of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, equals &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;The limit of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;, as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;The limit, as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;other (please provide)&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12626" LastEditorUserId="12626" LastEditDate="2019-09-17T20:50:12.273" LastActivityDate="2019-11-22T00:05:27.080" Title="How should students say in words the notation for a limit?" Tags="|undergraduate-education|calculus|limits|" AnswerCount="7" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17005" PostTypeId="2" ParentId="16999" CreationDate="2019-09-03T01:31:57.877" Score="7" Body="&lt;p&gt;One idea that comes to mind is the fact that a connected graph has a Eulerian circuit if and only if every vertex has even degree. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another idea that comes to mind is just the Fundamental Theorem of Calculus. If you have a rather boring video of a car's speedometer as the car travels from A to B, then you can figure out the total distance the car has traveled in getting from A to B. The &quot;list&quot; of speeds read off from the speedometer is local/instantaneous information. The net distance traveled is a global quantity.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-09-03T01:31:57.877" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17007" PostTypeId="2" ParentId="16997" CreationDate="2019-09-03T03:37:34.490" Score="1" Body="&lt;p&gt;My general idea of math results is that it only shows how good one is at that thing, and it's not very easily comparable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, you could be good at calculation and go extremely well in tests (that could be checked with a calculator) but be very weak on concepts. Likewise, you could be very good at concepts but not be good at, say, answering multiple choice questions. And, again, you could be good at multiple choice questions but then not know unique formulas and ingenious theorems that could help on a math contest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;People are good at different things and math, and especially evaluations in math, are different too.&lt;/p&gt;&#xA;" OwnerUserId="12626" LastActivityDate="2019-09-03T03:37:34.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17008" PostTypeId="2" ParentId="17004" CreationDate="2019-09-03T04:07:17.223" Score="0" Body="&lt;p&gt;1 &gt; 3 &gt; 2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I prefer the first wording.  You get more of the idea of  thinking about what x causes what y.  Yes, the second is equivalent, but it is awkward, to add the condition (approaching a) after the result.  Three is OK also.  Although I mildly prefer 1.  Maybe shorter.&lt;/p&gt;&#xA;" OwnerUserId="12630" LastActivityDate="2019-09-03T04:07:17.223" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17009" PostTypeId="2" ParentId="16995" CreationDate="2019-09-03T04:17:03.733" Score="1" Body="&lt;p&gt;Precalc courses differ a lot, so it's hard to say something is included or not.  After all there was a time when there didn't even exist a precalc course (different from a strong algebra 2, trig, and nalytic geometry sequence).  Then again there are some precalc course that spend a reasonable amount of time on theory of functions (and relations), inequalities, and include limits and even simple differentiation, anti-derivatives.  Basically an intro to calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sequence I had was:&#xA;*Pre-algebra (solving for single variable in first order equations)&#xA;*Algebra 1 (lotta x-y lines and maybe the quadratic equation)&#xA;*Geometry (lotta triangle proofs)&#xA;*Algebra 2 (logarithms, exponents, sequences, basically &quot;college algebra&quot;, except a very short intro to vectors (needed to support HS physics&#xA;*Trig (1 sem)&#xA;*Functions (1 sem)&#xA;*Analytic geometry&#xA;*AP Calculus (AB or BC)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The upper track kids did the sequence above in 7-12.  Taking &quot;freshman algebra&quot; as eighth graders and algebra 2 and trig being combined from 3 to 2 semesters in a course designed for that track.  The standard track started with algebra in year 9 (prealgebra in 8).  Finished with either functions (2nd semester senior year) or with analytic geometry (kids taking algebra 2 trig or maybe doubling up a semester somewhere...I know &quot;functions&quot; was required for AP Chem so that provided a driver for some kids who were not in 8th grade algebra.  I believe it was also acceptable to stop with geometry in year 10, for the lowest track.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;functions&quot; course corresponded to a strong pre-calc course as described above (but not some real analysis monstrosity).  And also maybe just gave some time/maturity/practice that would be helpful before going to the shock of college calculus (BC is very comparable to a classic college calc class).  Similarly for analytic geometry.  I mean in theory, you could go straight from trig to a classic calc course, with analytic geometry in the text also.  But I think you'd have a lot of attrition, given the sudden jump in difficulty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a single datum, the (strong) local public high school I went to in the 80s, did have limits in precalc.  And at a reasonably rigorous level, more than just &quot;derivatives are like tangents&quot;.  Talking about left and right hand limits and where a limit doesn't exist.  And the darned Le Hospital.  But without such a concentration that people become more interested in the rigor as opposed to the result). &lt;/p&gt;&#xA;" OwnerUserId="12630" LastEditorUserId="12630" LastEditDate="2019-09-03T04:36:52.190" LastActivityDate="2019-09-03T04:36:52.190" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17010" PostTypeId="2" ParentId="17004" CreationDate="2019-09-03T06:34:57.177" Score="3" Body="&lt;p&gt;I usually say&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;f(x) lähestyy L:ää, kun x lähestyy a:ta&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and sometimes I instead say it more colloquially as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;f(x):n raja on L, kun x on a.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The inverted&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Kun x lähestyy a:ta, f(x) lähenee L:ää&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is also fine and in use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These would correspond to 2) and 4) in English. I would avoid linguistic complexity, such as a side clause embedded in the sentence, since it might reduce clarity.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2019-09-03T06:34:57.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17011" PostTypeId="2" ParentId="16948" CreationDate="2019-09-03T10:03:31.163" Score="5" Body="&lt;p&gt;Absolute value is difficult for students because they have difficulty parsing and simplifying logical statements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the results of working with absolute value statements seem to actually &lt;em&gt;hide&lt;/em&gt; the inner mechanics of how the logic of an absolute value statement play out. Because of their piecewise nature, you must understand logical statements to work deeply with absolute values. Students often develop a superficial, rote approach to these expressions because they are simpler than actually describing what is going on (rote is also sufficient for most exercises that I have encountered).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best anecdote that I can offer was the following question, which was asked if me by a student in 11th grade: When you are solving the inequality abs(x-2)≤3, the solution is -1≤x≤5, which is an AND statement (-1≤x AND x≤5); but the solution process involves an OR statement (x&amp;lt;2 OR x≥2) to address both parts of the piecewise definition. Where does this change happen from OR to AND?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a question that really requires one to logically express the solution process clearly to answer (the answer is actually that the statement is &lt;em&gt;always&lt;/em&gt; an OR statement that just happens to be equivalent to the AND one... (x&amp;lt;2 AND x≥-1) OR (x≥2 AND x&amp;lt;5), where the outermost OR is the continuation of the two distinct possibilities from the solution process).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of the notions of the different &lt;em&gt;interpretations&lt;/em&gt; of absolute values are tools to aid students' abilities to intuit answers to problems involving absolute values, but do little to actually let students rigorously understand them. To do so, you would have to actually pick one as the definition and relate the others rigorously to that. This is an exercise that almost never happens in classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In truth, I think that students find absolute values difficult because the subject is legitimately difficult! Otherwise, why would we bother having so many ways to think about how to interpret such statements? These interpretations are shortcuts around the rather tedious legwork of working directly with the logic (my personal chosen definition). I think that most &lt;em&gt;teachers&lt;/em&gt; probably don't know absolute values as clearly as we would like them to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I offer the following questions as food for thought on the difficulty of absolute values:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When solving equations, as opposed to inequalities, it is more common in secondary school to find a countable number of solutions as opposed to an uncountable number, but the solution set for the equation abs(2x-2)-abs(x)=2-3x is [0,1]. Why can you get entire intervals of solutions to such an equation?&lt;/li&gt;&#xA;&lt;li&gt;How do you graph y=abs(x-abs(x-1))?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8275" LastActivityDate="2019-09-03T10:03:31.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17012" PostTypeId="2" ParentId="16997" CreationDate="2019-09-03T15:44:48.777" Score="4" Body="&lt;p&gt;&quot;Why it is said that Finland has a particularly good education system, but Finland's performance on international mathematics competitions is quite often at relatively intermediate level?&quot;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Because the first part is a comparison of population averages and the latter part is a comparison of population extremes.  If the distributions vary in shape, than you can't expect the inferences to be identical. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In addition, given there's a difference in population size, one can expect (all else equal), larger countries to perform better in Olympiads.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;P.s.  Your monicker is making me feel like quoting Highlander.&lt;/p&gt;&#xA;" OwnerUserId="12630" LastActivityDate="2019-09-03T15:44:48.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17013" PostTypeId="2" ParentId="17004" CreationDate="2019-09-04T00:10:30.573" Score="9" Body="&lt;p&gt;I say it the third way, for these reasons:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, from a notation point of view, the “&lt;span class=&quot;math-container&quot;&gt;$x\to a$&lt;/span&gt;” has to be written with the “&lt;span class=&quot;math-container&quot;&gt;$\lim$&lt;/span&gt;”, and no “&lt;span class=&quot;math-container&quot;&gt;$\lim$&lt;/span&gt;” can be written without it (without specifically saying what you mean by not having it), so it makes sense to put them together when you say it aloud. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, you could argue that &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\lim_{x \to a}$&lt;/span&gt; is an operation you perform on the function &lt;span class=&quot;math-container&quot;&gt;$f(x).$&lt;/span&gt; That is, you can change what &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; the limit is being found at and also what function you are doing it to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly the “&lt;span class=&quot;math-container&quot;&gt;$\lim_{x \to a} f(x)$&lt;/span&gt;” is a thing all by itself, and so it makes sense to say it all together.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="238" LastEditDate="2019-09-05T18:01:19.137" LastActivityDate="2019-09-05T18:01:19.137" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17014" PostTypeId="2" ParentId="17004" CreationDate="2019-09-04T01:20:12.410" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;As &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;First, we emphasize what is happening to the independent variable, then we explain the consequence.  I think that this phrasing is concise and easy to understand.  It is clean and efficient.  This is essentially (3), but I think that the sub-clause &quot;The limit...&quot; is unnecessary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, if we have a fixed function and want to consider limits at several points, it provides a consistent framework.  For example, consider the rational function&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ f : \mathbb{R}\setminus\{\pm 1\} \to \mathbb{R} : x \mapsto \frac{x+1}{x^2 - 1}. $$&lt;/span&gt;&#xA;As &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$\pm \infty$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.  On the other hand, as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt;, and as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is unbounded (in either the positive or negative direction, depending on whether the limit is taken from the left or the right).&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2019-09-04T01:20:12.410" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17015" PostTypeId="2" ParentId="17004" CreationDate="2019-09-05T13:24:19.297" Score="20" Body="&lt;p&gt;According to &lt;a href=&quot;http://englishlangkan.com/produk/E%20Book%20Handbook%20for%20Spoken%20Mathematics.pdf&quot; rel=&quot;noreferrer&quot;&gt;UCAR-10101: &lt;em&gt;Handbook for Spoken Mathematics,&lt;/em&gt; Lawrence A. Chang, Ph.D., page 38&lt;/a&gt;, a source for how to speak mathematics to sight-impaired students, we have:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to a} y = b$$&lt;/span&gt; is spoken as the &quot;limit as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;&quot;.  For the given expression, &lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to a} f(x) = L$$&lt;/span&gt; is spoken thusly: the &quot;limit as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is consistent with option &lt;span class=&quot;math-container&quot;&gt;$(3)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Alternatively, in analysis, from &lt;a href=&quot;http://www.mit.edu/~sanathd/analysis-notes.pdf&quot; rel=&quot;noreferrer&quot;&gt;ANALYSIS TAUGHT BY BJORN POONEN, NOTES BY SANATH DEVALAPURKAR&lt;/a&gt;, we have the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We write &lt;span class=&quot;math-container&quot;&gt;$$\lim_{x→a} f(x) =L$$&lt;/span&gt; to say that for every &lt;span class=&quot;math-container&quot;&gt;$\epsilon &amp;gt;0,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is eventually within &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;”.  This means that there exists &lt;span class=&quot;math-container&quot;&gt;$δ &amp;gt;0$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;|x−a|&amp;lt; δ,$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$|f(x)−L|&amp;lt; \epsilon$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;ADDED:  Please see &lt;a href=&quot;https://math.stackexchange.com/questions/35496/is-there-a-definitive-guide-to-speaking-mathematics?r=SearchResults&amp;amp;s=1|95.2213&quot;&gt;this mathematics.se post about &quot;speaking mathematics&quot;&lt;/a&gt;.  There are additional resources for those interested.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-09-07T13:39:18.617" LastActivityDate="2019-09-07T13:39:18.617" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17016" PostTypeId="2" ParentId="8252" CreationDate="2019-09-05T14:25:49.183" Score="1" Body="&lt;p&gt;My answer &lt;a href=&quot;https://matheducators.stackexchange.com/a/5556/11&quot;&gt;here&lt;/a&gt; to a different question has a link to some &quot;college-level contest&quot; problems that are &lt;strong&gt;arranged by topic&lt;/strong&gt; as you request. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue is that these contest problems are not really &quot;Algebra I&quot; exactly. So this may be helpful only to future readers, not to you personally, because they are sorted by topic just well enough that you will find that &quot;Algebra I&quot; is not well-represented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Topics in that link, as of 2019:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Induction&lt;/li&gt;&#xA;&lt;li&gt;Extremal Principle&lt;/li&gt;&#xA;&lt;li&gt;Inequalities&lt;/li&gt;&#xA;&lt;li&gt;Number Theory&lt;/li&gt;&#xA;&lt;li&gt;Modular Arithmetic&lt;/li&gt;&#xA;&lt;li&gt;Polynomials&lt;/li&gt;&#xA;&lt;li&gt;Pigeonhole Principle&lt;/li&gt;&#xA;&lt;li&gt;Sequences and Series&lt;/li&gt;&#xA;&lt;li&gt;Generating Functions&lt;/li&gt;&#xA;&lt;li&gt;Probability&lt;/li&gt;&#xA;&lt;li&gt;Games&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastActivityDate="2019-09-05T14:25:49.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17017" PostTypeId="2" ParentId="16999" CreationDate="2019-09-05T14:29:06.407" Score="2" Body="&lt;p&gt;Local-to-global is a big thing in algebra, starting with (Hasse-)Minkowski's result that if you have a solution to a quadratic form modulo all prime powers and in the reals, you also have one over the integers.  (See Theorem 2.4 &lt;a href=&quot;http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/MQ64047.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; for the Minkowskian statement without &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers, which is otherwise a bit difficult to find in a quick web search.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there are plenty of (simple) equations with solutions modulo &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, but no integer solutions.  There is a &lt;a href=&quot;https://mathoverflow.net/questions/47442/diophantine-equation-with-no-integer-solutions-but-with-solutions-modulo-every&quot;&gt;convenient list of these here&lt;/a&gt;, probably the easiest of which is &lt;span class=&quot;math-container&quot;&gt;$$(x^2-2)(x^2-17)(x^2-34)=0$$&lt;/span&gt;  Naturally, that example might be beyond/irrelevant to your students, but it's interesting that such things are possible.  Also, clearly if you have a integer solution you will have solutions modulo &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, so this is nontrivial.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-09-05T14:29:06.407" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17019" PostTypeId="2" ParentId="16984" CreationDate="2019-09-05T19:14:36.423" Score="0" Body="&lt;p&gt;If you're also interested in Network Science more broadly than Graph Theory, there is &lt;a href=&quot;https://sites.google.com/a/binghamton.edu/netscied/teaching-learning/network-concepts&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; list of &quot;Network Literacy: Essential Concepts and Core Ideas&quot; from the NSF-sponsored Network Science in Education group.&lt;/p&gt;&#xA;" OwnerUserId="12580" LastEditorUserId="1608" LastEditDate="2019-09-10T01:35:31.407" LastActivityDate="2019-09-10T01:35:31.407" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17020" PostTypeId="1" AcceptedAnswerId="17025" CreationDate="2019-09-06T13:15:44.957" Score="12" ViewCount="1493" Body="&lt;p&gt;In the math class I taught today I was asked a question, and I was unable to give a good answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem was as follows: A certain factory produces throat tablets. In each pack, there are from 48 to 52 tablets, with a given probability distribution in a table (I don't remember the exact probabilities; it was far from uniform). If we take two of these packs, what is the probability that there are 102 tablets in total?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The provided solution (as well as how I would've solved it myself) went on to say that there are three ways that this can happen:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The first box has 50 and the second box has 52&lt;/li&gt;&#xA;&lt;li&gt;The first box has 51 and the second box has 51&lt;/li&gt;&#xA;&lt;li&gt;The first box has 52 and the second box has 50&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For each of these three probabilities, the probability is calculated by multiplying the relevant probabilities. Then the three results are added up and that's the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One student asked me why there are three cases and not two. Why are the first and third cases separate? Why is there a first box and a second box, and not just two boxes?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another, more standard, example is when working with the sum of two thrown dice. Same thing here: Why can we declare that there is a first die and a second die (or a blue die and a red die)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that we can &quot;pretend to change&quot; the problem text to say that we pick first one box and count the tablets, and then pick the second box and count the tablets. But the problem as stated doesn't say that. According to the statement the two boxes are entirely equal. How can I justify breaking this symmetry? (Apart form saying &quot;See, it gives the answer you're supposed to get.&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this is not about the similar question in combinatorics. When asking &quot;how many ways are there to select three students from the class to form a committee&quot;, or something like that, I think many of them have understood (and I feel more confident in how to explain) when order matters and when it doesn't in such questions. And I am similarily confident in the probability questions that spring directly out form such scenarios (&quot;What is the probability that Alice and Bob are both chosen as part of the committee?&quot;).&lt;/p&gt;&#xA;" OwnerUserId="7624" LastEditorUserId="1853" LastEditDate="2019-09-09T23:33:22.787" LastActivityDate="2019-09-12T01:42:28.727" Title="Why do you need to distinguish between apparently identical objects in probability?" Tags="|probability|teacher-development|" AnswerCount="9" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17021" PostTypeId="2" ParentId="17020" CreationDate="2019-09-06T13:59:34.000" Score="3" Body="&lt;p&gt;The succinct answer is that there are two ways to choose &lt;span class=&quot;math-container&quot;&gt;$50$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$52$&lt;/span&gt; - if two boxes are chosen one could have &lt;span class=&quot;math-container&quot;&gt;$50$&lt;/span&gt; and other &lt;span class=&quot;math-container&quot;&gt;$52$&lt;/span&gt; or vice-versa - but this will be more apparent to a student if you write out the space of all &lt;span class=&quot;math-container&quot;&gt;$25$&lt;/span&gt; possible outcomes as a list of ordered pairs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Indistinguishable&quot; does not mean &quot;identical&quot;, just as &quot;isomorphic&quot; does not mean &quot;equal&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each pick of a box is the same as choosing a number from &lt;span class=&quot;math-container&quot;&gt;$\{1, 2, 3, 4, 5\}$&lt;/span&gt; (with some probabilities). So the problem is equivalent to the problem: what is the probability that two numbers chosen (repetition allowed) from &lt;span class=&quot;math-container&quot;&gt;$\{1, 2, 3, 4, 5\}$&lt;/span&gt; sum to &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;? Or equivalent to: what is the probability that the digits of a number with digits from &lt;span class=&quot;math-container&quot;&gt;$\{1, 2, 3, 4, 5\}$&lt;/span&gt; sum to &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;? But two indistinguishable boxes, each containing &lt;span class=&quot;math-container&quot;&gt;$\{1, 2, 3, 4, 5\}$&lt;/span&gt; are still not the same box - there are two boxes sitting on the table. The first digit chosen and second digit chosen are different digits. I could have called them the second digit and the first digit - that's a question of how I label them - but they are different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The terminology &quot;indistinguishable&quot; is misleading - it does not mean that one cannot physically distinguish the boxes - it means only that their contents are identified in some established way (&quot;isomorphism&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key thing here is really exchangeability. Let &lt;span class=&quot;math-container&quot;&gt;$X^{k}$&lt;/span&gt; be the set of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-tuples of elements of the set &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;. A probability distribution pn &lt;span class=&quot;math-container&quot;&gt;$X^{k}$&lt;/span&gt; is &lt;em&gt;exchangeable&lt;/em&gt; if it is invariant under permutations. In particular, any distribution that arises from &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; independent samples from &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is exchangeable. This exchangeability is the key thing. I have a canonical way of identifying the faces of two different dice, but they are still two different dice, and I can tell them apart. There are two ways to put them in order and I can call one first or second as I like - and nothing about probabilities depends on this labeling. However I decide to do it, I can get &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; on one and &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; on the other - and there are two ways this can happen because while indistinguishable, the dice are different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For convincing a student one can simulate the exercise - just write the numbers on pieces of paper - and make two selections with replacement - or better fill two shoeboxes with numbered papers and choose one from each. Better yet, one can do it on a computer.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2019-09-06T14:53:45.233" LastActivityDate="2019-09-06T14:53:45.233" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17022" PostTypeId="2" ParentId="17020" CreationDate="2019-09-06T17:14:09.290" Score="7" Body="&lt;p&gt;This is a very good question. The issue comes up frequently. I explain this using a toy model: throw two regular six-sided die. What is the probability that the sum is 3? With some physical modeling, you can become convinced that the answer is 2/36=5.5%, This corresponds to the two possibilities 1+2 and 2+1. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But why are these two possibilities distinct? We could run the experiment where one dice is red, the other blue. Now it becomes clear why {red-&gt;1,blue-&gt;2} is not the same outcome as {blue-&gt;1,red-&gt;2}. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the boxes of throat tablets in the original post, the boxes are different because they are composed of different atoms, which obey Fermi-Dirac statistics, not Bose-Einstein!&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-09-06T17:14:09.290" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17024" PostTypeId="2" ParentId="17020" CreationDate="2019-09-07T00:29:00.450" Score="2" Body="&lt;p&gt;I don't know from what principles you're teaching probability, but the answer I would give is the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When calculating probabilities, it is important  to remember the basic principles. We have a set of possible outcomes &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; and an event &lt;span class=&quot;math-container&quot;&gt;$E \subseteq S$&lt;/span&gt;. The probability that an outcome in our event is chosen is precisely &lt;span class=&quot;math-container&quot;&gt;$\frac{E}{S}$&lt;/span&gt;, *given that all outcomes in &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; are equiprobable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now when your student is unclear about something, ask them to write out all the possible outcomes. Depending on whether they choose to consider order they may have fewer or more. Then ask them to write out all possible outcomes in your event. Depending on whether they considered order they will have &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. Finally, ask them to tell you why they believe all events are equiprobable. If they chose not to consider order, they're in trouble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The advantage of this system is that it gives them a hard rule that they can always fall back on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; I do not necessarily advocate for the usage of the terminology I suggest here. If the idea of sets is new to them, then maybe explaining this with Venn diagrams is more helpful.&lt;/p&gt;&#xA;" OwnerUserId="12640" LastActivityDate="2019-09-07T00:29:00.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17025" PostTypeId="2" ParentId="17020" CreationDate="2019-09-07T00:55:11.280" Score="14" Body="&lt;p&gt;The answers provided here so far give lots of good tips but I think they're not addressing a key part of the question, which is &quot;&lt;em&gt;why do we need to count two events (50,52) and (52,50), instead of one event (50,52)?&lt;/em&gt;&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is that &lt;em&gt;you can do it either way&lt;/em&gt;. The textbook way is counting &lt;span class=&quot;math-container&quot;&gt;$$P(X=50 \cap Y=52) + P(X=52 \cap Y=50) + P(X=51 \cap Y=51) = P(X=50)P(Y=52) + P(X=52)P(Y=50) + P(X=51)P(Y=51) = P(X=50)P(X=52)\times 2 + P(X=50)^2$$&lt;/span&gt;&#xA;This approach assumes that the boxes are iid to express the task in terms of univariate distribution of a single box.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, the student's alternative is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$P(X+Y = 102) = P(one~box=50, another=52) + P(both=51)$$&lt;/span&gt;&#xA;This is entirely valid so far, and if you had a distribution defining probabilities of these events, you could continue. But the point of such tasks is usually that you are given a univariate distribution, or need to define it from a text-based setting (&quot;fair dice&quot;). It is this step that introduces the factor of 2, because there are two elementary outcomes that result in &lt;span class=&quot;math-container&quot;&gt;$(one~box=50, another=52)$&lt;/span&gt;.&lt;br&gt;&#xA;I suppose this could be demonstrated more clearly by a task where an initial part provides the joint distribution, but a second part requires to do the same but derive the distribution first.&lt;/p&gt;&#xA;" OwnerUserId="12639" LastActivityDate="2019-09-07T00:55:11.280" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17027" PostTypeId="2" ParentId="17020" CreationDate="2019-09-07T13:21:04.863" Score="1" Body="&lt;p&gt;You could explain that the first box always goes to Alice, while the second goes to Bob. Then it becomes obvious that Alice getting 50 and Bob getting 52 is different then Alice getting 52 and Bob getting 50, so much so that one of them could die if they get the wrong amount. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, Alice and Bob can both get 51 in only one possible way, and the effects on their throats remain the same even if the packages get switched by mistake in transport. That's why 51-51 isn't counted twice.&lt;/p&gt;&#xA;" OwnerUserId="10413" LastActivityDate="2019-09-07T13:21:04.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17030" PostTypeId="2" ParentId="17020" CreationDate="2019-09-07T13:50:19.217" Score="1" Body="&lt;p&gt;Challenge him with the following: let‘s think it through with a soccer match. In how many ways can the match end, such that the sum of the goals is 3? According to his logic only two possible ways (1, 2) or (0, 3) - I‘m pretty sure the fans won‘t agree. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same goes for the colored dice - how many ways can they sum up to 4? Thus what’s the probability for those 2 dice to sum up to 4? Does he think that removing the color changes the probabilty to get 4?&lt;/p&gt;&#xA;" OwnerUserId="12647" LastEditorUserId="12647" LastEditDate="2019-09-08T13:38:14.873" LastActivityDate="2019-09-08T13:38:14.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17032" PostTypeId="2" ParentId="17020" CreationDate="2019-09-07T20:53:02.547" Score="4" Body="&lt;p&gt;I don't think I can improve on juod's excellent answer, but there are a few points worth elaborating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, physical objects are &lt;strong&gt;always&lt;/strong&gt; distinguishable.  (I am here ignoring the phenomenon of identical particles in quantum mechanics, the issue there being that we really have to rethink the concept of physical object when talking about the electrons in a two-electron quantum state or the photons in a two-photon quantum state.)  So it's not actually &quot;pretending&quot; when we artificially introduce a distinction between the &quot;first pack&quot; and the &quot;second pack&quot;; a natural distinction already exists between physically different ojects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, one may, if one chooses, ignore the distinction between the two packs by forgetting which is which, or by not recording this information in the first place.  This requires that one frame the situation in terms of sets of two objects, and that one modify the mathematical description accordingly.  Making this choice will &lt;strong&gt;not&lt;/strong&gt; affect the answer you get.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third, there is an implicit assumption in the problem as stated that ought to be made explicit.  This assumption is that the joint probabilities, that is, the probabilities associated with various configurations of two packs, can be computed from the one-pack probabilities.  This really is an assumption, one which may or may not hold, depending on the details of the production process and the method of sampling.  For example, suppose that the the packaging equipment is such that if one pack is overfull, the next pack produced will have a higher probability of also being overfull.  And suppose that the sampling method is to examine two successive packs coming off the assembly line.  In this situation the probability of two overfull packs is not simply related to the probability of single packs being overfull.  In this problem, you are meant to assume that this isn't an issue, or that the sampling is done in such a way that the problem is ameliorated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fourth, how is it that even the single-pack probabilities are assigned?  In principle one might deduce these probabilities from a detailed inspection of the packaging equipment, but in practice the probabilities will be empirically determined.  Any real production process will only ever produce finitely many packs.  If the question is concerned with the packs produced in a particular time period, the probabilities boil down to ratios, &lt;span class=&quot;math-container&quot;&gt;$p_{48}=N_{48}/N$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$p_{49}=N_{49}/N$&lt;/span&gt;, and so on, where &lt;span class=&quot;math-container&quot;&gt;$N=N_{48}+N_{49}+\ldots+N_{52}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N_i$&lt;/span&gt; is the number of packs produced with &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; tablets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's now do the computation with and without distinguishing the packs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(with distinguishing) The probability the pack I open first has 52 tablets and the pack I open second has 50 tablets is &lt;span class=&quot;math-container&quot;&gt;$(N_{52}/N)(N_{50}/(N-1))$&lt;/span&gt;, and similarly for the other configurations.  The total probability is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\Pr(\text{102 tablets})=\frac{N_{52}N_{50}+N_{51}(N_{51}-1)+N_{50}N_{52}}{N(N-1)}\approx p_{52}p_{50}+p_{51}^2+p_{50}p_{52},&#xA;$$&lt;/span&gt;&#xA;where the approximation is valid when the numbers of packs are large.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(without distinguishing) The probability that the set of two packs contains a pack with 52 and a pack with 50 tablets is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\binom{N_{52}}{1}\binom{N_{50}}{1}}{\binom{N}{2}}=\frac{N_{52}N_{50}}{N(N-1)/2},&#xA;$$&lt;/span&gt;&#xA;while the probability that the set contains two packs of 51 tablets is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\binom{N_{51}}{2}}{\binom{N}{2}}=\frac{N_{51}(N_{51}-1)/2}{N(N-1)/2}.&#xA;$$&lt;/span&gt;&#xA;Combining gives an answer algebraically equivalent to the one before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your student's proposed method (combining the first and third cases; not distinguishing a first and second pack) gets the probability for case 2 right essentially &quot;by accident&quot;: there is a cancellation of two factors of &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; with the result that the joint probability of getting two packs of 51 equals the product of two one-pack probabilities.  No similar &quot;accident&quot; occurs in the case of one pack of 50 and one pack of 52: there is a factor of &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; in the denominator, but no corresponding factor of &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; in the numerator to cancel it.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastEditorUserId="1737" LastEditDate="2019-09-11T16:14:52.670" LastActivityDate="2019-09-11T16:14:52.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17033" PostTypeId="1" AcceptedAnswerId="17036" CreationDate="2019-09-07T21:36:47.827" Score="10" ViewCount="8789" Body="&lt;p&gt;This question made the rounds recently -&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$8÷2(2+2)=?$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I glanced at this, answered &quot;1&quot; and then saw the full article printed in the New York Times, &lt;a href=&quot;https://www.nytimes.com/2019/08/02/science/math-equation-pedmas-bemdas-bedmas.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Math Equation That Tried to Stump the Internet&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article concludes that 16 is the correct answer, citing that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$8÷2×4$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;when approached from left to right, results in 16. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My disagreement lies in the dismissal of the parentheses, as my explanation would be that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$8÷2(x+x)$&lt;/span&gt; would, as a first step, simplify to &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$8÷2(2x)$&lt;/span&gt; and then &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$8÷4x$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in which case, if I were to offer this last bit and asked for a value when x=2, few would argue that division comes first. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, my question - for those who agree with me, what is it about PEMDAS that misses this issue, that the number right before the parentheses multiplies the contents with a higher priority than the division to its left? How do we address that priority? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(If you agree with the articles' '16', I am happy to hear the reasoning.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The third option is also fine, that such a set of numbers and symbols is ambiguous, and a second set of parentheses is required for clarification. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. I don’t know how common it is outside the US. Or even in different locations within the US, for that matter. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-09-09T08:53:00.893" LastActivityDate="2020-03-04T03:31:08.600" Title="A PEMDAS issue request for explanation" Tags="|algebra|arithmetic-operations|" AnswerCount="12" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17035" PostTypeId="2" ParentId="17033" CreationDate="2019-09-07T22:47:49.980" Score="20" Body="&lt;p&gt;The issue is the implied multiplication in 8÷2(2). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2). If implicit multiplication has a higher priority than explicit operations, we would get what you want: 8÷4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hate these examples, because they lead people to think math is very silly. The conventions (of PEMDAS) are usually reflected in the typesetting conventions we use. You would never see someone using the ÷ symbol in a complicated expression like this. It would be written 8/2(2+2), and then it would feel right to do the division first. (I believe our subconscious has internalized PEMDAS as matching the typesetting, loose for + and -, and tighter for multiplication and division, and even tighter for exponentiation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to your 8÷4. If we write it as 8/4, can you see the ambiguity? Is that x on the top or bottom of the fraction?&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-09-07T22:47:49.980" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17036" PostTypeId="2" ParentId="17033" CreationDate="2019-09-08T00:09:37.843" Score="17" Body="&lt;p&gt;If the expression were, say, &lt;span class=&quot;math-container&quot;&gt;$48\div 4\times 12$&lt;/span&gt;, there would not be much disagreement (multiplication and division are performed from left to right).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KVUEI.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KVUEI.png&quot; alt=&quot;48dvt12&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But an expression such as &lt;span class=&quot;math-container&quot;&gt;$48\div 4(12)$&lt;/span&gt; results in disagreement because the parentheses could mean one of two different things:  a way of grouping or a way of multiplying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DlcVB.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DlcVB.png&quot; alt=&quot;48d4p12&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one interprets the parentheses as a set of grouping symbols, then the &quot;P&quot; in &quot;PEMDAS&quot; is used and the &quot;correct&quot; answer is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.  If one interprets the parentheses as a multiplication, then the &quot;M&quot; in &quot;PEMDAS&quot; is used and the &quot;correct&quot; answer is &lt;span class=&quot;math-container&quot;&gt;$144$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I know, there is no &quot;authority&quot; that declares which of these two interpretations is correct.  As you can see from the images above, even engineers working in the same calculator company do not seem to have a common interpretation.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2019-09-08T00:09:37.843" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17038" PostTypeId="1" CreationDate="2019-09-08T08:04:30.023" Score="1" ViewCount="107" Body="&lt;h1&gt;Regressive Learning&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;It's a really stressful situation. &lt;strong&gt;I can achieve but not retain expertise in maths problems.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;History&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;6 months back, I studied integration in Calculus at college. I learnt it all by myself as I was didn't attended college for days on end. I learnt it with great level of comprehension. In the beginning I solved all the exercises one by one, dealing with all kinds of varied techniques required by the various questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Over 6 months, I then progressed to the point that I could manage to do 85% of all the questions in a minute. Rarely did I take 1.5-2 minutes (on the really difficult ones). This mastery was achieved by reviewing and re-solving the exercises once a month for 5 months.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used to do all this timed practice like a race-against-time using a 1-minute hourglass.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;Amnesia&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Now, there came a Gap of 3 months between my college ending and university starting. Now I can't do any of the questions of integration except the simplest ones and even they would take me well over a minute.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This proves the point that &quot;practice&quot; and &quot;understanding&quot; is not all&#xA;  to learning maths. There must be something else too that teachers and&#xA;  students are overlooking.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;Future Concerns&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Now, unlike college, there will be 80 exercises instead of 8 at the university-level. What do I do, I cannot just practice maths-problems again and again and again every month &amp;amp; then fall &amp;amp; then go from prince to pauper after a couple of months gap in review (in terms of exam preparation).There are a lot of twisted techniques, a lot of rules and methods in solving problems just like a toolbox. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My question is this: Why does this happen to me &amp;amp; What can I do ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Any help in this regard is highly appreciated. Moreover, I believe I have come to the right place as the people here can surely help with their knowledge of mathematics, learning , education and pedagogy.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;A Note on Problem-Solving&lt;/h2&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Understanding&quot; means &quot;knowing what specific techniques or some&#xA;  combinations of them have to be applied in a specific orderly manner&#xA;  to solve the integrals&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Mathematical problem-solving is akin to programming, what really matters is proper, orderly application of logic and procedural operations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not so much about &quot;conceptual enlightenment&quot; &amp;amp; &quot;intuition&quot; &amp;amp; &quot;deep-learning&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since we are humans, we cannot store all these rules of if X, then Y, else Z.&#xA;Neither can we analyse all multiple possible approaches (brute-force method) without expending considerable time and effort. &lt;em&gt;That&lt;/em&gt;, my friends, is &lt;em&gt;The Problem&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12650" LastEditorUserId="12650" LastEditDate="2019-09-08T19:05:14.870" LastActivityDate="2019-09-09T02:15:07.083" Title="Retain problems and combat regression in learning" Tags="|mathematical-pedagogy|calculus|general-pedagogy|integration|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17042" PostTypeId="2" ParentId="17033" CreationDate="2019-09-08T15:07:36.037" Score="8" Body="&lt;p&gt;I am with you on this one. I feel like concatenation (implied multiplication) is of higher precedence than explicit division. For me &lt;span class=&quot;math-container&quot;&gt;$8:2x$&lt;/span&gt; means &quot;8 divided by 2 x'es&quot; - &lt;span class=&quot;math-container&quot;&gt;$8:(2x)$&lt;/span&gt; not &lt;span class=&quot;math-container&quot;&gt;$4x$&lt;/span&gt;. Replacing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$(2+2)$&lt;/span&gt; shouldn't change anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the formal answer is that it's undefined. There is no C for concatenation in PEMDAS. For me it should be PECMDAS.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;People in comments are suggesting &lt;em&gt;juxtaposition&lt;/em&gt; (positioning side by side) instead of &lt;em&gt;concatenation&lt;/em&gt; (joining strings together). PEJMDAS then. IMO both terms are very similar, but &lt;em&gt;juxtaposition&lt;/em&gt; is indeed more common in the math world.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I definitely don't agree that concatenation should be resolved to multiplication before applying PEMDAS. By such logic you could also start with resolving E and get &lt;span class=&quot;math-container&quot;&gt;$8:2^2=8:2×2=8$&lt;/span&gt; which is wrong. You are supposed to apply PEMDAS before resolving notation otherwise you are mixing order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A final but necessary note- just never write division inline, it's too ambiguous unless you divide by one term only at the very end of line. I always either divide under the bar or for simple cases just multiply by the inverse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Rethinking again I am convincing myself even more that this is not and shouldn't be answerable in terms of PEMDAS. PEMDAS is a tool to resolve sequence of operations out of a notation. If you add notation that is neither of PEMDAS, there is no answer. Doesn't even matter if it's concatenation, derivative, logarithm, applying function... Neither of that is answered by PEMDAS.&lt;/p&gt;&#xA;" OwnerUserId="766" LastEditorUserId="766" LastEditDate="2019-09-09T20:39:18.927" LastActivityDate="2019-09-09T20:39:18.927" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="17043" PostTypeId="2" ParentId="4351" CreationDate="2019-09-08T16:56:53.770" Score="-1" Body="&lt;p&gt;Another example is string theory as a candidate for the theory for quantum gravity. After numerous consistency checks (the so-called &quot;theoretical experiments&quot;) it still fails to be validated by the experimental data.&lt;/p&gt;&#xA;" OwnerUserId="12657" LastActivityDate="2019-09-08T16:56:53.770" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17044" PostTypeId="2" ParentId="17033" CreationDate="2019-09-08T18:50:27.077" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;My disagreement lies in the dismissal of the parentheses, as my explanation would be that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$8÷2(x+x)$&lt;/span&gt; would, as a first step, simplify to&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$8÷2(2x)$&lt;/span&gt; and then&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$8÷4x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It seems to me that you are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;ignoring the Left To Right rule,&lt;/li&gt;&#xA;&lt;li&gt;not making the implied multiplication between &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(2+2)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9544" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-09-10T01:06:03.987" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17045" PostTypeId="1" AcceptedAnswerId="17049" CreationDate="2019-09-08T19:15:51.823" Score="7" ViewCount="989" Body="&lt;p&gt;I would like to choose some random students in class to answer questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't want to make some students feel that they are targeted. So I used to use a software to choose random student name.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the problem is that who comes to class is a bit random. Many registered students do not show up. So the software often chooses name of student who is not there, which wastes a bit of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How do you choose students to answer questions?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2020-08-22T06:00:09.523" Title="How to choose random student to answer questions?" Tags="|class-participation|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17046" PostTypeId="2" ParentId="17038" CreationDate="2019-09-08T21:06:55.173" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I then progressed to the point that I could manage to do 85% of all the questions in a minute. Rarely did I take 1.5-2 minutes (on the really difficult ones).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It sounds like you were doing very simple, plug-and-chug problems.  For example, evaluate the integral from x = -3 to x = 4 of (4x³+3x²)dx.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problems that require detailed proofs, or examination of multiple aspects of a situation, require more than two minutes of work.  A &quot;solid understanding&quot; is evidenced by the ability to:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;explain and verify the inner workings of a technology,  &lt;/li&gt;&#xA;&lt;li&gt;notice the limitations of the technology, and  &lt;/li&gt;&#xA;&lt;li&gt;use the technology to solve problems one encounters.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Mathematical proofs are explanations and verifications.  They also provide many opportunities to notice the limitations of various theorems.  Can you re-construct a general proof for how a derivative is calculated?  For the derivative of a polynomial term?  For converting between various ways of expressing &lt;em&gt;e&lt;/em&gt;?  For the chain rule, product rule, and quotient rule?  If you never were able to do these things, I doubt that you had a &quot;solid understanding&quot; of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/14692&quot;&gt;Applications&lt;/a&gt; of mathematics to physics problems, engineering problems, and various real-world problems also provide opportunities to notice the limitations of various mathematical techniques.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2019-09-09T02:13:15.643" LastActivityDate="2019-09-09T02:13:15.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17047" PostTypeId="2" ParentId="17033" CreationDate="2019-09-08T22:27:27.290" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The third option is also fine, that such a set of numbers and symbols is ambiguous, and a second set of parentheses is required for clarification.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm going to suggest a variant of this which you might consider a fourth option: The &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; operator is at best confusing if not ambiguous, and should not be used. For a long time I actually thought it was a nelogism based on a fraction bar and placeholder dots, and derided it on that too, but apparently it does have much deeper history. Still, it's not used whatsoever by working mathematicians, engineers, physicists, etc. &lt;strong&gt;for very good reason&lt;/strong&gt;. The lesson here, in a math-education context, is that it shouldn't be used. Full fraction bars for division make it entirely unambiguous, and use of &lt;span class=&quot;math-container&quot;&gt;$/$&lt;/span&gt; is almost universally interpreted as binding looser than implicit multiplication (but should still probably be avoided in ambiguous contexts).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beyond that, this question makes great lesson for students in the potential ambiguity of notation, disagreements about meaning, and importance of &lt;strong&gt;defining&lt;/strong&gt; any notation you want to use for which there is not an obvious agreed-upon meaning.&lt;/p&gt;&#xA;" OwnerUserId="1146" LastActivityDate="2019-09-08T22:27:27.290" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17048" PostTypeId="2" ParentId="17033" CreationDate="2019-09-08T23:31:42.057" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Now, my question - for those who agree with me, what is it about&#xA;  PEMDAS that misses this issue, that the number right before the&#xA;  parentheses multiplies the contents with a higher priority than the&#xA;  division to its left? How do we address that priority?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The conventional order-of-operations in textbook math (or any math) simply don't prioritize juxtaposed multiplication over other types of multiplication. The ordering is simply (elementary operations only):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Exponents and radicals&lt;/li&gt;&#xA;&lt;li&gt;Multiplication and division&lt;/li&gt;&#xA;&lt;li&gt;Addition and subtraction&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Each stage associates left-to-right, and grouping symbols (parentheses) always prioritize whatever's inside them. One can also observe all kind of useful symmetries built in this list: Operations are in pairs, with inverses together on each line (really two sides of the same coin in each case). Operations on same-base-powers downshift one place in all cases (to exponentiate powers, multiply the exponents; to multiply powers, add the exponents, etc.). Operations distribute over any operation on line lower. Etc., etc. &lt;a href=&quot;http://www.madmath.com/2012/07/power-rules.html&quot; rel=&quot;noreferrer&quot;&gt;See more.&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this is the agreed-upon ordering, then the value of the expression in the OP is 16.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's say that we really, really want the value to be 1 instead. In this case we must add the extra, perceived (by the OP) rule that juxtaposed multiplication should come before other types of multiplying/division. We must insert a new stage:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Exponents and radicals&lt;/li&gt;&#xA;&lt;li&gt;Juxtaposed multiplications&lt;/li&gt;&#xA;&lt;li&gt;Other multiplications and division&lt;/li&gt;&#xA;&lt;li&gt;Addition and subtraction&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;With that, the result produced in the OP is 1. But note that all the previous symmetries are destroyed! We no longer dependably (and memorably) have two operations on each line, inverses are not entirely collected on one line, the &quot;shift down one place&quot; shortcut for operations on same-base powers is gone, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: you can design a rule which forces juxtaposed multiplications to happen at a special earlier stage. &lt;em&gt;But do you really want to destroy all the associated symmetries for that?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-09-08T23:31:42.057" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17049" PostTypeId="2" ParentId="17045" CreationDate="2019-09-09T00:28:52.383" Score="7" Body="&lt;p&gt;One of the classes I took that I remember most fondly was a Differential Equations class, in which the number of assigned homework problems equaled the number of students in the class. (Granted, there were only 18 students in the class.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The assigned classroom had blackboards on three sides of the room.  Typically, the classes would begin with the professor randomly assigning one assigned problems to each student in the class, after which we would all go to the board and write our solution to the assigned problem. No one was to &quot;sign&quot; their name to their solution, and if they had gotten stuck trying to complete the assigned problem, they could show their work, and write questions about how to proceed.  After doing so, the class would discuss the solutions and problem-solve any unsolved problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everyone, in other words, was targeted every class period, though, in fact, everyone felt responsible to come to class prepared. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not feasible in a lecture hall, of course.  In such large lectures, you can use your computer program, or preferably, Sue's suggestion of using &lt;span class=&quot;math-container&quot;&gt;$3\times 5$&lt;/span&gt; index cards for each student in the class. Proceed during lectures, using only the index cards not previously naming someone who was called upon.  Then, if time allows, start from the beginning.&lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2019-09-09T00:28:52.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17050" PostTypeId="2" ParentId="17038" CreationDate="2019-09-09T01:53:35.850" Score="4" Body="&lt;p&gt;&quot;Solving (routine) problems fast&quot; is a useful skill, but very far from what you really need. You have to be proficient in reading, understanding and criticizing proofs, come up with your own proofs. Be able to plan (and execute the plan) to attack ill-defined problems. Use tools, like a computer algebra system, Google or math.SE to ease over routine stuff/stuff you forgot/never learned.&#xA;Learn to learn efficiently. How to select what to study, find material on your level that agrees with your style.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="1147" LastEditDate="2019-09-09T02:15:07.083" LastActivityDate="2019-09-09T02:15:07.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17051" PostTypeId="1" CreationDate="2019-09-09T04:17:57.920" Score="12" ViewCount="1602" Body="&lt;p&gt;Examples. An &lt;strong&gt;indefinite integral&lt;/strong&gt; (or antiderivative) of &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int \cos = \sin.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;Edit: There has been much unexpected confusion with the above statement. I &lt;strong&gt;define&lt;/strong&gt; the above statement to mean precisely that an antiderivative of the cosine function (which has domain &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;) is the sine function (which has domain &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;). Or equivalently, the derivative of the sine function is the cosine function. This notation though is not what the question is about, so I hope you will not dwell on this. If you want to question the merits of this notation, that is perhaps for another question. If it really bothers you, you can just pretend the above is statement instead &lt;span class=&quot;math-container&quot;&gt;$\int \cos x = \sin x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\int \cos x \, \textrm dx = \sin x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\int \cos x \, \textrm dx = \sin x + C$&lt;/span&gt;. (I personally assign to these latter statements slightly different meanings. And yes, I do explain all of this very clearly and repeatedly to my students.)&lt;/sup&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;definite integral&lt;/strong&gt; (or &quot;area function&quot;) of &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\pi/2$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int^{\pi/2}_0 \cos = 1.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are distinct concepts. But by custom, we use the same symbol for them, because the Fundamental Theorem of Calculus tells us that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int^{\pi/2}_0 \cos = \left(\int \cos\right)\left(\frac{\pi}{2}\right)-\left(\int \cos\right)(0) = \sin \frac{\pi}{2}-\sin 0 = 1- 0=1.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This often causes students a great deal of confusion (see e.g. below &quot;Related (but distinct) discussions&quot;). In particular, students&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;don't see any difference between the indefinite and definite integrals;&lt;/li&gt;&#xA;&lt;li&gt;believe that integration is, by definition, the inverse of differentiation; and&lt;/li&gt;&#xA;&lt;li&gt;thus fail to understand that the significance of the FTC or see that it is more than a mere definition.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To reduce the above confusion, I have already tried to follow @PeteL.Clark's recommendation (in &lt;a href=&quot;https://matheducators.stackexchange.com/a/2364/378&quot;&gt;this answer&lt;/a&gt;) of using the term &lt;em&gt;antiderivative&lt;/em&gt; (rather than &lt;em&gt;indefinite integral&lt;/em&gt;) as much as possible.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I am now thinking of also trying the following approach:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;First introduce antidifferentiation. Never once mention the word &lt;em&gt;integral&lt;/em&gt;. Don't use &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; as our symbol for the antidifferentiation operator---instead, use some other symbol like &lt;span class=&quot;math-container&quot;&gt;$\square$&lt;/span&gt;. In which case, we'd write things like &lt;span class=&quot;math-container&quot;&gt;$\square \cos=\sin$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Introduce integration (i.e. the definite integral) as usual.&lt;/li&gt;&#xA;&lt;li&gt;Go through the FTC, explaining that integration turns out to be, in a certain precise sense, the &quot;same thing&quot; as antidifferentiation.&lt;/li&gt;&#xA;&lt;li&gt;Explain that because of the FTC (and custom), we'll now discard our &lt;span class=&quot;math-container&quot;&gt;$\square$&lt;/span&gt; symbol for the antidifferentiation operator and replace it with &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I don't think I've ever seen any writer using this approach (but please let me know if you have). So, what disadvantages might the above approach have?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Pete L. Clark also writes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At some point I slip up because after all we are using the same integral sign for both: I &lt;em&gt;almost&lt;/em&gt; wish we didn't. (Using almost identical notation for two &lt;em&gt;a priori&lt;/em&gt; incredibly different things which in the setting of the FTC become almost the same is one of the brilliant notational innovations of calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't quite understand the above quote. I understand that today, after centuries of use, it would be difficult and foolish to go against tradition and attempt to completely replace the &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; symbol for the indefinite integral/antiderivative with something else altogether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But why wouldn't it have been better if mathematicians had just decided to use a different symbol from the start? What is so brilliant about using the same symbol &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; for &quot;two &lt;em&gt;a priori&lt;/em&gt; incredibly different things&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that this has simply resulted in a great deal of confusion (at least among average students of calculus), without any obvious gain in convenience or understanding.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Related (but distinct) discussions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/5901/&quot;&gt;Explaining the symbols in definite and indefinite integrals&lt;/a&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/2338/&quot;&gt;Should we avoid indefinite integrals?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/4086/&quot;&gt;How useful/useless is the indefinite integral&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/2358/&quot;&gt;Which first, definite or indefinite integrals?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user12641" LastEditorDisplayName="user12641" LastEditDate="2019-09-11T01:24:36.247" LastActivityDate="2019-09-30T09:26:11.970" Title="A different symbol for the indefinite integral/antiderivative?" Tags="|calculus|notation|integration|derivative|" AnswerCount="5" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="17053" PostTypeId="2" ParentId="17033" CreationDate="2019-09-09T06:00:48.883" Score="4" Body="&lt;p&gt;This is one of the cases where a linguist has to admit that we don't actually &lt;em&gt;make&lt;/em&gt; the rules for a language.  We discover them, and codify them.  And there's always exceptions &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B079ZYZVBM&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;like&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;I before E&lt;/strong&gt;&lt;br&gt;&#xA;  Except when your foreign neighbor Keith receives eight beige counterfeit sleighs from feisty caffeinated weightlifters &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;English is &lt;em&gt;weird&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PEMDAS is a linguistic rule in the same vein.  There is no worldwide governing board that defines the typography and interpretation for these equations.  It's just how we got used to communicating with one another.  If you don't use PEMDAS, you increase the likelihood of miscommunication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, when it comes to multiplication after a division, in particular multiplication with an elided operator there are dialectical disagreements about the meaning.  Remember, all of these rules are designed to make communication easier.  Communication is easier without all those parentheses in the way, so if there is indeed an unambiguous way of parsing expressions without adding parentheses, we like to do so.  That's why we have PEMDAS in the first place.  It got rid of the need to write (2*x) + (3*y) = 0.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what about 1/xy?  There's an easier way to write (1/x)&lt;em&gt;y, mainly y/x, so that strongly suggests that the only reason one would write in this order is to imply 1/(x&lt;/em&gt;y). But then what about 2/3x?  That could be 2/(3x) or (2/3)x.  Writing fractions in front of a variable is an awfully common thing to do, so we typically don't burden it with extra parenthesis.  Some prefer to write that case as 2/3*x.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Elided operators seem to bind &quot;tighter&quot; than others, but not everyone agrees on this.  This tends to be the fundamental problem with this entire class of &quot;trick&quot; questions.  Most are very comfortable with agreeing upon &lt;span class=&quot;math-container&quot;&gt;$8\div 4\times x$&lt;/span&gt;, but get less comfortable with &lt;span class=&quot;math-container&quot;&gt;$8\div 4x$&lt;/span&gt;.  Situations like &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2x}$&lt;/span&gt; show up often enough to want to be able to phrase them with a short parentheses-free notation.  Whether this want is strong enough to change your interpretation is up to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what's the correct value of &lt;span class=&quot;math-container&quot;&gt;$8\div 2(2+2)$&lt;/span&gt;?  The correct value is a friend who's trying to be mean using ambiguous syntax's to look smart.  Alternatively, the correct answer is to go back to whoever wrote it and ask them to fix the ambiguous phrasing.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2019-09-09T06:00:48.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17054" PostTypeId="1" AcceptedAnswerId="17073" CreationDate="2019-09-09T08:56:51.797" Score="5" ViewCount="339" Body="&lt;p&gt;I am teaching final year high school students and needed to persuade them of the following fact:&lt;/p&gt;&#xA;&lt;p&gt;There exists an &lt;span class=&quot;math-container&quot;&gt;$n_0 &amp;gt; 0$&lt;/span&gt; such that for all &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;n_0$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$3(\log_2 n)^5 &amp;lt; \sqrt{n}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Plotting the functions is no use and so I would love a mathematical argument they could follow.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h1&gt;Update&lt;/h1&gt;&#xA;&lt;p&gt;The students know how to differentiate but are not confident with L'Hôpital's rule.  I thought about asking them to plug in values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; but it seems you need &lt;span class=&quot;math-container&quot;&gt;$n \geq 8760381590675189248$&lt;/span&gt; before the inequality is true!&lt;/p&gt;&#xA;" OwnerUserId="10432" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-09-27T08:51:59.233" Title="How to show $3(\log_2 n)^5 &lt; \sqrt{n}$" Tags="|secondary-education|" AnswerCount="7" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="17055" PostTypeId="2" ParentId="17051" CreationDate="2019-09-09T09:23:17.323" Score="12" Body="&lt;p&gt;First, yes, many teachers use that term, &quot;anti-derivative&quot; and &quot;integral&quot; only when a definite integral is in use. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For your examples of each, it seems to me that you offered a clear distinction between the two expressions, i.e. that the bounds is what what makes the integral definite. For me, that's where the explanation to students tends to reach a satisfying conclusion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For your desire to introduce a new symbol, please, don't. No matter how well intended, no good can come of it. If you use a textbook, it will confuse students who really need written material to follow the lectures. If you teach 100% from notes, they will still have an issue when &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a) moving on to the next course, and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) going on line for any further assistance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are already multiple symbols used for &lt;em&gt;derivative&lt;/em&gt;, with centuries of history. Best to not add to the confusion. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-09-11T12:32:42.870" LastActivityDate="2019-09-11T12:32:42.870" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17056" PostTypeId="2" ParentId="17033" CreationDate="2019-09-09T09:40:15.860" Score="4" Body="&lt;p&gt;I think (other than the fact that it's pretty much a deliberately ambiguous question) the thing that is missing is the concept of things being single terms, which ought to be evaluated first. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would interpret the expression as 1, because I'd consider 2(2 + 2) to be a single term that should be evaluated first before the answer is substituted back into the rest of the expression. If it had been written with a multiplication sign it wouldn't be a single term, so the answer would end up being 16. Similar to how 1 ÷ 2 would be treated differently to 1/2. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can also be expressed as implicit operations having higher priority than explicit ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I'm based in the UK, so likely to have come from a different perspective - I was certainly taught, when we got that far, that implicit had priority and thus you should treat terms as single items)&lt;/p&gt;&#xA;" OwnerUserId="12663" LastActivityDate="2019-09-09T09:40:15.860" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17057" PostTypeId="2" ParentId="17054" CreationDate="2019-09-09T12:48:09.970" Score="8" Body="&lt;p&gt;I would first try to convince them that they are &lt;strong&gt;capable of thinking about the problem&lt;/strong&gt;. Once you get past this hurdle, I think it will not be too bad.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To do so, start with a different example. Like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Show that there exists an &lt;span class=&quot;math-container&quot;&gt;$n_0$&lt;/span&gt; where for &lt;span class=&quot;math-container&quot;&gt;$n &amp;gt; n_0$&lt;/span&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt n &amp;gt; 100 + \log_7 n$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions to the students would be: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What kind of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is easy to plug in to the left side? (Perfect squares)&lt;/li&gt;&#xA;&lt;li&gt;What kind of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is easy to plug into the right side? (Powers of 7)&lt;/li&gt;&#xA;&lt;li&gt;Are there any perfect squares that are also powers of 7? (Yes, &lt;span class=&quot;math-container&quot;&gt;$7^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$7^4$&lt;/span&gt;, ...)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Try these. Try a bunch of them. The goal is to convince the students that they can play with the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then build them up to &lt;span class=&quot;math-container&quot;&gt;$n = 7^{2k}$&lt;/span&gt;. When you plug this in, you are asking whether for large enough &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$7^k &amp;gt; 100 + 2k$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is easier to believe, it happens for, for example, &lt;span class=&quot;math-container&quot;&gt;$k \geq 3$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For your example, &lt;span class=&quot;math-container&quot;&gt;$n = 2^{2k}$&lt;/span&gt; should work, but for beginning students, the realization that they can plug in &lt;span class=&quot;math-container&quot;&gt;$n = 2^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n = 2^4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n = 2^6$&lt;/span&gt; to see what is happening before abstracting to &lt;span class=&quot;math-container&quot;&gt;$n = 2^{2k}$&lt;/span&gt; would be my goal.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2019-09-09T12:48:09.970" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17058" PostTypeId="2" ParentId="16999" CreationDate="2019-09-09T15:06:28.707" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jordan_curve_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jordan Curve Theorem&lt;/a&gt;: If you have a closed curve, there is an &quot;inside&quot; and &quot;outside&quot;, and those are defined globally (two points are on different sides if &lt;em&gt;every&lt;/em&gt; path between them crosses the curve at least once), but one can test whether two points are on the same side with a somewhat local test of drawing a line between them. If the line intersects the curve an even number of times, they are on the same side, if odd, then they are on different sides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conservative Fields: Give them a diagram of a roller coaster, ask them to write an equation for the force at all points, then integrate over the whole path to get the ending velocity. If we pretend there's no friction, this should be the same as just using the difference in gravitational potential energy.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2019-09-09T15:06:28.707" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17059" PostTypeId="2" ParentId="17054" CreationDate="2019-09-09T15:11:48.343" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't know of an algebraic way of proving this--think you need calculus.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Just trying some numbers is a way to see that the gap between the functions widens with n (and still exists even for very low n).  Essentially the same point as just graphing the functions and/or their difference.  But useful to divide the problem into less than and greater than 2 (where the log base 2 hits 1, and also below 1, where the log goes negative, but radical stays positive).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I question &quot;why&quot; to push this for pre-calculus kids (or at least pre-pre-calculus).  This sort of topic belongs as a normal use of L'Hopital's equation.  There is no major loss in not exposing kids to it earlier.  If anything, maybe a gain in not confusing them.  Not every hill needs to be taken in a given year of the war.  Prioritize.  (If the kids know calc, show them with L'Hopital, of course.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you really get a hair up, you could find the minimum difference, as a max/min problem.  Take the derivative and set to zero.  It gives you a hairy equation in n--I don't think soluble algebraically.  But you can approximate it numerically.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12665" LastEditorUserId="12665" LastEditDate="2019-09-09T15:45:02.807" LastActivityDate="2019-09-09T15:45:02.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17060" PostTypeId="2" ParentId="17033" CreationDate="2019-09-09T15:40:14.150" Score="3" Body="&lt;p&gt;In the order of operations, one multiplication does not take precedence over another; all multiplications and divisions are performed from left to right.  A so-called &quot;concatenation&quot; with parentheses a(b) has the exact same meaning as &quot;a times b&quot;.  The parenthesis operation only takes precedence over other operations to evaluate what is inside.  After that, it becomes an M, and the P is done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is very unfortunate that Casio wasn't competent enough to parse an unambiguous context-free grammar correctly in one of their models.  That model should be recalled, because it is wrong.&lt;/p&gt;&#xA;" OwnerUserId="12667" LastActivityDate="2019-09-09T15:40:14.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17061" PostTypeId="2" ParentId="17045" CreationDate="2019-09-09T16:12:41.270" Score="6" Body="&lt;p&gt;When facing hard problems, use easy solutions!&lt;/p&gt;&#xA;&lt;p&gt;Step 1: Assign a number to each student in a systematic way (for example, from front-to-back, from-left-to-right attending to their seats)&lt;/p&gt;&#xA;&lt;p&gt;Step 2: Generate a random number between &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and the number of people in your class&lt;/p&gt;&#xA;" OwnerUserId="12399" LastEditorUserId="12399" LastEditDate="2020-08-21T09:53:54.303" LastActivityDate="2020-08-21T09:53:54.303" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17062" PostTypeId="2" ParentId="17033" CreationDate="2019-09-09T16:28:30.013" Score="2" Body="&lt;p&gt;Part of the difficulty is that order of operations is taught wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic rule of order of operations is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Operations are performed from left to right &lt;em&gt;unless&lt;/em&gt;...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first &lt;em&gt;unless&lt;/em&gt; is the existence of parentheses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now here's where another factor (pun unintentional) comes into play:  we write in lines.  Because of that, something has to come first as we read.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we should say is that multiplication and division are &lt;em&gt;coprecedent&lt;/em&gt;:  neither takes place before the other, at which point, the &quot;left to right &lt;em&gt;unless&lt;/em&gt;&quot; comes into play.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in &lt;span class=&quot;math-container&quot;&gt;$8 \div 2 \times 4$&lt;/span&gt;, we do the &lt;span class=&quot;math-container&quot;&gt;$8 \div 2$&lt;/span&gt; first, getting &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, then evaluating &lt;span class=&quot;math-container&quot;&gt;$4 \times 4$&lt;/span&gt;, getting &lt;span class=&quot;math-container&quot;&gt;$16$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, when we reduce order of operations to an acronym, we lump M and D into the same word, and since it's harder to pronounce PEDMAS, we say PEMDAS.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(A and S have the same coprecedence relationship:  &lt;span class=&quot;math-container&quot;&gt;$8 - 3 + 5 = 10$&lt;/span&gt;, not 0)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a more in-depth overview of order of operations:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/A3o555TzKb0?list=PLKXdxQAT3tCtutiLV9QYh_2lDuNphhhCP&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/A3o555TzKb0?list=PLKXdxQAT3tCtutiLV9QYh_2lDuNphhhCP&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11747" LastActivityDate="2019-09-09T16:28:30.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17063" PostTypeId="2" ParentId="16995" CreationDate="2019-09-09T16:36:10.717" Score="0" Body="&lt;p&gt;The closest approach (pun intentional) to a limit in precalculus is when you're dealing with asymptotes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/gOQffI21rpc?list=PLKXdxQAT3tCuJku9nTlRZgx_RjGZ7djMc&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/gOQffI21rpc?list=PLKXdxQAT3tCuJku9nTlRZgx_RjGZ7djMc&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The advantage here is that it really drives home the idea of the limit of a function as a prediction of what the function value will be.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Limits at finite values are harder.  You might be able to talk about rational functions, for example, but you'd have to really get students to understand that &lt;span class=&quot;math-container&quot;&gt;$\frac{x^{2} - 4}{x + 2} \neq x - 2$&lt;/span&gt;.  &lt;/p&gt;&#xA;" OwnerUserId="11747" LastActivityDate="2019-09-09T16:36:10.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17064" PostTypeId="2" ParentId="17045" CreationDate="2019-09-09T17:04:57.550" Score="6" Body="&lt;p&gt;Here are some options that might be feasible:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Use a deck of cards&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You'll need two identical decks of cards to do this. At the beginning of each class, hand out a card to each student from Deck 1. Then, when you want to choose a student, shuffle Deck 2 and pick a random card.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can either use a standard deck of playing cards, or make two sets of your own with just numbers on them. The second option means you can have any number you like, and also you can just hand them out in order and so know exactly how many people are there to choose from each time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Get them to log into a system/fill out a form&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they have devices with them, you could get them to log into a computer system (eg a Desmos activity or Kahoot) or fill out a form (eg a Google form) each class. Then you can take the list of people logged in and choose from those names randomly each time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One advantage of this is that you could make the filling out of the form part of the class. For example, you could ask them a formative question and collect responses from everyone, thus increasing participation from everyone. It could even make choosing a random student unnecessary, because you could choose from the several responses you have which you think the rest of the class needs to hear more about.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2019-09-09T17:04:57.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17066" PostTypeId="2" ParentId="17054" CreationDate="2019-09-09T17:58:38.400" Score="3" Body="&lt;p&gt;This may not be a satisfying answer, but you CAN do it graphically. If you draw &lt;span class=&quot;math-container&quot;&gt;$y= \frac{3(\log_2 x)^5}{\sqrt{x}}$&lt;/span&gt; in Desmos, then you can zoom out far enough to see that it begins to decrease after about x=20 000. Then you can REALLY zoom the x-axis out and zoom the y-axis in until you can see that the value drops below 1 after &lt;span class=&quot;math-container&quot;&gt;$9 \times 10^{18}$&lt;/span&gt;. &lt;a href=&quot;https://www.desmos.com/calculator/yb5ma3k7dj&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.desmos.com/calculator/yb5ma3k7dj&lt;/a&gt; The slow reveal here might be rather compelling to high school students actually.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might also be possible to convince them algebraically. Attempt to solve the inequality.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 3(\log_2(x))^5&amp;lt;\sqrt{x}\\ 9(\log_2(x))^{10}&amp;lt;x\\&#xA;9^\frac{1}{10}\log_2(x)&amp;lt;x^\frac{1}{10}\\&#xA;$$&lt;/span&gt;&#xA;Let &lt;span class=&quot;math-container&quot;&gt;$u=x^\frac{1}{10}$&lt;/span&gt; and then &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 9^\frac{1}{10}\log_2(u^{10})&amp;lt;u\\&#xA; 9^\frac{1}{10}\cdot 10\log_2(u)&amp;lt;u\\&#xA; \frac{\log_2(u)}{u}&amp;lt;\frac{1}{10&#xA;\cdot9^\frac{1}{10}}\\&#xA;$$&lt;/span&gt;&#xA;Since &lt;span class=&quot;math-container&quot;&gt;$\log_2(u)$&lt;/span&gt; grows &lt;em&gt;much&lt;/em&gt; slower than &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;, that means &lt;span class=&quot;math-container&quot;&gt;$\frac{\log_2(u)}{u}$&lt;/span&gt; gets smaller and smaller, so there will come a point where the left hand part is smaller than the right-hand part.&lt;/p&gt;&#xA;" OwnerUserId="1853" LastEditorUserId="1853" LastEditDate="2019-09-09T18:45:04.350" LastActivityDate="2019-09-09T18:45:04.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17068" PostTypeId="2" ParentId="17054" CreationDate="2019-09-09T18:27:08.320" Score="4" Body="&lt;p&gt;Since log base 2 appears, maybe look at the case &lt;span class=&quot;math-container&quot;&gt;$n=2^k$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$3(\log_2 2^k)^5$&lt;/span&gt; simplifies to &lt;span class=&quot;math-container&quot;&gt;$3k^5$&lt;/span&gt;, which is a polynomial in &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;. For the other quantity, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{n}$&lt;/span&gt; simplifies to &lt;span class=&quot;math-container&quot;&gt;$2^{k/2}$&lt;/span&gt;, which is exponential in &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;. Since exponential growth eventually dominates polynomial growth, the inequality is plausible. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-09-09T18:27:08.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17069" PostTypeId="2" ParentId="17033" CreationDate="2019-09-09T18:39:22.483" Score="5" Body="&lt;p&gt;This would be better as a comment but I don't have sufficient reputation. Daniel R. Collins asked for an example of a mathematical text writing something like &lt;span class=&quot;math-container&quot;&gt;$a/bc$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$a/(bc)$&lt;/span&gt;, and it might be useful as a counterexample to the strong claims that a mathematician would &lt;em&gt;never&lt;/em&gt; write such a thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On page 2 of &lt;em&gt;Geometry Revisited&lt;/em&gt; by Coxeter and Greitzer, in the context of the Law of Sines, they write that &lt;span class=&quot;math-container&quot;&gt;$\sin A=a/2R$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is the circumradius of the triangle &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt;. On the next page, they ask the reader to prove that the area of said triangle is &lt;span class=&quot;math-container&quot;&gt;$abc/4R$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So examples in pure math books by well-known mathematicians can be found, but the only reason I could easily point to this example is because I think it's the only one I've personally seen and I remember noticing the ambiguity right away. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;edit: I just happened to see the following expression on page 98 of &lt;em&gt;Basic Probability Theory&lt;/em&gt; by Robert Ash (free online):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{[6!/j!k!(6-j-k)!](1/6)^{j+k}(4/6)^{6-j-k}}{[6!k!(6-k)!](1/6)^k(5/6)^{6-k}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But since both numerator and denominator were written in a cleaner way right before this, and since there is only one sensible interpretation in the context of the problem, there is no ambiguity.  &lt;/p&gt;&#xA;" OwnerUserId="12671" LastEditorUserId="12671" LastEditDate="2020-03-04T03:31:08.600" LastActivityDate="2020-03-04T03:31:08.600" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17070" PostTypeId="2" ParentId="17033" CreationDate="2019-09-09T18:42:43.393" Score="2" Body="&lt;p&gt;At first I thought comments would suffice, instead of an answer, because there have been a couple good answers already. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it appears that there are a number of issues at hand, some of them regarding math education rather than mathematical notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, let me first make a couple notes about notation though, as part of the subject.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I would like to note, that the &lt;em&gt;original question could only arise in &quot;paper&quot; maths, which has the wonderful opportunity of its notation being able to be interpreted differently&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In computer programming, math expressions usually are not ambiguous at all. Every math expression is calculated the same way in the same programming language, and writing it one way you can only get one and the same answer with respect to operation precedence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The order of precedence is, as @Daniel-R-Collins wrote: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;all operations calculated started from left-to-right, subject to below precedence in order:&lt;/li&gt;&#xA;&lt;li&gt;operations within parenthesis taken as separate list of operations and calculated first, using this same set of rules.&lt;/li&gt;&#xA;&lt;li&gt;exponentiation (and some other operations less common in school equations, such as modulus)&lt;/li&gt;&#xA;&lt;li&gt;multiplication/division (usually denoted by &quot;*&quot; and &quot;/&quot; signs)&lt;/li&gt;&#xA;&lt;li&gt;addition/subtraction&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So, if you wanted 8÷2(2+2) to mean 16, you'd write it: &lt;code&gt;8/2*(2+2)&lt;/code&gt; = 16 (1)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you wanted it to mean 1, you'd have to write: &lt;code&gt;8/(2*(2+2))&lt;/code&gt; = 1 (2)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Computers don't care what you want. If you wrote (1), you'd always get 16, regardless of what you wanted [Note that there exist programming languages which do use eg. postfix notation for mathematical operations, such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Reverse_Polish_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reverse Polish Notation&lt;/a&gt;, which would write the above as:  &lt;code&gt;8 2 / 2 2 + *&lt;/code&gt; to get 16, and &lt;code&gt;8 2 2 2 + * /&lt;/code&gt; to get 1.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless &lt;strong&gt;I do not know any general purpose programming language, in which 8/2*(2+2) would be 1&lt;/strong&gt;. So at least this front has come to an agreement on the subject.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Next, as I wrote in comments already, there are five mathematical operations + parenthesis, which are described by the OP's PEMDAS rules. Five. &lt;strong&gt;There is no such mathematical operation as &quot;implicit multiplication&quot;. The only thing there is, is an implied multiplication &lt;em&gt;_sign_&lt;/em&gt; when writing mathematical equations on paper.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It can also be reasonably inferred, how this could have come about. Writing equations &lt;strong&gt;on paper&lt;/strong&gt; you would like to get rid of ·, which is a pesky little symbol that is taking space, seems to be more frequent than division, and is usually also tightly coupled to the expression, so it would frequently &lt;em&gt;be&lt;/em&gt; first in priority of expression. Also, on paper you do have the large horizontal bar for all your division needs, so you could simply stretch a luxurious &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{2(2+2)}$&lt;/span&gt; over your implied multiplication and avoid any confusion whatsoever.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a rule of thumb, if you turn something implicit into explicit, it should show what it was about. If you do not construct any imaginary &quot;implied multiplication&quot;, but take the &quot;implicit multiplication&quot; to simply mean &quot;implicit multiplication &lt;em&gt;sign&lt;/em&gt;&quot;, then what you get left from 8÷2(2+2) is exactly 8÷2·(2+2). &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Now we have come to the part of the matter, which keeps popping up in many comments. Forget about implicit multiplication, what about the division sign? &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As I read that in engineering (at least in some parts of the world), it is frequently expected that &lt;code&gt;1/something&lt;/code&gt; implies &lt;code&gt;1/(something)&lt;/code&gt;, using slash sign as a fraction bar. While I can't argue for or against it in particular field, I would be surprised if it didn't &lt;em&gt;only&lt;/em&gt; work for values, the correct form of which were &lt;em&gt;expected to be known&lt;/em&gt;. If you know that the modulating signal frequency in Hertz is equal to &lt;span class=&quot;math-container&quot;&gt;$$\frac{\omega_m}{2\pi}$$&lt;/span&gt; then it is expected that you could often write it as ωm/2π, and indeed it seems that people do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this agains seems to work in particular jargon of the field and &lt;strong&gt;on paper&lt;/strong&gt;. You can't willy nilly put long or short fraction bars when writing computer programs, therefore even engineers will have to use computer notation, when dealing with programming. Computer notation, again, to avoid ambiguity, deals with the matter simply by having a division sign that follows the same above listed PEMDAS rules. If you want to write &lt;span class=&quot;math-container&quot;&gt;$\frac{1/2\pi}{3+x}$&lt;/span&gt; on computer, you have to write &lt;code&gt;1/(2*pi)/(3+x)&lt;/code&gt;. No shortcuts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this leads me to the following conclusions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The way to calculate expressions such as 8/2(2+2) will feel more &quot;natural&quot; depending on how you were taught. In addition, in some fields, it is more expected to for division signs in electronic texts to be meant as long fraction bars on paper.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As described here and in @Daniel's answer it might be argued that it is more correct to use the form where 8/2(2+2)=16, simply because there are no no hidden expectations such as &quot;implicit multiplication&quot; and &quot;my division signs are in fact long fraction bars&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would be surprised if much of the world didn't expect their equations the 8/2(2+2)=16 way. The whole of the former USSR, for example, where nobody ever heard of implicit multiplication or fraction bars [we used &quot;:&quot; as division sign in notation and used parenthesis a lot more than fraction bars. Starting with some 5th grade it was more  frequent to see school examples as &lt;span class=&quot;math-container&quot;&gt;$15:(3+3)$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$\frac{15}{3+3}$&lt;/span&gt; ]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nevertheless, when it comes to math &lt;strong&gt;education&lt;/strong&gt;, the considerations should probably include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;whatever is &lt;em&gt;taught&lt;/em&gt; to people, they must be made &lt;em&gt;aware&lt;/em&gt; of the possibilities of different interpretation to notations;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;especially if they are taught that 8/2(2+2)=1 , they should be made aware that no computers will understand it this way, because exposure to computers is ubiquitous nowadays (I am not even talking programming, try doing this in Excel);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;whatever is taught to people in school will probably need to be in line with the &lt;em&gt;existing textbooks&lt;/em&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12668" LastEditorUserId="12668" LastEditDate="2019-09-10T05:27:41.553" LastActivityDate="2019-09-10T05:27:41.553" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17072" PostTypeId="2" ParentId="17054" CreationDate="2019-09-10T11:57:41.327" Score="6" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Plotting the functions is no use&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It still might be worth it, after proving the cross-over by one of the several&#xA;techniques suggested, to show the students a plot of the functions using software that permits&#xA;arbitrary ranges.&#xA;It helps the intuition to see the &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; function flatten while &lt;span class=&quot;math-container&quot;&gt;$\sqrt{n}$&lt;/span&gt;&#xA;steadily rises.&#xA;Here I'm echoing @DavidButlerUofA.&#xA;&lt;hr /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/DdCfX.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DdCfX.jpg&quot; alt=&quot;LogSqrt&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;&#xA;Plotted in Mathematica.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-09-10T12:34:34.260" LastActivityDate="2019-09-10T12:34:34.260" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17073" PostTypeId="2" ParentId="17054" CreationDate="2019-09-10T12:56:57.647" Score="6" Body="&lt;p&gt;From&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{n\to\infty}\frac{\log_2 n}{n} = 0&#xA;$$&lt;/span&gt;&#xA;we get (for any numbers &lt;span class=&quot;math-container&quot;&gt;$a,b &amp;gt; 0$&lt;/span&gt;)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{n \to \infty}\frac{ab\log_2 n}{n} = 0&#xA;$$&lt;/span&gt;&#xA;which is to say&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{n \to \infty}\frac{a\log_2 (n^b)}{n} = 0&#xA;$$&lt;/span&gt;&#xA;But write &lt;span class=&quot;math-container&quot;&gt;$m = n^b$&lt;/span&gt; to get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{m \to \infty}\frac{a\log_2 m}{m^{1/b}} = 0&#xA;$$&lt;/span&gt;&#xA;now for any &lt;span class=&quot;math-container&quot;&gt;$c &amp;gt; 0$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{m \to \infty}\frac{a^c(\log_2 m)^c}{m^{c/b}} = 0&#xA;$$&lt;/span&gt;&#xA;To do this problem, we want &lt;span class=&quot;math-container&quot;&gt;$a,b,c$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$c/b = 1/2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c=5$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a^c = 3$&lt;/span&gt;.  That is, &lt;span class=&quot;math-container&quot;&gt;$c=5, b=10, a=3^{1/5}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In general, any power of &lt;span class=&quot;math-container&quot;&gt;$\log n$&lt;/span&gt;, however big, is less than any positive power of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, however small.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(\log n)^{1000000} &amp;lt; n^{0.0000001}&#xA;$$&lt;/span&gt;&#xA;for large enough &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2019-09-10T13:03:35.233" LastActivityDate="2019-09-10T13:03:35.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17074" PostTypeId="2" ParentId="17051" CreationDate="2019-09-10T14:51:48.063" Score="7" Body="&lt;p&gt;I'll echo other responses with the same: Do NOT introduce made up notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've made the effort in my Calculus courses to follow your outline while avoiding any new symbols. Similar to user20311's comment, when I first cover antiderivatives, all questions are phrased as either &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Find an antiderivative of &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;Find &lt;span class=&quot;math-container&quot;&gt;$F(x)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}F(x) = \sin(x)$&lt;/span&gt;, or &lt;/li&gt;&#xA;&lt;li&gt;Find &lt;span class=&quot;math-container&quot;&gt;$F(x)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$F'(x) = \sin(x)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In fact, using capitalization to indicate antiderivative was Newton's original notation. (In the examples I've seen, Newton usually was working with a polynomial &lt;span class=&quot;math-container&quot;&gt;$a(x)$&lt;/span&gt; and its antiderivative &lt;span class=&quot;math-container&quot;&gt;$A(x)$&lt;/span&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, as you indicate, after covering Riemann Sums, using the &lt;span class=&quot;math-container&quot;&gt;$\int_a^b$&lt;/span&gt; sign for definite integration, and covering the FTC, then introduce the &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; for antiderivative (which was, of course, Leibniz's notation).&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2019-09-10T14:51:48.063" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17075" PostTypeId="2" ParentId="17051" CreationDate="2019-09-10T18:07:54.620" Score="6" Body="&lt;p&gt;Before inventing new notation, it is very important to learn the accepted notation and teach it to your students correctly. The question contains the following claim:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int \cos = \sin$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which is &lt;strong&gt;not a meaningful statement&lt;/strong&gt;. It is either incorrect, or unclear. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This notational issue is not a nitpick: it is a critical issue that, if it goes uncorrected, will prevent your students from succeeding in future courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Following your notation, the student who attempts to evaluate this integral:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$A = \int \cos(2x) \mathrm{d}x$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is not only more likely to arrive at the incorrect answer -- they will also be unable to see how their actions are in conflict with the &quot;fact&quot; you have told them, namely that &quot;&lt;span class=&quot;math-container&quot;&gt;$\int \cos = \sin$&lt;/span&gt;.&quot; And this is only the beginning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would urge you, therefore, to not only &lt;strong&gt;avoid inventing any new notation&lt;/strong&gt;, but to think deeply about why you don't like the standard notation and ask questions to colleagues or this site about why the notation is what it is. Specifically in this case, you should write your fact as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int \cos(x) \mathrm{d}x = \sin(x) + C$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;so that students who use the rule incorrectly can be pointed at the detail that the fact is very clear that the integral of cosine of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\mathrm{d}x$&lt;/span&gt; is the sine of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; up to a constant, and that it makes no claims about the cosine of &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;. &lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2019-09-10T18:07:54.620" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17078" PostTypeId="2" ParentId="17020" CreationDate="2019-09-11T18:29:59.943" Score="2" Body="&lt;p&gt;You didn't disclose the level you teach.&#xA;When I first ran into this issue (i.e. the need to explain this), I went to the example...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have 2 coins. There are 3 possible outcomes, 2 heads, One Head One Tail, 2 Tails. Are they each equally likely, each 1 in 3? I then walk them through the physical experiment. Even though we flip two coins at once (as another member stated) the coins are different, and a 'tree diagram' would show that the first flip, whether H or T, eliminates HH or TT, but still leaves us the 50/50 chance of one of each. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 2 coin model makes both an experiment, or diagrams on the board, easy to tackle. From that, the expansion to higher numbers is easier to explain. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-09-11T18:29:59.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17079" PostTypeId="2" ParentId="17020" CreationDate="2019-09-12T01:42:28.727" Score="1" Body="&lt;p&gt;I think there are a lot of good answers here, but one point of view that might convince some students is to use a simulation - not of the toy problems e.g. with dice suggested, but with this one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apologies in advance for my terrible R code - &lt;a href=&quot;https://sagecell.sagemath.org/?z=eJw9jcEKwjAQRO-F_sMcd1EwCYog7MdIaGTBbiFtQCj990ZSncsbZhgmTsWWIUPg-i5NGaRQg3949xWj71C1NmgCYS4jxUHfai_KxTRROGNUk-u98vmRW2BmiMC7wMce8X_0cyePrbUVR3hptzt_Iicp&amp;amp;lang=r&amp;amp;interacts=eJyLjgUAARUAuQ==&quot; rel=&quot;nofollow noreferrer&quot;&gt;live version&lt;/a&gt; - here I assume a uniform discrete distribution on 48 to 52.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;counter = 0&#xA;for (i in 1:100000) &#xA;    {&#xA;    if ( sum(ceiling(runif(2, min=47, max=52))) == 102)&#xA;    { counter = counter +1 }&#xA;    }&#xA;counter/100000&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Occasionally I get .122 or .118, but usually it's very, very close to 12 percent, which is indeed &lt;span class=&quot;math-container&quot;&gt;$3\times .2^2$&lt;/span&gt;.  Add another zero to make it better (but then it actually takes a few seconds).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This sort of argument seems to help some people accept the correct solution to the Monty Hall problem, for what it's worth.)&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-09-12T01:42:28.727" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17080" PostTypeId="1" CreationDate="2019-09-12T11:46:48.713" Score="7" ViewCount="439" Body="&lt;p&gt;I am a mathematics educator from Lithuania and I have recently realized that there is no separate word in our language for the word &quot;numeral&quot;. To be more precise there is no term to describe &lt;em&gt;the collection of symbols that uniquely identify a number&lt;/em&gt;. We only teach children the difference between a digit and a number. I believe that for this lack of terminology children are very likely to believe that a number is nothing more than a scribble on a piece of paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have checked that in the English language the word &quot;numeral&quot; is used in two contexts:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;In linguistics to mean &#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a numeral (or number word) in the broadest sense is a word or phrase that describes a numerical quantity. &lt;a href=&quot;https://en.wikipedia.org/wiki/Numeral_(linguistics)&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia on &quot;Numeral&quot; in linguistics&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;In mathematics to mean a collection of symbols to uniquely identify a number. &lt;a href=&quot;https://en.wikipedia.org/wiki/Numeral_system&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia on Numeral system&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My question: in your country and language, is there a dedicated word to express the idea &quot;a collection of symbols to uniquely identify a number&quot;? If so, could you please write that word and also add whether it differs from the word the linguists use to mean &quot;a word that describes a numerical quantity&quot;? If the answer is no, it would still be interesting to hear it.&lt;/p&gt;&#xA;" OwnerUserId="12680" LastEditorUserId="64" LastEditDate="2019-09-12T22:39:24.087" LastActivityDate="2023-10-28T19:45:42.010" Title="The word &quot;numeral&quot;, is it being taught and does the word exist for it in your language?" Tags="|terminology|notation|language-use|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17082" PostTypeId="2" ParentId="17080" CreationDate="2019-09-12T17:42:42.630" Score="3" Body="&lt;p&gt;In English, mathematically speaking, &quot;number&quot; represents, in the  &quot;broadest sense,&quot; a word or phrase that describes a numerical quantity.&quot;  That is, &lt;span class=&quot;math-container&quot;&gt;$0.5, \frac 12, \frac 48,$&lt;/span&gt; one-half, etc., all represent the same &lt;em&gt;number&lt;/em&gt;.  That is, the &lt;em&gt;number&lt;/em&gt; referenced here in my answer is represented by each and every member of the family of equivalent representations I list.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/Number#Numerals&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia, (English language), &quot;numerals&quot;&lt;/a&gt; are defined merely as &quot;the symbols used to represent numbers&quot;. So each of member of the collection &lt;span class=&quot;math-container&quot;&gt;$0.5, \frac 12, \frac 48,$&lt;/span&gt; and one-half, etc., are numerals, each representing the same unique number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The distinction between &quot;number&quot; and &quot;numeral&quot; is not often made in the U.S. educational system. So it is not surprising to me that you have not encountered this distinction to be a significant educational focus in your experience here in the U.S.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Very roughly speaking, one helpful analogy, in linguistics, might be between &lt;em&gt;syntax&lt;/em&gt; and &lt;em&gt;semantics&lt;/em&gt; (compared to &lt;em&gt;numerals&lt;/em&gt; and &lt;em&gt;numbers&lt;/em&gt;, respectively): The term syntax refers to grammatical structure (mathematical symbols used, or numerals), whereas the term semantics refers to the meaning of the vocabulary symbols arranged with that structure (in mathematics, the number being represented by any of a number of equivalent representative numerals). &lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-09-12T18:34:42.400" LastActivityDate="2019-09-12T18:34:42.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17083" PostTypeId="1" AcceptedAnswerId="17085" CreationDate="2019-09-12T18:27:24.990" Score="10" ViewCount="416" Body="&lt;p&gt;I work as a tutor for linear algebra.  Students will often have a problem that requires them to row reduce, and they'll make an arithmetic mistake somewhere.  Often they'll ask for my help finding their mistake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I often will go through the operations myself, but it takes a bit of time.  Do you have any recommendations for an online app that can check each row operation?  Ideally the app would:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Display previous matrices, and the row operations that led to each.&lt;/li&gt;&#xA;&lt;li&gt;Allow you to &quot;go back&quot; if you made a mistake.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I've found a couple ones, like this one: &lt;a href=&quot;https://www.zweigmedia.com/RealWorld/tutorialsf1/scriptpivotold.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.zweigmedia.com/RealWorld/tutorialsf1/scriptpivotold.html&lt;/a&gt;.  However, it doesn't print the previous versions of the matrix, and it doesn't let you &quot;go back.&quot;&lt;/p&gt;&#xA;" OwnerUserId="12682" LastEditorUserId="12682" LastEditDate="2019-09-17T18:15:07.420" LastActivityDate="2021-02-13T18:41:08.963" Title="Can anyone recommend good software for verifying row reduction steps?" Tags="|linear-algebra|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17084" PostTypeId="1" AcceptedAnswerId="17094" CreationDate="2019-09-12T20:12:57.913" Score="11" ViewCount="579" Body="&lt;p&gt;A pre-calculus book  (Precalculus ed 1 By Miller and Gerken),  presents trigonometry in the following order:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1- Angles&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2- Trigonometric functions defined on the unit circle&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3- Right triangle trigonometry&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4- Trigonometric functions of any angle&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not see the advantage of such a presentation. Topic 2 seems less familiar than 3; and 4 is essentially same as 2.  There must be a pedagogical reason for this rearrangement but I do not know what it is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some other texts have somewhat similar presentations. In the text by Dugopolski (4 ed) we have 1- Angles, 2- Sine and Cosine on unit circle. In the text by Neal, Gustafson, Hughes we have 1- Angles, 2- Unit circle and trigonometric functions. (That is, no mention of  &quot;sine=opposite/hypotenuse&quot; before the unit circle is brought into the picture.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The order I am familiar with (for example from the text by by Beecher, Penna, Bittinger) goes as follows&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A- Angles&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B- Right triangle trigonometry&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C- Trigonometric functions of any angle (and explain uses of unit circle here)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: Did the shift in order come from research? If so, what is behind it? &lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2019-09-13T15:11:19.880" LastActivityDate="2019-11-15T22:41:12.267" Title="Pedagogical considerations behind current order of presentation of trigonometry" Tags="|mathematical-pedagogy|textbooks|trigonometry|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17085" PostTypeId="2" ParentId="17083" CreationDate="2019-09-12T20:21:24.850" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://www.maa.org/book/export/html/117002&quot; rel=&quot;nofollow noreferrer&quot;&gt;There was a nice app&lt;/a&gt; at this &lt;a href=&quot;http://www.dangries.com/Flash/RowReducer/RowReducer.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;site&lt;/a&gt; by &lt;a href=&quot;https://dangries.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dan Gries&lt;/a&gt;. Unfortunately it required Adobe/Flash.&lt;/p&gt;&#xA;&lt;p&gt;You could set the matrix size and enter your data. Then the site provides you with three operations: swap two rows, add a multiple of a row to another, or multiply a row by a number. Just the basic elementary row operations. Student is expected to know the Gauss or Gauss-Jordan operation. So that one can perform the process while the app does the arithmetic. A life saver for linear algebra students and teachers, without giving away the final answer!&lt;/p&gt;&#xA;&lt;p&gt;Edit:&lt;/p&gt;&#xA;&lt;p&gt;The following apps are not Flash-based:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://viewer.nclab.com/42d820651a1e4e3da76815f70d97b187&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pavel Solin&lt;/a&gt;, NClab. You need to &lt;a href=&quot;https://nclab.com/login/&quot; rel=&quot;nofollow noreferrer&quot;&gt;register&lt;/a&gt; first then follow this link.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://pnp.mathematik.uni-stuttgart.de/igt/eiserm/lehre/Gael/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michael Eisermann&lt;/a&gt;, Stuttgart University.&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2021-02-10T18:17:21.730" LastActivityDate="2021-02-10T18:17:21.730" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17086" PostTypeId="1" AcceptedAnswerId="17088" CreationDate="2019-09-12T20:45:53.157" Score="4" ViewCount="1482" Body="&lt;p&gt;Context: This is for introductory linear algebra course, near the beginning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a sort of &quot;exit survey&quot; after one of my lectures, I would like to ask my students to try and define what &quot;unique&quot; is from a mathematical perspective. I would then talk about that next time. However, I am not sure&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;whether that's a good idea; or&lt;/li&gt;&#xA;&lt;li&gt;how to phrase the question so that I don't get answers like &quot;it means you're really special&quot;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="344" LastActivityDate="2019-09-13T19:44:20.240" Title="Asking students to define &quot;unique&quot;" Tags="|undergraduate-education|assessment|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="17087" PostTypeId="2" ParentId="17051" CreationDate="2019-09-12T20:52:20.133" Score="2" Body="&lt;p&gt;An alternate notation that I made up for derivative and antiderivative  of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$f^{\downarrow}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f^{\uparrow}$&lt;/span&gt;.  I experimented with them when trying to explain or write steps of tableau method or ladder method for multiple integration by parts. (This is hallowed ground so I will be offending many, but I go ahead and drop the variable and its differential, keeping in mind that this is for students of Calculus 1!) By mixing the notations one can write&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\int (fg) &amp;amp;= f g^{\uparrow} -\int (f^{\downarrow} g^{\uparrow}) \\&#xA;\int (fg) &amp;amp;= f g^{\uparrow} - f^{\downarrow}   g^{\uparrow \uparrow} + \int (f^{\downarrow \downarrow} g^{\uparrow \uparrow}) \\&#xA;\int (fg) &amp;amp;= f g^{\uparrow} - f^{\downarrow}   g^{\uparrow \uparrow} + f^{\downarrow \downarrow}   g^{\uparrow \uparrow \uparrow}  - \int (f^{\downarrow \downarrow \downarrow} g^{\uparrow \uparrow \uparrow})&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can call them &quot;down&quot; and &quot;up&quot; operations due to what happens to the exponent of &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; in the operation. So you have &quot;double down&quot; and &quot;double up&quot;, and so on.  Useful for students who might have a hard time with steps in &lt;span class=&quot;math-container&quot;&gt;$\int x^n e^{ax} dx$&lt;/span&gt;. You can push it and write &lt;span class=&quot;math-container&quot;&gt;$f^{\uparrow n}$&lt;/span&gt;.  &lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="8571" LastEditDate="2019-09-16T23:28:32.620" LastActivityDate="2019-09-16T23:28:32.620" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17088" PostTypeId="2" ParentId="17086" CreationDate="2019-09-12T20:58:52.207" Score="10" Body="&lt;p&gt;To avoid misinterpretations, I'd give it to them in a mathematical sentence and ask them to explain what they believe 'unique' means in this context. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For lack of knowing where you are in the course, I might give them an exit ticket such as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consider the sentence, &quot;The equation &lt;span class=&quot;math-container&quot;&gt;$x^3 - 2x + 3 = 0$&lt;/span&gt; has a unique&#xA;  solution.&quot; What do you think the word &quot;unique&quot; means in this context?&#xA;  Contrast this with the ways you might use &quot;unique&quot; in standard&#xA;  English.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, this sentence has a problem in that the word 'unique' can meaningfully only be interpreted in one way (or perhaps I'm uncreative); perhaps give them a mathematical sentence which is not quite so obvious.&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="8571" LastEditDate="2019-09-13T17:56:44.417" LastActivityDate="2019-09-13T17:56:44.417" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17090" PostTypeId="1" CreationDate="2019-09-13T00:08:52.140" Score="3" ViewCount="152" Body="&lt;p&gt;This book has been recommended to me by a guiding hand/mentor mathematician when I showed interest in his field, and he seemed to really like this book however he doesn't really know me and my mathematical maturity (could be over-estimating or under-estimating). &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Extra details:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read a little of Béla Bollobás' random graph theory and could follow perhaps a quarter of what I'd seen. I still mix up &quot;Hamiltonian Path&quot; and &quot;Eulerian Path&quot;, so I'm wondering if I need to re-master my intro. discrete maths content before reading The Petersen Graph, or could I just dive in and look up whatever I'm dodgy on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm aware that some books have innocuous titles like &quot;Graph Theory&quot; and turn out to be extremely dense and terse, taking several courses before you can make much progress. I'm a dodgy undergrad who's a bit rusty, and never gotten into combinatorics. I'm not quite expecting this book to be a transformative experience, but I hope for considerable enrichment (are these expectations too high?). I've done the basics of complex analysis, group theory and linear algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've perused the first pages of many maths books, only to be left in the dust when things go from 1 to 100 real fast. Is there a good way to see whether this might happen without actually reading the book until you hit a wall (not enough background material for instance)? If one doesn't know the particular books in question, it can be tricky to open Spivak Calculus expecting a watered-down high-school style expose rather than an intro to analysis. &lt;/p&gt;&#xA;" OwnerUserId="3252" LastEditorUserId="672" LastEditDate="2019-09-15T09:46:01.667" LastActivityDate="2019-09-15T09:46:01.667" Title="Is &quot;The Petersen Graph&quot; by Holton and Sheehan a good text for graph-theory students? Any requisite knowledge for mastering the text?" Tags="|textbooks|prerequisites|graph-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17091" PostTypeId="2" ParentId="16889" CreationDate="2019-09-13T00:20:12.393" Score="2" Body="&lt;p&gt;Ugh.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are few things worse than having to prepare different versions of an exam for different cohorts.   It's almost impossible to make the exams comparable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A better solution is to make the exams &quot;calculator independent.&quot;  In other words,  it should make no difference if you have a fancy calculator or none at all. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's an example (I've used something like this on assignments): given the value of a function at a few points (but not the function itself), approximate the integral.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or: given the graph of the derivative, find where the local max and min of the function exists. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I haven't taught vector calculus, so I don't have good examples off the top of my head,  but you get the idea: the questions don't have anything you can input into a calculator,  so it doesn't matter if you're using one)&lt;/p&gt;&#xA;" OwnerUserId="11747" LastActivityDate="2019-09-13T00:20:12.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17092" PostTypeId="2" ParentId="16882" CreationDate="2019-09-13T00:26:11.833" Score="2" Body="&lt;p&gt;To answer the question of why you need f...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... have them consider the region between two graphs.   Unless you have a way to distinguish between the different y values,  you're going to be hopelessly lost. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you don't need to use f... you could use subscripts:  &lt;span class=&quot;math-container&quot;&gt;$y_{1}, y_{2}$&lt;/span&gt; (and in fact,  that's how graphing calculators handle it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it's nice to use &lt;span class=&quot;math-container&quot;&gt;$R(x)$&lt;/span&gt; to represent revenue and &lt;span class=&quot;math-container&quot;&gt;$C(x)$&lt;/span&gt; to represent cost, instead of having to remember that revenue was &lt;span class=&quot;math-container&quot;&gt;$y_{1}$&lt;/span&gt; and cost was &lt;span class=&quot;math-container&quot;&gt;$y_{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="11747" LastActivityDate="2019-09-13T00:26:11.833" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17093" PostTypeId="2" ParentId="16784" CreationDate="2019-09-13T00:29:13.057" Score="-1" Body="&lt;p&gt;I'm a big fan of number bonds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They're a nice visual representation of the idea that you're breaking the number apart (especially if you actually show how some objects go into one bubble and the rest go into the other).&lt;/p&gt;&#xA;" OwnerUserId="11747" LastActivityDate="2019-09-13T00:29:13.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17094" PostTypeId="2" ParentId="17084" CreationDate="2019-09-13T14:09:47.887" Score="3" Body="&lt;p&gt;I've noticed this trend as well, and it's baffling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only justification I can see for it is that one of the main topics in precalculus is &quot;functions,&quot; so they introduce the sine and cosine functions first.  Then they say &quot;Hey, guess what, these apply to triangles.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this strikes me as being completely backward:  they should introduce the sine and cosine &lt;em&gt;relations&lt;/em&gt; first, and then say &quot;Hey, we can generalize these to functions.&quot;  It's how I usually introduce the topic:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/nyXgUWoAka0?list=PLKXdxQAT3tCt84Mw-dP8Yjlv83MwET7f1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/nyXgUWoAka0?list=PLKXdxQAT3tCt84Mw-dP8Yjlv83MwET7f1&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11747" LastActivityDate="2019-09-13T14:09:47.887" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17095" PostTypeId="1" CreationDate="2019-09-13T15:30:43.163" Score="17" ViewCount="799" Body="&lt;p&gt;Have there been any major &lt;em&gt;content&lt;/em&gt; (not pedagogical) changes in the basic US high school mathematics curriculum since the mid-1990's? More specifically, if I wanted to become a tutor of high school math today as someone who learned his secondary-level math skills back in the 1990's, would a review of my previous curriculum provide a decent foundation, or would there be skills or concepts that young people are learning that I would likely never have heard of?&lt;/p&gt;&#xA;&lt;p&gt;As a background, I completed US high school mathematics (up through basic Calculus) in the 1990's. Back then, there was a heavy focus on hand-drawing graphs, along with some level of acceptance of black-and-white 8-bit Z80 graphing calculators as an aid to comprehension. I later completed university level coursework (at least through Linear Algebra) in the early 2000's. My understanding is that students nowadays are drawing fewer hand-graphs and using more and more technical modeling technology, but does this primarily represent changes in &lt;em&gt;pedagogy&lt;/em&gt;, or are there &lt;em&gt;concepts&lt;/em&gt; that high schoolers are learning today that would not have been covered in the 1990's?&lt;/p&gt;&#xA;&lt;p&gt;I'm not asking about becoming a full classroom teacher (which would obviously require a strong foundation in modern pedagogy). I'm looking for something along the lines of &amp;quot;Oh, nearly every Algebra II class nowadays covers Smooran's Theorem of Transdynamic Coordination, but it was first discovered in 2008 so you probably never studied it.&amp;quot;, or something slightly weaker. I'm not asking about general pedagogical changes like online discussion boards or video chat that we really didn't have access to.&lt;/p&gt;&#xA;&lt;p&gt;Examples:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Change in pedagogy&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Everyone's submitting assignments online now instead of writing them out by hand. Here's a tutorial you can go through to learn how to type algebraic equations into MathPedagogyPortal2022 XL for Windows. Why did you buy all of those #2 pencils and packs of graph paper? You must be ancient.&lt;/li&gt;&#xA;&lt;li&gt;Oh, everything's pretty much the same, except that we don't put a bar across sevens anymore and &amp;quot;subtraction&amp;quot; is now called &amp;quot;subtractification&amp;quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Change in content&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The National Council of Teachers of Mathematics rolled out a new curriculum in 2015 centered around the squiggle and ploop operators as fundamental building-blocks of all other mathematical structures. Students now must reduce most equations, graphs, proofs, and even numbers themselves to squiggle and ploop notation to receive full credit, and homework assignments and tests are always expressed in squiggles and ploops rather than with numbers, variables, operators, or equations like they were in your time. The theorems establishing squiggle and ploop are found in a 2008 paper by Smith and Jones (cite) which is now required reading in third grade and must be learned in its entirety in order to advance to fourth grade. You can go to this website (link) to earn your Adult Remedial Certificate in Squiggle and Ploop Math, don't bother applying to be a tutor until you have it!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7481" LastEditorUserId="7481" LastEditDate="2022-05-27T16:34:20.243" LastActivityDate="2022-05-27T16:34:20.243" Title="Would a 1990's educated person need additional content knowledge to tutor high school mathematics today?" Tags="|secondary-education|tutoring|history|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17096" PostTypeId="1" AcceptedAnswerId="17099" CreationDate="2019-09-13T16:46:56.410" Score="30" ViewCount="8286" Body="&lt;p&gt;I'm having a &quot;challenge&quot; in my geometry classes getting students to avoid using rulers as measuring devices in constructions.  As natural as that usage is, they're only supposed to use them to connect points to form line segments and extend line segments.  It's very understandable when they use them to mark off a line segment congruent to a given line segment.  But then they extend the idea to a sort of &quot;guess and check&quot; when they really should be using a compass instead.  At the same time, I don't want to criticize their ingenuity or suggest that tools shouldn't be used as fully as possible to solve problems when appropriate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what I'm leaning towards now is having 25 or so unmarked straightedges to have students use, so they are constrained to using them as Euclid intended.  I haven't ever seen them sold as such, so I'm wondering if people have any suggestions about common &quot;blank ruler-like&quot; products that could be used as substitutes.  (I would like to have an actual product for them to use, and not, say, using the edge of another textbook or an unsharpened pencil or something like that.)  Alternatively, I might get something like a simple piece of moulding cut into foot-long segments and sanded down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any experience or suggestions along these lines?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, here's an example of a brilliant but illegal construction I'd like to avoid seeing:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0p3bm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0p3bm.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, we bisect &lt;span class=&quot;math-container&quot;&gt;$\angle BAC$&lt;/span&gt; by swinging an arc with &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; at the center that intersects the angle at &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;.  Then, &lt;strong&gt;using the ruler to measure &lt;span class=&quot;math-container&quot;&gt;$DE$&lt;/span&gt;, construct &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; as the midpoint&lt;/strong&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$\overline{AF}$&lt;/span&gt; is our angle bisector.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;ETA: I visited my local craft store and found 10&quot; craft sticks in a pack of 50 for $7.  The edges are rounded like popsicle sticks, but they're long enough that students will be able to draw any line segments they want with just the straight part.  I think this is going to work really well for me.  Thanks to everyone for the suggestions!&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;ETA after the constructions unit: I'm really glad I steered away from the rulers.  For instance, when we discussed as a class how to determine whether every point on the perpendicular bisector of a line segment was the same distance from the two endpoints of the line segment, I asked &quot;What tool do we use in constructions to verify that two line segments are the same length?&quot; and everyone held up their compasses.  That surely wouldn't have happened if they had a ruler on their desk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few downsides along the way.  First, not being lacquered, the craft sticks got written on.  Sometimes, students would play with them in ways that made them more splintery than I'd like to have classroom supplies.  Finally, and most surprising, both the students and I were constructing clearly parallel lines that were clearly skew to the degree that I'm not sure they're as uniformly straight as you would think.  Not a huge deal, because I was verifying their work with a compass anyways, but it was a bit of a bummer.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2019-10-27T20:38:24.003" LastActivityDate="2023-08-18T17:47:19.047" Title="How to give my students a straightedge instead of a ruler" Tags="|geometry|constructions|" AnswerCount="12" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="17098" PostTypeId="2" ParentId="17096" CreationDate="2019-09-13T17:15:05.840" Score="11" Body="&lt;p&gt;Try shopping for a &quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B000I1THFA&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;paint guide&lt;/a&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These products have:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a straight plastic or metal edge&lt;/li&gt;&#xA;&lt;li&gt;a grippable portion that is not on the edge&lt;/li&gt;&#xA;&lt;li&gt;no ruler-like markings&lt;/li&gt;&#xA;&lt;li&gt;low prices.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0027AEIFY&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Drafting triangles&lt;/a&gt; also have unmarked edges that are intended to be drawn along, but their accurate angles are too useful in your intended application.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2019-09-13T17:28:50.903" LastActivityDate="2019-09-13T17:28:50.903" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17099" PostTypeId="2" ParentId="17096" CreationDate="2019-09-13T17:29:08.407" Score="18" Body="&lt;p&gt;I have lots of ideas, which, it turns out, aren't going to fit into a comment.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If you don't need something too long, but want something that is pretty uniform and consistent, &lt;a href=&quot;https://www.google.com/search?q=popsicle+sticks&quot; rel=&quot;nofollow noreferrer&quot;&gt;popsicle sticks&lt;/a&gt; (also sold as craft sticks) might work.  The are pretty durable, but relatively cheap and, ultimately, disposable.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Others&lt;sup&gt;[1]&lt;/sup&gt; have suggested &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B071V82ZZD&quot; rel=&quot;nofollow noreferrer&quot;&gt;paint stirrers&lt;/a&gt; (which you are often given &lt;em&gt;gratis&lt;/em&gt; at hardware stores when you are buying other things).  I originally dismissed this idea, as they tend to be fairly rough cut and made from soft pine, which may not hold an edge for very long.  However, some time with a belt sander (and possibly the application of some wood lacquer) might make this a very viable option.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=wood+molding&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wood molding&lt;/a&gt; or other small pieces of wood can be cut down to size.  I would suggest that the best strategy here is to go to the hardware / lumber store and see what they have.  You might find something you like.  This might be a little more expensive, but you can get the pieces cut down to whatever length you like (or do it yourself if you have the tools).  If you go this route, you should probably use a harder wood (such as oak); it will hold an edge longer.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Going very low tech, take a piece of &lt;span class=&quot;math-container&quot;&gt;$8\frac{1}{2} \times 11$&lt;/span&gt; inch printer paper, and fold it in half the long way two or three times (therefore obtaining a piece of paper which is &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt; inches long and one or two inches wide.  If you put a sharp crease into the paper, it should last for an hour or two.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Pieces of cardboard, &lt;a href=&quot;https://en.wikipedia.org/wiki/Paperboard&quot; rel=&quot;nofollow noreferrer&quot;&gt;paperboard&lt;/a&gt;, or &lt;a href=&quot;https://en.wikipedia.org/wiki/Card_stock&quot; rel=&quot;nofollow noreferrer&quot;&gt;cardstock&lt;/a&gt;.  Cardstock (which is, basically, a subset of paperboard) can be obtained for cheap from most office supply stores, or can be cut out of cereal boxes (take a cereal box, a good ruler, and (preferably) an X-acto knife, and cut the box into one inch wide pieces).  Larger pieces can be cut down to size with a paper cutter or X-acto knife.  If you need something more durable, cut several pieces to size, then glue them together.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are wealthy, there is always a stainless steel straight-edge, like &lt;a href=&quot;https://www.globalgilson.com/stainless-steel-straight-edges&quot; rel=&quot;nofollow noreferrer&quot;&gt;this guy&lt;/a&gt;. That's what grants are for, right?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;[1] Tip o' the hat to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/17096/how-to-give-my-students-a-straightedge-instead-of-a-ruler/17098?noredirect=1#comment44745_17096&quot;&gt;André 3000&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/17096/how-to-give-my-students-a-straightedge-instead-of-a-ruler/17098?noredirect=1#comment44750_17098&quot;&gt;Darrel Hoffman&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2023-08-18T17:47:19.047" LastActivityDate="2023-08-18T17:47:19.047" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="17100" PostTypeId="2" ParentId="17096" CreationDate="2019-09-13T17:39:14.807" Score="1" Body="&lt;p&gt;You could use a &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Beam_compass&quot; rel=&quot;nofollow noreferrer&quot;&gt;beam compass&lt;/a&gt; and only let them use one side of the compass as the straightedge.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-09-13T17:39:14.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17101" PostTypeId="2" ParentId="17086" CreationDate="2019-09-13T19:44:20.240" Score="2" Body="&lt;p&gt;I think that a linear algebra course is the perfect venue to have students develop their understanding of the mathematical concept of &quot;uniqueness.&quot; A formative assignment such as the one you suggest is a good, initial step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later on, as the course progresses, students might even come to appreciate the mathematical phrase &lt;em&gt;unique up to unique isomorphism&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an example in linear algebra. A vector space of dimension &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; over a field &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is unique, because any two such objects are isomorphic by a linear transformation. However, they are not unique up to unique isomorphism. If instead, you work in the category of finite dimensional &lt;strong&gt;vectors spaces with an ordered  basis&lt;/strong&gt; over a fixed field &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, then two vector spaces of the same dimension are unique up to unique isomorphism. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-09-13T19:44:20.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17102" PostTypeId="2" ParentId="17084" CreationDate="2019-09-13T20:56:57.057" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I find it easier to think about a single number (x or y of unit circle) than a ratio.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also helpful with angles greater than 180 or negative.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I learned it this way back in the early 80s...so it's not like some totally new fangled approach.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Just because it seems strange to you or you get some agreement on this message board, doesn't validate your opinion.  Maybe you'd feel as I do if you'd learned it this way first.  It's not like double blind trials of a drug.  It's just discomfort and opinion.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12690" LastActivityDate="2019-09-13T20:56:57.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17103" PostTypeId="2" ParentId="17090" CreationDate="2019-09-13T21:11:04.443" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;There are no Amazon reviews.  Doesn't seem like a popular title.  (Just an indicator, not a Euclidean proof...but a negative note.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The preface says that it is approaching teaching all of graph theory via this one graph as motivation.  Seems rather non-standard.  Again, not Euclidean proof, but a negative indicator.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are self studying AND a weak student (really either of those, but especially with both), I recommend to go with the easiest, clearest text possible.  You can always go back later and repeat with a harder one.  But start with easiest path.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I reccommend this text:  &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486678709&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709/&lt;/a&gt; Very easy--in fact the criticism is that it doesn't cover enough.  But that's a feature, not a bug, at this point.  Very well reviewed (in number and score).  Cheap.  Old and somewhat standard in approach.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12690" LastActivityDate="2019-09-13T21:11:04.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17104" PostTypeId="2" ParentId="17095" CreationDate="2019-09-14T09:19:34.517" Score="4" Body="&lt;p&gt;The most profound shift that the Common Core introduced is scattering statistics into the standards across most of the grade levels.  For instance, in my state (New York), high school freshmen are expected to know how to calculate linear regressions from a two-column table and discuss the degree of correlation between the two factors.  Also, upperclassmen should be able to take data from a normally-distributed variable and express a 95% confidence interval for it.  I would venture to say that a reasonable and motivated person could pick up everything in an afternoon from watching Khan Academy videos.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beyond that, though, there is a shift in the gray area between content and pedagogy.  That is, we are often less interested in calculating the answer (especially when we can get tech to calculate it for us) and now more interested in correctly interpreting the answer and applying it to make decisions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, consider a table showing for each of &lt;span class=&quot;math-container&quot;&gt;$n=10$&lt;/span&gt; students the amount of time a student spent studying for a test and the grade they got on the test.  The first part of the question asks us to calculate the correlation coefficient &lt;span class=&quot;math-container&quot;&gt;$$r=\frac{(n\Sigma x_iy_i)-\Sigma x_i\Sigma y_i}{\sqrt{n\Sigma x_i^2-(\Sigma x_i)^2}\sqrt{n\Sigma y_i^2-(\Sigma y_i)^2}}$$&lt;/span&gt; and then interpret the meaning of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;.  In our day, the first part would have been worth 8 points and the second worth 2.  Nowadays those would be reversed and the students would probably never even see the formula.  On the other hand, one would expect the student would at least understand and agree with the following response.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Based on a correlation coefficient &lt;span class=&quot;math-container&quot;&gt;$r=-0.96$&lt;/span&gt;, the data suggests a strong negative correlation between time spent studying for a test and the grade on that test.  However, it cannot be concluded from this data that the studying is the cause of the test grade.  A controlled experiment would need to be conducted to establish causality.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-09-14T09:19:34.517" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17106" PostTypeId="2" ParentId="17095" CreationDate="2019-09-14T15:31:17.980" Score="6" Body="&lt;p&gt;I gave an earlier &lt;a href=&quot;https://matheducators.stackexchange.com/a/6088/&quot;&gt;&lt;strong&gt;answer&lt;/strong&gt;&lt;/a&gt; [to a rather different question] in which I pointed to the &lt;a href=&quot;http://www.jmap.org/JMAP_REGENTS_EXAM_ARCHIVES.htm&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Regents Exam Archives&lt;/strong&gt;&lt;/a&gt;. One approach that you could take would be to look over a few tests from the 1990s as compared to now to see whether you sense disconnects. Of course, the Regents are rather geographically specific; so, I am not sure how well this will apply to your specific situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Others have commented a bit about the inclusion of some topics from Statistics (e.g., into Algebra II). You may find relevant questions to this end by looking, specifically, at the Algebra II Regents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The major shift that I have seen as pertains to the Common Core State Standards is the declaration of a focus on mathematical modeling. This is a feature of CCSS that contrasts with earlier policy or curricular documents, but, candidly, I am not aware of large-scale shifts by US K-12 institutions towards meaningfully teaching the content knowledge that would be necessary to support math modeling. You can look into some of the resources developed by &lt;a href=&quot;https://www.comap.com/product/textbooks/index.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;COMAP&lt;/strong&gt;&lt;/a&gt;, but I would be surprised if you encountered students who are taking classes that broach topics like voting, scheduling, cake cutting, or required the relevant content knowledge.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2019-09-14T15:31:17.980" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17107" PostTypeId="2" ParentId="17084" CreationDate="2019-09-14T15:38:10.037" Score="7" Body="&lt;p&gt;I disagree with the notion that the unit circle approach preceding the triangle approach should be, if it is contextualized historically, &quot;baffling.&quot; To this end, I suggest two pieces if you are wondering about how the unit circle gets itself into the pedagogical broaching of a subject that etymologically appears to be the study of triangles/three sided figures' measurement [&quot;trigonometry&quot;].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, Bressoud's &quot;Historical Reflections on Teaching Trigonometry&quot; [MAA &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/historical-reflections-on-teaching-trigonometry&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;; direct PDF &lt;a href=&quot;https://www.maa.org/sites/default/files/images/upload_library/46/NCTM/mt2010-09-106a.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;link&lt;/strong&gt;&lt;/a&gt;]. Citation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Bressoud, D. M. (2010). Historical Reflections on Teaching Trigonometry. &lt;em&gt;The Mathematics Teacher, 104&lt;/em&gt;(2), 106-112.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Second, for a longer read, there is a Ph.D. dissertation that I cited in an earlier &lt;a href=&quot;https://matheducators.stackexchange.com/a/1838&quot;&gt;&lt;strong&gt;answer&lt;/strong&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Van Sickle, J. (2011). A History of Trigonometry Education in the United States: 1776-1900 (Doctoral dissertation, Columbia University). &lt;a href=&quot;https://academiccommons.columbia.edu/doi/10.7916/D8G166T7&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastActivityDate="2019-09-14T15:38:10.037" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17108" PostTypeId="2" ParentId="17096" CreationDate="2019-09-14T16:30:15.073" Score="17" Body="&lt;p&gt;A cheap Venetian blind with thin plastic slats can be cut up with ordinary scissors and will yield many straightedges of any length you choose. The slight bow in the blade will flatten out when pressed against paper, and makes it easier for fumble-fingered people like me to pick up the straightedge from a flat surface.&lt;/p&gt;&#xA;" OwnerUserId="12695" LastEditorUserId="12695" LastEditDate="2019-09-14T16:46:22.213" LastActivityDate="2019-09-14T16:46:22.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17109" PostTypeId="2" ParentId="17096" CreationDate="2019-09-14T19:57:39.887" Score="6" Body="&lt;p&gt;Take an ordinary piece of paper and fold it. This must be the simplest way to produce an edge that is absolutely straight. A folded paper can be used to verify the straightness of a ruler, which depends on manufacturing precision.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You would be showing how to improvise a useful tool from readily available materials and you might be able to teach some geometry by considering why the folded edge must be straight.&lt;/p&gt;&#xA;" OwnerUserId="12696" LastActivityDate="2019-09-14T19:57:39.887" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17110" PostTypeId="2" ParentId="17096" CreationDate="2019-09-14T23:03:12.230" Score="2" Body="&lt;p&gt;Stock metal comes to mind if you want something slightly better than plastic/paper type things - in the u.s for example this can be acquired at hardware stores and cut to size relatively easily, here is a link for reference: &lt;a href=&quot;https://www.homedepot.com/p/Everbilt-3-4-in-x-48-in-Aluminum-Flat-Bar-with-1-8-in-Thick-801917/204273967&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.homedepot.com/p/Everbilt-3-4-in-x-48-in-Aluminum-Flat-Bar-with-1-8-in-Thick-801917/204273967&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5923" LastActivityDate="2019-09-14T23:03:12.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17111" PostTypeId="1" CreationDate="2019-09-15T02:54:53.933" Score="2" ViewCount="164" Body="&lt;p&gt;Are there any math journals or magazines with math problems for middle school or university ?&lt;/p&gt;&#xA;" OwnerUserId="12484" LastActivityDate="2019-09-15T23:44:47.993" Title="Are there any math journals or magazines with math problems for middle school or university?" Tags="|journals|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17112" PostTypeId="2" ParentId="17096" CreationDate="2019-09-15T04:07:29.417" Score="4" Body="&lt;p&gt;I hope you like this one...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;String&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2019-09-15T04:07:29.417" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17116" PostTypeId="1" AcceptedAnswerId="17118" CreationDate="2019-09-15T21:46:09.737" Score="-1" ViewCount="192" Body="&lt;p&gt;I'm not sure if this is the proper place to ask this question, but I figured I'll try anyway. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I have the option of taking a Calculus AP class, and am thinking of pursuing a career in either genetic counseling, occupational therapy, or special education, is it worth it? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would such a course help me out?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(An AP test is an exam that tests a person for college level knowledge of a particular subject, and if a person passes the exam, they will be given college credit as if they had actually taken the courses. It is often taken after a year of a high-school class in that subject.)&lt;/p&gt;&#xA;" OwnerUserId="12529" LastEditorUserId="238" LastEditDate="2019-10-09T21:48:39.860" LastActivityDate="2019-10-09T21:48:39.860" Title="Questions about the calculus AP" Tags="|calculus|exams|" AnswerCount="1" CommentCount="7" ClosedDate="2019-09-16T23:34:49.727" ContentLicense="CC BY-SA 4.0" />
  <row Id="17117" PostTypeId="2" ParentId="17111" CreationDate="2019-09-15T23:44:47.993" Score="5" Body="&lt;p&gt;The various journals listed in the MSE question &lt;a href=&quot;https://math.stackexchange.com/q/129326/&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; can help with the university portion. These include, but are not limited to, the journals:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The American Mathematical Monthly (AMM)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;College Mathematics Journal (CMJ)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematics Magazine&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Math Horizons (I especially like this journal)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The MSE link above contains descriptions of each; I would say that the AMM problems are generally quite difficult, and/but the Math Horizons problems could be fun to try at a variety of math st/ages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As mentioned in a comment, there are also some NCTM journals. Previously, there were three different journals that covered up through [something like] the second year of college; but, all of those journals have since been combined to form a single publication that launches in January 2020:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.nctm.org/Publications/Mathematics-Teacher_-Learning-and-Teaching-PK-12/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Mathematics Teacher: Learning and Teaching PK-12 (MTLT)&lt;/strong&gt;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I assume that this journal will contain problems, as its predecessors all did; to this end, you may wish to check back issues of NCTM's:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematics Teaching in Middle School (MTMS)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Mathematics Teacher (MT)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Each of these has, at the least, a set of calendar problems. You can also find a good number of articles around problems; for example, I contributed a selection of problems in an article for MT about roots of polynomials (&lt;a href=&quot;https://www.researchgate.net/publication/318100979&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;link without paywall&lt;/strong&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, although it is not a journal or magazine, if you are simply interested in problems, then there are lots of resources online. One possibility - since your original question mentions middle school - is to look at past competitions from MathCounts; you can find their problems archived &lt;a href=&quot;https://www.mathcounts.org/programs/competition-series/past-competitions&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. At the other end of the spectrum, you can find very challenging problems in the archive of Putnam Problems maintained &lt;a href=&quot;https://kskedlaya.org/putnam-archive/&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt;. Both of these linked sites also contain sample solutions.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2019-09-15T23:44:47.993" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17118" PostTypeId="2" ParentId="17116" CreationDate="2019-09-16T01:12:10.113" Score="3" Body="&lt;p&gt;I'm teaching AP Calculus for the first time this year, and my students are all asking the same question as you.  Here's what I tell them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're going to college next year, and if you will be required to take at least a semester of calculus there (which is a pretty safe bet for nearly everyone but do your own research), AP Calc will do one of two things for you:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You'll take the AP exam and get a good enough grade to place out of a semester of calculus.  Congratulations, you're already one class closer to your degree!&lt;/li&gt;&#xA;&lt;li&gt;You won't get a good enough grade in the AP exam or decide not to take it altogether.  You'll take Calculus in your first semester of college.  However, you'll retain enough from your AP class that the course will be a breeze for you.  Congratulations, you're now the best friend of all of your classmates and have one less thing to stress you out in your transition to college life!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(The second may not seem like a big prize.  Having been through college, honestly I think it's the better deal.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The downsides and how to cope with them?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It's not really an easy course, especially when not taught by a great teacher.  As people have said in the comments, lean on the Khan Academy and an AP study guide to help explain things when you didn't get either the skills or the motivation behind them from your class.  From a quick glance, it seems like you know how to ask good questions on Math SE, so that's there for you as well.&lt;/li&gt;&#xA;&lt;li&gt;Some students want to protect their GPA in their final year so as not to look bad when applying to college.  Honestly, colleges appreciate it when you take risks, and they'll see that you're taking an AP class.  And maybe your school has the sense (as mine does) to give a GPA boost for taking AP classes because they know that the reward of being challenged in a useful class is better than sleepwalking through an easy A elective.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-09-16T01:12:10.113" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17119" PostTypeId="1" CreationDate="2019-09-16T11:13:00.187" Score="3" ViewCount="121" Body="&lt;p&gt;I am looking for a list of ODEs to use as examples in the teaching of a numerical methods course for engineers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for first and second order examples - the more applied (to engineering) the better, and the harder (or impossible) to solve analytically the better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here I list some the examples that I do I know about (in no particular order - many have easily found exact solutions.):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Object under variable gravity and drag with a variable drag coefficient.&lt;/li&gt;&#xA;&lt;li&gt;Oscillators: undamped, subject to and not subject to external forces.&lt;/li&gt;&#xA;&lt;li&gt;Temperature in a rod: insulated (transient) and uninsulated.&lt;/li&gt;&#xA;&lt;li&gt;Pendulum: Real and Approximate&lt;/li&gt;&#xA;&lt;li&gt;Van der Pol Oscillator&lt;/li&gt;&#xA;&lt;li&gt;Newton Cooling&lt;/li&gt;&#xA;&lt;li&gt;Euler-Bernoulli Beam Equations&lt;/li&gt;&#xA;&lt;li&gt;Charge on a capacitor with and without inductance.&lt;/li&gt;&#xA;&lt;li&gt;Height of a liquid.&lt;/li&gt;&#xA;&lt;li&gt;Salt concentrations.&lt;/li&gt;&#xA;&lt;li&gt;Coupled temperatures.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="584" LastEditorUserId="584" LastEditDate="2019-09-16T13:45:28.600" LastActivityDate="2019-09-16T15:50:43.590" Title="Applied ODEs for Numerical Methods" Tags="|undergraduate-education|examples|concept-motivation|differential-equations|engineering-mathematics|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17120" PostTypeId="2" ParentId="17096" CreationDate="2019-09-16T12:28:35.850" Score="2" Body="&lt;p&gt;Take a first step by forbidding straightedges also. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This will make them understand that &lt;a href=&quot;https://en.wikipedia.org/wiki/Mohr%E2%80%93Mascheroni_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;every construction possible with straightedge and compass is also constructible with compass alone (Mohr-Mascheroni)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can introduce with &lt;a href=&quot;https://en.wikipedia.org/wiki/Napoleon%27s_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Napoleon's problem&lt;/a&gt;: Divide a circle in 4 equals arcs. First with straightedge and compass and show that the ruler method you mention is not accurate enough. Then you show that the last draw of a straight line, may be replaced by a (more precise) draw of the compass with the approriate radius you just have on hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This will open the appetite to replicate the basic Euclidean constructions. When your students are convinced of the Mohr-Mascheroni, then give back the ruler and mention that it is a far less precise instrument than a compass to &quot;copy&quot; a distance. A ruler is just good enough to mesure a distance, or to draw a straight line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point your students should prefer starightedge and compass to rulers.&lt;/p&gt;&#xA;" OwnerUserId="12713" LastActivityDate="2019-09-16T12:28:35.850" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17121" PostTypeId="2" ParentId="17096" CreationDate="2019-09-16T12:50:25.690" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://www.screwfix.com/p/carpet-cover-door-strip-aluminium-0-9m-x-36mm/71602&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aluminium carpet joining strips&lt;/a&gt; are another hardware store option.  They make nice straight edges for use with craft knives and would also serve well here.  They're a better shape than cheap metal bar stock.&lt;/p&gt;&#xA;" OwnerUserId="8180" LastActivityDate="2019-09-16T12:50:25.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17123" PostTypeId="2" ParentId="17084" CreationDate="2019-09-16T13:06:36.520" Score="0" Body="&lt;p&gt;It all depends on the background (i.e. the axiomatic assumtions your students are able to cope with).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recevied, in 1973, my trigonomtry in the same order, but at the time I was already quite familar with the right triangle geomerty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My father, who studied trigonometry for surveing purpose, not mathematics, did recived it after an introduction to computation of right and not right angle triangle. The trigonoeric circle was just a memory support to remeber the signs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I received again trigonometry for graduation. The order was : (1) polar coordinates, (2) holomorphic functions, (3) complex exponential. I do not remember of triangle geometry but of Fourier analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During my PHD, I was again exposed to trigonometry. This time the order was: (1) Lie group, (2) Linear representation, (3) Eliptic functions. I then discoved by myselft (4) the triangle geometry in not euclidian spaces which is tantamount as trigomonetry on an hyperbola, parabola or elipse (circle).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SO, I'd tend to say thjat the order does not really matter, as long as they can easily rembember the basic formulas. And, for this, in my opinion, the unit circle is the best memory support.&lt;/p&gt;&#xA;" OwnerUserId="12713" LastActivityDate="2019-09-16T13:06:36.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17124" PostTypeId="2" ParentId="17119" CreationDate="2019-09-16T13:11:38.883" Score="3" Body="&lt;p&gt;I would consider to add two items to the list, both from a systems slant:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Predator prey relations.  The behavior can be graphically investigated, but the actual solution function is not analytically soluble.  Any solid ODE book will cover this.  E.g. Speigel's&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0130400971&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Applied-Differential-Equations-Murray-Spiegel/dp/0130400971#customerReviews&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Xenon transients in a nuclear reactor.  Long term equilibria can be calculated (limited of time as infinity), but the function of time is not analytically soluble.  The standard problem is response to a power change (assume a homogenous reactor).  In actuality, very large civilian reactors can have strange oscillations in the geometry and time dimension.  But I would avoid this as it is a PDE and too complicated.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;See:  &lt;a href=&quot;http://mafija.fmf.uni-lj.si/seminar/files/2016_2017/Seminar_-_Xenon_oscillations_rev3.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://mafija.fmf.uni-lj.si/seminar/files/2016_2017/Seminar_-_Xenon_oscillations_rev3.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Edit adding derivation]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Xe is a reactor &quot;poison&quot; (absorbs neutrons in competition with uranium).  Iodine is a precursor of Xe (look at the periodic table.)  For each of them, the &lt;span class=&quot;math-container&quot;&gt;$\frac{\mathrm{d}}{\mathrm{d}t}$&lt;/span&gt;Concentration = production rate - loss rate.  Xe has two PR and two LR.  I (iodine) has one of each.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm{d}I}{\mathrm{d}t}=PR-LR=aPWR-bI$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Addition is correlated to power as a certain percent of the fissions generate this fission product.  Loss rate is just exponential decline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm{d}}{\mathrm{d}t}Xe=PR-LR=PR_1+PR_2-LR_1-LR_2$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm{d}}{\mathrm{d}t}Xe=cPWR+bI-dXe-ePWR$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Xe has an extra production term as it is produced directly from fission and also from Iodine decay.  It has an extra loss term as it goes away both by decay and by &quot;burnout&quot; (the poisoning reaction).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, where I write PWR, it really should be the local neutron flux.  Which is related to overall power but not perfectly linearly (there is leakage at the reactor border for instance).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, that the Xe situation while it has the trappings of mechE (power reactor) and physics (radioactive decay), it is really a chem E problem in sheep's clothing.  As you are looking at concentration.  Many feed/bleed problems with salt tanks and the like live in ODE world.  And I would consider to included one or two.  Have a lot of application in chemistry, environment, etc.  I also find concentration somewhat intuitive (more so than bending beams), but maybe that's me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comments:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not an expert on this topic, but FWIW, your (2) is the very standard core of analytical ODE course.  Unless the forcing function is quite unusual, the problem is soluble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe your (3) is typically a PDE, not ODE, problem.  Perhaps some of the others as well.  However there area  lot of good PLATE (not rod) heat transfer problems that are ODEs.  (I seem to recall them as analytically soluble, but there may be some that are not.)&lt;/p&gt;&#xA;" OwnerUserId="12715" LastEditorUserId="12255" LastEditDate="2019-09-16T15:50:43.590" LastActivityDate="2019-09-16T15:50:43.590" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17125" PostTypeId="2" ParentId="17051" CreationDate="2019-09-16T15:12:28.533" Score="0" Body="&lt;p&gt;It's the whole subject of the Bourbaki's book IV Integration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First generalize your notion of derivative: a derivative is &lt;strong&gt;any&lt;/strong&gt; function &lt;span class=&quot;math-container&quot;&gt;$d:A \rightarrow A$&lt;/span&gt; of the space &lt;span class=&quot;math-container&quot;&gt;$A=\mathcal{C} ^ \infty(\mathbb{R})$&lt;/span&gt; such that &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$d(f+g) = d(f)+d(g) $&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$d(\lambda f) = \lambda f $&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$\lambda \in C^{te}) \cong R \subset A$&lt;/span&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$d(fg) = f d(g) + d(f) g $&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The first two mean the operator &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is linear, the third is called the &lt;em&gt;Liebnitz rule&lt;/em&gt;. Please note that this definition is purely algebraic, and this was the goal of Weil: to be able to make differential geometry on &lt;span class=&quot;math-container&quot;&gt;$Z/pZ$&lt;/span&gt; (or their topological completions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main result is to show that every derivative is of the form &lt;span class=&quot;math-container&quot;&gt;$df = f'(x) d\mu(x)$&lt;/span&gt; for a suitable &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; (note that &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; is rarely the identity &lt;span class=&quot;math-container&quot;&gt;$\mu(x)=x$&lt;/span&gt;). More rigorously, the work aims to establish for which classes of algebras &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; the above is true. But suffice to say here that it is at least true for &lt;span class=&quot;math-container&quot;&gt;$A=\mathcal{C} ^ \infty(\mathbb{C})$&lt;/span&gt;, thus for &lt;span class=&quot;math-container&quot;&gt;$A=\mathcal{C} ^ \infty(\mathbb{R})$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is true, because by a fundamental theorem any holomorphic function has an antiderivative, so it is possible (but not easy) to find &lt;span class=&quot;math-container&quot;&gt;$\mu(x)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f = \int {d(f)(x) d\mu(x)}$&lt;/span&gt;. Note that the &lt;em&gt;measure&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$d\mu$&lt;/span&gt; is found after the derivative &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is given, by globalizing (this is the hard part) along all &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. Not moreover that &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; is rarely a function &lt;span class=&quot;math-container&quot;&gt;$\in A$&lt;/span&gt; (it is a &lt;em&gt;distribution&lt;/em&gt; defined except on a countable subset), nor unique (the two distributions can differ on a countable subset). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it exists. So your approach is perfectly legal. You can use any distribution, in particular &lt;span class=&quot;math-container&quot;&gt;$\mu(x)=x$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$d\mu(x)=dx$&lt;/span&gt;, and you can make this assumption implicit: all your formulas will be correct, as long as they only involves only algebraic relationships.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In facts, you will see that calculus is often made easier and clearer by using only the above stated rules. This is especially true in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R^n}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you'll have too look at the meaning of &lt;span class=&quot;math-container&quot;&gt;$d/dx$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\int (\cdot) dx$&lt;/span&gt;, you'll have many choices: Bourbaki's view of distributions generalizing functions, integral in the sense of Borel or Lebesgue, Liebnitz' infinitesimals, non-standard analysis (Robinson ultrapower or Nelson Internal Set Theory). There are all different semantics, which surface to the same syntax: any algebraic or calculus that can be made in one theory is feasible in the others and gives the same result(s).   &lt;/p&gt;&#xA;" OwnerUserId="12713" LastEditorUserId="8571" LastEditDate="2019-09-16T23:33:30.993" LastActivityDate="2019-09-16T23:33:30.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17126" PostTypeId="2" ParentId="17095" CreationDate="2019-09-16T15:28:30.983" Score="0" Body="&lt;p&gt;Some years ago, I did stumble on a book of exam of my grand-father, and was stunned by the problems: they were on subjects totally different as of today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two examples: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A tree of height 20 feet has a circumference of 3 feet. There is an arrowroot vine which winds seven times around the tree and reaches the top. What is the length of the vine? [Answer: 29 ft.]&lt;/li&gt;&#xA;&lt;li&gt;A wooden log is encased in a wall. If we cut part of the wall away, at a depth of 1 inch, the width of the exposed log measures 1 foot. What is the diameter of the log? [Answer: 37 in.]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of course these are variants of the 15th problem of the &lt;em&gt;Jiuzhang suanshu&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PLP89.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PLP89.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, as for your question: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Yes, I would have been able to solve these questions and train someone to do so. &lt;/li&gt;&#xA;&lt;li&gt;No: I would not have been able to invent the appropriate type of problems to prepare someone for the exam.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12713" LastEditorUserId="12713" LastEditDate="2019-09-16T15:37:59.703" LastActivityDate="2019-09-16T15:37:59.703" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17127" PostTypeId="2" ParentId="17096" CreationDate="2019-09-16T15:41:57.133" Score="1" Body="&lt;p&gt;Tongue Depressor - They are six inches long and you can get a pack of 100 on amazon for about five dollars.  You can also get them at your local Walmart or Hobby Lobby.&lt;/p&gt;&#xA;" OwnerUserId="12716" LastActivityDate="2019-09-16T15:41:57.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17128" PostTypeId="1" CreationDate="2019-09-16T18:07:07.830" Score="2" ViewCount="1690" Body="&lt;p&gt;I am looking for a math textbook. I'm 15 and I'd like to complete &#xA; algebra 2 geometry and perhaps something about probability/ number theory or trigonometry would be nice too. Later I wanna do precalculus and calculus. The important thing is I want the textbook to truly be challenging and going well beyond what's usually taught in high school (honestly the harder it is, the better). The main focus should be problem solving although they don't have to be aimed straight at competetive math. Also it doesn't have to be 1 textbook covering it all (not that there is one probably).&lt;/p&gt;&#xA;" OwnerUserId="12717" LastEditorUserId="12717" LastEditDate="2019-09-16T19:08:20.110" LastActivityDate="2022-07-25T11:22:20.810" Title="High school maths textbook for talented students" Tags="|secondary-education|algebra|textbooks|geometry|reference-books|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17129" PostTypeId="2" ParentId="17128" CreationDate="2019-09-16T19:22:40.307" Score="6" Body="&lt;p&gt;You will find textbooks for all of these at &lt;a href=&quot;https://artofproblemsolving.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;artofproblemsolving.com&lt;/a&gt;. They formed in order to help students like you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have used their number theory book, while tutoring a young student (who was great at math), and I loved it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;===&#xA;Edited (9/28/19) to add:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another good resource is Henri Picciotto's &lt;a href=&quot;https://www.mathed.page/my-books/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathed.page&lt;/a&gt;. You'll find free pdf's there for at least two of his books: &lt;em&gt;Geometry Labs&lt;/em&gt; and &lt;em&gt;Algebra: Themes, Tools Concepts&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, in learning geometry, if you prefer to do constructions digitally (I do!), &lt;a href=&quot;http://www.euclidthegame.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;euclidthegame&lt;/a&gt; is a great way to do that. I also learned a lot by playing with constructions at &lt;a href=&quot;https://sciencevsmagic.net/geo/&quot; rel=&quot;nofollow noreferrer&quot;&gt;sciencevsmagic.net&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2019-09-30T01:38:44.223" LastActivityDate="2019-09-30T01:38:44.223" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17130" PostTypeId="2" ParentId="17096" CreationDate="2019-09-16T23:45:37.720" Score="0" Body="&lt;p&gt;If you have a budget for this then drafting triangles are a easy solution, and they give you physical examples of two common triangles (more useful in introductory trig or analytic goemetry then in pure Euclidian geometry, but still).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By design they have nice, crisp edges designed exactly to use for line drawing.&lt;/p&gt;&#xA;" OwnerUserId="1631" LastActivityDate="2019-09-16T23:45:37.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17131" PostTypeId="1" CreationDate="2019-09-17T06:17:25.400" Score="0" ViewCount="150" Body="&lt;p&gt;Does there exist a textbook in Norwegian (bokmål and nynorsk should both be fine) that covers the more advanced mathematics (needed for calculus) from videregående (secondary school, includes both gymnasium and vocational lines), as well as basic calculus up to the level of integration and differentiation of functions in one dimension?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We are currently using Claudia Neuhauser's 'Calculus for biology and medicine', which has the significant flaw of being in English. Also, it does not cover the basics very well. The style is very verbose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The course this is for is targeted for students of biology, geology, chemistry, etc. Many have very weak mathematics background; they have not taken the more advanced mathematics courses in videregående. I presume the additional cognitive load from using a book in a foreign language does not help matters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A book written in Danish or Swedish might also help.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorDisplayName="user507" LastEditDate="2019-10-19T23:48:45.457" LastActivityDate="2019-10-19T23:48:45.457" Title="Calculus book for basic calculus and repetition from videregående (Norwegian high school)" Tags="|undergraduate-education|textbooks|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17132" PostTypeId="2" ParentId="17131" CreationDate="2019-09-17T11:21:30.177" Score="1" Body="&lt;p&gt;Difficult for us to research it for you.  But you presumably have access to Norwegian libraries colleagues booksellers and know the language.  I would try to research it yourself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One other idea for how to skin the cat is just to pick a less verbose English text.  Maybe Schaums Outline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit.   Reading your question again it sounds like they could use some precalc also.  Frank Ayres first year college math is a good schaums for a review of all precalc and has a very simple intro to calc also.  If you did that plus the regular schaums calc they would be covered. Price is probably twenty for each. So cheaper than most standard texts overall.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And the students will love you because schaums is written to help students learn.  Not to impress professors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry for typos.  On my phone.  &lt;/p&gt;&#xA;" OwnerUserId="12472" LastEditorUserId="12472" LastEditDate="2019-09-17T12:44:51.943" LastActivityDate="2019-09-17T12:44:51.943" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17133" PostTypeId="2" ParentId="14015" CreationDate="2019-09-17T12:40:04.327" Score="1" Body="&lt;p&gt;Both these methods suffer from the same problem: they work from right to left, despite our reading numbers from left to right. That's why children have trouble with long sums.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I discovered as a small child that I could always be the first kid in class to put my hand up when the teacher wrote a subtraction (or addition) sum on the blackboard, by working from left to right, so that (if necessary) I could do the sum in my head while giving the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;.69 from 1.00 is (round the 60 up to 70 and take from 100 = 30; 9 from 10, no need to round the last digit up, is 1) hence thirty-one. It worked beautifully when I worked weekends as a sales clerk as a teenager.&lt;/p&gt;&#xA;" OwnerUserId="12719" LastActivityDate="2019-09-17T12:40:04.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17134" PostTypeId="2" ParentId="17096" CreationDate="2019-09-17T14:56:00.937" Score="0" Body="&lt;p&gt;There are painting straight edges available for about $7 apiece. Here's one I found &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B01NCZKTZQ&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;on Amazon&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/T6n5e.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/T6n5e.png&quot; alt=&quot;12 inch painting straight edge&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might find them cheaper (and possibly smaller, when used for trim work/etc) at your local hardware store, and they generally don't come with ruled markings because they're intended to get covered in paint anyways. Additionally, unlike many of the other suggestions, these have a handle side that's intended to be gripped and held steady.&lt;/p&gt;&#xA;" OwnerUserId="12721" LastActivityDate="2019-09-17T14:56:00.937" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17135" PostTypeId="2" ParentId="14015" CreationDate="2019-09-17T18:23:19.130" Score="2" Body="&lt;p&gt;I have been taught both methods. By different teacher, and at different age, although I do not remember in which method first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today I am using both indifferently, sometimes changing of method in the middle of the subtraction. I noted a tiny preference for the Austrian method, but I think it is purely esthetic, because most of the time the choice is totally unconscious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no harm in teaching both methods, and let the student decide.&lt;/p&gt;&#xA;" OwnerUserId="12722" LastActivityDate="2019-09-17T18:23:19.130" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17136" PostTypeId="2" ParentId="17004" CreationDate="2019-09-17T19:21:57.710" Score="2" Body="&lt;p&gt;You are interchangeable, like peas in a pod. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cauchy and Weierstrass were usually saying &quot;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; &lt;em&gt;becomes arbitrarily close to&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;&quot;, with the qualifier &quot;&lt;em&gt;as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&lt;/em&gt;&quot;, sometimes before, sometimes after, sometimes implied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They also followed Leibnitz and Lagrange to talked about a quantity &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; &lt;em&gt;becoming infinely close to&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;, when &lt;span class=&quot;math-container&quot;&gt;$|x-a|$&lt;/span&gt; is &lt;em&gt;infinitely small&lt;/em&gt; or an &lt;em&gt;infinitesimal&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12722" LastActivityDate="2019-09-17T19:21:57.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17137" PostTypeId="2" ParentId="1834" CreationDate="2019-09-18T23:48:38.973" Score="3" Body="&lt;p&gt;If the students are weak, lots and lots of tangibles:  get them into the habit of drawing pictures and moving stuff around (&quot;paper is cheap&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, integer arithmetic can either be a bunch of rules, or you can use the &quot;chip model&quot; to really get across certain ideas of how it works:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/EA8j7V677Z4?list=PLKXdxQAT3tCuSrZNwMUIXbfmpmYyS9bfy&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/EA8j7V677Z4?list=PLKXdxQAT3tCuSrZNwMUIXbfmpmYyS9bfy&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplication of polynomials can be a bunch of rules, or you can use the &quot;area model&quot; to perform products:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/7tK2DjxZQeY?list=PLKXdxQAT3tCvNbJUuFSqhXPfQ_53yskfg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/7tK2DjxZQeY?list=PLKXdxQAT3tCvNbJUuFSqhXPfQ_53yskfg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Absolute value can be a bunch of rules, or you can use its geometric meaning as &quot;distance between&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/jl9BtnBFRJA?list=PLKXdxQAT3tCvNbJUuFSqhXPfQ_53yskfg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/jl9BtnBFRJA?list=PLKXdxQAT3tCvNbJUuFSqhXPfQ_53yskfg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Graphing in the coordinate plane can either be a bunch of rules, or you can use the idea of it as a set of directions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/OYCfmHpYXlA?list=PLKXdxQAT3tCvNbJUuFSqhXPfQ_53yskfg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/OYCfmHpYXlA?list=PLKXdxQAT3tCvNbJUuFSqhXPfQ_53yskfg&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11747" LastActivityDate="2019-09-18T23:48:38.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17138" PostTypeId="1" CreationDate="2019-09-19T12:35:14.310" Score="12" ViewCount="414" Body="&lt;p&gt;I have had two students so far with a documented disability called &lt;em&gt;dyscalculia&lt;/em&gt;. However, I am a strong proponent in the idea that our primary goal as educators should be the development of something called a &lt;em&gt;growth mindset&lt;/em&gt; in our students.&lt;/p&gt;&#xA;&lt;p&gt;I am hoping to hear from someone who is familiar with both of these topics. I am not interested in any statements about &amp;quot;kids these days&amp;quot; or the &amp;quot;sad state of our society&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The most specific definition of dyscalculia I have found is &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3514770/&quot; rel=&quot;noreferrer&quot;&gt;difficulty acquiring basic arithmetic skills&lt;/a&gt;, but if that is the working definition, then students would not request extra time on exams in classes that allow sophisticated calculators. In my experience, the students and staff have all applied the term much more generally, which is what you will find if you look &lt;a href=&quot;https://www.understood.org/en/learning-attention-issues/child-learning-disabilities/dyscalculia/understanding-dyscalculia&quot; rel=&quot;noreferrer&quot;&gt;outside medical sources&lt;/a&gt;, for example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kids with dyscalculia also have trouble remembering math facts. Or they may understand the logic behind math, but not how or when to apply what they know to solve math problems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you go deeper, you find even more aggressive statements like &amp;quot;Developmental dyscalculia tends to &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/15497272/&quot; rel=&quot;noreferrer&quot;&gt;run in families&lt;/a&gt;, possibly because of a genetic predisposition.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This all conflicts pretty directly with the idea of a growth mindset, because the student is told by medical professionals, then their parents, and then support staff on campus that they are &lt;strong&gt;intrinsically bad at math&lt;/strong&gt; and even that it &lt;strong&gt;runs in the family&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I find that most of my success in math education comes from instilling a growth mindset: communicating the idea that anyone can improve at math with work and focus. But when I am directly contradicting all the other students' authority figures, I do not see a way to succeed. The pattern so far is that I cannot advance the student toward a growth mindset, and they inevitably fail their course of study.&lt;/p&gt;&#xA;&lt;p&gt;What can/should a math educator do in this situation?&lt;/p&gt;&#xA;&lt;p&gt;Edit: A &amp;quot;growth mindset&amp;quot; can be defined this way, see &lt;a href=&quot;https://web.archive.org/web/20190302233150/http://onedublin.org/2012/06/19/stanford-universitys-carol-dweck-on-the-growth-mindset-and-education/&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In a fixed mindset students believe their basic abilities, their intelligence, their talents, are just fixed traits. They have a certain amount and that's that, and then their goal becomes to look smart all the time and never look dumb.&lt;/p&gt;&#xA;&lt;p&gt;In a growth mindset students understand that their talents and abilities can be developed through effort, good teaching and persistence. They don't necessarily think everyone's the same or anyone can be Einstein, but they believe everyone can get smarter if they work at it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-09-22T15:41:21.070" Title="Instilling a Growth Mindset in a Student Diagnosed With Dyscalculia" Tags="|dyscalculia|" AnswerCount="2" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="17139" PostTypeId="1" AcceptedAnswerId="17140" CreationDate="2019-09-20T07:23:56.120" Score="1" ViewCount="359" Body="&lt;p&gt;I am doing Ph.D in Mathematics, I feel I lack few of the skills, if I can improve those skills I think I can do better as a Math scholar. I need some suggestion on these following(below I am talking about problems or theorems in University level text books or some competitive exams),&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When I try to solve a problem(or learning theorems) I just go by my intuition and creativity, sometimes that works, many times confused and got stuck and restart again. Is there any formal way of approaching the problems?&lt;/li&gt;&#xA;&lt;li&gt;Similarly, on making arguments I without knowing I use my intuition as a correct statement(many times it is a false) here and there in the flow of arguments. How to practice to go neatly?&lt;/li&gt;&#xA;&lt;li&gt;This is also same point as the (2) but while doing with pen and paper(problems in differential equation, I mean kind of algorithmic way problem solving), I do silly mistakes and skipping steps(doing a intermediate step in mind). How to overcome this?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12728" LastEditorUserId="12728" LastEditDate="2019-09-20T07:50:28.510" LastActivityDate="2019-09-20T14:32:33.923" Title="How to improve mathematical skills(University level)?" Tags="|undergraduate-education|proofs|mathematical-analysis|undergraduate-research|problem-solving|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17140" PostTypeId="2" ParentId="17139" CreationDate="2019-09-20T13:23:47.753" Score="3" Body="&lt;p&gt;I can only help with (3).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A.  This behavior is not unusual and not just with intuitionists.  It's good to be able to do things in your head, but you need to &quot;know your head&quot; and when you will have issues. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;B.  In general, when doing pen and paper you should try to write down all the steps prone to an error.  Of course there's a balancing point.  But clearly, you are doing too many computations in your head as per the result.  SHOW the algebra. If you do it often, you will at least get fast at writing it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C. Work lots of drill problems and do those that have answers available (to check yourself).  You need volume to build automaticity. (Find and read the Feynman quote about  this &quot;bing bing bing&quot;.  Calculus, and diffyQs is just integral calc on steroids, is an opportunity to do a lot of algebra...to solidify that skill.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;D.  WhenEVER you make an error (of any sort, missing a constant, sign, anything), this means you have to do the whole problem over again.  Yes, even if you instantly get why you did it wrong.*  When reworking the problem, flip a little switch inside yourself and work the problem &quot;as if from scratch&quot;.  Do the whole thing, not copying your old solution or just the part wrong.  But the whole thing as one piece from start to finish and performed correctly.  Do it as if forcing yourself to do a musical piece over again after a memory error, or a gymnastics routine over again after a single error.  (If you are punished, even mildly, for errors in algebra, by having to do more work, you will learn to stop making them.  Very similar to physical conditioning feedback in sports coaching.  I know you think of math in men as some spark of God to Adam in the painting...and in some ways it is...but in other ways we are dogs trained by Skinner and Pavlov.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  While I don't have the complete answer for 1 and 2, I think building some stronger algebraic muscles (like long series and the like) during 3, will rub off and help you on 1 and 2.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Don't worry, your &quot;spark&quot; of intuition ability won't go away because you get a better mechanical ability.  In fact, you may find yourself able to address some situations more easily because of having worked so many problems that you see patterns.  &quot;Tricks&quot; and complications can repeat themselves.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I wouldn't also say that training mechanically will build the genius puzzle solver (you need to solve lots of puzzles to build that muscle).  But it won't hurt, will help what it's meant to help and will even mildly help your intuition.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So don't be lazy.  Don't be proud.  Buckle down.  An easy one is to just do &quot;all the homework problems&quot;, not just the assigned one.  Don't even worry too much about time efficiency.  It will be hard at first.  But as you build the calculational muscles, you will find yourself getting faster and stronger.  And even the time efficiency will become decent (not a bad thing to have when tested).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*I advise to work the entire &quot;set&quot; of homework (say the section, if covering one section a night) and then check all the results.   And rework the missed parts.  It gives you a teensy bit of time in between the attempts so that you get more value from them.  Also, it forces you to address the entire set of homework without the crutch of reinforcement at each problem.  But still, you get rapid reinforcement by seeing it that evening, vice days later.  I think you will also find yourself starting to play the internal game of trying for 100% perfection on the section homework.  Obviously when you start doing that, or close to it, you have corrected the issue in (3).&lt;/p&gt;&#xA;" OwnerUserId="12730" LastEditorUserId="12730" LastEditDate="2019-09-20T14:32:33.923" LastActivityDate="2019-09-20T14:32:33.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17141" PostTypeId="2" ParentId="17138" CreationDate="2019-09-20T18:14:42.210" Score="7" Body="&lt;p&gt;To lay this out at the front, I do not have have any medical or therapeutic credentials -- I just have certification in both secondary mathematics and special education.  I will also be a bit US-centric in spots, as that is where my certification is.  For the sake of my international colleagues, an IEP (Individualized Education Program) is an annual plan for tracking the supports, test accommodations, and goals that are appropriate for each student with an identified disability in the US.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] because the student is told by medical professionals, then their parents, and then support staff on campus that they are &lt;strong&gt;intrinsically bad at math&lt;/strong&gt; and even that it &lt;strong&gt;runs in the family&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is a notion that we really need to confront as educators.  Your instinct about growth mindset is completely correct and appropriate.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Neither dyscalculia nor any other learning disability is, in itself, an excuse to not master mathematics.  What it means is that these students require extra time and support to develop individual strategies that their peers might not need to master the skills to ensure success in education.  In some cases, it might allow a student to use a calculator on a test that isn't assessing arithmetic skills.  It might involve getting pulled out to spend time learning Touch Math or something similar for students who need a tactile-kinetic activity to supplement memory.  The idea is that their IEP is going to be written to indicate how to address each student, but the idea is that the plan is developed to lead to success and should not ever be thought of is what they are doing while the other students are succeeding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Especially for these students, it's all the more critical that they also develop emotional skills to keep from writing themselves off.  As you say, they get enough of that from family and &quot;friends&quot;, and it's really easy to conclude that something is hard for you because you're not meant to be doing it.  But making sure that they're always noticing their progress and rewarding them for their effort more than their results is a great strategy no matter what challenges your students come into your room with.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-09-20T18:14:42.210" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17142" PostTypeId="2" ParentId="17138" CreationDate="2019-09-20T18:54:18.567" Score="-6" Body="&lt;p&gt;Chris, I do think it's true that students differ in their aptitude.  Assuming everyone is a tabula rasa would be naive.  We all aren't going to be Michael Jordan, no matter the training.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I do think the &quot;diagnosis&quot; with a fancy sounding malady is a bit of pseudoscience.  And you are correct to recognize that it's giving a fancy name for &quot;bad at math&quot;.  (Why is something better with a fancy name?  Education is so prone to sound fancy fads!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I do question if the &quot;diagnosis&quot; (as if we had put the kids in a spectroscope and perfectly analyzed them) can be off.  Especially with younger kids.  There can be a lot of confounding factors in human performance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, I agree that the diagnosis can lead to a bit of &quot;give up&quot; by the kids.  Versus &quot;I guess I gotta shoot more baskets than the next guy&quot;.  Also the testing allowances (extra time) are essentially letting the worse kids throw from a closer free throw line (versus training them harder, or even redirecting them to other pursuits).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a practical standpoint, you need to think how you do most good without going so torn up that you throw up your hands.  My advice would just be to be a bit thick skinned and somewhat ignore the pschobable diagnoses.  Don't take it on.  But also don't let it affect you.  Teach kids the best you can.  Que sera, sera.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously for things like extra time, you just have to allow it since those are the rules.  For kids that really don't need the math skills to move on to career, it might not be the end of the world that they are (to a degree) being social promoted.  For kids that do need the skills, they will get weeded out later on.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Big picture, don't get too amped up about the issue.  This is a practical optimization problem.  Don't let the 99 sheep get eaten by wolves, while you try to save one.  Bible was wrong here.  You can't save them all.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Triage baby.  Some will die regardless of assistance.  Some will get better on own regardless.  Some can be saved with medical help.  If the surgeon's time is limited...&lt;/p&gt;&#xA;" OwnerUserId="12730" LastActivityDate="2019-09-20T18:54:18.567" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17143" PostTypeId="2" ParentId="17080" CreationDate="2019-09-21T19:45:36.020" Score="6" Body="&lt;p&gt;In the Yoruba Language, we use the word òǹkà to refer to tokens for representing numbers. Thus, it would correspond to the word numeral. The word òǹkà literally translates as 'that which is used for reckoning.' The prefix on (the n is a nasal) means thing, and the other part, ka, means to reckon, count, calculate, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand the word number would correspond to iye, which may also mean amount or quantity.&lt;/p&gt;&#xA;" OwnerUserId="9204" LastActivityDate="2019-09-21T19:45:36.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17144" PostTypeId="1" AcceptedAnswerId="17148" CreationDate="2019-09-23T15:13:59.683" Score="6" ViewCount="3181" Body="&lt;p&gt;I am considering using &lt;a href=&quot;https://en.wikipedia.org/wiki/S._L._Loney&quot; rel=&quot;nofollow noreferrer&quot;&gt;S. L. Loney&lt;/a&gt;'s &lt;a href=&quot;https://books.google.com/books?id=grQUWYejIq4C&amp;amp;printsec=frontcover&amp;amp;dq=plane%20trigonometry%20loney&amp;amp;hl=zh-TW&amp;amp;sa=X&amp;amp;ved=0ahUKEwiI8vjpm-fkAhUHXKwKHRphCn4Q6AEIKTAA#v=onepage&amp;amp;q=plane%20trigonometry%20loney&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Plane Trigonometry&lt;/a&gt; as the textbook for my course in trigonometry and would like to ask for opinions about the book. This books is very odd and there might be pros and cons because of its oldness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Possible Pros:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It is a classical textbook and stands the test of time.&lt;/li&gt;&#xA;&lt;li&gt;The copyright has expired yet the book is still available. Students can obtain the book at a very cheap price.&lt;/li&gt;&#xA;&lt;li&gt;No more newer editions. So I do not need to adjust my syllabus every few years.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Possible Cons:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Some notation may be obsolete, such as the old notation for factorials:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/jOZ6r.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jOZ6r.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The author probably assumed no calculators (or computation software), which is now widely used in this course.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The language used in late 1800s may sound archaic to some students.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Overall do you consider this book still as a reasonable option for a course on Trigonometry today?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: The course is an undergraduate course with full name &lt;em&gt;Pre-Calculus Trigonometry&lt;/em&gt;. Its description is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Introduction to the elementary trigonometric functions using the&#xA;  functional approach, simple identities, identities using the &#xA;  summation,  half  arc,  and  double  arc  formulas,  inverse  and &#xA;  composite  functions,  sketching  of  the  elementary functions&#xA;  emphasizing phase shift, period, and amplitude, and the solution of&#xA;  right and obtuse angles.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The prerequisite for this course is called College Algebra with the following description:   &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This  course  provides  students  an  opportunity  to  gain  algebraic&#xA;  knowledge  needed  in  many  different  fields  such  as engineering,&#xA;  business, education, science, computer technology, and mathematics.&#xA;  Graphical, numerical, symbolic, and verbal methods support the study&#xA;  of functions and their corresponding equations and inequalities.&#xA;  Students will study linear, quadratic, rational, exponential,&#xA;  logarithmic, inverse, composite, radical, and absolute value&#xA;  functions; systems of equations  and  inequalities  modeling  applied &#xA;  problems;  and  curve  fitting  techniques.  There  will  be &#xA;  extensive  use  of graphing calculators.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the students in this course may or may not have any knowledge of trigonometry. &lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="238" LastEditDate="2019-09-24T18:51:07.003" LastActivityDate="2020-01-30T00:23:55.060" Title="Is Plane Trigonometry by S. L. Loney still good as a textbook today?" Tags="|textbooks|trigonometry|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17145" PostTypeId="2" ParentId="17144" CreationDate="2019-09-23T19:03:15.370" Score="6" Body="&lt;p&gt;I think the language is too much of a turnoff for class use.  If you want a cheap, concise text, I would go with the Schaum's:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0071795359&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Schaums-Outline-Trigonometry-5th-Problems/dp/0071795359#reader_0071795359&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you've gone to the bother of printing and binding the free Loney, it's almost a wash with just buying the Schaum's.  The Schaum's will be pretty direct (not a longwinded doorstop). And very good at dividing things into topics for the kids to learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for your target audience, kids taking trig in college in this age are (in general, statistically, blabla) not that strong.  With that in mind try to be direct and user friendly.  The &lt;em&gt;æ&lt;/em&gt; ligatures right away are a turnoff for Loney.  It is just too old fashioned in the writing.&lt;/p&gt;&#xA;" OwnerUserId="12736" LastEditorUserId="12255" LastEditDate="2019-09-24T12:53:02.217" LastActivityDate="2019-09-24T12:53:02.217" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17146" PostTypeId="1" AcceptedAnswerId="17150" CreationDate="2019-09-24T01:49:24.773" Score="17" ViewCount="6682" Body="&lt;p&gt;I have a question regarding tutoring a calculus student. They need to prove trig identities such as &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{1-\sin x}+\frac{1}{1+\sin x}=2\sec^2x.$$&lt;/span&gt; Doing this kind of problem is very tedious and time consuming. Is it really so necessary to focus on this for the learning of calculus or can this part be done more superficially?  &lt;/p&gt;&#xA;" OwnerUserId="12529" LastEditorUserId="238" LastEditDate="2019-09-24T18:51:17.517" LastActivityDate="2019-09-26T18:21:04.413" Title="How important is knowledge of trig identities for use in Calculus" Tags="|calculus|tutoring|trigonometry|" AnswerCount="9" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="17147" PostTypeId="2" ParentId="17146" CreationDate="2019-09-24T02:44:05.553" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Algebraic skills to do manipulations in general are important.  And this one is not that hard.  Good practice.  Get dirty and do it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There's some applications when you get to trig subs of quadratic radicals and the like.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12737" LastActivityDate="2019-09-24T02:44:05.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17148" PostTypeId="2" ParentId="17144" CreationDate="2019-09-24T05:10:56.873" Score="10" Body="&lt;p&gt;I started my career as an archaeologist before I ended up in mathematics.  I say this in order to emphasize that I am interested in trying to understand how people thought in the past&amp;mdash;the usual &quot;text&quot; that an archaeologist reads is the collection of artifacts which are left behind, but there is also a very active field called Historical Archaeology which seeks to associate historical records with a &quot;ground truth&quot;.  From the point of view an &quot;archaeologist&quot; (or historian) of mathematics, I think that texts such as Loney may be interesting, and well worth reading.  However, I would recommend &lt;strong&gt;against&lt;/strong&gt; using such a text in an introductory class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The cons which you mention are significant, but there are another couple of issues which should give you further pause:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Looking over the table of contents, it appears that much of the focus is on computation.  For example, starting around page 106, there are many pages spent on computing the sines and cosines of angles using the angle sum and half-angle formulae.  My experience is that modern exposition concerns itself more with the formulae themselves (as these recur in calculus), and largely elides explicit computation.  Such computation is, perhaps, useful as an exercise, but a CAS can typically do the job faster and more accurately.  In general, my preference would be to use a book which places much less emphasis on computation or which emphasizes the way in which modern computers can aid computation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There are a lot of topics in that text which are kind of archaic, or which are wholly inappropriate for a modern precalculus class.  For example, most of Chapters X and XI are not relevant in a modern classroom (there is no reason for students to be taught how to read a log table, for example).  Much of Chapter XV seems to focus on aspects of geometry which, for better or worse, are not typically part of the standard US precalculus curriculum (some of it might show up in the high school curriculum, but a lot of it isn't part any standard curriculum prior to upper division courses in geometry (or, perhaps, math competition prep courses)).  These topics certainly have some interest, but if the goal is to prepare students for a standard calculus curriculum, then they don't do any favors to the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then there is Part II.  Almost nothing in Part II is part of the precalculus curriculum (and nothing in Part II is mentioned in the course description in the question).  It starts with series representations of the logarithm and exponential (though using notation which is difficult to parse by modern standards&amp;mdash;the funky factorial and the lack of Sigma notation, for example), moves on to a couple of limits, then jumps into complex analysis.  The entire second half of the book is, in most modern classrooms, covered in courses on calculus and complex analysis.  It doesn't belong in a precalculus class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact that Part II is inappropriate for a precalculus class isn't a big deal&amp;mdash;Part I certainly contains enough material for a semester&amp;mdash;but I think that it might be better to select a book which is more narrowly focused on the topics which you actually need to cover.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In the question, it is noted that the language &quot;may sound archaic to some students&quot;.  I think that this fails to capture the magnitude of the problem&amp;mdash;I think that students are likely to bounce &lt;em&gt;hard&lt;/em&gt; off of a topic which is presented with unfamiliar notation (and, by the way, unfamiliar notation which they won't ever see again) written in a form of English which is distinctly old fashioned.  They inconsistent typesetting also does the book no favors (but now I'm being catty).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am struggling for an analogy which is not hyperbolic, but I think that the following works:  you don't teach students Russian by asking them to read &lt;em&gt;Война и Мир&lt;/em&gt; (&lt;em&gt;War and Peace&lt;/em&gt;) or &lt;em&gt;Евгений Онегин&lt;/em&gt; (a novel by Pushkin) in the original pre-reform language right out of the gate.  Get your students to read something from the 20th or 21st Century, first (maybe something like &lt;a href=&quot;https://en.wikipedia.org/wiki/One_Day_in_the_Life_of_Ivan_Denisovich&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Один День Ивана Денисовича&lt;/em&gt;&lt;/a&gt;&amp;mdash;the language is modern and pretty accessible, and it is still a classic).  Anyone who wants to become a scholar of Russian literature should probably get familiar with pre-reform works eventually, but that isn't the place to start.  Start with modern Russian, and work back.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, students of mathematics should start with a modern presentation and then, if they want to study the history of mathematics, start trying to tackle the classics (presuming that Loney is, indeed, a &quot;classic&quot;, and not simply &quot;old&quot;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Basically, my recommendation would be to find another text.  I don't think that Loney is appropriate for a modern introductory audience.  At best, you might use it supplement the course (for example, one of my critiques of the text, above, is that if over-emphasizes computation, at least by modern standards; however, there appear to be a lot of exercises in the text, which might prove useful).  Moreover, there is nothing wrong with reading a book such a Loney's and drawing inspiration from it (personally, I have gotten a lot of milage in my precalc classes out of Gelfand's &lt;a href=&quot;https://www.springer.com/gp/book/9780817635336&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Method of Coordinates&lt;/em&gt;&lt;/a&gt; and Klein's &lt;a href=&quot;https://www.springer.com/gp/book/9783662494400&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Elementary Mathematics from an Higher Standpoint&lt;/em&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, I also don't have a lot of advice about which book you &lt;em&gt;should&lt;/em&gt; use.  There are a lot of books out there with titles like &lt;em&gt;Precalculus (with Trigonometry!)&lt;/em&gt; and &lt;em&gt;Trigonometry for the Precalculus Student&lt;/em&gt; and whatnot.  Most of these books are 1,000 pages long, weigh 10 lbs., and cost $200+.  They make good doorstops, but are otherwise too dilute and broad to be of much use.  I'm also not a huge fan of the &lt;em&gt;Schaum's Outlines&lt;/em&gt; as course texts.  They are useful for exercises, but leave a lot to be desired &lt;em&gt;vis-à-vis&lt;/em&gt; exposition (of course, that is kind of the point&amp;mdash;they are &lt;em&gt;outlines&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps consider one of the open source texts that are out there?  For example, the &lt;a href=&quot;https://openstax.org/details/books/algebra-and-trigonometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax &lt;em&gt;Algebra and Trigonometry&lt;/em&gt;&lt;/a&gt; or the &lt;a href=&quot;https://scholarworks.gvsu.edu/cgi/viewcontent.cgi?referer=&amp;amp;httpsredir=1&amp;amp;article=1012&amp;amp;context=books&quot; rel=&quot;nofollow noreferrer&quot;&gt;Open Textbook Library's &lt;em&gt;Trigonometry&lt;/em&gt;&lt;/a&gt;.  I suspect that these books will suffer from many of the same problems as the $200+ tomes, but at least they are free. :\&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2019-11-04T13:03:50.310" LastActivityDate="2019-11-04T13:03:50.310" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17149" PostTypeId="2" ParentId="17146" CreationDate="2019-09-24T05:49:35.733" Score="1" Body="&lt;p&gt;Once the student figures out the process - start from one side of what is to be proven, use known formulas to get the other side, done (or start from both sides or do some other variation) - the rest is getting intuition about which formula to use in the process and when. I guess this intuition might be useful if they need to evaluate tricky integrals analytically, or if they do something very trigonometric later in their life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If getting good scores from exams is important to the student, then the importance of trigonometry depends on the instructor and the syllabus. We are probably ill advised to make guesses on those, especially since there is no country specified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it is worth, I have never been taught or needed any trigonometric functions but sin, cos, tan and their inverses.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2019-09-25T05:44:47.180" LastActivityDate="2019-09-25T05:44:47.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17150" PostTypeId="2" ParentId="17146" CreationDate="2019-09-24T07:58:26.703" Score="33" Body="&lt;p&gt;The specific identity &#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}\tag{A}&#xA;\tfrac{1}{1 - \sin{x}} + \tfrac{1}{1 + \sin{x}} = 2\sec^{2}{x}&#xA;\end{equation}&lt;/span&gt;&#xA;as such is probably not often encountered, but simplifications akin to &lt;span class=&quot;math-container&quot;&gt;\begin{equation}\tag{B}&#xA;\tfrac{1}{1-t} + \tfrac{1}{1 + t} = \tfrac{2}{1 - t^{2}}&#xA;\end{equation}&lt;/span&gt;&#xA;occur frequently. For example, integration via partial fractions requires undoing such a simplification, and this manipulation is impossible to understand for someone who does not understand the forward operation being undone. In the same spirit, seeing the formal similarity between (A) and (B) is relevant when it comes to making changes of variables in integrals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Understanding that the identity (A) on the one hand involves the general algebraic identity (B) and on the other hand uses the trigonometric identity &lt;span class=&quot;math-container&quot;&gt;$\sin^{2}{x} + \cos^{2}{x} = 1$&lt;/span&gt;, and understanding how to separate these two statements, is useful for developing the sort of calculational skills that are generally necessary for making progress in calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In more mercantile terms, experience teaching calculus suggests that students who cannot make manipulations such as (A) are unlikely to pass a university calculus course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, characterizing (A) as &lt;em&gt;very tedious and time consuming&lt;/em&gt; seems to me simply wrong, as it is neither. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-09-24T07:58:26.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17151" PostTypeId="2" ParentId="17146" CreationDate="2019-09-24T12:47:40.423" Score="6" Body="&lt;p&gt;There are many calculus textbooks that use no trig.  They may be called &quot;Calculus for Business&quot; or &quot;For Biology&quot; or &quot;For Social Science&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;random example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MyasX.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MyasX.jpg&quot; alt=&quot;book&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course math, physical science, and engineering, definitely use parts of calculus connected with trig functions.  I would have thought that Business would be interested in cyclic phenomena, but what do I know?&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2019-09-24T12:47:40.423" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17152" PostTypeId="2" ParentId="17146" CreationDate="2019-09-24T13:38:26.253" Score="15" Body="&lt;p&gt;Due to low enrollment, my AP Calc class was filled with the students who otherwise would have taken Pre-Calc this year.  So you can imagine that &quot;How much do you &lt;strong&gt;&lt;em&gt;really&lt;/em&gt;&lt;/strong&gt; need to know to see the bigger picture in calculus?&quot; has been on my mind lately.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's where my thoughts have fleshed out in regards to trig so far.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Periodic behavior is widespread throughout science, engineering, and the humanities, and sinusoidal functions are typically used to model periodic behavior.  So simply ignoring trig is not an option.&lt;/li&gt;&#xA;&lt;li&gt;There is absolutely prior knowledge that students should have about trig that will be essential to understand key ideas.  For instance, you cannot understand the derivative of sine without knowing that &lt;span class=&quot;math-container&quot;&gt;$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$&lt;/span&gt;.  You're not going to understand how to calculate &lt;span class=&quot;math-container&quot;&gt;$\int\frac{dx}{\sqrt{1-x^2}}$&lt;/span&gt; without understanding inverse trig functions.&lt;/li&gt;&#xA;&lt;li&gt;Beyond that, there really is a lot of fluff that doesn't seem to serve any real-world purpose.  I'm frankly on the fence about whether secant, cosecant, and cotangent should largely go the way of &lt;a href=&quot;https://blogs.scientificamerican.com/roots-of-unity/10-secret-trig-functions-your-math-teachers-never-taught-you/&quot; rel=&quot;noreferrer&quot;&gt;versine and exsecant&lt;/a&gt;.  Esoteric identities like the one the OP posts can have limited utility, but it seems to be largely an exercise in algebraic manipulation rather than authentic problem-solving.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the end, of course, a student needs to know enough trig identities to be prepared for their final exam, and trig has enormous utility in the real world.  But I am sympathetic to the argument that the twentieth century advanced algebra curriculum is not wholly authentic preparation for twenty-first century calculus.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2019-09-26T18:21:04.413" LastActivityDate="2019-09-26T18:21:04.413" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17153" PostTypeId="2" ParentId="17146" CreationDate="2019-09-24T16:04:59.633" Score="3" Body="&lt;p&gt;Trigonometric substitutions are useful for solving many integrals in closed form and learning how to solve integrals is a major part of most calculus courses. Often more than half of university-level &quot;Calculus II&quot; is concerned with integration techniques. Without trigonometric identities, it may not be obvious how to solve &lt;span class=&quot;math-container&quot;&gt;$\int \left( \frac{1}{1- \sin x} + \frac{1}{1+\sin x} \right)~\mathrm{d}x$&lt;/span&gt;. However, &lt;span class=&quot;math-container&quot;&gt;$\int \sec^2 x~\mathrm{d}x$&lt;/span&gt; is included in many integration tables and happens to be simply &lt;span class=&quot;math-container&quot;&gt;$\tan x + C$&lt;/span&gt;. &lt;/p&gt;&#xA;" OwnerUserId="12740" LastActivityDate="2019-09-24T16:04:59.633" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17154" PostTypeId="2" ParentId="17146" CreationDate="2019-09-24T17:08:46.323" Score="7" Body="&lt;p&gt;From a historical perspective, knowing these identities used to be somewhat more important than now.  Prior to the invention of logarithms, people who needed to do lots of sophisticated calculations [esp. astronomers] resorted to a technique know as 'prostapharesis'.  This involved combining certain trigonometric identities to produce equations (e.g) having a product of trig functions on one side and a sum or difference of trig functions on the other.  This allowed people to transform a multiplication into an addition or subtraction, like logarithms do (but in a somewhat more cumbersome manner).&lt;/p&gt;&#xA;" OwnerUserId="12741" LastActivityDate="2019-09-24T17:08:46.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17155" PostTypeId="2" ParentId="17146" CreationDate="2019-09-24T21:15:55.053" Score="3" Body="&lt;p&gt;The core of concepts of calculus: that is Differentiation and Integration can be defined rigorously and intuitively without any reference to trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be functionally able to use these tools well also requires a very strong command over algebra which other posters have commented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trigonometry is in some sense a “nice to know” since it lets you apply calculus to many more real life problems that would otherwise be inaccessible but care should be taken in my opinion to separate it from the core ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a personal note: I learned Euler’s formula before I learned the double angle formula or the angle sum formulas due to rather unusual educational circumstances. I was still able to do fine in all my classes as I could use these tools to derive and confirm whatever trig I needed on the fly and I essentially learned Trig very passively this way but it was never a conceptual problem. The fact that this worked and despite popular belief I was never “crushed” by AP or High School or College or Grad Level courses for my lack of traditional trig foundation is I think supporting evidence for what I outlined above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trig is a VERY IMPORTANT tool that lets us apply the math. It’s not particularly necessary or useful to get to the foundations of calculus, or understand where and when calculus is necessary.&lt;/p&gt;&#xA;" OwnerUserId="1909" LastActivityDate="2019-09-24T21:15:55.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17156" PostTypeId="2" ParentId="17146" CreationDate="2019-09-25T08:47:00.127" Score="1" Body="&lt;p&gt;IMO knowing basic identities such as &lt;span class=&quot;math-container&quot;&gt;$\sin^2x + \cos^2x=1$&lt;/span&gt; should be as fundamental as knowing &lt;span class=&quot;math-container&quot;&gt;$12\times 12 = 144$&lt;/span&gt; (without needing to reach for a calculator). That is not to say that students should know &lt;em&gt;all&lt;/em&gt; the identities by rote, but they should at least &lt;em&gt;recognise&lt;/em&gt; the identities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Categorising &quot;prove trig identities such as &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{1−\sin x}+\frac{1}{1+\sin x}=2\sec^2x$&lt;/span&gt; as very tedious and time consuming&quot; is quite the overstatement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basic algebra allows you to reformulate &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{1−\sin x}+\frac{1}{1+\sin x}=2\sec^2x$$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$$\frac{1+\sin x}{1−\sin^2x}+\frac{1-\sin x}{1-\sin^2x}=2\sec^2x$$&lt;/span&gt; which gives &lt;span class=&quot;math-container&quot;&gt;$$\frac{2}{\cos^2x}=2\sec^2x$$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Along with &lt;span class=&quot;math-container&quot;&gt;$\sin^2x+\cos^2x$&lt;/span&gt;, they should also then know, or at least recognise,&lt;br&gt;&#xA;  &lt;span class=&quot;math-container&quot;&gt;$\sec^2x=\frac{1}{\cos^2x}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\csc^2x=\frac{1}{\sin^2x}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;QED.&lt;/p&gt;&#xA;" OwnerUserId="12747" LastEditorUserId="12747" LastEditDate="2019-09-25T11:55:37.097" LastActivityDate="2019-09-25T11:55:37.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17157" PostTypeId="1" CreationDate="2019-09-25T12:27:23.393" Score="5" ViewCount="349" Body="&lt;p&gt;I would like to study university level mathematics on my own. I found rigorous books on single variable calculus and topology and learned those. But what would one recommend for several variable calculus? I found a lecture notes but there were many details omitted and for example Taylor polynomial was given only a special case and there was no material included to address how to estimate the error in Taylor's polynomial. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Previously I have read &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=1&amp;amp;ved=2ahUKEwj1_cyLhezkAhXypYsKHXK2ACMQFjAAegQIABAC&amp;amp;url=https%3A%2F%2Fwiki.helsinki.fi%2Fdownload%2Fattachments%2F58366795%2FAnalyysi1H-S.pdf&amp;amp;usg=AOvVaw0v676jGNvse4lPbCSL0iWB&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book&lt;/a&gt; (the link is to a .pdf file) and some lecture notes by Olli Martio. I was unable to find those note from the Internet. Also I read the book Topologia by Jussi Väisälä. Those are Finnish books but English material would be fine.&lt;/p&gt;&#xA;" OwnerUserId="12750" LastEditorUserId="8571" LastEditDate="2019-09-25T16:02:51.660" LastActivityDate="2020-02-27T21:54:35.423" Title="Is there a rigrous and relatively complete book/lecture notes for selfstudying several variable calculus?" Tags="|calculus|textbooks|self-learning|lecture-notes|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17158" PostTypeId="2" ParentId="7385" CreationDate="2019-09-25T14:52:38.240" Score="0" Body="&lt;p&gt;I think there is nothing wrong with reasoning using differentials/infinitesimals in introductory calculus and physics classes provided the nature of the reasoning is made clear and explicit. More precisely, my contention is that in general such uses serve to motivate definitions rather than to hide hard-to-formalize details of proofs, and it is operationally irrelevant whether such motivational uses are formally justifiable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I am conflating differential/infinitesimal here because, although in general I would not conflate these notions, in the context of the question that was asked, it seems to me that a differential &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; is being treated as something &lt;em&gt;very small but not zero&lt;/em&gt; by which it makes sense to divide, so the while the notation is for &lt;em&gt;differentials&lt;/em&gt; the sense is for &lt;em&gt;infinitesimal&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, when I teach these things, I do not write &lt;span class=&quot;math-container&quot;&gt;$dx/dy$&lt;/span&gt; to mean divide the very small &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; by the very small &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt;. I write instead &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Delta y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Delta x/\Delta y$&lt;/span&gt; which I speak of as &lt;em&gt;change in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;change in &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&lt;/em&gt; and treat as small, but nonzero, quantities. I do this because I don't want students to get in the habit of operating too freely with the Leibniz notation, not because I don't like the notation, but because experience suggests that free manipulation of such notation causes a lot of difficulties for students inclined to interpret everything as a series of formal manipulations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the specific example of the arc length of a curve, it is important to keep in mind that without calculus one has no &lt;em&gt;definition&lt;/em&gt; of the arclength of a general curve. One knows what one means by the length of a line segment (having fixed beforehand a length scale), but extending this even to graphs of polynomials is not obvious (it is exactly the sort of problem that gave rise to calculus). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The naive approach, which turns out to work, is to approximate a given continuous curve (but what does &lt;em&gt;continuous&lt;/em&gt; mean without some notion of limit/infinitesimal? - so if you like restrict to the graphs of polynomials or real analytic functions or some such class of curves whose members are de facto continuous but where this does not have to be proved) by a polygonal curve joining some points along the curve that partition the curve into subcurves. It is natural to view the length of this polygonal curve as an approximation to the as of yet undefined length of the curve in question. Doing so one obtains a sum plus some errors terms, and the sum in question can be interpreted as a Riemann sum approximating some so far unformalized integral. Moreover, its form does not depend on the chosen partition (of course its value does). The error terms can be controlled formally using the Taylor approximation, or can be treated as infinitesimals in some hand wavy way - it really doesn't matter - because one is not going to &lt;em&gt;prove&lt;/em&gt; that in the limit the errors are negligible and the remaining sum converges. Rather, one supposes the limit exists, or, more properly, calls the curve &lt;em&gt;rectifiable&lt;/em&gt; provided the limit of the sums exists, in which case that limit is precisely the integral for which the sums were viewed as approximations. Hence one says the curve is rectifiable if the corresponding integral exists (its precise form is irrelevant to this discussion), and when it does calls its value the &lt;em&gt;length&lt;/em&gt; of the curve. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That this is a reasonable definition is not something to be proved beyond the almost trivial observation that applied to the primitive objects used to motivated the definition (that is to line segments) it recovers the primitive notion of length, and in some other essentially &lt;em&gt;physical&lt;/em&gt; examples, such as the circumference of a circle, that it yields the known (empirically) answer. Consequently, it matters not at all that some informal notion of infinitesimals was used in motivating the definition. Its correctness (beyond its self-consistency) is a thing judged by empirical and aesthetic standards external to the mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, most uses of infinitesimals in introductory calculus and physics courses follow this paradigm. The purpose of their use is not to provide an informal or heuristic verification of something that could otherwise be proved. Rather, it is to motivate the extension of some primitive concept, such as length, area, work, circulation, etc. to a class of geometric objects for which the primitive notion is by itself inadequate. The correctness of this extension is judged by its utility or mathematical beauty, or whatever other external criteria are appropriate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One &lt;em&gt;can&lt;/em&gt; go astray in this sort of reasoning, and this does happen when it is applied in more complicated physical contexts where there may be more than one natural way to realize the approximation of whatever thing needs to be computed (defined! to compute something one first has to define what it is one computes) and they may lead to different limiting interpretations, with different results. In physical contexts one is correct, one is not, but this is judged comparison with experiment (something like this underlies the discrepancy between the vakonomic versus nonholonomic interpretations of velocity dependent constraints in classical mechanics - one of which is generally incorrect for real physical systems). In purely mathematical contexts one might (if one gets lucky!) obtain two different theories, and, while one might be judged more interesting than the other, it might be hard to call one more correct than the other.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-09-25T14:52:38.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17160" PostTypeId="2" ParentId="15536" CreationDate="2019-09-25T15:28:01.650" Score="0" Body="&lt;p&gt;There are a few examples in Physics. &lt;a href=&quot;https://en.wikipedia.org/wiki/Lagrangian_mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrangian Mechanics&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12399" LastActivityDate="2019-09-25T15:28:01.650" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17161" PostTypeId="1" AcceptedAnswerId="17164" CreationDate="2019-09-25T21:14:19.647" Score="32" ViewCount="7566" Body="&lt;p&gt;I have some high school students which seem to be &lt;em&gt;afraid&lt;/em&gt; of making mistakes. They are hesitant to make exercises in class because they want their course notes to be super clean, without any mistakes. The following has &lt;em&gt;often&lt;/em&gt; happened in my class. A student writes (when &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is a positive real) &lt;span class=&quot;math-container&quot;&gt;$\sqrt{16a^2+9a^2} = 4a+3a = 7a$&lt;/span&gt;. When the correct solution is shown and the mistake the student made is discussed, the student erases his mistake and writes down the correct solution. However, when studying the contents, he is not reminded of the mistake. (Which would be very valuable)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of a nice quote which shows that there is great learning potential in making mistakes (and figuring out why!). (Especially in math)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know of the following quote by prof. Francis Su. A nice quote, but it's more about the value of persistence.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Struggling is a good thing… it’s where learning happens, it’s what we professors are always doing in our research… the struggle is the most interesting place to be.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5898" LastEditorUserId="238" LastEditDate="2019-09-29T14:03:16.753" LastActivityDate="2020-11-21T00:01:21.587" Title="Quote to show students don't have to fear making mistakes" Tags="|secondary-education|students-mistakes|big-list|" AnswerCount="23" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17162" PostTypeId="2" ParentId="17161" CreationDate="2019-09-25T21:52:03.963" Score="5" Body="&lt;p&gt;If you know what you are doing, then you are wasting your time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anonymous &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-09-25T21:52:03.963" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17164" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T00:13:05.687" Score="25" Body="&lt;ul&gt;&#xA;&lt;li&gt;Johann Wolfgang von Goethe: &quot;By seeking and blundering we learn.&quot;&#xA;&lt;a href=&quot;https://german.stackexchange.com/a/54543/40038&quot;&gt;Original German&lt;/a&gt;, 1825.&lt;/li&gt;&#xA;&lt;li&gt;Albert Einstein: &quot;Anyone who has never made a mistake has never tried anything new.&quot; (However, attribution to Einstein is weak. See &lt;a href=&quot;https://quoteinvestigator.com/2014/12/16/no-mistakes/&quot; rel=&quot;nofollow noreferrer&quot;&gt;quoteinvestigator.com&lt;/a&gt;.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jo_Boaler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jo Boaler&lt;/a&gt;:&#xA;&quot;When I have tutored people in math, I've always started by saying, 'By the way, I just want you to know that I love mistakes the most. They are the time that your brain grows, when you really learn, so it's really great to make mistakes.' [...] people immediately relax and breathe a sigh of relief and are much more willing to jump in to problems and persist longer.&quot; &lt;a href=&quot;https://www.mathmammoth.com/lessons/value_of_mistakes.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cited here&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Keith Devlin: &quot;Learning occurs when we get something wrong and have to correct it.&quot; From &lt;a href=&quot;https://www.mathvalues.org/masterblog/student-teaching-evaluations-are-effective-but-not-in-the-way-you-think&quot; rel=&quot;nofollow noreferrer&quot;&gt;Devlin's Angle&lt;/a&gt;, Oct 1st 2019.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-10-06T00:46:59.507" LastActivityDate="2019-10-06T00:46:59.507" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17165" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T08:18:03.487" Score="10" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;“There is no man,” he began, “however wise, who has not at some period&#xA;  of his youth said things, or lived in a way the consciousness of which&#xA;  is so unpleasant to him in later life that he would gladly, if he&#xA;  could, expunge it from his memory. And yet he ought not entirely to&#xA;  regret it, because he cannot be certain that he has indeed become a&#xA;  wise man — so far as it is possible for any of us to be wise — unless&#xA;  he has passed through all the fatuous or unwholesome incarnations by&#xA;  which that ultimate stage must be preceded. I know that there are&#xA;  young fellows, the sons and grandsons of famous men, whose masters&#xA;  have instilled into them nobility of mind and moral refinement in&#xA;  their schooldays. They have, perhaps, when they look back upon their&#xA;  past lives, nothing to retract; they can, if they choose, publish a&#xA;  signed account of everything they have ever said or done; but they are&#xA;  poor creatures, feeble descendants of doctrinaires, and their wisdom&#xA;  is negative and sterile. &lt;strong&gt;We are not provided with wisdom, we must&#xA;  discover it for ourselves, after a journey through the wilderness&#xA;  which no one else can take for us, an effort which no one can spare&#xA;  us&lt;/strong&gt;, for our wisdom is the point of view from which we come at last to&#xA;  regard the world. The lives that you admire, the attitudes that seem&#xA;  noble to you are not the result of training at home, by a father, or&#xA;  by masters at school, they have sprung from beginnings of a very&#xA;  different order, by reaction from the influence of everything evil or&#xA;  commonplace that prevailed round about them. They represent a struggle&#xA;  and a victory. I can see that the picture of what we once were, in&#xA;  early youth, may not be recognisable and cannot, certainly, be&#xA;  pleasing to contemplate in later life. But we must not deny the truth&#xA;  of it, for it is evidence that we have really lived, that it is in&#xA;  accordance with the laws of life and of the mind that we have, from&#xA;  the common elements of life, of the life of studios, of artistic&#xA;  groups — assuming that one is a painter — extracted something that&#xA;  goes beyond them.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Marcel Proust, here: &lt;a href=&quot;https://ebooks.adelaide.edu.au/p/proust/marcel/p96w/complete.html&quot; rel=&quot;noreferrer&quot;&gt;https://ebooks.adelaide.edu.au/p/proust/marcel/p96w/complete.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is about a wise man who has committed foolish acts in his youth. Perhaps a little bit wordish (hey, it's Proust), but you might extract something like the part I have highlighted.&lt;/p&gt;&#xA;" OwnerUserId="12757" LastActivityDate="2019-09-26T08:18:03.487" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17166" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T09:00:43.297" Score="13" Body="&lt;p&gt;&quot;I have not failed. I've just found 10,000 ways that won't work.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Many of life's failures are people who did not realize how close they were to success when they gave up.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thomas A. Edison&lt;/p&gt;&#xA;" OwnerUserId="12486" LastEditorUserId="12486" LastEditDate="2019-09-26T09:27:26.120" LastActivityDate="2019-09-26T09:27:26.120" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17167" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T09:37:22.710" Score="5" Body="&lt;p&gt;&quot;I hope that in this year to come, you make mistakes. Because if you are making mistakes, then you are making new things, trying new things, learning, living, pushing yourself, changing yourself, changing your world. You're doing things you've never done before, and more importantly, you're doing something.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Neil Gaiman&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Failure is simply the opportunity to begin again, this time more intelligently.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Henry Ford&lt;/p&gt;&#xA;" OwnerUserId="12486" LastActivityDate="2019-09-26T09:37:22.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17168" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T09:57:22.183" Score="4" Body="&lt;p&gt;&quot;Human is the only animal that trips twice over the same stone&quot; - Anonymous&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Next time you trip over a stone, instead of stepping over it, place a big flashing sign that will remind you where you fell to the ground. The next time you travel the same path you will remember your past mistake and be able to avoid the hazard&quot; - Me&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is trying to be funny to convince them to mark their own mistakes so they can learn from them, instead of just getting over them. Please adjust to proper English where needed.&lt;/p&gt;&#xA;" OwnerUserId="12760" LastActivityDate="2019-09-26T09:57:22.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17169" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T11:51:49.450" Score="5" Body="&lt;p&gt;Is a quote about failing in topic? I think so. Here we go.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://booksonthewall.com/wp-content/uploads/2016/10/Samuel-Beckett-quote-Fail-better.png&quot; alt=&quot;quote as image&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Samuel Beckett&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More about this quote on &lt;a href=&quot;https://booksonthewall.com/blog/samuel-beckett-quote-fail-better/&quot; rel=&quot;noreferrer&quot;&gt;booksonthewall&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9652" LastActivityDate="2019-09-26T11:51:49.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17170" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T13:35:50.873" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The road to wisdom? — Well, it's plain&lt;br&gt;&#xA;  and simple to express:&lt;br&gt;&#xA;          Err&lt;br&gt;&#xA;          and err&lt;br&gt;&#xA;          and err again&lt;br&gt;&#xA;          but less&lt;br&gt;&#xA;          and less&lt;br&gt;&#xA;          and less.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;— &lt;a href=&quot;https://www.goodreads.com/quotes/7169960-the-road-to-wisdom-the-road-to-wisdom----well&quot; rel=&quot;nofollow noreferrer&quot;&gt;Piet Hein&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12762" LastEditorUserId="12762" LastEditDate="2019-09-28T08:42:22.283" LastActivityDate="2019-09-28T08:42:22.283" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17171" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T13:47:17.000" Score="1" Body="&lt;p&gt;The &lt;a href=&quot;https://www.uky.edu/~eushe2/Pajares/OnFailingG.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;But They Did Not Give Up&lt;/a&gt; site contains a large number of quotes and anecdotes regarding successful people from many different walks of life who achieved success only after many repeated failures.  For example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Albert Einstein did not speak until he was 4-years-old and did not read until he was 7. His parents thought he was &quot;sub-normal,&quot; and one of his teachers described him as &quot;mentally slow, unsociable, and adrift forever in foolish dreams.&quot; He was expelled from school and was refused admittance to the Zurich Polytechnic School. He did eventually learn to speak and read. Even to do a little math.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Thomas Edison's teachers said he was &quot;too stupid to learn anything.&quot; He was fired from his first two jobs for being &quot;non-productive.&quot; As an inventor, Edison made 1,000 unsuccessful attempts at inventing the light bulb. When a reporter asked, &quot;How did it feel to fail 1,000 times?&quot; Edison replied, &quot;I didn’t fail 1,000 times. The light bulb was an invention with 1,000 steps.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12763" LastEditorUserId="12763" LastEditDate="2019-09-27T13:48:11.230" LastActivityDate="2019-09-27T13:48:11.230" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17172" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T15:09:08.760" Score="10" Body="&lt;p&gt;There was a &lt;a href=&quot;https://academia.stackexchange.com/a/130560/30987&quot;&gt;post over at academia&lt;/a&gt;  where somebody essentially said that after starting by investigating &lt;em&gt;other people's&lt;/em&gt; mistakes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Eventually I got better &amp;mdash; I started making my own mistakes.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As I said over there, I liked that a lot and it stuck with me. Making your own mistakes is a sign of growing up.&lt;/p&gt;&#xA;" OwnerUserId="11894" LastEditorUserId="11894" LastEditDate="2019-09-30T09:29:38.647" LastActivityDate="2019-09-30T09:29:38.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17173" PostTypeId="2" ParentId="17054" CreationDate="2019-09-26T16:23:13.000" Score="2" Body="&lt;p&gt;That &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; will be big.  To solve:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\log_2(n)&amp;lt;\frac{n^\frac{1}{10}}{c}\quad\text{with $c=\sqrt[5]{3}$}.$$&lt;/span&gt;&#xA;As &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is argument of &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; and of a tenth root a well, an educated Ansatz for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; would be a number which covers both, that is, start with &#xA;&lt;span class=&quot;math-container&quot;&gt;$$n=2^{10m}.$$&lt;/span&gt;&#xA;The inequality then becomes &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 10m&amp;lt;\frac1c\cdot2^m.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As &lt;span class=&quot;math-container&quot;&gt;$c\approx1$&lt;/span&gt; we are convinced -- and a trivial induction proves it -- that all &lt;span class=&quot;math-container&quot;&gt;$m\geq7$&lt;/span&gt; will solve the inequality.&lt;br&gt;&#xA;From here &lt;span class=&quot;math-container&quot;&gt;$n=2^{70}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;NB: In case you want to show that any real &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;2^{70}$&lt;/span&gt; will do, consider &lt;span class=&quot;math-container&quot;&gt;$f(x)=\frac{2^x}{c}-10x$&lt;/span&gt;. We know that &lt;span class=&quot;math-container&quot;&gt;$f(7)&amp;gt;0$&lt;/span&gt;. Now show that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is strictly increasing for &#xA;&lt;span class=&quot;math-container&quot;&gt;$$x&amp;gt;\frac{\ln\left(\frac{10}{\ln(2)}c\right)}{\ln(2)}&#xA;\approx2.9.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="639" LastEditorUserId="639" LastEditDate="2019-09-27T08:51:59.233" LastActivityDate="2019-09-27T08:51:59.233" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17174" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T19:12:14.040" Score="2" Body="&lt;p&gt;Here are a few of my favorites:&lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/OWKvf.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OWKvf.jpg&quot; alt=&quot;FAIL Definition&quot;&gt;&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/40smh.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/40smh.jpg&quot; alt=&quot;Learning Flow Chart&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12766" LastActivityDate="2019-09-26T19:12:14.040" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17175" PostTypeId="2" ParentId="17161" CreationDate="2019-09-26T21:41:36.290" Score="6" Body="&lt;p&gt;The one I'm using is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An expert is a person who has found out by his own painful experience all the mistakes that one can make in a very narrow field.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Attributed to &lt;a href=&quot;https://en.wikiquote.org/wiki/Niels_Bohr&quot; rel=&quot;noreferrer&quot;&gt;Niels Bohr&lt;/a&gt;, quoted by Edward Teller, in &lt;em&gt;Dr. Edward Teller's Magnificent Obsession&lt;/em&gt; by Robert Coughlan, in &lt;em&gt;LIFE&lt;/em&gt; magazine &lt;a href=&quot;http://books.google.de/books?id=I1QEAAAAMBAJ&amp;amp;pg=PA62&quot; rel=&quot;noreferrer&quot;&gt;(6 September 1954), p. 62&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6275" LastActivityDate="2019-09-26T21:41:36.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17177" PostTypeId="2" ParentId="17161" CreationDate="2019-09-27T04:33:53.430" Score="1" Body="&lt;p&gt;Sometimes life's going to hit you in the head with a brick. Don't lose faith. - Steve Jobs&lt;/p&gt;&#xA;" OwnerUserId="12769" LastActivityDate="2019-09-27T04:33:53.430" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17178" PostTypeId="2" ParentId="17161" CreationDate="2019-09-27T06:25:07.317" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;In the remaining sections of this paper we briefly discuss various occurrences of the stability and pinching phenomena in differential geometry. The results we present are, for the most part, not new and we do not provide the detailed proofs. (These can be found in the papers cited in our list of references). What may be new and interesting for non-experts is an exposition of the stability/pinching &lt;em&gt;philosophy&lt;/em&gt; which lies behind the basic results and methods in the field and which is very rarely (if ever) presented in print. (This common and unfortunate fact of the lack of an adequate presentation of basic ideas and motivations of almost any mathematical theory is, probably, due to &lt;strong&gt;the binary nature of mathematical perception: either you have no inkling of an idea or, once you have understood it, this very idea appears so embarrassingly obvious that you feel reluctant to say it aloud; moreover, once your mind switches from the state of darkness to the light, all memory of the dark state is erased and it becomes impossible to conceive the existence of another mind for which [the] idea appears non-obvious.&lt;/strong&gt;)&amp;quot; --Mikhail Gromov, &lt;a href=&quot;https://www.ihes.fr/%7Egromov/wp-content/uploads/2018/08/stability_and_pinching.pdf&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Stability and Pinching&amp;quot;&lt;/a&gt; pp. 64-65. Bold emphasis added.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This quote (from an absolutely towering figure of modern mathematics) pretty much guides my entire pedagogical philosophy. In particular, we must make a conscientious effort as both instructors and students of mathematics to combat this binary nature of mathematical perception. As instructors, we need the empathy and humility to remember that a topic was not always blindingly obvious and that we too once struggled with it. As students, we need the perseverance to push through until the current struggle becomes blindingly obvious in hindsight.&lt;/p&gt;&#xA;&lt;p&gt;I do what I can to encourage this metacognition. Much of mathematics education can feel like an endless treadmill, of being perpetually stuck in the dark of new ideas and concepts. Seek out time to reinforce and reflect upon topics that once seemed impossible: &amp;quot;Remember when multiplication tables / equations of parabolas and lines / the product rule / linear differential equations / ... seemed hard? This too shall pass...&amp;quot;. As grating as we might find it when students write &lt;span class=&quot;math-container&quot;&gt;$\sqrt{a+b} = \sqrt{a} + \sqrt{b}$&lt;/span&gt;, who among us didn't make this mistake at least once upon our own journeys? While we are tasked with correcting this mistake, don't fall into the trap of believing it to be trivial or the sign of an inferior intelligence. Embrace and encourage acknowledgement of mistakes.&lt;/p&gt;&#xA;&lt;p&gt;That even an &lt;a href=&quot;https://en.wikipedia.org/wiki/Mikhail_Leonidovich_Gromov&quot; rel=&quot;noreferrer&quot;&gt;Abel prize winning mathematician&lt;/a&gt; can feel this way about mathematical understanding only further hammers this point home. If even Gromov can admit being embarassed about not grasping a subject at first pass, then what shame should the rest of us possibly bear? The only crimes would be to either lie to ourselves about having ever struggled or to write off those who struggle with it.&lt;/p&gt;&#xA;&lt;h3&gt;Edit: Another quote&lt;/h3&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Understanding what and why did not work may be more instructive than celebrating our successes.&amp;quot; ~ Mikhail Gromov&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am having trouble finding a rigorous citation for this quote, but I will edit if I can track it down.&lt;/p&gt;&#xA;" OwnerUserId="7260" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-09-27T06:38:46.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17179" PostTypeId="2" ParentId="17161" CreationDate="2019-09-27T07:19:49.907" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Good decisions come from experience, and experience come from bad decisions.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Unknown&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="12771" LastActivityDate="2019-09-27T07:19:49.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17181" PostTypeId="1" AcceptedAnswerId="17204" CreationDate="2019-09-27T15:17:12.407" Score="11" ViewCount="5604" Body="&lt;p&gt;I am a teaching assistant for an intro programming course. One assignment asked for the averages of a certain ratio, but most students, rather than returning&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\text{sum of all ratios}}{\text{total ratios}},$$&lt;/span&gt;&#xA;gave&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{\text{sum of the numerators used to calculate those ratios}}{\text{sum of the denominators used to calculate those ratios}}. $$&lt;/span&gt;&#xA;That isn't the same, but I can't put to words why it isn't the same; I just know not to do it and can't explain to these students &lt;em&gt;why&lt;/em&gt; it gives the wrong result. What's the intuitive way to explain it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ratios in question were mileages per gallon. The question wanted the average mpg of all trips, and asked for miles driven and gallons of fuel used separately.&lt;/p&gt;&#xA;" OwnerUserId="12777" LastEditorUserId="12777" LastEditDate="2019-09-27T19:33:27.787" LastActivityDate="2019-10-03T09:58:50.857" Title="How to explain that the sums of numerators over sums of denominators isn't the same as the mean of ratios?" Tags="|undergraduate-education|students-mistakes|statistics|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17182" PostTypeId="2" ParentId="17181" CreationDate="2019-09-27T15:33:25.097" Score="14" Body="&lt;p&gt;It actually depends on exactly what you're asking. Or even what you SHOULD be asking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want the average profitability of all the 500+ operators in the Permian, you could just average all the profit margin percentages.  This is taking the ratios (profit/revenue) for each company and averaging them.  It corresponds to your expected (mean) profit margin if you just picked an operator at random.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If instead, I take all the profits and divide by all the revenues, this would give me the average profitability of the INDUSTRY (operators in the Permian).  Often this is actually the question you are asking, or should be asking.  IOW, you want the revenue-weighted average profit margin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same point would apply if you were, say doing sampling of different size demographic categories and were interested in the total population polling estimate.  You need to weight by size of the buckets.  (Or just take the totals, which is mathematically equivalent.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since we don't know exactly what you asked, or how precise you were, it's hard to say if the students were wrong.  Of course, they may have been.  But I would just check.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Responding to your edited-in update on the question.  There's actually still some ambiguity about what you are (or should be) asking.  But if I had to guess, the student's way is more likely giving the desired answer.  (IOW, ratio the totals, rather than average the ratios.)  IOW, &quot;average fuel efficiency&quot; should be &quot;gallon&quot; weighted.  That's the one impacting your pocketbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, you may find that there's some correlation of fuel efficiency with trip length.  The engine being warmed up, operates more efficiently. Also, highway speeds may be more efficient than slow speeds.  Also the issue of frequent stops and starts.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if I were selecting a car, I would want the one that has the better average fuel efficiency (total miles/total gallons).  Ideally with something approximating my type of usage.  But I definitely wouldn't want to skew things by saying one &quot;10 mile, 1 gallon&quot; trip mean the same as one &quot;100 mile, 5 gallon&quot; trip in terms of the importance to my pocketbook.&lt;/p&gt;&#xA;" OwnerUserId="12778" LastEditorUserId="12778" LastEditDate="2019-09-27T16:50:30.687" LastActivityDate="2019-09-27T16:50:30.687" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17183" PostTypeId="2" ParentId="17161" CreationDate="2019-09-27T18:07:29.257" Score="10" Body="&lt;p&gt;&lt;strong&gt;M&lt;/strong&gt;istakes &lt;strong&gt;A&lt;/strong&gt;llow &lt;strong&gt;T&lt;/strong&gt;hinking to &lt;strong&gt;H&lt;/strong&gt;appen&lt;/p&gt;&#xA;" OwnerUserId="12529" LastActivityDate="2019-09-27T18:07:29.257" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17184" PostTypeId="2" ParentId="17181" CreationDate="2019-09-27T19:14:14.290" Score="12" Body="&lt;p&gt;I like guest's answer. To elaborate, here is a possible question to ask them.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You take two trips in your car:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Trip 1 is a 100 mile drive that takes you 2 hours.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Trip 2 is a 200 mile drive that takes you 1 hour.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(a) What is the average speed of your car?&lt;/p&gt;&#xA;&lt;p&gt;(b) What is the average speed on an average trip?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The answer to (a) is &lt;span class=&quot;math-container&quot;&gt;$\frac{100 \text{ miles} + 200 \text{ miles }}{2 \text{ hours } + 1 \text{ hour }} = \frac{300 \text{ miles }}{3 \text{ hours }} = 100 \text {mph}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The answer to (b) is &lt;span class=&quot;math-container&quot;&gt;$\frac{\frac{100 \text{ miles }}{2 \text{ hours }} + \frac{200 \text{ miles }}{1 \text{ hour }}}{2} = \frac{50 \text{ mph} + 200 \text{ mph} }{2} = 125 \text {mph}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You need to be very clear about the question if you prefer one of these answers.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-09-29T02:22:33.823" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17185" PostTypeId="2" ParentId="17181" CreationDate="2019-09-27T23:31:14.043" Score="18" Body="&lt;p&gt;One observation is that  &lt;em&gt;(sum of numerators)&lt;/em&gt; divided by &lt;em&gt;(sum of denominators)&lt;/em&gt; is not well defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, let's work with the two ratios &lt;span class=&quot;math-container&quot;&gt;$a=\frac01$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b=\frac11$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ratio of the sum of numerators to sum of denominators is &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, we can also write &lt;span class=&quot;math-container&quot;&gt;$a=\frac03$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b=\frac22$&lt;/span&gt;. Now the ratio is &lt;span class=&quot;math-container&quot;&gt;$\frac25$&lt;/span&gt;, which is not equal to &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt;! &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-09-27T23:31:14.043" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17186" PostTypeId="1" CreationDate="2019-09-28T07:03:44.977" Score="9" ViewCount="932" Body="&lt;p&gt;I am writing a story for young people about the history of the development of the cubic formula and complex numbers, partly because it has so much drama and partly because it's amusing that complex numbers were born of this almost farcical drama.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read the chapter on it in William Dunham's book, &lt;em&gt;Journey Through Genius&lt;/em&gt; (chapter 6, Cardano and the solution of the cubic). I've also read a number of online accounts. (Two that seemed helpful: &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/how-tartaglia-solved-the-cubic-equation-the-first-solutions&quot; rel=&quot;nofollow noreferrer&quot;&gt;an MAA account&lt;/a&gt;, and &lt;a href=&quot;https://www-history.mcs.st-and.ac.uk/history/Biographies/Tartaglia.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;information from the University of Saint Andrews&lt;/a&gt;.) I have a pretty good idea of what happened, but there are details that differ in different accounts. And it's hard to tell which sources are most accurate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question I have now is whether anyone uses the cubic formula nowadays? Is it used in the development of other mathematical ideas? My question seems to lie between math and math education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edited to add:&#xA;My story is coming along. I would like, at some point, to consult with a math historian who can verify that I've told the story as accurately as possible. &lt;strong&gt;How would I find a math historian who knows about these developments and might be interested in checking out my manuscript?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2019-10-14T02:47:24.000" LastActivityDate="2019-10-31T13:22:48.533" Title="Does anyone use the cubic formula these days?" Tags="|algebra|history|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17187" PostTypeId="2" ParentId="17157" CreationDate="2019-09-28T08:36:14.580" Score="3" Body="&lt;p&gt;Possibly not helpful to you since to my knowledge it is only available in French, but I have a marvelous recollection of the &quot;Petit guide calcul différentiel à l'usage de la licence et de l'agrégation&quot; by François Rouvière.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2019-09-28T08:36:14.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17188" PostTypeId="2" ParentId="16882" CreationDate="2019-09-28T08:50:57.243" Score="4" Body="&lt;p&gt;Many answers already, so I'll keep this one short: it has been realized by researchers in didactic that one difficulty in the concept of function is that it changes status: at first each function is considered as a &lt;em&gt;process&lt;/em&gt; (a &lt;em&gt;verb&lt;/em&gt; in @ΦDev's answer); they meet several of them, each being akin to a (unitary) operation, not very different from addition or multiplication. Then at some point we start &lt;em&gt;studying&lt;/em&gt; functions rather than &lt;em&gt;using&lt;/em&gt; them, i.e. we ask student to think of them as a mathematical &lt;em&gt;objects&lt;/em&gt;, in the same sense as e.g. numbers or geometric figures: functions will have or not have properties (monotony, continuity, etc.), will be subjected to operations (sums, differentiation, etc.), will be considered in relation to each other (inequalities, asymptotic comparison, primitive-derivative from each other, etc. ), etc. This is the point where one really need a variable name for a function, i.e. to write it as &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; instead of something specific such as &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt;, ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It could help to embrace this shift in the point of view on functions to overcome this difficulty. When I teach math-teachers-to-be, I use the case of &lt;em&gt;relations&lt;/em&gt; to have them go through the same process: they already now &lt;span class=&quot;math-container&quot;&gt;$\le$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt;, etc. but are not used to consider an unspecified relation and work on it as an object.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2019-09-28T08:50:57.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17190" PostTypeId="2" ParentId="17181" CreationDate="2019-09-28T11:36:09.813" Score="1" Body="&lt;p&gt;Do you want students to think or not?  According to your logic, when I do a 300 mile trip with a car and a day later reverse it in the drive way, the average mpg value of those two days can be calculated by taking the mpg value of the first day and averaging it with the (likely much higher) mpg value of the second day that saw almost no mileage and almost no gas use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't use &quot;textbook examples&quot; for meaningless calculations and then complain that the students were unable to make the calculations as meaningless as you wanted them done.  That's teaching them neither mathematics nor its proper application.&lt;/p&gt;&#xA;" OwnerDisplayName="user12785" LastActivityDate="2019-09-28T11:36:09.813" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17191" PostTypeId="2" ParentId="17186" CreationDate="2019-09-28T14:09:21.207" Score="4" Body="&lt;p&gt;Sue:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;(Tiny help, but maybe something.)  This came up in a Google search.  &lt;a href=&quot;http://archives.math.utk.edu/ICTCM/VOL10/C003/paper.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://archives.math.utk.edu/ICTCM/VOL10/C003/paper.pdf&lt;/a&gt;  The first page has a couple small motivating stories/quotes about why people are fascinated by cubics.  You could trace back the two quotes to the original sources and see if there's anything else that's helpful above the quotes.  But even as is, they give some mild motivation.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A.  The Kac reference is from his autobiography.  I don't have a copy of it, but you should be able to get it by interlibrary loan.  This link gives a little more on the pull of the cubic for Kac:  &lt;a href=&quot;http://lab.rockefeller.edu/cohenje/assets/file/136KacEnigmasOfChanceAnnProb1986.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://lab.rockefeller.edu/cohenje/assets/file/136KacEnigmasOfChanceAnnProb1986.pdf&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I founf a copy of the Kac bio.  It's pretty motivating, actually.  He wrote an all new derivation of Cardano, and got it published, as a 16 year old.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[And I am no Kac, but I share/d a similar desire to at least forward derive the equation, as we do with completing the square for the quadratic.  I didn't spend the summer doing it though...I guess that's why I'm not a mathematician.   But surely the urge to do so is common.  We had a recent discussion about repeated root solution to linear constant homo diffy qs...and I, with others, was bugged by how the common training/motivation is to &quot;just try this solution, it might work&quot; and then you prove it does.  I checked several texts and found that going back to at least ~1900 (e.g. Granville), this is a common way to present the idea.  However, it is not needed...Thomas/Finney derives the repeated root solution &quot;forward&quot;.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B.  I have a copy of the Feynman book.  There is really not much more on the cubic than within that quote.  Although a page or two more on the general issue of &quot;current&quot; (early 80s) Greek education over-venerating the ancient (in art, science, mechanics).&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;I don't see analytical solution of the cubic as a normal part of current science/engineering studies.  And I have a pretty broad chemistry/physics/engineering training.  A few cubic polynomials do occasionally occur in practical problems but even then people solve them by estimation.  For example, my AP Chemistry text from the 1980s has an occasional cubic in the equilibrium chapter problems (just a few and at the end and harder).  But the counsel was always to use estimation to find it, to within sig figs.  In contrast analytical solution of quadratics is extremely common, to the extent that bsq-4ac is iconic (hard grooved into memory).  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12788" LastEditorUserId="12788" LastEditDate="2019-09-28T19:04:12.717" LastActivityDate="2019-09-28T19:04:12.717" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17192" PostTypeId="1" CreationDate="2019-09-28T15:01:16.317" Score="6" ViewCount="204" Body="&lt;p&gt;The following question was in a high school teacher's guide: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f\colon\mathbb{R}\rightarrow\mathbb{R}$&lt;/span&gt; defined by&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$f(x)=\begin{cases} x &amp;amp; x\in\mathbb{R}\setminus\mathbb{Q}\\&#xA;2x &amp;amp; x\in\mathbb{Q}&#xA;\end{cases}$$&lt;/span&gt;&#xA;  Is &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; Riemann integrable in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}?$&lt;/span&gt;&#xA;  A student claims that this function is (Riemann) integrable in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. Which Theorem would you use to check the validity of such a claim?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My question is: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How would you explain to (high school) students (i suppose intuitively) that this function is not Riemann integrable?&lt;/p&gt;&#xA;" OwnerUserId="12789" LastEditorUserId="238" LastEditDate="2019-09-29T14:02:20.770" LastActivityDate="2019-09-29T22:53:05.553" Title="(Riemann integrability) How do you explain this to a high school student?" Tags="|mathematical-pedagogy|calculus|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17193" PostTypeId="2" ParentId="17186" CreationDate="2019-09-28T18:46:35.207" Score="7" Body="&lt;p&gt;It seems unlikely that the Cardano formula has even been of serious analytic use, i.e., used to approximate roots of a cubic. At least since the inception of calculus, Newton's method can be used to do so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my thinking, the modern perspective of Cardano's formula is an &lt;em&gt;algebraic&lt;/em&gt; perspective, not analytic, after Galois. Polynomials of degree 2, 3, 4 are solvable using rational operations and radicals precisely because the &quot;symmetry group&quot; of the roots, i.e., the &lt;em&gt;Galois group&lt;/em&gt; of such a polynomial, has a certain algebraic structure called &quot;solvability.&quot; The Galois group of a cubic is &lt;span class=&quot;math-container&quot;&gt;$S_3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$A_3$&lt;/span&gt;, i.e., the symmetric or alternating group. Both groups have the special algebraic structure of &quot;solvability.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Galois group of a polynomial of degree 5 or higher need not have this algebraic structure. Hence there cannot exist a universal &quot;Cardano-like formula for fifth degree polynomials.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the Cardano formula should be viewed as historical motivation for what eventually became the Abel–Ruffini theorem of the nineteenth century, which asserts that the &lt;em&gt;general&lt;/em&gt; polynomial of degree 5 or higher cannot be solved by radicals, and which explains why polynomials of degree 2, 3, and 4 can be always solved by radicals.  &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-09-28T18:46:35.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17195" PostTypeId="2" ParentId="17192" CreationDate="2019-09-28T23:29:16.337" Score="4" Body="&lt;p&gt;I would ask a student to justify their claim. How did they arrive at the conclusion that the function is integrable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a couple of problems that the student should have notices, given that usually they only know the fundamental theorem of calculus (and then they look for an antiderivative)&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Infinite domain of integration (Riemann definition does not apply)&lt;/li&gt;&#xA;&lt;li&gt;Lack of antiderivative (so they could not have applied the FTC)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Hence they would have been forced to evaluate upper and lower sums. But again there is the problem of infinite domain. Did they use the definition of improper integral? Or consider the integral over a closed interval.&#xA;What were their lower/upper sums?&lt;/p&gt;&#xA;" OwnerUserId="10125" LastActivityDate="2019-09-28T23:29:16.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17196" PostTypeId="2" ParentId="17192" CreationDate="2019-09-29T00:12:24.543" Score="3" Body="&lt;p&gt;As for theorems: Riemann integrable functions need to be almost everywhere continuous. This function is only continuous at zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for explanations for a high school students, I would do two Riemann sums one using the maximum over intervals and the other using minimums. They plainly have different limits, so the integral is not well defined.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2019-09-29T00:12:24.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17197" PostTypeId="2" ParentId="16882" CreationDate="2019-09-29T04:30:47.307" Score="0" Body="&lt;p&gt;You could start with two tables of values for x (the input variable) and y (the output value) in both. To start, each should represent a permutation on say, the set {1, 2, 3, 4, 5}, but don't use the word &quot;permutation.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Label one &quot;Table &lt;em&gt;A&lt;/em&gt;&quot;, the other &quot;Table &lt;em&gt;B&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For each line in Table &lt;em&gt;A&lt;/em&gt;, introduce the notation &lt;em&gt;A&lt;/em&gt;(1)=, &lt;em&gt;A&lt;/em&gt;(2)=, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, for Table &lt;em&gt;B&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then ask, what is &lt;em&gt;B&lt;/em&gt;(&lt;em&gt;A&lt;/em&gt;(1)), &lt;em&gt;B&lt;/em&gt;(&lt;em&gt;A&lt;/em&gt;(2)) etc.? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then ask what is &lt;em&gt;A&lt;/em&gt;(&lt;em&gt;B&lt;/em&gt;(1)), &lt;em&gt;A&lt;/em&gt;(&lt;em&gt;B&lt;/em&gt;(2)), etc.?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Be sure to test the examples before class. You don't want any weird coincidences that you would have to explain away.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastActivityDate="2019-09-29T04:30:47.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17198" PostTypeId="1" CreationDate="2019-09-29T08:53:12.337" Score="12" ViewCount="226" Body="&lt;p&gt;This is a question I asked on [Academia.se]. It did not get an answer, so I am re-posting it here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the country where I live, university students studying mathematics usually attend lectures, consultation with their lecturers (if they have questions relating to the material beingtaught), and tutorial/practice classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Years ago there was a change in the way tutorials are run. Now students work in groups of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; in front of white boards, and the tutor walks around checking and commenting on the solutions written by students (on the white boards). (Previously the students would have asked questions, and the tutor would have solved the problems on a white board).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The change was implemented on the basis of some research that suggested  greater learning benefits for the students through more active participation and peer-consultation in problem solving. If anyone is familiar with, or can give a reference to, this research article/s, please feel free to provide that here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd be interested to learn how tutorials, or active participation in problem solving with peers in mathematics classics, are carried out in other countries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The benefit of the above practice class is that the student is `forced' to participate. Or at least one student from each group is, since there are always students who either don't attend or simply stand and contribute very little to the group discussion (even though they are encouraged to take turns and help each other understand, this does not necessarily occur).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also hard to tell how much this helps an average student learn mathematics. Have there been any studies on this?&lt;/p&gt;&#xA;" OwnerUserId="10125" LastEditorUserId="238" LastEditDate="2019-09-29T14:03:04.760" LastActivityDate="2022-01-03T22:55:07.610" Title="tutorial active learning" Tags="|undergraduate-education|tutoring|class-participation|classroom-interaction|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17199" PostTypeId="2" ParentId="17128" CreationDate="2019-09-29T15:20:54.093" Score="5" Body="&lt;p&gt;To supplement Sue's great suggestion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another advantage of the &lt;a href=&quot;https://artofproblemsolving.com/&quot; rel=&quot;noreferrer&quot;&gt;artofproblemsolving.com&lt;/a&gt; is that it provides a whole lot of support for students like you, whether or not you purchase or use the textbooks available there.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You'll find, e.g., see &lt;a href=&quot;https://artofproblemsolving.com/community&quot; rel=&quot;noreferrer&quot;&gt;AoPS Community&lt;/a&gt;, which provide a number of online forums in which all students can participate, ask questions, e.g., about homework problems in various topics; and you can even help other students by providing solutions to their questions.  There are also opportunities to engage socially with other bright students through many of these formats, learn about post-highschool opportunities, etc.. In addition, there are links to contests, and tools to search the site and ask for help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, see &lt;a href=&quot;https://artofproblemsolving.com/resources&quot; rel=&quot;noreferrer&quot;&gt;AoPS Resources&lt;/a&gt;, which provides &quot;a wide variety of free resources for avid problem-solvers&quot;, including&#xA; - Math Training and Tools, including Videos&#xA; - News and Information&#xA; - Just for Fun&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, you might be interested in participating (free of cost) in &lt;a href=&quot;https://artofproblemsolving.com/school/mathjams&quot; rel=&quot;noreferrer&quot;&gt;AOPS Math Jams&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Math Jams are free online discussions hosted by Art of Problem Solving for a variety of purposes, including:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Informational sessions about Art of Problem Solving classes.&lt;/li&gt;&#xA;  &lt;li&gt;Review of the AMC and AIME competitions.&lt;/li&gt;&#xA;  &lt;li&gt;Introductions to new topics in math and other areas of study.&lt;/li&gt;&#xA;  &lt;li&gt;Special events, such as our annual discussions with admissions officers and professors from outstanding universities such as MIT, Carnegie Mellon, Stanford, and Duke.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Much of the AoPS curriculum, including for Intermediate Algebra (Alg.II), Intro to Number Theory, PreCalculus, and Calculus, while they offer new paperback hard-copies for sale, they also provide online textbooks and solution, all of which are available for relatively reasonable prices (far less than the prices of college texts) and are highly competitive, if not cheaper, than highly reputable non-AoPS texts for the topics in which you are interested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, in the AoPS Bookstore, you can click on &lt;a href=&quot;https://artofproblemsolving.com/store/list/other-products&quot; rel=&quot;noreferrer&quot;&gt;Other Products&lt;/a&gt; available, and if you scroll down to towards the bottom, you'll find &lt;em&gt;Other Books&lt;/em&gt; with a few lower priced high quality alternatives, but not in all the subjects you list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two books offered in the &lt;strong&gt;Other Books&lt;/strong&gt; list, &lt;em&gt;Intro to Inequalities&lt;/em&gt; and &lt;em&gt;Intro to Number Theory,&lt;/em&gt; are largely taken from the book &lt;a href=&quot;https://researchportal.bath.ac.uk/en/publications/introductions-to-number-theory-and-inequalities&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Introductions to Number Theory and Inequalities,&lt;/em&gt;&lt;/a&gt;, also produced by the UKMT.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;You might want to review some texts, in &lt;a href=&quot;https://www.google.com/search?client=firefox-b-1-d&amp;amp;ei=q7uQXe7XNYGOsQWlr6Yo&amp;amp;q=introduction+to+number+theory+pdf&amp;amp;oq=Introduction+to+Number+Theory&amp;amp;gs_l=psy-ab.1.1.0l10.9253.19946..22515...0.3..0.275.3899.0j20j4......0....1..gws-wiz.......0i71j33i299j33i160j0i22i30j0i22i10i30.6qL1Q1gS82E&quot; rel=&quot;noreferrer&quot;&gt;pdf format, in elementary number theory&lt;/a&gt;, and a few with more advanced (largely undergraduate) texts in elementary number theory. You might find one or more to be helpful to you, with no cost to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, with respect to Geometry:  This depends on whether you've you've covered any geometry before now. For the basics in high school geometry, you might like &lt;a href=&quot;https://en.wikibooks.org/wiki/High_School_Geometry&quot; rel=&quot;noreferrer&quot;&gt;High School Geometry, Wikibooks&lt;/a&gt;. A nice supplement to this free text would be &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1627340289&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometry by Construction: Object Creation and Problem-solving in Euclidean and Non-Euclidean Geometries&lt;/a&gt;, which sells for $25 on Amazon.com.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible (freely available textbook in pdf format) you might use to address Algebra II AND Trigonomtry can be found &lt;a href=&quot;https://www.sewanhakaschools.org/cms/lib/NY01001491/Centricity/Shared/Algebra%202%20and%20Trigonometry.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. It does presume any potential student has mastered basic geometry (e.g., as covered in the first pdf text that I list for geometry). To learn analytic geometry (addressed in this text), and to learn basic trigonometry, basic geometry is really a prerequisite, or corequesite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another Algebra II text, used in an Honors &lt;a href=&quot;https://canvas.instructure.com/courses/1045286/wiki&quot; rel=&quot;noreferrer&quot;&gt;Algebra II course&lt;/a&gt;, with each chapter of the text available freely for download in pdf format, can be found &lt;a href=&quot;https://canvas.instructure.com/courses/1045286/pages/online-textbook-algebra-ii&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. You'll find additional supplementary material to download at linked at the bottom of page at the bottom of the &lt;a href=&quot;https://canvas.instructure.com/courses/1045286/wiki&quot; rel=&quot;noreferrer&quot;&gt;Algebra II course page first linked above.&lt;/a&gt;. This text has a trigonometry unit, as well as a unit in probability and statistics.&lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2019-09-29T15:20:54.093" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17200" PostTypeId="2" ParentId="17181" CreationDate="2019-09-29T15:29:53.813" Score="2" Body="&lt;p&gt;Allow me to offer another example:&lt;br&gt;&#xA;Imagine you and your best friend both want to buy a new smart phone. The phone you have chosen will cost you 300€ but your friend chooses a phone that will cost as much as 600€! Luckily, you have two vouchers that will give you a discount:&lt;br&gt;&#xA;The first voucher will give you the cheaper phone for free, if you buy two phones.&lt;br&gt;&#xA;The second voucher will give you 50% off your entire purchase.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which voucher do you choose?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ratios are a tricky thing, because the ratio of two large numbers can be the same as the ratio of two small numbers. But in most real life applications, the number of elements used to determine these ratios (i.e. averages) are just as important. Because an average over a great number of elements (e.g. people in a survey) is more meaningful than an average over a select few. Therefore, they can not always be compared one-to-one.&lt;/p&gt;&#xA;" OwnerUserId="12792" LastActivityDate="2019-09-29T15:29:53.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17201" PostTypeId="2" ParentId="17192" CreationDate="2019-09-29T19:49:11.720" Score="5" Body="&lt;p&gt;As @Adam noted, a function is Riemann integrable if and only if it is almost everywhere continuous. Giving a proof in terms of the Darboux integral (upper and lower sums) would also be rigorous. However, all of these might be a bit mysterious to a high school calculus student. The main pedagogical point to make with a problem like this would be along the lines of noting that not every function is integrable and that we need to pay attention to hypotheses and definitions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I would take the following approach when trying to explain &lt;em&gt;intuitively&lt;/em&gt; why this function isn't Riemann integrable&amp;mdash;citing a theorem from measure theory feels a bit unsatisfying. Basically, if we're going to move into the weird word of measure theory, let's start by just dipping in a toe (the indicator function on the rationals), gaining some intuition, and then building towards the function given in the problem.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Show that between any two rational numbers there is an irrational number and vice versa. If you're comfortable with measure theory, you could discuss the measures of the rationals and irrationals. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Analyze the indicator function of the rationals:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\chi_\mathbb{Q} (x) := \begin{cases} 1 &amp;amp; x \in \mathbb{Q}, \\&#xA; 0 &amp;amp; x \in \mathbb{R}\setminus \mathbb{Q}. \end{cases}$$&lt;/span&gt;&#xA;Discuss that it is indeed a well-defined function, that it is nowhere continuous, and is not Riemann integrable. The numbers here are much easier to deal with than the original function! You could even show how to create a sequence of Riemann sums on the interval &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; such that the limit is any number between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Repeat for the function &lt;span class=&quot;math-container&quot;&gt;$g(x) = x\cdot \chi_\mathbb{Q}(x)$&lt;/span&gt;. Be sure to note that this function is continuous at only a single point! Alternatively, see if they can come up with this one on their own--ask for a well-defined function &lt;span class=&quot;math-container&quot;&gt;$g:\mathbb{R} \to \mathbb{R}$&lt;/span&gt; that is continuous at only a single point.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally we can analyze the given function &lt;span class=&quot;math-container&quot;&gt;$f(x) = x+ x\cdot\chi_\mathbb{Q}(x)$&lt;/span&gt;, observing that it has many of the same properties as the previous two functions. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As a coda, note that &quot;infinitely many disconintuities&quot; is not the same concept as &quot;almost everywhere (dis)continuous&quot;. The canonical example would be &lt;a href=&quot;https://en.wikipedia.org/wiki/Thomae%27s_function&quot; rel=&quot;noreferrer&quot;&gt;Thomae's function&lt;/a&gt;, defined by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x) := \begin{cases} \frac{1}{q} &amp;amp; x=\frac{p}{q}\in\mathbb{Q} \text{ in lowest terms, with $p \in \mathbb{Z}$ and $q \in \mathbb{N}$,} \\ 0 &amp;amp; x \in \mathbb{R}\setminus \mathbb{Q}. \end{cases}.$$&lt;/span&gt;&#xA;This function has the following curious properties:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It is indeed a well-defined function.&lt;/li&gt;&#xA;&lt;li&gt;It is discontinuous at every rational number and continuous at every irrational number (straightforward &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proof).&lt;/li&gt;&#xA;&lt;li&gt;It is Riemann integrable! It's easiest to just draw boxes, but a rigorous proof is reasonably straightforward as well. Intuitively, there are only so many spikes in the function over any given height and these get contained in narrower and narrower rectangles.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you introduce Thomae's function, you will invariably be asked if there is a function that is instead continuous at all the rationals and discontinuous at all the irrationals. It can be worth having them think about it for a few days, but the answer is no as this would contradict the Baire Category Theorem. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7260" LastEditorUserId="8571" LastEditDate="2019-09-29T22:53:05.553" LastActivityDate="2019-09-29T22:53:05.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17202" PostTypeId="2" ParentId="17181" CreationDate="2019-09-30T01:49:51.447" Score="2" Body="&lt;p&gt;Think of an example with two ratios: 1/3 and 4/5.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you add the numerators, and divide this by the sum of the denominators, you get (1 + 4)/(3 + 5) = 5/8. Now, think about what is happening with the denominators - the denominator of the first ratio should only act on the first numerator. But instead, when you add the ratios in this way, the denominator of the first ratio is acting on both numerators. Likewise for the second denominator. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, when you &lt;em&gt;average&lt;/em&gt; the two ratios, like this&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1/3 + 4/5) / 2 = (5/15 + 12/15) / 2 = 17/30&lt;/p&gt;&#xA;&#xA;&lt;p&gt;the denominators of the ratios work together to act on the numerators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Division has a higher order of precedence than addition. Thus, division should be carried out first, rather than combining the ratios.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this makes sense and helps you assist them.&lt;/p&gt;&#xA;" OwnerUserId="12794" LastActivityDate="2019-09-30T01:49:51.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17203" PostTypeId="1" CreationDate="2019-09-30T07:13:47.633" Score="5" ViewCount="286" Body="&lt;p&gt;I am teaching mathematical proof to kids (10th grade) and am of the opinion that proofs of theorems are a good place to start, where almost all of mathematics' important players come together. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On one side, we have logical structures like propositions, consequence, rules of inferences, propositional logic/calculus, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we have formal system, axiomatic system of natural numbers say, on the other side, which has structures of math like axioms, primitive notions/functions, theorems, lemmas etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then we have proof itself where there are various techniques like direct proof/ indirect proof, reasoning (deductive reasoning), using various axioms, theorems etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kids usually struggle with every one of these concepts, let alone all of them together. It is difficult to get the whole picture and all the moving parts. So this place (proof) seems like a good place to show all of this in action. I could make &lt;a href=&quot;https://venngage.com/blog/what-is-an-infographic/&quot; rel=&quot;nofollow noreferrer&quot;&gt;infographics&lt;/a&gt; myself, but it would take some time. So I wanted to know whether someone &lt;em&gt;already&lt;/em&gt; has done such and is willing to share?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if I were to create such an infographic, how do you suggest I begin, and what, in your opinion, &lt;em&gt;must&lt;/em&gt; it include?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[Edit]&lt;/strong&gt;: I uploaded an initial version of infographic to github, please click &lt;a href=&quot;https://github.com/shukash/Mathematical-Proof-Infographic/blob/master/Mathematical%20Proofs.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; to see. I VERY EAGERLY wait for your INVALUABLE feedback/suggestions to improve this infographic...&lt;/p&gt;&#xA;" OwnerUserId="12610" LastEditorUserId="12610" LastEditDate="2019-10-30T06:27:27.627" LastActivityDate="2019-10-30T06:27:27.627" Title="Are there mathematical proof info-graphics?" Tags="|proofs|logic|discrete-math|visual-proof|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="17204" PostTypeId="2" ParentId="17181" CreationDate="2019-09-30T07:59:20.400" Score="4" Body="&lt;p&gt;The problem is with the question, not with the students' answers. The question is ambiguous and I think the students' answer is actually much better than yours.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose I drive a thousand miles at 25mpg and you drive one mile at 35mpg. What's the average fuel efficiency? Your answer is 30mpg but I honestly can't think of any situation in which that is a meaningful or useful number. The students' answer is 25-and-a-bit mpg, which is a good measure of how far it can be expected to travel on one gallon of fuel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analogously, if I buy a thousand apples at 50c each and you buy one apple at 40c, you're claiming that the average price per apple is 45c, and your students are claiming that it's a hair under 50c.&lt;/p&gt;&#xA;" OwnerUserId="1285" LastActivityDate="2019-09-30T07:59:20.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17207" PostTypeId="2" ParentId="17161" CreationDate="2019-09-30T20:10:05.973" Score="0" Body="&lt;p&gt;I'm not sure if I understand your point here: your students want to keep their course notes clean and don't want to mix correct calculations with wrong ones. You, however, want to add wrong calculations, in order for them to learn from those mistakes, and you are looking for quotes to support your point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I, however, see that your students have a valid point. You do too. So, instead of forcing one point of view, I'll try to mix both:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You can tell your students to write wrong calculations in their course, but they need to do this in another colour (green, for instance). That way, their course (written in blue or black) will still be completely correct, taking into account that the green text needs to be understood in another way (as the way NOT to calculate something).&lt;/li&gt;&#xA;&lt;li&gt;You can tell your students to flip their course notes, and write the wrong calculations in the backside of their notes. Like this, the wrong calculations will still be in their notes but in such a way that it does not disturb the rest of their notes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2019-09-30T20:10:05.973" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17208" PostTypeId="2" ParentId="17203" CreationDate="2019-09-30T21:04:52.093" Score="4" Body="&lt;p&gt;&lt;strong&gt;Math With Bad Drawing&lt;/strong&gt; has some images that approach an info-graph (and in general is just  a great website for math education), for example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathwithbaddrawings.com/2015/07/01/infinity-plus-one-please-check-your-intuitions-at-the-front-desk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathwithbaddrawings.com/2015/07/01/infinity-plus-one-please-check-your-intuitions-at-the-front-desk/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are some good geometry ones, especially around old compass and straight-edge constructions but that wont really give you rules of inference. For instance&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.noborubitoy.com/?p=119&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.noborubitoy.com/?p=119&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, you could look into &lt;strong&gt;flowchart proofs&lt;/strong&gt;, but I haven't found many graphically nice examples. Here is an example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/teaching-and-learning/flowcharting-proofs&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/teaching-and-learning/flowcharting-proofs&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll be following this question, I've been looking for some decent proof visualizations as well. &lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2019-09-30T21:04:52.093" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17209" PostTypeId="2" ParentId="17203" CreationDate="2019-09-30T22:17:06.053" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Kids usually struggle with every one of these concepts, let alone all of them together. It is difficult to get the whole picture and all the moving parts. So this place (proof) seems like a good place to show all of this in action.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm afraid there is no &quot;complete picture&quot; of all the facets of logic and proof that you mention, interrelating in one infograph.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some examples of infographs for some components you'd like to include in a combined infograph:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://infograph.venngage.com/p/196211/laws-of-logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Laws of logic&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://infograph.venngage.com/p/102614/math-and-logic-thinking&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math and Logic Thinking&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.informationisbeautiful.net/visualizations/rhetological-fallacies/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Logical fallacies&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=logic+diagram+flowcharts&amp;amp;tbm=isch&amp;amp;source=univ&amp;amp;client=firefox-b-1-d&amp;amp;sa=X&amp;amp;ved=2ahUKEwjyg6T3w_nkAhUxh-AKHeyJCNAQ7Al6BAgCECQ&amp;amp;biw=1366&amp;amp;bih=642&quot; rel=&quot;nofollow noreferrer&quot;&gt;logic diagram flowcharts&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you decide to work on this project yourself, &lt;a href=&quot;https://www.google.com/search?client=firefox-b-1-d&amp;amp;q=tools+for+infographics&quot; rel=&quot;nofollow noreferrer&quot;&gt;here are some tools for creating infographs&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2019-09-30T22:17:06.053" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17211" PostTypeId="2" ParentId="17161" CreationDate="2019-10-01T22:30:59.597" Score="4" Body="&lt;p&gt;I appreciate Ben Orlin's &amp;quot;Math with Bad Drawings&amp;quot; &lt;a href=&quot;https://mathwithbaddrawings.com/2017/09/20/the-state-of-being-stuck/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;post&lt;/strong&gt;&lt;/a&gt; about Andrew Wiles, for which the theme is not &lt;strong&gt;mistakes&lt;/strong&gt; but rather &lt;strong&gt;being stuck&lt;/strong&gt;. I have the sense that it contains some relevant content for the question that is asked here; for example, consider the following excerpt (&lt;strong&gt;emphasis&lt;/strong&gt; added by me):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For example, take Wiles’ musings on the value of forgetfulness. “I think it’s bad to have too good a memory if you want to be a mathematician,” Wiles said. “You need to forget the way you approached [the problem] the previous time.”&lt;/p&gt;&#xA;&lt;p&gt;It goes like this. You try one strategy on a problem. It fails. You retreat, dispirited. Later, having forgotten your bitter defeat, you try the same strategy again. Perhaps the process repeats. &lt;strong&gt;But eventually—again, thanks to your forgetfulness—you commit a slight error, a tiny deviation from the path you’ve tried several times. And suddenly, you succeed.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Wiles has a nifty analogy for this: it’s like a chance mutation in a strand of DNA that yields surprising evolutionary success.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8OjML.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8OjML.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“If you remember all the false, failed attempts before,” said Wiles, “you wouldn’t try. But because I have a slightly bad memory, I’ll try essentially the same thing again, and then I’ll realize I was just missing this one little thing.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-10-01T22:30:59.597" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17212" PostTypeId="1" CreationDate="2019-10-01T23:46:53.647" Score="4" ViewCount="190" Body="&lt;p&gt;Someone asks the following question:&#xA;&lt;em&gt;Determine the values for the real numbers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; such that&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to0}\frac{ae^x+b\cos x}{x}=2$$&lt;/span&gt;&#xA;The students directly apply de LHospital's rule for the quotient. I asked them to justify why by applying this rule is sufficient and i got no answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How would you explain this to a student?&lt;/p&gt;&#xA;" OwnerUserId="12800" LastActivityDate="2019-10-02T20:13:21.147" Title="(use of de L'Hospital's rule) How would you explain this limit to high school students?" Tags="|limits|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17213" PostTypeId="2" ParentId="13460" CreationDate="2019-10-02T00:00:34.427" Score="1" Body="&lt;p&gt;It is possible for an expression to be undefined. If you work in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;, you can consider &lt;span class=&quot;math-container&quot;&gt;$2 - 3$&lt;/span&gt; to be undefined. We just have meaningless expressions. Similarly when working in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, we can also consider the expression &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0}$&lt;/span&gt; to be a meaningless expression. &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0}$&lt;/span&gt; is considred undefined because &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0}$&lt;/span&gt; means the number that if you multiply by 0 you get 1 but there is no such number. If that doesn't work, here's an alternate explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, we could define &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; with addition, multiplication, and division in such a way to satisfy the desirable properties. Suppose we defined &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0}$&lt;/span&gt; to actually represent a number. Then we find that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0} \times 0 = 0$&lt;/span&gt; but then the property that &lt;span class=&quot;math-container&quot;&gt;$\frac{x}{y} \times y = x$&lt;/span&gt; when ever &lt;span class=&quot;math-container&quot;&gt;$\frac{x}{y}$&lt;/span&gt; is defined would be lost. Then maybe we could instead define that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0} \times 0 = 1$&lt;/span&gt; inventing an infinite number and defining &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0}$&lt;/span&gt; to equal that number. Then we would find that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0} \times 0 = 1 \neq 2 = (\frac{1}{0} \times 0) + (\frac{1}{0} \times 0) = \frac{1}{0} \times (0 + 0)$&lt;/span&gt; so the distributivity of multiplication over addition would be lost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If after explaining all that, the person who was wondering how &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0}$&lt;/span&gt; could be undefined still knows how to derive the contradiction but cannot make any sense of it and is all confused, they may eventually figure out the answer to their own question. They may realize that it is a matter of how you define the number system and its operations and in order to define it in a way that avoids other problems, you have to define the system in such a way that multiplication by 0 always gets you 0 and saying &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0}$&lt;/span&gt; is another way of saying the number that if you multiply by 0 gets you 1 but there is no such number so &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0}$&lt;/span&gt; is a meaningless expression according to that definition.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastActivityDate="2019-10-02T00:00:34.427" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17214" PostTypeId="2" ParentId="17212" CreationDate="2019-10-02T00:14:14.297" Score="4" Body="&lt;p&gt;Remind the students of the &lt;a href=&quot;https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule&quot; rel=&quot;nofollow noreferrer&quot;&gt;requirements for L'Hôpital's rule&lt;/a&gt;. Use the requirements as stated in your textbook--the requirements may have been simplified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The students should quickly see that one requirement is that the limit of the numerator of the fraction must be zero or infinity. The textbook may leave out infinity--if infinity is included, point out the limit of this particular numerator cannot be infinity. Then have the students find the limit of the numerator and set that to zero to get another requirement on &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, go back to the definition (of L'Hôpital's rule), which is a major problem solving technique. You could emphasize the importance of checking conditions by showing the old &quot;proof&quot; that zero equal one (the proof uses division by zero).&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2019-10-02T00:14:14.297" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17215" PostTypeId="1" CreationDate="2019-10-02T02:52:07.510" Score="5" ViewCount="211" Body="&lt;p&gt;In prob/stats, students mostly get the idea of mean/expected value, and variance/s.d. slowly sinks in.  Even the third and fourth moments (about the mean) have &quot;standard&quot; interpretations - skewness and kurtosis.  The latter is something one can at least think of geometrically, e.g in terms of the tails.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have ideas for a good, student-centered geometric interpretation (or other intuitive one) for higher moments?  I realize that there may not be any great ones - see for instance &lt;a href=&quot;https://stats.stackexchange.com/questions/123251/intuition-for-moments-about-the-mean-of-a-distribution&quot;&gt;this answer&lt;/a&gt; - but even a nice graphic describing 3rd versus 5th moment, or how the moments change as they go higher (even versus odd) would be useful.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Yes, I did get this question.  To be fair, it's a math major course, not intro stats for social sciences or something.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="11" LastEditDate="2019-10-05T00:51:29.900" LastActivityDate="2019-10-14T19:51:47.763" Title="How to explain even higher moments" Tags="|statistics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17216" PostTypeId="1" AcceptedAnswerId="17218" CreationDate="2019-10-02T03:28:01.960" Score="4" ViewCount="218" Body="&lt;p&gt;I feel very confused when teaching students how to sketch a parametric curve like &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x(t)=e^{-t}t; y(t)=t^{2}+t.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here students are supposed to know derivatives, increasing-decreasing,... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In literature, I haven't found any example for this type of problems.&lt;/p&gt;&#xA;" OwnerUserId="12801" LastEditorUserId="1147" LastEditDate="2019-10-03T02:58:49.017" LastActivityDate="2019-10-03T13:56:16.677" Title="How to teach sketching a parametric curve?" Tags="|calculus|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17217" PostTypeId="2" ParentId="17216" CreationDate="2019-10-02T05:32:40.630" Score="3" Body="&lt;p&gt;This is a standard part of the calculus course.  Most books will have an explanation, examples, and problems.  (Which is how to learn the topic, or any topic: explanation (or general case), examples, drill.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I looked at a year old text (Granville) and it has this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf&lt;/a&gt; (page 79-83).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But so does Thomas Finney, Swokowski, etc. Just go to the library and browse some different texts.  (Of the three I hold, I like the 80s Thomas Finney the best...most applied.  But Granville emphasizes curve tracing more and Swokowski emphasizes ideas about smoothness and uniqueness more.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps you need to sit down and work some of the problems yourself, to get more comfortable teaching it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  Not sure what you mean by &quot;literature&quot; (on teaching or on the topic, itself).  But if you're looking for applications, the common one is kinematics (e.g. artillery shell physics problems).&lt;/p&gt;&#xA;" OwnerUserId="12802" LastActivityDate="2019-10-02T05:32:40.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17218" PostTypeId="2" ParentId="17216" CreationDate="2019-10-02T08:18:00.000" Score="6" Body="&lt;p&gt;In general there is no standard method for sketching parametric curves unless the paremeterization has some particular form (which the example considered here does not, so far as I can tell). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sketching a parametric curve is like graphing a function. Actually, it directly generalizes graphing a function, because the graph of &lt;span class=&quot;math-container&quot;&gt;$f(t)$&lt;/span&gt; is the trace of the parameterized curve &lt;span class=&quot;math-container&quot;&gt;$(t, f(t))$&lt;/span&gt;. To graph &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, one finds its zeros, its asymptotic behavior, its critical points, etc. To sketch a parameterized curve, one does the same with each coordinate function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this example &lt;span class=&quot;math-container&quot;&gt;$(x(t), y(t)) = (e^{-t}t, t^{2} + t)$&lt;/span&gt; one should focus on the simple and qualitative features. First, when &lt;span class=&quot;math-container&quot;&gt;$t = 0$&lt;/span&gt; the curve passes through the origin. When &lt;span class=&quot;math-container&quot;&gt;$t \to \infty$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x(t) \to 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y(t) \to + \infty$&lt;/span&gt;, while when &lt;span class=&quot;math-container&quot;&gt;$t \to -\infty$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x(t) \to -\infty$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y(t) \to + \infty$&lt;/span&gt;. In this case &lt;span class=&quot;math-container&quot;&gt;$y(t)$&lt;/span&gt; has a unique critical point, when &lt;span class=&quot;math-container&quot;&gt;$t = -1/2$&lt;/span&gt;, which is a local minimum, and &lt;span class=&quot;math-container&quot;&gt;$x(t)$&lt;/span&gt; has a unique critical point, when &lt;span class=&quot;math-container&quot;&gt;$t = 1$&lt;/span&gt;, which is a local maximum. Also &lt;span class=&quot;math-container&quot;&gt;$x(t)$&lt;/span&gt; is positive for positive &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$y(t)$&lt;/span&gt; changes sign at &lt;span class=&quot;math-container&quot;&gt;$t = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t = -1$&lt;/span&gt;. Thus the curve passes through the origin, &lt;span class=&quot;math-container&quot;&gt;$(-e, -1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(-e^{1/2}/2, -1/4)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(e^{-1}, 1)$&lt;/span&gt;, crossing the horizontal axis at &lt;span class=&quot;math-container&quot;&gt;$(-e, -1)$&lt;/span&gt;, staying always above &lt;span class=&quot;math-container&quot;&gt;$(-e^{1/2}/2, -1/4)$&lt;/span&gt; staying always to the left of &lt;span class=&quot;math-container&quot;&gt;$(e^{-1}, 1)$&lt;/span&gt;, and staying always to the right of the vertical axis when &lt;span class=&quot;math-container&quot;&gt;$t &amp;gt; 0$&lt;/span&gt;, and one knows how it behaves as &lt;span class=&quot;math-container&quot;&gt;$t \to \pm \infty$&lt;/span&gt;, one knows that &lt;span class=&quot;math-container&quot;&gt;$x(t) &amp;gt; 0$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$t &amp;gt; 0$&lt;/span&gt;. All this information is enough to make a qualitatively correct sketch of the curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general there is not much else one can do but:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Look at where the curve goes for particular values of &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Find the asymptotic behavior of each coordinate function.&lt;/li&gt;&#xA;&lt;li&gt;See where the coordinate functions change signs.&lt;/li&gt;&#xA;&lt;li&gt;See where the coordinate functions have critical points, and find the types of these critical points. &lt;/li&gt;&#xA;&lt;li&gt;Look at convexity of the coordinate functions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-10-02T08:18:00.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17219" PostTypeId="2" ParentId="17212" CreationDate="2019-10-02T08:24:21.560" Score="3" Body="&lt;p&gt;L'Hopital's rule is a horrid thing that should be banned from introductory calculus courses because it can and is used by those who do not understand it, and the problems it is used to solve are generally better solved by other means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question should be asked only of students who understand the Taylor approximations of the exponential and cosine. Then, since&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;ae^{x} + b\cos{x} &amp;amp;= a( 1+  x + \text{terms quadratic in $x$}) + b(1 + \text{terms quadratic in $x$}) \\&#xA;&amp;amp; = a + ax + b + \text{terms quadratic in $x$}&#xA;\end{align*}&lt;/span&gt;&#xA;it follows that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{ae^{x} + b\cos{x}}{x} = \frac{a+b}{x} + a + \text{terms linear in $x$}$$&lt;/span&gt;&#xA;and so to obtain &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x \to 0$&lt;/span&gt; it must be that &lt;span class=&quot;math-container&quot;&gt;$b = -a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a = 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course this is not a rigorous or careful argument, but all the &lt;em&gt;understanding&lt;/em&gt; is there.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-10-02T08:24:21.560" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17220" PostTypeId="1" CreationDate="2019-10-02T13:07:35.167" Score="-1" ViewCount="153" Body="&lt;p&gt;There are many university ranking consultancies trying to compare one leading maths department with another and to conclude which one is better. Although this doesn't seem to be a very reasonable thing to do (since you cannot compare apples and oranges), many &lt;strong&gt;high school&lt;/strong&gt; students still see such ranking as an important factor when deciding where to go. An example of ranking can be found &lt;a href=&quot;https://www.topuniversities.com/university-rankings/university-subject-rankings/2017/mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;here.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: why, for example, is MIT placed above Oxford (and many other examples) &lt;strong&gt;in most rankings&lt;/strong&gt;? Although different organisations give vastly different rankings, MIT seems to be &lt;strong&gt;always&lt;/strong&gt; above Oxford.  Is there any real difference between their &lt;strong&gt;undergraduate&lt;/strong&gt; education or is it just that the criteria of rankings are unfair? (&lt;strong&gt;NB&lt;/strong&gt; DO NOT focus too much on MIT and Oxford; they are just examples.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to limit the discussion to &lt;strong&gt;undergraduate&lt;/strong&gt; education only: &lt;strong&gt;does such ranking really help when choosing which college to go to?&lt;/strong&gt; Also, be aware that I am NOT talking about the overall ranking of universities; I am just talking about the ranking of &lt;strong&gt;math departments&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe this question is not very opinion-based - it can be answered &lt;em&gt;objectively&lt;/em&gt; by a detailed examination of the course contents and teaching methods, and how those are related to rankings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware that those rankings are not solely about undergraduate teaching, but chances are that high school students care about the rankings more than anyone else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarise, &lt;strong&gt;are rankings helpful in assessing the relative merits of undergraduate education at the maths department of different institutions?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="9339" LastEditDate="2019-10-04T08:25:28.407" LastActivityDate="2019-11-24T19:01:55.230" Title="How helpful are university subject rankings when choosing a place to study math?" Tags="|undergraduate-education|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17221" PostTypeId="2" ParentId="17212" CreationDate="2019-10-02T18:02:13.260" Score="1" Body="&lt;p&gt;Define &lt;span class=&quot;math-container&quot;&gt;$f(x):=ae^x+b\cos x,\,g(x)=x$&lt;/span&gt;. As &lt;span class=&quot;math-container&quot;&gt;$x\to0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\color{blue}{g\to0}$&lt;/span&gt; while &lt;span class=&quot;math-container&quot;&gt;$\color{blue}{g^\prime\to1\ne0}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\frac{f}{g}\to2$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$\color{blue}{f\to2\times0=0}$&lt;/span&gt;. The blue conditions are what need to be checked for the rule's antecedent.&lt;/p&gt;&#xA;" OwnerUserId="5794" LastActivityDate="2019-10-02T18:02:13.260" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17222" PostTypeId="1" CreationDate="2019-10-02T18:31:16.933" Score="0" ViewCount="142" Body="&lt;p&gt;I want to emphasize the aspect of environmental education in my math class. Now I'm reasoning whether to do that with linear inequalities or terms with two variables - these are our next topics. The problem is more or less that I'm lacking a good idea. I would like to treat climate change or waste in the environment, but haven't got a good example where the terms or inequalities are a means to model the facts. So I hope you can give me a hint for such an example.&lt;/p&gt;&#xA;" OwnerUserId="10036" LastEditorUserId="11" LastEditDate="2019-10-05T00:51:10.943" LastActivityDate="2019-10-05T04:42:02.127" Title="Examples for environmental topics in the context of terms or linear inequalities" Tags="|algebra|examples|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17223" PostTypeId="2" ParentId="17222" CreationDate="2019-10-02T19:57:25.523" Score="2" Body="&lt;p&gt;A new car has a carbon footprint of &lt;span class=&quot;math-container&quot;&gt;$15.25$&lt;/span&gt; tons of CO2.  Each year you drive a car, you contribute &lt;span class=&quot;math-container&quot;&gt;$4.6$&lt;/span&gt; tons  of CO2.  If you want to buy a new car, and emit less than &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; tons of CO2 total over your lifetime, how many years could you drive the car?  Would it be realistic to drive your car for that many years?  What other assumptions in this model might be unrealistic?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could spice it up a bit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each tree you plant can absorb 48 pounds of carbon per year.  You want to be able to drive the rest of your life with a clean conscience.  About how many trees do you need to plant over your lifetime?  How many acres of forest would that be?  In your explanation, be sure to clarify any assumptions you make.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2019-10-02T19:57:25.523" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17225" PostTypeId="2" ParentId="17216" CreationDate="2019-10-03T02:13:38.180" Score="6" Body="&lt;p&gt;When I introduce parametric curves in Calculus 3, I like to bring my &lt;em&gt;Etch A Sketch&lt;/em&gt; to class. It is a drawing toy from about 1960. The knob on the left encodes &lt;span class=&quot;math-container&quot;&gt;$x(t)$&lt;/span&gt;, and the knob on the right &lt;span class=&quot;math-container&quot;&gt;$y(t)$&lt;/span&gt;. A coordinated person can compose a circle. This proves to be a memorable segue to a short but longer discussion about Lissajous figures. There are many tactile ways to experience Lissajous figures, and good apps on the web to explore them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Etch A Sketch seems rather silly and retro, but when I meet former students a few years after the class, they always remember this. One told me that when he took a physics class a year later and used and oscilloscope in a lab, he immediately thought of the Etch A Sketch and understood the oscilloscope device. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ksyCm.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ksyCm.jpg&quot; alt=&quot;Etch a Sketch picture from Wiki&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, in class one needs to move on quickly to the calculus of parametric curves with things like &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}=\frac{\frac{dy}{dx}}{\frac{dx}{dt}}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ds=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\, dt$&lt;/span&gt;. But there is lots of fun physics lurking under parametric curves, so taking some time to plant a future memory connection is fruitful. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-10-03T02:13:38.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17226" PostTypeId="2" ParentId="17181" CreationDate="2019-10-03T09:58:50.857" Score="0" Body="&lt;p&gt;I think that the most interesting application of (sum of numerators/sum of denominators) “addition” is to continued fractions. For example, if you want to calculate the continued fraction expansion of the square root of 2,start with 1/0 and 0/1 and “add” them in the following way:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/i8qV7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/i8qV7.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the top row you put the results of the “adding” that are greater then the square root of 2 , in the bottom row the results that are less then the square root of 2, and you always “add” the last two results from the different sides of the square root of 2 . The pattern of 1 down, 2 up, 2 down, 2 up, 2 down, 2 up, … is the continued fraction expansion of the square root of 2 :&#xA;&lt;a href=&quot;https://i.stack.imgur.com/mz1xm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mz1xm.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10261" LastActivityDate="2019-10-03T09:58:50.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17227" PostTypeId="2" ParentId="17216" CreationDate="2019-10-03T12:27:56.333" Score="1" Body="&lt;p&gt;I find it very useful to draw random functions of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; on two distinct sets of axes, and then have the students try to draw the parameterized curve in the &lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt; plane.  This forces them to think about what they are doing in a very direct way, without any algebra or function notation getting in the way.  I spend a considerable amount of time on this activity (1 or 2 50 minute lessons if possible, and I include such a question on the exam).&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2019-10-03T12:27:56.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17228" PostTypeId="2" ParentId="17216" CreationDate="2019-10-03T13:56:16.677" Score="3" Body="&lt;p&gt;I once taught as shown below (before computer graphics were widely available), but have stopped. Beyond making a table of points and connecting the dots and knowing a few basic examples (circle, cartesian graph, some polar-coordinate graphs), such intricate skill with sketching parametrized curves seems to add little to the education of the students (imo &amp;amp; my department's).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, there's a 2 x 2 grid of graphs, starting from the top left and proceeding clockwise: y vs. t, y vs. x, t vs. x, t vs. t.  Note that the t-axis in the top-left is horizontal and in the bottom-right is vertical.  Then pick a t value in the bottom left.  Use a ruler or the edge of a piece of paper to find the x, y coordinates in the bottom-right, top-left plots (respectively), and transfer them to the top-right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DDyJH.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DDyJH.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addendum:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematica code for the graphics (may be run on wolframcloud.com, but dynamic updating is run server-side and performs slowly in a browser):&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;ClearAll[x, y, t, pr, plot];&#xA;With[{x = Exp[t] t, y = t^2 + t, tdom = {t, -2, 2}},&#xA; plot[a_, b_, t0_] := &#xA;  ParametricPlot[{a, b}, tdom, PlotRange -&amp;gt; {pr[a], pr[b]}, &#xA;   GridLines -&amp;gt; {{a /. t -&amp;gt; t0}, {b /. t -&amp;gt; t0}},&#xA;   GridLinesStyle -&amp;gt; LightRed,&#xA;   Epilog -&amp;gt; {Red, Point[{a, b} /. t -&amp;gt; t0]}];&#xA; pr[t] = {-2, 2}; pr[x] = {-1, 3}; pr[y] = {-1, 3};&#xA; Manipulate[&#xA;   With[{&#xA;     xt = plot[x, t, t0],&#xA;     ty = plot[t, y, t0],&#xA;     tt = plot[t, t, t0],&#xA;     xy = plot[x, y, t0]},&#xA;    Graphics[{&#xA;      Inset[ty, {0, 6}],&#xA;      Inset[tt, {0, 0}],&#xA;      Inset[xy, {6, 6}],&#xA;      Inset[xt, {6, 0}]&#xA;      }]&#xA;    ],&#xA;   {t0, -2, 1}]&#xA; ]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="498" LastActivityDate="2019-10-03T13:56:16.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17229" PostTypeId="5" CreationDate="2019-10-03T18:21:09.163" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2019-10-03T18:21:09.163" LastActivityDate="2019-10-03T18:21:09.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17230" PostTypeId="4" CreationDate="2019-10-03T18:21:09.163" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2019-10-03T18:21:09.163" LastActivityDate="2019-10-03T18:21:09.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17232" PostTypeId="1" CreationDate="2019-10-04T03:06:07.493" Score="7" ViewCount="212" Body="&lt;p&gt;In undergraduate maths study, there are three main areas: analysis, algebra, and geometry. (There are of course other small topics as well, but they don't have to be learnt by every student.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have discovered that, in most standard undergraduate curricula, there are more analysis topics than modern algebra topics. (Many words can be put before &quot;analysis&quot;, including &quot;real&quot;, &quot;complex&quot;, &quot;functional&quot;, &quot;harmonic&quot;, etc.) Geometry, on the other hand, only forms a very small portion of the courses taken. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why this is the case?&lt;/strong&gt; Why is analysis more &quot;important&quot; than algebra or anything else?&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="238" LastEditDate="2019-10-04T13:21:01.723" LastActivityDate="2019-10-05T04:02:50.727" Title="Why is there an emphasis on analysis courses in undergrad progams?" Tags="|undergraduate-education|mathematical-analysis|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17233" PostTypeId="2" ParentId="17220" CreationDate="2019-10-04T05:30:15.950" Score="1" Body="&lt;p&gt;Rankings are imprecise (e.g. may differ by source).  In addition, gradation of number 2 versus number 4 (or the like) is rather a nuance, when you consider the restriction of range.  More meaningful is top 5 verus top 50 or 500 or the like.  Also, any ranking of schools from one country to another is problematic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, prestige of a school seems to correlate to selectivity more than quality of the instruction.  So, yes, you can collect a nice brand (with value, agreed), but don't kid yourself that it is much more than that.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally rankings are not the most important thing in the world.  Lots of people go to state schools and do fine in life.  &lt;/p&gt;&#xA;" OwnerUserId="12810" LastEditorUserId="12810" LastEditDate="2019-10-04T05:49:25.450" LastActivityDate="2019-10-04T05:49:25.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17234" PostTypeId="2" ParentId="17232" CreationDate="2019-10-04T06:31:17.017" Score="4" Body="&lt;h1&gt;Analysis is useful&lt;/h1&gt;&#xA;&lt;p&gt;Physics uses analysis. Engineering uses analysis.&lt;/p&gt;&#xA;&lt;p&gt;Continuous models are widespread everywhere, and typically they look like differential equations, if there is time involved.&lt;/p&gt;&#xA;&lt;p&gt;Finance uses lots of analysis.&lt;/p&gt;&#xA;&lt;p&gt;Note that a lot of modern geometry is differential geometry, which builds on a foundation of analysis.&lt;/p&gt;&#xA;&lt;p&gt;Certainly, other parts of mathematics are also used, but they seem to not be as fundamental yet. Note also that many students learn statistics for the same reason - it is used widely.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-10-04T06:31:17.017" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17236" PostTypeId="2" ParentId="17232" CreationDate="2019-10-05T04:02:50.727" Score="2" Body="&lt;p&gt;(Too long for a comment and it is kind of a soft question anyways)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not so sure that your assumption of an imbalance is valid.  Maybe it is.  But would be better if demonstrated (or at least explored) first. Otherwise, we end up finding an explanation for a phenomenon that doesn't actually exist.  Plus the exploration would probably inform the answer, even if assumption shown correct.  (For example perhaps there are differences in schools with/without grad programs.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I looked at the course curriculum for USNA math and it seemed like there were about equal classes in algebra and analysis (maybe even more in algebra). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.usna.edu/Academics/Majors-and-Courses/course-description/SM.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.usna.edu/Academics/Majors-and-Courses/course-description/SM.php&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were two semesters of abstract algebra and three of linear algebra.  (Different difficulty &quot;tracks&quot; also, but I'm just talking the sequence.)  The formal analysis only included two semesters of real analysis and one of complex analysis.  Again with a confusing multiplicity of tracks, but just the three real semester courses.  No functional analysis as the institution is undergrad only.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess you could call the normal computational normal calculus courses, &quot;analysis&quot;.  Or ODEs, PDEs.  But I'm not so sure this is valid.  There's definitely a different feel for what people think of or use as calculus (solving problems) versus real analysis (theory).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, if you're going to count the foundational conventional calculus semesters, then I can always bring in the four (4!) semesters of high school algebra that students typically take in the US of A.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the actual explanation, I would say that &quot;calculus&quot; (and diffyQs is calculus on steroids) is useful in science and engineering.  But this is a different thing than the highfalutin math major &quot;analysis&quot; of your question.  So I think there are more likely other drivers for the math major interest in the topicm, other than physics applications.  Complex analysis, has way less application than calc 1-3 and diffyscrews. (Pedants:  I'm not saying there is never a single application, but not enough to make the topic a general STEM need and a required course.)  Real analysis even more so not needed, not required to be taken, etc. etc.  So don't blame your functional analysis courses on science and engineering.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note also, that HIGH SCHOOL algebra is a HUGE useful topic in science and engineering...freshman chem is essentially one giant algebra word problem.  (In a similar manner to how calculus and diffyQs are work horses.)  But did you even think about that when discussing algebra in the context of your question?&lt;/p&gt;&#xA;" OwnerUserId="12813" LastActivityDate="2019-10-05T04:02:50.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17237" PostTypeId="2" ParentId="17222" CreationDate="2019-10-05T04:23:20.397" Score="1" Body="&lt;p&gt;I suggest looking at some intermediate general science texts. Often there will be some simple computational problems involving very minor* algebra.  You might be able to find some that appeal to you although you will have to see what part of the algebra sequence they correspond to.  Green stuff has been of interest going back at least to the seventies.  And often 7th-8th grade general science (sort of like pre chemistry and pre physics, the way pre algebra is pre algebra) have either part or most of the curriculum around environmental topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would just caution you to keep it rather simple.  Word problems can be hard, in general.  And you don't want to pick ones that segue too much into needing to teach (or even explain) general science.  But you can still find examples.  But please make them extra easy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, you may be able to repurpose an example or exercise to use the physical example but repurpose it to your chapter of algebra.  For example they talk about biocentration (DDT getting worse as you go up the food chain) and in the context of ratios, but you can think how to steal the basic science concept and make up an example with an inequality or the like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Things like ratios, percents, unit conversions, etc.  And while these may be rather minor in math land, having facility with this sort of simple calculation is very important as a scientist, office worker, etc.&lt;/p&gt;&#xA;" OwnerUserId="12813" LastEditorUserId="12813" LastEditDate="2019-10-05T04:42:02.127" LastActivityDate="2019-10-05T04:42:02.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17238" PostTypeId="1" AcceptedAnswerId="17239" CreationDate="2019-10-05T09:50:38.833" Score="6" ViewCount="927" Body="&lt;p&gt;Why is it popular to teach modulus via the example of mod 12 and analogue clocks rather than rectangles or tables that have a finite number of columns in each row, and infinitely many rows?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's natural to identify an origin row that has the number one in the far left-hand column of the origin row, for people who write from left to right.  Then it can be indicated that the table extends infinitely far upward by filling in the blanks with zero and negative values, and the table also extends infinitely many rows down, just as for a Cartesian coordinate system.  Thus, every integer appears somewhere in the table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Focus on the clock analogy tends to restrict attention to mod 12, but of course it does also suggest going beyond integer values altogether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We could define mod &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt; as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;t is congruent to u (mod &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt;) &#xA;  if and only if&#xA;  &lt;span class=&quot;math-container&quot;&gt;$\sin(t) = \sin(u)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos(t) = \cos(u)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then we can generalize beyond &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt; to an arbitrary positive real number &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; by selecting a radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$2\pi r = m$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is that why the clock analogy is used?&lt;/p&gt;&#xA;" OwnerUserId="12816" LastEditorUserId="11" LastEditDate="2019-10-05T12:40:11.883" LastActivityDate="2019-10-05T12:40:11.883" Title="Why is it popular to teach modulus via the example of mod 12 and analogue clocks?" Tags="|mathematical-pedagogy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17239" PostTypeId="2" ParentId="17238" CreationDate="2019-10-05T12:25:19.393" Score="15" Body="&lt;p&gt;The answer is simple, practical, and historical. Analog clocks and watch faces were everywhere. We learned to tell time early on, by first or second grade, (age 6 or 7), so the leap to understanding Modulo-12 wasn't too tough. What's 9+6? 3, of course. And for the smart student who resists, insisting it's 15, the teacher offers the fact that military time actually calls this 15, but for regular citizens, whose clocks only show 1-12, we need to answer '3'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With respect to the rest of your question, offering trig, that was likely a pleasant extension of using the clock face, not the original reason. I recall, once we were comfortable with the clock concept, we moved on to other bases, also illustrated by a circle to maintain that familiarity. And the addition of multiple 'clock' faces to help show how each place represented the next order of magnitude in that base arithmetic.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-10-05T12:25:19.393" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17240" PostTypeId="2" ParentId="176" CreationDate="2019-10-05T19:25:00.430" Score="6" Body="&lt;p&gt;There are many aspects to this, and there are already some good answers here.  Just on one point:  I never heard of the &quot;expert blind spot&quot; before, but based on my own experience I have often told people that the hardest thing about teaching is remembering what it was like before you understood something.&lt;/p&gt;&#xA;" OwnerUserId="1214" LastActivityDate="2019-10-05T19:25:00.430" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17241" PostTypeId="1" CreationDate="2019-10-06T01:17:22.747" Score="3" ViewCount="1435" Body="&lt;p&gt;I have been looking everywhere for the name of this form of equation of a line &lt;span class=&quot;math-container&quot;&gt;$y=m(x-x_0)+y_0$&lt;/span&gt;. It's not quite point-slope. It's the same form you would write in the linear approximation of a function, &lt;span class=&quot;math-container&quot;&gt;$L(x) = f'(x_0)(x-x_0)+f(x_0)$&lt;/span&gt;. It seems important enough to warrant its own name. Does anyone have a clue ?&lt;/p&gt;&#xA;" OwnerUserId="12821" LastActivityDate="2019-10-06T17:35:15.230" Title="What is the name of the form of the line equation $y = m(x-x0)+y0$" Tags="|terminology|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17242" PostTypeId="2" ParentId="17241" CreationDate="2019-10-06T02:13:34.433" Score="8" Body="&lt;p&gt;Wolfram calls it the &quot;&lt;a href=&quot;http://mathworld.wolfram.com/Point-SlopeForm.html&quot; rel=&quot;noreferrer&quot;&gt;point slope form&lt;/a&gt;&quot;, you just have &lt;span class=&quot;math-container&quot;&gt;$y_0$&lt;/span&gt; on the rhs rather than lhs.&lt;/p&gt;&#xA;" OwnerUserId="12822" LastEditorUserId="64" LastEditDate="2019-10-06T02:29:43.680" LastActivityDate="2019-10-06T02:29:43.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17243" PostTypeId="2" ParentId="17241" CreationDate="2019-10-06T10:25:41.523" Score="3" Body="&lt;p&gt;Several Pearson calculus textbooks, such as &lt;em&gt;Calculus: Graphical, Numerical, Algebraic&lt;/em&gt; for high school and &lt;em&gt;Calculus and Analytic Geometry&lt;/em&gt; for university use what you show for the point-slope form. The latter puts the &lt;span class=&quot;math-container&quot;&gt;$y_0$&lt;/span&gt; term first, but I believe the former puts it last as you do.&lt;/p&gt;&#xA;" OwnerUserId="1905" LastActivityDate="2019-10-06T10:25:41.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17244" PostTypeId="2" ParentId="17241" CreationDate="2019-10-06T13:11:34.393" Score="1" Body="&lt;p&gt;Definition of slope of a line, given two known points on the line, say &lt;span class=&quot;math-container&quot;&gt;$(x_2,y_2), \;(x_1, y_1)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is the slope of the line:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\dfrac {y_2 - y_1}{x_2-x_1} = m$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this is equivalent to the equation: &lt;span class=&quot;math-container&quot;&gt;$(y_2-y_1) = m(x_2-x_1)$&lt;/span&gt;, which bears close resemblance to the next form of an equation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Once one calculates slope, or it is given, one can use either of the following &lt;a href=&quot;https://www.mathwarehouse.com/algebra/linear_equation/point-slope-form-of-a-line.php_&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;point-slope forms&lt;/strong&gt;&lt;/a&gt; of the equation of a line, and one of the points on the line, call it &lt;span class=&quot;math-container&quot;&gt;$(x_0, y_0)$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.mathwarehouse.com/algebra/linear_equation/point-slope-form-of-a-line.php_&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$(y-y_0) = m(x-x_0)$&lt;/span&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$y= m(x-x_0) - y_0$&lt;/span&gt; (This is algebraically equivalent to the former, in that one merely subtracts &lt;span class=&quot;math-container&quot;&gt;$y_0$&lt;/span&gt; from each side of the point-slope form of a line.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If one knows the slope of a line, &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; (the y-intercept of the line given by the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-value of the line when &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;, i.e., &lt;span class=&quot;math-container&quot;&gt;$(0, b)$&lt;/span&gt;), one can use the &lt;a href=&quot;https://www.mathwarehouse.com/algebra/linear_equation/slope-intercept-form.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;slope-intercept&lt;/strong&gt; form of an equation&lt;/a&gt;, given by:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$y= mx+b$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is particularly helpful when the slope of the line is determined, and one point on the line is given by &lt;span class=&quot;math-container&quot;&gt;$(0, b)$&lt;/span&gt; is known.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Finally, what is often called the &lt;a href=&quot;https://www.mathwarehouse.com/algebra/linear_equation/standard-form-equation-of-a-line.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;standard form&lt;/strong&gt; of the equation of a line&lt;/a&gt;, is given by: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ax + by=c$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This equation is equivalent to the point-slope form of the equation given by &lt;span class=&quot;math-container&quot;&gt;$$y = m(x-x_0) - y_0 \\ \\ \iff (\frac 1m)y= x - x_0 - (\frac 1m)y_0\\ \\ \iff x+ (-\frac ym) = \frac{y_0}m-x_0\\ \\ \iff a=1, b =-\frac 1m, c= \frac{y_0}m - x_0.$$&lt;/span&gt; More generally, &lt;span class=&quot;math-container&quot;&gt;$(a,b,c)$&lt;/span&gt; can be any ordered triple of the form &lt;span class=&quot;math-container&quot;&gt;$\Big(d(1), d\left(-\frac 1m\right), d\left(\frac{y_0}{m}-x_0\right)\Big),$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is any constant.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One of the reasons I catalogue each of the above forms of equations of a line is to make evident, that given any of the equations given above, including the equation for calculating slope, if needed, a student should be able to obtain any other form of the equation of a line. &lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-10-06T17:35:15.230" LastActivityDate="2019-10-06T17:35:15.230" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17245" PostTypeId="1" CreationDate="2019-10-06T14:41:36.017" Score="8" ViewCount="357" Body="&lt;p&gt;Every good teacher knows that giving an &lt;em&gt;informal explanation&lt;/em&gt; (that is, an explanation not based on strict definitions and maybe making use of metaphors, intuition, or everyday language) of a mathematical concept can be really helpful to students. And every good teacher has at least one of these informal explanations for each of the main mathematical concepts they teaches. However, I believe that some of these informal explanations happen to actually cause misunderstanding of the mathematical concept, to the point that it would be much better avoiding them (or replacing them with better informal explanations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it would be interesting to list some examples of these &lt;em&gt;bad&lt;/em&gt; informal explanations. Here my example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bad:&lt;/strong&gt; &lt;em&gt;&quot;Continuous functions are those functions that can be drawn without lifting the pencil from the paper.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I heard this one many times from many teachers, and I find it causes misconceptions for multiple reasons: 1) It is actually about the arc connectedness of the graph of the function, not the continuity 2) It convinces the students that many continuous functions are not continuous because they can't be drawn without lifting the pencil, like 1/x on its domain of definition 3) It has no connection with the delta-epsilon definition of continuity that the student need to learn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make my point more clear, I also give an example of a &lt;em&gt;good&lt;/em&gt; informal explanation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Good:&lt;/strong&gt; &lt;em&gt;&quot;The concept of group encodes the properties of transformations of an object. The identity element corresponds to 'doing nothing', the inverse element to 'reverting the transformation', associative laws means... etc.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are your examples of informal explanations that cause misconceptions?&lt;/p&gt;&#xA;" OwnerUserId="12824" LastEditorUserId="238" LastEditDate="2019-10-06T15:01:48.307" LastActivityDate="2019-10-09T07:15:12.877" Title="Examples of informal explanations that cause misconceptions" Tags="|teaching|intuition|big-list|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="17246" PostTypeId="1" CreationDate="2019-10-06T15:14:12.383" Score="3" ViewCount="658" Body="&lt;p&gt;Hi I've just discovered mathseducators stackexchange.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a maths tutor in the UK, I am irritated that some of my students - particularly GCSE and sometimes below - use the &lt;a href=&quot;https://en.wikipedia.org/wiki/FOIL_method#Table_as_an_alternative_to_FOIL&quot; rel=&quot;nofollow noreferrer&quot;&gt;table method&lt;/a&gt; for expanding brackets, rather than &quot;each term in the first bracket gets multiplied by each term in the second bracket&quot;, the latter method being what I'm familiar with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also prefer my method because:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) expanding brackets using my method is quicker. Drawing a table is unnecessary and tedious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) It is easier to see that my method can be derived from repeated applications of the Distributive Law (although students usually don't care about stuff like this).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) It is easier to simplify algebraic expressions, for example showing where the &quot;2b&quot; comes from when completing the square.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe it's the way they are teaching in schools nowadays, but the table method seems to me to be unnatural and unnecessary. My students insist on using it, however, and any attempts to get them to use my method are shut down because they are so used to their method being the &quot;right one&quot;. They know theirs works, so they don't need my method. But then, for example, teaching some of them how/why &quot;completing the square&quot; works is more difficult if they only use their table method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I guess my question is, should I try to get students to use my method and/or abandon their method, or should I accept their method and get over it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think what makes this question different to usual &quot;should I accept their method (assuming it works) or get them to use my method?&quot; questions is that, they have probably used their table method for many years and for them, it is probably a fundamental part of multiplication.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastActivityDate="2019-10-07T09:22:54.887" Title="&quot;Table&quot; method for expanding brackets vs &quot;each term in the first bracket gets multiplied by each term in the second bracket&quot;" Tags="|algebra|tutoring|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17247" PostTypeId="2" ParentId="17246" CreationDate="2019-10-06T16:15:28.297" Score="5" Body="&lt;p&gt;To demonstrate the distributive law, we often use an area model.  Then &lt;span class=&quot;math-container&quot;&gt;$(2+4)\times(5+7+9)$&lt;/span&gt; can be visually decomposed into the sum of 6 subproducts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather than actually drawing a grid, you might eventually start just labeling the edges and the products, without really caring about the relative sizes of the pieces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, this ``table method'' could appear as a final abstraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So students may prefer this table method because it is the end result of a logical progression.  Your more algebraic approach may feel more disconnected from the way they are actually reasoning about these problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a worthy goal to show them your alternative approach.  The more abstract method of applying algebraic laws formally is a very useful mathematical skill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just introduce it for what it is:  an alternative method which will help them build some new skills.  They may not be as comfortable with it at first, but promise (and enumerate) the rewards of this new and harder path.  More importantly, explore the connections between the two approaches!  Show how each line of algebra can be connected to a statement about areas, or to an aspect of the table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2019-10-06T16:15:28.297" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17248" PostTypeId="2" ParentId="17246" CreationDate="2019-10-06T17:15:09.167" Score="6" Body="&lt;p&gt;I run into this issue frequently. As a high school in-house math tutor, students visit to show me their quiz/test scores and ask about their work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The FOIL method is fine, if it works for the student. For those who are prone to making mistakes, I show them the Box method (call it what you will, that just my name for it). The benefit, if any, to this method is that it feels less prone to error, as whatever number of terms there are, each cell must be filled in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me, the ends justify the means. When a student returns, and offers evidence of success, or, at the time I offer this method, blurts out “that makes sense, why doesn’t my teacher teach it this way?” I can only trust my gut, that students are not a homogeneous bunch, and that we don’t all learn the same way. Again, call FOIL (distributing) the preferred method, but keep an open mind, there’s more than one way to present a concept. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So I guess my question is, should I try to get students to use my&#xA;  method and/or abandon their method, or should I accept their method&#xA;  and get over it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To clearly answer this - If students are successfully using a particular method, pushing them away from it can be harmful. There are teachers for whom &quot;the method&quot; is very important. For example, when teaching the various ways to solve two equations in two unknowns. I've seen tests where the instructions rigidly request each method to be demonstrated. If the teacher feels their instructions are being ignored, you may have created an issue that was avoidable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On reflection, offering a student what you feel is a superior (in this case, faster) method, also depends on the student. Those functioning at a higher level may always be seeking tips and tricks to improve their skills. At the lower level, I'd highly suggest mirroring the teacher's method. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-10-07T09:22:54.887" LastActivityDate="2019-10-07T09:22:54.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17249" PostTypeId="2" ParentId="17246" CreationDate="2019-10-06T17:47:39.587" Score="6" Body="&lt;p&gt;I teach in the U.S. at a community college. Although I prefer distributing without drawing an area box when I'm doing math myself, I often show the box in class to help students see how things work. When we use an area model to help students visualize the workings of the distributive property, it makes much more sense for many students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, the box seems to help them understand completing the square. You may want to watch a few of James Tanton's videos on this. (&lt;a href=&quot;https://www.youtube.com/watch?v=OZNHYZXbLY8&amp;amp;t=43s&quot; rel=&quot;noreferrer&quot;&gt;Completing the Square - Part I is here&lt;/a&gt;.) The 2b term is the diagonal of the area box.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found that his method of &lt;a href=&quot;https://www.youtube.com/watch?v=eDwi96DCdTc&quot; rel=&quot;noreferrer&quot;&gt;deriving the quadratic formula&lt;/a&gt;, which avoids fractions, is much easier for students to understand than the method as usually presented in textbooks. He completes the square with his area boxes in this video also. I think it is lovely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is much evidence that having a visual understanding of algebraic procedures deepens understanding for many students. (Possibly for everyone?) I understand that Jo Boaler has done some work on this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may help you to shift if you discover how many of your students don't really get the distributive property, especially when it is used in more complicated situations&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2019-10-06T18:02:10.747" LastActivityDate="2019-10-06T18:02:10.747" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17251" PostTypeId="1" AcceptedAnswerId="17254" CreationDate="2019-10-06T21:46:33.197" Score="0" ViewCount="483" Body="&lt;p&gt;I took Calc I in preparation for the CLEP. I did not learn about solids of revolution. Is this something that is normally part of Calc I and that most Calc II classes will expect that I can do already? I know that not all courses are the same, but is there any general answer for this question?&lt;/p&gt;&#xA;" OwnerUserId="12529" LastEditorUserId="12529" LastEditDate="2019-10-13T00:45:02.443" LastActivityDate="2019-10-13T00:45:02.443" Title="Is &quot;Volume of Solids of Revolution&quot; a part of Cal I or Cac II" Tags="|calculus|usa|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17252" PostTypeId="2" ParentId="17251" CreationDate="2019-10-06T21:56:25.957" Score="5" Body="&lt;p&gt;There's no clear definition here.  When schools find themselves in a situation where they have too much material to put into a single semester of a course, they just split it into two (or more) and add a number to distinguish each section, e.g. Calculus 1 versus Calculus 2 and Algebra 1 versus Algebra 2.  These are educational labels, not mathematical ones, so what content falls into each category is going to vary by school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I would expect solids of revolution to be in the second semester class and it's far enough in that I would be surprised to see a school fit it into the first semester.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2019-10-06T21:56:25.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17253" PostTypeId="2" ParentId="17251" CreationDate="2019-10-06T23:24:18.937" Score="1" Body="&lt;p&gt;It is likely that Calculus I, as taught in College, covers volumes of solids of revolution; it may not (or may) be taught in standard Calculus classes (or AP Calulus) in high school.  If you hope to take Calculus II in college, after having studying Calculus I in high school, and have not covered volumes of solid of revolution, I'd suggest you look at the following tutorials from &lt;a href=&quot;http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lamar University Calculus I course&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To study these applications of the integral on your own, you can visit &lt;em&gt;Paul's Notes: Calculus I&lt;/em&gt; tutorials on two different methods taught: &lt;a href=&quot;http://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithRings.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Method of Washers, or rings&lt;/a&gt;, and &lt;a href=&quot;http://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithCylinder.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Method of Cylinders&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also helpful is &lt;a href=&quot;https://www.mathsisfun.com/calculus/solids-revolution-disk-washer.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this tutorial&lt;/a&gt;, and &lt;a href=&quot;https://www.mathsisfun.com/calculus/solids-revolution-shells.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Method of Shells&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-10-07T13:43:05.307" LastActivityDate="2019-10-07T13:43:05.307" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17254" PostTypeId="2" ParentId="17251" CreationDate="2019-10-07T00:57:49.247" Score="4" Body="&lt;p&gt;As noted by other questions on this website (and in other answers), the course names &quot;Calculus I/II/II/A/B/C/whatever&quot; are not well defined: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/16903/is-there-a-more-telling-name-for-calculus-2&quot;&gt;[1]&lt;/a&gt;, &lt;a href=&quot;https://matheducators.stackexchange.com/questions/3939/naming-of-calculus-courses&quot;&gt;[2]&lt;/a&gt;, etc.  These are course titles, not mathematical terms.  This implies that the curriculum for such a class (or sequences of classes) is also not well defined.  Instead, you should look carefully at the syllabi or course descriptions of the courses that you are expecting to take or have taken.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Broadly speaking, undergraduate calculus courses tend to cover the following topics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Limits and Continuity:  this typically includes a &quot;naive&quot; approach to limits (with, perhaps, some mention of &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;, though these are not typically covered in much detail) and some basic results on continuity, including the intermediate value theorem;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Differential Calculus in One Variable: the definition of the derivative, as well as results for computation (the power rule, the chain rule, L'Hospital's rule, etc.);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Integral Calculus in One Variable: the definition of the Riemann integral from Riemann sums, the fundamental theorem of calculus (which links integrals and derivatives), and a god-awful long time on techniques of integration, including &quot;&lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-substitution&quot; (the single variable change-of-variables formula), integration by parts, trigonometric substitution, etc.;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sequences and Series: basic definitions and power series are typically covered here, as well as Taylor's theorem and various tests for convergence;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Multivariable Calculus: integration and differentiation in multiple variables&amp;mdash;the big results are the change of variables formula and the Green/Gauss/Stokes/Divergence theorems (whatever you want to call these theorems, which are all basically corollaries of each other); and&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Applications&quot;:  various &quot;applications&quot; are typically scattered throughout the curriculum, including models of ballistic flight, some simple differential equations, and (wait for it...) volumes of revolution.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In most of the curricula with which I am familiar, volumes of revolution are typically covered with techniques of integration.  In a semester system, they often show up about half-way through the second semester of college calculus, after a bunch of trigonometric substitutions.  This might be considered &quot;Calculus II&quot;, but might also be called &quot;Calculus IB&quot;.  On the other hand, in a quarter system, volumes of revolution often show up at the beginning of the third quarter, in what might be called &quot;Calculus IC&quot; or &quot;Calculus III&quot; (depending on the institution).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, again, read the relevant syllabi or course descriptions.  If you see anything like&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;applications (or possibly techniques) of integration, or&lt;/li&gt;&#xA;&lt;li&gt;volumes or solids of revolution,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;in the description of the course you intend to take, then you are probaby &lt;em&gt;not&lt;/em&gt; expected to know the topic going in.  On the other hand, if you see these terms in the descriptions of the &lt;em&gt;prerequisites&lt;/em&gt; for the course you intend to take, then you will need to bone up before you enroll.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2019-10-07T13:49:47.200" LastActivityDate="2019-10-07T13:49:47.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17255" PostTypeId="2" ParentId="17251" CreationDate="2019-10-07T03:28:30.853" Score="0" Body="&lt;p&gt;I looked at 3 books I had.  The two from the 80s (Thomas Finney and Swokowski) seemed to have it in calc 1.  Granville (old text) had it in calc 2.  The first two had it at chapter 5-6 of 14-15 total (before calc 3 topics).  Granville had it chapter 9.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main differentiator is if you have it right after introduction of simple integration or after learning more elaborate integration techniques.  Benefit of the former is more bang for the buck in calc 1 (if never taking calc 2).  Presumably the benefit of the latter is more elaborate integrals within the drill.&lt;/p&gt;&#xA;" OwnerUserId="12825" LastActivityDate="2019-10-07T03:28:30.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17256" PostTypeId="1" CreationDate="2019-10-07T03:31:54.923" Score="21" ViewCount="3013" Body="&lt;p&gt;I'm worried that I'm bad at realizing when a question I've written requires little or no conceptual understanding to answer. Like, when I'm writing a question for a homework assignment or exam, &lt;em&gt;I'll&lt;/em&gt; be thinking of it conceptually because that's how I've learned mathematics. But I don't know how to ensure that the question demands that a student think conceptually. Here's a silly minimal example calculus question to illustrate what I mean: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose that a truck's distance from you in meters at a time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; seconds after the big bang is given by the function &lt;span class=&quot;math-container&quot;&gt;$p(t) = \sin(t)+42 + \mathrm{e}^t$&lt;/span&gt;. What is the velocity of the truck &lt;span class=&quot;math-container&quot;&gt;$t=19$&lt;/span&gt;? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I wrote this question thinking that you need to understand the whole &quot;velocity is the derivative of position&quot; idea to answer. But really a student could answer this question by noticing that it has a function &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; in it, take that function's derivative because that's the only thing you do to functions in an differential calculus class, and then plug in &lt;span class=&quot;math-container&quot;&gt;$t=19$&lt;/span&gt; because why else would that &lt;span class=&quot;math-container&quot;&gt;$t=19$&lt;/span&gt; be there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can I learn to write questions that &lt;em&gt;require&lt;/em&gt; a student to think conceptually/deeply to answer? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or if you want to ask this question from a pessimistic/adversarial perspective, &lt;em&gt;how can I learn to write questions that'll thwart some learners' habits of cramming immediately before tests or exams so they can robotically reproduce what they've crammed, without ever really having understood it?&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested in resources that address this, but I'm not sure such resources exist, so sage advice is very welcome. :) &lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="238" LastEditDate="2019-10-07T14:09:37.627" LastActivityDate="2019-10-09T20:29:53.687" Title="How can I learn to write better questions to test for conceptual understanding?" Tags="|undergraduate-education|reference-request|exams|test-design|" AnswerCount="8" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17257" PostTypeId="2" ParentId="17256" CreationDate="2019-10-07T10:22:30.737" Score="0" Body="&lt;p&gt;If your assumption on how the students may get the right answer, maybe rephrase the question in a way where doing that is the wrong answer. For instance, you could ask for acceleration rather than velocity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, I don't think many students will get the right answer by following the logic you mentiond without having an understanding on what's behind!&lt;/p&gt;&#xA;" OwnerUserId="12399" LastActivityDate="2019-10-07T10:22:30.737" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17259" PostTypeId="2" ParentId="17256" CreationDate="2019-10-07T17:24:39.693" Score="6" Body="&lt;p&gt;I agree with @BrendanW.Sullivan's comment.  That is, when teaching an undergraduate course, like calculus, students need more than procedural knowledge.  For a deeper understanding, and efforts to evaluate such, students should be asked on exams to answer a few &quot;free form&quot; questions, like the one Brendan suggested. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good question to ask following any question you ask of students in class, in questions you assign for homework, and/or in questions you construct for an exam: &lt;strong&gt;&quot;How/Why does your solution make sense to you?&quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful, and why it is useful.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Further, you want to convince your students to realize that success in your course (and math in general), requires more than just &quot;getting the desired answer by mere procedural algorithms.&quot;  To enhance this sort of learning (conceptual learning), begin modeling in class, to help your students learn the sorts of questions they should ask themselves to develop a deeper, conceptual understanding of material you are covering; ask them conceptual questions about recently covered material, and ask for input from the students, perhaps even asking students to work in pairs or groups of four, e.g., to answer such a question, review group-work, offering hints, or, if necessary, answer conceptual-oriented questions in class, so that students have some familiarity with that mode of learning, and can better, then, replicate it in an exam.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do the above regularly and consistently, your students will be more familiar questions to assess conceptual understanding in an exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://www.researchgate.net/profile/Marietjie_Potgieter/publication/255664944_Undergraduate_Students%27_Performance_and_Confidence_in_Procedural_and_Conceptual_Mathematics/links/54216f6e0cf203f155c6af4b/Undergraduate-Students-Performance-and-Confidence-in-Procedural-and-Conceptual-Mathematics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Teaching procedural &lt;em&gt;and&lt;/em&gt; conceptual knowledge&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also see &lt;a href=&quot;https://www.researchgate.net/publication/261080807_Assessing_conceptual_understanding_in_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Assessing conceptual understanding in Mathematics&lt;/a&gt;, where you can download the full pdf.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-10-09T20:29:53.687" LastActivityDate="2019-10-09T20:29:53.687" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17260" PostTypeId="2" ParentId="17256" CreationDate="2019-10-07T17:47:29.597" Score="16" Body="&lt;p&gt;Agreeing with comments and other posts: If you want more conceptual answers, give them less details in the set-up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using your velocity problem, here are a couple of examples of making it more conceptual:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Suppose that a truck's distance from you in meters at a time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; seconds after the big bang is given by the function &lt;span class=&quot;math-container&quot;&gt;$p(t)$&lt;/span&gt;. What does &lt;span class=&quot;math-container&quot;&gt;$p'(19)$&lt;/span&gt; tell you (and what are its units)?&lt;/li&gt;&#xA;&lt;li&gt;Suppose that a truck's distance from you in meters at a time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; seconds after the big bang is given by the function &lt;span class=&quot;math-container&quot;&gt;$p(t)$&lt;/span&gt;. What does it mean if &lt;span class=&quot;math-container&quot;&gt;$p'(a)&amp;lt;0$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the absence of details, the students cannot just 'plug and chug' naively and are forced to actually think about the pertinent concepts.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2019-10-07T17:47:29.597" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="17261" PostTypeId="2" ParentId="17256" CreationDate="2019-10-07T17:57:58.403" Score="0" Body="&lt;p&gt;One way is to use simple, atypical mathematical objects, fringe cases, non-examples and counterexamples, etc.&lt;/p&gt;&#xA;" OwnerUserId="9204" LastActivityDate="2019-10-07T17:57:58.403" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17262" PostTypeId="2" ParentId="17256" CreationDate="2019-10-07T19:37:15.320" Score="8" Body="&lt;p&gt;Asking students to &lt;em&gt;explain&lt;/em&gt; why something happens can be useful for assessing understanding, although it is often harder to grade and works best with many demonstrations before the exam.  (Students need to know what your expectations for a thorough explanation are.)  I have found that asking students to critique a process will sometimes help me assess their conceptual understanding.  I give them a sample problem and solution, and ask them to analyze the work.  As an example, after introducing the indefinite integral but before discussing techniques like &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-substitution I will give my students the following problem. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A student is trying to find the following antiderivative &lt;span class=&quot;math-container&quot;&gt;$$\int&#xA;   xe^{x^2}\;dx$$&lt;/span&gt; and comes up with &lt;span class=&quot;math-container&quot;&gt;$$\int xe^{x^2}\;dx =&#xA;   2e^{x^2}+C.$$&lt;/span&gt; Is the student's work correct? Convince me&#xA;either way.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Answering this correctly requires students to (1) understand that differentiation of the antiderivative should result in the original integrand, (2) differentiate correctly, and (3) state a conclusion.  Simply finding the derivative of &lt;span class=&quot;math-container&quot;&gt;$2e^{x^2}+C$&lt;/span&gt; isn't enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also second Aeryk's suggestions, and would add that asking about units can be a good way to go, such as &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$r(t)$&lt;/span&gt; represents the rate in grams per minute at which a&#xA;bacterial colony is growing at time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, what are the units of&#xA;&lt;span class=&quot;math-container&quot;&gt;$\int_2^7 r(t)\;dt$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12605" LastActivityDate="2019-10-07T19:37:15.320" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17263" PostTypeId="2" ParentId="17256" CreationDate="2019-10-07T20:22:21.080" Score="5" Body="&lt;p&gt;As stated in other comments, try not to &quot;invite&quot; your students to merely apply rules they do know. In the specific example you mention, I would prefer a multiple choice question of the form:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A truck's distance in metres from you as a function of time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; (in seconds) is given by a smooth function &lt;span class=&quot;math-container&quot;&gt;$p:[0,+\infty)\to\mathbb{R}$&lt;/span&gt;. Knowing that &lt;span class=&quot;math-container&quot;&gt;$p(19)=12$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$p'(19)=24$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p''(19)=3$&lt;/span&gt;, the velocity of the truck at &lt;span class=&quot;math-container&quot;&gt;$t=19s$&lt;/span&gt; is:&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;12 m/s,&lt;/li&gt;&#xA;  &lt;li&gt;24 m/s,&lt;/li&gt;&#xA;  &lt;li&gt;3 m/s?&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;Accordingly, explain your answer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Under such a framework, one cannot avoid some conceptual processing - or exhibit, at least, the absence of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, the more abstract the context of a question is, the more possible it is that a student invokes conceptual processing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, when you invoke some context - e.g. motion - do it for a reason and not just so as to make the question more &quot;fancy&quot;.  I've made this mistake several times myself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider, for instance, the following questions, which both refer to the same issue:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Find an antiderivative &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$f(x)=4x+3$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$F(0)=3$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f(t)=4t+3$&lt;/span&gt; describe the way a particle's acceleration changes over time. Is it possible that the particle has a zero velocity at some time? From all the antiderivatives of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, which one describes a particle that starts from inertia and constantly accelerates with positive acceleration?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The second involves much more in-depth knowledge of the relation between a function and its antiderivative than the first one.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastEditorUserId="12829" LastEditDate="2019-10-08T18:25:36.060" LastActivityDate="2019-10-08T18:25:36.060" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17264" PostTypeId="2" ParentId="17256" CreationDate="2019-10-07T20:32:11.210" Score="0" Body="&lt;p&gt;&lt;strong&gt;The &quot;deeper&quot; or at least off-beaten path solution should be much quicker than a standard but boring approach&lt;/strong&gt; to create a system of equations or calculate a derivative or something like that. Give two-three problems and limit time so that they can solve them in time if they apply some sort of a trick. You'll get a lot of angry students :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an middle-school example, no derivatives, it can be solved with a system of linear equations:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Every day an engineer arrives to a station at 8 a.m. by train. Exactly&#xA;  at the same time a car, sent from a factory, drives up to the station,&#xA;  picks up the engineer and takes him to the factory. One day the&#xA;  engineer arrived at 7 a.m., decided not to wait for the car, and&#xA;  started walking towards the car. When the car met the engineer, it&#xA;  picked him up, turned back and arrived to the factory 20 minutes&#xA;  earlier than usual. For how long did the engineer walk? Consider the&#xA;  speeds of the engineer and the car constant.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or you can solve it literally in 30 seconds if you think about what is happening, and use 3rd grade arithmetic.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2019-10-07T20:40:40.133" LastActivityDate="2019-10-07T20:40:40.133" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17265" PostTypeId="2" ParentId="17245" CreationDate="2019-10-07T22:12:55.897" Score="5" Body="&lt;p&gt;I shall post my humble and incomplete list of bad explanations I've given or heard over the years:&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;A function is continuous if you can draw its graph without lifting your pencil&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;As you mentioned this is bad, but, depending on the level of the student, it can be a reason for big or small misunderstandings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a high school level, this simply ignores the fact that ony needs the domain of the function to be connected so as to imply that the graph of the function is also connected - which is the closest I can get to &quot;draw by pencil&quot; in terms of strict mathematics. However, as mentioned in the comments, &lt;span class=&quot;math-container&quot;&gt;$x\sin\dfrac{1}{x}$&lt;/span&gt; blows this story up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At an undergraduate level I consider such a description useless, since e.g. any sequence is a continuous function under the &lt;span class=&quot;math-container&quot;&gt;$\varepsilon-\delta$&lt;/span&gt; definition - &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt; has no real accumulation points - so, this explanation only causes some trouble. Also, the notion of &quot;drawing with pencil&quot; get's pointless if one considers a function which is &lt;a href=&quot;https://en.wikipedia.org/wiki/Weierstrass_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;everywhere continuous and nowhere differentiable&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possibly better explanation is that involving the high-school &quot;definition&quot; of continuity at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to x_0}f(x)=f(x_0).$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Under that definition, I prefer to explain to my students that continuity may be seen as the case when the estimation of the value of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; based on its values close to it (in other words &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{x\to x_0}f(x)$&lt;/span&gt;) agrees with the actual value of the function itself. Usually, an example of a smooth flow of some liquid is a nice add on such an interpretation of continuity - always as an introductory approach; pathologic cases are left out under such a scope. Also, note that such an approach refers mainly to high-school students.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Integration is the opposite of differentiation&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Well, this almost deterministically leads to the misunderstanding that if &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; is the derivative of some function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; is integrable. Evidently, this is not true, either because &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; may not be bounded, as in &lt;span class=&quot;math-container&quot;&gt;$f(x)=\dfrac{1}{x^2}\sin\dfrac{1}{x^2}$&lt;/span&gt;, which is a nice high-school level example, or because &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; may be too bad, as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Volterra%27s_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volterra&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Pompeiu_derivative&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pompeiu&lt;/a&gt; functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To dig a little deeper, this sentence seems to lead to the more crucial misunderstanding that a function having an anti-derivative is integrable, as well. Well, this makes the whole stuff pretty much messy, since, neither having an antiderivative implies integrability nor the opposite is correct - take any piecewise constant and discontinuous function for the opposite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, while when it comes to &quot;technique&quot;, (Riemannian) integration seems to be the inverse action of differentiation, it actually isn't. The way I've decided to tackle this is using the following question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it true that every derivative of a function is integrable on any closed interval of its domain?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After some discussion, counter-examples etc - always depending on the level of the students - the Fundamental Theorem of Calculus emerges as a necessity to put some order in the chaos we've created - if the derivative is continuous, then integration is exactly the inverse of differentiation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another approach would be to introduce the &lt;a href=&quot;https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henstock-Kurzweil integral&lt;/a&gt; into high shcool... (kidding)&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;An indefinite integral is the set of all antiderivatives&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Well, this is possibly the most &quot;true&quot; sentence one could write about the symbol &lt;span class=&quot;math-container&quot;&gt;$\int f(t)dt$&lt;/span&gt;, but I've seen it often cause much trouble. Consider the following equation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int f(t)+g(t)dt=\int f(t)dt+\int g(t)dt.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pretty much neat. But, are we adding sets on the RHS? This is a natural question that may emerge, especially if one fails to mention somehow that indefinite integrals also posess some algebraic structure - since &lt;span class=&quot;math-container&quot;&gt;$\int f(t)dt$&lt;/span&gt; is not merely a set of functions but an equivalence class of functions - hence the algebraic structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible way of tackling it, if one wants to avoid talking about vector spaces etc in a high school context, would be to define the indefinite integral as an arbitrary antiderivative of a function, however this may cause more trouble than that it solves.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;The definite integral &lt;span class=&quot;math-container&quot;&gt;$\int_a^bf(t)dt$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f\geq0$&lt;/span&gt; is the area under the curve&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I was a fan of this until I found out where the flaw is. The definite integral of a non-negative integrable function &lt;em&gt;is not&lt;/em&gt; the area, but it may be &lt;strong&gt;interpreted&lt;/strong&gt; as such - in a similar fashion, it is not the work of some force moving a particle, but it may be interperted as such.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make myself clear, take the following inequlity:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_1^5\frac{1}{x}dx&amp;lt;\sum_{k=1}^4\frac{1}{k}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we take the definite integral on the LHS to be area and the sum of the RHS to be a number, then the LHS is measured into square units, while the LHS in simple units, making the use of &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; between them nonsense. Yet, the RHS can be intepreted as area - e.g. a partial Darboux upper sum - and make both quantities &quot;comparable&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, the definite integral is interpretable,under some circmustances, as area, as well as every number - it would seem naive to try to compare my height with my weight, even if both of them are numbers :P.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;A differentiable function's graph has no edges and/or sharp points&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Well, this is a classic. Take, for instance &lt;span class=&quot;math-container&quot;&gt;$$f(x)=x^2,\ x\in[0,1].$$&lt;/span&gt; Isn't &lt;span class=&quot;math-container&quot;&gt;$A(1,1)$&lt;/span&gt; an edge of the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;? Isn't &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; differentiable there? The crucial point is that the point under discussion should have an &lt;span class=&quot;math-container&quot;&gt;$x-$&lt;/span&gt;coordinate that is internal point of the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. Provided that, this is also a nice intuitive reasoning of why we prefer to discuss differentiability on open sets.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;A function is differentiable at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; if its graph has a tangent line there&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;And how about the case when &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has a vertical tangent line &lt;span class=&quot;math-container&quot;&gt;$(x=x_0)$&lt;/span&gt; at some point &lt;span class=&quot;math-container&quot;&gt;$(x_0,f(x_0))$&lt;/span&gt;? In this case, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not differentiable at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; since:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=\pm\infty,$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;yet it has a tangent line at &lt;span class=&quot;math-container&quot;&gt;$(x_0,f(x_0))$&lt;/span&gt;. This could be nicely coupled with the previous one as the two most misleading interpretations of derivative. I mean, the tangent line function is the historical problem that leads to the definition of the derivative, but tangent lines themselves are not equivalent to differentiability - differentiability implies the existence of a tangent line and not vice versa.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Two functions are equal when they share the same domain and have the same formula&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Well, this is practical when it comes to high-school excercises, but it is unutterably misleading. A function may not have a &quot;formula&quot;, in the sense that not all functions can be expressed in terms of elementary functions.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;If the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; lies below that of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$f&amp;lt;g$&lt;/span&gt;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Intuitively, this seems correct. However, a common misunderstanding is, as I see it, that the inequality:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sup\{f(x)\}&amp;lt;\inf\{g(x)\}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;does not imply that &lt;span class=&quot;math-container&quot;&gt;$f,g$&lt;/span&gt; are comparable. For instance, consider:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x)=1-\sqrt{-1-x},\ g(x)=\sqrt{x}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$D_f\cap D_g=\varnothing$&lt;/span&gt;, we cannot compare them, yet the former inequality holds. So, &quot;below&quot; seems to vague in some cases - note however that the inverse implication (&lt;span class=&quot;math-container&quot;&gt;$f&amp;lt;g$&lt;/span&gt; implies that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; lies below &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;) is more clear, since we have established that &lt;span class=&quot;math-container&quot;&gt;$f,g$&lt;/span&gt; are comparable.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;The depictions of asymptotic behaviour of functions&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Well, this is not some explicit explanation but an effect that seems to take place due to &quot;overfitting&quot;. Most students depict the notion of a function asymptotically approaching a straight line in some way that implies monotonicity near &lt;span class=&quot;math-container&quot;&gt;$\pm\infty$&lt;/span&gt;. However, as, for instance, it happens with &lt;span class=&quot;math-container&quot;&gt;$f(x)=e^{-x}\sin x$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt;, no such depiction of asymptotic behaviour is a priori correct - e.g. in the aforementioned example, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; constantly oscillates as &lt;span class=&quot;math-container&quot;&gt;$x\to+\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;We add functions as we do with numbers: &lt;span class=&quot;math-container&quot;&gt;$(f+g)(x)=f(x)+g(x)$&lt;/span&gt;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Well, this equality never made sense to me as a high-schoold student. Are we saying something non-obvious? I mean, how else should addition between functions be defined? What I did not take into account was that what takes place in this expression is &quot;function over-loading&quot;, as a programmer would call it. The LHS &lt;span class=&quot;math-container&quot;&gt;$&quot;+&quot;$&lt;/span&gt; is a an addition that refers to the set of all functions while the RHS &lt;span class=&quot;math-container&quot;&gt;$&quot;+&quot;$&lt;/span&gt; refers to the usual addition of the reals. That, what we are actually doing here is defining a new function, &lt;span class=&quot;math-container&quot;&gt;$f+g$&lt;/span&gt;, pointwisely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just wanted this included in the list, since I think it is never given the attention it needs, at least based on personal experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I may edit this list later on to add some more vague explanations.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="12829" LastEditorUserId="12829" LastEditDate="2019-10-08T07:49:47.617" LastActivityDate="2019-10-08T07:49:47.617" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17266" PostTypeId="2" ParentId="17245" CreationDate="2019-10-07T23:11:29.050" Score="5" Body="&lt;p&gt;For a much lower-level topic, consider explaining to beginning algebra students why &quot;like terms&quot; can be combined. On a few occasions, I have resorted to reasoning with students that &lt;strong&gt;adding algebraic expressions is like adding quantities with units&lt;/strong&gt;. [Our curriculum begins with units and geometry before algebra, so this is usually safe ground in my class.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they already believe that &lt;span class=&quot;math-container&quot;&gt;$5cm$&lt;/span&gt; + &lt;span class=&quot;math-container&quot;&gt;$7cm$&lt;/span&gt; makes physical sense, and &lt;span class=&quot;math-container&quot;&gt;$5cm^2$&lt;/span&gt; + &lt;span class=&quot;math-container&quot;&gt;$7cm$&lt;/span&gt; does not, then we can make an argument that &lt;span class=&quot;math-container&quot;&gt;$5x + 7x$&lt;/span&gt; can reasonably be called &lt;span class=&quot;math-container&quot;&gt;$12x$&lt;/span&gt;, but &lt;span class=&quot;math-container&quot;&gt;$5x^2 + 7x$&lt;/span&gt; cannot be combined nicely. However, I always regret this line of conversation, because I feel I have to backpedal for the sake of honesty. Since students recognize that &lt;span class=&quot;math-container&quot;&gt;$5cm^2 + 7cm$&lt;/span&gt; isn't even computable (forgoing the possibility that &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; have convenient units), they will sometimes say that &lt;span class=&quot;math-container&quot;&gt;$5x^2 + 7x$&lt;/span&gt; is also a &quot;bad&quot; expression that cannot represent anything real. This is obviously a problem. [Hence, I avoid this informal explanation if possible.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-10-07T23:11:29.050" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17267" PostTypeId="2" ParentId="17245" CreationDate="2019-10-08T00:29:34.160" Score="8" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;A point (say, in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;) is a vector. Vectors and points are really no different. They are both &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-tuples in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The difference between two points (in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;) is a vector, but a vector has&#xA;no fixed position. Points are positions in space. Vectors&#xA;are displacements. It makes no sense to add two points, but it&#xA;does make sense to add two vectors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In spaces other than &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;, points and vectors can be&#xA;quite different.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-10-09T00:19:36.793" LastActivityDate="2019-10-09T00:19:36.793" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17269" PostTypeId="2" ParentId="14585" CreationDate="2019-10-08T07:23:05.037" Score="1" Body="&lt;p&gt;Let some &lt;span class=&quot;math-container&quot;&gt;$k&amp;gt;1$&lt;/span&gt; - since for &lt;span class=&quot;math-container&quot;&gt;$k=1$&lt;/span&gt; we have nothing to prove (&lt;span class=&quot;math-container&quot;&gt;$x=x$&lt;/span&gt;). Starting from the fact that &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;b∧ c&amp;lt;d\Rightarrow a+c&amp;lt;b+d$&lt;/span&gt; we arrive to the fact that for any &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$x+x&amp;gt;0$&lt;/span&gt;. Similarly, we can have &lt;span class=&quot;math-container&quot;&gt;$x+x+x&amp;gt;0$&lt;/span&gt; and, so on, applying this &quot;trick&quot; &lt;span class=&quot;math-container&quot;&gt;$k-1$&lt;/span&gt; times we arrive to:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\underbrace{x+x+\ldots+x}_{k-1\text{ times}}&amp;gt;0.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, add &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to both sides and we are done:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\underbrace{x+x+\ldots+x}_{k\text{ times}}&amp;gt;x\Leftrightarrow k\cdot x&amp;gt;x.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, we have proved that for &lt;span class=&quot;math-container&quot;&gt;$k\geq1$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$k\cdot x\geq x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, since &lt;span class=&quot;math-container&quot;&gt;$x\mid y$&lt;/span&gt; we have that &lt;span class=&quot;math-container&quot;&gt;$y=k\cdot x$&lt;/span&gt; for some positive integer &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, which implies that: &lt;span class=&quot;math-container&quot;&gt;$y\geq x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, in cases where a constructive proof is easy and possible, I usually avoid proof by contradiction.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastEditorUserId="12829" LastEditDate="2019-10-08T18:26:38.957" LastActivityDate="2019-10-08T18:26:38.957" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17270" PostTypeId="2" ParentId="17245" CreationDate="2019-10-09T07:15:12.877" Score="2" Body="&lt;p&gt;&lt;strong&gt;Modular arythmetic works just like a clock&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've seen several students expect &quot;half-hours&quot; making an appearance in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}_{12}$&lt;/span&gt;. This leads them to be confused about statements like &lt;span class=&quot;math-container&quot;&gt;$5^{-1}=5$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$5^{-1}=0:12$&lt;/span&gt; &lt;/p&gt;&#xA;" OwnerUserId="12399" LastActivityDate="2019-10-09T07:15:12.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17271" PostTypeId="2" ParentId="17256" CreationDate="2019-10-09T18:58:55.430" Score="2" Body="&lt;p&gt;What one has to do to test for conceptual understanding is hard to state in terms of general principles (although Polya's books on Plausible Reasoning do a pretty good job of addressing the issue) and maybe is best addressed via examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one example. Consider a cubic polynomial in one variable that is increasing as a function of its argument. Choose one that is not easily factored. One could ask to find its critical points and decide their nature. Or one could ask how many real roots the polynomial has. The first question is mechanical. One has to know the definition of critical point and some test to decide the type of a critical point. To answer the second question one has to realize that for the indicated polynomial to have more than one root, it must have a local minimum at which the value is negative. Then one has to do the same mechanical operations as in the first question to decide whether this is the case. The way the problem is posed forces much more serious thinking. Usually one has to ask both sorts of questions because the student who cannot solve the first has no hope of solving the second.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example. Students can be asked to evaluate an integral whose evaluation can be effected by making a change of variables. An alternative, which both isolates the change of variables issue and avoids recipes is the following. Give them the value of an integral such as &lt;span class=&quot;math-container&quot;&gt;$\int_{-\infty}^{\infty}e^{-x^{2}/2}\,dx$&lt;/span&gt; and ask them to find the value of &lt;span class=&quot;math-container&quot;&gt;$\int_{-\infty}^{\infty}e^{-3x^{2}}\,dx$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\int_{0}^{\infty}e^{-x^{2}/2}\,dx$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\int_{0}^{\infty}e^{-3x^{2}}\,dx$&lt;/span&gt; (the last is slightly harder than the first two, combining the issues present in both).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have often used variations of both of these examples (and there are many more). They are effective for discriminating among those who understand more and those who merely have mastered formal manipulations. On the other hand, a test full of questions like these will not yield good results in most classrooms in most universities (it would be too hard). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These sorts of questions force deeper thinking without requiring that the student make it explicit, something which is out of reach for all but the most interested and enthusiastic students. Understanding, at whatever level, has to be evaluated operationally. At an introductory level it rarely works to ask &quot;explain why ...&quot; When many students can answer this it is usually the case that they are parroting phrases repeated often by the teacher, and then not much is being learned. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-10-09T18:58:55.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17273" PostTypeId="1" CreationDate="2019-10-09T22:08:41.533" Score="19" ViewCount="5830" Body="&lt;p&gt;I am preparing an introductory course on Game Theory for children (between 10 and 17 years old). In the course description, I want to include a prisoner's dilemma in order to catch children's attention and persuade them to attend the course. Therefore, I would like to find a short, direct and attractive-to-children formulation, a &quot;slogan&quot; for my course. The problem is that all of the formulations I have found falls into one of the following issues:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It includes a reference to &quot;+18&quot; content (death, crimes, etc.).&lt;/li&gt;&#xA;&lt;li&gt;It is formulated &lt;em&gt;capitalistically&lt;/em&gt;, making money the main protagonist.&lt;/li&gt;&#xA;&lt;li&gt;It is a complex and enmeshed formulation, or it is too unreal for a kid's mind.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Can anyone suggest a formulation of the prisoner's dilemma that does not fall into any of these issues?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="12839" LastEditDate="2019-10-11T07:55:19.420" LastActivityDate="2020-09-12T04:18:34.527" Title="Prisoner's dilemma formulation for children" Tags="|secondary-education|course-design|middle-school|resource-request|" AnswerCount="8" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="17274" PostTypeId="2" ParentId="13150" CreationDate="2019-10-09T23:31:53.860" Score="0" Body="&lt;p&gt;I'm not a teacher. I don't know how most students really learn. This is just an idea and it should be taken with a grain of salt. It's the job of teachers to not rely on this answer, and do their own research based on this answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it will work so much better for the teacher to give the students a worked example in class. They should pick two fractions one of which has 4 as a denominator and the other of which has 6 as a denominator both of which are in lowest terms. Next, they should convert both fractions into another form with 12 as the denominator. For example, they could be shown that &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} + \frac{5}{6}$&lt;/span&gt; can be reexpressed as &lt;span class=&quot;math-container&quot;&gt;$\frac{9}{12} + \frac{10}{12}$&lt;/span&gt;. Then they could teach them that the way to do it is to find the lcm of the denominators of each of the fractions and then convert each fraction to the form whose denominator is the lcm of the original denominators. They should word it as the lcm of the original denominators. If they call it the lowest common denominator, some students might have forgotten what &quot;lowest common denominator&quot; means and be confused. It's so straight forward to just tell them to convert each fraction to the form where the denominator is the lcm of the original denominators. I think students will be very capable of teaching themselves how to do that. On the other hand, I think that if the teacher replaces the instructions to do it with the instructions on how to do it, some students will see it as a big long list of instructions on how to calculate a sum and not understand what they're saying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By picking that problem specifically or something like it, both fractions will be in lowest terms. Not only that but also neither denominator is a multiple of the other denominator nor is the lcm of them their product. That way, the students will learn how to do it when neither denominator is a multiple of the other nor is the lcm of them their product and the original fractions are in lowest terms.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="11718" LastEditDate="2019-10-10T00:48:09.920" LastActivityDate="2019-10-10T00:48:09.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17275" PostTypeId="2" ParentId="14332" CreationDate="2019-10-09T23:40:27.450" Score="2" Body="&lt;p&gt;This is the only way I've ever seen it taught or read it described in textbooks - except in my classes.  I just teach students to multiply the numerator and denominator of the first fraction by the denominator of the second and vice versa.  No prime factorizations, least common multiples, etc. are necessary.  This is based on the idea that you don't need the &lt;em&gt;least&lt;/em&gt; common denominator.  You just need &lt;em&gt;a&lt;/em&gt; common denominator - any one will do.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You mentioned &quot;roadblocks to learning&quot; and I definitely think is a big part of where people's negative feelings about fractions start.  The procedure for finding the least common multiple can be lengthy and involves multiple steps.  In my opinion, it's counterproductive to graft this lengthy algorithm onto the beginning of what is already a multi-step process.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2019-10-09T23:40:27.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17276" PostTypeId="2" ParentId="6089" CreationDate="2019-10-10T02:06:53.273" Score="0" Body="&lt;p&gt;I'm not a teacher. I have an idea on how to do it but it should be taken with a grain of salt. If you tell her that she needs to learn another method because gets the result faster, she might not listen or understand. She might be like &quot;How can it be done faster.&quot; However, it is true that there is another reason to learn the other method. That's for the sake of knowing how to do it by that method and showing the work. I think it's better if as a teacher, you tell her that reason and then give her a simple task of solving a problem such as &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} + \frac{5}{6}$&lt;/span&gt; by writing &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12}$&lt;/span&gt; and writing it that way on the test. Maybe she's so used to computing the value of a fraction by using long division that it's not obvious to her that &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} = \frac{9}{12}$&lt;/span&gt; until after she computes both of them by long division. Maybe she's really smart and realized that the value of a fraction can be defined using long division and that until you compute the value of both fractions using long division, you don't have a proof that they're equal according to that definition. It's just like questioning the axiom of choice is smart. Unfortunately, if that's the case, thinking for herself won't get her the marks. I would love it if I could get the way the world works changed legally so that students are taught using a student centered approach and the student is not pushed to agree that it's obvious that &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} = \frac{9}{12}$&lt;/span&gt;. Hopefully, if you teach her that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4} = \frac{1}{12} + \frac{1}{12} + \frac{1}{12}$&lt;/span&gt;, she will understand why that's true. Then it will be easy for her to see why &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} = \frac{9}{12}$&lt;/span&gt;. If she still can't understand why &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4} = \frac{1}{12} + \frac{1}{12} + \frac{1}{12}$&lt;/span&gt;, and she's not willing to perform calculations on fractions the way she was taught when she doesn't understand why it works, I think it's not worth pushing her to do it and she should have a right to make her own choice to not answer test questions whose answer requires that type of calculation and not get the marks.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastActivityDate="2019-10-10T02:06:53.273" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17277" PostTypeId="2" ParentId="17273" CreationDate="2019-10-10T05:31:47.850" Score="16" Body="&lt;p&gt;Here's a silly example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Give students collections of the same type of thing, where each collection contains &quot;good&quot; objects and &quot;bad&quot; objects -- for example, a stack of Pokemon cards with both &lt;a href=&quot;https://www.pokemon.com/us/pokemon-tcg/pokemon-cards/bw-series/bw9/117/&quot; rel=&quot;nofollow noreferrer&quot;&gt;rare&lt;/a&gt; and &lt;a href=&quot;https://www.pokemon.com/us/pokemon-tcg/pokemon-cards/xy-series/xyp/XY143/&quot; rel=&quot;nofollow noreferrer&quot;&gt;common&lt;/a&gt; cards. We might assume common cards are worth &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and rare cards are worth &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;. Have ready another stack of cards that are all common -- call this the &quot;Free Stack&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students pair up and play a trading game, as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students want to build a nice deck of cards (in the sense of overall point value), and they can get new cards by trading.&lt;/li&gt;&#xA;&lt;li&gt;A trade is made when both students select a card from their own deck, placing it &lt;em&gt;face down&lt;/em&gt; in front of them. When the cards are revealed simultaneously, the following takes place:&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If both cards are rare, then the cards are exchanged and each student takes a card from the Free Stack, adding it to their own deck. This ends the trade.&lt;/li&gt;&#xA;&lt;li&gt;If both cards are common, then the cards are exchanged and the trade ends.&lt;/li&gt;&#xA;&lt;li&gt;If the players offer different types of cards (one common and one rare), then the player who offered the common card gets both cards. This ends the trade.&lt;/li&gt;&#xA;&lt;/ol&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It is obviously advantageous for players to &quot;cooperate&quot; and both offer rare cards. They trade those cards, but they both get +1 from the free card from the Free Stack.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not helpful or harmful if they both offer common cards, so both players get +0 for that option.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If they offer different types, then the winner gets +5 because they received the rare card, and the loser gets -5 because they lost the rare card.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-10-25T20:43:42.380" LastActivityDate="2019-10-25T20:43:42.380" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="17278" PostTypeId="2" ParentId="17273" CreationDate="2019-10-10T11:03:33.330" Score="8" Body="&lt;p&gt;I like Nick C's idea more than modifying the typical formulation.  The notion of snitching on a friend, regardless of the severity of the &quot;crime&quot;, has real-world ramifications beyond the punishment put out by the authorities.  Depending on your student population, that is possibly going to spur a conversation that will overshadow the objectives of your lesson.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's another non-crime formulation that students will be familiar with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You and a fellow student are assigned to work on a team project that will give a significant grade in your class.  You break the work in half and both agree to do your part over the weekend.  Even though it is a major grade, the amount of work for each half of the project is reasonable considering the consequences of a good grade.  &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If you both do your best work, you will both get 100% but both of you have to do some work over the weekend.  If neither of you commit to the project, you will both get 70% but you can both have a free weekend.  If one of you commits but the other doesn't, then you will both get an 80% on the project but one of you will work over the weekend while the other doesn't.  Your partner won't know whether you decided to do your share of the project until Monday morning, and you won't know what your partner decided.  Will you do your share of the project?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This isn't a pure formulation of the Prisoner's Dilemma, because the outcome is being measured in two different currencies (grades and free time) which different students may value differently.  But I think any sort of &quot;tragedy of the commons&quot; scenario applied to two players would play out like the Prisoner's Dilemma.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-10-10T11:03:33.330" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17279" PostTypeId="2" ParentId="17273" CreationDate="2019-10-10T15:06:13.783" Score="2" Body="&lt;p&gt;Use of performance-enhancing drugs in sports is a good example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, why the objection to money and capitalism? Real-life actors value money very much and it affects their behavior greatly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very realistic and applicable example involving money is what happens when a group of friends goes dining, depending on whether each friend pays for themselves or the bill gets split equally. Suppose a pair of friends are offered a cheap snack that costs 5 units of money or an expensive dish for 25. The friends consider the dish overpriced, valuing it at 20. The snack is fairly priced in their opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the friends agree to split the bill evenly, the payoff matrix is as follows:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{array}{|l|c|c|}&#xA;\hline&#xA;&amp;amp; \text{Snack} &amp;amp; \text{Dish} \\ \hline&#xA;\text{Snack} &amp;amp; 0,0 &amp;amp; -10,5 \\ \hline&#xA;\text{Dish} &amp;amp; 5,-10 &amp;amp; -5,-5 \\ \hline&#xA;\end{array}&#xA;$$&lt;/span&gt;&#xA;which is typical of the prisoner’s dilemma.&lt;/p&gt;&#xA;" OwnerUserId="10394" LastEditorUserId="10394" LastEditDate="2019-10-10T22:49:22.750" LastActivityDate="2019-10-10T22:49:22.750" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17280" PostTypeId="2" ParentId="17273" CreationDate="2019-10-10T16:16:25.993" Score="14" Body="&lt;p&gt;I found out about the Prisoners' Dilemma as a kid from &lt;a href=&quot;https://www.penguinrandomhouse.com/books/288991/the-science-of-harry-potter-by-roger-highfield/&quot; rel=&quot;noreferrer&quot;&gt;a book about the Harry Potter phenomenon&lt;/a&gt;, which had a chapter about the problem, but presented as a story about Harry and Draco being accused of breaking school rules. Each was offered the same deal as in the original problem, formulated with House Points being taken away instead of a prison sentence. Maybe you could write a story using these or other characters that kids would be familiar with?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternately, the existence of crime and prison sentences are well-known to 10-17-year-olds, so you might not have to modify it at all. Maybe you could play up the second-person formulation (&quot;You are accused of a crime...&quot;) to make it more engaging.&lt;/p&gt;&#xA;" OwnerUserId="12845" LastActivityDate="2019-10-10T16:16:25.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17282" PostTypeId="2" ParentId="17273" CreationDate="2019-10-10T20:56:12.687" Score="2" Body="&lt;p&gt;Perhaps:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You and another classmate are together in an obstacle course.&lt;/p&gt;&#xA;&lt;p&gt;If you both make it to the end within a minute, you each get a free day off from school. If just one of you makes it to the end in a minute, they get a whole week off.&lt;/p&gt;&#xA;&lt;p&gt;You know that if you had the other student's help, you could easily finish in a minute, but without it, you won't finish in time. Likewise, the other student would complete the course with your assistance, but not without it.&lt;/p&gt;&#xA;&lt;p&gt;You don't know what the other student will do - even if you helped them, they might not return the favor.&lt;/p&gt;&#xA;&lt;p&gt;Do you help them?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8957" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-10-11T15:06:19.267" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17283" PostTypeId="2" ParentId="17273" CreationDate="2019-10-10T21:18:27.017" Score="3" Body="&lt;p&gt;It sounds like you wish to protect your students from the violence and greed of the adult world, while still making the dilemma real enough to keep them engaged.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To that effect, I offer two solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One, replace prison with detention.  Make the crime something like using cell phones in class or throwing spitballs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two, have them arbitrarily grade each others' work (obviously don't make it their actual grades.  It's up to you whether you want to let them think they're grading each other).  If they both grade each other well, they both get high grades.  If one grades the other poorly, but the other grades the first well, then the first gets a higher score, and the second gets 50%.  And whatever numbers you want to formulate in there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;as a side note... I don't think money and prison are taboo, or too much for kids to handle.  But I agree that they're kinda clichéd, and somewhat boring because we're desensitized and removed, except for the poor kids for whom the issue is far too close to home&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Edit&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Following the original prisoner's dilemma, no matter what the opponent does, ratting them out gives you a lighter sentence.  Not the best ethics lesson, but this is mathematics.  So...&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If they're ratting you out, not ratting them out would give you a E, ratting them out would give you a C.&lt;/li&gt;&#xA;&lt;li&gt;If they're not ratting you out, ratting them out would give you an A,&#xA;and not ratting them out would give you a B.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12851" LastEditorUserId="12851" LastEditDate="2019-10-10T21:38:19.757" LastActivityDate="2019-10-10T21:38:19.757" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17284" PostTypeId="2" ParentId="14464" CreationDate="2019-10-11T11:47:25.873" Score="1" Body="&lt;p&gt;Not an answer; just a tangential remark.&#xA;In general it is not an easy problem to reconstruct a&#xA;polygon from various sets of data. If the data does not&#xA;uniquely determine the polygon, it would not be easy &#xA;to &quot;generate a credible figure,&quot; to quote the OP, compatible&#xA;with the partial reconstruction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the paper below reconstructs a polygon from its &quot;visibility angles,&quot; which do uniquely determine the polygon.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Chen, Danny Z., and Haitao Wang. &quot;An improved algorithm for reconstructing a simple polygon from the visibility angles.&quot; In &lt;em&gt;International Symposium on Algorithms and Computation&lt;/em&gt;, pp. 604-613. Springer, Berlin, Heidelberg, 2011.&#xA;  &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0925772112000181&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elsevier link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/GoiKk.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GoiKk.gif&quot; alt=&quot;Rec1&quot;&gt;&lt;/a&gt;&amp;nbsp;&lt;a href=&quot;https://i.stack.imgur.com/apKQM.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/apKQM.gif&quot; alt=&quot;Rec2&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Fig. 1: Visibility angles at a vertex &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;. &amp;amp; Fig. 3.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;Polygon reconstruction problems arise from the need for&#xA;a mobile robot to navigate an environment using sensor/scanner measurements to avoid wall collisions. The&#xA;Chen-Wang paper above cites 10 papers exploring &#xA;a variety of reconstruction problems.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-10-11T11:47:25.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17285" PostTypeId="2" ParentId="17273" CreationDate="2019-10-11T18:22:28.213" Score="1" Body="&lt;p&gt;Perhaps the problems of &quot;make up&quot; and &quot;traffic&quot; could also be used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Make-up&lt;/strong&gt;: Assuming that the chemicals etc., damage your skin in the long run, we may state that &lt;em&gt;constant make-up&lt;/em&gt; is &quot;bad&quot;. And contrarily no make-up is better in the long run since your skin remains &quot;naturally beautiful&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given any two individuals with the option of applying make-up or &lt;em&gt;going natural&lt;/em&gt; almost everyone will succumb to the lure of make-up. Even though everyone is better off without it in the long run, everyone thinks they'll look better (read attractive) than everybody else - this &quot;race to the bottom&quot; is highly profitable for companies at the expense of the consumers, so to speak. Only a &lt;em&gt;tight&lt;/em&gt; cooperation amongst consumers can break out of this cycle and that's very hard to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Traffic&lt;/strong&gt;:  Let's say you're in a country where people are &lt;em&gt;semi-literate&lt;/em&gt; and if marginally violating a traffic &lt;em&gt;law/discipline&lt;/em&gt; gets them going faster, they are going to violate it. Example, lane discipline or &quot;right of way&quot; (i.e., whose turn is it). Those flouting the rules will seem to &quot;move faster&quot; through the system and slowly everyone will do it leading to major traffic issues. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why did I emphasize &lt;em&gt;semi-literate&lt;/em&gt;? You can get the cooperation of the individuals (players in the game) by having them take exams or disseminate information that decreases the likelihood of such traffic violations leading to overall improvement in traffic conditions. Hence, you'll find traffic a lot better in Europe or Canada or USA vs. say the Indian sub-continent, certain African countries and the like.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could try the same with candies - bring a box of candies to class along with some &quot;health treats&quot;. Ask each student to take a single item from the box - either a candy or a healthy option. The candy should be more attractive. To do this though you'll have to have some good candy/chocolates and not some crappy candy i.e., Candy &gt; healthy treat by taste/value. You could add a twist by saying that if you take one candy, the next person after you has the option to take an extra candy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another possibility is allowing them to &quot;throw paper balls&quot; in the middle of the class, without penalty or fear of punishment. The option is of a cleaner class vs. a messy one. Even if &lt;em&gt;one&lt;/em&gt; student does it, the whole class will devolve into it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal of the Prisoner's dilemma IMHO is simply this: selfishness is worse than cooperation in the long run, even though the former may come out ahead in the short run. Given this POV one could construct numerous examples to show case prisoner's dilemma in a classroom setting.&lt;/p&gt;&#xA;" OwnerUserId="3456" LastEditorUserId="3456" LastEditDate="2019-10-11T18:28:00.203" LastActivityDate="2019-10-11T18:28:00.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17286" PostTypeId="2" ParentId="17273" CreationDate="2019-10-11T20:56:46.460" Score="2" Body="&lt;p&gt;My suggestion would be to look at the existing literature.&lt;/p&gt;&#xA;&lt;p&gt;Blake, Rand, Tingley, and Warneken (2015) &amp;quot;introduce a novel implementation of the repeated Prisoner's Dilemma (PD) designed for children to examine whether repeated interactions can successfully promote cooperation in 10 and 11 year olds.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Dealing with younger children (ages 6-11), Fan (2000) reports on a study of &amp;quot;children’s behavior in a prisoner’s dilemma game.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;A classic treatment is Tedeschi, Hiester, and Gahagan (1969), which modified the Prisoner's Dilemma Game for a study of children ages 8 to 10.&lt;/p&gt;&#xA;&lt;p&gt;Multiple papers have been written on how children with autism do or do not cooperate in the Prisoner's Dilemma, if that's a concern.&lt;/p&gt;&#xA;&lt;p&gt;A journal search for &lt;code&gt;&amp;quot;prisoner's dilemma&amp;quot; children &amp;quot;game theory&amp;quot;&lt;/code&gt; came back with about 1000 references. You could limit that to recent studies, or include additional search terms such as &amp;quot;teenager&amp;quot; if you wish. Pick a few studies and read the methodology section for ideas.&lt;/p&gt;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&lt;p&gt;Blake, P. R., Rand, D. G., Tingley, D., &amp;amp; Warneken, F. (2015). The shadow of the future promotes cooperation in a repeated prisoner's dilemma for children. &lt;em&gt;Scientific Reports (Nature Publisher Group), 5&lt;/em&gt;, 14559. &lt;a href=&quot;https://doi.org/10.1038/srep14559&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1038/srep14559&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Fan, C.-P. (2000). Teaching children cooperation — An application of experimental game theory. &lt;em&gt;Journal of Economic Behavior &amp;amp; Organization, 41&lt;/em&gt;(3), 191–209. &lt;a href=&quot;https://doi.org/10.1016/s0167-2681(99)00072-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1016/s0167-2681(99)00072-4&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Tedeschi, J. T., Hiester, D., &amp;amp; Gahagan, J. P. (1969). Matrix values and the behavior of children in the Prisoners Dilemma Game. &lt;em&gt;Child Development, 40&lt;/em&gt;(2), 517. &lt;a href=&quot;https://doi.org/10.2307/1127419&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.2307/1127419&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3593" LastEditorUserId="3593" LastEditDate="2020-09-12T04:18:34.527" LastActivityDate="2020-09-12T04:18:34.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17288" PostTypeId="1" CreationDate="2019-10-12T15:40:00.407" Score="3" ViewCount="148" Body="&lt;p&gt;I am about to teach a second year undergraduate class on applied differential equation (first time) and, while I won't have time to go into the details, I wanted to show my students the difference between the propagation in even dimensions and the propagation in odd dimensions. Namely, the fact that in even dimensions one gets ripples but not in odd dimensions. Are there simple solutions in one (or 3) dimension and two dimensions that I can use to illustrate this? Thanks&lt;/p&gt;&#xA;" OwnerUserId="4603" LastActivityDate="2019-10-12T15:40:00.407" Title="The propagation of the wave equation in even versus odd dimension" Tags="|undergraduate-education|examples|differential-equations|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17289" PostTypeId="2" ParentId="17161" CreationDate="2019-10-12T19:41:51.683" Score="0" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;“Success is not final, failure is not fatal: it is the courage to continue that counts.”&#xA;  ― Winston S. Churchill &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“There is only one thing that makes a dream impossible to achieve: the fear of failure.”&#xA;  ― Paulo Coelho, The Alchemist &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“Failure is the condiment that gives success its flavor.”&#xA;  ― Truman Capote &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="238" LastActivityDate="2019-10-12T19:41:51.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17291" PostTypeId="2" ParentId="14261" CreationDate="2019-10-13T02:03:15.773" Score="1" Body="&lt;p&gt;Permit me to echo @guest's emphasis on &quot;business analysis.&quot;&#xA;Hardly &quot;academic references,&quot; to quote the OP, but...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a company,&#xA;&lt;a href=&quot;https://www.kepner-tregoe.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kepner-Tregoe&lt;/a&gt;, whose&#xA;focus is problem solving:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Problem solving is at the core of human evolution. It is the methods we use to understand what is happening in our environment, identify things we want to change and then figure out the things that need to be done to create the desired outcome. Problem solving is the source of all new inventions, social and cultural evolution, and the basis for market based economies. It is the basis for continuous improvement, communication and learning. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Another, &lt;a href=&quot;https://ccmit.mit.edu/problem-solving/&quot; rel=&quot;nofollow noreferrer&quot;&gt;CC/MIT&lt;/a&gt;&#xA;(MIT Office of Digital Learning):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why is problem solving important? Good problem solving skills empower you not only in your personal life but are critical in your professional life. In the current fast-changing global economy, employers often identify everyday problem solving as crucial to the success of their organizations. For employees, problem solving can be used to develop practical and creative solutions, and to show independence and initiative to employers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And a Huffpost article, &lt;a href=&quot;https://www.huffpost.com/entry/the-importance-of-problem_b_190514&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Importance of Problem-Solving&lt;/a&gt;,&#xA;by Ken Watanabe, who has emphasized problem-solving instead&#xA;of memorization:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It's important to realize that being a problem solver isn't just an ability; it's a whole mind-set, one that drives people to bring out the best in themselves and to shape the world in a positive way. Rather than accepting the status quo, true problem solvers are constantly trying to proactively shape their environment. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-10-13T02:11:25.343" LastActivityDate="2019-10-13T02:11:25.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17292" PostTypeId="1" CreationDate="2019-10-13T12:36:14.903" Score="7" ViewCount="1648" Body="&lt;p&gt;I'm thinking about teaching calculus by firstly introducing the asymptotic notations (big-Oh, little-oh, and &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt;), secondly explaining their &quot;arithmetic&quot; (things like how to sum little-oh's and similar), and then doing everything else (for example, the derivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; would be defined as the real number &lt;span class=&quot;math-container&quot;&gt;$f^\prime(x_0)$&lt;/span&gt;, if it exists, such that &lt;span class=&quot;math-container&quot;&gt;$f(x) = f(x_0) + f^\prime(x_0) (x - x_0) + o(x - x_0)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \to x_0$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that, of course, while explaining little-oh's I would implicitly explain the definition of limit to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, asymptotic notations are usually introduced to students after most of the theory have been shown, and mostly as a tool to compute limits. So I wonder if this alternative approach have been tried before and could be fruitful. Some pro/cons I see:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pro:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Asymptotics notations give a concise language to express many concepts (&quot;bounded&quot; is &lt;span class=&quot;math-container&quot;&gt;$O(1)$&lt;/span&gt;, &quot;infinitesimal&quot; is &lt;span class=&quot;math-container&quot;&gt;$o(1)$&lt;/span&gt;, the expression for the derivative that I already mentioned...) and work with them in an almost mechanical way.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students have to learn them anyway soon or later&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cons:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Asymptotic notations could be a bit difficult to understand at first, because they are not reflexive (things like &lt;span class=&quot;math-container&quot;&gt;$o(x^2) = o(x)$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$o(x) \ne o(x^2)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \to 0$&lt;/span&gt;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12824" LastEditorDisplayName="user507" LastEditDate="2019-10-14T21:41:03.677" LastActivityDate="2019-10-14T21:41:03.677" Title="Teaching asymptotic notations at the beginning of calculus" Tags="|mathematical-pedagogy|calculus|notation|" AnswerCount="5" CommentCount="15" ClosedDate="2019-10-15T00:45:57.977" ContentLicense="CC BY-SA 4.0" />
  <row Id="17293" PostTypeId="2" ParentId="17292" CreationDate="2019-10-13T15:10:47.620" Score="10" Body="&lt;p&gt;When I was an undergraduate, the big and little oh notations were taught to me in a first-year math course ostensibly aimed at physics students. The class was strong - several of us are now mathematics and physics professors in universities - and the attempt was, to my mind, moderately unsuccessful (although one could make the counterpoint that I still remember it clearly 27 years later). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the problem is that the big and little oh notation/terminology is a good example of not very good notation/terminology. Notation and terminology work best when they signal their own meanings, and when they easily distinguish similar cases. The big and little oh are too similar to easily distinguish mentally, and when written on the blackboard they often are genuinely indistinguishable. It's better to say simply &quot;bounded&quot; and &quot;infinitesimal&quot; than to write O(1) and o(1)! Particularly in a first-year class, one is unlikely to need to discuss more than these two special cases anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a classroom setting, with reference to Taylor approximations these notations are usually used as a way of formalizing physicists's dropping of higher order terms in a way that makes mathematicians feel good about themselves for not breaking the rules, rather than in a way that makes anything clearer to students. For students early in their careers, I think it's usually better to take the time to write out in more detail what such notation encodes. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-10-13T15:10:47.620" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17294" PostTypeId="2" ParentId="17292" CreationDate="2019-10-13T16:06:05.853" Score="7" Body="&lt;p&gt;I've never known or needed that notation in my life.  And I've used lots of calculus for science and engineering.  So one of your assumptions is wrong.  I think it would be more efficient to teach a normal calculus class.  Leave the esoteric abstractions for later, and for the kids who will really need them (a small subset of the typical calculus population, and not always already sorted in freshman year).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even for the kids who will eventually need it, I suspect it is more efficient to let them get a working knowledge of the practical calculus first.  I don't understand this urge to cram real analysis into calculus class.  It ignores practical psychological pedagogy.  (It's hard to learn something in its hardest fashion first.)&lt;/p&gt;&#xA;" OwnerUserId="12868" LastActivityDate="2019-10-13T16:06:05.853" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17295" PostTypeId="2" ParentId="17292" CreationDate="2019-10-13T18:17:33.940" Score="8" Body="&lt;p&gt;I think that there may be advantages to introducing the idea of asymptotics early, and I think that it might be interesting to experiment a bit with curriculum by bringing in asymptotics in calculus.  However, there are caveats:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think that students need to have a &lt;em&gt;solid&lt;/em&gt; understanding of limits and continuity, first.  I don't necessarily mean that the students need to be rattling off &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proofs at the drop of a hat, but they should have a good intuition about what limits are, how they behave (e.g. the limit of a sum is the sum of the limits, &amp;amp;c.), and how limits and continuity are related.  Most importantly, the students should demonstrate a clear understanding of the &quot;Squeeze Theorem&quot;, since a lot of the discussion of asymptotics reduces to applications of the Squeeze Theorem.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Asymptotics should be introduced gently, without reliance on notation.  For example, when teaching precalculus, a fair amount of time is spent discussing how to graph rational functions.  When considering the behaviour of a graph at infinity, our current curriculum suggests a notation like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{(x+2)^2(3x+4)(7-x)^5}{(x-2)^4(5x-2)^2}&#xA;= \frac{-3x^8 + \text{junk}}{25x^6 + \text{junk}},&#xA;$$&lt;/span&gt;&#xA;where &quot;&lt;span class=&quot;math-container&quot;&gt;$\text{junk}$&lt;/span&gt;&quot; would be more rigorously stated as &quot;lower order terms&quot;, or could be written in terms of big-Oh notation.  In precalculus, we are pretty hand-wavy about what this actually means (though we can make it more rigorous if we &lt;em&gt;really&lt;/em&gt; need to).  In a calculus class, this could (and should) be made much more rigorous, as the students are expected to know how to take limits.  Examples like this could motivate the introduction of special notation for asymptotics.  Further motivation comes from L'Hospital's rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any event, the underlying point is that the idea of asymptotics should arise naturally, and come from a desire to simplify the notation for the purposes of computation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do &lt;em&gt;not&lt;/em&gt; start using ambiguous or abusive notation until your students are very comfortable with the basics.  Equality should &lt;em&gt;always&lt;/em&gt; be a reflexive relation.  If you mean that &lt;span class=&quot;math-container&quot;&gt;$f(x) \in O(x)$&lt;/span&gt;, then write that.  Don't write &lt;span class=&quot;math-container&quot;&gt;$f(x) = O(x)$&lt;/span&gt;.  Calculus students are already trying to assimilate a lot of new information and ideas.  Calculus is a hard class for many students, particularly when we consider that it is a terminal class for many students (at my current institution, maybe half of the students in calculus end up taking further mathematics course, and less than 10% end up taking upper division mathematics).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was argued in comments by the original asker that asymptotic notation comes up a lot in analysis and computer science.  Suppose that we accept this claim&amp;mdash;so what?  The vast majority of students in every elementary calculus class I've ever taught are neither mathematics nor CS majors, and the math and CS majors who are present are likely to learn everything they need to about notation for asymptotics when it actually comes up in future classes.  Note that I am not arguing (here) against teaching Landau notation&amp;mdash;rather, I am arguing that this is not a good argument in favor of teaching it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My recommendation in these kinds of classes is to introduce as little notation and jargon as is humanly possible.  Don't ask students to use ambiguous notation. &#xA;Don't introduce notation which you are not going to use.  Don't introduce terms that you aren't going to use later.  Make sure that the exposition is clear and unambiguous.  Save abuses of notation for later.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Calculus is a prerequisite for other classes at many, many institutions.  Future instructors are going to assume certain knowledge, based on an understanding of a curriculum which is pretty well standardized across the United States (and, presumably, elsewhere, as well).  If you decide to introduce asymptotics with the associated big-o and little-o notations, make sure that you do so without losing out on any of the other curriculum which your students are going to need to know for future classes.  Remember that you, as an instructor, are a member of a community of educators&amp;mdash;you are responsible not only to your students, but also to your colleagues, who are going to have to teach your students in future classes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2019-10-13T18:17:33.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17296" PostTypeId="2" ParentId="17292" CreationDate="2019-10-14T07:14:18.207" Score="1" Body="&lt;p&gt;I agree with most of what's already been said so far, but wanted to add one point---that the answer is surely going to depend on your audience. Of course, this is going to be people who have never learnt calculus before so their mathematical maturity and exposure to such things is definitely going to be limited, but depending on where you're teaching the ability of students to pick new things up is going to differ widely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If, in your opinion, you are teaching a bunch of exceptional undergrads/college students who can pick anything up easily and with little risk of permanent confusion, I don't see why you shouldn't teach this notation. Even if, like the other answers said, it might not be of crucial importance, it is certainly good-to-know especially for those who would later pursue deeper studies in math, physics or (especially) computer science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if you don't think too highly of the ability level of (most of) your students, then I think this is a horrible idea. When there are so many other things new to students which are already going on in a typical calculus class, adding things which risk even more confusion is the worst thing you can do. Even as someone who is reasonably more experienced than your typical calculus freshman student, seeing things like &lt;span class=&quot;math-container&quot;&gt;$o(x^2)=o(x)$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$o(x)\neq o(x^2)$&lt;/span&gt; can still make me uncomfortable until I take a bit more time to stare at the equation to figure out what is the intended meaning. Adding this confusing (or perhaps even contentious: see Xander Handerson's answer and comments) notation to a class which already may be struggling with things like taking derivatives and manipulating integrals does not seem like the best idea to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, always have your audience in mind. If your class is made up of stellar, amazing students, then by all means go for it. Most likely though, most of them are not---in which case big and little oh notation are best left to a course &lt;em&gt;after&lt;/em&gt; calculus.&lt;/p&gt;&#xA;" OwnerUserId="10206" LastActivityDate="2019-10-14T07:14:18.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17297" PostTypeId="2" ParentId="14038" CreationDate="2019-10-14T08:46:53.047" Score="3" Body="&lt;p&gt;Like others, I feel this is too much. It turns the question into almost a mini research project (assuming there is some reasonable amount of checking needed).  It's one thing to just scan through ten choices, but then if not immediately clear (in a false, false, true, false...series), than having to look at so many options and compare them is difficult--remember the human working memory (immediate cognition) probably can consider something like three to seven items at a time.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if they are silly and don't really need checking, than there's little benefit of so much &quot;chaff&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then after running such a project-like enterprise, you still have a very digital result.  And one with zero credit for partial work or for the second best of 10 options.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, because of the added complexity, it means you can carry fewer total questions.  So you're getting less statistical smoothing (of multiple tries).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, I fail to see the benefit of so many options.  Does one really have situations with ten likely options?  And/or isn't it usually a bit of work even to get four or five (and test taking manuals explaining how often a couple of weaker choices or non-independent choices can be eliminated, even if the final choice of the smaller set not clear).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally students will not be familiar with it and you will be confounding anxiety about the material itself with anxiety about an unusual testing mechanism.  &lt;/p&gt;&#xA;" OwnerUserId="12870" LastEditorUserId="8571" LastEditDate="2019-10-14T17:12:57.197" LastActivityDate="2019-10-14T17:12:57.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17298" PostTypeId="2" ParentId="17292" CreationDate="2019-10-14T13:50:35.860" Score="0" Body="&lt;p&gt;Don't. The language is highly imprecise, confusing, contrary to the precise usage you hopefully will be teaching them, and in my experience not useful for doing calculus.&lt;/p&gt;&#xA;" OwnerUserId="1146" LastActivityDate="2019-10-14T13:50:35.860" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17299" PostTypeId="2" ParentId="17215" CreationDate="2019-10-14T17:43:44.290" Score="7" Body="&lt;p&gt;I must confess I'm not an educator, but I like this question and at the very least I can answer with the intuitive picture I use in my own head.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th central moment &lt;span class=&quot;math-container&quot;&gt;$\mu_n$&lt;/span&gt; of a random variable &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is defined by: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mu_n \equiv E[(X-E[X])^n]$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what follows we will take the measure space of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; to be the real numbers, since that's the easiest to visualize and the most common thing to deal with in statistics. Now, with this knowledge we can rewrite the moment as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mu_n = \int_{-\infty}^{\infty} (x-E[X])^nf_X(x)dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$f_X$&lt;/span&gt; is the pdf of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;. To help complete the mental image, it is important to keep in mind that &lt;span class=&quot;math-container&quot;&gt;$E[X]$&lt;/span&gt; is a constant and &lt;span class=&quot;math-container&quot;&gt;$f_X$&lt;/span&gt; is a normalized function that is non-negative. So, our integrand looks like some single term polynomial centered at the mean of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, multiplying the non-negative pdf of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;. This is a lot clearer once you see what I'm talking about, so I will show some photos and explain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Second Moment (and even moments in general):&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a picture illustrating the integral I was talking about for the second moment using some pdf I made up. The dotted line is the pdf, the dashed line is the polynomial term, and the solid line is the total integrand:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8wsT3.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8wsT3.png&quot; alt=&quot;second moment&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keep in mind that the moment is defined as the integral of the solid curve. There's a couple of things worth noting here:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The moment weights parts of the pdf that are further from the mean heavier. For instance, the right bump in the integrand has more area underneath it than the left one, even though the left bump of the pdf is more likely.&lt;/li&gt;&#xA;&lt;li&gt;It's not pictured, but hopefully it's clear that higher moments will weight things further from the mean even more heavily.&lt;/li&gt;&#xA;&lt;li&gt;For even moments, the polynomial term is non-negative, so the integrand cannot be negative anywhere. This means that there is no &quot;cancellation&quot;, as we'll see there is for odd moments.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Summarizing these points, we see that even moments give us a measure of how far from the mean we can expect our random variable to fall. As we go to higher and higher moments, these will give more and more weight to parts of our distribution that lie farther from the mean. So, if for whatever reason there's a bump in our pdf super far from the mean, or a tail that doesn't decay as quickly, it will lead to large higher order moments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Third Moment (and odd moments in general):&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an analogous picture for the third moment of the same pdf:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KjWr5.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KjWr5.png&quot; alt=&quot;third moment&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can see that odd moments work just like even moments for bullets one and two in the previous section. However, now there's an added complication-- our polynomial is negative for some values, so the integral from the left side of the mean will cancel the integral from the right side to some extent. Thus, odd moments no longer just measure how far things are distributed from the mean. They also provide a measure of the asymmetry of the pdf-- a pdf perfectly symmetric about the mean will have null odd moments. This gives them more information than even moments in some sense, but the drawback is that they're harder to interpret taken alone-- a third moment close to zero could mean that you have a perfectly symmetric, widely distributed pdf, or a narrowly distributed, highly asymmetric pdf. As before, higher order moments will accentuate features that lie far from the mean, so a tail that decays faster on one side than on the other will make a large impact in higher order even moments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why standardized moments?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the previous discussion took place using raw moments, but a lot of times the standardized moment is of more interest, ie&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\tilde{\mu_n} = \mu_n/\sigma^n$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sigma = \sqrt{E[(X-E[X])^2]}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is the standard deviation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two main reasons for this: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It combines the knowledge of two different moments into one. This is especially useful for odd moments, as it can help distinguish between the two degenerate examples given in the previous section: the asymmetric, narrow distribution would have a high skewness while the nearly symmetric, wide distribution would have a small one, even though they both have a small third moment.&lt;/li&gt;&#xA;&lt;li&gt;It is scale invariant. It is straightforward to show that &lt;span class=&quot;math-container&quot;&gt;$\sigma_{\lambda X} = \lambda \sigma_X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mu_n(\lambda X)=\lambda^n \mu_n(X)$&lt;/span&gt; If &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; is a real number scaling &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;. Together, this means &lt;span class=&quot;math-container&quot;&gt;$\tilde{\mu}_n(\lambda X)=\tilde{\mu}_n(X)$&lt;/span&gt;. This is an important property because in real life, we often are taking statistics of objects with units, and transformations between units are described by scaling them. You can describe the heights of people in a sample using feet or centimeters, but either way they're describing the same thing and so we would like our statistical measures to not change.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;TL;DR:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, it's impossible to give a completely general yet useful interpretation for what any one moment means. There will always be vastly different distributions that have the same moment, because fundamentally you're trying to describe something with an infinite number of degrees of freedom (the pdf) with a single number. However, there are three rough guidelines that help give a bit of intuition:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Even moments tell you something about how far things are distributed from the mean.&lt;/li&gt;&#xA;&lt;li&gt;Odd moments tell you something about how asymmetric the distribution is about the mean (skewed left = negative, right = positive, and zero = symmetric).&lt;/li&gt;&#xA;&lt;li&gt;Higher order moments put more of an emphasis on how the distribution behaves further from the mean, so tails and anomalous pdf bumps far away matter more than the bulk of the distribution that sits close to the mean.   &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12874" LastEditorUserId="12874" LastEditDate="2019-10-14T19:51:47.763" LastActivityDate="2019-10-14T19:51:47.763" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17300" PostTypeId="2" ParentId="12927" CreationDate="2019-10-14T18:48:20.123" Score="3" Body="&lt;p&gt;I suggest reading the AMS blog posts about &lt;em&gt;active learning&lt;/em&gt; [which I think can serve as a suitable substitute&lt;span class=&quot;math-container&quot;&gt;$^{\star}$&lt;/span&gt; for what you refer to as &amp;quot;student-centered&amp;quot;] for which the 2015 series can be found here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://blogs.ams.org/matheducation/tag/active-learning-series-2015/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://blogs.ams.org/matheducation/tag/active-learning-series-2015/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is one post that you might start with:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://blogs.ams.org/matheducation/2015/09/10/active-learning-in-mathematics-part-i-the-challenge-of-defining-active-learning/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://blogs.ams.org/matheducation/2015/09/10/active-learning-in-mathematics-part-i-the-challenge-of-defining-active-learning/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Excerpt:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;...&lt;em&gt;if the experiments analyzed here had been conducted as randomized controlled trials of medical interventions, they may have been stopped for benefit&lt;/em&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So strong is the evidence supporting the positive effects of active learning techniques in postsecondary mathematics and science courses that Freeman et al made the statement above in their 2014 Proceedings of the National Academy of Science (PNAS) article &lt;a href=&quot;https://www.pnas.org/content/111/23/8410.abstract&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Active learning increases student performance in science, engineering, and mathematics&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You may, in fact, find that reading through papers on &lt;em&gt;active learning&lt;/em&gt; is more productive than the search term &lt;em&gt;student-centered&lt;/em&gt;. See also the AMS Notices (2017) piece &lt;a href=&quot;https://www.ams.org/publications/journals/notices/201702/rnoti-p124.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;here&lt;/strong&gt;&lt;/a&gt; with citation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Braun, B., White, D., Bremser, P., Duval, A. M., &amp;amp; Lockwood, E. (2017). What Does Active Learning Mean for Mathematicians? &lt;em&gt;Notices of the American Mathematical Society, 64&lt;/em&gt;(2), 124-129.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\star$&lt;/span&gt;: Here is a later blog excerpt RE: active learning and student-centered approaches [&lt;strong&gt;emphasis&lt;/strong&gt; added]:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A &lt;strong&gt;student-centered&lt;/strong&gt; instructional approach places less emphasis on transmitting factual information from the instructor, and is consistent with the shift in models of learning from information acquisition (mid-1900s) to knowledge construction (late 1900s). This approach includes&lt;/p&gt;&#xA;&lt;p&gt;more time spent engaging students in &lt;strong&gt;active learning&lt;/strong&gt; during class&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-10-14T18:48:20.123" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17302" PostTypeId="1" CreationDate="2019-10-14T23:53:03.187" Score="11" ViewCount="814" Body="&lt;p&gt;In the precalculus curriculum I am teaching (using Stewart's book Precalculus: Mathematics for Calculus, 7th ed.), we do a bit of polar graphing, which includes discussion of symmetry on polar graphs. We teach the students to test for symmetry in standard ways:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If the equation is invariant under the transformation &lt;span class=&quot;math-container&quot;&gt;$(r,\theta)\mapsto(r,-\theta)$&lt;/span&gt;, then we have symmetry across the polar axis. This is equivalent to the rectangular transformation &lt;span class=&quot;math-container&quot;&gt;$(x,y)\mapsto(x,-y)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If the equation is invariant under the transformation &lt;span class=&quot;math-container&quot;&gt;$(r,\theta)\mapsto(r,\pi-\theta)$&lt;/span&gt;, then we have symmetry across the line &lt;span class=&quot;math-container&quot;&gt;$\theta=\frac{\pi}{2}$&lt;/span&gt;. This is equivalent to the rectangular transformation &lt;span class=&quot;math-container&quot;&gt;$(x,y)\mapsto(-x,y)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If the equation is invariant under either of the the transformations &lt;span class=&quot;math-container&quot;&gt;$(r,\theta)\mapsto(r,\pi+\theta)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(r,\theta)\mapsto(-r,\theta)$&lt;/span&gt;, then we have symmetry across the pole. This is equivalent to the rectangular transformation &lt;span class=&quot;math-container&quot;&gt;$(x,y)\mapsto(-x,-y)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These are all great, but then there was a homework question about the graph given by &lt;span class=&quot;math-container&quot;&gt;$r^2=\sin\theta$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It's clear that we can replace &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$-r$&lt;/span&gt;, so we have symmetry number 3. Additionally, &lt;span class=&quot;math-container&quot;&gt;$\sin\theta=\sin(\pi-\theta)$&lt;/span&gt;, we have symmetry number 2.&lt;/p&gt;&#xA;&lt;p&gt;The function &lt;em&gt;appears&lt;/em&gt; to fail the test for symmetry number 1, because &lt;span class=&quot;math-container&quot;&gt;$\sin(-\theta)=-\sin\theta$&lt;/span&gt;. However, any function with two of these symmetries automatically has the third. (Together with the identity transformation, we're just looking at the Klein 4-group here.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/B0Z8c.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/B0Z8c.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Symmetry number 1 &lt;em&gt;can be&lt;/em&gt; detected algebraically, but to do so, you have to use the equivalent transformation &lt;span class=&quot;math-container&quot;&gt;$(r,\theta)\mapsto(-r,\pi-\theta)$&lt;/span&gt;, which is clearly just the composition of symmetries 2 and 3. Similarly, symmetry number 2 could also be detected by the composition of 1 and 3: &lt;span class=&quot;math-container&quot;&gt;$(r,\theta)\mapsto(-r,-\theta)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Anyway, the online WebAssign homework was only counting symmetries 2 and 3 right. I emailed them, and they say they've replaced the question, and I'll see the new version next time I make this assignment. Great, but that leaves math/teaching questions unanswered.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Has anyone else had this issue come up, and how did you handle it?&lt;/li&gt;&#xA;&lt;li&gt;What is the easiest way to account for the failure of this equation, or one like it, when testing for symmetry number 1?&lt;/li&gt;&#xA;&lt;li&gt;If we applied two tests for each symmetry (one with &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and one with &lt;span class=&quot;math-container&quot;&gt;$-r$&lt;/span&gt;), are there any equations that would still &amp;quot;get away&amp;quot; undetected?&lt;/li&gt;&#xA;&lt;li&gt;Is it sufficient to state (A) that any function with two of these symmetries automatically has the third, and (B) any such function will pass two of the traditional tests, thus letting us know?&lt;/li&gt;&#xA;&lt;li&gt;Is there anything else students should be on the lookout for, when trying to obtain symmetries of polar graphs from analyzing equations in &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2034" LastEditorUserId="2034" LastEditDate="2024-03-15T20:44:55.910" LastActivityDate="2024-03-19T10:12:00.727" Title="Symmetry in polar functions - how to explain" Tags="|algebra|trigonometry|graphing|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17303" PostTypeId="2" ParentId="17186" CreationDate="2019-10-15T04:18:07.530" Score="6" Body="&lt;p&gt;Here's another computer-graphics example, which may or may not count as &quot;nowadays&quot;. When I was in college 25 years ago I started a project to write a ray-trace renderer, which I then continued to expand after college. Ray-tracing is an elegant model that just recently has gained the hardware support to make it feasible in real time, e.g.: &lt;a href=&quot;https://developer.nvidia.com/rtx/raytracing&quot; rel=&quot;noreferrer&quot;&gt;https://developer.nvidia.com/rtx/raytracing&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ray-tracing model that I'm familiar with requires geometrically coding primitives like a cube, sphere, plane, etc., and then designers can put those pieces together to make more sophisticated compound objects in space. As part of my expansion, I also added the ability for my code to handle the primitive for a torus (thinking that would be useful for space-station animations and the like). Since a ray can intersect a torus in as many as 4 locations, geometrically this requires solving a 4th-degree equation, i.e., a quartic. The mechanism to do that would be, in the general case, reduce it to a cubic in &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; and then call a function I wrote to solve that cubic -- i.e., an explicit implementation of the cubic formula. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nGOdV.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nGOdV.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-10-15T04:18:07.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17304" PostTypeId="2" ParentId="17186" CreationDate="2019-10-15T04:48:24.623" Score="3" Body="&lt;p&gt;(Second answer for follow-on question from Sue about contacting math historians.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry, to be so simplistic, Sue, but I would just reach out* to some general math historians of any stripe (at a university that has those) and just correspond.  Either with whoever you get or by trying to work your way to one with the specific subject matter expertise you want.  I would think any math historian (even if not an expert on this period) would know some right questions to ask, that would benefit you.  And then he might be able to say, &quot;talk to Jane Umptifratz&quot; if you want someone especially strong on this exact period.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, you don't mention it, so apologies if this is also obvious, but a quick Amazon search shows several books on the derivation of the cubic/quintic, which after all is a fascinating story of human conflict as well as interesting mathematics:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0691183678&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Secret-Formula-Mathematical-Renaissance-Uncovered/dp/0691183678&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0387360263&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Tales-Mathematicians-Physicists-Simon-Gindikin/dp/0387360263&lt;/a&gt; (chapter within)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B01JUPJMHC&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Cubic-Wars-Tartaglia-Cardano-Ferrari-ebook/dp/B01JUPJMHC&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also try reaching out to some people who are math populizers:  John Derbyshire or one of the fellows who wrote a Fermat Last Theorem book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, I suggest to look at the article on algebra in the Britannica as well as the articles on the protagonists.  Often the person chosen to write a Britannica article has deep knowledge (much more so than the space of the article contains), so you could try to correspond with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, with all these people be pleasant and don't expect them to help you and don't worry if some ignore you.  Just move on to the next.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*In this day and age, it is relatively easy to Google someone's email or even phone or to find a way to contact them through their department or publisher or the like.  (Even in the old days, I just got on the phone and called around.)&lt;/p&gt;&#xA;" OwnerUserId="12870" LastActivityDate="2019-10-15T04:48:24.623" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17305" PostTypeId="2" ParentId="17302" CreationDate="2019-10-16T06:39:00.613" Score="9" Body="&lt;p&gt;The problem underlying the discussion in the question can be summarized as that it is necessary to choose a branch cut to define a complex logarithm (or arctangent).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a mistake and pedagogically a bad practice to allow negative values of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a mistake because pairs &lt;span class=&quot;math-container&quot;&gt;$(r, \theta)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; possibly negative have no right to be called &lt;em&gt;coordinates&lt;/em&gt;. &lt;em&gt;Coordinates&lt;/em&gt; is a synonym for diffeomorphism (of whatever regularity). A system of coordinates on a domain gives a way to label uniquely each point of the domain. The uniqueness is essential for any labeling system to be properly called &lt;em&gt;coordinates&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is bad practice pedagogically because it generates unnecessary confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The map &lt;span class=&quot;math-container&quot;&gt;$(x, y) \to (\sqrt{x^{2} + y^{2}}, \arctan(y/x))$&lt;/span&gt; is a diffeomorphism from &lt;span class=&quot;math-container&quot;&gt;$(0, \infty)\times (—\pi, \pi]$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^{2}\setminus\{(x, y): x \geq 0\}$&lt;/span&gt;. Its inverse is the restriction to its image of the complex exponential, written in real coordinates. The branch cut along some axis is needed to obtain a diffeomorphism. The choice of the negative real axis as the branch is customary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Allowing negative values of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; introduces redundancy. It destroys the unique labeling property. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The equation &lt;span class=&quot;math-container&quot;&gt;$r^{2} = \sin \theta$&lt;/span&gt; has no solutions when &lt;span class=&quot;math-container&quot;&gt;$\theta \in (\pi, 2\pi)$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The picture drawn in the question is wrong because negative values of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; should not be allowed. Those points on the curve with &lt;span class=&quot;math-container&quot;&gt;$r &amp;lt; 0$&lt;/span&gt; do not correspond to points on the curve &#xA;&lt;span class=&quot;math-container&quot;&gt;$$(x^{2} + y^{2})^{3/2} = y.$$&lt;/span&gt; &#xA;The curve should only be the top part of the curve indicated. Of the symmetries considered only the map &#xA; - &lt;span class=&quot;math-container&quot;&gt;$(r, \theta) \to (r, \pi - \theta)$&lt;/span&gt; (reflect across the vertical axis)&#xA;is valid. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The map &lt;span class=&quot;math-container&quot;&gt;$(r, \theta) \to (r, \pi + \theta)$&lt;/span&gt; &lt;em&gt;is not a symmetry&lt;/em&gt; of the equation. If &lt;span class=&quot;math-container&quot;&gt;$\sin \theta$&lt;/span&gt; is positive, then &lt;span class=&quot;math-container&quot;&gt;$\sin(\pi + \theta)$&lt;/span&gt; is negative, and there is no solution. When negative values of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; are allowed, then &lt;span class=&quot;math-container&quot;&gt;$(-r, \theta) \sim (r, \pi + \theta)$&lt;/span&gt; so, for a point &lt;span class=&quot;math-container&quot;&gt;$(r, \theta)$&lt;/span&gt; of the curve with &lt;span class=&quot;math-container&quot;&gt;$r &amp;gt; 0$&lt;/span&gt;, the &quot;solution&quot; &lt;span class=&quot;math-container&quot;&gt;$(-r, \theta) \sim (r, \theta + \pi)$&lt;/span&gt; is not a solution! Put another way, the equation &lt;span class=&quot;math-container&quot;&gt;$r^{2} = \sin \theta$&lt;/span&gt; is &lt;em&gt;not well defined&lt;/em&gt; on the quotient of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}\setminus\{0\} \times (0, 2\pi)$&lt;/span&gt; by the identification &lt;span class=&quot;math-container&quot;&gt;$(r, \theta) \sim (-r, \pi + \theta)$&lt;/span&gt;. The solutions with &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; negative &lt;em&gt;do not correspond&lt;/em&gt; to any points in the &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt;-plane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The two-lobed curve drawn in the question corresponds to the Cartesian equation &lt;span class=&quot;math-container&quot;&gt;$$(x^{2} + y^{2})^{3} = y^{2},$$&lt;/span&gt; which corresponds to the &lt;em&gt;two&lt;/em&gt; polar equations &lt;span class=&quot;math-container&quot;&gt;$r^{2} = \pm \sin \theta$&lt;/span&gt;. This curve &lt;em&gt;does&lt;/em&gt; admit all the symmetries discussed in the question.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2019-10-17T05:53:08.067" LastActivityDate="2019-10-17T05:53:08.067" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17307" PostTypeId="1" CreationDate="2019-10-16T18:53:05.587" Score="9" ViewCount="1862" Body="&lt;p&gt;I am planning to give my students a take-home examination on ODE. The main topic that I would like to cover is Linear Differential Equations of Order Greater than One. For example, I will give my students question such as &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Solve the following Differential Equation:&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$y''+y'=x^2+2x.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But my concern is, if the students are allowed to answer such questions in a take-home examination, there is almost no reasonable way that I can prevent them from using some online ODE solvers, such as &lt;a href=&quot;https://www.wolframalpha.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WolframAlpha&lt;/a&gt; or &lt;a href=&quot;https://www.symbolab.com/solver/ordinary-differential-equation-calculator&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since some solvers not only give the solutions but also the details, it will defeat the purpose of this take-home examination. I can make &quot;no Differential Equation Solvers&quot; as a rule for this take-home examination, but I am not sure how to monitor my students and thus enforcing the rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surely, having an in-class exam solves the problem. But if it is not possible, is there a way to reasonably test ODE's in the format of a take-home examination?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I gave them the take-home exam asking them to sign to promise that they have not used any unauthorised resources when finishing the exam. From my observation, all of them seem to have finished the exam honestly. By the way, I work in a small religious college where most students have very high moral values. &lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2019-11-01T21:10:39.693" LastActivityDate="2020-02-27T21:36:27.180" Title="Take-Home Examination on Ordinary Differential Equations?" Tags="|undergraduate-education|exams|differential-equations|test-design|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17310" PostTypeId="2" ParentId="17307" CreationDate="2019-10-16T23:27:14.177" Score="18" Body="&lt;p&gt;This is not really a math problem, it's a social problem. Some schools, such as West Point and Cal Tech, have their own honor code systems for this sort of thing. From what I understand, they work very well. However, most schools do not have any such system. Social scientists and psychologists have studied what factors promote or prevent cheating. Cheating is promoted by the following factors:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a perception that everyone does it&lt;/li&gt;&#xA;&lt;li&gt;having a single, high-stakes exam, or a small number of such exams, rather than systematic, ongoing assessment&lt;/li&gt;&#xA;&lt;li&gt;a lack of self-efficacy.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;At a school that doesn't have an honor code, it's going to be hard to avoid the issue that everyone will think they have to cheat in order to keep up with others who are cheating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You don't say in your question how many exams you plan to give this term. For example, I'm giving 8 short exams over the course of 15 weeks in one of my classes this semester. If I was having the problem, as you describe, of running low on time, I would just cut one exam and do 7 exams. If you're doing only a couple of high-stakes exams, such as a midterm and a final, then you have one of the risk factors in place, and you're asking for trouble if you make the midterm a take-home.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you're kidding yourself if you think that the only issue is that students will use software to solve problems. They can also get help from human beings online. For example, there is a service called Chegg that many students on my campus use to cheat in STEM courses. They sign up for a recurring credit card charge, and in return they get full solutions to textbook problems and can also get help on problems they post. We have even had students use Chegg on their phones during a test to get live help, when the teacher wasn't vigilant enough about preventing access to phones. I believe they were both cooperating with each other and getting help from people off campus. You can't get out of this problem by constructing problems that are AI-proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your problem is lack of time, one good option is not to give in to the assumption that students will never read a textbook and must be spoon-fed every fact and method. Tell them what chapters of the book they're responsible for, and focus class time on helping them over any hard parts.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-10-16T23:56:20.183" LastActivityDate="2019-10-16T23:56:20.183" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17311" PostTypeId="2" ParentId="17307" CreationDate="2019-10-17T02:47:35.143" Score="7" Body="&lt;p&gt;Comment-answer, but too long for a comment:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you are thinking about this wrong.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tests are some of the MOST valuable hours in a course.  They are high stakes performances (like in music or sports).  Preparation for them drives a lot of learning.  Then the actual execution and subsequent feedback is often much more valuable training than routine drill problems, because of caring about the result.  You should maximize test utility versus another hour of lecture (which probably has less learning impact even than an hour of them working drill problems). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, by using a take home you are trying to get an extra hour of their time.  In addition, take homes are very hard to control from assistance (book, roommate, online).  Can you really expect them to be rigid about a time limit when taking a test at home?  If unlimited, then now you're taking more of their time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, take home tests often lead to instructors giving non ideal questions (ones that are too much like research projects for instance).  This is not the case here--your question is fine.  However, your urge to ask some sort of strange, hard to cheat on question may already be driving you away from what would be the normal best question in a controlled setting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, ODE books have too much content for a one quarter or one semester course.  It is very normal not to cover it all. So if you're already shaving some lessons off, just shave one more off.  And learn the core of the topic well.  And good tests are key to good learning.  Don't shave this 2nd order ODE with constant coefficients off.  It's the key part of the course!  Very high gain.  But surely there is some other lesson (existence, Wronskians, predator prey, transforms, Sturm Louisville, series, etc.) you can skip to allow use of class time for examination.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I love all the topics and would love to have you cover everything.  But you need to prioritize.  You got a lifeboat with 15 spots and 25 people, pick the best 15 to live.  2nd order constant coefficient ODE, with a forcing function, will be seen all over the place in physics, engineering, chemistry applications.  So it lives.  Some other topics go in the water.  And the test time needs to live.  It's &quot;high gain&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In theory, you could cover more content by telling the kids to learn a lesson sans lecture.  But I get the impression you're not training superstars here.  They will be unhappy without the scaffolding of some lecture time and Q&amp;amp;A time on the topic.  So, just treat this as an optimization problem and do a little less, but do it well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Moderator may cut my other answer.]&lt;/p&gt;&#xA;" OwnerUserId="12888" LastActivityDate="2019-10-17T02:47:35.143" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17312" PostTypeId="2" ParentId="14038" CreationDate="2019-10-17T03:12:26.177" Score="-1" Body="&lt;p&gt;Ten answers is a bit much. I agree with the other answers given, and there is also the test-taking anxiety factor. There are students out there who may know the material but simply find tests hard to take. For those students, ten possible answers can be incredibly overwhelming. Fewer possibilities will help these students push themselves to actually answer the questions given - showing their true level of understanding.   &lt;/p&gt;&#xA;" OwnerUserId="12529" LastActivityDate="2019-10-17T03:12:26.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17313" PostTypeId="2" ParentId="14038" CreationDate="2019-10-17T11:31:38.043" Score="1" Body="&lt;p&gt;I thought that the New York State Regents math exams had an elegant strategy back in the 80's (and probably before, but this was when I was a student).  They had a normal answer sheet with room for twelve multiple choice answers, but questions 13-24 had a short line where students would pencil in their answers.  These questions were designed to have simple answers like &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$3\sqrt7$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$2x-4$&lt;/span&gt;.  (There were constructed responses later in the test that would require students to show their work.)  Grading would presumably be only marginally more difficult than multiple choice, but you could be confident that students were constructing their own answers instead trying to game the multiple choice system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could definitely do something like that in a Calculus class.  For instance:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f(x)=x\sin x$&lt;/span&gt;, what is &lt;span class=&quot;math-container&quot;&gt;$f'(\frac\pi3)?$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That's definitely a Part 1 type of question where you don't want to dive through their work, but multiple choice answers (no matter how many options) are still going to give them undesirable clues like &quot;Oh, many of those answers have two terms, I didn't think about the Product Rule until I saw that&quot;.  Let them do their work on a scrap piece of paper and fill in a short line, and you'll give them a more valid assessment without making your grading a headache.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-10-17T11:31:38.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17315" PostTypeId="2" ParentId="1348" CreationDate="2019-10-17T16:53:20.947" Score="0" Body="&lt;p&gt;I have no teaching experience so I don't know the answer. Here's a guess on a method that will work. Take the problem &quot;What is &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} \times \frac{5}{6}?$&lt;/span&gt;&quot; Then explain how once you know the answer to &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} \times \frac{1}{6}$&lt;/span&gt;, you can then use that to calculate &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} \times \frac{5}{6}$&lt;/span&gt;. Then you could ask &quot;How do you calculate &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} \times \frac{1}{6}?$&lt;/span&gt;&quot; It's by first calculating &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4} \times \frac{1}{6}$&lt;/span&gt; which is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{24}$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} \times \frac{1}{6}$&lt;/span&gt; = &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{24}$&lt;/span&gt; which is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{8}$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} \times \frac{5}{6} = \frac{5}{8}$&lt;/span&gt;. Then you could give them a division problem but not tell them how to do it and let them teach themselves how to figure out the answer. They'll end up smarter in the long run that way.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="11718" LastEditDate="2019-10-18T13:26:03.030" LastActivityDate="2019-10-18T13:26:03.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17317" PostTypeId="2" ParentId="17095" CreationDate="2019-10-18T00:15:42.890" Score="2" Body="&lt;p&gt;I've notices that rigid motion transformations, rotations, translations, and reflections, is getting more attention now in geometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ive seen that topics from John Tukey's &quot;Understanding Robust and Exploratory Data Analysis&quot; have made it into the classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I saw one class where they were teaching voting theory, which I had no idea how to do.&lt;/p&gt;&#xA;" OwnerUserId="5144" LastActivityDate="2019-10-18T00:15:42.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17319" PostTypeId="1" CreationDate="2019-10-18T20:18:53.840" Score="14" ViewCount="1234" Body="&lt;p&gt;It seems to me that we all might benefit from an answer to this question, since math departments must defend their performance within their institutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I imagine there will be standard answers like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Cultivate a culture of diversity, where everyone can be a &quot;math person&quot;, and growth mindset is emphasized. &quot;Launch&quot; instead of &quot;weed&quot;.&lt;/li&gt;&#xA;&lt;li&gt;Have applied and pure offerings, as well as a robust math education presence in the department.&lt;/li&gt;&#xA;&lt;li&gt;Do outreach and actively recruit (what are the most effective ways to do this?)&lt;/li&gt;&#xA;&lt;li&gt;Help students find internships, jobs and careers&lt;/li&gt;&#xA;&lt;li&gt;Have a super passionate, engaged faculty who share the excitement for mathematics all the time!!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Any other concrete articles or suggestions would be very helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know there are articles about doing this in the literature (maa publications, etc). My reason for posting this is that I think math professors may have personal &quot;best of&quot; lists of these that they could share.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2019-10-22T23:05:55.853" LastActivityDate="2019-10-23T22:31:41.810" Title="How do you attract more math majors at a liberal arts college math department?" Tags="|undergraduate-education|liberal-arts|" AnswerCount="8" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="17320" PostTypeId="1" CreationDate="2019-10-18T21:40:42.193" Score="4" ViewCount="115" Body="&lt;p&gt;I have been doing a little bit of experimenting when it comes time to review with the class in preparation for the final exam.  The last handout I have been giving my students has usually been a compilation of problems similar to the ones they have been accustomed to seeing throughout the course.  Through the use of textbooks, I can save some time by not having to come up with the questions myself, but the textbooks sometimes do not use the same language that I use in class or on prior written assignments.  Recently, I have been considering the option of either:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A)&lt;/strong&gt;  a list of twenty-five possible questions, out of which eight to ten will be on the final exam, or&lt;br&gt;&#xA;&lt;strong&gt;B)&lt;/strong&gt; the exact eight to ten questions that will be on the final exam, with the particulars blacked out (particulars being numbers, equations, functions, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Benefits of A:&lt;/strong&gt;  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;allows students to strengthen all core skills required to say that they have a satisfactory understanding of all course material  &lt;/li&gt;&#xA;&lt;li&gt;if all twenty-five questions are completed to perfection, the student should have no problem reiterating the solution come exam day (barring test anxiety, time management problems, and other factors)&lt;/li&gt;&#xA;&lt;li&gt;hedging bets (we all know this problem when we studied stats):  what is the minimum number of questions I should be completely clear on how to do if I want to have a good chance of scoring at least an &lt;span class=&quot;math-container&quot;&gt;$80\%$&lt;/span&gt;, etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Benefits of B:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;allows students to focus solely on the exact type of question they should expect&lt;/li&gt;&#xA;&lt;li&gt;gives students the opportunity to predict how much time should be spent for each question, and, by extension, how much time can be spent studying for each topic&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To current students:  if you were given the option, which would you prefer?  Can you think of other ways that each option can benefit you?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To teachers:  which option have you personally used with your students before?  If neither, do you see yourself employing either of these methods when it comes to exam preparation for your students?&lt;/p&gt;&#xA;" OwnerUserId="12720" LastActivityDate="2019-10-19T00:05:35.550" Title="In preparation for exams: question bank or questions with omitted particulars?" Tags="|secondary-education|exams|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17321" PostTypeId="2" ParentId="17320" CreationDate="2019-10-19T00:05:35.550" Score="6" Body="&lt;p&gt;I prefer A. It essentially says these 25 questions represent the whole course.  Knowing them, you know it all.  The 8 selected will be broad, but by nescessity (time constraint) won't cover whole course.  So if you only give B, yuu're basically leaving too much of the course as untested.  (In that people won't prepare to know anything other than what is on the test.)  I mean even on professional certifications, there is a lot of material which can't all be covered in the exam.  If you gave only the 8 questions, you'd be allowing people to blow off too many topics int heir prep.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In comments, there was discussion of making number 15 versus 25, but I would lean more in the opposite direction, so you cover more possible material.  About one per lesson.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;FYI:  In my mind I'm thinking of a diffyQ or integral calc course, wher each topic is a new technique.  If you've mastered a question from each lesson, of &quot;medium&quot; level difficulty, you've mastered the course.  Of course, you could have lessons that cover a couple subtopics.  Or could imagine questions that synthesize lessons.  But I think one per lesson/lecture/text-section is a reasonable amount for a review.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be fine with the obscuring specifics in addition, or better yet, giving specifics, but telling them that the format of the question will be repeated with different specifics.  (Thus they can use the question as a drill problem still...but have some need to be able to do more than regurgitate come the final...at least minor differences.)&lt;/p&gt;&#xA;" OwnerUserId="12888" LastActivityDate="2019-10-19T00:05:35.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17322" PostTypeId="2" ParentId="17319" CreationDate="2019-10-19T05:28:26.517" Score="4" Body="&lt;p&gt;My advice is to emphasize the &quot;get a job&quot; aspect.  The university is filled with the dreamy beauty of knowledge stuff.  (And that's fine.  But you need to distinguish yourself and provide additional info that kids might not have.)  Do you even know what percent of graduating kids go into what fields?  How easy/hard it is to get jobs (like some sort of semiquantitative analysis)?  Is there a lot of &quot;option value&quot; to the degree?  Is it relatively recession proof.  Etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Donno the exact story, but if you've done a little bit of work to figure this out and then tout it on the department website at least it would show you care.  If story is mixed feel free to play up the positive aspects...everyone is rushing the froshes after all.  I just wouldn't be too overtly comparitive.  But find some subtle way to show that your kids won't end up as English major baristas.  And saying you care, doesn't show you care.  Showing work (analysis) you did...shows you care.  ;-)  Heck...you might even learn something interesting.  I mean even though you're a God-like professor, sir.  ;-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do also think it is important to have an applied track or two.  And show clearly (with a matrix) what the difference is.  I.e. get out of real analysis but have to take DOE...or whatever.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also quite like the idea of an applied track that can be portrayed as business friendly.  All these little kids want to be consultants.  They shouldn't.  But the world is sort of heading that way.  It doesn't need to be as hard core as a finance/econ/business degree.  But something where they can take a few classes with business application (OR, options pricing, etc.) would be helpful. Think about how to position this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Caveat:  Just ideas.  I'm not a teacher.&lt;/p&gt;&#xA;" OwnerUserId="12906" LastEditorUserId="12906" LastEditDate="2019-10-19T05:44:30.173" LastActivityDate="2019-10-19T05:44:30.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17323" PostTypeId="2" ParentId="17319" CreationDate="2019-10-19T08:40:24.433" Score="4" Body="&lt;p&gt;This is a fairly minor issue, but probably worth a mention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a technical university there was (and probably still is) a question about how to attract more female students. One aspect was the diversity in promotional material. The outcome was that&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;male students reacted equally well to diverse promotional material than to material with mostly white male students&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;female students greatly preferred more diverse promotional material.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Hence, to the extent that the university has a &quot;modern&quot; website with lots of pictures of smiling people or people in &quot;classroom&quot; situations and in front of blackboards etc., it is a good idea to have diverse people as students and teachers, and to have groups of mixed students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the results might not generalize to USA culture, so one should be a little bit careful here.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2019-10-19T08:40:24.433" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17324" PostTypeId="2" ParentId="17319" CreationDate="2019-10-19T14:36:49.020" Score="8" Body="&lt;p&gt;This might go over differently at a liberal arts school, but we've had some success inviting students to an optional, uncredited program where they work in groups on fun problems from various topics in math (ideally associated with upper level courses, so we can say, &quot;and if you liked this, you should take...&quot;), supervised by two older undergrads.  (There are similar programs at a number of other schools - we got ours from Columbia, and one of our grad students went to Northwestern and started it there.  The name that gets used is usually &quot;Emerging Scholars Program&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The premise is that some of the big factors that keep students away from math are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Not knowing what a major is actually like, and thinking it will all be more calculus and algebra,&lt;/li&gt;&#xA;&lt;li&gt;Not knowing other people in the major (this may be a bigger issue in a larger school),&lt;/li&gt;&#xA;&lt;li&gt;Seeing a lot of posturing and competitive, macho culture from the students who excelled in math in high school.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;We've found that the number of students who later enroll in the major from the program is pretty high, though we haven't done anything rigorous enough to determine how many would have enrolled without it.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-10-19T14:36:49.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17325" PostTypeId="2" ParentId="17319" CreationDate="2019-10-19T17:07:36.423" Score="4" Body="&lt;p&gt;There is quite a bit of literature under the banner&#xA;&quot;humanizing mathematics&quot; (or cognate phrases). I am just beginning to read this literature, so I can do no more than point&#xA;to a few references. This seems a slightly different emphasis than your first bullet on &quot;a culture of diversity.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Luis A. Leyva. &quot;Toward humanizing undergraduate mathematics education: A re-imagining through historical perspectives in mathematics.&quot;&#xA;  AMS blog. May 2018. &lt;a href=&quot;https://blogs.ams.org/inclusionexclusion/2018/05/29/toward-humanizing-undergraduate-mathematics-education-a-re-imagining-through-historical-perspectives-in-mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMS link&lt;/a&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Hottinger, Sara N. &lt;em&gt;Inventing the mathematician: Gender, race, and our cultural understanding of mathematics&lt;/em&gt;. SUNY Press, 2016.&#xA;  &lt;a href=&quot;https://www.sunypress.edu/p-6186-inventing-the-mathematician.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;SUNY Press link&lt;/a&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Bonner, Emily P., and Thomasenia L. Adams. &quot;Culturally responsive teaching in the context of mathematics: A grounded theory case study.&quot; &lt;em&gt;Journal of Mathematics Teacher Education&lt;/em&gt; 15, no. 1 (2012): 25-38.&#xA;  &lt;a href=&quot;https://link.springer.com/article/10.1007/s10857-011-9198-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-10-19T17:07:36.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17326" PostTypeId="2" ParentId="17319" CreationDate="2019-10-20T01:51:51.113" Score="2" Body="&lt;p&gt;One very brief answer is to have multiple entry points to the major, or even just to taking more courses.  That is to say, at many colleges the typical Calc I-II-III entry sequence may not be attended by a very large number of students who would love to take more math.  So have clear options for them.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;To elaborate into something not so brief, consider a university where perhaps there will be a biocalculus option, a business calculus, a finite math, or even a &quot;no math required&quot; option in many areas that might still entice people to math.  Having very clear pathways of &quot;what your next math course is if you liked this one&quot; helps a lot with allowing people to at least become a math minor, or even just a &quot;math tag-along&quot; person.  That may mean having a couple different linear algebra courses, or a &quot;calc II for people without calc I&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, having options for people who might not even get a minor, which fulfill additional core/distribution requirements, or having a couple different non-calculus options for people to start taking even if they didn't have a particularly strong high school experience can be useful.  Not everyone will be ready for these courses, but often students who did not have calculus in high school come out of the woodwork when they take some initial math course and experience success.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Naturally, the smaller your college (or at least your department), the harder it will be to do this; we are quite small so it's been difficult to develop such courses.  Even so, we probably get half or more of our majors as people who &lt;em&gt;add&lt;/em&gt; math or switch to it starting sophomore year by realizing they missed math.  I admit that many of those are people who started in related science fields, but definitely not all - and we have a number who start in math then drop it for something else (&quot;easier&quot;? not always) so it is good that we have them come in!    If most of your load is consumed with having multiple sections of the required service courses and then upper-level major courses, this is a tough row to hoe.  But a sympathetic dean might make it possible to try it out if you can point out the benefits of having a stronger program to them.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-10-20T01:51:51.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17327" PostTypeId="2" ParentId="1857" CreationDate="2019-10-20T14:18:23.407" Score="4" Body="&lt;p&gt;Can I recommend my two-volume book, Elementary School Mathematics for Parents and Teachers? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.worldscientific.com/worldscibooks/10.1142/9696&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.worldscientific.com/worldscibooks/10.1142/9696&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.worldscientific.com/worldscibooks/10.1142/10014&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.worldscientific.com/worldscibooks/10.1142/10014&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wrote it while I was designing a professional development program for primary school teachers and was baffled by the fact that I couldn't find a textbook covering the primary school curriculum in a way that is accessible to teachers (and parents).&lt;/p&gt;&#xA;" OwnerUserId="12910" LastActivityDate="2019-10-20T14:18:23.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17328" PostTypeId="2" ParentId="14547" CreationDate="2019-10-20T16:28:06.227" Score="1" Body="&lt;p&gt;This is an excellent example of a question which leaves both children and adults baffled. This is a truly meaningless question. Let's start with the fact that you add and multiply numbers whereas you join together sets (groups) of objects. The clause &quot;multiply to join groups&quot; may mean &quot;use multiplication to find the total number of objects when joining together groups of items&quot;, in which case I guess that the expected answer is &quot;only if all the groups are of equal size&quot;. But the fact that I am speculating about the intent of the question says it all.  &lt;/p&gt;&#xA;" OwnerUserId="12910" LastActivityDate="2019-10-20T16:28:06.227" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17329" PostTypeId="2" ParentId="14616" CreationDate="2019-10-20T20:00:31.367" Score="3" Body="&lt;p&gt;I would actually start with a &quot;real life&quot; question: there are two groups of children. In the first group the there are 6 children having 2,4,4,7,11 and 14 cookies respectively. In the second group there are 8 children having 2,2,3,4,7,7,12 and 15 cookies respectively. In which group a child has more cookies? This question brings up naturally adding up the numbers and dividing by the number of children. I would insist on using a number of children larger than 2 or 3, because statistical indexes are of interest only where there is a lot of data. &lt;/p&gt;&#xA;" OwnerUserId="12910" LastActivityDate="2019-10-20T20:00:31.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17330" PostTypeId="2" ParentId="17161" CreationDate="2019-10-21T06:15:16.307" Score="6" Body="&lt;p&gt;There are plenty of quotes here...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you're trying to increase willingness to make and learn mistakes, I doubt that quotes will change a lot of minds, because a quote will be a tiny trickle of words pushing in the direction you want versus a raging river pushing the other way. Even Michael Jordan's quote about being cut from his high school basketball team or a Jo Boaler video just isn't going to make a lick of difference against the forces pushing against you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few considerations:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How much of their letter grades for the course are explicitly about making mistakes and learning from them? E.g. &quot;Exams are 50% of your mark, quizzes are 25% of your mark, and the other 25% of your mark will be how you recorded and learned from your mistakes.&quot; Most teachers don't do this. Every student sees the gap between what teachers &lt;em&gt;say&lt;/em&gt; and what they &lt;em&gt;incentivize&lt;/em&gt;. The normal reaction to this is to ignore the words and internalize the &quot;obvious&quot; message from the incentives: &quot;You'll get high marks for correct answers and low marks for mistakes so don't make mistakes!&quot; I am aware that many teachers are not allowed to have grading schemes that incorporate work habits. And I am also aware that the message you want to get across is to make the mistakes &lt;em&gt;before&lt;/em&gt; the stakes are high, but...&lt;/li&gt;&#xA;&lt;li&gt;A huge share of math students, in the short-term, &lt;em&gt;can't&lt;/em&gt; really learn from their mistakes because they lack the background knowledge to do so. If you're highly confused by the difference between &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x+2$&lt;/span&gt;, then how do you learn from your mistakes in a 9-step algebra problem? You can't. But you've probably been told to learn from such mistakes a thousand times, all to no avail. The normal reaction to this is to believe that most attempts to &quot;learn from mistakes&quot; in math class are a total waste of time and to be annoyed by your teacher who tells you to waste a huge amount of your own time for the thousandth time. Frustrated students will just say &quot;I don't care what this means. Just show me the one right way to do it.&quot; [This problem is obviously solvable by backfilling knowledge gaps, but that's an enormous effort in and of itself.]&lt;/li&gt;&#xA;&lt;li&gt;What happens to the social status of students who make mistakes publicly? &lt;/li&gt;&#xA;&lt;li&gt;Do you exemplify learning from mistakes as a teacher? For example, do you video record what happens in your classroom, then share that with students, parents, colleagues, administrators, etc. all in order to find out how to improve? Do you try to make sure lots of people find all of your mistakes as a teacher? Would you focus your recording efforts on your least favorite lessons and topics to teach? Most teachers don't do this.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Overall, if it's a quote vs incentives, weak prior knowledge, countless bad experiences, lessened social status, and &quot;do as I say not as I do&quot;, it's pretty obvious which side will win out.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2020-05-26T21:41:32.197" LastActivityDate="2020-05-26T21:41:32.197" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17331" PostTypeId="1" CreationDate="2019-10-22T12:42:38.720" Score="9" ViewCount="271" Body="&lt;p&gt;edit: Thanks for all your answers so far.&lt;br&gt;&#xA;I have decided to develop my own solution, both because it is fun and I can then form it exactly as I want to. Once I am done (which might take some time, because I first have to learn some Python for it^^), I might share it on github if it turns out well enough (will edit here with a link, but don't expect anything for at least a few months).&lt;br&gt;&#xA;As mentioned in the answers and comments, people might find this question through google and your answers might help them, so I will not accept one answer now but instead leave everyone the chance to add other tools they know about.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Original question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was just recently thinking about how to best organize exercises to later reuse them. Sometimes you have a nice idea or see something nice (online/in a book), but you are not teaching a fitting class at that moment. Or maybe you just want to reuse some exercises (maybe with altered numbers) when teaching the same course again a few years later.&lt;br&gt;&#xA;I am especially thinking about the case where there is no textbook containing exercises, but instead the instructor has to come up with questions themselves (themself?^^).&lt;br&gt;&#xA;Yes, of course you can simply write them down somewhere, but wouldn't it be nice to have a database where you simply search for &quot;Gauss algorithm&quot; to get all exercises for that, or where you filter by &quot;Analysis, week 5&quot; tag to show you all exercises you had on this worksheet the last few times you read this course?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking of maybe coding up my own solution for the problem, but before investing time into that I just wanted to check if someone here knows of a good (free/open source) tool for that; if possible also with LaTeX support.&lt;/p&gt;&#xA;" OwnerUserId="8296" LastEditorUserId="8296" LastEditDate="2019-10-23T11:01:45.213" LastActivityDate="2019-10-23T11:18:20.833" Title="Exercise database" Tags="|exercises|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17333" PostTypeId="2" ParentId="17161" CreationDate="2019-10-22T13:58:17.157" Score="0" Body="&lt;p&gt;Although (as with several other answers) this one is more about persistence, I think it's relevant enough (&quot;depressingly slowly&quot; sure implies intermittent failure to me) to post.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a &lt;a href=&quot;https://allpoetry.com/T.T.T.&quot; rel=&quot;nofollow noreferrer&quot;&gt;poem by Piet Hein&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;T.T.T.&#xA;&#xA;Put up in a place&#xA;where it's easy to see&#xA;the cryptic admonishment&#xA;T.T.T.&#xA;&#xA;When you feel how depressingly&#xA;slowly you climb,&#xA;it's well to remember that&#xA;Things Take Time.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;(Amazingly, you can even buy plaques with a shorter version of the Danish original (?) of this on Etsy, with the famous picture that goes with it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/JJeic.jpg&quot; alt=&quot;picture from Etsy of TTT from www.etsy.com/il-en/listing/654730812/mint-true-danish-modern-piet-hein &quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-10-22T13:58:17.157" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17334" PostTypeId="2" ParentId="17331" CreationDate="2019-10-22T21:30:34.323" Score="4" Body="&lt;p&gt;WebWork maintains an &lt;a href=&quot;http://webwork.maa.org/wiki/Open_Problem_Library&quot; rel=&quot;nofollow noreferrer&quot;&gt;Open Problem Library&lt;/a&gt;. It is not necessarily &quot;easy&quot; to construct your own problems on WebWork, but the system is constructed with purposes like these in mind. In particular, if you have set up a WebWork server, you can maintain a local problem database, with a file structure, course structure, tags, difficulty levels, and so on. In general, WebWork is used to make online homework feasible, and also to make it possible to randomly generate the numbers used in each problem so that each student has slightly different calculations, but it could also be used for the purpose you're describing here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The languages in which problems are written is PG and PGML, and tutorials/examples are located on the WebWork wiki, currently located &lt;a href=&quot;http://webwork.maa.org/wiki/Problem_Templates&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://webwork.maa.org/wiki/Writing_Your_Own_Homework_Problems_Using_PGML&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2019-10-22T21:30:34.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17335" PostTypeId="2" ParentId="17319" CreationDate="2019-10-23T03:20:06.833" Score="7" Body="&lt;p&gt;For both options below, first and foremost the Math Department should be institutionally &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Friendly&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In my book this means letting students try to do what they want. Err on the side of bending policy to help students with unusual paths. Don't make students retake things they've already taken if at all possible. But, do give warnings if your institution has above average content in the course in question, this sort of advice requires some individualized care and attention to detail. On a personal note, this is one of the things which pushed me away from engineering and towards math and physics as an undergraduate. Both the math and physics departments had faculty who cared to advise me personally and I never felt appreciated or challenged in a positive fashion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible &lt;em&gt;high road&lt;/em&gt; to growing the major: (here I echo the excellent suggestions of the OP)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Share scholarship in general education courses. Don't succumb to teaching the course from the book. Instead, spend a summer to read old books, brush up on the history and write your own notes with which to teach the course. Also, write your own homework which is shockingly dissimilar from the math your general audience has seen throughout previous course work. Prove things like it matters. If you do this right you can attract elite students from majors outside math who thirst for analysis which is not surface level. Once upon a time, I taught such a course and it helped me earn us a new math major who went on to do significant work in graduate school and beyond.&lt;/li&gt;&#xA;&lt;li&gt;Generally, try to make the major as serious as possible. Build two or three semester sequences which dive deeper into core subjects. The goal here is not to weed out students, the goal is to teach students a broad and reasonably mature picture of math which sets them up for success in their future endeavors.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Downside, you will scare some students in the above approach. Certainly the weaker students need individualized reassurance in the midst of the above style. But, if you have a relatively small class size with a few good students then it can work. Also, if you have no students who endorse the program of excellence then it creates an unfortunate dynamic where excellence on the part of faculty is perceived as unreasonable behavior by the students. When the philosophy of the faculty and students are unaligned there is real danger for damaged morale for all parties. Upside, the new students you add the major will be sincere in their focus on math since sincere mathematics is what drew them in to begin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A possible &lt;em&gt;low road&lt;/em&gt; to growing the major:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;maximize the number of electives in the Degree Completion Plan (DCP)&lt;/li&gt;&#xA;&lt;li&gt;minimize the difficulty in required courses in the DCP&lt;/li&gt;&#xA;&lt;li&gt;maximize the choices of math electives so students can avoid subdisciplines they don't care for. For example, provide a path around too much probabilty or statistics. Or, provide a path around applied math for pure folks. Or provide a path around pure math for applied folks etc... general idea: don't make people do things they don't want to do.&lt;/li&gt;&#xA;&lt;li&gt;don't require physics or computer science, allow many courses to fill these slots in the DCP.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In summary, make the major as easy as possible. Upside to this approach: you will get more students. Also, with so much flexibility in scheduling, there is a good chance the students taking a given course actually want to be there. Downside, these new &quot;math&quot; majors may not really want to take all your math classes. Also, you will need to warn students that your &quot;major&quot; doesn't really prepare them for graduate math. Also, the lack of computer science and general lack of well-rounding in physics, chemistry etc. may give some employers pause on the applied side. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the low road better describes the position we are pushed towards when there is fear of downsizing and such. The high road is not easily maintained.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2019-10-23T03:20:06.833" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17336" PostTypeId="2" ParentId="17331" CreationDate="2019-10-23T11:18:20.833" Score="0" Body="&lt;p&gt;I think you may be able to accomplish the same ends, easier/better by using old tech methods.  I suggest to have a personal library that includes a few different versions of texts.  Often perusing/parsing them, you can get ideas better than a search retrieve on a database.  In addition, you get a feel for how much texts are different from each other or how much they copy each other.  This is useful for understanding the terrain of past work. It can be very illuminating to see how much problems repeat, are carried forward, etc.  (and this is not &quot;wrong&quot; per se, but good to know).  It can also be useful to see the few problems that really differ in some way.  (Especially for stronger students, contest math, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have problems you generate yourself or that you find in books, you don't own, I suggest to just make paper copies (of what is of specific interest/use) and then keep those in file folders.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, of course, there are disadvantages of paper--most things are not 100% better.  For instance, with paper, it's harder to cross reference (using several tags for a problem).  Instead you have to have a file heirarchy.  Still, I suspect the effort to fit things into a single heirarchy also has a benefit, in making you think.  For that matter, I have a pretty damned good memory, so can generally find anything, even if the interest at the time is from a secondary attribute, not the main one.  Also, of course, it's not as convenient for printing.  (That said, you can type it when you need to and save a lot of typing never needed.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure some Internet person (especially the kind that are more computer centric and thus hang out at places like &quot;stack exchange&quot;) will say &quot;lack of space&quot;, but that's really a false economy unless you're in a prison cell.  Like a really small prison cell.  It is extremely easy to get bookcases and to organize files into record boxes or file cabinets.  Honestly, this is a false economy if you think about value of your time, etc.  Paper is cheap, bookcases are cheap.  Time is valuable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  Before you slam the asnwer with the dislikes, realize that there is more than one way to skin a cat and it's good for someone in the future, thinking about this problem to have competing ideas. &lt;/p&gt;&#xA;" OwnerUserId="12919" LastActivityDate="2019-10-23T11:18:20.833" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17337" PostTypeId="2" ParentId="17319" CreationDate="2019-10-23T15:26:29.440" Score="5" Body="&lt;p&gt;Here is one very easy thing that has gotten good results at the University of Michigan: Every term , a few weeks before the end of the term, our director of undergraduate studies sends an e-mail to the math faculty saying roughly &quot;if you are teaching a class with a lot of undergraduate non-math majors, please choose 1-3 students in it who you think would do well as math majors, and &lt;b&gt;tell them so&lt;/b&gt;.&quot; Students who receive this feedback very often become math majors and, while correlation is not causation, we do a survey of our senior majors when they graduate which includes the question &quot;why did you choose to be a math major&quot;, and they often mention this interaction.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2019-10-23T15:26:29.440" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17338" PostTypeId="2" ParentId="17319" CreationDate="2019-10-23T22:31:41.810" Score="6" Body="&lt;p&gt;I wrote to a professor of mathematics at a liberal arts college that has, in recent years, found great success in attracting more math majors. Here are lightly edited excerpts from the two emails I received in response:&lt;/p&gt;&#xA;&lt;p&gt;Initial email:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Excellent teaching, welcoming classrooms, caring attitudes, support in and out of the classroom, etc. Don't make it a tough-guy &amp;quot;who is awesome enough to make it through the crucible of the math major&amp;quot; thing. Make it tough, but provide students with the support they need to succeed --- even, and &lt;em&gt;especially&lt;/em&gt;, the at-promise students. [Have math professors of different backgrounds - race, gender, etc - teaching students.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unprompted follow-up:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Accurate placement of students at the lower level is important, too. Too many well-prepared students try to weasel their way into lower-level courses (to get an easy A), while too many underprepared students try to shoot for the stars by trying to push upwards into higher-level starting courses (putting them at great risk of crashing and burning). So gentle but firm placement is key, BUT THEN, you ALSO need the instructors of the courses to teach at the appropriate level for the course. If a Calc 2 instructor gets excited that there are students in the class who are really getting the material so well, and starts talking about everything at a higher level to continue to inspire those students (who are actually the lazy weasels who &lt;em&gt;should&lt;/em&gt; have been taking multivariable calc instead of a class they already got an A+ in in high school), then the students who actually &lt;em&gt;belong&lt;/em&gt; in the class get lost and think, &amp;quot;I guess I'm just no good at math&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So it needs to be a team effort.&lt;/p&gt;&#xA;&lt;p&gt;Oh, I thought of another thing: personal notes on exams when you hand them back, to invite struggling students to office hours, to congratulate students doing well, and to suggest to students doing well that they should take the next math class, or maybe even consider a math major.&lt;/p&gt;&#xA;&lt;p&gt;Basically, all the time, actively make math and math classes a welcoming place where students feel supported. Challenged, but supported.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="262" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-10-23T22:31:41.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17339" PostTypeId="2" ParentId="13628" CreationDate="2019-10-24T05:35:55.700" Score="1" Body="&lt;p&gt;I will dodge the &quot;should&quot; part of the question, and try to address the &quot;objective&quot; issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to teach a class with fewer proofs and more applications,&quot;Serge Lang's book is not the way to go. The math books that are most concerned with &quot;applications&quot; are the ones written by and for engineers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Serge Lang's Linear Algebra book, like all his other books, is more about &quot;why&quot; than about &quot;what.&quot; That is, they have a lot of proofs and relatively few real world applications.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastActivityDate="2019-10-24T05:35:55.700" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17340" PostTypeId="2" ParentId="15293" CreationDate="2019-10-25T10:10:24.013" Score="3" Body="&lt;p&gt;Your question relates to a 5-year old child. This should be taken into account (e.g., no point in talking about algebraic manipulations). Parentheses are here to help us; and if they do, there is no reason to remove them. I would still tell a child that we usually prefer expressions that have as fewer symbols are possible, and explain that sometimes, parentheses are unavoidable, but if your 5-year old likes the expression (3+5)+2 better than 3+5+2, this is OK. A dialog like &quot;we first add 3 and 5 and then, whatever we got we add to 2&quot; is quite valuable. Rather than trying to eliminate parentheses, I'd rather explore the value of an expression with and without them.&lt;/p&gt;&#xA;" OwnerUserId="12910" LastActivityDate="2019-10-25T10:10:24.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17341" PostTypeId="2" ParentId="15293" CreationDate="2019-10-26T10:17:53.383" Score="2" Body="&lt;p&gt;The following aren't the same thing:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The meaning of an expression&lt;/li&gt;&#xA;&lt;li&gt;An option for computing the value of the expression&lt;/li&gt;&#xA;&lt;li&gt;A rigidly imposed sequence of instructions to be obeyed to compute the value&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The following example may help you more than your student:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Define x = 1 if there are infinitely many twin primes, and x = (- 1) otherwise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If what is inside parentheses &quot;shall be computed first&quot;, then how are you going to compute the value of the following expression: (1 - x) + (1 + x)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The instructions, if taken literally, require that we begin by resolving the twin primes conjecture, but nobody has achieved that.  Nor is it necessary.  Both roads go to the same result: 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for your question &quot;why shall it be done as 10 − (5 − 3) = 10 − 5 + 3 = 5 + 3 = 8&quot;, I think we can omit the how it &quot;shall be done&quot; command, and focus on recognizing that a - (b - c) = (a - b) + c is true for various kinds of numbers a, b, and c. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us suppose we are asked to determine the value of:&#xA; (a - (b - c)) - ((a - b) + c).&lt;br&gt;&#xA;We're back to a situation like the twin primes conjecture above, because that value can be determined to be zero, without any need to know the value of a, b, or c.  Thus, there is no need to begin by computing (b - c) and (a - b).  There may also be no possibility of computing (b - c) and (a - b) if we aren't provided with any information about the numbers a, b, or c.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parentheses could be introduced in connection with operation symbols so that we write an expression like (a + b).  If the plus symbol &quot;+&quot; is always accompanied by  parentheses, then ambiguities won't arise.  However, it will quickly seem to be a boring chore to write all parentheses, and that will motivate some consideration of when and how to omit parentheses without introducing ambiguities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea of parentheses can be conveyed by starting with a similar notation in linguistics that allows us to distinguish between sentences that have the same surface structure, but different deep structures or tree diagrams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The fruit flies like a banana.  (Here &quot;flies&quot; is a verb and &quot;like a banana&quot; is an adverb phrase)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The fruit flies like a banana. (Here &quot;fruit&quot; is an adjective in the noun phrase &quot;fruit flies&quot;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;You can do something similar with the following sentence:&#xA;&quot;British Left Waffles on Falkland Islands.&quot;  (I used initial capitals on most words in order to avoid having to choose between the past tense verb &quot;left&quot; and the political noun &quot;Left.&quot;)&lt;/p&gt;&#xA;" OwnerUserId="12816" LastEditorUserId="12816" LastEditDate="2019-10-26T10:24:02.453" LastActivityDate="2019-10-26T10:24:02.453" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17342" PostTypeId="2" ParentId="13628" CreationDate="2019-10-26T10:45:55.220" Score="4" Body="&lt;p&gt;I understand why you think a &amp;quot;pure&amp;quot; mathematics course should avoid &amp;quot;applications&amp;quot;. But applications don't have to be in the &amp;quot;applied mathematics&amp;quot; sense of the term; they can be examples of how studying one area of mathematics helps you understand another, or even just be interested in another. For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To fully understand things like Cauchy-Schwarz, you need an intuition for quadratic functions;&lt;/li&gt;&#xA;&lt;li&gt;Understanding projection in vector spaces helps you understand least-squares fitting, provided you realise a set of functions can be a vector space;&lt;/li&gt;&#xA;&lt;li&gt;Linear regression, as long as it has multiple predictors, lets you work with &lt;em&gt;any&lt;/em&gt; model of the form &lt;span class=&quot;math-container&quot;&gt;$y=\sum_ia_if_i(x)$&lt;/span&gt;, or even one that's transformable to that, such as a power law;&lt;/li&gt;&#xA;&lt;li&gt;Least-squares regression (again dependent on quadratics) is equivalent to MLE with Gaussian errors, and the study of Normal distributions also hinges on &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, and it's worth understanding why;&lt;/li&gt;&#xA;&lt;li&gt;If you also think of differential operators as vectors, it makes sense not only of the solution spaces (and dimensions thereof) of homogeneous linear differential equations, but also how we can construct rival orthonormal bases for them;&lt;/li&gt;&#xA;&lt;li&gt;In many problems, the possible terms are limited to a surprisingly small set of options just by the need for indices to contract, and in particular to achieve not just a law in terms of vectors, matrices etc., but to have such laws have the right kind of invariance;&lt;/li&gt;&#xA;&lt;li&gt;Once you think of polynomials as vectors, you'd love to have an inner product with respect to which monomials are orthonormal, and before you know it you're defining &lt;span class=&quot;math-container&quot;&gt;$\langle f,\,g\rangle:=\frac{1}{2\pi i}\oint_{|z|=1}f^\ast g\frac{dz}{z}$&lt;/span&gt;, which leads you into complex analysis;&lt;/li&gt;&#xA;&lt;li&gt;Sooner or later, anyone who thinks about vector spaces and complex analysis will think about topology, and how much of the geometry of vectors survives there, and why certain parts don't.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I could go on.&lt;/p&gt;&#xA;&lt;p&gt;Now, obviously you don't have time to flesh out all that in lessons when someone asks &lt;em&gt;why&lt;/em&gt; we study vectors. But if you remind yourself of points like this before you enter the classroom, you might find one or two worth mentioning.&lt;/p&gt;&#xA;&lt;p&gt;One of my favourite things about mathematics is that not only can two seemingly disparate areas have interesting connections, but to explain a connection you might need a third or even a fourth area. For example, why do the squares in &lt;span class=&quot;math-container&quot;&gt;$\color{brown}{\text{number theory}}$&lt;/span&gt; have reciprocals summing to something connected to &lt;span class=&quot;math-container&quot;&gt;$\color{limegreen}{\text{geometry}}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\color{red}{\frac16}\color{\limegreen}{\pi}^\color{blue}{2}$&lt;/span&gt;? One can explain the &lt;span class=&quot;math-container&quot;&gt;$\color{blue}{2}$&lt;/span&gt; in terms of calculus, but one can explain the &lt;span class=&quot;math-container&quot;&gt;$\color{red}{\frac16}$&lt;/span&gt; in terms of combinatorics. (In particular, that's the right way to think about it if you want to carry it over to &lt;span class=&quot;math-container&quot;&gt;$\zeta(4)=\frac{\pi^4}{90}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\zeta(6)=\frac{\pi^6}{945}$&lt;/span&gt; etc.)&lt;/p&gt;&#xA;" OwnerUserId="5794" LastEditorUserId="5794" LastEditDate="2020-08-20T18:59:26.707" LastActivityDate="2020-08-20T18:59:26.707" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17343" PostTypeId="2" ParentId="14547" CreationDate="2019-10-26T13:00:06.670" Score="15" Body="&lt;h1&gt;Here are the two previous pages from those materials (a pre-publication version found with a Google search):&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;&lt;br/&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/9PFlx.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9PFlx.png&quot; alt=&quot;a page from a math textbook&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/XD87G.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XD87G.png&quot; alt=&quot;a page from a math textbook&quot;&gt;&lt;/a&gt;&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;And here is the page containing the homework problem in question:&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;&lt;br/&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7QHnt.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7QHnt.png&quot; alt=&quot;a page from a math textbook&quot;&gt;&lt;/a&gt;&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;The intent now seems pretty clear. Students know that you can join groups by &lt;em&gt;adding&lt;/em&gt; them, but in the case where the groups are equal in size (e.g. five bags, each with nine goldfish), you can also use &lt;em&gt;multiplication&lt;/em&gt; to join the groups.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-10-26T13:00:06.670" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17344" PostTypeId="1" CreationDate="2019-10-27T01:15:31.937" Score="22" ViewCount="7202" Body="&lt;p&gt;There are lots of discussions out there about whether &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is a prime number (such as &lt;a href=&quot;https://math.stackexchange.com/questions/120/why-is-1-not-a-prime-number&quot;&gt;this one&lt;/a&gt;) and even about zero (such as &lt;a href=&quot;https://math.stackexchange.com/questions/539174/is-zero-a-prime-number&quot;&gt;this question&lt;/a&gt;, though note &lt;a href=&quot;https://math.stackexchange.com/questions/3698/why-doesnt-0-being-a-prime-ideal-in-mathbb-z-imply-that-0-is-a-prime-num&quot;&gt;zero does generate a prime ideal&lt;/a&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt; by the standard abuse of terminology ever since Kummer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I haven't seen a discussion of the related question on this site - namely: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How should one talk about the question of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; being prime (or composite, or neither) with primary or middle school children?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm particularly interested in the question about zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fundamental theorem of arithmetic uniqueness rationale is probably a little heady for them (though &lt;a href=&quot;https://bookstore.ams.org/mbk-90/&quot; rel=&quot;noreferrer&quot;&gt;this book gives it a great try&lt;/a&gt;) or even college students at times.  But I also don't really like the idea of saying &quot;because we said so&quot; when so much of school math feels like this to students. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Semi-sarcastic, but still possible final thoughts:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you think you can adapt Conway's notion of factoring all integers via &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;/&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; as &quot;prime powers&quot;, that would be great.  Also on topic here would be discussion of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/6107/a-non-unique-factorization-of-integers/6110#6110&quot;&gt;non-uniqueness of factorization&lt;/a&gt; for this age group.  But I think that those are probably asking a bit much.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2019-10-28T03:28:43.717" LastActivityDate="2019-11-01T04:13:10.943" Title="Explaining why (or whether) zero and one are prime, composite or neither to younger children" Tags="|primary-education|definitions|middle-school|prime-numbers|" AnswerCount="15" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="17345" PostTypeId="2" ParentId="17344" CreationDate="2019-10-27T09:39:36.803" Score="10" Body="&lt;p&gt;FYI:  here's some pro and con:  &lt;a href=&quot;http://primefan.tripod.com/Prime1ProCon.html&quot; rel=&quot;noreferrer&quot;&gt;http://primefan.tripod.com/Prime1ProCon.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One was originally considered prime.  It is prime with the most convenient (&quot;natural&quot;) definition.  It got excluded from prime-ness because many other higher theorems would be complicated by leaving it as prime.  Essentially &quot;prime&quot; -&gt; &quot;prime*&quot;.  The definition of primeness was tweaked to exclude one.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This definition change is glossed over with the &quot;different from itself&quot; formalism that doesn't sound as awkward as saying &quot;except one&quot;.  But clearly the reason for the change and the effect of it was the same as if we had adjusted the definition with a suspicious sounding &quot;except one&quot;.  I personally think just saying &quot;except one&quot; is a little more direct and revealing.  (Within the definition of prime.  I'm OK with the change to prime*.  But let's be real...we did it to vote one off the island.  If it didn't simplify a lot of higher math statements, we would not have made the change.  Certainly wouldn't have made the change if it complicated them!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I personally think a too dogmatic &quot;one is not a prime...how dare you think that...you are just wrong&quot; stance is too harsh to give to the child.  Just being honest and saying they tweaked the definition because it makes later math simpler is more honest and less upsetting, even though it leaves an impression of capriciousness.  &lt;/p&gt;&#xA;" OwnerUserId="12930" LastActivityDate="2019-10-27T09:39:36.803" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17346" PostTypeId="2" ParentId="17344" CreationDate="2019-10-27T13:12:33.583" Score="54" Body="&lt;p&gt;&quot;Because we said so&quot; &lt;em&gt;is&lt;/em&gt; a bit of a conversation closer, I agree.  But &quot;Because some people agreed a long time ago to define it that way so we could have conversations where we all understood each other.  Does that seem like it would be a good idea?&quot; is both more inclusive and more correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't even think it's that hairy to talk through the FTA with anyone old enough to understand primes.  Have everybody take out a sheet of paper and express, say, &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; as a product of the smallest numbers possible (with repetitions being okay).  You take a sheet of paper and write &lt;span class=&quot;math-container&quot;&gt;$1\cdot 2\cdot 3\cdot 1\cdot 1\cdot 2$&lt;/span&gt;.  Then you can discuss what everyone's paper has in common and if there are any differences.  Then point out that people a while ago realized that everyone's papers would have a lot more in common if they would just agree that &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; isn't a prime number, and that's what lead to the convention being established. &lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="238" LastEditDate="2019-10-31T18:32:50.067" LastActivityDate="2019-10-31T18:32:50.067" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17348" PostTypeId="2" ParentId="17344" CreationDate="2019-10-27T17:07:00.453" Score="34" Body="&lt;p&gt;There was a multiplication table posted on the wall.  Like this&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{alignat}4&#xA;1 &amp;amp;\quad 2 &amp;amp;\quad 3 &amp;amp;\quad 4 &amp;amp;\quad\cdots\\&#xA;2 &amp;amp;\quad 4 &amp;amp;\quad 6 &amp;amp;\quad 8 &amp;amp;\quad\cdots\\&#xA;3&amp;amp;\quad 6 &amp;amp;\quad 9 &amp;amp;\quad 12 &amp;amp;\quad\cdots\\&#xA;4&amp;amp;\quad 8 &amp;amp;\quad 12 &amp;amp;\quad 16 &amp;amp;\quad\cdots\\&#xA;\vdots&amp;amp;\quad \vdots &amp;amp;\quad \vdots &amp;amp;\quad \vdots &amp;amp;\quad\ddots\\&#xA;\end{alignat}&lt;/span&gt;&#xA;but going up to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Some questions to ask:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What numbers appear in this table somewhere?  All of them.  (Advanced language: all postive integers) because they are all in the first row.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What numbers appear only once in this table?  Just the number &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.  Any other number appears at least twice, once in the first row and once in the first column (and possibly elsewhere).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What numbers appear exactly twice in this table?  The numbers &lt;span class=&quot;math-container&quot;&gt;${}\ge 2$&lt;/span&gt; that do not appear except in the first row and first column.&lt;br&gt;&#xA;&lt;strong&gt;Definition&lt;/strong&gt; these are called &quot;prime numbers&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What numbers appears three or more times in this table?  All the numbers in the table when you omit the first row and the first column.&lt;br&gt;&#xA;&lt;strong&gt;Definition&lt;/strong&gt; these are called &quot;composite numbers&quot;.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2019-10-27T17:07:00.453" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17349" PostTypeId="2" ParentId="17344" CreationDate="2019-10-27T17:58:19.347" Score="4" Body="&lt;p&gt;If you build each number &lt;em&gt;n&lt;/em&gt; using &lt;em&gt;n&lt;/em&gt; square blocks in rectangular configurations, there are multiple configurations for each composite number. (4 is 4 by 1 or 2 by 2.) The primes are the ones that can only be built as a 1 by &lt;em&gt;n&lt;/em&gt; rectangle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems clear that 0 would be neither prime nor composite, when looked at this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The easiest way to understand why we don't call 1 prime is that when we factor a number like 12 down to primes, we like having just one answer. (Which is often true in mathematics.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edited (10-29-19) to add:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was asked &quot;Why not teach that the factorization is &lt;em&gt;1&lt;sup&gt;&lt;/sup&gt;×23×32&lt;/em&gt;?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The prime factorization describes how to break the number down into factors, emphasis on 'down'. 1 doesn't break it down into smaller factors, so it's not useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The preceding question, &quot;Why is having one answer a good thing?&quot;, is harder to answer. My answer to it, for now, is a bit of an exploration of thoughts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it feels natural to want one answer, but I'd have to have more experience with young kids to know whether it seems natural to them. Perhaps for me it comes from thinking about functions, and wanting just one thing to come out, when you put something in. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that square root can give even a good math student trouble, because it normally has one answer, but when we square root both sides of an equation, we put that plus or minus in front to give two answers. My student assistant had trouble with that when he was tutoring. He wanted to put the plus or minus in front when he was checking an answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know if that helps answer this question, but I hope that it shows that things that can have more than one answer get confusing for students. (In fact, one thing many people like about math is that there is one right answer.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2019-10-29T15:16:22.413" LastActivityDate="2019-10-29T15:16:22.413" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17350" PostTypeId="2" ParentId="17344" CreationDate="2019-10-28T02:45:43.753" Score="5" Body="&lt;p&gt;A good way to lead to the uniqueness of prime factorization and the convention that &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is not a prime is to build  factor trees (that's common in elementary school these days in fourth grade, sometimes third grade).&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;       24                24                24&#xA;     8    3           6      4           2    12&#xA;   2   4            3   2   2  2            3    4&#xA;      2  2                                      2  2&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The leaf labels always turn out the same, up to order, and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; never shows up.&lt;/p&gt;&#xA;&#xA; &lt;hr&gt;&#xA;&#xA;&lt;p&gt;I did this once in a math club. Later in the day, in the classroom,&#xA;Alejandro, who's in the club, volunteered the definition &quot;A number that only &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and itself go into, except that &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is not a prime.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The teacher asked &quot;Why isn't &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; a prime?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Because Dr. Bolker says so.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So appeal to authority often wins over thought.&lt;/p&gt;&#xA;" OwnerUserId="4984" LastEditorUserId="4984" LastEditDate="2019-10-28T17:44:59.613" LastActivityDate="2019-10-28T17:44:59.613" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17351" PostTypeId="2" ParentId="17344" CreationDate="2019-10-28T05:24:23.830" Score="8" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How should one talk about the question of 1 or 0 being prime ... with primary or middle school children?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Depending on what you did &lt;em&gt;before&lt;/em&gt; you will have an easy or a hard task:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the children were told:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A prime number is a natural number which cannot be divided by other numbers than by 1 and by itself.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;... you will have problems explaining why 1 is not a prime number because 1 is a natural number that cannot be divided by any other number than by 1 and by itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if they were told:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A prime number is a natural number which can be divided by exactly two numbers: By 1 and by itself.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;... it will be easy to explain why 1 is not a prime number: 1 is only divisible by 1, so it is not divisible by exactly two numbers, but by only one number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;This means&lt;/strong&gt; that the key is that the children are told a more or less &lt;strong&gt;correct definition&lt;/strong&gt; of the word &quot;prime number&quot;; otherwise you will later have problems explaining why 1 is not a prime number.&lt;/p&gt;&#xA;" OwnerUserId="7627" LastActivityDate="2019-10-28T05:24:23.830" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17352" PostTypeId="2" ParentId="17344" CreationDate="2019-10-28T11:07:21.813" Score="3" Body="&lt;p&gt;We don't need the full FTA upfront if we limit our discussion for the moment to obvious examples of what's necessary for a factorization to be unique. The FTA provides analogous sufficiency conditions they'll probably guess on their own, even if they don't know how it's proven. You can say,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; isn't considered prime because then there wouldn't be unique prime factorizations of anything, but because &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; isn't prime, it's not composite either because it has no prime factors. &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; isn't prime because every integer is a factor of it, but because &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; isn't prime, it's not composite either because you can't write it as a product of prime numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The worst reaction a child can have to that is, &quot;Oh, so you're saying if only we &lt;em&gt;do&lt;/em&gt; start prime numbers at &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, everything from that point will either be divisible only by itself and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, or will have a unique factorization in terms of such things?&quot; And you can say, &quot;yes, that can be proven, but it's a bit heavy for now; and in fact you don't need to consider those two different cases, if you'll say the prime factorization of a prime number is just that prime number&quot;. The best reaction they can have is to figure out the reply on their own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if a child does want to know how any of this is proven, you can probably make that fairly accessible by exploiting their intuition, rather than yammering on about strong induction, but that's another issue.&lt;/p&gt;&#xA;" OwnerUserId="5794" LastActivityDate="2019-10-28T11:07:21.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17353" PostTypeId="2" ParentId="17344" CreationDate="2019-10-28T15:24:05.037" Score="1" Body="&lt;p&gt;(If your context does not include negative numbers, turn all the negatives below positive.  This almost won't change the discussion.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Everything&lt;/em&gt; divides zero, so zero can't be prime.  &lt;span class=&quot;math-container&quot;&gt;$0 \cdot 7 = 0$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; divide &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.  &lt;span class=&quot;math-container&quot;&gt;$0 \cdot -8 = 0$&lt;/span&gt; means minus eight also divides zero.  Can we see that &lt;em&gt;everything&lt;/em&gt; divides zero, so zero is very far from being prime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Primes are numbers that are divisible by exactly two different positive numbers.  (Note that this also holds true for negative integers that are prime.)  Every number is divisible by one, so that must be one of the positive divisors of a prime.  Every number is divisible by its magnitude (&quot;itself&quot; if only talking about positives), so that must be the other positive divisor of a prime.  Non-primes must have more positive divisors.  If we take all the positive numbers bigger than one, take them in pairs and multiply them together, we get all the non-primes.&lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;2 \cdot 2 &amp;amp;= 4, 2 \cdot 3 = 6, 2 \cdot 4 = 8, \dots \\&#xA;3 \cdot 2 &amp;amp;= 6, 3 \cdot 3 = 9, 3 \cdot 4 = 12, \dots&#xA;\end{align*}&lt;/span&gt;&#xA;(This could be a good time to remind/discuss multiples of a number and to remind/discuss commutativity of multiplication to reduce redundant calculations.)&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;4 \cdot 4 &amp;amp;= 16, 4 \cdot 5 = 20, 4 \cdot 6 = 24, \dots \\&#xA;5 \cdot 5 &amp;amp;= 25, 5 \cdot 6 = 30, 5 \cdot 7 = 35, \dots&#xA;\end{align*}&lt;/span&gt;&#xA;Here might be a good time to point out that the smallest number we get in each of these lists is the square of the number used in every product on that row.  And the products get larger as we go to the right.  So is it possible that there are any composites less than &lt;span class=&quot;math-container&quot;&gt;$25$&lt;/span&gt; we have missed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's list our composites up to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;4 = 2 \cdot 2  &amp;amp;\text{, so $2$ also divides $4$.}  \\&#xA;6 = 2 \cdot 3  &amp;amp;\text{, so $2$ also divides $6$.}  \\&#xA;8 = 2 \cdot 4  &amp;amp;\text{, so $2$ and $4$ also divide $8$.}   \\&#xA;9 = 3 \cdot 3  &amp;amp;\text{, so $3$ also divides $9$.} &#xA;\end{align*}&lt;/span&gt;&#xA;This means the ones we did not produce in the table above, &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; must be prime -- they are only divisible by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can test this by checking each one for divisibility by smaller numbers.  For two, there is nothing to check since there are no smaller positive numbers between one and two, so two is prime.  For three, we see that two does not divide three, so three is prime.  For five, we check two, three, and four, and discover five is prime.  (This is a good time to notice that if four divides five, then two divides five, so we &lt;em&gt;really&lt;/em&gt; only need to test for divisibility by primes.)  We easily check that seven has no divisors among two, three, four, five, and six.&#xA;(This could be a good time to discuss that we only need to test divisors whose square is smaller than seven, otherwise the cofactor is smaller and we have already checked the smaller potential divisors.)&lt;/p&gt;&#xA;" OwnerUserId="5800" LastEditorUserId="376" LastEditDate="2019-10-28T17:46:24.620" LastActivityDate="2019-10-28T17:46:24.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17354" PostTypeId="2" ParentId="15242" CreationDate="2019-10-28T16:32:59.103" Score="1" Body="&lt;p&gt;I wonder if one could run a geometry course through GeoGebra? This requires laptops or workstations, but the software is free.&#xA;I am not finding the ideal textbook based on GeoGebra, but here are two links&#xA;that give some sense of what is available.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.geogebra.org/m/ebpKQ5C7#chapter/529&quot; rel=&quot;nofollow noreferrer&quot;&gt;Malin Christersson's&#xA;GeoGebra-book&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.geogebra.org/m/NUtDnGgC&quot; rel=&quot;nofollow noreferrer&quot;&gt;GeoGebra Geometry App: Beginner Tutorials with Lesson&#xA;Ideas&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/l9M0V.png&quot; width=&quot;500&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Snapshot from MalinC's GeoGebra-book.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;This seems too advanced for your purposes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Venema, Gerard. &lt;em&gt;Exploring advanced Euclidean geometry with GeoGebra&lt;/em&gt;. MAA, 2013.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-10-28T16:32:59.103" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17355" PostTypeId="2" ParentId="17344" CreationDate="2019-10-28T18:03:56.827" Score="5" Body="&lt;p&gt;This one is really very simple.  First, tell them what a prime number is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A prime number has exactly two different factors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(If they don't know what factors are, and they ask about primes, the correct answer is &quot;well, first you have to know about factors...&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that definition, it is very easy to figure out 0 and 1.  Is 1 a prime?  No, because it only has one factor.  Is 0 prime?  No, because every number is a factor of 0.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the next question is likely to be &quot;why is that the definition?&quot;  or &quot;what's their purpose?&quot;  or some such.  The answer to that is also simple - every number bigger than one is made up of prime factors.  And, for every number, there is exactly one combination of primes that makes that number.&lt;/p&gt;&#xA;" OwnerUserId="12948" LastActivityDate="2019-10-28T18:03:56.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17356" PostTypeId="2" ParentId="17344" CreationDate="2019-10-28T20:31:01.423" Score="1" Body="&lt;p&gt;When I was at school, after being taught about integer division, I was told:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;A natural number is prime if it has exactly 4 integer divisors&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then 2 is prime, as it can be divided by -2, -1, 1, and 2. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then one can be divided by -1 and 1, and those are only 2 divisors. Hence not prime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Zero can be divided by anything but itself, yielding zero, hence not exactly 4 options.&lt;/p&gt;&#xA;" OwnerUserId="12951" LastActivityDate="2019-10-28T20:31:01.423" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17357" PostTypeId="2" ParentId="7528" CreationDate="2019-10-28T23:47:29.920" Score="4" Body="&lt;p&gt;Ever notice the &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;-point star at the base of a pumpkin stem?&#xA;This one's pentagonal symmetry is especially evident:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/ddxjv.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ddxjv.jpg&quot; alt=&quot;Pump_both&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Two views of the same pumpkin.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-10-28T23:47:29.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17358" PostTypeId="1" AcceptedAnswerId="17364" CreationDate="2019-10-29T00:28:08.763" Score="5" ViewCount="608" Body="&lt;p&gt;For example, &lt;a href=&quot;https://www.springer.com/gp/book/9780817639143&quot; rel=&quot;noreferrer&quot;&gt;Trigonometry&lt;/a&gt; was written by Wolf-Prize winner &lt;a href=&quot;https://en.wikipedia.org/wiki/Israel_Gelfand&quot; rel=&quot;noreferrer&quot;&gt;Israel Gelfand&lt;/a&gt;, one of the top mathematicians in the 20th century. I am wondering if other world-class mathematicians have written textbooks on some elementary undergraduate courses such as College Algebra, Trigonometry, Pre-calculus, Calculus, Linear Algebra, ODE as I would love my students to read textbooks written by the best mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2019-11-01T22:40:48.900" Title="Which textbooks on College Algebra, Trigonometry, Pre-calculus, Calculus, Linear Algebra, ODE are written by world-class mathematicians?" Tags="|undergraduate-education|calculus|textbooks|trigonometry|precalculus|" AnswerCount="2" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="17359" PostTypeId="2" ParentId="17344" CreationDate="2019-10-29T01:39:44.160" Score="1" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp; \begin{array}{cccccccccccccccccccc}&#xA;&amp;amp; &amp;amp; &amp;amp; &amp;amp; &amp;amp; &amp;amp; &amp;amp; &amp;amp; &amp;amp; 840 \\[12pt]&#xA;= {} &amp;amp; &amp;amp;  &amp;amp; &amp;amp; &amp;amp; 28 &amp;amp; &amp;amp; &amp;amp; &amp;amp; \times &amp;amp; &amp;amp; &amp;amp; &amp;amp; &amp;amp; 30 \\[12pt]&#xA;= {} &amp;amp; &amp;amp; &amp;amp; &amp;amp; 4 &amp;amp; \times &amp;amp; 7 &amp;amp; &amp;amp; &amp;amp; \times &amp;amp; &amp;amp; &amp;amp; &amp;amp; 5 &amp;amp; \times &amp;amp; 6 \\[12pt]&#xA;= {} &amp;amp; &amp;amp; &amp;amp; 2 &amp;amp; \times &amp;amp; 2 &amp;amp; \times &amp;amp; 7 &amp;amp; &amp;amp; \times &amp;amp; &amp;amp; &amp;amp; 5 &amp;amp; \times  &amp;amp; 2 &amp;amp; \times &amp;amp; 3 &#xA;\end{array} \\[10pt]&#xA;&amp;amp; = 1\times2\times1\times1\times2\times7\times1\times5\times2\times1\times1\times3\times 1\times 1 \\[10pt]&#xA;&amp;amp; = 1\times1\times1\times2\times \cdots&#xA;\end{align}&lt;/span&gt;&#xA;Once you start taking out &lt;span class=&quot;math-container&quot;&gt;$1\text{s,}$&lt;/span&gt; you can keep doing that without adding any more information about the factorization of the number you started with. So the number &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; plays a different role from the role of a number that you factor and the role of a number that you end up with when you're done factoring.&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2019-10-29T16:07:11.017" LastActivityDate="2019-10-29T16:07:11.017" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17360" PostTypeId="2" ParentId="17344" CreationDate="2019-10-29T02:20:43.900" Score="2" Body="&lt;p&gt;I would start by showing them, on paper, what they already know - that in the context of multiplication the number 1 is useless. It is the identity function. It simply reflects the original number. It is a mirror. &quot;And just like your reflection in a mirror is not a real person, neither is 1 a real number when multiplying (not to be confused with a &lt;em&gt;Real&lt;/em&gt; number.) Since primes are only in the context of multiplication 1 isn't relevant and should not be considered in the family of primes.&quot; I would further add that if they pursue very advanced mathematics in college there are other, more formal reasons to not let it be in the family of prime numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, zero changes every number to itself. &quot;Just like a black hole absorbs matter and energy, zero, when multiplying, destroys every other number. You have lost any information in the equation when you multiply by zero. Prime numbers are useful in solving real world problems like making your text messages unreadable to everyone except your friend. Using zero as a prime would destroy the data and not allow your friend to read your text. It doesn't work as a prime number.&quot;&lt;/p&gt;&#xA;" OwnerUserId="11917" LastActivityDate="2019-10-29T02:20:43.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17361" PostTypeId="1" CreationDate="2019-10-29T03:13:12.117" Score="3" ViewCount="132" Body="&lt;p&gt;The idea is that a student who is reading from beginning to end will always be provided with a train of thought.  The student will have no motivation to stop, and memorize an unexplained product of thought.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In many cases, it may be excessively ambitious to hope for students of multivariable calculus to learn to avoid making mistakes that were made by the original creators/discoverers, so it would make sense -- if some parts of the actual historical path of discovery/invention are used -- to include consideration of ideas that don't work, although it may not be immediately obvious that they don't work.  Trains of thought don't proceed by the shortest path from beginning to end, but will naturally involve some dead-ends and changes of direction, but the changes of direction occur only after somebody detects a potential error, and confirms that there is an error.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's true that students who have learned to succeed by memorizing may fail to read carefully, and may memorize some errors, but perhaps there is more to be learned from errors than from unexplained products of thought that were created by means of mysterious, undisclosed trains of thought.&lt;/p&gt;&#xA;" OwnerUserId="12816" LastEditorUserId="12816" LastEditDate="2019-10-29T03:18:52.737" LastActivityDate="2019-10-30T17:08:37.787" Title="Are there any textbooks on multivariable calculus that introduce all non-trivial definitions and all non-trivial proofs with a &quot;first draft&quot;?" Tags="|reference-request|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17363" PostTypeId="2" ParentId="17358" CreationDate="2019-10-29T12:04:28.793" Score="2" Body="&lt;p&gt;This may not qualify, as it is a stretch to classify this as a &quot;textbook.&quot;&#xA;It arose from a series of lecture Klein gave to high-school teachers.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Klein, Felix. &lt;em&gt;Elementary Mathematics from an Advanced Standpoint: Geometry&lt;/em&gt;. Transl. from the Third German Ed. by ER Hedrick and CA Noble. Dover, 1939.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/LVbE5.jpg&quot; width=&quot;150&quot; /&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;I love this (1939) opening &quot;Gentlemen!&quot; Preface sentence:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Gentlemen! The course of lectures which I now begin will be an&#xA;  immediate continuation of, and supplement to, my course last Winter.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How times have changed! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am less familiar with the content&#xA;of the other volumes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Klein, Felix. &lt;em&gt;Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis&lt;/em&gt;. Vol. 1. Courier Corporation, 2004.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-11-01T22:40:48.900" LastActivityDate="2019-11-01T22:40:48.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17364" PostTypeId="2" ParentId="17358" CreationDate="2019-10-29T13:45:03.413" Score="6" Body="&lt;p&gt;Books about middle school algebra and trigonometry written by someone like Gelfand are rare (I don't know other examples; for example V.I. Arnold's books nomially for children are aimed at very special children, their author's claims to the contrary), but there are many undergraduate textbooks written by superb mathematicians. I list below some that occur to me now. However, most of those I list are aimed at a level probably a bit above that which interests the OP.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Vladimir Arnold wrote an excellent textbook on ordinary differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Peter Lax wrote a (moderately advanced) book on linear algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Serge Lang wrote textbooks on calculus, linear algebra, complex analysis, and some other undergraduate subjects. I don't particularly like his books, but that is a different matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cliff Taubes wrote a book called &quot;Modeling differential equations in biology&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While not exactly a textbook, the book of Persi Diaconis and Brian Skyrmes called &quot;Ten great ideas about chance&quot; is an excellent introduction to probabilistic thinking directed at a general undergraduate audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Marcel Berger wrote some textbooks directed at future math teachers that treat classical affine and projective geometry (they have been translated into English). They would be hard for a US educated audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Elias Stein has coauthored with Rami Shakarchi a series of books on real, complex, fourier, and functional analysis. The only one I've read is the book on Fourier analysis, and it is excellent and pedagogical, the book I always wished I had encountered as a student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yuri Manin wrote a linear algebra textbook (however, it is not afraid to discuss functors).&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-10-29T13:45:03.413" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17365" PostTypeId="2" ParentId="16713" CreationDate="2019-10-29T14:05:13.943" Score="3" Body="&lt;p&gt;I have been learning soroban on my own and find that it uses a totally different part of the brain than doing the sums and subtractions on paper. I read somewhere that it uses the part of the brain used in graphic tasks. I am one of those people who many moons ago did poorly in arithmetics but excelled in algebra, geometry, etc. I think that the whole idea of creating more pathways in the brain at an early age can only help later academically. I would be curious to know if two groups, one having learned to use the soroban, and another not, when reaching a new discipline that is more graphic in nature like geometry, would perform better than the group not exposed to the soroban method. I find the soroban process to be very interesting, I do find that decimal place order is very important in soroban and it requires the use of one's memory well. It is not just mechanical like the use of a calculator. The pushing of the beads is more methodical than the pushing of buttons in a calculator. One has to keep track of units and remainders while using a different direction of bead movement for positive and negative entries. It demands a rigor often necessary for solving mathematical problems. I find doing soroban worksheets very relaxing and satisfying. The idea of being able to perform soroban calculations without a physical soroban seems questionable in terms of an expected outcome. It seems like that outcome requires a lot of training and it is a skill that is not held long term.&lt;/p&gt;&#xA;" OwnerUserId="12955" LastEditorUserId="12955" LastEditDate="2019-10-29T14:19:13.083" LastActivityDate="2019-10-29T14:19:13.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17366" PostTypeId="2" ParentId="4482" CreationDate="2019-10-29T16:57:12.613" Score="5" Body="&lt;p&gt;Chris. I know you are really excited about this idea, but I'm very worried about a critical omission from your list of considerations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I notice that you teach at a college with a lot of faculty members. One of the biggest blind spots that many faculty (and apparently, you) have is &lt;strong&gt;which publisher resources their fellow faculty use&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Some use online homework as the core component of the coursework. &lt;/li&gt;&#xA;&lt;li&gt;Some use publisher-provided Powerpoint slides as the core component of their lectures. &lt;/li&gt;&#xA;&lt;li&gt;Some use publisher-provided Test Banks as the core component of their exams. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The point here is not whether those resources are a good idea, but rather whether your colleagues are accustomed to using them and will be blindsided if they disappear when you switch to an OER textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I noticed that nowhere in your question did you mention these kinds of resources. I would strongly recommend that you do research in this direction and include it as potentially the most important factor in your decision. Especially in your situation, where many faculty (did you even consider dual-credit instructors?) will be affected by the decision of a small core group, you could cause significant long-term disruption and animosity by making a change without taking the issue of educational resources seriously.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2019-10-29T16:57:12.613" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17367" PostTypeId="1" CreationDate="2019-10-29T16:59:19.937" Score="4" ViewCount="172" Body="&lt;p&gt;I am planning to teach (unofficially, I am a Grad student) a course in real analysis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aim of the course is to understand the convergence of Fourier series. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to start with the notion of sequence of real numbers, their convergence, metric space, convergence of sequence in metric space and so on...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has any one tried such a course with the time bound of 20 lectures, 1.5 hours each? Any suggestions are welcome. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to teach derivatives and integration. 30 hours may be less for usual course in real analysis that start with definition of convergence of sequence of real numbers to reach till the definition of Fourier series. What I had in mind is some of the topics can be skipped without breaking the flow. So, I am asking what would be a sequence of topics that starts with notion of limit, continuity and ends with convergence of Fourier series. This is for the benifit of students who take physics as major in their undergraduate. They might not take more courses in analysis in their undergraduate but use the setup of Fourier series without being sure if that makes sense or not.  &lt;/p&gt;&#xA;" OwnerUserId="12627" LastEditorUserId="12627" LastEditDate="2019-10-30T01:26:52.520" LastActivityDate="2019-10-31T01:14:27.313" Title="Road map to teach undergrads a first course in real analysis that concludes with convergence of fourier series" Tags="|teaching|mathematical-analysis|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17368" PostTypeId="1" AcceptedAnswerId="17373" CreationDate="2019-10-29T22:35:22.500" Score="-3" ViewCount="229" Body="&lt;p&gt;First, I will simply observe that it seems to be standard practice, in elementary set theory, to define relations to be sets of ordered pairs.  If we had the option of introducing a &quot;symmetric relation&quot; to be not a particular kind of relation, but instead a set of &lt;strong&gt;&lt;em&gt;unordered&lt;/em&gt;&lt;/strong&gt; pairs, then the commitment to using a set of unordered pairs to represent equality would remove the option of having a non-symmetric equality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the whole point of this thread is that the option of having a non-symmetric equality might be a convenient thing to have, so the initial structural decision to represent &lt;strong&gt;&lt;em&gt;all&lt;/em&gt;&lt;/strong&gt; relations via sets of &lt;strong&gt;&lt;em&gt;ordered&lt;/em&gt;&lt;/strong&gt; pairs may have unanticipated benefits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, let us suppose that students are provided with the following premise:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \forall t \in Q \left( t \neq 0 \Rightarrow \left( \frac{t}{t} = 1 \right) \right)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us suppose that students are required to accept any conclusions that can be deduced -- via valid reasoning -- from the above premise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, in particular, we can replace the conditional ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$t \neq 0 \Rightarrow \left( \frac{t}{t} = 1 \right)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... with its contrapositive, because the original conditional is logically equivalent to its contrapositive:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\left( \frac{t}{t} \neq 1 \right) \Rightarrow t = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, the original premise that was provided to students is logically equivalent to the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \forall t \in Q \left( \left( \frac{t}{t} \neq 1 \right) \Rightarrow t = 0 \right)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, given that zero is a rational number, we have ...&#xA;&lt;span class=&quot;math-container&quot;&gt;$ 0 \in Q$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So -- in asserting the original premise and obligating students to accept all conclusions that can be deduced from it -- teachers are requiring students to accept (via the substitution &lt;span class=&quot;math-container&quot;&gt;$t = 0$&lt;/span&gt;) the following statement as true:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \frac{0}{0} \neq 1 \Rightarrow 0 = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, one way to avoid the above difficulties is to revise the premise.  In particular, if we have a set S, such that ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \forall x \left( x \in S \iff \left( x \in Q \land x \neq 0 \right) \right)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... then the teacher may formulate the following proposed alternative premise:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \forall t \in S \left( \frac{t}{t} = 1\right)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We see a difference between two approaches:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Assert a potentially problematic statement as a premise to be accepted, and if any difficulties are encountered, then build a layer of interpretations and prohibitions to attempt to prevent the problems from being exposed, or to prevent people from giving serious consideration to the problems when they are exposed; and&lt;/li&gt;&#xA;&lt;li&gt;Revise the premise to refrain from asserting what we don't wish to assert&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Now, all of the above was merely a prelude or introduction to the main point of this thread.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We could imagine having a non-symmetric equality relation, where we interpret the assertion ...&#xA;&lt;span class=&quot;math-container&quot;&gt;$$t = u$$&lt;/span&gt; &#xA;...standing alone as a premise to be accepted -- after we have replaced the letter &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; with some term and replaced the letter &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; with some term -- to mean that we are authorized, if we see the term &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; in a statement, to replace it with &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;, but where the reverse substitution is not necessarily authorized.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given such an approach to equality, we could imagine having:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \Bigl( \Bigl( (0, t) \in = \Bigr) \land \Bigl( (t, 0) \notin = \Bigr) \Bigr)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... where we are making no distinction between the equality relation and its attempted representation as a set of ordered pairs, going so far as to replace the conventional sequence of symbols &quot;&lt;span class=&quot;math-container&quot;&gt;$0 = t$&lt;/span&gt;&quot; (analogous to the sequence [subject verb object] in a natural, spoken language) with the sequence of symbols: &lt;span class=&quot;math-container&quot;&gt;$(0,t) \in =$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, suppose that we refrain from asserting ...&lt;span class=&quot;math-container&quot;&gt;$$\left( \frac{t}{t} = 1\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... unless we have &lt;span class=&quot;math-container&quot;&gt;$\Bigl( (t, 0) \notin = \Bigr)$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given a non-symmetric equality, we would be permitted to nevertheless have &lt;span class=&quot;math-container&quot;&gt;$$\Bigl( (0, t) \in = \Bigr)$$&lt;/span&gt; ... and although it may be somewhat confusing given established habits, we could write that as:  &lt;span class=&quot;math-container&quot;&gt;$$ 0 = t $$&lt;/span&gt; ...even though we would be committed to remembering that the only reason we were authorized to assert ... &lt;span class=&quot;math-container&quot;&gt;$$\left( \frac{t}{t} = 1\right)$$&lt;/span&gt; ... was that we have &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; restricted to values such that &lt;span class=&quot;math-container&quot;&gt;$ t \neq 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Question&lt;/em&gt;&lt;/strong&gt;: Could a non-symmetric equality relation play a positive role in teaching mathematics, such as -- if nothing else -- by displaying in explicit form what some students might be silently thinking, and that they should understand the pitfalls of?&lt;/p&gt;&#xA;" OwnerUserId="12816" LastEditorUserId="143" LastEditDate="2019-10-30T21:46:57.140" LastActivityDate="2019-10-30T21:46:57.140" Title="A role for a non-symmetric equality relation in teaching mathematics?" Tags="|reference-request|student-motivation|logic|inquiry-based-learning|" AnswerCount="1" CommentCount="2" ClosedDate="2019-10-30T21:48:04.853" ContentLicense="CC BY-SA 4.0" />
  <row Id="17369" PostTypeId="1" CreationDate="2019-10-30T00:24:11.713" Score="-2" ViewCount="122" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; be the set of all undergraduate mathematical courses in the US and define a binary relation &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; such that for elements &lt;span class=&quot;math-container&quot;&gt;$a,b\in A$&lt;/span&gt; (that is, &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are undergraduate mathematical courses), &lt;span class=&quot;math-container&quot;&gt;$a\leq b$&lt;/span&gt; if and only if either &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt;, or there exists finitely many courses &lt;span class=&quot;math-container&quot;&gt;$a_1,\ldots,a_n$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$a=a_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b=a_n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a_{i}$&lt;/span&gt; is a prerequisite of &lt;span class=&quot;math-container&quot;&gt;$a_{i+1}$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$i=1,\cdots, n-1$&lt;/span&gt;. For example, under the above definition, Pre-calculus &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; Real Analysis and Linear Algebra &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; Galois Theory. Also, Differential Equations and Point-set Topology are not comparable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Under the definition above, it is easy to verify that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is a finite partially ordered set. May I know where I can find a chart which gives the partial order structure in &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2019-10-30T02:47:54.450" Title="Where can I find the partial order relation of prerequisites of undergraduate courses in the United States?" Tags="|undergraduate-education|usa|set-theory|prerequisites|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17370" PostTypeId="2" ParentId="17369" CreationDate="2019-10-30T02:47:54.450" Score="4" Body="&lt;p&gt;The US has different courses at different schools.  Sometimes with same name but differences in content or prereqs.  It would be more meaningful to sketch this tree for a given school.  Or do a few schools.  That should give you some feel for the general lay of the land.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I suggest to sketch it yourself, using a course catalog.  You'll learn more doing a little work yourself.  Also the course catalog will have course descriptions.  Useful since same course may have different names at different schools (or visa versa).  Not to say it is totally arbitrary of course. You'll see some stereotypical similarities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an example:  &lt;a href=&quot;https://www.undergradcatalog.registrar.vt.edu/1920/math.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.undergradcatalog.registrar.vt.edu/1920/math.html&lt;/a&gt;  Note that there are some remedial classes at the beginning which have no prereqs.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also note that it is possible to have a &quot;corequisite&quot;, which means that taking the coreq at same time is adequate (before is fine too).  Or to have prereqs that are not just math courses (e.g. department chair permission or interview or placement exam or the like).  Furthermore, it's less likely with math but certainly common for other departments to have prereqs that are out of department.  IOW calculus before physics.  Or physics before physical chemistry.  Etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is another example:  &lt;a href=&quot;https://www.usna.edu/Academics/Majors-and-Courses/course-description/SM.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.usna.edu/Academics/Majors-and-Courses/course-description/SM.php&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can select a few more on your own.  Most schools have this stuff open and online nowadays.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oh...and one more complexity: there are often a couple or more similar courses (e.g. enriched or specialized in some way) on the same content.  So for the last link, if you look at SM 465 Advanced DiffyQs, it has a prereq of &quot;(SM212 or SM222) [different versions of ODE courses] and (SM331 or SM331H or SM333). [different versions of real analysis courses]&lt;/p&gt;&#xA;" OwnerUserId="12962" LastActivityDate="2019-10-30T02:47:54.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17371" PostTypeId="2" ParentId="17344" CreationDate="2019-10-30T04:15:26.017" Score="0" Body="&lt;p&gt;Primes are called primes because all other integers above 1 are &lt;em&gt;(multiplicatively)&lt;/em&gt; &quot;built out of them.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can't build anything else (multiplicatively) out of 1s. No matter how many 1s you multiply together.&lt;/p&gt;&#xA;" OwnerUserId="5756" LastActivityDate="2019-10-30T04:15:26.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17372" PostTypeId="1" CreationDate="2019-10-30T12:51:37.347" Score="21" ViewCount="1860" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;How to deal with a student who refuses to learn?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've met a few of those over the years as a a private-class math teacher. They don't want to learn anything about the subject. Some of them are just not motivated, some have serious gaps in their past education, such that whatever I'm teaching makes very little sense for them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They offer resistance to being given explanations, as if everything (or most of what) I'm saying is unnecessary and excessive. They are usually only permissive of the solution to problems that are known to be on their school's exams, and prefer step-by-step instructions free from any context/theory/mathematical properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;To give some context:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I usually deal with either very young students who have some respect for older people authority (and come to think of it, this still exists), or pre-admittance exam students, who have well accepted the fact that they need to learn the subjects they are being taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But recently, I've been teaching this college student from a humanities field who needs to pass a class on high-school level mathematics. He's older than I'm used to, and confident he'll never need math in his &quot;real&quot; life. Hence, the problem I've been used to face reached a whole new level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Things I usually do, but aren't working:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Instruct the student to show some important result (such as proving Pythagorean theorem), the proof being written by his own hands usually puts a whole new meaning to theorems and formulas.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give application examples about finance (because everyone is usually interested in money).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain that understanding things makes it easier to remember and harder to forget.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain that other subjects/classes will require knowledge in mathematics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&#xA;There have been a few very good answers, but some of them and a few comments made me realize another point: Yes, I get that I usually can teach only the step-by step solution as a short term solution, this should allow the student to pass the test. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem I'm struggling right now is that over ten years worth of teachers have used this kludge with this student already. The result is that I'm expected to make a college student be able to solve high-school level problems while being oblivious to elementary school math and refusing to learn it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Think about this: The student needs to pass tests on arithmetic progressions and trigonometry, but forbids you to explain how to simplify fractions or solve first degree equations.&lt;/p&gt;&#xA;" OwnerUserId="12951" LastEditorUserId="12951" LastEditDate="2019-11-04T12:37:50.540" LastActivityDate="2022-08-22T16:51:42.640" Title="Teaching a student who refuses to learn" Tags="|secondary-education|student-motivation|tutoring|struggling-students|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17373" PostTypeId="2" ParentId="17368" CreationDate="2019-10-30T14:37:46.707" Score="6" Body="&lt;p&gt;I believe the answer to this question is &quot;No&quot;.  Getting students to utilize the equal sign correctly is enough of a headache as it is;  they do not understand regular equality.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This sounds like a potentially interesting project in mathematical logic, but I do not see any pedagogical uses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I cannot support this answer with any evidence because you have just invented this concept now, and there have never been any experiments on using this to teach students.  So I can only give you my &quot;gut reaction&quot; from my experience as a teacher.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="238" LastEditDate="2019-10-30T20:40:09.980" LastActivityDate="2019-10-30T20:40:09.980" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17374" PostTypeId="1" AcceptedAnswerId="17377" CreationDate="2019-10-30T14:42:01.263" Score="-2" ViewCount="136" Body="&lt;p&gt;In teaching elementary set theory, suppose we refrain from emphasizing historical decisions that were made in theory construction.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a danger that students may see the mathematical language of set theory not as a resource to help formulate a given idea, but as a framework that restricts the range of ideas that can be considered coherent and meaningful?  Is there a danger that the set theory could become a bed of Procrustes for their thoughts?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12816" LastEditorUserId="1608" LastEditDate="2019-10-30T19:49:08.780" LastActivityDate="2019-10-30T19:49:08.780" Title="Should the limits of one system of elementary set theory be the limits of a student's mathematical world?" Tags="|mathematical-pedagogy|concept-motivation|intuition|" AnswerCount="2" CommentCount="1" ClosedDate="2019-10-31T01:21:12.757" ContentLicense="CC BY-SA 4.0" />
  <row Id="17375" PostTypeId="2" ParentId="17374" CreationDate="2019-10-30T15:25:11.603" Score="6" Body="&lt;p&gt;Set theory has no important implications for 99% of normal mathematics, and the difference between, say, ZFC and NF has even less. Mathematics isn't generally written in the language of ZFC, it's written using more basic ideas and notation that are the same in other foundational systems. If I'm teaching freshman calculus and I talk about &lt;span class=&quot;math-container&quot;&gt;$\{x|x&amp;gt;0\}$&lt;/span&gt;, there is no implication of ZFC; the thing I'm describing would have been understandable to Euclid, if he had been willing to think of a ray as a collection of points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you work in foundations of mathematics and teach set theory to upper-division math majors or math grad students, then what's stopping you from simply adopting the philosophy you seem to prefer, which is to clearly motivate the choices of axioms and definitions by contrasting them with examples from other systems of other ways that it could have been done? I think I probably would have enjoyed it if the set theory course I took had been taught that way.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-10-30T15:31:28.160" LastActivityDate="2019-10-30T15:31:28.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17377" PostTypeId="2" ParentId="17374" CreationDate="2019-10-30T16:35:20.693" Score="3" Body="&lt;p&gt;This question is mixed with a lot of speculation and commentary, so I'm going to try to focus on the questions.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Should the limits of one system of set theory be the limits of a student's mathematical world?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;No.  The idea that a system of set theory should be a limit to mathematical ideas seems like poor pedagogy (as arbitrary restrictions usually are), and is totally at odds with mathematical practice.  I've never heard of anyone teaching such a way, but if someone is, I'm skeptical that the benefits outweigh the disadvantages.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In teaching elementary set theory, while refraining from emphasizing historical decisions that were made in theory construction, is there a danger that students may see the mathematical language of set theory not as a resource providing apparatus to help formulate a given idea, but as a framework that restricts the range of ideas that can be considered coherent and meaningful?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course it's always conceivable that students will misunderstand something, but I'm skeptical of the value of obsessing about every conceivable misunderstanding that might happen.  My experience has been that constantly trying to head off confusion before it happens just leads to students being more confused by all the digressions.  Holding and then discarding misunderstandings is a natural part of the learning process; it's more useful to teach with enough dialog so that if students have this-or another-misunderstanding, there's an opportunity to learn that they think this and address the matter then.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In particular if students are taught precisely one and only one system of set theory -- with enough application of it to mathematics that it becomes like their first language, and they know of other languages of set theory primarily from hearsay -- then is there a danger that the set theory they learn could become a bed of Procrustes for the thoughts that sprout from their minds, like arms or legs growing from the body?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I suppose anything is possible, but this seems fairly unlikely.  The jump from &quot;ideas can be formalized in set theory&quot; to &quot;all ideas must be formalized in set theory all the time&quot; is pretty big, and I don't see any reason an ordinary mathematics education should lead students to make such a leap.  I also don't see any reason that learning &lt;em&gt;other set theories&lt;/em&gt; is relevant; for most topics where there are issues of how to formalize them in set theory, those issues are going to persist in other systems of set theory as well.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-10-30T16:35:20.693" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17378" PostTypeId="2" ParentId="4408" CreationDate="2019-10-30T16:46:32.100" Score="3" Body="&lt;p&gt;I am grateful to Mark Conger for finding a video of this presentation and getting the University of Michigan to digitize it. It isn't produced in the style of math youtube content, because that was presumably not the goal of QED TV, which I do not know the history of. But it is found! I finally have a link to the video! Here it is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The video spends a lot of time on preliminaries, which you can skip by going to about 13:40.&lt;/li&gt;&#xA;&lt;li&gt;Since you have likely proved FTC before, I would recommend you start watching at 17:55, which is when the vegetables appear.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://umich.app.box.com/s/5lwbiroohcfwfk6yy13wqrz6yqadq76r&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://umich.app.box.com/s/5lwbiroohcfwfk6yy13wqrz6yqadq76r&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The vegetables are used to represent various things that appear in the proof:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/VBGau.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VBGau.jpg&quot; alt=&quot;Legend of Vegetables&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And the final product when h has approached zero (Carrot has approached Asparagus) looks like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Whqfi.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Whqfi.png&quot; alt=&quot;h has approached zero; the banana is squeezed.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2019-10-30T16:46:32.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17379" PostTypeId="2" ParentId="17372" CreationDate="2019-10-30T16:53:47.367" Score="13" Body="&lt;p&gt;If you are a private tutor, hired by an undergrad or adult student, or hired by the parents of a student in 6-12 (middle school/high-school), then I'd suggest that when you meet with a &quot;client&quot; as a potential tutor, that you develop a contract with the student and/or parents to make clear your expectations: what is the minimum level of participation/effort that you expect of the student you may agree to tutor?  Making this explicit ahead of engagement, then, allows you to determine when the &quot;contract is broken&quot; if such participation and effort does not materialize.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;things you usually do&quot; are spot on.  The only thing I would add is working from the beginning, to evaluate each student, inform each tutee (and/or their parents) of how you plan to address their needs, and what you need in return from the students (and/or their parents). Hence the &quot;contract&quot; I suggest.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, try as early as possible to build a &quot;team&quot; mentality with the tutee: spend a few minutes, particularly at the start, trying to get to know the students you are tutoring, what they are passionate about (helps with respect to using applications in your tutoring), so you can present yourself as being there to help them accomplish their goals. Offer yourself as a &quot;coach&quot;, someone there to help them better their performance.  But, explain to them that you can be most helpful to them, and are willing to spend the time and effort to help, but it won't work only they too commit some time and effort (training). Try to convey to your students that making mistakes in NOT a problem: the best of mathematicians make mistakes!  Sometimes a formerly unsuccessful student wrt math may appear completely disinterested in math, and like &quot;it doesn't matter at all to them&quot;.  Most of the time, that attitude has emerged, defensively, after previous shaming, or lack of success in earlier math classes.  Instead, many such students focus on and build their self-esteem around being a great athlete, or dancer, or artist, or musician, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes, for pre-college students, they might not be yet able to relate to your efforts wrt #4.  Try to show how math can be interesting to them, &lt;em&gt;right now&lt;/em&gt;: use football stats, or a book like &quot;How to lie with statistics&quot;, or budgeting (similar to your financing example) for something they are currently passionate about (clothes, concerts, a new smartphone, sports equipment, vidio games, dates, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is clear, from your post, that you care about your work, and want very much to help the students you work with. That's a huge plus for you. But you have the right to negotiate the &quot;terms of engagement, as much as the person paying you has.  The best teachers/professors I've &lt;em&gt;ever&lt;/em&gt; had the pleasure to work with expected A LOT from students, but their expectations were reasonable, and they also brought to the table patience, dedication, excitement, and encouragement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Added]: I should also suggest that in the &quot;contracts&quot; you set up with clients, for both your benefit, and the benefit of the student you will potentially tutor, that you write a sentence or two emphasizing that it is possible that there may be no &lt;em&gt;immediate results&lt;/em&gt; of your work with the student: Often times, a lot of time has elapsed between when the student was struggling, and when tutoring begins.  Hence, immediate results (in homework success or passing a test) after one or two or three teaching-sessions is likely unrealistic.  But success in renewed hope and effort from the student is an even more positive outcome for the future success of the student.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-10-30T17:40:34.753" LastActivityDate="2019-10-30T17:40:34.753" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17380" PostTypeId="2" ParentId="17361" CreationDate="2019-10-30T17:08:37.787" Score="3" Body="&lt;p&gt;One well-known work that might interest you is Schey's &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0393925161&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Div Grad Curl and All That&lt;/a&gt;.  It discusses elementary facets of electrostatic fields as a motivation for developing key ideas in vector calculus.  There is many points in the &quot;narrative&quot; where they describe why different strategies for solving a problem wouldn't work, which gives some sense of false paths that theorists go through.  And it has helped generations of students to get an intuitive knowledge of how the formulas are used in applied engineering.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-10-30T17:08:37.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17381" PostTypeId="2" ParentId="17186" CreationDate="2019-10-30T20:34:41.083" Score="3" Body="&lt;p&gt;I've got an idiosyncratic addition to these answers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a) Regarding contemporary mathematical uses of the Cardano formula, &lt;a href=&quot;https://arxiv.org/abs/1810.00492&quot; rel=&quot;nofollow noreferrer&quot;&gt;I coauthored an article&lt;/a&gt; that &quot;uses&quot; it! The role it plays in the paper is essentially as inspiration: we noticed that the set of roots of a family of polynomials we cared about had the same &lt;em&gt;form&lt;/em&gt; as the roots of a &quot;&lt;a href=&quot;https://fermatslasttheorem.blogspot.com/2006/11/depressed-cubic.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;depressed&lt;/a&gt;&quot; cubic as given by the Cardano formula, and our search for why that was happening led us to a deeper understanding of our family of polynomials. (They turned out to be generalized &lt;a href=&quot;https://www.encyclopediaofmath.org/index.php/Lucas_polynomials&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lucas polynomials&lt;/a&gt;!) As an addendum, we note that the same Cardano(/del Ferro/Tartaglia) solution method for the cubic can also be used to solve the polynomials in this family.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aside: the article has been accepted for publication in the &lt;em&gt;American Mathematical Monthly&lt;/em&gt; and will come out sometime next year. The review process led to a substantial revision that I believe vastly improves the article, but I can't post it because Taylor &amp;amp; Francis owns it, so we have to make do with the arXiv preprint till publication. That said, the treatment and use of the Cardano formula aren't really changed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) Regarding historians of mathematics, do you know &lt;a href=&quot;https://twitter.com/MBarany&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michael Barany&lt;/a&gt;? He is a historian of mathematics active (and generally responsive) on Twitter, so he may be able to point you toward people with the relevant expertise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(c) I am not a historian of math, but I have &lt;a href=&quot;https://researchinpractice.wordpress.com/2010/05/01/the-history-of-algebra-part-i-negative-numbers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;read Cardano's &lt;em&gt;Ars Magna&lt;/em&gt;&lt;/a&gt; (in English translation), so if you need a fact-check on something that's in there, I might know the answer. Email me? Otoh, it was a library book and it was a decade ago, so maybe this is a dumb idea. And I'm unlikely to know anything about the del Ferro/Tartaglia/Cardano/Ferrari(/Bombelli??) saga that you don't already know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(d) &lt;strong&gt;Addendum:&lt;/strong&gt; this is of possible relevance, although it is not an answer to the question and you probably already know it. Picking up on Dave Renfro's comments on the OP, my understanding is that one of the Cardano formula's important &lt;em&gt;historical&lt;/em&gt; roles was in getting mathematicians to take the complex numbers seriously. I learned about this from the first chapter in Tristan Needham's book &lt;em&gt;Visual Complex Analysis&lt;/em&gt;, and Needham also points to John Stillwell's book &lt;em&gt;Mathematics and Its History&lt;/em&gt;. Imaginaries are first discussed in print in Cardano's book but in a context unrelated to the cubics, and the discussion is &lt;em&gt;highly&lt;/em&gt; speculative and Cardano is really unsure there is any point to what he is saying. (See the link above on &lt;em&gt;Ars Magna&lt;/em&gt; which is a blog post I wrote that discusses this toward the end.) However, 30 years later, &lt;a href=&quot;https://en.wikipedia.org/wiki/Rafael_Bombelli&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rafael Bombelli&lt;/a&gt; discovered that certain real cubics with real roots, for which the Cardano(/del Ferro/Tartaglia) formula appears to &quot;break&quot; by requiring the extraction of a square root of a negative, can &lt;em&gt;still be solved by the formula&lt;/em&gt; by going ahead and just treating the extraction of the root of a negative as something you're allowed to do: all the weird imaginary numbers you get that way cancel out of the final answer, and the final answer is the correct real root. According to Needham, this was the real impetus for mathematicians starting to take complex numbers seriously. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I guess this is just registering a perhaps-lesser-known contribution that the cubic formula made to math history. I suppose while I'm brainstorming things of this type that you already know, I should mention the better-known fact that it [and Ferrari's quartic formula, also published in &lt;em&gt;Ars Magna&lt;/em&gt;] are what set off the search for the solution to the quintic, which is what led to the development of Galois theory, therefore group theory, therefore abstract algebra as a whole, so between complex numbers and abstract algebra, I guess pretty much all of modern math is downstream from the Cardano formula somehow!)&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="140" LastEditDate="2019-10-31T13:22:48.533" LastActivityDate="2019-10-31T13:22:48.533" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17382" PostTypeId="1" AcceptedAnswerId="17383" CreationDate="2019-10-30T21:02:26.603" Score="2" ViewCount="251" Body="&lt;p&gt;This seems like a stupid question .  I just don’t understand why the algebra textbooks I see don’t really address this with students.  I boy that I am tutoring brought it up and I was slightly embarrassed because I couldn’t give a quick and logical explanation for why he got it wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;b$&lt;/span&gt;.  If &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;a$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;b$&lt;/span&gt;, if we combine them we can write this as &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$b&amp;lt;x&amp;lt;a$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a&amp;gt;x&amp;gt;b$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are these “wrong”?  And if so, why?&lt;/p&gt;&#xA;" OwnerUserId="4728" LastActivityDate="2019-11-01T18:14:20.343" Title="Why not write “or” inequalities as $a&gt;x&gt;b$?" Tags="|algebra|" AnswerCount="3" CommentCount="4" ClosedDate="2019-11-01T17:49:06.590" ContentLicense="CC BY-SA 4.0" />
  <row Id="17383" PostTypeId="2" ParentId="17382" CreationDate="2019-10-30T21:35:56.673" Score="12" Body="&lt;p&gt;The convention is that &quot;&lt;span class=&quot;math-container&quot;&gt;$b&amp;lt;x&amp;lt;a$&lt;/span&gt;&quot; means &quot;&lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is less than &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &lt;strong&gt;and&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is less than &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&quot; What you suggest is only wrong in that it goes against the shorthand that everyone (?) has already agreed on. &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2019-10-30T21:35:56.673" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17384" PostTypeId="2" ParentId="17367" CreationDate="2019-10-30T22:51:41.007" Score="0" Body="&lt;p&gt;This is just an idea, not a complete answer.  And may be wrongheaded, but doing what I can.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might look at the approach in courses called &quot;sequences and series&quot;.  For instance, see here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.usna.edu/MathDept//_files/documents/courses/fall-2019-2020/SM333_Fall20.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.usna.edu/MathDept//_files/documents/courses/fall-2019-2020/SM333_Fall20.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there are some things missing, you may need to add them.  Or to find things to prune.  But at least it gives you a skeleton to start with.  Also, note this looks like 60 hours of instruction, not 30.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;FYI: USNA has three tracks for real analysis. A standard one (with a regular approachable real analysis text), an honors one with Baby Rudin, and then this one (which looks easiest and most applied). The honors track also allows for a followon second semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, I recommend to look at the textbook Advanced Engineering Mathematics by KReyszig and the three chapters entitled Fourier Series; Sequences and Series; and Power Series, Taylor Series and Laurent Series.  I generally find Kreyszig to be very time efficient in conveying useful content and not making it rely on things when it doesn't need to.  (Maybe still not all you need, but I suspect those three chapters could be covered in 30 hours.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  Kudos for the specificity of the time duration.  But would also be good to give some other factors.  What level students (in terms of prereqs and brains).  Also, what is driving this course description, is it some sort of specialized math track or service to another department?  (May be insights from that.)&lt;/p&gt;&#xA;" OwnerUserId="12971" LastActivityDate="2019-10-30T22:51:41.007" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17385" PostTypeId="2" ParentId="17367" CreationDate="2019-10-31T01:14:27.313" Score="3" Body="&lt;p&gt;I can think of two books which might provide a roadmap, though you would have to flesh it out a lot depending on your prerequisites and the timeline you propose.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Frank Morgan's &lt;a href=&quot;https://bookstore.ams.org/real&quot; rel=&quot;nofollow noreferrer&quot;&gt;Real Analysis&lt;/a&gt; has a minimum of prereqs and a maximum of topology and series.  Now, it doesn't construct the real numbers (they are just infinite series of decimals, if I recall correctly), and saying it is terse in exposition is an understatement.  But if you know what you are doing and choose sections carefully, you can definitely get to not just Fourier series convergence, but also Lebesgue integration.&lt;/li&gt;&#xA;&lt;li&gt;David Bressoud's &lt;a href=&quot;https://bookstore.ams.org/text-10/&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Radical Approach to Real Analysis&lt;/a&gt; instead takes a historical view of the subject, both starting &lt;em&gt;and&lt;/em&gt; ending with Fourier series.  You may find &lt;a href=&quot;https://www.maa.org/node/107161&quot; rel=&quot;nofollow noreferrer&quot;&gt;this review&lt;/a&gt; helpful.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In any case, whether or not you adopt a particular text, getting them from the library should give you a few sense of how to approach this goal.  If I had to do it, I'd try the historical approach, but the direct one also has a charm.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-10-31T01:14:27.313" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17386" PostTypeId="2" ParentId="17382" CreationDate="2019-10-31T01:21:10.940" Score="7" Body="&lt;p&gt;As a different view, I would say that this is &quot;wrong&quot; in the sense that we usually expect &lt;em&gt;transitivity&lt;/em&gt; with (many of) our relations.  E.g. if I write &lt;span class=&quot;math-container&quot;&gt;$a=b=c$&lt;/span&gt; then usually we would say &lt;span class=&quot;math-container&quot;&gt;$a=c$&lt;/span&gt; as well.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are certainly counterexamples for more general relations, such as if &lt;span class=&quot;math-container&quot;&gt;$aRb$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; share a hobby, so that &lt;span class=&quot;math-container&quot;&gt;$aRbRc$&lt;/span&gt; wouldn't necessarily mean &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; share a hobby (maybe &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; just has lots of hobbies).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But we don't use notation that has an unambiguous interpretation as an ordering for that.  And &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; definitely has that interpretation, so unless you want &lt;span class=&quot;math-container&quot;&gt;$$x&amp;lt;a&amp;lt;b&amp;lt;x&amp;lt;a&amp;lt;b&amp;lt;x\cdots$$&lt;/span&gt; then the notation you suggest is too open to misinterpretation.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Interesting side note: this ambiguity is actually &lt;em&gt;relied&lt;/em&gt; on in notation for outcomes of voting methods in the literature.  In that case, &lt;span class=&quot;math-container&quot;&gt;$A\succ B\succ C\succ A$&lt;/span&gt; has meaning when &lt;span class=&quot;math-container&quot;&gt;$A\succ B$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; defeats &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; in a head-to-head vote - because we don't always get transitive election results.  So sometimes our notation has to be careful depending upon who is reading it.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2019-10-31T02:05:12.083" LastActivityDate="2019-10-31T02:05:12.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17387" PostTypeId="2" ParentId="17382" CreationDate="2019-10-31T09:11:47.160" Score="3" Body="&lt;p&gt;Although, because there is no &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that satisfies both inequalities simultaneously, it would not be unreasonable to give the expression &lt;span class=&quot;math-container&quot;&gt;$1 &amp;lt; x &amp;lt; 0$&lt;/span&gt; the interpretation that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is not between &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, something that could equally well be written as &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$x &amp;lt; 0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 1$&lt;/span&gt;&lt;/em&gt;, this is definitely &lt;em&gt;not&lt;/em&gt; standard practice and is surely more confusing for students than simply writing &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$x &amp;lt; 0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 1$&lt;/span&gt;&lt;/em&gt;. Standard practice is to interpret the expression &lt;span class=&quot;math-container&quot;&gt;$1 &amp;lt; x &amp;lt; 0$&lt;/span&gt; as an absurdity, because standard practice is to interpret &lt;span class=&quot;math-container&quot;&gt;$p &amp;lt; q &amp;lt; r$&lt;/span&gt; as implying &lt;span class=&quot;math-container&quot;&gt;$p &amp;lt; r$&lt;/span&gt;, which is false if &lt;span class=&quot;math-container&quot;&gt;$p = 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r = 0$&lt;/span&gt;. Departing from standard practice causes problems. A student who learns nonstandard notations will have a hard time reading textbooks, learning from other teachers, and communicating about mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many possible notations. It is bad practice as a student and as a teacher to invent new notations when there are well-established standard notations. Sometimes it is necessary to improve on existing notation, but this needs real justification and should be done rarely. This is a pragmatic matter, because experience shows that (unnecessary) proliferation of notation generates far more confusion than it remedies.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2019-10-31T19:30:24.503" LastActivityDate="2019-10-31T19:30:24.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17388" PostTypeId="1" AcceptedAnswerId="17389" CreationDate="2019-10-31T14:07:50.293" Score="3" ViewCount="220" Body="&lt;p&gt;I am curious if you guys learn more from Hard problems, or do you learn more from easy problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I heard that you learn a lot more from doing harder problems, because you have to learn all of the easier mathematical processes along the way.&lt;/p&gt;&#xA;" OwnerUserId="12976" LastEditorUserId="238" LastEditDate="2019-11-23T02:08:34.547" LastActivityDate="2019-11-23T02:08:34.547" Title="What kind of problem do you learn most from (hard problems or easy problems)?" Tags="|self-learning|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17389" PostTypeId="2" ParentId="17388" CreationDate="2019-10-31T15:03:45.120" Score="2" Body="&lt;p&gt;I think a lot of people here and in research mathematics overemphasize the value of harder problems, discovery learning, etc.  It appeals to them since they are the tip of the spear in intellect.  And it's more intrinsically interesting to someone who already knows the stuff (them).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also many of them have never done sports coaching and learning the benefit of easy, progressive training, with only very occasional all out efforts (mostly for building emotional fortitude, not skill).  You can learn a LOT by imitation and practice.  In fact our brains are probably wired for it--look at how kids learn. Of course the real analysis jocks don't get that since they think in terms of legalistic formal frameworks and subconsiously think this is how the human animal (and it is an ANIMAL) learns.&lt;/p&gt;&#xA;" OwnerUserId="12971" LastActivityDate="2019-10-31T15:03:45.120" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17391" PostTypeId="2" ParentId="17344" CreationDate="2019-10-31T22:46:02.467" Score="1" Body="&lt;p&gt;The problem here is defining primes in their own right rather than defining them in terms of factorization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with a number like 30.  Writing 30 = 2 x 15 tells us something new; writing 30 = 1 × 30 doesn't.  Writing 30 = 2 x 3 x 5 tells us something new again, whereas writing 30 = -2 x -15 doesn't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once we get to 30 = 2 x 3 x 5, we can't break down any of the components any further in a way that tells us anything new.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now &lt;em&gt;define&lt;/em&gt; primes as those numbers we can't break down any further in a way that tells us anything new.&lt;/p&gt;&#xA;" OwnerUserId="623" LastActivityDate="2019-10-31T22:46:02.467" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17392" PostTypeId="2" ParentId="17344" CreationDate="2019-11-01T04:13:10.943" Score="4" Body="&lt;p&gt;I'm kind of restating what other answers have said, but I wanted to practice expressing it in the clearest, most concise way I could think of. (Coincidentally, this came up with my partner tonight, so I got a test-run with it, and got an entirely satisfying result). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider only natural numbers (i.e, positive integers). It seems like the &lt;em&gt;number&lt;/em&gt; of divisors for different numbers is interesting and important. The following terms indicate how many distinct divisors a number has:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Unit&lt;/strong&gt;: A number that has exactly 1 divisor. Only the number 1 satisfies this criterion. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Prime&lt;/strong&gt;: A number that has exactly 2 divisors. Numbers such as 2, 3, 5, 7, etc. are in this category. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Composite&lt;/strong&gt;: A number that has 3 or more divisors. Numbers such as 4, 6, 8, 9, etc. are in this set.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In short, we don't call &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; &quot;prime&quot; because it has a unique number of distinct divisors; just a single one. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-11-01T04:13:10.943" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17393" PostTypeId="1" CreationDate="2019-11-01T14:17:55.953" Score="4" ViewCount="701" Body="&lt;p&gt;The context is explaining to calculus students how to integrate rational functions by using partial fractions decomposition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As we all know, partial fractions decomposition is a method to write every rational function as a sum of rational functions of the following three types:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(1)\text{ polynomial}, \quad(2)\;\frac{A}{(ax + b)^n}, \quad(3)\;\frac{Ax + B}{(ax^2 + bx + c)^n},$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a positive integer and &lt;span class=&quot;math-container&quot;&gt;$A,B,a,b,c$&lt;/span&gt; are real numbers such that &lt;span class=&quot;math-container&quot;&gt;$a \neq 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b^2 - 4ac &amp;lt; 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After having done it many times, I can say that I have no trouble explaining how to do partial fractions decomposition and that it is usually understood quite well by the students. Also, I have no trouble explaining how to integrate (1), (2), and for &lt;span class=&quot;math-container&quot;&gt;$n = 1$&lt;/span&gt; (3), which are the cases that usually appear in exams. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I always find difficult explaining how to integrate (3) for a general &lt;span class=&quot;math-container&quot;&gt;$n \geq 1$&lt;/span&gt;. This of course can be done by induction on &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, using the third-last and second-last formulas &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_integrals_of_rational_functions#Integrands_of_the_form_xm_/_(a_x2_+_b_x_+_c)n]&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, but those formula are essentially impossible to memorize. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a better method, that requires little memorization for the students?&lt;/p&gt;&#xA;" OwnerUserId="12824" LastActivityDate="2019-11-22T01:08:42.827" Title="The hardest case of integration by partial fractions" Tags="|integration|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17394" PostTypeId="2" ParentId="17388" CreationDate="2019-11-01T16:11:50.327" Score="8" Body="&lt;p&gt;One way to address this issue is with more open-ended problems that have a &quot;low floor&quot; (easy entry) and &quot;high ceiling&quot; (hard aspects that can be addressed but don't need to be).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes this comes in a series of related problems. See the &lt;a href=&quot;https://www.jrmf.org/activities&quot; rel=&quot;noreferrer&quot;&gt;Julia Robinson Mathematics Festival activities&lt;/a&gt; for many great examples (click on 'activities library' for a pdf).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other times there can be one problem that a group of students can work on. One example that comes to mind is the &quot;squareness&quot; problem described &lt;a href=&quot;https://mathmamawrites.blogspot.com/2011/05/escape-from-textbook-group-met-on.html&quot; rel=&quot;noreferrer&quot;&gt;here (on my blog)&lt;/a&gt;. Henri Picciotto discusses this idea in the context of &quot;rich activities&quot; &lt;a href=&quot;https://blog.mathed.page/2013/11/24/enrichment/&quot; rel=&quot;noreferrer&quot;&gt;on this blog page&lt;/a&gt;. And that term takes us to a wealth of rich problems, called rich starting points (RISP) in the UK, &lt;a href=&quot;http://www.s253053503.websitehome.co.uk/risps/index.html&quot; rel=&quot;noreferrer&quot;&gt;at this site&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of these examples are K12. But these concepts can also be used at higher levels. &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-11-01T16:11:50.327" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17397" PostTypeId="2" ParentId="17388" CreationDate="2019-11-01T19:50:27.683" Score="6" Body="&lt;p&gt;I don't think that &quot;hard&quot; and &quot;easy&quot; are the right categories when it comes to answer the question: &quot;What kind of problems do you learn most from?&quot; Indeed &quot;hard&quot; and &quot;easy&quot; are really subjective and many people might find &quot;hard&quot; the &quot;easy&quot; problems I learn most from and vice versa.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since I guess that purpose of the question is giving students problems from which they can learn most, I think a better distinction is between:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Imperative problems:&lt;/strong&gt; Find &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;... Prove that... Check the condition...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Question problems:&lt;/strong&gt; Is it true that...? Does exist ...? Can we compute ...?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Althought both categories of problems are important, I believe that one learns much more from the second one (which, unfortunately, unsurprisingly, is also the more neglected in tests and exercises).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, on the one hand, when solving imperative problems, the target is clear from the beginning and usually the student has only to apply some algorithmic methods in the right way. What he learns is applying this methods better and better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, when solving question problems, neither the target nor the methods are clear from the beginning, and the student has to make many tries and guesses. What he learns is developing a better intuition of the subject.&lt;/p&gt;&#xA;" OwnerUserId="12824" LastActivityDate="2019-11-01T19:50:27.683" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17398" PostTypeId="1" CreationDate="2019-11-01T21:17:48.167" Score="6" ViewCount="675" Body="&lt;p&gt;I am teaching my daughter, who is currently about &lt;span class=&quot;math-container&quot;&gt;$46$&lt;/span&gt; months old, additions. She is very curious and asks a lot of &lt;em&gt;good&lt;/em&gt; questions. For example, when I told her that &lt;span class=&quot;math-container&quot;&gt;$2+6=8$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4+4=8$&lt;/span&gt;, she asked me the following question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why are they the same?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Surely I know the logical answer this question using &lt;a href=&quot;https://en.wikipedia.org/wiki/Peano_axioms&quot; rel=&quot;noreferrer&quot;&gt;Peano Axioms&lt;/a&gt; and the definition of natural numbers. But she has not learnt the Peano Axiom at this age.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how to answer my three-year-old daughter's question that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why &lt;span class=&quot;math-container&quot;&gt;$2+6$&lt;/span&gt; is the same as &lt;span class=&quot;math-container&quot;&gt;$4+4$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2019-11-01T22:28:16.987" LastActivityDate="2019-11-16T07:36:46.193" Title="How to answer a three-year-old the question &quot;Why is $2+6$ the same as $4+4$&quot;?" Tags="|gifted-students|arithmetic|preschool-education|student-question|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17399" PostTypeId="1" CreationDate="2019-11-02T01:11:33.740" Score="22" ViewCount="7873" Body="&lt;p&gt;In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined. For example, without the definition the fundamental group, it is almost impossible to teach anything serious about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as far as real numbers were concerned, they were taught very early without any clear definition was even given. Definitions of real numbers using say Dedekind cut were never introduced until in say Mathematical Analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pedagogically speaking, how is it even possible to teach about a mathematical concept (real number) before clearly defining it?   &lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2022-07-23T23:25:18.440" Title="Why is it possible to teach real numbers before even rigorously defining them?" Tags="|mathematical-pedagogy|mathematical-analysis|real-numbers|" AnswerCount="16" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="17400" PostTypeId="2" ParentId="17399" CreationDate="2019-11-02T01:24:10.807" Score="6" Body="&lt;p&gt;I believe that, unlike many other mathematical concepts (especially more abstract ones), even fairly young people have a basic understanding of distance, including fractions to any degree of units of measurement (e..g., inches, feet, miles, etc., in the imperial measurement system and/or centimetres, metres, kilometres, etc., in the metric measurement system).&lt;/p&gt;&#xA;&lt;p&gt;By using things like a number line, you can explain &amp;amp; demonstrate various properties related to real numbers without needing to first give a clear &amp;amp; &amp;quot;rigorous&amp;quot; definition of them. For example, you can at least &amp;quot;show&amp;quot;, although not necessarily &amp;quot;rigorously&amp;quot; prove (with this depending to a certain extent on what you consider to be &amp;quot;rigorous&amp;quot;), concepts like that addition is commutative, adding negative numbers is the same as subtracting the positive value of them, subtracting a negative number is the same as adding the corresponding positive value, etc.&lt;/p&gt;&#xA;&lt;p&gt;There are various online resources which can help you to use number lines to teach real number concepts. One quite good basic one is Algrebra1Coach's &lt;a href=&quot;https://algebra1coach.com/1-3-real-numbers-and-the-number-line/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Real Numbers and the Number Line&lt;/a&gt;, and a somewhat more rigorous &amp;amp; advanced one is LibreText's &lt;a href=&quot;https://math.libretexts.org/Bookshelves/Algebra/Book%3A_Beginning_Algebra_(Redden)/1%3A_Real_Numbers_and_Their_Operations/1.1%3A_Real_numbers_and_the_Number_Line&quot; rel=&quot;nofollow noreferrer&quot;&gt;Real numbers and the Number Line&lt;/a&gt;. In addition, a more specific resource is teachoo's &lt;a href=&quot;https://www.teachoo.com/6995/879/Locating-irrational-number-on-number-line/category/Irrational-Numbers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Locating irrational number on number line&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12978" LastEditorUserId="12978" LastEditDate="2022-02-20T23:50:56.900" LastActivityDate="2022-02-20T23:50:56.900" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17401" PostTypeId="2" ParentId="17398" CreationDate="2019-11-02T01:36:33.663" Score="20" Body="&lt;p&gt;I'm nearly &lt;em&gt;sure&lt;/em&gt; I did this with my child when she was young.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, establish that she understands that a number, like three, is equal to &lt;span class=&quot;math-container&quot;&gt;$1+1+1$&lt;/span&gt;. Hold three fingers up and ask her &quot;how many is this&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/HthuC.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HthuC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then spread them out and ask the same question. Are we adding &lt;span class=&quot;math-container&quot;&gt;$1+1+1$&lt;/span&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3kLwo.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3kLwo.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try holding up eight fingers (keep your thumbs down, for example) and ask her to count them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1xGsj.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1xGsj.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then hold them together, but move your hands apart. What numbers do you see added? What is the total?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ubowj.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ubowj.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then make a V with one hand, holding two fingers apart from the other six. What numbers are being added? What is the total?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/II6F2.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/II6F2.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then hold three fingers apart from the other five. What numbers are being added? What is the total?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NZJdA.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NZJdA.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, hold one finger apart from the other seven. What numbers are being added? What is the total?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ug4zc.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ug4zc.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then you can talk about different ways to add numbers to make &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;. This time, have her act it out with her hands.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-11-02T01:36:33.663" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17402" PostTypeId="2" ParentId="17399" CreationDate="2019-11-02T05:07:53.597" Score="68" Body="&lt;p&gt;Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of calculus predated formal definitions of the real numbers (Dedekind cuts, Cauchy sequences) and a formal definition of continuity, individuals must first make use of real numbers for years, before they are capable of even understanding the need for a formal definition.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thoreau wrote, “If you have built castles in the air, your work need not be lost; that is where they should be. Now put the foundations under them.” This is how mathematical knowledge expands. First we build castles in the air, and foundations afterwards. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-11-02T05:07:53.597" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17404" PostTypeId="2" ParentId="17399" CreationDate="2019-11-02T06:33:39.170" Score="44" Body="&lt;p&gt;It is possible to teach real numbers in elementary school before even rigorously defining them by using what H. Wu (&quot;&lt;a href=&quot;https://www.ams.org/journals/notices/201103/rtx110300372p.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Mis-Education of Mathematics Teachers&lt;/a&gt;,&quot; &lt;em&gt;Notices of the AMS&lt;/em&gt;, vol. 58, no. 3, p. 376) calls the &lt;em&gt;Fundamental Assumption of School Mathematics&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In terms of the nitty-gritty of classroom instruction, real numbers are handled in K–12 by what is called the &lt;strong&gt;Fundamental Assumption of School Mathematics&lt;/strong&gt; (FASM; see p. 101 of [Wu2002] and p. 62 of [Wu2008b]).  It states that any formula or weak inequality that is valid for all rational numbers is also valid for all real numbers.  For example, in the seventh grade, let us say, the formula for the addition of fractions,&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd},$$&lt;/span&gt;&#xA;  where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; are whole numbers, can be (and should be) proved to be valid when &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; are rational numbers.  By FASM, the formula is also valid for all real numbers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;.  Thus high school students can write, without blinking an eye, that&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{3}}=\frac{\sqrt{3}+2\sqrt{2}}{\sqrt{2}\sqrt{3}},$$&lt;/span&gt;&#xA;  even if they know nothing about what &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt{2}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}\sqrt{3}$&lt;/span&gt; means.  If this seems a little cut-and-dried and irrelevant, consider the useful identity&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{1−x}+\frac{1}{1+x}=\frac{2}{1−x^2}$$&lt;/span&gt;&#xA;  for all real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  If &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is rational, this identity is easily veriﬁed (see preceding addition formula).  But the identity implies also&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{1−\pi}+\frac{1}{1+\pi}=\frac{2}{1−\pi^2}.$$&lt;/span&gt;&#xA;  Without FASM, there is no way to conﬁrm this equality in K–12, so its validity is entirely an article of faith in school mathematics. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastActivityDate="2019-11-02T06:33:39.170" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17405" PostTypeId="2" ParentId="17399" CreationDate="2019-11-02T12:26:56.870" Score="17" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;But as far as real numbers were concerned, they were taught very early&#xA;  without any clear definition was even given.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Were they? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say that usually no teacher really speaks of real numbers to students until the first calculus lesson. In almost all pre-calculus exercises, students only meet rational numbers, in the form of fractions or decimal expansions. The only exceptions are some roots of integers, like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;, and the number &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, when doing geometry. Therefore, algebraic speaking, before calculus students are not working with real numbers, but they are working in a field extension like &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}(\pi, \sqrt{2}, \sqrt{3}, \dots)$&lt;/span&gt; or something a little bigger.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fundamental property of real numbers, that distinguish them from said field extensions of the rationals, is their completeness, that is, &lt;span class=&quot;math-container&quot;&gt;$\sup A$&lt;/span&gt; exists for every bounded &lt;span class=&quot;math-container&quot;&gt;$A \subseteq \mathbb{R}$&lt;/span&gt;. I doubt students ever uses or see this before calculus.&lt;/p&gt;&#xA;" OwnerUserId="12824" LastActivityDate="2019-11-02T12:26:56.870" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17406" PostTypeId="2" ParentId="17372" CreationDate="2019-11-02T13:29:38.420" Score="11" Body="&lt;p&gt;This is how most students perceive math tests. Whether it's fair or not, this &lt;em&gt;is&lt;/em&gt; the perception and it is the &lt;em&gt;normal&lt;/em&gt; response to a broken math education system.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Imagine you are a teenager and your driving test for your full license is in a week. The Department of Motor Vehicles is massively understaffed, so if you fail, you can't book another test for a year. You hire a driving instructor to help you with step-by-step instructions for parallel parking, how to not forget to shoulder-check before changing lanes, etc. But when you go for your driving lesson, you're instead told to learn about the static vs kinetic friction of tires on the road, mechanical engineering principles that underlie the internal combustion engine, and civil engineering theories about traffic. When you resist and ask for step-by-step instructions you know will be on the test, your driving instructor responds:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;...the proof [of civil engineering traffic formulae and Newton's Laws of Motion] being written by [your] own hands usually puts a whole new meaning to theorems and formulas.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Obviously, you and I know that real math, by definition, requires understanding, while safe driving only requires good physical habits. The student, though, may have experienced &lt;em&gt;nothing&lt;/em&gt; but pseudo-math (100% step-by-step instructions) for more than a decade. In their minds, the proof of the irrelevance of abstractions is the fact that they're not tested or incentivized, the same way a driving test excludes conceptual applications of Newtonian physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given how prevalent this sad misperception is, one must blame the math education system, years of bad teaching and awful testing. So, keep in mind when you say that you want them to &quot;understand&quot;, to know &lt;em&gt;why&lt;/em&gt; symbolic mathematical statements are true or false, they may have &lt;em&gt;no idea&lt;/em&gt; what you're talking about while facing enormous incentives to just focus on step-by-step instructions. In their minds, every moment you spend on Pythagorean Theorem proof is meaningless, ivory tower gibberish akin to learning the history of petroleum engineering research the day before the only driving test they can book for the next year. The idea that understanding will help them in the long-run is just incredibly foreign and &quot;obviously&quot; false, an irrational leap of faith that requires doing worse on an imminent, high-stakes test. In fact, they may have experienced the &lt;em&gt;opposite&lt;/em&gt; of this many times! Some will have tried to seek deep understanding, only to fail (because of prior knowledge gaps), then gotten burned on a test where rote memorization would have earned them a higher grade.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Against such entrenched delusion (&quot;Math is just steps and rules!&quot;) and terrible incentives (&quot;The math test is just steps and rules!&quot;), it's no surprise that your arguments for deep understanding  don't have much effect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not going to BS you and say that I have a great solution for this. I've talked to dozens of teachers, been to countless math education workshops and conferences, read zillions of books and websites, etc. and am unaware of anybody who claims to have solved this problem, especially when one doesn't control how abstract the assessments are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest thing to a solution I've come up with is the following toolbox. It often does not work because students are missing years of prior knowledge and the teachers' math tests are such #$@&amp;amp;*, but it's the best I've got.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Empathy. Tell them the story of the driving test and do so with great passion and anguish. Tell it better than they could. Earn an Oscar for your storytelling. Ask them if it's a fair comparison of their experience, motives, and incentives. This will massively boost your credibility while making them feel much more understood. (If you don't like the driving analogy, you could also make one up where a consultant insists an employee spend a ton of time on tasks for which the boss does not pay them or promote them for, when the annual review and promotions are in a week.)&lt;/li&gt;&#xA;&lt;li&gt;Safe Grades. At this point, they may be a little more open to things you have to say. Tell them they've been doing grammar exercises while you want to teach them poetry. Ask them what minimal grade they'd need to be earning before being willing to explore other aspects of math. Ask them how hard they're willing to work to achieve that Safe Grade.&lt;/li&gt;&#xA;&lt;li&gt;Deal or No Deal? If that willingness is way beneath what you consider necessary to earn the safe grade, then tell them to find another tutor. You're here posting on Mathematics Educators Stack Exchange, suggesting you are more caring and skillful than average, but that caring and skill is not necessary to help students rote memorize for tests. Tell them to move along. If they are willing to earn that Safe Grade, then giddy up! Get that over with ASAP, then you'll have time to do real math. :)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;P.S.: Use statistics to convince students that they'll need math. &quot;We're all going to get sick one day. What should you know about medical research? You seem to care about politics. How do you know if policies are working?&quot;&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2020-01-08T09:39:47.717" LastActivityDate="2020-01-08T09:39:47.717" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17407" PostTypeId="2" ParentId="17372" CreationDate="2019-11-02T13:49:37.657" Score="2" Body="&lt;p&gt;My advice is to accept 3/4 of a loaf.  Tailor your instruction to cover exactly this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;solution to problems that are known to be on their school's exams, and prefer step-by-step instructions free from any context/theory/mathematical properties.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is really still useful content to cover and better than nothing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I wouldn't kill yourself in terms of work and maybe charge a little higher, since it is more of a hassle for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He's an adult learner who just needs the minimum in this darned math class to support humanities stuff he likes better.  Just get him through.  It should not be your objective to instill some turnaround in his attitude towards math.  Heck if he just progresses and has some basic wins, that may slightly improve his attitude towards the subject.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one thing I would not tolerate is disrespect or refusal to take training (after a discussion on his objectives and an attempt to coach him in that direction).  If there is still a problem at that point, then just cut him loose and tell him to try someone else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  Think of this as very similar to personal training in the gym.&lt;/p&gt;&#xA;" OwnerUserId="12981" LastActivityDate="2019-11-02T13:49:37.657" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17408" PostTypeId="2" ParentId="17399" CreationDate="2019-11-02T16:08:45.313" Score="14" Body="&lt;p&gt;At German universities, one of the first lectures in mathematics is &quot;Analysis 1&quot; which is a kind of &quot;rigorous calculus&quot; and there one always proceeds more or less like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We start with an axiomatic approach to the real numbers. In short: The real numbers are a complete, ordered, Archimedean field. In practice we first introduce the axioms of a field, then the axioms of an order, and then, with the help of the Archimedean axiom, we introduce limits by epsilon/delta. Then you observe that there are rational sequences which should be converging, but there is no limit and some axiom of completeness is introduced (often &quot;every bounded set has a least upper bound&quot;, sometimes &quot;every Cauchy sequence converges&quot;, sometimes &quot;the intersection of nested closed  intervals is non-empty&quot;) and from then on we work with the real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes, but not always, some construction of the real number is given, but in most cases, this is skipped. It has been like this when I studied in the 1990s and it still is like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this is difficult for some students, but the majority of the students goes through this quite well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I like this approach. You introduce some objects not by exactly saying &lt;strong&gt;what they are&lt;/strong&gt;, but by stating a complete list of &lt;strong&gt;what you can do with them&lt;/strong&gt;. (Tim Gowers wrote a great chapter on this in &quot;A very short introduction to mathematics*.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, let me ask: If you want everything defined, where do you start? Define the natural number using set theory? But how to define a set? At some point you will be back the defining objects through their properties somehow.&lt;/p&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2019-11-02T16:08:45.313" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17409" PostTypeId="2" ParentId="17398" CreationDate="2019-11-02T19:10:09.227" Score="6" Body="&lt;p&gt;Blocks work well for thinking about addition. Have her count out 8 blocks, and then ask her about all the addition problems that have 8 blocks as the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A lovely children's book which looks at all the sum pairs for 7 is &lt;em&gt;Quack and Count&lt;/em&gt;, by Keith Baker. (You can &lt;a href=&quot;https://www.abebooks.com/servlet/SearchResults?isbn=0152928588&amp;amp;cm_sp=mbc-_-ISBN-_-all&quot; rel=&quot;noreferrer&quot;&gt;buy it used here&lt;/a&gt;.) It has luscious pictures, a driving rhythm, and a lovely storyline. (“Slipping, sliding, having fun, 7 ducklings, 6 plus 1.”) &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-11-02T19:10:09.227" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17410" PostTypeId="2" ParentId="17372" CreationDate="2019-11-02T22:19:58.713" Score="10" Body="&lt;p&gt;Well, there is a recent and excellent book about this question: &lt;a href=&quot;https://books.google.com.br/books?id=y0WCDQAAQBAJ&amp;amp;printsec=frontcover&amp;amp;hl=pt-BR&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Why Students Resist Learning: A Practical Model for Understanding and Helping Students&lt;/a&gt; by Anton O. Tolman, Janine Kremling and Anton O. Toman.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9Tthh.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9Tthh.jpg&quot; alt=&quot;Why Students Resist Learning: A Practical Model for Understanding and Helping Students&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The authors say resistance in learning may be a joint consequence of several factors, including resistance from teachers and schools. Here is a resistance framework proposed by them:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VokPo.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VokPo.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Metacognition is a very effective tool to help people with their resistance, accordingly the authors.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2019-11-02T22:19:58.713" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17411" PostTypeId="2" ParentId="7528" CreationDate="2019-11-02T23:07:15.950" Score="3" Body="&lt;p&gt;Parabolas in a fountain at Parque das Águas, Cuiabá, Brazil.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/88ha7.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/88ha7.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2019-11-02T23:07:15.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17412" PostTypeId="2" ParentId="12003" CreationDate="2019-11-03T00:37:23.903" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I ... find myself troubled by lacking proper examples of why a sophisticated theory can actually turn out to be beneficial ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I agree this is troubling. &quot;I feel your pain,&quot; as the saying goes.&#xA;I would like to suggest that &#xA;computer graphics can serve as a source for a subset of the examples you seek: to motivate polynomials, and motivate finding roots of polynomials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every time you print text in a particular font (e.g., &lt;a href=&quot;https://en.wikipedia.org/wiki/PostScript_fonts#Type_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Postscript Type-1&lt;/a&gt;) on a laser printer you are using&#xA;cubic polynomials. That every printer in the world is using cubic polynomials every day can&#xA;be a compelling story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finding roots of such polynomials is a common&#xA;need in computer graphics, e.g., to compute the highlights on a shiny&#xA;3D surface, like an apple.&#xA;The challenge is to intersect a light-ray with a cubic surface patch.&#xA;Search for ray-tracing Bézier, or B-spline, or NURBS surfaces to see more examples of this calculation.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/LPc0d.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LPc0d.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;https://www.youtube.com/watch?v=hqgmLmIgLSo&quot; rel=&quot;nofollow noreferrer&quot;&gt;JTrace&lt;/a&gt;.)&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-11-03T01:43:25.253" LastActivityDate="2019-11-03T01:43:25.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17413" PostTypeId="2" ParentId="17399" CreationDate="2019-11-03T01:48:01.923" Score="9" Body="&lt;p&gt;Introductory mathematics is not done with formal definitions. This is because most people at the corresponding age (kids) don't know what the words &quot;&lt;em&gt;formal&lt;/em&gt;&quot; or &quot;&lt;em&gt;definition&lt;/em&gt;&quot; actually means. This only come to play much later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You start teaching kids mathematics by showing them how to count things up to ten using their fingers. Then, you teach addition with little simple problems like &quot;&lt;em&gt;John had five apples. Mary gave him two more. How many apples does John have now?&lt;/em&gt;&quot; - You teach by making associations of numbers and math with phenomena that are easily recognizable and verifiable in their world and day-to-day activities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only after they are already somewhat used to counting numbers and doing addition, subtraction, multiplication and division and they already got a few lessons on sets of oranges, sets of cats, sets of some small numbers and union and intersection of those is that the concept of the set of natural numbers (&lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;) are introduced to them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some time further, you introduce the concept of fractional numbers and negative numbers. And again, this is associated with things easily recognizable in the mundane world like &quot;&lt;em&gt;two and a half pizzas&lt;/em&gt;&quot; or &quot;&lt;em&gt;a half orange&lt;/em&gt;&quot; or &quot;&lt;em&gt;John have five dollars and promised to give seven to Jane, so he is lacking two&lt;/em&gt;&quot;. So far, no one gave them a formal definition of those things, these are only show up later on when the kids are (or should be) firmly familiar with the concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The (in)formal definition of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt; only comes in after the kids did played a bit with negative numbers. Only after the kids are already familiar with fractions is that you explain that &quot;&lt;em&gt;every fraction is representable with a ratio between two integers numbers&lt;/em&gt;&quot; and you &quot;&lt;em&gt;call all those numbers that can be represented in this way as the set of rational numbers denoted as &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;&lt;/em&gt;&quot;. Gosh, that was an informal definition, but it is sufficient and more than enough to be understood by kids. No kid would easily and quickly understand something like &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q} = \{\frac{p}{q} | p \in \mathbb{Z} \land q \in \mathbb{Z}^*\}$&lt;/span&gt; because that notation and the rules governing them are alien-like for most of them even if they were already presented and used to all of the involved elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some time later, when teaching square roots and geometry, things like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3}$&lt;/span&gt; show up. You quickly tell them that these aren't rational numbers and that there is no way to represent them as a ratio of two integers. Again, that is an informal definition, but is enough. You might even give a further step by telling them that any square root of a prime number is irrational because that if that was not the case, by finding a ratio of two integer numbers that squared give the supposed-to-be prime number would mean that it would be composite (i.e. a proof by contradiction, but still informal). Then, you present the symbol &lt;span class=&quot;math-container&quot;&gt;$\mathbb{I}$&lt;/span&gt; to represent them. Also, soon &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; will also show up to join the irrational numbers group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, you present the real numbers just as &quot;&lt;em&gt;the union of rational and irrational numbers&lt;/em&gt;&quot;. This is a very simple and informal, although very precise and sufficient definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the answer is that people can learn about the real numbers without formal definitions by simply working to them with informal definitions and correlations with real-world mundane concepts and then by building up higher-level concepts on top of lower-level concepts. &lt;strong&gt;Formal rigorous definitions are then usable only for people that are already skilled enough in math to be able to make sense of them.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="12985" LastEditorUserId="238" LastEditDate="2019-11-03T04:08:34.460" LastActivityDate="2019-11-03T04:08:34.460" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17414" PostTypeId="2" ParentId="17399" CreationDate="2019-11-03T03:16:16.867" Score="3" Body="&lt;p&gt;&quot;In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no reason, this has to be true.  It is just asserted.  Two examples have been given where significant content can be learned without strict axiomatic buildup of the subject: arithmetic and calculus.  We could come up with others (set theory, topology, etc.).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: I am not disputing that the axiomatic buildup can also be beneficial at times, very much so.  But we should not just cite the original statement as some sort of doctrinal statement.  After all, we are discussing practical pedagogy in that statement, not the axiomatic structures of math themselves.  I think we would want evidence, statistics, explication before just going with the statement as true.  &lt;/p&gt;&#xA;" OwnerUserId="12981" LastActivityDate="2019-11-03T03:16:16.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17415" PostTypeId="2" ParentId="17399" CreationDate="2019-11-03T04:50:27.440" Score="4" Body="&lt;p&gt;This answer may seem off-topic, since this is &quot;Math Ed&quot; SE. But a similar question can be asked about many scientific concepts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on observations of the world around us animals (including humans) have built up heuristic models of physics, biology and psychology. In order to be a useful creature, one must have some predictive models, even if they are not formal (in the sense of formal mathematics). Someone that is good at sports, e.g. basketball, has some intrinsic sense of Newtonian physics. The math that describes how a ball flies through the air uses real numbers, but knowledge of throwing a ball doesn't require any knowledge of real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, to (maybe) answer the OP, when we teach math (or physics or biology), we are often telling students: &quot;You know how when you throw a basketball you have to control the angle and the force? Well, this is how to write a formula that tells you for any basketball shot, the ideal angle and force to use!&quot; We are connecting a formalism to a concept they already know, not building up a formalism from the bottom up.&lt;/p&gt;&#xA;" OwnerUserId="12987" LastActivityDate="2019-11-03T04:50:27.440" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17416" PostTypeId="1" CreationDate="2019-11-03T10:13:53.043" Score="4" ViewCount="315" Body="&lt;p&gt;I was teaching Fourier transform for engineering students. Since I didn't want to go into rigourous proofs during class, I often use intuition, just give students an idea to persuade them with the theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if &lt;span class=&quot;math-container&quot;&gt;$$F(x):=\int_{-\infty}^{+\infty}\frac{\cos(wx)}{1+w^2}dw,$$&lt;/span&gt; then we can differentiate &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; by taking the derivative under the integral sign on the right hand side. For example, this is exactly what I did when I gave my students an explanation (idea)  for the formula &lt;span class=&quot;math-container&quot;&gt;$(\mathcal{F}(f))'(w)=-i\mathcal{F}(xf(x))$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem now is that, for the above integral (which we encounter a lot in the course), we can easily check that (using software) &lt;span class=&quot;math-container&quot;&gt;$$F(x)=\pi e^{-x},\forall x\in\mathbb{R},$$&lt;/span&gt; on the other hand, if we differentiate under the integral sign, students then would expect that &lt;span class=&quot;math-container&quot;&gt;$$\pi e^{-x}=\int_{-\infty}^{\infty}\frac{\omega \sin(wx)}{1+w^{2}}dw,\forall x\in\mathbb{R}.$$&lt;/span&gt; However, this is not true, as we can check that this is only true for &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the problem now is: well,... (I bit myself in front of engineering students, who were already terrified with Maths). I understand that in general differenting under the integral sign is not always legitimate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did you teach and go around this in your teaching? How to teach the formula: &lt;span class=&quot;math-container&quot;&gt;$$(\mathcal{F}(f))'(w)=-i\mathcal{F}(xf(x))$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;when students easily face and get confused with the above confusing matter?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="12801" LastEditorUserId="12801" LastEditDate="2019-11-03T13:28:33.120" LastActivityDate="2019-11-16T04:09:17.553" Title="The trick didn't like me (teaching Fourier transform)" Tags="|mathematical-analysis|classroom-management|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17417" PostTypeId="2" ParentId="17399" CreationDate="2019-11-03T14:43:11.977" Score="2" Body="&lt;p&gt;Since real numbers include whole numbers, rational numbers, and irrational numbers (almost the entire landscape) I’m uncertain why you think teaching does not occur.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does it not begin very early with counting and learning numbers? It isn’t until later that these numbers are labeled as &lt;em&gt;real&lt;/em&gt;, when needing to distinguish from imaginary as introduced in pre-algebra(?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question seems to extend into psychology and philosophy; but one could also make the point that intuitive concepts need little-to-no teaching. We don’t need to know how to spell before we say words (or even write words) and we don’t need to know their definitions before they are used - especially not their exact definitions.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Note: I’m uncertain if your question is something nuanced to mathematics that applies beyond my level of understanding of it. If that is the case I will delete this answer.&lt;/p&gt;&#xA;" OwnerUserId="12991" LastActivityDate="2019-11-03T14:43:11.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17418" PostTypeId="2" ParentId="17399" CreationDate="2019-11-03T15:24:28.293" Score="1" Body="&lt;p&gt;Read Dirk's answer - we take the axioms of rational numbers (which are just fractions of integer numbers, so easy to understand) and get quite a few nice theorems. We even have Cauchy sequences (which are kind of sequences with limits but not quite, but the difference was never mentioned).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my Analysis I, the professor then went to prove the Intermediate Value Theorem. (If f is continuous, a &amp;lt; b, f(a) &amp;lt; 0 and f(b) &gt; 0, then there is an x, a &amp;lt; x &amp;lt; b, with f(x) = 0). So he defines the set of all y with f(y) &amp;lt;= 0, says this set is non-empty and has an upper bound, therefore it has a least upper bound x, and because f is continuous, f(x) can't be either less than or greater than 0, so f(x) = 0.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Muttering starts among the students and gets louder. At last someone gets up and says &quot;That proof is not right. How do you know that this set has a least upper bound?&quot; And you hopefully noticed that what the professor boldly claimed was &lt;em&gt;exactly&lt;/em&gt; the missing axiom that makes the difference between rational numbers and real numbers, and that's what the professor says. &quot;Axiom #10 of real numbers: Every non-empty set with an upper bound has a least upper bound&quot;. And immediately afterwards he proved what is now obvious: Every Cauchy sequence has a limit.&lt;/p&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2019-11-03T15:24:28.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17419" PostTypeId="2" ParentId="17399" CreationDate="2019-11-04T08:29:57.043" Score="5" Body="&lt;p&gt;You are confusing definitions and models.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &lt;em&gt;definition&lt;/em&gt; of real numbers is a set of axioms they obey. Different such sets exist, but they can be shown to be equivalent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &lt;em&gt;model&lt;/em&gt; of real numbers is a construction like Dedekinds, which is a set of sets of sets etc. that is carefully constructed so that they obey one such set of axioms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To work with real numbers you really only need to know what axioms apply.  You use them to construct theorems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This does not mean that models are useless, but they are approaching the subject from a different angle.&lt;/p&gt;&#xA;" OwnerUserId="5818" LastActivityDate="2019-11-04T08:29:57.043" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17420" PostTypeId="2" ParentId="17398" CreationDate="2019-11-04T09:31:12.517" Score="5" Body="&lt;p&gt;A &lt;em&gt;good&lt;/em&gt; answer to this question is one that (is correct, and) &lt;em&gt;she&lt;/em&gt; finds convincing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As kids are growing up and making sense of the world around them, &lt;em&gt;experimentation&lt;/em&gt; is often one of their key sources of truth. Perhaps, experimentation can also serve as a reliable and effective gateway to the abstract world of mathematical truths!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's one possible roadmap that leads convincingly to arithmetical truths.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Experiment with an Object&lt;/strong&gt;: Take &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; blocks, add &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; blocks. Count the total number of blocks to discover that the result is &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; blocks. Therefore, &lt;span class=&quot;math-container&quot;&gt;$(2 \mbox{ blocks}) + (6 \mbox{ blocks}) = (8 \mbox{ blocks})$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Discover that these results apply to all Other Objects&lt;/strong&gt;: Experiment to discover that this rule applies to other objects such as apples, almonds, chairs, tables, toys, candy and anything else you can think of. Perhaps she would be willing to take the leap with you to agree that this would apply to other things that she has not yet experimented with, such as ships, planets and whales. Therefore, &lt;span class=&quot;math-container&quot;&gt;$(2 \mbox{ things}) + (6 \mbox{ things}) = (8 \mbox{ things})$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;General result&lt;/strong&gt;: Since these results are valid irrespective of the object or thing being added, it is therefore acceptable (and more efficient) to write our experimental result concisely as: &lt;span class=&quot;math-container&quot;&gt;$2+6 = 8$&lt;/span&gt; without reference to any object in particular.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Soon we point out that, in particular, these rules also apply to fingers (just like any other object), which are always at hand(!) so we can use them no matter what it is that we are in fact trying to add.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key feature of the approach just outlined is that the act of writing &lt;span class=&quot;math-container&quot;&gt;$2+6 = 8$&lt;/span&gt; is a symbolic recording (by pencil on paper, or chalk on board) of an experimental truth that we have discovered by exploring the world around us. The question of logically or abstractly proving it, therefore, does not arise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we arrive at the question, &quot;what is &lt;span class=&quot;math-container&quot;&gt;$4+4$&lt;/span&gt;?&quot;, we must now resort to experimenting with our fingers (or any other object) to discover the result of this addition. Having discovered the result, we can record it symbolically as &lt;span class=&quot;math-container&quot;&gt;$4+4 = 8$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, we can get creative with our experiments. For example, to count the number of dots below, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;: : : : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;we can count them as &lt;/p&gt;&#xA;&#xA;&lt;p&gt;: &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; : : : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;or as &lt;/p&gt;&#xA;&#xA;&lt;p&gt;: : &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; : :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This provides an intuitive justification that &lt;span class=&quot;math-container&quot;&gt;$2+6$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4+4$&lt;/span&gt; are really just two different ways of visualizing &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I can't help pointing out that the following is a staggering gap in her present education:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But she has not learnt the Peano Axioms at this age.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I hope you will remedy it at the earliest! ;)&lt;/p&gt;&#xA;" OwnerUserId="12996" LastEditorUserId="12996" LastEditDate="2019-11-07T11:33:30.260" LastActivityDate="2019-11-07T11:33:30.260" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17421" PostTypeId="2" ParentId="17372" CreationDate="2019-11-04T19:20:17.607" Score="14" Body="&lt;p&gt;This is not an answer, but an assertion that what you are experiencing is not something new. Here are some quotes from a &lt;a href=&quot;http://toomandre.com/my-articles/engeduc/ARUSSIAN.PDF&quot; rel=&quot;nofollow noreferrer&quot;&gt;1993 article&lt;/a&gt; of an Estonian math prof, who moved to the U.S. in the early 1990s, so the problem is at least thirty years old. Some say that the commoditization of universities started from the Reagan times.&lt;/p&gt;&#xA;&lt;p&gt;This quote is not directly related to your student's situation, after all the author's students studied some watered-down version of calculus, not high-school math, still I think the general idea applies.&lt;/p&gt;&#xA;&lt;p&gt;I guess, the blame can be distributed between subpar school math programs, inadequate preparation of school teachers, constant pressure of high-stake tests, as well the students' expectation that good grades by themselves are a measure of success.&lt;/p&gt;&#xA;&lt;p&gt;Bold in quotations is by me.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When I had taught in Russia, I was thanked for teaching my students to be&#xA;humans, to behave reasonably in unusual situations. But I met a lot of&#xA;resistance from some of my American undergraduate students, especially&#xA;when I tried to give them something unexpected. &lt;strong&gt;On tests they wanted&#xA;to do practically the same as what they had done before&lt;/strong&gt;, only with&#xA;different numerical data.&lt;/p&gt;&#xA;&lt;p&gt;It is a common opinion that the United States of America supports&#xA;democracy. Democracy always was connected in my mind with good&#xA;education for all people, and I knew that American thinkers also&#xA;believed in this connection. Thus, when I came to this country, I&#xA;expected to have rich opportunities to teach students to think&#xA;critically, independently, and creatively and to solve non-standard&#xA;problems without hindrance from authorities.&lt;/p&gt;&#xA;&lt;p&gt;But ... when I started to teach so-called &amp;quot;business calculus&amp;quot; to&#xA;undergraduates, I got into absolutely new situation. All my ideas&#xA;about teaching students to think became completely out of place. Never&#xA;before had I seen so many young people in one place who were so&#xA;reluctant to meet challenges and to solve original problems. &lt;strong&gt;All they&#xA;wanted were high grades&lt;/strong&gt;, and they wanted to get them with a conveyor&#xA;belt regularity.&lt;/p&gt;&#xA;&lt;p&gt;I found out that every technical calculation, which I was used to&#xA;ignoring, was a considerable obstacle for my students. It took a&#xA;considerable amount of time for me to understand &lt;strong&gt;how poor they were in&#xA;basic algebraic calculations&lt;/strong&gt;. ... I had to learn by trial and error,&#xA;how much of elementary mathematics was taboo in the business calculus&#xA;course. Not at once I realized that I was lecturing about exponential&#xA;functions to students who were not required to know about geometrical&#xA;progressions. ... Another mistake made by me was to include a&#xA;trigonometrical function in a test problem. I could not imagine that&#xA;students who take &amp;quot;calculus&amp;quot; were not supposed to know trigonometry,&#xA;but it was the case.&lt;/p&gt;&#xA;&lt;p&gt;At one of my lectures of business calculus, when asked why I gave&#xA;problems unlike those in the book, I answered: &amp;quot;Because I want you to&#xA;know elementary mathematics.&amp;quot; I expected to convince students by this&#xA;answer. In Moscow a university student who was told that he or she did&#xA;not know elementary mathematics got ashamed and checked into the&#xA;matter immediately. Elementary mathematics was normally taught to&#xA;children who looked like children. Now imagine my astonishment when&#xA;right after my answer an imposing train of well-grown adults stood up&#xA;and tramped out. They decided (correctly) that &lt;strong&gt;they could graduate&#xA;from the university without knowing elementary mathematics&lt;/strong&gt;. And that&#xA;they would easily find a lecturer who would teach them from the text.&lt;/p&gt;&#xA;&lt;p&gt;The basic principle of market economy is that everybody tries to get&#xA;as much as possible paying as little as possible. Nothing wrong with&#xA;this when paying for goods. But some students apply the same rule to&#xA;learning: &lt;strong&gt;they seem to think that they BUY grades and PAY for them by&#xA;learning&lt;/strong&gt;. And they try to pay as little as possible.&lt;/p&gt;&#xA;&lt;p&gt;The advantages enjoyed by Americans are the results of real competence&#xA;and real efforts of previous generations, whose heritage is now&#xA;getting devalued as a results of the bureaucratic character of the&#xA;education system. And &lt;strong&gt;someday ignorant people with degrees and&#xA;diplomas may want power according to their papers rather than real&#xA;competence&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2022-08-22T16:51:42.640" LastActivityDate="2022-08-22T16:51:42.640" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17423" PostTypeId="1" CreationDate="2019-11-05T02:07:56.887" Score="3" ViewCount="428" Body="&lt;p&gt;In most of books on elementary algebra, intermediate algebra and college algebra, the degree of the non-zero polynomial &lt;span class=&quot;math-container&quot;&gt;$$f(x)=a_nx^n+\cdots a_1x+a_0$$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$a_n\neq 0$&lt;/span&gt; is defined to be &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I am not sure this is well defined. Namely, shall we prove to the students that &lt;em&gt;if the same &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; has another expression &lt;span class=&quot;math-container&quot;&gt;$b_mx^m+\cdots+b_1x+b_0$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$b_m\neq 0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$m=n$&lt;/span&gt;&lt;/em&gt; before we can actually &lt;strong&gt;define&lt;/strong&gt; the degree of the polynomial? Otherwise we say we are defining the degree but the definition is illogical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to be that at the level of elementary algebra, intermediate algebra and college algebra, this proof may be a bit too complicated to the students. On the other hand, without the proof the definition of the degree of polynomials is not even logically established.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="238" LastEditDate="2019-11-23T02:07:22.597" LastActivityDate="2019-11-23T02:07:22.597" Title="Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?" Tags="|undergraduate-education|algebra|definitions|polynomials|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17424" PostTypeId="2" ParentId="17399" CreationDate="2019-11-05T04:28:33.763" Score="1" Body="&lt;p&gt;Think of the naturals instead of the reals.  We learn to count around age four.  We learn a lot of properties of arithmetic, like addition, subtraction, multiplication, division, prime numbers, unique factorization, etc. without reference to any axioms.  This is just how numbers work.  Similarly we learn how reals work before we see a careful construction of the reals.  I saw and understood the proof that &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2$&lt;/span&gt; was irrational before I saw a construction of the reals.  In a sense the careful construction of the reals is wasted effort.  We wind up proving they behave as we already know they behave.  Having done that, we don't reference the construction any more, we get on with life proving things based on the properties (that used to be asserted, now are theorems) we know and love.&lt;/p&gt;&#xA;" OwnerUserId="12340" LastActivityDate="2019-11-05T04:28:33.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17425" PostTypeId="1" CreationDate="2019-11-05T06:28:42.733" Score="12" ViewCount="787" Body="&lt;p&gt;There are a lot of &quot;proofs&quot; of the identity &lt;span class=&quot;math-container&quot;&gt;$e^{ix}=\cos x+i\sin x$&lt;/span&gt; in textbooks, using either differential equations or power series. However, I find those proofs often misleading, because it appears to me that Euler's identity is just a beautiful &lt;em&gt;definition&lt;/em&gt; rather than a theorem. We have good reasons to define &lt;span class=&quot;math-container&quot;&gt;$e^{ix}$&lt;/span&gt; in this way, and those &quot;proofs&quot; tells us why we should define &lt;span class=&quot;math-container&quot;&gt;$e^{ix}$&lt;/span&gt; this way, but at the end of the day, this is just a definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be more precise: there are various ways to define &lt;span class=&quot;math-container&quot;&gt;$e^{ix}$&lt;/span&gt;. One very common definition is &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;e^z=1+z+\frac{z^2}{2!}+\ldots&#xA;$$&lt;/span&gt;&#xA;If we use this as a definition, then certainly we can &lt;strong&gt;prove&lt;/strong&gt; Euler's identity as a theorem. Alternatively, we can prove Euler's identity by assuming&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{de^z}{dz}=e^z,&#xA;$$&lt;/span&gt;&#xA;but this approach requires formal treatment of complex derivatives, holomorphic functions, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, what we are doing is just to prove one property of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; from another. So, why don't we accept  &lt;span class=&quot;math-container&quot;&gt;$e^{ix}=\cos x+i\sin x$&lt;/span&gt; as a definition in schools?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another reason why it might be seen a definition is that we are extending the &lt;strong&gt;definition&lt;/strong&gt; of &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="9339" LastEditDate="2019-11-05T06:35:12.523" LastActivityDate="2022-08-12T06:54:32.740" Title="Should Euler's formula $e^{ix}=\cos x+i\sin x$ be seen as a definition rather than something to prove?" Tags="|calculus|complex-numbers|" AnswerCount="6" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="17426" PostTypeId="2" ParentId="17425" CreationDate="2019-11-05T12:56:20.417" Score="16" Body="&lt;p&gt;I think it is worth acknowledging that there is a huge list of equivalent definitions of the complex exponential.  Any presentation should be explicit about the fact that choosing one definition over others is a choice of convenience, or taste, rather than being canonical.  Showing that all of these definitions are equivalent is the real work.  Here are some definitions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The unique entire function which agrees with the real exponential on the real axis (best to cover this after the identity theorem).&lt;/li&gt;&#xA;&lt;li&gt;The unique solution to the differential equation &lt;span class=&quot;math-container&quot;&gt;$f'(z) = f(z)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$f(0) = 1$&lt;/span&gt; .&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$e^z = \displaystyle \lim_{n \to \infty} \left(1+\frac{z}{n}\right)^n$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$e^z = 1+z+z^2/2!+z^3/3! + \dots$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$e^{x+iy} = e^x(\cos(y) + i\sin(y))$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There are some really beautiful interconnections between these definitions such as 3 just being Euler's Method applied to 2, 5 arising from 4, 4 arising from 1 and 2, etc.  I think this exploration is worthwhile, and shows the &quot;inevitability&quot; of the complex exponential.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I view definition 5. as the least motivated (if it is pulled out of a hat), but the easiest to &quot;work with&quot;.  So I might use several of the other definitions as &quot;motivation&quot;, then &quot;derive&quot; 5, and end up adopting 5 as the &quot;official definition&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also view the interplay between the complex exponential and the complex logarithm to be at the heart of an introduction to complex analysis.  Every monomial other than &lt;span class=&quot;math-container&quot;&gt;$z^{-1}$&lt;/span&gt; has an entire primitive, and so integration theory (in some sense) reduces to the study of moving around on the Riemann surface of the logarithm, which is really beautiful.  &lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2019-11-07T13:28:38.400" LastActivityDate="2019-11-07T13:28:38.400" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17427" PostTypeId="2" ParentId="17423" CreationDate="2019-11-05T13:52:35.900" Score="6" Body="&lt;p&gt;This: &quot;On the other hand, without the proof the definition of the degree of polynomials is not even logically established.&quot; is not quite right.  What is needed to establish the definition is the &lt;em&gt;fact&lt;/em&gt; that the degree is well-defined, but this fact is stated (implicitly) by stating the definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The expectation that all facts be &lt;em&gt;proven&lt;/em&gt; is out of place in a course at the level of elementary/intermediate/college algebra - it is neither useful nor appropriate to expect that all facts will be rigorously proven in courses at this level.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-11-05T13:52:35.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17428" PostTypeId="2" ParentId="17423" CreationDate="2019-11-05T19:15:03.007" Score="3" Body="&lt;p&gt;In general, I agree with @Henry Towsner on the fact that the proofs should not always be presented in an elementary course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I have to disagree on the implicit &quot;well-definition property&quot; of any definition. Such a definition would require that some sort of uniqueness property has been proved, which cannot - or should not - be done prior to the definition of &lt;em&gt;the&lt;/em&gt; degree of a polynomial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if you want to find a &quot;balance&quot; between formal and school mathematics, why not use the following rephrasing of the definition you proposed:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A polynomial &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; is said to be of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$n\in\mathbb{N}$&lt;/span&gt; if it is non-trivial if it can be represented in the form:&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$p(x)=a_n x^n+\ldots+a_1x+a_0,\ a_n\neq0.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, you may avoid uniqueness which could be either discussed with the class or be taken for granted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.:&lt;/strong&gt; In case you have talked about equality between polynomials, proving that a polynomial's degree is unique is almost obvious for a student.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2019-11-05T19:15:03.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17429" PostTypeId="2" ParentId="17425" CreationDate="2019-11-05T19:38:26.803" Score="6" Body="&lt;p&gt;I'll put out there that it might not be helpful to have this as a definition because it wasn't discovered in that way.  That is to say, &lt;a href=&quot;http://eulerarchive.maa.org/hedi/HEDI-2007-08.pdf&quot; rel=&quot;noreferrer&quot;&gt;Euler discovered this formula&lt;/a&gt; by clever manipulation of &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt; in ways that might not quite pass muster today in terms of rigor, but were still pretty cool.  According to the linked article, it used 3. in Steven's answer; apparently Newton was aware of the Taylor series for &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; as early as the 1660's.  To call this definition unmotivated (assuming you aren't using the other ones yet in a given course) is an understatement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So while you &lt;em&gt;could&lt;/em&gt; do something like you propose, and while it probably is pedagogically useful to use it as a provisional definition the first time you introduce it, I would beware saying, &quot;And look, all the usual properties can be proved from this&quot; rather than saying later in a course, &quot;Now let's recall a more annoying, but good, definition of &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; from a year or two ago, and show that our intuitive formula for &lt;span class=&quot;math-container&quot;&gt;$e^{ix}$&lt;/span&gt; actually does hold using that definition.&quot;&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-11-05T19:38:26.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17430" PostTypeId="2" ParentId="17425" CreationDate="2019-11-06T09:19:22.987" Score="5" Body="&lt;p&gt;A guiding principle for operations extension is the principle of permanence: &lt;strong&gt;a definition of an operation should be extended from a restricted domain to a wider one in such a way as to conserve the crucial properties of the operation&lt;/strong&gt;. For example, the crucial algebraic properties of exponentiation: a#(b+c) = a#b a#c, (a#b)#c = a#(bc) and (ab)#c = a#c b#c compels us to define the exponentiation in Z and Q in the usual way, e.g. a#(-n) = 1/a#n because a#n a#(-n) = a#(n-n) = a#0 = 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may say that you prove a#(-n) = 1/a#n by using the principle of permanence or you may say that you define a#(-n) = 1/a#n in accordance with the principle of permanence. I prefer the second formulation, but it really does not matter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To extend the exponentiation to R you should count continuity of exponentiation among its crucial properties and then define it on R in such a way as to conserve the crucial properties: its algebraic properties and the continuity. What you get is the usual exponentiation on R.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to extend it to C, these properties are not enough. They almost compel you to Euler’s formula. Namely, they compel you to the formula e#it = cos(ct) + i sin(ct) in which c can be any real constant. I proved that in &#xA;&lt;a href=&quot;https://www.researchgate.net/publication/267107387_Peacock&amp;#39;s_principle_and_Euler&amp;#39;s_equation&quot; rel=&quot;noreferrer&quot;&gt;this paper&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you take d(e#z)/e#z = e#z or d(e#it)/dt = ie#it  or e#z = 1 + z + z#2/2 + … as the property to be preserved you are compelled to c = 1 (i.e. to Euler’s formula) and all these moves are common, but I proved in the same paper that it is enough to take the differentiability of exponentiation as the property to be preserved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We may conclude: by the principle of permanence of the algebraic properties and the continuity we are almost compelled to Euler’s formula. We are definitely compelled to it if we extend the principle to the differentiability.&lt;/p&gt;&#xA;" OwnerUserId="10261" LastEditorUserId="1608" LastEditDate="2019-11-06T12:39:03.450" LastActivityDate="2019-11-06T12:39:03.450" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17431" PostTypeId="2" ParentId="17398" CreationDate="2019-11-06T16:56:25.617" Score="1" Body="&lt;p&gt;Here is a written explanation to accompany &lt;a href=&quot;https://matheducators.stackexchange.com/a/17401&quot;&gt;Nick C's excellent 'hand-waving' answer&lt;/a&gt;. =)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2+6 \quad = \quad 1+1 \quad + \quad 1+1+1+1+1+1$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; &amp;emsp; &amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$ = \quad 1+1+1+1+1+1+1+1$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;emsp; &amp;emsp; &amp;emsp; &lt;span class=&quot;math-container&quot;&gt;$ = \quad 1+1+1+1 \quad + \quad 1+1+1+1 \quad = \quad 4+4$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explain that adding the same numbers always gives the same result no matter which order you add them, so we can add the ones up in any way we like. Also explain that we define &lt;span class=&quot;math-container&quot;&gt;$2 = 1+1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4 = 1+1+1+1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$6 = 1+1+1+1+1+1$&lt;/span&gt;, so if we choose the right order to add the ones up, we can get either &lt;span class=&quot;math-container&quot;&gt;$2+6$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$4+4$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2019-11-06T16:56:25.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17432" PostTypeId="2" ParentId="17425" CreationDate="2019-11-06T18:42:30.323" Score="1" Body="&lt;p&gt;First, we start with those equations for &lt;a href=&quot;https://en.wikipedia.org/wiki/Sine#Series_definition&quot; rel=&quot;nofollow noreferrer&quot;&gt;sine&lt;/a&gt; and &lt;a href=&quot;https://blogs.ubc.ca/infiniteseriesmodule/units/unit-3-power-series/taylor-series/the-maclaurin-expansion-of-cosx/&quot; rel=&quot;nofollow noreferrer&quot;&gt;cosine&lt;/a&gt; with an infinite number of terms:&lt;/p&gt;&#xA;&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA; \sin x &amp;amp;= \left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \cdots \right) \\[8pt]&#xA; \cos x &amp;amp;= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) \\&#xA;\end{align}&lt;/span&gt;&#xA;&lt;p&gt;We also take &lt;a href=&quot;https://en.wikipedia.org/wiki/Exponential_function#Formal_definition&quot; rel=&quot;nofollow noreferrer&quot;&gt;a formula for the exponential function&lt;/a&gt; also with an infinite number of terms:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \exp x = e^{x} = \sum_{k = 0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;That &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt; is the exponential function. Both its domain and co-domain are real &lt;strong&gt;so far&lt;/strong&gt;. Thus, if you feed in a real number to that function, it will spit out another real number.&lt;/p&gt;&#xA;&lt;p&gt;Those series can be proved independently of the Euler's theorem, without relying on arbitrary-looking definitions.&lt;/p&gt;&#xA;&lt;p&gt;But, what happens if you feed the function a multiple of &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;? Let's make a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; that does that job:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ f(x) = \exp(ix) = e^{ix} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You feed in a real number to the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, it gives a non-real number to the &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt; function and spits out whatever the &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt; gives as an answer. So, this proceeds to:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ f(x) = \exp(ix) = e^{ix} = 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \cdots $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, considering that &lt;span class=&quot;math-container&quot;&gt;$i^2=-1$&lt;/span&gt;, that &lt;span class=&quot;math-container&quot;&gt;$i^3=-i$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$i^4=1$&lt;/span&gt;, this can be rewritten as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ f(x) = \exp(ix) = e^{ix} = 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + \frac{x^8}{8!} + \cdots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then, we split it into two sums, one without the &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;s and one with them:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ f(x) = \exp(ix) = e^{ix} = \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Replacing those power series with the sine and cosine power series:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ f(x) = \exp(ix) = e^{ix} = \cos x + i\sin x$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Or, simply:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$e^{ix} = \cos x + i\sin x$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Finally, what is &lt;span class=&quot;math-container&quot;&gt;$f(\pi)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align}&#xA;f(\pi) &amp;amp;= \exp(i\pi) = e^{i\pi} = \cos \pi + i\sin \pi\\&#xA;f(\pi) &amp;amp;= \exp(i\pi) = e^{i\pi} = -1 + 0i\\&#xA;f(\pi) &amp;amp;= \exp(i\pi) = e^{i\pi} = -1\\&#xA;\end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Or simply:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$e^{i\pi} = -1$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;BTW, this is exactly the &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler%27s_formula#Using_power_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;power series&lt;/a&gt; formula.&lt;/p&gt;&#xA;&lt;p&gt;About the definition, you could argue that the formula &lt;span class=&quot;math-container&quot;&gt;$i = \sqrt{-1}$&lt;/span&gt; is an arbitrary definition. However, the thing that actually is &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; exists independently of what we call it, how we describe it or how we define it. So, someone just gave the name &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; for it.&lt;/p&gt;&#xA;&lt;p&gt;Defining something is roughly saying that you give a name to something and a description of this thing. You could define &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; in many ways, and all of them would result in the same thing. You could define it as &lt;em&gt;&amp;quot;let's call &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; as the thing that multiplied by itself results in &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;, even if it does not actually exist&amp;quot;&lt;/em&gt;. Or, much more simply, you write the definition with &lt;span class=&quot;math-container&quot;&gt;$i = \sqrt{-1}$&lt;/span&gt;, where the variable to the left of the equals sign is the name that you are giving to something and the expression in the right side is the description of that thing.&lt;/p&gt;&#xA;&lt;p&gt;However, the case of &lt;span class=&quot;math-container&quot;&gt;$e^{ix} = \cos x + i\sin x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$e^{i\pi} = -1$&lt;/span&gt; is very different. They are not simple arbitrary definitions and were obtained mathematically from calculations. Defining something is very different from obtaining or inferring something.&lt;/p&gt;&#xA;&lt;p&gt;You could argue that the equals sign is ambiguous in this case since it doesn't show clearly if the equality is the result of an inference process or simply because someone used it to give a name to something. This is why a few people use &lt;span class=&quot;math-container&quot;&gt;$a := b$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$a \stackrel{\text{def}}{=} b$&lt;/span&gt; for the defining equals, but most people simply don't do that or rely on the textual context to tell the difference or simply don't care at all. Thus, the case of &lt;span class=&quot;math-container&quot;&gt;$i = \sqrt{-1}$&lt;/span&gt; is a definition, while &lt;span class=&quot;math-container&quot;&gt;$e^{ix} = \cos x + i\sin x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e^{i\pi} = -1$&lt;/span&gt; are the results of inference processes.&lt;/p&gt;&#xA;&lt;p&gt;Finally, this all shows that those formula can be obtained without needing to resort to integral calculus nor to derivatives nor to using some previously made-up definition of how to extend the exponential function into the &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; realm. All you need to have is the power-series formula for sine, cosine and exponential, which can be proved independently beforehand, all in the &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; domain. In fact, this formula tells us exactly how to extend the exponential function into the &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; realm without having to resort on something made-up. We just fed in a non-real number instead of a real one to the exponential formula and saw if it works and what the heck was the result.&lt;/p&gt;&#xA;" OwnerUserId="12985" LastEditorUserId="376" LastEditDate="2022-08-12T06:54:32.740" LastActivityDate="2022-08-12T06:54:32.740" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17433" PostTypeId="2" ParentId="17425" CreationDate="2019-11-07T04:22:29.163" Score="11" Body="&lt;p&gt;In my view Euler's Formula &lt;span class=&quot;math-container&quot;&gt;$e^{ix} = \cos(x)+ i \sin(x)$&lt;/span&gt; and the resulting extension ( this is (5.) in Steven Gubkin's excellent complex answer ) of &lt;span class=&quot;math-container&quot;&gt;$e^{x+iy} = e^x( \cos y + i \sin y)$&lt;/span&gt; is not a good definition for the exponential because it is special to the context of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt;. In other algebras the interplay between the exponential function and ordinary exponential on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; is much removed from Euler's Formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast, if we use the series definition of &lt;span class=&quot;math-container&quot;&gt;$e^z = 1+z+\frac{1}{2}z^2+ \cdots $&lt;/span&gt; then we can pretty easily prove all the usual features of the exponential function hold as a consequence of power series calculus (which makes sense for commutative real unital associative algebras of finite dimension). In particular,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \begin{align*}&#xA;e^0 &amp;amp;= 1\\&#xA;e^z e^w &amp;amp;= e^{z+w}\\&#xA;(e^z)^{-1} &amp;amp;= e^{-z}\\&#xA;\frac{d}{dz}e^z &amp;amp;= e^z &#xA;\end{align*}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm pretty sure these can also be extended to the noncommutative case modulo the law of exponents (see BCH relation). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just to make the point more explicitly, let me show what the analog of Euler's Formula is for several of my favorite low dimensional novel algebras:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Hyperbolic Numbers&lt;/strong&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathcal{H} = \mathbb{R} \oplus j \mathbb{R}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$j^2=1$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;-maginary exponential (sorry, you find a better term) given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ e^{jx} = \cosh(x)+j \sinh(x) $$&lt;/span&gt;&#xA;For fun, notice &#xA;&lt;span class=&quot;math-container&quot;&gt;$$e^{-jx}e^{jx} = 1 = ( \cosh(x)-j \sinh(x) )( \cosh(x)+j \sinh(x) ) = \cosh^2(x)-\sinh^2(x)$$&lt;/span&gt; &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Three Hyperbolic Numbers&lt;/strong&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathcal{H}_3 = \mathbb{R} \oplus k \mathbb{R} \oplus k^2 \mathbb{R}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$k^3=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\zeta = x+ky+k^2z$&lt;/span&gt; is a typical three hyperbolic variable. Here:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ e^{kx} = l(x)+km(x)+k^2c(x) $$&lt;/span&gt;&#xA;where by my personal preference I call &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; the Larry function, &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; the Moe function and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; the Curly function. These can be defined by series:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ l(x) = 1+x^3/3!+x^6/6!+\cdots $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ m(x) = x+x^4/4!+x^7/7!+\cdots, $$&lt;/span&gt; &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ c(x) = x^2/2+x^5/5!+x^8/8!+\cdots. $$&lt;/span&gt; &#xA;One of my students (Daniel Freese) noticed that there is an analog of the Pythagorean Identity for sine and cosine which constrains l,m,c as follows (my students refuse to use my three stooges terminology for some reason)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ l^3+m^3+c^3-3lmc = 1 $$&lt;/span&gt;&#xA;In fact, there is such an identity for each real algebra generated by some unit &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$K^n = \pm 1$&lt;/span&gt;. This general result reproduces the usual result for sine, cosine and cosh and sinh as well as the 3-stooges identity above and other much uglier relations between the component functions of &lt;span class=&quot;math-container&quot;&gt;$e^{Kx}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;An algebraist would complain that I already gave this answer, but, they're wrong. The &lt;strong&gt;direct product algebra&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$\mathcal{A} = \mathbb{R} \times \mathbb{R}$&lt;/span&gt; is funny in that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (a,b)(c,d) = (ac,bd) $$&lt;/span&gt;&#xA;makes &lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt; the unity of the algebra. So, this looks a bit different, here:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ e^{(1,0)x} = (e^x, 0) \qquad \&amp;amp; \qquad e^{(0,1)y} = (0,e^y) $$&lt;/span&gt;&#xA;So, the analog of the complex exponential is &lt;span class=&quot;math-container&quot;&gt;$e^{(x,y)} = (e^x,e^y)$&lt;/span&gt;. There is an isomorphism between &lt;span class=&quot;math-container&quot;&gt;$\mathcal{H}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathcal{A}$&lt;/span&gt; which can be used to connect this exponential to the one I wrote with cosh and sinh. In fact, this is another really cool aspect of the exponential; it plays nice with algebra isomorphisms.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Dual Numbers&lt;/strong&gt; here &lt;span class=&quot;math-container&quot;&gt;$\Gamma = \mathbb{R}\oplus \varepsilon \mathbb{R}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\varepsilon^2=0$&lt;/span&gt;. In this case the exponential is suprisingly simple:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ e^{\varepsilon x} = 1+ \varepsilon x $$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I've written a few papers about this sort of math and they're on the ArXiV. Perhaps some day I will find a journal where they may also find a home. My student (Nathan BeDell) also wrote a couple papers exploring what you might call precalculus for algebras. &lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="117" LastEditDate="2019-11-07T17:13:14.797" LastActivityDate="2019-11-07T17:13:14.797" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="17435" PostTypeId="2" ParentId="17425" CreationDate="2019-11-07T23:04:12.370" Score="3" Body="&lt;p&gt;The statement &lt;span class=&quot;math-container&quot;&gt;$e^{ix}=\cos x+i\sin x$&lt;/span&gt; can only be proven if the operation &lt;span class=&quot;math-container&quot;&gt;$e^z,z\in\mathbb C$&lt;/span&gt; is already defined, otherwise &lt;span class=&quot;math-container&quot;&gt;$e^{ix}$&lt;/span&gt; makes no sense. Students are likely to see &lt;span class=&quot;math-container&quot;&gt;$e^{ix}=\cos x+i\sin x$&lt;/span&gt; before they see any other definition of exponentiating complex numbers, so the short answer to your question is &lt;strong&gt;yes&lt;/strong&gt;, this identity should be treated as a definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What you're really showing with any &quot;proof&quot; of this identity is that this is the only reasonable extension of the exponential function to the complex numbers. More precisely, it is the only extension of the exponential function that preserves certain properties that we expect the exponential function should have.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is analogous to claiming that &lt;span class=&quot;math-container&quot;&gt;$x^{1/2}=\sqrt{x}$&lt;/span&gt;. If we have only defined exponentiation &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n\in\mathbb N$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x^{1/2}$&lt;/span&gt; makes no sense, and we can't &quot;prove&quot; it is equal to &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt;. What we &lt;em&gt;can&lt;/em&gt; prove is that if we would like to extend exponentiation to rational numbers and also preserve the property &lt;span class=&quot;math-container&quot;&gt;$x^a\cdot x^b=x^{a+b}$&lt;/span&gt;, then we must have &lt;span class=&quot;math-container&quot;&gt;$x^{1/2}\cdot x^{1/2}=x^{1/2+1/2}=x$&lt;/span&gt;, and therefore we must have &lt;span class=&quot;math-container&quot;&gt;$x^{1/2}=\sqrt{x}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, if we want to extend the function &lt;span class=&quot;math-container&quot;&gt;$f(x)=e^x$&lt;/span&gt; to complex numbers, &lt;em&gt;and&lt;/em&gt; we want to preserve the Taylor expansion property &lt;span class=&quot;math-container&quot;&gt;$e^x=1+x+\frac{1}{2}x^2+\dots$&lt;/span&gt;, then the only possible definition turns out to be &lt;span class=&quot;math-container&quot;&gt;$e^{ix}=\cos x+i\sin x.$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="9649" LastActivityDate="2019-11-07T23:04:12.370" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17438" PostTypeId="2" ParentId="17398" CreationDate="2019-11-08T17:36:30.150" Score="2" Body="&lt;p&gt;None of the answers so far seem to give a name for the kind of concept in the inquiry. What you're asking is &quot;how to make 8&quot; or more generally &quot;&lt;strong&gt;making numbers&lt;/strong&gt;&quot;. See &lt;a href=&quot;https://www.amazon.co.uk/Making-Numbers-Using-manipulatives-arithmetic/dp/0198375611&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://hal.archives-ouvertes.fr/hal-01938938/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;strong&gt;Making numbers&lt;/strong&gt;&quot; is usually taught late kindergarten or early grade 1. In this case, you might ask the school how they teach it and then teach it to your child the same way that that is taught (that the school will soon, but maybe not very soon, depending on curriculum, teach your child)? I think this is usually done using marbles or blocks to show that 6 marbles + 2 marbles = 4 marbles + 4 marbles. Thus, the kindergarten student (or early grade 1 student) learns to understand the &lt;strong&gt;uniqueness&lt;/strong&gt; of the number 8 (the &lt;strong&gt;existence&lt;/strong&gt; of the number 8 is of course known to the student): namely that the &quot;8&quot; as the combination of 6 and 2 is the same as the &quot;8&quot; as the combination of 4 and 4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of course this understanding is visual via the marbles and not of course an understanding that a grade 2 or 3 student would have already acquired.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(And of course you don't necessarily use the word &quot;unique&quot;. Let me know if ever you think of a way to introduce the word &quot;unique&quot; in this context.)&lt;/p&gt;&#xA;" OwnerUserId="12899" LastEditorUserId="238" LastEditDate="2019-11-11T20:42:49.503" LastActivityDate="2019-11-11T20:42:49.503" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17439" PostTypeId="2" ParentId="17399" CreationDate="2019-11-09T05:54:53.957" Score="3" Body="&lt;p&gt;You seem to think that calculations are impossible without definitions of terms. That is not true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, mathematical definitions were deduced from mathematical practice. That is, the practice came first, the definitions later.&lt;/p&gt;&#xA;" OwnerDisplayName="user13012" LastActivityDate="2019-11-09T05:54:53.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17440" PostTypeId="2" ParentId="17423" CreationDate="2019-11-12T00:11:47.823" Score="1" Body="&lt;p&gt;Soon after introducing polynomials, students learn to add and subtract polynomials. Notice that if &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; are polynomials with degrees &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; respectively, and if &lt;span class=&quot;math-container&quot;&gt;$f(x)-g(x)=0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$n=m$&lt;/span&gt;. It follows that the degree of a polynomial is well-defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the proof that degree is well-defined is not difficult at all. On the other hand, students at this level are typically not able to see the need to establish that it is well-defined. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-11-12T00:11:47.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17441" PostTypeId="1" AcceptedAnswerId="17445" CreationDate="2019-11-12T02:58:32.470" Score="7" ViewCount="770" Body="&lt;p&gt;I read this is the mathematics educators stack exchange so hopefully this is the right place for this question.  I was curious what is your favorite math toys, manipulatives, math games, or tools to use in your math class.  Pretend your principal has said you could spend 1500 dollars and you teach secondary math. &lt;/p&gt;&#xA;" OwnerUserId="9155" LastEditorUserId="376" LastEditDate="2019-11-12T10:48:08.143" LastActivityDate="2019-12-14T22:13:43.523" Title="Favorite secondary math manipulatives?" Tags="|secondary-education|manipulatives|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17442" PostTypeId="2" ParentId="17441" CreationDate="2019-11-12T03:47:39.330" Score="5" Body="&lt;p&gt;I don't know if this is the right dollar amount, but I think getting access to a 3d printer and making some useful manipulatives &lt;em&gt;of your choice&lt;/em&gt; with it would be cool.  That is, if you are in a large district and have enough time to invest in trying them out - I don't think it's a one-off process!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, &lt;a href=&quot;http://www.3dprintmath.com&quot; rel=&quot;noreferrer&quot;&gt;Henry Segerman's website&lt;/a&gt; has some absolutely amazing examples - and the code is all available on &lt;a href=&quot;https://www.thingiverse.com/thing:1569912&quot; rel=&quot;noreferrer&quot;&gt;Thingiverse&lt;/a&gt;, which is the go-to site for 3d printing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And once you start seeing these, it's hard to stop looking for more!  There are a &lt;em&gt;lot&lt;/em&gt; of things out there, so you don't necessarily have to reinvent the wheel at all.  Again, the caveats are important - this is a significant investment that is only worth it if you have people who would do it.  But if you do, &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/10511970.2018.1445675&quot; rel=&quot;noreferrer&quot;&gt;the sky's the limit&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-11-12T03:47:39.330" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17443" PostTypeId="2" ParentId="17441" CreationDate="2019-11-13T08:28:25.473" Score="1" Body="&lt;p&gt;As a no-cost option that gives you so many capabilities besides the conventional black/white-board, I always use geogebra. Not only as a demonstration tool, but also as a functionality for the students to investigate almost anything in precalculus or algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By simply adding, e.g. some moving graphics you can easily explain why, say, &lt;span class=&quot;math-container&quot;&gt;$e^x\approx\left(1+\frac{x}{n}\right)^n$&lt;/span&gt; for large values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; or several other interesting but difficult to formally approach in secondary school mathematics.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2019-11-13T08:28:25.473" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17444" PostTypeId="2" ParentId="17441" CreationDate="2019-11-13T13:51:59.717" Score="2" Body="&lt;p&gt;I might think about some iconic objects of the past that excite interest, perhaps mainly in geometry.  Drafting table (probably $500 for a professional one, but you can check second hand). French curve as Rust mentioned.  Slide rules.  Parallel rulers and 10 point dividers (good for poking people when bored...I would know).  Sextant.  Abacus.  Towers of Hanoi puzzle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Different types of scientific graph paper along the walls or in framed pictures (and have a file of sheets as well): Regular, thick/thin for, semilog (and with decibel markings), log-log, triangular phase diagram, hex sheets, circular maneuvering boards.  Posters with different geographic projections shown and explained.  2-D space group Escher tile posters. Random tables, used as wallpaper (log, trig, Bessel, etc.).  Maybe some posters showing different functions/relations (doesn't need to be stuff they are learning...this is more for art and for intrigue): Bessel, epicycle, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Handleable models of platonic solids and sphere packing (AB, ABC) and point group examples. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Get stuff that is sturdy and old and looks like it has seen the blood and sweat of combat.  Part of the charm.  Dress the place up.  Make it look like a Hogwarts classroom.  All sorts of mysterious instruments of mathematical torture arrayed on the walls and shelves.&lt;/p&gt;&#xA;" OwnerUserId="13023" LastEditorUserId="13023" LastEditDate="2019-11-13T13:57:54.283" LastActivityDate="2019-11-13T13:57:54.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17445" PostTypeId="2" ParentId="17441" CreationDate="2019-11-13T17:05:35.537" Score="4" Body="&lt;p&gt;Secondary math is a pretty broad subject for something as individualized as the use of manipulatives to enhance teaching, but this is a list of things I'd love to have if I were teaching every subject in the same year for some reason.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A two-pan balance scale and a bunch of clearly-labeled weights&lt;/strong&gt; (Algebra I) - I feel that it can be really evocative how an equation like &lt;span class=&quot;math-container&quot;&gt;$3x+4=16$&lt;/span&gt; can lead to &lt;span class=&quot;math-container&quot;&gt;$3x=12$&lt;/span&gt; and from there to &lt;span class=&quot;math-container&quot;&gt;$x=4$&lt;/span&gt; if you can manipulate labeled and unlabeled weights on a two-pan balance.  The notion that doing the same thing to both sides to an equation is really brought home by seeing that that's what it takes for a balance to remain ... balanced.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A wall-sized slide rule&lt;/strong&gt; (Algebra II) - Logarithms had a clearly useful purpose fifty years ago that helped us to put humans in space.  Slide rules bring that home.  One of the sad stories of my teaching life is that my mother was at an estate sale and saw a giant promotional slide rule that was on sale for \$50.  (Apparently, in the old days, if you bought a classroom full of handheld slide rules, you'd get a wall model for free.)  My mom thought it looked cool but didn't buy it for me.  Those things are usually $500, and I'd love to have one.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A collection of 3D shapes&lt;/strong&gt; (Geometry) - This one at least is affordable, and something I actually have in my room.  They're a set of plexiglass polyhedra like a cube, rectangular prism, pyramid, cylinder, cone, hemisphere, and so on.  There is a little hole in each one and they tend to share radii and heights when it would be convenient to use water to show that, for instance, the volume of a cone is &lt;span class=&quot;math-container&quot;&gt;$\frac13$&lt;/span&gt; the volume of a cylinder with the same height and radius.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A similar triangle &quot;convincer&quot;&lt;/strong&gt; (Geometry) - I've never seen one of these before, but I'd really like my father to make a simple two piece jigsaw puzzle for me.  (A 3-d printer would probably rock at this too. One of the key triangle similarity proofs is that a right triangle is split into two similar triangles by the altitude to the hypotenuse. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SUVyM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SUVyM.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I think it's hard for students to see because the common angle has to be &quot;reflected&quot;.  With manipulatives, it's easy to flip the piece over and see that it fits all of the angles and is the same size.  I've never seen one of these for sale, but again I'd appreciate having it in my room.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="238" LastEditDate="2019-12-14T22:13:43.523" LastActivityDate="2019-12-14T22:13:43.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17446" PostTypeId="2" ParentId="17423" CreationDate="2019-11-13T18:42:49.437" Score="14" Body="&lt;p&gt;I'm going to rewrite this answer to clarify what I think the issue is. I think the OP is imagining a different definition of the ring &lt;span class=&quot;math-container&quot;&gt;$k[x]$&lt;/span&gt; than most answerers are. Here are two reasonable definitions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$k[x]$&lt;/span&gt; is the ring of formal expressions of the form &lt;span class=&quot;math-container&quot;&gt;$\sum_{j=0}^{\infty} p_j x^j$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$p_j \in k$&lt;/span&gt; and we require that &lt;span class=&quot;math-container&quot;&gt;$p_j$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; sufficiently large. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Polynomial_ring&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; for the details.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let's define &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Poly}(k)$&lt;/span&gt; to be the ring of functions &lt;span class=&quot;math-container&quot;&gt;$k \to k$&lt;/span&gt; which are of the form &lt;span class=&quot;math-container&quot;&gt;$x \mapsto \sum_{j=0}^d p_j x^j$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$p_j \in k$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In any book on modern algebra which is sophisticated enough to have careful definitions (Herstein, Artin, Dummit and Foote...), the definition of &lt;span class=&quot;math-container&quot;&gt;$k[x]$&lt;/span&gt; will be something like the first one. With this definition, it is very clear that degree is well defined because equality of polynomials is defined to mean equality of coefficients.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the OP talking about a polynomial having two different representations makes me think that he or she is imagining a definition where that might be possible, and Definition 2 strikes me as the most likely. Indeed, if &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is the field with &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; elements, then &lt;span class=&quot;math-container&quot;&gt;$x^q$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; represent the same function in &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Poly}(k)$&lt;/span&gt;, even though one is a polynomial of degree &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; and the other has degree &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. This is one of the reasons that Definition 1 is the standard definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a course which is sophisticated enough to have careful definitions, it would be appropriate to prove that the obvious map &lt;span class=&quot;math-container&quot;&gt;$k[x] \to \mathrm{Poly}(k)$&lt;/span&gt; is an isomorphism for &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; infinite. I could also imagine a course where the ground field is always &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; and one might prove that &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}[x] \cong \mathrm{Poly}(\mathbb{R})$&lt;/span&gt; without bringing up that finite fields exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But we don't have to raise the issue when we define degree, because Definition 1 is the standard definition of &lt;span class=&quot;math-container&quot;&gt;$k[x]$&lt;/span&gt; and this avoids the issue. Also, as Henry Townser says, many courses are at too low a level to reach this level of precision.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2019-11-14T16:52:02.103" LastActivityDate="2019-11-14T16:52:02.103" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17447" PostTypeId="2" ParentId="17416" CreationDate="2019-11-14T03:51:13.320" Score="1" Body="&lt;p&gt;Interesting integral. &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ f(w)=\frac{\cos(wx)}{1+w^2} $$&lt;/span&gt;&#xA;has poles at &lt;span class=&quot;math-container&quot;&gt;$w = \pm i$&lt;/span&gt;. The residue at &lt;span class=&quot;math-container&quot;&gt;$w=i$&lt;/span&gt; is given as follows:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \text{Res}_{w=i}\left(\frac{\cos(wx)}{1+w^2}\right) = \text{Res}_{w=i}\left(\frac{\cos(wx)}{(w+i)(w-i)}\right) = \frac{\cos(ix)}{i+i} = \frac{\cosh(x)}{2i} $$&lt;/span&gt;&#xA;So, if I use the usual half-circle contour &lt;span class=&quot;math-container&quot;&gt;$C = [-R,R] \cup C_R^+$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$[-R,R]$&lt;/span&gt; is the line-segment on the real-axis connecting &lt;span class=&quot;math-container&quot;&gt;$w=-R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$w=R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C_R^+$&lt;/span&gt; is the CCW-oriented half-circle &lt;span class=&quot;math-container&quot;&gt;$|w|=R$&lt;/span&gt; which goes from &lt;span class=&quot;math-container&quot;&gt;$w=R$&lt;/span&gt; back to &lt;span class=&quot;math-container&quot;&gt;$w=-R$&lt;/span&gt; then we note the Residue Theorem indicates:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$  \frac{2\pi i\cosh(x)}{2i} = \int_{C} \frac{\cos(wx)dw}{1+w^2} = \int_{[-R,R]} \frac{\cos(wx)dw}{1+w^2}+\int_{C_R^+} \frac{\cos(wx)dw}{1+w^2}$$&lt;/span&gt;&#xA;as &lt;span class=&quot;math-container&quot;&gt;$R \rightarrow \infty$&lt;/span&gt; you can &lt;strong&gt;not&lt;/strong&gt; argue the half-circle integral vanishes as Paul Garret pointed out... hence I do &lt;strong&gt;not&lt;/strong&gt; get to conclude&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \pi \cosh(x) = \int_{-\infty}^{\infty} \frac{\cos(wx)dw}{1+w^2} $$&lt;/span&gt;&#xA;because the half-circle integral does not nicely simplify in the limit. &lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2019-11-16T04:09:17.553" LastActivityDate="2019-11-16T04:09:17.553" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17448" PostTypeId="1" AcceptedAnswerId="17449" CreationDate="2019-11-14T17:51:35.317" Score="10" ViewCount="1291" Body="&lt;p&gt;My school is hyped about the promise of co-curricular education and they are giving  the math and science teachers paid days off to develop lesson plans that synergize our learning goals.  I'm on board with this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that I'm teaching geometry and my science grade-level peer is teaching chemistry.  I'm confident that she could use my help reminding students about the kind of algebra that could help with balancing equations or doing stochiometry, and statistics refreshers would probably also be awesome.  But I can't think of much of anything that would use plane geometry on the Euclidean or Cartesian plane.  And while my experience of chemistry is several paradigm shifts ago, I can't imagine what she would teach that would help my students get better at proofs or constructions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have experience or references that could help me to bring some productive ideas to our collaborations?&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-11-16T14:06:15.827" Title="Co-curricular lessons between geometry and chemistry?" Tags="|secondary-education|geometry|curriculum|physical-sciences|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17449" PostTypeId="2" ParentId="17448" CreationDate="2019-11-14T21:20:47.920" Score="19" Body="&lt;p&gt;One angle you could look at is molecular geometry. Not really my subject area but a couple of examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Organic molecules can have different chiralities. That means that while one is the mirror image of another you cant rotate one molecule to the other. The reasons for this are pretty deep mathematically, but chemically give rise to interesting things as well: for example the smell of lemon and orange are chiral opposites. &lt;a href=&quot;https://en.wikipedia.org/wiki/Chirality&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Chirality&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bond structures restrict the geometry of molecules. As an example, can get them to make Buckyballs and prove that carbon structures have to take certain forms &lt;a href=&quot;https://en.wikipedia.org/wiki/Buckminsterfullerene&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Buckminsterfullerene&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Geometry of crystalline structures. Combining the two above, did you know that we've classified all of the regular tessellations? What does that say about solid structures like crystals made out of different compounds? Of course, the 2D and 3D versions are different but you can start with 2D (we have vertices that can attach to 3 edges and vertices that can attach to 4, what kinds of regular crystals can we make?) and then extend to 3d. On the chemistry side, the and specific molecules determines the material properties of the solid. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Finally, there's something about bond angles, but that's almost more physicsy. This sounds like a really interesting area to mine though! &lt;/p&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="9019" LastEditDate="2019-11-15T21:52:17.833" LastActivityDate="2019-11-15T21:52:17.833" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17450" PostTypeId="2" ParentId="17448" CreationDate="2019-11-15T01:34:18.387" Score="4" Body="&lt;p&gt;This is perhaps more molecular biology than it is chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are some accessible planar geometric questions suggested by the &#xA;&lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/9541869&quot; rel=&quot;nofollow noreferrer&quot;&gt;H-P (hydrophobic-hydrophilic) model of protein (amino acid) folding&lt;/a&gt;, &#xA;which could&#xA;be explored with simple manipulatives (such as K'nex).&#xA;For example, which proteins in this model have a unique minimum energy&#xA;folding?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/V0Lnz.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/V0Lnz.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Figure from &lt;a href=&quot;http://www.gfalop.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometric Folding Algorithms: Linkages, Origami, Polyhedra&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-11-15T01:34:18.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17451" PostTypeId="2" ParentId="17416" CreationDate="2019-11-15T07:10:23.907" Score="1" Body="&lt;p&gt;It's certainly beneficial for students (even if thy don't like maths) to be convinced that some manipulations are only valid under some conditions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe the following example is both significant and short enough for a rapid illustration:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Derivation under the integral sign is (&quot;formally&quot;):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{d}{dt} \int  f(x,t) dx = \int \frac{\partial f(x,t)}{\partial t} dx&#xA;$$&lt;/span&gt;&#xA;Derivation is essentially a limit:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\partial f(x,t)}{\partial t} = \lim_{h\rightarrow 0}\frac{ f(x,t+h)-f(x,t)}{h} &#xA;$$&lt;/span&gt;&#xA;Derivation under the integral sign consists of permuting the &quot;limit&quot; and &quot;integration&quot; operations:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim \int = \int \lim&#xA;$$&lt;/span&gt;&#xA;However this operation is not innocuous and is only valid under certain conditions, a counterexample being:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{t\rightarrow 0}\int_0^\infty te^{-tx}dx=1\neq 0 = \int_0^\infty \lim_{t\rightarrow 0}te^{-tx}dx&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="13027" LastEditorUserId="13027" LastEditDate="2019-11-15T15:17:31.063" LastActivityDate="2019-11-15T15:17:31.063" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17452" PostTypeId="2" ParentId="17448" CreationDate="2019-11-15T17:18:50.317" Score="6" Body="&lt;p&gt;Pick your battles.  Don't expect to have synergy in every place.  But where you do have synergy, exploit that, call it a win, and move on.  Concentrate on the partial fullness of the glass, not the partial emptyness.  For that matter, you don't have time to totally redesign each course from the ground up in a way new to man.  Nor do you want to screw up the basic stuff (in each course) that they need to learn no matter what.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;But certainly geometry of molecules is a NORMAL, MAJOR part of the chemistry course.  See chapter 10 (&quot;Covalent bonding&quot;) in 1980 Masterton, Stanitski, Slowinski.  Or the equivalent chapter in whichever text your chem colleague is using.  [It's about halfway throught the year, but so what.  Give you time to concentrate on your classes themselves, not try to start complicated joint immediately.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I suggest to add a small joint segment on point groups.  A small addition of content. I remember doing this as a lab in my AP chem class (without geometry collaboration) as the chem instructor didn't think our normal chem text covered it enough, but she liked it because she was an inorganic structural chemist.  Fortunately our lab text had a nice lab that covered the topic.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this topic will be descriptional (you're not going to have character tables for IR stretching modes...and nyet on teaching group theory).  But you can learn the basic symmetry operations (planes, rotations, inversions, etc.) and can build a few models with toothpicks and foam and/or diagnose a few common objects and molecules.  At that point, you declare victory.  It's a solid exposure which was taught a lot about an important topic to people unaware of the topic (steep part of the learning curve).  The kids will get it heavier, better in college inorganic chem class (if chem majors).  But it's enough for now. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you did an hour of lecture (probably joint) and then an hour of lab (joint if you want or just let the chem guy supervise the toothpickinig), that would cover it fine.  Kids would hand it written lab report with a few questions to answer, as normal for chem labs.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;These two should be enough to declare victory, not screw up your classes, not confound the kids, and keep the admins happy.  And you and the lab geek can go have a beer afterwards and watch the footie.&lt;/p&gt;&#xA;" OwnerUserId="13033" LastEditorUserId="13033" LastEditDate="2019-11-15T17:24:19.373" LastActivityDate="2019-11-15T17:24:19.373" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17453" PostTypeId="2" ParentId="17084" CreationDate="2019-11-15T22:28:54.390" Score="2" Body="&lt;p&gt;I have been teaching trigonometry for many years and, surprisingly, did not know of this issue. I read the responses and the references therein, and I am putting my thoughts on this issue here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is how to approach trigonometry,  right triangles first and then unit circle; or circle first and then triangle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the answer depends on the level of preparation of students, in particular how comfortable are they with the concept of functions when the section on trigonometry starts. At that point we may encounter two category of students, at one extreme are the ones who struggle with basic  algebra, and at the other extreme there are students who are comfortable with a question like &quot;Let &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2+3x+7$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \ge -3/2$&lt;/span&gt;. Find &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt; as well as its domain and range&quot;.  Does the same book  or approach work for both groups? I doubt it. I  think my students will do better starting with a brief intro using the triangle approach, polar coordinates, general circle, and then the unit circle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just as I would not advocate teaching logarithms in precalculus as the area under a hyperbola, I will also hesitate to start with the unit circle. Unit circle is pretty and offers a clean definition but much of that neatness is lost to my students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are decided advantages and disadvantages of triangle approach (T) and circle approach (C). Here are some items I can recall.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1- In (T) the angle takes a back seat to the triangle. The angles are lost in the fray of so many pieces of the triangle. In (C) the focus is on the angle and the function aspect of trig entities comes to light. The student cannot miss the fact that the terminal side of the angle is rotating like the handle of a clock. The angle is prominent from the start, hence the function view becomes central. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2- In (T) you are not tied to a coordinate system, an angle need not be in the standard position, and you do not worry about an orientation for angles. But you are limited to acute angles. In (C) you get the general definition and the find the signs easily.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3- In (T) You may call your angle &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Note that this is highly confusing in the unit circle presentation. Most books even avoid writing  &lt;span class=&quot;math-container&quot;&gt;$y=\cos x$&lt;/span&gt;, when they introduce cosine. You are essentially asking students to get accustomed to the following picture. Compare that to how students are used to seeing  &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WzaMx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WzaMx.png&quot; alt=&quot;defining &lt;span class=&quot;math-container&quot;&gt;$\cos x $&lt;/span&gt;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4- In (T) no triangle side is artificially set to be 1. This makes it easier to do some basic trig identities and substitutions. Say, you want to do a trig substitution for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x^2+2x+5}$&lt;/span&gt;. This is less error prone for students when you avoid fractions and use a triangle with 'whole' parts.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;5- In (C) you can visually approximate trig functions because you set the denominator to be 1. This takes the mystery out of trig functions. Students can eyeball a diagram and have an estimate for a trig function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;6- It is important to emphasize that the input and output of most mathematical functions are dimensionless quantities, and as such, when these functions are used in applications, their input and output are ratios of similar physical quantities. One of the rare examples of this issue that comes up in precalculus is the Richter Scale formula for earthquakes  &lt;span class=&quot;math-container&quot;&gt;$\dfrac{R}{R_0}=Log \dfrac{E}{E_0}$&lt;/span&gt;. It would be attractive to simplify the formula by taking denominators to be 1 but then we risk leaving the wrong impression that one can take the logarithm of energy. In the same fashion, when teaching &lt;span class=&quot;math-container&quot;&gt;$y=\cos x$&lt;/span&gt; we need to emphasize that both &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are ratios. Otherwise students may erroneously think one can take cosine of distance,  time,  the physical angle itself, or any other physical quantity, and that the output is another physical quantity such as distance. This lack of familiarity with the issue of units can be seen in even advanced math students who take PDEs and evaluate some lengthy integrals related to Fourier series. If they incorrectly come to  an answer such as &lt;span class=&quot;math-container&quot;&gt;$L+L^3$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; is supposed to be some  length, they do not see the obvious unit mismatch. &lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2019-11-15T22:41:12.267" LastActivityDate="2019-11-15T22:41:12.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17454" PostTypeId="1" AcceptedAnswerId="17460" CreationDate="2019-11-15T23:49:21.750" Score="40" ViewCount="4952" Body="&lt;p&gt;I would like to impress college students (undergraduates in the U.S.)&#xA;that there is more to planar geometry beyond what they learned in high school. I would like to show them beautiful theorems they likely never encountered. They surely know that triangle angle&#xA;bisectors meet in a point, but quite likely they do not know&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Morley%27s_trisector_theorem&quot; rel=&quot;noreferrer&quot;&gt;Morley's trisection theorem&lt;/a&gt;:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/YkDDq.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YkDDq.gif&quot; alt=&quot;Zach&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;(Image from &lt;a href=&quot;http://blog.zacharyabel.com/2012/03/many-morley-triangles/&quot; rel=&quot;noreferrer&quot;&gt;Zach Abel&lt;/a&gt;.)&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;But this is, in some sense, my sole example. So:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. What are some other beautiful and perhaps surprising&#xA;  theorems in plane geometry that students would not likely have &#xA;  encountered in a high-school geometry curriculum?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;I am grateful for the many wonderful answers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Theorems of Ceva and Menelaus and Desargues&lt;/li&gt;&#xA;&lt;li&gt;Theorems of Ptolemy and Pappus&lt;/li&gt;&#xA;&lt;li&gt;Steiner's porism&lt;/li&gt;&#xA;&lt;li&gt;Van Aubel's theorem &lt;/li&gt;&#xA;&lt;li&gt;Circle power theorems&lt;/li&gt;&#xA;&lt;li&gt;Inversions in a circle&lt;/li&gt;&#xA;&lt;li&gt;Poncelet's porism &lt;/li&gt;&#xA;&lt;li&gt;The Euler line&lt;/li&gt;&#xA;&lt;li&gt;Pappus's Hexagon theorem&lt;/li&gt;&#xA;&lt;li&gt;Napoleon's theorem&lt;/li&gt;&#xA;&lt;li&gt;Chapple: altitudes concur&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-11-18T21:43:47.360" LastActivityDate="2023-12-05T18:00:57.140" Title="Beautiful planar geometry theorems not encountered in high school" Tags="|undergraduate-education|geometry|" AnswerCount="13" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17455" PostTypeId="2" ParentId="17454" CreationDate="2019-11-16T00:44:15.137" Score="16" Body="&lt;p&gt;Although I was shown the theorems of Ceva and Menelaus in high school, I don't think they're part of the standard curriculum. And I consider them beautiful, especially when considered together so that their similarities become apparent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd also be inclined to nominate Desargues's theorem, especially in view of the observation that the Desargues configuration exhibits the theorem 10 times over: You can start with any of the configuration's 10 points, consider the configuration's 3 lines through that point, find two triangles in the configuration with their vertices on those 3 lines, find (still in the configuration) the intersections of corresponding sides of the triangles, and see that those three intersection points lie on one of the configuration's lines.&lt;/p&gt;&#xA;" OwnerUserId="326" LastEditorUserId="238" LastEditDate="2019-11-23T02:19:20.223" LastActivityDate="2019-11-23T02:19:20.223" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17456" PostTypeId="2" ParentId="17423" CreationDate="2019-11-16T03:22:05.193" Score="2" Body="&lt;p&gt;I do not believe that this is a concern that would surface, or be worth surfacing, in the courses named by this question's title. In my reading, the question is analogous to worrying about whether you can ask about e.g. the thousands place of &lt;span class=&quot;math-container&quot;&gt;$7521$&lt;/span&gt;: To do so assumes that the base ten representation of &lt;span class=&quot;math-container&quot;&gt;$7521$&lt;/span&gt; is unique, or, in particular, that &quot;thousands place&quot; is well-defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[In fact, this cooked-up analog might be viewed as an example of the main question: Write coefficients as &lt;span class=&quot;math-container&quot;&gt;$a_k$&lt;/span&gt; and evaluate the polynomial at &lt;span class=&quot;math-container&quot;&gt;$x=10$&lt;/span&gt; to get a number represented in base ten, i.e., consider the polynomial &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{R} \rightarrow \mathbb{R}$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$x \mapsto 7x^3 + 5x^2 + 2x + 1$&lt;/span&gt;, and observe that &lt;span class=&quot;math-container&quot;&gt;$f(10) = 7521$&lt;/span&gt;, where the &quot;biggest place&quot; is the thousands place, i.e., &lt;span class=&quot;math-container&quot;&gt;$\deg f = 3$&lt;/span&gt;.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Anyway:&lt;/strong&gt; I don't imagine any of the named courses would need to specify that &quot;thousands place&quot; is well-defined; so, aiming for a type of rigor that ensures elementary/intermediate/college algebra students see &quot;degree&quot; is well-defined appears to me at least as unnecessary.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2019-11-16T03:22:05.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17457" PostTypeId="2" ParentId="17454" CreationDate="2019-11-16T05:22:41.400" Score="13" Body="&lt;p&gt;Not quite as beautiful as Morley’s trisection theorem, but here are two I never saw in high school but which I find beautiful:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ptolemy’s theorem, which says that for a cyclic quadrilateral with vertices on a circle, the sum of the products of opposite sides is equal to the product of the diagonals.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/5NfKd.jpg&quot; alt=&quot;diagram of Ptolemy’s Theorem, showing that for a cyclic quadrilateral in a circle, the sum of the products of opposite side lengths is equal to the product of the diagonals  &quot; title=&quot;diagram of Ptolemy’s Theorem, showing that for a cyclic quadrilateral in a circle, the sum of the products of opposite side lengths is equal to the product of the diagonals&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A theorem of Pappus, which shows how to construct the arithmetic mean, geometric mean and harmonic mean of two lengths in one semi-circular diagram.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/5JyU5.jpg&quot; alt=&quot;diagram of a theorem of Pappus, showing the construction of the arithmetic, geometric and harmonic means in one figure&quot; title=&quot;diagram of a theorem of Pappus, showing the construction of the arithmetic, geometric and harmonic means in one figure&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-11-17T18:24:27.997" LastActivityDate="2019-11-17T18:24:27.997" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17458" PostTypeId="2" ParentId="17398" CreationDate="2019-11-16T07:36:46.193" Score="1" Body="&lt;p&gt;Start with two equations that are the same in every way, like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 + 2 = 4 and 2 + 2 = 4&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, add &quot;4&quot; more to each of the two equations, one at a time, so you'll get 8 in each equation. But do it different ways in each of the two equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1.) Add all of the &quot;4&quot; more to the second &quot;2&quot; in the first equation: 2 + 6 = 8 (and 2 + 2 = 4)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2.) Now split up the &quot;4&quot; you're adding to the second equation into two &quot;2's&quot; and add one &quot;2&quot; to each &quot;2&quot; in the second equation: (2 + 6 = 8 and) 4 + 4 = 8&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is similar to the first answer someone gave above, but without using fingers. I explained it like this (a similar question - not this identical question) years ago to my five year old son and again to my four year old niece.&lt;/p&gt;&#xA;" OwnerUserId="13034" LastActivityDate="2019-11-16T07:36:46.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17459" PostTypeId="2" ParentId="17448" CreationDate="2019-11-16T14:06:15.827" Score="1" Body="&lt;p&gt;Are proofs still part of the geometry curriculum?  (Some of my math colleagues have mentioned they've been downplayed over the last decade or so.) A good Chemistry answer looks an awful lot like a good proof; same sort of logic flow.  (As does a good programming solution, if you're looking for another connection.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be a bit simple, but you could talk about why volumetric measurements get turned into linear ones.  (E.g. a graduated cylinder is designed to keep volume proportional to the height of the liquid.)  The measurement difficulties of accurately measuring spherical volumes compound errors.  (I.e. a 5% error on 100 mL measured in a G.C. means something way different than a 5% error on 100 mL on sphere's radius.)  So an activity could be to measure a small ball's volume (maybe by using string, maybe calipers), and then submerging it (if your colleagues have a large enough graduated cylinder) and discussing any discrepancies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long shot, but are logarithms part of your curriculum?  It might be a stretch for the Chem teacher (depending on how deep they get into acids and bases), but pH is all about logs.  (In case you don't know, pH = -Log([H+]), where [H+] means the concentration of Hydrogen ions in solution.)  (I would juuuust got to this in my Honors Chemistry classes, so I don't know if your colleague will.)  Plus, there is the pOH scale --&gt; pOH = -Log([OH-]).  For acids and bases near 25C, pH + pOH = 14. and [H+] * [OH-] = 10^-14.  (This might be really pushing for the Chem teacher, but who knows.)&lt;/p&gt;&#xA;" OwnerUserId="13037" LastActivityDate="2019-11-16T14:06:15.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17460" PostTypeId="2" ParentId="17454" CreationDate="2019-11-16T21:41:40.967" Score="39" Body="&lt;p&gt;Given that I had a good time coercing Mathematica to give me the pictures below I might as well promote my comment to an answer.&lt;/p&gt;&#xA;&lt;p&gt;My favorite such a result is the so called &lt;em&gt;Steiner's porism&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is a statement about the relation of two circles, one inside the other. Such as these two:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ANCCj.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ANCCj.png&quot; alt=&quot;two circles, one inside the other&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;We get a so called Steiner chain of circles from this by first drawing a circle that is tangent to both the given circles. We have a lot of freedom in choosing exactly where we draw this new circle (=the first circle of the chain). Then we add a fourth circle (=the second of the cain) that is tangent to the two circles we started with as well as the freshly added third circle. Next we draw a circle tangent to both the two circles we started with as well as the fourth circle, then we draw a circle tangent to circles number 1,2 and 5 etc. We end up with a picture like the one below, where a chain of seven circles, all red, all tangent to the two we started with.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/creJU.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/creJU.gif&quot; alt=&quot;a Steiner chain of length seven&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You observe that this chain has a very special property. The seventh circle of the chain happens to be tangent to the first circle of the chain. This says something very special about the relation of the two black circles we started with. Normally it does not happen, but I designed the picture very carefully to make it so.&lt;/p&gt;&#xA;&lt;p&gt;Initially you might expect that whether this miracle happens depends on where we drew that first circle of the chain. The somewhat surprising result is that this is not the case. Behold! Here's another chain of seven circles. This time all blue. And it also closes after the seventh circle in the chain has been added.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/sUlkf.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sUlkf.gif&quot; alt=&quot;another chain of length seven&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Steiner's porism states that this is always the case. The Steiner chain either always closes (with the last circle tangent to the first) after a fixed number of steps (depending on the relation of the two black circles), possibly only after a number of laps, or it never does. Irrespective of the choice of the position of the first circle of the chain. So for our two black circles the chain always closes after the seventh.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The proof is surprisingly simple. That is, after you have covered the basic properties of circle inversions (a theme suggested also by Steven Gubkin in the comments). Namely,&#xA;you surely agree that the result would be trivial, if the two black circles were concentric. The point is that we can always find an inversion that makes that happen. Furthermore, inversions map circles to circles (or lines as a degenerate case), and preserve tangency.&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2021-05-17T14:36:19.987" LastActivityDate="2021-05-17T14:36:19.987" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17461" PostTypeId="1" AcceptedAnswerId="17464" CreationDate="2019-11-17T11:20:44.933" Score="2" ViewCount="195" Body="&lt;p&gt;I'm starting a repetition with my students in 5th grade after they learned in elementary school how to sum up larger natural numbers (also 5- to 6-digits) by writing down that calculation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As introductory example that has a connection to their every day life, I search for a good exercise that just needs them to add up several natural numbers without being too easy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I thought about an amusement park where there is a roller coaster and the park needs to calculate the degree of capacity utilization but am lacking a formulation that is consistent and just needs them to add some numbers up, since a decrease of the number of uses forces a subtraction.&#xA;Do you have a better idea?&lt;/p&gt;&#xA;" OwnerUserId="10036" LastEditorUserId="238" LastEditDate="2019-11-18T15:36:17.950" LastActivityDate="2019-11-19T11:31:24.050" Title="Introductory exercise for the addition of large natural numbers" Tags="|student-motivation|examples|teaching|arithmetic-operations|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17464" PostTypeId="2" ParentId="17461" CreationDate="2019-11-17T22:36:28.960" Score="2" Body="&lt;p&gt;(Posting my comment as @ChrisCunningham suggests.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How about: The number of days to their next birthday? This could involve: number of days remaining in this month, plus the number of days in the next month, the next, ..., until the number of days in their birthday month. Or, similarly: The number of days from now until Christmas: &lt;span class=&quot;math-container&quot;&gt;$13$&lt;/span&gt; in November&#xA;plus &lt;span class=&quot;math-container&quot;&gt;$25$&lt;/span&gt; in December: &lt;span class=&quot;math-container&quot;&gt;$13+25=38$&lt;/span&gt;. Capitalizing on their excitement over special days. &lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-11-17T22:36:28.960" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17465" PostTypeId="2" ParentId="17393" CreationDate="2019-11-17T23:55:34.137" Score="4" Body="&lt;p&gt;Left to my devices, for case (3.) I'd use a trig. substitution or a u-substitution. First, put the expression into the following form:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int\frac{ A(x-\alpha)+B}{[(x-\alpha)^2+\beta^2]^n} \, dx$$&lt;/span&gt;&#xA;The first part isn't bad, nearly all books cover this part,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int \frac{ A(x-\alpha)\, dx}{[(x-\alpha)^2+\beta^2]^n} = \int\frac{A \, du}{2u^n} $$&lt;/span&gt;&#xA;using a &lt;span class=&quot;math-container&quot;&gt;$u=(x-\alpha)^2+\beta^2$&lt;/span&gt; substitution. So, that part isn't hard. On the other hand, setting &lt;span class=&quot;math-container&quot;&gt;$x-\alpha = \beta \tan \theta$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$(x-\alpha)^2+\beta^2 = \beta^2(\tan^2 \theta+1)=\beta^2 \sec^2 \theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dx = \beta \sec^2 \theta d \theta$&lt;/span&gt; thus&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int \frac{ Bdx}{[(x-\alpha)^2+\beta^2]^n} =  \int \frac{ B\beta \sec^2 \theta \, d \theta}{[\beta^2 \sec^2 \theta]^n} = B\beta^{2n-1} \int \cos^{2n-2} \theta \, d\theta.$$&lt;/span&gt;&#xA;In the case &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt; we just face &lt;span class=&quot;math-container&quot;&gt;$\int d\theta = \theta$&lt;/span&gt; and so we recover the usual arctangent solution that is often presented as something to memorize. If &lt;span class=&quot;math-container&quot;&gt;$n=2,3,...$&lt;/span&gt; then we can just face the integral of cosine using whichever method you prefer. Personally, I like direct battle via repeated substitution of &lt;span class=&quot;math-container&quot;&gt;$\cos^2 \theta = \frac{1}{2}(1+ \cos 2 \theta)$&lt;/span&gt;, but I understand others are partial to the iterative formulas for integration of powers of cosine. In summary, the integral is really not that bad, it's just tedious if the polynomial in the denominator is quartic or higher order. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this said, it's probably a bit much for a test unless it was belabored in the homework. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2019-11-17T23:55:34.137" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17466" PostTypeId="2" ParentId="17454" CreationDate="2019-11-18T13:19:49.767" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Van_Aubel%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Van Aubel's theorem&lt;/a&gt; comes to mind for me:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On each side of a planar quadrilateral (can be non-convex), construct a square (all external or all internal to the quadrilateral). If we construct line segments between the centers of opposite squares, then the two line segments are orthogonal to each other and have equal length.&lt;/p&gt;&#xA;" OwnerUserId="1372" LastEditorUserId="5794" LastEditDate="2019-11-19T19:54:27.193" LastActivityDate="2019-11-19T19:54:27.193" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17467" PostTypeId="2" ParentId="17393" CreationDate="2019-11-18T16:09:18.973" Score="1" Body="&lt;p&gt;I have 3 calculus books and all of them cover this.  Thomas Finney and Swokowski show the method that James Cook showed (complete the square and trig substitution).  Granville shows the reduction formula that Renfro referred to, saying that students should practice it a bit to be familiar with it and able to use it out of integral tables, but not required to memorize it.  (I prefer the Thomas/Finney explanation the best.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thomas Finney shows it well, but has only a single drill problem (only has 20 for all of partial fractions lesson).  Swokowski has a bit more drill, couple problems.  Granville has eight drill problems for this technique.  However, Thomas Finney and Swokowski also have &quot;end of chapter&quot; drill problem sections of over 100 problems, in addition to the &quot;end of lesson&quot; problems.  And a few more drill problems on this topic, mixed in there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't mean to sound harsh, but I just read the textbook and worked some homework.  It's not any different for a teacher versus a student in building skills.  I promise that if you work 10 problems, you'll know it cold.  This is a slightly esoteric technique, but that's it.  Also, it does give some practice in using other algebraic manipulations.  So it's got some integrative (pun intended) benefit.  And building manipulative muscles is not a bad thing for stronger STEM students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is nothing wrong with (a) not teaching this; (b) just teaching it and not testing it, maybe giving a single homework problem; or (c) teaching it and drilling it hard, until mastery.  It all depends on how much time you have to cover how much material, how strong your kids are, and what you want to emphasize.  But first learn it yourself.&lt;/p&gt;&#xA;" OwnerUserId="13049" LastEditorUserId="238" LastEditDate="2019-11-19T21:29:13.347" LastActivityDate="2019-11-19T21:29:13.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17468" PostTypeId="2" ParentId="17454" CreationDate="2019-11-18T17:11:35.360" Score="3" Body="&lt;p&gt;A triangle's altitudes concur.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that this theorem wasn't known until &lt;a href=&quot;https://www.cut-the-knot.org/triangle/Chapple.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chapple&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related to this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt;'s orthocentre, &lt;span class=&quot;math-container&quot;&gt;$A, B, C, H$&lt;/span&gt; are an orthocentric system, i.e. the straight line between any two of them is perpendicular to that between the other two; and each is the orthocentre of the triangle formed by the other three.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt;'s circumradius, &lt;span class=&quot;math-container&quot;&gt;$AH=2R\cos\angle BAC$&lt;/span&gt;. (And similarly for &lt;span class=&quot;math-container&quot;&gt;$BH$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$CH$&lt;/span&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="6983" LastActivityDate="2019-11-18T17:11:35.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17469" PostTypeId="2" ParentId="17454" CreationDate="2019-11-18T17:30:29.193" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pappus%27s_hexagon_theorem&quot; rel=&quot;noreferrer&quot;&gt;Pappus's Hexagon Theorem&lt;/a&gt; has a particularly nice diagram associated with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/zO9iV.png&quot; alt=&quot;Pappus&amp;#39;s Hexagon Theorem&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; can be any three points along the line &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, while &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; can be any three points along the line &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;. The theorem states that the points &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; at the intersections all lie along a common line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This theorem was known in antiquity. What is particularly cool is that we can apply the &lt;a href=&quot;https://en.wikipedia.org/wiki/Duality_(projective_geometry)&quot; rel=&quot;noreferrer&quot;&gt;duality principle&lt;/a&gt; of projective geometry (which allows us to exchange the roles of points and lines) to get a completely new theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RTakU.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RTakU.png&quot; alt=&quot;Dual to Pappus&amp;#39;s Theorem&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic principles of duality are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;line &lt;span class=&quot;math-container&quot;&gt;$\leftrightarrow$&lt;/span&gt; point&lt;/li&gt;&#xA;&lt;li&gt;intersection of two lines &lt;span class=&quot;math-container&quot;&gt;$\leftrightarrow$&lt;/span&gt; line through two points&lt;/li&gt;&#xA;&lt;li&gt;three lines meeting at a point &lt;span class=&quot;math-container&quot;&gt;$\leftrightarrow$&lt;/span&gt; three points along the same line&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the dual version of the theorem, the lines &lt;span class=&quot;math-container&quot;&gt;$GA$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$GB$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$GC$&lt;/span&gt; take on the roles of the points &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; in the original, while the lines &lt;span class=&quot;math-container&quot;&gt;$Ha$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$Hb$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Hc$&lt;/span&gt; take on the roles of the points &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;. The points &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; take on the roles of the lines &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; from the original: just as &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; all lay on the line &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; in the original, so do the lines &lt;span class=&quot;math-container&quot;&gt;$GA$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$GB$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$GC$&lt;/span&gt; meet at the point &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we continue to apply these principles to the first diagram, then we arrive at the second diagram. The dual theorem is that the three coloured lines &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; all meet at a single point &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt;, just as the three points &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; all lay on a common line &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; in the original diagram.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe (though I am not sure) that this second version was discovered in exactly this way, by applying duality to the first version over a thousand years after the first version was discovered.&lt;/p&gt;&#xA;" OwnerUserId="1294" LastActivityDate="2019-11-18T17:30:29.193" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17470" PostTypeId="2" ParentId="17454" CreationDate="2019-11-18T17:52:28.787" Score="4" Body="&lt;p&gt;I'd recommend literally anything in &lt;a href=&quot;https://amazon.co.uk/dp/0883856395&quot; rel=&quot;nofollow noreferrer&quot;&gt;either&lt;/a&gt; of &lt;a href=&quot;https://amazon.co.uk/dp/0883856190&quot; rel=&quot;nofollow noreferrer&quot;&gt;these&lt;/a&gt; books, but I'm also partial to &lt;a href=&quot;https://en.wikipedia.org/wiki/Napoleon%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Napoleon's theorem&lt;/a&gt;. In particular, &lt;a href=&quot;https://www.mathpages.com/home/kmath270/kmath270.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;we can prove&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The equilateral triangles erected outward on the sides &lt;span class=&quot;math-container&quot;&gt;$a,\,b,\,c$&lt;/span&gt; of an area-&lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; triangle have orthocetres at the vertices of an equilateral triangle of squared side length &lt;span class=&quot;math-container&quot;&gt;$\frac{a^2+b^2+c^2}{6}+\frac{2\Delta}{\sqrt{3}}$&lt;/span&gt;;&lt;/li&gt;&#xA;&lt;li&gt;If we erect inward instead, the resulting equilateral triangle has squared side length &lt;span class=&quot;math-container&quot;&gt;$\frac{a^2+b^2+c^2}{6}-\frac{2\Delta}{\sqrt{3}}$&lt;/span&gt;;&lt;/li&gt;&#xA;&lt;li&gt;The equilateral triangles with squared side lengths &lt;span class=&quot;math-container&quot;&gt;$\frac{a^2+b^2+c^2}{6}\pm\frac{2\Delta}{\sqrt{3}}$&lt;/span&gt; differ in area by &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The smaller such triangle having area &lt;span class=&quot;math-container&quot;&gt;$\ge0$&lt;/span&gt; is &lt;a href=&quot;https://en.wikipedia.org/wiki/Weitzenb%C3%B6ck%27s_inequality&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weitzenböck's inequality&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5794" LastActivityDate="2019-11-18T17:52:28.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17471" PostTypeId="2" ParentId="17454" CreationDate="2019-11-19T00:32:05.920" Score="11" Body="&lt;p&gt;For the &quot;beautiful&quot; aspect, you might be inspired by other geometry theorems as illustrated by the 1960's painter &lt;a href=&quot;https://americanhistory.si.edu/collections/object-groups/mathematical-paintings-of-crockett-johnson&quot; rel=&quot;noreferrer&quot;&gt;Crockett Johnson&lt;/a&gt;. For example, here is the &lt;a href=&quot;https://www.si.edu/object/nmah_694672&quot; rel=&quot;noreferrer&quot;&gt;Morley Triangle&lt;/a&gt;, which is the first place I ever saw that particular theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3hXjC.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3hXjC.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is his painting of &lt;a href=&quot;https://en.wikipedia.org/wiki/Pascal%27s_theorem&quot; rel=&quot;noreferrer&quot;&gt;Pascal's Hexagon&lt;/a&gt;: When the opposite sides of a irregular hexagon inscribed in a circle are extended, they meet in three points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ds9wk.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ds9wk.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cross-Ratio in a Conic:  &quot;French mathematician Michel Chasles introduced a result, which is the subject of this painting. He considered two points on a conic section (such as an ellipse) that were both linked to the same four other points on the conic. He found that lines crossing both pencils of rays had the same cross ratio. Moreover, a conic section could be characterized by its cross ratio.This opened up an entirely different way of describing conic sections.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mwg6o.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mwg6o.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-11-19T00:32:05.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17472" PostTypeId="1" CreationDate="2019-11-19T10:28:20.540" Score="1" ViewCount="373" Body="&lt;p&gt;Many years ago when learning complex maths we used complex maths as an example in the quadratic equation to find real roots. My nephew is struggling to deal with complex maths as his teacher is teaching it as an academic exercise. I am tutoring himn and my background is engineering so I prefer real examples and using complex maths to solve the quadratic equation for real roots seems to me to be a useful example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone recall any real roots that require the use of complex maths to calculate? Or did I imagine it?&lt;/p&gt;&#xA;" OwnerUserId="10006" LastEditorUserId="238" LastEditDate="2019-11-23T02:15:24.220" LastActivityDate="2019-11-23T02:15:24.220" Title="Quadratic equations using complex math but with no imaginary roots" Tags="|complex-numbers|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17473" PostTypeId="2" ParentId="17461" CreationDate="2019-11-19T11:31:24.050" Score="2" Body="&lt;p&gt;Another example is that if you have lets say &lt;span class=&quot;math-container&quot;&gt;$30$&lt;/span&gt; guests coming to your home, you want to buy cups for the tea or coffee or whatever but these cups come  in dozens, i.e. there is only bags of &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; cups, so how many bags you need? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They start for example &lt;span class=&quot;math-container&quot;&gt;$12+12=24$&lt;/span&gt; not enough cups, &lt;span class=&quot;math-container&quot;&gt;$12+12+12=36$&lt;/span&gt; now we have enough cups, so we need &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; bags.&lt;/p&gt;&#xA;" OwnerUserId="12486" LastActivityDate="2019-11-19T11:31:24.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17474" PostTypeId="2" ParentId="17472" CreationDate="2019-11-19T19:46:32.063" Score="2" Body="&lt;p&gt;(Quarter loaf and too long for a comment)&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In integration of rational functions, using method of partial fractions, you can often have individual terms that have denominators which are complex-rooted quadratics. So this is a touchpoint, maybe. Yet the integral itself is still a real valued function.  (Whether definite or indefinite integral.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Obviously many aspects of EE use complex numbers to show phase angle and the like.  You know ELI the ICE man.  3 phase y-delta all that.  A basic EE course of 1-2 semesters is standard for all engineers (even mechanical, etc.) Of course there are more complex (pun intended) aspects of EE courses that use complex algebra or even calculus (signal processing, spectrum analysis stuff).   &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Most engineers (of whatever slant, but definitely mechanical and EE) have to take a control systems class.  This ends up having at its core the solution of the second order constant coefficient diffyQ (with forcing function).  The solution of the homo part of this problem ends up using complex numbers in the underdamped case. (or sin, cos...but it is very useful to kind of &quot;see&quot; this problem both in terms of sin/cos or in terms of complex exponentials.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Note:  Maybe this helps your neph to feel better.  However, I still suspect it is going to be a hard slog for him.  The real value of the complex applications (mostly) comes from the very first chapter of a complex analysis text (where you revisit algebra two complex algebra, do the Argand diagram, solve for the square root of i, etc.).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, there are applications of the analysis part of complex analysis also, but they will not be needed for anything in the stereotypical undergrad engineering curriculum...are more for grad students.  And even here, I bet your instructor is not going to teach things in a &quot;how to solve integrals&quot; manner, but with a lot of emphasis on proofs.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the kid would have been better off in a complex variables (engineering emphasis) class than this theory ball-buster.  But even that would have been overkill for what he really needs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My best advice is to just tell him to go whole hog and do things how the teacher wants.  Fatalistically.  He can always try to construct meaning from it later on his own.  But if he struggles too much with that now, it will derail him.  Tell him to &quot;get proofy&quot;.  Camp out on that bastard's office doorstep for extra instruction help.  Ask why &quot;five times&quot; until he gets it.  Be PERSISTENT.  And do lots of homework.  Maybe find easier, transitional books if the text is also a ball buster.  Read the text AHEAD of lecture and work the problems (at least try to) ahead of lecture.  [Standard advice for mastering any subject, but especially needed to pull out all the study skills stops, since the class is more difficult than normal.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck to the young man.  He will need it.&lt;/p&gt;&#xA;" OwnerUserId="13049" LastEditorUserId="13049" LastEditDate="2019-11-19T19:54:20.143" LastActivityDate="2019-11-19T19:54:20.143" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17475" PostTypeId="1" CreationDate="2019-11-20T02:22:27.850" Score="10" ViewCount="159" Body="&lt;p&gt;I was talking recently with my daughters about non-planar graphs, like &lt;span class=&quot;math-container&quot;&gt;$K_{3,3}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$K_5$&lt;/span&gt;, and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg&quot; rel=&quot;noreferrer&quot;&gt;7 bridges of Königsberg&lt;/a&gt;. They got pretty interested in it, and seemed to catch on to the core ideas. Then we tried to dive into the &lt;a href=&quot;https://sketchfab.com/3d-models/3d-petersen-graph-f56321a6a30945feba7dd5699b91e30e&quot; rel=&quot;noreferrer&quot;&gt;Petersen graph&lt;/a&gt; and the concept of &lt;a href=&quot;https://en.wikipedia.org/wiki/Linkless_embedding&quot; rel=&quot;noreferrer&quot;&gt;linkless embedding&lt;/a&gt; in 3 dimensions. This wasn't so successful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea of planarity can be illustrated pretty well on paper, but in 3 dimensions, one needs some sort of modeling toy to show what we're talking about in terms of vertices and edges that link together in cycles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are face-oriented toys out there for exploring geometry in 3D, like &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B01NAACMU8&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;these&lt;/a&gt; which allow you to build certain polyhedra by attaching rigid triangular and square faces to each other edge-to-edge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also rigid edge-oriented toys, like &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B000BW5FSE&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt; (unavailable), which offer more freedom in arrangement of vertices and edges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But we want to explore graph theory in a way that's not limited to edges that are all &lt;strong&gt;straight&lt;/strong&gt; and all the same length. In fact, sometimes experimenting with different &quot;routes&quot; for an edge is key to learning principles of graph theory. What are some good options for this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B07KX7FMB9&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;This flexible rod play set&lt;/a&gt; would work, I suppose, though the joints supplied don't look very much like vertices. Something cheap would be nice. &lt;a href=&quot;https://www.science20.com/standup_physicist/quaternions_third_grader-77169&quot; rel=&quot;noreferrer&quot;&gt;Pipe cleaners and clay&lt;/a&gt; seem to be a popular approach, as in the picture below, but I wonder what kind of clay is stiff enough to maintain a 3D configuration of more than a few vertices. Maybe pipe cleaners should be stuck into little foam balls? Any better ideas? They don't have to be brightly-colored pieces that are marketed as toys.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/d3pcT.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/d3pcT.jpg&quot; alt=&quot;graph hypercomplex&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I moved this question from math SE on the advice of someone there.)&lt;/p&gt;&#xA;" OwnerUserId="13054" LastActivityDate="2019-11-20T02:22:27.850" Title="toys/manipulatives for exploring graph theory" Tags="|manipulatives|graph-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17476" PostTypeId="1" CreationDate="2019-11-20T14:59:00.760" Score="8" ViewCount="142" Body="&lt;p&gt;I would like to teach the &lt;a href=&quot;https://en.wikipedia.org/wiki/German_tank_problem&quot; rel=&quot;noreferrer&quot;&gt;German Tank Problem&lt;/a&gt; to bright students with little prerequisites. They are 16 year old high school students with a lot of maths competition experience, but no prior knowledge of statistics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody have a source that explains the reasoning behind the frequentist and bayesian approach in an accessible manner?&lt;/p&gt;&#xA;" OwnerUserId="13058" LastEditorUserId="13058" LastEditDate="2019-11-20T15:37:59.437" LastActivityDate="2019-11-20T15:37:59.437" Title="Good Source for German Tank Problem" Tags="|statistics|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17477" PostTypeId="2" ParentId="17472" CreationDate="2019-11-20T15:00:20.490" Score="6" Body="&lt;p&gt;Per request, I am expanding my comment into an answer. Given a quadratic equation, &lt;span class=&quot;math-container&quot;&gt;$ax^2+bx+c=0$&lt;/span&gt;, the quadratic formula &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$&lt;/span&gt;&#xA;will only involve complex numbers exactly when the discriminant &lt;span class=&quot;math-container&quot;&gt;$b^2-4ac&amp;lt;0$&lt;/span&gt;. In this case the two roots come in a conjugate pair and there are no real solutions. However, when &lt;span class=&quot;math-container&quot;&gt;$b^2-4ac \geq 0$&lt;/span&gt;, there is no need to invoke complex numbers. Solutions may involve square roots, but they are of positive numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if we wish to solve a &lt;em&gt;cubic&lt;/em&gt; equation, the corresponding cubic formula will involve complex numbers even when all of the roots are real. Let's look at Cardano's formula for the depressed cubic, &lt;span class=&quot;math-container&quot;&gt;$x^3+px+q=0$&lt;/span&gt;. (depressed=no quadratic term) In this case&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x = \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4}+\frac{p^3}{27}}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}$$&lt;/span&gt;&#xA;(See &lt;a href=&quot;https://www.encyclopediaofmath.org/index.php/Cardano_formula&quot; rel=&quot;noreferrer&quot;&gt;https://www.encyclopediaofmath.org/index.php/Cardano_formula&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the cubic discriminant &lt;span class=&quot;math-container&quot;&gt;$D=-27q^2-4p^3$&lt;/span&gt; is positive, the roots are real and distinct, but the computation involves complex numbers in the intermediary steps. (Look at the square roots.) If I recall correctly, this formula is the first occurrence of the complex numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an addition, complex numbers often show in the context of some kind of system which oscillates or rotates. It might be worthwhile for your nephew to look at examples of the form &lt;span class=&quot;math-container&quot;&gt;$i^n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ \frac{ i^n + (-i)^n}{2}$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n=0,1,2,\ldots$&lt;/span&gt; It would be good practice and illustrates an important point.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2019-11-20T15:00:20.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17479" PostTypeId="2" ParentId="17454" CreationDate="2019-11-20T21:06:59.513" Score="9" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;h3&gt;Fundamental theorem of transformational plane geometry&lt;/h3&gt;&#xA;&lt;p&gt;A transformation &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; is an isometry iff &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; can be expressed as a composition of three or fewer reflections.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;h3&gt;Laisant's theorem&lt;/h3&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gdOag.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gdOag.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Given a triangle &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt;, extend the two sides &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$AC$&lt;/span&gt; as necessary to point &lt;span class=&quot;math-container&quot;&gt;$A_c$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; at distance &lt;span class=&quot;math-container&quot;&gt;$|BC|$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C_b$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$AC$&lt;/span&gt; at distance &lt;span class=&quot;math-container&quot;&gt;$|BC|$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; such that both points are on the same side of &lt;span class=&quot;math-container&quot;&gt;$BC$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. Then the direction of the line &lt;span class=&quot;math-container&quot;&gt;$A_bA_c$&lt;/span&gt; is independent of &lt;span class=&quot;math-container&quot;&gt;$BC$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;h3&gt;Dandelin spheres&lt;/h3&gt;&#xA;&lt;p&gt;An ellipse is a plane section of a cone. It is possible  to fit one sphere into the cone to touch the plane, between the plane and the vertex, and another sphere to touch the plane and the cone on the other  side.&lt;/p&gt;&#xA;&lt;p&gt;Dandelin, a professor of mechanics at Liege University, proved that the  two spheres touch the ellipse at its foci, and that the directrices of the ellipse  are the lines in which the cutting plane meets the planes of the circles in which the spheres touch the cone.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nnPVm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nnPVm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;h3&gt;Eyeball theorem&lt;/h3&gt;&#xA;&lt;p&gt;The tangents to each of two circles from the centre of the other are drawn. Then the line segments &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$XY$&lt;/span&gt; are equal in length.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8Kd6i.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8Kd6i.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(See also &lt;a href=&quot;https://mathoverflow.net/q/152192/6094&quot;&gt;the MathOverflow &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; eyeball image&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;h3&gt;Platonic solids&lt;/h3&gt;&#xA;&lt;p&gt;There are only five convex regular polyhedra, known as the Platonic solids.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;h3&gt;Dijkstra's theorem&lt;/h3&gt;&#xA;&lt;p&gt;In any triangle &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$\mathrm{sgn}(\alpha+\beta-\gamma)=\mathrm{sgn}(a^2+b^2-c^2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here &lt;span class=&quot;math-container&quot;&gt;$\mathrm{sgn}$&lt;/span&gt; is the signum function. It is easy to deduce from the law of cosines but Dijkstra &lt;a href=&quot;https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD975.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;proved it&lt;/a&gt; more simply. The pythagorean theorem is a special case.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="501" LastEditorUserId="15884" LastEditDate="2023-12-05T18:00:57.140" LastActivityDate="2023-12-05T18:00:57.140" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17480" PostTypeId="1" AcceptedAnswerId="17484" CreationDate="2019-11-21T02:30:29.637" Score="1" ViewCount="116" Body="&lt;p&gt;My daughter’s 5th grade teacher gives some pretty impossible questions and I don’t think she understands the material she’s teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, this question has no context from previous questions and short of using integration or something off the wall contrived, I know of no other method than length * width * height to calculate volume of a rectangular prism. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: My daughter says they haven’t talked about physical methods like pouring water into the rectangular prism to measure the volume.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/cjLOc.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cjLOc.jpg&quot; alt=&quot;Homework question&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don’t think this is just me; I have a BS in Applied Math, a MS in Physics, I’m working on my PhD and I taught high school math for a couple years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How and to whom do I tactfully bring up the issues I see? She seems nice so I don’t want her to get a reprimand or anything, I just feel like the kids aren’t getting the opportunity they should be.&lt;/p&gt;&#xA;" OwnerUserId="13060" LastEditorUserId="13060" LastEditDate="2019-11-21T02:50:10.113" LastActivityDate="2019-11-21T14:15:35.250" Title="How to address opportunities for improvement with a teacher" Tags="|primary-education|examples|teaching|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17481" PostTypeId="1" CreationDate="2019-11-21T08:25:38.383" Score="11" ViewCount="3151" Body="&lt;p&gt;In Norway we will have a new national mathematics curriculum for all ages including high school beginning august 2020. A fundamental change is the new focus on so called algorithmic thinking. In practice this means that we are going to add the use of Python to explore and solve mathematical problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of the examples we (teachers) have been exposed to in courses are simply traditional types of problems that are shoehorned into programming exercises, even though they would be easier to solve in a traditional way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think we need to modify our intuition for what kinds of mathematical problems that will become relevant in this new setting. But it is difficult to break out of old ways of thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'm asking is, does anyone here know of (resources of) mathematical problems that are difficult to impossible to solve with &quot;traditional school mathematics&quot;, but relatively easy to handle with programming (even Excel)?&lt;/p&gt;&#xA;" OwnerUserId="1957" LastEditorUserId="238" LastEditDate="2019-11-22T21:20:36.953" LastActivityDate="2019-11-22T21:20:36.953" Title="Algorithmic thinking problems" Tags="|course-design|technology-in-education|examples|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17482" PostTypeId="2" ParentId="17481" CreationDate="2019-11-21T13:34:11.633" Score="11" Body="&lt;p&gt;One well-known source is &lt;a href=&quot;https://projecteuler.net/archives&quot; rel=&quot;noreferrer&quot;&gt;Project Euler&lt;/a&gt;.  The concept behind it is that each problem is mathematical and designed to be solved by an efficient algorithm on a &quot;normal&quot; computer in less than a minute.  The early problems are all extremely accessible.  As the problems go on, they become (in my mathematical opinion) far more esoteric from either a mathematical or algorithmic standpoint, but there are more than enough of the former problems to either provide or generate ideas that could satisfy a class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that they are generally not expecting problems will be solvable in Excel, but some of them certainly could be.  For instance, the second problem in the list asks for the sum of all of the Fibonacci numbers less than four million that are even.  It took me a little digging into the formula documentation, but I did that in Google Sheets.  ^_^  Python will be a completely excellent and authentic language for solving the problems -- you could even make a decision about whether to include things like number theory libraries or if you want to code your own library.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-11-21T13:34:11.633" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17483" PostTypeId="2" ParentId="17481" CreationDate="2019-11-21T13:45:21.983" Score="5" Body="&lt;p&gt;Challenging question! Two ideas.&#xA;(1) Calculate the Greatest Common Divisor of two natural numbers, not&#xA;so easily accomplished by hand on moderately large numbers.&#xA;The &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Euclidean_algorithm&quot; rel=&quot;noreferrer&quot;&gt;Euclidean algorithm&lt;/a&gt;&#xA;could be used to illustrate recursion/induction. Here is Python3 code:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;trace = True # True turns on tracing prints.&#xA;def GCD( a, b ):&#xA;    '''Returns the Greatest Common Divisor to a &amp;amp; b.&#xA;       Assumes a &amp;amp; b are both positive ints.&#xA;    '''&#xA;    if trace:&#xA;        print( '=&amp;gt;GCD(',a,',',b,')' )&#xA;&#xA;    if a == b:&#xA;        return a&#xA;    elif a &amp;gt; b:&#xA;        return GCD( a-b, b )&#xA;    elif b &amp;gt; a:&#xA;        return GCD( a, b-a )&#xA;############################&#xA;=&amp;gt;GCD( 24 , 116 )&#xA;=&amp;gt;GCD( 24 , 92 )&#xA;=&amp;gt;GCD( 24 , 68 )&#xA;=&amp;gt;GCD( 24 , 44 )&#xA;=&amp;gt;GCD( 24 , 20 )&#xA;=&amp;gt;GCD( 4 , 20 )&#xA;=&amp;gt;GCD( 4 , 16 )&#xA;=&amp;gt;GCD( 4 , 12 )&#xA;=&amp;gt;GCD( 4 , 8 )&#xA;=&amp;gt;GCD( 4 , 4 )&#xA;############################&#xA;4&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;(2) Any result that employs randomness. For example,&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Buffon%27s_needle_problem&quot; rel=&quot;noreferrer&quot;&gt;Buffon's Needle&lt;/a&gt;&#xA;approximation to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. Here is Python3 code:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;from math import sin,cos,pi&#xA;from random import uniform&#xA;&#xA;def CreateNeedle():&#xA;    '''Returns y-coords of endpts of unit-length needle'''&#xA;    ang = uniform( 0, 2*pi )&#xA;    x2,y2 = cos( ang ),sin( ang )&#xA;    # x1=0&#xA;    y1 = uniform(-1,1)&#xA;    y2 = y2 + y1&#xA;    return y1,y2&#xA;&#xA;def Buffon( n ):&#xA;    '''Throws n needles, counts hits across&#xA;       x-axis'''&#xA;    hits = 0&#xA;    for i in range( n ):&#xA;        y1,y2 = CreateNeedle( )&#xA;        if (y1&amp;lt;0 and y2&amp;gt;0) or (y2&amp;lt;0 and y1&amp;gt;0):&#xA;            hits += 1&#xA;    print( hits,'out of',n,':n/hits=',n/hits,'::',pi )&#xA;&#xA;Buffon( 1000000 )&#xA;############################&#xA;318405 out of 1000000 :n/hits= 3.1406541982694995 :: 3.141592653589793&#xA;############################&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-11-21T14:09:43.080" LastActivityDate="2019-11-21T14:09:43.080" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17484" PostTypeId="2" ParentId="17480" CreationDate="2019-11-21T14:02:03.443" Score="1" Body="&lt;p&gt;There are two questions here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The easier question is describing volume.  There are two separate ways to measure volume.  In metric terms, one standard measure would be liters and the other would be cubic meters.  They are related in metric: one liter is 1000 cubic centimeters.  Both of them (and their &quot;American&quot; counterparts) are extremely common and important to know about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other question is how to confront a teacher who isn't really on top of the material.  This looks like an example of inquiry-based education, where the students have to think about examples in their life where they measure volume and then think about it to generate their own knowledge instead of passively getting facts from the teacher.  I think it's a great technique and one that is rightfully gaining popularity in education.  The downside is that some teachers think that you can just hand out a worksheet and kick back while they fill it out.  In reality, teachers have to work even harder to engage individual learners to think for themselves if they aren't sure how to engage with a lesson, and probably not all teachers are trained in how to do that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly, your first approach should be to talk to the teacher.  Talking to parents is a positive on our year-end review!  You might get a sense from there of how high up the problem goes.  Maybe the math department at the school doesn't care, or the entire school, or the entire district, or the entire state.  And going to a principal with a solution is definitely better than going to a principal with a problem, if you and the teacher can figure out how to make things better but it requires extra training or resources or whatever.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2019-11-21T14:15:35.250" LastActivityDate="2019-11-21T14:15:35.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17485" PostTypeId="2" ParentId="17481" CreationDate="2019-11-21T19:30:47.967" Score="4" Body="&lt;p&gt;An &quot;old school&quot; answer (nearly 60 years old now!) which works for any age range is turtle graphics, which is (are?) implemented as a Python module.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can only guess how much of your curriculum is officially labelled &quot;geometry&quot;, but it will certainly teach algorithmic thinking, and also be fun. &lt;/p&gt;&#xA;" OwnerUserId="7677" LastActivityDate="2019-11-21T19:30:47.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17486" PostTypeId="2" ParentId="17481" CreationDate="2019-11-21T19:44:09.253" Score="7" Body="&lt;p&gt;Any puzzle game which requires students to plan the entire problem before executing it might help. Also, there are physical puzzles which can be solved algorithmically much more neatly than if they use 'trial and error'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In English, one resource which I have not seen mentioned yet is &lt;a href=&quot;https://code.org/student/elementary&quot; rel=&quot;noreferrer&quot;&gt;Code.org&lt;/a&gt; which has themed coding puzzles for all ages. Other puzzles, including physical puzzles, which promote algorithmic thinking include the &lt;a href=&quot;https://www.turingtumble.com/&quot; rel=&quot;noreferrer&quot;&gt;Turing Tumble game&lt;/a&gt;. And surely even elementary students can understand the problem statement of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tower_of_Hanoi&quot; rel=&quot;noreferrer&quot;&gt;Towers of Hanoi&lt;/a&gt; and understand that trial-and-error makes moving a large tower difficult, but that it's possible to construct an algorithm (and a recursive one at that!) which lets them move it consistently. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, board games are an excellent avenue for learning conditional reasoning, as the rules are enforced by other players. Conditional reasoning is absolutely necessary to develop algorithmic thinking past &quot;this is the order of steps to do things.&quot; Obvious examples include &lt;a href=&quot;https://boardgamegeek.com/boardgame/51/ricochet-robots&quot; rel=&quot;noreferrer&quot;&gt;Ricochet Robots&lt;/a&gt; or any other sort of abstract strategy game. I found a lot of reviews for child-appropriate games on &lt;a href=&quot;https://fathergeek.com/tag/abstract-strategy-games/&quot; rel=&quot;noreferrer&quot;&gt;fathergeek.com&lt;/a&gt;.  However, even a &quot;silly&quot; card game like &lt;a href=&quot;https://boardgamegeek.com/boardgame/258/fluxx&quot; rel=&quot;noreferrer&quot;&gt;Fluxx,&lt;/a&gt; with its constantly-changing rules, can lead to entertaining logical loops and conditionals. I acknowledge that these resources are, again, in English, but my hope is that they give enough inspiration to find equivalent resources in other languages. Students writing their own board games with rules that include conditional statements and loops could also be meaningful. &lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2019-11-21T19:44:09.253" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17487" PostTypeId="2" ParentId="17481" CreationDate="2019-11-21T22:38:18.317" Score="2" Body="&lt;p&gt;These may be too hard, but the ACM's International Collegiate Programming Contest (ICPC) has a set of past programming problems that require algorithmic thinking. I took a class in college where we basically just worked on these for 3 hours a week. It was really good problem-solving experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://icpc.baylor.edu/worldfinals/problems&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://icpc.baylor.edu/worldfinals/problems&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6447" LastActivityDate="2019-11-21T22:38:18.317" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17488" PostTypeId="1" CreationDate="2019-11-21T22:58:45.633" Score="5" ViewCount="153" Body="&lt;p&gt;As I was being observed today, an administrator asked me for a practical application of parabolas. I responded by talking about objects in free-fall. Afterwards as I was re-thinking this conversation it occurred to me that an object in orbit is also in a sort of free-fall, but its path is not parabolic but elliptical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And a question began to form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can only turn to these contexts to illustrate conic sections because we live in the modern world. And math books often illustrate conic sections with technological examples like satellite dishes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But during the very long interval between the first ancient Greek discussions of conic sections and the work of Kepler and Newton, what did everyone think that conic sections were good for? Were they interested in them purely as ideas, without thought of utility? That would be oddly beautiful and somewhat surprising.&lt;/p&gt;&#xA;" OwnerUserId="8258" LastEditorUserId="8258" LastEditDate="2019-11-26T14:42:48.157" LastActivityDate="2019-11-26T14:42:48.157" Title="What were the applications of conic sections before Newton?" Tags="|history-of-math|" AnswerCount="1" CommentCount="2" ClosedDate="2019-11-22T21:34:04.043" ContentLicense="CC BY-SA 4.0" />
  <row Id="17489" PostTypeId="2" ParentId="17004" CreationDate="2019-11-22T00:05:27.080" Score="2" Body="&lt;p&gt;Either 1 or 3, for the reasons given in the other answers.&#xA;But I would also note that &quot;as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&quot; need not be set aside by commas (or a break in speech), as it is &lt;em&gt;not&lt;/em&gt; optional in the sentence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides the arrangement of the words, it is worth noting that ISO 80000-2, &lt;em&gt;Mathematical signs and symbols to be used in the natural sciences and technology&lt;/em&gt; (2009), lists &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &lt;em&gt;tends to&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&quot; as the verbal equivalent for the notation &lt;span class=&quot;math-container&quot;&gt;$x \rightarrow a$&lt;/span&gt;, in contrast to &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &lt;em&gt;approaches&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&quot;&lt;/p&gt;&#xA;" OwnerUserId="13064" LastActivityDate="2019-11-22T00:05:27.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17492" PostTypeId="2" ParentId="17481" CreationDate="2019-11-22T09:26:04.123" Score="1" Body="&lt;p&gt;The de Casteljau algorithm for generating polynomially parameterized curves in the Beziér representation admits a simple geometric interpretation in terms of the control polygon that is easily implemented in Python. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More precisely, a polynomially parameterized curve in the affine plane or affine three-space can be represented in the form &lt;span class=&quot;math-container&quot;&gt;$P(t) = \sum_{k = 0}^{n}\binom{n}{k}t^{k}(1-t)^{n-k}p_{k}$&lt;/span&gt; where the &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; points &lt;span class=&quot;math-container&quot;&gt;$p_{0}, \dots, p_{n}$&lt;/span&gt; are the &lt;em&gt;control points&lt;/em&gt;. The de Casteljau algorithm constructs &lt;span class=&quot;math-container&quot;&gt;$P(t)$&lt;/span&gt; in the following way. First one considers the &lt;em&gt;control polygon&lt;/em&gt; which is the polygonal curve whose sides join &lt;span class=&quot;math-container&quot;&gt;$p_{i}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$p_{i+1}$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$0 \leq i &amp;lt; n$&lt;/span&gt;. One subdivides each of these sides into two subsegments of relative lengths &lt;span class=&quot;math-container&quot;&gt;$1-t$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;. There results &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; points &lt;span class=&quot;math-container&quot;&gt;$b^{1}_{0}, \dots, b^{1}_{n-1}$&lt;/span&gt; given explicitly by &lt;span class=&quot;math-container&quot;&gt;$b^{1}_{i} = (1-t)p_{i} + tp_{i+1}$&lt;/span&gt;. One repeats the process with the &lt;span class=&quot;math-container&quot;&gt;$b^{1}_{i}$&lt;/span&gt; in place of the &lt;span class=&quot;math-container&quot;&gt;$p_{i}$&lt;/span&gt;. At each stage the number of points is reduced by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, so at stage &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; one has &lt;span class=&quot;math-container&quot;&gt;$b^{k}_{0}, \dots b^{k}_{n-k}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$b^{k}_{i} = (1-t)b^{k-1}_{i} + tb^{k}_{i+1}$&lt;/span&gt;. It is not hard to see that &lt;span class=&quot;math-container&quot;&gt;$P(t) = b^{n}_{0}$&lt;/span&gt;, the point obtained at the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th stage. This algorithm can be used to generate the curve by running it for &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; chosen from some partition of &lt;span class=&quot;math-container&quot;&gt;$[0, 1]$&lt;/span&gt;. The polygonal curve used at each stage can be drawn (by hand or using the computer) and the algorithm can be implemented in Python in a few lines. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The explicit expression for &lt;span class=&quot;math-container&quot;&gt;$P(t)$&lt;/span&gt; is not needed in any of this. One could simply start with the algorithm as a way of producing a curve from the data of an ordered collection of points. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a low degree curve it can be instructive for students to compare generating the curve by plugging values into the explicit expression for &lt;span class=&quot;math-container&quot;&gt;$P(t)$&lt;/span&gt; and by running the de Casteljau algorithm. These are two very different ways of constructing the curve on a computer. &#xA;Students often have never thought about how functions or graphics are represented in a computer or calculator. It can be very instructive for them to see that it is not necessarily achieved by substituting values. Also they often do not realize that what looks like a curve on the computer screen is really a discrete approximation of the mathematician's idealized continuous curve (everything is discrete on a computer!). Moreover this method (or, better, modifications and improvements of its analogue for splines, known as the Cox-de Boor algorithm, are used in practice in computer assisted geometric design (CAD)). &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-11-22T09:26:04.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17493" PostTypeId="2" ParentId="17481" CreationDate="2019-11-22T11:52:51.260" Score="2" Body="&lt;p&gt;This isn't a resource, but a fun algorithmic problem that I remember solving back in 7th grade:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider a long loop of train wagons, &lt;span class=&quot;math-container&quot;&gt;$1 ... N$&lt;/span&gt; for some unknown &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; where wagon &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; is connected to wagons &lt;span class=&quot;math-container&quot;&gt;$i-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$i+1$&lt;/span&gt;. You can walk from one wagon to its neighbors. In every wagon, there is a lightbulb connected to a switch. You start in a wagon and want to figure out how many wagons there are in the loop. If the lightbulbs all started out as turned off, this would be easy: Turn on the light in all wagons you pass and count the number of wagons until you find lights that are already turned on. But there is a catch; to begin with, the state of every lightbulb is randomly chosen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are you still able to find &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;? How many steps from a wagon to the next (or previous) one do you have to take? Can you minimize this number by making some modifications to your algorithm?&lt;/p&gt;&#xA;" OwnerUserId="13065" LastActivityDate="2019-11-22T11:52:51.260" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17494" PostTypeId="2" ParentId="17488" CreationDate="2019-11-22T13:16:54.407" Score="0" Body="&lt;p&gt;From Wikipedia on &lt;a href=&quot;https://en.wikipedia.org/wiki/Menaechmus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Menaechmus&lt;/a&gt;, ~350 BC:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Menaechmus is remembered by mathematicians for his discovery of the conic sections and his solution to the problem of doubling the cube. Menaechmus likely discovered the conic sections, that is, the ellipse, the parabola, and the hyperbola, as a by-product of his search for the solution to the Delian problem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Doubling_the_cube&quot; rel=&quot;nofollow noreferrer&quot;&gt;Delian problem&lt;/a&gt; is the problem of constructing the side length of a cube which has double the volume&#xA;of a given cube. If the given cube is a unit cube, the problem is to construct&#xA;&lt;span class=&quot;math-container&quot;&gt;$2^\frac{1}{3}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-11-22T17:29:08.337" LastActivityDate="2019-11-22T17:29:08.337" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17496" PostTypeId="2" ParentId="17481" CreationDate="2019-11-22T15:26:58.510" Score="1" Body="&lt;p&gt;Although some of the puzzles might be appropriate only for extra credit, have a look at &lt;em&gt;Algorithmic Puzzles&lt;/em&gt; by Anany and Maria Levitin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ad copy for this book gives a good description:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While many think of algorithms as specific to computer science, at its&#xA;  core algorithmic thinking is defined by the use of analytical logic to&#xA;  solve problems. This logic extends far beyond the realm of computer&#xA;  science and into the wide and entertaining world of puzzles. In&#xA;  Algorithmic Puzzles, Anany and Maria Levitin use many classic&#xA;  brainteasers as well as newer examples from job interviews with major&#xA;  corporations to show readers how to apply analytical thinking to solve&#xA;  puzzles requiring well-defined procedures.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13068" LastActivityDate="2019-11-22T15:26:58.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17497" PostTypeId="1" CreationDate="2019-11-23T00:38:54.003" Score="18" ViewCount="5698" Body="&lt;p&gt;What college mathematics topics can be taught in such a way that causes students to physically move around?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Background:&#xA;Recently, it was suggested to me by a faculty member (from another academic department) who visited my class that there are positive effects from having students physically move around during class. So, I wonder what mathematics topics are taught in such a way, and what are the students actually doing? Is this movement connected to the math they are learning, and do you find it beneficial?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I could just ask my students to &quot;stand up and stretch&quot; halfway through class, I would &lt;em&gt;not&lt;/em&gt; classify that as being directly tied to learning a math concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for things you have done (or would like to do) to this end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: I would most benefit from examples at the community college level (arithmetic through calculus), but I'm sure this is practiced at every level of math.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-11-23T01:18:44.157" LastActivityDate="2019-11-26T09:52:49.497" Title="What mathematics activities get students physically moving?" Tags="|interactive-teaching|community-colleges|classroom-interaction|" AnswerCount="14" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17498" PostTypeId="2" ParentId="17497" CreationDate="2019-11-23T00:40:11.520" Score="29" Body="&lt;p&gt;To start things off, some moments my students move around during class:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I have given students tape measures and had them determine how much they would spend at the paint store if they wanted to paint the walls and ceiling of the lecture room (while projecting two images on the overhead: one of a paint can label, showing the number of square feet per gallon; and another of an ad from the paint store, showing the cost per gallon).&#xA;Many of my beginning algebra students do not have a good sense of size when various units are used (particularly &lt;em&gt;metric&lt;/em&gt;), so actually measuring the room makes them directly confront a length like 18.3 meters (they must walk it).&lt;/li&gt;&#xA;&lt;li&gt;I have drawn a chalk line outside, labeled with its length, and had students time each other walking the length of the line. Then they use their personal walking speed to determine the length of a much longer, unlabeled chalk line, as well as the area of a large concrete quad area.&#xA;Again, most students don’t know things like how fast they walk, so they learn something about themselves. It becomes an easy story for later examples where we talk about speed: “Remember when you walked outside?”&lt;/li&gt;&#xA;&lt;li&gt;I have given students tape measures and mirrors to have them find unknown heights (of trees or buildings) by using shadows or reflections (similar triangles).&#xA;Students get to choose an object to indirectly measure, and they must work together to make sure things are set up properly. [e.g. If a mirror is used, is it sitting flat on the ground?]&lt;/li&gt;&#xA;&lt;li&gt;I very often start class by having students write solutions to homework problems on the chalkboards around the room.&#xA;This is mainly a way to get the writing process finished quickly, so we can discuss homework and I can give feedback on notation. Their physical movement is incidental, but students have given me positive feedback on the practice.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="470" LastActivityDate="2019-11-23T00:40:11.520" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="17500" PostTypeId="1" CreationDate="2019-11-23T07:01:46.587" Score="0" ViewCount="100" Body="&lt;p&gt;Which is the best tutorial available online to learn the recurrence relation concept and its solution in discrete mathematics, in a systematic way?&lt;/p&gt;&#xA;" OwnerUserId="13071" LastActivityDate="2019-11-26T15:29:26.467" Title="Need to learn recurrence relation discrete mathematics" Tags="|engineering-mathematics|discrete-math|computer-science|recursion|" AnswerCount="2" CommentCount="0" ClosedDate="2019-11-24T03:20:20.203" ContentLicense="CC BY-SA 4.0" />
  <row Id="17501" PostTypeId="2" ParentId="244" CreationDate="2019-11-23T08:06:07.197" Score="2" Body="&lt;p&gt;Looking at my father's War Manual textbooks (published ~39, versions of older texts), it seems like the sequence was much like it is now:&lt;/p&gt;&#xA;&lt;p&gt;*Algebra&lt;/p&gt;&#xA;&lt;p&gt;*Geometry (very similar content to what I had in the 80s)&lt;/p&gt;&#xA;&lt;p&gt;*College Algebra (almost exactly matches Algebra 2, even has a &amp;quot;refresher chapter&amp;quot; at the beginning to cover contents of &amp;quot;high school algebra&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;*Planar trigonometry AND spherical trigonometry&lt;/p&gt;&#xA;&lt;p&gt;*Analytic geometry (conic sections, rotations and translations)&lt;/p&gt;&#xA;&lt;p&gt;*Calculus (very similar to the AP course I took in the 1980s, even to including almost identical chapters on sequences/series and on ODEs)&lt;/p&gt;&#xA;&lt;p&gt;Note the placement of geometry is same as the current US stereotypical pattern, not mixed like &lt;a href=&quot;https://en.wikipedia.org/wiki/Saxon_math&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saxon&lt;/a&gt; or (some) Common Core.&lt;/p&gt;&#xA;&lt;p&gt;Probably only the first two courses were expected to occur in high school, but smarter students might be accelerated and encounter some of the other materials.&lt;/p&gt;&#xA;&lt;p&gt;The presence of spherical trig makes that a lot harder topic.  The curriculum does seem to lack vectors, which seems unfortunate to me.  Also there is no &amp;quot;pre-calc&amp;quot; in the sense of a course where you learn a little bit of differentiating and antiderivatives prior to the AP calculus to make that transition softer.  (But then school systems seem to vary a lot now on how the treat the path after algebra 2 and before AP calc, even now.  There is little clear definition of what should be in a &amp;quot;precalculus&amp;quot; course.)&lt;/p&gt;&#xA;" OwnerUserId="13072" LastEditorUserId="2083" LastEditDate="2023-01-17T07:36:54.070" LastActivityDate="2023-01-17T07:36:54.070" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17502" PostTypeId="2" ParentId="17497" CreationDate="2019-11-23T12:41:52.000" Score="4" Body="&lt;p&gt;One activity I've seen done at the high school level is a &lt;strong&gt;scavenger hunt&lt;/strong&gt;.  You come up with a group of questions with numeric answers, and then you print out a bunch of papers to hang around the room (or hallway or wherever).  One paper has the answer to #&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and the question to #&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, one has the answer to #&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and the question to #&lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, all the way around to one having the answer to #&lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; and the question to #&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a sample of two such cards from an &lt;a href=&quot;https://www.teacherspayteachers.com/Product/Solving-Equations-Scavenger-Hunt-157386&quot; rel=&quot;nofollow noreferrer&quot;&gt;equation-solving scavenger hunt&lt;/a&gt; where you'd solve the equation on the first card, see that the answer was &lt;span class=&quot;math-container&quot;&gt;$x=9$&lt;/span&gt;, and then search for the second card to find the next equation to solve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/7g27P.png&quot; width=&quot;240&quot; /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/1DC4X.png&quot; width=&quot;240&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;The students are given a sheet of paper where they have to list the card they started and and where they went from there.  The cards have a cycle to them, so students can start at any point in the cycle and still get the entire activity.  Grading it is pretty much a breeze, because you're just checking to make sure that they visited all the cards in the correct order (although you could also obviously give them a longer sheet to fill out where they would have to show all their work).  The activity also has a great separate formative assessment feel to it, because you can see the questions where there are larger crowds of students and understand that those are the topics you need to address more in the review.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the kind of activity that makes teacherspayteachers.com such a fabulous resource.  For most math skills, someone has already gone through the effort of doing the layout for all the cards and it's worth a buck or two to save yourself the same effort.  (Props to &lt;a href=&quot;https://www.teacherspayteachers.com/Product/Solving-Equations-Scavenger-Hunt-157386&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Duncan&lt;/a&gt; for making the free hunt that the two above cards came from.)  You might have to make your own calculus-based activities, but if you start with lower-level skills you can get a feel for the general format first.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps!  Students generally seem to enjoy getting up and seeing the room from different angles, and I don't think I've ever heard of a teacher who regretted it as an activity every now and then.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="511" LastEditDate="2019-11-23T14:07:10.913" LastActivityDate="2019-11-23T14:07:10.913" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17503" PostTypeId="2" ParentId="17497" CreationDate="2019-11-23T16:09:56.793" Score="4" Body="&lt;p&gt;A technique I use a lot is to use software to put students in random groups, then have them do active learning activities in those groups. These activities are sometimes the &quot;conceptest&quot; technique created by Mazur, or think-pair-share. My set of conceptest activities for freshman calc are &lt;a href=&quot;http://lightandmatter.com/fund/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; (click through to &quot;active learning resources&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The conceptest technique involves showing a conceptual, multiple-choice question. Students think silently for a minute or two, then vote. (I have them hold up 1 finger for A, 2 for B, etc.) If there is a clear consensus for the right answer, we discuss it briefly and move on. If not, then students discuss it in groups. The theory is that the right answer is supposed to &quot;win&quot; in debate. Then we vote again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although this technique can be done without having students get up and move, I find that having them move around like this helps a lot. If I don't assign groups, then inevitably many students will not participate in discussion. If I try to assign permanent groups, then we get problems because of absences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this technique, often there will be situations in which, e.g., everyone in group 2 initially votes for C, while everyone in group 3 votes for A. In this situation, I swap some students between groups so that each group will have something to debate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I'm doing several of these questions, I usually also re-randomize the groups between questions. This is optional, but it helps, for example, with situations where one group is composed entirely of weak students. That group doesn't have to languish and keep failing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reasons for doing all this don't necessarily have anything to do with getting blood flowing or reducing boredom, but it certainly also has that effect, and students seem to appreciate that. I think it also creates a nice social atmosphere in the class, because everyone knows everyone else, and they all know they need to maintain good relations.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-11-23T16:16:27.637" LastActivityDate="2019-11-23T16:16:27.637" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17504" PostTypeId="2" ParentId="17497" CreationDate="2019-11-23T19:35:46.477" Score="13" Body="&lt;p&gt;The &lt;a href=&quot;https://www.youtube.com/watch?v=ywWBy6J5gz8&quot; rel=&quot;noreferrer&quot;&gt;Hungarian Quicksort Dance&lt;/a&gt; demonstrates a computer science algorithm with dance. It's pretty advanced, but the idea is to have your students physically act out algorithms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps something similar can be done with a number line. Line your students up and have one walk down the line to demonstrate addition and subtraction, or take big steps for multiplication. Demonstrate complex numbers by having them move perpendicular to the line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometry is particularly good, get out some chalk and string and physically work it out on the floor. Surveyor and navigation techniques serve well. For example, roll a hoop one revolution and measure the resulting distance to show how the radius, circumference, and pi relate. Or demonstrate its utility for navigation with basic triangulation to determine the distance to an object. Or demonstrate mathematical proofs with geometry, like &lt;a href=&quot;https://www.youtube.com/watch?v=BRRolKTlF6Q&quot; rel=&quot;noreferrer&quot;&gt;why zero is so perilous&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus is the mathematics of change and motion, and realizing this (sadly post university) gave me a new appreciation for the utility of calculus. Demonstrate calculus with some change and motion. I can't think of one off the top of my head, but you might find inspiration in &lt;a href=&quot;https://www.goodreads.com/book/show/316793.A_Tour_of_the_Calculus&quot; rel=&quot;noreferrer&quot;&gt;A Tour of the Calculus&lt;/a&gt;, the flowery writing reviews complain about is an advantage here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Numberphile occasionally does physical exercises to demonstrate geometry. For example, &lt;a href=&quot;https://www.youtube.com/watch?v=ZNiRzZ66YN0&quot; rel=&quot;noreferrer&quot;&gt;calculating Pi with Pies&lt;/a&gt;. You don't have to use pies, any set of equal size circles. Or &lt;a href=&quot;https://www.youtube.com/watch?v=6_yU9eJ0NxA&quot; rel=&quot;noreferrer&quot;&gt;higher dimensional geometry with darts&lt;/a&gt;. Or &lt;a href=&quot;https://www.youtube.com/watch?v=l7lP9y7Bb5g&quot; rel=&quot;noreferrer&quot;&gt;math-based card tricks&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're discussing graph theory, physically make the graph. For big demonstrations I've used &lt;a href=&quot;https://www.knex.com/Tinkertoy/&quot; rel=&quot;noreferrer&quot;&gt;Tinker Toys&lt;/a&gt; to build the graph and Post-It Notes to label them. For finer ones Post-It Notes on whiteboard (nodes) and drawing lines between with marker works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The physicality of the &lt;a href=&quot;https://www.youtube.com/watch?v=5TkIe60y2GI&quot; rel=&quot;noreferrer&quot;&gt;famous Numberphile butcher paper&lt;/a&gt; helps with engagement. Rather than having students do exercises at their seats, you can distribute butcher paper and markers to groups of students to gather together and work out exercises together. In my own classes I've taped the paper to the wall so students have to physically get up, move around, stand, and talk to each other. Using colored markers makes it more enjoyable. And it allows me to less intrusively keep an eye on how each group is progressing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some folks to get inspiration from.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.youtube.com/channel/UCoxcjq-8xIDTYp3uz647V5A&quot; rel=&quot;noreferrer&quot;&gt;Numberphile&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.youtube.com/user/standupmaths&quot; rel=&quot;noreferrer&quot;&gt;Matt Parker&lt;/a&gt; the Standup Mathematician.&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In particular his &lt;a href=&quot;https://www.youtube.com/watch?v=1wAaI_6b9JE&quot; rel=&quot;noreferrer&quot;&gt;Things To See And Hear In The Fourth Dimension&lt;/a&gt; book and talk.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13075" LastActivityDate="2019-11-23T19:35:46.477" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17505" PostTypeId="2" ParentId="17500" CreationDate="2019-11-23T21:34:47.410" Score="3" Body="&lt;p&gt;I liked this &lt;a href=&quot;http://discrete.openmathbooks.org/dmoi2/sec_recurrence.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discrete Mathematics: An Open Introduction&lt;/a&gt;, by Oscar Levin, for generating functions, so I'm guessing it will be good for recurrence relations.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="238" LastEditDate="2019-11-23T22:11:19.817" LastActivityDate="2019-11-23T22:11:19.817" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17506" PostTypeId="2" ParentId="17500" CreationDate="2019-11-23T21:53:33.873" Score="1" Body="&lt;p&gt;Better yet, I very much liked &lt;a href=&quot;https://www.math.upenn.edu/~wilf/gfology2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;generatingfunctionology&lt;/a&gt; by Herbert Wilf; it is the go-to text for what you are seeking.  It addresses generating functions, and considerable help for understanding recurrence relations. (And it's free from the author, and downloadable as a pdf.!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, you'll find extensive information at Ronald L. Graham, Donald E. Knuth, and Oren Patashnik's &lt;a href=&quot;https://www.csie.ntu.edu.tw/~r97002/temp/Concrete%20Mathematics%202e.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Concrete Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For beginner information, there are a number of sources and tutorials online.   For example, see &lt;a href=&quot;https://brilliant.org/discrete-mathematics/?subtopic=recurrence-relations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brilliant.org, recurrence relations&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another potential online source for introductory material is located &lt;a href=&quot;https://www.tutorialspoint.com/discrete_mathematics/discrete_mathematics_recurrence_relation.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-11-26T15:29:26.467" LastActivityDate="2019-11-26T15:29:26.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17507" PostTypeId="2" ParentId="17497" CreationDate="2019-11-24T01:55:47.273" Score="4" Body="&lt;p&gt;How about drawing an ellipse, a parabola, and a hyperbola, using string&#xA;and a straightedge:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/HrMK4.png&quot; width=&quot;300&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/qqsJD.png&quot; width=&quot;300&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/JmXfm.png&quot; width=&quot;300&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Snapshots from &lt;a href=&quot;https://www.youtube.com/watch?v=mldZ_7QwLvs&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathLapse&lt;/a&gt;:&#xA;&quot;pin-and-string conics.&quot;&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-11-24T01:55:47.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17508" PostTypeId="1" CreationDate="2019-11-24T04:45:06.570" Score="8" ViewCount="275" Body="&lt;p&gt;As you might know already evaluating learning outcomes is very difficult. For example, student feedback is not always reliable; there are studies showing inverse relations between student learning and student ratings (eg. a course that managed to challenge students). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One potential idea is to have the final exam (or even midterms) in the math course written by someone else other than the instructors. Here are some pros:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;The instructors' bias&lt;/em&gt;. By having the final exam questions designed by someone else, we  substract the frequent issue of (consciously/unconsciously) training the students to pass the final exam. Ultimately the goal of the course is to help the students tackle outside-world challenges using the techniques from the course. &lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Fitting grades into a bell curve bias&lt;/em&gt;. Another frequent issue is being pressured from the department to adjust the test difficulty so that the grades fit the bell curve. So this often leads to unrealistically difficult tests that are less about testing a student's understanding and more about lowering the average grade. &lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Smaller percentage grade and instructor performance&lt;/em&gt;: The main goal here is to test student learning in an independent way and thus instructor performance. So we can have the final exam grade worth less. We can allow the instructor to spread the grade weight across the learning modules (more quizzes less big tests) to encourage active learning throughout the course. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Here are some cons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;some Course flexibility lost&lt;/em&gt;. By &quot;offshoring&quot; the final exam, the course material will have to be suited within some strict guidelines. Meaning that instructors will not be able to test new ideas that they presented during the course. &lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Too much emphasis on the final exam&lt;/em&gt;. This might lead to too much emphasis on the final exam, and less on active learning throughout the course. But that is an existing problem with the current paradigm and if the final exam relative percentage is not too high, that issue can be ameliorated. &lt;/li&gt;&#xA;&lt;li&gt;&quot;Who will design the final exam?&quot; The main issue here is to have the designer of the exam be independent of the course and that can be hard since the professor faculty is quite small. It might introduce a lot of unnecessary politics. Ideally, the process could be automated using technology (i.e. some algorithmic way of sampling exam questions from a database)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Any feedback, references or anecdotes are welcomed. Do you see any problems with this idea? Is the existing way (exams designed by the instructors) better than what suggested?&lt;/p&gt;&#xA;" OwnerUserId="8714" LastEditorUserId="8714" LastEditDate="2019-11-24T04:53:51.327" LastActivityDate="2019-11-24T19:20:46.683" Title="Suggested idea: Course exams written by someone else other than the instructors/coordinators. Good or Bad?" Tags="|exams|test-design|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17509" PostTypeId="2" ParentId="17454" CreationDate="2019-11-24T11:17:31.063" Score="4" Body="&lt;p&gt;Take a circle of radius 1. Inscribe a regular triangle in this circle. Inscribe a circle in this triangle. Inscribe a square in it. Inscribe a circle, regular pentagon, circle, regular hexagon and so forth. The radius of the limiting circle is called the Kepler-Beoucamp constant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Bu6lR.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Bu6lR.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13080" LastActivityDate="2019-11-24T11:17:31.063" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17510" PostTypeId="1" CreationDate="2019-11-24T11:32:31.853" Score="10" ViewCount="333" Body="&lt;p&gt;I am going to teach 3D vectors soon to my A-level students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last time I taught it a lot of them struggled to visualise the angle between two vectors in 3D space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had an idea that I could get them all 3D glasses and find some sort of immersive 3D animation of vectors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any software out there to create 3D animations (using the red and blue layers)? Or any existing videos/resources that can be viewed in 3D with the glasses?&lt;/p&gt;&#xA;" OwnerUserId="13081" LastActivityDate="2019-11-25T23:11:25.897" Title="3D vectors with 3D glasses?" Tags="|vectors|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17511" PostTypeId="2" ParentId="17508" CreationDate="2019-11-24T17:32:54.570" Score="4" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Your point about less weighting of the final exam is made as both a con and a pro (not logically consistent).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The issue of less flexibility to alter the curriculum is not purely a con, can be a pro. I know this will raise hackles of community of instructors here who enjoy flexibility and see themselves as Robin Williams in Dead Poets.  But there can be a real benefit from the student point of view and from the customer (school, military, employer, government) point of view in making sure that specific things are covered.  (You don't see the bar or PE exam as instructor-specific!)  And teachers may vary in how they lag, accelerate, or even push their own research (that feature can be a bug!)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another benefit can be motivational.  Jaime Escalante has written about this.  He was in it together with his kids to beat the tests (hopefully not by cheating).  And the presence of an external, seriously designed exam was very motivational because it was a somewhat objective bar to clear.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's not just the bell curve--you can do much more serious psychometrics when you have large test populations taking thoughtful independently designed tests (not ad hoc by individual profs for individual sections).  Can do things like cross year comparisons.  Can really try to estimate what it means to have &quot;A&quot;, &quot;B&quot;, etc. knowledge of the subject (for example what level/frequency of problem solving).  I.e. trying to set a bar, not just bell curve the kids.  See point 5 also, however. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think this is convenient for most single instructor courses. (I.e. not for majors courses or random humanities electives).  But it does make a lot of sense when you have a lot of different sections taking the same thing.  Consider the AP exams, for instance.  Or at military academies, there were many required courses (non majors) that had essentially the entire class taking them, but in separate sections of ~30 students (like high school).  So if you have 35 sections (say 15-20 instructors), than that is no big deal to have one of them (or a separate person) design the exams (not just the final or midterm, but intermediate exams, every 4 weeks in a 16 week semester).  Examples of this would be general chemistry (2 sems), physics (2 sems), calc 1-3, ODEs, EE (2 sems), world history (2 sems), naval history, CS, all the prodev courses (navigation, leadership, etc.). Also many of the engineering courses that all engineers take (statics, &quot;engine math&quot;, etc.) before going into major specifics.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Note that it's pretty normal if the test is independent for the grading to be independent (A grades B's students and visa versa) or more commonly A and B grade problem 1.  C and D grade problem 2.  Etc. And of course because of the scale of the effort a thoughtful key can be constructed. (AP even has a system where if graders encounter a problem, for example an unexpected solution approach, they can appeal it to an expert.)  Note however, for large classes, this sort of systematic grading can be done even if the test was set by the instructor.  For example Professor A. B. Sentmind at Enormous State U., teaching freshman chemistry in a 900 seat lecture hall. (Will have an army of TAs working as an organized team to grade the tests.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As far as Crowell's point (see comment below headpost) about instructor cheating, it's actually harder for them to do this when the exam is standardized and grading consolidated.  (Not impossible, harder...than if they do everything in isolation.)  In addition, the same objection about &quot;instructors might cheat&quot; if you measure them to assess performance applies to students as well.  Yet we still try to assess them.  Just the chance of some cheating should not stop attempts to measure things. (Should we stop having companies do SEC financial reports because some have cheated in the past?  No.  There is still a net benefit, even if not perfection, in having standardized reporting as per GAAP.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="13083" LastEditorUserId="13083" LastEditDate="2019-11-24T18:02:25.897" LastActivityDate="2019-11-24T18:02:25.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17512" PostTypeId="2" ParentId="17508" CreationDate="2019-11-24T19:20:46.683" Score="4" Body="&lt;p&gt;I am a high school teacher with three preps this year.  Two of them have a &quot;final&quot; that are written by an external party (a Geometry final from the New York State Board of Regents and an AP Calculus test written by College Board).  I don't mind either of them.  I fully admit that I &quot;teach to the test&quot;, in the sense that I trust the test to be well-designed and fully cover the domain of standards that I agreed to meet when I took on the class.  In return, the students who pass the class have an extra form of accreditation that says that they have a mastery of the domain that is worth something of value (either an advanced HS diploma or the potential to be placed out of college Calculus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To address the pros and cons you mention in your post:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Instructor bias: absolutely a problem.  If I hate optimization problems in Calc, I might decide to not teach them.  If I adore them, I might spend five weeks on it instead of two.  Neither of those decisions would serve my students, who are presumably learning an authentic survey of introductory Calculus so that they can use it in future courses or other career development.&lt;/li&gt;&#xA;&lt;li&gt;Fitting grades into a bell curve bias.  Not a problem that happens at my pay grade.  I suppose I am relieved that there is an external source assessing my students so that I am not only relieved of the high-level design of the syllabus but also of the level of mastery required for &quot;passing&quot;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Smaller percentage grade and instructor performance: I'm not sure I understand this point.  I don't think the design of my unit tests would be impacted in any way if I became responsible for my final exams.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some course flexibility lost.  This might betray my philosophy of education to some degree, but I am not certain that teachers have a broad enough view of their students' needs to deserve authority to decide what material they should teach in a course.  Sure, if you're teaching a doctoral student, then go for broke.  But if you're teaching, say, an undergraduate accounting class and spending time not ultimately preparing the students for the accreditation exams or their professional lives, then you're off the reservation.  (That may have come across as harsher against our professionalism than I intended.  We know how our field is changing, and deserve a voice in the direction of our course development.  But if we cannot convince a guiding committee that we are correct, we're on shaky ground when it comes to overriding them.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Who will design the final exam?&quot; If it's a course that is part of a sequence, I might suggest that the exam should be written by next semester's instructor.  This is definitely a stakeholder in measuring what the students are taking away from your class.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-11-24T19:20:46.683" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17513" PostTypeId="2" ParentId="17497" CreationDate="2019-11-25T01:06:43.817" Score="4" Body="&lt;p&gt;I took a small math class over the summer (about 11 people) where instead of lecturing the whole time (as is customary for many college math classes), the professor gave us some problems once in a while that we would do in groups. The act of forming groups, talking, and then explaining the solutions to the professor (at the group) I think allowed for a lot more interaction/movement than most math lectures. Also, if you are in the position where you do have time to have students come to the board to present solutions to groupwork/homework, that would be even better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the professor's insistence in having us work together in class while the professor made her way around the room made such a strong impression on me that if/when I become a professor I will find some way to carve time to do those sorts of things into the curriculum.&lt;/p&gt;&#xA;" OwnerUserId="13084" LastActivityDate="2019-11-25T01:06:43.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17514" PostTypeId="2" ParentId="17497" CreationDate="2019-11-25T05:27:31.310" Score="3" Body="&lt;p&gt;For example when explaining the properties of a perpendicular bisector of a segment, you can take the students to a hall, place 2 sheets on the ground (that will be our segment) and then give students a measuring tape and ask them to find a spot that is equidistant from both extremities of the segment (the 2 sheets) and another student find another spot and si on... they will understand how every point on the perpendicular bisector is equidistant from the extremities of the segment, that it is a straight line, passes through the midpoint, ....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By a similar manner you can make them draw some conics and circles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing you can do is putting cardboards in the corners of your class, each with some questions on it (say 3). Divide the class into 4 groups (each group of 3 for our example) and a student from each group goes up to the carboard to solve one question and come back high-five another group member that will go up to solve a second question, ...&lt;/p&gt;&#xA;" OwnerUserId="12486" LastActivityDate="2019-11-25T05:27:31.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17515" PostTypeId="2" ParentId="17497" CreationDate="2019-11-25T11:20:19.053" Score="5" Body="&lt;p&gt;To build intuition for the Cartesian plane, assign axes to the room (or to students, depending on how many you have and how they're arranged in the room). Then you can do things like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Stand up if your &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-coordinate is 0, 1, 2 etc.&lt;/li&gt;&#xA;&lt;li&gt;Stand up if your &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-coordinate is less than five.&lt;/li&gt;&#xA;&lt;li&gt;Stand up if your &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-coordinate is less than or equal to five.&lt;/li&gt;&#xA;&lt;li&gt;Stand up if your &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-coordinate equals your &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-coordinate.&lt;/li&gt;&#xA;&lt;li&gt;Everyone stand somewhere in the room so that your &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-coordinate will be equal to twice your &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-coordinate.&lt;/li&gt;&#xA;&lt;li&gt;Everyone stand somewhere in the room so that your &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-coordinate is equal to twice your &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-coordinate, plus 2.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can of course extend this to more complex equations, inequalities, loci etc. I find students often confuse &lt;span class=&quot;math-container&quot;&gt;$y=k$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=k$&lt;/span&gt; in terms of which produces a horizontal versus vertical line; this makes that quite intuitive.&lt;/p&gt;&#xA;" OwnerUserId="1295" LastActivityDate="2019-11-25T11:20:19.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17516" PostTypeId="1" CreationDate="2019-11-25T11:54:11.147" Score="3" ViewCount="427" Body="&lt;p&gt;I am majoring in aerospace engineering.&#xA;I love math a lot, especially calculus. I have a lot of free time right now and I want to learn more stuff in calculus that would be helpful for my major but I am confused where to start. Do you have any recommendations with good books or resources?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The things I have studied in mathematics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;limits in single variable &lt;/li&gt;&#xA;&lt;li&gt;differentiation single variable &lt;/li&gt;&#xA;&lt;li&gt;integration single variable &lt;/li&gt;&#xA;&lt;li&gt;some series and sequences &lt;/li&gt;&#xA;&lt;li&gt;linear algebra &lt;/li&gt;&#xA;&lt;li&gt;complex analysis &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13067" LastEditorUserId="376" LastEditDate="2019-12-13T15:19:55.677" LastActivityDate="2019-12-23T14:58:35.443" Title="What do I study in calculus beyond the minimum required for undergraduate engineering?" Tags="|reference-request|textbooks|reference-books|engineering-mathematics|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17517" PostTypeId="2" ParentId="17497" CreationDate="2019-11-25T12:05:20.230" Score="4" Body="&lt;p&gt;Take a look at &lt;a href=&quot;https://csunplugged.org/en/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Computer Science Unplugged&lt;/a&gt;. They have a well curated list of activities ready to use with students, quite a few of which include physical activity. You might need to adjust them a bit for older audience, but they are definitely worth the attention.&lt;/p&gt;&#xA;" OwnerUserId="13086" LastActivityDate="2019-11-25T12:05:20.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17518" PostTypeId="2" ParentId="14749" CreationDate="2019-11-25T12:18:47.827" Score="5" Body="&lt;p&gt;There can be no standard algorithm to solve &lt;em&gt;angle hunting&lt;/em&gt; problems. You have to build this skill in students by exposing them to simpler problems and gradually increasing the complexity. I suppose the difficulty of these problems can be guaged by the number of properties/constructions required to solve them completely. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key thing is to begin by working backwards i.e., identify the triangle containing the missing angle. From there keep working your way backwards trying to find the other missing angles or their sums. Some of the properties that could be used in doing this are (not necessarily in the order of difficulty):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Sum of angles in a triangle is 180&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sum of supplementary angles is 180&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sum of complimentary angles is 90&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Vertically opposite angles are equal&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Corresponding angles between parallel lines are equal&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Drop an orthogonal to form a right angled triangle&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Angle subtended by a diameter on a semicircle is 90&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Angle subtended by a chord at centre is twice the angle subtended in the major arc&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Properties of congruent triangles and similar triangles&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The list can't be exhaustive and the properties really depend on the level of students. As I said in the beginning these are open ended problems which can be mastered only through practice. Start with easy and move on to complicated problems.&lt;/p&gt;&#xA;" OwnerUserId="2191" LastActivityDate="2019-11-25T12:18:47.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17519" PostTypeId="2" ParentId="17497" CreationDate="2019-11-25T12:50:29.053" Score="4" Body="&lt;p&gt;My favourite is addition and multiplication by walking. This is my go-to activity when I talk at schools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$+2$&lt;/span&gt; is two steps forward. Then we get to negative numbers, and &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt; is three steps back. Multiplication scales and rotates- so multiplication by &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; turns you backwards (or rotates your &quot;true north&quot;), and &lt;span class=&quot;math-container&quot;&gt;$\times 2$&lt;/span&gt; does something twice. Now we can do &lt;span class=&quot;math-container&quot;&gt;$(3 \times -1) + 4$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, complex numbers. &lt;span class=&quot;math-container&quot;&gt;$+ i$&lt;/span&gt; is a step to the left. &lt;span class=&quot;math-container&quot;&gt;$\times i$&lt;/span&gt; is a &lt;span class=&quot;math-container&quot;&gt;$\pi/\,2$&lt;/span&gt; rotation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Very quickly, kindergarten children can do &lt;span class=&quot;math-container&quot;&gt;$(((3 \times i) + 4) \times i) - 3$&lt;/span&gt;. But they will experience it as a little walk around the classroom.  &lt;/p&gt;&#xA;" OwnerUserId="185" LastActivityDate="2019-11-25T12:50:29.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17520" PostTypeId="2" ParentId="17516" CreationDate="2019-11-25T13:38:43.263" Score="7" Body="&lt;p&gt;There is a big old textbook &lt;em&gt;Advanced Engineering Mathematics&lt;/em&gt; by Kreyszig.  Maybe looking at its table of contents &lt;a href=&quot;http://bcs.wiley.com/he-bcs/Books?action=contents&amp;amp;itemId=0470458364&amp;amp;bcsId=6767&quot; rel=&quot;nofollow noreferrer&quot;&gt;HERE&lt;/a&gt; will show you what mathematics he thought was useful for engineering students.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;(copied from the &lt;a href=&quot;http://bcs.wiley.com/he-bcs/Books?action=contents&amp;amp;itemId=0470458364&amp;amp;bcsId=6767&quot; rel=&quot;nofollow noreferrer&quot;&gt;web site&lt;/a&gt;)  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;WILEY&lt;br&gt;&#xA;  Kreyszig:&lt;em&gt;Advanced Engineering Mathematics&lt;/em&gt;, 10th Edition&lt;br&gt;&#xA;  Table Of Contents&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Chapter 1: First-Order ODEs&lt;br&gt;&#xA;  Chapter 2: Second-Order Linear ODEs&lt;br&gt;&#xA;  Chapter 3: Higher Order Linear ODEs&lt;br&gt;&#xA;  Chapter 4: Systems of ODEs. Phase Plane. Qualitative Methods&lt;br&gt;&#xA;  Chapter 5: Series Solutions of ODEs. Special Functions&lt;br&gt;&#xA;  Chapter 6: Laplace Transforms&lt;br&gt;&#xA;  Chapter 7: Linear Algebra: Matrices, Vectors, Determinants. Linear Systems&lt;br&gt;&#xA;  Chapter 8: Linear Algebra: Matrix Eigenvalue Problems&lt;br&gt;&#xA;  Chapter 9: Vector Differential Calculus. Grad, Div, Curl&lt;br&gt;&#xA;  Chapter 10: Vector Integral Calculus. Integral Theorems&lt;br&gt;&#xA;  Chapter 11: Fourier Series, Integrals, and Transforms&lt;br&gt;&#xA;  Chapter 12: Partial Differential Equations (PDEs)&lt;br&gt;&#xA;  Chapter 13: Complex Numbers and Functions&lt;br&gt;&#xA;  Chapter 14: Complex Integration&lt;br&gt;&#xA;  Chapter 15: Power Series, Taylor Series&lt;br&gt;&#xA;  Chapter 16: Laurent Series. Residue Integration&lt;br&gt;&#xA;  Chapter 17: Conformal Mapping&lt;br&gt;&#xA;  Chapter 18: Complex Analysis and Potential Theory&lt;br&gt;&#xA;  Chapter 19: Numerics in General&lt;br&gt;&#xA;  Chapter 20: Numeric Linear Algebra&lt;br&gt;&#xA;  Chapter 21: Numerics for ODEs and PDEs&lt;br&gt;&#xA;  Chapter 22: Unconstrained Optimization. Linear Programming&lt;br&gt;&#xA;  Chapter 23: Graphs. Combinatorial Optimization&lt;br&gt;&#xA;  Chapter 24: Data Analysis. Probability Theory&lt;br&gt;&#xA;  Chapter 25: Mathematical Statistics&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2019-12-23T11:17:01.230" LastActivityDate="2019-12-23T11:17:01.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17521" PostTypeId="2" ParentId="17497" CreationDate="2019-11-25T16:08:43.410" Score="2" Body="&lt;p&gt;w.r.t. &quot;Hungarian Quicksort Dance&quot; - mentioned by @Schwern in his answer - Stephenson's book &lt;a href=&quot;https://en.wikipedia.org/wiki/Anathem&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Anathem&lt;/em&gt;&lt;/a&gt; has a section where a group of physicists do the following at a sort of community &quot;open house&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Three fraas and two suurs sang a five-part motet while twelve others milled around in front of them. Actually they weren't milling; it just looked that way from where we sat. Each one of them represented an upper or lower index in a theorical equation inolving certain tensors and a metric. As they moved to and fro, crossing over one another's paths and exchanging places wile traversing in front of the high table, they were acting out a calculation on the curvature of a four-dimensional manifold, involving various steps of symmetrization, antisymmetrization, and raising and lowering of indices. Seen from above by someone who didn't know any theorics, it would have looked like a country dance.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've always wanted to see it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(In his book the word &quot;theorics&quot; means &quot;physics&quot;.)&lt;/p&gt;&#xA;" OwnerUserId="6656" LastActivityDate="2019-11-25T16:08:43.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17522" PostTypeId="2" ParentId="17510" CreationDate="2019-11-25T16:29:48.737" Score="2" Body="&lt;p&gt;If you do want this effect (some of the comments are right that there may be other ways to achieve your pedagogical goal), Sage Math has a 3d viewer called &quot;Jmol&quot; which has this ability built in.  Try &lt;a href=&quot;https://sagecell.sagemath.org/?z=eJwrTkxPtVJI16jQqdS0Lc7M06iIM9KujDPS5OUqBksV5OSXGKdopOsAleia6phq6mhUQhllmanlqUW2Slm5-TlKmgBBExW9&amp;amp;lang=sage&amp;amp;interacts=eJyLjgUAARUAuQ==&quot; rel=&quot;nofollow noreferrer&quot;&gt;this live example&lt;/a&gt; from &lt;a href=&quot;http://doc.sagemath.org/html/en/prep/Symbolics-and-Basic-Plotting.html#basic-3d-plotting&quot; rel=&quot;nofollow noreferrer&quot;&gt;the documentation&lt;/a&gt;, and right-click (Command-click on Mac) to get &quot;Style&quot;, then &quot;Stereographic&quot;, which will give you a number of options for this kind of viewing.  &lt;a href=&quot;https://sagecell.sagemath.org/?z=eJwrSyzSUK_QqdSpUtfk5SrIyS-JL0tNLskvik_LTM1JMU7R0NCt1NGt0qnQ1FHQqNDRNdYx1tTRqIQxqiAMTQAoWhPw&amp;amp;lang=sage&amp;amp;interacts=eJyLjgUAARUAuQ==&quot; rel=&quot;nofollow noreferrer&quot;&gt;Another example&lt;/a&gt; in &lt;a href=&quot;http://doc.sagemath.org/html/en/prep/Advanced-2DPlotting.html#vector-fields&quot; rel=&quot;nofollow noreferrer&quot;&gt;the documentation which mentions this&lt;/a&gt; is a vector field plot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that in the future you will need to use &lt;code&gt;viewer=&quot;jmol&quot;&lt;/code&gt; in any Sage 3d plot command as the default viewer is changing.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2019-11-25T16:29:48.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17523" PostTypeId="2" ParentId="17516" CreationDate="2019-11-25T16:51:55.943" Score="5" Body="&lt;p&gt;Are you an undergrad majoring in aerospace engineering, or are you interested prerequisites  from graduate programs in aerospace engineering?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In either case, &lt;a href=&quot;https://ceas.uc.edu/content/dam/aero/docs/graduate/Graduate%20Math%20Class%20Requirements%20in.pdf&quot; rel=&quot;noreferrer&quot;&gt;the following syllabus may help you consider courses as an undergrad so that you can round out strictly calculus coursework.&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an undergrad program, &lt;a href=&quot;http://www.sjsu.edu/ae/programs/bsae/bsae_prerequisite/&quot; rel=&quot;noreferrer&quot;&gt;note the following syllabus&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For great references on aerospace engineering, see Penn State's list &lt;a href=&quot;https://guides.libraries.psu.edu/aerospaceengineering&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-11-25T16:59:55.500" LastActivityDate="2019-11-25T16:59:55.500" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17524" PostTypeId="2" ParentId="17497" CreationDate="2019-11-25T17:32:45.247" Score="2" Body="&lt;h2&gt;Graph traversals&lt;/h2&gt;&#xA;&lt;p&gt;Many students do a lot of graph doodling in discrete mathematics courses, at least at first before they master the theoretical techniques. Instead of having your students draw the graphs on paper, have them lay them out on the floor or ground. Having access to a gym or an outdoor area helps, but isn't truly necessary if you move desks and chairs out of the way. Students then walk along various paths to simulate various traversals. They can also drop &amp;quot;bread crumb&amp;quot; tokens along the way to track where they have been.&lt;/p&gt;&#xA;" OwnerUserId="7481" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-11-25T17:32:45.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17525" PostTypeId="1" CreationDate="2019-11-25T17:47:24.267" Score="4" ViewCount="366" Body="&lt;p&gt;I've spoken to several people who attended US universities in the decades before I was born, and I was somewhat surprised to find that it seemed to be common (based on the anecdotes I received) for non-STEM majors to top out at what is now considered high school algebra (e.g. things like competing the square, and basic conic sections).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would have been a typical or average mathematics curriculum for a US university student around 1950 who was &lt;em&gt;not&lt;/em&gt; majoring in mathematics, a research science, or engineering? Would a student majoring in, say, English, art history, or theology have expected to have gone further than basic algebra?&lt;/p&gt;&#xA;" OwnerUserId="7481" LastActivityDate="2019-12-22T19:09:14.567" Title="How much math would a non-STEM major have studied in 1950?" Tags="|undergraduate-education|history|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="17527" PostTypeId="2" ParentId="17510" CreationDate="2019-11-25T21:57:16.743" Score="3" Body="&lt;p&gt;I remembered GeoGebra doing this several years ago, but I thought that functionality died with the last &quot;classic&quot; version of the program. But, I see it can still be done via the web app.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Go to the &lt;a href=&quot;https://www.geogebra.org/3d&quot; rel=&quot;nofollow noreferrer&quot;&gt;3D GeoGebra web app&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;In the input bar, type in some vectors. To type a vector, the notation is: Vector((starting point), (ending point))&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/sNUJs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sNUJs.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You should see a typical 3D rendering of the vectors.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/aGkUC.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aGkUC.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Now right-click somewhere on the graphing window, and you should see a pop-up window. Select the last option for &lt;strong&gt;Settings&lt;/strong&gt;: &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/yVCrK.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yVCrK.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You should now get another pop-up/side-scroll window. Select the last tab for &lt;strong&gt;Projection&lt;/strong&gt;, and click the button that looks like a pair of glasses.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JrXX5.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JrXX5.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Now the image should render as an anaglyph for red/cyan glasses.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LTCxt.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LTCxt.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This works with other functions as well. For example,  &lt;span class=&quot;math-container&quot;&gt;$z = \frac{6xy}{e^{0.2x^2 + 0.2y^2}}$&lt;/span&gt;. [I don't have a pair of 3D glasses, so I cannot tell you how well this works in practice.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KKwst.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KKwst.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Edit&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I forgot to mention the &quot;animation&quot; part of your question. You can rotate the image by clicking and dragging the image, but getting the image to automatically rotate around, say, the z-axis, requires a line or two of code and doesn't seem to run as smoothly as just clicking and dragging.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2019-11-25T23:11:25.897" LastActivityDate="2019-11-25T23:11:25.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17528" PostTypeId="1" CreationDate="2019-11-25T21:59:37.710" Score="11" ViewCount="507" Body="&lt;p&gt;L'Hôpital's rule for the indeterminate form &lt;span class=&quot;math-container&quot;&gt;$\frac00$&lt;/span&gt; at finite points can be given a nice intuitive explanation in terms of local linear approximations.  See for instance &lt;a href=&quot;https://activecalculus.org/single/sec-2-8-LHR.html&quot; rel=&quot;noreferrer&quot;&gt;this textbook&lt;/a&gt; or &lt;a href=&quot;https://openstax.org/books/calculus-volume-1/pages/4-8-lhopitals-rule&quot; rel=&quot;noreferrer&quot;&gt;this one&lt;/a&gt;.  And for limits as x approaches &lt;span class=&quot;math-container&quot;&gt;$\pm\infty$&lt;/span&gt;, we can just let &lt;span class=&quot;math-container&quot;&gt;$u = 1/x$&lt;/span&gt; and take &lt;span class=&quot;math-container&quot;&gt;$\lim_{u\to 0^+} \frac{f(1/u)}{g(1/u)}$&lt;/span&gt;, use the ordinary L'Hôpital's rule to get &lt;span class=&quot;math-container&quot;&gt;$\lim_{u\to 0^+} \frac{f'(1/u) \cdot(-1/u^2)}{g'(1/u)\cdot(-1/u^2)}$&lt;/span&gt; and cancel the &lt;span class=&quot;math-container&quot;&gt;$(-1/u^2)$&lt;/span&gt;'s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I've looked through half a dozen textbooks and not found even the barest hint of an explanation for why L'Hôpital's rule should also apply to the indeterminate form &lt;span class=&quot;math-container&quot;&gt;$\frac\infty\infty$&lt;/span&gt;.  Even textbooks (like those linked above) that give an intuitive explanation for the &lt;span class=&quot;math-container&quot;&gt;$\frac00$&lt;/span&gt; case simply state the &lt;span class=&quot;math-container&quot;&gt;$\frac\infty\infty$&lt;/span&gt; case as an unjustified fact.  Is there any intuitive explanation or (perhaps nonrigorous) calculation that explains this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best I've been able to come up with is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{1/g(x)}{1/f(x)} = \lim_{x\to a} \frac{-g'(x)/g(x)^2}{-f'(x)/f(x)^2} = \lim_{x\to a} \frac{f(x)^2 g'(x)}{g(x)^2 f'(x)} = \left(\lim_{x\to a} \frac{f(x)}{g(x)}\right)^2 \left(\lim_{x\to a} \frac{g'(x)}{f'(x)}\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;using L'Hôpital for &lt;span class=&quot;math-container&quot;&gt;$\frac00$&lt;/span&gt; and assuming that the limits of &lt;span class=&quot;math-container&quot;&gt;$f/g$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g'/f'$&lt;/span&gt; exist.  Then if both these limits are also nonzero, we can cross-cancel to get &lt;span class=&quot;math-container&quot;&gt;$\lim (f/g) = \lim(f'/g')$&lt;/span&gt;.  I'd prefer a more geometric explanation, but the real problem is that the assumptions that &lt;span class=&quot;math-container&quot;&gt;$\lim(f/g)\neq 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\lim(g'/f')$&lt;/span&gt; exists fail even in a simple case like &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\lim_{x\to \infty} \frac{x}{e^x}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more rigorous form of this argument was discussed &lt;a href=&quot;https://math.stackexchange.com/q/147356/91608&quot;&gt;on MSE&lt;/a&gt;, with the conclusion that it's not possible to deduce the &lt;span class=&quot;math-container&quot;&gt;$\frac\infty\infty$&lt;/span&gt; rule from the &lt;span class=&quot;math-container&quot;&gt;$\frac00$&lt;/span&gt; one in general.  I'm not looking for a rigorous proof, so I'm hoping the situation may be easier.  Can anyone give an informal justification of the &lt;span class=&quot;math-container&quot;&gt;$\frac\infty\infty$&lt;/span&gt; rule that applies when the limit equals 0 and would be comprehensible to calculus students?&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2020-01-05T23:41:44.700" Title="An intuitive explanation of l'Hôpital's rule for ∞/∞" Tags="|calculus|limits|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17529" PostTypeId="2" ParentId="17497" CreationDate="2019-11-26T03:48:32.123" Score="2" Body="&lt;h1&gt;Teaching the addition and subtraction of negative numbers by walking along a number line.&lt;/h1&gt;&#xA;&lt;p&gt;The idea of the exercise is that you draw out a number line on the floor, and have your students stand next to it on the zero position, facing the positive direction. You then have them calculate additions and subtractions by walking along the line; whenever they encounter a &amp;quot;-&amp;quot; symbol, they turn around to face the other way before walking and then turn around to face the positive direction again once they stop.&lt;/p&gt;&#xA;&lt;p&gt;So, for instance, if you tell the students to work out &amp;quot;5+-3--2&amp;quot;, they'd take five steps forward, turn around, take three steps forward, turn around to face the positive direction, then turn around twice and walk two steps forward, so that they're at the position on the number line corresponding to &amp;quot;4&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="12989" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-11-26T03:48:32.123" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17530" PostTypeId="2" ParentId="17528" CreationDate="2019-11-26T05:02:32.530" Score="8" Body="&lt;p&gt;Your approach is in fact geometrical, if we see all this happening on the Riemann sphere, i.e., the one-point compactification of &lt;span class=&quot;math-container&quot;&gt;${\bf C}={\bf R}^2$&lt;/span&gt;. Briefly, we have the usual coordinate &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;${\bf C}$&lt;/span&gt;, and the coordinate &lt;span class=&quot;math-container&quot;&gt;$u=\displaystyle{\frac{1}{z}}$&lt;/span&gt; on a neighborhood of point at infinity. Calculus on this neighborhood is conducted in terms of the coordinate &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; by the transformation &lt;span class=&quot;math-container&quot;&gt;$du=-\frac{1}{z^2}\,  dz$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now in the case of &lt;span class=&quot;math-container&quot;&gt;$\displaystyle{\lim_{z\to a}f(z)}=\infty$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\displaystyle{\lim_{z\to a}g(z)}=\infty$&lt;/span&gt;, with certain assumptions on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, we can think of &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(z)$&lt;/span&gt; as coordinates around the point at infinity. In this framework, your derivation of the L'Hôpital rule for the case &lt;span class=&quot;math-container&quot;&gt;$\infty/\infty$&lt;/span&gt; is quite geometrical. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason why a function like &lt;span class=&quot;math-container&quot;&gt;$h(z)=e^z$&lt;/span&gt; does not fit into this framework is because &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; has an essential singularity at infinity, so &lt;span class=&quot;math-container&quot;&gt;$h(z)$&lt;/span&gt; fails spectacularly as a coordinate in a neighborhood of infinity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course I understand that an instructor cannot take this geometrical framework into a calculus class and expect it to be helpful. But this does explain why your approach is the correct and  natural approach. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2019-11-26T11:02:26.207" LastActivityDate="2019-11-26T11:02:26.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17531" PostTypeId="2" ParentId="17497" CreationDate="2019-11-26T09:52:49.497" Score="1" Body="&lt;p&gt;As you say College-level: Simulate field biologists' jobs, for any statistics class (or for data input for mathematical modelling). Basically any physically spaced situation you can recreate, and choose a mathematical level appropriate for your group (how science-heavy is your course?; US college = university; UK college = 16--18y olds; etc).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my background: a capture-mark-release-recapture methodology to estimate population properties (popsize, sexratio, individual characteristics, clusters/subgroups, ... ): Say you have 10 students, split them in two groups. Space a 10x10 nontransparent containers in the room, each representing a search area. They contain whatever you're trying to study --- marbles of 3-4colours and 2-3 sizes, or even just text. In the first phase, half the students get to each sample 10 random containers and mark the contents -- which is like ringing birds, or tagging any animal, -- followed by a second phase where the other half of students is in charge of randomizing the contents across boxes (should they stay in clusters? should they change independently or are some 'paired for life'? do some stay together? With 100 containers and 50 red balls, 30 small blue balls, 10 bigger blue balls, how many containers expected empty? the second group knows these numbers, the first is trying to estimate them; the second group can calculate the expected numbers the first group will find). Then third phase 1st group students sample another 10 boxes each (if the students capture 24 red first round, and 26 second round then you expect 12 are marked). So it's all confidence intervals and Student-tests etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Containers can be simple origami boxes from another lesson (harder to make them look uniform); or nontransparent paper cups; or ... .&lt;/p&gt;&#xA;" OwnerUserId="13095" LastActivityDate="2019-11-26T09:52:49.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17532" PostTypeId="2" ParentId="17528" CreationDate="2019-11-26T15:31:34.650" Score="4" Body="&lt;p&gt;This is from a story: The Hodja was escorting Diba and Rushkady to a pancake-eating contest.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Look&quot;, said Diba.  &quot;Let's say you get a head start at the pancake eating contest.&#xA;  If I eat pancakes twice as fast as you, eventually I'll have eaten almost&#xA;  twice as many pancakes as you'll have done.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;You'll never eat more pancakes than me!&quot; Rushkady shouted.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;I meant hypothetically.  If a function grows at twice the rate as another,&#xA;  it will approach being twice as great.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;Hmm. I guess I can see it when the functions are linear.&#xA;  If, say, the number of pancakes I eat is &lt;span class=&quot;math-container&quot;&gt;$f(t)=2t$&lt;/span&gt;&#xA;  and the number you eat is &lt;span class=&quot;math-container&quot;&gt;$g(t) = t + 100$&lt;/span&gt;&#xA;  (the &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; being from &lt;em&gt;your&lt;/em&gt; head start!),&#xA;  then the limit of &lt;span class=&quot;math-container&quot;&gt;$2t/(t+100)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; goes to infinity.&#xA;  But are you sure it works when the rate isn't constant?&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Diba thought a while.  &quot;Yes, I think so.  If you eat pancakes&#xA;  faster at some times and slower at others, as long as &lt;em&gt;I&lt;/em&gt;&#xA;  eat pancakes exactly twice as fast as &lt;em&gt;you&lt;/em&gt;,&#xA;  I'll eventually approach eating twice as many.&#xA;  I mean, every time you eat one, I'll eat two.  If you eat&#xA;  ten, I'll eat twenty.  And so on. When you've &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; pancakes beyond your head start, I will have eaten &lt;span class=&quot;math-container&quot;&gt;$2t$&lt;/span&gt;, so it's just like the first case.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;I guess.  But you're saying that the limit of &lt;span class=&quot;math-container&quot;&gt;$f(x)/g(x)$&lt;/span&gt;&#xA;  is the same as the &lt;em&gt;limit&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$f'(x)/g'(x)$&lt;/span&gt;.  One might&#xA;  not be eating exactly twice as fast as another.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;OK, suppose &lt;span class=&quot;math-container&quot;&gt;$f'(x)/g'(x)$&lt;/span&gt; is increasing toward &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; as a limit.&#xA;  Then eventually &lt;span class=&quot;math-container&quot;&gt;$f'(x)/g'(x)$&lt;/span&gt; will be at least &lt;span class=&quot;math-container&quot;&gt;$1.9$&lt;/span&gt;.&#xA;  That means &lt;span class=&quot;math-container&quot;&gt;$f(x)/g(x)$&lt;/span&gt; will approach at least &lt;span class=&quot;math-container&quot;&gt;$1.9$&lt;/span&gt;.&#xA;  But eventually &lt;span class=&quot;math-container&quot;&gt;$f'(x)/g'(x)$&lt;/span&gt; will be greater than &lt;span class=&quot;math-container&quot;&gt;$1.99$&lt;/span&gt; and remain so.&#xA;  Then &lt;span class=&quot;math-container&quot;&gt;$f(x)/g(x)$&lt;/span&gt; will approach at least &lt;span class=&quot;math-container&quot;&gt;$1.99$&lt;/span&gt;, in the long run.&#xA;  And so on, toward &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;Oh.  I see now.  The limits will be the same.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;I don't know.  Somehow I think&#xA;  it only works if the limit of the derivatives exists,&quot;&#xA;  said The Hodja.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;But the bus had arrived at the pancake eating contest,&#xA;  and Diba's and Rushkady's thoughts turned to wondering&#xA;  whether either of them could eat half as fast as The Hodja.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The Hodja's comment is illustrated by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\underset{t\to \infty }{\text{lim}}&#xA; \frac{t+\cos t}&#xA;      {t+\sin t}\,.$$&lt;/span&gt;&#xA;I suppose your textbook has its own examples.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2019-11-28T15:57:20.073" LastActivityDate="2019-11-28T15:57:20.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17533" PostTypeId="1" CreationDate="2019-11-26T17:07:04.953" Score="2" ViewCount="253" Body="&lt;p&gt;The numerator is structured like:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(-b)\pm\sqrt{b^2- 4ac}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it confusing or acceptable to distinguish between the following two things?&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;An idiom; and&lt;/li&gt;&#xA;&lt;li&gt;What is or seems to be a compositionally transparent and faithful representation of meaning.  Observe that if we replace expressions (a,b) with either &quot;open_interval_(a,b)&quot; or &quot;ordered_pair_(a,b)&quot;, then we have taken a step towards disambiguation.  However,  that is technically not a matter of direct disambiguation of individual morphograph-like vocabulary items, unless we actually introduce &lt;strong&gt;four&lt;/strong&gt; symbols: open_interval_left_bracket, open_interval_right_bracket, ordered_pair_left_bracket, ordered_pair_right_bracket.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Should &quot;plus or minus&quot; be introduced as an operation, or as an informal notation that is a memory aid, with set theory not being involved?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose that we consider an operation on class variables in set theory: could we get something &lt;strong&gt;like&lt;/strong&gt; an unordered pair of proper classes that isn't merely the empty set, or does this get into an area of taboo, like ancient taboos associated with zero, or more recent taboos associated with infinity or infinitesimals?&lt;/p&gt;&#xA;" OwnerUserId="12816" LastEditorUserId="238" LastEditDate="2019-11-26T18:11:23.720" LastActivityDate="2019-11-26T23:10:20.047" Title="How should one approach the concept of &quot;plus or minus&quot;, such as in the numerator of the quadratic formula?" Tags="|mathematical-pedagogy|algebra|notation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17534" PostTypeId="2" ParentId="17533" CreationDate="2019-11-26T17:36:54.747" Score="7" Body="&lt;p&gt;When solving a quadratic equation, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ax^2+ bx+ c = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;we use shorthand for the two solutions, to include both &lt;span class=&quot;math-container&quot;&gt;$$x_1 = \frac {-b +\sqrt{b^2-4ac}}{2a}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and &lt;span class=&quot;math-container&quot;&gt;$$x_2 = \frac{-b - \sqrt{b^2 -4ac}}{2a}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, the shorthand, &lt;span class=&quot;math-container&quot;&gt;$$x_i = \frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I.e., the solutions to &lt;span class=&quot;math-container&quot;&gt;$ax^2+bx+c = 0$&lt;/span&gt; are given by &lt;span class=&quot;math-container&quot;&gt;$$x\in \left\{ \frac {-b +\sqrt{b^2-4ac}}{2a},\frac{-b - \sqrt{b^2 -4ac}}{2a}\right\}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$\pm$&lt;/span&gt; is shorthand, and is not itself an arithmetic operation. So set theory is not involved.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-11-26T18:19:00.647" LastActivityDate="2019-11-26T18:19:00.647" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17535" PostTypeId="2" ParentId="17533" CreationDate="2019-11-26T23:10:20.047" Score="7" Body="&lt;p&gt;Here's why you should take great care when considering &lt;span class=&quot;math-container&quot;&gt;$\pm$&lt;/span&gt; as an operator. It's not unusual to see a sentence of the form&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We deduce that &lt;span class=&quot;math-container&quot;&gt;$A=\pm B$&lt;/span&gt; and hence that &lt;span class=&quot;math-container&quot;&gt;$C=D\pm E$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This isn't simply saying that both (&lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is either &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-B$&lt;/span&gt;) and (either &lt;span class=&quot;math-container&quot;&gt;$C=D-E$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$C=D+E$&lt;/span&gt;). When two &lt;span class=&quot;math-container&quot;&gt;$\pm$&lt;/span&gt;'s appear in the same sentence it is implied that they are both to be read together, in this case as either (&lt;span class=&quot;math-container&quot;&gt;$A=B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C=D+E$&lt;/span&gt;) or (&lt;span class=&quot;math-container&quot;&gt;$A=-B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C=D-E$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it makes more sense here to read &lt;span class=&quot;math-container&quot;&gt;$\pm$&lt;/span&gt; as shorthand.&lt;/p&gt;&#xA;" OwnerUserId="6359" LastActivityDate="2019-11-26T23:10:20.047" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17536" PostTypeId="1" CreationDate="2019-11-28T01:09:14.860" Score="4" ViewCount="491" Body="&lt;p&gt;I would be grateful for a comparison between &#xA;the capabilities of &lt;a href=&quot;https://www.cinderella.de/tiki-index.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cinderella&lt;/a&gt; and&#xA;&lt;a href=&quot;https://www.geogebra.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geogebra&lt;/a&gt;, for teaching&#xA;at all levels, but especially at the college/university-level.&#xA;I became a reasonably adept user of Cinderella,&#xA;but have only dabbled in GeoGebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is my naive, nonexpert sense of where matters might stand:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Cinderella is more mathematically sophisticated, but&#xA;  now less actively supported, and with a smaller user base than GeoGebra. So it might be wise, pragmatically, to &quot;switch&quot; to GeoGebra.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To support my sense that &quot;Cinderella is more mathematically sophisticated&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=y7Ori_ehpLUC&amp;amp;pg=PA34&amp;amp;lpg=PA34&amp;amp;dq=cinderella+and+complex+numbers&amp;amp;source=bl&amp;amp;ots=jjDxzAtZ7y&amp;amp;sig=ACfU3U34SagOUF2wvULRH8cUJdquqVTBHw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiAmJWT2IvmAhURd98KHSbuBicQ6AEwC3oECA0QAQ#v=onepage&amp;amp;q=cinderella%20and%20complex%20numbers&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Cinderella Manual&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Cinderella's mathematical kernel is implemented entirely over the complex numbers&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://wiki.geogebra.org/en/Complex_Numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;GeoGebra manual&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;GeoGebra does not support complex numbers directly&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-11-28T01:33:09.690" LastActivityDate="2020-09-28T12:29:17.047" Title="Cinderella vs. GeoGebra" Tags="|geometry|software|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17537" PostTypeId="1" AcceptedAnswerId="17538" CreationDate="2019-11-28T16:18:17.677" Score="2" ViewCount="133" Body="&lt;p&gt;I've just discovered &quot;calculated questions&quot; in Moodle and I'm trying to create a simple one where I would be asking the student to find the probability that an observation from a standard normal population be greater than variable &lt;span class=&quot;math-container&quot;&gt;$\left\{z\right\}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I then define the range of &lt;span class=&quot;math-container&quot;&gt;$\left\{z\right\}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$[-3,3]$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; decimal places.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now since Moodle doesn't have erf implemented I have been trying to emulate it using Jack D'Aurizio's answer &lt;a href=&quot;https://math.stackexchange.com/q/1172532&quot;&gt;here&lt;/a&gt;, and otherwise Abramowitz and Stegun's first approximation given &lt;a href=&quot;https://en.wikipedia.org/wiki/Error_function#Approximation_with_elementary_functions&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, which results in &lt;span class=&quot;math-container&quot;&gt;$$P(z_i&amp;gt; z)\approx \frac{1-\sqrt{1-e^{-0.619495805z^2}}}2$$&lt;/span&gt; for the former, for instance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am facing the problem that &lt;strong&gt;Moodle refuses to calculate this when &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; is less than &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;&lt;/strong&gt; (I get the output &lt;strong&gt;&lt;em&gt;NaN&lt;/em&gt;&lt;/strong&gt;) and as for the values where it will output something, &lt;strong&gt;the outputs don't seem to depend much on &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;&lt;/strong&gt;, I get only four different possible outputs which are &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$0.005$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$0.035$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$0.102$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two examples copy-pasted from Moodle: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;1-(1+sqrt(1-exp(-0.619495805*((40.14-11.7)/4.3)^2)))/2 = 0.035&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;1-(1+sqrt(1-exp(-0.619495805*((33.18-13)/4.1)^2)))/2 = 0.102&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where here &lt;span class=&quot;math-container&quot;&gt;$\left\{z\right\}$&lt;/span&gt; was actually calculated via &lt;span class=&quot;math-container&quot;&gt;$$\left\{z\right\}=\frac{\left\{x\right\}-\left\{m\right\}}{\left\{s\right\}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know how to work around this and get actual proper values for the normal inverse function?&lt;/p&gt;&#xA;" OwnerUserId="12754" LastActivityDate="2019-11-28T22:25:22.690" Title="How to emulate erf and/or the Normal Inverse function in Moodle?" Tags="|software|probability|specific-software|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17538" PostTypeId="2" ParentId="17537" CreationDate="2019-11-28T22:25:22.690" Score="1" Body="&lt;p&gt;I've never used Moodle, but according to &lt;a href=&quot;https://docs.moodle.org/38/en/Calculated_question_type&quot; rel=&quot;nofollow noreferrer&quot;&gt;moodle.org&lt;/a&gt; you cannot use &lt;code&gt;^&lt;/code&gt; for powers; instead use &lt;code&gt;pow({foo}, 2)&lt;/code&gt;.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2019-11-28T22:25:22.690" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17539" PostTypeId="1" AcceptedAnswerId="17546" CreationDate="2019-11-29T03:54:17.010" Score="18" ViewCount="4018" Body="&lt;p&gt;Before complex numbers are introduced in senior high school courses, should we emphasise that solutions (e.g. to quadratic equations) are &lt;strong&gt;real&lt;/strong&gt; solutions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we do, then when non-real numbers finally arrive, students are (in theory) already somewhat primed for the fact that there are different kinds of numbers. This is already established by distinguishing natural numbers, integers, rational numbers, and irrational numbers. But without the context of complex numbers, saying that the set of rationals and irrationals make the set of real numbers means that no proper meaning is given to the term “real”. And therefore, I speculate, students will either dismiss the word real, or conclude that we’ve finally reached the &lt;em&gt;real&lt;/em&gt; set of all numbers (integers and rationals are not &lt;em&gt;really&lt;/em&gt; all the numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder whether we shouldn’t instead talk about &lt;strong&gt;all possible numbers on the continuous number line&lt;/strong&gt;. We can still say just “numbers” for short, but this longer qualification has immediately meaningful value (to Year 11/12 students) and doesn’t create any misconceptions about the word “real” that have to be undone when “imaginary” numbers are eventually presented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be clear, I’m asking about the language we should be using well before imaginary and complex numbers are introduced, and whether it matters much. Maybe most students who are beginning to learn complex numbers can probably handle a simple explanation about the terminology, and how everyday English meanings and mathematical meanings are not the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: And to be clearer, I think what I wanted to ask was about how best to help students avoid developing misconceptions around the term “real numbers”. But as usual, I was consumed in my thinking at the time. To be honest, I have learned (over the last two years) to introduce formal terms early, even if the larger context of those terms remains to be taught much later.&lt;/p&gt;&#xA;" OwnerUserId="7046" LastEditorUserId="7046" LastEditDate="2019-12-03T18:42:14.097" LastActivityDate="2019-12-03T19:49:25.223" Title="Should high school teachers say “real numbers” before teaching complex numbers?" Tags="|mathematical-pedagogy|terminology|real-numbers|" AnswerCount="8" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="17540" PostTypeId="2" ParentId="17399" CreationDate="2019-11-29T04:00:12.223" Score="1" Body="&lt;p&gt;… because students learning more about real numbers at university bring their understanding of real numbers that they developed through high school. Which is, &lt;strong&gt;real numbers are the set of rational and irrational numbers: all possible numbers on a single continuous number line&lt;/strong&gt;. And that’s a correct working schema to begin learning more about real numbers and develop formal definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In both learning and history, informal understandings typically preceded rigorous formal definitions.&lt;/p&gt;&#xA;" OwnerUserId="7046" LastActivityDate="2019-11-29T04:00:12.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17541" PostTypeId="2" ParentId="17539" CreationDate="2019-11-29T04:25:34.800" Score="2" Body="&lt;p&gt;In my opinion if some numbers are &quot;imaginary&quot; it doesn't mean they don't exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is needed to distinguish somehow between the two kinds of numbers, the &quot;real&quot; that we easily see im our daily life and that make sense to everyone, and the &quot;imaginary&quot; numbers (or complex in general) that are there in life, real of course with no doubt, but they're not easy to &quot;see&quot; in the daily normal life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think it is just the job of the teacher to emphasize the idea of complex numbers and explain their importance in many fields, I don't think there is a problem in the naming.&lt;/p&gt;&#xA;" OwnerUserId="12486" LastEditorUserId="6724" LastEditDate="2019-11-30T20:28:39.680" LastActivityDate="2019-11-30T20:28:39.680" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17542" PostTypeId="2" ParentId="17539" CreationDate="2019-11-29T07:14:55.030" Score="4" Body="&lt;p&gt;Regarding &lt;strong&gt;all possible numbers on the continuous number line&lt;/strong&gt;, in my opinion you're overthinking this. The vast majority of students aren't concerned with what the numbers are called, but how to solve the problems they have for homework and on tests. As for the use of &lt;strong&gt;real&lt;/strong&gt; being unnecessary in the absence of knowing about complex numbers, I disagree. For them, the word &lt;strong&gt;number&lt;/strong&gt; by itself could mean a positive integer, an integer, a rational number, or any real number, depending on context, and most good writing will not rely on context, but rather use the appropriate qualifying word. In particular, by saying &lt;strong&gt;real number&lt;/strong&gt; we are telling the reader/listener that &quot;anything goes&quot; so to speak, no restrictions. Regarding why we use the word &quot;real&quot;, students naturally might wonder about this, so tell them this word has been in use for a few hundred years to distinguish (if needed) from imaginary/complex numbers, something they'll learn about in a later math class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I say &quot;overthinking&quot;, and here I'm thinking of my U.S. school classmates from the early to mid 1970s, is that surely nearly all have heard about complex numbers being one of the differences in solving quadratic equations in Algebra 1 vs. in Algebra 2. Whether eating with friends in the cafeteria, attending daily sports practices after school, on long bus/van trips for various extra-curricula events after school, sitting and talking with people in homeroom or elsewhere after getting to school but before the first homeroom bell rings, ... (the list is endless) --- students talk. And talk. Often about teachers (their quirks, how hard/easy certain ones are, etc.), and sometimes even about what is covered in their classes (especially while quickly trying to complete unfinished homework). Anyone who is the least bit curious about whether real numbers consist of all possible numbers will surely have heard someone mention complex numbers at some time. And for the very few who have an interest in math, they surely would have seen complex numbers show up when flipping through someone's more advanced mathematics text, flipping through more advanced texts in their school library, looking up &quot;number&quot; in encyclopedias in the library, from their high school math teacher if asked, etc. And all this discussion was restricted to before the internet and our present nearly instant and total access to knowledge about most anything a student is likely to think of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think your last sentence is on the mark. Students will have seen many uses of words in math and in other subjects that differ from their everyday use. There are many words used in math classes that they will have seen and used before, but which have much more precise and specific meanings in math, such as &quot;variable&quot;, &quot;constant&quot;, &quot;average&quot;, &quot;positive&quot;, &quot;degree&quot;, &quot;origin&quot;, &quot;exponential growth&quot;, &quot;similar&quot;, etc. So they will be familiar with this sometimes mismatch between everyday meaning and subject-specific meaning for some words, and it's simply a matter of saying that &quot;real number&quot; is another example, mentioning briefly how &quot;real&quot; came to be in use, although I suspect that most any student today who is remotely interested in the &quot;how&quot; could easily find out with a 5 second google search leading to a Wikipedia article.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, to answer your question at the beginning, I'd say &quot;yes&quot;, and briefly explain why &lt;strong&gt;real&lt;/strong&gt; is used. In my case, I often said that complex numbers live in a plane, but for us we're only finding solutions that live on the (real) number line.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2019-11-29T07:14:55.030" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17544" PostTypeId="2" ParentId="17539" CreationDate="2019-11-29T17:28:40.453" Score="-1" Body="&lt;p&gt;I wouldn't. You feel the gap because you know what's coming.  But qualifying like that before the students have the context leads to bafflement.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's sort of a routine issue where people who already know the stuff want to present it perfect (complete).  But this is not pedagogically sound.  Suited to careful math explanation, but not to learning.  &lt;/p&gt;&#xA;" OwnerUserId="13106" LastActivityDate="2019-11-29T17:28:40.453" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17545" PostTypeId="2" ParentId="17539" CreationDate="2019-11-29T18:42:32.090" Score="7" Body="&lt;p&gt;For 95% of high school students, this sort of thing is of no interest. But:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The 5% do need to be served well and helped to achieve their potential.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The 95% may find such things &lt;em&gt;confusing&lt;/em&gt; if they are never explained, so it makes sense to offer them at least some brief explanation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even the 5% are in no position at this point to understand fully what is meant by the real number system. This would require appreciating the completeness property of the reals, which is really not even treated very carefully or completely in most calculus classes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I think the way most algebra 1 and algebra 2 textbooks treat this is pretty reasonable in view of these facts. They introduce all the elementary axioms of the reals, but not completeness. They say that there is a number system, called the real numbers, which satisfies these axioms, and which also includes our old friends such as &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; (because students have almost certainly encountered these things before high school). At appropriate points in the development of the subject, they mention briefly that these numbers do exist in the real number system, but that we can't prove that based on the axioms we've listed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They may or may not mention &quot;real&quot; vis a vis &quot;complex,&quot; or provide any brief characterization of the complex numbers as including &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;. I think it would be fine to do this, but it should be very brief. Students should not be led to believe, incorrectly, that a symbol like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; has meaning just because we say it does. For the 5%, it might not hurt to point out to them that a system including &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; is not going to be compatible with the real-number axioms about ordering.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I was writing a textbook at this level, I would probably provide an appendix on the complex numbers, and then in the main text just point to the appendix. Then students who are in the 5% have something they can look at if they're curious, and they'll get a careful, clear presentation rather than something brief and garbled. If you're a teacher who is using a textbook that doesn't do this, you could point them to something like James Nearing's nice treatment in ch. 3 of this free online book:  &lt;a href=&quot;http://www.physics.miami.edu/~nearing/mathmethods/&quot; rel=&quot;noreferrer&quot;&gt;http://www.physics.miami.edu/~nearing/mathmethods/&lt;/a&gt; .&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-11-29T18:53:10.557" LastActivityDate="2019-11-29T18:53:10.557" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17546" PostTypeId="2" ParentId="17539" CreationDate="2019-11-29T19:30:50.333" Score="24" Body="&lt;h3&gt;Short Answer&lt;/h3&gt;&#xA;&lt;p&gt;You should not avoid use of the term &lt;em&gt;real numbers&lt;/em&gt;.  This is a term-of-art in mathematics, and it is important for students to learn the correct jargon.  However, this technical term should be introduced as such—emphasize that &amp;quot;real&amp;quot; in mathematics does not mean the same thing that &amp;quot;real&amp;quot; means in everyday vernacular English.&lt;/p&gt;&#xA;&lt;h3&gt;Long Answer&lt;/h3&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://matheducators.stackexchange.com/a/17542/8571&quot;&gt;answer provided by Dave L Renfro&lt;/a&gt; is quite excellent.  I would, however, like to approach things from a slightly different point of view.&lt;/p&gt;&#xA;&lt;p&gt;First off, you should absolutely refer to the set of real numbers as the set of real numbers.  I think that hiding this language, or trying to get around it, has the potential to lead to great confusion down the road.  As Dr.(?) Renfro points out, simply using the word &amp;quot;number&amp;quot; is ambiguous, and &amp;quot;an element of the continuous number line&amp;quot; is clunky.  Moreover, students will encounter the proper names of various sets of numbers in the future, so it seems wise to introduce the correct terminology early on.&lt;/p&gt;&#xA;&lt;p&gt;That being said, I entirely understand the concern—I cannot count the number of times that I have had a student in a college algebra or precalculus class tell me that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; doesn't exist because &amp;quot;it is imaginary.&amp;quot;  The terms &amp;quot;real&amp;quot; and &amp;quot;imaginary&amp;quot; are, I think, poor terms which reflect the biases of Enlightenment natural philosophers (i.e. proto-mathematicians).  However, for better or worse, we are stuck with these terms-of-art for the long run.&lt;/p&gt;&#xA;&lt;p&gt;So... what can we do about it?&lt;/p&gt;&#xA;&lt;p&gt;My strategy is to emphasize (early and often) that the words &amp;quot;real&amp;quot; and &amp;quot;imaginary&amp;quot; are used in mathematics to refer to specific sets of numbers, and that the use of these words is technical, and is different from the way that these words are used in vernacular English.  Often, I try to make this point via a spiel such as the following (this usually chews up most of a 50 minute lecture, as there are examples at every step of the process, plus time for back-and-forth with the students):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I am a strict 5-ist.  I do not believe that there is any actual number larger than &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;—in fact, there are only five numbers which actually exist:  &lt;span class=&quot;math-container&quot;&gt;$1,2,3,4,5$&lt;/span&gt;.  Every other number is a fiction created by people.  And I think that &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; are a little suspicious—I believe that they exist, but only just barely.&lt;/p&gt;&#xA;&lt;p&gt;Don't believe me?&lt;/p&gt;&#xA;&lt;p&gt;Try this: have a friend pick out some number of essentially indistinguishable objects (pennies, marbles, popsicle sticks, etc).  Then have this friend arrange these objects in no particular manner—in fact, have them just drop them from a short height so that they fall at random.  Close your eyes while your friend does this, then, once they have arranged the objects, open your eyes and determine how many objects there are as fast as you can.  If there are &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; or fewer objects, you will likely be able to determine the number almost instantaneously.  If there are &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; or more objects, you will likely have to count them—at the very least, it will likely take you significantly longer to determine the number.  Studies with functional MRI have shown that &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; is about the upper limit for the number of objects that the vast majority of human beings can instantaneously recognize, regardless of how they are arranged.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, any &amp;quot;whole number&amp;quot; bigger than &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; is obviously fiction.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, mathematics would be pretty boring and useless if we couldn't use numbers larger than &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;, so we have to start inventing some new numbers.  One of the nice properties of the numbers &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; is that we can always &amp;quot;add one more&amp;quot; and get a bigger number.&lt;/p&gt;&#xA;&lt;p&gt;So what do we get when we &amp;quot;add one&amp;quot; to &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;The utterly fictional number &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;!&lt;/p&gt;&#xA;&lt;p&gt;And what if we &amp;quot;add one&amp;quot; to that?&lt;/p&gt;&#xA;&lt;p&gt;We get &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;, which is also a pretend number.&lt;/p&gt;&#xA;&lt;p&gt;And so on.&lt;/p&gt;&#xA;&lt;p&gt;By this process, we get a whole infinite set of pretend numbers which, out of some kind of sadism, we call the &lt;strong&gt;natural numbers&lt;/strong&gt;.  I don't see anything &amp;quot;natural&amp;quot; about these numbers, but here we are.  These are the counting numbers.  These numbers can be used to count objects, and can be added (to, for example, count the number of objects in two different groups, then lump them together) or multiplied (to, for example, combine several identical groups of objects).  Indeed, for a very long time, these numbers were sufficient to do most of the things that humans needed to do.  Of course, people invented these numbers, and they are completely imaginary, but that doesn't prevent them from being useful.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, these so-called &amp;quot;natural&amp;quot; numbers don't let us handle debts very well.  For example, if you have a sheep that you want to sell me for &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; chickens, but I only have &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; chickens, you might give me the sheep anyway, but expect me to give you &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; more chickens in the future.  So, how many chickens do I have after this interaction?  Clearly, I have no chickens, but I still owe you &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; chickens.  I have a &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-chicken debt!  The natural numbers are no good for describing this situation, so we have to invent a new set of numbers, called the &lt;strong&gt;integers&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The integers consist of all of the natural numbers, plus &amp;quot;negative&amp;quot; natural numbers, as well as a special number called &amp;quot;zero&amp;quot;.  These &amp;quot;negative numbers&amp;quot; and &amp;quot;zero&amp;quot; are completely fictional, and the product of fevered human imaginations, but they are tremendously &lt;em&gt;useful&lt;/em&gt;, so I guess we're stuck with them.&lt;/p&gt;&#xA;&lt;p&gt;Of course, there are still things that you can't do with integers.  For example, if I split a pie evenly among six friends, how many pies does each friend get?  They clearly get more than &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; pies each, but fewer than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; pie each.  But there are no integers between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, so the integers don't cut the mustard.&lt;/p&gt;&#xA;&lt;p&gt;Enter the &lt;strong&gt;rational numbers&lt;/strong&gt;.  A &lt;strong&gt;ratio&lt;/strong&gt;nal number is the &lt;strong&gt;ratio&lt;/strong&gt; of two integers.  The are called &amp;quot;rational&amp;quot; because they are thought of as ratios—this has nothing to do with their being &amp;quot;based on or in accordance with reason or logic&amp;quot;.  Indeed, they are completely unreasonable and fictitious, but they serve a useful role, so I suppose we should keep them.&lt;/p&gt;&#xA;&lt;p&gt;For most practical purposes, the rational numbers are actually good enough. Calculators and other computers really only deal with certain rational numbers, and nearly every computation that you will ever do in your life is going to be done using rational numbers.  However, mathematicians are often interested in constructing more powerful tools and techniques for describing the world.  This often requires the use of more esoteric kinds of numbers, such as &amp;quot;algebraic numbers&amp;quot; (you need these if you want to give meaning to &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;), and &amp;quot;computable numbers&amp;quot; (these things are quite esoteric).&lt;/p&gt;&#xA;&lt;p&gt;In order to do calculus, we require a &lt;em&gt;continuum&lt;/em&gt; of numbers.  What this actually means is somewhat technical, but it leads to the introduction of the so-called &lt;strong&gt;real numbers&lt;/strong&gt;.  The &amp;quot;real&amp;quot; numbers include all of the rationals (which include the integers (which include the naturals))), the algebraic numbers, the computable numbers, and a bunch of &amp;quot;filler&amp;quot; which is really hard to describe.  Of course, the &amp;quot;real&amp;quot; numbers are not at all real.  Again, they are a human invention, and are just as &amp;quot;real&amp;quot; (or &amp;quot;unreal&amp;quot;) as the integers.  They are a very useful fiction.&lt;/p&gt;&#xA;&lt;p&gt;Finally, it should be noted that the &amp;quot;real&amp;quot; numbers are often (and incorrectly, in my opinion) contrasted with the &lt;strong&gt;imaginary numbers&lt;/strong&gt;.  &amp;quot;Imaginary&amp;quot; numbers are just as real as the &amp;quot;real&amp;quot; numbers (or, again, just as fictitious), but are called &amp;quot;imaginary&amp;quot; for historical reasons.  In some sense, we invented the integers so that we could subtract, and we invented the rationals so that we could divide.  Similarly, we invented the imaginary numbers (or, really, the &lt;strong&gt;complex numbers&lt;/strong&gt;) so that we could take roots (specifically, roots of negative numbers).&lt;/p&gt;&#xA;&lt;p&gt;The moral of the story is that the terms &amp;quot;real&amp;quot; and &amp;quot;imaginary&amp;quot; are &lt;em&gt;technical&lt;/em&gt; terms in mathematics.  Both the &amp;quot;real numbers&amp;quot; and the &amp;quot;imaginary numbers&amp;quot; are a set of numbers which are defined via technical mathematical construction.  These are equally &amp;quot;valid&amp;quot; types of numbers, and each kind of number is just as real or imaginary as the rest.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'll note that this approach to numbers is far from Platonic (and I know that there are a lot of Platonists out there pulling their hair out in my direction right now).  If you are uncomfortable with a non-Platonist point of view, you might want to consider a different spiel. ;)&lt;/p&gt;&#xA;&lt;p&gt;A more relevant reference for this discussion might be the book &lt;a href=&quot;http://www.cogsci.ucsd.edu/%7Enunez/web/FM.PDF&quot; rel=&quot;noreferrer&quot;&gt;Where Mathematics Comes From&lt;/a&gt; by Lakoff and Nuñez.  There are fair criticisms of this text, but the early chapters and discussions of how the brain understands quantity are interesting.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I do tend to test these concepts.  For example, I often give exam questions of the form &amp;quot;True / False; justify your answer&amp;quot;.  One such question is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The square root of negative one (that is, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;) does not exist.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My intended answer is something like &amp;quot;False:  &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; does not have a square root in the real numbers, but it does have a square root in the complex numbers&amp;quot;, though I have given credit to students who say that it is true (e.g. &amp;quot;True:  the square root of negative one is complex number, and the complex numbers don't really exist.&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Another pair of T/F questions which test a related concept:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The equation &lt;span class=&quot;math-container&quot;&gt;$x^2 + 4 = 0$&lt;/span&gt; has no solutions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;vs&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The equation &lt;span class=&quot;math-container&quot;&gt;$x^2 + 4 = 0$&lt;/span&gt; has no real solutions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The expected answer to the first question is something like &amp;quot;False: the equation has complex solutions,&amp;quot; while the expected answer to the second is something like &amp;quot;True:  there is no real number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$x^2 = -4$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2019-12-02T00:27:28.970" CommentCount="23" ContentLicense="CC BY-SA 4.0" />
  <row Id="17547" PostTypeId="2" ParentId="17539" CreationDate="2019-11-29T21:34:00.710" Score="6" Body="&lt;p&gt;I recall having been taught different classes of numbers (in maths at school) way before we were introduced to complex numbers. Main reason was to distinguish &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;natural numbers&lt;/li&gt;&#xA;&lt;li&gt;integers&lt;/li&gt;&#xA;&lt;li&gt;rational numbers&lt;/li&gt;&#xA;&lt;li&gt;finally ... real numbers&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It's reasonable to teach students things like the coverage of numbers on the number line: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why are real numbers continuous while rationale ones are not?&lt;/li&gt;&#xA;&lt;li&gt;Cardinality and set theory (German: Maechtigkeit, I hope the translation is correct).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And, finally, it is reasonable to teach quite basic things like why pi (&lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;) is not 3, 3.14, or 22/7, and that those are just approximations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conclusion: Yes, tell the students. There will be some who feel they won't need it, but that's a poor reason to not teach the ones who are willing to learn. Additionally, students who will have an issue of thinking that &quot;imaginary numbers do not exist&quot; can also question the existence of real numbers. After all, they are all just constructions based on mathematical axioms.&#xA;And for those thinking &quot;but imaginary numbers are not real&quot; - well they are correct :)&lt;/p&gt;&#xA;" OwnerUserId="13107" LastEditorUserId="238" LastEditDate="2019-11-30T01:40:57.257" LastActivityDate="2019-11-30T01:40:57.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17548" PostTypeId="2" ParentId="17095" CreationDate="2019-11-30T01:04:05.067" Score="1" Body="&lt;p&gt;I am going to give an answer that will probably bore people to death; I am going to give an Australian perspective. Disclaimer: the Australian mathematics school curriculum has had enormous changes from the 1970's onward. Australian states also have huge differences in methods of assessment. The following will not be representative of all maths education in Australia. [yes we say maths, not math]. I am also going to oversimplify some things to make it easier to explain.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I completed high school in 1992. Problem solving was not a main part of the curriculum. It did happen. But it was an extension activity. Either in a problem solving competition such as the Westpac maths competition. Or as an extension class for brighter kids. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not a maths teacher. My only involvement since 1992 was in 2008; I did some volunteering to help kids with their maths and English. The kids brought some of their school maths problems along. I found that problem solving has changed completely; it's now taught as a main part of the curriculum to all kids from late primary school [US elementary school?]. I had a girl who was 11 at my table and her school problem solving task was harder than anything that I had been set in my 80's/90's school life!&#xA;I don't think that I could, or should, ever teach maths to any kids. I don't think that I could ever bridge such a change in problem solving emphasis. I'd be useless to kids. They need someone who has had problem solving as part of being educated; that could never be me. I doubt if I could be retrained. That's not even considering the computer aspects of problem solving in maths. In my day it it was rare to write a computer program to solve a maths problem. [actually it's more complex than that because in the 1970's maths in Australia did a better job of teaching computers for maths than it did in the 90's.But that is a whole other discussion.] Kids would want help with computer programs in maths. Again, I'd be useless. &lt;/p&gt;&#xA;" OwnerUserId="13108" LastActivityDate="2019-11-30T01:04:05.067" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17549" PostTypeId="1" CreationDate="2019-11-30T01:36:39.657" Score="4" ViewCount="384" Body="&lt;p&gt;This is a concise description of what happens if I attempt any problem solving task with numbers:&#xA;[1] I never work out where to start. &#xA;[2] If by some miracle [and I'm talking Moses parting the waves here] I work out where to start, then there will be some point where I won't work out which area of maths to use. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found no answer to [1] or [2]. I did maths at university and in one unit I passed with 55%; I got 90% for the bookwork and almost nothing for the problem solving. In one problem solving task I actually got zero; I never worked out where to start. That was following bad marks at high school maths because of the problem solving component. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can problem solve with things that are not numbers. I run Linux and have nobody else to help me out. I sort out problems with scanners, printers, bootloaders and many other aspects. I also get scores off the charts for the blocks component of the IQ test. I have always been reluctant to get any help with maths problem solving. Then I got the response from my family &quot;it's OK if you're thick&quot;. Then I would scream back &quot;I am not thick&quot;. That's one megaton H bomb - maths is so bound up in &quot;being thick&quot; if you can't do it.   &lt;/p&gt;&#xA;" OwnerUserId="13108" LastActivityDate="2019-12-02T15:50:36.313" Title="Are some people unteachable at mathematical problem solving?" Tags="|problem-solving|" AnswerCount="3" CommentCount="3" ClosedDate="2019-12-03T21:30:45.480" ContentLicense="CC BY-SA 4.0" />
  <row Id="17550" PostTypeId="1" CreationDate="2019-11-30T01:43:29.403" Score="5" ViewCount="187" Body="&lt;p&gt;I am looking for long-term (over the course of many semesters) strategies, including specific types of in-class activities, for developing the abilities of students to come up with intermediate steps in solving multi-step problems.  Ideal answers to this questions will have references that propose strategies and activities as well as study (either qualitatively or quantitatively) their effectiveness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me describe the problem I would like to address.  Students in a junior/senior level abstract algebra course have just learned the proof of Lagrange's Theorem (that the cosets of a subgroup &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; each have the size of the subgroup and partition the group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;) and the definition of a normal subgroup (in terms of equality of left and right cosets).  They are now asked to prove that, if the order of a subgroup is half the order of the group, the subgroup must be normal.  It turns out that most students, even only counting students that without a doubt understand the proof of Lagrange's Theorem and the definition of a normal subgroup, are simply unable to do it by themselves.  Almost all students need to be given the hint that, given &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; not in the subgroup, they should prove that both &lt;span class=&quot;math-container&quot;&gt;$aH$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Ha$&lt;/span&gt; are equal to &lt;span class=&quot;math-container&quot;&gt;$G\setminus H$&lt;/span&gt;.  Once given the hint, most students are able to both see why the hint is true and why it solves the problem for them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to emphasize that the difficulty is not in lack of knowledge or understanding of the concepts involved, but rather in not having the ability to come up with connecting facts.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not an issue that really can be solved within the context of the course, because, in the context of the course, the natural thing is simply to tell the students the hint and have them memorize it.  The issue is to develop the students' ability to come up with this hint by themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Presumably, this ability should be developed over the entire course of the students' intellectual (and not only mathematical) experiences starting with elementary school or earlier.  However, as a college professor, I am particularly interested in knowing about strategies and activities for earlier college courses, starting with calculus, but especially in elementary discrete mathematics, linear algebra, and introduction to higher mathematics courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am most interested in strategies and activities that help almost all students, not strategies that are particularly effective for the best students.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2019-11-30T02:24:42.257" Title="Developing abductive (problem-solving) skills" Tags="|undergraduate-education|problem-solving|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17551" PostTypeId="2" ParentId="17550" CreationDate="2019-11-30T02:24:42.257" Score="1" Body="&lt;p&gt;I believe an appropriate strategy that will help students achieve the goals you state is &lt;strong&gt;&lt;em&gt;inquiry based learning (IBL)&lt;/em&gt;&lt;/strong&gt; (in math, and ideally, other subjects, as well). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Outcomes were evaluated in this study/research on &lt;a href=&quot;https://www.fitchburgstate.edu/uploads/files/AcademicAffairs/IBLmathReportALL.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;IBL in mathematics at the college level&lt;/a&gt;, which provides background describing IBL in mathematics, as well as outcomes from a two-year experiment, from student and faculty surveys, as well as students' performance outcomes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find many more resources on IBL by searching &lt;em&gt;inquiry based learning in mathematics; college&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, at the college level, I have found that incorporating, at the Freshman level, having students acquire and read &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=12&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwiWvOH87pDmAhVOd6wKHew2A-cQFjALegQIARAB&amp;amp;url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FHow_to_Solve_It&amp;amp;usg=AOvVaw0YssPpsboyk5WWzQRDClkY&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How to Solve It,&lt;/em&gt; by George Pólya&lt;/a&gt; has been very helpful, along with supplementing an initial proof based course with &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=3&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwjmrN3f75DmAhVM2qwKHQ7WBSwQFjACegQIAxAC&amp;amp;url=http%3A%2F%2Fusers.metu.edu.tr%2Fserge%2Fcourses%2F111-2011%2Ftextbook-math111.pdf&amp;amp;usg=AOvVaw0KUDVh8Sw_FgU92Rb2IPdd&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How to Prove It&lt;/em&gt; by Daniel Velleman (pdf)&lt;/a&gt;.  &lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2019-11-30T02:24:42.257" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17552" PostTypeId="2" ParentId="17549" CreationDate="2019-11-30T02:34:21.150" Score="7" Body="&lt;p&gt;You might want to learn more about a learning disorder called &lt;a href=&quot;https://en.wikipedia.org/wiki/Dyscalculia&quot; rel=&quot;nofollow noreferrer&quot;&gt;dyscalculia&lt;/a&gt;.  Students (or people in general), who are diagnosed with this learning disorder, are not &quot;thick&quot;, nor &quot;stupid&quot;.  An analogy you might be more familiar with is &lt;em&gt;dyslexia&lt;/em&gt;, in which students struggle with reading and writing, but are in no way &lt;em&gt;inherently&lt;/em&gt; &quot;thick&quot;, nor &quot;stupid&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You seem to be quite capable, but struggle with numbers and calculations.  Over time you may have developed anxiety in relation to seeing numbers involved in any sort of math.  I'd suggest you read more about dycalculia, to see whether the description &quot;fits&quot; with your experience, or not.  If it does, there are professionals who can work with you to help you demonstrate your potential.  There are online support groups, and helpful information on the internet, as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just don't equate &quot;mathematical problem solving skills&quot; with &quot;numerical computation skills&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, it sounds like you are fully capable of mathematical problem solving.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Added:&lt;/em&gt; Reading through your question once more, I note your statement, &quot; I have always been reluctant to get any help with maths problem solving.&quot;  You and your conviction that you can't solve math problems, and your reluctance to get any help, are your biggest obstacles to further progress in math problem solving. There is absolutely no shame in seeking assistance.  And your family is not helping you move forward.  &lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-12-02T15:50:36.313" LastActivityDate="2019-12-02T15:50:36.313" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17553" PostTypeId="2" ParentId="17539" CreationDate="2019-11-30T05:53:46.353" Score="1" Body="&lt;p&gt;&lt;strong&gt;I'll take the dissenting opinion: don't use the phrase &quot;Real Number&quot; until you're prepared to teach what an imaginary number is.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Today in science, we're going to be using an optical microscope to&#xA;  look at culture slides.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;... which &lt;em&gt;immediately&lt;/em&gt; begs the question from any remotely curious student:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Wait, so there are non-optical microscopes?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And you now have two choices:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Option A - &quot;Yes, but we're not going to learn about them today&quot; / &quot;Yes, it's called an electron microscope.  But we don't learn about them until next semester.&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Which feels unfulfilling to the student.  You quashed their curiosity, and they're probably wondering why you bothered saying &quot;Optical&quot; in the first place (if you weren't going to teach them the alternatives.)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Option B - Actually talk for awhile on what an electron microscope is and explaining its basics.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But that's the problem - you &lt;em&gt;weren't&lt;/em&gt; planning on talking about Electron Microscopes in your lesson.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to the math example - if you're not willing to at least go into a little about imaginary numbers at the time, you probably shouldn't use the phrase 'Real Number'.  (And, honestly, imaginary numbers is a subject that can take awhile for students to get their heads around - it's not something you should try to do a quick little 5-minute blurb about.)  All in all, by throwing out &quot;Real Numbers&quot; before you're ready to go into Imaginary Numbers, you'd basically be opening a door that you're hoping nobody is curious enough to try to go through.&lt;/p&gt;&#xA;" OwnerUserId="13109" LastActivityDate="2019-11-30T05:53:46.353" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17554" PostTypeId="1" AcceptedAnswerId="17555" CreationDate="2019-11-30T08:36:23.783" Score="9" ViewCount="781" Body="&lt;p&gt;I sometimes teach calculus in English whereas it's not my native language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, during a course about antiderivatives, how do you (orally) pronounce &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which are the best?&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&quot;the function small &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;&quot; vs. &quot;the function big &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;&quot; ?&lt;/li&gt;&#xA;&lt;li&gt;&quot;the function lowercase &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;&quot; (or just &quot;the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;&quot;) vs. &quot;the function uppercase &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;&quot; ?&lt;/li&gt;&#xA;&lt;li&gt;Is this example correct: &lt;em&gt;&quot;The function capital &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is an antiderivative of the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&quot;&lt;/em&gt; ?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7558" LastEditorUserId="5273" LastEditDate="2019-12-01T17:08:25.150" LastActivityDate="2019-12-01T17:08:25.150" Title="How to (or should one) distinguish between lowercase and uppercase letters orally when lecturing?" Tags="|calculus|teaching|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17555" PostTypeId="2" ParentId="17554" CreationDate="2019-11-30T11:09:18.860" Score="13" Body="&lt;p&gt;When I need to distinguish &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, I choose&#xA; - &quot;small eff&quot; vs. &quot;big eff&quot; / &quot;capital eff&quot;. I find &quot;uppercase/lowercase eff&quot; to be a bit awkward; it doesn't run off the tongue as easily as the other options.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I might not distinguish them orally every single time. The flow of the lecture would determine when I would choose to emphasise the difference.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; occur in separate sections of the discussion, then I usually do not emphasise the difference. If needed, then I distinguish them once or twice when &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is introduced and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is left behind (or vice-versa), and leave it at that.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; occur in the same section of the discussion, then I make sure to distinguish them orally whenever they occur together in a result or an equation. If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has been playing the predominant role in the discussion, then I continue to call it &quot;eff&quot; and I call &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; as &quot;big eff&quot; or &quot;capital eff&quot;. Conversely, if &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; has been predominant, then I call it &quot;eff&quot; and I call &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; as &quot;small eff&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It is also probably better not to overdo it, but where the line should be drawn depends on the needs of your students. For example, I might say something like&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Okay, now take the function &quot;small eff&quot;. We want to know where &quot;eff&quot; is differentiable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If the students are keeping up with the discussion, then it should be immediately clear to them that &quot;eff&quot; means &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; in this case. If some students are lagging behind (still figuring out what was written on board 1 whereas the discussion is on board 3), then they might get confused by such a sentence since they are juggling between two inputs (the board and my words).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if the choice of notation is good and consistent, then after a certain point of time one can drop the distinction between &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; when speaking. As a related example, sometimes we need to talk about a family of functions, and we might use the notation &lt;span class=&quot;math-container&quot;&gt;$\mathscr{F}$&lt;/span&gt; (which I speak as &quot;script F&quot; / &quot;curly F&quot;). Suppose we often have to choose a function from the collection &lt;span class=&quot;math-container&quot;&gt;$\mathscr{F}$&lt;/span&gt;, that is, we often write &lt;span class=&quot;math-container&quot;&gt;$f \in \mathscr{F}$&lt;/span&gt; or some such variant. Then, after the first one or two instances of using &quot;script eff&quot;, we can say&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Let &quot;eff&quot; belong to &quot;eff&quot;&lt;/em&gt;, or &lt;em&gt;Let &quot;eff&quot; be an element of &quot;eff&quot;&lt;/em&gt;, or &lt;em&gt;Let &quot;eff&quot; be a function in &quot;eff&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;without any ambiguity, because the &quot;grammar&quot; is clear: none of the following make sense:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f \in f, \qquad \mathscr{F} \in f, \qquad \mathscr{F} \in \mathscr{F}.&#xA;$$&lt;/span&gt;&#xA;So, it is clear what we really mean, even without distinguishing orally. If your choice of notation is good and consistent, then your students will gradually become aware of this grammar, and you will be able to drop the oral distinctions after a point of time. At what point of time—that will depend on your students and how quickly they become comfortable with the material.&lt;/p&gt;&#xA;" OwnerDisplayName="user7990" LastActivityDate="2019-11-30T11:09:18.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17556" PostTypeId="1" CreationDate="2019-11-30T13:58:53.550" Score="18" ViewCount="6543" Body="&lt;p&gt;I sometimes see people arguing against concentrating too many resources in high school maths competition (such as IMO) training. Their reasons they give are usually the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Competitions are a very skewed representation of the subject. They only cover small areas like number theory, combinatorics, &quot;ancient&quot; parts of modern algebra, and the obsolete Planar Euclidean geometry. Major areas (analysis, modern algebra, and modern geometry) are not covered at all.&lt;/li&gt;&#xA;&lt;li&gt;Nobody disagrees that 16th-century &lt;strong&gt;tricks&lt;/strong&gt; (&lt;strong&gt;NOTE:&lt;/strong&gt; I mean tricks, not big theorems like Pythogras) of Euclidean geometry are not of much use to any professional mathematicians today. Some even doubt if it should be part of the high-school curriculum at all. In general, competition stuff are irrelevant to the research of most of the professional mathematicians in analysis, algebra and geometry (and of course, more irrelevant for other people). &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT: the competition stuff might be the application of an advanced theory, but never part of the advanced theory itself.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Although students are likely to learn some knowledge beyond high-school if they manage to get to IMO level, in general, for most students who are training for competitions, the contests tend to limit them within the scope of high-school contents. This, again, creates a biased picture of the subject in their mind.&lt;/li&gt;&#xA;&lt;li&gt;Some people do not like the fact that maths is made into a &quot;sport&quot; - the long hours of training are laborious, but does not necessarily lead to deep thoughts.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;All those reasons (especially points 1 and 2) appear to me that they are implying the following thing (although this is rarely explicitly said) - students who are good at maths would better not pursue their interests through competitions; &lt;strong&gt;instead, they can learn higher level university-level maths in advance.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It appears that in some countries, intensive competition training is unnecessary, as the resources used in contests is more than sufficient to allow promising students to learn undergrad math in advance (if less attention is paid to contests).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know it is slightly opinion-based, but there doesn't seem to be similar discussions before, so should students learn higher-level university-level maths in advance or do contests?&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="9339" LastEditDate="2019-11-30T23:01:48.597" LastActivityDate="2019-12-18T12:04:08.670" Title="Learn university maths or train for high school competitions: which is better?" Tags="|undergraduate-education|secondary-education|math-contests|" AnswerCount="7" CommentCount="24" ContentLicense="CC BY-SA 4.0" />
  <row Id="17557" PostTypeId="2" ParentId="17556" CreationDate="2019-11-30T16:13:27.820" Score="4" Body="&lt;p&gt;I don't know the answer, but clearly understand the question...which means it is a great question.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it will depend both on the student (interests, abilities) and the situation.  For the situation, it probably includes quality, but also pedagogy (efficient approach) as well as fun factor.  Consider the difference between just having a library card and hitting the stacks for some math books or having this generation's Jaime Escalante to entertain you in AP Calculus class.  (I have described the advancement option, but same thing might apply for the situation variable of IMO.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The IMO is better for motivation/fun.  The advanced classes are better for usefulness.  I haven't a clue how to decide which is more critical, but would maybe try to do a bit of both, allowing student interest to dictate how much time goes where.  Since either direction may not succeed, given distractors, perhaps following the one of interest is more likely to end up not being an abandoned experiment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other variable that the question seems to interestingly ignore is NON-MATH uses of the time.  Advancement in other studies (I realize this forum is full of ex math majors, but they shouldn't assume all &quot;likes math&quot; kids will go into math, versus physics or engineering, etc.).  Or time spent on non-academic pursuits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My gut feel is the IMO thing is probably better.  Ideally, it will teach participant some humility, as they will come up on people far better very quickly.  And they will still pick up advanced classes fine.  &lt;/p&gt;&#xA;" OwnerUserId="13110" LastEditorUserId="238" LastEditDate="2019-11-30T20:39:46.853" LastActivityDate="2019-11-30T20:39:46.853" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17559" PostTypeId="2" ParentId="17556" CreationDate="2019-11-30T18:26:10.570" Score="14" Body="&lt;p&gt;&lt;strong&gt;Short answer:&lt;/strong&gt; The skewed content is not a good reason for avoiding IMO-style contest training, because if the training is done right then the students will be led to explore mathematics and would never have a mistaken picture that mathematics is mostly about IMO topics. (I of course compare between good IMO-style training and good teaching of university-level mathematics, because there is no point comparing lousy training with anything.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Long answer:&lt;/strong&gt; (Note that the question has been edited slightly so my responses were to the original points.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe there are concrete mathematical pedagogical issues relevant to your question. A major point is that IMO-style mathematics is a significant step up from the blindly rote-learnt mechanical 'math' that pervades many high-school education systems (both in the syllabus and in the delivery). This is because there is a proper focus on &lt;strong&gt;rigorous logical reasoning&lt;/strong&gt; in the form of proofs. I think that this is the most important reason to value IMO-style problems, and should be properly emphasized in any competent training on IMO-style problems (whether or not with the goal of participating in contests):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Students should be taught logical reasoning in conjunction with how it is used to solve IMO-style problems (starting with easy problems but with fully rigorous solutions). The objective is that they should be able to carefully, accurately and confidently identify what exactly is &lt;strong&gt;proven&lt;/strong&gt;, what they &lt;strong&gt;wish to prove&lt;/strong&gt;, and what &lt;strong&gt;assumptions&lt;/strong&gt; they used, throughout the solving process, and similarly perform the same analysis at any point in someone else's proof.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students should be trained to write logically valid proofs, not just to write handwaving arguments or a messy bunch of statements or equations with imprecise logical structure. If any deductive steps are skipped in a proof, the student must be able to produce upon being asked a correct subproof to justify the jump.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students should also be trained to analyze others' proof attempts and correctly verify them or identify logical errors or gaps in the argument. Even if the teacher claims that a proof is correct, but actually is wrong, the students should be able to confidently reject the teacher's claim. In other words, an incorrect argument from authority must lose to the students. This is a litmus test that the students actually know what they are doing.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I also want to address several misconceptions revealed by some points in your question about competitive mathematics. You are definitely right on certain points, such as the fact that contest mathematics tend to be restricted to a small 'syllabus'. This is both a disadvantage (as you noted) and an advantage, because contest problems are usually much more interesting than 'standard' mathematics. But in my experience not all your points hold.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Nobody disagrees that 16th-century tricks of Euclidean geometry are not of much use to any professional mathematicians today. Some even doubt if it should be part of the high-school curriculum at all.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Well, here are some that I think professional mathematicians ought to know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The compass-and-straightedge construction of the square-root of a given ratio of lengths is directly relevant to the proof that the constructible reals are precisely those that can be obtained by a sequence of quadratic extensions. This result is an important one in the history of Galois theory, and I think it should be taught in any introductory course to field theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another one is the proof of the Pythagoras theorem, which is needed to give a real motivation for the modern notion of Euclidean spaces. Related are all the applications of circle and conic geometry in classical mechanics and optics, which is very much a part of applied mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet another often overlooked tool in geometry is inversion, which is closely related to the Mobius transformations in complex analysis, as well as the more general conformal mappings.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In general, competition stuff are irrelevant to the research of most of the professional mathematicians (and of course, more irrelevant for other people).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Let's look at the IMO syllabus specifically, which roughly covers elementary number theory, combinatorics, graph theory, inequalities, Euclidean geometry and functional equations. In my own experience, all the concepts in number theory, combinatorics and graph theory in the IMO syllabus also show up in any university-level courses in those topics, and these form foundations for higher-level courses. It is true that many things in the other topics do not show up in university courses, but still many do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the AM-GM inequality is a basic one that every mathematician must know, and higher mathematics also relies on some IMO-level inequalities such as Jensen's, Bernoulli's, Cauchy-Schwarz, Chebychev, Holder's, and the power mean inequality. The concept of smoothing is also a very important and productive one, as are the various other generic techniques commonly used in IMO including homogenization, re-parametrization, splitting of the domain, even if we scrupulously avoid whacking techniques that many IMO students learn such as Lagrange multipliers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have given some examples of Euclidean geometry applications above, but one major advantage of learning Euclidean geometry is that it is a beautiful and vast playground suitable for learning rigorous logical reasoning with focus on &lt;strong&gt;propositional logic&lt;/strong&gt;, as almost all Euclidean geometry problems can be solved in pure propositional logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for functional equations, I would say that some of it (such as learning the solution of well-known functional equation &lt;span class=&quot;math-container&quot;&gt;$f(x+y) = f(x)+f(y)$&lt;/span&gt; for continuous &lt;span class=&quot;math-container&quot;&gt;$f : \mathbb{R} → \mathbb{R}$&lt;/span&gt;) provides the IMO student a glimpse of &lt;strong&gt;real analysis&lt;/strong&gt;. The concept of transforming the function itself is also a very important one (such as in reducing the functional equation &lt;span class=&quot;math-container&quot;&gt;$f(x·y) = f(x)·f(y)$&lt;/span&gt; for continuous &lt;span class=&quot;math-container&quot;&gt;$f : \mathbb{R}^+ → \mathbb{R}$&lt;/span&gt; to the previous one), and such techniques show up in all branches of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although students are likely to learn some knowledge beyond high-school if they manage to get to IMO level, in general, for most students who are training for competitions, the contests tend to limit them within the scope of high-school conte[s]ts. This, again, creates a biased picture of the subject in their mind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You may be right that training for only high-school contests, if they do not involve proof questions, would create an inaccurate picture of mathematics. However, in many countries there are high-school contests with final stages involving proof questions. It would be sad if students merely trained to get some results on only multiple-choice or short-answer questions. As for IMO level, most IMO participants I ever got to know did not ever stay within the 'confines' of IMO-style mathematics, but on their own delved deep into other topics in mathematics as well.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Some people do not like the fact that maths is made into a &quot;sport&quot; - the long hours of training are laborious, but does not necessarily lead to deep thoughts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I do not agree with excessive training in only one thing, whether it is mathematics or not, if other important subjects are neglected. On the other hand, long hours of training is typically necessary to become an expert at anything. Some people say that an average person needs roughly 1000 hours to become reasonably good at something, and 10000 hours to become an expert, and I think it is roughly right.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;students who are good at maths would better not pursue their interests through competitions; instead, they can learn higher level university-level maths in advance.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have some gripes with contest mathematics (such as the fact that many of them including the IMO use problems that have appeared before), but I would say that there is much benefit in learning and even training for IMO-style mathematics. As I mentioned earlier, contest mathematics is usually far more interesting than modern mathematics to a high-school student, so there is ample motivation to explore.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, in my opinion, it is a rather 'clean' playground. Earlier I mentioned Euclidean geometry as being suitable for learning propositional logic. Even the other IMO topics are more suitable than modern mathematics for learning first-order reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, it is easy to learn induction, strong induction, structural induction, well-ordering, and the extremal principle in the context of combinatorics and graph theory, and there are numerous cute problems involving these tools in their solutions. I personally consider full first-order induction as a key part of &lt;strong&gt;logical reasoning&lt;/strong&gt;, but sadly numerous undergraduate math majors cannot use it properly.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="1550" LastEditDate="2019-12-01T09:53:33.173" LastActivityDate="2019-12-01T09:53:33.173" CommentCount="20" ContentLicense="CC BY-SA 4.0" />
  <row Id="17560" PostTypeId="2" ParentId="17556" CreationDate="2019-11-30T22:51:19.767" Score="21" Body="&lt;p&gt;I'm not sure whether your question aims at educators or at the students themselves, but my answer actually would be very similar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having as well as being a good, perhaps gifted student is a treasure. It's fun both to teach and to be one. The subject is fun. Conversations are fun. New angles are appearing which were not obvious, perhaps not even to the teacher. If the talent is substantial it is likely that the student will pursue a higher education and later a profession which will be both reasonably fulfilling and reasonably lucrative. The only thing that can go wrong is that the student loses interest, that the subject is no fun any longer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My advice is to let the student choose. Tease them, make offers, prepare material, let them participate in an olympiad, and also expose them to college-level math through tutoring or simply through text books. And then let the student choose their topics and their way, and support them without overtaxing them. The future will be bright no matter what; perceived usefulness for a career should be secondary, not least because we don't know the future (&quot;leave those large prime numbers alone and study something useful!&quot;).&lt;/p&gt;&#xA;" OwnerUserId="11894" LastActivityDate="2019-11-30T22:51:19.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17561" PostTypeId="2" ParentId="17556" CreationDate="2019-12-01T13:31:50.353" Score="12" Body="&lt;p&gt;One rather major argument in favour of contest-style mathematics is its ability to cultivate problem-solving abilities in students while not requiring much difficult machinery. It is of course undeniable that the ways of thinking in contest math are much different from the ways of thinking in university mathematics, but what both have in common is a requirement to be able to observe patterns, make deductions based on known facts, deduce results systematically and formally, et cetera. The problem solving abilities which contest math can cultivate can carry over naturally to university mathematics, and I think that is one of the main values in contest math for those seeking higher education in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So regarding the point that &quot;tricks&quot; used in contest math (e.g. most of Olympiad Euclidean geometry) are irrelevant in university math---that's true, but I think it rather misses the point. If one wants to investigate mathematics as a subject, obviously one should not restrict themselves to Olympiad topics or Olympiad methods. It's a bit like trying to solve a Number Theory question of research interest using only elementary contest-level mathematics; it's completely hopeless. The point of contest math is, instead, as a greenhouse or playground of sorts, to perfect one's mental acumen before being introduced to more advanced tools and machinery to tackle mathematics in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to the question of which is better: I think I concur with Peter that a student in this position should follow wherever their passion and interest takes them. And they should do so with the recognition that whichever they choose to go for (or even better, a healthy mix of both) will benefit their own mathematical ability. Choosing to do contest math is not as useless as OP seems to imply.&lt;/p&gt;&#xA;" OwnerUserId="10206" LastEditorUserId="10206" LastEditDate="2019-12-10T00:24:13.693" LastActivityDate="2019-12-10T00:24:13.693" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17562" PostTypeId="2" ParentId="17539" CreationDate="2019-12-01T18:57:00.363" Score="4" Body="&lt;p&gt;Here's something that the students might easily grasp and that could also be entertaining:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What's &quot;natural&quot; about the natural numbers?&lt;/li&gt;&#xA;&lt;li&gt;What's &quot;rational&quot; about the rational numbers?&lt;/li&gt;&#xA;&lt;li&gt;What's &quot;real&quot; about the real numbers?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Every one of these different sets of numbers is a mathematical idealization of something encountered in the &quot;real&quot; world. None of them is more &quot;natural&quot;, &quot;rational&quot;, or &quot;real&quot; than the others, and none is less so, they just serve different mathematical purposes for modelling the real world. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These words that we mathematicians use in a technical sense are just being borrowed from ordinary human language, because we mathematicians are humans after all. And each of these borrowed words has its flaws, because we mathematicians are humans after all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we should try to keep these things in mind as we study new number systems, such as the complex numbers, which are not particularly more &quot;complex&quot; than the natural, rational, and real numbers, not after you get used to them, anyway.&lt;/p&gt;&#xA;" OwnerUserId="7258" LastEditorUserId="7258" LastEditDate="2019-12-03T19:49:25.223" LastActivityDate="2019-12-03T19:49:25.223" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17563" PostTypeId="2" ParentId="17556" CreationDate="2019-12-01T19:20:33.187" Score="7" Body="&lt;p&gt;One thing that hasn't been mentioned yet is the social aspect. There are always some people who like to stay alone, but for the large majority, meeting like-minded people and engaging in some sort of activity with them is much more fun than sitting at home and staring at textbooks. Especially at the higher level, where some traveling is involved, competitions are much more than just exams, they are a chance to meet like-minded people for a weekend and spend some time with them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Contrast that with doing university maths early. Even if there is a nearby university where you can attend some classes, you'll be the odd one out. You might make some friends as well, but fundamentally you are in a different phase of your life. They might go out for a party after class, you'll go home to your parents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would not advise against doing university maths early, if the student is really interested in doing so. But even in the best case scenario, where the work can directly be turned into credits and the student does not have to bore himself through repetition of known topics at university, there is not that much gain in progressing through university one or more years earlier. Being ahead of people of the same age is a big thing at high school level, with huge implications and big discussions when students want to skip a grade. But being ahead at the end of university is almost negligible. A 27 year old who just finished a maths PhD is not fundamentally different from a 30 year old who just finished a maths PhD.&lt;/p&gt;&#xA;" OwnerUserId="8988" LastActivityDate="2019-12-01T19:20:33.187" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17564" PostTypeId="2" ParentId="17549" CreationDate="2019-12-01T19:43:07.553" Score="4" Body="&lt;p&gt;Most of real mathematics is not about numbers at all. If you can solve hardware and software problems that are essentially puzzles then you could probably do better at mathematics if you wanted to and needed to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You seem to be beyond university. If you don't need mathematics in your life but want to use your problem solving skills in some more abstract settings, search the web  for mathematical puzzles and work on the ones that look like fun.&lt;/p&gt;&#xA;" OwnerUserId="4984" LastActivityDate="2019-12-01T19:43:07.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17565" PostTypeId="2" ParentId="17549" CreationDate="2019-12-01T20:56:00.087" Score="5" Body="&lt;p&gt;As written, this seems overly caught up in your personal experience, but I think there's a real question here.  A lot of people make it to various stages of education having concluded that they're incapable of learning math, and teachers do need to figure out how to handle students who believe that they just can't learn to do math.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;More than most subjects, math carries a lot of cultural baggage about how associated it is with being &quot;smart&quot;.  That means a lot of people have a lot of emotions tied up in their experience of doing math, and those emotions often get in the way of actually doing it.  Math class isn't therapy, and we can only do so much about this on our own, but classes need to be set up in a way that gives students space to try things that don't work and to be in the process of learning something while still not fully understanding it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A lot of students pick up some really bad habits from their early mathematics education.  (For just one example, my students (mostly first-year college students) frequently think that if you look at a math problem and don't immediately know what to do with it, that means they don't know how to solve that problem.)  Helping students identify and break these habits can help a lot, though - especially because of the previous item - sometimes students have a lot of resistance to changing the way they think about math.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Learning math inevitably requires some un-learning: we first learn things in a narrow, often very rote way, and then later have to re-learn more conceptual and flexible approaches.*  But a lot of people have trouble with that step.  Students often approach math with a mindset &quot;the goal is to get to the final answer as quickly as possible&quot;, which makes it hard to revisit old &quot;easy&quot; material in light of new ideas.  (And, again, that emotional baggage gets in the way: once students identify as having understood something, they're often reluctant to let go of that and realize there's more to understand.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But all that shouldn't discount: neurological differences are real.  The way we teach math has some reliance on the way most people's brains learn to automatically do certain kinds of symbol parsing.  Some people have brains that either don't do this, or do it differently.  A lot of people say, &quot;I just can't do math&quot; and mean &quot;the way I learned to do math is totally failing me&quot;.  But some people really do have brains that work differently; we're at a very early stage of recognizing this,  and still at the level of saying &quot;I guess some people just have dyscalculia&quot;.  I'm optimistic that in the next decade or so, we'll begin to see alternate approaches to teaching math that are accessible to at least some people for whom math is currently genuinely inaccessible.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;*This site sees a lot of hand-wringing around this, so I want to emphasize that I don't think this is intrinsically a problem: I think it's an inevitable part of learning a difficult, abstract subject that we sometimes have to learn fragments of it incompletely, use them as stepping stones to harder things, and then reconceptualize the stepping stone.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-12-01T20:56:00.087" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17567" PostTypeId="2" ParentId="4437" CreationDate="2019-12-02T17:38:00.143" Score="3" Body="&lt;p&gt;I want to expand a little on quid's answer to call attention to a couple of specific fields of math where script fonts definitely helped me pick things up.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;One big one is topology.  The go-to notation for that is &lt;span class=&quot;math-container&quot;&gt;$(X,\mathscr T)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is the base set and &lt;span class=&quot;math-container&quot;&gt;$\mathscr T$&lt;/span&gt; is the collection of sets that are open in &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.  Here, script letters indicate collections of sets, captal letters indicate sets, and lowercase letters indicate points.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another is axiomatic linear algebra.  Here, &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; might indicate a vector, &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; a scalar, &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; a linear mapping (or a matrix, which is the same thing), and &lt;span class=&quot;math-container&quot;&gt;$\mathcal V$&lt;/span&gt; a linear space. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And, of course, high school geometry uses lowercase letters for distances, capital letters for points, Greek letters for angles, and script lowercase letters like &lt;span class=&quot;math-container&quot;&gt;$\ell$&lt;/span&gt; for lines.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The big win of using different typefaces for different types from both a practical and pedagogical perspective is that you can see a variable and know off the bat what kind of &quot;thing&quot; it is.  So if I'm in linear algebra and someone asks me if &lt;span class=&quot;math-container&quot;&gt;$\mathcal U\cap\mathcal V$&lt;/span&gt; is finite-dimensional, I can leap to &quot;Ah, those are linear spaces, so finite-dimensionality is a natural thing to wonder about.&quot;  Mathematical fields can often get by with just partitioning the alphabet (like in calculus where a, b, and c are fixed parameters, f, g, and h are functions, i, j, and k are unit vectors, x, y, z are real numbers and so on), but there is an elegance to being able to say &quot;Choose &lt;span class=&quot;math-container&quot;&gt;$V\in\mathcal V$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$v\in V$&lt;/span&gt;&quot; that makes the typefaces worth mastering mathematical handwriting.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-12-02T17:38:00.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17568" PostTypeId="2" ParentId="4437" CreationDate="2019-12-02T18:08:54.950" Score="3" Body="&lt;p&gt;Now that schoolchildren in the US no longer learn script handwriting, I conjecture that the usage of script letters in mathematics will decline.  Eventually, the usage may converge to a level similar to the usage of Fraktur letters in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;SO, to answer the question: If you are teaching young people (who have rarely or never seen script letters before), then DO NOT use script letters in your teaching.&lt;/p&gt;&#xA;&lt;p&gt;Anecdote.  I was working with some 5th graders.  I was careful to print everything I wrote (no cursive).  But my small letter &lt;em&gt;d&lt;/em&gt; turned out to have a loop in its stem.  So they were laughing about that, trying to write &lt;em&gt;d d d&lt;/em&gt; in my style,  and not paying attention to what I was saying.&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="7930" LastEditDate="2022-01-30T20:14:49.863" LastActivityDate="2022-01-30T20:14:49.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17569" PostTypeId="2" ParentId="13482" CreationDate="2019-12-02T22:53:53.400" Score="1" Body="&lt;p&gt;The answers in the comments are probably the best (in particular @James S. Cook's advice to talk to a professor at a local university if you have such a connection) but here's at least a list to get you started: Given that you've got quite a range of interested there I would suggest going through the MAA's list of Math programs: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/programs-and-communities/member-communities/students/student-resources/high-school&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/programs-and-communities/member-communities/students/student-resources/high-school&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular HCSSIM offers some sessions on continuous topics, but the focus isn't &quot;Learn Analysis.&quot; Johns Hopkins Center For Talented Youth has a topology course but that might be a little lower level then you're interested in:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://cty.jhu.edu/summer/grades7-12/intensive/catalog/math.html#topo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cty.jhu.edu/summer/grades7-12/intensive/catalog/math.html#topo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Otherwise, if you're already thinking about these topics, you might want to look at a local university and see if they're offering any summer courses or math camps. Finally, if you are on your way to college at the end of the summer you should check if your school has an early entry program for declared majors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This definitely does sound like something that should exist.&lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2019-12-02T22:53:53.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17570" PostTypeId="1" CreationDate="2019-12-03T09:26:56.537" Score="6" ViewCount="434" Body="&lt;p&gt;I am a senior teacher at this school. We have to select the textbooks for the upcoming session. I am proposing that we have to select books (in maths) that focus more on conceptual understanding and not just on procedural fluency. The curriculum mapped books are supplied by NCERT (I am based in India) &lt;a href=&quot;http://ncert.nic.in/textbook/textbook.htm&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, developed by a team of experts. The books have been developed keeping in mind the basic philosophy that learning should be connected to the child's environment.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I am having a hard time convincing the teachers who are used to working with books by private publishers, which focus only on procedural fluency and not at all on conceptual understanding.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What strong arguments can be put forward  to help convince them that the need of the hour is to go for books (and subsequent teaching) that are connected to the real-world and children's lives? I welcome any helpful advice. &lt;/p&gt;&#xA;" OwnerUserId="13103" LastEditorUserId="238" LastEditDate="2019-12-03T21:06:59.610" LastActivityDate="2019-12-08T02:02:19.050" Title="How do I convince my teachers that a book on maths must focus on conceptual understanding?" Tags="|mathematical-pedagogy|primary-education|concept-motivation|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17572" PostTypeId="2" ParentId="17570" CreationDate="2019-12-03T12:29:52.107" Score="6" Body="&lt;p&gt;I presume you're not asking for research references, because Googling &quot;procedural fluency vs conceptual understanding&quot; will get you to a page of Google Scholar before you could finish typing out that entire search.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose I would lean into how effective pure procedural fluency has been in the past.  I'm in my early 50's, and I was taught to do long calculations by hand (including extracting square roots and using log tables in Algebra 2) because &quot;you won't always have a calculator with you.&quot;  That prophecy clearly did not age well -- I even had a scientific calculator in classes when I was an undergraduate.  And now, of course, everyone can do math on their phones which are in their pockets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;How do you prepare students to live in a world where machines are better at arithmetic than we are?&lt;/em&gt;  I'm only certified in secondary education, so I'll leave the details to you and your peers.  But I think a large portion of that would be a greater conceptual understanding that would lead them to know which operations to use for different circumstances and a stronger awareness of how to use the results of those calculations to guide our decision-making processes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You're going to get pushback, because your teachers have dedicated their careers to the algorithmic piece in the middle.  Nobody likes being told that their training is obsolete.  But if they're willing to adapt, they can create learning that will serve their students throughout their lives.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-12-03T12:29:52.107" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17573" PostTypeId="2" ParentId="17570" CreationDate="2019-12-03T21:03:04.187" Score="6" Body="&lt;p&gt;I would suggest inviting, or &quot;recruiting&quot; the teachers you are leading &lt;em&gt;to help you&lt;/em&gt; with working out the direction in which you hope to move from an emphasis on &quot;procedural efficacy&quot; toward an emphasis on &quot;conceptual understanding.&quot;  Try to frame it as a &quot;team project&quot;, explicitly recognizing their experience and first-hand perspectives in the class-room. Don't frame the changes in terms of &lt;em&gt;their&lt;/em&gt; failures; rather frame things as you and they being at an exciting time in math education, based on much research, that all of you can participate in helping your students master mathematics, not just procedurally, but also conceptually!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I emphasize the above mode of approaching your teachers, because very few &quot;team members&quot; warm up to anything they are told is going to be &quot;shoved down their throats&quot;, whether they like it or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.g., start asap setting up a &quot;workshop&quot;, break it into groups of teachers, and have each group evaluate relevant studies/research, and/or possible alternative concept-based texts, to report back to the whole group to summarize.  Ask them for examples in their own teaching, in which student's seemed to grasp conceptual understanding beyond just mere procedural efficacy, even when having used the former texts.  Chances are there are already a few teachers who have springboarded off the given text, at the time, perhaps discovering greater mastery among their students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to do such workshops and updates on the new changes, and text selections, in the &lt;em&gt;least threatening manner to them as possible.&lt;/em&gt;  Be enthusiastic, respect their work, express confidence in the newer approach, and express utmost confidence that they have all the support they need (more workshops, etc.), and adequate experience, to make the new approach successful: More rewarding for both the teachers, and the students.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2019-12-03T21:10:57.263" LastActivityDate="2019-12-03T21:10:57.263" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17574" PostTypeId="2" ParentId="17454" CreationDate="2019-12-07T00:32:02.340" Score="7" Body="&lt;p&gt;I'm posting this for Joseph Malkevitch, who had difficulty while posting.&#xA;He wanted to include one of his favorite theorems, &#xA;the &lt;a href=&quot;https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem&quot; rel=&quot;noreferrer&quot;&gt;Wallace-Bolyai-Gerwien theorem&lt;/a&gt;,&#xA;that says that any two polygons of the same area have a common dissection.&#xA;See this impressive app developed by Satyan Devadoss's students:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://dmsm.github.io/scissors-congruence/&quot; rel=&quot;noreferrer&quot;&gt;Scissors Congruence app&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/NJk7b.png&quot; width=&quot;360&quot; /&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-12-07T01:25:39.227" LastActivityDate="2019-12-07T01:25:39.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17575" PostTypeId="1" CreationDate="2019-12-07T02:08:23.353" Score="3" ViewCount="368" Body="&lt;p&gt;Do you know any textbook with problems+solutions to support teaching of linear programming and reductions, and in particular, cover standard and slack forms, formulation of problems as linear programs, simplex method and algorithm, duality, etc.?&lt;/p&gt;&#xA;" OwnerUserId="13136" LastEditorUserId="238" LastEditDate="2019-12-07T17:17:35.183" LastActivityDate="2019-12-13T20:12:07.733" Title="How to teach linear programming and reductions?" Tags="|undergraduate-education|textbooks|teaching|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17576" PostTypeId="2" ParentId="17575" CreationDate="2019-12-07T03:15:46.887" Score="6" Body="&lt;p&gt;This does not directly address your question (and so is not an answer). But I consult this textbook when touching upon&#xA;linear programming when teaching Algorithms, and I find it&#xA;an impressive, unusually concise (~200 pages) text.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://www.springer.com/us/book/9783540306979?gclid=EAIaIQobChMIyqvji4Sk5gIVUPDACh07xgrZEAQYASABEgLLmfD_BwE&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matousek, Jiri, and Bernd Gärtner. Understanding and using linear programming. &lt;em&gt;Springer Science &amp;amp; Business Media,&lt;/em&gt; 2007.&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;The book is an introductory textbook mainly for students of computer science and mathematics.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/X8yPi.png&quot; width=&quot;280&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;The Klee-Minty cube (p.76).&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;I especially appreciate the authors connecting the linear&#xA;algebra to geometry.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-12-13T20:12:07.733" LastActivityDate="2019-12-13T20:12:07.733" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17577" PostTypeId="2" ParentId="17575" CreationDate="2019-12-07T14:06:00.827" Score="2" Body="&lt;p&gt;A simple Google search (linear programming problems and solutions textbook) yields this result as the first hit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.springer.com/gp/book/9783540417446&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.springer.com/gp/book/9783540417446&lt;/a&gt; (Linear Optimization and Extensions:&#xA;Problems and Solutions; Authors: Alevras, Dimitris, Padberg, Manfred W.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This appears to be exactly what you are looking for, based on reading the intro as well as just the title itself.  An alternate approach is to use the first six chapters of the Schaum's Outline on Operations Research (looking at the table of contents, viewable on line using the &quot;see inside&quot; feature at Amazon, shows they cover your topics).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Google search also shows that this question has been asked several times on the main (non-ed) math site.  Several other texts have been suggested.  Worth a look, although you will find many people who just suggest the text they had or who ignore the problems/solutions characteristic, a requirement for you.  Still, worth a quick look if the two previous solutions are not what you want.&lt;/p&gt;&#xA;" OwnerUserId="13139" LastEditorUserId="13139" LastEditDate="2019-12-07T14:17:24.150" LastActivityDate="2019-12-07T14:17:24.150" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17578" PostTypeId="2" ParentId="17570" CreationDate="2019-12-08T02:02:19.050" Score="5" Body="&lt;p&gt;(This entire post is speculation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think your teachers think that they are teaching conceptually &lt;em&gt;already&lt;/em&gt; and that the students &lt;em&gt;already&lt;/em&gt; have a deep understanding. You need to burst that bubble not with logic or reasoning, not with evidence or statistics, but with a sharp and emotional surprise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to do that: Take out a video recorder and do some interviews with students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask them to calculate this exactly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(\frac{17}{15})(29)=\_\_\_ $&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This should show procedural fluency.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then ask them to state whether or not the answer is greater than 800 in the following &lt;em&gt;without doing any calculations&lt;/em&gt;. Point the video camera at the students' faces. Ask them to explain their thinking aloud ,then ask them to rate their confidence. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(\frac{213}{5178})(799.1146)=\_\_\_ $&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This will show lack of comprehension in many students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For other ideas, check out Dan Meyer's post &quot;&lt;a href=&quot;https://blog.mrmeyer.com/2018/what-does-fluency-without-understanding-look-like/&quot; rel=&quot;noreferrer&quot;&gt;What Does Fluency Without Understanding Look Like?&lt;/a&gt;&quot;&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2019-12-08T02:02:19.050" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17579" PostTypeId="1" AcceptedAnswerId="17580" CreationDate="2019-12-09T19:55:14.533" Score="2" ViewCount="172" Body="&lt;p&gt;A friend of mine is planning to create an educational video dealing with Game theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;His teaching strategy is to introduce the subject via video games. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, he told me that the game &quot;Fortnite&quot; has, as drawback, to lead to some sort of &quot; Nash Equilibrium&quot; (in which gamers are induced to protect themselves around walls, and to adopt a &quot; wait and see&quot; attitude, any initiative having costs greater than possible gains). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not want to claim that my friend's claim is correct ( for I have absolutely no understanding , either of game theory or of video games). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is simply : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) could video games provide a good illustration of game theoretical concepts&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) are there references on this possible link between the two fields? &lt;/p&gt;&#xA;" OwnerDisplayName="user12295" LastEditorDisplayName="user12295" LastEditDate="2019-12-17T21:39:32.900" LastActivityDate="2019-12-17T21:39:32.900" Title="Using video games as an introductory tool for game theory? Can video games illustrate game theoretic concepts?" Tags="|games|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17580" PostTypeId="2" ParentId="17579" CreationDate="2019-12-09T23:59:23.793" Score="3" Body="&lt;p&gt;The table of games detailed in the Wikipedia article&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_games_in_game_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;List of games in game theory&lt;/a&gt;&#xA;could be useful. One column shows the number of &#xA;pure-strategy Nash equilibria for each game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Battle_of_the_sexes_(game_theory)&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Battle of the Sexes&lt;/a&gt; has two such Nash equilibria, as well as a mixed-strategy Nash equilibrium.&#xA;But, as the article says, &quot;This presents an interesting case for game theory since each of the Nash equilibria is deficient in some way.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example, &lt;a href=&quot;https://en.wikipedia.org/wiki/Unscrupulous_diner%27s_dilemma&quot; rel=&quot;nofollow noreferrer&quot;&gt;The diner's dilemma&lt;/a&gt;,&#xA;has a single pure-strategy Nash equilibrium.&#xA;It constitutes an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-player prisoner's dilemma:&#xA;&quot;the expensive meal is strictly dominant and thus the unique Nash equilibrium.&quot; Theory for this game has been compared to experiment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the answer to &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(1) could video games provide a good illustration of game theoretical concepts&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is &lt;strong&gt;&lt;em&gt;Yes&lt;/em&gt;&lt;/strong&gt;, as long as you interpret &quot;video game&quot; as just&#xA;a game with a graphics interface.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't respond knowledgeably your question (2) concerning &#xA;references. I will just mention that game theory is definitely used&#xA;in game &lt;em&gt;design&lt;/em&gt;:&#xA;&lt;a href=&quot;https://www.gamedesigning.org/learn/game-theory/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Game Theory in Video Game Design&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2019-12-10T11:48:51.870" LastActivityDate="2019-12-10T11:48:51.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17583" PostTypeId="2" ParentId="15264" CreationDate="2019-12-11T20:07:24.547" Score="0" Body="&lt;p&gt;How about learning to manipulate the formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_S\|\partial_v\times\partial_w\|dv\wedge dw$$&lt;/span&gt;&#xA;for students of vector calculus?&#xA;They, then, could be able calculate the area of the sphere, toroid, ellipsoid, ovoid in a well structured vector language, and volumes, with the aid of surface integral like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\iiint_Rdx\wedge dy\wedge dz=\iint_Sz\ dy\wedge dz$$&lt;/span&gt;&#xA;coming from the Divergence Theorem, for a bounded region &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; with bounding surface &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observe that modern books are struggling to advance to the Cartan's Calculus thematics and techniques. &lt;/p&gt;&#xA;" OwnerUserId="5582" LastActivityDate="2019-12-11T20:07:24.547" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17585" PostTypeId="2" ParentId="17556" CreationDate="2019-12-12T04:57:59.000" Score="7" Body="&lt;p&gt;I personally think taking advanced math would be way more fun than preparing for some contest largely based on tricks. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not convinced these contests are very revealing of much for most students. I've had students who went on to be successful in graduate school in nontrivial pure math and yet scored barely above the noise in one of these contests. On the other hand, his peer who was of similar intelligence and mathematical creativity took the contest and smoked it. Why did the second student kill it ? Simple, he was schooled in high school for the math Olympiad. Of course contests do attract elite students and one can always give evidence to correlate contests with excellence later in mathematical life. I doubt that excellence has much to do with the contests. I think it has everything to do with the contestants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My advice, focus your attention on what is most interesting. If you don't like contest math, find something else interesting in which to invest your time. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2019-12-12T04:57:59.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17586" PostTypeId="2" ParentId="13693" CreationDate="2019-12-12T08:15:09.047" Score="-3" Body="&lt;p&gt;Calculus with analytic geometry by George Simmons addresses differentials and their treatment in the chapter 5 of the second edition. &lt;/p&gt;&#xA;" OwnerUserId="13148" LastActivityDate="2019-12-12T08:15:09.047" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17587" PostTypeId="2" ParentId="17516" CreationDate="2019-12-13T13:40:35.013" Score="0" Body="&lt;p&gt;I'm going to plug my own book, &quot;Calculus from the Ground Up,&quot; for several reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It spends quite a bit of time showing you how to use calculus to create new formulas, which is important for engineering.&lt;/li&gt;&#xA;&lt;li&gt;It shows how to use the intuition behind the rules to build up new rules as needed.&lt;/li&gt;&#xA;&lt;li&gt;It shows how the intuition behind the math can be helpful for problem-solving in general.&lt;/li&gt;&#xA;&lt;li&gt;It focuses on differentials instead of derivatives, which is what physicists/engineers tend to do.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;It will be a review of the concepts for you, but I think it will also show it to you in a new, helpful way, that will be especially useful as you go forward.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2019-12-13T13:40:35.013" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17588" PostTypeId="2" ParentId="17528" CreationDate="2019-12-13T14:07:35.213" Score="2" Body="&lt;p&gt;We are looking for a rule to give us L'Hosptial's rule that says that &lt;span class=&quot;math-container&quot;&gt;$\frac{f(x)}{g(x)} = \frac{f'(x)}{g'(x)}$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$f(x) = \infty$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x) = \infty$&lt;/span&gt;.  So, let's simply say, &quot;what if we were asked about the opposite, &lt;span class=&quot;math-container&quot;&gt;$\frac{g(x)}{f(x)}$&lt;/span&gt;?&quot;  Clearly, &lt;span class=&quot;math-container&quot;&gt;$\frac{g(x)}{f(x)} = \frac{1}{\frac{f(x)}{g(x)}}$&lt;/span&gt;.  So, if we can find &lt;span class=&quot;math-container&quot;&gt;$\frac{g(x)}{f(x)}$&lt;/span&gt; we can just flip it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what is &lt;span class=&quot;math-container&quot;&gt;$\frac{g(x)}{f(x)}$&lt;/span&gt;?&lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{g(x)}{f(x)} = \frac{\frac{1}{f(x)}}{\frac{1}{g(x)}} = \frac{\frac{1}{\infty}}{\frac{1}{\infty}} = \frac{0}{0}$$&lt;/span&gt;&#xA;That means that we can use L'Hospital to find &lt;span class=&quot;math-container&quot;&gt;$\frac{g(x)}{f(x)} = \frac{g'(x)}{f'(x)}$&lt;/span&gt;.  Since &lt;span class=&quot;math-container&quot;&gt;$\frac{g(x)}{f(x)} = \frac{1}{\frac{f(x)}{g(x)}}$&lt;/span&gt; then we can simply flip the fraction and get &lt;span class=&quot;math-container&quot;&gt;$\frac{f(x)}{g(x)} = \frac{f'(x)}{g'(x)}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastEditorUserId="9594" LastEditDate="2019-12-13T20:05:17.057" LastActivityDate="2019-12-13T20:05:17.057" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17589" PostTypeId="2" ParentId="17516" CreationDate="2019-12-13T15:48:29.420" Score="4" Body="&lt;p&gt;You do not mention differential equations, but this is a very useful topic in engineering, aerospace included, and the list of topics/courses you have studied mean that you would be well prepared for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A relevant book is Goodwine's &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1441979182&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Engineering Differential Equations&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This book is a comprehensive treatment of engineering undergraduate differential equations as well as linear vibrations and feedback control. While this material has traditionally been separated into different courses in undergraduate engineering curricula. This text provides a streamlined and efficient treatment of material normally covered in three courses. Ultimately, engineering students study mathematics in order to be able to solve problems within the engineering realm. Engineering Differential Equations: Theory and Applications guides students to approach the mathematical theory with much greater interest and enthusiasm by teaching the theory together with applications. Additionally, it includes an abundance of detailed examples. Appendices include numerous C and FORTRAN example programs. This book is intended for engineering undergraduate students, particularly &lt;strong&gt;aerospace and mechanical engineers&lt;/strong&gt; and students in other disciplines concerned with mechanical systems analysis and control. Prerequisites include basic and advanced calculus with an introduction to linear algebra.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[emphasis mine]&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2019-12-13T15:48:29.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17590" PostTypeId="1" AcceptedAnswerId="17598" CreationDate="2019-12-13T16:52:51.780" Score="10" ViewCount="6228" Body="&lt;p&gt;I tried uploading a gif, but was unable to do so.  What I can do, is share a link to the gif &lt;a href=&quot;https://i.pinimg.com/originals/7e/cd/01/7ecd01ceb18938344a84b21617e648c6.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  (SE software seems to have allowed me to share the link, but not upload it.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What it shows, initially, is the times table for 9, without answers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{matrix} 1\times 9 = \\2\times 9 = \\ 3\times 9 = \\ 4\times 9 = \\ 5\times 9 = \\ 6\times 9 = \\ 7 \times 9 = \\ 8\times 9 = \\ 9\times 9 = \end{matrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the student proceeds by starting from &lt;span class=&quot;math-container&quot;&gt;$2\times 9$&lt;/span&gt; with the initial digit 1, then counts up from one as she moves down this column in the times table, to get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{matrix} 1\times 9 = \\2\times 9 = 1\\ 3\times 9 = 2 \\ 4\times 9 = 3\\ 5\times 9 = 4\\ 6\times 9 = 5 \\ 7 \times 9 = 6\\ 8\times 9 = 7\\ 9\times 9 = 8\end{matrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then they show the student counting backwards (starting from 1 at the bottom of the list, up to 9 at the top of the list, in the unit digit's place, to get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{matrix} 1\times 9 = \; 9\\2\times 9 = 18\\ 3\times 9 = 27 \\ 4\times 9 = 36\\ 5\times 9 = 45\\ 6\times 9 = 54 \\ 7 \times 9 = 63\\ 8\times 9 = 72\\ 9\times 9 = 81\end{matrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've never seen this before. Is this primarily a &quot;rote computational trick&quot; for elementary students to learn single-digit multiplication by &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;, or is there an explanation that elementary students can understand as to &quot;why this works&quot;?  I have done graduate research in secondary ed, and undergraduate math ed, and teach at a University, so I am not well versed wrt math education at the primary ed level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure what I think about this &quot;trick&quot;;  I would hope at least, for primary grade students, that subsequently, why this works for 9, would be explained to students. It isn't entirely symmetrical, either, as the student proceeds, with the ten's digit from the top &lt;span class=&quot;math-container&quot;&gt;$1\times 9$&lt;/span&gt; being blank (or zero), then descending by adding 1 down to 8, to complete the ten's digit.  Then she starts from the last row to enumerate unit digits, moves upward each step, counting from 1 to 9.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;ALSO, because it doesn't seem to be clear to all answerers,&lt;/strong&gt; I am firstly interested in how to help primary students learn the multiplication tables.  There are clever tricks, but often times students cannot understand WHY they work, until some future time.  I'm reluctant to demand rote memorization, so if there are reasonable tricks, accompanied with explanations that make sense to students first learning their multiplication tables, I would like to know those explanations.&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="-1" LastEditDate="2020-01-02T16:56:20.007" LastActivityDate="2020-05-28T03:40:53.837" Title="Is this primarily a &quot;rote computational trick&quot; for multiplication by 9?" Tags="|primary-education|arithmetic|multiplication-table|" AnswerCount="7" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="17591" PostTypeId="2" ParentId="17590" CreationDate="2019-12-13T17:36:49.000" Score="20" Body="&lt;p&gt;Yes. This is also a trick that you can do on your fingers, too.  For instance, let's say you wanted to calculate &lt;span class=&quot;math-container&quot;&gt;$9\times3$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lRZa3.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lRZa3.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hold out your hands and bend your third finger down as shown.  So nine fingers are &quot;up&quot; (fingers up, &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;, finger #3 down. (9x3). You have two fingers to the left of the bent finger and seven to the right, indicating the product of &lt;span class=&quot;math-container&quot;&gt;$27$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explaining this to students will depend on their prior knowledge and your standards.  I would probably note that &lt;span class=&quot;math-container&quot;&gt;$9=10-1$&lt;/span&gt;, and so adding nine to a number is the same as adding one to the ten's place and subtracting one from the unit's place (as long as there was something to subtract in the unit's place).  &lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="238" LastEditDate="2019-12-14T16:45:16.863" LastActivityDate="2019-12-14T16:45:16.863" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17592" PostTypeId="2" ParentId="17590" CreationDate="2019-12-13T18:40:21.520" Score="33" Body="&lt;p&gt;Anything that is just a trick leads to students having wrong ideas about what math is. But methods that help students see the patterns can help them learn the multiplication facts, along with getting a better feel for what's going on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd call this a way to think about 9s. (There are many.) This method shows that you add 10 for each new nine, and then take away 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplying by 4 is doubling and doubling again. Multiplying by 5, they should notice that odd numbers times 5 result in a unit digit of 5 and even numbers time 5 result in a unit digit of 0. Exploring patterns while learning these results is vital.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2019-12-13T18:40:21.520" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17597" PostTypeId="2" ParentId="17590" CreationDate="2019-12-14T03:34:14.360" Score="11" Body="&lt;p&gt;To add on to the other answers, the reason this works is because we use the decimal system, a.k.a. the base-10 system, for our everyday maths. The multiples of the number that is one less than the base results in a phenomenon where the second digit increases at the same rate as the first digit decreases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;9 * 1 = 09\\&#xA;9 * 2 = 18\\&#xA;9 * 3 = 27\\&#xA;9 * 4 = 36\\&#xA;9 * 5 = 45\\&#xA;etc...&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As the above implies, you see this with different multiples in different base-&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; systems. For example, the multiples of 7 in octal (base-8):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;7 * 1 = 07\\&#xA;7 * 2 = 16\\&#xA;7 * 3 = 25\\&#xA;7 * 4 = 34\\&#xA;7 * 5 = 43\\&#xA;etc...&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... and &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; in hexadecimal (base-16):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;F * 1 = 0F\\&#xA;F * 2 = 1E\\&#xA;F * 3 = 2D\\&#xA;F * 4 = 3C\\&#xA;F * 5 = 4B\\&#xA;F * 5 = 5A\\&#xA;F * 6 = 69\\&#xA;etc...&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Technically binary (base-2) does it too, though it's less interesting to look at:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;1 * 1 = 01\\&#xA;1 * 2 = 10&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I cheated a little there to keep it formatted nicely as 2 doesn't exist in the binary system, but hopefully the point is still conveyed.)&lt;/p&gt;&#xA;" OwnerUserId="13152" LastEditorUserId="13152" LastEditDate="2020-05-28T03:40:53.837" LastActivityDate="2020-05-28T03:40:53.837" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17598" PostTypeId="2" ParentId="17590" CreationDate="2019-12-14T14:42:29.090" Score="7" Body="&lt;p&gt;Note &lt;span class=&quot;math-container&quot;&gt;$9 = 10-1$&lt;/span&gt;  so:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;5 \times 9 = 5 \times (10-1) = 50 - 5 = 45,&#xA;$$&lt;/span&gt;&#xA;and the same for all the others:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;8 \times 9 = 8 \times (10-1) = 80 - 8 = 72.&#xA;$$&lt;/span&gt;&#xA;This works for &lt;span class=&quot;math-container&quot;&gt;$k \times 9$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$1 \le k\le 10$&lt;/span&gt;.&lt;br&gt;&#xA;Although we always have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; k \times 9 = (k-1)\times 10 + (10-k) ,&#xA;$$&lt;/span&gt;&#xA;this is the final decimal answer only when &lt;span class=&quot;math-container&quot;&gt;$1 \le k \le 10$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;After the kids do this, ask them to try to come up with a similar rule for multiplication by &lt;span class=&quot;math-container&quot;&gt;$99$&lt;/span&gt;.  (That one will work for &lt;span class=&quot;math-container&quot;&gt;$1 \le k \le 100$&lt;/span&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2019-12-14T14:42:29.090" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17599" PostTypeId="2" ParentId="17590" CreationDate="2019-12-14T15:31:45.520" Score="3" Body="&lt;p&gt;One neat trick that might help for some of your students is the &lt;em&gt;digital root&lt;/em&gt; of a number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a number, the sum of its digits is taken. For example,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;sumDigits(12345) = 1 + 2 + 3 + 4 + 5 = 15&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the result is not a single-digit number, keep repeating the process:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;dr(12345) = 1 + 2 + 3 + 4 + 5 = 15\\&#xA;= dr(15) = 1 + 5 = 6&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The base case is a single-digit number: its digital root is itself. So &lt;span class=&quot;math-container&quot;&gt;$dr(3) = 3$&lt;/span&gt;, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A useful trick for remembering the 9x tables specifically is that the digital root of any positive multiple of 9 is always 9!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;dr(9) = 9\\&#xA;dr(18) = 1 + 8 = 9\\&#xA;dr(27) = 2 + 7 = 9\\&#xA;dr(36) = 3 + 6 = 9\\&#xA;dr(45) = 4 + 5 = 9\\&#xA;dr(54) = 5 + 4 = 9\\&#xA;dr(63) = 6 + 3 = 9\\&#xA;dr(72) = 7 + 2 = 9\\&#xA;dr(81) = 8 + 1 = 9\\&#xA;...\\&#xA;dr(909) = 9 + 0 + 9 = dr(18) = 1 + 8 = 9&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="13155" LastActivityDate="2019-12-14T15:31:45.520" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17601" PostTypeId="2" ParentId="17590" CreationDate="2019-12-15T00:09:19.103" Score="1" Body="&lt;p&gt;In answer to the question &quot;I've never seen this before&quot; I knew this 'trick' with the 9x table and the one with 11x table also mentioned here when I went to primary school about 50 years ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will expand on the 11x as it has already been posted. This is mainly for the teacher but I was about 10 years old when I first learnt it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With 11 you start with:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&#xA;1 × 11 = 11&#xA;2 × 11 = 22&#xA;3 × 11 = 33&#xA;.....&#xA;When you get to:&#xA;&#xA;10 × 11 = 110&#xA;11 × 11 = 121&#xA;12 × 11 = 132&#xA;13 × 11 = 143&#xA;14 × 11 = 154&#xA;15 × 11 = 165&#xA;16 x 11 = 176&#xA;&#xA;The pattern emerges.&#xA;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Later you can multiply very large numbers with a 'trick'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;216354 x 11 = 2379894 which you can do in your head in seconds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You write down the first number (2) then add the next two numbers together and write down the answer, repeat for the following pairs and at the end write down the (4).&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&#xA;2   1     6     3     5     4    &#xA;2 2+1=3 1+6=7 6+3=9 3+5=8 5+4=9  4&#xA;2     3     7     9     8     9  4&#xA;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;If they add to 10 or more it is not so easy to do in your head as you have to carry the 10.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;276354 x 11 = 3039894&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&#xA;2     7     6      3     5     4&#xA;2   2+7=9 7+6=13 6+3=9 3+5=8 5+4=9 4&#xA;2       9     13     9     8     9 4&#xA;3(2+1)  0(9+1) 3     9     8     9 4&#xA;3       0      3     9     8     9 4&#xA;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;There are a few other nice 'tricks'.&lt;/p&gt;&#xA;" OwnerUserId="13161" LastActivityDate="2019-12-15T00:09:19.103" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17602" PostTypeId="1" CreationDate="2019-12-16T16:21:22.090" Score="4" ViewCount="206" Body="&lt;p&gt;A recent question (@Namaste) made me realize that it would be good to pull together the best resources for learning the multiplication facts. When seen as a rote memory task, this can turn students off or give students the wrong impression of what math is. When seen as finding patterns, it can be joyful and provide deep learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My son loved doubling numbers when he was young. I think many kids enjoy this.  That's the 2's. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are images that go with certain facts; like 2x6 is the shape of an egg carton, which holds 12 eggs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2s and 5s are easy because of their relation to 10. And, though it's less obvious, &lt;a href=&quot;https://matheducators.stackexchange.com/questions/17590/ive-never-seen-this-before-is-this-primarily-a-rote-computational-trick-for&quot;&gt;9s lovely patterns exist because of its relation to 10, as seen in @Namaste's post&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4 is doubling twice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What are good strategies for seeing 3s, and 6s through 8s? Also, what are good images for individual facts?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="238" LastEditDate="2019-12-17T01:19:22.910" LastActivityDate="2019-12-17T08:16:04.887" Title="Resources for Learning Multiplication Facts" Tags="|primary-education|teaching|multiplication-table|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17603" PostTypeId="2" ParentId="17602" CreationDate="2019-12-16T17:30:34.773" Score="0" Body="&lt;p&gt;When it was my turn to learn the multiplication table, one of the broadcast television stations in the United States would play animated music videos between the cartoon shows on Saturday morning.  This was Schoolhouse Rock.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://www.youtube.com/playlist?list=PLnx6r9S_SJ7I_Msib-Nj-zgROaicmyqA8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Multiplication Rock&lt;/a&gt; videos continue to this day.  They are generally 70's American folk rock, although they sample a wide variety of musical genres.  The Grammar Rock songs like &quot;Conjunction Junction&quot; are more famous, but I personally think that they have aged relatively well.  (I am also in my 50's, so don't take my musical tastes as gospel.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, I think the pedagogical tack they were taking was rote memorization by getting these earworms stuck in your head.  At the same time, there are often visual aids like your egg carton example that can spur mental connections.  And they also describe things like the distributive property and the tricks for 9, 10, and 11.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I doubt they're right for everyone, but it certainly helped me as a kid with auditory intelligence to memorize nine songs instead of one table.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2019-12-16T17:30:34.773" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17604" PostTypeId="2" ParentId="17602" CreationDate="2019-12-16T18:02:05.527" Score="3" Body="&lt;p&gt;As for 3's, I always found it easy as a student to think it as &lt;span class=&quot;math-container&quot;&gt;$3=2+1$&lt;/span&gt;. So, in order to multiply by 3, one has to double and then to add what they've found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$3\times7=2\times7+7=14+7=21,$$&lt;/span&gt;&#xA;which is much easier, since douling is carried out relatively easy by most students and addition is, in general, easier than multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, multiplication by 6 is made easy since &lt;span class=&quot;math-container&quot;&gt;$6=5+1$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, if &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}$&lt;/span&gt; is the set of all numbers whose multiplication table is easy, then &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; have relatively easy multiplication tables, for any &lt;span class=&quot;math-container&quot;&gt;$n\in\mathcal{N}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Summarizing the presented tricks, we have that the multiplication tables of:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$0,1,2,3,4,5,6,9,10$$&lt;/span&gt;&#xA;are easy or relatively easy to find/explain/remember. Consider that the tables of &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; are almost complete but for &lt;span class=&quot;math-container&quot;&gt;$7\times8$&lt;/span&gt;. So, the problem of rote learning the multiplication table from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; has been reduced to rote learning &lt;span class=&quot;math-container&quot;&gt;$7\times8$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastEditorUserId="12829" LastEditDate="2019-12-17T08:16:04.887" LastActivityDate="2019-12-17T08:16:04.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17605" PostTypeId="1" CreationDate="2019-12-16T19:05:22.690" Score="27" ViewCount="10473" Body="&lt;p&gt;In many U.S. middle schools and high schools, &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; are treated differently, with the intent that &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; is equivalent to &lt;span class=&quot;math-container&quot;&gt;$\log_{10}$&lt;/span&gt;. However, in undergraduate courses and in the academic world, &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; always means &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; is rarely, if ever used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a compelling reason to teach younger students the &quot;notation convention&quot; of &lt;span class=&quot;math-container&quot;&gt;$\log = \log_{10}$&lt;/span&gt;? Or should logarithms be taught always with a base so that students don't fall into this notation trap when they grow up?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to be that there is no need to teach the notation &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt;, and we should just teach &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; and use &lt;span class=&quot;math-container&quot;&gt;$\log_{10}$&lt;/span&gt; if we need base-ten logarithm.&lt;/p&gt;&#xA;" OwnerUserId="13175" LastEditorUserId="13175" LastEditDate="2019-12-16T19:11:32.267" LastActivityDate="2019-12-19T17:48:03.997" Title="Should we stop differentiating between ln and log?" Tags="|notation|logarithm|" AnswerCount="6" CommentCount="26" ContentLicense="CC BY-SA 4.0" />
  <row Id="17606" PostTypeId="2" ParentId="17605" CreationDate="2019-12-16T20:48:45.870" Score="46" Body="&lt;p&gt;Unrelated to US, in Germany the notation through school and university was quite consistent:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; using base &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; using base &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log_x$&lt;/span&gt; using base &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I may not know enough about the US education system, but if the definitions are not clearly homogeneous throughout the country, I would teach all three at school plus add a note that in some places the notations might differ.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, telling students that a certain notation is set in stone, when it really isn't, might lead to more confusion later.&lt;/p&gt;&#xA;" OwnerUserId="13107" LastEditorUserId="238" LastEditDate="2019-12-17T15:01:15.507" LastActivityDate="2019-12-17T15:01:15.507" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="17607" PostTypeId="2" ParentId="17605" CreationDate="2019-12-16T20:54:13.207" Score="22" Body="&lt;p&gt;You are wrong about undergraduate courses always treating &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt;. To my memory, all of my undergrad chemistry and physics (not just general, but majors texts), engineering, calculus, diffyQs, and engine maths books use &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt;\ for natural log and log for base 10.  Add onto that everything I've seen professionally in oil exploration and the military.  I have about 100 books like that on my shelves.  Just picked ten of them at random and none of the ten used &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt;\ for Napierian logarithm.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note also Sue's point on calculator buttons.  This is a huge, common, commercial usage pattern.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, I have occasionally seen the use of &quot;log&quot; to mean natural log, usually being able to detect it by something being &quot;off&quot; in the explication.  I reviewed a recent paper from two writers from &lt;a href=&quot;https://en.wikipedia.org/wiki/Federal_Reserve&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Fed&lt;/a&gt;, who were using &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt;\ for natural log (without clarification) and recommended that they note their usage at the beginning.  (I didn't need them to change their preferred notation, but did need them to clarify what they were doing.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I find most equations to be based on natural logs anyhow.  So in that case, use of &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt;\ avoids any ambiguity.  If you are planning to use &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; for base &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; logs, it is probably also best to clarify your usage initially.  Or if you insist on using &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean natural log, clarify that on first usage.  (But I would avoid, as less frequent.)&lt;/p&gt;&#xA;" OwnerUserId="13176" LastEditorUserId="143" LastEditDate="2019-12-19T17:48:03.997" LastActivityDate="2019-12-19T17:48:03.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17608" PostTypeId="2" ParentId="17605" CreationDate="2019-12-16T21:20:01.000" Score="19" Body="&lt;p&gt;In mathematics, &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; means natural logarithm.  So, as you become a mathematician, sometime during that process you must learn this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That &lt;a href=&quot;https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;standards document&lt;/a&gt; is for natural sciences, not for mathematics.  I guess mathematicians are too independent-minded for such things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would be seen, for example, in complex analysis and in real analysis.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt;  Of course we do not claim &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; is &lt;strong&gt;always&lt;/strong&gt; used for natural logarithm (except perhaps among mathematicians).  But neither can it be claimed that &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; is &lt;strong&gt;never&lt;/strong&gt; used for natural logarithm.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Example:&lt;br&gt;&#xA;W. Rudin, &lt;em&gt;Principles of Mathematical Analysis&lt;/em&gt; ... 3rd edition, page 181.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Another example:&lt;/em&gt; the &lt;a href=&quot;http://www.csci.csusb.edu/dick/c++std/cd2/lib-numerics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;C++ draft standard&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;&#xA;  template complex log(const complex&amp;amp; x);&lt;br&gt;&#xA;  Notes:&#xA;    the branch cuts are along the negative real axis.&lt;br&gt;&#xA;  Returns:&#xA;    the complex natural (base e) logarithm of x, in the range of a strip&#xA;    mathematically unbounded along the real axis and in the interval [-i&#xA;    times  pi,  i times pi ] along the imaginary axis. When x is a nega-&#xA;    tive real number, imag(log(x)) is pi.&lt;br&gt;&#xA;  template complex log10(const complex&amp;amp; x);&lt;br&gt;&#xA;  Notes:&lt;br&gt;&#xA;    the branch cuts are along the negative real axis.&#xA;  Returns:&#xA;    the  complex  common   (base   10)logarithm   of   x,   defined   as&#xA;    log(x)/log(10).&lt;br&gt;&#xA; &lt;/code&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="238" LastEditDate="2019-12-19T11:13:08.817" LastActivityDate="2019-12-19T11:13:08.817" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="17609" PostTypeId="2" ParentId="17605" CreationDate="2019-12-17T08:41:38.360" Score="15" Body="&lt;p&gt;My perspective on this (as a British computer scientist) is slightly different to the others already mentioned: my default expectation is that &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; is the natural logarithm, &lt;span class=&quot;math-container&quot;&gt;$\lg$&lt;/span&gt; is the binary logarithm, and &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; is used when the base is unimportant. Since I usually see logarithms inside a Landau &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt;, the irrelevancy of the base is the most common case.&lt;/p&gt;&#xA;" OwnerUserId="8005" LastActivityDate="2019-12-17T08:41:38.360" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17610" PostTypeId="2" ParentId="17605" CreationDate="2019-12-17T11:22:12.850" Score="21" Body="&lt;p&gt;I would argue that we should &lt;em&gt;never&lt;/em&gt; use &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\log_{10}$&lt;/span&gt; anymore, only warn that this was historically often done. Sticking to the ISO convention is probably safest:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{aligned}&#xA;   \log_{e} \equiv \ln&#xA;\\ \log_2 \equiv \operatorname{lb}&#xA;\\ \log_{10} \equiv \lg&#xA;\end{aligned}$$&lt;/span&gt;&#xA;I personally would use of the shorthands only &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt;, and write out all other logarithms explicitly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further I would remark that most programming languages use &lt;code&gt;log&lt;/code&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt;. This does make some sense, because&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In a programming context you have lots more variable names so standard functions don't stand out the way they do in maths formulas; in particular something two-letter like &lt;code&gt;ln&lt;/code&gt; would be not as clear to read.&lt;/li&gt;&#xA;&lt;li&gt;The natural logarithm is by far the most important in serious applications. Computer scientists may disagree (in discrete maths, base-2 tends to turn up more), but then, they mostly write logarithms in big-O notation and there the basis doesn't actually matter because it is equivalent to a constant factor.&lt;br&gt;The reason base-10 used to be common is that it makes calculation by hand easier, especially with a slide rule. But this is completely obsolete; today the base-10 logarithm is essentially irrelevant.&lt;/li&gt;&#xA;&lt;li&gt;&lt;code&gt;exp&lt;/code&gt; is generally understood to mean &lt;span class=&quot;math-container&quot;&gt;$x\mapsto e^x$&lt;/span&gt;, so using &lt;code&gt;log&lt;/code&gt; for &lt;span class=&quot;math-container&quot;&gt;$\log_e$&lt;/span&gt; gives a nice symmetry.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Still, in a maths context I'd continue to write &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; for the natural logarithm, and only ever use &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; with an explicit base annotation.&lt;/p&gt;&#xA;" OwnerUserId="6902" LastEditorUserId="6902" LastEditDate="2019-12-17T13:20:54.157" LastActivityDate="2019-12-17T13:20:54.157" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17611" PostTypeId="1" CreationDate="2019-12-17T21:57:44.863" Score="3" ViewCount="176" Body="&lt;p&gt;Many of my professors are experts in their field, but they are just not able to teach. I mean: most of the times they are confusing, don't follow a logic sequence in their speech or keep a superior level than the audience. They also answer to questions telling something different from the point of the question. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that I am not a genius and math is complicated but I am sure that many of you have experienced what I mean, I just cannot figure out why the more you are expert in something the less you are able to teach it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously I am generalising, please add &quot;usually&quot; in every sentence.    &lt;/p&gt;&#xA;" OwnerUserId="13196" LastEditorUserId="238" LastEditDate="2019-12-17T22:08:09.280" LastActivityDate="2019-12-17T23:24:39.537" Title="Why teaching undergraduate-level mathematics is so complicated?" Tags="|undergraduate-education|teaching|struggling-students|student-question|" AnswerCount="1" CommentCount="5" ClosedDate="2019-12-18T06:01:46.793" ContentLicense="CC BY-SA 4.0" />
  <row Id="17612" PostTypeId="2" ParentId="17611" CreationDate="2019-12-17T22:59:01.750" Score="7" Body="&lt;p&gt;There are many reasons, but the classic explanation is that professors (especially at research universities) are picked (and compensated) for research ability versus teaching efficiency.  (In general...caveat hawks.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably a secondary reason is that mathematics tends to be a field with a high emphasis on logic and precision.  However EDUCATION is more of a practical art (with aspects of psychology, leadership, tribal lore, efficiency, trade-offs, decisions with imperfect information, etc.) that are very different from Euclidean proofs. So, it would not surprise me that this issue occurs a lot in math, as a field.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one counteracting tendency is the need for large amounts of service teaching for engineers and natural and even applied scientists (e.g. econ).  This has some forcing to drive more effective teaching.  On the gripping hand, if you're in a majors course...:-(&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that none of this is new to the world.  You might also ask many other questions about non-ideal behavior (in many places, other than math ed).  Part of becoming an adult is realizing there are a lot of imperfect people/places out there. Don't get too exercised by it.  Just do what you need to do, to optimize things for you.  Find out which profs are good, buy Schaum's Outlines, etc.  Maybe too late for you (but not others), but one attraction of &quot;liberal arts schools&quot; is more emphasis on teaching.  You can always &quot;go to Berkeley&quot; for grad school.   &lt;/p&gt;&#xA;" OwnerUserId="13197" LastEditorUserId="238" LastEditDate="2019-12-17T23:24:39.537" LastActivityDate="2019-12-17T23:24:39.537" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17613" PostTypeId="2" ParentId="17605" CreationDate="2019-12-18T08:53:24.413" Score="8" Body="&lt;p&gt;This question is more difficult to answer than it appears. Part of the difficulty is that usage varies from area to area, from country to country, and from teacher to teacher. Part of the problem is that the answer might depend on the audience being taught. The summary of what follows is that: a. when teaching one should be up front about and make one's conventions clear; b. when teaching anyone other than future mathematicians one should either use &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; or be very clear that one is using &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt; (&lt;em&gt;very clear&lt;/em&gt; means repeating endlessly); c. that the issue in professional practice is one of context - in mathematics there is not much need for other bases, while in contexts where the reference scale is important and it is natural to use some integer base such as &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;, it is natural and convenient to write &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean the logarithm in this base; d. in purely mathematical contexts &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; means the natural logarithm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, what is the &lt;em&gt;logarithm&lt;/em&gt;? Among professional mathematicians, &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; means an inverse of the exponential function &lt;span class=&quot;math-container&quot;&gt;$e^{z} = \sum_{k \geq 0}\tfrac{z^{k}}{k!}$&lt;/span&gt;. More precisely, one calls any solution of &lt;span class=&quot;math-container&quot;&gt;$e^{z} = w$&lt;/span&gt; a &lt;em&gt;logarithm&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt;, and one writes &lt;span class=&quot;math-container&quot;&gt;$\log w$&lt;/span&gt; for any such number. If &lt;span class=&quot;math-container&quot;&gt;$z = x + iy$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; solves the equations &lt;span class=&quot;math-container&quot;&gt;$e^{x} = |w|$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e^{iy} = w/|w|$&lt;/span&gt;. The equation &lt;span class=&quot;math-container&quot;&gt;$e^{x} = |w|$&lt;/span&gt; has a &lt;em&gt;unique&lt;/em&gt; solution given by the &lt;em&gt;real logarithm&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$\log|w|$&lt;/span&gt;. In general, &lt;span class=&quot;math-container&quot;&gt;$\log w = \log |w| + i \arg w$&lt;/span&gt; where the definition of &lt;span class=&quot;math-container&quot;&gt;$\arg$&lt;/span&gt; involves the choice of a branch cut in the complex plane. Here I have just paraphrased the definitions from the book of Ahlfors, which must be a reference as standard as any.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a conceptual standpoint, over the real numbers, what are naturally defined are the exponential, as the aforementioned power series or as the solution of the initial value problem &lt;span class=&quot;math-container&quot;&gt;$\dot{x} = x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x(0) = 1$&lt;/span&gt;, and the real logarithm, as the integral &lt;span class=&quot;math-container&quot;&gt;$\log x = \int_{1}^{x}\tfrac{1}{t}\,dt$&lt;/span&gt;. These functions are inverses, and proving that is somehow the heart of a mathematically oriented calculus class. Which is the better starting point is perhaps a matter of preference. Personally, I like to start with the definition of the real logarithm as an integral, but on this point I change my mind every time I teach calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If one goes through these proofs, one sees that working with another &lt;em&gt;base&lt;/em&gt; is not terribly natural as it amounts to sticking some ad hoc constant factor in the initial value problem or integral. From this point of view, one has to do some work to define &lt;span class=&quot;math-container&quot;&gt;$a^{b}$&lt;/span&gt; (or equivalently, &lt;span class=&quot;math-container&quot;&gt;$\log_{a}$&lt;/span&gt;), and this work amounts to the standard functional relations for either the exponential or the logarithm (the integral definition of the logarithm makes these straightforward). From a &lt;em&gt;mathematical viewpoint&lt;/em&gt; there is no need to introduce any other notation than &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt;, nor is there often need to work with logarithms in other bases. Consequently, the mathematician often has difficulty seeing any need for introducing such things at all, and resists using notations such as &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt;, and in some cases cannot understand why engineers and students and other lower forms of life like economists insist on writing such barbarities all the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The change in the base of the logarithm is natural when one thinks about scales, measurement, and the like, concepts where are nonmathematical or extramathematical in the sense that there is no logical necessity for them (one can always write &lt;span class=&quot;math-container&quot;&gt;$\log(b)/\log(a)$&lt;/span&gt;). Moreover, since &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is irrational, it is &lt;em&gt;not&lt;/em&gt; a good choice operationally as a scale. Numbers such as &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; are better, more useful, more operational choices (are there any others?). In a context where &lt;span class=&quot;math-container&quot;&gt;$\log_{2}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\log_{10}$&lt;/span&gt; is used repeatedly, and no other choice is used, it is natural and convenient to drop the subscript and write simply &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean the logarithm in the given base, and this is precisely what has happened quite generally. In such a context it sometimes becomes necessary to compute, and when doing so, the &lt;em&gt;natural logarithm&lt;/em&gt; somehow creeps back because everyone writes &lt;span class=&quot;math-container&quot;&gt;$e^{x}$&lt;/span&gt;, and something like &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; becomes standard to distinguish the &lt;em&gt;natural logarithm&lt;/em&gt; (or does the notation abbreviate &lt;em&gt;Naperian logarithm&lt;/em&gt;?) from the other in common use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When teaching small children, it is impossible to speak of initial value problems and integrals, much less branch cuts and arguments, and there is no reasonable way to define &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt;. On the other hand, ever small child knows &lt;span class=&quot;math-container&quot;&gt;$3^{2} = 9$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$10^{2} = 100$&lt;/span&gt;, so it makes sense to define things like &lt;span class=&quot;math-container&quot;&gt;$\log_{2}9 = 3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\log_{10}(100) = 2$&lt;/span&gt;, and more generally, logarithms with integer bases of integers make sense and are easy to define, and this is the natural starting point for working with logarithms when dealing with children. In such a context, it is entirely &lt;em&gt;unnatural&lt;/em&gt; to write &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt;, since on the one hand one is not mentioning &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, and on the other hand one cannot make any sense of it in any case. Worse, in high school, one tries to make sense of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; via some limiting procedure that usually starts from understanding exponentials/logarithms with rational bases. So high school teachers find the insistence of professional mathematicians that there is one true logarithm, &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt;, to be a bit difficult, and write &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\log_{10}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt;, when they get around to trying to give some meaning to the latter expression (if they ever do that at all). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course there are countries where &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; is written for &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt; all along, and there are probably other countries where something else is written entirely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then along comes some signal processing engineer who wants to write &lt;span class=&quot;math-container&quot;&gt;$\lg$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\log_{2}$&lt;/span&gt; because it is all she uses, and &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; already means &lt;span class=&quot;math-container&quot;&gt;$\log_{10}$&lt;/span&gt; ... As a professional (hopefully) mathematician, this unnecessary barbarous new notation makes my blood boil, but latex already has the command &lt;em&gt;\lg&lt;/em&gt; defined so it must be in common use whether I know it or not ... And electrical engineers outnumber mathematicians, so I am fighting a losing battle - and for what gain?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now the student who has learned that &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$\log_{10}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt; gets to the university and the math professor keeps writing &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; and it gets very confusing. The professor has three approaches:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Write &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt; and not worry about the matter further.&lt;/li&gt;&#xA;&lt;li&gt;Observe that mathematicians use &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt;, acknowledge that this can be confusing at first, and use &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Observe that mathematicians use &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt;, but write &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; because it causes less confusion and the engineering students in the room are never going to study complex analysis or Fourier series anyway (some of them should, but that's a different matter).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;1 Is quite common, but not a good idea. In the past I always did 2, and think it is a perfectly defensible position. In recent years I have tended towards &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, simply because it costs me nothing and I think costs the student nothing either (I am not teaching mathematics students, rather engineers). If I use &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean &lt;span class=&quot;math-container&quot;&gt;$\log_{e}$&lt;/span&gt;, which I sometimes do because for me this usage is reflexive, I always mention that it means the base &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; log not the base &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; log. Students assimilate this rapidly. Were I teaching complex numbers, of course I would use &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to mean the natural logarithm; in that context there's no other practice that makes much sense. Were I teaching in a context (physics, engineering, etc.) where the values of numbers had interpretations that mattered, then I would always specify the base, and would use &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; when referring to the natural base.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="238" LastEditDate="2019-12-19T00:08:24.703" LastActivityDate="2019-12-19T00:08:24.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17614" PostTypeId="2" ParentId="17590" CreationDate="2019-12-18T09:00:32.930" Score="1" Body="&lt;p&gt;I was taught the multiplication table for single digit numbers multipled by &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; based on two observations:&#xA;a. the digits of the result sum to &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;&#xA;b. the first digit of &lt;span class=&quot;math-container&quot;&gt;$k \times 9$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$k-1$&lt;/span&gt;. &#xA;This sort of &lt;em&gt;trick&lt;/em&gt; is helpful because it makes the multiplication tables more interesting and easier to memorize. In particular the student feels proud for having mastered early the &lt;em&gt;big&lt;/em&gt; number &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For me as a child it was also one my first encounters with math (rather than arithmetic). I remember spending some time (probably with some parental urging) trying to understand why a. and b. are true, and why they are special to &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that there are similar &lt;em&gt;tricks&lt;/em&gt; for most of the multiplication tables - the digits of a multiple of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; sum to a multiple of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, the second digit of a multiple of &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; is even. For integers less than &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt;, only &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; admits no tricks ... &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-12-18T09:00:32.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17616" PostTypeId="2" ParentId="17556" CreationDate="2019-12-18T12:04:08.670" Score="5" Body="&lt;p&gt;Talented students with interest are much better served by learning more math than by being trained at contest math. Where contest math may play a positive role is in getting talented students interested in other kinds of math, something that might otherwise be difficult to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice students are often guided/directed to contest math rather than learning more advanced math for a number of essentially practical reasons, among which are: the teachers the student knows do not feel comfortable guiding/directing/teaching more advanced math with which they themselves do not feel sufficiently comfortable to teach; the books and other resources to which the student has access are not adequate (no decent library) and the student lacks the autonomy to filter and search herself; teachers have a professional interest in developing teams that do well in math contests; the social aspect of the math team. There are of course other reasons. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of these reasons are more compelling than others. The social, team aspect can be a solid reason for some students. The inadequacy of library resources was in the past a serious conditioning factor that now is far less relevant because good resources are available via the web. However, the searching/filtering problem is a serious problem, perhaps more serious now than in the past (if there was a library, it was likely to have some decent books among the few it had). The most serious practical problem is that there is not available a teacher/guide who feels confident acting as such for such a student. It is difficult to learn more advanced math without guidance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Said all that, I think it is socially more valuable to train students to do more advanced math than it is to train them to do contest math, and students who manifest interest in learning more math are not always well served by being steered to contest math. Here I admit that my personal experience in this regard was very negative. In high school I was interested in learning more math, got a hold of some books on groups and rings from the library (the local library had a lot of the Carus mathematical monographs published by the AMS), and tried to learn from them. I struggled because of lack of guidance. The teacher who could have acted as a guide tried to steer me into contest math, I think both because he felt uncomfortable with groups and rings and the like and because he was interested in having a strong math team. I disliked contest math and didn't particularly like the accompanying social atmosphere, and the whole set of circumstances wound up driving me away from studying math for a 3 or 4 year period that ended only in the middle of my university studies. I wound up a mathematician anyway, and I still like algebra, so I probably would have been better served by learning more math and following my interests. However, the practical problems are serious - looking back I don't see who else around me then could have provided the necessary guidance, and I lacked the autonomy to do it all myself (some students have this autonomy and become really good mathematicians if they somehow get access to the resources necessary to develop their inclinations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the summary is - guide the student where the student wants to go - if the student enjoys contest math, then it can be an excellent vehicle for learning about other kinds of math - but if the student, although showing talent and interest, does not like contest math, better to try to provide access to other stimulating resources. Extreme mathematical talents often manifest themselves early, sometimes in the early teens, and it's a shame if they are not developed.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2019-12-18T12:04:08.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17617" PostTypeId="1" AcceptedAnswerId="17618" CreationDate="2019-12-18T14:26:30.730" Score="6" ViewCount="380" Body="&lt;p&gt;Is the distinction between cardinal numbers and ordinal numbers taught as part of&#xA;mathematics (as opposed to part of learning the language distinction between&#xA;&quot;one&quot; and &quot;first&quot;) in pre-college or early-college math instruction outside&#xA;of set theory?&#xA;If so, where is it used? Where is it needed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am asking because I don't see a need for the distinction until one encounters&#xA;infinite sets, say, ordering the algebraic numbers. In my own college teaching I&#xA;need to distinguish between countably infinite &lt;span class=&quot;math-container&quot;&gt;$\aleph_0$&lt;/span&gt; and uncountably infinite &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{c}$&lt;/span&gt;, but rarely do I use &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;. It would help if I&#xA;understood where the cardinal vs. ordinal distinction is used/needed.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-12-27T01:01:38.063" Title="Cardinal vs. ordinal: When learned? When needed?" Tags="|real-numbers|infinity|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17618" PostTypeId="2" ParentId="17617" CreationDate="2019-12-18T15:36:21.457" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Is the distinction between cardinal numbers and ordinal numbers taught as part of mathematics (as opposed to part of learning the language distinction between &quot;one&quot; and &quot;first&quot;) in pre-college or early-college math instruction outside of set theory? If so, where is it used? Where is it needed? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm not aware of any early-college courses in which the distinction (or either notion individually) is taught systematically.  I've seen it come up in &quot;introduction to proof&quot; courses, and I could imagine it appearing in a discrete math course, but syllabi for those courses tend to vary a lot between institutions and instructors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any sufficiently advanced logic course, whether or not it includes set theory, is going to have to cover both notions, but that's getting out of what I'd call &quot;early-college&quot;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In my own college teaching I need to distinguish between countably infinite ℵ0 and uncountably infinite , but rarely do I use . It would help if I understood where the cardinal vs. ordinal distinction is used/needed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have trouble imagining where the distinction itself is useful.  The question is where one wants to work with ordinals at all; I think the main reason people usually introduce ordinals is because they need to do transfinite induction.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2019-12-18T15:36:21.457" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17619" PostTypeId="2" ParentId="17617" CreationDate="2019-12-18T15:48:51.167" Score="2" Body="&lt;p&gt;I don't know if this is always in a set theory course or not, but it makes a difference when you define the natural numbers using the Peano axioms (which is ordinal) or the Frege construction (which is cardinal).&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2019-12-26T17:48:39.470" LastActivityDate="2019-12-26T17:48:39.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17620" PostTypeId="1" CreationDate="2019-12-18T21:30:51.353" Score="6" ViewCount="288" Body="&lt;p&gt;I imagine there must exist a fair amount of literature and discussion about the idea of somehow redesigning college math courses, and the entire college math curriculum, to be self-paced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question: What are the most notable works that have been done along these lines? What should I read before attempting to implement my own self-paced courses and curriculum?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am also interested in hearing your own thoughts and ideas about whether or not self-paced math courses are a good idea in a college setting, and how they might be implemented.&lt;/p&gt;&#xA;" OwnerUserId="9633" LastEditorUserId="238" LastEditDate="2019-12-21T19:41:18.240" LastActivityDate="2019-12-22T16:06:12.400" Title="Redesigning college math courses and curriculum to be self-paced" Tags="|undergraduate-education|self-learning|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17622" PostTypeId="2" ParentId="17620" CreationDate="2019-12-20T22:05:44.110" Score="6" Body="&lt;p&gt;Usually, &quot;self-paced&quot; or &quot;personalized learning&quot; courses are managed via technology. An interesting review of the current state of the industry was published yesterday (as I write this) at Technology Review by Natalie Wexler:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A 2019 report from the National Education Policy Center at the&#xA;  University of Colorado on personalized learning—a loosely defined term&#xA;  that is largely synonymous with education technology—issued a sweeping&#xA;  condemnation. It found “questionable educational assumptions embedded&#xA;  in influential programs, self-interested advocacy by the technology&#xA;  industry, serious threats to student privacy, and a lack of research&#xA;  support.”&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Judging from the evidence, the most vulnerable students can be harmed&#xA;  the most by a heavy dose of technology—or, at best, not helped. The&#xA;  OECD study found that “technology is of little help in bridging the&#xA;  skills divide between advantaged and disadvantaged students.” In the&#xA;  United States, the test score gap between students who use technology&#xA;  frequently and those who don’t is largest among students from&#xA;  low-income families. A similar effect has been found for “flipped”&#xA;  courses, which have students watch lectures at home via technology and&#xA;  use class time for discussion and problem-solving. A flipped college&#xA;  math class resulted in short-term gains for white students, male&#xA;  students, and those who were already strong in math. Others saw no&#xA;  benefit, with the result that performance gaps became wider.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Observing a case where a young student gave up on trying to add 8 and 3 and instead starting doodling on the iPad that was quizzing them, the author further writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If Kevin had been asked to combine 8 and 3 by a teacher rather than an&#xA;  iPad, there’s a greater chance he would have been interested in trying&#xA;  to do it. “It’s different when you’re learning from a person and you&#xA;  have a relationship with that person,” cognitive psychologist Daniel&#xA;  Willingham has said. “That makes you care a little bit more about what&#xA;  they think, and it makes you a little bit more willing to put forth&#xA;  effort.”...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In addition to sapping motivation, technology can drain a classroom of&#xA;  the communal aspect of learning. The vision of some ed tech advocates&#xA;  is that each child should sit in front of a screen that delivers&#xA;  lessons tailored to individual ability levels and interests, often on&#xA;  subjects chosen by the students themselves. But a vital part of&#xA;  education is different kids bouncing their ideas off each other...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.technologyreview.com/s/614893/classroom-technology-holding-students-back-edtech-kids-education/&quot; rel=&quot;noreferrer&quot;&gt;Full story here.&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-12-20T22:05:44.110" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17623" PostTypeId="2" ParentId="17525" CreationDate="2019-12-22T03:03:47.333" Score="2" Body="&lt;p&gt;This is a little later than the 1950's, but it's been a while since your question was asked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I used to teach math at the Ohio State University in the early 1980's and even then, anyone getting a B.A., who were basically the students with non-STEM majors, were only required to take one basic algebra course to complete their degree. If I recall correctly, the B.A. majors could even substitute a certain Philosophy course in Logic for the algebra course and therefore not take any math at all in order to obtain their B.A.'s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the present, various places I teach part-time in my retirement allow non-STEM students to obtain their degrees after passing one, single fluff course in math made up of various topics like very basic statistics, very basic geometry, basic algebraic equations, and basic exponential equations.&lt;/p&gt;&#xA;" OwnerUserId="13034" LastActivityDate="2019-12-22T03:03:47.333" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17624" PostTypeId="2" ParentId="17620" CreationDate="2019-12-22T03:42:36.627" Score="6" Body="&lt;p&gt;Here are some problems I see with asynchronous self-paced instruction. Let me begin by first setting forth some assumptions I have about what &lt;em&gt;self-paced&lt;/em&gt; likely entails.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Instruction is automated: the content is largely produced asynchronously to your use. You are not directly interacting with a teacher. Instead, you are likely watching little tutorial videos and/or reading.&lt;/li&gt;&#xA;&lt;li&gt;Grading of the course is automated: you probably are using some system to either give short answer or multiple choice responses which can be scored by a computer. It is very unlikely you are submitting complete arguments or writing proper work for a human to properly evaluate&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Both of the above features are almost essential since they keep the cost of such instruction at a minimum. Unfortunately, you get what you pay for in this case because:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Lack of human interaction robs you of the chance to have unique experiences in the classroom with your peers. Jokes, shared suffering, asking questions and/or seeing your peers ask questions... these are not going to happen with automated instruction.&lt;/li&gt;&#xA;&lt;li&gt;Temptation to game the system: when you know the homework is just a computer game then it is awful tempting to treat it as such. On the other hand, if the homework is given in real time by a professor who expresses actual interest in your maturation as critically thinking students then it is much less likely you ignore the homework.&lt;/li&gt;&#xA;&lt;li&gt;Inability to identify outliers: part of the job of a professor is to identify excellence or diamonds in the rough. This is one of the more subtle parts of teaching, to see past bad preparation to see the student for what they could be rather than what they appear to be in the present. I cannot fathom an automated system which does such a service justice.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now, perhaps none of these things matter since the math we are thinking about is just a general education requirement. Even so, I know some of our best Math majors came from majors where math was just a general education requirement. If that gen-ed course had been automated, they might never had the chance to forge a relationship with a professor passionate about math. I think that is a great loss since such non-traditional lateral entries to the math major are some of our most creative students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should make an important caveat from what I say above. Most of my criticism is based on the automation and lack of professional guidance in the education. In fact, I am all for self-paced instruction provided it is &lt;strong&gt;guided by actual professors in a individually crafted fashion&lt;/strong&gt;. Furthermore, I think it works best when there is a single test at the end of the study, like a &quot;CLEP&quot;. This path is not for all students. Students must be self-motivated and above average for this to work. Ideally in the summer when there are not a lot of other distractions etc.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2019-12-22T16:06:12.400" LastActivityDate="2019-12-22T16:06:12.400" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17625" PostTypeId="1" CreationDate="2019-12-22T05:28:06.200" Score="6" ViewCount="276" Body="&lt;p&gt;I’m a homeschool teacher of a nine-year-old, and we sometimes have one-on-one ‘number talks’ (a.k.a. 'math talks') similar to the activity used in primary school classrooms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of this process involves writing down the thinking process used to solve a particular problem. For example, my child might explain the steps they used to  solve 5 x 25 in their head. In turn, I write the process down on paper as they watch. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is there a standard notation / recording technique for writing down thinking processes used in number talks? Or is there a resource that may provide some helpful ideas in this regard?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my brief experience, I’m finding that my written diagrams or notations are not ideal. Admittedly, I will draw operators, numerals, and arrows on-the-fly, and they don’t always clearly communicate the process, especially when looking at them a few days later. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’m particularly concerned about this because one of my longer term goals of these number talks is to encourage good written communication about math, and I’m not always being a good role model. &lt;/p&gt;&#xA;" OwnerUserId="13229" LastEditorUserId="13229" LastEditDate="2019-12-24T21:49:43.463" LastActivityDate="2019-12-25T22:06:04.063" Title="Are there standard notations for 'number talks' / ‘math talks?'" Tags="|primary-education|notation|board-use|mathematical-discourse|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17626" PostTypeId="2" ParentId="17516" CreationDate="2019-12-22T09:24:56.747" Score="1" Body="&lt;p&gt;I recommend learning multivariable calculus. I think the book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0393925161&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Div, Grad, Curl and All That&lt;/a&gt; is a pretty good short book on this topic. I also think that the book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1108420419&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Electrodynamics&lt;/a&gt; by Griffiths has a good intuitive explanation of vector calculus. If you want a more rigorous exposition, Folland's book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0130652652&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Advanced Calculus&lt;/a&gt; is good.&lt;/p&gt;&#xA;" OwnerUserId="2049" LastActivityDate="2019-12-22T09:24:56.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17627" PostTypeId="2" ParentId="17516" CreationDate="2019-12-22T14:46:49.643" Score="3" Body="&lt;p&gt;Here is the stereotypical undergraduate engineering math curriculum in the US.  Of course every person/school is different, but this is a useful reference point for initial awareness. (The last sentence is actually covered by the single word &quot;stereotypical&quot; but some of the caveat Na...police need to be told twice.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SEM 1: limits, differential calculus, start of integral calc&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SEM 2: more integral calc (trickier methods), chapter on series, chapter on diffyQs.  YOU ARE HERE.*&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SEM 3: multivariable calculus&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SEM 4: Ordinary differential equations (some repetition of chapter from SEM 2, but more detail).  Typically most (even basic/applied texts) have a bit too much to cover in a semester and later topics (especially systems of equations, Laplace transform, and series methods) will get an abbreviated (even omitted) coverage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SEM 5: &quot;Engineering Mathematics&quot;  or &quot;Methods for Engineers&quot;.  This is the most vague, but a typical course would be about a half semester of linear algebra (matrices) and a half semester of PDEs/Fourier series.  Both of these are topics that could easily get a full semester each, even for an applied engineer look...but few schools give that much time.  [You do have a huge slew of foundational mechE courses like statics, dynamics, fluids, thermo before you even get into your aero majors courses where you learn lift and drag--there is just limited time in the schedule.]  Although some will do a separate formal linear algebra requirement of a semester, in which case, the engine math course may go deeper into PDEs.  I have also sometimes seen physicists (but not engineers) doing a two semester &quot;methods for physicists&quot; which is basically like two semesters of engine math.  Typically undergrad engineers (especially non EEs) will not have complex analysis, even &quot;for engineers&quot; as part of their required sequence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is possible to get specialized texts for all of the semester 3 to 5 topics.  However, the Kreyszig text covers them very well.  It was written by a mathematician and actually teaches math, tricks and concepts/derivations (not just formulas in passing while learning some physical topic), but by someone from the prehistoric era when mathematicians liked engineers and physicists.  It's a very good middle ground between the sort of unsatisfying explication you get within a technical topic (by people not that great at math OR teaching math) and the proof-centric (but not caring about tricks/techniques) approach of the pure math types.  Within it, you will have a clear &quot;about semester&quot; coverage of the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Multivariable (&quot;vector&quot;) calculus&lt;/li&gt;&#xA;&lt;li&gt;ODEs (K reverses the order and has it first...nothing wrong with that)&lt;/li&gt;&#xA;&lt;li&gt;Fourier series and PDEs&lt;/li&gt;&#xA;&lt;li&gt;Linear algebra&lt;/li&gt;&#xA;&lt;li&gt;Complex analysis [This is &quot;extra&quot; above the minimum normally required for aero.]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you pound all that out (and I mean do it like a martial artist, learning skill after skill, doing each drill), you'll have a very, very strong basic coverage.  Better than the normal engineer.  Of course you can go to even greater applied depth with specific texts and/or do theory side trips.  But I would start with this baseline FIRST. Can build on the foundation later, if needed/time.  [You do need to learn slip and drag and Reynolds number and such sometime!]  I would just use Kreysig since it is a single book, to lay this baseline.  At least it is self consistent.  While still also being very easy to pick certain sections to work independently.  [Other competitor books probably fine too.  But, stay the hell away from Arfkin and Weber...it's a mess--see the Amazon reviews (and I have it)...physicists trying to write math methods and reads like a laundry list, not an integrated teaching book, really.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some other math topics, not on the main calculus path, but very useful for engineers.  I have listed them in priority order.  (Also I would prioritize (1) and (2) below BEFORE complex analysis, for an aero engineer.):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Probability and statistics (&quot;probes and shafts&quot;).  Do a basic survey course.  K has a chapter on this, if you want the fast version.  But maybe worth it to do a simple, applied text/semester on its own.  This has some of the &quot;ODE problem&quot; in that the &quot;texts that sell&quot; have more like 1.5-2 semesters of work (yes, even in a simple applied introduction) and teachers will skip many sections to breeze through the topic.  And it is possible to go into much more depth on TOP of what a basic intro would do--people do Ph.Ds in it!  (C'est la vie.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Design of Experiments, Quality Methods, etc. May have a non-intuitive name.  But the key concept is (grossly simplifying) how to design and evaluate partial factorial examinations of complex, poorly understood phenomena (multivariable and nonlinear, on steroids...i.e. real life, engineering, business, farming, social topics like crime and engineering, etc.).  The classic text here is Box, Hunter, Hunter but there are other good ones as well.  May be better to take an actual class here, versus self study, as you get access to a lab and perhaps university copy of helpful statistical computer program.  If you are going to go get a job in industry, this might even be more useful than the basic probes and shafts...and there is a little overlap.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Numerical methods.  This is probably overkill for the average engineer to do a specialization in.  But if you are going into grad school would be nice to actually spend a concentrated time on the different methods and understanding their error characteristics and computation efficiency.  K has a chapter or two here, but I would probably look at a course or self standing text.  But again, if you never get to it, you can still probably hack away as needed ad hoc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Perturbation methods.  Bender and Orzag is the classic text.  Really not needed unless you are a very high end (mathematically inclined) aero Ph.D.  But it is sort of &quot;more calculus&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Few other text recommendations.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A.  I am a huge fan of Schaum's Outlines.  Very clearly written (not too much text...get to the point, simple wording).  Lots of drill problems AND solutions.  And implicit &quot;teach by do&quot; approach.  They are economical, not like the normal textbook market.  Order one whenever you have an issue with a regular text, hard to understand teacher, or just want more drill problems.  Can do it ad hoc as needed by topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B.  I am a big fan of the Michael Lindberg engineer license prep text books.  The EIT reference manual essentially covers an entire undergrad engineer curriculum (all the statics/fluids mechE crap, but even basics in circuits, chemistry, physics, math, econ etc.) WITH DRILL PROBLEMS [spend the extra few bucks and buy the complete solutions manual also].  All in one book (with thin paper admittedly).  I have &quot;literally&quot; run circles around engineers (despite lacking a BSE) because of all the quick reference info in this book.  (OK, figuratively running the circles, but I was extremely tempted to warhoop around their desks.)  I don't know about aero, but the Lindberg mechE prep book (I have it) is of similar quality to the EIT volume.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*From your question (i.e. not knowing the landscape), comment answers, and questions on the main math site, it looks like you are about at the end of semester 2.&lt;/p&gt;&#xA;" OwnerUserId="13231" LastEditorUserId="13231" LastEditDate="2019-12-23T14:58:35.443" LastActivityDate="2019-12-23T14:58:35.443" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17628" PostTypeId="2" ParentId="17625" CreationDate="2019-12-22T17:46:38.410" Score="4" Body="&lt;p&gt;The notation used on the board in math lectures should be the same as that found in any other mathematical writing. That is: Get a good book, read it &lt;em&gt;very&lt;/em&gt; carefully, and follow that same notation precisely. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This strengthens connections and prepares the student to read outside texts and articles on their own (which should be one of the principal goals of any educational program, in my opinion). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't have any other textbooks available right now, then I recommend the &lt;a href=&quot;https://openstax.org/subjects/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax Math&lt;/a&gt; series. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-12-22T17:46:38.410" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17629" PostTypeId="2" ParentId="17525" CreationDate="2019-12-22T19:09:14.567" Score="6" Body="&lt;p&gt;I'll quote a few short things from the (fantastic!) articles shared in comments by Dan Fox and user1527. Morris Kline in 1954 wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What have we been feeding the liberal arts students? The almost&#xA;  universal diet has been college algebra and trigonometry.  I believe&#xA;  that these courses are a complete waste of time...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Kline proceeds to outline a plan for a historically-minded math-survey/appreciation course for liberal arts students, revolving around his textbook, &lt;em&gt;Mathematics in Western Culture&lt;/em&gt; (published 1953, after having taught 3 sections of such a course at NYU in the prior two years). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently the idea of a survey/appreciation course had been floating around for at least some decades before that, because Tucker writes in his 2015 historical survey:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While the early MAA educational activities focused on secondary school&#xA;  preparation in mathematics, there was a 1928 MAA report (MAA [22])&#xA;  addressing complaints about the first two years of college&#xA;  mathematics; also see Schaaf [31]. It acknowledged calls to offer a&#xA;  survey course of mathematical ideas with historical aspects for&#xA;  non-technical students...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Tucker notes that calculus didn't become a standard freshman college subject until a major disruption in the 1950's, when the Cold War response to Sputnik -- and observations that WWII was won largely by pure mathematicians working in ballistics, signaling, rocketry, cryptography, and nuclear design -- caused a massive social call and glorification of math and physics. Physics faculty started expecting calculus usage in freshman physics courses, and math departments followed suit. Up until that time, a student taking first-year courses would likely have only college algebra and trigonometry available, from my reading here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On that latter major revolution that Tucker describes in the 1950's, he points out that in the early 1960's, as many as 5% of incoming college students were interested in being mathematics majors, while by 1975 it had dropped to 1.1%, and has stayed at around that level ever since (partly explained by the spin-off of computer science and statistics majors). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2019-12-22T19:09:14.567" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17630" PostTypeId="2" ParentId="17625" CreationDate="2019-12-23T17:39:23.600" Score="2" Body="&lt;p&gt;Most elementary school teachers in the USA get their teaching certification by majoring in &quot;elementary education&quot; as an undergraduate. Curriculum for such programs typically includes a sequence of courses that focus on both the content of mathematics taught in elementary school and how to guide children in their learning. I think this guidance is what you refer to as &quot;math talks.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps you would find some answers by reading about what is called &quot;Cognitively Guided Instruction.&quot; For an open-source textbook that might be helpful, look here&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://open.umn.edu/opentextbooks/textbooks/mathematics-for-elementary-teachers&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://open.umn.edu/opentextbooks/textbooks/mathematics-for-elementary-teachers&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2019-12-23T17:39:23.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17631" PostTypeId="1" CreationDate="2019-12-23T22:55:45.327" Score="9" ViewCount="292" Body="&lt;p&gt;A problem I often encounter while introducing students to equations is that of changing the conceptual image of the equation symbol &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; from &quot;results to&quot; to &quot;is equal to&quot;. To be more precise:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;In the expression &lt;span class=&quot;math-container&quot;&gt;$2+3=5$&lt;/span&gt;, equality is often conceptualized by students as a predicate that denotes that the quantity on the LHS results to the quantity on the RHS. So, under this view, the reflexive property of equality as an equivalence relation is ignored.&lt;/li&gt;&#xA;&lt;li&gt;In the expression &lt;span class=&quot;math-container&quot;&gt;$2x+4=5-3$&lt;/span&gt;, equality is/should be concpetualized as &quot;the LHS is equal to the LHS and vice versa&quot; - more or less in a sense similar to that of &quot;balancing&quot; on the two sides of a libra.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The first notion of equality is often and strongly interferring with the second one, something which could be realized e.g. due to students having dfficulties in recognizing &lt;span class=&quot;math-container&quot;&gt;$x=3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3=x$&lt;/span&gt; as equaly valid mathematical expressions of the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do realize that this problem comes alongside with the introduction of mathematic notation that is structurally different from the way natural language is structured. Namely, a typical sentece has the form:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\text{Subject - Verb - Object}$$&lt;/span&gt;&#xA;where the object usually is the receiver of the subject's action(s) or their result(s). So, &lt;span class=&quot;math-container&quot;&gt;$x=3$&lt;/span&gt; is naturally translated to &quot;(The subject) &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; equals (the object) &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;&quot;. However, it would seem odd to translate &lt;span class=&quot;math-container&quot;&gt;$3=x$&lt;/span&gt; to &quot;(The subject) &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; equals (the object) &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, do you have any suggestions/activities that would help students nurture themselves to equality as an equivalence relation against maintaining the false image of equality as a &quot;results&quot; process?&lt;/p&gt;&#xA;" OwnerUserId="12829" LastEditorUserId="12829" LastEditDate="2019-12-28T13:55:46.473" LastActivityDate="2019-12-28T13:55:46.473" Title="Equality as &quot;makes&quot; vs equality as &quot;equals&quot;" Tags="|secondary-education|primary-education|equal-sign|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="17632" PostTypeId="1" AcceptedAnswerId="17633" CreationDate="2019-12-24T00:00:02.860" Score="15" ViewCount="3481" Body="&lt;p&gt;I'm halfway through my first year of teaching AP Calculus to high school seniors.  It's been going generally well, but I'm feeling like I really could have done better getting them into the Chain Rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I started with it the same basic way that I did with the Product and Quotient Rules -- showing that the rule worked for elementary polynomials and could save us some calculation time.  But, in retrospect, the Chain Rule is such a fundamental part of much of the rest of differentiation that I feel like there could have been more that would help them understand how it works and how the concept ties together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is the u-substitution notion a good idea?  Our class is generally much more comfortable with the &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; notation, and as a result I stayed away from the &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$&lt;/span&gt; format.  Instead, I did a lot of hand-waving around the &quot;inside function&quot; and the &quot;outside function&quot; that hasn't taken hold in all my students as well as I might have hoped.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any suggestions about what works in your calculus classrooms?&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2019-12-24T00:05:24.503" LastActivityDate="2020-02-16T04:11:42.287" Title="How should I introduce the Chain Rule" Tags="|secondary-education|calculus|curriculum|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17633" PostTypeId="2" ParentId="17632" CreationDate="2019-12-24T00:41:27.187" Score="30" Body="&lt;p&gt;I just start with constant rates of change, where it's pretty blazingly obvious that the chain rule works. E.g., Jane hikes 3 kilometers in an hour, and hiking burns 70 calories&#xA;per kilometer. At what rate does she burn calories?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Our class is generally much more comfortable with the &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; notation, and as a result I stayed away from the &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$&lt;/span&gt; format.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is your opportunity to help them overcome that irrational prejudice by showing them an application where the Leibniz notation is clearly superior. It's not as though learning the Leibniz notation is optional for, e.g., engineering majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also like to do the example of &lt;span class=&quot;math-container&quot;&gt;$x=A\cos bt$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; both have units of meters, &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is in seconds, and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; has units of inverse seconds. I explicitly act this out with a heavy object and elicit the interpretations of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. Then I take the derivative and intentionaly omit the factor of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; coming from the &quot;derivative of the inside stuff.&quot; I then point out that the result is obviously wrong, both because it has the wrong units and because it doesn't depend on the frequency, which it should.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-12-24T01:05:03.023" LastActivityDate="2019-12-24T01:05:03.023" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17634" PostTypeId="2" ParentId="17632" CreationDate="2019-12-24T10:09:38.043" Score="11" Body="&lt;p&gt;What is difficult about the chain rule is the function concept, more specifically the composition of functions. Notation that hides or leaves implicit the composition of functions causes a great deal of confusion for students. However, the fundamental issues are not the notation used (all choices are messy to some extent), but what the use of the notation leaves implicit or to be inferred, and the extent to which these expectations are unrealizable by beginning students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing potentially confusing (and I think not just for students) about the Leibniz notation is that in &lt;span class=&quot;math-container&quot;&gt;$\tfrac{dy}{dx} = \tfrac{dy}{du}\tfrac{du}{dx}$&lt;/span&gt; it is not clear that &lt;span class=&quot;math-container&quot;&gt;$\tfrac{dy}{du}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tfrac{du}{dx}$&lt;/span&gt; are both viewed as functions of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and that, in the case of &lt;span class=&quot;math-container&quot;&gt;$\tfrac{dy}{du}$&lt;/span&gt;, this moreover means that this notation really indicates the composition &lt;span class=&quot;math-container&quot;&gt;$\tfrac{dy}{du}(u(x))$&lt;/span&gt;. That is, the Leibniz notation, at least as commonly used, &lt;em&gt;hides&lt;/em&gt; the composition of functions. The notation &lt;span class=&quot;math-container&quot;&gt;$\tfrac{dy}{du}$&lt;/span&gt; appears to indicate a function of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;, and it is implicit from context that this function of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; is viewed as a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Too much is left implicit, to be inferred from context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One part of a solution is to make explicit all functional compositions. With the Leibniz notation this can become quite messy, particularly if higher derivatives are involved. For example, &lt;span class=&quot;math-container&quot;&gt;$\tfrac{d}{dx}(y \circ u) = (\tfrac{dy}{du}\circ u )\tfrac{du}{dx}$&lt;/span&gt; indicates more clearly the functional compositions that occur, although it still does not indicate the dependence on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Explicitly adding this notation becomes somewhat ugly -  &lt;span class=&quot;math-container&quot;&gt;$\tfrac{d}{dx}(y \circ u)(x) = (\tfrac{dy}{du}\circ u )(x)\tfrac{du}{dx}(x)$&lt;/span&gt; - but perhaps this is nonetheless preferable to begin. (Once students understand what they are doing, the explicit indication of the functional compositions becomes a bother, and it becomes clarifying to omit it, but at first I think the situation is reversed.) (I am not saying I &lt;em&gt;like&lt;/em&gt; any of these notations - on the contrary I generally avoid the Leibniz notation - Also, a cleaner, functorial notation would be something like &lt;span class=&quot;math-container&quot;&gt;$D(u^{\ast}y) = u^{\ast}(Dy)D(u)$&lt;/span&gt;, where the pullback is defined by &lt;span class=&quot;math-container&quot;&gt;$u^{\ast}y = y \circ u$&lt;/span&gt;, but such a presentation of the chain rule as a cocyle identity is simply not viable for most students as usually educated.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could write alternatively &lt;span class=&quot;math-container&quot;&gt;$(y \circ u)^{\prime}(x) = (y^{\prime}\circ u)(x)u^{\prime}(x)$&lt;/span&gt;, and this is in many senses easier to read. What can be confusing for the student is that operationally the prime requires taking the derivative with respect to different variables (&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in one instance, &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; in the other). Formally this is &lt;em&gt;not&lt;/em&gt; a problem as variable names are really just placeholders that indicate the sequencing of compositions (the derivative is the derivative, whatever one chooses to call the argument), but it can be the essence of the difficulties that students have. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, it is also one thing that can be problematic with the Leibniz notation - the Leibniz notation attaches too much significance to variable &lt;em&gt;names&lt;/em&gt;. The derivative of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; is not the derivative with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, it is the derivative of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; with respect to its argument, whatever name one gives to that argument. Fixation on variable names and their magical qualities is a quite natural, one might say primitive, human tendency, but it is also part of what needs to be overcome to understand properly the chain rule. Precisely one of the confusing aspects of &lt;span class=&quot;math-container&quot;&gt;$\tfrac{dy}{dx} = \tfrac{dy}{du}\tfrac{du}{dx}$&lt;/span&gt; is that, since its right-hand side must be a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for the equality to have sense, the expression &lt;span class=&quot;math-container&quot;&gt;$\tfrac{dy}{du}$&lt;/span&gt;, which the notation apparently indicates is a function of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;, has to be considered as a function of &lt;span class=&quot;math-container&quot;&gt;$u(x)$&lt;/span&gt;, that is with &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; viewed as a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and this aspect is hidden notationally, so has to be inferred. For those with experience, the notational higiene compensates for leaving something implicit, but for students it can be a source of serious confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the best tack is to make all of this as explicit as possible (obviously, in a language more accessible to students than that which I am using here), in particular indicating clearly what the difficulties are, where they occur, what is left implicit and what is not, whatever notation one choose to use. Operational rules of thumb that refer to &lt;em&gt;inside&lt;/em&gt; and &lt;em&gt;outside&lt;/em&gt; function will not help if they are not accompanied by precise explanation that makes clear what they intend to summarize and elide, although of course they can help when students are first sufficiently prepared to properly interpret them (however, in my experience this sort of informal summary works only with the most engaged students).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Diagrams can help. I am not sure how to make decent diagrams in mathjax, so I won't try here, but what I have in mind is a directed graph with three vertices and three arrows. The vertices represent the domains/codomains and the arrows represent the functions. The diagram can be labeled with the variable and function names. What it helps makes clear is that &lt;span class=&quot;math-container&quot;&gt;$y^{\prime}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; have the same domain (it is the codomain of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;, while &lt;span class=&quot;math-container&quot;&gt;$(y \circ u)^{\prime}$&lt;/span&gt;, must have the same domain as &lt;span class=&quot;math-container&quot;&gt;$y \circ u$&lt;/span&gt;. Accompanying computations by such diagrams, and repeating this a fair number of times can help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A fundamental example, useful for other reasons, that should be clarifying in the context of the chain rule, is to take the derivative of the sine function viewed as a function of degrees. A student who can do this correctly, and write correctly to what it corresponds in whatever abstract functional notation (Leibniz or otherwise) has understood the chain rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, a reflection. Many of the difficulties students have in calculus reduce to a failure to understand the abstract function concept. This concept is difficult and it is quite modern (In some sense it postdates calculus by one or two centuries). Its difficulty becomes apparent in any context requiring change of variables (chain rule, change of domain in integrals). Much of the problem is that it is often treated as something simple, requiring little explanation, or given explanation that is not precise. Better to treat the difficult topics directly and plainly than to look for ways to avoid them. &lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2019-12-25T11:13:18.740" LastActivityDate="2019-12-25T11:13:18.740" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="17635" PostTypeId="2" ParentId="17631" CreationDate="2019-12-24T15:48:19.597" Score="1" Body="&lt;p&gt;Grammatically, I suspect that part of the problem is that the verbs &quot;to make&quot; and &quot;to equal&quot; are transitive verbs (they take a direct object).  Thus there is real distinction in the grammatical structure between the left-hand side and the right-hand side of an equation when read aloud.  To see the effect of this, consider what happens when you elide the direct object (the right-hand side of an equation):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Two plus three makes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Makes what?  What does two plus three make?  Where is the end of the sentence?  If you replace &quot;makes&quot; with &quot;equals&quot;, you have the same problem&amp;mdash;the sentence requires a direct object in order make sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence I would suggest using a entirely different grammatical structure.For example, use &quot;is&quot; as a synonym for &quot;equals&quot;.  Note that the following is (in fact) a complete sentence:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Two plus three is.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here, &quot;two plus three&quot; is an object which exists, on its own.  The sentence makes sense without a direct object&amp;mdash;indeed, the verb &quot;to be&quot; does not take a direct object at all.  The sentence simply declares that this thing, two plus three, exists.  The notation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 2+3 = 5 $$&lt;/span&gt;&#xA;may be expanded to &quot;two plus three &lt;strong&gt;is&lt;/strong&gt; five&quot;, which provides additional information about the nature of two plus three.  Grammatically, the word &quot;five&quot; here is a predicate noun, which means that it falls into a somewhat distinct grammatical category from a direct object (the predicate noun is not being acted upon, but rather completes the verb phrase beginning with &quot;is&quot;).  You can add verbosity for clarity:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Two plus three is exactly the same as five.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2019-12-24T19:41:40.757" LastActivityDate="2019-12-24T19:41:40.757" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17636" PostTypeId="1" CreationDate="2019-12-24T15:55:06.833" Score="1" ViewCount="124" Body="&lt;p&gt;I would like to know how to teach geometric patterns in secondary school. I want to elaborate worksheets, which could include different kinds of strategies related to this topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there resources or references about it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you help me, please?&lt;/p&gt;&#xA;" OwnerUserId="1789" LastEditorUserId="238" LastEditDate="2019-12-24T17:56:34.017" LastActivityDate="2019-12-24T22:46:26.830" Title="How to teach geometric patterns?" Tags="|secondary-education|geometry|" AnswerCount="1" CommentCount="3" ClosedDate="2020-01-09T21:00:56.537" ContentLicense="CC BY-SA 4.0" />
  <row Id="17637" PostTypeId="2" ParentId="17636" CreationDate="2019-12-24T22:46:26.830" Score="2" Body="&lt;p&gt;As a generic rule for teaching patterns - geometric and/or algebraic (for instance, when introducing sequences/functions) - you could follow the following steps:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Find a problem that is or can be related with/to the pattern you want to discuss.&lt;/li&gt;&#xA;&lt;li&gt;Give the students the problem and discuss about the ground term of the pattern.&lt;/li&gt;&#xA;&lt;li&gt;Ask them to find some of the first terms of the pattern (e.g. the first four).&lt;/li&gt;&#xA;&lt;li&gt;Then, dependeing on the specific pattern, you can ask for several properties of the pattern for some larger numbers, so as to make them seek for a wider normality/rule and not stick to conducting simple conjectures/ad hoc manipulations.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Let me provide you with a small example, so as to make myself clear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our problem is to find how many pieces we would need to cover our towns central square (which happens to be square), with &lt;a href=&quot;https://www.google.com/imgres?imgurl=https%3A%2F%2Fwww.featurepics.com%2FStockImage%2F20081019%2Fpaving-stone-wave-pattern-stock-picture-937149.jpg&amp;amp;imgrefurl=https%3A%2F%2Fwww.featurepics.com%2Fonline%2FPaving-Stone-Wave-Pattern-937149.aspx&amp;amp;docid=qKg9V3uNgHJvAM&amp;amp;tbnid=wtl4-A79K9-eAM%3A&amp;amp;vet=12ahUKEwi1l52Prc_mAhUusaQKHcPHAc04ZBAzKAQwBHoECAEQBg..i&amp;amp;w=1300&amp;amp;h=930&amp;amp;itg=1&amp;amp;client=firefox-b-d&amp;amp;bih=1054&amp;amp;biw=1736&amp;amp;q=patter%20road%20covering&amp;amp;ved=2ahUKEwi1l52Prc_mAhUusaQKHcPHAc04ZBAzKAQwBHoECAEQBg&amp;amp;iact=mrc&amp;amp;uact=8&quot; rel=&quot;nofollow noreferrer&quot;&gt;this pattern&lt;/a&gt;. You could also ask about how many of the final pieces will be whole and how many will be half - depends on the dimensions of the square you have provided.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the way to find these results, you can gradually build up a deeper understanding of the patter that occurs - which, in this example is a pretty simple one - by using questions as the ones presented above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S.: If this was not what you asked for, please provide more context/info in your question! :)&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2019-12-24T22:46:26.830" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17638" PostTypeId="2" ParentId="17632" CreationDate="2019-12-24T23:10:17.807" Score="10" Body="&lt;p&gt;The intuition here is basically that of Ben Crowell's answer, and that kind of intuitive explanation might be worth going through first. What I want to show is the kind of activity you can explore with students to investigate how it works in a &quot;less than completely obvious&quot; situation, when at least one of the rates of change itself keeps changing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to approach differentiation is in terms of &quot;sensitivity&quot; - the derivative &lt;span class=&quot;math-container&quot;&gt;$f'(x_0)$&lt;/span&gt; measures the sensitivity of the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; to small changes in in its input, about &lt;span class=&quot;math-container&quot;&gt;$x = x_0$&lt;/span&gt;. In particular, &lt;span class=&quot;math-container&quot;&gt;$f(x_0 + \Delta x) \approx f(x_0) + f'(x_0) \Delta x $&lt;/span&gt;. In addition to thinking about this graphically, one can investigate numerically with a suitable function (depending on the students' level of prior knowledge, perhaps one they already know the result of differentiating even if they can't prove it from first principles) e.g. the difference between &lt;span class=&quot;math-container&quot;&gt;$f(4)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(4.001)$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chain rule is just about extending this idea to the sensitivity of &lt;em&gt;composite functions&lt;/em&gt;, i.e. how sensitive is &lt;span class=&quot;math-container&quot;&gt;$f(g(x))$&lt;/span&gt; to changes in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;? This is clearly going to depend on how sensitive &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; is to a small change in its input, but then also on how sensitive &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is to a change in its input... moreover the change in the input to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not just &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$\Delta u = \Delta g(x) \approx g'(x) \Delta x$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$u = g(x)$&lt;/span&gt; is the input to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. So overall we reach &lt;span class=&quot;math-container&quot;&gt;$$fg(x_0 + \Delta x) = f(u_0 + \Delta u) \approx f(u_0) + f'(u_0) \Delta u \approx fg(x_0) + f'(g(x_0)) g'(x) \Delta x$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again this can be investigated numerically by students given an appropriate pair of functions and a set of values to play with (I have seen this work quite well by getting all students to use the same functions but &quot;sharing out&quot; which values to input), for example with &lt;span class=&quot;math-container&quot;&gt;$f(u) = u^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x) = 3x + 1$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$fg(x) = (3x + 1)^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f'(x) = 3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g'(u) = 2u$&lt;/span&gt;. A student might work with &lt;span class=&quot;math-container&quot;&gt;$x_0 = 5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Delta x = 0.001$&lt;/span&gt;; they tabulate that &lt;span class=&quot;math-container&quot;&gt;$u_0 = g(x_0) = 16$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$u_0 + \Delta u = g(x_0 + \Delta x) = 16.003$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$\Delta u$&lt;/span&gt; = 0.003; this can be seen to match &lt;span class=&quot;math-container&quot;&gt;$g'(x_0) = 3$&lt;/span&gt; multiplied by &lt;span class=&quot;math-container&quot;&gt;$\Delta x = 0.001$&lt;/span&gt; (for more complicated functions this would only be an approximation, of course). In further columns of the table, the student might tabulate &lt;span class=&quot;math-container&quot;&gt;$f(u_0) = fg(x_0) = 256$&lt;/span&gt;; &lt;span class=&quot;math-container&quot;&gt;$f(u_0 + \Delta u) = fg(x_0 + \Delta x) = 256.096009$&lt;/span&gt;; &lt;span class=&quot;math-container&quot;&gt;$\Delta f(u) = 0.096009$&lt;/span&gt;; &lt;span class=&quot;math-container&quot;&gt;$f'(u_0) = 2 u_0 = 32$&lt;/span&gt; and finally &lt;span class=&quot;math-container&quot;&gt;$f'(u_0) \Delta u = 32 \times 0.003 = 0.096$&lt;/span&gt; which is reassuringly close to the value obtained for &lt;span class=&quot;math-container&quot;&gt;$\Delta f(u)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5640" LastActivityDate="2019-12-24T23:10:17.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17639" PostTypeId="1" CreationDate="2019-12-25T20:11:47.680" Score="5" ViewCount="283" Body="&lt;p&gt;I write content and conduct workshops for an education firm and also in schools where I try to make them realise how beautifully mathematics and rational thinking complement each other (on elementary level at least). For instance I team them up and design setups where they solve problems like &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How many pluses we should put between the digits of &lt;span class=&quot;math-container&quot;&gt;$987,654,321$&lt;/span&gt; to get a total of &lt;span class=&quot;math-container&quot;&gt;$99$&lt;/span&gt; and where?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;OR&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How to place numbers from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; such along sides of a triangle such that sum of numbers along each side of triangle is &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Students enjoy these problems and teachers appreciate this idea. But I cannot convince parents about how solving such problems can be useful for students in their academics because at the end all that matters for them is their child perform good in their academics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is how can I convince parents who do not know much mathematics and logic but are conscious about their child's academic that such mathematical puzzles will enhance academic performance of their children and is not a waste of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope I made myself clear. Also, I do not know suitable tag for this question so if you know some please add them to question.&lt;/p&gt;&#xA;" OwnerUserId="7654" LastEditorUserId="13100" LastEditDate="2019-12-30T13:17:37.687" LastActivityDate="2019-12-30T13:17:37.687" Title="How to convince parents that Mathematical puzzles/games help students in their academics too" Tags="|mathematical-pedagogy|games|math-puzzle|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17640" PostTypeId="2" ParentId="17625" CreationDate="2019-12-25T22:06:04.063" Score="3" Body="&lt;p&gt;As you realize, this&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In my brief experience, I’m finding that my written diagrams or&#xA;  notations are not ideal. Admittedly, I will draw operators, numerals,&#xA;  and arrows on-the-fly, and they don’t always clearly communicate the&#xA;  process, especially when looking at them a few days later.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is part of your problem. I suspect you may have difficulty reconstructing your own trains of thought (let alone those of the kids) when recorded this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Writing down one's thinking process is very difficult. There is no standard model. I suggest words in short sentences, with lots of &quot;so I know that&quot; and &quot;because&quot; and &quot;I remember&quot;. Keep the narrative glue. Keep symbols to the minimum necessary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Focus the need for explanation to the hard parts of the current discussion. Don't make kids &quot;justify their thinking&quot; at every step. If you're teaching multidigit multiplication you can just assume that the kids know the multiplication table and  can do the necessary addition routinely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Welcome different ways to solve problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only tangentially related, but a site you might enjoy:&#xA;&lt;a href=&quot;https://www.explodingdots.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.explodingdots.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4984" LastActivityDate="2019-12-25T22:06:04.063" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17641" PostTypeId="1" CreationDate="2019-12-26T03:32:09.307" Score="15" ViewCount="2088" Body="&lt;p&gt;The title might be a bit not specific, so let me give an example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In China, Japan, Korea, etc, there is a type of problem about chickens (or crane, or anything with two legs) and rabbits (4 legs) in the same cage:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Some chickens and rabbits are in the same cage. There are a total of &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; heads and a total of &lt;span class=&quot;math-container&quot;&gt;$28$&lt;/span&gt; legs. How many chickens are there and how many rabbits?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the same as solving the system&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{cases}&#xA;x+y=10, \\ 2x+4y=28.&#xA;\end{cases}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, we can teach this in the following two ways:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Teach how to solve a system of linear equations first, and write down the equations for this problem. Solve the equation.&lt;/li&gt;&#xA;&lt;li&gt;Spend some time with trial and error, before teaching the equations.&lt;/li&gt;&#xA;&lt;li&gt;Teach a method with does NOT involve any algebra. Example: assume all of them are chickens, then we get &lt;span class=&quot;math-container&quot;&gt;$28-2\times 10=8$&lt;/span&gt; extra legs, which means there are &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; rabbits.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Method 1 is the most efficient way, in the sense that it doesn't waste any time, and gives students the maximum amount of content. However, some argue that 2 or 3 is better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Those who support 2 argue that it helps students understand &lt;em&gt;why&lt;/em&gt; the equations should work and that it could help develop the idea of deduction (Modus ponens, Gen). However, those reasons are farfetched. It is hard to come up with the idea of equations after doing trial and error for days. Mathematicians would also agree that this problem is about calculations rather than deduction and proofs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reasons to support 3 is more simple - it would be nice and &lt;em&gt;easy&lt;/em&gt; to avoid teaching new concepts like equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to the title: should we teach this problem in fast ways (1 and 3) or the slow way (2)?&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="8571" LastEditDate="2019-12-27T20:13:12.293" LastActivityDate="2019-12-28T14:28:02.573" Title="Should we teach simple content quickly or slowly?" Tags="|secondary-education|" AnswerCount="7" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17642" PostTypeId="2" ParentId="17641" CreationDate="2019-12-26T08:59:37.600" Score="6" Body="&lt;p&gt;Well, for me, it depends on what you are aiming to give to your students with this problem. That is, are you using this as an introduction to linear systems? Then, for sure go for a more in depth treatment that lets the students see the necessity of introducing linear systems - however, this problem may be just a part of this introduction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If, on the other hand, you aim just to present this problem to your class, then you may want to let your students find their own way of solving the problem. In general, problem solving aims to allow students explore their own capabilities in finding ways to solve problems, so, from my perspective, there's no reason to restrict them to just one way - always depending on their age and previous knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summing up, it's a matter of what your goals are as a mathematics instructor alognside with the level of the students you have in your class.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2019-12-26T08:59:37.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17643" PostTypeId="2" ParentId="17641" CreationDate="2019-12-26T09:10:40.180" Score="12" Body="&lt;p&gt;(In this answer I'm taking 2 to mean that the students spend some time on trial and error, trying to work out how to solve the equations before they're told the recipe.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think for many secondary school teachers there isn't enough time to choose 2. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I would never choose 3. It teaches them a trick to solve a single question, but doesn't develop their mathematical ability. Some similar style might be applicable when preparing students for maths competitions. Or letting the students find that for themselves would help them develop their problem solving skills. But telling them outright would, I believe, never be a good choice for the majority.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 is probably the standard choice for most schools. It has the advantage of working relatively well, and just about fitting into the time available. It's also what the majority of students are likely to want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But by preference I try to use 2. I believe it helps students to understand why the methods work, and that makes it easier to remember what to do, and potentially to adapt the method yourself to new contexts, rather than being reliant on the algorithm taught. Also, I believe that it would give students a better understanding of what maths is. Most leave school thinking that maths in about running a fixed algorithm to get a number. But that's not really maths, it's being a computer. The experimenting yourself and trying to find a solution is more what maths should be about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I said, most school teachers don't have the time to make this decision, or at least don't think they do (some people have suggested that actually building the students' understanding buys back the time later). But this is a real tension in designing a university level curriculum, where there is more freedom to choose how much content gets covered. Personally I believe very much in keeping the content low enough that there is time to develop the students' understanding and problem-solving skills. There are many people who advocate including lots of content. I think in many cases these people don't realise how little of what gets 'taught' actually makes it into the students' heads.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2019-12-26T09:10:40.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17644" PostTypeId="2" ParentId="17639" CreationDate="2019-12-26T09:42:42.183" Score="2" Body="&lt;p&gt;Well, at first, it is very useful for a parent to be &lt;em&gt;transparent&lt;/em&gt; as far as the goals of your teaching methods are concerned. You should not just teach something because it once has worked for someone - whether that someone is you or some of your students etc. Deciding to introduce an activity in a class is a result of some considerations made about this &lt;em&gt;specific&lt;/em&gt; class and, as a result, you need to make that clear to the parents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apart from that, using personal examples may also help them see the problem from your viewpoint. But, do not be too specific. If all your examples are of the form &quot;I remember that this has/had helped me there&quot; then you seem like you draw all you techniques from your personal experience, which is not always considered to be a good practice. Instead, try to find examples beyond your current ones and integrate them to your teaching routine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, being confident and sure about your methods, as well as well informed about them and aware of the reasons you use them is the best way to convince a parent about your suitability as a mathematics instuctor. It is not the theoretical background you have, but the sense of confidence you create to them.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2019-12-26T09:42:42.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17645" PostTypeId="2" ParentId="17639" CreationDate="2019-12-26T13:54:30.407" Score="2" Body="&lt;p&gt;A spoon full of sugar makes the medicine go down.  Children (and adults!) are not silicon robotic philosophical drill machines.  We are organic, social animals.  Engaging our sense of play helps us engage with topics.  (I speculate that there is something instinctive related to our hunter gatherer origins and our social collaboration/mating behavior that makes us this way.  We also have emotions...and stuff.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like the Boris answer with the comment of looking to more than just yourself as an example (although it is still one to start with).  You could also add the example of Richard Feynman, looking with huge fondness at his high school algebra team, well after being a Nobel Prize winner.  And also having a love of riddles and safe cracking and Mayan hieroglyphs and the like.  It's still an appeal to authority, but at least a second one, and one that seems to click with many, many people.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=P1ww1IXRfTA&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=P1ww1IXRfTA&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also seems like a pattern that some of the creative mathematicians and physicists and the like seem to have been interested in games and puzzles and contests.  Look at John Nash inventing Hex.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hex_(board_game)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Hex_(board_game)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13241" LastEditorUserId="13241" LastEditDate="2019-12-26T14:16:17.170" LastActivityDate="2019-12-26T14:16:17.170" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17646" PostTypeId="2" ParentId="17641" CreationDate="2019-12-26T14:30:00.457" Score="1" Body="&lt;p&gt;Solving these sorts of problems is extremely common in algebra, of which students get two years--three if you count pre-algebra, and then in later math classes and in many science and engineering classes (particularly chemical stoichiometry).  Given this commonness, I absolutely think students should experience this sort of problem in MANY different ways.  I.e. absolutely take the time (there really is a slew of time versus content in &quot;algebra one&quot;).  Seeing a problem like this in many different ways is helpful just as it is good to picture a function in many ways...it helps us come to terms with it.&lt;/p&gt;&#xA;" OwnerUserId="13241" LastActivityDate="2019-12-26T14:30:00.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17647" PostTypeId="2" ParentId="17641" CreationDate="2019-12-26T17:37:48.943" Score="10" Body="&lt;p&gt;If the goal of this problem is to introduce students to systems of equations, I would avoid introducing notation and computation without first building up to it.   One of the biggest problems that I have with my (first year college) students is that they are very good at symbol manipulation, but they have no idea what the symbols mean or why they are manipulating them.  Students need a much stronger basic understanding of why they are doing the things that they are doing.   That being said, I don't think that the three alternatives given in the question are the only possible alternatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that the main idea is to somehow capture the relationship between rabbits, chickens, heads, and feet.  As such, a good way to start is with the inverse problem:  I tell you how many rabbits and chickens I have, and you tell me how many feet and heads there are.  With small numbers, this isn't too difficult (you can count on your fingers and toes!), but if there are 103 chickens and 83 rabbits, it is a bit harder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea is to get students to the point of understanding that there is a relationship between these variables:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\text{heads} &amp;amp;= \text{rabbits} + \text{chickens} \\&#xA;\text{feet} &amp;amp;= (4\times\text{rabbits}) + (2\times\text{chickens}). \end{align}&lt;/span&gt;&#xA;Note that there is also a secondary objective here:  get students to grok the idea of a function of two variables (of course, to a midddle- or high-school audience, we might not use the notation and language of functions, but the &lt;em&gt;idea&lt;/em&gt; is still there).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before actually asking them to solve the problem, I would introduce other similar kinds of setups (A theater sells tickets on the ground floor for \$80 and balcony tickets for \$55 dollars; how much money do they make if they sell &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; ground floor and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; balcony tickets?  Hot dogs are sold in packages of 12, while hot dog buns are sold in packages of 8; if you buy 4 packages of hot dogs and 6 packages of buns, how many of each (dogs and buns) do you have?  Follow-up question:  how many hot dogs can you serve (you need a bun for each hot dog)?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once they are comfortable answering these kinds of questions (which, in an early secondary setting, should take one or two 50 minute lessons, with some homework for extra practice), we can start talking about solving systems.  I would let the students struggle to figure out how to solve the problems on their own before introducing too much notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, after some struggle, we can abstract away the references to rabbits and hot dog buns, and introduce notation.  For example, in the original problem, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp;\text{heads} = \text{rabbits} + \text{chickens}&#xA;&amp;amp;\implies H &amp;amp;= R + C \\&#xA;&amp;amp;\text{feet} = (4\times\text{rabbits}) + (2\times\text{chickens})&#xA;&amp;amp;\implies F &amp;amp;= 4R + 2C \\&#xA;&amp;amp;\text{there are 10 heads} &amp;amp;\implies H &amp;amp;= 10 \\&#xA;&amp;amp;\text{there are 20 feet} &amp;amp;\implies F &amp;amp;= 28.&#xA;\end{align}&lt;/span&gt;&#xA;If the students have determined general methods for solving these problems, we can help them to see how the notation enables this and makes it faster.  If they haven't, then this is a good time to introduce general methods.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2019-12-26T17:37:48.943" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17648" PostTypeId="2" ParentId="17617" CreationDate="2019-12-26T23:39:37.727" Score="3" Body="&lt;p&gt;Here in the US, the &lt;em&gt;words&lt;/em&gt; began to be commonly taught in the early grades during the &quot;New Math&quot; era. I remember in first and second grade (ca. 1972) being taught this distinction and thinking that it was dumb. In this context, kids were taught that &quot;cardinal&quot; numbers were for counting how many things you had, while &quot;ordinal&quot; numbers were numbers like &quot;first,&quot; &quot;second,&quot; etc., which told you an item's position in a list. This may have mostly died out along with other goofy and ill-considered features of New Math, but if you google, you can still find materials online such as worksheets where kids are taught the two terms and asked to match up 3 with &quot;third,&quot; etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of course if some piece of silliness like this makes it into the lower-grade curriculum, it becomes something that has to be taught to preservice elemetary teachers as part of their education coursework. They don't have any idea of the context where this distinction would actually be useful, with infinite sets and so on. They just parrot it like all the other crap they are forced to parrot in the classes taught by a university's education department. So if your question is when this distinction is learned and when it's needed, one answer is that it may be learned when people are studying to get a teaching credential, and needed when they need to take a multiple-choice test that proves they know enough math to be elementary school teachers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There may be a more general phenomenon here, which is the perceived need to make isomorphisms more explicit, even when nothing is gained thereby. For example, Euclid didn't need separate concepts of angle and measure-of-an-angle, but it seems to be standard these days in high school geometry books to treat these as separate concepts, and to introduce a postulate saying that there is a  mapping from angles to real numbers, which has certain properties.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2019-12-27T01:01:38.063" LastActivityDate="2019-12-27T01:01:38.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17649" PostTypeId="1" CreationDate="2019-12-27T00:26:00.597" Score="26" ViewCount="2174" Body="&lt;p&gt;What research has been done on how much and in what circumstances various non-lecture types of teching are effective with regards to student knowledge and performance? Meta/review studies preferred over single case studies, of course. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am thinking of methods like learning stations, flipped classroom, discovery learning, project work, group learning, even the more common seminars and lab work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When and in what doses are these methods provide higher results than traditional lectures and exercise (problem solving) sessions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that nearly any method has it's place for some topics and to some amount. But is there any single method that can achieve same results when replacing most of the lessons in a course? I am thinking in parallels with biology - if there are 30 lessons in the course, a single field trip will be more useful than 2 more lessons. But replace all of the lessons with field work and you'll have spent a lot of time and leave with many gaps in knowledge. It seems that the concept of diminishing returns is somehow at play here, isn't it?&lt;/p&gt;&#xA;" OwnerUserId="766" LastActivityDate="2020-01-01T16:16:20.873" Title="What is the quantitative data on effectiveness of &quot;modern&quot; teaching methods?" Tags="|reference-request|general-pedagogy|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17650" PostTypeId="2" ParentId="17649" CreationDate="2019-12-27T01:16:54.257" Score="13" Body="&lt;p&gt;Re active learning, the best meta-analysis of the evidence that I know of is Freeman et al., &quot;Active learning increases student performance in science, engineering, and mathematics,&quot; &lt;a href=&quot;http://www.pnas.org/content/early/2014/05/08/1319030111&quot; rel=&quot;noreferrer&quot;&gt;http://www.pnas.org/content/early/2014/05/08/1319030111&lt;/a&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Active learning is not the opposite of fully guided instruction. The successful active learning techniques for which I've seen good empirical evidence are mostly done using fully guided instruction. Students have a textbook, and the textbook presents the principles, gives examples, and so on. Students just don't get the textbook spoon-fed to them again during class time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Active learning is not the same as &quot;edutainment&quot; practices such as the use of electronic clickers. Clickers are neither necessary nor sufficient for implementing active learning well.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2019-12-27T01:16:54.257" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17651" PostTypeId="2" ParentId="17649" CreationDate="2019-12-27T02:03:31.233" Score="8" Body="&lt;p&gt;A supplement to @Ben's &lt;em&gt;PNAS&lt;/em&gt; article citation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hodges, Linda C. &quot;Contemporary Issues in Group Learning in Undergraduate Science Classrooms: A Perspective from Student Engagement.&quot; &lt;em&gt;CBE—Life Sciences Education&lt;/em&gt; &lt;strong&gt;17&lt;/strong&gt;, no. 2 (2018): es3.&#xA;  &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5998321/&quot; rel=&quot;noreferrer&quot;&gt;Journal link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The focus is on group work. At my college we have a study group&#xA;of faculty exploring issues arising with group work, for example,&#xA;how best to form groups. The Hodges paper has a section entitled&#xA;&quot;How does the way groups are formed affect student outcomes?&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Should students self-select into groups or should instructors form groups? How do group size and composition affect the functioning and learning in the group? Should group composition be permanent or changing throughout a term?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Another section asks, &quot;How do learning spaces impact group work?&quot;---a particular focus of our study group.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-12-27T02:03:31.233" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17652" PostTypeId="2" ParentId="17641" CreationDate="2019-12-27T09:42:15.577" Score="9" Body="&lt;p&gt;I largely agree with Xander, but wanted to add my own lesson/unit plan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First off, solving two equations in two unknowns is not a simple concept and is worth devoting time so that students &lt;em&gt;understand&lt;/em&gt; how it works and how to apply it in situations where they aren't just handed two equations to solve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yeah, start off with some inquiry.  Since my students were really well-behaved in the last unit and deserve a treat, I'll even give them some modeling clay and tiny pegs to play with.  So they'll be building people (figures with two pegs for legs) and horses (figures with four pegs per legs).  (I'm switching from chickens and rabbits to people and horses because I think they're easier to model with clay.) I'll divide the class into groups -- half of the groups are trying to find all the possible ways to create a scene with 10 heads and the other half are trying to find all the possible scenes with 28 legs.  Fortunately, the unit before solving systems of equations is graphing linear equations, so each team has to create a chart and graph expressing their findings.  When they're done, we superimpose the two graphs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9kVfQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9kVfQ.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This gives us the opportunity to reinforce the fundamental concept of analytic geometry (which fortunately is another standard we needed to cover this year): the solution set of an equations in two unknowns can be graphed and there is a correspondence between points on that graph in our domain (in this case, just non-negative integers) and individual solutions to the equation.  In this case, we get to add the new lesson that the intersection points of two different graphs correspond to solutions of the system of equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From here, we can leverage this problem throughout the rest of the unit to discuss strategies for solving systems of equations if we didn't have modeling clay or graph paper.  The substitution method can be introduced by noting that the equations of the two lines in slope-intercept form are &lt;span class=&quot;math-container&quot;&gt;$H=-P+10$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$H=-\frac 12P+28$&lt;/span&gt; and we can equate the two RHS and solve for &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; to find the common solution.  And, with any luck, one of the students came up with your method 3 in the initial lesson, which is noting that we can subtract &lt;span class=&quot;math-container&quot;&gt;$2H+2P=20$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$4H+2P=28$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$2H=8$&lt;/span&gt;, which is the elimination method when you express it in algebraic language.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Reasons to support 3 is more simple - it would be nice and easy to avoid teaching new concepts like equations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On the contrary, &quot;teaching new concepts like equations&quot; is our job.  If we don't train introductory algebra students to think about systems of equations because simple problems can be solved with arithmetic, then what will they do next year when the third variable is introduced? No, method 3 is attractive because it allows us to observe that our intuitive strategies can be expressed in the algebraic language in ways that will continue to work when our intuition fails us.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2019-12-27T21:35:38.180" LastActivityDate="2019-12-27T21:35:38.180" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17653" PostTypeId="2" ParentId="17641" CreationDate="2019-12-27T11:58:21.267" Score="4" Body="&lt;p&gt;As a non-mathematician and not a teacher (but parent of two children,) I'd like to suggest you avoid &quot;trial and error.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;German math classes have this thing they do.  Kids in second or third grade (about 7 or 8 years old) are given assignments that can easily be solved with a system of equations, but which the kids can only &quot;solve&quot; by guessing.  I saw this with my kids, and the children of friends.  I assume the idea is to make practice adding and subtracting less boring, but what it does is to cause the kids to hate math - and it generally drives the parents batty, too (average parents don't know algebra.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The task given to the child looks like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Bm31c.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Bm31c.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The number in each &quot;brick&quot; is the sum of the two bricks it sits on.  The children have to fill in the empty bricks to make all the sums fit.  The given numbers are not always in the corners - any three bricks can be used, but the three corner task is the worst.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The solution to the example looks like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/blSgI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/blSgI.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kids can only get that solution by repeatedly guessing or systematically trying numbers in the empty blocks until they get an answer that fits.  They get frustrated, and it gives them the idea that math is just randomly doing things until you get an answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can systematically solve it, but you need concepts that children at that age haven't seen yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you must use trial and error, do it yourself in front of the class as a demonstration.  Make a couple of attempts.  That shows how inefficient guessing is.  Then tell them that it's a bad idea to repeatedly guess at possible answers, and introduce the systematic way to solve whatever type of problem you are working on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explain how to get from the description of the task to a mathematical equation.  That is often the &quot;missing link.&quot;  When I was a kid, most kids hated the &quot;story problems&quot; because they couldn't see how to get from the description to the equations we had learned.  From my own children, I know that they are never explicitly taught how to break down a task into a solvable equation - you either manage it on your own, or you go through school failing on story problems and hating math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math shouldn't be that hard.  It's just applying rules and following them to their conclusion.  Poor explanations and reliance on trial and error make it harder than it has to be, and makes kids hate it.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;There is a general solution for the bricks assignment.  It isn't taught to the kids at any point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wrote it out and discussed it with my daughter's math teacher when it came up.  The teacher wasn't interested - the assignments came out of the book, and that was all the teacher was interested in.  Kiddos do what they're told, and get the correct answers after playing guessing games, or fail the assignment.  My daughter still hates math.&lt;/p&gt;&#xA;" OwnerUserId="4800" LastActivityDate="2019-12-27T11:58:21.267" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="17654" PostTypeId="2" ParentId="17649" CreationDate="2019-12-27T13:01:32.113" Score="8" Body="&lt;p&gt;The Education Endowment Foundation has a 'Teaching and Learning Toolkit' which aims to assess the effectiveness of various teaching methods or types of intervention, including collaborative learning, altering teaching for individual learning styles, and small group learning. It is focused on teaching 5–16 year olds, and not specific to mathematics education, but it is all evidence-based and (in my opinion) persuasively presented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is available here: &lt;a href=&quot;https://educationendowmentfoundation.org.uk/evidence-summaries/teaching-learning-toolkit&quot; rel=&quot;noreferrer&quot;&gt;https://educationendowmentfoundation.org.uk/evidence-summaries/teaching-learning-toolkit&lt;/a&gt; and the citation is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Higgins, S., Katsipataki, M., Kokotsaki, D., Coleman, R., Major, L.E.,&#xA;  &amp;amp; Coe, R. (2014). &lt;em&gt;The Sutton Trust-Education Endowment Foundation&#xA;  Teaching and Learning Toolkit&lt;/em&gt;. London: Education Endowment&#xA;  Foundation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1295" LastActivityDate="2019-12-27T13:01:32.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17655" PostTypeId="2" ParentId="17649" CreationDate="2019-12-27T16:44:41.240" Score="15" Body="&lt;p&gt;Consider a paper from this year: Setren, et. al., &lt;a href=&quot;https://linkprotect.cudasvc.com/url?a=http%3A%2F%2Fwww.edworkingpapers.com%2Fsites%2Fdefault%2Ffiles%2Fai19-113.pdf&amp;amp;c=E,1,YtCmzoS4f0kqCRc9majAvzs2fSgKxgL7As6tJzDesrhMn4-ILlfb9YQEVrybSZ2Z2nZ7524s7vkq4AntP_O5509BVRZjR36IP71pQ6IZZvCTqXuwe8zy9_S3&amp;amp;typo=1&quot; rel=&quot;noreferrer&quot;&gt;&quot;Effects of the Flipped Classroom: Evidence from a Randomized Trial&quot;&lt;/a&gt;, Annenberg Institute at Brown University (2019). In their introduction, the authors write:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Despite the proliferation of the flipped classroom, little&#xA;  well-identified evidence exists on its impact on student learning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In this study a large (N = 1328 students), randomized trial was conducted at the West Point military academy in the United States, in economics and math classes. Their findings were that there is a small short-term boost in the math classes, none in economics classes, and those effects became negligible by the end of the course. About half of the students in flipped classes did the advance preparation. Flipped classrooms also increased inequity, in the sense that high-performing, white, male students did better, while weaker and minority students had no improvement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The authors speculate that the short-term benefit in math classes may be due largely to the excitement level of the instructors trying a new method. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In comments, the question has been raised whether the model of using videos for flipped classrooms, as tested in the study, is a fair one or not. Indeed, this seems to be the standard and widespread practice. From the bibliography, Schaffhauser, Dian and Rhea Kelly, 2016, &lt;a href=&quot;https://campustechnology.com/articles/2016/10/12/55-percent-of-faculty-are-flipping-the-classroom.aspx&quot; rel=&quot;noreferrer&quot;&gt;“55 Percent of Faculty Are Flipping the Classroom”&lt;/a&gt;, Campus Technology:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fifty-five percent of the survey respondents said they are somewhere&#xA;  along the spectrum of flipping all or some of their courses, in which&#xA;  they ask their students to view videos or some other digital matter&#xA;  online before coming to school and then use class time for other&#xA;  activities, such as hands-on and team projects or discussions.&#xA;  Twenty-five percent intend to introduce the flipped model into their&#xA;  courses over the next year or are exploring that possibility.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2019-12-30T02:36:22.643" LastActivityDate="2019-12-30T02:36:22.643" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17656" PostTypeId="2" ParentId="433" CreationDate="2019-12-27T17:58:47.160" Score="3" Body="&lt;p&gt;My opinion on this topic has vacillated a little through the years, but for the most part, I've always spent time on background information about mathematical and statistical information because I firmly believe that doing so encourages an increase in student understanding and  recall.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe it also helps students feel less stressed in my classes, because there are brief moments in the middle of difficult classes when they can relax and listen to their teacher tell short, relevant stories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe it also helps them feel an attachment toward me or with me which encourages them to do everything I want them to do to succeed as students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes I suspect that an attachment between the students and the teacher is very important. It seems as though some university level and college level teachers I know and speak to don't even think about attachment at all or even know it exists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing that matters is that background information about topics in math can be presented well or shoddily, and if it's done shoddily, students may feel you're just presenting useless information as part of a checklist, for example. But if you do it well, then maybe you might sit back and let the students know that the following information is to help them understand a new topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, an introduction to the origin of my opinion seems appropriate. I was from a family with no college education in it prior to me. Therefore, there was simply no one in my family nor school who could give me academic guidance back then. However, I just absolutely loved school, especially math and the sciences, and by the time I was eleven or so, I had decided I was going to go into a math-heavy career in adulthood. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was all working out well, and I was doing well, enjoying my classes and tolerating those I didn't enjoy, until my sophomore  year in high school. That year nearly changed the course of my entire life. My interest in math slammed to a complete halt two or three months into a full year geometry course. The teacher believed in a theorem-proof-theorem-proof-theorem-proof approach,  I won't expound, but I decided I was wrong about math. Remember, &#xA;I was a fourteen year old, young for my grade, with no guidance. I believe there are many with little guidance who we teach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next year, as a junior, I backed into a slower paced, single semester, half year, precalculus and trigonometry course intending to change my college goals. I decided to do that instead of taking the two semester, full year trigonometry and calculus course my sophomore geometry teacher, the one who taught in a way I hated, had placed me. I did so because I had changed my mind about math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The half year course was taught by the calculus teacher. I'll never forget him. I'll never forget his name. He died suddenly that Christmas, so he was only my teacher for one semester. He loved math. Here's how he started the course:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He started by talking about Euclid. Well, actually, by talking about the way the Greeks questioned assumptions. We talked about approximating areas and how the Greeks squared circles, then irrational numbers, then the question of when two lines are parallel, about whether two lines always have to cross somewhere, and about whether two lines ever cross, no matter how far out in space you go.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being 15, I thought his questions were stupid and jumped right in, and he enjoyed my reaction. When I saw his twinkle of smugness, which he was trying to hold back, about five or ten minutes of class had passed, and we got serious about noneuclidian geometry. That night, I was drawing lines on balls, trying to make them straight. Pretty quickly, I had figured it out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Years later, I was introducing everything that way, right down to  number systems in my introductory algebra classes. I teach at Xavier University. I might start a section on Complex Numbers by explaining why numbers were needed by Egyptians at all - first, they needed natural numbers so they could count things: 1, 2, 3, 4, .... But they realized you might have none at all, so they added zero and a new number system, the Whole numbers, was created by building on the Natural numbers: 0, 1, 2, 3, ....  And so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel that it's appropriate to place this reason for my development of my opinion here. It seems highly relevant. &lt;/p&gt;&#xA;" OwnerUserId="13034" LastEditorUserId="13034" LastEditDate="2019-12-27T18:19:45.343" LastActivityDate="2019-12-27T18:19:45.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17657" PostTypeId="2" ParentId="17641" CreationDate="2019-12-28T14:02:09.540" Score="4" Body="&lt;p&gt;I think best learning happens, when students discover the rules on their own. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But students shouldn't be forced to re-invent the wheel, so to say.  Because this is too difficult and frustrating for them.  Original discoveries in math took people hundreds or even thousands of years to do.  So, it's unreasonable to expect ignorant students to do something like this in a few minutes, or hours at most. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students can discover the rules for themselves by looking at several solved examples.  And then they can become competent in using these rules by solving a lot of similar problems and checking if they get the answer right or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think starting with specific examples and discovering abstract rules in them mimics how original math discoveries were made.  It's more natural and easier for people to learn this way.  But if you learn abstract ideas first and then try to apply them to specific examples, then this is like thinking backwards in terms of learning and discovering.  It's much more difficult and awkward for people to learn this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And practicing with immediate feedback about how well or poorly you've done is probably the most effective way gain competence is solving a class of problems. Because without immediate feedback, students often deceive themselves into believing that they've learned it.  And they only discover to their dismay on a test, that they haven't learned it nearly as well as they thought they did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;True knowledge is when you know both what you know and what you don't know.  It's meta-knowledge and not just knowledge. And only practicing with immediate feedback provides this kind of knowledge. &lt;/p&gt;&#xA;" OwnerDisplayName="user13255" LastEditorDisplayName="user13255" LastEditDate="2019-12-28T14:28:02.573" LastActivityDate="2019-12-28T14:28:02.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17658" PostTypeId="2" ParentId="17639" CreationDate="2019-12-28T23:32:54.050" Score="4" Body="&lt;p&gt;Oh this is subtle and tough, in my opinion. It seems to me (and forgive the appalling over-generalizations I am about to present) ...&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In some vague but emotionally potent sense, math is seen as vital for &quot;success&quot; in life. Many parents fear that without a solid math experience, their children will be denied a vital opportunity. (Good for them for feeling that math is important!)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Many people across the globe equate math with computation, and thus success in math is perhaps innately defined as &quot;getting answers to many computation questions, with speed, being correct essentially the first time each tried.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Assessments dictate classroom culture and practice. As much as we say we value &quot;grit&quot; and &quot;growth&quot; and &quot;play with ideas&quot; and &quot;agility of thought&quot;, etc., if, in the end, assessment only assesses timed test taking, then that only reinforces the image and practice of point 2. Parents may see that &quot;unusual&quot; activity as worthwhile, but the YES BUTs that follow are right on point given this context. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The counter YES BUT argument educators might offer: &quot;But do you really expect a store clerk to get out pencil and paper and do the long subtraction algorithm the traditional school curriculum emphasizes to compute 10.00 - 8.37 when making change for a $8.37 item from a ten dollar bill, or do you want him/her be able to think through it?&quot; is irrelevant. Emotions are potent. Yes, in the 21st-century we need to be teaching thinking and process and agility and flexibility of thought, just as puzzles do -- and not teach algorithms just as rote algorithms. (Nothing against teaching algorithms with thinking.) A clever citizen of this century, if really truly needing the answer 376 x 97, say, would just pull out her/his smartphone and ask Siri. (And perhaps have a sense if the answer seems about right.) We educators know the ultimate goal of math education, these days in particular, is not about getting answers to computation problems, actually.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But emotions of fear and change are potent,  and &quot;different&quot; is scary and dangerous - especially when it comes to opportunity in life for my child. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My only approach to answering your question is essentially going through the five points I just outlined; gently, of course, acknowledging and genuinely honouring the honest human emotion and process.          &lt;/p&gt;&#xA;" OwnerUserId="13256" LastActivityDate="2019-12-28T23:32:54.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17659" PostTypeId="2" ParentId="17639" CreationDate="2019-12-29T00:56:20.863" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;OP&lt;/strong&gt;: &quot;such mathematical puzzles will enhance academic performance&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Such puzzles will. But to my mind, what is more important is fostering&#xA;an aesthetic sense, an appreciation of the &lt;em&gt;beauty&lt;/em&gt; of mathematics,&#xA;and of &lt;em&gt;understanding&lt;/em&gt; something beautifully intricate.&#xA;Concerning beauty, Bertrand Russell famously said,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere...&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Concerning understanding, William Thurston said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, in reply to the OP's question,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;OP&lt;/strong&gt;: &quot;how can I convince parents ...?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would suggest: Emphasize appreciation of (a) beauty, and of (b) understanding,&#xA;rather than an instrumental focus on academic performance.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2019-12-29T00:56:20.863" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17660" PostTypeId="2" ParentId="17632" CreationDate="2019-12-29T23:39:37.073" Score="0" Body="&lt;p&gt;If you teach the chain rule with Leibniz notation I recommend &lt;a href=&quot;https://matheducators.stackexchange.com/a/10584/590&quot;&gt;this suggestion&lt;/a&gt; of Steven Gubkin. It makes computations more explicit and straightforward, and students pick it up fairly well in my experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the remainder I'll address some of the subtleties involved with derivative notation, the function concept and how that relates to the chain rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's start with notation. Many books suggest that when &lt;span class=&quot;math-container&quot;&gt;$y=f(x)=x^2$&lt;/span&gt;, all of the following mean the same:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f', \quad  \frac{df}{dx},\quad  \frac{dy}{dx}, \quad \frac{df(x)}{dx}, \quad \frac{d(x^2)}{dx}, \quad y', \quad (f(x))' , \quad (x^2)',\quad f'(x), \quad \frac{df}{dx}(x).&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since we all agree that &lt;span class=&quot;math-container&quot;&gt;$y\neq f$&lt;/span&gt;, the first two of these,&#xA;&lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dx}$&lt;/span&gt;,&#xA;cannot possibly mean the same as the rest. I'll discuss in a moment why the second, &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dx}$&lt;/span&gt;, is nonsense notation, but let's first look at the rest of the list. By virtue of &lt;span class=&quot;math-container&quot;&gt;$y=f(x)=x^2$&lt;/span&gt; and the principle that we can substitute equals for equals, we see that no 3. 4. and 5. must indeed all denote the same&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ &#xA;\frac{dy}{dx}=\frac{df(x)}{dx}= \frac{d(x^2)}{dx}.&#xA;$$&lt;/span&gt;&#xA;(The middle one should be parsed as &lt;span class=&quot;math-container&quot;&gt;$\frac{d(f(x))}{dx}$&lt;/span&gt;, but the parenthesis are omitted.) By the same reason no. 6. 7. and 8. should all denote the same&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;y' = (f(x))'= (x^2)'&#xA;$$&lt;/span&gt;&#xA;&lt;em&gt;if&lt;/em&gt; that notation where sensitive. I'll argue that it isn't and should be avoided, in particular when using the chain rule. No. 9, &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;, is perfectly fine, while the last one, &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dx}(x)$&lt;/span&gt;, should be discarded for the same reason as the second one, &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dx}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what's wrong with &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dx}$&lt;/span&gt;? If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is truly a function in the modern sense, namely the function &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{R}\to\mathbb{R}$&lt;/span&gt; that squares every input, then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is agnostic of the &lt;em&gt;name of its input variable&lt;/em&gt;. In particular, the function that for every &lt;span class=&quot;math-container&quot;&gt;$x\in\mathbb{R}$&lt;/span&gt; satisfies &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt; is exactly the same as the one which for every &lt;span class=&quot;math-container&quot;&gt;$y\in \mathbb{R}$&lt;/span&gt; satisfies &lt;span class=&quot;math-container&quot;&gt;$f(y)=y^2$&lt;/span&gt;. So if we allowed &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dx}$&lt;/span&gt; we should also allow &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dy}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dt}$&lt;/span&gt; etc. Anything could be placed in the denominator and it should all denote the same. The notation is redundant and misleading. Corollary: do not write &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dx}$&lt;/span&gt; for the derivative of a modern function. Simply write &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does that mean we should never apply Leibniz notation to functions? No. For instance when &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; depends on a parameter, say &lt;span class=&quot;math-container&quot;&gt;$f(x)=ax^2$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{da}$&lt;/span&gt; is meaningful. Which raises the question of when exactly to use Leibniz notation. This is more subtle, as can be seen from &lt;a href=&quot;https://mathoverflow.net/q/115416/745&quot;&gt;these&lt;/a&gt; &lt;a href=&quot;https://mathoverflow.net/q/307947/745&quot;&gt;two&lt;/a&gt; discussions, but the summary is: &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}$&lt;/span&gt; operates on functions &lt;strong&gt;of&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and not on functions. Examples of &lt;em&gt;functions of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/em&gt; are &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;, while &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is &lt;em&gt;not&lt;/em&gt; a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally what's the problem with writing &lt;span class=&quot;math-container&quot;&gt;$y'$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(f(x))'$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x^2)'$&lt;/span&gt;? Observe that here we are applying &lt;span class=&quot;math-container&quot;&gt;$(\;)'$&lt;/span&gt; to functions &lt;em&gt;of&lt;/em&gt; something and not to functions. But this notation is incomplete, in that it doesn't make explicit with respect to which variabel to differentiate. This is best illustrated with the chain rule: Suppose for instance &lt;span class=&quot;math-container&quot;&gt;$y=t^3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t= \cos \phi$&lt;/span&gt;, then obviously &lt;span class=&quot;math-container&quot;&gt;$y=(\cos \phi)^3$&lt;/span&gt;. Now what should &lt;span class=&quot;math-container&quot;&gt;$y'$&lt;/span&gt; mean? Is it &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dt}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{d\phi}$&lt;/span&gt;? You might think that it becomes clear once we write &lt;span class=&quot;math-container&quot;&gt;$(t^3)'$&lt;/span&gt; resp. &lt;span class=&quot;math-container&quot;&gt;$\left((\cos \phi)^3 \right)'$&lt;/span&gt;. But since &lt;span class=&quot;math-container&quot;&gt;$t^3=(\cos \phi)^3$&lt;/span&gt; we would violate the principle of substituting equals for equals, if those two expressions had a different meaning. Sounds like very bad notation to me. Corollary: Do not write &lt;span class=&quot;math-container&quot;&gt;$y'$&lt;/span&gt; for the derivative of variable quantity with respect to another variable. Always use Leibniz notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could of course insist on the convention that &lt;span class=&quot;math-container&quot;&gt;$y'$&lt;/span&gt; always denotes derivative with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. But this seems like bad practice from a didactical perspective. For one we would be using the same prime notation to denote two different things: derivative of a modern function as in &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; vs derivative of a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as in &lt;span class=&quot;math-container&quot;&gt;$y'$&lt;/span&gt;. But we agree that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; are objects of different types and (hopefully) want our students to understand that. Moreover in areas where most students will apply calculus (physics, engineering, economy etc), almost no variable is called &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, so the convention would be of little use.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2019-12-29T23:39:37.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17661" PostTypeId="2" ParentId="17649" CreationDate="2019-12-30T17:03:54.603" Score="6" Body="&lt;p&gt;I would like to echo Peter Saveliev's point in the comments:  You are asking about &quot;higher results&quot; but you do not specify what qualities you are interested in measuring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you care about grades?  Performance on standardized tests of content knowledge?  Student success in courses which depend on the given course as a prerequisite?  Student perception of how interesting the material is?  Likelihood that the student will sign up for optional successor courses?  Ability to speak coherently about the material?  Student perception of themselves as an independent authority on the subject matter?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a million distinct questions you can ask.  Different instructors have radically different values, and &quot;success&quot; can mean something very different to different instructors.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2019-12-30T17:03:54.603" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17662" PostTypeId="1" CreationDate="2019-12-31T13:39:32.617" Score="5" ViewCount="224" Body="&lt;p&gt;I read that- &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“The tragedy of Ramanujan was not that he died young, but that his&#xA;  genius was misdirected, side-tracked, and to a certain extent :&#xA;  distorted.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The years between 18 and 25 are the critical years in a&#xA;  mathematician’s career, and the damage had been done. Ramanujan’s&#xA;  genius never had again its chance of full development.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;He would probably have been a greater mathematician if he could have&#xA;  been caught and tamed a little in his youth, and He would have&#xA;  discovered more that was new, and that, no doubt, of greater&#xA;  importance.” - G.H. Hardy&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I want to know what Hardy meant, so I can understand the problem he is referring to when someone don't get proper education. So my question is -&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How it was exhibited/expressed/come to know that Ramanujan  was misdirected, side-tracked, and  distorted?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What Hardy meant by &quot;caught and tamed a little in his youth&quot;?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="13263" LastEditorUserId="13263" LastEditDate="2020-01-01T13:59:40.120" LastActivityDate="2020-01-01T16:41:19.963" Title="Misdirected, Side-Tracked, and Distorted Ramanujan: Problems in Basic Math Education?" Tags="|mathematical-pedagogy|self-learning|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17663" PostTypeId="2" ParentId="17662" CreationDate="2019-12-31T16:59:18.970" Score="8" Body="&lt;p&gt;I suspect that you might get more informed answers from the History of Science and Mathematics SE.  You're sure to find more people who are more learned about the relationship between Hardy and Ramanujan than I am.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I suspect that Hardy is talking about Ramanujan's relative inability to contribute to the mathematical progress of his day because he (Ramanujan) was not exposed to it during his formative years as an emerging mathematician.  Ramanujan's results were self-inspired, so he wound up not generally building on pre-existing structures of mathematical knowledge.  Certainly, people since have incorporated his work into our collective mathematical schema, but Hardy seems to be regretting that Ramanujan himself wasn't able to participate and respond to this growth.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2020-01-01T16:41:19.963" LastActivityDate="2020-01-01T16:41:19.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17665" PostTypeId="2" ParentId="17649" CreationDate="2020-01-01T15:16:31.307" Score="4" Body="&lt;p&gt;What follows is a critique of the paper cited in &lt;a href=&quot;https://matheducators.stackexchange.com/a/17655/8756&quot;&gt;another answer&lt;/a&gt; here: Setren, et. al., &quot;Effects of the Flipped Classroom: Evidence from a Randomized Trial&quot;, Annenberg Institute at Brown University (2019). That paper doesn't, in fact, refute the value of the flipped classroom in any way and is an example of poor research design. In fact, it says nothing about a flipped classroom by any reasonable definition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm adding this for the benefit of the O.P. who shouldn't be misled by &quot;research&quot; that purports to show that the flipped classroom is a poor educational practice, when it actually discusses something else entirely. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, the design of the experiment in the paper by Setren et. al. (the authors) is fatally flawed. I would not have let any of my doctoral students continue with this research, though a number might have been interested in the topic. Fundamentally, if you can predict the outcome of a research project simply from the design, then it isn't really research. I don't doubt the sincerity of the authors but this seems like a situation of &lt;em&gt;confirmation bias&lt;/em&gt; rather than a true search for truth regarding the flipped classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most serious issue with the paper, other than methodological failure, is an incorrect conception of the flipped classroom. The authors seem to have the mistaken impression that the &quot;flipped classroom&quot; is about &lt;em&gt;where&lt;/em&gt; things take place. This is a serious misconception that prejudices any conclusion from the study. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, the flipped classroom is about &lt;em&gt;what&lt;/em&gt; takes place when the students and instructors are face to face. If they aren't different from the standard model, then nothing positive can be expected to happen. In a standard lecture the students are relatively passive, though they may take notes and, perhaps, ask questions. In a flipped classroom the students are active, typically in groups with collaborative educational activities intended to reinforce learning. Students come together in a flipped classroom to do things that can &lt;em&gt;only&lt;/em&gt; be done face to face. Things that don't require interpersonal communication (not just broadcasting) are done otherwise. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this experiment, the students were given videos to watch on their own, along with readings. No effective channel to ask questions is described, and neither are exercises based on that preparation. This alone is a flaw. It advantages people used to passively watching lectures, but it disadvantages those who need more help and have questions along the way. Separating the questions (next day in class) from the video is, itself a flaw. Students can, of course, make a written note of questions for answering later, and the best and most advantaged students are likely to do that, but only if they have enough experience with the format to recognize that it is essential. Students who try to rely on memory alone to bridge the gap between the video and the question period are disadvantaged. (See below for comments on lack of training.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In most educational settings there is an expectation that students will spend twice as much time on a course outside the classroom as inside it. This seems to have been missing from the experimental setup as described. The videos were short, in particular (a good thing). But other preparation is also needed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, active learning seems to have been actively discouraged. This again, advantages those students who don't need much instruction and are comfortable with relatively passive &lt;em&gt;consumption&lt;/em&gt; of material. This is a poor way to generate insight, but most especially in those students who are otherwise disadvantaged by the educational system they have been faced with. The flipped classroom is actually an attempt to address this issue, hence group work. Group work is reported in the experiment, but mostly in the context of students solving problems, rather than more general learning and discussion. This again, seems to suggest a misconception. Students in a standard setting will, themselves, solve their exercises out of class partly in isolation and partly in a group context. But the flipped classroom, used properly, permits more opportunities since the instructor is present and can guide the activities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper seems to suggest that video is a key to the flipped classroom. This is not actually true, though it is a common misconception. Students are given some way to prepare for class, but it needn't be videos. It could be text, such as PDFs prepared by the instructor. It could involve interactive simulations that let the students engage with the material. And it could (and should) involve some exercises based on that preparation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best flipped classrooms will also provide a way to get questions asked and answered before class, perhaps via a mailing list. Students might be required to write a one or two paragraph summary of the readings (or videos) as a sort of &quot;ticket&quot; for attendance in the subsequent face to face session. None of this was provided or expected in the experiment. With video alone, students are likely to be passive. But it need not be that way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a better course design, the flipped classroom will provide the instructor some feedback, prior to the class, that follows up on the out of class activities. If there is a mailing list, for example, the questions posed there are seen by other students and by the instructor. Even if they aren't answered on the list itself, the instructor can prepare the next class period in response to student concerns. The instructor, then, can spend less time and effort on things that the students already grasp well, and can focus on more essential concerns. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper seems to suggest that students work on exercises (alone or in groups) when in class with some chance for feedback, but I doubt that it is especially effective, just due to lack of time and lack of instructor resource. A sole instructor and a T.A. can only work with a few students at a time, leaving the other students bored and stuck. This is why pairing and group work is suggested, and normally required, for the flipped classroom. Students help one another with questions and explorations, not just exercises, freeing up instructors for the most difficult cases. Research has been done on this in the situation of paired programming in a communal space. The U.S.M.A. has strict ethical rules about joint work, though the paper doesn't say if these prohibitions against assistance were enforced here or not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The faculty who actually carried out the experiment were all somewhat experienced in standard lecture formats and it was noted in the paper that they had little or no experience with other formats. Most of the instructors, however, were relatively inexperienced faculty members. As junior military officers, their own educational background was most likely on a standard model, with which they have excelled. Also, they had likely taught the target courses, or similar courses, in a standard format, similar to their own education. The experimental design doesn't describe that they were given any specific training in the flipped classroom. Nor were they permitted to deviate from the designed format. The paper does, however, admit that the students did better in those sections in which the instructor was more enthusiastic about the flipped classroom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper also seems to admit that it may have been a flaw in the design that each instructor taught both a standard model section and a flipped section. Military officers follow orders, of course, but they don't have to accept the validity of what they are commanded to do. The whole process was, in fact, an imposition on them, as none were volunteers. This leads to uniformity of a certain kind, of course, but not to good educational outcomes for students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, even if we somehow decide that all of this is to be forgiven, and that the flipped classroom model was well run (thus validating the data itself), there is still a fundamental flaw that would prevent us from drawing out generalized conclusions from the study: all of the subjects are from a highly advantaged subset of the US population. The U.S.M.A. is a highly selective undergraduate college in which students gain entry by proving that they excel in the standard educational process. They are used to lecture/exercise/exam based courses and have a history of success in such situations and most have little experience with alternate educational structures. Moreover, most of them come from dominant cultural and economic situations since admittance requires the recommendation of a member of Congress. They are not a cross section of students in any case, and especially, neither are their educational backgrounds. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, the experimental period was short, apparently between one and three weeks, with no one in the treatment group, student and instructor alike, given an opportunity to become comfortable or familiar with the process, while the control group had a long history of such experience with standard practice. Moreover, the authors report that students in the experimental group weren't very vigilant in doing the necessary preparation for class. This is likely due to their expectation from past experience that it was the classroom in which the material would be adequately &quot;covered&quot;. But it was not. So, many seem to have entered the classroom unprepared. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be natural to expect a bit of chaos in the experimental group as the experiment was carried out in the middle of an otherwise standard course. All of a sudden, the rules of engagement are changed. This alone would lead one to suspect lower achievement over that lesson, as students have been dropped into a new world for a few weeks, not knowing how to act or what to expect, other than that they were supposed to &quot;follow orders&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the experimental design, the outcome was foreordained. Put students into a new situation that might seem a bit chaotic, and they aren't likely to do very well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the following &lt;em&gt;thought experiment&lt;/em&gt; in which I suspect that you can easily predict the outcome. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we had a large group of students who had spent their entire lives learning in a collaborative group environment in which they have access outside the classroom to a large variety of resources, but who spend their face to face time in collaborative learning. They can, and are encouraged to, prepare for class by reading, etc, and are also expected to bring questions with them. But there are also exercises they can do to test their own understanding prior to class. The face to face time is spent in reinforcement of the ideas and feedback on attempts. Students are encouraged to speak and offer ideas and other students are encouraged to comment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, take the experiment as described in the paper, but flip the treatment and control groups. the course proceeds as normal for them (flipped) until we reach the start of the experiment. Otherwise run it as before, but in this case all of the students instructors are thoroughly familiar with the above described methods, but not so much with lecture to a relatively passive group, followed by exercises to be done at home. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you suppose the outcome of the new experiment would be. Isn't it pretty obvious?&lt;/p&gt;&#xA;" OwnerUserId="8756" LastEditorUserId="216" LastEditDate="2020-01-01T16:16:20.873" LastActivityDate="2020-01-01T16:16:20.873" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17666" PostTypeId="2" ParentId="1820" CreationDate="2020-01-01T16:01:46.003" Score="4" Body="&lt;p&gt;Anecdotally, my father's generation (1950s-60s, UK) all learned logarithms in primary school and used them extensively for calculations. All children had their own slide rule which they carried with them and used in classes other than maths classes. There were no electronic calculators. My father became a radar engineer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My generation (1980s-90s, UK) only touched on logarithms around age 14. We used calculators. My father explained to me early on what logarithms were and what they were useful for in a way that made sense to me at a young age (&quot;they turn multiplications into additions, so the calculations are easier. I guess you kids don't need them as you have calculators.&quot;). This still left me with an air of mystery regarding logarithms and that they were an 'old-fashioned' way of doing things. Such a shame. I wish I'd learned logarithms back then. Now I'm a software engineer and logarithms are often useful. I often discover new ways to apply them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am resolved to make sure my kids learn logarithms even if I have to teach them myself. That and dimensional analysis.&lt;/p&gt;&#xA;" OwnerUserId="13266" LastActivityDate="2020-01-01T16:01:46.003" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17668" PostTypeId="2" ParentId="220" CreationDate="2020-01-02T16:46:30.323" Score="7" Body="&lt;p&gt;A simple consequnce of:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Postage Stamp Problem&lt;/strong&gt;, which states that for any two relatively prime positive integers &lt;span class=&quot;math-container&quot;&gt;$m,n$&lt;/span&gt;, the greatest integer that cannot be written in the form &lt;span class=&quot;math-container&quot;&gt;$am + bn$&lt;/span&gt; for nonnegative integers &lt;span class=&quot;math-container&quot;&gt;$a, b$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$mn-m-n$&lt;/span&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is that every natural number greater or equal to &lt;span class=&quot;math-container&quot;&gt;$mn-m-n+1$&lt;/span&gt; can be writen in a form &lt;span class=&quot;math-container&quot;&gt;$am+bn$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And for some concrete &lt;span class=&quot;math-container&quot;&gt;$m,n$&lt;/span&gt; it is a nice problem to solve in HS: Say &lt;span class=&quot;math-container&quot;&gt;$m=3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n=5$&lt;/span&gt;, then prove that every natural number greater than &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; can be writen in a form &lt;span class=&quot;math-container&quot;&gt;$3a+5b$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proof:&lt;/strong&gt; Base cases: &lt;span class=&quot;math-container&quot;&gt;\begin{eqnarray}8 &amp;amp;=&amp;amp; 3\cdot 1+5\cdot 1\\&#xA;9 &amp;amp;=&amp;amp; 3\cdot 3+5\cdot 0\\&#xA;10 &amp;amp;=&amp;amp; 3\cdot 0+5\cdot 2&#xA;\end{eqnarray}&lt;/span&gt;&#xA;and now go from &lt;span class=&quot;math-container&quot;&gt;$n\to n+3$&lt;/span&gt;: If &lt;span class=&quot;math-container&quot;&gt;$n=3a+5b$&lt;/span&gt; then &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{eqnarray}n+3 &amp;amp;=&amp;amp;  3\underbrace{(a+1)}_{a'}+5b\\&#xA; &amp;amp;=&amp;amp; 3\cdot a'+5\cdot b&#xA;\end{eqnarray}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="10484" LastActivityDate="2020-01-02T16:46:30.323" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17669" PostTypeId="1" AcceptedAnswerId="17679" CreationDate="2020-01-02T23:54:29.077" Score="20" ViewCount="2998" Body="&lt;p&gt;I just inherited two slide rules from my grandfather-in-law, one wood with smooth action despite nearly a century without&#xA;use.&lt;sup&gt;1&lt;/sup&gt; (I used a K+E slide rule myself as an undergraduate in the 1970's.) It struck me that the conversion of multiplication/division to addition/subtraction via the&#xA;logarithmically ruled markings might serve as a&#xA;memorable physical instantiation of &lt;span class=&quot;math-container&quot;&gt;$\log ab = \log a + \log b$&lt;/span&gt;, &#xA;or &lt;span class=&quot;math-container&quot;&gt;$x^a x^b = x^{a+b}$&lt;/span&gt;,&#xA;etc. One of the slide rules came with instructions on how to take the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th root of a number!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've never taught logarithms, but I wonder if those&#xA;who have, ever bring in to the classroom slide rules as &quot;props&quot;?&#xA;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;&lt;a href=&quot;https://www.sliderulemuseum.com/Lawrence.htm&quot; rel=&quot;noreferrer&quot;&gt;Lawrence Engineering Service, Peru, Indiana&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-01-03T19:19:27.717" LastActivityDate="2020-01-04T21:14:59.850" Title="Does anyone teach logarithms via slide rules?" Tags="|logarithm|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17670" PostTypeId="2" ParentId="17669" CreationDate="2020-01-03T02:21:07.820" Score="11" Body="&lt;p&gt;A few thoughts:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Joe Pasquale from UCSD has done similar things: &#xA;&lt;a href=&quot;https://cseweb.ucsd.edu/~pasquale/SlideRuleTalkLasVegas14.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://cseweb.ucsd.edu/~pasquale/SlideRuleTalkLasVegas14.pdf&lt;/a&gt; and &#xA;&lt;a href=&quot;https://cseweb.ucsd.edu/~pasquale/FreshmanSeminarF03/&quot; rel=&quot;noreferrer&quot;&gt;https://cseweb.ucsd.edu/~pasquale/FreshmanSeminarF03/&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If the students have basic science training, they've already internalized scientific notation and base 10 logs, ex: &lt;span class=&quot;math-container&quot;&gt;$10^2 \times 10^3 = 10^5$&lt;/span&gt;. That's a simpler version of what slide rules do, and could be a bridge.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students can easily make their own paper slide rules, that might make the class more engaging and fun. Here's an easy version: &lt;a href=&quot;https://www.instructables.com/id/Make-a-simple-paper-slide-rule/&quot; rel=&quot;noreferrer&quot;&gt;https://www.instructables.com/id/Make-a-simple-paper-slide-rule/&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13269" LastActivityDate="2020-01-03T02:21:07.820" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17671" PostTypeId="1" CreationDate="2020-01-03T14:24:25.230" Score="7" ViewCount="488" Body="&lt;p&gt;In some universities, high schools, and summer programs, students are required to do their own research project in maths and write their own essays/research papers. At the same time, however, many other schools and colleges only require them to take examinations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In high school and the first a few years of university, it is almost impossible to come up with something new in maths (unless you are a genius:), and those research papers are usually just presentations of existing mathematical knowledge. Although writing papers develop research skill, it is likely to be very time-consuming for both students and teachers marking them. If they do exams instead, a lot of time could be saved, which means students have more freedom and can study more content. But of course, content is not everything. Doing a summary of existing knowledge doesn't seem to be more related to problem-solving skills than exams, either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, is it better to just do the exams and learn as fast as possible, or slow down and do some research projects?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer to this question is very likely to be &quot;it depends&quot;, but I am still asking this because there might be other reasons for/against doing mathematics research projects when one still doesn't have very much knowledge.&lt;/p&gt;&#xA;" OwnerUserId="9339" LastActivityDate="2020-01-04T13:33:00.077" Title="Doing research projects when one's knowledge is limited: is it preferable?" Tags="|undergraduate-education|secondary-education|undergraduate-research|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17672" PostTypeId="2" ParentId="17669" CreationDate="2020-01-03T14:40:29.003" Score="9" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;but I wonder if those who have, ever bring in to the classroom slide rules as &quot;props&quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the Olden Days (before hand-held calculators) I remember that we had (at Ohio State) a big demonstration slide rule.  It was maybe 6 feet long, on a stand with wheels.  So the instructor would wheel it in to the front of the classroom, and demonstrate calculations on it.  The students, in their seats, would try to replicate what was done on their own slide rules.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this was not for teaching logarithms, but for teaching the use of the slide rule.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2020-01-03T14:40:29.003" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17673" PostTypeId="2" ParentId="17669" CreationDate="2020-01-03T16:24:07.830" Score="8" Body="&lt;p&gt;(too-long comment) I think it's a nice adjunct, but I would not introduce the topic that way.  Introduce it after teaching of rational (math meaning) exponents and roots in a rational (well thought out meaning) progression.  I.e. on the board, with some discussion of concepts they learned in exponents that are also used in logarithms.  In addition to the slide rules, it is nice to hand out a sheet to each student of log-log and semi-log graph paper.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, a cheap handout of log TABLES is nice.  And not because we are going to force them to use tables (scientific calculators have been cheap and ubiquitous since late 70s).  But because it is easy to see, right in front of your face, how logarithms work and some useful ideas like mantissa and characteristic.  From a table (which is essentially a printed out Excel spreadsheet!), you can see at a glance, many different values.  Not possible with calculator.  I also think it's an easier step to understand (a table) than the sliding nomograph like properties of a slipstick.  A nice set of tables to print and distribute is the ones that were handed out for AP chemistry through the mid-90s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this is not to say don't show the slide-rule.  Sure, do it.  But don't make it central to the teaching.  More of a little enrichment exposure that perhaps clicks with some minority of the monsters.&lt;/p&gt;&#xA;" OwnerUserId="13241" LastActivityDate="2020-01-03T16:24:07.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17674" PostTypeId="2" ParentId="17671" CreationDate="2020-01-03T16:33:49.987" Score="3" Body="&lt;p&gt;Opinion:  I think they get enough exposure to writing reports in other subjects.  Think the time is better used for drill and exams, than for writing a report.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one benefit might be learning to search the literature, but even that is probably much better done in history or science where, although rather difficult, the papers are still comprehensible.  (Not so in math.)&lt;/p&gt;&#xA;" OwnerUserId="13241" LastActivityDate="2020-01-03T16:33:49.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17675" PostTypeId="2" ParentId="17671" CreationDate="2020-01-03T20:09:35.120" Score="4" Body="&lt;p&gt;Perhaps a research project could be replaced by an independent study project. The idea is to learn more math, in a more independent way. The student would still have to produce something to show what they've learned (although if the teacher found it easier, they could instead require an oral explanation of the content learned).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I learned a lot on the independent project where I worked from the (tiny) book Surreal Numbers, by Knuth. &lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-01-03T20:09:35.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17676" PostTypeId="2" ParentId="17671" CreationDate="2020-01-04T13:33:00.077" Score="8" Body="&lt;p&gt;I think it is very possible to have real, meaningful research projects at all levels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, I had 2 of my students in Calc 2 work on a project which started with the idea &quot;Can we define a tangent &lt;strong&gt;circle&lt;/strong&gt; instead of a tangent line?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They started by finding a formula for the center and radius of a circle given three non-colinear points:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/3aty1fcgdw&quot; rel=&quot;noreferrer&quot;&gt;https://www.desmos.com/calculator/3aty1fcgdw&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They then applied this formula to the points &lt;span class=&quot;math-container&quot;&gt;$(a,f(a))$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(a+h,f(a+h))$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(a-h,f(a-h))$&lt;/span&gt; of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/xowdomvxq3&quot; rel=&quot;noreferrer&quot;&gt;https://www.desmos.com/calculator/xowdomvxq3&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They then needed to take the limit of the expressions for the center and radius which they had as &lt;span class=&quot;math-container&quot;&gt;$h \to 0$&lt;/span&gt;.  As luck would have it, we covered Taylor series a few days before they needed it to take these limits!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/vfgzav8ngx&quot; rel=&quot;noreferrer&quot;&gt;https://www.desmos.com/calculator/vfgzav8ngx&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I then let them know that they had rediscovered &quot;curvature&quot; and the &quot;osculating circle&quot; of a plane curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This really worked out their algebra and calculus skills, and felt like real research.  They would get stuck for a week at a time, experiment, try it in Desmos, see it didn't work, go back to the drawing board.  I think it was an extremely valuable experience for them.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-01-04T13:33:00.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17677" PostTypeId="1" CreationDate="2020-01-04T20:13:56.720" Score="3" ViewCount="172" Body="&lt;p&gt;I'm teaching a class where statistics is not the main topic, but I would like to introduce the idea that if you take &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; measurements of a variable that are independent, identically distributed, and uncorrelated, then the mean has a standard deviation that scales down like &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt{n}$&lt;/span&gt;. I thought it would be fun if there were some impressive or compelling real-world example of this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Samples of things that I imagined along these lines: --&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Maybe someone has a webcam somewhere that shows their robot repeatedly flipping a coin, with running stats.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Long-term observation of Brownian motion.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maybe there is a publicly accessible government database that has collected large amounts of some kind of relevant data over time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would think that the examples with the largest &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and the best statistical properties would probably be from experiments in nuclear or particle physics, but this seems low in charisma and not as compelling and understandable.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I would be equally happy with something more of the flavor of the sum of random variables having a linearly increasing variance, which is mathematically the same thing, although not necessarily the same thing in terms of psychology/fun/impressiveness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A vaguely similar type of statistical demonstration would be the famous University of Queensland &lt;a href=&quot;https://en.wikipedia.org/wiki/Pitch_drop_experiment&quot; rel=&quot;nofollow noreferrer&quot;&gt;pitch drop experiment&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-01-04T20:23:16.987" LastActivityDate="2020-01-04T21:20:18.507" Title="Fun, impressive, or compelling examples of scaling of the standard deviation like $1/\sqrt{n}$?" Tags="|statistics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17678" PostTypeId="2" ParentId="17677" CreationDate="2020-01-04T20:39:35.000" Score="7" Body="&lt;p&gt;If you have a bunch of identical dice, (I recommend non-standard dice; as of writing, mathsgear is a good source of interesting ones), you can just pass out dice to students -- then have them collect a lot of data. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do this with my general education statistics students. My experience is it takes most of a class period, but you can get the class data up on the board as a histogram and they can see the standard deviation go down AND see how much it goes down by.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prerequisites:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The students need to know how to find the population mean and standard deviation for a die they are looking at.&lt;/li&gt;&#xA;&lt;li&gt;The students need to know that the inflection point on a normal distribution occurs one standard deviation away from the mean. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Outline:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pick a die and walk through calculating the mean and standard deviation for a single die roll. With a standard die you can write &lt;span class=&quot;math-container&quot;&gt;$\mu_x = 3.5$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sigma_x \approx 1.71$&lt;/span&gt;.  &lt;/li&gt;&#xA;&lt;li&gt;Pass out a die of that type to each student.&lt;/li&gt;&#xA;&lt;li&gt;Task the class with calculating &lt;span class=&quot;math-container&quot;&gt;$\overline{x}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$n = 10$&lt;/span&gt; several times. I have been asking groups of four students to find 10 values of &lt;span class=&quot;math-container&quot;&gt;$\overline{x}$&lt;/span&gt;, so each group must execute 100 rolls each. If your class has 8 groups, you can get 80 data points (You could try asking each individual student to find some values of &lt;span class=&quot;math-container&quot;&gt;$\overline{x}$&lt;/span&gt; but my experience is that you will not get every student to actually figure out what is going on this way). &lt;/li&gt;&#xA;&lt;li&gt;Tell students to write all their values of &lt;span class=&quot;math-container&quot;&gt;$\overline{x}$&lt;/span&gt; on the board. &#xA;Because you have picked &lt;span class=&quot;math-container&quot;&gt;$n = 10$&lt;/span&gt;, you have nice round numbers for your &lt;span class=&quot;math-container&quot;&gt;$\overline{x}$&lt;/span&gt;'s (Note: you will be tempted to choose &lt;span class=&quot;math-container&quot;&gt;$n=9$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$n=16$&lt;/span&gt; but the payoff for &lt;span class=&quot;math-container&quot;&gt;$n=10$&lt;/span&gt; is higher).&lt;/li&gt;&#xA;&lt;li&gt;Draw the bar graph of all the data points. Notice how almost-bell-shaped it is. Find the inflection points of the shape. &lt;em&gt;Small but serious issue: &lt;span class=&quot;math-container&quot;&gt;$\overline{X}$&lt;/span&gt; is not really normally distributed since &lt;span class=&quot;math-container&quot;&gt;$n = 10$&lt;/span&gt;. My experience is that things work out close enough.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Note that the inflection points are &lt;span class=&quot;math-container&quot;&gt;$\frac{\sigma_\overline{x}}{\sqrt{n}}$&lt;/span&gt; away from &lt;span class=&quot;math-container&quot;&gt;$\mu_\overline{x}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Key question to ask students:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Are the &lt;span class=&quot;math-container&quot;&gt;$\overline{x}$&lt;/span&gt;'s more or less spread out than the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2020-01-04T21:20:18.507" LastActivityDate="2020-01-04T21:20:18.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17679" PostTypeId="2" ParentId="17669" CreationDate="2020-01-04T21:14:59.850" Score="6" Body="&lt;p&gt;I have my college algebra students create slide rules each semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Key pedagogical question:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is the point of having an algebra student construct a slide rule?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Possible answers:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The goal is to convince the students that things like &lt;span class=&quot;math-container&quot;&gt;$\log 5$&lt;/span&gt; are &lt;em&gt;real numbers&lt;/em&gt;. If you ask a student what &lt;span class=&quot;math-container&quot;&gt;$\log 5$&lt;/span&gt; is, you often get the answer that it is a function, or a procedure, or even a &lt;em&gt;question&lt;/em&gt;. Having students physically work with expressions like &lt;span class=&quot;math-container&quot;&gt;$\log 5$&lt;/span&gt; could help them better conceptualize what the data type of &lt;span class=&quot;math-container&quot;&gt;$\log 5$&lt;/span&gt; is [citation needed; and I have none].&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's neat. Show students neat things.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Materials:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Yardsticks (here yardsticks are actually better than metersticks)&lt;/li&gt;&#xA;&lt;li&gt;Tape that is dry-erase&lt;/li&gt;&#xA;&lt;li&gt;Dry erase markers&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Prep:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Put tape on the yardsticks so that half the yardstick is covered but not the other half. The tape needs to wrap around covering both sides. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cr9wO.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cr9wO.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Outline:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Split the class into groups and give each group one prepared yardstick.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask half the class (the weaker groups) to put the yardstick so that &quot;1&quot; is on the left. They will make a mark on their yardstick labeled &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; at the position &lt;strong&gt;&lt;span class=&quot;math-container&quot;&gt;$\log x$&lt;/span&gt; feet&lt;/strong&gt; for &lt;span class=&quot;math-container&quot;&gt;$x = 1, 10, 100, 2, 20, 200, 3, 30, 300, \ldots, 1000$&lt;/span&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rkrx5.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rkrx5.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ask the other half of the class (the stronger half -- this is tricky) to flip the yardstick over so the &quot;1&quot; is on the right. They will make a mark on their yardstick labeled &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; at the position &lt;strong&gt;&lt;span class=&quot;math-container&quot;&gt;$3 - \log x$&lt;/span&gt; feet&lt;/strong&gt; for the same values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UW4sd.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UW4sd.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Then if you combine two of these sticks, you have a slide rule.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Key question to ask students:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How far is it from the end of the stick to the mark labeled &quot;5&quot;?&lt;/li&gt;&#xA;&lt;li&gt;How far is it from the end of the stick to the mark labeled &quot;4&quot;?&lt;/li&gt;&#xA;&lt;li&gt;How far is it from the end of the stick to the mark labeled &quot;20&quot;?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Key thing to emphasize from student answers&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log 5$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\log 4$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\log 20$&lt;/span&gt; are numbers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Then of course show them that you can now physically see &lt;span class=&quot;math-container&quot;&gt;$\log 4 + \log 5 = \log 20$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then pull out the actual slide rule you brought in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then show them the clip from the movie Apollo 13 where the engineers frantically calculate with a slide rule. &lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2020-01-04T21:14:59.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17680" PostTypeId="1" CreationDate="2020-01-05T15:31:43.657" Score="1" ViewCount="115" Body="&lt;p&gt;I see on the webpage of a high school math summer program, SuMac, that they will cover some algebraic topology in a period of several weeks. And they covered every aspect of this subject, including Homology, Homotopy, Cohomology and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am a bit curious about how this is actually done in practice. After all, almost all algebraic topology texts assume quite a bit of knowledge about topologic and metric spaces. There are a lot of theorems and proofs to do even before starting algebraic topology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know why they are soooo efficient in teaching such difficult contents?&lt;/p&gt;&#xA;" OwnerUserId="9339" LastActivityDate="2020-01-05T15:31:43.657" Title="How to teach year 3 undergraduate courses to high school students?" Tags="|undergraduate-education|secondary-education|topology|summer|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17681" PostTypeId="2" ParentId="220" CreationDate="2020-01-05T16:57:54.130" Score="13" Body="&lt;p&gt;Here is another one:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{blue}{\text{Prove that the power of $13$ can be writen as a sum of two squares}}.&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I will give two proofs of it. First one is more involved and includes the following identity: &lt;span class=&quot;math-container&quot;&gt;$$(a^2+b^2)(x^2+y^2)= (ax+by)^2+(bx-ay)^2$$&lt;/span&gt; yet the second one takes step 2 and it is much more elegant.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;1.st proof&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Base: &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$13 = 2^2+3^2$&lt;/span&gt; and we are done.&lt;/p&gt;&#xA;&lt;p&gt;We know that &lt;span class=&quot;math-container&quot;&gt;$13^n = a^2+b^2$&lt;/span&gt; for some integers &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt;. Then:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{eqnarray}13^{n+1} &amp;amp;=&amp;amp; 13\cdot 13^n\\&#xA; &amp;amp;=&amp;amp; (2^2+3^2)(a^2+b^2)\\&#xA; &amp;amp;=&amp;amp; (\underbrace{2a+3b}_x)^2+(\underbrace{3a-2b}_y)^2\\&#xA;&amp;amp;=&amp;amp; x^2+y^2&#xA;\end{eqnarray}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;2.nd proof&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Base cases &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$$13^1 = 2^2+3^2$$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n=2$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$$13^2 = 169 =25+144=5^2+12^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now we go from &lt;span class=&quot;math-container&quot;&gt;$n\to n+2$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{eqnarray}13^{n+2} &amp;amp;=&amp;amp; 13^2\cdot 13^n\\&#xA; &amp;amp;=&amp;amp; 13^2(a^2+b^2)\\&#xA; &amp;amp;=&amp;amp; (\underbrace{13a}_x)^2+(\underbrace{13b}_y)^2\\&#xA;&amp;amp;=&amp;amp; x^2+y^2&#xA;\end{eqnarray}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="10484" LastEditorUserId="10484" LastEditDate="2023-03-10T15:56:11.883" LastActivityDate="2023-03-10T15:56:11.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17682" PostTypeId="2" ParentId="220" CreationDate="2020-01-05T22:33:21.507" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/q/3316466/272831&quot;&gt;For what natural &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; does there exist a square composed of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; squares?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: 1, 4, and 6 are valid, but one cannot construct a square from 2, 3, or 5 squares.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xAVgW.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xAVgW.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof:&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; Construct squares composed of 1, 6, and 8 squares. Then show by induction that if you can construct a square composed of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; squares, then you can construct a square composed of &lt;span class=&quot;math-container&quot;&gt;$n+3$&lt;/span&gt; squares simply by cutting one of the squares into 4 more.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2020-01-05T22:33:21.507" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17683" PostTypeId="2" ParentId="12271" CreationDate="2020-01-05T23:25:16.350" Score="2" Body="&lt;p&gt;Disclaimer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answer shares the same ideals as &lt;a href=&quot;https://matheducators.stackexchange.com/a/12840/6220&quot;&gt;James S. Cook's answer&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Nowadays, I must be honest that I usually don't memorize any of the trig identities, save for possibly the Pythagorean identity because it appears so often. Instead I focus on remembering:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$e^{i\theta}=\cos(\theta)+i\sin(\theta)\\\cos(\theta)=\Re(e^{\pm i\theta})=\frac{e^{i\theta}+e^{-i\theta}}2\\\sin(\theta)=\pm\Im(e^{\pm i\theta})=\frac{e^{i\theta}-e^{-i\theta}}{2i}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From these three identities alone, one can derive not only all of the standard identities, but many other niche identities that are not obvious at first glance. Common examples include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The general formulas for &lt;span class=&quot;math-container&quot;&gt;$\cos^n(\theta)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin^n(\theta)$&lt;/span&gt; in terms of multiple angles via the binomial theorem.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The general formulas for &lt;span class=&quot;math-container&quot;&gt;$\cos(n\theta)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin(n\theta)$&lt;/span&gt; as factoring problems aside from the usual &quot;repeatedly applying sums of angles&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sums of trig functions over angles in arithmetic progression via geometric series.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Furthermore, the procedure is straightforward algebra. Consider the example identity, supposing we don't get division by zero:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\cos[(n+1)\theta]-\cos(n\theta)}{\cos(\theta)-1}=\frac{\sin[(n+\frac12)\theta]}{\sin(\frac12\theta)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd wager that trying to prove this identity with trig identities is a very troublesome task. However, with the complex exponentials, this problem is as easy as simply multiplying both sides out. You don't have to pick out which identities to use or how to use them because everything flows through the algebra you've built up.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2020-01-05T23:25:16.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17684" PostTypeId="2" ParentId="17528" CreationDate="2020-01-05T23:41:44.700" Score="0" Body="&lt;p&gt;We observe the proof on &lt;a href=&quot;https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule#Proof_of_L%27H%C3%B4pital%27s_rule&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;. The intuition behind it is Cauchy's mean value theorem, which states that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{f(a)-f(b)}{g(a)-g(b)}=\frac{f'(c)}{g'(c)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for some &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; between &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. From here all we need to do is divide by &lt;span class=&quot;math-container&quot;&gt;$g(a)/g(a)$&lt;/span&gt; to get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\frac{f(a)}{g(a)}-\frac{f(b)}{g(a)}}{1-\frac{g(b)}{g(a)}}=\frac{f'(c)}{g'(c)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and let &lt;span class=&quot;math-container&quot;&gt;$a\to x$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\lim_{a\to x}|g(a)|=\infty$&lt;/span&gt;. This gives us&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{a\to x}\frac{f(a)}{g(a)}=\frac{f'(c)}{g'(c)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for some &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; between &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. But by taking &lt;span class=&quot;math-container&quot;&gt;$b\to x$&lt;/span&gt;, we get the desired result by continuity/squeeze. In my opinion, I find this to be reasonably intuitive, while also being rigorous.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2020-01-05T23:41:44.700" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17685" PostTypeId="2" ParentId="12868" CreationDate="2020-01-06T02:26:36.653" Score="3" Body="&lt;h2&gt;Round. Check. Correct.&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;What we need is &lt;span class=&quot;math-container&quot;&gt;$436\div48$&lt;/span&gt;. Rounding to remove the last digit reduces the problem to &lt;span class=&quot;math-container&quot;&gt;$44\div5$&lt;/span&gt;. This gives us an &lt;em&gt;educated&lt;/em&gt; guess of either &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From here we then multiply and check, either one. Take &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; for an example. &lt;span class=&quot;math-container&quot;&gt;$48\times8=384$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$436-384=52$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By looking at what we have left, we know that we need only one more &lt;span class=&quot;math-container&quot;&gt;$48$&lt;/span&gt; to go in, since our estimate was accurate enough to reduce the numbers down to things we can easily work with. By adding one more onto &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;, and subtracting &lt;span class=&quot;math-container&quot;&gt;$48$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$52$&lt;/span&gt;, we get the first step:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align} \require{enclose}&#xA;    9\hphantom{^4~8} \\&#xA;    48 \enclose{longdiv}{4~3~6^4~8}&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From which we can easily see the answer is &lt;span class=&quot;math-container&quot;&gt;$91$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2020-01-06T02:26:36.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17686" PostTypeId="1" CreationDate="2020-01-06T02:29:25.820" Score="11" ViewCount="7782" Body="&lt;p&gt;tl;dr I'm interested in opinions on short division. Below I discuss my experience dealing with young children and long division versus &lt;a href=&quot;https://en.wikipedia.org/wiki/Short_division&quot; rel=&quot;nofollow noreferrer&quot;&gt;short division&lt;/a&gt;. For those that don't know of it, &lt;a href=&quot;https://www.wikihow.com/Do-Short-Division&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikiHow&lt;/a&gt; has a nice explanation on short division.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Recently I've been asked to help some children around the age of 10 with math, and almost immediately I find that they struggle with division. They take a long time to divide numbers and they always use long division, or some variant of it. I've even seen one person divide as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array} {r|l} \require{enclose}&#xA;    3 \enclose{longdiv}{1~8~6} \\&#xA;    \underline{{}-~1~5~0} &amp;amp; 5~0 \\&#xA;    3~6 \\&#xA;    \underline{{}-~3~0} &amp;amp; 1~0 \\&#xA;    6 \\&#xA;    \underline{{}-~6} &amp;amp; \underline{\hphantom{5~}2} \\&#xA;    0 &amp;amp; 6~2&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which just frustrates me. Excuse my rant. But I would expect people to know that one can directly see that the answer is &lt;span class=&quot;math-container&quot;&gt;$62$&lt;/span&gt; by seeing that &lt;span class=&quot;math-container&quot;&gt;$18\div3=6$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$6\div3=2$&lt;/span&gt;. Furthermore I ask them why they use &lt;span class=&quot;math-container&quot;&gt;$50$&lt;/span&gt; on the first step instead of &lt;span class=&quot;math-container&quot;&gt;$60$&lt;/span&gt;, and why not just not have the zero and do it digit-by-digit and they say they noticed &lt;span class=&quot;math-container&quot;&gt;$50$&lt;/span&gt; &quot;worked&quot; and that they've been taught this way at school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Poor dividing aside, long division inherently takes a long time to do, as the name implies, and when given a limited amount of time to do an assignment or assessment in class, time consuming division can really hurt. Likewise, there is what is known as short division, which generally takes much less time but requires more mental calculations. But it would appear that no-one knows about short division explicitly, at least out of everyone I've talked to. I myself actually &quot;figured out&quot; short division at a young age from examples like &lt;span class=&quot;math-container&quot;&gt;$186\div3$&lt;/span&gt; by simply skipping steps, but didn't even know it was called this before writing this question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Out of all of the children I've helped with division, the only ones that struggle now are the ones that don't have their multiplication tables memorized. Dividing by multi-digit numbers is still a process, but it's not the kind of thing I would expect someone to be able to do easily. Dividing by single-digit numbers occurs often though, for example reducing fractions. Some of the kids with stronger mental arithmetic skills that I've shown this to can now do this process digit-by-digit mentally, so that they don't write anything out when doing &lt;span class=&quot;math-container&quot;&gt;$\frac{38}{56}=\frac{19}{28}$&lt;/span&gt; for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering the usefulness of short division, it strikes me as surprising that this isn't well-known and taught more! Does anyone have any experience with teaching short division?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I may've glossed over why, exactly, I find short division not only less space consuming, but also easier and faster. Firstly, this does not require a huge amount of mental arithmetic. In fact it only requires being able to compute very small divisions several times over.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The essence of short division is that we are just repeatedly finding what goes into something and what the remainder is, which most of the kids I've dealt with are capable of doing mentally, as far as what's required for short division. Note that I'm not talking about problems with divisors larger than say 11.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take, for example, &lt;span class=&quot;math-container&quot;&gt;$847\div3$&lt;/span&gt;. Most kids I've worked with would know right off the bat that &lt;span class=&quot;math-container&quot;&gt;$8\div3=2\rm~R~2$&lt;/span&gt;. So they write a 2 on top and 2 by the 8. Then they know that &lt;span class=&quot;math-container&quot;&gt;$24\div3=8\rm~R~0$&lt;/span&gt;. So they write an 8 on top and 0 by the 4. Finally they know &lt;span class=&quot;math-container&quot;&gt;$7\div3=2\rm~R~1$&lt;/span&gt;. This gives the final answer: &lt;span class=&quot;math-container&quot;&gt;$847\div3=282\rm~R~1$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These steps are not difficult at all as long as the divisor is at most 11 or 12. In fact most of the kids find this easy to do. For larger divisors I wouldn't expect someone to do this mentally with short division. I would expect them to use long division as usual for something such as &lt;span class=&quot;math-container&quot;&gt;$847\div13$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also helps with situations such as the aforementioned fraction reduction such as &lt;span class=&quot;math-container&quot;&gt;$\frac{38}{56}=\frac{19}{28}$&lt;/span&gt;. It's not too hard to see that 2 goes in, but with long division writing this out can take a while. Hence why this is also faster in such cases.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="6220" LastEditDate="2020-01-07T01:58:32.880" LastActivityDate="2023-10-19T15:59:15.393" Title="Is short division taught these days and if not, why not?" Tags="|secondary-education|primary-education|arithmetic-operations|division|mental-math|" AnswerCount="8" CommentCount="23" ContentLicense="CC BY-SA 4.0" />
  <row Id="17687" PostTypeId="2" ParentId="16668" CreationDate="2020-01-06T14:04:00.877" Score="3" Body="&lt;p&gt;I would point out that integration is the reverse of differentiation, a pretty straightforward process. But often times the reverse of an easy algorithm is surprisingly hard!&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Solving a linear system of equations vs. plugging a value into linear equations&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Factoring vs. expanding (polynomial and prime factoring)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finding the inverse of a function at a point vs. computing the function at a point (&lt;span class=&quot;math-container&quot;&gt;$f^{-1}(a)$&lt;/span&gt; vs. &lt;span class=&quot;math-container&quot;&gt;$f(a)$&lt;/span&gt;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;some P vs. NP problems&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Even on a basic level, most people find division harder than multiplication. Radicals are harder than exponents (natural exponents at least). Differentiation and integration are no different.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="6220" LastEditDate="2020-01-06T15:25:11.943" LastActivityDate="2020-01-06T15:25:11.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17688" PostTypeId="2" ParentId="17686" CreationDate="2020-01-06T14:09:44.297" Score="31" Body="&lt;p&gt;It's pretty rare these days for anyone to actually do division by hand. Most people reach for a calculator. Given those realities, I would question whether it even makes sense that we spend such a vast amount of time teaching young children even one algorithm for division. Maybe we should postpone it, deemphasize it, or replace long division with a &lt;em&gt;slower&lt;/em&gt; or more space-consuming algorithm that's conceptually simpler and easier to learn. For example, we could teach kids to make an initial estimate of the result, multiply, find the error (which may be positive or negative), and then either decide whether their estimate is good enough or needs to be refined. Or we could teach them to do a binary search. We're no longer living in 18th century London, where human computation is the only option, so it has to be efficient, and every little kid has to be prepared for a life as a clerk or a bookkeeper who uses a quill pen.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I myself actually &quot;figured out&quot; short division at a young age from examples like 186÷3 by simply skipping steps, but didn't even know it was called this before writing this question.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is typical of the type of person who wants to understand why math works, is comfortable with numbers, and has good mental habits involving sense-making and pre-estimation. Most kids never develop these habits and attitudes. Our main goal should be to try to inculcate these in kids. Where I live (California), my kids actually did get explicit instruction in pre-estimating the results of arithmetic problems. They had worksheets where they were forced to do it. They grumbled about its being extra work, but I thought it was great.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're in the habit of pre-estimating, and also of checking your results afterward, then it's almost inevitable that you'll start to notice that some examples like 186/3 can be done without setting pencil to paper.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Out of all of the children I've helped with division, the only ones that struggle now are the ones that don't have their multiplication tables memorized.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is unfortunately a deeper problem. For many kids, being forced to memorize their multiplication tables represents the point in there lives where math goes from being interesting to being horrible drudgery. This is true even for many kids who are smart and good at math. There is the danger that you win the battle but lose the war. The kid becomes proficient at arithmetic but learns to hate math. Again, I wonder whether the problem isn't just that we spend too much time on it at too early a stage. Maybe kids could learn the basic algorithms of arithmetic while having access to a wall chart of the times tables, or a copy on their desk. After a year or two of this, they would have learned most of the facts by osmosis. Personally, I'm 54 years old, and I still do 6x8 by breaking it down into 2x3x8.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-01-06T14:54:36.060" LastActivityDate="2020-01-06T14:54:36.060" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17689" PostTypeId="2" ParentId="17686" CreationDate="2020-01-06T17:33:49.423" Score="8" Body="&lt;p&gt;If you are tutoring, it's important to value whatever algorithms work. Your frustration with the (new to me) lovely algorithm you show concerns me. It shows why each step makes sense, which is much better than the form of long division I learned.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;In fact a lot of kids find it [short division] amazing to learn.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If you show an individual a new process at just the right moment (when it really makes their work easier), it will be more exciting for them, and easier to remember. A tutor who pays attention to their students has more flexibility that way than a teacher who has to give lessons to maybe 30 kids at once.&lt;/p&gt;&#xA;&lt;p&gt;There is a sensible sequence: after kids get passably good at multiplication, they learn small division problems, then long division, done either as you show above, or stacked, showing all of the steps in a meaningful way. Once kids really understand that, then you show them the beauty of short division.&lt;/p&gt;&#xA;&lt;p&gt;To get to your main question, though, it could be that short division isn't taught, because the makers of the &amp;quot;Common Core State Standards&amp;quot; may have left it out. [I have mixed opinions about these standards. The 8 standards for mathematical practice are lovely. The individual content standards have many problems. The name seems like an oxmoron to me - if they are common across the nation, then they can't really be state standards, can they?]&lt;/p&gt;&#xA;&lt;p&gt;I looked at &lt;a href=&quot;http://www.corestandards.org/Math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this document&lt;/a&gt;, and found this for 6th grade:&#xA;CCSS.Math.Content.6.NS.B.2&#xA;Fluently divide multi-digit numbers using the standard algorithm.&lt;/p&gt;&#xA;&lt;p&gt;The implication is that there is one standard algorithm. I'd say there are about 3 standard algorithms (or more), counting all of the ways to do long division, and then the one way to do short division. This statement, along with a failed search, made me think that perhaps short division is not taught.&lt;/p&gt;&#xA;&lt;p&gt;Can any long-term elementary teachers here confirm my suspicion?&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-10-19T15:59:15.393" LastActivityDate="2023-10-19T15:59:15.393" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17690" PostTypeId="2" ParentId="17686" CreationDate="2020-01-06T20:35:12.123" Score="7" Body="&lt;p&gt;Not a teacher here, but I noticed when my kids went to school there was far less emphasis than I remember on techniques that require above average insight or intuition. I think there's more pressure these days toward making sure most students achieve a predetermined minimum performance, and less on helping high-performing students stretch their capabilities. A couple of my kids' teachers complained to me that I was teaching them things outside their curriculum. Just anecdotal evidence, but there could be a grain of truth.&lt;/p&gt;&#xA;" OwnerUserId="13286" LastActivityDate="2020-01-06T20:35:12.123" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17691" PostTypeId="2" ParentId="17686" CreationDate="2020-01-07T14:33:27.813" Score="2" Body="&lt;p&gt;Doing a quick google search for &quot;why do students struggle with division?&quot; made me realize myself why long division is difficult for a lot of students. Common reasons claimed include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The long division algorithm is long. Remembering the steps can be difficult.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The long division algorithm may not be intuitive. We can tell them that it's just &quot;how many groups of 5 go into 260&quot; but they don't understand what this actually means, especially with more digits involved.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We get lost focusing on the procedure too much. Again, they know we want groups of 5 that go into 260, but they don't understand how the steps get them there.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One site suggested trying out short division, and another suggested trying out estimation. For short division, they claimed that long division can make simple problems become overly complex, such as the &lt;span class=&quot;math-container&quot;&gt;$186\div3$&lt;/span&gt; example. The advocate for estimation said that estimation compels the student to think on their own and build their own number sense and intuition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also think that if they're taught to multiply 5 by 50 and then subtract the result from 260, they might get overwhelmed by the sizes of the numbers. If they're taught to look at the first digits only, divide, multiply, subtract, and drop down the next digit, the numbers become disconnected and lose meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Estimating helps in the first situation above. It let's students figure out how to avoid getting overwhelmed by all the numbers as well as how to handle larger divisors. It can also involve something similar to the method of adding/subtracting of bridging 10s. A nice example I found was dividing 700 by 20. By first recognizing how many times 20 went into 100 first, the problem can become more intuitive, and this is applicable in life. When pulling out money to pay for something, we often think in larger groups that are more easily managed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Short division helps in the second situation above. It avoids dropping all the digits down by instead moving the remainders up, and it simplifies the &quot;divide multiply subtract&quot; step down to make students realize that it's just smaller division problems that they can do easily. By emphasizing these steps as one whole step, the process can begin to make more sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, I also find short division intuitive as well. It gives meaning to the usage of remainders on small numbers when going to larger numbers, since we're taking out large groups, and then splitting up the remainders. A nice example I found online was splitting 103 skittles between 4 people. Have students try to do this, and then step in and suggest sharing groups of 10 skittles at a time. How many skittles are left after all the groups of 10 are shared? This is generally what all division algorithms boil down to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is short division faster? Maybe. In my experience, yes for most. A lot of students are capable of thinking in the larger step of &quot;how many times does it go in &lt;em&gt;and&lt;/em&gt; what's the remainder?&quot;, when the quotient and divisor are less than 10. If they can do that, then multiplying it out and subtracting no longer becomes necessary to explicitly write out. And it certainly saves space, but that shouldn't be the concern to begin with.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="6220" LastEditDate="2020-01-07T14:40:29.370" LastActivityDate="2020-01-07T14:40:29.370" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17693" PostTypeId="1" CreationDate="2020-01-07T16:09:08.060" Score="18" ViewCount="6015" Body="&lt;p&gt;I am a first year undergraduate student, currently in second semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So basically I learnt most of the first year stuff in high school, so I have a lot of free time in this year (currently in second semester), and thus I'm self studying math in the free time. In the first sem I tried reading some measure theory and lebesgue integration, some galois theory and some algebraic topology (I &lt;em&gt;think&lt;/em&gt; I had the prequisites -- I did half of baby Rudin, most of Artin and Topology, Willard in high school). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here are my problems:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I tend to pick up &lt;em&gt;very&lt;/em&gt; little &quot;big picture&quot; intuition while self studying. I was learning measure theory from Stein Shakarchi, and my experience - it's hard to describe in words - was like I can follow the theorems as a standalone basis, I don't have much difficulty solving the exercises but I was heavily bogged down in the details so I could neither see the big picture nor any &lt;em&gt;non-trivial&lt;/em&gt;/&lt;em&gt;deep&lt;/em&gt; connection between what's done in a page and what was done say three pages ago. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I have an &lt;em&gt;extremely&lt;/em&gt; terrible memory. My memory is so bad is that while I recall working on an abstract algebra book till Fundamental theorem of Galois Theory around in October, now I forgot what's a simple field extension (or even worse, I forgot the exact statement of Fundamental Theorem of Galois theory ! Only thing I remember is that under some conditions on &lt;span class=&quot;math-container&quot;&gt;$K \subset L$&lt;/span&gt; it gave a one to one correpsondence between intermediate fields between &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; and the subgroups of &lt;span class=&quot;math-container&quot;&gt;$Gal(L/K)$&lt;/span&gt;, and one direction is not hard but the other requires linear independence of characters)! This is extremely frustrating, when you forget what you learn one/two months ago completely.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I was a relatively good problem solver in high school (did quite well in national/regional math olympiads) but my problems solving skills don't seem to be that good in college level math. For example, I tried proving &lt;em&gt;Hilbert's &lt;strong&gt;Weak&lt;/strong&gt; Nullstellensatz&lt;/em&gt; (when I was studying it from Artin) couldn't prove it even though the proof is &quot;short&quot;. If you say it's short but tricky -- well I couldn't even prove the fact that the characters are linearly independent (I recall trying for like 30 minutes and then being frustrated since it seemed a pretty easy proof and then seeing the proof). Due to #2 and lack of mathematical maturity, I tend to give a shot at the proof of some theorem myself before seeing the solution, but this way consumes a huge amount of time and also I fail to prove the theorems when it's nontrivial in most of the cases.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So basically I still have plethora of time in this semester (just started; four months long) and I plan to self study some complex analysis, manifolds and algebraic number theory but the points above are heavily discouraging me to self study math or pursue math in general. Whenever I start to read a book I feel depressed as the pointlessness of me studying a book since (a) I will not have any big-picture understanding of it and (b) I will forget what I did after a while anyway (if I don't use it).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also I am currently the topper of my batch (which consists of one IMO medalist and many national olympians) -- but that's solely because I knew most of the course materials beforehand so I didn't need to put in too much effort outside the class to understand the stuff. As I mentioned in the points above, it's hard for me to get used to and not forget new and hard math so sometimes I feel insecure about performing extremely terribly in the upper division courses which I didn't know a priori (basically I wouldn't have any edge in those courses), so this is discouraging me more to study maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Am I making any obvious mistake while self studying ? &#xA;Is there any global change which I should make to end up with much better understanding/retention of the material ? &#xA;Would be a wise choice for me to not stay in academia and move on to CS/other applied math courses ?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: People can suggest actually sitting in the lectures of the courses I want to self study, but that's not feasible for me: While most profs are friendly towards second year people or higher auditing any course (basically sitting in the lecture w/o crediting it), they're mostly cold towards first year people sitting in the lectures. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Crossposted from Math SE]&lt;/p&gt;&#xA;" OwnerUserId="10161" LastActivityDate="2020-01-28T18:43:06.880" Title="Lack of intuition, retention while self studying" Tags="|self-learning|intuition|" AnswerCount="7" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17694" PostTypeId="2" ParentId="17686" CreationDate="2020-01-07T17:10:51.870" Score="3" Body="&lt;p&gt;Both of my sons are learning division right now (or rather, just finished the section), one in a public school using Eureka Math (in 4th grade level math), one in a Montessori (in Primary, at 1st grade age).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The public school focused on long division certainly, but (either because of the curriculum, or more likely because of the teacher, as I don't remember seeing this in Eureka's curriculum online) they definitely taught something akin to short division as well.  In fact, they taught about four different ways to do division - which turned out helpful for my son, who's extremely gifted at math but for some reason struggled with long division.  None of them were explicitly short division, certainly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One method he was taught was a kind of estimation, that seems sort of like what you showed above (and disliked) - but it was &lt;em&gt;extremely&lt;/em&gt; helpful for him.  Showing a slightly more complex example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2954 ÷ 16: &#xA;  16 x 200 = 3200 (200 16's is close to the number)&#xA;     - 320 = 2880 (20 16's less, gets you closer, now a little below)&#xA;     + 64  = 2944 (now we're right there)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;200 - 20 + 4 = 184 (r10)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He would do these in a few seconds, and instantly realize the right estimation to make, often faster than I could do the division (and I'm very gifted at mental math myself).  Again, he's highly gifted at mental math, so I don't know if this is common for other students or if other students were even taught this (as opposed to him figuring it out), but it works very well for him.  It's also not that far from short division in some ways, either, though certainly not the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Montessori of course still uses the &quot;racks and tubes&quot; method, which is basically long division with beads.  A great way to teach children I think who don't intuit long division immediately.&lt;/p&gt;&#xA;" OwnerUserId="1330" LastActivityDate="2020-01-07T17:10:51.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17695" PostTypeId="2" ParentId="17693" CreationDate="2020-01-07T17:23:24.803" Score="18" Body="&lt;p&gt;Perhaps you should seek texts that emphasize the high-level viewpoint that you are missing in the details of the more advanced texts. Three examples: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(1) Bressoud, David M. &lt;em&gt;A radical approach to Lebesgue's theory of integration&lt;/em&gt;. Cambridge University Press, 2008. &lt;a href=&quot;https://www.maa.org/press/maa-reviews/a-radical-approach-to-lebesgues-theory-of-integration&quot; rel=&quot;noreferrer&quot;&gt;MAA Review&lt;/a&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(2) Hajime Sato. &lt;em&gt;Algebraic Topology: An Intuitive Approach&lt;/em&gt;.  Transl: Kiki Hudson. Transl. of Math.&#xA;  Mono., V. 183. AMS, 1999.&#xA;  &lt;a href=&quot;https://www.maa.org/press/maa-reviews/algebraic-topology-an-intuitive-approach&quot; rel=&quot;noreferrer&quot;&gt;MAA Review&lt;/a&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(3) Ghrist, Robert W. &lt;em&gt;Elementary applied topology&lt;/em&gt;. Vol. 1. Seattle: Createspace, 2014.&#xA;  &lt;a href=&quot;https://www.ams.org/journals/bull/2020-57-01/S0273-0979-2018-01647-9/home.html&quot; rel=&quot;noreferrer&quot;&gt;AMS Review&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you grasp &quot;the big picture,&quot; then perhaps your retention will improve, as&#xA;then the concepts are pinned to a mental map rather than floating free.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-01-08T01:02:51.193" LastActivityDate="2020-01-08T01:02:51.193" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17696" PostTypeId="1" CreationDate="2020-01-07T20:51:34.750" Score="6" ViewCount="197" Body="&lt;p&gt;When matrix multiplication is introduced, it is usually introduced with an additional variable: Given two multiplicable matrices &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, one defines the product &lt;span class=&quot;math-container&quot;&gt;$C=AB$&lt;/span&gt; to be the matrix given by some formula for the coordinates of &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$c_{ij} = \dots$&lt;/span&gt;). Instead one could also define directly &lt;span class=&quot;math-container&quot;&gt;$(AB)_{ij} = \dots$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$(AB)_{ij}$&lt;/span&gt; are the entries of the product matrix. Every book I checked uses the longer notation, so is there a reason to use the long version? Which version is more beneficial for students?&lt;/p&gt;&#xA;" OwnerUserId="13299" LastEditorUserId="376" LastEditDate="2020-01-07T21:10:26.240" LastActivityDate="2020-01-08T08:37:36.843" Title="Notation in the definition of matrix multiplication" Tags="|notation|linear-algebra|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="17697" PostTypeId="2" ParentId="17693" CreationDate="2020-01-07T23:43:01.900" Score="12" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You need to pick pedagogically appropriate texts.  Not the Rudin ballbusters.  Pick ones that have explanations and were written for students with occasional imperfections in their previous knowledge.  Don't go too much off of what people say on the net is teh ultimate book to use.  SINCE IT ISN'T WORKING FOR YOU.  Find other texts that you can handle.  Obviously the tougher texts will be made more accessible if you know some of the material already and then only have to deal with the shitty pedagogy.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also, you need to have a source of problems that includes more progression (some easy problems, not just the skullcrackers, at least a section of easy, section of medium, section of hard).  As a self studier, it is important to have the answers to the problems as well.  (Ideally, worked solutions, but at a minimum answers...so you have a feedback loop.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I also question moving straight into real analysis versus doing other topics first (e.g. differential equations). At a minimum, I think you'd find it easier and thus a better choice to self study.  Could then do real analysis with the benefit of instruction.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You don't read math, you work it.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As far as memory and concepts and the like, this is related to (1) and (4).  You are doing insufficient drill problems, particularly insufficient basic ones.  Saying &quot;I got it&quot; is not sufficient, if you don't &quot;got it&quot; a few days later.  Grind the groove in.  You need to be Umbridge's pen and your mind needs to be Harry Potter's hand.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="13241" LastActivityDate="2020-01-07T23:43:01.900" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17698" PostTypeId="2" ParentId="17693" CreationDate="2020-01-08T03:49:16.767" Score="6" Body="&lt;p&gt;This is probably not a real answer, but its too long for a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last year I spent some time working on Stein and Shakarchi's Volume 2 with several advanced math majors. I actually think that text actually belongs in the list Joseph O'Rourke sets forth in his answer. In fact, they do make a large effort to explain some general themes and intuition before burying you under some mountain of inequalities etc. Now, I did not spend as much time as I really should have to deeply understand it, but I did find I could at least hack through proofs in real time with the help of the students who often see things more clearly. Certainly there are particular portions of the text where the intuition is just much more lucid. That said, it seems to me there is much more attention to intuition and motivation in Stein and Shakarchi than many other older texts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I do think you should read something a bit more pedestrian before hacking through their complex analysis)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, I think the comment by Dave Renfro is on point. In my experience, I really only understand something when I go through the trouble of rephrasing it in my own words. The process of writing it down for yourself is a good way to help remember a lot more. This is actually quite connected to &quot;guest's&quot; admonition to &quot;work&quot; the book, not just to read it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would add, I think you need to pick an example from each topic and make it your own. I've noticed, some of the greatest teachers when asked about a particular circle of topics will always come back to the same example. If you can find that one example which collects all the ideas and their interconnection then it can be very helpful in remembering the rest of the story (theorems, definitions etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Larger point, don't be too discouraged. It sounds like you're on the right path. There is always and ebb and flow to the excitement of learning. I doubt you'll find the same satisfaction outside math. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-01-08T03:49:16.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17699" PostTypeId="2" ParentId="17696" CreationDate="2020-01-08T08:37:36.843" Score="4" Body="&lt;p&gt;In class I define the matrix product by &lt;span class=&quot;math-container&quot;&gt;$(AB)_{ij} = \sum_{k = 1}^{q}A_{ik}B_{kj}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$p \times q$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$q \times r$&lt;/span&gt;. In my experience what causes the most confusion is not the subscripts indicating the component, but the meaning of the summation notation. Students do not identify the written formula with &lt;span class=&quot;math-container&quot;&gt;$(AB)_{ij} = A_{i1}B_{1j} +A_{i2}B_{2j} + \dots + A_{iq}B_{qj}$&lt;/span&gt;. Also they are confused by the difference between dummy indices such as &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, and numbers such as &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, etc. Worse still, &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is used to represent a definite number, and not a dummy index (its value is not fungible). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for defining &lt;span class=&quot;math-container&quot;&gt;$C_{ij}$&lt;/span&gt; and then defining &lt;span class=&quot;math-container&quot;&gt;$AB = C$&lt;/span&gt; (in this order, not the other) is a sort of logical fussiness that is misguided at the level where these things are taught. I am not trained in logic so I don't know how to state this carefully, but the issue is that a priori the product &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; is undefined, so referring to its components is somehow unsanitary. The point is that one is not supposed to refer to the components of something not yet specified. The solution is to write &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; for the putative undefined product, specify &lt;em&gt;its&lt;/em&gt; components and a posteriori declare that the matrix so defined is the product of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;. I suppose this is formally more correct (in some sense I am not able to verbalize), but for students who are struggling to understand the operational meaning of the sum involved in the definition, it is at best a distracting subtlety, and a worst a source of additional confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students in an introductory linear algebra class, including many of those who pass the class, never fully assimilate the definition of matrix product, nor are capable of recapitulating it completely correctly at the semester's end (those who pass can all realize it operationally, but that is a different thing). One sees this starkly if one asks them to write a program to calculate the matrix product. Writing loops that implement the sum is a problem that separates those who will pass an introductory programming class from those who won't. Many students who do fine in linear algebra struggle with writing loops to implement the matrix product.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2020-01-08T08:37:36.843" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17700" PostTypeId="1" CreationDate="2020-01-08T09:13:49.223" Score="5" ViewCount="233" Body="&lt;p&gt;Are there any more mathematical journals or websites with the &quot;problems and solutions&quot;? such as , the &lt;em&gt;American Mathematical Monthly&lt;/em&gt; and &lt;em&gt;Crux Mathematicorum&lt;/em&gt; &lt;/p&gt;&#xA;" OwnerUserId="12484" LastEditorUserId="8571" LastEditDate="2020-01-08T18:17:46.147" LastActivityDate="2020-01-08T18:17:46.147" Title="Are there any more mathematical journals or websites with the &quot;problems and solutions&quot;?" Tags="|reference-request|problem-solving|journals|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17701" PostTypeId="2" ParentId="17700" CreationDate="2020-01-08T13:42:03.030" Score="2" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/07468342.2017.1403145?journalCode=ucmj20&quot; rel=&quot;nofollow noreferrer&quot;&gt;The College Mathematics&#xA;Journal&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://pme-math.org/pme-journal-problem-department&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pi Mu Epsilon&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.mathproblems-ks.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathProblems Journal&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-01-08T13:42:03.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17702" PostTypeId="2" ParentId="17700" CreationDate="2020-01-08T13:42:09.513" Score="8" Body="&lt;p&gt;Problems columns I found (in 2009) useful to challenge undergraduates.  Some of them may no longer be current.  But even those may be interesting to consult past issues in your school library!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics Magazine   &lt;a href=&quot;https://www.maa.org/press/periodicals/mathematics-magazine&quot; rel=&quot;noreferrer&quot;&gt;https://www.maa.org/press/periodicals/mathematics-magazine&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematical Mayhem   &lt;a href=&quot;https://www.facebook.com/pg/Mathematical-Mayhem-1253467234689081/&quot; rel=&quot;noreferrer&quot;&gt;https://www.facebook.com/pg/Mathematical-Mayhem-1253467234689081/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;College Math J    &lt;a href=&quot;https://www.maa.org/press/periodicals/college-mathematics-journal/the-college-mathematics-journal&quot; rel=&quot;noreferrer&quot;&gt;https://www.maa.org/press/periodicals/college-mathematics-journal/the-college-mathematics-journal&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nieuw Archief  &lt;a href=&quot;http://www.nieuwarchief.nl/homeitems.php?onderwerp=6&quot; rel=&quot;noreferrer&quot;&gt;http://www.nieuwarchief.nl/homeitems.php?onderwerp=6&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math Spectrum   &lt;a href=&quot;http://www.appliedprobability.org/content.aspx?Group=ms&amp;amp;Page=ms&quot; rel=&quot;noreferrer&quot;&gt;http://www.appliedprobability.org/content.aspx?Group=ms&amp;amp;Page=ms&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2020-01-08T13:55:28.800" LastActivityDate="2020-01-08T13:55:28.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17705" PostTypeId="2" ParentId="17693" CreationDate="2020-01-08T15:50:27.840" Score="8" Body="&lt;p&gt;Your question sounds to me like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I have a terrible memory&lt;/li&gt;&#xA;&lt;li&gt;I can't intuitively understand some of the new things i come across&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I've seen this plenty of times, but not in maths. I've seen this in linguistics.&#xA;Languages as a whole, aren't built from logic. They just kind of happened, and we attributed some logic to them.&#xA;Due to this, any adult language learner, trying to learn a second language for the first time will have to Memorize thousands of root words before reaching a good deal of fluency.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can someone memorize so much?&#xA;The solution : &lt;strong&gt;Spaced Repetition.&lt;/strong&gt;&lt;br/&gt;&#xA;&lt;a href=&quot;https://www.fluentin3months.com/spaced-repetition/&quot; rel=&quot;noreferrer&quot;&gt;https://www.fluentin3months.com/spaced-repetition/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to the question of how often, I prefer short doses, with long breaks between them, on a daily basis.&#xA;A soft rule is an hour a day : 15 minutes in the morning, 30 minutes in the afternoon and 15 minutes in the evening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, i believe that one can only be fluent in a language when they understand it intuitively.&#xA;That, is less a question of intelligence and good memory, and more a question of practicing the above, for a length of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How much time : &lt;strong&gt;About 2-3 months.&lt;/strong&gt; &lt;br/&gt;&#xA;Personally, from studying Japanese over the past 2 years, i found words and characters become intuitive after 3(ish) months.&#xA;Between you and me, i believe that intuition and habits stem from the same thing, as both are reflexive behaviors. Here's a solid article on habit formation:&lt;br/&gt;&#xA;&lt;a href=&quot;https://jamesclear.com/new-habit&quot; rel=&quot;noreferrer&quot;&gt;https://jamesclear.com/new-habit&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to think of a new piece of knowledge like a single piece of grain: &lt;br/&gt;&#xA;Plant it in the spring, tend to it during the summer, harvest it in the autumn. &#xA;Never harvest too early and never expect to harvest without tending to the crops.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, how do language learners study new words and phrases : &lt;strong&gt;Flashcards&lt;/strong&gt;.&lt;br/&gt;&#xA;&lt;a href=&quot;https://apps.ankiweb.net/&quot; rel=&quot;noreferrer&quot;&gt;https://apps.ankiweb.net/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You need to make flashcards that are 1 to 1.&#xA;Questions need to have a single output. For example, in english, the word 'set' can have 464 meanings. &#xA;Sticking all of those into the back of one card would be terrible. However, creating 464 cards, with 464 situations where the word set is used to convey it's meaning, that'd be better.&#xA;After learning all 464, you may not know the exact linguistic description of the word, but you would end up with an intuitive understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, do take this all with a pinch of salt, as this is completely from my own experience.&lt;/p&gt;&#xA;" OwnerUserId="13304" LastActivityDate="2020-01-08T15:50:27.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17706" PostTypeId="1" AcceptedAnswerId="17956" CreationDate="2020-01-09T07:23:10.883" Score="13" ViewCount="471" Body="&lt;p&gt;I find that my community-college students are usually very hazy on the status and meaning of chained equality statements (or other relational statements). This seems like a really critical element of mathematical grammar that is almost never directly discussed in books or lectures, and not assessed or tested in any way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On that point, it occurred to me that I'm not entirely sure (and have no idea where to find) a standard prescription for how to read such statements out loud. For example, given: &lt;span class=&quot;math-container&quot;&gt;$$a = b = c$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could imagine reading this aloud as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&quot;,&lt;/li&gt;&#xA;&lt;li&gt;&quot;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&quot;,&lt;/li&gt;&#xA;&lt;li&gt;&quot;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, which is equal to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&quot;,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;or probably a variety of other constructions. What is most conventional, and what would be most clear for an audience struggling with these statements? Consider also cases that grow very long, possibly broken up over many lines on a single page. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13484/framework-for-compound-inequalities/13485&quot;&gt;Framework for Compound Inequalities&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2021-11-18T14:05:25.260" Title="How to read chained equalities out loud?" Tags="|undergraduate-education|algebra|lecturing|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17707" PostTypeId="2" ParentId="17706" CreationDate="2020-01-09T11:33:47.557" Score="0" Body="&lt;p&gt;I think how you read (and write) them depends on how they are being used. &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x^4 - y^4 = (x^2- y^2)(x^2 + y^2) = &#xA;(x - y )(x+y)(x^2 + y^2)  &#xA;$$&lt;/span&gt;&#xA;is fine but pedagogically&#xA; &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;x^4 - y^4 &amp;amp;= (x^2- y^2)(x^2 + y^2)\\&#xA;&amp;amp;=&#xA;(x - y )(x+y)(x^2 + y^2)  &#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;is better.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More important is dealing with this common misuse of chained equals when solving equations:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2x+6 = 10 = 2x = 4 = x = 2&#xA;$$&lt;/span&gt;&#xA;where every other equals sign is really a logical implication:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2x+6 = 10  &#xA;$$&lt;/span&gt;&#xA;then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; 2x = 4  &#xA;$$&lt;/span&gt;&#xA;and so&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  x = 2 .&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That common error suggests avoiding chained equalities even when they are correct.&lt;/p&gt;&#xA;" OwnerUserId="4984" LastActivityDate="2020-01-09T11:33:47.557" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17708" PostTypeId="2" ParentId="17706" CreationDate="2020-01-09T14:07:48.603" Score="8" Body="&lt;p&gt;When an inexperienced student sees &lt;span class=&quot;math-container&quot;&gt;$a=b=c$&lt;/span&gt;, I'd assume that both &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b=c$&lt;/span&gt; are clear but the transitivity that yields &lt;span class=&quot;math-container&quot;&gt;$a=c$&lt;/span&gt; might not be obvious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's why I'd focus on this hidden equality when reading it out loud, by not just reading the equation (your first option), but rather saying &quot;a, b, and c are all equal&quot; or &quot;a, b, and c are the same [number/term/equation/set...]&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For longer terms between the equal signs, I'd also emphasize that all of them are equal (borrowing from Ethan's example): &quot;The following terms are all equal: x to the fourth power minus y to the fourth power and the difference of ... times the sum of ... and finally ...&quot;&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2020-02-26T19:11:51.413" LastActivityDate="2020-02-26T19:11:51.413" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17709" PostTypeId="2" ParentId="13484" CreationDate="2020-01-09T14:18:22.277" Score="1" Body="&lt;p&gt;&quot;By default&quot;, all chained inequalities can be considered as illegal, because from an computer scientist's point of view and assuming that &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; is left-associative:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;b&amp;lt;c$&lt;/span&gt; simplifies to &lt;span class=&quot;math-container&quot;&gt;$\text{\{True or False\}} &amp;lt; c$&lt;/span&gt; which does not make sense (&quot;type error&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get around this, one defines the allowed combinations and their &quot;expanded forms&quot; as shown in Daniel's quote.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2020-01-09T14:18:22.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17710" PostTypeId="1" CreationDate="2020-01-09T15:06:51.633" Score="8" ViewCount="633" Body="&lt;p&gt;I want to know if there are any numerical bases that are notably well-suited for humans to learn and use at an elementary or grade-school level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that different numerical bases (i.e. decimal/base-ten, senary/base-six, ternary/base-three, dozenal/base-twelve) have different patterns and quirks and tricks. Many historic cultures used bases other than decimal (some have even hung around to modern times, like how we divide days into 24 hours and hours into 60 minutes), and most of them did quite well for their time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a similar question on this site, &lt;a href=&quot;https://math.stackexchange.com/questions/382774/&quot;&gt;What could be better than base 10?&lt;/a&gt;, but the question and its answers do not address my main question: ease of use for humans just starting to learn basic mathematics, while still remaining reasonably efficient for advanced mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; I'm not trying to suggest the world change to something other than the decimal system, or start teaching different bases to elementary schoolers. I'm just curious as to how other systems compare if we imagine parallel universes where each base has the same global presence, inertia, and educational/social infrastructure that is currently enjoyed by base-ten in our own universe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Primary Considerations&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ease of mental arithmetic (addition, subtraction, multiplication, division)&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In particular, prevalence of shortcuts/patterns that can be used to simplify mental calculation&lt;/li&gt;&#xA;&lt;li&gt;Multiplication tables are easy to learn, either because they're small or because they have intuitive patterns&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Compactness, in two contradicting categories that need a compromise:&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Numbers don't get long too quickly, to save time and space when writing&lt;/li&gt;&#xA;&lt;li&gt;Doesn't use too many symbols, to simplify learning&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Examples of poor compromising: Numbers stay really short in base-one-hundred-and-twenty, but it uses a ton of symbols. Base-two only uses two symbols, but numbers get really long really fast.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bonus Points&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The most common/basic fractions terminate (1/2, 1/3, 1/4)&lt;/li&gt;&#xA;&lt;li&gt;Interesting mathematical properties beyond simple arithmetic&lt;/li&gt;&#xA;&lt;li&gt;Many factors, like how dozenal divides evenly into halves, thirds, quarters, and sixths&lt;/li&gt;&#xA;&lt;li&gt;Simple conversion to/from binary, for binary computers&lt;/li&gt;&#xA;&lt;li&gt;Simple conversion to/from balanced ternary, for balance-scale math (or balanced ternary computers)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; Cross-posted to &lt;a href=&quot;https://math.stackexchange.com/questions/3493639/&quot;&gt;Mathematics Stack Exchange&lt;/a&gt; as suggested by @JohnOmielan. &lt;/p&gt;&#xA;" OwnerUserId="13314" LastEditorUserId="12255" LastEditDate="2020-04-21T14:38:09.610" LastActivityDate="2022-12-24T22:11:19.240" Title="Is there a numerical base that is in any way “better” for simple mathematical calculations than others?" Tags="|mathematical-pedagogy|primary-education|number-theory|arithmetic|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17711" PostTypeId="2" ParentId="17710" CreationDate="2020-01-09T18:11:25.813" Score="10" Body="&lt;p&gt;Clearly there is no historical data that addresses this question&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I want to know if there are any numerical bases that are notably&#xA;  well-suited for humans to learn and use at an elementary or&#xA;  grade-school level&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;since we have ten fingers and humans have learned only decimal arithmetic for everyday use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just finished four weekly sessions with fifth graders, learning arithmetic on Siff (the planet of the Six-fingered-folk) where, of course, numbers are written in (our) base 12. They invented new symbols and names for 10 and 11 and new names for 12, 144 and 1728 (10, 100 and 1000 on Siff). The game we played was that they were to learn the arithmetic operations from scratch, as if they were Sifflings, not convert back and forth to decimal. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The material progressed from counting through addition and subtraction,  multiplication and fractions, decimals and percentages, all in a new language, roughly covering the work of grades 1-5.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We rediscovered is that arithmetic is hard. It takes a lot of practice to develop what the elementary school curriculum calls &quot;number sense&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, in answer to (part of) your question. I think that everyday arithmetic would be a little bit easier in base 12 than in our base 10.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can play here: &lt;a href=&quot;https://www.cs.umb.edu/~eb/heath.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.cs.umb.edu/~eb/heath.pdf&lt;/a&gt; , &lt;a href=&quot;http://www.dozenal.org/&quot; rel=&quot;noreferrer&quot;&gt;http://www.dozenal.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4984" LastActivityDate="2020-01-09T18:11:25.813" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17713" PostTypeId="2" ParentId="17693" CreationDate="2020-01-09T22:36:27.250" Score="2" Body="&lt;p&gt;For (3), it happens a lot for me. If the theorem is nontrivial, maybe you should try to spot the nontrivial part and read only that part and try to prove the rest. Don't feel too guilty for not solving the problem, just view this as picking up some new tricks to your bag. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For (2), I agreed with most the answers here. The method I used is taking notes. &#xA;When the terminology comes up again, try to see if you remember it before &#xA;looking back to the notes. The last thing is that do exercises so that you&#xA;can repeat previous thing a lot of time. If you forget, you go and look back. &#xA;Again, don't feel bad because you don't remember (as I did before), because it will happen a lot. The main thing is that when you reread it, you need &#xA;to check for yourself that your learning speed has to be faster. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For (1), I don't have much experience in this. The method I want to use&#xA;in the future is that: try to spot the main theorems in the section you are reading, look up the internet ...&lt;/p&gt;&#xA;" OwnerUserId="13317" LastActivityDate="2020-01-09T22:36:27.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17715" PostTypeId="2" ParentId="17693" CreationDate="2020-01-10T03:20:15.230" Score="0" Body="&lt;p&gt;Start meditating. Your memory will improve, along with your energy, mood, concentration, etc. I use the Headspace app. A growing body of research suggests this is due to the fact that meditation reduces inflammation and furthermore that inflammation ay well be a main cause of numerous mental illness. Other things that reduce inflammation are hydration and exercise to name a couple. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Might not be the answer you were looking for, but for me and for many people I know who are now converts, meditation has been nothing short of life changing. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I have two science degrees and a healthy scepticism of pseudo-scientific remedies, but this  works. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13318" LastActivityDate="2020-01-10T03:20:15.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17716" PostTypeId="2" ParentId="13484" CreationDate="2020-01-14T01:51:29.050" Score="3" Body="&lt;p&gt;A statement such as &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;b&amp;lt;c$&lt;/span&gt; could be useful if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; are not constants. I would interpret it as &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;b\ \mathrm{and}\ b&amp;lt;c$&lt;/span&gt;, i.e. &lt;span class=&quot;math-container&quot;&gt;$b&amp;lt;\min(a, c)$&lt;/span&gt;. In general, I'd interpret a statement of the form &lt;span class=&quot;math-container&quot;&gt;$a★b‡c$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$a★b\ \mathrm{and}\ b‡c$&lt;/span&gt;. I'm pretty sure this is also how Python does it, for example.&lt;/p&gt;&#xA;" OwnerUserId="10665" LastEditorUserId="10665" LastEditDate="2020-05-14T02:08:32.127" LastActivityDate="2020-05-14T02:08:32.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17717" PostTypeId="1" CreationDate="2020-01-14T07:40:15.327" Score="1" ViewCount="181" Body="&lt;p&gt;Awhile back I was very weak with my trigonometry so I came to this site asking for help, and it turned out the few answers I got made a huge difference. I excelled at the trigonometry section in my precalc course and now I am enrolled in calculus 1. In fact, during my break I ended up teaching myself a lot of differential and integral calculus techniques (not much theory though, which is where my current class will make the difference). After this exposure I'm more confident that I am capable of getting my degree, but I have one obstacle - credit requirements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I transferred from a community college with no intentions to ever major in math and even no intentions to go to university, so I did not take many upper division courses. Having transferred 63 credits, it certainly makes my studies much easier and less stressful (in fact at this point I'm no longer required to take any general studies so only math courses are required of me), but unfortunately I am worried about the upper division credits required for my math major. I've completed 72 credits, with 10 more in progress (3 of which are upper division). However, a whopping 42 upper division credits is required of me after this semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my case, I want to avoid taking classes that are not purely-math related because I know I will do only best dedicating my entire focus on my major-related classes. I also hate most other classes and have to force myself to endure them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From this point on there are 21 credits worth of upper division math courses that are either already required or critical for a math major to take that I intend to take, this leaves me with another 21 credits worth of upper division courses of any subject. If I'm interpreting my major map right, I would have to take up to 5 to 6 more math courses than the average math major at my uni. This might not sound like a lot but I suspect its a huge deal at the upper-division levels (ring theory, abstract algebra, complex analysis, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What should I do? Should I suck it up and take unrelated upper division courses to avoid having so many advanced math classes - or is going all out on a pure math schedule not a bad idea? &lt;/p&gt;&#xA;" OwnerUserId="12501" LastActivityDate="2020-01-14T21:41:40.997" Title="Math undergrad courses" Tags="|undergraduate-education|curriculum|struggling-students|" AnswerCount="1" CommentCount="18" ClosedDate="2020-01-18T21:24:06.570" ContentLicense="CC BY-SA 4.0" />
  <row Id="17718" PostTypeId="1" CreationDate="2020-01-14T15:43:56.353" Score="2" ViewCount="146" Body="&lt;p&gt;I am now teaching Calculus of several variables this semester. In apllications of integrals, the problem of finding the work done in moving an object under a force &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is one of the most common problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's assume that an object is moving from a point A to a point B (on a horizontal line) under a constant force &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; pointing an angle &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; (w.r.t. the horizontal line). Then in most textbooks (Calculus), the work done in this case &lt;strong&gt;is defined by&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$W=|F||AB|\cos(\alpha)$&lt;/span&gt;. &lt;strong&gt;By accepting this definition,&lt;/strong&gt; then one can meaningfully find a way to evaluate the work done in moving an object along a curve (in 2D or 3D) and under a non-constant force (this leads to line integrals).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When teaching this, the most uncomfortable fact is &quot;why do we have such a definition of the work&quot; in the first place. Since the word &quot;work&quot; should say something, it might give  confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question&lt;/strong&gt;: how do we give intuition to the definition of work above? For example, how do we give an explanation for the fact that the work is negative if &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is greater than 90 degrees?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks so much for any hints. &lt;/p&gt;&#xA;" OwnerUserId="12801" LastActivityDate="2020-01-15T02:06:28.923" Title="How to teach integrals motivated by the work done in moving an object?" Tags="|mathematical-pedagogy|calculus|physical-sciences|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17719" PostTypeId="2" ParentId="17718" CreationDate="2020-01-14T17:23:22.503" Score="2" Body="&lt;p&gt;The best way to illustrate the sense of &lt;span class=&quot;math-container&quot;&gt;$W=|F||AB|\cos(\alpha)$&lt;/span&gt; is to show that it makes sense that each of the factors is proportional to the work done.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If you double the amount of force to an object to move it a certain distance, it doubles the amount of work needed.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you double the distance to move an object by applying a certain force, it doubles the amount of work needed.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you apply the force in the same direction as the direction the object moves, you are being completely efficient in your work.  The more your work is applied downward, the more effort you are wasting pushing the object into the ground instead of in the direction you want.  At the end, if you are pointing straight down, you are doing no work no matter how much force you expend.  And you if your force is in the opposite direction, you are doing &quot;negative work&quot; in the sense of moving the object further from your goal.  These intuitive facts turn out to be modeled precisely by the cosine of the angle between the direction of your work and the direction the object is to move.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-01-14T17:23:22.503" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17720" PostTypeId="2" ParentId="17717" CreationDate="2020-01-14T21:04:32.330" Score="5" Body="&lt;p&gt;A lot depends on what you plan to use the bachelor's for.  If you plan to go to grad school, than go crazy and take a bunch of hard upper div classes.  I really caution AGAINST grad school in pure math though (for you, based on your evident aptitude level).  Even for stars, this can be quite a daunting pyramid--how many of the people here are tenured profs at R1 universities?  I think it will be even worse for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you plan to use the bachelor's for just &quot;having a degree&quot; (e.g. to get sales manager versus salesman, to go &quot;mustang&quot; in the military, etc.), than it matters little, provided you pass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also possible that you may do something intermediate to the above.  Go work for the NSA or an insurance company or the like.  Or want to keep your options open.  There is also high school teaching--but having a bachelor's in math is plenty.  You don't need to have the hardest one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have not already done so, I strongly recommend to take a calculus-based (check on this) general physics class.  Don't need to go any further.  But this is really key application area you should know if you ever teach math in high school or college.  Drives me crazy to see sooper smart peeps here that don't know this (major reasons for them having their jobs!).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Edit:  see comments below on chemistry...can be unpleasant.]  Similarly, but not as critical would be to do a first year course in chemistry.  There are a lot of applications of algebra in chemistry (stoichiometry, equilibrium, baby thermo, kinetics).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some other classes outside of math that may be useful (but check the book/teacher to make sure you will enjoy it): economics and technical writing.  If you could pick up a decent accounting class that would be useful also.  The vast majority of college educated people encounter money things on the job, even if they are not financial analysts or in the finance department.  It's good to have had some exposure to basics of balance sheet and income statement (and ideally to NPV or &quot;engineering economics&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Within math (if the former suggestions just make you lose heart), consider some applied topics: design of experiments, operations research, etc.  Yeah, it's not the pure math you wanted but at least closer than my previous suggestions.  And won't be so wicked hard as taking a slew of upper division or baby grad school math topics simultaneously.  Also more useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also make sure to take a class in ODEs and a class in PDEs.  Shocks me to find people here who are math Ph.D.s and never took this stuff.  The normal, calculational emphasis, class is fine.  You don't need to take the most theoretical one (bad idea for first viewing).  I would also avoid classes with too heavy an emphasis on computers, numeric estimation, or linear algebra.  (Nothing wrong with any of those three...it's just that &quot;enriched&quot; classes like this usually seem harder and have more issues than just the standard fifty to hundred year old stereotypical class.)  Also stay away from any diffyQs class with too much emphasis on modeling or applications. (Little bit is fine, but learn the stuff in basic equations first, versus with a lot of physical interpretation.  Would you want to take a high school algebra class that started with word problems versus with the equations?)&lt;/p&gt;&#xA;" OwnerUserId="13241" LastEditorUserId="13241" LastEditDate="2020-01-14T21:41:40.997" LastActivityDate="2020-01-14T21:41:40.997" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17721" PostTypeId="2" ParentId="17718" CreationDate="2020-01-14T21:34:51.483" Score="0" Body="&lt;p&gt;I would start with the scalar.  Work = force times distance.  Starting with the scalar makes things easier, more intuitive.  (Not a math logic point, just a psychological one.)  You can then easily generalize, perhaps without showing a derivation and say &quot;well this is what work is in these trickier situations (vectors, non constant force, more dimensions, etc.)  At least they see the similarity of the look of the equation to what they know from high school basic physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as why W=F*d, you can devolve to &quot;this is the definition&quot;.  Or probably make some argument of how work is just another name for energy.  (The simplest and most intuitive example of work done is moving an object up vertically in a gravity field.  The work done is equal to the potential energy created.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that said, I would really try to avoid getting too tied up in the physics of the thing, especially when you are a little uncomfortable or unfamiliar with it yourself.  I suspect a lot of what bothers you, here, is your own questions, not those of the students.  But you are going to transfer that to them and bog the class down with a bunch of physics talk when they really ought to be practicing calculations. (This is frequent the commenters/questioners here want to explain to the class something that really bugs them from not being explained versus something that bugs the class from the omission.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember, they will have other classes (most will) that go over this from more of a physics background and less of a math one.  For a student taking both (at or near same time) it can be useful to see the same concept from two slants.  One more physical and one more calculational.  The physics class makes sense when you say &quot;I've seen that math before, can concentrate on physical meaning of what is happening&quot; (and conversely in math class).  But I would try to keep math class as more calcuational versus deep digressions to physics.  This supports the physics class better.&lt;/p&gt;&#xA;" OwnerUserId="13241" LastActivityDate="2020-01-14T21:34:51.483" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17722" PostTypeId="2" ParentId="17718" CreationDate="2020-01-15T02:06:28.923" Score="2" Body="&lt;p&gt;I mainly teach physics, only a little math on the side. If I was teaching this math topic and wanted to spend 3 minutes giving the appropriate physical motivation, I would do something like the following.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is something called energy. It comes in various forms. Food has energy, which is what we're talking about when we talk about calories. Heat, sound, and light all have energy. When a material object like this bowling ball is moving, then it has energy because of its motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I have this bowling ball, which I borrowed from the physics department's stockroom, and I'm going to roll it down the aisle of the classroom. Please pick up your legs and backpacks in case I don't do it perfectly straight. The ball started from rest, and then I made a force on it, which transferred energy into it. The amount of this energy transfer is called work. When the force is constant, we have &lt;span class=&quot;math-container&quot;&gt;$W=Fd$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, Sally, the ball is near your desk. Could you please roll it back up the aisle toward me at a speed similar to the one I used? Now I slow the ball to a stop with my foot. This time the work I did was negative: I took energy &lt;em&gt;out&lt;/em&gt; of the ball. This makes sense because if we pick some direction to call positive, then the force and the distance will have opposite signs, and their product will be negative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now when the force is not constant, we can't just multiply the distance by &quot;the&quot; force. This is a pattern that you'll see over and over again in applications of calculus. When we need to generalize a relationship involving multiplication to the case where one of the factors is varying, we do that using an integral.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-01-15T02:06:28.923" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17723" PostTypeId="1" AcceptedAnswerId="17726" CreationDate="2020-01-15T11:21:45.960" Score="9" ViewCount="4162" Body="&lt;p&gt;The most common way I have seen Euler's formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;re^{i\theta} = r(\cos\theta+i\sin\theta)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;introduced in a classroom environment is to substitute &lt;span class=&quot;math-container&quot;&gt;$i\theta$&lt;/span&gt; into the series expansion of the exponential function, and then notice that this can be rearranged into the sum of the series expansions for &lt;span class=&quot;math-container&quot;&gt;$\cos\theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$i\sin\theta$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this requires that the students are familiar with the series expansion of these three functions, which is often taught when discussing Taylor series. This requires an understanding of derivatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen other ways to introduce Euler's formula that rely on differential equations, however this also requires an understanding of derivatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm searching for a way to introduce Euler's formula, that does not require any calculus. The students are on an engineering course, and will have only seen algebraic manipulation, functions (including trigonometric and exponential functions), linear algebra/matrices and have just been introduced to complex numbers. They will see a fair amount of calculus (including Taylor's series) later in the course, but I would like to avoid saying &quot;Please just accept this for now, and we will revisit it later&quot; if possible.&lt;/p&gt;&#xA;" OwnerUserId="10057" LastActivityDate="2022-08-13T10:26:35.277" Title="Are there any proofs of Euler's Formula that do not rely on calculus?" Tags="|calculus|complex-numbers|" AnswerCount="8" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="17724" PostTypeId="2" ParentId="17723" CreationDate="2020-01-15T13:21:22.120" Score="9" Body="&lt;p&gt;There is a proof of de Moivre's formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(\cos\theta + i \sin\theta) ^n = \cos(n\theta) + i \sin(n\theta), \qquad n \in \mathbb Z&#xA;$$&lt;/span&gt;&#xA;by induction (for &lt;span class=&quot;math-container&quot;&gt;$n &amp;gt; 0$&lt;/span&gt;) and symmetry.  Maybe that is the best we can do without calculus.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some textbooks (not assuming calculus) use a notation &lt;span class=&quot;math-container&quot;&gt;$\mathrm{cis}\;\theta$&lt;/span&gt; meaning &lt;span class=&quot;math-container&quot;&gt;$\cos\theta + i\sin\theta$&lt;/span&gt; and do all the calculations with it.  The addition formula for &lt;span class=&quot;math-container&quot;&gt;$\mathrm{cis}\; \theta$&lt;/span&gt; combines the two addition formulas for &lt;span class=&quot;math-container&quot;&gt;$\cos\theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin \theta$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2020-01-15T13:27:32.460" LastActivityDate="2020-01-15T13:27:32.460" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17725" PostTypeId="2" ParentId="17723" CreationDate="2020-01-15T13:25:22.207" Score="2" Body="&lt;p&gt;I believe Euler's identity can be reached via &lt;a href=&quot;https://en.wikipedia.org/wiki/De_Moivre%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Moivre's Formula&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\cos(nx)+i\sin(nx)=\left( \cos(x)+i\sin(x)\right)^n$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I am not finding a clear exposition of this route, as often Euler's identity is used to prove De Moivre's, whereas here we're seeking the reverse.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/De_Moivre%27s_formula#Proof_by_induction&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia says&lt;/a&gt;, &quot;The truth of de Moivre's theorem can be established by using mathematical induction.&quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-01-15T13:45:05.787" LastActivityDate="2020-01-15T13:45:05.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17726" PostTypeId="2" ParentId="17723" CreationDate="2020-01-15T14:19:11.347" Score="21" Body="&lt;p&gt;Can you define &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; for real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; without calculus?  You need to take a limit, I think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you allow &quot;informal limit reasoning&quot;, then there is such an argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to define &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; for real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is by the limit &lt;span class=&quot;math-container&quot;&gt;$(1+\frac{x}{n})^n$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; grows without bound.  This is often motivated with a compound interest story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Substituting &lt;span class=&quot;math-container&quot;&gt;$i\theta$&lt;/span&gt; in for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; into this definition we have the limit of &lt;span class=&quot;math-container&quot;&gt;$(1+\frac{i\theta}{n})^n$&lt;/span&gt; as n grows without bound.  &lt;span class=&quot;math-container&quot;&gt;$1+\frac{i\theta}{n}$&lt;/span&gt; is geometrically very close to &lt;span class=&quot;math-container&quot;&gt;$\cos(\frac{\theta}{n})+i\sin(\frac{\theta}{n})$&lt;/span&gt; (wishy washy, but can be made precise.  This precision is &quot;differential calculus&quot;).  By De Moivre's formula, &lt;span class=&quot;math-container&quot;&gt;$e^{i\theta}$&lt;/span&gt; should be very close to &lt;span class=&quot;math-container&quot;&gt;$\cos(\theta)+i\sin(\theta)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="511" LastEditDate="2020-01-15T14:23:24.653" LastActivityDate="2020-01-15T14:23:24.653" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17727" PostTypeId="1" AcceptedAnswerId="17730" CreationDate="2020-01-15T15:35:42.093" Score="8" ViewCount="1418" Body="&lt;p&gt;I am a math teacher and I have been for a decade now. One of the foundations of my philosophical approach to teaching has to do with Synthesis.  For the purposes of this query, I consider Synthesis to be defined as &lt;em&gt;The process of putting discrete ideas together to form a new idea&lt;/em&gt;.  Consider that you could teach someone to dig holes, and you could teach that same person to pour concrete.  Then, they might employ synthesis to put those two skills together; they dig a hole and then pour in concrete to make the foundation for a pylon.  This is technically different from either skill that you taught them, but it represents a combination of the skills in question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a math teacher, I teach students about fundamental skills and concepts. The domain of a relation represented as a graph is the set of the x-coordinates of the points on the graph. You can translate a graph two units to the right using an algebraic form.  To synthesize these two fundamentals, I might offer an equation of a transformed graph and ask about the domain of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, here's the rub! I ask this, but I've never asked them to do this specific task in class. That is, I have never given them: &quot;The function f has domain (2,4). The function g is defined by g(x)=f(x-2). What is the domain of g?&quot; as an in-class problem. But, I have taught what domain means and what transformations are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it okay for me to ask this sort of question on a graded assessment?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that the answer is a resounding 'Yes!' but so many people have questioned my stance on this - people whom I respect - that now I'm wondering if I'm just being stubborn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be clear:&#xA;I teach skill/concept A.&#xA;I teach skill/concept B.&#xA;Skill C requires the use of both A and B in concert with one another.&#xA;I ask a question requiring skill C on a quiz but have never asked a skill C question to these students before.&#xA;Am I being unfair?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My argument typically focuses around the idea that sufficiently advanced mathematics requires students to write proofs, and those proofs are typically unique to that specific theorem (even though many structural elements will be similar to other proofs, there will be details that are different). So, advanced mathematics &lt;em&gt;requires&lt;/em&gt; its students to assemble concepts from different places to answer questions in an open-ended format.  Hence, I think it should be a necessary part of any assessment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But again, am I just being stubborn?&lt;/p&gt;&#xA;" OwnerUserId="8275" LastActivityDate="2020-01-16T14:13:55.133" Title="Acceptability of creative questions in assessments" Tags="|secondary-education|exams|general-pedagogy|test-design|critical-thinking|" AnswerCount="4" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="17728" PostTypeId="2" ParentId="17727" CreationDate="2020-01-15T16:23:33.653" Score="9" Body="&lt;p&gt;My advice is to minimize the amount of such synthesis required.  Don't make it a large fraction of your tests, if at all.  Teach the students the methods you expect them to display on the exam. Not something requiring some spark of creativity.  Program for success.  Creativity is tough in general and even tougher under test conditions.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you push too much creativity, you're basically testing IQ more and more and subject matter mastery less and less.  (I acknowledge there is always some mixture of the two, talking relative amounts.)  The smart kids have enough advantages as it is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not to say there is no value in synthesis or connections.  And I quite enjoyed the piling example. But it may be more appropriate to derivations, lesson plans, etc.  (Consider, would you really expect kids to derive the next lesson, not yet covered/worked, under test conditions?)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe a way to indulge yourself is to restrict these synthesis questions to a single &quot;extra credit&quot; question on your tests.  Keeps the smart kids interested.  And psychologically will be more acceptable to all the students (that it is &quot;extra&quot;, not core).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:  Given you are teaching pre-calc (inferred) with excellent students, some degree of puzzley synthesis may be allowable or even enjoyable.  But I would still caution you to not overdo it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are really going to be using a lot of &quot;never seen it before&quot; questions on tests, than you need to spend the time to train the students with a lot of puzzley drill so they exercise that faculty.  Probably also need to make sure they have very solid knowledge of the different aspects of all the topics also (to support handling later synthesis).  I just wonder if you have time for that sort of thing or if some of the basic calculational method familiarity will then suffer by not being covered enough.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider also that sometimes acceleration may be more beneficial than enrichment.  This is a somewhat contrarian stance.  But I got WAY more benefit in a STEM career from having a very solid AP Calculus BC background, than all the tricky stuff in &quot;functions&quot; (pre-calc) class.  Of course, it's a balance.  But just at least consider this concern.  &lt;/p&gt;&#xA;" OwnerUserId="13328" LastEditorUserId="13328" LastEditDate="2020-01-15T16:38:56.603" LastActivityDate="2020-01-15T16:38:56.603" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17729" PostTypeId="2" ParentId="17727" CreationDate="2020-01-15T17:42:35.970" Score="10" Body="&lt;p&gt;I would frame this issue a little differently than you have.  I think it's unreasonable, at least in the context of courses which aren't well into a math major, to ask students to do something they have not been taught to do.  That is, the problems on an assessment should be the same as problems they've seen already.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The catch is that &quot;the same&quot; is actually a really ambiguous term, and it depends on how abstractly you think about the steps - and, more importantly, how abstractly it's appropriate to ask your students to think about them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make that concrete, suppose that in class we did one of those calculus optimization problems where someone's building a fence out of a couple materials with different prices, and you want to maximize area given a certain budget.  &quot;The same&quot; task could mean:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;literally the same problem,&lt;/li&gt;&#xA;&lt;li&gt;the same words, but the numbers have changed,&lt;/li&gt;&#xA;&lt;li&gt;the same setup, but this time we want to minimize price with a fixed area,&lt;/li&gt;&#xA;&lt;li&gt;the same setup, but the geometry of the fence is a little different,&lt;/li&gt;&#xA;&lt;li&gt;a similar setup, but this time the geometry of the fence is changed in a way that means the algebra needed to solve the problem looks different,&lt;/li&gt;&#xA;&lt;li&gt;a different word problem that leads to similar equations,&lt;/li&gt;&#xA;&lt;li&gt;a different word problem that leads to unrelated equations (i.e. any optimization word problem).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;An assessment is typically going to have some range of levels of abstraction, and what that range should be depends on the specifics of the class (and maybe the specific assessment - I might ask for more flexibility in approaching a subject on a unit test than on a quiz in the middle or on a final exam).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Getting the top of that range right is difficult, especially because it's easy for us, as specialists, to underestimate how unfamiliar a problem will look to our students.  When we aim too high, students start to see math as an endless series of inexplicable magic tricks they could never reproduce.  When we aim too low, we end up with cookbook math, where students stress themselves out memorizing fifty algorithms and then plow their way through the exam with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To bring all that around to your specific question, I suspect that you're currently aiming a little too high, and you're struggling with that because the only alternative you're considering is too low.  At the level of class you're describing, I think you probably should be asking them to solve problems where you haven't shown them how to do that exact task in class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you want them to synthesize ideas from class, you probably need to have &lt;em&gt;taught them how to synthesize those ideas&lt;/em&gt; in a pretty explicit way: if you want them to solve a problem by combining A and B, you probably need to have done a problem in class where they needed to combine A and D, and also a problem where they needed to combine E and B.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2020-01-16T14:13:55.133" LastActivityDate="2020-01-16T14:13:55.133" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17730" PostTypeId="2" ParentId="17727" CreationDate="2020-01-15T18:25:32.823" Score="11" Body="&lt;p&gt;I think the given example is highly appropriate.  You cannot cover every possible combination of ideas in class.  Students display understanding of a concept (rather than &quot;recipe following&quot;) by showing the ability to adapt at least a little bit to novel conditions.  I think the problem you gave is a great homework problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally like homework to be a place for students to grow their understanding, which means engaging with novel and creative ideas.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exams, with their time pressure, are not appropriate places for such questions however.  I like to think of exams as &quot;basic content mastery checks&quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-01-15T18:25:32.823" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17732" PostTypeId="2" ParentId="17723" CreationDate="2020-01-15T22:07:45.503" Score="4" Body="&lt;p&gt;You could show that, if &lt;span class=&quot;math-container&quot;&gt;$A^{ix} = f(x) + i g(y)$&lt;/span&gt; is defined in a way that it satisfies &lt;span class=&quot;math-container&quot;&gt;$A^{ix+iy} = A^{ix} A^{iy}$&lt;/span&gt; and other standard properties of the exponential function, then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; must satisfy the addition and subtraction formulas for cosine and sine, so they have to be &lt;span class=&quot;math-container&quot;&gt;$\cos(kx)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin(kx)$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To preserve properties of exponential when extending to imaginary exponents:&lt;br&gt;&#xA;- &lt;span class=&quot;math-container&quot;&gt;$A^0i = 1$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$f(0) + ig(0) == 1 + 0i$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$f(0) = 1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g(0) = 0$&lt;/span&gt;  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$A^{x+y} = A^x * A^y$&lt;/span&gt; ==&gt;&lt;br&gt;&#xA;-- &lt;span class=&quot;math-container&quot;&gt;$A^{i(x+y)} = A^{ix+iy} = A^{ix} * A^{iy}$&lt;/span&gt;&lt;br&gt;&#xA;-- &lt;span class=&quot;math-container&quot;&gt;$f(x+y) + ig(x+y) = (f(x) + ig(x))*(f(y)+ig(y))$&lt;/span&gt;&lt;br&gt;&#xA;-- &lt;span class=&quot;math-container&quot;&gt;$f(x+y) + ig(x+y) = f(x)f(y) + if(x)g(y) + ig(x)f(y) - g(x)g(y)$&lt;/span&gt;&lt;br&gt;&#xA;-- &lt;span class=&quot;math-container&quot;&gt;$f(x+y) + ig(x+y) = (f(x)f(y) - g(x)g(y)) + i(f(x)g(y) + g(x)f(y))$&lt;/span&gt;&lt;br&gt;&#xA;-- &lt;span class=&quot;math-container&quot;&gt;$f(x+y) == f(x)f(y) - g(x)g(y)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x+y) = f(x)g(y) + g(x)f(y)$&lt;/span&gt;  &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also, the distinction between &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-i$&lt;/span&gt; is arbitrary, so when extending a real-valued function &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; to a complex-valued one, it is natural that &lt;span class=&quot;math-container&quot;&gt;$h(\overline{z}) = \overline{h(z)}$&lt;/span&gt;  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\overline{A^{ix}} = A^\overline{ix}$&lt;/span&gt;&lt;br&gt;&#xA;-- &lt;span class=&quot;math-container&quot;&gt;$f(x) - ig(x) = A^{-ix}$&lt;/span&gt;&lt;br&gt;&#xA;-- &lt;span class=&quot;math-container&quot;&gt;$f(x) - ig(x) = f(-x) + ig(-x)$&lt;/span&gt;&lt;br&gt;&#xA;-- &lt;span class=&quot;math-container&quot;&gt;$f(x) == f(-x)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g(x) = -g(-x)$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x + (-x)) = f(0)$&lt;/span&gt;&lt;br&gt;&#xA;-- &lt;span class=&quot;math-container&quot;&gt;$f(x)f(-x) - g(x)g(-x) = 1$&lt;/span&gt;&lt;br&gt;&#xA;-- &lt;span class=&quot;math-container&quot;&gt;$f(x)f(x) + g(x)g(x) = 1$&lt;/span&gt;  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But showing that for &lt;span class=&quot;math-container&quot;&gt;$A = e$&lt;/span&gt;, the functions taking radian measure work probably isn't possible.  (As others mentioned, defining &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; without calculus is problematic.)&lt;/p&gt;&#xA;" OwnerUserId="13331" LastEditorUserId="13331" LastEditDate="2020-01-22T13:48:27.067" LastActivityDate="2020-01-22T13:48:27.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17733" PostTypeId="2" ParentId="16668" CreationDate="2020-01-16T05:35:17.013" Score="4" Body="&lt;p&gt;Because &lt;strong&gt;integrals are &lt;em&gt;not&lt;/em&gt; something to be &quot;solved&quot; by default, but a tool of mathematical expression in their own right.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you express &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; as something else? Probably nothing simpler. You can approximate it, but that's about it. Likewise, why should you expect to be able to express some arbitrary integral, say&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{0}^{1} e^{\sin x}\ dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;as something simple? But of course, generating arbitrarily precise approximations to either is trivial, because it basically amounts in both cases to cutting off the limiting definitions at a finite number of steps, or finding some more efficient procedure that is equivalent thereto, and hence is systematic. In a sense, numerical integration is already there within the definition of integration itself, as a limit of the Riemann sum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real problem this relates to is a) not emphasizing enough the distinction between a mathematical object and a &lt;em&gt;representation&lt;/em&gt; of that object, and b) thinking that there is &quot;one true and correct&quot; representation. When you &quot;solve&quot; an integral, what you're really doing is trying to &lt;em&gt;create a representation&lt;/em&gt; of the value (or function, if it's an indefinite integral) it denotes, with something that is either not itself an integral, or doesn't have to be one, at least (already one can suggest that the natural logarithm is, in effect, the first &quot;non-elementary&quot; integral, as it is the first one to take one outside the field operations of the real numbers, but we call it elementary because tradition).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But not only may such representations not exist, but even if they do, they might not be the &lt;em&gt;best&lt;/em&gt; representations to use in a given situation. For example, this indefinite integral&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int \sqrt[3]{\tan x}\ dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;famously starred in a YouTube video by a maths professor going under the handle &quot;blackpenredpen&quot;, because while it can be &quot;solved&quot; symbolically with &quot;elementary functions&quot;, the expression is a long, crazy, awful mess, so bad that if you encountered it in a problem somewhere, I would actually think it preferable to simply leave it as an integral, with a small note that the integral has an elementary symbolic representation, but it is far too cumbersome to use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, this extends to the whole idea that you should always &quot;simplify&quot; everything. Actually - no. What you should really seek is to put things into a form that makes their intent the most clear, and/or that makes them the most pleasing aesthetically. In many cases, that may involve simplification, but not all. Same for reducing fractions, etc. - I've seen &lt;em&gt;tons&lt;/em&gt; of times when that an important pattern in some sequence of fractions would be seen far more clearly if one did &lt;em&gt;not&lt;/em&gt; reduce them. And likewise, I would &lt;em&gt;not&lt;/em&gt; call the expression coming from the integral immediately above a &quot;simplification&quot; - more like a &quot;complication&quot; to me. Yes, it gets rid of the integral, but it's very difficult to decipher any useful meaning from it. Moreover, one can in fact also argue it doesn't even do that: tangent, and other trigonometric functions, ultimately need some kind of integral-like process to define them, so you're in effect &quot;massaging&quot; some integrals into other integrals, that are worse and more verbose and not at all clean. Same with my very first example. There are ways to express it using various fascinating and obscure mathematical functions, but in terms of sheer elegance, I think simply leaving it as the integral actually is the best justice. There's no need, as you in effect have hinted at, for such expressions for doing calculations, when powerful numerical integrators exist already.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once you stop seeing integrals as things &lt;em&gt;needing&lt;/em&gt; to be &quot;solved&quot;, any more than &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; &quot;needs&quot; some kind of &quot;solving&quot; to be done to it when you encounter it, then questions like these reveal themselves to be rather unnecessary, or their answer in fact quite trivial: Integrals are really a &lt;em&gt;much, *much* more powerful&lt;/em&gt; mathematical operation than the ones you have had on hand before you are introduced to them, and they should be thought of as opening up whole &lt;em&gt;new&lt;/em&gt; vistas of expressibility of things you could not express before, and &lt;em&gt;that&lt;/em&gt; is the meaning of &quot;why they're hard&quot;. Reducing them to things for &quot;solving&quot; is to, in a way, insult this power, and suggest they are an obstacle, instead of a facility. Integration is not hard. Trying to tame the power it represents, and shrink it down to what you call &quot;familiar&quot;, is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And so terms like &quot;Exact forms&quot;, &quot;closed forms&quot;, etc. are things that I think should be banished, too, for the same reason. There is nothing &quot;inexact&quot; about any of the above expressions. We should instead talk of various kinds of forms - representation by integrals, by arithmetic functions, by trig/log functions, by codified integral-defined functions (&lt;span class=&quot;math-container&quot;&gt;$\mathrm{erf}$&lt;/span&gt;, etc.), by infinite sums, and so forth and not get hung up on any one in particular as &quot;better&quot; or &quot;worse&quot; but to get a good feel for the pros, cons and utility of each. Maths is really a language to express, not a puzzle to crack. Puzzles are what you express &lt;em&gt;in&lt;/em&gt; maths, just as you can express them in human language. And ideally, just as we should strive for clarity in expressing ourselves in our human languages, we should do the same for our mathematical language, as well.&lt;/p&gt;&#xA;" OwnerUserId="7650" LastEditorUserId="7650" LastEditDate="2020-01-17T04:04:53.700" LastActivityDate="2020-01-17T04:04:53.700" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17734" PostTypeId="2" ParentId="17727" CreationDate="2020-01-16T11:22:43.240" Score="2" Body="&lt;p&gt;A compromise approach could be to give a problem (with parts), as you might on a guided worksheet.  Such as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has domain &lt;span class=&quot;math-container&quot;&gt;$(2,4)$&lt;/span&gt;.  We define &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$g(x) = f(x-2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) Is &lt;span class=&quot;math-container&quot;&gt;$g(3)$&lt;/span&gt; defined?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) Is &lt;span class=&quot;math-container&quot;&gt;$g(5)$&lt;/span&gt; defined?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) What is the domain of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might even give a follow up problem (perhaps as extra credit) along the lines of:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has domain &lt;span class=&quot;math-container&quot;&gt;$[3,7)$&lt;/span&gt;.  Define &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$g(x)=f(2x-5)$&lt;/span&gt;.  What is the domain of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2020-01-16T11:22:43.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17736" PostTypeId="1" CreationDate="2020-01-16T16:25:49.570" Score="7" ViewCount="751" Body="&lt;p&gt;I was having a discussion with a colleague who is in the process of writing some curriculum, and we ended up having a discussion about what &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; (with all the standard restrictions) meant. We couldn't quite agree. Here's where each of us landed.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;One of us thought that &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; was strictly a fraction. That this is, strictly speaking, one number. If needed, and depending on the context, you could use the binary operation of division by using the numerator and denominator as inputs to map &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; to its decimal representation.&lt;/li&gt;&#xA;&lt;li&gt;One of us thought that &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; could be interpreted as either a fraction or as division, it depends on context.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I haven't been able to find any clear distinction between the nuance of these two. I'm curious to know if:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Anyone knows of a spot where a distinction has been made in the literature&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;and/or&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;strong&gt;Which of the two definitions (or third definition not listed) of &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; above do you adhere to?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="267" LastActivityDate="2020-01-21T19:09:15.383" Title="How is $\frac{a}{b}$ interpreted?" Tags="|definitions|fractions|division|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17737" PostTypeId="2" ParentId="17736" CreationDate="2020-01-16T16:45:41.470" Score="11" Body="&lt;p&gt;A US specific answer:  The Common Core State Standards define &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{b}$&lt;/span&gt; by saying it is one of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; equal parts making up a whole &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.  &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; is then defined as &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; of these.  Connecting &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$a \div b$&lt;/span&gt; requires some reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance &lt;span class=&quot;math-container&quot;&gt;$\frac{5}{3}$&lt;/span&gt; of a candy bar means you take your one candy bar, divide it into 3 equal sized pieces.  5 pieces of that size (which requires more than one candy bar to form!) is &lt;span class=&quot;math-container&quot;&gt;$\frac{5}{3}$&lt;/span&gt; of a candy bar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5 \div 3$&lt;/span&gt; might come up in this context:  I have 5 candy bars, and three children.  I want each child to receive an equal share.  One way to think about it is to break each candy bar into thirds, so I have &lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt; thirds.  Then each child can receive &lt;span class=&quot;math-container&quot;&gt;$\frac{5}{3}$&lt;/span&gt; of the candy bar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While we take a lot of these basic equivalences for granted, I do think that having a single consist meaning, and arguing the equivalences (at first) is likely to lead to deeper understanding than just declaring that the same set of symbols can have a whole host of (apparently) distinct meanings.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-01-17T18:30:23.187" LastActivityDate="2020-01-17T18:30:23.187" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17738" PostTypeId="2" ParentId="17736" CreationDate="2020-01-16T19:17:58.127" Score="4" Body="&lt;p&gt;By definition, a rational number is a number that can be expressed as the quotient of two integers. This quotient is called fraction and is written as &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt;. Hence, division and fraction are the same, at least in the context of dividing integers and turning them into rationals. This concept is expanded in middle school for dividing irrationals and polynomials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everyone whom I know uses division and fractions interchangeably depending on whichever is simpler in a particular case, but fractional form is the preferred one. I do the same, in fact, this is how I was taught in school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say, &lt;span class=&quot;math-container&quot;&gt;$5 \div 3 \times 6 \div 4 \div 10 + 3 \div 8 \times 2$&lt;/span&gt; turns into &lt;span class=&quot;math-container&quot;&gt;$\frac{5 \times 6}{3 \times 4 \times 10} + \frac{3 \times 2}{8} = \frac{1}{4} + \frac{3}{4} = 1 $&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2020-01-16T19:50:46.423" LastActivityDate="2020-01-16T19:50:46.423" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17739" PostTypeId="1" CreationDate="2020-01-17T09:21:03.793" Score="8" ViewCount="2420" Body="&lt;p&gt;I am in a research project where one of our jobs is improving the first-year university experience for our students. One of the topics we are looking into is changing the way we teach our introductory course in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;In this regard I have two questions -- that I hope are not too broad&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If you were given 100% control of how the first semester mathematics course were taught, what would you do and why?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there any sources, research articles, papers etc of what works? I have found plenty that talk about the difficulties in designing such a course, but not many &amp;quot;solutions&amp;quot;. Is PBL a good way to go, designing oral exams? Projects instead of exams?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;EDIT: This is a course taken by all of our engineers, and also a handful of mathematicians. Usually about 140 engineers and 10 mathematicians and some physics students.&lt;/p&gt;&#xA;" OwnerUserId="596" LastEditorUserId="7365" LastEditDate="2022-11-17T22:15:59.500" LastActivityDate="2022-11-17T22:15:59.500" Title="Teaching Calculus I to engineers" Tags="|undergraduate-education|calculus|course-design|education-research|" AnswerCount="7" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="17740" PostTypeId="2" ParentId="17739" CreationDate="2020-01-17T14:01:14.577" Score="10" Body="&lt;ol&gt;&#xA;&lt;li&gt;Given 100% control, I would have one-to-one instruction.  One instructor meeting individually with each student.  That instructor can change the approach, the speed, the order of topics, the method of instruction, based on that individual student.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;But of course hiring (and training) enough instructors for that is probably way beyond any reasonable budget.  (It is possible that in the futture such quality instruction will be provided for us by computers; but I do not think we are there yet.)   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably no single method of instruction is &quot;the best&quot; for all of your 140 students.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[The question 1 may be closed as &quot;opinion based&quot;, but here are my opinions.  Question 2 deserves some answers, though.]&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2020-01-17T14:01:14.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17742" PostTypeId="2" ParentId="17739" CreationDate="2020-01-17T18:41:38.393" Score="13" Body="&lt;p&gt;Without directly answering your question, you don't seem to have the background you need to be &quot;improving&quot; the undergraduate experience yet, and have some work to do.  I think you're right in sensing that your question is too general&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't talk about the &quot;difficulties in forming such a course&quot;, and spend your initial time finding out what aspects of the first year experience you're trying to fix.  Is it too many students fail calculus I?  Is it that students don't know calc well enough when they're done to support the engineering curriculum?  Is is that too many students drop engineering as a major and go to other disciplines? Is it that attendance in the class is poor?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might have a number of stakeholders that you need to question -- maybe students, advisors, faculty,...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can spend an awful lot of effort revamping a curriculum that doesn't touch the real issues.  You minimize wasted resources by going through the exercises of characterizing the issues, and prioritizing what problems need to be fixed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, let me, as an engineering prof, try to guess about the kinds of situations that you might be trying to improve.  The biggest issues I see concerning our calc sequence is that some of our first year students have plenty of on ramp issues, and might not be ready to take calculus.  I think we &lt;em&gt;owe&lt;/em&gt; these students the support that they need to thrive in our programs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own experience is that there isn't going to be ONE course that can address the needs of all the first year students at the same time.  If I was trying to create ONE class to support all the students, it would have multiple scaffolding options that students can select, according to their needs, but students that don't have a good foundation would need to put in an awful lot of extra work, right when they're transitioning from a high school environment.  Tough path, but maybe some of the newer tools available through learning management systems can be of some help (not holding my breath).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We've landed on replacing our Calc I/II sequence with a somewhat decelerated three course sequence for students we feel would benefit, based upon placement calculations from their applications packages, with many students completing the third course during the summer.   This allows a fairly usual engineering sequence, though there are some issues with the timing of Physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This works to some extent.  I always wonder if the better option would be a semester of pre-calc for students that would benefit, followed by Calc I and Calc II&lt;/p&gt;&#xA;" OwnerUserId="13341" LastEditorUserId="7930" LastEditDate="2020-01-22T19:41:57.903" LastActivityDate="2020-01-22T19:41:57.903" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17743" PostTypeId="2" ParentId="17739" CreationDate="2020-01-17T18:43:59.943" Score="1" Body="&lt;p&gt;Select and use an open textbook. Students who wish to use a physical textbook will be able to get a copy run off and hole-punched at a local print shop, then put in a binder, for far less than the cost of a conventional textbook. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will save your students a pile of money and there are enough available options that you should be able to find something that suits your needs. This has benefits to student learning other than supplying beer money; financial stress is real, and impedes scholarship. Steps can and should be taken to alleviate said stress where possible. I don't have sources off-hand, but I suspect research is available to back that up. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.ubc.ca/~CLP/about/&quot; rel=&quot;nofollow noreferrer&quot;&gt;This is what's used&lt;/a&gt; at my alma mater; the program has been quite successful, and textbooks for derivative and integral calculus are available. All questions have hints, answers, and fully-worked solutions. It is available under a CC BY-NC-SA 4.0 license, which I believe will permit usage at other institutions: their contact information is available on the page linked for clarification. &lt;/p&gt;&#xA;" OwnerUserId="13342" LastEditorUserId="13342" LastEditDate="2020-01-17T23:10:15.473" LastActivityDate="2020-01-17T23:10:15.473" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17744" PostTypeId="2" ParentId="17739" CreationDate="2020-01-17T19:01:28.767" Score="2" Body="&lt;p&gt;I'm an engineer who really likes and uses mathematics on work and daily life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Myself, I would try to make the course as much self-contained in classes, but still stick to some written material the students have access to. This allows the subject to be more complex and extensive while keeping down the difficulty settings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd prove theorems as much as feasible in class, and show both synthetic and practical examples, i.e. both a simple equation that showcases the point you are making and a rather complex but realistic problem. epsilon-delta proofs are important for creating a sound basis, but engineers often dislike them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, students need to learn how to use computers. So a few classes on software that computes limits, derivatives and integrals is handy as well. Mathematica comes to mind, but there are free tools for this purpose. Symbolic integration warrants the comment &quot;learn this first, so you'll not get addicted to numeric integration&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even better, Automated Differentiation has even better applications in industry and science (AdiMat is one example). Maybe your students don't need to know how this works, but they should know it exists and what it can or cannot achieve. I've once used one of these tools to differentiate a function defined by over 300 lines of code, with respect to 7 variables. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students would thank you for providing them with these tools early in their graduation. Others might just consider this &quot;extra work&quot;, forget how to use them and move on with life. The challenge is that your students are likely not to know anything about programming at this point in their courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if I taught all of this, usual examples of derivation and integration should be shown as &quot;this is simple enough you need to get a hang of it&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also a few functions such as the error function, should be introduced as well.&lt;/p&gt;&#xA;" OwnerUserId="12951" LastActivityDate="2020-01-17T19:01:28.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17745" PostTypeId="2" ParentId="17739" CreationDate="2020-01-17T20:44:38.743" Score="5" Body="&lt;p&gt;It's about the learning experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An issue with project based learning is the need for objective grading isn't going away. Since the first semester of mathematics is what serves to filter out under performing students there is a desire to have an unmovable bar. Without it you get grade inflation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This isn't to say there can't be projects. You just have to ensure that the student work is objectively and individually gradable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So rather then do anything to change how the students are graded I'd emphasize changes to their learning experience.  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In the age of youTube any lecturer who doesn't encourage questions will find themselves outclassed. Be interactive or die.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The best tutor isn't the teacher. It's someone who just took the same class. If you don't have a &lt;a href=&quot;https://library.spalding.edu/mathlab&quot; rel=&quot;noreferrer&quot;&gt;math lab&lt;/a&gt; create one. Populate it with second semester tutors. Encourage everyone, regardless of performace, to seek tutoring&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teach study skills aimed directly at math itself, the class in question, and the test.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3.1 If the tests are closed book and closed notes teach the students the fine art of recreating a sheet of formulas from memory on a blank sheet of paper.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3.2 If the tests are open notes teach the students the fine art of organizing all the formulas on a blank sheet of paper so they're easy to use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3.3 If the tests are open notes and open book teach them 3.2 anyway so they don't spend forever flipping through the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3.4 As for project based learning, consider this story: way back in my statistics days our instructor gave fantastically hard quizzes for 5 minutes in every class. The whole class was bombing them. So a group of us poured over the quiz we just failed for an hour after each class. After that hour we usually had the quiz figured out which we checked with the instructor. Our group got the only A's in the class because we worked together on quizzes that gave us no credit. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3.5 Encourage your tutors to go a step beyond. When I worked as a math lab tutor I was asked to lecture a number of test preparation sessions. As an undergrad this is wonderful experience that teaches you the material like nothing else, because you know you're gonna get asked.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3305" LastActivityDate="2020-01-17T20:44:38.743" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17746" PostTypeId="2" ParentId="17739" CreationDate="2020-01-18T01:27:07.287" Score="-1" Body="&lt;p&gt;I'm not in education, so take my answer with a grain of salt. I am merely approaching this from a perspective of someone who is really frustrated with how mathematics is taught. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics is a tool we use to understand the universe. If you want to make math more engaging, it would be useful to actually show how one could use math to solve problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, say you're teaching a class simple rates. I think in the current paradigm a teacher/prof would ask a question like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Given a tank of 5L, at a rate of 100mL/min, how long will it take to fill the tank?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;then go on to explain how the units work with this problem and whatnot. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pretty boring right? I think a better way to approach people with this question is to actually bring a tank of water and start to fill it with water. This will naturally make people wonder how long it will take to fill the tank. From there, the teacher/prof could ask:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How would you solve this problem? Any thoughts? &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The educator would then guide the class through to the solution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me that's captivating. I believe that by the end of that class, you will definitely understand rates, and see how math is useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think most concepts in math are not taught this way because it is challenging to come up with examples, and you may require physical examples. &lt;/p&gt;&#xA;" OwnerUserId="13346" LastActivityDate="2020-01-18T01:27:07.287" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17747" PostTypeId="1" AcceptedAnswerId="17750" CreationDate="2020-01-18T11:14:53.887" Score="14" ViewCount="2229" Body="&lt;p&gt;My son is in high school (France, 2&lt;sup&gt;nde&lt;/sup&gt;) and I was watching how he solves math exercices. This led me to the following question: &lt;strong&gt;when are students expected to plug in actual values in their calculations?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a background, I am a physicist by education and was always of the opinion that the symbolic calculations should be dragged as far as possible. The actual numbers are used at the very end and it is not the most important part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that the way math is taught today is different. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will take a concrete example: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have three points &lt;span class=&quot;math-container&quot;&gt;$M(7;-2)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$N(0;t)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P(3;1)$&lt;/span&gt;. Find &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; so that they are aligned.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would have done it by generalizing the points (&lt;span class=&quot;math-container&quot;&gt;$M(x_M;y_M)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$N(x_N;y_N)$&lt;/span&gt;, ...) and finding co-linear vectors with the calculations being done on those generalized points. It is only at the very end, having a general formula, that I would have used the actual values of &lt;span class=&quot;math-container&quot;&gt;$M(7;-2)$&lt;/span&gt;, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My son is telling me that they do the calculations directly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Which approach is expected at a high school level?&lt;/strong&gt; The example below is a simple &quot;training&quot; exercise, but when the problems get longer I see that he is still using the numbers early on, without any generalization.&lt;/p&gt;&#xA;" OwnerUserId="8076" LastEditorUserId="12619" LastEditDate="2020-01-18T12:02:26.073" LastActivityDate="2020-01-20T23:58:55.627" Title="How much symbolic calculations before plugging in actual values?" Tags="|secondary-education|exercises|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17748" PostTypeId="2" ParentId="17747" CreationDate="2020-01-18T12:00:59.640" Score="17" Body="&lt;p&gt;Speaking from an American perspective, your son's approach strikes me as much more natural.  For instance, to solve your problem&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have three points &lt;span class=&quot;math-container&quot;&gt;$M(7;-2)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$N(0;t)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P(3;1)$&lt;/span&gt;. Find &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; so that they are aligned.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;my students would start by observing that the slope (gradient) of &lt;span class=&quot;math-container&quot;&gt;$\overline{MN}$&lt;/span&gt; would have to be the same as that of &lt;span class=&quot;math-container&quot;&gt;$\overline{MP}$&lt;/span&gt; and from there go straight to &#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{t-(-2)}{0-7}=\frac{1-(-2)}{3-7}\\\frac{t+2}{-7}=\frac{3}{-4}\\-4(t+2)=-21$$&lt;/span&gt;&#xA; and finish solving that equation to find the value of &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;.  By contrast, your approach would look more like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{y_N-y_M}{x_N-x_M}=\frac{y_P-y_N}{x_P-x_N}\\(y_N-y_M)(x_P-x_N)=(y_P-y_N)(x_N-x_M)$$&lt;/span&gt;&#xA;which has us multiplying out two pairs of binomials and then rearranging the subsequent eight terms in an effort to solve for &lt;span class=&quot;math-container&quot;&gt;$y_N$&lt;/span&gt; in terms of the other five variables and then turning it from there into a relatively complex arithmetic problem.  Not only is it easier to multiply numbers than bimonials, it is far easier to avoid or at least spot the simple arithmetic or bookkeeping errors.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This was driven home to me in my calculus class a few weeks ago.  We were working on a problem along the lines of&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find the slope of the tangent to the curve &lt;span class=&quot;math-container&quot;&gt;$x^3+y^3=9xy$&lt;/span&gt; at the point &lt;span class=&quot;math-container&quot;&gt;$(4,2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the first step is to implicitly differentiate both sides with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$$3x^2+3y^2y'=9xy'+9y$$&lt;/span&gt;  My expectation was then that students would do some algebra to get to &lt;span class=&quot;math-container&quot;&gt;$$y'=\frac{9y-3x^2}{3y^2-9x}=\frac{3y-x^2}{y^2-3x}$$&lt;/span&gt; and then plug in the variables.  But a student piped up and suggested that it was much less effort if you substitute in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; immediately after the differentiation, and he was absolutely right.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This is not to say that there isn't an important place for the strategy you describe.  We wouldn't have valuable tools like the quadratic formula or Heron's formula if we didn't stop and say &quot;Okay, just for once let's solve this problem where all of the values are arbitrary variables instead of specific numbers.&quot;  But I always have to really slow down when I do those derivations to make sure that I'm not making mistakes along the way.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-01-18T12:00:59.640" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17750" PostTypeId="2" ParentId="17747" CreationDate="2020-01-18T15:06:58.727" Score="11" Body="&lt;p&gt;Calculate when you want an answer.  Solve algebraically when you seek patterns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the problem was &quot;show that for any two points, you can find a third point along the y-axis that is collinear with them,&quot; then symbolic logic is the right way to go.  But if you have the points, just plugging them in simplifies the problem dramatically and makes  it easier to visualize by drawing a picture.  Drawing a picture of the generic case is hard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing calculations opens up the door to simplifications that may not have been there in the symbolic algebra.  There may be some very ugly &lt;span class=&quot;math-container&quot;&gt;$ab^2\ln(\sin c)$&lt;/span&gt; term that causes so much ugliness, but was really nothing more than a combination of constants that equals 2.  There may be shortcuts here.  Sometimes the numbers provide very clean  relationships, such as when two numbers turn out to be pretty much a multiple of 2 pi from each other.  You can leverage symmetries that only appear in the special cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There may also be shortcuts because you can avoid pathological cases which make the math monsterous.  If you'll note, in my first paragraph, I suggested somethign to show &lt;em&gt;which was false&lt;/em&gt;.  I didn't include any wording to account for there being a vertical line between the two points that never crossed the y axis.   Nor did I consider the case where both points are on the y axis, such that every third point qualified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have to be careful.  Consider this famous &quot;proof&quot;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;let a and b be equal nonzero values: &lt;span class=&quot;math-container&quot;&gt;$a = b$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;multiply by a: &lt;span class=&quot;math-container&quot;&gt;$a^2 = ab$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;subtract &lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$a^2 - b^2 = ab - b^2$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;factor: &lt;span class=&quot;math-container&quot;&gt;$(a + b)(a - b) = b(a - b)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;divide by &lt;span class=&quot;math-container&quot;&gt;$a-b$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$a + b = b$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Substitute b for a, as they are equal: &lt;span class=&quot;math-container&quot;&gt;$b + b = b$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Combine: &lt;span class=&quot;math-container&quot;&gt;$2b = b$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Divide by b: &lt;span class=&quot;math-container&quot;&gt;$2 = 1$&lt;/span&gt; QED&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is a rather famous false proof.  One has to remember that division by &lt;span class=&quot;math-container&quot;&gt;$a-b$&lt;/span&gt; is only defined if &lt;span class=&quot;math-container&quot;&gt;$a-b$&lt;/span&gt; is nonzero, and in this construction &lt;span class=&quot;math-container&quot;&gt;$a-b$&lt;/span&gt; is always zero.  If we used numbers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Let a and b both equal 3: &lt;span class=&quot;math-container&quot;&gt;$3 = 3$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Multiply by a (which is 3): &lt;span class=&quot;math-container&quot;&gt;$9 = 9$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;subtract &lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt; (which is 9): &lt;span class=&quot;math-container&quot;&gt;$0 = 0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;factoring is a bit irrelevant with the individual numbers&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Divide by &lt;span class=&quot;math-container&quot;&gt;$a - b$&lt;/span&gt; (which is 0): &lt;span class=&quot;math-container&quot;&gt;$\frac{0}{0}=\frac{0}{0}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Wow, we found that problem fast!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way we do math has changed over the years, due to the quantity of data that is available and the amount of processing power we have available.  The focus has indeed drifted away from the symbolic manipulation.  That mirrors the way everybody does math in real life these days, with calculators literally a tap away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Myself, I'm not a physicist.  I'm an engineer.  We'll take the symbolic logic &lt;em&gt;just&lt;/em&gt; far enough, and then plug it into a simulation.  Why?  You really don't want to try to solve some of our equations these days by hand.  In fact, most interesting ones are recognized to not have a closed form solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The approach I advocate today is to bring in symbology when you want to use it as a shortcut.&lt;/strong&gt;  For example, consider the finding of roots of quadratic equations.  Teach students to complete the square, with numbers.  Teach them to do this over and over.  When they get tired of it, then you point out that you can put on your symbolic algebra hat, sharpen your pencil (no mistakes!), and prove that you can solve these problems by deriving the quadratic equation once and then merely plugging in values from there!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More than once, while Engineering, I relied on symbolic manipulation to show that some pathological case we were spending lots of money on can never possibly occur because the math shows it can never occur.  I did that because it was the shortcut.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This approach is consistent with a story I was told from how they did apprenticeships in German machine shops.  Obviously the Germans are known worldwide for their precise machining with powerful tools that drive tolerances as close to 0 as one can get.  When an apprentice is brought into one of these shops, they are handed a length of square bar stock.  They are handed a file and told &quot;make it circular.&quot;  And the plug away at the problem, learning how to make a circle.  Then they are invited to learn what circular &lt;em&gt;really&lt;/em&gt; means, showing them the ridiculous tolerances one may have, and hand  techniques that can be done to make them circular.  Then, once they have a circular bar that passes standards, they are told to make it square.  And they file away for a bit, before they are taught what square &lt;em&gt;really&lt;/em&gt; means.  Only then are they introduced to the lathe and the mill, which are machine tools designed to solve circularness and squareness for all sorts of circumstances.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2020-01-18T15:06:58.727" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17751" PostTypeId="2" ParentId="17747" CreationDate="2020-01-18T20:34:10.113" Score="3" Body="&lt;p&gt;You can't generalize a single, isolated question, which is what your high school question about collinear points is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course you can &lt;em&gt;invent&lt;/em&gt; a set of similar problems, create a general solution for the whole set, and then solve the particular problem given - but why do all that unnecessary work? If your child's homework set contains several similar questions, would you expect them to create the same (or slightly different) general solutions for each one?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the same as a general principle in computer software design, known as YAGNI - If &quot;you ain't gonna need it,&quot; don't waste your time dealing with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The situation is different if you already have a general tool that you can use. For example, if you already know the formula for solving a general quadratic, plugging in the known values to solve the equation in front of you may be quicker than completing the square, or factorizing the equation by inspection - or it may not, depending both on the equation and on the individual student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At university level, it is much more likely that students &lt;em&gt;do&lt;/em&gt; know (or should know!) some general tools that are relevant - Newton's laws of motion, Lagrangian or Hamiltonian dynamics, conservation laws, special or general relativity, calculus of variations, or whatever. So it is more likely that it is worth carrying a &quot;symbolic&quot; solution further, especially if the question is about getting insight into the &lt;em&gt;general&lt;/em&gt; way the system behaves for a range of parameter values, not just a numerical value for one set of conditions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At any level, there is a trade off between pure computation and mathematical theory. For example, understanding the solution of a set of linear equations using determinants and Cramer's rule can give valuable theoretical insight, but calculating a numerical solution to any system of order &gt; 2 by Cramer's rule is the &lt;em&gt;wrong&lt;/em&gt; way to do it, because it is less efficient than simple elimination. And the apparently smart idea of calculating the solution of a system of equations by writing it in matrix form and explicitly inverting the matrix numerically is also the wrong way, because &lt;em&gt;factorizing&lt;/em&gt; the matrix numerically is almost always more efficient and more numerically stable than inverting it - though sadly, I have met dozens of PhD level students who think they are experts using MATLAB or Mathematica, but have never learned those basic facts of numerical life!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;If the only tool you have is a hammer, every problem looks like a nail.&quot; That is not what students should be learning!&lt;/p&gt;&#xA;" OwnerUserId="7677" LastEditorUserId="7677" LastEditDate="2020-01-18T20:39:40.643" LastActivityDate="2020-01-18T20:39:40.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17752" PostTypeId="2" ParentId="17747" CreationDate="2020-01-18T20:49:20.227" Score="6" Body="&lt;p&gt;I don't think this is a nationalistic difference (I'm in the US), but I also don't think your example is optimal. As an example where the correct technique is more well defined, let's say we have a physics problem like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A bug starts from rest and accelerates with constant acceleration for 0.53 s, traveling 1.37 m. Find the bug's acceleration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would consider it wrong wrong &lt;em&gt;wrong&lt;/em&gt; to solve this by first writing down the equation &lt;span class=&quot;math-container&quot;&gt;$x=(1/2)at^2$&lt;/span&gt;, then plugging in numbers, then solving for &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. As you say, a competent person will do the algebra first and then plug the numbers in at the last step. Below is a section from my syllabus for a physics course where I present this kind of thing, including this example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ca3yY.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ca3yY.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The unfortunate reality today is that many physics instructors use computerized grading of homework, and they implement this in such a way that they never actually see their students' written work. Therefore their students never get feedback on this kind of thing.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-01-19T02:55:21.960" LastActivityDate="2020-01-19T02:55:21.960" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17753" PostTypeId="1" CreationDate="2020-01-18T21:50:52.637" Score="4" ViewCount="100" Body="&lt;p&gt;I have been independently considering two edX courses in mathematics.  The first, &lt;a href=&quot;https://www.edx.org/course/data-science-probability&quot; rel=&quot;nofollow noreferrer&quot;&gt;a course on probability theory drawing from a financial crisis case study&lt;/a&gt;, appeared to me plausibly comparable in scope to an actual undergraduate class in probability theory.  However, the second, &lt;a href=&quot;https://www.edx.org/course/differential-equations-fourier-series-and-partial&quot; rel=&quot;nofollow noreferrer&quot;&gt;a course on partial differential equations emphasizing Fourier Series&lt;/a&gt;, looks like it might be much narrower in scope than an actual undergraduate class in PDEs.  Glancing at the program which the latter belongs to, I wonder if it's more akin to the brief introduction to PDEs sometimes given at the end of an ODE class.  This, needless to say, has caused me to question the probability theory course as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How does the scope and depth of an edX course compare to that of an average undergraduate class?  Is the class more comparable to an entire edX program instead of an individual course?  Does it depend on the specific program/course?  If so, what criteria can be used to decide whether a particular course, such as the two I'm considering, would be sufficient to use as a self-study alternative to an undergraduate class on the subjects?&lt;/p&gt;&#xA;" OwnerUserId="13350" LastActivityDate="2020-01-18T23:51:43.973" Title="EdX Courses for Self-Study" Tags="|probability|online-instruction|differential-equations|resource-request|resources|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17754" PostTypeId="2" ParentId="17753" CreationDate="2020-01-18T23:26:06.483" Score="3" Body="&lt;p&gt;(In before the close!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd say the PDE course looks more like a traditional course than the prob/stats course.  Look at the hours expected, for example (~6.5 versus ~1.5), each times 8 weeks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The PDE course looks like a solid half to two thirds of a semester of a normal, engineering support course.  You cover a couple of the 3-4 major equations.  And get Fourier series.  Looks like a couple chapters out of Kreyszig.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The prob/stats course lists a bunch of topics but it's easy to do that in stats.  I could do a 2-3 hour lecture for Six Sigma students and briefly hit a bunch of topics also.  Looks like that is what you are getting, plus the case example.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Of course do what is more interesting to you.  And if anything the stats/finance course will probably be more easy to treat as an avocation and more passively as a lecture watcher.  Yeah...working the problems still needed for real learning.  But in PDE course, you'll be absolutely lost without working.  In the sexier finance-stats course you can still float along and enjoy it.  (Given the very real possibility of distractions, you might see this &quot;bug&quot; of the data analysis course as more of a &quot;feature&quot;.)  But really...do what turns you on.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In terms of the general questions:  you are just looking at the difference of &quot;extension courses&quot; versus traditional courses.  This is a difference that is more general than the Harvard source, than Internet modality.  Extension courses are, on purpose, lighter than normal courses, and are geared to working professionals.  They're not kids looking to get a bachelor's.  They want to do a little bit at night to get exposed to something or to help them in a new role they have taken at work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The PDE course is really a bit &quot;heavy&quot; for an extension course.  That finance case study is more what a normal extension course would look like.  (I.e. lighter than the PDE course and less mimicking a traditional topic.)&lt;/p&gt;&#xA;" OwnerUserId="13328" LastEditorUserId="13328" LastEditDate="2020-01-18T23:51:43.973" LastActivityDate="2020-01-18T23:51:43.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17756" PostTypeId="1" CreationDate="2020-01-19T07:16:40.927" Score="5" ViewCount="1023" Body="&lt;p&gt;Seeing as this is the math educator site, perhaps someone can help me out:   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking to become a math teacher, but I am having a hard time figuring out which math courses I need to be taking. Any insight?   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Right now I am at the undergraduate level. The courses I am looking at include: Calc 1,2, and 3, and Statistics. They also include Linear Algebra, Mathematical Modeling, College Geometry, and Discrete Math - but I'm not sure which of those make sense to take.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like I said, I am trying to get to Secondary Mathematics Education. I can't ask an advisor because some of these courses are not offered by my college and I'll have to transfer them in. I am also not on campus and I find that the advisors are not as helpful over the phone.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be a bit more specific: &lt;a href=&quot;https://www.nystce.nesinc.com/content/docs/NY_fld04_objs.pdf&quot; rel=&quot;noreferrer&quot;&gt;This&lt;/a&gt; is the link to the material on the competency test for New York. Because I have not taken the courses yet, I do not know which classes it reflects. If anyone has taken the test and understands which areas the test is testing, that would be very insightful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any insight is appreciated!&lt;/p&gt;&#xA;" OwnerUserId="12529" LastEditorUserId="12529" LastEditDate="2020-01-19T16:42:29.603" LastActivityDate="2020-01-20T16:38:18.323" Title="What math courses should I take in order to become a secondary math educator?" Tags="|undergraduate-education|secondary-education|self-learning|usa|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17757" PostTypeId="1" CreationDate="2020-01-19T13:43:33.513" Score="6" ViewCount="343" Body="&lt;p&gt;Can someone please post some (relatively easy, high school level) combinatorial problems which can be solved with polynomials (but not generating functions).&lt;/p&gt;&#xA;&lt;p&gt;Here is an example of one such problem:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;We have &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; different numbers &lt;span class=&quot;math-container&quot;&gt;$a_1,\dotsc,a_n, b_1,\dotsc,b_n$&lt;/span&gt;. A table &lt;span class=&quot;math-container&quot;&gt;$n\times n$&lt;/span&gt; is divided on &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; unit cells and in cell &lt;span class=&quot;math-container&quot;&gt;$(i,j)$&lt;/span&gt; we write a number &lt;span class=&quot;math-container&quot;&gt;$a_i+b_j$&lt;/span&gt;. Suppose that all products of numbers written in cells in each column are the same. Prove that then all products of numbers written in cells in each row are the same.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Idea for a solution: Observe a polynomial &lt;span class=&quot;math-container&quot;&gt;$$P(x) = (x+a_1)\dotsb(x+a_n)-(x-b_1)\dotsb(x-b_n)$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="10484" LastEditorUserId="10484" LastEditDate="2021-03-25T14:21:24.800" LastActivityDate="2021-03-25T14:21:24.800" Title="Combinatorial problems which can be solved with polynomials" Tags="|secondary-education|problem-solving|math-contests|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17758" PostTypeId="2" ParentId="17747" CreationDate="2020-01-19T17:03:42.923" Score="2" Body="&lt;p&gt;I think the difference between the two approaches is the goals. Your solution is exactly what I would do, but I'm an engineer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In high school, the goal of learning is usually more about understanding the general concept and gaining some practice with the mechanics. I would think that plugging in the values early on is &quot;easier&quot; to grasp the ideas than solving algebraicaly for the general case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However in first year university the goals shift and you're no longer expected to just loosely grasp the ideas but you are expected to have a deep understanding. At that stage I would say that solving for the general case would be more expected. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In retrospect after years of having done similar problems in the general case solving the singular case becomes a trivial matter, but you (and I) have years of experience doing this. In high school they may have only solved a handful of problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you're first introduced to a topic many like to see it work, algebra tends to abstract the result away. It's not until you've gained som experience and practice that you really see the elegance in it. Solving the problem numerically the first few times demonstrates the concept, then the next step could be for the teacher to show how the process can be abstracted into a general form. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A really good teacher (in my opinion) would show first how the problem might come up in real world use (showing that there is an application). Then let the students struggle with solving it on their own, not showing them the algebra but allowing them to try it on their own. For a practical problem that you only need once, numerical solutions are typically the go to first step. Then after they have struggled and gotten familiar, showing them that if you take a step back from the specific case and look at it generally you can solve it more elegantly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I consider this to be a better approach as it allows the students to see why they should care and then why it's worth while to not rush into a problem. But also allows them to both struggle on their own and then also guides them to a general and more elegant solution. The process can then be applied to many different subject areas. Math, physics, engineering, programming etc.&lt;/p&gt;&#xA;" OwnerUserId="13356" LastActivityDate="2020-01-19T17:03:42.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17759" PostTypeId="2" ParentId="17756" CreationDate="2020-01-19T18:34:13.497" Score="3" Body="&lt;p&gt;You want to be a high school math teacher in the state of New York. I searched on &quot;high school math teacher new york state requirements&quot;, and found &lt;a href=&quot;https://www.mathteacheredu.org/new-york/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this site&lt;/a&gt;.  I'm a bit leery of this site, because I can't tell who is responsible for it. But the information seems reasonable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will want to talk to someone at a teacher preparation program. &lt;a href=&quot;http://eservices.nysed.gov/teach/certhelp/BaseExtReqSet.do?certCredential=989&amp;amp;WIN_TYPE=null&quot; rel=&quot;nofollow noreferrer&quot;&gt;(This government site&lt;/a&gt; lists the requirements. And &lt;a href=&quot;http://www.nysed.gov/COMS/RP090/IRPS2C&quot; rel=&quot;nofollow noreferrer&quot;&gt;this site&lt;/a&gt; lists the schools that have the teacher preparation programs approved by the state.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-01-19T18:34:13.497" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17760" PostTypeId="2" ParentId="17756" CreationDate="2020-01-19T20:14:34.953" Score="5" Body="&lt;p&gt;If you want to teach high school math, then of course you'll need to know high school math. And if you want to get a teaching credential, you'll need to take whatever courses your state says are necessary. Beyond that, it's just a matter of getting a deeper understanding of the material, rather than just knowing enough go through the steps. Real Analysis helps you understand the theory behind Calculus, Algebra helps you with Algebra of course. For Geometry, this tends to be when schools start doing a lot of proofs, so you might want to take a Logic course. You could also take a course in non-Euclidean Geometry, such as Projective Geometry. Many high schools are including Statistics as an option, so you might want to take a course in that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should also take a look at what sort of pay scales school districts that you're planning on working for offer. Many give a higher salary to people with a Master's degree. Keep in mind that they also generally have a &quot;professional development&quot; requirement, in which you have to take a certain amount of classes each year, so if you find that you have a gap in your education, you can count it towards that requirement.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2020-01-19T20:14:34.953" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17761" PostTypeId="2" ParentId="17756" CreationDate="2020-01-19T20:22:19.600" Score="2" Body="&lt;p&gt;I got my NYS certification in Math 7-12 and Generalist Special Education 7-12 (with the grades 5-6 extension in both) about two and a half years ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The test you are describing is what Sue's website calls the Content Specialty Test in Mathematics.  (I also know it by that name, and didn't realize that the name had changed.)  Your basic coursework will leave you more than prepared for it.  If you went to high school in New York, then the test is 80% a mixture of the multiple choice from the three mathematics Regents exams and the AP Calculus AB exam.  The other 20% is pedagogical content knowledge.  That is an essay question that will look something like &quot;Sam solved the following problem in the following way.  Identify what Sam did right and wrong, and design a strategy to address and correct his mistake.&quot;  A sufficiently above-average student could pass the test easily before graduating high school.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The courses you've described sound great.  It's a good survey of introductory modern mathematics, and will cover you for the next few paradigm shifts in mathematical education that you will endure over the course of your career.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will also need to see to your non-math classes, though.  The one class that you will absolutely need is Adolescent Psychology.  Beyond that, make sure you have a broad sampling of the rest of the liberal arts core.  I tried to short-cut my way to initial certification halfway through grad school and was denied because I had never taken a fine arts course, of all things.  New York has a reputation of being the fussiest for teacher cert in the US, but it also has the highest median pay to cover for what they expect from us.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-01-19T20:22:19.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17765" PostTypeId="2" ParentId="17756" CreationDate="2020-01-20T16:38:18.323" Score="5" Body="&lt;p&gt;Expanding upon my comments to the question, now that I have time to:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of the courses you listed as looking at are ones that I think every high school math teacher should take and be exposed to. Calculus and Statistics are also courses that you may very well be expected to teach at the high school level, so these make the most sense of your list to take first. Elements of Linear Algebra, Mathematical Modeling, College Geometry, and Discrete Math may all appear in high school courses as well, though likely not in their entirety. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Down the line, I would encourage you to take courses in Number Theory and Group Theory, since I think they provide excellent insights into high school math content and that even a cursory knowledge of those will help an Algebra teacher understand the content more deeply. Similarly, Real Analysis would make sense after your completed your Calculus sequence. It is also typical in some high schools in New York State to have math teachers fill in for other STEM courses in the building when necessitated. You may want to take other math adjacent courses like Computer Science, Accounting, or another hard science you're interested in. In particular, if I were in your position, I'd look to take a few Computer Sciences classes, since I think we're going to start seeing more and more of these classes offered at the secondary level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are firmly planning on becoming a secondary math teacher, then you might want to get a jump start on the masters degree that will be required if you plan on teaching in New York State. A pedagogy or psychology class, whether it was for secondary mathematics or not, would be valuable I would think. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at the linked material for the competency test, it seems to me to be very much aligned to the Common Core State Standards -- the competencies listed on page three are almost exactly the same. Aside from the Calculus and Pedagogical Content Knowledge sections, I would be inclined to predict that you know most of the other content already from your high school education if you went to high school in New York State and took the three Regents and passed with decent scores. While I myself have not taken the Certification Exam in NYS, I know many people who have who have done so with little more classroom math exposure beyond high school math. The competency test, in general, is not something that I think you should be worried about if you are considering the classes you have listed; it is perhaps the easiest part of gaining your teaching certification in NYS.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there are courses or professors that you're interested in at your school, that might drive your choices. If you take a course with a teacher you admire, then you may decide to take more courses with that person. Follow your interests and pursue your passions, but don't get so focused that you ignore opportunities to take other classes that might expand your thinking. I think it's great that you have the idea that you want to be a high school math teacher, but lots of people also change their minds while in college because of the classes they take and the people they encounter, and that's perfectly okay too.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2020-01-20T16:38:18.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17767" PostTypeId="2" ParentId="17747" CreationDate="2020-01-20T23:58:55.627" Score="2" Body="&lt;p&gt;My own experience in math classes (as a student in Germany) and tutoring my peers would lead me to the following conclusion regarding your question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students have problems with the generalized formulas, because they find it rather unintuitive to calculate with &quot;letters&quot; rather than numbers and they would frequently ask for real world examples and applications for these mathematical problems. They had an easier time following examples with real numbers/values/problems.&#xA;I can totally see where you are coming from and I would say that there is much value in teaching the generalized formulas, but they were very daunting to most of my peers and did throw many of them off of maths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the main job of highschool maths is to get the students interested in solving these kinds of problems, seeing the value of learning maths (which in my experience was pretty low among my peers) and equipping them with the tools to solve more complicated problems, though the extend of the last point was always debated among the teachers at my school, because it was very easy to scare people off that were not interested in the first place or to just lose their attention during class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it is a fine balance for our education system between stimulating the interest of pupils and giving them in-depth/generalized solutions to those kinds of problems I'd say. And more in-depth courses for pupils that are either very apt at maths or just interested in learning more should always be an option.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for tutoring I always seemed to have to most success in starting off with real world examples/real numbers and providing the generalized formulas afterwards, most of my &quot;students&quot; got the gist of it with this approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also Diesel has a good point that coming from real world examples and letting pupils figure the more general approaches/formulas out themselves will give them the chance to develop analytical skills on their own, which can be very gratifying and motivation plays a big part in learning.&lt;/p&gt;&#xA;" OwnerUserId="13362" LastActivityDate="2020-01-20T23:58:55.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17768" PostTypeId="2" ParentId="501" CreationDate="2020-01-21T00:40:31.517" Score="2" Body="&lt;p&gt;I suggest to research the placement tests and psychrometry of USMA, USNA, USAFA, etc. They have a very long experience in doing these tests and assessing how well the resultant students do in their courses.  You could even reach out to someone in one of the departments and see what insights they give you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, whenever you are looking for large data sets of assessment tests, performance history etc., it's worthwhile to look at the military.  Both lower end like the ASVAB and up to more detailed assessments (pilot aptitude).  They're not perfect of course but they do tend to have large populations, reasonably stable quality of entrants and stable course demands.  It's much more statistically insightful than random one off studies you see in many ed journal articles, with low statistics or poor controls.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See here for some general practice in calculus:  &lt;a href=&quot;https://www.usna.edu/MathDept/academics/placement.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.usna.edu/MathDept/academics/placement.php&lt;/a&gt;  [but I think a detailed search/outreach would get you more.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note:  They're basically sorting people into pre-calc (a small but not insignificant fraction of the class) versus standard engineering calculus.  So you may need an easier test for sorting people into lower levels of pre-calc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For validation of calculus, I would go by the standard 4-5 of AB gets you calc1 and 4-5 of BC gets you calc 1 and 2.  For kids that didn't take AP but want to validate, just have some assessment that's similar (probably last year's calc final or the like).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I probably wouldn't waiver kids getting out of where the placement put them and would keep it simple by ignoring grades, SATs, etc.  Of course if you want to create something more complicated you can.  There's an argument that you get more correlation to end results by including more indicators.  But I doubt you have really trained the formula well enough for that.  And also psychologically it's just easier to tell the kids you're driving it off the test performance (put up or shut up!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you were going to waiver to let kids go higher, what you want to do is let kids that are really smart (but didn't learn things properly) get past the hurdle. More than the ardent workers but not as smart.  Since shmartie is more likely (if he gets unlazy) to survive a course he is unprepared for.  But really, I just wouldn't waiver.  Too hard to justify.  Keep It Simple, Stupid.&lt;/p&gt;&#xA;" OwnerUserId="13363" LastActivityDate="2020-01-21T00:40:31.517" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17769" PostTypeId="2" ParentId="501" CreationDate="2020-01-21T04:14:17.103" Score="6" Body="&lt;p&gt;In recent literature, this issue would be referenced as &lt;strong&gt;multiple measures placements&lt;/strong&gt;. There's been quite a bit of work in this area in recent years, and a number of places cite positive results (in terms of shortened time to gain more credits, etc.). The displayed effect is usually stronger for English, and weaker for math. It certainly makes intuitive sense to me that using more data for decision-making ought to be more robust. However, I'm a bit suspicious of some of the findings because of ulterior motives by the institutes pushing them; usually these are joint with efforts to reduce time in school for students, reduce funding for developmental education, reduce costs for placement-testing, etc. For example, Columbia University's Teacher's College has been at the forefront of the push to end developmental classes in colleges. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's an &lt;a href=&quot;https://www.insidehighered.com/news/2018/09/20/new-report-encourages-use-multiple-measurements-student-placement&quot; rel=&quot;nofollow noreferrer&quot;&gt;overview article at Inside Higher Ed&lt;/a&gt; (2018). At SUNY 2016-2017 a disjunctive rule (&quot;or&quot;) was used; students could qualify for certain classes with either a high Accuplacer score, or high school gpa (HSGPA), or state exam, or high school rank. Reported results are 14% of student were placed in a higher math course, and 41% in a higher English course. Those who did get placed higher than the placement exam would have received supplemental &quot;corequisite&quot; support programs throughout the semester. This resulted in 12% more students completing credit-bearing English classes in the first semester, and a 3% increase for math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;California in 2017 passed a law requiring community colleges to use multiple measures for placements. One paper supporting this was Ngo, et. al., &lt;a href=&quot;https://pullias.usc.edu/wp-content/uploads/2013/10/Multiple_Measures_Brief.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Course Placement In Developmental Mathematics: Do Multiple Measures Work?&quot;&lt;/a&gt;, USC Rosier, 2013. The practice at the time was to use a placement test as the basis, and then add or subtract some points based on survey questions and HSGPA. The overall results are broadly similar to the SUNY findings. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the practice on the ground has changed rapidly between the time the OP asked this question (2014) and when I'm writing this answer (2020). For example, CUNY (where I work) has both announced the complete dissolution of developmental courses (in English, arithmetic, basic algebra, etc.), and also eliminated all system-wide placement testing. The current theory from the central university follows the SUNY practice above; any of several disjunctive criteria will qualify students for courses (SAT, or HSGPA, or state exam, or passing the class in high school, etc.). As noted, this should cut costs for placement testing and developmental education (namely, from classes that 80% of students would take to zero). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, among the difficulties this poses is that very close to half of the students at my institution are foreign transfer students, and do not have standard high school transcripts, SAT scores, or state exams. The CUNY policy in these cases seems to promote accepting &lt;em&gt;self-reported&lt;/em&gt; high school grades for this purpose. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among the presentations that were distributed to us to support these decisions was one by John J. Hetts of the &lt;a href=&quot;https://www.edresults.org/partners&quot; rel=&quot;nofollow noreferrer&quot;&gt;Educational Results Partnership&lt;/a&gt; (ERP), a non-profit working in this area and funded by corporate partners such as the Gates Foundation, Chevron, AT&amp;amp;T, etc. Hetts has a number of variations of a slide deck under the title &quot;Let Icarus Fly&quot;; &lt;a href=&quot;http://bit.ly/CCAMM2&quot; rel=&quot;nofollow noreferrer&quot;&gt;one can be found here&lt;/a&gt;. This gives a glowing report in favor of multiple measures (or disjunctive measures as I'd call the SUNY/CUNY practice), quickened student time-to-credential, and reduced costs. However, some of the findings seem suspect. E.g., writing in favor of self-reported HSGPA, he writes that, &quot;Under-reporting was 2-4X as common as over-reporting.&quot; (p. 45). But if we inspect the table on the prior page supporting this statement, we see that what under-reporting occurs is effectively negligible (e.g., in the topmost bracket, –0.04 point on average, for 1% of total GPA, N = 599), while the over-reporting is hugely significant (e.g., in the lowest bracket, +0.82 on average, or 80% of the whole GPA, N = 85). So: The students with the worst grades in high school show the highest propensity and payoff for misreporting grades to their advantage. If that wasn't an obvious outcome at the outset, one could reflect on principles such as the &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Dunning-Kruger-Effect-en.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dunning-Kruger effect&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Campbell&amp;#39;s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Campbell's Law&lt;/a&gt;, or the &lt;a href=&quot;http://freakonomics.com/2011/01/19/those-cheating-teachers-a-new-freakonomics-marketplace-podcast/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Freakonomics podcast&lt;/a&gt; episode on institutional-scale cheating in schools which pointed out what &quot;school administrators were unable (or unwilling?) to foresee... when a rule change gives people incentive to behave badly, a small share of them inevitably will&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary: Using multiple measures for college placements (such as placement exam, high school GPA, etc.) has rapidly become widespread in recent years, and on its face should be a reasonable idea. However, the ulterior motives of the organizations pushing these findings leaves one somewhat skeptical of the very positive results that they assert (and some of the interpretations seem downright disingenuous). Moreover, the findings have been very rapidly taken up to counter-intuitively abolish all placement testing and courses like basic algebra at many large colleges. Keep in mind that even the most positive proponents show the most significant acceleration in English programs, and small or negligible accelerations in math programs. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-01-22T18:28:41.387" LastActivityDate="2020-01-22T18:28:41.387" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17770" PostTypeId="2" ParentId="220" CreationDate="2020-01-21T04:47:22.650" Score="6" Body="&lt;h1&gt;Putnam 1963&lt;/h1&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\mathbb N$&lt;/span&gt; be the set of positive integers, and let &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb N\to\mathbb N$&lt;/span&gt; be a strictly increasing function such that &lt;span class=&quot;math-container&quot;&gt;$f(2)=2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(m)f(n)=f(mn)$&lt;/span&gt; for all positive integers &lt;span class=&quot;math-container&quot;&gt;$m,n$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\gcd(m,n)=1$&lt;/span&gt;. Find all such functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;h1&gt;Watered down version:&lt;/h1&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\mathbb N$&lt;/span&gt; be the set of positive integers, and let &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb N\to\mathbb N$&lt;/span&gt; be a strictly increasing function such that &lt;span class=&quot;math-container&quot;&gt;$f(3)=3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(m)f(n)=f(mn)$&lt;/span&gt; for all positive integers &lt;span class=&quot;math-container&quot;&gt;$m,n$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\gcd(m,n)=1$&lt;/span&gt;. Show that &lt;span class=&quot;math-container&quot;&gt;$f(n)=n$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;h1&gt;Proof:&lt;/h1&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; Note that if &lt;span class=&quot;math-container&quot;&gt;$f(m)=m$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$f(n)=n$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$n&amp;lt;m$&lt;/span&gt; since &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is strictly increasing. Thus &lt;span class=&quot;math-container&quot;&gt;$f(2)=2$&lt;/span&gt;. We now prove by induction that &lt;span class=&quot;math-container&quot;&gt;$f(2^i+1)=2^i+1$&lt;/span&gt;. It holds for &lt;span class=&quot;math-container&quot;&gt;$i=1$&lt;/span&gt;. Suppose it holds for &lt;span class=&quot;math-container&quot;&gt;$i=j$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$f(2^{j+1}+2)=f(2)f(2^j+1)=2^{j+1}+2$&lt;/span&gt;. Thus &lt;span class=&quot;math-container&quot;&gt;$f(2^{j+1}+1)=2^{j+1}+1$&lt;/span&gt;. Hence it holds for &lt;span class=&quot;math-container&quot;&gt;$i=j+1$&lt;/span&gt;. Hence it holds for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. Q.E.D.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Alteration for the Putnam problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; One simply needs to show &lt;span class=&quot;math-container&quot;&gt;$f(3)=3$&lt;/span&gt;, then the rest follows as above. Observe that &lt;span class=&quot;math-container&quot;&gt;$$f(3)f(5)=f(15)&amp;lt;f(18)=f(2)f(9)&amp;lt;f(2)f(10)=f(2)^2f(5)\\\implies f(3)&amp;lt;4\implies f(3)=3$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The alterations to the problem are to avoid the convoluted extra base case (focus on the main induction) and to provide the solution &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to make the end goal more clear. It is interesting to note that one has to be more creative than usual, as a backwards induction argument is used. However the solution is still not too complicated to figure out.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastEditorUserId="8571" LastEditDate="2020-04-18T20:02:53.073" LastActivityDate="2020-04-18T20:02:53.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17771" PostTypeId="2" ParentId="17736" CreationDate="2020-01-21T07:42:41.447" Score="5" Body="&lt;p&gt;A key idea in maths education, that at least becomes more visible at university level, is that people tend to start thinking of new ideas as &lt;em&gt;processes&lt;/em&gt;, but to do more advanced maths they need to move on to thinking of the same things as &lt;em&gt;objects&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, we are taught to think of &lt;span class=&quot;math-container&quot;&gt;$2+3$&lt;/span&gt; as an instruction to take &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and add &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; to it. Most people will write &lt;span class=&quot;math-container&quot;&gt;$2+3=5$&lt;/span&gt; to mean 'when I follow the instruction given the thing I end up with is &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But mathematically it is more correct to say that &lt;span class=&quot;math-container&quot;&gt;$2+3$&lt;/span&gt; is a number, and &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; is a number, and &lt;span class=&quot;math-container&quot;&gt;$2+3=5$&lt;/span&gt; is the statement that these two numbers are in fact the same number. This interpretation is rarely taught at school, meaning most people never get past thinking of maths as 'following some instructions to arrive at a number'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you are thinking that way, it is very hard to ask questions about the properties of &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;. Instructions aren't things we tend to think about the properties of. But if &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; changes to being an object (a binary operation), it makes more sense to ask about it being associative, commutative etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Going back to the original question, &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; is both a fraction and a division. More precisely, fractions and division are two different ways of thinking about the same mathematical thing. The key word you use is &lt;em&gt;interpreted&lt;/em&gt;. There is one mathematical object, but more than one way of thinking about it, and it is good to be prepared to think about it in different ways in general (although in a specific case there may be a good reason to pick one interpretation over the other).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside, you might like to consider &lt;span class=&quot;math-container&quot;&gt;$\frac{2x+3}{4-x}$&lt;/span&gt;. This is a number if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a number (ignoring details). But if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a formal variable then this is not a number. It's still a fraction, just in a different ring of fractions. It is still the result of division, or an instruction to divide. For practical purposes though, what will mostly help is to be prepared to think of it as &lt;em&gt;one thing&lt;/em&gt;, so you don't have to use up too much brain space thinking about it (see cognitive load theory).&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2020-01-21T07:42:41.447" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17772" PostTypeId="2" ParentId="17736" CreationDate="2020-01-21T19:09:15.383" Score="1" Body="&lt;p&gt;Adding another view to the point, one could view a fraction &lt;span class=&quot;math-container&quot;&gt;$\dfrac{a}{b}$&lt;/span&gt;, as a symbol indicating a change of measure unit. More precisely, talking about integers, at first, what does e.g. &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; mean? It is &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; of some unit/unitary quantity. For instance, &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; cats, or &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; pies etc. This can lead us to introduce the notion of &lt;span class=&quot;math-container&quot;&gt;$\dfrac{7}{1}$&lt;/span&gt;, as the symbol that denotes exactly the above; 7 instances of some (whole) unit (1). Then, we extend this notation to &lt;span class=&quot;math-container&quot;&gt;$\dfrac{7}{2}$&lt;/span&gt;, meaning that we talk about 7 objects of a unit seperated in two parts. Similarly, we can write &lt;span class=&quot;math-container&quot;&gt;$\dfrac{7}{3},\dfrac{7}{4}$&lt;/span&gt; etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this context, a fraction is not seen as the result of division but more as a shorthand for a process - changing measurement units. Under this view, division is &quot;hidden&quot; as a process within the fraction symbol.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One advantage of viewing fractions as shorthands for the &quot;measurement unit shift&quot; process is that it this definition stands in the middle of a purely procedural and a purely declarative perception of the fraction. It is neither entirely a process nor entirely a number - so, it remains easy to lean towards one of the two views of the fraction (number or division operator) whenever needed.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2020-01-21T19:09:15.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17774" PostTypeId="1" CreationDate="2020-01-21T21:08:30.317" Score="3" ViewCount="204" Body="&lt;p&gt;I am presenting an enrichment session on 3D geometry and quadric surfaces to able 15-year-old secondary school students. They know algebra but not calculus. They have learned about equations of parabolas and circles but not other conic sections such as ellipses and hyperbolas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking of explaining what 3D geometry is (&lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;-axis will be unfamiliar), showing a few examples such as the sphere and the cone, and then getting students to match the equations of surfaces with their graphs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any nice ideas for how to introduce this topic or for anything else that could be covered in this session?&lt;/p&gt;&#xA;" OwnerUserId="8900" LastEditorUserId="8900" LastEditDate="2020-02-04T19:56:08.257" LastActivityDate="2020-02-04T19:56:08.257" Title="Introducing quadric surfaces in high school" Tags="|secondary-education|geometry|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17775" PostTypeId="2" ParentId="17774" CreationDate="2020-01-22T13:18:01.623" Score="3" Body="&lt;p&gt;This may be a bit of a stretch, but once you introduce&#xA;the sphere, you could at least mention how to shade a sphere&#xA;for 3D graphics:&#xA;if &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is the angle between the light source ray and the normal&#xA;to the sphere at point &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, then shade &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; with&#xA;a light intensity proportional&#xA;to &lt;span class=&quot;math-container&quot;&gt;$\cos \theta$&lt;/span&gt;. This is known as &#xA;&lt;a href=&quot;https://www.scratchapixel.com/lessons/3d-basic-rendering/introduction-to-shading/diffuse-lambertian-shading&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lambertian shading&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when &lt;span class=&quot;math-container&quot;&gt;$\theta =0$&lt;/span&gt;, the light is head-on, and the&#xA;intensity is maximized. When &lt;span class=&quot;math-container&quot;&gt;$\theta \rightarrow 90^\circ$&lt;/span&gt;,&#xA;the intensity goes to zero:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/TqTIl.png&quot; width=&quot;250&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;Of course the same holds for any smooth surface, but&#xA;the surface normals for a sphere are the easiest to visualize.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many sources for this calculation, e.g.,&#xA;&lt;a href=&quot;https://courses.washington.edu/arch481/1.Tapestry%20Reader/4.Rendering/4.Shading/2.Flat%20Shading.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Flat Shading&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-01-22T13:18:01.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17776" PostTypeId="2" ParentId="17774" CreationDate="2020-01-22T15:17:02.577" Score="1" Body="&lt;p&gt;I think it's too early to be showing this sort of topic and discussing the connection to equations.  So resist that mathy temptation to only think of the topic analytically.  These kids haven't even done the equation of an ellipse yet, let alone analytical geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given you have to do this topic, the way to approach it is descriptively, not analytically.  Do something where you show the different shapes, give their name, perhaps indicate how they can be created by rotation (descriptively or with a movie, NOT via equations).  Any applications or instances in nature, architecture, machinery, fiction, the arts would be good also.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like the idea to do an activity, but I'm having a hard time thinking of one that is not hokey.  But my initial poor brainstorm:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*recognition test at period end&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*flash card recognition game&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*clay modeling (of 3-D and 2-D shapes)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to share the equations, I would do it in some en passent manner, where you are not requiring the kids to understand them.  Like just a random detail included in the descriptions, but not emphasized or explained.  Maybe even say &quot;you'll learn these in college&quot;.  But I'm not sure even of that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can make it so they have fun and recognize the names of the different new animals, then it will have been a good trip to the zoo.  Think of this more as an interest awakener than a skill builder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry I don't have the complete peeled-grape edible answer for you. It's a tough spot really.  But I think if you go with too much of an algebraic approach (which is the natural instinct of people with sophistication, like yourself) it will be a mistake, given your audience's algebraic newbieness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  It would also be helpful if you told us more of the basics (and considered them yourself in terms of how the constraints affect the solution space):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*where exactly are the kids in development?  (Will they be doing ellipse equations soon?)  Algebra 1 or 2?  (I realize that's US-centric.)  You actually gave us a fair amount of info, but anything else on the audience would help.  E.g. AGE. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*How long is your session?  (A useful tip:  try to end early, NOT late.  Both students/teacher will appreciate it.  Classic way to give good speeches!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*How bright are the kids?  Above average, average, below?  Affects how difficult you can be and also some remarks about college prep, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Maybe tweak (i.e. change) the topic?  Think a little outside the box.  The topic you mentioned looks like it was picked as a CLASS of equations.  This is a very normal way for you to think.  But is not really appropriate for the students yet.  Maybe do some general descriptive topic in solid geometry (e.g. weird shapes and examples in real life, see the excellent thread in this forum on that topic.  Extra credit if you use your Google-fu to find it.)  Another very fun (and useful) descriptive topic is 2-D and 3-D symmettry.  No, don't do any discussion using matrices or abstract algebra.  Just list the types of operations.  Then students can classify objects (real or pictures), to list their elements at least.  You can even do easy hands on toothpick/styrofoam ball (or clay) activities.  Another fun topic is the platonic solids, including higher dimension types (DESCRIPTIVE treatment, remember).&lt;/p&gt;&#xA;" OwnerUserId="13375" LastEditorUserId="13375" LastEditDate="2020-01-22T17:49:31.727" LastActivityDate="2020-01-22T17:49:31.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17777" PostTypeId="2" ParentId="17723" CreationDate="2020-01-22T15:21:21.847" Score="8" Body="&lt;p&gt;The title of the question asks for a proof, but the actual question seems to be more about providing some motivation. For students at this level, I would aim more for motivation that they will understand rather than a proof that they will just see as a mysterious magic trick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's how I motivate it. I have the students work through &lt;span class=&quot;math-container&quot;&gt;$i^0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$i^1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$i^2$&lt;/span&gt;, ... with me, while I stand at the board and plot each point in the complex plane as we go along. With college students who have had a previous, brief intro to complex numbers, I find that many are surprised by the fact that it goes around in a circle, and they volunteer that it's cool.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Optionally, do a second example in which you plot &lt;span class=&quot;math-container&quot;&gt;$(\sqrt i)^n$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once they've seen this, it should be clear that you can take any number &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; on the unit circle, and when you do &lt;span class=&quot;math-container&quot;&gt;$b^r$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is real, you're going around the unit circle with an angle proportional to &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;. Clearly it would be convenient to find the base such that the constant of proportionality is 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For students at this level, who have not even officially learned limits, I would just jump from that to stating Euler's formula without proof. If this is a precalculus class, then as preparation for calculus I think it would be valuable to have them see an informal discussion of a limit like &lt;span class=&quot;math-container&quot;&gt;$\lim_{n\rightarrow\infty} (1+x/n)^n=e^x$&lt;/span&gt;, but I think that would be valuable in the more familiar context with real numbers. If there are one or two students in the class who could really absorb something like this as applied to complex numbers, then it might be appropriate to provide them with some kind of written treatment that they could read if they're interested, at their leisure when they have time to think it over.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-01-22T15:21:21.847" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17778" PostTypeId="1" CreationDate="2020-01-22T21:25:22.277" Score="8" ViewCount="408" Body="&lt;p&gt;This semester I have decided to give students a simple question or two at the beginning of every calculus class that will trap them into making the most common errors that we all know...e.g. the freshman dream. There is a list/partial classification of such errors &lt;a href=&quot;https://sites.calvin.edu/scofield/courses/materials/tae/tae.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;here.&lt;/a&gt; They generally involve crimes with square roots, fractions, function composition, lines (thinking they have written the equation for a tangent line that is not a straight line at all, for example), exponents, logarithms and strange other beliefs such as all functions....especially trig functions...being linear.&lt;/p&gt;&#xA;&lt;p&gt;I always have imagined that someone would have constructed a nice list of simple &amp;quot;trap&amp;quot; problems that instructors could use to construct the kinds of short quizzes that I'm looking to give, but surprisingly there isn't a really good repository of quick, but good, problems that aim to isolate the most common student crimes.&lt;/p&gt;&#xA;&lt;p&gt;As a specific restriction, for example, I'd really like to avoid problems that just say &amp;quot;simplify&amp;quot; or &amp;quot;rationalize&amp;quot;, because these commands reinforce the cookbook thinking that got students into this mess in the first place. I would, however, like the problems to be short...i.e. take a competent student around 5 minutes to complete.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Where is a good source of short problems (not of &amp;quot;simplify&amp;quot; or &amp;quot;rationalize&amp;quot; type) that are &amp;quot;traps&amp;quot; to elicit common algebra/trig/precalculus student blunders? I'm interested in online sources, or your favorite problems listed as answers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I think it is important to point out, due to the valid concern expressed by Daniel R. Collins and echoed by others, that I am asking for these sorts of questions &lt;em&gt;precisely because&lt;/em&gt; telling students about pitfalls seems to inevitably lead to them falling in precisely the pits you told them not to fall in. The philosophy of the question is that questions and exercises answered by the students are just that...questions and exercises answered by the students. Certainly nobody is claiming that if a student solves an exercise that elicits a particularly pernicious pitfall that the student will suffer a spectacular fall from grace and will then be unable to avoid making the mistake! The idea is that problems that show these mistakes to their would-be solver will be memorable in the right way.&lt;/p&gt;&#xA;&lt;p&gt;This said, if indeed tricky problems like this do lead to &amp;quot;black hole errors&amp;quot; then this would explain the shenanigans I keep seeing on people's papers...&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2022-04-28T11:19:50.670" LastActivityDate="2022-04-28T11:19:50.670" Title="Algebra/trig/precalculus review questions that elicit common student errors" Tags="|algebra|students-mistakes|precalculus|fractions|remedial-courses|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="17779" PostTypeId="2" ParentId="17778" CreationDate="2020-01-23T02:52:50.590" Score="4" Body="&lt;p&gt;A common example is that students think that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{ab}=\sqrt a \sqrt b$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt;; try tricking them by saying that &lt;span class=&quot;math-container&quot;&gt;$1=\sqrt{1}=\sqrt{-1\cdot -1} = \sqrt{-1}\sqrt{-1} = i\cdot i = -1$&lt;/span&gt;. Similarly students will not often realize that &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sqrt{x^2}$&lt;/span&gt; is NOT &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, but rather &lt;span class=&quot;math-container&quot;&gt;$|x|$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Extraneous solutions (i.e. solutions that pop out when solving but are actually incorrect) are also tricky; students usually do not check: &lt;a href=&quot;https://en.wikipedia.org/wiki/Extraneous_and_missing_solutions&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Extraneous_and_missing_solutions&lt;/a&gt;; &lt;a href=&quot;http://mathmistakes.info/facts/AlgebraFacts/learn/ctm/extra.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://mathmistakes.info/facts/AlgebraFacts/learn/ctm/extra.html&lt;/a&gt;. A cool example of this is if you have some sort of equation like &lt;span class=&quot;math-container&quot;&gt;$\sin(x) + \cos(x) = 1.2$&lt;/span&gt;, you can solve it by making it into &lt;span class=&quot;math-container&quot;&gt;$\sin(x) = 1.2-\cos(x) \implies 1-\cos^2(x) = 1.44 - 2.4\cos x + \cos^2x$&lt;/span&gt; and solving it like a quadratic equation. However, students are often so elated that this &quot;trick&quot; produced a solution to an equation which at first doesn't look solvable that they do not check their solutions to make sure the signs are in fact correct and that it doesn't go outside bounds of &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos(x)$&lt;/span&gt;; i.e. they might get that &lt;span class=&quot;math-container&quot;&gt;$\cos(x) = 1.1$&lt;/span&gt; and not realize that it is impossible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When working with limits students see&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to 2}\frac{(x-2)(x+1)}{(x-2)} = 3$$&lt;/span&gt;&#xA;but students may not realize that &lt;span class=&quot;math-container&quot;&gt;$f(x) = \frac{(x-2)(x+1)}{(x-2)}$&lt;/span&gt; is NOT defined at &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope these are useful.&lt;/p&gt;&#xA;" OwnerUserId="13084" LastActivityDate="2020-01-23T02:52:50.590" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17780" PostTypeId="2" ParentId="17778" CreationDate="2020-01-23T05:17:16.450" Score="6" Body="&lt;p&gt;I made a handout with 10 true false questions, titled Algebra Temptations. I've put it on google drive &lt;a href=&quot;https://drive.google.com/file/d/1IaG-oW5N7gseXqSIJ0xvfP2tH7i0tSX7/view?usp=sharing&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. I have students work in groups to decide which are true. I think it makes a good activity, though there is the risk mentioned in another answer of reinforcing the wrong idea.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-01-23T05:17:16.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17781" PostTypeId="2" ParentId="17723" CreationDate="2020-01-23T09:29:21.937" Score="1" Body="&lt;p&gt;A guiding principle for operations extension is the principle of permanence: a definition of an operation should be extended from a restricted domain to a wider one in such a way as to conserve the crucial properties of the operation.&lt;/p&gt;&#xA;&lt;p&gt;The crucial algebraic properties of exponentiation: &lt;span class=&quot;math-container&quot;&gt;$a^{b+c}=(a^b)(a^c)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(a^b)^c=a^{bc}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(ab)^c=(a^c)(b^c)$&lt;/span&gt; compels us to define the exponentiation in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt; in the usual way, e.g. &lt;span class=&quot;math-container&quot;&gt;$a^{-n}=\frac{1}{a^n}$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$(a^n)(a^{-n})=a^{n-n}=a^0=1.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;To extend the exponentiation to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; you should count continuity of exponentiation among its crucial properties and then define it on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; in such a way as to conserve the crucial properties: its algebraic properties and the continuity. What you get is the usual exponentiation on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://www.researchgate.net/publication/267107387_Peacock%27s_principle_and_Euler%27s_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt; I show that if a continuous function &lt;span class=&quot;math-container&quot;&gt;$\exp: \mathbb{C} \to \mathbb{C}$&lt;/span&gt; satisfies the three algebraic properties I listed above, and also satisfies &lt;span class=&quot;math-container&quot;&gt;$\exp(\bar{z}) = \overline{\exp(z)}$&lt;/span&gt; then it must be of the form &lt;span class=&quot;math-container&quot;&gt;$\exp(x+iy) = e^x (\cos(cy) + i\sin(cy))$&lt;/span&gt; for some real constant &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.  However, these properties alone do not allow us to determine that &lt;span class=&quot;math-container&quot;&gt;$c=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The fact that &lt;span class=&quot;math-container&quot;&gt;$c=1$&lt;/span&gt; iff that function is (complex) differentiable is a trivial consequence of Cauchy-Riemann conditions.&lt;/p&gt;&#xA;&lt;p&gt;To conclude: by the principle of permanence of the algebraic properties and the continuity we are almost compelled to Euler’s formula. We are definitely compelled to it if we extend the principle to the (complex) differentiability.&lt;/p&gt;&#xA;" OwnerUserId="10261" LastEditorUserId="117" LastEditDate="2022-08-13T10:26:35.277" LastActivityDate="2022-08-13T10:26:35.277" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17782" PostTypeId="2" ParentId="501" CreationDate="2020-01-23T09:35:17.290" Score="3" Body="&lt;p&gt;In Spain such placement is impossible, because university curricula are fixed in the first few years (there are no choices to be made). On the other hand, entrance into a given degree program is based on a mix of high school grades and performance on (state level) exams administered at the end of the final year of high school (these cover high school calculus and linear algebra). In a sense the placement is made at the level of the degree program. Where I teach there are 7 different engineering degrees all of which have essentially the same first year courses (the calculus course is the same for all) but which have quite different entrance requirements. Looking at ten years of data (so about 6000 students in total) student performance in essentially the same class is strongly linked to the degree program, so implicitly strongly linked to entry level, that is to performance on the exams taken at the end of high school and grades in high school. A reasonable inference is that these exams are effective in discriminating readiness. The conclusion is that a well designed exam should facilitate placement, particularly if coupled with information about performance in relevant high school courses. The test to which I refer covers a sort of low level approximation of what students will encounter in the university. For example, with respect to calculus, it tests operational ability more than conceptual understanding, so a student who has done well on it at least knows the integrals and derivatives of polynomials and trigonometric functions and the like. Such a student is prepared for a university calculus class. A rule of thumb is that one wants to test something like the material one will teach in the first two or three weeks of a course. A student who can handle that material will probably do well in the course, and a student who cannot should probably take a predecessor course. &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2020-01-23T09:35:17.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17783" PostTypeId="1" CreationDate="2020-01-24T16:25:46.550" Score="3" ViewCount="1239" Body="&lt;p&gt;This is long so...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;TLDR:&lt;/strong&gt; Proposing the math community steer away from the misleading term &lt;em&gt;unique&lt;/em&gt;, with respect to functions and algebraic operations. Instead, use &lt;em&gt;unambiguous&lt;/em&gt;. Why not? Analysis below.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I just started to get back into math after a year off, and after cracking open Pinter's book on intro abstract algebra I was reminded of a phrasing issue I've always had. The phrase &quot;uniquely defined&quot; or &quot;uniquely mapped&quot;. For example, the operation &lt;span class=&quot;math-container&quot;&gt;$a*b$&lt;/span&gt; defined by normal multiplication &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; is considered &lt;em&gt;uniquely&lt;/em&gt; defined—the operation takes any ordered pair in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; and produces one, and only one value in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. As Pinter describes &lt;em&gt;uniquely defined&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In other words, the value of &lt;span class=&quot;math-container&quot;&gt;$a*b$&lt;/span&gt; must be given unambiguously.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After a few moments, it becomes clear to the experienced math reader that this is a defining property shared with functions—an operation can't have more than one output. We get it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as I was shaking off the cobwebs, I remembered this terminology being a small barrier for me when I first started learning about functions and operations (and other objects), one that, I'm ashamed to admit, slowed me down once again. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reflecting over the past hour, I've begun to view my learning friction as the result of poor &lt;em&gt;encoding&lt;/em&gt;—the the word &lt;em&gt;unique&lt;/em&gt; doesn't align with intuition. When I first read the word unique I interpret it as well... the normal word &lt;em&gt;unique&lt;/em&gt;. As Merriam-Webster defines it.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;being the only one&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Applying that to function/operation context, there's a few interpretations to be had, but one the first and most natural one that comes to my mind is that &lt;em&gt;each ordered pair is assigned a value unique to that ordered pair&lt;/em&gt;. Which is stronger definition of uniqueness. Clearly, the example above, with normal multiplication, does not meet this stronger definition, since &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(b,a)$&lt;/span&gt; map to the &lt;em&gt;same&lt;/em&gt; value under normal multiplication on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. In other words, &lt;span class=&quot;math-container&quot;&gt;$(4, 3)$&lt;/span&gt; is not the &quot;only one&quot; since it shares a value with &lt;span class=&quot;math-container&quot;&gt;$(2,6)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(3, 4)$&lt;/span&gt;, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I acknowledge that in this function/operation context there is an argument that mappings in their &lt;em&gt;entirety&lt;/em&gt; are always &lt;em&gt;unique&lt;/em&gt;, since we're using ordered values—one unique ordered pair, for example, maps to a value. But this is not how &lt;em&gt;uniqueness&lt;/em&gt; is portrayed or described in textbooks, in textbooks emphasis is on function output or operation value being singular. And I further find this justification for the term &lt;em&gt;unique&lt;/em&gt; to be weak, since it leans on the uniqueness of ordered values and not the combining process of the elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, mathematical terms and definitions are not meant to align perfectly with normal language. So I'm not arguing that we restrict math terminology to only language aligned perfectly with intuition, however we do (and should) always make an effort to name things appropriately. I think that here, the term &lt;em&gt;unique&lt;/em&gt; violates naming principles and is misleading. I propose that Pinter's own terminology better. Let's call this defining property of functions and operations &lt;strong&gt;&lt;em&gt;unambiguousness&lt;/em&gt;&lt;/strong&gt;. E.g., functions and operations are &lt;em&gt;unambiguous&lt;/em&gt;. &lt;em&gt;Unambiguity&lt;/em&gt; seems like the word we actually mean, which is probably why Pinter wrote it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wrote this because, anything that can be done to remove learning friction should be.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;So... My Question&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Can you see a good reason to not shift gears and teach functions/operations as &lt;em&gt;unambiguous&lt;/em&gt;. Any obvious faults? And of course, I'd be telling students about classic terminology.&lt;/p&gt;&#xA;" OwnerUserId="13380" LastActivityDate="2020-01-25T20:51:54.480" Title="The term &quot;unique&quot; for functions and operations" Tags="|terminology|abstract-algebra|functions|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17784" PostTypeId="2" ParentId="17783" CreationDate="2020-01-24T17:22:01.100" Score="5" Body="&lt;p&gt;I personally use the following terminology:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A relation &lt;span class=&quot;math-container&quot;&gt;$R \subset A \times B$&lt;/span&gt; is said to be &lt;strong&gt;single-valued&lt;/strong&gt; if &lt;span class=&quot;math-container&quot;&gt;$(a,b_1) \in R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(a,b_2) \in R$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$b_1 = b_2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A relation &lt;span class=&quot;math-container&quot;&gt;$R \subset A \times B$&lt;/span&gt; is said to be &lt;strong&gt;total&lt;/strong&gt; if for all &lt;span class=&quot;math-container&quot;&gt;$a \in A$&lt;/span&gt; there exists &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$(a,b) \in R$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A relation which is both single valued and total is a function.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-01-24T17:22:01.100" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17785" PostTypeId="2" ParentId="17783" CreationDate="2020-01-24T20:17:19.550" Score="14" Body="&lt;p&gt;I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students have trouble with the notion of a function &lt;em&gt;because it's hard&lt;/em&gt;.  The way they're going to get a handle on it is by struggling with it, encountering the hard parts of the definition, and finding and eliminating their misconceptions about it.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Reflecting over the past hour, I've begun to view my learning friction as the result of poor encoding—the the word unique doesn't align with intuition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think you should be skeptical of this reflection.  When we're learning, there's a common tendency to credit the very last thing that makes the idea fall into place, and miss that all the seemingly aimless struggling that came before it was actually what prepared us for that last insight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe if functions had been described to you using &quot;unambiguous&quot; rather than &quot;unique&quot;, you would have understood them sooner - but I think it's more likely that you would have needed about the same amount of thinking about the idea to sort through all the possible misunderstandings, and some other aspect would appear to have been the last insight you needed.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2020-01-24T20:17:19.550" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17786" PostTypeId="2" ParentId="17783" CreationDate="2020-01-25T10:46:02.103" Score="6" Body="&lt;p&gt;In Germany, we already do this. A function is introduced as an unambiguous mapping in 7th grade (~13 years). While I don't have any data on this, I doubt that German students do significantly better due to this choice of words.&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2020-01-25T20:51:54.480" LastActivityDate="2020-01-25T20:51:54.480" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17788" PostTypeId="2" ParentId="17723" CreationDate="2020-01-26T05:07:28.983" Score="2" Body="&lt;p&gt;I think calculus is not technically needed if you are merely interested in algebraic features of the imaginary exponential. In particular, let us note that:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \cos (\theta + \beta) = \cos (\theta) \cos (\beta) - \sin( \theta) \sin(\beta) $$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \sin (\theta + \beta) = \sin(\theta) \cos (\beta)+ \sin( \beta) \cos (\theta). $$&lt;/span&gt;&#xA;These may both be derived without calculus (provided you use a geometric definition of sine and cosine). Then, since &lt;span class=&quot;math-container&quot;&gt;$i^2=-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a+ib=c+id$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$a=c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b=d$&lt;/span&gt; we can package the equations together as one wonderful complex equation:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\cos(\theta + \beta)+i\sin(\theta + \beta) &amp;amp;= \cos \theta \cos \beta +i^2 \sin \theta \sin\beta +i \left( \sin \theta \cos \beta+ \sin \beta \cos \theta \right) \\&#xA;&amp;amp;= (\cos \theta + i \sin \theta)(\cos \beta +i \sin \beta)&#xA;\end{align}&lt;/span&gt;&#xA;Thus, if we simply introduce the &lt;strong&gt;notation&lt;/strong&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ e^{i \theta} = \cos \theta + i \sin \theta $$&lt;/span&gt;&#xA;then the law of trigonometry above is simply written:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ e^{ i (\theta+ \beta)} =  e^{ i\theta}e^{ i \beta} $$&lt;/span&gt;&#xA;Moreover, we can solve &lt;span class=&quot;math-container&quot;&gt;$e^{i \theta} = \cos \theta + i \sin \theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e^{-i \theta} = \cos (-\theta) + i \sin (-\theta)  = \cos \theta - i \sin \theta$&lt;/span&gt; and obtain from adding and subtracting equations the elegant algebraic formulations of sine and cosine:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \cos \theta = \frac{1}{2}\left( e^{i \theta}+e^{-i\theta} \right) \qquad \&amp;amp; \qquad \sin \theta = \frac{1}{2i}\left( e^{i \theta}-e^{-i\theta} \right) $$&lt;/span&gt;&#xA;from which we may explicitly and algebraically derive nearly any trig. identity.&#xA;Alternatively, introduce &lt;span class=&quot;math-container&quot;&gt;$e^{i\theta}$&lt;/span&gt; as a notation for &lt;span class=&quot;math-container&quot;&gt;$(\cos \theta , \sin \theta)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$e^{i(\theta+\beta)} = e^{i \theta}e^{i\beta}$&lt;/span&gt; can be derived by direct geometric reasoning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fine, but... why use the notation &quot;e&quot; for &lt;span class=&quot;math-container&quot;&gt;$e^{i \theta}$&lt;/span&gt; ? I would simply say that calculus is best used to explain this because the exponential is an object which properly belongs to calculus. Furthermore, I use the notation &lt;span class=&quot;math-container&quot;&gt;$e^{i \theta}$&lt;/span&gt; because it is standard notation which you will continue using once you know more math. Since math is a language it is good to learn the words even if we cannot stomach the etymology at a given juncture.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-01-26T05:07:28.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17790" PostTypeId="2" ParentId="17739" CreationDate="2020-01-26T15:09:46.530" Score="1" Body="&lt;p&gt;For a Calculus I class, I usually:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Don't teach limits first.  Limits, as they are currently taught, lead to confusion.  You don't do anything with them except prove the derivative.  After that they are largely forgotten, basically leaving students feeling they wasted their time, &lt;em&gt;and&lt;/em&gt; they are quickly forgotten.  Instead, teach limits at the &lt;em&gt;end&lt;/em&gt; to justify the the fundamentals of calculus, and show additional uses with infinity.  &lt;/li&gt;&#xA;&lt;li&gt;Focus on differentials instead of derivatives.  Most of calculus is simplified if your focus is on differentials.  It unifies single-variable, implicit, and multivariable versions of calculus in an intuitive way, and leverages students' knowledge of algebra.&lt;/li&gt;&#xA;&lt;li&gt;Get rid of logarithmic differentiation.  There is a simple rule that can be used in place of logarithmic differentiation.  Using this rule makes &lt;em&gt;all&lt;/em&gt; of Calculus I derivatives a simple application of rules. If you aren't familiar, the rule is &lt;span class=&quot;math-container&quot;&gt;$d(x^y) = yx^{y - 1} dx + \ln(x)x^y\,dy$&lt;/span&gt;.  Its a combination of the power rule and exponent rule (and actually points towards an interesting property of partial derivatives).&lt;/li&gt;&#xA;&lt;li&gt;Treat the integral as primarily an infinite sum of infinitely small values.  Areas under the curve are an &lt;em&gt;application&lt;/em&gt; area of the integral, not the main thing.  This unifies the usage of integration across all the different geometric uses.  You are simply finding an infinitely small geometric figure and summing all the values.  In &quot;area under the curve&quot;, the figures are boxes, &lt;span class=&quot;math-container&quot;&gt;$y\cdot dx$&lt;/span&gt;, in revolutions around &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, the figures are circles, &lt;span class=&quot;math-container&quot;&gt;$\pi\,y^2dx$&lt;/span&gt;.  A lot of curriculum tries to get tricky, by moving parts of this (like &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;) outside of the integral.  The problem is that this makes it &lt;em&gt;less&lt;/em&gt; obvious to students what is happening - a simple sum of regular geometric shapes/formulas.&lt;/li&gt;&#xA;&lt;li&gt;When you do introduce limits, the hyperreal version of limits greatly reduces the cognitive load compared to epsilon-delta proofs, &lt;em&gt;and&lt;/em&gt; the students are more likely to remember how to do them.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;See the paper &quot;Simplifying and Refactoring Introductory Calculus&quot; for more details.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2020-01-26T15:09:46.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17791" PostTypeId="2" ParentId="12" CreationDate="2020-01-26T15:13:57.680" Score="8" Body="&lt;p&gt;My book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1944918027&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus from the Ground Up&lt;/a&gt; focuses on differentials, and uses it to provide a unification of process and simplification of understanding of a lot of different parts of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To read about the thought process that led to the book you can see &lt;a href=&quot;https://arxiv.org/pdf/1811.03459.pdf&quot; rel=&quot;noreferrer&quot;&gt;this arXiv link&lt;/a&gt;; the focus on differentials that you are asking for led naturally to a refactoring of the way introductory calculus is presented. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Differences from other books:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The arXiv link gives some important information, but I'll repeat some of it here.  First of all, the focus of the &lt;em&gt;entire&lt;/em&gt; book is on differentials.  We do a lot of derivatives, but the focus is always on differentials, and for several important reasons.  First, it unifies several important practices into a single system - single-variable, multivariable, and implicit differentiation all has the exact same process.  Second, it makes the different geometric integrals more obvious.  The integral is presented as a sum of infinitesimals, not as an area under the curve (which becomes merely one of the application areas).  The integral simply sums up whichever geometry is being used. &lt;span class=&quot;math-container&quot;&gt;$\int y\,dx$&lt;/span&gt; for summing areas of rectangles, &lt;span class=&quot;math-container&quot;&gt;$\int \pi y^2\,dx$&lt;/span&gt; for summing volumes of cylinders, and &lt;span class=&quot;math-container&quot;&gt;$\int \sqrt{dx^2 + dy^2}$&lt;/span&gt; for summing arc lengths.  The way you are asked to memorize them is exactly what the geometry states. For instance, many books want you to have the volume of cylinders as &lt;span class=&quot;math-container&quot;&gt;$\pi \int y^2\,dx$&lt;/span&gt;.  That's correct, but moving the &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; outside means that it no longer &lt;em&gt;looks like&lt;/em&gt; the volume of a cylinder equation for students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, the book includes a rule that seems to have gone missing for doing differentials of the form &lt;span class=&quot;math-container&quot;&gt;$u^v$&lt;/span&gt;.  For those who don't know (because it is missing in most modern books), &lt;span class=&quot;math-container&quot;&gt;$d(u^v) = vu^{v - 1}du + \ln(u)u^vdv$&lt;/span&gt; (I wish the font for &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; had a more distinct look here, but oh well).  Many books teach &quot;logarithmic differentiation&quot; for this, but it is wholly unnecessary.  Just like all the other differentials, all you need is the rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also try to include additional life lessons that we can learn from calculus.  For instance, in the discussion of Taylor polynomials, I discuss how this can be used as a template for solving impossible problems (not only in math but anywhere).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I wanted to make a note on the second differential, because it came up in the discussion of Keisler.  I don't make a big deal about it (I put it in the Appendix), but I actually introduce a form for the second derivative that makes the chain rule for the second derivative work algebraically.  Generally, in the text, I avoid this situation by simply introducing a variable for the first derivative, and then take the derivative of that variable.  However, in the appendix I show that second differentials can be made algebraic by making the second derivative &lt;span class=&quot;math-container&quot;&gt;$\frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2}$&lt;/span&gt;.  If that looks strange to you, you can derive it for yourself by simply taking the derivative of &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt;.  Note that &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; is a quotient, so you would use the quotient rule to take the derivative of it.  This leads to differentials that are 100% algebraically manipulable.  Most texts focusing on differentials don't tell you either the problem nor the solution for using second differentials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The structure of the book differs from &quot;Calculus Made Easy&quot; in that it &lt;em&gt;starts&lt;/em&gt; with derivatives, since a slope is more intuitive for people coming from algebra.  Unlike Keisler, it saves discussion of limits for the end of the book.  Essentially, it gives you the intuition and the toolset &lt;em&gt;first&lt;/em&gt;, and then, at the end, goes into a bit more formally the underpinnings of what makes it work.  I find that students prefer this approach.  Like Keisler, I use the hyperreal numbers (though I don't formally introduce them until the last third of the book, which focuses on the infinite).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, I always try to write things in such a way as to focus the student on the intuitions behind everything, so that learning calculus doesn't just teach them calculus, but it &lt;em&gt;improves their thinking&lt;/em&gt;.  For instance, when talking about the other geometric uses of the integral (volumes from cylinders, volumes from shells, arc lengths, etc.), I gave a general mental mechanism that is used to generate all of these.  (a) the problem can be estimated by a formula, (b) the problem can be divided into subproblems, (c) each subproblem must have the same form as (a), (d) the result must be attainable by &lt;em&gt;adding&lt;/em&gt; the results of the subproblems, and (e) increasing the number of subdivisions &lt;em&gt;improves&lt;/em&gt; the accuracy of the estimation method.  The goal here is to show the students how the thought process works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, my student's also love the fact that I show where all of their formulas that they learned in previous math classes come from.  I show how to derive the interest rate formula, the volume of a cone formula, and the volume of a sphere formula.  In fact, that's another aspect of the book - I teach how to derive &lt;em&gt;formulas&lt;/em&gt;.  We use calculus to derive the vertex formula for quadratics, and a homework problem is deriving the vertex formula for cubics.  I tell students that calculus is the &quot;where babies come from&quot; of math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;NOTE - I edited this to include more details about the book and what makes it different because I was requested to below.  Sorry if this comes off as more of an advertisement than was intended.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastEditorUserId="9594" LastEditDate="2020-01-29T12:59:38.253" LastActivityDate="2020-01-29T12:59:38.253" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="17792" PostTypeId="2" ParentId="12776" CreationDate="2020-01-26T21:57:51.950" Score="3" Body="&lt;p&gt;As Benjamin Dickman, I am also not aware of any texts or other resources aimed directly at high-school (or earlier) teachers, &lt;strong&gt;but&lt;/strong&gt; there are a number of APOS oriented studies at these levels that should be accessible and may contain some concrete ideas on structuring e.g. lesson plans. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a small selection of texts that you and your colleagues may find interesting:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I. Arnon, P. Nesher, R. Nirenburg &lt;a href=&quot;https://link.springer.com/article/10.1023/A:1017998922475&quot; rel=&quot;nofollow noreferrer&quot;&gt;Where do Fractions Encounter their Equivalents? – Can this Encounter Take Place in Elementary-School?&lt;/a&gt;, International Journal of Computers for Mathematical Learning, 6 (2), 2001.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;E. Dubinsky, R. Wilson, &lt;a href=&quot;https://eric.ed.gov/?id=EJ995529&quot; rel=&quot;nofollow noreferrer&quot;&gt;High school students' understanding of the function concept&lt;/a&gt;, Journal of Mathematical Behavior, 32 (1), 2013.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;E. Dubinsky, K. Weller, I. Arnon, &lt;a href=&quot;https://link.springer.com/article/10.1080/14926150902817381&quot; rel=&quot;nofollow noreferrer&quot;&gt;Preservice Teachers’ Understanding of the Relation Between a Fraction or Integer and Its Decimal Expansion&lt;/a&gt;, Canadian Journal of Science, Mathematics and Technology Education, 9 (1), 2009.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;K. F. Hollebrands, &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/S073231230300004X&quot; rel=&quot;nofollow noreferrer&quot;&gt;High school students’ understandings of geometric transformations in the context of a technological environment&lt;/a&gt;, Journal of Mathematical Behavior, 22 (1), 2013.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;T. Tossavainen, &lt;a href=&quot;https://scholarworks.umt.edu/tme/vol6/iss3/10/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Who can solve &lt;span class=&quot;math-container&quot;&gt;$2x=1$&lt;/span&gt;? An analysis of cognitive load related to learning linear equation solving&lt;/a&gt;,&#xA;The Mathematics Enthusiast, 6 (3), 2009.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="56" LastActivityDate="2020-01-26T21:57:51.950" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17793" PostTypeId="1" CreationDate="2020-01-27T14:52:46.407" Score="1" ViewCount="152" Body="&lt;p&gt;Already knowing that the famous quotation &quot;there is no royal road to mathematics&quot;, I believe that the most efficient and best way to learn mathematics is to make it intuitive to oneself, at least to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I can't give any intuitive meaning to given statement, then I can't do anything. Likewise, if I can &quot;see&quot; how it is going on, then I can prove or disprove something easily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this is why I want to know the way to assimilate mathematical concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've tried to prove any theorem on book I've been reading without peeking up the book, and solve exercises as many as possible time permits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But no such methods immediately give me an intuition... only time permits me to obtain it...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently, I've read two opinion related to this topic on stackexchange, Paul Garrett said that one need to &quot;rehearse&quot; the idea(though I couldn't understand what &quot;rehearse&quot; means really) and Yuval Filmus said that if one wants to assimilate the idea, one needs to reflect on what one's learnt and ask oneself interesting question and try to answer on one's own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somehow I guess they were talking about seemingly similar method. what is it? any advice will be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="13388" LastEditorUserId="13388" LastEditDate="2020-01-28T00:43:04.097" LastActivityDate="2020-01-29T07:14:12.067" Title="How do I assimilate mathematical concept?" Tags="|self-learning|intuition|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17794" PostTypeId="2" ParentId="17793" CreationDate="2020-01-27T16:33:00.580" Score="5" Body="&lt;p&gt;I agree that insight sometimes takes time, but don't dismiss the benefit of exercises.  Sometimes, the insight monster will lurk in the background in your subconscious.  If you do exercises, it may come out and give you the revelations later, not directly from the drill, but somehow aided by the time/exposure.  If you solely chase insight and don't build your computational muscles, you probably will be inefficient at gaining mastery of either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest to read some of the writings of Richard Feynman.  He was definitely in favor of insight (not mindless drill).  But he also had huge value for sort of building a repertoire of tricks.  Also, he felt that certain sorts of manipulations should become automatic and easy from practice (calculus and even algebra).  Having these down cold (&quot;bing, bing, bing&quot;) makes it easier to look at physics (perhaps for you higher math), without getting confused by errors.  In addition, at one point he felt weak in classical E&amp;amp;M (and needed it for something).  So he went and got a standard textbook and worked every single problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bottom line:  I would be worried by the spending four years and not describing your plan more strategically and then within that &quot;as much exercises as time permits&quot;.  I'd be very worried you are spinning your wheels and wasting a productive time of your life.  These years are golden.  Be savvy with how you spend them.&lt;/p&gt;&#xA;" OwnerUserId="13375" LastActivityDate="2020-01-27T16:33:00.580" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17795" PostTypeId="2" ParentId="17693" CreationDate="2020-01-28T06:09:06.327" Score="4" Body="&lt;p&gt;Regarding retention, doing lots of problems can help with this. Also, asking lots of questions about what you’re reading in the book, and trying to figure them out. You mentioned field extensions, so one question might be:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“It’s a bit counterintuitive that this book spends more time talking about field extensions than about fields themselves. What’s so special about them?”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An answer might include things like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A field, by itself, isn’t a vector space. In order to call something a vector space, you need to be able to say what field it’s a vector space over, and we need the smaller field in the field extension for that to be defined. For example, even a basic statement like &quot;the complex numbers are two dimensional&quot; isn't really a statement about the complex numbers, it's a statement about the field extension &lt;span class=&quot;math-container&quot;&gt;$[\mathbb{C}: \mathbb{R}]$&lt;/span&gt;; when we look at the complex numbers in isolation we have no notion of dimension.&lt;/li&gt;&#xA;&lt;li&gt;We often want to look at fields created by adding the roots of a polynomial to some other field, in which case it’s useful to distinguish between the “new” field and the “old” one.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Another question might be: &#xA;“This book defines a field extension as a pair of fields and an inclusion map from one to the other. But isn’t that pretty much the same thing as taking a subfield of the bigger field? What’s the point of explicitly having an inclusion map?”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And an answer to that might be something like “Well, sometimes we want to construct a bigger field from a smaller field, in a way that the smaller field embeds canonically into the larger field, but technically isn’t set-theoretically the same as its image in the big field (e.g. taking the splitting field of a polynomial). In this case, the definition with the inclusion map makes things a bit more convenient if we want to be super rigorous.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other questions might be something like “this proof seems unnecessarily complicated, is it possible to do it this simpler way?” (usually the answer is no, but you’ll still get a better understanding of why the complexity is necessary), or “why is this condition necessary in this definition? What breaks without it?”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I partially agree with Jhal's answer; the part I agree with is the last sentence about taking it with a pinch of salt. What you want, presumably, is deep understanding, not rote memorization of many superficial facts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the weak Nullstellensatz, I think trying to prove it yourself was a good idea, but I wouldn’t get discouraged about not being able to do it. I think the root of the problem there may simply be that math is hard (there is a reason Hilbert is famous you know). And you still learn a lot when you try to prove something and don’t succeed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding big picture: you seem to be trying a bottom-up approach, where you expect the big picture to emerge as you read the book. You might want to try the top-down approach, a complementary strategy: instead of expecting the big picture to emerge as you read, you start off with some vague understanding of the big picture before reading, and reading the book is how it becomes more detailed. Here are a few concrete suggestions to help you do this: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Pay attention to what you are curious about. Ask lots of questions and let those questions guide your learning. That way, you'll never feel like what you're reading is pointless, it's always relevant to some question you are curious about. I consider it a major advantage of self-teaching that it gives you more freedom to learn this way; by all means you should take advantage of it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to find a high-level explanation of the topic. Wikipedia is sometimes good for this. If you know someone who already understand the topic well, you can also try asking them questions about it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Skim over a part of the book before reading it, just looking at the statements of the theorems and definitions without looking at any of the proofs. This can give you a sense of the general trajectory of that section. Sometimes, skipping ahead in the book can also help, e.g. if there is some big important theorem you are aiming towards, you might be able to get some vague idea about how some of the concepts introduced earlier relate to it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Continue to try proving theorems from the book yourself sometimes. Even if you fail, you will end up with a better appreciation of the proof when you finally read it. You will better see why it has to be done that way. Also, this will be great for your problem-solving skills. Another thing to try is briefly glancing over a proof in the book to get hints / a general outline of the proof, before you try to prove the theorem yourself, using the hints you glanced from the book. This is like an in-between strategy that’s easier than proving it all on your own but still a better exercise than memorizing the proof. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Finally, while I hope that something I’ve said here will be useful to you, it important to remember that math is difficult, and requires lots of patience and persistence, and there is no advice anyone can give you that will change that.&lt;/p&gt;&#xA;" OwnerUserId="13390" LastEditorUserId="13390" LastEditDate="2020-01-28T18:43:06.880" LastActivityDate="2020-01-28T18:43:06.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17796" PostTypeId="1" CreationDate="2020-01-28T20:11:19.633" Score="2" ViewCount="119" Body="&lt;p&gt;For example, there is a book titled  &lt;em&gt;&quot;Synopsis of Elementary Results in Pure and Applied Mathematics: Containing Propositions, Formulae, And Methods Of Analysis, With Abridged Demonstrations&quot; (Cambridge Library Collection - Mathematics) by George Shoobridge Carr&lt;/em&gt;, which I am told consists of theorem without proof, mostly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there similar type of book on number theory, abstract algebra which mainly consists of theorems/results?&lt;/p&gt;&#xA;" OwnerUserId="13263" LastActivityDate="2020-01-28T20:11:19.633" Title="Is There Book on Collection of Theorems?" Tags="|mathematical-pedagogy|self-learning|reference-books|" AnswerCount="0" CommentCount="1" ClosedDate="2020-02-01T04:47:29.123" ContentLicense="CC BY-SA 4.0" />
  <row Id="17797" PostTypeId="1" CreationDate="2020-01-29T03:27:23.663" Score="2" ViewCount="182" Body="&lt;p&gt;This question might not suit the site, if so I apologise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to regularly test my maths skills, essentially finding out if I'm getting better or worse at a maths in general and then plot the data. The rational behind this is I'd like to find out if things like regular exercise, changes to diet, etc. make a measurable different &lt;em&gt;to me&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My concern is although there are ways of testing my analytical and mathematics abilities using some form of test, whatever test I use I will inevitably get better &lt;em&gt;at taking the test&lt;/em&gt; outside of any changes I make to my diet etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only method I can think of to work around this is to make no changes for some length of time, and test myself during this period, thereby being able to identify the degree to which I get better just by taking the tests multiple times - but this is flawed because it's unlikely to be a linear process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand the standard for this is to have multiple test subjects and a control group, but can someone do this type of work alone?&lt;/p&gt;&#xA;" OwnerUserId="13392" LastActivityDate="2020-01-31T12:07:01.350" Title="Measuring ones mathematics ability regularly - how to avoid getting better at the tests" Tags="|testing|test-design|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17798" PostTypeId="2" ParentId="17793" CreationDate="2020-01-29T07:14:12.067" Score="2" Body="&lt;p&gt;Expanding on the point already mentioned by guest's answer, I think you should &lt;strong&gt;stop focussing on building intuition&lt;/strong&gt;. Instead, focus your time on solving the exercises even if, by the time you solve them, you don't feel like you've gained much in terms of intuition. Just solve the problem and move on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This might seem very counterintuitive, but the point is that intuition is not an easy thing to build (at least, not for most people, for most topics). You don't, for example, magically obtain a strong intuition for how finite groups behave immediately after being introduced to the definition of groups. Things just don't work that way. Instead, intuition is slowly developed after attempting some (occasionally large) number of computational exercises, where you really work with whatever object you're studying &quot;hands on&quot; and get a feel for how they behave. This active process of manipulation is completely different from just reading proofs in your textbook, and cannot in my opinion be replaced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very practical way to go about doing this is by working through all the proofs you encounter in your textbook (or alternatively, the proofs presented by your professor in class). For textbooks, before reading through the proofs for each result, you can (and should) attempt to first understand what the result means, by for example writing down a few examples, or if that's not possible by rewriting on scrap paper the statement of the result and trying to gain some intuitive understanding for what the result claims. Then, you might attempt to come up with the proof yourself without looking at the proof presented in the book: this is often surprisingly easy once you've obtained a clear intuition for what the result is saying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, this is not always possible in practice. And there are some textbooks where the learning curve is so steep that it is practically impossible to come up with proofs of any of the claimed results on your own without reference to the provided argument (especially since, perhaps, a key idea that is used in the argument has not been introduced yet). However, at least the first step of understanding results---even if it means slowing down and taking an hour for a single page in the book---will be very instructive and lead to a significant increase in your learning quality and effectiveness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In conclusion: stop thinking about building intuition. Actually do it by working with the objects you're studying, either through the (possibly computational or &quot;tedious&quot;) exercises that are provided, or by working through and understand the results presented in your text/class. &lt;/p&gt;&#xA;" OwnerUserId="10206" LastActivityDate="2020-01-29T07:14:12.067" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17799" PostTypeId="1" CreationDate="2020-01-29T10:13:26.773" Score="4" ViewCount="633" Body="&lt;p&gt;In Euclid's &lt;em&gt;The Elements&lt;/em&gt;, Book 1, Proposition 4, he makes the assumption that one can create an angle between two lines and then construct the same angle from two different lines. I do not see anywhere in the list of definitions, common notions, or postulates that allows for this assumption. When teaching my students this, I do teach them congruent angle construction (with straight edge and compass), but do not understand why Euclid is allowed to make this logical leap. Thanks for any help you can give.&lt;/p&gt;&#xA;" OwnerUserId="13394" LastEditorUserId="376" LastEditDate="2020-01-29T11:33:02.507" LastActivityDate="2020-02-02T03:22:23.777" Title="Euclid Book 1 Proposition 4" Tags="|geometry|proofs|" AnswerCount="2" CommentCount="4" FavoriteCount="0" ClosedDate="2020-02-04T14:37:41.913" ContentLicense="CC BY-SA 4.0" />
  <row Id="17800" PostTypeId="2" ParentId="17799" CreationDate="2020-01-29T12:24:42.973" Score="7" Body="&lt;p&gt;I have found &lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/java/elements/toc.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this site&lt;/a&gt; to be an invaluable online resource for commentary.  Long story short, nobody in history ever wrote an authoritative second edition of &lt;em&gt;Elements&lt;/em&gt;, so we're still &quot;stuck&quot; with Euclid's original thoughts. Undeniably a work of genius that deserves its place in history, but it does have a few rough patches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI4.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; are David Joyce's thoughts on I.4.  It's a bit of a hot mess.  I personally think SAS should have been taken as the definition of congruence, another term that Euclid fails to define.  He talks about superposition, which one should probably interpret like our current notion of a series of rigid motions.  So it evidently isn't an angle construction so much as &quot;sliding&quot; &lt;span class=&quot;math-container&quot;&gt;$\triangle ABC$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\overline{AB}$&lt;/span&gt; lines up with &lt;span class=&quot;math-container&quot;&gt;$\overline{DE}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\angle BAC$&lt;/span&gt; lines up with &lt;span class=&quot;math-container&quot;&gt;$\angle EDF$&lt;/span&gt; and &quot;proving by intimidation&quot; that &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; must fall on &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; because where else could it be?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not his best proof by any stretch.  But I like to remind my students that it's impossible to build a dictionary where every word is defined in terms of previous words in the dictionary, and it's even difficult to imagine trying to do such a thing.  Given that mindset and the fact that Euclid gets up to full speed as quickly as he does, I find myself able to &quot;forgive&quot; some of the early proofs like this.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2020-01-29T17:05:50.957" LastActivityDate="2020-01-29T17:05:50.957" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17801" PostTypeId="2" ParentId="17799" CreationDate="2020-01-29T16:10:01.773" Score="1" Body="&lt;p&gt;Let the two original lines be &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$AC$&lt;/span&gt;, with the angle to recreate as &lt;span class=&quot;math-container&quot;&gt;$\angle BAC$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw a line parallel to &lt;span class=&quot;math-container&quot;&gt;$AC$&lt;/span&gt; that passes through &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; and is not &lt;span class=&quot;math-container&quot;&gt;$AC$&lt;/span&gt;. Call this line &lt;span class=&quot;math-container&quot;&gt;$DE$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$DE$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; intersect (&lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;), drop the perpendicular of &lt;span class=&quot;math-container&quot;&gt;$DE$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$AC$&lt;/span&gt;, and call this point &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$AG$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$GF$&lt;/span&gt; can be used to recreate the angle.&lt;/p&gt;&#xA;" OwnerUserId="5071" LastActivityDate="2020-01-29T16:10:01.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17802" PostTypeId="2" ParentId="17144" CreationDate="2020-01-30T00:23:55.060" Score="1" Body="&lt;p&gt;Although, I do not have any recommendation for trigonometry text and most of the arguments for &amp;amp; against seems valid here. But I do want to point out, SL Loney is widely used as a reference textbook in India for Trignometry &amp;amp; Coordinate geometry. Any student who is serious about qualifying the university entrance exams must solve part 1 of the Loney's trigonometry. I still use this book to teach trigonometry to my kids who are in US high school. &lt;/p&gt;&#xA;" OwnerUserId="13398" LastActivityDate="2020-01-30T00:23:55.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17803" PostTypeId="2" ParentId="220" CreationDate="2020-01-30T03:30:05.870" Score="1" Body="&lt;p&gt;Prove that, for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;, we have  &lt;span class=&quot;math-container&quot;&gt;$x+1 \neq x$&lt;/span&gt;. (Requires use of proof by contradiction.)&lt;/p&gt;&#xA;&lt;p&gt;Use only the following properties of addition on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Addition is closed on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x+1 \neq 1$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$x+1 = y+1$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x=y$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Principle of Mathematical Induction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;OUTLINE&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Base case&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1+1 \neq 1$&lt;/span&gt; follows directly from (2).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Inductive step&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x \neq x+1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Suppose to the contrary that &lt;span class=&quot;math-container&quot;&gt;$x+1 = x+1+1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Obtain the contradiction &lt;span class=&quot;math-container&quot;&gt;$x=x+1$&lt;/span&gt; by applying (3).&lt;/p&gt;&#xA;&lt;p&gt;Conclude (by contradiction) that &lt;span class=&quot;math-container&quot;&gt;$x+1 \neq x+1+1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Conclude that for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x \neq x+1$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$x+1 \neq x+1+1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorDisplayName="user7990" LastEditDate="2020-09-08T21:57:07.090" LastActivityDate="2020-09-08T21:57:07.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17804" PostTypeId="2" ParentId="17757" CreationDate="2020-01-31T02:08:07.180" Score="8" Body="&lt;p&gt;An interesting example is &lt;a href=&quot;https://en.wikipedia.org/wiki/Sicherman_dice&quot; rel=&quot;noreferrer&quot;&gt;Sicherman dice&lt;/a&gt;: A pair of 6-sided dice, with positive number of eyes on each face that are &lt;em&gt;not&lt;/em&gt; the classic 1..6 ones; if you throw them, the distribution of total eyes is the same as for two regular dice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To construct them, one die is represented by:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{align*}&#xA;   D(z)&#xA;     &amp;amp;= z + z^2 + z^3 + z^4 + z^5 + z^6 \\&#xA;     &amp;amp;= z (1 + z) (1 - z + z^2) (1 + z + z^2)&#xA;\end{align*}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Throwing two dice gives the distribution:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{equation*}&#xA;   D^2(z)&#xA;     = z^2 (1 + z)^2 (1 - z + z^2)^2 (1 + z + z^2)^2&#xA;\end{equation*}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we want two dice, call them &lt;span class=&quot;math-container&quot;&gt;$D_1(z)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$D_2(z)$&lt;/span&gt;, such that &lt;span class=&quot;math-container&quot;&gt;$D^2(z) = D_1(z) \cdot D_2(z)$&lt;/span&gt;. The respective polynomials have to satisfy several conditions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;We want that all faces have some eyes, i.e., the constant term of the polynomial has to be zero. We need to assign a &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; factor to each.&lt;/li&gt;&#xA;&lt;li&gt;The coefficients have to be integers (number of faces with each number of eyes). It is a fun fact that if a polynomial with integer coefficients factors, the primitive factors have integer coefficients (from Gauss' lemma). So this isn't a restriction.&lt;/li&gt;&#xA;&lt;li&gt;The number of faces of the dice has to be six. This is the sum of the coefficients in the polynomial, which is just &lt;span class=&quot;math-container&quot;&gt;$D_i(1)$&lt;/span&gt;. The respective factors at &lt;span class=&quot;math-container&quot;&gt;$z = 1$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. We have to give a &lt;span class=&quot;math-container&quot;&gt;$1 + z$&lt;/span&gt; and a &lt;span class=&quot;math-container&quot;&gt;$1 + z + z^2$&lt;/span&gt; to each, we can shuffle the &lt;span class=&quot;math-container&quot;&gt;$1 - z + z^2$&lt;/span&gt; factors around.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thus the &lt;em&gt;only&lt;/em&gt; solution (except for switching the dice, and classical dice) is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{align*}&#xA;  D_1(z)&#xA;    &amp;amp;= z (1 + z) (1 + z + z^2) \\&#xA;    &amp;amp;= z + 2 z^2 + 2 z^3 + z^4 \\&#xA;  D_2(z)&#xA;    &amp;amp;= z (1 + z) (1 + z + z^2) (1 - z + z^2)^2 \\&#xA;    &amp;amp;= z + z^3 + z^4 + z^5 + z^6 + z^8&#xA;\end{align*}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This translates into dice with faces marked &lt;span class=&quot;math-container&quot;&gt;$\{1, 2, 2, 3, 3, 4\}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\{1, 3, 4, 5, 6, 8\}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2020-02-11T01:09:38.877" LastActivityDate="2020-02-11T01:09:38.877" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17805" PostTypeId="2" ParentId="14712" CreationDate="2020-01-31T02:50:07.653" Score="0" Body="&lt;p&gt;Be careful, &quot;Did you understand step 5?&quot; will get an uniform &quot;Aye!&quot;. &lt;em&gt;Check&lt;/em&gt; if they understood. Ask &lt;em&gt;for what&lt;/em&gt; this step was taken, &lt;em&gt;what&lt;/em&gt; alternatives were, &lt;em&gt;why&lt;/em&gt; this step was selected, &lt;em&gt;what&lt;/em&gt; can be done if this approach fails. Chaining together the various steps is much more important that getting the &quot;rational decomposition of &lt;span class=&quot;math-container&quot;&gt;$P(z) / Q(z)$&lt;/span&gt;&quot; right (your friendly computer algebra system takes care of routine chores).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-01-31T02:50:07.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17806" PostTypeId="1" CreationDate="2020-01-31T04:37:39.320" Score="4" ViewCount="415" Body="&lt;p&gt;I am constructing a learning sequence for middle grade students designed to convince them that empirical arguments (arguments by example) are not sufficient in mathematics. To motivate this, I am using Stylianides' three-task sequence. However, I believe Stylianides' narrative of a pattern which falls apart at a large &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a bit sophisticated for middle school learners. I'm wondering if others can help me think of another narrative showing a conjecture that falls apart at some unexpected &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. I'm including &lt;a href=&quot;https://books.google.com.ph/books?id=7xdoDwAAQBAJ&amp;amp;lpg=PP1&amp;amp;pg=PA25#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stylianides' narrative&lt;/a&gt; below:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A group of mathematicians was exploring outputs for the expression &lt;span class=&quot;math-container&quot;&gt;$1+1141n^2$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a natural number. After evaluating the expression for several natural numbers and looking for any patterns that seemed to be emerging, the mathematicians made the conjecture that this expression never returns a square number. To test their conjecture, they wrote a computer program to evaluate the expression for all natural number inputs and to stop when the output tested as a perfect square. Then they left the computer to its work.&lt;/p&gt;&#xA;&lt;p&gt;They kept checking in with the computer’s work, and as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; grew bigger and bigger without an output of a square number, they became more and more confident that their conjecture was true. Imagine their surprise when they returned to the computer and found that it had stopped running the program, meaning that the program had an output of a square number. They were even more surprised to find that this expression does return a square number, but not until &lt;span class=&quot;math-container&quot;&gt;$$n=30\,693\,385\,322\,765\,657\,197\,397\,208.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7550" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-02-08T16:50:27.853" Title="Patterns that unexpectedly fall apart at large $n$" Tags="|proofs|middle-school|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17807" PostTypeId="2" ParentId="17806" CreationDate="2020-01-31T09:06:35.473" Score="9" Body="&lt;p&gt;It might work better, with this age level, not to be concerned about how large n is when the apparent pattern falls apart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My favorite example is the problem of making all possible straight-line segments between n points on a circle, and then counting the regions. 2 points makes 2 regions, 3 makes 4, 4 makes 8, and 5 makes 16. It sure looks like doubling...&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-01-31T09:06:35.473" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17808" PostTypeId="2" ParentId="17797" CreationDate="2020-01-31T11:52:12.863" Score="2" Body="&lt;p&gt;This is an interesting question!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One option could be testing the &lt;strong&gt;quality of how&lt;/strong&gt; you do math, for example:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Count how many careless mistakes you are making when doing maths that you usually do;&lt;/li&gt;&#xA;&lt;li&gt;Measure how long you can stay focused doing maths while you are still productive and you are not easily distracted. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I think the interventions you mention enhance your cognitive ability in general, and consequently you are capable of learning more maths and focus longer on more challenging problems, thus your mathematical ability improves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These measures also allow you to set the process of testing easier and more aligned with the goals you want to achieve in mathematics. For example, to measure focusing time you could just read maths books you would like to read anyway and stop the timer the first time you get distracted from the book. &lt;/p&gt;&#xA;" OwnerUserId="13399" LastEditorUserId="13399" LastEditDate="2020-01-31T12:07:01.350" LastActivityDate="2020-01-31T12:07:01.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17809" PostTypeId="2" ParentId="17806" CreationDate="2020-01-31T12:04:36.593" Score="4" Body="&lt;p&gt;This happened to me once on a topic that should be accessible to middle-school students.  As an undergraduate, I formed the following conjecture:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An &lt;em&gt;antiprime&lt;/em&gt; (also called a &lt;a href=&quot;https://en.wikipedia.org/wiki/Highly_composite_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;highly composite number&lt;/em&gt;&lt;/a&gt;) is a positive integer that has more divisors than any number less than it. The first few antiprimes are &lt;span class=&quot;math-container&quot;&gt;$$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360,...$$&lt;/span&gt; &lt;strong&gt;Conjecture&lt;/strong&gt;: For every antiprime &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;1$&lt;/span&gt;, there is a prime &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$p\mid n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n/p$&lt;/span&gt; is an antiprime.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This would be a really useful fact, because you could make a list of all antiprimes recursively by just multiplying the ones you know times all of their prime factors plus their first unused prime factors and just check those instead of having to check every single number.  But I was never able to find a proof for my conjecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many years later, I asked on the xkcd math forums if anyone could help me with my proof.  That website is gone now and my post wasn't archived, but someone came up with a counterexample instead of a proof!  Thanks to &lt;a href=&quot;https://math.stackexchange.com/questions/3529219/find-a-counterexample-for-every-antiprime-n1-there-is-a-prime-divisor-p-s&quot;&gt;kind people in MSE&lt;/a&gt;, that counterexample can be recreated:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$362279431624673937974303738230488502933082643722886373107941760000$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yeah, just trusting that it works out for small numbers wasn't a winning strategy for me here.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2020-01-31T15:50:42.290" LastActivityDate="2020-01-31T15:50:42.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17810" PostTypeId="2" ParentId="17806" CreationDate="2020-01-31T13:38:57.747" Score="0" Body="&lt;p&gt;Suppose you put n distinct points on the circumference of a circle then draw segments connecting each point to every other point.  What's the maximum number of regions into which the circle is divided?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 point has nothing to connect to so you get 1 region, i.e. the whole circle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 points make a chord which divides the circle into two regions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3 points make a triangle which gives you four regions, the interior of the triangle and three curved sections outside the triangle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4 points make a quadrilateral with its two diagonals for a total of eight regions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now look for a pattern&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 point =&gt; 1 region&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 points =&gt; 2 regions&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3 points =&gt; 4 regions&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4 points =&gt; 8 regions&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Inductive reasoning could lead you to conclude that the number of regions is equal to 2^{n-1}.  However, the pattern breaks down when n = 6 and you only get 31 regions rather than the expected 32.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2020-01-31T13:38:57.747" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17811" PostTypeId="1" CreationDate="2020-01-31T14:00:37.670" Score="1" ViewCount="111" Body="&lt;p&gt;In the video by the well-known physicist Water Lewin &lt;a href=&quot;https://www.youtube.com/watch?v=cB8GNQuyMPc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link Here&lt;/a&gt;,&#xA;in which he demonstated a series of experiments of rolling objects. Many are quite surprising. At the end of the video, the attended students asked him to give some explanation for the phenomenon. But he just said that it is not easy! and needs &quot;Calculus&quot; and then only gave a very brief idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not a physics student but I am very interested to learn how Calculus comes into play here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could anyone give some explanations based on more the Mathematics aspect (like Lewin suggested) for his experiments!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for any help.&lt;/p&gt;&#xA;" OwnerUserId="12801" LastActivityDate="2020-02-01T02:32:59.453" Title="How to use Calculus to give an explanation for a video by Lewin" Tags="|calculus|physical-sciences|" AnswerCount="1" CommentCount="1" ClosedDate="2020-02-01T06:34:40.687" ContentLicense="CC BY-SA 4.0" />
  <row Id="17812" PostTypeId="2" ParentId="17806" CreationDate="2020-01-31T15:17:16.027" Score="9" Body="&lt;p&gt;I second &lt;a href=&quot;https://matheducators.stackexchange.com/a/17807/80&quot;&gt;Sue Van Hattum's suggestion&lt;/a&gt; that you should not be so concerned with &lt;em&gt;how large the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is where the pattern eventually fails&lt;/em&gt;. I'll go one step further and recommend an example where that &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is not only fairly &quot;small&quot; but also such a situation where the students can see &lt;em&gt;why&lt;/em&gt; that &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; makes the pattern fail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider the function &lt;span class=&quot;math-container&quot;&gt;$f(n) = n^2+n+41$&lt;/span&gt;. Which values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; make the output &lt;span class=&quot;math-container&quot;&gt;$f(n)$&lt;/span&gt; a &lt;strong&gt;prime number&lt;/strong&gt;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You and the students could complete a group task to check &lt;span class=&quot;math-container&quot;&gt;$f(1)=43$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(2)=47$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(3)=53$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(4)=61$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(5)=71$&lt;/span&gt; are all prime. You can then ask whether they think this will keep going. You can then &quot;wow&quot; them with a much larger value, showing that &lt;span class=&quot;math-container&quot;&gt;$f(20)=461$&lt;/span&gt;, for example, is also prime. (This could lead you into some tangential discussions about primality testing and factorization methods and so on.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But after all that, I believe you could easily convince the students (or Socratically guide them into realizing on their own!) that &lt;span class=&quot;math-container&quot;&gt;$f(41)=1763$&lt;/span&gt; is &lt;strong&gt;not prime&lt;/strong&gt;. You can get them to verify that &lt;span class=&quot;math-container&quot;&gt;$41\times 43=1763$&lt;/span&gt;, and you can do the easy algebra with the function: &lt;span class=&quot;math-container&quot;&gt;$f(41)=41^2+41+41=41(41+1+1)=41\times 43$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(At that point, you could even work with them to realize &lt;span class=&quot;math-container&quot;&gt;$f(40)=40^2+40+41 = 40(40+1)+41 = 40(41)+41=41(40+1)=41^2$&lt;/span&gt; is not prime, either!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this will be more instructive, at least as a &lt;em&gt;first example&lt;/em&gt; of such a situation. Afterwards, you could use more &quot;exotic&quot; examples like the one you mentioned. But I believe the students will feel more comfortable with the ideas after having seen this example.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2020-01-31T15:17:16.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17814" PostTypeId="2" ParentId="17811" CreationDate="2020-02-01T02:32:59.453" Score="2" Body="&lt;p&gt;To understand rolling without slipping we need to understand the kinetic energy falls into two modes; linear kinetic energy and rotational kinetic energy. For a rigid body if the velocity of the center of mass &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}mv^2$&lt;/span&gt; gives the linear KE. On the other hand, analyzing the motion of the little bits of mass rotating around the center of mass allows us to capture that collective energy by &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}I \omega^2$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is the moment of inertia and &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; is the angular velocity about the center of mass. Generally it is more complicated and you need an inertia tensor to capture the story, but when the rotational motion is just around one axis then we get the usual Freshman stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ok, so, &#xA;&lt;span class=&quot;math-container&quot;&gt;$$KE = \frac{1}{2}mv^2+ \frac{1}{2}I \omega^2 $$&lt;/span&gt;&#xA;Conservation of energy for a rigid body starting from rest gives us that the initial gravitational potential energy &lt;span class=&quot;math-container&quot;&gt;$PE = mgh$&lt;/span&gt; is matched by the final &lt;span class=&quot;math-container&quot;&gt;$KE$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rolling without slipping means that &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; are connected; &lt;span class=&quot;math-container&quot;&gt;$v = \omega R$&lt;/span&gt; hence we find:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ mgh = \frac{1}{2}mv^2+ \frac{1}{2}I (v/R)^2 $$&lt;/span&gt;&#xA;hence&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ mgh = \frac{1}{2}(m+ I/R^2)v^2 $$&lt;/span&gt;&lt;br&gt;&#xA;and so,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ v = \sqrt{\frac{2mgh}{m+I/R^2}} $$&lt;/span&gt;&#xA;We find larger &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; iimplies smaller &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;. Calculus shows &lt;span class=&quot;math-container&quot;&gt;$I = \frac{1}{2}mR^2$&lt;/span&gt; for a solid cylinder whereas &lt;span class=&quot;math-container&quot;&gt;$I = mR^2$&lt;/span&gt; for a cylindrical shell. Another common example somewhere between these extremes is the solid sphere with &lt;span class=&quot;math-container&quot;&gt;$I = \frac{2}{5}mR^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the explanation is largely algebra provided we have already developed the basic rules for rotational kinematics. That said, calculus is naturally employed to derive the moment of intertia formulas.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-02-01T02:32:59.453" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17815" PostTypeId="2" ParentId="17806" CreationDate="2020-02-01T11:26:55.433" Score="1" Body="&lt;p&gt;Perhaps it is a good idea to start with truly simple examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The simplest example accessible to children is the notion that every odd number is prime. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another is to calculate &lt;span class=&quot;math-container&quot;&gt;$\cos(\pi/n)$&lt;/span&gt;. For &lt;span class=&quot;math-container&quot;&gt;$n = 1, 2, 3, 4$&lt;/span&gt; it works. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is simple in principle to construct a (degree &lt;span class=&quot;math-container&quot;&gt;$N-1$&lt;/span&gt;) polynomial that takes &quot;nice&quot; values at the first &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; integers by solving an interpolation problem. Of course the polynomial that results may not be &quot;nice&quot;, but with some fiddling it should be possible to find a serviceable example this way.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2020-02-01T11:26:55.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17819" PostTypeId="1" CreationDate="2020-02-04T13:58:31.333" Score="2" ViewCount="156" Body="&lt;p&gt;In any standard development of measure theory in several well-known textbooks, the use of almost everywhere (a.e.) defined functions are first seen in the statement of Fubini's theorem, which states, in particular (under standard notation) that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int f(x, y) d\mu(x) = g(y)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is defined almost everywhere. Now &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; becomes an a.e. defined function. It is indeed true that the development carried out so far in the book can be formulated in terms of a.e. functions, and not lose much, but still, I think it deserves a remark, if not a full-blown methodology that uses only a.e. functions right from the beginning. This latter suggestion is justifiably not adopted, because it gets in the way of communicating the main ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, most sources I've seen completely avoids even mentioning this fact (one exception I've seen is Tao's &lt;em&gt;Analysis II&lt;/em&gt;). Why is this the case?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt;: I've also not seen any &lt;em&gt;advanced&lt;/em&gt; book seriously developing the whole theory consistently using a.e. defined functions (one exception is Fremlin's monograph). Why is this stuff left unformalized? Is it just because its a notational hassle, or is there some proper reason?&lt;/p&gt;&#xA;" OwnerUserId="13411" LastEditorUserId="8571" LastEditDate="2020-02-04T20:23:01.920" LastActivityDate="2020-02-04T20:23:01.920" Title="Why are a.e. defined functions rarely mentioned in elementary books?" Tags="|mathematical-analysis|graduate-education|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17820" PostTypeId="2" ParentId="220" CreationDate="2020-02-04T14:32:20.573" Score="3" Body="&lt;p&gt;Hmm, I'm surprised than no one mentioned de Moivre's formula: &lt;span class=&quot;math-container&quot;&gt;$$(\cos \phi +i\cdot \sin \phi)^n= \cos (n\phi) + i\cdot\sin (n\phi)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is an excellent exercise to prove it. We must also use the addition theorems for sin and cos.&lt;/p&gt;&#xA;" OwnerUserId="10484" LastEditorUserId="10484" LastEditDate="2021-01-10T13:06:24.637" LastActivityDate="2021-01-10T13:06:24.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17821" PostTypeId="2" ParentId="16709" CreationDate="2020-02-04T16:33:16.763" Score="1" Body="&lt;p&gt;You could look at &lt;em&gt;separable differential equations&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given a differential equation of the form&#xA; &lt;span class=&quot;math-container&quot;&gt;$$g(y)y'(x) = f(x),$$&lt;/span&gt; integrating between &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; and x on both side and integration by substitution with &lt;span class=&quot;math-container&quot;&gt;$u = y(x)$&lt;/span&gt; gives&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{x_0}^xg(y)y'(x) dx = \int_{x_0}^xf(x)dx $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{y(x_0)}^{y(x)}g(u) du = \int_{x_0}^xf(x)dx,$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which allows to solve the differential equation, provided we can find antiderivatives &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; and invert &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;. It is probably possible to find a physical context where a separable equation appears if you want to insist on the &quot;real-life&quot; aspect.&lt;/p&gt;&#xA;" OwnerUserId="12752" LastActivityDate="2020-02-04T16:33:16.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17823" PostTypeId="1" CreationDate="2020-02-05T18:48:09.900" Score="11" ViewCount="624" Body="&lt;p&gt;When we talk about: &lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to{c}}f(x)=L.$$&lt;/span&gt; Is there a formal name for the number &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;I know that the notation means &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; is the limit of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&amp;quot;. It just would be nice to be able to refer to &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&amp;quot; separately without saying &amp;quot;what &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches&amp;quot;. I tried to look at various textbooks but didn't find anything.&lt;/p&gt;&#xA;" OwnerUserId="1914" LastEditorUserId="8571" LastEditDate="2023-08-17T02:36:28.697" LastActivityDate="2023-08-17T02:36:28.697" Title="Terminology for parts of limit notation" Tags="|calculus|notation|terminology|limits|" AnswerCount="3" CommentCount="11" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17824" PostTypeId="1" CreationDate="2020-02-06T14:28:28.013" Score="1" ViewCount="96" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\DeclareMathOperator{\lcm}{lcm}$&lt;/span&gt;There are many applications for finding the GCD or LCM of two numbers.  I'm now interested in finding anything that could be used to illustrate the following mathematical relationship.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$r = \lcm(r_1, r_2)$&lt;/span&gt;.  Find a divisor &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$r/c$&lt;/span&gt; is a multiple of &lt;span class=&quot;math-container&quot;&gt;$r_1$&lt;/span&gt; but not a multiple of &lt;span class=&quot;math-container&quot;&gt;$r_2$&lt;/span&gt; (or a multiple of &lt;span class=&quot;math-container&quot;&gt;$r_2$&lt;/span&gt; but not a multiple of &lt;span class=&quot;math-container&quot;&gt;$r_1$&lt;/span&gt;).  Notice there are cases in which no such &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; can be found.  For example, when &lt;span class=&quot;math-container&quot;&gt;$r_1 = r_2 = \lcm(r_1, r_2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any interesting motivation that [can] be put on this?&lt;/p&gt;&#xA;" OwnerUserId="13418" LastEditorUserId="13418" LastEditDate="2020-02-11T03:31:09.090" LastActivityDate="2020-02-11T03:31:09.090" Title="Is there a game or analogy for searching for a number $R$ that is multiple of some number $A$ but not multiple of another $B$?" Tags="|concept-motivation|games|" AnswerCount="0" CommentCount="9" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17825" PostTypeId="1" AcceptedAnswerId="17828" CreationDate="2020-02-06T23:44:44.587" Score="2" ViewCount="1453" Body="&lt;p&gt;If a masters program (Fordham in math education) requires &quot;at least one full year of calculus&quot; does that mean just calc one or more than that?&lt;/p&gt;&#xA;" OwnerUserId="12529" LastActivityDate="2020-02-08T17:11:12.450" Title="What does one full year of calculus mean?" Tags="|graduate-education|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17827" PostTypeId="2" ParentId="17825" CreationDate="2020-02-07T00:09:40.570" Score="11" Body="&lt;p&gt;Asking random math educators about the policies of a specific graduate school makes absolutely no sense at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The page you were looking at has this contact information at the bottom. Use it!&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Adolescence Mathematics Contacts&#xA;45 Columbus Avenue&#xA;2nd Floor&#xA;New York, NY 10023&#xA;Tel: 212-636-6400&#xA;Fax: 212-636-7106&#xA;Email: gse_admiss@fordham.edu&#xA;Dr. Alesia Moldavan&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="11" LastActivityDate="2020-02-07T00:09:40.570" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17828" PostTypeId="2" ParentId="17825" CreationDate="2020-02-07T02:11:17.640" Score="8" Body="&lt;p&gt;In the US, the stereotypical &quot;one year of calculus&quot;, means &quot;calculus one&quot; and &quot;calculus two&quot; (semester system). It would rather approximate what is in AP Calculus BC, except perhaps without graphing calculator use.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would include basic differentiation and applications of differentiation (max/min, rel. rates), integrals, special techniques of integration (parts, partial fractions, substitutions, applications like volumes.  Usually there will be a small section on series tacked on at the end.  And very often a small section on simple ODEs (but not all the content in the semester ODE course, just a quick chapter).  It WILL NOT (stereotypically) include multivariable calculus (from the Nick C answer).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the Fordham course catalog, this is their standard &quot;full year&quot; (not a year and half!) of calculus:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MATH 1206. Calculus I. (4 Credits)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus for science and math majors. Functions, limits, continuity, Intermediate Value Theorem. The derivative and applications, antiderivatives, Riemann sums, definite integrals, the Fundamental Theorem of Calculus. Four-credit courses that meet for 150 minutes per week require three additional hours of class preparation per week on the part of the student in lieu of an additional hour of formal instruction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Attributes: ENVS, MCR, NEUR.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MATH 1207. Calculus II. (4 Credits)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A continuation of MATH 1206. The definite integral, area, volumes, work. Logarithm, inverse functions, techniques of integration, Taylor polynomials. Four-credit courses that meet for 150 minutes per week require three additional hours of class preparation per week on the part of the student in lieu of an additional hour of formal instruction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prerequisites: MATH 1206 or MATH 12AB or MATH 12BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[So yes, Fordham is a good example of the stereotype.  Also, note that Math 2004, multivariable calc, is a 2-level course (implicitly the 2 means second year).  So would not be part of a &quot;full year&quot;, would be part of a year and a half.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://bulletin.fordham.edu/undergraduate/mathematics/?_ga=2.115156231.1200530478.1581040617-621694673.1581040617#coursestext&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fordham University Math Course Overview&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13420" LastEditorUserId="64" LastEditDate="2020-02-08T17:11:12.450" LastActivityDate="2020-02-08T17:11:12.450" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17829" PostTypeId="1" AcceptedAnswerId="17833" CreationDate="2020-02-07T03:22:10.020" Score="32" ViewCount="3188" Body="&lt;p&gt;How important is enforcing standard mathematical language and/or notation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today, I was questioned by a writing instructor as to how vital it is to correct students when they explain something using nonstandard mathematical language. The example we discussed was correcting a student who says “x two” to indicate &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; (instead of “x to the power of two” or “x squared”). I said that my primary role is to first determine what a student means, and if they mean &lt;span class=&quot;math-container&quot;&gt;$x\times x$&lt;/span&gt;, then I will almost &lt;em&gt;always&lt;/em&gt; correct the language as saying such things may be misinterpreted as &lt;span class=&quot;math-container&quot;&gt;$x \times 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question came to me in the context of a discussion with other faculty about inclusive teaching practices, where a method shared involved allowing classes some freedom in creating and using their own language for items/principles/etc discussed in the topic. The perception was that allowing (and even encouraging) students to do this would build community in the classroom and let students attach some personal meaning to the items discussed. The instructors speaking favorably about this were from certain Humanities departments (writing, film studies, history). I was the only instructor from math or science in this conversation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is of interest to me because while my students must be able to communicate mathematics with others once they leave my class, I wonder if allowing some creativity here might do something positive for their overall experience studying math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have definitely had moments in certain classes where a student will (usually as a joke) propose a name for something — a solution technique, a common pitfall, a type of function — and the class adopts it. This happens when I don’t know of any standard terminology for it, so this class name may stick around for the term. I’m actually really happy when this happens organically. What I haven’t done is encourage playfulness with things that already have names.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So — is there any place in mathematics for student-created terminology and/or notation? How have you (or would you) allow it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I understand what a huge setup for failure this would be if we didn’t enforce any language or notation — students would go to some other class and be completely lost, not recognizing basic things as typically presented. If you don’t like this idea in principle, that is totally fine. I am just wondering if there’s any place in modern mathematics for this kind of freedom for students.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-02-09T15:41:32.660" Title="Allowing nonstandard mathematical language and/or notation" Tags="|undergraduate-education|secondary-education|notation|terminology|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17830" PostTypeId="2" ParentId="17829" CreationDate="2020-02-07T17:37:52.883" Score="1" Body="&lt;p&gt;It's your job to teach standard notation and terminology, and to correct your students if they get it wrong. If they get it wrong repeatedly, it's your job to correct them repeatedly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student comes out of an algebra class saying &quot;x two&quot; for &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;, then that student has been done an enormous disservice. They will be marked as mathematically illiterate by any person they encounter in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would actually argue for much stricter enforcement of style, but this is more a matter of opinion. I would correct students who wrote any of the following, as a matter of style:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sin(\theta),\qquad \frac{a}{\frac{b}{c}},\qquad x2y.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They should also be learning the Greek alphabet, e.g., that &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt; makes the &quot;r&quot; sound and is not p, and so on.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-02-07T17:37:52.883" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17831" PostTypeId="2" ParentId="17829" CreationDate="2020-02-07T18:16:03.087" Score="8" Body="&lt;p&gt;Part of teaching mathematics is teaching how to communicate mathematics - both how to understand it, and how to write it clearly.  Some of that is obviously complementary to teaching students how to solve math problems and understand mathematics; for instance, students who can't understand a question aren't going to be able to answer it correctly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it's not obvious that we should expect these skills to be learned perfectly in parallel.  My understanding is that when teaching writing, the current approach prioritizes having students first learn to write fluidly, and only much later starts pushing students to improve spelling and grammar.  I suspect our expectations in math should be similar: we should expect student's ability to write math conventionally to lag, possibly significantly, what they understand, &lt;em&gt;and that's absolutely fine and developmentally normal&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Learning how to do mathematics and learning how to express mathematics really are two separate things, and it can be confusing to students to have them too mixed together.  Students often can't tell the difference between mathematical reasoning and arbitrary notational rules, so being overly strict about how things are written distracts students from learning the math, and feeds the perception that math is a collection of unmotivated rules imposed by the teacher.  (Particularly since many notational choices are motivated by concerns that come up much later, so students aren't in a good position to appreciate them when they're being introduced.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We should certainly teach the standard notation, and we should certainly expect students to be able to read it.  But allowing them to discuss and sometimes write it in non-standard terminology, as long as they understand that it's non-standard, could sometimes be beneficial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where I've had the time in a class (which is rarely), I've even been trying to make the distinction more explicit: to be more flexible about notation most of the time, but have a few assignments where the assignment is specifically to take a problem they've already solved and write it up more formally for their classmates.  That gives students a change to practice correct notation, and emphasizes that correct writing is primarily about communicating with peers, not about satisfying the rules a teacher has laid down.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2020-02-07T18:16:03.087" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17832" PostTypeId="2" ParentId="17829" CreationDate="2020-02-07T20:42:55.173" Score="6" Body="&lt;p&gt;It is of course important to give your students a good understanding of the customs and conventions that are commonly used in mathematics, but I think that creativity, imagination and flexibility of mind are just as important - as is the courage to think 'crazy' thoughts, such as &quot;what is &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}?$&lt;/span&gt;&quot;, &quot;is infinity always the same?&quot; and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maths is in some ways closer to art than to the 'hard sciences', and there is scope for a lot of playfulness. Mathematical jokes are very valuable in making the subject enjoyable, especially if they illustrate some clever, not obvious point; eg. something that is beguilingly plausible, but actually impossible. Or, from topology in particular, a counterexample to something that seems intuively true - there is a famous book about this: &quot;Counterexamples in Topology&quot; by Lynn Steen and J. Arthur Seebach, Jr.&lt;/p&gt;&#xA;" OwnerUserId="13424" LastActivityDate="2020-02-07T20:42:55.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17833" PostTypeId="2" ParentId="17829" CreationDate="2020-02-07T22:53:30.480" Score="22" Body="&lt;p&gt;I think, while teaching, the principal way to judge mathematical language is not whether it's &lt;em&gt;standard&lt;/em&gt;, but whether it's &lt;em&gt;effective&lt;/em&gt; communication. This difference applies principally to communication that's more substantive than &quot;read an equation out loud&quot; where there's only really one right way and not much opportunity for change - but, even on the small level, efficacy is a way easier standard for students to internalize than an arbitrary standard set of language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, it is worth correcting &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; two&quot; because this language does not convey what operation is being used - and it might even convey a lack of understanding of the fact that the notation &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; refers to an operation. I think a reasonable response to such notation as a teacher is to point out that the student had something (exponentiation!) in mind that they didn't succeed in communicating and to give them the tool to communicating that (e.g. the language &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; squared&quot; or &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to the power of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;&quot; or however you want to phrase it). I would give similar feedback to a student who was manipulating a large expression and wrote something like &lt;span class=&quot;math-container&quot;&gt;$e^{(x+1)^2=x^2+2x+1}$&lt;/span&gt; because I would want them to understand that they are really separately noting the equation &lt;span class=&quot;math-container&quot;&gt;$(x+1)^2=x^2+2x+1$&lt;/span&gt; and then substituting that into a larger expression - and to ensure that they don't treat equality signs merely as the way to express a chain of simplifications. These sorts of language issues are usually easy to resolve, but should get the same attention that a mathematical issue would since they reflect mathematical structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a student said something like &quot;We take &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and plus it with &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;&quot;, I might point out that the usual way to say that is &quot;We add &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot;, but this is a lot lower of a priority than the previous example - I'd be inclined to let it slide in spoken mathematics, but would correct it in anything written. A similar example is that a student once, while explaining her proof to me, used the phrase &quot;we take &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and two-thirds it over to &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&quot; to refer to a weighted average - which was perfectly clear while we were both looking over her diagram of the process and more enjoyable than using technical language, even if I wouldn't like it in writing. These are more explicitly issues of poor notation - which should be dealt with wherever good notation is expected, but not confused with more important conceptual issues. (Just to emphasize: I am saying that there can comfortably be two standards of communication here - in class, it's often worth setting aside conventions to avoid distracting the class from the big picture and to avoid possibly alienating students purely based on language, but it's important that you at least &lt;em&gt;sometimes&lt;/em&gt; ask students for composed work so that they have an opportunity to learn conventional phrasings and don't later feel like they were denied this knowledge)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This said, looking at small uses of language misses the point: the primitive concepts of mathematics have fixed names and students should learn to use these primitives properly. The question of nonstandard language is not about terms like &quot;squared&quot; but rather about the concepts students might wish to build on top of them. If you expect students to be able to communicate effectively, that means that they have to explain - in a human language - what they're doing and that means that suddenly we should be talking about students producing sentences such as &quot;We begin by isolating &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&quot; and putting these sentences together into paragraphs (along with equations and formal manipulations) - and then explaining what they're up to at a high level, in the same sense that a writing teacher would demand &quot;topic sentences&quot;. There's suddenly &lt;em&gt;a lot&lt;/em&gt; of room for creativity once you ask students to communicate at this level - and there's room for idiosyncrasy too as a class comes across methods and explanations that appeal particularly to them. Focussing on these larger blocks of language is also something that I've found to help weaker students since it gives them the tools to &lt;em&gt;explain&lt;/em&gt; understanding that may have been hard won for them - and the ability to explain is something that they may not have felt included in before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There really is a danger of a teacher overreaching into this territory of higher abstraction - I had plenty of frustration as a student when teachers insisted that a concept not only be &lt;em&gt;correct&lt;/em&gt;, but always understood and phrased in the teacher's language; it's better to push the student to be able to explain their mathematical thinking process - for instance, productive comments might look like &quot;Your equations are correct, but hard to follow; could you include more writing about why you did the manipulations you did?&quot; or &quot;This would be clearer if you included a worked example&quot; or &quot;Could you draw a diagram of this step to help the reader?&quot; or &quot;You could phrase this more clearly by writing this last equation and then saying 'by taking the square of both sides'.&quot; The goal in such teaching would be that every student can communicate in a way that is clear - and while this involves standard notation for the details of mathematical rigor, there's not much to prescribe &lt;em&gt;beyond&lt;/em&gt; that.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As a small side note, there is &lt;em&gt;some&lt;/em&gt; language such as the names of various theorems where one could argue that the standard names might be better replaced by non-standard ones - and where using the standard notation doesn't actually reflect a better conceptual understanding. For instance, if you refer to Bezout's identity as &quot;axby&quot;, you suddenly are using more descriptive notation than standard mathematics &lt;em&gt;and&lt;/em&gt; you get to say a fun word. If someone in your class makes an insightful question, you can call it &quot;So and so's conjecture&quot; to give some ownership where it is due until the class arrives at whatever conclusion there is to be had. I put this as a footnote because, while I've found this to work well when teaching higher mathematics to high schoolers over summer (although it sometimes annoys students with more prior knowledge than their classmates), I've also had experiences as a student where teachers have found a flashy new way of teaching like this and implemented all the superficial furnishings of it - like using different names for things - but not actually done anything to make students feel the sort of ownership reflected in a class-specific vocabulary - and these experiences have been frustrating and felt insincere (or worse: have allowed a subset of the class to create notation that bewilders and excludes others). It's not something I would do for teaching more basic content or for teaching a class with an especially large range of comfort with mathematics. Basically: even in cases where the language &lt;em&gt;isn't&lt;/em&gt; important, that's not where a mathematics teacher should look to improve their teaching - rather, they should look to the larger picture of communication.&lt;/p&gt;&#xA;" OwnerUserId="3229" LastEditorUserId="3229" LastEditDate="2020-02-08T05:27:21.013" LastActivityDate="2020-02-08T05:27:21.013" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17834" PostTypeId="2" ParentId="17829" CreationDate="2020-02-08T04:30:54.077" Score="6" Body="&lt;p&gt;I very much appreciate these questions. I am in favor of a very strict style; or in other words, helping students to avoid errors and confusions in the standard language as much and as early as possible. The main thing that sticks in my mind is that the longer a bad practice persists, the &quot;stickier&quot; it is, and the harder it is to fix later on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comparing (a) learning a thing right the first time, vs. (b) learning a thing wrong and fixing it later, item (b) takes roughly three times as much effort as (a). That's because (b) entails (1) learning the thing wrong initially, (2) working to lose the erroneous idea/muscle memory, and (3) finally learning the thing right. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anecdote: I was teaching a remedial algebra class and reviewing the arithmetic order of operations. As part of the discussion, it became clear to me and a young woman of color that she had flat-out been taught something totally incorrect at the high school level. She burst into tears and fled from the classroom, enraged that, &quot;she'd been taught everything wrong and had to learn it all over again&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, that was an outlier reaction, but it's always stuck with me. She's right, and I respect her high level of intellectual honesty. Not bothering to get these fundamentals right early on is something like a brutality to the students in question. Not caring about it is an equity issue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's compare to an analogous trend a few decades ago in English instruction; to get rid of directed instruction in phonics, in favor of a more flexible, exploratory, &quot;whole word&quot; learning of vocabulary. In short, the results were that such a removal of direct instruction was clearly detrimental. From a meta-study reported by Brady, Susan A. &quot;Efficacy of phonics teaching for reading outcomes.&quot; &lt;em&gt;Explaining individual differences in reading: Theory and evidence&lt;/em&gt; (2011): 69-96:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Overall, research reviewed in the NRP report indicates that students&#xA;  taught with systematic phonics instruction have better reading scores,&#xA;  whether measured at the end of the training period or at the end of&#xA;  the school year of instruction (Cohen's d = .44), Systematic phonics&#xA;  instruction was found to produce better reading growth than all of the&#xA;  types of nonsystematic or nonphonics instruction (i.e., basal&#xA;  programs, whole-language approaches, regular curriculum, whole word&#xA;  curriculum, and miscellaneous programs). Further, systematic phonics&#xA;  was found to be effective whether taught through individual tutoring&#xA;  (d = .57), through small groups (d = .43), or to the whole class (d =&#xA;  .39).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My impression is that likewise, the elimination of directed grammar instruction and correction is now producing evidence of serious corrosive effects; but I'm not up on that research and would need to go digging for specifics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary, all the signs I can gather -- both from in-class experiences and decisive evidence from language-instruction research -- point toward it being better to get the fundamentals right sooner rather than later.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-02-09T04:01:56.520" LastActivityDate="2020-02-09T04:01:56.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17835" PostTypeId="2" ParentId="17686" CreationDate="2020-02-08T23:28:46.037" Score="4" Body="&lt;p&gt;Long division is a useful thing to teach because at some point, polynomial long division is going to be something that someone is going to want to teach them, and it helps if they've seen something similar before. Thus, we don't want to replace it with anything. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question, then, becomes whether we want to add a second algorithm, and I can't see any particularly good reason to do so (and it needs to be a particularly good reason, given the limited availability of time), and indeed several reasons not to: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;As you yourself demonstrate, capable students with a strong number sense will pick it up naturally by just skipping steps, so will benefit little from being taught it explicitly, whereas instead investing that time into developing that number sense will place more people into this category. &lt;/li&gt;&#xA;&lt;li&gt;For those students without such a strong number sense, teaching them two different algorithms won't magically give them the strong number sense needed for short division to actually be practically useful. The student from your first example would still not see that 60 and 3 are the relevant factors, regardless of the algorithm in question. &lt;/li&gt;&#xA;&lt;li&gt;If we &lt;em&gt;did&lt;/em&gt; want to invest time into teaching fast and efficient mental arithmetic algorithms, there are &lt;em&gt;many&lt;/em&gt; higher-priority algorithms to teach before short division (addition/subtraction/multiplication is more common, anzan-based algorithms are faster).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="11771" LastActivityDate="2020-02-08T23:28:46.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17836" PostTypeId="1" CreationDate="2020-02-09T00:25:47.170" Score="9" ViewCount="405" Body="&lt;p&gt;Contrasting 2D and 3D in my field (Discrete &amp;amp; Computational Geometry&lt;sup&gt;1&lt;/sup&gt;) is essential. For example, every 2D polygon can be triangulated (with vertex-to-vertex diagonals), but not every 3D polyhedron can be similarly tetrahedralized.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I experience significant push-back when I move to 3D&#xA;(with undergraduates in the US). Typical student reaction: &quot;Whoa!&quot;&#xA;(Most have not had Calc III.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. How do you ease the 2D&lt;span class=&quot;math-container&quot;&gt;$\rightarrow$&lt;/span&gt;3D transition in your teaching?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I use &lt;a href=&quot;https://matheducators.stackexchange.com/a/7975/511&quot;&gt;polydrons&lt;/a&gt;,&#xA;but still it's a scary transition for many students.&#xA;And I have to prepare them for the leap to dimensions &lt;span class=&quot;math-container&quot;&gt;$&amp;gt; 3$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;&#xA;&lt;sub&gt;&lt;a href=&quot;https://www.crcpress.com/Handbook-of-Discrete-and-Computational-Geometry-Third-Edition/Toth-ORourke-Goodman/p/book/9781498711395&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Handbook of Discrete and Computational Geometry&lt;/em&gt;&lt;/a&gt;. Chapman and Hall/CRC, 2017.&#xA;&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-02-09T16:15:32.033" LastActivityDate="2020-02-09T16:15:32.033" Title="Fear of 3-dimensions" Tags="|undergraduate-education|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17837" PostTypeId="2" ParentId="17836" CreationDate="2020-02-09T03:41:59.890" Score="8" Body="&lt;p&gt;After teaching Calculus 3 for many semesters, your question certainly resonates. I think it is empowering to coach students in how to sketch surfaces in 3D. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their tendency from curves in 2D is to first draw the &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; axes, and then superimpose the curve. That's how I sketch. But for surfaces in 3D, it's best to first draw the surface, and afterwards, superimpose the &lt;span class=&quot;math-container&quot;&gt;$(x,y,z)$&lt;/span&gt;-axes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start with a hyperboloid of one-sheet, then progress to planes, spheres, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At first blush, this is a trivial matter. But learning how to sketch surfaces builds confidence.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2020-02-09T03:41:59.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17838" PostTypeId="2" ParentId="17836" CreationDate="2020-02-09T04:03:34.250" Score="1" Body="&lt;p&gt;(long comment)  I think you just have to give them time. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Solidify your algorithms and the rest of it within the 2-D world.  Don't mention 3-D at all then.  Lay a solid foundation on the tools themselves.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then move to 3-D and start by doing some refresher on solid geometry itself (for some first time as solid geometry gets very short shrift in schools versus all the SAS triangle stuff).  Don't even do the computer crap, but just solid geometry.  And I would do a first lesson that was just &quot;names of animals in the zoo&quot; (pyramids, prisms, platonic solids, etc.)  Give them a homework or in class assignment to label the creatures.  Then you can do a little stuff with vertices and like in another lesson (but still just solid geometry).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then do whatever computer stuff you are doing with the cells or whatever it is in 3 dimensions.  And give this some time.  And don't you dare mention 4-D.  Keep focused on the immediate hurdle.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then go to 4-D.  Show a little pictures of shapes and the like for the cool factor.  Don't dwell on it. But for many it will be first time seeing the polytopes.  Wiki has a good descriptive article.  Maybe don't have to do as much remedial here as it's not so important they know all the names of the creatures here.  But do a little orientation before getting into the calculational work.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And then get into whatever math you do with them.  And how to keep track of the stuff analytically (I guess with coordinates with 4 numbers).  Stick to 4 at this time until it's down solid.  No 5 or higher.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then generalize the methods already taught in 4 space to 5, 6 or whatever.  Should be simple if they got the 4 down.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;P.s.  I find you very engaging and suspect your students love you.  But I also kind of get the impression you are a little &quot;hard&quot; and don't do enough progression or expect too much in terms of preparation.  Just a comment, not an insult.  If it resonates great, if not, ignore.  Of course a mature mathematician or very strong new student can handle multiple dimensions at first exposure. But I think this is not your demographic.  So be much more step by step progressive and take the time needed.  [I don't think there is some simple &quot;aha&quot; trick to get the higher dimension awareness.  Need to take the time.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.  I do think that hand drafting (especially isometric view, but even plan and sides) gives students a lot of practice in thinking 3-D.  Unfortunately technical drawing has been rather deprecated in the modern curriculum.  There are great YouTube videos on this stuff though.  I don't see how you can use this insight, but just mentioning it because I do think it helps with thinking spatially.  But also requires large amounts of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.s.  Just scanned the book.  That looks really hard for a startout (even the intro) for kids who have only had frosh calculus and may not be top of the pack.  Also, a handbook seems too long and too unsynthesized (multiple authors are a problem even for specialist encyclopedia tomes...would prefer to read Greenwood and Earnshaw to some cobbled together set of articles by 65 chemists) to be the right way to instruct new students.  They need an easier, integrated approach.  Then they can go buy that handbook if they care enough, or really if they work in the field.&lt;/p&gt;&#xA;" OwnerUserId="13427" LastEditorUserId="13427" LastEditDate="2020-02-09T04:25:39.713" LastActivityDate="2020-02-09T04:25:39.713" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17839" PostTypeId="1" CreationDate="2020-02-09T14:37:34.183" Score="5" ViewCount="169" Body="&lt;p&gt;I'm going to participate in a course (as a teacher) where I'm suppose to teach high school math to high school students in about 15 days during the year. Each class has about 1 hour long. Now, I think it is quite impossible to really get deep into the concepts in so little time, and so I got the following idea: I could gather some texts/articles discussing advanced high school subjects and try to discuss them together with the class. My intention is to provide different experiences and to develope critical thinking. The problem is that I know very little material on this matter, so I'm looking for some references and tips on where to find appropriate material. Can someone help me, please?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; Let me add more information to my post. This course is not a regular one. It is actually a voluntary set of lectures taking place at the university and aimed to help high school students to pass the 'vestibular', which is a brazilian test (similar to the american SAT's) students have to do in order to get accepted in public universities. The (math) content of the test is basically all high school  math taught in Brazil. This includes basic set theory, basic notions of functions, trigonometry, plane and analytic geometry, algebra of matrices and so on. Thus, our course is aimed to help these students with a set of prep lectures. Each teacher can choose the topics he wants to present. In my case, I'll probably teach about functions and first and second order degree equations but these topics are really demanding and 15 one-hour classes are not sufficient to cover all material, so I'm thinking about alternative ways to approach it. Because students often like class discussions, I thought about providing some discussions based on alternative material (not regular books, which are too long to get at any point in 15 days). &lt;/p&gt;&#xA;" OwnerUserId="13428" LastEditorUserId="13428" LastEditDate="2020-02-09T16:53:16.423" LastActivityDate="2020-02-27T20:47:09.290" Title="Tips and References for a 15 days Course on Math" Tags="|secondary-education|reference-request|education-research|critical-thinking|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17840" PostTypeId="2" ParentId="17829" CreationDate="2020-02-09T15:41:32.660" Score="6" Body="&lt;p&gt;This reminds me of my question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/15298/amplitude-of-tan-and-cot-functions&quot;&gt;“Amplitude” of Tan and Cot functions&lt;/a&gt; which referenced a non-standard use of the word amplitude. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math has a language, one that should make communication on this topic pretty clear. When discussing notation or language, I frequently resort to the example &quot;You are on the phone with a friend, pre facetime, audio only. As you describe your equation or geometric construct, will your friend be able to write down exactly what you are describing? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Rusty commented, I hear x two as &lt;span class=&quot;math-container&quot;&gt;$x_{2}$&lt;/span&gt; as well, although I often pronounce the latter as &quot;X sub two&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question that might remain is how much to push for perfection. I respect Ben, but it wouldn't have occurred to me that sin(x) isn't perfect notation. I think we agree that 3 minus 4 is -1 and is pronounced &quot;negative 1&quot;, but do we correct a student who answers &quot;minus 1&quot;?  That's not rhetorical, I'll answer. No, I don't. When in front of a class of 20 High School students, 16 of whom aren't raising their hands and I need to push them to do so, I'm not going to choose that level of pedantry and risk having the student go silent. I do, however, take the opportunity as some other time to show them that the TI84 we commonly use in class has a minus key and also a negative key. That can help get the point across. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, there should be room for a bit of 'inside jokes' or phrases the class might use. When reviewing the 45/45/90 triangle ratios, a student observed &quot;it's faster when going from hypotenuse to leg to simply multiply by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}/2$&lt;/span&gt; than to divide by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; and rationalize. I was happy to see this aha moment, and the class named that manipulation &quot;the Sam trick&quot; when describing the steps later on. Obviously, that isn't going to spread around, and the students know it's a shortcut, if even that. (And I know that we shouldn't fear the radical in the denominator, but I am subbing this semester, not my call).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-02-09T15:41:32.660" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17841" PostTypeId="2" ParentId="17839" CreationDate="2020-02-09T18:45:09.943" Score="3" Body="&lt;p&gt;You might engage them in billiards on an elliptical table&#xA;(1st ref below), or on a circular table but aiming to hit&#xA;a second ball (2nd ref below).&#xA;Nice analytic geometry coupled with intuitive situations.&#xA;See this &lt;a href=&quot;https://www.youtube.com/watch?v=4KHCuXN2F3I&quot; rel=&quot;nofollow noreferrer&quot;&gt;Numberphile video&lt;/a&gt;.&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/ZiI4S.png&quot; width=&quot;400&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Fig.1 (detail) in Reznick et al.&#xA;&lt;/sup&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/HDfPq.png&quot; width=&quot;200&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Fig.2 in Drexler-Gander.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Reznik, Dan, Ronaldo Garcia, and Jair Koiller. &quot;Can the Elliptic Billiard Still Surprise Us?.&quot; &lt;em&gt;The Mathematical Intelligencer&lt;/em&gt; (2019): 1-12.&#xA;  &lt;a href=&quot;https://arxiv.org/abs/1911.01515&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv preprint abstract&lt;/a&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Michael Drexler and Martin Gander,&#xA;  &quot;Circular Billiard.&quot;&#xA;  &lt;a href=&quot;https://www.unige.ch/~gander/Preprints/Billiard.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-02-09T22:00:35.690" LastActivityDate="2020-02-09T22:00:35.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17842" PostTypeId="2" ParentId="7654" CreationDate="2020-02-10T00:01:48.483" Score="1" Body="&lt;p&gt;Most problems I see (in classes after the first calculus courses) are basic misunderstandings in what is allowed to do with equations (and even worse with inequalities). There are also misconceptions on the meaning of strings of equations/inequalities, i.e., &lt;span class=&quot;math-container&quot;&gt;$p^2 &amp;lt; p^2 + u \le p^2 + p &amp;lt; p^2 + 2 p + 1 = (p + 1)^2$&lt;/span&gt; (here &lt;span class=&quot;math-container&quot;&gt;$1 \le u \le p$&lt;/span&gt;). There are problems with the concept of a variable (i.e., it &lt;em&gt;doesn't&lt;/em&gt; vary, it is a fixed, perhaps unknown value or some value we don't want to nail down for the sake of generality; note that this are two different notions). &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-02-10T00:01:48.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17843" PostTypeId="2" ParentId="7837" CreationDate="2020-02-10T10:01:59.910" Score="2" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\times n$&lt;/span&gt; is an action. It moves each point on the number line to a different point. &lt;span class=&quot;math-container&quot;&gt;$\div n$&lt;/span&gt; is the opposite action, meaning: &quot;&lt;span class=&quot;math-container&quot;&gt;$\times n$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\div n$&lt;/span&gt;&quot; doesn't move any points at all. It's the same as doing nothing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's start with an action and rewrite it in several steps:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\div \frac{a}{b}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;the opposite of &lt;span class=&quot;math-container&quot;&gt;$\times \frac{a}{b}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;the opposite of &quot;&lt;span class=&quot;math-container&quot;&gt;$\times a$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\div b$&lt;/span&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\times b$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\div a$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\times \frac{b}{a}$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="13431" LastEditorUserId="13431" LastEditDate="2020-02-10T10:18:01.020" LastActivityDate="2020-02-10T10:18:01.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17844" PostTypeId="2" ParentId="17839" CreationDate="2020-02-10T18:03:19.980" Score="2" Body="&lt;p&gt;My advice:  Look at the test itself.  If possible get some statistics on common errors.  If not, ask other teachers or at least use your intuition.  I.e., make an intelligent guess, &quot;Bayesian estimate&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then design something (lecture and practice together) to hit the common mistakes.  Don't reteach theory, concepts, etc. in some sort of organic manner.  You don't have time for that and it will be a turnoff.  But you will have their attention if you say (and mean it) that you will help them fix the most common mistakes and improve exam score.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your approach should not be &quot;you don't understand this, let me clarify theory to you&quot;.  (The mistaken pedagogy of most of this site.)  But more like...&quot;hey watch out for this snake that wants to bite you.&quot;  Maybe a little concept explanation, but not much.  Be more practical instead.  (Weaker students learn more from practice and correction than by explication.)  Try to make the practice approximate the tests as much as possible.  If old exams are public, mine those directly.  If not, mimic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I had a bunch of cool enrichment ideas for you, like JOR.  But that is not what these kids need.  See Dave Renfro's comments.&lt;/p&gt;&#xA;" OwnerUserId="13427" LastEditorUserId="123" LastEditDate="2020-02-27T20:47:09.290" LastActivityDate="2020-02-27T20:47:09.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17845" PostTypeId="2" ParentId="5648" CreationDate="2020-02-10T20:16:24.147" Score="0" Body="&lt;p&gt;&quot;divided by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot; means &quot;times &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; inverse&quot;. &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; inverse&quot; is the number such that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; inverse = 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;0 times anything = 0. 0 times nothing = 1. Nothing is 0 inverse. &quot;0 inverse&quot; does not refer to anything. &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; times 0 inverse&quot; does not refer to anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;x divided by 0&quot; does not refer to anything.&lt;/p&gt;&#xA;" OwnerUserId="13431" LastActivityDate="2020-02-10T20:16:24.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17846" PostTypeId="2" ParentId="562" CreationDate="2020-02-11T15:28:24.757" Score="4" Body="&lt;p&gt;You only need &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$f^\prime$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}\Psi$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; is an &lt;strong&gt;expression&lt;/strong&gt; whose value depends on that of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, is the rate of change of the value of &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d}{dx}\Psi = \lim_{h \rightarrow 0} \frac{f\left(x+h\right) - f\left(x\right)}{h}\ \text{where}\ f = k \mapsto \left(\Psi,\ \text{given}\ x=k\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; is the rate of change of a &lt;strong&gt;dependent variable&lt;/strong&gt; (&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;) with respect to an independent variable (&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;). Even though &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is simply an expression whose value depends on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (making &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}y$&lt;/span&gt; synonyms), &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; is nice because its form reminds us of the intuition that comes from &lt;span class=&quot;math-container&quot;&gt;$\frac{\Delta y}{\Delta x}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{dy}{dx}= \lim_{h \rightarrow 0} \frac{f\left(x+h\right) - f\left(x\right)}{h}\ \text{where}\ f = k \mapsto \left(y,\ \text{given}\ x=k\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f^\prime$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a &lt;strong&gt;function&lt;/strong&gt; with one argument, is the rate of change of change of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;'s output with respect to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;'s input.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f^\prime = k \mapsto \lim_{h \rightarrow 0} \frac{f\left(k+h\right) - f\left(k\right)}{h}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some caveats:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dx}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx} f$&lt;/span&gt; are meaningless because &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a function, not a variable or an expression. The value of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; does not change with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;; the value of &lt;span class=&quot;math-container&quot;&gt;$f\left(x\right)$&lt;/span&gt; changes with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. So use &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx} f\left(x\right)$&lt;/span&gt; instead. Note that &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx} f\left(x\right)$&lt;/span&gt; is a synonym of &lt;span class=&quot;math-container&quot;&gt;$f^\prime\left(x\right)$&lt;/span&gt;. Also note that &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx} f\left(a\right)=0$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$f\left(a\right)$&lt;/span&gt; is not an expression that depends on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Likewise, &lt;span class=&quot;math-container&quot;&gt;$y^\prime$&lt;/span&gt; is meaningless because &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is a variable, not a function. &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; could depend on any number of variables, so there's no unambiguous way to write &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as a function. But you could theoretically write: &lt;span class=&quot;math-container&quot;&gt;$\left( k \mapsto \left(y,\ \text{given}\ x=k \right)\right)^\prime\left(x\right)$&lt;/span&gt; which is a synonym of &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13431" LastEditorUserId="13431" LastEditDate="2020-02-12T00:18:01.373" LastActivityDate="2020-02-12T00:18:01.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17847" PostTypeId="2" ParentId="13353" CreationDate="2020-02-11T23:16:51.300" Score="1" Body="&lt;p&gt;&amp;quot;Explain _____&amp;quot; is an open-ended question. Someone who understands a concept and can easily explain it face-to-face, while being prompted with clarifying questions, might struggle when faced with a written question and a blank page. Knowing where to begin is half the battle. It's a form of writer's block.&lt;/p&gt;&#xA;&lt;p&gt;Someone who &lt;em&gt;has&lt;/em&gt; an insight might not &lt;em&gt;recognize&lt;/em&gt; that a question is asking for a crystallized expression of that specific insight. As a student, I've had my fair share of short-form essay questions that were frustrating because, although my answer proved that I understood what I needed to understand, I had no way to tell if I was presenting the information in exactly the way the grader was looking for. (Especially questions that asked what are the steps in a process, where the division into steps is arbitrary.) Resistance to &amp;quot;explain _____&amp;quot; questions does not necessarily indicate resistance to conceptual learning.&lt;/p&gt;&#xA;&lt;p&gt;The good news is that there are often ways to test someone's conceptual understanding without asking them to explain it ex nihilo. You just have to get them to apply their understanding in a way that reveals any gaps. And they have to understand and accept that they'll be tested on questions where there's no rote technique to arrive at the answer. This acceptance naturally follows if:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The test/assignment is perceived as fair, i.e. there is an objectively true and uncontroversial answer and the student will get full credit for providing that answer.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There are opportunities to practice for the types of questions that will be encountered on tests. Most importantly, the practice problems should reveal the exact conceptual weaknesses that the student needs to work on. This might require the teacher to review the student's practice work and point the student in the right direction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So my answer is to find the middle ground. Teach to the test. Choose problems that play to the strengths of the test format &lt;em&gt;and&lt;/em&gt; require understanding and insight. Use tests/assignments as diagnostics to identify conceptual gaps and motivate the student to fill in those gaps.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For example, you mentioned: &amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$h\left(x\right)$&lt;/span&gt; is a function and I know its graph, explain how to find the graph of &lt;span class=&quot;math-container&quot;&gt;$h^{−1}\left(x\right)$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Instead, try this: present three different graphs, and ask the student to draw approximately the inverse of each curve. The graphs are followed by a question: &amp;quot;How did you know what to draw?&amp;quot; This is an example of the more general questioning method of: instead of asking how someone &lt;em&gt;would&lt;/em&gt; solve a problem, ask them to solve a problem and then ask how they &lt;em&gt;did&lt;/em&gt; solve it.&lt;/p&gt;&#xA;&lt;p&gt;Another thing you can do is pose problems that are trivial to solve if the student understands a concept, but would be impossible to solve (or impractical to solve within the time constraint) otherwise.&lt;/p&gt;&#xA;&lt;p&gt;Example: say you want to test them on the concept that &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt; is an isomorphism from &lt;span class=&quot;math-container&quot;&gt;$\left(\mathbb{R},+\right)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\left(\mathbb{R}_{&amp;gt;0},\times\right)$&lt;/span&gt;. You want to verify that the student not only has memorized the rule that &lt;span class=&quot;math-container&quot;&gt;$e^{x+y} = e^x e^y$&lt;/span&gt;, but also intuitively grasps that &lt;span class=&quot;math-container&quot;&gt;$\left(\mathbb{R},+\right)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\left(\mathbb{R}^+,\times\right)$&lt;/span&gt; are essentially the same structure. A problem could be:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Find the value of &lt;span class=&quot;math-container&quot;&gt;$e^1 \times e^2 \times \cdots \times e^{100}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The only feasible way to solve this on a test is to recognize that &lt;span class=&quot;math-container&quot;&gt;$e^1 \times e^2 \times \cdots \times e^{100} = e^{1+2+\cdots+100} = e^{100 \times 101 \div 2} = e^{5050}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Another problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vt1Is.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vt1Is.png&quot; alt=&quot;number line&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The bottom half of this line is the standard real-valued number line. The top half of the line is empty.&lt;/p&gt;&#xA;&lt;p&gt;Consider any two arrows that begin at the center and point either left or right. Their &lt;em&gt;composition&lt;/em&gt; is the combined arrow you get when the first arrow begins at the center and the second arrow begins where the first arrow ends.&lt;/p&gt;&#xA;&lt;p&gt;We want to fill in the top half with real numbers. If there's a point on the line equal to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; on the bottom half and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; on the top half, we call that point &lt;span class=&quot;math-container&quot;&gt;$\left(x, y\right)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If the first arrow stops at &lt;span class=&quot;math-container&quot;&gt;$\left(x_1, y_1\right)$&lt;/span&gt; and the second stops at &lt;span class=&quot;math-container&quot;&gt;$\left(x_2, y_2\right)$&lt;/span&gt;, then we want their composition to stop at &lt;span class=&quot;math-container&quot;&gt;$\left(x_1 + x_2, y_1 y_2 \right)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We also want each top-half number to be greater than all top-half numbers to its left.&lt;/p&gt;&#xA;&lt;p&gt;Is this possible? If not, then explain why. If it is possible, then describe how to fill in the top half and prove that it's correct. Which number is at the center?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13431" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-04-16T04:18:41.170" CommentCount="0" CommunityOwnedDate="2020-02-11T23:16:51.300" ContentLicense="CC BY-SA 4.0" />
  <row Id="17848" PostTypeId="2" ParentId="17632" CreationDate="2020-02-12T04:17:39.200" Score="0" Body="&lt;p&gt;There's a delightfully simple visual intuition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine you have the curve of &lt;span class=&quot;math-container&quot;&gt;$y=f\left(x\right)$&lt;/span&gt; drawn for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Put your pencil on the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept. Start moving to the right at 1 unit per second, but keep your pencil on the curve. At time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, you're at the point &lt;span class=&quot;math-container&quot;&gt;$\left(t, f\left(t\right)\right)$&lt;/span&gt; and your vertical velocity is &lt;span class=&quot;math-container&quot;&gt;$f^\prime \left(t\right)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now imagine there's a screen in front of you that displays &quot;&lt;span class=&quot;math-container&quot;&gt;$x=$&lt;/span&gt;&quot;, followed by a real number. Internally, it uses a clock and a function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; to decide which number to display: &lt;span class=&quot;math-container&quot;&gt;$x = g\left(t\right)$&lt;/span&gt;. Currently, &lt;span class=&quot;math-container&quot;&gt;$t=0$&lt;/span&gt;. When you press &quot;start&quot;, the clock starts running, and the number starts changing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Put your pencil on the curve at &lt;span class=&quot;math-container&quot;&gt;$\left(x, f\left(x\right)\right)$&lt;/span&gt; according to the display. Press &quot;start&quot;, watch the number, and keep your pencil on the curve for the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-value shown.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your vertical velocity is &lt;span class=&quot;math-container&quot;&gt;$f^\prime\left(x\right) g^\prime\left(t\right)$&lt;/span&gt;. It's the slope of the hill times your horizontal velocity of attacking the hill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that because &lt;span class=&quot;math-container&quot;&gt;$x=g\left(t\right)$&lt;/span&gt;, your vertical velocity is &lt;span class=&quot;math-container&quot;&gt;$f^\prime\left(g\left(t\right)\right)g^\prime\left(t\right)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also note that because &lt;span class=&quot;math-container&quot;&gt;$f^\prime\left(x\right)$&lt;/span&gt; is the rate of change of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$g^\prime\left(t\right)$&lt;/span&gt; is the rate of change of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, you can write your vertical velocity as: &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx} \frac{dx}{dt}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The rest of the challenge is knowing how to apply this rule. Specifically, knowing how to decompose a function, and recognizing when a function can be decomposed into two functions whose derivatives you already know. I think that just takes practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But while you're practicing, it's helpful to stay rooted in the above intuition. And it's helpful to remember that the equivalence between &lt;span class=&quot;math-container&quot;&gt;$f^\prime\left(g\left(t\right)\right)g^\prime\left(t\right)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx} \frac{dx}{dt}$&lt;/span&gt; comes from the fact that they are simply two different ways of writing &lt;span class=&quot;math-container&quot;&gt;$f^\prime\left(x\right) g^\prime\left(t\right)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$y=f\left(x\right)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=g\left(t\right)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13431" LastEditorUserId="13431" LastEditDate="2020-02-12T04:24:43.590" LastActivityDate="2020-02-12T04:24:43.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17849" PostTypeId="2" ParentId="17632" CreationDate="2020-02-12T05:40:57.393" Score="3" Body="&lt;p&gt;This is kind of a &quot;how I did it,&quot; but I think that it's something worth writing up somewhere, and it's too long for a comment. Here's my approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I last taught calculus, I began with a graphed function f(x) which had no simple formula (piecewise linear is sufficient), and we sketched its derivative where it was defined. Then I asked them about what the derivative of f(x-1) should be. After some reminder of what f(x-1) means and how it leads to a translation of the function, most students agree that it should be f'(x-1). &quot;Does this always happen for graph transformations?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we took it a little further using f(2x) and I asked students what they thought should happen to the derivative of a function as you performed the horizontal compression that f(2x) causes. They immediately see that the derivative should be compressed too. But we graph f(2x) and notice that there's something wrong about its slopes compared to f(x). The rise/run values at corresponding points are different: the &quot;rise&quot; stayed the same, but the corresponding &quot;run&quot; got cut in half, making the derivative double in value. So we came up with the formula d/dx f(2x) = f'(2x)*2 . The f'(2x) was needed to make the derivative &quot;match up in x&quot; with the stretches performed on the original function, and the *2 was needed to account for the change in steepness that happens because of the stretch. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This motivated the chain rule enough that we then could ask &quot;What is the formula for the derivative of f(g(x)) in general?&quot; This I did not prove, but it was enough to comment that you'd have to use f'(g(x)) to get the derivative to match with horizontal transformations, and g'(x) to deal with the steepness changing, giving the final formula d/dx f(g(x))= f'(g(x))g'(x). The analogous notation dy/du du/dx is introduced simultaneously and compared/contrasted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we went to computing the derivatives of functions like sin(x^3) where the students practiced identifying the outer and inner functions as well as computing the derivatives, comparing/contrasting the two uses of different notation and noticing that the results were the same either way we did it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's worth noting that I particularly emphasized the notion of function composition at the start of the course and had assigned multiple homework problems leading up to that point where students were expected to perform compositions and describe transformations, under the guise of &quot;let's make sure you know prerequisite content,&quot; and the priming activity at the start of the class was to graph a function transformation. I think this fit in one to two hour-long college class periods.&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2020-02-13T17:59:17.243" LastActivityDate="2020-02-13T17:59:17.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17852" PostTypeId="1" CreationDate="2020-02-12T14:00:02.920" Score="8" ViewCount="418" Body="&lt;p&gt;I am recently wondering whether it is a good idea to read old (or very old) books in mathematics. In high school level, some people suggest students who are interested in geometry to read Euclid's &lt;em&gt;Elements&lt;/em&gt;. Euclid does have some beautiful theorems and proofs, but the way they are originally stated in &lt;em&gt;Elements&lt;/em&gt; are much more awkward than modern versions of the same theorems/proofs. For example, when proving that a triangle with two angles (say, &lt;span class=&quot;math-container&quot;&gt;$ABC,ACB$&lt;/span&gt;) equal must be isoceles, Euclid constructed &lt;span class=&quot;math-container&quot;&gt;$AC'=AB$&lt;/span&gt; along the side &lt;span class=&quot;math-container&quot;&gt;$AC$&lt;/span&gt; and showed that if &lt;span class=&quot;math-container&quot;&gt;$C'$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; does not coincide, then there is a contradiction. This is clearly more awkward than the modern version of (essentially) the same proof, which does NOT involve any construction of new lines or points, but just use the fact that triangles &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ACB$&lt;/span&gt; are congruent (note that the order of &lt;span class=&quot;math-container&quot;&gt;$A,B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; matters, so &lt;span class=&quot;math-container&quot;&gt;$AB=AC, BC=CB$&lt;/span&gt;, and so on). People back then seem to have a hard time accepting that a triangle is congruent to itself. This is sort of an &quot;automorphism&quot;. (Well, the stabilizer subgroup of the set &lt;span class=&quot;math-container&quot;&gt;$\{A,B,C\}$&lt;/span&gt; in the group of all isometries of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^2$&lt;/span&gt; is of order 2.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same thing is also true for university-level texts. An old text (by &quot;old&quot;, I actually mean something like before 1980s) could be quite good if the subject discussed in the book is well-established. However, sometimes, a branch of mathematics might be slightly &quot;reworded&quot; over time, with some possibly significant shifts in its emphasis. To give an example, Allen Hatcher remarks in the preface of his Algebraic Topology book the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In a sense, the book&#xA;  could have been written thirty years ago since virtually all its content is at least that&#xA;  old. However, the passage of the intervening years has helped clarify what the most&#xA;  important results and techniques are.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now, here is the problem: &lt;strong&gt;how to tell if a book is too old to be used today?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="9339" LastActivityDate="2020-05-20T05:20:17.400" Title="Reading old or ancient texts in mathematics" Tags="|textbooks|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17853" PostTypeId="2" ParentId="17852" CreationDate="2020-02-12T15:41:38.403" Score="3" Body="&lt;p&gt;For me, generally, using any book that is easy to use, for a new topic is the ideal strategy.  Worrying too much about missing some particular insight is counter-productive versus just moving forward.  There will always be a tension of completeness versus ease of first entry.  I just think you are better off going forward and then filling in, versus worrying too much about learning it all perfectly (or at least as the Internet answerers define things) to start.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of practical implications, I would draw the line where the language or notation is too difficult.  I think that is earlier than the 80s.  You see huge amounts of Dover paperbacks or Schaum's Outlines from the 50s that still get a lot of high reviews on Amazon from students (often dissatisfied with their prescribed modern, longer, texts).  I find them very accessible, sometimes more so.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anything, I would probably draw the line in the 30s (approximately).  Although some earlier books (e.g. Granville calculus) were very readable and thus used into the 60s.  And I find Granville still accessible now, kind of the first modern book.  But some older books I find hard to use because of language (Edwards Treatise on Integral Calculus, Taylor DiffyQs, etc.)  I would draw the rough age cut later for a text in &quot;British&quot; than a text in &quot;American&quot; (but maybe the reverse for subjects of the Queen).  This is not to say not lots of fun nuggets inside old books.  But just that starting with such a text for first exposure may be non-optimal.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also on your specific topic, I think learning high school geometry from a normal text, not Euclid, is more time-efficient and more likely to be tolerable (meaning you won't get frustrated and give up).  I would leave the direct Euclid study for later, if you ever really care enough.  Or you can go to St. Johns and do it in Ancient Greek.  Then again, that school remains rather idiosyncratic...so not like it's caught on.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course if you learn geometry deeply or become a teacher of it, you will eventually see different texts and have some perspective on which is good at what.  Which is missing what (and being complete is not always a feature, can be a bug).  That is a wonderful thing about an experienced teacher that they have this perspective.  But not what a neophyte should concentrate on, now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But really you should just see how things work for you.  You can tell if you are turned off/on by a text.  It's a free country, still (sorta, barely).  So pick something you like.  For instance, it is charming to see how many people from India still use books from British math writers from 1860-1930 and get a lot out of them and presumably don't struggle with the older language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But at the end of the day, if you use a non-optimal book and learn it perfectly (work every problem of the homework, etc.) than you will do WAY better than someone who dabbles (or even just works at a non-mastery level) in a book that is optimal.  It's like sports.  Yeah, having a great coach is nice and helpful.  But still, if you are a good athlete and train diligently, that is way more important than facilities or who your coach is (versus other athletes).&lt;/p&gt;&#xA;" OwnerUserId="13427" LastActivityDate="2020-02-12T15:41:38.403" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17854" PostTypeId="1" CreationDate="2020-02-12T18:43:31.763" Score="10" ViewCount="190" Body="&lt;p&gt;I am currently reflecting two educational experiences that have made me question if there's research around this particular topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) A few years ago, I was an adjunct instructor for a Trigonometry class and a Calculus 1 class. I taught those classes for a few semesters and I had several students come up to me on the first day and say something along the lines of &quot;I haven't been in a math class in forever. I graduated high school, went out and got a job for the past 10 years and now I want to change my career. They said I have to take this class but I don't remember much of anything about math.&quot; Some of those students did struggle, but I was actually surprised at how many of those students ended up earning top marks, pointing connections out to me that I had not even thought of, or thinking of novel ways to solve a problem that wasn't in the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Another anecdotal point is that I once had to be a TA for a summer class aimed at advanced high school students. The instructor of the course told the class about his personal journey: He was never 'good at school' and barely scraped through 9th grade. He dropped out in the middle of 10th grade and got a job. After a while, he eventually wanted to get a GED but feared having to relearn the math. But all of sudden, the math...just made sense. Some stuff he struggled with of course, but he was able to figure it out and progress. Currently he has a PhD in math and has such great intuition for complicated problems that I don't even understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So both of those stories made me think about the way we try to motivate math. Often times we are faced with the &quot;when am I ever going to use this?&quot; question. I am honest in my response. I say something like &quot;honestly, you probably won't. But getting an understanding of &lt;em&gt;this&lt;/em&gt; problem, will help develop logic and problem solving skills you'll need for &lt;em&gt;future&lt;/em&gt; unknown problems.&quot; Society tries to advertise math to students by saying that &quot;studying math will develop critical thinking skills needed elsewhere.&quot; But the two stories I shared have me thinking in the opposite direction: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;QUESTION 1&lt;/strong&gt;&lt;br&gt;&#xA;  Is there research that shows if the critical thinking skills and&#xA;  experience that people develop when dealing with the real life issues&#xA;  that come with work, families, finances, etc. will help them be&#xA;  'better at learning math' than someone who just finished high school?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And I know my experience in story 1) might not be typical. I've seen plenty of 30, 40, and 50 years-olds taking very remedial math classes and not understanding fractions.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;QUESTION 2&lt;/strong&gt;&lt;br&gt;&#xA;  Is there any research on what helps returning adult learners more&#xA;  successful in math?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2020-02-18T09:05:59.750" Title="Adult Learners Returning to College Level Math" Tags="|education-research|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17856" PostTypeId="2" ParentId="17854" CreationDate="2020-02-13T12:53:14.940" Score="5" Body="&lt;p&gt;In the &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Encyclopedia of Mathematics Education&lt;/em&gt;.&#xA;  Lerman, Stephen, ed. Dordrecht: Springer, 2014.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;there is an article entitled&#xA;&quot;Adults Learning Mathematics&quot;&#xA;by Diana Coben and John O’Donoghue &#xA;(&lt;a href=&quot;https://link.springer.com/referenceworkentry/10.1007%2F978-94-007-4978-8_5&quot; rel=&quot;noreferrer&quot;&gt;Springer link&lt;/a&gt;),&#xA;which says&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Adults learning mathematics is a young field of study and research, emerging towards the end of the twentieth century&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The article has ~70 references, so it might respond to your &lt;strong&gt;Q2&lt;/strong&gt; (research),&#xA;but scanning it I do not see that it addresses your interesting &lt;strong&gt;Q1&lt;/strong&gt;&#xA;(life skills &lt;span class=&quot;math-container&quot;&gt;$\rightarrow$&lt;/span&gt;? math skills).&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-02-13T13:16:54.847" LastActivityDate="2020-02-13T13:16:54.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17857" PostTypeId="2" ParentId="17852" CreationDate="2020-02-13T15:25:18.357" Score="7" Body="&lt;p&gt;This has been a question for many, many years.  Dodgson (i.e. Lewis Carroll) goes to &lt;a href=&quot;https://en.wikisource.org/wiki/Euclid_and_His_Modern_Rivals&quot; rel=&quot;noreferrer&quot;&gt;typically wacky lengths&lt;/a&gt; to defend using a two-thousand-year-old text (see a &lt;a href=&quot;https://www.jstor.org/stable/3621985&quot; rel=&quot;noreferrer&quot;&gt;helpful review here&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Honestly, it really depends a lot upon the subject matter.  Some people are still using Granville's Calculus (apparently), but try to read an algebraic geometry text from 80 years ago with a recent education and you will have lots of trouble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think that while, as a beginning instructor, recent books may be a good idea, as one gains experience using older texts may be suitable &lt;em&gt;if&lt;/em&gt; the older text lends insight in some useful way, or is simply no worse and no better than modern texts (again, particularly in calculus this is likely to be true).  But it is a judgment call, and really dependent upon content.  Obviously disciplines like game theory and operations research didn't even exist a hundred years ago ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, properly annotated, older texts can &lt;em&gt;enhance&lt;/em&gt; understanding, because one might see how something came to be, not just handed a formula.  In that regard, I recommend the &lt;a href=&quot;https://blogs.ursinus.edu/triumphs/&quot; rel=&quot;noreferrer&quot;&gt;TRIUMPHS project&lt;/a&gt;, which has dozens of primary source materials put into a form that &quot;modern&quot; students can really benefit from, going all the way from the Babylonians, Greeks, and the Nine Chapters to near today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, one should note it also depends on the goals of a course.  For me, teaching a linear algebra or number theory course without talking about quite modern computational applications would be dereliction of duty, so if I were to choose a text (new or old) which didn't have some resources on this, I would need to create my own or use web resources appropriately.  But not everyone would agree with me on this, and for them using in some cases quite old texts could be perfectly acceptable.  A similar comment applies for things like exercises, which tend not to be a big factor in the pre-industrial textbook industry!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To sum up, as long as one as instructor is aware of the modern approaches, notations, and results, and can effectively connect them to the text and translate them for the students, there shouldn't be an arbitrary time limit, though for some topics the further back you go, the less realistic this will be.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2020-02-13T15:25:18.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17859" PostTypeId="2" ParentId="8011" CreationDate="2020-02-13T22:44:55.873" Score="-2" Body="&lt;p&gt;Pronouncing &lt;span class=&quot;math-container&quot;&gt;$-x$&lt;/span&gt; as &quot;negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot; leads to confusion because &quot;negative&quot; as an adjective means &quot;less than zero&quot;. A red car is red; an even number is even; but negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; might not be negative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If math had evolved differently, we could have been using different symbolism today. Here's how the two symbolisms would translate:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{array}{rcl}&#xA;2 &amp;amp; : &amp;amp; \overset{\rightarrow}{2}\\&#xA;-2 &amp;amp; : &amp;amp; \overset{\leftarrow}{2}\\&#xA;-x &amp;amp; : &amp;amp; \overset{\rightleftharpoons}{x}\\&#xA;x-y &amp;amp; : &amp;amp; x + \overset{\rightleftharpoons}{y}&#xA;\end{array}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could show this to your students, tell them they can mentally translate between the two if it helps them, and pronounce &quot;&lt;span class=&quot;math-container&quot;&gt;$-x$&lt;/span&gt;&quot; / &quot;&lt;span class=&quot;math-container&quot;&gt;$\overset{\rightleftharpoons}{x}$&lt;/span&gt;&quot; as &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; flipped&quot; to emphasize that it's what you get by flipping the number line.&lt;/p&gt;&#xA;" OwnerUserId="13431" LastActivityDate="2020-02-13T22:44:55.873" CommentCount="3" CommunityOwnedDate="2020-02-14T07:14:28.320" ContentLicense="CC BY-SA 4.0" />
  <row Id="17860" PostTypeId="2" ParentId="5794" CreationDate="2020-02-13T23:21:10.047" Score="0" Body="&lt;p&gt;The &quot;&lt;span class=&quot;math-container&quot;&gt;$-x$&lt;/span&gt;&quot; symbolism obscures the concept. Some &lt;a href=&quot;https://matheducators.stackexchange.com/a/17859/13431&quot;&gt;different symbolism&lt;/a&gt; is more transparent:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{array}{rcl}&#xA;2 &amp;amp; : &amp;amp; \overset{\rightarrow}{2}\\&#xA;-2 &amp;amp; : &amp;amp; \overset{\leftarrow}{2}\\&#xA;-x &amp;amp; : &amp;amp; \overset{\rightleftharpoons}{x}\\&#xA;x-y &amp;amp; : &amp;amp; x + \overset{\rightleftharpoons}{y}&#xA;\end{array}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\rightleftharpoons$&lt;/span&gt; flips the arrow under it.&#xA;&lt;span class=&quot;math-container&quot;&gt;$\overset{\rightleftharpoons}{x}$&lt;/span&gt;&#xA;is where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; moves to when you flip the number line. Flipping something twice is the same as doing nothing to it. So&#xA;&lt;span class=&quot;math-container&quot;&gt;$\overset{\rightleftharpoons}{\left(\overset{\rightleftharpoons}{x}\right)}$&lt;/span&gt;&#xA;is what you get when you move &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to where it already is (i.e. it's &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Multiplying by &lt;span class=&quot;math-container&quot;&gt;$\overset{\rightleftharpoons}{x}$&lt;/span&gt; is scaling the number line by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and then flipping it. Therefore, &#xA;&lt;span class=&quot;math-container&quot;&gt;$\overset{\rightleftharpoons}{x} \times \overset{\rightleftharpoons}{y}$&lt;/span&gt;&#xA;means: start at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, flip, scale by &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, flip. There are two ways to think about this:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The number line was flipped twice, so it wasn't really flipped at all. What we did was the same as: start at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, scale by &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Start at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, scale by &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, flip&quot; is the same as &quot;Start at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, flip, scale by &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&quot;. Therefore,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{array}{rcl}&#xA;x\times\overset{\rightleftharpoons}{y} &amp;amp; = &amp;amp; \overset{\rightleftharpoons}{x}\times y\\&#xA;\overset{\rightleftharpoons}{x}\times\overset{\rightleftharpoons}{y} &amp;amp; = &amp;amp; \overset{\rightleftharpoons}{\left(\overset{\rightleftharpoons}{x}\right)}\times y\\&#xA; &amp;amp; = &amp;amp; x\times y&#xA;\end{array}$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Draw a number line, label it with numbers in the style of &#xA;&lt;span class=&quot;math-container&quot;&gt;$\overset{\leftarrow}{2}$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$\overset{\rightarrow}{2}$&lt;/span&gt;,&#xA;and label a pair of points as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and&#xA;&lt;span class=&quot;math-container&quot;&gt;$\overset{\rightleftharpoons}{x}$&lt;/span&gt;.&#xA;Show what happens when the number line is flipped and scaled. The notation can be discarded when the student is convinced that &lt;span class=&quot;math-container&quot;&gt;$-(-(x))=x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(-x) \times (-y) = x \times y$&lt;/span&gt;. It's a temporary communication tool, not something to be tested on.&lt;/p&gt;&#xA;" OwnerUserId="13431" LastEditorUserId="13431" LastEditDate="2020-02-14T02:12:21.207" LastActivityDate="2020-02-14T02:12:21.207" CommentCount="4" CommunityOwnedDate="2020-02-14T08:58:00.413" ContentLicense="CC BY-SA 4.0" />
  <row Id="17861" PostTypeId="2" ParentId="17632" CreationDate="2020-02-13T23:59:18.373" Score="1" Body="&lt;p&gt;3blue1brown has an excellent &lt;a href=&quot;https://www.youtube.com/watch?v=YG15m2VwSjA&quot; rel=&quot;nofollow noreferrer&quot;&gt;video&lt;/a&gt; about this. (Chain rule discussion starts at 8:40.)&lt;/p&gt;&#xA;" OwnerUserId="13431" LastActivityDate="2020-02-13T23:59:18.373" CommentCount="0" CommunityOwnedDate="2020-02-13T23:59:18.373" ContentLicense="CC BY-SA 4.0" />
  <row Id="17862" PostTypeId="2" ParentId="2279" CreationDate="2020-02-14T03:00:41.017" Score="4" Body="&lt;p&gt;3blue1brown makes the best math videos in existence in my opinion, and he uses a Python library called &lt;a href=&quot;https://github.com/3b1b/manim&quot; rel=&quot;nofollow noreferrer&quot;&gt;manim&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13431" LastActivityDate="2020-02-14T03:00:41.017" CommentCount="0" CommunityOwnedDate="2020-02-14T03:00:41.017" ContentLicense="CC BY-SA 4.0" />
  <row Id="17863" PostTypeId="1" CreationDate="2020-02-14T03:32:04.537" Score="5" ViewCount="882" Body="&lt;p&gt;I've asked a question on Math Stack Exchange, but it was suggested it might be a better idea to post it on this Educators instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the question link:&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/3545998/intuition-behind-sin-theta-when-introducing-this-to-high-school-students&quot;&gt;https://math.stackexchange.com/questions/3545998/intuition-behind-sin-theta-when-introducing-this-to-high-school-students&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Briefly, the question was:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose I'm introducing trig, and I ask students to imagine trying to figure out the height of a tree, if you know how far you are from the base of the tree and you estimate the angle to the top of the tree.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you take the tree &lt;strong&gt;example&lt;/strong&gt; from above, we have the adjacent side and the angle. Now:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The definition of &lt;span class=&quot;math-container&quot;&gt;$\tan(\theta)$&lt;/span&gt; is the missing quantity we wanted in&#xA;  the first place. The ratio of the opposite side and the adjacent side.&#xA;  But how does &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt; go and calculate the ratio when I give it an&#xA;  angle? Is it just some magic computer?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think it's possible to convince them - once I have this ratio, I can find the length of the missing side: &lt;span class=&quot;math-container&quot;&gt;$\text{Opposite} = \tan(\theta)\times \text{Adjacent}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Clarification&lt;/em&gt;: Thanks to those in the comments for responding and suggesting clarifications.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Yes, how can we explain to you how a calculator computes, say, tan(&lt;span class=&quot;math-container&quot;&gt;$62^∘$&lt;/span&gt;)? And I'm also looking for ideas for how to then explain that to students?&lt;/li&gt;&#xA;&lt;li&gt;A method on how it might be computed, which is a little more general then asking them to imagine a infinitely large table of values where we've taking heaps of right angled triangles and calculated their ratios&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13446" LastEditorUserId="128" LastEditDate="2020-02-14T15:37:48.410" LastActivityDate="2020-02-19T11:21:08.353" Title="How to intuitively understand how the trig ratios are calculated" Tags="|secondary-education|concept-motivation|trigonometry|intuition|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17864" PostTypeId="1" AcceptedAnswerId="17867" CreationDate="2020-02-14T08:12:30.330" Score="38" ViewCount="9409" Body="&lt;p&gt;I teach many high school students, and all of them complain about being unable to fully understand mathematical concepts. I try to show them the joy of learning and deepen their understanding through games and quizzes. Yet they still feel it is too tedious to understand the theories. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of the students say they don’t like mathematics because they only learn for examinations. How do I show them the true beauty of mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I teach high school students, so I am interested in the high school syllabus only, like algebra, trigonometry, and geometry.&lt;/p&gt;&#xA;" OwnerUserId="13448" LastEditorUserId="11" LastEditDate="2020-02-21T15:17:01.390" LastActivityDate="2023-08-15T18:53:15.890" Title="How do I show students the Beauty of Mathematics?" Tags="|secondary-education|student-motivation|teaching|" AnswerCount="18" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="17865" PostTypeId="2" ParentId="17863" CreationDate="2020-02-14T16:00:49.737" Score="8" Body="&lt;p&gt;If you are teaching this at an introductory level, then the algorithm that calculators use today is going to go far over their heads.  (It might go over MY head!)  The story of how we developed increasingly accurate trig tables over the course of history would be an interesting topic of inquiry for advanced algebra / pre-calculus / calculus, but at the beginning you should focus on the key ideas for the beginning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A constructivist introduction to trig would be to put aside the scientific calculator for a while.  So you estimate the height of the tree by drawing a right triangle with the same angle of elevation, noticing that the triangle is similar to the real-world triangle by AA, and using a proportion to solve for the unknown height.  Then you might extrapolate from that.  What if your job was estimating the height of 50 trees a day?  Drawing and measuring all of those triangles would take up a lot of your day, especially when you would find yourself redrawing the same triangle when you had the same angle of elevation.   So realizing that the only needed input value was the angle of elevation and the only needed output value was the ratio of the two sides, you can replace all of that triangle drawing with a simple table of angle to ratio.  So that is a brand new &lt;strong&gt;function&lt;/strong&gt; that we just created to address a real-world problem!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you feel like sticking on with the constructivist perspective, you can then move on to having the class construct a rudimentary class-wide trig table by having different teams measure the three principal ratios for triangles ranging from &lt;span class=&quot;math-container&quot;&gt;$5-85-90$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$45-45-90$&lt;/span&gt; in five degree increments.  You can attach names to the functions at this point and start to develop observations that are important.  For instance, it is no accident that the sine of an angle is the same as the &lt;strong&gt;co&lt;/strong&gt;sine of the &lt;strong&gt;co&lt;/strong&gt;mplementary angle.  You can then have the class start solving real-world trig problems, except still using their own constructed trig tables and a four-function calculator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From here, it's just a simple step forward to say that this is such an obviously important trio of functions that people throughout history have struggled to measure them accurately.  Because of engineering, navigation, and ballistics, it's not at all an overstatement to claim that the balance of global power has shifted based on having a more modern trig table than your rivals!  Ancient folks like Hipparchus and Ptolemy used geometry (see below).  The development of calculus made Taylor series a reality that Charles Babbage and Ada Lovelace used to develop to precursors to the first mechanical computers.  And modern computers and calculators can essentially compute the trig values of very precisely measured angles to any necessary degree of precision.  But I think the key idea is that we can use the calculator in the same way that students used the table they developed and have a very reliable measurement that they don't have to treat as &quot;magic&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;That said, I have long thought that it would be an interesting honors project for a sufficiently motivated precalculus student to try and create a portion of the sine table to a certain decimal precision without technology in a process inspired by Ptolemy's calculations (which may have been Hipparchus' method as well, although Hipparchus' work is lost to history because of the jerks who destroyed the library at Alexandria).  Ptolemy's function measured the length of a chord subtended by a central angle of a given size.  That function &lt;span class=&quot;math-container&quot;&gt;$\operatorname {crd}$&lt;/span&gt; is not in use today, but it is the same as &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; except for scaling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key ideas are noting that &lt;span class=&quot;math-container&quot;&gt;$\sin 30^\circ=0.5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin 36^\circ=\frac{\sqrt{10-2\sqrt5}}4$&lt;/span&gt; (which can be shown with a little trig and factoring a particular cubic polynomial).  Then you basically set up the sine subtraction and half angle formulas in terms of sine alone.  Doing this and learning the old-school strategy for calculating square roots, you can calculate &lt;span class=&quot;math-container&quot;&gt;$\sin 6^\circ$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sin 3^\circ$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sin 1.5^\circ$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\sin 0.75^\circ$&lt;/span&gt;.  Finally, interpolate the final two values to estimate &lt;span class=&quot;math-container&quot;&gt;$\sin 1^\circ$&lt;/span&gt;, and then you can combine them with other known values to estimate the entire sine table to the nearest degree.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's a lot, but it obviously within the realm of achievement.  It would be interesting to see how far a student or team of students could get even if they used a spreadsheet to do the work arithmetically.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2020-02-14T17:29:21.297" LastActivityDate="2020-02-14T17:29:21.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17866" PostTypeId="2" ParentId="17864" CreationDate="2020-02-14T16:38:04.887" Score="4" Body="&lt;p&gt;I'd look at the books by Dunham (&quot;Journey through genius&quot;, &quot;The mathematical universe&quot; are more or less general, he wrote several others) or at the outstanding &quot;Proofs from THE BOOK&quot; by Aigner/Ziegler. On Quora there is a section on &quot;Beautiful mathematics&quot;, mostly elementary stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably a search for &quot;beautiful mathematics&quot; will net a selection of blogs by enthusiastic mathematicians. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-02-14T16:38:04.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17867" PostTypeId="2" ParentId="17864" CreationDate="2020-02-14T17:33:03.437" Score="37" Body="&lt;p&gt;To expand on my comment, I found that high school kids like watching YouTube videos. (I mean, they don't have to do any work right? Just sit and listen.) These are a few of my go to channels to pull mathematical ideas from. I try to show them short clips that might motivate them to think abuot math in a different way, not only just &quot;plug it into the formulas.&quot; That could be anything that makes them wonder what college level math is like, why certain patterns are true, some counter-intuitive examples, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/user/numberphile&quot; rel=&quot;noreferrer&quot;&gt;Numberphile&lt;/a&gt;: Good for more recreational type math. Think of Fibonacci and Golden Ratio type stuff. It's like, just math for fun and neat patterns, but they go a little deeper on some topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw&quot; rel=&quot;noreferrer&quot;&gt;3Blue1Brown&lt;/a&gt;: GREAT animations, focuses on the intuition behind a lot of deeper math. Might be a bit advanced though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/channel/UC9yt3wz-6j19RwD5m5f6HSg&quot; rel=&quot;noreferrer&quot;&gt;ThinkTwice&lt;/a&gt;: Great animations as well, shows a lot of &quot;proofs with no words&quot; and really helps connect the algebraic equations to the geometric visuals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/channel/UCpCSAcbqs-sjEVfk_hMfY9w&quot; rel=&quot;noreferrer&quot;&gt;Zach Star&lt;/a&gt; (formerly called Major Prep): He has some videos like &quot;What Mathematicians still can't solve,&quot; &quot;surprising facts from math,&quot; and even a few videos called &quot;Dear students, this is why you're learning about _____&quot; (although that last example is more aimed at Calculus students)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;VSauce just happened to have a good &lt;a href=&quot;https://www.youtube.com/watch?v=skvnj67YGmw&quot; rel=&quot;noreferrer&quot;&gt;Brachistochrone video&lt;/a&gt;, but I'm not sure what else he has that is directly math related.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Steve's Gubkin's answer talks about being in prison and painting. The painting analogy reminds me of &lt;a href=&quot;https://www.youtube.com/watch?v=Yexc19j3TjE&quot; rel=&quot;noreferrer&quot;&gt;this interview&lt;/a&gt; that tried to answer the question &quot;why does everyone hate math.&quot; The mathematician creates the analogy of an art class. In art class, you were surely shown the paintings of Picasso, Van Gogh, Da Vinci, Michelangelo, etc. They tried to inspire you by showing you the greatest hits of all time. How many math teachers show 'the greatest math hits' of all time? Tell your students about Fermat's Last Theorem, the Four Color Theorem, The Riemann Hypothesis, solving the Cubic Formula. Show them that these greatest hits weren't discovered over night, but instead, after several decades (or centuries even) of tough work. This might show them that math is something other than just a call-and-response game of &quot;read the question, then plug the numbers into a formula.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 2: After reading the comments, I remembered another channel worth mentioning. &lt;a href=&quot;https://www.youtube.com/user/misterwootube/featured&quot; rel=&quot;noreferrer&quot;&gt;Eddie Woo&lt;/a&gt; is a pretty famous math educator. He teaches high school and posts a lot of his class lectures on YouTube. I wouldn't be so interested in the class lectures, but he has some clips that talk about why learning math is interesting and the deeper side of math. &lt;a href=&quot;https://www.youtube.com/watch?v=19c4c3SwtS8&quot; rel=&quot;noreferrer&quot;&gt;Here's one&lt;/a&gt; that talks about the motivation behind complex numbers and a bit of the history of the cubic formula. It's one of those mis-attributed moments where a lot of teachers say &quot;we use complex number to solve all quadratics&quot; when in reality, complex numbers only came about in search of the cubic formula and our inability to reconcile the fact that expressions like &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{2-2i}+\sqrt[3]{2+2i}$&lt;/span&gt; can actually be a &lt;em&gt;real&lt;/em&gt; number. So, I wouldn't use this channel to show to students, but I would use it to see how someone else motivates a particular topic when I have a hard time finding a motivation for it.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastEditorUserId="8527" LastEditDate="2020-02-17T13:23:47.727" LastActivityDate="2020-02-17T13:23:47.727" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="17868" PostTypeId="2" ParentId="17863" CreationDate="2020-02-14T17:41:24.040" Score="0" Body="&lt;p&gt;There are a lot of approaches to approximating sine and cosine which would be sensible to a high school audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One method is to use the double angle formula repeatedly, together with small angle approximations.  For instance, if you want to approximate &lt;span class=&quot;math-container&quot;&gt;$\sin(\theta)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos(\theta)$&lt;/span&gt;,  approximate &lt;span class=&quot;math-container&quot;&gt;$\sin(\theta/64) \approx \theta/64$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos(\theta/64) \approx 1$&lt;/span&gt;.  Then use the double angle formulas repeatedly until you have an approximation for &lt;span class=&quot;math-container&quot;&gt;$\sin(\theta)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos(\theta)$&lt;/span&gt;.  If you need more precision, use a larger power of two in the denominator (but be prepared to pay with more time calculating).  In principle you could do all of these calculations by hand:  you only need the 4 basic operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think (but I am not sure) that this is the original reason people cared about the angle sum formulas. &lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-02-14T17:41:24.040" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17869" PostTypeId="2" ParentId="17864" CreationDate="2020-02-14T17:50:02.310" Score="46" Body="&lt;p&gt;Imagine you are put in jail.  You are forced to paint a painting every day for 10 years.  You have no choice in the subject:  one month you paint dogs, another month you paint horses, another month you paint lampposts.  The prison guard verbally chastises you when your painting is not up to their standard.  If you doodle something on your own, outside of the directed painting, you are similarly chastised.&lt;/p&gt;&#xA;&lt;p&gt;Failure to paint a sufficient number of paintings of sufficient &amp;quot;quality&amp;quot; will result in your sentence being extended by a year or two.&lt;/p&gt;&#xA;&lt;p&gt;In this context, how can we inspire you to see the beauty in painting?&lt;/p&gt;&#xA;&lt;p&gt;So my answer to this question is:  Dismantle the oppressive system which is causing the damage.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-05-20T14:54:07.100" LastActivityDate="2021-05-20T14:54:07.100" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="17870" PostTypeId="1" AcceptedAnswerId="17871" CreationDate="2020-02-14T17:55:49.213" Score="5" ViewCount="304" Body="&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; What are your arguments &lt;strong&gt;in favor of&lt;/strong&gt; the big array of different notations used in the context of undergraduate vector calculus: line integrals, surface integrals (of scalars and fields), Stoke's and divergence theorems, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because I personally believe that, even from the first introduction of the students to multivariable integration, we should abandon all that mess with creatures that look like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;{\oint\!\!\oint}_S  \bar{F} \, dS \,&#xA;$$&lt;/span&gt;&#xA;and &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\iiint_{V }  (\nabla \cdot \mathbf{F}) \, dV  = \oint\!\! \oint_{\partial V} \mathbf{F} \cdot d \mathbf{S}&#xA;$$&lt;/span&gt;&#xA;in favor of replacements like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_S \vec{F} \,&#xA;$$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int _V \nabla \cdot F = \int_{\partial V} F \ .&#xA;$$&lt;/span&gt;&#xA;A few points in support of my viewpoint:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) What do the &lt;span class=&quot;math-container&quot;&gt;$dr$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$ds$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\cdot \vec{ds}$&lt;/span&gt;, in case of line integrals, and &lt;span class=&quot;math-container&quot;&gt;$dS$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d\sigma$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cdot \vec{dS}$&lt;/span&gt;, in case of surface integrals, really mean? They discourage the question of whether the integrals are intrinsic, because, for instance, &lt;span class=&quot;math-container&quot;&gt;$dr$&lt;/span&gt; finds meaning as &lt;span class=&quot;math-container&quot;&gt;$|r'(t)| \, dt$&lt;/span&gt; &lt;em&gt;only after&lt;/em&gt; a parameterization of the curve. So, is &lt;span class=&quot;math-container&quot;&gt;$dr$&lt;/span&gt; meaningful as an intrinsic &quot;length element&quot;? Is &lt;span class=&quot;math-container&quot;&gt;$dS$&lt;/span&gt; intrinsic? (Adding notation does not answer the real question of the nature of such integrations.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) What is that stupid loop over integrals &lt;span class=&quot;math-container&quot;&gt;$\oint$&lt;/span&gt;? Just to say that the curve/surface is closed?! One would know the curve/surface they are integrating over is closed anyway. Such integrals are always preceded by assumptions beforehand that give the context: &lt;em&gt;Let &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; be a closed region in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;... Let &lt;span class=&quot;math-container&quot;&gt;$S \subset \mathbb{R}^3$&lt;/span&gt; be a smooth oriented surface and &lt;span class=&quot;math-container&quot;&gt;$\bar{F}$&lt;/span&gt; a vector field...&lt;/em&gt; So, even the simplest notation will unambiguously tell us what is being integrated over what.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) If we use successive &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; signs because the domain is 2 or 3 dimensional then are you going to keep adding new &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-dimension? See Wiki's Vector Calculus article for &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\underbrace{ \idotsint_{V \subset \mathbb R^n} }_{n} (\nabla \cdot \mathbf{F}) \, dV &#xA;={\underbrace{ \oint \!\cdots\! \oint_{\partial V}}_{n-1}}  \mathbf{F} \cdot d \mathbf{S}&#xA;$$&lt;/span&gt;&#xA;Furthermore, multiple integral signs might be confused with iterated integrals as in Fubini's theorem. Yes, by the same reasoning, I like &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_{[0,1]^2} f \quad \text{or} \quad \int_{[0,1]^2} f \, dx\,dy&#xA;$$&lt;/span&gt; &#xA;over &#xA;&lt;span class=&quot;math-container&quot;&gt;$$\iint_{[0,1]^2} f \, dx\,dy .&#xA;$$&lt;/span&gt;&#xA;Don't even ask about &lt;span class=&quot;math-container&quot;&gt;$\int_0^1\int_0^1 f \, dx\,dy$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2&lt;/strong&gt;: I know later on many people switch to the shorter notation, but isn't it about time to save our new generations the pain of having to learn a ton of nonsense notation conventions by saying goodbye to some &quot;notation and terminology [that] was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis&quot;?! (Wiki, same article.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 3&lt;/strong&gt;: Are there pedagogical benefits to sticking to the old notation? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe we should throw the words &quot;Hausdorff measures&quot; into the air, just like we use &quot;area/surface element&quot;, and agree that &lt;span class=&quot;math-container&quot;&gt;$ d\mathcal{H}^1 $&lt;/span&gt; is for integration over 1-D objects, curves, straight or bent, &lt;span class=&quot;math-container&quot;&gt;$d\mathcal{H}^2$&lt;/span&gt; is for integration domains that are 2D, (curved) sheets in space, etc. Then for instance&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_C  f(x) \, d\mathcal{H}^1 (x) \quad \text{and} \quad \int_S f(x) \, d\mathcal{H}^2(x).&#xA;$$&lt;/span&gt;&#xA;Then every time you look at an integral you will immediately see the dimension of the object your are integrating over?!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;If you can talk from experience teaching Calculus to undergraduates please do so.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="8272" LastEditorUserId="8272" LastEditDate="2020-02-15T00:03:29.917" LastActivityDate="2020-02-15T21:11:20.823" Title="Undergraduate Vector Calculus Notation Mess" Tags="|notation|history|vector-calculus|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17871" PostTypeId="2" ParentId="17870" CreationDate="2020-02-14T22:38:47.163" Score="7" Body="&lt;p&gt;This is a service course for students who are mostly engineering majors. Therefore any drastic change in notation like this is likely to be a bad idea. Leaving out &lt;span class=&quot;math-container&quot;&gt;$d\textbf{S}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dV$&lt;/span&gt; would be particularly unfortunate, since leaving out the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; is such a common student &lt;em&gt;mistake&lt;/em&gt; anyway in freshman calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, any notation that has the wrong units is a non-starter. As a typical application, we have Gauss's law for electric fields,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int \textbf{E}\cdot d\textbf{S}=4\pi k q,$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is the Coulomb constant and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is the enclosed charge. The SI units on the left are&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\text{N}}{\text{C}}\cdot\text{m}^2,$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and these match the units on the right. Without the &lt;span class=&quot;math-container&quot;&gt;$d\textbf{S}$&lt;/span&gt;, the units don't match. Your students' physics and engineering instructors are pounding it into them over and over that they need to check their units, and that if an equation is wrong in terms of dimensional analysis, then it's guaranteed to be a mistake. Neither the students nor these instructors are going to be happy if you start undoing that careful training.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other things you mention, however, are more trivial differences in notation, and not particularly standardized anyway in the physics and engineering literature. Nobody is going to care whether you write &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\oint$&lt;/span&gt; , and whether you write &lt;span class=&quot;math-container&quot;&gt;$\iint$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I know later on many people switch to the shorter notation, but isn't it about time to save our new generations the pain of having to learn a ton of nonsense notation conventions [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This hinges on your opinion that these are &quot;nonsense.&quot; Getting the notation to correctly show the units is certainly not &quot;nonsense,&quot; although many mathematicians might not be aware of it.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there pedagogical benefits to sticking to the old notation? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you're proposing a radical change in notation, then the burden of proof is on you to show that there are pedagogical benefits to &lt;em&gt;changing&lt;/em&gt; the notation, and that these benefits are huge.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Maybe we should tell them about Hausdorff measures first&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Please be realistic. At my school, almost none of the students who take this course are math majors. About half of them have serious deficits in basic algebra and arithmetic, such as not being able to simplify the expression &lt;span class=&quot;math-container&quot;&gt;$2/(3/4)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="8571" LastEditDate="2020-02-15T03:35:30.213" LastActivityDate="2020-02-15T03:35:30.213" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17872" PostTypeId="2" ParentId="17864" CreationDate="2020-02-15T06:45:03.560" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;“The Mechanical Universe,” is a critically-acclaimed series of 52 thirty-minute videos covering the basic topics of an introductory university physics course.&lt;/p&gt;&#xA;&lt;p&gt;Each program in the series opens and closes with Caltech Professor David Goodstein providing philosophical, historical and often humorous insight into the subject at hand while lecturing to his freshman physics class. The series contains hundreds of computer animation segments, created by Dr. James F. Blinn, as the primary tool of instruction. Dynamic location footage and historical re-creations are also used to stress the fact that science is a human endeavor.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/playlist?list=PL8_xPU5epJddRABXqJ5h5G0dk-XGtA5cZ&quot; rel=&quot;noreferrer&quot;&gt;YouTube&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I used to watch this as a kid, and although I (still) understand none of the math, I learned that the entire cosmos can be explained mathematically. The professor is as engaging as the 70s animations are dated, but the synopsis above explains why someone who could probably care less can become enraptured.&lt;/p&gt;&#xA;" OwnerUserId="8148" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-02-15T06:45:03.560" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17875" PostTypeId="2" ParentId="17864" CreationDate="2020-02-15T14:48:29.027" Score="8" Body="&lt;p&gt;As a child I found maths extremely boring, the biggest problem was that what I was being taught had no relevance to my life as a child, after all apart from counting and spending pocket money what possible use was it to me?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Until I discovered the equation for calculating the optimal size of loud speakers for my bedroom. Finally maths that I could relate to and use, suddenly maths wasn't boring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my advice to you would be to try and find some way to make the maths relevant and something your pupils can relate to in their lives.&lt;/p&gt;&#xA;" OwnerUserId="13454" LastActivityDate="2020-02-15T14:48:29.027" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17876" PostTypeId="1" CreationDate="2020-02-15T16:11:01.170" Score="4" ViewCount="617" Body="&lt;p&gt;Other than the standard math module for Python3, is there another library out there that uses degrees by default (as opposed to radians)? I am teaching students to use turtle (which uses degrees by default) to draw figures and they need to use trigonometry to compute angles/sides. Right now they are only familiar with degrees. Ideas?&lt;/p&gt;&#xA;" OwnerUserId="13451" LastActivityDate="2020-02-15T16:27:07.277" Title="Python programming - math library that uses degrees by default" Tags="|technology-in-education|trigonometry|" AnswerCount="1" CommentCount="1" ClosedDate="2020-02-18T17:38:06.150" ContentLicense="CC BY-SA 4.0" />
  <row Id="17878" PostTypeId="2" ParentId="17876" CreationDate="2020-02-15T16:27:07.277" Score="6" Body="&lt;p&gt;May I suggest giving them a bit of header code? i.e. Instead of using&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;from math import sin&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;try&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;import math&#xA;&#xA;def sin(theta):&#xA;    return math.sin(math.radians(theta))&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2020-02-15T16:27:07.277" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17879" PostTypeId="2" ParentId="17864" CreationDate="2020-02-15T18:10:51.670" Score="4" Body="&lt;p&gt;I'm having a lot of success right now in that area with a class I'm teaching, in which I have several assignments over the course of the quarter that center around self-directed learning. It takes some scaffolding to make it work well (how do you ask a good mathematical question? What counts as a good answer?) but once I got that done, a lot of the students really caught fire - last quarter, they turned in two five-page papers discussing a question they developed and the way they answered it. A few even taught me something I'd never known before!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, doing that came at the cost of a fair bit of class time - if you're teaching in a high-school environment, or a traditional college course, you probably have enough curriculum to get through that you don't have time for a major assignment that isn't directly covering the curriculum. And grading something like this is a bit of a nightmare. But you can do miniature versions of it - a day or two of class where students have a choice of which problems to work on might be successful, especially if you set aside some time after to reflect with them (&quot;which problems were easier? Which were more interesting? Look at this weird little detail, isn't that funny?&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's a prevailing theme here, and it's the same as what a couple of answers have suggested before: there's no beauty or fun without choice. The more choice you can put into the hands of the students, the better they'll understand the material and the more they'll appreciate it. I make a habit of building this into everyday activities, whenever possible; for example, when demonstrating a problem for the students, I aim to show not just the &lt;em&gt;best&lt;/em&gt; way to solve a problem, but a few alternatives that are a little less efficient but might &quot;feel&quot; better for them.&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2020-02-15T18:10:51.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17880" PostTypeId="2" ParentId="17864" CreationDate="2020-02-15T19:45:30.043" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Many of the students say they don’t like mathematics because they only learn for examinations. How do I show them the true beauty of mathematics?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How about testing them on that beautiful content on the examinations?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;EDIT: And/Or give a wider variety of math assessments that count for grades. Projects, presentations, performances, role playing, games, discussions, etc., all of which can more easily incorporate the true beauty of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2020-02-17T21:14:52.217" LastActivityDate="2020-02-17T21:14:52.217" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17881" PostTypeId="2" ParentId="17870" CreationDate="2020-02-15T20:44:20.457" Score="5" Body="&lt;p&gt;I think you are being harsh in your criticism of the classical notation. Of course, at the mathematician's end of the spectrum, the notation you promote towards the end of your question has merit. But I have taught vector calculus for many years, and find the classical notations that provoke you do in fact help learners decode theorems and calculations. These classical notations tell stories that capture the interest of calculus students, and make connections between the words in theorems and symbolism. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students who take vector calculus in the USA are from a mixture of majors, such as physics, chemistry, engineering, and mathematics. Some are double majors. Many take a course in electrodynamics from the physics department, and need to see connections with vector calculus taught in the mathematics curriculum. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In physics, they might see Maxwell's equations in integral form, while in vector calculus, they see the theorems of Gauss and Stokes. Maintaining reasonably consistent notation between the two courses is important. And I don't think my physics colleagues would take warmly to a suggestion that they reform their notation to one that mathematicians recommend. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematicians do not have a body that oversees notations, such as the Académie Française, which oversees matters related to the French language. A top-down movement to reform the notation of vector calculus could never get a foothold. It would only happen by random, incremental evolution. Aspiration to such a reform reminds me of the aspiration of an international language such as Esperanto.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An important quality of mathematics is its ubiquity with applications. We should not aspire to burn the bridge between vector calculus and classical electromagnetism taught by our colleagues in physics departments. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2020-02-15T21:11:20.823" LastActivityDate="2020-02-15T21:11:20.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17882" PostTypeId="2" ParentId="17632" CreationDate="2020-02-16T04:11:42.287" Score="1" Body="&lt;p&gt;The chain rule is one of the areas where teaching using differentials (instead of derivatives) shines.  If you are not aware, instead of teaching the &quot;derivative&quot; as the fundamental operation of calculus, you teach the differential.  When you differentiate with the differential, there is no preferred variable with respect to differentiation.  So, your rule, instead of being &lt;span class=&quot;math-container&quot;&gt;$y = x^n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx} = nx^{n-1}$&lt;/span&gt;, the rule is instead &lt;span class=&quot;math-container&quot;&gt;$dy = nx^{n - 1}\, dx$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has several advantages.  First, it is much more symmetric.  You are always doing the same thing to both sides, and always doing the same thing in all situations.  You can still &lt;em&gt;solve&lt;/em&gt; for the derivative (by just dividing by &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; in many cases), but the &lt;em&gt;operation&lt;/em&gt; is the differential.  The rules then become such like &lt;span class=&quot;math-container&quot;&gt;$d(nu) = n\,du$&lt;/span&gt;.  Note that there are no extraneous variables here (like &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;), so there is less going on.  This means that it is &lt;em&gt;easier&lt;/em&gt; to apply this rule in a multivariable situation.  Example: &lt;span class=&quot;math-container&quot;&gt;$y = x + z^2$&lt;/span&gt;.  The differential is &lt;span class=&quot;math-container&quot;&gt;$dy = dx + 2z\,dz$&lt;/span&gt;.  I can then solve for any derivative I want.  This makes related rates, implicit differentiation, and the like super-easy because you &lt;em&gt;aren't&lt;/em&gt; adding any new rules, you are just applying algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And that's also the case with the chain rule.  The rule for &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt;, for instance, is &lt;span class=&quot;math-container&quot;&gt;$d(\sin(u)) = \cos(u)\,du$&lt;/span&gt;.  Now, any rule you apply has to match the formula exactly (but can use any variable we want).  So, if we have &lt;span class=&quot;math-container&quot;&gt;$d(\sin(x^2))$&lt;/span&gt;, this doesn't match our rule exactly.  But, we can &lt;em&gt;make&lt;/em&gt; it match our rule exactly with a variable substitution.  We can say, &lt;span class=&quot;math-container&quot;&gt;$q = x^2$&lt;/span&gt;.  Now our problem is &lt;span class=&quot;math-container&quot;&gt;$d(\sin(q))$&lt;/span&gt; which becomes &lt;span class=&quot;math-container&quot;&gt;$\cos(q)\, dq$&lt;/span&gt;.  We can easily back-substitute &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$\cos(x^2)\,dq$&lt;/span&gt;, but we still have the pesky &lt;span class=&quot;math-container&quot;&gt;$dq$&lt;/span&gt; to take care of.  However, we also have a new equation describing &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; which we can differentiate in order to get a value for &lt;span class=&quot;math-container&quot;&gt;$dq$&lt;/span&gt;.  If we differentiate &lt;span class=&quot;math-container&quot;&gt;$q = x^2$&lt;/span&gt;, the result is &lt;span class=&quot;math-container&quot;&gt;$dq = 2x\,dx$&lt;/span&gt;.  Therefore, we will replace &lt;span class=&quot;math-container&quot;&gt;$dq$&lt;/span&gt; in our result, giving &lt;span class=&quot;math-container&quot;&gt;$\cos(x^2)\,2x\,dx$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing it this way, the chain rule barely qualifies as a rule.  It's just a natural mathematical tool to make a substitution to transform an equation to be manageable under the rules that we already understand.  There's no &quot;special&quot; rule called &quot;the chain rule&quot;, it is just the natural extension of applying algebra to differentials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Side note - differentials have fewer problematic cases.  For instance, if you take the derivative with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x = 1$&lt;/span&gt;, you will get &lt;span class=&quot;math-container&quot;&gt;$1 = 0$&lt;/span&gt;.  However, if you take the differential, you will get &lt;span class=&quot;math-container&quot;&gt;$dx = 0$&lt;/span&gt;, which is actually true.  It will also be more obvious when you try to transform it into a derivative why it is problematic (&lt;span class=&quot;math-container&quot;&gt;$\frac{dx}{dx}$&lt;/span&gt; becomes &lt;span class=&quot;math-container&quot;&gt;$\frac{0}{0}$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2020-02-16T04:11:42.287" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17883" PostTypeId="2" ParentId="17864" CreationDate="2020-02-16T08:52:50.980" Score="2" Body="&lt;p&gt;Péter Rózsa's classic book, Playing with Infinity is an excellent intro of maths to non-mathematicians. Perhaps it would help. MAA review: &lt;a href=&quot;https://www.maa.org/press/maa-reviews/playing-with-infinity&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/maa-reviews/playing-with-infinity&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1850" LastActivityDate="2020-02-16T08:52:50.980" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17884" PostTypeId="2" ParentId="17864" CreationDate="2020-02-16T09:42:05.373" Score="4" Body="&lt;p&gt;Simple: You do not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people like maths and will see beauty in it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Others will not consider it beautiful no matter how useful it is or no matter how much you show them maths that you personally consider 'beautiful'.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;A personal anecdote:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally speaking I very much disliked maths at school, despite being told I was good at it.&lt;br&gt;&#xA;Like fellow user @SteveSmith, one of my primary reasons for disliking it was that I saw no use for it.&lt;br&gt;&#xA;Later in life I took up programming and suddenly found that I actually had a use for all those formerly useless things I despised.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Over time my hatred of maths lessened as I found more uses for it, to the point that I now find most of it tolerable or am willing to attempt things if I have a particular use for them, and in some cases I've even gone out of my way to learn things that I don't need to know (e.g. group theory) purely because I've found it interesting.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, despite there being a number of things about maths that I find useful or interesting,&lt;br&gt;&#xA;I have never found any part of maths that I'd honestly consider 'beautiful'.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So my advice to you is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't even bother trying to show your class that maths is 'beautiful'.&lt;br&gt;&#xA;The people you convince will only be the ones who would naturally like maths.&lt;br&gt;&#xA;For the rest, if you truly want to convince them to take an interest then don't try to convince them that maths is beautiful, try to demonstrate to them that maths is relevant to their lives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you cannot manage that, then try to find ways to make maths less painful for them - demonstrate ways to break problems down into simpler problems, demonstrate rules like additive and multiplicative identity and inverse elements and how those things can be used to reduce complex problems to simpler forms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Failing that, just give up on trying to convince them to like maths.&lt;br&gt;&#xA;Let them reluctantly get through the class and achieve whatever grade they achieve.&lt;br&gt;&#xA;For some people, it's simply not possible to convince them to like maths.&lt;/p&gt;&#xA;" OwnerUserId="957" LastActivityDate="2020-02-16T09:42:05.373" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17885" PostTypeId="1" AcceptedAnswerId="17889" CreationDate="2020-02-16T13:11:33.040" Score="40" ViewCount="20294" Body="&lt;p&gt;In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for many high school students, especially those who are keen on pursuing the humanities, they do not see any point in studying Mathematics for the rest of their schooling life, or how any concept in Mathematics could possibly be applied in their future work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is Mathematics a compulsory subject for high school students, especially those who are clearly studying in Humanities streams?&lt;/p&gt;&#xA;" OwnerUserId="13448" LastActivityDate="2020-02-18T14:59:42.770" Title="Why is learning mathematics compulsory?" Tags="|secondary-education|curriculum|" AnswerCount="14" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17886" PostTypeId="2" ParentId="17885" CreationDate="2020-02-16T14:18:20.107" Score="30" Body="&lt;p&gt;The OP may be interested in the controversial 2012 article by Andrew Hacker in the NYTimes: &#xA;&lt;a href=&quot;https://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html&quot; rel=&quot;noreferrer&quot;&gt;Is Algebra Necessary?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one reply, by Peter Flom:&#xA;&lt;a href=&quot;https://medium.com/q-e-d/is-algebra-necessary-a-reply-to-andrew-hacker-fce5b7e1d99c&quot; rel=&quot;noreferrer&quot;&gt;A reply to Andrew Hacker&lt;/a&gt;. His closing remarks:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is algebra necessary? In the strict sense, no. You can live without it. You can also live without art, music, literature or sports. Would you want to?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And here is a response from the MAA (Math Assoc Amer):&#xA;&lt;a href=&quot;https://www.maa.org/press/periodicals/maa-focus/should-algebra-be-required&quot; rel=&quot;noreferrer&quot;&gt;Denny Gulick&lt;/a&gt;. It includes a point-by-point rebuttal to Hacker's five main points.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-02-16T15:56:15.733" LastActivityDate="2020-02-16T15:56:15.733" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17887" PostTypeId="2" ParentId="17885" CreationDate="2020-02-16T14:56:34.963" Score="12" Body="&lt;p&gt;Underwood Dudley answers the question &amp;quot;What is mathematics education for?&amp;quot; in &lt;a href=&quot;https://www.ams.org/notices/201005/rtx100500608p.pdf&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt; from 2010 (&lt;em&gt;Notices of the American Mathematical Society&lt;/em&gt;, vol. 57, no. 5, pp. 608-613) (even though the title of the article is &amp;quot;What Is Mathematics For?&amp;quot;).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So that there is no confusion, let me say that by “mathematics” I mean algebra, trigonometry, calculus, linear algebra, and so on: all those subjects beyond arithmetic.  There is no question about what arithmetic is for or why it is supported. Society cannot proceed without it. Addition, subtraction, multiplication, division, percentages: though not all citizens can deal ﬂuently with all of them, we make the assumption that they can when necessary. Those who cannot are sometimes at a disadvantage.&lt;/p&gt;&#xA;&lt;p&gt;Algebra, though, is another matter. Almost all citizens can and do get through life very well without it, after their schooling is over. Nevertheless it becomes more and more pervasive, seeping down into more and more eighth-grade classrooms and being required by more and more states for graduation from high school. There is unspoken agreement that everyone should be exposed to algebra. We live in an era of universal mathematical education.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He concludes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What mathematics education is for is not for jobs. It is to teach the race to reason. It does not, heaven knows, always succeed, but it is the best method that we have. It is not the only road to the goal, but there is none better.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-02-16T14:56:34.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17888" PostTypeId="2" ParentId="17885" CreationDate="2020-02-16T15:54:12.887" Score="89" Body="&lt;p&gt;Questions like this, or variants (from students, the notorious &quot;when will I use this in real life&quot;) seem to be pretty common, and I'm always a little surprised, because the unstated premise - that high school is supposed to teach students things narrowly tailored to their future career - is so obviously false.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's obviously false because &lt;em&gt;almost no&lt;/em&gt; academic subjects in a conventional high school curriculum are widely applicable career skills.  The actual essentials for functioning in ordinary society - literacy, basic arithmetic, a minimal ability to write - are covered by middle school, if not elementary school.  (I'm distinguishing academic subjects, because, at least in the US, it's common for high schools to also offer more practical classes.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've never understood why math is the target of these questions so much more frequently than other subjects.  Many fewer people &lt;em&gt;need&lt;/em&gt; social studies in their professional life than need high school level math, but we understand that we don't teach social studies primarily as a career skill: we teach them because they're necessary to being an informed citizen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can get through a day at most jobs without knowing anything about history.  What you can't do without some knowledge of history is make any sense of the news.  Math has a similar role: we live in a world in which math plays a central role in nearly everything that happens around us - it's central to the technology we all use on a daily basis, to understanding the flood of scientific (especially health related) information that surrounds us, and to the decisions that affect us being made by corporations and governments all the time.  We teach math because understanding what's going on in the world requires some basic fluency in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are good arguments that current high school curricula don't do that as well as they should.  Most high school curricula do, especially in college-oriented tracks, mix that with more specialized material needed for students going into STEM fields.  So I don't mean this as a defense of any particular curriculum.  But I do mean this as a rejection of the premise: the math curriculum, like the rest of the high school curriculum, will never make sense if the main question you ask is &quot;how will I use this in my job&quot;.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2020-02-16T15:54:12.887" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="17889" PostTypeId="2" ParentId="17885" CreationDate="2020-02-16T16:03:56.340" Score="75" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Why is Mathematics a compulsory subject for high school students, especially those who are clearly studying in Humanities streams?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A kid at age 14 is not ready to make irrevocable decisions that will affect them for the rest of their life. That's why we don't let them get married. I have a friend who, at age 30, decided to apply to grad school in sociology, and she is now tenured at a research university. A huge obstacle for her when she was re-entering school was that she needed to learn enough statistics. She had had a fine high school and college education, but hadn't concentrated much on math. If she had simply been encouraged to stop studying math completely at age 14, then I can't imagine how she could have ever gotten over this hurdle at age 30.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The kid may want to study the humanities in college but then, say, go into business. If they don't understand enough math to get through a watered-down 9th grade algebra class, then they aren't going to be able to do the relevant quantitative reasoning.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The reason why a country like the US has universal, free, and compulsory education is not (just) that it helps kids get jobs and boosts the economy. Education is also necessary in order to have a functioning democracy. Voters who can't do basic algebra are going to be severely handicapped in making decisions about issues like nuclear power and global warming.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When you provide multiple tracks for students through the educational system, it can have nasty side-effects. In the US, there was a time when African-American and Latino students were routinely sent into one academic track, while white kids were put on a more demanding one. It's pretty common for kids to get put in less demanding classes because of superficial issues like poor handwriting. For these reasons, I think it's better simply to provide classes at a variety of &lt;em&gt;levels&lt;/em&gt;, but not to classify kids into different categories.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If you want to criticize the practice of forcing kids to take math, then I think there are some more appropriate targets for criticism:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Efforts to force kids to take algebra at lower and lower ages, such as attempts in California to make all kids take algebra in 8th grade.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Requirements that college biology majors take a full year of calculus (including stuff like doing integrals using trig substitutions) and calculus-based physics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Unrealistic government requirements that encourage public schools to pretend that all students are succeeding at a high level in math, when in fact many aren't succeeding at all.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-02-16T16:03:56.340" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17890" PostTypeId="2" ParentId="17885" CreationDate="2020-02-16T17:59:38.270" Score="2" Body="&lt;p&gt;The simple answer I would give is that there is a good chance they go into the business world at SOME TIME in the future of their life.  Perhaps they will get an MBA at some point to help them move up.  And you'd laugh at how deer in the headlights the English major types were with finance math.  And how the STEM majors ate it up like Valentines Day heart candy,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As such, no they won't be doing algebra, per se.  But some basic training in algebra is helpful to be more numerate when looking at spreadsheets or presentation slides where percentages, ratios, compounding, etc. occur.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think you could cut the cord before trig/calc, though.  Of course that means they can't handle high school physics.  So they won't know that either.  But high school chem is fine with algebra 2.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I disagree on the counterpoint that STEM people can skip humanities.  You absolutely do need a knowledge of writing (more and more).  Also a knowledge of your culture from history.  I would draw the line on lib'rul college foolishness like Rigoberto Menchu.  But classical Western Civ (or Chinese for a Chinese) is absolutely something an educated...person should know (to read the newspaper, discuss politics, etc.)&lt;/p&gt;&#xA;" OwnerUserId="13465" LastActivityDate="2020-02-16T17:59:38.270" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17891" PostTypeId="2" ParentId="17885" CreationDate="2020-02-16T19:51:16.103" Score="6" Body="&lt;p&gt;Math teaches reasoning using scenarios that are exact enough for there to be concrete right and wrong answers.  Many areas of life require reason, but mathematics forces students to actually engage in the discipline of finding the &lt;em&gt;right&lt;/em&gt; answer, because there is only one right answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, some people think that we can teach reasoning by doing mock trials, and that this is a much more practical mode of teaching reasoning that people can relate to.  It may be more relatable, but, the problem is that it is not always clear in such a scenario why certain answers are right or wrong.  In other words, it allows bad reasoning to be perpetuated as long as it is done sufficiently glibly.  In math, no matter how well I speak, I will not convince my math teacher that X + 5 = 7 means that X = 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I give presentations about mathematics, I always include a slide that says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Math gives you practice on core reasoning skills with concrete&#xA;  problems that have definite answers in order to achieve mastery of the&#xA;  reasoning process which will allow you to apply the skill to fuzzier&#xA;  problems where answers are not as certain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are also many analogs to specific mathematical techniques which I have (informally) found that math-trained people are more likely to do well in.  One is in reconfiguring processes.  In math, we train people to do prime number factoring.  This is exactly the type of skill needed when figuring out what is needed to reconfigure a process.  You have to be able to break down the process into its primary components, and rebuild it using those.  Being able to look at something and &quot;see&quot; what basic building blocks it is composed of is essential to thought.  Again, math gives us practice in these core reasoning skills on concrete problems that have definite answers so that we can achieve mastery of the reasoning process which allows us to apply the skill (with confidence) to fuzzier problems where the answers are not as certain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are parts of math which are needed just to provide grist to the logical reasoning process.  There are parts of math which are good direct (or indirect) analogs to specific reasoning processes that will be used in the future.  There are parts of math which are just part of being an educated human.  And, for sure, there are parts of math that we should probably stop cluttering the textbooks with.  But the latter are few and far between.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example - I think it was Hacker that complained about logarithms and exponents being taught to everyone.  That is literally the most useful thing that math teaches that isn't evident from normal everyday life.  Knowing how exponentiation looks like is a key component of understanding (a) debt, (b) interest, (c) investing, which are things that &lt;em&gt;everyone&lt;/em&gt; will likely be involved in at some point in life. If you don't understand how debt grows exponentially, you won't understand its danger.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Final note - I do think we should teach math with the reasoning skills we want students to know more clearly in mind (and even should share them with students).  I think this would improve (a) the student's attachment to the curriculum, (b) the quality of the curriculum itself, and (c) the public's understanding of why the curriculum is there.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2020-02-16T19:51:16.103" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17892" PostTypeId="2" ParentId="17864" CreationDate="2020-02-16T21:13:38.967" Score="3" Body="&lt;p&gt;I agree with anjama's comment: give them applied examples. &lt;strong&gt;Use something like the free Godot engine to create simple &quot;games&quot; (read: interactive apps) that demonstrate concepts of trigonometry, geometry, and linear algebra.&lt;/strong&gt; It's really quite easy to get started with Godot if you know some basic programming, and once the kids see a &lt;em&gt;game engine&lt;/em&gt;, and see how math can be directly used to &lt;em&gt;make video games&lt;/em&gt;, you can bet at least some of them will be excited.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If any of my teachers had shown me anything like that, I would have shat myself with excitement. &quot;I can use this math to make games on my computer at home?!&quot; Instead we went outside and used a clinometer to measure the height of trees ....&lt;/p&gt;&#xA;" OwnerUserId="13467" LastActivityDate="2020-02-16T21:13:38.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17893" PostTypeId="2" ParentId="17885" CreationDate="2020-02-16T22:19:48.660" Score="7" Body="&lt;h1&gt;Citizenship&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;The health of a democracy depends on the electorate being thoughtful and &lt;em&gt;informed&lt;/em&gt;.  And said citizens can only be informed if they understand the information around them (e.g., the news).  Very few policies boil down to: &quot;A is morally right and B is morally wrong.&quot;  Those issues were mostly settled long ago or established in the Constitution.  The issues facing most legislators are questions like: &quot;Is A better than B?  By how much?  What are the costs involved?  How much uncertainty do we have?&quot;  And while we will always take qualitative values into consideration, the difference between A and B often comes down to &lt;em&gt;quantities&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only a few people debate whether we should &lt;em&gt;have&lt;/em&gt; a minimum wage.  Most folks instead debate: &lt;em&gt;how large&lt;/em&gt;?  To do this reasonably, we need studies with results, which means &lt;em&gt;statistics&lt;/em&gt;.  The very process of conducting an election is rife with statistics.  Which means, at the very least, every citizen should have a solid understanding of basic statistics, and the stronger that understanding, the better.  But it's hard to get a foundational knowledge of statistics without some algebra, and even calculus is helpful in understanding continuous distributions.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Economics&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Related to the citizenship issues above, but worth mentioning on its own, is the centrality of economics to every citizen.  We all need to know how to spend money wisely (except the few of us born into extreme wealth).  It's hard to answer a basic question like: &quot;Which is cheaper: the 16 oz jar for \$3.88 or the 24 oz jar for $4.96?&quot; without knowing some algebra.  And even if you can do this math without having passed an algebra course, you are doing algebra whether you call it such or not.  So students would probably be more motivated if algebra were taught in the context of personal finance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And that's not even getting into more difficult questions like: &quot;Should I pay an extra $10,000 on my mortgage to take off a point?&quot;  Or: &quot;How much should I invest in my 401(k) vs. directly buying ETFs?&quot;  Anyone who lives a life without math lives a life without money.  Capitalism is rife with value judgments of a very precise numerical nature.  Even computing the tip at a restaurant or coffee shop requires math skills.  Formulating a retirement strategy at the start of your career requires more than simple arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Computer Science&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;As a professional software engineer, I obviously think that programming is valuable (but the salary : education ratio of CS also speaks quite loudly).  I think most of us recognize that the future is automation.  Virtually every scientific field is inundated with software and automation.  Every startup wants to call itself a &quot;tech company&quot; (including obviously not-tech companies like mattress-maker Casper).  But every corporation benefits from tech-savvy employees, even if that just means knowing how to put together a few formulas in Excel to aid planning and scheduling.  The ubiquity of personal electronics, from laptops to smart phones, means that the youngest generation is practically born as &quot;power users&quot;, especially relative to older generations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the very least, computer science needs combinatorics.  And for combinatorics, you need algebra.  Eventually, some very basic scripting capability will be required of virtually all white collar jobs.  So I would double down and say that not only &lt;em&gt;should&lt;/em&gt; math be required education for all high schoolers, but &lt;em&gt;also&lt;/em&gt; computer science.  Anyone who says we don't need math and CS is living in the past.&lt;/p&gt;&#xA;" OwnerUserId="13468" LastActivityDate="2020-02-16T22:19:48.660" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17894" PostTypeId="2" ParentId="17885" CreationDate="2020-02-16T22:50:28.397" Score="11" Body="&lt;p&gt;Because&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;many high school students, especially those who are keen on pursuing&#xA;  the humanities, they do not see any point in studying Mathematics for&#xA;  the rest of their schooling life, or how any concept in Mathematics&#xA;  could possibly be applied in their future work.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;shows they have no clue what mathematics is worth, therefore it needs to be compulsory to protect them from themselves.&lt;/p&gt;&#xA;" OwnerUserId="13472" LastActivityDate="2020-02-16T22:50:28.397" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17895" PostTypeId="2" ParentId="17885" CreationDate="2020-02-17T01:03:12.277" Score="5" Body="&lt;p&gt;Math, taught well, teaches modes of thought, not facts. A person who is exposed to a wider variety of modes of thought will be more prepared to critically consider new ideas. It's the same reason we teach history and literature - I can't honestly say that I've had reason to use my middle-school knowledge of Macbeth for anything, but the tools of critical reading and strategies for considering interpretations of a text have been useful for spotting articles that are meant to provoke a certain feeling and bias the reader. I've never needed to know when my hometown was founded, but a basic understanding of what kind of historical facts we know and what evidence we have of them helps me distinguish between historical fact and conspiracy theory. And I've never needed to be able to do algebra (except, of course, in my capacity as a math instructor) but the ability to mix systematic thought with creative strategizing -- the skills that come out of long algebra problems, especially word problems -- allows me to manage my finances, cook a decent meal, organize my closet, or schedule a vacation. The ability to pay attention to key details without losing the big picture allows me to follow detailed instructions well (have you ever assembled a piece of Ikea furniture?). The ability to reason precisely, as in a geometry proof, helps me spot when other people's arguments are appealing to emotion rather than reason.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be honest, of all the skills I learned at or below the high-school level, math seems like the &lt;em&gt;most&lt;/em&gt; useful.&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2020-02-17T01:03:12.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17896" PostTypeId="2" ParentId="17885" CreationDate="2020-02-17T01:37:23.970" Score="18" Body="&lt;p&gt;Many people think that the clientele of our public school system is &lt;em&gt;the students&lt;/em&gt;. Others act as if the clientele were the &lt;em&gt;parents&lt;/em&gt; of the students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Those people are wrong. The client is the &lt;em&gt;citizenry of his state and nation&lt;/em&gt;, who need the electorate to be educated and informed, in order to secure the benefits of a well-ordered government. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question, &quot;How does the study of mathematics&lt;sup&gt;1&lt;/sup&gt; benefit the student at any point in his life?&quot; is irrelevant. The student has not contracted for the education he is receiving, and has no right to dictate its form. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proper question to ask of publicly funded universal compulsory education is, &quot;How does the study of mathematics benefit the state, nation, and society that the student will eventually be called upon to supervise and control?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the studies, that contribute to a citizen's ability to keep his republic, also augment his ability successfully to pursue happiness, that is a fortunate serendipity. But the educational purposes of the student do not trump the educational purposes of the state. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the answer to &quot;Why is learning mathematics compulsory?&quot; is &quot;Because the people have decided that you need to know mathematics in order to be a worthy citizen.&quot; &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;For &quot;mathematics&quot; substitute any subject that the &lt;em&gt;people&lt;/em&gt; consider necessary to an educated voter. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="13476" LastEditorUserId="13476" LastEditDate="2020-02-18T01:59:05.443" LastActivityDate="2020-02-18T01:59:05.443" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17898" PostTypeId="2" ParentId="17885" CreationDate="2020-02-17T03:43:43.500" Score="16" Body="&lt;p&gt;I'll make some observations from a Western perspective (which may or may resonate with a Chinese student, say). Consider: Mathematics has been at the forefront of all education forever -- since long before the idea of compulsory education itself was conceived. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/Education_in_ancient_Greece&quot; rel=&quot;noreferrer&quot;&gt;Classical Athens&lt;/a&gt; (~420 BC):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;More focused fields of study included mathematics, astronomy,&#xA;  harmonics and dialectic – all with an emphasis on the development of&#xA;  philosophical insight. It was seen as necessary for individuals to use&#xA;  knowledge within a framework of logic and reason... Wealth played an&#xA;  integral role in classical Athenian Higher Education. In fact, the&#xA;  amount of Higher Education an individual received often depended on&#xA;  the ability and the family's willingness to pay for such an&#xA;  education... Women and slaves were also barred from receiving an&#xA;  education... The people in a pythagorean society were known as&#xA;  &lt;em&gt;mathematikoi&lt;/em&gt; (μαθηματικοί, Greek for &quot;learners&quot;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadrivium&quot; rel=&quot;noreferrer&quot;&gt;system for higher education was described by Plato&lt;/a&gt;, and used as the structure for university studies through medieval and Renaissance Europe:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The quadrivium consisted of arithmetic, geometry, music, and&#xA;  astronomy. These followed the preparatory work of the trivium,&#xA;  consisting of grammar, logic, and rhetoric. In turn, the quadrivium&#xA;  was considered the foundation for the study of philosophy (sometimes&#xA;  called the &quot;liberal art par excellence&quot;) and theology. The quadrivium&#xA;  was the upper division of the medieval education in the liberal arts,&#xA;  which comprised arithmetic (number), geometry (number in space), music&#xA;  (number in time), and astronomy (number in space and time).&#xA;  Educationally, the trivium and the quadrivium imparted to the student&#xA;  the seven liberal arts (essential thinking skills) of classical&#xA;  antiquity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that in each of these systems, mathematics comes first. In the quadrivium, all of the components are actually numerical; including music, which one might consider today as a humanities subject. I'll emphasize again of this &lt;em&gt;all-numerical&lt;/em&gt; higher education course: &lt;strong&gt;&quot;the quadrivium was considered the foundation for the study of philosophy (sometimes called the 'liberal art par excellence') and theology&quot;&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, why do these connections and foundations seem hard to understand today? I might suggest that there's been a historical progression in which education and mathematics have become so monumentally important -- an &lt;em&gt;existential issue&lt;/em&gt; for any modern nation (say, since the military conflicts in the 19th and 20th centuries) -- that every nation is nearly frantic to produce a maximal number of STEM professionals, scientists, and engineers. These nations have demanded that everyone have some amount of compulsory schooling, with emphasis on the mathematics that would be a pipeline towards engineering and the like (specifically: algebra leading to calculus). Unfortunately, this systemic stress has cracked the educational institutions into producing a Kabuki theater version of mathematics -- students can pass relatively mindless algorithmic tests, without understanding or insightful explanations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On that latter progression, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Campbell&amp;#39;s_law&quot; rel=&quot;noreferrer&quot;&gt;Campbell's Law&lt;/a&gt; (thinking here in terms of the assessment, &quot;it would be good if students in our schools could get high test scores in math&quot;): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The more any quantitative social indicator is used for social&#xA;  decision-making, the more subject it will be to corruption pressures&#xA;  and the more apt it will be to distort and corrupt the social&#xA;  processes it is intended to monitor.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Perhaps the major missing component, which has set mathematics education on a foundation of sand, is the lack of prior training in &lt;em&gt;logic&lt;/em&gt; (see above: in the classic trivium, grammar-logic-rhetoric were prerequisites to the all-mathematics higher education). Without this, mathematics teaching tends to have the appearance of a huge list of random and disconnected non-meaningful procedures, instead of the training in reasoning and persuasive speaking which mathematics was long held out to be (and which you'll hear many mathematics educators proclaim, although it's not generally understandable by recipients of the current educational system). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In summary: Mathematics really is about connections and developing provably correct deductions from available information we have about the world. That's not a rhetorical flourish; the field is really a concrete crystallization of all we know about &lt;a href=&quot;https://www.azquotes.com/quote/574070&quot; rel=&quot;noreferrer&quot;&gt;necessary conclusions&lt;/a&gt;. Historically it was synonymous with higher education itself, neither of which were considered appropriate for most people. It's unfortunate that in an attempt to pretend that most people in a modern nation understand it, the logical-reasoning component has been largely wrung out of the system, rendering the whole curriculum as not making very much sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A question I always ask U.S. non-STEM people who express frustration with their math education: How did you feel about your high-school &lt;em&gt;geometry&lt;/em&gt; class, with its emphasis on proofs and justifications? I find that very frequently people respond with, &quot;Oh, that's the only math class I liked; I really enjoyed it!&quot; And my follow-up is to say (again in the U.S. system): &lt;em&gt;that's the only course in high school that counts as real math from a mathematician's perspective&lt;/em&gt; (as a prover of theorems). If you like and feel confident with the work in a geometry class, then that's the best possible sign that you're a real mathematical thinker, even if you didn't get that message previously. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll end with an open question, to pivot from the original query: Granted that mathematics is the core subject matter of all education in general -- why is education itself now compulsory in modern advanced nations? Maybe it shouldn't be? That's a perhaps more essential and much harder question (and unfortunately not appropriate for S.E. Math Educators). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-02-18T02:55:53.470" LastActivityDate="2020-02-18T02:55:53.470" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17899" PostTypeId="2" ParentId="17864" CreationDate="2020-02-17T03:55:33.320" Score="2" Body="&lt;p&gt;A person doesnt really need to know &lt;strong&gt;all&lt;/strong&gt; of the why in math only what relates to them. Teach people how it relates to them like providing real world examples (for example determining the height of a tree). Show them that you can do cool stuff with math. As far as understanding, be open to tutor a student through something. Generally too though, you never appreciate that you have learned something till &lt;strong&gt;after&lt;/strong&gt; you have taken the course. During the course, it can feel like slave labor. And one last thing, show your enthusiasm and the students will be encouraged to be positive as well.&#xA;Unfortunately, math is not a well viewed subject with high school students.&lt;/p&gt;&#xA;" OwnerUserId="13480" LastActivityDate="2020-02-17T03:55:33.320" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17900" PostTypeId="1" CreationDate="2020-02-17T04:56:52.530" Score="37" ViewCount="2714" Body="&lt;p&gt;I am interested in finding examples of poorly written proofs that exemplify the types of mistakes made by undergraduate students in their first year or two of writing proofs. I am interested both in examples of poor writing that introduce logical errors into the proof as well as those that just make the proof hard to read.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware that there is already &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2206/example-bad-proofs&quot;&gt;a question on this site&lt;/a&gt; asking for examples of bad proofs. I find the answers there kind of unsatisfactory (one answer simply says to look at student assignments and the other just consists of a broken link), but perhaps it's because there aren't many great examples of poorly written proofs on the internet. Also, that question is only focused on proofs that are wrong or &quot;could be simplified&quot; whereas I am also interested in more general problems with the way proofs are written.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In case there are no good examples online of poorly written proofs, I would like to try to generate some myself. Ideally, for each general type of mistake that undergraduates are prone to make when writing proofs, I want to write a bad proof that isolates and exemplifies that mistake. So I am interested in compiling a &quot;taxonomy&quot; of ways that proof writing by students can go wrong. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's my current taxonomy:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Backwards proofs:&lt;/strong&gt; Proofs that start from the desired conclusion and work backwards until they arrive at the premises. Not only can this be confusing to read, it is easy to introduce errors when some step of the proof is not reversible.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;The &quot;obvious&quot; worst case:&lt;/strong&gt; It is popular to warn students against &quot;proof by example&quot; but I have found that students are usually already aware that an example is not a proof... unless of course they think that the example they chose is obviously the worst case scenario without bothering to justify it.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;The tangle:&lt;/strong&gt; The proof that keeps going and going, introducing a complicated series of manipulations, definitions, hand-waving, etc. until somehow arriving at the right conclusion. Often some step in the middle is wrong (or possibly nonsensical) but it's hard to find because it's buried under a pile of more-or-less-correct but irrelevant details. A special favorite of students taking exams.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;The mysterious stranger:&lt;/strong&gt; Some object is used without ever being introduced or defined. Often there is a reasonable way to fill in the definition of the object but the proof becomes a guessing game of what the student meant the object to be. Sometimes this is caused by a student trying to use a variable outside of its proper scope. In that case it can also lead to the student giving the same name to two distinct objects.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Inappropriate types:&lt;/strong&gt; The student claims that some ideal is an element of the ring. Or tries to divide by a set of real numbers. Etc. The point is that they are trying to do something with an object which is of the wrong type to do it with. Or sometimes they are trying to perform a typecast that cannot automatically be done in an unambiguous way like if they try to apply a function of type &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \to \mathbb{R}$&lt;/span&gt; to a complex number. Often in these cases the student does have in mind a way to do the typecast (although sometimes they have just misunderstood a definition or don't even really grok that mathematical objects have types) but problems occur when there are multiple mutually incompatible ways to do it.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;The incantation:&lt;/strong&gt; You've just learned a big, important sounding theorem in class so you should probably use it on the homework, right? You don't quite understand it or know when it applies, but it seems like you should be using it so you just go ahead and invoke it anyways. It's pretty common for students to invoke big theorems without checking that the hypotheses of the theorem hold and sometimes in situations where it doesn't even make sense to invoke the theorem.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Implication confusion:&lt;/strong&gt; The student is supposed to prove &lt;span class=&quot;math-container&quot;&gt;$X \implies Y$&lt;/span&gt; but instead proves &lt;span class=&quot;math-container&quot;&gt;$\lnot X \implies \lnot Y$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$Y \implies X$&lt;/span&gt; or tries (and typically fails) to prove &lt;span class=&quot;math-container&quot;&gt;$X\land Y$&lt;/span&gt;. Or the student is told to assume that &lt;span class=&quot;math-container&quot;&gt;$X \implies Y$&lt;/span&gt; and then concludes that &lt;span class=&quot;math-container&quot;&gt;$Y \implies X$&lt;/span&gt;. Most common when the statements of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; are a bit complicated or involve implications as well.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Which leads me to my question.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What other common pitfalls of student proof writing should be added to this taxonomy? What are some poorly written proofs that exemplify one of these pitfalls (either one that I have already listed or a new one)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Feel free to argue with my current list. Also, if anyone wants to add an answer to the old question on &quot;bad proofs&quot; I'd be very happy.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; After discussing this with one of my friends, I've added two more entries to the taxonomy.&lt;/p&gt;&#xA;" OwnerUserId="13482" LastEditorUserId="13482" LastEditDate="2020-02-17T18:24:33.783" LastActivityDate="2021-05-14T15:24:00.503" Title="Taxonomy of bad proofs" Tags="|undergraduate-education|mathematical-pedagogy|proofs|students-mistakes|writing|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="17902" PostTypeId="1" AcceptedAnswerId="17944" CreationDate="2020-02-17T09:13:46.070" Score="7" ViewCount="877" Body="&lt;p&gt;The standard scalar product on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; is defined via&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\vec a\cdot\vec b := a_1b_1+a_2b_2+a_3b_3$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, it can be expressed in a more geometrical way through the lengths of the vectors and the angle &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; they enclose via&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\vec a\cdot\vec b = |\vec a| |\vec b|\cos\phi$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I struggle to establish a link between the two expressions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possibility is to consider the second expression as a definition for the angle &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;, since it can be shown that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{\vec a\cdot\vec b}{|\vec a| |\vec b|}\in[-1;1]$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and thus can be written as the cosine of some angle. But this does not explain why the angle in question is the one enclosed by the two vectors, we could also try to go with the sine of some angle rather than a cosine according to this argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another possibility is to introduce, say, cylindrical coordinates and consider two unit vectors &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\vec a = (\cos\phi, \sin\phi, 0),\quad \vec b = (1,0,0)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which clearly enclose the angle &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; and compute the scalar product according to its definition which gives us &lt;span class=&quot;math-container&quot;&gt;$\cos\phi$&lt;/span&gt; as a result, then argue that the scalar product is also proportional to the lengths of both vectors involved and finally show that it is invariant under rotations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both approaches require rather heavy machinery considering that the scalar product is being introduced in school already.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is: What is a good way to explain to school students (11th grade, that is, around 17 years old) why the two expressions for the scalar product are equivalent?&lt;/p&gt;&#xA;" OwnerUserId="3542" LastEditorUserId="8571" LastEditDate="2020-02-20T22:10:44.927" LastActivityDate="2020-02-22T16:35:16.283" Title="How do you plausibly explain that the geometric and the coordinate expressions for the scalar product are equivalent?" Tags="|vectors|analytic-geometry|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17903" PostTypeId="1" CreationDate="2020-02-17T09:23:24.963" Score="2" ViewCount="60" Body="&lt;p&gt;I have a &lt;a href=&quot;https://math.stackexchange.com/questions/3549328/intuition-about-degenerate-conics&quot;&gt;question on MSE&lt;/a&gt; that maybe can be better posed here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is about degenerate conics, and especially  the case of two parallel lines, as  in the equation &lt;span class=&quot;math-container&quot;&gt;$ ^2+2+^2=1$&lt;/span&gt;.&#xA;Usually, teaching  the conics as sections of  a double cone , we say that  degnerate conics are generated when  the plane passes  thorough the vertex of the cone. So the position of the intersecting plane is the ''cause'' of degeneracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it seems that this is not true in the case of two parallel lines,  where we need a different kind of degeneracy, that is the fact that the center of the conic goes to infinity,  so that the cone becomes a cylinder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this fact  teached  in common textbooks or courses? And what can be its interesting consequences?&lt;/p&gt;&#xA;" OwnerUserId="5314" LastActivityDate="2020-02-17T14:53:41.993" Title="On a special degenerate conic" Tags="|secondary-education|analytic-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17904" PostTypeId="2" ParentId="17863" CreationDate="2020-02-17T10:22:09.903" Score="3" Body="&lt;p&gt;The common way to introduce how trig functions are calculated numerically is via Taylor/Maclaurin series. These are used extensively in a lot of engineering and physics based applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this requires a knowledge of calculus, and isn't very useful in the context of basic trigonometry. It also isn't useful for giving students a &quot;feel&quot; of how these functions work, or how they might have developed them themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, let me introduce you to a more elegant tool from a more civilised time: The Unit Circle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nzmeI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nzmeI.png&quot; alt=&quot;A unit circle.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This particular version is my own presentation, created using the excellent pynomo package. Anyone is free to use or adapt it as they wish.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is just a very simple diagram consisting of a circle with a radius of 1 arbitrary unit. It has a horizontal and vertical axis, and it has an angular scale marked round the outside in revolutions, degrees and radians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Introduce it to students by showing them a triangle with the hypotenuse deliberately chosen to be one unit long. Assuming that they have been taught:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(\theta) = \frac{Opposite}{Hypotenuse}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(\theta) = \frac{Adjacent}{Hypotenuse}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then it should be easy to see for a triangle with a hypotenuse of 1:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(\theta) = \frac{Opposite}{1} = Opposite$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(\theta) = \frac{Adjacent}{1} = Adjacent$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means that for the special case where a right angled triangle has a hypotenuse of 1, the horizontal line will have a length equal to the cosine of the angle and the vertical line will have a length equal to the sine of the angle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A circle of radius 1 represents all points that are 1 unit away from the origin. So if we draw a circle, and then draw a radial line, then the horizontal coordinate of where it meets the circumference will be the cosine of the angle, and the vertical coordinate will be the sine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can demonstrate this with the unit circle as follows. Suppose you want to find the sine and cosine of 20 degrees (&lt;span class=&quot;math-container&quot;&gt;$\pi/9$&lt;/span&gt; radians). Lay a ruler on the unit circle such that it passes through the centre of the circle, and the 20 degree mark:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Lynak.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Lynak.png&quot; alt=&quot;Unit circle demonstration&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, simply read off the horizontal and vertical coordinates where the ruler passes through the circle. you will find that the vertical axis has a coordinate of about 0.34, and the horizontal coordinate of 0.94. If you want to check on a calculator, you will find:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(20 ^{\circ}) \approx 0.9397$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(20 ^{\circ}) \approx 0.3420$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to find the tangent of an angle, you can either employ the trig identity:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or you can place a 2nd ruler tangent to the point where the first ruler crosses the unit circle, and measure the number of units between the tangent point and the projection of the horizontal axis:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/g0J25.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/g0J25.png&quot; alt=&quot;Trig construction.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This also helps introduce why it's called the &quot;tangent&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like using this approach because it is such a simple construction where everything is apparent. There is no &quot;magic&quot; in it, and you can measure any sine and cosine very quickly. You can also have your students play about with it in order to get a feel of how these functions behave. It allows you to introduce inverse functions by asking them  what angle is associated with a given &lt;span class=&quot;math-container&quot;&gt;$\sin(\theta)$&lt;/span&gt;. You can also introduce a bit of maths history about why &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt; are referred to as circular functions.&lt;/p&gt;&#xA;" OwnerUserId="10057" LastEditorUserId="10057" LastEditDate="2020-02-17T10:28:05.853" LastActivityDate="2020-02-17T10:28:05.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17905" PostTypeId="2" ParentId="17902" CreationDate="2020-02-17T12:03:11.683" Score="17" Body="&lt;p&gt;Cosine rule!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Think of vectors &lt;span class=&quot;math-container&quot;&gt;$\vec a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec b$&lt;/span&gt; as two sides of a triangle, with tails at a common vertex. The remaining side is given by &lt;span class=&quot;math-container&quot;&gt;$\vec a - \vec b$&lt;/span&gt;. Then cosine rule gives us &lt;span class=&quot;math-container&quot;&gt;$|\vec a|^2 + |\vec b|^2 - 2|\vec a||\vec b|\text{cos}\,\theta=|\vec a-\vec b|^2$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is the angle between &lt;span class=&quot;math-container&quot;&gt;$\vec a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec b$&lt;/span&gt;. Now write out &lt;span class=&quot;math-container&quot;&gt;$|\vec a|^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$|\vec b|^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|\vec a - \vec b|^2$&lt;/span&gt; in terms of their orthogonal components, and the result follows after a little bit of algebra (which is accessible to 11th graders).&lt;/p&gt;&#xA;" OwnerUserId="9455" LastActivityDate="2020-02-17T12:03:11.683" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17906" PostTypeId="2" ParentId="17885" CreationDate="2020-02-17T12:24:36.890" Score="1" Body="&lt;p&gt;Mathematic notation and numbers are fundamental to basic communication and problem solving.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our existence depends on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also very fun. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this question would come up less often if we presented kids with problems to solve, and then they realize that they need tools. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My physics teacher derived formulas from observation and testing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Memorising tables sucks. &#xA;recognizing a pattern can be fun. &lt;/p&gt;&#xA;" OwnerUserId="13494" LastActivityDate="2020-02-17T12:24:36.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17907" PostTypeId="2" ParentId="17885" CreationDate="2020-02-17T13:20:22.287" Score="4" Body="&lt;p&gt;To give a German-centric answer which would certianly apply to other countries as well: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The German Abitur is the right to study any subject at a university (Allgemeine Hochschulreife). As such the pupils need to be taught everything such that they need in order to be able to succeed. After all, they may want to change the subject after finishing school and you do not want them to have to decide at an early age.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other countries, a similar concept exists at different levels. As such, a broad education is necessary. Which is why me, as a mathematician, needed to study history and social studies, which I clearly did not have any interest in. &lt;/p&gt;&#xA;" OwnerUserId="13496" LastActivityDate="2020-02-17T13:20:22.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17908" PostTypeId="2" ParentId="17903" CreationDate="2020-02-17T13:46:01.910" Score="3" Body="&lt;p&gt;The general quadric surface in projective space is ruled by two families of lines. These are smooth surfaces. Think about a hyperboloid of one sheet, where is is easy to visualize the two families of lines. You can &quot;pinch&quot; this surface at an &quot;ordinary&quot; point to get the cone, or you can &quot;pinch&quot; at one of the surface's points at infinity to get the cylinder. This pinching at infinity of the hyperboloid of one sheet amounts to widening the waist, to morph the hyperboloid into a cylinder.  The lines on the cone through the vertex are nothing more than the lines on the cylinder, all of which meet at the cylinder's vertex positioned at infinity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps there is cognitive conflict caused by an erroneous assumption that the cylinder is smooth, whereas the cone is clearly not smooth at the vertex. But the cylinder is not smooth. It has a vertex at infinity.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2020-02-17T14:53:41.993" LastActivityDate="2020-02-17T14:53:41.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17910" PostTypeId="2" ParentId="17900" CreationDate="2020-02-17T18:44:21.380" Score="16" Body="&lt;p&gt;Perhaps related to &quot;The Tangle&quot; is what I call &quot;Wishful Thinking&quot;. This most often happens when the student has a correct algebraic expression/equality and knows the correct final expression/equality, but makes great (or incorrect) leaps to get from one to the other, e.g. in proofs by induction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also the &quot;Using What You Are Proving&quot; (maybe similar to your &quot;Backwards Proof,&quot; but more subtle). Example: in proving two elements &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; commute, they use &lt;span class=&quot;math-container&quot;&gt;$(ab)^2 = a^2b^2$&lt;/span&gt;. Or in an onto proof where they assume a general element of the range has the form of an image.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2020-02-17T18:44:21.380" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17912" PostTypeId="2" ParentId="5794" CreationDate="2020-02-17T19:26:34.587" Score="0" Body="&lt;p&gt;I tried the following approach recently in a college algebra course (I kept insisting everything in math has a &lt;em&gt;reason&lt;/em&gt;, and then a student asked about the reason for the rule of multiplication of negative numbers).   The approach seemed to go over well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, I made it concrete.  Instead of asking why &lt;span class=&quot;math-container&quot;&gt;$(-x)(-y) = xy$&lt;/span&gt;, I focused on why &lt;span class=&quot;math-container&quot;&gt;$(-4)(-3) = 12$&lt;/span&gt;.  Then I drew the following table  &lt;span class=&quot;math-container&quot;&gt;$$\begin{bmatrix} &amp;amp;|&amp;amp; 3 &amp;amp; 2 &amp;amp; 1 &amp;amp;0 &amp;amp; -1 &amp;amp; -2 &amp;amp; -3\\ 4\times &amp;amp;|&amp;amp; 12&amp;amp;8&amp;amp;4&amp;amp;0\end{bmatrix}$$&lt;/span&gt; where the top row is just a list of consecutive integers and the bottom row is what you get by multiplying the top row by 4.  (Apologies for the bad formatting, the tabular environment seems to not be supported.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked the class to ignore the top row and just focus on the bottom row to see the pattern.  They more or less immediately saw that each entry comes from the previous by subtracting &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;.  They were then able to complete the table:  &lt;span class=&quot;math-container&quot;&gt;$$\begin{bmatrix} &amp;amp;|&amp;amp; 3 &amp;amp; 2 &amp;amp; 1 &amp;amp;0 &amp;amp; -1 &amp;amp; -2 &amp;amp; -3\\ 4\times &amp;amp;|&amp;amp; 12&amp;amp;8&amp;amp;4&amp;amp;0 &amp;amp; -4 &amp;amp; -8 &amp;amp; -12\end{bmatrix}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This &quot;established&quot; the rule (positive)&lt;span class=&quot;math-container&quot;&gt;$\cdot$&lt;/span&gt; (negative) &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; negative.  Having this at our disposal, I now drew the table for multiplication by &lt;span class=&quot;math-container&quot;&gt;$-4$&lt;/span&gt;:  &lt;span class=&quot;math-container&quot;&gt;$$\begin{bmatrix} &amp;amp;|&amp;amp; 3 &amp;amp; 2 &amp;amp; 1 &amp;amp;0 &amp;amp; -1 &amp;amp; -2 &amp;amp; -3\\ -4\times &amp;amp;|&amp;amp; -12&amp;amp;-8&amp;amp;-4&amp;amp;0\end{bmatrix}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once again ignoring the top row, the evident pattern on the bottom row is that each item is obtained from the previous item by adding &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;.  Then it is obvious how to complete the table:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{bmatrix} &amp;amp;|&amp;amp; 3 &amp;amp; 2 &amp;amp; 1 &amp;amp;0 &amp;amp; -1 &amp;amp; -2 &amp;amp; -3\\ -4\times &amp;amp;|&amp;amp; -12&amp;amp;-8&amp;amp;-4&amp;amp;0 &amp;amp; 4 &amp;amp; 8 &amp;amp; 12\end{bmatrix}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And what do you know - a negative times a negative is a positive!&lt;/p&gt;&#xA;" OwnerUserId="149" LastActivityDate="2020-02-17T19:26:34.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17913" PostTypeId="2" ParentId="17885" CreationDate="2020-02-17T19:50:50.380" Score="2" Body="&lt;p&gt;We are allowing people with less than even basic numeracy to leave school.&lt;br&gt;&#xA;We are allowing people to enter university with not only no knowledge of basic algebra or calculus but also with no realisation that basic algebra and calculus are essential tools when dealing with subjects such as history or social sciences. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A social scientist or historian needs a basic understanding of exponential growth or decay - if only enough to provide a platform for more learning when the need latterly becomes obvious. Higher levels of knowledge can be far more easily be built on a basic foundation than being obtained with no prior introduction.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You buy 5 items that cost 0.30 currency units each.&lt;br&gt;&#xA;You give the shop assistant a 5 currency unit note.&lt;br&gt;&#xA;They pick up a calculator to multiply 0.3 by 5 and then to subtract this answer from 5.&lt;br&gt;&#xA;They give you 3 currency units in change.&lt;br&gt;&#xA;They wonder why you look at them in what used to be astonishment until it happened many many many times.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's just the most basically basic of numeracy.&lt;br&gt;&#xA;Now for some basic mind bending with algebra and geometry ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we are releasing people into the world who cannot calculate 5 - 0.3 * 5 (as we are) then we are stopping far short of a reasonable target. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen social science papers which are utter unmitigated and irredeemable balderdash BECAUSE the writer has no grasp of basic mathematical concepts, and doesn't know that they don't know&lt;/p&gt;&#xA;&#xA;&lt;p&gt;STAY AT SCHOOL*.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;*Failing to gain a minimum broad education is functionally failing to 'stay at school' &lt;/p&gt;&#xA;" OwnerUserId="13518" LastEditorUserId="13518" LastEditDate="2020-02-18T10:43:56.030" LastActivityDate="2020-02-18T10:43:56.030" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17917" PostTypeId="2" ParentId="17902" CreationDate="2020-02-18T00:32:15.540" Score="6" Body="&lt;p&gt;Here's a way to do it by computing the length of &lt;span class=&quot;math-container&quot;&gt;$\vec{a} + \vec{b}$&lt;/span&gt; in two different ways: one of which is purely symbolic and the other uses some geometric knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This argument is somewhat similar to Stephan Kubicki's argument, just with a &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; instead of a &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\vec{a} + \vec{b}$&lt;/span&gt; can be defined geometrically and component-wise in a straightforward way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First a word on notation, &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; means the vector &lt;span class=&quot;math-container&quot;&gt;$\vec{a}$&lt;/span&gt; to reduce visual clutter. Additionally, I'll be using &lt;span class=&quot;math-container&quot;&gt;$\langle x, y \rangle$&lt;/span&gt; consistently to represent the dot product &lt;span class=&quot;math-container&quot;&gt;$x \cdot y$&lt;/span&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The very first thing I think is to define the inner product (101). In this definition, &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is the number of dimensions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \langle a, b \rangle \stackrel{\text{def}}{=\!=} \sum_ {k =1} ^ n a_kb_k \tag{101} $$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to motivate this definition, one can show that the squared length of a vector is the dot product of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; with itself (102).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ |a|^2 = \langle a , a \rangle \tag{102} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As another bit of motivation, you can also show that two vectors are perpendicular if and only if their dot product is zero and work through a couple of examples (111).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \text{$x$ and $y$ are perpendicular} \iff \langle x, y \rangle = 0 \tag{111} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that out of the way, we can ask about the squared length of &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt; (103).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ |a+b|^2 \tag{103} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is equivalent to the dot product of &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt; with itself (103a).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \langle a + b, a + b \rangle \tag{103a} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, let's look at this problem symbolically. You can distribute over the left and right arguments (104)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \langle a +b, a+b \rangle = \langle a, a \rangle + \langle a, b \rangle + \langle b, a \rangle + \langle b, b \rangle \tag{104} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it's straightforward to show that &lt;span class=&quot;math-container&quot;&gt;$\langle a, b \rangle = \langle b, a \rangle$&lt;/span&gt;, giving (104a).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \langle a +b, a+b \rangle = \langle a, a \rangle + 2\langle a, b \rangle + \langle b, b \rangle \tag{104a} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, ask students to picture computing &lt;span class=&quot;math-container&quot;&gt;$| a + b | $&lt;/span&gt; by splitting &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; into two vectors, one of which is parallel to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; (let's call it &lt;span class=&quot;math-container&quot;&gt;$b_\text{sam}$&lt;/span&gt;) and one of which is perpendicular to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; (let's call it &lt;span class=&quot;math-container&quot;&gt;$b_\text{dif}$&lt;/span&gt;) (105). For this sort of argument I think a picture would help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ | a + b | = \sqrt{ |a + b_\text{sam}|^2 + |0 + b_\text{dif}|^2 } \tag{105} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is the angle between &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, then we can rewrite this expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ | a + b | = \sqrt{ (|a| + |b|\cos{\theta})^2 + (|b|\sin{\theta})^2 } \tag{105a} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next we can expand it out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ | a + b | = \sqrt{ (|a||a| + 2|a||b|\cos{\theta} + |b||b|\cos{\theta}\cos{\theta}) + (|b||b|\sin{\theta}\sin{\theta}) } \tag{105b} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can exploit the fact that &lt;span class=&quot;math-container&quot;&gt;$\cos{\theta}\cos{\theta} + \sin{\theta}\sin{\theta} = 1 $&lt;/span&gt; (105c).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ |a+b| = \sqrt{|a||a| + 2|a||b|\cos{\theta} + |b||b|} \tag{105c} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Square both sides (105d)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ |a+b||a+b| = |a||a| + 2|a||b|\cos{\theta} + |b||b| \tag{105d} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Use the dot product instead of the squared length (105e).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \langle a+b, a+b \rangle = \langle a, a \rangle + 2|a||b|\cos{\theta} + \langle b, b \rangle \tag{105e} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next we compare the (105e) and (104a), reproduced below for convenience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \langle a +b, a+b \rangle = \langle a, a \rangle + 2\langle a, b \rangle + \langle b, b \rangle \tag{104a} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \langle a+b, a+b \rangle = \langle a, a \rangle + 2|a||b|\cos{\theta} + \langle b, b\rangle \tag{105e} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, as desired, &lt;span class=&quot;math-container&quot;&gt;$ \langle a, b \rangle = |a||b|\cos{\theta} $&lt;/span&gt; .&lt;/p&gt;&#xA;" OwnerUserId="13521" LastEditorUserId="13521" LastEditDate="2020-02-20T08:35:52.687" LastActivityDate="2020-02-20T08:35:52.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17919" PostTypeId="2" ParentId="17864" CreationDate="2020-02-18T02:24:06.163" Score="3" Body="&lt;p&gt;If you mean visual beauty, audio visualizers like &lt;a href=&quot;https://www.youtube.com/watch?v=9TbLJI7ja4s&quot; rel=&quot;nofollow noreferrer&quot;&gt;MilkDrop&lt;/a&gt;, &lt;a href=&quot;https://www.youtube.com/watch?v=Y-6cH3trmHs&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friktal&lt;/a&gt;, and &lt;a href=&quot;https://www.youtube.com/watch?v=fpViZkhpPHk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Particles&lt;/a&gt; can be interesting. As of course game engines like &lt;a href=&quot;https://www.unrealengine.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Unreal&lt;/a&gt; and the &lt;a href=&quot;https://www.youtube.com/watch?v=O4T7pIs--LA&quot; rel=&quot;nofollow noreferrer&quot;&gt;demoscene&lt;/a&gt; that tends to combine them into incredibly small programs like &lt;a href=&quot;https://www.youtube.com/watch?v=Y3n3c_8Nn2Y&quot; rel=&quot;nofollow noreferrer&quot;&gt;64 KiB&lt;/a&gt; and even &lt;a href=&quot;https://www.youtube.com/watch?v=XBNyhIY61Yw&quot; rel=&quot;nofollow noreferrer&quot;&gt;8 KiB&lt;/a&gt; or &lt;a href=&quot;https://twitter.com/kometbomb/status/1360666861415890945&quot; rel=&quot;nofollow noreferrer&quot;&gt;a single tweet&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;One of those demo guys wrote &lt;a href=&quot;https://www.smashingmagazine.com/2011/10/quick-look-math-animations-javascript/&quot; rel=&quot;nofollow noreferrer&quot;&gt;an article about automating boring math&lt;/a&gt;. (In JSFiddle you may need to change the JavaScript load type to &amp;quot;No wrap - bottom of &lt;code&gt;&amp;lt;body&amp;gt;&lt;/code&gt;&amp;quot;.)&lt;/p&gt;&#xA;&lt;p&gt;Also great is &lt;a href=&quot;https://twitter.com/wonderofscience/status/1171426852340207617&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carl Sagan's explanation of how the ancient Greeks knew the Earth was round and calculated its circumference over 2,000 years ago&lt;/a&gt;, which can segue to rocket science via &lt;a href=&quot;https://www.kerbalspaceprogram.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kerbal Space Program&lt;/a&gt; and &lt;a href=&quot;https://www.theverge.com/2019/10/30/20939147/deepmind-google-alphastar-starcraft-2-research-grandmaster-level&quot; rel=&quot;nofollow noreferrer&quot;&gt;machine learning&lt;/a&gt; for &lt;a href=&quot;https://www.youtube.com/watch?v=mHeBQbSm1Rk&amp;amp;list=PLkVJPA-i8f5iDnSKaWCymxczifndJdgNT&quot; rel=&quot;nofollow noreferrer&quot;&gt;less collateral damage&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It would be nice to draw possible career paths now and it appears that modeling is at the frontier of the &lt;a href=&quot;https://www.flickr.com/photos/95869671@N08/32264483720&quot; rel=&quot;nofollow noreferrer&quot;&gt;landscape&lt;/a&gt;/&lt;a href=&quot;https://www2.stetson.edu/%7Emhale/logic/tree.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;math tree&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vAlYH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vAlYH.png&quot; alt=&quot;math tree&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I conceive of mathematics as a fantastic citrus tree: the three main&#xA;branches of oranges, limes, and lemons representing the major fields&#xA;of algebra, geometry, and analysis. Each part of the upper canopy&#xA;takes advantage of what is below it. Thus, number theory has two great&#xA;approaches, one algebraic and the other analytic. Topology makes use&#xA;of all three fields, as does the very different field of mathematical&#xA;modeling.&lt;/p&gt;&#xA;&lt;p&gt;All trees have woody parts, foliage, and fruit. The trunk and branches&#xA;absorb water and nutrients from the soil. The foliage absorbs energy&#xA;from the sun, using it to convert the nutrients to usable food, and&#xA;allowing the tree to grow. The fruit allows the tree to reproduce. I&#xA;have represented the foundations of mathematics as the trunk of the&#xA;tree, the support and food supply. The foliage and fruit could&#xA;represent the two aspects of pure and applied mathematics: which is&#xA;which, do you think? - &lt;a href=&quot;https://www2.stetson.edu/%7Emhale/index.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Margie Hale&lt;/a&gt;,&#xA;Professor of Mathematics, Emeritus&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even if people decide math isn't for them, their computers can still use it to &lt;a href=&quot;https://boinc.berkeley.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;save the world&lt;/a&gt;, &lt;a href=&quot;https://www.smbc-comics.com/comic/math-education&quot; rel=&quot;nofollow noreferrer&quot;&gt;save money&lt;/a&gt;, or make &lt;a href=&quot;https://www.youtube.com/watch?v=jVD67pMdv9k&quot; rel=&quot;nofollow noreferrer&quot;&gt;ever-changing&lt;/a&gt; &lt;a href=&quot;https://www.youtube.com/watch?v=O5RdMvgk8b0&quot; rel=&quot;nofollow noreferrer&quot;&gt;art&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=tgni7Vrez8E&quot; rel=&quot;nofollow noreferrer&quot;&gt;An infinite universe&lt;/a&gt; arising from &lt;a href=&quot;https://youtu.be/7MotVcGvFMg&quot; rel=&quot;nofollow noreferrer&quot;&gt;a simple formula&lt;/a&gt; is sure to inspire, such as this &lt;a href=&quot;https://twitter.com/MathsTown/status/1203632013648904192&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Red Roots&amp;quot;&#xA;Triangle Inequality Average Mandelbrot fractal&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bfER4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bfER4.jpg&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=xP5-iIeKXE8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Likewise&lt;/a&gt; &lt;a href=&quot;https://www.youtube.com/watch?v=dQJ5aEsP6Fs&quot; rel=&quot;nofollow noreferrer&quot;&gt;cellular automata&lt;/a&gt; like &lt;a href=&quot;https://www.youtube.com/watch?v=C2vgICfQawE&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conway's Game of Life&lt;/a&gt; which &lt;a href=&quot;https://en.wikipedia.org/wiki/Turing_completeness&quot; rel=&quot;nofollow noreferrer&quot;&gt;can compute anything&lt;/a&gt;, &lt;a href=&quot;https://www.youtube.com/watch?v=8unMqSp0bFY&quot; rel=&quot;nofollow noreferrer&quot;&gt;even itself&lt;/a&gt;, other &lt;a href=&quot;https://en.wikipedia.org/wiki/Artificial_life&quot; rel=&quot;nofollow noreferrer&quot;&gt;articifial life&lt;/a&gt; like &lt;a href=&quot;https://www.youtube.com/watch?v=QbUPfMXXQIY&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boids&lt;/a&gt; in &lt;a href=&quot;https://www.youtube.com/watch?v=bqtqltqcQhw&quot; rel=&quot;nofollow noreferrer&quot;&gt;3D&lt;/a&gt;, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Emergence&quot; rel=&quot;nofollow noreferrer&quot;&gt;emergent&lt;/a&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/a/18157/13523&quot;&gt;multidimensional&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Snake_(video_game_genre)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Snake&lt;/a&gt; in &lt;a href=&quot;https://www.youtube.com/watch?v=7-97RhAZhXI&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lenia&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In short, make &lt;a href=&quot;https://www.youtube.com/watch?v=B1J6Ou4q8vE&quot; rel=&quot;nofollow noreferrer&quot;&gt;math&lt;/a&gt; &lt;a href=&quot;https://amitness.com/2019/08/math-for-programmers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;relatable&lt;/a&gt; and &lt;a href=&quot;https://www.youtube.com/watch?v=wfctJRQSPc4&quot; rel=&quot;nofollow noreferrer&quot;&gt;useful&lt;/a&gt; to &lt;a href=&quot;https://www.youtube.com/watch?v=5M2RWtD4EzI&quot; rel=&quot;nofollow noreferrer&quot;&gt;them&lt;/a&gt;. As &lt;a href=&quot;https://www.youtube.com/watch?v=6avJHaC3C2U&quot; rel=&quot;nofollow noreferrer&quot;&gt;this excellent presentation&lt;/a&gt; quotes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Clouds are not spheres, mountains are not cones, coastlines are not&#xA;circles, and bark is not smooth, nor does lightning travel in a&#xA;straight line.&amp;quot; - Benoit Mandelbrot&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yet:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The cosmos is full beyond measure of elegant truths; of exquisite&#xA;interrelationships; of the awesome machinery of nature.&amp;quot; - Carl Sagan&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But most importantly:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Subjects should not be introduced without explaining the purpose or&#xA;reason, or without giving any way in which the material could be&#xA;really used to discover something interesting. I don't think it is&#xA;worthwhile teaching such material.&amp;quot; - &lt;a href=&quot;https://en.wikipedia.org/wiki/New_Math&quot; rel=&quot;nofollow noreferrer&quot;&gt;Richard Feynman&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I, for one, would like to learn &lt;a href=&quot;https://www.youtube.com/watch?v=XUy8lELWhJg&quot; rel=&quot;nofollow noreferrer&quot;&gt;how to simply scratch my own holograms.&lt;/a&gt; and if i were mechanically inclined, &lt;a href=&quot;https://www.youtube.com/watch?v=A_BlNA7bBxo&quot; rel=&quot;nofollow noreferrer&quot;&gt;build a robot to automate that&lt;/a&gt;. And/or do some &lt;a href=&quot;https://www.futurity.org/photo-wake-up-moving-images-2084262-2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;weird science&lt;/a&gt; for &lt;a href=&quot;http://thebestpageintheuniverse.net/c.cgi?u=math&quot; rel=&quot;nofollow noreferrer&quot;&gt;real men&lt;/a&gt; instead of &lt;a href=&quot;https://www.reddit.com/r/funny/comments/o7vip/still_waiting/&quot; rel=&quot;nofollow noreferrer&quot;&gt;despairing over strange symbols and formulas&lt;/a&gt; that still fail to explain &lt;a href=&quot;https://www.youtube.com/watch?v=YWsK6rmsKSI&quot; rel=&quot;nofollow noreferrer&quot;&gt;how a bicycle works&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13523" LastEditorUserId="13523" LastEditDate="2023-08-15T18:53:15.890" LastActivityDate="2023-08-15T18:53:15.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17923" PostTypeId="1" CreationDate="2020-02-18T12:26:47.080" Score="5" ViewCount="166" Body="&lt;p&gt;The Mathematics Assessment Project (MAP) has created lesson plans for high school lessons in maths based around formative assessment. They make the following remark about marking students' work in this &lt;a href=&quot;https://www.map.mathshell.org/download.php?fileid=1608&quot; rel=&quot;noreferrer&quot;&gt;teacher guide&lt;/a&gt; for a lesson about substitution.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We suggest that you do not score students’ work. The research shows&#xA;  that this will be counter-productive, as it will encourage students to&#xA;  compare their scores and distract their attention from what they can&#xA;  do to improve their mathematics.` &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would like references to research that supports this claim.&lt;/p&gt;&#xA;" OwnerUserId="3407" LastActivityDate="2020-02-18T21:27:07.373" Title="Scoring students' work is counter-productive" Tags="|reference-request|education-research|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17924" PostTypeId="1" CreationDate="2020-02-18T13:06:54.750" Score="6" ViewCount="306" Body="&lt;p&gt;An &lt;a href=&quot;https://matheducators.stackexchange.com/a/17889/766&quot;&gt;answer&lt;/a&gt; on this site mentions that it would be more appropriate to criticize&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Efforts to force kids to take algebra at lower and lower ages, such as attempts in California to make all kids take algebra in 8th grade.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What exactly is this &quot;algebra&quot; that they have to learn?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As another person in the comments, I don't find starting algebra at the eight grade outrageous, but maybe the topics behind the general term are a bit different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've had what's called &quot;algebra&quot; here since grade 7. We had equation and inequality solving, single-argument functions, linear functions, monomials and polynomials. And in grade 8 we got to factoring polynomials (&lt;em&gt;short multiplication&lt;/em&gt; formulae) and solving quadratic equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do people in the US normally have at that age? And what is in the California attempt?&lt;/p&gt;&#xA;" OwnerUserId="766" LastActivityDate="2020-02-18T13:06:54.750" Title="What is the controversial 8th grade algebra mentioned on this answer?" Tags="|curriculum|" AnswerCount="0" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="17925" PostTypeId="2" ParentId="11310" CreationDate="2020-02-18T14:01:27.200" Score="2" Body="&lt;p&gt;I once tutored someone and simple linear equations was part of the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example I came up with related to the student's own life was saving up to buy a smartphone. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem can be made into different variations, with a basic version such as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I want to buy a phone which costs $400. With a weekly income of a, how many weeks do I need to save to be able to buy it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This gives you an equation like &lt;code&gt;y = ax + b&lt;/code&gt; where &lt;strong&gt;y&lt;/strong&gt; = 400  (target price),  &lt;strong&gt;a&lt;/strong&gt; = your weekly income,  &lt;strong&gt;x&lt;/strong&gt; the number of weeks to save and &lt;strong&gt;b&lt;/strong&gt; the starting money. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can adjust the problem by changing their weekly salary or working more hours, and/or changing the amount they have today in their bank account. You could also change the target variable, for example setting a fixed number of weeks and checking what price I'll be able to pay for a phone by that time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can also increase the problem scope by introducing a (real) factor that the price of the phone will decrease over time, giving us a system of two equations, where the price of the phone over time might be for example  400 - 20x, and we'll solve the system for where these two lines will intersect. (When will I be able to buy it, and what will the price be then?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the price won't go to zero in reality, perhaps the price function should rather be a quadratic function with some reasonable minimum. This would also perhaps model the price drop more realistically.&lt;/p&gt;&#xA;" OwnerUserId="6423" LastEditorUserId="6423" LastEditDate="2020-02-18T14:06:42.117" LastActivityDate="2020-02-18T14:06:42.117" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17926" PostTypeId="1" CreationDate="2020-02-18T14:12:42.107" Score="4" ViewCount="243" Body="&lt;p&gt;Could we say it has one when substituting the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; value into the original equation? Obviously, to solve the trascendental equation one has to operate the two terms with logarithms, reducing the equation to &lt;span class=&quot;math-container&quot;&gt;$\log((x-10)/(x-3))=3$&lt;/span&gt; and finally having &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approximately equal to &lt;span class=&quot;math-container&quot;&gt;$2.633$&lt;/span&gt;. Now how can we explain a student that the value obtained, if substituted into the original equation, yields both a LHS and a RHS not in reals?&lt;/p&gt;&#xA;" OwnerUserId="6338" LastActivityDate="2020-02-27T18:48:10.453" Title="Has the equation $\log(x-10)=3+\log(x-3)$ a *substitutable* solution in $\mathbb{R}$?" Tags="|secondary-education|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17927" PostTypeId="2" ParentId="17926" CreationDate="2020-02-18T14:42:13.130" Score="1" Body="&lt;p&gt;Title question:  Uh...yes.  Approximately 2.633.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Now how can we explain a student that the value obtained, if substituted into the original equation, yields both a LHS and a RHS not in reals?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By just showing them?  It's not a requirement that intermediate parts of some other equation are real.  Just that x is.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But consider, that you could also think of the problem as having started with the log of the fraction.  Thus breaking it out into the separate logs would be the manipulation creating the strange result.  Maybe there are many ways to generate such oddities, starting from something non-weird.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;FWIW: You can also have real roots of cubics that depend on complex numbers in their intermediate solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  I hope your students are well-drilled on the basics.  I see many questioners/answers at this site who are fascinated by the &quot;oolies&quot; (harder problems, definitional distinctions) while simultaneously teaching a lot of kids who really need better drill in the basics. &lt;/p&gt;&#xA;" OwnerUserId="13542" LastActivityDate="2020-02-18T14:42:13.130" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17928" PostTypeId="2" ParentId="17902" CreationDate="2020-02-18T14:48:05.903" Score="0" Body="&lt;p&gt;If you can get across the intuition that the scalar product is rotation invariant, and then that dotting with xhat or yhat pulls out the coordinate, after rotating a so that it is xhat this  is the definition of cosine. (Xhat and yhat are the unit length vectors in the coordinate directions.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the rotation invariance, one intuitive approach to it breaks down into: calculating the scalar product between orthogonal* and parallel unit  vectors, and noting that by distributivity this is sufficient to characterize the scalar product, that the concepts of orthogonal and parallel unit vectors are rotation invariant, and that rotation is a linear transformation. Of course, some of this would have to be phrased in a lot more detail for a high school audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*For the scalar product between two orthogonal unit vectors, write one of them as (cos(a), sin(a)) and the other as (cos(a + pi), sin(a + pi)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another important simplification I made is that one can work in the plane. You don't have to explain why this is the case, I think. &lt;/p&gt;&#xA;" OwnerUserId="8029" LastActivityDate="2020-02-18T14:48:05.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17930" PostTypeId="2" ParentId="17926" CreationDate="2020-02-18T15:30:52.450" Score="3" Body="&lt;p&gt;No, you can see that &lt;span class=&quot;math-container&quot;&gt;$\log(x-10) &amp;lt; 3 + \log(x-3)$&lt;/span&gt; by checking the derivatives: &lt;span class=&quot;math-container&quot;&gt;$$0 &amp;lt; \frac{1}{x-10} &amp;lt; \frac{1}{x-3}$$&lt;/span&gt;&#xA;and the value at any point where they are both defined. i.e. &lt;span class=&quot;math-container&quot;&gt;$\log(x-10)$&lt;/span&gt; starts below &lt;span class=&quot;math-container&quot;&gt;$3+\log(x-3)$&lt;/span&gt; and grows slower.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, if you either expand the domain by rewriting the equation as &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \log | x-10 | = 3+\log|x-3| $$&lt;/span&gt;&#xA;or by allowing &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; to take negative real inputs at the expense of throwing off extra &lt;span class=&quot;math-container&quot;&gt;$\pi\sqrt{-1}$&lt;/span&gt; terms, then &lt;span class=&quot;math-container&quot;&gt;$$x = \frac{3e^3 - 10}{e^3 - 1}$$&lt;/span&gt;&#xA;becomes a solution. In the absolute value case, we actually pick up yet another solution, as you can see in the graph below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1l09B.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1l09B.png&quot; alt=&quot;Graph of log|x-10| and 3+log|x-3|&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2020-02-18T19:56:10.403" LastActivityDate="2020-02-18T19:56:10.403" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17931" PostTypeId="2" ParentId="17864" CreationDate="2020-02-18T15:58:08.817" Score="3" Body="&lt;p&gt;Mathematics is a very beautiful subject that is being taught in a totally wrong way, causing it to be perceived as a very boring and ugly subject. One of the biggest problems with math education is the totally flawed idea that one has to focus almost exclusively on the elementary basics and make sure these are mastered 100% before one can move in to the next level. No other subject is taught in this way, although physics comes close and we see the same sort of problems with that subject too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine that we would teach language in the same way as math. Then children would not read books, write essays etc. in school. They would instead learn lots and lots of grammar, exceptions to the grammar rules etc., and after several years still not have progressed far enough to apply what they've learned to simple conversations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Music taught in this way would degenerate to practicing playing simple notes on musical instruments over and over again, never to move toward playing an actual piece of music. That would be considered to be &quot;advanced university level music&quot;, and you need to first have mastered the basics good enough to qualify for that. There would be no better way to make children hate music class!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, how can we best remedy the problem with math teaching? I.m.o., the best way is to design a new curriculum that is based on using the tools children like to play with. Computers should be used a lot more. The best way for children to learn mathematics with pleasure is by teaching them computer programming. In trials it has been shown that primary school children have no problems learning to code in languages like C++.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of subjecting children to boring math problems, one can give them assignments where they have to write code to tackle a problem. They'll then be exposed to the rigorous mathematical logic at a far younger age, and yet they'll have a lot of fun. Most of the the feedback they get when they make a mistake will come quite promptly from their own computers, when the compiler complains about errors. The children working on a project will then both practice their logical reasoning skills and yet not get bored by having to do that, as their goal isn't to get to just any code that the compiler will accept, but to finish the project they are working on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, mathematics as we traditionally learn it does need to be presented differently so that it can be presented in this context. For example, instead of introducing calculus via smooth curves, integration as area under a curve etc., one can just as well introduce this topic via coarse graining. Unlike the concepts use in traditional calculus teaching, children encounter coarse graining all the time. Every time they look at a picture on their computer screens, they are looking at a coarse grained representation of the data that makes up the picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Limits are then about computing properties of data in a scaling limit, and calculus is about doing commutations directly with the quantities defined in the scaling limit. This way of introducing calculus is i.m.o. not just easier to master, it's also more consistent with what the continuum really is. It's not fundamental, it's an artifact of having taken a prior scaling limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is something that can be obvious to five-year-olds zooming into a smooth pictures and then seeing pixels. In contrast, we are indoctrinated by the wrong idea about a fundamental continuum, and we then need to unlearn this and do computations properly when Nature tells us we're making a mistake, &lt;a href=&quot;http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;see e.g. here on page 12&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1856" LastActivityDate="2020-02-18T15:58:08.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17932" PostTypeId="1" CreationDate="2020-02-18T16:34:43.147" Score="1" ViewCount="195" Body="&lt;p&gt;In trying to find lecture-length videos of epsilon-delta proofs, I've found that almost all of them just start with the definition and then work through the algebra to get the answer. In effect, it reduces the problems to &quot;do you have the &lt;strong&gt;pre-calculus algebra&lt;/strong&gt; to solve the question?&quot; and all the questions are basically the same template-like format but with different numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure if viewers would know &lt;em&gt;why&lt;/em&gt; the definition is the way it is or &lt;em&gt;why&lt;/em&gt; the proof is structured that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When teaching &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proofs, what are you strategies to make sure students truly understand the concepts (&lt;em&gt;beyond mere calculations&lt;/em&gt;)? Of course you can't completely escape computation and manipulation, but I don't want my students to be computers who can only do problems they've seen before; I want them to be free-thinkers who can expand beyond what I've shown them.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;My answer/suggestion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give a concrete example of what I'm talking about, in my own video at &lt;a href=&quot;https://www.youtube.com/watch?v=QkxUE6w8qZk&quot; rel=&quot;nofollow noreferrer&quot;&gt;25:25&lt;/a&gt;, I made a part that shows some creativity in picking the &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; relationship for any line of &lt;span class=&quot;math-container&quot;&gt;$mx+b$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before that section, I proved that you could have &lt;span class=&quot;math-container&quot;&gt;$\frac{\epsilon}{\left|m\right|}=\delta$&lt;/span&gt; but then you'd need to do another case for &lt;span class=&quot;math-container&quot;&gt;$m=0$&lt;/span&gt;. But if &lt;span class=&quot;math-container&quot;&gt;$\delta=\frac{\epsilon}{\left|m\right|+1}$&lt;/span&gt; then you wouldn't have to do 2 cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, something like that shows a little bit of creative thinking can cut down on computation and show that one recognizes that the &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; found in the stratch work is just the maximum value, and that smaller (positive) ones can work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some of the creative ways you make &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proofs more than just computation?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorUserId="8571" LastEditDate="2020-02-18T20:44:07.907" LastActivityDate="2020-02-18T20:47:47.993" Title="Making epsilon-delta proofs not just precalculus" Tags="|undergraduate-education|calculus|mathematical-analysis|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17933" PostTypeId="2" ParentId="6031" CreationDate="2020-02-18T18:44:54.780" Score="0" Body="&lt;p&gt;A very general method for &lt;em&gt;any&lt;/em&gt; polinomial is the rational root test: Any rational zero of &lt;span class=&quot;math-container&quot;&gt;$a x^2 + b x + c$&lt;/span&gt; must be of the form &lt;span class=&quot;math-container&quot;&gt;$p / q$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is a factor of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;  a factor of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; (don't forget signs!). This affords a factor &lt;span class=&quot;math-container&quot;&gt;$q x - p$&lt;/span&gt;, for a quadratic the other must be &lt;span class=&quot;math-container&quot;&gt;$(a / q) x - (c / p)$&lt;/span&gt;. As a bonus point, the (simple) proof of the test and the resulting factorization in this case exercise some non-trivial manipulations.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-02-18T18:44:54.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17934" PostTypeId="2" ParentId="17932" CreationDate="2020-02-18T20:42:09.723" Score="2" Body="&lt;p&gt;I think you should consider that for most calculus students, they will have a limited exposure to e-d and then move on to massive amounts of other calculations.  And that will serve the vast amount of physicists, chemists, engineers, geologists, etc. just fine.  Even from a TIME usage perspective, going deeper and more intuitive into e-d may be a waste of time.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth, I'm an experienced person in STEM (not math) and have never regretted not having a deeper feeling for e-d.  It didn't hold me back.  As far as I remember, my intuitive concept of it was that one gets smaller and smaller and another gets smaller and smaller and then there's an inequality or something.  And you can sort of grok how that would pin the beast in and not let it escape the corral.  But still a butt-load of algebra to slog through.  (Good enough for concept.  And then rock the algebra.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But really for the purpose of the rest of the course, it wouldn't even matter if you had blown off the e-d proof stuff.  I mean when you are off finding turning points of polynomials or volumes of rotation, who cares about e-d?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I would not &quot;dis&quot; the value of practicing algebra.  There will be other areas where tedious algebra is needed (series solution to diffyQs for instance).  Some facility with tracking a multi-step calculation down the pages is a good thing to develop.  Heck, solidifying algebra was like a side benefit of calculus itself (Richard Feynman makes the same point).  And the ability to do detailed calculations is a frequent need in chemistry (equilibrium, stoichiomentry) or engineering (hydraulic circuits, power cycles, electrical circuits) or physics (all over the place).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, it's sometimes useful to learn something mechanically and later develop the intuitions.  (Or maybe at least DEEPEN the initial intuitions, later.)  E.g. quantum mechanics.  Although this point does not completely apply since I don't think it is a useful allocation time for students to go deeper into e-d AT THIS STAGE in their courses.&lt;/p&gt;&#xA;" OwnerUserId="13542" LastEditorUserId="13542" LastEditDate="2020-02-18T20:47:47.993" LastActivityDate="2020-02-18T20:47:47.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17935" PostTypeId="2" ParentId="17923" CreationDate="2020-02-18T21:27:07.373" Score="2" Body="&lt;p&gt;Scoring too strictly might be counter-productive if the students are too young?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not sure if this is what you're looking for but in Karen Elizabeth Kebles's &lt;a href=&quot;https://search.proquest.com/docview/1787831395/fulltextPDF/E50D3C9138AD49FEPQ/&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; &quot;What's in an A? A Quantitative Study on the Grading Perceptions of Middle School and High School Math Teachers&quot; If you skip down to page 114 to see the results of &quot;What are middle school and high school math teachers’ perceptions of their grading practices in one mid-Atlantic state?&quot; the middle school teachers seem to be more critical about grades for teaching than high school teachers.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-02-18T21:27:07.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17936" PostTypeId="2" ParentId="17778" CreationDate="2020-02-19T00:33:45.837" Score="1" Body="&lt;p&gt;I wouldn't necessarily call them &quot;trap&quot; questions, but I wouldn't be surprised if a vocabulary quiz on middle/high school math could be difficult or low scoring.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask students to define or to give examples for &quot;degenerate triangle&quot;, &quot;extraneous solution&quot;, &quot;rational number&quot;, &quot;principle square root&quot;, &quot;inverse function&quot;, &quot;quotient&quot;, &quot;numerator&quot;, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even some true or false questions could pose difficulties, like &quot;if positive integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is not prime, then it is composite&quot; or &quot;prime numbers are all odd&quot;&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-02-19T00:33:45.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17937" PostTypeId="2" ParentId="17926" CreationDate="2020-02-19T04:02:57.847" Score="3" Body="&lt;p&gt;The way I explain it is that the log rule that you're using - the claim that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\log_ax - \log_ay = \log_a\frac{x}{y}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is &lt;em&gt;only&lt;/em&gt; correct if &lt;span class=&quot;math-container&quot;&gt;$\log_ax$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\log_ay$&lt;/span&gt; both exist. This is pretty clear if you prove the rule from the relevant exponent law - you have to start by writing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as powers of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, which can't be done if they're nonpositive. So, essentially, in the process of solving we made a mistake by using a rule that wasn't correct. Now, one way to fix it would be to be more careful: every time you use the rule, add a caveat of &quot;unless &lt;span class=&quot;math-container&quot;&gt;$x \leq 0$&lt;/span&gt;&quot; or what have you, and deal with that as a separate case. But that's needlessly complicated, so the way we usually approach it is to just pretend that the rule always works, and let the verification process tell us when it doesn't.&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2020-02-19T04:02:57.847" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="17938" PostTypeId="2" ParentId="17863" CreationDate="2020-02-19T08:07:09.450" Score="0" Body="&lt;p&gt;Like the other answers, I'd also suggest lots of computations for many triangles of increasing steepness. This way students could gradually figure out what the curves are like, what the domains and ranges are, and where it increases, decreases, etc. They could sketch approximations of the functions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, for example with &lt;span class=&quot;math-container&quot;&gt;$\tan(62)$&lt;/span&gt;, even if they don't know what exactly it is, they could at least tell if it is greater or less than similar values like &lt;span class=&quot;math-container&quot;&gt;$\tan(60)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\tan(65)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For degenerate triangles or triangles in other quadrants, I'd start off with a unit circle, note how ratios stay the same for similar triangles, and how &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; values are related to side lengths. This way they can work with &quot;&lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;&quot; side lengths as well as &quot;negative&quot; ones. (Relating it to the sketches, they'd be able to know in quadrants the ratios are positive or negative)&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-02-19T08:07:09.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17939" PostTypeId="2" ParentId="17863" CreationDate="2020-02-19T11:05:38.943" Score="0" Body="&lt;p&gt;You have to observe that &lt;span class=&quot;math-container&quot;&gt;$$ Opposite = \frac{Opposite}{Adjacent}*Adjacent = \frac{Opposite}{Hypothenuse}*\frac{Hypothenuse}{Adjacent}*Adjacent = \frac{\sin(\theta)}{\cos(\theta)} *Adjacent= \tan(\theta)*Adjacent$$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you start with an unsual math-trick and multiply by one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you can concentrate on &lt;span class=&quot;math-container&quot;&gt;$\frac{Opposite}{Adjacent} =\frac{\sin(\theta)}{\cos(\theta)}$&lt;/span&gt;. So you have to understand that when scaling an triangle, the angles and the ratio of the sides stays the same.&lt;/p&gt;&#xA;" OwnerUserId="13554" LastEditorUserId="13554" LastEditDate="2020-02-19T11:21:08.353" LastActivityDate="2020-02-19T11:21:08.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17940" PostTypeId="2" ParentId="12200" CreationDate="2020-02-19T19:09:10.430" Score="1" Body="&lt;p&gt;Defining a function as a set of pairs is a lot more accessible than it might seem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Think of a directory in a building, which lists people's names and the room number of each person's office. (Assume everyone's name is different, but people might share an office.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The directory spells out an assignment, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, of names to numbers. Name &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is presented next to room number &lt;span class=&quot;math-container&quot;&gt;$d(x)$&lt;/span&gt;. In this context, we're not talking about who the people are, or the physical layout of the directory, or whether it's even true. We're just talking about the raw, uninterpreted assignment of names to numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The statement &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt; is one way of writing an assignment of numbers to numbers. Another way would be to write an infinitely long directory. Looking up &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; in that directory would give you the same answer that a calculator would give you if you pressed &lt;code&gt;1 0 × 1 0 =&lt;/code&gt; on a calculator. In fact, a graph is a sort of directory where you can approximately &quot;look up&quot; an entry by choosing a horizontal coordinate and looking at the associated vertical coordinate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a complete directory existed for &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, you could find all its zeroes with a reverse-lookup. Alternatively, you could just solve &lt;span class=&quot;math-container&quot;&gt;$x^2=0$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; algebraically. But you can't do that with every function. Sometimes you can only approximate the zeroes numerically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that we're treating the assignment itself as a mathematical object, independently of how it's written down and independently of what its entries might refer to. Thinking about it as a bunch of entries is useful in many ways. It's the tool that calculus uses to study rates of change. The value of a thing changing with time is modeled mathematically as a set of entries that assign time-values to thing-values, and its rate of change at a point in time is defined by its relationship to the surrounding entries.&lt;/p&gt;&#xA;" OwnerUserId="13431" LastEditorUserId="13431" LastEditDate="2020-02-19T19:24:02.143" LastActivityDate="2020-02-19T19:24:02.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17941" PostTypeId="2" ParentId="12200" CreationDate="2020-02-19T21:02:28.707" Score="0" Body="&lt;p&gt;One thing I do when introducing functions is to draw a blob (set) on the left and a blob (set) on the right. Put some points in each with labels (at first arbitrary symbols, later numbers). Call the left the &quot;Domain&quot;, the right the &quot;Range&quot; (*) and say that a function tells us where elements in the domain go to in the range. I'll illustrate this by drawing an arrow from a point on the left to a point on the right. Then continue adding arrows from domain elements to range elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on the situation I might ask if it's okay to draw two arrows out of the same domain point (hoping for someone to say &quot;No, then I don't know which arrow to take.&quot;) or I just tell them that only one arrow is allowed to come out of each domain element. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I might ask if it's okay to draw two arrows into the same range point. Somewhere along the line, they realize that that's fine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, I explain that &lt;strong&gt;the function is the collection of arrows&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, depending on the situation, I might do an example where the function in &quot;squaring&quot;. I'll draw arrows &lt;span class=&quot;math-container&quot;&gt;$0 \to 0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$1 \to 1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2 \to 4$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-1 \to 1$&lt;/span&gt;, etc. maybe even &lt;span class=&quot;math-container&quot;&gt;$x \to x^2$&lt;/span&gt;. After doing a few arrows with the blobs, I point out that it makes more sense to have a (vertical) number line on left and another on the right since numbers have order instead of floating around in a blob. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then I'll redraw the diagram as arrows from spots on the left number line to spots on the right number line. At some point, I ask &quot;what about &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&quot; Then I point out that there's a lot of arrows and the drawing is getting messy. So I take the left (vertical) number line and turn it horizontal and overlap zeroes. Instead of drawing arrows from spots on the horizontal axes to spots on the vertical axes, I say that instead we just plot a point on the plane. Stitching all those infinite points together gives the graph. Now our condition that only one arrow can come out of a domain point becomes that a vertical line can only intersect the graph in one spot, etc. etc. etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(*) Yes, I know that there's a difference between Range and Co-domain. But I don't introduce this nuance in the introduction of functions and stick to calling both the Range.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2020-02-19T21:02:28.707" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17942" PostTypeId="2" ParentId="604" CreationDate="2020-02-20T15:26:54.767" Score="14" Body="&lt;p&gt;The reasons for not distinguishing &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; are historical.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;For roughly 250 years, from the first use of the word &amp;quot;function&amp;quot; in mathematics (by Leibniz ~1680) until ~1930, the official (and only) definition of the word was: &lt;em&gt;A quantity &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is called a &lt;strong&gt;function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/strong&gt;, if its value is uniquely determined as soon as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; assumes a value.&lt;/em&gt; Look into any book from that period. You might find small variations, but we can all agree that according to those definitions &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; are functions of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;It was normal to drop the &amp;quot;of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; part when it was clear from the context. So people would simply call &lt;span class=&quot;math-container&quot;&gt;$x^2,\cos x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; functions. This is familiar from everyday language, where we also drop the &amp;quot;of&amp;quot; 's. We might for instance speak of &amp;quot;the owner of the car&amp;quot; and later just say &amp;quot;the owner&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;The notation &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; for an arbitrary function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; was introduced by Johann Bernoulli some 40 years after Leibniz introduced the word &amp;quot;function&amp;quot;. Bernoulli decided to call &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; the &lt;em&gt;characteristic of the function&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, as did Euler and Lagrange after him. But this terminology didn't catch on, since people didn't feel the need to treat &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; as an object by itself before ~1890. So they never talked about &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; in isolation. They were satisfied with treating &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; as an object, and for that they had the name &amp;quot;function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;. (Actually Bernoulli, Euler and Lagrange wrote &lt;span class=&quot;math-container&quot;&gt;$f\,x$&lt;/span&gt;, without parenthesis. Parenthesis became part of the standard notation only around 1790. In a sense there was nothing revolutionary about Bernoulli writing &lt;span class=&quot;math-container&quot;&gt;$f\,x$&lt;/span&gt; for an arbitrary function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, since people were already writing &lt;span class=&quot;math-container&quot;&gt;$r\,x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$l\,x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s\,x$&lt;/span&gt; for the &lt;em&gt;root&lt;/em&gt;, the &lt;em&gt;logarithm&lt;/em&gt; and the &lt;em&gt;sine&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; long before.)&lt;/li&gt;&#xA;&lt;li&gt;Around 1840, Jacobi introduced a confusing notation in relation to partial derivatives, with the good intention of causing less confusion. You can read more about it &lt;a href=&quot;https://mathoverflow.net/a/278927/745&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://mathoverflow.net/questions/336309&quot;&gt;here&lt;/a&gt;. Probably inspired by this notation he writes &lt;span class=&quot;math-container&quot;&gt;$y(x)$&lt;/span&gt; for what should be &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; in the same article (using different letters). As far as I know, this is the first implicit but conscious use of the &lt;span class=&quot;math-container&quot;&gt;$y=y(x)$&lt;/span&gt; abuse. Soon after that (1850) one finds Cayley write &lt;span class=&quot;math-container&quot;&gt;$\Omega = \Omega(r,v,y)$&lt;/span&gt; explicitly as an abbreviation for &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$r,v,y$&lt;/span&gt;&amp;quot; and many others adopted this convention. Nevertheless, I haven't seen Euler and several other important mathematicians (before and after Jacobi) confuse &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;. For them that would have been like confusing &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt;. I'm also certain Jacobi understood the difference. He probably just didn't foresee the confusion his notation might generate.&lt;/li&gt;&#xA;&lt;li&gt;Around 1890 Dedekind, Cantor, Peano and Frege all isolated the &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; as an independent mathematical object and gave definitions. But each of them decided to give it a different name:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Dedekind called &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; a &lt;em&gt;mapping&lt;/em&gt; and kept the name function for &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Cantor called &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; an &lt;em&gt;allocation&lt;/em&gt; and kept the name function for &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Peano called &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; the &lt;em&gt;prefix sign for a function&lt;/em&gt; (in line with the original name &lt;em&gt;characteristic of a function&lt;/em&gt;) and kept the name function for &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Only Frege decided to call &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; a &lt;em&gt;function&lt;/em&gt; and claimed that variable quantities like &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; have no place in mathematics. (See his 1904 article &amp;quot;What is a function?&amp;quot;.) Frege was well aware of the distinction between &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, but he was also aware of the already existing abuse &lt;span class=&quot;math-container&quot;&gt;$y=y(x)$&lt;/span&gt;. Since he convinced himself that variable quantities like &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; denote nothing, he concluded that mathematicians must be referring to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; when they speak of functions. Be that as it may, the fact that among these four mathematicians only Frege decided to call &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; a function suggest it was a bad choice.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;Probably due to Freges influence among logicians, the name &amp;quot;function&amp;quot; for &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; was adopted by mainstream mathematics during the rise of set theory and logicism (1900-1940). At the same time the notion of &lt;em&gt;something being a function of something else&lt;/em&gt; disappeared from the official mathematical language. Today many people still speak of functions &amp;quot;of something&amp;quot;, but are probably unaware that they're not talking about modern functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt; We've been calling &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; a function for much longer that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and it's officially wrong only since ~1940, by some unfortunate choice of names and notations. It's hard to get rid of this heritage.&lt;/p&gt;&#xA;" OwnerUserId="590" LastEditorUserId="590" LastEditDate="2021-07-06T12:13:47.997" LastActivityDate="2021-07-06T12:13:47.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17944" PostTypeId="2" ParentId="17902" CreationDate="2020-02-21T03:28:36.623" Score="2" Body="&lt;p&gt;I'll hazard a somewhat physicsy answer. Consider two vectors &lt;span class=&quot;math-container&quot;&gt;$\vec{A}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec{B}$&lt;/span&gt;. Without loss of generality, choose coordinates where the positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis aligns with &lt;span class=&quot;math-container&quot;&gt;$\vec{A}$&lt;/span&gt;. Hence, &lt;span class=&quot;math-container&quot;&gt;$\vec{A} = \langle A, 0 \rangle$&lt;/span&gt;. Suppose &lt;span class=&quot;math-container&quot;&gt;$\vec{B}$&lt;/span&gt; makes counter-clockwise angle &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$\vec{A}$&lt;/span&gt; and denote &lt;span class=&quot;math-container&quot;&gt;$\vec{B} = \langle B_1, B_2 \rangle$&lt;/span&gt;. See the picture below:&lt;a href=&quot;https://i.stack.imgur.com/t1FCx.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/t1FCx.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently &lt;span class=&quot;math-container&quot;&gt;$B_2 = B \sin \theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B_1 = B \cos \theta$&lt;/span&gt;. Notice that &lt;span class=&quot;math-container&quot;&gt;$B = \sqrt{B_1^2+B_2^2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A = \sqrt{A_1^2+A_2^2}$&lt;/span&gt; where we have assumed &lt;span class=&quot;math-container&quot;&gt;$A_1 &amp;gt;0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A_2=0$&lt;/span&gt; hence &lt;span class=&quot;math-container&quot;&gt;$A = A_1$&lt;/span&gt;. In total,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ A_1B_1+A_2B_2 = AB\cos \theta. $$&lt;/span&gt;&#xA;Naturally, if you don't wish to begin with the rather greedy step of choosing coordinates to make the problem trivial then we'd have to fight through the change of coordinates in essence. Perhaps I will return to this and add that later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit (2-22-2020)&lt;/strong&gt; next, let us suppose &lt;span class=&quot;math-container&quot;&gt;$\vec{A}$&lt;/span&gt; is at angle &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; measured CCW from the positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis. See below:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/gicIt.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gicIt.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can use the usual adding-angles formulas for sine and cosine to expand the geometric expressions &lt;span class=&quot;math-container&quot;&gt;$A_1 = A\cos( \alpha + \theta )$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A_2 = A \sin( \alpha + \theta)$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ A_1 = A\cos( \alpha + \theta ) = A (\cos \alpha \cos \theta- \sin \alpha \sin \theta) $$&lt;/span&gt; &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ A_2 = A\sin( \alpha + \theta ) = A (\cos \alpha \sin \theta+ \sin \alpha \cos \theta) $$&lt;/span&gt; &#xA;Now we have all we need to investigate the geometric content of the algebraic expression &lt;span class=&quot;math-container&quot;&gt;$A_1B_1+A_2B_2$&lt;/span&gt;, let's see what happens:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;A_1B_1+A_2B_2 &amp;amp;= (A\cos \alpha ) B (\cos \alpha \cos \theta- \sin \alpha \sin \theta) \\ \notag&#xA;&amp;amp; \qquad + (A\sin \alpha) B (\cos \alpha \sin \theta+ \sin \alpha \cos \theta) \\ \notag&#xA;&amp;amp;= AB( \cos^2 \alpha \cos \theta -\cos \alpha \sin \theta + \cos \alpha \sin \theta + \sin^2 \alpha \cos \theta) \\ \notag&#xA;&amp;amp;= AB(\cos^2 \alpha+\sin^2 \alpha) \cos \theta \\ \notag&#xA;&amp;amp;= AB \cos \theta.&#xA;\end{align}&lt;/span&gt;&#xA;Naturally this more complicated (and unphysicsy) argument collapses to my initial physicsy argument when we set &lt;span class=&quot;math-container&quot;&gt;$\alpha = 0$&lt;/span&gt;. I think students would find the way &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; disappears quite rewarding if they could discover it for themselves.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2020-02-22T16:35:16.283" LastActivityDate="2020-02-22T16:35:16.283" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17945" PostTypeId="2" ParentId="17902" CreationDate="2020-02-21T05:13:48.310" Score="3" Body="&lt;p&gt;Suppose we take as the definition of the dot product that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\textbf{a}\cdot\textbf{b} = ab\cos\phi. \qquad (1)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's then fairly straightforward to show that the dot product is bilinear, i.e., that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(p\textbf{a}+q\textbf{b})\cdot\textbf{c}=p\textbf{a}\cdot\textbf{c}+q\textbf{b}\cdot\textbf{c}\qquad (2)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(and likewise for the right-hand factor). Properties (1) and (2) are both clearly independent of how we rotate or translate our coordinate system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From property (1), &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{x}}\cdot\hat{\textbf{x}}=1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{x}}\cdot\hat{\textbf{y}}=0$&lt;/span&gt;, and so on. And then from property (2), the coordinate expression for the dot product follows.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-02-21T05:13:48.310" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17946" PostTypeId="1" CreationDate="2020-02-22T16:39:28.767" Score="0" ViewCount="171" Body="&lt;p&gt;i hope i can get some help on how to get better at high school maths i find them very difficult compared to middle school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whats the big difference so i can work on it ?&lt;/p&gt;&#xA;" OwnerUserId="13573" LastActivityDate="2020-02-22T22:19:26.930" Title="I find high school math very hard compared to middle school?" Tags="|secondary-education|middle-school|" AnswerCount="2" CommentCount="0" ClosedDate="2020-02-22T23:13:34.793" ContentLicense="CC BY-SA 4.0" />
  <row Id="17947" PostTypeId="2" ParentId="17946" CreationDate="2020-02-22T17:15:04.390" Score="2" Body="&lt;p&gt;You're starting to move into more adult learning when you get into high school.  In elementary, they spend huge amounts of time for very slow learning and do lots of review.  In high school, it's more cut and dried...topic a, lesson 1, topic b, lesson 2, etc.  You HAVE TO KEEP UP or you'll get killed.  (Intermediate school is halfway in between.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can probably handle the high school math, but you will need to start working more.  Consider it like a sport.  You need to train.  And it is a tougher thing because you are growing up into a man.  High school football is a whole different feeling than peewees.    &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Do the homework every night.  (At least all assigned.  If possible, just work ALL in the section.  It's like running more laps...you will get better wind.)  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Preview the lesson that you will get the next day, by reading the textbook.  This will seem hard at first, but your body (oops, mind) will get trained to enduring the work.  When you read, pay SPECIAL attention to the example problems or derivations and work them in your notebook.  Math is like sports, more about doing than passive reading.  (languages, physics and chemistry are similar to math albeit not as extreme; history, English, and biology are more passively readable).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Go to your teacher every day after school and get help.  If you don't have a specific question, no problem, sit in a desk in front of her and read the book and work problems.  Be a little badger about getting help.  Just keep biting at it until you get what you need.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Do all three of these and report back here, weekly, with your progress.  I am sure that within three weeks of working like this, you will be kicking ass instead of getting kicked.&lt;/p&gt;&#xA;" OwnerUserId="13574" LastActivityDate="2020-02-22T17:15:04.390" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17948" PostTypeId="2" ParentId="17946" CreationDate="2020-02-22T22:13:40.717" Score="0" Body="&lt;p&gt;Is there a particular topic in high school math that you find hard?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my experience, I found that middle school mathematics was mostly computation and/or algebraic manipulation. For example, learning about exponents, solve simple linear equations, expand and factor polynomials, and the mathematical topics were compartmentalized without really connecting them. Everything was somewhat straightforward where you could almost type questions (as you see them) into a calculator and get the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, you'd learn how to factor &lt;span class=&quot;math-container&quot;&gt;$f(x)=y=x^2+6x+5=(x+1)(x+5)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;High school math, however, I found was more about generalization and/or connecting different topics in math. So, using our example, if we wanted to sketch the graph &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; we could &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Find the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis intercepts by doing the additional step of setting set &lt;span class=&quot;math-container&quot;&gt;$f(x)=0$&lt;/span&gt; and realizing that the product of those two factors would need to be zero (&lt;span class=&quot;math-container&quot;&gt;$x+1=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x+5=0$&lt;/span&gt;) and then you'd get &lt;span class=&quot;math-container&quot;&gt;$x=-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-5$&lt;/span&gt;. &lt;/li&gt;&#xA;&lt;li&gt;Learning that the line of symmetry is in between the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis intercepts. So &lt;span class=&quot;math-container&quot;&gt;$x=-3$&lt;/span&gt;. And also learning that the vertex lies on the line of symmetry and so the max/min would be when &lt;span class=&quot;math-container&quot;&gt;$x=-3$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$f(-3)=-4$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Find the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept, by setting &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; and getting &lt;span class=&quot;math-container&quot;&gt;$y=5$&lt;/span&gt; &lt;/li&gt;&#xA;&lt;li&gt;Connect the intercepts &lt;span class=&quot;math-container&quot;&gt;$(-1,0), (-5,0), (0,5)$&lt;/span&gt; and the min &lt;span class=&quot;math-container&quot;&gt;$(-3,-4)$&lt;/span&gt; with a smooth curve to sketch the quadratic.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Later, you could generalize further and have the quadratic &lt;span class=&quot;math-container&quot;&gt;$ax^2+bx+c, a \neq 0$&lt;/span&gt; but you could still figure out the general shape and specific points based on &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, questions get longer with multiple parts and they aren't as straightforward computation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The three specific things that always helped me and what I always tell my students:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Vocabulary&lt;/strong&gt; is going to be very important since terms will give you a lot of information that will help you understand what a question is asking. It would be good to make a vocab list &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Geometry&lt;/strong&gt; or visual interpretation will make the algebraic manipulation more &quot;concrete&quot;. It starts with lines and slopes and intercepts, but then continues on with quadratics, lines of symmetry, vertexes, and so on.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Step-by-step&lt;/strong&gt; from the basics so that you're not calculating without thinking, but that you're thinking about &lt;em&gt;why&lt;/em&gt; you're doing each step. Break down one hard questions into lots of easy questions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-02-22T22:19:26.930" LastActivityDate="2020-02-22T22:19:26.930" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17949" PostTypeId="1" CreationDate="2020-02-23T03:05:52.693" Score="11" ViewCount="315" Body="&lt;p&gt;While teaching Function Transformations, I found the verbal descriptions of stretch and squeeze really confusing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So for &lt;span class=&quot;math-container&quot;&gt;$y = f(x)$&lt;/span&gt;,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = 2f(x)$&lt;/span&gt; is said to stretch &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; vertically by a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = \frac{1}{2}f(x)$&lt;/span&gt; is said to squeeze &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; vertically by a factor of &lt;kbd&gt;?&lt;/kbd&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I found in textbooks and online both &lt;kbd&gt;&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;&lt;/kbd&gt; and &lt;kbd&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;&lt;/kbd&gt; and can not decide which one is right.  The former makes more sense to me since &quot;squeeze&quot; has already indicated the opposite direction of stretch marked by the exponent &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$2^{-1} = \frac{1}{2}$&lt;/span&gt;.  For me, this is also compatible with translation where&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = f(x) + 2$&lt;/span&gt; is said to shift &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; up by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; units;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = f(x) + (-2)$&lt;/span&gt; is said to shift &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; down by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; units.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note how the word &quot;down&quot; has indicated the opposite direction of &quot;up&quot; marked by the negative sign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only way I can make sense of the use of &lt;kbd&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;&lt;/kbd&gt; is to think the phrase &quot;a factor of &lt;kbd&gt;?&lt;/kbd&gt;&quot; refers to the factor in front of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, hence &quot;by a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;&quot; for stretch and &quot;by a factor of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;&quot; for squeeze.  But then we should avoid words like &quot;stretch&quot; and &quot;squeeze&quot; that already indicate directions (away from or toward the axis), shouldn't we?  I found the word &quot;dilate&quot; is used in some textbooks and online articles.  But &lt;a href=&quot;https://www.mathwords.com/c/compression.htm&quot; rel=&quot;noreferrer&quot;&gt;Mathwords&lt;/a&gt; says it is wrong and English has no words to cover both stretch and squeeze.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or should I replace &quot;by a factor of &lt;kbd&gt;?&lt;/kbd&gt;&quot; with &quot;by a multiple of &lt;kbd&gt;?&lt;/kbd&gt;&quot; or &quot;&lt;kbd&gt;?&lt;/kbd&gt; times&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what description do you use and why?&lt;/p&gt;&#xA;" OwnerUserId="7564" LastEditorUserId="7564" LastEditDate="2020-02-24T12:23:15.837" LastActivityDate="2020-02-25T10:07:01.123" Title="Confusing verbal descriptions of function transformations" Tags="|terminology|functions|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17950" PostTypeId="2" ParentId="17949" CreationDate="2020-02-23T06:49:57.253" Score="2" Body="&lt;p&gt;I've found terminology differs as well, but this is how I think of the phrase &quot;by a factor of&quot; for &lt;span class=&quot;math-container&quot;&gt;$af(x)$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;|a|&amp;lt;1$&lt;/span&gt; then it is a vertical &quot;squeeze&quot; &lt;em&gt;to get a factor of&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; between the new and old &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; values&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$|a|&amp;gt;1$&lt;/span&gt; then it is a vertical &quot;stretch&quot; &lt;em&gt;to get a factor of&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And then a point &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; becomes &lt;span class=&quot;math-container&quot;&gt;$(x,ay)$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is the new scale or ratio, vertically-speaking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whenever I teach functions, I honestly just explain that the terminology differs and can be confusing, and that students should use the terms that fit the book we're using or the test they're preparing for.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-02-23T07:02:06.627" LastActivityDate="2020-02-23T07:02:06.627" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17951" PostTypeId="1" CreationDate="2020-02-23T14:38:53.553" Score="3" ViewCount="249" Body="&lt;p&gt;In this short question, I would like to ask whether it is possibly good to teach quantifier before logical connectives in a logic introduction lecture?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know there is a relationship between them but my question is based on my observation that students usually already get the concept of existential quantifier. Before introducing them to more complex forms of statement, maybe they can understand basic form of quantifier statements first (of course, after introducing open and closed sentence). This is just an idea. Let me know if anyone has done this because I have never seen one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you very much!&lt;/p&gt;&#xA;" OwnerUserId="13576" LastActivityDate="2020-11-22T16:07:30.770" Title="Teaching Quantifiers Before Logical Connectives" Tags="|undergraduate-education|curriculum|logic|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17952" PostTypeId="1" AcceptedAnswerId="18155" CreationDate="2020-02-24T00:00:44.377" Score="23" ViewCount="1741" Body="&lt;p&gt;I've encountered the following misunderstanding.&#xA;I pose a question (to undergraduates in the U.S.), for example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; be a polygon of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices.&#xA;Is it true that every triangulation of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;&#xA;has the same number of triangles?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This question depends on what constitutes a &amp;quot;triangulation,&amp;quot; but&#xA;assume the student know that. The answer is &lt;em&gt;Yes&lt;/em&gt;:&#xA;every triangulation of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; consists of &lt;span class=&quot;math-container&quot;&gt;$n-2$&lt;/span&gt; triangles.&lt;/p&gt;&#xA;&lt;p&gt;Here is the problem I encounter. The students apparently&#xA;don't understand that &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; be a polygon&amp;quot; means,&#xA;let your mind run over all possible polygons, so &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is&#xA;an &amp;quot;arbitrary&amp;quot; polygon, in that it can be anything that&#xA;fits the definition of a polygon (which the students also know). They wonder, well, maybe &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is a convex polygon,&#xA;and should I answer specific to convex polygons?&lt;/p&gt;&#xA;&lt;p&gt;This example doesn't quite illustrate the problem because&#xA;the answer is always &lt;em&gt;Yes&lt;/em&gt;. But when the answer is sometimes &lt;em&gt;Yes&lt;/em&gt;, sometimes &lt;em&gt;No&lt;/em&gt;, they seem to get confused&#xA;over the quantifier. I think it may come down to the&#xA;meaning of the phrase: &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; be a &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;.&amp;quot; Let &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; be&#xA;a point in the plane &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt;—meaning&#xA;&lt;em&gt;any&lt;/em&gt; point in the plane, an &amp;quot;arbitrary&amp;quot; point in the plane. Let &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; be a polygon, meaning &lt;em&gt;any&lt;/em&gt; polygon.&#xA;Let &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; be a triangulation of a set of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; points.&#xA;Does every triangulation of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; points have the same number of triangles? (&lt;em&gt;No&lt;/em&gt;, not always.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Have you encountered this confusion in your teaching? If so, how do you circumvent it?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="5563" LastEditDate="2023-05-19T23:20:10.923" LastActivityDate="2023-09-15T09:52:20.433" Title="Can we avoid confusion over using &quot;let&quot; as a quantifier?" Tags="|undergraduate-education|mathematical-pedagogy|logic|vocabulary|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="17953" PostTypeId="1" CreationDate="2020-02-24T14:42:00.300" Score="8" ViewCount="509" Body="&lt;p&gt;One of my high school students who has ZERO knowledge on complex numbers and the modulus function has showed me the following algebra:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(16)^{\frac{1}{2}}=(16)^{\frac{2}{4}}=((16)^2)^{\frac{1}{4}}=((-16)^2)^{\frac{1}{4}}=(-16)^{\frac{2}{4}}=(-16)^{\frac{1}{2}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Hence &lt;span class=&quot;math-container&quot;&gt;$$16=-16 \qquad\text{and so}\qquad 1=-1.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Should we impose that &lt;span class=&quot;math-container&quot;&gt;$$(a^m)^n=a^{mn}$$&lt;/span&gt; only when &lt;span class=&quot;math-container&quot;&gt;$a \gt 0$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="5666" LastEditorUserId="8571" LastEditDate="2020-03-10T17:39:01.120" LastActivityDate="2020-03-10T18:06:38.693" Title="How should I convince a student who proved $1=-1$" Tags="|secondary-education|algebra|powers|exponent|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17954" PostTypeId="2" ParentId="17953" CreationDate="2020-02-24T15:35:22.037" Score="7" Body="&lt;p&gt;Obviously the correct mathematical answer is to show how the exponent rules actually work, and when they do not work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So please don't accept this answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, the educational answer is to see that student is using the fact that &lt;span class=&quot;math-container&quot;&gt;$1^2 = (-1)^2$&lt;/span&gt; in the critical middle step. Show them the corresponding fact for cubes, because it's crazy, and might expand their mathematical playground:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 1^3 = \bigg( \frac{-1 + i\sqrt{3}}{2} \bigg)^3 $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See what they can &quot;prove&quot; next. your student has an imagination; show them the rules and when they work -- but also keep that spark alive by showing them neat things.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2020-02-24T15:35:22.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17955" PostTypeId="1" CreationDate="2020-02-24T15:46:05.810" Score="4" ViewCount="438" Body="&lt;p&gt;Over the past years teaching freshmen calculus I've repeatedly seen students make the following type of error:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose they have to express some quantity &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, when the relation between &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; has been given by other means (like a graph, a table of values, verbally etc.). Suppose a correct answer is&#xA;&lt;span class=&quot;math-container&quot;&gt;$y = x^2+1$&lt;/span&gt; (I wouldn't mind if they add an &lt;span class=&quot;math-container&quot;&gt;$=f(x)$&lt;/span&gt; in between.)&#xA; A significant amount of students (I estimate 5%) will write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;y=f(x^2+1) % \quad \text{ or  simply } \quad f(x^2+1)&#xA;$$&lt;/span&gt;&#xA;(The letter &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; not having been used in the problem statement previously.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are engineering students in Switzerland and they do this in the first week of the first semester, but also towards the end of the first semester after I've addressed the issue in class. From school they should in principle be familiar with a correct use of function notation &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't understand where this comes from and how to fix it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is this error well-know? Has it been studied?&lt;/li&gt;&#xA;&lt;li&gt;Are there explanations for it and suggested ways of addressing it?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="590" LastActivityDate="2020-03-02T15:00:26.993" Title="Students writing $f(x^2+1)$ when they probably mean $f(x)=x^2+1$" Tags="|students-mistakes|notation|functions|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17956" PostTypeId="2" ParentId="17706" CreationDate="2020-02-24T17:16:10.287" Score="4" Body="&lt;p&gt;I'll suggest that the best verbalization is the last one in the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, which is equal to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here's why I think so: It's the closest to a literal symbol-by-symbol reading of the chained equation, while still being a grammatically-correct English sentence. Note that this matches the top-voted (but not accepted) answer to the &lt;a href=&quot;https://english.stackexchange.com/questions/525126/how-to-say-abc&quot;&gt;analogous closed question on SE English Language &amp;amp; Usage&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Contrast this with a purely symbolic reading, like &quot;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&quot;. Unfortunately, this doesn't qualify as valid English, being an example of a &lt;a href=&quot;https://academicguides.waldenu.edu/writingcenter/grammar/runonsentences&quot; rel=&quot;nofollow noreferrer&quot;&gt;run-on sentence&lt;/a&gt;. There are two separate valid statements here (&quot;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;&quot; and &quot;&lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&quot;), but in English we need a conjunction to join them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, recall that we're looking for a direct reading, not an interpretation. In this regard, &quot;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; are the same number&quot; is an interpretation, not a literal verbalization; for example, the verb clause (&quot;are the same&quot;) only appears once, whereas the analogous relation (&quot;&lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;&quot;) appears twice, it's in a different order, etc. Consider also that such an interpretation/summary would not extend well to chained relations using a variety of symbols, such as &lt;span class=&quot;math-container&quot;&gt;$a = b &amp;lt; c = d \le e$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The root of the problem is that algebraic chained relations have a structure that simply isn't permitted (as a run-on sentence) in English. Personally I choose to verbalize something that is proper English, but as close to the written symbols as possible, by inserting the single word &quot;which&quot; -- thereby turning the relations after the first into nonrestrictive relative clauses (&lt;a href=&quot;https://www.merriam-webster.com/dictionary/which&quot; rel=&quot;nofollow noreferrer&quot;&gt;see definition three here&lt;/a&gt;). &lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-02-24T22:42:48.873" LastActivityDate="2020-02-24T22:42:48.873" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17957" PostTypeId="2" ParentId="17955" CreationDate="2020-02-24T22:16:35.527" Score="1" Body="&lt;p&gt;If I continually saw this mistake then I'd try to get the idea of &lt;span class=&quot;math-container&quot;&gt;$y=f(x)=x^2+1$&lt;/span&gt; more solidified: &quot;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and that function can be represented by an equation.&quot; I'd probably try to incorporate functions and transformations into the first week of homework so that students know the difference between &lt;span class=&quot;math-container&quot;&gt;$f(2x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)=2x$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that I might try is briefly mention the idea of multivariable functions to clarify that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; takes certain variables, say, &lt;span class=&quot;math-container&quot;&gt;$x, y$&lt;/span&gt; and this means it is &lt;span class=&quot;math-container&quot;&gt;$f(x,y)$&lt;/span&gt;. It not that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; takes the equation involving &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder if issue also occurs when you have polar functions? Would they consider &lt;span class=&quot;math-container&quot;&gt;$r=r(\theta)=2cos(\theta)$&lt;/span&gt; to be something like &lt;span class=&quot;math-container&quot;&gt;$r(2cos(\theta))$&lt;/span&gt;? I'd also wonder what they'd think parametric equations would be like. How would they consider &lt;span class=&quot;math-container&quot;&gt;$x=x(t)=cos(t), y=y(t)=sin(t)$&lt;/span&gt;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I'd bring up polar and parametric equations is to see if they understand function notation for that. If they do, then maybe that knowledge could be applied to understanding &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-02-24T22:16:35.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17958" PostTypeId="2" ParentId="17953" CreationDate="2020-02-25T02:17:48.870" Score="6" Body="&lt;p&gt;When I asked the &lt;a href=&quot;https://math.stackexchange.com/questions/1628759/what-are-the-laws-of-rational-exponents&quot;&gt;What are the Laws of Rational Exponents?&lt;/a&gt; question on SE Mathematics, I was largely thinking about this context; teaching at the level of high school or early (remedial) college math. While it wasn't the top-voted, &lt;a href=&quot;https://math.stackexchange.com/q/1639018&quot;&gt;my answer there&lt;/a&gt; represents my best thinking about the status of this issue in classes at that level. As I wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Regarding the example in the question, most everyone agrees that&#xA;  &lt;span class=&quot;math-container&quot;&gt;$(-1)^{2 \cdot \frac{1}{2}} \ne ((-1)^2)^\frac{1}{2}$&lt;/span&gt;, if both sides&#xA;  are simplified in the standard order of operations; and this&#xA;  highlights the fact that the identity &lt;span class=&quot;math-container&quot;&gt;$(a^r)^s$&lt;/span&gt; = &lt;span class=&quot;math-container&quot;&gt;$a^{rs}$&lt;/span&gt; is not true&#xA;  unrestrictedly. Exactly what restrictions need to be honored depend on&#xA;  the definitions in use in a particular textbook.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Just to expand on the last line there; part of the conversation around these questions usually dredges up the fact that there is a standard definition for fractional exponents in real-numbers, and another one in complex-numbers, and that these two definitions actually disagree with each other (e.g.: the value of &lt;span class=&quot;math-container&quot;&gt;$(-8)^{1/3}$&lt;/span&gt;, on which numerous academic articles have been written). As a result, real-number theorists and complex-numbers theorists have a tendency to start arguing with each other on these questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: I assume you are working from some standard textbook. That textbook must have &lt;em&gt;some&lt;/em&gt; restrictions around the &lt;span class=&quot;math-container&quot;&gt;$(a^m)^n = a^{mn}$&lt;/span&gt; identity (although they vary between books and contexts). I would recommend you read your book very carefully, note the restriction used there, and apply it conscientiously in your classroom practice. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-02-25T02:17:48.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17959" PostTypeId="2" ParentId="17949" CreationDate="2020-02-25T10:07:01.123" Score="1" Body="&lt;p&gt;I tend to be very strict on how I define things in this area - I like to keep things formal in preparation for University study. I'll explain everything I use and why I believe it to be effective - and use your examples. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$ y = f(x) $&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$ y = 2f(x) $&lt;/span&gt; is said to be a vertical stretch of scale factor &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; parallel to the y-axis.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$ y = \frac{1}{2} f(x) $&lt;/span&gt; is said to be a vertical stretch of scale factor &lt;span class=&quot;math-container&quot;&gt;$ \frac{1}{2} $&lt;/span&gt; parallel to the y-axis. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I always clarify the type of stretch as when listing stretch as opposed to later - I find it helps clarify for examiners when marking. I always use scale factor instead of factor as I find factor can be ambiguous, a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; could imply multiply or divide by this.  I clarify that it is parallel to the y-axis, not &quot;in&quot; the y-axis. Finally, I define both these ideas of &quot;squeezes&quot; and &quot;compression&quot; as stretches and make it explicitly clear to students that if the scale factor of the stretch &lt;span class=&quot;math-container&quot;&gt;$a $&lt;/span&gt; is such that &lt;span class=&quot;math-container&quot;&gt;$ 0&amp;lt; |a| &amp;lt; 1 $&lt;/span&gt; then it will have this &quot;shrinking&quot; affect. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to translations I often repeat this process of emphasising the type of translation when stating it. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$ y = f(x) + 2 $&lt;/span&gt; is said to be a vertical translation of 2 units upwards. &lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$ y = f(x) + (-2) $&lt;/span&gt; is said to be a vertical translation of 2 units downwards. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I often find mixing the idea of translation with &quot;shift&quot; is confusing, and stick to one terminology and use it strictly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These definitions are now what I use teaching the Canadian PEI curriculum as in the UK we don't say upwards/downwards/left/right when describing translations we would describe the following as: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$ y = f(x+4) + 3 $&lt;/span&gt; as being a translation being described by the vector &lt;span class=&quot;math-container&quot;&gt;$ \begin{pmatrix} -4 \\ 3 \end{pmatrix} $&lt;/span&gt;. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This gets rid of confusing terminology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use these because they provide a similar structure for all transformations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, looking at reflections, my students can look at these as &quot;a type of&quot; stretch and it helps when it comes to combinations of transformations too. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps. &lt;/p&gt;&#xA;" OwnerUserId="13583" LastActivityDate="2020-02-25T10:07:01.123" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17960" PostTypeId="1" AcceptedAnswerId="17961" CreationDate="2020-02-25T12:49:14.773" Score="6" ViewCount="2329" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2cU0F.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2cU0F.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you draw the altitude to the right triangle as shown, it is easily seen that &lt;span class=&quot;math-container&quot;&gt;$$\triangle KLM\sim\triangle KNL\sim\triangle LNM.$$&lt;/span&gt;  This in turn leads to several interesting proportional relations like &lt;span class=&quot;math-container&quot;&gt;$$\frac{KN}{KL}=\frac{KL}{KM} \qquad\text{and}\qquad \frac{KN}{LN}=\frac{LN}{NM}.$$&lt;/span&gt;  These turn out to be crucial to what is arguably the canonical geometric proof of the Pythagorean theorem.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YxaDlm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YxaDlm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Quick version: since &lt;span class=&quot;math-container&quot;&gt;$AC^2=AD\cdot AB$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$BC^2=BD\cdot AB$&lt;/span&gt;, the two red regions have the same area and the two green regions have the same area.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm looking for strategies to help my students master this concept.  Is there a special name for this key geometric observation that would help my search, like how Thales' theorem was important enough to get its own name?  My personal experience, even as a gifted student, is that it can be hard to see the similarity when you have to reflect one of the triangles to visualize the dilation compared to a similarity diagram with a triangle and one of its midsegments, for instance.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-02-26T18:08:10.053" Title="What are strategies for teaching that the altitude of a right triangle creates two similar triangles?" Tags="|secondary-education|geometry|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17961" PostTypeId="2" ParentId="17960" CreationDate="2020-02-25T13:09:11.883" Score="12" Body="&lt;p&gt;I've always seen it referred to as the &quot;Right Triangle Altitude Theorem&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may take a bit of practice, but I start by declaring &lt;span class=&quot;math-container&quot;&gt;$\angle M$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\angle K$&lt;/span&gt; complementary. Then I draw a mark to indicate &lt;span class=&quot;math-container&quot;&gt;$\angle NLM$&lt;/span&gt; and show that it is also complementary to &lt;span class=&quot;math-container&quot;&gt;$\angle M$&lt;/span&gt;. Thus &lt;span class=&quot;math-container&quot;&gt;$\angle NLM$&lt;/span&gt; is congruent to &lt;span class=&quot;math-container&quot;&gt;$\angle K$&lt;/span&gt; and so on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do this visually by getting 2 large pieces of white cardboard, and draw the triangle as you show, but then have a second layer taped on top, same triangle, but cut along line LN. So I can remove the two smaller triangles and then orient all 3 to the visually pleasing 'right angle on bottom view'. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-02-25T13:31:17.140" LastActivityDate="2020-02-25T13:31:17.140" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17962" PostTypeId="2" ParentId="17953" CreationDate="2020-02-25T13:43:51.233" Score="9" Body="&lt;p&gt;&lt;em&gt;Should we impose that &lt;span class=&quot;math-container&quot;&gt;$(a^m)^n=a^{mn}$&lt;/span&gt; only when &lt;span class=&quot;math-container&quot;&gt;$a \gt 0$&lt;/span&gt;?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe you should tell your student that he/she have discovered by himself/herself the proof that the rule &lt;span class=&quot;math-container&quot;&gt;$(a^m)^n=a^{mn}$&lt;/span&gt; cannot be true with &lt;strong&gt;negative bases and rational exponents&lt;/strong&gt;, and this is a great achievement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try to explain that he/she proved that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If the said rule is true for negative bases and rational exponents, then &lt;span class=&quot;math-container&quot;&gt;$1=-1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As we know that &lt;span class=&quot;math-container&quot;&gt;$1\neq -1$&lt;/span&gt;, the rule cannot be true. This is why the restriction &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;0$&lt;/span&gt; appears in the books and why you will impose it.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;Edit&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Maybe I'm wrong but, in view of the title's question (&quot;How should I convince a student...&quot;), I suppose that the question is not about the rule &lt;span class=&quot;math-container&quot;&gt;$(a^m)^n=a^{mn}$&lt;/span&gt; itself (which, I believe, the OP knows), but on how to approach it with the student.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The essence of my suggestion is: tell to the student what he/she did right (a mathematical discovery) instead of focusing on the mistake (wrong application of a rule that, probably, the student didn't know yet).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Of course, the bold text above is not as general as possible. But this is only the first step (see my comments below). Afater that, the professor could guide the student to further investigations in order to discover other forms of the rule.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13585" LastEditorUserId="13585" LastEditDate="2020-03-08T12:53:50.643" LastActivityDate="2020-03-08T12:53:50.643" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="17963" PostTypeId="2" ParentId="17960" CreationDate="2020-02-25T14:20:50.527" Score="7" Body="&lt;p&gt;You could present the concept that, ignoring scale, a right triangle can be defined entirely by a single number: its smallest interior angle. Call this the triangle's &quot;describing number.&quot; If you want to dramatize this, you could specify some angle like 30 degrees and ask students to cut triangles out of paper that have this describing number. Comparing all the triangles, it will be obvious that they're all similar, but of different sizes. Some may have to be flipped over.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the proof you're presenting, where you dissect a right triangle into two smaller right triangles, you could ask where those triangles get their describing numbers. Clearly they can only depend on the parent triangle's describing number. This is not by itself a proof that their describing numbers are equal to that of the parent, but when you then prove that they are, I don't think it should be surprising at all.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-02-25T14:20:50.527" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17964" PostTypeId="1" CreationDate="2020-02-25T15:03:49.083" Score="9" ViewCount="325" Body="&lt;p&gt;I’m currently interested in textbooks, especially the ones in math and physics that are used at the high school, undergraduate and graduate levels and, given the experience of the people on this website in using and teaching from these textbooks I’d appreciate your opinion on the following questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) How do you choose which textbooks to use for your courses? (Which criteria do you employ? E.g. appropriate selection of exercises, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) How have textbooks in your discipline(s) changed over the years (e.g. legibility, use of figures/examples, number of exercises given, how challenging they are, use of online resources,…)? (if they have changed at all) and do you find these changes beneficial or detrimental?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Do (and how) your criteria for choosing a textbook change when choosing an undergraduate or graduate textbook?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I welcome all of your comments regarding these and other questions related that you may deem relevant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you all for your time.&lt;/p&gt;&#xA;" OwnerUserId="13584" LastActivityDate="2020-02-29T22:23:30.277" Title="Evaluating textbooks in math and physics" Tags="|undergraduate-education|secondary-education|textbooks|graduate-education|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17966" PostTypeId="2" ParentId="41" CreationDate="2020-02-25T19:40:20.327" Score="2" Body="&lt;p&gt;Those of you who have been fortunate to see &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/J%C3%A1nos_Pach&quot; rel=&quot;nofollow noreferrer&quot;&gt;János Pach&lt;/a&gt;&#xA;give a chalk-talk (or its electronic&#xA;equivalent) will appreciate that his beautiful handwriting has been converted into a&#xA;downloadable font by &lt;a href=&quot;http://luc.devroye.org/fonts-23975.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Luc Devroye&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/LFuAE.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LFuAE.png&quot; alt=&quot;JanosFont&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-02-25T19:40:20.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17967" PostTypeId="2" ParentId="17964" CreationDate="2020-02-26T00:17:44.557" Score="7" Body="&lt;p&gt;1) Primarily I choose textbooks based on the needs of the class (and the students's backgrounds and goals) and what it is preparing for.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For high school, if it's for AP tests or IB tests at the end of the year, I'll very likely try to find textbooks (or materials) that explicitly state they help prepare students for that. The best books contain questions that are similar to the actual tests, or at least cover the topics necessary. Not to say I'm teaching to the test, but, for example, I need students to be good with AP-style multiple choice since there is a large multiple choice section on the test. &lt;/li&gt;&#xA;&lt;li&gt;For undergraduate, a deciding factor would depend on if it's math for math majors or non-math majors. Book for math majors would need to have lots of definitions, proofs, and questions that can't be solved simply by calculation, and would need to be discussed during office hours. Books for non-math majors, I'd probably look for books that worked-out solutions since they'd probably focus quite a lot on computation and seeing the steps will help them self-study if they get mistakes.&lt;/li&gt;&#xA;&lt;li&gt;For graduate, I'd probably stick to classic texts that have been used for decades for the core content and suggested readings but I'd honestly expect students at that level to go to the library and look up additional content on their own.&lt;/li&gt;&#xA;&lt;li&gt;Personally, I find graduate level texts easier to navigate since there's comparably less of them. For Real Analysis, for example, it seems that 95%+ of instructors in North America use the same 10 books by Rudin, Royden, Folland, Stein + Shakarchi, Kolmogorov + Fomin, Wheeden + Zygmund, Leib + Loss, etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;2) I've only been teaching for just over 10 years but in my times I've found that books have become more colourful and more full of visuals. Meta-book changes are mostly technological in the sense of online resources that have links to videos or applets that demonstrate concepts, or questions that need a computer or calculator to be solved. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In terms of difficulty, I feel that many newer books are getting easier but that might be due to how many are for introductory courses and are helping bridging the gap between modern students and classic texts. For example, if you had to use Rudin's Principles of Mathematics in 3rd year after only using, say, Stewart's Calculus in 1st and 2nd years, then that would be extremely steep learning curve, and you'd probably need to have an additional course (or book) about mathematical proof. But, now there are books like Abbot's Understanding Analysis which provide modern students more of a stepping stone to classic texts. Eventually, if they get higher up, students will eventually encounter the classic difficult texts, but it probably won't be as much of a &quot;baptism by fire&quot; as it could have been.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;3) Generally, I like books that either have the first chapter be &quot;As you &lt;em&gt;should&lt;/em&gt; know...&quot; for non-math major courses and/or the last chapter be &quot;As you &lt;em&gt;will&lt;/em&gt; know...&quot; for math major courses&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For non-math majors, with introductory calculus books, my favourites contain pre-calculus and high school algebra at the front. It eases them into the course and it's allows them to know what is required (or assumed) knowledge. It also probably shows that it will be a computation-heavy course which don't need too much proof.&lt;/li&gt;&#xA;&lt;li&gt;For math majors, with introductory real analysis books, my favourites contain a chapter on measure theory at the end. It obviously won't be on the course outline, but I do like how I can teach with an end in mind and say, &quot;when we're nearly done this book, this is what your next course will be about.&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-02-26T00:17:44.557" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17968" PostTypeId="2" ParentId="17964" CreationDate="2020-02-26T05:04:35.673" Score="4" Body="&lt;p&gt;I teach at a community college, so all of what I have to say applies to the first two years of undergraduate study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been trying for years to get my department to consider OER (Open Educational Resources) aka books that are Creative Commons instead of copyright. Our students are usually on very tight budgets, and cheaper books means less hours at a low-paying job and more study time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has finally happened, and it's a mixed blessing. They picked the OpenStax books for Calculus, and for Algebra. They aren't very good (imo), but free is a great start. The draw was that they have online homework available. (I don't use it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found that it's nearly impossible to know how good a book will be before actually using it for a semester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am using &lt;a href=&quot;https://faculty.uml.edu/klevasseur/ads2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this (free) book&lt;/a&gt; for Discrete Math. It doesn't have enough exercises, but there's another free book that I can supplement with (&lt;a href=&quot;http://discrete.openmathbooks.org/dmoi3.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). I might change over the next time I teach this course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are interested in free textbooks, AIM (American Institute of Mathematics) has an &lt;a href=&quot;https://aimath.org/textbooks/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Open Textbook Initiative&lt;/a&gt;, with books they've approved. I often start there.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-02-26T05:04:35.673" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17969" PostTypeId="2" ParentId="17960" CreationDate="2020-02-26T10:09:15.863" Score="4" Body="&lt;p&gt;If you have established that the sum of the interior angles of a triangle is always 180°, you're basically done. Each of the small triangles shares one angle with the original one; the second angle is a right angle. So the third one needs to be the same as the &quot;missing&quot; angle. All three angles being the same, the triangles have to be similar.&lt;/p&gt;&#xA;" OwnerUserId="3590" LastActivityDate="2020-02-26T10:09:15.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17971" PostTypeId="2" ParentId="17960" CreationDate="2020-02-26T12:37:52.743" Score="2" Body="&lt;p&gt;The similitude is stated in &lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/java/elements/bookVI/propVI8.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid's book VI, proposition 8&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is not exactly the same statement, but two metric relations that are equivalent to those similitudes are known in Italy as &lt;em&gt;Euclid's first and second theorem&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://it.wikipedia.org/wiki/Primo_teorema_di_Euclide&quot; rel=&quot;nofollow noreferrer&quot;&gt;Primo teorema di Euclide&lt;/a&gt;: &lt;span class=&quot;math-container&quot;&gt;$KL^2 = KN \cdot KM$&lt;/span&gt;;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://it.wikipedia.org/wiki/Secondo_teorema_di_Euclide&quot; rel=&quot;nofollow noreferrer&quot;&gt;Secondo teorema di Euclide&lt;/a&gt;: &lt;span class=&quot;math-container&quot;&gt;$LN^2 = KN \cdot KM$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So there would be at least some historical motivation to call that similitude &lt;em&gt;Euclid's theorem&lt;/em&gt;, even though the name is already taken for the &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;statement&lt;/a&gt; (in a completely different field) that there are an infinite number of primes.&lt;/p&gt;&#xA;" OwnerUserId="4930" LastActivityDate="2020-02-26T12:37:52.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17972" PostTypeId="1" CreationDate="2020-02-26T12:57:29.987" Score="4" ViewCount="123" Body="&lt;p&gt;What difficulties can be met while teaching the &quot;less than yearly compounded interest&quot; concept? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on my own (learning) experience, an objection arose when I was presented with the formula: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$A = A_0 \left(1+\frac{r}{n}\right)^{nt}.$$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The objection is : &quot;why should the rate be diminished (that is, divided by &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;) on the sole ground that the period is shorter?&quot; &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Is this objection frequent? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In which confusion does this objection originate? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How to teach this concept in order to avoid this miscomprehension? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user12116" LastEditorUserId="8571" LastEditDate="2020-02-26T13:38:51.090" LastActivityDate="2020-02-26T13:38:51.090" Title="How to explain the &quot;less than yearly compounded interest&quot; concept?" Tags="|students-mistakes|applied-mathematics|exponential|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17973" PostTypeId="2" ParentId="17951" CreationDate="2020-02-26T13:18:37.220" Score="1" Body="&lt;p&gt;A strategy I once adopted while teaching logic for philosophers at the undergraduate ( 1st year) level &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) introducing connectives with the motivation of explaining set operations and set relations &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) working with connectives on non explicitly quantified open sentences to prove some set theoretic basic laws ( using truth tables or natural deduction ) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) finally introducing explicit quantification to make the proofs rigorous. &lt;/p&gt;&#xA;" OwnerDisplayName="user12116" LastActivityDate="2020-02-26T13:18:37.220" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17974" PostTypeId="2" ParentId="17926" CreationDate="2020-02-26T15:53:33.010" Score="4" Body="&lt;p&gt;These kinds of problems typically show up (in the US, at least) in precalculus classes, hence appeals to the derivative or branches of the complex logarithm, while correct, miss the mark (in my opinion).  My presumption is that by the time a student knows any calculus or complex analysis (1) they should know better (or, at least, they should know that they should know better; if you remind them that the logarithm of a negative number is not real, they should immediately recognize that there is an issue) and (2) these kinds of problems are a bit artificial&amp;mdash;while issues with the domains of logarithms pop up, they rarely do so in the manner presented in the question (what is much more common, I think,  is for sign errors to crop up when the argument of the logarithm is smaller than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When working through such a problem with students in precalculus classes, I tend to remain vigilant throughout the process regarding the restrictions on the possible values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  For example, in the given problem, I would explain as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find all real solutions (if any) to the equation&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$ \log(x-10) = 3 + \log(x-3), $$&lt;/span&gt;&#xA;  where &lt;span class=&quot;math-container&quot;&gt;$\log : (0,\infty) \to \mathbb{R}$&lt;/span&gt; is the real natural logarithm&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;By definition, the logarithm is a function on the positive real numbers.  Thus this equation only &quot;makes sense&quot; if &lt;span class=&quot;math-container&quot;&gt;$x-10 &amp;gt; 0$&lt;/span&gt; and if &lt;span class=&quot;math-container&quot;&gt;$x-3 &amp;gt; 0$&lt;/span&gt;.  That is, we are trying to find a real number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; which has the following three properties:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log(x-10) = 3 + \log(x-3)$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x - 10 &amp;gt; 0$&lt;/span&gt;, and&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x - 3 &amp;gt; 0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Conditions 2 and 3 simplify to &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 10$&lt;/span&gt;:  since &lt;span class=&quot;math-container&quot;&gt;$x - 10 &amp;lt; 0$&lt;/span&gt;, it follows from additive cancelation (that is, if follows from &quot;adding &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; to both sides&quot;) that &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 10$&lt;/span&gt;.  In other words, &lt;span class=&quot;math-container&quot;&gt;$x \in (10, \infty)$&lt;/span&gt;.  Similarly, it must also be the case that &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 3$&lt;/span&gt;, and so &lt;span class=&quot;math-container&quot;&gt;$x \in (3, \infty)$&lt;/span&gt;.  As both conditions must be satisfied, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; must be in &lt;em&gt;both&lt;/em&gt; intervals, which means that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is in the intersection:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x \in (10,\infty) \cap (3, \infty) = (10,\infty). $$&lt;/span&gt;&#xA;Hence &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 10$&lt;/span&gt;.&lt;sup&gt;[1]&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp;\log(x-10) = 3 + \log(x-3) &amp;amp;\text{and} &amp;amp;&amp;amp;&amp;amp; x &amp;gt; 10 \\&#xA;&amp;amp;\qquad\iff \log\left(\frac{x-10}{x-3}\right) = 3 &amp;amp;\text{and} &amp;amp;&amp;amp;&amp;amp; x &amp;gt; 10 \\&#xA;&amp;amp;\qquad\iff \frac{x-10}{x-3} = \mathrm{e}^3 &amp;amp;\text{and} &amp;amp;&amp;amp;&amp;amp; x &amp;gt; 10 \\&#xA;&amp;amp;\qquad\iff x-10 = (x-3)\mathrm{e}^3 &amp;amp;\text{and} &amp;amp;&amp;amp;&amp;amp; x &amp;gt; 10 \\&#xA;&amp;amp;\qquad\iff x - \mathrm{e}^3 x =(1-\mathrm{e}^3) x = 10 - 3\mathrm{e}^3 &amp;amp;\text{and} &amp;amp;&amp;amp;&amp;amp; x &amp;gt; 10 \\&#xA;&amp;amp;\qquad\iff x = \frac{10 - 3\mathrm{e}^3}{1 - \mathrm{e}^3} \approx 2.633  &amp;amp;\text{and} &amp;amp;&amp;amp;&amp;amp; x &amp;gt; 10.&#xA;\end{align}&lt;/span&gt;&#xA;It is not possible for this final condition to be satisfied, therefore the original equation has no real solution.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;[1]  In the precalculus class that I generally teach, some very elementary set theory and the notation for intervals is established early on.  In a course where this is not the practice, the discussion of Conditions 2 and 3 can be handled in whatever manner is appropriate for the class, e.g. shaded pictures of number lines, other algebraic techniques, etc.  The point is to determine restrictions on the possible values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as early as possible in the process.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2020-02-27T18:48:10.453" LastActivityDate="2020-02-27T18:48:10.453" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17975" PostTypeId="2" ParentId="17960" CreationDate="2020-02-26T18:08:10.053" Score="0" Body="&lt;p&gt;At secondary school level, label one angle 'x' and work your way around the shape labelling each angle relative to the first one you chose. You'll get a variety of 'x' and '90-x' angles. Then a concluding sentence to say that the angles correspond and so the triangles are similar.&lt;/p&gt;&#xA;" OwnerUserId="13592" LastActivityDate="2020-02-26T18:08:10.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17976" PostTypeId="2" ParentId="14672" CreationDate="2020-02-26T18:20:05.850" Score="3" Body="&lt;p&gt;My mathematical teaching experience is limited to helping my kids with their homework, but have you considered getting her to draw the problems?  Pizzas, chocolate bars, stickmen, whatever - depending on the context.&lt;/p&gt;&#xA;" OwnerUserId="13594" LastActivityDate="2020-02-26T18:20:05.850" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17977" PostTypeId="2" ParentId="17706" CreationDate="2020-02-27T07:34:40.750" Score="2" Body="&lt;p&gt;Personally, I often add a variety of different expressions. I think doing so not only makes the presentation marginally less dry, but also perhaps increases the fluency with which students will be able to deal with symbols by understanding their meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, instead of just &quot;equals&quot; I would often use expressions like &quot;the same as&quot;, &quot;is nothing but&quot;, or &quot;is equivalent to&quot;. Similarly, when dealing with things like logical equivalences, instead of always using &quot;if and only if&quot;, I've found it natural to say things like &quot;which happens exactly when&quot;, but also &quot;is equivalent to&quot; and &quot;the same as&quot; when the meaning is clear.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea that we can easily switch between the different expressions makes it clear that equality is really a relation that just claims that two things are equal and equivalent, and is not some kind of voodoo symbol that is devoid of any meaning. This could also aid in solving the problem of some students thinking of the equal sign as some indication of &quot;the next step&quot;, leading to problems like &lt;span class=&quot;math-container&quot;&gt;$2x=10=x=5$&lt;/span&gt; occurring. If you read this out in the fluent and natural way, one gets &quot;&lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; is equal to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; which is the same as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&quot;, and any student would (hopefully) be able to catch the error here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This verbalisation makes the most sense when talking through the process of solving a problem or going through a derivation, of course. It would be unnatural and unnecessary to use this sort of thing in a problem statement by saying things like &quot;the number of real roots of the polynomial &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; is nothing but &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;&quot;. It is also probably unnatural when discussing a long chain of equal terms without any logical relation, e.g. &quot;twice of the number of apples that Alan has is the same as thrice the number of apples that Bob has which is equal to seven times the number of apples David has&quot; could be confusing; in this case I would go with Jasper's suggestion. Else, in the case where there is clear logical connection between terms in the equality, varying the expression from just &quot;equals&quot; can in my experience be very helpful.&lt;/p&gt;&#xA;" OwnerUserId="10206" LastActivityDate="2020-02-27T07:34:40.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17980" PostTypeId="1" CreationDate="2020-02-27T15:54:22.840" Score="6" ViewCount="125" Body="&lt;p&gt;Does anybody know if there is a website/database/... on textbooks adoption in the US or some other country? (math/science textbooks)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be interesting to see which textbooks are (and have been) most adopted in various courses over time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="13584" LastEditorUserId="13584" LastEditDate="2020-02-27T18:12:01.227" LastActivityDate="2020-02-29T17:31:28.987" Title="Data on textbook adoptions in universities (math/science)" Tags="|undergraduate-education|textbooks|education-research|graduate-education|resource-request|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17981" PostTypeId="2" ParentId="11736" CreationDate="2020-02-27T17:05:38.887" Score="0" Body="&lt;p&gt;I would suggest this : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Always make sure you are able to read the symbols and the sentences in &lt;strong&gt;&lt;em&gt;spoken mathematics&lt;/em&gt;&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(a) = 5$&lt;/span&gt; --&gt; &quot; &lt;em&gt;the image of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; under function f is 5&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$b=a^{-1}$&lt;/span&gt; --&gt; &quot; b is the multiplicative inverse of a&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$b=a*c$&lt;/span&gt;  --&gt; &quot; b is the image of the ordered pair (a,c) under the operation &lt;span class=&quot;math-container&quot;&gt;$*$&lt;/span&gt;&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;{{1,2},3 } --&gt; &quot; the set having as elements the set {1,2} and the number 3 &quot; ( this is therefore a set with 2 elements) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;(A--&gt;B)--&gt;C&lt;/em&gt;&quot;  --&gt; &quot; IF (A implies B) THEN C (is true)&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Always make sure you know to what &lt;strong&gt;&lt;em&gt;category&lt;/em&gt;&lt;/strong&gt; every object involved in the sentences belong ( is it a sentence, number, a function, a relation, an operation)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:  AInterB is a set ( not a sentence alledgedly meaning that set A and set B intersect or have anything in common) ; provided function f is differentiable at 2, &lt;span class=&quot;math-container&quot;&gt;$f'(2)$&lt;/span&gt; is a number ( not a function); (4 - 4) is something, namely, the number 0, and not a &quot; nonentity&quot;, or a &quot; nothing&quot;, &quot; the (indefinite integral) from 2 to 5 of (t²)dt&quot; is not a function, but a number , etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) Reword every theorem as a practical rule , that is, &lt;strong&gt;&lt;em&gt;rephrase every theorem ( if possible)  as a method to prove things&lt;/em&gt;&lt;/strong&gt; : &quot; IF i know that .... THEN  I know that ...&quot;, or &quot; TO PROVE ... IT IS SUFFICIENT TO PROVE ...&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) find the contrapositive of each theorem ( everytime the theorem is an &quot;if..then..&quot; statement, or can be turned into such a conditional statement ) and express the contrapositive -  in the same way as in (3) - as a proof method. &lt;/p&gt;&#xA;" OwnerDisplayName="user12116" LastEditorDisplayName="user12116" LastEditDate="2020-02-27T17:31:50.313" LastActivityDate="2020-02-27T17:31:50.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17982" PostTypeId="2" ParentId="17307" CreationDate="2020-02-27T21:36:27.180" Score="2" Body="&lt;p&gt;Leave more conceptual questions for take home. They have more time, can think things over. Encourage using technology, ask for a sketch of how they think the solution to something hairy looks like (Asymptotes? Increasing? Maxima?). Explore e.g. the effect of initial values on the plot of the solution.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-02-27T21:36:27.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17983" PostTypeId="2" ParentId="17157" CreationDate="2020-02-27T21:54:35.423" Score="3" Body="&lt;p&gt;I like the various &lt;a href=&quot;http://www.williamchen-mathematics.info/ln.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;lecture notes&lt;/a&gt; by William Chen a lot. Clear and reasonably complete, if introductory.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-02-27T21:54:35.423" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17984" PostTypeId="1" CreationDate="2020-02-28T16:55:01.797" Score="24" ViewCount="2612" Body="&lt;p&gt;In one textbook I use for College Algebra, the author teaches that one should interchange &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; when looking for inverse functions. For example, the inverse function of &lt;span class=&quot;math-container&quot;&gt;$$y=2x+2$$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$$y=0.5x-1.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a calculus textbook the author does not teach interchanging variables. For example, the inverse function of &lt;span class=&quot;math-container&quot;&gt;$$y=2x+2$$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$$x=0.5y-1.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surely there is no substantial difference between using &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as the variable and using &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as the variable for a function. I am a bit concerned about the two different teachings my students received from two different courses as students may get confused. Should we stop teaching &quot;interchange &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&quot; when finding the inverse function?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;p.s. If we do teach &quot;interchange &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&quot; when finding the inverse function&quot;, the inverse function of &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$C=\frac{5}{9}(F-32)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;should be &lt;span class=&quot;math-container&quot;&gt;$$C=\frac{9}{5}F+32,$$&lt;/span&gt;where in the inverse function, &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; stands for the temperature in Celsius while &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; stands for the temperature in Fahrenheit. Isn't this confusing?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2020-02-28T18:50:09.323" LastActivityDate="2020-12-27T08:47:17.110" Title="Should we stop teaching &quot;interchange $x$ and $y$&quot; when finding the inverse function?" Tags="|undergraduate-education|algebra|definitions|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17985" PostTypeId="2" ParentId="17984" CreationDate="2020-02-28T17:34:15.053" Score="4" Body="&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$y=2x+2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=0.5y-1$&lt;/span&gt; have the same graph on the x-y plane, I am hesitant to call them inverse functions.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Clearly, it's just a matter of making a distinction between independent variables and dependent variables.  It is both traditional and sensible to treat &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as the default independent variable and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as the default dependent variable.  Based on that, I don't have a problem teaching that an inverse function switches the roles of the independent and dependent variables and that is why we exchange the variable names.  If an author had a bee in their bonnet that we shouldn't assume the dependence of a variable they used to describe it, I suppose they're welcome to fight for that hill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also the tradition to use &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as the dummy independent variable in function notation.  So I have no difficulty in saying that &lt;span class=&quot;math-container&quot;&gt;$f(x)=e^x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)=\ln x$&lt;/span&gt; are inverse functions, and would be unnecessarily confused if an author felt compelled to switch one of those dummy variables.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-02-28T17:34:15.053" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17986" PostTypeId="2" ParentId="17984" CreationDate="2020-02-28T18:16:58.750" Score="4" Body="&lt;p&gt;I usually introduce the idea of inverse functions by linking it to the idea of basic composite functions&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For your example, I'd ask: ``What are the (intermediary) functions done to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$2x+2$&lt;/span&gt;?&quot;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Given &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, we'd multiply by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and then we'd add &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;So it's like we have &lt;span class=&quot;math-container&quot;&gt;$y=f\left(g\left(x\right)\right)=2x+2$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$f(x)=x+2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)=2x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Then to get &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$y=f\left(g\left(x\right)\right)$&lt;/span&gt;, we'd have to undo those &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; functions. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The inverse of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is what undoes &lt;span class=&quot;math-container&quot;&gt;$x+2$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(x)=x-2$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Likewise, &lt;span class=&quot;math-container&quot;&gt;$g^{-1}(x)=\frac{x}{2}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And introduce that &lt;span class=&quot;math-container&quot;&gt;$f^{-1}\left(f(x)\right)=x$&lt;/span&gt;, which is basically &quot;whatever is done to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is undone by &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(x)$&lt;/span&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;So if &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(x) $&lt;/span&gt; was applied to &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, we'd have &lt;span class=&quot;math-container&quot;&gt;$f^{-1}\left(f\left(g\left(x\right)\right)\right)=g\left(x\right)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Algebraically, &lt;span class=&quot;math-container&quot;&gt;$[2x+2]-2=2x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Then if we apply &lt;span class=&quot;math-container&quot;&gt;$g^{-1}(x)$&lt;/span&gt;, we'd get &lt;span class=&quot;math-container&quot;&gt;$g^{-1}\left(g(x)\right)=x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Algebraically, &lt;span class=&quot;math-container&quot;&gt;$\frac{2x}{2}=x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you're teaching younger students (like middle school grade?), I might use a sort of &quot;last in; first out&quot; analogy. Like &quot;when you are getting ready to walk out the door, first you'd put your sock on and, lastly, put on your shoe. Coming home, you'd need to take off your shoe, and then take off the sock&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously a big downside is that it only works for very simple functions but I do like it since it always keeps the idea of an inverse function linked to the function itself, rather than just swapping variables and solving.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found it most helpful when teaching students new to trigonometry since, for example, &lt;span class=&quot;math-container&quot;&gt;$\sin(\theta)=\frac{O}{H}$&lt;/span&gt; and you want to solve for &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To get &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; alone, you need to &quot;undo&quot; that &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; function that is done to &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;To undo &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; you need &lt;span class=&quot;math-container&quot;&gt;$\sin^{-1}$&lt;/span&gt;, and since it's an equation what is done to one side you need to do it to the other.&lt;/li&gt;&#xA;&lt;li&gt;Algebraically, &lt;span class=&quot;math-container&quot;&gt;$\sin^{-1}\left(\sin(\theta)\right)=\sin^{-1}\left(\frac{O}{H}\right)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\longrightarrow \theta = \sin^{-1}\left(\frac{O}{H}\right)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-02-28T18:51:31.680" LastActivityDate="2020-02-28T18:51:31.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17988" PostTypeId="2" ParentId="17984" CreationDate="2020-02-29T04:07:36.133" Score="5" Body="&lt;p&gt;It's important to clarify the difference between the way mathematicians do things, and the way scientists do things. In math, we typically reserve &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for the independent variable (aka input) and the horizontal axis, while &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is reserved for the dependent variable (aka output) and the vertical axis. With these variable choices, we need to swap variables in the process of finding the inverse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In science, variables generally stand for something meaningful. &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; means Fahrenheit, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; means Celsius, and the relationship can be written with either one as input. So we can think of &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; as a function of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$F(c) = \frac{9}{5}\cdot c + 32, $$&lt;/span&gt;&#xA;or of &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; as a function of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;: &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ C(f) = \frac{5}{9}\cdot(f-32). $$&lt;/span&gt;&#xA;In this case, there is of course no swapping, in going from the first equation to its inverse, meaning is properly preserved by writing &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$F(c)$&lt;/span&gt;, and then solving for &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it's important for algebra students to see the science way, which is more concretely meaningful. Eventually, they need to understand inverse functions done with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, also.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2020-02-29T19:33:53.567" LastActivityDate="2020-02-29T19:33:53.567" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17989" PostTypeId="1" CreationDate="2020-02-29T13:33:46.523" Score="12" ViewCount="1361" Body="&lt;p&gt;An important feature of mathematics is that it is relatively easy (compare to many other subjects) to know whether or not one's understanding is correct. There are plenty of ways to check: one can simply do exercises and check the answers. Even if a student has some false beliefs or misunderstanding, it is quite likely that after looking for proofs/counterexamples, the student will realise the problem.&lt;/p&gt;&#xA;&lt;p&gt;This seems to make mathematics more suitable for self-study than many other subjects. A student need to receive regular tutorials on how to play the piano or to write stories/essays because they need to be sure they are on the right track, but for mathematics, a student could probably learn by themselves for a long time. (This is, of course, under the assumption that they understand the material from which they learn.) And indeed, there are many self-taught mathematicians in the world.&lt;/p&gt;&#xA;&lt;p&gt;So it is natural to ask the following question: as a result of what I have described above, how is the role of students and teachers in the learning of mathematics different from the role of teachers and students in subjects such as Literature, Music, (or &lt;strong&gt;even science subjects such as&lt;/strong&gt; Chemistry and Biology)?&lt;/p&gt;&#xA;&lt;p&gt;This question applies, of course, to both school level and university level learning.&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="15832" LastEditDate="2021-04-12T15:36:36.517" LastActivityDate="2021-04-12T15:36:36.517" Title="About the effectiveness of self-studying maths (compared with other subjects)" Tags="|undergraduate-education|secondary-education|self-learning|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17990" PostTypeId="2" ParentId="17980" CreationDate="2020-02-29T17:19:58.493" Score="4" Body="&lt;p&gt;The &lt;a href=&quot;https://blog.opensyllabus.org/about-the-open-syllabus-project/&quot; rel=&quot;noreferrer&quot;&gt;Open Syllabus Project&lt;/a&gt; presents a large amount of data about books listed on syllabuses of courses taught around the world (especially in the USA and not only about math/science). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that it is not possible to ensure that the presented textbooks are indeed the most adopted (or bought or sold or used or read). However, they probably are (at least) the most &quot;officially suggested&quot; ones. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to illustrate, I compiled below the top 3 textbooks for some particular searches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://opensyllabus.org/results-list/titles?size=50&amp;amp;findWorks=calculus&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Calculus, entire period, all schools:&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Calculus, James Stewart - Multiple Editions.&lt;/li&gt;&#xA;&lt;li&gt;Essential Calculus,&#xA;James Stewart -&#xA;Multiple Editions.&lt;/li&gt;&#xA;&lt;li&gt;Thomas' Calculus,&#xA;Joel Hass &amp;amp; Maurice D. Weir -&#xA;Multiple Editions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://opensyllabus.org/results-list/titles?size=50&amp;amp;publishedYearStart=2010&amp;amp;publishedYearEnd=2020&amp;amp;findWorks=calculus&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Calculus, published in the last 10 years, all schools:&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Calculus,&#xA;William L. Briggs &amp;amp; Lyle Cochran &amp;amp; Bernard Gillett -&#xA;Multiple Editions.&lt;/li&gt;&#xA;&lt;li&gt;Calculus: Early Transcendentals, &#xA;Soo Tang Tan -&#xA;Brooks/Cole, 2010.&lt;/li&gt;&#xA;&lt;li&gt;Calculus for Scientists and Engineers: Early Transcendentals,&#xA;William L. Briggs -&#xA;Pearson, 2013.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://opensyllabus.org/results-list/titles?size=50&amp;amp;schools=8263517077504&amp;amp;findWorks=calculus&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Calculus, entire period, MIT:&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Multivariable Calculus,&#xA;Bruce H. Edwards &amp;amp; Ron Larson -&#xA;Multiple Editions.&lt;/li&gt;&#xA;&lt;li&gt;Calculus With Analytic Geometry,&#xA;George Finlay Simmons -&#xA;Multiple Editions.&lt;/li&gt;&#xA;&lt;li&gt;Calculus,&#xA;James Stewart -&#xA;Multiple Editions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://opensyllabus.org/results-list/titles?size=50&amp;amp;classYearStart=1990&amp;amp;classYearEnd=1995&amp;amp;usState=CO&amp;amp;findWorks=calculus&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Calculus, 1990 to 1995, institutions of Colorado:&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Calculus and Analytic Geometry,&#xA;George B. Thomas -&#xA;Multiple Editions.&lt;/li&gt;&#xA;&lt;li&gt;Calculus,&#xA;Tom M. Apostol -&#xA;Multiple Editions.&lt;/li&gt;&#xA;&lt;li&gt;Calculus for Business and Economics,&#xA;Robert L. Childress -&#xA;Pearson / Prentice Hall, 1972.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://opensyllabus.org/results-list/titles?size=50&amp;amp;openAccess=true&amp;amp;findWorks=calculus&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Calculus, entire period, Open License/Access:&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Calculus: Early Transcendentals,&#xA;David Guichard -&#xA;Multiple Editions.&lt;/li&gt;&#xA;&lt;li&gt;Calculus,&#xA;Edwin Prine Herman &amp;amp; Gilbert Strang -&#xA;Multiple Editions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="13585" LastEditorUserId="13585" LastEditDate="2020-02-29T17:31:28.987" LastActivityDate="2020-02-29T17:31:28.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17991" PostTypeId="2" ParentId="17964" CreationDate="2020-02-29T22:23:30.277" Score="5" Body="&lt;p&gt;For about 25 years, in upper-division undergrad as well as graduate courses, I have created notes to accompany all courses I give. Yes, this takes some energy. Also, by the time I started this, I'd been teaching such courses for almost 20 years, so had some experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Especially after it became possible to create typeset notes as fast as one could type, the classic problems about &quot;following/not-following the book&quot; could be made to evaporate. Instead of worrying about the impression &quot;the assigned text&quot; would create in students' minds, I could create notes that would follow my intended lectures+discussions. And revise as needed, upon consideration of students' reactions!!!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another part of my motivation was the ridiculous prices of math textbooks... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;... paired with the exaggerated but deliberate encyclopedic-ness of &quot;popular&quot; textbooks: to be a widely accepted text, one must not omit any of the favorite (if idiosyncratic) topics of any instructors. In contrast, if one is not aiming for universal adoption, one &lt;em&gt;can&lt;/em&gt; omit less-important things, having the desirable effect that what remains is arguably more important. (The requirements of a good text are not the same as those of a good reference...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, I do not create any sort of unlimited number of exercises, especially not make-work ones, but try to focus on genuinely important issues. And I write out complete discussions of these. Yes, year-to-year, this certainly ruins the &quot;surprise&quot; or &quot;secret&quot; aspects, but that seems to me inescapable. The fact that the number of really important basic questions in grad-level math is not unlimited is not a reason to create &quot;new&quot; questions that are silly. (And I think that it is wildly inefficient and silly to pretend that everyone should reprove everything for themselves, e.g.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I've noticed that being the author of course notes (or actual physical books) gives an instructor greater credibility in the eyes of students... and even if they're &quot;notes&quot; rather than a physical book, creation of the thing does demonstrate commitment and effort.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2020-02-29T22:23:30.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17992" PostTypeId="2" ParentId="17989" CreationDate="2020-03-01T07:50:03.267" Score="3" Body="&lt;p&gt;The way I view my teaching is that I'm more of a guide and organizer. I can help guide students through the various topics in math but it's up for the student to figure things out for themselves. I can also organize material so that students are can figure things out in a systematic way. At the end of the day, I can't do the thinking for the students, but I can make (or do my best to make) the environment most effective for self-study learning. This way, I can be confident the student will do well without me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of my techniques in the classroom are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Going over the course outline, so students know when to expect things&lt;/li&gt;&#xA;&lt;li&gt;Briefly talking about the textbook and its topics that will be used, so students know what they'll be learning. (I do my best to describe the course in ways someone who hasn't done the course understands)&lt;/li&gt;&#xA;&lt;li&gt;Talk about learning objectives and what the student has to do, which is basically do readings before class, self-mark homework, come to class with questions, etc&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If everything goes well, classes shouldn't be lectures where I'm speaking &lt;em&gt;to&lt;/em&gt; the student about what material they need to learn. Rather, classes are discussions about the material where I'm talking &lt;em&gt;with&lt;/em&gt; the student who has self-studied the material. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't honestly answer how math education is different than that of other subjects (since I don't teach them) but I'd suppose it would be harder to &quot;build up&quot; the content step-by-step without a more involved navigator. Also, where they'd navigate would be different depending on student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if I was teaching Euclidean Geometry using &lt;em&gt;The Elements&lt;/em&gt;, virtually everything would build up step-by-step and I, as teacher, would just be there for clarification or elaboration. I could imagine a gifted student who would seemingly recreate Euclid Elements on their own, as if I wasn't needed. Further, since that math has a universality to it, differences between the student, the teacher, and translations of the text, will probably not make too much difference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast, if I was teaching, say, gender, political, or religious studies, it would be extremely different in terms of my personal involvement, and the outcomes of the students. &lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-03-01T07:50:03.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17993" PostTypeId="2" ParentId="17989" CreationDate="2020-03-01T11:48:19.393" Score="3" Body="&lt;p&gt;For me, teaching has some universal principles which are the same for all fields, be it maths, physics, music or literature. More precisely, two core values are,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;assisting students in an investigative learning procedure is a common ground for teaching in general, since, knowledge that is doscovered (partially or as a whole) by a student is more persistent and stable than knwoledge that is simply revealed to them by a &quot;source of knowledge&quot; - teacher, book, internet etc.&lt;/li&gt;&#xA;&lt;li&gt;diversifying the lesson, according to the needs of each student. In the 21st century it is mandatory to take into consideration the special characteristics of each student in order to effectively teach. Students mhave access to so many sources and come from vastly different backgrounds making it indeed hard and ineffective to try to conform all of them to a &quot;universal&quot; way of teaching.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As for the differences, as a math and music teacher (theory and practice) I have to say that they are lying on the differences of each field. For instance, teaching music theory one often needs to explain why is something that way in e.g. classical music theory. To do so, a discussion in the class about the era of the under discussion music genre and a relevant research on the web may help students detect the reasons behind the phenomenon at hand. All the abovementioned procedure demands and develops high level congitive skills such as synthetic, analytic as well as critical thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the above skills are also necessary in and trained by studying mathematics, mathematics pose many more challenges. One reason behind it is the fact that mathematical formalism, when compared e.g. to musical one, is often far more distant from the initial &quot;stimulating phenomenon&quot; - in terms of the abstraction needed to arrive at it. For example, the emergence of a certain rule in classical music theory can be explained, most of the times, by looking at the ethics and aesthetics of the time, while the notion of &quot;number&quot; being simultaneously a ground notion in school maths and a result of a high level abstraction is more hard to deeply understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, while mathematics seems like the perfect field for self-study - the algorithm of self-study is indeed described easily and seems &quot;computationally&quot; efficient - this is not the case for me. The intuition behind the key notions of mathematics and the consequtive abstraction in order to arrive to them consist a constant challenge for the learner. Hence, a teacher is needed in most of the cases so as to channel the student's efforts towards the right directions and let them see the intuition behind it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me close this with an example of how difficult could self-learning be. Imagine a student that has arrived at the notion of the &quot;tangent line of a circle&quot; (N1) and is about to be introduced to differential calculus and, more specifically, to the notion od the &quot;tangent line of a function's graph&quot; (N2) - having, of course, knowledge of limits and all the prerequisite knowledge. The transition for N1 to N2 requires an abstraction from the circle's tangent line to that of a more general curve. However, letting the student find the most adequate property of N1 to generalize so as to arrive to N2 may lead to several problematic generalizations - e.g. considering a tangent line as the one that cuts the curve at exactly one point or meddling with the graphs convexity and so on. At this point, a teacher who can orientate the student is indeed needed.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2020-03-01T11:48:19.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17994" PostTypeId="2" ParentId="17989" CreationDate="2020-03-01T14:53:52.307" Score="14" Body="&lt;p&gt;The question is extremely broad, since it talks about &quot;both school level and university level learning,&quot; and doesn't make any distinctions between students with different levels of intellectual development and maturity. The brain of a kid at age 7 is doing really different things intellectually than the brain of an adult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly there are people who can self-learn math. For example, Abraham Lincoln rode around the court circuit, staying at lodging houses at night and studying Euclid, which he thought was a perfect model for presenting a logical legal argument. But most people aren't as smart or motivated as Abraham Lincoln, and most people learn most of their math as kids and teenagers, so their brains aren't as developed as Lincoln's was at that point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I spend a lot of time every week working with my community college students in my office hours on mathematical problem solving. Most of these students need a &lt;em&gt;lot&lt;/em&gt; of guidance.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An important feature of mathematics is that it is relatively easy (compare to many other subjects) to know whether or not one's understanding is correct.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For the students I interact with, this is mostly not true at all. Figuring out how to check your own answer is actually a very high-level skill. Few of them have this skill. As a typical real-world example, I have a student this semester who has had two years of calculus, but does computations like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\left(\frac{3}{2}\right)^2=\frac{6}{4}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there ways to check this answer and realize that it's wrong? Of course. For example, one could simplify the result and observe that it's the same size as the number that was being squared, but that doesn't make sense, because we expect &lt;span class=&quot;math-container&quot;&gt;$x^2&amp;gt;x$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;1$&lt;/span&gt;. Or they could compute &lt;span class=&quot;math-container&quot;&gt;$1.5^2$&lt;/span&gt; on a calculator and compare. But this student doesn't &lt;em&gt;have&lt;/em&gt; the kind of high-level understanding that would be required in order to think of these checks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my experience, the vast majority of people who learn math in school are pretty similar to the student I've described above in terms of their conceptual understanding. Their fluency with computation may be better, but they fundamentally imagine math as a black box, not as something that they can make sense out of. You could call this type of student &quot;concept-blind.&quot; Concept-blind students are not going to succeed with self-study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is actually probably needed is &lt;em&gt;more&lt;/em&gt; interaction with teachers who are themselves competent in math, who give proper attention to conceptual understanding, and who don't let students slide through to the next level without understanding what they're doing. How to make this happen, I don't know. At the K-6 level, there are far too many teachers who are themselves concept-blind. In the community college system here in LA, students tend to shop around on web sites like ratemyprofessor to look for professors who are &quot;easy,&quot; which means that they're systematically avoiding the kind of appropriate guidance they need.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-03-01T15:01:00.080" LastActivityDate="2020-03-01T15:01:00.080" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17995" PostTypeId="2" ParentId="17984" CreationDate="2020-03-01T22:46:55.497" Score="1" Body="&lt;p&gt;Let me take a moment to look at an example which involves more variables. Consider the map defined by:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x,y) = \left( \frac{x}{x^2+y^2},\frac{-y}{x^2+y^2}\right)$$&lt;/span&gt;&#xA;for &lt;span class=&quot;math-container&quot;&gt;$(x,y) \neq (0,0)$&lt;/span&gt;. Let us study how to find the inverse map. We seek &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt; such that &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ f(f^{-1}(u,v)) = (u,v) \qquad \&amp;amp; \qquad f^{-1}(f(x,y))=(x,y) $$&lt;/span&gt;&#xA;for all appropriate &lt;span class=&quot;math-container&quot;&gt;$(u,v)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$f: dom(f) \rightarrow range(f)$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$f^{-1}: range(f) \rightarrow dom(f)$&lt;/span&gt;. In this example we should eventually be convinced that &lt;span class=&quot;math-container&quot;&gt;$dom(f)=range(f) = \mathbb{R}^2-\{ (0,0)\}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pragmatically, to calculate the formula for the inverse we can set &lt;span class=&quot;math-container&quot;&gt;$f(x,y) = (u,v)$&lt;/span&gt; and solve the following for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \left( \frac{x}{x^2+y^2},\frac{-y}{x^2+y^2}\right) = (u,v) \ \ (*) $$&lt;/span&gt;&#xA;Remember, we assume &lt;span class=&quot;math-container&quot;&gt;$(x,y) \neq (0,0)$&lt;/span&gt; hence multiplication by &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2$&lt;/span&gt; is a reasonable step. &#xA;&lt;span class=&quot;math-container&quot;&gt;$$(x,-y) = (u(x^2+y^2), v(x^2+y^2)) $$&lt;/span&gt;&#xA;which means &lt;span class=&quot;math-container&quot;&gt;$ x = u(x^2+y^2)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-y = v(x^2+y^2)$&lt;/span&gt;. Suppose &lt;span class=&quot;math-container&quot;&gt;$u,v \neq 0$&lt;/span&gt;, we can return to the cases &lt;span class=&quot;math-container&quot;&gt;$u=0, v\neq 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$u \neq 0, v=0$&lt;/span&gt; after we've finished the interesting case. We already can rule out the case &lt;span class=&quot;math-container&quot;&gt;$u=v=0$&lt;/span&gt; since that forces &lt;span class=&quot;math-container&quot;&gt;$x=y=0$&lt;/span&gt; in view of the above equations. Now solve both equations for &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x^2+y^2 = \frac{x}{u} = \frac{-y}{v} \ \ (**)$$&lt;/span&gt; &#xA;Next, consider&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ u^2+v^2 = \left(\frac{x}{x^2+y^2} \right)^2+ \left(\frac{-y}{x^2+y^2} \right)^2 = \frac{x^2+y^2}{(x^2+y^2)^2} = \frac{1}{x^2+y^2} \ \ \Rightarrow x^2+y^2 = \frac{1}{u^2+v^2}. $$&lt;/span&gt;&#xA;Great, now with the above and (**) we derive:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{x}{u} = \frac{1}{u^2+v^2} \qquad \&amp;amp; \qquad \frac{-y}{v} = \frac{1}{u^2+v^2} $$&lt;/span&gt;&#xA;Thus,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x = \frac{u}{u^2+v^2} \qquad \&amp;amp; \qquad y = \frac{-v}{u^2+v^2} $$&lt;/span&gt;&#xA;We find,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ f^{-1}(u,v) = \left( \frac{u}{u^2+v^2}, \frac{-v}{u^2+v^2} \right) $$&lt;/span&gt;&#xA;But, what about &lt;span class=&quot;math-container&quot;&gt;$u=0, v\neq 0$&lt;/span&gt; ? Let's investigate if the formula above works in such a context,&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;f(f^{-1}(0,v)) &amp;amp;= f\left( \frac{0}{0^2+v^2}, \frac{-v}{0^2+v^2} \right) \\&#xA;               &amp;amp;= f(0, -1/v) \\&#xA;               &amp;amp;= \left(\frac{0}{0^2+(-1/v)^2}, \frac{-(-1/v)}{0^2+(-1/v)^2}\right) \\&#xA;               &amp;amp;= \left(0, \frac{1/v}{1/v^2}\right) \\&#xA;               &amp;amp;= \left(0,v \right). &#xA;\end{align}&lt;/span&gt;&#xA;Similar calculations show that &lt;span class=&quot;math-container&quot;&gt;$f(f^{-1}(u,0)) = (u,0)$&lt;/span&gt;. Indeed, you can check, &lt;span class=&quot;math-container&quot;&gt;$f(f^{-1}(u,v) = (u,v)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$(u,v) \neq (0,0)$&lt;/span&gt;. Amusingly, pretty much identical calculations reveal that &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(f(x,y)) = (x,y)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$(x,y) \neq 0$&lt;/span&gt;. In fact, since &lt;span class=&quot;math-container&quot;&gt;$dom(f) = range(f)$&lt;/span&gt; we might as well say &lt;span class=&quot;math-container&quot;&gt;$f=f^{-1}$&lt;/span&gt; in this case. But, if I intend to keep distinct the copies of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; in which &lt;span class=&quot;math-container&quot;&gt;$dom(f)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$range(f)$&lt;/span&gt; reside then writing &lt;span class=&quot;math-container&quot;&gt;$f=f^{-1}$&lt;/span&gt; is not a good idea. I suspect a notation such as:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ f: dom(f) \subset \mathbb{R}^2_{xy} \rightarrow range(f) \subset \mathbb{R}^2_{uv} $$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ f^{-1}:  range(f) \subset \mathbb{R}^2_{uv} \rightarrow  dom(f) \subset \mathbb{R}^2_{xy} $$&lt;/span&gt;&#xA;is helpful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fine, full disclosure, &lt;span class=&quot;math-container&quot;&gt;$f(z) = 1/z$&lt;/span&gt; hence &lt;span class=&quot;math-container&quot;&gt;$w=1/z$&lt;/span&gt; yielding &lt;span class=&quot;math-container&quot;&gt;$z =1/w$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(w) = 1/w$&lt;/span&gt; is the better way to see that the reciprocal map on the punctured plane is it's own inverse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If students in the precalculus are taught to solve for the independent variable in terms of the dependent variable then it is a fairly easy transition to higher dimensions where we simply solve for the independent variable&lt;strong&gt;s&lt;/strong&gt; in terms of the dependent variable&lt;strong&gt;s&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2020-03-01T23:26:44.393" LastActivityDate="2020-03-01T23:26:44.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17996" PostTypeId="2" ParentId="11736" CreationDate="2020-03-02T04:19:58.437" Score="0" Body="&lt;p&gt;The real problem is the way that math books are written.  Usually, they are written proof first.  Then, maybe, something about the intuition behind it.  When possible, aim for better books.  Math books should be intuition-first, then the proof.  This makes the proof much easier to follow.  Otherwise you are looking at lemmas and theorems that are not connected to anything (and therefore harder to learn/remember).  If, instead, we could see where we are going, learning it would be much easier.  I say this in case there is someone writing math books - help us out by stating the intuition first, lay out the path of the proof, and &lt;em&gt;then&lt;/em&gt; do the proof!  We will all be thankful for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in short, the answer is find the best books.  I will say that, if time is not an issue, what I tend to do is to read a book until I stop understanding it.  Then, I put it back on the shelf for a time (month or year).  Then, I pick it back up and read some more.  Usually, by this time, the ideas have formed themselves better in my head, and I get a fresh look.  I'm not stuck on the same thought pattern as before, and my brain has been working on it while I'm doing other things.  This makes me much more likely to get past my prior issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If time is an issue, here is my suggestion - do multiple readings.  The first time through, try to go through the whole book even if you don't understand it.  This will help you see where the author is trying to go.  Then, go through it to try to find the author's big picture points.  Finally, go through it slowly to pick out the finer details.  I think this will wind up being paradoxically much faster than just slogging your way through a page at a time.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2020-03-02T04:19:58.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17997" PostTypeId="2" ParentId="17989" CreationDate="2020-03-02T07:19:14.947" Score="3" Body="&lt;p&gt;The assumption is that math is more suitable for self study because the learner can know when he has gotten something right.  In fact there is a wide range of material with answers, but it is not always clear which problems to do.  Doing all of them will need too much time.  Doing some runs the risk of missing a subtle concept.  A good teacher will help the learner figure out which problems to do.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If a learner does a problem and gets it wrong, the learner can't always find out why, leaving them unable to learn from their mistakes as well as stuck and frustrated.  At times the learner may get the right answer by a method that worked in this case but not in all cases, causing them to learn the wrong things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many students who study on their own, may learn the algorithms without the higher understanding that would be useful as they continue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good teacher, will help a student who is stuck by pointing them to the next step. Those who look at a solution will never get just the step that they are stuck on, but more than that and will miss out on valuable practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student may think they understand something because they got a few problems right, but not do enough practice to cement their learning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have addressed why I think a teacher is useful in mathematics given it's uniqueness in problem solving and available answers.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-03-02T07:19:14.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17998" PostTypeId="2" ParentId="17955" CreationDate="2020-03-02T07:25:10.067" Score="1" Body="&lt;p&gt;Suggest teaching students about composition of functions.&#xA;Show them that if &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2+1$&lt;/span&gt;, then &#xA; &lt;span class=&quot;math-container&quot;&gt;$f(2x)= (2x)^2 +1$&lt;/span&gt;  and&#xA; &lt;span class=&quot;math-container&quot;&gt;$f(x^2 + 1)$&lt;/span&gt; = &lt;span class=&quot;math-container&quot;&gt;$(x^2+1)^2+1$&lt;/span&gt;&#xA;If they do enough problems with composition of functions, they will be less likely to make this mistake again. This should be more effective than explanation about what the notation means.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-03-02T07:25:10.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17999" PostTypeId="1" AcceptedAnswerId="18005" CreationDate="2020-03-02T14:53:59.423" Score="4" ViewCount="236" Body="&lt;p&gt;We're going to make a new math course for kids as intermediary between middle and high school with math profile (for preparation to entrance exams to high school), and before the main part (arithmetic, addition, multiplication, Minkowski addition, Fundamental theorem of arithmetic with proofs) we'd like to put some content about proofs tactics, how to proof theorems, formal logic and basic terms like ring, field, group, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would you recommend to look at among introductory courses as a preparation material to such classes?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that it sounds like a broad question, but I'was trying to look through some materials like Coursera's &quot;&lt;a href=&quot;https://www.coursera.org/learn/mathematical-thinking&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical thinking&lt;/a&gt;&quot; and it seems not to be enough for this goal.&#xA;Can you help me with collection of materials for kids, which are ready to take and systemized?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you all!&lt;/p&gt;&#xA;" OwnerUserId="13610" LastActivityDate="2020-05-13T15:48:57.100" Title="What would you recommend for the math thinking course for school?" Tags="|mathematical-pedagogy|proofs|mathematics-in-daily-life|visual-proof|critical-thinking|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18000" PostTypeId="2" ParentId="17955" CreationDate="2020-03-02T15:00:26.993" Score="3" Body="&lt;p&gt;For me, the standout problem here is this: relations are verbs. Without a relation/verb, you don't have a statement, you have the equivalent of a sentence fragment. What the student has written in this case isn't even wrong, it's just malformed nonsense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try mightily to get students to at least see, at first pass, that a piece of writing without any verb/relation is clearly malformed. For example, here's a &lt;a href=&quot;http://www.automatic-algebra.org/chainedrelations.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;quick quiz on reading chained relations&lt;/a&gt; where that's really the point I'm trying to get across (expect to see relations for meaning to be expressed, from which conclusions can be made). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in the last few days I was informed: (a) by a middle-school teacher that lessons on grammar are prohibited, (b) by a college speech instructor that they never correct grammar or syntax, (c) by a national educational advisor that no students can read textbooks anymore. So if that's true my desire to leverage grammar knowledge of natural language may not help. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-03-02T15:00:26.993" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18001" PostTypeId="1" CreationDate="2020-03-02T21:40:28.990" Score="3" ViewCount="663" Body="&lt;p&gt;I read the question &lt;a href=&quot;https://math.stackexchange.com/questions/1593091/how-to-explain-why-study-prime-numbers-to-5th-graders&quot;&gt;https://math.stackexchange.com/questions/1593091/how-to-explain-why-study-prime-numbers-to-5th-graders&lt;/a&gt; and according to the body of the question, some students sigh. Also according to &lt;a href=&quot;https://psychology.stackexchange.com/questions/20867/why-cant-we-learn-what-we-dont-like&quot;&gt;https://psychology.stackexchange.com/questions/20867/why-cant-we-learn-what-we-dont-like&lt;/a&gt;, it's sometimes hard for people to learn stuff that's not interesting for them. I don't know which education system the person who asked about teaching prime numbers was talking about, so I decided I don't care to determine which one it is and will just ask about the education system in United States because the fact that some education system has that problem is a good sign that the American one might have that problem as well whether or not it's the same education system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think some schools just expect students to figure out what the teacher means by what they're saying and some students just cannot figure out what the teacher means and really struggle with their attempt to. I would love to get an answer by a teacher who is trying to teach prime numbers to elementary school students about what's happening with their attempt to teach them prime numbers. I would like them to tell me what teaching style they're using. I would also like to know what they're having trouble teaching the students in a way that they can understand. Are they trying to get the students to come up with an explanation all on their own of what the fundamental theorem of arithmetic is to teach them how to think for themselves?&lt;/p&gt;&#xA;" OwnerUserId="11718" LastEditorUserId="2083" LastEditDate="2021-02-08T07:40:06.683" LastActivityDate="2021-02-08T07:40:06.683" Title="How actually are prime numbers taught in elementary school in United States and how easily do students learn what they're being taught about them?" Tags="|mathematical-pedagogy|primary-education|problem-solving|usa|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18002" PostTypeId="2" ParentId="18001" CreationDate="2020-03-02T23:53:48.950" Score="3" Body="&lt;p&gt;I relate prime numbers to prime factorizations of big numbers and usually ask the students, &quot;which is easier to work with? Large numbers all at once, or small numbers one at a time?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it's elementary school, I'd think of arithmetic of fractions.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do operations with highly composite numbers in two ways: one without prime factorization, and one with. I think elementary school is mostly computation and so being able to do that in much more efficient ways will probably look very impressive and meaningful.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Another thing I might do is talk about the Fundamental Theorem of Arithmetic (FTAr) and why &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; isn't considered a prime due to how it would mean different prime factorizations of the same number (or how you'd never be finished a factor tree).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, to make the students feel they're learning something that has a lot of meaning, you could show them a section from a university-level algebra book that has the FTAr and say they're doing things that even the &quot;big kids&quot; and adults use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd suppose that the biggest difficulty would be having students adopt Prime factorization (and cancellation) and not resort back to laborious multiplication and divisions.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-03-02T23:53:48.950" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18003" PostTypeId="1" CreationDate="2020-03-04T01:20:21.240" Score="11" ViewCount="252" Body="&lt;p&gt;There is a tradition in the use of orthogonal projections to represent 3D objects in printed school math textbooks. On the other hand, perspective projections represent better the way as we &quot;see&quot; real objects with our eyes. Take a look to compare:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/P61ay.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/P61ay.png&quot; alt=&quot;cubes in perspective and orthogonal projections&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people argue that orthogonal projections are better because it is easier to draw with them by hand. But, today, we have technology that allows the construction of 3D objects in our smartphones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, our team is writing a (free) mathematics textbook for High School and this question appeared: is there any research or study that support the choice of a given projection in textbooks?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any suggestions are welcome!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks, Humberto. &lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2020-03-06T09:22:08.257" Title="2D drawings of 3D objects in printed school textbooks: orthogonal or perspective?" Tags="|secondary-education|textbooks|geometry|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18004" PostTypeId="2" ParentId="18001" CreationDate="2020-03-04T03:18:07.167" Score="6" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I would love to get an answer by a teacher who is trying to teach prime numbers to elementary school students about what's happening with their attempt to teach them prime numbers. I would like them to tell me what teaching style they're using.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This may not be exactly what you're looking for but just the other day I introduced prime numbers to my first grade (7 year old) daughter. The setting was more relaxed and comfortable for her than a classroom but I don't see why the same technique wouldn't work in class.&lt;/p&gt;&#xA;&lt;p&gt;What we did was to see how many rectangles we could make with a given amount of square Legos. I wrote the numbers 1 through 13 or so vertically and then we wrote the number of rectangles and the dimensions of the rectangles next to them. If we could only make one rectangle, eg &lt;span class=&quot;math-container&quot;&gt;$1\times 7$&lt;/span&gt;, we circled the number. I wanted to go up to at least 12 so she would see a case where three rectangles were possible. Now, the number 1 ended up being circled and I quickly mentioned that because 1 is so special we don't usually bother calling it a prime, but that's just a technicality that I don't think is very important now (or ever for that matter). With older students it's worth discussing though.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;edit:&lt;/strong&gt; This year she's using the Investigations 2017 Student Activity Book for Grade 2 and there's an activity called &amp;quot;Building Rectangles&amp;quot; (pp. 109-111). It's exactly what I described above (even ending on the number 12) except it doesn't make any mention of primes. Nevertheless, I think it's clear that that's what it's leading up to, perhaps in a later grade, so I'm pretty confident that primes are introduced this way in at least some schools.&lt;/p&gt;&#xA;&lt;p&gt;As for the second part of your question, I'm sorry to say that I still can't be of much help there!&lt;/p&gt;&#xA;" OwnerUserId="12671" LastEditorUserId="12671" LastEditDate="2021-02-06T03:55:56.247" LastActivityDate="2021-02-06T03:55:56.247" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18005" PostTypeId="2" ParentId="17999" CreationDate="2020-03-04T17:49:15.370" Score="5" Body="&lt;p&gt;Not sure how to answer the question apart from suggesting a book about mathematical proof that can be found free online, and that is on the easier side of understanding. It's actually been adopted by a lot of universities and so I'm certain it is a quality textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.people.vcu.edu/~rhammack/BookOfProof/&quot; rel=&quot;noreferrer&quot;&gt;https://www.people.vcu.edu/~rhammack/BookOfProof/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know if it's for kids or grade school but quite a lot of it should be useful to develop your own questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, if you want good free books, here is a list: &lt;a href=&quot;https://aimath.org/textbooks/approved-textbooks/&quot; rel=&quot;noreferrer&quot;&gt;https://aimath.org/textbooks/approved-textbooks/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-03-04T17:49:15.370" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18007" PostTypeId="2" ParentId="18003" CreationDate="2020-03-05T15:13:29.530" Score="0" Body="&lt;p&gt;Stick with the orthogonal projection.  Your students are going to see that much more in courses before/after your course.  And in literature in general.  It is mildly disconcerting to see the alternate projection when so used to the standard one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, that this argument applies even if the alternate projection is &quot;better&quot;.  And the same thing applies with the many other questions we have with people obsessed with changing notation or the like.  Or on Wikipedia with changing English grammar, spelling, etc. to something they think makes more sense (but is non standard).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many areas you can still innovate and improve.  One is cost, of course.  But I would challenge you to push for quality composition (ZERO typos, grammar errors, etc.).  It is frustrating to use even purchased texts that contain errors.  And unfortunately, amateur texts tend to be even worse on average.  Make it an objective to be four dot oh!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another area is quality pedagogy.  Many amateur texts suffer from poor explanation (for NEW LEARNERS, not math colleagues!), even worse than the norm in professional texts, which have issues also.  They tend to read like the grad student or new faculty explaining things to himself, not to his audience (with awareness of new learner limitations).  Same thing for exercises/answers/examples/hints.  (Squared! Since &quot;doing&quot; is how we learn, not monograph explication.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, you should have a sophisticated knowledge from assessing several other books of ways to skin the cat.  This doesn't mean you can never have a new idea.  But it is important if you're going to do &quot;one more attempt at an X text&quot; that you have looked at more than one.  In other words, not just trying the project from scratch, with no perspective, learning as you go.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And resist the temptation (probably prone by personality to proofy &lt;em&gt;mathematicians&lt;/em&gt;, but not people-savvy &lt;em&gt;educators&lt;/em&gt;) of wanting to convert the world back to Beta from VHS.&lt;/p&gt;&#xA;" OwnerUserId="13615" LastEditorUserId="13615" LastEditDate="2020-03-05T19:00:18.527" LastActivityDate="2020-03-05T19:00:18.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18008" PostTypeId="1" CreationDate="2020-03-05T17:01:55.717" Score="10" ViewCount="469" Body="&lt;p&gt;I am currently teaching a course in &quot;vector analysis&quot;, following Colley's book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far we have reviewed multivariable calculus (a prereq for the course), and discussed: the derivative in general; the gradient, vector fields, divergence and curl; flow lines and conservative fields; arc length and the TNB frame; Kepler's laws (with complete proofs). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Logically speaking, I would like to move on to the change of variables formula for integration; but we have one more class meeting before spring break, and I'd like to present some &quot;lighter&quot; fare. Nothing suitable comes to mind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any &quot;light and easy&quot; and &quot;fun&quot; topic I could present to my students, involving some of the background I mentioned above?&lt;/p&gt;&#xA;" OwnerUserId="13616" LastActivityDate="2021-12-01T18:07:09.683" Title="A fun, one-day topic for a vector analysis course" Tags="|vector-calculus|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18009" PostTypeId="2" ParentId="17864" CreationDate="2020-03-05T20:24:14.757" Score="2" Body="&lt;p&gt;As traditional museums try to show the beauty of paintings and sculptures, museums of mathematics also try to do the same with Mathematics. Here are two of them:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.erlebnisland-mathematik.de/en/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erlebnisland Mathematik | Technische Sammlungen Dresden &lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wOerw.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wOerw.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://momath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MoMath | National Museum of Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Cptxe.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Cptxe.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There also big math festivals for the general public: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nationalmathfestival.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;National Math Festival&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SammA.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SammA.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://festivaldamatematica.org.br/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Festival da Matemática in Brazil&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is possible to organize such mini museums in schools at low cost:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=mqSURLypcNw&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cabinet of Mathematics 1&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=nDy7-evpy7g&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cabinet of Mathematics 2&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also would like to mention the posters provided by AMS:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/publicoutreach/mathmoments/mathmoments&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Moments | American Mathematical Society&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pWN9D.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pWN9D.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As different people like different artistic genres, different students may like different aspects of mathematics: puzzles, generative art, math olympic problems, math circles etc. It is necessary to know your students first to present them aspects of Mathematics that engage them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a very simple game that I've used to illustrate what Mathematics is (the study of patterns). Here is the article:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nctm.org/uploadedFiles/Conferences_and_Professional_Development/Tips_for_Teachers/Game-of-9-Cards-complete.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;What is The Name of This Game? Article by John E. Mahoney published in Mathematics Teaching in The Middle School (NCTM)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The three books Mathematical Gems I, II and III by Ross Honsberger have nice examples showing the beauty (gems!) in Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="8828" LastEditDate="2020-03-06T15:13:15.077" LastActivityDate="2020-03-06T15:13:15.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18010" PostTypeId="2" ParentId="1816" CreationDate="2020-03-05T20:31:28.620" Score="0" Body="&lt;p&gt;If you read Portuguese, here is the &lt;a href=&quot;https://umlivroaberto.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Projeto Livro Aberto de Matemática&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2020-03-05T20:31:28.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18011" PostTypeId="2" ParentId="17864" CreationDate="2020-03-06T01:27:58.410" Score="2" Body="&lt;p&gt;I just read a nice piece, Suber's &lt;a href=&quot;https://legacy.earlham.edu/~peters/writing/infapp.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;A crash course in the mathematics of infinite sets&quot;&lt;/a&gt;. No &quot;higher&quot; math required, just sharp thinking. Suprising results, with simple, clear path to them. I believe this is mathematics at it's best.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-03-06T01:27:58.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18012" PostTypeId="2" ParentId="18003" CreationDate="2020-03-06T09:22:08.257" Score="2" Body="&lt;p&gt;I &lt;em&gt;for some reason&lt;/em&gt; seem to aesthetically like perspective projection more, but, now that I think about it, I always sketch on the board using orthogonal perspective if I have straight edges. (Though, I might use perspective projection if I'm trying to show an osculating plane?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know what kind of book you're making but if it is very easily real-world application and could include photographs of real-world objects, then you might consider perspective projection since then you could superimpose the &lt;em&gt;real-world-accurate&lt;/em&gt; line drawings/sketch over the photograph and then have the sketch next to it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/F2tdP.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/F2tdP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't researched the topic yet but if your &quot;front&quot; and &quot;back&quot; are important, then it might be beneficial to consider dotted lines or even no lines for more clear perspectives of what vertex is closest to the viewer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DRYre.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DRYre.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-03-06T09:22:08.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18013" PostTypeId="2" ParentId="18008" CreationDate="2020-03-06T16:38:31.163" Score="5" Body="&lt;p&gt;As the students are by now familiar with the cross product,&#xA;it is a natural question of how it might generalize to higher dimensions.&#xA;It is surprising that the cross product only exists in dimensions&#xA;&lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;: The &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;-dimenstional cross-product is the only&#xA;&quot;bilinear product of two vectors that is vector-valued, orthogonal, and has the same magnitude as in the 3D case.&quot;&lt;sup&gt;1&lt;/sup&gt;&#xA;This could take you into very interesting territory.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Massey, W. S. &quot;Cross products of vectors in higher dimensional&#xA;Euclidean spaces.&quot; &lt;em&gt;American Mathematical Monthly&lt;/em&gt; (1983): 697-701.&#xA;(&lt;a href=&quot;http://www.jstor.org/discover/10.2307/2323537?uid=3739256&amp;amp;uid=2&amp;amp;uid=4&amp;amp;sid=21103780384137&quot; rel=&quot;nofollow noreferrer&quot;&gt;JSTOR link&lt;/a&gt;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Seven-dimensional cross product: &lt;a href=&quot;https://en.wikipedia.org/wiki/Seven-dimensional_cross_product&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&#xA;link&lt;/a&gt;.&lt;sup&gt;1&lt;/sup&gt;&#xA;&quot;Just as the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-dimensional cross product can be expressed in terms of the &#xA;quaternions, &#xA;the &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;-dimensional cross product can be expressed in terms of the octonions.&quot; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Seven-dimensional_cross_product&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; on &#xA;the &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;-dimensional cross product.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-03-07T01:00:54.370" LastActivityDate="2020-03-07T01:00:54.370" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18014" PostTypeId="2" ParentId="18008" CreationDate="2020-03-06T16:41:36.333" Score="0" Body="&lt;p&gt;I have couple point five suggestions that are different ideas from the previous comment/answers:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Do an application that uses the techniques they have already learned.  This won't open a new topic math-wise before the break.  And gives natural review.  And can be made self-contained and defined in scope.  Note, when &lt;em&gt;I say&lt;/em&gt; &quot;application&quot;, I &lt;strong&gt;don't&lt;/strong&gt; mean a derivation into another math topic.  I mean something in engineering or physics or chemistry or business or the like.  In particular, look for something in data science or computers or the like (a Google search shows entire texts around this topic).  Note, the point is really just a review and self contained.  It's not like you have to turn them into radio engineers...but a little practice in applied word problems (regardless of exact field) is a good change of mental frame.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do a general review/drill day.  Make it interactive by having an exercise sheet and a plan for how to march through the hour.  Note that if you say (and mean it!) that the hour will help them with upcoming exams, that will get their attention.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;2.5.  (Related to 2, but little funner but less didactic.) &quot;Vector Calc Jeopardy&quot;:  very easy on the chalkboard--draw the grid, have questions ready, divide the room into teams, appoint a scorekeeper and answer-spotter (from the teams raising hands), insist that they answer &quot;in the form of a question&quot;, etc.  It's a bit more of a social interaction and a nice head out to the holidays...&lt;/p&gt;&#xA;" OwnerUserId="13621" LastActivityDate="2020-03-06T16:41:36.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18017" PostTypeId="2" ParentId="17953" CreationDate="2020-03-06T19:27:56.737" Score="0" Body="&lt;p&gt;(My new and hopefully improved answer)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Should we require that &lt;span class=&quot;math-container&quot;&gt;$(a^m)^n=a^{mn}$&lt;/span&gt; only when &lt;span class=&quot;math-container&quot;&gt;$a \gt 0$&lt;/span&gt; ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That might &quot;solve&quot; the problem in some sense, but we do have legitimate cases with negative bases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Example:&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$((-2)^2)^3 =(-2)^{2\times 3}=(-2)^6=64$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According user &quot;Gae. S.&quot; at &lt;a href=&quot;https://math.stackexchange.com/questions/3576301/is-am-in-mathbbr-and-an-in-mathbbr-sufficient-for-amn-amn#3576356&quot;&gt;Math SE&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$a^m, a^n \in \mathbb{R}$&lt;/span&gt; is a &lt;em&gt;sufficient&lt;/em&gt; condition for &lt;span class=&quot;math-container&quot;&gt;$(a^m)^n=a^{mn}$&lt;/span&gt; when using real-number arithmetic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the Power of a Power Rule (on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;) can be restated: &lt;span class=&quot;math-container&quot;&gt;$(a^m)^n = a^{mn}$&lt;/span&gt; if we have &lt;span class=&quot;math-container&quot;&gt;$a^m, a^n \in \mathbb{R}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may provide a more satisfying resolution for this precocious student: Since &lt;span class=&quot;math-container&quot;&gt;$(-16)^{1/4} \notin \mathbb{R}$&lt;/span&gt;, we cannot infer that &lt;span class=&quot;math-container&quot;&gt;$((-16)^2)^{\frac{1}{4}}=(-16)^{\frac{1}{2}}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2020-03-10T18:06:38.693" LastActivityDate="2020-03-10T18:06:38.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18019" PostTypeId="2" ParentId="14932" CreationDate="2020-03-08T16:24:35.930" Score="1" Body="&lt;p&gt;To provide a source... I brought this term to the LINCT professional development team (Andrew Sanfratello and others) and the New York State Transition curriculum in 2014. I learned it from the late Nicholas Branca who had led professional developments in New York City back in 2003 and 2004. It can be a powerful tool in a wide range of problems. Al Cuoco and others will label it as &quot;delayed evaluation.&quot;&lt;/p&gt;&#xA;" OwnerUserId="12033" LastActivityDate="2020-03-08T16:24:35.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18021" PostTypeId="2" ParentId="18008" CreationDate="2020-03-08T20:59:54.600" Score="7" Body="&lt;p&gt;If we study the tangent line to &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$(a,f(a))$&lt;/span&gt; as given by:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ y = L^a_f(x) = f(a)+f'(a)(x-a) $$&lt;/span&gt;&#xA;then you can verify that a little &lt;strong&gt;input&lt;/strong&gt; interval &lt;span class=&quot;math-container&quot;&gt;$(a-h,a+h)$&lt;/span&gt; is transferred to the &lt;strong&gt;output&lt;/strong&gt; interval &lt;span class=&quot;math-container&quot;&gt;$(f(a)-f'(a)h,f(a)+f'(a)h)$&lt;/span&gt; by the linearization &lt;span class=&quot;math-container&quot;&gt;$L^a_f$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt;. What is the significance of the derivative in this viewpoint ? Observe the length of the input interval &lt;span class=&quot;math-container&quot;&gt;$(a-h,a+h)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$2h$&lt;/span&gt; whereas the length of the output interval is &lt;span class=&quot;math-container&quot;&gt;$2f'(a)h$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The value of the derivative tells us how the function stretches out the input. It gives a scale factor by which the input length is scaled to the output length.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A simple example, &lt;span class=&quot;math-container&quot;&gt;$f(x)=2x$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$f([0,1]) = [0,2]$&lt;/span&gt;. The unit-interval &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; is stretched by a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; to give the length two interval &lt;span class=&quot;math-container&quot;&gt;$[0,2]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/fxUmy.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fxUmy.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Most functions do not allow us to think about length in this global sense, but it is clear in this linear case. For a function with a variable derivative we have to think infinitesimally to see the scaling. This discussion ultimately leads to the formulation of arclength, so we can either return to it later or tie into that now (depending on your class).&lt;/p&gt;&#xA;&lt;p&gt;Ok, great. Functions of one variable are pretty easy to visualize since two dimensions we can draw etc. But, what about a function which takes in two variables and outputs two variables ? That gives us 4 possibly independent degrees of freedom. We will not be able to view it in terms of graphs as it would require four dimensions (which I personally cannot directly visualize). So what to do ? One natural solution is to look at two separate planes where one plane is the arena of inputs and the other plane is the arena of outputs for the map. Let's say &lt;span class=&quot;math-container&quot;&gt;$T: dom(T) \subseteq \mathbb{R}^2_{xy} \rightarrow \mathbb{R}^2_{uv}$&lt;/span&gt; meaning that we agree to use &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; as Cartesian coordinates for the inputs and &lt;span class=&quot;math-container&quot;&gt;$u,v$&lt;/span&gt; as Cartesian coordinates for the outputs. Previously we used &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; which has &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as independent and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as dependent. Now, for &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;, we think of &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; as independent and &lt;span class=&quot;math-container&quot;&gt;$u,v$&lt;/span&gt; as dependendent. We could describe &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; in terms of two equations:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ u = T_1(x,y) \qquad \&amp;amp; \qquad v = T_2(x,y) $$&lt;/span&gt;&#xA;for all &lt;span class=&quot;math-container&quot;&gt;$(x,y) \in dom(T)$&lt;/span&gt;. In such a case we can write &lt;span class=&quot;math-container&quot;&gt;$T=(T_1,T_2)$&lt;/span&gt; to indicate that the map &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; has &lt;strong&gt;component functions&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$T_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$T_2$&lt;/span&gt; (this sentence should be skipped if kids look bewildered already).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$S \subset dom(T)$&lt;/span&gt; is mapped by &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$T(S)$&lt;/span&gt; then how does the area of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; relate to the area of &lt;span class=&quot;math-container&quot;&gt;$T(S)$&lt;/span&gt; ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We should expect that this question cannot be nicely answered for nonlinear maps. Also, the area of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is tough to calculate for general subsets. It follows that a good starting point to make progress on the above question is the most simple nontrivial case. We ought to study &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; being a rectangle and &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; being a linear map. To say &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is linear means we can write:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ T(x,y) = \left[\begin{array}{cc} a &amp;amp; b \\ c &amp;amp; d \end{array} \right]\left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} ax+by \\ cx+dy \end{array} \right] = (ax+by, cx+dy). $$&lt;/span&gt;&#xA;the matrix &lt;span class=&quot;math-container&quot;&gt;$\left[\begin{array}{cc} a &amp;amp; b \\ c &amp;amp; d \end{array} \right]$&lt;/span&gt; is known as the &lt;strong&gt;standard matrix&lt;/strong&gt; of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; and a common shorthand notation is simply &lt;span class=&quot;math-container&quot;&gt;$[T] = \left[\begin{array}{cc} a &amp;amp; b \\ c &amp;amp; d \end{array} \right]$&lt;/span&gt;. Each choice of &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; matrix uniquely determines a linear map and vice-versa. We ought to expect the entries of &lt;span class=&quot;math-container&quot;&gt;$[T]$&lt;/span&gt; somehow inform how the area of a rectangle &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is modified to give the area of &lt;span class=&quot;math-container&quot;&gt;$T(S)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;At this point some student will complain that we've moved past rectangles and we care only about parallelograms since we are all itching to use the cross-product to calculate area. Furthermore, the students demand concrete definitions of parallelograms at this point so this whole exercise need not be vague. Since we have to give the audience what it wants let us define the &lt;strong&gt;parallelogram with sides &lt;span class=&quot;math-container&quot;&gt;$\vec{A},\vec{B}$&lt;/span&gt; based at &lt;span class=&quot;math-container&quot;&gt;$\vec{r}_o$&lt;/span&gt;&lt;/strong&gt; by:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \mathcal{P}_{\vec{r}_o}(\vec{A},\vec{B}) = \vec{r}_o+ \{ s\vec{A}+t\vec{B} \ | \ 0 \leq s,t \leq 1 \} $$&lt;/span&gt;&#xA;Here we allow parallelograms to collapse to lines or even points. In other words, our parallelograms are possibly degenerate. If the parallelogram is based at the origin then we omit &lt;span class=&quot;math-container&quot;&gt;$\vec{r}_o$&lt;/span&gt;. For example, the unit-square&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \mathcal{P}( (1,0),(0,1) ) = \{ (s(1,0)+t(0,1) \ | \ 0 \leq s,t \leq 1 \} = [0,1]^2 $$&lt;/span&gt;&#xA;Next, we ought to draw a picture to show that &lt;span class=&quot;math-container&quot;&gt;$\vec{A}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec{B}$&lt;/span&gt; align with the sides with common vertex at &lt;span class=&quot;math-container&quot;&gt;$\vec{r}_o$&lt;/span&gt;. It follows that the area of  &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}_{\vec{r}_o}(\vec{A},\vec{B})$&lt;/span&gt; is given by:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \text{area}( \mathcal{P}_{\vec{r}_o}(\vec{A},\vec{B})) = \| \vec{A} \times \vec{B} \| $$&lt;/span&gt;&#xA;Since this is getting a bit long, let me merely outline the remainder for the time being:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Show that &lt;span class=&quot;math-container&quot;&gt;$ \mathcal{P}_{\vec{r}_o}(\vec{A},\vec{B})$&lt;/span&gt; maps under &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; to a new parallelogram &lt;span class=&quot;math-container&quot;&gt;$ \mathcal{P}_{T(\vec{r}_o)}(T(\vec{A}),T(\vec{B}))$&lt;/span&gt;. I strongly encourage the use of vector notation here!&lt;/li&gt;&#xA;&lt;li&gt;Notice that &lt;span class=&quot;math-container&quot;&gt;$T(\mathcal{P}_{\vec{r}_o}(\vec{A},\vec{B})$&lt;/span&gt; has sides &lt;span class=&quot;math-container&quot;&gt;$T(\vec{A})$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$T(\vec{B})$&lt;/span&gt;. Thus to calculate the area of &lt;span class=&quot;math-container&quot;&gt;$T(\mathcal{P}_{\vec{r}_o}(\vec{A},\vec{B}))$&lt;/span&gt; we ought to calculate the magnitude of &lt;span class=&quot;math-container&quot;&gt;$ T(\vec{A}) \times T(\vec{B}) $&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Compare the result above with &lt;span class=&quot;math-container&quot;&gt;$\| \vec{A} \times \vec{B} \|$&lt;/span&gt;. You should be able to find a number &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\| T(\vec{A}) \times T(\vec{B}) \| = m\| \vec{A} \times \vec{B} \|$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Spoiler Alert: &lt;span class=&quot;math-container&quot;&gt;$m = | ad-bc|$&lt;/span&gt; where&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\text{det}([T]) = \text{det}\left[\begin{array}{cc} a &amp;amp; b \\ c &amp;amp; d \end{array} \right] = ad-bc. $$&lt;/span&gt;&#xA;is the &lt;strong&gt;determinant&lt;/strong&gt; of the linear transformation &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;. Notice, if the determinant of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is zero then the area of the mapped parallelogram is likewise zero. Linear maps only preserve the full dimension of their domain if they have nonzero determinant.&lt;/p&gt;&#xA;&lt;p&gt;The absolute value in &lt;span class=&quot;math-container&quot;&gt;$m = |ad-bc|$&lt;/span&gt; is needed since the determinant can be negative. If you study it then you'll see the significance of the sign is that positive determinant maps preserve the orientation. Orientation for a pair of vectors is geometrically understood in terms of CCW rotation; &lt;span class=&quot;math-container&quot;&gt;$\{ \vec{A}, \vec{B} \}$&lt;/span&gt; is &lt;strong&gt;positively oriented&lt;/strong&gt; if vector &lt;span class=&quot;math-container&quot;&gt;$\vec{B}$&lt;/span&gt; can be reached by rotating &lt;span class=&quot;math-container&quot;&gt;$\vec{A}$&lt;/span&gt; in a CCW fashion (less than &lt;span class=&quot;math-container&quot;&gt;$180^o$&lt;/span&gt;). You can check, if &lt;span class=&quot;math-container&quot;&gt;$\text{det}(T) &amp;lt;0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\{ \vec{A}, \vec{B} \}$&lt;/span&gt; is positively oriented then &lt;span class=&quot;math-container&quot;&gt;$\{ T(\vec{A}), T(\vec{B}) \}$&lt;/span&gt; will be negatively oriented in the sense that the vector &lt;span class=&quot;math-container&quot;&gt;$T(\vec{B})$&lt;/span&gt; is reached from a CW rotation of &lt;span class=&quot;math-container&quot;&gt;$T(\vec{A})$&lt;/span&gt; by some &lt;span class=&quot;math-container&quot;&gt;$\theta \leq 180^o$&lt;/span&gt;. The map pictured below has positive determinant:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/qxWmI.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qxWmI.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For a classroom talk, you probably should suggest notation for components of &lt;span class=&quot;math-container&quot;&gt;$\vec{A}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec{B}$&lt;/span&gt; so the students can calculate and share progress with one another. Alternatively, you could have them calculate a special case rather than attacking the derivation in full generality.&lt;/p&gt;&#xA;&lt;p&gt;Finally, this leads us to the next question: what is the analog of the linearization of &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; ? How can we linearize something like &lt;span class=&quot;math-container&quot;&gt;$F(x,y) = (x^2+y^2, xy)$&lt;/span&gt; at some point in the plane ? We'll see that an affine map &lt;span class=&quot;math-container&quot;&gt;$L_F^a = \vec{a}+d_aF$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$d_aF$&lt;/span&gt; is the &lt;strong&gt;differential&lt;/strong&gt; of &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; gives the linearization of &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;. Moreover, &lt;span class=&quot;math-container&quot;&gt;$[d_aF]=J_F(a)$&lt;/span&gt; is the &lt;strong&gt;Jacobian&lt;/strong&gt; matrix. The determinant of this Jacobian matrix tells us how an infinitesimal area scales under the map &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; just as &lt;span class=&quot;math-container&quot;&gt;$f'(a)$&lt;/span&gt; tells us how the length of a small line-segment is stretched under the map &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. Good news: the story for maps on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;2$&lt;/span&gt; is no different. But, we'll have to learn how to calculate determinants of bigger matrices... or... calculate some wedge products :)&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="470" LastEditDate="2021-12-01T18:07:09.683" LastActivityDate="2021-12-01T18:07:09.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18022" PostTypeId="1" CreationDate="2020-03-09T08:58:57.403" Score="3" ViewCount="186" Body="&lt;p&gt;I work as a private math tutor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a student, she is 10 years old. Her mother has asked me to provide assistance in preparation for the admission process to the eight-year high school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My student is extremely unmotivated to study math. Last week she told me that she has lost all her papers containing examples, that I gave her as a homework. Her mother has found all the papers later - the examples there were not computed. In addition, when I come to their house, my client, at least in the last couple of lessons, always comes late and sits late (5-7min) to her chair.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My analysis is that she has no desire for acquiring new knowledge in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;question: What can I do to motivate my young client to study?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have talked to one friend, who already has a child, and he told me, that if a child is ordered to do something, it can either work or the child can create even a stronger resistance than before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="13628" LastActivityDate="2020-03-16T02:46:56.110" Title="How to motivate my ten year old math student" Tags="|algebra|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18023" PostTypeId="1" CreationDate="2020-03-09T16:21:40.913" Score="2" ViewCount="1002" Body="&lt;p&gt;Taylor series (or Maclaurin Series) are the only way to get values for some functions, such as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{t^2} dt = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{n!(2n+1)}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, while it is easy to show the students convergence of any such series (using the ratio or root tests) and have them find the radius and interval of convergence, the students might legitimately ask: &quot;it converges to a number (if inside the interval of convergence), but how do we know this convergence is to the actual number this function should give if we can't check by another means?&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a series like &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt;, we can get the value expected using trigonometry for comparison to the series expansion and show they match, and use this to compare the accuracy of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; terms of the expansion. For a function where you cannot simply compare the actual value to the series resultant because the series is the only way to get a result, how do you show that the series converges to the actual function value and not another value? How do you establish how good the approximation is for the first n terms of the series? Especially for functions with huge radii of convergence, why should the students expect derivative information taken around a single number to give accurate values extremely far from that number? How do you give students an intuition for the number of terms needed to get an accurate approximation vs distance from the expansion point?&lt;/p&gt;&#xA;" OwnerUserId="13629" LastEditorUserId="8571" LastEditDate="2020-03-09T17:01:14.990" LastActivityDate="2020-03-12T04:58:23.673" Title="Proof that convergent Taylor Series converge to actual value of function" Tags="|calculus|series|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18024" PostTypeId="2" ParentId="18022" CreationDate="2020-03-09T19:51:45.910" Score="1" Body="&lt;p&gt;One summer I tutored the two sons of my chiropractor (a trade). One enjoyed math and wanted to improve. The other didn't enjoy it. I offered them both a few problems to try between sessions, but I knew it wasn't likely that they would do them. For the unmotivated student, I searched for puzzles he'd like. He enjoyed logic and got good at it. I kept his mom informed of what we were doing, and she was ok with not doing the regular math. My goal was to help him enjoy it more. I was somewhat successful at that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student is really resistant, you may not be able to change that.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-03-09T19:51:45.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18025" PostTypeId="2" ParentId="18023" CreationDate="2020-03-09T22:11:09.143" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Taylor series (or Maclaurin Series) are the only way to get values for some functions&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is not true in the example you give. There is at least one other way to get the value of erf, which is to do numerical integration of the integral you wrote down as a definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;BTW, you don't need to say &quot;Taylor series (or Maclaurin Series),&quot; because a Maclaurin series is a Taylor series.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;how do we know this convergence is to the actual number this function should give if we can't check by another means?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Well, if the only information you have about this function is its Taylor series, then you &lt;em&gt;can't&lt;/em&gt; determine whether the Taylor series converges to the correct value (at a point inside its radius of convergence) -- because you have no other information about its correct value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure there are functions we can define such that nobody on earth can prove any nontrivial facts about exact values of the function. For example, let &lt;span class=&quot;math-container&quot;&gt;$F_n$&lt;/span&gt; be the nth Fibonacci number, and define the function&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x)=\sum \frac{x^n}{F_n!}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This function is analytic everywhere on the real line. I don't know, maybe someone can prove something about some exact value of this function other than the trivial fact that &lt;span class=&quot;math-container&quot;&gt;$f(0)=1$&lt;/span&gt;, but since I made this example up essentially at random, it seems unlikely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But very few real-world examples are like this. In most cases, we have some reason why we're interested in this function, which implies other things about it. E.g., some books do define &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; in terms of its Taylor series, but then they prove things like &lt;span class=&quot;math-container&quot;&gt;$(e^x)'=e^x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e^{x+y}=e^xe^y$&lt;/span&gt; based on that definition. This gives you a body of facts that can all be correlated with one another.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Especially for functions with huge radii of convergence, why should the students expect derivative information taken around a single number to give accurate values extremely far from that number?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In general, they should &lt;em&gt;not&lt;/em&gt; expect this. Analytic functions are in some sense just a infinitesimal subset of the set of all functions. (WP gives the following more rigorous statement of this fact: &quot;And in fact the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions.&quot;) But many of the important functions we use a lot in math are ones that have nice properties, and the nice properties are the reason we study those functions to study. Once such nice property is if a function is analytic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are certainly techniques for proving whether certain Taylor series converge to certain values, but they may not be appropriate to teach in a second-semester freshman calc course. For example, if I'm remembering my long-ago complex analysis correctly, then the function &lt;span class=&quot;math-container&quot;&gt;$1/(x^2+1)$&lt;/span&gt; is going to have a Taylor series about &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; that converges to the correct value throughout its radius of convergence of 1, and this is because it's formed by the composition of functions that are analytic except at &lt;span class=&quot;math-container&quot;&gt;$x=\pm i$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can certainly say things about the error incurred by truncating a Taylor series, e.g., putting bounds on this error. But I don't think this has much to do with your question, since functions like &lt;span class=&quot;math-container&quot;&gt;$\exp(-1/x^2)$&lt;/span&gt; would have small bounds on the truncation error, but the error relative to the desired value is large.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-03-09T22:16:25.093" LastActivityDate="2020-03-09T22:16:25.093" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18026" PostTypeId="2" ParentId="18008" CreationDate="2020-03-10T02:00:20.243" Score="5" Body="&lt;p&gt;Based on a suggestion I once found here on MESE, my class last term proved (via Green’s theorem) the existence of a linear &lt;a href=&quot;https://en.wikipedia.org/wiki/Planimeter&quot; rel=&quot;nofollow noreferrer&quot;&gt;planimeter&lt;/a&gt;, showing how it could mechanically measure the area of a shape.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we were done with the proof, I brought out the one I built at home from scrap wood:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kTuZs.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kTuZs.jpg&quot; alt=&quot;Homemade linear planimeter made of wood&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Figure 1:  The red arm freely pivots on a block that is constrained to move linearly (sliding between two fixed pieces of wood, shown on the left). At the end of the arm is a piece of wire used to trace around a closed, oriented curve. On the arm is a wheel (also wood) that turns when the arm pivots so that the wire moves parallel to the oriented curve.&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/sWZ5M.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sWZ5M.jpg&quot; alt=&quot;Linear planimeter measuring wheel&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Figure 2:  Blow-up of the wheel, which I marked-off in tenths of a rotation. Note the red mark used to count whole rotations.&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;To test it, I drew a circle with a compass, and had a student measure it and determine its area with &lt;span class=&quot;math-container&quot;&gt;$\pi r^2$&lt;/span&gt;. I used the planimeter, having another student count the net change in number of rotations of the wheel. Students were surprised that our answers for the area were within 2% of each other. Then, we tested it with a print-out of a region between the x-axis and a complicated curve. [I prepared this sheet in advance, scaling it nicely so the planimeter would be able to trace around it.] We all agreed that computing the area by hand would be very difficult, so we used Wolfram|Alpha. Again, the planimeter's value of the area was close to Wolfram|Alpha's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I brought out a polar planimeter I got on Ebay and the class oohed-and-ahed at the improved precision and accuracy when we used it to find the area of the circle and the &quot;complicated curve&quot; area. We did not go through the details of proving why the pivot could be on a fixed circle for the polar planimeter (as opposed to a fixed line in the linear case), but I suggested this as an interesting exploration for those who wanted to further explore the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QECVW.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QECVW.jpg&quot; alt=&quot;Polar planimeter&quot;&gt;&lt;/a&gt;&#xA;Figure 3:  $10 vintage planimeter from Ebay&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sketch of the proof that students worked through (with some help):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First, we drew an oriented curve that passed near a straight axis. [The axis represents the straight path that the planimeter's pivot slides on.] From the axis, segments of fixed length &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; are drawn to points on the curve. These segments represent the fixed distance from the planimeter's pivot to the point of the wire at the end of the arm. This wire will trace along every point &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; on the curve.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QeLhH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QeLhH.png&quot; alt=&quot;Oriented curve with vector V connecting point from x axis to point x,y on the curve. Vector F is perpendicular to vector V.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Figure 4:  Set-up for creating a vector field&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Now, we created our vector field by first considering a vector &lt;span class=&quot;math-container&quot;&gt;$\vec{v}$&lt;/span&gt; that points from some arbitrary point &lt;span class=&quot;math-container&quot;&gt;$(a, 0)$&lt;/span&gt; to the point &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; on the curve. Our vector &lt;span class=&quot;math-container&quot;&gt;$\vec{F}(x,y)$&lt;/span&gt; will be a unit vector, perpendicular to &lt;span class=&quot;math-container&quot;&gt;$\vec{v}$&lt;/span&gt;, and oriented in the same direction as the curve. Since &lt;span class=&quot;math-container&quot;&gt;$(a, 0)$&lt;/span&gt; is assumed to be on a straight line, with the Pythagorean theorem, students found that &lt;span class=&quot;math-container&quot;&gt;$\vec{F}(x,y) = \langle \frac{-y}{L}, \frac{\sqrt{L^2-y^2}}{L} \rangle$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Next, they determine that &lt;span class=&quot;math-container&quot;&gt;$curl \vec{F}$&lt;/span&gt; is constant.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, they use Green's Theorem to compute &lt;span class=&quot;math-container&quot;&gt;$L \int_C \vec{F}\cdot d\vec{r}$&lt;/span&gt;, seeing that it represents the area of the region enclosed by the curve &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, and is calculated by: &lt;span class=&quot;math-container&quot;&gt;$L \times ($&lt;/span&gt;how far the wheel rolled&lt;span class=&quot;math-container&quot;&gt;$)$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion&lt;/strong&gt;:  I liked this application as it was right at their ability level skill-wise (using Green's Theorem), and because it pushed them conceptually (considering &lt;em&gt;and&lt;/em&gt; writing the equation of a vector field that measures something physical, other than the typical examples explored in class: electricity/magnetism, gravity).&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2020-03-10T17:59:38.873" LastActivityDate="2020-03-10T17:59:38.873" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18028" PostTypeId="2" ParentId="18023" CreationDate="2020-03-10T14:42:24.033" Score="3" Body="&lt;p&gt;If the power series &lt;span class=&quot;math-container&quot;&gt;$\sum_{j=0}^\infty a_j z^j$&lt;/span&gt; converges to some function &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt;, then the Maclaurin series of &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\sum_{j=0}^\infty a_j z^j$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the converse need not hold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It could happen that the Maclaurin series of a function &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\sum_{j=0}^\infty a_j z^j$&lt;/span&gt;, but &lt;span class=&quot;math-container&quot;&gt;$\sum_{j=0}^\infty a_j z^j$&lt;/span&gt; converges to some function other than &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt;.  Calculus textbooks often use the example&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(z) = \begin{cases} \exp\left(\frac{-1}{z^2}\right),\qquad &amp;amp;z&amp;gt;0\\&#xA;0,\qquad &amp;amp;z \le 0\end{cases}.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2020-03-11T10:05:36.730" LastActivityDate="2020-03-11T10:05:36.730" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18029" PostTypeId="2" ParentId="18023" CreationDate="2020-03-10T16:08:17.193" Score="2" Body="&lt;p&gt;Functions whose Taylor series converge to the original function are called analytic. How do you know if a function is analytic? For teaching basic calculus it probably suffices to know that &lt;span class=&quot;math-container&quot;&gt;$\exp,\cos,\sin$&lt;/span&gt; are analytic and that analytic functions are closed under sums, products, division, composition, inversion, derivation and anti-derivation. In particular all elementary functions are analytic at every point in the interior of their domain of definition and the function from you example is analytic.&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Analytic_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Analytic_function&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="590" LastEditorUserId="590" LastEditDate="2020-03-11T00:58:03.610" LastActivityDate="2020-03-11T00:58:03.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18030" PostTypeId="1" AcceptedAnswerId="18107" CreationDate="2020-03-10T17:09:27.493" Score="18" ViewCount="6737" Body="&lt;p&gt;Like a lot of people, my school is in the position of having to move their courses online. Does anyone know a good whiteboard app that allows simple latex entry? The closest I've been able to find is Twiddla&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.twiddla.com/t0xq6n&quot; rel=&quot;noreferrer&quot;&gt;https://www.twiddla.com/t0xq6n&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but that still requires entering the equation in another window and then dragging it into position. My ultimate preference would be ability to click anywhere, type latex as normal, and have the text appear at that spot.  I'm open to variations but please restrict suggestions to apps/sites/programs that are latex compatible and able to be drawn on as a whiteboard.&lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2022-05-24T02:00:22.710" Title="Online Whiteboard Application with Simple Latex Support" Tags="|online-instruction|computer-programs|latex|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18031" PostTypeId="1" CreationDate="2020-03-12T03:39:24.523" Score="74" ViewCount="9785" Body="&lt;p&gt;Many universities, including mine, are now requiring we teach our courses online because corona.  How shall we do this?  Let’s brainstorm here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some challenges:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;My school provides limited online resources.  I have access to Blackboard Collaborate.  My only cameras at home are those on my MacBook, iPad and iPhone.  I don’t have access to an online whiteboard (I think).  How am I to write out all the math?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am a lowly paid adjunct.  I do not want to spend inordinate time typing up technical documents.  I do not have a scanner at home for written notes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How are we to give exams?  I cannot trust these students with a take-home.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I’m sure there’s more.  I’ll contribute these later when I remember them.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Suggestions?  Many of us are in this boat.  Let’s get the ball rolling.&lt;/p&gt;&#xA;" OwnerUserId="13635" LastEditorUserId="262" LastEditDate="2020-04-25T20:35:58.883" LastActivityDate="2021-09-03T02:06:05.073" Title="How shall we teach math online?" Tags="|undergraduate-education|calculus|algebra|technology-in-education|online-instruction|" AnswerCount="17" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="18032" PostTypeId="2" ParentId="18023" CreationDate="2020-03-12T04:58:23.673" Score="0" Body="&lt;p&gt;I agree with your assertion this issue is not dealt with in many introductory calculus texts for a variety of reasons. There are two rather different concepts of convergence at play:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Convergence of the Taylor series centered at &lt;span class=&quot;math-container&quot;&gt;$x_o$&lt;/span&gt; let's say &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \sum_{n=0}^{\infty} a_n(x-x_o)^n$&lt;/span&gt;. The Interval of Convergence (IOC) is the set of all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for which numerical series &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \sum_{n=0}^{\infty} a_n(x-x_o)^n$&lt;/span&gt; converges. For a given &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and set of coefficients we have numerous tools to decide the IOC. Ratio or Root Tests cover much ground here.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Convergence of the Taylor series to the function for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; near &lt;span class=&quot;math-container&quot;&gt;$x_o$&lt;/span&gt;; let &#xA;&lt;span class=&quot;math-container&quot;&gt;$\displaystyle T_k(x) = \sum_{n=0}^{k} a_n(x-x_o)^n$&lt;/span&gt; does &lt;span class=&quot;math-container&quot;&gt;$T_k \rightarrow f$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$k \rightarrow \infty$&lt;/span&gt; ? Actually, what sort of convergence is this ? In truth, this is not the sort of convergence we actually give careful discussion of in Calculus II (in the typical US curriculum). This is convergence of a sequence of &lt;strong&gt;functions&lt;/strong&gt;. The work around for Calculus II is Taylor's Theorem with remainder. Essentially, it says&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ f(x) = T_k(x)+R_k(x) $$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$R_k(x) = \frac{f^{(k+1)}(c)}{(k+1)!}(x-x_o)^{k+1}$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; between &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_o$&lt;/span&gt;. The Mean Value Theorem is the basic example of this result and the proof of the general result can follow a very similar path. So, we get to trade the original question of convergence for the easier task of somehow arguing &lt;span class=&quot;math-container&quot;&gt;$R_k(x) \rightarrow 0$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$k \rightarrow \infty$&lt;/span&gt; for appropriate &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. It is a pain. It is certainly harder than the series convergence and divergence analysis which already kills the given audience. So, many books try to downplay this topic. Personally, I try to prove cosine or sine is analytic because I have typically built sine and cosine from radian measure and limits etc... all without power series and I just need &lt;span class=&quot;math-container&quot;&gt;$-1 \leq  \cos \theta, \sin \theta \leq 1$&lt;/span&gt;. I think the argument for the exponential function is not too bad. But generally, unless I was teaching an honors Calculus II it is something I touch on then walk away from saying we will just assume the function in question is analytic. If the students are shown the example given by Gerald Edgar then they can appreciate the distinction in concepts of convergence. If you show them how to prove analyticity then perhaps they can appreciate why you don't ask them to show it usually. We have many other more pressing issues at this point in calculus: ability to apply convergence tests logically, ability to find Taylor series via non-rediculous methods, mastery of geometric series techniques...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Convergence of a sequence of functions is usually dealt with in a later analysis course where we consider pointwise convergence as well as uniform convergence. Anyway, the problem for Calculus II is we cannot even compare &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to its Taylor series unless we have a definition of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; which allows us to make the comparison. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One rather slippery solution here:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is defined by a given power series then it necessarily converges to its Taylor expansion. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The above may seem tautological, but if you use power series to formulate the definition of a function then it really is just that simple. However, more generally, to even hope to answer if &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is analytic we need some formulation or information about &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; which allows us to compare &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to its Taylor series. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, if you want to see how to bound the error in a given Taylor series for an analytic function then I have a few examples worked out and visualized in my notes. &lt;a href=&quot;http://www.supermath.info/OldschoolCalculusII.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;See Pages 204-208 or so&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Often we can use the alternating series estimation theorem to get around the sort of arguments I face in the notes. For example, &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \sin(1) = 1- \frac{1}{6}+ \frac{1}{5!}+ \cdots $$&lt;/span&gt;&#xA;is an alternating series hence &lt;span class=&quot;math-container&quot;&gt;$\sin(1) = 5/6$&lt;/span&gt; to within an error of &lt;span class=&quot;math-container&quot;&gt;$1/5!$&lt;/span&gt;. Many problems allow this simple run-around. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder, have you thought about showing the derivative of a power series is in fact the derivative of the function as well ? This is actually quite technical and I skipped it for years without a student ever noticing... or at least without noticing and saying something. Salas Hille and Etgen actually does have arguments for this. But, most schools choose lighter fare for their course. After all, retention, enrollment and worse yet what if engineering starts their own course (gulp). &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-03-12T04:58:23.673" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18033" PostTypeId="2" ParentId="18031" CreationDate="2020-03-12T06:05:42.463" Score="10" Body="&lt;p&gt;I like You Tube for posting videos. Once you get started it's pretty simple. There are various levels of privacy possible which you can read about. If your school has a convenient way to post videos and you have broadband (we're talking about 1-1.5 Gb files here, do NOT use HD resolution or worse yet the 4k resolution...). Given all that, basically the thing to do is to find a way you can comfortably write for a long time. I like writing on a desk or table in this context. You really don't want to fuss much with the camera or require any sort of post processing. I found making a wood frame to mount a small camcorder[&lt;span class=&quot;math-container&quot;&gt;$\star$&lt;/span&gt;] above my writing worked well. Key is the camcorder/camera needs to allow mounting on a tripod. I just use the tripod mount to fix the camera to my frame. The videos I posted here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.supermath.info/DifferentialGeometry.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;My Differential Geometry Online Course&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[&lt;span class=&quot;math-container&quot;&gt;$\star$&lt;/span&gt;] (Canon, about \$250 because I needed the external mic jack for classroom applications, you can probably get a more basic model for around \$150 if you don't need to connect a wireless mic)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can post a picture tomorrow of the set-up if you're interested. Added 3-20-2020,&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Lvwiu.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Lvwiu.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/zBqHK.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zBqHK.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/BunRt.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BunRt.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The trouble with cell-phone cameras is they have the nasty habit of timing out because apparently battery life is an important design consideration in phones. I'm sure a sufficiently clever person could find a way to hang a cell-phone camera in the same fashion as I use the camcorder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if you like standing and writing. The easy fix is the white board at Home Depot or Lowes for about \$ 15 for a &lt;span class=&quot;math-container&quot;&gt;$4' \times 8'$&lt;/span&gt; piece. It erases easy and you can hang it with removable velcro tabs which are pretty easy to mount. I can put one up with a friend in a class room which lacks proper writing space in about 15 minutes. It would take longer to do it level :)&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2020-03-20T22:03:05.520" LastActivityDate="2020-03-20T22:03:05.520" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18034" PostTypeId="2" ParentId="18031" CreationDate="2020-03-12T09:55:15.717" Score="6" Body="&lt;p&gt;Well,  since I have also been obliged to teach from home due to CoVID-19 these days, I will describe here a possible solution to your problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a fast and cheap solution - I have been granted no access to any platform, unfortunately - I use the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Skype, as a platform to communicate with my students and conduct the major part of the lesson.&lt;/li&gt;&#xA;&lt;li&gt;Handwriting notes apps - like the open source Xournal++ for ubuntu/fedora/debian distributions. I simply share my screen (through skype) and then write my notes - you can use your mouse, keyboard and/or a stylus as an input device. You can also export the notes you are taking as a pdf - as well as in various formats - so as to share it with your class.&lt;/li&gt;&#xA;&lt;li&gt;Also, as for organizing students' work - e.g. handing out tasks etc - you can use some existing e-class platform - e.g. Google classroom - which can help you schedule students' tasks etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of course, you can enhance these with the extra features provided by Blackboard collaborate - although I find some features of this specific platform quite uncomfortable and dysfunctional.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2020-03-12T09:55:15.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18037" PostTypeId="2" ParentId="18031" CreationDate="2020-03-12T15:07:13.530" Score="18" Body="&lt;p&gt;Already put two comments but ideas keep coming to me so I'll just package them here. Keep in mind I'm not an educator, I'm just trying to think of practical solutions to the problem as a whole.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another answer recommends YouTube to upload source material but I feel this might be inadequate interaction. Other alternatives:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Discord&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently in response to COVID, they've upped their free screenshare capacity to 50. The first approach could be assemble a small top-down view from of a portable whiteboard that you write on during the explanation. Something like &lt;a href=&quot;https://m.media-amazon.com/images/M/MV5BMjg4ZjQyZjktZGQxZC00N2VhLWE0NzAtMTlkYzM1NDY0YWY1XkEyXkFqcGdeQXVyNzgxNDk0NTI@._V1_.jpg&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. You'd have to position a portable webcam to achieve this. Discord also has slack-like text and speech channels that you could maybe host discussions about dedicated topics. E.g. &lt;em&gt;Algebra Channel&lt;/em&gt; or &lt;em&gt;Calculus&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue with discord is that while you can stream/screenshare to those present, the stream media isn't archived which can be useful for future reference. However a service that does:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Twitch.tv&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although primarily used for entertainment purposes, I think you might be able to use it in your situation. If you still have access to your faculty building you could use an empty lecture theatre's whiteboard and effectively &quot;stream&quot; a lecture free to Twitch. Set up your laptop on a desk at the front, start streaming and conduct the lesson as normal. This has the added benefit that Twitch archives your past broadcasts so your students can use it as a reference repository in the future.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another potentially useful feature is that there is a chat sidebar so that your students can ask questions which you can answer in real time should that be necessary. It's not the most formal of arrangements but needs must.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are concerned about privacy, I'm pretty certain you can restrict public access to private links you can distribute to your students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. I find it most strange that your faculty isn't helping you financially nor advising you on the faculty process of teaching going forward.&lt;/p&gt;&#xA;" OwnerUserId="13639" LastEditorUserId="13639" LastEditDate="2020-03-12T15:19:43.183" LastActivityDate="2020-03-12T15:19:43.183" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18038" PostTypeId="2" ParentId="18031" CreationDate="2020-03-12T15:31:19.237" Score="3" Body="&lt;p&gt;To enhance my last comment, this is what I was talking about with Blackboard Collaborate Ultra. I did a trial session with some of the other faculty yesterday and it seemed to work alright. We come &quot;back&quot; from spring break next week, so I haven't had a chance to try it with students yet. It might be worthwhile to either do a training session with students before class, or resign yourself to the first 10 minutes of regular class being lost to getting-familiar-with-tech.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/fljNK.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fljNK.png&quot; alt=&quot;Blackboard Collaborate Ultra Share features&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2020-03-12T15:31:19.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18039" PostTypeId="2" ParentId="18031" CreationDate="2020-03-12T16:31:31.777" Score="24" Body="&lt;p&gt;Instructor at University of Washington here - we were one of the early closures, so I feel like we're starting to get the hang of it. Here's what I'm using:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Zoom:&lt;/strong&gt; Zoom is similar to Skype, with better support for many-participant calling and additional features. It has a built-in whiteboard you can write or type on, mechanics for allowing students to &quot;raise hands&quot;, and a &quot;breakout room&quot; option that allows you to split the class into small groups. The free version only allows 40-minute sessions, as I recall, which is a little too short for a class (though I would imagine you could do a 40 minute lecture followed by a 10-20 minute mini-assignment); the Pro version allows sessions of any length. Some schools are able to provide Pro accounts to faculty, but I can't imagine most are -- I also work at a local community college, and they don't have enough Zoom licenses for the faculty.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://www.instructure.com/canvas/?newhome=canvas&quot; rel=&quot;nofollow noreferrer&quot;&gt;Canvas&lt;/a&gt; and Panopto:&lt;/strong&gt; Canvas was already an essential component of all my classes, but it's now extra useful. It links into Panopto easily, which is a platform for recording lectures -- if you have a way to write on a screen, it can record that, or you can use your own physical whiteboard. Or even just write on paper and hold it up, I guess. Canvas also has a functionality called &quot;Conferences&quot;, which (based on my cursory inspection) seems to function a lot like Zoom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Exams:&lt;/strong&gt; I plan to administer exams through Canvas, which handles timing for me. That does have the disadvantage that I have no way to ensure that they don't make use of the Internet or similar resources, or talk to each other during the exam. To ameliorate that, I'm adjusting my exam questions to lean more conceptual than technical -- utilities like Symbolab can calculate the roots of a polynomial, for example, but they can't write a few sentences explaining the interpretation of the roots in context. I'm also planning to have the exams open for only a fixed window -- for a two-hour exam, I plan to have the Canvas exam available for a total of two and a half hours, reducing the risk that students might complete the exam and then communicate the questions or their answers to other students. I also will be requiring students to submit their work (the process they used to get the answer) as image files or PDFs after the exam.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Teamwork:&lt;/strong&gt; The biggest advice I have here is &lt;em&gt;stay in touch with other faculty at your institution.&lt;/em&gt; This is a confusing time for the students, and confusion does not help with learning. Anything you can do to maintain consistency will help -- if all of the faculty are using similar techniques for online teaching, that's likely to be more helpful to the students than if every instructor digs up their own alchemy of random online tools, no matter how effective those tools might be.&lt;/p&gt;&#xA;" OwnerUserId="10081" LastEditorUserId="13778" LastEditDate="2020-04-08T13:27:11.833" LastActivityDate="2020-04-08T13:27:11.833" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="18040" PostTypeId="2" ParentId="18031" CreationDate="2020-03-12T22:15:33.910" Score="2" Body="&lt;p&gt;We're on Blackboard as well, here's some workflow's we've been experimenting with to do everything through that interface:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lectures&lt;/strong&gt;: Blackboard Collaborate Ultra works pretty well here. Just remember to turn your mike on, and to go into the settings once you've joined the session, click &quot;Notification Settings&quot; and turn on Browser Popup Notifications for things like hand raising and chat. That will allow you to use a different application to present the material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the actual presentation, log into Blackboard Collaborate Ultra with your computer as a guest user (use the link) and then log in as a presenter (using your normal account through blackboard) on your iPad. You can then draw in the whiteboard on the iPad while monitoring the chat and speaking through your computer. If you don't like Collaborate Ultras application, go to &quot;Share Application/Screen&quot; and share your full screen, then navigate to your favorite application. The idea here is that you're using the iPad as white board (and sharing it with the class) while using computer to monitor the class conversation/questions. So far this seems pretty effective.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're more comfortable with Latex, you can just use the computer, and use&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.twiddla.com/t0xq6n&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.twiddla.com/t0xq6n&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, you can hand write your notes out on the iPad, save them as a PDF, and them upload them to Collaborate Ultra and go through them with the students during class time, answering questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Homework&lt;/strong&gt;: Use the assignments. Blackboard will allow students to upload PDF, Word Docs, etc.  If students don't have scanners have them use their phone and take a nicely framed picture. You can add notes to PDFs on the file itself, or add a grade explanation under the score. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Quizzes&lt;/strong&gt;: Blackboard has fairly good instructions for building quizzes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://help.blackboard.com/Learn/Instructor/Tests_Pools_Surveys/Create_Tests_and_Surveys&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://help.blackboard.com/Learn/Instructor/Tests_Pools_Surveys/Create_Tests_and_Surveys&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To administer them, announce the time they will become visible and give the students a finite amount of time to complete them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't want to create questions on a quiz through blackboard, hand write them with the iPad and save them as a Pdf. Then, only make the assignment file available at the beginning of class, and tell them that they have to upload it by the end of class. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're very worried about cheating, do it during class time and have them log into Blackboard Collaborate Ultra or Panopto with their video on (although smart cheaters will be able to get around this). Set the quiz time for 15 minutes and have it close 15 minutes after it becomes visible. And then, don't worry about the fact that there are ways to game this, most students won't cheat so don't try so hard to ferret out cheaters that you punish the students really trying. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Exams&lt;/strong&gt;: I don't know if we have a good solution for this yet. Keep in touch with your administration and hope they come up with something. Plan for a take home exam, if possible, or to do a one-on-one Q&amp;amp;A's during finals week. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck everyone, and keep the ideas flowing!&lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2020-03-12T22:15:33.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18041" PostTypeId="2" ParentId="18031" CreationDate="2020-03-13T00:46:50.240" Score="1" Body="&lt;p&gt;Though I taught electronic maintenance and simple math years ago, I am now a casual student.  I actually clicked on the link to this discussion just out of curiosity.  I may have some small insight into the problems you are now facing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently found an online course that was free to take if I did not want a certificate.  I thought it would help me bridge the gap between the last classes I took in early 80s and today's math.  In my mind I see it as the difference between knowing that 1 + 1 = 2 and knowing why.  That may not be a good description for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I started the very first lesson and dropped the course.  I think I understood what was being presented.  However, it was a video lecture.  I do not have reliable internet.  The course provided a transcript of the lecture, and I thought I would just read the transcript.  However, the text did not include the content of what was, I assume, displayed as a slide or an electronic chalk board.  Without that, I could not do the course work, take quizzes, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope there is some value in this.&lt;/p&gt;&#xA;" OwnerUserId="13642" LastActivityDate="2020-03-13T00:46:50.240" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18042" PostTypeId="2" ParentId="18031" CreationDate="2020-03-13T02:25:06.737" Score="6" Body="&lt;p&gt;In some sense this question is waaaay too broad, but it is attracting a useful collection of hints, and it's super topical for thousands of college math instructors (likely to be followed by primary/secondary ones), so here are a few things which I don't see mentioned yet, collated from the far too much time  spent on this subject today.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Web/Doc cam.  There are USB webcams that are very reasonably priced which you just plug into a computer, and which could easily be mounted (with popsicle sticks if you had to) to create a faux document camera.  (Or you could ask a dept. to buy a small doc cam for about $100 for faculty to share, or use one on campus if that is an option though those don't necessarily plug into your computer - many will accept SD cards.)  You could probably even use your phone to do this, if you could find a plug that let you direct the input directly to your computer (I do not know if that is possible).  The advantage to this whole suggestion is that one doesn't need a whiteboard or good resolution or anything to film people doing math ... it's just on paper.  (&lt;a href=&quot;https://mooculus.osu.edu&quot; rel=&quot;noreferrer&quot;&gt;Mooculus&lt;/a&gt; uses just paper for a lot of its videos.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://obsproject.com&quot; rel=&quot;noreferrer&quot;&gt;Open Broadcaster Software (OBS)&lt;/a&gt; is available for all platforms - I was even able to download one for my ancient Mac. You can have it export mp4 files (see YouTube for many tutorials) and upload to your LMS, assuming that you have one provided.  This means in particular that you don't have to try to use something fancier or pricier.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For many subjects, there are zillions of videos online for just solving problems, so you might be able to make a short video of yourself explaining what they don't say instead, just using your phone.  (Or TikTok if you prefer that.)  I like 3Blue1Brown for linear algebra, for instance.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are having trouble with student access to books, or if scanning is prohibitively annoying for various reasons, there is a &lt;a href=&quot;https://aimath.org/textbooks/&quot; rel=&quot;noreferrer&quot;&gt;growing number of open texts in mathematics&lt;/a&gt; which may cover the content you want - in which case, depending on what OS you have, you could even make a screencast.  (There are many ways to do this, a number of which are free - you still have to have somewhere to host it, though I really hope your institution is providing that at least.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Good luck, and my apologies for any attempts at humor above that fell flat - you have to laugh or you cry sometimes, right?  But we're in this together, keep asking and advocating.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2020-03-13T02:25:06.737" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18043" PostTypeId="2" ParentId="18031" CreationDate="2020-03-13T02:48:44.803" Score="12" Body="&lt;p&gt;I gave a presentation to my department about this today. Like you, as I see in a comment, I am also at a CUNY math department. I haven't done all-online classes before, but I've used Blackboard heavily for ~20 years and have had a hybrid (partly online) class for the last two years.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I have access to Blackboard Collaborate. My only cameras at home are&#xA;those on my MacBook, iPad and iPhone. I don’t have access to an online&#xA;whiteboard (I think). How am I to write out all the math?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Blackboard Collaborate Ultra includes a built-in online whiteboard. Click on the bottom-right button to expand a menu; click on the 3rd box icon, and you get the &amp;quot;Share Content&amp;quot; menu. The topmost item here is the &amp;quot;Virtual Whiteboard&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ToUMB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ToUMB.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this tool you can type text or draw with the manual pencil tool (in different colors). Math symbols are not supported by the typed text tool, so you'll have to draw them manually. Some people might try that with the mouse or a touch screen. Best practice would be to get yourself a &amp;quot;graphics/drawing tablet&amp;quot; -- low-end version on Amazon for about $30. (My department is working on a purchase order at this time.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I am a lowly paid adjunct. I do not want to spend inordinate time&#xA;typing up technical documents. I do not have a scanner at home for&#xA;written notes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For documents, worst-case scenario, I think, is that you could take any existing documents, or hand-write materials, take a photo with your phone, and upload that. Better would certainly to prepare materials digitally or scan them. I have an inexpensive &amp;quot;portable&amp;quot; scanner I bought for uses such as these which has worked fine for me.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How are we to give exams? I cannot trust these students with a&#xA;take-home.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The truth is that you'll basically have to operate on the honor system at this time; there simply isn't any other reasonable choice (without in-person proctoring). There are systems in some places that require testing students to be watched by webcam, but (a) you probably can't procure that system through your college at this time, (b) you can't expect all students will have webcams, and (c) there are privacy concerns that would be wise for you to avoid.&lt;/p&gt;&#xA;&lt;p&gt;Blackboard has an automatic testing facility (works best for multiple-choice questions; automatically graded, and students can't enter special symbols on essay-type questions). Or you could provide assignment sheets in PDF form to email back, or upload as an &amp;quot;Assignment&amp;quot; feature.&lt;/p&gt;&#xA;&lt;p&gt;One thing that has worked well for me is to prepare tests using the Pearson TestGen application (includes testbanks for many major textbooks), and then there's a few-click process to (a) export from TestGen, (b) import as a Blackboard question Pool, and (c) create a Test from those questions in Blackboard.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I would personally recommend that you stick to whatever tools are provided, managed, and vetted by your college (in our case at CUNY: Blackboard). Using outside third-party tools have privacy, security, and possibly FERPA problems.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A year-and-a-half after this Q&amp;amp;A was originally posted, I can also offer this depository of resources I developed for our department during the pandemic, and hosted at the CUNY Academic Commons site -- including documents, handouts, and video walkthroughs specific to using the Blackboard system for math &amp;amp; computing courses.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://teachmathonline.commons.gc.cuny.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;CUNY Academic Commons: Teaching Math and Computing Online&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-09-03T02:06:05.073" LastActivityDate="2021-09-03T02:06:05.073" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18044" PostTypeId="2" ParentId="18031" CreationDate="2020-03-13T03:08:53.150" Score="1" Body="&lt;p&gt;This is best done via an email list. You write up lecture notes using LaTeX, produce a PDF file which you email to all your students. You can ask your students to submit homework to you via email. This way of teaching is far more efficient than teaching in the traditional way or the live online methods, as you can prepare the lecture notes whenever it suits you while the students can study when it suits them. &lt;/p&gt;&#xA;" OwnerUserId="1856" LastActivityDate="2020-03-13T03:08:53.150" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18045" PostTypeId="2" ParentId="18031" CreationDate="2020-03-13T12:30:56.483" Score="6" Body="&lt;p&gt;I am surprised that a school would effectively say &quot;go figure it out&quot;. You ask for &quot;brainstorming&quot;... here are my thoughts..&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You haven't quite defines your goal, although I did hear, loud and clear, you'd like 'free' or close to it. Still, there are a number of outcomes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Live video&lt;/strong&gt; - I believe there are many options, but in general, a multiuser teleconference. You do this at the school, on a whiteboard, students dial in. You use your phone (battery concern? you keep it plugged in.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Recorded video&lt;/strong&gt; - The iPad will let you screen record audio/video. A stylus can be had for cheap, since the Apple pencil is about the $100 you wish to avoid. For math, I am thinking this is a good video approach, as there's no need to see you up at a board. You can decide whether to edit or not. There are a number of apps to just let you write notes, the recorder feature is part of the OS. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Written notes&lt;/strong&gt; - like a text book. Again, there are many 'scanner' apps for the iPhone/iPad and you can just use those. I taught the last couple months of a precalc (HS) class a couple years ago, and while I actually taught the class live, I made the prep notes I used available to the students that day, after the class, so they weren't looking ahead. This was well received, as they paid more attention during class, with less time furiously copying off the board. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 comments - It seems to me, a school should have a common approach. My HS is currently meeting to discuss how we will remote teach. The goal is to be consistent, so students aren't needing to understand how to access, say 5 platforms, for multiple classes. If \$100 (only. of course, there's a limit) is the difference between doing your job well, and half-assing it, I'd highly recommend you think about this. If not for the students, then for your own career path. Projecting an attitude of &quot;I'll do whatever it takes&quot; will reward you far more than a single $100 investment in your own tech. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-03-13T12:30:56.483" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18046" PostTypeId="2" ParentId="18031" CreationDate="2020-03-13T16:42:42.687" Score="15" Body="&lt;p&gt;The case for &lt;a href=&quot;https://webwork.maa.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;WeBWorK&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;tl;dr&lt;/strong&gt; - use WebWork as an easy way of giving students as much practice solving problems as they can handle.  &lt;strong&gt;Learn by doing&lt;/strong&gt; lots of problems with a tight feedback loop.  It engages the game-playing, obsessive nature in us.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Main use case - Homework engine&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;The two biggest features of a VLE/LMS are the presentation of materials and facilitating assessment.  WebWork was specifically developed to provide formative and summative assessment for mathematics in a way that eases the burden on the instructor and allow the student to attempt problems as many times as they wish before the deadline.  By immediately seeing which answers are incorrect while the problem is still fresh in their minds, students can try multiple approaches until they find the correct solution.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;it has a library of ~30,000 math and over 1000 physics questions and includes such &lt;a href=&quot;https://github.com/openwebwork/webwork-open-problem-library/blob/master/OpenProblemLibrary/Textbooks&quot; rel=&quot;nofollow noreferrer&quot;&gt;textbooks&lt;/a&gt; as Stewart's Calculus&lt;/li&gt;&#xA;&lt;li&gt;it randomizes problem parameters (different numbers, drawing from question pools, MCQ ordering) so that each student gets an individual question which:&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;hinders copying&lt;/li&gt;&#xA;&lt;li&gt;promotes student collaboration&lt;/li&gt;&#xA;&lt;li&gt;immediate feedback and configurable number of attempts&lt;/li&gt;&#xA;&lt;li&gt;allows the instructor to re-assign a problem set and students get new numbers&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;understands derivatives&lt;/li&gt;&#xA;&lt;li&gt;hints and solutions available after cut-off dates&lt;/li&gt;&#xA;&lt;li&gt;Student statistics shows you current state of progress.  i.e. 19 attempts signals difficulty with a particular problem which could benefit from intervention&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;email student/instructor buttons (is this a good thing in your use case?)&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;LaTeX typesetting makes the Hardcopy version of problems look good on the page for those who cannot always work online.&lt;/li&gt;&#xA;&lt;li&gt;25 years of development with a good community&lt;/li&gt;&#xA;&lt;li&gt;backing of MAA, wide adoption in the US&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h3&gt;Content delivery&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;It has the basic delivery mechanism for presenting course content as webpages or PDF, but it would be better suited to pairing with your existing LMS/VLE environment, Blackboard and Moodle, either as a weblink&#xA;or through the LTI bridge.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Extensible&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Integrates with &lt;a href=&quot;https://webwork.maa.org/wiki/GeoGebraWeb1&quot; rel=&quot;nofollow noreferrer&quot;&gt;GeoGebra&lt;/a&gt; and other tools.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Open source&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;You can run this on your PC providing your IT networking team will allow it through the firewall.  It has a Docker image which could be run from the cloud.  There might be hosting options available&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Examinations&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;To answer Stephen's comment below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you consider an examination as a problem set completed under a time limit with only one submission, have a look at the documentation for &#xA;&lt;a href=&quot;https://webwork.maa.org/wiki/Gateway_Tests_and_Quizzes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gateway Quizes&lt;/a&gt; .&#xA;The access restrictions should let you set out a decently administered exam, given that the answers will be different for each student reducing the problem of copying, eg. you have 2 hours to complete the quiz which must be finished before the deadline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have more determined cheaters and need to verify their identity, the &lt;em&gt;proctored&lt;/em&gt; quiz options permit you to require students to enter an examination center, suitably distanced in time and space, where a masked Proctor will enter the online key to allow the student to begin their exam.&lt;/p&gt;&#xA;" OwnerUserId="13647" LastEditorUserId="13647" LastEditDate="2020-04-17T15:06:20.780" LastActivityDate="2020-04-17T15:06:20.780" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18047" PostTypeId="1" CreationDate="2020-03-14T08:35:11.447" Score="3" ViewCount="497" Body="&lt;p&gt;I'm looking for a good textbook introduction to probability/statistics that a first/second year undergrad math student could use! I'd like a book that emphasizes theory over procedure. I'd prefer an open-source textbook, but I'm willing to pay if you believe it is worth the money.&lt;/p&gt;&#xA;" OwnerUserId="12501" LastActivityDate="2021-03-02T20:56:23.743" Title="Best resources for probability/statistics textbooks" Tags="|self-learning|statistics|reference-books|probability|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18048" PostTypeId="1" AcceptedAnswerId="18049" CreationDate="2020-03-14T10:34:49.823" Score="24" ViewCount="7471" Body="&lt;p&gt;When taking a MOOC in calculus the exercises contain 5 options to select from. I then solve the question and select the option that matches my answer. Obviously only one of the options is correct. But there are (quite a few) times where my solution is wrong even though it is one of the available options. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, how does a maths teacher know how to create wrong options that are as plausible as the correct answer?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found these two good questions but I am wondering how would an educator come up with wrong answers in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/16573/what-are-some-common-ways-students-get-confused-about-finding-an-inverse-of-a-fu/16576#16576&quot;&gt;What are some common ways students get confused about finding an inverse of a function?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/12045/how-are-students-messing-up-in-this-khan-academy-surface-area-problem-right-tr&quot;&gt;How are students messing up in this Khan Academy surface area problem? (right triangular prism: 3-4-5 triangular base, height of 11)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13650" LastEditorUserId="8175" LastEditDate="2020-03-15T08:18:49.107" LastActivityDate="2021-04-19T04:03:05.003" Title="How does a teacher come up with plausible wrong answers for multiple choice tests?" Tags="|students-mistakes|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18049" PostTypeId="2" ParentId="18048" CreationDate="2020-03-14T17:25:18.770" Score="44" Body="&lt;p&gt;I freelance as an item writer, someone who writes questions for standardized tests.  When making up alternate choices, I always have to justify my reasons for the &quot;wrong answers&quot; or distractors.&#xA;Here are some strategies I use.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I focus on common misconceptions for students at that grade level.This is easier after years of classroom experience.        &lt;/li&gt;&#xA;&lt;li&gt;In a multi-step problem I often use the answer to an intermediate step.  &lt;/li&gt;&#xA;&lt;li&gt;When calculator use isn't allowed, I use answers with common&#xA;calculation errors.&lt;/li&gt;&#xA;&lt;li&gt;I look for errors students might make if they don't read the problem carefully.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-03-14T17:25:18.770" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18050" PostTypeId="2" ParentId="10758" CreationDate="2020-03-15T01:27:58.380" Score="8" Body="&lt;p&gt;I once read a suggestion to explain the difference between local and global maxima by using the example of Mount Everest (global maximum of the height function) and K2 (local maximum). I consider this a bad example because it gives two erroneous impressions: (1) local maxima almost achieve the global maximum value (because K2 is almost as high as Everest) and (2) local maxima occur at locations very near the global maximum (because K2 is essentially right next to Everest). A much better example of a local maximum of height would be Mount Brighton (a ski hill about 50 miles from Ann Arbor --- built on a landfill, if I remember correctly).  Students need to learn that, if you want a global maximum and you find a local maximum instead, you can be wildly wrong.&lt;/p&gt;&#xA;" OwnerUserId="326" LastEditorUserId="470" LastEditDate="2021-03-27T14:34:22.190" LastActivityDate="2021-03-27T14:34:22.190" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18051" PostTypeId="2" ParentId="18048" CreationDate="2020-03-15T01:37:51.067" Score="21" Body="&lt;p&gt;If a teacher has taught the course before, and has asked questions that are free-response (not multiple-choice), then the teacher can look at the incorrect answers previously given by the students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If not, then the teacher can ask other teachers who have had this experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Errors that &quot;appear to stem from consistent application of a faulty method, algorithm, or rule&quot; are called &quot;systematic errors&quot; (or &quot;bugs&quot;) in the literature.  Use these terms to search the mathematics education literature.  (See my answer &lt;a href=&quot;https://matheducators.stackexchange.com/a/10879/77&quot;&gt;here&lt;/a&gt;, for example.)&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2020-03-15T01:37:51.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18052" PostTypeId="1" CreationDate="2020-03-15T10:05:34.610" Score="3" ViewCount="133" Body="&lt;p&gt;Recently, I was watching &lt;a href=&quot;https://www.youtube.com/watch?v=CKl1B8y4qXw&quot; rel=&quot;nofollow noreferrer&quot;&gt;this video&lt;/a&gt; and I began thinking about how much my arithmetic skills have declined in recent years due to over reliance of calculators in upper year (high school) math courses. I've tried to keep these skills practised (e.g doing simple calculations in my head during commute, trying to do homework w/o a calculator, etc). I have also always had an interest in how past mathematicians, scientists and engineers performed calculations (Although this part may be a question for the Math and Science History SE) I am looking for resources pertaining to arithmetic, specifically algorithms/techniques/tricks to performing calculations on paper/mentally (potentially from a historical perspective?). Thanks in advance. &lt;/p&gt;&#xA;" OwnerUserId="13403" LastActivityDate="2020-03-15T10:37:50.467" Title="Mental/&quot;Paper and Pencil&quot; Arithmetic" Tags="|self-learning|arithmetic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18053" PostTypeId="2" ParentId="18052" CreationDate="2020-03-15T10:23:43.503" Score="3" Body="&lt;p&gt;You might be interested in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Trachtenberg_system&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trachtenberg system&lt;/a&gt; of rapid mental calculation.  The Wikipedia page lists a summary of its multiplication and addition methods.  More information can be found at &lt;a href=&quot;https://trachtenbergspeedmath.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trachtenberg Speed Math&lt;/a&gt;, where a book about it is for sale.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2020-03-15T10:23:43.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18055" PostTypeId="1" CreationDate="2020-03-15T20:13:45.150" Score="6" ViewCount="216" Body="&lt;p&gt;(I've generalized the original question as @BrendanW.Sullivan suggests.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would appreciate recommendations for a whiteboard app for a tablet using an e-pencil. For me: an iPad, using an Apple pencil. For others, please specify the hardware &amp;amp; OpSys &amp;amp; software.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just want to be able to draw in colors and erase easily, like a physical whiteboard. The whiteboard will be shown to my (remote) students; I don't need them to also access the whiteboard. Other posted questions seek special capabilities (e.g., LaTeX support). My needs are more basic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Free would be nice but my school will reimburse me so I can also purchase. What I see online is a bewildering variety of products not easily evaluated. &lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-03-15T21:51:06.893" LastActivityDate="2020-03-15T21:51:06.893" Title="Tablet whiteboard app w e-pencil" Tags="|undergraduate-education|software|interactive-teaching|online-instruction|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18056" PostTypeId="2" ParentId="18022" CreationDate="2020-03-16T02:40:43.247" Score="1" Body="&lt;p&gt;Welcome to my world...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’m also a private maths tutor and I’ve had this situation quite a few times, so I can sympathise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, there are quite a few things you can try. It’s your job as a maths tutor to try to see what works i.e. what motivates and engages them- what makes them “tick”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Try: visual learning/including shapes when explaining things (e.g. shapes when teaching fractions really helps).&#xA;Also asking the student what their interests are and then trying to relate the maths to their interests often works.&#xA;But that’s just 2 things. There’s lots of things you can try. Search google and ME SE for more decent advice/articles on what else you can do to get them engaged in lessons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, just keep doing those things- the things that actually work for that individual student, even if you find them unusual. It’s really important &lt;em&gt;they&lt;/em&gt; don’t find the lessons boring, even if you find the maths basic and “obvious”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But most importantly, if it’s been a few lessons and you are trying lots of different things to get them engaged but they still are not caring, then you should &lt;em&gt;not blame yourself&lt;/em&gt;- you’re doing all you can and  they are not caring. That’s not your fault. This sounds like the advice Robin Williams gives in that scene from “Good Will Hunting”, but it’s true. Leave the lesson, go home and at the end of the day say, “well I did my part”. Sometimes the kid just isn’t in the right place mentally to learn a bunch of maths. Maybe they have ADHD or some other learning difficulty. Maybe they’re just a kid and they are thinking about the crazy stuff kids think about. Maybe they just hate maths and they’re totally blocking learning anything related to maths out. Who knows. You’re not a psychiatrist (although sometimes it can feel like you are lol).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a maths tutor, part of your job is to be a detective trying to figure out how you can help the student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously you should always be communicating with the parents. Be straightforward and tell the parents, “he/she was not engaging at all in today’s lesson...”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But don’t spend 10 hours a week worrying about helping 1 student who isn’t being engaging or responsive or giving any real feedback. That’s their problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And by the way, loads of students (probably about half of my students)- even ones who do well in lessons - try to avoid doing homework - basically due to laziness. There’s not much you can do other than remind them (and the parents) it’s for their own benefit- which is certainly true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was actually thinking today- how much does a student improve when they do their homework compared to when they do no homework? I estimate around 3-10 times more (faster) improvement when they do their homework compared to when they do no homework. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in conclusion- try your best, but don’t beat yourself up about it if what you try doesn’t work- you’re not going to be able to help everyone all the time. And it’s probably&#xA;(usually) not your fault. And you just have to accept this fact and move on.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2020-03-16T02:46:56.110" LastActivityDate="2020-03-16T02:46:56.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18057" PostTypeId="2" ParentId="18031" CreationDate="2020-03-16T12:55:05.137" Score="4" Body="&lt;p&gt;I'm a full-time faculty member at a community college in NJ. I've been teaching for 12 years but have never taught an online course. When our college left for spring break I was left scrambling for how to make use of the tools that I had at that moment: my course notes and textbooks, my laptop, and my iPhone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With these tools I write, by hand, a short lecture with each slide representing a different point or step in solving a problem. I use my iPhone's scanning app to take a photo of each slide that I want. At the end I get a merged PDF. I then use Zoom to share my screen with the PDF pulled up (in full-screen mode) and record as if I were in class. The end result looks &quot;good enough&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In total for a 30 minute recorded lecture I spent about an hour doing the write up by hand and then 30 minutes doing the recording. It's time consuming but it's the best that I can do with the tools that I have.&lt;/p&gt;&#xA;" OwnerUserId="8559" LastActivityDate="2020-03-16T12:55:05.137" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18058" PostTypeId="1" CreationDate="2020-03-17T00:18:20.067" Score="8" ViewCount="833" Body="&lt;p&gt;This is similar to another question about teaching online, but I want to focus on something specific: what are the most popular setups for recording math video lectures? What's the easiest and best way to do it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B079DLTG9F&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;document camera&lt;/a&gt;, and I was planning to record lectures using Zoom -- this would allow students to see simultaneously a video of what I'm writing (and what I'm pointing to with my hands) and my face as I'm talking / explaining. I think both of those things are important. The problem with this plan is that the Zoom servers seem to be swamped, with like 24 hr processing times before a video can be viewed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the most popular setups? Which software is most popular?&lt;/p&gt;&#xA;" OwnerUserId="2049" LastActivityDate="2020-03-27T02:10:42.583" Title="Most popular setups for recording video lectures" Tags="|online-instruction|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18059" PostTypeId="1" AcceptedAnswerId="18061" CreationDate="2020-03-17T06:36:33.247" Score="2" ViewCount="226" Body="&lt;p&gt;Imagine a math problem that consists in doing all sum operations you can in 2 minutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then imagine these are the correct operations each student has done (each list entry represents a different student, it is ordered to keep it simple):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;4 correct&lt;/li&gt;&#xA;&lt;li&gt;7 correct&lt;/li&gt;&#xA;&lt;li&gt;10 correct&lt;/li&gt;&#xA;&lt;li&gt;11 correct&lt;/li&gt;&#xA;&lt;li&gt;12 correct&lt;/li&gt;&#xA;&lt;li&gt;12 correct&lt;/li&gt;&#xA;&lt;li&gt;15 correct&lt;/li&gt;&#xA;&lt;li&gt;15 correct&lt;/li&gt;&#xA;&lt;li&gt;17 correct&lt;/li&gt;&#xA;&lt;li&gt;22 correct&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;One way of evaluating it would be based on the overall students performance (maybe &lt;code&gt;&amp;lt; 10 not passed, &amp;gt; 16 excellent&lt;/code&gt;). But I don't see fair evaluating a student based on another student performance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way I though would be to set each &quot;milestone&quot; myself. For example &lt;code&gt;&amp;gt; 6 passed, &amp;gt; 18 excellent&lt;/code&gt; but I was not the test author and students didn't know about that &quot;milestones&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, do you have any idea of what could be a fair option to evaluate students in an exercise of this kind? (It is a real case.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="13672" OwnerDisplayName="Hagyn" LastActivityDate="2020-03-18T21:22:01.237" Title="How to evaluate students in a &quot;do all you can&quot; exercise?" Tags="|teaching|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18060" PostTypeId="2" ParentId="18059" CreationDate="2020-03-17T09:16:45.653" Score="3" Body="&lt;p&gt;I find it unfair to give the student a task where they don't know what they need to achieve to get the best grade. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would ask another colleague(s) to do the task. These colleagues should be expert but haven't seen the task before. Then I take their average to be equivalent to the full grade. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the exam sheets, I inform the students how many correct operations they need to do to get the full grade. &lt;/p&gt;&#xA;" OwnerUserId="13673" LastActivityDate="2020-03-17T09:16:45.653" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18061" PostTypeId="2" ParentId="18059" CreationDate="2020-03-17T11:25:28.420" Score="2" Body="&lt;p&gt;You should try to have a feel for what level of skill you want.  Now, this is not an easy problem.  Much more exploratory and Bayesian.  (But actual teaching/pedagogy is different than Euclidean if-then logic of math itself.)  There are ways to estimate this.  Previous class performance.  What you think they need for the next lesson or year. Your own performance.  Or the excellent suggestion of having another teacher try the task (could just try a few numerate, but non teacher adults also).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would help a bit if you told us what sorts of addition, you gave.  But assuming they were no longer than two digit problems, I would say something like 20 is reasonable.  Kids can learn speed at easy operations.  But first try it yourself and see how fast you can do it.  Just the act of actually doing the test will give you insights, different than purely observing.&lt;/p&gt;&#xA;" OwnerUserId="13675" LastActivityDate="2020-03-17T11:25:28.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18062" PostTypeId="2" ParentId="18048" CreationDate="2020-03-17T11:52:32.370" Score="-1" Body="&lt;p&gt;So. I started working as a teaching assistant for a course, and the professor showed me &quot;test generating&quot; software. It comes down to the fact that the editor of a textbook creates various open, multiple choice, and true/false questions and answers, and you - using the software - can just click &quot;generate X number of MC questions&quot;. No need to think about it at all. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to create a midterm, I just downloaded the software, hit &quot;generate&quot; and got all the questions and answers. I know this does not exactly answer your question, but I did not know this existed and it completely changed the &quot;they must think so hard about all of this&quot; idea I had in my head and I had to share this. Especially as I was shown this only AFTER coming up with questions and answers for the previuos midterms.&lt;/p&gt;&#xA;" OwnerUserId="13676" LastActivityDate="2020-03-17T11:52:32.370" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18063" PostTypeId="2" ParentId="18058" CreationDate="2020-03-17T13:06:40.663" Score="3" Body="&lt;p&gt;I use a combination of PowerPoint for the lecture content and my &quot;2 in 1&quot; laptop with a stylus for writing.  PowerPoint has an option to record the lecture which is what I use for the actual recording part.  I then do any after the fact editing in Camtasia.  Camtasia also has options for things like recording your screen and recording from both the screen and a camera at the same time.  It keeps the two recordings synchronized so it's easy to do an insert from the camera overlaying what you're writing so viewers can see both you and your text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that said, when I'm recording lectures, I just use PowerPoint and don't include a recording of my face.  I used to but then I realized that what I really wanted students focused on was what I was writing and saying.  My facial expressions and hand gestures were just a distraction.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2020-03-17T13:06:40.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18065" PostTypeId="2" ParentId="8150" CreationDate="2020-03-17T20:22:24.343" Score="1" Body="&lt;p&gt;Feel free to copy and paste my rewrite of these 2 r/ELI5 posts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://old.reddit.com/r/explainlikeimfive/comments/1qpdsv/eli5_what_are_logarithms_actually_doing_when/cdf3tsk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;By user cambridge_ms, 2013 Nov 15&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you multiply a number by itself a few times, you can express that number in terms of an &lt;strong&gt;exponent&lt;/strong&gt;. The exponent is a little number to the upper right of the number that says, &quot;this is how many times the number has been multiplied by itself&quot;. So, &lt;span class=&quot;math-container&quot;&gt;$2 \times 2 \times 2 = 2^3$&lt;/span&gt;. The number 2 has the exponent 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;logarithm&quot; is the reverse of this operation. When we ask, &quot;what is &lt;span class=&quot;math-container&quot;&gt;$\log_28$&lt;/span&gt;?&quot; we are asking, &quot;what is the base 2 logarithm of the number 8&quot;, or, &quot;how many times did we multiply 2 &lt;strong&gt;(the base)&lt;/strong&gt; to get the number 8&quot;. The answer to this question is the &lt;strong&gt;exponent&lt;/strong&gt; from above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$2 \times 2 \times 2 = 2^3 = 8$&lt;/span&gt; (that is, the &lt;strong&gt;base&lt;/strong&gt; 2 multiplied by itself 3 times equals &lt;strong&gt;base&lt;/strong&gt; 2 to the &lt;strong&gt;exponent&lt;/strong&gt; 3 power equals 8), we know that the &lt;span class=&quot;math-container&quot;&gt;$\log_2(8) = 3$&lt;/span&gt;. How many times did we multiply 2 to get the number 8? 3 times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What this means, is that we can use logarithms to express HUGE numbers relative to each other in a much smaller space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imagine a graph that graphs data ranging from 2 to the number 1,048,576. That's going to be a graph with an enormous Y-axis scale (1 million is way bigger than 2).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if we took the base-2 logarithm of every data point, we'd end up with a y-axis that only goes from 1 to 20 (&lt;span class=&quot;math-container&quot;&gt;$\log_2(2) = 1; \log_2(1048576) = 20$&lt;/span&gt;). Thus, we often display data using a log scale because it allows us to relate HUGE numbers to each other on a scale that is much easier to understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://old.reddit.com/r/explainlikeimfive/comments/12u4df/eli5_the_logarithmic_scale/c6y8bsk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;By TheBB, Nov 8 2012&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you measure something. By its very nature, this thing that you measure produces only positive numbers: this is a requirement to use a logarithmic scale. You get these measurements:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1, 3, 500 000, 0.00005&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ouch. That's quite some variation there. If you plot this on a normal scale, these measurements will look unsightly. One point (500 000) is way off somewhere, and the rest sort of clustered around zero. It'll be impossible for you to look at this plot and appreciate that there's a huge difference between 1 and 0.00005.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not a contrived example. I made it up, but lots of natural phenomena are like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what do we do? Use a logarithmic scale! Take &lt;a href=&quot;https://www.rapidtables.com/calc/math/Log_Calculator.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;the logarithm of those numbers&lt;/a&gt; above: &lt;span class=&quot;math-container&quot;&gt;$0, 0.477, 5.699, -4.301$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you plot these smaller numbers on a normal scale, things are much clearer. There's a tiny one, a huge one, and two middling ones. &lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2020-03-17T20:22:24.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18066" PostTypeId="2" ParentId="18058" CreationDate="2020-03-17T20:34:14.033" Score="3" Body="&lt;p&gt;I use Zoom because that is the default software I'm supposed to use for the company I work for. You mentioned 24 hours processing times, but what I've noticed is that most of the live videos and teaching I record actually ends up processing within a few hours (like anywhere between 1-6 hours approximately). Using the document camera works fine but what I use instead is have a digital blank copy of the page I want to show on my 2-in-1 laptop itself, and write on screen exactly the way as I expect my students to write in their notes. I've also used iPad to write as well by logging in as another user on Zoom. You should use mouse to point to things using the built-in pointers/tools that come with Zoom and also in PowerPoint itself.&lt;/p&gt;&#xA;" OwnerUserId="13678" LastEditorUserId="13678" LastEditDate="2020-03-17T21:38:57.113" LastActivityDate="2020-03-17T21:38:57.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18067" PostTypeId="2" ParentId="18048" CreationDate="2020-03-17T20:48:09.873" Score="2" Body="&lt;p&gt;One thing common, especially at middle school/high school levels is to list the unknown (for example, x = 3) as one of the options, especially when the question actually asks you to solve for something else that requires knowing x. At the middle/high school levels, several students have a habit of thinking that &quot;x is the answer, or you're done once your find x&quot;. In such questions, listing x = 3 (or whatever the unknown actually is) as one of the answer choices tends to throw students off. This is how I come up with wrong answers sometimes.&lt;/p&gt;&#xA;" OwnerUserId="13678" LastActivityDate="2020-03-17T20:48:09.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18068" PostTypeId="2" ParentId="18058" CreationDate="2020-03-17T21:02:44.663" Score="4" Body="&lt;p&gt;I just spent ~3 hrs making a 4-min video :-). A few notes that may help those in similar software situations:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;MacOS, &lt;em&gt;Catalina&lt;/em&gt;. Others ignore. &lt;/li&gt;&#xA;&lt;li&gt;I used &lt;em&gt;QuickTime&lt;/em&gt; to record a subset of my screen (&quot;Screen Recording&quot;). &lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Apparently there is no way to pause &lt;em&gt;QuickTime&lt;/em&gt;. So I recorded, stopped, recorded, stopped. And eventually stitched the ~8 recordings into one (using &lt;em&gt;QuickTime&lt;/em&gt; edit: drag each into the 1st).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Re @XanderHenderson's point: There is an option to track the mouse cursor (or not).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;Upload to &lt;em&gt;YouTube&lt;/em&gt;. I recommend ignoring your IT's advice to use Moodle or Blackboard. &quot;YouTube re-encodes every upload at multiple resolutions so your viewers can...&quot; etc. They already have a sophisticated setup that your local IT cannot match. Select &quot;unlisted&quot; and send your students the URL. They are already YouTube experts.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Concerning processing time (OP: &quot;Zoom servers seem to be swamped, with like 24 hr processing times before a video can be viewed.&quot;). &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Save on your local computer, so you are not limited by Zoom's cloud&#xA;storage. &lt;/li&gt;&#xA;&lt;li&gt;Use YouTube.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Caveat: I am not an expert. Learning day-by-day.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-03-18T01:33:13.500" LastActivityDate="2020-03-18T01:33:13.500" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18069" PostTypeId="2" ParentId="16761" CreationDate="2020-03-17T21:35:46.263" Score="1" Body="&lt;p&gt;I would recommend a 2-in-1 laptop rather than a Surface Book due to the price, but of course, that comes with pros-and-cons. Office 365 or Office 2019 with Word, PowerPoint, etc. should be great for not only writing but also saving the notes you write, thereby allowing you to switch screens without the worry of losing the notes you wrote on screen upon switching. I use my Lenovo Yoga to record/share live screen sessions teaching/online, while being able to switch screens, takes notes, record using Zoom, etc.&lt;/p&gt;&#xA;" OwnerUserId="13678" LastActivityDate="2020-03-17T21:35:46.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18070" PostTypeId="1" AcceptedAnswerId="18081" CreationDate="2020-03-17T22:15:07.583" Score="4" ViewCount="1050" Body="&lt;p&gt;I think using the traditional compass with those styluses that can literally be used to hurt or accidentally hurt someone are very dangerous. Most people don't use these in day-to-day life anyways, right? The school I used to work for formerly used &quot;&lt;a href=&quot;https://www.eaieducation.com/Product/534435/Slide_N_Measure%C2%AE_Compass_Assorted_Colors_-_Set_of_10.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Slide N' Measure&lt;/a&gt;&quot; compasses instead of the pointy ones. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/F3bxS.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/F3bxS.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the &lt;a href=&quot;https://www.eaieducation.com/Product/530058/SAFE-T_Compass%C2%AE_-_Set_of_30.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Safe-T compass&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Mz3kx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Mz3kx.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here's an image of the traditional ones:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qMHx3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qMHx3.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Shouldn't we simply get rid of the other pointy ones like in the last picture above from schools?&lt;/strong&gt; &lt;/p&gt;&#xA;" OwnerUserId="13678" LastEditorUserId="13678" LastEditDate="2020-03-18T00:06:28.993" LastActivityDate="2022-03-11T01:29:20.187" Title="Should we stop using traditional compass in schools &amp; start/encourage adopting compasses like &quot;Slide N Measure&quot; or &quot;Safe-T&quot; compasses instead?" Tags="|secondary-education|geometry|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18071" PostTypeId="2" ParentId="18070" CreationDate="2020-03-17T22:28:11.910" Score="6" Body="&lt;p&gt;No.  It removes the ability for so much delightful classroom mischief. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only surpassed by the use of ten pointed dividers by navy QMs to jab each other:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AZ5xT.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AZ5xT.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are you going to do next?  Eliminate dodge ball?  Don't wrap the kids in bubble wrap.&lt;/p&gt;&#xA;" OwnerUserId="13679" LastActivityDate="2020-03-17T22:28:11.910" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18073" PostTypeId="1" CreationDate="2020-03-18T07:22:58.410" Score="5" ViewCount="197" Body="&lt;p&gt;In Korea, basically every teaching material for 10th grade math(about the level of precalculus) contains this kind of exercises in their &lt;strong&gt;first&lt;/strong&gt; treatment of complex numbers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Evaluate &lt;span class=&quot;math-container&quot;&gt;$f(x)=4x^4-8x^3+3x^2+10$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x= \frac{3+\sqrt2 i}{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The standard answer goes like this: &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a zero of the real polynomial &lt;span class=&quot;math-container&quot;&gt;$g(x)=4x^2-12x+1$&lt;/span&gt;, and dividing &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; we get &lt;span class=&quot;math-container&quot;&gt;$f(x)=g(x)(x^2+x+1)+x-1$&lt;/span&gt;. Plug in &lt;span class=&quot;math-container&quot;&gt;$x= \frac{3+\sqrt2 i}{2}$&lt;/span&gt; and we get &lt;span class=&quot;math-container&quot;&gt;$\frac{1+\sqrt2 i}{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have three reasons against having students at the level solve this type of problem. I would like to see what the community has to say about it.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It is more like an algebraic gymnastics than a useful tool for future learning. I don't see the students making use of this in any setting of math learning before they get into undergraduate level courses like Complex Analysis, and even then it won't be too late to pick up this kind of trick.&lt;/li&gt;&#xA;&lt;li&gt;Going back and forth between (mostly rational) real coefficient polynomials and their complex values can blur students' vision of functional domains and codomains when they have absolutely no idea about complex functions.&lt;/li&gt;&#xA;&lt;li&gt;Appropriating polynomial division to that end can mislead students to think of polynomials as something to break down to get to numerical 'answer', rather than mathematical objects on their own with proper operations.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I would like to add something about field extension, about how arbitrarily plugging in complex numbers to real(rational) polynomials could deter students' understanding later on, but that's probably too much when thinking about school math curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what do you think? Is it OK to keep this type of exercise in Precalculus level textbooks?&lt;/p&gt;&#xA;" OwnerUserId="6519" LastActivityDate="2020-03-19T21:12:13.380" Title="Is evaluating a Real Polynomial at a Complex Value a suitable task for Precalculus students?" Tags="|algebra|textbooks|precalculus|complex-numbers|polynomials|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18074" PostTypeId="2" ParentId="18073" CreationDate="2020-03-18T10:51:26.080" Score="1" Body="&lt;p&gt;Personally, I agree with your viewpoints on the topic. Apart from what you mention, such excercises do not unveil the reasons behind the emergence of complex numbers - which, at first glance, are a quite counter-intuitive entity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I would prefer an introduction based on a more historical context such as some cubic equations that need Tartaglia's formulae - i.e. &quot;depressed&quot; cubic equations - and then some equations that need Cardano's general formula. Thus, students have to manipulate complex numbers so as to arrive to even real solutions of the initial equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another approach that I have followed in classes that have been taught about vector plane geometry is initiating a discussion about how one could, alongside the typical vector addition on the Cartesian plane, define a multiplication. After some discussion, we end up that multiplication seen as rotation is a suitable choice for such an extension. Then, roation is written, using some trigonometry, analytically in terms of the vector's coordinates. At this point, the notion of &lt;span class=&quot;math-container&quot;&gt;$i=\sqrt{-1}$&lt;/span&gt; appears as a very useful convention so as to extend the already known properties of the reals to the new &quot;complex&quot; plane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above may sound too theoretical, but I do not invoke that much &quot;hard-coding&quot;. Instead of verifying multiplication properties using trigonometry etc I prefer using some images like the ones below - which show how rotation satisfies the distributive law etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5Kgr0.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5Kgr0.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2020-03-18T10:51:26.080" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18075" PostTypeId="2" ParentId="18073" CreationDate="2020-03-18T11:32:33.983" Score="0" Body="&lt;p&gt;I think some algebraic gymnastics can be a good thing.  They will need to keep track of such things when they do series (moderately in 10th grade and then in more detail for calculus and ODEs).  Some &quot;muscles&quot; for this sort of thing are not bad.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, I think the &quot;concept lovers&quot; make a mistake in not allowing for some approach to learning a topic first, more mechanically.  Or at least alternately more mechanically.  There is a saying about quantum mechanics, that you just get used to it over time, rather than thinking about it intuitively immediately.  Certainly just lots of manipulation of i (sort of a strange quantity, philosophically) is a way of getting used to it.  Just as very significant practice/exposure was needed in elementary school with negative numbers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do think if this is the very first problem, than that's not appropriate.  (Start with definition of i and simple quadratics.  But I doubt it is very first problem, since the students know what i is.)  But within an overall chapter?  It's OK.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, nothing prevents later learning/thinking about i in different ways: phase angle, impedance versus resistance, etc.  But it is nice to already be sort of familiar with our old friend i as just that thing that makes solutions of otherwise impossible quadratics (and other polynomials).&lt;/p&gt;&#xA;" OwnerUserId="13679" LastActivityDate="2020-03-18T11:32:33.983" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18076" PostTypeId="1" CreationDate="2020-03-18T12:46:05.457" Score="1" ViewCount="261" Body="&lt;p&gt;There are some general questions about potential topics for undergraduate seminars:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/q/11963/13681&quot;&gt;topics for an undergraduate Math seminar&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/q/14514/13681&quot;&gt;Undergraduate Math Seminar topic&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am looking for topics for a 15-hour &lt;em&gt;undergraduate mathematics seminar&lt;/em&gt;, to be attended by both Master-level &lt;strong&gt;mathematics educators&lt;/strong&gt; or Bachelor-level &lt;strong&gt;mathematics undergraduates&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prerequisites would ideally be limited to linear algebra, real analysis and elementary geometry (some Euclidean and hyperbolic geometry), which everyone will have seen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this question I would like to ask about topics which are directly relevant to the &quot;life&quot; of a mathematics educator, for example&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;deepening understanding of topics that are usually tacitly assumed or brushed over when entering university because &quot;everyone knows this from high school&quot; (e.g. how to really construct the real numbers, what is a set really)&lt;/li&gt;&#xA;&lt;li&gt;relation to high school curriculum (calculus, Euclidean geometry, probability theory, ...)&lt;/li&gt;&#xA;&lt;li&gt;relation to daily life (voting theory, mental arithmetic, ...)&lt;/li&gt;&#xA;&lt;li&gt;relation to recreational ese topics would be imathematics (origami mathematics, ...)&lt;/li&gt;&#xA;&lt;li&gt;etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(This eliminates quite a few fun topics — for example I don't see how knot theory, or fractals, or singular value decomposition (and compression of images, say) would be directly relevant to teaching high school mathematics — of course I don't mean to imply that that these topics shouldn't be learned by mathematics educators!)&lt;/p&gt;&#xA;" OwnerUserId="13681" LastEditorUserId="13681" LastEditDate="2020-03-22T11:47:23.960" LastActivityDate="2020-03-23T12:39:28.820" Title="Topics for undergraduate seminar for mathematics educators" Tags="|undergraduate-education|secondary-education|mathematics-in-daily-life|seminar|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18077" PostTypeId="2" ParentId="18059" CreationDate="2020-03-18T21:22:01.237" Score="2" Body="&lt;p&gt;If your students are 7 years old, I believe you are asking the wrong question. At this age I would not try to be fair, but try to motivate them. To me this has much to do with evaluating each student with respect to their efforts. So (to me) the question is &lt;strong&gt;not&lt;/strong&gt; &quot;how much does student A know compared to student B&quot;, but rather &quot;how much effort did student A put into learning the current subject&quot;. Since math exams at the age of 7 are unable to achieve this goal, I would try to be generous. In addition I would ask myself: Do I motivate my student by telling them, that the marks are given w.r.t. the other students? If you are not telling them this fact, then it really doesn't matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm teaching at the university level. Unfortunately, the classes are too large to know every student personally, so at first sight it seams to be hard to judge on an &quot;effort scale&quot;. However, my students have to work on problem sheets and use an online platform to hand in their solutions. These &quot;problem sheet grades&quot; partially add to the grade achieved in the final exam. However, what the students don't know is that I use these &quot;problem sheet grades&quot; to estimate their effort. In every exam there are extra points, which could be argued either side. Either I am generous and give half a point for a partial solution, or the solution is insufficient.  Thus, even at this stage (20+ years old) I do not try to be fair, but rather try to motivate them by judging them on an &quot;effort scale&quot;. &lt;/p&gt;&#xA;" OwnerUserId="13682" LastActivityDate="2020-03-18T21:22:01.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18078" PostTypeId="1" AcceptedAnswerId="18079" CreationDate="2020-03-18T23:35:22.013" Score="8" ViewCount="630" Body="&lt;p&gt;Due to COVID-19, I have been planning to transition to online teaching (which, of course, includes online testing as well). The LMS that we use is Blackboard which is integrated with Proctorio. However, they work well for multiple choice problems not free response math problems. Any ideas on how to conduct online testing for course like Calculus in which we cannot give take-home exams as the answers are easily available online.&lt;/p&gt;&#xA;" OwnerUserId="13683" LastEditorUserId="13683" LastEditDate="2020-03-19T04:14:35.937" LastActivityDate="2020-03-19T15:10:35.640" Title="How to conduct online testing for Calculus?" Tags="|undergraduate-education|calculus|online-instruction|testing|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18079" PostTypeId="2" ParentId="18078" CreationDate="2020-03-19T15:10:35.640" Score="3" Body="&lt;p&gt;Well, one idea would be that of shifting to evaluating students through projects. For instance, there could be created a pool of project topics which will be randomly assigned to students. This may be way off the usual written examination, it makes it, however, necessary for the students to process the knowledge they have obtained conceptually and not only &quot;copy-and-paste&quot; from the internet or each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some sample topics related to calculus could be the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The critical comparison of the several ways in which one may express the completeness axiom - e.g. through suprema or using Cantor's principle or using Bolzano-Weierstass's theorem orthrough Heine-Borel theorem etc.&lt;/li&gt;&#xA;&lt;li&gt;Several proofs of Bolzano's theorem for continuous functions - related to the previous one.&lt;/li&gt;&#xA;&lt;li&gt;A discussion upon the definitions of integrable functions using Riemann and Darboux sums - equivalence etc. This could be extended to include Riemann-Stieltjes integral.&lt;/li&gt;&#xA;&lt;li&gt;The creation of a cognitive map connceting all the theorems of calculus the students have been exposed so far - e.g. in terms of which can be proved using some other(s).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A drawback of such a way of examination is that it is usually focused on some specific part of the topics discussed in a calculus course. To treat this, one could split each student's evaluation to two or more projects, coming from different areas of calculus.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2020-03-19T15:10:35.640" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18081" PostTypeId="2" ParentId="18070" CreationDate="2020-03-19T21:53:05.493" Score="14" Body="&lt;p&gt;I started my class year with a combination of traditional compasses and a set of slide-and-measure compasses that last year's teacher bought at the end of the year before giving the class to me.  I was extremely underwhelmed by their performance.  Among the issues:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The pencil hole would slide while you were tracing an arc, which mostly defeats the purpose of the entire enterprise&lt;/li&gt;&#xA;&lt;li&gt;Our workbooks were spiral-bound, and these compasses can only draw an arc if the entire compass can lay flat on the page, frustrating my students to no end&lt;/li&gt;&#xA;&lt;li&gt;Being made of plastic, they would fall apart in the students' hands.  (It is possible that the other teacher bought particularly cheap models, but it's not clear to me how the device could be made to last given the best intentions.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'd buckle in and live with it if stabbings were a frequent event in my classroom.  But it simply isn't.  And if a student were to stick a compass into a classmate, I would be infinitely more prone to hold the student accountable instead of the compass.  As others have noted, the general education environment is full of nigh-equally pointed objects.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you're dealing with a cohort of students with emotional disabilities, then by all means you should set up your classroom to restrict the ability to cause harm to oneself or others.  Barring that, they are a bad solution to a non-existent problem.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-03-19T21:53:05.493" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18083" PostTypeId="1" CreationDate="2020-03-20T19:44:06.230" Score="10" ViewCount="195" Body="&lt;p&gt;I suspect I'm not the only one here who finds himself with a dormant math circle due to quarantine.  I'm interested in two kinds of suggestions regarding this situation:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;suggestions for running a math circle remotely, to the extent that the best practices are different from what one would do for a class, as recorded in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18031/how-shall-we-teach-math-online&quot;&gt;How shall we teach math online?&lt;/a&gt; (e.g., how to recreate the participation-, exploration-, and activity-heavy nature of a math circle compared to a perhaps more typical math class); and&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;suggestions for good resources for math circle participants when the circle co-ordinators are not able to devote the resources to running an offline circle.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I'm in situation #2, but I assume that #1 is of interest, and the questions seem closely enough related to put them together.  The circle of which I'm a part is for US 5–9 graders (roughly 10–14 years old); I'm open to suggestions at any level, but, if that makes it too unfocussed, then we can focus on that age range.&lt;/p&gt;&#xA;" OwnerUserId="2070" LastEditorUserId="2070" LastEditDate="2020-03-22T14:34:44.077" LastActivityDate="2020-03-24T04:07:22.297" Title="Math circles while under quarantine" Tags="|online-instruction|math-circle|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18084" PostTypeId="2" ParentId="18058" CreationDate="2020-03-20T21:41:29.810" Score="4" Body="&lt;p&gt;I have an old Wacom Bamboo tablet and pen and a microphone. I use xournal (&lt;a href=&quot;http://xournal.sourceforge.net&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://xournal.sourceforge.net&lt;/a&gt;) and record what I'm writing and saying using a screencasting program called vokoscreen, starting and stopping occasionally so I get a bunch of video files. I use flowblade (&lt;a href=&quot;https://jliljebl.github.io/flowblade/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://jliljebl.github.io/flowblade/&lt;/a&gt;) to splice these together and then use ffmpeg to compress them. I upload to YouTube, though I'm looking into also uploading to alternative sites that are accessible from within China.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once I've got a video ready, I watch it back and type up a transcript (with timings) in LaTeX: this ensures that the splicing didn't introduce any errors (e.g. including bloopers in the final cut) and also gives students a transcript. Technically, it's not LaTeX that I write, but a shorthand I devised a few years ago which I call lazy latex (lzl). I have a Python script (available here: &lt;a href=&quot;https://github.com/jde27/lzl&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://github.com/jde27/lzl&lt;/a&gt;) which outputs HTML from the lzl file, in particular turning tikz into embedded images and rendering maths using MathJax. Then I end up with an HTML page with the notes and the video embedded at the top.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may sound like a big faff to set up, but once it's up and running you can record quite quickly: I made eight videos today. In the past I used this to produce a flipped lecture course on algebraic topology:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://jde27.uk/tg&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jde27.uk/tg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and am in the middle of doing the same for my linear algebra lectures for next term (for coronavirus-reasons)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://jde27.uk/la/01_matrices.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jde27.uk/la/01_matrices.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My advice: if you're trying to produce videos like this, don't aim for perfection, trying to rerecord every little glitch. It takes long enough as it is. Only rerecord if you think your explanation is going on too long/is too complicated/is really bad/misses some really crucial point. The occasional &quot;typo&quot; or slip of the tongue doesn't matter too much and gives people something to ask questions about in comments.&lt;/p&gt;&#xA;" OwnerUserId="13028" LastEditorUserId="13028" LastEditDate="2020-03-20T21:48:30.420" LastActivityDate="2020-03-20T21:48:30.420" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18085" PostTypeId="2" ParentId="10895" CreationDate="2020-03-21T06:47:00.370" Score="2" Body="&lt;p&gt;When I use the term &amp;quot;mathematician&amp;quot;, I set a pretty high bar.  By way of example,&#xA;I have been working in mathematics for more than a decade now:  I started as a secondary math teacher (after finishing a BA in mathematics), then went back for a masters (the local school district was undergoing a hiring freeze following the financial crisis of 2008, so it was return to academia, or continue moving from one long-term substituting gig to another for an indefinite period of time), and ended up pursuing a PhD (which will be done in another month or three).  I have a paper in a good journal, have presented at a number of important conferences in my field, and have even been flown to the UK to give a talk.  Yet I don't really consider myself to be a mathematician—I only have one good pub, and no PhD yet.&lt;/p&gt;&#xA;&lt;p&gt;All of that being said, I would never presume to tell someone else that they are not a mathematician—it simply isn't my place.  I (obviously?) have sympathy for the point of view that a middle school mathematics instructor is not a mathematician, but I am not going to attempt to impose my point of view on others.  This just seems arrogant to me.  If a middle school math instructor claims to be a mathematician, great.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, I can see the problems that this might cause.  I think, however, that the correct approach is to try to change the minds of your &lt;em&gt;students&lt;/em&gt;, rather than to try to change he behaviour of your colleagues.  For example, you suggest that students say things such as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Being a mathematician seems a bit boring and easy. My Grade 9 teacher is a mathematician and it really doesn't seem like an interesting or difficult occupation. There's only so much you can learn about fractions and trigonometry.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A reasonable response might be&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Your grade 9 teaching is &lt;em&gt;one kind&lt;/em&gt; of mathematician.  Her job is to take very difficult ideas and explain them in an approachable manner.  It is, perhaps, a good sign that her job seems easy and uninteresting, as this indicates that she is doing an excellent job of making the material seem easy and approachable.&lt;/p&gt;&#xA;&lt;p&gt;Of course, the are different kinds of mathematicians.  There are, for example, mathematicians who spend their time working on difficult mathematical puzzles which require a lot of heavy abstraction.  These &amp;quot;pure mathematicians&amp;quot; typically work at universities (though not always).  As a student, you might encounter them as an instructor in a classroom, but this is only a small part of their job.  Mathematicians of this type spend the majority of their time engaged in research, which involves a lot of explaining their ideas to others in papers and conference talks.&lt;/p&gt;&#xA;&lt;p&gt;Another type of mathematician might work in &amp;quot;industry&amp;quot;.  These mathematicians solve practical, real world problems.  For example, these kinds of &amp;quot;applied mathematicians&amp;quot; might work for health organizations, where they may be tasked with understanding how diseases (such as the coronavirus) spread.  Such mathematicians are also involved in understanding climate change, and in designing rocket systems which might one day take people to Mars (and beyond!).  There are even applied mathematicians working in the CIA and NSA, where they seek design (and break) better methods for sending secret methods from one place to another.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you don't like my examples of other kinds of mathematicians, pick your own examples.  The main point is that a middle school teacher is just one &lt;em&gt;kind&lt;/em&gt; of mathematician.  If students don't find that kind of mathematics interesting, there are other ways of engaging with the field.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-03-21T14:10:01.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18086" PostTypeId="2" ParentId="18031" CreationDate="2020-03-21T08:42:45.637" Score="5" Body="&lt;p&gt;My university (I am in Spain) has Microsoft Teams integrated with student accounts, so I use this. My classes are two hours. I open a chat (there are 60-70 students). (Zoom for free only allows 40 minutes and I am not sure it can handle the 60-70 students; a colleague is using the free video mixing software OBS and broadcasting class via Youtube, but learning to us OBS is nontrivial.) I do not show my face. I speak over whatever I am showing. Class is a mix of microsoft's blackboard, which I use as a traditional blackboard via a drawing tablet, and graphics and code generated by or written in various applications (depends on the class I am teaching) such as matlab, C, geogebra, python, etc. (Before I realized I could use microsoft's blackboard I was using the freely available Mypaint and Autosketchpad as blackboards). I mimic what I would do in a classroom, with certain modifications to adapt to the format. Students can ask questions (controlling interruptions and excessive question askers is more difficult than in person, and engaging the passive and shy is a problem I have not solved). I use a headset (10 euros) and drawing tablet (the one I bought cost 70 euros and works fine); these I went out and bought the first day all this happened. The university does not provide us such resources. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I record class and put the video at the student's disposition. I also record (outside of class) myself solving exercises and make these videos available to students. Prepreparing quality videos requires a lot (a lot) of work. A virtue of using a drawing tablet is that it really mimics a traditional blackboard (even improves upon - more colors, neater writing, better drawings, no chalkdust, everything is saved for later consultation) and does not require the fine tuning that a preprepared video (MOOC style) does. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Time will tell how all this works but my initial experience is relatively good. I've had some technical problems - computer freezing up - because I'm working out of home. I run two computers at once, using one as a monitor the way stage performers do (both are old and slow, so one serves as a backup also, and I have had to use it). Also the internet connection is not what it is at the university. Uploading gigas of videos takes a lot of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assessment is a difficult problem. The university has more or less suspended all assessment until it can be figured out what can be done. Exams with the students off site are considered useless because of the ease of obtaining assistance, collaborating, cheating, etc. As it seems possible that presencial examination will be not be possible in this semester (we are in Spain) other options are being explored. So called secure systems for online exams have been considered but they are not considered effective and they are very expensive. Indications are that we are headed towards a system of online oral exams. This will be very time consuming (in some cases four or five professors have to examine 600 students), and there are concerns about guaranteeing uniformity/homogeneity as well as concerns about the availability to students of resources to realize the exams. Something will have to be done, and some imperfections will have to be accepted.  &lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2020-03-21T08:42:45.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18088" PostTypeId="2" ParentId="18031" CreationDate="2020-03-22T13:30:00.173" Score="3" Body="&lt;p&gt;My home teaching setup. Still to be tested. &lt;a href=&quot;https://i.stack.imgur.com/Cc6Zy.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Cc6Zy.jpg&quot; alt=&quot;My home teaching setup&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10168" LastActivityDate="2020-03-22T13:30:00.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18089" PostTypeId="2" ParentId="18076" CreationDate="2020-03-22T17:04:38.833" Score="4" Body="&lt;p&gt;There are deep mathematical questions everywhere.  I am fairly certain that, given enough time to prepare, I could take &lt;em&gt;any&lt;/em&gt; topic in mathematics, no matter how elementary, and present a 60 minute talk that is both (1) accessible to a lay-audience and (2) concludes with several fairly deep&amp;mdash;possibly even research-level&amp;mdash;questions.  By way of example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My go-to talk when I am asked to speak to high school students, non-math major undergraduates, or a general lay audience is on dimensions.  It starts with a series of fairly simple examples, which are built up slowly:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;&lt;p&gt;If a one-dimensional object of length &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; is scaled by a factor of &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, the scaled object has length &lt;span class=&quot;math-container&quot;&gt;$Ls$&lt;/span&gt;.  This can be justified non-rigorously by appealing to &quot;obviousness&quot; for a line segment, then generalizing to the perimeter of a polygon.  If one wants to take a digression into a little bit of calculus, the more general result follows by approximating a (rectifiable) curve with segments.  Otherwise, appeal to intuition.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;If a two-dimensional object of area &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is scaled by a factor of &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, the scaled object has area &lt;span class=&quot;math-container&quot;&gt;$As^2$&lt;/span&gt;.  This is &quot;obvious&quot; for a square, and any &quot;reasonable&quot; region can be approximated by squares.  One can also appeal to formulae for the ares of standard examples (triangles, disks, etc).&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;If a three-dimensional object of volume &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is scaled by a factor of &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, the scaled object has volume &lt;span class=&quot;math-container&quot;&gt;$Vs^3$&lt;/span&gt;.  Again, start with the &quot;obvious&quot; example of a cube, and work from there.  I like to use Minecraft as an example of how three-dimensional objects can be approximated by cubes (voxels, in that context).&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;From these examples, we might conclude that the dimension of an object is the exponent that shows up when that object is scaled.  If an object has measure &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; (whatever &quot;measure&quot; means: non-rigorously, the measure of a one-dimensional object is length, of a two-dimensional object is area, and so one&amp;mdash;this hints, however, at measure theory, which is a deep mathematical idea already) and is scaled by a factor of &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, then the scaled object will have measure &lt;span class=&quot;math-container&quot;&gt;$Ms^n$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is the dimension of the object.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;At this point, we can introduce the &lt;a href=&quot;https://en.wikipedia.org/wiki/Sierpinski_carpet&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sierpinski carpet&lt;/a&gt;.  After showing students how the Sierpinski carpet is constructed, we can ask about the dimension of the carpet.  Students typically assert that it must be two-dimensional, but we can show them that it isn't!  Suppose that we start with a carpet of measure &lt;span class=&quot;math-container&quot;&gt;$M=1$&lt;/span&gt; unit (again, whatever &quot;measure&quot; means&amp;mdash;it is a little unclear in this context).  If we scale this carpet up by a factor of &lt;span class=&quot;math-container&quot;&gt;$s=3$&lt;/span&gt;, it can be precisely covered by &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; copies of the original carpet.  This means that the scaled carpet has measure &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; units.  But remember that the measure of the scaled carpet &lt;em&gt;should be&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$Ms^n = 3^n$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is the dimension of the original carpet.  Now for the scary part: a logarithm!  A little bit of computation reveals that&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$ 8 = 3^n \implies n = \log_3(8) \approx 1.893. $$&lt;/span&gt;&#xA;  Hence the Sierpinski carpet is approximately &lt;span class=&quot;math-container&quot;&gt;$1.893$&lt;/span&gt;-dimensional!&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Now the talk can be concluded in a number of ways, generally by asking questions.  Because it is my area of research, I typically start asking about whether or not this notion of dimension is reasonable.  For example:&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;&lt;p&gt;Most shapes cannot be covered by smaller copies of themselves (they are not &quot;self-similar&quot;), so can the computation for the Sierpinski carpet be generalized at all?&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;The notion of &quot;measure&quot; is quite nebulous and inexact&amp;mdash;can we pin it down at all? (Welcome to measure theory!)&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;The Sierpinski carpet example shows that dimensions need not be integer valued.  Given a real number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, can we make a shape which is &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-dimensional?  What if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is negative?  What if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; isn't real at all, but is complex?!&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;  &#xA;  &lt;p&gt;Notice that any one of these questions could, with a little more background, form the foundation of a PhD thesis project.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would suggest that the main thrust of a seminar of the type described in the question here should be to expose potential secondary instructors to this kind of thinking.  These instructors should understand that very simple ideas can lead to deep questions&amp;mdash;they need to know that the material taught in high school fits into a larger context, and they need to have enough knowledge of that context in order to answer difficult questions posed by really clever students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also suggest that the actual topics taught in such a seminar are utterly irrelevant.  The goal should be to get the seminar attendees to &lt;em&gt;think mathematically&lt;/em&gt;.  My example is fractals, 'cause that's what I know.  But the other topics dismissed by the question-asker are also fertile ground:  a few years ago, I attended a great talk by Gilbert Strang on singular value decomposition which fairly neatly falls into the category of lectures I've described.  I think that knot theory is a great way to engage in mathematical reasoning, as it provides a path for asking the fairly deep question of how we can tell things apart.  Thus I would be careful about dismissing any particular topic as &quot;not relevant&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My recommendation would be to pick several topics which are of interest to either the student in the seminar, or the organizers.  Pick a couple of research topics, and spend some time getting at the deep questions.  Pick an example which could benefit from some coding (SVD?  the Collatz conjecture?).  Mine the content of, for example &lt;a href=&quot;https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw&quot; rel=&quot;nofollow noreferrer&quot;&gt;3Blue1Brown&lt;/a&gt; for interesting topics.  Have the organizers talk about their own mathematical research.  &lt;em&gt;Any&lt;/em&gt; topic is fair game, if it is approached correctly.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2020-03-22T17:04:38.833" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18090" PostTypeId="2" ParentId="18083" CreationDate="2020-03-22T20:58:40.787" Score="4" Body="&lt;p&gt;&lt;strong&gt;Question #1:&lt;/strong&gt; This thought comes from teaching college classes, but may help a math circle too. You can communicate more often online, so you can keep them playing with ideas by sending out daily puzzles. Or you can send out a few puzzles related to your topic a day or two before &quot;meeting&quot;, to get them thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question #2:&lt;/strong&gt; For 5th to 9th graders, assuming not much facility with algebra, but some, and assuming they enjoy math.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Geometric construction (as mentioned in my comment): here (&lt;a href=&quot;https://sciencevsmagic.net/geo/&quot; rel=&quot;nofollow noreferrer&quot;&gt;sciencevsmagic.net/geo&lt;/a&gt;) and here (&lt;a href=&quot;http://www.euclidthegame.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;euclidthegame.com&lt;/a&gt;). The two sites have some overlap but very different presentations and choices of challenges. Euclidthegame uses geogebra, so you can try your constructions there too; 25 levels. Sciencevsmagic does not; 10 shapes, 4 challenges for each. (Euclidthegame has comments, which sometimes contain helpful hints, sometimes contain wrong info, and sometimes are not suitable for kids.)&lt;/li&gt;&#xA;&lt;li&gt;Modular arithmetic can be fun, and might make questions about Pythagorean triples accessible. You can look up Pythagorean triples, and you'll find lots of information. Or you could start with &lt;a href=&quot;https://mathmamawrites.blogspot.com/2009/12/pythagorean-triples.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;my blog post&lt;/a&gt; on it. Not modular but related: I ran a math circle for families in my home, where we used base 3. &lt;a href=&quot;https://mathmamawrites.blogspot.com/2010/01/richmond-math-salon-and-base-three.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's my blog post&lt;/a&gt; about that.&lt;/li&gt;&#xA;&lt;li&gt;Rodi Steinig does amazing math circles and has blogged extensively about them. Some will translate to online. Perhaps &lt;a href=&quot;https://talkingsticklearningcenter.org/data-interpretation-and-analysis/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this voting one&lt;/a&gt; would be interesting?&lt;/li&gt;&#xA;&lt;li&gt;The &lt;a href=&quot;https://jrmf.org/activities&quot; rel=&quot;nofollow noreferrer&quot;&gt;Julia Robinson Mathematics Festival&lt;/a&gt; has tons of great problems. (Click on the Activities button and wait for the pdf to download.)&lt;/li&gt;&#xA;&lt;li&gt;The &lt;a href=&quot;https://www.mathcircles.org/activities&quot; rel=&quot;nofollow noreferrer&quot;&gt;National Association of Math Circles&lt;/a&gt; has a list of activities too.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-03-22T20:58:40.787" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18091" PostTypeId="1" CreationDate="2020-03-23T06:11:06.493" Score="9" ViewCount="2243" Body="&lt;p&gt;Locked up in my self-distancing isolation in NYC, I'm reminded of how much I really like Frankel's book, which contains a wealth of beautiful geometry and topology from the standpoint of a mathematically minded physics student. It contains so much pure mathematics from a first year graduate course in differential geometry with so many pictures, but even more wonderful, it contains many applications of this material to physics, particularly classical mechanics and electromagnetism. Since I've always believed mathematics-REAL mathematics, with proofs!-and the physical sciences should never be separated, I naturally found the book an amazing find. Interestingly, as I've asked around, I've found that mathematicians and mathematics students generally like the book considerably more then the physicists the book was written for!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's my question: Despite it's beauty, as it was written for advanced physics students, it would be a bit problematic to use as a text for pure mathematics graduate students. While it certainly covers enough pure geometry for such a course, Frankel isn't as careful with his proofs as one would like for such a course. Indeed, he admits as much in his preface. I'd like to know what pure mathematics text would the geometers on the exchange would recommend as a supplement to &quot;tighten up&quot; such a course. Ideally, one wouldn't want that book to be too long and comprehensive, like Spivak, since it would repeat most of the material in Frankel. I was considering using Frank Warner's book. Ordinarily, I'd be horrified to recommend it as the sole text for a course, but as a supplement to Frankel, it might be ideal as the 2 books seem to complement each other beautifully! I'd love to try to use Conlon's beautiful book also, but doesn't seem to have enough range. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do the geometers here think? Any other recommendations? &lt;/p&gt;&#xA;" OwnerUserId="7033" LastActivityDate="2020-04-06T03:45:41.957" Title="A Question about Theodore Frankel's &quot;The Geometry of Physics&quot;" Tags="|graduate-education|physical-sciences|differential-geometry|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18092" PostTypeId="2" ParentId="18076" CreationDate="2020-03-23T12:39:28.820" Score="1" Body="&lt;p&gt;I would do something on point group symmetry or 2-D space groups.  (Allows to bring in some Escher and the like.)  Keep it a little &quot;art-y&quot; and emphasize examples, description, and recognition.  Can bring to a close with some remarks about group theory, but I wouldn't even try to definitively describe the topic in this abstract manner (not enough time and too hard).  But at least letting them know it exists.  (Sort of how we can tell people that only 4th and lower polynomials are all soluble to radicals...without proving it, but just saying somebody did prove it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Resist any tendency to assume these math teachers have had/understand real analysis and want to go deeper down that hole, getting even &quot;better&quot; at it.&lt;/p&gt;&#xA;" OwnerUserId="13698" LastActivityDate="2020-03-23T12:39:28.820" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18093" PostTypeId="2" ParentId="18083" CreationDate="2020-03-24T04:07:22.297" Score="0" Body="&lt;p&gt;At a technological level, teleconferencing systems like Zoom, Google Meet, etc. often have &quot;virtual whiteboards&quot; which can be shared by a group of participants in a meeting.  If the participants have touch screens and preferably a stylus, they can write on the virtual whiteboard.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Access to touch screens and styluses is not as common as I'd wish, but you might be surprised by how many of your participants can do this with (e.g.) an iPad or a Google Chromebook.  &lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2020-03-24T04:07:22.297" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18094" PostTypeId="1" CreationDate="2020-03-24T12:36:10.727" Score="3" ViewCount="218" Body="&lt;p&gt;I would like to teach math online I'm wondering what is the tools for mathematical presentation.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For example, I tired Powerpoint with MathType but with no luck &lt;/li&gt;&#xA;&lt;li&gt;Beamer for mathematical presentation but I found it very exhaustive&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.mathcha.io/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathcha.io/&lt;/a&gt;   Online Mathematics Editor&#xA;a fast way to write and share mathematics&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/To2qB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/To2qB.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="489" LastEditorUserId="489" LastEditDate="2020-03-24T15:54:29.650" LastActivityDate="2020-03-24T15:54:29.650" Title="Tools for Mathematical presentation" Tags="|interactive-teaching|online-instruction|presentation|" AnswerCount="0" CommentCount="3" ClosedDate="2020-03-27T19:32:14.080" ContentLicense="CC BY-SA 4.0" />
  <row Id="18095" PostTypeId="2" ParentId="18091" CreationDate="2020-03-24T13:01:01.257" Score="0" Body="&lt;p&gt;Feeling bad for you with no answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might (or might not!) be opposite to the direction you need, but you take a look at the Kreysig book in this area.  He was a mathematician but with a very strong sympathy for engineers and physicists. And in any case, it is very strong pedagogically (progression of ideas based on student learning not math efficiency, answers to exercises in the back, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486667219&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Differential-Geometry-Dover-Books-Mathematics/dp/0486667219&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13702" LastActivityDate="2020-03-24T13:01:01.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18096" PostTypeId="2" ParentId="4379" CreationDate="2020-03-24T17:36:43.583" Score="1" Body="&lt;p&gt;A short term fix that misses a lot of pieces is just a video conferencing app with a camera aimed down at a pad of paper.  This is certainly the fastest way to push equations across the internet.&lt;/p&gt;&#xA;&lt;p&gt;Mount the camera on a cardboard 'baseball cap' on your monitor.  The camera has to be upside down to its usual orientation.  A flex lamp or spring cantilever lamp can also be used.&lt;/p&gt;&#xA;&lt;p&gt;If you are tutoring a bunch of students, something like Google Hangouts may be in order.&lt;/p&gt;&#xA;&lt;p&gt;As you finish pages, snap a pic with a smart phone.  Ideally this uploads immediately to a shared folder.&lt;/p&gt;&#xA;&lt;p&gt;Later tag images to facilitate finding them again.&lt;/p&gt;&#xA;&lt;p&gt;Use a black fine point marker.  Pen or pencil don't show up well.  You may find that using coloured paper to reduce the contrast between pen and paper will make your photos more readable.&lt;/p&gt;&#xA;&lt;p&gt;Lots of PDF graph papers on line for quick changes.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Added several years later:&lt;/p&gt;&#xA;&lt;p&gt;The gotcha with this is the image reversal.  On lots of telconference systems your local view is reversed right for left.  So a camera aimed down gives &lt;em&gt;you&lt;/em&gt; mirror writing, but sends a proper image to the other end.&lt;/p&gt;&#xA;&lt;p&gt;You need to look at your paper, not at your screen&lt;/p&gt;&#xA;" OwnerUserId="2114" LastEditorUserId="2114" LastEditDate="2020-09-06T02:47:00.353" LastActivityDate="2020-09-06T02:47:00.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18097" PostTypeId="1" CreationDate="2020-03-24T19:00:01.190" Score="5" ViewCount="239" Body="&lt;p&gt;Like many other math teachers, I am teaching remotely in Spring term.  I teach Math 60 (Intro Algebra) and Math 95 (Intermediate Algebra) at a community college.  In the age of math solving technology (like Photomath), I want to create more conceptual or more open-ended test questions, so the student can't just point their phone at an equation and have the phone solve it for them.  (Yes, I'm being quite cynical here!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some of the topics I teach:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Solve basic linear equations, like 8x + 5 = 6x - 5 &#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Factoring&lt;/li&gt;&#xA;&lt;li&gt;Add rational expressions, like 1/x + 1/(x+1)&lt;/li&gt;&#xA;&lt;li&gt;Solve rational equations.&lt;/li&gt;&#xA;&lt;/ol&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;One idea I have is giving two different proposed solutions to an equation, like #1 above, where each proposed solution is either correct, has a small error, or has a big error.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any creative ideas?&lt;/p&gt;&#xA;" OwnerUserId="3265" LastActivityDate="2020-03-24T20:40:41.953" Title="help thinking of conceptual or open-ended Intro Algebra and Intermediate Algebra test questions" Tags="|test-design|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18098" PostTypeId="2" ParentId="18097" CreationDate="2020-03-24T20:40:41.953" Score="3" Body="&lt;p&gt;I taught algebra at a community college for several years (recently switched to CS), and this was largely my goal for weekly quizzes (given in online multiple-choice format via Blackboard). Roughly half the questions were directly computational or symbolic manipulations, with the other half an attempt at being conceptual. I've taken my quiz source document for the intermediate algebra class and cut it down to most of the conceptual prompts/examples here:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Name the result of the operation.&#xA;1) The result of a subtraction operation is called a ____. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Identify the correct word.&#xA;2) Any written combination of numbers and operations (with no equal sign) is called a(n) ____. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Write the sentence as an equation, using n to represent the unknown number.&#xA;5) Eight times the difference of 12 and a number is -56.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Identify the false statement regarding signed numbers.&#xA;6) Select the statement regarding signed numbers which is false.&#xA;A) adding a positive and a negative may result in a number of either sign&#xA;B) multiplying two negatives results in a positive number&#xA;C) multplying a positive and a negative results in a negative number&#xA;D) adding two negatives results in a positive number&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Characterize the set of numbers.&#xA;11) When a rational number is expressed in decimal notation, which of the following is not a possibility?&#xA;A) an infinite, non-repeating decimal      B) a terminating decimal&#xA;C) a repeating decimal      D) a non-terminating decimal&lt;/p&gt;&#xA;&#xA;&lt;p&gt;List all numbers from the given set that are of the indicated type.&#xA;12) Rationals in {9, -15, 0, (2/3), 0.overbar(5), 0.47, π}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Identify the property illustrated by the statement.&#xA;13) (4 ∙ 3) ∙ 2 = 4 ∙ (3 ∙ 2)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Find the inverse.&#xA;14) Find the multiplicative inverse of 3(8/9).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analyze the form of the expression.&#xA;15) Why can we not say that this expression is &quot;simplified&quot;?&#xA;3((2/3)x + 8) + (6/15)x&#xA;A) fractions not reduced      B) parentheses not removed&#xA;C) like terms not combined      D) all of the above&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Identify the incorrect rule of exponents.&#xA;16) According to the Fundamental Rule of Exponents, which of the following shortcuts on same-base powers is incorrect?&#xA;A) exponents add powers      B) multiplying adds powers&#xA;C) division subtracts powers      D) radicals divide powers&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Identify the correct rule of distribution.&#xA;17) According to the General Distribution Rule, which of the following operations will distribute piecewise?&#xA;A) exponents over addition      B) radicals over subtraction&#xA;C) radicals over multiplication      D) exponents over subtraction&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Identify the polynomial by its degree and number of terms.&#xA;21) 15s^2 + 9s + 6&#xA;A) quadratic monomial      B) linear trinomial      C) cubic binomial      D) quadratic trinomial&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Identify the well-defined real number.&#xA;31) Which of the following is well-defined as a real number?&#xA;A) sqrt(-9)      B) (6/0)      C) 0 to power of ((-3))      D) (0/8)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Find any real number for which the rational expression is undefined.&#xA;32) (25x to power of ((2)) - 64/2x - 8)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indicate the quadrant in which the point belongs.&#xA;46) (11, -7)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-03-24T20:40:41.953" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18099" PostTypeId="2" ParentId="18031" CreationDate="2020-03-25T10:33:08.337" Score="2" Body="&lt;p&gt;My approach:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lectures&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I use a tablet laptop at home, which I can write on with a bluetooth pen. I use to OneNote app to actually write thngs down, though there are many other options.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To do an online lecture, I use Google Meet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students (about 50 of them) gather online in the meeting room (a link I send them 15 minutes before the lecture starts).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I share my screen with them and ask them to turn off their cameras and microphones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a couple of slides or multi-choice questions that I show while waiting for everone to &quot;arrive&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the lecture starts, I start &lt;em&gt;recording&lt;/em&gt; the meeting, I switch to OneNote and deliver the lecture, writing down everything like I would on a whiteboard. I have another computer open by my side, which shows the screen as the students can see it. Importantly, I also have the chat window open on that screen and tell students that they can type in questions, which I can then address directly as I go along.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end of the lecture, I close the Google Meeting, send the students a link to the recording of the lecture and upload a pdf of the OneNote file (sometimes with added annotation that I type) to the Blackboard site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have to say, so far, so good. It actually works better than a regular lecture in terms of getting the students to ask questions. I think they prefer to type stuff in rather than raise their hand and ask something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Homework&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just set the problems on blackboard and provide the solutions later. Students would normally have had access to a &quot;problem class&quot;, so I will schedule an addiitonal class to go over some of the problems in the same way as described above for the lectures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Exams&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The exam for my module will be a timed test. Students will be sent the test at a particular time on a particular day 9they are told in advance !). The exam is also made available on Blackboard at the same time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Students must answer the questions within a limited time. They must scan their solutions (they have been provided with instructions about how to do this using mobile phone apps like &quot;camscanner&quot;) and then submit a single pdf file of their attempts through the &quot;Turnitin&quot; submission system by a certain time on the same day. We will have an &quot;emergency&quot; phone number for students to call if they are having problems scanning/submitting, in which case we may accept emailed solutions, jpegs, or as a last resort, students will be asked to post their solutions that day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clearly there is opportunity for collaboration and online use. Effectively the exam has to be open-book, so the questions cannot be too straightforward or easily googled. Students are remined that questions of authenticity will result in themselves having to defend their work in a viva.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is probbly the least-worst system. We did think about different questions for each student, but this would be just too much work and potentially unfair on students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We will be doing a dry run before any proper test/exam is done.&lt;/p&gt;&#xA;" OwnerUserId="13709" LastActivityDate="2020-03-25T10:33:08.337" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18100" PostTypeId="2" ParentId="18058" CreationDate="2020-03-25T17:56:34.250" Score="4" Body="&lt;p&gt;I will share what I have been doing. I cannot claim this is popular, nor that it is necessarily the best, feasible option for everyone. But I know that when I started searching online for suggestions, I could not find a full description of anyone's method. So I'd like to at least share mine here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;tl;dr: I make slides in Beamer (LaTeX). I use an app on a tablet device to record my voice and the screen as I write on the slides with a stylus. I then share these videos with students, along with a link to download the annotated slides. Here is an example: &lt;a href=&quot;https://youtu.be/QeCYkvoXudg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/QeCYkvoXudg&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I use Beamer to make some slides to present during the video. You could use Powerpoint, Google Slides, or anything similar. But, it's important for what I do later that I have a &lt;strong&gt;pdf file&lt;/strong&gt; to write on.&lt;/li&gt;&#xA;&lt;li&gt;I have some slides with &quot;all information&quot;, like definitions and theorems, or pictures of graphs, etc. I have some slides with much more blank space; perhaps the statement of a problem is on the top and the rest is blank, to leave space for writing.&lt;/li&gt;&#xA;&lt;li&gt;I sometimes include a url to a desmos graph. While recording the video later, I can switch to the desmos app on my tablet and show that graph and demonstrate things with it. I say, &quot;You can pause the video and visit this url to see the same graph.&quot;&lt;/li&gt;&#xA;&lt;li&gt;I now have a pdf file. I open the &lt;a href=&quot;https://play.google.com/store/apps/details?id=com.viettran.INKredible&amp;amp;hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Inkredible&lt;/strong&gt; app&lt;/a&gt; on my tablet and import that pdf file. (I have a Samsung Galaxy Tab A tablet. I chose Inkredible because it allows me to load a pdf, has easy-to-use writing tools, looks good in landscape mode, allows me to switch pages easily &lt;em&gt;without scrolling&lt;/em&gt;, and is not expensive.)&lt;/li&gt;&#xA;&lt;li&gt;I fire up the &lt;a href=&quot;https://play.google.com/store/apps/details?id=com.hecorat.screenrecorder.free&amp;amp;hl=en_US&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;AZ Screen Recorder&lt;/strong&gt; app&lt;/a&gt;. You can use the app to record a specified area of the screen, but I just have it record the whole screen. (Turn off pop-up notifications from all other apps first!) It is also easy to &lt;em&gt;pause recording&lt;/em&gt;. So, I sometimes pause, write/type some annotations on the screen, then un-pause and say, &quot;Now you can see I've written...&quot;&lt;/li&gt;&#xA;&lt;li&gt;While recording, I use different colors of pens and highlighters to write and point to information. Sometimes, while talking, I will highlight something on the slide, and then immediately hit &quot;undo&quot;. This serves to emphasize the information while I am talking, but it will not appear as an annotation on the finished pdf file.&lt;/li&gt;&#xA;&lt;li&gt;While recording, I sometimes jump over to the desmos or WolframAlpha apps to show a graph or a calculation, or even to demonstrate how to use those apps.&lt;/li&gt;&#xA;&lt;li&gt;When I'm done, I click &quot;Stop recording&quot;. I upload the video to my Dropbox folder, and I use the Inkredible app to share the annotated pdf to that same folder. I then share these files with my students (through our LMS or on YouTube).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;For me, the most work goes into &lt;strong&gt;planning the slides and graphs ahead of time&lt;/strong&gt;, and then &lt;strong&gt;deciding what to say and write while recording&lt;/strong&gt;. I often start the video and mess up somewhere and start over. This happens once or twice for each video. But these are also lessons that I've been giving for a while. Don't be worried if you start and stop multiple times.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2020-03-25T17:56:34.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18101" PostTypeId="2" ParentId="18058" CreationDate="2020-03-27T01:11:54.350" Score="1" Body="&lt;p&gt;I usually present (live) lectures by using Beamer to prepare a PDF document with slides that are about half-completed, and I then use PDF Expert on an iPad to project this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The slides contain things like the definitions (which I don't want to waste time writing out) but then I illustrate, explain and give examples using an Apple Pencil.  Students can download the half-completed slides in advance, and I publish the annotated PDF at the end of each week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I now do the same thing with Zoom - more precisely, I run Zoom on my 6-year old MacBook Air and after greeting the students, I screen-share the iPad screen via Air Play and just annotate the slides as usual. Students are free to raise a virtual hand (though so far they seem reluctant to do so). I simply use the Zoom record feature to record the entire thing to the MacBook Air, and then upload the resulting file to our LMS at the end of the lecture. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have done it like this for a few reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Minimal change for the students - I stick to the regular lecture times and lecture for the same amount of time etc, and they can ask questions. I hope this contributes to establishing a routine for the students working from home.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't find it easy to give a lecture without immediate feedback as I rely heavily on body language, facial expressions, actual questions to let me decide on the spot if a concept needs another example, or a definition needs more explanation etc. I ask a few students to leave a webcam on so that I can see them and hope that they are representative of the class. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I want the students to see me and interact with me while actually working problems - calling me out when I make mistakes, watching me puzzle things out. Too many of my students seem to believe that solutions / proofs magically materialise in their lecturers' heads fully-formed and polished. So when they cannot just &quot;see&quot; the answer to a problem, they simply assume that the magic is not working rather than actually getting out some paper and pencil and start working things out.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(Selfish reason here) I don't want to create a unit that can easily be run in its entirety without me. My university was already out of money before this virus started, and now it will be losing tens of millions of dollars. We know from experience that literally the only financial thing the university has in its control is staff numbers. So when this is over, unless some magical money tree appears, the university will be laying off hundreds of staff. I'd rather not create my own replacement just yet.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4795" LastEditorUserId="4795" LastEditDate="2020-03-27T02:10:42.583" LastActivityDate="2020-03-27T02:10:42.583" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18104" PostTypeId="1" CreationDate="2020-03-27T16:23:09.870" Score="6" ViewCount="974" Body="&lt;p&gt;Due to the coronavirus outbreak, and other local problems, we are seeing the need of developing most of our teaching online. Just &quot;do a class like always&quot; (but with no blackboard) via some network application or other doesn't cut it, not by far. Experimentally, giving readings to be discussed online nets you a whooping 5% participation (if that many). So we have to rethink classes, from the ground up. That means restructuring the subject matter, create new content (not just slides!). A particular issue is the design of online/offline evaluations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One particular problem is how much work is this for an &quot;average&quot; college level course? How should we approach this problem (particularly what can we do right now, under virtual quarantine, and with very little infrastructure, no real access to specialists)?&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2024-01-04T18:57:26.280" Title="How much does it cost to develop an online course?" Tags="|undergraduate-education|technology-in-education|online-instruction|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18105" PostTypeId="2" ParentId="18104" CreationDate="2020-03-27T16:34:36.877" Score="8" Body="&lt;p&gt;The Lone Ranger had escaped many dangers, but this time, the situation was hopeless.  Back to the wall with 200 wild Indians surrounding him.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He turned to his faithful sidekick, Tonto, and said &quot;what will we do now?&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tonto replied, &quot;who's 'we', paleface?&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The needs of the administrators, students, teachers are all different.  In other words, &quot;who's 'we'?&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assuming you are the teacher, I would go ahead and do something reasonable that doesn't require more time than previous work.  If they (customers or administrators) want more, they need to pay more.  And they won't.  So do enough to keep people off your back (that you did something) and don't get fired.  But don't do some heroic unpaid effort.  Remember, if they really REALLY needed that, they would pay for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, this is not going to last forever.  So don't invest huge time (or even dollars if an administrator) on a temporary situation.&lt;/p&gt;&#xA;" OwnerUserId="13718" LastActivityDate="2020-03-27T16:34:36.877" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18106" PostTypeId="2" ParentId="18104" CreationDate="2020-03-27T19:17:26.210" Score="7" Body="&lt;p&gt;I am using zoom, and getting better attendance sometimes than in my face-to-face classes.&lt;/p&gt;&#xA;&lt;p&gt;Modified tools for getting interaction:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I ask students to put a number from 1 to 5 in the chat to rate their understanding.&lt;/li&gt;&#xA;&lt;li&gt;I ask for a brave volunteer to work with me sometimes, and I walk that one student through, asking them to give me a next step, etc.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Presentation: I did spend about $130 on a 5' x 3' whiteboard (Staples) that is currently propped on two chairs, but will soon be installed on my living room wall.&lt;/p&gt;&#xA;&lt;p&gt;ETA on 1-4-2024: I see that someone is still reading this. (It was written early in the pandemic, and is no longer how I do things.)&lt;/p&gt;&#xA;&lt;p&gt;I taught online for 3 1/2 years, and then retired because it was way more work and less satisfying (students did not always turn their camera on, participated much less, and cheated more).&lt;/p&gt;&#xA;&lt;p&gt;I am not yet sure that any online course can help students learn as well as an in-person course does. The sense of community is lost. Online learning through gamified systems does seem to work well. That doesn't need a teacher or a class. I enjoy using Duo to learn languages and &lt;a href=&quot;https://artofproblemsolving.com/alcumus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alcumus&lt;/a&gt; (part of Art of Problem Solving) to practice math. I'm not sure how well it does when the student needs to learn new concepts. I also love &lt;a href=&quot;https://sciencevsmagic.net/geo/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page&lt;/a&gt; for working on geometry constructions.&lt;/p&gt;&#xA;&lt;p&gt;It turned out that an ipad with apple pencil does work well for me. I use goodnotes with that, and love it.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2024-01-04T18:57:26.280" LastActivityDate="2024-01-04T18:57:26.280" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18107" PostTypeId="2" ParentId="18030" CreationDate="2020-03-28T06:48:54.377" Score="7" Body="&lt;p&gt;Wanted to make this a comment but don't have enough reputation.&lt;/p&gt;&#xA;&lt;p&gt;There's &lt;a href=&quot;https://app.scratchwork.io&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scratchwork&lt;/a&gt; and &lt;a href=&quot;https://www.worldwidewhiteboard.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WorldWideWhiteboard&lt;/a&gt; that do what you want. They allow you to write in latex and it appears automatically on the board.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Update: 3 Nov 2020: &lt;a href=&quot;https://whiteboard.fi/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://whiteboard.fi/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13720" LastEditorUserId="13720" LastEditDate="2020-11-03T08:04:22.350" LastActivityDate="2020-11-03T08:04:22.350" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18109" PostTypeId="2" ParentId="18104" CreationDate="2020-03-28T19:38:50.100" Score="3" Body="&lt;p&gt;I have/had a few courses on Udemy and have a few hundred YT videos, so I've been down this road a couple times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My current set up:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Blue Yeti microphone, \$130.  You can probably get away with a built-in laptop microphone.  I needed something better because of Udemy's requirements.&lt;/li&gt;&#xA;&lt;li&gt;Presentations in PowerPoint and/or &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt;, \$0.  Well, PowerPoint might not be free but there are free alternatives to it.  &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; costs \$0 but a ton of time if you have no experience with it.&lt;/li&gt;&#xA;&lt;li&gt;Screen recording with Camtasia Studio, \$200.  I might be wrong on the price, it's been so long I can't remember, but I do know I thought the price was worth it for what I was doing.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Previously I used a Monoprice graphic tablet to write in MS Paint.  The tablet was about \$60.  Looks like it's currently sold out, but I'm sure there are comparable ones available.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach part time at a local community college and this hasn't been much of a strain on most of our math and computer science courses.  If your class requires a ton of in-person discussions or has a bunch of group work then it might be difficult, but otherwise, putting it online &quot;almost as-is&quot; is good enough.  It doesn't need to be a flawless work of art.  I can't put it any better than &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18104/how-much-does-it-cost-to-develop-an-online-course#comment48448_18104&quot;&gt;Ben did already in his last sentence here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, reach out to other instructors.  Is anybody helping anybody else there?  Both departments I'm in have been very active with communication and tips.  I get that each school is different, but it still strikes me as very odd that there would be any teacher anywhere who felt like they were being left to their own devices.&lt;/p&gt;&#xA;" OwnerDisplayName="user6648" LastActivityDate="2020-03-28T19:38:50.100" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18110" PostTypeId="1" CreationDate="2020-03-29T13:07:16.013" Score="9" ViewCount="3582" Body="&lt;p&gt;I run a German website for mathematics education. We produce lessons, each of which contains various kinds of media, such as an introduction, videos, wikis, software, worksheets, and online tests.&#xA;We have started to produce more material for teachers and they mostly ask &lt;strong&gt;for worksheets&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trying to understand the &quot;future of education&quot;, I am trying to find out if our planned production of ~1000 worksheets makes actually sense. If worksheets should be abandoned in (let's say) 10 years, then this immense work could be unnecessary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked several (older) teachers, most of them stated they want to continue using worksheets because:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;different structure of worksheets&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;structured writing on paper&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;prepare for online tests&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;better for complex exercises&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;do not have to copy exercises from screen, print worksheet&quot;&lt;/li&gt;&#xA;&lt;li&gt;&quot;better presentation of coordinate systems and tables&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Does anyone has experience if very modern schools are using still worksheets? Or can you predict the future of printed worksheets?&lt;/p&gt;&#xA;" OwnerUserId="10339" LastEditorDisplayName="user507" LastEditDate="2020-03-29T13:55:12.270" LastActivityDate="2020-03-31T10:22:23.653" Title="The Future of Worksheets - will they still be used or abandoned?" Tags="|teaching|writing|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18111" PostTypeId="2" ParentId="18110" CreationDate="2020-03-29T14:16:38.580" Score="7" Body="&lt;p&gt;It sounds like you're producing high school math material, whereas I'm teaching mostly physics (and a little math) at a community college in California. Hopefully there is enough overlap to make my experience helpful to you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do a variety of paper and pencil exercises with my students, some of which could be characterized as worksheets. Below is an example of a typical exercise, which students would do together in small groups. Most of my students have the (free) book in PDF form only. Probably 25% buy the book in print. (It costs $12.) Quite a few are using the PDF on a phone, although I try to convince them not to do that. There are desktop computers in the room, but not enough that every student could sit at their own computer. Students cannot print from these computers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given this set of circumstances, I often do print out the exercises for my students. For the exercise below, for example, I would print out one copy per group, in order to encourage them all to focus on the activity together. I tell them to try to agree on what to write down on the paper. They also have it in their copy of the book if they want it later for reference or if they want to transcribe their own copy of their group's results. While they're working, I do laps around the room and look at what each group has agreed to write down. If they're wrong, I give them feedback. The work doesn't count toward their grade.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;structured writing on paper&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes. If I just popped up the figure on the screen and asked them to do this activity, without a form on paper for them to write on, they would scribble random, disorganized stuff, and I would have no way to check them and give them feedback.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;do not have to copy exercises from screen, print worksheet&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes. If I didn't push them to use a paper copy, many of them would be trying to do this off of the tiny screens on their smartphones, which would be ridiculous. If this was high school, then there would be the added issue that many schools don't want kids using their phones in class, so many kids do not have any digital device. There is also the theoretical issue that some kids might not have any device, but I don't encounter that at my school. (I actually don't own a cell phone myself, which my students consider a bizarre eccentricity.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EEMij.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EEMij.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-03-29T14:16:38.580" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18112" PostTypeId="2" ParentId="18110" CreationDate="2020-03-29T14:39:13.663" Score="16" Body="&lt;p&gt;It's hard to tell what will be the changes in 10 years.  Maybe we will all have jacks in our heads.  Then again, some things change slower. (Where's my flying car?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of the reasons for getting rid of worksheets have been around for 20-40 years and they still have high demand.  They still have various advantages.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most importantly, if you look at the NPV-DCF calculation (use the beta for publishers), almost all the value of this investment is paid back in the first 10 years anyways.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, I would proceed and meet the current market demand.  Doesn't stop a little playing around with electronic apps, games, etc. so you are ready for any changes.  But I advise you to put most investment in current demand.  (If some more modern app gets traction, of course, feed it capital at that point.)&lt;/p&gt;&#xA;" OwnerUserId="13724" LastEditorUserId="13724" LastEditDate="2020-03-29T14:59:22.670" LastActivityDate="2020-03-29T14:59:22.670" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18113" PostTypeId="2" ParentId="14831" CreationDate="2020-03-30T06:03:45.160" Score="2" Body="&lt;p&gt;For kids, I always use the &lt;strong&gt;pie (or pizza) example&lt;/strong&gt;, cutting it into pieces. This way it is very visual and easy to understand that 1/2 is the same as 2/4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FtwOG.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FtwOG.gif&quot; alt=&quot;circle fraction half&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the &lt;a href=&quot;https://www.matheretter.de/wiki/bruche-kreis&quot; rel=&quot;nofollow noreferrer&quot;&gt;fraction program&lt;/a&gt; you can do any &quot;cutting&quot; of a circle interactively. Just set 1/2 and then 2/4. (Disclaimer: I am the developer of the website, it is in German.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Following an example how you can do n out of 4 with a pizza: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/p2qTk.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/p2qTk.png&quot; alt=&quot;Pizza fractions&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10339" LastActivityDate="2020-03-30T06:03:45.160" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18114" PostTypeId="2" ParentId="14831" CreationDate="2020-03-30T16:37:52.973" Score="-1" Body="&lt;p&gt;From your question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As per the kid, 2/4 is more than 1/2 since in first case the boy gets 2 candies out of 4 and in second case he gets 1 candy out of the 2. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Getting 2 candies out of 4 is because 4/2 = 2. But 2/4 means 2 cookies and 4 hungry little kids. 1/2 is 1 cookie and 2 hungry little kids. Ask your kid what they prefer now.&lt;/p&gt;&#xA;" OwnerUserId="9295" LastActivityDate="2020-03-30T16:37:52.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18115" PostTypeId="2" ParentId="18110" CreationDate="2020-03-30T18:08:51.880" Score="6" Body="&lt;p&gt;Worksheets are useful as a fill-in-the blanks forms to sign up for a credit card or for a car registration, but are &lt;a href=&quot;https://medium.com/@ReflectiveObserver/worksheets-are-evil-ab2d3f61310b?source=friends_link&amp;amp;sk=34d2a96b6deaf00ad5a06830353ce6b1&quot; rel=&quot;noreferrer&quot;&gt;detrimental in education&lt;/a&gt;. They instill the thinking that all information needed to solve a problem is presented on the same worksheet the students need to fill out. They supply information and problems in piecemeal fashion. They discourage skills needed to work with textbooks. They do not foster methodical approach to solving problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not to mention that printed worksheets can be used only for a specific exercise, unlike blank paper that can be used for anything. Thus, worksheets are environmentally unfriendly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sadly, worksheets are here to stay, if not in physical paper format then in computerized forms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given a choice, I would always use a traditional textbook and blank paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/olFpn.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/olFpn.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2020-03-30T18:15:04.953" LastActivityDate="2020-03-30T18:15:04.953" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18116" PostTypeId="1" AcceptedAnswerId="18143" CreationDate="2020-03-30T20:16:16.217" Score="3" ViewCount="193" Body="&lt;p&gt;We're planning to run the project for first two grades of the elementary school kids, in which we want to facilitate the spatial thinking development along with the regular arithmetic course and make it easier for them to get into advanced math in further educational levels. What would you recommend to start with and which well-recognized and trusted textbooks, approaches, methodologies, and quizzes do you know what may ease this task? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We're located in Moscow as a company, but going to work way beyond the region, besides, anyway, I wanted to make sure it's possible in general for the considerably early age (6-9 y.o.) from the psychological point of view, first of all, without binding to the local realia of the region.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two disputable articles I've read up to the moment in this direction:&#xA;&lt;a href=&quot;https://www.researchgate.net/publication/313588375_The_Development_of_Spatial_Skills_in_Elementary_School_Students&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.researchgate.net/publication/313588375_The_Development_of_Spatial_Skills_in_Elementary_School_Students&lt;/a&gt;&#xA;&lt;a href=&quot;http://ling.umd.edu/~jlidz/Teaching/F05Seminar/feigensondehaenespelke04.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ling.umd.edu/~jlidz/Teaching/F05Seminar/feigensondehaenespelke04.pdf&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I checked the OECD report about the early STEM preparation and correlation with spatial thinking and math skills development:&#xA;&lt;a href=&quot;https://www.oecd.org/officialdocuments/publicdisplaydocumentpdf/?cote=EDU/WKP(2017)10&amp;amp;docLanguage=En&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.oecd.org/officialdocuments/publicdisplaydocumentpdf/?cote=EDU/WKP(2017)10&amp;amp;docLanguage=En&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you for help :-)&lt;/p&gt;&#xA;" OwnerUserId="13610" LastEditorUserId="13610" LastEditDate="2020-04-06T18:59:09.593" LastActivityDate="2020-04-06T18:59:09.593" Title="The spatial thinking course for primary school - what to use?" Tags="|mathematical-pedagogy|primary-education|mathematics-in-daily-life|critical-thinking|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="18117" PostTypeId="2" ParentId="18110" CreationDate="2020-03-31T10:22:23.653" Score="2" Body="&lt;p&gt;My daughter is at a newly-built primary school in the UK, well-equipped with smartboards and the rest of the latest technology; they also use app-based and online tools for homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However at this age at least, they make near-daily use of worksheets in maths and other core subjects; many if not most of the ones I've seen are from external providers.  Her teachers over the last couple of years have been of a range of ages and career stages, and all have used worksheets extensively.  &lt;/p&gt;&#xA;" OwnerUserId="8180" LastActivityDate="2020-03-31T10:22:23.653" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18118" PostTypeId="2" ParentId="10716" CreationDate="2020-03-31T13:26:59.993" Score="2" Body="&lt;p&gt;You can find articles and videos on teaching math to low vision and blind students from the Texas School for the Blind and Visually Impaired. &lt;a href=&quot;https://www.tsbvi.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tsbvi.edu/&lt;/a&gt;&#xA;In particular, you can find informative videos from Susan Osterhaus on that site.  She is an instructor with 25+ years of teaching experience with blind and visually impaired students mostly in K-12.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The tsvi site also has manipulatives and other resources for sale.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally the blindmath listserv has information - mostly on how to address reading math. The listserv is essentially a forum for math students (principally college level) and professionals to address different questions and issues.&lt;/p&gt;&#xA;" OwnerUserId="13733" LastEditorUserId="13733" LastEditDate="2020-03-31T21:28:22.883" LastActivityDate="2020-03-31T21:28:22.883" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18119" PostTypeId="1" CreationDate="2020-03-31T20:11:28.697" Score="3" ViewCount="87" Body="&lt;p&gt;What are good and free online testing platforms for mathematics? Is there any who can easily display math formulas? Any allowing LaTeX code? Apparently the only way to insert formulas in Google Forms is with pictures, weschool doesn't even allow to insert image in the answer, do we have better choices?&lt;/p&gt;&#xA;" OwnerUserId="5601" LastActivityDate="2020-03-31T20:11:28.697" Title="Online testing platform" Tags="|testing|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18120" PostTypeId="1" CreationDate="2020-04-02T04:52:19.770" Score="28" ViewCount="3543" Body="&lt;p&gt;This is my first post so bear with me, but something I've been thinking about lately is: Why didn't I ever question the relationship between the derivative and the integral when I was taking calculus?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me explain what I mean: In most courses, the derivative is introduced as the slope of a curve at a point, or the &quot;instantaneous rate of change&quot;. Then the integral is introduced as the area under a curve. Then students are told that these two things &quot;undo&quot; each other (the fundamental theorem of calculus). So now I'm wondering, why didn't I ever question why these &quot;undo&quot; each other? It's not intuitive at all. For example, addition and subtraction, multiplication and division, logarithms and exponentials; all of these things I can intuitively understand why they &quot;undo&quot; each other. But how does something that represents the area under a curve &quot;undo&quot; something that represents an instantaneous rate of change? What is the connection between those two concepts? Is there a better way to explain these concepts that makes the fundamental theorem of calculus more intuitive?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking forward to hearing some of y'alls responses. Thanks!&lt;/p&gt;&#xA;" OwnerUserId="13742" LastActivityDate="2022-02-22T20:56:15.960" Title="What is the best way to intuitively explain the relationship between the derivative and the integral?" Tags="|undergraduate-education|calculus|" AnswerCount="10" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="18121" PostTypeId="2" ParentId="18120" CreationDate="2020-04-02T11:56:39.153" Score="36" Body="&lt;p&gt;You start by noticing that the Riemann sums (multiplication followed by addition) and the difference quotients (subtraction followed by division) undo  each other. Their limits -- the integral and the derivative -- still undo each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Added: The first sentence is a part of what is called “Discrete Calculus” &lt;a href=&quot;https://en.wikipedia.org/wiki/Discrete_calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Discrete_calculus&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10168" LastEditorUserId="10168" LastEditDate="2020-04-06T15:28:46.090" LastActivityDate="2020-04-06T15:28:46.090" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="18122" PostTypeId="2" ParentId="18120" CreationDate="2020-04-02T19:29:29.667" Score="4" Body="&lt;p&gt;When I first began teaching Calculus, I realized that I really didn't understand the Fundamental Theorem. So I looked for something that would help me have a deep understanding, that would also help me help students to see it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found a lovely project, which I have modified over the years. Here are links to &lt;a href=&quot;https://drive.google.com/file/d/1C7KXgvGsltKyY2xIXN-613scdu9JyWeS/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;the pdf&lt;/a&gt; and to &lt;a href=&quot;https://drive.google.com/file/d/1J7hlT_UfsDpdt2ELoh7g96__rjowxxAo/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;a .doc version&lt;/a&gt; (in which the formulas are messed up, because google docs couldn't read them). If you use this, you'll want to change the part near the end where I reference the textbook.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-04-02T19:29:29.667" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18123" PostTypeId="2" ParentId="18120" CreationDate="2020-04-02T21:03:12.767" Score="10" Body="&lt;p&gt;Derivative of integral is the original function:&#xA;Let &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; be the integral function of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$F(x)$&lt;/span&gt; is the area under the graph from zero to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&#xA;For small &lt;span class=&quot;math-container&quot;&gt;$h&amp;gt;0$&lt;/span&gt; the difference &lt;span class=&quot;math-container&quot;&gt;$F(x+h)-F(x)$&lt;/span&gt; is the area of a narrow vertical strip.&#xA;The width is &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; and height approximately &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&#xA;As &lt;span class=&quot;math-container&quot;&gt;$h\to0$&lt;/span&gt;, this means &lt;span class=&quot;math-container&quot;&gt;$F'(x)=f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you like thinking in terms of infinitesimals, write &lt;span class=&quot;math-container&quot;&gt;$dF=F(x+dx)-F(x)=f(x)dx$&lt;/span&gt; and divide by the differential.&#xA;This can be a cleaner way to think as the limit process is left implicit in a way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Integral of derivative is the original function:&#xA;The integral is roughly&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_0^x&#xA;f'(x)dx&#xA;=&#xA;\sum_{k=1}^n&#xA;f'(x_k)(x_k-x_{k-1})&#xA;$$&lt;/span&gt;&#xA;and the derivative is roughly&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f'(x_k)&#xA;=&#xA;\frac{f(x_k)-f(x_{k-1})}{x_k-x_{k-1}}.&#xA;$$&lt;/span&gt;&#xA;Combine these and you get a telescoping sum and the desired result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the way it makes intuitive and graphical sense to me.&#xA;The link is not immediate; that's why it's a great theorem.&lt;/p&gt;&#xA;" OwnerUserId="2074" LastEditorUserId="60" LastEditDate="2020-04-03T04:38:59.823" LastActivityDate="2020-04-03T04:38:59.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18124" PostTypeId="2" ParentId="18120" CreationDate="2020-04-02T21:07:02.007" Score="20" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/sE84j.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sE84j.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(this is from my calculus notes, see page 233 of: &lt;a href=&quot;http://www.supermath.info/OldschoolCalculusII.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.supermath.info/OldschoolCalculusII.pdf&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-04-02T21:07:02.007" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18125" PostTypeId="2" ParentId="18120" CreationDate="2020-04-03T05:06:27.803" Score="5" Body="&lt;p&gt;I gave a similar post to this one touching on this on Math.StackExchange.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, the way I would go about it is to say that there is a very easy way by which one can think of at least Riemann integration (the usual definition given in a &quot;most courses&quot; calculus course) as an inverse of differentiation &lt;em&gt;by construction&lt;/em&gt;: that is, the relationship between the two is not an &quot;accident&quot;, but design, so that given one, you could fairly easily be led to the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And to do that, I'd first suggest getting rid of the whole &quot;slope&quot; vs &quot;area&quot; business altogether - if anything, those are best given as &lt;em&gt;theorems&lt;/em&gt; to be &lt;em&gt;proved&lt;/em&gt;, after you have other, free-standing &lt;em&gt;definitions&lt;/em&gt; of a tangent line and an area which, by the way, can be done, but just too-often aren't, in favor of various hand-waving arguments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, the relevant idea is &lt;em&gt;change&lt;/em&gt;: the derivative of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, i.e. &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt;, stands for a kind of &quot;sensitivity&quot; measure. Suppose that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; were like a kind of meter or instrument, with a knob attached to it, and a readout of some kind. The knob attached is the function's input argument, typically denoted &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. The indicator on the readout is the return value, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is set at some value of interest &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;, then likewise the readout will be at &lt;span class=&quot;math-container&quot;&gt;$f(x_0)$&lt;/span&gt;. Now suppose you &quot;wiggle&quot; the knob &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; back and forth a little bit, and you see how the needle &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; responds to that small impulse. For a continuous function, the size of the output's wiggle will be smaller in absolute terms the smaller you make the input, but the &lt;em&gt;proportionate&lt;/em&gt; size, i.e. how much it wiggles &lt;em&gt;relative to&lt;/em&gt; how much you wiggle the input value, may not be. When we say &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differentiable, what that means is that there exists at each point &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; a proportionality factor &lt;span class=&quot;math-container&quot;&gt;$f'(x_0)$&lt;/span&gt; such that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x \pm \underbrace{dx}_\mbox{&quot;wiggle&quot; in $x$}) \approx f(x_0) \pm \underbrace{[f'(x_0)\ dx]}_\mbox{&quot;wiggle&quot; in $f(x)$}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;so that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mbox{proportional &quot;wiggle&quot;} = \frac{\mbox{&quot;wiggle&quot; in $f(x)$}}{\mbox{&quot;wiggle&quot; in $x$}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;so long as the change &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; is suitably small&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Integration, then, goes the other way. Suppose that I am &lt;em&gt;given&lt;/em&gt; now, not the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, but only its derivative, &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt;, and want to &lt;em&gt;find&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. First off, one should observe that since &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; only deals with &lt;em&gt;changes&lt;/em&gt; in the input, to start, we actually need one more piece of information, and that is some sort of initial value, i.e. &lt;span class=&quot;math-container&quot;&gt;$f(0)$&lt;/span&gt;. Suppose this to be given as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Starting at &lt;span class=&quot;math-container&quot;&gt;$f(0)$&lt;/span&gt;, suppose we apply a small, but nonzero, change &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; to the input, i.e. we ask, &quot;given &lt;span class=&quot;math-container&quot;&gt;$f(0)$&lt;/span&gt;, what is &lt;span class=&quot;math-container&quot;&gt;$f(0 + \Delta x)$&lt;/span&gt;, to the best we can do?&quot;  Well, since we know &lt;span class=&quot;math-container&quot;&gt;$f(0)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt;, then since &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; is a &lt;em&gt;small&lt;/em&gt; change or &quot;wiggle&quot;, we can say that &lt;em&gt;approximately&lt;/em&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(0 + \Delta x) \approx f(0) + [f'(0)\ \Delta x]$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which is just what you see above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, suppose we take &lt;em&gt;another&lt;/em&gt; step of &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt;. We're now going from &lt;span class=&quot;math-container&quot;&gt;$x = 0 + \Delta x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x = (0 + \Delta x) + \Delta x$&lt;/span&gt; (or &lt;span class=&quot;math-container&quot;&gt;$2\ \Delta x$&lt;/span&gt;, but I find writing it this way makes it clearer what is going on - &quot;simpler&quot; isn't necessarily &quot;better&quot;). At this second step, likewise, treating &lt;span class=&quot;math-container&quot;&gt;$0 + \Delta x$&lt;/span&gt; as a prior input in and of itself, we have&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f([0 + \Delta x] + \Delta x) \approx f([0 + \Delta x]) + [f'([0 + \Delta x])\ \Delta x]$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which, combining with the previous expression, becomes&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f([0 + \Delta x] + \Delta x) \approx f(0) + [f'(0)\ \Delta x] + [f'([0 + \Delta x])\ \Delta x]$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and it is not hard to then continue this process so that we see for &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; steps,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(0 + N[\Delta x]) \approx f(0) + \sum_{i=0}^{N-1} f'(0 + i[\Delta x])\ \Delta x$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or, if we set &lt;span class=&quot;math-container&quot;&gt;$x_i := 0 + i[\Delta x]$&lt;/span&gt;, we can say more neatly as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(0 + N[\Delta x]) \approx f(0) + \sum_{i=0}^{N-1} f'(x_i)\ \Delta x$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and to be more general, if we take &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; steps of suitable, perhaps different, sizes &lt;span class=&quot;math-container&quot;&gt;$\Delta x_i$&lt;/span&gt; to reach from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; some fixed point &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x_0) \approx f(0) + \sum_{i=0}^{N-1} f'(x_i)\ \Delta x_i$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and then we consider what happens as the steps become arbitrarily fine, at which point we &lt;em&gt;hope&lt;/em&gt; - and need to &lt;em&gt;prove&lt;/em&gt; - that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x_0) = f(0) + \left[\lim_{||\Delta|| \rightarrow 0}\ \sum_{i=0}^{N-1} f'(x_i)\ \Delta x_i\right]$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which leads us to define this new operation, given by the limit on the right...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{0}^{x_0} f'(x)\ dx := \lim_{||\Delta|| \rightarrow 0}\ \sum_{i=0}^{N-1} f'(x_i)\ \Delta x_i$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="7650" LastActivityDate="2020-04-03T05:06:27.803" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18126" PostTypeId="2" ParentId="18120" CreationDate="2020-04-03T09:58:26.000" Score="1" Body="&lt;p&gt;Well, for me, the nicest intuition comes from physics. If &lt;span class=&quot;math-container&quot;&gt;$F=F(x)$&lt;/span&gt; is some force applied to an object &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; casuing &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; to move, where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is its displacement, when the work &lt;span class=&quot;math-container&quot;&gt;$W(x)$&lt;/span&gt; produced by that force from &lt;span class=&quot;math-container&quot;&gt;$x_0=0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; metres is given by:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$W(x):=\int_0^xF(s)ds.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, what if we ask what is the rate of change of that work with respect to displacement? At some certain point &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, the work provided to or substracted by that object &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; is determined by the force &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; applied on it. So, one can intuitively expect that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$W'(x)=\left(\int_0^xF(s)ds\right)'=F'(x).$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, the larger the force, the faster the energy flows from the one applying the force towoards the object. The less the force, the slower that energy flows.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2020-04-03T09:58:26.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18128" PostTypeId="2" ParentId="18120" CreationDate="2020-04-03T12:10:17.717" Score="5" Body="&lt;p&gt;My answer also comes from physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say p(t) is the position of an object in the time t. For concreteness, suppose you are traking a truck and the truck is going forward on a road from A to B. (i.e., all derivatives are positive)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is very natural to graph p(t), and derive it, arriving at v(t), the speed of the truck for each time. After all, the derivative is just taking a small time increment &lt;span class=&quot;math-container&quot;&gt;$dt$&lt;/span&gt;, calculating the corresponding &lt;span class=&quot;math-container&quot;&gt;$ds$&lt;/span&gt; (a space increment) and dividing&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, when I look at the graph of v(t), I can also obtain (almost everything) from p(t). Say I am at the position 100 km in the time t=0. After a &lt;span class=&quot;math-container&quot;&gt;$dt$&lt;/span&gt;, I shall be at &lt;span class=&quot;math-container&quot;&gt;$p(t+dt) = 100+dt\cdot v(t)$&lt;/span&gt;. And I can do it again and again&#xA;&lt;span class=&quot;math-container&quot;&gt;$p(t+dt+dt+dt) = 100+dt\cdot v(t)+dt\cdot v(t+dt)+dt\cdot v(t+2\cdot dt)+dt\cdot v(t+3\cdot dt)$&lt;/span&gt; (notice that after some &lt;span class=&quot;math-container&quot;&gt;$dt$&lt;/span&gt;s, I use the updated speed)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that (taking a limit) is an integral!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only weird thing is my assuption that I am at position 100km at time t=0. The graph of v(t) cannot tell me that. Intuitively, it does tell me how fast I am going, and therefore can tell me how much how much I walked, but it cannot tell me at which specific part of the road &lt;strong&gt;I started&lt;/strong&gt;. That is that weird constant +c that appears on the indefinite integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then again, we dont use the indefinite integral much, do we? We are more interested in summing &lt;span class=&quot;math-container&quot;&gt;$dt$&lt;/span&gt;*v(t) for ranges. Say, from t=3 to t=10. That gives me the amount I walked between those two times. So, mercifully, the +c disappears in most use cases: to know how much I walked, there is no need to know where I started.&lt;/p&gt;&#xA;" OwnerUserId="1662" LastActivityDate="2020-04-03T12:10:17.717" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18130" PostTypeId="2" ParentId="9760" CreationDate="2020-04-04T12:53:16.360" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://gameaboutsquares.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gameaboutsquares&lt;/a&gt; may help spatial thinking. The beginning is suitable for the young, and it gets progressively more difficult. &lt;/p&gt;&#xA;" OwnerUserId="8660" LastActivityDate="2020-04-04T12:53:16.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18131" PostTypeId="2" ParentId="6163" CreationDate="2020-04-05T00:22:39.710" Score="2" Body="&lt;p&gt;I like this article that helps explain why studying triangles is important. &quot;But triangles aren't just mathematically significant, they are also fundamental to the way we build our environments, both physical and virtual.&quot;&#xA;&lt;a href=&quot;https://wild.maths.org/power-triangles&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://wild.maths.org/power-triangles&lt;/a&gt;&#xA;From my experience teaching, Common Core is attempting to raise the ability of students' analytical and reasoning ability by requiring more explanations (sentences) along with the numbers that pop out at the end--What do the numbers actually mean in the context of the (story) problem?&lt;/p&gt;&#xA;" OwnerUserId="9750" LastActivityDate="2020-04-05T00:22:39.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18132" PostTypeId="2" ParentId="18120" CreationDate="2020-04-05T00:27:53.857" Score="2" Body="&lt;p&gt;If you want more clearly &quot;undoing&quot; operations, define &quot;stacking&quot; as the following: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take some &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt;. Chop the curve into rectangles of width &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; and height &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;. (There's some leeway as to take &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; at the left side, right side, middle, minimum, maximum, etc. The rest of my description will be right side.) Now take each rectangle and put the bottom of each rectangle at the top of the previous rectangle. For instance, if you have &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Delta x = 0.1$&lt;/span&gt;, you'd have a rectangle with lower left corner at the origin, and top left at &lt;span class=&quot;math-container&quot;&gt;$(0.1, 0.01)$&lt;/span&gt;. Then the next rectangle would have lower left corner at &lt;span class=&quot;math-container&quot;&gt;$(0.1, 0.01)$&lt;/span&gt;, and upper right at &lt;span class=&quot;math-container&quot;&gt;$(0.2, 0.05)$&lt;/span&gt;, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Define &quot;unstacking&quot; as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take some &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt;. Chop the curve into rectangles of width &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; and height &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;. Now, take each rectangle and shift it vertically down the height of the previous rectangle. For instance, if &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt;,  the first rectangle will have height &lt;span class=&quot;math-container&quot;&gt;$0.01$&lt;/span&gt;. The second one will have height &lt;span class=&quot;math-container&quot;&gt;$0.04$&lt;/span&gt;, so for the unstacked version, we move it down &lt;span class=&quot;math-container&quot;&gt;$0.01$&lt;/span&gt;, leaving its new height as &lt;span class=&quot;math-container&quot;&gt;$0.03$&lt;/span&gt;. Another way of phrasing it is for each &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, take the rectangle whose bottom left corner is at &lt;span class=&quot;math-container&quot;&gt;$(x, f(x))$&lt;/span&gt; and top right corner is &lt;span class=&quot;math-container&quot;&gt;$(x+\Delta x, f(x+\Delta x))$&lt;/span&gt;. Then create a chart out of all of those rectangle, with all of them moved down so their bottom is on the x-axis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For &quot;well-behaved&quot; functions (I believe that uniform continuity is a sufficient condition), as &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; goes to zero, stacking becomes integration, and unstacking become differentiation (with the proper scaling).&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2020-04-05T00:27:53.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18133" PostTypeId="2" ParentId="17864" CreationDate="2020-04-05T00:58:02.313" Score="2" Body="&lt;p&gt;Students are hard customers to satisfy, especially after having less than optimal math instruction for so many years, and probably having negative self-talk about their math ability. Make it real. Make it hands-on. Make it immediate and &quot;in their face&quot; as much as possible. What do high school kids do in their daily lives? They play video games. They ride skateboards. They play musical instruments. They play sports.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Elder Scrolls V: Skyrim, Fortnite, Clash of Clans: how these programs use triangles and other formulas to calculate what is happening. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Skateboarding: Bring a skateboard in to class, or ask one of your students to demonstrate tricks, and then break down the math behind the physics.&#xA;&lt;a href=&quot;https://www.tested.com/science/physics/43749-the-physics-of-landing-a-1080-skateboarding-trick/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tested.com/science/physics/43749-the-physics-of-landing-a-1080-skateboarding-trick/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Musical instruments: play a popular song. Isolate one note. Show the frequency response on a graph. Extract a piece of mathematical information relating to the appropriate math level. Have a student who plays an instrument give a demonstration. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sports: Have a someone who plays baseball throw a ball as hard as he/she can, and see if the class can determine the mathematical formula. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's almost reverse from &quot;normal&quot; teaching--observe the physical phenomena first, then determine the mathematical approximation.&lt;/p&gt;&#xA;" OwnerUserId="9750" LastActivityDate="2020-04-05T00:58:02.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18135" PostTypeId="2" ParentId="18120" CreationDate="2020-04-05T15:00:01.673" Score="3" Body="&lt;p&gt;Intuitively, the fundamental theorem of calculus states that &quot;&lt;strong&gt;the total change is the sum of all the little changes&lt;/strong&gt;&quot;. &lt;span class=&quot;math-container&quot;&gt;$f'(x) dx$&lt;/span&gt; is a tiny change in the value of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. We sum up all these little changes to get the total change &lt;span class=&quot;math-container&quot;&gt;$f(b) - f(a)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I elaborated on this explanation here: &lt;a href=&quot;https://math.stackexchange.com/a/1537836/40119&quot;&gt;https://math.stackexchange.com/a/1537836/40119&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2049" LastActivityDate="2020-04-05T15:00:01.673" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18136" PostTypeId="1" AcceptedAnswerId="18144" CreationDate="2020-04-05T23:43:34.367" Score="11" ViewCount="4293" Body="&lt;p&gt;I introduced the hypercube (to undergraduate students in the U.S.) in the&#xA;context of generalizations of the Platonic solids, explained its&#xA;structure, showed it rotating.&#xA;I mentioned &lt;a href=&quot;https://matheducators.stackexchange.com/a/1824/511&quot;&gt;Alicia Stott&lt;/a&gt;, who discovered the &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; regular&#xA;polytopes in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^4$&lt;/span&gt; (discovered after Schläfli).&#xA;I sense they largely did not grasp what is the hypercube, let alone&#xA;the other regular polytopes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd appreciate hearing of techniques for getting students to&#xA;&quot;grok&quot; the fourth dimension.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-04-05T23:56:17.287" LastActivityDate="2022-06-29T20:20:59.933" Title="Ideas for explaining 4D and higher dimensions" Tags="|undergraduate-education|geometry|" AnswerCount="11" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="18137" PostTypeId="2" ParentId="18136" CreationDate="2020-04-06T00:19:49.153" Score="15" Body="&lt;p&gt;I would just say you have a square in 2D, and a cube is the similar shape in 3D, then what is the next shape in 4D.  Then show them the cube in cube view and cross like fold out.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't jump to Schlegel diagrams and the rotating pictures on Wiki so fast...they are confusing.  Do like I said instead.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think showing that transition from 2d to 3D, gets them thinking there must be some thing in 4D.  Just doing that makes it simpler than talking about all the Platonic solids.  Which themselves most people don't know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then next show them the Dali painting:  &lt;a href=&quot;https://en.wikipedia.org/wiki/Crucifixion_(Corpus_Hypercubus)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Crucifixion_(Corpus_Hypercubus)&lt;/a&gt;  Mention that 4d has been intriguing to arts and writers (mention Flatland and how the sphere comes and explains 3D to the plane figure square.  And then when the square says, fine, I bet there are higher dimensions than you, the sphere gets mad and runs off.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point here is not to be having them list axes of symmetry or character groups or the like.  It's to get them comfortable that there's some 4d stuff...and it's weird and hard to think about it.  But at least they are in the mood to think about it!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you are a super nice guy and wicked smart.  But too &quot;hard&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, if it becomes important to show all the polytopes than you will need to backtrack and discuss Platonic solids.  Unless you played D&amp;amp;D, you don't know them as much.  I mean how many non-chemists know there are C3 axes in the dodecahedron?  But start with square-cube-tesseract before doing the Platonic solids.  (They're harder than you think, even though all 3D.)  Square-cube-tesseract is powerfully intuitive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. I am so tempted to make the Smith/Wellesly/Radcliff proverb a part of this...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.s.  &lt;a href=&quot;https://en.wikipedia.org/wiki/%22%E2%80%94And_He_Built_a_Crooked_House%E2%80%94%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/%22%E2%80%94And_He_Built_a_Crooked_House%E2%80%94%22&lt;/a&gt;  &quot;Stating that it &quot;was, for many readers, the first introduction to four-dimensional geometry that held any promise of comprehensibility&quot;, Carl Sagan in 1978 listed &quot;—And He Built a Crooked House—&quot; as an example of how science fiction &quot;can convey bits and pieces, hints and phrases, of knowledge unknown or inaccessible to the reader&quot;.&quot;&lt;/p&gt;&#xA;" OwnerUserId="13762" LastEditorUserId="13762" LastEditDate="2020-04-08T12:54:22.447" LastActivityDate="2020-04-08T12:54:22.447" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18138" PostTypeId="2" ParentId="18136" CreationDate="2020-04-06T00:39:57.503" Score="11" Body="&lt;p&gt;I highly recommend &quot;Flatland The Movie.&quot; Your institution should be able to purchase it. You can find a free trailer on the internet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was young, I read the book &quot;Flatland: A Romance of Many Dimensions,&quot; probably in high school, and it made me &quot;grok&quot; the fourth dimension. &lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2020-04-06T00:39:57.503" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18139" PostTypeId="2" ParentId="18091" CreationDate="2020-04-06T03:45:41.957" Score="6" Body="&lt;p&gt;I've owned the revised first edition of Frankel's&#xA;&lt;em&gt;The Geometry of Physics: an Introduction&lt;/em&gt; at least since I was a graduate student. The texts I suggest in this answer are largely based on my personal library.&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;Part I: Manifolds, Tensors, and Exterior Forms&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;p&gt;Contains 6 chapters which construct the canvas on which the later part of the text plays out. I'll sketch the content then follow with my recommended readings to complement Frankel:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Chapter 1: &lt;strong&gt;Manifolds and Vector Fields&lt;/strong&gt; (submanifolds, manifolds, construction of tangent vectors at a point, vector fields and flows in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Chapter 2: &lt;strong&gt;Tensors and Exterior Forms&lt;/strong&gt; (covectors, the differential, differential forms, tangent and cotangent bundles, pull-backs, tensors, exterior algebra, how it works in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;, orientation, interior products and contraction)&lt;/li&gt;&#xA;&lt;li&gt;Chapter 3: &lt;strong&gt;Integration of Differential Forms&lt;/strong&gt; (integration of p-forms in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;, line and surface integrals, manifolds with boundary, partitions of unity, inducing orientations on boundary, integration of psuedo-n-forms on n-manifold, Maxwell's Equations)&lt;/li&gt;&#xA;&lt;li&gt;Chapter 4: &lt;strong&gt;The Lie Derivative&lt;/strong&gt; ( of vector fields, of forms, how to differentiate an integral, application to Hamiltonian mechanics )&lt;/li&gt;&#xA;&lt;li&gt;Chapter 5: &lt;strong&gt;Poincare Lemma and Potentials&lt;/strong&gt; ( generalized Stokes' theorem, closed forms, exact forms, complex analysis, the converse to Poincare Lemma, finding potentials)&lt;/li&gt;&#xA;&lt;li&gt;Chapter 6: &lt;strong&gt;Holonomic and Nonholonomic Constraints&lt;/strong&gt; (Frobenius integrability condition, distributions of vector fields and forms, the Frobenius Theorem, foliations and maximal leaves, systems of Mayer-Lie, holonomic and nonholonomic constraints, thermodynamics via Caratheodory)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Ok , let me pause with this &lt;em&gt;partial&lt;/em&gt; recap of the table of contents to point towards a few things I'd recommend to partner with the above:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;John M. Lee's &lt;em&gt;Introduction to Smooth Manifolds&lt;/em&gt; is great for in-depth coverage of material in Chapters 1, 2, 3 and 4 and parts of 5 and 6.&lt;/li&gt;&#xA;&lt;li&gt;Jeffrey M. Lee's &lt;em&gt;Manifolds and Differential Geometry&lt;/em&gt; also covers nearly all the material in Chapters 1-6 of Frankel. I think Chapter 10 contains the nuts and bolts of the proof of Poincare's Lemma, which I think is an improvement on the sketch in Flander's &lt;em&gt;Differential Forms with Application to the Physical Sciences&lt;/em&gt; ( incidentially, engineers at Purdue in the 1960's were apparently pretty awesome). Then Chapter 11 in Jeffrey Lee's text has a really nice uninterrupted study of the question and solution which is offered by Frobenius theorem. In my teaching, I found the differential forms and exterior product formula in this text to be a useful supplement where I got lazy.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;h2&gt;&lt;strong&gt;Part II: Geometry and Topology&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;p&gt;This part of Frenkel is largely concerned with the addition of structure to manifolds which capture the concept of geometry in the Riemannian or semi-Riemannian sense.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 7: &lt;strong&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; and Minkowski Space&lt;/strong&gt; (Frenet frames in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;, 4-vectors and Minkowski Space, Electromagnetism in differential form notation on Minkowski Space)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 8: &lt;strong&gt;Geometry of Surfaces in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;&lt;/strong&gt; (first and second fundamental forms, the Weingarten equations, principal curvatures, Gaussian curvature, mean curvature, Gauss map, Brouwer degree and fixed point theorem, Gauss-Bonnet Theorem, first variation of area, soap bubbles and minimal surfaces, Gauss's Theorem Egregium, geodesics, intrisic derivative, parallel displacement of Levi-Civita)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 9: &lt;strong&gt;Covariant Differentiation and Curvature&lt;/strong&gt; (covariant derivative or affine connection, coordinate frames, curvature of affine connection, torsion-free connections, the Riemann connection, Cartan's Exterior Covariant Differential, exterior covariant derivative of vector field or form, Cartan's structural equations, exterior covariant derivative of vector-valued form, curvature 2-forms, change of basis and gauge transformations, curvature forms in Riemannian manifolds, classical differential geometry ala Gauss recovered from Cartan's structure equation viewpoint, parallel displacement and curvature on surface, flat metrics, horizontal distributions, Riemann's theorem on flatness and construction of local frame)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 10: &lt;strong&gt;Geodesics&lt;/strong&gt; (vector fields along surface, geodesics, Hamilton's Principle in the Tangent Bundle, Hamilton's Principle in Phase Space, Jacobi's Principle of &amp;quot;Least&amp;quot; Action, closed geodesics and periodic motion, Geodesics Spiders and the Universe [aka detecting geometry from an intrinsic viewpoint]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 11: &lt;strong&gt;Relativity, Tensors, and Curvature&lt;/strong&gt; (calculus on curved space necessary for detailed understanding of Einstein's field equations, identities galore, Hilbert's variational approach to GR, curvature, sectional curvature, geometry of Einstein's equations, three dimensional version of Gauss's awesome theorem, remarks on Schwarzschild's solution)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 12: &lt;strong&gt;Curvature and Topology: Synge's Theorem&lt;/strong&gt; (second variation of arclength, Jacobi fields, conjugate points, Synge's Theorem stating closed geodesics are unstable in an even-dimensional orientiable manifold with positive sectional curvatures, also the application of Synge's Theorem towards simple connectivity as well as rigid body mechanics)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 13: &lt;strong&gt;Betti Numbers and De Rahm's Theorem&lt;/strong&gt; (singular chains and boundaries, singular homology groups, cycles and boundaries and homology and Betti numbers, homology groups of manifolds such as real projective space and torii, De Rahm's Theorem)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 14: &lt;strong&gt;Harmonic Forms&lt;/strong&gt; (the &lt;span class=&quot;math-container&quot;&gt;$\ast$&lt;/span&gt; operator, scalar product on the exterior algebra, the codifferential operator, divergence in curved space, Maxwell's equations in curved space, Hilbert Lagrangian, Laplace operator on forms, harmonic forms on closed manifolds, Hodge's Theorem on solving Poisson's equation on a closed Riemannian manifold, Bochner's Theorem on vanishing Betti number, tangent and normal differential forms, Hodge's Theorem for Tangential Forms, existence of electric field subject to given boundary potential as special case of general result on harmonic field subject appropriate boundary conditions, relative homology, Hodge's Theorem for Normal Forms,  Morse's Theory of Critical Points, Morse's Theorem)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And now for my recommendations. Following the order of the chapters,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Misner Thorne and Wheeler's &lt;em&gt;Gravitation&lt;/em&gt; is still one of my favorites for reading about differential forms on Minkowski Space. Of course, there are a lot of books which help with Chapter 7 in Frenkel, I just have a certain nostalgia for MTW.&lt;/li&gt;&#xA;&lt;li&gt;Barrett O'Neill's &lt;em&gt;Elementary Differential Geometry&lt;/em&gt; revised 2nd edition does a great job of explaining the genesis of Cartan's method where you can actually see things. I like this book for a companion to Chapter 8 of Frenkel. Of course this material can be found many many other places.&lt;/li&gt;&#xA;&lt;li&gt;Jeffrey Lee's text covers a lot of the same ground as Part II. I think it also does a good job of giving a more abstract treatment of what is done in O'Neill's more elementary text.&lt;/li&gt;&#xA;&lt;li&gt;Gregory L. Naber's &lt;em&gt;Topology, Geometry and Gauge Fields: Foundations&lt;/em&gt; discusses some of the topology touched on in Part II of Frenkel, but its real use is in providing more discussion of monopoles and their connection to geometry and topology. Also this has connections on fiber bundles in a later chapter with an aim towards application in physics.&lt;/li&gt;&#xA;&lt;li&gt;Gregory L. Naber's &lt;em&gt;Topology, Geometry and Gauge Fields: Interactions&lt;/em&gt; here he covers integration on manifolds, De Rahm cohomology and there is an appendix on Seiberg Witten theory which I am jealous to understand (my student is currently borrowing this text and I need to force him to teach it to me in repayment)&lt;/li&gt;&#xA;&lt;li&gt;Shigeyuki Morita's &lt;em&gt;Geometry of Differential Forms&lt;/em&gt; I probably should have this listed for Part I of Frenkel as well. This little book covers much and what I have read of it made a lot of sense. It has proofs of De Rahm's Theorem as well as the Hodge Decomposition. There is much to see here.&lt;/li&gt;&#xA;&lt;li&gt;Lawrence Conlon's &lt;em&gt;Differentiable Manifolds&lt;/em&gt; can be useful to get a quick big picture of some of the math in Part II of Frenkel. In particular, I recall the Cartan frame calculus is written at an interesting level in Conlon. Some attention is paid to the interplay between calculus on bundles and topology of the basespace.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;h2&gt;&lt;strong&gt;Part III: Lie Groups, Bundles, and Chern Forms&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;p&gt;This part of Frenkel is largely concerned with the addition of structure to fiber bundles which capture the concept of local symmetry and thus lead to the natural formulation of Gauge theory on curves space&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 15: &lt;strong&gt;Lie Groups&lt;/strong&gt; (Lie groups, invariant vector fields and forms, one parameter subgroups, Lie algebra of a Lie group, exponential map, examples of Lie algebra, covering G with one-parameter subgroups?, subgroups and subalgebras, commutators of matrices, left invariant vector fields generate right translations)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 16: &lt;strong&gt;Vector Bundles in Geometry and Physics&lt;/strong&gt; ( vector bundles, fiber coordinates, transition functions, local trivialization, normal bundle to submanifold, Poincare's Theorem and the Euler Characteristic, Hopf's Theorem, connections in a vector bundle, covariant derivative, curvature, complex vector space, structure group of bundle, complex line-bundles, The Electromagnetic Connection, Weyl's principle of Gauge invariance, global potentials, the Dirac Monopole, Aharonov-Bohm Effect)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 17: &lt;strong&gt;Fiber Bundles, Gauss-Bonnet, and Topological Quantization&lt;/strong&gt; (fiber bundles, principal bundles, frame bundles, action of structure group on principal bundle, coset spaces, transitive actions, free actions, stability, isotropy, little subgroup, homogeneous space, Grassmann manifolds, Chern's proof of the Gauss-Bonnet-Poincare-Theorem, Gauss-Bonnet as an Index Theorem, generalizations of Gauss-Bonnet, hermitian line-bundles, index, Chern forms, intersection number, topological quantization condition, Berry Phase, monopoles and the Hopf Bundle)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 18: &lt;strong&gt;Connections and Associated Bundles&lt;/strong&gt; (Maurer-Cartan form, Lie-algebra valued forms on a manifold, Maurer-Cartan equation, anticommutator, connections in a principal bundle, &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;-frames, horizontal distribution, principal bundle, representation, associated bundle through representation, connections in associated bundles, Adjoint bundle, sections of vector bundle, curvature of Ad bundle)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 19: &lt;strong&gt;The Dirac Equation&lt;/strong&gt; (the groups &lt;span class=&quot;math-container&quot;&gt;$SO(3)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$SU(2)$&lt;/span&gt;, rotation group, Lie algebra &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{su}(2)$&lt;/span&gt;, Pauli matrices, &lt;span class=&quot;math-container&quot;&gt;$SU(2)$&lt;/span&gt; is topologically the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-Sphere, adjoint map from &lt;span class=&quot;math-container&quot;&gt;$SU(2)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$SO(3)$&lt;/span&gt; in detail, spinors and rotations of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;, Hamilton and quaternions, Clifford algebras, Dirac as squareroot of d'Alembertian, Lorentz group, &lt;span class=&quot;math-container&quot;&gt;$SU(2)$&lt;/span&gt; is deformation retract of &lt;span class=&quot;math-container&quot;&gt;$SL(2, \mathbb{C})$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$SO(3)$&lt;/span&gt; is a deformation retract of &lt;span class=&quot;math-container&quot;&gt;$L_0$&lt;/span&gt;, Dirac algebra, Dirac Spinors, Dirac Operator, spinor bundle, spin connection)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 20: &lt;strong&gt;Yang-Mills Fields&lt;/strong&gt; (tensorial nature of Lagrange's equations, Noether's Theorem for internal symmetries, Noether Principal relating symmetries and conservation laws, Dirac Lagrangian, building scalars from spinors, Weyl's gauge invariance revisited, Electromagnetic Lagrangian, quantization of the field: photons, Heisenberg nucleon, Yang-Mills nucleon, field strength, quarks, gluons, charge, compact groups and Yang-Mills action, Yang-Mills equation, Yang-Mills analogy with electromagnetism, instantons, pure gauges, instanton winding number, instantons and the vacuum, tunneling and independent vacua )&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 21: &lt;strong&gt;Betti Numbers and Covering Spaces&lt;/strong&gt; (bi-invariant forms, Cartan p-forms, bi-invariant Riemannian metrics, geodesics as one-parameter subgroups or their translates, harmonic forms in the bi-invariant metric, bi-invariant forms are harmonic w.r.t. the bi-invariant metric, Weyl's theorem on vanishing Betti number, Cartan's Theorem for the existence of a nontrivial harmonic 3-form, Poincare's fundamental group &lt;span class=&quot;math-container&quot;&gt;$\pi_1(M)$&lt;/span&gt;, homotopy of loops, simply connected, covering space, universal covering space, orientable covering, lifting paths, universal covering group, Theorem of S.B. Myers, connection of bi-invariant metric, Weyl's theorem about finite fundamental group)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chapter 22: &lt;strong&gt;Chern Forms and Homotopy Groups&lt;/strong&gt; (Yang-Mills &amp;quot;winding number&amp;quot;, winding number in terms of field strength, Chern-Simons 3-form, Chern Forms on &lt;span class=&quot;math-container&quot;&gt;$U(n)$&lt;/span&gt; bundle, Theorem of Chern and Weil, homotopy, covering homotopy, topology of &lt;span class=&quot;math-container&quot;&gt;$SU(n)$&lt;/span&gt;, higher homotopy groups, homotopy groups of spheres, exact sequences of groups, boundary homomorphism, relation between homotopy and homology groups, Hurewicz theorem, some computations of homotopy groups, Hopf map, Hopf fibration, Chern forms as obstructions, Chern's integral)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;My recommendations from the books I happen to have on hand:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Once again &lt;em&gt;Topology, Geometry and Gauge Fields: Foundations&lt;/em&gt; and &lt;em&gt;Topology, Geometry and Gauge Fields: Interactions&lt;/em&gt; by Gregory L. Naber. Also, Shigeyuki Morita's &lt;em&gt;Geometry of Differential Forms&lt;/em&gt; has much to help with later chapters in Frenkel.&lt;/li&gt;&#xA;&lt;li&gt;Mikio Nakahara's &lt;em&gt;Geometry, Topology and Physics&lt;/em&gt; has a lot of discussion of monopoles and instantons, Yang-Mill's equations and more.&lt;/li&gt;&#xA;&lt;li&gt;David Bleecker's &lt;em&gt;Gauge Theory and Variational Principles&lt;/em&gt; builds the framework needed to write Yang-Mills equations in spacetime. Plenty of details on representations of the Lorentz group, connections on fiber bundles etc. This appears in the Bibliography of Frankel with good reason.&lt;/li&gt;&#xA;&lt;li&gt;Stephen Bruce Sontz's &lt;em&gt;Principal Bundles: The Classical Case&lt;/em&gt; this text hits a lot of the high points in Frankel. It is also unusual in its approach to using some category theory to streamline discussion. It builds enough to do Yang-Mills, I don't see much Topology or index theory, but it is intended as more of an introductory treatment. There is certainly an audience for this text. It would have been invaluable to me at one point in my history.&lt;/li&gt;&#xA;&lt;li&gt;Lewis H. Ryder's &lt;em&gt;Quantum Field Theory&lt;/em&gt; in particular Section 3.3 gives a window into how Physicist's came to demand the transformation rule for a potential one-form in electromagnetism. It helps to appreciate the derivation of that same transformation rule arising naturally from the geometry of principal fiber bundles. Of course there is also plenty on Lorentz groups, Pauli matrices and all that good stuff.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;h2&gt;&lt;strong&gt;Other Texts of Note:&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;p&gt;Similar books which I also tend to glance in for topics in Frankel:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Paul Renteln's &lt;em&gt;Manifolds, Tensors, and Forms&lt;/em&gt; I think it is fair to say Renteln is a shorter more disciplined version of Frenkel. It has similar cross-disciplinary goals. Renteln has a lot of wisdom and experience in both math and physics and that text is a treasure.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;M. Gockeler and T. Schucker's &lt;em&gt;Differential Geometry, Gauge Theories, and Gravity&lt;/em&gt; is a relatively short Cambridge monograph (230 pages) so you can surmise the detail. This has great sentimental value to me in that my advisor recommended it as a text to understand why Maxwell's Equations were consistent with Special Relativity. I read it, waiting to find that in there, well... I think its not there (I wanted what is in Resnik's text, gory partial derivative change of variable stuff), but it introduced me to this wild world of abstract bundles and beautiful geometry. No regrets.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;R.W.R. Darling's &lt;em&gt;Differential Forms and Connections&lt;/em&gt; I also read this as an undergraduate and I think a lot of it slipped past me. However, I think this is where I first learned of the flux and work form maps which are essential for understanding the interplay between the differential forms and vectors on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;. I probably would learn a lot if I reread this book now.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bernard Schutz &lt;em&gt;Geometrical Methods of Mathematical Physics&lt;/em&gt; on occasion I find formulas in this book are really helpful.&lt;/p&gt;&#xA;&lt;p&gt;Well, I'm sure there are more to add. It occurs to me that Frankel's text may well be the Kevin Bacon for math physics books I treasure most.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="128" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-04-06T03:45:41.957" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18140" PostTypeId="2" ParentId="18136" CreationDate="2020-04-06T10:06:12.363" Score="6" Body="&lt;p&gt;The best explanation of the general concept I've encountered so far is the introduction to a 4D game, Miegakure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea of extra dimensions is described in the jump from 2D to 3D first, which makes it much easier to visualize and extrapolate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=9yW--eQaA2I&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=9yW--eQaA2I&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13764" LastActivityDate="2020-04-06T10:06:12.363" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18141" PostTypeId="1" CreationDate="2020-04-06T12:19:00.023" Score="6" ViewCount="903" Body="&lt;p&gt;It's easy to recognize visually when an orthogonal coordinate system has its axes in the same scale. See, for instance, the following image. But I'm trying to write down a &lt;strong&gt;precise definition&lt;/strong&gt; of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After searching the usual channels (Google Scholar, Google Books), my impression is that it seems this knowledge is implicit in High School teaching, that is, the explanation given is mainly visual: the teacher shows an example of an orthogonal coordinate system with the same scale on both axes and another example with different scales. Done. Some people define &quot;the axes are on the same scale if they have the same unit&quot; but, then, what does it mean &quot;to have the same unit&quot;? I'm looking for a precise definition accessible to Secondary School students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my three questions are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Do you know a precise definition for &quot;axes on the same scale&quot; accessible to Secondary School students?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Do you know some school textbook or a scientific article/book where such definition is presented or discussed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) What do you think about this definition: &quot;We say that an orthogonal coordinate system of the plane has the x and y axes in the same scale if the segment joining (0, 0) to (1, 0) has the same length as the segment joining (0, 0) and (0, 1) when measured with the same ruler.&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While in Analytic Geometry it is always assumed the axes are on the same scale, when studying functions or Statistics, axes on different scales is a necessity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/dYcQn.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dYcQn.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2020-04-22T13:21:23.657" Title="How to define &quot;axes with the same scale&quot; in Secondary/High School?" Tags="|secondary-education|definitions|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18142" PostTypeId="2" ParentId="18141" CreationDate="2020-04-06T13:42:31.937" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;We say that an orthogonal coordinate system of the plane has the x and y axes in the same scale if the segment joining (0, 0) to (1, 0) has the same length as the segment joining (0, 0) and (0, 1) when measured with the same ruler.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think there is a problem with this definition, which has to do with the difference between how mathematicians think about numbers and how scientists and engineers do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose that your graph shows the growth of a plant. We graph its height &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; versus the time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;. The height is in mm and the time is in days. Mathematicians think of the units as being part of the definition of the variables, while scientists and engineer think of them as being part of the value of the variables. On a graph labeled &quot;&lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; (mm),&quot; a mathematicians says that the &quot;(mm)&quot; defines &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, while a scientist says that the &quot;(mm)&quot; is a unit to be applied to all the numbers on the scale, so that we don't need to write 1 mm, 2 mm, 3 mm, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think like a physicist, so if I try to apply your definition, I say, &quot;The point (1,0) doesn't exist on this graph. The unitless 1 isn't a possible value for the variable &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;.&quot; An example of a point that does exist is (1 day,0).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if you want to be compatible with the way scientists and engineers think, I would do something like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We say that an orthogonal coordinate system of the plane has the x and y axes in the same scale if the axes have compatible units and the segment joining (0, 0) to &lt;span class=&quot;math-container&quot;&gt;$(a, 0)$&lt;/span&gt; has the same length as the segment joining (0, 0) and &lt;span class=&quot;math-container&quot;&gt;$(0, a)$&lt;/span&gt; when measured with the same ruler.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;By this definition, we could have square graph paper with 1 cm boxes on the x axis and 10 mm boxes on the y axis, and that would be OK.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-04-06T13:42:31.937" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18143" PostTypeId="2" ParentId="18116" CreationDate="2020-04-06T14:35:00.513" Score="2" Body="&lt;p&gt;Well, this is not a systematic course proposal, but a list of useful references I've collected about the subject:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A must read for your staff team (it brings arguments from neuroscience): &lt;a href=&quot;https://www.amazon.com.br/dp/B07KW5F7R9/ref=dp-kindle-redirect?_encoding=UTF8&amp;amp;btkr=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mind in Motion: How Action Shapes Thought&lt;/a&gt; by Barbara Tversky.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.frontiersin.org/research-topics/1167/perspective-taking-building-a-neurocognitive-framework-for-integrating-the-social-and-the-spatial&quot; rel=&quot;nofollow noreferrer&quot;&gt;Perspective Taking: building a neurocognitive framework for integrating the “social” and the “spatial”&lt;/a&gt; by Kessler et al.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.springer.com/gp/book/9783658206314&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Relationship between Language and Spatial Ability: An Analysis of Spatial Language for Reconstructing the Solving of Spatial Tasks&lt;/a&gt; by Mizzi Angel.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.springer.com/gp/book/9783319443843&quot; rel=&quot;nofollow noreferrer&quot;&gt;Visual-spatial Ability in STEM Education: Transforming Research into Practice&lt;/a&gt; by Myint Swe Khine.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1138792039&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Spatial Reasoning in the Early Years: Principles, Assertions, and Speculations&lt;/a&gt; by Brent David and The Spatial Reasoning Study Group.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://my.vanderbilt.edu/smpy/files/2013/02/Wai2009SpatialAbility.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Spatial Ability for STEM Domains: Aligning Over 50 Years of Cumulative Psychological Knowledge Solidifies Its Importance&lt;/a&gt; by Jonathan Wai, David Lubinski, and Camilla P. Benbow.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.math.yorku.ca/~whiteley/cmesg.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The  Decline  and  Rise  of  Geometry  in  20th  Century&lt;/a&gt; by Walter Whiteley.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://twitter.com/davidkbutleruoa/status/1001662373348818945&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jenga Views&lt;/a&gt; by David Butler. File is &lt;a href=&quot;https://drive.google.com/file/d/1rPl5rv1C6QgpSEtEKHkFjKIbpYHmhhO7/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://upcommons.upc.edu/bitstream/handle/2099/1045/st14-09-a6-ocr.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Twelve Exercises On Spatial Perception&lt;/a&gt; by Janos Baracs.&lt;/li&gt;&#xA;&lt;li&gt;While concrete manipulatives are highly recommended for early ages (see the Jenga Views activity, 3D printer projects and &lt;a href=&quot;https://en.wikipedia.org/wiki/Perplexus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Perplexus 3D Maze&lt;/a&gt;), I would like to suggest the use of a 3D dynamic geometry software like GeoGebra 3D (&lt;a href=&quot;https://www.geogebra.org/download&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geogebra.org/download&lt;/a&gt;). In particular, there is a module for augmented reality using GeoGebra for smartphones that allows unique experiences. Please, see the videos: &lt;a href=&quot;https://youtu.be/l2B-RS6xrvg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/l2B-RS6xrvg&lt;/a&gt;, &lt;a href=&quot;https://youtu.be/_rnzLbF_WKo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/_rnzLbF_WKo&lt;/a&gt;, &lt;a href=&quot;https://youtu.be/e_HoHZPSdTQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/e_HoHZPSdTQ&lt;/a&gt;, &lt;a href=&quot;https://youtu.be/QC7Yg6-D2lA&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/QC7Yg6-D2lA&lt;/a&gt;, &lt;a href=&quot;https://youtu.be/YjT7dQsuZg8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/YjT7dQsuZg8&lt;/a&gt;. Moreover, it would be a great exercise if your students use the software to buid by themselves their own AR constructions in GeoGebra 3D.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MjUDP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MjUDP.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This is a 3D game that connects Spatial Thinking with Languages: &lt;a href=&quot;http://www.cdme.im-uff.mat.br/html5/triplets/triplets-html/triplets-en.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trip-Lets&lt;/a&gt;. Here a children playing with it: &lt;a href=&quot;https://youtu.be/n5ymEpqvfSQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/n5ymEpqvfSQ&lt;/a&gt;. The game also appeared in the &quot;Geometry and Imagination&quot; at ICM 2018 at Rio de Janeiro. There are version in English, French, German, Spanish and Portuguese. If you want, we may try to provide a version in Russian.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eAl7s.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eAl7s.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2020-04-06T14:35:00.513" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18144" PostTypeId="2" ParentId="18136" CreationDate="2020-04-06T14:54:44.487" Score="9" Body="&lt;p&gt;I strongly recommend the film &lt;a href=&quot;http://www.dimensions-math.org/Dim_E.htm&quot; rel=&quot;noreferrer&quot;&gt;Dimensions&lt;/a&gt; by Jos Leys, Étienne Ghys and Aurélien Alvarez. It's free! The main tools used by the authors to explain the several dimensions are cross sections and steregraphic projections. The animation is very didactic, building the ideas in 2D and 3D as preparation for 4D. There are dubbed versions in Deutsch, American English, Français, Español, Italiano, 日 本語 and Pусский.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are the first four chapters of the film:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dimension 2: &lt;a href=&quot;https://youtu.be/6cpTEPT5i0A&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/6cpTEPT5i0A&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ud8v8.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ud8v8.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dimension 3: &lt;a href=&quot;https://youtu.be/AhM9JH5GNiI&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/AhM9JH5GNiI&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/flOqy.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/flOqy.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dimension 4: &lt;a href=&quot;https://youtu.be/nz0ku71x22A&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/nz0ku71x22A&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ojr5J.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ojr5J.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="8828" LastEditDate="2020-04-06T16:23:04.430" LastActivityDate="2020-04-06T16:23:04.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18146" PostTypeId="2" ParentId="18136" CreationDate="2020-04-06T16:27:40.810" Score="1" Body="&lt;p&gt;I would posit it's impossible for most humans to truly grok higher dimensions. I've heard of extremely rare exceptions but I'm going to ignore them here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having an intuitive understanding of some fact about geometry requires having a &lt;em&gt;mental image&lt;/em&gt; of what is going on. You can't have 4D mental images. Can &lt;em&gt;you&lt;/em&gt; mentally &quot;see&quot; a tesseract? Not the common 3D bastardization of it but the real thing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anything, trying to represent 4D objects for 3D brains on a 2D medium seems counter-productive. The classical 3D representation of a tesseract is actually misleading. A better way to put it would be to film a cube changing colors over time, or to show 2 cubes while saying that there is a unit distance in the 4th dimension between the &quot;same&quot; vertices of the two cubes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, complete visual representations are just the wrong tool for this. You might sort of get something out of it for the 4th dimension, but it becomes hopeless beyond that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For inspiration, here's a 3blue1brown video which did an OK job giving intuition about a higher dimension problem, though a completely different one: &lt;a href=&quot;https://youtu.be/zwAD6dRSVyI&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/zwAD6dRSVyI&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13579" LastActivityDate="2020-04-06T16:27:40.810" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18147" PostTypeId="2" ParentId="18136" CreationDate="2020-04-06T18:04:05.653" Score="9" Body="&lt;p&gt;That reminds me of my first college programming course where they drew a square picture of a 2D array, a cube for a 3D array, and then said 4D arrays were very hard to understand. But I'd already made 4D arrays that were fine, since they weren't representing points in 4d space. I'd played a computer dungeon game where you had continent, province, dungeon, and floor. 4D. They're not orthogonal to each other, exactly, but the general concept of N dimensions is simple enough, that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A questionaire on a bad dating site could be: True/False, you enjoy: Hiking, Cooking, Dancing, Travel. That has the properties of a 4D cube, right? Each set of answers is like a corner, with Hamming distance of a most 4. Adding more questions increases the dimension. The number of corners and edges blow up, but conceptually, a dozen dimensions is simple.&lt;/p&gt;&#xA;" OwnerUserId="13766" LastActivityDate="2020-04-06T18:04:05.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18148" PostTypeId="2" ParentId="18136" CreationDate="2020-04-06T19:10:50.093" Score="3" Body="&lt;p&gt;For more than four dimensions, I would consider treating each pixel in a gray scale image as a &quot;dimension&quot; and its brightness as the value of the corresponding &quot;coordinate&quot;. Then, ℝ&lt;sup&gt;&lt;i&gt;m&lt;/i&gt; x &lt;i&gt;n&lt;/i&gt;&lt;/sup&gt; is just the set of all images (including photos) of &lt;i&gt;m&lt;/i&gt; pixels by &lt;i&gt;n&lt;/i&gt; pixels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a video on YouTube which explores this approach (and a few geometric interesting aspects of these high dimension spaces): &lt;a href=&quot;https://www.youtube.com/watch?v=BePQBWPnYuE&quot; rel=&quot;nofollow noreferrer&quot;&gt;My understanding of the Manifold Hypothesis | Machine learning&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5848" LastActivityDate="2020-04-06T19:10:50.093" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18149" PostTypeId="2" ParentId="18136" CreationDate="2020-04-06T22:34:02.997" Score="13" Body="&lt;p&gt;My suggestion would be to demystify the concept and try to disassociate it from spatial interpretations at the first approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; dimensions is just having &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; variables. As one professor I had liked to repeat, a grocery store owner who is trying to  maximize gains by selling oranges, apples, bananas and peaches with some constraint in his storage is attempting to solve a &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;-dimensional problem without even thinking about it. This is much easier to use for motivation. Then, from there, &lt;em&gt;if it is useful/needed/appreciated&lt;/em&gt;, one can introduce spatial visualizations like projections into space/plane etc and geometric structures like the inner product etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found (personally) that relying on &quot;materializing&quot; higher dimensions as an initial approach often backfires, both for the understanding of students (those who get interested frequently veer into philosophical aspects rather than mathematical ones) and their motivation (those who do not get interested will just act as if it is useless and/or nonsensical and/or obscure).&lt;/p&gt;&#xA;" OwnerUserId="4817" LastActivityDate="2020-04-06T22:34:02.997" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18150" PostTypeId="1" CreationDate="2020-04-07T05:22:56.810" Score="3" ViewCount="324" Body="&lt;p&gt;I have been searching for the International Math Olympiad(IMO) qualification process in Pakistan and I found out that students take the NMTC(National Mathematics Talent Contest) paper which to get into you have to have certain marks on your O levels. Then 50 of the top students are chosen which are invited to a training camp. The people are narrowed down until 5 are left. However, according to my knowledge, some parts have changed and I couldn't find reliable sources to confirm what has changed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I am asking for is essentially the qualification process and the sources which are important for the information.&lt;/p&gt;&#xA;" OwnerUserId="13769" LastActivityDate="2024-02-26T07:23:44.533" Title="IMO selection process in Pakistan" Tags="|math-contests|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18151" PostTypeId="2" ParentId="18136" CreationDate="2020-04-07T09:58:39.897" Score="0" Body="&lt;p&gt;I invented higher dimensions at an early age and have scored &quot;off the charts&quot; in spatial insight on any test. Since I've made numerous &quot;infinite dimensional&quot; puzzles (because 3 dimensional ones were too boring).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is how I grasp the concept of a hypercube:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each hyper cube of N dimensions exists of 2^N points. Half of those points (aka 2^(N-1)) form a N-1 dimensional hypercube, as does the other half. There a N ways to pick such a pair *). One such half is a copy of the other, merely translated the side L into a dimension perpendicular to the N-1 dimensions that those hypercubes exist in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*) Each corner point has (for example) coordinates 0 or 1 for each dimension: each point is represented by a vector like [0,1,1,0,0,1,0,0,0,1] where every permutation of 0 and 1's occur (leading to the 2^N points). Chose any coordinate and separate the points into two groups: one where that coordinate is 0 and one where that coordinate is 1. Hence, N choices. The remaining N-1 coordinates are again a vector of 0 and 1's that contain every permutation; so they are obviously also hypercubes, of one dimension less.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence you can &quot;build up&quot; a hypercube from lower dimensions as follows: start with a point. Translate this point over a distance L. Note how it doesn't matter in WHICH direction, even though you have 3 dimensions to pick from (when restricting yourself still to 3D space). The point &quot;draws&quot; a line while being translated, giving you a line piece. The number of points have doubled: from 1 point to 2 points. Now you have a 1D hypercube.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next translate this line piece (1D hypercube) in any direction perpendicular to the previous used direction (even in 3D space this still allows choice, but which choice you make doesn't matter: all not used dimensions are equivalent), over a distance L. This doubles the points again, and each point draws a line again while being translated (in the end ask the students to find the formula for the number of lines as function of N). Next translate the resulting 2D hypercube (the square) over a distance L perpendicular to the square. This draws four more lines and doubles the number of points from 4 (one square) to 8 (original square plus copy).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, translate the 3D hypercube over a distance L in a direction perpendicular to all previously used 3 dimensions. Note that there are infinite dimensions, but which direction you choose is not important, as long as it is per perpendicular to the used dimensions. The result of that is that new lines that are being drawn during the translation of the copy all are perpendicular to the orginal hypercube and thus all make an angle of 90 degrees with every previous drawn line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And so on: make a copy of the N-dimensional hypercube, translate it over a distance L perpendicular to all previous used dimensions, making all 2^N points draw 2^N extra lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note how every dimension is symmetrical: there are N axis, on each axis there are two opposite N-1 dimensional hypercubes: the &quot;outsides&quot; that limit the hypercube on that dimension (aka there are 2N outsides).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some students will grasp it.  Let them form groups were students that got it explain in their own words to other students how they see it and how they grasped it. It can help to have someone else explain it (in different words).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a puzzle that I made:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given a hypercube of N dimension in an N dimensional space. If you paint the 2N outsides of the hypercube from a pallet of k colors, how many under rotation different permutations can you make?  For example, N=2, k=2 gives: AAAA, AAAB, AABB, ABAB, ABBB and BBBB, so 6 different permutations (rotation of the squares is rotations of the strings here). N=2, k=3 gives 24 different permutations. What is the general formula? Don't look it up cause I have it published on the net somewhere :p&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More abstract, but certainly important, are the coordinate vectors with all permutations of 0's and 1's.  You could explain that if you add more zero's but never change those zero's - then they don't matter. Aka:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;0,0,0,0,0,0&#xA;0,0,0,1,0,0&#xA;0,0,1,0,0,0&#xA;0,0,1,1,0,0&#xA;0,1,0,0,0,0&#xA;0,1,0,1,0,0&#xA;0,1,1,0,0,0&#xA;0,1,1,1,0,0&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;spans a 3D cube (in 6D space, but that doesn't matter at all).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Likewise you could keep a coordinate at 1 (or whatever) as long as it doesn't change, it isn't used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Making a copy then is easy: copy the table and change one of the unused 0's into a 1.  Both are 3D cubes as explained before, but they are translated by a distance 0,0,0,0,0,1 (or whatever coordinate you changed), and together now form a 4D hypercube.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question for the class: what if you correlate the coordinates? Ie, you pick two columns and only use 0,1 or 1,0 and never 0,0 or 1,1. Then that one column counts as 1 bit.  This way you can ALSO make 2^N vectors of every &quot;permutation&quot;, but using more than N (changing) coordinates (answer: a hyperblock; unless you &lt;em&gt;only&lt;/em&gt; use pairs, for example,&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;0,1,0,1,0,1&#xA;0,1,0,1,1,0&#xA;0,1,1,0,0,1&#xA;0,1,1,0,1,0&#xA;1,0,0,1,0,1&#xA;1,0,0,1,1,0&#xA;1,0,1,0,0,1&#xA;1,0,1,0,1,0&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;is a perfect 3D cube, in 6D space).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;unrelated maybe, but a neat invention of me:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;0,0,0,0,0,1&#xA;0,0,0,0,1,0&#xA;0,0,0,1,0,0&#xA;0,0,1,0,0,0&#xA;0,1,0,0,0,0&#xA;1,0,0,0,0,0&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Is an N-dimensional hyper tetrahedron in N+1 dimensions.&#xA;Isn't it amazing how simple the coordinates become if you add one dimension?!&#xA;Try to write the coordinates down using only N dimensions :p (if at all possible!).&lt;/p&gt;&#xA;" OwnerUserId="13770" LastEditorUserId="13770" LastEditDate="2020-04-07T10:25:45.210" LastActivityDate="2020-04-07T10:25:45.210" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18152" PostTypeId="2" ParentId="18141" CreationDate="2020-04-07T14:34:15.427" Score="0" Body="&lt;p&gt;This is a good question, and brings up some subtle issues. A clue is to observe the tacit assumption, made when drawing the two graphs in the original post, that the axes are perpendicular. We draw the axes to appear perpendicular, even though orthogonality might have no intrinsic sense in the underlying mathematics or physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we start with is a pair of one-dimensional vector spaces &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, with not particular basis designated. Think of &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; as &quot;time&quot; and &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; a &quot;temperature.&quot; We form the two-dimensional vector space &lt;span class=&quot;math-container&quot;&gt;$V=L\oplus M$&lt;/span&gt;, which still has no designated basis. Now here is the key point: we select non-zero vectors &lt;span class=&quot;math-container&quot;&gt;$l\in L,\ m\in M$&lt;/span&gt;. This is the point where we choose &quot;units&quot; like &lt;em&gt;seconds&lt;/em&gt; and &lt;em&gt;degrees Celsius&lt;/em&gt;. Furthermore, we identify &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$({\bf R}^2,\langle\cdot,\cdot\rangle)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\langle\cdot,\cdot\rangle$&lt;/span&gt; is the canonical inner product on &lt;span class=&quot;math-container&quot;&gt;${\bf R}^2$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So at this point, we have identified &lt;span class=&quot;math-container&quot;&gt;$V=L\oplus M$&lt;/span&gt;, which was endowed with  &lt;em&gt;neither&lt;/em&gt; an inner product or a basis, with the canonical inner product space &lt;span class=&quot;math-container&quot;&gt;$({\bf R}^2,\langle\cdot,\cdot\rangle)$&lt;/span&gt;. This is what allows us to graph in the Cartesian plane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is interesting to observe that the notion of &lt;em&gt;area&lt;/em&gt; under the curve has meaning, and the arena for this is the tensor product &lt;span class=&quot;math-container&quot;&gt;$L\otimes M$&lt;/span&gt;, which is a one-dimensional vector space. Now think of &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; as &quot;distance&quot; and &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; as &quot;force&quot; (scalar). If we measure distance with the units of &lt;em&gt;feet&lt;/em&gt; and force in &lt;em&gt;pounds&lt;/em&gt;, then area is &quot;work&quot; and has units of &lt;span class=&quot;math-container&quot;&gt;$\hbox{foot}\cdot\hbox{pounds}$&lt;/span&gt;. This mathematical quantity (area) lives in the tensor product &lt;span class=&quot;math-container&quot;&gt;$L\otimes M$&lt;/span&gt;, which inherits a basis from the choice of units in &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, i.e., foot and pound.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now--how to bring this to secondary/high school? Evidently, this perspective is too abstract for that. I suggest a brief mention of &lt;em&gt;oblique&lt;/em&gt; coordinates, and the move on quickly!&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2020-04-16T01:21:18.403" LastActivityDate="2020-04-16T01:21:18.403" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18153" PostTypeId="2" ParentId="18136" CreationDate="2020-04-07T19:24:47.533" Score="2" Body="&lt;p&gt;Here's how i understood it, am 18 so correct me if I'm wrong&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This site was cool --&gt; &lt;a href=&quot;https://4dtoys.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://4dtoys.com/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think its important to have a physical understanding/visualization of the dimensions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1D point would be like a bead on a string.&#xA;2D shape is like a hockey puck on a hockey table, the plane.&#xA;3D shape is something like an apple or a cube in a room.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;note how a 3d shape like an apple is made up of many 2d slices/apple crossections, and how a 2d shape is made out of many lines, and that lines are a bunch of points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A super important part of these dimensions is that when a 1D shape is stuck along one dimension, another shape cannot exist at the same point in that one dimension without overlap. Basically two beads cannot both be at the 1 inch mark, 2 hockey pucks cannot both lay in the same place while still laying flat on their plane, and two people cannot take up the same space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only way to have two objects lay in the same x dimensions is to add another dimension. Like put the bead on another string, or stack the puck on top of the other. From a certain perspective, these two objects are now totally overlapping, when in reality they are separate thanks to the additional dimension&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what about 3D objects? Like the apple and the cube?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simply pick up the cube, and put the apple where the cube was. Now they have taken up the same overlapping space without actually overlapping. The 4th dimension here is time. If you &quot;remove&quot; that dimension then the apple and cube would overlap.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;oh, and stuff can disappear if it escapes your observable dimension. like how the apple slices get bigger and smaller 3d stuff can get bigger, smaller, and disappear in this dimension, but really still exist in another. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And in the same way you can slice a cube in different ways to create shapes other than more rectangular prisms, these 3D slices of 4D things can be more than cubes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Idk, this stuff really is wack. But the idea is there. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think stuff like the Klein bottle needs this 4th dimension to exist since it overlaps itself in the 3rd dimension, and basically the parts that would've overlapped exist in different dimensions?&lt;/p&gt;&#xA;" OwnerUserId="13775" LastEditorUserId="13775" LastEditDate="2020-04-07T20:06:00.353" LastActivityDate="2020-04-07T20:06:00.353" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18155" PostTypeId="2" ParentId="17952" CreationDate="2020-04-08T06:47:46.857" Score="20" Body="&lt;p&gt;Many logicians that I have spoken to have concurred with my assessment that this is an issue of the misleading use of &quot;let&quot;. Many teachers use this word in two &lt;strong&gt;very different&lt;/strong&gt; and &lt;strong&gt;incompatible&lt;/strong&gt; ways. The first is universal quantification, as in your example. The second is existential instantiation, as in &quot;Let &lt;span class=&quot;math-container&quot;&gt;$z = \exp(x+y)$&lt;/span&gt;. Then [blah blah] about &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;.&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The solution is simple. Do not use the bare word &quot;let&quot; for universal quantification, because it is not even technically the most precise way of expressing the desired meaning in English. Here are some much preferable alternatives:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Take/consider any polygon &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. ...&lt;/li&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; be any given polygon. Then ...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The key word is &quot;&lt;strong&gt;any&lt;/strong&gt;&quot;, which is needed to precisely convey the universal quantification. If you want to express universal quantification in a single sentence, you can use the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Given any polygon &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, ...&lt;/li&gt;&#xA;&lt;li&gt;For every polygon &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, ...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note that the English word &quot;any&quot; behaves differently from &quot;every&quot;. For example, &quot;If for any &lt;span class=&quot;math-container&quot;&gt;$x∈S$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$f(x)∈T$&lt;/span&gt;, then ...&quot; actually means &quot;For every &lt;span class=&quot;math-container&quot;&gt;$x∈S$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$f(x)∈T$&lt;/span&gt; then ...&quot; and not &quot;If for every &lt;span class=&quot;math-container&quot;&gt;$x∈S$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$f(x)∈T$&lt;/span&gt;, then ...&quot;. So to minimize confusion it is better to stick with &quot;for every&quot; in single-sentence universal quantification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I often hear the excuse that people have been using &quot;let&quot; in the confusing way for so long already, and that students have to learn to interpret it anyway. That is an excuse, because we have no reason not to use &lt;strong&gt;precise&lt;/strong&gt; words in our teaching. As you yourself pointed out &lt;em&gt;emphatically&lt;/em&gt;, using the word &quot;any&quot; would make things so much clearer. So we should use it! After students have acquired a proper grasp of logical reasoning (and not before that), facilitated by precise teaching, we can then tell students that some textbooks actually use &quot;let&quot; in the confusing manner, and they will have no trouble with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related to this, students understand quantifiers far better when they are explained in terms of &lt;a href=&quot;https://math.stackexchange.com/a/1782071/21820&quot;&gt;game semantics&lt;/a&gt;. Similarly for teaching &lt;a href=&quot;https://matheducators.stackexchange.com/a/10034/1550&quot;&gt;structural induction&lt;/a&gt;. These are also perfectly in line with the use of the phrasing &quot;given any ...&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2020-04-08T06:47:46.857" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="18156" PostTypeId="1" CreationDate="2020-04-08T09:22:00.247" Score="4" ViewCount="200" Body="&lt;p&gt;I know Dr. Martin Greenhow and his team of Brunel University London have developed online questions and I have used some of these at the following url: &lt;a href=&quot;http://maths-for-all.co.uk/engineering-mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://maths-for-all.co.uk/engineering-mathematics/&lt;/a&gt; Are there any other online questions like these that students can use for testing themselves at these difficult times for them at home?&lt;/p&gt;&#xA;" OwnerUserId="1567" LastActivityDate="2020-04-09T11:18:55.663" Title="Are there any online question bank of mathematics questions?" Tags="|mathematical-pedagogy|online|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18157" PostTypeId="2" ParentId="18136" CreationDate="2020-04-08T11:33:16.223" Score="1" Body="&lt;p&gt;Use arrays (aka &lt;a href=&quot;https://www.doitpoms.ac.uk/tlplib/tensors/what_is_tensor.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;tensors&lt;/a&gt; in math lingo) to store the values of a place:&lt;/p&gt;&#xA;&lt;p&gt;0 dimensions / zero rank (or order) tensor:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;a = 1&#xA;v = a&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;These simple values are also known as &lt;em&gt;scalars&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;1 dimension / first rank tensor:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;a = [1, 2, 3, ...]&#xA;v = a[x]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;These lists of values are also known as &lt;em&gt;vectors&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;2 dimensions / second rank tensor:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;a = &#xA;[&#xA;    [1, 2, 3, ...],&#xA;    [1, 2, 3, ...],&#xA;    [1, 2, 3, ...],&#xA;    ...&#xA;]&#xA;v = a[x][y]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;These tables are also known as rows by columns &lt;em&gt;matrices&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;3 dimensions / third rank tensor:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;a = &#xA;[&#xA;    [&#xA;        [1, 2, 3, ...],&#xA;        [1, 2, 3, ...],&#xA;        [1, 2, 3, ...],&#xA;        ...&#xA;    ],&#xA;    [&#xA;        [1, 2, 3, ...],&#xA;        [1, 2, 3, ...],&#xA;        [1, 2, 3, ...],&#xA;        ...&#xA;    ],&#xA;    [&#xA;        [1, 2, 3, ...],&#xA;        [1, 2, 3, ...],&#xA;        [1, 2, 3, ...],&#xA;        ...&#xA;    ],&#xA;    ...&#xA;]&#xA;v = a[x][y][z]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;4 dimensions / 4th rank tensor / 4D array/matrix/object with 2+ elements per dimension and numbered integer indices:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;a = &#xA;[&#xA;    [&#xA;        [&#xA;            [1, 2, ...],&#xA;            [1, 2, ...],&#xA;            ...&#xA;        ],&#xA;        [&#xA;            [1, 2, ...],&#xA;            [1, 2, ...],&#xA;            ...&#xA;        ],&#xA;        ...&#xA;    ],&#xA;    [&#xA;        [&#xA;            [1, 2, ...],&#xA;            [1, 2, ...],&#xA;            ...&#xA;        ],&#xA;        [&#xA;            [1, 2, ...],&#xA;            [1, 2, ...],&#xA;            ...&#xA;        ],&#xA;        ...&#xA;    ],&#xA;    ...&#xA;]&#xA;v = a[i1][i2][i3][i4]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Etc. Note the self-similarity.&lt;/p&gt;&#xA;&lt;p&gt;When you scale the dimensional axes, the amount of data changes by the power of the number of axes. Doubling a 1D array doubles the data (2&lt;sup&gt;1&lt;/sup&gt; = 2), doubling a 2D array squares the data (2&lt;sup&gt;2&lt;/sup&gt; = 4), doubling a 3D array cubes the data (2&lt;sup&gt;3&lt;/sup&gt; = 8), and so on.&#xA;To save data, one could simply store the coordinates on those axes, as a dimension is &amp;quot;a measurable extent of a particular kind, such as length, breadth, depth, or height&amp;quot; according to the dictionary.&lt;/p&gt;&#xA;&lt;p&gt;Other self-similar things are fractals, so named because they have &lt;a href=&quot;https://www.youtube.com/watch?v=gB9n2gHsHN4&quot; rel=&quot;nofollow noreferrer&quot;&gt;a fractional dimension&lt;/a&gt;. Consider a Sierpiński triangle. Doubling its size leads to a data increase of 3, so 2&lt;sup&gt;D&lt;/sup&gt; = 3, so its dimension is log&lt;sub&gt;2&lt;/sub&gt;(3) = 1.58496250072.&lt;/p&gt;&#xA;" OwnerUserId="13523" LastEditorUserId="13523" LastEditDate="2022-06-29T20:20:59.933" LastActivityDate="2022-06-29T20:20:59.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18158" PostTypeId="1" AcceptedAnswerId="18160" CreationDate="2020-04-09T11:09:36.747" Score="-1" ViewCount="116" Body="&lt;p&gt;I am really not sure how to tackle a 200 page huge script with a lot of information about the topic linear algebra. Writing down the definitions would not make sense in my opinion, because if I want to look up something, I probably would take a look in my script.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would it not be better to just write down the name of the definition and concepts one has to learn on a paper such that you learn by solving problems? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to look up things that you have to know, you can take a look on the paper and see how you feel about it? So writing down huge part of the script would not be efficient. It would be more efficient when you take that time to practice?&lt;/p&gt;&#xA;" OwnerUserId="13782" LastActivityDate="2020-04-09T12:30:31.193" Title="How to start tackling a 200 page big script?" Tags="|self-learning|linear-algebra|" AnswerCount="1" CommentCount="2" ClosedDate="2020-04-09T14:45:58.253" ContentLicense="CC BY-SA 4.0" />
  <row Id="18159" PostTypeId="2" ParentId="18156" CreationDate="2020-04-09T11:18:55.663" Score="4" Body="&lt;p&gt;The Open University offers &lt;a href=&quot;https://www.open.ac.uk/courses/maths&quot; rel=&quot;nofollow noreferrer&quot;&gt;courses&lt;/a&gt; in Mathematics and Associated subjects at a range of levels.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their site has short, marked quizzes to rate your current ability as well as useful study resources.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Select one from Certificate, Diploma and Degree.&lt;/li&gt;&#xA;&lt;li&gt;Choose a combination, mathematics or a combined option. &lt;/li&gt;&#xA;&lt;li&gt;Scroll down and click on the Entry Requirements tab.&lt;/li&gt;&#xA;&lt;li&gt;Select the self-assessed quiz. There are study resources to prepare.&lt;/li&gt;&#xA;&lt;li&gt;Choose which level 1, 2, 3 and complete the quiz!&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8660" LastActivityDate="2020-04-09T11:18:55.663" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18160" PostTypeId="2" ParentId="18158" CreationDate="2020-04-09T12:30:31.193" Score="3" Body="&lt;p&gt;Each one learns in different ways. You need to find one that works for you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not all texts are equally agreeable to everybody. As you seem to struggle with the current one, make a rough list of the topics it covers and look for alternatives that cover the same ground and are easier to understand (or at least help you get a grip to help in diving into your text). Linear algebra is standard fare, I'm sure a search nets scores of free books and lecture notes. I'm partial to Treil's &lt;a href=&quot;https://www.math.brown.edu/~treil/papers/LADW/LADW.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Linear Algebra done Wrong&lt;/a&gt;, but that's just me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take a bird's view look at a chapter, see what it is about, what it's more important results are. Perhaps try your hand at some of the easier problems. Then dig in, look at definitions. Try to figure out &lt;em&gt;why&lt;/em&gt; the stuff is defined the way it is. Make up examples of the things defined, and non-examples (as contrast). Check any results that are proved, see how they are used. Once you've got a grip on the use and importance, check out the proof. See what it proves, make sure it checks out (proofs use the previous results, often in not-so-obvious ways, so understanding them is good training). Maybe just look at the more important results first, sort of spiraling in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reading math of any type is &lt;em&gt;not&lt;/em&gt; like reading a novel. The writing is &lt;em&gt;much&lt;/em&gt; denser, a large part of it devoted to discuss stuff that isn't obvious or known beforehand. Don't expect to advance fast. Just keep moving. If it gets too tough, look for other sources. Then, maybe, retake the thread where you left.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do the exercises, &lt;em&gt;after&lt;/em&gt; solving (or at least giving them a decent shot) look up solutions. Again, there are many hundreds of linear algebra courses being offered all over the world, many publish homework/exams, often with full solutions.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-04-09T12:30:31.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18161" PostTypeId="2" ParentId="18141" CreationDate="2020-04-12T01:11:11.940" Score="1" Body="&lt;p&gt;Great question!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And great diagrams: it suddenly occurs to me that a mathematical circle may appear pictorially as an ellipse if the scales are different. This doesn't mean we're talking about a non-circular ellipse; it just means we're drawing the circle under question in a particular way. Honestly, this had never occurred to me before!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your question, how about: The axes of a graph are said to have the &lt;em&gt;same scale&lt;/em&gt; if and only if we can rotate said graph in such a way that (a) the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; on the horizontal axis moves to the position of the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; on the vertical axis and (b) the new origin coincides with the old origin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, you'll have to break down the definition into parts and talk through some examples before students will understand what's being described. Those diagrams will definitely help!&lt;/p&gt;&#xA;" OwnerUserId="13800" LastActivityDate="2020-04-12T01:11:11.940" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18162" PostTypeId="1" CreationDate="2020-04-13T22:37:14.743" Score="5" ViewCount="461" Body="&lt;p&gt;My institution (in USA) is planning to introduce a minor in mathematics and here are the proposed course requirements:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Calculus I&lt;/li&gt;&#xA;&lt;li&gt;Calculus II&lt;/li&gt;&#xA;&lt;li&gt;Calculus III or Differential Equations&lt;/li&gt;&#xA;&lt;li&gt;Any two of the following three: Linear Algebra, Bridge Course to Abstract Mathematics and Elementary Statistics II.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I personally feel uncomfortable with the proposal: In my humble understanding, Calculus and Linear Algebra are of fundamental importance to mathematics and I tend to believe that both Calculus III and Linear Algebra should be required for a mathematics minor. Differential Equations may be an elective; I do not consider Differential Equations as necessarily more important than Real Analysis, Complex Analysis, Abstract Algebra, Functional Analysis and Topology. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is, what course are usually considered as required for minor in mathematics?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2020-04-14T14:55:06.123" LastActivityDate="2020-04-14T14:55:06.123" Title="Mathematics minor requirements" Tags="|undergraduate-education|usa|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="18163" PostTypeId="2" ParentId="18162" CreationDate="2020-04-14T01:58:49.583" Score="-2" Body="&lt;p&gt;Consider an opposite perspective (unless you were only looking for affirmation).  Maybe you learn something by getting challenged.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Differential equations has huge application in other fields.  I'm fine with the ODE course at same level as the calc 3 course.  Probably even prefer it if forced to pick.  It's very similar to integral calculus (calc 2) in thought pattern and I've used it more than calc 3 as an engineer and scientist.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Linear Algebra is consistently overrated on this site by the theory brigade. I think it comes from I donno being more abstract or something.  Yes, there's applications too (but then they're not interested in them either).  And for a practical matter, most engineers/scientists can get the limited amount they need in a quick chapter on matrices, within a math methods course that mostly hits PDEs (time wise).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the topics (Real Analysis, Functional Analysis, Complex Analysis) are relatively more advanced than ODEs and I think less important than basic ODEs.  Also, I think the practice in series solution of ODEs is good to have prior to these courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an ideal world, sure the kids would take a gazillion courses and be super advanced.  But clearly this is a much more constrained situation than that.  (Think linear programming haha.)  So, I find the math minor proposed to be extremely practical and normal. I'm on their side and against your side.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before you dismiss my scientist/engineer slant consider that they are also time constrained.  So the courses that are key for them may also be key for time/teacher constrained students at remote Alpine Lutheran colleges.  ;-)&lt;/p&gt;&#xA;" OwnerUserId="13762" LastActivityDate="2020-04-14T01:58:49.583" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18164" PostTypeId="2" ParentId="18162" CreationDate="2020-04-14T04:09:17.640" Score="4" Body="&lt;h3&gt;Comparison to other programs&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;While implementing a new minor, it might make sense to see what other institutions are doing.  Thus it would behoove you to look at the catalogs of other institutions to see what they do.  I would recommend looking at a few top-flight schools (UCLA, Stanford, Cornell, MIT, Michigan, etc) to get a sense of what they do, then look at several schools in the same tier as your own institution.  If you don't have the faculty, students, or courses to offer something at least as good as institutions similar to your own, then you probably aren't doing your students any good by offering a minor in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Making a comparison to institutions where I have knowlege, my impression of the proposed minor is that it seems a little anemic:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;At my bachelor and masters institution, &lt;a href=&quot;https://catalog.unr.edu/preview_program.php?catoid=35&amp;amp;poid=65027&quot; rel=&quot;nofollow noreferrer&quot;&gt;University of Nevada, Reno&lt;/a&gt;, the requirements for a mathematics minor are three semesters of calculus (a year of single variable calculus, and a semester of multivariable calculus), followed by three semesters of upper division mathematics.  The upper division courses may be chosen freely from among all offerings at the university, though a lower division &quot;intro to proof writing&quot; course is a prerequisite for many (though not all) of the upper division classes, so a typical mathematic minor ends up being something like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Calculus (12 semester credits)&lt;/li&gt;&#xA;&lt;li&gt;Intro to Proof Writing (3 semester credits)&lt;/li&gt;&#xA;&lt;li&gt;upper division coursework (9 semester credits)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;br /&gt;This is a total of 24 semester credits, and requires (more or less) that a student take one mathematics course for each of six or seven semesters (i.e. if one takes only one mathematics course at each opportunity, the minor will take about three years to complete).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, there is a &lt;a href=&quot;https://catalog.unr.edu/preview_program.php?catoid=35&amp;amp;poid=65028&quot; rel=&quot;nofollow noreferrer&quot;&gt;statistics minor&lt;/a&gt; offered by the department, which has similar requirements, but is more narrowly focused.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At my current institution, &lt;a href=&quot;https://studentdocs.ucr.edu/registrar/UCR_Catalog_2019-2020.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;University of California Riverside&lt;/a&gt; (.pdf, see page 404), the requirements for a mathematics minor are five quarters of calculus (a year of single variable calculus and two quarters of multivariable calculus), followed by six quarters of upper division classes.  Again, the upper division classes may be taken freely from among everything offered, though there are some restrictions (there is a limit on the number of &quot;reading courses&quot; which may be taken, and a limit on the number of courses which can be applied to another program in addition to the mathematics minor).  Again, the &quot;bridge course&quot; is a lower division class, and does not count toward the minor, though is recommended before taking upper division classes. A typical course list is something like&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Single Variable Calculus (12 quarter credits)&lt;/li&gt;&#xA;&lt;li&gt;Multivariable Calculus (8 quarter credits)&lt;/li&gt;&#xA;&lt;li&gt;Introduction to Discrete Structures (the bridge course, 4 quarter credits)&lt;/li&gt;&#xA;&lt;li&gt;upper division coursework (24 quarter credits)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;br /&gt;This is a total of 48 quarter credits, and requires (more or less) that a student take one mathematics course for each of twelve quarters (i.e. if one takes only one mathematics course at each opportunity, the minor will take about three years to complete).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;By comparison, the proposed curriculum requires a year and a half of lower-division coursework (which is on par with UNR and UCR), but only a year of upper-division coursework (which is less than either UNR or UCR), and this extra year may include a bridge course (which is considered lower-division at the above cited institutions).  The proposed minor is also quite narrow, and is likely to overlap significantly with coursework that is already being taken for other majors (for example, CS and physics majors are often required to take calculus, differential equations, and linear algebra).&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;What is the goal?&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;In addition to comparing your proposed program to other programs, you might want to consider what your goal is.  I am of the opinion that the goal of a minor in any field is to get students to take classes which they would not otherwise take.  A cynical reason for this is to boost enrollment in classes which might not otherwise run, but which instructors are keen to teach.  This is also a way of bolstering enrollment in courses for math majors, but this seems not to be a problem in the current context, as the asker's institution appears not to offer such a major.  A less cynical (and perhaps more meaningful and important) rationale is to broaden the horizons of students by getting them to take courses which expose them to some of the depths of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From this point of view, it seems reasonable to remove the bridge course from among those that count towards the minor.  A bridge course is a prerequisite (perhaps) for upper division coursework, but is not, in and of itself, an upper-division class.  It would also be reasonable to add something akin to &quot;Only &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; of the upper-division courses may count toward major requirements,&quot; which is a way of forcing students to expand their horizons a bit.  If a student finishes their major and notices &quot;Oh, hey! I completed a math minor, too!  I should put that on my transcript!&quot;, then I think that you have failed to provide a compelling program.  What you want students to say is &quot;Oh, hey!  If I take just one (or two) more math classes, &lt;em&gt;then&lt;/em&gt; I can get a minor in math!  I should do that!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you do not have the faculty, courses, or students to sustain these requirements, then it might not be worth offering a mathematics minor.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;TL;DR&lt;/h3&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The lower-division offerings seem fine, and are inline with other institutions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Consider dropping the bridge course from the list of courses which count towards the minor, particularly if students are only going to be required to take two courses beyond their lower-division calculus / differential equations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Consider requiring three upper-division courses (rather than just two).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Consider expanding the list of upper-division courses which will satisfy the minor.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Consider adding a clause to the minor requirements limiting the number of courses which may be counted both towards a student's major and the mathematics minor.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If your college or program are so small that you cannot sustain a mathematics minor which offers more upper-division courses which are not required for some other major, consider abandoning the idea of implementing a mathematics minor.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2020-04-14T13:41:52.577" LastActivityDate="2020-04-14T13:41:52.577" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18165" PostTypeId="1" CreationDate="2020-04-14T16:24:26.100" Score="6" ViewCount="214" Body="&lt;p&gt;Teaching first year undergraduates, I've noticed that what gives them the most trouble is simple computations like factoring, expanding, handling fractions, powers, especially when variables and other parameters come into play.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here in France these things are supposedly taught in middle school and high school but the degree of mastery that our first year students have of these topics can be very low.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm currently in the process of designing a new &quot;calculus 1&quot; course for those students and would like to include in the first two weeks some material to have student work their computational muscles. However I'm afraid if those exercises look too much like high school the students might get bored. So I'd like to give them exercises that look like small computational challenges but rely only on these elementary precalculus topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example I like is the classical 2 squares or 4 squares identities (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler%27s_four-square_identity&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the question is :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question :&lt;/strong&gt; What are some good resources of reasonably challenging problems of mostly computational problems which involves only the pre-calculus tools listed above ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Disclaimer : I did browse MESE but didn't find anything, though I might just use the wrong keywords. Also I'm not too familiar with MESE and hope this question is not too broad.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Context&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I'm an assistant professor in a university in France. I need to design a calculus course for first year students in Maths, Engineering, Physics, Computer Science and Chemistry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Up until now, calculus topics where scattered in with more formal analysis and algebra courses, which had the unfortunate side-effect that student came to know that they could focus on the computational techniques to have a decent grade rather than get hands with definitions and proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;French high school pupils do have a basic exposure to calculus, they know how to compute simple limits (though they have no idea of the definition) and derivatives of not too complicated (they don't know the chain rule for instance, but know how to differentiate &lt;span class=&quot;math-container&quot;&gt;$u^\alpha$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$e^u$&lt;/span&gt; for instance), they can study the variations of functions using the derivative and have seen that there is a thing called &quot;integral&quot; that you can compute using antiderivatives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;University curriculum in France is not centralized so we are not tied in the topics we can cover.&lt;/p&gt;&#xA;" OwnerUserId="400" LastEditorUserId="376" LastEditDate="2020-06-01T08:25:22.980" LastActivityDate="2020-06-01T08:25:22.980" Title="Resources for improving computational skills at the high school/university transition" Tags="|undergraduate-education|secondary-education|calculus|resource-request|calculations|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18166" PostTypeId="1" AcceptedAnswerId="18167" CreationDate="2020-04-14T16:28:27.997" Score="0" ViewCount="1853" Body="&lt;p&gt;When assigning a late penalty for assignments, is that penalty subtracted from the total or does it reduce the student's score by that percentage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ex: Assignment is worth 10 points and the late penalty is 20%&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student A hands it in late, getting all questions correct (base score of 10); they get 80% in either circumstance&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student B hands it in late, getting half of the questions correct (base score of 5); do they get 3/10 (subtract 20% of the total from their score) or do they get 4/10 (reduce their score by 20% of it's original value)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Student C hands it in late, getting only 1 question right (base score of 1); do they get -1/10, 0/10, or 0.8/10?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which is the most reasonable late penalty? Or at least which one has better outcomes?&lt;/p&gt;&#xA;" OwnerUserId="13814" LastEditorUserId="13814" LastEditDate="2020-04-15T14:42:47.820" LastActivityDate="2020-04-17T02:20:53.660" Title="When assigning a late penalty for assignments is that penalty subtracted from the total or does it reduce the score by that percentage" Tags="|grading|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18167" PostTypeId="2" ParentId="18166" CreationDate="2020-04-15T00:27:08.130" Score="2" Body="&lt;p&gt;The most common is the straight X% off, not the proportional one.  I would also say it is the fairest. Plus it's simpler. (Obviously, once you are at zero, don't go negative though.  But I have had assignments zero-ed from being several days late.)&lt;/p&gt;&#xA;" OwnerUserId="13816" LastActivityDate="2020-04-15T00:27:08.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18168" PostTypeId="2" ParentId="7528" CreationDate="2020-04-15T03:56:41.990" Score="3" Body="&lt;p&gt;Synergia, built with congruent space-filling hendecahedra:&#xA;&lt;a href=&quot;https://exhibitcolumbus.org/exhibition/synergia&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://exhibitcolumbus.org/exhibition/synergia&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ref: &lt;a href=&quot;https://www.jstor.org/stable/3618509&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/3618509&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;… and a unremarkable regular icosahedron in a playground near my neighborhood:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/2qmGb.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="13817" LastEditorUserId="13817" LastEditDate="2020-04-15T21:38:43.173" LastActivityDate="2020-04-15T21:38:43.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18169" PostTypeId="2" ParentId="5648" CreationDate="2020-04-15T14:41:34.810" Score="0" Body="&lt;p&gt;You might explain, at any level, that for any real number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$0x=0$&lt;/span&gt;. So the equation &lt;span class=&quot;math-container&quot;&gt;$0x=0$&lt;/span&gt; has no unique solution. And &lt;span class=&quot;math-container&quot;&gt;$0x=1$&lt;/span&gt;, for example, has no solution whatsoever. For these reasons, neither &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt; (with no unique solution) nor &lt;span class=&quot;math-container&quot;&gt;$1/0$&lt;/span&gt; (with no solution) can be real numbers.    &lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2020-04-15T17:12:57.723" LastActivityDate="2020-04-15T17:12:57.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18170" PostTypeId="2" ParentId="18141" CreationDate="2020-04-15T18:24:10.240" Score="0" Body="&lt;p&gt;Since the axes you're talking about are real things on paper or on a screen, I think it's easy and I'd go with something very similar to your proposal:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A coordinate system has &lt;em&gt;axes with the same scale&lt;/em&gt; if every number on one axis has the same distance to the origin as the same number on the other axis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think we can get around the &quot;with the same ruler&quot; part because using different rulers to measure the length of segments on paper doesn't make sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By using the term &quot;number&quot;, we make clear that the concept is useless for most &quot;physical&quot; graphs with different units on both axes. For physical graphs with same units on both axes, one could substitute number with &quot;quantity&quot; in thw definition above.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2020-04-15T18:24:10.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18171" PostTypeId="1" CreationDate="2020-04-15T21:56:21.757" Score="2" ViewCount="128" Body="&lt;p&gt;Extra credit questions sometimes motivate students to study more. Occasionally I find several valuable questions which are all good as extra credit question choices but I have to limit the number of points that a student can possible receive from doing extra credit work. Say the full mark of a homework set is &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; and there are &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; questions which are suitable as extra credit questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here my goal is:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Encourage students to try as many of them as possible.&lt;/li&gt;&#xA;&lt;li&gt;Do not give a student more than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; point for all the extra credit work.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Here I am asking if the following is a good strategy:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the student attempted &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; out of the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; extra credit questions, the student will receive up to &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; points for his best finished question, up to &lt;span class=&quot;math-container&quot;&gt;$1/4$&lt;/span&gt; points for his second best finished question, etc. In general, I will give the student up to &lt;span class=&quot;math-container&quot;&gt;$1/2^m$&lt;/span&gt; points for the &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;-th best answered question. So a motivated student can try most extra credit questions out of the set of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; questions and no student can receive more than one point for his extra credit work.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2020-04-21T17:34:34.087" Title="Multiple extra credit questions in a homework set" Tags="|undergraduate-education|student-motivation|homework|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18172" PostTypeId="2" ParentId="18166" CreationDate="2020-04-17T02:20:53.660" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Which is the most reasonable late penalty?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would argue that the most reasonable late penalty is to simply get no grade on the assignment at all (i.e., score is null, or 100% penalty). This makes the most efficient use of the instructor's time.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Instructor does not have to chase students after the fact to submit late assignments.&lt;/li&gt;&#xA;&lt;li&gt;Instructor does not have to re-set grading framework (pull out books, answer sheet, programming test suite, software grade book website, etc.) to restart grading.&lt;/li&gt;&#xA;&lt;li&gt;Instructor does not have to spend time debating/calculating the percent off penalty, as shown in this question.&lt;/li&gt;&#xA;&lt;li&gt;Instructor can spend added time working on improved presentations for students who did submit the assignment.&lt;/li&gt;&#xA;&lt;li&gt;Instructor can scan assignments for plagiarism (by hand, TurnItIn, MOSS, etc.) in a single batch process, and not re-do the step at a later date. &lt;/li&gt;&#xA;&lt;li&gt;Instructor can immediately distribute answers, results, and feedback the next day, so as to capitalize on the lessons while they're fresh. &lt;/li&gt;&#xA;&lt;li&gt;The motivation for students submitting the assignment on time is maximized. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-04-17T02:20:53.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18174" PostTypeId="1" CreationDate="2020-04-17T21:50:55.363" Score="12" ViewCount="3716" Body="&lt;p&gt;I am an undergraduate secondary math education major. In &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; weeks I have to give a &lt;a href=&quot;https://www.mathmammoth.com/lessons/number_talks.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Number Talk&lt;/a&gt; in my math ed class on the problem &quot;&lt;span class=&quot;math-container&quot;&gt;$3.9$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$7.5$&lt;/span&gt;&quot;. I need to come up with as many different solution methods as possible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what I have come up with so far:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The most common way: multiply the two numbers &quot;vertically&quot;, ignoring the decimal, to get &lt;span class=&quot;math-container&quot;&gt;$2925$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{array}&#xA;{}\hfill {}^6{}^439\\&#xA;\hfill \times\  75 \\\hline&#xA;\hfill {}^1 195 \\&#xA;\hfill +\ 273\phantom{0} \\\hline&#xA;\hfill 2925&#xA;\end{array}&lt;/span&gt; Since there are two numbers that are to the right of the decimal, place the decimal after the &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; to get the answer &lt;span class=&quot;math-container&quot;&gt;$29.25$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Write both numbers as improper fractions: &lt;span class=&quot;math-container&quot;&gt;$$3.9= \dfrac{39}{10}$$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$$7.5=\dfrac{75}{10}$$&lt;/span&gt;Then multiply &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{39}{10}\cdot\dfrac{75}{10}$$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$\dfrac{2925}{100}$&lt;/span&gt; which simplifies to 29.25.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use lattice multiplication. This is a very uncommon method that I doubt the students will use, and I need to review it myself before I consider it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$3.9$&lt;/span&gt; is very close to &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, we could instead do &lt;span class=&quot;math-container&quot;&gt;$4\cdot7.5=30$&lt;/span&gt; and then subtract &lt;span class=&quot;math-container&quot;&gt;$0.1\cdot7.5=0.75$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$30 - 0.75=29.25$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similarly, since &lt;span class=&quot;math-container&quot;&gt;$7.5$&lt;/span&gt; rounds up to &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;, we can do &lt;span class=&quot;math-container&quot;&gt;$3.9\cdot 8=31.2$&lt;/span&gt; and then subtract .&lt;span class=&quot;math-container&quot;&gt;$5\cdot 3.9=1.95$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$31.2-1.95=29.25$&lt;/span&gt; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Are there any other possible methods the students might use? (&lt;strong&gt;Note:&lt;/strong&gt; they are junior college math ed students.) Thanks!&lt;/p&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="13657" LastEditDate="2020-04-18T20:14:53.007" LastActivityDate="2020-04-23T10:06:38.813" Title="Different ways to multiply decimals" Tags="|arithmetic-operations|arithmetic|" AnswerCount="13" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18175" PostTypeId="2" ParentId="18174" CreationDate="2020-04-17T22:45:39.220" Score="7" Body="&lt;p&gt;Estimate the result. Multiply the numbers without paying attention to the decimal point. Place the decimal point so that the result is close to the estimate.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-04-17T22:45:39.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18176" PostTypeId="2" ParentId="18174" CreationDate="2020-04-18T01:13:23.017" Score="8" Body="&lt;p&gt;Combining your last two methods: &lt;span class=&quot;math-container&quot;&gt;$3.9*7.5 = 4*8 - 0.1*8 - 4 * 0.5 + 0.1*0.5$&lt;/span&gt;, which can be thought of as computing the area of the big rectangle below, cutting off the two extra strips along the edge, then adding back in a copy of the corner piece since you have removed it twice, instead of once.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NtDJV.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NtDJV.png&quot; alt=&quot;rectangles&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2020-04-18T01:13:23.017" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18177" PostTypeId="2" ParentId="18031" CreationDate="2020-04-18T04:01:23.740" Score="3" Body="&lt;p&gt;Here I broach only the case in which students have consistent internet access; the more general case is clearly much more difficult. Moreover, this answer is written for the present context, i.e., a sudden shift to remote learning (perhaps &lt;a href=&quot;http://thejosevilson.com/on-disaster-distance-learning-in-new-york-city/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;disaster distance learning&lt;/strong&gt;&lt;/a&gt; is a more apt name) due to a pandemic. This is &lt;strong&gt;not&lt;/strong&gt; an answer about teaching mathematics online using the full resources of, say, an Office of Distance Learning. Finally, I am writing essentially about teaching (at least) secondary school; the task at hand is, again, significantly complicated by a need to have students accompanied by an adult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My advice for right now is to simplify drastically, and to make your goals fourfold:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Ensure that you are checking in with students, even if their work is not done;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To provide enough work for those who want to work on mathematics to have materials;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To ensure that if all students complete all work that it will not be too much for you, as the instructor, to handle;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To create a routine and structure in a time of great uncertainty.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Concretely, I am using a daily schedule of &lt;strong&gt;DO&lt;/strong&gt;, &lt;strong&gt;WATCH&lt;/strong&gt;, &lt;strong&gt;READ&lt;/strong&gt;. For a senior level mathematics course, I have made my ongoing notes available online: the class is called &lt;a href=&quot;https://docs.google.com/document/d/1OptDDFPekLbM7buZ893QkyVRfFnAVcbbR7o0c9iyE4o/edit&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Problem Solving &amp;amp; Posing&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a separate suggestion for how this routine could be carried out in the context of, say, a course on Algebra 2 for which the goals centered are the four listed above:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Set up Google Classroom as a place where students can turn in problems; they can do the problems by typing them out, or by writing them out and uploading pictures. So, the &lt;strong&gt;DO&lt;/strong&gt; here would be whatever list of problems you intended on covering before going remote; but, be sure that you are satisfying &lt;em&gt;both&lt;/em&gt; number 2 and number 3 in the list of goals above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the &lt;strong&gt;WATCH&lt;/strong&gt; component, assign consecutive videos related to the problems from Khan Academy. Is this ideal? No, it is not; but, we are operating out of a non-ideal world, and this is a simple structure that you can set up for students to operate within.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the &lt;strong&gt;READ&lt;/strong&gt; component, I recommend the set of MAA published anecdotes &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/ebooks/pdf/LivingProof_WEB.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Living Proof&lt;/strong&gt;&lt;/a&gt; (PDF). I have been asking students to read one anecdote in a day (about 2 pages) and then to google the person about whom they read the following day. You could alternate this across Mon/Tue and Wed/Thu, then assign something different to read on Fridays/weekends (perhaps nothing at all, depending on bandwidth). There are 41 anecdotes in the linked book; therefore, this structure would last for about 20 weeks. Is this ideal? Actually, I think these stories are powerfully humanizing for a discipline that is too often viewed in a dehumanized, decontextualized manner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have a synchronous component too, then I suggest using that time to go over the problems. Asynchronously, I also suggest creating a shared document that all students in a given class can edit: in particular, they can ask questions and/or answer classmates' questions. If you can, download Screencastify for a nice way of going over how problems can be solved: you can solve them on the computer (I've used &lt;a href=&quot;https://www.bitpaper.io/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;bitpaper&lt;/strong&gt;&lt;/a&gt;) while recording yourself speaking. I imagine it will be better for many students to hear a familiar voice, and you can clarify the steps you are going through as you work out e.g. questions left unanswered in the shared document.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a number of routines (notice/wonder, number strings, which one doesn't belong, attribute-listing/what-if-not-ing, slow reveal graphs, etc) that you could begin to introduce at a later point for live/synchronous sessions; however, I think the &lt;strong&gt;DO&lt;/strong&gt;, &lt;strong&gt;WATCH&lt;/strong&gt;, &lt;strong&gt;READ&lt;/strong&gt; structure is worth implementing and tinkering with before deciding to enact more demanding pedagogical approaches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Start simple and be kind to yourself.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2020-04-18T04:01:23.740" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18178" PostTypeId="1" AcceptedAnswerId="18184" CreationDate="2020-04-18T04:28:10.407" Score="6" ViewCount="951" Body="&lt;p&gt;I am planning to sign up for an undergraduate &quot;course&quot; in point-set topology next semester. It is really an &quot;independent study&quot; in that this course will not have any lectures. It will just have two tests and a final exam. I plan to follow a YouTube playlist on Topology to solidify my understanding of the subject. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since it is an &quot;independent study,&quot; the instructor of this independent study has asked me if I preferred to read a certain topology textbook. So, I researched the books on Topology and decided upon &lt;em&gt;Topology: A first Course&lt;/em&gt; by Munkres. However, last semester I had done a similar &quot;independent study&quot; in Abstract Algebra for which I had studied &lt;em&gt;A First Course in Abstract Algebra&lt;/em&gt; by Fraleigh which is highly recommended online. However, I found that for an &quot;independent study,&quot; this was was rather stingy with examples, although this might just be me. I was still getting acquainted with axiomatic math courses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you believe that the book by Munkres would be a good choice for an &quot;independent study&quot; in Topology? Is it generally skimpy on examples like Fraleigh's book? Would you recommend some other book on Topology? P.S. I would hate to go for a book that is either less challenging or less mathematically rigorous that Mukres's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Potentially relevant information&lt;/em&gt;: So far, the axiomatic courses I have completed include Real Analysis (based on first six chapters from Rudin), Mathematical Statistics (based on Hogg, Craig's first seven chapters), Abstract Algebra (based on Fraleigh), and a course in Complex Variables (based on Churchill's first four chapters). Any suggestions and/or advice are deeply appreciated!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: I am planning to pursue a doctoral degree in economics for which I have been advised that I should take a course in Topology.  &lt;/p&gt;&#xA;" OwnerUserId="13834" LastEditorUserId="13834" LastEditDate="2020-04-18T16:46:32.733" LastActivityDate="2021-06-17T15:46:58.957" Title="Textbooks for an independent study in point-set topology" Tags="|undergraduate-education|textbooks|self-learning|topology|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="18179" PostTypeId="2" ParentId="13846" CreationDate="2020-04-18T08:48:29.077" Score="4" Body="&lt;p&gt;Addressing the point about the course not being sufficiently connected to applications, you might be interested in Boyd &amp;amp; Vandenberghe's recent &lt;a href=&quot;http://vmls-book.stanford.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares&lt;/a&gt; (VMLS). The book is available for free online, with permission from the publisher. It includes numerous applications, some of them from business/economics. Some of the applications in the first half of the book, which focuses on vectors and matrices, are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Leontief input-output models&lt;/li&gt;&#xA;&lt;li&gt;portfolio sector exposures&lt;/li&gt;&#xA;&lt;li&gt;supply chain dynamics&lt;/li&gt;&#xA;&lt;li&gt;conversion from purchase quantity to purchase currency unit (dollar, euro, etc.) matrix&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Eigenvalues and eigenvectors are not dealt with in VMLS, but you say that these are not covered until the course in the second year, so that shouldn't be a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You don't mention least squares specifically, but if you do cover them or decide to introduce them, VMLS devotes the entire third part of the book (chapters 12-19) to this topic, starting with basic least squares and ending with constrained nonlinear least squares. Some applications from business/economics in these chapters include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;advertising purchases&lt;/li&gt;&#xA;&lt;li&gt;predicting return of an asset from the return of the whole market&lt;/li&gt;&#xA;&lt;li&gt;time series trends and predictions&lt;/li&gt;&#xA;&lt;li&gt;regression models&lt;/li&gt;&#xA;&lt;li&gt;fraud detection&lt;/li&gt;&#xA;&lt;li&gt;marketing demographic classification&lt;/li&gt;&#xA;&lt;li&gt;product demand shaping&lt;/li&gt;&#xA;&lt;li&gt;minimum cost trading to achieve target sector exposures&lt;/li&gt;&#xA;&lt;li&gt;portfolio optimization&lt;/li&gt;&#xA;&lt;li&gt;finding equilibrium prices&lt;/li&gt;&#xA;&lt;li&gt;internal rate of return&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you want to include programming in the course, there are companions available for using Julia and Python.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(If the authors sound familiar by any chance, they also wrote the more advanced book &lt;a href=&quot;https://web.stanford.edu/~boyd/cvxbook/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Convex Optimization&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2020-04-19T12:11:45.563" LastActivityDate="2020-04-19T12:11:45.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18180" PostTypeId="2" ParentId="18174" CreationDate="2020-04-18T15:12:40.590" Score="2" Body="&lt;p&gt;I would do &lt;span class=&quot;math-container&quot;&gt;$$3.9\times7.5=\frac{3.9\times30}4=\frac{39\times3}4=\frac{(40-1)\times3}4=\frac{40\times 3}4-\frac34=30-\frac34.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="475" LastEditorUserId="475" LastEditDate="2020-04-18T18:36:50.693" LastActivityDate="2020-04-18T18:36:50.693" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18181" PostTypeId="2" ParentId="18178" CreationDate="2020-04-18T19:44:50.450" Score="7" Body="&lt;p&gt;If you're going to be an economist, even doing highly theoretical work, then the vast majority of the field of topology is going to be of zero relevance to you. Very little of it has any practical application in the sciences. What we actually use in the sciences is mostly just the topology of manifolds, which is a tiny, tiny piece of the larger picture of topology. For a sample of the kind of thing that is studied in topology but that is almost certain not to help you as an economist, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Long_line_(topology)&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Long_line_(topology)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if you just think topology would be a cool, fun thing to learn about, that's fine, have fun! But given that you're an undergrad econ major, you could also choose to focus more narrowly. (I assume the structure of this course allows you to have some say in what topics you learn.) A really fun book with a narrower focus is Topology Now!, by Messer and Straffin.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I am planning to pursue a doctoral degree in economics for which I have been advised that I should take a course in Topology. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm extremely skeptical of this claim. It sounds like you just like math, so if you do it, do it for that reason.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-04-18T19:44:50.450" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18182" PostTypeId="2" ParentId="18174" CreationDate="2020-04-18T19:51:51.970" Score="3" Body="&lt;p&gt;Write out the numbers using the &quot;expanded form&quot;, then use the distributive property of multiplication over addition to evaluate the product.  Consistently using powers of &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; throughout, this is&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;3.9 \cdot 7.5&#xA;&amp;amp;= \left(\color{red}{3}\cdot 10^\color{red}{0} + \color{blue}{9}\cdot 10^\color{blue}{-1} \right) \left( \color{orange}{7}\cdot 10^\color{orange}{0} + \color{green}{5}\cdot 10^\color{green}{-1} \right) \\&#xA;&amp;amp;= \left(\color{red}{3}\cdot \color{orange}{7} \cdot 10^{\color{red}{0}+\color{orange}{0}} \right)&#xA; + \left( \color{red}{3}\cdot \color{green}{5}\cdot 10^{\color{red}{0}+(\color{green}{-1})} \right) \\&#xA; &amp;amp;\qquad\qquad+ \left( \color{blue}{9}\cdot \color{orange}{7}\cdot 10^{\color{blue}{-1}+\color{orange}{0}} \right)&#xA; + \left( \color{blue}{9}\cdot \color{green}{5}\cdot 10^{\color{blue}{-1}+(\color{green}{-1})} \right) \\&#xA;&amp;amp;= \left( 21\cdot 10^{0} \right) + \left( 15\cdot 10 ^{-1}\right) + \left( 63 \cdot 10^{-1} \right) + \left( 45 \cdot 10^{-2} \right) \tag{$\ast$} \\&#xA;&amp;amp;= \left( 21\cdot 10^{0} \right) + \left( 1.5\cdot 10 ^{0}\right) + \left( 6.3 \cdot 10^{0} \right) + \left( 0.45 \cdot 10^{0} \right) \tag{$\ast$} \\&#xA;&amp;amp;= 29.25 \cdot 10^{0} \\&#xA;&amp;amp;= 29.25.&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are some advantages to this approach&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It emphasizes the place-value notation&amp;mdash;when we write a number, each digit represents some power of ten.  This would be an appropriate thing to emphasize in late elementary school in the US (it hits on a &lt;a href=&quot;http://www.corestandards.org/Math/Content/5/NBT/A/3/a/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grade 5 Common Core Standard (CCS)&lt;/a&gt;; there is also an important connection to scientific notation here).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It emphasizes the distributive property of multiplication over addition.  This is part of the &lt;a href=&quot;http://www.corestandards.org/Math/Content/6/EE/A/3/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grade 6 CCS&lt;/a&gt;, and is an excellent way to practice skills that are important later in algebra. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;We don't have to track the decimal point.  Of course, this comes at a cost, as we have to keep track of powers of ten.  &lt;a href=&quot;https://en.wikipedia.org/wiki/There_ain%27t_no_such_thing_as_a_free_lunch&quot; rel=&quot;nofollow noreferrer&quot;&gt;TANSTAAFL&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;However, there is a lot of extra notation running around, particularly at the two marked lines.  An alternative, though similar, approach would be to skip the exponential notation and work with the decimals:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;3.9\cdot 7.5&#xA;&amp;amp;= (\color{red}{3} + \color{blue}{0.9}) ( \color{orange}{7} + \color{green}{0.5} ) \\&#xA;&amp;amp;= \left( \color{red}{3} \cdot \color{orange}{7} \right)&#xA; + \left( \color{red}{3} \cdot \color{green}{0.5} \right)&#xA; + \left( \color{blue}{0.9} \cdot \color{orange}{7} \right)&#xA; + \left( \color{blue}{0.9} \cdot \color{green}{0.5} \right) \\&#xA;&amp;amp;= 21 + 1.5 + 6.3 + 0.45 \\&#xA;&amp;amp;= 29.25.&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2020-04-18T19:51:51.970" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18184" PostTypeId="2" ParentId="18178" CreationDate="2020-04-18T20:31:55.560" Score="6" Body="&lt;p&gt;Munkrese is a good book for introductory Topology in part because it has a lot of background at the start. If I recall correctly, the first 70 pages aren't so much Topology as they are just basic abstract math background. You can get a copy pretty cheaply and when you ask questions about it there are lots of people who have also studied it and can help you if you ask. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fraleigh is a good book, but I think you'd benefit from having a copy of Dummit and Foote's 3rd edition to look at when questions about algebra arise. Similarly, it's good to supplement Munkrese with something else. I like &lt;em&gt;Topology&lt;/em&gt; by Marco Manetti, &lt;a href=&quot;https://www.google.com/books/edition/Topology/89zyCQAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;for example see here&lt;/a&gt;, but there are dozens of good texts written in the past 20 or so years. Topology, much like Complex Analysis, is a topic which many Mathematicians love and as such find time to write their own book about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Topology is likely to play a larger role in economics in the future as big data plays a larger role in the function of corporations and governments. Topological Data Analysis(TDA) requires all that fancy schmancy math that is sometimes poo-poo'd around here. TDA is very modern and still very much under development. Looking for large scale patterns in data is a problem which requires the full sophsitication of modern topology as well as a good grounding in mathematical statistics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Topology is also more and more important to the design of semiconductors from what I've heard from those who know much more than me...&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-04-18T20:31:55.560" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18185" PostTypeId="2" ParentId="18174" CreationDate="2020-04-18T22:24:57.843" Score="10" Body="&lt;p&gt;There's a fun method which I've seen referred to as Russian peasant multiplication or ancient Egyptian multiplication. (I don't know if these names have a historical basis.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you think about how the algorithm works behind the scenes, it's essentially multiplying the two numbers in base 2, so it could have more than one pedagogical use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This technique is for integers; you'll still need to handle the placement of the decimal point separately, as usual.  (You could actually keep the decimal points in column 2, but you'd still have to adjust the placement of the decimal point in the product, so it's probably easier just to work with integers throughout.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the method:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a) Put each number at the top of a column.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) In column 1, divide by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; repeatedly, ignoring any remainders, until you reach &lt;span class=&quot;math-container&quot;&gt;$1.$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(c) In column 2, multiply by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; repeatedly. Stop when column 2 is the same height as column 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(d) Cross off every row where the number in column 1 is even.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(e) Add the numbers in column 2 that you haven't crossed off.  That's the product.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Here's your example of &lt;span class=&quot;math-container&quot;&gt;$39\times75,$&lt;/span&gt; worked out using this method:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&#xA;39     75&#xA;19    150&#xA; 9    300&#xA; &lt;s&gt;4    600&lt;/s&gt;&#xA; &lt;s&gt;2   1200&lt;/s&gt;&#xA; 1   2400&#xA;     ----&#xA;     2925&#xA;&lt;/pre&gt;&#xA;" OwnerUserId="13838" LastEditorUserId="13838" LastEditDate="2020-04-19T02:35:19.403" LastActivityDate="2020-04-19T02:35:19.403" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18186" PostTypeId="1" AcceptedAnswerId="18187" CreationDate="2020-04-19T07:10:39.933" Score="3" ViewCount="158" Body="&lt;p&gt;Would it be possible to get a complete list of math textbooks written jointly by Dr. Mary Dolciani and Dr. Robert Sorgenfrey? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm particularly interested in focusing on how they collaborated while writing their textbooks. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far, all I can find is this list: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;PreAlgebra: An Accelerated  Course&lt;/li&gt;&#xA;&lt;li&gt;Algebra Structure and Method Book 1&lt;/li&gt;&#xA;&lt;li&gt;Algebra and Trigonometry Structure and Method Book 2&lt;/li&gt;&#xA;&lt;li&gt;Introductory Analysis &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;But absolutely nothing on the details of their collaboration. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kindest regards, thank you.&lt;/p&gt;&#xA;" OwnerUserId="216" LastEditorUserId="216" LastEditDate="2020-04-19T07:23:27.877" LastActivityDate="2020-04-19T08:15:57.503" Title="Dolciani and Sorgenfrey textbook collaboration" Tags="|secondary-education|textbooks|big-list|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18187" PostTypeId="2" ParentId="18186" CreationDate="2020-04-19T08:15:57.503" Score="4" Body="&lt;p&gt;Try the google search &lt;a href=&quot;https://www.google.com/search?q=Dolciani+Sorgenfrey+math+worldcat&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dolciani + Sorgenfrey + math + worldcat&lt;/a&gt;, where &quot;worldcat&quot; is the keyword for the type of information you're looking for. The top hit for me is &lt;a href=&quot;https://www.worldcat.org/title/mathematics-structure-and-method-course-2/oclc/24060829/editions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Showing all editions for 'Mathematics : structure and method course 2'&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, the authors of these books varied over time and Sorgenfrey's name was not included in the original mid 1960s versions, nor on any of these 4 books I went through while I was in high school. (I went through all 4 books from Aug. 1973 to about March 1975, and the books themselves were variously dated from 1968 to maybe 1971 or 1972.) Indeed, the earlier versions had mostly different authors aside from Dolciani and slightly different titles (e.g. &quot;Modern School Mathematics&quot; was part of the title), but I suspect the content wasn't all that much different, maybe more pictures and updated real world references in the later versions. Also, I suspect that rather than a true co-author in the traditional sense, Sorgenfrey played more of the role of an adviser who double-checked certain content aspects. For all of Dolciani's books, &lt;a href=&quot;https://www.worldcat.org/search?q=au%3ADolciani%2C+Mary+P.&quot; rel=&quot;nofollow noreferrer&quot;&gt;see this worldcat page&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2020-04-19T08:15:57.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18188" PostTypeId="2" ParentId="7528" CreationDate="2020-04-19T13:13:38.633" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Two Swedish architects named Bigert &amp;amp; Bergstrom showcased their newest creation… the Solar Egg. This 16-foot tall egg-shaped structure is actually a sauna which burns wood to create heat.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/YYrBR.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YYrBR.jpg&quot; alt=&quot;SolarEgg&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Image from &lt;a href=&quot;https://themindcircle.com/mirrored-golden-egg-sauna-sweden/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&#xA;&lt;/sup&gt;&#xA;&lt;hr /&gt;&#xA;The tesselation appears to be irregular.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-04-19T13:13:38.633" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18189" PostTypeId="2" ParentId="18174" CreationDate="2020-04-19T13:23:20.077" Score="3" Body="&lt;p&gt;Trachtenberg methods: this depends on calculating the answer one digit at a time, starting at the units. Maybe this is what you call &quot;lattice&quot;. This was in vogue around 1960 -- maybe it gets rediscovered every generation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Units can only come from 9 x 5. Write down 5, carry 4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tens can come from 3 x 5 and 9 x 7, i.e. 78, plus carry. Write down 2, carry 8.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hundreds come from 3 x 7 plus carry, so 29. Write 9, carry 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thousands leaves only the carry. Write down 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On paper, it is customary to put a dot over each digit of the top number as it is used, until you get proficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are specific methods for particular multipliers, but they are special cases of this method.&lt;/p&gt;&#xA;" OwnerUserId="13841" LastActivityDate="2020-04-19T13:23:20.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18190" PostTypeId="2" ParentId="18174" CreationDate="2020-04-20T04:17:42.550" Score="3" Body="&lt;p&gt;You can do questions like this in your head pretty easily which is a great skill to teach students. Since &lt;span class=&quot;math-container&quot;&gt;$3.9$&lt;/span&gt; is close to &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$7.5$&lt;/span&gt; is easy to add to itself. So &lt;span class=&quot;math-container&quot;&gt;$3.9\times 7.5 = 4\times 7.5 - 0.1\times7.5$&lt;/span&gt;.&#xA;&lt;span class=&quot;math-container&quot;&gt;$4\times 7.5$&lt;/span&gt; is just doubling it twice. &lt;span class=&quot;math-container&quot;&gt;$7.5\times 2=15$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$15\times 2=30$&lt;/span&gt;. Then subtract &lt;span class=&quot;math-container&quot;&gt;$0.1\times 7.5=0.75$&lt;/span&gt;. So that's &lt;span class=&quot;math-container&quot;&gt;$29.25$&lt;/span&gt;.&#xA;A similar process can be used to do most questions like this in your head.&lt;/p&gt;&#xA;" OwnerUserId="13846" LastEditorUserId="13846" LastEditDate="2020-04-23T10:06:38.813" LastActivityDate="2020-04-23T10:06:38.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18191" PostTypeId="2" ParentId="18171" CreationDate="2020-04-20T04:42:30.710" Score="3" Body="&lt;p&gt;I agree with Xander’s comment; there’s no need to introduce complicated grading schemes, it just makes everyone’s life difficult. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would even go further and say that extra credit is generally only ever attempted by students who were already going to get an excellent grade anyway, so I don’t even make extra credit worth a significant amount of points. My rule of thumb is that I don’t offer more than 1 extra credit for 20 normal questions, and I make them worth the same as all other problems.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes I offer “challenge questions” which aren’t graded at all, but only when I think the statement of the question seems actually interesting to students. &lt;/p&gt;&#xA;" OwnerUserId="13193" LastEditorUserId="13193" LastEditDate="2020-04-21T17:34:34.087" LastActivityDate="2020-04-21T17:34:34.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18192" PostTypeId="2" ParentId="18174" CreationDate="2020-04-20T07:39:09.340" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QyJRr.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QyJRr.png&quot; alt=&quot;Slide rule&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This slide rule is set to show the same multiplication problem, &lt;span class=&quot;math-container&quot;&gt;$3.9\times7.5=29.25.$&lt;/span&gt;  (Click on the image for a larger view showing enough detail to see what's going on.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is equivalent to computing the product by adding logarithms, which is yet another way to do it:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$3.9\times7.5=e^{\ln 3.9 + \ln 7.5}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="13838" LastEditorUserId="13838" LastEditDate="2020-04-20T07:55:52.273" LastActivityDate="2020-04-20T07:55:52.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18193" PostTypeId="2" ParentId="18174" CreationDate="2020-04-20T15:32:42.287" Score="2" Body="&lt;p&gt;Consider the following solution. &lt;span class=&quot;math-container&quot;&gt;\begin{align*}7.5\times3.9&amp;amp;=(0.1\times75)\times(0.1\times39)\\\\&amp;amp;=0.01\times75\times39\\\\&amp;amp;=75\%\,\,\text{of}\,\,39\quad(\text{as}\,\,\%\,\,\text{means}\,\,1/100=0.01)\tag{1}\\\\&amp;amp;=\frac34\times39\quad\small{\left[=\frac{4-1}4\times39=\left(\frac44-\frac14\right)\times39\right]}\tag2\\\\&amp;amp;=\left(1-\frac14\right)\times39\quad\small{\left[=1\times39-\frac14\times39\right]}\\\\&amp;amp;=39-\frac{39}4\\\\&amp;amp;=39-\frac{36+3}4\quad\small{\left[=39-\left(\frac{36}4+\frac34\right)\right]}\\\\&amp;amp;=39-\left(9+\frac34\right)\\\\&amp;amp;=39-9-\frac34\\\\&amp;amp;=30-0.75\\\\&amp;amp;=29.25\end{align*}&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; As an aside, it may be of interest to mention that &lt;span class=&quot;math-container&quot;&gt;$75\%\times39=39\%\times75$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; Comparing the length of this solution to &lt;a href=&quot;https://matheducators.stackexchange.com/a/18180/8598&quot;&gt;Martin's&lt;/a&gt;, it shows the importance of choosing whether to express &lt;span class=&quot;math-container&quot;&gt;$3=4-1$&lt;/span&gt; (as in this answer), or &lt;span class=&quot;math-container&quot;&gt;$39=40-1$&lt;/span&gt; which yields a quicker solution.&lt;/p&gt;&#xA;" OwnerUserId="8598" LastActivityDate="2020-04-20T15:32:42.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18194" PostTypeId="2" ParentId="18174" CreationDate="2020-04-20T20:20:47.673" Score="2" Body="&lt;p&gt;&lt;strong&gt;Do it in slices.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We teach young kids that multiplication is just repeated addition.  What's 5x4?  It's just 5, added 4 times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So do the same with the decimal numbers. 3.9 x 7.5.  First, let's use 3.9 added 7 times.  You can't add it an eighth time, because that's over 7.5 - we only need to add it 0.5 times now.  So how would you add it 0.5 times?  Divide it by 10, and add it five times.  So you'd get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3.9 + 3.9 + 3.9 + 3.9 + 3.9 + 3.9 + 3.9 + 0.39 + 0.39 + 0.39 + 0.39 + 0.39&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Express Each Term As A Rational Fraction&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every non-infinite decimal value can be expressed as a rational fraction.  0.153258 is just 153258/1000000.  And you can multiply two rational fractions without worrying about decimal places.  Then, finally at the end, you can reduce it back into decimal form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3.9 is 39/10.  7.5 is 15/2 (or 75/100, if you want.)  Multiply those fractions together, and you get (39x15)/(10x2) - or reduced down 117/4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Jokingly ask them to convert it to binary and let them discover the joy of floating point errors.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... hey, you asked for as many ways as possible, right?&lt;/p&gt;&#xA;" OwnerUserId="13109" LastActivityDate="2020-04-20T20:20:47.673" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18195" PostTypeId="2" ParentId="18171" CreationDate="2020-04-20T20:27:43.207" Score="1" Body="&lt;p&gt;Some comments, which, together, constitute something like an answer:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;As I said in a comment, experience has shown me that students react badly to any grading scheme which they see as &amp;quot;nonstandard.&amp;quot;  You can fight these battles with students, and you might win, but it comes at a cost:  every single nonstandard thing you do in a class is going to be a battle, and if you try to fight more than one or two of those battles in a given term, your students are going to start checking out pretty quickly.  So, before you implement such a grading scheme, ask yourself, &amp;quot;Is &lt;em&gt;this&lt;/em&gt; the hill I want to die on?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Your answer may be &amp;quot;yes&amp;quot;.  Maybe you are teaching a bog-standard curriculum out of a bog-standard text using bog-standard lectures.  If so, then a &amp;quot;weird&amp;quot; grading scheme may be worth it.  Otherwise, you might want to reconsider.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is it that you hope to gain from such an extra credit scheme?  It might benefit the high achieving students (if the problems are enriching, they'll learn more or better), but it won't effect their grades, and they would probably have done the problems, anyway.  It probably won't benefit the low-achieving students, because they won't bother doing the problems.  As such, I am of the opinion that offering extra credit for extra problems is a waste of time and effort.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students don't like fractions of points.  If you are going to implement some kind of exponential decay, I would suggest inflating the value of everything, and assigning only whole number point-values.  For example, give 10 points per problem (for a total of 100 points).  Then make the first extra credit problem worth 5 points, the next worth 2, and every one after that worth 1 point up to a total of 10.  This is roughly the same scheme you proposed (with questions 2 and 3 getting a little less weight, and questions 4 and 5 getting a little more weight—everything after that is less than epsilon in either scheme).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An alternative is to &amp;quot;lie&amp;quot; with big numbers.  I have long offered extra credit to students who type up their homework assignments.  Every assignment is worth 3 points (1 point for attending the discussion workshop where they first encounter the homework problems in small groups, 1 point for turning in a completed assignment, and 1 point for giving (mostly) correct solutions).  A student can earn another point by typing their work (this is a WHOPPING 25% bonus).&lt;sup&gt;[1]&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;The trick is that homework only constitutes 5–10% of their final grade, which means that if a student types every single assignment, then they get an extra 1.25–2.5% towards their final grade.  On a standard grading scale (&lt;span class=&quot;math-container&quot;&gt;$x \ge 90\% = A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$80\%\le x &amp;lt; 90\% = B$&lt;/span&gt;, and so on), this extra credit might be the difference between a B+ and an A-.  It &lt;em&gt;looks like&lt;/em&gt; a huge bonus, because it is large &lt;em&gt;per assignment&lt;/em&gt;, but the ultimate value of that extra credit is low, because it is in a category which doesn't count very highly.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;[1] I said above that I don't offer extra credit for extra problems.  However, I do often offer extra credit for performing tasks which are tangential to the main content of the class.  Learning to type mathematics is a useful skill, but not everyone needs to learn it.  However, for the students who are willing to learn, I feel they should be rewarded.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-04-20T20:27:43.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18196" PostTypeId="2" ParentId="18174" CreationDate="2020-04-20T20:28:05.127" Score="3" Body="&lt;p&gt;Pull out your slide rule.  Align the second &quot;1&quot; on the C scale with the &quot;3.9&quot; on the D scale.  Move the slide to the &quot;7.5&quot; on the C scale and read &quot;2.92&quot; off the D scale.  Observe that there are two decimal places in the problem, and write down the answer as approximately 29.2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(If you want more precision, get a bigger slide rule.  A standard 10-inch rule is only good for about three and a half digits.)&lt;/p&gt;&#xA;" OwnerUserId="346" LastActivityDate="2020-04-20T20:28:05.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18197" PostTypeId="2" ParentId="18174" CreationDate="2020-04-20T22:21:05.830" Score="2" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\left\lfloor \frac{\left(x+y\right)^2}{4} \right\rfloor - \left\lfloor \frac{\left(x-y\right)^2}{4} \right\rfloor = xy.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answer uses the floor function (which essentially means to get rid of the decimal). You can then quickly consult a table to find 0.25*x^2. &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;X, 0.25*x^2&#xA;&#xA;1, 0.25&#xA;2, 1&#xA;3, (9/4)&#xA;4, 4&#xA;5, (25/4)&#xA;6, 9&#xA;7, (49/4)&#xA;8, 16&#xA;9, (81/4)&#xA;10, 25&#xA;11, (121/4)&#xA;12, 36&#xA;13, (169/4)&#xA;14, 49&#xA;15, (225/4)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplication&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplication&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13851" LastEditorUserId="13851" LastEditDate="2020-04-22T23:19:17.840" LastActivityDate="2020-04-22T23:19:17.840" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18198" PostTypeId="2" ParentId="17710" CreationDate="2020-04-21T04:24:24.367" Score="5" Body="&lt;p&gt;I am thinking that, if you had someone who was an expert in this field, they would observe that the immediate concept of counting does not logically imply running out of numerals and having to invent the idea of “ten”.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They would then observe that different values for “ten” result in different artefacts under processes such as addition and multiplication… and choosing one would be about choosing which type of artefact one wanted.  For instance, a prime number such as 7 or 29 would have features that are over the author’s head, such as might interest a cryptographer.  Conversely, a number with several prime factors, such as [our] 30 (=2*3*5) or 12 (=2*2*3) or 6 (=2*3) yields interesting patterns when multiplying and dividing (for instance).  (There is some discussion of these features in the page linked in the question.)  They would also observe that base 2 is interesting on account of having only one numeral other than 0.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this person would suggest that what the OP is really thinking is that it would be interesting to teach young students about the artefacts that arise differentially across different bases.  &lt;/p&gt;&#xA;" OwnerUserId="13853" LastEditorUserId="13853" LastEditDate="2020-05-01T06:00:30.993" LastActivityDate="2020-05-01T06:00:30.993" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18199" PostTypeId="2" ParentId="12298" CreationDate="2020-04-21T21:10:01.433" Score="1" Body="&lt;p&gt;oh I know this is old but I can't not comment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your child didn't take a numeracy test. They took a mathematics test. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a difference. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Its such a shame that the beauty of mathematics is continually overshadowed by boring routine manual calculation that serves little purpose in a digital age. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course children need to know about Roman Numerals. The failure of the curriculum is that it doesn't include Mayan Numbers, Egyptian Numbers, Babylonian Numbers. The invention of zero (which is one of the greatest inventions of civilisation) is hidden in the small print of the curriculum. &#xA;I do also think that they shouldn't have to learn about it in primary school when they haven't developed the critical thinking skills to be able to compare and contrast these different numbers systems. &lt;/p&gt;&#xA;" OwnerUserId="13858" LastActivityDate="2020-04-21T21:10:01.433" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18200" PostTypeId="1" AcceptedAnswerId="18201" CreationDate="2020-04-22T00:15:14.997" Score="9" ViewCount="2436" Body="&lt;p&gt;Asking for a friend, this seems generally useful given #distanceLearning right now. Below is her message in a quote block followed by my response.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I am using Google Classroom, G-sheets, G-Docs, G-Forms, to teach my&#xA;  8th graders remotely. &lt;br/&gt;These are the issues I need help addressing.&lt;br/&gt;&lt;br/&gt;&#xA;  1. I need an efficient way to view student responses in a homework assignment. I am already using G-Forms, but It doesn't have the capacity for certain things.&lt;br/&gt;&#xA;  2. Students need a way to type their equations, and math steps into their assignment. Math symbols are not always accepted on the differing google platforms. &lt;br/&gt;&lt;br/&gt; Here is the assignment I am working on right now. It is the Pythagorean Spiral. &lt;br/&gt;I can easily upload the &lt;a href=&quot;https://bcfdbf88-739e-4b4a-9cbf-b283ab8e2169.filesusr.com/ugd/7e0fdd_372e03fbb39840df80ae325b739c12ad.pdf&quot; rel=&quot;noreferrer&quot;&gt;pdf instructions&lt;/a&gt; to the Google Classroom assignment. &lt;br/&gt;I can screencastify a&#xA;  quick tutorial to post on the assignment. &lt;br/&gt;My problem is the&#xA;  table/chart that the students will type their answers into. &lt;br/&gt;I am&#xA;  attaching that chart (its on the second page) so you can see what I am&#xA;  talking about. &lt;br/&gt;I have considered re-creating a fillable table in a&#xA;  Sheet template for each student, but the cells don't accept equations,&#xA;  only formulas. &lt;br/&gt;I have considered a Google Doc, with a recreated&#xA;  fillable table. I can insert equations, but I don't know how students&#xA;  can type equations and radical symbols, etc. They need an accessible&#xA;  way to type an equation easily. &lt;br/&gt;Ideally I would like their work to&#xA;  dump into something like a spreadsheet, like a Google form does.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here's that &lt;a href=&quot;https://www.youtube.com/watch?v=MVnHNhOmngg&quot; rel=&quot;noreferrer&quot;&gt;youtube vid&lt;/a&gt; I saw.&lt;br/&gt;&lt;br/&gt;&#xA;Some thoughts from our conversation,&lt;br/&gt;- Having students write the table by hand and take pictures &lt;br/&gt;- Annotate the pdf&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My proposed solution:&lt;br/&gt;&#xA;Students could do the table in google sheets and include two columns, one for the formula  &quot;3*SQRT(2)&quot; and one for the value &quot;=3*SQRT(2)&quot;&lt;br/&gt;&lt;br/&gt;&#xA;Here is what it would look like:&lt;br/&gt;&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;|---------------------|------------------|------------------|------------------|&#xA;|        LEG 1        |       LEG 2      |    HYPOTENUSE    |       VALUE      |&#xA;|---------------------|------------------|------------------|------------------|&#xA;|          3          |         3        |    3*SQRT(2)     |   4.24264068733  |&#xA;|---------------------|------------------|------------------|------------------|&#xA;|          3          |     3*SQRT(2)    |    3*SQRT(3)     |   5.196152423    |&#xA;|---------------------|------------------|------------------|------------------|&#xA;|          3          |     3*SQRT(3)    |    3*SQRT(4)     |         6        |&#xA;|---------------------|------------------|------------------|------------------|&#xA;|          3          |     3*SQRT(4)    |                  |                  |&#xA;|---------------------|------------------|------------------|------------------|&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I've shared the question on &lt;a href=&quot;https://twitter.com/E_Rushton/status/1252718370799026176?s=20&quot; rel=&quot;noreferrer&quot;&gt;Twitter&lt;/a&gt;,&#xA;Math Educator StackExchange and on Reddit (mathteachers, matheducation). I've included this point for interested readers to see additional responses. &lt;br/&gt;&lt;/p&gt;&#xA;" OwnerUserId="1693" LastEditorUserId="1693" LastEditDate="2020-04-22T22:52:26.140" LastActivityDate="2020-04-25T00:39:33.350" Title="Do you have an efficient way to view student responses to remote assignments allowing them to type math symbols and steps? (Middle + High school)" Tags="|homework|grading|assessment|" AnswerCount="7" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18201" PostTypeId="2" ParentId="18200" CreationDate="2020-04-22T00:52:38.363" Score="20" Body="&lt;p&gt;In a Google doc, one can &quot;Insert/Equation&quot; (marked by &lt;span class=&quot;math-container&quot;&gt;$\pi^2$&lt;/span&gt;). Then tiny pull-down menus appear in a top bar:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Er35Y.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Er35Y.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using these menus, I just typed this nonsense:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7F7EY.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7F7EY.png&quot; alt=&quot;equation&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-04-22T12:53:15.050" LastActivityDate="2020-04-22T12:53:15.050" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18202" PostTypeId="2" ParentId="18200" CreationDate="2020-04-22T12:17:06.730" Score="16" Body="&lt;p&gt;Consider to just review scanned or photographed handwritten homework.  Yes, this is not as easy as looking at typed work, but consider:  would you require typed work normally?  So why now?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your main objective is just a completion grade (or something fast like an overall plus/check/minus grade), this should be sufficient.  I would argue against ever doing detailed parsing of drill homework anyway.  A completion grade is sufficient (or even not grading homework at all, if you have frequent tests).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While requiring kids to do typed work (and even special typed work given the need for symbols) appeals to the LATEX luvverz, I consider it an additional hurdle to just working the drill work itself.  I suspect your completion rate will be significantly lower than if you allowed handwritten work.  Is the ease of grading and/or higher feedback you give for those who do this typed work worth the loss from kids that don't do the drill at all?  (This is a practical optimization question, not a quantitative math problem.)&lt;/p&gt;&#xA;" OwnerUserId="13859" LastActivityDate="2020-04-22T12:17:06.730" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18203" PostTypeId="2" ParentId="18141" CreationDate="2020-04-22T13:09:38.080" Score="0" Body="&lt;p&gt;I find the first sentence of the Wiki article on Cartesian coordinates sufficient for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I am timid of my rigor righteousness when someone where asks for &quot;precise&quot;, even with added bold to hammer the request harder.  ;-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;A Cartesian coordinate system (UK: /kɑːˈtiːzjən/, US: /kɑːrˈtiʒən/) is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Cartesian_coordinate_system&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Cartesian_coordinate_system&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  Another answer commented on the preference for perpendicularity.  The one area I see this waived a lot is in crystallography (or properties of crystals) where having coordinate systems that are non-perpinducular (or non equal in length) makes sense as it corresponds to the anisoptropic substance.  So the norm is absolutely to use a monoclinic coordinate system with monoclinic crystals.  Not gyrate around with a cubic coordinate system and do all the conversions.&lt;/p&gt;&#xA;" OwnerUserId="13859" LastEditorUserId="13859" LastEditDate="2020-04-22T13:21:23.657" LastActivityDate="2020-04-22T13:21:23.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18204" PostTypeId="1" AcceptedAnswerId="18212" CreationDate="2020-04-22T14:05:57.497" Score="3" ViewCount="197" Body="&lt;p&gt;... in the literature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I an wondering is there a (considered) list of &quot;fundamental mathematical skills&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure can I give a solid definition of &quot;fundamental mathematical skill&quot;. What I mean by a fundamental skill would certainly include &quot;pattern spotting&quot;, &quot;visualisation&quot;, and what I would call &quot;substitution&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A skill that would perhaps also be a fundamental skill would be &quot;estimation&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However something like &quot;algebraic manipulation&quot; would not be a fundamental skill. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A soft definition might be to say that things that belong to the realm of &quot;mathematical thinking&quot; might be fundamental skills, while things that belong to the realm of &quot;mathematical knowledge&quot; would not. I appreciate this is very far from a clear demarcation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Context:&lt;/strong&gt; Colleagues and I are thinking about developing a diagnostic test for new students. We are thinking around including questions that might test these fundamental skills.&lt;/p&gt;&#xA;" OwnerUserId="584" LastActivityDate="2020-04-24T15:39:44.873" Title="Has a List of Fundamental Mathematical Skills been compiled?" Tags="|undergraduate-education|mathematical-pedagogy|reference-request|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18205" PostTypeId="2" ParentId="18200" CreationDate="2020-04-22T19:14:51.157" Score="5" Body="&lt;p&gt;I put this as a comment, but maybe it deserves an answer to prevent it from being dismissed outright. My comment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why not use LaTeX? It might seem a little overbearing (and admittedly I've used it very seldom myself), but it's more than capable of doing all of it and it's not as complicated as it seems. It's typically just a simple Google search for latex using &lt;em&gt;X&lt;/em&gt; symbol has always been enough; the first Google result shows me exactly how to do what I want. &lt;a href=&quot;http://latex-project.org&quot; rel=&quot;noreferrer&quot;&gt;LaTeX Project&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To expand a bit more, I wanted to demonstrate. In fact, fun enough, StackExchange uses a form of LaTeX, so I can do it quite easy!&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$x(t) = x_0 × (1 + r) ^ t$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$log(x(t)) = x_0t + log(1 + r)$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac {log(1000000) - log(46000)} {log(1 + 0.12)} = t$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;is literally typed out as&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;\$\$x(t) = x_0 × (1 + r) ^ t\$\$&lt;/li&gt;&#xA;&lt;li&gt;\$\$log(x(t)) = x_0t + log(1 + r)\$\$&lt;/li&gt;&#xA;&lt;li&gt;\$\$\frac {log(1000000) - log(46000)} {log(1 + 0.12)} = t\$\$&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Saying this is easy isn't generous enough. And in fact, for middle school to high school, everything I've put here is probably all you'll ever need to type out. Maybe for high school calculus you might have to do integrals, so here's that just to prove it works&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_a^b \! f(x) \, \mathrm{d}x$$&lt;/span&gt;&#xA;which is typed&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\$\$\int_a^b ! f(x) \, \mathrm{d}x\$\$&lt;/p&gt;&#xA;" OwnerUserId="13860" LastActivityDate="2020-04-22T19:14:51.157" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18206" PostTypeId="2" ParentId="18200" CreationDate="2020-04-22T19:51:36.553" Score="2" Body="&lt;p&gt;Similar to another answer, but different enough to be worth answering in its own right:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We're about to start marking undergrad (physics) exams as typed text plus handwritten equations and hand-drawn diagrams, photographed and uploaded.  This would seem to be a reasonable hybrid approach, and access to cameras has never been better&lt;/p&gt;&#xA;" OwnerUserId="8180" LastActivityDate="2020-04-22T19:51:36.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18207" PostTypeId="2" ParentId="18204" CreationDate="2020-04-22T20:36:22.980" Score="6" Body="&lt;p&gt;This is not really an answer, but is too long for a comment. BTW, the downvote wasn't mine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried constructing something similar to this about 25 years ago, with some guidance from my mother, who worked in standardized testing. The details of the list of skills were different in my case, but it was a similar situation. The problem I would expect you to run into is the following (which is what happened for me).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All questions were correlated with one another, and all were correlated with students' grades. However, these correlations were all relatively weak -- some weaker than others. And there was no real evidence that the discrete list of skills we had in mind were actually separate skills. If they really were separate skills, then you would expect that when you did a covariance of question &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; with question &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, there would be a strong correlation when &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; were designed to measure the same skill, and a weak correlation when &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; were designed to measure different skills. This didn't happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I can't predict what your results would actually be, but by default I would kind of expect results such as the ones I found. And I think then what's happening is that you're simply measuring some over-all measure of intelligence, academic preparation, and reading comprehension. This is the kind of experience that led Spearman to originally postulate the existence of some kind of general intelligence or &quot;&lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;&quot; parameter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although IQ testing is basically pseudoscience, there is a kernel of truth in it, which is that basically all intellectual abilities tend to be  correlated with one another, and it's difficult to disentangle separate factors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possible way to approach your problem might be to look around at what well-constructed tests already exist, and give those tests to your students. Then if you have a hypothesis that there is some well-defined skill like estimation, look for questions that seem to you to prove estimation, and see if they really do correlate strongly with each other, weakly with other questions, and strongly with students' grades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The advantage of starting with other people's published tests is that you get the advantage of all their previous work in making sure the test works well. For example, they will have already eliminated questions that are not useful because they're too easy (everybody gets them right), too hard (nobody does better than random guessing), or not valid (don't correlate with anything external). For a multiple-choice test, there is a huge amount of work that can go into finding good distraction answers that probe actual student misunderstandings.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-04-22T20:43:16.170" LastActivityDate="2020-04-22T20:43:16.170" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18208" PostTypeId="2" ParentId="18174" CreationDate="2020-04-23T04:57:49.820" Score="2" Body="&lt;p&gt;This is similar to Xander Henderson's approach, but the calculations have been made slightly easier by using two FOIL multiplications instead of just one:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$3.9 \times (3.8 + 3.7)$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$=3.9 \times 3.8 + 3.9 \cdot 3.7$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$=(4 - 0.1)(4-0.2)+(4-0.1)(4-0.3)$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$=16-0.8-0.4+0.02+16-1.2-0.4+0.03$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$=16+16-0.8-1.2-0.4-0.4+0.02+0.03$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$=(16+16)-(0.8+1.2)-(0.4+0.4)+(0.02-0.03)$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$=32-2-0.8+0.05$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$=29.2+0.05$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$=29.25$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="13864" LastEditorUserId="13864" LastEditDate="2020-04-23T08:22:22.947" LastActivityDate="2020-04-23T08:22:22.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18209" PostTypeId="1" CreationDate="2020-04-23T08:15:20.767" Score="8" ViewCount="218" Body="&lt;p&gt;Recently I asked a question on math.stackechange about the &lt;a href=&quot;https://math.stackexchange.com/questions/3618177/what-are-most-ways-to-differentiate-the-same-function&quot;&gt;most ways to differentiate the same function&lt;/a&gt; and it didn't seem to generate any interest - rather, the reason why I'd ask such a question was the highest voted comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, there are many reasons why you would want to do the same question different ways, or to give questions to students that could be done in different ways - &lt;em&gt;off the top of my head...&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Checking your work since the more different ways you do the same question then the more support you have for the correct answer. &lt;/li&gt;&#xA;&lt;li&gt;Different approaches highlight different topics, which then could provide insight into other questions of said topic. You'll see the pro's and con's of each approach.&lt;/li&gt;&#xA;&lt;li&gt;Flexibility when it comes to doing future questions since you don't always have to do it the &quot;expected&quot; way, which could be laborious. With the intuition and the insight, &quot;shortcuts&quot; could be made.&lt;/li&gt;&#xA;&lt;li&gt;Ability to accept that there are multiple ways to do the same question and that open-mindedness could help you interact with others who solve things differently from you.&lt;/li&gt;&#xA;&lt;li&gt;Emphasizing that process is as important as final answer, and that an answer &lt;em&gt;in itself&lt;/em&gt; doesn't mean we need to stop thinking about the problem.&lt;/li&gt;&#xA;&lt;li&gt;Showcasing mathematical consistency in how seemingly separate methods of doing questions, or how different branches of math, actually relate to each other.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Unfortunately most of those points might be paraphrases of each other, but the fact remains that I've always tried different methods to do questions and I've always smiled it when my students give me approaches that I didn't think of. Even if said approach doesn't ultimately get the answer wanted, I love how the their thinking and knowledge wasn't restricted to current lesson.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is, &lt;strong&gt;do you teach different proofs or calculations of same question?&lt;/strong&gt; And if you do it in calculus or real analysis, what questions yield the most different methods of solving?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-05-31T00:00:14.990" Title="Do you teach different proofs or calculations of same question?" Tags="|calculus|proofs|concept-motivation|problem-solving|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18210" PostTypeId="2" ParentId="18200" CreationDate="2020-04-23T09:55:13.553" Score="3" Body="&lt;p&gt;If you want to use LaTeX you should check out www.mathcha.io. It's a very user-friendly editor. It allows for LaTeX /HTML/pdf export.&lt;/p&gt;&#xA;" OwnerUserId="13867" LastActivityDate="2020-04-23T09:55:13.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18211" PostTypeId="2" ParentId="18200" CreationDate="2020-04-23T13:35:45.473" Score="3" Body="&lt;p&gt;If you are open to using solutions beyond GSuite, you could consider tools like GoFormative or Edulastic which have easier equation editing features. I would rather use their ‘Show your Work’ question type which is basically a simple whiteboard students can draw or scribble on. I use GoFormative through Google Classroom.&lt;/p&gt;&#xA;" OwnerUserId="13869" LastActivityDate="2020-04-23T13:35:45.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18212" PostTypeId="2" ParentId="18204" CreationDate="2020-04-24T15:39:44.873" Score="2" Body="&lt;p&gt;Common Core includes &lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Standards for Mathematical Practice&lt;/a&gt;. Those may be a good description of what you're looking for. I have seen a few other good lists of mathematical thinking skills. I enjoyed thinking about how I might use &lt;a href=&quot;http://www.withoutgeometry.com/2010/09/habits-of-mind.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one, put together by Avery Pickford&lt;/a&gt;, in my teaching. (You might find more good thinking in the comments.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-04-24T15:39:44.873" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18213" PostTypeId="1" AcceptedAnswerId="18222" CreationDate="2020-04-24T15:43:57.213" Score="12" ViewCount="1536" Body="&lt;p&gt;&lt;em&gt;(I hope the question in in scope, please see &lt;a href=&quot;https://matheducators.meta.stackexchange.com/questions/669/are-questions-around-simplification-in-scope&quot;&gt;my question on Meta&lt;/a&gt; about that)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My 15 yo son (2&lt;sup&gt;nde&lt;/sup&gt; in France, this is the first year of the equivalent of a High School) is going through basic statistics. One of the exercises in his book made me wonder about why some simplifications are made in an exercise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I would love to be corrected and learn something, the core of my question is rather whether this exercise is&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;simplified because it will help to understand something later (and one can live with the simplification)&lt;/li&gt;&#xA;&lt;li&gt;simplified because this is the right thing to do (this is the moment where I would learn something about statistics)&lt;/li&gt;&#xA;&lt;li&gt;a bad exercise (and the problem is over)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The problem&lt;sup&gt;(1)&lt;/sup&gt; is :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 2016 women were asked about their age at the time they got married.&#xA;  The table below contains their answers:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \begin{array} {|r|r|}\hline Age &amp;amp; 20 \le A \lt  30 &amp;amp; 30 \le A \lt &#xA;40 &amp;amp; 40 \le A \lt  50 &amp;amp; 50 \le A \lt  60 &amp;amp; 60 \le A \lt  70 \\ \hline&#xA;Amount &amp;amp; 10 &amp;amp; 35 &amp;amp; 15 &amp;amp; 2 &amp;amp; 1 \\ \hline  \end{array} $$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Calculate the mean age of marriage for these women.   Calculate the&#xA;  standard deviation of this series.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The answer then proceeds with&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In order to determine the mean, we assume that a person who is between 20 and 30 years old is 25 years old.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then the problems is solved as expected for a series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The assumption is, I believe, wrong&lt;/strong&gt;, it depends on the distribution within the range. There are no reasons for the distribution to be symmetric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very similar problem was given later, where the range were apartment surfaces. For the range 0 to 20 m&lt;sup&gt;2&lt;/sup&gt;, I expect that I should have stated that the average is 10 sqm, which makes even less sense as there obvious limits on the lower size of an apartment and the distribution will be heavily skewed to the right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My problem with these exercises is that they make an artificial issue (= having to decide on the value to use in the ranges) while it is easy to have exercises such as &lt;em&gt;&quot;satisfaction level from 1 to 10 over a month, calculate the average &amp;amp; deviation&quot;&lt;/em&gt; or &lt;em&gt;&quot;high jump - which athletes are more consistent in their results?&lt;/em&gt; (some would have low and high results, some more around the average for instance)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My core questions are:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Was there in the exercise I quoted a reason to do the assumption? &lt;/li&gt;&#xA;&lt;li&gt;Is this a good assumption? &lt;/li&gt;&#xA;&lt;li&gt;In what will it help later in their curriculum?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;(1)&lt;/sup&gt; &lt;em&gt;Déclic 2&lt;sup&gt;nde&lt;/sup&gt; Math, Programme 2019, Hachette. Ex. 3 p. 311&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="8076" LastActivityDate="2020-04-27T09:00:55.340" Title="Are these assumptions in statistics correct or beneficial?" Tags="|secondary-education|statistics|" AnswerCount="8" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18214" PostTypeId="2" ParentId="18200" CreationDate="2020-04-24T16:35:39.187" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://edulastic.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edulastic&lt;/a&gt;&#xA;will do all of the things you're looking for and is free. You can embed a YouTube video into the question for students to watch, then create a numeric question with a table. There's a math editor with a big range of mathematical symbols both for building questions and student answers. Edulastic automatically grades the work and you can see it in real time. There's also an &quot;Express Grader&quot; which shows student answers and/or scores in a spreadsheet type format (and you can download to csv). It syncs with Google Classroom, so you can share assignments with students there if you use that. Hope this helps!&lt;/p&gt;&#xA;" OwnerUserId="13877" LastEditorUserId="511" LastEditDate="2020-04-25T00:39:33.350" LastActivityDate="2020-04-25T00:39:33.350" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18215" PostTypeId="2" ParentId="18213" CreationDate="2020-04-24T16:55:13.140" Score="24" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The assumption is, I believe, wrong, it depends on the distribution within the range. There are no reasons for the distribution to be symmetric.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It's not wrong, it's an approximation. It's equivalent to the rectangle rule for approximating an integral: &lt;a href=&quot;https://en.wikipedia.org/wiki/Numerical_integration&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Numerical_integration&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's good in my opinion that your kid's text does an example like this where you have to come up with an approximation. Students should be exposed to the realities of life, not sheltered from those realities by only having them work on sanitized examples that fit perfectly with some theoretical framework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be good if the book correctly described it as an approximation.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-04-24T16:55:13.140" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18216" PostTypeId="2" ParentId="18213" CreationDate="2020-04-24T20:11:27.293" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think it's a good example, given the students ability.  Better to give them simpler examples and just work with crunching numbers.  Since they're just encountering the topic for the first time, no reason to draw in concepts like good/bad/middling assumptions in data analysis.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nevertheless, I don't think it's the end of the world that your boy had a suboptimal question given to him.  Many commenters here are logicians by training and mindset, therefore expecting perfection, often in areas not of the most importance.  But life goes on.  C'est la vie. ;-)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I deal with datasets like this, for practical use, at times.  For instance, here is another similar data set.  EIA data on L48 oil production by EIA gravity bin:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.eia.gov/dnav/pet/pet_crd_api_adc_mbblpd_m.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.eia.gov/dnav/pet/pet_crd_api_adc_mbblpd_m.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see that 40-45 is the largest bin, but that the data is non-normal (skew).  And it's even fuglier if you try looking at an individual state.  (The data is only collected in 5 degree bins so that is as good as it gets, absent doing your own, very expensive survey.  And then the stat data is obscured for reasons of commercial confidentiality given some states are dominated by a small number of producers, buyers.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, you have the confusion of a greater than or less than bin at the end of the distributions that is not an exact 5 degree bin, but extends to 0 or infinity (in theory, but practically to 8 or 70 or so in terms of known crude strains).  I think I used 2.5 degrees below the LT or above the GT as an approximation for those bins.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have actually crunched this data, a couple ones.  One is to just assume midpoint.  The next, probably better, is to linearly weight versus the surrounding bins.  However, the answer I got was within a half degree of the simpler assumption.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could even do higher order fits, but I would caution against it given the small degrees of freedom.  For that matter, I think even the linear weighting introduces an unfortunate complexity when discussing the answer and perhaps not worth the &quot;chaff&quot; given that a half degree is not important functionally in terms of insight.  For that matter basic trend analysis over time is also insensitive to the choice of within bin average.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Note that API gravity is actually sort of a bastard's reciprocal of density, so you can't/shouldn't average it per barrel anyways.  That said, you can convert to densities and back again.  And it doesn't &quot;change the story&quot; in terms of useful insights either.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But all this sort of minutia/thinking is a distraction to kids that should just be learning to add the stuff and divide, for now.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="13859" LastActivityDate="2020-04-24T20:11:27.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18217" PostTypeId="2" ParentId="18209" CreationDate="2020-04-25T01:43:25.297" Score="4" Body="&lt;p&gt;My goal is not only for my students to recognize that there are multiple methods for solving a problem, but more importantly, for them to be able to identify the &lt;em&gt;appropriate&lt;/em&gt; method(s) for solving each problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, once a proof of a particular method is learned, it becomes a tool that can be used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specific to differentiation, the reason why I would use different methods is because each method is an important technique &lt;strong&gt;on its own&lt;/strong&gt;.  At some point in the course, I expect my students to able to identify the tediousness or ridiculousness of using certain methods to answer a particular problem.  I would not, for example, expect a student to check his work for differentiating a product of powers of three functions by expanding the product first and applying the power rule to each term.  Rather, going through the work using &lt;strong&gt;the same method&lt;/strong&gt; is sufficient (though, teaching how to properly check work is a different subject entirely).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a few cases where I require students to use a specific method.  For example, one of my recent test questions say&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Using the first principles definition of the derivative, show that &lt;span class=&quot;math-container&quot;&gt;$$\frac{d}{dx}\tan x=\sec^2x.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When the derivative of the tangent function was taught, we used the knowledge of trigonometric identities and other derivative rules (i.e. &lt;span class=&quot;math-container&quot;&gt;$\tan x=\sin x/\cos x$&lt;/span&gt;, use quotient rule).  The reason why I would ask the question above is because it is an application of limits (one of the other topics taught in this course), and not because it is &lt;em&gt;just&lt;/em&gt; a different method to use.&lt;/p&gt;&#xA;" OwnerUserId="12720" LastActivityDate="2020-04-25T01:43:25.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18219" PostTypeId="2" ParentId="18213" CreationDate="2020-04-25T11:51:24.527" Score="2" Body="&lt;p&gt;The approximation isn't bad — in fact, it's quite reasonable, if the binned data is all we have:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Follow-up exercise for advanced students (and curious parents): show that the answer you got using the approximation must be within ±5 years of the true average, regardless of how the actual ages of marriage are distributed within the ranges.  Can you provide any intuitive arguments for why the actual error is likely to be considerably less than that?  (Try to come up with at least two reasons.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As for the pedagogical merit of using such an approximation in this exercise, I would say it depends on how it's presented in the broader context of the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's good that students are exposed to such approximations, because in the real world we have to make them all the time.  If we couldn't do that, and be reasonably confident that the answer is still close to correct, statistics would be all but useless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, I would say that it's not good if the students are simply told to use that particular approximation without being given any intuition on &lt;em&gt;why&lt;/em&gt; it's a reasonable or safe thing to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, if the student is left with the same impression as you seem to have been, i.e. that the approximation is being made &lt;em&gt;just because that's what you're instructed to do&lt;/em&gt; in this math class, even though it &lt;em&gt;intuitively feels wrong&lt;/em&gt; to them, it can contribute to the all too common feeling that the mathematics one learns in school is somehow divorced from reality, just a collection of meaningless rote formulae only useful for passing the exam.*&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, either the textbook or the teacher of the class (or, preferably, both) would've taken a moment to mention that taking the midpoint of the range is indeed an approximation, and that the average obtained using it won't be exactly right, but that it also won't be too far from the true average, either.  They could also use something like the follow-up exercise I suggested above to illustrate that, perhaps as a shared in-class discussion activity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the context of such a discussion, it could also be good to discuss why, in practice, data from surveys like this is often binned into such wide ranges, and why someone might feel uncomfortable providing e.g. their exact birthdate or the date of their marriage in a survey like this.  I would also recommend emphasizing the fact that, in practice, all data is approximate, and that even if we knew the ages of marriage in the exercise down to, say, one year, that would still be an approximation — just a more fine-grained one than grouping them in 10-year ranges.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As a slight tangent, it might also be worth noting that, if we followed the common everyday practice of rounding ages &lt;em&gt;down&lt;/em&gt; to a whole year before taking the average, that would introduce a systematic bias to the average.  For statistical purposes, a person known to be at least 22 but less than 23 years old should really have their approximate age counted as 22.5 years.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, in case your son's math textbook and/or their teacher haven't explicitly brought this up, this is also a good opportunity to do it yourself while going through the exercises with your son.  You could even bring it up in a discussion with the teacher — not confrontationally, but just mentioning that you found the suggested assumptions in this particular exercise a bit confusing, and that it might be something worth discussing in class, if they haven't already done so.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;*) A topic that I've previously commented on here in the context of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5877/should-word-problems-be-reasonable/5883#5883&quot;&gt;unreasonable word problems&lt;/a&gt;.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="526" LastActivityDate="2020-04-25T11:51:24.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18220" PostTypeId="1" CreationDate="2020-04-25T13:32:21.693" Score="6" ViewCount="1433" Body="&lt;p&gt;In USA, there is a course called College Algebra and a course description may look like the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This  course  provides  students  an  opportunity  to  gain  algebraic  knowledge  needed  in  many  different  fields  such  as engineering, business, education, science, computer technology, and mathematics. Graphical, numerical, symbolic, and verbal methods support the study of functions and their corresponding equations and inequalities. Students will study linear, quadratic, rational, exponential, logarithmic, inverse, composite, radical, and absolute value functions; systems of equations  and  inequalities  modeling  applied  problems;  and  curve  fitting  techniques.  There  will  be  extensive  use  of graphing calculators.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How about in other places such as UK, Russia, India, China, Japan, Brazil, Germany, etc.? What course will be roughly equivalent to College Algebra in other countries?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not expecting a complete answer. A few examples will be greatly appreciated.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="4791" LastEditDate="2020-04-25T14:08:12.050" LastActivityDate="2021-04-28T12:15:40.967" Title="Courses equivalent to College Algebra in other countries?" Tags="|undergraduate-education|secondary-education|algebra|usa|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18221" PostTypeId="2" ParentId="18213" CreationDate="2020-04-25T14:26:48.863" Score="7" Body="&lt;p&gt;It is worth considering that, if the ages would have been recorded as integers, rather than intervals, the assumption would have still been wrong in a similar but less obvious way. That is, a 25 year old and a 25.25 year old would have been grouped together and the 'binning' will affect the mean to some extent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any finite presentation of numerical data, such as ages, will bin them into imprecise groups and will produce the same issue. The important thing to recognize is that the end result has error bars, and we can calculate them! You can simply replace the midpoint with the lower and upper ends of the intervals to get the minimum and maximum possible means.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;while it is easy to have exercises such as &quot;satisfaction level from 1 to 10 over a month, calculate the average &amp;amp; deviation&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Though these seem numerical, one could argue that these are actually categorical data and that means and deviations are not meaningful. For example, imagine the 3 state satisfaction scale happy :) neutral :| sad :(  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the mean of :) and :| ?&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2020-04-25T14:26:48.863" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18222" PostTypeId="2" ParentId="18213" CreationDate="2020-04-25T14:59:20.847" Score="6" Body="&lt;p&gt;This is a real-life situation: Sometimes you receive data in groups (bins) like this; that's a pretty common result from using automatable multiple-choice survey forms. There is no way to retrieve the original discrete data points, or the distribution of the data in the bins. So what are you going to do?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Formulas for generating statistics from grouped data like this are very standard. Here's the presentation in Weiss, &lt;em&gt;Introductory Statistics&lt;/em&gt;, Sec. 3.2:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xSjeq.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xSjeq.png&quot; alt=&quot;Weiss: Grouped-Data Formulas&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note the text that says, &quot;these formulas yield only approximations to the actual sample mean and sample standard deviation&quot;, which is accurate and addresses the OP's point. Each term &lt;span class=&quot;math-container&quot;&gt;$x_i$&lt;/span&gt; &quot;denotes class midpoint&quot;, matching the OP's given exercise. In some sense, this represents the average of all possible distributions over the unknown individual bin data, so it's the best approximation we can make in the general case (or at least a reasonable default model, such that the burden of proof is on one suggesting some different improved model). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this some kind of error or mistake? No; the whole essence of statistics is to take limited data about the world, and make some kind of estimate or approximation to the larger truth that we can't see. Deductive reasoning is relatively easy; inferential reasoning is much harder, and that's exactly what the field of statistics attempts to formalize. Recall the famous words of Bishop Joseph Butler (1736):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Probable Evidence, in its very nature, affords but an imperfect kind&#xA;  of Information; and is to be considered as relative only to Beings of&#xA;  limited Capacities. For nothing which is the possible object of&#xA;  Knowledge, whether past, present, or future, can be probable to an&#xA;  infinite Intelligence; since it cannot but be discerned absolutely as&#xA;  it is in itself, certainly true, or certainly false: but to us,&#xA;  Probability is the very Guide of Life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-04-25T14:59:20.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18223" PostTypeId="2" ParentId="18220" CreationDate="2020-04-25T16:04:38.097" Score="2" Body="&lt;p&gt;[Too long to fit in comments and addressing your meta-questions, versus just a square peg for square question response.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's basically the same thing as &quot;Algebra 2&quot; in US high school.  The funny thing is it is an anachronism to call it College Algebra.  Prior to WW2, it was common that US high schools only had first year algebra and geometry.  Thus &quot;Algebra 2&quot; was a college course as was trigononmetry.  And College Algebra really was college algebra.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the stereotypical norm now is for kids to do 9th grade algebra one (lines mostly), 10th geometry, 11th algebra two (logs, exponents, etc.), 12th trig and other precalculus.  Calculus is the normal first year college course. Of course some kids are accelerated and do calc in HS.  But it is still considered college material. Thus the AP test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A kid who takes College Algebra in college is remedial now.  He should be nominally taking first semester calculus as he starts freshman year if he is neither remedial nor accelerated track.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  Some kids will even have had some calculus in HS (it is probably the most commonly accelerated subject) in either an AP class or some easier version.  However, many of them still have to take calc as freshmen as they really didn't master it in high school (can't pass an AP, can't pass a college-specific placement exam) but just got some exposure.  Some weaker ones even end up in precalculus although they were in a &quot;calculus class&quot; in HS.&lt;/p&gt;&#xA;" OwnerUserId="13859" LastEditorUserId="13859" LastEditDate="2020-04-25T18:57:13.630" LastActivityDate="2020-04-25T18:57:13.630" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18224" PostTypeId="2" ParentId="18213" CreationDate="2020-04-25T16:42:18.407" Score="3" Body="&lt;p&gt;The situation you give is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A woman got married when she was between 20 and 30 years old.  At what age did she get married?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But let me give a different situation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A woman is between 20 and 30 years old.  What is her age?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Given no information, assume the value is in the middle.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the situation I gave, a good estimate is 25 (midway).  Why not, say, 22?  Because we have no idea if the person is relatively young (nearer 20).  Why not, say, 27?  Because we have no idea if the person is relatively old (nearer 30).  We do not know &quot;the distribution within the range.&quot;  Is the distribution asymmetric?  If so, should you assume it is positively skewed?  Why not assume it is negatively skewed?  There is no information.  So we assume the skewness is zero (in the middle).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;If there is information, then the middle might not be a good assumption.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I say the height of a certain person is between 5 feet and 15 feet, &quot;common sense&quot; tells us that 10 feet is not a good assumption.  This is because in our experience, there are no people who are ten feet tall.  There &lt;em&gt;is&lt;/em&gt; information in this case, even though it is not explicitly stated.  Experience tells us that the majority of people who are 5 or more feet tall are not 7 or more feet tall, so probably a good guess for the height is 6 feet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;In the situation that you gave, what is the information?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The woman got married when she was between 20 and 30 years old.  Other women got married at an older age.  And, it seems (from your data), that no other women got married when they were less than 20 years old.  So a person with &quot;common sense&quot; might assume that &quot;the middle&quot; in this case is more than 25;  there are fewer women who got married at a younger age and more women who got married at an older age.  Okay, so it's more than 25.  Is it 26?  27?  How is a student expected to know?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You assume that the student &lt;em&gt;doesn't&lt;/em&gt; know.  You assume that there is no information.  You assume that the value is in the middle.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2020-04-25T16:47:36.027" LastActivityDate="2020-04-25T16:47:36.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18225" PostTypeId="1" CreationDate="2020-04-25T19:28:49.027" Score="6" ViewCount="243" Body="&lt;p&gt;Andreas Vohns in his article &lt;a href=&quot;https://link.springer.com/article/10.1007/s13138-016-0086-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fundamental Ideas as a Guiding Category in Mathematics Education—Early Understandings, Developments in German- Speaking Countries and Relations to Subject Matter Didactics&lt;/a&gt; says: &amp;quot;&lt;em&gt;In its day-to-day regime the mathematics classroom is mainly focused on students’ mastery of specific knowledge and skills currently at hand. But do they see the bigger picture? Do they get an appropriate idea of what mathematics is essentially about? &lt;strong&gt;Fundamental ideas&lt;/strong&gt; have been a regularly proposed way to outline the bigger picture.&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The second version of the Brazilian Common Core (BNCC) puts the following pairs of Fundamental Ideas in Mathematics: variation and constancy; certainty and uncertainty; movement and position; relationships and interrelationships.&lt;/p&gt;&#xA;&lt;p&gt;Nilson José Machado, a Brazilian scholar, in turn, proposes the following Fundamental Ideas in Mathematics: equivalence; order; proportionality; interdependence; measurement; invariance; variance; periodicity; randomness; problematicity; optimization.&lt;/p&gt;&#xA;&lt;p&gt;I'm looking for other authors who have addressed the subject of Fundamental Ideas in Mathematics (including countries other than Germany and Brazil).&lt;/p&gt;&#xA;&lt;p&gt;In particular, shouldn't &amp;quot;recursion&amp;quot; be included in the list of Fundamental Ideas?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="6053" LastEditDate="2020-11-02T14:27:19.213" LastActivityDate="2020-11-02T14:27:19.213" Title="Fundamental Ideas in Mathematics" Tags="|resource-request|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18226" PostTypeId="2" ParentId="18220" CreationDate="2020-04-25T19:38:52.590" Score="5" Body="&lt;p&gt;I am going to list the topics from the Russian Algebra/PreCalc school program, and you decide for yourself. The material below is mandatory for all students. I did not include optional material.&lt;/p&gt;&#xA;&lt;p&gt;Russia has 11-year grade school; it used to have 10-year until late 1990s. No K class. The age to enroll 1st grade is between 6.5 and 8 years old.&lt;/p&gt;&#xA;&lt;p&gt;There is no &amp;quot;college algebra&amp;quot; in Russia. The lowest university math course is calculus.&lt;/p&gt;&#xA;&lt;p&gt;8 grade: [14-16 years]&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Simple functions&lt;/li&gt;&#xA;&lt;li&gt;Functions and graphs&lt;/li&gt;&#xA;&lt;li&gt;Functions &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=\frac 1 x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Square roots&lt;/li&gt;&#xA;&lt;li&gt;Quadratic and rational equations&lt;/li&gt;&#xA;&lt;li&gt;Linear, quadratic and reciprocal functions&lt;/li&gt;&#xA;&lt;li&gt;Systems of linear equations&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;9 grade:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Inequalities&lt;/li&gt;&#xA;&lt;li&gt;Linear one-variable inequalities&lt;/li&gt;&#xA;&lt;li&gt;Quadratic one-variable inequalities&lt;/li&gt;&#xA;&lt;li&gt;Rational inequalities&lt;/li&gt;&#xA;&lt;li&gt;Power&lt;/li&gt;&#xA;&lt;li&gt;Function &lt;span class=&quot;math-container&quot;&gt;$y^n$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;n-th root&lt;/li&gt;&#xA;&lt;li&gt;Sequences&lt;/li&gt;&#xA;&lt;li&gt;Arithmetic progression&lt;/li&gt;&#xA;&lt;li&gt;Geometric progression&lt;/li&gt;&#xA;&lt;li&gt;Intro to theory of probability, statistics and combinatorics&lt;/li&gt;&#xA;&lt;li&gt;Approximations&lt;/li&gt;&#xA;&lt;li&gt;Descriptive statistics&lt;/li&gt;&#xA;&lt;li&gt;Permutations, combinations, partitions&lt;/li&gt;&#xA;&lt;li&gt;Experiments, outcomes, events&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;10 grade:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Roots, powers, logarithms&lt;/li&gt;&#xA;&lt;li&gt;Trigonometric formulas and functions&lt;/li&gt;&#xA;&lt;li&gt;Sine and cosine&lt;/li&gt;&#xA;&lt;li&gt;Tangent and cotangent&lt;/li&gt;&#xA;&lt;li&gt;Trigonometric identities&lt;/li&gt;&#xA;&lt;li&gt;Trigonometric functions&lt;/li&gt;&#xA;&lt;li&gt;Trigonometric equations and inequalities&lt;/li&gt;&#xA;&lt;li&gt;Intro to probability theory&lt;/li&gt;&#xA;&lt;li&gt;Probability&lt;/li&gt;&#xA;&lt;li&gt;Conditional probability&lt;/li&gt;&#xA;&lt;li&gt;Frequency of events&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;11 grade&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Functions, derivative, integral&lt;/li&gt;&#xA;&lt;li&gt;Functions and their graphs&lt;/li&gt;&#xA;&lt;li&gt;Limit of a function, continuity&lt;/li&gt;&#xA;&lt;li&gt;Inverse functions&lt;/li&gt;&#xA;&lt;li&gt;Derivative&lt;/li&gt;&#xA;&lt;li&gt;Application of derivative&lt;/li&gt;&#xA;&lt;li&gt;Antiderivative and integral&lt;/li&gt;&#xA;&lt;li&gt;Equations, inequalities, and systems of them&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="12201" LastEditDate="2021-04-28T12:15:40.967" LastActivityDate="2021-04-28T12:15:40.967" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18227" PostTypeId="2" ParentId="18220" CreationDate="2020-04-25T20:35:23.567" Score="6" Body="&lt;p&gt;In the UK, what you describe lies somewhere in between GCSE and A-Level Maths. Both of these exams are intended for secondary school students. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is unusual for these topics to be taught at the university level, and if they are taught, it is often not by the Maths department. For example, the Economics department might have such a course for their first year undergraduates with insufficient maths background. A small number of British universities offer a 'Foundation Year' before beginning an undergraduate degree. A Foundation Year for a scientific degree would cover these topics. &lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2020-04-25T20:35:23.567" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18228" PostTypeId="1" CreationDate="2020-04-25T21:14:44.183" Score="5" ViewCount="218" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6SMcv.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6SMcv.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Linguistics, recursion is the ability to amend one sentence to another, combining them in different ways. Example: &quot;After I saw the movie, I went to a cafeteria and then I ate a pizza.&quot; Without recursion, we would have to use three phrases, something like: &quot;I saw the movie. Then I went to the cafeteria. I ate a pizza.&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no consensus (the topic is controversial), but some linguists (Chomsky is one of them) argue that recursion is a particularity of human language. Only humans use recursion (the Pirahã language of an Amazonian tribe can be a counterexample). [Note: this phrase is very recursive!]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, on the one hand, recursion is an attribute of human language. On the other hand, recursion makes sentences longer and more complex.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the dilemma: you're going to write a math textbook for Basic School. On the one hand, it seems that using simple sentences (without recursion) would be easier for the student to understand what is being addressed. On the other hand, shouldn't the student be familiar with recursive texts, including those dealing with Mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be possible to build a scale over the school years of the level of recursion that can be used to have a balance between understanding of the mathematical concept to be treated and, at the same time, doing this by practicing the linguistic attribute of recursion which is a linguistic characteristic of humanity?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, are there mathematical topics that can only be described via recursion? I think that every sentence of the form A =&gt; B is recursive: A is a sentence, B is a sentence and =&gt; combines the sentences, so A =&gt; B is (linguistic) recursive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of any study that deals with what is expected, in terms of the development of linguistic resources (including recursion, verb tenses, ...), throughout the school years? What language resources could (should) be used in each school grade?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a text that makes an introduction to recursion in linguistics and the comparison with the use of the term in Mathematics: &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/abs/10.1002/wcs.131&quot; rel=&quot;noreferrer&quot;&gt;https://onlinelibrary.wiley.com/doi/abs/10.1002/wcs.131&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2020-04-26T01:49:52.553" Title="Linguistic recursion issues while writing Mathematics textbooks for Basic School" Tags="|language-use|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18232" PostTypeId="2" ParentId="18213" CreationDate="2020-04-27T03:33:12.703" Score="2" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Was there in the exercise I quoted a reason to do the assumption?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Yes, as pointed out in the other answers, it gives a simple way to answer the question and is clearly the intended solution.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Is this a good assumption?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;No. Because you don't have the actual data, the mean and standard deviation are actually random variables, so it is not even clear what the question is asking. In order to calculate the expected value of the mean and standard deviation, you need to make some assumptions about how the data are generated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, you could use the maximum entropy assumption, and assume that each sample of ages is uniformly distributed in the given range. (There are strong arguments that this is what you should do if you have zero knowledge.) This will give the same formula for the expected mean as in the question, but the expected standard deviation will be smaller than the standard deviation you get from assuming that all the ages are equal to the midpoint of the range.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, you can assume that the ages are randomly sampled from a particular underlying distribution, say a beta distribution in this example, and then use inference to fit this distribution, from which you can get the expected mean and standard deviation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you point out in your example of apartment area, your choice of distribution will depend on your knowledge of the subject matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said this, I admit that in a lot of cases, you will see that the naive formulas do actually give results that are pretty close to a more statistically-valid approach. &lt;a href=&quot;https://stats.stackexchange.com/questions/60256/standard-deviation-of-binned-observations&quot;&gt;There are some examples on Crossvalidated.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;In what will it help later in their curriculum?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In real life, it is very common to have anonymised data. Statistical agencies often bin data like this in order to reduce the danger of people being identifiable (e.g. if I have your birth year, it is easier for me to steal your identity than if I only know your age to within 10 years).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, it is very likely that the students will have to deal with data like this in real life. Therefore, it is good to give them a quick and dirty strategy for dealing with it, rather than just giving up or doing something excessively complicated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the standard deviation part of the question could be improved by mentioning that in practice, statisticians would probably use something like &lt;a href=&quot;https://en.wikipedia.org/wiki/Sheppard%27s_correction&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sheppard's Correction&lt;/a&gt; to get a more accurate estimate of the standard deviation.&lt;/p&gt;&#xA;" OwnerUserId="814" LastActivityDate="2020-04-27T03:33:12.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18233" PostTypeId="2" ParentId="18213" CreationDate="2020-04-27T09:00:55.340" Score="1" Body="&lt;p&gt;As some other answers have mentioned, this kind of assumptions may not be statistically valid but are nevertheless useful in their simplicity and convenience for calculation. A more accurate term for them is &quot;heuristic&quot;. Note that if nothing at all is known about the distribution, then it is an unbiased heuristic. However, if we expect the underlying distribution to be continuous, then it is clearly incorrect, but there is no easy way to get the 'correct' answer. For example, if the distribution is known to be normal, we can estimate the true mean by finding the best parameter &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; such that the probability of observing the given data has maximum likelihood. But if the distribution is not known, we still need a heuristic since there are many continuous distributions that would yield the same data on average.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2020-04-27T09:00:55.340" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18234" PostTypeId="2" ParentId="18225" CreationDate="2020-04-27T17:47:05.840" Score="5" Body="&lt;p&gt;Researchers Sebastian Rezat, Mathias Hattermann and Andrea Peter-Koop have published a book &quot;&lt;em&gt;Transformation - A Fundamental Idea of Mathematics Education&lt;/em&gt;&quot;. You can find the link &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1461434882&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a description of the book:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The diversity of research domains and theories in the field of&#xA;  mathematics education has been a permanent subject of discussions from&#xA;  the origins of the discipline up to the present. On the one hand the&#xA;  diversity is regarded as a resource for rich scientific development on&#xA;  the other hand it gives rise to the often repeated criticism of the&#xA;  discipline’s lack of focus and identity. As one way of focusing on&#xA;  core issues of the discipline the book seeks to open up a discussion&#xA;  about fundamental ideas in the field of mathematics education that&#xA;  permeate different research domains and perspectives. The book&#xA;  addresses transformation as one fundamental idea in mathematics&#xA;  education and examines it from different perspectives. Transformations&#xA;  are related to knowledge, related to signs and representations of&#xA;  mathematics, related to concepts and ideas, and related to instruments&#xA;  for the learning of mathematics. The book seeks to answer the&#xA;  following questions: What do we know about transformations in the&#xA;  different domains? What kinds of transformations are crucial? How is&#xA;  transformation in each case conceptualized?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It mentions transformation as a fundamental idea to mathematics education but transformations are very much relevant to mathematics in general. &lt;/p&gt;&#xA;" OwnerUserId="47" LastEditorUserId="47" LastEditDate="2020-04-27T21:58:03.543" LastActivityDate="2020-04-27T21:58:03.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18235" PostTypeId="1" CreationDate="2020-04-27T22:23:38.387" Score="8" ViewCount="726" Body="&lt;p&gt;I have undertaken the teaching a business calculus course for this semester (spring II).&lt;br /&gt;&#xA;The various assesments for the students, include quizzes/hw/midterms/final exams, adjusted with suitable weights, and a project which counts for 10% of the total grade.&lt;br /&gt;&#xA;The project we have been using for the last couple of semesters revolves around break-even analysis and  includes:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;designing and pricing a product,&lt;/li&gt;&#xA;&lt;li&gt;organizing a poll&lt;/li&gt;&#xA;&lt;li&gt;utilizing Excel to interpolate the data from the poll in order to construct the demand function,&lt;/li&gt;&#xA;&lt;li&gt;constructing cost-revenue-profit functions,&lt;/li&gt;&#xA;&lt;li&gt;optimizing the revenue and the profit,&lt;/li&gt;&#xA;&lt;li&gt;comparisons between the various otpimizations,&lt;/li&gt;&#xA;&lt;li&gt;critical questions on the procedures, ... etc&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This semester (spring II) we have decided to update the various assesments, starting by redesigning the project for the course.&lt;/p&gt;&#xA;&lt;p&gt;So my &lt;strong&gt;question&lt;/strong&gt; is the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Are there some good ideas for projects suitable for a 1st year business calculus course?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some remarks to be taken into account:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I am interested either in some references (papers/online etc) or in some explicit description of ideas.&lt;/li&gt;&#xA;&lt;li&gt;The redesigned project will have an upgraded weight (smt like 15%-20% of the total grade).&lt;/li&gt;&#xA;&lt;li&gt;One of the weaknesses i see (in the resources we have already been using) has to do with the fact that our current project actually stops at methods of differential calculus (1st/2nd derivative criteria for optimization, etc). Since our syllabus also includes integral calculus + multivariable calculus (partial derivatives+optimization of multivariable functions), i would like -ideally- some project which makes full use of the mathematical arsenal developed during the course.&lt;/li&gt;&#xA;&lt;li&gt;it would be interesting if we could include some module related to to the law of supply/demand and the market equilibrium (instead of break-even analysis) or maybe combine both.&lt;/li&gt;&#xA;&lt;li&gt;It would also be interesting and to a certain extend desirable -but not the main thing- if we could include the use of technology: for example utilizing a CAS like &lt;a href=&quot;https://www.wolframalpha.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WolframAlpha&lt;/a&gt; would be really nice!&lt;/li&gt;&#xA;&lt;li&gt;The course will be delivered online. It is the first time we are doing this (actually during Spring I, half of the course was delivered online, due to the well-known circumstances). I am not sure if this affects my question (i leave it to the potential answerers to judge), but it is not the main point of the question.&lt;/li&gt;&#xA;&lt;li&gt;FWIW, the main textbook we are using right now is &lt;a href=&quot;https://www.vitalsource.com/en-uk/products/applied-calculus-for-business-economics-and-the-laurence-hoffmann-gerald-v0077156943&quot; rel=&quot;nofollow noreferrer&quot;&gt;Applied Calculus, Hoffmann/Bradley/Sobecki/Price, 11th edition&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So, i am open to interesting ideas and/or references.&lt;br /&gt;&#xA;Thank you in advance!&lt;/p&gt;&#xA;" OwnerUserId="13896" LastEditorDisplayName="user507" LastEditDate="2021-08-04T17:59:13.760" LastActivityDate="2021-08-04T17:59:13.760" Title="Ideas and/or references for projects for a business calculus course" Tags="|undergraduate-education|calculus|course-design|technology-in-education|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18236" PostTypeId="2" ParentId="18235" CreationDate="2020-04-27T23:25:50.860" Score="2" Body="&lt;p&gt;&quot;So, i am open to interesting ideas and/or references.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More in the idea realm than the &quot;everything you asked for&quot;, but take a look at petroleum economics.  Especially useful if you are in a petroleum producing country.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ref1:  &lt;a href=&quot;http://investors.clr.com/presentations&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://investors.clr.com/presentations&lt;/a&gt; (open the MAR2018 presentation): pages 15 and 17 show a simple optimization problem.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also make a simple single well economics model and do various optimizations with that.  These are your familiar break even calculations. For instance the trade-off of a fancier well completion versus accelerating or increasing the EUR.  The one nice feature is that it's not just increasing revenue.  The acceleration impact also can be very important in improving NPV.  Shale well models are actually done with months, not just with the year convention of most NPV models.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I would just build the well NPV model yourself (simpler).  But if you want some references (some overly detailed), see here:  &lt;a href=&quot;https://www.google.com/search?q=oil+well+NPV+model+excel&amp;amp;hl=en&amp;amp;sxsrf=ALeKk03E2qKbEjB_q-5qZ_tTpxPMjsOtuA:1588029457546&amp;amp;source=lnms&amp;amp;tbm=vid&amp;amp;sa=X&amp;amp;ved=2ahUKEwixxtbd3onpAhUHg3IEHddiCbAQ_AUoAnoECAsQBA&amp;amp;biw=1920&amp;amp;bih=966&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=oil+well+NPV+model+excel&amp;amp;hl=en&amp;amp;sxsrf=ALeKk03E2qKbEjB_q-5qZ_tTpxPMjsOtuA:1588029457546&amp;amp;source=lnms&amp;amp;tbm=vid&amp;amp;sa=X&amp;amp;ved=2ahUKEwixxtbd3onpAhUHg3IEHddiCbAQ_AUoAnoECAsQBA&amp;amp;biw=1920&amp;amp;bih=966&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm really not a fan of CAS and worry that you're combining too much here (weaker students, basic calculus concepts, a project, and business insights...and then ALSO CAS).  But if you must needs involve it, some of the more complicated decline curves in modified Arps models, are probably fun to play with with CAS.  Also, you're doing integration when looking at EUR.  And that's a concept you wanted to bring into the course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=blvorGxNtxM&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=blvorGxNtxM&lt;/a&gt; is a decent overview.  But there are a lot of other articles and videos out there that give you an overview of this topic.&lt;/p&gt;&#xA;" OwnerUserId="13897" LastActivityDate="2020-04-27T23:25:50.860" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18237" PostTypeId="1" AcceptedAnswerId="18243" CreationDate="2020-04-28T03:11:30.800" Score="9" ViewCount="550" Body="&lt;p&gt;Does anyone know a good geometrical representation of the fact that &lt;span class=&quot;math-container&quot;&gt;$\DeclareMathOperator{lcm}{lcm}\DeclareMathOperator{gcf}{gcf}\lcm(a,b) \gcf(a,b) = ab$&lt;/span&gt;? Because &lt;span class=&quot;math-container&quot;&gt;$\lcm$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\gcf$&lt;/span&gt; are abstract concepts, it's often difficult to provide insight to the geometry-inclined student and one can easily get lost in mathematical paths that are not familiar. As I see, defining &lt;span class=&quot;math-container&quot;&gt;$\lcm(a,b) = \min (\langle a \rangle \cap \langle b \rangle)$&lt;/span&gt; is not enough for the formation of the idea around &lt;span class=&quot;math-container&quot;&gt;$\lcm$&lt;/span&gt;, which should include the concept of the &lt;em&gt;join&lt;/em&gt; for &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; in the lattice &lt;span class=&quot;math-container&quot;&gt;$(\mathbb{N},|)$&lt;/span&gt;, as oppose to &lt;span class=&quot;math-container&quot;&gt;$\gcd(a,b)$&lt;/span&gt;, which is their &lt;em&gt;meet&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me give one (poor) example: If a rectangle has sides &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\gcf(a,b)$&lt;/span&gt; is the size of the largest square that can evenly cover the rectangle and, counting the number of squares and multiplying by the length of size of the square, one gets the result. Nevertheless, this is not a geometrical operation, this multiplication does not represents clearly any area or segment, and so it's a poor illustration of the kind of appropriate visualization I seek.&lt;/p&gt;&#xA;" OwnerUserId="2206" LastEditorUserId="8571" LastEditDate="2020-04-28T14:13:12.900" LastActivityDate="2020-09-11T19:26:39.383" Title="Geometrical interpretation of the identity $\operatorname{lcm}(a,b) \operatorname{gcf}(a,b) = ab$" Tags="|number-theory|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18238" PostTypeId="2" ParentId="18237" CreationDate="2020-04-28T04:19:20.477" Score="3" Body="&lt;p&gt;I'll hazard a probably uninteresting answer. Here's an example:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/VNJhv.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VNJhv.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's run through how this goes in general. Suppose &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; are positive integers such that they share prime-power divisors &lt;span class=&quot;math-container&quot;&gt;$p_1^{r_1},\dots, p_m^{r_m}$&lt;/span&gt; (we assume each prime power is as large as is possible). Furthermore, suppose&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ a = p_1^{r_1+s_1}\cdots p_m^{r_m+s_m}a' \qquad \&amp;amp; \qquad &#xA;   b = p_1^{r_1+t_1}\cdots p_m^{r_m+t_m}b'&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$a',b'$&lt;/span&gt; are not divided by any of the primes &lt;span class=&quot;math-container&quot;&gt;$p_1,\dots , p_m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s_1,\dots, s_m,t_1, \dots, t_m \geq 0$&lt;/span&gt;. Notice that &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ gcd(a,b) = p_1^{r_1}\cdots p_m^{r_m} $$&lt;/span&gt;&#xA;Also,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ a = p_1^{r_1}\cdots p_m^{r_m}p_1^{s_1}\cdots p_m^{s_m}a' \qquad \&amp;amp; \qquad &#xA;   b = p_1^{r_1}\cdots p_m^{r_m}p_1^{t_1}\cdots p_m^{t_m}b'&#xA;$$&lt;/span&gt;&#xA;The least common multiple of &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; needs to carry all the prime powers in the factorizations of &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt;. However, the lcm should not carry more prime power factors than are needed. In particular, we see &lt;span class=&quot;math-container&quot;&gt;$p_1^{r_1}\cdots p_m^{r_m}$&lt;/span&gt; only needs to appear once. In my current notation we find:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;lcm(a,b) &amp;amp;= p_1^{r_1}\cdots p_m^{r_m}p_1^{s_1}\cdots p_m^{s_m}a'p_1^{t_1}\cdots p_m^{t_m}b'  \\&#xA;&amp;amp;= \left(p_1^{r_1}\cdots p_m^{r_m}p_1^{s_1}\cdots p_m^{s_m}a' \right) \left(p_1^{t_1}\cdots p_m^{t_m}b' \right) \\&#xA;&amp;amp;= a \left( p_1^{t_1}\cdots p_m^{t_m}b' \right) \\&#xA;&amp;amp;= \left(p_1^{s_1}\cdots p_m^{s_m}a'\right)\left(p_1^{r_1}\cdots p_m^{r_m}p_1^{t_1}\cdots p_m^{t_m}b'\right)  \\&#xA;&amp;amp;= \left(p_1^{s_1}\cdots p_m^{s_m}a'\right)b.&#xA;\end{align}&lt;/span&gt;&#xA;it is clear from the above calculation that &lt;span class=&quot;math-container&quot;&gt;$p_1^{r_1}\cdots p_m^{r_m}p_1^{s_1}\cdots p_m^{s_m}a'p_1^{t_1}\cdots p_m^{t_m}b'$&lt;/span&gt; is a multiple of both &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. Minimality requires further analysis which I omit here. In any event, it is clear that:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;ab &amp;amp;= \left( p_1^{r_1}\cdots p_m^{r_m}p_1^{s_1}\cdots p_m^{s_m}a' \right)\left( p_1^{r_1}\cdots p_m^{r_m}p_1^{t_1}\cdots p_m^{t_m}b' \right) \\&#xA;&amp;amp;= \left( p_1^{r_1}\cdots p_m^{r_m}\right)\left(p_1^{r_1}\cdots p_m^{r_m}p_1^{s_1}\cdots p_m^{s_m}a'p_1^{t_1}\cdots p_m^{t_m}b' \right) \\&#xA;&amp;amp;= gcd(a,b)lcm(a,b).&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Geometrically: if we take an &lt;span class=&quot;math-container&quot;&gt;$a \times b$&lt;/span&gt; rectangle and form another rectangle of the same area whose height is &lt;span class=&quot;math-container&quot;&gt;$gcd(a,b)$&lt;/span&gt; then the length of the squished rectangle is exactly the &lt;span class=&quot;math-container&quot;&gt;$lcm(a,b)$&lt;/span&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-04-28T04:19:20.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18239" PostTypeId="2" ParentId="18237" CreationDate="2020-04-28T13:36:38.800" Score="6" Body="&lt;p&gt;The OP describes a geometrical illustration using the fact that the GCD is the side of the greatest square that tiles a &lt;span class=&quot;math-container&quot;&gt;$A \times B$&lt;/span&gt; rectangle.&#xA;I don't understand why the following is not a &quot;geometrical operation.&quot; But perhaps it is slightly different than what the OP had in mind, since he claimed &quot;this multiplication does not represent...any area...,&quot; but it does below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SYryB.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SYryB.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can even find the GCD square with Euclid's algorithm using a compass and straightedge.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2020-04-28T13:36:38.800" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18240" PostTypeId="1" CreationDate="2020-04-28T16:07:18.677" Score="4" ViewCount="191" Body="&lt;p&gt;For an online assignment application where you can configure different versions of a single assignment/test to have different questions, should it be possible to also have the different versions have different number of questions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Details:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have an online assignment/test application that lets you make assignments with different versions. These versions can have different numeric values in the questions and can even have different questions between them (chosen from a common pool of questions for that assignment).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it also reasonable to have different numbers of questions in the different versions? Or should the number of questions be enforced to be the same across all versions of a particular assignment/test?&lt;/p&gt;&#xA;" OwnerUserId="13814" LastActivityDate="2020-04-29T14:30:41.687" Title="For an online assignment application, should different versions of the same assignment be allowed to have different numbers of questions?" Tags="|assessment|online-instruction|online|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18241" PostTypeId="2" ParentId="18240" CreationDate="2020-04-28T19:08:27.487" Score="6" Body="&lt;p&gt;When I write an assessment, I am trying to understand how well the students have mastered some particular body of material.  In principle, a &quot;perfect designer&quot; could write multiple assessments with wildly different questions or numbers of questions, and consistently obtain valid results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I am not a perfect designer&amp;mdash;nor, frankly, is anyone else.  Therefore when I design assessments, one of the things that I do in order to ensure that the assessment will be consistent (at least) is to ask every student the same questions.  If there is some systemic flaw in my design, at least it will have the same effect on all of the students, and can be (hopefully) corrected for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there are other practical concerns which sometimes compel instructors to write multiple assessments for the same material.  The most common reason, I suspect, is to prevent cheating, and it is my assumption that this is the rationale which motivates the question.  In this situation, the usual practice is to write multiple versions of the same assessment so that students cannot simply copy off of each other.  In this setting, you have already given up on 100% consistency.  However, there are still things which can be made to reduce the chances of errors biasing the assessment in favor of (or against) students who take any particular version of the assessment.  The chief technique is to write one exam, then modify questions by, for example, changing some of the constants.  Asking completely different questions may also prevent cheating, but it runs the risk of interfering with the consistency of the assessment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, after all of that, my short answer is &lt;strong&gt;no&lt;/strong&gt;, I do not think that one should create different versions of assessments with variable numbers of questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, I think that there are questions which one can ask to arrive at an answer of their own:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;What is the goal of the assessment?&lt;/strong&gt; Is it formative (designed to check in on student progress toward understanding) or summative (designed to determine whether or not the students have mastered the material and assign a grade)?  If the assessment is summative, then I would suggest that consistency is much more important&amp;mdash;the stakes are higher for the students, and the risk of constructing an inconsistent assessment is commensurably higher.  On the other hand, if the assessment is summative, it might make sense to give wildly different assessments, on the principle that you are polling or sampling the class in some way, and that the students taking one version of the exam are representative of the whole class.  This might be a good way to catch errors in teaching without making any one student feel overwhelmed.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;What is the goal of writing multiple versions of the assessment?&lt;/strong&gt;  If the goal is to prevent cheating, then it seems to me that consistency is desirable.  As such, it seem like poor practice to give wildly different versions of an assessment to different students.  On the other hand, maybe the assessment is something that students will be building on in the future in some kind of group activity, e.g. you give two very different assessments, then ask students to work in groups to share solutions to these different versions.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;What is lost by writing multiple versions?&lt;/strong&gt;  This relates some to the first two questions. How much time are you going to lose (both writing the assessment and grading it later)?  Will you lose consistency?  validity?  Are you potentially introducing errors in one version of the assessment and not the other? do these errors matter?  etc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;What is gained by writing multiple versions?&lt;/strong&gt;  Again, consider this in the context of the answers to the first two questions.  Will you really prevent any cheating?  Are you really concerned that cheating is going to take place (in a precalculus class where most of the students are non-majors and are only taking the class because they are required to, cheating might be a big problem; in an upper division class for mathematics majors, cheating is likely much less prevalent).  Are you exposing students to a wider range of material?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2020-04-29T12:40:31.773" LastActivityDate="2020-04-29T12:40:31.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18243" PostTypeId="2" ParentId="18237" CreationDate="2020-04-29T01:30:29.507" Score="9" Body="&lt;p&gt;&#xA;I don't see anything wrong with what you were trying to say. In fact, I think that is a good way of saying it. Let's draw some pictures with some concrete numbers. Let's take &lt;span class=&quot;math-container&quot;&gt;$a=6$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b=9$&lt;/span&gt;. Then we can draw an &lt;span class=&quot;math-container&quot;&gt;$a\times b$&lt;/span&gt; grid as shown below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/x5Z47.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/x5Z47.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&#xA;&#xA;&lt;p&gt;Now as you said we can show the greatest common divisor &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is the side length of the largest square that can evenly cover the rectangle. This is shown in the following picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gDnF2.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gDnF2.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&#xA;Below, the area of the black region, which contains only the diagonal of each &lt;span class=&quot;math-container&quot;&gt;$d \times d$&lt;/span&gt; subsquare and is therefore &lt;span class=&quot;math-container&quot;&gt;$1/d$&lt;/span&gt; of the total area, i.e., &lt;span class=&quot;math-container&quot;&gt;$ab/d$&lt;/span&gt;, illustrates the least common multiple.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ll0bw.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ll0bw.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&#xA;&#xA;&lt;p&gt;That the black area is a multiple of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; can be seen from the fact that they comprise an integer number of rows and columns as seen in the following gifs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/aQdQH.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aQdQH.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/8of2A.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8of2A.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These pictures are only an illustration and do not prove that the quantities shown are in fact the greatest common divisor and least common multiple.&lt;/p&gt;&#xA;" OwnerUserId="7479" LastActivityDate="2020-04-29T01:30:29.507" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18244" PostTypeId="2" ParentId="18240" CreationDate="2020-04-29T12:26:03.033" Score="1" Body="&lt;p&gt;I took the computerized GRE 25 years ago and it had different numbers of questions because it used an algorithm to hone in on how strong you were.  Basically it would keep pecking at you until it honed in on a score, then stop.  So, sure &quot;it's possible&quot;.  Even &quot;has been done&quot;.  And a generation ago.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[But I'm sure a lit search (gentle hint/nudge) would show examples of variable length testing from much older dates, even pre-computers.  It's not like it's something that is hard to imagine.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But like Xander, I think you are not asking the question properly if you don't describe more what is the purpose of your application (for assessment testing, grading, learning or some mix of those).  As this may influence how you decide to construct the thing.&lt;/p&gt;&#xA;" OwnerUserId="13897" LastEditorUserId="13897" LastEditDate="2020-04-29T14:30:41.687" LastActivityDate="2020-04-29T14:30:41.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18246" PostTypeId="1" CreationDate="2020-04-29T19:04:21.653" Score="17" ViewCount="4624" Body="&lt;p&gt;What kind of exposure to complex numbers can you expect in mathematics majors at American colleges?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I teach at a very large public university. It occurred to me that it is possible to graduate in their math majors with the exposure to complex numbers being exactly two lessons in calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems a bit odd to me as far as math majors are concerned. For example, complex eigenvalues are never mentioned at my university.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this normal by US standards? &lt;/p&gt;&#xA;" OwnerUserId="8895" LastEditorUserId="376" LastEditDate="2020-04-30T06:00:35.137" LastActivityDate="2020-06-17T03:01:24.140" Title="(How) Do American undergraduate math programs teach complex numbers?" Tags="|undergraduate-education|usa|complex-numbers|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18247" PostTypeId="2" ParentId="18246" CreationDate="2020-04-29T21:00:44.417" Score="11" Body="&lt;p&gt;At my institution we do not discuss complex numbers at all in calculus, but we assume that students are somehow familiar with them in differential equations and linear algebra (to analyze real-linear mappings with complex eigenvalues).  A study of single variable holomorphic functions is an upper level course, and most math majors will graduate without ever meeting a holomorphic function.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-04-29T21:00:44.417" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18248" PostTypeId="2" ParentId="18246" CreationDate="2020-04-29T22:43:55.513" Score="6" Body="&lt;p&gt;The basics of complex numbers are really covered in high school algebra.  (Basics means easy/simple/prioritized, though.)  Complex numbers come up occasionally in other topics (as needed, often real numbers is all that is needed).  But obviously roots of the second order linear homo ODE is one area in math.  As is reactance versus resistance in physics and EE (ELI the ICE man).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess in a way it is sad that math majors don't have a required unit in complex analysis, but you have to realize it is a BS degree, not a grad degree.  And there needs to be some space for kids that are NOT going to go to math grad school.  A LOT OF SPACE for kids like that.  They are more the norm than the exception.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And we can clearly think of other topics that in some way feel like they should be required (e.g. a PDE class), but that are not a strict requirement, but an elective instead.  But as always with these sorts of questions on MESE, what I find lacking is people saying what they will get rid of, when they consider to add a requirement.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I look at the math courses at USNA, which of the required math classes should I get rid of if I make complex analysis required?  The one baby real analysis class?  Probability and statistics?  I don't think so.  I think a math major lacking a basic prob/stat course is a bigger gap than missing complex analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMA.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMA.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But realistically a lot of them will probably select a complex analysis as one of their electives anyhow.  Not all, but many.  And let some freedom exist also.  Not everything needs to be required.&lt;/p&gt;&#xA;" OwnerUserId="13897" LastActivityDate="2020-04-29T22:43:55.513" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18250" PostTypeId="2" ParentId="18246" CreationDate="2020-04-30T01:12:56.173" Score="28" Body="&lt;p&gt;The following is anecdotal, but based on experience as a student and instructor in American high schools in three states, as well as my undergraduate and graduate experiences.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;High School:&lt;/strong&gt;  In the high school curriculum, complex numbers first appear in algebra courses.  The usual curriculum first introduces complex numbers as a way of interpreting the quadratic formula when the discriminant is negative, i.e. as a way of giving meaning to the result&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ ax^2 + bx + c = 0&#xA;\iff x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a} $$&lt;/span&gt;&#xA;in the case that &lt;span class=&quot;math-container&quot;&gt;$b^2 - 4ac &amp;lt; 0$&lt;/span&gt;.  Good instructors will interpret this graphically, and note how the graph sits above or below the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis if the discriminant is negative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps motivated by this, there is typically a short chapter or section (1&amp;ndash;2 weeks of lecture, maybe) on the complex numbers.  This might consists of a discussion of how complex numbers may be added, subtracted, multiplied, and divided, but likely not much more than this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also worth noting that the complex numbers are often treated as some kind of mysterious, vague, not really useful or &quot;real&quot; system.  They are a &quot;trick&quot; used to solve certain problems, but are only really useful as a trick.  This is, of course, incorrect, but is reflective of the fact that most high school teachers in the US have not taken much mathematics beyond calculus (e.g. it is &lt;em&gt;possible&lt;/em&gt; to teach high school mathematics with two semesters of calculus, a handful of watered-down upper division courses in mathematics, and not much else).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The high school curriculum usually tops out in a course titled &quot;Precalculus&quot; (or something similar).  This course is typically a hodge-podge of mathematical concepts:  real functions and the associated notation, some discussion of limits and continuity, basic trigonometry, exponential and logarithmic functions, simple linear algebra (e.g. vector and matrix operations), and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The complex number system sometimes comes up again in this class, often in the context of vectors.  After a discussion of trigonometric functions, vectors in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; are introduced, and students are shown how to give vectors in their polar form.  At this point, you might see the complex plane introduced, and the students might be given a few exercises converting back and forth between the Cartesian and polar forms of a complex number, i.e.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ z = a + ib = r \big(\cos(\theta) + i\sin(\theta)\big). $$&lt;/span&gt;&#xA;Some instructors might go a little deeper than this, but I think that such discussion is uncommon.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Lower Division College Classes:&lt;/strong&gt;  The complex numbers typically don't come up very much in lower division college classes, which include precalculus (which is generally much the same as the high school course, but with an accelerated calendar), calculus (of one and several variables), linear algebra, and differential equations.  Some instructors might discuss the complex numbers in a differential equations class, in the context of complex roots of the characteristic polynomial.  However, the discussion is usually elided, and students are instructed to write their solutions in terms of trigonometric functions.  For example, the differential equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ u'' + u = 0 $$&lt;/span&gt;&#xA;has characteristic polynomial &lt;span class=&quot;math-container&quot;&gt;$r^2 + 1 = 0$&lt;/span&gt;, which has roots &lt;span class=&quot;math-container&quot;&gt;$\pm i$&lt;/span&gt;.  Hence a general solution to this equation is of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ u(t) = C_1 \mathrm{e}^{it} + C_2 \mathrm{e}^{-it}, $$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$C_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C_2$&lt;/span&gt; are arbitrary constants (which will ultimately depend on some initial condition).  However, via the principle of superposition (linear combinations of solutions are also solutions), the existence of a minimal spanning set of solutions, and the facts that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \cos(t) = \frac{\mathrm{e}^{it} + \mathrm{e}^{-it}}{2}&#xA;\qquad\text{and}\qquad&#xA;\sin(t) = \frac{\mathrm{e}^{it} - \mathrm{e}^{-it}}{2i}, $$&lt;/span&gt;&#xA;it is possible to rewrite the general solution as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ u(t) = k_1 \cos(t) + k_2 \sin(t), $$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$k_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$k_2$&lt;/span&gt; are constants.  Again, it is normal to spend very little time on the complex analysis, and instructors often assume that students already know the materials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The complex numbers might also come up in linear algebra in the context of complex eigenvalues.  However, again, the discussion is often very brief (if it exists), and most of the examples students are given involve only real eigenvalues.  As the main goal of lower division linear algebra is often to get to the diagonalization of a square matrix (rather than the Jordan canonical form, or, better yet, singular value decomposition), complex eigenvalues are often ignored as something too difficult to deal with.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Upper Division College Courses:&lt;/strong&gt;  There are upper division classes where complex numbers become a major focus of the course or are important examples:  complex analysis (duh?), Fourier and/or harmonic analysis, proofs-driven courses in differential equations and linear algebra, abstract algebra, etc.  However, these courses are typically elective courses for mathematics majors.  The general assumption in these courses is that the students are (1) bound for graduate school, probably in mathematics or some closely related field, (2) already familiar with the complex number system from previous courses (which courses? ¯\_(ツ)_/¯ ), and (3) relatively mathematically mature and therefore capable of filling in gaps in their knowledge on their own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, upper division courses in which a more comprehensive treatment of the complex number system might appear (e.g. complex analysis or differential equations) are typically electives, and not required even for mathematics majors.  It is therefore &lt;em&gt;very&lt;/em&gt; common for students to complete an undergraduate degree in mathematics knowing next to nothing about the complex numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In short, the answer to the original question is &quot;No, based on my (anecdotal) experience, it is not at all uncommon for US institutions to graduate mathematics majors who have had very little experience with the complex number system.&quot;&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2020-04-30T01:12:56.173" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18251" PostTypeId="2" ParentId="18246" CreationDate="2020-04-30T02:07:20.090" Score="8" Body="&lt;p&gt;I teach physics and occasionally a little math at a community college in California. My students come from all over the place, so I think their exposure to math is somewhat of a representative sample of what kids in the US learn in high school and the first couple of years of college. In our three-semester physics survey course for STEM majors, I review complex numbers with my students in both the second semester electricity and magnetism class (so they can use them for complex impedances and such) and in the third semester (for use in the baby quantum mechanics that we do).&lt;/p&gt;&#xA;&lt;p&gt;I say &amp;quot;review&amp;quot; because I have never seen a student who had had no previous exposure to the topic. I think they must see it in 9th grade algebra when they learn the quadratic formula. Also, the catalog description for my school's trigonometry class says that it includes de Moivre's theorem, clearly implying that they are already expected to have seen complex numbers before that point.&lt;/p&gt;&#xA;&lt;p&gt;However, their exposure seems always to have been very shallow and poorly motivated. They do not recognize notations for magnitude and argument, and they don't understand the interpretation of multiplication and division in terms of the polar representation. A calculation like &lt;span class=&quot;math-container&quot;&gt;$\operatorname{arg}(1/(1+i))$&lt;/span&gt; takes them a long time to figure out. They tend to cling to the cartesian representation even when it's the wrong tool for the job.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I teach at a very large public university. It occurred to me that it is possible to graduate in their math majors with the exposure to complex numbers being exactly two lessons in calculus.&lt;/p&gt;&#xA;&lt;p&gt;This seems a bit odd to me as far as math majors are concerned. For example, complex eigenvalues are never mentioned at my university.&lt;/p&gt;&#xA;&lt;p&gt;Is this normal by US standards?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So no, in my experience this does not seem normal or even possible here in California, because the state tries to make every kid take algebra in 9th grade, and they get some basic exposure to complex numbers there.&lt;/p&gt;&#xA;&lt;p&gt;It also seems weird to me that students could graduate as math majors without ever seeing complex eigenvalues. This could not have happened at my undergrad school (UC Berkeley). I still have my textbook from my lower-division linear algebra course there, and it barely mentions complex eigenvalues. However, a math major there is required to take a core of four upper-division required courses, which are analysis, linear algebra, abstract algebra, and complex analysis. Three out of four of these deal  with complex numbers, and the upper-division lin alg course would certainly cover complex eigenvalues. Maybe there are vast differences between more and less selective schools in the US in terms of their requirements for an undergraduate math degree.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-04-30T02:23:41.087" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18253" PostTypeId="1" CreationDate="2020-04-30T22:16:08.427" Score="6" ViewCount="2537" Body="&lt;p&gt;I am currently a 15-year-old in the 9th grade, and I am studying to test out all of my high school math classes, which is something I didn't realize I could do in the 8th grade and would have if I had. I contacted faculty, and if all goes well, I'll test out of Geometry, Algebra II, Precalculus, and AP Calculus BC. I also hope that once at college, I can test out of Linear Algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have also been studying Japanese for a few years, and know the meaning of all the required Jōyō Kanji, although I lack vocabulary otherwise. Because of this background, I plan to Major in Pure Mathematics, and Minor in Japanese Studies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I end up taking classes at Delta, (a nearby community college) I would not be able to take Topology, Complex Analysis, Real Analysis, or Abstract Algebra. Instead, I would take Multivariable Calculus, Linear Algebra (if I fail to test out of it), and an ODE course, as Delta only has a Transfer Math program, and no Japanese language program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been told that my high school will also pay for my college classes while enrolled in high school, although the only people they have ever done this for went to Delta during high school, and they were seniors in the 12th grade. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That being said, I have a few general questions about college that fall under the umbrella of the title.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Would it make sense to apply for colleges as anything other than a Senior, or duel enroll in non-community colleges?&lt;/li&gt;&#xA;&lt;li&gt;What will writing a thesis be like for someone with no Math paper writing experience?&lt;/li&gt;&#xA;&lt;li&gt;I have below-average grades in everything but Math, how much will this affect me?&lt;/li&gt;&#xA;&lt;li&gt;As a high school freshman, I've yet to take the SAT, how much will this affect me?&lt;/li&gt;&#xA;&lt;li&gt;For the math people reading this, how do you recommend studying math on my own?&lt;/li&gt;&#xA;&lt;li&gt;My mother doesn't want me to test out of my math classes, because &quot;they teach me hard work,&quot; and because she wants me to raise my other grades first before doing something &quot;fun&quot;. How do I deal with that?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Calc 3, ODEs, and LA are all solid classes to take. Nothing wrong with them.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am currently studying these (and enjoying them, no less!) but the main reason these deterred me was that there are many online resources to self-study these with, whereas there are significantly fewer resources for the more pure subjects.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This post was moved from Academia, as Undergraduate admissions are not allowed there. However, the mods who closed my question directed me here. I wasn't sure if it would be appropriate, but because this site allows posts about &quot;the process of learning mathematics,&quot; I thought it worth a try.&lt;/p&gt;&#xA;" OwnerUserId="13911" LastEditorUserId="-1" LastEditDate="2020-05-01T17:03:10.407" LastActivityDate="2020-05-02T00:33:21.417" Title="What will my academic path look like after testing out of high school math classes?" Tags="|undergraduate-education|self-learning|usa|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18254" PostTypeId="2" ParentId="18253" CreationDate="2020-04-30T22:42:57.737" Score="6" Body="&lt;p&gt;Calc 3, ODEs, and LA are all solid classes to take.  Nothing wrong with them.  Completely useful even if you move into pure math.  For one thing a decent chance you may teach them!  Also, it is good to build your manipulative skills.  Finally, it keeps options open in case, you end up going into other fields (as those are common prereqs for most STEM majors).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Get the rest of your grades up.  Colleges care about that stuff.  It shows work ethic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take the SATs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may end up going to college early, although I wouldn't go immediately.  A little physical maturing wouldn't hurt you either.  Huge difference of 16 yo versus 14 yo. &lt;/p&gt;&#xA;" OwnerUserId="13912" LastActivityDate="2020-04-30T22:42:57.737" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18255" PostTypeId="2" ParentId="18253" CreationDate="2020-04-30T23:50:59.423" Score="10" Body="&lt;p&gt;So you dual enroll at the community college while you're in high school, and you finish up in two more years. While you're at it, consider some computer science courses. Also, physics is a lovely course. (I was put into chemistry and hated it. I would expect almost anyone who loves math to at least like physics.) Do you have a high school counselor you trust for non-math recommendations? If you figure out which teachers you'd like best, I bet you could do well in other courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For other higher-level math courses before you leave high school, consider Group Theory and Number Theory at &lt;a href=&quot;https://artofproblemsolving.com/school/catalog&quot; rel=&quot;noreferrer&quot;&gt;Art of Problem Solving, which offers these classes online&lt;/a&gt;. Art of Problem Solving also has materials for studying on your own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In two years, you should be ready for the college of your choice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Note to mom: People work hardest at what they love. I am a community college teacher, and I've published a book. I'm also a single mom. I have never gotten good at doing the things I don't like to do. I do grade my students' homework, which is my least favorite task. If your son is this advanced in math, he's been working hard at it. He will slowly get better over his lifetime at doing the parts he doesn't like to do. If he's passing his other courses, and excelling at math, that sounds reasonable to me.]&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-04-30T23:50:59.423" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18256" PostTypeId="2" ParentId="18253" CreationDate="2020-05-01T22:48:30.690" Score="3" Body="&lt;h1&gt;The Color of Grass&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;My guess is that your excitement to race through school is partly due to a low valuation of your current high school experience, including boredom and social isolation.  College and the real world surely look like places where &quot;the grass is definitely greener.&quot;  It's possible that you will never value other human relationships as highly as the average person, and thus, racing through the phases of life is an optimal strategy for you.  But I would bet that at some point, you &lt;em&gt;will&lt;/em&gt; value those relationships much more than you do now, and find yourself in an awkward position.  You see, high school and college &lt;em&gt;are&lt;/em&gt; stressful and chaotic times, socially speaking.  But they are also essential to building your social skills, because you will likely never be in a situation where you have so much close contact with people your age.  Once you get into the working world, the spread of ages increases, and the level of social connection decreases.  It gets harder to meet people the older you get.  If you compress this time by 50% or more, you will likely look back on it one day and think: &quot;Perhaps I should have spent more time meeting people and socializing back then.  There were so many opportunities.&quot;&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Completeness&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;It's easy to focus on the things we're good at, and avoid the things we are not.  That is human nature.  But if I had to guess, your mom is quite possibly worried about the scenario I'm describing above, but not explaining it as clearly as she could.  Can you survive as a pure math prodigy?  Well, there's &lt;a href=&quot;https://en.wikipedia.org/wiki/Grigori_Perelman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grigori Perelman&lt;/a&gt; (I did enjoy his biography).  But unless you want to become a hermit, you will need to become somewhat well-rounded.  At the very least, you will need to be able to communicate clearly with others, especially in written form.  And if you choose to go into industry rather than academia, you will almost certainly need to learn some application of mathematics.  I strongly second Sue's suggestion to study some Computer Science.  You may find that programming suits your personality well (and gives you a lot of career choices later on).  Odds are, no matter what path you choose, having some programming skill will be useful to indispensable in the future, so this effort will certainly reward dividends.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's good that you mentioned the SAT.  Obviously, having poor grades in non-math subjects does not bode well for potential scores.  You don't need a 1600 to get into good schools, but performing poorly on the reading/writing section will not work in your favor.  As I said before, communication will be an essential skill for the rest of your life.  Spending an adequate effort developing this skill by reading broadly and practicing your writing is an investment that will only yield strong positive returns.  There is nothing more frustrating than having a brilliant idea, but not being able to explain it to others.  If you're slacking in your English classes, just pretend that English is a somewhat broken formal language, and use computer science skills to turn it into a math problem.  That might help with your motivation.  But reading outside of math is also important to become a complete human being.  It's good to go deep (learning as much about one topic as possible), but it's also important to go broad (learning a little bit about as many topics as possible).  Yes, you can do both; and there is no right answer about the optimal balance.  But you will find that the most successful people in any field (including pure mathematics) almost all have a broad range of interests and pursuits, even if they have gone very deeply into one of them.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Learning&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;I find that the best way to learn something new is to pick a goal and work towards it.  Try to find some concept that interests you, and learn everything you need to understand the state of the art in that area.  Sometimes you will encounter an enticing side trail, which you should follow.  Sometimes, you will discover an even more interesting problem that you didn't know about when you started, and you will shift your focus.  The important point is keeping yourself motivated.  If you're getting stuck on a concept, it means you have gone too far, and need to go back and study the fundamentals.  Nailing the fundamentals is essential in every field and skill.  But practicing them for their own sake is not terribly fun.  That's why it's helpful to &quot;go deep&quot; first, and find a problem that motivates the need for stronger fundamentals.  This then gives you the motivation to back and learn something better that may have seemed boring at first glance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where to get ideas?  Well, StackExchange is a pretty decent place to start.  Just browse random questions on the Mathematics SE (which I assume you already do), and look for ones where the answer involves concepts you haven't encountered yet.  Chase the ones which sound most interesting, and your plate will be full in no time.  It's not unusual to find math-heavy questions on Physics or Computer Science or [Electrical] Engineering or Aviation, so it's worthwhile checking those out as well (and helps with the &quot;going broad&quot; strategy).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the reasons that going broad is such a valuable strategy is in the nature of learning itself.  Ultimately, the abstract reasoning which separates humans from most other animals comes down to generalizing across distinct ideas.  It's about finding connections.  It's about looking at an apple and a tennis ball and a rock and thinking about what makes them different, but also what makes them the same.  Your brain will do this for you all on its own, as long as you feed it lots of data.  And the more diverse the data, the more connections you will discover.  Many fruitful ideas come from cross-field collaborations, where people studying ostensibly unrelated topics end up finding an unexpected connection that helps them understand better.  It seems hard to believe that your pure mathematics skills could be improved by studying biology or aviation or economics, until you realize that partial differential equations well describe many phenomena in all those fields, and having physical instances to visualize and reason over strengthens your mathematical intuition about the equations themselves.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Conclusion&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;If I were you, I'd make this deal with your mom:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mom, I'm learning math much faster than my high school classes can possibly teach me.  Sitting through them is a waste of my time when I could be productively learning maths several years ahead.  On the other hand, I recognize that I have not been as diligent in my other subjects as I could be, but that these subjects will be more important in the future than I have given them credit for.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Thus, I propose this bargain: you let me test out of all the math I can and take whatever college math courses are available to me, and I will raise all my other grades to at least a B.  If, at any point, my grades fall below a B, I will withdraw from my college math courses.  Is that fair?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Finally, I would recommend that you try to socialize with a broad distribution of your high school peers.  You surely find that you have little in common with many of them, but learning the strengths and shortcomings of your fellow humans and members of society is most certainly a skill that you will use for the rest of your life; and the better you are at it, the happier you will be, even if you end up cloistered in an ivory tower inscribed with Greek letters.  That includes playing a sport, if you don't already.  &quot;Balance&quot; is about developing the body as well as the mind (and physical fitness improves your brain function).  Team sports are especially good, because few activities teach you how to cooperate with others (including exercising and following leadership) than playing on a team.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;IMO, your best strategy is to learn as much math as you can, anywhere you can, while still spending 4 years in high school and 4 years in an undergrad program where you can fully develop your non-math skills and exploit the wealth of social opportunities that you will almost certainly never again encounter in later life.  Good luck!&lt;/p&gt;&#xA;" OwnerUserId="13468" LastActivityDate="2020-05-01T22:48:30.690" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18257" PostTypeId="2" ParentId="18253" CreationDate="2020-05-02T00:33:21.417" Score="0" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Would it make sense to apply for colleges as anything other than a Senior, or duel enroll in non-community colleges?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;yeah you can definitely look into applying for colleges a year early. I've heard of it and I think some people had good experiences doing it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for dual-enrolling in non-community colleges: what do you mean by dual enroll?  You could definitely take classes at non-community colleges, but you probably don't have to go through an application process for it.  See if your high school can help set it up.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For the math people reading this, how do you recommend studying math on my own?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Not a math person (CS), but you should do shit that you like.  Do you like math competitions, like math olympiads?  If so, do those.  They're nice in that a lot of high school+ math is about learning procedures to solve math problems, but competitions and puzzles may involve more creative thinking, which is closer to what the fun part of math is: figuring out how to solve new problems.  &lt;/p&gt;&#xA;" OwnerUserId="13924" LastActivityDate="2020-05-02T00:33:21.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18259" PostTypeId="1" CreationDate="2020-05-03T18:40:44.490" Score="20" ViewCount="3868" Body="&lt;p&gt;I'm 20 and I never learned math properly in my highschool. I found my interest in math last year and I've been searching for books since then.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried Khan academy but I have difficulty paying attention to videos. I prefer reading over video courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how do I search for proper math books for self-study? I've tried GoodReads but unable to get any recommendations over there for school math.&lt;/p&gt;&#xA;" OwnerUserId="13932" LastActivityDate="2020-05-04T20:48:12.563" Title="What does it take to find a good math book for self study?" Tags="|textbooks|self-learning|reference-books|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18260" PostTypeId="2" ParentId="18259" CreationDate="2020-05-03T19:38:39.807" Score="4" Body="&lt;p&gt;Try Schaum's Outlines.  They are designed to be review material and drill manuals.  Also, the text tends to be simpler writing.  This is much more efficient for the adult learner and self studier (than the loquacious doorstops that are designed to appeal to professors and textbook selection committees, rather than to the learner).  Check on Amazon and see the reviews.  I prefer the older ones, but the newer ones still decent.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Frank Ayers First Year College Math is an excellent review of all high school math, in one text:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B009NNXQF6&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/First-College-Mathematics-Frank-Ayres/dp/B009NNXQF6&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, you must not &quot;read&quot;.  You must work problems.  But just look at the section and work the drill problems.  Remember studying math is a lot like music or sports.  You need to drill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck:  You may surprise yourself in what comes back and in what knowledge you firm up.&lt;/p&gt;&#xA;" OwnerUserId="13912" LastActivityDate="2020-05-03T19:38:39.807" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18261" PostTypeId="2" ParentId="17710" CreationDate="2020-05-03T19:50:09.337" Score="2" Body="&lt;p&gt;The concept of zero and ten was a huge advancement! Roman numerals did not run out of numerals but it was nearly impossible to compute something like MXVII divided by IXXI. They had to use precomputed books of multiplication and division tables to look up what today is trivial in an exponential based number system.&lt;/p&gt;&#xA;&lt;p&gt;Teaching &lt;strong&gt;another&lt;/strong&gt; number base is an opportunity to teach the eponential principles involved instead of the mechanics that, &lt;em&gt;let's be honest&lt;/em&gt;, most people never move beyond.&lt;/p&gt;&#xA;&lt;p&gt;At the elementary level, base 3 &amp;amp; 6 make for convenient comparisons of repeating base 10 decimals like 1/3. The classic 1 = 0.999... is instantly resolved in base 3 or 6.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Quickly in your head, compute:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Base 7&lt;/strong&gt; (4356.5512) divided by &lt;strong&gt;Base 10&lt;/strong&gt; (49) and provide the answer in &lt;strong&gt;Base 7&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The intent here was to recognize that 49 is is 100 base 7, so one need only shift the period (radix point) two places to the left to get 43.565512&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="10376" LastEditorUserId="10376" LastEditDate="2022-12-24T22:11:19.240" LastActivityDate="2022-12-24T22:11:19.240" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18262" PostTypeId="2" ParentId="18259" CreationDate="2020-05-04T01:17:54.780" Score="8" Body="&lt;p&gt;Permit me to recommend&#xA;the answers to this MO question:&#xA;&lt;a href=&quot;https://mathoverflow.net/q/31879/6094&quot;&gt;Are there other nice math books close to the style of Tristan Needham?&lt;/a&gt;.&#xA;The emphasis here is on readable, interesting expositions,&#xA;a rather different focus from Schaum's Outlines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is &lt;a href=&quot;https://math.stackexchange.com/a/85244/237&quot;&gt;my comment on the Needham book (&lt;em&gt;Visual Complex Analysis&lt;/em&gt;) in particular&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Here is one figure from the book, p.135:&#xA;  &lt;br /&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;img src=&quot;https://i.stack.imgur.com/HpbKO.png&quot; alt=&quot;Figure 13&quot;&gt;&lt;br /&gt;&#xA;  You can almost guess the theorem from the figure:  The two spheres &lt;span class=&quot;math-container&quot;&gt;$S_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$S_2$&lt;/span&gt; are&#xA;  orthogonal iff the two circles &lt;span class=&quot;math-container&quot;&gt;$C_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C_2$&lt;/span&gt; are orthogonal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-05-04T01:17:54.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18263" PostTypeId="2" ParentId="18259" CreationDate="2020-05-04T05:04:18.527" Score="4" Body="&lt;p&gt;Here is a list of four easy-to-read, enjoyable Math books that can get you started on a journey toward a true quantitative understanding and connection with the world, which it sounds like you might be seeking right now.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Math With Bad Drawings, by Ben Orlin&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Secrets of Mental Math, by Arthur Benjamin&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Flatland, by Edwin Abbott&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Factfulness, by Hans Rosling&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These four books, or even just one of them if it is read carefully, can help you develop that basic number sense that a person needs in order to really understand math. When it comes right down to it, basic number sense will get a person really far in math and it's impossible to get very far without that basic number sense. Too many of my students lack it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After reading a selection of these, you can choose specific topics like geometry, statistics, probability, algebra, or calculus, and delve into each topic you choose more deeply by reading and working through &quot;For Dummies&quot; and &quot;Schaums Outlines&quot; books, which present the material in different, sometimes refreshing ways. They usually offer you practice problems to help you firm up what you've learned, too.&lt;/p&gt;&#xA;" OwnerUserId="13034" LastEditorUserId="13034" LastEditDate="2020-05-04T05:12:06.110" LastActivityDate="2020-05-04T05:12:06.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18264" PostTypeId="2" ParentId="18259" CreationDate="2020-05-04T05:15:38.247" Score="8" Body="&lt;p&gt;If you're looking to fill in a deficient high-school education then I might recommend the OpenStax series, which has free, open-resource books for everything from arithmetic through calculus and statistics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://openstax.org/subjects/math&quot; rel=&quot;noreferrer&quot;&gt;https://openstax.org/subjects/math&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-05-04T05:15:38.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18265" PostTypeId="2" ParentId="18259" CreationDate="2020-05-04T06:25:57.500" Score="4" Body="&lt;p&gt;Although you can find some good hints through recommendations, there's nothing like scanning a book shelf (in real life or virtually), taking a peek into a book all by yourself, reading the preface (where the author details whom is the book to), maybe reading the 1st chapter and trying the exercises. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This way you'll find something that's accessible and interesting for you, something that fits into your time constrains and matches your other interests.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schaum's books were mentioned above. But also take a look at other serious publishers like: &lt;strong&gt;McGraw-Hill&lt;/strong&gt;, &lt;strong&gt;Dover&lt;/strong&gt; (for example, Concepts of Modern Mathematics by Ian Stewart), &lt;strong&gt;Cambridge University Press&lt;/strong&gt; (i.e. How to Think Like a Mathematician by Kevin Houston), or &lt;strong&gt;Oxford University Press&lt;/strong&gt; (How to Study as a Mathematics Major by Lara Alcock).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Noitice about Schaum's Outlines, prompted by comments on another answer, bold mine:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Many titles feature noted authors in their respective fields, such as&#xA;  Murray R. Spiegel and Seymour Lipschutz. Originally designed for&#xA;  college-level students as a &lt;strong&gt;supplement to standard&lt;/strong&gt; course &lt;strong&gt;textbooks&lt;/strong&gt;,&#xA;  each chapter of a typical Outline begins with only a &lt;strong&gt;terse&lt;/strong&gt; explanation&#xA;  of relevant topics, followed by many fully &lt;strong&gt;worked examples&lt;/strong&gt; to&#xA;  illustrate common problem-solving techniques, and ends with a set of&#xA;  further exercises where usually &lt;strong&gt;only brief answers&lt;/strong&gt; are given and not&#xA;  full solutions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As said, take a look and decide for yourself if the book you have in hands is what you need/want. Different books have different purposes. You might well train your mechanical abilities or solve general problems or dig into shaping your math thinking, or give a try to some specific, normally not viewed in high-school, math field, like game theory or graph theory.&lt;/p&gt;&#xA;" OwnerUserId="237" LastEditorUserId="237" LastEditDate="2020-05-04T20:48:12.563" LastActivityDate="2020-05-04T20:48:12.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18267" PostTypeId="1" AcceptedAnswerId="18269" CreationDate="2020-05-04T14:01:37.817" Score="14" ViewCount="5270" Body="&lt;p&gt;Are there some fun results in set theory to set as &lt;em&gt;landmarks&lt;/em&gt; while introducing to kids?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, while introducing graph theory to kids, I could explain isomorphism via a pentagon and pentagram, introduce the Eulerian graph via the 7 bridges puzzle, show &lt;span class=&quot;math-container&quot;&gt;$V+F-E = 2$&lt;/span&gt; as it is fun itself, and explain the Euler characteristic via drawing a utility graph on a coffee cup. These are funny results, but also linked up as an interesting journey.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there similar funny results to show the basics of set theory? The barber paradox could be one, but it's something to explain &quot;why we need proper defined set theory&quot;, other than &quot;real set theory stuff after drawing the boundary&quot;.&lt;/p&gt;&#xA;" OwnerUserId="9059" LastEditorUserId="238" LastEditDate="2020-05-06T11:06:57.343" LastActivityDate="2020-05-07T18:28:20.743" Title="Fun set theory for kids" Tags="|secondary-education|primary-education|set-theory|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18268" PostTypeId="2" ParentId="18267" CreationDate="2020-05-04T14:43:40.343" Score="9" Body="&lt;p&gt;Questions about infinity are one way to go. e.g. 'Are there more natural numbers or even natural numbers?'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Intuition says there are more natural numbers (&lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;) than even natural numbers (&lt;span class=&quot;math-container&quot;&gt;$2\mathbb{N}$&lt;/span&gt;), but division by two gives a bijection from &lt;span class=&quot;math-container&quot;&gt;$2\mathbb{N} \to \mathbb{N}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is counterintuitive enough to drive discussion and really puts a focus on definitions. For example, you could ask someone who disagrees to put forth another definition of size and contrast it with cardinality.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2020-05-04T14:43:40.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18269" PostTypeId="2" ParentId="18267" CreationDate="2020-05-04T14:58:11.490" Score="22" Body="&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=Uj3_KqkI9Zo&quot; rel=&quot;noreferrer&quot;&gt;Hilbert's Hotel&lt;/a&gt; is a nice thought experiment for explaining results about cardinality of infinite sets and the aleph numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have also used plastic bags to explain the difference between &lt;span class=&quot;math-container&quot;&gt;$\varnothing, \{\varnothing\}, \{\varnothing,\{\varnothing\},\{\{\varnothing\}\}\}$&lt;/span&gt; etc. to kids. Let an empty plastic bag represent the empty set. Then a plastic bag containing only an empty plastic bag is &lt;span class=&quot;math-container&quot;&gt;$\{\varnothing\}$&lt;/span&gt;, and so on.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastEditorUserId="8900" LastEditDate="2020-05-05T10:51:10.473" LastActivityDate="2020-05-05T10:51:10.473" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18270" PostTypeId="2" ParentId="18267" CreationDate="2020-05-04T16:00:44.107" Score="11" Body="&lt;p&gt;I found the formula connecting the union and intersection of two sets useful at school.&lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$n(A\cup B) + n(A\cap B) = n(A) + n(B)$$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say you wish to find how many numbers from 1- 1000 inclusive are multiples of 10 or 25. This may be phrased as which money amounts up to $10 can be made just from dimes or just from quarters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let A be the set of amounts that can be made from dimes, there are 1000/10 = 100 of these.&lt;br&gt;&#xA;Let B be the set of amounts that can be made from quarters, there are 1000/25 = 40 of these.&lt;br&gt;&#xA;The amounts that can be made from both are multiples of 50, so there are 1000/50 = 20 of these.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is then:&lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$n(A\cup B) + n(A\cap B) = n(A) + n(B)$&lt;/span&gt;&lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$n(A\cup B) + 20 = 100 + 40$&lt;/span&gt;&lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$n(A\cup B) = 120$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="8660" LastEditorUserId="8660" LastEditDate="2020-05-04T16:06:11.980" LastActivityDate="2020-05-04T16:06:11.980" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18271" PostTypeId="1" AcceptedAnswerId="18292" CreationDate="2020-05-04T16:37:50.823" Score="3" ViewCount="297" Body="&lt;p&gt;I will be teaching an introductory analysis course (see topics below) and I need some source-code Latex slides or PPT slides, and am willing to choose my textbook based on these slides (rather than, traditionally, the other way around).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: apparently people want to chime in on teaching philosophy. Many publishers include pre-made lecture slides, which go along with the book, that instructors can edit to their needs (hence my request for open source slides). Examples: Pearson includes sets for many math and physics titles; Cengage includes sets with their math titles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am simply looking for introductory analysis slides; and by extension, the textbook these slides would fit with. The slides are more important than the textbook; hence why I consider the slides the first priority, and the textbook can follow from that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Topics: Sets of Natural Numbers, Rational Numbers, Real Numbers, Completeness Axiom, Sequences, Limit Theorems for Sequences, Monotone Sequences and Cauchy, Sequences, Series, Alternating Series and Integral Tests, Continuity, Properties of Continuous Functions, Sequences and Series Functions, Uniform Convergence, Differentiation and Integration of Power Series, Mean Value Theorem, L’Hopital Rule, Taylor Theorem, Integration)&lt;/p&gt;&#xA;" OwnerUserId="11880" LastEditorUserId="11880" LastEditDate="2020-05-07T01:46:49.140" LastActivityDate="2020-05-07T21:33:06.103" Title="Introductory Analysis lecture slides" Tags="|mathematical-analysis|lecture-notes|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="18272" PostTypeId="2" ParentId="4351" CreationDate="2020-05-04T20:34:58.263" Score="6" Body="&lt;p&gt;Here is another example, inspired by this video: &lt;a href=&quot;https://www.youtube.com/watch?v=OxDhN2XRSzs&quot; rel=&quot;noreferrer&quot;&gt;Solving a crazy iterated floor equation&lt;/a&gt;, where assuming a solution exists leads to the wrong conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solve the equation &lt;span class=&quot;math-container&quot;&gt;$$x \lfloor x \lfloor x \rfloor \rfloor=4,$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\lfloor \; \rfloor$&lt;/span&gt; denotes the floor function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an approximation &lt;span class=&quot;math-container&quot;&gt;$x \approx \sqrt[3] 4=1.587 \dots$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f(x)=x \lfloor x \lfloor x \rfloor \rfloor$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$f(1)=1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(2)=8$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is an increasing function, such an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (if it exists!) must satisfy &lt;span class=&quot;math-container&quot;&gt;$1 &amp;lt; x &amp;lt;2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$m=\lfloor x \lfloor x \rfloor \rfloor$&lt;/span&gt;, an integer. Then &lt;span class=&quot;math-container&quot;&gt;$xm=4$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$x = \frac 4 m$&lt;/span&gt;. This gives &lt;span class=&quot;math-container&quot;&gt;$1 &amp;lt; \frac 4 m &amp;lt; 2$&lt;/span&gt;, and therefore &lt;span class=&quot;math-container&quot;&gt;$$2 &amp;lt; m &amp;lt; 4.$$&lt;/span&gt; Since &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is an integer we must have &lt;span class=&quot;math-container&quot;&gt;$m=3$&lt;/span&gt;, and hence our candidate solution is &lt;span class=&quot;math-container&quot;&gt;$$x = \frac 4 m = \frac 4 3.$$&lt;/span&gt;&#xA;But we can easily check&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac 4 3 \lfloor \frac 4 3 \lfloor \frac 4 3 \rfloor \rfloor=\frac 4 3 \neq 4,$$&lt;/span&gt;&#xA;and the solution is therefore not valid.&lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2020-05-04T20:34:58.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18273" PostTypeId="2" ParentId="18267" CreationDate="2020-05-04T22:54:35.167" Score="2" Body="&lt;p&gt;Depending on what you mean by 'kids':&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe you could explain the formulas &lt;span class=&quot;math-container&quot;&gt;$|A \dot{\cup} B| = |A| + |B|$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|A \times B| = |A||B|$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|Maps(B,A)| = |A|^{|B|}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This let's you see all sorts of arithmetic identities as identifications between sets of maps, which can be more intuitive than pure arithmetic (or differently intuitive). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance &lt;span class=&quot;math-container&quot;&gt;$|A|^{|B| + |C|} = |A|^{|B|} |A|^{|C|}$&lt;/span&gt; would come from thinking about how to define functions on a disjoint union, and &lt;span class=&quot;math-container&quot;&gt;$(|A|^{|B|})^{|C|} = |A|^{|B||C|}$&lt;/span&gt; from thinking about how &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; valued functions on a &lt;span class=&quot;math-container&quot;&gt;$B \times C$&lt;/span&gt; are the same thing as &lt;span class=&quot;math-container&quot;&gt;$Maps(C,A)$&lt;/span&gt; valued functions on &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; (Is it ever too early to learn about &lt;a href=&quot;https://en.wikipedia.org/wiki/Currying&quot; rel=&quot;nofollow noreferrer&quot;&gt;currying&lt;/a&gt;?).&#xA;You can also explain the distributive property this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably these identities best posed as exercises to think about over a long period of time, but I'm not sure of how to pose them as riddles that don't feel forced...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A related thing to discuss would be &lt;span class=&quot;math-container&quot;&gt;$\mathscr{P}(A) = Maps(A, \{0,1\})$&lt;/span&gt;. You can use this to explain why &lt;span class=&quot;math-container&quot;&gt;$2^n = \sum_{k = 0}^n {n \choose k }$&lt;/span&gt;... related to this, another place where elementary set theory shines is in doing combinatorial calculations in a careful and unambiguous way - reasoning on the level of words gets confusing for me, its better to build sets and prove relationships among them.&lt;/p&gt;&#xA;" OwnerUserId="8029" LastEditorUserId="8029" LastEditDate="2020-05-06T23:46:04.917" LastActivityDate="2020-05-06T23:46:04.917" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18274" PostTypeId="2" ParentId="18267" CreationDate="2020-05-04T23:49:42.523" Score="4" Body="&lt;p&gt;Is it possible to have a bag of stones, and every time you arrange them on the ground in a spiral and count them, you get 64, but whenever you arrange them to fill a triangular region and count them then you get 63?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can play the devil's advocate until the child gets their proof in shape.&#xA;(&quot;No, that's impossible!&quot;  &quot;Well, why would it be impossible?  It's easy to imagine.&quot;  &quot;But it has to be the same number!&quot;  &quot;What does that mean?  Why would it have to be the same?&quot;  etc.)  When their argument is solid, you capitulate and they enjoy their success.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It helps with thinking about bijections and the meaning of equal-sized sets, and it also helps convey the feeling that proofs are how we can really be sure of things (even if the word &quot;proof&quot; will wait until they are older).&lt;/p&gt;&#xA;" OwnerUserId="7999" LastActivityDate="2020-05-04T23:49:42.523" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18275" PostTypeId="1" CreationDate="2020-05-05T01:15:07.413" Score="27" ViewCount="8573" Body="&lt;p&gt;This semester, I was forced to learn LaTeX for my Real Analysis class. The professor wanted all homework assignments to be typed in LaTeX in order to produce &quot;high-quality&quot; work. At first I was annoyed because I had no choice but to learn a new coding language in a short time frame. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I found that LaTeX was not all that difficult for me to learn. I have actually come to appreciate how intuitive the commands are. I would say it took me about 2 months to become comfortable with LaTeX (using it on and off). Before this semester I was not really into coding and I'm definitely not an expert at using LaTeX, but I at least know the basics pretty well. I still have to google what command to use every now and then. Lately I've been transferring all of my handwritten tutoring notes to LaTeX so that they are easier to read. Not to mention I can always go back and edit my notes whenever I want, which is not so easy to do when the notes are handwritten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read &lt;a href=&quot;https://matheducators.stackexchange.com/questions/504/how-should-latex-be-taught-to-university-students&quot;&gt;this post&lt;/a&gt; that discusses how LaTeX is handled at the college level. I've been wondering if high school students should be taught LaTeX (particularly the students who want to have a career in any math-related field). It could be an important skill for them to have because from my understanding, most, if not all, mathematicians seem to know LaTeX as if it is a prerequisite for their field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that learning LaTeX is not like learning a more, for lack of a better word, intensive programming language such as Java or Python (please convince me of otherwise if I'm being neglectful). Therefore I suspect that if LaTeX is currently taught in schools, it is not taught for a full semester. Again, I could be wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It could be beneficial for students to learn LaTeX for when they have to write a math research paper or type up a homework assignment on proofs later on. My idea is that the earlier they are exposed to LaTeX, they can master it sooner rather than just knowing the basics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there are downsides as well. I wonder if the students would find it worth their time to learn LaTeX. Some might find it boring because it's similar to Google Docs or Microsoft Word in terms of purpose. There are also tons of free resources online, so one could argue that it is better for the students to learn it on their own. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Overall, I'm curious if LaTeX is taught to high school students at all. If not, should they learn it? &lt;/p&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="376" LastEditDate="2020-05-05T09:06:58.393" LastActivityDate="2020-05-08T00:22:38.120" Title="Should LaTeX be taught in high school?" Tags="|secondary-education|course-design|curriculum|computer-programs|latex|" AnswerCount="10" CommentCount="11" ClosedDate="2024-01-07T12:10:59.550" ContentLicense="CC BY-SA 4.0" />
  <row Id="18276" PostTypeId="2" ParentId="18267" CreationDate="2020-05-05T03:32:30.393" Score="3" Body="&lt;p&gt;Richard Schwartz's picture book &lt;a href=&quot;http://www.richardevanschwartz.com/farm.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Life on the Infinite Farm&lt;/em&gt;&lt;/a&gt; illustrates three funny bijections between infinite sets. The first two are the classic bijections from &lt;span class=&quot;math-container&quot;&gt;$\{\bullet\} \sqcup \mathbb{N}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$10\mathbb{N}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;. The third is a mind-bending bijection from the edges of an infinite binary tree &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; to the edges of &lt;span class=&quot;math-container&quot;&gt;$B \sqcup B$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The pictures, like the bijections, are zany, colorful, and a little disturbing. I really like the narrative's pacing. It sets up three conflicts that leave you wondering, &quot;How's our pal gonna wriggle out of &lt;em&gt;this&lt;/em&gt; mess?&quot; Then it lets you stew for a while before the bijection swoops in and saves the day. It reminds me of what Dan Meyer says about &lt;a href=&quot;https://blog.mrmeyer.com/2017/presentation-math-is-power-not-punishment/&quot; rel=&quot;nofollow noreferrer&quot;&gt;aspirin and headaches&lt;/a&gt;—though with infinite sets, the aspirin is often a headache of its own.&lt;/p&gt;&#xA;" OwnerUserId="5119" LastActivityDate="2020-05-05T03:32:30.393" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18277" PostTypeId="2" ParentId="18267" CreationDate="2020-05-05T03:55:44.630" Score="5" Body="&lt;p&gt;A great way to get folks engaged is to bring up a contentious question. &lt;a href=&quot;https://math.stackexchange.com/q/235081/16063&quot;&gt;Here's one&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What's the 0th power of a whole number? In particular, what's the 0th power of 0?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Like Lorenzo Najt &lt;a href=&quot;https://matheducators.stackexchange.com/a/18273/5119&quot;&gt;mentioned&lt;/a&gt;, you can settle this pretty convincingly if you treat numbers as abstractions of finite sets. I think &lt;a href=&quot;https://math.stackexchange.com/a/1475935/16063&quot;&gt;my write-up&lt;/a&gt; at Math.StackExchange is pretty kid-friendly, and could be made even more so.&lt;/p&gt;&#xA;" OwnerUserId="5119" LastActivityDate="2020-05-05T03:55:44.630" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18278" PostTypeId="2" ParentId="18275" CreationDate="2020-05-05T15:26:49.267" Score="57" Body="&lt;p&gt;I'm a LaTeX user, but I'll stake out a devil's advocate position against this proposal. Reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Quality of mathematical thinking neither causes nor results from using a certain piece of software.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Tech ed belongs in tech ed. K-12 education should mainly be about enriching people's intellectual lives and creating the level of education that makes it possible to have a functioning republic with universal suffrage. If kids are going to learn how to fix a fuel injector or use a particular piece of software, that's great, but that's tech ed, which is a separate thing from academic programs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Schools should not be telling kids what software to use. I would consider it an intolerable thing if my school district required every kid to use Windows at home, and by the same token they should not be requiring them to use some other software at home, even free, open-source software that I personally like.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Beginners at coding need good error messages. Tex's error messages are infamously awful.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For many kids, the only computer they have access to at home is a cell phone, and in many cases the only smartphone they have access to is one that belongs to a parent.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-05-05T15:26:49.267" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="18279" PostTypeId="2" ParentId="18275" CreationDate="2020-05-05T15:58:00.927" Score="15" Body="&lt;p&gt;I don't think it should be a base skill for students in general.  Have held jobs in engineering, chemistry, military, and finance and never needed it.  Nor did my colleagues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Didn't need it for a thesis or science papers either.  Just MS Word was fine.  (I think I did enable the MSFT equation editor since it helps with typesetting sub and super scripts on same letters.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, there may be areas of math or CS where it is more critical.  But no reason why those who do need it, can't get it then.  And even there, I suspect the &quot;converts&quot; will overemphasize the importance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want general utility, I would push MSFT Office on the kids way before latex.  Many jobs where those are very normal tools for white collar workers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I think you have to consider that time is limited and that many students aren't mastering core curriculum.  Of course, lack of learning X, doesn't rule out trying to teach Y.  But it should at least be a consideration.  Also, time certainly is limited and the topics (not only math related) we might want to add into a curriculum are a &quot;big set&quot;.&lt;/p&gt;&#xA;" OwnerUserId="13948" LastEditorUserId="13948" LastEditDate="2020-05-05T16:05:44.413" LastActivityDate="2020-05-05T16:05:44.413" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18280" PostTypeId="2" ParentId="18275" CreationDate="2020-05-05T21:57:13.697" Score="29" Body="&lt;p&gt;This is not an answer to the posed question, but only an &#xA;anecdote. This semester, teaching US college students (&lt;em&gt;Discrete &amp;amp; Computational Geometry&lt;/em&gt;), I prepared all my assignments in LaTeX, and made available a &lt;code&gt;.zip&lt;/code&gt; file of the &lt;code&gt;.tex, .bbl, .bib, Figure/&lt;/code&gt; directory constituting&#xA;the assignment. Students could submit assignment answers in any form&amp;mdash;from LaTeX &lt;code&gt;.pdf&lt;/code&gt;, to a cell phone photo of handwritten answers, to physical handwritten &#xA;sheets (prior to the pandemic). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I gave no LaTeX instructions, but linked to resources&#xA;(MacOs, Windows, Linux).&#xA;There was a monotonic increase&#xA;in students editing the assignments in LaTeX over the semester. They learned LaTeX by seeing how I used it.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-05-05T21:57:13.697" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18281" PostTypeId="2" ParentId="18275" CreationDate="2020-05-06T07:28:40.403" Score="9" Body="&lt;p&gt;I only want to give anecdotical advice since I happened to have been taught LaTeX in &lt;em&gt;regular&lt;/em&gt; school (10th grade in the German system, iirc). However, I attended a school with a MINT focus and especially maths, also this was not part of the actual curriculum but the teacher &quot;showed&quot; us LaTeX and the interested students that we were promptly picked it up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I think that is key: &lt;em&gt;Make people curious&lt;/em&gt; (generally a thing I strive for when teaching) about something they can learn. This is in my experience quite easy by showing them the results and the &quot;magic&quot; that it does.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second thing is giving &lt;em&gt;good resources&lt;/em&gt;. Proper LaTeX resources are difficult to find and many issues with LaTeX only surface when writing non-English texts as. If you ask me, nowadays everyone should use LuaLaTeX+polyglossia+csquotes+fontspec+OTF fonts as well as LaTeXmk (or similar) and not pdfLaTeX+babel+inputenc+fontenc+T1 fonts as the latter is only viable when you only use Latin characters without many diacritics. Many &quot;problems&quot; with LaTeX could be eradicated if the defaults where more sane and the resources not wrong or really old. Thus &lt;em&gt;pointing to good resources&lt;/em&gt; and providing &lt;em&gt;proper sample documents&lt;/em&gt; that are not horribly over-engineered but pleasant to read is definitely a way to go.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Primary Education&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;These things, however, apply to teaching LaTeX in general. Primary schools have a different aim though (enabling to partake in society), than higher education (preparing for job or general higher knowledge in case of universities).  LaTeX itself is not critical for the former.  However, there &lt;em&gt;are&lt;/em&gt; reasons why one could evaluate teaching it, just as I was taught it:  If you teach a class of &lt;em&gt;interested&lt;/em&gt; students (not necessarily in maths IMHO) then showing LaTeX to them can be a way of enabling them to better choose what software they want to use and why.  Most students get out of school with seeing Microsoft Office as the only real way forward not even knowing or considering alternatives, thus not being able to make an informed decision based on what values they prefer (be it technical aspects, licensing, privacy, not supporting a monopoly, ...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, while there might be less technical reason to show them LaTeX, there might be other reasons that are more about showing students something &quot;outside the box&quot;.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Secondary Education&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;High schools have different aim as well, and I think we should separate between trade schools and universities. While the former is strictly preparation for later jobs, the second emphasizes free thinking and research. This, to me, is an important reason why one should show LaTeX to university students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When teaching maths in high school we also have the technical reasons&#xA; of LaTeX being &quot;the&quot; tool for the job. I'd definitely recommend showing it in high-school, while leaving choice.&lt;/p&gt;&#xA;" OwnerUserId="13956" LastEditorUserId="13956" LastEditDate="2020-05-06T07:34:29.157" LastActivityDate="2020-05-06T07:34:29.157" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18282" PostTypeId="1" AcceptedAnswerId="18290" CreationDate="2020-05-06T07:47:00.690" Score="1" ViewCount="176" Body="&lt;p&gt;I am currently marking a set of math assignment at university, and I am wondering about the following general point: in a question such as &quot;Find the solutions to the equation &lt;span class=&quot;math-container&quot;&gt;$x^2 - 3 = 0$&lt;/span&gt;&quot;, does the term &quot;find&quot; &lt;em&gt;in and of itself&lt;/em&gt; imply that the reasoning that led to finding the solution must be laid out as part of a perfect answer? I.e. does the phrasing of the question imply that &lt;span class=&quot;math-container&quot;&gt;$x = \pm\sqrt{3}$&lt;/span&gt; is insufficient to completely and appropriately solve the problem at hand?&lt;/p&gt;&#xA;" OwnerUserId="13958" LastEditorUserId="8900" LastEditDate="2020-05-06T11:20:15.540" LastActivityDate="2020-05-06T23:10:41.180" Title="Does the &quot;Find&quot; in questions such as &quot;Find the solution&quot; imply the process of finding must be laid out?" Tags="|grading|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18283" PostTypeId="1" CreationDate="2020-05-06T09:56:36.677" Score="1" ViewCount="391" Body="&lt;p&gt;I am writing up a small text about (increasing) exponential functions versus linear functions. I want to make the point that it is not true that if an exponential is bigger than a linear function that it will not always stay that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make my text more readable, I want to label the images with some relatable things that grow exponentially versus linearly. Now, I have two problems: First, I have a really, really hard time to think about things not covid-related. Secondly, I want to have an example where we &lt;em&gt;want&lt;/em&gt; the exponential growth to exceed the linear growth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have thought about repopulation of depleted environments, but the examples feel forced. Interest on a bank account is certainly &quot;good&quot; exponential growth, but I do not see a natural linear growth for comparison.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, do you have very common recognizable examples of exponential versus linear growth where we are happy that the exponential growth exceeds the linear one?&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2020-05-06T15:44:42.577" Title="Examples for &quot;good&quot; exponential growth versus linear growth" Tags="|examples|exponential|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18284" PostTypeId="2" ParentId="951" CreationDate="2020-05-06T10:05:55.987" Score="1" Body="&lt;p&gt;I do not see how any of these examples are confirmation bias.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;if the teacher suspects that the student has some problem, he will try to help before checking if it really is that problem;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Not checking is worse than confirmation bias. You do not even think that your assumptions need confirmation.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;if the teacher thinks that the child does not understand some concept, he/she will try to prove that it indeed is the case;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Confirmation bias might come into it while trying to prove it, but not necessarily so.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the teacher may want to stress why his/her solution is better than some other solution presented by the student (perhaps not even accepting it).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There is no confirmation bias here. Again, there is the assumption that one's opinion does not need confirmation at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, of course, confirmation bias is a big problem. The most common one is asking if everything is understood so far and being satisfied with one student answering &quot;yes&quot;.&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2020-05-06T10:05:55.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18285" PostTypeId="2" ParentId="18283" CreationDate="2020-05-06T13:13:03.003" Score="3" Body="&lt;p&gt;One example for &quot;good&quot; exponential growth is cryptographics. Linear key length growth vs. exponential growth for the effort to break the key. In this example we are kept safe by the exponential growth.&lt;/p&gt;&#xA;" OwnerUserId="13961" LastActivityDate="2020-05-06T13:13:03.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18286" PostTypeId="2" ParentId="18283" CreationDate="2020-05-06T15:35:37.537" Score="1" Body="&lt;p&gt;Chain reactions in explosives.  I mean, when you want one of those.  After all, people develop and use them.  Without the chain reaction and resultant exponential growth, you don't get the desired effect. (Linear would be too slow and/or require too high of an initial input, because of less amplification.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Chain_reaction&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Chain_reaction&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also avalanche breakdown in Geiger counters:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Geiger_counter#Principle_of_operation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Geiger_counter#Principle_of_operation&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Townsend_discharge&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Townsend_discharge&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The extreme amplification is needed to detect and signal individual particles.&lt;/p&gt;&#xA;" OwnerUserId="13966" LastEditorUserId="13966" LastEditDate="2020-05-06T15:44:42.577" LastActivityDate="2020-05-06T15:44:42.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18287" PostTypeId="2" ParentId="18282" CreationDate="2020-05-06T18:56:10.957" Score="2" Body="&lt;p&gt;I think instead of trying to get some legalistic interpretation of the words (that will apply to all classes/instructors), you have to look at the individual teacher/class expectations.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, it is understood on AP tests, that the questions are &quot;essay questions&quot;.  So find the general solution for y'' + 2y' + y = 25x requires a solution that shows the method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is Navy Nuclear Power School, where the acronym &quot;NWNC&quot; stood for &quot;no work, no credit&quot;.  All it takes is a couple marking of that, to drive the requested behavior.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think in general, most teachers and classes require that students show the process for getting the answer.  (For the legalistic pedants...Yes, and more than guessing.  Yes, more than plugging the answer back in and showing it works.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in any case, the way to resolve this is to ask the instructor what he expects.  Then mark versus that expectation.  There's no way we can anticipate every instructor's quirks.  Nor any reason to think that every instructor wants cats skinned in the same fashion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  And yes, it is a judgment call what is a complete answer.  And yes, it is more work to have to look at different methods (by the student) to answer a given question,  So?  This is how the normal, non-Euclidean world works.  Not everything is an exact equation, when doing complex practical tasks, like grading.&lt;/p&gt;&#xA;" OwnerUserId="13966" LastActivityDate="2020-05-06T18:56:10.957" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18288" PostTypeId="2" ParentId="18282" CreationDate="2020-05-06T19:09:27.380" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;does the term &quot;find&quot; in and of itself imply that the reasoning that led to finding the solution must be laid out as part of a perfect answer&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;No, definitely not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The conventions for what's expected in an answer very wildly across individual schools, classes, and professors.  No single word or phrase is going to convey what's expected of students dealing with a new assignment; if you want particular information included in the answers, students need to be told that explicitly.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2020-05-06T19:09:27.380" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18290" PostTypeId="2" ParentId="18282" CreationDate="2020-05-06T23:10:41.180" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Does the “Find” in questions such as “Find the solution” imply the process of finding must be laid out?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;No.  No single word or phrase is ever going to convey the idea that a student should solve the problem and explain the solution process in some particular level of detail.  An entirely appropriate response to &quot;Find all solutions to [equation]&quot; is &quot;Okay, I found all of the solutions.  Now what?&quot;  Alternatively:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xBw3B.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xBw3B.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You cannot, therefore, rely on a single word or phrase to convey your intentions.  Rather, you must establish expectations, both early and often.  Such expectations should become part of the culture of the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you expect students to &quot;show their work,&quot; tell them so.  Start early in the term by giving examples of correctly written solutions, and non-examples of poor solutions.  Mark formative assessments with a critical eye, and point out anywhere that steps you consider vital are skipped.  When giving formative assessments, include language which clearly communicates to the students what the expectation is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in a precalculus course which I have taught multiple times, students are told over and over again that they must show how they arrived at a solution, but that they are not required to simplify.  The syllabus for that course (which is very nearly a textbook) contains hundreds of problems, about a third of which have completely worked out solutions in the style expected on exams.  Prior to each exam, the students are given a sample exam&amp;mdash;significant numbers of office hours and a small amount of class time are devoted to working through those problems, including how they should be written up.  Finally, every exam contains an instruction of the form&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;You &lt;em&gt;must&lt;/em&gt; justify your answers&amp;mdash;describe the process which leads to your solution.  You will not receive credit for correct answers if insufficient work is shown.  You will not lose points for correct statements (even if they are not relevant to the problem), so while keeping argument simple is a valuable skill, it is a good idea to err on the side of writing more, rather than less, on this exam.  You do not need to simplify your results.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2020-05-06T23:10:41.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18292" PostTypeId="2" ParentId="18271" CreationDate="2020-05-07T01:54:19.743" Score="2" Body="&lt;p&gt;I trudged through Pearson's site and found that Lay's &lt;em&gt;Analysis with an Intro. to Proof&lt;/em&gt; has PowerPoint slides on their resources page. You'll have to get an account with them to download the files, but I think that the process is straightforward for anyone who is verifiably a faculty member.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://www.pearson.com/us/higher-education/program/Lay-Analysis-with-an-Introduction-to-Proof-5th-Edition/PGM279803.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.pearson.com/us/higher-education/program/Lay-Analysis-with-an-Introduction-to-Proof-5th-Edition/PGM279803.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know nothing about this book or the slides, pro or con. It does look like the only such book on Pearson's site. (Many others have problem solutions, but no slides.) You should try other publishers such as Cengage or Wiley as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that the reason that you are getting push-back about this is that lecturing from slides is a fairly atypical approach for what is usually an upper-level proof-based course and is often taught by someone whose research is or was in the area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like others, I wouldn't advocate this approach in a normal semester. However, in the comments is is mentioned that this is a 5 week online summer course. It doesn't seem comparable to a normal semester at all; you hopefully have made the learning goals a bit more achievable. Best of luck to you and your students.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2020-05-07T21:33:06.103" LastActivityDate="2020-05-07T21:33:06.103" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18293" PostTypeId="2" ParentId="18275" CreationDate="2020-05-07T06:20:30.543" Score="5" Body="&lt;p&gt;I'd suggest &lt;em&gt;exposing&lt;/em&gt; students to LaTeX, perhaps by having them doing a assignment/project with it, perhaps in a group, but not requiring them to learn much about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would seem to have several advantages:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It gives students an idea of a tool that they might later use.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It increases their understanding of what's out there, broadening their general knowledge about the world.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It lays the groundwork for those who might later use something like it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Generally, I'm of the opinion that educational-exposure is underrated.  I perceive there to be a lot of value in showing students what's out there and giving them a brief opportunity to play with it, plus the ability to follow-up if they're interested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I'd suggest showing them LaTeX in a low-pressure setting.  Perhaps:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Provide type-set documents, showing them the source code and output without making them work for it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give them the opportunity to experiment with it, perhaps more free-form or challenging them to be creative.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Talk to them about ways and places in which LaTeX can be useful.  For example, in composing documents for publication and in writing stuff on websites that have &lt;span class=&quot;math-container&quot;&gt;$\mathrm{\TeX}$&lt;/span&gt; enabled.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Likewise talk to them about alternatives, e.g. the type-setting in Microsoft Word, Wolfram Mathematica, and other document-composition or/and mathematical environments.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Contextualize what LaTeX is by talking about markdown-vs.-presentation.  Give other common examples of markup/presentation languages, e.g. HTML.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8943" LastEditorUserId="8943" LastEditDate="2020-05-07T06:26:07.190" LastActivityDate="2020-05-07T06:26:07.190" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18295" PostTypeId="2" ParentId="18267" CreationDate="2020-05-07T12:11:30.400" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;Starting with some mind blowing paradoxes like asking the famous Russel's set paradox- does a set, that consists of all the sets that don't consist of themselves, consists of itself? &#xA;(If it does - it doesn't, if it doesn't - it does). &lt;/li&gt;&#xA;&lt;li&gt;Establishing the term of bijective functions, and with finite examples explaining the motivation to define two equivalent sets as two that have a bijection between them. &lt;/li&gt;&#xA;&lt;li&gt;A very cool and intuitive definition for an infinite set - a set which is equivalent to a strict subset of itself. It is nice to demonstrate with the Natural numbers and the Even numbers. &lt;/li&gt;&#xA;&lt;li&gt;After the bijection term is well-established, it is time for Cantor's Diagonal Argument, to show that there could not be a bijection from the Natural numbers to the unit segment. I think this might just be the best 'party trick' to use when you want someone to get enthusiastic about set theory, though I find it necessary to talk a bit about bijections first.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="13977" LastEditorUserId="13977" LastEditDate="2020-05-07T18:28:20.743" LastActivityDate="2020-05-07T18:28:20.743" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18296" PostTypeId="2" ParentId="18275" CreationDate="2020-05-07T15:24:19.320" Score="6" Body="&lt;h1&gt;No, teach Markdown&lt;/h1&gt;&#xA;&lt;p&gt;&lt;strong&gt;Separation of content&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This and future generations of students should learn to &lt;strong&gt;separate content from formatting.&lt;/strong&gt; IT professionals have been very successful in applying this concept for over a decade. However, absent marketing and strong commercial interests have kept wider audiences from applying this very useful principle. Focusing on content instead of tinkering with format, is also extremely helpful in &lt;strong&gt;combating procrastination&lt;/strong&gt; or fear of the empty page. Any &lt;strong&gt;distraction free&lt;/strong&gt; plain text editor will do to create content in Markdown.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Writing content in Markdown&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The most student friendly way to write content happens to be the very same markup language used over here at StackExchange; i.e. &lt;a href=&quot;https://daringfireball.net/projects/markdown/syntax#overview&quot; rel=&quot;nofollow noreferrer&quot;&gt;Markdown&lt;/a&gt;.&#xA;Writing in Markdown has two advantages over writing directly in &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; or (X)HTML:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Markdown is extremely easy to learn.&lt;/li&gt;&#xA;&lt;li&gt;Markdown does not require closing text environments with any tags or encapsulating it within any accolades.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Moreover, Markdown inherited &lt;strong&gt;the &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; way of entering mathematical formulae.&lt;/strong&gt;&#xA;Numerous other proponents of the use of Markdown in a teaching environment have exposed these arguments in more detail:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=hpAJMSS8pvs&quot; rel=&quot;nofollow noreferrer&quot;&gt;Academic Writing in Markdown&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://blog.kdheepak.com/writing-papers-with-markdown.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Writing Technical Papers with Markdown&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.simongriffee.com/notebook/happier-people-computers-bigorgname/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Happier People and Computers in BigOrgName&#xA;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://programminghistorian.org/en/lessons/sustainable-authorship-in-plain-text-using-pandoc-and-markdown&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sustainable Authorship in Plain Text using Pandoc and Markdown&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hamwaves.com/markdown/en/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Content Writing in Markdown&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Document conversion with Pandoc&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Markdown would not be as versatile as it is today, without the document conversion command tool &lt;a href=&quot;https://pandoc.org/MANUAL.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pandoc&lt;/a&gt;. Pandoc converts any Markdown file in a single go into a &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; PDF, MS Word™, LibreOffice Write, &lt;a href=&quot;https://wiki.contextgarden.net/Main_Page&quot; rel=&quot;nofollow noreferrer&quot;&gt;ConTeXt&lt;/a&gt; or even XHTML or HTML5 document, to name but a few. In such a workflow, &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; , apart from its math, gets reduced to essentially a templating language; which is not a bad thing.&lt;/p&gt;&#xA;&lt;p&gt;Pandoc can also convert Markdown to XHTML. This may take things even further than &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt;, as it allows for &lt;a href=&quot;https://tex.stackexchange.com/a/340365/26348&quot;&gt;unattended CSS typesetting&lt;/a&gt;. Such a feat is &lt;a href=&quot;https://tex.stackexchange.com/q/162489/26348&quot;&gt;not possible&lt;/a&gt; with &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt;. More information about CSS paged media can be found here:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://thomaspark.co/2015/01/pubcss-formatting-academic-publications-in-html-css/&quot; rel=&quot;nofollow noreferrer&quot;&gt;PubCSS: Formatting Academic Publications in HTML &amp;amp; CSS&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.princexml.com/samples/&quot; rel=&quot;nofollow noreferrer&quot;&gt;PrinceXML samples&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://alistapart.com/article/boom/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Printing a Book with CSS: Boom!&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hamwaves.com/workflow/en/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Markdown Conversion Workflow&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Free, Libre &amp;amp; Open Source Software&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Pandoc is written as open source software by Prof. John MacFarlane of Berkeley.&#xA;It runs locally on the PC and is freely available on all platforms (MS Windows™, Apple Mac OS, GNU/Linux,…)&#xA;This is important, as &lt;strong&gt;not every household has money to spare on an annual Microsoft Office 365™ license subscription&lt;/strong&gt; or on the latest hardware. Schools can even save the substantial amounts of annually recurring Microsoft licensing fees if they choose to run their (aged?) PCs on a GNU/Linux distribution, like for example &lt;a href=&quot;https://xubuntu.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Xubuntu LTS&lt;/a&gt;. Pandoc even runs on the cheap, educational and energy efficient &lt;a href=&quot;https://www.raspberrypi.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Raspberry Pi&lt;/a&gt;. As such, it can easily be managed in a classroom using &lt;a href=&quot;http://pinet.org.uk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;PiNet&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For the same reason one does not spray paint a giant red &lt;em&gt;Coca Cola&lt;/em&gt; logo all over the whole schoolyard, educational institutions should stop spoon feeding their students commercial solutions of Microsoft, Apple or Google. The reason is simple: &lt;strong&gt;it enslaves.&lt;/strong&gt; Some arguments can be found here:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://medium.com/backchannel/why-i-m-saying-goodbye-to-apple-google-and-microsoft-78af12071bd&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why I’m Saying Goodbye to Apple, Google and Microsoft&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.antipope.org/charlie/blog-static/2013/10/why-microsoft-word-must-die.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why Microsoft Word must Die&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Ending the publishing oligopoly&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is a well known that the publication of many peer-reviewed scholarly journals is in the hands of but a few publishing houses. These publishers most often charge prohibitive fees for journal subscriptions and article reprints. This hampers researchers world-wide across different economies.&lt;/p&gt;&#xA;&lt;p&gt;Most of the power of these publishing houses stems from their (outsourced) knowledge about typesetting. Behind the scenes, this typesetting is often actually done with &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If we teach our students the right free and open tools today, one may hope that those pursuing an academic career will one day be capable to edit their own peer-reviewed scholarly journals. Pandoc Markdown plays a crucial role in this pursuit, as is evidenced by these articles:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://pierog.it/en/2018/03/markdown-workflow/&quot; rel=&quot;nofollow noreferrer&quot;&gt;A pandoc-based layout workflow for scholarly journals&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://sachsmc.github.io/pandoc-journal-templates/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pandoc templates for the major statistics journals&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13962" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-05-07T20:08:46.880" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18297" PostTypeId="1" CreationDate="2020-05-07T19:37:55.087" Score="37" ViewCount="3461" Body="&lt;p&gt;What are some interesting mathematical things you have learned while grading student work (or marking, if you prefer)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is final exams time here, so if anyone can help cast a more positive light on the grading experience, it would be most welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answers can be things that students wrote, or inspired by something a student wrote, or just something we learned during the grading process in some way. For example, clever proofs that students came up with; nice counterexamples or insights; interesting new questions inspired while grading; even just something you looked up to find out if a student's work was valid. However, for an answer to be interesting, it should be something beyond just a different way to solve a problem.&lt;/p&gt;&#xA;" OwnerUserId="12530" LastActivityDate="2022-07-25T14:00:42.737" Title="Interesting things you learned while grading?" Tags="|grading|big-list|" AnswerCount="10" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18298" PostTypeId="2" ParentId="18275" CreationDate="2020-05-07T20:11:04.290" Score="11" Body="&lt;p&gt;This is clearly an opinion based question. I will answer based on my experience and opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;No, LaTeX should not be taught at high school.&lt;/strong&gt; It is a skill that is costly to learn with essentially no benefit at high school level. Even at college level, it is not particularly useful. Homeworks are typically submitted handwritten which works perfectly fine. And for note taking LaTeX is far too slow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The big benefit of LaTeX kicks in when it comes to thesis writing. When that happens, (either very late in UG career or during graduate studies) LaTeX can be picked up as part of the thesis writing experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am a big fan of LaTeX. I love it. It's a very powerful tool, but it's a tool for a specific job that high school students (and even college students) are not required to do.&lt;/p&gt;&#xA;" OwnerUserId="12108" LastActivityDate="2020-05-07T20:11:04.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18299" PostTypeId="2" ParentId="18275" CreationDate="2020-05-07T20:40:33.287" Score="2" Body="&lt;p&gt;My feeling is LaTeX could be part of a very gentle introduction to computer programming. Maybe. Or it could be taught by a math instructor if the students have already seen HTML. Otherwise there are plenty of other things that could be taught in its place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you want to very slowly work up to coding -- like &quot;AP Programming Principles&quot; does. Learning any mark-up language is an easy warm-up. Web-page HTML is the simplest. After that, LaTeX could be the second. It's still just a mark-up language, but different: the {}'s look much more like computer code, as do the cascading cryptic errors. Now you can point to 2 concrete skills: making a web page, and making a good-looking pdf. And they understand the general concept of mark-up. When they see the syntax of Java or Python -- just more mark-up.&lt;/p&gt;&#xA;" OwnerUserId="13766" LastActivityDate="2020-05-07T20:40:33.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18300" PostTypeId="2" ParentId="18297" CreationDate="2020-05-07T23:21:15.240" Score="49" Body="&lt;p&gt;I once asked students to find the derivative of &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt; (with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;). One student figured that if the exponent were a constant then the answer would be &lt;span class=&quot;math-container&quot;&gt;$xx^{x-1}$&lt;/span&gt; which is to say &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt;, while if the base were constant the answer would be &lt;span class=&quot;math-container&quot;&gt;$x^x\log x$&lt;/span&gt;, so she added the two together to get &lt;span class=&quot;math-container&quot;&gt;$x^x+x^x\log x$&lt;/span&gt;. I was just about to mark the answer as wrong, when I realized that she had arrived at the correct answer – and, later, realized that it wasn't a coincidence, her unorthodox method actually works in a more general setting. &lt;/p&gt;&#xA;" OwnerUserId="13981" LastActivityDate="2020-05-07T23:21:15.240" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18301" PostTypeId="2" ParentId="18275" CreationDate="2020-05-07T23:27:54.007" Score="4" Body="&lt;p&gt;First, LaTeX is a markup language and should be taught as such; trying to use it as a programming language would be an exercise in frustration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, I believe that all HS students should learn a markup language as an alternative to &lt;a href=&quot;http://foldoc.org/WYSIAYG&quot; rel=&quot;nofollow noreferrer&quot;&gt;WYSIAYG&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third, while I routinely use LaTeX both for papers and for short documents, I am not convinced that it is suitable as a student's first markup language. Something like DocBook, HTML or Markdown might be more appropriate. The original documentation hasn't been updated in ages, the newer documentation is scattered and the error messages are terrible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fourth, with all its faults LaTeX is very powerful and I would like to see HS students learn it after they are confortable with an easier markup language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fifth, LaTeX was not nearly as painful for me as ms word; every time I tried simple things like copying items from an ordered list it totally garbled the formatting. &lt;/p&gt;&#xA;" OwnerUserId="13980" LastActivityDate="2020-05-07T23:27:54.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18302" PostTypeId="1" AcceptedAnswerId="18303" CreationDate="2020-05-07T23:40:55.277" Score="10" ViewCount="502" Body="&lt;p&gt;When I teach math classes, one goal I have in mind is to help students develop the cluster of thinking skills usually called analytical skills or critical thinking skills.   And I think that math classes can help develop analytical and critical thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These skills include being able to apply mathematical reasoning, but also other skills: to be able to take big problems and break them down in to smaller one, to be able to combine two ideas to create a third idea, to be able to generalize a concept to apply it to similar situations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there currently accepted measures of how good learners are at these skills?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I really think math and science courses help develop these skills, I would love to not only learn more about them, but to be able to point to research that says these skills are developed by quality math and science instruction.  Can I get some leads on that research?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own Google Scholar searches come up with mostly noise level results. &lt;/p&gt;&#xA;" OwnerUserId="8642" LastEditorUserId="8642" LastEditDate="2020-05-07T23:57:15.267" LastActivityDate="2020-05-13T12:44:17.130" Title="Is there research to back up the claim that math classes help develop analytical skills?" Tags="|teaching|mathematical-analysis|critical-thinking|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18303" PostTypeId="2" ParentId="18302" CreationDate="2020-05-08T00:01:32.650" Score="10" Body="&lt;p&gt;You may be interested in Academically Adrift: Limited Learning on College Campuses, Arum and Roksa, 2011. Also summarized in &lt;a href=&quot;http://www.newyorker.com/arts/critics/atlarge/2011/06/06/110606crat_atlarge_menand&quot; rel=&quot;noreferrer&quot;&gt;http://www.newyorker.com/arts/critics/atlarge/2011/06/06/110606crat_atlarge_menand&lt;/a&gt; . They have a lot of discussion of something called the Collegiate Learning Assessment, which is a standardized test of critical thinking. They find that students in the liberal arts do significantly improve their critical thinking abilities over the course of their college education, while the gains for students in most other majors, such as business and engineering, are smaller. Science majors tend to be more similar to liberal arts majors, which might support your claim, but I don't have anything in my notes about math majors specifically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They find that greater learning is correlated with faculty's high expectations, high standards, and approachability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's possible that the differences between majors really just measure preexisting differences between the student populations that go into those majors. Also, many people who are initially STEM majors in college end up switching to easier majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another book with useful material on this topic is Valen E. Johnson, Grade Inflation: A Crisis in College Education. His main topic is grade inflation and the corrosive effects of student evaluations of teaching, but he also has some carefully constructed statistical studies that show huge differences between the level of difficulty of different majors (a point that is also supported by Arum and Roksa). He rules out interpretations saying that the different groups of students are equally smart, but just at different things. Math is one of the majors that does come out to be objectively harder according to his data, and therefore one would expect math students to be more able and to improve their critical thinking skills more while in college.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-05-08T00:09:35.220" LastActivityDate="2020-05-08T00:09:35.220" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18304" PostTypeId="2" ParentId="18275" CreationDate="2020-05-08T00:22:38.120" Score="5" Body="&lt;p&gt;Dave from Boyinaband had a &lt;a href=&quot;https://www.youtube.com/watch?v=8xe6nLVXEC0&quot; rel=&quot;noreferrer&quot;&gt;song&lt;/a&gt; about all the stuff he found useless in school. While I disagree with some particular things, his opinion in general is right. There are too many things taught that aren't needed by most students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Latex might be good for maths professors. How many people go to that field? Not a lot. Those who go there can learn Latex easily (or use something else). Teaching this in high school is pointless. Time is limited, and it could be used better for almost anything. (e.g. whatever he mentions, or Python)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;strong&gt;no&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13474" LastActivityDate="2020-05-08T00:22:38.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18306" PostTypeId="1" AcceptedAnswerId="18308" CreationDate="2020-05-08T13:21:09.463" Score="13" ViewCount="2513" Body="&lt;p&gt;I'm due to deliver a session on graph theory for 16–17-year old students (UK sixth formers) as a taster of what studying mathematics at university is like. What would you recommend as content, and a 'route' through the content for such a talk? There will be a short set of exercises to accompany the talk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My aim is to mimic some of the features of an undergraduate lecture, especially in terms of starting with formal definitions and writing proofs of formal statements.&lt;/strong&gt; I'm not specifically looking for it to be dry but on this occasion I am willing to sacrifice being engaging and dynamic to some extent!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At present I have the definitions of a graph/subgraph, walks/paths/cycles, trees and planar graphs, but it feels a bit flimsy, for want of a better word. The section on planar graphs is only there in order to enable a question about proving the Euler faces/edges/vertices theorem and since I want to leave that for the exercises I don't really have much else to say about planar graphs in general.&lt;/p&gt;&#xA;" OwnerUserId="1295" LastEditorUserId="1295" LastEditDate="2020-05-08T21:51:05.927" LastActivityDate="2020-05-11T14:55:53.343" Title="Content for a 40-minute lecture on graph theory for high schoolers" Tags="|secondary-education|curriculum|graph-theory|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18307" PostTypeId="2" ParentId="18306" CreationDate="2020-05-08T14:08:52.910" Score="5" Body="&lt;p&gt;You might include results on coloring plane graphs:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Art_gallery_problem&quot; rel=&quot;noreferrer&quot;&gt;Art Gallery&#xA;Theorem&lt;/a&gt;: &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-coloring.&lt;/li&gt;&#xA;&lt;li&gt;Using Euler's theorem to prove that there must be a vertex of degree at most &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;From there to &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;-coloring plane graphs.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;-coloring is more difficult, but there is a nice exposition in &lt;a href=&quot;https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK&quot; rel=&quot;noreferrer&quot;&gt;Proofs from THE BOOK&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Finally, sketch a history of the &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;-color theorem.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;a href=&quot;https://i.stack.imgur.com/JBygZ.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JBygZ.jpg&quot; alt=&quot;DanTeague&quot;&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;sup&gt;&#xA;Image from &lt;a href=&quot;https://www.slideserve.com/egil/the-infamous-five-color-theorem&quot; rel=&quot;noreferrer&quot;&gt;Dan Teague&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-05-08T14:08:52.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18308" PostTypeId="2" ParentId="18306" CreationDate="2020-05-08T14:09:48.937" Score="20" Body="&lt;p&gt;I suggest you discuss the &lt;a href=&quot;https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg&quot; rel=&quot;noreferrer&quot;&gt;Seven bridges of Königsberg problem&lt;/a&gt; (the problem that essentially started the field of graph theory), then discuss the &lt;a href=&quot;https://en.wikipedia.org/wiki/Three_utilities_problem&quot; rel=&quot;noreferrer&quot;&gt;Three utilities problem&lt;/a&gt;.  For each, discuss the problem first, then introduce the definitions, then perhaps give a sketch of a proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first problem allows you to introduce the concepts of vertex, edge, walk, and path; the second allows you to talk about complete bipartite graphs, and planar (and nonplanar) graphs.  Both involve proving that something is impossible, something that I think is a very useful skill.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2020-05-08T14:09:48.937" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="18309" PostTypeId="2" ParentId="18306" CreationDate="2020-05-08T14:21:41.100" Score="6" Body="&lt;p&gt;I would try to keep it heavy on sugar and light on medicine.  In particular emphasize visual representations more than precise terms that students don't know (or even worse, symbol soup).  Maybe something like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=iW_LkYiuTKE&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=iW_LkYiuTKE&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth, I find this very similar to the issue of alkane isomers.  Just got done Zoom teaching my 8 year old nephew with a Tinkertoy-like college chem modeling set.  This is a topic that he can handle well before having all the grounding of bonding theory, orbitals, etc.  He picked it up no problem and now knows more about butanes (two isomers) and pentanes (3 isomers) than 95% of oil and gas executives (who have &quot;NGLs&quot; as one of their income statement items!)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point (in addition to the cool similarity of chemical isomers and &quot;trees&quot;) is that this is an &quot;easy&quot; entree to a tough topic (chemistry) and can be done sans lots of prerequisites.  So I LOVE that you are covering graph theory with the kids as it is generally easier--yes, of course, at the real &quot;coal face&quot; of research it's brutal hard and you have to know all sorts of hard stuff.  But the neophyte can appreciate a lot of that topic in a way that he could not get much out of say stochastic calculus.&lt;/p&gt;&#xA;" OwnerUserId="13989" LastActivityDate="2020-05-08T14:21:41.100" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18310" PostTypeId="2" ParentId="18306" CreationDate="2020-05-08T19:07:11.843" Score="5" Body="&lt;p&gt;I always introduce graph theory with the Königsberg Bridges, as Joel Reyes Noche suggested in his answer. Years ago, I went to a wonderful math talk that used this problem to describe 5  or more stages of what I'll call mathematizing. There was the physical problem. Then drawing it to be able to &quot;walk&quot; the bridges with a pencil, instead of for real. Then moving to a graph of it. Then building theory about all graphs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am teaching discrete math. To introduce planar graphs, I actually started with models of platonic solids. (I only had the right polydron pieces to build two of them, and then I grabbed a die for a third. I asked students if there could be more.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I noticed your title called this a lecture and your text called it a session. The more you can get them to participate and tell you what should happen, the better (ie less lecture, more dialogue and activity). I hope you are able to use this as a chance to inspire.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-05-08T19:07:11.843" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18311" PostTypeId="2" ParentId="18306" CreationDate="2020-05-08T19:54:01.883" Score="12" Body="&lt;p&gt;I study pursuit-evasion games on graphs, so I will recommend using &lt;strong&gt;the cops &amp;amp; robbers game&lt;/strong&gt; as a way to introduce graph theoretic terminology, concepts, and examples. It should also keep the tone informal and recreational, which will do far more (I believe) to actually inspire the students to study more mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below are the rules of the game, so you know them. You could present them at the top of the talk in a fun way, and use a specific example of a graph &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; (something simple like a small tree or a cycle). You can even use volunteers from the audience as the two players to demonstrate a sample game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also suggest that it's okay to use vague terms like &quot;dots&quot; and &quot;connections&quot; (instead of vertices and edges) to start with. You can introduce the formal terminology as the audience realizes that we need such words to describe some concepts. So, I highly suggest an activity in the middle of the talk where the audience breaks into small groups to play the game with each other. If you haven't already used words like &lt;em&gt;vertex&lt;/em&gt; and &lt;em&gt;path&lt;/em&gt; and &lt;em&gt;cycle/circuit&lt;/em&gt;, the audience will hopefully realize they need them and invent them right there (or you could guide them into doing so).&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Specify a graph &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; to be the playing board. There are two players: a cop and a robber.&lt;/li&gt;&#xA;&lt;li&gt;The cop gets to choose a vertex to start on.&lt;/li&gt;&#xA;&lt;li&gt;Knowing the cop's starting location, the robber then gets to choose a starting location.&lt;/li&gt;&#xA;&lt;li&gt;The cop gets to move first. They may travel along an edge to an adjacent vertex, or they may choose to stay put. They can &quot;see&quot; where the robber is at all times.&lt;/li&gt;&#xA;&lt;li&gt;The robber gets to move next. Likewise, they may travel along an edge to an adjacent vertex, or they may choose to stay put (and sometimes that is the strategically sound choice). They can &quot;see&quot; where the cop is at all times.&lt;/li&gt;&#xA;&lt;li&gt;The turns go back and forth like that. If at any point, the cop lands on the robber's vertex, the cop wins. If, instead, the robber can &lt;em&gt;always evade&lt;/em&gt; the cop (formally: there exists a &lt;em&gt;winning strategy&lt;/em&gt;) then we say the robber wins. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are some natural questions that arise, and you can pose these in your talk, or work with the audience to come up with them together:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;On which graphs &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; will the cop win? On which will the robber win instead? This question was answered in the early 1980s by Nowakowski &amp;amp; Winkler: &lt;a href=&quot;https://en.wikipedia.org/wiki/Cop-win_graph&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Cop-win_graph&lt;/a&gt;  You could use simple examples like a tree (cop wins), cycle (robber wins as long as length is at least 4), Petersen graph (robber wins), etc.&lt;/li&gt;&#xA;&lt;li&gt;Let's say we have a &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; where we know the cop wins. How many moves will it take? If both sides play optimally, how long can the robber draw out the process of losing? This is the concept of &lt;em&gt;capture time&lt;/em&gt; and it has been well studied but there are lots of open problems.&lt;/li&gt;&#xA;&lt;li&gt;Let's saw we have a &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; where we know the cop loses. What if we allowed &lt;em&gt;two cops&lt;/em&gt;, and they can move independently at the same time? On which graphs will these two cops win? For example, cycles require 2 cops; but something like the Petersen graph requires more than 2. &lt;/li&gt;&#xA;&lt;li&gt;What if we have more than one cop but we let only one move at a time? (Kinda like chess where you have a team of pieces but can only move one piece per turn.) This is know as the &lt;em&gt;lazy cops&lt;/em&gt; variant. Does that change any of the examples in #3 we just explored? For example, cycles still require 2 cops but the &lt;em&gt;capture time&lt;/em&gt; will be longer. And some graphs have different results! For example, consider a graph whose vertices are the squares of a &lt;span class=&quot;math-container&quot;&gt;$3\times 3$&lt;/span&gt; chessboard and whose edges represent the possible moves of a rook (so vertices are connected by an edge if they're in the same row or column). Then 2 cops can win on this graph, but they both need to be able to move simultaneously; if you only allow one to move at a time, then you need &lt;em&gt;3 lazy cops&lt;/em&gt; to win. Some students and I managed to prove that this graph is, in fact, &lt;em&gt;the smallest possible example&lt;/em&gt; of such a phenomenon: &lt;a href=&quot;https://arxiv.org/abs/1606.08485&quot; rel=&quot;noreferrer&quot;&gt;https://arxiv.org/abs/1606.08485&lt;/a&gt; &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="80" LastActivityDate="2020-05-08T19:54:01.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18315" PostTypeId="1" AcceptedAnswerId="18316" CreationDate="2020-05-09T22:15:47.467" Score="12" ViewCount="904" Body="&lt;p&gt;I am looking for some examples of when proof by contradiction is used in a problem with more than one case. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In all the elementary examples, there are only two options (eg rational/irrational, infinite/finite), so you assume the opposite, show it cannot be true and then conclude the result. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I remember from my undergraduate study situations where there were three or more options and proof by contradiction was used to eliminate one or more of these. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not remember any of these examples anymore, but I was wondering if I could have some examples of statements (ideally elementary, but more advanced works) that follow such a style in their proof. This would be good to have to show students I teach typical contexts in which this may arise. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you&lt;/p&gt;&#xA;" OwnerUserId="14007" LastActivityDate="2020-05-10T01:11:44.153" Title="Proof by contradiction - more than one case" Tags="|proofs|examples|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18316" PostTypeId="2" ParentId="18315" CreationDate="2020-05-09T23:12:58.447" Score="18" Body="&lt;p&gt;&lt;strong&gt;(1)&lt;/strong&gt;&#xA;Here is a &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-case proof from &lt;a href=&quot;http://zimmer.csufresno.edu/~larryc/proofs/proofs.contradict.html&quot; rel=&quot;noreferrer&quot;&gt;Larry Cusick&lt;/a&gt;'s webpages:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt;. There are no rational number solutions to the equation &lt;span class=&quot;math-container&quot;&gt;$x^3 + x + 1 = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Proof&lt;/em&gt;. (Proof by Contradiction.) Assume to the contrary there is a rational number &lt;span class=&quot;math-container&quot;&gt;$p/q$&lt;/span&gt;, in reduced form, with &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; not equal to zero, that satisfies the equation. Then, we have &lt;span class=&quot;math-container&quot;&gt;$p^3/q^3 + p/q+ 1 = 0$&lt;/span&gt;. After multiplying each side of the equation by &lt;span class=&quot;math-container&quot;&gt;$q^3$&lt;/span&gt;, we get the equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$p^3 + p q^2 + q^3 = 0 \;.$$&lt;/span&gt;&#xA;There are three cases to consider. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) If &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; are both odd, then the left hand side of the above equation is odd. But zero is not odd, which leaves us with a contradiction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) If &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is even and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is odd, then the left hand side is odd, again a contradiction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) If &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is odd and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is even, we get the same contradiction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fourth case&amp;mdash;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; even and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; even&amp;mdash;is not possible because we assumed that &lt;span class=&quot;math-container&quot;&gt;$p/q$&lt;/span&gt; is in reduced form. This completes the proof.&#xA;&lt;hr /&gt;&#xA;&lt;strong&gt;(2)&lt;/strong&gt;&#xA;Another &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-case example may be found in Euclid's &lt;em&gt;Elements&lt;/em&gt;,&#xA;Book I, Prop.6,&#xA;which uses the &lt;em&gt;law of trichotomy&lt;/em&gt; for lines:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Proposition&lt;/strong&gt;. If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI6.html&quot; rel=&quot;noreferrer&quot;&gt;David Joyce&lt;/a&gt;'s webpage:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;If &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; does not equal &lt;span class=&quot;math-container&quot;&gt;$AC$&lt;/span&gt;, then one of them is greater.&quot; There are three cases: &lt;span class=&quot;math-container&quot;&gt;$AB &amp;lt; AC$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$AB = AC$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$AB &amp;gt; AC$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-05-10T01:11:44.153" LastActivityDate="2020-05-10T01:11:44.153" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18317" PostTypeId="2" ParentId="18306" CreationDate="2020-05-09T23:40:47.513" Score="3" Body="&lt;p&gt;Here's a drawing of a house&lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/9bm7G.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9bm7G.gif&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;br&gt;&#xA;Can you draw it without lifting your pencil?  What about if the roof is removed?  What about two houses side-to-side?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Here’s the floor plan for a house with some doors (some of which lead to the outside)&lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/G94DY.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/G94DY.gif&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;br&gt;&#xA;Is it possible to draw a path that walks through every door, without going through any door twice?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Then you can talk about how graph theory is used in GPS systems for navigation, for packet routing on the Internet, and even in video games for path finding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iwLjJ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iwLjJ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6120" LastActivityDate="2020-05-09T23:40:47.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18318" PostTypeId="1" AcceptedAnswerId="18320" CreationDate="2020-05-10T11:27:39.347" Score="21" ViewCount="6084" Body="&lt;p&gt;In the UK, calculus taught in secondary school focuses mainly on computation of derivatives and integrals and solving simple differential equations. There is a small amount of discussion about limits and the definition of the derivative, but students are not required to know things like: proofs of the product, quotient and chain rules, riemann integration, FTC, MVT, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examination questions (national exams) usually test ability to perform calculations rather than deeper understanding, so from a results standpoint, there is not much incentive for teachers to spend a long time motivating concepts or discussing proofs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also realise there has to be a balance between a rigorous presentation of the material and student enthusiasm and interest. There is a risk that students might become discouraged if things are done too formally. For example, I do not think there would be much value in showing students the &lt;span class=&quot;math-container&quot;&gt;$\varepsilon-\delta$&lt;/span&gt; definition of a limit. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How rigorous should high school calculus be? Which things are better left for later and why? &lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2020-07-02T23:38:37.437" Title="How rigorous should high school calculus be?" Tags="|secondary-education|mathematical-pedagogy|calculus|" AnswerCount="6" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="18319" PostTypeId="2" ParentId="18318" CreationDate="2020-05-10T13:36:16.997" Score="6" Body="&lt;p&gt;Students who go on to be math majors will get a later course in theoretical calculus.  The vast majority of science and technology students (i.e. non math majors) will NEVER have such a class.  And not need it, either (to support their mechE, chemistry, etc. majors courses).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I think the current approach is fine.  The math majors get taken care of with rigor, later.  The others don't, but don't need it.  This is more efficient than cramming rigor into people who will never need it.  And it preserves option flexibility for students not sure if they will study math or science, after high school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also add that it is not pedagogically clear that rigor prior to manipulation is the best way to learn a difficult topic, given the inherent imperfections of &quot;meat computers&quot;.  For instance, would you try to force first graders to learn formal properties of numbers before arithmetic?  Would you make high school algebra students learn Galois theory before solving quadratics or factorable higher polynomials?  Do I have to have a perfect handstand (which is non-trivial on rings, try it) before I'm allowed to just swing circles on the apparatus (as a beginning gymnast).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think the current approach is just ducky.  They even really get SOME exposure to the theory topics, but are not required to master them.  Maybe a little bit like how LaPlace transforms are handled within the time restrictions of a typical ODE class.  This is done as an exposure, with some basic translation back and forth.  However, transform mastery or derivations are really only done by the EEs and systems engineers, who use that topic a lot.  And they do that in more specialized classes, later.  But for the mechEs, at least they have briefly seen it.  So if some minority of them (say doing controls work) needs to dive in more later, they'll at least have sort of heard of it before before diving into more difficult/detailed work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that said, I took AP BC in the early 80s at a rather competitive public US high school.  So we did see the epsilon delta.  And I was fine with it, but I was spending an inordinate amount of time on calc class.  But most of the (pretty strong) class hated it and had other demands on their time from tough courses in chem, English, etc.  And the calc teacher said, you won't need any of this for the rest of the course when we do partial fractions and related rates and all that jazz. And she was right.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;OK, yeah, yeah, evidence by anecdote.  But the point really is that very few people will need/benefit from the rigor push, when learning high school calculus.  And it is important to think of the overall audience. NOT &quot;well, I could handle it&quot; or &quot;well, I like it&quot;, but instead think about the audience and their needs/desires, which may be DIFFERENT from yours.&lt;/p&gt;&#xA;" OwnerUserId="14012" LastEditorUserId="14012" LastEditDate="2020-05-10T14:03:37.477" LastActivityDate="2020-05-10T14:03:37.477" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18320" PostTypeId="2" ParentId="18318" CreationDate="2020-05-10T14:36:47.420" Score="54" Body="&lt;p&gt;Not very rigorously at all, but that doesn't (and shouldn't) mean just memorizing calculations.  (I should add that I'm basing this on my experience teaching calculus to non-major college students, but I think the relevant issues are similar.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematicians have a bad habit of conflating rigor with conceptual understanding.  A lot of this seems to come out of mathematical training, where we internalize that something hasn't been said clearly until it's been said in the rigorous language of formal mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for non-mathematicians (and, secretly, most mathematicians) that's not really how understanding works, and it's &lt;em&gt;really&lt;/em&gt; not true for high school students.  Most high school students don't have the background to understand or make sense of proofs, so teaching calculus &quot;rigorously&quot; isn't really a step to conveying the concepts: it's play-acting, where the teacher goes through some steps to satisfy the teacher's internal sense of what a justification is, while the students watch.  (Witness what happens to − in a lot of non-major calculus courses which keep them: students do fill-in-the-blank &quot;proofs&quot; where they &quot;prove&quot; specific values of limits from a small class of functions, because they don't actually understand how to work with the quantifiers, but they can memorize how to do the specific calculations.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That doesn't mean the only thing left to do is plug-and-chug calculations.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;There are non-rigorous conceptual explanations.  These have real explanatory value, even when they're not precise or might not cover cases which aren't going to be considered in the course.  (People did calculus with informal infinitesimals for centuries!)  Even for students who are laser focused on the exam, these can be useful because understanding at this level makes it easier to memorize, problem solve, and catch one's own mistakes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There's a lot of space for laying the foundations of more rigorous math for students who might go on to study more later.  I'm always astounded at how much we talk about covering &lt;em&gt;proofs&lt;/em&gt; in calculus for students who don't even know what a &lt;em&gt;theorem&lt;/em&gt; is: students at this level often struggle with applying theorems with the form &quot;if ... then ...&quot;.  They treat the conclusion as true in all situations, or they use the converse instead, or they decide they can't use it in a situation where the variables have different letters.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;One of the reasons there's not much point in dealing with rigorous proofs at this level is that students don't know what they're for.  Mathematicians care about proofs because they solve a problem; but students don't know what the problem is, so they're not likely to care about the solution.  One of the most useful things a calculus course can do to prepare students for later math is to introduce the problem: that we have a lot of intuition for very nice functions like low degree polynomials, and only some of it extends to more complicated functions (inverse trig functions, rational functions, and so on).  Theorems like IVT, MVT, and FTC which apply to many but not all of the functions students know, provide a chance to begin thinking about those issues.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2020-05-10T14:36:47.420" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18321" PostTypeId="2" ParentId="18318" CreationDate="2020-05-10T19:15:12.170" Score="8" Body="&lt;p&gt;The question really is: how individualized is your high-school calculus course ? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have a kid who reads Godel Escher Bach and is fascinated by formal logic etc then probably you should offer a pretty rigorous calculus with &lt;span class=&quot;math-container&quot;&gt;$\epsilon \delta$&lt;/span&gt; and basic topology and attempts at proofs of most theorems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If on the other hand, your student intends to be an engineer then your focus might be more on how to apply Theorems and how to interpret derivatives and integrals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All students of calculus should be taught the major theorems of calculus (IVT, MVT, EVT and FTC I and II etc.) Furthermore, I think all students of calculus should be taught how calculus extends what we knew from algebra alone. For example, the binomial series vs. the binomial theorem. Compare and contrast. I suspect graphing and interpreting the derivative is largely short-changed in our current calculus because interpreting derivatives requires a mastery of inequalities we're afraid to test in the general population. Calculating a derivative is easy. Finding which &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; make &lt;span class=&quot;math-container&quot;&gt;$df/dx &amp;gt;0$&lt;/span&gt; is almost always much harder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the problems I just described are not what we would consider &quot;proofs&quot; formally speaking, but they require logical analysis which is more difficult than epsilon-delta proof &lt;span class=&quot;math-container&quot;&gt;$\lim_{x \rightarrow 1}(3x-7) = -4$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what is the audience ? Can they factor a polynomial without flinching ? Can they solve a quadratic inequality ? Do they have a clear understanding of basic function graphing ? What about trigonometry, do they have mastery of identities and domains and solution sets to reasonable trigonometric equations ? Depending on these questions we may or may not be able to offer a real version of calculus. We may have to be careful to not offend the audience by teaching them prerequisite material in the midst of this wild new world of limits, derivatives and integrals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as teaching to the test goes, it would be nice if there was a lot less predictability in the test. It ought to be a goal to make the only way to prepare for the test is to teach the whole of the subject. This includes proofs. There ought to be a reward for students who do more than just learn routine calculations. But, until that day, it is crucial for teachers to teach as if there was no test. Otherwise, well, look at the schools in America...&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-05-10T19:15:12.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18322" PostTypeId="2" ParentId="18306" CreationDate="2020-05-11T13:57:54.830" Score="2" Body="&lt;p&gt;Typical topics are the Dijkstra and Floyd algorithms to find the shortest way between two nodes in a graph (used e.g. for NPC wayfinding in games, navigation etc.) or typical problems of IT theory (e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Graph_coloring&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Graph_coloring&lt;/a&gt; vertex coloring).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could also cover neural net types that are build upon a graph-like structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually, if I have to teach my university class about graph theory, the simple properties of the graph (undirected, directed, connected, fully connected, path etc.) takes up to 15 minutes. So I would cover Dijkstra or Floyd (because way finding in graphs is very important), maybe program a maze solver and be done with it. That gives a quick summary of what graphs are and one specific usage case.&lt;/p&gt;&#xA;" OwnerUserId="14022" LastEditorUserId="376" LastEditDate="2020-05-11T14:55:53.343" LastActivityDate="2020-05-11T14:55:53.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18323" PostTypeId="1" CreationDate="2020-05-11T23:49:50.653" Score="-4" ViewCount="119" Body="&lt;p&gt;I typically will write my own slides for a course, and then may make a screen recorded video of me talking over the slides. Now I'm planning on doing a screen recording over slides the publisher has included with our textbook, and posting the video to our learning platform (D2L, Moodle, etc.). I do have to edit the slides before doing this. Is there any legal issue here at any level? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What if I remove the annoying copyright at the bottom of the (title) slide(s)?&lt;/p&gt;&#xA;" OwnerUserId="11880" LastActivityDate="2020-05-12T14:10:56.523" Title="Legality of posting dictated video of publisher's slides" Tags="|presentation|" AnswerCount="1" CommentCount="8" ClosedDate="2020-05-12T16:44:23.057" ContentLicense="CC BY-SA 4.0" />
  <row Id="18324" PostTypeId="2" ParentId="18323" CreationDate="2020-05-12T14:10:56.523" Score="1" Body="&lt;p&gt;Some relevant sections from &lt;a href=&quot;https://www.copyright.gov/title17/92chap5.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.copyright.gov/title17/92chap5.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(c) Fraudulent Copyright Notice.—Any person who, with fraudulent&#xA;  intent, places on any article a notice of copyright or words of the&#xA;  same purport that such person knows to be false, or who, with&#xA;  fraudulent intent, publicly distributes or imports for public&#xA;  distribution any article bearing such notice or words that such person&#xA;  knows to be false, shall be fined not more than \$2,500.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(d) Fraudulent Removal of Copyright Notice.—Any person who, with&#xA;  fraudulent intent, removes or alters any notice of copyright appearing&#xA;  on a copy of a copyrighted work shall be fined not more than \$2,500.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am not a lawyer, but you should consider that 'not more than \$2500' might be interpreted as a per incident cap. i.e. per student, or worse, per download.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the publisher will probably give you wide latitude fine with their material, but you should contact them for details and permission.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2020-05-12T14:10:56.523" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18325" PostTypeId="2" ParentId="15263" CreationDate="2020-05-12T20:20:04.637" Score="4" Body="&lt;p&gt;Nick C mentions in a comment that there are a handful of Great Books colleges that currently use Euclid in their course of studies. I went to one of these and we devoted our entire freshman year of math (8 semester-hours total) to the &lt;em&gt;Elements&lt;/em&gt;. No specific edition was required but almost everyone used the Heath translation as found in either the multi-volume Dover paperback set, or the Brittanica Great Books collection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The general procedure was this: the teacher assigned a handful of propositions at the end of each class period; we had to learn them before the next class; and then in class the teacher would call students at random to demonstrate each of the assigned propositions at the board and field questions from the rest of the class. Then there would be open discussion of the prop.  Using this method we covered a great deal of the &lt;em&gt;Elements&lt;/em&gt;; it's been almost 20 years so I don't remember in a lot of detail but glancing over a summary of the text I'd say we worked through a good 75% of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We also spent a fair amount of time discussing the front matter of each book (definitions, postulates, etc.) and we had to write a paper or two as well as essay-format final exams at the end of each semester; in all of these we had to demonstrate a theoretical understanding of the geometrical concepts we had been working with throughout the year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the answers above suggest that the 'demonstrate propositions' approach to geometry is basically just rote memorization, but my experience was that after the first few propositions of Book I, it's more or less impossible to do this sort of thing by rote; if you don't understand why each step of each proposition follows from the previous one, and if you don't have a bigger-picture grasp of the overall logic and structure of each book, you will probably fail this sort of class -- because eventually you just won't be able to get through a proposition at the board, and you certainly won't be able to speak or write intelligently about it even if you can reproduce it mechanically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I have no idea how this neo-approach compares to historical approaches.&lt;/p&gt;&#xA;" OwnerUserId="14033" LastActivityDate="2020-05-12T20:20:04.637" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18326" PostTypeId="1" CreationDate="2020-05-12T21:44:03.640" Score="7" ViewCount="238" Body="&lt;p&gt;We all know the (apparently verified&lt;sup&gt;1&lt;/sup&gt;) anecdote recounting &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/George_Dantzig&quot; rel=&quot;nofollow noreferrer&quot;&gt;George Dantzig&lt;/a&gt;&#xA;arriving late to a lecture (by Jerzy Neyman), and later solving two open&#xA;problems written on the board, mistaking them for homework.&#xA;My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. Are there other examples of a similar misunderstanding&#xA;  that led to the solution of an unsolved problem?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Perhaps even in an exam, where an exam question's novel solution led to&#xA;a new perspective on the topic?&#xA;This possibility inspired by &lt;a href=&quot;https://matheducators.stackexchange.com/q/18297/511&quot;&gt;Interesting things you learned while grading?&lt;/a&gt;.&#xA;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;&#xA;Mentioned in &lt;a href=&quot;https://supernet.isenberg.umass.edu/photos/gdobit.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dantzig's obituaries&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-05-13T00:13:54.213" LastActivityDate="2020-05-22T22:18:49.097" Title="Solving open problems through a misunderstanding" Tags="|history|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18327" PostTypeId="1" CreationDate="2020-05-13T04:45:01.787" Score="21" ViewCount="697" Body="&lt;p&gt;I recently taught an introduction to real analysis. I assigned a (covid-induced) take-home final, which included the question: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Define the set S by&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\bigcup_{n=1}^\infty \left\{&#xA;\frac{a}{2^n}\colon 0\le a\le 4^n, a\in\mathbb{Z} \right\}.&#xA;$$&lt;/span&gt;&#xA;  Identify &lt;span class=&quot;math-container&quot;&gt;$\bar S$&lt;/span&gt;, showing that &lt;span class=&quot;math-container&quot;&gt;$\bar S$&lt;/span&gt; is what you claim. [ Hint: for any real number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and any positive integer &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, there exists &lt;span class=&quot;math-container&quot;&gt;$a \in \mathbb Z$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$kx \in [a,a+1)$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$|x−\frac a k|&amp;lt;\frac 1 k$&lt;/span&gt;.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Shockingly (to me), I received 9 essentially identical  solutions that all contained the same serious, and somewhat subtle error.&#xA;I first suspected a &quot;homework help&quot; site, and indeed found that the question had been posted and answered on chegg (a colleague has subscribed to that web site, and showed me the solution that was posted there, which was identical to the incorrect solution). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, that's not the end of the story! Another student, when challenged said she &quot;had never heard of chegg&quot;, and claimed she had found it in some materials she had been studying &quot;before the exam&quot;. She then gave me a link to a &lt;a href=&quot;https://github.com/navidgator/SphereMesh/blob/master/Math_Analysis/PR1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;github repository&lt;/a&gt; (see question 19) that appears to be dated 3 years ago that contained almost exactly my question (that I thought I was the first to invent!) -- with the same incorrect solution as appears on chegg. Given the quite distinctive formatting of the solutions and the distinctive error that they all contained, I am convinced this is not a case of two people independently making the same mistake.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;My question&lt;/strong&gt;: How were the &quot;cheggspert&quot; (i.e. the person employed by chegg to solve students' exam questions) and my student able to locate the github repository containing this question? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Needless to say, I harbour the hope that understanding how this kind of cheating takes place will give me ways to prevent it in the future. &lt;/p&gt;&#xA;" OwnerUserId="14035" LastEditorUserId="5220" LastEditDate="2020-06-03T22:36:24.227" LastActivityDate="2020-06-04T16:41:41.227" Title="How do I track down the sources of solutions which students have used to cheat on exams?" Tags="|assessment|cheating|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="18328" PostTypeId="2" ParentId="18318" CreationDate="2020-05-13T08:12:23.577" Score="0" Body="&lt;p&gt;I think that high school calculus is not supposed to be too rigorous. Calculus is a subject introduced in high school, and if it is too theoretical, then students  can have a hard time understanding it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is pretty reasonable to think that proofs of rules of differentiation(like chain, quotient, product etc.) must be told. But as far as course is concerned one is not required to do much with the proofs of the rules. Important thing is to understand their applications in various domains of mathematics( and physics too) and being able to solve problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also its not like that proofs aren't taught for any rule. They are certainly taught and sometimes derivation is more focused upon than the rule itself, especially in integration and differential equation part. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also proofs are meant to be actually studied, if one goes on to major in mathematics in their college years. But at high school levels it isn't very much needed. &lt;/p&gt;&#xA;" OwnerUserId="14038" LastActivityDate="2020-05-13T08:12:23.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18329" PostTypeId="2" ParentId="18318" CreationDate="2020-05-13T09:12:52.803" Score="0" Body="&lt;p&gt;Speaking as an A Level physics teacher having seen what maths students do, they do stuff like quotient rule and chain rule. Hell, I even teach exponential decay with caclculus if I have a mathematically inclined class. But to get a C/4/5 grade in the edexcel maths papers you needed 21%. To get a 6/7/B you needed about 45-50%. In the UK schools I've worked at, they start the GCSE content in year 9 because of the extra content and rigour to cover. Add to all of this the peverse incentive to achieve good inspection grades and GCSE results, you have a factor in the teach to the test mentality and reduced time to teach important mathematical foundations. There are wider issues with recruitment itself. Being so desperate for maths teachers, I've seen economists and computer scientists recruited as head of maths. Sometimes I've even seen no pure mathematicians in a department. As much as I love calculus, any curriculum changes need to be thought out carefully. &lt;/p&gt;&#xA;" OwnerUserId="14039" LastActivityDate="2020-05-13T09:12:52.803" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18330" PostTypeId="2" ParentId="17852" CreationDate="2020-05-13T12:12:21.087" Score="3" Body="&lt;p&gt;I can only answer from my own experience but for me reading Euclid in college was life-changing. I had always been &quot;good&quot; at math -- went as far as I could go in high school (Calculus BC) and did very well in it. But I also never entirely saw the point, besides challenging myself, since I didn't have any intention of going into a math-intensive profession. (I am in fact a software developer but the business code that I write doesn't involve much math)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reading Euclid opened up a very different perspective -- for the first time I saw math as worth knowing in its own right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like to reflect on the way that Euclid approaches the Pythagorean theorem vs the way one typically encounters it in high-school geometry. The conventional approach might spend a little time on a proof or two, but seems mainly aimed at providing students with a useful tool for calculating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid, on the other hand, presents the theorem as the culmination of Book I of the &lt;em&gt;Elements&lt;/em&gt;. The first 46 propositions lead directly to it, and then once he has proven the theorem (prop. 47) and its converse (prop. 48), he's done, and moves on to another branch of geometry in Book II. There are no calculation exercises that follow because, it seems, Euclid is mainly interested in helping us to &lt;em&gt;see&lt;/em&gt; the theorem. In other words, the &lt;em&gt;Elements&lt;/em&gt; presents a fundamentally speculative approach to math rather than a practical one. And this is beautiful and delightful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to answer your bottom-line question: Any text that can open a student's eyes to the beautiful and delightful world of speculative math is worth studying, no matter how old it is.&lt;/p&gt;&#xA;" OwnerUserId="14033" LastActivityDate="2020-05-13T12:12:21.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18331" PostTypeId="2" ParentId="18302" CreationDate="2020-05-13T12:44:17.130" Score="2" Body="&lt;p&gt;The idea that math-study imparts certain fundamental intellectual skills was taken for granted for most of the history of Western education, and reaches back at least to Plato, who (reportedly) refused to accept students who had not mastered geometry. &quot;Let no one ignorant of geometry enter here&quot; was supposedly inscribed over the entrance to his Academy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In late Antiquity and the Medieval period the term 'liberal arts' designated an elementary course of studies in language arts, logic, and math. The majority of the traditional liberal arts (four of the seven) were mathematical, and mastery of all seven arts was considered an essential pre-requisite to higher studies in philosophy, theology, or one of the professions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You were probably asking for a different sort of literature than what I'm about to suggest, but it wouldn't hurt to become familiar with the older tradition, if you're not already.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some ancient and medieval texts that discuss the educational theory are Aristotle's &lt;em&gt;Nicomachean Ethics&lt;/em&gt;, Boethius's &lt;em&gt;De Trinitate&lt;/em&gt;, Aquinas's commentaries on both of the former as well as a few questions in his &lt;em&gt;Summa Theologiae&lt;/em&gt;, and Hugh of St. Victor's &lt;em&gt;Didascaliae&lt;/em&gt;. And since the &quot;Great Books&quot; revival of the mid-20th century there has been an enormous amount written about these topics. A heavily-footnoted survey of all of this is Clark &amp;amp; Jain, &lt;em&gt;The Liberal Arts Tradition&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14033" LastActivityDate="2020-05-13T12:44:17.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18332" PostTypeId="2" ParentId="17999" CreationDate="2020-05-13T15:48:57.100" Score="2" Body="&lt;p&gt;Have them work through a few chunks of Euclid's &lt;em&gt;Elements&lt;/em&gt;. Then they are doing proof, rather than reading about logic in a theoretical way, which is probably too abstract for kids in the age group you're targeting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The theory of logic will make much more sense to them after they have had an opportunity to practice it, and the whole point of Euclid's text is to walk you step by step through the practice of geometry. In my experience children of that age respond extremely well to the actual text of Euclid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I described how I learned Euclid &lt;a href=&quot;https://matheducators.stackexchange.com/a/18325/14033&quot;&gt;here&lt;/a&gt; -- it was a college course but the general approach should be possible for younger students as long as the pace is adjusted.&lt;/p&gt;&#xA;" OwnerUserId="14033" LastActivityDate="2020-05-13T15:48:57.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18333" PostTypeId="1" AcceptedAnswerId="18339" CreationDate="2020-05-14T04:15:35.513" Score="5" ViewCount="206" Body="&lt;p&gt;&lt;a href=&quot;https://redd.it/9pwsv9&quot; rel=&quot;noreferrer&quot;&gt;This r/math post spurred this question&lt;/a&gt;. To concretize the meaning of &quot;ace&quot;, assume&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;success means earning professorial tenure at a Top 100 world university.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;your liked stronger branch doesn't exactly relate to your weaker branch.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;E.g. can you ace Stochastic Calculus and Financial Mathematics, while flubbing Abstract Algebra? &lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2020-05-15T01:39:17.677" Title="Can you ace one branch of math, while bumbling another branch?" Tags="|self-learning|" AnswerCount="3" CommentCount="4" ClosedDate="2020-05-21T16:05:50.300" ContentLicense="CC BY-SA 4.0" />
  <row Id="18334" PostTypeId="2" ParentId="18333" CreationDate="2020-05-14T11:38:50.073" Score="3" Body="&lt;p&gt;Absolutely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math is such a hugely diverse field that it's been over a century since anyone could reasonably claim to understand the whole thing.  That kind of diversity creates subbranches and specialties that have absolutely nothing to do with each other.  To get an idea of how diverse the field is, take a look at &lt;a href=&quot;https://www.youtube.com/watch?v=OmJ-4B-mS-Y&quot; rel=&quot;nofollow noreferrer&quot;&gt;this great video&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4845" LastActivityDate="2020-05-14T11:38:50.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18335" PostTypeId="2" ParentId="18333" CreationDate="2020-05-14T12:20:02.583" Score="4" Body="&lt;p&gt;It often takes many years to develop a strong foundation and an accurate intuition in an area of mathematics to become steadily productive in that area. By that point, one might be proficient in, say, elliptic curves and modular forms, but relatively unschooled in, say, combinatorics on words. So I agree with @G.Allen: The answer to the OP's question is &lt;em&gt;Yes&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a broader issue not easily identified with expertise in particular subfields of mathematics, discussed in Gower's &lt;em&gt;The Two Cultures of Mathematics&lt;/em&gt;. The two cultures&#xA;are, loosely,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;the distinction between mathematicians who regard their central&#xA;  aim as being to solve problems, and those who are more concerned with building and understanding theories.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But this takes us far from the OP's question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;Gowers, William Timothy. &quot;The two cultures of mathematics.&quot; &lt;em&gt;Mathematics: Frontiers and Perspectives&lt;/em&gt; (V. Arnold et al., eds) (2000): 65-78. &lt;a href=&quot;https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-05-15T00:41:21.167" LastActivityDate="2020-05-15T00:41:21.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18336" PostTypeId="2" ParentId="18297" CreationDate="2020-05-14T13:41:42.017" Score="7" Body="&lt;p&gt;I gave an advanced course on Probability that contained some ergodic theory. In exercises, I outlined the usual proof of the equidistribution of &lt;span class=&quot;math-container&quot;&gt;$e^{in\theta}$&lt;/span&gt; on the circle, for &lt;span class=&quot;math-container&quot;&gt;$\theta/\pi$&lt;/span&gt; irrational. The proof I knew was generalizing equidistribution from indicators of intervals to arbitrary (say, continuous) functions and then using Fourier transform.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then one of the students pointed out the following elementary solution. Assume that &lt;span class=&quot;math-container&quot;&gt;$I,J$&lt;/span&gt; are half-open intervals on the circle, and &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is longer than &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt;. Then, you can write &lt;span class=&quot;math-container&quot;&gt;$I=I_1\sqcup I_2$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$I_2$&lt;/span&gt; is a translation of &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt; that follows &lt;span class=&quot;math-container&quot;&gt;$I_1$&lt;/span&gt; counterclockwise. Let &lt;span class=&quot;math-container&quot;&gt;$n_1$&lt;/span&gt; be the first time &lt;span class=&quot;math-container&quot;&gt;$\exp(i\theta n)$&lt;/span&gt; belongs to &lt;span class=&quot;math-container&quot;&gt;$I_1$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$n_2$&lt;/span&gt; is the first time after &lt;span class=&quot;math-container&quot;&gt;$n_1$&lt;/span&gt; that it belongs to &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt;.  Then, &lt;span class=&quot;math-container&quot;&gt;$\exp(i(n+n_2)\theta)\in J$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$\exp(i(n+n_1)\theta)\in I$&lt;/span&gt;, which readily implies &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{1}{N}\#\{n\leq N:\exp(in\theta)\in J\}=\frac{1}{N}\#\{n_2\leq n\leq N:\exp(in\theta)\in J\}+o(1)\leq \frac{1}{N}\#\{n\leq N:\exp(in\theta)\in I\}+o(1).&#xA;$$&lt;/span&gt;&#xA;This means that  &lt;span class=&quot;math-container&quot;&gt;$\liminf$&lt;/span&gt; of the quantity on the right is greater than &lt;span class=&quot;math-container&quot;&gt;$\limsup$&lt;/span&gt; of the quantity on the left. From this and additivity of density the result easily follows.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastEditorUserId="10349" LastEditDate="2020-05-29T19:05:25.890" LastActivityDate="2020-05-29T19:05:25.890" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18338" PostTypeId="1" CreationDate="2020-05-14T21:25:47.560" Score="19" ViewCount="3901" Body="&lt;p&gt;I hope the essence of the question is clear from the title. There are obvious advantages to making the linear map the central notion of a linear algebra course: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the notion can be illustrated with nice, intuitive geometric (or otherwise) examples, motivating the whole subject; &lt;/li&gt;&#xA;&lt;li&gt;the composition of maps is a very natural thing, from which the definition of multiplication of matrices flows smoothly;&lt;/li&gt;&#xA;&lt;li&gt;many results, such as the structure of a solution space to a linear system, of spectral theorem for self-adjoint operators, can be nicely visualized;&lt;/li&gt;&#xA;&lt;li&gt;many proofs (e.d., associativity/distributivity of the matrix product) become simple and conceptual, replacing unenlightening computations;&lt;/li&gt;&#xA;&lt;li&gt;the notion of determinant can be given its geometric meaning (or, if you dare, the true meaning as the action on top exterior degree), and the reason for the product rule becomes obvious;&lt;/li&gt;&#xA;&lt;li&gt;After all, that's just &lt;em&gt;right&lt;/em&gt;, mathematically and physically: the primary object is the linear space and the linear map, the basis, and thus the matrix, is a matter of choice, secondary, technical devices.&lt;/li&gt;&#xA;&lt;li&gt;Many people actually complain that they &quot;never get the linear algebra&quot;, that it's all abstract nonsense for them, so why not try and equip them with the way &lt;em&gt;we, mathematicians&lt;/em&gt;, understand it?  &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Yet I have the impression that a typical undergrad linear algebra course, especially a &quot;service&quot; course, never goes this way. For example, Strang's textbook first introduces &quot;the concept of linear map&quot; on page 400, after most of the theory has been covered, almost as to make sure it's left off any actual course due to lack of time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can  also see some reasons to actually prefer the matrix approach:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the formal manipulation are easier to teach (and, especially, easier to grade);&lt;/li&gt;&#xA;&lt;li&gt;before, the students have been taught Math as a set of formal rules applied to numerical examples, so they are well prepared to be fed more of those and are unable to learn to reason with abstract notions such as &quot;linear space&quot; and &quot;linear map&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;However, neither of these reasons are satisfactory. We could say that if the students are unable to grasp these concepts, we would serve them better by training them in that, rather than teaching them the singular value decomposition without a proof. We could also say that if the conceptual understanding is so valuable for them, one should actually allocate some resources into that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, are there any actual reasons to prefer the matrix approach? &lt;em&gt;Why&lt;/em&gt; is it so popular?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; since many understood this post as advocating throwing matrices away altogether, let me tell that this is not the case. I admit that matrices are important. What I don't understand is the pedagogical value of, for example, exposing students to matrix multiplication before (or without ever) discussing compositions of linear maps. &lt;/p&gt;&#xA;" OwnerUserId="10349" LastEditorUserId="238" LastEditDate="2020-05-17T19:18:17.247" LastActivityDate="2021-03-18T00:38:33.637" Title="Why do some linear algebra courses focus on matrices rather than linear maps?" Tags="|proofs|linear-algebra|" AnswerCount="8" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18339" PostTypeId="2" ParentId="18333" CreationDate="2020-05-15T01:39:17.677" Score="7" Body="&lt;p&gt;I think it matters what you mean by flub.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a couple specific exceptions, but I think any of my colleagues in my department could teach any of our undergraduate courses in an emergency.  Would they be the best person at it?  No.  Could they do a competent job for an emergency situation and be helpful to almost all the students in learning the material, even perhaps as they are just reading the textbook a week ahead of the students?  Yes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are general mathematical skills of reading and understanding theorems, proofs, and definitions, and coping with abstract concepts.  They are usually enough to get through any undergraduate material, and almost all mathematicians have mastered these skills in a general context well enough to learn and even teach all undergraduate material on their own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are of course exceptions, but they are unusual.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a higher level, I wouldn't feel at all comfortable teaching a first-year graduate class in all but a few areas of mathematics, and I think I would not have a reasonable chance of learning some of these areas in a reasonable amount of time (but other areas I could), and I would not be able to do decent research in them even if I took as much time as I needed to retrain.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2020-05-15T01:39:17.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18340" PostTypeId="2" ParentId="18338" CreationDate="2020-05-15T01:56:36.757" Score="9" Body="&lt;p&gt;You might know (or not) enough computer science to know there are such things as functional programming languages.  These are programming languages (the most popular are probably Scheme, ML, and Haskell) where loops and variables are avoided, computation is largely done by recursion, and, most importantly, functions are treated like data and can be passed as parameters to other functions.  (For example, they have a library function, usually called something like 'map', which takes a unary function f and a list L as input, and returns a list whose contents are (in order) f(l) for each element l in L.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is an important and useful style of programming, but probably a majority of computer science departments spend less than 3 weeks (in their programming languages course) on it over an entire degree.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why?  They have accepted that most of their students will never be able to think of a function as an actual object.  Their students will not be able to distinguish between a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and its abstract application &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, even after spending a few weeks working with this idea with a skilled instructor.  (To be fair, our notation in calculus is terrible for this.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are way overestimating the cognitive abilities of an average person.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2020-05-15T01:56:36.757" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18341" PostTypeId="2" ParentId="18338" CreationDate="2020-05-15T02:00:51.800" Score="22" Body="&lt;p&gt;Welcome Kostya! The mapping view is definitely important, but I don't think it's supreme. For me here's how I think about it. There are three ways to think about (basic) linear algebra: As a theory of matrices, linear maps and systems of linear equations. It is a fascinating result that when you formalize these three views the mathematical objects are isomorphic! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But each of the three views is useful for different kinds of applications, even very similar ones. If I'm trying to solve a system of linear equations, say to fix the constants for an IVP in a system of ODE's, I don't personally see how the linear transform view helps at all, whereas the systems of linear equations view is very useful. On the other hand, if I'm trying to construct a discrete dynamical system the mapping view is incredibly clear. If I put on my statisticians hat and try to fit&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\hat{y} = XW$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; by minimizing &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;RSS(\hat{y}) = (y - \hat{y})^T(y - \hat{y}) = \sum_{i=1}^N (y-\hat{y})^2&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;then it makes much more sense to think of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; as matrices and vectors. Neither of the other views are immediately useful when defining matrix derivatives (unless we want to talk about differential forms). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certain problems are easy to define under one view and not under others. I personally try to teach all three views so that students have a wider toolkit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, sometimes with linear algebra there is a beautiful trick you can pull where you take a problem that is most clearly defined under one view and you switch to another view and extract some deep insight about its solution. Eigenvectors and linear maps are an excellent example of this. But it's the fluidity here that makes it useful, not the rigidity of fixing ourselves to a single view. &lt;/p&gt;&#xA;" OwnerUserId="9019" LastEditorUserId="376" LastEditDate="2020-05-15T06:16:07.573" LastActivityDate="2020-05-15T06:16:07.573" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18342" PostTypeId="1" CreationDate="2020-05-15T09:18:03.960" Score="9" ViewCount="1310" Body="&lt;p&gt;In his book &quot;Surely you're joking Mr. Feynman&quot;, Richard Feynman relates the following story. As he was supervising a group of calculators for Manhattan project, he at some point gave them a lecture on what they were actually calculating. He claimed that this improved their performance - even though their job was, essentially, that of arithmetic gates in a computational scheme, so the lecture wasn't directly useful in their work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is there evidence for a similar effect in teaching mathematics?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, suppose I want to teach Gauss elimination. Would students learn better if I chose examples from some actual real-world problems? Or, when teaching determinants, many students are happy when presented with the &quot;volume&quot; interpretation, but does it actually enhance learning of the usual algebraic/computational aspects? When teaching integration or ODEs, is it beneficial to choose examples relevant to physics/biology/...? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally, is providing a context/big picture helpful or a waste of time when teaching &quot;hard skills&quot;? Disregarding the &quot;added value&quot; of the students knowing the big picture, being able to apply the skills, etc., does the learning of skills per se improve? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm mostly interested in rigorous research answering this, but anecdotal evidence is also welcome.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2020-05-18T22:25:23.943" Title="&quot;Feynman effect&quot; in teaching mathematics" Tags="|reference-request|student-motivation|concept-motivation|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18343" PostTypeId="2" ParentId="18342" CreationDate="2020-05-15T11:26:00.507" Score="4" Body="&lt;p&gt;I think it's useful but can be overdone and/or can make things more difficult.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I actually think it is easier to learn ODEs first, in the mechanical way (e.g. solving the linear second order equation).  Initial practice should be on more algebraic problems, not applied.  This is because word problems are harder.  After mastering the process with equations, you can do some applications, to see what the process is good for, and for some physical insights (especially the forcing function) and really just for more practice volume.  Even here, some examples seem to click more than others.  I find the kinematic examples more intuitive than the electrical ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the reasons I find most PDE texts hard is that they start immediately with a physical problem and teach PDE manipulation in the context of solving that problem.  Rather than the ODE approach which is more how we learned to manipulate algebra to &quot;solve for x&quot; and integrals to &quot;solve for y&quot;, before moving on to applied problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note also, that there can be a big difference between saying nothing about usefulness--for some math teachers, not even knowing applications of the equations, sometimes even proud of the ignorance--and doing detailed application practice.  An intermediate approach is to say something like &quot;this is useful in XYZ field or course&quot;.  Just assert it.  You don't need to prove it.  If done truthfully and especially by a teacher with some familiarity with other STEM classes, this just serves as motivation.  If challenged, you should know enough to back yourself up, but don't need to hone the whole lecture around use of technique A in field B.&lt;/p&gt;&#xA;" OwnerUserId="14049" LastEditorUserId="14049" LastEditDate="2020-05-15T12:16:00.927" LastActivityDate="2020-05-15T12:16:00.927" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18344" PostTypeId="2" ParentId="18220" CreationDate="2020-05-15T16:07:01.763" Score="2" Body="&lt;p&gt;In Spain these things are generally not taught at the university level and there is no course comparable to what you call College Algebra, as these topics form part of the high school curriculum. Some degree programs offer a &quot;Zero Course&quot; that reviews mathematical material from high school, but such a course usually focuses on graphing functions using basic calculus.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2020-05-15T16:07:01.763" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18345" PostTypeId="2" ParentId="7394" CreationDate="2020-05-15T19:29:51.393" Score="0" Body="&lt;p&gt;If you are searching for a website where you can mathematics worksheets for the students of primary class (grade 1 to 5), then I am suggesting you a website where you can get mathematics worksheets for class 1 to 5. I have a daughter who study in class 2. I am taking &lt;a href=&quot;https://homework.selfstudymantra.com/p/maths-homework-for-class-2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maths Homework for Class 2&lt;/a&gt; from here because it provides daily homework with solution also. So this site might be helpful for you also.&lt;/p&gt;&#xA;" OwnerUserId="14051" LastActivityDate="2020-05-15T19:29:51.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18346" PostTypeId="2" ParentId="18338" CreationDate="2020-05-15T21:36:26.533" Score="4" Body="&lt;p&gt;One reason is &lt;strong&gt;curriculum inertia&lt;/strong&gt;—canonical math courses change slowly, perhaps best measured on a timescale of generations. For multiple centuries old material such as calculus and elementary algebra, the presentations have reached an equilibrium, but linear algebra is &quot;new&quot; enough that the education system is still figuring it out (the same thing can be said about discrete mathematics). Inertia is also how I would explain that synthetic geometry is still being taught to teachers, when all others have dropped it in favour of other material, but that's a digression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;@nate's point about three views of a common theory and fluidity of formalisation is definitely an important one in understanding the character of linear algebra—I for one always found least squares optimisation to be mysterious until I realised it was an orthogonal projection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the claims made in the question regarding the superiority of the map view should be nuanced:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Matrix multiplication arises more naturally from matrix–vector multiplication than from composition of linear maps. One could of course argue that matrix–vector multiplication &lt;em&gt;is&lt;/em&gt; applying a linear map, but that's an advanced view; quite often you organise a bunch of calculations as a matrix–vector product before you become aware that the vector can be thought of as living in a vector space. The take-home message for an educator is probably that &lt;em&gt;a matrix should be thought of as something poised to act on a vector&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Taking a geometric view of determinants doesn't require defining them primarily for linear maps, and in fact it can be an obstruction, even if &lt;span class=&quot;math-container&quot;&gt;$\det(AB)=\det(A)\det(B)$&lt;/span&gt; follows immediately that way. The catch is that when starting from the linear map point of view you impose a distinction between rows and columns of a matrix—columns are geometric objects, rows are just arrays of numbers—whereas with determinants it is actually closer at hand to view the rows as geometric objects (at least if you wish to make a geometric interpretation of row operations, which you should: adding a multiple of a row to another row is just skewing the set whose volume the determinant measures, so of course the determinant remains unchanged). This is of course related to one downside of the linear map view you forgot to mention:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The transpose is a whole lot more mysterious in the linear map view than in the array of numbers view. Abstractly the transpose can be understood as the contravariant functor mapping a vector space to its dual—if &lt;span class=&quot;math-container&quot;&gt;$A\colon V \longrightarrow W$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$A^\mathrm{T}\colon W^* \longrightarrow V^*$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$A^\mathrm{T}(w^*)(v) = w^*\bigl( A(v) \bigr)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$v \in V$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$w^* \in W^*$&lt;/span&gt;—but that's really not the way one wants to describe it on a first linear algebra course.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Frankly, I find that the mainstream of linear algebra textbook authors are still struggling to free themselves of the historical bias that determinants are about solving linear equation systems.&lt;/p&gt;&#xA;" OwnerUserId="14052" LastActivityDate="2020-05-15T21:36:26.533" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18347" PostTypeId="2" ParentId="18338" CreationDate="2020-05-16T18:25:58.460" Score="-1" Body="&lt;p&gt;Let me make the case that linear maps between vector spaces is the only way that linear algebra should be taught.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rather old &quot;definition&quot; of a matrix is a set of symbols that &quot;transform&quot; according to a rule involving indices. But this &quot;definition&quot; is just the change of basis formula in a vector space for a linear map. The indices refer to a particular basis. Thus, the confusion - what &quot;transforms&quot; is the &lt;strong&gt;representation&lt;/strong&gt; of a linear map with respect to the old and new basis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ditto for the old &quot;definition&quot; of a tensor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;definitions&quot; are theorems - showing how the matrix representation of a linear map changes when the vector space basis is changed. The definition of a linear map is fundamental. The matrix representation is not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simmonds &lt;a href=&quot;https://smile.amazon.com/dp/038794088X/ref=cm_sw_em_r_mt_dp_U_Y2cWEbJ6KVE76&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Brief on Tensor Analysis&lt;/a&gt; has a side-by-side discussion of the coordinate-free representation (linear and multi-linear maps) and the older representation via indices. The coordinate-free discussion using vector spaces as the underlying algebraic structure is not only cleaner, but it is much easier to understand.&lt;/p&gt;&#xA;" OwnerUserId="14058" LastEditorUserId="14058" LastEditDate="2020-05-18T12:33:16.043" LastActivityDate="2020-05-18T12:33:16.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18348" PostTypeId="2" ParentId="18338" CreationDate="2020-05-17T00:20:07.853" Score="1" Body="&lt;p&gt;i like your approach but… it look on a way after whole journey—it’s probably quite difficult to imagine to you how you would approach the topic seeing it for a first time. You probably would have some nice geometric insight… for lines, planes, maybe for the space; this is somewhat enough to stop there (as e.g. cited strang generally stops at 2×2 matrices).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;it’s beautiful to see all of those, whole picture; it is only a matter of where to start: i, my own, started with roughly systems of linear equations and jumped quickly to matrices, and rather stopped there trying to find what’s what geometrically later on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;this is the real question: where to start to provide most natural feeling, backstory, understanding, tools but also teach something more than mere knowledge—yet it be geometric or algebraic! in most courses there’re also other factors as time, courses attended so far, courses that depend on this knowledge (which part!?) etc. all this shall be taken into account when assessing ’which approach is best and why’—i’d really like to agree with you, i even sympathise with you but it’s rather obvious for me that you didn’t teach this topic from the very beginning any freshman! (don’t feel bad: me too! :D)&lt;/p&gt;&#xA;" OwnerUserId="14059" LastActivityDate="2020-05-17T00:20:07.853" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18349" PostTypeId="2" ParentId="18338" CreationDate="2020-05-17T22:44:16.670" Score="2" Body="&lt;p&gt;As someone who is currently teaching an introductory linear algebra course and who likes matrices, let me tell you why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, here's why I like coordinates: if I want to tell you a specific position in the plane, the easiest way to do so is to tell you its coordinates. The idea that you can convert positions into tuples of numbers is extraordinarily powerful. It makes programming computers to do geometric things (ray tracing, flight simulation, anything) much easier. It's hard to imagine how modern algebraic geometry could have developed beyond lines and conics without this notion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, why do I like matrices? Because they provide coordinates on the space of linear maps. Again, this is an extraordinarily powerful jump: just by writing down four numbers, I get a transformation of the plane. By writing down 25, I get a transformation of 5-dimensional space, which maybe I hadn't even imagined was a thing until now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without this device, even in 2D, I'd need to say something like &quot;a shear by this amount with invariant direction pointing in the (17,63)-direction&quot; or what have you. You may say that we never need to dirty our hands with explicit transformations like this, but in my research, when I think about almost toric Lagrangian fibrations and mutations of polygons, messy, explicit 2-by-2 matrices make my life an order of magnitude easier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For students, this idea that you can encode a &lt;em&gt;transformation&lt;/em&gt; with a finite list of numbers should be an empowering first step along the path to thinking of a transformation as a mathematical object in its own right. And the subsequent yoga of determinants and eigenthings give them the tools for manipulating these objects. You have to keep making the connection back to geometry so that they don't forget that what they're doing is more than symbol-crunching, but the symbol crunching helps them to internalise the idea that transformations are tangible things they can deal with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you start by telling them that a linear map is a function satisfying certain axioms, it will take them longer to figure out the domain in which they're working. Sure you can give them examples that are easy to explain (rotations, reflections, projections, whatnot) but the advantage of matrices is that you give them the potential to generate &lt;em&gt;all&lt;/em&gt; examples for themselves and play with them.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;composition of maps is a very natural thing,&#xA;  from which the definition of multiplication of&#xA;  matrices flows smoothly&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I wholeheartedly agree, but I also think you don't need to introduce linear maps first to explain it this way. This is exactly how I define matrix multiplication in my course: once you've told them how a matrix defines a transformation, you can ask them to figure out the formula for the matrix defining the composition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to see the approach I take, my notes and videos are available here: &lt;a href=&quot;http://jde27.uk/la&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jde27.uk/la&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13028" LastActivityDate="2020-05-17T22:44:16.670" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18350" PostTypeId="2" ParentId="18338" CreationDate="2020-05-17T23:19:03.920" Score="2" Body="&lt;p&gt;&lt;strong&gt;Context:&lt;/strong&gt; I taught Linear Algebra as a junior level course for a mixed audience (math, engineers, science, education) who had already had a proof technique course and all three semesters of calculus including vectors and their analytic geometry. Of course, there were always a few math-education majors whose curriculum path forbade them to have the third semester of calculus (sad, these kids may well teach highschool math in the not too distant future, we really should require the math major for high school math instruction in the USA... I digress)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I try to teach with a balanced approach because I do see the merit in what you say. But, I also know that matrices are incredible useful and are truly interesting on their own. Here is the quick break-down of the course I taught a bunch of times:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 1:&lt;/strong&gt; what is the matrix ? Matrix addition, multiplication, matrix algebra, blocks and applications. I introduce all the basic component notation here and prove things like &lt;span class=&quot;math-container&quot;&gt;$(AB)^T=B^TA^T$&lt;/span&gt;. On a personal level, it gives me great joy for this reflects my personal interest in tensor calculation. Needless to say, students have mixed feelings about my exuberance for index notation. As the years went on, I find myself drifting more and more into column-based arguments. But, that is basically the take-away message notationally, to verify a matrix identity you can focus at the matrix, column or row, or component level. Each viewpoint has its merits. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 2:&lt;/strong&gt; Gaussian elimination and elementary matrices. I spend a day on row-reduction, a day on intepreting solution sets and a day on how row reductions can be implemented by left multiplication of an elementary matrix. I might slip an application in here somewhere. I do not attempt to &quot;prove&quot; the uniqueness, but I do emphasize the ideas of forward and backwards pass. If I had more time, I'd work LU-decomposition in here. In contrast to Week 1, almost everything here is a matrix or column level notation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 3:&lt;/strong&gt; Inverse matrices, spanning, linear independence and the Column Correspondance Property (CCP). There is a good chunk of theoretical matrix algebra to cover here. I try to prove why left inverse implies right inverse in this context. The many equivalent characterizations of invertibility give a nice theorem to continue adding to as other ideas come in later. If I did Week 2 correctly, I've already showed them how to solve multiple systems with the same coefficient matrix so the usual magic trick for calculation the inverse is easily understood. Spanning and LI are new ideas, but the matrix calculations are the same we've been doing. Notice I focus attention here just on column matrix spanning and LI. The abstract version comes later.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 4:&lt;/strong&gt; Determinants motivated from volume. Laplace Expansion by minors and usual calculation tricks. Application to eigenvectors introduced (I have maybe one homework question on eigenvectors here just to prime the pump for later). Then, Week 4 ends with Quiz 1 and time for questions about the homework solutions which I provided.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 5:&lt;/strong&gt; Classroom interaction! (Test 1). Then we move on to abstract vector space definition and examples galore, subspace test and the theory of spanning and LI for abstract vector space. Many of my examples of abstract vector spaces are based on &lt;strong&gt;matrices&lt;/strong&gt;. It's good they have lots of experience and we've already introduced the language to handle them efficiently &lt;span class=&quot;math-container&quot;&gt;$A = \sum_{i,j} A_{ij}E_{ij}$&lt;/span&gt; etc. I introduce functions of vector spaces as an example of vector spaces. If you wish, I've introduced linear transformations here.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 6:&lt;/strong&gt; bases and coordinate maps, theory of dimension, linear transformations and their subspaces. I've tried various approaches over the years here. However, usually I use an argument which boils down to calculating the trace and using &lt;span class=&quot;math-container&quot;&gt;$tr(I_n)=n$&lt;/span&gt; as well as &lt;span class=&quot;math-container&quot;&gt;$tr(CD)=tr(DC)$&lt;/span&gt; to prove the number of elements in a basis is unique. There are many ways to get at this, and I always regret whatever I do since we could spend so much more time here to really get into all the methods. Also, by the end of the week I'm feeling guilty about all the properties of linear transformations I've forgotten to prove. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 7:&lt;/strong&gt; On restriction, extension and isomorphism. I try to impress on them how amazing it is to define a map on an infinity of points by its values on a handful of inputs. Linearity is hugely simplifying. The concept of defining a linear map by linear extension off a basis is introduced and used to formulate various isomorphisms. Then we return to linear transformations and introduce the concept of the matrix of a linear transformation. I typically spend a day showing how to calculate this in abstract case.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 8:&lt;/strong&gt; coordinate change for vectors and transformations. I draw pictures to derive the formulas then emphasize how these things simplify in special cases like column vectors or usage of the standard basis in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;. I have a few truly complicated examples in my notes which I'll project without actually working out. The point of the example is to share the motivation for the study: coordinate change allows us to find the most beautiful formulation of a given linear transformation. (incidentally, I think the under-emphasis of linear transformations in some curricula make coordinate change even harder to understand, even so, this is a difficult topic for most kids)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 9:&lt;/strong&gt; quotient vector space and the first isomorphism theorem, direct sum decompositions. (I try to prove things about cosets carefully and just sketch the idea of the direct sum and how invariant subspaces make the matrices nice). The depth of this week depends on the particular audience. I also try to take some time to contrast concept of null space for a matrix vs. kernel of a linear transformation. The coordinate maps are isomorphisms which transfer us between these various worlds. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Spring Break&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 10:&lt;/strong&gt; Quiz 2 and Test 2, then the end of week we introduce Eigenvectors. Throughout the discussion of Eigenvectors I bounce back and forth between the e-vector of a matrix vs. e-vector of a linear transformation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 11:&lt;/strong&gt; Eigenvectors continued, Jordan form. I don't prove everything here, although I do try to prove LI results about eigenvectors. I'll introduce notation for Jordan form and give examples, but I'm not going to show algorithmically how to find the basis nor prove its existence.  Then I spend a day on complexification of a real vector space along with the concept of a complex eigenvector.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 12:&lt;/strong&gt; continuing from complexification we get the so-called Real Jordan Form which is what is actually needed to understand applications. Then, the rest of the week we dive into Inner Product Spaces and Euclidean Geometry. I try to spend a little time here talking about the various choices we have for norms and how the &quot;circle&quot; can be a square or diamond.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 13:&lt;/strong&gt; beauty of orthonormal bases, GS-algorithm, closest vector problem and orthogonal subspace theorems, application to least squares.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 14:&lt;/strong&gt; orthonormal diagonalization and the Spectral Theorem, sometimes I get into the proof of the Spectral Theorem, it depends how tired I am at this point.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 15:&lt;/strong&gt; application to real quadratic forms, application to calculus of many variables. Matrix exponential and solution to system of DEqns leveraging all we know about e-vectors, complex e-vectors and the real Jordan form.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Week 16:&lt;/strong&gt; multilinear algebra. Ok, I probably should cover the Singular Value Decomposition here, or the QR decomposition or something else. But, I should be allowed to have fun at least one week in a semester, right ? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In summary, I think your idea for teaching the course is fine, but you will still need to teach the matrix calculations somewhere because they come up in examples. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any event, we should all teach from the heart. So, take your own advice before any of ours.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comment following initial post on 5-18-2020 by Kostya:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...you start by introducing matrix multiplication, transposed matriced etc., by just imposing a set of formal rules upon students; then, presumably, drill the students on those rules... What is the pedagogical benefit of that? If you need to teach both matrix multiplication (which looks technical and poorly motivated to the beginner) and composition of linear maps (which is natural and very easy to motivate), why not first do the latter and then the former?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The pedagogical benefit of introducing notation is that it gives me a language which allows me to efficiently and smoothly communicate general examples. Matrix multiplication gives me a way to convert a system of linear equations into a single matrix equation. There is motivation from that alone for the matrix-column multiplication. hen, going beyond matrix-column vector products,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ Ax_1=b_1, Ax_2=b_2 , \dots , Ax_s = b_s \Leftrightarrow A[x_1|x_2|\cdots|x_s]=[b_1|b_s|\cdots |b_s] $$&lt;/span&gt;&#xA;So, thinking about multiple systems of equations with the same coefficient matrix naturally leads to the concept of matrix multiplication. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be honest, I don't motivate matrix multiplication when I define it. I just put it out there and start showing how it works. I take a more pragmatic approach, I do tell them that the initial definition is made so that matrix multiplication will fit with composition of linear maps. But, that is just a comment. I circle back to it later and show it explicitly once we later introduce linear maps. Then I circle back again later still and show it still makes sense with the extra baggage of coordinates (&lt;span class=&quot;math-container&quot;&gt;$T: V_{\beta} \rightarrow W_{\delta}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$S: W_{\delta} \rightarrow U_{\gamma}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$[T]_{\beta, \delta}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$[S]_{\delta, \gamma}$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$[S \circ T]_{\beta, \gamma} = [T]_{\beta, \delta}[S]_{\delta, \gamma}$&lt;/span&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, yes, I do think students should be made aware that matrix multiplication can be defined by the necessity of adhering to the mechanics of function composition. But, on the other hand, I don't want to be talking about function composition while I'm focused on how to solve equations and interpret their solution sets. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, initially, I do want to share some enthusiasm for how we can use matrices to construct other objects. For example, typically I have them study the product of matrices of the form &lt;span class=&quot;math-container&quot;&gt;$\left[ \begin{array}{cc} a &amp;amp; -b \\ b &amp;amp; a \end{array} \right]$&lt;/span&gt;. Connecting that matrix with corresponding linear map would require much greater sophistication at this point in the discussion. &lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2020-05-18T14:41:36.463" LastActivityDate="2020-05-18T14:41:36.463" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18351" PostTypeId="1" CreationDate="2020-05-18T16:02:36.187" Score="4" ViewCount="357" Body="&lt;p&gt;My job is starting to have me delve into categories that require things like regression analyses on data sets, essentially I'm being introduced to &amp;quot;Data Science&amp;quot; type material. Coming from a computer science background however, I'm aware of how easy it is to misapply statistics.&lt;/p&gt;&#xA;&lt;p&gt;My only course on statistics was over a decade ago and it didn't include any of the theoretical underpinnings that actually explained &lt;em&gt;why&lt;/em&gt; the formulas worked, which would help me understand when to use method X vs method Y.  I'm dreadfully afraid of creating false findings.&lt;/p&gt;&#xA;&lt;p&gt;I'm looking for teaching resources that would hopefully bridge the gap from &amp;quot;I can do X, Y, or Z&amp;quot; but would give enough math to help me understand when and where to use what.  If there's anything like Roger Penrose's &amp;quot;The Road to Reality&amp;quot; for statistics, that would help.  Otherwise any suggestions of what a good course plan for self-study would look like would be welcome.&lt;/p&gt;&#xA;&lt;p&gt;[EDIT] Clarifications:&lt;/p&gt;&#xA;&lt;p&gt;Having a comp sci background, I know for example that if the problem before me seems like a graph-theoretic problem, I can reach for my copy of &amp;quot;Graph Theory and its Applications&amp;quot; from Gross/Yellen.&lt;/p&gt;&#xA;&lt;p&gt;Physics isn't my forte, but when I want to leverage the mathematics I do have, I can reach for &amp;quot;The Road to Reality&amp;quot; by Roger Penrose.&lt;/p&gt;&#xA;&lt;p&gt;And if it's dealing with Data structures, I can check out &amp;quot;The Art of Computer Programming&amp;quot; from Donald Knuth, and for algorithms--although it's a textbook--there's also Cormen's &amp;quot;Introduction to Algorithms,&amp;quot; or even Skeina's book &amp;quot;The Algorithm Design Manual.&amp;quot;  All of these works provide enough theory and proofs to make it pretty clear &lt;em&gt;why&lt;/em&gt; these tools work.  I'm fishing for items in this same category for Statistics, but judging by the comments, there isn't quite this level of organization and unity in statistics?&lt;/p&gt;&#xA;" OwnerUserId="676" LastEditorUserId="8571" LastEditDate="2020-08-27T12:29:40.587" LastActivityDate="2021-07-13T08:06:10.643" Title="Is there a 'statistics theory' course plan for practitioners?" Tags="|self-learning|statistics|reference-books|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="18353" PostTypeId="2" ParentId="18342" CreationDate="2020-05-18T22:25:23.943" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Generally, is providing a context/big picture helpful or a waste of time when teaching &quot;hard skills&quot;? Disregarding the &quot;added value&quot; of the students knowing the big picture, being able to apply the skills, etc., does the learning of skills per se improve?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From an emotional perspective, it can be helpful. It shows why what you're trying to learn is important. It provides motivation and a breath of fresh air when you are deep down in theorems, lemmas, propositions, and definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From learning and cognitive perspectives, that depends on how you present that big picture and how you keep it up over the course. From the point of view of &lt;a href=&quot;https://en.wikipedia.org/wiki/Cognitive_load&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cognitive Load Theory&lt;/a&gt; (CLT) the aim of learning is to acquire schemas in the long-term memory, which in turn improves the ability to process new information, and that in return improve problem solving. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A schema is generally domain-specific and consists of all knowledge related to that domain in long-term memory. That knowledge includes, but is not limited to, the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Definitions;&lt;/li&gt;&#xA;&lt;li&gt;Important theorems and propositions;&lt;/li&gt;&#xA;&lt;li&gt;General problem-solving strategies related to that domain, such as the tricks of the trade for proving certain statements, what theorems get generally used for certain results, etc;&lt;/li&gt;&#xA;&lt;li&gt;Important examples and counterexamples.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If that big picture does not contribute to the schema acquisition it may become one of those &quot;nice to know&quot; things but that won't directly contribute to anything but motivation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sources:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0521677580&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jan Plass, Roxana Moreno, Roland Brünken (Editors) - Cognitive Load Theory&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://link.springer.com/article/10.1007/s10648-019-09465-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Sweller, Jeroen J. G. van Merriënboer &amp;amp; Fred Paas - Cognitive Architecture and Instructional Design: 20 Years Later.&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="47" LastActivityDate="2020-05-18T22:25:23.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18354" PostTypeId="1" AcceptedAnswerId="18355" CreationDate="2020-05-19T22:27:28.783" Score="3" ViewCount="170" Body="&lt;p&gt;I hope this is the right place for posting this, but if not, please let me know!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently took a second class in Python programming which, toward the end, also taught a little bit of SQL. As it turns out, SQL was the part of the class that I enjoyed the most. So, over the summer, I am thinking of acquiring a Microsoft certification in SQL. However, before engaging myself in that, I wanted to ask if this certification is likely to benefit my career plans directly or indirectly (I am planning to pursue a Masters in Statistics, with probably some data science component in a couple years).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do statisticians use SQL in their work? Is it reasonable to assume that SQL might become prevalent in statisticians' work in 10 or so years? &lt;/p&gt;&#xA;" OwnerUserId="13834" LastEditorUserId="13834" LastEditDate="2020-05-19T23:15:05.277" LastActivityDate="2020-05-19T23:15:05.277" Title="Is SQL relevant to statisticians' work?" Tags="|graduate-education|statistics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18355" PostTypeId="2" ParentId="18354" CreationDate="2020-05-19T23:05:11.073" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Do statisticians use SQL in their work? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes! I would say it is already prevalent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Four indications:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;One is how SQL queries are now easily accessed from within R: &lt;a href=&quot;https://db.rstudio.com/getting-started/database-queries/&quot; rel=&quot;noreferrer&quot;&gt;Database Queries with R&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You are apparently already familiar with Python and SQL interfaces, e.g., &lt;a href=&quot;https://www.python-course.eu/sql_python.php&quot; rel=&quot;noreferrer&quot;&gt;SQLite&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a DataCamp course &lt;a href=&quot;https://www.datacamp.com/courses/introduction-to-sql&quot; rel=&quot;noreferrer&quot;&gt;Introduction to&#xA;SQL&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are on the job market as a statistician, anecdotally it is now common to require SQL knowledge and experience. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-05-19T23:05:11.073" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18356" PostTypeId="2" ParentId="17852" CreationDate="2020-05-20T05:20:17.400" Score="1" Body="&lt;p&gt;I think this depends a lot on the book and the author.  Some books from the 80s already read as old-fashioned and out-of-date, while some older books were really ahead of their times.  It also depends on the field, since fields go through times of change which can make earlier books obsolete.  The earliest book that I would happily use as a text in a normal mathematics class is Dirichlet's Lectures on Number Theory from 1863.  It's truly a joy to read, and other than using quadratic forms instead of ideals, it's modern enough for advanced undergraduates.&lt;/p&gt;&#xA;" OwnerUserId="345" LastActivityDate="2020-05-20T05:20:17.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18357" PostTypeId="1" CreationDate="2020-05-20T06:52:37.273" Score="4" ViewCount="436" Body="&lt;p&gt;The New Math curriculum built math eduction on set theory.  Have there been any attempts to do  something similar with category theory?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was fortunate to grow up in a relatively enlightened spacetime, such that I got &lt;a href=&quot;https://en.wikipedia.org/wiki/New_Math&quot; rel=&quot;nofollow noreferrer&quot;&gt;New Math&lt;/a&gt; in my earliest primary education. It Worked For Me in the sense that, as a child, math seemed both interesting and straightforward. Though I retained a continuing interest in math (which I tend to attribute to that primary-math pedagogy), my formal study ended with the undergraduate engineering curriculum (differential equations and calculus-based probability) and the more-fascinating-to-me introduction to discrete math taught in undergraduate informatics (misnamed &quot;computer science&quot; in too much of the world).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a longer-standing interest in logics and the history and philosophy of logic, and started learning about attempts to &quot;found mathematics&quot; (where &lt;em&gt;found&lt;/em&gt; is a verb, not an adjective) via philosophy of math. Hence when I learned about efforts to found math on set theory, I wasn't able to &quot;go too deep&quot; (lacking much pure math), but found it fairly tractable--that was how I was first taught math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I later learned a bit more about category theory, which I had previously encountered (less than a bit, and much too briefly) in a programming-language-theory class. Lately I've been reading &lt;a href=&quot;https://archive.org/details/conceptualmathem00lawv&quot; rel=&quot;nofollow noreferrer&quot;&gt;a bit more about category theory&lt;/a&gt;; I seem to understand it well enough (though of course without testing, that's just my &lt;a href=&quot;https://en.wikipedia.org/wiki/Qualia&quot; rel=&quot;nofollow noreferrer&quot;&gt;quale&lt;/a&gt;), but (aside from the applications of CT to sets) I still find CT less immediate or intuitive than set theory. My hypothesis is, even though I didn't get &lt;em&gt;taught&lt;/em&gt; formal ST until college (which, in my case, was decades after secondary school), New Math wired some deep and persistent neural networks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which motivates my question: has anyone attempted a primary-math pedagogy mostly based on category theory, similar to the way New Math was mostly based on set theory?  If so, does it have any empirical advantages (in terms of educational outcomes) other than making category theory more intuitive?&lt;/p&gt;&#xA;" OwnerUserId="14074" LastEditorUserId="376" LastEditDate="2020-05-20T12:45:41.847" LastActivityDate="2020-05-30T16:15:27.213" Title="Have there been attempts to base early math education on category theory?" Tags="|primary-education|new-math|category-theory|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18358" PostTypeId="1" CreationDate="2020-05-20T11:51:49.940" Score="7" ViewCount="1984" Body="&lt;p&gt;If you were to &quot;map&quot; mathematics onto a tree structure where the top is &quot;Mathematics&quot;, and then below it the different branches, then sub-branches, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you suggest is a good structure, for educational purposes?  The idea is this - if I have a student learning a topic, then all I would have to do is look at all the sub-branches (and theirs) below that topic in the tree, to represent everything the student needs to know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm certain there will be overlaps, but I'm okay with this - looking for some insight on this, or relevant articles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you think that such a hierarchy does not even exist, please let me know what you think the right structure is for educational purposes, and if you know where to find it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many thanks!&lt;/p&gt;&#xA;" OwnerUserId="7960" LastEditorUserId="7960" LastEditDate="2020-05-21T05:07:09.680" LastActivityDate="2020-06-16T13:48:51.773" Title="Looking for a HIERARCHY of math subjects" Tags="|mathematical-pedagogy|self-learning|problem-solving|topology|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18359" PostTypeId="2" ParentId="18358" CreationDate="2020-05-20T17:00:41.190" Score="7" Body="&lt;p&gt;(1) Here is &lt;a href=&quot;https://www2.stetson.edu/~mhale/logic/tree.htm&quot; rel=&quot;noreferrer&quot;&gt;Margie Hale's&lt;/a&gt; tree:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xuVjl.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xuVjl.gif&quot; alt=&quot;HaleTree&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) And here is &lt;a href=&quot;https://www.quora.com/Can-somebody-show-me-the-map-of-mathematics-that-shows-how-each-component-of-mathematics-is-connected-to-one-another&quot; rel=&quot;noreferrer&quot;&gt;Gaspard Sagot's hierarchy&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/fvv9t.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fvv9t.png&quot; alt=&quot;SagotHier&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-05-20T17:00:41.190" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18360" PostTypeId="2" ParentId="18358" CreationDate="2020-05-21T03:04:03.267" Score="-1" Body="&lt;p&gt;I'm not sure that a complete hierarchical categorization of math is possible even if we ignore the incompleteness therom. For example arithmetic can be derived from lambda calculus or geometry or incrementation, recursion and inversion; but I learned those subjects from teachers who assumed I new arithmetic and this is just a simple case.&lt;/p&gt;&#xA;" OwnerUserId="1766" LastActivityDate="2020-05-21T03:04:03.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18361" PostTypeId="1" CreationDate="2020-05-21T04:19:34.740" Score="1" ViewCount="238" Body="&lt;p&gt;Students like to categorize &lt;em&gt;notations&lt;/em&gt; to pin down their understanding of exactly what these &lt;em&gt;notations&lt;/em&gt; stand for. Thus, given the expressions &lt;span class=&quot;math-container&quot;&gt;$f(x_{0})=f(x)|_{x\leftarrow x_{0}}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x=x_0+h$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$lim_{x\rightarrow x_{0}}$&lt;/span&gt;, they like that the &lt;em&gt;notation&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is in the category of &lt;code&gt;(global) variables&lt;/code&gt; and that the &lt;em&gt;notation&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is in the category of &lt;code&gt;(local) variables&lt;/code&gt;. And, while they do understand that the &lt;em&gt;notation&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; stands for a number that is fixed &quot;for the duration&quot;, they want to know what is the (name of the) category of &quot;notations&quot; that &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; falls in.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastEditorUserId="4815" LastEditDate="2020-05-24T20:08:40.317" LastActivityDate="2020-05-24T20:08:40.317" Title="What to call a symbol that denotes an &quot;undisclosed&quot; given number?" Tags="|notation|terminology|" AnswerCount="1" CommentCount="11" ClosedDate="2020-05-25T18:59:39.377" ContentLicense="CC BY-SA 4.0" />
  <row Id="18362" PostTypeId="2" ParentId="13669" CreationDate="2020-05-21T04:31:45.697" Score="1" Body="&lt;p&gt;I have long used &lt;code&gt;size&lt;/code&gt; in place of &lt;code&gt;absolute value&lt;/code&gt; and thus &lt;code&gt;large-in-size&lt;/code&gt; and &lt;code&gt;small-in-size&lt;/code&gt; as attributes of a number.&lt;/p&gt;&#xA;" OwnerUserId="4815" LastActivityDate="2020-05-21T04:31:45.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18364" PostTypeId="1" AcceptedAnswerId="18370" CreationDate="2020-05-21T14:32:30.533" Score="8" ViewCount="2047" Body="&lt;p&gt;I am looking for 3d-math animation software that is specifically meant for&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Visualization: That is geometry-based software that can aid in making visualizations as in &lt;a href=&quot;https://www.youtube.com/watch?v=x7d13SgqUXg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=x7d13SgqUXg&lt;/a&gt; (on the sphere-eversion process).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Presenting concepts: That is software that can help in making videos as in &lt;a href=&quot;https://www.youtube.com/watch?v=rB83DpBJQsE&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=rB83DpBJQsE&lt;/a&gt; (on divergence curl), or &lt;a href=&quot;https://www.youtube.com/watch?v=r18Gi8lSkfM&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=r18Gi8lSkfM&lt;/a&gt; (on Fourier Transform).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So my question is which mathematical software is freely available/cheap, that can help in achieving the above task?&lt;/p&gt;&#xA;" OwnerUserId="14082" LastEditorUserId="376" LastEditDate="2020-05-25T11:42:47.970" LastActivityDate="2020-05-28T00:14:55.203" Title="Math 3d animation software" Tags="|software|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18365" PostTypeId="2" ParentId="18361" CreationDate="2020-05-21T16:42:56.163" Score="5" Body="&lt;p&gt;In this instance I would probably say &quot;Fix an (arbitrary) point &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;&quot; and carry on.  The idea is it's initially unknown but after its introduction stays constant which this sounds like it conveys pretty nicely, and I don't think it's worth splitting philosophical hairs over what is a &quot;variable&quot; vs. &quot;parameter&quot;, especially in the face of how scope and discharging of variables are rarely precisely dealt with in typical mathematical writing.&lt;/p&gt;&#xA;" OwnerUserId="4679" LastActivityDate="2020-05-21T16:42:56.163" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18366" PostTypeId="2" ParentId="18357" CreationDate="2020-05-21T20:37:10.510" Score="1" Body="&lt;p&gt;Great question!  Yes, this was Piaget's project after initially working with more classic structuralism (e.g. his text titled &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1138854484&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Structuralism&lt;/em&gt;&lt;/a&gt;). He explicitly mentions Mac Lane's work in his book on understanding functions, &lt;a href=&quot;https://www.springer.com/gp/book/9789027712424&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Epistemology and Psychology of Functions&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I discuss this a bit in my &lt;a href=&quot;https://academiccommons.columbia.edu/download/fedora_content/download/ac:202849/content/Koehler_columbia_0054D_13634.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;dissertation&lt;/a&gt;, but am not familiar with other work that compares mathematical styles and models for children's thinking.    &lt;/p&gt;&#xA;" OwnerUserId="7003" LastEditorUserId="8660" LastEditDate="2020-05-23T03:50:30.783" LastActivityDate="2020-05-23T03:50:30.783" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18367" PostTypeId="2" ParentId="18364" CreationDate="2020-05-22T08:28:06.790" Score="4" Body="&lt;p&gt;I have not personally made videos, although I have used &lt;a href=&quot;https://inkscape.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maple&lt;/a&gt; (not free) to create many visualizations and &lt;a href=&quot;https://inkscape.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inkscape&lt;/a&gt; (free) for some diagrams. I usually prefer to write mathematics by hand, but from time to time I have used PowerPoint for slides when many visuals are required that would be difficult or time-consuming to draw by hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am impressed by the materials &lt;a href=&quot;https://www.math.upenn.edu/~ghrist/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Ghrist&lt;/a&gt; has produced. If you are interested in knowing more about how he creates them, check out his &lt;a href=&quot;https://www.profghristmath.com/2019/08/10/about-creating-graphics-video/&quot; rel=&quot;nofollow noreferrer&quot;&gt;blog post&lt;/a&gt; but be aware that you could be in for a steep learning curve. To summarize a little of what he writes, he advises investing the money and time if you want to create pro-quality graphics. Also, learn basic graphics, design principles and compositing. You might also need to learn coding and/or game engines, especially if you want to simulate systems. You could get started using &lt;a href=&quot;https://processing.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Processing&lt;/a&gt;, and there are always languages such as &lt;a href=&quot;https://www.python.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Python&lt;/a&gt; and &lt;a href=&quot;https://julialang.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Julia&lt;/a&gt;. A list of 3D animation software can be found &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_3D_animation_software&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, some of which is free.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2020-05-22T08:28:06.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18368" PostTypeId="1" AcceptedAnswerId="18374" CreationDate="2020-05-22T21:17:01.070" Score="30" ViewCount="3360" Body="&lt;p&gt;I tutor a 16 year old girl. As far as I can tell, she has average talent and interest in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, her knowledge of math is that of a 10 year old or even below. She knows the basic operations on positive integers below 100, but that's mostly it. She has problems sorting negative numbers. She has insufficient understanding of fractions: she would understand what eg. 3/4 would mean graphically, but does not understand that it's the same number as 6/8, and can not do operations with fractions. She can not solve simple worded exercises like &quot;A new phone costs double as an old one. If the new costs 200$, how much does the old one cost?&quot; that is, she can not translate the words to mathematical operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She would say things like 0.9 + 0.1 = 0.10, or that 3/4 and 6/8 are different numbers, or that -1 &amp;lt; -2, which I suppose at some point are normal part of development, and these struggles eventually lead to development of more abstract concepts of numbers (given proper guidance and enough patience, of course).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I can tell, she has average cognitive abilities (intelligence, verbality, memory etc). I suppose that the reason she has fallen so much behind is that she went to an extremely weak elementary school and somehow no one raised a red flag before.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been teaching her once a week for half a year online (we're 1000 miles apart).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We made some progress, but it is extremely slow. I realized that she completely misses the necessary concepts, and therefore we need to take it slow and work on hands-on direct experience and not force abstractions, just as someone would go about teaching a smaller child.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is, that most teaching material for her level focuses on much younger kids, so I really struggle to come up with activities that are suited to her age, especially ones that can be done online (unsurprisingly, she isn't very interested in coloring pies). She doesn't hate math and she is neither super interested, nor super disinterested - she is, as far as I can tell, a typical teenager. She isn't particularly lazy either - if I give her homework, she puts in effort to do it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that motivation is highly individual, so there might not be a general answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I am curious, if there are any resources or methods that are aimed at teenagers who have, for one reason or another, fallen behind in math. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, that our native language isn't English, and she does not speak English much at all. That said, I'm definitely interested in English books and websites etc. it just means I have to adapt them.&lt;/p&gt;&#xA;" OwnerUserId="14091" LastEditorUserId="14091" LastEditDate="2020-05-23T09:13:09.563" LastActivityDate="2020-05-26T17:02:31.343" Title="How to motivate an adolescent who has fallen behind in conceptual development?" Tags="|secondary-education|" AnswerCount="6" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="18369" PostTypeId="2" ParentId="18326" CreationDate="2020-05-22T22:18:49.097" Score="3" Body="&lt;p&gt;William Tutte apparently introduced the concept of the thickness of a graph. I have a memory of reading or being told that the concept of the coarseness of a graph arose because Paul Erdos misunderstood the definition of thickness.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2020-05-22T22:18:49.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18370" PostTypeId="2" ParentId="18364" CreationDate="2020-05-22T22:30:01.403" Score="5" Body="&lt;p&gt;The math videos creator &lt;a href=&quot;https://www.3blue1brown.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;3blue1brown&lt;/a&gt; has a webpage and in the &lt;a href=&quot;https://www.3blue1brown.com/faq&quot; rel=&quot;nofollow noreferrer&quot;&gt;FAQ section&lt;/a&gt; he goes into detail about how we went about creating his videos. Here's his answer for the question &quot;What do you use to animate your videos?&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I create most of the animations programmatically, using a python&#xA;  library named &quot;&lt;a href=&quot;https://github.com/3b1b/manim&quot; rel=&quot;nofollow noreferrer&quot;&gt;manim&lt;/a&gt;&quot; that I've been building up.  I’ve also used&#xA;  Grapher for a number of 3d animations.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It’s open-source on github, and a &lt;a href=&quot;https://www.reddit.com/r/manim/&quot; rel=&quot;nofollow noreferrer&quot;&gt;small community&lt;/a&gt; has emerged of&#xA;  people who use it, but before you dive in I feel like a warning is in&#xA;  order.  This is something I put together with my own use case in mind,&#xA;  never really meant to be a professional product that’s consistently&#xA;  maintained for others.  It's not that I want to discourage others from&#xA;  doing similar things, quite the contrary I love when that happens, but&#xA;  often my workflow and development with manim can make it more&#xA;  difficult for an outsider to learn than other better-documented&#xA;  animation libraries, like Matplotlib and Mathematica, or other tools&#xA;  built for math visualization like Geogebra, Desmos, Grapher and more.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I also get worried when I hear people ask things like “how do I sync&#xA;  up narration into manim”.  This is just a tool for spitting out the&#xA;  individual mathematical animations to be edited together later.  For&#xA;  goodness sake, use traditional video editing software for as much as&#xA;  you can!&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;There are aspects of producing videos with a self-made tool like this&#xA;  that I find quite pleasing, but which are pleasing precisely because&#xA;  it's my own.  It enforces a uniqueness of style, for example, which is&#xA;  by its very nature a benefit that can't be shared.  There's also a&#xA;  certain freedom in being able to tear up the guts of the tool every&#xA;  now and then when I feel a change is in order, since backward&#xA;  compatibility needs are very limited when you only care about videos&#xA;  moving forward.  Not exactly the best practice from a collaborative&#xA;  standpoint.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The channels I know who took inspiration from 3b1b and did it best,&#xA;  like &lt;a href=&quot;https://www.youtube.com/channel/UCKzJFdi57J53Vr_BkTfN3uQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Primer&lt;/a&gt; or &lt;a href=&quot;https://www.youtube.com/channel/UC8bvGUlaTahF0lHhMsZFhfw&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jazon Jiao&lt;/a&gt;, found or created the tooling that worked&#xA;  best for their particular content.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="47" LastEditorUserId="376" LastEditDate="2020-05-25T11:44:19.867" LastActivityDate="2020-05-25T11:44:19.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18371" PostTypeId="2" ParentId="18368" CreationDate="2020-05-23T00:12:30.607" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://beastacademy.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beast Academy&lt;/a&gt; &lt;strong&gt;is&lt;/strong&gt; for younger kids, but she might still like it. The format is &quot;guide books&quot; that are like graphic novels, where the beast students are taught by beast teachers, along with practice books. You can see examples on their website. It has levels 2 to 5. But it was designed for gifted kids, and it offers plenty of interesting challenge, whatever level you're at, along with excellent conceptual instruction. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;She could look over the website, and if she likes it, she could sign up for their online version, which would allow her to move through the levels at whatever pace works for her.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will be using it with an adult student this summer, who just failed my calculus class. I told him he has lots of holes from what I can tell and filling in the holes will help him succeed when he retakes calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you suggest this to her, and it does work for her, I'd like to know. I'm trying to get clarity on how helpful it is for adults (and teens).&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-05-23T00:12:30.607" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18372" PostTypeId="1" CreationDate="2020-05-23T05:35:03.580" Score="4" ViewCount="156" Body="&lt;p&gt;I would like to have a comparison  or a big picture of how and why the approach for teaching math varies from primary (or pre &#xA;primary) to middle to higher classes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand at every level one need to learn concepts and procedures, so why I see so much compartmentalization for teaching math at different levels?/&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I accept that in pre-primary classes focus should be on learning how to learn and learning using 5 senses. But is there anything more than this that  nessicitates&#xA; a different approach especially when moving to upper primary, then to middle, secondary and higher and finally to university? I would like to know about those approaches and contrast the same to have a big picture. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am new to maths teaching, so kindly bear with me if you find  this question too naive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks and regards&lt;/p&gt;&#xA;" OwnerUserId="13103" LastEditorUserId="13103" LastEditDate="2020-05-23T05:51:08.067" LastActivityDate="2020-08-25T12:54:36.277" Title="Teaching Approach at primary, middle and higher level" Tags="|undergraduate-education|secondary-education|primary-education|middle-school|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18373" PostTypeId="2" ParentId="18368" CreationDate="2020-05-23T08:43:19.513" Score="6" Body="&lt;p&gt;A warning:  I have read some of the books I mention here but I have not tried them on actual students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend getting your student some of &lt;a href=&quot;https://mckellarmath.com/&quot; rel=&quot;noreferrer&quot;&gt;Danica McKellar's books&lt;/a&gt;.  Your student might want to start with &quot;&lt;em&gt;Math Doesn't Suck&lt;/em&gt;&quot; which is targeted at ages 9-12 and covers fractions, decimals, and percentages.  If this is too difficult, then try &quot;&lt;em&gt;Do Not Open This Math Book&lt;/em&gt;&quot; (ages 6-9, addition and subtraction) then &quot;&lt;em&gt;The Times Machine!&lt;/em&gt;&quot; (ages 8-10, multiplication and division).  After she understands arithmetic, let her try &quot;&lt;em&gt;Kiss My Math&lt;/em&gt;&quot; (ages 11-13, pre-algebra), then &quot;&lt;em&gt;Hot X: Algebra Exposed&lt;/em&gt;&quot; (ages 12-14, Algebra 1), then &quot;&lt;em&gt;Girls Get Curves: Geometry Takes Shape&lt;/em&gt;&quot; (ages 14-16, geometry).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get some idea why I think Danica's books are quite different from the others, see my blog post &lt;a href=&quot;https://joelreyesnoche.wordpress.com/2015/03/17/math-doesnt-suck/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.  She writes in a very clear way, and her books are actually perfect for parents trying to understand how to help their children with their homework.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My PhD dissertation was about conceptual and procedural ways of teaching mathematics, and I was curious about which of the two approaches Danica favored.  I was actually very pleasantly surprised to see that she managed to explain things using a very good combination of both.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my blog post about &quot;&lt;em&gt;Math Doesn't Suck&lt;/em&gt;&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I think the inspirational messages spread throughout the book are the most important thing about it. Here’s the last part of the final paragraph of the final section (“Great Expectations”) of Chapter 5:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Every homework problem you think you can’t do—but then through determination, you solve—every time you exercise your brain and your beauty, inside and out, you’re becoming the young woman you aspire to be. I’m here to tell you from personal experience that you can be a glamour girl &lt;em&gt;and&lt;/em&gt; a smart young woman—who can &lt;em&gt;certainly&lt;/em&gt; do math.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2020-05-23T14:48:24.500" LastActivityDate="2020-05-23T14:48:24.500" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18374" PostTypeId="2" ParentId="18368" CreationDate="2020-05-23T11:57:51.717" Score="11" Body="&lt;p&gt;One of the skills that math education (or any education, for this purpose) includes is the ability of the student to learn on his own. Judging by your description, your student lacks this skill. I have also found from my experience with teaching teens that these students often also lack retention skills and do not ask meaningful questions during the classtime. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of that, consider starting with a more frequent schedule (three to four times a week), then gradually decreasing to weekly once the student learns to learn on their own, to retain previous concepts and come up with questions. &lt;/p&gt;&#xA;" OwnerUserId="6275" LastActivityDate="2020-05-23T11:57:51.717" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18375" PostTypeId="1" CreationDate="2020-05-23T14:08:36.847" Score="1" ViewCount="265" Body="&lt;p&gt;I am trying to compile a document of preparation questions to use in mock interviews with students applying to study Maths at top UK Universities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have many questions already, but it is always good to have more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what would be your favourite questions to ask students? These would have to be fairly short (whole question solvable in less than half an hour) with multiple stages or areas for further exploration/extension so that able candidates can get stretched. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Topics of interest are: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Number Theory Problems&lt;/li&gt;&#xA;&lt;li&gt;Graph Sketching Problems&lt;/li&gt;&#xA;&lt;li&gt;Complex Number Problems&lt;/li&gt;&#xA;&lt;li&gt;Calculus Problems&lt;/li&gt;&#xA;&lt;li&gt;Statistics&lt;/li&gt;&#xA;&lt;li&gt;Mechanics Problems&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of course there are other fields, which I would appreciate your favourite questions about, and questions may even overlap, but I've put this list to hopefully get a broad coverage of the topics. &lt;/p&gt;&#xA;" OwnerUserId="14007" LastActivityDate="2020-05-23T14:32:02.713" Title="Undergraduate Interview Maths Questions" Tags="|undergraduate-education|" AnswerCount="1" CommentCount="1" ClosedDate="2020-05-28T09:48:00.080" ContentLicense="CC BY-SA 4.0" />
  <row Id="18376" PostTypeId="2" ParentId="18375" CreationDate="2020-05-23T14:32:02.713" Score="1" Body="&lt;p&gt;Tom Bowler's booklet &lt;a href=&quot;https://www.drfrostmaths.com/resource.php?rid=261&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; has more than 100 questions like this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also think that the first part of some STEP I/II questions would be suitable for a mock interview. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few other questions I am fond of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sketch &lt;span class=&quot;math-container&quot;&gt;$y=\dfrac{\ln x}{x}$&lt;/span&gt; and hence determine which of &lt;span class=&quot;math-container&quot;&gt;$e^{\pi},\pi^e$&lt;/span&gt; is larger.&lt;/li&gt;&#xA;&lt;li&gt;Ignoring issues of convergence, work out the value of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\ldots}}}}}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2020-05-23T14:32:02.713" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18377" PostTypeId="2" ParentId="18372" CreationDate="2020-05-23T17:15:40.157" Score="4" Body="&lt;p&gt;It's a very broad question (to do the comparison you want).  And made even more difficult when you consider that best pedagogy for any of the stages is not agreed on.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal opinion is that much of the methodology should be similar, because humans are similar.  Much more than people think.  There's probably some social sensibilities that are different.  For example, teaching skiing I can't treat adults or pre-schoolers how I treat 5th graders (10 year old boys are the bomb...I would semi-literally go to war with them).  But a lot of the drills and practice required are same (much more than different).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I went to a school that taught college classes &quot;just like high school&quot;:  similar class sizes, teachers rewarded for teaching not research, homework drill problems rather than project-style questions, being called to the board, more frequent tests versus just a midterm/final, etc.  In talking with sisters that went to normal schools, I thought it was a total gyp what they were getting.  From a student perspective, this sort of support/attention/methodology was much more time efficient in training us than the college lecture halls at Enormous State Universities or non-native English speaking grad students whose real priority was their thesis work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think a lot of the rationale for &quot;not spoonfeeding&quot;, etc. ignores human behavioral traits.  They are generally, instead, excuses for the economic setup involved in US colleges.  (Which is why I recommend to kids not to go to an R1 school but go to a liberal arts school on a ROTC scholarship instead.)  It's even worse in graduate school courses in terms of the &quot;go teach yourself&quot; excuses, grade inflation, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s.  &quot;It never gets easier.  You just go faster.&quot;  -Greg LeMond.&lt;/p&gt;&#xA;" OwnerUserId="14049" LastEditorUserId="14049" LastEditDate="2020-05-23T17:53:50.837" LastActivityDate="2020-05-23T17:53:50.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18378" PostTypeId="2" ParentId="18368" CreationDate="2020-05-23T23:08:28.917" Score="7" Body="&lt;p&gt;Most people tend to only understand math when they can translate it to the real world, something they can understand. You've given several examples of theoretical concepts that look like magic to someone who can't translate it into something meaningful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But most of these (still somewhat elementary) concepts can easily be shown using real world examples.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;She has insufficient understanding of fractions: she would understand what e.g. 3/4 would mean graphically, but does not understand that it's the same number as 6/8&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Take a pizza, cut it into 4 pieces. Ask her how many slices she wants (let's say she says 2). Then continue cutting the pizza in 8 slices, and give her two of those slices. Wait for her to complain, or if she doesn't, ask her if she received as much pizza as she asked her for. Try to get her to explain to you that you didn't give her what she asked for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The core idea to convey is that 2 slices when there were only 4 slices is exactly as much pizza as 4 slices are when there are 8 slices. Therefore, 2/4 = 4/8.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While a pizza may be impractical for a tutoring session, the same applies to a small cake or a chocolate bar.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;and can not do operations with fractions&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My elementary school teacher used water and measuring jugs (each containing a liter, but each jug was marked in 4 parts, or 5 parts, or ...) to first prove to us that e.g. 2/4 + 1/5 = 7/10. At this point, it was no longer a wild magical claim, but it was provably the case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At that point, it became easier to try and understand (using unintuitive mathematical tricks) how we could've figured out that this was the case. You can do this in steps:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First, using two jugs that are divided into 4 parts, prove that 2/4 + 2/4 = 4/4. She should be able to predict that when she sees it done with the jugs.&lt;/li&gt;&#xA;&lt;li&gt;Then, using a 4-part-jug and a 2-part-jug, prove to her that 2/4 = 1/2. Pour the liquid from one jug to the other to prove the point.&lt;/li&gt;&#xA;&lt;li&gt;Then ask her what 2/4 + 1/2 is going to be. Try to avoid asking her for a fraction, instead try to get her to state whether it's going to fill the jug or not.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;To this day, I still use those jugs when doing fractions in my head. It really stuck with me because it's a visual and tangible representation of something that is otherwise just some arbitrary number magic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, you can use the pizza/cake/chocolate bar to come up with similar real life examples.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;She can not solve simple worded exercises like &quot;A new phone costs double as an old one. If the new costs 200$, how much does the old one cost?&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Try to come up with simpler and simpler worded exercises, to the point of balantly easy maths. Try to figure out the point where she starts getting stuck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A new phone costs double as an old one. If the new one costs \$200, how much does the old one cost?&lt;/li&gt;&#xA;&lt;li&gt;A new phone costs \$10 more than an old one. If the new one costs \$200, how much does the old one cost?&lt;/li&gt;&#xA;&lt;li&gt;A new phone costs \$10 more than an old one. If the old costs \$200, how much does the new one cost? (this is slightly easier as the two sentences map more clearly to each other)&lt;/li&gt;&#xA;&lt;li&gt;You intended to buy a \$150 phone, but you ended up buying the fancier \$200 phone. How much did you overspend?&lt;/li&gt;&#xA;&lt;li&gt;You're going to buy a \$100 phone, but you want to also buy the \$20 phone case that goes with it. How much money should you bring to the shop?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If she really struggles with visualizing this, she may be more comfortable talking about an actual purchase she made instead of a theoretical one. Ask her about something she's bought or is wanting to buy. You can get her to do the same puzzle solving depending on the situation)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If her parents paid for &lt;em&gt;some&lt;/em&gt; of it, but not all, ask her how much it cost, how much her parents paid, and how much she paid for herself.&lt;/li&gt;&#xA;&lt;li&gt;If she's still saving up to buy it, ask her how much it costs and how much she has already saved. Then ask her how much she still needs to save.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Whatever you end up talking about, then follow up with a theoretical example that exactly mirrors the real life situation she already understands. This will very much telegraph to her that these two cases (the one she knows and the one she doesn't) are effectively the same but with different numbers/words.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing this repeatedly will eventually teach her to translate an unknown situation into a known one.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Stick to what they know&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;It's a cliché example, but in The Office there is a character called Kevin Malone, whose main characteristic is being allround dumb. It's established at one point that he can't count money, but he can count food. If ask him to count pies instead of dollars (or salads), the question becomes answerable to him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/XpM90vEn8_g?t=51&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's the scene in question&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's a really stupid joke, but it actually gets at the heart of how people do math: &lt;strong&gt;visualization&lt;/strong&gt;. People are intuitively able to visualize that which they understand, and no one is able to visualize math without having been taught how to translate the abstract into the real (and equal), or vice versa. That is your job as a math educator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your task is to find the visualization that works for your student. That's going to be a highly personal process. Relate it to her hobbies and past experiences. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ask her things the already (subconsciously) understands, no matter how trivial, and then present more theoretical examples that are blatantly the same situation as the one she already understands. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The goal is to get her to realize that these seemingly different situations are the same. The more blatant the similarity, the faster she'll figure it out.&lt;/p&gt;&#xA;" OwnerUserId="13169" LastEditorUserId="11" LastEditDate="2020-05-26T17:02:31.343" LastActivityDate="2020-05-26T17:02:31.343" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18379" PostTypeId="1" CreationDate="2020-05-24T06:58:12.953" Score="4" ViewCount="149" Body="&lt;p&gt;I'm taking integral calculus at the moment. I was understanding everything quite well until we started learning about finding volume of a solid of revolution. I understand the concept, but practicing it has been stressful to say the least. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think if there was an app that takes a function and its boundaries, to then rotate about some axis for me then I would have a much better time understanding how to inspect what things like inner/outer radius, height, etc might be. Plus, I'd like to have very nice image examples to import into my digital notes so that I can review my notes and get a much better grasp of what's happening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any such tool I could use?&lt;/p&gt;&#xA;" OwnerUserId="12501" LastActivityDate="2020-05-30T03:35:54.067" Title="Graphing program for conceptualizing calculus" Tags="|calculus|computer-programs|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18380" PostTypeId="2" ParentId="18368" CreationDate="2020-05-24T10:24:37.640" Score="3" Body="&lt;p&gt;Based on my own experience as a learner (and occasionally 'teacher'), I can think of a couple of other possibilities:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;She may objectively be able to understand maths and even intelligent, but lacking the confidence to believe that she can. I remember it from myself, on occasion, but also from children I have helped a few times. If that is the case, then she needs a lot of encouragement, and guidance along the lines of 'what do you think?' - the sort of questions that provoke discussion and opinion.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Personally, one thing I found incredible demotivating was the lack of deeper insight. In my elementary education it was mostly about the mechanics of how to count, add, subtract, etc, and what I realised later was that I crave deep understanding: I derived enormous pleasure from learning what a natural number is (a representative of a class of isomorphic sets), from understanding the reason why multiplication of multidigit number work the way it does (~ a number such as &lt;span class=&quot;math-container&quot;&gt;$5348$&lt;/span&gt; is actually the sum &lt;span class=&quot;math-container&quot;&gt;$8 \times 10^0+4 \times 10^1+5 \times10^2+5 \times 10^3$&lt;/span&gt; - if you use the sum-rule, you get to the right method).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Modern school education is still far too focused on rote-learning of rules, just get students up to a certain level of function, rather than encouraging interest and independent opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps or perhaps inspires you in some way.&lt;/p&gt;&#xA;" OwnerUserId="13424" LastActivityDate="2020-05-24T10:24:37.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18381" PostTypeId="2" ParentId="18368" CreationDate="2020-05-24T12:30:09.140" Score="1" Body="&lt;p&gt;Age should not be confused with ability. Cognitive understanding of mathematical principles is indicative of higher-order thinking which in turn is ability and skills based. The fact that this student is 16 is irrelevant in terms of ability or skills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding ability, ability is defined by an understanding of key foundational concepts and the development if skills is in turn dependent on this. The skill needed to solve word sums is dependent on an ability to understand the core fundamentals of plus, minus, add and divide. Shortcomings in ability is addressed by skill. Just because you struggle to understand mathematics naturally should not be an inhibitor to learning the skill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same logic applies to things like emotional intelligence. Just because you struggle to communicate naturally does not mean that you will always struggle, it simply means that you need to acquire a skill that others naturally posses. Obviously this is a generalised statement and there are exceptions. Dr Andre Vermeulen at &lt;a href=&quot;https://neurolink.company/cms/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Neuro-Link&lt;/a&gt; is an expert with regards to this and some of the material on their website might aid you in developing a learning plan which encourages the development of Neuro-agility.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a number of possible contextual explanations to what you are experiencing with this student however have you considered the possibility of a repressed problem? As an example, a person who is colour blind accepts their reality as being THE reality. They do not know that they are colour blind as they think that is the norm and they will not say it either. They only discover that they are colour blind through others around them or through testing or in most cases, by chance. Other examples of this include something as basic as eye-sight. If you struggle to see and you have accepted it as being the reality chances are you will not address the problem because to you it is not a problem. Perhaps you should advise your student to have their eyes tested if possible?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also see that one of the answers here makes reference to self-learning. Indeed that is an incredibly important point and it is a shortcoming in the majority of education approaches in the world. Most education systems promote learning through the idea of content delivery. I.E. a teacher stands in front of a class and gives info that the class must absorb and this is accepted as being &quot;learning&quot;. The students are seldomly &lt;em&gt;really&lt;/em&gt; encouraged to self-learn and textbooks are written so as to be able to replace the teacher. The problem with this is that people are not taught to learn for themselves, you listen in class, if you do not understand you read the textbook and just apply the steps laid out therein. I.E. learning is taken to mean understanding.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arguably the best math program in the world which encourages self-learning is &lt;a href=&quot;https://www.kumon.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kumon&lt;/a&gt;. In fact if you read through the &lt;a href=&quot;https://www.kumon.com/about-kumon/history&quot; rel=&quot;nofollow noreferrer&quot;&gt;bio&lt;/a&gt; on their website, Toru Kumon the creator of the Kumon method, developed Kumon to be a program based around self-learning. I would strongly suggest you advise this student to join Kumon in addition to you tutoring her. This approach might yield the best results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The last thing I want to touch on is attitude. Yes it is the cliche explanation for the root of many problems however its effect on the development of ability and skill is vastly underestimated. I have seen this happen in a number of cases, people start off at school with the idea of &quot;Math is difficult and I struggle with it&quot; and never are they taught otherwise. By the time they finish school they have convinced themselves &quot;I will never be able to become and engineer/accountant/scientists et cetera because I can't do math&quot;. In a couple of years they went from saying &quot;I struggle&quot; to &quot;I can't&quot; purely because of attitude. The most powerful outlook on the understanding of math is to say &quot;I don't understand it YET&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you spend your entire life trying to understand something, you have not failed, you have had an entire life of learning and more than that you can't ask for.&lt;/p&gt;&#xA;" OwnerUserId="14099" LastActivityDate="2020-05-24T12:30:09.140" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18382" PostTypeId="2" ParentId="18379" CreationDate="2020-05-24T15:15:48.997" Score="12" Body="&lt;p&gt;I suggest Geogebra.  There are some great conceptual activities there that will help you with volume.  My favorite creator is Tim Brezezinski.  The following link will take you to an activity that does what you want:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.geogebra.org/m/YpqytNph#material/BZWTCPfd&quot; rel=&quot;noreferrer&quot;&gt;https://www.geogebra.org/m/YpqytNph#material/BZWTCPfd&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a wider range of activities that look at volume of many kinds in depth you can go to &lt;a href=&quot;https://www.geogebra.org/m/YpqytNph#chapter/163298&quot; rel=&quot;noreferrer&quot;&gt;https://www.geogebra.org/m/YpqytNph#chapter/163298&lt;/a&gt; which is part of his Geogebra calculus &quot;book&quot;&lt;/p&gt;&#xA;" OwnerUserId="14100" LastActivityDate="2020-05-24T15:15:48.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18383" PostTypeId="2" ParentId="18364" CreationDate="2020-05-25T21:10:23.873" Score="4" Body="&lt;p&gt;You could try VPython (&lt;a href=&quot;https://vpython.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://vpython.org/&lt;/a&gt;). It's free and does 3D animation. It was built for teaching physics, and can visualize vector fields, graphs, various primitive objects, etc. as well as animating them, and letting you interact with them.&lt;/p&gt;&#xA;" OwnerUserId="5256" LastEditorUserId="5256" LastEditDate="2020-05-28T00:14:55.203" LastActivityDate="2020-05-28T00:14:55.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18384" PostTypeId="2" ParentId="18364" CreationDate="2020-05-26T22:39:21.070" Score="3" Body="&lt;p&gt;There are many good options for the &quot;presentation&quot; one.  If you are doing multivariate calc stuff, then &lt;a href=&quot;https://www.monroecc.edu/faculty/paulseeburger/calcnsf/CalcPlot3D/&quot; rel=&quot;nofollow noreferrer&quot;&gt;CalcPlot3D&lt;/a&gt; is a great option.  (I also recommend many others such as &lt;a href=&quot;https://www.sagemath.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sage Math&lt;/a&gt;, which created &lt;a href=&quot;https://nilesjohnson.net/hopf.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; amazing graphic using Tachyon for image production, but CP3D is the easiest to start with.)&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2020-05-26T22:39:21.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18385" PostTypeId="2" ParentId="17161" CreationDate="2020-05-27T16:37:04.027" Score="0" Body="&lt;p&gt;I had a poster of Jake the Dog (from Adventure Time) in my high school geometry classroom, with this phrase:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9jHnI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9jHnI.png&quot; alt=&quot;Jake the dog saying &amp;quot;Being Bad at Something is the First Step to being Sorta Good at Something&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also gave frequent closed-notes quizzes, and then made my students correct their quizzes on an error analysis sheet like this one (&lt;a href=&quot;https://mathequalslove.blogspot.com/2015/08/error-analysis-sheet-types-of-errors.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/z22Qh.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/z22Qh.png&quot; alt=&quot;Error analysis sheet&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had a poster with a description of different types of errors, in order to encourage them to analyze why they made the mistake they made and how to avoid it in the future. I didn't use the same types of errors as the above blog, instead I used the following: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rFPx8.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rFPx8.png&quot; alt=&quot;description of types of errors&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2020-05-27T16:37:04.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18390" PostTypeId="2" ParentId="17823" CreationDate="2020-05-29T03:33:36.790" Score="5" Body="&lt;p&gt;I vote for Pedro. We should call the point at which the limit is taken the limit point. In contrast, the value obtained by the limit (if it exists) is the limit's value. In particular, &#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \lim_{x \rightarrow c} f(x) = L $$&lt;/span&gt;&#xA;has limit point &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and the value of the limit is &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;. This terminology keeps with the usual usage of the term &lt;em&gt;value&lt;/em&gt; for outputs of the function. In addition, while the term &lt;em&gt;limit point&lt;/em&gt; does have a more abstract topological definition, I don't think there is much danger of confusion. &lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-05-29T03:33:36.790" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18391" PostTypeId="2" ParentId="18379" CreationDate="2020-05-30T03:35:54.067" Score="1" Body="&lt;p&gt;You seem like a real go getter, so I would suggest learning a real language that will generate some nice graphs and is going to come in handy down the line.  Maybe try Python?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I made you a little starter notebook for solids of revolution here: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://colab.research.google.com/drive/19hdZziUxA3RoXg0OEG2ktX1elQWQrfZW&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://colab.research.google.com/drive/19hdZziUxA3RoXg0OEG2ktX1elQWQrfZW&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should get an idea of how to generate a 3d plot with Python here.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2020-05-30T03:35:54.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18392" PostTypeId="2" ParentId="18357" CreationDate="2020-05-30T16:04:03.397" Score="1" Body="&lt;p&gt;Dan Ghica has a &lt;a href=&quot;https://danghica.blogspot.com/2019/09/discovering-category-theory-with.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;blog post&lt;/a&gt; on discovering category theory for primary school children by way of knot theory. Quite charming, I think, albeit not really an answer to your question. Still, I thought it might be worth noting here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I came across it when reading Paweł Sobociński’s &lt;a href=&quot;https://graphicallinearalgebra.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graphical Linear Algebra blog&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2020-05-30T16:15:27.213" LastActivityDate="2020-05-30T16:15:27.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18393" PostTypeId="2" ParentId="18165" CreationDate="2020-05-30T23:48:33.833" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://artofproblemsolving.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Art of Problem Solving&lt;/a&gt; has plenty of problems involving pre-calculus tools, some of them quite challenging.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the best way forward might not be computationally challenging exercises. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You said they lack mastery. Please correct me if I'm wrong but I interpret this as they cannot reliably perform said computations and use said pre-calculus tools in a consistent, effortless, and automatized way. This is probably because their practice was done in 'blocks', so they used those tools a lot at once and then never again, or a long time after the initial practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to improve their computational skills, my advice would be to incorporate frequent tests (spaced repetition) with questions involving a wide range of tools (interleaving). They should not know in advance what kind of tools they would be using so that you improve their performance on a steady basis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There should be room for error, of course, so you could drop a couple of lowest scores to improve morale and give them a chance to make mistakes without overburdening their grades. Even if you repeat the types of questions they faced in high school their knowledge and automation of those tools should improve over the semester (and for everything afterwards).&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2020-05-30T23:48:33.833" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18394" PostTypeId="2" ParentId="18209" CreationDate="2020-05-31T00:00:14.990" Score="2" Body="&lt;p&gt;Yes, whenever possible. Teaching different proofs or different solutions to the same problem has several benefits:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There is less cognitive load because the problem is already known and a solution has already been found;&lt;/li&gt;&#xA;&lt;li&gt;You strengthen students' schemas by creating a connection between existing domains, possibly unconnected until then;&lt;/li&gt;&#xA;&lt;li&gt;You exhibit an application of a very known result or a set of principles in an innovative way, expanding the set of tools students' have at their disposal.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some care is needed: the new solutions should involve knowledge somewhat ready at the students' concept toolbox. If it's not you should be prepared to unwind some of the theory and prepare them to assimilate the new solution. Else you risk them simply overlooking the new ways, in which case you may get questions like &quot;can I just know solution A?&quot; and students' minds will be tuned off everything you do.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2020-05-31T00:00:14.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18395" PostTypeId="1" CreationDate="2020-05-31T03:55:04.997" Score="16" ViewCount="1143" Body="&lt;p&gt;Most of the stuff I'm finding online about ungrading are either general descriptions of its virtues, or personal accounts from instructors &lt;em&gt;from subjects other than math&lt;/em&gt;. Does anyone know any resources pertaining to ungrading specific to math classes?   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For anyone unfamiliar, &lt;a href=&quot;https://www.jessestommel.com/how-to-ungrade/&quot; rel=&quot;noreferrer&quot;&gt;ungrading&lt;/a&gt; is a buzzword being tossed about for assessing students' progress without focusing on quantitative feedback or letter grades, but instead having the students reflect on their own learning progress. Here's a &lt;a href=&quot;https://matheducators.stackexchange.com/q/712/5220&quot;&gt;mildly related popular question on this site&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2020-06-04T16:50:39.430" LastActivityDate="2024-01-31T14:16:36.313" Title="What are some resources for &quot;ungrading&quot; in a math class?" Tags="|reference-request|course-design|grading|assessment|resource-request|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18396" PostTypeId="2" ParentId="18297" CreationDate="2020-06-01T09:08:24.033" Score="6" Body="&lt;p&gt;An answer I saw a few times while marking a particular question was&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\ln(x+1)=\ln(x)+\ln.$$&lt;/span&gt;&#xA;I think this explains the 'everything is linear' phenomenon: everything is &lt;em&gt;linear&lt;/em&gt; because everything is &lt;em&gt;multiplication&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2020-06-01T09:08:24.033" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18397" PostTypeId="1" AcceptedAnswerId="18400" CreationDate="2020-06-01T09:33:48.630" Score="7" ViewCount="996" Body="&lt;p&gt;I am UK-based but I guess this issue currently affects all maths educators and has probably been addressed by those how have been delivering courses through online channels for the past few years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say I wish to set a maths assignment for students in the age-range 16-18. In the UK these students would be taking A-Level which in the US I guess is SAT or college-entrance standard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say the question is a binomial expansion or an integration by parts where in a hand-written answer you would expect to see some lines of working.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Currently I am distributing question papers as pdfs attached to emails or via links to cloud storage. This is what I did pre-lockdown. For these questions the only approach I have found to be reasonable for students is they hand write their solutions and photograph or scan their answers then email them to me. So I receive either pdfs or image files (jpg or png) with their solutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of the above is as before. The problem is the marking and feedback. I have to use &quot;mark up&quot; tools (is that the correct term?) built into packages such as Foxit Reader (pdf), Adobe Acrobat DC (pdf) , Adobe Photoshop (jpg/image), or Gimp (jpg/image) so I can &quot;electronically&quot; write on their solutions. This is (very) time-consuming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the alternatives of converting questions to multiple choice format is both time-consuming and not suitable for these &quot;algebraic working&quot; style questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what online tools do people use to set, mark and feedback this type of  A-Level / pre-college question that speeds up the setting and marking process when performed remotely?&lt;/p&gt;&#xA;" OwnerUserId="7954" LastEditorUserId="470" LastEditDate="2020-06-01T16:01:48.553" LastActivityDate="2020-06-03T02:15:19.300" Title="Efficient methods to receive and grade online mathematics assessments?" Tags="|secondary-education|assessment|online-instruction|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18398" PostTypeId="2" ParentId="18397" CreationDate="2020-06-01T15:16:15.707" Score="4" Body="&lt;p&gt;We use Gradescope, &lt;a href=&quot;https://www.gradescope.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.gradescope.com/&lt;/a&gt; Basically, you upload the scans to Gradescope, and as you make comments on one student's paper, you can assign point values to those comments and re-use the comments on every other student's paper.  If you decide at some point that you want to change the point value, you can just do it, and all of the students' grades will be updated accordingly.  We've found we can give much better feedback using this system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This video gives an idea of the basic functionality: &lt;a href=&quot;https://www.youtube.com/watch?v=F2e9AK7pNFY&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=F2e9AK7pNFY&lt;/a&gt;  &lt;/p&gt;&#xA;" OwnerUserId="7709" LastActivityDate="2020-06-01T15:16:15.707" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18399" PostTypeId="2" ParentId="18397" CreationDate="2020-06-01T15:37:40.460" Score="4" Body="&lt;p&gt;I am recommending &lt;a href=&quot;https://www.myopenmath.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MyOpenMath&lt;/a&gt; (MOM) for this. It is a free, open source, online course management system for mathematics and other quantitative fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Note: There are other systems out there, such as &lt;a href=&quot;https://courses1.webwork.maa.org/webwork2&quot; rel=&quot;nofollow noreferrer&quot;&gt;WeBWorK&lt;/a&gt;, which have similar functionality. I am recommending MOM for its quick set-up for those transitioning to systems like this.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MyOpenMath supports &quot;file upload&quot; for student submissions (if you want to have students hand-write and then take pictures of their work). However, I recommend using a 100% typed format -- both for their work and your comments. And right before everything went online this term, a &quot;show work&quot; feature was added to MOM.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say you want to see a student explain how to find &lt;span class=&quot;math-container&quot;&gt;$\int x \sin(x)dx$&lt;/span&gt;. If you enable &lt;em&gt;show work&lt;/em&gt;, your question might look like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PkaTj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PkaTj.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: Your students will have to type their work. See the math template tools in the problem for writing math content. A good first assignment is one that teaches them how to type mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In your grade book, you will be able to type your feedback to individual students, using the same math template tools they had for their work. [Copy/paste can be a huge time saver for commenting on common mistakes or using a feedback template.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other benefits of this kind of system:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You get to see exactly how long a student took to answer the problem.&lt;/li&gt;&#xA;&lt;li&gt;You can author your own problems. Problems can be algorithmic if you want students to potentially have different versions of a problem.&lt;/li&gt;&#xA;&lt;li&gt;Free to students.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2020-06-01T18:27:23.287" LastActivityDate="2020-06-01T18:27:23.287" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18400" PostTypeId="2" ParentId="18397" CreationDate="2020-06-01T17:38:31.390" Score="11" Body="&lt;p&gt;My technique is pretty low-tech. I distribute solutions to the homework after it's due, so students can mostly tell what they did right or wrong by looking at the solutions. Then I reply to each student's email with any additional comments that they need in order to get feedback that they can't get just by reading the solutions. E.g.,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;#37 -- What went wrong here was that in part b, you tried to use your&#xA;  two equations to eliminate both x and y, but the equations weren't&#xA;  independent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This actually seems to go faster than writing comments on a paper in red pen. For one thing, many students make similar mistakes, so I can just cut and paste.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-06-01T17:38:31.390" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18401" PostTypeId="2" ParentId="18297" CreationDate="2020-06-01T20:21:11.043" Score="1" Body="&lt;p&gt;An interesting one I saw on a student exam many years ago.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The sequence diverges because the Cauchy criterion is dissatisfied.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="127" LastActivityDate="2020-06-01T20:21:11.043" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18402" PostTypeId="2" ParentId="18397" CreationDate="2020-06-01T22:14:39.970" Score="9" Body="&lt;p&gt;If you have access to a device capable of touch input (like a tablet), then I would highly recommend using a stylus to annotate the pdf document. It should take almost the same time as if you were marking assignments on paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Foxit PDF and Gimp have in-built support for touch input. OneNote is also good for mobile and tablet devices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't have access to a stylus, and cannot get one due to the lockdown then you can easily make a DIY one:&#xA;&lt;a href=&quot;https://www.instructables.com/id/DIY-Capacitive-Stylus/&quot; rel=&quot;noreferrer&quot;&gt;Instructables&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.wikihow.com/Make-a-Stylus&quot; rel=&quot;noreferrer&quot;&gt;WikiHow&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14129" LastActivityDate="2020-06-01T22:14:39.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18403" PostTypeId="1" CreationDate="2020-06-02T10:58:55.427" Score="0" ViewCount="99" Body="&lt;p&gt;An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the applications and uses of unreducible fractions? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A classical one is the proof that square root of 2 is irrational (&lt;a href=&quot;https://en.wikipedia.org/wiki/Irreducible_fraction#Applications&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Irreducible_fraction#Applications&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there other uses of it in Basic School (High School or earlier) besides to express a fraction in simplified terms as answer to a math problem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related question: when the concept should be first presented to students? K4? K5? Only in High School? Why?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="8828" LastEditDate="2020-06-03T15:41:24.930" LastActivityDate="2020-06-03T15:41:24.930" Title="Applications of unreducible fractions in Basic School" Tags="|fractions|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18404" PostTypeId="2" ParentId="18403" CreationDate="2020-06-02T14:16:10.443" Score="7" Body="&lt;p&gt;Lots of different fractions all represent the same rational number.  So if someone hands you two fractions, it can be difficult to tell if they represent the same number or not.  For instance, you might need to do some work to determine if the fractions &lt;span class=&quot;math-container&quot;&gt;$\frac{18399}{30665}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{20991}{34985}$&lt;/span&gt; represent the same number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, all rational numbers have only one reduced fraction representation!  So you can detemine equality of rational numbers by finding the reduced fraction representation.  In the example above, both fractions reduce to &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{5}$&lt;/span&gt; so we can be sure they are equal.  Even more interesting is that we can be sure two rational numbers are not equal if they do not have the same reduced fraction representation.  For instance, we can instantly tell that &lt;span class=&quot;math-container&quot;&gt;$\frac{23}{11}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{27}{13}$&lt;/span&gt; are not equal because they are both reduced, but not identical as fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the most basic application of reduced fractions is in making rational number equality easy to detect.  This is often why instructors request their students to report their answer as a reduced fraction:  it makes the answer easy to check!  Otherwise, you would need to either reduce the fraction to check the equality, or use the definition of rational number equality to decide equality (&lt;span class=&quot;math-container&quot;&gt;$a/b = c/d \leftrightarrow ad=bc$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-06-02T16:25:59.183" LastActivityDate="2020-06-02T16:25:59.183" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18405" PostTypeId="2" ParentId="18397" CreationDate="2020-06-02T19:17:55.377" Score="3" Body="&lt;p&gt;An iPad with an Apple pencil- I can hand write comments and corrections on the .pdf and then upload it back to the LMS so that the student can see the feedback.  &lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2020-06-02T19:17:55.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18406" PostTypeId="1" AcceptedAnswerId="18410" CreationDate="2020-06-03T00:03:33.647" Score="3" ViewCount="161" Body="&lt;p&gt;How do I refresh advanced math I learned at a graduate level?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I once was able to do the full solution of a particle in a parabolic well and other advanced math, however 20 years later I'm struggling to follow again Eugene Butkov: Mathematical Physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, after 20 years I need for professional reasons to solve PDE's again and all that college knowledge that I had would be handy.&lt;/p&gt;&#xA;" OwnerUserId="14133" LastActivityDate="2020-06-03T14:55:12.477" Title="Refreshing math knowledge" Tags="|self-learning|vector-calculus|differential-equations|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18407" PostTypeId="2" ParentId="18397" CreationDate="2020-06-03T02:15:19.300" Score="2" Body="&lt;p&gt;This may be a case where falling back to more primitive technology makes things easier.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Print out (on paper) each of the pages the student uploaded.&lt;/li&gt;&#xA;&lt;li&gt;Mark and comment on the paper copies as you normally would.&lt;/li&gt;&#xA;&lt;li&gt;Re-scan the page or pages from each student as a separate document, and mail that document back to the student.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This may not work well if you have a small home printer or similar that doesn't have a sheet feeder for the scanner, where you probably have to do further post-scan editing on the computer to recombine the pages into a separate document for each student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Standard office printer/copier/scanners usually (when well configured) allow you to place one or more pages in the input feeder and press a button to create a single document from all the pages, stored for later retrieval over the network. This is key; you want to be able quickly to scan in a few dozen separate documents, getting exactly one for each student, since then you processing after that is simply to select each document and mail it back to the student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, if you don't have a scanner and network system that itself provides good workflow, you may be able to develop fast post-scan computer-side workflow as a substitute:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If the your scanner can provide individual images, you may be able to just dump these all into a folder and attach them directly to messages back to the student. This does run the risk that you miss an image when returning multi-page documents.&lt;/li&gt;&#xA;&lt;li&gt;If your scanner consolidates many images into a PDF, there are tools for rapidly separating pages out of a PDF into individual PDFs.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14134" LastActivityDate="2020-06-03T02:15:19.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18408" PostTypeId="1" AcceptedAnswerId="18413" CreationDate="2020-06-03T03:16:28.667" Score="2" ViewCount="3158" Body="&lt;p&gt;(1) When do students in Canada learn about the four triangle centres (centers), circumcenter, incenter, orthocenter, and centroid? In the USA (more precisely, Indiana), the math curriculums are by topics and the 9th grade students who completed Algebra 1 can take Geometry in which they learn about them. I looked up some textbooks for Grade 9 to 12 in Canada and couldn't find about them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) How could students in Canada (Ontario) take both MCV4U Calculus and Vectors and  MDM4U Math for Data Management in their 12th grade? MHF4U Advanced functions should be taken before MCV4U. MCF3M Functions and applications should be taken before MDM4U. It seems that the 12th graders should take all three MHF4U, MCV4U, and MDM4U in their last year in high school. I think in USA (Indiana) students take Algebra 2 in Grade 10, Precalculus in Grade 11, and both Calculus and Statistics in Grade 12. In Algebra 2 students are introduced to the early part of Precalculus including the essential functions. It seems that MHF4U still covers Precalculus that is offered in the 12th grade at the same time with MCV4U Calculus.&lt;/p&gt;&#xA;" OwnerUserId="6370" LastEditorUserId="6370" LastEditDate="2020-06-03T14:33:34.487" LastActivityDate="2020-06-03T19:14:52.153" Title="Curriculum in USA vs. Canada" Tags="|secondary-education|curriculum|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18409" PostTypeId="2" ParentId="18408" CreationDate="2020-06-03T04:03:45.607" Score="7" Body="&lt;p&gt;The centroid, circumcenter, and orthocenter are concepts that I teach in Grade 10.  It is part of the analytic geometry unit, and is where we relate knowledge on slope and distance formulas to geometric concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A textbook that includes this topic is &lt;a href=&quot;https://school.nelson.com/principles-of-mathematics-10-student-book-student-text-online-pdf-files/&quot; rel=&quot;noreferrer&quot;&gt;Principles of Mathematics 10&lt;/a&gt; by Nelson.  Here is a set of &lt;a href=&quot;http://www.nelson.com/secondarymath/principles10/documents/PM10TR_Chapter2_TeachingNotes.pdf&quot; rel=&quot;noreferrer&quot;&gt;sample teaching notes&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference:   &lt;a href=&quot;http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf&quot; rel=&quot;noreferrer&quot;&gt;Grade 10 math curriculum (Ontario, Canada)&lt;/a&gt;.  &lt;/p&gt;&#xA;" OwnerUserId="12720" LastActivityDate="2020-06-03T04:03:45.607" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18410" PostTypeId="2" ParentId="18406" CreationDate="2020-06-03T14:29:22.400" Score="4" Body="&lt;p&gt;Don't underestimate how much your knowledge and skills can fade when unused, even after a couple of years, let alone twenty. I've experienced this personally when trying to study advanced topics for which I had the prerequisites on paper from many years ago, but which were no longer fresh in my memory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My general advice is first to check the immediate prerequisites for the area you are trying to refresh. If your recall of those is fairly solid, then find an introductory book, lecture notes - maybe your own old lecture notes if you still have them - or video lectures on your target area (in your case, PDEs) and get to work. Be patient with yourself, as it may take some time to get up to speed. As you proceed, you may find yourself recognizing topics and even recalling definitions, examples or methods that you once learned. Take heart and be encouraged!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you find yourself hazy on the prerequisites, spend some time on those first. How much time is a balancing act. Ideally perhaps, you would study the prerequisites thoroughly, but time may not permit that, or you might simply get bogged down and lose motivation. Monitor how you are going and when you feel somewhat ready, return to your target area. With any luck, you will now be able to make some headway, dipping back into the prerequisites when you need to. (In your case, you might want to revisit some Calculus, Linear Algebra, ODEs and maybe Numerical Analysis.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2020-06-03T14:55:12.477" LastActivityDate="2020-06-03T14:55:12.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18411" PostTypeId="2" ParentId="2336" CreationDate="2020-06-03T15:38:18.183" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Identify what you want you'd like to teach the students for the class. Maybe more importantly, identify the &lt;em&gt;goals&lt;/em&gt; of the course. What do you want the students to &lt;em&gt;know&lt;/em&gt; and what do you want them to be able to &lt;em&gt;do&lt;/em&gt; once the course is over? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Write up a reasonable calendar for the course. Put the lessons you want to give in a reasonable order and designate which days will correspond to each lesson. You may need to alter your goals for the course at this point upon noticing you don't have enough time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Each day in the course, just &lt;em&gt;teach&lt;/em&gt; the students the thing you set out to teach them. There's no need to have an immutable structure to each day (a 5 minute warm-up, followed by 27 minutes of lecture, then a 12½ minute breakout exercise, …); use your training as a teacher, and your own intuition, to teach them as best as you can each day. Some topics will require more lecturing than others to setup. Some topics will require very little scaffolding, and you can have the students explore during the class meeting. Some topics are better discussed in groups, whereas some are best thought about individually. Be flexible.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2020-06-03T15:38:18.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18412" PostTypeId="1" CreationDate="2020-06-03T16:32:02.047" Score="4" ViewCount="274" Body="&lt;p&gt;After seeing no direct responses to &lt;a href=&quot;https://matheducators.stackexchange.com/q/18395/5220&quot;&gt;this question&lt;/a&gt;, I'll instead be more direct myself. &lt;em&gt;Ungrading&lt;/em&gt; is a buzzword being tossed about for assessing students' progress without focusing on quantitative feedback or letter grades, but instead emphasizing students' reflect on their own learning progress. Much of the discussion around ungrading is from instructors in the humanities. But math is different. At the very least, it's much tougher for students to assess their own correct understanding and learning progress in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How must the usual ideas of ungrading be adapted to work in a math class? How would the structure of a math class where the students receive no quantitative feedback have to differ from such a class in the humanities? Answers to this question certainly depend on the maturity of the students, so let's focus on undergraduate college students. For more information on ungrading, check out &lt;a href=&quot;https://www.jessestommel.com/how-to-ungrade/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jesse Stommel's &lt;em&gt;How to Ungrade&lt;/em&gt;&lt;/a&gt; or &lt;a href=&quot;https://www.alfiekohn.org/article/case-grades/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alfie Kohn's &lt;em&gt;The Case Against Grades&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2020-06-03T16:32:02.047" Title="How must the &quot;ungrading&quot; idea be adapted to work in a math class?" Tags="|undergraduate-education|course-design|grading|assessment|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18413" PostTypeId="2" ParentId="18408" CreationDate="2020-06-03T19:14:52.153" Score="1" Body="&lt;p&gt;To answer (2): MHF4U is a co-requisite of MCV4U so there nothing stopping a student taking all 3 courses at the same time. Further more, many Ontario high school operate on semester bases - a student could take MHF4U in first semester and MCV4U in second semester in Grade 12.&lt;/p&gt;&#xA;" OwnerUserId="14136" LastActivityDate="2020-06-03T19:14:52.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18414" PostTypeId="1" CreationDate="2020-06-04T03:28:54.790" Score="2" ViewCount="166" Body="&lt;p&gt;I am currently reading Spivak Calculus on Manifolds and Munkres Analysis on Manifolds.  I am looking for a more advanced text, especially on vector fields as they relate to the great conserved fields of physics.  Thanks for your help.&lt;/p&gt;&#xA;" OwnerUserId="14137" LastActivityDate="2020-06-05T10:22:10.050" Title="Advanced textbook for vector fields" Tags="|textbooks|mathematical-analysis|vector-calculus|" AnswerCount="1" CommentCount="2" ClosedDate="2020-06-10T23:57:11.843" ContentLicense="CC BY-SA 4.0" />
  <row Id="18415" PostTypeId="2" ParentId="18048" CreationDate="2020-06-04T04:54:25.033" Score="2" Body="&lt;p&gt;If the answer requires a complicated formula, I would present several &amp;quot;permutations&amp;quot; of the formula, only one of which is right. Example:&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f(x) = \frac{u(x)}{v(x)}$&lt;/span&gt;, what is &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;? (Quotient Rule.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Answers:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;a) &lt;span class=&quot;math-container&quot;&gt;$u'(x)v(x) + v(x)u'(x)$&lt;/span&gt;. Product rule formula, wrong. &lt;br&gt;&#xA;b) &lt;span class=&quot;math-container&quot;&gt;$u'(x)v(x) - v(x)u'(x)$&lt;/span&gt;. Correct numerator, no denominator, wrong. &lt;br&gt;&#xA;c) &lt;span class=&quot;math-container&quot;&gt;$\frac{u'(x)v(x) + v(x)u'(x)}{(v(x))^2}$&lt;/span&gt;. Correct denominator, wrong numerator, wrong. &lt;br&gt;&#xA;d) &lt;span class=&quot;math-container&quot;&gt;$\frac{u'(x)v(x) - v(x)u'(x)}{(v(x))^2}$&lt;/span&gt;. Correct denominator, correct numerator, correct.&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="-1" LastEditDate="2021-04-19T04:03:05.003" LastActivityDate="2021-04-19T04:03:05.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18416" PostTypeId="1" CreationDate="2020-06-04T07:09:16.497" Score="6" ViewCount="734" Body="&lt;p&gt;We recently received feedback from our 7 year old daugther's school teacher. One of the things mentioned was that our daughter still counts her fingers when she does addition and subtraction. The teacher said that it is important to find other strategies which will save time and effort.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She doesn't have any trouble identifying ones, tens, hundreds, thousands, and she does addition and subtraction on paper when we stack the numbers on top of each other. So, the understanding of numbers and positions are in place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I understand that counting fingers may not be the fastest way to do addition or subtraction, I'm not quite sure which / if other strategies  are available to her / us?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answers to &lt;a href=&quot;https://matheducators.stackexchange.com/q/5866/14139&quot;&gt;a similar post on this site&lt;/a&gt;, although about younger children, seems to agree that using fingers are perfectly fine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I'm curious to know if there are any &quot;standard&quot; ways out there to teach our daughter to do subtraction and addition in her head - without inventing our own system for that.&lt;/p&gt;&#xA;" OwnerUserId="14139" LastEditorUserId="376" LastEditDate="2020-06-07T05:45:50.400" LastActivityDate="2020-06-11T04:47:26.163" Title="What are other strategies for a 7 year old for addition and subtraction besides counting fingers?" Tags="|primary-education|teaching|arithmetic|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18417" PostTypeId="2" ParentId="18414" CreationDate="2020-06-04T11:13:31.073" Score="1" Body="&lt;p&gt;This may not move in the directions of your greatest interests, but it is one option.&#xA;(Not my field so I cannot assess comparable books.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Eschrig, Helmut. &lt;em&gt;Topology and Geometry in Physics&lt;/em&gt;. Vol. 822. Springer Science &amp;amp; Business Media, 2011.&#xA;  &lt;a href=&quot;https://www.springer.com/gp/book/9783540231257&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer link&lt;/a&gt;.&#xA;  &lt;a href=&quot;https://projecteuclid.org/download/pdf_1/euclid.jgsp/1495677632&quot; rel=&quot;nofollow noreferrer&quot;&gt;Review by Jan Jerzy Sławianowski&lt;/a&gt;.&#xA;  Publisher summary: &quot;Written as a set of largely self-contained extensive lectures, the book gives an introduction to topological concepts in gauge theories, BRST quantization, chiral anomalies, supersymmetric solitons and noncommutative geometry.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ytTrV.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ytTrV.jpg&quot; alt=&quot;Cover&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Note: Not to be confused with &lt;em&gt;Topology and Geometry for Physicists&lt;/em&gt; by Charles Nash and Siddhartha Sen.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-06-05T10:22:10.050" LastActivityDate="2020-06-05T10:22:10.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18418" PostTypeId="2" ParentId="18327" CreationDate="2020-06-04T16:41:41.227" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How were the &quot;cheggspert&quot; (i.e. the person employed by chegg to solve&#xA;  students' exam questions) and my student able to locate the github&#xA;  repository containing this question?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here's a possible timeline:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A student posts the question to Chegg.&lt;/li&gt;&#xA;&lt;li&gt;The Cheggster, who probably keeps a trove of solutions to real analysis problems, digs up the solution and sells it to the student.&lt;/li&gt;&#xA;&lt;li&gt;Student shares it with other students.&lt;/li&gt;&#xA;&lt;li&gt;One of the students searches the internet for the text of &lt;strong&gt;the answer from Chegg&lt;/strong&gt; and stumbles on the Github repo, points to that as the source.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This may work if the solution is easier to search for than the text of the problem. [i.e. More typed words in the solution]&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-06-04T16:41:41.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18419" PostTypeId="2" ParentId="12382" CreationDate="2020-06-04T17:21:48.500" Score="1" Body="&lt;p&gt;I think the answer to this question boils down to how concerned you are about &quot;&lt;em&gt;how far we will get&lt;/em&gt;&quot;. Like, it's certainly easier to set a pace if you're lecturing, but you only need to do this if you have a mandated list of &lt;em&gt;topics to cover.&lt;/em&gt; Without such a mandate though, you can run the class exactly as you prefer; that's your prerogative as the instructor. :) Then using &lt;em&gt;all&lt;/em&gt; the class-time for this is depends on what you're comfortable with. Do you need to have a class meeting to hold an exam, or are you comfortable with the students taking an assessment on their own? Do you need to need to have the whole class meet to give a lecture, or can you record a lecture for them to watch (&lt;a href=&quot;https://en.wikipedia.org/wiki/Asynchronous_learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;asynchronously&lt;/a&gt;) on their own?&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2020-06-04T17:21:48.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18421" PostTypeId="1" CreationDate="2020-06-05T02:11:21.973" Score="6" ViewCount="1605" Body="&lt;p&gt;I am teaching 4th-grade kids. The topic is Fraction. Basic understanding of a fraction as a part of the whole and as part of the collection is clear to the kids. Several concrete ways exist to teach this basic concept. But when it comes to fraction addition/subtraction I could not find a way that teaches it concretely.&lt;br&gt;&#xA;Of course, teaching fraction addition &amp;amp; subtraction of the form 3/2 + 1/2 is easy. But what about 3/2+ 4/3?&lt;br&gt;&#xA;It is where we start talking about the algorithm (using LCM), which makes the matter less intuitive and more abstract which I am trying to avoid in the beginning. I believe all abstract concepts should come after the concrete experience.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So teachers do you have any suggestions? &lt;/p&gt;&#xA;" OwnerUserId="13103" LastActivityDate="2020-06-23T09:59:56.010" Title="Concrete way to teach addition and subtraction of fractions" Tags="|mathematical-pedagogy|primary-education|fractions|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18422" PostTypeId="2" ParentId="18395" CreationDate="2020-06-05T02:43:28.640" Score="2" Body="&lt;p&gt;One of my favorite resources has been the shell center’s mathematics assessment project. They have both lessons and more traditional assessment scenarios, however the rubrics provided focus on ways to give formative feedback rather than a strictly point based approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.map.mathshell.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.map.mathshell.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2020-06-05T02:43:28.640" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18423" PostTypeId="2" ParentId="18421" CreationDate="2020-06-05T05:21:35.477" Score="1" Body="&lt;p&gt;Use a &lt;em&gt;language&lt;/em&gt; approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not meant to be a snide remark; I came across a &lt;a href=&quot;https://www.youtube.com/watch?v=V6yixyiJcos&quot; rel=&quot;nofollow noreferrer&quot;&gt;particular TED talk&lt;/a&gt; a while ago that illustrated how math is just another language and we must first learn the proper syntax.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, we know that adding fractions require the use of common denominators, so I would first introduce the topic of equivalent fractions.  If the concept of common denominators is new to the student, perhaps using something concrete like denominations of coins can help bridge that gap.&lt;/p&gt;&#xA;" OwnerUserId="12720" LastActivityDate="2020-06-05T05:21:35.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18424" PostTypeId="2" ParentId="18421" CreationDate="2020-06-05T09:40:44.500" Score="9" Body="&lt;p&gt;Use a piece of paper as your whole.   To teach &lt;span class=&quot;math-container&quot;&gt;$3/2 + 4/3$&lt;/span&gt; do the following.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Give each child/group of children 6 pieces of paper.&lt;/li&gt;&#xA;&lt;li&gt;One piece of  paper should be left as a whole - the students can&#xA;write 1 whole on the paper.&lt;/li&gt;&#xA;&lt;li&gt;Have the students fold 2 of the pieces in half lengthwise, label&#xA;each half as &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt;, and cut out the halves.&lt;/li&gt;&#xA;&lt;li&gt;Have the students fold 2 pieces of papers in thirds widthwise, label&#xA;each third as &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; and cut out the thirds.&lt;/li&gt;&#xA;&lt;li&gt;Have the students take &lt;span class=&quot;math-container&quot;&gt;$3/2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4/3$&lt;/span&gt; and try to add them. You will&#xA;have 2 wholes and &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt;  Discuss  what to do with the&#xA;leftover pieces...&lt;/li&gt;&#xA;&lt;li&gt;Next take the last piece of paper and fold in sixths by folding in&#xA;half lengthwise and in thirds widthwise.  Label each piece &lt;span class=&quot;math-container&quot;&gt;$1/6$&lt;/span&gt; and&#xA;cut out the pieces.  &lt;/li&gt;&#xA;&lt;li&gt;Put the some of the sixths from part 6 on the leftover pieces in&#xA;part 5 to show that &lt;span class=&quot;math-container&quot;&gt;$1/2=3/6$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1/3 = 2/6$&lt;/span&gt;.  Together they make&#xA;&lt;span class=&quot;math-container&quot;&gt;$5/6$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Hope this works for you.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-06-05T09:40:44.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18425" PostTypeId="2" ParentId="18416" CreationDate="2020-06-05T09:52:02.917" Score="4" Body="&lt;p&gt;I have no problem with her counting on her fingers but here are 2 other strategies that she may find useful.  These 2 strategies assume she can add 1 or subtract 1 without her fingers.  If not she should learn that adding or subtracting 1 is just counting up or down.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If she knows her doubles e.g. 3 + 3 or 8 + 8, then when she adds 3 +&#xA;4 or 8 + 9, she could realize that the new equations are just&#xA;doubles plus one. Even if she doesn't know a lot of doubles, she&#xA;probably knows that 5 + 5 is 10.&lt;/li&gt;&#xA;&lt;li&gt;When adding 10 + 7, she should recognize that you just have to put 7&#xA;in the ones place (although) she may not call it that) to get 17. &#xA;Once she knows the trick for adding 10's, she can add 9's by adding&#xA;10 and taking away 1.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There may be other strategies listed in her book - it is worth&#xA;    looking or asking the teacher to suggest strategies. She should also know some of her facts by heart.  In particular it is good to learn doubles and sums to 10 by heart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When counting on her fingers, encourage her to start from one of the numbers and count up, so that if she is adding 3 + 9, she shouldn't count 3 and then another 9.  Instead she should start at 9 and then count up 3 more using her fingers (10, 11, 12).  This is faster and shows a greater understanding.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-06-05T09:52:02.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18426" PostTypeId="2" ParentId="18421" CreationDate="2020-06-05T20:27:48.943" Score="4" Body="&lt;p&gt;I like the cutting paper approach the best because it is cheap and easy, and it forces students to confront basic fractions by folding/cutting them out. But here’s another thought...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For you to demonstrate: have several clear-plastic, cylindrical cups, each marked off in different fractions of a whole (maybe start with 1/2’s, 1/3’s, 1/4’s, 1/6’s, 1/8’s, 1/10’s and 1/12’s). Have another larger cylinder marked off in whole numbers of the smaller cups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Write the problem &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2} + \frac{4}{3}$&lt;/span&gt; and have students make a guess/estimate, and then use the 1/2 and 1/3 cups to fill up with colored liquid or rice (which is why you will demonstrate this). Add the liquid to the large cylinder to see that “yeah, it’s between 2 and 3”, as the students should have guessed. Then work backward, pouring away whole numbers worth of smaller cups until you have less than a whole cup left In the large cylinder. Now try the smaller cups one-by-one until you find one in which the remaining liquid fills up to a marker line. Finally, declare the answer as the mixed number &lt;span class=&quot;math-container&quot;&gt;$2 \frac{5}{6}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I might do this with two nice examples (where I have the appropriately-marked cups) and then try the first one out again, but pour each of the original fractions into the 1/6’s cup before adding them together in the large cylinder. Lots of things you could do with these. &lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-06-05T20:27:48.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18427" PostTypeId="1" CreationDate="2020-06-07T16:24:42.127" Score="4" ViewCount="184" Body="&lt;p&gt;It's hard to be more descriptive with the title without making it excessively long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'm wondering is this: what is the &quot;right attitude&quot; or &quot;right psychology&quot; when confronted with a mathematical question that one is not able to solve or get right (within a reasonable amount of time)? Maybe this sounds like a silly question, but I've struggled with it for a long time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm an independent-minded person. I get more satisfaction from work when I am able to do it by myself. And mathematics is a huge body of knowledge that requires the student to depend on everybody else's work. Obviously that's the case with pretty much every discipline; there's always a certain amount that we always have to learn from others. But what makes it more difficult in math is that there is a culture or expectation of self-reliance. &quot;The proof is left to the reader&quot; is pretty much a meme. Math educators frequently and deliberately omit information. Also math is highly intuitive. It's conceivable that a person &lt;em&gt;could&lt;/em&gt; get pretty far working alone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've had students who get frustrated at themselves for not being able to solve a problem. On one hand, I feel like this attitude isn't healthy. The frustration will just break down their confidence. On the other hand, I completely understand it and feel the same way sometimes. What advice would you offer those kinds of students?&lt;/p&gt;&#xA;" OwnerUserId="14151" LastEditorUserId="14151" LastEditDate="2020-06-08T13:47:13.303" LastActivityDate="2020-06-08T23:56:31.753" Title="self-reliance and psychology in learning math" Tags="|psychology|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18428" PostTypeId="2" ParentId="18427" CreationDate="2020-06-07T23:04:43.710" Score="1" Body="&lt;p&gt;This is an extreme example, but perhaps shows that there is &#xA;no &quot;right attitude,&quot; but rather there is&#xA;a range of approaches&#xA;depending on your cognitive skills and inclinations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Richard Feynman was renowned for teaching himself algebra, trigonometry, and calculus at &#xA;a young age, and then of course following with a brilliant career.&#xA;Krass says in his book&lt;sup&gt;1&lt;/sup&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Feynman needed to fully understand every problem he encountered by starting from scratch, solving it in his own way and often in several different different ways.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://infoproc.blogspot.com/2013/11/feynmans-cognitive-style.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Steve Hsu&lt;/a&gt;&#xA;writes that &quot;it was often easier for [Feynman] to invent his own solution than to read through someone else’s lengthy paper.&quot;&#xA;And Hsu discusses the rumor(?) that Feynman never understood the conventional&#xA;formulation of QED even after it was established that his approach was equivalent to Schwinger's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feynman was extreme in his need (and ability) to re-derive everything.&#xA;Most are more comfortable grasping what others have accomplished and building upon that. &#xA;There is a tradeoff between depth of understanding (if you derive everything from&#xA;scatch) and the breadth of what mathematics you learn (if you build upon&#xA;results you accept because you believe with effort you could derive them).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;Krauss, Lawrence M. &lt;em&gt;Quantum Man: Richard Feynman's Life in Science&lt;/em&gt; (Great Discoveries). WW Norton &amp;amp; Company, 2011. Quote from p.7.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-06-08T11:39:51.270" LastActivityDate="2020-06-08T11:39:51.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18429" PostTypeId="2" ParentId="18416" CreationDate="2020-06-08T05:37:46.127" Score="3" Body="&lt;p&gt;In short: a standard expectation is that children of about your daughter's age should have memorized all sums of one-digit numbers. For example, here's the language from the &lt;a href=&quot;http://www.corestandards.org/Math/Content/2/OA/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Common Core Standards for Grade 2&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Add and subtract within 20.&lt;/strong&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;CCSS.Math.Content.2.OA.B.2 &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Fluently add and subtract within 20 using mental strategies. By end&#xA;  of Grade 2, know from memory all sums of two one-digit numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Nowadays this memorized, instant-recall knowledge is often referred to as &quot;addition facts&quot;. It's a fundamental expectation to how our base-10 number system is designed to support operations like adding and multiplying large numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was your daughter's age, the customary way of learning this was to get a set of flash cards with all of the  combinations of one-digits addends and practice with those daily. Nowadays there are many more options (such as online games, phone apps, etc.) &lt;a href=&quot;https://www.helpingwithmath.com/by_subject/addition/addition-subtraction-facts01.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's one site with suggestions.&lt;/a&gt; You can internet search for the phrase &lt;a href=&quot;https://www.google.com/search?q=learn%20addition%20facts&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;learn addition facts&quot;&lt;/a&gt; for many more. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-06-08T05:37:46.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18430" PostTypeId="1" CreationDate="2020-06-08T08:04:53.150" Score="1" ViewCount="310" Body="&lt;p&gt;Recently, there is an idea sparkling out from my mind due to the COVID-19 outbreak. In the past, we typically attend a course, complete homework and assignment, then finish the semester with a final exam. This takes around 4 courses per semester and 4 years for a whole degree. But I feel we can speed up this process with the aids of the online lecture, as provided from Quality of Videos Lectures and Lectures vs Textbooks. Scott Young had successfully done this by completing a 4-year computer science degree through his MIT Challenge. Thus, I believe online lectures are beneficial for the following reasons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It helps people to accumulate the background to do research as fast as possible. As we know, mathematics is becoming more and more structured and complicated. To overcome this, we have to learn much faster and more effectively. Typically, online lectures are around 30 hours where people can watch all of them in a week (with note-taking and understanding). The best thing is we can always rewatch the part we don't understand and search for appropriate reference (using Google) to understand the idea better. Then we take another week to complete exercises and do final revision using Feynman techniques (teaching to yourself). We can complete a course ideally in two weeks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's see how many courses do we need minimally to be able to read research materials in mathematics. We need Analysis I, Analysis II, Linear Algebra, Abstract Algebra, Point Set Topology, Measure Theory, Complex Analysis, Functional Analysis, Commutative Algebra, Algebraic Topology, Algebraic Curves, Differential Geometry, Representation Theory... Let say around 20 courses to reach the level. Then we only need 40 weeks for the whole program. To be more relaxing, one year is sufficient for us to have the background to do research, of course with the aids of online lectures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is my central idea: &#xA;Self-studying by watching online lectures and then supplementing with exercises and textbook is much more effective than to read the textbook from pages to pages. It's like playing a new board game. Online lecturers (people who know board game) give the main idea of study (game) and then the students (players) do the exercises (play the games) after listening to the main rules. If there is any problem, they consult the reference book (player guide).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you guys think about this? Is learning online a better option for the future? I feel this is an interesting topic to discuss and I hope to get some opinions on this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\textbf{Edit}:$&lt;/span&gt; I have to mention that doing this way only gives sufficient background to pursue research faster, but to consolidate the knowledge, the students have to read research papers, read more references, do more exercises on that particular research topic. For instance, people doing algebraic geometry should nevertheless do all the exercises in Hartshorne, read more advanced topics etc. When they face any difficulties, go back to the relevant prerequisites and consolidate them. (It should be easy to trace back since they have completed all courses)&lt;/p&gt;&#xA;" OwnerUserId="14154" LastEditorUserId="14154" LastEditDate="2020-06-08T08:10:59.610" LastActivityDate="2020-07-12T23:39:03.980" Title="Future way of learning mathematics (towards graduate level)" Tags="|self-learning|" AnswerCount="2" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="18431" PostTypeId="2" ParentId="18427" CreationDate="2020-06-08T14:30:02.943" Score="5" Body="&lt;p&gt;I don't think telling people how they should feel is generally helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am sure that there are lots of people who are &quot;successful&quot; mathematicians who have all kinds of different emotional relationships with the subject.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might imagine one who is primarily motivated by fame and the approval of peers, another who gets really angry when they face a problem they cannot solve and attacks the problem relentlessly until it is vanquished, another who finds great joy in tinkering with puzzles and sharing their insights with others, another who is primarily motivated by supporting the community of mathematicians and would always network while solving a problem, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of these perspectives might be healthier than others in terms of long term emotional well-being, but I am not sure any one of them is &quot;correct&quot; from the perspective of &quot;success&quot; in our academic and social systems.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-06-08T14:30:02.943" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18432" PostTypeId="2" ParentId="18430" CreationDate="2020-06-08T23:50:11.577" Score="2" Body="&lt;p&gt;If you are pushing videos so hard, you haven't even read Scott's story carefully.  He says a textbook with problems is far more important than online lectures (which are not that much needed).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.scotthyoung.com/blog/2012/11/13/why-lectures/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.scotthyoung.com/blog/2012/11/13/why-lectures/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as the &quot;who will check the work&quot; comment (below OP's question), the answer is to use books with answers for the drill so you can check your self for mistakes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There:  two changes that fundamentally reduce the demand for math educators.  Now we won't have to pay them all double six figures.  (Oops.)  ;-)&lt;/p&gt;&#xA;" OwnerUserId="14158" LastActivityDate="2020-06-08T23:50:11.577" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18433" PostTypeId="2" ParentId="18427" CreationDate="2020-06-08T23:56:31.753" Score="0" Body="&lt;p&gt;Life is finite and time has a value to it.  You should use the most efficient manner of learning possible.  This means to utilize more traditional read/practice/test strategies using the most pedagogically efficient texts.  Occasionally, you can deepen your understanding by challenging yourself with derivations and the like.  But don't make it your main approach.  Life is short.&lt;/p&gt;&#xA;" OwnerUserId="14158" LastActivityDate="2020-06-08T23:56:31.753" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18434" PostTypeId="2" ParentId="11971" CreationDate="2020-06-09T03:49:11.750" Score="0" Body="&lt;p&gt;Before starting calculus you must know the following topics.                         &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Relations and functions. &lt;/li&gt;&#xA;&lt;li&gt;Trigonometric functions. &lt;/li&gt;&#xA;&lt;li&gt;Combinatorics.&lt;/li&gt;&#xA;&lt;li&gt;Sequences and series.&lt;/li&gt;&#xA;&lt;li&gt;Infinite series (exponential series, geometric series, logarithmic series, and binomial theorem for any index).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14159" LastEditorUserId="8571" LastEditDate="2020-06-09T20:09:44.337" LastActivityDate="2020-06-09T20:09:44.337" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18435" PostTypeId="1" CreationDate="2020-06-09T09:23:08.297" Score="0" ViewCount="84" Body="&lt;p&gt;I would like to construct a good worksheet for my students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am thinking of creating an instruction that will cover all problems related to identity equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is how I initially constructed the worksheet:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Determine the values of a, b, c, and d that will make the following equation an identity.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;equation 1&quot; - the variables that they need to solve is a and b only&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;equation 2&quot; - the variables that they need to solve is a, b, and c only&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;equation 3&quot; - they need to solve the variables a, b, c, and d.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Notice that for no. 1 and 2, not all the variables a, b, c, and d are asked to find.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it still okay to use this instruction or should I write each number separately like this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solve the following:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Given that &quot;equation 1&quot; is an algebraic identity in x, what are the values of a and b?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given that &quot;equation 2&quot; is an algebraic identity in x, what are the values of a, b, and c?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given that &quot;equation 3&quot; is an algebraic identity in x, what are the values of a, b, c, and d?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8410" LastActivityDate="2020-06-09T09:23:08.297" Title="Better instruction for an identity equation worksheet" Tags="|assessment|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18437" PostTypeId="2" ParentId="18416" CreationDate="2020-06-10T02:34:13.527" Score="2" Body="&lt;p&gt;Typical strategies include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Memorizing sums to 10, e.g. &lt;span class=&quot;math-container&quot;&gt;$7 + 3 = 10$&lt;/span&gt;, then using that equation to determine &lt;span class=&quot;math-container&quot;&gt;$6 + 3 = \_\_$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$7+4=\_\_$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Memorizing the doubles, e.g. &lt;span class=&quot;math-container&quot;&gt;$7 + 7 = 14$&lt;/span&gt;, then using that to determine &lt;span class=&quot;math-container&quot;&gt;$8+7=\_\_$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Decomposing/Recomposing. e.g. &lt;span class=&quot;math-container&quot;&gt;$9+8=10+7=17$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Using single digit facts to determine sums within 20. e.g. Since &lt;span class=&quot;math-container&quot;&gt;$3+6=9$&lt;/span&gt;, then we can easily figure out &lt;span class=&quot;math-container&quot;&gt;$13+6=\_\_$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If these strategies do not seem obvious to your daughter after seeing them in symbolic form, then you should use ten-frames, linking cubes, and number tracks to help her visualize them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.pagat.com/invented/auction.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Auction games&lt;/a&gt; are a fun way to practice a lot of this.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2020-06-10T02:34:13.527" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18438" PostTypeId="2" ParentId="18421" CreationDate="2020-06-10T02:40:56.507" Score="4" Body="&lt;p&gt;Suggestion: Do not bother having students do calculations with &lt;em&gt;two&lt;/em&gt; fractions until they can do a comparable operation with a fraction and a whole number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example: Before teaching, say, &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}-\frac{4}{3}=\_\_$&lt;/span&gt;, ensure they can answer something like &lt;span class=&quot;math-container&quot;&gt;$4 - \frac{1}{5}=\_\_$&lt;/span&gt;. Make sure all students can draw that on a number line and with area diagrams. This is how you ensure that their knowledge of fractions will integrate with their knowledge of whole numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once students are ready to add and subtract fractions that already have common denominators, ensure they can do so on a number line and with area diagrams and with &lt;a href=&quot;https://duckduckgo.com/?t=ffab&amp;amp;q=fraction%20tiles&amp;amp;iax=images&amp;amp;ia=images&quot; rel=&quot;nofollow noreferrer&quot;&gt;fraction tiles&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, when it comes time to do &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}+\frac{4}{3}=\_\_$&lt;/span&gt;, you can:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Put the halves tiles and the thirds tiles together&lt;/li&gt;&#xA;&lt;li&gt;Pretend to be horribly confused by how much that is&lt;/li&gt;&#xA;&lt;li&gt;Estimate it to be a little under 3 wholes&lt;/li&gt;&#xA;&lt;li&gt;Demonstrate how sixths tiles come to the rescue.   :)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2020-06-10T02:40:56.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18440" PostTypeId="1" CreationDate="2020-06-10T20:38:51.133" Score="3" ViewCount="723" Body="&lt;p&gt;I am a student from CS background. I have been following &quot;Discrete Mathematics and its Applications&quot; by Kenneth Rosen, though it is a good book, but it does not cover group theory. I would like to concentrate on the following topics and if required I am prepared to study more to build my foundations to understand the topics which I said shall be concentrating on. The topics are as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Algebraic Structures and Morphism: Algebraic Structures with one Binary Operation, Semi Groups, Monoids, Groups, Congruence Relation and Quotient Structures, Free and Cyclic Monoids and Groups, Permutation Groups, Substructures, Normal Subgroups, Algebraic Structures with two Binary Operation, Rings, Integral Domain and Fields.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would like to learn about a textbook which deals with them starting from introductory level. As I would be using the book for self study, I request the language of the book to be quite easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is &quot;Elements of Discrete Mathematics&quot; by C.L. Liu a good choice?&lt;/p&gt;&#xA;" OwnerUserId="14166" LastActivityDate="2020-06-14T05:15:05.307" Title="Textbook to study group theory as a part of Discrete Mathematics" Tags="|reference-request|textbooks|reference-books|discrete-math|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18441" PostTypeId="2" ParentId="1330" CreationDate="2020-06-10T21:08:34.513" Score="2" Body="&lt;p&gt;Isn't the &quot;functional equation road&quot; the same thing as the &quot;differential equations road&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can take an axiomatic approach to defining sine and cosine. See Apostol's Calculus book, page 95. Many other authors have done a similar thing and the background isn't too steep. You simply declare that there exist two functions satisfying a few properties (which can be geometrically motivated). Then later in your course you prove the existence portion in a rigorous fashion. I don't see anything wrong with this.&lt;/p&gt;&#xA;" OwnerUserId="13694" LastActivityDate="2020-06-10T21:08:34.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18442" PostTypeId="2" ParentId="18416" CreationDate="2020-06-11T04:47:26.163" Score="2" Body="&lt;p&gt;Play games: dice, board games, card games. So many of them include counting and &quot;adding on&quot;. Or an online game like &lt;a href=&quot;https://www.mathplayground.com/number_bonds_20.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Play store, where you pretend to buy things. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw a circle numbered 1 to 10, and draw lines between the pairs that add to 10. Then decorate it. If she likes this, do a circle numbered 1 to 12, and connect the pairs that add to 12. Make up your own rules. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea is to make friends with the numbers, so it's not tedious. As Montessori said, &quot;Play is children's work.&quot;&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-06-11T04:47:26.163" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18443" PostTypeId="2" ParentId="18440" CreationDate="2020-06-11T16:14:55.937" Score="5" Body="&lt;p&gt;To repeat my comments:&#xA;I can recommend Fraleigh's classic introduction.&#xA;It is easy to read for beginners, with many exercises, from easy to difficult, on which self-learners could check themselves.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;John B. Fraleigh &#xA;  &lt;em&gt;A First Course in Abstract Algebra&lt;/em&gt;, 7th Edition.&#xA;  Pearson, 2002.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/gd3eu.jpg&quot; height=&quot;200&quot; /&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;Because it's been around so long (1971?), there are many opportunities to find inexpensive used copies.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-06-12T00:03:06.473" LastActivityDate="2020-06-12T00:03:06.473" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18444" PostTypeId="1" AcceptedAnswerId="18447" CreationDate="2020-06-11T20:12:10.857" Score="5" ViewCount="1196" Body="&lt;p&gt;I am a student, in my last year of school(17 years old)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was about 13 years old I fell into the &lt;a href=&quot;https://artofproblemsolving.com/news/articles/avoid-the-calculus-trap&quot; rel=&quot;noreferrer&quot;&gt;calculus trap&lt;/a&gt; by starting off learning trigonometry on my own, when I was supposed to factor equations or solve basic probability questions. Being good at math(fast arithmetic skills, could grasp new concepts easily etc.) I came across an interschool math competition at that time which was purely based on geometry. I studied by myself for that comp, mainly the properties of circles, triangles and as I earlier mentioned, was introduced to trigonometry. Even though I didn't score well in that comp, I was mesmerized studying those new topics which led me to discover more and more, and finally started learning calculus just one year after that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I soon realized that I have fallen into a large hole with no end, though going deeper and deeper was becoming more interesting, it started affecting my performance in other subjects. I also realized  that instead of starting with calculus, I could have solved much harder problems related to my school curriculum and hence could score better in Olympiads. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately I am still in that hole, going deeper and deeper, studying more abstract, higher level concepts. But I will be passing out of my school next year and will get admission in a nice college aiming to study higher maths and everything will become normal. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My main question is that should I encourage my juniors to do the same thing which I did? Or should I guide them to study maths in a more systematic way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://artofproblemsolving.com/news/articles/avoid-the-calculus-trap&quot; rel=&quot;noreferrer&quot;&gt;link&lt;/a&gt; that I had provided is from art of problem solving and hence I found it a bit harsh towards my situation, so I decided to ask here for guidance. I also read &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7718/effects-of-early-study-of-advanced-books&quot;&gt;this&lt;/a&gt; question but that is not exactly my case, I was thorough in my study, also took help from my teachers if I did not understand anything.&lt;/p&gt;&#xA;" OwnerUserId="14170" LastActivityDate="2020-06-14T12:34:04.010" Title="Falling into the calculus trap" Tags="|secondary-education|student-motivation|student-question|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18445" PostTypeId="2" ParentId="18440" CreationDate="2020-06-11T20:44:19.433" Score="7" Body="&lt;p&gt;The list of topics you want to study corresponds rather to abstract algebra than group theory. You did not say why you are interested specifically in group theory, but I believe that acquaintance with various algebraic structures in addition to groups would be beneficial for any person learning mathematics beyond school mathematics. Therefore, abstract algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wish to recommend&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gallian, Joseph. &lt;em&gt;Contemporary Abstract Algebra&lt;/em&gt;. 9th ed., Cengage Learning, 2017.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is suitable for beginners because it has many examples explained in great detail, sometimes even tediously. Also it contains many exercises. The author included some applications. The book's level of generality is low for my tastes, but it is okay for a beginner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The phrase “discrete mathematics” is not a useful keyword for searching books on mathematics. As you can read in the Wikipedia, the scope of Discrete Mathematics is defined by what it does not contain (Analysis). There is no clear-cut set of notions that Discrete Mathematics is about. Its best definition is “mathematics for computer science” which, naturally, is a lot of things.&lt;/p&gt;&#xA;" OwnerUserId="7992" LastEditorUserId="7992" LastEditDate="2020-06-12T12:32:40.727" LastActivityDate="2020-06-12T12:32:40.727" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18446" PostTypeId="2" ParentId="18440" CreationDate="2020-06-12T01:38:43.667" Score="-2" Body="&lt;p&gt;It's possible you just need the group theory, not the abstract algebra first.  Would be good to look into what you need, before trying to satisfy it.  The math types here will assume you need A before B.  But for us scientists it's not always needed to roll how they do.  As a practical example consider the books for chemists and spectroscopists on group theory (e.g. Cotton's).  Very far from a &quot;have to learn abstract algebra in order&quot; approach.  But give you the insights needed to understand why different shaped molecules have different numbers of vibrational modes (and consequent spectrums).&lt;/p&gt;&#xA;" OwnerUserId="14158" LastActivityDate="2020-06-12T01:38:43.667" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18447" PostTypeId="2" ParentId="18444" CreationDate="2020-06-12T18:23:45.973" Score="8" Body="&lt;p&gt;Echoing @AndreasBlass' remark, and having experienced somewhat similar episodes, it is already precarious enough to make such choices _for_oneself_. So, to directly answer your question: I think &quot;no, do not encourage others to (too violently) disconnect from the math curriculum at school&quot;. I don't think it's about problem-solving versus calculus, at all. And, no, I'm not a fan of typical school math curricula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If nothing else, given the way many kids think about things (as I did, for sure), there is the risk of providing a convenient rationalization for disconnection and non-conformity that is not replaced by anything constructive. And then it's hard to get back in sync even if one wants to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.g., when I learned about the expressibility of trig functions in terms of exponentials, via (basic things about) complex numbers, the game/challenge of proving trig identities lost its charm. I could not make myself care very much. But the instructor for that class did not care about &quot;better alternatives&quot;, so, after some earlier class-room debacles, I decided to play along to a sufficient degree to avoid trouble. And &quot;physics without calculus&quot; seems ridiculous, too, but, ... and &quot;economics without calculus&quot;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I'd recommend maintaining at least two threads: one following your own curiosity and &quot;genuine&quot; mathematics and science (rather than school curricula), but another to maintain &quot;presentability&quot;, in effect showing that you can understand the ambient social constructs and cooperate with other people at least minimally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No, I am no fan of conformity for its own sake! :)  But it is certainly very convenient for &lt;em&gt;individuals&lt;/em&gt; to be &lt;em&gt;able&lt;/em&gt; to &quot;code switch&quot;, to survive. Or &quot;to survive long enough to get to a position to have less obligation to conform&quot;? A tricky balance, for sure.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2020-06-12T18:23:45.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18448" PostTypeId="2" ParentId="18444" CreationDate="2020-06-12T23:25:30.890" Score="8" Body="&lt;p&gt;The article at artofproblemsolving seems silly to me. The author's idiosyncratic opinion seems to be that students who are ready to take calculus should refrain from taking calculus and instead do math contests. People are all different, and there is not just one appropriate path for a mathematically precocious student. Some people might want to take calculus &lt;em&gt;and&lt;/em&gt; do math contests -- it's not like they're mutually exclusive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One good argument for learning calculus early is that at many universities, the quality of instruction in first-semester calculus is  atrocious. It might be better to learn it on your own or in an AP high school class than to be subjected to that.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-06-12T23:25:30.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18449" PostTypeId="1" CreationDate="2020-06-13T16:33:25.767" Score="12" ViewCount="467" Body="&lt;p&gt;Now that everyone has had the experience of teaching math in an online/remote/synchronous/asynchronous format, and looking forward to more of this in the Summer and Fall terms, how do we change our practices?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the recent question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18031/how-shall-we-teach-math-online&quot;&gt;How Shall We Teach Math Online&lt;/a&gt;, the main issues posed were:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Limited online resources and technology&lt;/li&gt;&#xA;&lt;li&gt;Limited time/pay for creating online content&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The comments and answers generated more issues to consider:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How to conduct exams/quizzes&lt;/li&gt;&#xA;&lt;li&gt;How to hold video meetings &amp;amp; office hours&lt;/li&gt;&#xA;&lt;li&gt;Video creation setups&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What have been the takeaways from our experience teaching remotely or online this term? Are any new best-practices emerging with respect to how we create-and-deliver content, engage with our classes, give exams or make the transition-to-remote easier for students?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[This question is broad, so I will accept its closure, but a lot has happened in the last 3 months in math instruction. I think we could all learn from our collective experience and hear what works in this new format.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-06-14T00:46:41.223" Title="How, now, shall we teach math online?" Tags="|undergraduate-education|secondary-education|exams|assessment|online-instruction|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18450" PostTypeId="2" ParentId="18449" CreationDate="2020-06-13T18:22:17.357" Score="7" Body="&lt;p&gt;I taught all of my classes in an asynchronous (fully online) format this term.&lt;/p&gt;&#xA;&lt;p&gt;Based on ideas posted here, I chose the following routine:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Content Delivery&lt;/strong&gt;:  I made videos with my phone and posted them to YouTube for the &amp;quot;lecture&amp;quot; content. I printed copies of activities I wrote, and worked through them in the video, periodicaly asking students to &amp;quot;pause the video and try this&amp;quot;. I posted the pdf of the activity sheet on a lesson page in Blackboard (our LMS), followed by the embedded YouTube video I shot and embedded GeoGebra graphs where appropriate. I also updated a list of &amp;quot;recommended textbook problems&amp;quot; for each lesson.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Assessments&lt;/strong&gt;:  Each video lesson was followed by a &lt;a href=&quot;https://www.myopenmath.com/info/aboutus.php&quot; rel=&quot;noreferrer&quot;&gt;MyOpenMath&lt;/a&gt; &amp;quot;concept check&amp;quot; I wrote (between 5 and 15 questions related to the lesson), and each week had either a quiz (again MyOpenMath), a group assignment (students collaborated to solve more in-depth problems and type solutions up) or an exam (also MyOpenMath, with essay problems). The quizzes and &amp;quot;concept checks&amp;quot; had no time limit, due only at the end of the term.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Exams&lt;/strong&gt;:  The first exam was given with a 2.5-hour time limit. It was given through MyOpenMath and had problems with randomized elements (numbers, variable names, graphs, names of people, etc.). Grades were as I might have expected from a face-to-face course.&lt;/p&gt;&#xA;&lt;p&gt;The 2nd exam and Final exams were also given with MyOpenMath, with a longer time limit and the use of essay questions. Due to the longer time limit, I caught four students posting some-to-all of the questions to Chegg, and it was easy to tell exactly who posted them because of the randomized elements (e.g. &amp;quot;Mikaela painted a house in 18.5 hours...&amp;quot; vs. &amp;quot;Robert painted a fence in 3.75 hours...&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Meeting Students&lt;/strong&gt;: I opened &lt;a href=&quot;https://slack.com/&quot; rel=&quot;noreferrer&quot;&gt;Slack&lt;/a&gt; workspaces for each class, making it a requirement that students joined. Students really liked it, and I was just a little afraid we would go over the 10,000 messages allowed in the free plan. [We used about 4000 for a class of 30.] I also used Zoom for office hours and meeting students one-on-one for help.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Takeaways&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;p&gt;When surveyed, students were very happy with the videos, online concept checks and the use of Slack. I will continue with YouTube as it is easy to use, it creates a decent first run at captions (which can be edited) and the view-count suggests students made use of them. I will also continue to use MyOpenMath and Slack for assessments and communication, as they provide immediate feedback on work and (at least the perception of) immediate access to the instructor.&lt;/p&gt;&#xA;&lt;p&gt;With this first experience behind me, I will be changing the way exams are delivered. Students will still use MyOpenMath for exams, with the following restrictions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Exams will be taken on Zoom so that I can see what students are writing.&lt;/li&gt;&#xA;&lt;li&gt;They will have a shorter time-limit.&lt;/li&gt;&#xA;&lt;li&gt;I will require all hand-written work uploaded at the end of the exam.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Unfortunately, giving students too much time on an exam is a window fit for defenestration.&lt;/p&gt;&#xA;&lt;p&gt;I will also be conducting an exit-interview for each student, via Zoom, in which I want to hear what they know from the course. I will create and publish to them, in advance, a rubric outlining how I will assess them in the interview. The purpose is to get a sense of what basics they actually understand.&lt;/p&gt;&#xA;&lt;p&gt;[I teach at a community college in the United States.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:41.637" LastActivityDate="2020-06-14T00:46:41.223" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18451" PostTypeId="2" ParentId="18440" CreationDate="2020-06-14T05:15:05.307" Score="3" Body="&lt;p&gt;I'll echo @XanderHenderson in the comments that Hungerford's &lt;em&gt;Abstract Algebra: An Introduction&lt;/em&gt; is a really nice textbook. It's what I had as an undergraduate, and I'm currently re-reading it for maybe the third time. It's one of my favorite, well-written, clear texts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that he's committed to a pedagogy that starts from the most familiar/concrete and gets subsequently more abstract, hence the sequence goes: Modular Arithmetic, then Rings, then Groups. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-06-14T05:15:05.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18452" PostTypeId="2" ParentId="18444" CreationDate="2020-06-14T05:39:38.433" Score="2" Body="&lt;p&gt;I will also chime in and say that the argument on the linked Art of Problem Solving site is unpersuasive, and somewhat misses a broader point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, the &lt;em&gt;real&lt;/em&gt; point of the mathematical discipline is to identify patterns in systems and prove their correctness (hopefully in an insightful, persuasive, explanatory style). The &quot;trap&quot; that I would identify is the standard calculus-track being about numerical calculations for grades K-14, and then mathematics majors needing to switch tracks to proof-based courses, which is the essence of the profession of mathematics. It's really a huge irritation that so many students are &quot;tricked&quot; into thinking that being good at following rules for calculations means that majoring in math is a good choice. (E.g., in reality, computers can compute any such K-14 exercise instantaneously, so humans doing calculations is not really in itself of any practical use.) I may argue that &quot;problem solving&quot; in terms of applications with numerical answers at the end is not any better in this regard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my top suggestion would be to supplement the time between high school geometry and the 2nd year of college with some kind of practice at reading and writing proofs (number theory being a common starting sandbox). I might recommend Richard Hammack's free, open-textbook &lt;a href=&quot;http://www.people.vcu.edu/~rhammack/BookOfProof/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book of Proof&lt;/a&gt; as a good starting point, even though it's written for a college audience -- if others have better suggestions I'd like to hear them. &lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-06-14T05:39:38.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18453" PostTypeId="2" ParentId="18444" CreationDate="2020-06-14T12:34:04.010" Score="4" Body="&lt;p&gt;[VERY LONG ANSWER, needs patience to read through]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel this is a problem many students who are good at maths face. They understand the simple tricks and patterns which are present in the school syllabus and so it is simple for them and after some practise and memorisation they are done. Then they seek out more maths and find out about topics like trigonometry and calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing which I feel is a big factor causing the drive and a part of the problem which many people ignore is the 'need to be a genius'. Growing up we all hear stories about Einstein and his legendary E=mc^2 formula which he dreamt up because his brain was just so big. In mathematics we have role models like Euler, Gauss and Ramanujan who seemingly picked out amazing results from thin air. These stories make up our perception about scientists who are introverted geniuses which seem to know everything except how to talk to people and conform to normality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So students such as myself begin to rebel against the system and find out more advanced maths on their own. However, this is much more difficult than just following the school curriculum. Even if the topic is within your intellectual capacity, having no one to explain it to you is very discouraging and you give up easily. Since you are talking about studying high level mathematics in school you definitely would have experienced finding out about some topic or the other in which you have no idea how to even process the proofs and theorems related to it after a certain basic point. This is very frustrating because you have an image of being good at maths in your mind and you cannot meet it if you don't immediately understand your textbook or whatever you are studying. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This then becomes a trap: you don't study topics which you don't immediately understand or have good intuition for, and so you keep going down and down into deep rabbit holes where you go into one sub-topic after another without pausing at any level to expand your knowledge to related topics and building a firm base before you go on into more specialization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is why there is a need to tell students to be firmly confident in their own stupidity. They don't get things immediately and make like 10 mistakes while solving a question, but they can explore and work harder until they are proficient in that topic. Not only their own stupidity, they should know that everyone else is also stupid. Even Euler, Gauss and Ramanujan were stupid, in the sense that each one of them must have struggled with some topic or the other, and they must have felt frustrated and incompetent many times because of that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best way I have found to overcome this inferiority complex is to let students make something original of their own. If you know about the process of making an entirely new discovery (new to them, maybe not to the world) without always relying on thought patterns and tricks which are just programmed into you after solving many school and Olympiad level problems (which do not test your mathematical ability accurately), you learn to appreciate many things. You realise how random and arbitrary making progress in a problem is. You can be stuck for days on a single lemma but come up with a one liner that completely resolves it while having a bath. It is also extremely non-linear, which means you can take long detours without coming close to the correct method. However, you also realise that progress in a small or big form will always come if you try hard for long enough (and take enough breaks to reset).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you realise that every one of those geniuses went through this same random, frustrating but highly satisfying process every time they solved a hard problem, the illusion of being smart only if you are lightning fast in solving and understanding is quickly broken. The only thing you need to make an independent, worthwhile discovery is lots of studying and lots of thinking. That is how maths in the real world is done. You may never get even close to a level like that of Euler, but it is unreasonable to hold such an expectation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think you should encourage your juniors to explore maths: not just topics which are at a higher level, but to truly &lt;em&gt;explore&lt;/em&gt; and wonder about things and try to find out more about them. Learning a new topic should be an interest rather than a habit or a drive. It is psychologically very unhealthy and dangerous to have such an unfounded fear of failure and lack of confidence in your abilities. Expanding your horizons above school, competitions and Olympiads helps with that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, when I proved the formula for the sum of a geometric series in 9th class months before it was taught, I felt much more proud and confident in my abilities than when I solved much harder Olympiad type problems which tested not my independent thinking but my ability to remember and apply obscure formulas and patterns. It also probably helped me much more in developing mathematical thinking than those problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for school, unfortunately, it is a necessary part of life and you have to devote some time to study school maths and other topics according to what marks you want. It is not all  bad: if you look close enough, there are many things to be explored in school mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, when talking about studying other subjects, you should have practicality in mind and know about the consequences of your actions when you avoid studying for Olympiads or school. If you are okay with sacrificing that to fulfill your interest in maths, only then should you do that, otherwise you should search for a compromise.&lt;/p&gt;&#xA;" OwnerUserId="14177" LastActivityDate="2020-06-14T12:34:04.010" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18454" PostTypeId="2" ParentId="17454" CreationDate="2020-06-14T15:44:48.023" Score="2" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Hyperboloid&quot; rel=&quot;nofollow noreferrer&quot;&gt;hyperboloid of one sheet&lt;/a&gt; appears to be very curvaceous everywhere, and yet it is actually a ruled surface (i.e., can be traced out from a straight line moving in space).&lt;/p&gt;&#xA;" OwnerUserId="14178" LastActivityDate="2020-06-14T15:44:48.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18457" PostTypeId="2" ParentId="18358" CreationDate="2020-06-15T23:50:38.470" Score="1" Body="&lt;p&gt;Here's an interesting attempt in both text and &#xA;interactive images. Not a hierarchy but rather a &quot;map.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;The Map of Mathematics&lt;/strong&gt;.&#xA;  A project by &lt;em&gt;Quanta Magazine&lt;/em&gt;. Text by Kevin Hartnett.&#xA;  Design and visualizations by Kim Albrecht and Jonas Parnow.&#xA;  Feb. 2020.&#xA;  &lt;a href=&quot;https://www.quantamagazine.org/the-map-of-mathematics-20200213/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quanta link&lt;/a&gt;.&#xA;  &lt;hr /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;img src=&quot;https://i.stack.imgur.com/Uaofd.png&quot; width=&quot;200&quot; /&gt;&#xA;  &lt;br /&gt;&#xA;  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;  &lt;sup&gt;&#xA;  The Riemann zeta-function.&#xA;  &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&#xA;&lt;p&gt;The top-level map is &lt;a href=&quot;https://www.quantamagazine.org/the-map-of-mathematics-20200213/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. (Need to scroll a bit and click on &#xA;&lt;span class=&quot;math-container&quot;&gt;$\fbox{The Map}$&lt;/span&gt;.)&#xA;Hover over a topic to see connections between fields.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-06-16T13:48:51.773" LastActivityDate="2020-06-16T13:48:51.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18458" PostTypeId="1" CreationDate="2020-06-16T07:10:54.793" Score="1" ViewCount="259" Body="&lt;p&gt;I'm looking for advanced workbooks and exercises for working in class (math high school/undergraduate level) covering the following topics (or some of them):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Logic and sets (propositional calculus, predicates, relations between predicates and sets, constructing of sets)&lt;/li&gt;&#xA;&lt;li&gt;Addition and subtraction (associativity, commutativity, additive identity, etc.), multiplication and division, comparison&lt;/li&gt;&#xA;&lt;li&gt;Natural numbers&lt;/li&gt;&#xA;&lt;li&gt;The Pythagorean theorem visually&lt;/li&gt;&#xA;&lt;li&gt;The Binomial theorem visually&lt;/li&gt;&#xA;&lt;li&gt;Commensurability, the Euclidian theorem&lt;/li&gt;&#xA;&lt;li&gt;Straight-line mathematics (motion, shift, the composition of shifts, group).&lt;/li&gt;&#xA;&lt;li&gt;Reflection&lt;/li&gt;&#xA;&lt;li&gt;Cayley table for line shifts&lt;/li&gt;&#xA;&lt;li&gt;Chasles' theorem (lemma about three nails).&lt;/li&gt;&#xA;&lt;li&gt;Circle movements, group of circle movements, Chasles' theorem&lt;/li&gt;&#xA;&lt;li&gt;Integers, ring.&lt;/li&gt;&#xA;&lt;li&gt;Greatest common divisor and the Euclidian theorem&lt;/li&gt;&#xA;&lt;li&gt;Prime numbers and the Fundamental theorem of arithmetic&lt;/li&gt;&#xA;&lt;li&gt;The object symmetry, the symmetry of the equilateral triangle, the symmetry of the regular polygon.&lt;/li&gt;&#xA;&lt;li&gt;Subgroups of the circle movements.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Can you help me with authors/keywords in your country? We have already put up the theoretical material, we need only to collect exercises for them.&lt;/p&gt;&#xA;&lt;p&gt;Thank you for your help in advance!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;UPD:&lt;/strong&gt; Question needs an additional clarification: we've put together the theoretical material for advanced high school kids / undergrads, you can check it here: &lt;a href=&quot;https://github.com/nkrishelie/mathempire/blob/master/250/250le%C3%A7ons.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://github.com/nkrishelie/mathempire/blob/master/250/250le%C3%A7ons.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's in Russian, but you can use Google Translate on PDF document to get the basic idea. Actually, the problem is that we have one-two problems per chapter, but we need to get some more covering our theory with practice. So, I need a collection of books/articles (just name and title) with collections of problems matching topics we try to cover. Something like this. The most critical for us now is to cover &lt;strong&gt;Chapters 9 - 15&lt;/strong&gt;. &lt;strong&gt;Permutations, linear algebra, continuum, algebraic numbers and elements of analysis.&lt;/strong&gt; For the beginning of the book I've already found some materials to work on.&lt;/p&gt;&#xA;&lt;p&gt;The goal for all of this is simple: to collect better problems and avoid cheating by using non-googleable materials from other countries.&lt;/p&gt;&#xA;" OwnerUserId="13610" LastEditorUserId="13610" LastEditDate="2020-07-10T10:27:33.393" LastActivityDate="2020-07-10T10:27:33.393" Title="Workbooks for advanced high school math topics" Tags="|geometry|homework|logic|arithmetic-operations|set-theory|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18459" PostTypeId="1" CreationDate="2020-06-16T15:06:35.580" Score="8" ViewCount="291" Body="&lt;p&gt;Does anyone know of any studies or have personal experience dealing with difficulties (if any) faced by students studying mathematics if they come from countries which use languages written from right-to-left or top-down? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been wondering about this recently because I have been working on supplemental study guides. It occured to me that mathematics covering everything from arithmetic to calculus is heavily viewed in terms of a left-to-right perspective (e.g. digits with the highest place value are the leftmost ones in a numeral and are thus read first; the positive x-axis extends to the right; a line with positive slope is seen as one which goes up when viewed from left-to-right)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then again there are many accomplished mathematicians who come from countries which speak and write these languages so maybe it's not that difficult of a conceptual jump after all. I think I for one would have difficulty though...&lt;/p&gt;&#xA;" OwnerUserId="1914" LastActivityDate="2020-07-13T00:11:38.327" Title="Math Education for Students who use Right-to-Left Written Languages" Tags="|mathematical-pedagogy|reference-request|education-research|english-language-learners|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18460" PostTypeId="2" ParentId="18246" CreationDate="2020-06-17T03:01:24.140" Score="0" Body="&lt;p&gt;My background includes attending one liberal arts college for undergrad; graduate school at a relatively small university; and two years as a mathematics professor at another liberal arts college.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I wouldn't describe only two lessons as &quot;normal,&quot; neither would I describe it as surprising. What I would &lt;em&gt;expect&lt;/em&gt; is a few weeks of complex numbers between a couple of early or intermediate courses, typically ones labelled &quot;Discrete Math&quot;, &quot;Linear Algebra&quot;, or &quot;Differential Equations&quot; or the like. Then there will be an advanced course in Complex Analysis, but it will probably be optional and not necessary for math majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm actually a bit surprised that complex numbers were mentioned in a calculus class; I'm used to the only mention of them at that level being something of the &quot;and square roots of negative numbers is something we will save for a more advanced course&quot; variety.&lt;/p&gt;&#xA;" OwnerUserId="14185" LastActivityDate="2020-06-17T03:01:24.140" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18461" PostTypeId="2" ParentId="18459" CreationDate="2020-06-17T04:30:32.657" Score="9" Body="&lt;p&gt;I have some personal experience.&lt;/p&gt;&#xA;&lt;p&gt;I taught in a school that has Israeli students whose families had moved to the US. These children had Hebrew as their native language. Hebrew is written right to left.  I never saw these students struggle with math because they learned to write from right to left.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore all students in my school studied both Hebrew and English, learning to write both right to left and left to right.  No one seemed to struggle with direction in math class which was taught in English.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2020-07-10T05:12:59.490" LastActivityDate="2020-07-10T05:12:59.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18462" PostTypeId="2" ParentId="18458" CreationDate="2020-06-18T14:11:56.360" Score="2" Body="&lt;p&gt;This seems like a quite peculiar assortment of topics, and I do not imagine that any existent text addresses all of these.  I think you will need to either write your own materials (if you want a coherent treatment), or just cobble together textbook treatments from a diversity of sources.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-06-18T14:11:56.360" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18463" PostTypeId="1" AcceptedAnswerId="18464" CreationDate="2020-06-18T14:48:58.937" Score="31" ViewCount="7561" Body="&lt;p&gt;I am a college instructor who's just had an outbreak of academic dishonesty connected to students posting take-home exam problems on a platform called Chegg. Chegg collects a membership fee from students and uses it to pay contractors to answer their homework and exam problems, no questions asked.&lt;/p&gt;&#xA;&lt;p&gt;I could editorialize on the line of Chegg being a scurrilous company that lives off of liquidating whatever trust and integrity still exist in academia while exploiting contractors who seem to be mainly from the developing world, but more important is that I currently know of no other websites where students can as easily receive complete solutions to problems quickly.&lt;/p&gt;&#xA;&lt;p&gt;For instance, on &lt;em&gt;&lt;a href=&quot;https://math.stackexchange.com/tour&quot;&gt;Mathematics&lt;/a&gt;&lt;/em&gt; (Stack Exchange site), &amp;quot;problem statement questions&amp;quot; are generally closed immediately, while other platforms I'm aware of are mainly forums with a small population of volunteer participants capable of solving problems beyond the high school curriculum.&lt;/p&gt;&#xA;&lt;p&gt;Are there other sites comparably problematic to Chegg that other instructors keep an eye on for students getting complete solutions to problems?&lt;/p&gt;&#xA;" OwnerUserId="5233" LastEditorUserId="851" LastEditDate="2020-06-20T18:32:40.067" LastActivityDate="2021-11-25T03:27:13.110" Title="What websites allow students to purchase solutions to problems?" Tags="|cheating|" AnswerCount="4" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="18464" PostTypeId="2" ParentId="18463" CreationDate="2020-06-18T15:38:27.470" Score="26" Body="&lt;p&gt;Some of Chegg's competitors are Studyblue, Course Hero, Slader, and Cramster. However, Chegg is the market leader by a mile.&lt;/p&gt;&#xA;&lt;p&gt;It's a little difficult to sort out which of these businesses actually sell solutions, because they don't admit they do that. For example, the NY Times did a &lt;a href=&quot;https://www.nytimes.com/2020/06/12/business/chegg-dan-rosensweig-corner-office.html&quot; rel=&quot;noreferrer&quot;&gt;softball interview&lt;/a&gt; recently with Chegg's CEO, where they asked, &amp;quot;Many teachers believe that their students are using Chegg as a means by which to cheat. Is this a problem? And if so, what are you doing about it?&amp;quot; He answered with lies and evasions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It’s always been a problem for colleges. Let’s face it: Students have always found a way, whether it’s in fraternities, or whether they go to Google. But Chegg is not built for that. We have built technology that removes copyrighted material before it even gets posted. If we’re notified by a professor or a school that there’s copyrighted material, it immediately gets flagged and then removed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(None of this is true. Chegg &lt;em&gt;is&lt;/em&gt; built for that, and their entire business model is based on illegal use of copyrighted material, i.e., the questions themselves. The publishers tried to get Chegg to take down copyright-violating material, and ultimately gave up because it was like whac-a-mole.)&lt;/p&gt;&#xA;&lt;p&gt;I would actually love to see a careful, impartial analysis of which web sites do what and how their business models work, but I haven't been able to find one. It's pretty difficult to know exactly what they're doing unless you pay for an account yourself. But just anecdotally, in my department the issues are 100% Chegg. It's especially bad now that we have the campus quarantined, so that all tests have to be take-home. One of my colleagues found all of his finals on Chegg last semester. He ended up not grading the finals and assigning projects instead. The consensus in my department seems to be that starting next semester we'll start just doing a half-hour, synchronous quiz every week, instead of big exams. This will hopefully make it harder for students to post exam questions on Chegg and get answers in time.&lt;/p&gt;&#xA;&lt;p&gt;In comments, Peter Saveliev wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;They are just exposing some pre-existing problems in academia.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't think this is true at all. When I started teaching in 1996, problems with students cheating on homework were minor and intermittent. It was easy to nip it in the bud if you cared enough to read any student work. Today it's a qualitatively different environment. In classes where professors continue to count homework as a significant portion of students' grades, cheating is nearly universal. It's very corrosive, because students come to believe that they can't compete on an even playing field if they don't cheat.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-06-26T18:04:00.667" LastActivityDate="2020-06-26T18:04:00.667" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18466" PostTypeId="2" ParentId="18463" CreationDate="2020-06-19T01:38:12.513" Score="20" Body="&lt;p&gt;If you are teaching algebra or calculus courses, &lt;strong&gt;Mathway&lt;/strong&gt; and &lt;strong&gt;Symbolab&lt;/strong&gt; have algorithms that allow paying users to see full solutions with steps shown. The algorithms often do strange things that no human would do, which allows you to catch students who use them.&lt;/p&gt;&#xA;&lt;p&gt;For example I gave the following question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Use the comparison test to show that this series converges: &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty \frac{n+4}{n^3 + 4}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Symbolab* starts by comparing the fraction to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{(2n+1)^2}$&lt;/span&gt; (???).&lt;/p&gt;&#xA;&lt;p&gt;A few of my students made the same comparison. That gave me enough evidence to start looking into their other answers, which of course were also copied from various online sources. This was enough evidence to open academic integrity cases against them.&lt;/p&gt;&#xA;&lt;p&gt;*: I think it was Symbolab. I let my subscriptions run out for the summer, so I can't check it again right now.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2020-06-19T02:54:35.323" LastActivityDate="2020-06-19T02:54:35.323" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18468" PostTypeId="1" CreationDate="2020-06-19T07:59:19.023" Score="3" ViewCount="558" Body="&lt;p&gt;[I originally posted this is the Mathematics Stack Exchange and was told to post it here instead]&lt;/p&gt;&#xA;&lt;p&gt;This might seem like an odd topic for this forum, but I'm losing my mind. I am about eight months away from completing a Bachelor's degree in Mathematics. I am what you would call a &amp;quot;nontraditional&amp;quot; student - I am an adult who decided to change careers and I'm pursing my current degree online. My school, despite having a physical campus, has zero support in the realm of undergraduate research or mentorship. I've tried connecting with a number of my instructors, but as the program is exclusively online, it seems that they are not particularly interested in helping.&lt;/p&gt;&#xA;&lt;p&gt;I hope to attend graduate school where I would also study Mathematics - I would opt for on-campus learning for this. Since I have no support, I have no idea how to prepare for the application process nor do I have any primer for how to jump into the world of mathematical research. I have a very high GPA and have aced all of my math classes, but this seems trivial when reading about how difficult it is to get into a graduate math program. I recently applied for a short-term research program for undergraduates, for which I was soundly rejected (I can only assume what about my background seemed inadequate). I've tried perusing papers to get a sense of the research landscape, but all of it seems well beyond my own abilities. I'm in quite the low place and it seems that things are quite hopeless. Does anyone here have any insight that might point me in the right direction?&lt;/p&gt;&#xA;" OwnerUserId="14191" LastEditorDisplayName="user507" LastEditDate="2020-06-19T15:35:48.127" LastActivityDate="2020-06-19T23:37:56.420" Title="Grad school after doing an online bachelor's degree without support for undergraduate research" Tags="|online-instruction|undergraduate-research|advising|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18469" PostTypeId="1" CreationDate="2020-06-19T09:24:32.567" Score="3" ViewCount="191" Body="&lt;p&gt;Once, when I Was doing calculus 2, someone challenged me to calculate and prove the Gaussian integral, with a few hints, and a few days, I managed to.&lt;br /&gt;&#xA;It was a great feeling to solve a “deep” multi step long  math problem.&lt;br /&gt;&#xA;My question:&lt;br /&gt;&#xA;Do any of you know of resources for problems of this kind?&lt;br /&gt;&#xA;I’m looking for an online resource, in the level of calculus 2.&lt;br /&gt;&#xA;As a person planning on being a pure mathematician, I know that I will be presented with a problem and I will need to find an approach to solve it, most calculus textbooks provide a step by step explanation on the problem, detailing someone else’s reasoning.&lt;br /&gt;&#xA;When I see someone else’s reasoning it makes sense but I always wondered if I was presented with the problem what I would have done.&lt;/p&gt;&#xA;" OwnerUserId="14192" LastEditorUserId="14192" LastEditDate="2020-06-21T07:53:54.610" LastActivityDate="2020-06-21T07:53:54.610" Title="Help finding good deep mathematics problems" Tags="|problem-solving|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18470" PostTypeId="2" ParentId="18463" CreationDate="2020-06-19T10:14:24.120" Score="7" Body="&lt;p&gt;Not organised sites like Chegg but I try to keep an eye on several sources.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Freelance &amp;quot;teachers&amp;quot; who solve exercises, sometimes somehow disguised as classes - classes where your exercises are solved. They can be found by Googling the name of the subject and &amp;quot;exams&amp;quot;, the name of the institution or the exercise. It's easier in languages other than English because the first pages of results in English get flooded with banks of exercises and other legitimate resources. A few examples in Spanish: &lt;a href=&quot;https://www.milanuncios.com/anuncios/examen-estadistica.htm&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;exams by WhatApp&amp;quot;&lt;/a&gt;, &lt;a href=&quot;https://www.superprof.cl/resuelven-pacs-estadistica-matematicas-uoc.html&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;we solve assessment exercises&amp;quot;&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;WhatApp groups (or similar media): Although that isn't a commercial service solving exams, I've seen class WhatsApp groups where results circulate freely and exercises that are supposed to be solved individually (including exams) became a team assignment. Some independent tutors often take part on these groups in order to publicise their for pay classes - with different levels of spam.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;However, keeping an eye on those groups - and even sneaking into WhatsApp groups or forums where professors aren't expected to be - just served me to confirm the obvious fact that students cheat when they can cheat. How to solve that is probably out of the scope of the question.&lt;/p&gt;&#xA;" OwnerUserId="7178" LastActivityDate="2020-06-19T10:14:24.120" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18471" PostTypeId="2" ParentId="18468" CreationDate="2020-06-19T11:30:09.657" Score="2" Body="&lt;p&gt;Are you in the US?&lt;/p&gt;&#xA;&lt;p&gt;It is probably true that you will not get into grad school in math at Princeton or MIT with no undergrad research.  But many other US universities will gladly admit you with: good grades, good recommendations from undergraduate instructors, and good results on the Graduate Record subject exam in math.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2020-06-19T11:30:09.657" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18472" PostTypeId="2" ParentId="18468" CreationDate="2020-06-19T15:35:20.787" Score="1" Body="&lt;p&gt;How are your test scores?  General as well as subject.  The good thing about those is they identify smart people from less fancy schools.&lt;/p&gt;&#xA;&lt;p&gt;Oh...and just apply.  Mention your work background.  I got into a top science grad school with no undergrad research as I did college in the military.  But was a smart cookie.  And had a TAD that was sort of research related.  No publication but showing an interest.&lt;/p&gt;&#xA;&lt;p&gt;It's not as hard as you think. They need more people every year.  Grad school at MIT is easier to get into than undergrad. Just realize the job prospects can be worse.&lt;/p&gt;&#xA;" OwnerUserId="14190" LastActivityDate="2020-06-19T15:35:20.787" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18473" PostTypeId="2" ParentId="18468" CreationDate="2020-06-19T15:51:59.723" Score="5" Body="&lt;p&gt;You could look around for a program like &lt;a href=&quot;http://sfp.caltech.edu/programs/surf/eligibility_requirements&quot; rel=&quot;nofollow noreferrer&quot;&gt;CalTech's SURF&lt;/a&gt;, which takes visiting students from other undegraduate institutions. However, I suspect that it's too late to do those for this summer or (since you're graduating) next summer.&lt;/p&gt;&#xA;&lt;p&gt;So failing that, my suggestions would be the following:&lt;/p&gt;&#xA;&lt;p&gt;(1) Line up the best recommendations you can from the people you've been taking classes from, who sound like they're all adjuncts being paid poverty wages. These recommendations will basically say not much more than &amp;quot;A student, did all the work.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;(2) Figure out a self-directed project of your own that interests you and is within your ability. It doesn't have to be cutting-edge, publishable work at the forefront of the field. For example, when I was an undergrad, I got interested in the problem of what happens if you take a polyhedron and shave off its vertices to form new faces. What happens if you repeat the process? Does it always converge to a spherical shape? I wrote some code to compute examples and made some pretty pictures that I hung on the wall in my dorm room. Now this was not at all publishable work. However, for someone in your situation, a project of this type is something to point to that shows your ability to work creatively and independently.&lt;/p&gt;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-06-19T17:46:41.180" LastActivityDate="2020-06-19T17:46:41.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18474" PostTypeId="2" ParentId="18468" CreationDate="2020-06-19T18:09:45.653" Score="7" Body="&lt;p&gt;It seems like some of the other answers are aiming at PhD programs.  I would suggest (as your question on academia.sx suggests) that you may wish to look at a Master's program (at a non-PhD-granting institution).  Obviously not all have the same quality, but a lot of them have high-quality coursework, &lt;em&gt;with assistantships&lt;/em&gt; of various kinds.  I regularly see emails saying &amp;quot;we still have positions available&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;And many of their faculty are going to be quite involved in the research world, and be interested in bringing you into research, and be able to write quality letters of recommendation - which is likely to be what you are missing right now for admission, if your grades are great and your GRE is acceptable. (The issue of the GRE subject test not necessarily measuring readiness for research is a separate one, but for now is part of the ecosystem.)&lt;/p&gt;&#xA;&lt;p&gt;This might not work for you - for instance, you may have geographical considerations to keep in mind that make this harder - but at any rate is worth considering.  And I'm sorry your current institution is not supporting you, at least morally, in pursuing your adventure. That is a real shame.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2020-06-19T18:09:45.653" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18475" PostTypeId="2" ParentId="18469" CreationDate="2020-06-19T20:11:44.993" Score="2" Body="&lt;p&gt;razivio, look at Joseph Edwards Treatise on Integral Calculus froim 1920 (can find pdf on the web).  Full of challenge problems from Cambridge Tripos exams.  (Warning, stay in the earlier part of the text as that is calc 2.  But he also has calc 3 and really complex analysis in there also.)&lt;/p&gt;&#xA;" OwnerUserId="14158" LastActivityDate="2020-06-19T20:11:44.993" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18476" PostTypeId="2" ParentId="18468" CreationDate="2020-06-19T23:37:56.420" Score="1" Body="&lt;p&gt;The sad fact is that your mathematics preparation may very well be inadequate for graduate school.  My department admits with some frequency people from small regional public universities or unknown private liberal arts colleges whom their faculty recommend as the best student they've had in 10 years, and sometimes we find their preparation woefully inadequate (and sometimes they are quite good).  The truth is that many universities simply do not have any students who are capable of succeeding in a course that would prepare students for grad school, and hence don't teach such courses.  This is not a matter of the subject matter covered, but rather of the creativity and depth of understanding demanded of the students.&lt;/p&gt;&#xA;&lt;p&gt;Can you come up with a proof on your own for the following statement?  Let G be a finite group and H a subgroup of G, with |H| = (1/2) |G|.  Prove H must be normal.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2020-06-19T23:37:56.420" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18477" PostTypeId="1" AcceptedAnswerId="18522" CreationDate="2020-06-20T19:36:32.407" Score="33" ViewCount="2975" Body="&lt;p&gt;Many of us our coming off our first semester of required-online classes; and at some of our institutions we are preparing for what is most likely a required-online semester in the fall. (That is: The first time we know in advance of the all-online requirement, and with some experiential knowledge of how things progressed in the spring.)&lt;/p&gt;&#xA;&lt;p&gt;Assume that we're going to be giving math tests online in the future. In my department there are a wide variety of opinions about how best to deploy those tests, and we've passed around competing published opinion papers on the matter (mostly from a time when courses were not all-online by necessity; i.e., prior publications assume students registered for online courses as a special case). Most of the conversation revolves around how to tamp down on cheating during those tests. Granted that, what are the best parameters under which to give tests? Some points that would be addressed in the best answer:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Short vs. long availability period?&lt;/strong&gt; E.g.: Some instructors want the testing to occur within the  normal class hour, and require students to begin within a 15 minute window. Others look to give students maximum flexibility, say, a 6-hour or day-long window in which to take the one-hour test (so as to work around other courses, family/work obligations, etc.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Short vs. long timer?&lt;/strong&gt; A short timer on the test may reduce opportunities for cheating, but likewise make a student more desperate and likely to resort to cheating; perhaps a longer timer would reduce that stress.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Present questions all at once or one at a time?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Randomize question order?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Prohibit backtracking?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Require academic integrity statement?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Now, my own situation involves teaching math and computing in the context of a community college STEM degree program; e.g., at the level of college algebra through calculus III, linear algebra, and discrete mathematics. Traditionally these tests are largely computational -- a small number of proof-based questions are asked at the uppermost levels, and it is a minority of our students who can answer any of them (e.g.,  making all questions proof-based is not an option at this level). Hopefully this question can cover an array of similar-level courses at other institutions.&lt;/p&gt;&#xA;&lt;p&gt;The online system we use is Blackboard, but likewise I hope this question is appropriate for any  online platform (e.g., my bulleted points above are basically the Blackboard test deployment options, excepting the last one).&lt;/p&gt;&#xA;&lt;p&gt;The larger institution has made an executive decision to not license or employ any online proctoring system (e.g., video surveillance of students taking tests), so that is not an option in our case.&lt;/p&gt;&#xA;&lt;p&gt;Note some related but distinct questions on ME.SE that have been recently asked:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/18449/how-now-shall-we-teach-math-online&quot;&gt;&amp;quot;How, now, shall we teach math online?&amp;quot;&lt;/a&gt; This is a much broader question about all aspects of online teaching, which only touches on testing in passing; the present question is solely focused on how to deliver online tests.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/18463/what-websites-allow-students-to-purchase-solutions-to-problems&quot;&gt;&amp;quot;What websites allow students to purchase solutions to problems?&amp;quot;&lt;/a&gt; This is specific about tracking websites that support cheating on homework and tests; the present question is about how best the instructor can structure tests.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/18078/how-to-conduct-online-testing-for-calculus&quot;&gt;&amp;quot;How to conduct online testing for Calculus?&amp;quot;&lt;/a&gt; This question is specific to calculus; and more critically, has a selected answer which is to not give tests in the first place, which therefore addresses none of the points in the present question.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So: What are the best-practice parameters with which to deploy an online test in the current environment (all classes by necessity being fully online)?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note: Suggestions to refrain from giving tests are out of scope for the current question, and will not be selected as the answer. Please address any such possible practices in a separate question.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-06-21T02:35:45.890" LastActivityDate="2021-03-04T23:22:27.780" Title="What are the best practices for giving online tests?" Tags="|undergraduate-education|testing|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18478" PostTypeId="2" ParentId="18477" CreationDate="2020-06-20T20:02:40.993" Score="11" Body="&lt;p&gt;&lt;strong&gt;Generally about cheating in tests&lt;/strong&gt;&lt;br /&gt;&#xA;if you are trying to prevent cheating, trying to limit the student's ability to do it is very hard and mostly counterproductive, as it can introduce frustration , specificity in an online setting.&lt;br /&gt;&#xA;Instead, I recommend to minimize the student's motivation to cheat, after the first time it is just get's easier,  if you design tests to not be frustrating you'll get a much better result instead of trying to fight the student.&lt;br /&gt;&#xA;Hard &lt;span class=&quot;math-container&quot;&gt;$\not \Rightarrow$&lt;/span&gt; Frustrating, even the opposite, giving the student's three hard integrals is &lt;em&gt;mostly&lt;/em&gt; better then 20 easy ones.&lt;br /&gt;&#xA;&lt;strong&gt;Short vs. long availability period&lt;/strong&gt;&lt;br /&gt;&#xA;I’m on the Long availability period side, a student in the online setting might have difficulties to attend during specific times and thus it’d be better to let them have time.&lt;br /&gt;&#xA;&lt;strong&gt;Long timer vs. Short timer&lt;/strong&gt;&lt;br /&gt;&#xA;I’m on the long timer side, from my experience, students cheat either out of habit or desperation.&lt;br /&gt;&#xA;in the long timer scenario, the student is less likely to be desperate and resort to cheating.&lt;br /&gt;&#xA;&lt;strong&gt;Backtracking&lt;/strong&gt;&lt;br /&gt;&#xA;if you prohibit backtracking you are just annoying the student, there isn't something more frustrating then realizing you were wrong and can't fix it.&lt;br /&gt;&#xA;&lt;strong&gt;Randomize question order&lt;/strong&gt;&lt;br /&gt;&#xA;It really doesn't matter from the student's perspective and they'l probably not even notice it.&lt;br /&gt;&#xA;&lt;strong&gt;Require academic integrity statement&lt;/strong&gt;&lt;br /&gt;&#xA;from my experience, this  won't deter a student from cheating.&lt;br /&gt;&#xA;&lt;strong&gt;General tips to minimize cheating&lt;/strong&gt;&lt;br /&gt;&#xA;A majority of online resources cannot answer “why?” Questions.&#xA;So, for example, having a question such as &lt;span class=&quot;math-container&quot;&gt;$\int \frac{dx}{\sin(x)}$&lt;/span&gt; Could make a student just put the integral into WolframAlpha and get the solution.&lt;br /&gt;&#xA;On the other hand, if we add an intermediary question such as: “Why does substituting &lt;span class=&quot;math-container&quot;&gt;$u=\sin(x)$&lt;/span&gt; not work?”&#xA;WolframAlpha can’t help the student cheat anymore.&lt;br /&gt;&#xA;This question is especially good because most answers will use the &lt;span class=&quot;math-container&quot;&gt;$\cot(x) + \csc(x)$&lt;/span&gt; trick, which would light a red flag as the answer expected will probably be the &lt;span class=&quot;math-container&quot;&gt;$\cos(x)$&lt;/span&gt; substitution and partial fractions.&lt;/p&gt;&#xA;" OwnerUserId="14192" LastEditorUserId="14192" LastEditDate="2020-07-23T10:07:27.583" LastActivityDate="2020-07-23T10:07:27.583" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18479" PostTypeId="2" ParentId="18477" CreationDate="2020-06-21T18:49:37.200" Score="17" Body="&lt;p&gt;&lt;strong&gt;On The Timer&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I tried a long timer (about 3-5 days) for my exams and final exam last semester. The reason for this is that my favorite assignments from college -- the ones where I learned the most -- were challenging assignments that had a long availability period and allowed incremental work over the course of several days.&lt;/p&gt;&#xA;&lt;p&gt;However, an unbelievably large number of my students used Chegg, Mathway, Symbolab, and Wolfram|Alpha to cheat. I gave F's for academic integrity violations to almost 25% of my students. That's just the ones where I had evidence that I felt was strong enough to win an appeal. Many more clever students may have used the resources without being so obvious about it.&lt;/p&gt;&#xA;&lt;p&gt;Some students probably got a lot of value out of the long-timer assignment, the same way that I did when I was in school. If you go this route, be prepared to buy a subscription to Chegg at a minimum.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2020-06-21T18:49:37.200" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="18480" PostTypeId="2" ParentId="18477" CreationDate="2020-06-21T21:35:15.967" Score="-4" Body="&lt;p&gt;Do standard time exams in the normal course period.  Keep as much the same and stable vise loosely goosey.&lt;/p&gt;&#xA;&lt;p&gt;Email it as a pdf.  Let the kids send handwritten work back, scanned.&lt;/p&gt;&#xA;&lt;p&gt;Avoid the grad student impulse to give project style questions.  Do normal problems similar to the drill work. Take it as a victory if they learn that.&lt;/p&gt;&#xA;&lt;p&gt;Ask them to box their answers. If you want to make it easy on yourself, do short frequents questions and just grade the final result.  Students can see the key after the exam to see where they went wrong.  If still confused they can request a session to go over it.  Not grade arguing but to learn.&lt;/p&gt;&#xA;&lt;p&gt;Do a nonverbise integrity statement.  Doesn't hurt.&lt;/p&gt;&#xA;" OwnerUserId="14205" LastActivityDate="2020-06-21T21:35:15.967" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18481" PostTypeId="2" ParentId="18477" CreationDate="2020-06-22T08:56:37.850" Score="6" Body="&lt;p&gt;&lt;strong&gt;Randomize questions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Other answers are good, but I think a very important aspect on online quizzes has not been covered: questions should be different for every student - as different as resources and fairness allow.&lt;/p&gt;&#xA;&lt;p&gt;Questions can be randomized in several ways:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Random parameters.&lt;/li&gt;&#xA;&lt;li&gt;Slightly different versions of the same question (e.g. one version with permutations with repeat and another version without repeat).&lt;/li&gt;&#xA;&lt;li&gt;Mathematically equivalent questions with apparently different settings (e.g. one question about husbands and wives and another about locks and keys).&lt;/li&gt;&#xA;&lt;li&gt;Actually different questions. If possible, a large pool of independent questions taken at random.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In online tests (and even more in online homework) my ideal goal is to make the students need to know statistics (or to learn them, in homework) even to able to cheat. This way, I need to minimize the &amp;quot;cheatable&amp;quot; part, ideally making it indistinguishable from learning the subject. For example, I like questions about computing a probability in a normal distribution because when parameters are different the only thing that can be copied is the procedure, which is what I want them to learn.&lt;/p&gt;&#xA;&lt;p&gt;However, I must admit that preparing random quizzes is a lot of work.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit in response to comment:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The comment asks how to implement random quizzes for problems with answers that can't be expressed as a number with decimals.&lt;/p&gt;&#xA;&lt;p&gt;Some ways that worked for me (in Moodle):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.wiris.com/en&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wiris quizzes&lt;/a&gt;, that allow for answers including formulas and can check fore mathematical equivalency. For example it allows the answer to be &lt;span class=&quot;math-container&quot;&gt;$\frac{3 \cdot \pi}{2}$&lt;/span&gt; or even a function. Since Wiris is a commercial extension to Moodle, it is not installed in all institutions, but there may also be equivalent extension from other vendors.&lt;/li&gt;&#xA;&lt;li&gt;Multiple option questions: The answer can be as complex as you want, because in the end the student just needs to choose between a finite set of answers, and that can be easily graded by Moodle.&lt;/li&gt;&#xA;&lt;li&gt;Transforming the answer to a decimal number. If, instead of asking the derivative of a function, we ask for the value of the derivative at a given point, the answer is a real number and we just need to evaluate the answer against a given tolerance. However, I know that pure mathematicians usually dislike this approach.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To generate those kinds of questions I use three different tools:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Moodle &lt;a href=&quot;https://docs.moodle.org/39/en/Calculated_question_type&quot; rel=&quot;nofollow noreferrer&quot;&gt;calculated questions&lt;/a&gt;. Quite easy but very little mathematical capabilities.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.r-exams.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;R package exams&lt;/a&gt;. You can do in the question anything you can do in R. Great for statistics and numerical analysis, useful but not so great for calculus or algebra.&lt;/li&gt;&#xA;&lt;li&gt;Wiris, which has built-in random capabilities.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And of course, few automatic quizzes will outperform the teaching and assessing effect of a hand-graded essay or a long problem. However, automatic quizzes are great at easing logistics and introducing randomness.&lt;/p&gt;&#xA;&lt;p&gt;We have also used random data in hand-graded exercises, but those are slow to grade and randomness needs to be limited. In the end all the students have the same problem with some different parameters and teachers can grade using a table of results.&lt;/p&gt;&#xA;" OwnerUserId="7178" LastEditorUserId="7178" LastEditDate="2020-07-23T16:33:12.627" LastActivityDate="2020-07-23T16:33:12.627" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18482" PostTypeId="1" CreationDate="2020-06-22T21:52:59.707" Score="17" ViewCount="2957" Body="&lt;p&gt;I feel it may be a good idea to introduce some related open problems in a calculus course. Surely I am not expecting my students to solve any one of them, though I cannot say it is absolutely impossible; but I think it is good to let the students see some unsolved problems which may motivate them to love mathematics or even to start some undergraduate research. I am only interested in open problems that:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the statement of which can be understood by an average calculus student;&lt;/li&gt;&#xA;&lt;li&gt;It is related to some material in calculus.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;As an example, after teaching the scalar product of vectors, I may introduce the following (open as of 22/06/2020) problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Does there exist &lt;span class=&quot;math-container&quot;&gt;$668$&lt;/span&gt; vectors &lt;span class=&quot;math-container&quot;&gt;$v_1,\ldots,v_{668}$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^{668}$&lt;/span&gt; such that each coordinate of each vector is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v_i\cdot v_j=0$&lt;/span&gt; for every distinct &lt;span class=&quot;math-container&quot;&gt;$i,j$&lt;/span&gt;?&#xA;The number &lt;span class=&quot;math-container&quot;&gt;$668$&lt;/span&gt; can be replaced by some other numbers including &lt;span class=&quot;math-container&quot;&gt;$716, 892, 1004, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is related to &lt;a href=&quot;https://en.wikipedia.org/wiki/Hadamard_matrix&quot; rel=&quot;noreferrer&quot;&gt;Hadamard Matrix&lt;/a&gt; and I have rephrased the problem so that it is understandable to an average calculus III student.&lt;/p&gt;&#xA;&lt;p&gt;Any other examples of open problems that can be suitably introduced in a calculus course?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="851" LastEditDate="2020-06-23T12:00:45.517" LastActivityDate="2020-06-24T19:54:20.243" Title="What are some of the open problems that can be suitably introduced in a calculus course?" Tags="|undergraduate-education|calculus|student-motivation|undergraduate-research|vectors|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18483" PostTypeId="2" ParentId="18482" CreationDate="2020-06-23T01:04:41.390" Score="12" Body="&lt;p&gt;It takes a lot of browsing to find problems somehow related to calculus or analysis, but this is a great MathOverflow list: &lt;a href=&quot;https://mathoverflow.net/q/100265/6094&quot;&gt;Not especially famous, long-open problems which anyone can understand&lt;/a&gt;. Here are a few from&#xA;that list:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Are there an infinite number of primes &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; such that the repeating&#xA;part of the decimal expansion of &lt;span class=&quot;math-container&quot;&gt;$1/p$&lt;/span&gt; has length &lt;span class=&quot;math-container&quot;&gt;$p-1$&lt;/span&gt;?&#xA;&lt;a href=&quot;https://mathoverflow.net/a/100287/6094&quot;&gt;Link&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Does there&#xA;exist a point in the unit square whose distance to each of the four&#xA;corners is rational? &lt;a href=&quot;https://mathoverflow.net/a/208678/6094&quot;&gt;Link&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Is the sequence &lt;span class=&quot;math-container&quot;&gt;$(3/2)^n \bmod 1$&lt;/span&gt; dense in the unit interval? &lt;a href=&quot;https://mathoverflow.net/a/102048/6094&quot;&gt;Link&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-06-23T01:04:41.390" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18484" PostTypeId="1" CreationDate="2020-06-23T01:07:05.133" Score="14" ViewCount="1594" Body="&lt;p&gt;In schools, many students learn the usage of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\therefore$&lt;/span&gt;&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\because$&lt;/span&gt;&amp;quot; in proofs. Such three-dot notation are popular in many high-school books and exams, but are almost never used in university-level texts. (It seems that, at degree level, this notation only appears in some books about mathematical logic.)&lt;/p&gt;&#xA;&lt;p&gt;Very often, it is somewhat awkward to use  &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\therefore$&lt;/span&gt;&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\because$&lt;/span&gt;&amp;quot; for proofs, because modus ponens, the most commonly used principle of deduction, contains &lt;strong&gt;three&lt;/strong&gt; parts, while &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\therefore$&lt;/span&gt;&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\because$&lt;/span&gt;&amp;quot; are just &lt;strong&gt;two&lt;/strong&gt; symbols. Modus ponens states that from &lt;span class=&quot;math-container&quot;&gt;$A\Rightarrow B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; we could deduce &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, so the three parts are: &lt;span class=&quot;math-container&quot;&gt;$A\Rightarrow B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We will of course write &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; after &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\therefore$&lt;/span&gt;&amp;quot;, but it is a good question where to put &lt;span class=&quot;math-container&quot;&gt;$A\Rightarrow B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. We may either put both &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A\Rightarrow B$&lt;/span&gt; after &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\because$&lt;/span&gt;&amp;quot;, or put &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; after &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\because$&lt;/span&gt;&amp;quot; and &lt;span class=&quot;math-container&quot;&gt;$A\Rightarrow B$&lt;/span&gt; in brackets after &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\therefore B$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In the end, the three-dot notation does not make the logic structure entirely clear. &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\therefore $&lt;/span&gt;&amp;quot; clearly indicates the conclusion, but the meaning of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\because$&lt;/span&gt;&amp;quot; is not entirely clear - it could be either a theorem &lt;span class=&quot;math-container&quot;&gt;$A\Rightarrow B$&lt;/span&gt; or a condition &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. Sometimes, &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is too &lt;strong&gt;long&lt;/strong&gt; (takes too many words) to be written out fully, which causes confusion.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Is there any better alternative to the three-dot notation?&lt;/strong&gt; It is, after all, completely clear to just write everything in words.&lt;/p&gt;&#xA;" OwnerUserId="9339" LastActivityDate="2020-06-24T23:31:00.353" Title="The use of &quot;$\therefore$&quot; and &quot;$\because$&quot;" Tags="|undergraduate-education|secondary-education|proofs|notation|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18485" PostTypeId="2" ParentId="18482" CreationDate="2020-06-23T01:25:16.057" Score="10" Body="&lt;p&gt;This is a bit obvious I think, but when you introduce sequences and their notation in either an algebra or calculus class, you should certainly show students the &lt;a href=&quot;https://en.wikipedia.org/wiki/Collatz_conjecture#Statement_of_the_problem&quot; rel=&quot;noreferrer&quot;&gt;Collatz Conjecture&lt;/a&gt; as one of the examples.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2020-06-23T14:27:08.957" LastActivityDate="2020-06-23T14:27:08.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18486" PostTypeId="2" ParentId="18482" CreationDate="2020-06-23T01:37:37.377" Score="16" Body="&lt;p&gt;It's still not known whether &lt;span class=&quot;math-container&quot;&gt;$$\zeta(5) = \sum_{n=1}^\infty \frac{1}{n^5}$$&lt;/span&gt; is a rational number.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2020-06-23T01:37:37.377" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18487" PostTypeId="2" ParentId="18421" CreationDate="2020-06-23T09:59:56.010" Score="0" Body="&lt;p&gt;Another way to represent addition and subtraction of two fractions is by subdividing rectangles both horizontally and vertically.&lt;/p&gt;&#xA;&lt;p&gt;Say you want to show how to add &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{5}$&lt;/span&gt;. Draw three equal sized rectangles. Subdivide the first in threes with horizontal lines. Subdivide the second rectangle in fifths with vertical lines. The students probably understand these representations of the fractions already.&lt;/p&gt;&#xA;&lt;p&gt;The third rectangle is used to represent the sum. Subdivide that rectangle both horizontally and vertically to get fifteens. Now you must ensure that the students understand that this is actually fifteenths, and that they can see how many fifteenths are covered by &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{5}$&lt;/span&gt;. The first two rectangles should make this obvious to most students.&lt;/p&gt;&#xA;&lt;p&gt;This representation of the sum of fractions naturally shows how adding thirds and fifths leads to fifteenths. That is, it clearly shows how a common denominator solves the problem. It also shows why the common denominator is found by multiplication.&lt;/p&gt;&#xA;&lt;p&gt;The representation is not perfect, though. It does not naturally explain the &lt;em&gt;least&lt;/em&gt; common denominator, it does not extend to three or more summands without adding dimensions, and it is awkward with improper fractions.&lt;/p&gt;&#xA;" OwnerUserId="8357" LastActivityDate="2020-06-23T09:59:56.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18488" PostTypeId="2" ParentId="18482" CreationDate="2020-06-23T13:12:21.907" Score="11" Body="&lt;p&gt;You probably get Euler's constant &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; when you do the integral test comparing &lt;span class=&quot;math-container&quot;&gt;$\sum\frac1n$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\int\frac{dx}{x}$&lt;/span&gt;.  Then you can remark that it is unknown whether &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; is rational.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2020-06-23T13:12:21.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18489" PostTypeId="1" CreationDate="2020-06-23T13:30:51.510" Score="3" ViewCount="796" Body="&lt;p&gt;&lt;em&gt;The Art of Mathematics&lt;/em&gt; by Sung-Dae Hong is the standard high-school mathematics textbook in South Korea. The series gets new editions and reprints since 1966. Wikipedia has a &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Art_of_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;page&lt;/a&gt; for it.&lt;/p&gt;&#xA;&lt;p&gt;Had it ever been translated into English? I've checked the Library of Congress and Amazon, without success. Maybe there's a manuscript that has not been officially published?&lt;/p&gt;&#xA;" OwnerUserId="14210" LastActivityDate="2020-06-23T21:48:02.497" Title="English translation of Sung-Dae Hong's The Art of Mathematics" Tags="|secondary-education|textbooks|reference-books|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18490" PostTypeId="2" ParentId="18489" CreationDate="2020-06-23T21:48:02.497" Score="2" Body="&lt;p&gt;I found a series of textbooks on Amazon entitled &lt;em&gt;Oh Math!&lt;/em&gt; with subtitles which seem to match The Art of Mathematics by Hong. These were written by Chris Cho. I can't see too much with the preview, but it seems they are half Korean half English. It might be what you want. At a minimum, take a look:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1520530714&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;The English Korean OH! MATH Probability-Statistics textbook&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, searching &amp;quot;OH! MATH&amp;quot; or &amp;quot;OH MATH&amp;quot; does not help.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-06-23T21:48:02.497" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18491" PostTypeId="2" ParentId="18484" CreationDate="2020-06-24T11:08:18.403" Score="14" Body="&lt;p&gt;&lt;em&gt;Is there any better alternative to the three-dot notation?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The usual general advice is to use words instead of symbols.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that you are explaining the subject to a friend on a long walk in the woods, with no paper available; fall back on symbolism only when it is really necessary.&lt;/p&gt;&#xA;&lt;p&gt;(Paul Halmos, &lt;em&gt;How to Write Mathematics&lt;/em&gt;, p. &lt;a href=&quot;https://books.google.com.br/books?id=6UypbkweqiwC&amp;amp;pg=PA40&amp;amp;dq=%22The+best+notation+is+no+notation%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiFh93ZpJrqAhUnGbkGHeZqDV0Q6AEwAXoECAUQAg#v=onepage&amp;amp;q=%22The%20best%20notation%20is%20no%20notation%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;40.&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This applies particularly  to the three-dot notation.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Do not misuse the implication operator ⇒ or the symbol ∴. The former is employed only in symbolic sentences; the latter is not used in higher mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Bad: a is an integer ⇒ a is a rational number.&lt;br /&gt;&#xA;Good: If a is an integer, then a is a rational number.&lt;br /&gt;&#xA;Bad: ⇒ x = 3.&lt;br /&gt;&#xA;Bad: ∴ x = 3.&lt;br /&gt;&#xA;Good: hence x = 3.&lt;br /&gt;&#xA;Good: and therefore x = 3.&lt;/p&gt;&#xA;&lt;p&gt;Bad Theorem. n odd ⇒ 8|n² − 1.&lt;br /&gt;&#xA;Bad proof.&lt;br /&gt;&#xA;n odd ⇒ ∃j ∈ Z, n = 2j + 1;&lt;br /&gt;&#xA;∴ n² − 1 = 4j(j + 1);&lt;br /&gt;&#xA;∀j ∈ Z, 2 | j(j + 1) ⇒ 8 | n² − 1&lt;/p&gt;&#xA;&lt;p&gt;This is a clumsy attempt to achieve conciseness via an entirely symbolic exposition.Combining words and symbols and adding some short explanations will improve&#xA;readability and style.&lt;/p&gt;&#xA;&lt;p&gt;(Franco Vivaldi, &lt;em&gt;Mathematical Writing&lt;/em&gt;, p. &lt;a href=&quot;https://books.google.com.br/books?id=wpQvBQAAQBAJ&amp;amp;pg=PA4&amp;amp;lpg=PA4&amp;amp;dq=%22Do+not+misuse+the+implication+operator%22&amp;amp;source=bl&amp;amp;ots=V-975rPdP-&amp;amp;sig=ACfU3U28Qiy2NtmWQIjDF2_IryVyg9G5NQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjKw4G-p5rqAhUHIbkGHYm4BcgQ6AEwAHoECAQQAQ#v=onepage&amp;amp;q=%22Do%20not%20misuse%20the%20implication%20operator%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;4&lt;/a&gt; and &lt;a href=&quot;https://books.google.com.br/books?id=wpQvBQAAQBAJ&amp;amp;pg=PA132&amp;amp;lpg=PA132&amp;amp;dq=%22This+is+a+clumsy+attempt+to+achieve+conciseness%22&amp;amp;source=bl&amp;amp;ots=V-975rPdMU&amp;amp;sig=ACfU3U0nawqc1HLq5t9jslplPAcXaaJIhA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjK58Ksp5rqAhUBIrkGHd1oC5MQ6AEwAHoECAIQAQ#v=onepage&amp;amp;q=%22This%20is%20a%20clumsy%20attempt%20to%20achieve%20conciseness%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;132&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13585" LastActivityDate="2020-06-24T11:08:18.403" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18493" PostTypeId="2" ParentId="18484" CreationDate="2020-06-24T19:55:27.857" Score="2" Body="&lt;p&gt;The context isn't entirely clear so I'll assume this is about teaching. Then, I support Pedro's answer but also want to add that doing both verbal and symbolic versions may be a good idea. For example:&lt;/p&gt;&#xA;&lt;p&gt;Theorem. A polynomial has a higher order than another if and only if its degree is higher.&lt;/p&gt;&#xA;&lt;p&gt;In other words, for any two polynomials &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, we have:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$P=o(Q) \ \Longleftrightarrow\ \deg P&amp;lt;\deg Q\, .$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="10168" LastEditorUserId="10168" LastEditDate="2020-06-24T23:31:00.353" LastActivityDate="2020-06-24T23:31:00.353" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18494" PostTypeId="1" AcceptedAnswerId="18496" CreationDate="2020-06-24T20:23:33.700" Score="13" ViewCount="5369" Body="&lt;p&gt;I teach in a regional university.&#xA;In my department, students take their &amp;quot;proof course&amp;quot; (a course that sole focus on writing proofs) in the &lt;em&gt;third&lt;/em&gt; or even &lt;em&gt;fourth&lt;/em&gt; year.&#xA;All the courses before that have minimum proof component.&#xA;E.g., even linear algebra is taught without requiring students to produce non-trivial proofs.&lt;/p&gt;&#xA;&lt;p&gt;Is this normal?  What are the common practices?&#xA;Do students in other universities learn proofs in first or second year?&lt;/p&gt;&#xA;" OwnerDisplayName="user13395" LastEditorUserId="376" LastEditDate="2020-06-25T11:13:10.510" LastActivityDate="2020-06-30T21:41:35.890" Title="When do college students learn rigorous proofs?" Tags="|undergraduate-education|proofs|course-sequence|" AnswerCount="11" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="18495" PostTypeId="2" ParentId="18494" CreationDate="2020-06-24T21:02:35.933" Score="3" Body="&lt;p&gt;Naval Academy has their proofs class spring of sophomore year.  It is required for either applied math or pure math.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMA.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMA.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMP.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMP.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;They have it before linear algebra.&lt;/p&gt;&#xA;" OwnerUserId="14158" LastEditorUserId="11" LastEditDate="2020-06-30T21:41:35.890" LastActivityDate="2020-06-30T21:41:35.890" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18496" PostTypeId="2" ParentId="18494" CreationDate="2020-06-24T22:36:39.050" Score="13" Body="&lt;p&gt;In my experience (U.S.), that's on the boundary between 2nd and 3rd year -- either the end of sophomore year or the start of junior year.&lt;/p&gt;&#xA;&lt;p&gt;Two years ago I did a survey of Associate in Science (2-year) Mathematics degree programs. It's not common to have a dedicated only-proofs course, but I think many use a Discrete Mathematics course as a vehicle where proof-writing is taken seriously for the first time, and one of the core focus points of the course (I could be biased, but that's how it's used at my institution, following the Rosen text; previously Ross/Wright with similar themes). About half (6 of 12) of the programs I looked at have either a Discrete Mathematics or dedicated Introduction to Proofs course.&lt;/p&gt;&#xA;&lt;p&gt;That also matches my own undergraduate experience, where the Introduction to Proofs course was again taken either sophomore or junior year.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-06-27T16:51:58.690" LastActivityDate="2020-06-27T16:51:58.690" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18498" PostTypeId="2" ParentId="18494" CreationDate="2020-06-25T07:37:37.277" Score="23" Body="&lt;p&gt;From day one.&lt;/p&gt;&#xA;&lt;p&gt;In my experience in Germany, proofs are taken seriously from day one, or even before that. We had a voluntary prep course before the first semester that was half a repetition of calculus (which is a part of the high school curriculum here) and half an introduction to proofs.&lt;/p&gt;&#xA;&lt;p&gt;And the very first homework assignments in analysis and linear algebra already included some simple proofs as well.&lt;/p&gt;&#xA;" OwnerUserId="2082" LastActivityDate="2020-06-25T07:37:37.277" CommentCount="23" ContentLicense="CC BY-SA 4.0" />
  <row Id="18499" PostTypeId="2" ParentId="18494" CreationDate="2020-06-25T15:28:07.427" Score="6" Body="&lt;p&gt;In the UK, students usually learn proofs in the first year of a mathematics degree. My experience is similar to Sumyrda's answer.&lt;/p&gt;&#xA;&lt;p&gt;They also gain some exposure to proof techniques before university in A-Level Mathematics and Further Mathematics, which include proof by contradiction, trig proofs, elementary algebraic proof and proof by induction.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2020-06-25T15:28:07.427" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18500" PostTypeId="2" ParentId="18494" CreationDate="2020-06-25T17:59:05.437" Score="2" Body="&lt;p&gt;I can only speak to my personal experience, but during my time in Undergrad there was a dedicated proof writing class (&amp;quot;Introduction to Higher Mathematics&amp;quot;) that was coded as a 330 course. All courses coded above 330 required 330 as a prerequisite, while everything below it did not. Courses below 330 were Calculus 1, 2, 3; Linear Algebra, and (I believe) Differential Equations -- along with a ton of other options for non-STEM majors. During my time there, I think all of the math majors took 330 after taking Calc 2. Sometimes after Calc 3, or Linear Algebra, or Differential Equations (or in the same semester as these).&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2020-06-25T17:59:05.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18501" PostTypeId="2" ParentId="17989" CreationDate="2020-06-26T01:06:59.097" Score="2" Body="&lt;p&gt;Ok firstly I need to debunk your misconception regarding mathematics being easier to self learn than other subjects.&lt;/p&gt;&#xA;&lt;p&gt;Firstly, mathematics does not have an experimental component for the most part, therefore peer review for content online is a natural outcome.&lt;/p&gt;&#xA;&lt;p&gt;Secondly, you seem to be of the generation that takes the internet for granted in some respects. As a follow on lemma of my first point, this technology enabled  this outcome to occur in a historically rapid fashion.If you wish to have my point proven in the most impactful way possible, disconnect your internet connection, and continue your pursuit. If tertiary education is free in your country, drop out. It shouldn't be long before you are exhausted and drunk in a public bath, much like the best of the best millennia ago. We are so fortunate in modern times, it's hard for me to find words to describe it.&lt;/p&gt;&#xA;&lt;p&gt;Thirdly, the entire concept of self-taught is a fallacy. Abandon a child in an unpopulated area, and granted he or she survives a few months, when you find them they will be very much hunter gather in psychology.&lt;/p&gt;&#xA;&lt;p&gt;Thanks for your input in the community.&lt;/p&gt;&#xA;" OwnerUserId="14222" LastActivityDate="2020-06-26T01:06:59.097" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18502" PostTypeId="2" ParentId="18494" CreationDate="2020-06-26T07:04:36.087" Score="7" Body="&lt;p&gt;From my experience in French &lt;a href=&quot;https://en.wikipedia.org/wiki/Classe_pr%C3%A9paratoire_aux_grandes_%C3%A9coles&quot; rel=&quot;nofollow noreferrer&quot;&gt;Classes Préparatoires&lt;/a&gt;, we learn proofs during year one without a specific course about it, just while we learn calculus and linear algebra, starting on day one (we actually start a bit in high school).&lt;/p&gt;&#xA;&lt;p&gt;Maths in these classes are very rigorous, and everything that is taught gets proven (with few exceptions), even requiring to re-define all types of numbers from the &lt;a href=&quot;https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZFC set&lt;/a&gt; theory in first year.&lt;/p&gt;&#xA;&lt;p&gt;However, these classes are quite elitist and may not reflect what happens in all French universities.&lt;/p&gt;&#xA;" OwnerUserId="14224" LastEditorUserId="14224" LastEditDate="2020-06-27T13:25:10.397" LastActivityDate="2020-06-27T13:25:10.397" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18503" PostTypeId="2" ParentId="18494" CreationDate="2020-06-26T12:15:02.100" Score="3" Body="&lt;p&gt;I'm in Scotland and did rigorous proofs in the first year of my degree (Physics) as I sat the full first year maths syllabus. Although, in the final year of high school we were introduced to some simple proofs such as sqrt(2) irrational, and had discussed logic and different proof methods such as contradiction and proof by induction.&lt;/p&gt;&#xA;" OwnerUserId="14225" LastActivityDate="2020-06-26T12:15:02.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18504" PostTypeId="1" AcceptedAnswerId="18507" CreationDate="2020-06-26T15:17:29.550" Score="4" ViewCount="798" Body="&lt;p&gt;I am tutoring a student preparing to take Calculus 1 at a university.  This student hasn't taken precalculus for a year, so I have been drilling him on definitions, rules, and theorems from a college level algebra course and precalculus.  We were discussing types of functions.  The following problem was brought up on an online quiz:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Classify the following function:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(z)=5e^z+3$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, I've visited many sites and they all seem to conclude that the following is the definition of an exponential function:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x)=ab^x \qquad \text{or} \qquad f(x)=ab^{cx+d}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;with suitable restrictions on constants &lt;span class=&quot;math-container&quot;&gt;$a,b,c,d$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So why isn't this function &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt; above considered an exponential function?  Certainly the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; represents only a shift of the exponential function up by three units.  When these types of shifts are applied to polynomials, rational functions, trigonometric functions, they are still considered of that type.  So why the change for exponential function?  Why would a vertical shift be excluded in the definition?&lt;/p&gt;&#xA;" OwnerUserId="4728" LastEditorUserId="8571" LastEditDate="2020-06-26T19:53:42.670" LastActivityDate="2023-11-08T12:44:39.683" Title="Why is a translated exponential function considered an exponential function?" Tags="|functions|definitions|exponential|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="18505" PostTypeId="2" ParentId="18463" CreationDate="2020-06-26T16:11:25.923" Score="5" Body="&lt;p&gt;one that hasn't been mentioned is &lt;strong&gt;Slader&lt;/strong&gt;. I only learned about it through this article on cheating in a Princeton math course:&#xA;&lt;a href=&quot;https://www.dailyprincetonian.com/article/2020/05/princeton-teaching-assistant-math-department-slader-mat202-academic-integrity-cheating-covid&quot; rel=&quot;noreferrer&quot;&gt;https://www.dailyprincetonian.com/article/2020/05/princeton-teaching-assistant-math-department-slader-mat202-academic-integrity-cheating-covid&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10289" LastActivityDate="2020-06-26T16:11:25.923" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18506" PostTypeId="2" ParentId="18504" CreationDate="2020-06-26T16:22:02.250" Score="2" Body="&lt;p&gt;I say the key descriptor of a exponential function is &lt;em&gt;constant multiplicative rate of change&lt;/em&gt;, much as the descriptor of a linear function is &lt;em&gt;constant additive rate of change&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The function &lt;span class=&quot;math-container&quot;&gt;$f(x)=a(1.5)^x$&lt;/span&gt; increases by 50% when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; increases by 1:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{f(x+1)}{f(x)} = \frac{a(1.5)^{x+1}}{a(1.5)^x} = 1.5$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But adding a non-zero constant changes that:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{f(x+1)}{f(x)} = \frac{a(1.5)^{x+1}+c}{a(1.5)^x+c} \neq 1.5$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, if you define an exponential function by &amp;quot;constant multiplicative/percent rate of change&amp;quot;, then you can't shift it.&lt;/p&gt;&#xA;&lt;p&gt;[This is how I would explain it to an algebra student. I think the derivative argument would be great for a calculus student.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2020-06-26T17:04:49.657" LastActivityDate="2020-06-26T17:04:49.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18507" PostTypeId="2" ParentId="18504" CreationDate="2020-06-26T16:23:18.680" Score="9" Body="&lt;p&gt;To start with an opinion, I think that this classification exercise is kind of silly.  The student is being asked to put functions into some categories without having a clear idea about what those categories mean or are used for.  We introduce definitions and categorizations in order to help us understand abstract ideas.  A definition without the underlying motivation is quite hard to grasp.  As such, this is an exercise in rote regurgitation—I fail to see its value. :\&lt;/p&gt;&#xA;&lt;p&gt;That being said, I would argue that there is (up to scaling by a constant) only &lt;em&gt;one&lt;/em&gt; exponential function:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x \mapsto \mathrm{e}^{x}. $$&lt;/span&gt;&#xA;This function is the unique solution to the initial value problem&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{cases}&#xA;u' = u \\&#xA;u(0) = 1.&#xA;\end{cases}$$&lt;/span&gt;&#xA;The defining characteristic of the natural exponential function is that it is its own derivative.  More generally, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ b^x = \mathrm{e}^{\log(b) x}&#xA;\implies \frac{\mathrm{d}}{\mathrm{d}x} b^x = \log(b) \mathrm{e}^{\log(b) x} = \log(b) b^x. $$&lt;/span&gt;&#xA;Thus a function &lt;span class=&quot;math-container&quot;&gt;$x \mapsto b^x$&lt;/span&gt; has the property that it is proportional to its own derivative.  This is what I take to be the defining characteristic of an exponential function.  That is, the rate at which an exponential function changes is equal to (or, at least, &lt;em&gt;proportional to&lt;/em&gt;) the value of the function.&lt;/p&gt;&#xA;&lt;p&gt;Any function of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x \mapsto a \mathrm{e}^{bx} \tag{1}$$&lt;/span&gt;&#xA;has this property.  Functions of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x \mapsto a \mathrm{e}^{bx} + k \tag{2}$$&lt;/span&gt;&#xA;do &lt;em&gt;not&lt;/em&gt; have this property.  Because this property is important, it is reasonable to classify these two types of functions differently.  Functions of type (1) are exponential, and functions of type (2) are not.&lt;/p&gt;&#xA;&lt;p&gt;The problem here is that exponential functions are transcendental.  You can't really discuss transcendental functions without relying on concepts from analysis (limits, continuity, differentiability, etc).  As such, the most important feature of an exponential function (it is proportional to its own derivative) is inaccessible to a student who has not taken calculus.  Of course, this renders the question &amp;quot;is this an exponential function or not?&amp;quot; completely mysterious to a precalculus student.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2020-06-26T16:23:18.680" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18508" PostTypeId="2" ParentId="18504" CreationDate="2020-06-26T17:05:14.373" Score="1" Body="&lt;p&gt;&lt;em&gt;Now, I've visited many sites and they all seem to conclude that the following is the definition of an exponential function: &lt;span class=&quot;math-container&quot;&gt;$f(x)=ab^x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)=ab^{cx+d}$&lt;/span&gt; with suitable restrictions on constants &lt;span class=&quot;math-container&quot;&gt;$a,b,c,d$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;These definitions are not good (unless the restrictions are &lt;span class=&quot;math-container&quot;&gt;$a=1$&lt;/span&gt; in the first case and &lt;span class=&quot;math-container&quot;&gt;$ab^d=1$&lt;/span&gt; in the second). A reasonable definition of &amp;quot;exponential function&amp;quot; should imply that it satisfies the basic rule of exponents &lt;span class=&quot;math-container&quot;&gt;$a^na^m=a^{n+m}$&lt;/span&gt;. That is, for an exponential function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; the property&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x+y)=f(x)f(y)\tag{1}$$&lt;/span&gt;&#xA;should be valid because it is this property that characterize the concept of &amp;quot;exponential&amp;quot;. Translated exponential functions should not be considered exponential functions due to the same reason.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Remark 1:&lt;/em&gt; Under suitable hypotheses, it is possible to prove that the only functions that satisfy &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; have the form &lt;span class=&quot;math-container&quot;&gt;$f(x)=a^x$&lt;/span&gt; (with &lt;span class=&quot;math-container&quot;&gt;$a=f(1))$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Remark 2:&lt;/em&gt; Usually, a definition is a matter of taste. Therefore, it is not wrong to define anything you want as &amp;quot;exponential function&amp;quot;. Probably, it will only be unusual and not convenient.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Remark 3:&lt;/em&gt; In science and engineering, functions that &amp;quot;behave&amp;quot; as exponential functions as all types mentioned in this post are usually called functions of &amp;quot;exponential order&amp;quot; (however, the concept of &amp;quot;exponential order&amp;quot; includes many other types of functions).&lt;/p&gt;&#xA;" OwnerUserId="13585" LastEditorUserId="13585" LastEditDate="2020-06-27T04:23:48.677" LastActivityDate="2020-06-27T04:23:48.677" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18509" PostTypeId="2" ParentId="18504" CreationDate="2020-06-26T19:00:18.580" Score="4" Body="&lt;p&gt;The working definition I have in my head doesn't fit the more rigorous definitions others have put in their answers. I think of exponential growth and decay as being constant percentage growth or decay from or toward an asymptote. My favorite example is temperature of an object, which is shifted with the ambient temperature being the asymptote. I use y = a*b^x + c.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-06-26T19:00:18.580" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18511" PostTypeId="2" ParentId="18494" CreationDate="2020-06-27T11:01:49.883" Score="-1" Body="&lt;p&gt;Yes.from the first day we learned proofs.&#xA;But i dunno which approach i correct.&#xA;We had a course called fundementals of mathematics which introduced us to mathematical proof and college level math. Also in every course including calculus we used to study proofs for every theorem.&#xA;But it was frustrating and hard for us as freshmen and led to disappointment for some students. I guess it's better to have a lighter program which teaches how to THINK right instead of introducing students to a lot of proofs.then u can teach them what you think suits best for them.&lt;/p&gt;&#xA;" OwnerUserId="14232" LastActivityDate="2020-06-27T11:01:49.883" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18512" PostTypeId="2" ParentId="18494" CreationDate="2020-06-27T23:04:30.327" Score="0" Body="&lt;p&gt;I think, in the US, it is the norm for classes focusing on proofs to be nominally at the 3rd-year level. This is because the third year of college is generally the first year that students have committed to their academic specialty.&lt;/p&gt;&#xA;&lt;p&gt;More advanced students can still take those classes in the first or second year. This may end up being the norm for students expecting to go on to an advanced degree.&lt;/p&gt;&#xA;" OwnerUserId="13126" LastActivityDate="2020-06-27T23:04:30.327" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18513" PostTypeId="2" ParentId="18494" CreationDate="2020-06-29T01:15:29.390" Score="6" Body="&lt;p&gt;This answer is meant to supplement &lt;a href=&quot;https://matheducators.stackexchange.com/a/18496/8571&quot;&gt;Daniel R. Collins' answer&lt;/a&gt;, which is excellent—my goal is to draw a little more distinction between the &amp;quot;levels&amp;quot; of education and training in the US.&lt;/p&gt;&#xA;&lt;h3&gt;Short Answer&lt;/h3&gt;&#xA;&lt;p&gt;In the United States, &amp;quot;proofs based&amp;quot; courses (and formal proofs in general) are typically regarded as topics in &amp;quot;higher mathematics&amp;quot;, and are taught to mathematics majors (rather than a more general audience).  Undergraduates in the US typically don't specialize into a major until their third year of college.  Thus most US undergraduates never take course in mathematical proof, and mathematics majors typically don't see such topics until their third year.&lt;/p&gt;&#xA;&lt;h3&gt;Academia in the US&lt;/h3&gt;&#xA;&lt;p&gt;Eduction in the United States is roughly broken into three levels:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Primary and secondary eduction&lt;/strong&gt;, which consists of grade school (Kindergarten to 5th or 6th grade; ages 5 to 11–12), middle school (or junior high; grades 5–6 through 9–10; ages 11–12 to 14–15), and high school (grades 9–10 through 12; ages 14 to 18 or so).  The following descriptions are based on (1) my experience as a public school student in three states as a child, (2) the experiences of my siblings in two other states, and (3) my training as a classroom instructor in Nevada (in particular, I have probably spent more time reading and learning the Common Core than most folk, though I would imagine that there are folk &lt;em&gt;here&lt;/em&gt; more knowledgable than I—I welcome their comments).  Thus it is anecdotal, but (I think) broad enough to make some general statements.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In grade school, students are taught by in a single classroom by a single teacher throughout most of the day, with (perhaps) excursions to other classrooms for specialized instruction in art, must, PE, technology, and so on.  Instruction at this level is &lt;em&gt;very&lt;/em&gt; general—with respect to mathematics, it is mostly basic arithmetic, plus related &amp;quot;life skills&amp;quot; like how to count currency, tell time, and use a ruler.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In middle school, students start to take classes from more than one instructor—they will often have a &amp;quot;homeroom&amp;quot; class, where organization and directed studying take place, but will move as a cohort from one classroom to another, where they receive specialized instruction in mathematics, social studies (history, civics, etc), science, language, and so on.  At this stage of instruction, students might be exposed to some very minimal mathematical reasoning, but it is typically informal.  The emphasis is on rote memorization and algorithms.&lt;/p&gt;&#xA;&lt;p&gt;I'll note for clarity that not every middle school has the same structure; indeed, the structure of middle schools is quite variable.  The common theme is that these institutions are transitional places where students go from being children in grade school to being young adults (teenagers...) in more rigorous high schools.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In high school, students start to have some choice in the classes which they take, and when they take them.  Typically, math, language, science, and social studies classes are compulsory every term, but students take 6–8 classes every term, and there is quite a bit of flexibility in how those extra classes are filled out.  There are also generally multiple &amp;quot;tracks&amp;quot; which can be taken.  In short, there is some specialization which takes place in high school.&lt;/p&gt;&#xA;&lt;p&gt;It used to be that students were first introduced to proofs (in the style of Euclid, via the study of geometry) in the 10th or 11th grade (15–17 years old).  My experience of these classes is that they are very scripted and algorithmic—students learn how to write &amp;quot;&lt;a href=&quot;https://en.wikibooks.org/wiki/Geometry/Proof#Two-Column_Proof&quot; rel=&quot;noreferrer&quot;&gt;two column proofs&lt;/a&gt;&amp;quot;.  My impression is that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Integrated_mathematics&quot; rel=&quot;noreferrer&quot;&gt;integrated math&lt;/a&gt; approach has gained significant traction in the US.  This approach downplays mathematical proof, though doesn't completely eliminate it.&lt;/p&gt;&#xA;&lt;p&gt;In any event, American students very rarely see very much in the way of mathematical reasoning before they graduate from high school.  American high schools are &lt;em&gt;very&lt;/em&gt; general—there is very little specialization in high school. Beyond a handful of students who take Advanced Placement or &amp;quot;dual-enrollment&amp;quot; classes, most students take essentially same curriculum, and there is very little &amp;quot;tracking&amp;quot; (e.g. there are not really &amp;quot;college tracks&amp;quot; and &amp;quot;vocational tracks&amp;quot; in US schools, which &lt;em&gt;I think&lt;/em&gt; is different from much of how the rest of the world operates—my understanding is that, for example, European students often start to specialize into academic topics as early as age 16).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Tertiary education&lt;/strong&gt; (or post-secondary education, or post-high school education), which consists of college-level studies through the completion of a bachelor's degree.  I think that a lot of the confusion about the American education system is related to the transition from high school to college.  As noted above, high school is very general, so the first two years of college are generally devoted to more general studies which are part of a classical liberal education.  There are typically two tracks:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Many students spend their first two years of post-secondary education at &amp;quot;community&amp;quot; or &amp;quot;junior&amp;quot; colleges.  These institutions teach general education classes, which are often taught by folk with masters degrees who are not required to conduct research.  Community colleges generally offer only 2 year &lt;a href=&quot;https://en.wikipedia.org/wiki/Associate_degree&quot; rel=&quot;noreferrer&quot;&gt;associate's degrees&lt;/a&gt;, which are not terribly specialized, and are seen as preparation for work, i.e. the majority of community college attendees obtain a degree and enter the workforce (and leave academia).  However, a significant number to transfer to bachelor's granting institutions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The other track is to matriculate at a bachelor's granting institution right out of high school.  The first two years of study at such an institution are typically very similar to what a community college offers, but the classes tend to be larger, and taught either by PhD'd faculty, or graduate teaching assistants.&lt;br /&gt;&lt;br /&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Whether students start at a bachelor's granting institution or not, the first two years of post-secondary education tend to be fairly general.  These &amp;quot;lower division&amp;quot; classes are meant to give students a broad foundation of knowledge, and to help them determine a course of specialization.&lt;/p&gt;&#xA;&lt;p&gt;After a student's second year of post-secondary education, they are expected to declare a major.  That is, they are expected to pick a field in which they will specialize.  In most majors, there is a major change in the type and style of courses offered at the &amp;quot;lower division&amp;quot; (which are taught to non- and potential-major) and at the &amp;quot;upper division&amp;quot; (which are taught primarily to majors).  For example, lower division courses in anthropology tend to focus very broadly on the four fields (cultural, linguistics, physical, and archaeology), whereas upper division classes will be more specialized (people's and cultures of southeast Asia; archaeology of the Puebloan southwest, etc).&lt;/p&gt;&#xA;&lt;p&gt;In mathematics, the transition is marked by a change from &amp;quot;cookbook&amp;quot; classes (e.g. introductory calculus) to &amp;quot;proofs based&amp;quot; classes (e.g. introduction to analysis).  It is common in American institutions to offer some kind of course which is meant to offer a transition to higher mathematics as part of this advancement from lower- to upper-division classes, though the nature of this transition is far from uniform.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;At my bachelor's institution, the first proofs based course that many students took was undergraduate real analysis (limits, continuity, elementary metric topology, differentiation, and integration from a more formal point of view).  This class was mostly attended by students in their third year (calculus and differential equations were prerequisites).  This class had a very high fail rate (as it amounted to tossing novice students into the mathematical deep end), so the institution eventually started offering a specific transitional class which focused on proof techniques and logic as applied to set theory (the class essentially followed the skeleton of Halmos' book &lt;em&gt;Naive Set Theory&lt;/em&gt;, though I think that some other text was used).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At my PhD institution, there was also a transitional class, called &amp;quot;discrete mathmatics&amp;quot;, which is offered (typically) at the end of a student's second year of post secondary education.  The class is meant to teach the basics of proof via combinatorics, modular arithmetic, and naive set theory, and is a &amp;quot;recommended&amp;quot; prerequisite for all upper division mathematics classes.  My impression is that this is not an uncommon approach.&lt;br /&gt;&lt;br /&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In any event, mathematics majors (i.e. those students who have declared that they will pursue a bachelor's level specialization in mathematics) and minors (i.e. those students who are specializing in something other than mathematics, but are taking on a secondary specialization in math) are typically first exposed to &amp;quot;formal&amp;quot; proofs in the transition from lower- to upper-division coursework, which roughly corresponds to the end of their second year of college, or the beginning of their third year.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Post-graduate education&lt;/strong&gt;, which consists of masters and doctoral level training.  My feeling is that an American masters or doctoral degree (particularly in mathematics) is quite similar to a masters or doctorate awarded by a European (or other) institution.  As I don't think there is much difference, and as post-graduate students are expected to know how to read and write a proof from their first day, I'll stop here.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2020-06-29T01:15:29.390" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18514" PostTypeId="1" CreationDate="2020-06-29T18:18:59.870" Score="4" ViewCount="142" Body="&lt;p&gt;If you could build an ideal online homework platform for mathematics, what combination of features should it have in order to be effective at levels from high school through undergraduate, and why? Please post what features an ideal, effective online homework platform would have for use in math class at a high school to undergraduate level. Ideally, you would support why those features are important, with some theoretical (education theory or mathematical) or practical reasoning.&lt;/p&gt;&#xA;&lt;p&gt;This question is not asking about &lt;a href=&quot;https://matheducators.stackexchange.com/questions/552/online-homework-vs-paper-homework&quot;&gt;whether online homework or paper homework should be used&lt;/a&gt;. Nor is it asking about existing homework platforms. I think that this makes the question unique on this site.&lt;/p&gt;&#xA;&lt;p&gt;I am not posting this for my own market research purposes - I think it is fairly clear from my history here that I teach classes and do not have any incentive or energy to create such a platform. Instead, noting the glut of &lt;em&gt;bad&lt;/em&gt; online platforms, I hope to combine our collective knowledge here in a publicly visible arena, in hopes that people who do attempt to create these platforms have an easily accessible list of must-haves, with at the very least keywords that point towards research support.&lt;/p&gt;&#xA;&lt;p&gt;Best answer will be awarded to the most complete list with meaningful reasons for each item. I do plan to answer this question myself when time allows.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Some example features that are in my personal list of ideals, with my reasoning being &amp;quot;practical purposes&amp;quot; and/or research-supported tactics italicized:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Must be able to select problem sets from an easy-to-navigate problem bank, for practical purposes (teachers should not spend too much time searching for problems to assign).&lt;/li&gt;&#xA;&lt;li&gt;Must be able to randomize the numbers in each problem, for practical purposes (students can work together on the same problem but still have to do their own work).&lt;/li&gt;&#xA;&lt;li&gt;Should identify commonly-confused concepts and have pre-made problem sets contrasting the types of problems for &lt;em&gt;interleaving&lt;/em&gt; purposes.&lt;/li&gt;&#xA;&lt;li&gt;Should make it easy for the teacher to promote &lt;em&gt;spaced repetition&lt;/em&gt; (e.g. when creating a problem set, it could auto-suggest problems similar to two problems from the previous week and one problem from two weeks before).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="724" LastActivityDate="2020-06-29T18:18:59.870" Title="Ideal Features of Online Homework Platform?" Tags="|homework|online-instruction|big-list|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18516" PostTypeId="2" ParentId="18494" CreationDate="2020-06-30T16:41:24.027" Score="3" Body="&lt;p&gt;At my school, UIUC (&lt;a href=&quot;https://illinois.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://illinois.edu/&lt;/a&gt;), we have a dedicated proofs course for CS majors (CS 173: Discrete Mathematics) and a dedicated proofs course for math majors (Math 347: Fundamental Mathematics). CS majors will typically take CS 173 in the first or second semester of their first year, and Math majors will take Math 347 before the end of their second year, as it required or highly recommended for almost all math classes after Calc 3.&lt;/p&gt;&#xA;" OwnerUserId="14206" LastActivityDate="2020-06-30T16:41:24.027" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18517" PostTypeId="2" ParentId="18318" CreationDate="2020-06-30T20:31:19.003" Score="0" Body="&lt;p&gt;In UK 40 years ago the calculus was really rigorous first studying limits and series, we did virtually proved all results, except perhaps some integrals but we also had to memorize all formulas as we did not have formula sheet in the exams like they have today in UK A level. In Spain it is still like UK traditional syllabus from back then. I was shocked to see the huge changes in UK A level nowadays compared to back then. There has been even more radical change in UK physics eliminating almost all math. BUT after teaching the modern “international” UK syllabus (Pearson and Cambridge) I am a partly converted. The formula sheets and getting to the point skipping some tricky proofs is more fun. And usually there is a simple way to intuitively “get” the answer Eg. Looking at the x cubed graph can anyone imagine which power might work as the gradient function? Lots of kids can do that!&lt;/p&gt;&#xA;" OwnerUserId="14245" LastEditorUserId="14245" LastEditDate="2020-07-02T23:38:37.437" LastActivityDate="2020-07-02T23:38:37.437" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18518" PostTypeId="1" CreationDate="2020-07-01T19:34:25.163" Score="7" ViewCount="1809" Body="&lt;p&gt;As an undergrad student of applied mathematics, I have something to say that make's me ashamed of myself. I suck at proving things in mathematics and i know that if I don't get better in doing this right now, the cost will be greater in the future.&lt;/p&gt;&#xA;&lt;p&gt;I really want to get better at proving things, so I ask you folks can suggest how I can get better at proving: any links, resources?&lt;/p&gt;&#xA;" OwnerUserId="14247" LastEditorUserId="238" LastEditDate="2020-07-02T21:38:17.360" LastActivityDate="2020-07-20T08:35:16.737" Title="How to get better at proofs" Tags="|undergraduate-education|proofs|undergraduate-research|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18519" PostTypeId="2" ParentId="18518" CreationDate="2020-07-01T21:27:41.347" Score="6" Body="&lt;p&gt;Daniel Solow, &lt;em&gt;How to Read and Do Proofs&lt;/em&gt;.  Wiley, 6th Edition, 2013.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2020-07-01T21:27:41.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18520" PostTypeId="2" ParentId="18518" CreationDate="2020-07-01T22:36:12.477" Score="9" Body="&lt;p&gt;A great start to doing proofs is working  through Daniel Velleman's &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0521675995&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How to Prove it: A Structured Approach&lt;/em&gt;, 2nd Edition.&lt;/a&gt;.  I've used it many times in teaching, usually as a supplementary text.&lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2020-07-01T22:36:12.477" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18521" PostTypeId="2" ParentId="18518" CreationDate="2020-07-02T07:55:49.600" Score="5" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;As a general introduction, the book &lt;em&gt;Theorems, Corollaries, Lemmas, and Methods of Proof&lt;/em&gt; by Richard J. Rossi can be useful. For example, how to prove that a sequence converges? A detailed explanation is given on &lt;a href=&quot;https://books.google.com.br/books?id=kSwVGbBtel8C&amp;amp;pg=PA168&amp;amp;lpg=PA168&amp;amp;dq=%22Algorithm+for+Proving+a+Sequence+Converges%22&amp;amp;source=bl&amp;amp;ots=z52a8N-axM&amp;amp;sig=ACfU3U1UCI4mLwRTjGEtIW9Xa2Mk-udvIA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjl8eSFsPzpAhX5JrkGHYP8CfAQ6AEwAHoECAEQAQ#v=onepage&amp;amp;q=%22Algorithm%20for%20Proving%20a%20Sequence%20Converges%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;pages 168-170&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As a general rule, to get better we have to do a lot of proofs. A lot of sentences to be proved can be found in problem books on the subject you want to get better. For example, &lt;a href=&quot;https://books.google.com.br/books?id=SY-_COzW4toC&amp;amp;printsec=frontcover&amp;amp;dq=problem+book+in+linear+algebra+halmos&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjU4K_via7qAhUQIbkGHeBuB1wQ6AEwAHoECAEQAg#v=onepage&amp;amp;q=problem%20book%20in%20linear%20algebra%20halmos&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; for linear algebra and &lt;a href=&quot;https://books.google.com.br/books?id=IQhGAAAAQBAJ&amp;amp;printsec=frontcover&amp;amp;dq=a+problem+book+in+real+analysis&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwi43JrEiq7qAhXmGbkGHbOxCWwQ6AEwAHoECAUQAg#v=onepage&amp;amp;q=a%20problem%20book%20in%20real%20analysis&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; for real analysis.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13585" LastActivityDate="2020-07-02T07:55:49.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18522" PostTypeId="2" ParentId="18477" CreationDate="2020-07-02T11:30:30.580" Score="13" Body="&lt;p&gt;Here is my experience:&lt;/p&gt;&#xA;&lt;p&gt;Throughout the term I gave several quizzes with&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;long timer&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;short availability&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;questions all at once&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;academic integrity statement&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;few modifications&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The average score was around &lt;strong&gt;80 %&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For the final exam, I gave a quiz with majority (around 2/3) of the questions very similar to those given throughout the term (or even the same questions, but reworded). However, the conditions were different:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;short timer&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;questions one at a time&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;randomized question order&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;prohibit backtracking&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;random questions&lt;/strong&gt; (a random question from each category with 1 to 4 similar questions)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;no academic integrity statement&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Besides this, I was tracking the progress of students to make sure that there are no students waiting for someone else to answer their question. This led to quizzes being mostly individual.&lt;/p&gt;&#xA;&lt;p&gt;This time, the average score was &lt;strong&gt;45 %&lt;/strong&gt;. (And of course, there were many examples of students failing questions which they had correctly answered before.)&lt;/p&gt;&#xA;&lt;p&gt;I should admit that cheating is a real plague here, starting from primary school. I am trying to fight it as much as I can. For instance, besides the quiz, there were problems to solve in the final exam, which I made as variable as I could (the exam was in Statistics, so I used random samples). The students do know how much I despise cheating and that they would get F if they are caught, so most of them will prefer to return a blank paper if there is a chance to get caught cheating. And still there were 4 F's in the final exam because of cheating (out of around 40 students).&lt;/p&gt;&#xA;&lt;p&gt;So, speaking of quizzes, I am not sure that the second scheme is better, but I'd rather be using it. However, I understand that this may not be the best idea in other environments where cheating is not so ubiquitous as here.&lt;/p&gt;&#xA;" OwnerUserId="7528" LastEditorUserId="77" LastEditDate="2021-02-25T02:14:25.233" LastActivityDate="2021-02-25T02:14:25.233" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18523" PostTypeId="2" ParentId="14527" CreationDate="2020-07-02T12:06:22.087" Score="2" Body="&lt;p&gt;I use zoom combined with a wacom tablet to draw/write on the online whiteboard. To work really well I think you'd both need a tablet so you can both use the white board (as you would on a piece of paper or blackboard if you were in the same room). You an save the white board so he can look back on it afterwards too.&lt;/p&gt;&#xA;" OwnerUserId="14252" LastActivityDate="2020-07-02T12:06:22.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18524" PostTypeId="1" CreationDate="2020-07-02T20:11:11.733" Score="7" ViewCount="670" Body="&lt;p&gt;Inspired by the question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18494/when-do-college-students-learn-rigorous-proofs&quot;&gt;When do college students learn rigorous proofs?&lt;/a&gt;, I became curious when pupils in secondary schools learn about proofs, what kinds of proofs they are, how rigorously they are taught, do they learn any formal logic before or simultaneously with learning proofs?&lt;/p&gt;&#xA;&lt;p&gt;I am not focussed on American schools, I would like to learn how proofs are tackled around the world.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2020-07-04T23:02:05.493" LastActivityDate="2020-11-13T17:37:10.017" Title="Logic and proofs in secondary school" Tags="|secondary-education|mathematical-pedagogy|proofs|logic|middle-school|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18525" PostTypeId="2" ParentId="18518" CreationDate="2020-07-02T22:56:14.597" Score="3" Body="&lt;p&gt;I learned more from Lakatos than any other source&#xA;about what constitutes a proof, and:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the roller-coaster ride adjusting definitions to clarify the proof claim,&lt;/li&gt;&#xA;&lt;li&gt;perhaps realizing that a hidden lemma has a counterexample,&lt;/li&gt;&#xA;&lt;li&gt;the need to reformulate the proof statement in light of these ups-and-downs,&lt;/li&gt;&#xA;&lt;li&gt;and on and on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/fr1gg.png&quot; /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Proofs_and_Refutations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proofs and Refutations&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;And I &lt;em&gt;still&lt;/em&gt; have much to learn!&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-07-02T23:01:44.953" LastActivityDate="2020-07-02T23:01:44.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18526" PostTypeId="2" ParentId="18524" CreationDate="2020-07-04T23:01:35.077" Score="6" Body="&lt;p&gt;&lt;strong&gt;I can start with how it was/is done in Russia.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Logic was taught in Russian gymnasiums as a separate subject in late 19th century. When Bolsheviks came to power they pulled logic out of the curriculum. Logic was reinstated in Soviet schools in late 1940s only to be abolished again by the end of 1950s when Khruschev took over. AFAIK, there is no separate logic course in modern Russian schools, at least it is not mandated by the federal ministry of education, some specialized schools may teach it as an elective. There is a separate course of logic in universities, I don't know whether it is mandatory for everyone, or depends on a major.&lt;/p&gt;&#xA;&lt;p&gt;Proofs are common in Russian algebra and geometry courses (both subjects are mandatory, start from 7th grade and continue until the senior 11th grade). Most commonly they are &lt;em&gt;direct proofs&lt;/em&gt; based on the material learned earlier, so their application is obvious and does not require much of an explanation how the proof works. Traditionally, proofs in Russian schools are of &amp;quot;free text&amp;quot; variety, not a two-column system used in many U.S. schools.&lt;/p&gt;&#xA;&lt;p&gt;Terms &lt;em&gt;direct&lt;/em&gt;, &lt;em&gt;inverse&lt;/em&gt;, &lt;em&gt;converse&lt;/em&gt; and &lt;em&gt;contrapositive&lt;/em&gt; are introduced in 7th grade geometry in relation to theorems, not to abstract logical propositions. There is a simple explanation of why contrapositive theorem is equivalent to the original one, similar to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Contraposition#Simple_proof_by_contradiction&quot; rel=&quot;nofollow noreferrer&quot;&gt;one shown in Wikipedia&lt;/a&gt;. This fact is harnessed when using &lt;em&gt;proof by contradiction&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Older textbooks may have a separate, albeit tiny, section dedicated to proofs, like &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0977985202&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kiselev's Planimetry&lt;/a&gt;, which was used in Tsarist Russia, then was updated in 1938 and was used until 1956, when it was replaced with &amp;quot;New Math&amp;quot; textbooks. These new textbooks proved disastrous, so many teachers continued to covertly use Kiselev's textbook. The textbook was re-issued in 2004 and again in 2013. Here are relevant pages from the English translation, which includes problems; in Russian edition problems comprise a separate problem book.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/X41wP.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/X41wP.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/qMoNU.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qMoNU.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Other methods that feel more or less self-evident, are sometimes used: &lt;em&gt;proof by construction&lt;/em&gt;, &lt;em&gt;proof by exhaustion&lt;/em&gt;, &lt;em&gt;Dirichlet's box principle&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;9th grade algebra course introduces proof by &lt;em&gt;mathematical induction&lt;/em&gt;, although in some textbooks it is marked as optional.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2020-11-13T17:37:10.017" LastActivityDate="2020-11-13T17:37:10.017" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18527" PostTypeId="1" AcceptedAnswerId="18605" CreationDate="2020-07-05T03:01:34.450" Score="15" ViewCount="491" Body="&lt;p&gt;I want to try making some calculation-less questions about vector calculus identities that are solely based upon picture diagrams of vector fields, or fields that could be sketched out by hand. The purpose of this is to introduce the ideas so that students aren't just calculating and using formulas without thinking about what they mean.&lt;/p&gt;&#xA;&lt;p&gt;It seems that the most common analogies to these vector calculus identities are the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gradient is the steepest path up a mountain at a location.&lt;/li&gt;&#xA;&lt;li&gt;Divergence is the way dust would disperse (or contract) at a specific location. Usually questions involve trying to find out if the field acts like a source, a sink, or &amp;quot;zero&amp;quot; (or maybe there's a better description for zero divergence?)&lt;/li&gt;&#xA;&lt;li&gt;Curl is how a paddle wheel (at a location) would rotate. Usually questions involve comparing vectors contributing to clockwise and counterclockwise motions of the wheel, and seeing which is net positive.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I can't remember an analogy for the Laplacian but it seems that it could be understood as the &lt;a href=&quot;https://math.stackexchange.com/questions/50274/intuitive-interpretation-of-the-laplacian&quot;&gt;&amp;quot;average changing&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-07-05T03:15:52.060" LastActivityDate="2020-07-24T17:20:52.240" Title="Analogies for grad, div, curl, and Laplacian?" Tags="|calculus|vector-calculus|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18528" PostTypeId="2" ParentId="18458" CreationDate="2020-07-05T12:09:31.183" Score="0" Body="&lt;p&gt;For Advanced High School math, I'd consider looking at &amp;quot;IB Further Math&amp;quot; materials.&lt;/p&gt;&#xA;&lt;p&gt;If you look it up on Wikipedia, you'll find that it's been discontinued but if you search online for pdfs then you could find some materials that could cover your topics. For example, &lt;a href=&quot;https://www.math.ksu.edu/%7Edbski/writings/further.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-07-05T12:09:31.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18529" PostTypeId="2" ParentId="18518" CreationDate="2020-07-05T19:19:26.207" Score="6" Body="&lt;p&gt;Aside from any particular book, I'd say that you need a &lt;em&gt;human being&lt;/em&gt; reviewing and giving feedback on your proofs. This is a type of writing for consumption by other people. One of the main things is that a proof should be clear, explanatory, and insightful. Partly this criteria depends on the level of expertise of the expected audience.&lt;/p&gt;&#xA;&lt;p&gt;Now, I'm not entirely sure what is the best way to arrange this outside of class. Get a &amp;quot;proof pen pal&amp;quot; with whom you swap proofs and give feedback for how clear it was? An online study-group type situation? Post attempted proofs to SE Mathematics and ask for feedback/improvements? (Kind of like how SE Software Engineering, I think, accepts code-review posts).&lt;/p&gt;&#xA;&lt;p&gt;When you do that, bear in mind that's how actual professional articles get written. Someone writes a draft of one or more sections, they get sent around to colleagues for commentary, and the draft gets re-written and re-written until everyone gives the thumbs-up to it. So if you can get that process going, then you'll have an additional leg up on the people-skills that a math researcher needs.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-07-05T23:04:28.590" LastActivityDate="2020-07-05T23:04:28.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18530" PostTypeId="2" ParentId="10993" CreationDate="2020-07-06T18:11:25.923" Score="0" Body="&lt;p&gt;I would avoid accelerating any further. Be aware though that even without acceleration a talented child will have taught themselves or be able to infer without being taught much of the syllabus. The problem withh acceleration is that while it exposes the child to new material it doesn't necessarily provide intellectual challenge. A very bright child wil thrive off creative problem solving - this doesn't have to require much knowledge but will need lateral thinking and imagination.&lt;/p&gt;&#xA;&lt;p&gt;Definitely look into the maths challenges for their age groups and higher - if they're 11 now they can do the primary maths challenge and probably have a go at the junior ones too. I would encourage them also to learn some programming. Just a kahn academy course to learn the basics of a programming language (e.g. python) then they can try some Euler problems. There are places like the London SMP currently offering classes for maths and computing online which are outside of the UK curriculum.&lt;/p&gt;&#xA;" OwnerUserId="14252" LastActivityDate="2020-07-06T18:11:25.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18531" PostTypeId="1" CreationDate="2020-07-07T00:23:32.870" Score="4" ViewCount="268" Body="&lt;p&gt;I like the Common Core State Standards for Math as they focus on objectives that students need to reach. However, some standards have way too many parts for them.&lt;/p&gt;&#xA;&lt;p&gt;For instance, the standard &lt;a href=&quot;http://www.corestandards.org/Math/Content/HSA/REI/D/11/&quot; rel=&quot;nofollow noreferrer&quot;&gt;CCSS.MATH.CONTENT.HSA.REI.D.11&lt;/a&gt; is stated below. It would be ridiculous to teach this standard in an Algebra 1 class as students don't know logarithms yet. What do other teachers do in the way of teaching this standard for intersection of lines for Algebra 1? Is there such thing as reaching half a standard? What do others do?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12366" LastEditorUserId="11" LastEditDate="2020-07-13T17:07:31.563" LastActivityDate="2020-07-13T21:41:00.823" Title="Is There Such Thing as Reaching Half a Standard?" Tags="|secondary-education|usa|common-core|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18532" PostTypeId="1" CreationDate="2020-07-07T05:01:40.770" Score="3" ViewCount="245" Body="&lt;p&gt;(Cross posted at MSE: &lt;a href=&quot;https://math.stackexchange.com/questions/3742948/how-did-we-isolate-the-properties-of-identity-commutativity-associativity-and&quot;&gt;https://math.stackexchange.com/questions/3742948/how-did-we-isolate-the-properties-of-identity-commutativity-associativity-and&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;In elementary school, I remember learning about the basic algebraic properties of the integers like identities, commutativity, associativity, and distributivity, and not really thinking much about them (I mean, as a kid I thought they were obvious and not worth dedicating a month to, haha). Now that I'm starting abstract algebra, these four things pop up again, but this time around, these laws seem far more mysterious, perhaps because they are being used as some sort of &amp;quot;basis&amp;quot; for generating a &amp;quot;valid&amp;quot; algebraic structure, instead of just random facts about numbers.&lt;/p&gt;&#xA;&lt;p&gt;My question is this; I would expect there to be lots of formulas regarding elementary arithmetic, but somehow these four ideas generate everything. How could one trying to isolate algebraic properties of &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z$&lt;/span&gt; come up with this exact &amp;quot;basis&amp;quot;? Is there some kind of &lt;strong&gt;logical/algorithmic method we could use to systematically discover these laws and be sure that they encompass everything we care about&lt;/strong&gt; when it comes to elementary arithmetic?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;An idea I had was that if someone could tell a story about building arithmetic from the Peano axioms, like here: &lt;a href=&quot;https://www.math.wustl.edu/%7Ekumar/courses/310-2011/Peano.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.math.wustl.edu/~kumar/courses/310-2011/Peano.pdf&lt;/a&gt;, sort of like: ok we defined the operator &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; that takes in two things from &lt;span class=&quot;math-container&quot;&gt;$\mathbb N$&lt;/span&gt; and spits out one thing in &lt;span class=&quot;math-container&quot;&gt;$\mathbb N$&lt;/span&gt; recursively by saying &lt;span class=&quot;math-container&quot;&gt;$n+1 = \sigma(n)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n+\sigma(m)=\sigma(n+m)$&lt;/span&gt;. Now an example: we already defined &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;&amp;quot;, and let's define &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$2 = \sigma(1)$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$1+1=\sigma(1)=2$&lt;/span&gt;. Nice! How about &lt;span class=&quot;math-container&quot;&gt;$2+1$&lt;/span&gt;? Well, &lt;span class=&quot;math-container&quot;&gt;$2+1 = \sigma(2)$&lt;/span&gt; which we'll call &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. But what if I asked about &lt;span class=&quot;math-container&quot;&gt;$1+2$&lt;/span&gt;? Then the 1st rule won't help, but we can write &lt;span class=&quot;math-container&quot;&gt;$1+2=1+\sigma(1)=\sigma(1+1)=\sigma(2)=3$&lt;/span&gt;. Yay! But this was annoying because we know intuitively that switching the things around on the &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; operator doesn't change anything, so let's prove this property (which we'll call commutativity).&lt;/p&gt;&#xA;&lt;p&gt;However, I can't seem to shoehorn associativity or distributivity in a convincing manner, so perhaps this is the wrong approach.&lt;/p&gt;&#xA;&lt;p&gt;Another idea I had was like starting again from the Peano axioms and then saying like &amp;quot;ok, we rigorously defined numbers and addition and multiplication and induction. Let's do the age old Gauss integer sum problem from the Peano axiom framework!&amp;quot;. This problem immediately forces us to define addition for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; numbers (associativity), and then the end result involves &lt;span class=&quot;math-container&quot;&gt;$n(n+1)$&lt;/span&gt; so distributivity comes up naturally. However, this is kind of awkward (like it's awkward to shoehorn in Gauss's sum problem randomly in the middle discussing foundational arithmetic--at least it feels slightly unnatural in my eyes), so I don't know. Phrased another way, my complaints for this idea is that there arise two questions: &amp;quot;why should we consider this Gauss problem&amp;quot; and &amp;quot;why should this problem be all that is needed to develop every property we care about in arithmetic&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Criticisms and ideas are welcome!&lt;/p&gt;&#xA;" OwnerUserId="13084" LastEditorUserId="13585" LastEditDate="2020-07-07T18:49:11.240" LastActivityDate="2020-08-24T15:15:38.760" Title="How to naturally encounter the properties of identity, commutativity, associativity, and distributivity (to define rings)?" Tags="|abstract-algebra|intuition|arithmetic-operations|" AnswerCount="4" CommentCount="1" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18533" PostTypeId="2" ParentId="18532" CreationDate="2020-07-07T14:13:01.880" Score="5" Body="&lt;p&gt;This isn't exactly what you are looking for, but it's pretty close: the book &amp;quot;Surreal Numbers&amp;quot; by Donald Knuth builds up the theory of surreal numbers by telling a fictional story of two people discovering and building the correct axioms. You should check it out. It's a neat read, and if I remember correctly it actually has exercises too.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;From the Back Cover&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The book's primary aim, Knuth explains in a postscript, is not so much to teach Conway's theory as &amp;quot;&lt;strong&gt;to teach how one might go about developing such a theory&lt;/strong&gt;.&amp;quot; He continues: &amp;quot;Therefore, as the two characters in this book gradually explore and build up Conway's number system, I have recorded their false starts and frustrations as well as their good ideas. I wanted to give a reasonably faithful portrayal of the important principles, techniques, joys, passions, and philosophy of mathematics, so I wrote the story as I was actually doing the research myself.&amp;quot;...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastActivityDate="2020-07-07T14:13:01.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18534" PostTypeId="2" ParentId="10993" CreationDate="2020-07-07T14:59:54.283" Score="1" Body="&lt;p&gt;Since you are in the UK, I have a few specific suggestions below. I agree with existing answers and comments that there is little benefit in rapid acceleration.&lt;/p&gt;&#xA;&lt;p&gt;The Royal Institution hosts mathematics masterclasses in some areas. See &lt;a href=&quot;https://www.rigb.org/education/masterclasses&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.rigb.org/education/masterclasses&lt;/a&gt;&#xA;for more information. Sometimes the sessions are for students in specific age groups and sometimes local schools are asked to invite a small number of pupils, but it may be worth researching whether there are any masterclasses in your area.&lt;/p&gt;&#xA;&lt;p&gt;The UKMT has lots of books suitable for extension material for able students of all ages. Some books are specifically aimed at improving in their competitions. See &lt;a href=&quot;https://www.ukmt.org.uk/shop&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ukmt.org.uk/shop&lt;/a&gt;.&#xA;All Junior Maths Challenge and Junior Maths Olympiad papers going back to 1999 and 2003 resepctively are freely available on their website at the moment. I think it is quite common for JMC/JMO questions to be used for extension material in schools.&lt;/p&gt;&#xA;&lt;p&gt;I also highly recommend the Art of Problem Solving books. These are based on an American curriculum, but there is plenty of excellent extension material for students at all levels to delve more deeply into topics they have learnt. See &lt;a href=&quot;https://artofproblemsolving.com/store&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://artofproblemsolving.com/store&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2020-07-07T14:59:54.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18535" PostTypeId="1" CreationDate="2020-07-07T15:11:20.990" Score="1" ViewCount="60" Body="&lt;p&gt;I am an undergraduate student from the CS background.&lt;/p&gt;&#xA;&lt;p&gt;While reading a portion of the text &lt;em&gt;&amp;quot;Introduction to Algorithms&amp;quot; by Cormen et. al.&lt;/em&gt; I came across a section of &lt;strong&gt;Matroids&lt;/strong&gt;. I was unable to understand it properly as I have not been lucky enough to have a formal matroid course.&lt;/p&gt;&#xA;&lt;p&gt;I was recommended the following book &lt;em&gt;&amp;quot;Matroid Theory&amp;quot; by John Oxley&lt;/em&gt; in &lt;a href=&quot;https://math.stackexchange.com/a/3748367/606943&quot;&gt;this answer&lt;/a&gt;. Now I find the above book labelled as &lt;em&gt;&amp;quot;Oxford Graduate Text In Mathematics&amp;quot;&lt;/em&gt; ,so with my current knowledge of as an undergraduate CS student (just completed 2nd year) I do not quite know whether I shall be able to comprehend the text, more-over the text is not quite short (a total of 719 page in the second edition). Moreover with the current tightness of the present curriculum, I cannot simply afford to start a text and leave it half way being unable to understand (it shall cause frustration).&lt;/p&gt;&#xA;&lt;p&gt;So I request here to guide me as to whether I shall be able to understand the book given my current qualification. Moreover could a video lecture series be recommended to me so that I shall easily understand the concepts of matroid (in a considerably short time, given that I am planning to do self study and I shall not have any teacher support, without a teacher resolving doubts takes time).&lt;/p&gt;&#xA;" OwnerUserId="14166" LastActivityDate="2020-07-07T22:19:25.897" Title="Reference Request: Suitability of a text and corresponding video lecture series request for &quot;Matroid Theory&quot;" Tags="|reference-request|textbooks|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18536" PostTypeId="2" ParentId="18535" CreationDate="2020-07-07T22:19:25.897" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Wilson, Robin J. &amp;quot;An introduction to matroid theory.&amp;quot; &lt;em&gt;The American Mathematical Monthly&lt;/em&gt; 80, no. 5 (1973): 500-525.&#xA;&lt;a href=&quot;https://www.jstor.org/stable/2319608?seq=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;JStor link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is an elementary introduction, not even assuming much knowledge of graph theory. He explains enough transversal theory to proceed to matroids. He defines a matroid in several ways, so that you get different perspectives.&lt;/p&gt;&#xA;&lt;p&gt;Of course it is out-of-date, but it is quite accessible, and might allow you to then move on to more recent expositions.&#xA;For example,&#xA;this text is explicitly written for undergraduates:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gordon, Gary, and Jennifer McNulty. &lt;em&gt;Matroids: A Geometric Introduction&lt;/em&gt;. Cambridge University Press, 2012.&#xA;&lt;a href=&quot;https://www.cambridge.org/core/books/matroids-a-geometric-introduction/2DCAB3A148C84F5DB32E1E398AD7A6FD&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cambridge link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/5r7kh.jpg&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-07-07T22:19:25.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18537" PostTypeId="1" CreationDate="2020-07-07T22:44:54.563" Score="2" ViewCount="546" Body="&lt;p&gt;My main question is directly related to the title: &amp;quot;Do you mention that (in its domain) the empty function is everywhere continuous and everywhere discontinuous?&amp;quot; (and a similar question based upon differentiability)&lt;/p&gt;&#xA;&lt;p&gt;A secondary question I'd ask is, &amp;quot;do you mention the empty function at all?&amp;quot;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;My answer: I personally like counter-intuitive questions that really test one's knowledge of definitions, or at least make one aware of &lt;em&gt;what&lt;/em&gt; definition they're using. I might mention it as an aside &lt;em&gt;as something interesting to note&lt;/em&gt; but I don't go into it.&lt;/p&gt;&#xA;&lt;p&gt;(edit notes: updated based upon comment by DaveLRenfro)&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-07-09T13:08:28.870" LastActivityDate="2020-07-13T13:11:51.853" Title="Do you mention the continuity and the differentiability of the empty function" Tags="|undergraduate-education|calculus|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18539" PostTypeId="2" ParentId="18537" CreationDate="2020-07-08T01:41:13.647" Score="14" Body="&lt;p&gt;I think that this would be too much of a detour in a regular Calculus class.&lt;/p&gt;&#xA;&lt;p&gt;You would need to first establish that the empty function is even a function.  This requires a really pedantic reading of the definition of a function (a &lt;strong&gt;relation&lt;/strong&gt; from &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; which is &lt;strong&gt;total&lt;/strong&gt; and &lt;strong&gt;single-valued&lt;/strong&gt;:  now you have to really dig in to what those words mean).  This is beyond what most calculus students are prepared for:  in my experience most of them think of functions as algebraic expressions, and some of them think of them as &amp;quot;function machines&amp;quot;.  One in a thousand could give a formal definition of a function between two sets.  That usually comes up in an intro to proof course, or a discrete mathematics course.&lt;/p&gt;&#xA;&lt;p&gt;Understanding that the empty function is a function, and understanding that it is continuous and smooth, also uses the concept of &amp;quot;vacuous truth&amp;quot;.  Namely, for any predicate &lt;span class=&quot;math-container&quot;&gt;$P(x)$&lt;/span&gt;, the statement &lt;span class=&quot;math-container&quot;&gt;$\forall x \in \emptyset, P(x)$&lt;/span&gt; is true.   This is also a very subtle logical point which will just sound like mystical gobbledegook until the student has taken a logic/intro-proof course.  In such courses, I do encourage my students to be &amp;quot;scholars of the empty set&amp;quot;, and think about such questions as whether the empty graph is connected, whether the empty relation is an equivalence relation, etc.&lt;/p&gt;&#xA;&lt;p&gt;There are plenty of opportunities for the kind of pedantic definition lawyering we want our students to engage in in Calculus. For example, you can given them a formal definition of &amp;quot;increasing function&amp;quot; and ask them whether a constant function is increasing or not.&lt;/p&gt;&#xA;&lt;p&gt;TLDR:  Not appropriate for a Calculus class.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-07-08T01:48:24.753" LastActivityDate="2020-07-08T01:48:24.753" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18540" PostTypeId="2" ParentId="10993" CreationDate="2020-07-08T02:38:53.147" Score="0" Body="&lt;p&gt;I think there is a benefit to having alternate, conflicting answers to soft questions like this.  My take is to keep going with the kid.  Just ask for him to go with the kids a couple grades higher, when doing math.  If they don't have that because of the grade structure, than have him be independent study for a period (or see if high school is near the junior high and he can walk over for an hour).&lt;/p&gt;&#xA;&lt;p&gt;Note that ADM Rickover in his famous book on American education made strong arguments AGAINST what you are hearing here (conventional wisdom to do &amp;quot;enrichment&amp;quot;).  Basically his argument is that it is an inefficient use of the time of strong students.  Provided they are mastering the material like a decent of age kid would, than their time is better used by advancing them.  After all, you are not holding back the of age kids for a couple years to enrich them.  And all the &amp;quot;wonders of enrichment&amp;quot; pale compared to the wonders of a whole new topic, like Feynman moving to calculus from trig.&lt;/p&gt;&#xA;" OwnerUserId="14269" LastActivityDate="2020-07-08T02:38:53.147" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18541" PostTypeId="2" ParentId="10993" CreationDate="2020-07-08T17:25:27.847" Score="0" Body="&lt;p&gt;I see two options: acceleration or enrichment. Middle school is the time when acceleration becomes possible, at least in most U.S. schools. So, instead of Math 6 he would be taking Math 6/7, instead of Math 7 he would be taking Math 7/8, and instead of Math 8 he would be taking Algebra I or HS Math I in districts that have switched to integrated math.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Acceleration serves two goals: it attempts to rid the student from boredom, and allows taking more challenging courses, or more of them. It is also possible to combine courses in one year, say Geometry and Algebra II in grade 9, then precalc in grade 10, then AP Calc BC in grade 11, and then something else in grade 12. AP classes can earn college credits, allowing to save time and money in college.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Enrichment is best to be taken before high school, because there are fewer options for acceleration yet the schedule is not crazy. This is the time for math circles, for recreational math, for leisurely going into vectors and matrices, for presenting basics of derivative and integral, for studying mechanics and solving a bunch of word problems about a swimmer who swam up and down the river losing and then catching his hat and stuff like that. Also introduce basics of logic and proofs.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7930" LastActivityDate="2020-07-08T17:25:27.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18544" PostTypeId="1" CreationDate="2020-07-09T02:49:17.513" Score="1" ViewCount="203" Body="&lt;p&gt;I have a theory that if standardized grading were abolished for a pass/fail system, people would be more mathematically competent.&lt;/p&gt;&#xA;&lt;p&gt;Bear with me here. With graded homework, especially homeworks that come with diminishing returns (limited attempts that reduce points with every failing attempt) - I argue that makes an incentive to search up answers online (online calculators, Chegg, etc.). When people do this they set up themselves to fail or do poorly in their exams.&lt;/p&gt;&#xA;&lt;p&gt;You can argue graded homework encourages people to do homework, and I agree with this but the problem is it doesn't necessarily encourage them to do their homework without aid. There's a multitude of factors that contribute to poor studying and cheating, such as universities having credit requirements too steep for young adults to endure simultaneously with responsibilities like work, bills, parenthood, etc. (which are clearly designed to maximize tuition profit). I think with consideration to reduce economic and academic stress imposed on students as well as abolishing our standardized grading system, we could really maximize success in the classroom. We wouldn't have students landing in calc 2 with a poor understanding of differentiation just because they managed to pass calc 1 with a 70%, or students that rely on homework, quizzes, curves, etc., to pass just because they consistently do poorly on their exams. I think the fact they do poorly on their exams is an indication our grading system is not doing its job.&lt;/p&gt;&#xA;" OwnerUserId="12501" LastActivityDate="2020-07-09T07:13:30.770" Title="Would it make sense for math courses to be pass/fail?" Tags="|undergraduate-education|secondary-education|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18545" PostTypeId="2" ParentId="18544" CreationDate="2020-07-09T04:06:09.340" Score="3" Body="&lt;p&gt;Proficiency in any field requires putting some rather high number of hours of hard work (some 10,000 hours or so) into it. You have to align the incentives of would-be learners so they do put in that time. If the best way is just pass/fail or a grade system is anybody's guess, but perhaps a search will turn up rigorous studies on the matter.&lt;/p&gt;&#xA;&lt;p&gt;Note that I don't consider asking Google, looking up the solution in Wikipedia or having Wolfram Alpha compute the hairy formula &amp;quot;cheating&amp;quot;, those are &lt;em&gt;tools&lt;/em&gt; that you have to learn to use well to be proficient in your area. Those are raw inputs, how you use it is what is important.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-07-09T04:06:09.340" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18546" PostTypeId="2" ParentId="18544" CreationDate="2020-07-09T07:13:30.770" Score="2" Body="&lt;p&gt;You don't show why, not can I see how, your change to a digital outcome from a scaled one would change the behavior that concerns you, cheating on graded homework.&lt;/p&gt;&#xA;&lt;p&gt;In contrast, I can see how changes like eliminating graded homework and replacing it with in class orictored exams would reduce, not eliminate, but reduce, cheating. Because it is harder under those conditions.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, even if changing to pass fail helped reduce cheating, it would be important to consider other aspects of losing a scaled measurement.   Real world problems are multivariable, not x and y.   You need to consider various impacts of a change.&lt;/p&gt;&#xA;" OwnerUserId="14273" LastActivityDate="2020-07-09T07:13:30.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18547" PostTypeId="1" CreationDate="2020-07-09T10:40:31.443" Score="5" ViewCount="350" Body="&lt;p&gt;This question was prompted by looking at Khan Academy's website to see how a comprehensive lecture series could be done and often I see the word, &amp;quot;mastery&amp;quot;. To me, I'd think mastery is something like &amp;quot;being an expert in&amp;quot;, and that in turn means &amp;quot;being extremely capable and knowledgeable&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;However, when I look at what &amp;quot;mastery&amp;quot; means to Khan Academy, it seems to be completion of various topics and knowing computation. One of Khan's quotes in a Ted talk is, &amp;quot;&lt;em&gt;Due to the constraints of time and curriculum a student may get only 60% right, or 80% right or even 95%... so it creates a Swiss cheese gap in the foundation of your understanding of that subject.&lt;/em&gt; &amp;quot; Also, he often gives the analogy of a house being built and how if, for the second story not to fall, it needs a solid first story, and that needs a solid foundation.&lt;/p&gt;&#xA;&lt;p&gt;I'm probably being too nit-picky with word choice, but the more I look into the website, it seems that this is more &amp;quot;Completion Learning&amp;quot; rather than &amp;quot;Mastery Learning&amp;quot; because it not based on mathematical proof and rigor. Or, if it is mastery learning, it's mastery in the sense of computation and calculation of the basic ideas, but not of where those ideas come from. (And, btw, I'm not against computation and calculation but if that is what was meant then I'd rather have more accurate wording)&lt;/p&gt;&#xA;&lt;p&gt;If I look at one of his proof videos (which would be more of what I'd think &amp;quot;mastery&amp;quot; is), like his video of &amp;quot;&lt;a href=&quot;https://www.youtube.com/watch?v=0sCttufU-jQ&quot; rel=&quot;noreferrer&quot;&gt;formal definition of limits - Part 4&lt;/a&gt;&amp;quot;, from what I can tell, the algebraic manipulation seems to go the &amp;quot;wrong way&amp;quot; in terms of finding the &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; relationship, and the final &lt;span class=&quot;math-container&quot;&gt;$\left|x-a\right|&amp;lt;\delta$&lt;/span&gt; without the &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;$&lt;/span&gt; on the left side. My rhetorical questions are &amp;quot;does that video show mastery from the teacher?&amp;quot; and &amp;quot;would a student master that topic based on that video?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&lt;p&gt;If I teach a topic, then how do I know if I'm teaching &amp;quot;mastery&amp;quot; of that topic? (or am I more honest in saying &amp;quot;computation&amp;quot; and &amp;quot;calculation&amp;quot; of a topic) Or if I see a student's work, how would I know if they have mastered the topic's concepts and logic, or are just repeating &amp;quot;plug'n'chug&amp;quot; math with a sophisticated kind of rote memorization?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-07-10T03:57:45.407" Title="What is &quot;mastery&quot; in a mathematical topic?" Tags="|mathematical-pedagogy|proofs|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18549" PostTypeId="2" ParentId="18547" CreationDate="2020-07-09T14:20:19.820" Score="3" Body="&lt;p&gt;The Kumon Math Program defines mastery as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The ability to apply a skill or concept with the accuracy, speed and confidence to demonstrate a total command of the material. Mastery means that a skill or concept has become second nature and will be permanently retained. Mastery is the prerequisite for advancement and the goal of Kumon study. &lt;a href=&quot;https://www.kumon.com/chicago-northwest/aboutcenter&quot; rel=&quot;nofollow noreferrer&quot;&gt;[source]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Speed + Accuracy = Mastery&lt;br /&gt;&#xA;Material should be completed with a perfect score within a prescribed period of time. &lt;a href=&quot;https://www.kumonsunnyvale.com/kumonmath&quot; rel=&quot;nofollow noreferrer&quot;&gt;[source]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This &amp;quot;prescribed period of time&amp;quot; is called the Standard Completion Time, which is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A prescribed number of minutes to complete a particular worksheet with a perfect (or near perfect) score. Kumon determines-and continually reevaluates-SCT based on the actual performance of tens of thousands of students. Successfully completing worksheets within the SCT demonstrates that the student has mastered the material and is ready to move on. &lt;a href=&quot;https://www.kumon.com/chicago-northwest/aboutcenter&quot; rel=&quot;nofollow noreferrer&quot;&gt;[source]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In practice, &amp;quot;near perfect&amp;quot; usually means &amp;quot;at least 90%.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2020-07-09T14:20:19.820" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18550" PostTypeId="2" ParentId="18547" CreationDate="2020-07-10T02:10:19.770" Score="5" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think fighting for the definition of a word makes most sense.  After all, we could consider Robbie mastery versus Kahn Academy mastery (as different attributes).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For what it's worth, the Kahn Academy sense of mastery is a rather normal one.  The idea comes from the theory of building block automaticity.  That you have skills down to the sense that you can use them without struggle.  Our minds are not silicon, they have very limited RAM.  So if you struggle on basic reading, it becomes a chore to do the reading and then do higher processing.  Similarly, if you struggle on basic manipulations, than new concepts in physics or math are hard to work on.  Consider the issues for a kid working in calculus who is weak in trig or doesn't have an iconic recall of (-b +/-sqrt(bsq-4ac)/2a).  [Own goal on me, if I flubbed that formula!]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Here is an example of Richard Feynman discussing mastery in the sense of manipulation skill:&lt;/p&gt;&#xA;&lt;p&gt;So, this guy comes into my office and asks me to try to make everything straight that I taught him, and this is the best I can do. The problem is to try to explain the stuff that was being taught. So I start, now, with the review. I would tell this guy, “The first thing you must learn is the mathematics. And that involves, first, calculus. And in calculus, differentiation.”&lt;/p&gt;&#xA;&lt;p&gt;Now, mathematics is a beautiful subject, and has its ins and outs, too, but we’re trying to figure out what the minimum amount we have to learn for physics purposes are. So the attitude that’s taken here is a “disrespectful” one towards the mathematics, for sheer efficiency only; I’m not trying to undo mathematics.&lt;/p&gt;&#xA;&lt;p&gt;What we have to do is to learn to differentiate like we know how much is 3 and 5, or how much is 5 times 7, because that kind of work is involved so often that it’s good not to be confounded by it. When you write something down, you should be able to immediately differentiate it without even thinking about it, and without making any mistakes. You’ll find you need to do this operation all the time—not only in physics, but in all the sciences. Therefore differentiation is like the arithmetic you had to learn before you could learn algebra.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, the same goes for algebra: there’s a lot of algebra. We are assuming that you can do algebra in your sleep, upside down, without making a mistake. We know it isn’t true, so you should also practice algebra: write yourself a lot of expressions, practice them, and don’t make any errors. Errors in algebra, differentiation, and integration are only nonsense; they’re things that just annoy the physics, and annoy your mind while you’re trying to analyze something. You should be able to do calculations as quickly as possible, and with a minimum of errors. That requires nothing but rote practice—that’s the only way to do it. It’s like making yourself a multiplication table, like you did in elementary school: they’d put a bunch of numbers on the board, and you’d go: “This times that, this times that,” and so on—Bing! Bing! Bing!&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;I would also suggest that instead of debating the usage of the term mastery, you could also differentiate the subject.  In other words, if I master Thomas Finney calculus.  (I literally and not how millennials use that word did every homework problem in that text and scored perfect tests throughout high school and as far as I could tell on the AP exam.)  Now I would say that was automaticity and ease of use and Kahn-Feymnanian mastery OF the 1980s AP calculus class.  Of course if you want to say there should be some OTHER calculus course (with a bunch of real analysis thrown into the frosh calc course instead of waiting until junior year), then fine.  But you're really debating the subject, not the concept of mastery. After all mastering high school geometry certainly involved lots of proofs since that course was mostly proving geometric propositions (was actually weak on mensuration practice, I thought.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also, I think there are some reasonable differences in the depth of mastery expected of a student and of a teacher.  Both should have acquired automaticity at doing the homework drill problems.  But the teacher actually benefits from knowing things not really even PART of the course for the student. Advanced courses that use the skills from the course in discussion.  The history of the development of the subject.  And applications.  But it's not reasonable to expect the same thing of the student.  He just needs to master the subject itself.  Not a bunch of related matters outside the scope.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think nitpicking a mistake in one of his videos is relevant to the concept of student mastery and/or the preferred definition/usage of the term.  They are really not connected.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14279" LastEditorUserId="14279" LastEditDate="2020-07-10T03:57:45.407" LastActivityDate="2020-07-10T03:57:45.407" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18551" PostTypeId="1" CreationDate="2020-07-10T18:00:50.227" Score="1" ViewCount="282" Body="&lt;p&gt;I use textbook study and am planning on studying Spivak's Calculus, Mathematics It's Content, Methods, and Meaning, How to Prove it by Velleman, etc. However, I'm worried I lack the prerequisite knowledge gained from formal classes in algebra I &amp;amp; II, Trig and beyond. Does Basic Mathematics by Serge Lang, The Zakon Series on Mathematical Analysis, Basic Mathematics, The Kiselev Geometry Books, H.S. Hall's Algebra Series, Euclid's Elements, An Introduction to Logic and the Scientific Method by Cohen and Nagel, and Russel's Mathematical Philosophy cover everything necessary? Does anything need to be omitted as prerequisite or anything added to cover some information?&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="14283" LastActivityDate="2020-08-10T06:41:47.247" Title="What are the resources to learn prerequisite knowledge to latter High school and undergrad prep textbooks?" Tags="|secondary-education|textbooks|self-learning|middle-school|resource-request|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18552" PostTypeId="1" AcceptedAnswerId="18555" CreationDate="2020-07-10T21:09:30.220" Score="8" ViewCount="180" Body="&lt;p&gt;I am a high school mathematics teacher and I regularly create problems and their solutions for my students. It has always lingered in my mind that this content can also benefit others.&lt;/p&gt;&#xA;&lt;p&gt;What would be a good place for me to share it?&lt;/p&gt;&#xA;&lt;p&gt;Hosting own website would just be an island with no visitors. I believe that I ideally need a place with a regular stream of problem solving visitors, some of whom could then benefit from my content as well.&lt;/p&gt;&#xA;" OwnerUserId="12907" LastEditorUserId="12907" LastEditDate="2020-07-10T22:26:54.207" LastActivityDate="2020-07-11T06:23:11.880" Title="What is a good place for teachers to share self-created content?" Tags="|undergraduate-education|secondary-education|online-instruction|problem-solving|online|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18553" PostTypeId="2" ParentId="18459" CreationDate="2020-07-10T21:10:55.017" Score="2" Body="&lt;p&gt;Am Arabian Mathematician. In my country in all school we used to write math from right to left i scored very well in 12th grade 179/200. But in the universities we use English books so we write from  left to right it was hard for me to get through that in the first course I scored 65/100 by the time I used to that and now am PhD student in the US. So my opinion it is not that hard by practice 100 or more problems normal student should be fine.&lt;/p&gt;&#xA;" OwnerUserId="5993" LastEditorUserId="5993" LastEditDate="2020-07-13T00:11:38.327" LastActivityDate="2020-07-13T00:11:38.327" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18554" PostTypeId="2" ParentId="18551" CreationDate="2020-07-11T01:26:21.920" Score="2" Body="&lt;p&gt;Your list seems like overkill to me.&lt;/p&gt;&#xA;&lt;p&gt;As far as geometry, Kiselev is basically a rehash of Euclid, so I don't see the point in studying both. Just pick one. I don't think you need the solid geometry parts of either.&lt;/p&gt;&#xA;&lt;p&gt;If using Euclid: -- Euclid contains stuff like number theory done in an ancient style that is now only of historical interest, so if using Euclid, skip that. Euclid's definitions of terms like &amp;quot;line&amp;quot; and &amp;quot;point&amp;quot; are nonsense by modern standards, so skip them as well. (The Russell book will give you a better modern mathematician's intro to how definitions work.) Make sure to work from a well-annotated edition of Euclid. IIRC there's a very good free one online by Kirkpatrick. Some of Euclid's arguments have flaws (including his very first theorem).&lt;/p&gt;&#xA;&lt;p&gt;The Russell book spends a lot of time developing what we would now call set theory, but it predates the standard modern formulation of ZFC, so I'm not sure I'd recommend using it as an intro to that topic. I don't think any significant amount of the material from this book is needed for Spivak.&lt;/p&gt;&#xA;&lt;p&gt;I'd avoid making an affectation out of reading books by famous people dating to 100-130 years ago.&lt;/p&gt;&#xA;&lt;p&gt;For algebra, trig, and geometry, you can get all the preparation you need from any crappy high school books you can find at the library or half.com. The hard part about Spivak is going to be the that it's written for an audience of future mathematicians, so it doesn't dumb things down, and the problems are reputedly hard.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-07-11T01:26:21.920" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18555" PostTypeId="2" ParentId="18552" CreationDate="2020-07-11T06:23:11.880" Score="4" Body="&lt;p&gt;What is the medium of the content?&lt;/p&gt;&#xA;&lt;p&gt;If they are videos, then YouTube would probably be first option but if they are, say, pdfs, then you might have to make a blog and then upload to that there. It still might feel like an island at first but it will gain traffic from your students and then other people who find it in search results. There are also problem-solving forums which one could join, but the rules of posting content would need to be read beforehand.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-07-11T06:23:11.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18556" PostTypeId="1" CreationDate="2020-07-11T14:46:00.330" Score="2" ViewCount="136" Body="&lt;p&gt;I’m searching for an online resource, preferably free.&lt;br /&gt;&#xA;I refer to equations such as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A^3 -A+ I=0$$&lt;/span&gt;&#xA;Or even something like:&#xA;&lt;span class=&quot;math-container&quot;&gt;$${A^t}^2+A^2=I$$&lt;/span&gt;&#xA;I know that normal factorization methods don’t work as matrices aren’t a field.&lt;br /&gt;&#xA;Anyone knows about a resource, such as a book, about this topic?&lt;br /&gt;&#xA;I’m &lt;em&gt;not&lt;/em&gt; looking for a general linear algebra resource, I’m looking for a resource specifically about matrix equations.&lt;/p&gt;&#xA;" OwnerUserId="14192" LastEditorUserId="14192" LastEditDate="2020-07-11T15:25:58.473" LastActivityDate="2020-07-12T02:13:00.413" Title="Good resources on matrix equations" Tags="|resource-request|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18557" PostTypeId="2" ParentId="18556" CreationDate="2020-07-11T18:09:08.380" Score="1" Body="&lt;p&gt;Take a look at the Naval Academy course on Matrix Theory.  It emphasizes matrix manipulation more than proofs shmoofs.  And is not a general LA class.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.usna.edu/Users/math/hottovy/Teaching/SM261.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.usna.edu/Users/math/hottovy/Teaching/SM261.php&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are some worksheets and notes you could look at also on the SM261 page.&lt;/p&gt;&#xA;&lt;p&gt;Here is a link to Hefferon's LA text (used for the 261 class), which is progressive and FREE and emphasizes matrix manipulation at the start.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://joshua.smcvt.edu/linearalgebra/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://joshua.smcvt.edu/linearalgebra/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Other than that, I heartily recommend the chapter on matrices in Kreyszig's Advanced Engineering Mathematics.  It is very self-standing and gives you exactly what you want (learning to rock the matrices around the clock, not abstraction for the proof lovers).  It's not free, but you can get used copies (edition not important, I liked the 5th).  About 50% of the problems had answers in the back, so helpful for drill.  Also, you can just look at in the library (maybe even zox the chapter).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.thriftbooks.com/w/advanced-engineering-mathematics-8th-edition_erwin-kreyszig/263019/item/2112180/?mkwid=%7cdc&amp;amp;pcrid=395931773744&amp;amp;pkw=&amp;amp;pmt=&amp;amp;slid=&amp;amp;plc=&amp;amp;pgrid=80257881302&amp;amp;ptaid=pla-838159019236&amp;amp;gclid=EAIaIQobChMIvojsuOLF6gIVDbLICh29-QcgEAQYAiABEgL2FfD_BwE#isbn=0471862517&amp;amp;idiq=2112180&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.thriftbooks.com/w/advanced-engineering-mathematics-8th-edition_erwin-kreyszig/263019/item/2112180/?mkwid=%7cdc&amp;amp;pcrid=395931773744&amp;amp;pkw=&amp;amp;pmt=&amp;amp;slid=&amp;amp;plc=&amp;amp;pgrid=80257881302&amp;amp;ptaid=pla-838159019236&amp;amp;gclid=EAIaIQobChMIvojsuOLF6gIVDbLICh29-QcgEAQYAiABEgL2FfD_BwE#isbn=0471862517&amp;amp;idiq=2112180&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You might also look for a cheap used copy of Kolman Linear Algebra with Applications (I have third edition).  It emphasizes matrix manipulation much more than abstraction.  And gets right into it, versus building theory first. It looks like the recent editions are more abstract and there is a second author (normal scope creep that you see for commercial reasons as books age: appeals to professors who already know the stuff and sniff at ommissions and they select the books, but can be frustrating for students or self studiers).  But looking at my third edition on the shelf is very user friendly.&lt;/p&gt;&#xA;" OwnerUserId="14279" LastActivityDate="2020-07-11T18:09:08.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18558" PostTypeId="2" ParentId="18556" CreationDate="2020-07-12T02:13:00.413" Score="0" Body="&lt;p&gt;The first reference listed in &lt;a href=&quot;https://en.wikipedia.org/wiki/Matrix_polynomial&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; has a table of contents which looks appropriate:&lt;/p&gt;&#xA;&lt;p&gt;Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009) [1982]. Matrix Polynomials. Classics in Applied Mathematics.&lt;/p&gt;&#xA;&lt;p&gt;But as I commented above, you can do it for yourself if you think about the fact that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P^{-1}AP$&lt;/span&gt; will always be solutions to the same matrix polynomials. Then solve the equation for the case where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is in Jordan form. All other solutions will be conjugates of these type of solutions. (There will often be several Jordan forms which solve the equation, depending on multiplicity of roots, etc.)&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2020-07-12T02:13:00.413" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18559" PostTypeId="2" ParentId="15278" CreationDate="2020-07-12T15:50:46.757" Score="2" Body="&lt;p&gt;I don't know for sure.  But I think a part of the problem comes from notation.  I don't know how you've approached math education, but I find that students are often times confused by the distinction and overlap between the concept of division and the concept of fractions. I think the cause of this confusion is found in the notation we use and the order in which we teach the subject.&lt;/p&gt;&#xA;&lt;p&gt;We cant tell them that 10/2 = 5 is division and it requires that the top number be bigger than the bottom number... and explain to them that this is an &lt;em&gt;operation&lt;/em&gt; between two integers....&lt;/p&gt;&#xA;&lt;p&gt;... and then show them something like 2/7 and tell them this isn't an operation between integers, even though it looks like it, but an entire numeric entity all by itself.&lt;/p&gt;&#xA;&lt;p&gt;My suggestion has always been to restrict ourselves to the use of the obelus when talking about division, and leave the slash or horizontal bar for topics on fractions.&lt;/p&gt;&#xA;&lt;p&gt;Then after you explain division of fractions, you start showing them the relationship between division of integers and the entity known as a fraction. At this point the fraction bar and the obelus becomes the same object.&lt;/p&gt;&#xA;&lt;p&gt;It's my hope to convert the traditional &amp;quot;believe this in faith that they are the same thing because I said so&amp;quot; into a genuine epiphany that they can have themselves.&lt;/p&gt;&#xA;" OwnerUserId="14287" LastEditorUserId="6053" LastEditDate="2020-09-30T13:42:35.290" LastActivityDate="2020-09-30T13:42:35.290" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18560" PostTypeId="2" ParentId="18430" CreationDate="2020-07-12T23:39:03.980" Score="-2" Body="&lt;p&gt;It's so unclear. I cannot follow what your idea is. I understand what you're saying the result of your idea might be, learning the material better even when there's so much to learn in a limited amount of time. I am unable to figure out what your idea is that you think might lead to that result. In theory, I could try to learn more about the things you discussed; Quality of Videos Lectures; Lectures vs Textbooks; and Feymann techniques from the web pages I got by Googling them; &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/10872981.2018.1555434;&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tandfonline.com/doi/full/10.1080/10872981.2018.1555434;&lt;/a&gt; &lt;a href=&quot;https://academia.stackexchange.com/questions/52678/what-is-the-point-of-a-lecture-when-you-have-a-textbook;&quot;&gt;https://academia.stackexchange.com/questions/52678/what-is-the-point-of-a-lecture-when-you-have-a-textbook;&lt;/a&gt; and &lt;a href=&quot;https://fs.blog/2012/04/feynman-technique/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://fs.blog/2012/04/feynman-technique/&lt;/a&gt; and then see if I can figure out an answer to your question after studying them carefully that probably wasn't stated directly. However, I don't think my mind is going to be on that long enough to bother with. There may be some Stack Exchange users who when ever they encounter a question they deem not to have a satisfactory answer, focus only on answering the one question and take a really long time to give up on answering that question and will not move onto another question until they're either answered it or given up. Maybe attention to this question from 1 Stack Exchange user who takes a month to give up could do a better job of answering this question than attention from 30 Stack Exchange users who give up after a day, similar to my tendencies. Hopefully, you'll get an answer from somebody who focuses on one question at a time and takes a really long time to give up on that question.&lt;/p&gt;&#xA;&lt;p&gt;I know the question is an idea of how to adapt to the situation of being flooded with so much material, not a question of whether we should be flooded with so much material. However, I cannot answer that and am not sure the idea would even work anyway so I'll instead answer how I think it doesn't have to be this way, imposing mountains of material on us in a limited amount of time. I read on the internet that Finland has the best education system in the world. I also saw a YouTube video explaining how Finland teaches less material and explores it in more depth. Students were struggling to learn all the material even before COVID-19. I think it's better if the world makes a coordinated change where a lot of the material is moved to job specific training and jobs no longer rely on people having learned what they couldn't learn in school. It's better to have extra people able to do jobs.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastActivityDate="2020-07-12T23:39:03.980" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18561" PostTypeId="2" ParentId="18537" CreationDate="2020-07-13T13:11:51.853" Score="2" Body="&lt;p&gt;The answer by Steven Gubkin is good, but this thought is too long for a comment and is relevant, I think.&lt;/p&gt;&#xA;&lt;p&gt;I want to suggest that one should &lt;em&gt;be ready&lt;/em&gt; for such a question even in a low-level precalc or calc class - you really never know what to expect, and I have gotten &amp;quot;naive&amp;quot; questions in such contexts that were actually quite deep, though the students didn't know that!  (Then I recruit them to be math majors.)&lt;/p&gt;&#xA;&lt;p&gt;So although I don't think you should intentionally introduce such topics, you should have a game plan for this.  In similar contexts I usually say something like &amp;quot;In this class, that isn't a function, but when you take MATH 12345 we'll really dig into that and it will turn out to be the best function of all, even though it has no values!&amp;quot;  The first half is in a normal voice, the second half &lt;em&gt;sotto voce&lt;/em&gt;.  Most students will ignore the wacky math-crazed teacher and not worry about it, but the ones who want to know will come up after class and ask more about this MATH 12345 class you have mentioned.&lt;/p&gt;&#xA;&lt;p&gt;Lest one think this is concocted, I semi-routinely get questions in my non-major calculus class (which doesn't even have precalculus as a prerequisite in any form) about things like complex numbers, whether &lt;a href=&quot;https://www.goodreads.com/quotes/501477-i-had-a-feeling-once-about-mathematics---that-i&quot; rel=&quot;nofollow noreferrer&quot;&gt;vertical asymptotes go to infinity&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$\log(0)$&lt;/span&gt;, and functions defined at just one point.  It becomes a running gag in the class that &amp;quot;not in this class! but ...&amp;quot; and actually adds to the bonhomie a bit.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2020-07-13T13:11:51.853" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18563" PostTypeId="1" CreationDate="2020-07-13T23:27:36.723" Score="8" ViewCount="459" Body="&lt;p&gt;For sake of argument, consider that skill of a topic is spectrum from &amp;quot;new and learner&amp;quot; to &amp;quot;experienced and expert.&amp;quot; Where should you &lt;em&gt;relatively&lt;/em&gt; be in order to teach the topic effectively so that your students could do well in the long run?&lt;/p&gt;&#xA;&lt;p&gt;Omitting the situation where the student is more knowledgeable of the subject that the teacher themself, take the example where the teacher is more or less at the same level of the student. They would seem unprepared when a student asks a question that is beyond the scope of that current lesson or a question that demands more than a &amp;quot;Wikipedia-level&amp;quot; familiarity of the subject.&lt;/p&gt;&#xA;&lt;p&gt;Then again, take the other extreme where one could be a PhD in number theory but teaching addition to kindergarten. Let's assume that this person is also good with young kids and is effective with primary school education, I think it's safe to say that being so advanced is overkill.&lt;/p&gt;&#xA;&lt;p&gt;So if we say the teacher is more skilled in the topic than the student, the question becomes &lt;strong&gt;how much more skilled?&lt;/strong&gt; For example, could a pre-calculus teacher effectively teach the subject thoroughly without not knowing measure theory? It seems very possible (and is probably overwhelmingly common). In fact, they could probably not know analysis and still teach well. But could a pre-calculus teacher not know calculus?&lt;/p&gt;&#xA;&lt;p&gt;If we consider post-secondary teachers at colleges and universities as standard, they generally have a PhDs and they often teach 1st year courses. In the spectrum of Baccalaureate-Masters-Doctorate, then it seems that it's 2 &amp;quot;degrees&amp;quot; ahead is what's necessary?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-07-14T00:16:58.183" LastActivityDate="2020-07-19T02:16:38.850" Title="How much more skilled in the topic should you be in order to teach the topic?" Tags="|undergraduate-education|secondary-education|graduate-education|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18564" PostTypeId="2" ParentId="18563" CreationDate="2020-07-14T01:54:36.260" Score="6" Body="&lt;p&gt;I teach at community college. That means I have a masters degree. When I first taught statistics, I had never taken a statistics course. Luckily, it was before I adopted my son. I spent 60 hour weeks, studying from many different textbooks, and asking myself questions. (I proved some results that we would never prove in our stats class.) I was an ok teacher that first semester. (I never got great, not because of my limited content knowledge, but because of my limited enthusiasm for stats. You need to keep coming up with interesting data sets. Not fun for me.)&lt;/p&gt;&#xA;&lt;p&gt;When I teach Calc II, it would help if I were also teaching Calc III, so I could talk more about how the topics continue on. (I have never taught Calc III. I once learned it. But that was long ago.)&lt;/p&gt;&#xA;&lt;p&gt;To me, it's not so much what degrees you have, as how deep your understanding of the content of a course is, along with what it connects to in following courses. I'm not sure why you're asking this question, so I'm not sure how useful my answer is.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-07-14T01:54:36.260" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18565" PostTypeId="2" ParentId="17723" CreationDate="2020-07-14T03:56:09.427" Score="1" Body="&lt;p&gt;I think that Mathologer's explanation does not require calculus.  He uses the concept of limits and relies on geometric interpretation of complex multiplication.  It is the simplest explanation that I am aware of.&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=-dhHrg-KbJ0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=-dhHrg-KbJ0&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14292" LastActivityDate="2020-07-14T03:56:09.427" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18567" PostTypeId="1" CreationDate="2020-07-15T10:20:10.417" Score="21" ViewCount="6923" Body="&lt;p&gt;I am pursuing mathematics through distance education and I find that it takes me a long time to understand the concepts (e.g. sigma fields, measure theory, connected topological spaces, etc.).&lt;/p&gt;&#xA;&lt;p&gt;After I have understood the theorems, I practiced them one at a time writing them all. This already takes a lot of time, then when I visit to the earliest learned concept, I realize that I have forgotten the theorems (e.g. I have forgotten the Sylow theorems that I learned two weeks ago). There's just too much to learn and I seem to be in a circular loop.&lt;/p&gt;&#xA;" OwnerDisplayName="user14243" LastEditorUserId="376" LastEditDate="2020-07-15T11:11:00.787" LastActivityDate="2021-05-02T08:39:49.397" Title="How do I learn advanced mathematics without forgetting?" Tags="|self-learning|graduate-education|advising|" AnswerCount="8" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="18568" PostTypeId="2" ParentId="18567" CreationDate="2020-07-15T15:39:29.493" Score="2" Body="&lt;p&gt;I think it is important to do some drill EVEN (maybe especially) with advanced concepts.  This is because the concepts may be more strange and abstract.  So you need to do some basic work to get familiarity with it.  Quantum mechanics (or E&amp;amp;M) are rather non-intuitive and you just need to work with them.  Not everything is as much an &amp;quot;aha&amp;quot; realization as kinematics (or differentiating polynomials).&lt;/p&gt;&#xA;&lt;p&gt;From the question (&amp;quot;read&amp;quot;), it already sounds like you're concentrating too much on a passive, not active approach.  In addition, sometimes advanced topics are made harder (even then they need to be), because less pedagogical effort is put in by the writers (since it is for such a small audience).  So you get harder and less problems and less progressive (step by step) learning.  This is falsely talked about as &amp;quot;you're in uni now&amp;quot; or &amp;quot;you're in grad school, now&amp;quot;.  But really is a laziness on the TEACHING side of textbook writers (not the MATH side), by taking less effort.  Pedagogy is even harder than math.  An extreme example is writers of monographs who don't have any problems.  Basically they are teaching a universe of one, themselves, when writing the monographs.  [Many math Wiki writers are like this also...not really trying to teach, but to clarify for themselves!]&lt;/p&gt;&#xA;&lt;p&gt;What you need to do when you encounter these types of materials or courses is to (at a minimum) do all the problems they contain.  And also reach out for alternate sources.&lt;/p&gt;&#xA;&lt;p&gt;Even someone like Andrew Wiles, working with research papers and his own ideas, still felt the need to do little by-hand examples to build familiarity with what he was proposing!  Watch the Singh video.&lt;/p&gt;&#xA;&lt;p&gt;I would also caution you, on the practical side, that some level of forgetting and relearning is to be expected and may even have some positive aspects.  Just keep the reference materials next to you and go back to what you did before.  It will come back much quicker since you saw it before.  And you will know it better and retain it longer.&lt;/p&gt;&#xA;" OwnerUserId="14301" LastActivityDate="2020-07-15T15:39:29.493" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18569" PostTypeId="2" ParentId="18567" CreationDate="2020-07-15T15:49:08.577" Score="20" Body="&lt;p&gt;Memorization per se should not be the primary focus. When I learn something new, I type up notes on the computer in a reverse-indented outline format. Then as time goes on and my understanding improves, I edit the notes to reflect that. When I work an exercise or read a paper, I refer back to my notes. Memorization, to the extent that it happens, is just a side-effect of this process. If a school, teacher, or educational system puts too much emphasis on memorization in math classes, it's a sign of low quality. If you're stuck in such a class, and, e.g., must take closed-notes tests, then just think of that as a side issue and allocate some time the day before an exam to commit some things to short-term memory.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After I have understood the theorems, I practiced them one at a time writing them all.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm not sure here if you mean the statements of the theorems or their proofs, and whether writing means copying them down verbatim out of the book or reexpressing them in your own words. But basically this sounds like rote memorization, which is a bad idea.&lt;/p&gt;&#xA;&lt;p&gt;I tell my students that when they study a proof, they should mainly focus on its inputs and outputs, and try to understand what goes wrong when you try to change those. For example, Newton proved Kepler's laws mathematically. A good exercise for someone trying to understand a presentation of such a proof is to think about what happens if gravity is changed from a &lt;span class=&quot;math-container&quot;&gt;$1/r^2$&lt;/span&gt; force to some other exponent like &lt;span class=&quot;math-container&quot;&gt;$1/r^7$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-07-15T15:57:56.340" LastActivityDate="2020-07-15T15:57:56.340" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18570" PostTypeId="2" ParentId="18567" CreationDate="2020-07-15T17:39:52.947" Score="16" Body="&lt;p&gt;For context, I have a lot of experience self-learning mathematics. I spent a summer learning additional algebra, point-set topology, linear algebra, and analysis (to extend my undergraduate degree) before entering my current graduate program. This was sufficient to skip a literal year in the program. I am now well into the program and have to self-teach constantly, as very few people who know the material I am learning are available to tell me about it.&lt;/p&gt;&#xA;&lt;p&gt;From your description, you are not doing any activities that lead to true understanding or retention. That is, learning mathematics requires active thought, and you are passively writing and rewriting the same theorems. This does not lead to the processing that is necessary to retain them, and will ultimately lead to you taking more time to progress.&lt;/p&gt;&#xA;&lt;p&gt;Do the texts you are working from have exercises? If so, you should do a representative selection of exercises from each chapter before moving to the next. Does the text you are working from give summaries of proofs before or instead of the full detail of the proof? You should see if you can work out the full details of each proof before reading it, given a summary. Are you certain that your solutions are complete, not neglecting any details? If not, you should post your attempted solutions online on, say, math.stackexchange or r/learnmath for feedback.&lt;/p&gt;&#xA;&lt;p&gt;In general, effective learning cannot occur when the only thing you do is absorb information. You must practice using that information--memorization becomes the side effect. I recommend &lt;a href=&quot;https://www.learningscientists.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.learningscientists.org/&lt;/a&gt; as a resource to learn how to effectively learn. They suggest 6 effective strategies for learning. The most important tools in their toolbox, adapted to your situation, are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Recall practice. This it appears you are doing, but perhaps too late. You should attempt to recall any new information you learn at multiple stages. For example, say you just learned about the Sylow theorems. As soon as you write down those theorems, close the book and see if you can recall the conditions and conclusions of the theorems, as well as why they &amp;quot;make sense.&amp;quot; If you cannot, then you can't expect to remember them much later either. Then, while doing the exercises from the text, aim first to do them without any reference to the text. Only after some struggle should you permit yourself access to the text. This forces you to acknowledge what material you do not currently remember, and helps it stick in your memory. For direct recall of the theorems, you can make yourself simple decks in a program like Anki, which takes little time each day.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Spaced practice: You should aim to return to each topic a week later, and then a month later. This can be done by setting aside one day each week to be a &amp;quot;review the previous week&amp;quot; study day, where you pick exercises you skipped from each section the first go around. Then one day each month can be a &amp;quot;review the previous month&amp;quot; study day, where you do a similar thing. (Notice: This should not only be repetition of the theorems, but applying those theorems in new problems.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Elaborative practice: Any time you learn a new theorem, you should try to explain to yourself or someone else (even a &amp;quot;rubber duck&amp;quot;) how it fits in to the old material. Why do you care about it? What questions does it answer? What questions does it not answer?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Concrete examples: Any time you learn a definition, before learning anything else, you should try to come up with an example of a mathematical object that fits that definition, and one which does not. See if you can justify why.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Dual Coding: If you are learning an algebraic fact, can you come up with a visual example of that fact? For example, if you are learning about the group D_2n, can you both give algebraic rules for manipulating that group and have a visual example of an object with D_2n as its symmetry group? Where do the elements of the center of D_{2n} show up in terms of the visuals of the symmetry group? If you are asked to come up with the Sylow-2 subgroup(s) of D_{12} and, perhaps, their conjugacy classes in an exercise, can you determine what they are both algebraically and within your visual model?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;These strategies do necessarily take more time than simple reading, and your learning will be &amp;quot;slower&amp;quot; as a result -- but you will retain more of the old information. Placing understanding over speed as your primary goal will ultimately end up with you taking longer in the short run, but you will learn and retain faster over time, because you will have a stronger base to go back to.&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2020-08-22T00:36:23.657" LastActivityDate="2020-08-22T00:36:23.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18571" PostTypeId="2" ParentId="15278" CreationDate="2020-07-16T01:44:19.017" Score="1" Body="&lt;p&gt;Admittedly, a fraction has twice as many components as (say) an integer, and these components greatly increase the numbers of ways they might interact, in a combinatorial fashion. Consider an arbitrary binary operation on integers: &lt;span class=&quot;math-container&quot;&gt;$a \odot b$&lt;/span&gt;. With only two components to the arguments, there is only a single relation that needs consideration: the one between &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. On the other hand, consider fractions:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac a b \odot \frac c d$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this case, there are actually &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; relations that might need consideration: &lt;span class=&quot;math-container&quot;&gt;$(a, b), (a, c), (a, d), (b, c), (b, d), (c, d)$&lt;/span&gt;. Sextupling the cognitive load on the student is quite a leap forward. Of course, the relations that need consideration/manipulation are different for the various different operations of addition, subtraction, multiplication, division, etc., so there's a lot to digest and memorize there. (This is to say nothing of mixed numbers, with 6 components, for a total of 15 possible relations.)&lt;/p&gt;&#xA;&lt;p&gt;Consider the operation of addition on proper fractions: In most school situations this involves (1) factoring the denominators &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, (2) composing the least common denominator by inspecting those factors, (3) multiplying each component &lt;span class=&quot;math-container&quot;&gt;$a, b, c, d$&lt;/span&gt; appropriately to create the desired denominator in each fraction, (4) adding the modified numerators and writing over the common denominator, (5) factoring the new numerator again, and (6) possibly canceling like factors in the numerator and denominator (and then multiplying the remaining ones back together to simplify).&lt;/p&gt;&#xA;&lt;p&gt;So in summary: It seems by my estimation that the conceptual and mechanical work in handling fractions is about &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; times that of integers. Granted that &lt;a href=&quot;https://en.wikipedia.org/wiki/Working_memory#Capacity&quot; rel=&quot;nofollow noreferrer&quot;&gt;some theories of working memory&lt;/a&gt; suggest a capacity of perhaps 4 or 7 chunks (and less in small children), this may in fact be at or beyond the limit of many children's short-term memory to manage.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-07-17T01:52:25.260" LastActivityDate="2020-07-17T01:52:25.260" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18573" PostTypeId="2" ParentId="18477" CreationDate="2020-07-16T07:39:51.660" Score="3" Body="&lt;p&gt;The main issue with using proctoring software is that some students perceive this as being required to install spyware on their computer with all the features of tracking keystrokes etc., and when there is a recording of the exam session there are questions of where these  recordings are stored and who has access to them.&lt;/p&gt;&#xA;&lt;p&gt;At my institution we are moving toward live proctoring over Zoom without recording. However, proctoring through a front-facing camera is not very effective because students can easily communicate with others or access unauthorized materials on their other devices on their desktop if we cannot see their hands. For this reason, our preferred method for proctoring will be through the Zoom mobile app where the students sets up their cell phone as a side camera, clearly showing their workspace and with their computer screen tilted slightly toward the camera. Some schools are already doing this and have published detailed guidelines, including: Texas A&amp;amp;M University, Hong Kong University of Science &amp;amp; Technology, and KTH Royal Institute of Technology in Sweden. We expect that we may need to accommodate a small minority of students that cannot create a suitable environment for taking an exam at home in this way.&lt;/p&gt;&#xA;" OwnerUserId="14308" LastActivityDate="2020-07-16T07:39:51.660" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18575" PostTypeId="2" ParentId="18567" CreationDate="2020-07-17T09:30:39.697" Score="1" Body="&lt;p&gt;I believe you should follow these steps:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Understand what the theorem says, with some applications of it (which also means to do the exercises, also show your work to TAs)&lt;/li&gt;&#xA;&lt;li&gt;Try to prove it yourself and get stuck quickly&lt;/li&gt;&#xA;&lt;li&gt;Work through the proof and try to understand it&lt;/li&gt;&#xA;&lt;li&gt;Take a fresh sheet of paper and now try again to do the proof&lt;/li&gt;&#xA;&lt;li&gt;When you get stuck (which you will again) then read again (and repeat these steps)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;When working out a proof like this you usually notice that there are like one or two 'big' ideas at the kernel of the proof. Like in prooving (there are different ones of course) that from f'=g' follows f=g+c you use the mean value theorem. Now make a note next to the theorem which acts as a reminder what was the part at the kernel and not the technical parts which you can easily make up later.&lt;/p&gt;&#xA;&lt;p&gt;These ideas you need to remember, and this helps you to remember the applicability. Like in the example if you know you need to use mean value theorem it is immediately clear that the domains of the functions have to be an interval.&lt;/p&gt;&#xA;&lt;p&gt;Of course this is a lot of work. You do not do this in the first reading, only for some theorems. You should plan to go through the whole material at least 3 times (over the course of 6 month)&lt;/p&gt;&#xA;&lt;p&gt;TL;DR dont learn the theorem, learn the proofs&lt;/p&gt;&#xA;" OwnerUserId="11794" LastActivityDate="2020-07-17T09:30:39.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18576" PostTypeId="1" AcceptedAnswerId="18599" CreationDate="2020-07-17T13:40:25.590" Score="2" ViewCount="194" Body="&lt;p&gt;Pls I wonder what Ramanujan's results could be explained to middle school level audience, ie without using integral etc that is up to university curriculum?&lt;/p&gt;&#xA;&lt;p&gt;For example &lt;a href=&quot;https://en.wikipedia.org/wiki/Nested_radical#Ramanujan%27s_infinite_radicals&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ramanujan's infinite radicals&lt;/a&gt; could be explained easily&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 3=\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="9059" LastActivityDate="2020-07-22T12:59:25.520" Title="Ramanujan results for middle school?" Tags="|middle-school|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18577" PostTypeId="2" ParentId="18567" CreationDate="2020-07-17T15:27:24.073" Score="3" Body="&lt;p&gt;You don't describe engaging in conceptual &lt;a href=&quot;https://en.wikipedia.org/wiki/Chunking_(psychology)&quot; rel=&quot;nofollow noreferrer&quot;&gt;chunking&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You mention the Sylow theorems as an example and I will also use them as an example.  It is easy enough to memorize the theorem statements.  If you do not chunk the material in the proofs, then I challenge any claim that you understand the proofs.&lt;/p&gt;&#xA;&lt;p&gt;(Fix a positive prime integer, &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.)  The Sylow theorems examine the left-multiplication action of a group on its &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-power-sub&lt;em&gt;sets&lt;/em&gt; (establishing existence of Sylow &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-subgroups), the left-multiplication action of a group on its &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-subgroups (establishing uniqueness of the conjugacy class of Sylow &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-subgroups), and the conjugation actions of a group and its Sylow &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-subgroups on its Sylow &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-subgroups (to obtain counting/combinatorial data).&lt;/p&gt;&#xA;&lt;p&gt;That pattern -- existence, uniqueness, data/algorithm -- is so pervasive you should notice it quickly.  But it is invisible if you do not chunk the information in the theorems/proofs.  (This sequence of proof types is an example of one of the frequently repeated &amp;quot;tools&amp;quot; that I discuss further below.)&lt;/p&gt;&#xA;&lt;p&gt;Chunking is an abstraction process -- If we &amp;quot;hand wave&amp;quot; the lowest level details of this particular paragraph of a proof, express what this paragraph accomplishes in a single sentence.  (Note that &amp;quot;paragraph&amp;quot; should not be taken &lt;strong&gt;too&lt;/strong&gt; literally.  Sometimes a single mathematical idea projects onto several textual paragraphs.  Sometimes a &amp;quot;stream of consciousness&amp;quot; projects several ideas into a single textual paragraph.  The point is to capture the ideas one level of abstraction higher than that of the details.)  You are, in essence, recreating an outline of the proof process, starting only with the leaves of the tree structure.  This compresses a detailed proof into a shorter sequence of abstractions that are easier to remember.&lt;/p&gt;&#xA;&lt;p&gt;What you will find is that the same chunks appear in multiple proofs.  The process of chunking associates an idea with the detailed implementation of that idea.  With repetition, you should find your understanding of a step of a proof toggling between &amp;quot;this block of details looks very like 'let group act on its Sylow &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-subgroups'&amp;quot; and &amp;quot;let group act on its Sylow &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-subgroup&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;There is a next layer of chunking that comes with breadth of reading.  Continuing the example, when discussing [special subobjects] of [object], one may see &amp;quot;this block of details looks very like 'let [object] act on its [special subobjects under discussion]'&amp;quot;.  You eventually recognize the common theme: to understand some special subobjects, let the object act on them.  Now you have gone from a particular method to tease structure out of Sylow &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-subgroups to a general method that you have learned to apply to understand future [special subobjects].  The chunking has taken you from&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;an implementation of a tool applied to a specific subject&lt;/li&gt;&#xA;&lt;li&gt;to abstraction where you call out and name the tool&lt;/li&gt;&#xA;&lt;li&gt;to the tool (detached from its initial context).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Once you have the tool and have seen it used in a few contexts, your &amp;quot;reach&amp;quot; expands through the use of the tool.  Many proofs, in some sense, are tutorials on using such a tool.  Chunking is a method of recognizing a tool.  Like any skill, this takes practice.  When starting out you may need a few examples so that a tool's repeated use becomes obvious.  With practice, you will chunk at various levels of abstraction almost automatically, so you may recognize the use of a new tool the first time you see it.&lt;/p&gt;&#xA;" OwnerUserId="5800" LastActivityDate="2020-07-17T15:27:24.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18578" PostTypeId="1" CreationDate="2020-07-17T15:32:17.713" Score="6" ViewCount="271" Body="&lt;p&gt;I'm looking for any references that exist for what a good video length for an online math class should be.&lt;/p&gt;&#xA;&lt;p&gt;I am aware of these three papers but these are basically only for MOOCs - I'm looking for something in a mathematics class that's required - Ideally measuring long term learning but at this point anything is welcomed.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Effective educational videos By Cynthia J. Brame, CFT&#xA;&lt;a href=&quot;https://www.lifescied.org/doi/10.1187/cbe.16-03-0125&quot; rel=&quot;noreferrer&quot;&gt;https://www.lifescied.org/doi/10.1187/cbe.16-03-0125&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How Video Production Affects Student Engagement:An Empirical Study of MOOC&#xA;Video - Guo et al., 2014&#xA;&lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.714.195&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;noreferrer&quot;&gt;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.714.195&amp;amp;rep=rep1&amp;amp;type=pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Understanding In-Video Dropouts and Interaction Peaks inOnline Lecture Videos&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Guo et al., 2014&#xA;&lt;a href=&quot;https://dspace.mit.edu/bitstream/handle/1721.1/90413/Miller_Understanding%20in-video.pdf?sequence=1&amp;amp;isAllowed=y&quot; rel=&quot;noreferrer&quot;&gt;https://dspace.mit.edu/bitstream/handle/1721.1/90413/Miller_Understanding%20in-video.pdf?sequence=1&amp;amp;isAllowed=y&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;EDIT: I'm looking at the university [undergrad particularly] level. Yes I agree that static videos usually aren't best for learning (but say for algorithm demonstration; I find a video to be vastly more effective and efficient than a long prose describing the algorithm as both a learner and as an educator).&lt;/p&gt;&#xA;" OwnerUserId="12000" LastEditorUserId="5321" LastEditDate="2020-07-19T17:02:57.333" LastActivityDate="2020-07-22T00:22:06.713" Title="Research for Video Length for Math Videos" Tags="|undergraduate-education|videos|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18579" PostTypeId="2" ParentId="18567" CreationDate="2020-07-18T02:23:51.243" Score="4" Body="&lt;p&gt;For most people, including myself, there is pretty much no chance that you’ll remember most of the things you’ll see and do in math-land the first time ‘round. It’s just an inevitable relic of the fact that mathematics is... well... hard, and it isn’t something that the human brain was designed to do. The fact that you’ve gotten as far as you have is an accomplishment you should be proud of. You’ve already bridged the statistical gap!&lt;/p&gt;&#xA;&lt;p&gt;But, as you’ve clearly realized, mathematics is easily the farthest thing from a spectator sport. It gets easier, but there are always going to be challenges, that’s part of the fun. Anyway, back to the point: Math is hard to learn, and even harder to remember. How do you do it right? Take good notes!&lt;/p&gt;&#xA;&lt;p&gt;I can’t stress how important this really is. I take notes as if I was to forget everything the next day, because chances are I will. The better the notes, the more of the material will fall into place every time you go over it. Every time you go over stuff, add your immediate conclusions and thoughts. In this sense, you will be building up a better and better “imprint” of your own brain’s critical response to the material.&lt;/p&gt;&#xA;&lt;p&gt;Frame it in your own language, draw pictures, whatever you need to do, just make sure you’ve done everything correctly and that you haven’t sacrificed any of the concept in exchange for intuition. Intuition is great, but not for definitions. The whole point of math is to create a more functional, tangible language to describe things, and watch how the intuition unfolds. Not the other way around.&lt;/p&gt;&#xA;&lt;p&gt;However, us humans aren’t perfect, optimal and all-knowing holders of ultimate truth and beauty, we have the slightly more disappointing form of a messy set of poorly-calibrated neural networks that are really only capable of expanding conceptually on the relatively basic patterns that only arise after many years of consistent experiences. So the only reason we have wildly brilliant people like Dirac, Einstein, and Ramanujan among us is the fact that they were masters of framing mathematics, more-or-less, in terms of basic experience and intuition. So, as cheesy as I know this is going to sound, “The answer lies within YOU.” Only you know what you understand better than anything else. So, whatever that thing is for you, find it, because it’s going to be the most effective thing you’ll ever be able to use as an analogy.&lt;/p&gt;&#xA;&lt;p&gt;TL;DR: Mathematics is a beautiful subject, and the best way to learn it is to relate it to what you find to be your own core experiences.&lt;/p&gt;&#xA;" OwnerUserId="14319" LastEditorUserId="14319" LastEditDate="2020-07-18T16:27:54.410" LastActivityDate="2020-07-18T16:27:54.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18580" PostTypeId="1" CreationDate="2020-07-18T14:10:38.250" Score="2" ViewCount="1326" Body="&lt;p&gt;At what age are most children able/taught to convert between rational fractions and decimals?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;For example&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Convert 0.25 to a fraction consisting only of whole numbers.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is 3/4 expressed in decimal notation?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Notes&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Please answer with an age in &lt;em&gt;years&lt;/em&gt; rather than a country-specific measure such as &amp;quot;Grade N&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm interested in 'most' children rather than the exceptional prodigy.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;At what age (in relation to teaching an &amp;quot;average&amp;quot; class) is it realistic to expect young people to have the mathematical maturity to understand and carry out such computations?&lt;/p&gt;&#xA;" OwnerUserId="5798" LastActivityDate="2023-08-17T21:28:08.810" Title="At what age are most children able to convert between rational fractions and decimals?" Tags="|primary-education|arithmetic-operations|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18581" PostTypeId="2" ParentId="18580" CreationDate="2020-07-18T16:15:59.473" Score="2" Body="&lt;p&gt;This is a US Common Core Standards based answer.  The standards are grades based, because new knowledge builds on old, and that old knowledge is presumed to come from earlier grades.  In the US, most states do not &lt;em&gt;require&lt;/em&gt; children to enter until they are 6 years old (and many states don't require entrance until a child is 7), but children &lt;em&gt;usually&lt;/em&gt; start kindergarten at age 5.  Thus, in principle, any particular class of students might have an age range of 3–4 years.  Hence pinning down ages is a more difficult task than pinning down grades.  I've given approximate ages in my answer, under the assumption that a child starts kindergarten at age 5.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In the 3rd grade (approximately ages 8–9) the &lt;a href=&quot;http://www.corestandards.org/Math/Content/3/NF/&quot; rel=&quot;nofollow noreferrer&quot;&gt;standard&lt;/a&gt; suggests that students should be able to&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In 4th grade (approximately ages 9–10), the &lt;a href=&quot;http://www.corestandards.org/Math/Content/4/NF/&quot; rel=&quot;nofollow noreferrer&quot;&gt;standard&lt;/a&gt; suggests that students should be able to&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In the 8th grade (approximately ages 13–14), the &lt;a href=&quot;http://www.corestandards.org/Math/Content/8/NS/&quot; rel=&quot;nofollow noreferrer&quot;&gt;standard&lt;/a&gt; suggests that students should be able to&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The standards don't seem to ever demand that students fluently convert back and forth between fraction and decimal representations.  Instead, the standards mostly emphasize more general ideas of &amp;quot;number sense&amp;quot; (compare fractions, plot things on number lines, etc.).  Given easy access to technology, I don't find this surprising.&lt;/p&gt;&#xA;&lt;p&gt;In any event, based on these standards, it seems that an answer to your question is that students should be able to start familiarizing themselves with fractions and decimals around age 10, and that they should probably be maximally fluent (to the extent demanded by the standards) by around age 15.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2020-07-19T00:48:31.713" LastActivityDate="2020-07-19T00:48:31.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18582" PostTypeId="2" ParentId="18563" CreationDate="2020-07-18T19:19:10.923" Score="4" Body="&lt;p&gt;There is no generally applicable answer to your question.&lt;/p&gt;&#xA;&lt;p&gt;In my own career I often taught &lt;em&gt;while&lt;/em&gt; learning the material as my mathematics department began to offer more and more computer science. I learned the material a course at a time by signing up to teach it. I talked a lot with colleague mentors who knew the material.&lt;/p&gt;&#xA;&lt;p&gt;I always told my students in the first class of the term that we would be learning together. I think some weaker students thought about writing the governor to complain that their professor was unprepared.  The better ones knew we would have an adventure. On the first exam in my assembly language programming course one student earned an A and my grade was just A-. He had better answers to my questions than I did. We are still friends.&lt;/p&gt;&#xA;&lt;p&gt;Although I did not &amp;quot;know the material&amp;quot; my professional experience as a mathematician and a teacher helped me have a good sense of what was important, where the difficult ideas would occur, how to learn them myself, and how to stay far enough ahead of the class as the semester went on.&lt;/p&gt;&#xA;&lt;p&gt;At the other extreme of &amp;quot;how much more should the teacher know&amp;quot; the answer for elementary school teachers is &amp;quot;a lot more&amp;quot;. I have spent time in elementary schools helping the teachers see how the material they are talking about in class fits into a much more sophisticated abstract framework they may never have been exposed to. The good teachers are thrilled to see that.&lt;/p&gt;&#xA;" OwnerUserId="4984" LastEditorUserId="8571" LastEditDate="2020-07-19T02:16:38.850" LastActivityDate="2020-07-19T02:16:38.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18583" PostTypeId="2" ParentId="18580" CreationDate="2020-07-19T07:41:14.133" Score="3" Body="&lt;p&gt;There is no single answer because there are several semi-overlapping methods to convert fractions and decimals. Depending on the numbers, they require drastically varying amounts of prior knowledge.&lt;/p&gt;&#xA;&lt;p&gt;There are &lt;em&gt;orders of magnitude more nuance here than meets the eye&lt;/em&gt;, making this an area where many educators, materials, and educational systems make curse of knowledge mistakes, both with and without awareness. So prepare for an epic post. It's gonna be loooooong.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;But mastering the pedagogy here strikes right at the heart of why so many people hate math&lt;/em&gt;. If you need to cut back on time spent on surface area or bar graphs to do this, it's a worthy trade off. Frankly, f$%@ curriculum standards if you have to - mastering fraction, decimal, and percent conversions is far more important for the kids' futures.&lt;/p&gt;&#xA;&lt;h1&gt;Part 1: Converting Fractions to Decimals&lt;/h1&gt;&#xA;&lt;p&gt;Let's make the &lt;strong&gt;wildly optimistic assumption&lt;/strong&gt; that the students already know that a fraction is a number and can intelligently compare and contrast fractional numbers with whole numbers. Only then should you bother with decimal conversions. Otherwise you will really pound into their head that math is just a bunch of weird rules to memorize. This assumption alone is worth many threads. :-)&lt;/p&gt;&#xA;&lt;p&gt;But with that assumption in place:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Method 1: Apply Base 10 Conventions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{100}=0.03$&lt;/span&gt; because the 3 is in the hundredths place. Though this is is trivial to memorize, it is actually quite difficult for students to internalize that this is just a logical extension of the base 10 number system, but, unlike their prior knowledge, we're scaling &lt;em&gt;down&lt;/em&gt; here. Sadly, since many students, do not arrive in their fractions and decimal units understanding base numbers, all they can do is memorize this. In turn, that makes virtually all aspects of rational numbers extremely difficult to learn afterward.&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$\frac{36}{100}=\frac{30}{100}+\frac{6}{100}=\frac{3}{10}+\frac{6}{100}=0.36$&lt;/span&gt; because the 3 is in the tenths place and the 6 is in the hundredths place. This requires an understanding of base 10 and the ability to see how tenths and hundredths correspond  with number lines or 10x10 grids. They don't &lt;em&gt;have&lt;/em&gt; to have conceptualized equivalent fractions yet, but that's ideal.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Method 2: Use Equivalent Decimal Fractions Then Base 10 Conventions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{5}=\frac{6}{10}=0.6$&lt;/span&gt;. Note that this requires all the previous knowledge mentioned above and then some.&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$\frac{7}{5}=\frac{14}{10}=1\frac{4}{10}=1.4$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$\frac{7}{5}=1\frac{2}{5}=1\frac{4}{10}=1.4$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$\frac{7}{250}=\frac{28}{1000}=\frac{20}{1000}+\frac{8}{100}=\frac{2}{100}+\frac{8}{1000}=0.028$&lt;/span&gt;. Notice the number of steps required to do this rigorously and correctly. Also since it's kinda hard to see visually how two-hundred-and-fiftieths and thousandths correspond, or even hundredths and thousandths, students must have conceptualized equivalent fractions to do this.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Method 3: Interpret Fractions as Quotients of Integers&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{3}=8\div3=2.\bar{6}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{3}=2\frac{2}{3}=2+2\div3=2+0.\bar{6}=2.\bar{6}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$\frac{12}{3}=12\div3=4$&lt;/span&gt; [No decimal required! Remember to mix in plenty of whole numbers in ALL activities with students learning fractions and decimals as it is the &lt;em&gt;only&lt;/em&gt; way to ensure integration of new knowledge (rational numbers) with prior knowledge (whole numbers).]&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$\frac{16}{32}=16\div32=0.5$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Notice that Method 3 &lt;em&gt;assumes&lt;/em&gt; knowledge of fractions as a quotient. Tragically, this is usually just papered over in math classes and, for many students, becomes just another weird property to memorize. I'd say it's worth spending 3+ hours with students developing their intuition on why fractions can &lt;em&gt;always&lt;/em&gt; be considered a quotient and relating that to their prior knowledge of division. A context to motivate this is also useful. &amp;quot;In a medical study, &lt;span class=&quot;math-container&quot;&gt;$\frac{379}{581}$&lt;/span&gt; patients survived after taking a pill. &lt;span class=&quot;math-container&quot;&gt;$\frac{340}{521} $&lt;/span&gt; patients survived after surgery. Which treatment has the higher survival rate? How do you know? Can you just use number sense and intuition to answer this? Or other methods of comparing fractions? Explain.&amp;quot;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;After learning the methods in that order, students should then be required to:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Generate examples of fractions for which each method is most appropriate or to which multiple methods could apply, and carry out the conversions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Justify their reasoning with sketches of area models, number lines, and explicit references to base 10.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h1&gt;Part 2: Converting from Decimals to Fractions&lt;/h1&gt;&#xA;&lt;p&gt;If the students understand Method 1 above, then converting from terminating decimals to fractions is trivial and there's really only one method and its shortcut.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Method 4: Using Base 10 Conventions &amp;quot;Backward&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example using base 10 conventions: &lt;span class=&quot;math-container&quot;&gt;$1.309 = 1 + \frac{3}{10} + \frac{0}{100} + \frac{9}{1000} = 1 + \frac{300}{1000} + \frac{0}{1000} + \frac{9}{1000} = 1\frac{309}{1000}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example using shortcut: &lt;span class=&quot;math-container&quot;&gt;$1.309=?$&lt;/span&gt; &amp;quot;There are 3 digits after the decimal, so three zeroes in the denominator, so 1000.&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$1.309 = 1\frac{309}{1000}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Students should then demonstrate further conceptualization of scaling down with base 10 by converting fractions such as, say, &lt;span class=&quot;math-container&quot;&gt;$23.00901460$&lt;/span&gt; to fraction notation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Method 5: &amp;quot;Easy&amp;quot; Repeating Decimals&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Converting from repeating decimals to fractions is drastically harder and I'm unaware of any way to do this without either memorization of common repeating decimals OR the use of algebra.&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$0.\bar{6} = ?$&lt;/span&gt; Let: &lt;span class=&quot;math-container&quot;&gt;$0.\bar{6}=x$&lt;/span&gt; Then, we use 9 since there is only one digit that repeats and it's in the tenths column: &lt;span class=&quot;math-container&quot;&gt;$9x=6$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x=\frac{6}{9}=\frac{2}{3}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Method 6: &amp;quot;Hard&amp;quot; Repeating Decimals&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$0.0\overline{142857} = ?$&lt;/span&gt; Let: &lt;span class=&quot;math-container&quot;&gt;$0.0\overline{142857} = x$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$10x=0.\overline{142857}=\frac{1}{7}$&lt;/span&gt; (as recalled from memory) so  &lt;span class=&quot;math-container&quot;&gt;$x=\frac{1}{7}\div10=\frac{1}{70}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$0.0\overline{142857} = ?$&lt;/span&gt; Let: &lt;span class=&quot;math-container&quot;&gt;$0.0\overline{142857} = x$&lt;/span&gt;. Then, multiply &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by some number and... ugh, I hope kids aren't forced to do this &lt;em&gt;too often&lt;/em&gt;. In my opinion, understanding - and not just execution of - the long division algorithm is enough to prove that all repeating decimals must have an exact simple fraction representation.&lt;/p&gt;&#xA;&lt;p&gt;The strongest students will understand this from the age of maybe 12 onward, but many high school graduates won't really get this at all. Prerequisite knowledge of fraction arithmetic, base 10, long division, and algebra is so extremely varied that I don't see how anybody can know when an average class is ready to begin studying this.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So, perhaps a better question to ask is &amp;quot;At which age do average students have the prior knowledge in place so that they &lt;em&gt;can&lt;/em&gt; learn each of those methods?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;My answer to that would be:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;8 or 9 years old for Methods 1 and 4&lt;/li&gt;&#xA;&lt;li&gt;10 or 11 for Method 2&lt;/li&gt;&#xA;&lt;li&gt;12+ for Method 3&lt;/li&gt;&#xA;&lt;li&gt;13+ and a very good teacher for remaining methods&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;That being said, in a truly spectacular school system, all of this should be mastered by all students by the age of 12.&lt;/p&gt;&#xA;&lt;p&gt;One can only dream. :P&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2020-07-19T07:54:33.950" LastActivityDate="2020-07-19T07:54:33.950" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18584" PostTypeId="2" ParentId="18580" CreationDate="2020-07-19T14:18:11.353" Score="1" Body="&lt;p&gt;Translation: US 4th grade is ~10yo. US fifth is ~11yo.  (not at the year start, but sometime within it).  Again, that's averages and slightly to the high side.  Few kids slightly younger (birthday in the summer, not hitting the mark until summer after the grade.)&lt;/p&gt;&#xA;&lt;p&gt;My personal experience in US public school was that 5th grade was the standard spot for decimals.  And I don't recall breaking it up by years (like learning some decimals in 4th and then the rest later as Common Core seems to say).  Fractions are started earlier, although you don't get the whole topic (e.g. cross multiply to divide) in earlier grades.&lt;/p&gt;&#xA;&lt;p&gt;FWIW, my experience teaching sports (with decimal timing of races) to children was that 4th graders almost always did not understand decimal representation and that kids 5th grade (say 50% through year) and above were fluent with them.  It was uncanny how this played out, with even relatively sharp kids, and with all the decimal exposure on TV, watches, etc.  But it was just an issue of them not being exposed to it, since decimals are traditionally a 5th grade topic.&lt;/p&gt;&#xA;&lt;p&gt;As far as translation from fractions to decimals, like the easy examples you gave, that would be in fifth as well.  Not saying average kid is invariably perfect.  Or that there aren't laggards.  But .25 to a quarter is completely reasonable expectation for an average kid halfway through 5th grade in USA.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I think this is actually reasonably close to CC standard (which just approximates normal practice anyhow).  And I wouldn't get too wrapped up about slight differences of CC standard versus practice.  The US really does not have a national curriculum like France.  But sort of a loose defacto pattern of progression.  [Why you will hear people talk about &amp;quot;Algebra 1&amp;quot; in 9th grade and the like.]&lt;/p&gt;&#xA;" OwnerUserId="14326" LastActivityDate="2020-07-19T14:18:11.353" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18585" PostTypeId="2" ParentId="18580" CreationDate="2020-07-19T17:01:41.050" Score="1" Body="&lt;p&gt;In my experience as a teacher, many children are asked to memorize certain equivalent fractions before they are taught the computation.  Some of these include:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$0.25$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1/4$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$0.75$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3/4$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$0.2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1/5$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$0.8$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4/5$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Student often memorize these and then memorize their percent equivalents.  In this way they get away without ever really being able to do the calculations required.  They may be taught how to do it, but for too many children, the computation never sticks.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-07-19T17:01:41.050" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18588" PostTypeId="1" AcceptedAnswerId="18593" CreationDate="2020-07-21T03:12:30.657" Score="36" ViewCount="9262" Body="&lt;p&gt;For example, if you're teaching integration of &lt;span class=&quot;math-container&quot;&gt;$\int \frac{dx}{1+x^2}$&lt;/span&gt;, would you mention the common wrong answer of &lt;span class=&quot;math-container&quot;&gt;$\ln\left(1+x^2\right)+C$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&lt;p&gt;For myself, I very rarely mention common mistakes since I'd feel I'm possibly causing the very problem I'd seek to avoid. But if I do mention issues, I will emphatically state before and after that it's the wrong approach (and often, I'll just write it on at the edge of the board and then promptly erase it so it's not written down by students)&lt;/p&gt;&#xA;&lt;p&gt;One of the reason why I avoid warning of common mistakes is because I can imagine how a student could see the intuitive appeal in the answer, even though the logic is flawed.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorUserId="13211" LastEditDate="2020-08-19T21:25:38.457" LastActivityDate="2023-03-18T23:20:10.840" Title="How can teachers warn students about common mistakes without causing the student to make the mistake?" Tags="|mathematical-pedagogy|examples|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18590" PostTypeId="2" ParentId="18588" CreationDate="2020-07-21T14:12:06.763" Score="9" Body="&lt;p&gt;I don't think you have anything to be afraid of here, unless you spend &lt;em&gt;all&lt;/em&gt; of your time playing what-if games with common pitfalls, or peppering the board with false statements and not clarifying them as such. As a class conversation about mistakes-to-avoid or a way to encourage students to always check their answers, I think bringing these up can be really helpful.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One of the reason why I avoid warning of common mistakes is because I can imagine how a student could see the intuitive appeal in the answer, even though the logic is flawed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These are common errors, whether you introduce them or not. You want to teach your students to apply some &lt;a href=&quot;https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow&quot; rel=&quot;noreferrer&quot;&gt;slow thinking&lt;/a&gt; to check an answer or analyze an argument, and having some examples ready that test their ability to do this on the fly could benefit them.&lt;/p&gt;&#xA;&lt;p&gt;For my part, I always end a conversation of this type by writing &amp;quot;NO&amp;quot; or &amp;quot;FALSE&amp;quot; or &amp;quot;DON'T DO THIS&amp;quot; above the common pitfall we were discussing, so as to not leave something misleading on the board.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-07-21T14:12:06.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18593" PostTypeId="2" ParentId="18588" CreationDate="2020-07-21T18:41:23.090" Score="34" Body="&lt;p&gt;This is a 100% subjective opinion, but it is based on teaching in various venues for close to 20 years (although none of that teaching was pure math).  Also, my college calculus courses are close to 30 years behind me, so please excuse me if my examples aren't directly related.&lt;/p&gt;&#xA;&lt;p&gt;IMHO, one of the biggest mistakes in teaching is failure to compare and contrast items that are similar in some ways but different in others.  I mean, that's where students make the most mistakes, right?  It's failure to pick up on fine distinctions.&lt;/p&gt;&#xA;&lt;p&gt;I agree with the other replies that say &amp;quot;Yes, you should point out this common error.&amp;quot;  But I would go further than that.  Don't present it as the &amp;quot;wrong answer.&amp;quot;  Present it as the &amp;quot;right answer &lt;em&gt;to a different question&lt;/em&gt;.&amp;quot;  Then go into the details of why it's a different question and how students can tell the difference.&lt;/p&gt;&#xA;&lt;p&gt;As an alternative to that direct approach, homework assignments that mix and match the two types of questions can give students the &amp;quot;ah ha!&amp;quot; moment to see the difference.  That's how I learned it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;March 2023:&lt;/strong&gt; Based on comments, my attempt at describing &amp;quot;the right answer to a different question&amp;quot; wasn't successful.  Maybe I was trying to be too poetic. All I meant by that is that this hypothetical student &lt;em&gt;correctly&lt;/em&gt; recalled the formula &lt;span class=&quot;math-container&quot;&gt;$\int\frac{1}{x}dx = \ln|x| + c$&lt;/span&gt;. The problem is that she is &lt;em&gt;applying it&lt;/em&gt; in the wrong situation.&lt;/p&gt;&#xA;&lt;p&gt;My own calculus is rusty, so I hope I'm getting these details right and using appropriate terminology.  But she hasn't learned that &lt;em&gt;x&lt;/em&gt; in this situation can't be &lt;em&gt;any expression&lt;/em&gt;. It must be the variable of integration (as shown in d&lt;em&gt;x&lt;/em&gt;). It can't be a polynomial in &lt;em&gt;x&lt;/em&gt; and it certainly can't be a composite function of &lt;em&gt;x&lt;/em&gt;.  Those situations require different methods.  And it's comparing and contrasting those methods, explicitly calling out when they apply and when they don't apply, that helps students.  Sure, the requirements for using &lt;span class=&quot;math-container&quot;&gt;$\int\frac{1}{x}dx = \ln|x| + c$&lt;/span&gt; were probably covered when the formula  was first introduced. But because homework problems for that night were all meant to practice that same situation, she never had to practice determining if a problem met the requirements or not.  Also, it's not until after &lt;em&gt;other methods&lt;/em&gt; (appropriate for polynomials and composite functions) are covered that a student will be in a position to fully appreciate a comparison and contrast between the different methods.&lt;/p&gt;&#xA;&lt;p&gt;Some students will realize their thinking is muddled. They will go back to their notes, textbook, or office hours and &lt;em&gt;dig&lt;/em&gt; until they understand the boundaries between concepts with fine distinctions. Other students either never realize their thinking is muddled, or they don't know what to do about it. That's where helping them through a comparison and contract can be very helpful.&lt;/p&gt;&#xA;" OwnerUserId="5465" LastEditorUserId="5465" LastEditDate="2023-03-18T23:20:10.840" LastActivityDate="2023-03-18T23:20:10.840" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18594" PostTypeId="2" ParentId="18578" CreationDate="2020-07-22T00:22:06.713" Score="1" Body="&lt;p&gt;This data from &lt;a href=&quot;https://wistia.com/learn/marketing/optimal-video-length&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wistia&lt;/a&gt; supports several of the comments:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2lL4l.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2lL4l.png&quot; alt=&quot;Engagement&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;This chart is not specialized to mathematics.&#xA;Likely the attention-span constraints are pretty much universal, independent of content (but not independent of interactivity).&#xA;" OwnerUserId="511" LastActivityDate="2020-07-22T00:22:06.713" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18595" PostTypeId="2" ParentId="18588" CreationDate="2020-07-22T00:49:58.693" Score="14" Body="&lt;p&gt;Here's another approach when there is a common pitfall that you wish the students to avoid. After teaching the correct reasoning: present the error to the class and ask a student to identify, explain, and correct the mistake. Being able to correct others' mistakes shows a high-level of understanding, and students who make that same mistake might be able to realize and fix it after seeing how.&lt;/p&gt;&#xA;&lt;p&gt;Then, the students are forming an understanding of why a mistake is wrong, rather than thinking that it is actually correct.&lt;/p&gt;&#xA;" OwnerUserId="13211" LastActivityDate="2020-07-22T00:49:58.693" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18596" PostTypeId="1" CreationDate="2020-07-22T02:47:51.263" Score="9" ViewCount="281" Body="&lt;p&gt;I have a similar problem addressed in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5996/system-of-equations-generator&quot;&gt;System of Equations Generator&lt;/a&gt;. What I need is an automatic way of generating a system of equations with unique solutions, but the equations are not exclusively linear.&lt;/p&gt;&#xA;&lt;p&gt;The method of working backwards, by randomly assigning integer values to variables and then generating the coefficients and computing the determinant matrix is perfect for a system of linear equations of the form:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}x + y = 7\\3x - 2y = 6\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I can verify that the solution is unique with x = 4 and y = 3.&lt;/p&gt;&#xA;&lt;p&gt;Now I would like to add two more operators: multiplication and division. The expected equations would be like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}x + y = 4\\2x * y = 6\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;How to generate these systems of equations and verify if the solution is unique, assuming integer variables and up to 5 variables? I found a lot of papers about uniqueness in non-linear equations, but I guess my requirement is a little simpler.&lt;/p&gt;&#xA;" OwnerUserId="14335" LastActivityDate="2020-07-23T08:19:18.007" Title="Generating system of equations with unique solutions" Tags="|homework|arithmetic-operations|arithmetic|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18597" PostTypeId="2" ParentId="18588" CreationDate="2020-07-22T11:35:44.750" Score="4" Body="&lt;p&gt;You could perhaps use examples and non-examples.&lt;/p&gt;&#xA;&lt;p&gt;This is less about explicitly telling students what the common mistakes are, but more about where the theorem or result applies and where it doesn't.&lt;/p&gt;&#xA;&lt;p&gt;At high school, this might be showing students what is meant by angles in the same segment and what is not meant by angles in the same segment, for example.&lt;/p&gt;&#xA;&lt;p&gt;In the case of your example, it might be about giving examples of what looks like &lt;span class=&quot;math-container&quot;&gt;$\frac{kf'(x)}{f(x)}$&lt;/span&gt; and what does not. Discussion can then be focused about why thinking it would be a log integral is wrong and that students should consider other techniques.&lt;/p&gt;&#xA;" OwnerUserId="14007" LastActivityDate="2020-07-22T11:35:44.750" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18598" PostTypeId="2" ParentId="18596" CreationDate="2020-07-22T12:50:03.047" Score="10" Body="&lt;p&gt;&lt;strong&gt;Generating systems.&lt;/strong&gt; The same method that works for linear equations works also for polynomial equations. Starting with a solution in mind (in mathematics and computer science, we call this a &lt;em&gt;planted&lt;/em&gt; solution), generate in some way left-hand sides of equations, and then compute the corresponding right-hand sides. If you use more than one equation per variable then you are likely not to get additional solutions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Verifying uniqueness.&lt;/strong&gt; Modern computer algebra systems can find the complete set of solutions of a system of polynomial equations. Under the hood, they might use &lt;a href=&quot;https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gröbner bases&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As an example, in &lt;a href=&quot;https://www.sagemath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;sage&lt;/a&gt; you can write&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;sage: var('x, y')&#xA;(x, y)&#xA;sage: solve([x + y == 4, 2*x*y == 6], [x, y])&#xA;[[x == 1, y == 3], [x == 3, y == 1]]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Sage correctly found the two unique solutions.&lt;/p&gt;&#xA;&lt;p&gt;Assuming that sage is using Gröbner bases, here is what happens.&#xA;First, sage computes a Gröbner basis in elimination order.&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;sage: R.&amp;lt;x,y&amp;gt; = PolynomialRing(ZZ, 2, order = 'lex')&#xA;sage: I = ideal(x + y - 4, 2*x*y - 6)&#xA;sage: I.groebner_basis()&#xA;[x + y - 4, 2*y^2 - 8*y + 6]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;It solves the quadratic to find all possible values of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, substitutes them, and finds the corresponding values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. The same idea works with more variables, although the degrees can get quite large.&lt;/p&gt;&#xA;" OwnerUserId="5929" LastEditorUserId="5929" LastEditDate="2020-07-23T08:19:18.007" LastActivityDate="2020-07-23T08:19:18.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18599" PostTypeId="2" ParentId="18576" CreationDate="2020-07-22T12:59:25.520" Score="1" Body="&lt;p&gt;You can try the &lt;a href=&quot;https://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_identities&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rogers–Ramanujan identities&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The number of partitions of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; in which adjacent parts are at least 2 apart is the same as the number of partitions of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; in which each part ends with 1,4,6,9.&lt;/li&gt;&#xA;&lt;li&gt;The number of partitions of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; &lt;em&gt;without 1&lt;/em&gt; in which adjacent parts are at least 2 apart is the same as the number of partitions of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; in which each part ends with 2,3,7,8.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For example, taking &lt;span class=&quot;math-container&quot;&gt;$n=10$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Partitions in which adjacent parts are at least 2 apart:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;10 = 10 = 9 + 1 = 8 + 2 = 7 + 3 = 6 + 4 = 6 + 3 + 1&#xA;$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Partitions in which each part ends with 1,4,6,9:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;10 = 9 + 1 = 6 + 4 = 6 + 1 + 1 + 1 + 1 = 4 + 4 + 1 + 1 = 4 + 6\times 1 = 10 \times 1&#xA;$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Partitions without 1 in which adjacent parts are at least 2 apart:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;10 = 10 = 8 + 2 = 7 + 3 = 6 + 4&#xA;$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Partitions in which each part ends with 2,3,5,8:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;10 = 8 + 2 = 7 + 3 = 3 + 3 + 2 + 2 = 2 + 2 + 2 + 2 + 2&#xA;$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5929" LastActivityDate="2020-07-22T12:59:25.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18600" PostTypeId="2" ParentId="18596" CreationDate="2020-07-22T16:07:37.003" Score="5" Body="&lt;p&gt;If you are looking at intersecting the zero sets of two polynomials in two variables, then by Bezout's theorem you should anticipate that the number of solutions will be equal to the product of the degrees of the polynomials.  So in general, you will not get unique solutions.  You will need to be very careful to hide these extra solutions &amp;quot;at infinity&amp;quot; or in the complex plane, or as &amp;quot;double roots&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-07-22T16:07:37.003" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18603" PostTypeId="2" ParentId="18588" CreationDate="2020-07-23T22:05:11.827" Score="9" Body="&lt;p&gt;I constantly present wrong answers, identified as such, in my class.  My strategy for dealing with exactly the issue you present is that wrong statements, and only (intentionally) wrong statements, appear in red.  I hope that this is a convenient and reliable visual cue.  Students hate it, though.&lt;/p&gt;&#xA;&lt;p&gt;One approach that has given me reasonable success, and that students, while not praising, at least don't complain about, is this:  when I know students &lt;em&gt;will&lt;/em&gt; make a common mistake, I present them with an exam question saying:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Your friend thinks that &lt;span class=&quot;math-container&quot;&gt;$$\int \frac{\mathrm dx}{1 + x^2} \qquad\text{equals}\qquad \ln(1 + x^2).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(a) What mistake(s) did your friend make?&lt;/p&gt;&#xA;&lt;p&gt;(b) What is the correct solution?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Examples of this sort have to be chosen with some care—it is not always reasonable for all errors to expect the student to be able to answer (a) beyond &amp;quot;it's wrong&amp;quot;—but I think that this particular one fits well.&lt;/p&gt;&#xA;&lt;p&gt;(This is quite similar to @ElizaWilson's &lt;a href=&quot;https://matheducators.stackexchange.com/a/18595&quot;&gt;suggestion&lt;/a&gt;.  It avoids possible issues with fear of public speaking, but maybe just substitutes another kind of pressure.)&lt;/p&gt;&#xA;" OwnerUserId="2070" LastEditorUserId="8571" LastEditDate="2020-07-24T18:35:24.573" LastActivityDate="2020-07-24T18:35:24.573" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18604" PostTypeId="1" AcceptedAnswerId="18610" CreationDate="2020-07-24T10:34:06.980" Score="5" ViewCount="793" Body="&lt;p&gt;When teaching partial fractions, there's probably no way to escape the heavy algebra necessary for partial fractions, but I'm wondering how to introduce the idea in a way that is intuitive or geometric. (Like how you could introduce &lt;a href=&quot;https://en.wikipedia.org/wiki/Integration_by_parts#Visualization&quot; rel=&quot;nofollow noreferrer&quot;&gt;integration by parts&lt;/a&gt; by a picture before you do examples, could you do the same with partial fractions?).&lt;/p&gt;&#xA;&lt;p&gt;I've found &lt;a href=&quot;https://matheducators.stackexchange.com/a/4349/13544&quot;&gt;an example of an answer here&lt;/a&gt;. I could start teaching by showing the a simple graph, like &lt;span class=&quot;math-container&quot;&gt;$$\frac{2x}{x^2-1}$$&lt;/span&gt; and get &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{x-1} + \frac{1}{x+1},$$&lt;/span&gt; I could see how students could believe that each of those denominators contribute to a vertical asymptote but it doesn't seem easy to generalize to the overall concept of partial fractions.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-07-24T23:38:44.137" LastActivityDate="2020-07-30T13:49:37.427" Title="Intuition or geometry for Partial Fractions" Tags="|calculus|intuition|integration|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18605" PostTypeId="2" ParentId="18527" CreationDate="2020-07-24T12:38:56.687" Score="5" Body="&lt;p&gt;I motivate these grad, curl, and div for myself as being the things which would make the respective version of Stokes' theorem true infinitesimally.  You can read about this interpretation in the context of differential forms here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/614473/34287&quot;&gt;https://math.stackexchange.com/a/614473/34287&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Given that interpretation, we can see&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The gradient at a point &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is the vector which represents the map &lt;span class=&quot;math-container&quot;&gt;$df|_p$&lt;/span&gt;.  In other words &lt;span class=&quot;math-container&quot;&gt;$df|_p(\vec{v}) = \nabla  f|_p \cdot \vec{v}$&lt;/span&gt;.  You can see that the dot product is maximized when &lt;span class=&quot;math-container&quot;&gt;$\vec{v} = \nabla f|_p$&lt;/span&gt;, which justifies the &amp;quot;steepest slope&amp;quot; interpretation.&lt;/li&gt;&#xA;&lt;li&gt;Given a vector field &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, convert it into a covector field &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; via &lt;span class=&quot;math-container&quot;&gt;$\omega(\vec{v}) = F \cdot \vec{v}$&lt;/span&gt;.  Then the integral of the differential form &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; along a curve is the same as the &amp;quot;line integral&amp;quot; of the vector field &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;.   Following my answer above, you can define &lt;span class=&quot;math-container&quot;&gt;$d\omega$&lt;/span&gt; in terms of the line integral around little parallelograms. &lt;span class=&quot;math-container&quot;&gt;$d\omega$&lt;/span&gt; is a 2-form, but it can be converted into a vector field by a similar representation trick (this one only works in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;): define &lt;span class=&quot;math-container&quot;&gt;$\textrm{curl}(F)$&lt;/span&gt; as the vector for which &lt;span class=&quot;math-container&quot;&gt;$\textrm{curl}(F) \cdot (\vec{v} \times \vec{w}) = d\omega(\vec{v},\vec{w})$&lt;/span&gt;.  The RHS is maximized when the line integral around a small parallelogram defined by &lt;span class=&quot;math-container&quot;&gt;$\vec{v}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec{w}$&lt;/span&gt; is maximized and then the LHS would require that &lt;span class=&quot;math-container&quot;&gt;$\textrm{curl}(F)$&lt;/span&gt; be perpendicular to that parallelogram, and pointing in the direction given by the  right hand rule so as to make the integral positive.&lt;/li&gt;&#xA;&lt;li&gt;Given a vector field &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, convert it into a &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-form &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; via &lt;span class=&quot;math-container&quot;&gt;$\omega(\vec{v},\vec{w}) = \textrm{Det}(\vec{v},\vec{w},\vec{F}) = \vec{F} \cdot ( \vec{v} \times \vec{w})$&lt;/span&gt;.  The integral of &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; over a surface is the same as the surface integral of &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;.  We can define &lt;span class=&quot;math-container&quot;&gt;$d\omega(\vec{v},\vec{w},\vec{b})$&lt;/span&gt; in terms of the integral of &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; around the boundary of the small oriented parallelepiped defined by the three vectors.  This is the same as the surface integral of &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; over that same surface.  This is a top level form in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;, so it is a multiple of the volume form, i.e. &lt;span class=&quot;math-container&quot;&gt;$\textrm{div}(F)\textrm{Det}(\vec{v},\vec{w},\vec{b}) = d\omega(\vec{v},\vec{w},\vec{b})$&lt;/span&gt;. So the sign of the divergence can be &amp;quot;seen&amp;quot; according to whether most of the vectors &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; are pointing &amp;quot;into&amp;quot; or &amp;quot;out of&amp;quot; a small parallelepiped based at &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;You can replace the parallelogram with a circle and the parallelepiped with a sphere to obtain a more &amp;quot;symmetric&amp;quot; representation, but you loose the easy connection with the differential forms.  Also, it is much easier to decompose surfaces into small parallelograms (and solids into small parallelepipeds), which leads to the global Stokes' theorems being just &amp;quot;telescoping&amp;quot; sums, with cancellation of all interior terms.&lt;/p&gt;&#xA;&lt;p&gt;I think the real difficulty with motivating these things comes from the &amp;quot;unnatural&amp;quot; conversion of differential forms into scalar and vector fields, in ways which only work in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;.  They are somewhat arbitrary.  They are also unnecessary, as any work you want to do can be done with the differential forms themselves.&lt;/p&gt;&#xA;&lt;p&gt;These will let you draw pictures though! In each case, you have to imagine many different &amp;quot;test&amp;quot; domains of integration, and try to figure out which domain will maximize the integral.  The grad, curl, and div connect to those maximizing domains.&lt;/p&gt;&#xA;&lt;p&gt;EDIT:  I forgot to mention the laplacian, which doesn't really fit into this framework.  The interpretation you link to seems pretty much optimal though.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="127" LastEditDate="2020-07-24T17:20:52.240" LastActivityDate="2020-07-24T17:20:52.240" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18606" PostTypeId="2" ParentId="18588" CreationDate="2020-07-24T22:30:51.947" Score="3" Body="&lt;p&gt;Whether you mention the common mistake or not, it is likely to be made by some student. I think the best approach is to mention this mistake but you need to be very clear about why it is wrong, and show the thought process that arrives you at the correct answer. Many educators concern about the consequences of introducing the idea of a wrong answer because it may concern students. But in my experience when I was a student, often in my study I would come across these wrong answers and it would have been helpful to be able to talk myself out of them being right, by applying the same thought process that my teacher provided me with.&lt;/p&gt;&#xA;" OwnerUserId="14350" LastActivityDate="2020-07-24T22:30:51.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18607" PostTypeId="2" ParentId="18524" CreationDate="2020-07-24T22:46:43.227" Score="4" Body="&lt;p&gt;Here in Australia (NSW specifically) the highest level of high school maths in Year 12 has a topic on the logic and methods of proof. This includes general concepts of proof (symbolic logic, truth tables, the contrapositive, proof by contradiction, proof be counterexample, etc.) and some specific methods of proof. I think the best way to get a feel for what the questions are like is just to show you some. Here are two questions from previous exams using these concepts. I have also included a link to a small section of a textbook as an introduction of the logic of proof.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FYkRD.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FYkRD.png&quot; alt=&quot;Question 1&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BBx0V.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BBx0V.png&quot; alt=&quot;Question 2&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://drive.google.com/file/d/1HOej3PvDFn0MUCM3hBRrR2Ly2MizryCP/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://drive.google.com/file/d/1HOej3PvDFn0MUCM3hBRrR2Ly2MizryCP/view?usp=sharing&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14350" LastEditorUserId="14350" LastEditDate="2020-07-27T06:07:43.607" LastActivityDate="2020-07-27T06:07:43.607" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18609" PostTypeId="2" ParentId="18604" CreationDate="2020-07-25T16:09:17.533" Score="8" Body="&lt;h3&gt;Introduction&lt;/h3&gt;&#xA;&lt;p&gt;I wasn't taught the partial fractions decomposition (PFD) in calculus.  We didn't cover it in high school, and when I went to college, they assumed we all knew it.  Somehow it was when I read the proof in van der Waerden's &lt;em&gt;Modern Algebra&lt;/em&gt; that I understood why it was called partial fractions.  I looked at it again a couple days ago, and interestingly, his presentation is not exactly the idea I came away with.  But that's the way of learning: sometimes you learn it in your own way.  Suppose the degree of a polynomial &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is less than the degree of a product of two polynomials &lt;span class=&quot;math-container&quot;&gt;$pq$&lt;/span&gt;. We can consider a decomposition into fractions&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; f(x) = {g(x) \over p(x)\,q(x)} = {A(x) \over p(x)} + {B(x) \over q(x)}&#xA;$$&lt;/span&gt;&#xA;as partial if &lt;span class=&quot;math-container&quot;&gt;$A/p$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$B/q$&lt;/span&gt; can be further resolved into fractions (for instance if &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is not a power of an irreducible polynomial). The standard PFD can be found inductively: Take one of the factors &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; to be the power of an irreducible appearing in the denominator; find &lt;span class=&quot;math-container&quot;&gt;$A/p$&lt;/span&gt; and subtract from &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, leaving &lt;span class=&quot;math-container&quot;&gt;$B/q$&lt;/span&gt;; and repeat until no factors are left.  It has a procedural form similar to partial integration (now called &amp;quot;integration by parts&amp;quot; to avoid confusing the two).&lt;/p&gt;&#xA;&lt;p&gt;We'll summarize the takeaways to understanding PFD in the perspectives below.  They don't all meet the OP's goal of a picture, but they provide some insight into how things work. First of all, the PFD can be viewed as a weighted sum (with the weights &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; in the formula above). This can be connected to a weighted average, center of mass, barycentric coordinates, and interpolation.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The first section below uses the limit of an interpolation to derive a fractional part of the PFD, similar to the standard secant/tangent line development of the derivative in calculus.  This provides the best visualization, in my opinion, for the case of real poles of whatever order.&lt;/li&gt;&#xA;&lt;li&gt;The second looks at the relationship of the PFD to the center of mass in the case of two simple poles and how it breaks down with more than two.  Nonetheless, the center of mass provides, to my imagination, a nice visualization of balancing the weights.  It is, perhaps surprisingly, dualized: The weights &lt;span class=&quot;math-container&quot;&gt;$A,B$&lt;/span&gt;, of the poles from &lt;span class=&quot;math-container&quot;&gt;$1/p,1/q$&lt;/span&gt; (resp.) weight the complementary zero(s) of &lt;span class=&quot;math-container&quot;&gt;$q,p$&lt;/span&gt; (resp.). Thus the visualization only works when &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; are both linear and distinct.  For more than two simple poles, the barycentric Lagrange interpolation formula arises naturally.  It gives a way to think of the PFD as weighted sum of the &amp;quot;pole&amp;quot; functions &lt;span class=&quot;math-container&quot;&gt;$1/(x-a)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The third uses higher algebra and is connected to the Chinese Remainder Theorem.  This is the proper view, in my opinion. The coefficients &lt;span class=&quot;math-container&quot;&gt;$A,B,\dots$&lt;/span&gt; becomes coordinates equipped with coordinate projection operators.  It is not suitable for visualization, AFAICT.&lt;/li&gt;&#xA;&lt;li&gt;Finally, there is the way I have tended to look at the PFD most of the time, as a high-school algebra problem that plausibly should work out (and does, of course).  It's plausible because of the way one adds fractions. It's plausible because the system of equations you get has the same number of equations as unknowns.  Finally, that you decompose a complicated fraction into simpler parts convinced me of its usefulness.  The plausibility corresponds to an intuition of how the denominators work; but the intuition is algebraic, and therefore abstract, and does not lead to pictures.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Let's consider visualizing the PFD.  Say we have a curve going off to infinity at an asymptote, and how it goes to infinity is very important to the form of the PFD.  But, by eye, one cannot really see the differences precisely enough as the the curve goes to infinity, except whether the power in the denominator is even or odd.  So I think there is some limit to how well we can see what is going on.  I think we'll have to use our imagination.&lt;/p&gt;&#xA;&lt;p&gt;The algebraic structure of the partial fractions problem is not simple, except in the case of two simple poles.  It is complicated by the way that multiplication of polynomials is a sort of convolution and not a simple operation like addition and multiplication by a constant.  Just how to picture it is not easy. One could model linear and quadratic factors as lengths and areas, but it quickly gets out of hand.  Indeed, driven by Pappus' problem, Descartes went the other direction and introduced algebra into geometry to derive necessary relations that could not (easily) be pictured.&lt;/p&gt;&#xA;&lt;h3&gt;1. Approximating poles via interpolation and finite differences&lt;/h3&gt;&#xA;&lt;p&gt;&lt;strong&gt;1.1&lt;/strong&gt;&#xA;One approach, which leads to a picture, is an extension of the &lt;em&gt;Methodus Incrementorum&lt;/em&gt; (1715), of Brook Taylor, who used interpolation to derive Taylor's Theorem. Once, finite differences were a quite common analytical tool.  Euler based calculus on them.  One application is still quite common, if not universal: The use of the secant line to develop the notion of the tangent line and the derivative.  Some teachers may give a similar development of the second derivative in terms of evaluating the function at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x+h$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x+2h$&lt;/span&gt;, which corresponds to a quadratic interpolant.  Taylor's approach extends the notion of an interpolant approximating the tangent line to higher orders to develop the notion of Taylor series.&lt;/p&gt;&#xA;&lt;p&gt;The method may be adapted to determining a fractional part of the partial fractions decomposition.  The idea is to interpolate &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; near an asymptote &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; of order &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, where the interpolation has the form &lt;span class=&quot;math-container&quot;&gt;$$I(x)={p(x) \over (x-a)^k}$$&lt;/span&gt;&#xA;with the degree of the polynomial &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; less than &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;.  The interpolation conditions are for &lt;span class=&quot;math-container&quot;&gt;$h\ne0$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; f(x_j) = p(x_j)/(x_j-a)^k,\ \ x_j = a+jh,\ \ j=1,\dots,k\,.&#xA;$$&lt;/span&gt;&#xA;Equivalently, &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; is the interpolating polynomial that interpolates &lt;span class=&quot;math-container&quot;&gt;$(x-a)^kf(x)$&lt;/span&gt; at the &lt;span class=&quot;math-container&quot;&gt;$x_j$&lt;/span&gt;.  As &lt;span class=&quot;math-container&quot;&gt;$h \rightarrow 0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$p(x)\,/\,(x-a)^k$&lt;/span&gt; approaches the partial fractions part associated with &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; {A_1 \over x-a}+\cdots+{A_{k} \over (x-a)^{k}}\,.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MmXhM.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MmXhM.gif&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;A.&lt;/strong&gt;&#xA;On the left are shown the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; (blue) and interpolant &lt;span class=&quot;math-container&quot;&gt;$I(x)$&lt;/span&gt; (gold).  The interpolation points are shown in red (&lt;span class=&quot;math-container&quot;&gt;$h&amp;lt;0$&lt;/span&gt;).&#xA;&lt;strong&gt;B.&lt;/strong&gt;&#xA;On the right are shown the difference between the interpolant  &lt;span class=&quot;math-container&quot;&gt;$I(x)$&lt;/span&gt; and each of the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; (blue), the difference being zero at the interpolation points; and the difference between the interpolant &lt;span class=&quot;math-container&quot;&gt;$I(x)$&lt;/span&gt; and the fractional part &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; (gold), the difference approaching zero as &lt;span class=&quot;math-container&quot;&gt;$h\rightarrow0$&lt;/span&gt;.&lt;/em&gt; &lt;a href=&quot;https://www.desmos.com/calculator/luvjwmtueq&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link to Desmos version of graphic.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;1.2&lt;/strong&gt;&#xA;From (Newton) interpolation theory, one can derive a formula for the partial fractions decomposition, although it might not give a clear intuitive picture. If&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x) = {g(x) \over (x-a_1)^{n_1} \cdots (x-a_m)^{n_m}}\,$$&lt;/span&gt;&#xA;where the &lt;span class=&quot;math-container&quot;&gt;$a_r$&lt;/span&gt; are distinct (possibly complex) roots of the denominator and the degree of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is less than the degree of the denominator, then the partial fractions decomposition is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x)=\sum_r \sum_{k=0}^{n_r-1} {A_{r,k} \over (x-a_r)^{n_r-k}} \,,\quad&#xA;A_{r,k}=&#xA; {{d^k \over dx^k} \left[f(x)(x-a_r)^{n_r}\right]\Big|_{a_r} \over k!}$$&lt;/span&gt;&#xA;The appearance of the derivatives intuitively comes from the several points &lt;span class=&quot;math-container&quot;&gt;$a_r+jh$&lt;/span&gt; approaching the same root &lt;span class=&quot;math-container&quot;&gt;$a_r$&lt;/span&gt;, similar to what was mentioned in the introduction of this section.&lt;/p&gt;&#xA;&lt;h3&gt;2. Weighted sums and barycentric coordinates&lt;/h3&gt;&#xA;&lt;p&gt;&lt;strong&gt;2.1&lt;/strong&gt;&#xA;Let's say we have two distinct linear factors: &lt;span class=&quot;math-container&quot;&gt;$\alpha (x-c)\,/\,[(x-a)(x-b)]$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the weighted average of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; or center of mass: &lt;span class=&quot;math-container&quot;&gt;$c = (Ab + Ba)/(A+B)$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$\alpha (x-c)\,/\,[(x-a)(x-b)] = A/(x-a) + B/(x-b)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The linear factor in the numerator disappears, as in &lt;span class=&quot;math-container&quot;&gt;$\alpha\,/\,[(x-a)(x-b)]$&lt;/span&gt;, when &lt;span class=&quot;math-container&quot;&gt;$c=\infty$&lt;/span&gt;, that is, when &lt;span class=&quot;math-container&quot;&gt;$A+B=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thus we can think of &lt;span class=&quot;math-container&quot;&gt;$[A:B]$&lt;/span&gt; as (non-normalized) barycentric coordinates of the point &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; on the projective line, relative to the points &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;2.2&lt;/strong&gt;&#xA;With Lagrange interpolation, we can connect &lt;strong&gt;1&lt;/strong&gt; and &lt;strong&gt;2&lt;/strong&gt; in the case that &lt;span class=&quot;math-container&quot;&gt;$f(x)=g(x)/h(x)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$h(x)$&lt;/span&gt; having distinct (possibly complex) roots.  One can, if desired, restrict the class of problems further and just deal with distinct real roots.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$a_r$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$r=1,\dots,n$&lt;/span&gt; be distinct real or complex numbers.&#xA;Define the Lagrange polynomial and weights by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; \ell(x) = \prod_{r=1}^n (x-a_r)\,,\quad&#xA;  w_r =  {1 \over \prod_{j \ne r} (a_r-a_j)}\,.&#xA;$$&lt;/span&gt;&#xA;The polynomials &lt;span class=&quot;math-container&quot;&gt;$\ell(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h(x)$&lt;/span&gt;, having the same roots, differ by a constant factor equal to the leading coefficient of &lt;span class=&quot;math-container&quot;&gt;$h(x)$&lt;/span&gt;.  We can call this constant &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;.  Thus &lt;span class=&quot;math-container&quot;&gt;$h(x) = m\,\ell(x)$&lt;/span&gt;. One form of the barycentric Lagrange interpolation formula is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$g(x) = f(x)h(x) = \ell(x) \sum_{r=1}^n {w_r \over x-a_r}\,g(a_r)\,.$$&lt;/span&gt;&#xA;Therefore&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x) = {1 \over m} \sum_{r=1}^n w_r\,g(a_r)\,{1\over x-a_r}\,,$$&lt;/span&gt;&#xA;which is at once an interpolating formula between poles, a weighted sum, and the partial fractions decomposition.  See &lt;a href=&quot;https://matheducators.stackexchange.com/a/4344/498&quot;&gt;this answer by David Speyer&lt;/a&gt; for it worked out another way with four poles.&lt;/p&gt;&#xA;&lt;p&gt;The connections with the interpolation in &lt;strong&gt;1&lt;/strong&gt; and the weighted sum in &lt;strong&gt;2&lt;/strong&gt; are worth contrasting, because they are not analogies per se.  In &lt;strong&gt;1&lt;/strong&gt;, interpolation is used to resolve high-order poles, one at a time; here it was used to resolve distinct simple poles. The approaches are somewhat orthogonal or complementary to each other.  In &lt;strong&gt;2&lt;/strong&gt;, the focus was on the roots &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, which is possible only in the case of two distinct simple poles.  It comes from the weighted sum, though:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;{A \over x-a} + {B \over x-b} =&#xA; {A(x-b) + B(x-a) \over (x-a)(x-b)} =&#xA; (A+B)\,{x-(Ab + Ba)/(A+B) \over (x-a)(x-b)}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h3&gt;3. Via number theory and algebra&lt;/h3&gt;&#xA;&lt;p&gt;&lt;strong&gt;3.1&lt;/strong&gt;&#xA;Alternatively, an analogy with number theory: If &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are distinct primes, and you have a fraction &lt;span class=&quot;math-container&quot;&gt;$p\,/\,[a^j b^k]$&lt;/span&gt;, you can decompose it &lt;span class=&quot;math-container&quot;&gt;$p\,/\,[a^j b^k] = \alpha\,/\,a^j + \beta\,/\,b^k$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2|\alpha|\le a^j$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2|\beta|\le b^k$&lt;/span&gt; and perhaps expand &lt;span class=&quot;math-container&quot;&gt;$|\alpha| = \alpha_0+\alpha_1a+\alpha_2a^2+\cdots+\alpha_{j-1}a^{j-1}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|\beta|=\beta_0+\beta_1b+\beta_2b^2+\cdots+\beta_{k-1}b^{k-1}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$0\le\alpha_m&amp;lt;a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$0\le\beta_n&amp;lt;b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;3.2&lt;/strong&gt;&#xA;To extend the number theory analogy, I would have said that the right intuition for partial fractions is the Chinese Remainder Theorem, not complex analysis.  It is essentially an algebra problem, not a geometry/analysis problem.&lt;/p&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$f(x)=g(x)/h(x)$&lt;/span&gt; is a rational function in a polynomial ring &lt;span class=&quot;math-container&quot;&gt;$F[x]$&lt;/span&gt; over a field &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$h(x) = p(x)q(x)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; relatively prime.  There is no need to factor them into irreducibles, although it is done for the purposes of integration.  If we can decompose&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; f(x) = {g(x) \over p(x)\,q(x)} = {A(x) \over p(x)} + {B(x) \over q(x)}\,,&#xA;$$&lt;/span&gt;&#xA;we can continue decomposing each partial fraction inductively until the denominators are irreducible.&lt;/p&gt;&#xA;&lt;p&gt;The partial fractions decomposition can be viewed in terms of the isomorphism&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; F[x]/(pq) \cong F[x]/(p) \oplus F[x]/(q)&#xA;$$&lt;/span&gt;&#xA;given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; g \mapsto (A,B)&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; satisfy the congruences&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; g \equiv Aq\ (p),\quad g \equiv Bp\ (q) \,.&#xA;$$&lt;/span&gt;&#xA;Since &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; are relatively prime, we can solve for the projections &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; onto the &lt;span class=&quot;math-container&quot;&gt;$(p)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(q)$&lt;/span&gt; components as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; A \equiv (q)^{-1}_p\,g\ (p)\,,\quad&#xA; B \equiv (p)^{-1}_q\,g\ (q)\,,&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$(q)^{-1}_p$&lt;/span&gt; is the inverse of &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; mod &lt;span class=&quot;math-container&quot;&gt;$(p)$&lt;/span&gt;&#xA;and &lt;span class=&quot;math-container&quot;&gt;$(p)^{-1}_q$&lt;/span&gt; is the inverse of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; mod &lt;span class=&quot;math-container&quot;&gt;$(q)$&lt;/span&gt;.&#xA;These may be computed via Euclid's algorithm.&lt;/p&gt;&#xA;&lt;h3&gt;4. More polynomial algebra&lt;/h3&gt;&#xA;&lt;p&gt;&lt;strong&gt;4.1&lt;/strong&gt;&#xA;Treat &lt;span class=&quot;math-container&quot;&gt;$F[x]$&lt;/span&gt; as a graded algebra, by degree, and assume the linear algebra works out so that there is a always a unique solution to the partial fractions decomposition.&#xA;In the solution to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x) = {A(x) \over p(x)} + {B(x) \over q(x)}\,,$$&lt;/span&gt;&#xA;if the degree of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is less than the degree of &lt;span class=&quot;math-container&quot;&gt;$pq$&lt;/span&gt;,&#xA;the degrees of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; can turn out be anything less than the degrees of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;, respectively.&#xA;For &lt;span class=&quot;math-container&quot;&gt;$F = {\Bbb R}$&lt;/span&gt; the field of real numbers, we can consider the cases where &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is of the form &lt;span class=&quot;math-container&quot;&gt;$(x-a)^k$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(x^2+ax+b)^k$&lt;/span&gt;.&#xA;In both cases &lt;span class=&quot;math-container&quot;&gt;$A(x)/p(x)$&lt;/span&gt; can be put in standard form by some simple algebra:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; {A_1 \over x-a}+\cdots+{A_{k} \over (x-a)^{k}}&#xA;$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; {A_1 x + B_1 \over x^2+ax+b}+\cdots+{A_{k} x + B_k \over (x^2+ax+b)^{k}}&#xA;$$&lt;/span&gt;&#xA;The algebraic steps in the first case is to replace &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$a+u$&lt;/span&gt;, expand, and replace &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$x-a$&lt;/span&gt;.  In the second case, you have to successively reduce any power of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; greater than or equal to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; by replace factors &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$u-ax-b$&lt;/span&gt;, expanding all the while, until no more replacements are possible; then with the numerator expanded, replace &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$x^2+ax+b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As I said in the introduction, this is the way I looked at partial fractions intuitively as a young mathematician, even before I formally learned or connected it with &amp;quot;graded algebras&amp;quot;: how degrees of polynomials work is just obvious.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2020-07-30T13:49:37.427" LastActivityDate="2020-07-30T13:49:37.427" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18610" PostTypeId="2" ParentId="18604" CreationDate="2020-07-27T20:48:00.300" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I'm wondering how to introduce the idea in a way that is intuitive or geometric.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How about an introduction along the lines of adding two shapes together. Start with adding fractions to have a picture in mind:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4FLa3.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4FLa3.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For example, one could add the fractions &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{5}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{7}$&lt;/span&gt; by first drawing  a rectangle with width &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{5}$&lt;/span&gt; and height &lt;span class=&quot;math-container&quot;&gt;$1 = \frac{7}{7}$&lt;/span&gt;, and then a rectangle with width &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{7}$&lt;/span&gt; and height &lt;span class=&quot;math-container&quot;&gt;$1 = \frac{5}{5}$&lt;/span&gt;. Each of the smallest pieces (shaded red) has area &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{35}$&lt;/span&gt;. Then, the sum &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{5} + \frac{1}{7}$&lt;/span&gt; is just the combined areas of all of the small pieces: seven of the &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{35}$&lt;/span&gt;'s from the first picture, and five of the &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{35}$&lt;/span&gt;'s from the second picture. Result: &lt;span class=&quot;math-container&quot;&gt;$\frac{7+5}{35} = \frac{12}{35}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can change this picture for other examples, like the one you mentioned: &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x-1} + \frac{1}{x+1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VMQ2L.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VMQ2L.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Again, &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x-1}$&lt;/span&gt; is the area of a rectangle with width &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x-1}$&lt;/span&gt; and height &lt;span class=&quot;math-container&quot;&gt;$1 = \frac{x+1}{x+1}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x+1}$&lt;/span&gt; is the area of a rectangle with width &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x+1}$&lt;/span&gt; and height &lt;span class=&quot;math-container&quot;&gt;$1 = \frac{x-1}{x-1}$&lt;/span&gt;. It is easy to sum these together, as each small piece has area &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{(x-1)(x+1)}$&lt;/span&gt;. You just add the &lt;span class=&quot;math-container&quot;&gt;$x+1$&lt;/span&gt; of them from the first picture and the &lt;span class=&quot;math-container&quot;&gt;$x-1$&lt;/span&gt; of them from the second picture. Answer: &lt;span class=&quot;math-container&quot;&gt;$\frac{x+1+x-1}{(x-1)(x+1)} = \frac{2x}{(x-1)(x+1)}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, what you really want to do is motivate the reverse process -- partial fraction decomposition. While I don't know how to let the picture solve this problem, it does show the objects we're counting when trying to find &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\frac{A}{x-a} + \frac{B}{x-b}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, suppose you wanted to decompose &lt;span class=&quot;math-container&quot;&gt;$\frac{3x+4}{(x-1)(x+1)}$&lt;/span&gt; into partial fractions. From the picture, it is clear that we need some number of the smallest pieces from the left-hand side and the right-hand side. Since the fraction &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x-1}$&lt;/span&gt; is made of all &lt;span class=&quot;math-container&quot;&gt;$x+1$&lt;/span&gt; of the &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{(x-1)(x+1)}$&lt;/span&gt;'s, we will need some multiple &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; of them: &lt;span class=&quot;math-container&quot;&gt;$A(x+1)$&lt;/span&gt;. Similarly, &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x+1}$&lt;/span&gt; takes &lt;span class=&quot;math-container&quot;&gt;$x-1$&lt;/span&gt; of the &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{(x-1)(x+1)}$&lt;/span&gt;'s, so we will take some multiple &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; of those: &lt;span class=&quot;math-container&quot;&gt;$B(x-1)$&lt;/span&gt;. In total, we need &lt;span class=&quot;math-container&quot;&gt;$3x+4$&lt;/span&gt; of these little pieces, so we have an equation: &lt;span class=&quot;math-container&quot;&gt;$$A(x+1) + B(x-1) = 3x+4$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I agree it would be cool to have a picture method to actually find the new coefficients, but I suspect we can't have that. The integration-by-parts picture does a nice job of motivating that the formula should seem reasonable/intuitive, but of course it doesn't actually perform the calculus.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-07-27T20:48:00.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18611" PostTypeId="1" CreationDate="2020-07-31T02:44:38.567" Score="0" ViewCount="157" Body="&lt;p&gt;So I'm doing a freelance writing job, writing a script for a YouTube video about eigenvectors/values.  It took me a while to decide what the focus was going to be, but I finally settled on focusing on how all linear transformations can be thought of as a composition of rotation, scaling, and flipping/reflecting transformations and how eigenvectors are vital to be able to actually do such a decomposition (I'm thinking specifically of eigendecomposition for diagonalizable matrices and polar decomposition and SVD for general matrices, all of which rely on the existence if eigenvectors.  However, I'm not planning on getting into the mechanics of how to actually do these decompositions and even necessarily distinguishing between them, since the video is aimed at a general audience.  I will be explaining the basics of what a linear transformation is and what a vector is (probably just sticking to the &amp;quot;arrow on the euclidean plane&amp;quot; explanation though).&#xA;My problem is that I'm stuck on the introduction.  I have most of the body written (though it needs revision), but I'm unsure how to introduce the topic to a general audience in a way that will get people immediately interested. What I currently have for the intro is this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hey Crazies!  [That's how he always introduces videos on his channel] Have you ever wondered how we keep track of how things move? Like, REALLY thought about it?   Well, first we need to choose some point that ISN'T moving. This can be whatever we want -- like we've said previously, motion is relative. We call this point the origin and denote it as (0,0). The usual approach is to then the define three vectors of length one, one on the x axis, one on the y axis, and one on the z axis. Then, if we want to see what happens to other vectors are they're moving, we represent them in terms of these three &amp;quot;basis&amp;quot; vectors. (Animation of a particular example of this) . But sometimes, this isn't very convenient.  For example, let's say we have a transformation like this (animation of some linear transformation that rotates the standard basis vectors a lot) that's rotating those vectors all over the place. It's hard to even keep track of where they're going, let alone use them to keep track of where everything else is going! We need a better approach. Let's look at the other vectors. Are there any that AREN'T being rotated so much? Well there has to be at least ONE that's not being rotated at all -- the axis of rotation. Are there any others? If there are, we can use them to much more easily keep track of where all the other vectors are moving. Such vectors are called eigenvectors.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, the guy I'm writing it for said it's too abstract and I agree.  Plus, it doesn't really fit the theme I'm going with, which is eigenvectors as a way of breaking complicated transformations into several simple ones.&lt;/p&gt;&#xA;&lt;p&gt;I think what I need is a specific example/analogy of how eigenvectors can make things easier, but one that will make sense to a general audience who aren't necessarily familiar with linear algebra.  He suggested diagonalization and I do like that idea, as it relates very much to eigendecomposition -- after all, eigendecomposition is really just rewriting the matrix in its eigenbasis but with the change of basis matrix and its inverse still acting on it and the diagonalized form of the matrix is just the matrix written in it's eigenbasis.  But how do I introduce those ideas to a general audience as an introduction?  I need something that will get people immediately interested so they'll want to keep watching.&lt;/p&gt;&#xA;&lt;p&gt;Any advice is greatly appreciated!  :)&lt;/p&gt;&#xA;" OwnerUserId="14369" LastActivityDate="2020-07-31T02:44:38.567" Title="How can I introduce the idea of eigenvectors and matrix decompositions to a general audience in an engaging manner?" Tags="|algebra|concept-motivation|linear-algebra|eigenvalues|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18612" PostTypeId="1" CreationDate="2020-07-31T08:55:35.153" Score="5" ViewCount="359" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{Green}{\text{My Question in short}}$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Unprovable claim&lt;/strong&gt;: Someone who is familiar with 100 thinking-games, and plays 30 of them reasonably well, is more prepared to go to math than the average community.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The most important available facilities, are volunteers&lt;/strong&gt;. There are some people who travel to villages and deprived areas. I know one of them, They can teach simple games and hobbies to local educators so that children can enjoy a wider variety of games and hobbies, and the children can be taught and raised better. Note that almost all attempts are individual or team, that's why there is no significant money for this work.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Goal&lt;/strong&gt;: My goal is not to make games for kids. My goal is &lt;strong&gt;to create a booklet&lt;/strong&gt;/ notebook/ ... of suitable thinking-games for different age groups. If such a booklet is available to these volunteers, they can introduce local educators and children to better hobbies and thinking-games. I am looking for suitable thinking-games.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Properties of a suitable game&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;ol start=&quot;0&quot;&gt;&#xA;&lt;li&gt;Easy to make [The most important factor is the zero factor]&lt;/li&gt;&#xA;&lt;li&gt;Does not require mobile&lt;/li&gt;&#xA;&lt;li&gt;Easy to learn, and easy to play&lt;/li&gt;&#xA;&lt;li&gt;As attractive as possible&lt;/li&gt;&#xA;&lt;li&gt;Preferably durable&lt;/li&gt;&#xA;&lt;li&gt;Involving as many players as possible&lt;/li&gt;&#xA;&lt;li&gt;Can be played/done in a short time, playable/doable as short as possible&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;EDITED:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;About the age-range&lt;/strong&gt;: No specific age range can be specified. The age-range of children is between 4 to 15 years. The age distribution is almost uniform. I realize that this wide range includes a large number of age-ranges. I am not an expert, but I believe that the target society may need to be divided into 5-7 different age-groups.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{Green}{\text{End}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{Red}{\text{Details start here}}$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;I am looking for a significant number of thinking-games and board games that have these properties [however, even a single offer would be welcome]:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Properties of a suitable game in detail&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;ol start=&quot;0&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Easy to make&lt;/strong&gt;. Due to financial difficulties, it is not possible to buy games. But we can think of &amp;quot;building the games&amp;quot;. The shape and form of the beads and pieces should not matter, and their shape and form can be reduced to the simplest accessible state. Beads and pieces should be designed in a very simple shape and form. [For instance, bottle caps can be used as beads, smooth cut pieces of wood (cuboid woods) can be used as pieces. The design of the game screen, should be possible in a short time, for example 1 or 2 hours. If the beads and pieces have different sizes and forms, they should be distinguished by coloring, or marking, or some other simple ideas.]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Does not require mobile&lt;/strong&gt;. Children in the target community do not necessarily have access to a mobile or computer, so mobile and computer games may not necessarily be suitable suggestions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Easy to learn, and easy to play&lt;/strong&gt;. The game should be &amp;quot;simple&amp;quot; and &amp;quot;light&amp;quot; [or at most &amp;quot;light-medium&amp;quot;]. [The game should not have too many details; rules and restrictions should be simple. There is no sensitivity to the number of rules and restrictions, but it is better not to have too many]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Attractiveness[As attractive as possible]&lt;/strong&gt;: Completely abstract games may not be very fun for kids. Definitely, &lt;a href=&quot;https://en.wikipedia.org/wiki/Snakes_and_Ladders&quot; rel=&quot;nofollow noreferrer&quot;&gt;Snakes and Ladders&lt;/a&gt;/ &lt;a href=&quot;https://en.wikipedia.org/wiki/Mensch_%C3%A4rgere_Dich_nicht&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mensch ärgere Dich nicht&lt;/a&gt; does not have the beauty of chess, but during Snakes and Ladders/ Mensch/ ... more friendly interactions are formed. And these very mild social interactions are very effective in &amp;quot;appealing&amp;quot; the game and &amp;quot;motivating&amp;quot; players and kids to get back to it. A game that does not create social interactions between players will largely lose its appeal. In my opinion, games that are played on paper do not have much chance to attract a general audience. I think it is a lot of fun for kids to move the beads or pick up and touch the pieces. Another reason to avoid abstract games is that, &lt;strong&gt;most likely, there will be no teacher to do this with the children&lt;/strong&gt; (Probably, after the first few days, children have to play on their own, so &lt;strong&gt;games should be as attractive as possible&lt;/strong&gt; for children); and the game itself should be engaging for children. [*: I guess this section may not be agreed upon by everyone, ignore it if necessary]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Preferably durable&lt;/strong&gt;. In addition to being easy to build, we need to be able to build them at a reasonable cost and expect them to last at least a year. For example, card-based games will not be very suitable. In card-based games where the cards need to be unknown, if the cards are damaged, they become marked and known. Preferably games are not cards, as they are easily damaged.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Involving as many players as possible&lt;/strong&gt;. The more players can play at the a time, the better. Creating a team, for each turn or piece, is not an interesting idea, this idea has many problems in practice. (See my &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14226/targeted-group-game-for-8-or-9-players#comment35668_14243&quot;&gt;comment&lt;/a&gt; to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14226/targeted-group-game-for-8-or-9-players/14243#14243&quot;&gt;this answer&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Can be played in a short time, playable as short as possible&lt;/strong&gt;. It is better that the game-play is not too long.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The most important factor is the zero factor. After this factor, &lt;strong&gt;attractiveness&lt;/strong&gt; will definitely be the next priority. I am looking for games that satisfy the zero factor along with a significant number of these factors. Years ago, I asked a &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14226/targeted-group-game-for-8-or-9-players&quot;&gt;question&lt;/a&gt; on this site and I received a convincing and satisfying answer. During that question, I became acquainted with the &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14226/targeted-group-game-for-8-or-9-players/14244#14244&quot;&gt;Ricochet Robot and Can't Stop&lt;/a&gt;; in which Ricochet Robot clearly satisfies almost all of the above conditions, and Can't Stop is also very satisfying. Also, &lt;a href=&quot;http://jdh.hamkins.org/fun-with-orthoprojections/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ortho-Projections&lt;/a&gt; has almost all the conditions and the only problem is that it is not attractive for children (Also, one can find other games in Hamkins's page; see &lt;a href=&quot;http://jdh.hamkins.org/category/math-for-kids/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://jdh.hamkins.org/category/math-for-kids/page/2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). For instance, the &lt;a href=&quot;https://www.chiark.greenend.org.uk/%7Esgtatham/puzzles/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simon Tatham's Puzzles&lt;/a&gt; (&lt;a href=&quot;https://play.google.com/store/apps/details?id=name.boyle.chris.sgtpuzzles&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link to the game for mobile&lt;/a&gt;) are eligible for almost every factor, except that their biggest drawback is that it is either impossible or very difficult to play without a mobile phone or computer.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;EDITED:&lt;/p&gt;&#xA;&lt;p&gt;I forgot to mention popular games like &lt;a href=&quot;https://en.wikipedia.org/wiki/Nine_men%27s_morris&quot; rel=&quot;nofollow noreferrer&quot;&gt;9 men's morris&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Morabaraba&quot; rel=&quot;nofollow noreferrer&quot;&gt;12 Men&lt;/a&gt; and their simpler variations. Of course, these are not the only popular two-player games that are easy to make and play, which belong to this brilliant category (brilliant category of simple light classic board games). Each game in this brilliant category satisfies most of the above features, and their main drawback is that they are two-player, and may not be appealing to children.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;EDITED:&lt;/p&gt;&#xA;&lt;p&gt;I came across &lt;a href=&quot;https://mathoverflow.net/questions/281447/mathematical-games-interesting-to-both-you-and-a-5-year-old-child&quot;&gt;this MO post&lt;/a&gt;, but I have not read it yet, I think I should read it patiently. Probably, &lt;a href=&quot;https://mathoverflow.net/questions/13638/which-popular-games-are-the-most-mathematical&quot;&gt;this page&lt;/a&gt; may contain related content, or perhaps even &lt;a href=&quot;https://mathoverflow.net/questions/290318/which-popular-games-have-been-studied-mathematically&quot;&gt;this page&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I know some other games, but I am not an expert in the field of mathematical-pedagogy. I would welcome stronger connections [E-mail/ chat conservation/ video conversation/ ...] with educators to receive more advice from them. My main purpose in raising this issue here, is to find lots of games; preferably with the properties listed above.&lt;/p&gt;&#xA;&lt;p&gt;Consider that my goal is to get acquainted with a &lt;strong&gt;significant number of engaging and fun thinking-games, that reinforce aspects of mental, mathematical, and logical skills&lt;/strong&gt; in the form of games; however, &lt;strong&gt;even a single offer would be welcome&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8883" LastEditorUserId="8883" LastEditDate="2020-08-02T14:52:20.090" LastActivityDate="2020-08-03T11:04:58.760" Title="Appealing thinking games, that are easy to make; easy to learn, and easy to play" Tags="|mathematical-pedagogy|games|" AnswerCount="2" CommentCount="13" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18613" PostTypeId="2" ParentId="18612" CreationDate="2020-07-31T18:35:05.100" Score="1" Body="&lt;p&gt;&lt;strong&gt;Set&lt;/strong&gt; ( &lt;a href=&quot;https://en.wikipedia.org/wiki/Set_(card_game)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Set_(card_game)&lt;/a&gt; ) Set is my go to for primary school:  You could make your own version without too much trouble. It's about pattern matching, it's real time, as many students as you want can play it at once, and its very simple. 8 year olds can beat 16 years but it's completely skill/observation based. It seems to hit most of your criteria anyway. There's an online version here if you want to try it out: &lt;a href=&quot;https://smart-games.org/en/set/start&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://smart-games.org/en/set/start&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2020-07-31T18:35:05.100" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18616" PostTypeId="1" CreationDate="2020-08-03T03:43:29.450" Score="6" ViewCount="437" Body="&lt;p&gt;I am having problems understanding &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers/&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic number theory/&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic analysis. I have tried some notes on the internet, but these notes were not helpful.&lt;/p&gt;&#xA;&lt;p&gt;Can anyone suggest a book, lecture notes or video that gives a good introduction?&lt;/p&gt;&#xA;&lt;p&gt;For example, I find Jeffrey Stopple (A Primer of Analytic Number Theory), John Stillwell (Elements of Number Theory) and Ian Stewart's (Galois Theory) books comprehensible, as these books are written &lt;strong&gt;in quite a descriptive way, in a friendly elaborate manner&lt;/strong&gt;, so anything like these books will be good.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Postscript:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Is there any technical term for mathematical books which are descriptive and explain in an elementary way like the above-mentioned books of John Stillwell and Ian Stewart?&lt;/p&gt;&#xA;" OwnerUserId="13263" LastEditorUserId="376" LastEditDate="2020-08-03T11:43:49.847" LastActivityDate="2020-08-10T12:53:00.317" Title="Introductory book or other resource on $p$-adic numbers/number theory/analysis" Tags="|textbooks|self-learning|mathematical-analysis|reference-books|number-theory|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18617" PostTypeId="2" ParentId="18612" CreationDate="2020-08-03T11:04:58.760" Score="1" Body="&lt;p&gt;I am writing about 2 number sense games. Some of these are pencil and paper, but all have been received enthusiastically by my students.  In addition the students would play these games when I was not around.&lt;/p&gt;&#xA;&lt;p&gt;a)  Guess the number.  The enthusiasm for this game among my students always surprises me, but it has been a favorite for over 25 years.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;One person, is the &amp;quot;teacher&amp;quot;.  S/he picks a number - it can be any number.  S/he writes it down on a piece of paper and folds the paper up and puts it somewhere that everyone playing can see.&lt;/li&gt;&#xA;&lt;li&gt;The &amp;quot;teacher&amp;quot; needs a large paper to write on.  (or blackboard, or whiteboard, or anything).  The important things is that when s/he holds it up, everyone will see it.&lt;/li&gt;&#xA;&lt;li&gt;The &amp;quot;teacher&amp;quot; calls on people in order to guess the number.  After each number is guessed, the teacher writes it down at the top of the paper/board if it is larger than the number picked and says lower OR at the bottom of the board if it is smaller than the number picked and says higher.&lt;/li&gt;&#xA;&lt;li&gt;Subsequent high guesses are written at the top of the board under previous high guesses and subsequent low guesses are written at the bottom of the board on top of previous low guesses.&lt;/li&gt;&#xA;&lt;li&gt;After several guesses the board might look like this:&#xA;37,000,222&#xA;60,000&#xA;49,000&#xA;10,001&#xA;19&#xA;This space is empty and room for future guesses.&lt;br /&gt;&#xA;10&lt;br /&gt;&#xA;5&lt;/li&gt;&#xA;&lt;li&gt;When someone finally guesses the number, the teacher unfolds the paper with the number to show that the guesser is right.  Then the guesser becomes the &amp;quot;teacher&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;If a real teacher shows the class, the teacher, might pick a regular number, then a very large number, than a negative number, then a fraction.  This will give the students an idea of the range of possibilities.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;b) Create the numbers from 1 - 100&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;You will be making numbers using 5 dice.  Depending on the abilities of the students, you can use addition, subtraction, division, multiplication, exponents, parentheses, square roots, parentheses.  Students can all put two numbers &amp;quot;together&amp;quot; so that if one dice has 6 and one dice has 1, you can make 16 or 61.&lt;/li&gt;&#xA;&lt;li&gt;This can be played solitaire, or as a team, or competitively.  My students loved to play this as solitaire once they finished their classwork.&lt;/li&gt;&#xA;&lt;li&gt;Students start with papers where they make a list of numbers from 1 to 100 with room for their work next to each number. (IF they are playing as  team you only need one list per team) They also needs lot of scrap paper.&lt;/li&gt;&#xA;&lt;li&gt;The dice is spun (students take turns spinning).  Students must use all 5 numbers to make any number from 1 - 100.  They put their expression next to the number.&lt;/li&gt;&#xA;&lt;li&gt;First one to make all the numbers wins. This can take time to do - so teams may be preferable or you can end after 10 or 20 spins.  Then whoever made the most numbers wins or whoever made the smallest or largest number wins etc.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-08-03T11:04:58.760" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18618" PostTypeId="1" CreationDate="2020-08-03T11:38:15.143" Score="8" ViewCount="551" Body="&lt;p&gt;I have come across this question in a textbook&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How many factors are there in the term &lt;span class=&quot;math-container&quot;&gt;$5ab(x+y)$&lt;/span&gt;? State what they are&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is being praised because it encourages thinking, which it does. However, I'm not sure if there is really a clear-cut answer.&lt;/p&gt;&#xA;&lt;p&gt;One could argue that there are two factors &lt;span class=&quot;math-container&quot;&gt;$5ab$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x+y)$&lt;/span&gt;. But then I've also imagine arguments for &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x+y)$&lt;/span&gt; each being factors giving 4 in total. However, the problem with that is that each of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; could be further decomposed if they are not prime.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What do you think? What definition would you give to your students&#xA;of what a factor is in light of this question?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;They also call &lt;span class=&quot;math-container&quot;&gt;$5ab(x+y)$&lt;/span&gt; a &lt;em&gt;term&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;For me, it looks like a two-term expression.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;What definition would you give to your students as to what a term is?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14007" LastActivityDate="2020-08-05T16:48:50.700" Title="Definitions of factors and terms" Tags="|secondary-education|definitions|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18619" PostTypeId="2" ParentId="18618" CreationDate="2020-08-03T12:05:52.397" Score="3" Body="&lt;p&gt;I agree that it isn't a term.  A term is a monomial and this can be simplified to 2 monomials.  Certainly if I was collecting like terms, I would simplify this first.&lt;/p&gt;&#xA;&lt;p&gt;As for factors, I would count: 5, a, b, and (x+y).  The reason is that if I was looking for a common factor (if this was part of a long expression with other terms), I might factor out any one of these: 5, a, b, or (x+y).  I might also factor out a combination of factors, such as 5a or ab.  Since a and b are variables I wouldn't worry about their factorizations.&lt;/p&gt;&#xA;&lt;p&gt;I would be curious to see if others agree.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-08-03T12:05:52.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18620" PostTypeId="1" CreationDate="2020-08-03T12:21:32.300" Score="2" ViewCount="853" Body="&lt;p&gt;In some texts, a critical point is when the derivative exists and is zero, and a singular point is when the derivative does not exist. So I suppose, at &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$|x|$&lt;/span&gt; would have a singular point while &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; would have a critical point.&lt;/p&gt;&#xA;&lt;p&gt;I've usually seen critical points include both cases, but I'm wondering if there's an advantage to using the more precise exclusive definition?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user507" LastEditDate="2020-08-17T23:07:00.953" LastActivityDate="2020-08-17T23:07:00.953" Title="In single variable calculus, do you distinguish between critical and singular points?" Tags="|calculus|terminology|derivative|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18622" PostTypeId="2" ParentId="18620" CreationDate="2020-08-03T13:27:44.490" Score="5" Body="&lt;p&gt;The main reason we have the word &amp;quot;critical point&amp;quot; is for the first derivative test.  The statement is slightly easier if you only have to say &amp;quot;critical point&amp;quot; instead of &amp;quot;critical point or singular point&amp;quot;.  However, students might actually remember to check for singular points if they had a different word for this case.  So I can see both sides of the argument.&lt;/p&gt;&#xA;&lt;p&gt;An activity I do like to do is present several pictures, and get students to capture the behavior symbolically.  I draw pictures of a function which looks locally like &lt;span class=&quot;math-container&quot;&gt;$x \mapsto x^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x \mapsto x^3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x \mapsto |x|$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x \mapsto \sqrt[3]{x}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x \mapsto |\sqrt[3]{x}|$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x \mapsto x\sin(1/x)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x \mapsto x^2\sin(1/x)$&lt;/span&gt; (the point of interest for all of these functions is the origin, but the sketch I draw could be centered anywhere).  Students think of some of these graphs as smooth even when the function isn't differentiable (like &lt;span class=&quot;math-container&quot;&gt;$x \mapsto \sqrt[3]{x}$&lt;/span&gt;).  So we spend time just playing with these, writing limits of SDQs which represent the different situations, and play with &amp;quot;classifying&amp;quot; these different kinds of situations.  This is good preparation for understanding the diversity of different critical points which are possible.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-08-03T13:49:50.333" LastActivityDate="2020-08-03T13:49:50.333" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18623" PostTypeId="2" ParentId="18618" CreationDate="2020-08-03T14:00:47.567" Score="3" Body="&lt;p&gt;[I'm answering your questions in the order I usually introduce these words to beginning algebra students.]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;What definition would you give to your students as to what a term is?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In our in-house texts, we teach that &lt;em&gt;terms&lt;/em&gt; are:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Parts of expressions separated by addition or subtraction symbols, except those between &amp;quot;grouping symbols&amp;quot; or in implied groups.&lt;/em&gt; [We then go on to identify grouping symbols as including all forms of parentheses, () [] {}; radical symbol, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\phantom{x}}$&lt;/span&gt;; fraction bar; exponents]&lt;/p&gt;&#xA;&lt;p&gt;This would make &lt;span class=&quot;math-container&quot;&gt;$5ab(x+y)$&lt;/span&gt; a single-term expression, as the addition is happening inside of parentheses.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What definition would you give to your students of what a factor is in light of this question?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We use such problems in our texts, where the intent is for students to identify &lt;em&gt;factors&lt;/em&gt; as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Numbers or variables separated by multiplication symbols, whether explicitly-written (e.g. &lt;span class=&quot;math-container&quot;&gt;$\ast$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cdot$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;) or implied (e.g. concatenation or integer exponents)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This would give four factors in your expression:  &lt;span class=&quot;math-container&quot;&gt;$$5ab(x+y) \implies 5, a, b, x+y$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This definition (my wording) is given to students immediately before introducing the concept of &lt;em&gt;factoring&lt;/em&gt; an expression, such as &lt;span class=&quot;math-container&quot;&gt;$5a^3b(c+1)+7a^2b^2(c+1)$&lt;/span&gt;. They are taught to list out the factors in each term (there are two terms) so they can decide what can be factored (usually by expanding exponents and then circling or underlining the factors that match):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$5a^3b(c+1)+7a^2b^2(c+1)=5\underline{aa}\phantom{.}a\phantom{.}\underline{b}\phantom{.}(\underline{c+1}) + 7\underline{aa}\phantom{.}\underline{b}\phantom{.}b\phantom{.}(\underline{c+1}) = a^2b(c+1)\left(5a+7b\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-08-03T14:00:47.567" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18624" PostTypeId="2" ParentId="18618" CreationDate="2020-08-03T16:01:23.113" Score="0" Body="&lt;p&gt;In my experience (German high school teacher):&lt;/p&gt;&#xA;&lt;p&gt;A factor is an operand in a multiplication and I would identify 5, a, b and (x+y) as factors.&lt;/p&gt;&#xA;&lt;p&gt;For &amp;quot;term&amp;quot;, I'm aiming for a distinction between terms and equations. Thus a quite good definition of a term could be &amp;quot;Any legal sentence of mathematical symbols except equations&amp;quot;. Depending on the previous knowledge, relations or other things might have to be excluded.&lt;/p&gt;&#xA;&lt;p&gt;Another way to express this is that a term is something that describes a calculation in symbolic form.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps the only unambiguous way (but in my opinion, not helpful for &amp;quot;secondary-education&amp;quot;) is to define a term with a context free grammar:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A number is a term.&lt;/li&gt;&#xA;&lt;li&gt;A variable is a term.&lt;/li&gt;&#xA;&lt;li&gt;If a and b are terms, a+b is a term.&lt;/li&gt;&#xA;&lt;li&gt;... [expand to known operations] ...&lt;/li&gt;&#xA;&lt;li&gt;Nothing else is a term.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2020-08-04T13:03:33.503" LastActivityDate="2020-08-04T13:03:33.503" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18625" PostTypeId="1" CreationDate="2020-08-03T16:29:33.917" Score="9" ViewCount="623" Body="&lt;p&gt;I am looking for literature related to fear of notation in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;It is even heard that the font size  and font type make a reader reluctant to study mathematical literature, often lecture notes, book has hazily-appeared (poor formatting,also not well spaced, unpleasant font) writing, and difficult to understand description that takes a lot time, specially if the reader's native tongue is not English.&lt;/p&gt;&#xA;&lt;p&gt;Is there any literature that discuss the issue or talk about any solution? Thanks.&lt;/p&gt;&#xA;" OwnerUserId="13263" LastEditorUserId="13263" LastEditDate="2020-08-09T15:46:11.490" LastActivityDate="2020-08-09T16:42:00.080" Title="Fear of notation and hazily-appeared writing in Mathematics" Tags="|undergraduate-education|mathematical-pedagogy|reference-request|self-learning|notation|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18626" PostTypeId="2" ParentId="18618" CreationDate="2020-08-03T17:33:15.433" Score="1" Body="&lt;p&gt;To further complicate matters, note that definitions in two of the answers (so far) either say or at least could be interpreted to say that &lt;span class=&quot;math-container&quot;&gt;$5a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$5ab$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ab(x+y)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b(x+y)$&lt;/span&gt; are also factors. And I've probably used &amp;quot;factor&amp;quot; to also include things like &lt;span class=&quot;math-container&quot;&gt;$5b$&lt;/span&gt;, for example if I wanted to cancel a factor from the numerator and denominator of a fraction.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2020-08-03T17:33:15.433" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18627" PostTypeId="2" ParentId="18618" CreationDate="2020-08-03T19:09:51.390" Score="5" Body="&lt;p&gt;There are at least 16 factors: anything of the form &lt;span class=&quot;math-container&quot;&gt;$5^ia^jb^k(x+y)^l$&lt;/span&gt;, where the exponents are all either 0 or 1. You could in fact say that there are infinitely many factors. For instance, I could say that the expression contains a factor of &lt;span class=&quot;math-container&quot;&gt;$5/7$&lt;/span&gt;, and also a factor of &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;. The only moral I can see to this example is that the normal way we talk about words like &amp;quot;factor&amp;quot; is very general, and if we want to get more specific, we need to use more specific definitions. For example, if you consider the field of polynomials in 4 variables with integer coefficients, then factoring becomes more well defined. If that's the intended moral of the exercise, then I guess that's OK. But that doesn't mean we should demand an exact definition of &amp;quot;factor&amp;quot; for all contexts. There can and should be many different definitions of a general, flexible term like this.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-08-03T19:09:51.390" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18628" PostTypeId="2" ParentId="18618" CreationDate="2020-08-04T05:57:49.190" Score="2" Body="&lt;p&gt;&lt;strong&gt;&lt;em&gt;tl;dr&lt;/em&gt;–&lt;/strong&gt;  Probably best to focus on having students draw &lt;a href=&quot;https://en.wikipedia.org/wiki/Abstract_syntax_tree&quot; rel=&quot;nofollow noreferrer&quot;&gt;abstract-syntax-trees&lt;/a&gt;.  That said, an expression's &lt;strong&gt;&lt;em&gt;terms&lt;/em&gt;&lt;/strong&gt; are the addends of the top-level summation; if the explicit top-level operator isn't a summation, then we regard it as a single-addend summation, where the entire explicit expression is the only addend, and thus the only term.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h1&gt;Students might be better off focusing on &lt;em&gt;abstract-syntax-trees&lt;/em&gt;.&lt;/h1&gt;&#xA;&lt;p&gt;Probably best to refocus the question on drawing the &lt;a href=&quot;https://en.wikipedia.org/wiki/Abstract_syntax_tree&quot; rel=&quot;nofollow noreferrer&quot;&gt;abstract syntax tree&lt;/a&gt; for the expression instead.&lt;/p&gt;&#xA;&lt;p&gt;There are multiple correct answers.  For example,&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;  *&#xA; / \&#xA;5   *&#xA;   / \&#xA;  a   *&#xA;     / \&#xA;    b   +&#xA;       / \&#xA;      x   y&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;, which focuses on binary operators, or&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;   *&#xA;  / \&#xA; /| |\&#xA;5 a b +&#xA;     / \&#xA;    x   y&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;, where there's a single, quaternary (or &lt;span class=&quot;math-container&quot;&gt;$n\text{-ary})$&lt;/span&gt; product operator describable as a folding of the root binary operator from the first example with its subordinate product operators.&lt;/p&gt;&#xA;&lt;p&gt;There're two most correct answers to this question:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The factors are &lt;span class=&quot;math-container&quot;&gt;$\left\{5, \, ab\left(x+y\right)\right\}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The factors are &lt;span class=&quot;math-container&quot;&gt;$\left\{5, \, a, \, b, \, \left(x+y\right)\right\}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The first answer's a tad more technical, as it avoids assuming properties like commutativity and associativity, making it more general if we're constructing math.  The second answer'd probably be preferred if students are to assume commutativity and associativity.&lt;/p&gt;&#xA;&lt;p&gt;Regardless, as long as students can demonstrate an understanding of the abstract-syntax-tree behind an expression, that'd seem sufficient.  No need to push them toward adopting unnecessary notions of what a &amp;quot;&lt;em&gt;term&lt;/em&gt;&amp;quot; ought to be taken as in this context.&lt;/p&gt;&#xA;&lt;p&gt;Overall, it looks like a good question once we fix it up a bit.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h1&gt;Definition of &amp;quot;&lt;em&gt;term&lt;/em&gt;&amp;quot;.&lt;/h1&gt;&#xA;&lt;p&gt;As explained above, I don't think it's productive to get too focused on terminology.  Seems to promote limited, inflexible understandings.&lt;/p&gt;&#xA;&lt;p&gt;That said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;They also call &lt;span class=&quot;math-container&quot;&gt;$5ab(x+y)$&lt;/span&gt; a &lt;em&gt;term&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;For me, it looks like a two-term expression.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;What definition would you give to your students as to what a term is?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When you have any expression, e.g. &lt;span class=&quot;math-container&quot;&gt;$`` 5ab \left(x+y\right) &quot; ,$&lt;/span&gt; a &lt;strong&gt;&lt;em&gt;term&lt;/em&gt;&lt;/strong&gt; is a component of the top-level summation.  In this case, the top-level summation is trivial, and the single term is simply &lt;span class=&quot;math-container&quot;&gt;$`` 5ab \left(x+y\right) &quot; .$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In other words, if we draw the abstract-syntax-tree where there's a top-level-summation node, it's:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;   +&#xA;   |&#xA;   *&#xA;  / \&#xA; /| |\&#xA;5 a b +&#xA;     / \&#xA;    x   y&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;, where the the top-level &lt;span class=&quot;math-container&quot;&gt;$`` + &quot;\text{-node}$&lt;/span&gt; has a single element.  Since the &lt;em&gt;terms&lt;/em&gt; are the child nodes of the top-level-summation, that's just&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;   *&#xA;  / \&#xA; /| |\&#xA;5 a b +&#xA;     / \&#xA;    x   y&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;, which in linear notation is &lt;span class=&quot;math-container&quot;&gt;$`` 5ab \left(x+y\right) &quot; .$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is a well-defined concept, so it's not really ambiguous.  And the word &amp;quot;&lt;em&gt;term&lt;/em&gt;&amp;quot; is common in math-related texts, so it would be a good thing for students to know.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;Procedure for identifying an expression's terms.&lt;/h3&gt;&#xA;&lt;p&gt;To identify the terms in an expression:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Write the expression as an abstract-syntax-tree.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If the expression's top-level node &lt;strong&gt;&lt;em&gt;isn't&lt;/em&gt;&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$`` + &quot; ,$&lt;/span&gt; append a higher-level &lt;span class=&quot;math-container&quot;&gt;$`` + &quot;\text{-node}$&lt;/span&gt; which has the prior top-level node as a child-element.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The top-level node's child elements are the expression's &lt;strong&gt;&lt;em&gt;terms&lt;/em&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8943" LastEditorUserId="8943" LastEditDate="2020-08-05T16:48:50.700" LastActivityDate="2020-08-05T16:48:50.700" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18629" PostTypeId="1" CreationDate="2020-08-04T19:32:40.870" Score="6" ViewCount="411" Body="&lt;p&gt;At a previous institution, we used moodle for our learning management system, which has a fairly useful tool for creating quizzes that make it more difficult for the student to cheat.  In particular, you are able to use TeX code, as well as input &amp;quot;wildcards&amp;quot; so that you can add parameters to problems, and most importantly, you can put these wildcards &lt;em&gt;within the TeX code itself&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For example, I could type into moodle &amp;quot;Solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;:  &lt;span class=&quot;math-container&quot;&gt;$[k]x + 1 = 2$&lt;/span&gt;.&amp;quot;  Here, the parameter would be &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, I can decide on a range values of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; to consider, and moodle would give each student this problem with a random value of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; from my range.  The student would simply see: &amp;quot;Solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$3x + 1 = 2$&lt;/span&gt;.&amp;quot;, where &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; was one of the values of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; in my range.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;However, at my current institution, we use Blackboard, which allows use of TeX code and parameters, but doesn't allow you to use these parameters &lt;em&gt;inside&lt;/em&gt; the TeX code (as far as I know).&lt;/p&gt;&#xA;&lt;p&gt;Are you aware of any third-party software besides moodle that accomplishes all of these things:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Understands TeX code,&lt;/li&gt;&#xA;&lt;li&gt;Allows the use of parameters in some way to randomize numbers for each student,&lt;/li&gt;&#xA;&lt;li&gt;Allows of these parameters within the TeX code?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="14385" LastActivityDate="2020-08-19T19:35:28.520" Title="Do you know of any third-party math-friendly testing software?" Tags="|assessment|online-instruction|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18630" PostTypeId="2" ParentId="18629" CreationDate="2020-08-04T21:02:49.150" Score="3" Body="&lt;p&gt;I'll reiterate &lt;a href=&quot;https://matheducators.stackexchange.com/a/18399/470&quot;&gt;my recommendation&lt;/a&gt; of &lt;a href=&quot;https://www.myopenmath.com/info/aboutus.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;MyOpenMath&lt;/a&gt;. To your points:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It understands TeX (a simplified version, using &lt;em&gt;` math `&lt;/em&gt; instead of &lt;em&gt;\$ math \$&lt;/em&gt;)&lt;/li&gt;&#xA;&lt;li&gt;It sllows the use of parameters in some way to randomize numbers for each student&lt;/li&gt;&#xA;&lt;li&gt;It allows these parameters within the TeX code.&lt;/li&gt;&#xA;&lt;li&gt;It can be integrated into Blackboard (that's how I've been using it) using an LTI link. (&lt;a href=&quot;https://www.youtube.com/watch?v=HHxT2Aewj9k&quot; rel=&quot;nofollow noreferrer&quot;&gt;video&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Here are some problems I wrote, each with two randomized versions I got by clicking &amp;quot;New Version&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem #1&lt;/strong&gt;. [Same problem coding, initiated by random &amp;quot;seed&amp;quot; numbers. This problem took 6 lines of code to write, and you can get pretty specific with how you want students to enter their answers. I left it totally open to accept decimals, fractions or expressions, provided they are equivalent to the correct answer.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mQF5F.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mQF5F.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rX7pg.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rX7pg.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem #2&lt;/strong&gt;. [Again, same problem coding, initiated by random &amp;quot;seed&amp;quot; numbers. Note that you can randomize function names and randomly order the presentation of the problem.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/HblTR.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HblTR.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gK4U3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gK4U3.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem #3&lt;/strong&gt;. [Same problem coding, initiated by random &amp;quot;seed&amp;quot; numbers. For the answers involving expressions, you can enter any algebraic expression equivalent to the correct answer (e.g. x(3x-1/x) or 3x^2-1+sin(0). It has built-in graphing capabilities that can accept random elements. Again, you can specify how the answer must look, such as being factored, etc. This particular problem was &amp;quot;scaffolded&amp;quot; so a student must get the first part correct before moving on to the next part. Each part can have hints, etc.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OTzKB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OTzKB.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jRKeL.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jRKeL.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The math is almost TeX, just with slightly simplified commands (MOM was coded in php). For example, one can type `frac(3)(4)` or just `3/4` to get &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4}$&lt;/span&gt;. Or for integrals, you can type `int_5^(3x) g(t)dt` to get &lt;span class=&quot;math-container&quot;&gt;$\int_5^{3x} g(t)dt$&lt;/span&gt;. [Notice the use of ( ) instead of { }, as well as the lack of \ characters.]&lt;/p&gt;&#xA;&lt;p&gt;Anyway -- I hope this helps. This program has really saved me these past few terms.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[Edit] In case you wanted to see the code for that first problem:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;\$anstypes=&amp;quot;calculated,numfunc&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;\$answerformat[1]=&amp;quot;equation&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;\$a,\$b,\$c,\$d=nonzerodiffrands(-6,6,4)&lt;/p&gt;&#xA;&lt;p&gt;\$func=makexxpretty(&amp;quot;\$a x + \$b&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;\$answer[0]=\$a*\$c + \$b&lt;/p&gt;&#xA;&lt;p&gt;\$answer[1]=&amp;quot;x=(\$d-\$b)/\$a&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Then the problem text looks like:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Suppose `f(x)=\$func`.&lt;/p&gt;&#xA;&lt;p&gt;Evaluate `f(\$c)`.&lt;/p&gt;&#xA;&lt;p&gt;Answer: \$answerbox[0]&lt;/p&gt;&#xA;&lt;p&gt;Solve the equation `f(x) = \$d`. [Remember to give your answer in the form &amp;quot;x = ...&amp;quot;]&lt;/p&gt;&#xA;&lt;p&gt;Answer: \$answerbox[1]&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2020-08-04T21:15:14.193" LastActivityDate="2020-08-04T21:15:14.193" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18631" PostTypeId="2" ParentId="18625" CreationDate="2020-08-04T23:06:20.093" Score="3" Body="&lt;p&gt;This is not a direct answer to your question, but it is literature&#xA;that addresses at least one aspect of the issue you raise.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.stanford.edu/%7Ekdevlin/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Keith Devlin&lt;/a&gt; wrote an article entitled,&#xA;&amp;quot;The Symbol Barrier&amp;quot;:&#xA;&lt;a href=&quot;http://www.thedailyriff.com/articles/learning-math-the-symbol-barrier-891.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&#xA;The point of his article is to argue that video games can help&#xA;overcome &amp;quot;the greatest obstacle to good mathematical learning.&amp;quot;&#xA;But setting video games aside, he does discuss &amp;quot;the symbol barrier.&amp;quot;&#xA;Some quotes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the biggest obstacle to practical mastery of middle school math: the symbol barrier.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We teach mathematics symbolically because, for many centuries, symbolic representation has been the most effective way to record mathematics and pass on mathematical knowledge to others.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The one exception, where the brain does not require the aid of symbolic representations (...) is what for several years now I have been calling &amp;quot;everyday mathematics.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is with everyday mathematics that the symbol barrier emerges.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When ordinary people are faced with doing everyday math regularly as part of their everyday lives, they rapidly achieve a high level of proficiency (typically hitting that 98% mark). Yet their performance drops to the 35 to 40% range when presented with the same problems in symbolic form.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is simply not the case that ordinary people cannot do everyday math. Rather, they cannot do symbolic everyday math.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;That, my friend, is the symbol barrier.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is a link to a Powerpoint presentation&#xA;with more detail:&#xA;&lt;a href=&quot;https://web.stanford.edu/%7Ekdevlin/Presentations/Symbol%20barrier.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-08-04T23:15:10.057" LastActivityDate="2020-08-04T23:15:10.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18632" PostTypeId="1" AcceptedAnswerId="18634" CreationDate="2020-08-05T02:16:56.023" Score="4" ViewCount="198" Body="&lt;p&gt;When evaluating the limit of &lt;span class=&quot;math-container&quot;&gt;$f(x, y)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$(x_0, y_0)$&lt;/span&gt;, we should or should not consider only those &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; in the domain of &lt;span class=&quot;math-container&quot;&gt;$f(x, y)$&lt;/span&gt; ? I am confused by different practices of Calculus textbooks. Have anyone searched and found some authoritative opinion ?&lt;/p&gt;&#xA;&lt;p&gt;Thomas Calculus 14e §14.2 Example 2 (Page 802-803) &lt;span class=&quot;math-container&quot;&gt;$\lim_{(x, y) \to (0, 0)} \frac{x^2 - x y}{\sqrt{x} - \sqrt{y}}$&lt;/span&gt; considers only those &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; in the domain. The authors' answer (&lt;span class=&quot;math-container&quot;&gt;$\mathbf{0}$&lt;/span&gt;) is the same as &lt;a href=&quot;https://www.wolframalpha.com/input/?i=lim_%7B%28x%2C%20y%29%20-%3E%20%280%2C%200%29%7D%20%28x%5E2%20-%20x%20y%29%20%2F%20%28sqrt%28x%29%20-%20sqrt%28y%29%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;the answer by WolframAlpha&lt;/a&gt; . See &lt;a href=&quot;https://i.stack.imgur.com/vSqoW.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;textbook page 802&lt;/a&gt; and &lt;a href=&quot;https://i.stack.imgur.com/0kUyp.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;textbook page 803&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;p&gt;Larson Calculus 10e §13.2 Exercise 27 (Page 887) &lt;span class=&quot;math-container&quot;&gt;$\lim_{(x, y) \to (0, 0)} \frac{x - y}{\sqrt{x} - \sqrt{y}}$&lt;/span&gt; considers &lt;strong&gt;NOT&lt;/strong&gt; only those &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; in the domain. The authors' answer (&lt;strong&gt;DNE&lt;/strong&gt;) is &lt;strong&gt;NOT&lt;/strong&gt; the same as &lt;a href=&quot;https://www.wolframalpha.com/input/?i=lim_%7B%28x%2C%20y%29%20-%3E%20%280%2C%200%29%7D%20%28x%20-%20y%29%20%2F%20%28sqrt%28x%29%20-%20sqrt%28y%29%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;the answer by WolframAlpha&lt;/a&gt; (&lt;span class=&quot;math-container&quot;&gt;$\mathbf{0}$&lt;/span&gt;). See &lt;a href=&quot;https://i.stack.imgur.com/jez7z.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;textbook page 887&lt;/a&gt; and &lt;a href=&quot;https://i.stack.imgur.com/MIOJS.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;solution manual page 1268&lt;/a&gt; .&lt;/p&gt;&#xA;" OwnerUserId="14386" LastEditorUserId="14386" LastEditDate="2020-08-05T23:54:43.640" LastActivityDate="2020-08-06T01:56:25.030" Title="When evaluating the limit of $f(x, y)$ as $(x, y)$ approaches $(x_0, y_0)$, should we consider only those $(x, y)$ in the domain of $f$?" Tags="|calculus|limits|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18633" PostTypeId="1" AcceptedAnswerId="18635" CreationDate="2020-08-05T02:27:03.693" Score="6" ViewCount="170" Body="&lt;p&gt;I'm hunting online to find a remote math tutor for my 9 and 11 year olds. I have hired tutors in the past ranging from high school students to experienced teachers and my luck has varied.&lt;/p&gt;&#xA;&lt;p&gt;What are the specific questions that one should ask prospective tutors to determine if they are a good fit as a tutor for one's kids? i.e. what attributes would they have to make them appropriate remote math tutors?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/95/how-to-choose-test-future-tutors&quot;&gt;This question&lt;/a&gt; is similar but addresses it from an Educators perspective. I'm coming at this from a naive parent's perspective and I don't have the expertise that the participants in that question have.&lt;/p&gt;&#xA;" OwnerUserId="14387" LastActivityDate="2020-08-05T13:57:30.873" Title="How do I select an appropriate math tutor?" Tags="|tutoring|" AnswerCount="1" CommentCount="10" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18634" PostTypeId="2" ParentId="18632" CreationDate="2020-08-05T12:25:27.523" Score="9" Body="&lt;p&gt;Most definitions of a limit of a function only include the domain. See the &lt;a href=&quot;https://en.wikipedia.org/wiki/Limit_of_a_function#More_general_subsets&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; about limits or this &lt;a href=&quot;https://www.maa.org/press/maa-reviews/introduction-to-analysis&quot; rel=&quot;noreferrer&quot;&gt;textbook&lt;/a&gt; about Real Analysis (or the various textbooks in &lt;a href=&quot;https://math.stackexchange.com/questions/3780409/when-evaluating-the-limit-of-fx-y-as-x-y-approaches-x-0-y-0-shou&quot;&gt;this&lt;/a&gt; Math StackExchange answer from @twJizhan's comment).&lt;/p&gt;&#xA;&lt;p&gt;But this is only what the definition is, so I'll try to provide a few (informal) arguments about why the definition should be this way.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Generality&lt;/strong&gt;: Because we &lt;strong&gt;can&lt;/strong&gt; define limits in such a way that makes more functions have limits at more points in a reasonable manner, we might as well. Of course, we must be conscious of the domain and codomain of functions, but we should always be conscious of them anyway.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Notation&lt;/strong&gt;: It's quite weird to talk about the behavior of a function &lt;strong&gt;outside&lt;/strong&gt; of it's domain. Every function is undefined outside of its domain; to make a definition like this reasonable, we must have a notion of an &amp;quot;ambient space&amp;quot; that our function lives in. If we define limits in this way, we must not only keep track of the (co)domain of the function (not in the standard limit notation), but also this &amp;quot;ambient space&amp;quot;, also not in the notation.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Intuition&lt;/strong&gt;: An informal understanding of &lt;span class=&quot;math-container&quot;&gt;$\lim_{x \to p} f(x) = L$&lt;/span&gt; is that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; approximates &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; arbitrarily well when given inputs &amp;quot;around&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;. This informal understanding works just fine with only considering the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; (&amp;quot;of course&amp;quot; we can only give &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; inputs in its domain), but falls apart a little if we require that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; be defined in some (punctured) open ball around &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; w.r.t the ambient space—then even a function that &lt;strong&gt;does&lt;/strong&gt; approximate &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; arbitrarily well, in a well-defined way, will no longer have a limit at &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14206" LastEditorUserId="14206" LastEditDate="2020-08-06T01:56:25.030" LastActivityDate="2020-08-06T01:56:25.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18635" PostTypeId="2" ParentId="18633" CreationDate="2020-08-05T13:57:30.873" Score="5" Body="&lt;p&gt;To keep my answer specific, I'd probably say the biggest three factors in any personalized teacher would be (1) flexibility in the material (2) compatibility with students (3) inventiveness in the online classroom.&lt;/p&gt;&#xA;&lt;p&gt;For 1) since the teaching will be personalized, the teacher would need to know the material well enough to interact with spontaneous questions, and being able to cater the pace to how quick or slow the student. For the age range, it seems like early &amp;quot;middle school&amp;quot; (and so I suppose beginning algebra? Like introduction to variables and equations and solving them?), so you'd also need to have a tutor who is experienced with handling those kind of questions.&lt;/p&gt;&#xA;&lt;p&gt;(Speaking from my own experience, although I could do middle school math, I can't really teach it since I easily run out of ways to relate ideas to the student. So knowledge, &lt;em&gt;as important as it is&lt;/em&gt;, is only as good as being able to communicate topics to someone new to it. So I'd probably look for experience in the math topics that you're looking for.)&lt;/p&gt;&#xA;&lt;p&gt;For 2) you'd probably just have to get lucky with finding someone who works well with not only kids but also your kids. Teaching style might have something to do with it, but also just the general atmosphere when doing the class. Are the tutors primarily in it for the money? The work experience to get a different job? That they like kids and want to educate young minds?&lt;/p&gt;&#xA;&lt;p&gt;(Again, for myself, when I've tutored, it's usually because I like communicating about problem solving with other people. I like how both student and teach and learn about the topic.)&lt;/p&gt;&#xA;&lt;p&gt;For 3) since it's online, you'd need to somehow see how much set-up they have for online materials? Do they have pdfs of worksheets? Videos to watch? Tablet with pen to sketch or are they really good with a mouse and Paint? If they've ever had technical difficulties, then what happened?&lt;/p&gt;&#xA;&lt;p&gt;Also, since you're looking online, would this person know your local school system so they know what kind of math will be happening in your children's school - or is the hiring for general mathematics knowledge, and so international tutors could be an option?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-08-05T13:57:30.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18637" PostTypeId="1" CreationDate="2020-08-05T20:21:57.110" Score="2" ViewCount="91" Body="&lt;p&gt;Crossposting from the math.stackexchange (&lt;a href=&quot;https://math.stackexchange.com/questions/3781227/a-question-about-online-teaching-aids?noredirect=1#comment7786199_3781227&quot;&gt;Question&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Online classes will start again as the semester will start, I bought an XP-Pen deco 03 to help me with online teaching. I will be teaching commutative algebra next semester. I would like to ask what should be the best software (whiteboard software or any other) that goes well with this pen tablet and easy to write mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I explored google &lt;a href=&quot;https://jamboard.google.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jamboard&lt;/a&gt; it is good, but minimal for the choices (there is just one pen, etc ). On the plus side, I can invite all my students and teach totally on the board (although the audio is not streamed through it for which I have to fall back on something like google meet, in that case sharing the window is better). Another thing I don't like about it is the lack of full-screen mode.&lt;/p&gt;&#xA;&lt;p&gt;I would like to ask people here for some suggestions.&lt;/p&gt;&#xA;" OwnerUserId="5453" LastActivityDate="2020-08-05T20:21:57.110" Title="Teaching aid for online mathematics course" Tags="|mathematical-pedagogy|online-instruction|online|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18638" PostTypeId="1" AcceptedAnswerId="18639" CreationDate="2020-08-06T15:59:34.317" Score="4" ViewCount="1202" Body="&lt;p&gt;Without getting bogged down in details, I'll list the names only. It seems that the strategy I generally use is this:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Divergence test first&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is it a recognizable form? p-series or geometric?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a) No negative terms? Integral, direct comparison, limit comparison?&lt;br /&gt;&#xA;b) Possibly negative terms? Alternating series test, root test, ratio test?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Those are my main categories, and then each test in the category has its own favourite type of series to work with (like ratio test with factorials and nth powers, for example) (edit: btw, the tests in each group aren't in any particular order)&lt;/p&gt;&#xA;&lt;p&gt;But oddly, the most common thing I stress when teaching series is perseverance and practice since it's often about recognizing known patterns in unknown series&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorUserId="13585" LastEditDate="2020-08-07T14:32:23.457" LastActivityDate="2020-08-07T14:32:23.457" Title="What strategy for picking convergence tests for series do you teach?" Tags="|calculus|series|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18639" PostTypeId="2" ParentId="18638" CreationDate="2020-08-06T17:50:22.053" Score="15" Body="&lt;p&gt;I made this flowchart for my students last time I taught this stuff. Not the best visually, but I think it effectively conveys my thought process.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/A7F7f.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/A7F7f.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-08-06T17:50:22.053" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18640" PostTypeId="2" ParentId="18638" CreationDate="2020-08-06T20:10:24.170" Score="5" Body="&lt;p&gt;I don't teach this course, and I think the desire to have an algorithm this detailed may be an example of the kind of thing that people will start to feel because they're teaching the class but that people in the real world don't actually use. Students want to be told that everything is rule-based, because it makes them feel safer.&lt;/p&gt;&#xA;&lt;p&gt;When I encounter this kind of thing in a real-world context like physics, I basically reason by analogy, and that pretty much always works.&lt;/p&gt;&#xA;&lt;p&gt;For example, I know that &lt;span class=&quot;math-container&quot;&gt;$\sum n^{-1}$&lt;/span&gt; diverges, but is on the ragged edge of convergence. This tells me that an example like &lt;span class=&quot;math-container&quot;&gt;$\sum 1/n\ln n$&lt;/span&gt; may or may not converge (the extra log factor might tip it over the edge). However, if it does converge, it will converge so slowly that it would be useless to try to evaluate in any practical context. So in an application, I'm done. I don't pursue this example any further.&lt;/p&gt;&#xA;&lt;p&gt;I don't think people in the real world know a list of tests, and in particular they don't think in terms of an &amp;quot;integral test.&amp;quot; They think in terms of an integral &lt;em&gt;analogy&lt;/em&gt;. If someone asks me to explain how I know that &lt;span class=&quot;math-container&quot;&gt;$\sum n^{-1}$&lt;/span&gt; diverges, I'll say, &amp;quot;Oh, it diverges logarithmically.&amp;quot; Never mind that what produces a log is the integral, not the sum -- everybody knows what is meant by this.&lt;/p&gt;&#xA;&lt;p&gt;If for some reason I really wanted to know for sure whether &lt;span class=&quot;math-container&quot;&gt;$\sum 1/n\ln n$&lt;/span&gt; converged in theory, the first thing I would do would be to try it in a CAS and see what it said. If it said it diverged, I would believe it. If it said it evaluated to some exact expression, I would also believe it. If it spit out a decimal approximation, I would know not to trust it. When I did this in the CAS Maxima, for this example, it said it didn't know.&lt;/p&gt;&#xA;&lt;p&gt;OK, so if I really still cared, the next thing I would do would be to try the analogous integral, which in Maxima is &lt;code&gt;integrate(1/(x*log(x)),x,1,inf);&lt;/code&gt;. It tells me it diverges. OK, cool, I'm done if all I want to know is the answer.&lt;/p&gt;&#xA;&lt;p&gt;In the real-world context I probably don't care about an actual proof, but if I did, I'd see if the CAS could do the indefinite integral. It turns out that it can, and the result is &lt;span class=&quot;math-container&quot;&gt;$\ln\ln x$&lt;/span&gt;. This shows me that it diverges, but ever so gently. If I care about writing up a human-readable argument to this effect that doesn't depend on sofwtare, then I can check that the derivative of &lt;span class=&quot;math-container&quot;&gt;$\ln\ln x$&lt;/span&gt; really is &lt;span class=&quot;math-container&quot;&gt;$1/x\ln x$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-08-06T20:10:24.170" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18641" PostTypeId="2" ParentId="18638" CreationDate="2020-08-07T00:35:14.117" Score="3" Body="&lt;p&gt;First, let's ignore geometric series and &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-series because those are standard examples.&lt;/p&gt;&#xA;&lt;p&gt;For infinite series with positive terms, if you &lt;em&gt;really&lt;/em&gt; understand how sequences grow then almost all examples that occur in basic courses can be handled by the limit comparison test unless the sequence has factorials in it, in which case you use the ratio test.&lt;/p&gt;&#xA;&lt;p&gt;The root test is largely irrelevant in basic courses because no important series with positive terms requires it. A student who goes on to study real math will find out that in fact the root test is theoretically very important as the basic idea behind Hadamard's radius of convergence formula (using &lt;span class=&quot;math-container&quot;&gt;$\varlimsup$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\lim$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2020-08-07T00:35:14.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18642" PostTypeId="2" ParentId="16948" CreationDate="2020-08-07T05:24:47.413" Score="4" Body="&lt;p&gt;It is difficult because there are many things implicitly done behind the scenes:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Function definition with case analysis usually for the first time.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Solving problems, among which are equations and inequalities, usually involves case analysis for all variables and all their possible values and then checking if solution satisfies initial conditions, or in other words just doing &lt;strong&gt;Backtracking&lt;/strong&gt;, but teachers don't do analogies showing general solving strategy, and for example comparing absolute value problem to some real life situation. They just do example and that's it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;and&lt;/strong&gt; and &lt;strong&gt;or&lt;/strong&gt;, De Morgan's laws etc statements when solving equations and inequalities, also doing set operations that teachers don't explain well.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students don't have good algebra intuition - not explaining in details what happens with distance interpretation &lt;span class=&quot;math-container&quot;&gt;$|x-y|$&lt;/span&gt; &amp;quot;What happens for example when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is negative, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is negative and other cases&amp;quot;, or just seeing &lt;span class=&quot;math-container&quot;&gt;$-x$&lt;/span&gt; that is positive.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Seeing other properties like reverse triangle inequality might be intimidating.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Graphing on &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; plane and seeing some stroked lines for the first time and then doing mapping with algebra equation is a challenge too.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Convoluting even more for example adding some parameters and stating some questions about solutions without explaining overall problem solving strategy.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Resemblance with &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; and other stuff that other commenters mentioned.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14399" LastEditorUserId="238" LastEditDate="2020-09-21T00:09:20.590" LastActivityDate="2020-09-21T00:09:20.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18643" PostTypeId="2" ParentId="18638" CreationDate="2020-08-07T05:37:49.240" Score="6" Body="&lt;p&gt;I think this is one of those places where teaching a detailed strategy is a form of &amp;quot;teaching to the test&amp;quot; that is counterproductive for the students' intellectual development.&lt;/p&gt;&#xA;&lt;p&gt;It's important to emphasize that all the convergence tests have preconditions that must be satisfied before one can use the rule, and one should not use a test without explicitly checking that those preconditions are satisfied.&lt;/p&gt;&#xA;&lt;p&gt;As students advance in their studies (no matter what they study), and when students use their knowledge outside of the classroom, they will often be in situations where they have a problem that they haven't been taught how to solve, but they do know a number of different potential strategies for solving the problem and will have to pick one or more to try.  The problem of testing a series for convergence is a good problem on which to develop the general skill of approaching and solving this kind of problem.  Part of the point of the course is to give them practice, with guidance, at developing their own strategy for this kind of problem.&lt;/p&gt;&#xA;&lt;p&gt;Sine finding formulas for antiderivatives is a similar kind of problem, I tell students on the first day of Calc II that the main purpose of the course is to develop this general skill, and in fact that this main purpose is more important than any kind of problem they learn how to solve (which, after all, can all be done by our favorite symbolic computation software).&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2020-08-07T05:37:49.240" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18645" PostTypeId="2" ParentId="18638" CreationDate="2020-08-07T09:27:34.760" Score="0" Body="&lt;p&gt;I taught this subject only once (two years ago). In that occasion, I wrote an one page summary for my students in the following order.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Divergence test&lt;/li&gt;&#xA;&lt;li&gt;a) Geometric&lt;br /&gt;&#xA;b) p-series&lt;br /&gt;&#xA;c) Alternating&lt;/li&gt;&#xA;&lt;li&gt;Ratio&lt;/li&gt;&#xA;&lt;li&gt;Root&lt;/li&gt;&#xA;&lt;li&gt;Integral&lt;/li&gt;&#xA;&lt;li&gt;Comparison&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="13585" LastEditorUserId="13585" LastEditDate="2020-08-07T09:33:15.203" LastActivityDate="2020-08-07T09:33:15.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18646" PostTypeId="1" CreationDate="2020-08-07T10:55:09.523" Score="1" ViewCount="221" Body="&lt;p&gt;Context: I'm a Grade 11 IBDP student and very fond of math. Assuming that I get the spare time to self-study material alongside the syllabus, I wish to spend it constructively (the only step I've taken thus far in learning mathematics is to complete HS math courses on Khan Academy up to and including Integral Calculus). I want to tackle a heavy-duty calculus book like Spivak or Apostol before entering college (if you feel aspiring mathematicians in HS would be better served by other objectives, please share them in your answer).&lt;/p&gt;&#xA;&lt;p&gt;I decided against compiling a list of textbooks from various sources and followed the advice given in &lt;a href=&quot;https://www.physicsforums.com/threads/should-i-become-a-mathematician.122924/post-1006099&quot; rel=&quot;nofollow noreferrer&quot;&gt;this answer on Physics Forums&lt;/a&gt; (an arbitrary choice; I just thought it was a well-written answer), which mentioned the SMSG textbooks. I am, however, concerned about their obsolescence.&lt;/p&gt;&#xA;&lt;p&gt;Are these textbooks still relevant? Would I be better off learning different topics from a variety of publishers (like the books recommended on &lt;a href=&quot;https://www.ocf.berkeley.edu/%7Eabhishek/chicmath.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;this webpage&lt;/a&gt; or in &lt;a href=&quot;https://www.physicsforums.com/threads/did-i-choose-the-right-books-for-relearning-math-geometry-algebra-logic-and-proofs.988171/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this question&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="14401" LastEditorUserId="14401" LastEditDate="2020-08-07T11:14:55.407" LastActivityDate="2020-08-07T11:14:55.407" Title="Would the SMSG textbooks provide adequate extracurricular study?" Tags="|secondary-education|textbooks|self-learning|reference-books|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="18647" PostTypeId="2" ParentId="16948" CreationDate="2020-08-07T12:40:47.933" Score="19" Body="&lt;p&gt;My experience is that weak students latch onto absolute value of a number is always positive.  They are fine working with constants.  When you introduce a variable, it all falls apart.  To these students it is clear that:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lvert-x\rvert= x$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;After all the absolute value sign takes away the negative sign.  If you want students to understand the absolute value of an algebraic expression then you MUST get them past this hump.  All this is before you move onto word problems and inequalities.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="64" LastEditDate="2020-08-08T00:03:56.737" LastActivityDate="2020-08-08T00:03:56.737" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18648" PostTypeId="1" CreationDate="2020-08-07T15:58:29.790" Score="5" ViewCount="203" Body="&lt;p&gt;This fall I'll be teaching a standard &amp;quot;Introduction to Proofs&amp;quot; course and administration is advising us to be ready at any time to switch from in-person to fully online (or anything in between). I've been able to translate most of what I do into various modalities, but I'm having trouble with the memorization quizzes.&lt;/p&gt;&#xA;&lt;p&gt;In a typical face-to-face version of the course I give weekly 10-question, 10-minute quizzes that require students to have memorized important definitions and/or theorem statements. I find this essential to developing their proof-writing skills. But &lt;strong&gt;if I give this in an online modality, what's to prevent them from just looking up the definition/theorem?&lt;/strong&gt; (In my other courses I take the attitude that any online assessment is written to be open book/internet.)&lt;/p&gt;&#xA;&lt;p&gt;Perhaps there's a better/alternate way to get students to really absorb the definitions/theorems and place importance on having them mentally readily-accessible. Perhaps in the modern age, all memorization is moot. I'm open to radical viewpoints/suggestions.&lt;/p&gt;&#xA;&lt;p&gt;Note: I find myself rereading &amp;quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/18449/how-now-shall-we-teach-math-online&quot;&gt;How, now, shall we teach math online?&lt;/a&gt;&amp;quot; almost daily; mining for things I might have missed/forgotten and hoping that more people will post helpful ideas there.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2020-08-08T18:23:59.913" Title="Online Definition/Theorem Statement Quizzes" Tags="|assessment|online-instruction|definitions|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18650" PostTypeId="2" ParentId="18648" CreationDate="2020-08-07T21:54:40.270" Score="7" Body="&lt;p&gt;Here are some ideas:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;As Brendan says above, you can give an example or non-example, and ask the student to explain why (using the definition) it is an example or non-example.&lt;/li&gt;&#xA;&lt;li&gt;You can ask for examples satisfying certain properties (&amp;quot;Give an example of a relation on a three element set which is transitive and symmetric, but not reflexive&amp;quot;).&lt;/li&gt;&#xA;&lt;li&gt;You can ask for critiques of fake student responses:  &amp;quot;A student says that a number is prime if doesn't have any divisors.  Is this a correct definition?  If not, why not?&amp;quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-08-07T21:54:40.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18651" PostTypeId="2" ParentId="18625" CreationDate="2020-08-07T22:55:51.763" Score="2" Body="&lt;p&gt;Just a link to an &lt;a href=&quot;https://m.xkcd.com/2343/&quot; rel=&quot;nofollow noreferrer&quot;&gt;xkcd symbols joke&lt;/a&gt;,&#xA;posted just today [7Aug2020]:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8UxU4.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8UxU4.png&quot; alt=&quot;xkcd&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-08-07T22:55:51.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18652" PostTypeId="2" ParentId="18648" CreationDate="2020-08-07T23:14:12.910" Score="0" Body="&lt;p&gt;I've taught Linear Algebra a number of times. Teaching students to deal with proofs is an important part of that course. Luckily the textbook (Linear Algebra and its Applications 4th Edition, by David C. Lay) had true false questions in every section, and a bunch more at the end of each chapter. I used those to help students think carefully about the definitions and about proof.&lt;/p&gt;&#xA;&lt;p&gt;I think you could come up with good questions whose answers don't appear online.&lt;/p&gt;&#xA;&lt;p&gt;That will not prevent cheating, however. Students can get answers to newly posed questions pretty quickly through chegg and other cheating sites.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-08-07T23:14:12.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18653" PostTypeId="2" ParentId="18648" CreationDate="2020-08-08T15:48:44.810" Score="0" Body="&lt;p&gt;Conducting tests in general in an online setting is hard, but especially checking things that the students should remember.&lt;br /&gt;&#xA;Some students will unavoidably cheat, but trying to minimize student frustration with the exercise can help to minimize it.&lt;br /&gt;&#xA;Having a longer exercise may seem to increase the chance of the student cheating but they actually make the exercise less stressful and by that minimizes cheating.&lt;br /&gt;&#xA;Having the questions be less about memorizing a 2 line definition and more about remembering the concept can help, by, for example, instead of “define a ‘transitive relation’” you could use “is the negation of a transitive relation transitive? Explain.”.&lt;/p&gt;&#xA;" OwnerUserId="14192" LastEditorUserId="14192" LastEditDate="2020-08-08T18:23:59.913" LastActivityDate="2020-08-08T18:23:59.913" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18654" PostTypeId="2" ParentId="16948" CreationDate="2020-08-08T18:58:15.573" Score="12" Body="&lt;p&gt;In my experience, this is the first really clear example that students experience of dissonance between how something &lt;em&gt;looks&lt;/em&gt; and what it &lt;em&gt;is&lt;/em&gt;. One of the most common errors I see regarding absolute value - before bringing in variables, of course - looks like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$|5 - 6| = 5 + 6 = 11$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Whereas, of course, if I ask the student &amp;quot;what's &lt;span class=&quot;math-container&quot;&gt;$5 - 6$&lt;/span&gt;?&amp;quot; and then &amp;quot;okay, what's the absolute value of that?&amp;quot; they'll get the answer perfectly correct. The issue seems to be that they have a mental picture of &amp;quot;absolute values make negative signs positive&amp;quot;, and they see a negative sign in the expression &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$5 - 6$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;As experts, &lt;em&gt;we&lt;/em&gt; understand that the absolute value operates on &lt;em&gt;numbers&lt;/em&gt;, not on symbols; it cares whether the number it's given is negative, not whether the way the number is written includes a negative sign. But that's not an easy distinction to make if a student has grown up with the usual symbol-based approaches for problem-solving.&lt;/p&gt;&#xA;&lt;p&gt;This is a similar issue to what often happens with simplifying fractions - one common mistake looks like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{3 + 2}{6 + 2} = \frac{3}{6} = \frac12$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;A student who makes this mistake is thinking of cancellation as a symbolic action (&amp;quot;delete the same symbols from the top and bottom&amp;quot;) instead of an arithmetic one (&amp;quot;multiplying the numerator and denominator by the same number is the same as multiplying by one&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Many students I've worked with get around this problem with absolute value by just memorizing the rule &amp;quot;simplify inside the absolute value first&amp;quot;. That's correct, but it suddenly stops helping when variables come into play; that leads them to say things like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$|-x| = x$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The situation is made even worse when they're presented with the piecewise definition that other people have mentioned in the answers. My sense is that it's largely because they feel like they're being asked to accept a completely new meaning to a word that feels familiar; it's as if you were trying to convince them to accept that the word &amp;quot;elephant&amp;quot; now refers to a small bird. To the average student, &amp;quot;absolute value&amp;quot; means performing this very simple symbol-based process of removing negative signs; now you're telling them it's actually a complicated process of conditions and arithmetic!&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2020-08-08T18:58:15.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18655" PostTypeId="2" ParentId="16948" CreationDate="2020-08-09T05:51:45.340" Score="0" Body="&lt;p&gt;These are great answers to a great question.&lt;/p&gt;&#xA;&lt;p&gt;Although my answer might only be a rephrasing of what is already said, I think a main problem with absolute value is that it can be ambiguous (or have similar wording but different answers)..&lt;/p&gt;&#xA;&lt;p&gt;For example, if I say the difference between &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; then what is the answer?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is it &lt;span class=&quot;math-container&quot;&gt;$x-4 =3 \to 7$&lt;/span&gt;, &lt;em&gt;using the &amp;quot;difference&amp;quot;&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is it &lt;span class=&quot;math-container&quot;&gt;$|x-4|=3$&lt;/span&gt; and then it becomes &lt;span class=&quot;math-container&quot;&gt;$x-4=3 \to x= 7$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-(x-4)=3 \to x = 1$&lt;/span&gt; &lt;em&gt;using the absolute difference&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;To me, the second option seems more likely if you think geometrically about placement on the number line, but more likely it's the first answer since the question didn't state &amp;quot;absolute difference&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&lt;p&gt;Another thing I thought of for absolute value, is that it can be unnecessary like &lt;span class=&quot;math-container&quot;&gt;$|x^2|=x^2$&lt;/span&gt;, but then necessary with &lt;span class=&quot;math-container&quot;&gt;$|x| \neq x$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-08-09T06:24:16.423" LastActivityDate="2020-08-09T06:24:16.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18657" PostTypeId="2" ParentId="18551" CreationDate="2020-08-10T06:41:47.247" Score="1" Body="&lt;p&gt;I definitely recommend a book that carefully explains how to write proofs and various aspects of mathematical reasoning. I haven't read How to Prove It by Velleman, but I have heard good things about it.&lt;/p&gt;&#xA;&lt;p&gt;I am familiar with Mathematics: Its Content, Methods, and Meaning. It's more of a survey book of various aspects of mathematics, with an emphasis on Soviet mathematics (given the era it was written in). I wouldn't recommend it for learning much actual mathematics, though. I think that a well-written textbook would be more suitable for self-study.&lt;/p&gt;&#xA;&lt;p&gt;If you know trigonometry and know basic set notation (you can pick this up in any trigonometry textbook), then I think you can go ahead and start Spivak's calculus book.&lt;/p&gt;&#xA;" OwnerUserId="13249" LastActivityDate="2020-08-10T06:41:47.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18658" PostTypeId="2" ParentId="18616" CreationDate="2020-08-10T06:44:26.577" Score="2" Body="&lt;p&gt;There's a really nice book by Svetlana Katok called &amp;quot;p-adic analysis compared to Real.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="13249" LastActivityDate="2020-08-10T06:44:26.577" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18659" PostTypeId="2" ParentId="18616" CreationDate="2020-08-10T12:53:00.317" Score="8" Body="&lt;p&gt;A few recommendations:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Fernando Q. Gouvêa's &lt;a href=&quot;http://personal.colby.edu/personal/f/fqgouvea/padic.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers: An Introduction&lt;/em&gt;&lt;/a&gt;.  This text gives a good introduction to the &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic number system and the properties of the space of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers, vector spaces over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}_p$&lt;/span&gt;, and the metrically completed algebraic closure of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}_p$&lt;/span&gt;.  There is also some discussion of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic analysis near the end (e.g. power series and entire functions on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}_p$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I've used this book as a reference for a couple of undergraduate research projects on &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic Brownian motion, and I think that the text is an appropriate introductory text for a general undergraduate audience.  It would probably be best if they have already taken some real analysis, and familiarity with some basic complex analysis might help to motivate the material in the last chapter of the book, but nothing in the text really assumes that backgroud—a clever undergraduate with a solid calculus background and the ability to read and write proofs should be fine.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Neal Koblitz's &lt;a href=&quot;https://www.springer.com/gp/book/9780387960173&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic Numbers, &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic Analysis, and Zeta-Functions&lt;/em&gt;&lt;/a&gt; is a little more advanced, and is more interested in the number theoretic aspects of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic analysis.  Koblitz also introduces the reader to the &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers, but very quickly get's into some crunchier questions in algebraic geometry.  There are some solid discussions of the Riemann zeta function and related problems—think &amp;quot;Tate's thesis, but with more exposition&amp;quot; (kind of).&lt;/p&gt;&#xA;&lt;p&gt;I think that a student should have a relatively strong background in algebra before trying to tackle Koblitz.  The constructions in the text are generally in terms of algebraic ideas:  Koblitz works heavily with rings of polynomials with &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic coefficients, and relies on intuition about the structure of such spaces.  A background in algebraic geometry is not, strictly speaking, necessary, but knowing what a sheaf is wouldn't hurt (on the other hand, I suppose that one might take Koblitz's book as an introduction to these ideas, and use it to motivate students to learn algebraic geometry in greater detail).  One &lt;em&gt;might&lt;/em&gt; give this book to a clever undergraduate, but I think that it is probably more appropriate for a beginning graduate student.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;W. H. Schikof's &lt;a href=&quot;https://www.cambridge.org/core/books/ultrametric-calculus/FD8E1172062BF97D41430B559C895CCC&quot; rel=&quot;noreferrer&quot;&gt;*Ultrametric Calculus: An Introduction to &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic Analysis&lt;/a&gt; is a solid introduction to &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic analysis.  It feels like it fills the same role in introducing the theory to students as Apostol's &lt;em&gt;Mathematical Analysis&lt;/em&gt; or Spivak's &lt;em&gt;Calculus&lt;/em&gt;—it gives a relatively gentle introduction to the material, but without holding the student's hand.  The &amp;quot;meat&amp;quot; of the book consists of two chapters which give an overview of elementary calculus in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}_p$&lt;/span&gt;, but there is some discussion of ideas in functional analysis, too.&lt;/p&gt;&#xA;&lt;p&gt;I get the feeling that this text was written with undergraduates in mind.  I think that a student who has made it through the US-standard calculus curriculum could probably get started in a this book without too much difficulty, though if they have never seen a proof before, they will need help.  A student with some background in analysis should have no difficulty working through most of Koblitz's text.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2020-08-10T12:53:00.317" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18660" PostTypeId="1" AcceptedAnswerId="18661" CreationDate="2020-08-10T13:11:10.180" Score="11" ViewCount="403" Body="&lt;p&gt;I'd love to learn from anyone's recent experiences teaching online proof based math courses, especially those that have a large group of students who will be working asynchronously.  My usual proof based course includes some lecture, with a lot of student interaction and some group work/guided discovery.  I'm struggling with how to translate that online.&lt;/p&gt;&#xA;&lt;p&gt;How have you tried to present proofs online?&#xA;Have you been able to encourage group work in asynchronous classes?&#xA;What have you done for online exams which have to be taken in different time zones?&lt;/p&gt;&#xA;&lt;p&gt;Any advice or links to discussions elsewhere is appreciated. I've seen a lot of discussion of best practices for calc classes and below, but not so much about proof based classes.&lt;/p&gt;&#xA;" OwnerUserId="9542" LastEditorUserId="14024" LastEditDate="2020-08-17T05:34:59.240" LastActivityDate="2020-08-17T05:34:59.240" Title="Transitioning proof based math courses online" Tags="|undergraduate-education|mathematical-pedagogy|proofs|online-instruction|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18661" PostTypeId="2" ParentId="18660" CreationDate="2020-08-10T16:02:20.923" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Have you been able to encourage group work in asynchronous classes?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In linear algebra last term, I used our learning management system (Blackboard) to put students into groups that I created, and assigned each group to complete a set of problems and type up their work. Since we were using &lt;a href=&quot;https://slack.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Slack&lt;/a&gt; for class discussions/chat, I suggested that each group create their own Slack channel to collaborate on the work. I gave them deadlines for rough- and final-drafts, with some number of days in between each of those. [e.g. Assignment posted Sunday night, rough draft due by Friday, final draft due Sunday night.] Since I was also on Slack for questions, I was reading a lot of pre-rough drafts (which is more than I usually get in a face-to-face class).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How have you tried to present proofs online?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As a first online attempt, I made videos, stopping periodically to say &amp;quot;now pause your video to answer this part of the problem&amp;quot;. When I do this again, I would like to try a feature of an online homework system that allows a video to automatically pause and a graded question to appear. [e.g. &amp;quot;For this problem, we're going to use mathematical induction. As a reminder, what are the steps for a proof using this method?&amp;quot; Problem opens up with various, jumbled steps for an induction proof, etc.]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What have you done for online exams which have to be taken in different time zones?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Last term, I made exams available for a long period of time (e.g. 8AM-8PM) to accommodate this and other issues.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2020-08-12T13:27:44.973" LastActivityDate="2020-08-12T13:27:44.973" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18664" PostTypeId="2" ParentId="18660" CreationDate="2020-08-11T04:35:32.580" Score="2" Body="&lt;p&gt;Jennifer Quinn blogged about having oral final exams, at her blog &amp;quot;Math in the time of coronavirus&amp;quot;, &lt;a href=&quot;https://mathinthetimeofcorona.wordpress.com/2020/06/10/june-10-day-94-finals-part-i/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathinthetimeofcorona.wordpress.com/2020/06/10/june-10-day-94-finals-part-i/&lt;/a&gt;.&#xA;and the more detailed part 2&#xA;&lt;a href=&quot;https://mathinthetimeofcorona.wordpress.com/2020/06/11/june-11-day-95-finals-part-ii/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathinthetimeofcorona.wordpress.com/2020/06/11/june-11-day-95-finals-part-ii/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;She gave each student 20 minutes to answer two questions. She gave the class a list of 9 questions. The students could choose one question to answer, and she would choose the second.&lt;/p&gt;&#xA;&lt;p&gt;In her posts she discusses many aspects of this, including choosing the questions, and how long it took her (vs. how long it would have taken to grade &amp;quot;traditional&amp;quot; paper exams).&lt;/p&gt;&#xA;&lt;p&gt;I haven't tried this. I only read about it on her blog. But I'm going to try it in one of my classes this fall.&lt;/p&gt;&#xA;" OwnerUserId="12530" LastEditorUserId="10289" LastEditDate="2020-08-11T17:35:01.360" LastActivityDate="2020-08-11T17:35:01.360" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18665" PostTypeId="2" ParentId="16948" CreationDate="2020-08-11T05:31:40.213" Score="0" Body="&lt;p&gt;There is a lot of truth in many of the answers here, but they are all addressing this from a pure math standpoint. The logic of absolute value isn't hard, though - even if there are many examples here of weak prior knowledge that can cause problems.&lt;/p&gt;&#xA;&lt;p&gt;Absolute value is hard for students to learn because it is almost always taught in extremely abstract and boring ways that just repel the vast majority of people from math while powerfully discouraging sense-making.&lt;/p&gt;&#xA;&lt;p&gt;Here is an example from the ironically named &amp;quot;&lt;a href=&quot;https://www.mathsisfun.com/numbers/absolute-value.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math is Fun - Absolute Value&lt;/a&gt;&amp;quot; website&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eg5Al.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eg5Al.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;How fun and interesting is this? If we were to poll 1000 students, would 5 of them enjoy making meaning of this? Would even &lt;em&gt;one&lt;/em&gt; see absolute value as useful?&lt;/p&gt;&#xA;&lt;p&gt;I doubt it. The normal reaction almost assuredly would be something like &amp;quot;This is just more proof that math is just a bunch of weird rules and steps to obey. More rote crap to memorize.&amp;quot; With this as a foundation, there is almost no way to make sense of any of the other very clear and logical points made in this thread.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In order to get students &lt;em&gt;interested in the meaning&lt;/em&gt; of absolute value they need &lt;em&gt;context&lt;/em&gt; in which absolute value thinking is useful.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You are a purchasing agent at a pharmacy. You need to order a large&#xA;amount of a generic cancer drug and three different factories offer to&#xA;sell it to you. You ask for a sample of their pills to check their&#xA;quality. Each pill should have exactly 700mg of active ingredient, but&#xA;no factory is perfect.&lt;/p&gt;&#xA;&lt;p&gt;Factories A and B are brand new and not ready to produce enough pills&#xA;for you. They send you pills just to get your feedback. Factories C&#xA;and D are ready to sell to you now. They send a much larger sample of&#xA;pills for you to analyze.&lt;/p&gt;&#xA;&lt;p&gt;Your mission: (1) Recommend to your manager which factory to buy&#xA;from. (2) Justify your answer. (3) Develop a written process for evaluating&#xA;quality of all future pill samples.&lt;/p&gt;&#xA;&lt;p&gt;Do great work because your patients' lives depends on it!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://docs.google.com/spreadsheets/d/1A2fkSZHN_qeKOt4nUbD2nQWP1DAc6vk_5Azcsf0qFIU/edit?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pill Quality Case&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Instructions are on the first sheet. Data is on the second sheet. Reflect explicitly to consolidate is on the third sheet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here's a little snippet of data.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VyEA0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VyEA0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;First you can have them estimate the quality of each factory. B is clearly better than A.&lt;/p&gt;&#xA;&lt;p&gt;Now, you can guide them through calculating, totaling, and averaging errors for each factory, in which case a pill being, say, 200mg over and another pill being 200mg under exactly cancels. And whaddaya know: Factory A is perfect and Factory B sucks! Math beats common sense! According to the total and mean error, we should go with Factory A. Maaaaaaybe grandma will scream in unnecessarily excruciating pain and the baby will go into overdose, but according to mean error calculations, that is OK. lolz&lt;/p&gt;&#xA;&lt;p&gt;Then, the more outspoken ethical sticklers will say that they, in fact, don't want to torture grandma and kill a baby. (BUT, LIKE, MEAN ERROR CALCULATIONS YO, IT'S JUST MATH DUDE, GOTTA LISTEN TO DEM NUMBERS.)&lt;/p&gt;&#xA;&lt;p&gt;Hmm. Maybe 200mg under and 200mg over should &lt;em&gt;not&lt;/em&gt; cancel out when added up, because an overdosing baby doesn't really cancel out screaming grandma. So we &lt;em&gt;could&lt;/em&gt; if we were very careful, always make sure to do:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Error = 700mg - (measured active ingredient)   when the mg of active ingredient is less than 700mg&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;but&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Error = (mg of active ingredient) - 700mg    when the mg of active ingredient is greater than 700&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And that way, we get no negative numbers. No more errors that offset each other.&lt;/p&gt;&#xA;&lt;p&gt;Great!&lt;/p&gt;&#xA;&lt;p&gt;We just have to be really careful on every single calculation. Every. Single. Pill.&lt;/p&gt;&#xA;&lt;p&gt;That's pretty easy for Factories A and B.&lt;/p&gt;&#xA;&lt;p&gt;Who wants to do all those calculations for the hundreds of pills from Factories C and D? Hands up for volunteers! Hands up please!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ANE2P.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ANE2P.png&quot; alt=&quot;Uhh... anyone... anyone?&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If only there were an easier way to get rid of those damn negative errors... Wouldn't that be &lt;em&gt;awesome&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;At this point, you can introduce absolute value (and clicking/dragging formulae in spreadsheets) and every student knows why it's necessary and &lt;em&gt;what it means&lt;/em&gt;. You could go a little deeper here into this case, formally relate the two error formulae to absolute value, then gradually abstract away from this and towards all the other posts in this thread. It can now become obvious why &lt;span class=&quot;math-container&quot;&gt;$\lvert x-700 \rvert=400$&lt;/span&gt; must have two solutions: An absolute error of 400mg can be caused by overage or underage.&lt;/p&gt;&#xA;&lt;p&gt;This stuff can all &lt;em&gt;make sense&lt;/em&gt; because absolute value has meaning in their minds.&lt;/p&gt;&#xA;&lt;p&gt;Anyways, if you have any suggested updates for the case or other thoughts on how I teach absolute value, please reply. :-)&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2020-08-11T05:51:40.900" LastActivityDate="2020-08-11T05:51:40.900" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18666" PostTypeId="1" AcceptedAnswerId="18668" CreationDate="2020-08-11T22:24:53.870" Score="7" ViewCount="920" Body="&lt;p&gt;I'm on a semester break and thankfully I was recently provided with an opportunity to hire any private teacher I like to learn even more math while on break. However, I don't really know how to approach it. What are we going to do? They'll start to explain material that I can read in the book? Solve homework together? I think it's very counter-productive because I can learn more from my mistakes. Or should I come up with questions like &amp;quot;Could you explain why we use open sets in differential calculus?&amp;quot;. But I can just google them and get the answer I need.&lt;/p&gt;&#xA;&lt;p&gt;At the same time I really want to use this possibility since I was offered it. I don't want the opportunity go to waste and at the same time I feel like mathematics is a one man road. Maybe someone can provide me with an option on how to use the opportunity without being bored or counter-productive.&lt;/p&gt;&#xA;" OwnerUserId="14427" LastEditorUserId="143" LastEditDate="2020-08-12T22:37:04.240" LastActivityDate="2020-08-12T22:37:04.240" Title="How to structure undergraduate level private lessons?" Tags="|self-learning|private-lessons|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18667" PostTypeId="2" ParentId="18666" CreationDate="2020-08-11T23:26:28.263" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;They'll start to explain material that I can read in the book?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(1) Depending on where you are in your mathematical development (which you don't make clear in your posting), you might take on a challenging text, for example, Michael Spivak's &lt;em&gt;&lt;strong&gt;Calculus&lt;/strong&gt;&lt;/em&gt; (4th ed), and work it through with the guidance of a private teacher.&lt;/p&gt;&#xA;&lt;p&gt;(2) If not Calculus, then Gilbert Strang's &lt;em&gt;&lt;strong&gt;Linear Algebra&lt;/strong&gt;&lt;/em&gt;&#xA;(also available in &lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MIT Courseware&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;(3) If not Calculus or Linear Algebra, perhaps &lt;em&gt;&lt;strong&gt;Real Analysis&lt;/strong&gt;&lt;/em&gt; by Karl Stromberg.&lt;/p&gt;&#xA;&lt;p&gt;The point is to take a rigorous, challenging text and proceed through it methodically under the guidance of an experienced teacher.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-08-11T23:33:15.467" LastActivityDate="2020-08-11T23:33:15.467" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18668" PostTypeId="2" ParentId="18666" CreationDate="2020-08-12T01:36:17.760" Score="3" Body="&lt;p&gt;I agree that you need to do a lot of the heavy lifting.  Furthermore, I would try to use texts that are suitable for self-study, basically always.  Feel like the teacher-salary-justifiers try to push books that need an instructor more.  I.e. even with a paid teacher, I think you should pedagogically optimized materials.&lt;/p&gt;&#xA;&lt;p&gt;All that said, I do think a tutor can benefit you by challenging your knowledge with drill and interrogation.  Something like how tutorials work in Oxford or Cambridge or whatever.&lt;/p&gt;&#xA;&lt;p&gt;You can also, keep a notebook for your self studying and just write questions in the back to be discussed with the tutor.  It will probably end up being more time efficient and accurate and interactive than SE questioning.  You can plow through several in a session at once.  (Also pretty much how you should use office hours.)&lt;/p&gt;&#xA;" OwnerUserId="14429" LastActivityDate="2020-08-12T01:36:17.760" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18669" PostTypeId="2" ParentId="18666" CreationDate="2020-08-12T10:36:25.843" Score="4" Body="&lt;p&gt;My suggestion would be that you pick a book with some difficult exercises, attempt to work through the exercises yourself, and then your tutor can help you with the exercises you got stuck on. This works best if they give you hints which lead you to the solution, rather than just solving the exercise for you. The tutor can also check exercises you believe you have solved, as you might have overlooked certain cases or omitted important details.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2020-08-12T10:36:25.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18670" PostTypeId="2" ParentId="18660" CreationDate="2020-08-12T22:40:09.347" Score="1" Body="&lt;p&gt;To complement the other great answers, I'll add a helpful link to an MAA discussion of exactly this topic &lt;a href=&quot;https://www.youtube.com/playlist?list=PLevtNOOa6SZXH76yD-4bM2wixZJGtOBVb&quot; rel=&quot;nofollow noreferrer&quot;&gt;available here&lt;/a&gt; (along with two other helpful videos for online courses) which I recently discovered.&lt;/p&gt;&#xA;" OwnerUserId="9542" LastActivityDate="2020-08-12T22:40:09.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18671" PostTypeId="1" CreationDate="2020-08-13T11:11:09.317" Score="9" ViewCount="297" Body="&lt;p&gt;Not only do calculators have solving capabilities, but some computer programs or websites also provide step-by-step solutions to questions (&lt;a href=&quot;https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/&quot; rel=&quot;noreferrer&quot;&gt;here is WolframAlpha's&lt;/a&gt;). Although I understand a logical approach to doing questions, where one step leads to another (often due to algebraic manipulation), if a computer could print out a solution to a math test that is virtually indistinguishable from a student's then should that test be changed?&lt;/p&gt;&#xA;&lt;p&gt;Although there is the academic dishonesty side of things (where maybe one might use the computer to make it seem they know the calculations), there's also the issue that I might be teaching and testing the students in a skill-set that will be replaced. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Computer_(job_description)&quot; rel=&quot;noreferrer&quot;&gt;computer (occupation)&lt;/a&gt; is obsolete now and so what's the point of testing to see if a person could do what a computer could do (and most likely would do outside the classroom)?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-08-13T15:40:56.843" Title="If a computer could be programmed to do a math test, then should those tests be changed?" Tags="|mathematical-pedagogy|technology-in-education|calculators|computer-programs|calculations|" AnswerCount="1" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18672" PostTypeId="2" ParentId="18671" CreationDate="2020-08-13T13:51:27.453" Score="6" Body="&lt;p&gt;No.&lt;/p&gt;&#xA;&lt;p&gt;There is a sort of magical thinking implicit here that something needs to stay, be different between computers and people.  At the end of the day, we are &amp;quot;meat&amp;quot; computers.  And the computers themselves are getting more and more sophisticated in doing things that were hard for them previously.  But the existence of a soul (whether or not true) does not change the inherent fact that there is and will be an overlap of computer and human reasoning.&lt;/p&gt;&#xA;&lt;p&gt;Just because a horse runs faster, doesn't mean there is no value in running for a man.  Similarly as computers learn facial recognition, it does not obviate the need/benefit to humans in doing the same.&lt;/p&gt;&#xA;&lt;p&gt;Even the tasks that you (implicitly) and many others here deprecate as mere calculations are actually very important to new students, building their skills.  One struggles with the higher level skills, if one hasn't mastered the lower level ones.  So the idea that we'll just stop doing mechanical skills and do some sort of ETS neoliberal &amp;quot;concept&amp;quot; stuff is a chimera.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  What's really funny is seeing the same people who say &amp;quot;let the kids use WA&amp;quot; and &amp;quot;who cares about mechanical skills&amp;quot; complain when their remedial pre-calc kids don't know dem fractions...&lt;/p&gt;&#xA;&lt;p&gt;P.s.s.  I will cede you that cursive is on the way out.  But basic calculational skills?  Sorry...needed.  Been hearing this &amp;quot;don't need it&amp;quot; story since the 70s... from the calculator/computer pushers...and basic manipulation skills are still needed.&lt;/p&gt;&#xA;" OwnerUserId="14435" LastEditorUserId="14435" LastEditDate="2020-08-13T15:40:56.843" LastActivityDate="2020-08-13T15:40:56.843" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18673" PostTypeId="2" ParentId="18629" CreationDate="2020-08-13T16:26:34.633" Score="1" Body="&lt;p&gt;I'm the developer of the STACK online assessment system, which is a question type for the Moodle quiz.  STACK is used by about 1000 sites including some very large institutions including the University of Edinburgh, The Open University, Aalto, Loughborough University and the University of Birmingham.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://stack-demo.maths.ed.ac.uk/demo/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://stack-demo.maths.ed.ac.uk/demo/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;To answer your questions directly,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;STACK uses MathJAX to embed LaTeX mathematics environments (not document structure)&lt;/li&gt;&#xA;&lt;li&gt;STACK has full CAS support using Maxima which allows the use of parameters throughout the question.&lt;/li&gt;&#xA;&lt;li&gt;STACK allows these parameters within the TeX code.&lt;/li&gt;&#xA;&lt;li&gt;STACK is a question type for the Moodle quiz, and at the University of Edinburgh we integrate this into Blackboard using an LTI link.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;We published some comprehensive case studies (with example questions) last year: &lt;a href=&quot;https://stack-demo.maths.ed.ac.uk/demo/question/type/stack/doc/content/2019-cate-case-studies.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://stack-demo.maths.ed.ac.uk/demo/question/type/stack/doc/content/2019-cate-case-studies.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you would like to talk directly, drop me an email at Edinburgh.&lt;/p&gt;&#xA;&lt;p&gt;Chris Sangwin&lt;/p&gt;&#xA;" OwnerUserId="7787" LastActivityDate="2020-08-13T16:26:34.633" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18674" PostTypeId="1" CreationDate="2020-08-14T05:12:42.123" Score="0" ViewCount="891" Body="&lt;p&gt;I know pre-algebra is like a terminology thrown around to really basic stuff that are taught before high school algebra. Some stuff taught there are already considered as part of algebra in some sources. I just want to ask what are the topics considered to be part of this, whether they are common or come only in some situations.&lt;/p&gt;&#xA;&lt;p&gt;Yes, I know this will sound again as some opinion based thing and someone will point out that I should consider it on my own. I am asking here for what are the general stuff. Yes, even if it is based on your opinion but somewhat backed up by sources.&lt;/p&gt;&#xA;&lt;p&gt;I wanted to ask this to get a grasp on what topics should people know before taking algebra in high school and which part of those don't intersect much with high school algebra. Like for example, arithmetic are stuff that are emphasized in elementary. Sure, variables are used in algebra but the introduction of what the operations are were totally done in elementary.&lt;/p&gt;&#xA;" OwnerUserId="14439" LastActivityDate="2020-08-16T13:00:33.480" Title="What topics are considered to be part of pre-algebra?" Tags="|algebra|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18675" PostTypeId="2" ParentId="18674" CreationDate="2020-08-14T05:37:39.657" Score="0" Body="&lt;p&gt;As how precalculus is mainly the algebraic manipulation necessary to do calculus questions, I'd suppose that prealgebra is the arithmetic manipulation necessary to do algebra questions. Looking at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pre-algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia page&lt;/a&gt;, it seems that it's mostly ability to work with numbers and variables with basic operations of &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt;, (as well as basic roots and exponents). And this makes sense since how would a student balance equations well if they can't do the computations with the coefficients and constants.&lt;/p&gt;&#xA;&lt;p&gt;I think one thing that's really lacking for students are inequalities. They always seem to know how to solve for something with direct relationships, but unsure about indirect relationships. For example, triangle inequality is used a lot in mathematical proofs and probably comes up quite a bit in high school level math contests, but I don't recall seeing it very often in actual lessons or textbooks.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-08-14T05:37:39.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18676" PostTypeId="2" ParentId="18674" CreationDate="2020-08-14T09:24:04.130" Score="0" Body="&lt;p&gt;On a simple level, I remember,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pre-algebra: linear equations in one variable, 7x +10 = 5-3x&lt;/li&gt;&#xA;&lt;li&gt;first year HS algebra: lots of lines, y=mx+b&lt;/li&gt;&#xA;&lt;li&gt;second year HS algebra: conic sections, exponents, logs&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The manipulation work in pre-algebra to sort out and solve for the x, is a basic skill that is called on in many higher classes.  Often re-arranging the spinach, to see clearly enough to apply some rule, is half the battle.&lt;/p&gt;&#xA;&lt;p&gt;We had a heavy dose of inequality battling in a course called &amp;quot;functions&amp;quot;, which along with analytic geometry came after Algebra 2, and served as pre-calc.  I disagree with Robbie's comment about inequalities in the context of pre-algebra.  That topic should come later.&lt;/p&gt;&#xA;&lt;p&gt;[EDIT]  Note that some elements of algebra are included in pre-algebra.  Just as some derivatives and anti-derivatives are included in pre-calculus (&amp;quot;functions&amp;quot; and analytic geometry).  Well...at least in Fairfax County public schools of the 70s and 80s. And Algebra 2 covered all the logs and such that are part of &amp;quot;college algebra&amp;quot;.  And there was a full semester on trig also (after algebra 2), but before &amp;quot;functions&amp;quot;.  The GT kids took Algebra 2/Trig (which was 3 semesters in 2).&lt;/p&gt;&#xA;&lt;p&gt;But I would not rely too much on names.  Looking at textbooks, I see several pre-algebra texts that have the &amp;quot;solve for x&amp;quot; as key content.  It's like baby steps into algebra.  Then they throw a y in along with it in algebra 1.&lt;/p&gt;&#xA;&lt;p&gt;For example this is the first hit on google (that I got for a prealgebra textbook pdf).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.cbsd.org/cms/lib010/PA01916442/Centricity/Domain/2723/PreAlgebra%20textbook.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cbsd.org/cms/lib010/PA01916442/Centricity/Domain/2723/PreAlgebra%20textbook.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that they get right into solving for x in chapter 2 of the book.  (See page 91 for the classic adding things to both sides of the equation stuff.)  Granted, there's a bunch of pictures of girls with horses...don't rememember that when I did it.  But &amp;quot;solve for x&amp;quot;.  Classic pre-algebra.  Kid has exercised that before high school algebra 1.&lt;/p&gt;&#xA;" OwnerUserId="14442" LastEditorUserId="14445" LastEditDate="2020-08-16T13:00:33.480" LastActivityDate="2020-08-16T13:00:33.480" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18677" PostTypeId="2" ParentId="18674" CreationDate="2020-08-14T14:46:52.160" Score="1" Body="&lt;p&gt;In my experience, students need to have a firm grasp of fractions and order of operations to succeed in algebra.  Manipulating fractions with algebraic expressions can be difficult if you don't know how to manipulate them with numbers. Without a solid grounding in fractions, you can't simplify an algebraic fraction, add, subtract and multiply them, or find the domain.  Order of operations is necessary for evaluating expressions, using a calculator (where do those parentheses go?), and figuring out how to solve an equation by working backwards. I'm sure there are other important topics but these two seem essential and many students are weak.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-08-14T14:46:52.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18678" PostTypeId="1" CreationDate="2020-08-14T18:04:14.687" Score="3" ViewCount="164" Body="&lt;p&gt;This question isn't really about lecture notes in themselves, like these questions (&lt;a href=&quot;https://matheducators.stackexchange.com/questions/1114/hand-out-lecture-notes-or-not&quot;&gt;Hand out lecture notes or not?&lt;/a&gt; or &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10977/is-it-better-to-provide-students-with-guided-notes-or-to-have-them-write-their-o&quot;&gt;Is it better to provide students with guided notes or to have them write their own notes, or both?&lt;/a&gt;). Rather it's about, if you were to give out notes, would you give them out in an editable format (rather than, say, a static pdf)?&lt;/p&gt;&#xA;&lt;p&gt;I've sometimes given out class notes, but they've happened to always be pre-printed or pdf format &lt;em&gt;for students to print out&lt;/em&gt;. Either way, they can personalize the actual text. And I'm wondering if, in providing notes, I should also provide what made those notes, like an editable document (like Word) or give the LaTeX code.&lt;/p&gt;&#xA;&lt;p&gt;As how some people learn from their own notes, or from others', or how they learn from writing it by hand, or by computer font, it might be possible they could learn also by adapting and rephrasing things. All the time, I've reworked writings from textbooks for my own learning and teaching but it can be difficult if one has to start from the beginning. Although they might not reformat the equations due to not knowing how to do it, by providing the editable files, they could rework the words in a way that they understand. For example, they could add in ALL CAPS or ***'s around things that need more attention.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-09-19T22:38:19.447" Title="Sharing notes in an editable format?" Tags="|lecture-notes|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18679" PostTypeId="2" ParentId="18674" CreationDate="2020-08-14T20:06:51.293" Score="2" Body="&lt;p&gt;I think the most important concepts to have mastered before starting algebra are basic arithmetic and general reasoning. Fluency with negative numbers and the order of operations cannot be under-estimated as this prevents troubles with substitution. Students should understand what is meant by equality and inverse operations, i.e. that &lt;span class=&quot;math-container&quot;&gt;$70 + 2 = 72$&lt;/span&gt; is equivalent to &lt;span class=&quot;math-container&quot;&gt;$72 - 2 = 70$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the UK, students start algebra in primary school but tend to get their first proper exposure when they're around 11 or 12. At this stage, many still need more work on their numeracy. The difference is clear: students who have stronger numerical reasoning tend to be better at generalising to algebraic methods. Things like working out how much change does one get given a shopping list and prices or problems that work backwards to find 'what number did I start with' are examples of things young students would benefit from being confident with.&lt;/p&gt;&#xA;&lt;p&gt;I've seen teachers trying to teach algebraic expressions to students with poor numerical reasoning. For example, if there are 7 socks in a packet and I buy &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; packets, then it is obvious for many that there are &lt;span class=&quot;math-container&quot;&gt;$7x$&lt;/span&gt; socks in total, but for weak students that struggle with multiplication, the generalisation is harder.&lt;/p&gt;&#xA;" OwnerUserId="14007" LastActivityDate="2020-08-14T20:06:51.293" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18680" PostTypeId="1" CreationDate="2020-08-15T04:18:08.257" Score="1" ViewCount="277" Body="&lt;p&gt;Preferably I want those that contain the following topics:&lt;/p&gt;&#xA;&lt;p&gt;Solid Figures&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;Polyhedrons&#xA;Prisms&#xA;Pyramids&#xA;Prismatoid&#xA;Truncated Prisms&#xA;Cylinders&#xA;Cones&#xA;Spheres&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;I've been able to find good references for Plane Geometry. But for solid geometry, it feels so limited. I cannot find a comprehensive reference that really deals with all kinds of 3D figures and is made in a modern way with detailed explanations, flow and illustrations.&lt;/p&gt;&#xA;" OwnerUserId="14439" LastActivityDate="2020-08-15T10:31:12.697" Title="What books are good to study Solid Mensuration?" Tags="|geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18681" PostTypeId="2" ParentId="18680" CreationDate="2020-08-15T10:31:12.697" Score="2" Body="&lt;p&gt;If you are looking for self-study, I recommend this book:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B001CZ0V88&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Solid-Geometry-310-M-Welchons/dp/B001CZ0V88&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is modern in terms of language (I can read books from the 1940s without effort).  And there is about 1 illustration per page (small pages).  In addition, it's a thorough treatment.  And is addressed to students, not practitioners (i.e. is accessible).&lt;/p&gt;&#xA;&lt;p&gt;I don't have an in print recommendation, if you need to purchase several for a classroom.  (But I'm sure there are products sold, which you could research by contacting local textbook sales reps.)&lt;/p&gt;&#xA;" OwnerUserId="14445" LastActivityDate="2020-08-15T10:31:12.697" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18684" PostTypeId="1" AcceptedAnswerId="18692" CreationDate="2020-08-17T03:45:51.160" Score="3" ViewCount="243" Body="&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt; I am a graduate student in a mid-tier U.S. university, and I am struggling. I feel like I during my undergrad, I haven't aquired the neccesary skills to keep up with the high volume/pace of graduate classes. This is for three reasons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I got my undergrad degree at a low-tier institution, so the pace was not that high.&lt;/li&gt;&#xA;&lt;li&gt;I only went to school part time while working part time. I've never taken 4 classes in one semester; I usually took 2, a few times 3, and most often only one of them was a math class.&lt;/li&gt;&#xA;&lt;li&gt;For years now, I have been losing my passion for math. I still sometimes get a small enjoyment from studying it, and it's my favorite academic subject, but honestly it has just become a chore.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Now I am about to start my second year in graduate school, and I am freaking out. The first year was stressful, but I was able to get by since a lot of the material was review from undergrad. In this second year, the classes now only seem much harder and faster, but they will also be 100% new material. Additionally, I have to do a research project (which I do not feel prepared for), and on top of this, I have to worry about exams in May.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; I light of the background, I am looking to improve my skills of studying material at a fast pace and in a high volume. Are there any specific strategies for this?&lt;/p&gt;&#xA;&lt;p&gt;I have read that one crucial component of learning is to constantly revisit material. However, I'm afraid that this may be impossible for me, at the current rate. In other words, I think that the classes will be going so fast, and cover so much material, that it will take all my time and energy to just keep up with the new material.&lt;/p&gt;&#xA;" OwnerUserId="14454" LastActivityDate="2020-08-27T11:44:32.583" Title="Learning strategies for high volume/pace learning?" Tags="|self-learning|mathematical-analysis|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18685" PostTypeId="2" ParentId="18629" CreationDate="2020-08-17T19:58:15.623" Score="0" Body="&lt;p&gt;You might want to look at &lt;a href=&quot;https://webwork.maa.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;WeBWorK&lt;/a&gt;. It's open-source, supported by the MAA and NSF.&lt;/p&gt;&#xA;&lt;p&gt;I don't know if it has BlackBoard integration though.&lt;/p&gt;&#xA;" OwnerUserId="12530" LastActivityDate="2020-08-17T19:58:15.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18686" PostTypeId="2" ParentId="14846" CreationDate="2020-08-19T18:43:43.337" Score="0" Body="&lt;p&gt;To me, it seems like, maybe the child has a fear of mathematics. We can help him out by innovating the teaching methods. For example,&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;1. Make the class interesting, with objects, presentations&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;p&gt;Some theme-based presentations designed with contents focussing on logic development can help. This may also engage a child to solve a problem interestingly.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/3MUUv.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3MUUv.png&quot; alt=&quot;Teaching a child how to deal with numbers&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;2. Introducing to topics in a new way!&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;p&gt;We can give exposure to a topic with the help of real-world examples/applications, or by conducting some fun activities. Introducing topics in such a manner can work in two ways: first, the students would be eager to learn. Second, they'll understand why the topic is being introduced.&lt;/p&gt;&#xA;" OwnerUserId="14466" LastEditorUserId="14466" LastEditDate="2020-09-14T20:08:19.003" LastActivityDate="2020-09-14T20:08:19.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18687" PostTypeId="2" ParentId="18678" CreationDate="2020-08-19T18:53:04.943" Score="0" Body="&lt;p&gt;Many of the students find it easy to understand the self-written/made notes &amp;amp; many probably have the habit to write key-points or self-help points! So, providing a format which helps adding notes or pins could be useful. Also, the authenticity of your notes will be maintained.&lt;/p&gt;&#xA;" OwnerUserId="14466" LastActivityDate="2020-08-19T18:53:04.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18688" PostTypeId="2" ParentId="18629" CreationDate="2020-08-19T19:14:48.560" Score="0" Body="&lt;ul&gt;&#xA;&lt;li&gt;MATHPORTAL : &lt;a href=&quot;https://www.mathportal.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathportal.org/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This site provides free math calculators,formulas and lessons.On this site you could calculate many complex equations including Quadratic Equations, Polynomials etc.&#xA;You can select a calculator that's most convenient for you. It provides list of multiple calculators :&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Standard Deviation Calculator&lt;/li&gt;&#xA;&lt;li&gt;Quadratic equation solver&lt;/li&gt;&#xA;&lt;li&gt;Right triangle calculator&lt;/li&gt;&#xA;&lt;li&gt;Sine and Cosine Law Calculator&lt;/li&gt;&#xA;&lt;li&gt;Square Calculator&#xA;and many more....&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It's really helpful tool for calculations. Following images show the options you'll have on the homepages :&#xA;&lt;a href=&quot;https://i.stack.imgur.com/sLzC3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sLzC3.png&quot; alt=&quot;Image1&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/S71NB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/S71NB.png&quot; alt=&quot;Image2&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MkMqu.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MkMqu.png&quot; alt=&quot;Image3&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can explore many things. I suggest to try this for calculations once.&lt;/p&gt;&#xA;" OwnerUserId="14466" LastEditorUserId="14466" LastEditDate="2020-08-19T19:35:28.520" LastActivityDate="2020-08-19T19:35:28.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18689" PostTypeId="1" CreationDate="2020-08-20T06:11:39.703" Score="15" ViewCount="1033" Body="&lt;p&gt;I am preparing beamer slides for an online class, and I am unsure whether I should display different items in a single slide as they are discussed or all at once.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;To be more precise: I am teaching Linear Algebra. Most topics discussed during class consist of 3 or so parts, displayed in 1 or 2 slides, such as &amp;quot;definition&amp;quot;+&amp;quot;example&amp;quot;+&amp;quot;warning&amp;quot; +/or &amp;quot;theorem&amp;quot;, such as&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Definition&lt;/strong&gt;: Linear span&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Example&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$(1,2)$&lt;/span&gt; belongs to the linear span of &lt;span class=&quot;math-container&quot;&gt;$\{(0,1),(1,0)\}$&lt;/span&gt;, because &lt;span class=&quot;math-container&quot;&gt;$(1,2)=1\cdot (1,0)+2\cdot(0,1)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Warning&lt;/strong&gt;: &amp;quot;The linear span of the empty set is &lt;span class=&quot;math-container&quot;&gt;$\{0\}$&lt;/span&gt;&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Similarly, a proof of a theorem could consist of several different steps.&lt;/p&gt;&#xA;&lt;p&gt;I have the option to either display each slide fully right off the bat, or to display each part/step while I am explaining it. I am not sure if there is a better approach to how to prepare my class. Not preparing pauses would save me a little time.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If I display the whole content right at the beggining, this might make students want to read it all and not pay attention to the explanation of the subject.&lt;/li&gt;&#xA;&lt;li&gt;If I display the content step-by-step, this might create several &amp;quot;mini-surprises&amp;quot; throughout a single slide, which could make them lose focus for a second or two, while they're reading more-or-less the same thing that I am saying.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Is there a &amp;quot;better practice&amp;quot; when it comes to introducing these pause between parts of the material? Is there a preferred approach? Experience and constructive opinions on the subject are more than welcome.&lt;/p&gt;&#xA;" OwnerUserId="14410" LastEditorUserId="14410" LastEditDate="2020-09-04T13:52:15.703" LastActivityDate="2020-09-04T13:54:13.223" Title="Best form for slide / beamer presentation: display items in a slide as they are discussed or all at once?" Tags="|teacher-preparation|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18690" PostTypeId="2" ParentId="18689" CreationDate="2020-08-20T13:12:00.183" Score="2" Body="&lt;p&gt;My preference is to use &lt;code&gt;\pause&lt;/code&gt; or similar commands to review parts of slides little by little, but not to the extreme.&lt;/p&gt;&#xA;&lt;p&gt;As you said, if you review too all content at once, students get distracted. Also, too much content on one slide creates a dreadful/boring impression.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, beamer significantly slows down if you use too much pauses. That's because each pause generates a new PDF page. I had a slide show that took 2 minutes to compile. You can speed &lt;a href=&quot;https://tex.stackexchange.com/questions/27351/speed-up-beamer-compile-time&quot;&gt;things up a little bit&lt;/a&gt;. But it is quite fiddly to do this.&lt;/p&gt;&#xA;&lt;p&gt;I'd suggest that if you are familiar with &lt;a href=&quot;https://jupyter.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;jupyter&lt;/a&gt;, you can try &lt;a href=&quot;https://rise.readthedocs.io/en/stable/&quot; rel=&quot;nofollow noreferrer&quot;&gt;RISE&lt;/a&gt;, which is much easier to create incremental slide shows.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorDisplayName="user11702" LastEditDate="2020-08-20T13:49:48.797" LastActivityDate="2020-08-20T13:49:48.797" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18691" PostTypeId="2" ParentId="18689" CreationDate="2020-08-20T14:03:33.407" Score="13" Body="&lt;p&gt;A style which I really like is to have all the material on the slide, but have the material which you haven't reached yet in light grey. Then, as you move forward, advance your slides to turn light grey into black. That way, people who really want to see what is coming ahead can do that, but it is clear where their attention should be. Here are some &lt;a href=&quot;https://kskedlaya.org/slides/promys2020.pdf&quot; rel=&quot;noreferrer&quot;&gt;slides by Kiran Kedlaya&lt;/a&gt; in this style.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2020-08-20T14:03:33.407" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18692" PostTypeId="2" ParentId="18684" CreationDate="2020-08-20T22:40:31.803" Score="4" Body="&lt;p&gt;I do not believe there are any good such strategies, especially if you are in a Ph.D. program. I think your question is an important one because the comprehensive exam system at some places conveys some need to desperately learn lots of mathematics quickly, and as a result, many students feel this sort of pressure. The late Ken Appel told me once that nobody ever failed a comp on a hard question, but the grad students were almost always, in my opinion, scrambling to learn too much too quickly and then would fail on these easy questions because they were not able to slow down and adequately solidify the basics. In the end, I think most of the failing grad students were even more demoralized in the end because they felt that they &amp;quot;didn't even know&amp;quot; the basics when of course they could not when they were trying to learn everything all at once. I think there are a lot of grad students at a lot of places that really should not be in grad school, but universities have graduate programs and courses that need TAs and so many very weak students are let into grad school and spend a lot of valuable time working toward a degree in order to find themselves in a desperate career conundrum later on.&lt;/p&gt;&#xA;&lt;p&gt;The reality is that students with solid backgrounds are building on a stock of intuitions and habits that have been built over years of practice. Your question is like asking how to quickly become fluent in a new foreign language. There are ways to improve quickly, but I don't think anybody really thinks that any such method will, for most people, quickly build fluency.&lt;/p&gt;&#xA;&lt;p&gt;There is another strategy in this situation, though. Often when talking with some strong graduate students and mathematicians, you'll find that they disavow any proficiency with things, not in their areas of expertise, and deny having any ability to understand these things. Often these people do have the raw ability to learn more broadly, but instead seem to more completely embrace the understanding that their ability and time are limited and employ a &amp;quot;lens&amp;quot; approach to learning. They talk to people to find out quickly the way to think about the core fundamentals of a new area, and then develop this in a very targeted way. They master the basics unapologetically and stick close to what they know...gradually broadening the base of their knowledge.&lt;/p&gt;&#xA;&lt;p&gt;The trouble with this, sometimes, is that you can finish a Ph.D. in this way, but if your results are not spectacular, you will face very stiff competition in the academic job market from people with solid deep results and a broader knowledge base. So it's a long shot when you have a weak background, and you have to think very carefully about whether you are in love with the idea of being a mathematician, or if you love mathematics. The first situation happens a lot more than you'd think, and it is unsustainable.&lt;/p&gt;&#xA;&lt;p&gt;Another important thing to consider here is your health. If I were to tell you that you need to train to run a marathon in two weeks and you were by no means in shape for it, you'd probably not try to do it. Trying to push yourself to &amp;quot;survive&amp;quot; graduate school by trying to accumulate a bunch of things you aren't familiar with very quickly is similar, and just as trying to cram for a marathon would be bad for your health, doing something like this is likely to be very bad for your physical and mental health.&lt;/p&gt;&#xA;&lt;p&gt;It is hypothesized that people are generally happiest and most successful when their skills match their challenges. I don't think grad programs are one size fits all, and the fit is very important to consider carefully.&lt;/p&gt;&#xA;&lt;p&gt;A sprinkle of hope, though. Suppose you start with a good fit for a master's degree, and then focus on the basics, the rate at which you learn is probably roughly proportional to what you've already solidly learned and so exponential growth may make you more competitive than you originally ever thought you could be in a strong Ph.D. program. This probably works best, though, when you aren't thinking about &lt;em&gt;volume&lt;/em&gt; or &lt;em&gt;speed&lt;/em&gt; but are feeling safe enough to focus on what is in front of you.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="14466" LastEditDate="2020-08-27T11:44:32.583" LastActivityDate="2020-08-27T11:44:32.583" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18693" PostTypeId="2" ParentId="18689" CreationDate="2020-08-20T22:52:52.043" Score="7" Body="&lt;p&gt;Ultimately, I think that this is a matter of taste and presentation style.  You should create documents which allow you to present in a way that feels natural to you.&lt;/p&gt;&#xA;&lt;p&gt;My preference (and this is a &lt;em&gt;preference&lt;/em&gt;, I have no research to back me up) is to use &lt;code&gt;\pause&lt;/code&gt;, &lt;code&gt;\onslide&lt;/code&gt;, &lt;code&gt;\only&lt;/code&gt;, etc quite liberally.  I like to tell a story, which (for me) means having the space to spread things out a bit.  By way of example, my PhD defense consists of 30 frames, only 23 of which contain any real content (the remainder are things like the title slide, ToC, references, section headers, etc).  However, due to the liberal use of &lt;code&gt;\pause&lt;/code&gt; and friends, the compiled .pdf file is about 140 pages long.&lt;/p&gt;&#xA;&lt;p&gt;Of course, one should also keep in mind that what is on the slide should be disjoint from what you are saying.  For example, if I give a definition on a slide, I will talk about the &lt;em&gt;meaning&lt;/em&gt; of the definition—my students can read the thing; my goal in lecture is to give &lt;em&gt;intuition&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;That being said, beamer has a useful feature for (in some sense) doing both:  the &amp;quot;handout&amp;quot; option in the document class.  If you declare&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;\documentclass[handout]{beamer}&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;in your preamble, then your document will be compiled as if there were no pauses or other similar statements.  This can monkey with the spacing a bit if you use &lt;code&gt;\only&lt;/code&gt; at all, but otherwise works pretty well.  Generally, when I make presentations, I create two files.  The first is the &amp;quot;handout&amp;quot; version, a file called (for example) &lt;code&gt;handout.tex&lt;/code&gt; which contains precisely the following&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;\PassOptionsToClass{handout}{beamer}&#xA;\def\ishandout{1}&#xA;\input{slides}&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Then there is a second file called &lt;code&gt;slides.tex&lt;/code&gt; (the file input by the third line of &lt;code&gt;handout.tex&lt;/code&gt;) which is an ordinary beamer file.  Either file can be compiled as usual (e.g. run&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;$ pdflatex handout.tex&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;$ pdflatex slides.tex&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;from the command line).  Note that the flag &lt;code&gt;ishandout&lt;/code&gt; can also be used through the TeX document.  For example, maybe I want to write a proof out by hand while giving a presentation, but I want my students to have a typeset proof from the slides.  Then in the &lt;code&gt;slides.tex&lt;/code&gt; file, I can write&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;\ifdefined\ishandout&#xA;    &lt;span class=&quot;math-container&quot;&gt;\begin{proof}&#xA;        ...&#xA;    \end{proof}&lt;/span&gt;&#xA;\fi&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;The proof will be included in the handout, but not the slides.&lt;/p&gt;&#xA;&lt;p&gt;When teaching a class, I use the version with all of the pauses in it.  I can then upload the pause-less version to wherever it is that my students can find it (e.g. to Blackboard or Canvas or Moodle or Dropbox or Google Drive or my personal website or whatever).&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2020-08-20T23:51:01.063" LastActivityDate="2020-08-20T23:51:01.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18694" PostTypeId="2" ParentId="18689" CreationDate="2020-08-21T01:32:07.247" Score="2" Body="&lt;p&gt;One technique I've seen used which I like is, for a given topic, to have just all of the parts up to the current one being discussed shown. This way there's no temptation to read ahead since it's not shown, the students can see the previous parts to remind them of what they were (and, if for any reason they didn't read some of it earlier, they can do that now) and you can more easily refer back to any previous part(s) if need be.&lt;/p&gt;&#xA;&lt;p&gt;Using your example, the topic could be shown over &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; slides as:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Definition&lt;/strong&gt;: Linear span&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Definition&lt;/strong&gt;: Linear span&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Example&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$(1,2)$&lt;/span&gt; belongs to the linear span of &lt;span class=&quot;math-container&quot;&gt;$\{(0,1),(1,0)\}$&lt;/span&gt;, because &lt;span class=&quot;math-container&quot;&gt;$(1,2)=1\cdot (1,0)+2\cdot(0,1)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Definition&lt;/strong&gt;: Linear span&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Example&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$(1,2)$&lt;/span&gt; belongs to the linear span of &lt;span class=&quot;math-container&quot;&gt;$\{(0,1),(1,0)\}$&lt;/span&gt;, because &lt;span class=&quot;math-container&quot;&gt;$(1,2)=1\cdot (1,0)+2\cdot(0,1)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Warning&lt;/strong&gt;: &amp;quot;The linear span of the empty set is &lt;span class=&quot;math-container&quot;&gt;$\{0\}$&lt;/span&gt;&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A variant of this idea would be to have all of the parts after the one currently being discussed being greyed out instead of not being shown, similar to what &lt;a href=&quot;https://matheducators.stackexchange.com/users/51/david-e-speyer&quot;&gt;David E Speyer&lt;/a&gt;'s &lt;a href=&quot;https://matheducators.stackexchange.com/a/18691/12978&quot;&gt;answer&lt;/a&gt; suggests. I haven't seen this variant used before, but I don't think I'd like it as much since I would find the greyed out text to be distracting. I'm mentioning it here anyway in case you might like it better.&lt;/p&gt;&#xA;" OwnerUserId="12978" LastEditorUserId="12978" LastEditDate="2020-08-21T06:17:58.137" LastActivityDate="2020-08-21T06:17:58.137" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18695" PostTypeId="2" ParentId="17864" CreationDate="2020-08-21T04:57:18.307" Score="3" Body="&lt;p&gt;I can relate to your students, as the education system in most countries is very much like a straight-jacket, especially when it comes to math.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Some tips I can think of that might help&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;When introducing a topic, begin by motivating it with real world examples of where you might apply it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Don't just do the &amp;quot;drill and kill&amp;quot;. Instead, mix things up by having group assignments and projects. It needs a little creativity on your part to understand when it is appropriate to use these methods.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try and use heuristics and intuitive explanation where possible, as this help with bridging the gap between everyday thinking and the more formal stuff.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When introducing a problem, have an introduction where you work out what you expect using only intuition, and at the end of the problem, do a debrief where you check if your intuitions matched up.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have extra credit assignments that are more open-ended by nature, and encourage exploratory skills.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have a good amount of healthy competition between your students by dividing them up into teams, when possible.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I know these are not all directly related to showing them the beauty of math. The reason I think they are still important is that the beauty of math is multifaceted in nature. Sometimes, math is beautiful for its simplicity and elegance, and at other times, it is the sheer usefulness that draws one to it. Some students may be drawn to its practical power, while others may be drawn to its aesthetic elegance. Let them decide which parts they like most and for what reasons, but by following the above points, you will at least be giving them a chance to get proficient in the subject.&lt;/p&gt;&#xA;&lt;p&gt;REMEMBER: Sometimes, the math bug &lt;a href=&quot;https://barbaraoakley.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;can bite you later in life&lt;/a&gt;. You want to give them the best chance possible. The rest is up to them and Lady Luck.&lt;/p&gt;&#xA;" OwnerUserId="14471" LastActivityDate="2020-08-21T04:57:18.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18696" PostTypeId="2" ParentId="17045" CreationDate="2020-08-21T15:36:29.823" Score="1" Body="&lt;p&gt;I have a spreadsheet which I can use to select random students.  I have created a version of that spreadsheet as a &lt;a href=&quot;https://docs.google.com/spreadsheets/d/1XksqaHM-Jbi6XKILwUSOulVLlS1D3E1LdFZROdPOcwc/edit?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google sheet&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ns3aF.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ns3aF.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;To explain how this works:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Column A is a list of the students in the class.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Column B tracks attendance.  I take attendance at the beginning of every class, and enter that into my gradebook.&lt;sup&gt;[1]&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The cell C2 contains the function &lt;code&gt;=sort(A2:A,B2:B,TRUE)&lt;/code&gt;.  This (essentially) creates a sorted copy of column A, with the students who are present on top.  This function populates the rest of column C.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The cell D2 contains the function &lt;code&gt;=index(C2:C, randbetween(1,counta(B2:B)))&lt;/code&gt;, which randomly selects a student from among those present.  This is a little complicated:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;code&gt;counta(B2:B)&lt;/code&gt; returns the number of non-empty cells in column B, excluding the header cell&lt;/li&gt;&#xA;&lt;li&gt;&lt;code&gt;randbetween(1,n)&lt;/code&gt; returns a random integer between &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; (inclusive); in this case &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a count of the number of students who are present, so this function returns the index of some student who is present&lt;/li&gt;&#xA;&lt;li&gt;&lt;code&gt;index([list],k&lt;/code&gt; returns the &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-th element of [list]; in this case, the list is &lt;code&gt;C2:C&lt;/code&gt;, which is a list of all students in the class (plus a bunch of empty cells), sorted so that the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; entries are the students who are present; as &lt;span class=&quot;math-container&quot;&gt;$k \le n$&lt;/span&gt;, the &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-th element of [list] will be a student who is present&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The cell E2 contains a checkbox.  The default behaviour of Google Sheets (and MS Excel, and Apple's Numbers) is to refresh the spreadsheet every time any cell in the sheet is altered.  Thus checking (or unchecking) the box will rerun the random number generator, thus making a new selection.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The random number generator may select the same student more than once (there is nothing here which prevents this from happening).  If you want to ensure that you don't call on a student a second time, delete the &lt;code&gt;1&lt;/code&gt; in column B after you call on them.  This will cause the list in column C to be resorted, and will also select a new student.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;[1]  When I assign a failing grade or an administrative withdrawal, I am required to list the last date of attendance.  Thus, as a CYA measure, I track attendance in my gradebook.  When I record attendance in my gradebook at the beginning of class, I enter a &lt;code&gt;1&lt;/code&gt; if a student is present, and leave the appropriate cell blank otherwise.  It is pretty simple to copy that column over to this sheet at that time.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2020-08-21T15:36:29.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18697" PostTypeId="2" ParentId="17045" CreationDate="2020-08-22T06:00:09.523" Score="2" Body="&lt;p&gt;In my opinion there is never a random student. You pick a student from a group, a group is made up of student at same level of skillset and knowledge etc and within a group you sort of go by gut feel and more or less round robin way such that everyone gets asked questions. It is important for students to know that anyone can be asked to answer a question.&#xA;In my opinion choosing students is a bit of art and science. More of an art in my opinion.&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2020-08-22T06:00:09.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18698" PostTypeId="2" ParentId="18567" CreationDate="2020-08-22T07:13:42.160" Score="0" Body="&lt;p&gt;So I teach kids and this is what I do. For them two things are most important:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To know their bearings in Mathematical Universe&lt;/li&gt;&#xA;&lt;li&gt;Connection with &amp;quot;Real World&amp;quot; entities/concepts etc.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Whether I am teaching kids a new procedure division, multiplication, square root or concepts like sets, matrices etc. I make sure that they know/understand these new things in reference to things already known to them, otherwise they find it difficult to place the knowledge properly and are left hanging with no connection to reference points AND how/where it connects with this real world. This gives a place for kids to arrange that concept in their mind(bearing) and connection with real world helps them understand, formulate ideas and opinions about it and remember. And of course you cannot over emphasize Repetition. Practicing the procedure, reading and thinking about concept multiple times helps kids internalize it.&lt;/p&gt;&#xA;&lt;p&gt;The same applies to higher Mathematics....&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2020-08-22T07:13:42.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18702" PostTypeId="2" ParentId="18532" CreationDate="2020-08-22T14:56:55.143" Score="3" Body="&lt;p&gt;Part of the answer is that mathematicians stumbled upon structures that had an &amp;quot;addition&amp;quot; and a &amp;quot;multiplication&amp;quot; that behaved (more or less) like integers do (rationals, reals, complex numbers, the residue rings, matrices, polynomials, power series, rational functions, ...). So it made sense to just assume a set and operations with those properties (and nothing else) as axioms (i.e., as given, not to be questioned) and see what can be deduced from them. Bonus is that anything you prove for abstract rings automatically applies to each of the above.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-08-22T14:56:55.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18703" PostTypeId="2" ParentId="18532" CreationDate="2020-08-23T00:16:55.243" Score="0" Body="&lt;p&gt;This is not a direct response to your question.&lt;/p&gt;&#xA;&lt;p&gt;One of the hot topics in algebraic geometry today is known&#xA;as&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Tropical_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tropical geometry&lt;/a&gt;.&#xA;The addition and multiplication of polynomials is&#xA;replaced with&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{eqnarray}&#xA;x \oplus y &amp;amp; = &amp;amp;\min\{x,y\} \\&#xA;x \otimes y &amp;amp; = &amp;amp; x + y&#xA;\end{eqnarray}&lt;/span&gt;&#xA;in the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Tropical_semiring&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;tropical semiring&lt;/em&gt;&lt;/a&gt;.&#xA;Maybe you could use this to make these abstract properties&#xA;newly relevant?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/302GQ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/302GQ.jpg&quot; alt=&quot;Trop&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;Image from &lt;a href=&quot;http://www.mittag-leffler.se/langa-program/tropical-geometry-amoebas-and-polytopes-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mittag-Leffler&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-08-23T00:16:55.243" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18705" PostTypeId="2" ParentId="1" CreationDate="2020-08-23T18:12:02.920" Score="3" Body="&lt;p&gt;There are many answers already, but just speaking out my point of view. I am a college student. Usually what happens is the student who have uncleared concepts needs help but they don't reach out to professor in the class, may be because they are afraid of asking questions that seem easy to other students (as you already mentioned).&lt;/p&gt;&#xA;&lt;p&gt;In College, they don't have the good bonds with the tutor as they have in schools. So, if you try creating a &lt;strong&gt;friendly and supportive environment&lt;/strong&gt;, it may result in following good things :&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Students will love attending your classes.&lt;/li&gt;&#xA;&lt;li&gt;They'll learn the things with an understanding and not just for learning purpose.&lt;/li&gt;&#xA;&lt;li&gt;They'll start loving mathematics once they'll have a supportive tutor + cleared&#xA;concepts.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;A supportive tutor is always a plus point.&lt;/strong&gt;&#xA;It feels good when your tutor helps you out to untangle the tangled strings of concepts and logic.&lt;/p&gt;&#xA;" OwnerUserId="14466" LastActivityDate="2020-08-23T18:12:02.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18706" PostTypeId="1" AcceptedAnswerId="18707" CreationDate="2020-08-24T12:55:42.027" Score="1" ViewCount="108" Body="&lt;p&gt;I'm currently an undergraduate student who wants to do research on the pedagogy of formal logic. As a result, I wanted to know what are some challenges that instructors (or even students for that matter) encounter when teaching introductory formal logic (propositional, first-order, modal). Proofs are the obvious answer, since natural deduction may sometimes not come across as intuitive to students, or just the abstract nature of logic in general. Some students, such as computer science, mathematics, and engineering students may find formal logic easier since it relates to some of their topics (boolean logic, for example), whereas others that aren't as logically inclined may struggle. This is what I have so far. Does anyone have any stories or other examples? Thanks!&lt;/p&gt;&#xA;" OwnerUserId="14486" LastActivityDate="2020-08-24T14:03:13.587" Title="What are some of the struggles that come with teaching introductory formal logic?" Tags="|logic|undergraduate-research|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18707" PostTypeId="2" ParentId="18706" CreationDate="2020-08-24T14:03:13.587" Score="2" Body="&lt;p&gt;One struggle I've often run into is that logic seems to require a lot of preliminary material before one gets to interesting consequences: there's often a lot of &amp;quot;saying obvious things slowly and carefully&amp;quot; before reaching the interesting consequences.&lt;/p&gt;&#xA;&lt;p&gt;That sometimes leads students to get bored and check out or drop the course when they might have enjoyed it if they stayed longer.&lt;/p&gt;&#xA;&lt;p&gt;It also means there's usually a big jump in difficulty part way through the course.  But a lot of students get into their routine early in the semester (or other academic time unit) and get used to treating an easy course as not needing a lot of time, and adjust slowly or not at all when that changes.  (It can also mean that some students who aren't really sufficiently prepared decide they can handle the course because they find the early material very difficult but doable, but are then in too deep when the course gets harder.)&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2020-08-24T14:03:13.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18708" PostTypeId="2" ParentId="18532" CreationDate="2020-08-24T14:44:39.220" Score="3" Body="&lt;p&gt;I think associativity is the most natural property to encounter. This is because transformations are associative. For transformations &lt;span class=&quot;math-container&quot;&gt;$f,g,h$&lt;/span&gt; with compatible domains and ranges, &lt;span class=&quot;math-container&quot;&gt;$(f\circ g)\circ h=f\circ(g\circ h)$&lt;/span&gt;. Transformations are not &lt;em&gt;necessarily&lt;/em&gt; commutative.&lt;/p&gt;&#xA;&lt;p&gt;In your question, I sense you want to focus on motivating the foundational properties of the ring &lt;span class=&quot;math-container&quot;&gt;$(\mathbb{Z},+,\cdot)$&lt;/span&gt;. To see why we should postulate associativity, we only need to think about the situation of an iterated function &lt;span class=&quot;math-container&quot;&gt;$f:X\to X$&lt;/span&gt;. Evidently, modeling the notion of iterated functions leads one to the structure &lt;span class=&quot;math-container&quot;&gt;$(\mathbb{Z},+)$&lt;/span&gt;, which is associative and commutative.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;naturalness&amp;quot; of the distributive property will of course depend on how we introduce multiplication to our additive structure above. Typically this is done by modeling the idea of &amp;quot;repeated addition&amp;quot; beginning with  &lt;span class=&quot;math-container&quot;&gt;$$2\cdot x=x+x$$&lt;/span&gt;&#xA;From this point of view, multiplication is a mere notational convenience that builds on our ability to count. The distributive property of multiplication over addition is a consequence of the commutativity and associativity of our additive structure.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2020-08-24T15:15:38.760" LastActivityDate="2020-08-24T15:15:38.760" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18710" PostTypeId="2" ParentId="18372" CreationDate="2020-08-25T12:54:36.277" Score="0" Body="&lt;p&gt;Math seems difficult for most of the students. I'm a student at university. What makes mathematics easier for a student is concepts. Like, for instance, trigonometry lesson is introduced in 9th Grade and 90% of teachers don't explain what actually trigonometry is! Also, it's introduced as a difficult topic. According to me, if it's in the following pattern every student would love to solve the math problems :&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;1.Introducing the topic with real-life examples&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;If someone's going to teach me differentiation conducting some fun activity related to the topic, I would be eagerly waiting for the next class. Why this thing is necessary? What usually happens is students actually don't know the applications of the topics taught.&lt;/p&gt;&#xA;&lt;h2&gt;2.Explaining the theory and methods of solving&lt;/h2&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;You can introduce some basic theory to students, for instance, what d/dx symbolizes. It has two good things: your students will have some extra knowledge outside the textbook and secondly they'll know the importance of understanding the concepts rather than learning them. My favorite part in notes was: method of solving. My teacher added a stepwise method of solving in her notes. When students have stepwise instructions they would be able to solve the problems faster. Also, divide the lesson into subtopics!&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;3. Practice&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Giving sums to practice in after explaining the theory. Explain the theory and method of solving for a subtopic and then give 5-10 different types of sums that students will solve in your presence.&lt;/p&gt;&#xA;&lt;p&gt;This really helped me in building good concepts and improving my results. Also, if not needed don't disclose difficulty levels of lessons.&lt;/p&gt;&#xA;" OwnerUserId="14466" LastActivityDate="2020-08-25T12:54:36.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18712" PostTypeId="2" ParentId="18351" CreationDate="2020-08-25T14:16:10.427" Score="-1" Body="&lt;p&gt;I am a third-year student pursuing a bachelor of science with a major subject as statistics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Statistics can be categorized into 2 types:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Descriptive Statistics:&lt;/strong&gt; It is used for summarizing observations etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Inferential Statistics:&lt;/strong&gt; It is used for interpreting the meaning of the descriptive stats.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Firstly, search out which part of statistics is more useful to you, i.e., descriptive or inferential, and then learn the basics. I suggest learning the basics of statistics as without basics there's no use of other theories. For learning, you have two ways: online and offline. For online learnings, You could simply login to Coursera or Khan academy they have simple and good explanations of the topics. For offline, my suggestions on books are based on materials used by students in our college.&lt;/p&gt;&#xA;&lt;p&gt;Suggested Textbooks  for self-study:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Fundamentals of Mathematical Statistics, Sultan Chand&#xA;&amp;amp; Sons Pvt. Ltd. New Delhi.&lt;/strong&gt; &lt;a href=&quot;https://www.academia.edu/31552250/FUNDAMENTALS_OF_MATHEMATICAL_STATISTICS&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.academia.edu/31552250/FUNDAMENTALS_OF_MATHEMATICAL_STATISTICS&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Statistics, S. Chand and Company Ltd. New Delhi&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Mathematical Statistics with Applications, Pearson Education.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I hope this is helpful.&lt;/p&gt;&#xA;" OwnerUserId="14466" LastEditorUserId="14466" LastEditDate="2020-10-16T07:04:59.147" LastActivityDate="2020-10-16T07:04:59.147" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18713" PostTypeId="2" ParentId="18047" CreationDate="2020-08-25T15:37:46.177" Score="0" Body="&lt;p&gt;&lt;em&gt;&lt;strong&gt;&lt;a href=&quot;http://fstroj.uniza.sk/kam/orsansky/pdf/eng/basicsofstatisticalmethods.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://fstroj.uniza.sk/kam/orsansky/pdf/eng/basicsofstatisticalmethods.pdf&lt;/a&gt;&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The pdf of this book contains :&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Probability theory&lt;/li&gt;&#xA;&lt;li&gt;Random variable&lt;/li&gt;&#xA;&lt;li&gt;Significant continuous distribution of a random variable&lt;/li&gt;&#xA;&lt;li&gt;Descriptive statistics&lt;/li&gt;&#xA;&lt;li&gt;Estimates of parameters&lt;/li&gt;&#xA;&lt;li&gt;Testing statistical hypotheses&lt;/li&gt;&#xA;&lt;li&gt;Correlation Analysis&lt;/li&gt;&#xA;&lt;li&gt;Paired linear regression&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This book covers the maximum syllabus of the first/second year of undergrad mathematics/statistics syllabus. This book was suggested by my professors. I'm a third-year undergraduate statistics student. I hope this helps!&lt;/p&gt;&#xA;&lt;p&gt;While studying things online, I came across this book :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Naked Statistics: Stripping the Dread from the Data [ Book by Charles Wheelan ]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This book was successful in filling in the conceptual gaps of my understanding of statistics. So, if you are looking for some book that clears your concepts of statistics, this is a good read. I ordered the hard copy since I didn't found the opensource for this yet, but if I will get, I'll share the link here.&lt;/p&gt;&#xA;" OwnerUserId="14466" LastEditorUserId="14466" LastEditDate="2020-08-31T17:29:34.903" LastActivityDate="2020-08-31T17:29:34.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18716" PostTypeId="2" ParentId="4018" CreationDate="2020-08-25T17:49:26.057" Score="0" Body="&lt;p&gt;Sergei Treil &lt;em&gt;Linear Algebra Done Wrong&lt;/em&gt; is used in a first course for honors linear algebra at Brown, so could be useful for a second course at some other places.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://sites.google.com/a/brown.edu/sergei-treil-homepage/linear-algebra-done-wrong?authuser=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://sites.google.com/a/brown.edu/sergei-treil-homepage/linear-algebra-done-wrong?authuser=0&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10289" LastActivityDate="2020-08-25T17:49:26.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18717" PostTypeId="1" AcceptedAnswerId="18755" CreationDate="2020-08-25T23:54:45.773" Score="4" ViewCount="201" Body="&lt;p&gt;Most of the students I've encountered seem to have had the same sort of math education I've had, the standard lecture, book readings, homework problems and exams.&lt;/p&gt;&#xA;&lt;p&gt;In my own experience and in my education classes, this method has been superseded by more modern, active strategies where students learn by doing.&lt;/p&gt;&#xA;&lt;p&gt;Are there any studies that can give an estimate of how many K-12 math classes are taught in the traditional way and how many in more modern ways?&lt;/p&gt;&#xA;&lt;p&gt;I've been looking around, but my searches have returned noise level results.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;b&gt;Additional&lt;/b&gt;: &#xA;&lt;p&gt;This article might clarify some of the questions about what is a traditional lecture versus a more modern active learning approach.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.pnas.org/content/111/23/8410&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.pnas.org/content/111/23/8410&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8642" LastEditorUserId="8642" LastEditDate="2020-09-01T19:21:22.653" LastActivityDate="2020-09-01T19:21:22.653" Title="Is there a study on how often modern teaching methods versus traditional methods are used in a math classes?" Tags="|reference-request|interactive-teaching|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18718" PostTypeId="1" AcceptedAnswerId="18719" CreationDate="2020-08-26T06:38:02.670" Score="1" ViewCount="185" Body="&lt;p&gt;Due to the COVID pandemic,  classes at my school (small public liberal arts college) will be all online. I've chosen to try teaching &lt;strong&gt;asynchronously&lt;/strong&gt; (via pre-made video lectures) starting next week. I have 5 students in intro abstract algebra (senior level course) and 30 students in intro linear algebra (sophomore/junior level course). (Both are undergraduate courses).&lt;/p&gt;&#xA;&lt;p&gt;The following off-the-wall idea occurred to me about grading for the course, and I haven't really thought this through at all, and I wanted to hear feedback about it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I'm thinking of basing the entire course grade on roughly every-other-day emails (about anything about the course material) I ask students to send to me&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here are the details:&lt;/p&gt;&#xA;&lt;p&gt;I'll ask students to email me &lt;strong&gt;every other day&lt;/strong&gt; (or &lt;strong&gt;4 out of 7&lt;/strong&gt; days of the week they select in advance) telling me anything they &lt;em&gt;learned/thought/did&lt;/em&gt; that day or the previous day involving the course material: e.g. in their email they can include any one of the following (these are just examples):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;homework problem(s) they did (attach a photo, I'll provide feedback but not record a grade)&lt;/li&gt;&#xA;&lt;li&gt;tell me one or two ideas in a video they watched&lt;/li&gt;&#xA;&lt;li&gt;summarize the steps to do a type of problem&lt;/li&gt;&#xA;&lt;li&gt;ask in the email a question (or confusion, etc) they have about the course material (perhaps with some&lt;/li&gt;&#xA;&lt;li&gt;mention a question they read (or asked) on math.stackexchange.com on abstract or linear algebra and what they learned&lt;/li&gt;&#xA;&lt;li&gt;show me they can meet a `standard' from a list of standards (in the sense of standards based grading, but I won't actively enforce those standards) associated to the course (e.g. find the reduced row echelon form of a matrix). (Attach a photo).&lt;/li&gt;&#xA;&lt;li&gt;make up their own problem and solve it&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Please help me think of other possibilities or improved wording!&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now to the grading.&lt;/p&gt;&#xA;&lt;p&gt;One option is to go full ungrading mode and assign no points at all, and just give brief feedback (e.g words instead of points in terms of effort/output). And then ask students to give a grade to themselves at the end of the course requiring supporting evidence, telling them in advance  that these frequency and quality of the emails is one factor. However during the course students may not know exactly where they stand gradewise, while with points they have a better idea... but based on what I’m reading about points being bad, that points information could be problematic/misleading/demotivating&lt;/p&gt;&#xA;&lt;p&gt;I'll grade each email response on a scale of 0-7 points (even though I've been reading about ils out point based grading, and want to stop using points) as follows:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;0 points missed deadline to send email&lt;/li&gt;&#xA;&lt;li&gt;1 point for at least sending me an email, but nothing substantive (for example saying that you couldn't get a chance to work on anything today; I understand and appreciate that)&lt;/li&gt;&#xA;&lt;li&gt;2,3  minimal effort/output (in my opinion)&lt;/li&gt;&#xA;&lt;li&gt;4,5 decent effort/output&lt;/li&gt;&#xA;&lt;li&gt;6,7 significant thought or work done&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I reserve the dock points if they are falling significantly 'behind' in terms of the pace of the course, but will try to avoid using this except in extreme cases. e.g studen starting week 1 homework on week 14 is not going to earn much (although if they are reviewing then that is fine).&lt;/p&gt;&#xA;&lt;p&gt;I'll also ask students to (if they want) &lt;em&gt;tell me&lt;/em&gt; in the email  how many points they think they deserve. (I'll generally give whatever they ask for; if there is some disagreement I'll give what they ask and tell them put more effort next time).&lt;/p&gt;&#xA;&lt;p&gt;In terms of how this translates, here is the correspondence between course letter grade and daily average:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A 4.5&lt;/li&gt;&#xA;&lt;li&gt;B 3.5&lt;/li&gt;&#xA;&lt;li&gt;C 2.5&lt;/li&gt;&#xA;&lt;li&gt;D 1&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(and these cutoffs might be lowered in students' favor, I have no idea if they are too high).&#xA;So to pass one just needs to .&lt;/p&gt;&#xA;&lt;p&gt;One slightly thorny issue is dealing with cases where for various reasons (e.g religions, childcare, work) one can't do course work on certain days of the week (e.g. Saturday), but that's why I changed what initially was daily emails to every-other-day and then to each student can choose 4 out of the 7 days the week to email me.&lt;/p&gt;&#xA;&lt;p&gt;Brief oral midterm and final exams to make sure these grades are working appropriately (i.e. if someone is getting 7's but then can't answer on the spot questions similar to what they've been submitting&lt;/p&gt;&#xA;&lt;p&gt;Here's the &lt;strong&gt;why&lt;/strong&gt; behind the idea (which as I write this, seems a bit ill-conceived)&lt;/p&gt;&#xA;&lt;p&gt;The main difficulty with asynchronous courses I feel is that there is no structure to a student's day; and so it is easy to procrastinate/postpone working on it. See for example discussion &lt;a href=&quot;http://teaforteaching.com/146-lessons-learned-online/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://teaforteaching.com/146-lessons-learned-online/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;My 4-out-of 7 email assignment I feel from an (amateur) neuroscience prospective, provides some structure, and also spaces out the learning. The unconscious mind thinks about the material more often then and I imagine leads to better learning (but of course I have no evidence).&lt;/p&gt;&#xA;&lt;p&gt;I am hesitant to include homework and exams as part of the grade.&lt;/p&gt;&#xA;&lt;p&gt;First for homework, I am tired of students blindly copying answers to many many questions (from each other, from Chegg and their ilk, from college tutors) and turning in things they don't understand. This has been what has been happening in my face to face classes, although perhaps it is not the majority of students.&lt;/p&gt;&#xA;&lt;p&gt;As for standard hour long written exams, based on past experience (in face to face courses no less), I have no faith in students not cheating in &lt;strong&gt;exams&lt;/strong&gt; taken online, and I also don't believe in setting up surveillance state. However, from my own experience as a student, I feel such exams were crucial for forcing me to study and the pressure (e.g to be able to do problems closed book and under timed conditions) helped me learn to do the material. I previously gave an upper level undergraduate course with no exams, just homework, and some students just turned in solutions copied from elsewhere and simply did not retain learning anything. One possibility I'm toying with is &lt;strong&gt;optional&lt;/strong&gt; midterm/final written exams - if students want the opportunity it's their choice (doubt many students will take me up on it, and that's ok with me)&lt;/p&gt;&#xA;&lt;p&gt;(I do have the option to do synchronous sessions (90 minute long) during regularly scheduled class time, but I don't plan to do so (maybe during the scheduled class time I'll try once a week optional 15-30 minute chat/answer questions, etc just to make things feel less impersonal, but it won't be a 90 minute long lecture - although I fear that is what students will want).&lt;/p&gt;&#xA;&lt;p&gt;I am working on trying to be more compassionate towards students, and of course during this pandemic time having compassion is more important than ever.&lt;/p&gt;&#xA;&lt;p&gt;Anyways, all this is poorly thought out so I welcome criticisms.&lt;/p&gt;&#xA;" OwnerUserId="10289" LastEditorUserId="10289" LastEditDate="2020-08-26T07:17:09.947" LastActivityDate="2020-08-26T08:41:05.440" Title="asynchronous teaching and requiring frequent email updates from students, and having these as the only part of their grade" Tags="|grading|online-instruction|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18719" PostTypeId="2" ParentId="18718" CreationDate="2020-08-26T08:41:05.440" Score="5" Body="&lt;p&gt;With 35 students, 4 emails from each student per week, that's 20 emails per day on average from students - including weekends. If you don't read the emails on weekends, then it's 28 emails per day. I would be concerned about the amount of time it would take to read, respond, and make a grade judgement, and hence whether this is anything close to an efficient and effective way of using the time you have to help your students.&lt;/p&gt;&#xA;" OwnerUserId="9455" LastActivityDate="2020-08-26T08:41:05.440" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18720" PostTypeId="2" ParentId="14846" CreationDate="2020-08-26T10:42:18.760" Score="0" Body="&lt;p&gt;We can help a child by giving some tricks to solve things. For example, if a student is confused while performing operations on negative numbers, we can take the help of the number line.&#xA;Let's see how!&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Q: What is the addition of &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;What you can do in this case is: Draw a numbed line with &lt;span class=&quot;math-container&quot;&gt;$(-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6)$&lt;/span&gt;&#xA;The second step, ask the child what are the two numbers we have and what number he/she wants to take first. suppose he/she answers, &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt;, Then explain that we have to jump &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; units in a negative direction.  And adding &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; means going &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; more steps in the negative direction. which will lead to -5 (required answer).&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Few more Examples :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/yAnNu.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yAnNu.png&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;And for the equations with variables, here's another way :&lt;/h2&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Q: Find the vale of x : &lt;span class=&quot;math-container&quot;&gt;$2x+5=1$&lt;/span&gt;&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Solution :&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2x=(1-5)=-4$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$∴x=(-4/2)=-2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;These are the easy steps for 90% of the students but for the remaining 10%, we have to explain why &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; is subtracted from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; on the other side. We can explain this by explaining balance examples. For example, a weighing scale has &lt;span class=&quot;math-container&quot;&gt;$2x+5$&lt;/span&gt; apples on one side and one apple on the other. And &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; is the equal level between both sides. To balance you took five apples from both sides but, the side where we have &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; apple will become &lt;span class=&quot;math-container&quot;&gt;$-4$&lt;/span&gt; because you took one apple but you have to take &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; more apples!! And further you divide both weights by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;You can explain the concepts with the easiest examples you have. This will help a child to understand why we solve problems in this manner.&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="14466" LastEditorUserId="14466" LastEditDate="2020-08-26T10:49:05.270" LastActivityDate="2020-08-26T10:49:05.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18721" PostTypeId="1" CreationDate="2020-08-26T11:06:16.677" Score="4" ViewCount="576" Body="&lt;p&gt;I am interested in how you would encourage students to layout their working to a trigonometric equation.&lt;/p&gt;&#xA;&lt;p&gt;For instance, let's consider this problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Solve the equation &lt;span class=&quot;math-container&quot;&gt;$6\cos x - 8\sin x = 7$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; x &amp;lt; 2 \pi$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then I might write &lt;span class=&quot;math-container&quot;&gt;$$6 \cos x - 8 \sin x = -10 \sin \left(x - 0.6435... \right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and then solve &lt;span class=&quot;math-container&quot;&gt;$$-10 \sin(x - 0.6435...) = 7$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This re-arranges to &lt;span class=&quot;math-container&quot;&gt;$$\sin(x-0.6435...) = -0.7$$&lt;/span&gt; The principal value is &lt;span class=&quot;math-container&quot;&gt;$x - 0.6435... = -0.7753...$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The values in range are &lt;span class=&quot;math-container&quot;&gt;$x-0.6435... = \pi + 0.7753...$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x - 0.6435... = 2\pi - 0.7753...$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And if you solve these you get &lt;span class=&quot;math-container&quot;&gt;$x=4.56$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=6.15$&lt;/span&gt; to two decimal places.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;However, this has a LOT of decimals and I find that it is not the clearest way for a student to lay out their working.&lt;/p&gt;&#xA;&lt;p&gt;Of course we can work more exactly, but then that'd be quite difficult for lower ability high school students to understand rather than the formulaic approach.&lt;/p&gt;&#xA;&lt;p&gt;So how would you advise a student to layout their working for solving a trigonometric equation to keep it as simple and neat as possible?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit: in UK schools, solving equations of the form &lt;span class=&quot;math-container&quot;&gt;$a\cos x + b \sin x = c$&lt;/span&gt; is almost always encouraged to be done by first writing &lt;span class=&quot;math-container&quot;&gt;$a \cos x + b \sin x$&lt;/span&gt; in the form &lt;span class=&quot;math-container&quot;&gt;$R \sin (x + \alpha)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$R \cos (x + \alpha)$&lt;/span&gt;, depending on what is most suitable. That is why I skipped that step.&lt;/p&gt;&#xA;&lt;p&gt;My main point is I guess a general question in how do you avoid the excessive use of decimals in a problem like this? Of course, as I said above, you would work exactly and for me and you as skilled mathematicians this is easy. However, for the lower ability students, I wondered if anybody else had a neater way to lay something like this out that doesn't up the demand that much.&lt;/p&gt;&#xA;" OwnerUserId="14408" LastEditorUserId="14408" LastEditDate="2020-08-26T16:46:21.497" LastActivityDate="2020-08-27T12:25:46.923" Title="How to layout a solution to a trig equation?" Tags="|trigonometry|writing|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18723" PostTypeId="2" ParentId="18721" CreationDate="2020-08-26T14:40:02.490" Score="8" Body="&lt;p&gt;In most mathematics classes, we don't actually care about the solution to an exercise.  The point is to get students to practice with the concepts, and figure out how to communicate their thinking.  The solution you present has a number of problems for which I would likely deduct points:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The first step is to assert that &lt;span class=&quot;math-container&quot;&gt;$6 \cos(x) - 8 \sin(x) = -10 \sin(x - 0.6435\dotso)$&lt;/span&gt;.  First off, why?  How did you come to that conclusion?  Why is this true?  Second, it isn't really even true.  The ellipses kind of save the day, but what you seem to have done is used a calculator to compute something, which means that there is some roundoff error somewhere.  You don't have true equality.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Farther along, you assert that &amp;quot;The principle value is &lt;span class=&quot;math-container&quot;&gt;$x - 0.6435\dotso = -0.7753\dotso$&lt;/span&gt;&amp;quot;.  Again, why?  How did you get that?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In general, I would delay calculator based approximations until as late as possible.  If there are ugly expressions which you want to avoid working with, do as &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18721/how-to-layout-a-solution-to-a-trig-equation#comment51337_18721&quot;&gt;Andrew Chin suggests&lt;/a&gt;, and hide those expressions in a variable.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Much of what you have written is in the first person singular, but you finish in the second person.  It is also phrased somewhat passively.  I suppose that this is a stylistic choice, but I tend to take off (small fractions of) points for style.  Mathematics should be written actively, preferably in the imperative, and if one &lt;em&gt;must&lt;/em&gt; resort to pronouns, use the first person plural (e.g. &amp;quot;&lt;em&gt;We&lt;/em&gt; write...&amp;quot;, &amp;quot;&lt;em&gt;We&lt;/em&gt; solve...&amp;quot;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Near the end of the problem, you get two solutions by taking advantage of the fact that &lt;span class=&quot;math-container&quot;&gt;$\sin(\theta) = \sin(\pi - \theta)$&lt;/span&gt; for any angle.  This should be made explicit.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If I were to write up a solution to this problem, I would likely write something similar to the following:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Solve  &lt;span class=&quot;math-container&quot;&gt;$6\cos(x)- 8\sin(x) = 7$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; x &amp;lt; 2\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt;  By the angle addition formula the sine function,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 7 = 6\cos(x)- 8\sin(x) = A \sin(x+\alpha) = A\sin(\alpha)\cos(x) + A \cos(\alpha)\sin(x). \tag{*}$$&lt;/span&gt;&#xA;Equating coefficients on the left and right,&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\begin{cases}&#xA;6 = A\sin(\alpha), \\&#xA;-8 = A\cos(\alpha) \\&#xA;\end{cases}&#xA;&amp;amp;\implies&#xA;A = -\frac{8}{\cos(\alpha)}&#xA;\\&amp;amp;\implies&#xA;6 = -\frac{8}{\cos(\alpha)} \sin(\alpha)&#xA;\\&amp;amp;\implies&#xA;-\frac{3}{4} = \frac{\sin(\alpha)}{\cos(\alpha)} = \tan(\alpha).&#xA;\end{align}&lt;/span&gt;&#xA;This final equation is solved by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \alpha = \arctan\left(-\frac{3}{4}\right) + k\pi,\qquad k\in\mathbb{Z}.$$&lt;/span&gt;&#xA;With &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; even, the angle &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is in the fourth quadrant, and so corresponds to a right triangle with &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; at the origin, and vertices at &lt;span class=&quot;math-container&quot;&gt;$(4,0)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(4,-3)$&lt;/span&gt;.  This is a &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; right triangle with &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; adjacent to the leg of length &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, thus&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \cos(\alpha) = \frac{4}{5}. $$&lt;/span&gt;&#xA;By similar reasoning, if &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is odd, then &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; corresponds to an angle in quadrant II, and so&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \cos(\alpha) = -\frac{4}{5}. $$&lt;/span&gt;&#xA;Therefore&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \cos(\alpha) = (-1)^k \frac{4}{5}, \qquad k\in\mathbb{Z}. $$&lt;/span&gt;&#xA;Hence&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ A&#xA;= -\frac{8}{\cos(\alpha)}&#xA;= -\frac{8}{(-1)^k (4/5)}&#xA;= (-1)^{k+1} \frac{8\cdot 5}{4}&#xA;= (-1)^{k+1} 10.&#xA;$$&lt;/span&gt;&#xA;Substitute this into (*) and use the fact that &lt;span class=&quot;math-container&quot;&gt;$\arctan(-X) = -\arctan(X)$&lt;/span&gt; to get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 7 = 6\cos(x)- 8\sin(x) = (-1)^{k+1} 10 \sin\left(x-\arctan\left(\frac{3}{4}\right) + k\pi \right), \qquad k\in \mathbb{Z}. $$&lt;/span&gt;&#xA;Isolate the sine term and solve to obtain solutions:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align} &amp;amp;\sin\left(x-\arctan\left(\frac{3}{4}\right)+ k\pi \right) = (-1)^{k+1} \frac{7}{10} \\&#xA;&amp;amp;\qquad\implies x-\arctan\left(\frac{3}{4}\right)+ k\pi = \arcsin\left( (-1)^{k+1} \frac{7}{10} \right) \\&#xA;&amp;amp;\qquad\implies x = \arcsin\left( (-1)^{k+1} \frac{7}{10} \right) + \arctan\left(\frac{3}{4}\right)+ k \pi. \tag{**}&#xA;\end{align}&lt;/span&gt;&#xA;Observe that&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align} &amp;amp;x = \arcsin((-1)^{k+1} X) + k\pi \\&#xA;&amp;amp;\qquad \implies \pi - x = (k+1)\pi - \arcsin((-1)^{k+1} X) = (k+1)\pi + \arcsin((-1)^{k+2}X). \end{align}&lt;/span&gt;&#xA;If the first statement corresponds to an even &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, then the second corresponds to an odd &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, and vice versa.  Thus (**) gives all possible solutions to the original equation.&lt;/p&gt;&#xA;&lt;p&gt;To obtain decimal approximations, note that when &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is even, the solution is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x = \arcsin\left(  -\frac{7}{10} \right) + \arctan\left(\frac{3}{4}\right) + 2n \pi \approx -0.1319 + (6.2832)n, \qquad n\in\mathbb{Z}. $$&lt;/span&gt;&#xA;To obtain a solution between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt;, take &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt; to get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x \approx 6.1513.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;When &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is odd, the solution is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x = \arcsin\left( \frac{7}{10} \right) + \arctan\left(\frac{3}{4}\right)+ (2n+1) \pi \approx 1.4189 + 6.2832(2n+1), \qquad n\in\mathbb{Z}.$$&lt;/span&gt;&#xA;To obtain a solution between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt;, take &lt;span class=&quot;math-container&quot;&gt;$n=0$&lt;/span&gt; to get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x \approx 4.5605. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I will note that my solution is quite long.  There are likely a lot of places where it could be simplified and shortened—the above is, essentially, a stream of consciousness, which has not been edited much.  However, this is kind of how I expect students to turn in work: stream of consciousness.  The important thing is that each step of the computation is explained and justified.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2020-08-27T12:25:46.923" LastActivityDate="2020-08-27T12:25:46.923" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18724" PostTypeId="2" ParentId="18721" CreationDate="2020-08-26T18:22:37.020" Score="5" Body="&lt;p&gt;This was written &lt;strong&gt;after&lt;/strong&gt; the OP's Edit update, which in turn was made after Xander Henderson's answer.&lt;/p&gt;&#xA;&lt;h1&gt;Using &lt;span class=&quot;math-container&quot;&gt;$\;a \cos x + b \sin x \; = \;R \sin (x + \alpha)$&lt;/span&gt;&lt;/h1&gt;&#xA;&lt;p&gt;Assuming the student is allowed to use without verification the relevant formulas, maybe this would be how I'd want students to present their work. (Note: All decimal expansions are 3-digit truncations of the corresponding exact values.)&lt;/p&gt;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$\;6\cos x - 8\sin x = R\sin(x + \alpha)\;$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$R = -\sqrt{6^2 + 8^2} = -10$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\alpha = \arctan \left(-\frac{3}{4}\right) = -0.643 \ldots,$&lt;/span&gt; we have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$-10 \sin \left[x \; + \; \arctan\left(-\frac{3}{4}\right)\right] \;\; = \;\; 7$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sin \left[x \; - \; \arctan\left(\frac{3}{4}\right)\right] \;\; = \;\; -0.7$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Sine&lt;/em&gt; is negative only in Q3 and Q4, and because &lt;span class=&quot;math-container&quot;&gt;$\arcsin(-0.7) = -0.775 \ldots,$&lt;/span&gt; the &lt;span class=&quot;math-container&quot;&gt;$(x$&lt;/span&gt;-axis) reference angle for &lt;span class=&quot;math-container&quot;&gt;$x -  \arctan\left(\frac{3}{4}\right)$&lt;/span&gt; in these quadrants is &lt;span class=&quot;math-container&quot;&gt;$0.775 \ldots$&lt;/span&gt;. Therefore,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x \; - \; \arctan\left(\frac{3}{4}\right) \; = \; \pi + 0.775\ldots, \;\; 2\pi - 0.775\ldots $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x \; - \; \arctan\left(\frac{3}{4}\right) \; = \; 3.916 \ldots, \;\; 5.507 \ldots $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x \;\; = \;\; (3.916 \ldots) + (0.643 \ldots), \;\; (5.507 \ldots) + (0.643 \ldots) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x \;\; = \;\; 4.560 \ldots, \;\; 6.151 \ldots $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since &lt;strong&gt;all&lt;/strong&gt; solutions to &lt;span class=&quot;math-container&quot;&gt;$\;\sin (\text{stuff}) = -0.7\;$&lt;/span&gt; are given by adding &lt;span class=&quot;math-container&quot;&gt;$2n\pi$&lt;/span&gt; to the above two solutions (i.e. by adding integer multiples of &lt;span class=&quot;math-container&quot;&gt;$2\pi),$&lt;/span&gt; and only &lt;span class=&quot;math-container&quot;&gt;$n=0$&lt;/span&gt; gives values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; x &amp;lt; 2\pi,$&lt;/span&gt; the above two values are all solutions for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; x &amp;lt; 2\pi.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h1&gt;Using &lt;span class=&quot;math-container&quot;&gt;$\;\sin^2 x + \cos^2 x = 1$&lt;/span&gt;&lt;/h1&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 6\cos x \; = \; 7 + 8\sin x $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ (6\cos x)^2 \; = \; (7 + 8\sin x)^2 $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Because we've squared both sides of an equation, we will need to check for extraneous solutions at the end.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 36\cos^2 x \;\; = \;\; 49 + 112\sin x + 64\sin^2 x $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 36(1 - \sin^2 x) \;\; = \;\; 49 + 112\sin x + 64\sin^2 x $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 100\sin^2 x + 112\sin x + 13 \; = \; 0 $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \sin x \;\; = \;\; \frac{-112 \; \pm \; \sqrt{{112}^2 \; - \; 4(100)(13)}}{200} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \sin x \;\; = \;\; \frac{-112 \; \pm \; \sqrt{7344}}{200} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \sin x \;\; = \;\; -0.131 \ldots, \;\; -0.988 \ldots $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Sine&lt;/em&gt; is negative only in Q3 and Q4, and because &lt;span class=&quot;math-container&quot;&gt;$\arcsin(-0.131 \ldots) = -0.131 \ldots$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\arcsin(-0.988 \ldots) = -1.418 \ldots,$&lt;/span&gt; the &lt;span class=&quot;math-container&quot;&gt;$(x$&lt;/span&gt;-axis) reference angles for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in these quadrants are &lt;span class=&quot;math-container&quot;&gt;$0.131 \ldots$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1.418 \ldots$&lt;/span&gt;. Therefore,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x \;\; = \;\; \pi + 0.131 \ldots, \;\; \pi + 1.418 \ldots, \;\; 2\pi - 0.131 \ldots, \;\; 2\pi - 1.418 \ldots $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x \;\; = \;\; 3.273 \ldots, \;\; 4.560 \ldots, \;\; 6.151 \ldots, \;\; 4.864 \ldots $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since &lt;strong&gt;all&lt;/strong&gt; solutions to &lt;span class=&quot;math-container&quot;&gt;$\;\sin (\text{stuff}) = \text{constant}\;$&lt;/span&gt; are given by adding &lt;span class=&quot;math-container&quot;&gt;$2n\pi$&lt;/span&gt; to the above four solutions (i.e. by adding integer multiples of &lt;span class=&quot;math-container&quot;&gt;$2\pi),$&lt;/span&gt; and only &lt;span class=&quot;math-container&quot;&gt;$n=0$&lt;/span&gt; gives values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; x &amp;lt; 2\pi,$&lt;/span&gt; the above four values include all solutions for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; x &amp;lt; 2\pi.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Checking for extraneous solutions, we find that &lt;span class=&quot;math-container&quot;&gt;$3.273 \ldots$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4.864 \ldots$&lt;/span&gt; do not satisfy the original equation. Therefore, the solutions to the given equation such that &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; x &amp;lt; 2\pi$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$ x = 4.560 \ldots$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x = 6.151 \ldots$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2020-08-27T07:03:53.870" LastActivityDate="2020-08-27T07:03:53.870" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18725" PostTypeId="1" AcceptedAnswerId="18728" CreationDate="2020-08-26T22:17:25.167" Score="12" ViewCount="2971" Body="&lt;p&gt;How do you explain the concavity of a polynomial without any calculus?&lt;/p&gt;&#xA;&lt;p&gt;As the title suggests, I am struggling to explain when given a graph of a polynomial, how we determine when it is concave up or concave down without using any calculus or tangent lines. I need to teach this because some homework questions require this knowledge and I am given the homework to go over by the coordinator of the class.&lt;/p&gt;&#xA;&lt;p&gt;All the book does is just give examples, no reasoning nothing.&lt;/p&gt;&#xA;" OwnerUserId="14503" LastEditorUserId="5321" LastEditDate="2020-08-27T07:05:03.417" LastActivityDate="2020-08-28T21:15:51.820" Title="How do you explain concavity of a polynomial without any calculus?" Tags="|precalculus|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18726" PostTypeId="2" ParentId="18725" CreationDate="2020-08-26T22:42:34.453" Score="2" Body="&lt;p&gt;One non-rigorous starting point would be that a function that could “hold water” when poured from above is concave up. (This isn’t a very robust idea and breaks down quickly on sine, for example.)&lt;/p&gt;&#xA;&lt;p&gt;What about appealing to the apparent rate of change of the rate of increase of each function (of course this is calculus, but certain simple functions are intuitive).&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2020-08-26T22:42:34.453" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18728" PostTypeId="2" ParentId="18725" CreationDate="2020-08-26T22:51:02.090" Score="23" Body="&lt;p&gt;Here is a proposed definition:&lt;/p&gt;&#xA;&lt;p&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is said to be &lt;strong&gt;concave up&lt;/strong&gt; on an interval &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt; if for all &lt;span class=&quot;math-container&quot;&gt;$x,y \in [a,b]$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;y$&lt;/span&gt;, the line &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; connecting &lt;span class=&quot;math-container&quot;&gt;$(x,f(x))$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(y,f(y))$&lt;/span&gt; satisfies &lt;span class=&quot;math-container&quot;&gt;$L(t) \geq f(t)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$t \in (x,y)$&lt;/span&gt;.  It is &lt;strong&gt;concave down&lt;/strong&gt; if the final inequality is reversed.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-08-26T22:51:02.090" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="18729" PostTypeId="1" AcceptedAnswerId="18733" CreationDate="2020-08-27T07:21:29.837" Score="5" ViewCount="654" Body="&lt;p&gt;I found teaching &lt;a href=&quot;https://en.wikipedia.org/wiki/Taylor_series#Taylor_series_in_several_variables&quot; rel=&quot;nofollow noreferrer&quot;&gt;Taylor expansion for multivariable functions&lt;/a&gt; rather challenging. It is a bit complicated to prove and to to compute.  So what happened to me last year was that my students simply memorized the formula for expanding &lt;span class=&quot;math-container&quot;&gt;$f(x,y)$&lt;/span&gt; to the second degree. I did not dear to ask them for anything beyond that in the exam. Are there some good source/examples for explaining this topic?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I was teaching 2nd year undergraduates in various engineering (e.g. environment) programs. The topic is in the curriculum and I have to cover it.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorDisplayName="user11702" LastEditDate="2020-08-27T09:06:41.020" LastActivityDate="2020-08-27T22:08:00.690" Title="What is a good way to teach Taylor expansion of multi-variable calculus?" Tags="|calculus|vector-calculus|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18730" PostTypeId="2" ParentId="18729" CreationDate="2020-08-27T08:42:30.813" Score="2" Body="&lt;p&gt;I looked at some books I have.&lt;/p&gt;&#xA;&lt;p&gt;It's not clear to me how vital the multivariable Taylor topic even is. Swokowski does not cover it, for instance.  Neither does Kreyszig, although he does cover it for complex analysis.  So you could consider to stick with a cursory treatment or even cut it entirely.  After all, time is limited and there are plenty of other &amp;quot;calculus 3&amp;quot; topics.  [Your question would be helped if you confirmed what class you are teaching as well as how strong the students are.]&lt;/p&gt;&#xA;&lt;p&gt;Granville has a quick derivation using the just previously taught &amp;quot;Law of the Mean&amp;quot; for multivariable functions, within the multivariable calc (applications of partial derivatives) section.  There are also nine practice problems.  Note: He also takes an approach of only showing up to the second power and mentioning that the third or higher derivatives are &amp;quot;complicated&amp;quot;.  But does convey to the reader that they and higher derivatives are homogeneous.  He also discusses McLaurin series of multivariable, very briefly.  So, worth a look.  I have the 1941 War Department Edition.  See section 242.  (The earlier edition online pdfs of Granville unfortunately do not cover this topic.)&lt;/p&gt;&#xA;&lt;p&gt;Edit: saw your explanation.  My advice would be stick to the second derivative and just make the topic brief. Maybe this is a bit like &amp;quot;rotations&amp;quot; in analytic geometry, which I just remember being a mess of algebra, that you deal with once for the good of your soul, but that you don't internalize and use recurrently as a building block, like you do the quadratic equation, for instance.  I guess if you have time to kill you could go deeper and do the Granville Law of the Mean application also.  But who ever has time to kill?  Might be the better part of valor to just tell the students that this is a brief exposure, that it's required that they see it.  But not the most important animal in the zoo.&lt;/p&gt;&#xA;" OwnerUserId="14501" LastEditorUserId="14501" LastEditDate="2020-08-27T09:20:06.553" LastActivityDate="2020-08-27T09:20:06.553" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18731" PostTypeId="2" ParentId="18721" CreationDate="2020-08-27T10:24:32.970" Score="1" Body="&lt;p&gt;The two solutions already posted suggest using &amp;quot;arctan(-3/4)&amp;quot; in place of the lengthy decimal throughout. I suggest something slightly different, which is a valuable strategy whenever you have an annoying repetition: &lt;em&gt;give the repeated thing a name&lt;/em&gt;. So, something like this (where I have made a few incidental choices in different ways from the original solution; obviously you can use &lt;span class=&quot;math-container&quot;&gt;$\sin(x+\theta)$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\cos(x+\theta)$&lt;/span&gt; and obviously you can use &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; or whatever). I've actually found it convenient to give names to &lt;em&gt;two&lt;/em&gt; different quantities that arise in the solution&lt;/p&gt;&#xA;&lt;p&gt;We begin by writing &lt;span class=&quot;math-container&quot;&gt;$6\cos x-8\sin x=R\cos(x+\theta)$&lt;/span&gt;. Following the usual procedure we find &lt;span class=&quot;math-container&quot;&gt;$R=\sqrt{6^2+8^2}=10$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\theta=\tan^{-1}(8/6)=0.927...$&lt;/span&gt; .&lt;/p&gt;&#xA;&lt;p&gt;Now we must solve &lt;span class=&quot;math-container&quot;&gt;$10\cos(x+\theta)=7$&lt;/span&gt; or, equivalently, &lt;span class=&quot;math-container&quot;&gt;$\cos(x+\theta)=0.7$&lt;/span&gt;. One solution to this is &lt;span class=&quot;math-container&quot;&gt;$x+\theta=\cos^{-1}(0.7)=0.795...$&lt;/span&gt;; calling the RHS &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, all the solutions are &lt;span class=&quot;math-container&quot;&gt;$x=(2\pi n\pm y)-\theta$&lt;/span&gt; for integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. We are interested in solutions with &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;x&amp;lt;2\pi$&lt;/span&gt;, which means the only ones we need are &lt;span class=&quot;math-container&quot;&gt;$2\pi+y-\theta=6.151...$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2\pi-y-\theta=4.560...$&lt;/span&gt; .&lt;/p&gt;&#xA;&lt;p&gt;In practice, a student might do well to be more explicit about figuring out which integer multiples of &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt; and which choices of sign yield solutions in the given range. Something like this:&lt;/p&gt;&#xA;&lt;p&gt;... We are interested in solutions with &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;x&amp;lt;2\pi$&lt;/span&gt;. We see that &lt;span class=&quot;math-container&quot;&gt;$y-\theta$&lt;/span&gt; is a little smaller than 0, so this is too small but &lt;span class=&quot;math-container&quot;&gt;$2\pi+y-\theta$&lt;/span&gt; is in range (and &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt; more than that is too large); and &lt;span class=&quot;math-container&quot;&gt;$-y-\theta$&lt;/span&gt; is very roughly -2, so again this is too small but adding &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt; brings it into range (and adding &lt;span class=&quot;math-container&quot;&gt;$4\pi$&lt;/span&gt; makes it too large). So our solutions are &lt;span class=&quot;math-container&quot;&gt;$2\pi+y-\theta=6.151...$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2\pi-y-\theta=4.560...$&lt;/span&gt; .&lt;/p&gt;&#xA;&lt;p&gt;(Most students will probably need to compute the actual values rather than just saying &amp;quot;a little smaller than 0&amp;quot; and &amp;quot;very roughly -2&amp;quot;, but if I were writing a model answer then I might leave phrases like those in it, to encourage students to learn to do that sort of rough estimate in their heads. Contrariwise: Some students will need to write down three numerical values for each choice of sign, one &amp;quot;too small&amp;quot;, one &amp;quot;just right&amp;quot;, and one &amp;quot;too large&amp;quot;. There's something to be said for doing that in a model answer, too.)&lt;/p&gt;&#xA;" OwnerUserId="9376" LastActivityDate="2020-08-27T10:24:32.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18732" PostTypeId="2" ParentId="18725" CreationDate="2020-08-27T13:18:06.840" Score="9" Body="&lt;p&gt;As other answers have noted, a function is said to be &lt;a href=&quot;https://en.wikipedia.org/wiki/Convex_function&quot; rel=&quot;noreferrer&quot;&gt;convex&lt;/a&gt; (or &amp;quot;convex up&amp;quot;; I've never seen &amp;quot;concave up&amp;quot; before, although the meaning is obvious enough in context) if the line segment connecting any two points on its graph lies entirely &lt;em&gt;above&lt;/em&gt; (or on) the graph between those points, and concave (or &amp;quot;convex down&amp;quot; / &amp;quot;concave down&amp;quot;) if the line segment connecting any two points on its graph lies entirely &lt;em&gt;below&lt;/em&gt; (or on) the graph between those points.&lt;/p&gt;&#xA;&lt;p&gt;A rigorous algebraic definition, to complement this geometric description, is that a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is defined to be convex on a subset &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; of its domain if and only if, for all &lt;span class=&quot;math-container&quot;&gt;$a,b \in S$&lt;/span&gt; and all &lt;span class=&quot;math-container&quot;&gt;$t, s \in (0,1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$$t + s = 1 \implies t f(a) + s f(b) \ge f(ta + sb),$$&lt;/span&gt; and concave if the opposite inequality holds (i.e. if &lt;span class=&quot;math-container&quot;&gt;$-f$&lt;/span&gt; is convex).  Further, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is said to be &lt;em&gt;strictly&lt;/em&gt; convex (or concave) if the corresponding inequality is strict.&lt;/p&gt;&#xA;&lt;p&gt;(Note that the definition given above is often simplified by directly substituting &lt;span class=&quot;math-container&quot;&gt;$1 - t$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, but that somewhat obscures the underlying symmetry of the definition.  The symmetric form also generalizes more readily to the various forms of &lt;a href=&quot;https://en.wikipedia.org/wiki/Jensen%27s_inequality&quot; rel=&quot;noreferrer&quot;&gt;Jensen's inequality&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;The connection between these two definitions is that any &lt;span class=&quot;math-container&quot;&gt;$x \in (a,b)$&lt;/span&gt; can be written as the weighted average &lt;span class=&quot;math-container&quot;&gt;$x = ta + sb$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$t + s = 1$&lt;/span&gt; and both &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; are positive.  Then &lt;span class=&quot;math-container&quot;&gt;$(x, f(x))$&lt;/span&gt; is a point on the curve of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, while &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$y = t f(a) + s f(b)$&lt;/span&gt;, is the corresponding point on the straight line segment between the points &lt;span class=&quot;math-container&quot;&gt;$(a, f(a))$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(b, f(b))$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Notably, this definition (in either its geometric or algebraic form) does not require the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to be differentiable or even continuous (although it can be shown that a function which is convex on an open interval must necessarily be continuous on the whole interval and differentiable at all but at most countably many points on it).  Thus, it is more general than definitions based on derivatives and can be applied to more kinds of functions.  For example, the function &lt;span class=&quot;math-container&quot;&gt;$f(x) = |x|$&lt;/span&gt; is clearly convex on all of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; according to this definition, even though it's not differentiable at &lt;span class=&quot;math-container&quot;&gt;$x = 0$&lt;/span&gt;.  Thus, this definition, or something similar and equivalent, is usually taken as the fundamental definition of convexity, with more narrowly applicable ones like &amp;quot;a twice differentiable function is convex if its second derivative is non-negative&amp;quot; being proven as theorems.&lt;/p&gt;&#xA;" OwnerUserId="526" LastActivityDate="2020-08-27T13:18:06.840" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18733" PostTypeId="2" ParentId="18729" CreationDate="2020-08-27T22:08:00.690" Score="6" Body="&lt;p&gt;For 2nd year engineering students, you should not be wasting time trying to prove the expansion, except perhaps giving a heuristic argument for the quadratic approximation in the case of two variables. Stick to operational algebraic formulations and computations, which is the only thing they'll ever need.&lt;/p&gt;&#xA;&lt;p&gt;My go-to book for an elementary treatment of this topic (indeed, for an elementary treatment of many multivariable calculus topics) is &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0387964053&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Calculus of Several Variable&lt;/strong&gt;&lt;/a&gt; by Serge Lang (see the section titled &amp;quot;The general expression for Taylor's formula&amp;quot;). Don't let Lang's reputation in his many other books mislead you --- this book is excellent for students with a 2-semester background in single variable calculus. The examples are excellent, the explanatory material is well-written and comprehensible by students, the text is uncluttered both verbally and symbolically, and while the text is suitable for average level students who do not intend to major in math, future math majors will not be misled by incorrect or misleading statements.&lt;/p&gt;&#xA;&lt;p&gt;What follows is taken from a take-home assignment I gave to very strong high school students &lt;a href=&quot;https://www.lsmsa.edu/curriculum-detail?fromId=247482&amp;amp;LevelNum=1393&amp;amp;DepartmentId=24673&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; in a &lt;a href=&quot;https://en.wikipedia.org/wiki/AP_Calculus#AP_Calculus_BC&quot; rel=&quot;noreferrer&quot;&gt;post-BC calculus class&lt;/a&gt; that I taught several times in the mid to late 1990s. I didn't cover this topic in class (I &amp;quot;worked it in&amp;quot; by including it in a take-home assignment), but I can easily imagine using it as the basis for one class session devoted to this topic, and thus it might be of use to you. Of course, some simple examples would need to be included and probably some aspects of the problems below would not be appropriate in your case. Keep in mind these were near olympiad-level students, so I tried to work in enrichment topics as much as I could, something you should NOT do unless the enrichment topics are engineering applications.&lt;/p&gt;&#xA;&lt;p&gt;One formulation of the two-variable Taylor expansion is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \begin{array} ff(x,y) &amp;amp; = &amp;amp; \; &amp;amp; f(a,b) \\ \; &amp;amp; \; &amp;amp; + &amp;amp; \left[\,f_x(a,b) \cdot (x-a) \; + \; f_y(a,b) \cdot (y-b) \right] \, \\ \; &amp;amp; \; &amp;amp; + &amp;amp; \frac{1}{2!} \cdot \left[\,f_{xx}(a,b) \cdot (x-a)^2 \; + \; 2f_{xy}(a,b) \cdot (x-a)(y-b) \; + \; f_{yy}(a,b) \cdot (y-b)^2 \right] \, \\ \; &amp;amp; \; &amp;amp; + &amp;amp; \cdot \cdot \cdot&#xA;\end{array} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This corresponds to the following formulation of the one-variable Taylor expansion:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ f(x) \;\; = \;\; f(a) \;\; + \;\; f'(a) \cdot (x-a) \;\; + \;\; \frac{1}{2!} \cdot f''(a) \cdot (x-a)^2 \;\; + \;\; \cdot \cdot \cdot  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this formulation of the one-variable Taylor expansion, if we replace &lt;span class=&quot;math-container&quot;&gt;$(x-a)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; (hence, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is replaced with &lt;span class=&quot;math-container&quot;&gt;$a + \Delta x),$&lt;/span&gt; then we get an alternative formulation of the one-variable Taylor expression:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ f(a + \Delta x) \;\; = \;\; f(a) \;\; + \;\; f'(a) \cdot \Delta x \;\; + \;\; \frac{1}{2!} \cdot f''(a) \cdot \left(\Delta x \right)^2 \;\; + \;\; \cdot \cdot \cdot   $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The corresponding alternative formulation of the two-variable Taylor expansion is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \begin{array} ff\left(a + \Delta x, \; b + \Delta y\right) &amp;amp; = &amp;amp; \; &amp;amp; f(a,b) \\ \; &amp;amp; \; &amp;amp; + &amp;amp; \left[\,f_x(a,b) \cdot \Delta x \; + \; f_y(a,b) \cdot \Delta y \right] \, \\ \; &amp;amp; \; &amp;amp; + &amp;amp; \frac{1}{2!} \cdot \left[\,f_{xx}(a,b) \cdot \left(\Delta x \right)^2 \; + \; 2f_{xy}(a,b) \cdot \left(\Delta x \right) \left( \Delta y \right) \; + \; f_{yy}(a,b) \cdot \left( \Delta y \right)^2 \right] \, \\ \; &amp;amp; \; &amp;amp; + &amp;amp; \cdot \cdot \cdot&#xA;\end{array} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Using vector notation, we can express the alternative formulation of the two-variable Taylor expansion in a way that more closely parallels the alternative formulation of the one-variable Taylor expansion. Let&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \vec{X_{\,}} = \pmatrix{x \\ y}, \;\;\;\; \vec{X_0} = \pmatrix{a \\ b}, \;\;\;\; \Delta \vec{X_{\,}} = \pmatrix{\Delta x \\ \Delta y} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \begin{array} f\left[f\right]_{\vec{X_0} \; + \; \Delta \vec{X_{\,}}}  &amp;amp; = &amp;amp; \; &amp;amp; \left[f\right]_{\vec{X_0}} \\&#xA; \; &amp;amp; \; &amp;amp; + &amp;amp; \left[ \left( \vec{\nabla \;} \cdot \Delta \vec{X_{\,}} \right) (f) \right]_{\vec{X_0}} \, \\ \; &amp;amp; \; &amp;amp; + &amp;amp; \frac{1}{2!} \left[ \left( \vec{\nabla \;} \cdot \Delta \vec{X_{\,}} \right)^2 (f) \right]_{\vec{X_0}} \, \\ &#xA; \; &amp;amp; \; &amp;amp; + &amp;amp; \frac{1}{3!} \left[ \left( \vec{\nabla \;} \cdot \Delta \vec{X_{\,}} \right)^3 (f) \right]_{\vec{X_0}} \, \\ \; &amp;amp; \; &amp;amp; + &amp;amp; \cdot \cdot \cdot \, ,&#xA;\end{array} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where the subscripts refer to evaluations and &lt;span class=&quot;math-container&quot;&gt;$ \vec{\nabla \;} \cdot \Delta \vec{X_{\,}} $&lt;/span&gt; represents the (formal) dot product of &lt;span class=&quot;math-container&quot;&gt;$\vec{\nabla \;}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$\Delta \vec{X_{\;}}.$&lt;/span&gt; The various powers of &lt;span class=&quot;math-container&quot;&gt;$ \vec{\nabla \;} \cdot \Delta \vec{X_{\,}} $&lt;/span&gt; are to be expanded &amp;quot;algebraically&amp;quot;, meaning (for example) that the square of &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial}{\partial x}$&lt;/span&gt; represents &lt;span class=&quot;math-container&quot;&gt;$\frac{{\partial}^2}{\partial x^2},$&lt;/span&gt; and so on in the obvious manner. I say &amp;quot;obvious&amp;quot;, since you already have the expansion written out in terminology you should understand. Just make the obvious identifications needed for the expansions to be equal. The nice thing about writing the Taylor expansion this way is that if you decide to make everything three variables (or more), then the vector formulation remains the same.&lt;/p&gt;&#xA;&lt;h1&gt;Problem 1&lt;/h1&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; be a real-valued function of the three variables &lt;span class=&quot;math-container&quot;&gt;$x,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$y,$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z,$&lt;/span&gt; and put&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \vec{X_{\,}} = \pmatrix{x \\ y \\ z}, \;\;\;\; \vec{X_0} = \pmatrix{a \\ b \\ c}, \;\;\;\; \Delta \vec{X_{\,}} = \pmatrix{\Delta x \\ \Delta y \\ \Delta z} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Expand the following to obtain three-variable terms that correspond to the two-variable terms in the first two-variable Taylor expansion I gave.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$  \left[ \left( \vec{\nabla \;} \cdot \Delta \vec{X_{\,}} \right) (f) \right]_{\vec{X_0}} \;\; = \;\; \text{?}  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$  \left[ \left( \vec{\nabla \;} \cdot \Delta \vec{X_{\,}} \right)^2 (f) \right]_{\vec{X_0}} \;\; = \;\; \text{?}  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$  \left[ \left( \vec{\nabla \;} \cdot \Delta \vec{X_{\,}} \right)^3 (f) \right]_{\vec{X_0}} \;\; = \;\; \text{?}  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For the last expression, you will have to cube a trinomial. Here is a neat way to obtain the cube of a trinomial.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(A + B + C)^3 = (A+B+C)(A+B+C)(A+B+C)$&lt;/span&gt; can be algebraically expanded by adding together all products of the form &lt;span class=&quot;math-container&quot;&gt;$PQR,$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is a choice of a term in the left factor, &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; is a choice of a term in the middle factor, and &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is a choice of a term in the right factor. Thus, there will be &lt;span class=&quot;math-container&quot;&gt;$(3)(3)(3) = 27$&lt;/span&gt; terms added together, many of which will be the same (e.g. choosing &lt;span class=&quot;math-container&quot;&gt;$P = A,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$Q = A,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$R = B$&lt;/span&gt; gives the term &lt;span class=&quot;math-container&quot;&gt;$A^2B$&lt;/span&gt;; choosing &lt;span class=&quot;math-container&quot;&gt;$P = A,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$Q = B,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$R = A$&lt;/span&gt; also gives the term &lt;span class=&quot;math-container&quot;&gt;$A^2B).$&lt;/span&gt; As a consequence of this method of expansion, note that all the terms in the algebraic expansion will be of the third degree. These &lt;span class=&quot;math-container&quot;&gt;$27$&lt;/span&gt; terms can be organized into the following three types: terms that are the cube of a variable, terms that include the square of a variable, and the terms equal to &lt;span class=&quot;math-container&quot;&gt;$ABC.$&lt;/span&gt; Clearly, there is only one term equal to &lt;span class=&quot;math-container&quot;&gt;$A^3,$&lt;/span&gt; and similarly for &lt;span class=&quot;math-container&quot;&gt;$B^3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C^3.$&lt;/span&gt; Moreover, it is easy to see that there is a total of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; terms equal to &lt;span class=&quot;math-container&quot;&gt;$A^2B,$&lt;/span&gt; since there is a total of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; ways to choose exactly one &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and exactly two &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;'s. Similarly for the terms equal to &lt;span class=&quot;math-container&quot;&gt;$AB^2,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$AC^2,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$A^2C,$&lt;/span&gt; etc. Therefore,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ (A+B+C)^3 \; = \; A^3 + B^3 + C^3 + 3(A^2B + A^2C + AB^2 + B^2C + AC^2 + BC^2) + k \cdot ABC $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;for some positive integer constant &lt;span class=&quot;math-container&quot;&gt;$k.$&lt;/span&gt; You can find the value of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; by substituting &lt;span class=&quot;math-container&quot;&gt;$A=B=C=1$&lt;/span&gt; and solving for &lt;span class=&quot;math-container&quot;&gt;$k.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Problem 2&lt;/h1&gt;&#xA;&lt;p&gt;Use the 2nd order Taylor polynomial for functions of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; variables to find the quadratic approximation to&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$  f(x,y,z) \; = \; e^x \sin y \cos z \;\;\; \text{about the point} \;\;\; \pmatrix{a \\ b \\ c} = \pmatrix{0 \\ 0 \\ 0} .   $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h1&gt;Problem 3&lt;/h1&gt;&#xA;&lt;p&gt;Use the angle addition formula for SINE to show that &lt;span class=&quot;math-container&quot;&gt;$\; \sin y \cos z = \frac{1}{2}\sin(y-z) \; + \; \frac{1}{2} \sin(y+z).$&lt;/span&gt; This is one of the &lt;a href=&quot;http://www.sosmath.com/trig/prodform/prodform.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;product to sum&lt;/em&gt; formulas in trigonometry&lt;/a&gt;, which you might recall using in carrying out &lt;a href=&quot;https://www.google.com/search?q=trigonometry+%22product+to+sum%22+integration&quot; rel=&quot;noreferrer&quot;&gt;certain integrations&lt;/a&gt;. Now use this equality, and substitution into the one-variable Taylor series for the SINE function, to obtain an expansion of &lt;span class=&quot;math-container&quot;&gt;$\sin y \cos z$&lt;/span&gt; (up to and including third degree terms will be enough for what follows). Then multiply this expansion of &lt;span class=&quot;math-container&quot;&gt;$\sin y \cos z$&lt;/span&gt; by the 2nd order Taylor polynomial of &lt;span class=&quot;math-container&quot;&gt;$e^x,$&lt;/span&gt; namely &lt;span class=&quot;math-container&quot;&gt;$1 + x + \frac{1}{2}x^2,$&lt;/span&gt; keeping only terms of degree &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; or less. Show that the answer you get is the same as what you got in Problem 2.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2020-08-27T22:08:00.690" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18734" PostTypeId="2" ParentId="4379" CreationDate="2020-08-28T00:01:47.897" Score="2" Body="&lt;p&gt;Why not just share screen on your ipad via Zoom? That's how my professors do it these days. If they go into whiteboard mode, we can all write on the screens simultaneously as well and there is no lag whatsoever. I don't know the graphing capabilities of Notability but I imagine there are satisfactory apps for that.  It seems like what you really are looking for is the Zoom annotation feature.&lt;/p&gt;&#xA;&lt;p&gt;Edit: Oops, I just realized this was asked 5 years ago. Back when there were no good options.&lt;/p&gt;&#xA;" OwnerUserId="14497" LastActivityDate="2020-08-28T00:01:47.897" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18735" PostTypeId="2" ParentId="18725" CreationDate="2020-08-28T00:16:52.460" Score="4" Body="&lt;p&gt;You might want to discuss the etymology. There's &amp;quot;con&amp;quot;, which means &amp;quot;with&amp;quot;, and shows up in other words such as &amp;quot;converse&amp;quot; and &amp;quot;context&amp;quot;, and &amp;quot;cave&amp;quot;, which comes from &amp;quot;cavus&amp;quot;, meaning &amp;quot;hollow&amp;quot;, and shows up in words such as &amp;quot;cavity&amp;quot; and of course &amp;quot;cave&amp;quot;. So &amp;quot;concave&amp;quot; means &amp;quot;with hollow&amp;quot;. Concave down means the hollow is below the curve, and concave up means the hollow is above the curve.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2020-08-28T00:16:52.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18736" PostTypeId="1" CreationDate="2020-08-28T02:30:57.683" Score="4" ViewCount="594" Body="&lt;p&gt;I am writing an exercise for a precalculus homework assignment that deals with the topic of interval notation. The point of the exercise is to convert open, closed, and half open intervals described in English into interval notation. Here is a snippet of the exercise:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Write the following sets of numbers represented textually in &lt;strong&gt;interval notation&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(a) All real numbers between 5 and 7 that include 5 but don't include 7.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(b) All real numbers between 1 and 10 that include both 1 and 10.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So for example, the answer to part (a) would be &lt;span class=&quot;math-container&quot;&gt;$[5, 7)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I feel as though my wording is awkward and confusing in both the prompt and parts (a) and (b). Could anyone give me advice on re-wording or re-imagining this question to make it more clear?&lt;/p&gt;&#xA;" OwnerUserId="14512" LastActivityDate="2022-03-10T13:49:15.647" Title="How to word this exercise about converting &quot;English&quot; into interval notation?" Tags="|precalculus|language-use|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18737" PostTypeId="2" ParentId="18736" CreationDate="2020-08-28T05:28:01.573" Score="6" Body="&lt;p&gt;If you feel your instruction is unclear, then provide an example.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Write the following sets of numbers represented in interval notation.  For example, &amp;quot;all real numbers from 0 to 1 exclusive&amp;quot; in interval notation is &lt;span class=&quot;math-container&quot;&gt;$(0,1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a) All real numbers greater than or equal to 5 and less than 7.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;b) All real numbers from 1 to 10 inclusive.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastActivityDate="2020-08-28T05:28:01.573" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18738" PostTypeId="2" ParentId="18736" CreationDate="2020-08-28T06:53:26.477" Score="7" Body="&lt;p&gt;This is my suggestion.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Write each set of numbers in interval notation.&lt;/p&gt;&#xA;&lt;p&gt;(a) All real numbers between 5 and 7, including 5 but not including 7.&lt;/p&gt;&#xA;&lt;p&gt;(b) All real numbers between 1 and 10, including both 1 and 10.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I assume you are giving examples in class before you assign this.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5919" LastEditDate="2020-08-28T18:51:22.690" LastActivityDate="2020-08-28T18:51:22.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18739" PostTypeId="2" ParentId="18736" CreationDate="2020-08-28T08:07:33.587" Score="5" Body="&lt;p&gt;&lt;em&gt;Write the following sets of numbers represented textually in interval notation&lt;/em&gt; --- Note that this can be parsed in two different ways: &amp;quot;Write the following sets of numbers (represented textually in interval notation)&amp;quot; and &amp;quot;Write the following sets of numbers represented textually (in interval notation)&amp;quot;. The first asks for sets of numbers to be written, where the sets of numbers are given in a textually represented form that is written in interval notation. The second asks for sets of numbers to be written in interval notation, where the sets of numbers are given in a textually represented form. Of course, context prevents one from parsing this in the first way, but students who are very uncertain about what this is all about could easily parse this in the first way and be confused as to what it is you want.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For each of the following textual descriptions of a set of numbers, write the set of numbers in interval notation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(a) All real numbers between 5 and 7, including 5 and not including 7.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(b) All real numbers between 1 and 10, including 1 and including 10.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note that I've written (a) and (b) in a similar fashion (this is called &lt;a href=&quot;https://www.google.com/search?q=%22parallel+writing%22&quot; rel=&quot;noreferrer&quot;&gt;parallel writing&lt;/a&gt;), and there are 4 formats that the words after the comma can take: (1) including &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and including &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;; (2) including &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and not including &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;; (3) not including &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and including &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;; (4) not including &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and not including &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. For an individual class, being this meticulous is not particularly important, but if these questions are to appear on a widely taken standardized test in which the results are to be analyzed statistically, the use of parallel writing tends to decrease &lt;a href=&quot;https://www.google.com/search?q=%22irrelevant+variance%22+test+psychometrics&quot; rel=&quot;noreferrer&quot;&gt;irrelevant variance&lt;/a&gt; caused by differing reading skills and other factors that one would want to disentangle from the math skills being measured.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2020-08-28T08:07:33.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18740" PostTypeId="2" ParentId="18725" CreationDate="2020-08-28T17:23:41.163" Score="2" Body="&lt;p&gt;I think it's helpful for students to see an important concept like this from multiple points of view, so although a definition like Ilmari Karonen's is probably the best primary definition, here is one that would also be good as calculus prep.&lt;/p&gt;&#xA;&lt;p&gt;Suppose that for a certain point p on the graph of a polynomial, there is a unique linear function L that passes through p but doesn't cut through the graph at p. We call this a no-cut line.&lt;/p&gt;&#xA;&lt;p&gt;A no-cut line, when defined, is also the unique tangent line, meaning intuitively that it's the best linear approximation near p.&lt;/p&gt;&#xA;&lt;p&gt;A (nonlinear) polynomial has no-cut lines everywhere except possibly at a finite number of points, called inflection points. (Tangent lines can be defined at inflection points, but they are not no-cut lines.)&lt;/p&gt;&#xA;&lt;p&gt;In any interval not containing inflection points, we can define the polynomial's concavity. If the slope of the no-cut line is increasing on this interval, the concavity is up, if decreasing, then down.&lt;/p&gt;&#xA;&lt;p&gt;Remark: These definitions also carry over to many other functions, e.g., the sine and exponential. They do not work without modifications for less &amp;quot;well-behaved&amp;quot; functions such as discontinuous ones.&lt;/p&gt;&#xA;&lt;p&gt;The notion of &amp;quot;cuts through&amp;quot; is rigorous at the level of Euclid's Proposition I, which IMO is plenty rigorous enough for a high school class.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-08-28T17:32:31.480" LastActivityDate="2020-08-28T17:32:31.480" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18742" PostTypeId="1" CreationDate="2020-08-28T20:23:44.927" Score="15" ViewCount="2864" Body="&lt;p&gt;The textbook I am using to teach Calculus I includes in the exercises of most chapters a number of interesting real-world applications of the concepts from that chapter. However, the chapter on the derivative of the natural logarithm is remarkably abstract in its exercises.&lt;/p&gt;&#xA;&lt;p&gt;Are there not scenarios in which it would be useful to differentiate a logarithm to answer a real-world problem? Something to do with determining the stimuli needed to accomplish a particular exponential rate of growth?&lt;/p&gt;&#xA;&lt;p&gt;Or is differentiating natural logarithms primarily motivated by its usefulness for simplifying differentiation using logarithmic differentiation?&lt;/p&gt;&#xA;&lt;p&gt;Since a number of practical exponential growth and logarithm exercises problems revolve around population growth, I've been trying to contrive an example in those terms. But it feels very odd and abstract:&lt;/p&gt;&#xA;&lt;p&gt;Exponential function: What will be the population after x years?&#xA;Derivative of exponential: How quickly will the population be growing x years from now?&#xA;Logarithmic function: How many years will it take to reach a particular target population?&#xA;Derivative of logarithm: How much would increasing or decreasing a given target population affect the time it takes to reach it?&lt;/p&gt;&#xA;&lt;p&gt;Is that even really a practical question? How do I make it interesting?&lt;/p&gt;&#xA;&lt;p&gt;I've tried searching this question here and on Google, but haven't found anything. Thanks in advance for your help!&lt;/p&gt;&#xA;" OwnerUserId="14510" LastEditorUserId="14510" LastEditDate="2020-08-28T22:21:04.283" LastActivityDate="2020-09-04T15:27:53.760" Title="Are there direct practical applications of differentiating natural logarithms?" Tags="|calculus|examples|concept-motivation|logarithm|derivative|" AnswerCount="6" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="18743" PostTypeId="2" ParentId="18725" CreationDate="2020-08-28T21:15:51.820" Score="0" Body="&lt;p&gt;I would be prudent with &amp;quot;any 2 points&amp;quot;&#xA;in case of by example a sinus any 2 points far enough apart will give serious problems.&lt;/p&gt;&#xA;&lt;p&gt;to analyse a small part of the polynome:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;choose a relevant x0, calculate y0&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;chose x1 very close to but not on x0 and calculate y1 of the polynome&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;chose x2 very close but different to x0 and x1&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;T1 = (y1 - y0)/(x1-x0) gives a proxy to the tangent between x0 and x1&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;T2 = (y2 - y1)/(x2-x1) gives a proxy to the tangent between x1 and x2&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;T2 being bigger or smaller than T1 gives a suggestion for the convexity&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;if T1 is &amp;gt; T2 then the suggested part is concave&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;BUT&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;there will be the risk that we have the bad luck of working in a zone where convexity changes ! ( example =  the point x = pi of a sinus function )&lt;/p&gt;&#xA;&lt;p&gt;the make sure that this is not the case we can work with 5 of x points instead of 3,&lt;/p&gt;&#xA;&lt;p&gt;to make sure that&#xA;T1 &amp;gt; T2 &amp;gt; T3 &amp;gt; T4   for concave  or&#xA;T1 &amp;lt; T2 &amp;lt; T3 &amp;lt; T4   for convex    and not a mix,&lt;/p&gt;&#xA;&lt;p&gt;if we get something like T1 &amp;gt; T2 &amp;gt; T3 &amp;lt; T4 then convexity has changed somewhere in the zone x2 to x4 .....&lt;/p&gt;&#xA;" OwnerUserId="14518" LastActivityDate="2020-08-28T21:15:51.820" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18744" PostTypeId="2" ParentId="18742" CreationDate="2020-08-28T23:42:06.087" Score="6" Body="&lt;p&gt;I couldn't find a lot either.  Suggest playing with some logarithmic properties and constructing problems based on that.&lt;/p&gt;&#xA;&lt;p&gt;E.g. pH is log10 of the hydronium ion concentration.  Could ask how the pH changes with hydronium concentration addition (assume strong acid addition, to an unbuffered solution).  Of course this brings in chemistry, which weirds the kids out more than logs!&lt;/p&gt;&#xA;&lt;p&gt;Maybe play with decibels or Richter or NPV (discount rate).  [Sorry, I can't give you the fish, just a direction to some lakes.]&lt;/p&gt;&#xA;&lt;p&gt;All that said, I think applied problems are a bit of a double edged sword.  And I say this as one of the knuckledraggers usually fighting the theorists.  The issue with applied problems is they're usually harder.  &amp;quot;Word problems are hard.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="14519" LastActivityDate="2020-08-28T23:42:06.087" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18745" PostTypeId="2" ParentId="18742" CreationDate="2020-08-29T00:52:24.003" Score="5" Body="&lt;p&gt;Boltzmann's equation for entropy is &lt;span class=&quot;math-container&quot;&gt;$S=k\ln W$&lt;/span&gt;, and the second law of thermodynamics is all about &lt;em&gt;change&lt;/em&gt; in entropy. Maybe this is a place to start with your quest for a practical application of the derivative of a logarithmic function.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2020-08-29T00:52:24.003" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18746" PostTypeId="1" CreationDate="2020-08-29T11:19:37.037" Score="8" ViewCount="142" Body="&lt;p&gt;This fall, I'll be a reader (i.e. homework grader) for the first time, and the course is a second-level linear algebra course, which is likely the first proof-based course most of the students will experience. Most students will be in a math major (more applied than pure) but some will be from other STEM areas. Points will be awarded based on rubrics, but I feel like additional feedback is especially important and formative for a first proof course.&lt;/p&gt;&#xA;&lt;p&gt;Unlike grading as a TA, I won't be able to communicate with students outside of writing notes on their solutions, and students will not be allowed to contact me directly (they must go through the TA to ask for regrade requests, etc.). If I could talk the students, I'd be able to get their feedback and adjust as I go, but this will not be possible for me as a reader. What are some general best practices for grading and giving feedback in a way that lets students have a good experience?&lt;/p&gt;&#xA;" OwnerUserId="14521" LastActivityDate="2020-08-30T04:10:47.660" Title="Grading in a way that lets students have a good and formative experience" Tags="|homework|grading|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18747" PostTypeId="2" ParentId="18742" CreationDate="2020-08-29T21:27:40.563" Score="18" Body="&lt;p&gt;Have you thought about the fact that you’re asking this in the middle of a pandemic for which log plots are being used all over the place to visualize the growth of COVID cases?&lt;/p&gt;&#xA;&lt;p&gt;At any rate,&#xA;&lt;span class=&quot;math-container&quot;&gt;$${d \over dt} \ln f(t) = {f’(t) \over f(t)} = \text{relative growth rate}\,.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus where the graph is roughly straight with slope &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, we have the number of cases growing exponentially, proportional to &lt;span class=&quot;math-container&quot;&gt;$e^{mt}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;And when the graph is concave up, the growth is greater than exponential (and worrisome).&lt;/p&gt;&#xA;&lt;p&gt;When it’s concave down, it can be much harder to judge by eye.  The slope of the tangent when &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a power function &lt;span class=&quot;math-container&quot;&gt;$A t^k$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$k/t$&lt;/span&gt; — that is, they all look like log graphs (obviously, since &lt;span class=&quot;math-container&quot;&gt;$\ln A t^k = \ln A + k \ln t$&lt;/span&gt;). That’s why, as the pandemic went on and the growth of cases became subexponential, log graphs of daily new cases began to appear along with total cases.  For &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; measured in days, &lt;span class=&quot;math-container&quot;&gt;$f'(t)$&lt;/span&gt; is the daily rate, and the slope of &lt;span class=&quot;math-container&quot;&gt;$\ln f’(t)$&lt;/span&gt;, which is &lt;span class=&quot;math-container&quot;&gt;$f’’(t)/f’(t)$&lt;/span&gt;, when the log graph is an upward-trending line still indicates exponential growth, but linear growth (&lt;span class=&quot;math-container&quot;&gt;$f’’(t)$&lt;/span&gt; constant) yields a flat log graph of new cases (since &lt;span class=&quot;math-container&quot;&gt;$f’’(t)=0$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I think some of this can be understood without derivatives. For instance, increasing and decreasing are pretty elementary notions and easy to see.  But understanding the slope of a log plot beyond just when it is going up or down takes some analysis.  Caveat: Beware the students measuring the slope.  If they take you down this route, you might have to explain to them what to do with the powers of ten on the vertical axis, since that is what is usually shown (i.e., &lt;span class=&quot;math-container&quot;&gt;$\log_{10} f(t)$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\ln f(t)$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2020-09-04T15:27:53.760" LastActivityDate="2020-09-04T15:27:53.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18748" PostTypeId="2" ParentId="18742" CreationDate="2020-08-29T22:25:07.050" Score="5" Body="&lt;p&gt;A couple of direct applications:&lt;/p&gt;&#xA;&lt;p&gt;Showing that a power law appears on a log-log graph as a straight line with gradient equal to the exponent of the power law (although that can be done by other, probably easier, means too).&lt;/p&gt;&#xA;&lt;p&gt;Thinking about a wind turbine that's well within the turbulent regime of the atmospheric boundary layer, the average wind speed incident on that turbine will depend on its height according to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Law_of_the_wall&quot; rel=&quot;noreferrer&quot;&gt;logarithmic law of the wall&lt;/a&gt;, so the derivative of a logarithm is needed to tell you how much additional wind speed you can get by increasing the height of the turbine slightly.&lt;/p&gt;&#xA;&lt;p&gt;In addition, the inverse operation, i.e. integrating &lt;span class=&quot;math-container&quot;&gt;$1/x$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$\ln\left(x\right)$&lt;/span&gt;, turns up in several places in thermodynamics and fluid mechanics, e.g.&lt;/p&gt;&#xA;&lt;p&gt;Deriving the Poisson adiabat, starting from the ideal gas law, the non-flow energy equation, and the definition of &amp;quot;specific heat capacity at constant volume&amp;quot; (or &amp;quot;specific internal energy capacity&amp;quot;). (Batchelor, 2000, &lt;em&gt;An introduction to fluid dynamics&lt;/em&gt;, Cambridge University Press, pp. 43-45)&lt;/p&gt;&#xA;&lt;p&gt;Deriving an algebraic expression for the entropy of an ideal gas in terms of pressure and temperature, also starting from the ideal gas law, the non-flow energy equation, and the definition of &amp;quot;specific heat capacity at constant volume&amp;quot; (or &amp;quot;specific internal energy capacity&amp;quot;). (Adkins, 1983, &lt;em&gt;Equilibrium thermodynamics&lt;/em&gt;, Cambridge University Press, p. 119)&lt;/p&gt;&#xA;&lt;p&gt;Deriving the relationship between pressure gradient and flow rate, in isothermal compressible flow of an ideal gas in a pipe of uniform cross-section, starting from the continuity equation, the Euler momentum equation, the ideal gas law, and a dimensional analysis of the shear traction at the pipe walls. (Douglas et al., 2005, &lt;em&gt;Fluid mechanics&lt;/em&gt;, Pearson Prentice Hall, section 17.9)&lt;/p&gt;&#xA;&lt;p&gt;But the trouble is, the typical order in which concepts are presented in education systems is such that students are expected to know about the derivative of &lt;span class=&quot;math-container&quot;&gt;$\ln\left(x\right)$&lt;/span&gt; &lt;em&gt;before&lt;/em&gt; they know about any of the physical principles that are the starting points for those derivations, so these applications may not be particularly helpful if you're just introducing calculus for the first time.&lt;/p&gt;&#xA;" OwnerUserId="14522" LastActivityDate="2020-08-29T22:25:07.050" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18749" PostTypeId="2" ParentId="18746" CreationDate="2020-08-30T04:10:47.660" Score="1" Body="&lt;p&gt;My advice: grade them according to the rubric.  Leave instruction and feedback to the TA.  You have already been told that is the agreed method.  Don't circumvent it with long notes.  Just do your job and follow the plan.&lt;/p&gt;&#xA;" OwnerUserId="14519" LastActivityDate="2020-08-30T04:10:47.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18750" PostTypeId="2" ParentId="18742" CreationDate="2020-08-30T09:23:18.100" Score="14" Body="&lt;p&gt;Whenever we measure a quantity on a &lt;a href=&quot;https://en.wikipedia.org/wiki/Logarithmic_scale&quot; rel=&quot;noreferrer&quot;&gt;log scale&lt;/a&gt; (such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Richter_magnitude_scale&quot; rel=&quot;noreferrer&quot;&gt;Richter&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Decibel&quot; rel=&quot;noreferrer&quot;&gt;decibels&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Interval_(music)&quot; rel=&quot;noreferrer&quot;&gt;musical pitch&lt;/a&gt;, or a &lt;a href=&quot;https://en.wikipedia.org/wiki/Semi-log_plot&quot; rel=&quot;noreferrer&quot;&gt;log-plot axis&lt;/a&gt;), we are focusing attention on &lt;em&gt;relative&lt;/em&gt; variation in that quantity. If &lt;span class=&quot;math-container&quot;&gt;$y = \ln x$&lt;/span&gt;, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{dy}{dx} = \frac{1}{x}$$&lt;/span&gt;&#xA;and thus, for small finite changes,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\Delta y \approx \frac{\Delta x}{x}.$$&lt;/span&gt;&#xA;That is, an &lt;em&gt;absolute&lt;/em&gt; change in the logarithm equates to a &lt;em&gt;relative&lt;/em&gt; change in the original magnitude. For example, a 1% change in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; will change &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; additively by about 0.01.&lt;/p&gt;&#xA;&lt;p&gt;This property is what makes log scales useful. People are naturally attuned to relative changes (often using percentages to express them). The logarithm helps express the &amp;quot;impact&amp;quot; of a phenomenon, because to &amp;quot;move the needle&amp;quot; on the logarithm (&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;), the quantity (&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;) must change by a noticeable amount &lt;em&gt;compared to its own size&lt;/em&gt;. It is also very useful that the change in the logarithm does not depend on the physical unit in which &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is measured, because this unit cancels between &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Example 1:&lt;/em&gt; A chorus with 10 equal singers produces a sound level measured at 70.0 dB. What will be the approximate sound level after an 11th equal singer joins the chorus?&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Answer:&lt;/em&gt; Sound level is defined as &lt;span class=&quot;math-container&quot;&gt;$L = 10\,\mathrm{dB} \times \log_{10} P$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is a measure of sound energy. The derivative of this logarithmic function gives&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\Delta L \approx \frac{10\,\mathrm{dB}}{\ln 10}\, \frac{\Delta P}{P}.$$&lt;/span&gt;&#xA;Adding one more singer to a group of 10 means &lt;span class=&quot;math-container&quot;&gt;$\Delta P/P = 1/10$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$\Delta L \approx 0.4\,\mathrm{dB}$&lt;/span&gt;. Thus, the new sound level is about 70.4 dB. This illustrates that there is very little difference in perceived loudness between 10 and 11 singers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Example 2:&lt;/em&gt; How many semitones is the musical interval between 1000 Hz and 1100 Hz?&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Answer:&lt;/em&gt; The pitch in semitones is defined as &lt;span class=&quot;math-container&quot;&gt;$S = 12 \log_2 f$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is the frequency. The derivative of this logarithmic function gives&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\Delta S \approx \frac{12}{\ln 2}\, \frac{\Delta f}{f}.$$&lt;/span&gt;&#xA;With &lt;span class=&quot;math-container&quot;&gt;$\Delta f/f = 100/1000$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$\Delta S \approx 1.7$&lt;/span&gt;. The interval is about 1.7 semitones.&lt;/p&gt;&#xA;" OwnerUserId="14524" LastActivityDate="2020-08-30T09:23:18.100" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18751" PostTypeId="2" ParentId="18742" CreationDate="2020-08-30T13:32:21.937" Score="2" Body="&lt;p&gt;In physics, in particular statistical mechanics, it is very common to take derivatives of logarithms. The basic idea is this:&lt;/p&gt;&#xA;&lt;p&gt;Assume we have some system that has many possible states, labeled &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, and the probability of any given state is given by the Boltzmann distribution, so that &lt;span class=&quot;math-container&quot;&gt;$P_s \propto e^{-E(s)/T}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$E(s)$&lt;/span&gt; is the energy of the state and &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is the temperature of the system (expressed in units of energy, i.e. my &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; here is really &lt;span class=&quot;math-container&quot;&gt;$k_B T$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$k_B$&lt;/span&gt; is Boltzmann's constant with units of energy / temperature). The normalized probability is then given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$P_s = \frac{e^{-E(s)/T}}{\sum_{s'} e^{-E(s')/T}}$$&lt;/span&gt;&#xA;where in the denominator we sum over every state. We define the denominator to be the &lt;a href=&quot;https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;parititon function&lt;/a&gt;, and for convenience define the inverse temperature &lt;span class=&quot;math-container&quot;&gt;$\beta = 1/T$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Z(\beta) = \sum_{s} e^{-\beta E(s)}$$&lt;/span&gt;&#xA;Now consider we want to calculate some quantity of interest, for example the average energy of the system, &lt;span class=&quot;math-container&quot;&gt;$\langle E \rangle$&lt;/span&gt;. Well that is just given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\langle E \rangle = \sum_{s} E(s) P_s = \frac{\sum_s E(s) e^{-\beta E(s)}}{Z(\beta)}$$&lt;/span&gt;&#xA;Note that we &lt;em&gt;always have to divide by &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; to get the right normalized expectation value&lt;/em&gt;. With this in mind, we can write the above with a moment's thought as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\langle E \rangle = -\frac{\partial}{\partial \beta} \ln Z(\beta) = -\frac{1}{Z} \frac{\partial Z}{\partial \beta}$$&lt;/span&gt;&#xA;Moreover, we can calculate the heat capacity of the system (at constant volume to be precise),&#xA;&lt;span class=&quot;math-container&quot;&gt;$$C_v = \frac{\partial \langle E\rangle}{\partial T} = - \frac{\partial}{\partial T} \frac{\partial}{\partial \beta} \ln Z(\beta) = \frac{1}{T^2} \frac{\partial^2}{\partial \beta^2}\ln Z(\beta)$$&lt;/span&gt;&#xA;if we work this out we will find &lt;span class=&quot;math-container&quot;&gt;$T^2 C_v = \langle E^2 \rangle  - \langle E \rangle^2 = \mathrm{var}(E)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is generally of more use to work with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Helmholtz_free_energy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Helmholtz free energy&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; rather than the energy &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, which is defined by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Z = e^{-\beta F} \leftrightarrow F = -T \ln Z$$&lt;/span&gt;&#xA;The entropy is defined as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$S = \langle -\ln P_s \rangle = -\sum_s P_s \ln P_s$$&lt;/span&gt;&#xA;and again we can easily show that this is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$S = -\frac{\partial F}{\partial T}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;All of this is much more general than just thermodynamics. From a statistician's point of view, &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Moment-generating_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;generating function of moments of the distribution&lt;/a&gt;, while &lt;span class=&quot;math-container&quot;&gt;$\ln Z$&lt;/span&gt; is the generating function of &lt;a href=&quot;https://en.wikipedia.org/wiki/Cumulant&quot; rel=&quot;nofollow noreferrer&quot;&gt;connected moments&lt;/a&gt; (or cumulants) of the distribution. For a general case, consider some probability distribution for a random variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P(x)$&lt;/span&gt;. Then consider the quantity (let's just assume this converges, e.g. if &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is gaussian)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Z(t) = \int_{-\infty}^\infty dx\, e^{tx} P(x)$$&lt;/span&gt;&#xA;Expanding the exponential, we get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Z(t) = \sum_{n} \frac{t^n}{n!} \int_{-\infty}^\infty dx\, x^n P(x)=\sum_n \frac{t^n}{n!} \langle x^n \rangle$$&lt;/span&gt;&#xA;then clearly&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left.\frac{d^n Z}{dt^n}\right|_{t=0} = \langle x^n \rangle$$&lt;/span&gt;&#xA;For the connected moments, akin to &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; above, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$W(t) = \ln Z(t) = \sum_n \frac{t^n}{n!}\langle x^n \rangle_c$$&lt;/span&gt;&#xA;The first connected moment is the mean, the second is the variance, the third is the skewness, the fourth is the kurtosis, etc. (Note there is a nice insight here about why the normal distribution is special: it is fully characterized by only the first two connected moments, the mean and variance, and all of its higher connected moments are exactly zero). Once again,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\langle x^n \rangle_c = \left. \frac{d^n W}{dt^n}\right|_{t=0}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;A more complicated but exactly parallel method is used in Quantum Field Theory for the computation of scattering amplitudes, where &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; is the Feynman Path integral (note this looks very similar to the definition of the partition function),&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Z = \int \mathcal{D}\phi\, e^{i S[\phi]}$$&lt;/span&gt;&#xA;Using essentially the same derivative tricks as above leads to a series expansion for any expectation we want to compute where each term in the expansion can be represented by a Feynman diagram. Then the quantity &lt;span class=&quot;math-container&quot;&gt;$W = \ln(Z)$&lt;/span&gt; is the generator of the fully-connected, 1-particle-irreducible diagrams, where again the logarithm takes care of proper normalization in the calculation of probability amplitudes.&lt;/p&gt;&#xA;&lt;p&gt;In all of these cases, the importance of the logarithm is to ensure that on taking the derivative we get a properly normalized result, i.e. we divide by &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt;, which is the sum of all the &lt;em&gt;un&lt;/em&gt;-normalized probabilities.&lt;/p&gt;&#xA;" OwnerUserId="14526" LastEditorUserId="14526" LastEditDate="2020-08-30T13:49:48.643" LastActivityDate="2020-08-30T13:49:48.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18752" PostTypeId="1" CreationDate="2020-08-30T16:35:40.817" Score="5" ViewCount="434" Body="&lt;p&gt;Background: I am a lecturer in computer science, but my research is mostly theoretic and mathematic, so I ask here.&lt;/p&gt;&#xA;&lt;p&gt;I want to encourage my undergraduate students to become research students after their graduation (as master or Ph.D. students). To do this, I have to explain to them what it means to be a research student. As undergraduate students, they are used to just doing what we tell them to: we give them homework, they do the homework as requested, and get a grade. However, as research students they will have to be much more creative, bring more ideas of their own, take the initiative, try new directions, etc.&#xA;I started to write a short document describing to undergraduates how the life of a research student looks like, but then I figured that someone must have done this before.&lt;/p&gt;&#xA;&lt;p&gt;Do you know of a short text that explains to undergraduate students the meaning of doing research (as graduate students) in a way that is both accurate and attractive?&lt;/p&gt;&#xA;" OwnerUserId="13986" LastEditorUserId="13986" LastEditDate="2020-08-30T20:59:58.573" LastActivityDate="2020-09-05T10:39:07.880" Title="How to explain to undergraduate students what research means?" Tags="|undergraduate-research|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18753" PostTypeId="1" CreationDate="2020-08-30T17:35:13.567" Score="3" ViewCount="386" Body="&lt;p&gt;I am a maths teacher and I want to make maths explainer videos particularly like &lt;a href=&quot;https://www.youtube.com/watch?v=9dyK_op-Ocw&quot; rel=&quot;nofollow noreferrer&quot;&gt;this guy is doing.&lt;/a&gt; In fact, in my research, I came to know that manim is the latest tool for creating maths animations but it is hard and time-consuming (at least for me) because it requires python and long codes. But I am sure this guy is not using manim. If you know how one can make such videos please let me know.&lt;/p&gt;&#xA;&lt;p&gt;Possibly, he is using blender but I am not sure as I am new on blender.&lt;/p&gt;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="14528" LastEditorUserId="14528" LastEditDate="2020-09-02T08:38:19.843" LastActivityDate="2020-09-05T10:01:52.507" Title="How to make maths explainer videos?" Tags="|course-design|interactive-teaching|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18754" PostTypeId="2" ParentId="18752" CreationDate="2020-08-30T18:32:15.233" Score="5" Body="&lt;p&gt;Here's a useful article.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gareth E. Roberts.&#xA;&amp;quot;Conducting Mathematical Research with Undergraduates.&amp;quot; 2008.&#xA;&lt;a href=&quot;https://mathcs.holycross.edu/%7Egroberts/Papers/Primus.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Abstract&lt;/strong&gt; (extract). This&#xA;article discusses some of the key issues concerning undergraduate mathematical research:&#xA;selecting good research students, finding appropriate research questions, mentoring versus&#xA;collaboration, and presenting and publishing student work. Some useful professional and&#xA;financial resources supporting undergraduate research are also highlighted.&lt;/p&gt;&#xA;&lt;p&gt;Along the way, Gareth Roberts mentions that,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Joe Gallian, who has been running a highly successful REU&lt;sup&gt;1&lt;/sup&gt; program at the University&#xA;of Minnesota Duluth for over thirty years, spends the entire year leading up to his summer&#xA;program searching for good research problems [15].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You might be interested in this recent history by Gallian:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Joseph A. Gallian.&#xA;&amp;quot;A History of Undergraduate Research in Mathematics.&amp;quot; 2018.&#xA;&lt;a href=&quot;https://cpb-us-e1.wpmucdn.com/sites.providence.edu/dist/6/48/files/2018/02/Undergrad_Centennial-2c8g7kw.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Abstract&lt;/strong&gt; (extract). In this article we identify the key events that have led to&#xA;the current widespread acceptance of the importance of opportunities for undergraduates to&#xA;engage in research in mathematics.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;REU = Research Experiences for Undergraduates, a US National Science Foundation summer program.&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-08-31T00:41:26.630" LastActivityDate="2020-08-31T00:41:26.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18755" PostTypeId="2" ParentId="18717" CreationDate="2020-08-30T22:43:22.000" Score="1" Body="&lt;p&gt;I'm a physicist and only rarely get to teach math, but I suspect the situation is about the same in both fields. I've visited a fairly decent-sized sample of classes for adjunct and tenure-track review at the community college where I work. My observations from visiting these classes are of course not really what you asked for, but I suspect they're pretty representative of the US, because we're in a big metropolitan area (LA) and hire from all over the US.&lt;/p&gt;&#xA;&lt;p&gt;The evidence  supporting active learning in physics has been pretty definitive and largely unchanging AFAIK since about 1995, when I started teaching. See Von Korff for a meta-analysis. The evidence in math seems somewhat less definitive, but we've had other questions on this topic here, which you can look for.&lt;/p&gt;&#xA;&lt;p&gt;What I generally observe when I visit classrooms is that about 3/4 of faculty do nothing but straight lecture. The other 1/4 do mostly straight lecture but also make some cursory, ineffective use of practices that are spin-offs from research-based active learning, e.g., computer polling or gamified quizzes.&lt;/p&gt;&#xA;&lt;p&gt;My impression is that the evidence-based movement toward more effective pedagogy got going a little ca. 1995-2000 and then died out -- not because the evidence had changed but because of other factors:&lt;/p&gt;&#xA;&lt;p&gt;(1) Faculty don't know about the evidence and have never seen evidence-based practices modeled. In math, I think the evidence is also not as definitive as in physics.&lt;/p&gt;&#xA;&lt;p&gt;(2) The techniques take practice to do effectively. (I have also seen them modified and misapplied by touchy-feely social justice types in a way that degrades intellectual standards and disrespects the legitimate role of the written word.)&lt;/p&gt;&#xA;&lt;p&gt;(3) Many students will complain, which can be devastating for adjuncts and non-tenured faculty.&lt;/p&gt;&#xA;&lt;p&gt;(4) The advent of ratemyprofessor, mymathlab, and chegg have caused a breakdown in faculty's ability and willingness to realistically evaluate students' level of mastery, so bad results of traditional practices don't get perceived as bad results.&lt;/p&gt;&#xA;&lt;p&gt;I think evidence-based teaching in physics never got going at all in K-12, and survives today at the undergrad level only in some specific places that have their own educational research programs. Examples would be UW and possibly Harvard.&lt;/p&gt;&#xA;&lt;p&gt;I'm sure there are quite a few math teachers at the high school and community college level who spend some significant percentage of class time on techniques like think-pair-share and are very effective with them, but I don't think there's any significant number of people doing the kind of systematic, wholesale active learning approach that you may see advocated in your education classes. It would be interesting to see studies comparing the effectiveness of that kind of more modest mixed format with more aggressive active-learning techniques.&lt;/p&gt;&#xA;&lt;p&gt;Von Korff et al., &amp;quot;Secondary Analysis of Teaching Methods in Introductory Physics: a 50k-Student Study,&amp;quot; Am. J. Phys. 84, 969 (2016), &lt;a href=&quot;https://arxiv.org/abs/1603.00516&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/1603.00516&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-08-30T23:17:27.077" LastActivityDate="2020-08-30T23:17:27.077" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18756" PostTypeId="2" ParentId="18752" CreationDate="2020-08-31T02:40:06.113" Score="0" Body="&lt;p&gt;Research:&#xA;To boldly go to places no man has gone before...&lt;/p&gt;&#xA;&lt;p&gt;I work in IT and it is only after working for 13 or so years, one day FINALLY, PC became personal to me. I felt as if it is watching me through it's camera, watching my actions, trying to know me, learning me. I said to my laptop one day, if you are trying to get inside me, I'll get inside you. And that's when I became interested in Machine Learning and Artificial Intelligence, I wanted to see what a computer sees when He(it) looks at me.&lt;/p&gt;&#xA;&lt;p&gt;So Research has to be a personal quest. The subject becomes personal to you, you live it and it's fun. For example all my school life I struggled with Proofs in Mathematics. There are direct proofs, indirect proofs, proofs by contradiction, then there is logic, axioms, lemmas, corollaries and what not. So I decided to understand these mathematical proofs one day. I started studying and was so engrossed in the Logic part of it that I started living it. I was watching a sitcom and on listening to their dialogues I was getting irritated, I COULD NOT UNDERSTAND WHY?! Upon thinking I realised that since one can draw more than one meaning of what that person said, I was finding it irritating since logic says that unambiguously people MUST arrive at a single conclusion from premises. So I was living Logic during those times. I have the output &lt;a href=&quot;https://github.com/shukash/Mathematical-Proof-Infographic/blob/master/Mathematical%20Proofs.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So research is devotion, it's a way of life. You don't don a &amp;quot;Researcher's hat&amp;quot; when you go to work, it's an attitude. One has to be an explorer at heart....&lt;/p&gt;&#xA;&lt;p&gt;Ohh and by the way I found out how to make &amp;quot;Conscious Machines&amp;quot; NOT &amp;quot;Intelligent Machines&amp;quot; but Conscious machines, how to give them intuition...&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2020-08-31T02:40:06.113" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18757" PostTypeId="2" ParentId="2" CreationDate="2020-08-31T03:23:23.570" Score="0" Body="&lt;p&gt;I think proofs can be/should be taught as early as possible. I am not sure at what age exactly but as soon as they can understand the concepts maybe around 12-13 years old. So I keep giving them bits and pieces of it and if they are able to pickup the concepts then I teach them fully. One of the things that we need to do is to coherently represent all the moving parts in a Mathematical Proof. I always struggled with &amp;quot;the big picture&amp;quot; when I was studying. We also need to explain the underlying concepts in easy way. For example what is Logic? I tell my kids, Logic is a vehicle that takes your ordinary day today language and converts it into mathematics. Towards the big picture, I created an infographic which you might find useful. Have a look &lt;a href=&quot;https://github.com/shukash/Mathematical-Proof-Infographic/blob/master/Mathematical%20Proofs.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2020-08-31T03:23:23.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18758" PostTypeId="2" ParentId="8064" CreationDate="2020-08-31T09:41:15.547" Score="1" Body="&lt;p&gt;A graphic tablet would be a suitable input device for annotating a shared screen to teach online .&#xA;I know a very busy online math tutor who uses BitPaper and a XP-PEN Star G640 digital writing tablet every day.&lt;/p&gt;&#xA;" OwnerUserId="14529" LastActivityDate="2020-08-31T09:41:15.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18759" PostTypeId="2" ParentId="8064" CreationDate="2020-08-31T13:35:31.667" Score="1" Body="&lt;p&gt;I got a Wacom intuos small a couple of weeks ago after using an iPad for many months.&#xA;It is only a tiny fraction of the cost of the iPad but is so much better for what I do. I can easily annotate/write on online whiteboards, zoom whiteboard etc. It makes the job so much easier.I would strongly recommend it.&lt;/p&gt;&#xA;" OwnerUserId="10125" LastActivityDate="2020-08-31T13:35:31.667" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18760" PostTypeId="2" ParentId="8064" CreationDate="2020-08-31T17:04:37.017" Score="3" Body="&lt;p&gt;You can of course keep using iPad. One way to do this is to start a zoom meeting with your students and share your iPad screen. Then you can write in a note taking app such as GoodNotes.&lt;/p&gt;&#xA;&lt;p&gt;The draw back of this method for me is that sometimes I want to show some Mathematica generated 3d models. You can use Wolfram Player on iPad. But then you can not edit the notebook. In this case, I would use my laptop to join the meeting and share the screen. Then I turn on Zoom's annotation function. So I can draw on the shared screen from iPad on top the shared screen from my laptop.&lt;/p&gt;&#xA;&lt;p&gt;A tip for joining a Zoom meeting with two device is to mute the speaker and microphone of one device, otherwise you may get very sharp noise.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2020-08-31T17:04:37.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18761" PostTypeId="2" ParentId="8064" CreationDate="2020-08-31T22:36:20.633" Score="2" Body="&lt;p&gt;I use a 16&amp;quot; Wacom Cintiq Pro.  The advantage that the Cintiq line has over the Intuos line is that the Cintiqs act as secondary displays.  There are a number of reasons why I consider this to be a huge selling point.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I can see what I am writing.  That is, as I write, I see writing appear &amp;quot;on the page&amp;quot; under the pen.  This is in contrast to graphics tablets like the Intuos line, which are essentially mice with absolute (rather than relative) positioning.  The couple of times that I have used Intuos tablets, I found them quite awkward and difficult to use (though I image that one eventually adapts).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Having a second display makes it easier to stage work.  With a laptop and Cintiq, I can keep multiple browser windows, .pdfs, Maple worksheets, etc off-screen until I want to use them.  When I want to show them to students, I just drag them to the the Cintiq.  With a third display (such as a cheap television with an HDMI input), I can put my Brady Bunch Zoom squares on one display, my own notes on another, and whatever I am showing my students on a third.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The 16&amp;quot; Cintiq is small enough to be portable-ish—it is not quite convenient for work at a cafe (are we ever going to be able to work at cafes again?), but I can happily pack my laptop and Cintiq into my bag for traveling.  If you need something which is truly portable, Wacom makes smaller Cintiq tablets.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PPQoN.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PPQoN.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;[From left-to-right:  laptop, cheap television with an HDMI port, Cintiq tablet w/ display.]&lt;/p&gt;&#xA;&lt;p&gt;Going the other direction, I have also used a setup with a 27&amp;quot; Cintiq Pro, which was amazing.  At a cost of a couple of thousand dollars, I'll never get one without institutional funding, but if you don't need portability, it is a wonderful piece of hardware to have.&lt;/p&gt;&#xA;&lt;p&gt;I'll also note that, as much as I am shilling for Wacom, I have no connection to them.  In fact, before getting my own graphics tablet, I spent some time reading reviews of similar tablets from other manufacturers.  A good place to start might be with &lt;a href=&quot;https://www.colourmylearning.com/2019/03/best-budget-graphics-drawing-tablets-with-built-in-display-for-the-creative/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this list of similar tablets&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;With respect to software, I have found that Notability works reasonably well as a whiteboard.  The main disadvantage of Notability is that it produces monstrously large .pdf files—I suspect that it is exporting high resolution raster images rather than vectorized graphics.  I haven't really tried any other software, so I can't speak much to the relative advantages or disadvantages.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2020-08-31T22:36:20.633" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18763" PostTypeId="1" CreationDate="2020-09-01T16:01:47.127" Score="4" ViewCount="254" Body="&lt;p&gt;I am speaking about high school mathematics .&#xA;Students have attended a mathematics course . By the end thereof ,  students are supposed to be able to:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;find the limit of a real function f as “x” approaches to a given “a” and write the equations of vertical , horizontal and oblique asymptotes to the function’s curve  (but not curved asymptotes)&lt;/li&gt;&#xA;&lt;li&gt;use the squeeze theorem to find some limits&lt;/li&gt;&#xA;&lt;li&gt;find the derivative of a real function and deduce whether the function is monotonic or not  (depending on the sign of the derivative)&lt;/li&gt;&#xA;&lt;li&gt;write the equation of the tangent straight line to a curve at a given point&lt;/li&gt;&#xA;&lt;li&gt;determine the number of the solutions to an equation of the form f(x) = m (m is a real number) using the mean value theorem&lt;/li&gt;&#xA;&lt;li&gt;determine whether a real function is continuous or not at a given point&lt;/li&gt;&#xA;&lt;li&gt;draw the curve of a given function in an orthogonal system&lt;/li&gt;&#xA;&lt;li&gt;decide whether  a sequence is monotonic or not&lt;/li&gt;&#xA;&lt;li&gt;decide whether a sequence is an arithmetic one , a geometric one , or neither&lt;/li&gt;&#xA;&lt;li&gt;calculate the limit of a recursive sequence (of the form un+1 = f(un) )&lt;/li&gt;&#xA;&lt;li&gt;decide whether two sequences  are adjacent or not&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students are familiar with the following functions: the square root function , the absolute value function , exponential and logarithmic functions , polynomial functions , and trigonometric functions&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;12&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;decide whether three vectors are coplanar or not&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;calculate the dot product of two vectors&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;write a cartesian equation of a plane , sphere , cone and cylinder&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;write a parametric equation of a straight line , a half line , a line segment&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;find the barycenter of n weighted points&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;solve a system of three linear equations&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;write the algebraic, rectangular and exponential forms of a complex number&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;solve a second degree equation with complex variables  and/or coefficients&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;describe each of these geometrical transformations (rotation , homothety&#xA;, and translation ) using complex numbers&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Those are NOT all the skills which the students must have developed throughout  the course.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;After the course have ended and the students have mastered the skills above,  they are supposed to take a 3 hours test which is “the final exam for the entire course” .&lt;/p&gt;&#xA;&lt;p&gt;The questions is :&#xA;How the test is supposed to be designed?&lt;/p&gt;&#xA;&lt;p&gt;The problems from which the test is composed , should they be routine, typical ones which mimic the ones in the students’ textbooks?&#xA;Or new ones which need a lot of thinking and imagination, yet require the same knowledge provided by the students’ textbook?&lt;/p&gt;&#xA;&lt;p&gt;Some students who are accurate and do not make “arithmetic error” would find no difficulty solving any of the routine problems they are used to such as&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Given a function f defined on a set I : the students would find the limits of the function , the derivative thereof, determine the number of solutions to the equation f(x)=0, draw the graph of the function in an orthogonal system&lt;/li&gt;&#xA;&lt;li&gt;Given two complex numbers , the students would write both numbers in the exponential form , found  both the exponential and algebraic form of their product , deduce the trigonometric ratios of an angle (most probably , the argument of the product )&#xA;Etc.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;However, what benefit did such a test give? Did the test reflect enough the mathematical thinking of the students who took it?&lt;/p&gt;&#xA;&lt;p&gt;Would it be reasonable if ,for instance, out  of 100 students , 10 students took a perfect score ? It is possible that there exists 10 math “geniuses” within a group of 100?&lt;/p&gt;&#xA;&lt;p&gt;I hope the question became clearer after this edit.If still not clear, please notify in the comments.&lt;/p&gt;&#xA;&lt;p&gt;Please note that the test is NOT a multiple choice test.&lt;/p&gt;&#xA;" OwnerUserId="14534" LastEditorUserId="14534" LastEditDate="2020-09-06T12:15:11.013" LastActivityDate="2020-09-06T12:15:11.013" Title="How should mathematics tests be designed?" Tags="|exams|critical-thinking|" AnswerCount="1" CommentCount="34" ClosedDate="2020-09-05T13:38:37.580" ContentLicense="CC BY-SA 4.0" />
  <row Id="18765" PostTypeId="2" ParentId="18763" CreationDate="2020-09-03T12:25:02.703" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The problems from which the test is composed , should they be&#xA;routine, typical ones which mimic the ones in the students’ textbooks?&#xA;Or new ones which need a lot of thinking and imagination, yet require&#xA;the same knowledge provided by the text?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I recommend to go with the &amp;quot;routine&amp;quot; questions.  (Scare quotes intentional!)  (1) Don't underestimate the difficulty of &amp;quot;routine&amp;quot; to many students.  (1.5) ESPECIALLY under test conditions.  (2.)  You're covering a lot of material and some of it will be several months old.  (3.)  There is a value to the routine questions.  (4.) Separating the stars (small minority) from the above average (larger amount, but still a minority) should not replace the objective of measuring basic competency for the majority of the kids.  (4.5) And you'll get it anyway, since in a 3 hour test, there's room for stumbles.  It won't be so basic that the top 3 of 30 can't be differentiated from the next 6.  They'll separate.  Look at the literature on psychrometrics.  Look at &amp;quot;easy&amp;quot; tests like SAT-M.&lt;/p&gt;&#xA;&lt;p&gt;In fact, there's a real danger of running acrux if your questions are too tricky and non routine.  You're basically testing problem solving savvy more than mastery of the curriculum and even the sharpies can just get unlucky about figuring out some Euler trick or the like under exam conditions.  I think a routine 20 question test (one per objective) will be fine.&lt;/p&gt;&#xA;&lt;p&gt;Oh...and a lot of those objectives are pretty conceptual.  Not sure I agree with so much stress on that.  It is important to have kids that can do manipulations to support what they will need in physics and engineering homeworks.  It's not awful and there are some tactical objectives.  And maybe it's just written that way to look fancy and the questions are more recognizable.  Not a strong objection, just a little hair on my neck.  ;-)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;It's interesting to look at your MSE question, where you ask about making silly mistakes.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/3796357/how-can-i-prevent-silly-frustrating-mistakes-while-solving-math-problems&quot;&gt;https://math.stackexchange.com/questions/3796357/how-can-i-prevent-silly-frustrating-mistakes-while-solving-math-problems&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's one thing if you just want hard questions.  But hard questions (concept emphasis) when you don't have mastery of the basics?  Sorry, that is a big problem.  You NEED to have automaticity of the basics if you are going to do advanced math or physics or engineering (even decent undergrad level homework, especially junior year on, let alone research).&lt;/p&gt;&#xA;&lt;p&gt;Look at Olympic level gymnasts (I worked out with some back in the day).  They spent their time working on very hard D-level and above skills (full twisting double backs and such).  Remember asking one how to progress and he told me...you need to be solid on your basics...you don't have a 30 second handstand, you don't have good circles on horse, and your backhandspring has no distance.&lt;/p&gt;&#xA;&lt;p&gt;The movie The Karate Kid is an exaggeration of this concept, but it really is true in spirt in martial arts.  Well, at least boxing.  You need to have basic stance and protection and the like down solid.  And jab, jab, jab.  Yeah, you'll want those fancy combinations.  But learn to jab, baby.  It will keep you safe.&lt;/p&gt;&#xA;&lt;p&gt;Here is what Richard Feynman advised CALTECH students regarding mathematics (section 1-3 of Feynman Lectures Physics):&lt;/p&gt;&#xA;&lt;p&gt;So, this guy comes into my office and asks me to try to make everything straight that I taught him, and this is the best I can do. The problem is to try to explain the stuff that was being taught. So I start, now, with the review. I would tell this guy, “The first thing you must learn is the mathematics. And that involves, first, calculus. And in calculus, differentiation.”&lt;/p&gt;&#xA;&lt;p&gt;Now, mathematics is a beautiful subject, and has its ins and outs, too, but we’re trying to figure out what the minimum amount we have to learn for physics purposes are. So the attitude that’s taken here is a “disrespectful” one towards the mathematics, for sheer efficiency only; I’m not trying to undo mathematics.&lt;/p&gt;&#xA;&lt;p&gt;What we have to do is to learn to differentiate like we know how much is 3 and 5, or how much is 5 times 7, because that kind of work is involved so often that it’s good not to be confounded by it. When you write something down, you should be able to immediately differentiate it without even thinking about it, and without making any mistakes. You’ll find you need to do this operation all the time—not only in physics, but in all the sciences. Therefore differentiation is like the arithmetic you had to learn before you could learn algebra.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, the same goes for algebra: there’s a lot of algebra. We are assuming that you can do algebra in your sleep, upside down, without making a mistake. We know it isn’t true, so you should also practice algebra: write yourself a lot of expressions, practice them, and don’t make any errors. Errors in algebra, differentiation, and integration are only nonsense; they’re things that just annoy the physics, and annoy your mind while you’re trying to analyze something. You should be able to do calculations as quickly as possible, and with a minimum of errors. That requires nothing but rote practice—that’s the only way to do it. It’s like making yourself a multiplication table, like you did in elementary school: they’d put a bunch of numbers on the board, and you’d go: “This times that, this times that,” and so on—Bing! Bing! Bing!&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I would even say that you can use some of the advanced problems as an OPPORTUNITY to hone your rusty &amp;quot;routine&amp;quot; skills.  For example if you do a maximization problem (not the most conceptually hard, but still a little non routine in that it is multistep and may be a &amp;quot;word problem&amp;quot;) and you have the method down, but mess up a calculation.  Now, when you check your answer and recognize the mistake, DON'T just say no biggie and move on.  Instead, put the solution (or answer) away and do the problem again from scratch as if it were new.  This will quickly drill you to start NOT making so many errors and it will give you more practice in both the method and the manipulations.  And it will tend to improve your test speed.&lt;/p&gt;&#xA;" OwnerUserId="14519" LastEditorUserId="14519" LastEditDate="2020-09-04T11:43:23.407" LastActivityDate="2020-09-04T11:43:23.407" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18766" PostTypeId="1" AcceptedAnswerId="18767" CreationDate="2020-09-03T12:39:57.583" Score="58" ViewCount="5658" Body="&lt;p&gt;I teach future elementary educators mathematics content courses.&lt;/p&gt;&#xA;&lt;p&gt;We play a lot in class with tasks like &amp;quot;Write a variety of word problems which would require the student to multiply 2.3 by 1.4&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Often the questions which students produce are unintelligible, unanswerable, or target the wrong operation.&lt;/p&gt;&#xA;&lt;p&gt;Unintelligible:  &amp;quot;Bob has 2.3 pizza, and Jim has 1.4 pop.  How much all together?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Unanswerable:  &amp;quot;I have 2.3 dollars, and want to buy 1.4 cups of ice cream.  How much more money do I need to buy the ice cream?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Target the wrong operation: &amp;quot;I have 2.3 cups of flour and 1.4 cups of water.  If a recipe calls for 1 cup of each, how many recipes can I make?&amp;quot;  (The operation here is &lt;span class=&quot;math-container&quot;&gt;$\textrm{min}(2.3,1.4)$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;p&gt;No matter how much we practice this, when the exam comes around I still have a very large percentage of students writing these kinds of responses.  This is extremely concerning to me, to the point that I don't feel comfortable with students who would produce such responses becoming educators.&lt;/p&gt;&#xA;&lt;p&gt;Have others observed this phenomenon?  Is there any research on it?  Does anyone have success with fixing this problem?&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-09-05T22:41:33.630" Title="Future educators writing nonsense questions" Tags="|primary-education|teacher-preparation|" AnswerCount="4" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="18767" PostTypeId="2" ParentId="18766" CreationDate="2020-09-03T14:35:32.433" Score="49" Body="&lt;p&gt;You might try starting this kind of lesson with an assignment where you provide a list of different responses to the prompt &amp;quot;Write a variety of word problems which would require the student to multiply 2.3 by 1.4&amp;quot; and have students (perhaps in groups) arrange and rank them by clarity/mistakes/etc. Instead of having students start by writing their own word problems, by bringing your own list of good-and-bad examples, you're not setting students up to be ridiculed since you've created the set of responses.&lt;/p&gt;&#xA;&lt;p&gt;I've had colleagues do this kind of thing with other tasks, such as &lt;em&gt;learning how to show work&lt;/em&gt; in an intermediate algebra class. The first assignment is to take a stack of 8-10 teacher-created &amp;quot;solutions&amp;quot; to a word problem, read them with your group, and rank them by clarity and completeness. Then the class discussion revolves around lessons they can take from this when writing up their own solutions to word problems.&lt;/p&gt;&#xA;&lt;p&gt;[Edit on 9/4]: I think another benefit of this plan is that if you follow up with a full-class discussion about why certain of the responses were objectionable, you'll likely build a class-specific vernacular that students may find light/playful to use when identifying errors in their/other's writing: Your &amp;quot;unintelligible&amp;quot; may be dubbed &amp;quot;what a mess&amp;quot; or &amp;quot;I can't even&amp;quot;, etc. Since these descriptors aren't &lt;a href=&quot;https://matheducators.stackexchange.com/questions/17829/allowing-nonstandard-mathematical-language-and-or-notation&quot;&gt;standard&lt;/a&gt; mathematical terms, I personally would readily accept new language if it means the class is learning the topic at hand and can communicate about it (though I would surely use my own terms from time to time in discussion).&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2020-09-04T16:31:47.557" LastActivityDate="2020-09-04T16:31:47.557" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18768" PostTypeId="2" ParentId="18689" CreationDate="2020-09-04T13:54:13.223" Score="1" Body="&lt;p&gt;I did a survey with my students, asking them what is their preferred presentation method. Here are the results:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;27% preferred the &amp;quot;invisible content&amp;quot;, where the content is fully hidden and displays as the lesson goes on.&lt;/li&gt;&#xA;&lt;li&gt;42% preferred the &amp;quot;transparent content&amp;quot;, where the content is transparent and displays as the lesson goes on.&lt;/li&gt;&#xA;&lt;li&gt;31% preferred the &amp;quot;visible content&amp;quot;, where the whole content displays right at the beginning of each slide.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Based on that, I will use the &amp;quot;transparent content&amp;quot; style, as it is the preferred method among the students, and it seems to be a good middle-ground for the different possibilities.&lt;/p&gt;&#xA;" OwnerUserId="14410" LastActivityDate="2020-09-04T13:54:13.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18769" PostTypeId="1" CreationDate="2020-09-04T16:29:08.707" Score="1" ViewCount="1494" Body="&lt;p&gt;I'm sure Thompson's books were a fine series back in his time, but are they still worth recommending for, say, interested high-school students or prospective college students that want to brush up their math skills?&lt;/p&gt;&#xA;&lt;p&gt;How much has notation changed in +70 years? How much basic math education has made any progress? Is the scope of topics still up-to-date?&lt;/p&gt;&#xA;" OwnerUserId="237" LastActivityDate="2020-09-04T20:15:57.917" Title="Are books as old as J.E. Thompson's &quot;for the practical man&quot; series outdated?" Tags="|textbooks|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18770" PostTypeId="2" ParentId="18769" CreationDate="2020-09-04T19:23:22.300" Score="4" Body="&lt;p&gt;The topic hasn't changed (that) much.  And for a neophyte, they are fine, even accessible.  They won't replace deeper study, but are fine for a bright kid who wants to play with things a little before AP Calculus.  They're not the only option though.  Could just get Granville (reasonably readable prose, modern enough not to be annoying, and pleasantly consice (not in an abstract way but in not being too text filled, like many doorstops).  Also a Schaum's Outline is fine.  For that matter you can get regular texts for cheap through Amazon.  Used books are very cheap and any from the 50s on will be relatively modern in terms of treating limits and the like.  Whatever you do though, stay away from the hardest books usually recommended by people on MSE (Spivak, etc.) Those are better for those who know the topic already and/or are unlikely to be turned off.  Really, for the dabbler something like Caculus Made Easy might be fun.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Calculus_Made_Easy&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Calculus_Made_Easy&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14519" LastActivityDate="2020-09-04T19:23:22.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18771" PostTypeId="2" ParentId="18769" CreationDate="2020-09-04T20:15:57.917" Score="2" Body="&lt;p&gt;There are tons of good,accessible, calculus texts available online, like the introductory &lt;a href=&quot;http://www.williamchen-mathematics.info/ln.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;lecture notes by William Chen&lt;/a&gt;, and there is the &lt;a href=&quot;https://aimath.org/textbooks&quot; rel=&quot;nofollow noreferrer&quot;&gt;Open Textbook Initiative&lt;/a&gt; to look at...&lt;/p&gt;&#xA;&lt;p&gt;It is very hard to give blanket recommendations, the book I find engaging somebody else might find incomprehensible or plain boring. And the recommendation would depend very much on the background and interests of the prospective reader.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-09-04T20:15:57.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18772" PostTypeId="1" AcceptedAnswerId="18774" CreationDate="2020-09-04T20:53:50.540" Score="12" ViewCount="1615" Body="&lt;p&gt;I'm teaching calculus 1 online this term and anticipate being plagued by the perennial problem of cheaters. I have seen suggestions for how to arrange the testing time to accommodate for traditional tests (such as &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18477/what-are-the-best-practices-for-giving-online-tests&quot;&gt;this previous question&lt;/a&gt; and the questions it mentions) but I would rather improve my question-writing instead of impose onerous requirements on my students in terms of time. I would hope that by making these sorts of questions common in all assessment levels that this would also improve my students' learning, especially as they learn that I expect them to understand instead of merely regurgitate. I've also contemplated moving to projects, but have come to similar conclusions that it is difficult to evaluate students on procedural fluency with projects, and devising effective projects is probably going to take me more time than simply writing better questions.&lt;/p&gt;&#xA;&lt;p&gt;So, with that said, what are some examples of types of questions which are suited for an online exam? A requirement here is that WolframAlpha &amp;amp; other solving websites often do in an unnatural way, or in a way which we do not ask the students to do it. Or equivalently, what ways of formulating questions are there that are difficult to put in to WolframAlpha?&lt;/p&gt;&#xA;&lt;p&gt;The goal here is to make it less desirable to cheat (because it is difficult to do so) and to make it easier to catch those who cheat. I also do not think entirely avoiding mechanical differentiation problems is possible or desirable, as students need to develop the pattern-recognition to be able to learn integration in the next term, where it is possible we may end up going back face-to-face.&lt;/p&gt;&#xA;&lt;p&gt;Some types of problems that I can think of include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In a question, requiring logarithmic differentiation to simplify problems which would involve large numbers of products and quotients.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Differentiation of absolute value problems, if you ask for it to be done in the &amp;quot;piecewise&amp;quot; method instead of teaching them that the derivative of |x| is x/|x|&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finding the value of a constant which makes a function continuous, or perhaps even its derivative continuous.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;Abstract&amp;quot; Problems with chain rule, quotient rule, etc. that involve having a formula for one function and a simple f(x) for the other, represented as table or a graph.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Find the place where a made-up student made an error, and complete the problem from that point on correctly. Or, explain what is wrong with a specific step and what should be done to correct it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Of course, applications problems. But these do not necessarily test their knowledge of the symbol manipulation and the deeper algebraic fluency which should come from a well-taught calculus class.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I have difficulty coming up with lower-DOK questions in this same vein, however!&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2022-03-16T04:24:40.460" Title="Question formats for online tests, to deter cheating" Tags="|undergraduate-education|exams|online-instruction|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18773" PostTypeId="2" ParentId="18772" CreationDate="2020-09-04T22:09:21.357" Score="5" Body="&lt;p&gt;WolframAlpha is not your problem.  It really isn't.  The students who are cheating by and large aren't good enough to figure out WolframAlpha's syntax and interpret its answers.&lt;/p&gt;&#xA;&lt;p&gt;Your problem is that any of your students can buy a membership to Chegg.com and upload an image of a question.  Chegg.com his hired people in less developed countries who will do the problem in half an hour and upload the result, and it will be accessible to anyone with a Chegg.com membership who is searching for that question.  Then these students can just copy that answer.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2020-09-04T22:09:21.357" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18774" PostTypeId="2" ParentId="18772" CreationDate="2020-09-04T22:39:13.430" Score="9" Body="&lt;p&gt;For problems that are beyond just calculations, if you can algorithmically create different problems for each student that are similar in difficulty, this may help with the near-guarantee that someone will be posting all of your problems to Chegg. Also, it may be done with any of the problem ideas you list above.&lt;/p&gt;&#xA;&lt;p&gt;Here is an exam question I gave this term, together with a few different versions:&lt;/p&gt;&#xA;&lt;p&gt;Version 1:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EJa9c.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EJa9c.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Version 2:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/G24HR.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/G24HR.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Version 3:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/enX0s.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/enX0s.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think randomizing as much as possible is a good idea (up to the point where one student faces a much more difficult problem than another student).&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-09-04T22:39:13.430" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18775" PostTypeId="2" ParentId="18766" CreationDate="2020-09-05T00:02:22.190" Score="26" Body="&lt;p&gt;&lt;strong&gt;Note: This is an answer from a non-US perspective, after reading some remarks from @Rusty Core I fear it won't be helpful.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;How about you set certain standards and let sub-standard students fail your class? Just like in every other course the goal is to separate the wheat from the chaff.&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't want my children to be educated by teachers asking the kind of exam questions you provided as examples. If they don't improve until the re-exam they fail the class.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;This is extremely concerning to me, to the point that I don't feel comfortable with students who would produce such responses becoming educators.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Why concerning? Why are you so invested in these students, it's not your fault, that they don't study. Just fail them.&lt;/p&gt;&#xA;" OwnerDisplayName="user13921" LastEditorDisplayName="user13921" LastEditDate="2020-09-05T00:35:53.240" LastActivityDate="2020-09-05T00:35:53.240" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="18776" PostTypeId="2" ParentId="18753" CreationDate="2020-09-05T10:01:52.507" Score="2" Body="&lt;p&gt;I suspect that PowerPoint is used, as explained in &lt;a href=&quot;https://youtu.be/Ksw0LUVu-Gw&quot; rel=&quot;nofollow noreferrer&quot;&gt;this video&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13585" LastActivityDate="2020-09-05T10:01:52.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18777" PostTypeId="2" ParentId="18752" CreationDate="2020-09-05T10:39:07.880" Score="1" Body="&lt;p&gt;&lt;em&gt;How to explain to undergraduate students what research means?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;A nice, short and general explanation is given in &lt;a href=&quot;http://matt.might.net/articles/phd-school-in-pictures/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The illustrated guide to a Ph.D&lt;/a&gt;, by Matt Might. As a complement for the other suggestions, you could use it as an introduction for the discussion.&lt;/p&gt;&#xA;" OwnerUserId="13585" LastActivityDate="2020-09-05T10:39:07.880" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18778" PostTypeId="2" ParentId="18766" CreationDate="2020-09-05T17:30:45.967" Score="11" Body="&lt;p&gt;Partial answer regarding an &lt;strong&gt;approach to fix this problem&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;First: Don't tell them (criticism), but lead them to find out themselves (insight). Now comes the fun part. &lt;em&gt;Don't let them write just the questions&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Have them each write the &lt;strong&gt;question&lt;/strong&gt; on one sheet, and write the correct &lt;strong&gt;solution&lt;/strong&gt;, and a short (maybe single-sentence) &lt;strong&gt;explanation&lt;/strong&gt; for their solution (choice of operation, solution method) on a different sheet.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then have them swap just the question sheets and try to solve it :)&lt;/p&gt;&#xA;&lt;p&gt;If the question is intelligible, answerable, using the correct operations, &lt;em&gt;and&lt;/em&gt; the other person is sufficiently competent, he or she should not only be able to give the correct solution, but also to come up with a very similar explanatory statement.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In case the other person &lt;em&gt;isn't&lt;/em&gt; able to do that, you'll have demonstrated to them that &lt;em&gt;something&lt;/em&gt; isn't working yet... with no criticism involved.&lt;/p&gt;&#xA;" OwnerUserId="9279" LastActivityDate="2020-09-05T17:30:45.967" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18779" PostTypeId="2" ParentId="18766" CreationDate="2020-09-05T19:11:57.473" Score="10" Body="&lt;p&gt;You might start out by giving them a test in the kind of arithmetic they are supposed to teach. &lt;a href=&quot;https://www.theguardian.com/education/2010/feb/14/primary-teachers-fail-maths-tests&quot; rel=&quot;noreferrer&quot;&gt;This article&lt;/a&gt; in the Guardian from 2010 reported that many primary school teachers in the UK were unable to do the arithmetic required by the primary curriculum.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Fewer than four out of 10 of those who sat the test – designed for 11-year-olds – could calculate 2.1% of 400, and only a third answered correctly that 1.4 divided by 0.1 was 14. Overall, four out of ten scored 40% or below, only one got all the answers correct and the average mark was 12 out of 27 or 45%.&lt;/p&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;p&gt;The test, carried out for the Channel 4 documentary series Dispatches, included addition and multiplication sums, ­simple algebra and questions involving fractions, conversions and averages. ­Teachers performed well on some of the easier questions. For example, 97% were able to work out 2 x 5 - 4 = 6 and 75% knew that three sevenths of 21 was 9. But only six teachers (4%) knew that the answer to 2 divided by 0 was infinity.&lt;/p&gt;&#xA;&lt;p&gt;The test was set by Richard Dunne, a former teacher and the author of Maths Makes Sense.&lt;/p&gt;&#xA;&lt;p&gt;Dunne said teachers were &amp;quot;thoroughly dedicated&amp;quot;, but argued the test showed that more than half of them understood &amp;quot;so little maths that they cannot be conveying mathematics to their children in the classroom&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Aside: the question about infinity is maybe a little unfair, given that its not on the curriculum. But bright kids will be asking about it.)&lt;/p&gt;&#xA;&lt;p&gt;The problem seems to be that the kind of person who becomes a primary school teacher is often not strong on maths and science; the job tends to attract people who are stronger on the nurturing and caring side, who can bring an understanding attitude to a shoe full of wee. Many of them haven't done any maths since scraping through GCSE (i.e. age 16), and what understanding they had then has only decayed in the intervening years. So being a bit confused about what the technical term &amp;quot;multiplication&amp;quot; means is only to be expected.&lt;/p&gt;&#xA;" OwnerUserId="5194" LastActivityDate="2020-09-05T19:11:57.473" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18780" PostTypeId="1" CreationDate="2020-09-05T20:21:43.707" Score="4" ViewCount="109" Body="&lt;p&gt;Recently I am using &lt;a href=&quot;https://moodle.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moodle&lt;/a&gt; to create problems with randomized parameters. In Moodle, there is so-called Calculated questions that let me do this in a straightforward way. For example, I can simply type: &amp;quot;Evaluate &lt;code&gt;$$\int_{{a}}^{{b}}(x^{2}+{c}x+{d})dx$$&lt;/code&gt;&amp;quot; for which the output is something like: &amp;quot;Evaluate&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{2}^{7}(x^{2}-4x+3)dx.\text{&quot;}$$&lt;/span&gt;&#xA;Here the numbers &lt;span class=&quot;math-container&quot;&gt;$2,7,-4,3$&lt;/span&gt; are generated randomly by Moodle. Thus each student would receive a different problem. Of course, Moodle also allows me to input the answer for this problem: &lt;code&gt;(-2*pow({a},3) - 3*pow({a},2)*{c} - 6*{a}*{d} + {b}*(2*pow({b},2) + 3*{b}*{c} + 6*{d}))/6&lt;/code&gt; so that the students can check their work right after they type down the answer and I can also assess the student's work automatically.&lt;/p&gt;&#xA;&lt;p&gt;However, Moodle only allows me to set &lt;em&gt;one output value&lt;/em&gt; for each problem. For example, I would like to create a problem having several outputs, like: &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^{2}+\sqrt{9-x^{2}}+7$&lt;/span&gt;. The greatest integer number in the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\boxed{\text{a blank box to type the answer here}}$&lt;/span&gt; and the greatest integer number in the range of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\boxed{\text{a blank box to type the answer here}}$&lt;/span&gt;&amp;quot;. Here the numbers &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; are also generated randomly.&lt;/p&gt;&#xA;&lt;p&gt;I do not know how to do this in Moodle. Note that in Moodle there is so-called &lt;em&gt;Embedded answers (Cloze) question&lt;/em&gt; that allows multiple outputs, however it does not allow randomized parameters.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My question&lt;/strong&gt;: How do we create a question like in the above example &lt;em&gt;with randomized parameters&lt;/em&gt; and &lt;em&gt;several outputs&lt;/em&gt; for which we are also able to &lt;em&gt;assess&lt;/em&gt; the student's work automatically?&lt;/p&gt;&#xA;&lt;p&gt;For this, I mean that we can use any platform (not necessarily Moodle). Doing a research I found that we may use some PHP codes to do this in Moodle, see  &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=&amp;amp;ved=2ahUKEwiooLe-6NLrAhXVeisKHbDpAwcQFjAGegQIBxAB&amp;amp;url=http%3A%2F%2Fblogs.exeter.ac.uk%2Fonlineassessment%2Ffiles%2F2012%2F07%2FExamples-of-Mathematics-Eassessment-in-Moodle.docx&amp;amp;usg=AOvVaw1_hVNsUsVD3ouAV7-R2aJd&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. But I haven't got this work on my own, probably due  to a lack of PHP experience (I think with some suggestions it would be alright). Another way is to install a plugin from &lt;a href=&quot;http://www.wiris.com/en&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wiris&lt;/a&gt;, however I am not authorised to do this on my Moodle system.&lt;/p&gt;&#xA;" OwnerUserId="14551" LastActivityDate="2020-09-05T20:21:43.707" Title="Create a mathematical problem with randomized parameters" Tags="|assessment|online-instruction|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18782" PostTypeId="2" ParentId="18772" CreationDate="2020-09-06T03:44:20.273" Score="5" Body="&lt;p&gt;Thoughts:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Student can't go back.  You have to solve each question to the extent you are able, then move on.  If you are unable, you can mark it &amp;quot;return later&amp;quot;  But when you return you get a new version of the problem.  This effectively eliminates the Chegg problem, as there system has too much latency.  This requires submitting your first partial answer, then your second answer later.  After submitting both, you can view both, and choose which one is your real submission.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Distinct versions of the problem mean that for a class of 100 students you will have hundreds of potential versions of the problem.  Chegg works on &amp;quot;reselling&amp;quot; the answer.  That is one person asks the question, one answer is generated, but it's used by a bunch of different students.  This is very effective for textbook questions where there are tens of thousands of copies of popular textbooks, but is less effective with exams, and exams where each question is unique makes Chegg unprofitable.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It also makes Chegg more difficult to use.  Dr. Brown's Final Exam Question #3 isn't the same for all people, so for the Chegg user FINDING the same question even if you have only a limited field of questions becomes and issue.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If the finding is done by image comparison, then having the program have several possible layouts for the question adds more noise to the Chegg system.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Changing the order of questions will also throw a spanner in Chegg's gears. Now my #3 doesn't even look like your #3.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Requiring problems to be worked out means, at best they cannot send the Chegg answer straight away, but have to copy the answer before scanning it.  Otherwise each answer would be in a different handwriting.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Multiple choice questions are easier to randomize.  A test giving webpage can randomize both the order of questions, the order of answers, and possibly change the number of possible answers.  A clever one can leave &amp;quot;None of the above&amp;quot; as the last answer, and &amp;quot;All of the above&amp;quot; as the second last answer.  Again problems can be generated so that each test is unique.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This sort of approach is an easy fallout from well done computer mediated instruction.&lt;/p&gt;&#xA;&lt;p&gt;Last time I looked computer math instruction was slightly awful, with students moving thorugh a rigid lock step format.  A good CMI math program would generate problems of appropriate difficulty for the student's ability at that time, and be able to understand the student's work.  As the student improved, he'd be encouraged to use shortcuts, do multiple steps at once, but be flagged immediately when a shortcut was mis-applied or didn't work.&lt;/p&gt;&#xA;&lt;p&gt;It also would be able to analyze mistakes to see if there were bugs in the student's software.  (A study of the mistakes made by primary kids learning subtraction found that most errors were software errors.  E.g. Doing the borrow to add 10 to the top digit, but forgetting to take it form the left digit.  Or always subtracting the smaller digit from the larger regardless of positions.&lt;/p&gt;&#xA;" OwnerUserId="2114" LastEditorUserId="2114" LastEditDate="2021-02-24T19:37:31.470" LastActivityDate="2021-02-24T19:37:31.470" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18783" PostTypeId="1" AcceptedAnswerId="18789" CreationDate="2020-09-06T13:13:30.780" Score="3" ViewCount="448" Body="&lt;p&gt;For each mathematical subject on the undergraduate level there are many textbooks, often with quite different approaches to the subject. Some are just concise and rigorous, some focus on examples, some on the historical development of the subject, some on intuitive pictures, and so on. When designing a course, one of course wants to find the most illuminating explanations, examples, and pictures that help students to learn the subject. But how does one go about browsing through this vast literature to find them?&lt;/p&gt;&#xA;&lt;p&gt;Of course one can't read each text book, the time just isn't there. How much should one focus on exploring the literature for better explanations? Or should one focus on coming up with explanations and examples by oneself? (Maybe if one comes up with the explanations oneself rather than taking them from a text book one can convey them to the students more easily?)&lt;/p&gt;&#xA;&lt;p&gt;I admit this question is very much opion-based, but I would be interested if there's something like a 'consensus' among math educators.&lt;/p&gt;&#xA;" OwnerUserId="14543" LastActivityDate="2020-09-07T13:53:24.083" Title="How much literature research should one do when designing a course?" Tags="|undergraduate-education|textbooks|course-design|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18784" PostTypeId="1" AcceptedAnswerId="18802" CreationDate="2020-09-06T14:34:07.317" Score="2" ViewCount="666" Body="&lt;p&gt;My kid is soon 7 years old, he could understand fractions, linear equation and modulo operation. I've just taught him Chinese remainder theorem, looking to introduce some more basic number theory stuff.&lt;/p&gt;&#xA;&lt;p&gt;One of the topics sound interesting is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclidean_algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclidean algorithm&lt;/a&gt;, together with the &lt;a href=&quot;https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bézout's identity&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;p&gt;Given two coprime numbers &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, i.e. &lt;span class=&quot;math-container&quot;&gt;$(m,n)=1$&lt;/span&gt;, there exist &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ Pm-Qn=1$$&lt;/span&gt;. The explicit construction of Bézout's identity can be done via &lt;a href=&quot;https://en.wikipedia.org/wiki/Continued_fraction&quot; rel=&quot;nofollow noreferrer&quot;&gt;continued fraction&lt;/a&gt; that if &lt;span class=&quot;math-container&quot;&gt;$$\frac{m}{n}=[a_0;a_1,\dots,a_s],$$&lt;/span&gt;&#xA;then &lt;span class=&quot;math-container&quot;&gt;$$\frac{Q}{P}=(-1)^{s-1}[a_0;a_1,\dotsc,a_{s-1}].$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This construction method could be proved using &lt;a href=&quot;https://en.wikipedia.org/wiki/Continued_fraction#Properties&quot; rel=&quot;nofollow noreferrer&quot;&gt;Continued fraction's properties&lt;/a&gt;. However, I'm afraid this is a bit too much to my kid, with 2 sequences, and algebra manipulation.&lt;/p&gt;&#xA;&lt;p&gt;Is there a way I could adopt to prove it, using a simpler approach? Or is there another way to construct Bézout's identity, without continued fraction but also easy to understand?&lt;/p&gt;&#xA;" OwnerUserId="9059" LastEditorUserId="9059" LastEditDate="2020-09-19T17:04:12.413" LastActivityDate="2022-10-03T18:56:20.650" Title="How can I explain construction of the Bézout's identity to my kid?" Tags="|number-theory|" AnswerCount="5" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="18785" PostTypeId="1" CreationDate="2020-09-06T17:48:41.740" Score="3" ViewCount="969" Body="&lt;p&gt;I have found Engelmann’s book (mentioned in subject) to be extremely effective.  Is there an equivalent to this book for teaching Arithmetic?  I believe the overall approach or method is called Direct Instruction.&lt;/p&gt;&#xA;" OwnerUserId="14557" LastActivityDate="2022-08-07T21:06:16.213" Title="Is there an arithmetic book similar to “Teach Your Child to Read in 100 Easy Lessons” by Siegfried Engelmann?" Tags="|textbooks|primary-education|curriculum|teaching|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18786" PostTypeId="2" ParentId="18772" CreationDate="2020-09-06T19:39:05.233" Score="3" Body="&lt;p&gt;I hold the opinion that preventing or disincentivizing cheating is practically impossible.&lt;br /&gt;&#xA;If I want to quiz my students about partial fractions or trigonometric integrals or the sorts, I can’t prevent them from putting it into WolframAlpha or the like by just changing the phrasing a bit, because WolframAlpha can solve partial fraction integrals.&lt;br /&gt;&#xA;Instead, I think the best course of action is to make the students less frustrated and thus, less vulnerable to cheating.&lt;br /&gt;&#xA;Here are the things I see to reduce student frustration.&lt;br /&gt;&#xA;&lt;em&gt;&lt;strong&gt;Hard &lt;span class=&quot;math-container&quot;&gt;$\not \Leftrightarrow$&lt;/span&gt; frustrating&lt;/strong&gt;&lt;/em&gt;&lt;br /&gt;&#xA;Actually, quite the contrary, a student will be less frustrated by 3 hard integrals then 20 easy ones.&lt;br /&gt;&#xA;&lt;em&gt;&lt;strong&gt;Don’t fight&lt;/strong&gt;&lt;/em&gt;&lt;br /&gt;&#xA;Students who already cheat are practically non-recoverable.(after all, it only gets easier after the first time)&lt;br /&gt;&#xA;A student won’t &lt;em&gt;stop&lt;/em&gt; cheating because the he couldn’t backtrack in the test, but, if you do prevent backtracking, you are frustrating the students who don’t cheat.&lt;br /&gt;&#xA;I have elaborated on this here: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18477/what-are-the-best-practices-for-giving-online-tests/18478#18478&quot;&gt;What are the best practices for giving online tests?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14192" LastActivityDate="2020-09-06T19:39:05.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18787" PostTypeId="2" ParentId="18772" CreationDate="2020-09-07T04:15:31.087" Score="7" Body="&lt;p&gt;One of the things I'm currently testing is the addition of &amp;quot;find the error&amp;quot; based problems, and possible variations. For example (related but not the same topic) last semester I taught indefinite integrals, so in a partial test I decided to ask my students the following problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The integral &lt;span class=&quot;math-container&quot;&gt;$$\int \dfrac{3-x}{(2+x)^2}~dx$$&lt;/span&gt; can be solved in several ways. For example, if you try partial fractions, you can obtain &lt;span class=&quot;math-container&quot;&gt;$$-\dfrac{5}{x+2} - \ln |x+2| + C.$$&lt;/span&gt; But if you perform integration by parts, the answer is &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{x-3}{x+2} - \ln |x+2| + C.$$&lt;/span&gt; I ask my students to check both results, and to explain why the same integral is giving two 'different' solutions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Do I completely remove cheating? Absolutely no. But in this case, Wolfram solves the integral by partial fractions and coincidently a lot of my students felt overwhelmed by integration by parts.&lt;/p&gt;&#xA;" OwnerUserId="3380" LastEditorUserId="8571" LastEditDate="2020-09-07T13:19:38.837" LastActivityDate="2020-09-07T13:19:38.837" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18788" PostTypeId="2" ParentId="176" CreationDate="2020-09-07T06:12:09.330" Score="1" Body="&lt;p&gt;From my understanding if you aren't consciously learning to teach then you just &amp;quot;learn&amp;quot; things without really concentrating upon the various aspects associated with it. For example there are two distinct universes, mathematical and this real one. Mathematical Universe has things which are a representation of real world things. A teacher must know that for a student to understand and place a new knowledge properly in the Mathematical Universe, student must know the connections of this new thing with things ALREADY known, so that a student is able to properly place this new knowledge. For an experienced teacher it is very easy to move between Abstract Mathematical world and Real Physical World but not at all for a student. So the connection between the things in Mathematical Universe and Real Universe must be made very clearly such that student is able to follow his/her teacher in both the universes.&lt;/p&gt;&#xA;&lt;p&gt;Also if a student is not able to bring himself into the Mathematical Universe then the teacher must realise that and take Mathematical Knowledge to the Student's world and explain Math using things in student's world in a language student understands(language here means manner of speaking). Different students understand things differently based on so many things like personality, social background etc. Identifying all of this has got nothing to do with &amp;quot;knowledge of subject&amp;quot; rather a human art form.&lt;/p&gt;&#xA;&lt;p&gt;A person good in Mathematics may be lacking in soft skills required to communicate the knowledge to the student. That's why good knowledge of subject doesn't necessarily translate into good teacher of the subject...&lt;/p&gt;&#xA;&lt;p&gt;[EDIT]&lt;/p&gt;&#xA;&lt;p&gt;Because not all are good story tellers. It's the stories kids understand and like, so the stories we tell them must be easy &amp;amp; accurate.&lt;/p&gt;&#xA;" OwnerUserId="12610" LastEditorUserId="12610" LastEditDate="2021-09-12T05:12:50.403" LastActivityDate="2021-09-12T05:12:50.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18789" PostTypeId="2" ParentId="18783" CreationDate="2020-09-07T13:53:24.083" Score="4" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You should know the assigned text perfectly.  Read it.  Drilled it.  Four dot oh.  Don't ignore it.  After all, it was designed to mesh with the drill problems in it.  And if you have the choice, assign a structured text with good explanations, and drill problems.  NOT the approach of some liberal arts topic where you just opine on things and assign a bunch of articles or don't have a strong text as a backbone.  This will get you to the 80% level.  (Note: I'm not saying to recite the book, but you can compose examples that are different numerically or decide what to emphasize/not.  Based mostly on the flow of the text.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Know at least one other text reasonably well (probably the one you had as a student).  Use it as your cheat comparison.  Really, at that point, having two things to compare, you can see differences and similarities and are at the 90%.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you have one or two more, checked out from the library, that's enough. Don't feel you need to consult them often at all.  Just where you have a real question or if dissatisfied with 1&amp;amp;2.  (An example might be repeated roots in linear constant coefficient homo diffyQs.)  And if still dissatisfied, I would argue to &lt;strong&gt;move on&lt;/strong&gt;.  Remember teaching and learning are processes, not Euclidean proofs.  You have to prioritize your time and effort.  Think industrial engineering, not Andrew Wiles in an attic for 7 years.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14559" LastActivityDate="2020-09-07T13:53:24.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18790" PostTypeId="1" CreationDate="2020-09-07T16:09:13.200" Score="3" ViewCount="123" Body="&lt;p&gt;This might be a more appropriate question for math.stackexchange, but it's about a problem I'm considering giving my students, so here it goes.&lt;/p&gt;&#xA;&lt;p&gt;One of the later exercises in Section 7.4 of James Stewart's Calculus textbook (this section is on partial fraction decomposition) states the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The rational number &lt;span class=&quot;math-container&quot;&gt;$\frac{22}7$&lt;/span&gt; has been used as an approximation for the number &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; since the time of Archimedes. Show that &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_0^1 \frac{x^4(x-1)^4}{x^2+1}\,dx = \frac{22}7-\pi.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, this isn't a particularly difficult calculation; a fairly straightforward polynomial long division gives you that this integrand is &lt;span class=&quot;math-container&quot;&gt;$x^6 - 4x^5 + 5x^4 - 4x^2 + 4 - \frac{4}{x^2+1}$&lt;/span&gt;, and that's pretty easy to integrate. But the later exercises in Stewart are usually about some specific insight or deeper conceptual illustration. And mentioning the &lt;span class=&quot;math-container&quot;&gt;$\frac{22}7$&lt;/span&gt; approximation for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is pretty specific. But what insight is this supposed to give students? Is there a more clever way to solve this problem that uses geometry or numerical approximation in some specific way? Or is this really just a simple rational function integration problem with a little extra dressing? To put it bluntly, what's the point of this particular problem?&lt;/p&gt;&#xA;" OwnerUserId="14562" LastActivityDate="2020-09-07T19:24:06.300" Title="The purpose of a particular rational function integration exercise" Tags="|integration|" AnswerCount="1" CommentCount="8" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18791" PostTypeId="2" ParentId="18785" CreationDate="2020-09-07T17:37:42.617" Score="3" Body="&lt;p&gt;I decided to write my reply as an answer because I feel strong about both reading instruction and early math education. I hope that neither the original question nor this reply will be edited, because the question is very typical to how many approach education. But feel free to skip the rant below to the recommendation of a book.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Listicles in education&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The listicle-style title of the reference book is a tell-tale sign of a cookbook-style instruction. Listicles are fine for presenting a collection of disparate unrelated items, like 50 Italian pasta recipes or 100 best American poems. But when used in education, this approach forcefully squeezes or more often extends the subject matter and pedagogy into a Procrustes bed of arbitrary number of steps. This usually results in a bloated course with boring, repetitive and often nonsensical steps students have to follow.&lt;/p&gt;&#xA;&lt;p&gt;Engelmann's book is 400-page long. Does it really take this long to learn to read? The book refers to &amp;quot;Why Johnny Can't Read&amp;quot; by Rudolf Flesch as an inspiration. Well, Flesch's is a good book. The first two thirds of it is a seething critique of Look-Say / Whole Language / Balanced Literacy approach, and the last twenty pages or so is a compact yet very methodical primer to teach reading phonetically. His book is a much better choice both as an insight into phonetic reading instruction as well as a reading primer.&lt;/p&gt;&#xA;&lt;p&gt;Engelmann's book, on the other hand, is bloated with repetitive &amp;quot;direct instruction&amp;quot; script, but what ruins it completely in my opinion is non-standard grammar. To represent sounds the book uses symbols that are neither from English alphabet nor from International Phonetic Alphabet. The pronunciation guide recursively defines these funky symbols using the same symbols, like &amp;quot;ē&amp;quot; must be pronounced as &amp;quot;ēēē&amp;quot; (IPA symbol is /iː/), &amp;quot;sh&amp;quot; is pronounced as &amp;quot;shshsh&amp;quot; (IPA symbol is /ʃ/), &amp;quot;th&amp;quot; is pronounced as &amp;quot;ththth&amp;quot; (IPA symbol is /ð/ for voiced and /θ/ for voiceless consonant). Such nonsense. The symbol &amp;quot;ɪ&amp;quot; is used to represent the same sound as in the word &amp;quot;I&amp;quot; with complete disregard that in IPA the same symbol represents a sound like in &amp;quot;bit&amp;quot;. Instead of giving clear instructions how to read open syllables in words like &amp;quot;take&amp;quot; (for example, &amp;quot;letter 'e' at the end makes preceding vowel sound like it is spelled in alphabet&amp;quot;) Engelmann's book diminishes the terminating &amp;quot;e&amp;quot; and puts a bar over &amp;quot;A&amp;quot; to indicate that it is a long vowel, like &amp;quot;TĀKe&amp;quot;. &amp;quot;A&amp;quot; in the word &amp;quot;take&amp;quot; sounds like /eɪ/ not like /ɑː/, throwing off anyone who may think that the bar means simply elongated sound.&lt;/p&gt;&#xA;&lt;p&gt;What does the bar mean then? The book uses the bar for words like &amp;quot;for&amp;quot;, &amp;quot;shore&amp;quot; and &amp;quot;go&amp;quot;, writing them as &amp;quot;&lt;strong&gt;fōr&lt;/strong&gt;&amp;quot;, &amp;quot;&lt;strong&gt;shōr&lt;/strong&gt;e&amp;quot; and &amp;quot;&lt;strong&gt;gō&lt;/strong&gt;&amp;quot;, but is the &amp;quot;o&amp;quot; pronounced the same in these words? Regional dialects may be different, it can be something like /fɔr/, /ʃɔːr/ and /goʊ/ — these are three different sounds: &amp;quot;short o&amp;quot;, &amp;quot;long o&amp;quot;, and a diphthong. Between inconsistent symbols and non-standard grammar, how kids are supposed to switch to regular English? It does not have bars over letters. Does the book implicitly suggest that kids will memorize the words with the bar so they would be able to read them later without the bar? How is this better than explaining how the sound changes depending on other letters around?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RNKrj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RNKrj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are many books like this intended for math education. Their writers are often former elementary school teachers who have discovered a well-known practice on their own, and being happy of this amazing achievement they want to share their newly-found knowledge with everyone else, giving it a new fancy name in the process. Clearly, the job of a consultant in education pays better than of a regular teacher and comes with no burden of creating a year-long course or grading homework. But because just one or two successful practices &amp;quot;discovered&amp;quot; by those &amp;quot;experts&amp;quot; are not enough for a book, they invent intermediate steps to bloat the content, to add more pages, to get the clout of respectability through sheer weight and number of pages. (Does my previous sentence sound more trustworthy because of the repetition?)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My book recommendation&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I recommend &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0977985253&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arithmetic For Parents&lt;/a&gt; by Ron Aharoni as a good starting point in teaching your child the basics of arithmetics. Ron taught at college, but then decided to try teaching young kids in elementary school. He had to find pedagogical approach different from teaching in college, but this gave him an insight to how young kids should be taught. In short, from concrete to abstract, without skipping even the smallest step.&lt;/p&gt;&#xA;&lt;p&gt;Really, nothing new, most stuff in the book is common knowledge, but at least he does not give these centuries-old practices new names. It is not a cookbook, but it does have specific teaching advice. It does not suggest how to cut and arrange flash cards for addition facts or multiplication table, but it talks about what number is, what operations mean and why well-known algorithms work.&lt;/p&gt;&#xA;&lt;p&gt;Also I recommend taking a look at &lt;a href=&quot;https://www.youtube.com/channel/UC7uSOIJi4mA2CCNE15ddTCA&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math-U-See&lt;/a&gt; course. I have never bought any products from them, I just watched their videos on YouTube, and they served as inspiration to helping my own son learn math. I did not homeschool him, but I augmented whatever he had in school with my own instruction. For example, here is a &lt;a href=&quot;https://www.youtube.com/watch?v=dJaqQe5ZoIw&quot; rel=&quot;nofollow noreferrer&quot;&gt;Place Value workshop&lt;/a&gt; — mesmerizing!&lt;/p&gt;&#xA;&lt;p&gt;If you are not swamped by your day job and have time to devise math games and playdates and contests and whatnot, I recommend &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/082186873X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers&lt;/a&gt;, written by a Russian parent who ran a &amp;quot;math circle&amp;quot; for his children and children of his friends for several years.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2020-09-23T19:22:52.680" LastActivityDate="2020-09-23T19:22:52.680" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18792" PostTypeId="2" ParentId="18790" CreationDate="2020-09-07T18:42:17.417" Score="2" Body="&lt;p&gt;This is a classic example of transforming an approximation problem from the real line to the polynomial one.&lt;br /&gt;&#xA;Let’s image we found a really good approximation, &lt;span class=&quot;math-container&quot;&gt;$p(x)\approx\frac{4}{x^2+1}$&lt;/span&gt;.&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int^1_0 p(x)-\frac{4}{x^2+1} = p-\pi$$&lt;/span&gt;&#xA;Means that &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is a really good approximation of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;br /&gt;&#xA;The natural conclusion is to select a few points from &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{x^2+1}$&lt;/span&gt; in the range and use the polynomial gotten as &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; to get a &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; approximation.&lt;br /&gt;&#xA;In conclusion, if &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; is a good approximation of &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{x^2+1}$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\int^1_0 p(x)$&lt;/span&gt; is a good approximation of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;br /&gt;&#xA;Note: &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; need only be a good approximation in &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14192" LastEditorUserId="14192" LastEditDate="2020-09-07T19:24:06.300" LastActivityDate="2020-09-07T19:24:06.300" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18793" PostTypeId="1" AcceptedAnswerId="18794" CreationDate="2020-09-07T23:24:04.437" Score="9" ViewCount="3218" Body="&lt;p&gt;I am a math teacher for sixth graders and I am trying to think of some strategies to keep the students who finish their work quickly productively occupied. I would like to have a selection of activities - including games, projects, etc. Any ideas of what I can include? Maybe games like &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Set_(card_game)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Set&lt;/a&gt;&lt;/em&gt;? Or Q-bitz? What else can I include?&lt;/p&gt;&#xA;" OwnerUserId="12529" LastEditorUserId="851" LastEditDate="2020-09-09T17:39:25.420" LastActivityDate="2021-06-10T18:12:27.703" Title="Math activities for fast-finishers" Tags="|classroom-management|usa|games|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18794" PostTypeId="2" ParentId="18793" CreationDate="2020-09-08T02:02:17.113" Score="18" Body="&lt;p&gt;Stuff for fast finishers/gifted kids should be out of the stream --stuff that isn't what you're covering next week or next year.&lt;/p&gt;&#xA;&lt;p&gt;Look at puzzles -- especially geometrical ones.  Graph theory -- Bridges of Konigsburg, three houses, three wells.  Chess board problems.  There are lots of good puzzlebooks out there.  Logic puzzles too.&lt;/p&gt;&#xA;&lt;p&gt;Mechanical puzzles too.  Rubik's cube, sliding block puzzles. Ring and string puzzles.  You will have to figure out how to keep them from wandering away permanently.&lt;/p&gt;&#xA;&lt;p&gt;Scientific American had a whole series of books, mostly extracts of the columns of Martin Gardener of puzzles and activities.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Paper and scissors activities.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Properties of a mobius strip.&lt;/li&gt;&#xA;&lt;li&gt;hexaflexagon. (Use adding machine tape)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Number theory conjectures:&lt;/p&gt;&#xA;&lt;p&gt;Pick a positive whole number.  If odd, triple it and add 1.  If even cut in half.  Repeat.&lt;/p&gt;&#xA;&lt;p&gt;E.g. 3 -&amp;gt; 10 -&amp;gt; 5 -&amp;gt; 16 -&amp;gt; 8 -&amp;gt; 4 -&amp;gt; 2 -&amp;gt; 1&lt;/p&gt;&#xA;&lt;p&gt;Any number that results in 4-2-1 is said to be wonderous.  Are there any non-wonderous numbers?  Write a program to test numbers.  Is there any pattern to how many steps it takes to reach 4-2-1&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Can any even natural number be expressed as the sum of two primes? (You need to consider 1 a prime for it to work with 2.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Can any natural number be expressed as the difference of two cubes?  Why is this one intrinsically harder?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For something bigger, get them started on geometry, follow that up with coordinate geometry.&lt;/p&gt;&#xA;" OwnerUserId="2114" LastEditorUserId="2114" LastEditDate="2020-09-13T14:23:33.530" LastActivityDate="2020-09-13T14:23:33.530" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18795" PostTypeId="1" CreationDate="2020-09-08T05:15:27.597" Score="4" ViewCount="705" Body="&lt;p&gt;Here I mean a differential equation of the form &lt;span class=&quot;math-container&quot;&gt;$y'=f(x,y)$&lt;/span&gt; where for some &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$f(tx,ty)=t^\alpha f(x,y)$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;. I have no idea why this topic seems to appear in every ODE curriculum. Are there important applications of such equations, or is the idea perhaps simply to collect one a few cases in which we can actually solve some nonlinear equations?&lt;/p&gt;&#xA;" OwnerUserId="5233" LastEditorUserId="376" LastEditDate="2020-09-08T11:40:03.450" LastActivityDate="2020-09-16T02:56:46.387" Title="Why are &quot;homogeneous differential equations&quot; in the standard ODE curriculum?" Tags="|undergraduate-education|curriculum|differential-equations|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18796" PostTypeId="2" ParentId="18793" CreationDate="2020-09-08T10:13:01.547" Score="6" Body="&lt;p&gt;I taught gifted students and had many fast finishers.&#xA;Here are some of things we did (most require no preparation on your part but some require resources that you may not have):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I kept a supply of &lt;a href=&quot;https://www.thinkfun.com/&quot; rel=&quot;noreferrer&quot;&gt;Think Fun&lt;/a&gt; puzzles.  These are solitaire&#xA;brain teasers that the students enjoyed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some of my classes had access to tablets/computers and the students&#xA;were allowed to work om Khan Academy polishing their skills, by&#xA;working on what I assigned. There were also games on &lt;a href=&quot;https://illuminations.nctm.org/Search.aspx?view=search&amp;amp;type=ac&amp;amp;gr=6-8&quot; rel=&quot;noreferrer&quot;&gt;NCTM&#xA;Illuminations&lt;/a&gt; that I allowed them to play&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;My classes had hand-held &lt;a href=&quot;https://flashmaster.com/handheld-flashmaster/&quot; rel=&quot;noreferrer&quot;&gt;Flashmasters&lt;/a&gt; - the&#xA;students practiced their facts on them. Those students who used them&#xA;regularly developed amazing speed and accuracy. Note they now have&#xA;an app instead of the hand-held devices.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The students were&#xA;given 5 dice and they had to use the 5 numbers that they spun to&#xA;make a target number (usually 100).  They had to use each number&#xA;exactly once and were allowed to use: decimal points, the four&#xA;operations, exponents, parentheses, and factorial. They were also&#xA;allowed to combine two of the numbers to make a two digit number&#xA;(e.g. if they spun a 3 and a 6, they could make 36 or 63).  The&#xA;students had to write out their expression using order of&#xA;operations.  This was a big hit with the students. Alternatively you can the game 24 and they can record their answers using order of operations. 24 is quieter than dice.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The students worked on puzzles from &lt;a href=&quot;https://www.moems.org/&quot; rel=&quot;noreferrer&quot;&gt;MOEMs&lt;/a&gt; (math olympiads for&#xA;elementary and middle schools) and from &lt;a href=&quot;https://www.mathcounts.org/&quot; rel=&quot;noreferrer&quot;&gt;Math Counts&lt;/a&gt;.  These are&#xA;two different competitions for students of this age.  You might&#xA;consider entering a team in either competition or you can just give them&#xA;the problems for extra challenge.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I kept a collection of math books that students enjoyed in my classroom.  I would particularly recommend anything by Brian Bolt (e.g. &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B001CSBHCE&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Arcade&lt;/a&gt;), &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B00D6KEHWI&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Boy Who Loved Math&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0316117412&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;I Hate Math&lt;/a&gt; by Marilyn Burns (look for others by Marilyn Burns), &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/081262744X&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Cat in Numberland&lt;/a&gt;.  There are many others, but Brian Bolt writes great books of activities and if you get one book, I suggest you get one of his and assign some of the activities to your students.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Some words of caution - some of these activities are a lot of fun (e.g. Think Fun games)  and you don't want students to rush work so that they can do these. I would usually check their work before allowing them to enjoy these activities, but if you have a big class, that might not be possible.  You will have to judge in that case if you should avoid the activities that are the most fun.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-09-08T10:13:01.547" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18797" PostTypeId="2" ParentId="18795" CreationDate="2020-09-08T12:38:58.223" Score="2" Body="&lt;p&gt;My copy of Speigel (which has separate chapters on applications, after each theory/calculation chapter) has one problem in &amp;quot;geometry&amp;quot; (really ray optics, finding curve that gives parallel reflection of a point source) that results in a homo first order (non separable) ODE.  I didn't check overall first order applictions chapter, but it seems like there were very few such applications (I didn't see any more) as compared to applications resulting in integration by an integrating factor or separation of variables.&lt;/p&gt;&#xA;&lt;p&gt;So, yeah, I don't think it's some classic application like the 2nd order ODE with constant coefficients (control circuits, and many others).  More like just a trick to learn since you can.  With occasional, rare, reasons to pull it out of the toolbox.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I know that Rota's essay is engaging and often cited here, but would recommend you to take it with a grain of salt or at least think about what parts to agree/disagree with.  For examples, there are many physics or engineering derivations that include solution of an ODE with an integrating factor (I know this since I just went through the applicable applications chapter or Spiegel!)  On the other hand, he is dead on with the comments about 2nd order ODE with constant coefficients (which homo and nonhomo) is often included in calculus 2 texts (was when I took AP BC). On the gripping hand, he takes a sideways slice at the old Cambridge Tripos, that the winners never did anything (First this is silly to judge a set/system only by outright winners.  Second Wikipedia gives a quick &amp;quot;Many Senior Wranglers have become world-leading figures in mathematics, physics, and other fields. They include George Airy, John Herschel, Arthur Cayley, James Inman, George Stokes, Isaac Todhunter, Morris Pell, Lord Rayleigh, Arthur Eddington, J. E. Littlewood, Jayant Narlikar, Frank Ramsey, Donald Coxeter, Jacob Bronowski, Lee Hsien Loong, Kevin Buzzard, Christopher Budd, Ben Green, and John Polkinghorne.&amp;quot;, to which I'd argue adding Forsyth.)&lt;/p&gt;&#xA;&lt;p&gt;Also, check out the Amazon reviews for his textbook.  Very sloppy work.  Maybe a little more attention to detail and less attention to lofty philosophy or being new, would have helped Rota.  I suspect a cross-section of average ODE students would get more out of a Schaum's Outline than out of Rota's slipshod effort.  This isn't just a slam...the point is that for pedagogical efficiency, it is important to be accurate.  The more grist in the mill, the more it hiccups.  This, in contrast to the constant emphasis here on debating coverage.&lt;/p&gt;&#xA;&lt;p&gt;See also: &lt;a href=&quot;https://mathoverflow.net/questions/235698/undergraduate-ode-textbook-following-rota&quot;&gt;https://mathoverflow.net/questions/235698/undergraduate-ode-textbook-following-rota&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14568" LastEditorUserId="14568" LastEditDate="2020-09-08T13:39:41.410" LastActivityDate="2020-09-08T13:39:41.410" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18798" PostTypeId="2" ParentId="18793" CreationDate="2020-09-08T13:16:23.610" Score="11" Body="&lt;p&gt;One option is to assign tasks with a low floor, but high ceiling. That means that almost everyone has the background to start tinkering, and accomplish something, but there is enough complexity and variation that one could keep discovering new things for a very long time. This naturally differentiates instruction.  Engaging tasks of this form are hard to craft, and find, however.&lt;/p&gt;&#xA;&lt;p&gt;An example:  When teaching about area, you could assign the task &amp;quot;How many 2 inch by 3 inch rectangles can fit in a 60 inch by 81 inch rectangle?&amp;quot;.  Progression of associated tasks:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;For students just beginning to learn about area, they might use graph paper and count the number of rectangles by hand.&lt;/li&gt;&#xA;&lt;li&gt;More advanced students might jump immediately to using division.  For them, you can modify the problem to 61 inches by 84 inches.  Now the small rectangles do not tile evenly:  or do they?  Try different arrangements to see how many you can fit.  Still &amp;quot;easy&amp;quot; to answer if you allow &amp;quot;fractional tiles&amp;quot;, but not if you restrict yourself to whole tiles (note:  this will not be easy if knowledge of fractions still needs work, and hence is a good task in 6th grade).&lt;/li&gt;&#xA;&lt;li&gt;If a student solves the previous task to their satisfaction, you can ask them to change the size of the board and the size of the tile.  Can they find a formula or procedure to find the maximum number of tiles which fit?&lt;/li&gt;&#xA;&lt;li&gt;What if you have access to two different sized tiles?  What if your tiles or board are not rectangular?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;.&#xA;.&#xA;.&lt;/p&gt;&#xA;&lt;ol start=&quot;100&quot;&gt;&#xA;&lt;li&gt;This train of thought can keep you occopied for a whole lifetime. &lt;a href=&quot;https://www.ics.uci.edu/%7Eeppstein/junkyard/polyomino.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.ics.uci.edu/~eppstein/junkyard/polyomino.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Some students might spend their entire class period stuck at 1, or progressing from 1 to 2.  This would be time well spent.  The fast finishers would have plenty to chew on, which could occupy them for the rest of the class period, possibly the rest of the semester, or (depending on how hard these problems end up being) the rest of their lives as professional mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;I just came up with this, so I do not know the answer to question 3 or 4.  They have the flavor of something I should be able to solve with elementary number theory (linear diophantine equations), but I have never tackled these particular problems. In particular, the geometric considerations make this possibly much harder than I would anticipate (how do you decide when to lay a tile vertically or horizontally?).  It could certainly be fun to spend some time thinking about them whether you were a 6th grader or a professional mathematician.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-09-08T13:16:23.610" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18800" PostTypeId="1" CreationDate="2020-09-08T17:19:41.977" Score="6" ViewCount="304" Body="&lt;p&gt;There are some students in freshman calculus/even precalculus who compulsively use the three dots &lt;span class=&quot;math-container&quot;&gt;$\therefore$&lt;/span&gt; in every single step: &lt;a href=&quot;https://en.wikipedia.org/wiki/Therefore_sign&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Therefore_sign&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's not &amp;quot;wrong&amp;quot; really, it just feels odd. I was never taught this sign in any class growing up. It seems like so much extra work and it's not necessary. Where do students get this idea from? Should it be discouraged?&lt;/p&gt;&#xA;" OwnerUserId="14571" LastEditorUserId="3593" LastEditDate="2020-09-19T05:24:00.980" LastActivityDate="2020-09-19T05:24:00.980" Title="Where does the compulsive use of three dots come from and should it be discouraged?" Tags="|undergraduate-education|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18801" PostTypeId="1" AcceptedAnswerId="18805" CreationDate="2020-09-08T17:45:36.143" Score="29" ViewCount="4328" Body="&lt;p&gt;I teach an introductory discrete mathematics course at a community college to math and computing majors, usually in their sophomore year. As is common, it's partly used as the first foray into formal and proof-based math courses in the case that they later proceed to a bachelor's program.&lt;/p&gt;&#xA;&lt;p&gt;Probably the most abstract topic that we get to are binary relations, culminating in identifying and proving that things are equivalence relations. This is particularly hard for our students, and perhaps once or twice I've had students ask the question, &amp;quot;What's the point of learning equivalence relations?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Now, obviously I'm familiar with reading more advanced math texts, being told that something is an equivalence relation, and grokking the rather large number of facts that have just been communicated. But I'm accustomed to being able to answer these &amp;quot;why this?&amp;quot; questions pretty well with specific applications in most cases, whereas in this case I'm drawing a total blank. The book has no example of applications in this case (and I'm not sure what any such would look like).&lt;/p&gt;&#xA;&lt;p&gt;So: What's a &lt;strong&gt;good response to a student in their first semi-formal discrete math course&lt;/strong&gt; asking, &amp;quot;What's the point of learning equivalence relations?&amp;quot; Are there any good applications or use-cases where knowing something is an equivalence relation makes some particular work feasible or easier in some sense (esp. in computing)?&lt;/p&gt;&#xA;&lt;p&gt;(Side note: Up until this semester, I had one lecture on n-ary relations in the context of DBMS's, but I've had to cut that out due to credit-hour reductions in our program. Even when I had that there wasn't any explicit connection to the equivalence relation concept.)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-09-09T16:02:19.317" LastActivityDate="2021-12-20T02:02:15.820" Title="What's the point of learning equivalence relations?" Tags="|concept-motivation|community-colleges|discrete-math|computer-science|" AnswerCount="15" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18802" PostTypeId="2" ParentId="18784" CreationDate="2020-09-08T19:22:25.850" Score="6" Body="&lt;p&gt;I will answer with an example.  I seek the Bezout coefficients for 99 and 707.&lt;/p&gt;&#xA;&lt;p&gt;First I execute the Euclidean algorithm:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;707 &amp;amp;= 7 \cdot 99+14\\&#xA;  99  &amp;amp;= 7 \cdot 14+ 1 &#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, I will recursively &amp;quot;backtrack&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;1 &amp;amp;= 99-7\cdot 14\\&#xA;&amp;amp;= 99-7 \cdot (707-7 \cdot 99)\\&#xA;&amp;amp;=50 \cdot 99 - 7 \cdot 707&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-09-08T19:22:25.850" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18804" PostTypeId="2" ParentId="18801" CreationDate="2020-09-08T23:28:21.523" Score="-4" Body="&lt;p&gt;I wonder if it is not so much a direct application (in a coding problems) as it is very similar in thought process to being rigorous about your code structure.&lt;/p&gt;&#xA;&lt;p&gt;That, and I think CS likes to dress up the curriculum with some math, mostly not useful for B.S. students.  But it's just part of the dynamic.&lt;/p&gt;&#xA;&lt;p&gt;FWIW, a Google search found some (but not much...definitely have to weed through it for  genuine applications as opposed to &amp;quot;used in math&amp;quot; applications.  But you could sift through.&lt;/p&gt;&#xA;" OwnerUserId="14574" LastActivityDate="2020-09-08T23:28:21.523" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18805" PostTypeId="2" ParentId="18801" CreationDate="2020-09-08T23:53:03.437" Score="28" Body="&lt;p&gt;Perhaps emphasize to students the &lt;em&gt;spirit&lt;/em&gt; of equivalence relations. They partitions sets into equivalence classes--cutting down the amount of cases necessary to prove something.&lt;/p&gt;&#xA;&lt;p&gt;To illustrate this, take a geometry example first. &amp;quot;Is similar to&amp;quot; is an equivalence relation on the set of regular polygons (I'll omit a proof here). Now say I need to prove something about squares. If I don't use the &amp;quot;square&amp;quot; equivalence class, then I have to consider every possible square and make a general proof. But instead, if I frame my proof in such a way that respects the &amp;quot;is similar to&amp;quot; relation, then I no longer need to prove it for all squares, but just any single one. This cuts down the amount of cases significantly.&lt;/p&gt;&#xA;&lt;p&gt;In programming, we should constantly test our code in as many conditions as possible. But for many functions, there are a potentially infinite number of inputs--far too many to run unit tests for in reasonable time. Instead, if we look at our tests through the lens of equivalence classes, we can usually cut down our test cases into a finite (and often small) number of cases, and still feel confident that it will run correctly in all contexts. Granted, this often more easily said than done, but it usually leads to more coherent unit tests and frankly more coherent code.&lt;/p&gt;&#xA;&lt;p&gt;Say, going back to our regular polygons example, I am working on a geometry library. I have a function that calculates the angles of a regular polygon. Without thinking about equivalence classes, I would have to test EVERY square, EVERY regular pentagon, EVERY regular hexagon, etc... This is an uncountably infinite number of tests. No one has enough compute or patience for that.&lt;/p&gt;&#xA;&lt;p&gt;Instead, since I know that angles are respected by the equivalence classes under similarity (since that's part of the definition of similarity), I only need to check one of each shape, and I can be confident that it will work for every instance of that similar shape.&lt;/p&gt;&#xA;&lt;p&gt;Even better, we can actually use that mindset to guide our coding process! If we know that similarity preserves angle, we can simply acknowledge that in our code. Instead of calculating the angles between lines (which is complicated in general), we can simply use the properties of squares/pentagons/whatever, which is a much simpler process. If we're even more clever, we can reduce this down to a single arithmetic statement.&lt;/p&gt;&#xA;&lt;p&gt;So, equivalence relations partition sets of inputs into cases. In pure math, these cases guide our proofs, but even in coding, they can guide our testing. If we start through this lens, instead of retroactively applying it, it can even simplify our code.&lt;/p&gt;&#xA;" OwnerUserId="14520" LastEditorUserId="14520" LastEditDate="2020-09-09T21:22:54.480" LastActivityDate="2020-09-09T21:22:54.480" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18806" PostTypeId="1" CreationDate="2020-09-09T00:49:38.207" Score="2" ViewCount="140" Body="&lt;p&gt;The analogy for cross-sections is easy since we can think of how slices of bread can make up a loaf.&lt;/p&gt;&#xA;&lt;p&gt;But what would be the analogy for cylindrical shells?&lt;/p&gt;&#xA;&lt;p&gt;Regarding shapes, apparently there's something called a &amp;quot;bundt&amp;quot; cake, but I suppose a bagel or donut (aka torus) would be more well known. However, I'm not sure if idea of gradually building up the solid by cylinders works?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-09-10T00:35:33.673" Title="Analogy for cylindrical shells" Tags="|calculus|integration|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18807" PostTypeId="2" ParentId="18806" CreationDate="2020-09-09T01:12:06.233" Score="3" Body="&lt;p&gt;I usually reference how we are filling the volume with nested cans (they can all visualize metal food cans with the top and bottom removed) and that works pretty well.  Usually when I mention this, a student will ask if cylindrical shells are like Russian nesting dolls - which I think is a good analogy as well, but not all students may know of these dolls.&lt;/p&gt;&#xA;" OwnerUserId="14575" LastActivityDate="2020-09-09T01:12:06.233" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18808" PostTypeId="2" ParentId="18801" CreationDate="2020-09-09T01:18:12.543" Score="1" Body="&lt;p&gt;I always thought it was interesting that you could use equivalence relations to make &amp;quot;higher&amp;quot; definitions. You collect a bunch of objects which have a common attribute and then you formally define an abstract idea to be an equivalence class. For example, a geometric vector (the abstract idea) can be thought of as an equivalence class. In axiomatic geometry, the set of equivalence classes of congruent segments becomes a field (so an equivalence class can capture the idea of a number). Same thing with Cauchy sequences. You can also use equivalence classes to define the idea of direction.&lt;/p&gt;&#xA;" OwnerUserId="13694" LastEditorUserId="13694" LastEditDate="2020-09-09T02:52:10.280" LastActivityDate="2020-09-09T02:52:10.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18809" PostTypeId="2" ParentId="18793" CreationDate="2020-09-09T02:28:23.020" Score="2" Body="&lt;p&gt;As someone who was not too long ago in middle school, I have fond memories of working on AMC 8 or AMC 10 problems once I finished my school work. These are competitions for middle and beginning high school students that typically require much more thought than standard classes. These problems would also make for great discussions if you wanted to steer your class in that direction.&lt;/p&gt;&#xA;&lt;p&gt;There is also a series of books by Art of Problem Solving (Volume 1 and the introductory series would be appropriate at this level).&lt;/p&gt;&#xA;&lt;p&gt;Here are some links:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://artofproblemsolving.com/wiki/index.php/AMC_8_Problems_and_Solutions&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://artofproblemsolving.com/wiki/index.php/AMC_8_Problems_and_Solutions&lt;/a&gt;&#xA;&lt;a href=&quot;https://artofproblemsolving.com/wiki/index.php/AMC_10_Problems_and_Solutions&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://artofproblemsolving.com/wiki/index.php/AMC_10_Problems_and_Solutions&lt;/a&gt;&#xA;&lt;a href=&quot;https://artofproblemsolving.com/store/item/aops-vol1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://artofproblemsolving.com/store/item/aops-vol1&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user14577" LastActivityDate="2020-09-09T02:28:23.020" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18810" PostTypeId="2" ParentId="18801" CreationDate="2020-09-09T03:03:13.140" Score="16" Body="&lt;p&gt;In general, all applications are going to be more of the &amp;quot;here is what we have to check to make sure that our algorithm/theory/definitions work&amp;quot;. We don't usually encounter practical problems where we're given a completely arbitrary relation and have to check if it's an equivalence relation.&lt;/p&gt;&#xA;&lt;p&gt;Here are some of the more common equivalence relations that we might care about:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In graph theory, &amp;quot;being connected by a path&amp;quot; is an equivalence relation; proving that lets us say that graphs are divided into equivalence classes called &amp;quot;connected components&amp;quot;. This one might be the most relevant for students focused on computer science.&lt;/li&gt;&#xA;&lt;li&gt;A variant of this is communicating states in Markov chains, which also tell us a lot about the structure of a Markov chain, and save us work determining which states are transient/recurrent.&lt;/li&gt;&#xA;&lt;li&gt;Asymptotic analysis has a couple of equivalence relations: the stricter one where &lt;span class=&quot;math-container&quot;&gt;$f,g$&lt;/span&gt; are equivalent if &lt;span class=&quot;math-container&quot;&gt;$f \sim g$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\lim_{x \to \infty} \frac{f(x)}{g(x)} = 1$&lt;/span&gt;, and the looser one where &lt;span class=&quot;math-container&quot;&gt;$f \in \Theta(g)$&lt;/span&gt;. This might also be interesting for computer science applications.&lt;/li&gt;&#xA;&lt;li&gt;In linear algebra, matrices being similar is an equivalence relation; when we diagonalize a matrix, we choose a better representative of the equivalence class.&lt;/li&gt;&#xA;&lt;li&gt;Various quotient objects in abstract algebra and topology require having equivalence relations first. That's in particular what happens with modular arithmetic. But this might be too abstract to make good examples.&lt;/li&gt;&#xA;&lt;li&gt;Sometimes we use them in definitions to say &amp;quot;there are many ways to arrive at this thing; we don't care about which way we do it, because they're all basically the same&amp;quot;. That's what happens when we define real numbers via Cauchy sequences, or the fundamental group via closed loops.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Maybe it's equally important to look at examples of things we &lt;em&gt;wish&lt;/em&gt; were equivalence relations, but &lt;em&gt;aren't&lt;/em&gt;. For example, in a directed graph, the relation &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$v \sim w$&lt;/span&gt; if there is a path from &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt;&amp;quot; is not symmetric; if we try to patch it to &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$v \sim w$&lt;/span&gt; if there is a path from &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; or a path from &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;&amp;quot; then it's not transitive. As a result, it's much harder to take a directed graph and solve the problem of which vertices can reach which other vertices, compared to the undirected case.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Additionally, I think the definition of equivalence relations looks unmotivated, not just because students don't see any direct applications, but because the three properties (reflexive, symmetric, and transitive) seem arbitrary.&lt;/p&gt;&#xA;&lt;p&gt;So it's worth explaining that the reason these three properties exist is not just &amp;quot;these are the three properties of &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; that we liked&amp;quot;. They are exactly the things we need to check in order to be able to group things into equivalence classes, and that's the motivation for asking for them.&lt;/p&gt;&#xA;&lt;p&gt;Admit honestly that pretty much nobody cares about reflexivity, since you can always &amp;quot;patch&amp;quot; it by replacing relation &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; with relation &lt;span class=&quot;math-container&quot;&gt;$R'$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a \mathrel{R'} b$&lt;/span&gt; whenever &lt;span class=&quot;math-container&quot;&gt;$a \mathrel{R} b$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt;; also, it &lt;em&gt;almost&lt;/em&gt; follows from the other two properties, except in the awkward case where some object is not related to any other object, not even itself.&lt;/p&gt;&#xA;&lt;p&gt;For some students, it might help to mention that there's alternate (equivalent) definitions of equivalence relations. Hilbert used a two-property definition where we first check that &lt;span class=&quot;math-container&quot;&gt;$a \mathrel{R} a$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, then check that if &lt;span class=&quot;math-container&quot;&gt;$a \mathrel{R} b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b \mathrel{R} c$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$c \mathrel{R} a$&lt;/span&gt;, which is a concise but not very clear way of getting symmetric and transitive at once.&lt;/p&gt;&#xA;" OwnerUserId="9609" LastEditorUserId="9609" LastEditDate="2020-09-09T17:30:55.733" LastActivityDate="2020-09-09T17:30:55.733" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18811" PostTypeId="2" ParentId="18801" CreationDate="2020-09-09T04:21:06.847" Score="5" Body="&lt;p&gt;&lt;strong&gt;Dictionary/hash table&lt;/strong&gt; relies on equivalence to bucketize items.&lt;/p&gt;&#xA;&lt;p&gt;So knowing that one would never try to build a hashtable by a &lt;em&gt;distance&lt;/em&gt; between cities (objects on a plane): distance is not transitive.&lt;/p&gt;&#xA;&lt;p&gt;In real programming there is another common way to violate equivalence which is sort of implied in pure math/CS: &amp;quot;a == b hence in 5 minutes a == b too&amp;quot; can broken very easily. This is again directly leads to requirements of dictionary/hash table - equivalence between items must never change while items in that data structure.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Rotation&lt;/strong&gt; of a physical object or an observer in most cases considered to be equivalent by module 360 (or 2*Pi whatever you like). Combined with knowledge that addition and multiplication preserves that relationship one can safely say that both spinning 5 times by 361 degree or 5 times by 721 degree would result in an equivalent position. It is somewhat intuitive for +5 degrees, but far less obvious for multiplying. As result one does not need to normalize rotation on every step of they code:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;  angle = prevAngle * 4 + 32 + otherAngle&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;instead of something like following after every step of the operation:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;  normalizedOther = otherAngle &amp;lt; 0 ? &#xA;           otherAngle + 360 : otherAngle &amp;gt; 360 ?&#xA;           otherAngle - 360 : otherAngle;&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="14578" LastActivityDate="2020-09-09T04:21:06.847" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18812" PostTypeId="2" ParentId="18801" CreationDate="2020-09-09T06:36:56.260" Score="2" Body="&lt;ul&gt;&#xA;&lt;li&gt;Equivalence relations serve as an introduction to the types of &amp;quot;order-like&amp;quot; relations that one may impose on a set. Rather than directly discussing preorders without motivation, it's a lot nicer to first get the hang of equivalence relations, observe the &lt;em&gt;very surprising&lt;/em&gt; similarity between equivalence relations and partial orders (some analogy exists to the surprising similarity between exponential and periodic functions, and that's all I'll say about it) and &lt;em&gt;then&lt;/em&gt; abstract to preorders. In a sense, it's like teaching linear algebra early as motivation for abstract mathematics.&lt;/li&gt;&#xA;&lt;li&gt;More generally, equivalence relations are a particularly good way to introduce &lt;em&gt;the idea of a mathematical structure&lt;/em&gt; and perhaps even to the notion of &lt;a href=&quot;https://ncatlab.org/nlab/show/stuff,+structure,+property&quot; rel=&quot;nofollow noreferrer&quot;&gt;stuff, structure, property&lt;/a&gt;. An equivalence relation makes a set &amp;quot;less discrete&amp;quot;, reduces the distinctions between points. You see this idea appear over and over again, in &lt;a href=&quot;https://thewindingnumber.blogspot.com/2019/10/sigma-fields-are-venn-diagrams.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;probability theory&lt;/a&gt;, in topology, and even in geometry (you may notice the axioms for a metric look weirdly similar to those of equivalence relations). The general notion of a structure is a bit more general than this but still has to do with &amp;quot;imposing relations on points&amp;quot;, which is the basic moral.&lt;/li&gt;&#xA;&lt;li&gt;A lot of the things that you &amp;quot;thought&amp;quot; were equalities as a child, like remainders and improper integration, are actually equivalences. This is around the least significant reason to care about equivalence relations but a good criminal mastermind achieves multiple goals with one plot.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12912" LastActivityDate="2020-09-09T06:36:56.260" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18814" PostTypeId="2" ParentId="18801" CreationDate="2020-09-09T15:24:17.313" Score="6" Body="&lt;p&gt;If you have looked at modular arithmetic, then one possibility is: &lt;strong&gt;Give/recall some example of an algebraic argument in modular arithmetic; then point out that the argument is implicitly relying on the fact that congruence is an equivalence relation.&lt;/strong&gt;  So then you can explain: equivalence relations are designed to axiomatise what’s needed for these kinds of arguments — that there are lots of places in maths where you have a notion of “congruent” or “similar” that isn’t quite equality but that you sometimes want to use like an equality, and “equivalence relations” tell you what kind of relations you can use in that kind of way.&lt;/p&gt;&#xA;&lt;p&gt;Finding a good specific example argument will depend on what you covered in modular arithmetic.  Almost any algebraic argument will use “chains of equalities”, and so illustrate transitivity.  Going through the proofs you cover, it’s pretty likely that you can find one that uses symmetry.  Reflexivity is much less often used — but if your example illustrates two out of the three defining properties, then it will serve well enough as a “motivational application”.&lt;/p&gt;&#xA;" OwnerUserId="610" LastActivityDate="2020-09-09T15:24:17.313" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18815" PostTypeId="2" ParentId="18801" CreationDate="2020-09-09T16:19:58.807" Score="3" Body="&lt;p&gt;One of the more popular high-school proofs (&lt;a href=&quot;https://en.wikipedia.org/wiki/Pons_asinorum#Pappus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pappus's&lt;/a&gt;) that the base angles of an isosceles triangle are equal relies on an equivalence relation.  They typically don't call it that in high-school, but that's what it is.&lt;/p&gt;&#xA;&lt;p&gt;Indeed much of high-school geometry and trigonometry relies on this equivalence relation (the SSS equivalence).&lt;/p&gt;&#xA;" OwnerUserId="556" LastActivityDate="2020-09-09T16:19:58.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18816" PostTypeId="1" CreationDate="2020-09-09T19:05:56.157" Score="2" ViewCount="92" Body="&lt;p&gt;I am undergraduate math student who is interested in being a high school math teacher. I have been given an assignment to present to my class (for a total of about 20 minutes) a teaching tool or a teaching strategy that will enhance students’ (high school students) learning of a &lt;em&gt;particular&lt;/em&gt; secondary mathematics topic.&#xA;For example, I have a friend who is presenting &lt;a href=&quot;https://www.canfigureit.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.canfigureit.com/&lt;/a&gt;, which can help students better understand and set up geometric proofs.&lt;/p&gt;&#xA;&lt;p&gt;I don't know any websites/apps to teach mathematical concepts except for  &lt;a href=&quot;https://www.mathsisfun.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathsisfun.com/&lt;/a&gt;, but this website is very generic and can be used to teach many different topics.&lt;/p&gt;&#xA;&lt;p&gt;I was hoping that you can respond with websites/apps that you found helped high school students to discover a &lt;em&gt;specific&lt;/em&gt; mathematical concept/topic.&lt;/p&gt;&#xA;" OwnerUserId="14582" LastActivityDate="2020-09-09T19:05:56.157" Title="Math websites/apps for high school students" Tags="|secondary-education|mathematical-pedagogy|technology-in-education|concept-motivation|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="18817" PostTypeId="2" ParentId="18784" CreationDate="2020-09-09T21:55:00.640" Score="2" Body="&lt;p&gt;Simplest explanation I've seen is to take &lt;span class=&quot;math-container&quot;&gt;$a, b$&lt;/span&gt; integers and consider the set &lt;span class=&quot;math-container&quot;&gt;$\{u a + v b\}$&lt;/span&gt; for integer &lt;span class=&quot;math-container&quot;&gt;$u, v$&lt;/span&gt;. It is a bunch of integers, so it has to contain a smallest positive one, call it &lt;span class=&quot;math-container&quot;&gt;$d = u_0 a + v_0 b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now divide &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$a = q d + r$&lt;/span&gt;, by the Euclidean &amp;quot;algorithm&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$0 \le r &amp;lt; d$&lt;/span&gt;. You see that:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{align*}&#xA;   a&#xA;     &amp;amp;= (u_0 a + v_0 b) q + r \\&#xA;   r&#xA;     &amp;amp;= (1 - q u_0) a - q v_0 b &#xA;\end{align*}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; belongs to our set, is non-negative, and is smaller than it's smallest positive element &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;. Only possible value is &lt;span class=&quot;math-container&quot;&gt;$r = 0$&lt;/span&gt;. You can repeat the same argument for &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$d \mid a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d \mid b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now any number that divides both &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; must also divide &lt;span class=&quot;math-container&quot;&gt;$d = u_0 a + v_0 b$&lt;/span&gt;, i.e., &lt;span class=&quot;math-container&quot;&gt;$\gcd(a, b) \mid d$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now suppose &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is a common divisor of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, this means:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{align*}&#xA;   a&#xA;     &amp;amp;= c x_0 \\&#xA;   b&#xA;     &amp;amp;= c y_0 \\&#xA;   d&#xA;     &amp;amp;= u_0 x_0 c + v_0 y_0 c \\&#xA;     &amp;amp;= (u_0 x_0 + v_0 y_0) c&#xA;\end{align*}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So any such &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is largest possible.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-09-09T21:55:00.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18818" PostTypeId="2" ParentId="18806" CreationDate="2020-09-10T00:35:33.673" Score="2" Body="&lt;p&gt;How about telescopic collapsible camping cups?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/70YOi.jpg&quot; width=&quot;300&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&lt;p&gt;Image from &lt;a href=&quot;https://www.wish.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;wish.com&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;Similar to @BrendanSullivan's suggestion.&#xA;" OwnerUserId="511" LastActivityDate="2020-09-10T00:35:33.673" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18819" PostTypeId="2" ParentId="18801" CreationDate="2020-09-10T05:52:46.537" Score="11" Body="&lt;h1&gt;Equality vs. Identity&lt;/h1&gt;&#xA;&lt;p&gt;Alexei touched on this topic by mentioning hash tables, but I would like to spell it out more explicitly, because this is a critical and fundamental topic in software engineering, and essential for every programmer to know and understand.&lt;/p&gt;&#xA;&lt;p&gt;Every high level programming language has a mechanism for comparing two values for &amp;quot;equality&amp;quot;.  But every PL with reference types must also provide one more ability: comparing whether two references point to the same object.  In Java, D, C#, Kotlin, and many other similar languages, the &lt;code&gt;==&lt;/code&gt; operator tests for &lt;em&gt;identity&lt;/em&gt; (do these two references point to the same address?), while the &lt;code&gt;equals()&lt;/code&gt; method tests for &lt;em&gt;equivalence&lt;/em&gt; (may I regard these expressions as having the same value?).  Like I said, understanding this distinction and when to use which operator is absolutely essential to writing correct code.&lt;/p&gt;&#xA;&lt;p&gt;A non-programmer (especially a mathematician, perhaps), might assume that &lt;code&gt;==&lt;/code&gt; is the more useful function, because pure math and common experience can usually make do with identity (all instances of the number &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; are identical in math).  The reality is that in the majority of production code, &lt;code&gt;equals()&lt;/code&gt; far outnumbers &lt;code&gt;==&lt;/code&gt;.  That's because the majority of objects in production code are mutable and lack referential transparency.&lt;/p&gt;&#xA;&lt;h1&gt;Strings&lt;/h1&gt;&#xA;&lt;p&gt;To understand the above, we need look no further than strings.  In Java, strings are &lt;em&gt;immutable&lt;/em&gt;, but &lt;em&gt;still&lt;/em&gt; require comparison via &lt;code&gt;equals()&lt;/code&gt; rather than &lt;code&gt;==&lt;/code&gt;.  To see why, consider this Java code fragment:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;void areEqual() {&#xA;    String a = &amp;quot;hello&amp;quot;;&#xA;    String b = &amp;quot;hello&amp;quot;;&#xA;    if (a == b) System.out.println(&amp;quot;Same&amp;quot;);&#xA;    else System.out.println(&amp;quot;Different&amp;quot;);&#xA;}&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Now, as the naive reader expects, the function above will print &lt;code&gt;Same&lt;/code&gt;, but only because the strings in question are literals, and thus, the compiler will generate code which causes &lt;code&gt;a&lt;/code&gt; and &lt;code&gt;b&lt;/code&gt; to point to the same memory address.  One small change will break this code:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;void areEqual() {&#xA;    String a = &amp;quot;hello&amp;quot;;&#xA;    String b = &amp;quot;hel&amp;quot;;&#xA;    String c = b + &amp;quot;lo&amp;quot;;&#xA;    if (a == c) System.out.println(&amp;quot;Same&amp;quot;);&#xA;    else System.out.println(&amp;quot;Different&amp;quot;);&#xA;}&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;This function will print &lt;code&gt;Different&lt;/code&gt;, even though we could print out &lt;code&gt;a&lt;/code&gt; and &lt;code&gt;c&lt;/code&gt; and they would look the same to anyone who stared at them.  In this case, we defeat the string interning mechanism, and &lt;code&gt;c&lt;/code&gt; resolves to a different object than &lt;code&gt;a&lt;/code&gt;.  Under the covers, the first example might produce something like: &lt;code&gt;a == 0x1234abcd; b == 0x1234abcd&lt;/code&gt; while the second example might produce: &lt;code&gt;a = 0x1234abcd; b == 12349876&lt;/code&gt;.  So the &lt;em&gt;values&lt;/em&gt; of the strings are &lt;em&gt;equivalent&lt;/em&gt;, but the &lt;em&gt;addresses&lt;/em&gt; of the strings are quite distinct.&lt;/p&gt;&#xA;&lt;p&gt;This is important, because if one has a container of strings, and one wishes to see if some new string exists in that container, in virtually every real-world program, the programmer wants to know if an &lt;em&gt;equivalent&lt;/em&gt; string exists in the container (i.e., an object in the same equivalence class, or, the same sequence of characters), rather than the &lt;em&gt;identical&lt;/em&gt; string.&lt;/p&gt;&#xA;&lt;h1&gt;Business Objects&lt;/h1&gt;&#xA;&lt;p&gt;The discussion above might lead one to assume that identity vs. equality is just an esoteric implementation detail of programming languages with reference semantics, and that we can simply switch to calling &lt;code&gt;equals()&lt;/code&gt; everywhere and be done with it.  But it's not so simple.  Suppose we have a shopping website, and a large catalog with millions of items and their descriptions.  An item description might be a surprisingly complex composite object, consisting of a short text blurb in addition to structured data about the item (its weight, shipping box dimensions, manufacturer, etc.).  However, there may be duplicates within the catalog.  When this happens, we can save precious memory by reusing the same object when two descriptions are &lt;em&gt;equivalent&lt;/em&gt; even though they are not identical.  Now, if we have defined the naive &lt;code&gt;equals()&lt;/code&gt; method on the class &lt;code&gt;ItemDescription&lt;/code&gt; which compares every field one by one, then this seems like another boring application of &lt;code&gt;equals()&lt;/code&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But it's not that simple.  You see, data like this will almost certainly come from a database, and any good DBA will demand that every table contain a primary key, which will usually be a synthetic autoincrement value.  So, it is very, very likely, that there will exist two sets of records in the item catalog which are the same &lt;em&gt;up to their PKey&lt;/em&gt;.  An app which stores huge portions of the catalog does not want to waste memory storing these duplicates separately.  Thus, it is useful to define an &lt;em&gt;additional&lt;/em&gt; equality operator that detects this &amp;quot;equality-up-to-pkey&amp;quot;.  That is, we wish to traffic in the equivalence class of &lt;code&gt;ItemDescription - PKey&lt;/code&gt;.  One way to do this would be to define another method on the &lt;code&gt;ItemDescription&lt;/code&gt; class which implements this equivalence class, and use that comparator on a collection of unique &lt;code&gt;ItemDescription&lt;/code&gt;.  This will ensure there is only one copy of each equivalent &lt;code&gt;ItemDescription&lt;/code&gt; in the in-memory collection.  This kind of equivalence class occurs &lt;em&gt;frequently&lt;/em&gt; in the industrial programming world.&lt;/p&gt;&#xA;&lt;h1&gt;Inheritance&lt;/h1&gt;&#xA;&lt;p&gt;One of the defining characteristics of Object-Oriented Programming (OOP) is the mechanism of &amp;quot;inheritance&amp;quot;.  When a type &lt;code&gt;Child&lt;/code&gt; &amp;quot;inherits&amp;quot; from a type &lt;code&gt;Parent&lt;/code&gt;, we say that a &lt;code&gt;Child&lt;/code&gt; &amp;quot;is-a&amp;quot; &lt;code&gt;Parent&lt;/code&gt;.  This notion is neither identity nor the naive field-by-field notion of equivalence that is commonly assumed.  The &amp;quot;is-a&amp;quot; relation really means: &amp;quot;can be substituted for&amp;quot;.  That is, any code which expects a value of type &lt;code&gt;Parent&lt;/code&gt; will gladly accept a value of type &lt;code&gt;Child&lt;/code&gt;.  But what does that mean?  What if the &lt;code&gt;Child&lt;/code&gt; class introduces new fields which don't exist in &lt;code&gt;Parent&lt;/code&gt;?  Well, those fields are ignored.  When you pass a &lt;code&gt;Child&lt;/code&gt; in as a &lt;code&gt;Parent&lt;/code&gt;, only the &lt;code&gt;Parent&lt;/code&gt; portion of the object is considered.&lt;/p&gt;&#xA;&lt;p&gt;Here is a small code example:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;class Animal {&#xA;    protected String sound = &amp;quot;&amp;lt;gurgle&amp;gt;&amp;quot;;&#xA;    public Animal(String sound) {&#xA;        this.sound = sound;&#xA;    }&#xA;    ...&#xA;}&#xA;class Bird extends Animal {&#xA;    int wings = 2;&#xA;    public Bird() {&#xA;        super(&amp;quot;&amp;lt;tweet&amp;gt;&amp;quot;);&#xA;    }&#xA;    ...&#xA;}&#xA;class Pigeon extends Bird {&#xA;    String trait = &amp;quot;annoying&amp;quot;;&#xA;    ...&#xA;}&#xA;class Dog extends Animal {&#xA;    int legs = 4;&#xA;    public Dog() {&#xA;        super(&amp;quot;&amp;lt;woof&amp;gt;&amp;quot;);&#xA;    }&#xA;    ...&#xA;}&#xA;void tickle(Animal animal) {&#xA;    System.out.println(&amp;quot;You tickle the animal, and it goes: &amp;quot; + animal.sound);&#xA;}&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Note that you are free to pass an &lt;code&gt;Animal&lt;/code&gt;, &lt;code&gt;Bird&lt;/code&gt;, &lt;code&gt;Pigeon&lt;/code&gt; or &lt;code&gt;Dog&lt;/code&gt; to &lt;code&gt;tickle()&lt;/code&gt;.  It will accept any of them.  This fact alone illustrates that &amp;quot;is-a&amp;quot; defines an equivalence class over types, given that the type system requires an argument's type to match the type of the parameter.  Given that almost every major programming language with mutable data supports OOP features, one could say that equivalence classes are again pervasive in the type system.&lt;/p&gt;&#xA;&lt;h1&gt;Conclusion&lt;/h1&gt;&#xA;&lt;p&gt;There are many more examples of equivalence classes used commonly within professional software engineering, but hopefully this is enough to get you started.&lt;/p&gt;&#xA;" OwnerUserId="13468" LastEditorUserId="13468" LastEditDate="2020-09-10T23:02:48.293" LastActivityDate="2020-09-10T23:02:48.293" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18820" PostTypeId="1" CreationDate="2020-09-10T11:05:32.273" Score="3" ViewCount="116" Body="&lt;p&gt;Base 10 blocks (units, rods, flats and cubes) are a widespread manipulative material for teaching place value.&lt;/p&gt;&#xA;&lt;p&gt;However, piling rods on top of each other is not very stable, so it is not easy to build one single tower with all pieces that represent a number.&lt;/p&gt;&#xA;&lt;p&gt;As an alternative, one can use flat pieces for rods, cubes and beyond, exploiting the fact that 10 is a triangular number.&lt;/p&gt;&#xA;&lt;p&gt;Using this system, 9999 looks like this:&lt;a href=&quot;https://i.stack.imgur.com/EHLcU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EHLcU.png&quot; alt=&quot;9999 with vertical sum&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I would like to know whether this system has been described before.&lt;/p&gt;&#xA;" OwnerUserId="6718" LastEditorUserId="6718" LastEditDate="2020-09-11T10:11:15.613" LastActivityDate="2020-09-11T10:11:15.613" Title="References for visualizing numbers in base 10 flats" Tags="|reference-request|manipulatives|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="18821" PostTypeId="2" ParentId="18801" CreationDate="2020-09-10T12:09:53.977" Score="6" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;The notion of equivalence relation is one of the basic building blocks out of which all mathematical thought is constructed.&lt;/strong&gt; (Paul Halmos)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;What's the point of learning equivalence relations?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The concept of equivalence relation is a generalization of the concept of equality. Why is it good to know that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is equal to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;?  Because, in this case, all we know about &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is also true for &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, and vice versa (they are the same thing!). Analogously, why is it good to know that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is equivalent to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;? Because, in the case, &amp;quot;almost&amp;quot; all we know about &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is also true for &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, and vice versa (they are &amp;quot;almost&amp;quot; the same thing!). However, for any practical purposes, &amp;quot;almost&amp;quot; in this context is as good as if it was &amp;quot;all&amp;quot;. Why? Because, when it is needed, the equivalence relation is defined in a way that the elements of the equivalence class differ only with respect to irrelevant aspects (anyone we choose in the class will do the job).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example.&lt;/strong&gt; If we are interested only in the area below the graph, we can regard the functions&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x)=\left\{\begin{aligned}&#xA;1,&amp;amp;&amp;amp;0\leq x\leq 1\\&#xA;2,&amp;amp;&amp;amp;1&amp;lt; x\leq 2&#xA;\end{aligned}\right.\qquad \text{and}\qquad g(x)=\left\{\begin{aligned}&#xA;1,&amp;amp;&amp;amp;0\leq x&amp;lt; 1\\&#xA;2,&amp;amp;&amp;amp;1\leq x\leq 2&#xA;\end{aligned}\right.$$&lt;/span&gt;&#xA;as being the same function because a single point does not change the area. In this case, we define &lt;span class=&quot;math-container&quot;&gt;$f\sim g$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$f(x)\neq g(x)$&lt;/span&gt; only for a finite number of values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. This seemingly simple idea is actually used in more advanced contexts.&lt;/p&gt;&#xA;&lt;p&gt;[What I have in mind here is that &lt;span class=&quot;math-container&quot;&gt;$\int f g$&lt;/span&gt; is an inner product on the space of (equivalence class of) piecewise continuous functions (we can use it in the study of Fourier series (which can be used to solve partial differential equations (which can be used to solve &amp;quot;real-world&amp;quot; problems (thus we have an indirect/artificial application here. Of course, it is not satisfactory for the student but reflect the relevance of the subject)))).]&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;What's a good response to a student ... (esp. in computing)?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;You could try to elaborate on the last example of &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00207390410001686652&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Teaching equivalence relations using collaborative activities&lt;/em&gt;&lt;/a&gt; by Janet E. Mills:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the area of coding theory, one needs to know if two codes are doing essentially the same thing. The distance between code words is a critical factor in analysing a code. Let &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; be subsets of &lt;span class=&quot;math-container&quot;&gt;$Z^n_2$&lt;/span&gt; (i.e. codes). Define &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$Z^n_2$&lt;/span&gt; by by &lt;span class=&quot;math-container&quot;&gt;$S \sim T$&lt;/span&gt; if and only if there exists a one-to-one onto linear function &lt;span class=&quot;math-container&quot;&gt;$\varphi:S\to T$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$d(x, y) = d(\varphi(x), \varphi(y))$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x, y$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$d(x, y)$&lt;/span&gt; is the number of nonzero components in &lt;span class=&quot;math-container&quot;&gt;$x- y$&lt;/span&gt; (or the number of digits where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; differ, called the Hamming distance).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You could to try to define what a code is (mention message, encoder, channel, noise, decoder), define Hamming distance, define equivalence of codes (you can use permutation if &lt;span class=&quot;math-container&quot;&gt;$\varphi$&lt;/span&gt; is too abstract), exploit the fact that equivalent codes are &amp;quot;the same&amp;quot; (though &amp;quot;different&amp;quot;), for example, have the same error correction capability. Maybe this situation could be sufficiently simple, interesting and convincing.&lt;/p&gt;&#xA;" OwnerUserId="13585" LastEditorUserId="13585" LastEditDate="2020-09-10T12:18:39.257" LastActivityDate="2020-09-10T12:18:39.257" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18822" PostTypeId="2" ParentId="18801" CreationDate="2020-09-10T16:08:45.323" Score="-3" Body="&lt;p&gt;One of my replies to such a question would be &amp;quot;You know how most software doesn't work very well ? Well the people who wrote it also didn't see the point of studying basic logic and set theory&amp;quot;. Although I would take pleasure in the provoking tone of such a reply, yet I also believe it is a very important point.&lt;/p&gt;&#xA;&lt;p&gt;EDIT:&#xA;The kernel of any function is an equivalence relation on the domain of the function. In particular this applies to hashes. Why not learn a general idea that unifies this idea and all the many other concrete examples of equivalence relations ? If that seems too boring or arduous to the student then I don't have high hopes for the quality of the code they will eventually write.&lt;/p&gt;&#xA;&lt;p&gt;EDIT:&#xA;The Mr. Miyagi principle: Just paint the damn fence and tell me at the end of the movie whether you think it was a waste of time ! I'm sure that plumbing and roofing apprentices don't dare trouble their gnarled mentors with questions about the value of what they are being taught. Why should it be different for university students ? I suppose one reason might be that while there probably aren't many bad master plumbers nor bad master roofers, yet there are plenty of bad teachers. The answer to which is: Don't be a bad teacher.&lt;/p&gt;&#xA;" OwnerUserId="6893" LastEditorUserId="6893" LastEditDate="2020-09-10T16:31:08.227" LastActivityDate="2020-09-10T16:31:08.227" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18824" PostTypeId="2" ParentId="18801" CreationDate="2020-09-10T19:46:40.810" Score="2" Body="&lt;p&gt;It may be useful to show that some forms of genealogical relationships are equivalence relations and some aren't.  Consider the following for relationships:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A person #1 is M-related to #2 if both were borne to the same mother [one kind of half- or full sibling]&lt;/li&gt;&#xA;&lt;li&gt;A person #1 is F-related to #2 if both were sired by the same father [another kind of half-sibling]&lt;/li&gt;&#xA;&lt;li&gt;A person #1 is S-related to #2 if both were sired by the same father &lt;em&gt;and&lt;/em&gt; borne to the same mother [full siblings]&lt;/li&gt;&#xA;&lt;li&gt;A person #1 is H-related to #2 if both were sired by the same father &lt;em&gt;or&lt;/em&gt; borne to the same mother [half siblings]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Any two people who are M-related to the same person will be M-related to each other.  That would likewise be true for any F-related and S-related pairs of people, but would not be true for all possible pairs of H-related people who share a common H-relative.  Thus, the M-, F-, and S-relationships defined above are equivalence relations, but H-relationship is not an equivalence relation.&lt;/p&gt;&#xA;&lt;p&gt;An important thing about equivalence relationships is that it makes it possible to identify relationships between objects without having to individually compare the objects to each other.  If one has lists of people and all of the people within each list are M-related to each other, but one also knows that there is &lt;em&gt;at least one person&lt;/em&gt; in the first list to whom &lt;em&gt;at least one person&lt;/em&gt; on the second are not M-related, that would be sufficient to prove that &lt;em&gt;nobody&lt;/em&gt; on the first list is M-related to &lt;em&gt;anybody&lt;/em&gt; on the second.  By contrast, it would be possible to have disjoint lists of people, where all the people on each list were H-related to each other, and some people on the first list were not H-related to some on the second, but there would still be some people on the first list who were H-related to some people on the second.  Even if one knew that the two people were on different lists, that would prove nothing about any possible H-relationship between them.&lt;/p&gt;&#xA;" OwnerUserId="1740" LastActivityDate="2020-09-10T19:46:40.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18825" PostTypeId="2" ParentId="18801" CreationDate="2020-09-11T00:11:17.510" Score="2" Body="&lt;p&gt;This is a challenging question to answer in the way you want it answered, because the temptation is strong to say something like &amp;quot;Of course equivalence relations are interesting, every concept arises from an equivalence relations!&amp;quot; More precisely, the two fundamental ways of carving up the universe are (a) looking at just part of it at a time and (b) saying certain overly specific bits of it are &amp;quot;basically the same&amp;quot;, i.e. applying an equivalence relation. For instance:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;There's an equivalence relation on points in your retinal image given by &amp;quot;these points are part of the same object&amp;quot; which gives rise to visual perception, and the trickiness of calculating this equivalence relation in your brain gives rise to certain optical illusions. Similarly, for perception through other senses.&lt;/li&gt;&#xA;&lt;li&gt;Or, focusing more on math, much of the entire subject of high school algebra can be reduced to the study of equivalence classes of algebraic expressions under the equivalence relation &amp;quot;these expressions evaluate to the same number for every input.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Or elsewhere in science, &amp;quot;an element of matter&amp;quot; is an equivalence class of atoms under the equivalence relation &amp;quot;these atoms have the same number of protons in their nucleus.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Of course this is not limited to chemistry: numerous abstract concepts can be treated in this way, sometimes controversially. For instance, a species is an equivalence class of organisms under &amp;quot;can reproduce with each other&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;An academic field can (arguably!) be identified with an equivalence class of academics under the relation &amp;quot;these people agree they study the same thing&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;A nation's cuisine is, perhaps, an equivalence class of meals under the relation &amp;quot;the same people make this food&amp;quot; or alternatively, &amp;quot;these foods use the same characteristic spices, ingredients, and methods of preparations...&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;A triangle is a member of an equivalence class of polygons under the equivalence relation &amp;quot;these polygons have the same number of sides&amp;quot;, or alternatively, the concept &amp;quot;triangle&amp;quot; &lt;em&gt;is&lt;/em&gt; this equivalence class.&lt;/li&gt;&#xA;&lt;li&gt;An algorithm is an equivalence class of programs under the (sometimes difficult to assess!) relation of &amp;quot;these programs perform the same high-level steps&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;A family is an equivalence class of people...&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;OK, so every concept is an equivalence relation; you can generate examples &lt;em&gt;ad infinitum&lt;/em&gt; and painlessly. That said, this doesn't seem to be quite the kind of justification you're looking for, and I understand the sense that students might find this all a bit far out and vague. This is, I guess, the justification I &lt;em&gt;wish&lt;/em&gt; I were a good enough pedagogue to communicate convincingly.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A more concrete motivation for equivalence relations is to connect with the study of surjections and to combinatorics, via Bell numbers. This is most natural, of course, if you're already counting the surjections between two finite sets of fixed size at some point in the course, via inclusion-exclusion for instance. Indeed, equivalence relations are relation to surjections as subsets are to injections, once you identify an equivalence relation on &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; with the surjection from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to the set of equivalence classes. This is, incidentally, the quotient set that people seem to assume you can't tell community college students about that, though I'm not so sure--first, I tell my community college students about it, and second, we certainly talk about &lt;em&gt;individual&lt;/em&gt; equivalence classes! A hard problem this leads to, which kind of crowns all the emphasis on injections, surjections, and bijections: any function &lt;span class=&quot;math-container&quot;&gt;$f:A\to B$&lt;/span&gt; can be &lt;em&gt;uniquely&lt;/em&gt; written as a composite of three pieces: the standard map to the set of equivalence classes of an equivalence relation (namely the relation &lt;span class=&quot;math-container&quot;&gt;$f(a_1)=f(a_2)$&lt;/span&gt;, then a bijection, then the inclusion of a subset (namely the image.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="5233" LastActivityDate="2020-09-11T00:11:17.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18826" PostTypeId="1" CreationDate="2020-09-11T04:24:02.967" Score="2" ViewCount="159" Body="&lt;p&gt;I'm taking a Mathematical Structures course as a 2nd year math major. Our course textbook is &lt;em&gt;Discrete mathematics and Its Applications, by Rosen&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It's been very difficult finding good content on places like YouTube to make up for gaps in my lectures or to supplement my notes. I'm by no means asking for some quick 3 minute videos on problem solving, I'm not trying to neglect theory here. But I would greatly appreciate suggestions, whether it's an online course of bundled videos that costs a couple bucks or its a good youtube channel that covers everything I need to know.&lt;/p&gt;&#xA;&lt;p&gt;This class is kind of kicking my butt at the moment, some recommendations would be greatly appreciated.&lt;/p&gt;&#xA;" OwnerUserId="12501" LastActivityDate="2020-09-11T16:03:46.803" Title="Recommendation for Discrete Mathematics" Tags="|reference-request|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18827" PostTypeId="2" ParentId="18801" CreationDate="2020-09-11T04:36:52.047" Score="3" Body="&lt;p&gt;OP here: There's so many good answers here, this is possibly my favorite question that I've ever asked on SE! I'm so glad that I thought to post the question here.&lt;/p&gt;&#xA;&lt;p&gt;For comparison purposes, having considered all the great responses to date, I'll share how I've decided to revise my lecture notes. Observe that we have a crushing time constraint in the course, as about half the lecture time needs to be spent on cooperative exercises. As-is there's only about 30 minutes of lecture on binary relations, 30 on equivalence relations, and about 10 on modular arithmetic. So I don't have time to expand on new subject matter in full, but really need a &amp;quot;snappy&amp;quot; reference or retort to the original question. What I've done is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Worked in more explicit comparisons to the &amp;quot;natural language&amp;quot; meaning of the phrases, straight from a standard dictionary, so we can reflect on what these definitions are trying to pin down in our formal mathematics. For &amp;quot;relation&amp;quot; we see: &amp;quot;an existing connection; a significant association between or among things&amp;quot;. For &amp;quot;equivalent&amp;quot; we find: &amp;quot;equal in value, measure, force, effect, significance, etc.&amp;quot; -- that is, two things that are &amp;quot;the same&amp;quot; in some important way.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Included the quote by Halmos (1982): &amp;quot;It is one of the basic building blocks out of which all mathematical thought is constructed&amp;quot;. That is incredibly lovely, so glad to have that in my slides.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Exchanged the former lecture example from a rather arbitrary one (&lt;span class=&quot;math-container&quot;&gt;$aRb$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$a = b$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$a = -b$&lt;/span&gt;) to congruence modulo &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, as several people highlighted as being of critical importance. This also gives some extra precious minutes thinking about modular arithmetic at all, of which we're sorely lacking.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Now if a student still asks the question, I can at least riff a bit on the importance of identifying any two things as being &amp;quot;the same&amp;quot;, or being in the same category, in any way (to sciences in general) -- and then also mention that in later courses having this tool makes a wide variety of definitions and proofs easier to handle. That's a bit more abstract than I was hoping for, but it's a significant improvement to where I was previously.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-09-11T05:47:16.433" LastActivityDate="2020-09-11T05:47:16.433" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18828" PostTypeId="2" ParentId="11923" CreationDate="2020-09-11T12:27:37.087" Score="2" Body="&lt;p&gt;Benjamin Hutz has a recent book that could be appropriate: &lt;a href=&quot;https://bookstore.ams.org/amstext-31&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Experimental Introduction to Number Theory&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data. Each topic is motivated by a question to be answered, followed by some experimental data, and, finally, the statement and proof of a theorem. There are numerous opportunities throughout the chapters and exercises for the students to engage in (guided) open-ended exploration. At the end of a course using this book, the students will understand how mathematics is developed from asking questions to gathering data to formulating and proving theorems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Regarding languages, in the &lt;a href=&quot;https://bookstore.ams.org/cdn-1599806593742/amstext-31/%7E%7EFreeAttachments/amstext-31-pref.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;preface&lt;/a&gt; he mentions the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The book is not tied into any one computer algebra system, and there are several freely available. The systems SageMath and PARI/GP are two excellent freely&#xA;available systems. Similarly, Mathematica, Maple, or other commercial software&#xA;could also be used. Because there are many excellent tutorials for each of these&#xA;systems freely available, they will not be presented in this book.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="376" LastActivityDate="2020-09-11T12:27:37.087" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18829" PostTypeId="2" ParentId="18826" CreationDate="2020-09-11T16:03:46.803" Score="3" Body="&lt;p&gt;Sue has linked to a great book (which I hope she turns into an answer), and here are two more quality ones that are free, at least, though less inquiry-oriented.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://faculty.uml.edu/klevasseur/ads2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Doerr and Levasseur&lt;/a&gt;, intended for computer engineers as well as CS&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://appliedcombinatorics.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Keller and Trotter&lt;/a&gt;, a bit higher level but has some very good material&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;There are others at &lt;a href=&quot;https://pretextbook.org/catalog.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://pretextbook.org/catalog.html&lt;/a&gt; but I am not as familiar with those.  Unfortunately I don't think that any of these (yet) has embedded videos, though I bet that will come.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2020-09-11T16:03:46.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18830" PostTypeId="2" ParentId="12211" CreationDate="2020-09-11T19:13:36.337" Score="1" Body="&lt;p&gt;In my opinion, this can be explained in 2 ways  :&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;By telling the students that conditional probability is still a probability, so, the laws of probability apply to it as well.&lt;/li&gt;&#xA;&lt;li&gt;We can use Venn Diagrams to make the technical explanation easy.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ahUfG.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ahUfG.png&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;a href=&quot;https://i.stack.imgur.com/uTOLd.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uTOLd.png&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We took  &lt;span class=&quot;math-container&quot;&gt;$P(A'∩B)= P(B) - P(A∩B)$&lt;/span&gt; it the above step. But why? Because&#xA;&lt;span class=&quot;math-container&quot;&gt;$P(A'∩B)$&lt;/span&gt; is the dark gray region in the below Venn diagram [the region is an intersection of A' and B]. We could&#xA;compute that region by subtracting  &lt;span class=&quot;math-container&quot;&gt;$P(A∩B)$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$P(B)$&lt;/span&gt; We subtract &lt;span class=&quot;math-container&quot;&gt;$P(A∩B)$&lt;/span&gt; because the compliment of &lt;span class=&quot;math-container&quot;&gt;$P(A')$&lt;/span&gt; is every element of sample space without &lt;span class=&quot;math-container&quot;&gt;$P(A)$&lt;/span&gt; and the intersection of &lt;span class=&quot;math-container&quot;&gt;$P(A')$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$P(B)$&lt;/span&gt; will give the shaded region.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AVrou.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AVrou.png&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And by substituting the above value in the equation we'll get the desired&#xA;answer, as shown in the image below :&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7siIx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7siIx.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/rMBE7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rMBE7.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14466" LastEditorUserId="14466" LastEditDate="2020-09-11T20:06:16.993" LastActivityDate="2020-09-11T20:06:16.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18831" PostTypeId="2" ParentId="18237" CreationDate="2020-09-11T19:26:39.383" Score="-1" Body="&lt;p&gt;During lockdown we hoarded toilet papers, so I piled up some like this and explained &lt;span class=&quot;math-container&quot;&gt;$gcd\cdot lcm=ab$&lt;/span&gt; to my kid&lt;a href=&quot;https://i.stack.imgur.com/CWlKd.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CWlKd.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="9059" LastActivityDate="2020-09-11T19:26:39.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18832" PostTypeId="2" ParentId="12211" CreationDate="2020-09-11T21:36:38.350" Score="-1" Body="&lt;p&gt;After a football game, the team always send a member to a nearby pizza shop to pack food. Different member has different preference on different tastes of pizzas, but whoever was on the duty, sum of the possibility of his choosing each taste , shall be 1.&lt;/p&gt;&#xA;" OwnerUserId="9059" LastActivityDate="2020-09-11T21:36:38.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18833" PostTypeId="2" ParentId="12211" CreationDate="2020-09-12T07:17:17.600" Score="0" Body="&lt;p&gt;In the spirit of the answer by Tac-Tics, I like the picture with a unit square. One event splits the square horizontally, while the other splits vertically. Your second formula focuses on one strip only. Somehow given B means you can only be in one strip now, not in the whole square.&lt;/p&gt;&#xA;&lt;p&gt;You may have a look at figure 2, a geometric interpretation of Bayes' Theorem in &lt;a href=&quot;https://en.wikipedia.org/wiki/Bayes%27_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Bayes%27_theorem&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6718" LastActivityDate="2020-09-12T07:17:17.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18834" PostTypeId="2" ParentId="220" CreationDate="2020-09-12T14:09:43.553" Score="1" Body="&lt;p&gt;I wrote an article that explains mathematical induction in the easiest way possible using an easy-to-understand example. It was published in Medium's #1 Math Publication, so I guess it must be pretty good. Feel free to take a look.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://medium.com/cantors-paradise/a-comprehensible-introduction-to-mathematical-induction-aef1a61eedcf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://medium.com/cantors-paradise/a-comprehensible-introduction-to-mathematical-induction-aef1a61eedcf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The used example is: &lt;span class=&quot;math-container&quot;&gt;$$1 + 2 + \cdots (n-1) + n = \frac{n(n+1)}{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="14599" LastEditorUserId="470" LastEditDate="2020-09-12T15:37:12.760" LastActivityDate="2020-09-12T15:37:12.760" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18835" PostTypeId="2" ParentId="12211" CreationDate="2020-09-12T16:48:35.750" Score="2" Body="&lt;p&gt;Here is another way to look at this.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(A)+P(A^c)=1$&lt;/span&gt; aligns with the logical tautology &lt;span class=&quot;math-container&quot;&gt;$(A\text{ or }\lnot A)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The preposition&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$B\rightarrow(A\text{ or  }\lnot A)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;is also a logical tautology, and corresponds to&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(A\vert B)+P(A^c\vert B)=1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2020-09-12T16:48:35.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18836" PostTypeId="1" AcceptedAnswerId="18842" CreationDate="2020-09-13T00:31:34.110" Score="2" ViewCount="793" Body="&lt;p&gt;In high school we learn that the cube root of &lt;span class=&quot;math-container&quot;&gt;$-8$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;. Much later some of us learn about the single valued natural logarithm of a complex number, and that &lt;span class=&quot;math-container&quot;&gt;$w^z = e^{z\cdot Lz(w)}$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; are complex. That gives a different result for the cube root of &lt;span class=&quot;math-container&quot;&gt;$-8$&lt;/span&gt;. When would what we learn in high school not serve an undergraduate majoring in physics or engineering?&lt;/p&gt;&#xA;" OwnerUserId="7001" LastEditorUserId="8571" LastEditDate="2020-09-13T20:00:01.090" LastActivityDate="2020-09-14T14:15:03.583" Title="When does thinking $(-8)^{1/3} = -2$ result in problems for an undergraduates?" Tags="|undergraduate-education|algebra|complex-numbers|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18838" PostTypeId="2" ParentId="18836" CreationDate="2020-09-13T03:17:49.157" Score="9" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;That gives a different result for the cube root of -8.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It doesn't give a different result, it just gives two additional roots that are complex numbers, for a total of three roots.&lt;/p&gt;&#xA;&lt;p&gt;A physics or engineering student in the US probably first learns about complex numbers in high school, but never sees any interesting applications. Then in college classes they get such applications (possibly not until they get to upper-division classes), and then they do have to do a certain amount of unlearning of facts that hold for the reals but not the complex numbers. (This unlearning should of course have happened in high school when they first saw complex numbers, but it doesn't really.) These applications would include, for example, analyzing linear filters in an electrical engineering class.&lt;/p&gt;&#xA;&lt;p&gt;In my experience the particular issue you ask about has never been a problem for my physics students. The main problem with their high school background is that they have been drilled endlessly in the cartesian representation, so they have trouble transitioning to the polar representation, which is more natural and convenient for the applications we do. For example, it takes them 10 minutes to compute &lt;span class=&quot;math-container&quot;&gt;$1/i$&lt;/span&gt;, because they're doing it using some algorithm they've been taught for computing &lt;span class=&quot;math-container&quot;&gt;$(a+bi)/(c+di)$&lt;/span&gt;, rather than just visualizing &lt;span class=&quot;math-container&quot;&gt;$1/i$&lt;/span&gt; on the unit circle.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-09-13T03:17:49.157" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18839" PostTypeId="2" ParentId="14846" CreationDate="2020-09-13T04:28:46.857" Score="1" Body="&lt;p&gt;I noticed that you said that he needs reassurance and many of the comments suggest that he needs drill. I would therefore encourage you to use Khan Academy.  You can set up an account with you as his teacher.  This will give you an opportunity to assign practice in skills in a specific time frame and see his progress.  More importantly, he will have immediate feedback and reassurance which he seems to need.  He can also review concepts with short videos which might be considerably easier for him than learning to read a math book.&lt;/p&gt;&#xA;&lt;p&gt;As for math as an algorithm (i.e. a set of instruction) vs. understanding concepts, I don't believe it is one or the other. There is a spectrum in which we combine understanding with doing things by rote.  Some examples - although I have deep understanding of many concepts, there are many times when  I am not thinking about the concepts of why an algorithm works, but just plugging away at an algorithm. Some examples include: long division, taking the derivative of various polynomials, using the Pythagorean theorem etc.  You should explain the concepts but there should be emphasis on algorithms and how to do things. Most of us can't do math without these rules, formulas, and algorithms.&lt;/p&gt;&#xA;&lt;p&gt;There should also be emphasis on how to check work.  For example, plugging your answer back into the equation to see if it works.  Doing this will also give immediate feedback.&lt;/p&gt;&#xA;&lt;p&gt;Weak students often need significant driling to fully get order of operations.  I would emphasize that with your student.&lt;/p&gt;&#xA;&lt;p&gt;Hope some of these suggestions are helpful.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2020-09-13T04:28:46.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18840" PostTypeId="2" ParentId="18793" CreationDate="2020-09-13T15:38:21.857" Score="0" Body="&lt;p&gt;When I was in my high-school, I was a kid that solved all the questions quickly. This thing was good practice for me and I loved it when I was appreciated. But what I think personally is, healthy competition is always a good part of the learning process. I have a suggestion for the similar :&lt;/p&gt;&#xA;&lt;h2&gt;Conducting group activities&lt;/h2&gt;&#xA;&lt;p&gt;You can divide the class into groups and provide some points, say 10, if 3 students out of 5 in the group solves the question, their team scores 30 points.  This may help students to learn from other students of the class, also fun activities increase love for the subjects.&lt;/p&gt;&#xA;" OwnerUserId="14466" LastActivityDate="2020-09-13T15:38:21.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18841" PostTypeId="2" ParentId="18836" CreationDate="2020-09-14T02:32:50.573" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;When would what we learn in high school &lt;strong&gt;not serve&lt;/strong&gt; an undergraduate majoring in physics or engineering?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It would begin to be a problem when you enter any field where complex numbers appear. Obvious places where complex numbers appear would be in linear ODEs, Fourier transforms, etc. In quantum mechanics, wave functions are complex valued.&lt;/p&gt;&#xA;&lt;p&gt;In many of these situations, making the cube root single valued would mean that you miss &lt;strong&gt;real&lt;/strong&gt; valued solutions so some equations. Here's an analogue to your example:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y'''=-8y$$&lt;/span&gt;&#xA;As it turns out, the solutions are generated by those of the form &lt;span class=&quot;math-container&quot;&gt;$y=e^{rx}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is some constant. (I believe physicists call this an ansatz.) When you plug this form into the ODE, you get the requirement that &lt;span class=&quot;math-container&quot;&gt;$r^3=-8$&lt;/span&gt;. If you take the real valued solution you get &lt;span class=&quot;math-container&quot;&gt;$y=e^{-2x}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;That misses the other two real valued generators:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ y= e^{x} \cos\left(\sqrt{3}x\right) \quad \text{and} \quad y= e^{x} \sin\left(\sqrt{3}x\right)$$&lt;/span&gt;&#xA;which are formed from the complex root solutions: &lt;span class=&quot;math-container&quot;&gt;$y=e^{2 \omega_3 x}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=e^{2 \omega_3^2 x}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\omega_3 = e^{i\pi/3}=\cos(\pi/3)+i\sin(\pi/3)=\frac{1+i\sqrt{3}}{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you only had the real root, you would think that the system rapidly settles down to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. However, while this is possible, the generic behavior of a solution is an increasingly wild oscillation.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2020-09-14T02:32:50.573" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18842" PostTypeId="2" ParentId="18836" CreationDate="2020-09-14T13:41:51.363" Score="5" Body="&lt;p&gt;This is a complicated question, and there are a number of articles written on the topic in the math education literature. Here are some of the entries that I would recommend (taken from the bottom of &lt;a href=&quot;https://math.stackexchange.com/questions/1628759/what-are-the-laws-of-rational-exponents&quot;&gt;this answer&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Goel, Sudhir K., and Michael S. Robillard. &amp;quot;The Equation: &lt;span class=&quot;math-container&quot;&gt;$-2 = (-8)^\frac{1}{3} = (-8)^\frac{2}{6} = [(-8)^2]^\frac{1}{6} = 2$&lt;/span&gt;.&amp;quot; &lt;em&gt;Educational Studies in Mathematics&lt;/em&gt; 33.3 (1997): 319-320.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Tirosh, Dina, and Ruhama Even. &amp;quot;To define or not to define: The case of &lt;span class=&quot;math-container&quot;&gt;$(-8)^\frac{1}{3}$&lt;/span&gt;.&amp;quot; &lt;em&gt;Educational Studies in Mathematics&lt;/em&gt; 33.3 (1997): 321-330.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Choi, Younggi, and Jonghoon Do. &amp;quot;Equality Involved in 0.999... and &lt;span class=&quot;math-container&quot;&gt;$(-8)^\frac{1}{3}$&lt;/span&gt;&amp;quot; &lt;em&gt;For the Learning of Mathematics&lt;/em&gt; 25.3 (2005): 13-36.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Woo, Jeongho, and Jaehoon Yim. &amp;quot;Revisiting 0.999... and &lt;span class=&quot;math-container&quot;&gt;$(-8)^\frac{1}{3}$&lt;/span&gt; in School Mathematics from the Perspective of the Algebraic Permanence Principle.&amp;quot; &lt;em&gt;For the Learning of Mathematics&lt;/em&gt; 28.2 (2008): 11-16.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As the OP alludes to, there are in fact different definitions of the (single-valued) &lt;em&gt;principal root&lt;/em&gt; in the real and complex contexts. It's quite common on SE for there to be confusion, cross-talk, and frustration between specialists in the two different domains, each assuming a different context and definition. I suppose that interacting with any of those types of discussions could count as an example for the OP's purpose.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-09-14T13:41:51.363" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18843" PostTypeId="2" ParentId="18836" CreationDate="2020-09-14T14:15:03.583" Score="2" Body="&lt;p&gt;The equation &lt;span class=&quot;math-container&quot;&gt;$(-8)^{1/3}=-2$&lt;/span&gt; in isolation is taught in early algebra. Later, in precalculus, on learns about the Fundamental Theorem of Algebra. At this point, one starts to understand that this equation should be seen more generally in the context of roots of the polynomial &lt;span class=&quot;math-container&quot;&gt;$x^3+8=0$&lt;/span&gt;. One learns there are three roots, and falls back on the fact that &lt;span class=&quot;math-container&quot;&gt;$(-2)^3=-8$&lt;/span&gt; to find one such root. Using the factor theorem and polynomial division with &lt;span class=&quot;math-container&quot;&gt;$\frac{x^3+8}{x+2}$&lt;/span&gt;, one reduces the problem of finding the other two roots to using the quadratic equation to solve &lt;span class=&quot;math-container&quot;&gt;$x^2-2x+4=0$&lt;/span&gt;. Typically these two solutions in precalculus are left in the form &lt;span class=&quot;math-container&quot;&gt;$x=1\pm i\sqrt{3}$&lt;/span&gt;. This is where many students stop developmentally, as described in Ben Crowell's answer.&lt;/p&gt;&#xA;&lt;p&gt;Students going further in fields such as physics, engineering, and mathematics will develop further mathematically, and become comfortable with the polar representation of complex numbers. At this stage, the student will see the solutions of &lt;span class=&quot;math-container&quot;&gt;$z^3+8=0$&lt;/span&gt; best represented by &lt;span class=&quot;math-container&quot;&gt;$z=2\exp\left(\frac{(2k+1)i\pi}3\right)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$k=0,1,2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So as others have pointed out in comments and answers, teaching that &lt;span class=&quot;math-container&quot;&gt;$(-8)^{1/3}=-2$&lt;/span&gt; in elementary algebra is not going to be a problem for those who take more mathematics to become physicists, engineers, and mathematicians. In fact, teaching this arithmetic fact is the basic foundation for a future higher level understanding.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2020-09-14T14:15:03.583" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18844" PostTypeId="1" CreationDate="2020-09-15T04:39:48.277" Score="0" ViewCount="119" Body="&lt;p&gt;When I participated in the 2019 John Hopkins Math Tournament, the team round consisted of two separate topics, 40 minutes per round, and 3 people per team. My team suffered on the topology round. Only one team managed to get through more than a third of the round, and that was because one of their team members had taken topology before.&lt;/p&gt;&#xA;&lt;p&gt;Now that I'm in college and actually taking topology, I decided to revisit the round. The questions are easy*, but 40 minutes for 13 problems, some of which contain multiple parts or subproblems, is just not enough time. I'm sure that even my professor would struggle to scribble down an answer for each question in reasonable detail within the time limit. With that being said, my specific question is as such: &lt;strong&gt;How long should the team round be to ensure that students with knowledge of set theory, but little to no prior knowledge of topology, would be able to solve at least half of the questions?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://math.jhu.edu/%7Emathclub/assets/2019/%5Bproof1%5DPointSet.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2019 JHMT Topology Round&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;*In the sense that they serve as normal difficulty homework problems.&lt;/p&gt;&#xA;" OwnerUserId="11711" LastActivityDate="2020-09-15T04:39:48.277" Title="What should the time limit for this high school math competition topology round be?" Tags="|topology|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18845" PostTypeId="1" AcceptedAnswerId="18855" CreationDate="2020-09-15T13:35:21.580" Score="18" ViewCount="3009" Body="&lt;p&gt;Addition and multiplication are commutative. Denoting &lt;span class=&quot;math-container&quot;&gt;$\circ$&lt;/span&gt; as either such operation, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x \circ y = z \Leftrightarrow y \circ x = z.$$&lt;/span&gt;&#xA;Subtraction and division have a similar property, where&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x \circ y = z \Leftrightarrow x \circ z = y.$$&lt;/span&gt;&#xA;Is there a name for this latter property?&lt;/p&gt;&#xA;&lt;p&gt;I would like to teach my child, who has learned about commutativity and who has observed the second property casually. Giving the property a name would make it more explicit.&lt;/p&gt;&#xA;" OwnerUserId="3381" LastEditorUserId="376" LastEditDate="2020-09-15T16:45:56.500" LastActivityDate="2020-09-16T18:23:29.743" Title="Does this property of subtraction and division have a name?" Tags="|terminology|arithmetic|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18846" PostTypeId="2" ParentId="18845" CreationDate="2020-09-15T19:35:49.853" Score="11" Body="&lt;p&gt;I have never seen a name for this property specifically. When I was in grade school, I recall learning about &lt;a href=&quot;https://www.splashlearn.com/math-vocabulary/number-sense/fact-family&quot; rel=&quot;noreferrer&quot;&gt;Fact Families&lt;/a&gt;, which are generated by this property. The idea is that a fact family is all of the arithmetic equations generated by the same numbers.&lt;/p&gt;&#xA;&lt;p&gt;This property in particular is really just a consequence that subtraction is the inverse of addition which is commutative. You can see this (using algebra, so probably not appropriate for a young child) by adding y to both sides. You then have your commutativity example above. This is &lt;em&gt;not&lt;/em&gt; anti-commutativity as one comment said, although subtraction &lt;em&gt;is&lt;/em&gt; anti-commutative.&lt;/p&gt;&#xA;&lt;p&gt;As there is not a common name for this property (other than a reference to fact families), I agree with the suggestion @Ben Crowell to let your child give it a name.&lt;/p&gt;&#xA;" OwnerUserId="14520" LastActivityDate="2020-09-15T19:35:49.853" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="18847" PostTypeId="1" CreationDate="2020-09-15T22:54:22.813" Score="4" ViewCount="61" Body="&lt;p&gt;I am looking for papers that have the similar style to Hervé Lehning's  1989 &lt;em&gt;The American Mathematical Monthly&lt;/em&gt; article &lt;a href=&quot;https://www.lehning.eu/uploads/1/2/0/8/120804163/from_experimentation_to_proof_maa_1989.pdf&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;From Experimentation to Proof&amp;quot; (PDF link via lehning.eu)&lt;/a&gt;. It's like teaching-oriented papers.&lt;/p&gt;&#xA;&lt;p&gt;Here's what the author said about the style at the end of his paper:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I described this small example in such detail because I think we do not distinguish enough between research and proofs in our teaching. Of course, we all know that mathematical activity requires an experimental phase. But, too often we imitate Bourbaki's way of writing, which hides the research and the trial phases completely. So, some students see mathematics as a dead science whereas mathematics is very much alive.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14613" LastActivityDate="2020-09-15T22:54:22.813" Title="Looking for papers with teaching-oriented style" Tags="|undergraduate-education|teaching|writing|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18848" PostTypeId="2" ParentId="18845" CreationDate="2020-09-16T02:52:15.223" Score="1" Body="&lt;p&gt;I don't know if this word is used specifically to describe this phenomenon, but the term &amp;quot;complement&amp;quot; is used in general to refer to two things that combine to make some third thing, so this applies here. When we subtract &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, we're basically asking what &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;'s (additive) complement with respect to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is.&lt;/p&gt;&#xA;&lt;p&gt;Another term that could be considered to apply is &amp;quot;conjugate&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2020-09-16T02:52:15.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18849" PostTypeId="2" ParentId="18795" CreationDate="2020-09-16T02:56:46.387" Score="4" Body="&lt;h2&gt;Introduction&lt;/h2&gt;&#xA;&lt;p&gt;This is some sort of “apology” for teaching homogeneous ODEs.  I think there’s a certain beauty and simplicity to them.  That beauty is overlooked in most textbooks, which Rota sourly criticizes.  Even one of my favorite older textbooks (G. Simmons), which does a better job for conceptual and qualitative development than its contemporaries, merely frames homogeneous ODEs as “at the next level of complexity” above separable ones.  While Rota gives me the impression that taking or teaching introductory differential equations is hopeless, V.I. Arnold’s textbook has a much more exciting attitude toward a reformed differential equations course, but it is at too high a level for an introductory course in most US curricula.  Shirley Yap (&lt;a href=&quot;https://www.jstor.org/stable/10.4169/002557010x479947&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://www.maa.org/press/periodicals/visualizing-and-utilizing-symmetries-of-differential-equations&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) does a good job of starting with material within reach of second or third year students.&lt;/p&gt;&#xA;&lt;p&gt;One of the nice things about homogeneous ODEs is that it allows students to see mathematics at work.  In 35 years of teaching, I have taught differential equations only about a dozen times, but I have never taught an engineering student in it.  I have taught students who mainly do not become mathematicians or even mathematics majors, although in differential equation it might be mostly mathematics majors.  Nonetheless, they are students who find, sometimes to their surprise, that they appreciate how mathematics works.&#xA;At this level, the logic of a proof is ghostly to most students and does have any real life to it, until it can be attached to concrete examples.  Richard Hamming wrote that the purpose of computing is not numbers but insight.  If you show them, the student can see how mathematics works in homogeneous ODEs.&lt;/p&gt;&#xA;&lt;p&gt;There is a beauty to mathematics that is accessible to a child.  Ours brains a wired to make mathematics out of ordinary experience.  That beauty can grow continuously on through a Ph.D. program and into the life of a professional mathematician.  I agree with whoever says that the beauty a Ph.D. sees is difficult to translate to an average college student fresh out of high school.  Such students have their own ability to see beauty in mathematics.  The instructor should strive to understand that.  It is more difficult than harkening back to one’s days as a teenager, because you made your choices to pursue mathematics for reasons that probably do not apply to most of your students.  If you succeed in showing them the beauty, they may follow you in your wake for a little while. Eventually, one hopes, the things they are passionate about will guide them, and many will choose a path other than mathematics.&lt;/p&gt;&#xA;&lt;p&gt;It’s time to bring this introduction to a close.  It should be obvious to any teacher that they have to consider their actual students.   I’ll present some things that make homogeneous equations interesting.  Some of these ideas are really for the instructor and may be too much of a stretch for your students.  But the instructor may understand more than the student.  All the ideas are not applications, but they give reasons why homogeneous ODEs should be part of the curriculum, some of which reasons may be reasons why it is still a part of the curriculum.&lt;/p&gt;&#xA;&lt;h2&gt;Background&lt;/h2&gt;&#xA;&lt;p&gt;Before we start, let’s set up a few ways to look at homogeneous equations.  The first is the ODE in &lt;strong&gt;derivative form&lt;/strong&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y’=f(x,y) \tag{1}$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$f(tx,ty)=f(x,y)$&lt;/span&gt; is homogeneous of degree zero.&#xA;Taking &lt;span class=&quot;math-container&quot;&gt;$t=1/x$&lt;/span&gt;, we see &lt;span class=&quot;math-container&quot;&gt;$f(x,y)=f(1,y/x)={\tilde f}(y/x)$&lt;/span&gt; can be reduced to a function of one variable &lt;span class=&quot;math-container&quot;&gt;${\tilde f}(m)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$m=y/x$&lt;/span&gt;.&#xA;We can also express the ODE in &lt;strong&gt;differential form&lt;/strong&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$M(x,y)\,dx + N(x,y)\,dy = 0 \,, \tag{2}$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$f(x,y)=-M(x,y)/N(x,y)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; are defined up to a common factor and homogeneous of the same degree:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$M(tx,ty)=t^a M(x,y),\quad N(tx,ty)=t^a N(x,y) \,.$$&lt;/span&gt;&#xA;Finally we can write an equivalent &lt;strong&gt;autonomous system&lt;/strong&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\eqalign{&#xA; {dx\over dt} &amp;amp;= -N(x,y) \cr&#xA; {dy\over dt} &amp;amp;= \phantom{-}M(x,y) \cr&#xA;} \tag{3}&#xA;$$&lt;/span&gt;&#xA;In this set up, the (phase) velocity field &lt;span class=&quot;math-container&quot;&gt;$(-N,M)$&lt;/span&gt; is orthogonal to the vector field &lt;span class=&quot;math-container&quot;&gt;$(M,N)$&lt;/span&gt; in (2), and &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; are likewise defined up to a common factor.&#xA;Both fields are homogeneous and define tangent and normal directions to the phase curves at each non singular point.&lt;/p&gt;&#xA;&lt;p&gt;I will call the differential equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y’=f(x) \tag{4}$$&lt;/span&gt;&#xA;the &lt;em&gt;&lt;strong&gt;basic differential equation&lt;/strong&gt;&lt;/em&gt;, which is solvable by quadrature (that is, the solution can be expressed in terms of an integral):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y = \int f(x)\;dx + C \,.$$&lt;/span&gt;&#xA;Of course, the student may still be disappointed if they cannot calculate the integral.  The basic equation (4) is characterized by the fact that its direction/vector field admits the symmetry given by “vertical” translation; that is, the equation is invariant under the transformation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(x,y) \longmapsto (x, y+C) \,,$$&lt;/span&gt;&#xA;which translates the dependent variable &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and leaves the independent variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; fixed.  A way to say this graphically is that for all vertical lines &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt;, each solution curve crosses the line &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; with the same slope.  All symbolic methods of solving ODEs come down to transforming an ODE into such a form.  One takes a shortcut in separable ODEs because you can get from &lt;span class=&quot;math-container&quot;&gt;$dw = dy/g(y) = dx/f(x)$&lt;/span&gt; to the solution without actually writing out the transformation to the variable &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt;, which in some cases is hard to solve explicitly.  When you can see for an ODE of the general form (1) and a new variable &lt;span class=&quot;math-container&quot;&gt;$u=u(x,y)$&lt;/span&gt; that each solution curve crosses the curve &lt;span class=&quot;math-container&quot;&gt;$u=a$&lt;/span&gt; with the same slope, then you have found a way to solve the ODE.  In the case of separable equations, the slope varies according with the transformation to &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; that is skipped.  In the case of a homogeneous ODE, it is scaled as follows.&lt;/p&gt;&#xA;&lt;p&gt;One of the most common variable transformations to consider is &lt;span class=&quot;math-container&quot;&gt;$u = \ln x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$u = \ln y$&lt;/span&gt;.  It converts scaling by &lt;span class=&quot;math-container&quot;&gt;$t = e^C$&lt;/span&gt; into translation by &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;.  Translation and scaling (and perhaps its cousin, rotation) are very common symmetries in the sciences.  It is also connected to the differential &lt;span class=&quot;math-container&quot;&gt;$du = dx/x$&lt;/span&gt; representing relative growth.  Note the substitution &lt;span class=&quot;math-container&quot;&gt;$u = \ln x$&lt;/span&gt; is just one of a family of substitutions &lt;span class=&quot;math-container&quot;&gt;$u= \ln ty = \ln y+C$&lt;/span&gt; that solve the ODE; we tend to pick a convenient one.&lt;/p&gt;&#xA;&lt;h3&gt;Remarks on homogeneous equations and scaling&lt;/h3&gt;&#xA;&lt;p&gt;First, students like &lt;strong&gt;solving specific problems&lt;/strong&gt;.  Having some at hand is useful for building their confidence.  Homogeneous ODEs are fairly easy but nontrivial problems.  However, &lt;span class=&quot;math-container&quot;&gt;$e^{y/x}$&lt;/span&gt; is a what-the-heck-is-that sort of function bred only in the captivity of a diff. eq. book.&lt;/p&gt;&#xA;&lt;p&gt;Students do need to understand &lt;strong&gt;substitutions and transformations of ODEs&lt;/strong&gt;. And homogeneous ODEs are fairly easy but nontrivial problems in this context, too.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;strong&gt;physical science&lt;/strong&gt;, if homogeneous ODEs arise, then &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; normally have to have the same physical dimensions.  This is rare in science, but we will come back to this.  One exception is that reduction of order transforms the harmonic oscillator to a first-order homogeneous ODE; see the remarks on linear equations below.&lt;/p&gt;&#xA;&lt;p&gt;Autonomous systems from physical science in the form (3) sometimes become a homogeneous ODE of the form (1)/(2) when the dimension time is eliminated.  Such a physical problem came up in the last few days on Mathematica.SE in a pair of related Q&amp;amp;A &lt;a href=&quot;https://mathematica.stackexchange.com/questions/229656/two-first-order-non-linear-differential-equation-using-dsolve-to-obtain-the-sol&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://mathematica.stackexchange.com/questions/229818/dsolve-leaks-internal-error-messages-when-attempting-to-solve-two-first-order-au&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;strong&gt;geometry,&lt;/strong&gt;&#xA;it is more common that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; have the same dimensions, say, length.  One can find applications in geometry.  For instance, the orthogonal trajectories of a homogeneous ODE are the solutions of another homogeneous ODE; so both equations are solvable, provided one can compute the integrals that arise.&lt;/p&gt;&#xA;&lt;p&gt;Geometric interpretation of the substitution for homogeneous equations:&#xA;A homogeneous ODE in terms of &lt;span class=&quot;math-container&quot;&gt;$y(x)$&lt;/span&gt; can be written as a separable ODE in polar coordinates &lt;span class=&quot;math-container&quot;&gt;$dr/r = g(\theta)\,d\theta$&lt;/span&gt; in terms of the &lt;strong&gt;dimensionless angle&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; or in the form &lt;span class=&quot;math-container&quot;&gt;$dx/x = g(m)\,dm$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$m=\tan\theta$&lt;/span&gt;, in terms of the &lt;strong&gt;dimensionless slope&lt;/strong&gt; of the line through the origin and &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; (the standard substitution).&lt;/p&gt;&#xA;&lt;p&gt;The &lt;strong&gt;scaling transformation&lt;/strong&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$(x,y) \mapsto (tx,ty)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$t&amp;gt;0$&lt;/span&gt;, is a symmetry of a homogeneous vector field, and a homogeneous ODE is invariant under scaling:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ {d(ty) \over d(tx)} = f(tx, ty) \Longrightarrow {dy \over dx} = f(x,y)\,.$$&lt;/span&gt;&#xA;This means that the slope is constant along radial lines.  Therefore we can use the radial lines to construct a new independent variable &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;, such as the slope &lt;span class=&quot;math-container&quot;&gt;$z=m$&lt;/span&gt; or polar angle &lt;span class=&quot;math-container&quot;&gt;$z=\theta$&lt;/span&gt;, and the scaling symmetry implies the dependent &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; variable will appear in the transformed ODE as &lt;span class=&quot;math-container&quot;&gt;$du/u = g(z)\,dz$&lt;/span&gt;.  For instance, with &lt;span class=&quot;math-container&quot;&gt;$y=mx$&lt;/span&gt;, the ODE (1) becomes&#xA;&lt;span class=&quot;math-container&quot;&gt;$${dx \over x} = {dm \over f(1,m) - m}  \,.$$&lt;/span&gt;&#xA;With &lt;span class=&quot;math-container&quot;&gt;${x, y} = (r \cos\theta, r \sin\theta)$&lt;/span&gt;, the ODE (1) becomes&#xA;&lt;span class=&quot;math-container&quot;&gt;$${dr \over r} = {f(\cos \theta, \sin \theta) \sin \theta + \cos \theta \over f(\cos \theta, \sin \theta) \cos \theta - \sin \theta} \, d\theta\,.$$&lt;/span&gt;&#xA;My assertion here is that it can be &lt;em&gt;seen&lt;/em&gt; that these substitutions will reduce the ODE (1) to quadrature in the same way the basic equation (4) , or a separable equation, can be seen to be solved by quadrature.  What I mean is that students can be taught to see it.  They do have to learn the ideas in order to improve their perception.  The &lt;a href=&quot;http://www.sci.csueastbay.edu/%7Eyap/paper.dir/Loci.dir/intro.dir/intro.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;intro section&lt;/a&gt; to the web book, &lt;a href=&quot;https://www.maa.org/press/periodicals/visualizing-and-utilizing-symmetries-of-differential-equations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yap, &lt;em&gt;Visualizing and Utilizing Symmetries of Differential Equations&lt;/em&gt;&lt;/a&gt;, has some nice animations scaling a homogeneous ODE.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Now we come to ideas that go beyond a typical first course in ODEs in the US.  The first idea, quasi-homogeneous ODEs, is accessible and is less important than homogeneous ones, about which the OP already expresses doubts.  The rest provides a broader context, which the instructor may be able to use from time to time.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Quasi-homogeneous&lt;/strong&gt; or &lt;strong&gt;weighted homogeneous&lt;/strong&gt;&#xA;ODES are a generalization of homogeneous ones.  (I prefer “weighted” but “quasi,” while more opaque, seems more common.)  An ODE is quasi-homogeneous if it is invariant under the scaling transformation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(x,y) \longmapsto (t^a x, t^b y) \,.$$&lt;/span&gt;&#xA;Instead of &lt;span class=&quot;math-container&quot;&gt;$m = y/x$&lt;/span&gt;, the equation (1) is transformed into a separable ODE by &lt;span class=&quot;math-container&quot;&gt;$u=y^a/x^b$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$u = x^b/y^a$&lt;/span&gt;.  For instance, &lt;span class=&quot;math-container&quot;&gt;$dy/dx = (x+3y^2)/(y^3-2xy)$&lt;/span&gt; is quasi-homogeneous with weights &lt;span class=&quot;math-container&quot;&gt;$a = 2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b = 1$&lt;/span&gt;; substitute &lt;span class=&quot;math-container&quot;&gt;$x = uy^2$&lt;/span&gt; and one obtains a separable equation. Geometrical problems might be quasi-homogeneous if the variables have different dimensions such as length and area, but I can’t recall any.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Transformation of solutions.&lt;/strong&gt;&#xA;Symmetries of an ODE map a solution to a solution.  It is sometimes possible to construct the general solution from a particular solution by applying the symmetry group.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Dimensional analysis.&lt;/strong&gt;&#xA;Despite the fact that models in science deal with quantities whose dimensions are not homogeneous, dimensional analysis can sometimes be used to find dimensionless variables.  I don’t know any good examples at the intro level that are not already separable.  However there is one application that might be worth mentioning, &lt;a href=&quot;https://en.wikipedia.org/wiki/Blasius_boundary_layer&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Prandtl-Blasius problem of viscous flow&lt;/a&gt; over a flat plate.  It involves a system of PDEs in space.  Dimensional analysis can be used to reduce the problem to solving the Blasius equation &lt;span class=&quot;math-container&quot;&gt;$2y’’’ +y’’y=0$&lt;/span&gt;, which cannot be solved symbolically.  However, it is quasi-homogeneous (&lt;span class=&quot;math-container&quot;&gt;$a=-b=1$&lt;/span&gt;) and has other symmetries, so that just a single numerical solution can be mapped to any other solution symbolically without any further numerical methods.  (See Bluman and Anco, &lt;em&gt;Symmetry and Integration Methods for Differential Equations&lt;/em&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Infinitesimal generator.&lt;/strong&gt;&#xA;There is a connection between the symmetries of a differential equation and the substitution that transforms it into an ODE solvable by quadrature.  First we need to write the symmetries in the form of a one-parameter group of transformations, &lt;span class=&quot;math-container&quot;&gt;$(x,y) \mapsto (x^*(s), y^*(s))$&lt;/span&gt;, with say &lt;span class=&quot;math-container&quot;&gt;$(x^*(0),y^*(0))=(x,y)$&lt;/span&gt; being the identity transformation and the composition of the transformations for &lt;span class=&quot;math-container&quot;&gt;$s=s_1,s_2$&lt;/span&gt; being equal to the transformation for &lt;span class=&quot;math-container&quot;&gt;$s=s_1+s_2$&lt;/span&gt;.&#xA;For homogeneous scaling, we have &lt;span class=&quot;math-container&quot;&gt;$t=e^s$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x^*(s),y^*(s))=(e^s x,e^s y)$&lt;/span&gt;. Then the &lt;em&gt;infinitesimal generator&lt;/em&gt; is the differential operator&#xA;&lt;span class=&quot;math-container&quot;&gt;$$X = \left.\left({\partial x^* \over \partial s}\, {\partial \over \partial x} + {\partial y^* \over \partial s}\, {\partial \over \partial y} \right)\right|_{s=0} \,.$$&lt;/span&gt;&#xA;This is the directional derivative &lt;span class=&quot;math-container&quot;&gt;$v_0 \cdot \nabla$&lt;/span&gt; with respect to the (non-normalized) initial velocity vector &lt;span class=&quot;math-container&quot;&gt;$v_0$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;, where the “velocity” is given by &lt;span class=&quot;math-container&quot;&gt;$v_0=n \left.({\partial x^* / \partial s}, {\partial y^* / \partial s})\right|_{s=0}$&lt;/span&gt;.&#xA;Then the substitution we seek is one that satisfies&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Xu=0 \,.$$&lt;/span&gt;&#xA;Intuitively, this ensures that locally the curves &lt;span class=&quot;math-container&quot;&gt;$u=a$&lt;/span&gt; cross solutions curves transversely; recall the characteristic property of the basic equation.&#xA;For homogeneous scaling,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$X = x\,{\partial \over \partial x} + y\, {\partial \over \partial y} \,.$$&lt;/span&gt;&#xA;One can check that &lt;span class=&quot;math-container&quot;&gt;$u=y/x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$u=\theta=\arctan(y/x)$&lt;/span&gt;, and more generally any &lt;span class=&quot;math-container&quot;&gt;$u=u(y/x)$&lt;/span&gt; is a solution to &lt;span class=&quot;math-container&quot;&gt;$Xu=0$&lt;/span&gt;.&#xA;Thus there is a beautiful theory that geometrically connects up what may seem a hodgepodge of algebraic tricks for finding solutions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Other connections to scaling symmetries&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Linear homogeneous&lt;/strong&gt;&#xA;ODEs (&lt;span class=&quot;math-container&quot;&gt;$f(x,y)=P(x)\,y$&lt;/span&gt;) are invariant under the scaling &lt;span class=&quot;math-container&quot;&gt;$(x,y)\mapsto(t^0 x,ty)$&lt;/span&gt;.  The substitution with weights &lt;span class=&quot;math-container&quot;&gt;$a=0,b=1$&lt;/span&gt; is trivial, &lt;span class=&quot;math-container&quot;&gt;$u=x$&lt;/span&gt;, because the ODE is already separable into the form &lt;span class=&quot;math-container&quot;&gt;$dy/y = P(x)\,dx$&lt;/span&gt;.  For higher-order equations with constant coefficients, the substitution &lt;span class=&quot;math-container&quot;&gt;$u = \ln y$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$y=e^{u(x)}$&lt;/span&gt; suggested by the scaling leads to a differential equation for which &lt;span class=&quot;math-container&quot;&gt;$u(x) = m x + b$&lt;/span&gt; “obviously” yields an algebraic equation in which &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; does not matter.  (The standard method is obvious enough, at least after trying it once or twice, but people claim it’s an unmotivated trick.)  Second-order linear autonomous homogeneous ODES such as the damped harmonic oscillator equation become first-order nonlinear homogeneous ODES with the standard reduction of order transformation, &lt;span class=&quot;math-container&quot;&gt;$\dot x = v$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\ddot x = v \, dv/dx$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$a\,{d^2 x \over dt^2}+b\,{dx \over dt} +c\, x==0&#xA; \longmapsto a\,v\,{dv \over dx} + b\,v+c\,x==0 \,.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Euler equidimensional equation&lt;/strong&gt;&lt;/a&gt;&#xA;has an added symmetry.  Being linear, it is invariant under scaling &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.  But being “equidimensional,” it is also invariant under scaling &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  Unlike a homogeneous&#xA;equation, the equidimensional equation is invariant under scalings of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; separately.  Picking different substitutions for each, &lt;span class=&quot;math-container&quot;&gt;$x = e^u$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=e^{ku}$&lt;/span&gt;, the differential equation imposes a relation on them; that is, we can plug in &lt;span class=&quot;math-container&quot;&gt;$y=x^k$&lt;/span&gt;, the standard ansatz which comes from eliminating &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; from our two substitutions, to determine &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; by plugging into the ODE.&lt;/p&gt;&#xA;&lt;h2&gt;Summary&lt;/h2&gt;&#xA;&lt;p&gt;Pros:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Substitution is an important technique.  Practice is good.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Reduction to separable equations and finding implicit solutions are also important experiences for building confidence.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Homogeneity, scaling, the exponential function.  Scaling symmetries are common and are connected to the importance of logarithms and exponential functions in the mathematical sciences.  Homogeneous ODEs are a somewhat special case, but they illustrate the scaling symmetries in an accessible, calculable way.  They are certainly the easiest example (other than the basic equation) with which to show the role of symmetry.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Scaling symmetries are also important throughout differential equations.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Homogeneous functions are important in other branches of mathematics.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Cons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;There are few practical applications at the intro level.  Hence it is hard to motivate as an important general technique.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Homogeneous ODEs are a special case and perhaps not common enough to justify the time spent on them.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Introducing homogeneous ODEs merely for the sake of having something to solve seems stupid.  Except that when people are asked to do things for seemingly stupid reasons, some of them keep asking themselves why and discover things on their own, which is good but not good enough.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Few textbooks support this approach.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="498" LastActivityDate="2020-09-16T02:56:46.387" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18850" PostTypeId="2" ParentId="11950" CreationDate="2020-09-16T04:07:22.413" Score="0" Body="&lt;p&gt;I don't know how to explain it. I sometimes have a natural passion to do some of my own thinking from scratch independently of what I thought before. I kind of have an idea of how I could get a deep understanding from scratch about why the associative law of natural number addition is true and like it. The study of what I have the potential to do is kind of a topic in and of itself to see and notice things in and like the feeling of when my mind is on it. Maybe it really is for the things I see and notice right then.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastActivityDate="2020-09-16T04:07:22.413" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18851" PostTypeId="2" ParentId="15449" CreationDate="2020-09-16T04:23:26.373" Score="1" Body="&lt;p&gt;I know it's probably impossible for you to completely figure out what it really is that's going to make her happy and the best you can do is get some idea of it. If she could find a way to figure out more about her true feelings, she might get a better idea of it and then if this question explained it better, I might be able to get a good idea of what will suit her. If she has the brain that won't resist it too strongly, what might work best is for her to just keep doing what ever she feels like doing right then and not work so hard to make her future go a certain way because she will be happier if she can naturally live in the moment and have no to do lists and not work so hard to keep doors open to certain things for later and learn how to quickly adapt to what ever the choices are after she burns her bridges. I don't know what will work.&lt;/p&gt;&#xA;" OwnerUserId="11718" LastActivityDate="2020-09-16T04:23:26.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18852" PostTypeId="1" AcceptedAnswerId="18854" CreationDate="2020-09-16T08:29:58.270" Score="2" ViewCount="550" Body="&lt;p&gt;When I teach basic math I want to emphasize on it's power (algebraic part for starters) as a tool for solving certain problems you cannot solve with &lt;em&gt;naked&lt;/em&gt; brain, so that one models a problem with mathematical notation and then applies simple math (algebra) to solve it. I noticed that most of the tasks learners tend to crack up with just their brain, thus losing potential motivation to learn more abstract math which has to power you up for future more complicated problems.&lt;/p&gt;&#xA;&lt;p&gt;Are there some cool/fun/practical tasks to be used for modeling a system of 2 (maybe 3 at most) equations?&lt;/p&gt;&#xA;&lt;p&gt;P.S. For now I have something like this:&lt;br /&gt;&#xA;Allegra has a secret cake recipe everyone likes. It's made of just 2 ingredients and weights 600 grams. Alice wants to figure out the recipe so she sneaks into Allegra's granary and discovers that ingredient &lt;strong&gt;x&lt;/strong&gt; gets used up 4 times faster than ingredient &lt;strong&gt;y&lt;/strong&gt;. How can Alice figure out exact weights for ingredients in the recipe?&lt;/p&gt;&#xA;&lt;p&gt;which some still manage to &lt;em&gt;brute force&lt;/em&gt; (without algebra usage) &lt;code&gt;x = 480; y = 120&lt;/code&gt;&lt;/p&gt;&#xA;" OwnerUserId="14617" LastActivityDate="2024-01-30T16:03:59.363" Title="Practical case for solving with system of 2 equations" Tags="|algebra|teaching|problem-solving|" AnswerCount="8" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18853" PostTypeId="2" ParentId="18845" CreationDate="2020-09-16T08:59:12.723" Score="5" Body="&lt;p&gt;A helpful way to rewrite that statement would be (assuming subtraction for simplicity):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x - y - z ⇔ x - z - y$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We are observing how swapping y and z does not change the value of the expression. While it may initially look like there is a useful property behind this, the example is showing an easy case of what you are allowed to swap. Here is a visual representation of what swaps are allowed.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7qMa8.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7qMa8.png&quot; alt=&quot;Expression trees&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;These are expression trees. The filled in circles can be swapped with other circles that are filled in with the same color. The first tree shows the original case of &lt;span class=&quot;math-container&quot;&gt;$x - y - z$&lt;/span&gt; and shows how &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; can be swapped. The next tree represents &lt;span class=&quot;math-container&quot;&gt;$(a - b) - (c - d)$&lt;/span&gt;. The third tree is another full binary tree, but with twice as many variables and represents &lt;span class=&quot;math-container&quot;&gt;$((a - b) - (c - d)) - ((e - f) - (g - h))$&lt;/span&gt;. You can normalize the first tree into the second by replacing &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; in the original expression with &lt;span class=&quot;math-container&quot;&gt;$z - 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you just look at the first tree it may seem like you should be able to swap children on the right side, but once you start looking at the other trees, you should notice that the pattern is no longer quite as simple. One way to describe the pattern would be that a node's right child is able to swap with that node's other child's right child. This may be more restrictive than you initially expected when comparing this property to commutativity.&lt;/p&gt;&#xA;&lt;p&gt;I do not know a name for this property, but it is not as useful as commutativity due to the number of restrictions on being able to swap nodes. If you run into this you may want to try and find a different approach that can use commutativity such as converting subtraction into addition of negative number, or by trying to make as many nodes of the tree swapable by for example not allowing parenthesis to be used.&lt;/p&gt;&#xA;" OwnerUserId="14618" LastActivityDate="2020-09-16T08:59:12.723" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18854" PostTypeId="2" ParentId="18852" CreationDate="2020-09-16T16:05:41.443" Score="5" Body="&lt;p&gt;Here's one that students always enjoy.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A bottle and a cork cost 1 dollar and ten cents.   The bottle costs $1.00 more than the&#xA;cork.  How much does each cost?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Student often think that it is one dollar for the bottle and 10 cents for the cork. That's incorrect because \$1.00 isn't \$1.00 more than 10 cents but the answer is easily discovered with algebra.&lt;/p&gt;&#xA;&lt;p&gt;You might want to update it for items that are more realistically priced and more relevant to students' lives.  Even if you do, I am not sure you will consider it a practical problem.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="117" LastEditDate="2020-09-17T01:00:43.443" LastActivityDate="2020-09-17T01:00:43.443" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18855" PostTypeId="2" ParentId="18845" CreationDate="2020-09-16T18:23:29.743" Score="19" Body="&lt;p&gt;This is &amp;quot;left &lt;a href=&quot;https://en.wikipedia.org/wiki/Involution_(mathematics)&quot; rel=&quot;noreferrer&quot;&gt;involution&lt;/a&gt;&amp;quot;.  (&amp;quot;left&amp;quot; because it doesn't work when you try it on the right.)&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;x \circ y &amp;amp;= z  &amp;amp;  \\&#xA;x \circ (x\circ y) &amp;amp;= x \circ z  &amp;amp;  [\text{apply $x \circ -$}]  \\&#xA;y &amp;amp;= x \circ z  &amp;amp;  [\text{simplify the involution}]  \text{.}&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I would be shocked to see anyone use that term outside of some very narrow niches.  In ring theory, an involution is an &lt;a href=&quot;https://en.wikipedia.org/wiki/Antihomomorphism&quot; rel=&quot;noreferrer&quot;&gt;antiisomorphism&lt;/a&gt;, which brings us around to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Anticommutative_property&quot; rel=&quot;noreferrer&quot;&gt;anticommutativity&lt;/a&gt; mentioned in comments and other answers.&lt;/p&gt;&#xA;" OwnerUserId="5800" LastActivityDate="2020-09-16T18:23:29.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18856" PostTypeId="2" ParentId="10977" CreationDate="2020-09-16T20:09:13.123" Score="1" Body="&lt;p&gt;In my experience, guided notes seem to be correlated to poor learning. While preference for guided notes tends to split about 50/50, invariably the bottom half in terms of academic performance is dominated by people who use guided notes. Now is this causal? I'm not sure; it seems reasonable that &amp;quot;poor&amp;quot; students might prefer the easy-to-use notes over &amp;quot;good&amp;quot; students who want to do things their own way, so it could be that they are self-selecting. Either way, I have seen no benefit to using them at all.&lt;/p&gt;&#xA;" OwnerUserId="14622" LastActivityDate="2020-09-16T20:09:13.123" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18857" PostTypeId="2" ParentId="18852" CreationDate="2020-09-16T21:32:04.017" Score="3" Body="&lt;p&gt;Another common problem often used in school algebra classes are of the form, similar to the following:&lt;/p&gt;&#xA;&lt;p&gt;Maria and Juan are siblings.  The sum of their ages now is 16.  In four years, Maria will be twice as old as her brother Juan.&lt;/p&gt;&#xA;&lt;p&gt;What are Maria's and Juan's ages now?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Algebraically, we'd have the system of equations, with &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; representing Juan's current age, and &lt;em&gt;m&lt;/em&gt; representing Maria's current age:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align} j+m &amp;amp;= 16\\ \\&#xA;m+ 4 &amp;amp;=2(j+4) \end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="238" LastEditDate="2020-09-17T00:48:13.003" LastActivityDate="2020-09-17T00:48:13.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18858" PostTypeId="2" ParentId="18852" CreationDate="2020-09-16T23:33:53.130" Score="4" Body="&lt;p&gt;Algebra is a like a hammer that always works unlike arithmetics that may require inventive tricks, different for each problem. You may want to read a pertinent short story &lt;a href=&quot;https://www.ibiblio.org/eldritch/ac/tutor.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tutor by Anton Chekhov&lt;/a&gt;. The problem posed in the story can be solved arithmetically, algebraically and arithmetically with tools like abacus.&lt;/p&gt;&#xA;&lt;p&gt;Do you do word problems with your students? There are tons of word problems like a pool with pipes or two railroad trains or a guy rafting down or across a river that are solved with a system of two or three linear equations. These problems prepare elementary and middle-schoolers to algebra and physics.&lt;/p&gt;&#xA;&lt;p&gt;I think you using x and y in your problem is too on the nose, can you use actual names of components? I wonder what cake can be made just of two components unless it is a ready-made cake for which you need cake mix and water. Why not just calling them cake mix and water? Also, the word &amp;quot;faster&amp;quot; implies that there is a time component to this, which complicates the matters.&lt;/p&gt;&#xA;&lt;p&gt;Here is a basic word problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Two people left their respective towns simultaneously and started&#xA;walking over the same road that connected the towns towards each&#xA;other. The first pedestrian walked 24 miles until they met, walking at 4 miles per&#xA;hour. The other pedestrian walked at 5 miles per hour. What distance&#xA;have the second pedestrian walked before they met?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is a more fun one. Can be done either arithmetically (need to think a little) or algebraically (no need to think much, just use the hammer):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Every day an engineer arrives to a station at 8 a.m. by train. Exactly&#xA;at the same time a car, sent from a factory, drives up to the station,&#xA;picks up the engineer and takes him to the factory. One day the&#xA;engineer arrived at 7 a.m., decided not to wait for the car, and&#xA;started walking towards the car. When the car met the engineer, it&#xA;picked him up, turned back and arrived to the factory 20 minutes&#xA;earlier than usual. For how long did the engineer walk? Consider the&#xA;speeds of the engineer and the car constant.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A boat travels upstream the Mississippi river at full steam. When it passes Lexington&#xA;Bridge, a barrel with corn syrup falls overboard into the river. When the&#xA;loss is noticed 40 minutes later, the boat quickly turns back to chase&#xA;the barrel at full steam. The boat catches with the barrel at Great Western Bridge&#xA;four miles downstream. What is the speed of river current?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2020-09-17T00:31:04.543" LastActivityDate="2020-09-17T00:31:04.543" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18859" PostTypeId="2" ParentId="12211" CreationDate="2020-09-17T06:06:55.690" Score="1" Body="&lt;p&gt;Use concrete examples.&lt;/p&gt;&#xA;&lt;p&gt;Probability and its notation are so abstract to so many students that using context-free logic and symbols is all but guaranteed to not work with a huge share of students. They just don't trigger &lt;em&gt;any&lt;/em&gt; related knowledge in students' heads. One of the few truly good rules of thumb in teaching is that you should usually go from concrete to abstract and from prior knowledge to new knowledge. Going backwards rarely works.&lt;/p&gt;&#xA;&lt;p&gt;Starting here and going backwards rarely works.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(A \mid B) + P(A^C \mid B) = 1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Instead, how about:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1) The probability of a voter arriving at a voting station then filling in the ballot correctly is 87%. (a) What is the probability of a voter arriving at a voting station and NOT filling the form in correctly? How do you know? (b) Does the given info tell us anything at all about voters who vote by mail?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(2) Among students who start grade 12 before their 19th birthday, 6% fail to graduate within a year. (a) Among those students, what percentage DO not graduate within a year? How do you know? (b) Does this question tell us about students who start grade 12 courses as an adult?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(3) Use let statements and standard probability notation to express your thinking in parts (a) and (b) of the two questions above.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2020-09-17T06:06:55.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18860" PostTypeId="1" CreationDate="2020-09-17T10:20:52.447" Score="1" ViewCount="86" Body="&lt;p&gt;Highschool-courses, at least in my country, are structured in a way that you tackle with a higher level of a given subject after some time constraint. So, after one year, you get to a higher class where the questions are more difficult and concepts are more of a challenge to your skill level.&lt;/p&gt;&#xA;&lt;p&gt;Recently, I started thinking very deeply of how exactly we should teach students, and on that idea, I started browsing physics and mathematics stack exchange to see what the basic level questions which people have when they start studying a field (and also some self analysis).&lt;/p&gt;&#xA;&lt;p&gt;From my study (*),  I concluded that people have most confusion in the most basic concepts. For example, simple things such as resolution of vectors or the assumptions in the kinematic formula derivations are confusing to people. This made me think that perhaps we are rushing through the subject material more quickly than most people can grasp it.&lt;/p&gt;&#xA;&lt;p&gt;Or, maybe it is that people take much longer to grasp the basic things, so they have to figure it out on their own and it is something which each has to do their own.  I personally feel that this may not be the case but rather if you explain the correct basics to someone properly then they can do anything more complex easily.&lt;/p&gt;&#xA;&lt;p&gt;This phenomena came more clear to me when I started doing a college course, and started realizing that most of my confusion which I had with the experience of the same subject in high school was due to incomplete information. Surprisingly, the course I did had less complexity of application and more on the fundamentals which made me appreciate and understand the subject more. I couldn't understand why we were keeping the simple fundamentals for college and the complicated applications in high school!&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Therefore, I ask this question:&lt;/p&gt;&#xA;&lt;p&gt;How researched is this topic I am discussing? If yes, what are the conclusive findings on the best methods to deal with such hindrances in education? if not, then why aren't we researching this topic/not in large interest?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;*: I couldn't really get sample sizes and I found surveying people  individually to be a difficult task.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2020-09-17T10:20:52.447" Title="How long should we spend on teaching students the basics of subject?" Tags="|education-research|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18861" PostTypeId="1" CreationDate="2020-09-17T10:54:52.667" Score="5" ViewCount="636" Body="&lt;p&gt;I hope I am asking my question in the right forum.&lt;/p&gt;&#xA;&lt;p&gt;I am trying to introduce some mathematical problems (Better to be famous in the math community) to a group of senior high school students with a typical background in high school mathematics like (differentiation and applications - basic probability- basic plane geometry- little combinatorics, and graph theory and basic number theory and introductory linear algebra which are common to solve a system of linear equations.)&lt;/p&gt;&#xA;&lt;p&gt;Actually, my friends and I are trying to introduce mathematics to senior high school students; in between, we would like to present them with some famous math problems in which there were some unsuccessful attempts to solve them, but at last, they've been solved. Or famous wrong conjectures which were thought to be correct. Through this, we wanted to emphasize that even mathematicians at a high level can get wrong, and guessing wrong or failing an attempt shouldn't worry you in studying mathematics&lt;/p&gt;&#xA;&lt;p&gt;Thank you for helping me. I am also very grateful for other suggestions for our work, besides these.&lt;/p&gt;&#xA;" OwnerUserId="14626" LastEditorUserId="14626" LastEditDate="2020-09-17T11:15:10.347" LastActivityDate="2020-09-19T03:28:07.810" Title="What are some famous problems, which are not difficult to understand, for senior high school students" Tags="|undergraduate-education|secondary-education|examples|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18862" PostTypeId="2" ParentId="18861" CreationDate="2020-09-17T12:54:22.403" Score="8" Body="&lt;p&gt;Some problems that are easy to understand (but not understand the solution to):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem&quot; rel=&quot;noreferrer&quot;&gt;Fermat's Last Theorem&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Four_color_theorem&quot; rel=&quot;noreferrer&quot;&gt;Four Color Theorem&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Cubic_equation#Cardano%27s_formula&quot; rel=&quot;noreferrer&quot;&gt;Solving the Cubic&lt;/a&gt; and Quartic Equations&lt;/li&gt;&#xA;&lt;li&gt;The impossibility of the Quintic Equation&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedes%27s_cattle_problem&quot; rel=&quot;noreferrer&quot;&gt;Archimedes' Cattle Problem&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematics_of_Sudoku#Puzzles&quot; rel=&quot;noreferrer&quot;&gt;Sudoku problems&lt;/a&gt; (17 being the minimum amount of moves needed in order to be solvable)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Theorema_Egregium&quot; rel=&quot;noreferrer&quot;&gt;Theorema Egregium&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Heptadecagon&quot; rel=&quot;noreferrer&quot;&gt;Constructing a 17-gon&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Optimal_solutions_for_Rubik%27s_Cube&quot; rel=&quot;noreferrer&quot;&gt;Every Rubik's Cube can be solved in 20 or less moves&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://artofproblemsolving.com/wiki/index.php/1988_IMO_Problems/Problem_6&quot; rel=&quot;noreferrer&quot;&gt;1988 IMO Problem #6&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Unsolved problems are easy to understand the statement of:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Perfect_number&quot; rel=&quot;noreferrer&quot;&gt;Does an odd perfect pumber exist?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Twin_prime&quot; rel=&quot;noreferrer&quot;&gt;Twin Prime Conjecture&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Euler_brick&quot; rel=&quot;noreferrer&quot;&gt;Is there a perfect Euler Brick?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/P_versus_NP_problem&quot; rel=&quot;noreferrer&quot;&gt;P vs NP&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Collatz_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Collatz Conjecture&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Goldbach%27s_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Goldbach Conjecture&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Inscribed_square_problem&quot; rel=&quot;noreferrer&quot;&gt;Inscribed Square Problem&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Moving_sofa_problem&quot; rel=&quot;noreferrer&quot;&gt;Moving Sofa Problem&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Beal_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Beal Conjecture&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Congruent_number&quot; rel=&quot;noreferrer&quot;&gt;The Congruent Number Problem&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8527" LastEditorUserId="8527" LastEditDate="2020-09-17T14:53:55.350" LastActivityDate="2020-09-17T14:53:55.350" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18863" PostTypeId="1" AcceptedAnswerId="18864" CreationDate="2020-09-17T14:45:26.077" Score="7" ViewCount="4304" Body="&lt;p&gt;Previously I've explained some basic things of graphs to my kid, such as planar, &lt;span class=&quot;math-container&quot;&gt;$V-E+F=2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now when I introduce geometry, he asked, &amp;quot;Why we have to be so precise in Geometry?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Indeed, in graphs, vertices are just stones, and edges rubber bands; but things are different in Geometry, edges can't bend, vertices can't just shift.&lt;/p&gt;&#xA;&lt;p&gt;Why so? How could I explain the constraint?&lt;/p&gt;&#xA;" OwnerUserId="9059" LastActivityDate="2020-09-20T11:39:38.230" Title="Why we have to be so precise in Geometry?" Tags="|geometry|" AnswerCount="6" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="18864" PostTypeId="2" ParentId="18863" CreationDate="2020-09-17T16:41:34.877" Score="42" Body="&lt;p&gt;Because the aim of geometry is to study properties related to shape, size and length. Therefore, in the context of geometry, we cannot deform our objects because deformation changes these properties.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the properties in which graph theory is interested are independent of size, shape and length. Therefore, we do not care about them.&lt;/p&gt;&#xA;&lt;p&gt;(However, it is important to note that, from the mathematical point of view, we are &amp;quot;precise&amp;quot; in both.)&lt;/p&gt;&#xA;" OwnerUserId="13585" LastActivityDate="2020-09-17T16:41:34.877" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18866" PostTypeId="2" ParentId="18800" CreationDate="2020-09-17T22:24:28.017" Score="6" Body="&lt;p&gt;I don't know the sources... but (at my R1 university in math in the U.S.) many students from abroad do tend to use this. My comment on it is that it is all too easy to misread this. That is, it's not stable/robust from an information-theory point of view. &lt;em&gt;And&lt;/em&gt; I do advocate writing a &lt;em&gt;narrative&lt;/em&gt; (in natural language, e.g., English) rather than a parade of formal expressions without explanation other than formal justification. That is, I do not just want to see a computation, I want to first hear the &lt;em&gt;plan&lt;/em&gt; of the computation/deduction, and comments along the way about how the specific low-level details fit into that plan... etc.&lt;/p&gt;&#xA;&lt;p&gt;Sure, at an early stage, when people have just first gotten the idea of actual deduction and proof, being very atomic-detail-oriented is not a bad thing. But, as I also tell my students, especially grad students, at a certain point no one wants to see all the small steps, because they're willing to trust you and/or can fill them in by themselves. Instead, a more high-level/top-down &lt;em&gt;narrative&lt;/em&gt;, that &lt;em&gt;prescribes&lt;/em&gt; the atomic steps (modulo irrelevant details) replaces the line-by-line, eventually.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2020-09-17T22:24:28.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18867" PostTypeId="2" ParentId="18852" CreationDate="2020-09-18T02:36:36.790" Score="2" Body="&lt;p&gt;One of the most interesting word problems of all time, which broadened the human intellect in many ways, is the Archimedes cattle problem. There are many excellent books and articles about this--start with the wikipedia page. Also look for &amp;quot;The Sand Reckoner.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Archimedes is trying to explain that &amp;quot;infinity&amp;quot; is (conceptually) much more than just a very large number. He starts with, &amp;quot;&lt;em&gt;There are some, King Gelon, who think that the number of the sand is infinite&lt;/em&gt;..&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;He poses an innocent sounding word problem about the number of cattle in a herd with various colors. &amp;quot;&lt;em&gt;Compute, O friend, the number of the cattle ..&lt;/em&gt; He sets up a word problem that leads to seven linear equations in eight unknowns. The smallest solution is about 50 million. Archimedes says that if you can get this far, then &amp;quot;&lt;em&gt;thou art no novice in numbers.&lt;/em&gt;&amp;quot; So pat yourself on the back. But then he adds two more equations, which are non-linear, but still seem innocent. One is that the sum of two of the eight unknowns is a square, so &lt;span class=&quot;math-container&quot;&gt;$x+y=n^2$&lt;/span&gt;. As it turns out, the smallest solution for the size of the herd is then represented as a base-ten numeral with over 200,000 digits. This is an &amp;quot;incomprehensibly&amp;quot; large number. If you can solve this, he says, then &amp;quot;&lt;em&gt;then exult as a conqueror, for thou hast proved thyself most skilled in numbers.&lt;/em&gt;&amp;quot; The number is more than the number of grains of sand on the earth, indeed much, much more.&lt;/p&gt;&#xA;&lt;p&gt;So the point here is not that Archimedes is using the equations to solve a practical problem to help a certain person figure out how many cattle are in a herd. Instead, he is setting up a word problem that seems, on the surface, to be rather ordinary and not too ridiculous. But the solution! It is not infinite, but wow..it's a big number. In the Sand Reckoner, Archimedes uses this practical sounding word problem about counting cattle to discuss the size of the universe (3rd century BC!), how many grains of sand would fit into it, how to invent a system to name such an enormous number, and how even enormous numbers are not infinite. It's also a challenge to understand how Archimedes managed to contrive the problem to look innocent, but to have such a large solution. Maybe this mystery just comes down to the fact that he was one of the most brilliant persons to have ever lived.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2020-09-18T13:30:15.943" LastActivityDate="2020-09-18T13:30:15.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18869" PostTypeId="2" ParentId="18863" CreationDate="2020-09-18T17:52:11.140" Score="3" Body="&lt;p&gt;Geometry was invented because we needed to measure land. If you have a field and want to calculate how many cows can you fit in it (area), you will want to be as precise as possible.&lt;/p&gt;&#xA;" OwnerUserId="14637" LastActivityDate="2020-09-18T17:52:11.140" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18870" PostTypeId="1" CreationDate="2020-09-18T22:01:23.923" Score="1" ViewCount="94" Body="&lt;p&gt;I need the collection of exercises for such topics as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;maps and transformations, composition of maps&lt;/li&gt;&#xA;&lt;li&gt;homothety, rotation homothety, homothetic center&lt;/li&gt;&#xA;&lt;li&gt;similarities of the line and the plane&lt;/li&gt;&#xA;&lt;li&gt;free vector and vector space&lt;/li&gt;&#xA;&lt;li&gt;matrix arithmetic&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;for the beginning of the course of linear algebra.&lt;/p&gt;&#xA;&lt;p&gt;Can you recommend anything out of your experience as a good source?&lt;/p&gt;&#xA;" OwnerUserId="13610" LastEditorUserId="13610" LastEditDate="2020-09-19T06:31:20.220" LastActivityDate="2020-09-19T06:31:20.220" Title="Exercises for explaning homothety, homothetic center, similarity on line and plane, free vector and vector space" Tags="|undergraduate-education|algebra|geometry|linear-algebra|exercises|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18871" PostTypeId="1" AcceptedAnswerId="18877" CreationDate="2020-09-18T22:30:03.980" Score="-4" ViewCount="354" Body="&lt;p&gt;Numerous online resources parrot the usual proof by contradiction of the irrationality of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;. These all rely upon the assumption that the rational form (say, &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt;) is in its simplest representation, i.e., &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are co-prime, and then easily demonstrates that in for &lt;span class=&quot;math-container&quot;&gt;$(a/b)=\sqrt{2}$&lt;/span&gt; both &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; must be even, which contradicts the assumption that they are co-prime.&lt;/p&gt;&#xA;&lt;p&gt;However, in all of the renditions that I have seen, a reasoning step appears to me to be missing, or at least never stated. Specifically, the co-primacy of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, and thus that &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt; is in simplest form, is assumed but never appears to enter into the proof. If we simply don't assume that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are co-prime, then they could easily both be even, and there would be no contradiction. And since their co-primacy never appears to enter into the proof, there's no apparent reason for making that assumption.&lt;/p&gt;&#xA;&lt;p&gt;Of course, don't think that the proof is wrong. (I'm not insane!) I'm just looking for an explication of the missing assumption (or reasoning step?) that demands co-primacy of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, or, more likely, an explanation that there's nothing missing, but instead, I'm missing something! (I.e., I'm not insane, just stupid! :-)&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="14430" LastEditorUserId="470" LastEditDate="2020-09-19T00:07:54.773" LastActivityDate="2020-09-20T23:10:42.380" Title="Missing Step in Most Proofs of the Irrationality of $\sqrt{2}$" Tags="|proofs|prime-numbers|" AnswerCount="2" CommentCount="2" ClosedDate="2020-09-20T01:51:19.427" ContentLicense="CC BY-SA 4.0" />
  <row Id="18872" PostTypeId="2" ParentId="18871" CreationDate="2020-09-18T23:01:19.467" Score="3" Body="&lt;p&gt;You are correct in that the coprimality of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and  &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is not used in its full strength. It is adequate to merely assume that they are not both even. But since people are so used to reducing a rational number to lowest terms, making this assumption improves the readability of the proof. The unnecessary stronger assumption of coprimality is not used, but it does no harm, since it guarantees they are not both even.&lt;/p&gt;&#xA;&lt;p&gt;Thing of flow of proof as starting like this:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Before we get going, let us agree that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are coprime. If they are not, then before we get started, go off and reduce them so they are. I'll wait. Done? OK, let's get going...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Then after some thought, we end up with a conclusion that both are even, which is a contradiction.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2020-09-18T23:01:19.467" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="18873" PostTypeId="2" ParentId="18861" CreationDate="2020-09-19T03:28:07.810" Score="0" Body="&lt;p&gt;Euler conjectured that there are no &lt;a href=&quot;https://en.wikipedia.org/wiki/Mutually_orthogonal_Latin_squares&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graeco-Latin squares&lt;/a&gt; of size congruent to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; mod &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;. He sought a proof of this for size &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; in order to prove that the &lt;span class=&quot;math-container&quot;&gt;$36$&lt;/span&gt; officers problem had no solution, but was unable to find one. It was later found that he was right for size &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;, but not in general. For example, there exists a Graeco-Latin square of size &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2020-09-19T03:28:07.810" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18874" PostTypeId="2" ParentId="18863" CreationDate="2020-09-19T09:08:16.267" Score="3" Body="&lt;p&gt;Here's an example which shows what happens when we get sloppy with geometry: &lt;a href=&quot;https://en.wikipedia.org/wiki/Missing_square_puzzle&quot; rel=&quot;nofollow noreferrer&quot;&gt;The missing square triangle&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ngXx2.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ngXx2.gif&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can cut the pieces with your kid, and let them try to spot the problem. It might be hard to notice visually, so that's why we use precise fractions or angles.&lt;/p&gt;&#xA;&lt;p&gt;It's similar to the &lt;a href=&quot;https://www.youtube.com/watch?v=z7tRr49qZfo&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Infinite Chocolate Bar Trick&amp;quot;&lt;/a&gt;, but be careful not to disappoint your kid when the harsh truth is revealed!&lt;/p&gt;&#xA;" OwnerUserId="9380" LastEditorUserId="376" LastEditDate="2020-09-19T09:42:12.843" LastActivityDate="2020-09-19T09:42:12.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18875" PostTypeId="2" ParentId="18863" CreationDate="2020-09-19T13:40:47.843" Score="1" Body="&lt;p&gt;because when we say 'Geometry' in normal school terms we are referring to Geometry as studied by Euclid's book from ancient Greece, which was based on 5 axioms. it is also sometimes called 'straight edge and compass' geometry. And it was precise by it's design.&lt;/p&gt;&#xA;&lt;p&gt;The trick is with Euclid's geometry that you are not just studying shapes and lines, you are studying how logic and reason works. It shows how far you can go and what you can accomplish by starting with very good basic assumptions about something, and working out the conclusions based on logic and reason. Philosophy and Geometry used to be very closely related subjects in schools of ancient times.&lt;/p&gt;&#xA;&lt;p&gt;That is why we still study it even though non-Euclidean geometry has existed for 100 years and numerical approximation is much more important in the modern computerized world. In real life, engineers use computers to calculate tolerances within a precision, objects are never perfectly shaped. And in physics, Einstein's Field Equations are the most accurate known representation of the motion of bodies in the cosmos, and they are not precise in the sense that everything has to be approximated by calculation. There are no orbits that are perfect circles or ellipses, everything is approximated numerically in a computer. But we still study Euclid, kind of like why we still learn the alphabet, study Chess or Go, or still play cards or sports.&lt;/p&gt;&#xA;" OwnerUserId="13595" LastActivityDate="2020-09-19T13:40:47.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18876" PostTypeId="2" ParentId="18784" CreationDate="2020-09-19T15:49:21.033" Score="2" Body="&lt;p&gt;Probably you know all of the following, but just to have it on the record:&lt;/p&gt;&#xA;&lt;p&gt;Most seven year olds are not fluent with fractions, but if you have one that is especially skillful in manipulating them, he might be able to learn how to relate the Euclidean algorithm to the continued fraction, learn to manipulate continued fractions and to understand their properties, and eventually—but this may take a while—to see how the Bézout coefficients come out of the continued fraction. It all depends on whether he manages to stay interested during the process, which could be a lengthy one. Continued fractions are fascinating objects in their own right, and well worth learning about.&lt;/p&gt;&#xA;&lt;p&gt;Before proceeding, I wanted to mention that there seems to be a factor of &lt;span class=&quot;math-container&quot;&gt;$(-1)^{s-1}$&lt;/span&gt; missing in your expression for &lt;span class=&quot;math-container&quot;&gt;$\frac{Q}{P}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let's show that the &lt;span class=&quot;math-container&quot;&gt;$\gcd$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$6186$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3014$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, with the continued fraction and standard Euclidean algorithm shown side-by-side:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\frac{6186}{3014}&amp;amp;=2+\frac{158}{3014} &amp;amp; &amp;amp; 6186=2\cdot3014+158\\&#xA;&amp;amp;=2+\frac{1}{19+\frac{12}{158}} &amp;amp; &amp;amp; 3014=19\cdot158+12\\&#xA;&amp;amp;=2+\frac{1}{19+\frac{1}{13+\frac{2}{12}}} &amp;amp; &amp;amp; 158=13\cdot12+2\\&#xA;&amp;amp;=2+\frac{1}{19+\frac{1}{13+\frac{1}{6+\frac{0}{2}}}} &amp;amp; &amp;amp; 12=6\cdot2+0.\\&#xA;\quad&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Computing the convergents by brute force—we'll do it in a better way in a second—gives&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2=\frac{2}{1},\quad 2+\frac{1}{19}=\frac{39}{19},\quad 2+\frac{1}{19+\frac{1}{13}}=\frac{509}{248},\quad 2+\frac{1}{19+\frac{1}{13+\frac{1}{6}}}=\frac{3093}{1507}=\frac{6186}{3014}.&#xA;$$&lt;/span&gt;&#xA;Now for the better way. Evaluating the third convergent as an example, and keeping focus on the dependence of the convergent on the third term in the continued fraction, which has value &lt;span class=&quot;math-container&quot;&gt;$13$&lt;/span&gt; here, we see that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2+\frac{1}{19+\frac{1}{13}}=2+\frac{13}{19\cdot13+1}=\frac{2(19\cdot13+1)+13}{19\cdot13+1}=\frac{39\cdot13+2}{19\cdot13+1}.&#xA;$$&lt;/span&gt;&#xA;We observe that&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the dependence on the parameter &lt;span class=&quot;math-container&quot;&gt;$13$&lt;/span&gt; is of the form &lt;span class=&quot;math-container&quot;&gt;$x\mapsto\frac{ax+b}{cx+d}$&lt;/span&gt;;&lt;/li&gt;&#xA;&lt;li&gt;the integer coefficients are derived from the previous two convergents, &lt;span class=&quot;math-container&quot;&gt;$\frac{39}{19}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{1}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These features are true of all convergents, and it's not hard to understand why. Let's compute the fourth convergent by modifying the third convergent. The term &lt;span class=&quot;math-container&quot;&gt;$13$&lt;/span&gt; needs to be replaced by &lt;span class=&quot;math-container&quot;&gt;$13+\frac{1}{6}$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{39\left(13+\frac{1}{6}\right)+2}{19\left(13+\frac{1}{6}\right)+1}=\frac{(39\cdot13+2)\cdot6+39}{(19\cdot13+1)\cdot6+19},&#xA;$$&lt;/span&gt;&#xA;confirming that the dependence on the fourth term, &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;, is of the expected form, with coefficients given by the previous two convergents.&lt;/p&gt;&#xA;&lt;p&gt;To follow this, of course, your son would have to be very comfortable with dividing fractions, using the distributive, commutative, and associative laws, and things like that, which, in my experience, would be extremely rare in a seven year old. But this method avoids algebra, and I think is still convincing. I chose the numbers so that the terms in the continued fraction would be distinctive, allowing you to watch how they move around in the calculation, but you could choose more tractable numbers, and repeat the calculation on several small examples to make the same points.&lt;/p&gt;&#xA;&lt;p&gt;Continuing on, note that since&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2+\frac{1}{19}=\frac{2\cdot19+1}{1\cdot19+0},&#xA;$$&lt;/span&gt;&#xA;and since&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2=\frac{1\cdot2+0}{0\cdot2+1},&#xA;$$&lt;/span&gt;&#xA;the pattern of convergents can be continued backwards, with the same rule applying, to get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{0}{1},\quad\frac{1}{0},\quad\frac{2}{1},\quad\frac{39}{19},\quad\frac{509}{248},\quad\ldots,&#xA;$$&lt;/span&gt;&#xA;where the initial two convergents have the same values for every continued fraction.&lt;/p&gt;&#xA;&lt;p&gt;The next thing to learn is what happens when you cross multiply successive convergents and take the difference. Considering the convergents&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{39}{19},\quad\frac{509}{248}=\frac{39\cdot13+2}{19\cdot13+1},\quad\frac{3093}{1507}=\frac{(39\cdot13+2)\cdot6+39}{(19\cdot13+1)\cdot6+19},&#xA;$$&lt;/span&gt;&#xA;compute&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp;3093\cdot248-1507\cdot509\\&#xA;&amp;amp;\quad=\left[(39\cdot13+2)\cdot6+39\right]\left(19\cdot13+1\right) - \left[(19\cdot13+1)\cdot6+19\right]\left(39\cdot13+2\right)\\&#xA;&amp;amp;\quad=39\cdot(19\cdot13+1)-19\cdot(39\cdot13+2)=39\cdot1-19\cdot2=1\\&#xA;&amp;amp;\quad=-\left[(39\cdot13+2)\cdot19-(19\cdot13+1)\cdot39\right]\\&#xA;&amp;amp;\quad=-\left[509\cdot19-248\cdot39\right].&#xA;\end{align}&lt;/span&gt;&#xA;Examining this calculation should convince you that the difference of cross multiplications alternates in sign and has value &lt;span class=&quot;math-container&quot;&gt;$\pm1$&lt;/span&gt; for every pair of successive convergents. This implies, by the way, that the convergents must be fractions reduced to lowest terms, since any factor common to the numerator and denominator would be common to both terms in the expression above, and would therefore divide &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now if you rewrite the final convergent, &lt;span class=&quot;math-container&quot;&gt;$\frac{3097}{1507}$&lt;/span&gt;, as &lt;span class=&quot;math-container&quot;&gt;$\frac{6186}{3014}$&lt;/span&gt;, you see how the &lt;span class=&quot;math-container&quot;&gt;$\gcd$&lt;/span&gt; (up to a possible minus sign) comes about by cross multiplying and subtracting it with the second-to-last convergent.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2020-09-19T15:49:21.033" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18877" PostTypeId="2" ParentId="18871" CreationDate="2020-09-19T18:17:21.020" Score="-1" Body="&lt;p&gt;(An answer because this is a little long for a comment) Here is a proof which does not use coprimality, but is essentially the same: Suppose that &lt;span class=&quot;math-container&quot;&gt;$a/b = \sqrt{2}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are integers. Then &lt;span class=&quot;math-container&quot;&gt;$a=\sqrt{2}b$&lt;/span&gt; and so &lt;span class=&quot;math-container&quot;&gt;$a^2 = 2 b^2$&lt;/span&gt; where both sides are integers.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; be the number of factors of 2 in &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; be the number of factors of two in &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. By the fundamental theorem of arithmetic, &lt;span class=&quot;math-container&quot;&gt;$a^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2 b^2$&lt;/span&gt; have prime factorizations and, further, they must be the same. (This uniqueness is also used in defining &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;.) In particular, the number of factors of 2 must be equal. Thus &lt;span class=&quot;math-container&quot;&gt;$2n = 2m+1$&lt;/span&gt;, which cannot be satisfied by any integral &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; as it would imply that there is a number which is both even and odd. This is the desired contradiction.&lt;/p&gt;&#xA;&lt;p&gt;Note: You don't really need coprimality, the only place that it enters in here is that it implies that at least one of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is zero.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2020-09-20T23:10:42.380" LastActivityDate="2020-09-20T23:10:42.380" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18879" PostTypeId="2" ParentId="18863" CreationDate="2020-09-19T19:55:50.407" Score="1" Body="&lt;p&gt;When I was at school, both the teachers and the textbooks carefully avoided using the names &amp;quot;edge&amp;quot; and &amp;quot;vertex&amp;quot; in topology, using the names &amp;quot;arc&amp;quot; and &amp;quot;node&amp;quot; instead.  That way, no-one got confused by expecting the edges and vertices of 3D geometry to have the same properties as the topological objects.&lt;/p&gt;&#xA;&lt;p&gt;Has anyone got a copy of the ISO 80000-3 standard?  It might say something about this.&lt;/p&gt;&#xA;" OwnerUserId="14522" LastEditorUserId="14522" LastEditDate="2020-09-19T20:01:10.560" LastActivityDate="2020-09-19T20:01:10.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18880" PostTypeId="2" ParentId="18863" CreationDate="2020-09-19T20:21:09.550" Score="2" Body="&lt;p&gt;Mathematics &lt;em&gt;is&lt;/em&gt; about precision and abstraction. You abstract away (i.e., completely ignore) some aspects of your problem to simplify reasoning precisely about those that do interest you.&lt;/p&gt;&#xA;&lt;p&gt;Geometry (Euclid style, here) is about line lengths and angles and also areas. You disregard line or area widths, colors, textures; consider (mostly) straight lines and circles and their intersections&lt;/p&gt;&#xA;&lt;p&gt;Graph theory is about objects (vertices) and their connections (edges). You omit everything else.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-09-19T20:21:09.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18881" PostTypeId="2" ParentId="18678" CreationDate="2020-09-19T22:38:19.447" Score="1" Body="&lt;p&gt;I wouldn't share my notes' source code so lightly in this context, and the possible costs are greater than the almost inexistent benefits. There certainly are security concerns as @PhysicsMathsLove's comment shows, not to mention possible venues for plagiarism, and most students don't know &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; enough to typeset their own notes or simply can't be bothered to, as it's a very lengthy process.&lt;/p&gt;&#xA;&lt;p&gt;Many .pdf viewers allow you to insert comments without the need to edit the source code, so if it's just to write some quick notes I don't see the benefit in sharing the original code. Some programs such as &lt;a href=&quot;https://github.com/xournalpp/xournalpp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Xournal++&lt;/a&gt; even let you type &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; equations over a .pdf, so long as a valid TeX environment is installed in your system.&lt;/p&gt;&#xA;&lt;p&gt;However there are reasons why I would share source code: for a concrete example, one course I recently took required us to write solutions to a given list of exercises (mostly due to the online format forced upon us by the COVID pandemic), and this list was available in both .pdf and .tex source code precisely so students with less experience typesetting documents could read the source and learn from it. In this case the professor shared the source code &lt;em&gt;because it was necessary for the course to function properly&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;That said, many authors share the source code for their books: for instance, Jim Hefferon's &lt;a href=&quot;https://hefferon.net/linearalgebra/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Linear Algebra&lt;/a&gt; has become a standard reference for many courses and the source code is freely available on his webpage. In this case, any possible issues such as plagiarism, etc. are mitigated by the fact that Hefferon is a well-known professor with a public webpage, and by the fact that he has decided to publish and license his work freely (you may use his book under the GNU or Creative Commons 2.5 license).&lt;/p&gt;&#xA;" OwnerUserId="13754" LastActivityDate="2020-09-19T22:38:19.447" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18882" PostTypeId="1" CreationDate="2020-09-20T01:07:06.987" Score="10" ViewCount="279" Body="&lt;p&gt;I am interested in learning how a course in geometry is employed today&#xA;at undergraduate colleges/universities in the U.S.&#xA;On the one hand, such a course seems to serve as an optional (rarely required)&#xA;upper-level course&#xA;of variable content for mathematics majors.&#xA;But on the other hand, often future high-school teachers need to see&#xA;Hilbert-style&lt;sup&gt;1&lt;/sup&gt; planar geometry proofs. So for them, the emphasis on proofs and&#xA;rigor is essential. Whereas for the upper-level math majors, the course&#xA;could be differential geometry, algebraic geometry, discrete geometry, etc.,&#xA;according to the instructor's inclinations.&lt;/p&gt;&#xA;&lt;p&gt;I'd be interested to learn how this tension (if I've accurately captured it) is&#xA;currently resolved,&#xA;and what might serve as the content of a college geometry course.&#xA;Contrasts beyond the U.S. are welcome.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;&#xA;Gries D., Schneider F.B. (1993) &quot;Hilbert-style Proofs.&quot; In: &lt;em&gt;A Logical Approach to Discrete Math&lt;/em&gt;. Texts and Monographs in Computer Science. Springer, New York, NY. &#xA;&lt;a href=&quot;https://doi.org/10.1007/978-1-4757-3837-7_7&quot; rel=&quot;noreferrer&quot;&gt;DOI link&lt;/a&gt;.&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;PS. I wanted to cite these articles by Joseph Malkevitch (@JosephMalkevitch), advocating for discrete geometry, but&#xA;I am not finding public links. (Later) Author provided a link.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Malkevitch, Joseph. &amp;quot;The 'New Math' and Claims Discrete Mathematics is the New 'New Math'.&amp;quot; &lt;em&gt;Mathematics in School&lt;/em&gt; 40, no. 2 (2011): 8-10.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Malkevitch, Joseph. &amp;quot;Discrete Mathematics and Public Perceptions of Mathematics.&amp;quot; &lt;em&gt;Discrete Mathematics in the Schools&lt;/em&gt; 36 (1992).&#xA;&lt;a href=&quot;http://web.york.cuny.edu/%7Emalk/Mymaterials/public-perceptions.pdf&quot; rel=&quot;noreferrer&quot;&gt;author's PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;Added&lt;/em&gt; 21Sep2020: Thanks to Joe Malkevitch,&#xA;just saw this recommendation from CUPM (MAA Committee on the Undergraduate Program in Mathematics) on Geometry: &amp;quot;the curricular needs in geometry of future high school mathematics teachers are best satisfied within geometry courses for mathematics majors, not by a separate &lt;em&gt;Geometry for Teachers&lt;/em&gt; course.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-09-21T16:35:29.800" LastActivityDate="2020-09-21T20:22:45.173" Title="College undergraduate geometry courses" Tags="|undergraduate-education|geometry|discrete-math|differential-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18886" PostTypeId="1" AcceptedAnswerId="18888" CreationDate="2020-09-21T10:03:31.697" Score="1" ViewCount="286" Body="&lt;p&gt;I am teaching math to a 10 year old student. He learned that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{300}{100}=\require{cancel}\frac{3\cancel{00}}{1\cancel{00}}=3$$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{6000}{300}=\require{cancel}\frac{60\cancel{00}}{3\cancel{00}}=\frac{60}{3}=\frac{20+20+20}{3}=20$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But now I want to teach him how to compute &lt;span class=&quot;math-container&quot;&gt;$\frac{300}{200}$&lt;/span&gt;. Before I explain this he responded quickly that first we cancel out zeroes:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{300}{200}=\require{cancel}\frac{3\cancel{00}}{2\cancel{00}}=\frac{3}{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Well, his argument is correct but I want to teach him that &lt;span class=&quot;math-container&quot;&gt;$300=1\times 200+100$&lt;/span&gt; (&lt;a href=&quot;https://tex.stackexchange.com/a/274250/108724&quot;&gt;using this method&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;and remainder is &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt;. But in previous solution its remainder is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$3=1\times 2+1$&lt;/span&gt;). How can I explain this without fooling or tricking him?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt; Note that he started his education at the age of seven.&lt;/p&gt;&#xA;" OwnerUserId="14651" LastEditorUserId="5321" LastEditDate="2020-09-23T15:31:19.910" LastActivityDate="2020-09-23T15:31:19.910" Title="How can I teach $\frac{300}{200}$ to 10 years old students while s\he knows canceling the zeroes rule?" Tags="|mathematical-pedagogy|teaching|fractions|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18887" PostTypeId="2" ParentId="18886" CreationDate="2020-09-21T10:49:31.613" Score="1" Body="&lt;p&gt;Unless I'm misinterpreting the question, is this about division with remainders?&lt;/p&gt;&#xA;&lt;p&gt;I'd probably ask with &amp;quot;how many X are in Y?&amp;quot; and use &lt;a href=&quot;https://en.wikipedia.org/wiki/Highly_composite_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;composite numbers&lt;/a&gt; as my Y's to get lots of results&lt;/p&gt;&#xA;&lt;p&gt;For example, &amp;quot;How many 10's are in 60?&amp;quot;, &amp;quot;how many 2's in 60?&amp;quot;, &amp;quot;how many 15's in 60?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Then I'd ask, &amp;quot;how many 25's are in 60?&amp;quot; And then you might get something like &amp;quot;there are 2 full 25's in 60 but then you have some left over&amp;quot; And then you can ask, &amp;quot;how many are left over?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If you want applications of this, you could ask, if you have a 60 books and you have a bookshelf with 6 shelves, how many books on each shelf if you want the same number on each shelf. (I think with books you'd prevent decimal or fractional answers)&lt;/p&gt;&#xA;&lt;p&gt;Although I've never taught remainders, 300 and 200 seem to be big numbers? Not like I can recall learning this, but I'm sure I must have learned remainders by drawing something like 13 dots and then drawing circles around them so that there were five dots in each circle, and then realizing 3 were outside of any circle. If the student is new to remainders, maybe try numbers that could be dotted out?&lt;/p&gt;&#xA;&lt;p&gt;Continuing with that, then I'd have &lt;strong&gt;2 groups of 5&lt;/strong&gt; &lt;em&gt;and&lt;/em&gt; &lt;strong&gt;3 left over&lt;/strong&gt;, and connect that to the idea that &lt;span class=&quot;math-container&quot;&gt;$2 \times 5 + 3 = 13$&lt;/span&gt; is &amp;quot;the amount of groups&amp;quot; times &amp;quot;the amount in each groups&amp;quot; add &amp;quot;amount left over&amp;quot;.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-09-21T10:49:31.613" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18888" PostTypeId="2" ParentId="18886" CreationDate="2020-09-21T13:33:25.667" Score="5" Body="&lt;p&gt;&lt;em&gt;...his argument is correct but I want to teach him that &lt;span class=&quot;math-container&quot;&gt;$300=1\times 200+100$&lt;/span&gt; and reminder is &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt;. But in previous solution its reminder is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$3=1\times 2+1$&lt;/span&gt;). How can I explain this without fooling or tricking him?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;If I understood it correctly, the question is how to explain the fact that equivalent fractions leave different remainders. Well, you could try this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain that a fraction represents a part of a whole.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain that an equality between two fractions means that both fractions represent the same part.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Illustrate using pictures:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/j4aKc.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/j4aKc.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Conclude that the remainders of equal fractions are possibly different because the whole was divided into different parts: remainder &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\frac{300}{200}$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; parts of &lt;span class=&quot;math-container&quot;&gt;$200$&lt;/span&gt; small pieces while remainder &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}$&lt;/span&gt; means only &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; part of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; bigger pieces.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13585" LastEditorUserId="13585" LastEditDate="2020-09-21T15:02:11.933" LastActivityDate="2020-09-21T15:02:11.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18889" PostTypeId="2" ParentId="18882" CreationDate="2020-09-21T20:22:45.173" Score="6" Body="&lt;p&gt;I taught a geometry course which counted both for math majors (sophomore level), and was required for the math education program secondary education teacher majors, and we used the text:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Edwin Moise: &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0201508672&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elementary Geometry from an Advanced Standpoint&lt;/a&gt;, 3rd Edition,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;which was amply challenging for both cohorts of students.&lt;/p&gt;&#xA;&lt;p&gt;For math majors completing this course, I later taught a sequel course, *Differential Geometry&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2020-09-21T20:22:45.173" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18892" PostTypeId="2" ParentId="18852" CreationDate="2020-09-23T13:31:46.717" Score="3" Body="&lt;p&gt;Here's a problem that's so practical, I solved it on a piece of scrap wood in the middle of building a gate for my wooden picket fence. I used it in my intermediate algebra class and it went over really well.&lt;/p&gt;&#xA;&lt;p&gt;I have a section of pre-built picket fence that is irregularly sized (I had to cut an 8 foot panel in half to fit it in my hatchback). It is constructed of two horizontal rails with 9 pickets nailed to them.&lt;/p&gt;&#xA;&lt;p&gt;The rails are (say) 44 inches long, and my gate needs to be 41 3/4 inches wide, so I need to trim 2 1/4&amp;quot; from the section. But the pickets are not attached symmetrically; the rail sticks out beyond the last picket by 3&amp;quot; on the left side and by 4 1/2&amp;quot; on the right. Being a bit compulsive about symmetry, I want to trim some amount from each side so that the final gate is both 41 3/4&amp;quot; wide &lt;em&gt;and&lt;/em&gt; so that the rails extend the same amount on each side.&lt;/p&gt;&#xA;&lt;p&gt;To set up a system of two equations: Let &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; be the amounts I will cut off the left and right sides, respectively. Then we have:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ 3 - L = 4.5 - R $$&lt;/span&gt; (the rails should extend the same amount after trimming) and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ L + R = 2.25 $$&lt;/span&gt; (I want to trim a total of 2 1/4&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;This turned out to be a great problem for two reasons. 1) It can actually be solved with a single equation, by defining &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to be the length of rail &lt;em&gt;remaining&lt;/em&gt; after trimming. This is a valuable lesson about the importance of setting up the problem properly. [NB: I did it the two-variable way in my garage. Another important lesson: even experts don't see 'tricks' right off the bat]&lt;/p&gt;&#xA;&lt;p&gt;And 2), The process of actually solving it with algebra corresponds very very closely with how students inevitably solve it in their heads. They'll say: &amp;quot;I subtracted 1.5 inches from the longer (right) side to make the sides even, and also from the 2.25 amount, leaving 0.75 remaining. I split that in half, getting 3/8&amp;quot; more to be trimmed from each side.&amp;quot; These are, of course, the exact same operations done when solving the (single equation) version.&lt;/p&gt;&#xA;" OwnerUserId="14654" LastActivityDate="2020-09-23T13:31:46.717" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18893" PostTypeId="1" AcceptedAnswerId="18896" CreationDate="2020-09-23T17:13:00.640" Score="-1" ViewCount="201" Body="&lt;p&gt;I am talking about literally asking for kids' time/attention by offering them candies: not giving them tasks about summing real candies, etc.&lt;br /&gt;&#xA;I try to teach math from time to time to my relatives of different ages and noticed that the main problem in teaching is just capturing kids' attention. Most of the time kids do what you ask them to do, but express their tedium, though they get really excited when it's something about candies. However I have concerns about usage of candies when doing math.&lt;br /&gt;&#xA;From math history I noticed that most natural mathematicians knew the whole school math before even attending school, for example &lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#Early_years&quot; rel=&quot;nofollow noreferrer&quot;&gt;from Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gauss was barely three years old he corrected a math error his father made; and that when he was seven, he confidently solved an arithmetic series problem (commonly said to be 1 + 2 + 3 + ... + 98 + 99 + 100) faster than anyone else in his class of 100 students&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(from paper source I have info regarding last problem is that Gauss had read &lt;strong&gt;Number theory&lt;/strong&gt; textbook prior to school where exact same problem was discussed)&lt;br /&gt;&#xA;... and &lt;a href=&quot;https://iep.utm.edu/pascal-b/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pascal&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pascal's father intended to save his son from doing math too early, but Pascal young secretly read Euclid's Elements and gave 3 additional proofs of triangles inequalities at age of 5&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I guess kids can get motivated to do math just to help their parents to do their regular everyday job, but it's not an option for me, so the only left for me is to abuse the primal obsession for food ^^. Also, when there is more than one kid participating in doing math, competition factor will take place when kid who gets involved in a most energetic way is rewarded more as well.&lt;/p&gt;&#xA;&lt;p&gt;I can't materialize my concerns about such approach, but it somehow unpleasantly resembles me of the way people train animals. Is it an appropriate way to treat people and especially young ones? Are there any research regarding such topic or maybe some well-known statement about it?&lt;/p&gt;&#xA;" OwnerUserId="14617" LastActivityDate="2020-09-24T13:17:59.953" Title="Can I motivate kids to do math by giving them candies?" Tags="|student-motivation|preschool-education|" AnswerCount="1" CommentCount="22" ClosedDate="2020-09-30T00:29:25.260" ContentLicense="CC BY-SA 4.0" />
  <row Id="18895" PostTypeId="1" AcceptedAnswerId="18910" CreationDate="2020-09-24T12:09:17.980" Score="1" ViewCount="262" Body="&lt;p&gt;This is a question about the intuition of the rational number having measure zero. Let us consider followng proof:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$I = [0,1]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q = \mathbb Q \cap I$&lt;/span&gt; and let &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; be the Lebesgue measure. We know &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; is countable, so let &lt;span class=&quot;math-container&quot;&gt;$Q = \{q_1, q_2, q_3, \ldots\}$&lt;/span&gt;. Now consider the balls (intervals) &lt;span class=&quot;math-container&quot;&gt;$J_k = B_{b^{-k}/2}(q_k)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$b \in (0,1/2)$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$\lambda (J_k)= b^{-k}$&lt;/span&gt;. Now we have &lt;span class=&quot;math-container&quot;&gt;$Q \subset \bigcup_k J_k$&lt;/span&gt;, therefore&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lambda(Q) \leq \lambda(\bigcup J_k) \leq \sum_k \lambda(J_k) = \sum b^{-k} = b\frac{1}{1-b} \leq 2b$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So since &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; can be arbitrarily small, we can conclude that &lt;span class=&quot;math-container&quot;&gt;$\lambda(Q)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is a classic proof that the rationals have Lebesgue measure zero within the reals. Now intuitively we cover each rational number in &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; with an interval of positive length, and since &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; is dense in &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;, one might &amp;quot;feel&amp;quot; like the whole interval &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; should be covered by &lt;span class=&quot;math-container&quot;&gt;$\bigcup J_k$&lt;/span&gt;,&#xA;but at the same time we know that &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; can occupy at most &lt;span class=&quot;math-container&quot;&gt;$2b$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;. So intuitively there must be huge gaps if we e.g. choose &lt;span class=&quot;math-container&quot;&gt;$b&amp;lt;&amp;lt;1/2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This intuition seems to be misguided in the way that it would make sense for finitely many objects, but as soon as we have infinitely many objects it fails.&lt;/p&gt;&#xA;&lt;p&gt;So if someone were to ask about these &amp;quot;gaps&amp;quot;, can we still somehow find a &amp;quot;better&amp;quot; intuitive reasoning that somehow make sense of these gaps, or should we completely abandon this idea and reason that we ususally do not have very good intuition about infinite collections of things?&lt;/p&gt;&#xA;" OwnerUserId="1936" LastActivityDate="2020-09-27T16:43:30.330" Title="Intuition explanation about Lebesgue measure zero of the rational numbers" Tags="|intuition|series|infinity|" AnswerCount="1" CommentCount="4" ClosedDate="2020-09-29T21:51:07.397" ContentLicense="CC BY-SA 4.0" />
  <row Id="18896" PostTypeId="2" ParentId="18893" CreationDate="2020-09-24T13:17:59.953" Score="0" Body="&lt;p&gt;My experience is small rewards and games (e.g. with candy) spark interest from the kids, in any sort of teaching/coaching, so math should be no different.  It is not purely gluttony, but the sense of play with a reward, even small. Even with adults, it is very effective.  For instance a cheap bottle of champagne, even with corporate execs making several six figures, will perk them up in a training session, especially if competitive.  And the small reward somehow makes it more meaningful than just brag rights.&lt;/p&gt;&#xA;&lt;p&gt;Note: This is more of a practical &amp;quot;ed&amp;quot; question, ape-psychology than a &amp;quot;math&amp;quot; question.  But these are completely relevant here.  As the human in the loop is part of the issue.  Not just definitional correctness.&lt;/p&gt;&#xA;&lt;p&gt;I don't see any issue with the sweets either.  Then again, I brought jack knives to school as a little boy, because little boys love jack knives.  No Columbine threat... ;-)&lt;/p&gt;&#xA;" OwnerUserId="14658" LastActivityDate="2020-09-24T13:17:59.953" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18897" PostTypeId="1" CreationDate="2020-09-24T19:43:49.693" Score="3" ViewCount="155" Body="&lt;p&gt;Let us assume that we have 30 balls( 7 green, 10 black, 13 white).&lt;/p&gt;&#xA;&lt;p&gt;I was trying to explain to someone how we count the number of possibilities of getting 3 greens, 3 black, and 3 white balls in a situation where the order is not important, and no replacement is allowed.&lt;/p&gt;&#xA;&lt;p&gt;I did the following drawing.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/udRk6.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/udRk6.jpg&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And then, this person asked me: &amp;quot;Why do we divide each group of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$3!$&lt;/span&gt;, and not the whole group of &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$9!$&lt;/span&gt; ?&amp;quot;&#xA;Followed by: &amp;quot;And aren't we assuming in the drawing that we have (3g,3b,3w) when we would also want to count points for e.g. (b,w,w,g,b,g,w,b)?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I wasn't exactly sure how to answer. Maybe this question is better suited for Math.SE... I don't know.&lt;/p&gt;&#xA;" OwnerUserId="2029" LastEditorUserId="2029" LastEditDate="2020-09-24T20:06:08.897" LastActivityDate="2020-09-26T16:18:17.027" Title="Trying to explain(understand?) combinations" Tags="|secondary-education|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18898" PostTypeId="1" CreationDate="2020-09-25T04:45:52.410" Score="4" ViewCount="108" Body="&lt;p&gt;This is one of the problem taunting me over years while explaining probability.&lt;/p&gt;&#xA;&lt;p&gt;In most of the high-school as well as graduate textbooks, there are at most very few lines to deal with this problem.&lt;/p&gt;&#xA;&lt;p&gt;The problem is that the authors conveniently escape from explaining probabilities based on sample spaces (list of outcomes) to probability without talking about sample spaces at least once (deals with real-life events) after the definition of random variable comes in.&lt;/p&gt;&#xA;&lt;p&gt;There will be no stress on such transition in most of the text books. Some students ask about the explicit sample space and events as sets. I say that they are no more necessary. But it became hard for me to teach the transition.&lt;/p&gt;&#xA;&lt;p&gt;Some students randomly brings research papers containing random variables and asks for sample spaces and random experiments underlying.&lt;/p&gt;&#xA;&lt;p&gt;Is there any universal rule of getting sample space during teaching the transition? Or making the students to understand that they are not necessary.&lt;/p&gt;&#xA;" OwnerUserId="8052" LastEditorUserId="8052" LastEditDate="2020-09-25T09:59:57.333" LastActivityDate="2020-09-25T09:59:57.333" Title="Transition into a non-explicit sample space world" Tags="|probability|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18899" PostTypeId="2" ParentId="18784" CreationDate="2020-09-25T09:18:22.747" Score="0" Body="&lt;p&gt;Just to record what my kid and me went through.&lt;/p&gt;&#xA;&lt;p&gt;First of all, thank you everyone for the great ideas and help!&lt;/p&gt;&#xA;&lt;p&gt;We started from Euclidean Algorithm, explained by folding an A4 paper again and again:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jHg8P.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jHg8P.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then following the classic way, as Steven showed, to rely on Euclidean Algorithm's steps to derive Bézout's Identity:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/lcK2n.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lcK2n.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Just as Will said, this has to go through &amp;quot;dividing fractions, using the distributive, commutative, and associative laws&amp;quot;, it took him a while to see things such as &lt;span class=&quot;math-container&quot;&gt;$a-(b-c)d=a-bd+cd$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$ax-by+az=a(x+z)-by$&lt;/span&gt;, but it's a good practice.&lt;/p&gt;&#xA;&lt;p&gt;Also it took him quite some time to understand the substitution -- he preferred to compute things out instead of keep the &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; to reach the &lt;span class=&quot;math-container&quot;&gt;$ax+by$&lt;/span&gt; form.&lt;/p&gt;&#xA;&lt;p&gt;Then it's to show that continued fraction actually can serve as a short form of Euclidean Algorithm steps, basically they are all different ways to tell the same story:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/80E7r.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/80E7r.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Last it's to show that the continued fraction form from Euclidean Algorithm, after removing the inner most part, actually is just the one to compute Bézout's Identity. Or, bottom up, the Bézout's Identity &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are driven by the same mechanism driving &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This logic is mentioned by Will and Marty (&amp;quot;hop and skip&amp;quot; in ch 1 of his fabulous book &lt;a href=&quot;http://illustratedtheoryofnumbers.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Illustrated Theory of Numbers&lt;/a&gt;), also by &lt;a href=&quot;http://www.davidcgarlock.com/number-theory.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;David C Garlock&lt;/a&gt; (in ch 3 of his number theory book), and Andrew Granville ( in Appendix 1A. &amp;quot;Reformulating the Euclidean algorithm&amp;quot; of his book &lt;a href=&quot;https://www.amazon.co.uk/Number-Theory-Revealed-Andrew-Granville/dp/1470441586/ref=sr_1_1?dchild=1&amp;amp;keywords=by%20Number%20Theory%20Revealed%20-%20A%20Master%20Class%20%2B&amp;amp;qid=1601024855&amp;amp;sr=8-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Number Theory Revealed - A Master Class&lt;/a&gt;, very clear, but, alas, used matrix).&lt;/p&gt;&#xA;&lt;p&gt;By adopting ideas I drew a somewhat &amp;quot;tree&amp;quot; style diagram (it's not tree, but i don't know how to describe such shape) to represent the continued fraction, and a &amp;quot;simple tree&amp;quot; to only remark the driving force of the &amp;quot;tree&amp;quot;, then it's shown that the two continued fractions are the same except the inner most part.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ttqyf.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ttqyf.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then we are done.&lt;/p&gt;&#xA;" OwnerUserId="9059" LastActivityDate="2020-09-25T09:18:22.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18900" PostTypeId="1" AcceptedAnswerId="18902" CreationDate="2020-09-25T18:31:36.557" Score="9" ViewCount="2679" Body="&lt;p&gt;I tutor various math subjects online for a large tutoring company. I know that as a math tutor, it's my responsibility to be able to explain any concept in a way that makes sense to the student, or to be able to ask them the appropriate questions that will guide them through their homework.&lt;/p&gt;&#xA;&lt;p&gt;There are two types of sessions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;One-on-one sessions that are scheduled in advance.&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;Instant sessions&amp;quot; where I meet up with a student on the spot.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;For sessions that are scheduled in advance, I have found that asking the student (before the session) to send me the problems that they are stuck on will improve my performance as a tutor during the session. I don't have to worry about improvising because I will have already pinpointed the challenging concepts and taken notes.&lt;/p&gt;&#xA;&lt;p&gt;However, for &amp;quot;instant sessions&amp;quot;, I lose the advantage of being proactive. I have no idea what the student is going to want to review until after the session has already begun. &lt;strong&gt;Every now and then, a student will be stuck on a challenging math problem such that when I read through it, I know immediately that I do not understand it and that I will struggle to help them.&lt;/strong&gt; At that point, I feel that I have let the student down because it's literally my job to help them.&lt;/p&gt;&#xA;&lt;p&gt;When such a situation arises, I find myself Googling the problem hoping to find a solution. But this is hit or miss; some teachers write their own math problems (which is what they should do), and the wait-time means that the student starts to get suspicious that I am incapable of helping them.&lt;/p&gt;&#xA;&lt;p&gt;One idea that I have is to take a step back and remind the student of the basic concepts that (I think) the solution will involve. However, while this may be the &amp;quot;correct&amp;quot; approach from a math education standpoint, this may take extra time, and the student has to pay more money the longer the session goes. I fear that many students would become impatient, unless I could convince them that what I'm doing is relevant to solving the problem.&lt;/p&gt;&#xA;&lt;p&gt;So my question is, &lt;strong&gt;how does my idea sound? Are there better ways to respond in such a situation that I should consider?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;EDIT: I've tutored hundreds of different students, and I would say that around 90% of my sessions go well. I'm always trying to improve, which is why I'm asking this question.&lt;/p&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="13657" LastEditDate="2020-09-25T23:55:51.593" LastActivityDate="2020-09-26T04:31:38.597" Title="What to do when uncertain about a question while tutoring" Tags="|tutoring|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18901" PostTypeId="2" ParentId="18900" CreationDate="2020-09-25T18:34:46.163" Score="8" Body="&lt;p&gt;I think, especially in such a blatantly transactional interaction, you should be forthright with them, and allow them to make a decision.&lt;/p&gt;&#xA;&lt;p&gt;I would say &amp;quot;I don't know how to do this.  Do you want to try to figure it out together (no promises), or would you like to end the meeting and get a refund?&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-09-25T18:34:46.163" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18902" PostTypeId="2" ParentId="18900" CreationDate="2020-09-25T18:53:21.387" Score="14" Body="&lt;p&gt;I teach at a community college, and under normal circumstances (not covid), we have a walk-in tutoring room where students can get help from any of the faculty who do their office hours there rather than in their own offices. Most of the students I see in there are my own, but maybe 30% are students in other professors' classes.&lt;/p&gt;&#xA;&lt;p&gt;I would say that maybe 1 time in 1000 I draw a blank, and when that happens, I try to admit it honestly to the student. Typically this will be a situation where another professor is going into a lot of detail on a sub-topic that I don't teach and haven't thought about for a long time. Call this situation A. So I'll be saying things like, &amp;quot;I don't go into this level of detail when I teach this topic, so I don't know off the top of my head, but can you find the relevant discussion of this in your textbook or your notes?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Considerably more common, maybe 1 in 100, is the situation, call it situation B, where I am able to do the problem myself, but it's something that I don't teach, and therefore I end up doing a lousy job educationally of helping the student with the problem. E.g., I don't know what issues students have understanding a certain topic, so my efforts in helping them are not very good, because I don't realize until too late what conceptual issue is hanging them up.&lt;/p&gt;&#xA;&lt;p&gt;If you find yourself in situation A one time in 1000 and situation B one time in 100, then I would say that by my own standards, you're doing about as well as can be expected.&lt;/p&gt;&#xA;&lt;p&gt;However, if situation A is happening more frequently, like 1 time in 10, then you are probably tutoring a level of math or topics in math that you're not qualified to tutor. You should talk to your supervisor and ask to be assigned to lower-level work or work that isn't on those topics.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-09-25T18:53:21.387" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18903" PostTypeId="2" ParentId="18900" CreationDate="2020-09-26T04:31:38.597" Score="5" Body="&lt;p&gt;Depends on what the questions is, this shouldn't usually happen &lt;em&gt;especially&lt;/em&gt; if it's related to core knowledge in the course. Although it's pretty demanding, I'd say you need to know the course inside and out and be able to do virtually every question in the book you're using. (This is the reason why I say you need to be about 2 &amp;quot;degrees&amp;quot; ahead in the content. Like if you're tutoring high school, then you should probably be able to handle graduate level math, or if it's middle school, then you should be able to do undergrad)&lt;/p&gt;&#xA;&lt;p&gt;That said, it has happened that students randomly ask me things (usually in reference to applications, or some sort of contest math) and the answer isn't apparent to me right away since it's not related to the course content and the topic isn't usually developed upon in later courses. When that happens, I just say, &amp;quot;I'm not sure at the moment, but I'll get back to you.&amp;quot; and then I look it up or work it out on my own time and give them my answer.&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't think that a math teacher means that I'm an answer key, Rather, I hope it means that I can help you get to and understand the answer, and if I can't do that initially, then I'll find a way to do that eventually.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-09-26T04:31:38.597" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18904" PostTypeId="1" AcceptedAnswerId="18907" CreationDate="2020-09-26T04:49:59.693" Score="4" ViewCount="204" Body="&lt;p&gt;One way, which is the most obvious, is do sketches of 3d shapes that tend to be the ones that we can all draw (like rectangle, cone, cylinder, sphere, etc.)&lt;/p&gt;&#xA;&lt;p&gt;Another way is by analogy so even if we can't &lt;em&gt;really&lt;/em&gt; sketch the graph to any kind of precision or accuracy, there is enough analogy to &amp;quot;fill in the blanks&amp;quot; of what the sketch is lacking. Like how a skier might who wants to do down the hill fastest will go along the gradient.&lt;/p&gt;&#xA;&lt;p&gt;Akin to that above one is real-life pictures or diagram like contour maps for showing elevations, or heat maps. These could show 3d functions on a 2d surface.&lt;/p&gt;&#xA;&lt;p&gt;There might be a way of 2d-3d analogy in how, say, finding the area underneath is curve via rectangles is like finding the volume under a surface via rectangular prisms.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;What are ways you show &amp;quot;visuals&amp;quot; without having to type up equations that will be graphed by a computer program?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-09-27T00:42:37.960" Title="Low-tech ways of visualizing multivariable and vector calculus" Tags="|calculus|vector-calculus|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18905" PostTypeId="2" ParentId="18897" CreationDate="2020-09-26T16:18:17.027" Score="1" Body="&lt;p&gt;It looks to me like your student just has a different, but valid, idea for approaching the problem. If your student made both of the suggested changes, (1) including all ways of ordering the colors, and (2) dividing by &lt;span class=&quot;math-container&quot;&gt;$9!$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$(3!)^3$&lt;/span&gt;, they would get the same answer as you got.&lt;/p&gt;&#xA;&lt;p&gt;The main downside to your student's approach is that including all ways of ordering the colors complicates the solution somewhat. Let's review how this would go. In the first step, we want the number of ways of selecting and arranging nine balls such that there are three balls of each color. In order to ensure that the condition on colors holds, assign color labels to nine slots using the labels &lt;em&gt;G&lt;/em&gt;, &lt;em&gt;G&lt;/em&gt;, &lt;em&gt;G&lt;/em&gt;, &lt;em&gt;B&lt;/em&gt;, &lt;em&gt;B&lt;/em&gt;, &lt;em&gt;B&lt;/em&gt;, &lt;em&gt;W&lt;/em&gt;, &lt;em&gt;W&lt;/em&gt;, &lt;em&gt;W&lt;/em&gt;. We need to consider all possible ways of doing this; the multinomial coefficient &lt;span class=&quot;math-container&quot;&gt;$\binom{9}{3,3,3}=\frac{9!}{3!\,3!\,3!}$&lt;/span&gt; is the number of ways of doing so. Multiplying this by the product in the boxes in your drawing, which is the number of ways of putting balls of the appropriate colors in the labeled slots, produces the total number of arrangements of balls:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{9!}{3!\,3!\,3!}P(7,3)P(10,3)P(13,3).&#xA;$$&lt;/span&gt;&#xA;The second step is to divide this by &lt;span class=&quot;math-container&quot;&gt;$9!$&lt;/span&gt;, which gives your expression.&lt;/p&gt;&#xA;&lt;p&gt;The advantage of your method is that it eliminates the complication of having to consider different arrangements of the colors, but you need to convince your student that this is a legitimate procedure. Perhaps it would be better to address whatever issue it was that made you feel the need to present the solution in terms of a series of slots in the first place. Why not simply write down the final expression, &lt;span class=&quot;math-container&quot;&gt;$\binom{7}{3}\binom{10}{3}\binom{13}{3}$&lt;/span&gt;, directly? A student who has truly internalized both combinations and the multiplication principle should be able to see why that final expression is correct. On the other hand, a student who isn't fully comfortable with the multiplication principle is probably going to have trouble seeing why your procedure, which amounts to factoring out and ignoring the color arrangement aspect of the problem, works.&lt;/p&gt;&#xA;&lt;p&gt;To convince your student that the final combinations formula is valid, you could try using a tree, but you'll have to reduce the numbers in the problem greatly. You could try starting with four green and three black balls, and looking for the number of selections with one green and two black. The tree may help convince your student that the green selection is independent of the black selection. Label the balls &lt;span class=&quot;math-container&quot;&gt;$G1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$G2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$G3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$G4$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B3$&lt;/span&gt;. The branches in the first layer of the tree the branches are labeled &lt;span class=&quot;math-container&quot;&gt;$\{G1\}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\{G2\}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\{G3\}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\{G4\}$&lt;/span&gt;. Each of these four branches leads to three branches in the second layer of the tree, labeled &lt;span class=&quot;math-container&quot;&gt;$\{B2,B3\}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\{B1,B3\}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\{B1,B2\}$&lt;/span&gt;. The twelve outcomes are obtained by taking the union of the sets along the branches, e.g. &lt;span class=&quot;math-container&quot;&gt;$\{G2,B2,B3\}$&lt;/span&gt; when you take the second branch in the first layer and the first subbranch in the second layer. The main point you hope your student will see is that the choices for the black balls remain the same regardless of which green ball was chosen and therefore we need simply multiply the number of green selections by the number of black selections.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2020-09-26T16:18:17.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18906" PostTypeId="1" CreationDate="2020-09-26T16:40:14.923" Score="1" ViewCount="124" Body="&lt;p&gt;Benjamin Hutz, in Chapter 10 of his &lt;a href=&quot;https://bookstore.ams.org/amstext-31/&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Experimental Introduction to Number Theory&lt;/a&gt;, allows for the optional inclusion of discrete dynamical systems with a number-theoretic flavor in an undergraduate introduction to number theory.&lt;/p&gt;&#xA;&lt;p&gt;What are the possibilities to start the other way around, with an introductory course on discrete dynamical systems, such as Robert L. Devaney's &lt;a href=&quot;https://www.routledge.com/A-First-Course-In-Chaotic-Dynamical-Systems-Theory-And-Experiment/Devaney/p/book/9780367235994&quot; rel=&quot;nofollow noreferrer&quot;&gt;A First Course in Chaotic Dynamical Systems: Theory and Experiment&lt;/a&gt;, and include some relevant material from elementary number theory? What would be good topics or results to include? They would ideally be accessible without extremely lengthy excursions into number theory, but would show how number theory can arise, sometimes unexpectedly, when studying dynamics.&lt;/p&gt;&#xA;&lt;p&gt;The expected background would be at least first year calculus, preferably with some mathematical maturity gained from differential equations, second year calculus and/or linear algebra, but not necessarily abstract algebra, complex analysis or other higher courses. A prior course in number theory would not be a prerequisite.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2022-07-25T09:09:02.353" LastActivityDate="2022-07-25T09:09:02.353" Title="Number theory in an introductory course on discrete dynamical systems" Tags="|undergraduate-education|course-design|number-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18907" PostTypeId="2" ParentId="18904" CreationDate="2020-09-27T00:42:37.960" Score="6" Body="&lt;p&gt;I don't know if this is what you're looking for exactly, but I've run forms of this activity a few times when introducing partial/directional derivatives.&lt;/p&gt;&#xA;&lt;p&gt;Supply the class with materials: square grids printed on paper, scissors, compass (with attached pencil), ruler and protractor [Sometimes I forego the compass and just provide appropriately-sized circles printed on paper.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/u2Asf.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/u2Asf.jpg&quot; alt=&quot;Grid paper&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Give them the following directions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Measure the length of three squares from the grid, and multiply this by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;. Call your result &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use your compass to draw a circle with radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, and cut it out with scissors.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Qtj07.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Qtj07.jpg&quot; alt=&quot;Circle cut out from paper&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Mark a small point on the perimeter of the circle, labeling it #1.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;With a protractor, find and label another point #2 on the perimeter which is &lt;span class=&quot;math-container&quot;&gt;$105.5^{\circ}$&lt;/span&gt; away from point #1 (with the center of the circle as the vertex of that angle).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cut a straight line from mark #1 to the center of the circle.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Curve the paper so mark #1 coincides with mark #2.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Tape your surface together (inside and out), being sure not to crease or otherwise deform the surface.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Draw an x-axis and a y-axis on the grid paper provided, so the origin &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; is in the center of the grid.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Place your surface on the grid so the point is upward, directly above the point &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; on the plane.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mse0X.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mse0X.jpg&quot; alt=&quot;Paper cone sitting on grid paper&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Congratulations! You have just made the surface &lt;span class=&quot;math-container&quot;&gt;$z=f(x,y)=3-\sqrt{x^2+y^2}$&lt;/span&gt; which is part of (the bottom half of) a right, circular cone.&lt;/p&gt;&#xA;&lt;p&gt;Then a series of questions, such as:&lt;/p&gt;&#xA;&lt;p&gt;(a) What are the coordinates of the point of the cone?&lt;/p&gt;&#xA;&lt;p&gt;(b) If you walked straight up the side of the cone, what would be the slope?&lt;/p&gt;&#xA;&lt;p&gt;(c) What is &lt;span class=&quot;math-container&quot;&gt;$f(1,0)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;(d) Plot the point &lt;span class=&quot;math-container&quot;&gt;$(1,0,f(1,0))$&lt;/span&gt; on your surface.&lt;/p&gt;&#xA;&lt;p&gt;(e) From that point, if you kept your x-value fixed but moved slightly in the y-direction, what would you experience as you moved along the surface of the cone? Approximate your slope at that point.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XNhJn.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XNhJn.jpg&quot; alt=&quot;Point drawn on cone, labeled (1,0,f(1,0))&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Repeat parts (c)-(e) with the point &lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(1,2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/J3rbZ.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/J3rbZ.jpg&quot; alt=&quot;Three points drawn and labeled on cone&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Compare your approximations of the slope at each of those points.&lt;/p&gt;&#xA;&lt;p&gt;Now, Let's try this out algebraically. Since we're leaving the x value fixed, take the derivative of the function &lt;span class=&quot;math-container&quot;&gt;$z=f(x,y)=3-\sqrt{x^2+y^2}$&lt;/span&gt; by imagining that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a constant. [That is, just differentiate with respect to &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.]&lt;/p&gt;&#xA;&lt;p&gt;Etc. [Connect the dots to make a slice. What is this curve? How can you see the &amp;quot;slopes&amp;quot; on this curve? Repeat questions where we move in x and not in y.]&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;It just happens to be the simplest surface I can imagine students building with any level of precision for measurement or approximation of slopes. It might be the first time in years that they have used a compass or a protractor, and it gets them handling a surface they built.&lt;/p&gt;&#xA;&lt;p&gt;I'm sure there are other better examples of this kind of thing, but it has served me well for introductory purposes. There's probably a book of these types of activities, but this one just occurred to me to try once when I felt my students were too disconnected from the physicality of the topic.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-09-27T00:42:37.960" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18910" PostTypeId="2" ParentId="18895" CreationDate="2020-09-27T16:31:47.250" Score="5" Body="&lt;p&gt;Let me build on the idea of Steven Gubkin in his comments. One way to visualize this scenario is to use Ford circles. The standard picture is to plot a circle tangent to the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis at &lt;span class=&quot;math-container&quot;&gt;$\frac{p}{q}$&lt;/span&gt; with radius &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2q^2}$&lt;/span&gt; where we always assume &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; are relatively prime. This gives an intriguing family of circles with special tangency relationships. There is a crowd of circles with small radii just above the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis that can only be visualized in the mind's eye. These come from rational numbers with large denominators.&lt;/p&gt;&#xA;&lt;p&gt;Think about this as an open covering of  &lt;span class=&quot;math-container&quot;&gt;${\bf Q}\cap [0,1]$&lt;/span&gt;: push each circle downwards, so that it is centered on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis. The diameter of this depressed circle then marks an open interval of radius &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2q^2}$&lt;/span&gt; centered at &lt;span class=&quot;math-container&quot;&gt;$\frac{p}{q}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BLLMQ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BLLMQ.png&quot; alt=&quot;Ford Circles&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;An important visualization tool afforded by this: let &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;\alpha&amp;lt;1$&lt;/span&gt; and draw a vertical line into the Ford circles emanating from the point &lt;span class=&quot;math-container&quot;&gt;$(\alpha,0)$&lt;/span&gt; on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis. Then every intersection of this vertical line with a Ford circle corresponds to a rational approximation &lt;span class=&quot;math-container&quot;&gt;$\frac{p}{q}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; satisfying &lt;span class=&quot;math-container&quot;&gt;$|\alpha-\frac{p}{q}|&amp;lt;\frac{1}{2q^2}$&lt;/span&gt;. Of course it is difficult to actually see most of these intersections because the vast majority of Ford circles have small radius and blend into the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis.&lt;/p&gt;&#xA;&lt;p&gt;Dirichlet's approximation theorem states that for all  irrational &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, there are infinitely many rational numbers such that &lt;span class=&quot;math-container&quot;&gt;$|\alpha-\frac{p}{q}|&amp;lt;\frac{1}{2q^2}$&lt;/span&gt;. In visual terms, the vertical line emanating from &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;  interests infinitely many of the Ford circles. Stated in terms of the open covering of the rationals corresponding to the depressed Ford circles, each irrational &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is in infinitely many of the open intervals. It is important to note that the &amp;quot;total length&amp;quot; of these intervals is infinite. There are &lt;span class=&quot;math-container&quot;&gt;$\phi(q)$&lt;/span&gt; circles with radius &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2q^2}$&lt;/span&gt; corresponding to rationals &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{q}, \frac{2}{q},\cdots$&lt;/span&gt;.&#xA;Heuristically, &lt;span class=&quot;math-container&quot;&gt;$$\hbox{total length}=\sum_{\frac{p_i}{q_i}}\frac{1}{q_i^2}=\sum_{q_i}\frac{\phi(q_i)}{q_i ^2}\approx\sum_q \frac{1}{q}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But now for the next step: let us shrink the radii of the Ford circles from &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2q^2}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2q^{2+\epsilon}}$&lt;/span&gt;. Here, for purposes of plotting, we will take &lt;span class=&quot;math-container&quot;&gt;$\epsilon=\frac12.$&lt;/span&gt; So we will shrink radii to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2q^{5/2}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(In his comment, Steven Gubkin refers to &lt;span class=&quot;math-container&quot;&gt;$q^4$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$q^\frac{5}{2}$&lt;/span&gt;. This of course works, but I found that the circles got too small to be an effective plot.)&lt;/p&gt;&#xA;&lt;p&gt;Here is the corresponding visualization.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/pAnKH.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pAnKH.png&quot; alt=&quot;Modified Ford Circles&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Notice that as soon as &lt;span class=&quot;math-container&quot;&gt;$\epsilon&amp;gt;0$&lt;/span&gt;, the beautiful tangency relationships between the Ford circles is lost. We still have an open covering of the rational numbers in &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; by depressing these modified Ford circles.  But one starts to &amp;quot;see,&amp;quot; or least to anticipate, that something will change with the vertical lines and rational approximations to irrational numbers &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;. Indeed, we have Roth's Theorem, which tells us that if &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is an algebraic irrational number, then &lt;span class=&quot;math-container&quot;&gt;$\vert \alpha-\frac{p}{q}\vert&amp;lt;\frac{1}{q^{2+\epsilon}}$&lt;/span&gt; has at most finitely many solutions. Thus for algebraic irrational numbers, we know that the vertical line will intersect at most finitely many modified (shrunken) Ford circles. In terms of the corresponding open covering of the rationals that we get by depressing the modified Ford circles, such &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; can be in at most finitely many of the open intervals. Also, we can see more by summing the lengths of the intervals:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\hbox{total length}=\sum_{\frac{p_i}{q_i}}\frac{1}{q_i^{\frac52}}=\sum_{q_i}\frac{\phi(q_i)}{q_i ^{\frac52}}&amp;lt;\sum_q \frac{q}{q^{\frac52}}=\sum_q \frac{1}{q^{\frac32}}&amp;lt;\infty.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In summary, by shrinking the radii of the Ford circles, we maintain an open covering of the rationals in the unit interval, but can control their radii so that the total length of this open cover is less than 1. It follows that there must be uncountably many vertical lines that do not intersect any modified Ford circle. In terms of the corresponding open cover of the rationals in the unit interval, there are uncountably many irrationals that are not in any of the  subintervals.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2020-09-27T16:43:30.330" LastActivityDate="2020-09-27T16:43:30.330" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18911" PostTypeId="1" AcceptedAnswerId="18912" CreationDate="2020-09-28T01:06:09.547" Score="9" ViewCount="4753" Body="&lt;p&gt;I am teaching an introductory calculus course for high school juniors and seniors. It is not formally described as an AP Calculus course, but it is supposed to map roughly onto Calculus AB.&lt;/p&gt;&#xA;&lt;p&gt;The students needed more review of algebra, trigonometry, and precalculus than one might hope, and as a result, we are quite far behind, and risk not getting to integrals and the fundamental theorem of calculus, which are supposed to be the high point of this course. So I am looking for things to cut along the way.&lt;/p&gt;&#xA;&lt;p&gt;The most obvious candidate seems to be Newton's method. There is a section on it before the section on antiderivatives. It seems to be something of a side note.&lt;/p&gt;&#xA;&lt;p&gt;Am I going to regret cutting this? Does it serve some pedagogical purpose for introducing anti-derivatives or integrals? Is there something else that's going to depend on this method when they get to Calculus II?&lt;/p&gt;&#xA;" OwnerUserId="14510" LastEditorUserId="238" LastEditDate="2020-09-28T22:02:35.480" LastActivityDate="2021-12-28T10:54:16.620" Title="Can we skip Newton's Method?" Tags="|secondary-education|calculus|curriculum|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18912" PostTypeId="2" ParentId="18911" CreationDate="2020-09-28T11:52:27.643" Score="10" Body="&lt;p&gt;If time constraints are so dire that you risk not getting to cover integrals and the fundamental theorem of calculus, then I'd cut Newton's method (and probably much more) since I don't see how you could pass the test without knowing those former topics.&lt;/p&gt;&#xA;&lt;p&gt;But, if you cut Newton's method from the teaching, wouldn't it still come up in practice AP tests that the students do &lt;em&gt;assuming you'd have time to do it&lt;/em&gt;? If I did cut it, then I would have to definitely highlight it when it does come up on practice materials.&lt;/p&gt;&#xA;&lt;p&gt;What I might do is discuss the organization of the material in how you could streamline the course so that you definitely cover the most important things and then will come back to the &amp;quot;add-ons&amp;quot; (like Newtwon's method) if you have time. Or, you could make it a self-study unit?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-09-28T11:52:27.643" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18913" PostTypeId="2" ParentId="17536" CreationDate="2020-09-28T12:29:17.047" Score="3" Body="&lt;p&gt;As you say - GeoGebra is still actively developed. For example we've just added support for complex functions&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://wiki.geogebra.org/en/Reference:Changelog_6.0&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://wiki.geogebra.org/en/Reference:Changelog_6.0&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14673" LastActivityDate="2020-09-28T12:29:17.047" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18914" PostTypeId="2" ParentId="18911" CreationDate="2020-09-28T16:12:16.333" Score="10" Body="&lt;p&gt;AP classes have become much more common these days, and at many schools the result has been that very few students actually pass the AP exam with a grade that would allow them to skip the course in college. The trouble is that even if 90% of your students fall in this category, you also have a duty to serve the other 10%. Those students are going to be shortchanged if they miss out on this standard topic of freshman calculus.&lt;/p&gt;&#xA;&lt;p&gt;I would suggest that you just offer an optional 1-hour zoom session on a weekend for those students who want to have a little bit of instruction on this.&lt;/p&gt;&#xA;&lt;p&gt;If this is a high school class with 5 contact hours per week for 18 weeks, then I don't really understand how you can literally run out of time for all the standard topics of a freshman calculus course. An AP calculus course is a college course. That means that your students need to be working at the level of a college student, meaning that they don't need to be spoon-fed every topic and then hand-held on practicing it in excruciating detail.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-09-28T16:12:16.333" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18916" PostTypeId="2" ParentId="18911" CreationDate="2020-09-28T23:45:41.810" Score="3" Body="&lt;p&gt;Reasons to study Newton's method:&lt;br&gt;&#xA;-It's an application of derivatives&lt;br&gt;&#xA;-It is a good example of numerical methods&lt;br&gt;&#xA;-It can help strengthen understanding of relationship between derivative and tangent line&lt;br&gt;&#xA;-It's likely going to be on the AP test&lt;/p&gt;&#xA;&lt;p&gt;I've put these roughly in order of how important they are to your situation; as nice as it is to get greater conceptual understanding, what benefits your student from a practical point of view is passing the test. And if you use AP tests in your course as practice tests, you'll have to go through them and mark any questions on Newton's method as optional.&lt;/p&gt;&#xA;&lt;p&gt;You should consider making a list of the things you're cutting, finding self-study resources on those topics, and sending an email to students and/or parents explaining that the course isn't fully matching the AP test, and if they want to take the test at the end of the course, they should study these topics on their own.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2020-09-28T23:45:41.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18917" PostTypeId="1" AcceptedAnswerId="18918" CreationDate="2020-09-29T08:24:48.967" Score="7" ViewCount="1555" Body="&lt;p&gt;I am teaching a &amp;quot;linear algebra for engineers&amp;quot; course, and am currently building my lectures on Gram-Schmidt, QR-decomposition and least squares equation solving.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\bullet$&lt;/span&gt; I can motivate Gram-Schmidt perfectly well. Geometrically, if I have a tilted plane in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; and want to describe geometric figures in it, I want to work in orthonormal coordinates in order for lengths and angles to have their usual meaning. And, more abstractly, Gram-Schmidt leads to orthogonal polynomials, which come up in tons of PDE applications.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\bullet$&lt;/span&gt; I can motivate least-squares perfectly well, it is the first basic step in statistical modeling, and also key to solving linear equations with numerical errors.&lt;/p&gt;&#xA;&lt;p&gt;But I don't really see why students need &lt;span class=&quot;math-container&quot;&gt;$QR$&lt;/span&gt;-decomposition or, more precisely, why they need the matrix &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;. Orthogonal projection is given by &lt;span class=&quot;math-container&quot;&gt;$Q Q^T$&lt;/span&gt;. Least squares solving can be presented as &lt;span class=&quot;math-container&quot;&gt;$R^{-1} Q^T$&lt;/span&gt;, but the formula &lt;span class=&quot;math-container&quot;&gt;$(A^T A)^{-1} A^T$&lt;/span&gt; is also available. The latter is certainly nicer for hand computation; I'm not sure what is actually lurking inside computer algebra systems.&lt;/p&gt;&#xA;&lt;p&gt;Why is it worth teaching the students to compute &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, and to think about what it means?&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="376" LastEditDate="2020-09-29T08:45:04.933" LastActivityDate="2020-09-29T14:57:20.707" Title="Why does a first course in linear algebra teach QR-decomposition?" Tags="|linear-algebra|applied-mathematics|engineering-mathematics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18918" PostTypeId="2" ParentId="18917" CreationDate="2020-09-29T13:45:59.860" Score="7" Body="&lt;p&gt;I wouldn’t feel bad about leaving it out, but I think it’s a valuable conceptual example for understanding matrix algebra. Computing the QR decomposition is equivalent to applying Gram-Schmidt orthogonalization to the columns, and I think it’s really instructive to see how this corresponds exactly to the fact that Q is orthogonal and R is upper triangular (with positive diagonal): an iterative process has turned into a structural statement about a matrix factorization, which is a great example of what matrix algebra can do for you. Then this more abstract perspective is genuinely useful, because we can ask whether there are other, potentially better ways of computing this factorization, for example with improved numerical stability (the answer is yes). This question is much less natural if you view Gram-Schmidt orthogonalization as a specific process, rather than just one way of achieving an abstract goal. (Of course you could phrase the abstract goal without using matrix factorization, but this is a nice way to do it.)&lt;/p&gt;&#xA;&lt;p&gt;I also rather like the QR algorithm for eigenvalues. It’s very slick and simple, if you don’t get into the details of how to do it most efficiently or how to analyze the convergence, and it’s a great example of an iterative approximation algorithm. This is an important step up in sophistication from exact, finite algebraic algorithms.&lt;/p&gt;&#xA;&lt;p&gt;So my inclination would be to motivate the QR factorization as an example of the power of abstraction: by taking something familiar and turning it into a matrix factorization, we open up both new questions and new applications.&lt;/p&gt;&#xA;&lt;p&gt;As for why the students need R, one way I’d explain it to them is that it relates the new, orthonormal basis to the old basis. In some applications, you don’t care about this: the original basis had no special significance, and you’re happy to throw it away forever and replace it with the new basis. However, sometimes the original basis is important (either because of some intrinsic conceptual meaning, or just because it’s hard coded into some dataset), and then keeping track of how you transformed the basis becomes worthwhile.&lt;/p&gt;&#xA;" OwnerUserId="2018" LastActivityDate="2020-09-29T13:45:59.860" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18919" PostTypeId="2" ParentId="18911" CreationDate="2020-09-29T14:20:52.623" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;It is not formally described as an AP Calculus course, but it is supposed to map roughly onto Calculus AB.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It sounds like there isn't a clear definition for the level of the course or the content that needs to be covered.  It may be beneficial to poll the students to find out how many are planning on skipping Calculus II in college and moving straight to Calculus III (multi-dimensional applications) or other upper level courses.  If you find out that most of them will be taking Calculus II, then they will eventually learn integrals, while any applications of derivatives such as Newtons method will likely be missed.  However, If you find out that most plan to skip Calculus II, then they will miss the basics of integrals, which is critical for the more advanced courses.  This also applies to other Calculus II topics such as sequences, series, and summations that will likely be skipped.  Keep in mind that they will still be missing Newton's method and any other Calculus I content which will be skip.&lt;/p&gt;&#xA;&lt;p&gt;Also poll students on their field of study and career path.  Any students who are interested in a career in programming and computer science would greatly benefit from numerical methods applications such as Newton's method. This is especially true for schools that do not offer computer science courses.  In those cases, students rely on mathematics courses to get that experience.&lt;/p&gt;&#xA;" OwnerUserId="14683" LastEditorUserId="14683" LastEditDate="2020-09-29T19:05:44.993" LastActivityDate="2020-09-29T19:05:44.993" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18920" PostTypeId="2" ParentId="18917" CreationDate="2020-09-29T14:52:11.140" Score="5" Body="&lt;p&gt;Solving least squares problems by QR factorization is much more numerically stable than solving them by Cholesky factorization of the normal equations. This can easily be demonstrated on an ill-conditioned test problem.&lt;/p&gt;&#xA;" OwnerUserId="5841" LastEditorUserId="376" LastEditDate="2020-09-29T14:57:20.707" LastActivityDate="2020-09-29T14:57:20.707" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18921" PostTypeId="1" CreationDate="2020-09-30T01:15:45.030" Score="-2" ViewCount="63" Body="&lt;p&gt;I want to aproach the undertanding of the trigonometric function based on the concept of relacional undertanding, but I have problems to came up with and problemic situation for it. I mean I don´t want the first aproach to the students to be the typical unit circumference thing, I want they to comprenhend the how and why the trigonimetric function pop up the way they do.&lt;/p&gt;&#xA;&lt;p&gt;For example: I read about some formulas for some geometric objects, there is the formulas and no more; by the other hand, in other place the authors intuitively explaned the notion of area in a rectangle, ans they use it for came up with all the formulas that in the first paper only apear of nowhere (parallelogram, traprice, irregular trapezoid, triangle and even circle).&lt;/p&gt;&#xA;&lt;p&gt;If someone could recomend me some book to search about this, or give me some orientention. Thank for your time.&lt;/p&gt;&#xA;" OwnerUserId="14685" LastActivityDate="2020-09-30T04:21:15.477" Title="Relational understanding for a specific topic" Tags="|mathematical-pedagogy|education-research|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18922" PostTypeId="2" ParentId="18921" CreationDate="2020-09-30T04:21:15.477" Score="-1" Body="&lt;p&gt;[comment]&lt;/p&gt;&#xA;&lt;p&gt;Personally, I found the unit circle to be the most intuitive way to think about sin and cosine (and still think of them this way in my head, to this day).  Just feels less arbitrary and simpler than SOCATOA.  I mean of course, do what you want.  But realize, not everyone agrees that you are making things easier for the students.  Many would disagree.  P.s. see previous questions/answers/discussions on this topic, at this site.&lt;/p&gt;&#xA;" OwnerUserId="14687" LastActivityDate="2020-09-30T04:21:15.477" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18923" PostTypeId="1" CreationDate="2020-09-30T05:19:24.660" Score="0" ViewCount="793" Body="&lt;p&gt;Hello to all the professors who read this. I'm an electrical engineering undergrad student. I wanted to ask for advice on what I should learn beforehand to fully grasp the Laplace transform. I also want to know what the prerequisites for Fourier Analysis/ Fourier Series/ Fourier Transform and the prerequisites to work through &amp;quot;Trigonometric Series&amp;quot; by Zygmund. would be, now that I'm asking this question.&lt;/p&gt;&#xA;&lt;p&gt;Any book titles are warmly welcomed. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="14688" LastActivityDate="2020-09-30T05:19:24.660" Title="Prerequisites to study Laplace Transform completely?" Tags="|undergraduate-education|mathematical-pedagogy|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="18928" PostTypeId="1" AcceptedAnswerId="18929" CreationDate="2020-09-30T13:01:36.703" Score="5" ViewCount="332" Body="&lt;p&gt;My calculus students diligently learn Newton's Method, the Trapezoidal Rule, Simpson's Rule, and Euler's Method throughout their calculus careers, but the topics always feel like strange digressions into a different field instead of coherent parts of the narrative of the calculus course.&lt;/p&gt;&#xA;&lt;p&gt;I'm looking for any neat ways to apply these throughout calculus classes instead of following my normal approach, which is to cover them (or skip them when low on time) but fail to convince students that they really fit into our calculus courses.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2020-10-03T04:56:01.983" Title="What are some nice ways to incorporate Euler's Method or other numerical methods throughout calculus?" Tags="|applications|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18929" PostTypeId="2" ParentId="18928" CreationDate="2020-09-30T14:22:03.783" Score="6" Body="&lt;p&gt;Applying Euler's method to the differential equation &lt;span class=&quot;math-container&quot;&gt;$F'(x) = f(x)$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt; gives the approximation &lt;span class=&quot;math-container&quot;&gt;$$F(b) \approx F(a)+\sum_{i=1}^N f\left(a+\frac{i(b-a)}{N}\right) \frac{b-a}{N}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Taking the limit as &lt;span class=&quot;math-container&quot;&gt;$N \to \infty$&lt;/span&gt; yields the Fundamental Theorem.&lt;/p&gt;&#xA;&lt;p&gt;You can see this visually in the following Desmos app: &lt;a href=&quot;https://www.desmos.com/calculator/l4sikc9mcd&quot; rel=&quot;noreferrer&quot;&gt;https://www.desmos.com/calculator/l4sikc9mcd&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The piecewise linear approximation to F is exactly what is obtained by following a slope-field given by &lt;span class=&quot;math-container&quot;&gt;$y' = f(x)$&lt;/span&gt;, aka applying Euler's method.&lt;/p&gt;&#xA;&lt;p&gt;So Euler's method is a generalization of the Fundamental Theorem, and can actually be used to motivate the Fundamental Theorem.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-09-30T14:22:03.783" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18930" PostTypeId="1" CreationDate="2020-09-30T17:07:40.353" Score="1" ViewCount="222" Body="&lt;p&gt;In any undergraduate course on probability, one of the fundamental results that is discussed is the central limit theorem. Apart from the interesting mathematics needed to prove the result (most often introducing moment generating functions, and more subtly the idea that the distribution of a random variable &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; can be characterized by expectations &lt;span class=&quot;math-container&quot;&gt;$E[f(X)]$&lt;/span&gt; over a rich enough class of test functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;), the central limit theorem can be a good opportunity to teach students about the so-called universality of the Gaussian distribution.&lt;/p&gt;&#xA;&lt;p&gt;That being said, I would like my explanation of universality to come with one very important caveat, so that my students do not, like many people, become &amp;quot;unreasonably enthusiastic&amp;quot; about the Gaussian distribution. (Essentially the error I would like them to avoid is what is explained in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1325/impressive-common-misleading-interpretations-in-statistics-to-make-students-awar/1337#1337&quot;&gt;this&lt;/a&gt; answer to the question &amp;quot;Impressive common misleading interpretations in statistics to make students aware of&amp;quot;.)&lt;/p&gt;&#xA;&lt;p&gt;So my question would be as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; Are there well-documented examples of people who, likely because they erroneously assumed some distribution to be Gaussian, were led to a catastrophic outcome? (Perhaps in the insurance or financial industries, or other.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I feel that an interesting real-life example would help drive the point home that, if you want to argue something is Gaussian by invoking the CLT, you better be able to write it down as the sum of a large number of i.i.d. variables divided by square root of their number.&lt;/p&gt;&#xA;" OwnerUserId="14693" LastActivityDate="2020-11-01T19:40:02.727" Title="Cautionary tales for Gaussians/the central limit theorem?" Tags="|probability|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18931" PostTypeId="1" CreationDate="2020-09-30T19:02:04.423" Score="1" ViewCount="269" Body="&lt;p&gt;Can anyone please tell me if the AOPS (Art of Problem Solving) Geometry is more rigorous/difficult generally than a book called &amp;quot;Geometry for Enjoyment and Challenge&amp;quot; by authors Rhoad, Milauskas and Whipple&lt;/p&gt;&#xA;" OwnerUserId="14694" LastActivityDate="2020-10-28T18:06:19.967" Title="Which Geometry book is more rigorous/harder?" Tags="|geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18932" PostTypeId="2" ParentId="14108" CreationDate="2020-09-30T19:31:33.353" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://phys.org/news/2020-09-busy-pictures-hinder-ability-children.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Busy pictures hinder reading ability in children&lt;/a&gt;, according to study at Carnegie Mellon University. They find that an overly busy page with extraneous images can draw the reader's attention away from the text, resulting in lower understanding of content.&lt;/p&gt;&#xA;&lt;p&gt;I would argue the same is true for math: colors can give false impression about properties of objects like geometric figures, especially if they are presented using the same color again and again, like squares are always green and triangles are always red, etc. Using colored objects may also train the students to dismiss anything that is not colored, like on the image below, where is a triangle? Color may make unimportant attributes seem important.&lt;/p&gt;&#xA;&lt;p&gt;There are standard symbols that must be learned by students — and teachers — that allow to indicate, for example, that two angles are congruent or not, or that two lines are parallel or that two values are equal.&lt;/p&gt;&#xA;&lt;p&gt;The dumbing down of textbooks have been going for at least three decades. Meaningless pictures supposedly make textbooks &amp;quot;livelier&amp;quot;. Adding color to formulas and geometric drawings strives for the same &amp;quot;math is fun!&amp;quot; spirit.&lt;/p&gt;&#xA;&lt;p&gt;A textbook is not a fashion magazine, and students would be served better if they were presented with coherent text and clear and simple illustrations with no egregious mistakes.&lt;/p&gt;&#xA;&lt;p&gt;Here is a portion from Core Plus Mathematics I with colored squares. The book mentions the Pythagorean Theorem, but does not prove it. Instead it asks students to calculate area of the squares and of the triangle, thereby, supposedly, asserting the correctness of the theorem. I wonder how the students are supposed to find the sides of the squares built on the hypotenuse — just by measuring it with a ruler? If yes, why the dotted grid? Cannot use it anyway.&lt;/p&gt;&#xA;&lt;p&gt;The triangle is almost lost on this picture, because it is not colored. Is it important that squares are colored, and the triangles are not? Is it important that corresponding squares have the same color? The unit square has beige color, there are two other squares that are beige, does it mean they are also unit squares?&lt;/p&gt;&#xA;&lt;p&gt;This is not a third-grade textbook, this is a ninth-grade textbook.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PjgYY.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PjgYY.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2020-09-30T20:51:31.637" LastActivityDate="2020-09-30T20:51:31.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18933" PostTypeId="2" ParentId="1545" CreationDate="2020-09-30T20:33:26.960" Score="2" Body="&lt;p&gt;Following on the example above by Markus Klein:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A gardener has 20 meters of fence-material and wants to fence a rectangle shaped garden with maximal area. Furthermore, the garden will be located next to the house such that the house will serve as one of the four walls.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The problem then reads: Maximize &lt;span class=&quot;math-container&quot;&gt;$f(a,b) = a*b$&lt;/span&gt; subject to &lt;span class=&quot;math-container&quot;&gt;$2a + b = 20$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The answer &lt;span class=&quot;math-container&quot;&gt;$(a = 5, b = 10)$&lt;/span&gt; is not so obvious to students.&lt;/p&gt;&#xA;&lt;p&gt;(I realize this should probably be just a comment, but I am new here and don't yet have the ability to comment.)&lt;/p&gt;&#xA;" OwnerUserId="14695" LastActivityDate="2020-09-30T20:33:26.960" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18934" PostTypeId="2" ParentId="18930" CreationDate="2020-10-01T04:59:55.610" Score="-1" Body="&lt;p&gt;A real world, time I have encountered this, was doing process variation calculations (e.g. &amp;quot;Cpk&amp;quot;) for pharmaceutical analytical or mechanical quality measurements.  In fact, our issue was taking people who were doing zero annual analysis (not even trending, averaging, standard deviation, etc.) and moving to doing some control measurements, anything.&lt;/p&gt;&#xA;&lt;p&gt;But then after installing those, I got the criticism that FDA has fussed (rightfully in some cases) about non-normal variations.  So we had to do normality tests, to tie the bow, and say we had gone from a C to a B to an A.&lt;/p&gt;&#xA;&lt;p&gt;See for instance:  &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwjzjPvLzJLsAhUPsKQKHbf0ADsQFjAKegQIARAC&amp;amp;url=https%3A%2F%2Fscholarworks.waldenu.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D4800%26context%3Ddissertations&amp;amp;usg=AOvVaw3ozM0ECpaljyZdNqJD8YGB&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwjzjPvLzJLsAhUPsKQKHbf0ADsQFjAKegQIARAC&amp;amp;url=https%3A%2F%2Fscholarworks.waldenu.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D4800%26context%3Ddissertations&amp;amp;usg=AOvVaw3ozM0ECpaljyZdNqJD8YGB&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;for an overview.&lt;/p&gt;&#xA;&lt;p&gt;My understanding is there have been a few high profile product recalls where failure to check for normality was an issue (see the intro section of the thesis, I linked to).  Honestly we were worse than that...since we didn't even do any SPC.  but that's another issue...and just shows that there can be a value in doing some tests, even imperfect.  All that said, I ran the normality tests, so we didn't get fussed at, since at that point we were in the gunsights and giving the &amp;quot;went from a C to a B&amp;quot; argument wasn't going to work.  Just had to do it at A standards, since our facility was shut down for poor quality, patient injuries, etc.  :-(&lt;/p&gt;&#xA;" OwnerUserId="14696" LastActivityDate="2020-10-01T04:59:55.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18936" PostTypeId="1" CreationDate="2020-10-01T09:10:13.933" Score="0" ViewCount="134" Body="&lt;p&gt;Over the years of being a student or of being a teacher, it's occurred to me that I've never done every topic in a textbook. If it's a workbook that has a few introductory theory pages and then a few more pages devoted to working on practice problems &lt;em&gt;and has blank space for students to write in&lt;/em&gt;, then yes, I've completely covered everything in those books (both as a student, and in assigning it as a teacher). But I've never had such thoroughness when it comes to a textbook.&lt;/p&gt;&#xA;&lt;p&gt;I'm not asking if you &lt;strong&gt;should&lt;/strong&gt; try and cover everything, but I'm just wondering if any textbooks are have been completely utilized? If so, then which ones and what about them made it so that you could (or would) cover all topics?)&lt;/p&gt;&#xA;&lt;p&gt;(About level of textbook, I'm referring to post-secondary/high school (so I'm referring to college or university texts))&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorUserId="5321" LastEditDate="2020-10-01T13:05:49.553" LastActivityDate="2020-10-01T13:05:49.553" Title="Have you ever completely covered every topic in a textbook, and if so, which text?" Tags="|undergraduate-education|textbooks|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18937" PostTypeId="1" CreationDate="2020-10-01T15:21:57.840" Score="0" ViewCount="112" Body="&lt;p&gt;I run a small tutoring company, and I specialize in math and science. During the pandemic, I began tutoring online using our 3 year old HP Pavilion touchscreen laptop. It has served me well using just my finger, and Zoom's whiteboard function. Unfortunately, my 7 year old daughter dropped it recently, and now the touchscreen is only half functional (think half the screen space for whiteboard functionality). I am getting by, but it is severely hampering my workflow, and as a result, I find myself feeling very nervous with my students because I am worried that the capability of my laptop is not going to hold out through the hour session. I need something else, like, now.&lt;/p&gt;&#xA;&lt;p&gt;Years ago, I remember my dynamics professor used one of the first touchscreen 2-in-1 laptops in class, and to this day I think about the importance of functioning technology in supporting students in learning STEM subjects. I need a new laptop as soon as possible, but am overwhelmed by the options (my professor likely only had the one option since it was so new).&lt;/p&gt;&#xA;&lt;p&gt;What are you all using? How is it supporting your teaching capabilities. I've done research on several options but am having severe trouble deciding because cost is, of course, a factor for my small business. I prefer something that can securely handle the myriad files associated with running a small business, but with a nice big sturdy touchscreen that I can easily use to draw diagrams and equations on without feeling cramped. Help! Note: I do NOT need it to run specific kinds of higher level STEM software, as that is not business my specialty.&lt;/p&gt;&#xA;" OwnerUserId="14701" LastEditorUserId="376" LastEditDate="2020-10-01T15:37:08.473" LastActivityDate="2020-10-01T15:37:08.473" Title="Best Touchscreen Laptop for Tutoring During COVID Pandemic" Tags="|technology-in-education|tutoring|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18938" PostTypeId="2" ParentId="18928" CreationDate="2020-10-02T02:14:35.590" Score="1" Body="&lt;p&gt;I've had a relatively good experience doing some of these via spreadsheets. (Google sheets works for everyone who has a web browser.) You can get ahold of some interesting data --- such as the melt rate of glaciers in Greenland --- and cook up the relevant column to apply the &lt;code&gt;SUM()&lt;/code&gt; function to and that's your approximation to the integral. You can't let &lt;span class=&quot;math-container&quot;&gt;$h \to 0$&lt;/span&gt;, but that's what real data is like anyway.&lt;/p&gt;&#xA;&lt;p&gt;I also did this sort of thing in a lower-than-calc level college class for calculating annuities, Fibonacci numbers, etc. If nothing else, students got a toe-hold on some spreadsheet skills.&lt;/p&gt;&#xA;&lt;p&gt;Having taught a decent number of CS classes, I can see the temptation to try Python. I would avoid it at this level though; you want to teach math and not do tech support. Nor do you want to explain syntax, &lt;code&gt;for&lt;/code&gt; and &lt;code&gt;range&lt;/code&gt;, the different semantics of variables in math and cs, etc.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2020-10-02T02:14:35.590" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18939" PostTypeId="2" ParentId="18928" CreationDate="2020-10-02T06:30:27.007" Score="6" Body="&lt;p&gt;Calculus students usually see the theorem that a continuous function on a closed interval that takes on different signs at the intervals endpoints must have a zero in the interval. This can be proved rigorously via the bisection algorithm. In addition to being clearer than a more abstract proof, this can be used as a first introduction to numerical methods.&lt;/p&gt;&#xA;&lt;p&gt;The geometric interpretation of Newton's method requires an understanding of the relation between the derivative of a funciton and the line tangent to the graph of the function. Consequently, Newton's method fits well in the early discussion of derivatives. On the other hand, it gives an application of the derivative (to root finding) that might help motivate interest in derivatives.&lt;/p&gt;&#xA;&lt;p&gt;And so forth. Nearly every introductory topic can be related with numerical methods. That this is not customary everywhere reflects the inertial resistance to change in elementary curricula.&lt;/p&gt;&#xA;&lt;p&gt;(However, coding these algorithms should be left for a subsequent class devoted to that task. Many students struggle with coding loops and conditionals and this requires its own pedagogy. But it is easier if the students already understand the algorithms they are asked to code.)&lt;/p&gt;&#xA;&lt;p&gt;Calculus students usually acquire the mistaken notions that most integrals can be evaluated exactly and that exact expressions/formulas are useful. In fact most computation is numerical and most evaluation of exact expressions/formulas is also numerical. Particularly with students who will mostly use calculus on computers it is important to introduce them early to the issues related to practical computation of integrals and derivatives.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2020-10-02T06:30:27.007" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18940" PostTypeId="2" ParentId="18928" CreationDate="2020-10-03T04:56:01.983" Score="3" Body="&lt;p&gt;Analysts would be horrified, but I'm a combinatorialist, so when I teach calculus, I take the viewpoint that a function is first and foremost a table of data.  We have a bunch of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s, and for those &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s we have values &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.  (The &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s are not required to be evenly spaced.)  Sometimes the table comes from plugging &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s into a symbolic formula - in which case particularly nice things happen, or they are read from a graph, but the model of function I want my students to have in mind first is that of a table of data.&lt;/p&gt;&#xA;&lt;p&gt;Derivatives are first estimated with no limits (because we have discrete data) and integrals are first estimated numerically, both before any formulas are discussed.&lt;/p&gt;&#xA;&lt;p&gt;Computations are done in an Excel spreadsheet, so no sigma notation is necessary (though you get the equivalent in Excel's SUM() constructions).&lt;/p&gt;&#xA;&lt;p&gt;From this point of view, numerical methods are the most fundamental part of a calculus course.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2020-10-03T04:56:01.983" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18941" PostTypeId="1" AcceptedAnswerId="18946" CreationDate="2020-10-03T11:02:23.437" Score="2" ViewCount="881" Body="&lt;p&gt;Until this point I have felt inspired to study math and looked forward to my classes. But lately I've been having difficulty with my math skills, where I keep making errors over problems I had absolutely no problem with some months ago - errors I never thought I'd make so frequently. Mistreating constants as variables, neglecting coefficients, neglecting bounds, forgetting basic trig identities, forgetting integration techniques, etc.&lt;/p&gt;&#xA;&lt;p&gt;It doesn't help I landed a professor that is extremely strict. I'm thinking I might be hitting a temporary setback, a few bad months because of covid-19. But part of me is also concerned I'm losing my drive for math. The prospect of me having to retake a class has made me almost always sleep deprived and unable to focus and enjoy my study sessions like I did previously (because I already failed it once last summer). I used to be the student happily helping other students solve problems and trying to get them to get their aha moments on my own free time, but these days I've stopped interacting with my classmates because of these issues.&lt;/p&gt;&#xA;&lt;p&gt;I guess the question is, when do I know to call it quits? Do I change majors or drop out after having to retake the same class a couple times?  Do I chalk this down to a bad semester and a bad time to be a student?&lt;/p&gt;&#xA;" OwnerUserId="12513" LastActivityDate="2020-10-05T16:24:43.057" Title="Struggling with Math Skills and When to Quit" Tags="|student-motivation|struggling-students|" AnswerCount="3" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="18943" PostTypeId="1" CreationDate="2020-10-04T01:40:45.723" Score="3" ViewCount="240" Body="&lt;p&gt;A student has a english problem, should he get another chance and have another chance?&lt;/p&gt;&#xA;" OwnerUserId="14707" LastEditorUserId="14707" LastEditDate="2020-10-07T04:43:14.220" LastActivityDate="2020-10-07T04:43:14.220" Title="Should a student have another chance at a math test question when they have a english problem?" Tags="|secondary-education|testing|" AnswerCount="2" CommentCount="6" ClosedDate="2020-10-06T14:21:09.907" ContentLicense="CC BY-SA 4.0" />
  <row Id="18944" PostTypeId="2" ParentId="18943" CreationDate="2020-10-04T02:31:51.670" Score="1" Body="&lt;p&gt;Pretty open-ended questions since there aren't any specifics, so I'll give my scenarios:&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't allow another chance if it was mathematically-related word that would have been seen in a previous year. For example a high school student asking what a &amp;quot;numerator&amp;quot; is, or a &amp;quot;sum&amp;quot;, or a &amp;quot;factor&amp;quot; etc. (I don't think it's unfair to set a standard of require knowledge based on prerequisite courses)&lt;/p&gt;&#xA;&lt;p&gt;I might allow another chance if most students couldn't answer it, and the question couldn't be understood contextually. For example, I could make up words but I don't think they effect the math, like this: &amp;quot;A qwert has 10 werts and each wert has 5 erts. If I have 3 qwerts, how many erts do I have?&amp;quot; Although those are made-up words, I think the question could be done regardless. (In fact, since I use metric, this is how I usually think of imperial units words)&lt;/p&gt;&#xA;&lt;p&gt;Could we see the question?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-10-04T02:31:51.670" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18945" PostTypeId="2" ParentId="18941" CreationDate="2020-10-04T03:48:18.267" Score="4" Body="&lt;p&gt;It sounds like OP is in their first year, and likely first semester, of a college math program. One's first year in college has traditionally been a very large changeover point. College courses (good ones, at least) require a lot more study, rigor, and precision in writing -- especially in fields like math and computing. Admittedly, they're not for everyone. Furthermore, college life and courses require that the student take control of their own goals and become self-directed and reasonably disciplined without as much hand-holding or safety net supports from family or instructors. The combination of new challenges can be burdensome for many or most people.&lt;/p&gt;&#xA;&lt;p&gt;For your program, also bear in mind that the discrete math/discrete structures course is generally used as a &amp;quot;transition&amp;quot; and introduction-to-proof course, which is the gateway to the rest of the studies in the math discipline. This switch from calculation-based to proof-based courses is therefore a third big hurdle that you're facing right now. The calculus II course shouldn't be difficult for a math major, exactly, but it's not the most important thing. In the far future it's possible that you may use little to none of that content; but the concepts of logic and how to read and write a proof are essential skills used always by any math practitioner. Do you &lt;em&gt;like&lt;/em&gt; the discrete math content, does it fire your imagination? That's the best sign that the math major is the right choice (and that's what most later classes look like).&lt;/p&gt;&#xA;&lt;p&gt;For the math courses, I would encourage you to start learning to read the textbook itself. One of the most formative experiences I had was, in my first college calculus course, realizing that the classroom experience itself wasn't doing much for me (delivered at distance via closed-circuit television), and I started reading the book very carefully and slowly on my own. The fact that I could do this was a revelation and was the primary tool I relied on to get through an undergraduate math program. Arguably, my job today is basically to apply that same skill: read an arbitrary textbook and summarize it in digestible chunks for current students.&lt;/p&gt;&#xA;&lt;p&gt;For time-management skills, I recommend that you read the first 3 chapters of Eva Lantsoght, &lt;a href=&quot;https://www.dbooks.org/the-a-z-of-the-phd-trajectory-3319774255/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The A-Z of the PhD Trajectory&lt;/a&gt;. You're not going for a PhD (now), and neither am I, but recently I've found the time-management procedures there to be extremely helpful. Lantsoght says she developed most of them while she was flailing as an undergraduate science student.&lt;/p&gt;&#xA;&lt;p&gt;More generally, I would encourage you to use your undergraduate experience (esp. the early part) to take courses from as wide an array of disciplines and departments as possible, and see if any open your mind in ways you didn't expect. You likely at least have general-education requirements, and you should look at these as opportunities and essential college experiences. Do any of the areas excite and intrigue you more than mathematics? Possibly areas you'd never even heard of before now (many of which don't exist in the high school curriculum)? If so, then you should follow the path of greatest excitement (balanced by considerations of resource and career potential). You can change paths any time in life, but it's certainly easiest at the point where you are right now. Early undergraduate years should be a time of broad exploration.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-10-04T14:03:38.520" LastActivityDate="2020-10-04T14:03:38.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18946" PostTypeId="2" ParentId="18941" CreationDate="2020-10-04T11:54:47.540" Score="7" Body="&lt;p&gt;To add to Daniel Collins' thoughtful and considered &lt;a href=&quot;https://matheducators.stackexchange.com/a/18945/376&quot;&gt;advice&lt;/a&gt;, another helpful book could be Lara Alcock's &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0199661316&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to Study as a Mathematics Major&lt;/a&gt;. It offers tips on calculation procedures, transitioning to proof-based mathematics, dealing with others and time management.&lt;/p&gt;&#xA;&lt;p&gt;In my experience, many students hit a wall at some point. It can be very frustrating and cause you to doubt yourself, but it can also be a chance to take a breather, reflect, evaluate, and gather the courage to try again/keep going, possibly with an adjusted strategy. The wall may not be as insurmountable as it seems at first glance. Be patient with yourself and do not assume you are not cut out for studying mathematics. Ultimately, you might decide to chart a different course in your degree, or maybe not, but it's early days yet.&lt;/p&gt;&#xA;&lt;p&gt;Also, life circumstances are not to be underestimated. The Covid-19 pandemic has made life more challenging for many students and their family/friends. This is on top of all the other difficulties life can throw at you at times. Make sure you investigate (and use) the resources offered to you by the university: study skills training, counselling, tutors, TAs, etc.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2020-10-04T11:54:47.540" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="18947" PostTypeId="1" CreationDate="2020-10-04T13:30:54.717" Score="1" ViewCount="136" Body="&lt;p&gt;What are standard tests for mental visualization (image, representation) for kids?&#xA;So far I know about mental arithmetics and spatial rotation tests. Is there any other way to check mental vision ability?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As an example, here's a test I've created. One has to imagine and do manipulation over the mental image:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;imagine a &lt;strong&gt;white&lt;/strong&gt; square with clearly visible white borders on a black background&lt;/li&gt;&#xA;&lt;li&gt;now draw white vertical line in the middle of the square. What do you see now?&lt;/li&gt;&#xA;&lt;li&gt;now draw white horizontal line in the middle of the square. What do you see now? Which and how many parts do you see?&lt;/li&gt;&#xA;&lt;li&gt;now clear up the two lines you draw. Instead, draw two vertical lines which divide square vertically on equal parts, and draw two horizontal lines which divide square horizontally on equal parts. What do you see? How many parts are there?&lt;/li&gt;&#xA;&lt;li&gt;now use lines as guides and cut out corners of the resulting figure and remove those from pictures. What do you see? How many parts does it have?&lt;/li&gt;&#xA;&lt;li&gt;now attach &lt;strong&gt;red&lt;/strong&gt; squares to the corners (instead of previously cut ones)&lt;/li&gt;&#xA;&lt;li&gt;now split the figure into 3 parts using vertical lines you've imagined before. Let the side parts glide in opposite directions and rotate 90 degrees clockwise. After rotation, merge the the parts back into one figure&lt;/li&gt;&#xA;&lt;li&gt;now imagine two transformations: 1) rotate resulting figure 180 degrees and 2) mirror the figure. Are the two transformations equal (do they produce same image)?&lt;/li&gt;&#xA;&lt;li&gt;now name colors of parts of the image after 180 degrees rotation&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Here's how I'd like to interpret results:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;correct answers to all questions: child is capable for mental vis&lt;/li&gt;&#xA;&lt;li&gt;in the middle of test says &amp;quot;this is boring&amp;quot; or distracts: child has reduced capability for mental vis&lt;/li&gt;&#xA;&lt;li&gt;has problems with answering 3rd questions: child isn't capable of mental vis. This is a real problem for further education&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I haven't yet tried to run this test, but I'd like to know if there are other such, which can verify a child is doing mental visualization.&lt;/p&gt;&#xA;" OwnerUserId="14709" LastActivityDate="2020-10-04T13:30:54.717" Title="Mental visualisation ability test for each age?" Tags="|primary-education|meta-cognition|mental-math|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="18948" PostTypeId="1" AcceptedAnswerId="18949" CreationDate="2020-10-04T20:10:10.720" Score="13" ViewCount="11870" Body="&lt;p&gt;Looking up for definition for &lt;em&gt;whole numbers&lt;/em&gt; on Google yields a result which mentions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The whole numbers are also called the positive integers (or the nonnegative integers, if zero is included).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I was suspicious about this answer and I decided to dig more into this. I found a &lt;a href=&quot;https://www.quora.com/Can-negative-numbers-still-be-whole-numbers&quot; rel=&quot;noreferrer&quot;&gt;Quora answer&lt;/a&gt; which mentions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;According to American middle and high school textbooks, the set of whole numbers includes all positive integers and 0, and not anything&#xA;else. In this context negative numbers cannot be whole numbers. I have&#xA;no clue as to the origin of this poor usage.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Professional mathematicians (researchers) tend to use the term whole numbers somewhat more informally as a synonym for integers, with&#xA;“whole” meaning without fractional part. Certainly −1, −2, … have no&#xA;fractional part so they are whole. This is consistent with German&#xA;terminology where the formal name for integers is “ganze Zahlen”,&#xA;which literally means whole numbers, as well as French terminology&#xA;“nombre entier” (often simply “entier”), with entier being a cognate&#xA;to the English “entire” in the sense of whole. The best way to avoid&#xA;ambiguity is to be explicit about which integers you are referring to&#xA;by applying the appropriate adjective to “integer”. “Integer” by&#xA;itself includes positive, 0, and negative; positive integers means 1&#xA;and up; non-negative integers means 0 and up; negative integers means&#xA;−1 and down; non-positive integers means 0 and down. Do not use the&#xA;terms natural numbers and whole numbers.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;However, be careful with the concepts of positive and negative&#xA;(“positif” and “négatif”, respectively), as the French think of 0 as&#xA;both positive and negative, whereas English, German, and many other&#xA;languages regard 0 as neither positive nor negative.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Does anyone know why American textbooks decided to go this way?&lt;/p&gt;&#xA;&lt;p&gt;I ask because I’m using American textbooks to study maths and now I’m more curious about incorrect things in them. Yet, this is the first thing I found.&lt;/p&gt;&#xA;" OwnerUserId="14711" LastEditorUserId="376" LastEditDate="2020-10-06T09:47:32.207" LastActivityDate="2020-10-09T16:48:54.887" Title="Why don’t American school textbooks recognize negative numbers as whole numbers?" Tags="|textbooks|terminology|negative-numbers|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18949" PostTypeId="2" ParentId="18948" CreationDate="2020-10-05T02:49:26.460" Score="39" Body="&lt;p&gt;&amp;quot;The whole numbers&amp;quot; is not a term that professional mathematicians use to describe a certain set of numbers. The term is used in elementary education when fractions are introduced, so that one can distinguish between numbers that have a fractional part and numbers that don't. In the US, this happens in grades 3 or 4. As far as I can tell from the Common Core standards, negative numbers aren't introduced until around grade 6. So at the time when the term is introduced to kids, there is no question to address as to whether -1 is a &amp;quot;whole number,&amp;quot; because those kids don't know that there is any number called -1.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-10-05T02:49:26.460" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="18950" PostTypeId="2" ParentId="18948" CreationDate="2020-10-05T04:45:36.027" Score="20" Body="&lt;p&gt;I don't think that &amp;quot;textbooks&amp;quot; decided this, usage did. The term &amp;quot;integer&amp;quot; covers positive and negative,  so it would be redundant for whole numbers to refer to that category. And there is an argument to be made for the term linguistically: a negative number is sort of the opposite of having a whole thing.&lt;/p&gt;&#xA;&lt;p&gt;But ultimately, there's not much to be said for &amp;quot;why&amp;quot; questions when it comes to words. You imply that this is an &amp;quot;incorrect&amp;quot; thing, suggesting that you have a Platonic view of words, that there is some objectively &amp;quot;correct&amp;quot; word. But words are just conventions. It is a convention that negative numbers are not whole numbers, therefore negative numbers are not whole numbers.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2020-10-05T04:45:36.027" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18951" PostTypeId="1" CreationDate="2020-10-05T08:29:29.507" Score="2" ViewCount="98" Body="&lt;p&gt;I am teaching linear algebra as part of an information retrieval course, which now occurs online. I have a Wacom tablet and free drawing software, sketchBook for artists, so can draw circles ellipses etc. I need better software to draw cartesian coordinate systems, which includes grids and is preferably aimed at teaching mathematics. Can you advise whether there is such a package out there? An internet search has not brought one up. I know and use Mathematica, but it may be a little overkill to go that route. Perhaps a combination of Wacom and Mathematica would work. If you have examples of your own teaching of linear algebra (e.g. eigenvalues, eigenvectors, similar matrices, spectral decomposition etc) I'd much appreciate tips on how to set it up best.&lt;/p&gt;&#xA;" OwnerUserId="14713" LastEditorUserId="8571" LastEditDate="2020-10-05T11:50:50.350" LastActivityDate="2020-10-05T11:50:50.350" Title="Teaching linear algebra, wacom tablet display of coordinate system, eigenvectors, markov chains" Tags="|linear-algebra|online-instruction|software|vectors|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18953" PostTypeId="2" ParentId="18941" CreationDate="2020-10-05T16:24:43.057" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I guess the question is, when do I know to call it quits?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Having been in shoes, I will give you the advice I wish someone would have told me since the beginning: seek a mental health professional. Lack of motivation, sleep deprivation, inability to focus, no longer enjoying activities that used to be fun, retraction from social interactions, feeling like a &amp;quot;fraud&amp;quot; could all be symptoms of depression or anxiety.&lt;/p&gt;&#xA;&lt;p&gt;The COVID pandemic has been shown to exacerbate those issues. Don't quit before being sure they're being caused by math itself, instead of external factors. Once you get your mental health in order, the appreciation for learning (not because you need to &amp;quot;pass&amp;quot;, but because it makes life more fulfilling) should return gradually.&lt;/p&gt;&#xA;&lt;p&gt;It might mean you'll need to lower your expectations and take on a lighter load for a while. I don't know where you live, but most Western colleges offer counselling for those situations, since they're far more common than they're willing to admit. Try not to think of it as a a setback, but rather as a pit stop: when you're in it for the long run, some refuelling is bound to happen time to time.&lt;/p&gt;&#xA;" OwnerUserId="14720" LastActivityDate="2020-10-05T16:24:43.057" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18954" PostTypeId="2" ParentId="18948" CreationDate="2020-10-05T16:27:08.110" Score="26" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I’m more curious about incorrect things in them. Yet, this is the first thing I found.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There's absolutely nothing &amp;quot;incorrect&amp;quot; about this.&lt;/p&gt;&#xA;&lt;p&gt;As Dave L Renfro noted in a comment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;and whole number is rarely used as a precise designation outside of school mathematics&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;there is no agreed-upon rigorous definition of the term, and in fact it's largely viewed by professional mathematicians as a matter of philosophy of mathematics and of one's particular school/ideology whether you consider &amp;quot;whole numbers&amp;quot; to include negatives or even zero. Mathematicians writing rigorously and with a concern for not raising annoying pedantic rehashing of this topic will tend to write things out explicitly as &amp;quot;positive whole numbers&amp;quot;, &amp;quot;non-negative whole numbers&amp;quot;, etc. (albeit typically using the more rigorous term &lt;em&gt;integers&lt;/em&gt;, which is always understood to include negative numbers and zero). But more importantly, they'll take care to &lt;em&gt;define the terms they use&lt;/em&gt; in the context they're using them in.&lt;/p&gt;&#xA;&lt;p&gt;To me, the important takeaway for mathematical education is the idea that definitions vary by contexts and cultures, and that the underlying mathematical idea is independent of how a particular writer or teacher chooses to present it. Teachers who fail to acknowledge this and who insist (especially to young children) that certain definitions are &amp;quot;wrong&amp;quot; are doing a huge disservice to their students' appreciation of mathematics - it undermines the universality entirely and teaches them that this is just another subject where the teacher pompously thinks they're always right.&lt;/p&gt;&#xA;" OwnerUserId="1146" LastActivityDate="2020-10-05T16:27:08.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18955" PostTypeId="1" CreationDate="2020-10-05T18:32:01.680" Score="4" ViewCount="154" Body="&lt;p&gt;Is there any university program that offers an introductory statistical methods class that is bayesian -instead- of a frequentist one?&lt;/p&gt;&#xA;&lt;p&gt;Surely every intro to stats class in every program in the world, whether agronomy or economics or psychology or a service class by the stats department, is primarily frequentist based.&lt;/p&gt;&#xA;&lt;p&gt;And just as surely there are many (but not all) programs where later in the curriculum there are some specialty courses (requiring a few stats pre-reqs) that are titled 'Intro to Bayesian Statistics'.&lt;/p&gt;&#xA;&lt;p&gt;But &lt;strong&gt;are there at least some places somewhere that offer a bayesian class as &lt;em&gt;the&lt;/em&gt; intro statistical methods class?&lt;/strong&gt; (either as the only intro to stats -or- simply as an alternative first class to a frequentist intro)&lt;/p&gt;&#xA;&lt;p&gt;I would count an intro stats class that teaches bayesian methods in addition to frequentist ones.&lt;/p&gt;&#xA;&lt;p&gt;If such programs are not as rare as I imagine, then is there any indication of percentage?&lt;/p&gt;&#xA;&lt;p&gt;I am wondering how easy it is to have a bayesian methods course without first having a frequentist methods course.&lt;/p&gt;&#xA;" OwnerUserId="2134" LastActivityDate="2020-10-11T16:34:23.070" Title="Is there any university program that offers an introductory statistical methods class that is bayesian -instead- of a frequentist one?" Tags="|statistics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18956" PostTypeId="2" ParentId="18948" CreationDate="2020-10-06T00:31:47.157" Score="-2" Body="&lt;p&gt;The natural numbers are the counting numbers  1,2,3,....&#xA;Add zero  and then negative numbers and you have the integers.&#xA;Then come the rationals,   reals,  complex,  quaternions, and octonions.&#xA;Then you have run out of numbers.  There is NO whole number in those.&lt;/p&gt;&#xA;&lt;p&gt;So the answer is because there are nothing called 'whole' numbers in the math I studied at the uni.   Thus your textbook could define something new called whole numbers.  And apparently they defined it in a way you do not like.&lt;/p&gt;&#xA;" OwnerUserId="14725" LastActivityDate="2020-10-06T00:31:47.157" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18957" PostTypeId="1" CreationDate="2020-10-06T06:48:07.007" Score="3" ViewCount="176" Body="&lt;p&gt;I am struggling with teaching my 9th grade kid to solve math problems that are just outside of routine.&lt;/p&gt;&#xA;&lt;p&gt;For e.g.,&lt;/p&gt;&#xA;&lt;p&gt;An example problem given by math teacher at school.&lt;/p&gt;&#xA;&lt;p&gt;x, y, z are in geometric progression&lt;/p&gt;&#xA;&lt;p&gt;x + y + z = C1&lt;/p&gt;&#xA;&lt;p&gt;x * x + y * y + z * z = C2&lt;/p&gt;&#xA;&lt;p&gt;Find x, y, z.&lt;/p&gt;&#xA;&lt;p&gt;My solution (after a couple of failed attempts) for this involves recognizing some patterns&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;(x + y + z)^2  = ...&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Expressing x, y, z as  a/r, a, ar&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A bit of manipulation of [1] so that I can reuse x + y + z = C1 and eliminate r from the equation and find a.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;How does one teach students to independently solve these problems? Are students &amp;quot;wired&amp;quot; to solve these v/s not?&lt;/p&gt;&#xA;" OwnerUserId="14727" LastActivityDate="2020-10-09T05:26:11.543" Title="How do I teach my kid" Tags="|secondary-education|algebra|" AnswerCount="3" CommentCount="2" ClosedDate="2020-10-10T22:19:35.817" ContentLicense="CC BY-SA 4.0" />
  <row Id="18958" PostTypeId="2" ParentId="18948" CreationDate="2020-10-06T09:14:44.597" Score="12" Body="&lt;p&gt;You seem to describe &amp;quot;whole numbers&amp;quot; in this American usage as describing &lt;span class=&quot;math-container&quot;&gt;$\mathbb {N}$&lt;/span&gt;, the set of natural numbers, whereas you expected it to describe &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt;, the set of integers. As others have pointed out there is nothing &amp;quot;incorrect&amp;quot; about them, it's a language difference. Although it is worth knowing those language differences since especially in maths (or &amp;quot;math&amp;quot; is it is known in either the US or the UK can't remember or be bothered to check) it can really make trouble!&lt;/p&gt;&#xA;&lt;p&gt;Your quoted Quora answer brings French into it, for a French speaker it might be all the more confusing that &amp;quot;whole numbers&amp;quot; and &amp;quot;integers&amp;quot; both clearly derive from the word &amp;quot;entier&amp;quot;, which means &amp;quot;whole&amp;quot; and can be used for either set (&amp;quot;entiers relatifs&amp;quot; would be &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt; and &amp;quot;entiers naturels&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$\mathbb {N}$&lt;/span&gt;). If that is your situation I can understand your confusion all the more, but it is still merely a question of language convention, not being correct or not. It might also suggest a reason they started using &amp;quot;whole numbers&amp;quot; to describe &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;: since they used &amp;quot;integer&amp;quot; to describe &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt;, that freed up &amp;quot;whole number&amp;quot; to describe a different set.&lt;/p&gt;&#xA;&lt;p&gt;You might be interested in an English dictionary of mathematics, that might clarify a lot of those nuances for you. There are some differences between American and UK English that are positively awful, like there are two geometric features that have &lt;em&gt;exactly inverted names&lt;/em&gt; between the two. I cannot remember the exact example, I thought it was in the parallelogram family but I might be confusing with the trapezoid/trapezium difference, which is also a good example of inconsistent mathematical words between US English and others but doesn't seem to be a total inversion at least.&lt;/p&gt;&#xA;" OwnerUserId="14728" LastActivityDate="2020-10-06T09:14:44.597" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18959" PostTypeId="2" ParentId="18957" CreationDate="2020-10-06T10:22:12.130" Score="3" Body="&lt;p&gt;I should disclaim I haven't managed to solve this equation yet, even with your description. What were your failed attempts like? What was your own approach to this problem, and can't this approach help your son?&lt;/p&gt;&#xA;&lt;p&gt;Overall my approach for a problem like this would be:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;in general, to solve for N variables you need N independent equations. Here the problem gives 3 variables and 2 equations, this means it is either unsolvable or there is extra information in there that can be made into an equation. In this case, &amp;quot;x,y,z are in a geometric progression&amp;quot; which can be converted into your expression of x, y, z according to r and a.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You can kind of generalize that, many twisty logic problems have fun features like knowing that there is an answer, or that somebody found the answer this or that way, are vital missing clues to finding the actual answer. The general principle there is to extract information from the problem to find the answer, and so when stuck you might ask yourself &amp;quot;is there information given in this problem that I haven't used in finding my answer? If so, is there any way I can use this information alongside what I've already done?&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;to get back to equations, if you have N equations with N variables, you want to end up with 1 equation with 1 variable, which you can then solve easily and use that variable to find the other N ones. There are many specific techniques for this which your child might have learned and they can pick which one to choose. But in general it means transforming one equation until it looks like part of another one of the equations, at which point you can plug it in and you'll often have gotten rid of some complexity. In your case, I suppose squaring x + y + z brings it a bit closer to the equation with x&lt;em&gt;x+y&lt;/em&gt;y+z*z.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This is similar to the larger information issue, but a teacher will usually present a problem in the context of the larger lesson. Was this presented as one example of problems that can be solved through the techniques being learned right now? In that case it's worth trying those techniques even if it doesn't look like they fit. Did your own resolution of the problem end up using techniques your son did learn (or should have learned) recently? You say it's outside of routine, but maybe that's the &amp;quot;trap&amp;quot; or &amp;quot;hook&amp;quot; of the problem, it's actually in the routine but it's presented differently. Or was it presented as something more out-there or advanced, a fun optional challenge? If so maybe the teacher doesn't really expect the student to solve it in a systematic way, maybe it requires grinding effort (like many failed attempts) or a leap of lateral thinking that they expect some students to manage but not all. If so it might be worth not worrying about too much.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14728" LastActivityDate="2020-10-06T10:22:12.130" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18960" PostTypeId="2" ParentId="18943" CreationDate="2020-10-06T11:20:49.833" Score="5" Body="&lt;p&gt;You'll probably get a better answer after the question is closed, edited for clarity, and reopened.&lt;/p&gt;&#xA;&lt;p&gt;I have had multiple experiences of this for students for whom English is not their first language. &amp;quot;What is '&lt;a href=&quot;https://en.wikipedia.org/wiki/Ferris_wheel&quot; rel=&quot;noreferrer&quot;&gt;ferris wheel&lt;/a&gt;'&amp;quot;? Taking a trigonometry test and we were using the ferris wheel in a test question. ('We', but I did not author the test. I was just proctoring). No, I don't expect a student to know what a ferris wheel is, and in fact, schools in mt area of the US are trying to do a better job of cultural awareness. I proceeded to explain what this ride was, showed him a picture of one, and how it worked.&lt;/p&gt;&#xA;&lt;p&gt;Similar for a test question regarding probability where the teacher uses &lt;a href=&quot;https://en.wikipedia.org/wiki/Playing_card&quot; rel=&quot;noreferrer&quot;&gt;playing cards&lt;/a&gt;, the standard 52 card deck. When I was your age (in the late 1970's), there were no computers, no video games. Cards were ubiquitous. English language aside, students would have a near 100% chance of knowing what any reference to a card deck meant. Now, such an assumption would be wrong, and it's on the teacher to introduce the deck of cards as if no one had ever played with it.&lt;/p&gt;&#xA;&lt;p&gt;If the student didn't understand a math vocabulary word, and it's not language-related, I wouldn't adjust the grade. It's one test/quiz of many.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-10-06T11:26:01.757" LastActivityDate="2020-10-06T11:26:01.757" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18961" PostTypeId="2" ParentId="18948" CreationDate="2020-10-06T20:43:50.523" Score="0" Body="&lt;p&gt;I don't remember using the term &amp;quot;whole numbers&amp;quot; in High School. It's not a rigorously-defined mathematical term, but the idea of &amp;quot;whole [counting] numbers&amp;quot; as opposed to &amp;quot;fractions&amp;quot; is quite useful when you're young and the concepts are new.&lt;/p&gt;&#xA;&lt;p&gt;Older students would be learning the proper mathematical terms (natural numbers, integers, rationals, etc) and so expanding the definition of &amp;quot;whole numbers&amp;quot; to mean &amp;quot;integers&amp;quot; doesn't really buy you anything with most older students. In fact it would be lose-lose: you'd be changing the meaning of a word they've known for many years to mean something else, and you'd be encouraging them to use that word instead of the proper term that they will encounter if they pursue mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I think a key dividing line is practical/physical math and more theoretical math. I can physically hold 3 glasses, or 2/3 of a pie. Having -3.14159 pies is more abstract/theoretical. At some point -- probably in High School -- students will come to the place that they will either continue into more abstract topics (algebra, calculus, etc) which will require more precise definitions, or more practical math (what used to be called &amp;quot;home economics&amp;quot;: shopping, cooking, buying, etc) which to a large extent need only &amp;quot;whole numbers&amp;quot; and &amp;quot;fractions&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="848" LastEditorUserId="848" LastEditDate="2020-10-06T20:49:57.307" LastActivityDate="2020-10-06T20:49:57.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18962" PostTypeId="2" ParentId="18948" CreationDate="2020-10-07T00:51:01.410" Score="1" Body="&lt;p&gt;In elementary school in a Dutch-speaking country, we were taught the concepts of &lt;span class=&quot;math-container&quot;&gt;$\mathbb {N}$&lt;/span&gt;, the set of &amp;quot;natuurlijke getallen&amp;quot; (&amp;quot;natural numbers&amp;quot;) and &lt;span class=&quot;math-container&quot;&gt;$\mathbb {Z}$&lt;/span&gt;, the set of &amp;quot;gehele getallen&amp;quot; (&amp;quot;whole numbers&amp;quot;). At no point was there ever a hint that these terms were not unequivocal.&lt;/p&gt;&#xA;&lt;p&gt;I remember &lt;span class=&quot;math-container&quot;&gt;$\mathbb {Z}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbb {N}$&lt;/span&gt; then being used as examples to teach Venn diagrams, at which point it was emphasized again that the difference between the two sets was the negative numbers.&lt;/p&gt;&#xA;" OwnerDisplayName="user14737" LastEditorDisplayName="user14737" LastEditDate="2020-10-09T16:48:54.887" LastActivityDate="2020-10-09T16:48:54.887" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18963" PostTypeId="1" AcceptedAnswerId="18967" CreationDate="2020-10-07T04:41:53.603" Score="1" ViewCount="445" Body="&lt;p&gt;The question is John has locked a 4 digit combination lock with each of the numbers 0-9. He knows the numbers 1,4,6, appears exactly once, but he does not remember the position of the numbers and he is not sure of what the fourth number is.&lt;/p&gt;&#xA;&lt;p&gt;The student think the fourth number is the number in the fourth position of the combination lock. He thinks so because people do not say an order of the number you know or not know. So later I wanted him to try again with the right idea. There are two students who thought this way. The students first language is not English.  I would let them try again since its not a math issue and would you? Also can it also be interpreted this way?&lt;/p&gt;&#xA;" OwnerUserId="14707" LastEditorUserId="14707" LastEditDate="2020-10-08T06:42:41.510" LastActivityDate="2020-10-08T06:42:41.510" Title="Should students get another chance at a math question because of English troubles?" Tags="|secondary-education|testing|" AnswerCount="1" CommentCount="24" ContentLicense="CC BY-SA 4.0" />
  <row Id="18964" PostTypeId="1" AcceptedAnswerId="18966" CreationDate="2020-10-07T13:41:32.770" Score="2" ViewCount="90" Body="&lt;p&gt;(Repost from MO, where the question will eventually be closed.)&lt;/p&gt;&#xA;&lt;p&gt;This question is related to lectures I have to make concerning differential calculus in one variable, but the students are quite advanced, and prepare themselves to be high school teachers.&lt;/p&gt;&#xA;&lt;p&gt;Requiring a function to be defined on an interval is not sufficient (think of ↦1/), nor is it sufficient to only consider open subsets of &lt;span class=&quot;math-container&quot;&gt;$\mathbf R$&lt;/span&gt; (think of a function defined on &lt;span class=&quot;math-container&quot;&gt;$[0;1]$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;In principle, that no point is isolated suffices to define the derivative, but this may sound as a too far-fetched hypothesis.&lt;/p&gt;&#xA;&lt;p&gt;To add a bit of context, let me recall that the standard French curriculum for calculus is particularly touchy on hypotheses. For example, there are 3 Taylor formulas (Lagrange, Young, with integral remainder), each with its precise formulation, and, as a consequence, students are supposed to pay attention whether one assumes derivability on the interior only, or everywhere…&lt;/p&gt;&#xA;&lt;p&gt;So what do you feel is the reasonable context for a function defined over a subset of the reals, to discuss differentiability and higher derivatives of functions in 1 variable?&lt;/p&gt;&#xA;" OwnerUserId="14744" LastActivityDate="2020-10-07T14:23:36.287" Title="Appropriate context for teaching derivative (undergraduate/graduate)" Tags="|calculus|precalculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18965" PostTypeId="1" CreationDate="2020-10-07T13:56:31.463" Score="0" ViewCount="155" Body="&lt;p&gt;I am looking for a good, stable and cheap graphics tablet for online math education that you use to use and love to use. I need to prepare remote videos.&lt;/p&gt;&#xA;" OwnerUserId="14745" LastEditorUserId="376" LastEditDate="2020-12-09T07:56:59.447" LastActivityDate="2020-12-09T07:56:59.447" Title="Recommendation for cheap practical graphics tablet?" Tags="|technology-in-education|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="18966" PostTypeId="2" ParentId="18964" CreationDate="2020-10-07T14:23:36.287" Score="4" Body="&lt;p&gt;If you are following a textbook, I would strongly recommend adopting their convention.&lt;/p&gt;&#xA;&lt;p&gt;If you are writing your own lecture notes, there are several ways I see to proceed.&lt;/p&gt;&#xA;&lt;p&gt;My personal favorite approach (if your students are sophisticated) would be to have them suggest conditions, challenge them with situations where it seems reasonable for the &amp;quot;derivative&amp;quot; to be defined but which is not covered by their condition, then have them come up with a new condition which covers that case as well.  Repeat the process until the entire class (including yourself) are satisfied with the proposed definition.&lt;/p&gt;&#xA;&lt;p&gt;Alternatively, you could do this yourself and come up with a suitably general definition, and then adopt this as the definition for the course without any student input.&lt;/p&gt;&#xA;&lt;p&gt;An example definition of this sort:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\Omega \subset \mathbb{R}$&lt;/span&gt;, and let &lt;span class=&quot;math-container&quot;&gt;$p \in \Omega$&lt;/span&gt; be a limit point of &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;.  Let &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{\Omega} \to \mathbb{R}$&lt;/span&gt; be a function.  We say that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differentiable at &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;  iff &lt;span class=&quot;math-container&quot;&gt;$$\lim_{x \to p} \frac{f(x) - f(p)}{x-p} \textrm{ exists}$$&lt;/span&gt;  In this case we call this limit &lt;strong&gt;the derivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;&lt;/strong&gt;, denoted &lt;span class=&quot;math-container&quot;&gt;$f'(p)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is said to be differentiable on &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; if it is differentiable at every point of &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-10-07T14:23:36.287" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18967" PostTypeId="2" ParentId="18963" CreationDate="2020-10-07T14:41:50.080" Score="1" Body="&lt;p&gt;It's not an English language issue, but a poorly worded question.  Even for a native speaker, the question is ambiguous/confusing.&lt;/p&gt;&#xA;&lt;p&gt;I would have thought you meant fourth position, also.  Maybe I'm wrong, but it seems like you are asking for very legalistic grammatical precision if I am wrong.  Nothing like normal English comprehension.  (And I'm not even sure I am wrong.)&lt;/p&gt;&#xA;" OwnerDisplayName="user14746" LastActivityDate="2020-10-07T14:41:50.080" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18968" PostTypeId="1" CreationDate="2020-10-07T15:45:40.740" Score="8" ViewCount="903" Body="&lt;p&gt;My instinct is often that there should be some technical word for the common class of equations and inequalities. That is: statements that connect two numerical expressions, with a relational operator in the middle. Granted that &amp;quot;relation&amp;quot; collects the operators of equality, lesser-than, etc., it seems like there should likewise be a common word for the class of statements using those symbols.&lt;/p&gt;&#xA;&lt;p&gt;One halfway common name I see is &lt;a href=&quot;https://www.reference.com/world-view/mathematical-sentence-cc8d7fd426e18a0f&quot; rel=&quot;noreferrer&quot;&gt;mathematical sentence&lt;/a&gt;. I'm not sure how widespread this really is; and it seems to lack the feeling of a technical mathematical term. In fact, today on social media I saw a number of parents of school children react vehemently to the phrase, saying that it had to be obviously nonsense. (Here's a more &lt;a href=&quot;https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)&quot; rel=&quot;noreferrer&quot;&gt;technical definition&lt;/a&gt; that I doubt they'd be any happier about.)&lt;/p&gt;&#xA;&lt;p&gt;So: Is there any more widely-used convention in mathematical circles for a name of the common class of equations and inequalities? Or is &amp;quot;mathematical sentence&amp;quot; the best we have?&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2021-08-18T18:40:41.327" Title="What's the common word for equations and inequalities?" Tags="|language-use|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="18969" PostTypeId="2" ParentId="18968" CreationDate="2020-10-07T17:22:43.400" Score="5" Body="&lt;p&gt;I'd go with &amp;quot;simple relational sentence.&amp;quot;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;sentence&amp;quot; - as you have researched, this is the technical phrase for a well formed formula with no free variables.  What you describe here fits that bill&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;relational&amp;quot; - The particular kind of operator you are looking at is a &lt;a href=&quot;https://en.wikipedia.org/wiki/Relational_operator&quot; rel=&quot;noreferrer&quot;&gt;relational operator&lt;/a&gt;.  One might be more pedantic and call it an &amp;quot;ordering&amp;quot; operator, but I think there's a better intuitive sense that comes from &amp;quot;relational.&amp;quot;  The idea of ordering only comes when you consider sets of values and determine if they are ordered or not.&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;simple&amp;quot; - Presuming you want to exclude sentences like &amp;quot;x&amp;lt;3 ∧ y=4&amp;quot; which have a conjunction in them, you'll want to add an adjective like &amp;quot;simple.&amp;quot;  I like &amp;quot;simple&amp;quot; because its typically not a well defined math term.  Simple is always relative.  This gives you some wiggle room as to why your definition might not line up with anyone else's.  &amp;quot;Atomic&amp;quot; would probably be more precice, but I think that precision may harm rather than help.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I think that the qualifications on &amp;quot;sentence&amp;quot; do a good job of avoiding the concerns parents might have with &amp;quot;mathematical sentence.&amp;quot;  I would argue that &amp;quot;Three is less than four&amp;quot; is also a simple relational sentence, and I think that gives credence to the name in the minds of anyone who wants to think of &amp;quot;sentences&amp;quot; as a purely English concept (or other natural language).&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2020-10-07T17:22:43.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18970" PostTypeId="2" ParentId="18968" CreationDate="2020-10-07T18:39:04.807" Score="12" Body="&lt;p&gt;My phrase has always been math &amp;quot;statement&amp;quot;. Equations and inequalities clearly assert/state a relationship between two or more things.&lt;/p&gt;&#xA;&lt;p&gt;My go-to direction using this would be something like: &amp;quot;Clearly explain the meaning of each math statement below in the context of the problem you just read.&amp;quot; Then I list things like &lt;span class=&quot;math-container&quot;&gt;$f(4)=3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$g(7)&amp;gt;g(1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I still use the word &amp;quot;expressions&amp;quot; when there is no stated relationship between two things, such as &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\gcd(8,12)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$4==2$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2020-10-07T18:49:05.017" LastActivityDate="2020-10-07T18:49:05.017" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18971" PostTypeId="2" ParentId="14765" CreationDate="2020-10-07T23:27:37.823" Score="1" Body="&lt;p&gt;Answered in comments:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://brilliant.org/wiki/forward-backwards-induction&quot; rel=&quot;nofollow noreferrer&quot;&gt;brilliant.org/wiki/forward-backwards-induction&lt;/a&gt; – Bill Cook Nov 9 '18 at 3:35&lt;/p&gt;&#xA;&lt;p&gt;Maybe not quite what you're looking for, but have you heard of Cauchy induction? You prove your base case, then you prove that &lt;span class=&quot;math-container&quot;&gt;$P(n) \implies P(2n)$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$P(n) \implies P(n-1)$&lt;/span&gt;, which covers all cases. – Mike Pierce Nov 9 '18 at 16:12&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-10-07T23:41:08.150" LastActivityDate="2020-10-07T23:41:08.150" CommentCount="3" CommunityOwnedDate="2020-10-08T17:31:35.887" ContentLicense="CC BY-SA 4.0" />
  <row Id="18972" PostTypeId="2" ParentId="18968" CreationDate="2020-10-07T23:38:41.990" Score="4" Body="&lt;p&gt;Simplifying Cort's answer, what about &lt;strong&gt;relation&lt;/strong&gt;? This fits the lay meaning of the word, and the definition of &lt;em&gt;relation&lt;/em&gt; as an extension of the concept of function.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt; In response to Daniel's comment: I agree that &lt;em&gt;relation&lt;/em&gt; falls short of being the correct term. What is lacking is a modifier: What about &amp;quot;A relation of the form &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; is called a &lt;em&gt;numerical/quantal/quantitative relation&lt;/em&gt;&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="2172" LastEditDate="2020-10-10T15:57:58.277" LastActivityDate="2020-10-10T15:57:58.277" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18973" PostTypeId="2" ParentId="18957" CreationDate="2020-10-08T07:41:51.293" Score="1" Body="&lt;p&gt;Take a look at Polya's classic &amp;quot;How to solve it&amp;quot;.&#xA;What you are asking is &amp;quot;how do I teach creative problem solving&amp;quot;, and for that, sorry to say, there is no royal road.&#xA;In any case, it isn't fair to expect your kid to solve a problem you struggle with...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2020-10-08T07:41:51.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18974" PostTypeId="2" ParentId="18968" CreationDate="2020-10-08T23:44:42.353" Score="2" Body="&lt;p&gt;&lt;em&gt;Is there any more widely-used convention in mathematical circles for a name of the common class of equations and inequalities?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Well, following the &lt;em&gt;Dictionary of algebra, arithmetic, and trigonometry&lt;/em&gt; by S. Krantz, we can say that the equations (assertions of equality between mathematical expressions...) and inequalities (expressions that involve members of some partially ordered set...) are in the class of the &lt;strong&gt;expressions&lt;/strong&gt;. However, it seems this class is bigger than you want.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;expression&lt;/strong&gt; A mathematical statement, using mathematical quantities such as scalars, variables, parameters, functions, and sets, as well as relational and logical operators such as equality, conjunction, existence, union, etc. (&lt;a href=&quot;https://archive.org/details/dictionaryofalge00kran/page/72/mode/2up?q=expression+A+mathematical+statement%2C+using&quot; rel=&quot;nofollow noreferrer&quot;&gt;Krantz Dictionary&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In order to be more specific, you could use &lt;strong&gt;relational expression&lt;/strong&gt; (as defined in &lt;a href=&quot;https://books.google.com.br/books?id=wpQvBQAAQBAJ&amp;amp;pg=PA52&amp;amp;lpg=PA52&amp;amp;dq=%22This+is+a+relational+expression.+The+relational+operator%22&amp;amp;source=bl&amp;amp;ots=V007_iK9IS&amp;amp;sig=ACfU3U3OFuHcbSmxIpnwY_S9P7h89d9nHg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwj1_NT_mKbsAhVAILkGHYqpBEYQ6AEwAHoECAEQAg#v=onepage&amp;amp;q=%22This%20is%20a%20relational%20expression.%20The%20relational%20operator%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vivaldi's book&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="13585" LastEditorUserId="13585" LastEditDate="2020-10-08T23:57:08.957" LastActivityDate="2020-10-08T23:57:08.957" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18976" PostTypeId="2" ParentId="18957" CreationDate="2020-10-09T05:26:11.543" Score="3" Body="&lt;p&gt;My (serious, practical) advice is to teach your kid by&lt;/p&gt;&#xA;&lt;p&gt;(1) giving more progressive problems to solve (by progressive, I mean easier than this but not bone simple drill). If you were lifting weights, you wouldn't throw 300# on the bar to start.  Try for problems with half the complexity of this exercise, for instance.  As sort of 150# on the bar.&lt;/p&gt;&#xA;&lt;p&gt;(2) Use problems like this as an opportunity to explore, deepen knowledge, more than just the getting it without help.  IOW, don't simply look at this as a riddle that he gets or not.  But if necessary, share the answer (IOW, he doesn't get the aha of solving it without assistance...sure that would be nice...but nobody can figure out every riddle) and then explain the how to solve it and THEN repeat the exercise with a few similar examples (so he will get these problems in the future).  Dick Feynman was great at puzzles and riddles...but he mentioned that a key way to being good at riddles (if you can't just intuit it and make the leap without help) is just to have seen the problem before.  DON'T devalue the value in this.&lt;/p&gt;&#xA;" OwnerDisplayName="user14746" LastActivityDate="2020-10-09T05:26:11.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18977" PostTypeId="1" CreationDate="2020-10-09T10:05:24.390" Score="4" ViewCount="645" Body="&lt;p&gt;The question &lt;a href=&quot;https://matheducators.stackexchange.com/q/269/376&quot;&gt;What are some good mathematical applications to present in an abstract algebra course?&lt;/a&gt; asks about &lt;em&gt;mathematical&lt;/em&gt; applications of abstract algebra.&lt;/p&gt;&#xA;&lt;p&gt;What are some applications of abstract algebra &lt;em&gt;outside&lt;/em&gt; of mathematics? These could be included either in a traditional abstract algebra course or in an applied algebra course. Which abstract algebra textbooks have particularly good support for applications or have a focus on applied algebra?&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2022-02-26T02:24:20.497" Title="Applications of abstract algebra outside of mathematics and suitable textbook" Tags="|undergraduate-education|textbooks|abstract-algebra|applications|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="18979" PostTypeId="2" ParentId="18977" CreationDate="2020-10-10T08:09:35.983" Score="4" Body="&lt;p&gt;There are a lot of applications of group theory in chemistry and physics.  (I know the chemistry side of it better.)&lt;/p&gt;&#xA;&lt;p&gt;Point groups of molecules describe their symmetry and thus their spectroscopy (bending modes for IR spectra).  See for instance,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0471510947&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Chemical-Applications-Group-Theory-3rd/dp/0471510947&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;OR&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=BjknQc3bQ8o&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=BjknQc3bQ8o&lt;/a&gt; (note the character tables)&lt;/p&gt;&#xA;&lt;p&gt;OR&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=pEw0RKCANFs&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=pEw0RKCANFs&lt;/a&gt; (simple visual)&lt;/p&gt;&#xA;&lt;p&gt;Space groups are intrinsic to X-ray crystallography, which describes a huge amount of matter (even things you don't think of as crystals, like metals or ceramics or [bulk of] computer chips, are crystalline solids).  See for instance&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://it.iucr.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://it.iucr.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user14746" LastEditorDisplayName="user14746" LastEditDate="2020-10-10T15:53:22.923" LastActivityDate="2020-10-10T15:53:22.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18980" PostTypeId="2" ParentId="18977" CreationDate="2020-10-10T11:10:12.760" Score="3" Body="&lt;p&gt;Thanks in part to Jyrki Lahtonen's remarks on coding theory and cryptography, I have an applied algebra book to suggest:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.springer.com/gp/book/9783030440732&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra for Applications&lt;/a&gt;, Arkadii Slinko, 2nd ed., 2020 Springer&lt;/p&gt;&#xA;&lt;p&gt;The preface to the first edition states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This book originated from my lecture notes for the one-semester course which I&#xA;have given many times in The University of Auckland since 1998. The goal of this&#xA;book is to show the incredible power of algebra and number theory in the real&#xA;world. It does not advance far in theoretical algebra, theoretical number theory or&#xA;combinatorics. Instead, we concentrate on concrete objects like groups of points on&#xA;elliptic curves, polynomial rings and finite fields, study their elementary properties&#xA;and show their exceptional applicability to various problems in information handling.&#xA;Among the applications are cryptography, secret sharing, error-correcting,&#xA;fingerprinting and compression of information.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It does not replace a more traditional course on abstract algebra and indeed at The University of Auckland where Slinko teaches &lt;em&gt;Algebra and Applications&lt;/em&gt;, a conventional course, &lt;em&gt;Algebraic Structures&lt;/em&gt;, is also taught (see &lt;a href=&quot;https://www.math.auckland.ac.nz/en/about/our-courses/undergraduate-courses.html#6c042d5924659a88bf47339e5a6f8df1&quot; rel=&quot;nofollow noreferrer&quot;&gt;list of undergraduate courses&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;For an MAA review of the first edition by Mark Hunacek, see &lt;a href=&quot;https://www.maa.org/press/maa-reviews/algebra-for-applications&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/maa-reviews/algebra-for-applications&lt;/a&gt;. (For a mini review of the second edition mentioning the changes made, see &lt;a href=&quot;https://www.maa.org/press/maa-reviews/algebra-for-applications-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;Whether Slinko's book counts as &amp;quot;a book dedicated to algebra&amp;quot;, to borrow Jyrki Lahtonen's phrase, is debatable I suppose. I think Slinko tries to strike a balance between introducing the necessary algebra and giving sufficient space to the applications. I also think that in general, the line is blurred somewhat between algebra, number theory and applications. For example, some searching has revealed Niederreiter &amp;amp; Winterhof's &lt;em&gt;Applied Number Theory&lt;/em&gt;, which after a review of some number theory and algebra, covers cryptography, error-correcting codes, quasi-Monte Carlo methods, pseudorandom numbers and miscellaneous further applications. Another example is Hoffstein, Pipher &amp;amp; Silverman's &lt;em&gt;An Introduction to Mathematical Cryptography&lt;/em&gt;, which naturally focuses on cryptography, but introduces the necessary topics from number theory, algebra, probability and information theory to keep the book fairly self-contained. The contents of these books and Slinko's book clearly overlap to some extent.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2022-01-24T18:54:17.007" LastActivityDate="2022-01-24T18:54:17.007" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18981" PostTypeId="2" ParentId="18977" CreationDate="2020-10-10T15:26:17.247" Score="1" Body="&lt;p&gt;This is a classic article:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hayes, Brian. &lt;em&gt;Group Theory in the Bedroom, and other Mathematical Diversions&lt;/em&gt;. Hill and Wang, 2008.&#xA;&lt;a href=&quot;http://grouptheoryinthebedroom.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book link&lt;/a&gt;.&#xA;&amp;quot;Having run out of sheep the other night, I found myself counting the ways to flip a mattress. ...&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;           &#xA;&lt;img src=&quot;https://i.stack.imgur.com/2RCdE.jpg&quot; width=&quot;300&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;(The title refers to one article. The entire collection is worthwhile.)&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-10-10T15:26:17.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18983" PostTypeId="2" ParentId="18968" CreationDate="2020-10-11T12:47:40.633" Score="6" Body="&lt;p&gt;I taught 1st through 5th graders from 1990 - 2015 and we used the term: &lt;strong&gt;number sentence.&lt;/strong&gt;&lt;br /&gt;&#xA;Students were very comfortable with this phrase.&lt;/p&gt;&#xA;&lt;p&gt;Added on Oct 15, 2020:  I am currently writing questions for 4th grade achievement tests.  I just wrote a multiple choice question asking, &amp;quot;Which inequality is true?&amp;quot; and my editor asked me to revise it as &amp;quot;Which &lt;strong&gt;number sentence&lt;/strong&gt; is true?&amp;quot;&#xA;It would seem &lt;strong&gt;number sentences&lt;/strong&gt; are still preferred and used!&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2020-10-15T07:40:56.810" LastActivityDate="2020-10-15T07:40:56.810" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18984" PostTypeId="2" ParentId="18955" CreationDate="2020-10-11T16:34:23.070" Score="3" Body="&lt;p&gt;I taught an introductory statistical methods class that was wholly Bayesian a few times, on the Mechanical, Marine, and Materials Engineering programmes at the University of Plymouth, but I'm not sure I'd consider it a great success (the class had no summative assessment of its own, being assessed through the &amp;quot;analysis of evidence&amp;quot; element of the undergraduate dissertation, and the vast majority of students found non-statistical ways to demonstrate their &amp;quot;analysis of evidence&amp;quot; skills and didn't engage with the class at all).&lt;/p&gt;&#xA;&lt;p&gt;In my own student life (Natural Sciences programme at the University of Cambridge, with main specialism Experimental and Theoretical Physics), the introduction to statistical methods was through a second-year undergraduate course called &amp;quot;Experimental methods&amp;quot;, which was neutral between Bayesian and frequentist (the statistical elements of the class were based on a parameter estimation method which can be justified either from a Bayesian or a frequentist perspective, and a model assessment method which can be viewed either as a very rough approximation to a Bayesian model comparison or as a very rough approximation to a frequentist significance test - the class never referred to either the Bayesian or the frequentist foundational aspects), followed up by a class in the master's year called &amp;quot;Information theory, pattern recognition and neural networks&amp;quot;, which was the course from which David MacKay's book &lt;em&gt;Information theory, inference and learning algorithms&lt;/em&gt; grew, and which was thoroughly Bayesian.&lt;/p&gt;&#xA;" OwnerUserId="14522" LastActivityDate="2020-10-11T16:34:23.070" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18985" PostTypeId="1" CreationDate="2020-10-11T17:35:42.207" Score="2" ViewCount="204" Body="&lt;p&gt;Symmetry is an important concept in mathematics and it has a built-in aesthetic appeal.  The following shows how different types of symmetry relate to one another.&#xA;Start with the equation &lt;span class=&quot;math-container&quot;&gt;$f(x) = -x^2 + bx + c$&lt;/span&gt;.  Without having to complete the square, we can find the axis of symmetry. Rewrite the equation as &lt;span class=&quot;math-container&quot;&gt;$f(x) = x(b - x) + c$&lt;/span&gt;. Now find &lt;span class=&quot;math-container&quot;&gt;$f(b - x) = (b-x)x + c = f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$f(b-x) = f(x)$&lt;/span&gt;, we know that for every point &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; on the curve, there is a point &lt;span class=&quot;math-container&quot;&gt;$(b - x,y)$&lt;/span&gt; on the curve.  The midpoint of these two points on the curve is &lt;span class=&quot;math-container&quot;&gt;$(b/2, y)$&lt;/span&gt;.  The line &lt;span class=&quot;math-container&quot;&gt;$x=b/2$&lt;/span&gt; is an axis of symmetry.  We can generalize this by noticing that what made this work is that the equation for &lt;span class=&quot;math-container&quot;&gt;$f(x) = x(b-x) + c$&lt;/span&gt; is symmetric in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b -x$&lt;/span&gt;.  We can get similar results for &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2 + bx + c$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$f(x) = -x(-b - x) + c$&lt;/span&gt;. Reasoning as before, it follows that &lt;span class=&quot;math-container&quot;&gt;$x = -b/2$&lt;/span&gt; is an axis of symmetry.&lt;/p&gt;&#xA;&lt;p&gt;Consider the function &lt;span class=&quot;math-container&quot;&gt;$f(x) = \arctan(x) + \arctan(6-x)$&lt;/span&gt;, which is symmetric in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(6 - x)$&lt;/span&gt;.  We can say immediately that the line &lt;span class=&quot;math-container&quot;&gt;$x=3$&lt;/span&gt; is an axis of symmetry. One of the reasons that this works is that the function &lt;span class=&quot;math-container&quot;&gt;$g(x) = b - x$&lt;/span&gt; is self-inverse, &lt;span class=&quot;math-container&quot;&gt;$g(g(x))=x$&lt;/span&gt;. If we take any function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; that is symmetric in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and a self-inverse function &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;, we can say that &lt;span class=&quot;math-container&quot;&gt;$f(x) = f(g(x))$&lt;/span&gt;, although this will not usually result in an axis of symmetry.&lt;/p&gt;&#xA;&lt;p&gt;Finally, there is a way of talking about self-inverse functions in terms of symmetry. The equation &lt;span class=&quot;math-container&quot;&gt;$y = b - x$&lt;/span&gt; does not at first appear to have any symmetry, but if we write it as &lt;span class=&quot;math-container&quot;&gt;$x + y = b$&lt;/span&gt;, it is clear that it is symmetric in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.  The function for getting &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the same as the function for getting &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, causing &lt;span class=&quot;math-container&quot;&gt;$g(g(x)) = x$&lt;/span&gt;.  For every point &lt;span class=&quot;math-container&quot;&gt;$(x, g(x))$&lt;/span&gt; on a self-inverse curve there is a point &lt;span class=&quot;math-container&quot;&gt;$(g(x), x)$&lt;/span&gt; on the curve.  The midpoint is &lt;span class=&quot;math-container&quot;&gt;$((x+g(x))/2, (x+g(x))/2)$&lt;/span&gt;, which is on the line &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;.  The line &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt; is an axis of symmetry for all self-inverse curves.&lt;/p&gt;&#xA;" OwnerUserId="8590" LastEditorUserId="8571" LastEditDate="2020-10-15T12:44:29.677" LastActivityDate="2020-10-15T13:17:58.247" Title="Would this be a good way of teaching about symmetry?" Tags="|functions|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18986" PostTypeId="1" CreationDate="2020-10-11T18:15:06.213" Score="3" ViewCount="700" Body="&lt;p&gt;I always wanted to teach my siblings mathematics, and one, ten years of age, is particularly eager. For the purposes of specializing recommendations, I will add he can use arithmetic up to exponentiation. Over time, I have become increasingly afraid of his curiosity and love for mathematics being crushed by the school system.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, I am looking for a good way to teach him math without any of the terrible, &lt;em&gt;A Mathematician's Lament&lt;/em&gt; sort of stuff present in schooling. I was thinking of using &lt;em&gt;Art of Problem Solving&lt;/em&gt;, but I only have the Geometry textbook, which is predicated on a basic understanding of Algebra. I was thinking of teaching Knot Theory, seeing a Japanese textbook on teaching it to people his age, which he seems to be interested in after seeing multiple Numberphile videos on the topic.&lt;/p&gt;&#xA;&lt;p&gt;To make this question as answerable as possible, I'll state it as this:&lt;/p&gt;&#xA;&lt;p&gt;What topics (and textbooks on said topics) would be suitable for a proof/discovery oriented course with a 10-year-old?&lt;/p&gt;&#xA;" OwnerUserId="13911" LastActivityDate="2020-10-14T11:40:10.997" Title="Teaching Mathematics to a Younger Sibling" Tags="|secondary-education|mathematical-pedagogy|textbooks|primary-education|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18987" PostTypeId="2" ParentId="18986" CreationDate="2020-10-11T19:01:44.910" Score="4" Body="&lt;p&gt;You mention Art of Problem Solving. They have a great curriculum for kids called Beast Academy. You could get that for him, and just be there for when he has questions. It is a lot of fun for many of the kids using it.&lt;/p&gt;&#xA;&lt;p&gt;Also, you could play games with him. (You may have to squelch your desire to teach, just a bit.) I love Blokus and Katamino.&lt;/p&gt;&#xA;&lt;p&gt;There are also two online geometric construction sites that gamify it. sciencevsmagic.net/geo and euclidthegame.com.&lt;/p&gt;&#xA;&lt;p&gt;You might find other cool games and books to share with him at my blog: mathmamawrites.blogspot.com&lt;/p&gt;&#xA;&lt;p&gt;One book he might enjoy reading and discussing with you is The Number Devil.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-10-11T19:01:44.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18988" PostTypeId="2" ParentId="18985" CreationDate="2020-10-11T22:53:09.297" Score="4" Body="&lt;p&gt;When you're designing instruction, the lesson is the last thing you create.  You start with finding the standard that students need to meet.  How will you assess whether students have met that standard?  What do students need to know in order to succeed at that assessment?  What do your students know coming in to this lesson?  Only when you have answered all of those questions can you design a lesson that is custom-made for your students and their need to meet the specific standard.&lt;/p&gt;&#xA;&lt;p&gt;I think that most teachers have discovered first-hand that there are consequences for skipping steps along that path.  Probably fewer teachers have learned from those mistakes, because it's easy to blame students for not comprehending a lesson that was hard to understand and had no purpose.  For myself, I remember teaching a similar lesson the first month I taught Calculus when I thought I had prepared my students to understand difference quotients.  Lo and behold, I asked them what &lt;span class=&quot;math-container&quot;&gt;$f(x+h)$&lt;/span&gt; was and they pretty much replied &lt;span class=&quot;math-container&quot;&gt;$fx+fh$&lt;/span&gt; because they had never seen an expression inside function notation but they did know the distributive property.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-10-11T22:53:09.297" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="18989" PostTypeId="1" CreationDate="2020-10-12T02:46:56.530" Score="6" ViewCount="4854" Body="&lt;p&gt;My two main ones are Electrostatic force field &lt;span class=&quot;math-container&quot;&gt;$\mathbf{E}\left(\mathbf{r}\right)=\frac{Q}{4\pi\epsilon_0 \left|\left|\mathbf{r}\right|\right|^3}\mathbf{r}$&lt;/span&gt; and Gravitational force field, &lt;span class=&quot;math-container&quot;&gt;$\mathbf{F}\left(\mathbf{r}\right)=-\frac{GMm}{\left|\left|\mathbf{r}\right|\right|^3}\mathbf{r}$&lt;/span&gt; but I'd like to have some others that could be used throughout an entire course. With those two, I can talk about grad, div, and curl, and could also explain the mathematical results through what we'd expect through physics. For a student, they could possibly predict the answer through intuition, and then prove themselves correct through the math. And, for as a teacher, I like the idea of bringing up repeat examples since it helps link together different topics and make the course seem less like random exercises in computation.&lt;/p&gt;&#xA;&lt;p&gt;What are some fields that are well-known and could link together various vector calculus topics?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorUserId="2083" LastEditDate="2023-10-20T17:59:00.670" LastActivityDate="2023-10-20T17:59:00.670" Title="Examples of real-life vector fields for vector calculus" Tags="|undergraduate-education|vector-calculus|applied-mathematics|applications|physical-sciences|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="18990" PostTypeId="2" ParentId="18986" CreationDate="2020-10-12T06:44:20.363" Score="2" Body="&lt;p&gt;I think the best thing for a beginner is to focus on &lt;strong&gt;concrete interesting mathematics&lt;/strong&gt;. You definitely do not want to do any abstract stuff at this point unless it is motivated by some concrete problem. I recommend &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0199218439&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Nets, Puzzles and Postmen&amp;quot;&lt;/a&gt; by Peter Higgins, and later on &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0521675995&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;How to Prove It&amp;quot;&lt;/a&gt; by Daniel Velleman. The first book is a really unique book, in that it is written at the layman level but does not handwave any of the core mathematical ideas, and furthermore it has notes for advanced readers!&lt;/p&gt;&#xA;&lt;p&gt;I want to also note that playing &lt;strong&gt;combinatorial games&lt;/strong&gt; (e.g. Hex, Go, ...) can be very helpful for honing intuition of quantifiers, because if you think about it a winning strategy of maximal length &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is nothing but a true sentence with &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; alternating quantifiers. There are also many nice puzzles with very strong mathematical flavour such as &lt;a href=&quot;https://www.chiark.greenend.org.uk/%7Esgtatham/puzzles&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tatham's puzzles&lt;/a&gt; and &lt;a href=&quot;http://pleasingfungus.com/Manufactoria&quot; rel=&quot;nofollow noreferrer&quot;&gt;Manufactoria&lt;/a&gt;. These can very easily inspire an investigation into reductions (i.e. if I can solve one puzzle easily I can solve another one too) and also encourage thinking about NP-hardness (i.e. it is possible that it is easy to verify a solution but hard to find one), which in some sense is similar to hardness of mathematical proofs (i.e. it is easy to verify a formal proof but may be hard to find one or even know whether one exists).&lt;/p&gt;&#xA;&lt;p&gt;If your brother loves mechanical puzzles such as the Rubik's cube, you can happily explore various invariants with him as well. One very important invariant, namely parity of the pieces in each group (sides and corners), is crucial to &lt;a href=&quot;http://math.stackexchange.com/q/1096592/21820&quot;&gt;a complete understanding of solving it and many other permutation puzzles&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It may also be of interest to you that my interest in mathematics grew out of playing around with such concrete mathematics, and one example I clearly remember from my childhood was investigating the relation between the forward difference iterates and the original sequence. Although at that time I did not know any of the terms, I knew about Pascal's triangle, which was enough to find and understand the decomposition of any given polynomial sequence into a weighted sum of (diagonal) columns of Pascal's triangle, where the weights can be easily obtained from the forward difference iterates. You can take a look at &lt;a href=&quot;https://math.stackexchange.com/a/2319671/21820&quot;&gt;this post&lt;/a&gt; for a high-level explanation that I wrote up many many years later, but certainly the way to teach such stuff to your brother is via Pascal's triangle and &lt;em&gt;not&lt;/em&gt; the abstract operator viewpoint. Well, at least not so soon. =)&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2020-10-12T06:44:20.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18991" PostTypeId="2" ParentId="18989" CreationDate="2020-10-12T10:20:33.873" Score="9" Body="&lt;p&gt;Air speed/direction on a weather map) is a very intuitive one.  There's also other fluid velocity (and flux) vector fields in various chemE, mechE, and nukeE applications.&lt;/p&gt;&#xA;&lt;p&gt;I personally think the air speed is most intuitive as something where you really need speed and direction (i.e. a vector, not a scalar) and it's something people encounter in daily life.  Electrostatics is a little mysterious as I always find electrical things more &amp;quot;hidden&amp;quot; then mechanics or fluids.  But the nice thing is you can actually do demonstrations of magnetic fields using iron filings on the overhead projector (showing the hidden field).&lt;/p&gt;&#xA;&lt;p&gt;My concern a little about the two force fields, you listed is that many familiar  two point calculations can be done, rather more conveniently, using scalars to solve for forces, potential energy change etc.  I.e. it's not clear why we need to invoke vector calculus.  Granted, it is possible to complicate the problems.  But, I just would think to list some situations like air flow, where it's very clear the situation is more complex.&lt;/p&gt;&#xA;&lt;p&gt;[Edit: for Steve.  It's an interesting question, but I would be pretty hesitant about showing such an example to beginning general calc 3 students.  After all, most econ undergrads don't even have a calc 3 requirement, sometimes taking &amp;quot;business calculus&amp;quot; rather than even a normal calc 1/2 sequence. Note: it's very important to differentiate between the requirements of research level econ and typical undergrad work.  This in contrast to say engineering or physics, which are fairly mathematical even at the BS level.]&lt;/p&gt;&#xA;" OwnerUserId="14767" LastEditorUserId="14767" LastEditDate="2020-10-12T16:34:16.337" LastActivityDate="2020-10-12T16:34:16.337" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18992" PostTypeId="2" ParentId="18989" CreationDate="2020-10-12T14:37:29.260" Score="5" Body="&lt;p&gt;The two examples you give both have zero curl, which limits their usefulness. Examples that do have a curl would be:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;an electromagnetic wave&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the magnetic field of a wire, inside the wire&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the magnetic field of a slab of current, inside the slab&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the field of a point charge that is moving inertially.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The external magnetic field of a wire is also an interesting example, because it looks curly, but actually has a curl of zero.&lt;/p&gt;&#xA;&lt;p&gt;I would suggest not referring to your &lt;span class=&quot;math-container&quot;&gt;$1/r^2$&lt;/span&gt; examples by names like &amp;quot;the electrostatic field.&amp;quot; Many, many students have an unshakeable belief that the &lt;span class=&quot;math-container&quot;&gt;$1/r^2$&lt;/span&gt; equation applies to virtually any static field, including, say, the field of a dipole or of a charged plane. A &lt;span class=&quot;math-container&quot;&gt;$1/r^2$&lt;/span&gt; electric field is the field of a point charge at rest (or the external field of a spherically symmetric charge distribution).&lt;/p&gt;&#xA;&lt;p&gt;You use the word &amp;quot;force field,&amp;quot; but that isn't really right. Physicists don't say &amp;quot;force field.&amp;quot; And your examples of E and F aren't analogous. The gravitational field is &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, not &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, and it doesn't have units of force. The things physicists call fields are properties of empty space. They're not interactions between objects. I would present these as:&lt;/p&gt;&#xA;&lt;p&gt;The electric field of a point charge at rest:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathbf{E}\left(\mathbf{r}\right)=\frac{kQ}{\left|\left|\mathbf{r}\right|\right|^3}\mathbf{r}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The gravitational field of a point mass:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathbf{g}\left(\mathbf{r}\right)=\frac{GM}{\left|\left|\mathbf{r}\right|\right|^3}\mathbf{r}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(In newtonian gravity, which is what you're doing here, it doesn't matter if the mass is at rest.)&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-10-12T14:37:29.260" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18993" PostTypeId="1" CreationDate="2020-10-12T17:19:47.900" Score="1" ViewCount="248" Body="&lt;p&gt;Where can I find a good course on tensor/ricci calculus not focused on applications and physics?&lt;/p&gt;&#xA;&lt;p&gt;I've been running into lots of tensor-theoretic stuff in differential geometry, so I don't know if proof-based differential geometry courses constitute a &amp;quot;Tensor Calculus for Mathematicians&amp;quot;, or if they just happen to have the parts useful to differential geometry, like topology to analysis.&lt;/p&gt;&#xA;&lt;p&gt;As of now, it seems like the vast majority of tensor calculus books are directed at theoretical physicists learning general relativity, and as such, they lack that mathematical rigor I've recently grown to know and love.&lt;/p&gt;&#xA;&lt;p&gt;Also, textbooks are a preferred answer compared to video courses/lectures, as I find them easier to work through.&lt;/p&gt;&#xA;" OwnerUserId="13911" LastEditorUserId="13911" LastEditDate="2020-10-12T18:52:11.603" LastActivityDate="2020-10-13T08:46:12.317" Title="Where can I find a good course on tensor calculus not focused on applications and physics?" Tags="|textbooks|self-learning|differential-geometry|" AnswerCount="2" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18994" PostTypeId="2" ParentId="18993" CreationDate="2020-10-12T21:50:57.450" Score="1" Body="&lt;p&gt;You probably want more of a Riemannian geometry book. I'd probably go with Lee's &amp;quot;Riemannian Manifolds&amp;quot;, but he also has &amp;quot;Introduction to Smooth Manifolds&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;There is also Bott and Tu's &amp;quot;Differential forms in Algebraic Topology&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Munkres has a book &amp;quot;Analysis on Manifolds&amp;quot; which is not as dry as the class I took, where it was the text. (But that's not saying much.)&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="470" LastEditDate="2020-10-12T21:58:24.713" LastActivityDate="2020-10-12T21:58:24.713" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18995" PostTypeId="2" ParentId="18989" CreationDate="2020-10-12T23:35:06.207" Score="2" Body="&lt;p&gt;The &lt;em&gt;Hairy Ball Theorem&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/cgqnK.jpg&quot; width=&quot;400&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.wikiwand.com/en/Hairy_ball_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikiwand link&lt;/a&gt;:&#xA;&amp;quot;there is no nonvanishing continuous tangent vector field on even-dimensional &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-spheres.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-10-12T23:35:06.207" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="18996" PostTypeId="2" ParentId="18989" CreationDate="2020-10-13T01:03:17.210" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;The oscillation of species' populations due to predation.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/L6N0B.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/L6N0B.png&quot; alt=&quot;Vector field formed by population dynamics of two species&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;The convergence of machine learning models using gradient descent - this is a special vector field that's tuned by uniformly multiplying the field with a scalar (bonus points for involving a segue into machine learning theory and having them tune the vector field for most rapid descent)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;A couple examples:&#xA;&lt;a href=&quot;https://blog.paperspace.com/intro-to-optimization-momentum-rmsprop-adam/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://blog.paperspace.com/intro-to-optimization-momentum-rmsprop-adam/&lt;/a&gt;&#xA;&lt;a href=&quot;https://datascience-enthusiast.com/DL/Optimization_methods.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://datascience-enthusiast.com/DL/Optimization_methods.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;It might help to know the demographics of your target audience to tune the examples toward things they're most likely interested in.&lt;/p&gt;&#xA;" OwnerUserId="13060" LastActivityDate="2020-10-13T01:03:17.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18997" PostTypeId="1" AcceptedAnswerId="19000" CreationDate="2020-10-13T03:48:48.680" Score="2" ViewCount="199" Body="&lt;p&gt;I'm currently self studying a Linear Algebra text book due to the fact that I forgot the vast majority of it since I took the course 10 years ago. My general strategy is to take every single example, practice problem, and exercise and do it myself. If I get it wrong, either I try to find where I made a mistake or I basically do the Feynman Method so that I find where the flaw in my understanding is. Generally this approach has been agreeable with me, however I've hit an exercise set where of the 37 exercises given, I only got 17 of them correct in total. So I got about a 46%, which got me wondering, what should my minimum threshold be for 'passing' a section of work when self studying a math text with no outside assistance sans a solution sheet to check my answers against?&lt;/p&gt;&#xA;" OwnerUserId="14768" LastActivityDate="2020-10-13T09:22:36.497" Title="What should be the bare minimum wrong answer threshold when self studying a math text?" Tags="|self-learning|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="18998" PostTypeId="2" ParentId="18997" CreationDate="2020-10-13T04:36:34.350" Score="1" Body="&lt;p&gt;I don't agree that there should be a &lt;em&gt;passing score&lt;/em&gt; when it comes to studying using problem sets if you wish to simply grind through all the problems.  Should it be acceptable to have one wrong answer?  If so, why would having two wrong answers not be acceptable?  Three?  Out of how many problems?&lt;/p&gt;&#xA;&lt;p&gt;Assuming that your problem sets all feature the same type of problem, below could be a good strategy:&lt;/p&gt;&#xA;&lt;p&gt;Randomize the problems and answer them one at a time.  Your goal is to correctly answer some number of problems consecutively -- ten to twelve problems sounds about right, but you can adjust to your own goals.  If/when you make a mistake, start the count over and review the material to see where you had gone wrong.  When you get those ten to twelve problems correct in a row, end your session.&lt;/p&gt;&#xA;&lt;p&gt;The next session, repeat what was done above.  Then, if you are able to get three sessions in a row without making mistakes, you can be pretty confident you have a good handle on the material.&lt;/p&gt;&#xA;&lt;p&gt;If your problem sets feature different types of problems (e.g. computation, proof writing, etc.), then first separate your problems into categories, then repeat the above for each category.&lt;/p&gt;&#xA;" OwnerUserId="12720" LastActivityDate="2020-10-13T04:36:34.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18999" PostTypeId="2" ParentId="18993" CreationDate="2020-10-13T08:46:12.317" Score="0" Body="&lt;p&gt;I would like to add a classic:&lt;/p&gt;&#xA;&lt;p&gt;Bishop, R. and Goldberg, S. (1980). &lt;a href=&quot;https://store.doverpublications.com/0486640396.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tensor Analysis on Manifolds&lt;/a&gt;. Dover Publications.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2020-10-13T08:46:12.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19000" PostTypeId="2" ParentId="18997" CreationDate="2020-10-13T09:22:36.497" Score="5" Body="&lt;p&gt;My biggest advice is not just to &amp;quot;see where you went wrong&amp;quot; or &amp;quot;do the FM&amp;quot;.  But redo the entire missed problems as if you had not seen the answer.  Very important!&lt;/p&gt;&#xA;&lt;p&gt;IOW, after finding out your mistake and correcting it, put the old work aside, look only at the problem statement and work it fresh.  (I.e. in your mind, treat it as a fresh problem, write the entire calculation out.  Don't do it as sort of a regurgitation of the previous writing, but as if you were seeing it fresh.  This sounds like a silly game, but I find it really drives the ability to crack such problems in the future.  If  you don't do this, you won't really master the material.)&lt;/p&gt;&#xA;&lt;p&gt;In terms of numbers, of course &amp;quot;everything is relative&amp;quot;.  But to be responsive.  46% is obviously pretty concerning.  As a general rule, you should shoot for 90%.  There will be times, you fall short, especially on initial drill, so I wouldn't be upset at all about 80%+ (provided you DO force yourself to redo all missed problems, as I discuss above).  Anything below 60-70, I would seriously re-read the chapter, work all the examples (&amp;quot;fresh&amp;quot;), and, if possible, find additional drill problems (like from a competing text).&lt;/p&gt;&#xA;&lt;p&gt;Note:  the small &amp;quot;punishment&amp;quot; of having to redo all problems can be used to gamify the process.  This helps with getting some feedback to allow you to persist in the work and not get bored or give up.  So, you are shooting for 100% each time.  You won't get it much, but it can become a little apple you are chasing.  Similarly, you should &amp;quot;grade yourself&amp;quot; and assign a number grade and write it at the top of your drill work.  A for 90%, B for 80%, etc.  It sounds juvenile, but self study can be incredibly boring and monastic.  Having some element of play, even silly like this, can make it more endurable.  Like dieting or quitting smoking, the big danger with self study, is not the methodology (they all pretty much work), but that you give up.&lt;/p&gt;&#xA;&lt;p&gt;I would maybe give yourself a little bit of a lower hurdle if the problems are all ballbuster headscratchers (from one of these people who falsely deprecates the learning value of simple drill).  But if that's the case, you need to find a text with some simpler drill.&lt;/p&gt;&#xA;&lt;p&gt;You don't learn to do difficult sports by only practicing elite movements.  You need drill on the basics.  People are not never-forgetting silicon machines, they are animals, one step above P's dogs.  They need and benefit from drill.  This doesn't mean they can only do drill...they are capable of great things as well.  But they do benefit from automaticity of drill material, and the only way to get that is to work simple drill problems until acing them.&lt;/p&gt;&#xA;" OwnerUserId="14767" LastActivityDate="2020-10-13T09:22:36.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19001" PostTypeId="1" AcceptedAnswerId="19003" CreationDate="2020-10-13T19:53:14.613" Score="11" ViewCount="1945" Body="&lt;p&gt;The following &amp;quot;chain rule&amp;quot; is in my multivariable calculus course:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; depends on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, but &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; depend on &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dt} = \frac{\partial f}{\partial  x} \frac{d x}{d t} + \frac{\partial f}{\partial y} \frac{d x}{d t}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, it doesn't apply to this question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$f(x, y, t) = xyt$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$x = t$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y = t$&lt;/span&gt;. Use the chain rule to find &lt;span class=&quot;math-container&quot;&gt;$\frac{d f}{d t}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course &lt;span class=&quot;math-container&quot;&gt;$f(t) = t^3$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$\frac{d f}{d t}$&lt;/span&gt; should be &lt;span class=&quot;math-container&quot;&gt;$3t^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But if you apply the &amp;quot;chain rule&amp;quot; above, you end up with&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{d f}{d t} = (yt)1 + (xt)1 = 2t^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I need the result of my chain rule to look more like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{d f}{d t} = (yt)1 + (xt)1 + (xy) = 3t^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But I have no good notation for the missing &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt;&amp;quot; here. Is it &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{\partial f}{\partial t}$&lt;/span&gt;?&amp;quot;  That's awful -- I can't write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; depends on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, but &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; depend on &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dt} = \frac{\partial f}{\partial  x} \frac{d x}{d t} + \frac{\partial f}{\partial y} \frac{d x}{d t} + \frac{\partial f}{\partial t}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;... can I? How do I fix this?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2020-10-18T07:12:04.890" Title="What is the right notation to use in multivariable chain rules?" Tags="|notation|vector-calculus|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19002" PostTypeId="2" ParentId="19001" CreationDate="2020-10-13T20:46:43.937" Score="17" Body="&lt;p&gt;What's wrong with this?:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} \frac{dt}{dt}$$&lt;/span&gt;&#xA;with &lt;span class=&quot;math-container&quot;&gt;$\frac{dt}{dt}=1$&lt;/span&gt;. It's what you would get if you had &lt;span class=&quot;math-container&quot;&gt;$f(x,y,z)$&lt;/span&gt;, except that &lt;span class=&quot;math-container&quot;&gt;$z=t$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You do have to be a bit careful, though. You want to be clear that &lt;span class=&quot;math-container&quot;&gt;$f(x,t,t)=xt^2$&lt;/span&gt;  should either be thought of as &lt;span class=&quot;math-container&quot;&gt;$f(x,t)$&lt;/span&gt; or as the composition of &lt;span class=&quot;math-container&quot;&gt;$f(x,y,z)=xyz$&lt;/span&gt; and a curve in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;. (i.e. &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial f}{\partial t}$&lt;/span&gt; should be unambiguous at to which variable-slot is being differentiated.)&lt;/p&gt;&#xA;&lt;p&gt;The general principle here is that when you write down something like &lt;span class=&quot;math-container&quot;&gt;$f(x,y,t)=\cdots$&lt;/span&gt;, you are defining a formula which works for any possible values of &lt;span class=&quot;math-container&quot;&gt;$x,y,t$&lt;/span&gt; (where the formula is defined), regardless of any possible relations which might actually hold amongst these variables. This independence is  what you need to define the partial derivatives. Otherwise, you cannot do the &amp;quot;hold all other variables constant&amp;quot; procedure in the definition. So the chain rule formula should not really care that &lt;span class=&quot;math-container&quot;&gt;$x,y=t$&lt;/span&gt;; it only knows that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$x,y,t$&lt;/span&gt;. The formula shouldn't even really care that we are differentiating w.r.t. &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;. You just get &lt;span class=&quot;math-container&quot;&gt;$\nabla f \cdot \vec{v}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\vec{v}$&lt;/span&gt; is the vector of derivatives of the 1st, 2nd, and 3rd parameters whatever they may be called.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2020-10-13T21:10:10.663" LastActivityDate="2020-10-13T21:10:10.663" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19003" PostTypeId="2" ParentId="19001" CreationDate="2020-10-14T00:42:17.043" Score="12" Body="&lt;p&gt;Do you ever introduce the Jacobian matrix, or the derivative of a function at a point as a linear map?  This clarifies everything.&lt;/p&gt;&#xA;&lt;p&gt;If not, and you must restrict yourself to &amp;quot;traditional multivariable calculus&amp;quot; notation, you could write the chain rule as &lt;span class=&quot;math-container&quot;&gt;$(f \circ \gamma)'(t) = \nabla f|_{\gamma(t)} \cdot \gamma'(t)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In your example &lt;span class=&quot;math-container&quot;&gt;$f(x,y,t) = xyt$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$\nabla f|_{(x,y,t)} = \langle yt, xt, xy \rangle $&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\gamma(t) = (t,t,t)$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$\gamma'(t) = \langle 1, 1, 1\rangle$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We can now calculate &lt;span class=&quot;math-container&quot;&gt;$(f \circ \gamma)'(t) = \langle t^2, t^2, t^2 \rangle \cdot \langle 1, 1, 1 \rangle = 3t^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Note:  One must point out that the variable &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; appearing in the argument of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, the argument of &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;, and in the argument of &lt;span class=&quot;math-container&quot;&gt;$(f \circ \gamma)$&lt;/span&gt; are all &amp;quot;different&amp;quot;.  Function definitions are universally quantified, and quantified variables have a scope which is limited to the sentence which they quantify.  So even though they use the same letter, they are &amp;quot;unrelated&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I agree with Adam that the real difficulty here is paying attention to what the domain and codomain of your functions actually are, and also not confusing the scope of the variables in a function definition.  The Leibniz notation confuses the scope of the variables inherently, and this does lead to endless confusion.&lt;/p&gt;&#xA;&lt;p&gt;I have also seen &lt;span class=&quot;math-container&quot;&gt;$D_j f$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$f_j$&lt;/span&gt; used in place of &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial f}{\partial x_j} $&lt;/span&gt;, and this clears up some confusion.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-10-14T15:30:42.883" LastActivityDate="2020-10-14T15:30:42.883" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19004" PostTypeId="2" ParentId="19001" CreationDate="2020-10-14T07:03:20.533" Score="11" Body="&lt;p&gt;I guess you are aware of it, but since you don't say so explicitly: the chain rule on the top of your post does not apply to your example, since that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not a function of &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; but of &lt;span class=&quot;math-container&quot;&gt;$x,y,t$&lt;/span&gt;. Also, as written, you are confounding the modern concept of &amp;quot;function&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{R}^3\to\mathbb{R}$&lt;/span&gt; with the original notion of &amp;quot;function of&amp;quot;. This mix up leads to many of these &amp;quot;paradoxes&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;If you work with the modern concept of function, the best solution is to use the notation &lt;span class=&quot;math-container&quot;&gt;$D_if$&lt;/span&gt; suggested by Steven Gubkin (in fact, the &lt;span class=&quot;math-container&quot;&gt;$\partial/\partial x$&lt;/span&gt; operation makes no sense when applied to &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{R}^3\to \mathbb{R}$&lt;/span&gt;, since such an &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; does not depend on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;). The correct way of writing the chain rule, with &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{R}^n\to \mathbb{R}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(\gamma_1,\ldots,\gamma_n):\mathbb{R}^m\to\mathbb{R}^n$&lt;/span&gt;, is then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;D_j(f\circ(\gamma_1,\ldots,\gamma_n))= \sum_{k=1}^n \left((D_kf)\circ (\gamma_1,\ldots,\gamma_n)\right)\cdot D_j\gamma_k&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$j=1,\ldots, m$&lt;/span&gt;. This is unfortunately quite distant from the thing shown in most calculus texts.&lt;/p&gt;&#xA;&lt;p&gt;If, on the other hand, you work with the original idea of &amp;quot;functions of&amp;quot; (I've written elsewhere on MESE how to formalize that with modern functions on manifolds), then Adams answer is close to the truth. There is in principle nothing wrong with Leibniz notation &lt;span class=&quot;math-container&quot;&gt;$\partial z/\partial x$&lt;/span&gt;, except that it's not precise enough and you should also indicate which variables are held fixed, as in &lt;span class=&quot;math-container&quot;&gt;$(\partial z/\partial x)_y$&lt;/span&gt;. If you don't do this you eventually run into trouble, for instance if you try to write the chain rule with Leibniz notation in the context of reverse automatic differentiation (a special case of which is backpropagation in machine learning).&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2020-10-14T07:03:20.533" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19006" PostTypeId="2" ParentId="19001" CreationDate="2020-10-14T23:35:24.337" Score="5" Body="&lt;p&gt;There is, as others have said, nothing wrong with&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} \frac{dt}{dt}$$&lt;/span&gt;&#xA;Note that a partial derivative like &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial f}{\partial t}$&lt;/span&gt; is really to be understood as the derivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; with respect to &lt;em&gt;its third variable&lt;/em&gt;, more so than it is the derivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;. The fact that the first and second variable of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; are actually functions of the third is immaterial to the partial derivatives.&lt;/p&gt;&#xA;&lt;p&gt;An alternative is to instead use &lt;span class=&quot;math-container&quot;&gt;$f(x, y, z)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$x, y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; are all functions of &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, and we just happen to have &lt;span class=&quot;math-container&quot;&gt;$z(t) = t$&lt;/span&gt;. In that case, I think you will agree without issue that the chain rule says&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt}&#xA;$$&lt;/span&gt;&#xA;which after substitution turns out to be the same as the above.&lt;/p&gt;&#xA;" OwnerUserId="7624" LastActivityDate="2020-10-14T23:35:24.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19007" PostTypeId="1" AcceptedAnswerId="19016" CreationDate="2020-10-14T17:24:40.080" Score="5" ViewCount="1374" Body="&lt;p&gt;Currently a 5th year PhD student, and I've been fighting tooth and nail to teach one of our junior-year honors sections in undergraduate algebra next fall (desperately hopeful we'll be able to return to normal in-person lectures at that point).  Looks like it may come to fruition.  Some questions on texts.&lt;/p&gt;&#xA;&lt;p&gt;When I took undergraduate algebra, it was taught out of Herstein's classic &lt;em&gt;Topics in Algebra&lt;/em&gt; for a two-semester honors sequence. The reception was somewhat lukewarm from the students.  I definitely recall a point towards the end of the second semester (Galois theory) when about half the class was more-or-less completely lost, and from what I remember I was a bit lost myself but still managed to do well in the course.  I liked the book at the time, but I remember feeling like it was times a bit slick, bit generally a good read overall and I learned a ton.&lt;/p&gt;&#xA;&lt;p&gt;Recently, I've been made aware of another text by Dan Saracino titled &lt;em&gt;Abstract Algebra: An Introduction&lt;/em&gt;, and apparently it's well-regarded by some people I respect.  I've ordered a copy, but in the meantime I'm curious what others' opinions are on a comparison between the two, and experience with students' performance using both?  In previous years, the course has used Herstein (and I believe at one point Artin, another great option but with a very different style).&lt;/p&gt;&#xA;&lt;p&gt;The course is aimed at some pretty bright students, many of whom will pursue PhDs themselves, so I'm tempted to use Herstein (the problems are great, albeit at times very challenging, and it's one of the standards for this purpose).  At the same time, I'm always of the opinion that students learn best when the exposition is clear and well-motivated, and it sounds like Saracino is a good candidate for that.&lt;/p&gt;&#xA;&lt;p&gt;Any advice or anecdotes for people who have used both texts?  I'm very passionate about teaching and I'd like to put the students in a good position for future graduate work.  At the same time, I want the material to actually stick.  I guess everyone has their own style, but I prefer courses that follow a text pretty closely rather than relying only on lecture notes.&lt;/p&gt;&#xA;&lt;p&gt;For context, most students will be coming from a history of similar honors-type courses (including a recent course in Linear Algebra from Axler's excellent book, and an analysis course from baby Rudin, so they're well-versed in proof-writing and have a bit of maturity).  It sounds like about half of students are expected to have taken a general intro to abstract algebra course (basic group theory, rings, and vector spaces) and half will be seeing it for the first time.&lt;/p&gt;&#xA;" OwnerUserId="14774" OwnerDisplayName="jons_stupid_questions" LastEditorUserId="14774" LastEditDate="2020-10-18T18:16:59.310" LastActivityDate="2020-10-18T18:16:59.310" Title="Choice of textbook for an undergraduate abstract algebra course" Tags="|undergraduate-education|reference-request|textbooks|abstract-algebra|" AnswerCount="1" CommentCount="22" ContentLicense="CC BY-SA 4.0" />
  <row Id="19008" PostTypeId="2" ParentId="19001" CreationDate="2020-10-15T01:21:51.533" Score="2" Body="&lt;p&gt;One can fix this problem in another way - by expressing partial derivatives in a more natural form not involving quotients*.&lt;/p&gt;&#xA;&lt;p&gt;Given&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f = xyt$$&lt;/span&gt;&#xA;we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$df = yt\,dx + xt\,dy + xy\,dt$$&lt;/span&gt;&#xA;no matter how the variables &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; may or may not be related - and this simply reads:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The rate of change in &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$yt$&lt;/span&gt; times the rate of change in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; plus &lt;span class=&quot;math-container&quot;&gt;$xt$&lt;/span&gt; times the rate of change in &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt; times the rate of change in &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Any way you could imagine changing the variables &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, this is true - it doesn't matter if maybe there's some secret variable &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; that's out there but just ignored in the formula for &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and it doesn't matter if the given variables have some relationship among them or if some are actually constants. It's just true. It's also nice in the sense that the equation can be read out quite literally without assuming that everyone knows and agrees upon the meaning of &amp;quot;the rate of change of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If one notes that &lt;span class=&quot;math-container&quot;&gt;$x=y=t$&lt;/span&gt; then it does follow that &lt;span class=&quot;math-container&quot;&gt;$dx=dy=dt$&lt;/span&gt; so one can just substitute into the last equation to get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$df=yt\,dt+xt\,dt+xy\,dt=3t^2\,dt$$&lt;/span&gt;&#xA;as desired - although one could also get the result &lt;span class=&quot;math-container&quot;&gt;$df=3x^2\,dx$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$df=3y^2\,dy$&lt;/span&gt; or even &lt;span class=&quot;math-container&quot;&gt;$df = 3xy\,dt$&lt;/span&gt; if desired - there are lots of ways to correctly write this, each corresponding to some way to describe the rate of change of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; in terms of some other rates.&lt;/p&gt;&#xA;&lt;p&gt;Note that this is also quite flexible, since the original equation wasn't hiding anything - if I want to say &lt;span class=&quot;math-container&quot;&gt;$x=t^2$&lt;/span&gt; then I can get &lt;span class=&quot;math-container&quot;&gt;$dx=2t\,dt$&lt;/span&gt; and substitute those in for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;. Maybe I'd then decide that I want &lt;span class=&quot;math-container&quot;&gt;$y^3=t$&lt;/span&gt; and get &lt;span class=&quot;math-container&quot;&gt;$3y^2\,dy=dt$&lt;/span&gt;, which I could substitute in for &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dt$&lt;/span&gt;. I could even get fancier by asking that &lt;span class=&quot;math-container&quot;&gt;$x=ty$&lt;/span&gt; and seeing &lt;span class=&quot;math-container&quot;&gt;$dx=y\,dt+t\,dy$&lt;/span&gt; or even do this all with implicit equations. There is then no need to have a chain rule at all because the chain rule just becomes literal substitutions - corresponding to the idea that if I have some term of &amp;quot;the rate of change of &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;&amp;quot; and I happen to know that that rate of change is an expression of some others, I might as well use that expression.&lt;/p&gt;&#xA;&lt;p&gt;There's a bit of sleight of hand here** - &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is no longer being spoken of as a function - but this is a handy point of view to understand (if only because many problems go away once you have two ways to look at them!) and one that is often useful, but seldom taught outside of differential geometry.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;*I would argue that the notation &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial f}{\partial x}$&lt;/span&gt; is the source of the problem: in a single dimension, it makes sense that if you had &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt; you could write &lt;span class=&quot;math-container&quot;&gt;$dy = 2x\,dx$&lt;/span&gt; and reasonably say that this should mean &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx} = 2x$&lt;/span&gt; even if this doesn't formally make sense as a quotient.&lt;/p&gt;&#xA;&lt;p&gt;This just doesn't work in more dimensions - in this context &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial f}{\partial x}$&lt;/span&gt; means &amp;quot;write &lt;span class=&quot;math-container&quot;&gt;$df$&lt;/span&gt; as a sum of &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dt$&lt;/span&gt; and then tell me what the coefficient of &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; was&amp;quot; - which we can see to be &lt;span class=&quot;math-container&quot;&gt;$yt$&lt;/span&gt; above, but there's a glaring problem in this notation: it requires that we talk about &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dt$&lt;/span&gt;, but these variables are never mentioned in the partial derivative notation - it's easy to make the mistake then of writing&#xA;&lt;span class=&quot;math-container&quot;&gt;$$df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$$&lt;/span&gt;&#xA;which is essentially the trouble you are alluding to - it's not true because the change of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; doesn't depend only on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, even though the notation fails to recognize this.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps equally bad is that when one has relationships such as &lt;span class=&quot;math-container&quot;&gt;$x=y=t$&lt;/span&gt;, the notation encourages us to forget that these things are related - despite one could quite happily notice that &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial f}{\partial x}$&lt;/span&gt; is not equal to &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial (x^3)}{\partial x}$&lt;/span&gt;, despite the fact that the only difference between the two is a number of substitutions. This is a related issue to the previous one - the thing that changes between those expressions is the implicit basis used in this notation. The quantity &amp;quot;the change in &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; alone&amp;quot; starts to seem somewhat arbitrary when one hits examples like this - and it comes nowhere near the intuitive sense of saying &amp;quot;the rate of change in &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$3x^2$&lt;/span&gt; times the rate of change in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; - which would, of course, be true, although not to the exclusion of other things being true as well.&lt;/p&gt;&#xA;&lt;p&gt;I say this not to try to discredit a particular notation, but to point out what it means in full; this notation is sort of happiest when you have a function &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n\rightarrow\mathbb R$&lt;/span&gt; and everyone agrees on how to give the domain coordinates - but then you get the tricky business that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; needs to be regarded as a function &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^3\rightarrow\mathbb R$&lt;/span&gt; and we are considering its values along a single line of that - which would be represented as some path &lt;span class=&quot;math-container&quot;&gt;$\mathbb R\rightarrow\mathbb R^3$&lt;/span&gt; taking &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(t,t,t)$&lt;/span&gt;. As you observe, one needs to treat &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; with equal footing when working with &lt;span class=&quot;math-container&quot;&gt;$f(x,y,t)=xyt$&lt;/span&gt;, even though this feels counterintuitive when there are relations between them. Working things out this way gives the nice formulations with linear maps outlined in other answers - but this can be a bit tricky and is certainly not the only intuition we might apply to calculus. We just need to be careful what happens when we mix notation arising in different contexts.&lt;/p&gt;&#xA;&lt;p&gt;**Of course, worth pointing out: there is a similar sleight of hand in the question! If you define &lt;span class=&quot;math-container&quot;&gt;$f(x,y,t)=xyt$&lt;/span&gt; it does not make really sense to then say &lt;span class=&quot;math-container&quot;&gt;$x=y=t$&lt;/span&gt; because those are not variables, but parameters - one could define the same function with different parameter names; if we were really talking about functions, then I could rephrase the question as &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$f(a,b,c)=abc$&lt;/span&gt;. Suppose &lt;span class=&quot;math-container&quot;&gt;$x=y=t$&lt;/span&gt;, ...&amp;quot; where something is clearly off. The question is just &lt;em&gt;asking&lt;/em&gt; for confusion to arise between the notation that treats &amp;quot;functions of several variables&amp;quot; and the notation that treats &amp;quot;various related quantities.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="3229" LastEditorUserId="3229" LastEditDate="2020-10-15T01:27:27.090" LastActivityDate="2020-10-15T01:27:27.090" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19009" PostTypeId="2" ParentId="19001" CreationDate="2020-10-15T10:29:41.490" Score="6" Body="&lt;p&gt;To clear up confusion from this, when I teach I make a massive emphasis of the fact that&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;a function takes in an ordered list of inputs&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;(and not something like  'variables'). Then I use notation like &lt;span class=&quot;math-container&quot;&gt;$D_1f$&lt;/span&gt; to denote the partial derivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; with respect to its first input, &lt;span class=&quot;math-container&quot;&gt;$D_2f$&lt;/span&gt; the second input etc. I go on about how each of these is a new function and not a procedure that you apply to the expression that computes &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So in your example, there are 3 inputs to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. The chain rule as stated is clearly for a function that takes in two inputs. It therefore does not apply in a very literal sense, not because there's any actual subtlety.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;My favourite nasty example is to define functions &lt;span class=&quot;math-container&quot;&gt;$f, g : \mathbf{R}^2 \to \mathbf{R}$&lt;/span&gt; by something like &lt;span class=&quot;math-container&quot;&gt;$g(\alpha,r) = \alpha^2 + r$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(r,\alpha) = r^2 + \alpha$&lt;/span&gt; (I use something other than &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; deliberately to throw off any sense of the correct ordering). Observe that these are the same function in the sense that for any point &lt;span class=&quot;math-container&quot;&gt;$(x,y) \in \mathbf{R}^2$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$f(x,y) = g(x,y)$&lt;/span&gt;. So&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;g \equiv f,&#xA;$$&lt;/span&gt;&#xA;right? But of course if you differentiate naively:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\partial f}{\partial r} = 2r \\&#xA;\frac{\partial g}{\partial r} = 1.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="10001" LastEditorUserId="10001" LastEditDate="2020-10-18T07:12:04.890" LastActivityDate="2020-10-18T07:12:04.890" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19010" PostTypeId="2" ParentId="18985" CreationDate="2020-10-15T13:17:58.247" Score="3" Body="&lt;p&gt;To your actual question (would this be a good way to teach about...) my serious but sounding mean answer is...no.&lt;/p&gt;&#xA;&lt;p&gt;This is because when I read your explication, my eyes glassed over with all the detail, with the arguments inside sentences, etc. Furthermore, I also think you need to start at the simplest level first. y =x and y= -x maybe. Not with a quadratic. The second para is rather complex also.&lt;/p&gt;&#xA;&lt;p&gt;In summary, I think you are making the classic mistake of thinking something that appeals to you (a sophisticate) is the right way to instruct a novice. Bad pedagogy.&lt;/p&gt;&#xA;" OwnerUserId="14767" LastActivityDate="2020-10-15T13:17:58.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19011" PostTypeId="1" AcceptedAnswerId="19012" CreationDate="2020-10-17T15:07:41.513" Score="1" ViewCount="125" Body="&lt;p&gt;Today I came across the inclusion-exclusion principle for the first time. I believe I have understood it, however when I tried solving some questions on it, I got severely stuck. I couldn't solve any of them. Could you please suggest some questions of escalating difficulty, so that I can get a good grasp on the topic at hand?&#xA;One of the questions is this one :&lt;a href=&quot;https://math.stackexchange.com/questions/2765746/inclusion-exclusion-problem-sitting-arrangement&quot;&gt;https://math.stackexchange.com/questions/2765746/inclusion-exclusion-problem-sitting-arrangement&lt;/a&gt;&#xA;, It was in a Greek contest-math book and I also found it on this site.&lt;/p&gt;&#xA;&lt;p&gt;The other question I couldn't find it on the internet, so I'll translate it:&lt;/p&gt;&#xA;&lt;p&gt;Find the number of solutions to the equation &lt;span class=&quot;math-container&quot;&gt;$x_1+x_2+x_3=100$&lt;/span&gt;, if for every &lt;span class=&quot;math-container&quot;&gt;$3\ge i\ge1$&lt;/span&gt;, xi is a non negative integer with &lt;span class=&quot;math-container&quot;&gt;$40\ge x_i$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I have found the solutions to both of these questions, the reason which I am posting them as well, is to give a better understanding to the reader of the post , of what level of questions in inclusion-exclusion principle are giving me difficulty. Could you please suggest some resources on this principal (e.g. questions of escalating difficulty and theory)?&lt;/p&gt;&#xA;" OwnerUserId="14785" LastActivityDate="2020-10-17T18:42:25.160" Title="Questions relating to inclusion-exclusion principle" Tags="|self-learning|" AnswerCount="1" CommentCount="8" ClosedDate="2020-10-17T21:57:31.920" ContentLicense="CC BY-SA 4.0" />
  <row Id="19012" PostTypeId="2" ParentId="19011" CreationDate="2020-10-17T18:42:25.160" Score="3" Body="&lt;p&gt;See &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0883856158&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Mathematics of Choice. How to Count Without Counting&lt;/strong&gt;&lt;/a&gt; by Ivan Niven (1965). For what it's worth, problems like this can be relatively straightforward with the appropriate tools (found in the book I cited) and they can be quite difficult even with a reasonable knowledge of these tools, and sometimes it's not easy to tell in advance unless one is very experienced in solving these types of problems.&lt;/p&gt;&#xA;&lt;p&gt;I've solved quite a few problems like this in the past three years as part of my &amp;quot;day job&amp;quot; work, and even I sometimes can't tell very easily whether or not the problem can be solved easily with the right approach. Sometimes I've found it's simpler to just list all possibilities rather than try to determine a clever non-counting approach. If it's no more than &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$30,$&lt;/span&gt; then this is often the quickest approach (for me, at least); and if it becomes clear that there will be too many to reasonably list all of them, then sometimes by attempting to list all of them, you'll gain some insight as to &amp;quot;how to count without counting&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2020-10-17T18:42:25.160" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19013" PostTypeId="1" CreationDate="2020-10-17T23:49:53.623" Score="4" ViewCount="617" Body="&lt;p&gt;Suppose one were writing a book aimed at high-school students (and their teachers),&#xA;where &amp;quot;high-school&amp;quot; in the US means grades 9,10,11,12 (where college/university starts at 13).&#xA;The book is not a textbook, but rather &amp;quot;math enrichment.&amp;quot;&#xA;The book attempts to expose such students to proofs, and to research questions in mathematics.&#xA;I have questions about the best way to cite the research questions.&#xA;Here are four alternatives:&lt;/p&gt;&#xA;&lt;p&gt;(1) With introductory explanation, treat it just like a research paper, e.g., &amp;quot;this was&#xA;proved in [ABC99].&amp;quot; One point would be train students to understand such references and&#xA;how to access them (increasingly easy via the Internet).&lt;/p&gt;&#xA;&lt;p&gt;(2) Instead, use &lt;em&gt;Notes&lt;/em&gt;, gathered at the end of each chapter, something like this,&lt;sup&gt;1&lt;/sup&gt;&#xA;and then the referenced note says, &lt;sup&gt;1&lt;/sup&gt;This was&#xA;proved in [ABC99]. Again, advanced explanation would be needed to interpret &amp;quot;[ABC99].&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;(3) Have no interruptions in the text—no citations, no footnotes—but at the end of the chapter in a &lt;em&gt;Notes&lt;/em&gt; section,&#xA;explain the references and history etc, citing backwards to previous pages.&lt;/p&gt;&#xA;&lt;p&gt;(4) Similar to (3), but have all &lt;em&gt;Notes&lt;/em&gt; gathered at the end of the book. This seems to be the&#xA;current standard in academic treatises.&lt;/p&gt;&#xA;&lt;p&gt;I'd appreciate your opinions!&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-10-18T00:05:07.607" LastActivityDate="2020-10-18T02:39:06.497" Title="Advice: How to cite literature for high-school students?" Tags="|secondary-education|general-pedagogy|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19014" PostTypeId="2" ParentId="19013" CreationDate="2020-10-18T00:22:01.423" Score="1" Body="&lt;p&gt;My advice is to use the least obtrusive method, probably 4, from your list. And to also minimize the number of citations.  What you are considering a feature (oh goodie, we get footnotes) is really more of a bug or at least a pill for someone not already used to wading through research papers.  You've got enough headwinds already by combining two difficult topics (proofs and research areas) rather than picking one.  Don't make the citation density the camel-back straw.&lt;/p&gt;&#xA;" OwnerUserId="14767" LastActivityDate="2020-10-18T00:22:01.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19015" PostTypeId="2" ParentId="19013" CreationDate="2020-10-18T02:39:06.497" Score="8" Body="&lt;p&gt;Research papers require references to verify where your data comes from. Schoolkids don't care where you get your data, extra references or footnotes just irritate them. If the presented information is incorrect, you are the one who should bear the grilling.&lt;/p&gt;&#xA;&lt;p&gt;Regarding footnotes vs inline references, as a reader - and a former schoolkid as we all were - I prefer short inline notes where it makes sense, like &amp;quot;the theorem bla-bla says bla-bla. Because the proof is too long or complex or needs knowledge of this and that, which supposedly you don't possess yet, etc. we are not going to prove it here. If interested, see [ABC99] for proof&amp;quot;. I can read this quickly as I read the theorem and decide whether I want to see the proof. No context switching. [ABC99] will be fully defined at the end of the book in the list of the references.&lt;/p&gt;&#xA;&lt;p&gt;Footnotes tell me nothing aside that there is some additional information pertaining the prior sentence. What kind of information is it? Do I care? I would not know until I suspend my reading to look at the footnote. And by the way, I am used to Arabic numbers to mean footnotes, that is, notes on the same page. Notes at the end of the chapter or the book are usually denoted with Roman numbers, but I think there is no hard rule on that.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, unless you want to plop a whole proof into the footnote so it would take half a page in tiny typeface, I'd rather have in on the same page than at the end of the chapter or the book, because switching between the main text and the footnote does not require to leave the page. This requires careful formatting to ensure that footnote is always on the same page where it is referred to. Choosing between notes at the end of the chapter and the end of the book I prefer the latter, it is easier to flip to the back of the book than to search where the damn chapter ends (I don't use bookmarks).&lt;/p&gt;&#xA;&lt;p&gt;And please do not mention something that is defined two or twenty pages later or not defined at all, Core-Plus Math is notorious of this, very irritating. I expect to see definitions of all important things before these things are used.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastActivityDate="2020-10-18T02:39:06.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19016" PostTypeId="2" ParentId="19007" CreationDate="2020-10-18T18:16:48.390" Score="2" Body="&lt;p&gt;Thanks for the advice everyone. After looking at many options and talking with faculty, it's been decided that we're using Lang's Undergraduate Algebra. I was leaning heavily towards Artin the past few days, but I was actually pleasantly surprised when I flipped through Lang's Undergraduate Algebra... far more standard organization, great exercises, and exposition is clear. It wasn't even on my radar (I was only familiar with the graduate version...) but this was strongly suggested by my advisor and it looks really fantastic.&lt;/p&gt;&#xA;" OwnerUserId="14774" LastActivityDate="2020-10-18T18:16:48.390" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19017" PostTypeId="1" AcceptedAnswerId="19019" CreationDate="2020-10-19T18:41:58.223" Score="50" ViewCount="16210" Body="&lt;p&gt;When casually discussing with my 13 yo child about probabilities, he told me&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;there is a 50% chance to win at the lottery&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To what I said&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;no, there is a 1 chance over 90 million&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(I roughly estimated &lt;span class=&quot;math-container&quot;&gt;$_{7}^{49}\text{C}$&lt;/span&gt; which I think is more or less the lottery here)&lt;/p&gt;&#xA;&lt;p&gt;To what he replied&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;no: either you win, or you don't. That's the probability of the fact to win.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He is obviously wrong, me being the educated father and him the child with silly ideas. I am now sitting and thinking about a counter-argument.&lt;/p&gt;&#xA;&lt;p&gt;On the serious side, I am trying to quantify his answer from a mathematical perspective but I believe that the whole premise of his reasoning is wrong (but I am not sure where).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;: I am asking the question here and not on Math SE because it is in my opinion more a matter of how to explain math to children (and accessorily, to their parents), more than a question about probabilities.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note 2&lt;/strong&gt;: I should have made it clear that we are there after many discussions on probability so he understands the &amp;quot;number of positive outcomes&amp;quot; / &amp;quot;number of all possible outcomes&amp;quot;. What he said was more like an invitation for discussion about the concept of &amp;quot;winning or not, as a single event&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="8076" LastEditorUserId="-1" LastEditDate="2022-05-26T16:34:39.700" LastActivityDate="2023-08-08T19:36:50.510" Title="How to explain that winning the lottery is not a 50/50 distribution?" Tags="|probability|" AnswerCount="21" CommentCount="23" ContentLicense="CC BY-SA 4.0" />
  <row Id="19018" PostTypeId="2" ParentId="19017" CreationDate="2020-10-19T21:38:10.517" Score="49" Body="&lt;p&gt;I don't think that talking about probabilities formally would be to any benefit for your son. However, you could simulate a lottery at home, using a die. Say that a player wins if they guess right the next outcome. So, at each roll, there is a &lt;span class=&quot;math-container&quot;&gt;$1/6$&lt;/span&gt; probability that the player wins. Roll the dice many times - say 50 or 60 - and write down the number of wins. If your son is not Donald Duck's super-lucky cousin - whose name I cannot recall - then you will give him some food for thought.&lt;/p&gt;&#xA;&lt;p&gt;After your game, you could ask him something like: &amp;quot;Why didn't you win about half of the times but only &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;?&amp;quot; - substitute &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; with the number your son won in your game.&lt;/p&gt;&#xA;&lt;p&gt;Another approach could be to get out of house to some basketball court and take shots blind-folded. According to your son, about half of your shots should be on target, however, given your son is not Steph Curry, you will have again a chance to explain him how the relative size of the basket narrows down the chances that you hit the target about half of the times.&lt;/p&gt;&#xA;&lt;p&gt;Also, note that the two above situations are nice demonstrations of discrete (dice) and continuous probabilities (shots), in case it comes up in your discussions!&lt;/p&gt;&#xA;&lt;p&gt;P.S.: Congratulations for engaging in discussions about maths with your children! :)&lt;/p&gt;&#xA;" OwnerUserId="12829" LastEditorUserId="64" LastEditDate="2020-11-12T02:39:07.687" LastActivityDate="2020-11-12T02:39:07.687" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="19019" PostTypeId="2" ParentId="19017" CreationDate="2020-10-20T03:34:59.833" Score="67" Body="&lt;p&gt;Your child is using the Principle of Insufficient Reason, which states that if we have no information about something other than the set of possible outcomes, then we should assume that all outcomes are equally likely.&lt;/p&gt;&#xA;&lt;p&gt;This principle is behind basically all of statistics, probability theory, and statistical mechanics, although it is often disguised in some way.&lt;/p&gt;&#xA;&lt;p&gt;The Principle is not applicable in this situation, since we &lt;em&gt;do&lt;/em&gt; know how many winning tickets there are in the lottery, and how many tickets are possible. But we can still apply the Principle to the lottery, in the sense that we assume that each ticket is equally likely to win.&lt;/p&gt;&#xA;&lt;p&gt;Your child would be right if we knew nothing about the lottery, other than that there are two possible outcomes &amp;quot;win&amp;quot; or &amp;quot;lose&amp;quot;. However, we do know quite a lot about lotteries in real life, so even knowing that it &lt;em&gt;is&lt;/em&gt; a lottery already tells us that the two outcomes (win and lose) are probably not equally probable.&lt;/p&gt;&#xA;&lt;p&gt;So, your child's reasoning would be right (or at least justifiable) if you said, for example: &amp;quot;Let's play a game. Either you win or you lose. What is your chance of winning?&amp;quot;. Or if, for example, he was an alien who had just arrived on earth and had no idea what &amp;quot;lottery&amp;quot; meant, and someone said &amp;quot;Let's play the lottery. Either you win or you lose. What is your chance of winning?&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="814" LastActivityDate="2020-10-20T03:34:59.833" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19020" PostTypeId="2" ParentId="19017" CreationDate="2020-10-20T06:02:50.070" Score="17" Body="&lt;p&gt;Without going into the mathematics too deeply, I would say it boils down to this:&#xA;There is only one way of winning the lottery: guessing all the numbers correctly.&#xA;But there are &lt;em&gt;a lot&lt;/em&gt; more possible ways to lose. So the &amp;quot;whole premise of his reasoning is wrong&amp;quot;, as you say, because he is ignoring millions of possible outcomes!&lt;/p&gt;&#xA;" OwnerUserId="14800" LastEditorUserId="-1" LastEditDate="2021-01-14T12:47:17.733" LastActivityDate="2021-01-14T12:47:17.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19021" PostTypeId="1" CreationDate="2020-10-20T08:36:03.970" Score="1" ViewCount="213" Body="&lt;p&gt;If we accept that there's not much learning from doing the &amp;quot;same&amp;quot; questions, like find the derivative of &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$x^3$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$x^4$&lt;/span&gt; due to the algorithmic way of how it's done, then what happens when it applies to questions in a more subtle way?&lt;/p&gt;&#xA;&lt;p&gt;To use an example, I've always found implicit differentiation to be very algorithmic, to the point that comparing different examples will showcase the same general ideas &lt;em&gt;but with different functions&lt;/em&gt;. For example, if you look at example 5 of &lt;a href=&quot;https://tutorial.math.lamar.edu/classes/calci/implicitdiff.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul's online math notes&lt;/a&gt;, aren't those 3 specific examples are structurally the same, but have variations of functions. I could have given 3 steps in the algorithm as this:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Differentiate with respect to x&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Remember to use chain rule for y so that you get y'&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Solve for the derivative y'&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;For sake of argument, if I'm designing questions on implicit differentiation to test if students know a calculus concept, &lt;em&gt;rather than designing a test to see if they know how to algebraically manipulate things&lt;/em&gt;, then (1) should I do more creative questions, and (2) how would I do that?&lt;/p&gt;&#xA;&lt;p&gt;I think open ended questions and examples would be more helpful for conceptual understanding, like &amp;quot;at what angles do these curves intersect?&amp;quot; or &amp;quot;are these curves orthogonal?&amp;quot; but I won't specifically mention implicit differentiation. It might be likely that the answer is found via implicit differentiation if it's in the implicit differentiation section, but don't think I can do much about that. Or maybe &amp;quot;what's the relationship between &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, and the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;- and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercepts of any line tangent to the graph of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}+\sqrt{y}=\sqrt{c}$&lt;/span&gt;?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But anyway, in general, if I analyze a bunch of questions and see that they're all variations of the same structure, then should that structure be pointed out? (I find, if it is, then we can move onto more interesting and/or creative questions. Or, at least, move onto questions that focus on understanding the concepts at hand rather than seeing if the student knows the prerequisite algebraic manipulation skills)&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-10-21T01:39:32.233" LastActivityDate="2020-10-21T01:39:32.233" Title="Do you avoid examples or test questions that showcase an algorithmic plug'n'chug approach?" Tags="|mathematical-pedagogy|examples|exams|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19022" PostTypeId="2" ParentId="19017" CreationDate="2020-10-20T10:26:26.147" Score="28" Body="&lt;p&gt;I think so far best reaction is the top-voted comment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Have you asked him to explain what he thinks “probability” means?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'd address the topic from here. And as this is not a school environment when teacher has the truth and shares it with the student, I'd take the approach to try to understand him through questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What probability means for you?&lt;/li&gt;&#xA;&lt;li&gt;Are there situations where chance is not 50-50?&#xA;&lt;ul&gt;&#xA;&lt;li&gt;if yes, what are those?&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How are those different from lottery?&lt;/li&gt;&#xA;&lt;li&gt;Can we apply something similar also for lottery?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;if no, then maybe that's a time to start explaining that &amp;quot;probability&amp;quot; has a different meaning for him and for (at least majority of) rest of the world&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2&gt;For the updated question:&lt;/h2&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What he said was more like an invitation for discussion about the concept of &amp;quot;winning or not, as a single event&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Again, without teaching, you might start asking about what's the practical use of probability if you exclude &amp;quot;PROBABILITY&amp;quot; from it and just list outcomes.&lt;/p&gt;&#xA;&lt;p&gt;Or, just go back to definition: &amp;quot;probability for me is chance of winning. When I say 1 to 90M I mean I need to buy 90M tickets and I expect one of them will win. What 50-50 means for you?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;(if he still GENUINELY insists to 50% it would even worth price of 10 tickets for me to buy together than discuss why 5 of them was not winning. All other answers approach this from doing a different experiment, but that case it's all about explaining how those other events are similar instead of focusing on the lottery)&lt;/p&gt;&#xA;" OwnerUserId="14808" LastEditorUserId="14808" LastEditDate="2020-10-21T04:24:29.883" LastActivityDate="2020-10-21T04:24:29.883" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19023" PostTypeId="2" ParentId="19017" CreationDate="2020-10-20T11:27:54.013" Score="10" Body="&lt;p&gt;Taking the ed part of the question:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Don't feel like you have to convince the kid of everything immediately.  Give him time.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In particular, watch out for him just trolling you.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you do decide to engage him, you could do so by suggesting that if he really thinks it's 50-50, that you'll take the opposite side of the bet for 2:1 odds.  Ask him to put some of his lawn-mowing money down...and see if he really believes in his thesis.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(Lapsing into math), of course the probability of each outcome is not 50% just because there are two possibilities.  (back into ed)  Perhaps you can come up with similar bets (e.g. bet that the next house door you see will be open or closed...it's a binary result, but with unequal probabilities).  Pepper him with a couple examples like that.  But mind (1) and (2).  ;-)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P.s. Still feels more like a math question than a how to communicate question, given you didn't answer it yourself and then discuss the comms.  Also given the responses here.&lt;/p&gt;&#xA;" OwnerUserId="14767" LastActivityDate="2020-10-20T11:27:54.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19024" PostTypeId="2" ParentId="19017" CreationDate="2020-10-20T14:01:20.293" Score="-6" Body="&lt;p&gt;&amp;quot;there is a 50% chance to win at the lottery&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;In the context of his answer he's absolutely right&lt;/strong&gt; so instead of these other answers where you challenge his views ... &amp;quot;explain probability to me&amp;quot; ... you need to focus on understanding the context of his answer.&lt;/p&gt;&#xA;&lt;p&gt;He's right because the chance of you EVALUATING if that ticket is the right ticket is 50/50 ... 1/2 ... one half.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;First you guessed a set of numbers ... one out of millions ... then you evaluated if that guess was right ... they are two different events.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The scenario of the guess being right is different than the chance of guessing the right number.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are millions of other tickets so the chance of your evaluating ALL tickets being a winner or not is the number of tickets multiplied by 1/2.&lt;/p&gt;&#xA;&lt;p&gt;For example if I said I'm thinking of a number, a digit, between 0 and 9, guess what it is, at first you're 50/50 of being right or wrong ... 1 over 2 ... one guess where you were right or wrong ... right + wrong = 2 possibilities.&lt;/p&gt;&#xA;&lt;p&gt;But if you're wrong and you choose again the fraction is now 1 over 3 ... right + wrong, the possible results of your first choice, PLUS the wrong guess you already made, for a total of three outcomes for the one guess ... 1/3 ... and if you keep going you get to 1/10 ... the number of possible guesses for all the numbers that can be chosen is 1/10 ... 10%.&lt;/p&gt;&#xA;&lt;p&gt;So the chance of winning the lottery is figuring out how many possible guesses there can be ... not if the one you chose is right or wrong.&lt;/p&gt;&#xA;&lt;p&gt;If there are ten number in the lottery it's 1/10 ... 10% ... ten &amp;quot;per cent&amp;quot; is 10/&amp;quot;per cent&amp;quot; = 10/100 (cent =100) = 0.1.&lt;/p&gt;&#xA;&lt;p&gt;If there are a hundred numbers in the lottery it's 1/100 = 0.01&lt;/p&gt;&#xA;&lt;p&gt;If there are a thousand numbers in the lottery it's 1/1000 = 0.001&lt;/p&gt;&#xA;&lt;p&gt;Lets say you can choose six different numbers and each one can be between 0 and 99 - a hundred choices.&lt;/p&gt;&#xA;&lt;p&gt;The chance of choosing one of those six right is 1/100.&#xA;The chance of choosing the second of those six right is 1/99 ... because there's one less number to choose from.&#xA;The chance of choosing the third of those six right is 1/98 ... because there's again one less number to choose from.&lt;/p&gt;&#xA;&lt;p&gt;The chance to choose all six is 1/100 * 1/99 * 1/98 * 1/97 * 1/96 * 1/95 ...six SETS of choices, each SET of choices being one out of one hundred, one out of 99, etc.&lt;/p&gt;&#xA;" OwnerUserId="14814" LastEditorUserId="14814" LastEditDate="2020-10-20T14:58:15.420" LastActivityDate="2020-10-20T14:58:15.420" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19025" PostTypeId="2" ParentId="1655" CreationDate="2020-10-20T14:46:51.207" Score="0" Body="&lt;p&gt;This is a tricky one, since there's a combination of two things:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The student is probably genuinely interested in the math and wants to grapple with the material.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The student is disrupting the classroom, and interrupting the lesson for other students.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;On the one hand, you want to foster the student's interest and enjoyment of mathematics.  On the other, you can't allow a single student to disrupt the lecture repeatedly.  I think as others have suggested, a somewhat delicate approach is needed.&lt;/p&gt;&#xA;&lt;p&gt;I recommend that the next time the student does something along these lines, don't react angrily or allow them to fluster you, but instead in a very calm way slowly walk through the derivation and show why the student is incorrect.  If it can be done with elementary methods, use them.  But if not, don't necessarily shy away.  The goal is simply to get to the truth.  The student needs to recognize two things: (1) you, the instructor, are an expert on the material, (2) they, the student, still have a lot to learn.  Being slow and methodical will hopefully make the student realize just how disruptive they're being, and responding calmly will help defuse any desire they have to elicit a reaction from you.&lt;/p&gt;&#xA;&lt;p&gt;Having been an instructor for many freshman non-major math courses, in my experience many of the students are still very much like high-schoolers, unfortunately.  But they're at an age where they &lt;em&gt;want&lt;/em&gt; to be seen as more mature.  If you keep the level of classroom maturity high, they'll respond well.  But if you get dragged into a conflict with a student, it will only get worse.&lt;/p&gt;&#xA;&lt;p&gt;Lastly, I think it's important to remember that this is only one student among many.  If they're innocently asking many questions out of curiosity, it's definitely not malicious and shouldn't be treated as such.  But if you get the impression that they're trying to disrupt the class for their own enjoyment, that's a serious problem, since they're detracting from the learning experience of the entire class, and I would have zero patience for students like that.  If you get the impression it's the latter, I wouldn't feel badly about reporting the student to the professor.  Everyone else (or, more realistically, their parents) is paying good money for the opportunity to learn in that class.&lt;/p&gt;&#xA;" OwnerUserId="14774" LastActivityDate="2020-10-20T14:46:51.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19026" PostTypeId="2" ParentId="19017" CreationDate="2020-10-20T18:07:43.767" Score="5" Body="&lt;p&gt;I think the answer talking about the Principle of Insufficient Reason is a very good one, but I'll give a &amp;quot;keyword&amp;quot; that I've found helpful thinking about this kind of thing. And that is: &amp;quot;probability distribution&amp;quot;. &amp;quot;There are two outcomes, therefore it is 50/50&amp;quot; assumes that the outcomes follow a &lt;strong&gt;uniform distribution&lt;/strong&gt; (and that is in fact the assumption of the Principle of Insufficient Reason). But there are other kinds of probability distribution, like the Gaussian distribution, a Poisson distribution, and anything you could name, under which different outcomes do not in fact have the same probability. Another answer suggested illustrating this with dice, and that could be one way: use dice, or even just a coin, to figure out the frequencies of different events. For example, flipping a coin 100 times you'll find you get roughly 50/50 odds of heads and tails. But flip 2 coins 100 times and you'll find that the outcomes &amp;quot;two heads&amp;quot; and &amp;quot;two tails&amp;quot; will be less frequent than &amp;quot;one hand and one tail&amp;quot;. On the other hand if you distinguish the coins, then you end up with 4 possibilities (2 heads, 2 tails, A heads B tails, A tails B heads) and &lt;em&gt;those&lt;/em&gt; are all equally likely. And that itself points out that the probability distribution the events follow itself depends on how you parcel out the probability space, how you define an &amp;quot;event&amp;quot; - are you conflating the head-tails possibilities or distinguishing the two? In the former case you'll get an event that's twice as likely as the others, and the latter you'll get two events that are both as likely as any other. You can also look at different experiments and name the different probability distributions that emerge (I don't remember if my multiple coin example would involve a normal distribution or a binomial one but you can look it up).&lt;/p&gt;&#xA;&lt;p&gt;This notion of &amp;quot;probability distribution&amp;quot; can be a good way to put words to the notion that just because you have X possibilities, doesn't mean they all have to be equally likely... and that saying they are is in fact a very specific thing to say, something you'll only do under specific circumstances (either they actually are equally likely, or you have no information to say they aren't and thus follow the Principle of Insufficient Reason).&lt;/p&gt;&#xA;&lt;p&gt;And at that point, at future times this comes up (and I expect it would; I find probabilities very hard to keep a mental grasp on) you could answer something like &amp;quot;yes, the odds of winning are 50/50 if there are 2 possibilities &lt;em&gt;if they follow a uniform probability distribution&lt;/em&gt;. Do they?&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="14728" LastEditorUserId="14728" LastEditDate="2020-10-20T18:14:41.167" LastActivityDate="2020-10-20T18:14:41.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19027" PostTypeId="2" ParentId="19017" CreationDate="2020-10-20T18:42:44.380" Score="13" Body="&lt;p&gt;A slightly different approach:&lt;/p&gt;&#xA;&lt;p&gt;Let's say there are 100 lottery tickets in total and there is only one ticket that will win you the prize. If you don't buy any tickets at all, what's your chance of winning? No chance. Nothing out of a hundred is zero %. What if you could buy all the tickets? You are certain to win. You've got 100 out of 100 tickets. Your chance to win is 100 %. How many tickets do you have to buy to have a 50 % chance? How big is your chance when you buy just one ticket?&lt;/p&gt;&#xA;&lt;p&gt;Edit: yet another attempt to get into a non-mathematical mind:&lt;/p&gt;&#xA;&lt;p&gt;It's a banana or it's not a banana. If it's not a banana, it can be an apple, a pear, an orange or a chayote. If it's a banana it can only be a banana. So &amp;quot;banana&amp;quot; or &amp;quot;not banana&amp;quot; are not the same kind of thing. They are not equal. The same applies to &amp;quot;you win&amp;quot; or &amp;quot;you don't win&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="14823" LastEditorUserId="14823" LastEditDate="2020-10-21T21:48:02.160" LastActivityDate="2020-10-21T21:48:02.160" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19028" PostTypeId="2" ParentId="19017" CreationDate="2020-10-20T22:14:40.930" Score="7" Body="&lt;p&gt;Ask him whether the probability of winning is the same if you bought 1000 tickets rather than one ticket.&lt;/p&gt;&#xA;&lt;p&gt;Or, imagine a lottery with 100 tickets, of which only one was a winner. If 100 different people bought a ticket, how many would win? What does that suggest for the probability of any one ticket being a winner?&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2020-10-20T22:14:40.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19029" PostTypeId="2" ParentId="19017" CreationDate="2020-10-20T23:29:17.520" Score="5" Body="&lt;p&gt;Funny kid. He would be right if there was only winning and losing, but, given your numbers, there are actually winning and over 90 million varieties of losing. He's only calculating the probability of losing in one way.&lt;/p&gt;&#xA;&lt;p&gt;Wildly edited:&lt;/p&gt;&#xA;&lt;p&gt;This seems like a confusion in semantics. I say just tell him as above. But if he insists that the idea of winning or losing as a single event is useful one, there's a mathematical perspective.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Consider a smaller number, a 1/3 chance lottery with 3 choices. He can enumerate the total number of pairs he has to win against: every choice (3) multiplied by the number of pairs to compare for a choice (2). Then, consider the winning pairs, which is the total number of choices minus one. Don't spoil that it's 2 winning comparisons in 6 possible comparisons.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\dfrac{(n-1)}{n(n-1)} = \dfrac1n$&lt;/span&gt; if he wants, where &lt;em&gt;n&lt;/em&gt; is the number of lottery number combinations.&lt;/p&gt;&#xA;&lt;p&gt;From here you can suggest to him exactly what it &lt;em&gt;does&lt;/em&gt; mean for there to really be the two possibilities of winning or losing, where winning is a single event and so is losing; and the lengths he'd have to go to in order to find something like the probability of winning the lottery, while defining what that is if he doesn't know.&lt;/p&gt;&#xA;&lt;p&gt;It seems that in the context of the &lt;em&gt;right&lt;/em&gt; answer, from an outside point of view, he's using the principle of insufficient reason; he's simply made a logical error solving the right question. But also within the realm of possibility is that he's found a sort of glitch in the way you could understand the question that leads to some bizarre statements about the real thing.&lt;/p&gt;&#xA;" OwnerUserId="14827" LastEditorDisplayName="user22729" LastEditDate="2023-08-08T09:45:49.727" LastActivityDate="2023-08-08T09:45:49.727" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19030" PostTypeId="2" ParentId="19017" CreationDate="2020-10-21T00:06:27.590" Score="12" Body="&lt;h1&gt;Make it Personal&lt;/h1&gt;&#xA;&lt;p&gt;Take a marshmallow (or some small candy that you know he likes), show it to him, then put it into one hand behind your back and say: &amp;quot;If you pick the hand with the marshmallow, you can have it.  Now, what is the probability you will choose correctly?&amp;quot;  Then let him guess.&lt;/p&gt;&#xA;&lt;p&gt;Then, get a jar of marbles (like, 100+), pick one distinct marble, and say this: &amp;quot;If you pick this magic marble blindfolded in one try, I'll give you this giant candy bar.  If you don't, you have to mow the lawn, take out the garbage, and wash the dishes for a month.  What is the probability that you will win this bet?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If he insists it is not 50/50, then say: &amp;quot;But picking this marble is just like the lottery.  In fact, it's much easier than the lottery.  So why isn't it a simple 50/50 gamble?  We agreed that the marshmallow test was 50/50, so if the marshmallow test is 50/50, and the lottery is 50/50, why isn't the marble test 50/50?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Then you can reduce the marble test to 2 marbles, and add one marble at a time for him to learn concretely and intuitively how probability works.  The stakes make the cost of losing very high, so that he cannot be flippant about his answer.  At that point he knows he can't just guess.  He needs to be &lt;em&gt;right&lt;/em&gt;.  Guessing the odds on the lottery has zero cost to him, because as a minor, he isn't legally allowed to play.&lt;/p&gt;&#xA;" OwnerUserId="13468" LastActivityDate="2020-10-21T00:06:27.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19031" PostTypeId="2" ParentId="19017" CreationDate="2020-10-21T01:03:02.570" Score="6" Body="&lt;p&gt;I want to offer a game to play with your son that he would almost definitely understand and would impart the principles of probability (and the futility of gambling at the same time).&lt;/p&gt;&#xA;&lt;p&gt;First, get some tokens. These can be marbles, beads, painted rocks, Legos, bits of colored paper, or whatever else you have handy. Of the tokens, choose a defining characteristic that you can use to separate the tokens into two easily-discernable groups. For the sake of this explanation, I will choose glass beads that are either red or blue.&lt;/p&gt;&#xA;&lt;p&gt;In a small bag, put in 1 red token and some crazy number of blue tokens. Then offer the bag to your son, disallowing him to look inside and telling him only that the bag contains blue and red tokens and if he draws a red token, he wins. You could even sweeten the deal by having some candy as a reward, which will serve to further drive the point home later.&lt;/p&gt;&#xA;&lt;p&gt;Let him reach into the bag (again, without letting him look) and pull out a token. Regardless of whether he &amp;quot;wins&amp;quot;, tell him to put his token back in the bag, then mix up the contents of the bag and have him try again. Do this as many times as you think is necessary, or until he gets bored and/or frustrated of losing over and over again.&lt;/p&gt;&#xA;&lt;p&gt;When the game ends, sit him down and dump out the bag in front of him. Have him see that there are tons of blue tokens and only 1 red token. Demonstrate that, in a sea of blue tokens, the chances of him picking a red token at random is incredibly slim, so even if the only possible outcomes are either winning or losing, it's much more likely to lose than to win.&lt;/p&gt;&#xA;&lt;p&gt;Then explain that the lottery is the same way. In order to win the lottery, you need to buy a winning ticket (which is a red token) and not a losing ticket (which is a blue token). Explain how in the lottery there are millions of losing tickets to the point where he would have to find the red token not in a small bag, but in a huge garbage bag that was packed full of blue tokens but still only had one red piece. Ask him if he wants to play that game, and you shouldn't be surprised if he says no. :)&lt;/p&gt;&#xA;&lt;p&gt;One thing to make the &amp;quot;game&amp;quot; more like the lottery is to give him 10 pieces of candy to start with, and every time he loses, he has to give up a piece of candy, and every time he wins, he gets 5 pieces of candy. Even though he will lose far more often than he wins, encourage him to keep playing when he does win to drive home the feeling of gambling. What seems like a sweet deal at first will quickly turn into a no-win situation. Of course, if you go this route, be a good parent afterward and give him some candy anyway. ;)&lt;/p&gt;&#xA;" OwnerUserId="13152" LastEditorUserId="13152" LastEditDate="2021-01-14T00:47:37.867" LastActivityDate="2021-01-14T00:47:37.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19032" PostTypeId="2" ParentId="19017" CreationDate="2020-10-21T02:46:30.723" Score="-1" Body="&lt;p&gt;Standard joke aside. Explain with an info-graphic, and also describe the operation of the lottery properly. YouTube has plenty of these.&lt;/p&gt;&#xA;&lt;p&gt;It depends how many prizes are offered too, and how lots are sold.&#xA;Oh, you mean if I only have one lot, what are the chances of winning the major prize?&lt;/p&gt;&#xA;&lt;p&gt;Does winning have to satisfy a requirement to profit from the win? Buying exactly half the lots should give you a 50% chance of winning the major prize.&lt;/p&gt;&#xA;&lt;p&gt;If you spend as little as $100 on Lotto like games, you should have a fair chance of getting something back, finding the 0.50 depends upon the game.&lt;/p&gt;&#xA;&lt;p&gt;Of course if the lottery is conscription it may genuinely be 50%.&lt;/p&gt;&#xA;&lt;p&gt;This is why the Monty Hall problem is generally hated, because it is frequently not properly stated.&lt;/p&gt;&#xA;&lt;p&gt;The fallacy lies in the outcomes being mapped, and not their likelihoods.&lt;/p&gt;&#xA;&lt;p&gt;Try that 10 coin example with loaded coins, let's say depleted uranium on one face and polystyrene on the the other, maybe don't use the Russian Roulette example.&lt;/p&gt;&#xA;&lt;p&gt;I know that if I had to write code for two outcomes I might have to spend the same amount of effort upon each.&lt;/p&gt;&#xA;" OwnerUserId="12937" LastActivityDate="2020-10-21T02:46:30.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19033" PostTypeId="2" ParentId="19017" CreationDate="2020-10-21T13:36:21.340" Score="5" Body="&lt;p&gt;Counterexamples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Tomorrow rains or not. But it doesn't rain half of the days (at least, in my city).&lt;/li&gt;&#xA;&lt;li&gt;A treat (candy) may or may not like, but mostly likes.&lt;/li&gt;&#xA;&lt;li&gt;You can evaluate correctly or incorrectly a sum, but do not fail in 50% of them.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;... (choose the ones more near to your children).&lt;/p&gt;&#xA;" OwnerUserId="7441" LastEditorDisplayName="user22729" LastEditDate="2023-08-08T09:45:36.837" LastActivityDate="2023-08-08T09:45:36.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19034" PostTypeId="2" ParentId="19017" CreationDate="2020-10-21T17:00:56.717" Score="2" Body="&lt;p&gt;Present a deck of playing cards. Fan out to show all the cards. Red you win. Black you lose. fifty fifty right?&lt;/p&gt;&#xA;&lt;p&gt;But what happens if I do this:&lt;/p&gt;&#xA;&lt;p&gt;Lay one red card down.&lt;/p&gt;&#xA;&lt;p&gt;Lay nine black cards on top of it.&lt;/p&gt;&#xA;&lt;p&gt;Turn over and shuffle.&lt;/p&gt;&#xA;&lt;p&gt;Still think your odds are fifty fifty?&lt;/p&gt;&#xA;" OwnerUserId="3305" LastActivityDate="2020-10-21T17:00:56.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19035" PostTypeId="2" ParentId="19017" CreationDate="2020-10-21T17:39:55.403" Score="4" Body="&lt;p&gt;&lt;strong&gt;There's a way in which your kid has hit on an important attribute of probability.&lt;/strong&gt; When he says,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;no: either you win, or you don't. That's the probability of the fact to win.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He's describing a component that goes into computing probability: the number of possible outcome states. If instead he had said,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;no: either you win, or you don't. That's all the things that can happen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He would have been completely correct.&lt;/p&gt;&#xA;&lt;p&gt;The number of possible outcome states are important when he wants to describe a system with more than two outcomes; say, winning, losing, and tieing.&lt;/p&gt;&#xA;&lt;p&gt;Consider a game where two players have to guess heads or tails when a coin is flipped. There are now 3 game states to model.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$P(\text{&quot;a tie&quot;}) = 1\cdot\frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$P(\text{&quot;a win&quot;}) = (1\cdot\frac{1}{2}) \vee (1\cdot\frac{1}{2}) - P(\text{&quot;a tie&quot;}) = (1 - (1- \frac{1}{2})(1- \frac{1}{2})) - \frac{1}{4} = \frac{1}{2}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$P(\text{&quot;both lose&quot;}) = 1\cdot\frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;What's great about this game is you can easily enumerate all possible outcomes to check your math:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;H/T P1 P2 TIE WIN LOSE&#xA;H   H  H  1&#xA;H   H  T      1&#xA;H   T  H      1&#xA;H   T  T          1&#xA;T   H  H          1&#xA;T   H  T      1&#xA;T   T  H      1&#xA;T   T  T  1        &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;(&lt;code&gt;H/T&lt;/code&gt; is the result of the flip, &lt;code&gt;P1&lt;/code&gt; and &lt;code&gt;P2&lt;/code&gt; are the guesses of player 1 and 2, &lt;code&gt;TIE&lt;/code&gt;, &lt;code&gt;WIN&lt;/code&gt;, and &lt;code&gt;LOSE&lt;/code&gt; are marked with a 1 if that game state occured)&lt;/p&gt;&#xA;&lt;p&gt;Now, play the game with him and ask him to predict how many games he will win by guessing correctly while you guessed wrong (i.e. he wins but doesn't tie).&lt;/p&gt;&#xA;&lt;p&gt;What's nice about this formulation is that it offers a way for his intuition to be incorrectly bolstered by the &lt;span class=&quot;math-container&quot;&gt;$P(\text{&quot;a win&quot;}) = \frac{1}{2}$&lt;/span&gt; result, aligning with his preconception that &amp;quot;you either win or you don't&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Once you've played the game, reevaluate the game states. What's missing? Why did he win half as frequently as he thought he would? (Hint: while we have described all possible game states—winning, losing, and tieing—we haven't described all possible player states. What are all the possible states he may find himself in as a player? What are the probabilities of those?)&lt;/p&gt;&#xA;&lt;p&gt;Through exploring this with him he can come to understand that not all game states are equally likely. &lt;strong&gt;The heart of probability is about knowing what &lt;em&gt;can occur&lt;/em&gt; (which he already understands) as well as how &lt;em&gt;frequently it does occur&lt;/em&gt; (which he's yet to grok).&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This game is a great tool for exploring probability because with one coin and two players it's quite simple, but if you add players and coins it can get much more complicated to model how frequently someone will win by guessing the number of heads flipped (eventually leading to the need for game theory to describe dominant strategies and counterstrategies).&lt;/p&gt;&#xA;&lt;p&gt;Try it with two coins to show him a game where not only all game states have different probabilities, but all player states do as well.&lt;/p&gt;&#xA;" OwnerUserId="12129" LastActivityDate="2020-10-21T17:39:55.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19036" PostTypeId="1" CreationDate="2020-10-21T18:58:17.650" Score="2" ViewCount="67" Body="&lt;p&gt;I'm working on converting a course to make it free to the students, and I'm considering using Ximera.  What I haven't found online is documentation for/information on tips and best practices for randomizing values for problems in Ximera, and how well it will work. I have two main concerns:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I'd first like to find information on specifics on how to write randomized questions in Ximera in ways that will be easy enough to do so that we can have TAs and not overly tech-savvy instructors help write the questions/set up the course.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Secondly, does anyone know of a Ximera course already running somewhere that does use randomized values in many questions, and are there significant issues/delays/server overload problems with the actual day-to-day experience of the students in the course?  We have ~1500 students per semester taking this course that we want to convert from our current paid system.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;EDIT: this is for a College Algebra course.&lt;/p&gt;&#xA;" OwnerUserId="14841" LastEditorUserId="14841" LastEditDate="2020-10-21T21:55:50.960" LastActivityDate="2020-10-21T21:55:50.960" Title="Setting up a new Ximera course with randomized values in problems" Tags="|course-design|technology-in-education|homework|software|problem-design|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19037" PostTypeId="2" ParentId="19036" CreationDate="2020-10-21T19:10:12.603" Score="2" Body="&lt;p&gt;I worked with Jim Fowler and Bart Snapp on Ximera as a graduate student. It is still (to my knowledge) primarily being maintained by Jim Fowler.  I would recommend contacting him at his Ohio State email address.&lt;/p&gt;&#xA;&lt;p&gt;I am not sure if this is the case, but one design issue we ran into with randomization is how frequently edge cases come up, and the wide variability of problem difficulty.  We found it was generally easier to write a script to randomly generate 20 problems and then scratch the 3 or 4 of them which seemed unreasonable.&lt;/p&gt;&#xA;&lt;p&gt;At the time, this system needed some serious &amp;quot;tech savvyness&amp;quot; to operate, but that may have improved in the last 4 or 5 years.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-10-21T19:10:12.603" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19038" PostTypeId="1" CreationDate="2020-10-21T22:03:25.887" Score="3" ViewCount="125" Body="&lt;p&gt;I am working with a college student who was placed into a pre-college level math class. She's in a unit that covers a lot of unit conversions (simple ones, like 5 km = ___ miles or 3 yards = ___ cm ). She says she's confused by what the teacher showed her in class so I spent time working with her to reinforce the method they learned in class (the &amp;quot;Train Track Method&amp;quot; is what she calls it). I like the idea because to me it makes a lot of sense. Organizing the conversion this way helps the students focus on cancelling the units first, before filling in the numbers. For example, for 3 yards, I want to cancel the yards, so I know I'd multiply by feet/yards to get yards on the bottom of the fraction to cancel. Then to get to inches I'd multiply by inches/feet, then finally cm/inches. Then once all the units are cancelling out properly, then I can worry about filling in the numbers and handling them.&lt;/p&gt;&#xA;&lt;p&gt;Well, after two different 1-hour long meetings with her, she still isn't getting it. She says it's totally foreign to her. Every other student I have shown this idea to or practiced it with was able to get it and do very well on this class's unit conversion quiz. But she is still stumped. She emailed me asking me for other helpful resources for these kind of problems, but everything I Google seems to be a re-hashing of the same method, or the exact same explanations I was already saying to her.&lt;/p&gt;&#xA;&lt;p&gt;What are some good resources (handouts, webpages, videos, etc.) to help a student that is really struggling with basic conversions?&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2020-10-22T00:43:39.817" Title="Resources for Unit Conversions" Tags="|resource-request|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19040" PostTypeId="1" AcceptedAnswerId="19057" CreationDate="2020-10-22T00:38:08.330" Score="1" ViewCount="154" Body="&lt;p&gt;I am currently mentoring my little brother in mathematics. There is an issue with the pedagogy of unit rates. For example when given the following concept &amp;quot; 11.00 U.S. Dollars to 20 Planet X Dollars Find the Unit Exchange Rate.&amp;quot; I am aware that does eventually leads to unit rates which is the name of the lesson the math teacher is on. Which the individual has also explained. The issue is are there any resources on how to understand the unit rate, cost, or etc. that is being presented in this lesson outline? My attempts to help him with problem has been to grasp that the problem ultimately has to do with something like this &lt;span class=&quot;math-container&quot;&gt;$\frac{\mathrm{rate}}{1}=\frac{11.00}{20}$&lt;/span&gt;. Or are there any examples, or methods that can help drive the concept home?&lt;/p&gt;&#xA;&lt;h1&gt;Edit&lt;/h1&gt;&#xA;&lt;p&gt;This is arbitrary &amp;quot; 11.00 U.S. Dollars to 20 Planet X Dollars Find the Unit Exchange Rate.&amp;quot; Any rate either from U.S dollars to Planet X dollars or reverse is fine. Here is the specific one &amp;quot;11.00 U.S Dollars to 20 Planet X Dollars Find the Unit Rate Exchange for U.S Dollars to Planet X Dollars.&amp;quot; Again none of these specific problems are actually meaningful if the general concept is what is the underlying issue. Firstly, the definition of ratio, proportion, fraction, and decimal are all given before hand if the assessment of these problems are being presented.  This is a stack exchange about Math Educators so math educators in the domain of middle/elementary school would ideally know the issue. The Unit Rate might not make a lot of sense it college educator level, but it does to me a student who passed Calculus 1, 2, 3, and Differential Equations and is aiming to major in Mathematics, and finds uses to its applications.&lt;/p&gt;&#xA;" OwnerUserId="7250" LastEditorUserId="7250" LastEditDate="2020-10-23T20:58:56.380" LastActivityDate="2020-10-23T22:46:27.110" Title="Resources for Unit Rates" Tags="|mathematical-pedagogy|primary-education|middle-school|resources|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19041" PostTypeId="2" ParentId="19038" CreationDate="2020-10-22T00:38:36.377" Score="2" Body="&lt;p&gt;Could try section 0.9 of my book: &lt;a href=&quot;http://lightandmatter.com/lm/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://lightandmatter.com/lm/&lt;/a&gt; . The discussion question at the end of the section is meant to immunize against common mistakes. I go through each of these with my students and do a reality check to get them to see how absurd the results are, before they're even allowed to shout out what the actual mistake is.&lt;/p&gt;&#xA;&lt;p&gt;It sounds like she's American. You can also work with her to make sure she can visualize the various metric units look like. Many people in the US have no clue how big a centimeter is, how much weight a kilogram of beans is, and so on. If they don't know these things, then they can't check for reasonableness.&lt;/p&gt;&#xA;&lt;p&gt;Most likely what's happening with your student is that she doesn't really understand conceptually what multiplication and division mean. This probably needs to be dealt with at a more basic level, like Alice, Bob, and Carol sharing applies from a bag.&lt;/p&gt;&#xA;&lt;p&gt;Some people have an easier time with money, and you can then make the analogy between money and other units. E.g., if I change my dollars into pennies, do I end up with more pennies, or fewer? Then, if they get that: if I change my kilograms into grams, do I end up with more grams, or fewer? I tell a joke with this, which is that when we change to a bigger unit, the number gets smaller; it's compensation, like the guy who needs a really big pickup truck because he's compensating for ... something else that's really small.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-10-22T00:43:39.817" LastActivityDate="2020-10-22T00:43:39.817" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19043" PostTypeId="2" ParentId="5636" CreationDate="2020-10-22T08:38:42.997" Score="1" Body="&lt;p&gt;A recent book is &lt;a href=&quot;https://bookstore.ams.org/amstext-35&quot; rel=&quot;nofollow noreferrer&quot;&gt;Number Theory and Geometry: An Introduction to Arithmetic Geometry&lt;/a&gt; by Álvaro Lozano-Robledo, AMS 2019. It's particularly nice for students who want to see how number theory works in the context of curves, especially lines, conics and cubics. It might attract some to studying elliptic curves and arithmetic/diophantine geometry. The author has also written a more advanced, but still introductory &lt;a href=&quot;https://bookstore.ams.org/cdn-1599001847663/stml-58/&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt; on elliptic curves, modular forms and L-functions.&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://www.maa.org/press/maa-reviews/number-theory-and-geometry-an-introduction-to-arithmetic-geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/maa-reviews/number-theory-and-geometry-an-introduction-to-arithmetic-geometry&lt;/a&gt; for a review by Abbey Bourdon. Here is a short excerpt:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In Number Theory and Geometry: An Introduction to Arithmetic Geometry, standard topics are covered with the unifying goal of finding rational (or, at times, integral) points on curves. These kinds of problems can be traced back to the Greek mathematician Diophantus in his series of books called Arithmetica, and they have continued to inspire an active area of research today.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="376" LastActivityDate="2020-10-22T08:38:42.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19044" PostTypeId="1" CreationDate="2020-10-22T08:47:41.860" Score="5" ViewCount="294" Body="&lt;p&gt;Note: by &amp;quot;elementary&amp;quot; I mean &amp;quot;without using more advanced theory and tools&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Students are sometimes required or encouraged to solve very difficult problems using limited number of tools and machinery. It is not uncommon in competition-style exams that problems in algebraic/analytic number theory are solved using elementary number theory (i.e. without analysis or abstract algebra). Various other exams in the world also have a limit on the number of tools that can be used. For example, sometimes, people are required to prove something similar to mean value theorem for a specific given function (with an explicit expression) solely by very complicated algebraic computations, without Calculus, because the rigorous &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; definition of derivatives and limits are not yet taught.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What are the reasons why we wish to tackle hard problems with elementary methods? And is it beneficial or not?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;If we are building a theory from axioms, when we write proofs, we must only use things that are already proven. But in the situation I describe above, we are not building up a theory; instead, we are &lt;strong&gt;applying&lt;/strong&gt; some theory to solve a problem, so in this situation, what are the reasons why we sometimes limit our range of tools? What are some pros/cons of this?&lt;/p&gt;&#xA;&lt;p&gt;This might be, to some extent, a matter of taste, but it is still interesting to know reasons for this.&lt;/p&gt;&#xA;" OwnerUserId="9339" LastEditorUserId="9339" LastEditDate="2020-10-24T08:34:05.340" LastActivityDate="2020-11-16T10:54:56.933" Title="How important is it to come up with or learn an elementary solution?" Tags="|undergraduate-education|secondary-education|proofs|problem-solving|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19045" PostTypeId="2" ParentId="19017" CreationDate="2020-10-22T09:55:44.203" Score="4" Body="&lt;p&gt;You could explain the fact to him like that:&#xA;Each ticket is either a winning one or a losing one, and there is only one winning ticket.&#xA;How many tickets there are that you can buy? If the answer is more than two and there is only one winning ticket, then the probability is not 50/50, because if the total number of tickets is 100, then the chance that buying one you get the winning one is 1/100.&lt;/p&gt;&#xA;&lt;p&gt;A simple proof of that is that only a handful of people had ever won the lottery, not half of the players.&lt;/p&gt;&#xA;&lt;p&gt;If it was 50/50 the probability of a win, then half of the ticket sold must be a winning one, which is obviously not true, since the hypothesis is there is only one winning ticket.&lt;/p&gt;&#xA;&lt;p&gt;Also, if he was right, that must apply to each lottery player, and if he has a 50/50 chance of a win, also the other N players would have the same winning probability, and for that to be true either half of the tickets are winning tickets or the total probability is more than 100%, which cannot be.&lt;/p&gt;&#xA;" OwnerUserId="14848" LastActivityDate="2020-10-22T09:55:44.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19046" PostTypeId="1" AcceptedAnswerId="19063" CreationDate="2020-10-22T12:48:55.257" Score="7" ViewCount="305" Body="&lt;p&gt;The dimension theorem (the rank-nullity theorem) can be explained in many ways. I consider it as a consequence of the first isomorphism theorem/splitting lemma. When I teach undergrad matrix-theoretic linear algebra, I start with the equation &lt;span class=&quot;math-container&quot;&gt;$Ax=b,$&lt;/span&gt; and I tell my students that the dimension theorem basically says that the number of total variables equals the sum of the number of free variables and the number of &amp;quot;non-free&amp;quot; variables. They find this statement very easy. If I teach a &amp;quot;formal/proof-based&amp;quot; undergrad mathematics class, I tell my students that the dimension theorem basically tells us how much &amp;quot;stuff&amp;quot; we need to put inside the nullspaces to extend it to the given vector space.&lt;/p&gt;&#xA;&lt;p&gt;Today I found a very good analogy: In some sense, the dimension theorem is the linear algebraic analog of the Pigeonhole Principle. Note that for any finite set &lt;span class=&quot;math-container&quot;&gt;$A,$&lt;/span&gt; the function &lt;span class=&quot;math-container&quot;&gt;$f: A \rightarrow A$&lt;/span&gt; is injective iff surjective iff bijective. It's a consequence of the Pigeonhole Principle. The dimension theorem gives a similar kind of conclusion for a finite-dimensional vector space &lt;span class=&quot;math-container&quot;&gt;$V,$&lt;/span&gt; and any linear map &lt;span class=&quot;math-container&quot;&gt;$T: V \rightarrow V.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, could you help me by providing a couple of more analogies that can be explained to an undergrad junior? Thank you so much. Please stay safe.&lt;/p&gt;&#xA;" OwnerUserId="14850" OwnerDisplayName="James" LastEditorUserId="376" LastEditDate="2020-10-22T14:03:41.780" LastActivityDate="2020-10-24T17:19:59.303" Title="The dimension theorem and pedagogy" Tags="|undergraduate-education|linear-algebra|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19047" PostTypeId="2" ParentId="19017" CreationDate="2020-10-22T14:04:55.333" Score="3" Body="&lt;p&gt;I don't play the lottery. When someone asks why, I resort to the old joke:&#xA;&amp;quot;I figure my chances of winning are about the same whether I play or not&amp;quot;.&#xA;The point being that there isn't much difference between 0 and 0.000000001&lt;/p&gt;&#xA;&lt;p&gt;But my brother (who does play the lottery) objected: He said that if you don't play then you can't win, but if you do play then you can win. So he looked at it from a binary point of view: can't win or can win.  Like your child it seems.&lt;/p&gt;&#xA;&lt;p&gt;FWIW, he's a smart guy and understands probability, but figures a few dollars per year are a small price to pay for just the &lt;em&gt;possibility&lt;/em&gt; of winning.&lt;/p&gt;&#xA;" OwnerUserId="14854" LastActivityDate="2020-10-22T14:04:55.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19049" PostTypeId="1" CreationDate="2020-10-22T20:53:04.360" Score="1" ViewCount="266" Body="&lt;p&gt;I'm taking a discrete math course. The textbook we use is such a pain to read because the amount of material is very overwhelming. For example, chapter 1 alone is 115 pages. And every subsection, 1.1-1.8, is very important to the coursework (from propositional logic to intro to proofs), so it's not like I can just skip a couple sections.&lt;/p&gt;&#xA;&lt;p&gt;I'm trying to read everything and then summarize all of chapter 1 into one short chapter, but doing this feels like I'm neglecting a lot of the nuance that fills up the pages.&lt;/p&gt;&#xA;&lt;p&gt;How do I approach overwhelming course material without neglecting nuance and without burning myself out?&lt;/p&gt;&#xA;" OwnerUserId="12501" LastEditorUserId="12501" LastEditDate="2020-10-22T21:04:34.263" LastActivityDate="2020-10-24T05:18:56.673" Title="How to summarize textbook material?" Tags="|discrete-math|student-question|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19050" PostTypeId="2" ParentId="19049" CreationDate="2020-10-22T21:18:06.783" Score="1" Body="&lt;p&gt;I recommend to do some analysis of the situation.  Telling us how the long the chapter is means nothing if we don't know how fast you have to cover things.  Try to look at the situation as pages per day.&lt;/p&gt;&#xA;&lt;p&gt;Recommendations:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Work every day and do the required pages every day, to stay on track.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(Perhaps) just chalk this up as a course that is going to take more time than others.  And give it the time.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(This runs counter to the others, but you have to consider alternate strategies):  Find the minimum you need to do, by talking to people who took the course before.  There is a tendency I've noticed in academia of wanting to assign tomes, but then really not giving the time to cover them.  This is pretty different from high school or lower level uni courses, where you cover most of the material and really learn the topic.  The thing is you find people wanting the prestige of coverage without the work.  (Combined with the lack of attention to pedagogy, i.e. TEACHING, in colleges.)  This is sort of an unfortunate situation, but you need to at least identify if this is the case in your course (they don't really expect mastery).  In which case, you can usually find out what are the critical things to learn if you talk to the grapevine.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14861" LastActivityDate="2020-10-22T21:18:06.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19051" PostTypeId="2" ParentId="19049" CreationDate="2020-10-23T02:52:10.127" Score="2" Body="&lt;p&gt;I've never done this myself, but what you could do is make a some sort of tiered summarization where the major topics have blue tabs, then subtopics have red tabs, and then minor subtopics have green tabs. And as you do notes, the major ideas will be on the blue, the examples could be on red, and proofs and nuance could be green.&lt;/p&gt;&#xA;&lt;p&gt;This way, if you look at all blue tabs, for example, you can see all the major topics and have an idea of how things work, but if you want examples then you'll have to look at the red, &lt;em&gt;and so on&lt;/em&gt; Here, I never sacrificed anything in the course summary &lt;em&gt;overall&lt;/em&gt;, but the summary is tiered in regards to depth for how I'd want to study.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, study habits differ so much that what works for me may or may not work for you, but consider if the only thing overwhelming is sheer amount of content. Could it be complexity of the content? The amount of new vocabulary? The nuance and caveats? The arithmetic and algebraic manipulation? etc.. I've always found compartmentalization of overwhelming too-big topics into smaller manageable topics to be the best strategy.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-10-24T05:18:56.673" LastActivityDate="2020-10-24T05:18:56.673" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19052" PostTypeId="2" ParentId="14089" CreationDate="2020-10-23T15:11:33.060" Score="3" Body="&lt;p&gt;In the US, Cayley-Hamilton isn't typically taught in a first course in Linear Algebra (non proof-based, intended for students majoring in the sciences and engineering), but is typically taught in a second more advanced course for mathematics majors.  You can see this by looking at textbooks intended for these two different kinds of linear algebra courses. The textbook by Hoffman and Kunze that you mention is very commonly used in more advanced courses for mathematics majors.&lt;/p&gt;&#xA;&lt;p&gt;The recommendations of the Linear Algebra Curriculum Study Group have had a major influence on the content of the first course in linear algebra.  See&lt;/p&gt;&#xA;&lt;p&gt;Carlson, David, Charles R. Johnson, David C. Lay, and A. Duane Porter. &amp;quot;The Linear Algebra Curriculum Study Group recommendations for the first course in linear algebra.&amp;quot; The College Mathematics Journal 24, no. 1 (1993): 41-46.&lt;/p&gt;&#xA;" OwnerUserId="5841" LastEditorUserId="5841" LastEditDate="2020-10-23T15:37:49.010" LastActivityDate="2020-10-23T15:37:49.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19054" PostTypeId="2" ParentId="19017" CreationDate="2020-10-23T17:04:13.617" Score="4" Body="&lt;p&gt;There are already a lot of good answers here, but here's one way to put it in simple terms that even a younger child should be able to comprehend.&lt;/p&gt;&#xA;&lt;p&gt;Your son said &lt;code&gt;either you win, or you don't&lt;/code&gt; which means he's seeing two possible outcomes and then dividing 100% by 2. You could explain that there are actually many more possible outcomes because there are more ways to lose than ways to win.&lt;/p&gt;&#xA;&lt;p&gt;So if we take a simple lottery where you pick a number from 1 to 10 and one number is selected as the winning number, instead of &lt;code&gt;you either win, or you don't&lt;/code&gt; the situation is &lt;code&gt;either you pick the winning number, or you pick the first losing number, or you pick the second losing number, or you pick the third losing number, and so on...&lt;/code&gt;. This illustrates that there are more than two outcomes, even though nine of the ten outcomes are all losing. So instead of dividing 100% by 2, you should divide it by 10 to get a 10% chance of winning.&lt;/p&gt;&#xA;" OwnerUserId="14863" LastActivityDate="2020-10-23T17:04:13.617" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19055" PostTypeId="1" CreationDate="2020-10-23T20:17:50.580" Score="5" ViewCount="156" Body="&lt;p&gt;For the following linear differential equation&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a_ny^{(n)}+\cdots+a_1y'+a_0y=Q(x),$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;most books teach the method of undetermined coefficients, variation of parameters and Laplace transforms. Tenenbaum and Pollard's &lt;a href=&quot;https://www.pdfdrive.com/download.pdf?id=162256767&amp;amp;h=020d582a98dbb03f8ae1fc00c4b94774&amp;amp;u=cache&amp;amp;ext=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ordinary Differential Equations&lt;/a&gt; teaches all these methods, but also the inverse operator method and has about 40 pages on it. I have never learned this method as an undergraduate but once I tried to teach it, I loved this method very much and it seems to me that in many cases, this method will be easier to use than other methods.&lt;/p&gt;&#xA;&lt;p&gt;However, I do not find this method in many books on Differential Equations. Do people generally not teach this method in an undergraduate course? Why?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2020-10-24T13:57:39.387" LastActivityDate="2020-10-24T13:57:39.387" Title="Do people usually teach solving a linear differential equation by inverse operators in an undergraduate course?" Tags="|undergraduate-education|differential-equations|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19056" PostTypeId="2" ParentId="19055" CreationDate="2020-10-23T21:22:25.833" Score="1" Body="&lt;p&gt;I'm just an ex-student, not a teacher.&lt;/p&gt;&#xA;&lt;p&gt;But my hunch is that it's more user friendly after you know the other methods, rather than before them.  Sort of like the &amp;quot;tic tac toe&amp;quot; method of integration by parts.  It's good.  But makes more sense, if you've had to slog through the detailed algebra several times before.&lt;/p&gt;&#xA;&lt;p&gt;The issue then becomes one of time.  One of the issues with a standard ODE course is just getting everything covered.  If I had to pitch passsengers off the lifeboat, I would sacrifice operator method well before series solutions, Euler equation, etc.&lt;/p&gt;&#xA;&lt;p&gt;P.s. Spiegel does cover it (I think, am out to dinner...but check when I get home). Edit:  yes, he does.  Calls it an alternate method, treats it after the others.  Has several pages on it, but not the 40 page length of Tannenbaum.&lt;/p&gt;&#xA;" OwnerUserId="14864" LastEditorUserId="14864" LastEditDate="2020-10-23T22:19:44.600" LastActivityDate="2020-10-23T22:19:44.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19057" PostTypeId="2" ParentId="19040" CreationDate="2020-10-23T22:26:53.413" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Firstly, the definition of ratio, proportion, fraction, and decimal&#xA;are all given before hand if the assessment of these problems are&#xA;being presented.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If they have been given before, it would be nice to know those specific definitions.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is a stack exchange about Math Educators so math educators in the&#xA;domain of middle/elementary school would ideally know the issue.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One would hope so.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Unit Rate might not make a lot of sense it college educator level,&#xA;but it does to me a student who passed Calculus 1, 2, 3, and&#xA;Differential Equations and is aiming to major in Mathematics, and&#xA;finds uses to its applications.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The word &amp;quot;rate&amp;quot; is such an overloaded word and everyone bends it differently. To me there is a &lt;em&gt;ratio&lt;/em&gt;, which can be written using a colon or division sign or as a fraction. Equality of two ratios is a &lt;em&gt;proportion&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Ratio can be between same-dimension values, it this case the resulting ratio is dimensionless, like widescreen TV is 16 cm by 9 cm or 16 inches by 9 inches or just 16:9 or 1.78. Side note: every time I see how compression ratio of an internal combustion engine is expressed in American magazines, I smirk: it is something like 9.5:1 or 12:1, but never just 9.5 or 12.&lt;/p&gt;&#xA;&lt;p&gt;When ratio links two values having different dimensions, then the word &lt;em&gt;rate&lt;/em&gt; comes up, and we can say &amp;quot;per&amp;quot;, like pressure is force per area, although to me it is a bit weird to call it rate. If the unit in denominator is time, then &lt;em&gt;temporal rate&lt;/em&gt; comes up and we can have miles per hour. In colloquial speech people remove &amp;quot;temporal&amp;quot; and just say &amp;quot;rate&amp;quot;. Speed is rate. I am puzzled why so many people prefer saying &amp;quot;rate&amp;quot; instead of &amp;quot;speed&amp;quot; or &amp;quot;velocity&amp;quot;. But very often when people mean speed they say &amp;quot;rate of speed&amp;quot;, which is wrong, because rate of speed is acceleration, speed is rate of change of position. In elementary school they do it all the time.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Exchange rate&lt;/em&gt; is a special fiscal term. Do you expect middle schoolers to know what exchange rate is? Ok, supposedly they know, they were given a definition. It would be nice to see it, but you don't want to provide it. Now you introduce yet another term, &lt;em&gt;unit rate&lt;/em&gt;. Judging by your proportion, it is how much of one thing corresponds to just one item of another thing. Actually, it is how much of another thing corresponds to one item of chosen thing for which you are calculating this &amp;quot;unit rate&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In your proportion &lt;span class=&quot;math-container&quot;&gt;$\frac{\mathrm{rate}}{1}=\frac{11.00}{20}$&lt;/span&gt; what you wrote as &amp;quot;rate&amp;quot; is not rate, it is number of USD per 1 PlanetXD. It is money. It has dimension, USD. It is x: &lt;span class=&quot;math-container&quot;&gt;$\frac{\mathrm{x\ USD}}{\mathrm{1\ PlanetXD}}=\frac{\mathrm{11\ USD}}{\mathrm{20\ PlanetXD}}$&lt;/span&gt;. Rate is &lt;span class=&quot;math-container&quot;&gt;$\frac{\mathrm{x\ USD}}{\mathrm{1\ PlanetXD}} = \mathrm{x\ USD/PlanetXD}$&lt;/span&gt;. Rate has dimension of USD/PlanetXD. American dollars per one Planet X dollar. Numerically the same, but conceptually different.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2020-10-23T22:46:27.110" LastActivityDate="2020-10-23T22:46:27.110" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19058" PostTypeId="1" AcceptedAnswerId="19061" CreationDate="2020-10-24T00:15:26.610" Score="2" ViewCount="217" Body="&lt;p&gt;I am teaching Calculus (one and also several variables). I would like to create questions in which students would watch a short video and then answer a question related to the content of the video. In this way, students might find it interesting and appreciate the subject. The video is not necessarily related to Maths (directly), but at some point in the video we could ask a simple question related to Calculus, such as limits or derivatives.&lt;/p&gt;&#xA;&lt;p&gt;I would think that there are only a very few such videos on the internet. Nevertheless, could you recommend me a video?&lt;/p&gt;&#xA;&lt;p&gt;Thks.&lt;/p&gt;&#xA;" OwnerDisplayName="user14865" LastEditorDisplayName="user14865" LastEditDate="2020-10-24T00:26:12.507" LastActivityDate="2020-10-24T15:04:44.823" Title="A wow video to teach Calculus" Tags="|online-instruction|online|videos|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19059" PostTypeId="2" ParentId="14385" CreationDate="2020-10-24T00:23:57.223" Score="-2" Body="&lt;p&gt;I think y-intercept is a directed line segment. It is a vector in which it has a magnitude and direction. For example for the line y=2x-8, the y-intercept must be -8 meaning it is 8 units downward from the origin. The point involved is (0,-8) but it is not the y-intercept. It is only the point where the line crosses the y-axis. If you consider y-intercept as a point, what will happen to the equation of a line mentioned above. Will it be y=2x+(0,-8)?&lt;/p&gt;&#xA;" OwnerUserId="14866" LastActivityDate="2020-10-24T00:23:57.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19060" PostTypeId="2" ParentId="19058" CreationDate="2020-10-24T04:35:51.063" Score="1" Body="&lt;p&gt;For Calc II, there is &lt;a href=&quot;https://www.youtube.com/watch?v=jhdBDWZVXks&quot; rel=&quot;nofollow noreferrer&quot;&gt;a great scene&lt;/a&gt; from the movie Holes. A slightly shorter shovel is preferred, because you can dig a slightly smaller hole. How much smaller? (The movie is great.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="470" LastEditDate="2020-10-24T15:04:44.823" LastActivityDate="2020-10-24T15:04:44.823" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19061" PostTypeId="2" ParentId="19058" CreationDate="2020-10-24T13:31:27.563" Score="2" Body="&lt;p&gt;Dan Meyer used to do &amp;quot;&lt;a href=&quot;https://docs.google.com/spreadsheets/u/0/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsfqouLWNNC6Z4/pub?output=html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Three-Act Math Tasks&lt;/a&gt;&amp;quot; where he would start with a short video (followed by math questions, etc). Maybe you can re-purpose these videos for a calculus audience.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-10-24T13:31:27.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19062" PostTypeId="1" CreationDate="2020-10-24T14:56:19.900" Score="4" ViewCount="832" Body="&lt;p&gt;Years ago as a college freshman I was taking my first calculus course. Another freshman skipped it because he had calculus in Advanced Placement in high school. I mentioned we were learning the limit definition of a derivative&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \lim_{\Delta x\to 0} \frac{f(x+\Delta x) -f(x)}{\Delta x}$$&lt;/span&gt;&#xA;and he didn't know what I was talking about.&lt;/p&gt;&#xA;&lt;p&gt;Maybe I misunderstood (I hope I did!). But is it even possible to teach calculus without this? If so is that common in AP classes?&lt;/p&gt;&#xA;" OwnerUserId="14854" LastEditorUserId="15367" LastEditDate="2022-03-20T15:36:39.960" LastActivityDate="2022-03-20T15:36:39.960" Title="Teaching calculus in AP without the formal definition of the derivative" Tags="|calculus|teaching|usa|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19063" PostTypeId="2" ParentId="19046" CreationDate="2020-10-24T17:19:59.303" Score="1" Body="&lt;p&gt;It seems to me that a discussion of how to make any function &lt;span class=&quot;math-container&quot;&gt;$f: A \rightarrow B$&lt;/span&gt; into a bijection might be in order. First, we can deal with onto by replacing &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$f(A)$&lt;/span&gt;. So, let &lt;span class=&quot;math-container&quot;&gt;$g: A \rightarrow f(A)$&lt;/span&gt; be the function given by &lt;span class=&quot;math-container&quot;&gt;$g(x)=f(x)$&lt;/span&gt; for each &lt;span class=&quot;math-container&quot;&gt;$x \in A$&lt;/span&gt;. Next, we may need to make the domain smaller, for each non-empty fiber&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f^{-1} \{ b \} = \{ a \in A \ | \ f(a)=b \} $$&lt;/span&gt;&#xA;we should &lt;em&gt;select one point&lt;/em&gt;. Let the collection of all such points be &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;. Define &lt;span class=&quot;math-container&quot;&gt;$h: S \rightarrow f(A)$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$h(x)=f(x)$&lt;/span&gt; for each &lt;span class=&quot;math-container&quot;&gt;$x \in S$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$x,y \in S$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h(x)=h(y)$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$f(x)=f(y)$&lt;/span&gt; hence there exists &lt;span class=&quot;math-container&quot;&gt;$b \in B$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$x,y \in f^{-1}\{ b \}$&lt;/span&gt;. But, by construction of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$x=y$&lt;/span&gt;. That is, &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is injective.&lt;/p&gt;&#xA;&lt;p&gt;The innocent little step of &lt;em&gt;selecting one point&lt;/em&gt; is rather difficult for arbitrary functions. In contrast, for linear transformations each fiber of the domain is an affine subspace. In particular, &lt;span class=&quot;math-container&quot;&gt;$T: V \rightarrow W$&lt;/span&gt; has&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ T^{-1}\{ w \} = \{ x \in V \ | \ T(x)=w \} $$&lt;/span&gt;&#xA;If &lt;span class=&quot;math-container&quot;&gt;$x,y \in T^{-1}\{ w \}$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$T(x)=w=T(y)$&lt;/span&gt; hence &lt;span class=&quot;math-container&quot;&gt;$T(y-x)=0$&lt;/span&gt; and so &lt;span class=&quot;math-container&quot;&gt;$y-x \in \text{Ker}(T)$&lt;/span&gt;. That is, &lt;span class=&quot;math-container&quot;&gt;$y = x+z$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$z \in \text{Ker}(T)$&lt;/span&gt;. Indeed,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ T^{-1}\{ w \}  = x + \text{Ker}(T) $$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$T(x)=w$&lt;/span&gt;. This is a very interesting equation because it means all the fibers of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; are the same size as &lt;span class=&quot;math-container&quot;&gt;$\text{Ker}(T)$&lt;/span&gt;. Ok, it's better than that, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; have the same size, but they have different dimension. The dimension of the fibers of a linear map are all the same. Of course, the only way &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; can be injective is for the size of these fibers to shrink to a single element, in turn that means the kernel must be zero dimensional.&lt;/p&gt;&#xA;&lt;p&gt;The linear algebraic proof that a linear map &lt;span class=&quot;math-container&quot;&gt;$T: V \rightarrow W$&lt;/span&gt; has&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\text{dim}(V) = \text{dim}( \text{Ker}(T))+ \text{dim}( \text{Image}(T))$$&lt;/span&gt;&#xA;is anchored to basis extension arguments. The basis for the kernel in &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is extended to those vectors outside the kernel. Then, those vectors outside the kernel have an image in &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; which serves to generate &lt;span class=&quot;math-container&quot;&gt;$T(V)$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$V=W$&lt;/span&gt; then the only way the vectors outside the kernel can generate &lt;span class=&quot;math-container&quot;&gt;$T(V)=V$&lt;/span&gt; is for there to be no vectors in the kernel (except zero).&lt;/p&gt;&#xA;&lt;p&gt;I'm posting this in the hope that my answer might spur someone else to a better answer.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-10-24T17:19:59.303" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19064" PostTypeId="2" ParentId="19062" CreationDate="2020-10-25T08:38:54.597" Score="0" Body="&lt;p&gt;&lt;em&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt; At first, I want to highlight that I am not familiar with AP Calculus and the corresponding syllabus, so I will cover solely the mathematical dimension of this question.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, teaching calculus without limits, in general, means the you have at hand some other means of talking about approximations etc. The only way that comes to my mind to do so is by using &lt;em&gt;infinitesimals&lt;/em&gt; - i.e. positive quantities &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\varepsilon&amp;lt;x$&lt;/span&gt; for any real &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, using this and standard parts of hyperreal numbers, one may define the derivative of a function as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The derivative of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at some point &lt;span class=&quot;math-container&quot;&gt;$a\in D_f$&lt;/span&gt; is defined as the standard part of the following hyperreal:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{f(x+dx)-f(x)}{dx},$$&lt;/span&gt;&#xA;whenever this standard part exists/has sense.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, I find it mostly improbable that this was the one preferred in an introductory calculus class. However, one may &amp;quot;define&amp;quot; a notion as e.g. the derivative, &lt;em&gt;procedurally&lt;/em&gt; by describing the way one may find a functions derivative - in terms of algebraic manipulations.&lt;/p&gt;&#xA;&lt;p&gt;For instance, one way to do so is to take some &lt;span class=&quot;math-container&quot;&gt;$h\neq0$&lt;/span&gt; and then calculate the rate of change fraction:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{f(x+h)-f(x)}{h}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then, once the above fraction has been simplified enough, by setting &lt;span class=&quot;math-container&quot;&gt;$h=0$&lt;/span&gt; - just to avoid to mention any limits - you get the so wanted derivative.&lt;/p&gt;&#xA;&lt;p&gt;Even if the above approach has several - and from my viewpoint, enormous - faults in terms of sacrificing mathematical rigour in favour of a more procedural understanding, it still seems to me the only &amp;quot;reasonable&amp;quot; way in which one may avoid mentioning limits in a calculus class.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2020-10-25T08:38:54.597" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19065" PostTypeId="2" ParentId="16702" CreationDate="2020-10-25T17:52:09.907" Score="0" Body="&lt;p&gt;About a week ago I ordered 7 or 8 books from amazon (I do this 2 to 4 times a year, depending on how much I can afford to spend), and yesterday Wooton’s book arrived (bibliographic details below). This is the book that, in my answer to &lt;a href=&quot;https://matheducators.stackexchange.com/a/13106/745&quot;&gt;Where can I find primary sources from the New Math movement in the 60s?&lt;/a&gt;, I said I don’t believe I’ve ever looked at before. At some later time I might archive some information about Wooton’s book in that earlier answer. However, as I was reading through his book the past couple of days, I came across some comments that I thought would fit well in this thread.&lt;/p&gt;&#xA;&lt;p&gt;William Wooton (1919-1988), &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0006BMQOG&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;SMSG. The Making of a Curriculum&lt;/strong&gt;&lt;/a&gt;, Yale University Press, 1965, x + 182 pages.&lt;/p&gt;&#xA;&lt;h1&gt;&lt;strong&gt;REVIEWED BY:&lt;/strong&gt;&lt;/h1&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://science.sciencemag.org/content/150/3693/202.1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harry Merrill Gehman (1898−1981), &lt;strong&gt;Science&lt;/strong&gt; (NS) 150 #3693 (8 October 1965), p. 202.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/3613949&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bryan Thwaites (1923−___), &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 50 #374 (December 1966), pp. 403−404.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/41187280&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Marion Todd (1928−2015), &lt;strong&gt;Arithmetic Teacher&lt;/strong&gt; 14 #3 (March 1967), p. 232.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h1&gt;5-paragraph excerpt from pp. 26−29:&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A detailed description, here, of the outline or of the philosophy underlying the 9th grade text would involve mathematical considerations beyond the scope of this work. To show the nature of the dialogue involved, however, it seems worthwhile to look at one of the more fundamental problems the group faced. Most persons who, at one time or another, have been exposed to a course in elementary algebra have little difficulty recalling that central to the subject is the use of letters of the alphabet as symbols. In particular, the letter &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; comes easily to mind. It is around the use of such symbols that some of the “modern” controversy centers.” Traditionally, such symbols have been referred to as “unknowns,” “literal numbers,” “general numbers,” and “variables,” in some cases depending on context, in others depending on the whim of the user. The student has been told that these symbols are unknown numbers and that, in some cases, he can add these numbers &lt;span class=&quot;math-container&quot;&gt;$(x+x = 2x)$&lt;/span&gt; while, in other cases, he cannot (e.g., &lt;span class=&quot;math-container&quot;&gt;$x+y).$&lt;/span&gt; He has been instructed in ways of “finding the unknown,” as, for example: if &lt;span class=&quot;math-container&quot;&gt;$x+2 = 5,$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$x=3.$&lt;/span&gt; But, on the other hand, he has been told that he is not to try to “find &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” when writing such things as &lt;span class=&quot;math-container&quot;&gt;$x(x+2) = x^2 + 2x.$&lt;/span&gt; This is due to the fact that the usual manipulations in which he becomes involved when working with such symbol groups produce &lt;span class=&quot;math-container&quot;&gt;$\text{O} = \text{O},$&lt;/span&gt; which, though true, does not seem to tell him anything he wants to know about &lt;span class=&quot;math-container&quot;&gt;$x.$&lt;/span&gt; Worse, should he inadvertently apply his ingenuity to &lt;span class=&quot;math-container&quot;&gt;$x = x+1,$&lt;/span&gt; he arrives at the mystifying &lt;span class=&quot;math-container&quot;&gt;$\text{O} = 1,$&lt;/span&gt; which not only does not tell him anything about &lt;span class=&quot;math-container&quot;&gt;$x,$&lt;/span&gt; but causes him to call into question the sensibilities of anyone who finds interest in a discipline that deals with such absurdities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is one of the concerns of those seeking to revise the mathematics curriculum to make the meaning of such symbols clear to students, and to place their use on a sound logical foundation. Granting this, however, the best way to establish such a foundation is a matter of much controversy (hence, one reason for the lengthy discussions of the 9th grade subgroup). Present-day logicians, in their inquiries into the foundations of mathematics, have had occasion to use the notion of what they call a “placeholder,” and it was this viewpoint that UICSM had adopted. Briefly: a number is an abstraction. Nobody has ever heard, felt, or seen a number, but the body proper of mathematics stems from the fact that the human mind can conceive of such abstractions. Furthermore, in discussing numbers, symbols are used which are called, in some cases, numerals, and in others, pronumerals, placeholders, or variables. The logicians, having found it necessary to work at varying levels of abstraction, have come to view a symbol such as “&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;” not as a number, nor even as the name of a number, but rather as a representation of the name of a number. Another representation of the same name of the same number is “two.” A number has infinitely many names; for example, another name for the number whose name can be represented by “&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;” is the name represented by the symbolism “&lt;span class=&quot;math-container&quot;&gt;$1+1.$&lt;/span&gt;” A symbol such as “&lt;span class=&quot;math-container&quot;&gt;$x,$&lt;/span&gt;” then, can be viewed as holding a place in an expression such as “&lt;span class=&quot;math-container&quot;&gt;$x+5$&lt;/span&gt;” for a representation of a name of a number, hence the name “placeholder.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A step from the logicians are those who believe that the distinction between a name and its representation is unnecessary in all but the deepest discussions of foundations, and that it is sufficient to distinguish between the number itself and its name. Thus, the symbol “&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;” is the name of a number, just as the word “Tom” is the name of a person. At this level of abstraction, however, there are various ways of viewing symbols such as “&lt;span class=&quot;math-container&quot;&gt;$x.$&lt;/span&gt;” In one view, when one is considering the expression “&lt;span class=&quot;math-container&quot;&gt;$x+5,$&lt;/span&gt;” the symbol “&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” is conceived of as holding a place for the name of a number and is thus, conceptually, a “placeholder.” In another view, the symbol “&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” is used in the expression “&lt;span class=&quot;math-container&quot;&gt;$x+5$&lt;/span&gt;” &lt;em&gt;as&lt;/em&gt; the name of a number and is called a “variable.” Inherent in both viewpoints, of course, is the agreement that there is some specified set of numbers with which the symbol “&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” is associated.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although there is almost universal agreement on the importance of distinguishing between a number and its name, the best way, mathematically, and pedagogically, to view the use of a symbol such as “&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” is the center of much controversy. In view of the eminence of many proponents of each point of view, it would appear that, for the present, the way in which such symbols should be handled depends on the spirit in which the subject matter of algebra as a whole is handled, and this is chiefly a matter of who is doing the handling. It should be apparent, however, that the question is not a trivial one, since, in a sense, the way in which it is resolved determines to a greater or lesser extent the spirit in which the subject of algebra is taught. The question is essentially one of the “level of abstraction” and, barring gross misrepresentations, the argument reduces as much to a pedagogical matter as it does to a mathematical one. The problem of finding the level of abstraction appropriate to the cognitive readiness of the student is a very real one, and the definitive answer, if such exists, has not yet been found.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To return to the matter at hand, the question of how to treat symbols was one (though not the only one) of the causes for the lengthy pre-discussions held by the 9th grade group before beginning detailed outlines of chapters, and their final decision was not free from critical attack by proponents of alternative viewpoints, as was to be expected.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="745" LastActivityDate="2020-10-25T17:52:09.907" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19066" PostTypeId="2" ParentId="19062" CreationDate="2020-10-26T00:08:10.177" Score="6" Body="&lt;p&gt;The definition of a limit, and of the derivative in terms of a limit, are standard material that should be covered in all freshman calculus classes. A student who is never exposed to these definitions is being shortchanged educationally.&lt;/p&gt;&#xA;&lt;p&gt;Realistically, however, very few students in this type of class are intellectually capable of understanding these definitions or of doing even the most trivial epsilon-delta proofs, although they may understand the general concept at some more vague level, like &amp;quot;if you make the deltas small, you get the derivative.&amp;quot; They also tend to ignore this material or forget it because it is never used again once rules for differentiation are introduced.&lt;/p&gt;&#xA;&lt;p&gt;The problem with skipping this material would be that if you have 10% of the class who are going to be math majors or who are intellectually capable of doing this material, then you're not serving them well.&lt;/p&gt;&#xA;&lt;p&gt;It is possible to use other frameworks for calculus. It's relatively straightforward to do derivatives of polynomials without any fancy foundational definitions. This was done historically before Newton and Leibniz. I do this in sections 1.2.3-5 in my book &lt;a href=&quot;http://lightandmatter.com/fund/&quot; rel=&quot;noreferrer&quot;&gt;Fundamentals of Calculus&lt;/a&gt;. However, it's hard to use this approach to get off the ground with differentiation rules and derivatives of transcendental functions. That's what Newton and Leibniz accomplished. Until about 1900, students in the English-speaking world learned calculus using Newton's fluxions, and others learned it using Leibniz's infinitesimals. Infinitesimals were given a more secure logical foundation by Abraham Robinson and others ca. 1950 in non-standard analysis, and something similar can be done using nonclassical logic. The best-known freshman calc text to use NSA is one by Keisler, which is free online these days. For previous discussion of this, see &lt;a href=&quot;https://matheducators.stackexchange.com/questions/15195/would-teaching-nonstandard-calculus-in-an-introduction-calculus-course-make-it-e&quot;&gt;Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?&lt;/a&gt; .&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-10-26T00:08:10.177" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19067" PostTypeId="2" ParentId="19062" CreationDate="2020-10-26T16:29:06.607" Score="1" Body="&lt;p&gt;Rate of change definition of derivative is still covered as part of AP calculus. (Your question is actually &amp;quot;someone told me something that sounded strange&amp;quot; and...well...yes, you were to think it sounded strange.)&lt;/p&gt;&#xA;&lt;p&gt;See the official description of the AP course:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://apcentral.collegeboard.org/courses/ap-calculus-bc/course&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://apcentral.collegeboard.org/courses/ap-calculus-bc/course&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[In particular, see the second link on that page, a pdf document that gives an overview of the course and shows coverage of the rate of change limit definition of a derivative.]&lt;/p&gt;&#xA;&lt;p&gt;The next question (that you didn't ask, but some people are interested to discuss) is what about epsilon-delta proofs.  Those are very lightly covered in BC calculus.  And arguably have been rather lightly covered for a long time in traditional engineering school calculus courses.  But that's actually not what you asked about.&lt;/p&gt;&#xA;" OwnerUserId="14875" LastActivityDate="2020-10-26T16:29:06.607" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19068" PostTypeId="2" ParentId="19062" CreationDate="2020-10-26T18:54:12.590" Score="8" Body="&lt;p&gt;As a new AP Calculus teacher who just went through the certification process with College Board, I can expand on the answer that Ben Crowell (i.e. user507) gave.  Not only is it pretty irresponsible to teach introductory calculus without this formal definition of the derivative, but such a course would not be allowed to call itself &amp;quot;AP Calculus AB&amp;quot; on student transcripts.  Teachers and principals agree to follow the posted syllabus even when it covers material that is not covered on the Calculus AB exam.&lt;/p&gt;&#xA;&lt;p&gt;But the formal limit definition of derivatives is material that definitely is covered on the Calculus AB exam.  For instance, a student may be asked to calculate &lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to  e}\frac{\ln x-1}{x-e}$$&lt;/span&gt; without a calculator and should be able to apply the difference quotient definition to see that this is &lt;span class=&quot;math-container&quot;&gt;$f'(e)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$f(x)=\ln x$&lt;/span&gt; and understand from there that the limit is &lt;span class=&quot;math-container&quot;&gt;$1/e$&lt;/span&gt; in a few seconds.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2022-03-16T14:58:23.947" LastActivityDate="2022-03-16T14:58:23.947" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19069" PostTypeId="1" CreationDate="2020-10-26T19:47:51.500" Score="7" ViewCount="4916" Body="&lt;p&gt;In this weird pandemic school year, I'm doubly interested in technology integration to help my virtual (high school) students as much as my in-person students.  I've been particularly eager to get that working with compass-and-straightedge constructions.  Obviously, students need a little familiarity with holding a compass and making accurate markings without injuring themselves, but learning and practicing the steps of the necessary constructions could be more convenient with virtual tools instead of real ones.&lt;/p&gt;&#xA;&lt;p&gt;My problem is that I really don't like how Geogebra and other tools (like IXL and Euclidea) manage constructions.  There are two main drawbacks that I can identify.  The first is that they all use compasses to make circles instead of arcs.  For instance, here is the traditional look of a successful angle bisection versus the same construction where full circles are drawn.  Frankly, I find it barely recognizable.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rQJAdm.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rQJAdm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7hHX3m.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7hHX3m.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The second flaw is that the &amp;quot;rusty compass&amp;quot; style of setting a compass size and then drawing a number of arcs with that radius is not native to the apps.  Rather, the &amp;quot;compass&amp;quot; setting on all of these tools require you to set a radius (via two points or a line segment) for each circle drawn.  Both of these drawbacks keep Geogebra, IXL, and Euclidea from effectively simulating what a real-world compass-and-straightedge construction looks and feels like.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know of an online tool that would do a better job?  I'm imagine an interface where the compass operations would be changing the radius and making a circle.  The latter would have you pick the center, draw a faint complete circle, let you choose the arc(s) of the circle you wanted, and then erase the complete circle when you were done.  I seem to recall that the Smart Board construction interface is somewhere along these lines, but I'm looking for a sandbox that students can play with on their chromebooks or smart phones.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2024-03-21T03:10:48.587" Title="Is there online geometric construction software that models physical constructions?" Tags="|geometry|technology-in-education|constructions|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19070" PostTypeId="2" ParentId="16702" CreationDate="2020-10-26T22:08:47.657" Score="3" Body="&lt;p&gt;I would encourage you not to teach your students that an equals sign represents a question or implies that an action should be taken.  Many students already struggle to interpret the equals sign as relational (indicating equality or balance) instead of operational (indicating an action to be taken).  &lt;a href=&quot;https://alibali.psych.wisc.edu/wp-content/uploads/sites/371/2018/02/KnuthAlibaliHattikudurMcNeilStephens2008.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Importance of Equal Sign Understanding in Middle Grades&lt;/a&gt; by Knuth et al. discusses the difference, and notes that students with a relational view of the equals sign perform better at equation-solving than students with an operational view.&lt;/p&gt;&#xA;&lt;p&gt;In addition, when students are presented with an identity, such as &lt;span class=&quot;math-container&quot;&gt;$\sin^2t+\cos^2t=1$&lt;/span&gt; which is true for all real numbers &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, do we want our students to see the equals sign and assume their goal is to attempt to solve the equation for &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;I, too, have been searching for ways to help my students understand what seem to be contradictory uses of &amp;quot;variables&amp;quot; in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Susanna S. Epp &lt;a href=&quot;https://condor.depaul.edu/%7Esepp/VariablesInMathEd.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;suggests that we define variables as placeholders&lt;/a&gt;, and this is what I do now.  I would argue that &lt;strong&gt;we absolutely should tell students that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; has different meanings in different contexts&lt;/strong&gt;, and that it is the context we are given and the questions we are asked about the expression, equality, number sentence, mathematical phrase, etc that give meaning to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  The context and the questions indicate how we should use or manipulate the expression, equality, number sentence, etc that we are given.  In one context the variable could be a placeholder for finitely many numbers that make the equality (or inequality) true.  In another context, like the Pythagorean identity above, the variable could be a placeholder for &lt;em&gt;any&lt;/em&gt; real number.  The variable could represent a fixed quantity, or could represent a quantity that changes.  Variables can also be placeholders for angles and points and matrices and a myriad of other mathematical objects that are not real numbers.&lt;/p&gt;&#xA;&lt;p&gt;The values that can be represented by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in the single equation &lt;span class=&quot;math-container&quot;&gt;$(x^2+5)(x+1)(x-.5)=0$&lt;/span&gt; depend on context; are we looking for all real numbers, all complex numbers, or perhaps all integer values that satisfy the equality?&lt;/p&gt;&#xA;&lt;p&gt;Questions and context such as&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Find all real numbers for which the equality is true.&#xA;&lt;li&gt;Show the equality holds for all real numbers.&#xA;&lt;li&gt;Consider the points &lt;span class=&quot;math-container&quot;&gt;$P(x,y)$&lt;/span&gt; with integers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2=4$&lt;/span&gt;.&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;will tell us what the placeholders are saving places for.&lt;/p&gt;&#xA;&lt;p&gt;I have seen attempts to differentiate between and define symbols used for variables, parameters, and constants.  I like &lt;a href=&quot;http://patthompson.net/ThompsonCalc/section_3_1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this treatment from Calculus: Newton, Leibniz, and Robinson Meet Technology&lt;/a&gt; which uses multiple cylinders to illustrate the difference between a constant (&lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;) which doesn't change between cylinders, a parameter (the radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;) which is fixed in a given cylinder, but changes with different cylinders, and a variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; representing the depth of water in the cylinder, which can take on different values even with a specific cylinder.  Again, context matters.&lt;/p&gt;&#xA;&lt;p&gt;(I really notice which students have a robust understanding of variable and symbol use when we start using &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; to represent a permutation, and not the ratio of circumference to diameter they are used to.)&lt;/p&gt;&#xA;" OwnerUserId="12605" LastEditorUserId="12605" LastEditDate="2020-10-26T22:16:55.387" LastActivityDate="2020-10-26T22:16:55.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19071" PostTypeId="2" ParentId="19069" CreationDate="2020-10-27T00:22:51.543" Score="4" Body="&lt;p&gt;This looks promising but it is not free:&#xA;&lt;a href=&quot;https://www.mathspad.co.uk/i2/construct.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathspad.co.uk&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/sjImk.png&quot; width=&quot;400&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;You can experiment with the tools without creating (or paying for)&#xA;an account:&#xA;&lt;hr /&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/5gyUW.png&quot; width=&quot;400&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-10-27T14:03:54.747" LastActivityDate="2020-10-27T14:03:54.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19072" PostTypeId="2" ParentId="16702" CreationDate="2020-10-27T02:21:02.143" Score="2" Body="&lt;p&gt;I think perhaps the definitive discussion of this topic was written by Zalman Usiskin in &amp;quot;Conceptions of School Algebra and Uses of Variables&amp;quot;, originally published as a chapter in the 1988 NCTM Yearbook, &lt;em&gt;&lt;a href=&quot;https://www.worldcat.org/title/ideas-of-algebra-k-12-1988-yearbook/oclc/17745806&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Ideas of Algebra, K-12&lt;/a&gt;&lt;/em&gt; (A. F. Coxford and A. P. Shulte, eds), and later reprinted in the 1999 book &lt;em&gt;&lt;a href=&quot;https://books.google.com/books/about/Algebraic_Thinking_Grades_K_12.html?id=R7c_NAAACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebraic Thinking, Grades K–12: Readings from NCTM’s School-Based Journals and Other Publications&lt;/a&gt;&lt;/em&gt; (B. Moses, ed).&lt;/p&gt;&#xA;&lt;p&gt;Usiskin opens with an enumeration of different cases:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Consider these equations, all of which have the same form -- the product of two numbers equals a third:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$A=LW$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$40=5x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin x = \cos x \cdot \tan x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$1 = n \cdot (1/n)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y = kx$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Each of these has a different feel. We usually call (1) a formula, (2) an equation (or open sentence) to solve, (3) an identity, (4) a property, and (5) an equation of a function of direct variation (not to be solved). These different names reflect different uses to which the idea of variable is put. In (1), &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; stand for the quantities area, length, and width and have the feel of knowns. In (2), we tend to think of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as unknown. In (3), &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is an argument of a function. Equation (4), unlike the others, generalizes an arithmetic pattern, and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; identifies an instance of the pattern. In (5), &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is again an argument of a function, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; the value, and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; a constant (or parameter, depending on how it is used). Only with (5) is there the feel of “variability,” from which the term &lt;em&gt;variable&lt;/em&gt; arose. Even so, no such feel is present if we think of that equation as representing the line with slope &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; containing the origin.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The article is far too long and thorough to successfully summarize here, but here is a brief summary that I wrote in my own book, &lt;a href=&quot;https://www.routledge.com/Secondary-Mathematics-for-Mathematicians-and-Educators-A-View-from-Above/Weiss/p/book/9781138294677&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Secondary Mathematics for Teachers and Mathematicians: A View from Above&lt;/em&gt;&lt;/a&gt; (see p. 114 for context):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Usiskin (1999) distinguishes between multiple distinct notions of&#xA;“variable” in school algebra. He describes the use of variables as&#xA;pattern generalizers, unknowns (to be solved for), arguments (to be&#xA;substituted in to functions), parameters, and referent-free symbols&#xA;(“marks on paper”). Usiskin also calls attention to “the question of&#xA;the role of functions and the timing of their introduction”:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is&#xA;clear that these two issues relate to the very purposes for teaching&#xA;and learning algebra, to the goals of algebra instruction, to the&#xA;conceptions we have of this body of subject matter. What is not as&#xA;obvious is that they relate to the ways in which variables are used...&#xA;My thesis is that the purposes we have for teaching algebra, the&#xA;conceptions we have of the subject, and the uses of variables are&#xA;inextricably related. Purposes for algebra are determined by, or are&#xA;related to, different conceptions of algebra, which correlate with&#xA;the different relative importance given to various uses of variables.&#xA;(pp.8–9, emphasis in original)&#xA;Usiskin identifies four distinct&#xA;“conceptions of Algebra”, each corresponding to a different use of&#xA;variables. Algebra is, in his analysis, (1) a generalization of&#xA;arithmetic, (2) the study of procedures for solving certain kinds of&#xA;problems, (3) the study of relationships among quantities, and (4) the&#xA;study of structures. When we (for example) study the graph of a&#xA;polynomial function or inquire after its zeros, we are primarily&#xA;attending to uses (2) and (3); in that context, a variable stands for&#xA;an unspecified element of a replacement set. On the other hand, when&#xA;we factor polynomials we are primarily focusing on (4). In that&#xA;context, variables stand for indeterminates; that is, “marks on paper”&#xA;that are to be manipulated without attending to what they stand for.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2020-10-27T15:24:14.390" LastActivityDate="2020-10-27T15:24:14.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19073" PostTypeId="2" ParentId="19069" CreationDate="2020-10-27T22:27:55.777" Score="4" Body="&lt;p&gt;Have you checked out Math Open Reference? The site has animations that demonstrate how to do the constructions, in the style that you indicated. It won't let students DO the construction online; instead it shows them the steps necessary to do the construction. It also includes printable worksheets for them to practice doing it by hand as well as providing an explanation (proof) of why it works.&lt;/p&gt;&#xA;&lt;p&gt;The link below is for the amination of the angle bisector construction.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.mathopenref.com/constbisectangle.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathopenref.com/constbisectangle.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1253" LastActivityDate="2020-10-27T22:27:55.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19074" PostTypeId="2" ParentId="18931" CreationDate="2020-10-28T18:06:19.967" Score="3" Body="&lt;p&gt;AOPS is more comprehensive and mechanical - they don't provide the proofs. The other book is more rigorous as it is proof oriented&lt;/p&gt;&#xA;" OwnerUserId="10335" LastActivityDate="2020-10-28T18:06:19.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19075" PostTypeId="1" CreationDate="2020-10-28T21:29:03.547" Score="10" ViewCount="269" Body="&lt;p&gt;I've been thinking a bit of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/19021/do-you-avoid-examples-or-test-questions-that-showcase-an-algorithmic-plugnchug#comment52509_19021&quot;&gt;this comment&lt;/a&gt; and the following one:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;(Look at all the complaints we get about kids who can't combine&#xA;fractions!)&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;Because it is not about you. Not about Robbie. But&#xA;about the student. (This is a classic confusion I see on this board,&#xA;confusing what is easy/interesting for an individual, versus what is&#xA;needed for the class.&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And I think I just disagree with it. Referencing the second point, rather than making it about me or the student,&amp;quot; it's about the class.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If, for example, I'm teaching a calculus course, then it's a calculus course - not one on pre-calculus (which seems to be algebraic manipulation), and it's definitely not a pre-algebra course (which seems to be arithmetic with variables). Not every student has to be in every course or that particular course. But once they're in that course, I feel they really should learn that topic, rather than spend class time doing things they should already know &lt;em&gt;because such things were supposed to be covered in the prerequisites&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;With a set amount of class time, you have to pick and choose what to cover. And if I choose to review rather than focus on the current material, then it most likely means other things will be rushed or simply not covered. This results in students possibly passing the course but then messing them up for a future course where the one I taught was a prerequisite. (In effect, I pass on a problem to another teacher)&lt;/p&gt;&#xA;&lt;p&gt;Ultimately, I'm pretty strict on prerequisites since I can foresee bigger problems if I'm lax on them. I'd much rather have a student do a course later than try to rush when they're not ready. In fact, before registering for a course, students can see the required prerequisites and the minimum grade needed. (But then, of course, there might be the issue of inflated grades)&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-10-30T16:35:46.290" Title="How strict are you with prerequisites?" Tags="|prerequisites|" AnswerCount="1" CommentCount="8" ClosedDate="2020-10-31T23:29:38.447" ContentLicense="CC BY-SA 4.0" />
  <row Id="19076" PostTypeId="1" CreationDate="2020-10-29T01:31:16.610" Score="6" ViewCount="348" Body="&lt;p&gt;I am teaching Calculus III this semester and a student signed up for this course after completed Calculus I and II in a different institution.&lt;/p&gt;&#xA;&lt;p&gt;I quickly realised that this student does not understand many basic concepts from Calculus I and II. For example, she has never seen Polar Coordinates, does not understand Taylor's series, and cannot compute some simple integrals such as &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\int e^{-x^2}x\mathrm{d}x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If she does not care about her grade in Calculus III and does not study for it, I probably will just give her an &amp;quot;F&amp;quot; and move on. But the reality is, she admits that she had a horrible Calculus teacher before and she did not learn much in her Calculus I and II class, but she got passing grades anyway and now she is in my Calculus III class. She works very hard and keeps talking with me and e-mailing me about her questions and difficulties. While I should be helpful to my students and I am trying to be, I also feel I am spending too much of my time on this single student.&lt;/p&gt;&#xA;&lt;p&gt;When she seeks my help with concepts from Calculus I and II, I feel frustated as it seems I am paying the price for the incompetence of her previous calculus teacher. Should I tell her to retake, or at least sit in some Calculus I and II class? In general, is it a teacher's resonsibility to help a student with his/her problem from a prerequisite course?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2020-10-30T02:00:07.433" Title="Am I responsible to help a student who does not understand/know some prerequisites of a course?" Tags="|calculus|struggling-students|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19077" PostTypeId="1" CreationDate="2020-10-29T04:49:55.200" Score="0" ViewCount="567" Body="&lt;p&gt;What are the estimated hours for teaching AP Calculus AB for students aiming for a 5?&#xA;Similar question, how many hours of practice will the student need to put it.&lt;/p&gt;&#xA;&lt;p&gt;Some Clarifications based on comments:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I am asking for teaching (face to face) hours.&#xA;Not prep time to be put in by the teacher&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am  not teaching in a school. So I get to decide the time.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;(In terms of maths level, you can assume reasonable, but not high levels of development).&#xA;If they don't know certain foundation skills (trigo, log/exp, func), I am not counting those in the teaching hours for the AP - that needs to be taught separately.&lt;/p&gt;&#xA;" OwnerUserId="12570" LastEditorUserId="12570" LastEditDate="2020-11-01T11:08:16.093" LastActivityDate="2020-11-01T11:08:16.093" Title="Teaching Hours for AP Calculus AB" Tags="|calculus|teacher-preparation|" AnswerCount="1" CommentCount="4" ClosedDate="2020-11-01T12:58:00.920" ContentLicense="CC BY-SA 4.0" />
  <row Id="19078" PostTypeId="2" ParentId="19077" CreationDate="2020-10-29T10:52:13.610" Score="2" Body="&lt;p&gt;I can't answer it in any meaningful way since it varies so much, but what I can offer is what the makers of the AP Calc AB test say:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://apcentral.collegeboard.org/pdf/ap-calculus-ab-bc-course-and-exam-description-0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://apcentral.collegeboard.org/pdf/ap-calculus-ab-bc-course-and-exam-description-0.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;On pages 27-29, it has a list of topics and pacing&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Unit (class periods of 45-min)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;1 (22-23)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;2 (13-14)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;3 (10-11)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;4 (10-11)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;5 (15-16)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;6 (18-20)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;7 (8-9)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;8 (19-20)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So it's around 120 units of 45 min, so 90hrs of content? (But it seems rather long &lt;em&gt;or leisurely-paced&lt;/em&gt; to me?)&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-10-29T10:52:13.610" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19079" PostTypeId="2" ParentId="19076" CreationDate="2020-10-29T11:00:40.193" Score="0" Body="&lt;p&gt;To answer the title, I'd say you're not responsible since Calc I and II should be prerequisites for Calc III and so it's expected that she know that material. Not sure how she has never seen Polar or Taylor (sounds like her prior classes didn't finish in time?)&lt;/p&gt;&#xA;&lt;p&gt;I'd probably say &amp;quot;F&amp;quot; and retake, and/or sit in Calc I and II classes. The reason I say this is because I only foresee this getting worse. If she doesn't have the foundations of single-variable, then a lot of the integration and differentiation that is taken for granted in going to be too much. What happens when you get to cylindrical and spherical Coordinates, or double integrals, change of variables, etc.?&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2020-10-29T11:00:40.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19080" PostTypeId="2" ParentId="19076" CreationDate="2020-10-29T13:22:51.600" Score="6" Body="&lt;p&gt;It surprises me that you only have one student with this problem. I suppose you must teach at a school with highly selective admissions and high academic standards. I teach at a community college, and I would say that among our students who have taken first-year calculus, if the worst problems a student has are the ones you describe, then that student would be in the top 25%. Roughly half of our students have much more serious problems, e.g., they aren't clear on the difference between an integral and a derivative, or they notate the derivative of &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x^2 d/dx$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It's good that your student understands realistically that she is unprepared. It's your job to help her, but the help should be helping her to plan a crash program of self-study to learn the material she didn't learn before. I would send her an email outlining to her what this would look like: --&lt;/p&gt;&#xA;&lt;p&gt;Allocate about 2 hours per day, 7 days a week, to this for the rest of the semester. Go through your first-year calculus textbook and figure out which chapters you don't know. For each chapter, read it, then go back through and read it again, typing up notes on a computer (so that they're editable). These should be about half a page per chapter. Try to work the problems at the end of the chapter. Pick problems for which you have a way to check your own answer. Start with the easiest ones, and work your way up to the hardest ones. As your understanding improves, update your notes. If you get stuck, use the resources our campus provides. One of those resources is me, but I can't do the bulk of this, so you should mainly be using other resources: TA, campus math tutoring center, etc.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-10-29T13:45:45.930" LastActivityDate="2020-10-29T13:45:45.930" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19081" PostTypeId="2" ParentId="19076" CreationDate="2020-10-29T20:28:56.613" Score="2" Body="&lt;p&gt;Here is some complementary advice to that given by Ben Crowell.&lt;/p&gt;&#xA;&lt;p&gt;To prioritize the remediation effort, my advice to the student would be to start with the notes she takes in your lecture. Notes taken on the example problems should include both the formulaic steps of the problem but also some written terminology about the technique, &lt;strong&gt;especially if the justification for the manipulation is unclear.&lt;/strong&gt; Then, after the class, she can review the notes, use the terminology to search in the index for a section which explains the technique, and proceed as Ben suggested. If the section does not make sense, then go backward one section at a time until the discussion is comprehensible. After that's done, she could take a stab at the homework and follow up with you or another resource (TA, tutoring center, supplemental instruction, etc.) for an explanation to drive more index-driven research.&lt;/p&gt;&#xA;&lt;p&gt;This approach balances the need to re-learn the old material with the reality that your course is still moving along and takes advantage of work that's already being done (sitting in your lectures, taking notes, doing homework) to drive the prioritization of an otherwise daunting task.&lt;/p&gt;&#xA;&lt;p&gt;You may be doing this already, but one thing you could do to facilitate this would be to just mention the name of an old technique even if you do not write out all the details (e.g., &amp;quot;we evaluate this integral using trig substitution&amp;quot; and then just show the final answer). Name dropping the technique probably would not take much effort but would give confused students something to look up in the index.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastEditorUserId="1230" LastEditDate="2020-10-30T02:00:07.433" LastActivityDate="2020-10-30T02:00:07.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19082" PostTypeId="2" ParentId="19075" CreationDate="2020-10-30T16:35:46.290" Score="8" Body="&lt;p&gt;For several decades, this issue has plagued me, at all levels, from calculus to grad-level courses.&lt;/p&gt;&#xA;&lt;p&gt;First, my conclusion by this point is that very few people truly &amp;quot;have&amp;quot; the prerequisites for any course, even with an excellent performance in the prerequisites, if only because they do not assiduously review and practice in the interim. People forget lots of things over the summer, or during winter break.&lt;/p&gt;&#xA;&lt;p&gt;And my experience is that &lt;em&gt;explicit&lt;/em&gt; review is most often very inefficient, especially with lower-division undergrads, because they believe that they don't need the review (due to their good previous grade), and ignore what the instructor says &amp;quot;in review&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So, finally, I do not really-really assume that everyone has mastery of the prerequisites, but/and do many by-the-way reviews, or, really, as in @SueVanHattum's comment, I just spell out lots of details &amp;quot;from the prerequisites&amp;quot;, rather than assuming without comment that everyone remembers.&lt;/p&gt;&#xA;&lt;p&gt;I think the point is that the student populations that show up for courses mostly must be dealt with as-is, and it is unworkable to &amp;quot;hold to standards&amp;quot; that are not actually manifest in the rest of the system. The popular mythology in math classes of acting as though everyone has perfect recall of all their math classes so far is destructive.&lt;/p&gt;&#xA;&lt;p&gt;There's also the far-more-true motto that in many endeavors we don't really come to understand topic X until we take the course for which X is a prerequisite. :)&lt;/p&gt;&#xA;&lt;p&gt;So, no, I am not &amp;quot;strict in enforcing prerequisites&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2020-10-30T16:35:46.290" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19083" PostTypeId="1" CreationDate="2020-10-30T17:55:02.803" Score="4" ViewCount="642" Body="&lt;p&gt;Long ago, when I was the student, I encountered a bunch of stories that we were asked to illustrate with graphs. I seem to remember that it was a set of around 10 examples. One involved some two quantities, like height over time, for a child in a Ferris wheel; the graph was properly sinusoidal, but many of us had a lot of trouble accepting that. A second was a story about a child walking to school, returning home to get something left behind, and then proceeding on his way, for which some students confused the curve on the graph with a path on a map.&lt;/p&gt;&#xA;&lt;p&gt;Anyone know where I can get this activity or something of the sort? If it's amusing, that's a big plus for me.&lt;/p&gt;&#xA;" OwnerUserId="8258" LastActivityDate="2020-10-31T01:48:40.213" Title="Relationships amusingly difficult to graph" Tags="|examples|graphing|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19084" PostTypeId="2" ParentId="19083" CreationDate="2020-10-31T01:48:40.213" Score="3" Body="&lt;p&gt;Here's an example of @SueVanHattum's graphing stories.&#xA;&lt;a href=&quot;https://mathbitsnotebook.com/JuniorMath/Graphing/GRStoryGraphPractice.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &#xA;&lt;img src=&quot;https://i.stack.imgur.com/HWLyV.png&quot; width=&quot;500&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-10-31T01:48:40.213" CommentCount="0" CommunityOwnedDate="2020-10-31T01:48:40.213" ContentLicense="CC BY-SA 4.0" />
  <row Id="19085" PostTypeId="1" CreationDate="2020-10-31T14:04:51.943" Score="3" ViewCount="167" Body="&lt;p&gt;What is the clearest way to indicate various operations along the following lines:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x) = 3x$&lt;/span&gt;: The function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; multiplies its input by 3.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$g(x) = x-5$&lt;/span&gt;: The function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; decreases its input by 5.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$h(x) = 2^x$&lt;/span&gt;: The function &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;...Raises 2 to the power of its input? Exponentiates its input on a base of 2? Takes 2 to its input?&lt;/p&gt;&#xA;&lt;p&gt;In the first two cases, I made a choice of whether to refer to the operation's name (&amp;quot;multiply&amp;quot;) or a (hopefully) plain language action being performed (&amp;quot;decrease&amp;quot;). For the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, I could have said that it &lt;em&gt;triples its input&lt;/em&gt; or &lt;em&gt;increases its input by 200%&lt;/em&gt;, but these don't seem to generalize well for the purposes of communicating, and the latter is almost never obvious to students. Similarly, I could have said that the function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; subtracts 5 from its input, but I am not convinced this is any simpler than &lt;em&gt;&amp;quot;decreases its input by&amp;quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Is there a simplest verb phrase for exponents? What do you think is clearest for students if I want to maintain the form &amp;quot;The function &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; ____________&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2020-10-31T14:13:48.157" LastActivityDate="2020-11-08T01:30:42.893" Title="Clearest verb phrases for operations" Tags="|language-use|exponential|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19087" PostTypeId="2" ParentId="18930" CreationDate="2020-11-01T19:40:02.727" Score="2" Body="&lt;p&gt;You may want to lookup&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Donnelly, C., &amp;amp; Embrechts, P. (2010). The devil is in the tails: actuarial mathematics and the subprime mortgage crisis. Astin Bulletin, 40(1), 1-33.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It confronts the story of the use of the Gaussian Copula in the financial industry. The Gaussian Copula does not allow for co-variance of the included variables at the extremes of the support (in technical terms, the Gaussian Copula does not exhibit &amp;quot;tail-dependence&amp;quot;). This simply means that if we use the Gaussian Copula to model the dependence structure between variables, say, between the value of financial securities, then &lt;em&gt;by construction&lt;/em&gt; we have assumed that these values will not co-vary together at the extremes: in other words they will not go down all together... but which is what happened during the 2008 financial crisis.&lt;/p&gt;&#xA;&lt;p&gt;Of course this did not &lt;em&gt;cause&lt;/em&gt; the financial crisis of 2008, but it certainly did not contribute any warning sign either to the professionals and the practitioners. So we can fantasize that maybe, if said warning signs were given by the financial model, some mitigating actions may had been taken.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2020-11-01T19:40:02.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19088" PostTypeId="1" CreationDate="2020-11-02T16:59:51.717" Score="6" ViewCount="584" Body="&lt;p&gt;I am supposed to hold tutorial sessions for an undergraduate course in calculus. But the software provided by the university is not good at all. I saw the question suggesting to use Ziteborad but it is not what I am looking for.&lt;/p&gt;&#xA;&lt;p&gt;I am searching for some real time Latex compiler which would allow me to write what I am explaining on the fly without waiting to compile each time I add a sentence or a formula. Stack Exchange has an incorporated MathJax compiler whose functionality is similar to what I would like. Is there not a program or browser page which visualizes what I write in Latex code on a big page easy to present via screen sharing?&lt;/p&gt;&#xA;" OwnerUserId="14894" LastEditorUserId="376" LastEditDate="2020-11-02T18:44:05.537" LastActivityDate="2020-11-25T07:51:54.117" Title="Writing mathematics in real time for lectures using Latex" Tags="|technology-in-education|software|" AnswerCount="6" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19089" PostTypeId="1" AcceptedAnswerId="19090" CreationDate="2020-11-02T17:02:42.657" Score="9" ViewCount="2284" Body="&lt;p&gt;I'm writing up an activity where students are looking at pathlengths that follow along gridlines.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/miFJ5.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/miFJ5.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AqVSd.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AqVSd.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Is there a word or phrase that is commonly used to describe those paths, but doesn't include diagonals?&lt;/p&gt;&#xA;&lt;p&gt;I'll probably call them 'grid line paths', but if there is a common term, I'd like to align with that.&lt;/p&gt;&#xA;" OwnerUserId="8642" LastEditorUserId="376" LastEditDate="2020-11-02T20:31:02.193" LastActivityDate="2020-11-10T02:48:14.293" Title="Is there a name for paths that follow gridlines?" Tags="|geometry|terminology|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19090" PostTypeId="2" ParentId="19089" CreationDate="2020-11-02T17:04:03.487" Score="22" Body="&lt;p&gt;Generally, this math falls under the scope of what is commonly called &lt;em&gt;Taxicab Geometry&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I would use &lt;em&gt;taxicab path&lt;/em&gt; as a noun to describe the specific paths illustrated in the original question; whereas &lt;em&gt;taxicab geometry&lt;/em&gt; would be a term I'd use for the subset of mathematics covering these types of scenarios.&lt;/p&gt;&#xA;" OwnerUserId="267" LastEditorUserId="267" LastEditDate="2020-11-03T04:37:24.867" LastActivityDate="2020-11-03T04:37:24.867" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19091" PostTypeId="2" ParentId="19089" CreationDate="2020-11-02T17:58:31.933" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwigvsn1tuTsAhWaIDQIHa4vDV4QFjABegQIBBAC&amp;amp;url=http%3A%2F%2Fzimmer.fresnostate.edu%2F%7Ekhangt%2FRook-Queen.pdf&amp;amp;usg=AOvVaw14hacO4eXIoPt1_vuhLX33&quot; rel=&quot;nofollow noreferrer&quot;&gt;This paper&lt;/a&gt; calls a horizontal/vertical path a &lt;em&gt;rook path&lt;/em&gt;, for the movement of a rook on a chess board.&lt;/p&gt;&#xA;&lt;p&gt;If the paths connect the lower-left corner to the upper-right corner, this would be a &lt;a href=&quot;https://en.wikipedia.org/wiki/Lattice_path&quot; rel=&quot;nofollow noreferrer&quot;&gt;North-East Lattice Path&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Edit: If the path didn't need to go through points with integer coordinates, I would submit the name &amp;quot;Cardinal Paths&amp;quot; in reference to the cardinal directions of North, East, South and West.&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2020-11-04T22:01:52.627" LastActivityDate="2020-11-04T22:01:52.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19092" PostTypeId="1" CreationDate="2020-11-02T21:01:43.207" Score="1" ViewCount="658" Body="&lt;p&gt;Do students learn implicit equations (such as &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2-r^2 = 0$&lt;/span&gt;)&#xA;and parametric equations (e.g., &lt;span class=&quot;math-container&quot;&gt;$x=a t^2,\;y= 2 a t$&lt;/span&gt;)&#xA;in a first course in algebra,&#xA;which in the US would be early high school, maybe 9th grade?&#xA;Or not until pre-calculus or calculus, late high school or early college?&#xA;Perhaps parametric equations are not part of standard curricula?&lt;/p&gt;&#xA;&lt;p&gt;I'm trying to gauge how much it is reasonable for me to assume when writing for&#xA;high-school students.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="2083" LastEditDate="2020-11-20T09:02:30.497" LastActivityDate="2020-11-20T09:02:30.497" Title="When are students taught implicit and parametric representations of curves?" Tags="|secondary-education|precalculus|usa|analytic-geometry|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19093" PostTypeId="2" ParentId="19092" CreationDate="2020-11-02T21:13:14.333" Score="1" Body="&lt;p&gt;The community college where I teach puts an introduction (lines, circles, ellipses, parabolas) in precalculus, and then covers it again in vector calculus. So, this would be first year and again in second year of undergrad.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-11-02T21:13:14.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19094" PostTypeId="2" ParentId="19088" CreationDate="2020-11-02T22:59:37.887" Score="5" Body="&lt;p&gt;You might find &lt;a href=&quot;https://jupyter.org/&quot; rel=&quot;noreferrer&quot;&gt;jupyter notebooks&lt;/a&gt; useful, especially if you use either the &lt;a href=&quot;https://www.sympy.org/en/index.html&quot; rel=&quot;noreferrer&quot;&gt;Sympy&lt;/a&gt; or &lt;a href=&quot;https://www.sagemath.org/index.html&quot; rel=&quot;noreferrer&quot;&gt;Sage&lt;/a&gt; computer algebra systems. These notebooks allow you to enter mathjax enabled markdown on the fly; which can also be generated as the output of sage and sympy. For example, sympy shows a live example session on their site:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CigxM.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CigxM.png&quot; alt=&quot;example session&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Graphs are also straightforward, and can be quick to generate with some practice.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2020-11-02T22:59:37.887" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19096" PostTypeId="2" ParentId="19088" CreationDate="2020-11-03T13:55:03.327" Score="8" Body="&lt;p&gt;I believe Overleaf would be your best bet. I am currently taking a class right now where the lecturer writes up the contents of the lecture using overleaf and it's very feasible. Even though you don't want to wait around for it compile, Overleaf renders documents pretty fast, and can render without you needing to press the command to do so. I'd say there's also the added benefit that your students can see the latex code as it's being written so they can pick up how to use latex if they don't know it already.&lt;/p&gt;&#xA;" OwnerUserId="14883" LastActivityDate="2020-11-03T13:55:03.327" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19097" PostTypeId="2" ParentId="19088" CreationDate="2020-11-03T14:20:42.313" Score="2" Body="&lt;p&gt;I use &lt;a href=&quot;https://www.lyx.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lyx&lt;/a&gt; as my LaTeX &amp;quot;word processor&amp;quot;, and it's really good at real-time interpretation.  You can build your statements using a GUI sort of like Microsoft Equations back in the day or you can just type straight LaTeX.  The creators insist that it isn't WYSIWYG, but the distinction between WYS on the screen and WYG out of a printer is insignificant from my perspective.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-11-03T14:20:42.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19098" PostTypeId="2" ParentId="19089" CreationDate="2020-11-03T14:25:11.370" Score="10" Body="&lt;p&gt;In the field of micro/nano-lithography, such geometry would be called &lt;em&gt;Manhattan&lt;/em&gt; geometry; containing only two directions of edges orthogonal to each other. If diagonals were included, it would be classified as having &lt;em&gt;skew edges&lt;/em&gt;. Finally, if it were truly freeform, it would be called &lt;em&gt;curvilinear&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;a href=&quot;https://doi.org/10.1117/12.2243030&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1117/12.2243030&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14904" LastActivityDate="2020-11-03T14:25:11.370" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19099" PostTypeId="2" ParentId="19092" CreationDate="2020-11-03T15:07:43.830" Score="3" Body="&lt;p&gt;I am in a US high school.&#xA;The implicit equations look like &amp;quot;conics&amp;quot;, and are part of the junior (3rd year) class typically called Trigonometry with Algebra. Parametrics are part of the course labeled pre-calculus, which comprises either juniors who will take calculus as seniors, or as seniors in preparation of college calculus.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-11-03T16:12:36.217" LastActivityDate="2020-11-03T16:12:36.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19100" PostTypeId="2" ParentId="10697" CreationDate="2020-11-03T15:07:52.627" Score="2" Body="&lt;p&gt;Since it has not been mentioned by name in the other answers, I'll say that once we model the x and y coordinates of a point on the unit circle as &lt;span class=&quot;math-container&quot;&gt;$\left(\cos(\theta),\sin(\theta)\right)$&lt;/span&gt; (marked by a ray through the origin), &lt;strong&gt;I define the tangent as the slope&lt;/strong&gt; of that ray. Since they've found the slope of a line so many times before, my students end up drawing a pretty reasonable graph of the function &lt;span class=&quot;math-container&quot;&gt;$y=\tan(\theta)$&lt;/span&gt; before knowing the &lt;span class=&quot;math-container&quot;&gt;$\frac{\text{opposite}}{\text{adjacent}}$&lt;/span&gt; definition. [But they end up figuring out &lt;span class=&quot;math-container&quot;&gt;$\frac{\text{sine}}{\text{cosine}}$&lt;/span&gt; based on their slope definition: &lt;span class=&quot;math-container&quot;&gt;$\frac{\text{rise}}{\text{run}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Of course, this works because the text we use does everything with the unit circle&lt;span class=&quot;math-container&quot;&gt;${}^*$&lt;/span&gt; for a chapter before covering triangles.&lt;/p&gt;&#xA;&lt;p&gt;[* And I model the circle by pulling the front wheel off my bicycle and putting a sticky-note on the edge of the tire. Spinning the wheel and holding it so it faces the students and is either vertical or horizontal is my go-to for seeing the up-and-down or back-and-forth motion of the sticky-note point.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2020-11-03T17:10:17.217" LastActivityDate="2020-11-03T17:10:17.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19101" PostTypeId="2" ParentId="19088" CreationDate="2020-11-03T16:25:26.497" Score="2" Body="&lt;p&gt;I have used &lt;a href=&quot;https://www.onenote.com/Download&quot; rel=&quot;nofollow noreferrer&quot;&gt;Microsoft OneNote 2016&lt;/a&gt; (plus screensharing software) for online tutorial sessions in number theory during the past year. It has worked very well and I will continue to use it.&lt;/p&gt;&#xA;&lt;p&gt;You can enter plain text just like in Microsoft Word and switch to Math mode by pressing a shortcut ([&amp;quot;Alt&amp;quot; + &amp;quot;+&amp;quot;] in my case (German), but it might also be [&amp;quot;Alt&amp;quot; + &amp;quot;=&amp;quot;] or something else). In Math mode, you can essentially write LaTeX code which is transformed into mathematics upon pressing space. (This works similarly to autocorrection.) Some experience is required to use this swiftly and there are some differences to LaTeX (e.g. round brackets are used for logical grouping, not curly brackets). It is also not as powerful as LaTeX, but you can do all kinds of basic mathematics like greek letters, basic relational symbols, integrals, summation symbols, matrices etc. You can also add, remove and change math commands (go to Options&amp;gt;Proofing&amp;gt;Autocorrect options). For example, I added lots of LaTeX commands without backslashes for myself, so I can write &amp;quot;alpha&amp;quot; instead of &amp;quot;\alpha&amp;quot; to get &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;An example from my tutorial sessions:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DMcOY.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DMcOY.png&quot; alt=&quot;Example&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The formatting is not as great as in LaTeX, obviously.&lt;/p&gt;&#xA;&lt;p&gt;Further remarks:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The same math functionality is available in Microsoft Word, but OneNote is free and supports infinitely large pages. Further, if you have a touchscreen device with a pen, you can also make drawings next to your text as needed.&lt;/li&gt;&#xA;&lt;li&gt;Make sure you use OneNote 2016, not OneNote for Windows 10. These are two similar versions of OneNote (both are freely available), but OneNote 2016 is better suited for this usage.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14900" LastActivityDate="2020-11-03T16:25:26.497" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19102" PostTypeId="2" ParentId="10832" CreationDate="2020-11-04T03:25:04.737" Score="2" Body="&lt;p&gt;The word &lt;em&gt;imply&lt;/em&gt; has significantly different meanings in ordinary speech and mathematics, and has caused me some confusion in the past.&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://www.lexico.com/definition/imply&quot; rel=&quot;nofollow noreferrer&quot;&gt;this dictionary entry&lt;/a&gt;, the word indicates a &lt;em&gt;suggestion&lt;/em&gt; (rather than a definite conclusion) that some statement is true. This gives rise to expressions that cover different degrees of strength in a connection between statements - 'weakly imply', 'imply', 'strongly imply' etc.&lt;/p&gt;&#xA;&lt;p&gt;To my understanding, this usage of 'imply' only relates (in a way) to the &lt;em&gt;likelihood&lt;/em&gt; of something. In stark contrast, the &lt;a href=&quot;https://math.stackexchange.com/questions/2531122/what-does-imply-mean-in-maths&quot;&gt;usual math definition&lt;/a&gt; is of a &lt;em&gt;conclusive&lt;/em&gt; nature - A implies B means that whenever A is true, B is &lt;strong&gt;definitely&lt;/strong&gt; true. No clauses.&lt;/p&gt;&#xA;&lt;p&gt;In fact, the above dictionary entry goes on to state:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Imply and infer do not mean the same thing and should not be used interchangeably: see&#xA;&lt;a href=&quot;https://www.lexico.com/definition/infer&quot; rel=&quot;nofollow noreferrer&quot;&gt;infer&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;...and the provided definition of 'infer' is somewhat closer to the mathematical sense in which 'imply' is normally used.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.quora.com/Can-correlation-ever-imply-causation-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's an example&lt;/a&gt; of a Quora page (about correlation implying causation) in which the two meanings come into conflict. There's a wide variety of answers saying 'no', 'maybe' and 'yes', and I'm sure some of the misunderstanding stems from the dual meanings of 'imply'.&lt;/p&gt;&#xA;" OwnerUserId="14907" LastEditorUserId="14907" LastEditDate="2020-11-04T03:38:06.767" LastActivityDate="2020-11-04T03:38:06.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19103" PostTypeId="1" CreationDate="2020-11-04T04:45:36.643" Score="7" ViewCount="545" Body="&lt;p&gt;I'm currently working on some precalculus packages for students who need review. For inspiration, I'm looking at some prealgebra books and I'm wondering why isolating for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is taught before factoring.&lt;/p&gt;&#xA;&lt;p&gt;I agree that there are ways to solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; when it comes to linear and quadratic, as seen below:&lt;/p&gt;&#xA;&lt;p&gt;Solving for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for linear:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;2x+4 &amp;amp; =10 \\&#xA;2x &amp;amp; = 6 \\&#xA;x &amp;amp; = 3&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Solving for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for quadratic:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;x^2+5x+6 &amp;amp; =0 \\&#xA;\left(x+\frac{5}{2}\right)^2 -\frac{25}{4} + \frac{24}{4} &amp;amp; = 0 \\&#xA;\left(x+\frac{5}{2}\right)^2 &amp;amp; = \frac{1}{4} \\&#xA;x + \frac{5}{2} = &amp;amp; \pm \frac{1}{2} &#xA;\end{align*}&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x_1=-3, x_2=-2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But then, for cubic and quartic, it would be impractical.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, if it was solving by factoring from the start, it seems easier to generalize to higher degree polynomials.&lt;/p&gt;&#xA;&lt;p&gt;Factoring for linear:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;2x+4 &amp;amp; =10 \\&#xA;2x-6 &amp;amp; = 0 \\&#xA;2\left(x-3\right) &amp;amp; =0&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x=3$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Factoring for quadratic:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;x^2+5x+6 &amp;amp; =0 \\&#xA;\left(x+3\right)\left(x+2\right) &amp;amp; = 0&#xA;\end{align*}&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x_1=-3, x_2=-2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since the factoring version seems to be more flexible (since it could also apply to trigonometric functions), then why isn't that one taught first?&lt;/p&gt;&#xA;&lt;p&gt;For my goals, since it's review, I'd probably introduce the two methods alongside each other but then state that factoring is probably the preferred method when it comes to higher courses.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastActivityDate="2023-06-26T02:54:05.213" Title="Why is isolating for $x$ taught before factoring?" Tags="|algebra|precalculus|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19104" PostTypeId="2" ParentId="19103" CreationDate="2020-11-04T05:53:23.530" Score="7" Body="&lt;p&gt;Pedagogically speaking, factoring is a lot less intuitive than 'simple' rearrangement. For your example we have that, &lt;span class=&quot;math-container&quot;&gt;$$ 2x +4 =10. $$&lt;/span&gt; When first teaching Algebra, there are many nice and neat tricks/visualizations to understand the process of unraveling the equation to solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. A classic analogy is to see the equation as a kind of seesaw that's balanced and you need to do steps so that the see-saw is always balanced.&lt;/p&gt;&#xA;&lt;p&gt;A (perhaps insensitive) way that I was taught was to role play as a greedy family lawyer who had to 'divorce' &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; from it's current relationship with the numbers it is with by doing actions that oppose what holds their relationship together in the first place.&lt;/p&gt;&#xA;&lt;p&gt;So there are a lot of ways to explain this to someone who's first learning about something pretty abstract. In comparison, to get the solution by factoring does not have any nice analogy that can be used. In essence, we need to ask: what value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; must be satisfied such that the RHS is zero? Which to an untrained mind is an extra layer of abstraction that doesn't need to be added until they're already comfortable with manipulating equations.&lt;/p&gt;&#xA;&lt;p&gt;Further, it's even harder with the factoring method because the solution splits off into two 'branches'. Which is once again not immediately obvious to students why that should be the case. This problem can be swept under the rug with &lt;span class=&quot;math-container&quot;&gt;$\pm$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, I guess there are things that need to be untrained with the method of straight manipulation. So I agree that a greater class of problems can be solved a lot more straightforwardly by factoring, especially when working with &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Regardless, I think that usually people have a hard time learning Algebra for the first time and so we need to make this initial step up into abstraction as easy as possible.&lt;/p&gt;&#xA;" OwnerUserId="14883" LastEditorUserId="14883" LastEditDate="2020-11-05T00:07:28.400" LastActivityDate="2020-11-05T00:07:28.400" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19105" PostTypeId="2" ParentId="10832" CreationDate="2020-11-04T15:35:33.920" Score="0" Body="&lt;p&gt;In ordinary speech, the adjective &lt;strong&gt;random&lt;/strong&gt; describes something that occurs without a definite pattern or predictability.&lt;/p&gt;&#xA;&lt;p&gt;In mathematics, its meaning is, in a sense, inverted: random/stochastic processes &lt;em&gt;do&lt;/em&gt; have underlying structures over time/space and &lt;em&gt;can&lt;/em&gt; be used for forecasting.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/3893774/21813&quot;&gt;Two aspects of randomness&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="372" LastActivityDate="2020-11-04T15:35:33.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19106" PostTypeId="2" ParentId="19089" CreationDate="2020-11-04T18:08:05.490" Score="3" Body="&lt;p&gt;The term that I have always used for this is “&lt;em&gt;rectilinear&lt;/em&gt;”, that is, following the lines of a rectangular grid.&lt;/p&gt;&#xA;&lt;p&gt;Most of the definitions of rectilinear found on the internet are commonly non-mathematical and IMHO, not correct (they essentially define it the same as “linear” which is clearly not what it means in math).  However the Wikipedia article on &lt;a href=&quot;https://en.wikipedia.org/wiki/Rectilinear_polygon&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rectilinear Polygons&lt;/a&gt; provides the best explanations.&lt;/p&gt;&#xA;" OwnerUserId="556" LastEditorUserId="556" LastEditDate="2020-11-04T18:16:00.803" LastActivityDate="2020-11-04T18:16:00.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19107" PostTypeId="2" ParentId="19089" CreationDate="2020-11-04T19:57:08.577" Score="8" Body="&lt;p&gt;&lt;strong&gt;These are pretty much universally called lattice paths&lt;/strong&gt;, which Will Orrick comments on the question.&lt;/p&gt;&#xA;&lt;p&gt;A quick search on the arXiv reveals &lt;a href=&quot;https://arxiv.org/pdf/2007.05061&quot; rel=&quot;noreferrer&quot;&gt;a&lt;/a&gt; &lt;a href=&quot;https://arxiv.org/pdf/2008.02240&quot; rel=&quot;noreferrer&quot;&gt;large&lt;/a&gt; &lt;a href=&quot;https://arxiv.org/pdf/2009.02644&quot; rel=&quot;noreferrer&quot;&gt;number&lt;/a&gt; &lt;a href=&quot;https://arxiv.org/pdf/2009.04900&quot; rel=&quot;noreferrer&quot;&gt;of&lt;/a&gt; &lt;a href=&quot;https://arxiv.org/pdf/2010.09963.pdf&quot; rel=&quot;noreferrer&quot;&gt;papers&lt;/a&gt; from combinatorics and computer science using the term in this way.  (The main competition is from papers about path integrals on lattices, which gets unfortunately concatenated to &amp;quot;lattice path integral.&amp;quot;)  For examples from published literature, &lt;a href=&quot;https://scholar.google.com/scholar?hl=en&amp;amp;as_sdt=0%2C5&amp;amp;q=lattice+paths&amp;amp;btnG=&amp;amp;oq=lattice+path&quot; rel=&quot;noreferrer&quot;&gt;this Google Scholar search&lt;/a&gt; is a good start.&lt;/p&gt;&#xA;&lt;p&gt;The jargon is not exactly localized to combinatorics and comp sci either; &lt;a href=&quot;https://arxiv.org/pdf/1904.01495&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, for example, is a probability paper using the same terminology.&lt;/p&gt;&#xA;&lt;p&gt;Finally, Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Lattice_path&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; on lattice paths has picture with a path moving diagonally; this generalization is an unusual use of the term.&lt;/p&gt;&#xA;" OwnerUserId="14910" LastActivityDate="2020-11-04T19:57:08.577" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19109" PostTypeId="2" ParentId="19103" CreationDate="2020-11-05T05:35:44.480" Score="5" Body="&lt;p&gt;This is kind of an interesting question. Three observations spring to mind.&lt;/p&gt;&#xA;&lt;p&gt;First, you're really not going to short-circuit the need to present the basic inverses method of solving (the addition and multiplication properties). Your factoring examples have skipped presenting those steps, but they're still in there, e.g.:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2(x-3) = 0 \implies x - 3 = 0 \implies x = 3$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You definitely need to explain and justify that middle step (in which you'll add 3 to both sides of the equation). I mean: sometimes test-preparation materials can cheat on this, maybe just teach by rote that if you see &lt;span class=&quot;math-container&quot;&gt;$(x-3)$&lt;/span&gt; as a factor you'll get a solution of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, but that would be invalid mathematics and students would suffer later on with that kind of &amp;quot;faith based math&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So you still need to deliver the basic inverses technique even to finish off your examples of solving by factoring. If you expect to also teach the method of factoring and the zero-product property, then at that point it seems like an unnecessary detour just to solve a linear equation.&lt;/p&gt;&#xA;&lt;p&gt;Second, you can be sort of tricked by lots of &amp;quot;nice&amp;quot; examples that are being given to make life easy for the beginning students. Sure, many starting examples will have the constant term divisible by the linear coefficient (i.e.: factorable &lt;em&gt;in integers&lt;/em&gt;, which is another unstated assumption). But what about any other case? E.g.: &lt;span class=&quot;math-container&quot;&gt;$2x - 3 = 0$&lt;/span&gt;? Again, you immediately need both the addition and multiplication principles in order to finish that off. What about general numerical problems: arbitrary fractions for coefficients, arbitrary decimals, etc.?&lt;/p&gt;&#xA;&lt;p&gt;Note that many or most algebra books quickly exercise students on such general linear equations, ones that cannot be factored in integers. At this point you have a fairly nice general technique for solving linear equations of all sorts. For example, see &lt;a href=&quot;https://openstax.org/books/elementary-algebra-2e/pages/2-5-solve-equations-with-fractions-or-decimals&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;OpenStax Elementary Algebra&lt;/em&gt;, Section 2.5&lt;/a&gt;: &amp;quot;Solve Equations with Fractions or Decimals&amp;quot;, which comes immediately after the general strategy for solving linear equations by inverses.&lt;/p&gt;&#xA;&lt;p&gt;Third, many books and curricula also treat general linear inequalities at about the same time. That's pretty close to the same process, with one added trick (flip inequality direction if one multiplies by a negative number). It's even less clear what kind of trick you'd apply to jump over that &amp;quot;missing&amp;quot; step in your examples to handle this with an always-factoring approach. Again, see &lt;em&gt;OpenStax Elementary Algebra&lt;/em&gt;, Section 2.7, for these applications.&lt;/p&gt;&#xA;&lt;p&gt;(Note also that this curriculum then follows with graphing lines and solving linear equations before higher-degree objects are handled; this provides a spiral-type path where you get to revisit the ideas of solving equations, inequalities, and graphing, in progressively more advanced contexts -- which is often needed by such basic students.)&lt;/p&gt;&#xA;&lt;p&gt;In short, the general process for solving linear equations and inequalities can pretty quickly be presented, and in fact must be presented, even if you wanted to focus on factoring all the time (which therefore presents an unnecessary delay). So the student has a fairly nice package of tools to handle linear stuff, possibly numerically with calculator technology, even if it's not factorable in integers.&lt;/p&gt;&#xA;&lt;p&gt;In fact, for some students they may not progress any further at all in their math pathway. Consider in this case &lt;em&gt;&lt;a href=&quot;https://openstax.org/details/books/prealgebra-2e&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax Prealgebra&lt;/a&gt;&lt;/em&gt;: that work manages to cover solving linear equations, but never gets to any higher-degree work. For some students that will be the end of the line, and time spent on factoring will be an unhelpful delay and distraction. (A key point of debate for basic math skills at my institution has in fact been administrators arguing that non-STEM students don't need to learn factoring, for example.)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-06-26T02:54:05.213" LastActivityDate="2023-06-26T02:54:05.213" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19110" PostTypeId="2" ParentId="10832" CreationDate="2020-11-05T11:17:43.330" Score="1" Body="&lt;p&gt;Here is an example that is simple, but that causes many problems in math classes: &lt;strong&gt;or&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Example: Do you prefer coffee &lt;strong&gt;or&lt;/strong&gt; milk? In everyday English, the &amp;quot;or&amp;quot; is interpreted as to choose only one option: only coffee, only milk but not both. In Mathematics, the &amp;quot;or&amp;quot; allows both choices: only coffee, only milk, coffee and milk.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2020-11-05T11:17:43.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19111" PostTypeId="1" CreationDate="2020-11-05T13:55:05.360" Score="1" ViewCount="1548" Body="&lt;p&gt;I'm in 8th grade and just finished Algebra 2. What math would I do for the next 2 years? In what order would math I would do in 9th, 10th, 11th, and 12th grade. Thanks!&lt;/p&gt;&#xA;" OwnerUserId="14909" LastActivityDate="2020-11-06T03:32:46.027" Title="I'm in 8th grade and just finished Algebra 2. What math would I do for the next 4 years?" Tags="|secondary-education|algebra|" AnswerCount="1" CommentCount="7" ClosedDate="2020-11-12T21:17:03.310" ContentLicense="CC BY-SA 4.0" />
  <row Id="19112" PostTypeId="2" ParentId="19092" CreationDate="2020-11-05T14:38:50.480" Score="1" Body="&lt;p&gt;Neither topic is covered as a Common Core standard as such.&lt;/p&gt;&#xA;&lt;p&gt;Somewhere in middle school, students learn that a circle cannot be a graph of a function because it fails what we call the Vertical Line Test.  (I could look up the exact standard and grade level if you're curious about it.)  In Geometry (tenth or eleventh grade typically, but it can vary by district), students learn to calculate the standard and general equations for a circle based on its radius and the coordinate of its center and working backwards from there, but we don't describe it as an implicit function.  There are no common standards for pre-calculus at the moment, but people generally cover conics to some level of detail.  Implicit differentiation is covered in the AP Calculus standards (both AB and BC), but IME we don't miss the lack of prior knowledge.&lt;/p&gt;&#xA;&lt;p&gt;When I was in high school 35 years ago, we did cover parametric equations in what was essentially pre-calculus, and I believe differentiating parametric equations came along with studying the Chain Rule in Calc AB.  I can easily imagine that some college courses would still cover it, but it is no longer in the AP Calculus standards.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-11-05T14:38:50.480" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19113" PostTypeId="2" ParentId="19085" CreationDate="2020-11-05T14:59:53.440" Score="4" Body="&lt;p&gt;If the domain was the real numbers, I would say that it raises two to the power of its input. If the domain was positive integers, I would break your model and say that it multiplied together &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; copies of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; because I think it is more intuitive to describe what exponentiation actually does that to imply that it is as natural as addition and multiplication.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-11-05T14:59:53.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19114" PostTypeId="1" AcceptedAnswerId="19115" CreationDate="2020-11-05T16:23:34.990" Score="2" ViewCount="983" Body="&lt;p&gt;This seems like such a niche trick to teach students when factoring polynomials. Like, the polynomials I've seen textbooks ask students to &lt;a href=&quot;https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratics-multiplying-factoring/x2f8bb11595b61c86:factor-quadratics-grouping/a/factoring-by-grouping&quot; rel=&quot;nofollow noreferrer&quot;&gt;factor by grouping&lt;/a&gt; seem so cherry picked that I can't imagine this is a meaningful thing to show students.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For a quadratic &lt;span class=&quot;math-container&quot;&gt;$ax^2+bx+c$&lt;/span&gt; you have a see a clever way to partition that &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$d+e$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$ax^2+dx$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ex+c$&lt;/span&gt; have a common linear factor. But what's the point of asking a student to play this guess-and-check game over the guess-and-check game for &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$a(x-r)(x-s)$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;li&gt;For a cubic to be factorable by grouping I think it must be of the form &lt;span class=&quot;math-container&quot;&gt;$a(x-b)(x^2-c)$&lt;/span&gt;, where the quadratic factor can't have more than two summands.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So what's the motivation for considering this as a topic in the curriculum?&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2020-11-07T01:58:41.617" Title="What is the motivation for teaching Factoring by Grouping?" Tags="|algebra|curriculum|concept-motivation|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19115" PostTypeId="2" ParentId="19114" CreationDate="2020-11-05T17:03:13.797" Score="4" Body="&lt;p&gt;That is a great question that I struggled with the one year I taught Algebra 1.  I came to the same conclusion that factoring by grouping problems are inauthentic because they only work when the expression is built to work.&lt;/p&gt;&#xA;&lt;p&gt;The story that got me through it is that you can use factoring by grouping to factor a quadratic expression if you manage to come up with a clever decomposition of the linear term that makes it work.  (One of my main points of the unit is that factoring polynomials is the inverse of multiplying polynomials, so we want to find a strategy to undo the &amp;quot;box method&amp;quot; of multiplying binomials and collecting like terms.)&lt;/p&gt;&#xA;&lt;p&gt;So what is the clever decomposition of, say, &lt;span class=&quot;math-container&quot;&gt;$x^2+7x+12$&lt;/span&gt;?  Here (and since I was working with non-accelerated ninth graders I just told them that this was the trick rather than formally justifying it), the trick is to think of two numbers that add to 7 and multiply to 12.  These numbers are obviously 3 and 4, so we can decompose the expression into &lt;span class=&quot;math-container&quot;&gt;$x^2+3x+4x+12$&lt;/span&gt; and use the factoring by grouping strategy to come up with a factorization of &lt;span class=&quot;math-container&quot;&gt;$(x+3)(x+4)$&lt;/span&gt;.  So, at least for me, factoring by grouping is a &amp;quot;lemma&amp;quot; strategy that lets me solve the far more authentic problem of factoring quadratic expressions.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-11-05T17:03:13.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19117" PostTypeId="1" CreationDate="2020-11-06T01:44:44.377" Score="3" ViewCount="7533" Body="&lt;p&gt;Are differential equations considered calculus and included in a calculus class or is it its own class? Also, if it is its own class then what calculus classes does it come after?&lt;/p&gt;&#xA;" OwnerUserId="14909" LastEditorUserId="12255" LastEditDate="2020-11-06T12:32:06.933" LastActivityDate="2020-11-18T05:44:21.360" Title="Are differential equations considered calculus and included in a calculus class or is it its own class?" Tags="|calculus|differential-equations|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19118" PostTypeId="2" ParentId="19117" CreationDate="2020-11-06T01:56:24.153" Score="3" Body="&lt;p&gt;In the US, it has become common to introduce differential equations within the first year of calculus.  Usually, there is also an &amp;quot;Introduction to Ordinary Differential Equations&amp;quot; course at the sophomore level that students take after a year of calculus.  The details of this can vary a lot between institutions.&lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2020-11-06T01:56:24.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19119" PostTypeId="2" ParentId="19111" CreationDate="2020-11-06T03:32:46.027" Score="2" Body="&lt;p&gt;If you have finished Algebra 2 in November of 8th grade, you are probably very good at math. You may want to explore options outside of your school. (When I was in high school, I did not get good counseling regarding math.)&lt;/p&gt;&#xA;&lt;p&gt;There is a wonderful online organization called &lt;a href=&quot;https://artofproblemsolving.com/?utm_source=google&amp;amp;utm_medium=cpc&amp;amp;utm_campaign=aops_brand&amp;amp;cmid=1419410580&amp;amp;agid=54779698534&amp;amp;search_keyword=art%20of%20problem%20solving&amp;amp;search_type=b&amp;amp;netid=g&amp;amp;adid=272880760302&amp;amp;adpos=&amp;amp;dvt=&amp;amp;ptid=&amp;amp;gclid=CjwKCAiA4o79BRBvEiwAjteoYNa_P0t7tH1vJ1HHfjZpPsqqHi0X0Kzwnmz0Fo25HGmbBLeqX5avihoCsb4QAvD_BwE&quot; rel=&quot;nofollow noreferrer&quot;&gt;Art of Problem Solving&lt;/a&gt;. A class of theirs that you might enjoy that would not be found in high schools is Number Theory. But check out all of their courses, and see what you think. (They are not free, but they do very good work.)&lt;/p&gt;&#xA;&lt;p&gt;If you'd like to learn more about calculus before you hit that course (and it doesn't have to be terribly hard, depends on the textbook and the teacher and the student), you might want to check out a short book by W.W. Sawyer, &lt;em&gt;What is Calculus About?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-11-06T03:32:46.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19120" PostTypeId="2" ParentId="19114" CreationDate="2020-11-06T04:12:55.963" Score="3" Body="&lt;p&gt;This is a thought provoking question. Learning to think algebraically is a gradual process. It takes practice to see that &lt;span class=&quot;math-container&quot;&gt;$$ax+ay=a(x+y)$$&lt;/span&gt; is the same as &lt;span class=&quot;math-container&quot;&gt;$$(x+1)x+(x+1)y=(x+1)(x+y).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Eventually, some students will have enough algebraic fluency to work with something like &lt;span class=&quot;math-container&quot;&gt;$f(x+h)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2-x.$&lt;/span&gt; Such a successful student will have developed  fluency parsing algebraic expressions where the &amp;quot;atoms&amp;quot; are not just single-character variables such as &lt;span class=&quot;math-container&quot;&gt;$a, x, y$&lt;/span&gt;, but potentially more involved expressions such as &lt;span class=&quot;math-container&quot;&gt;$x+1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x+h$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Factoring by grouping is not a means to an end. If our sole purpose was to have students factor expressions, we could just teach students to use computer algebra system software, and we could ignore the goal of developing algebraic fluency.&lt;/p&gt;&#xA;&lt;p&gt;An analogy might be a child learning to read. The beginning reader can parse and write simple sentences such as &amp;quot;&lt;em&gt;The dog runs fast&lt;/em&gt;.&amp;quot; Some years later, the student will be like George Bernhard Shaw, writing complex sentences such as &amp;quot;&lt;em&gt;A life spent making mistakes is not only more honorable, but more useful than a life spent doing nothing&lt;/em&gt;.”&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2020-11-06T04:20:58.633" LastActivityDate="2020-11-06T04:20:58.633" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19121" PostTypeId="2" ParentId="19117" CreationDate="2020-11-06T11:54:58.370" Score="2" Body="&lt;p&gt;Both.&lt;/p&gt;&#xA;&lt;p&gt;Differential equations are defined in the second semester of calculus as a generalization of antidifferentiation and strategies for addressing the simplest types are addressed there.  But you do a more indepth analysis in a separate course that usually is called something like Introduction to Ordinary Differential Equations (ODE).  That course is independent from the typical third semester of calculus in three dimensions and you could take the two courses in parallel.  (There are also a separate category called partial differential equations, which has its own series of courses that will wait for after all of this.)&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2020-11-06T13:17:16.760" LastActivityDate="2020-11-06T13:17:16.760" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19123" PostTypeId="2" ParentId="19114" CreationDate="2020-11-07T01:58:41.617" Score="1" Body="&lt;p&gt;While it's pretty clear from your question what method you're talking about, the title isn't so clear. When I hear factoring by grouping, I think of something like &lt;span class=&quot;math-container&quot;&gt;$3x^2+2x+9x+6$&lt;/span&gt;, and not &lt;span class=&quot;math-container&quot;&gt;$3x^2+11x+6$&lt;/span&gt;. When the student has to first come up with a way to split the middle term I've heard it called, not surprisingly, factoring by &amp;quot;splitting the middle term&amp;quot; and also &amp;quot;the AC method&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;As for the AC method, I agree with Dave L Renfro's comment that one reason for teaching certain things is to reinforce basic skills. A little practice with some mental arithmetic is never a bad thing, although I guess they'd get that with pattern matching too. AC also includes what I would call factoring by grouping in the final steps, so that gets practiced as well. I've personally taught the method before because I had the time and freedom to do so, and because I think it's a cool method. However, if I had to choose between AC and, say, completing the square, I think I'd teach completing the square every time. AC is nice but omitting it is no big deal.&lt;/p&gt;&#xA;&lt;p&gt;For what I would call factoring by grouping, as in my first example &lt;span class=&quot;math-container&quot;&gt;$3x^2+2x+9x+6$&lt;/span&gt;, I would want all students to be able to factor this by inspection. There's at least one major result down the road that it's very useful for: proving the product rule by writing &lt;span class=&quot;math-container&quot;&gt;$$f(x+h)g(x+h)-f(x)g(x)=f(x+h)g(x+h)-f(x+h)g(x)+f(x+h)g(x)-f(x)g(x)$$&lt;/span&gt;&#xA;which is itself a great example of proof by &amp;quot;hey this term might be helpful to have so I'll just jam it in there and make up for it elsewhere to keep the equality&amp;quot;. (Another great example is completing the square, which doesn't have as much of a rabbit out of a hat feel as the product rule.)&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2020-11-07T01:58:41.617" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19124" PostTypeId="1" AcceptedAnswerId="19125" CreationDate="2020-11-07T16:22:40.047" Score="14" ViewCount="2845" Body="&lt;p&gt;To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and &amp;quot;identify opposite sides&amp;quot;. We can imagine gluing together one pair of opposite sides to get a cylinder, and then gluing together opposite ends of the cylinder to get a torus. (Provided that our material is stretchy enough, which isn't an issue topologists concern themselves with.)&lt;/p&gt;&#xA;&lt;p&gt;In the past, I've described this description of the torus by analogy with &lt;a href=&quot;https://en.wikipedia.org/wiki/Pac-Man&quot; rel=&quot;noreferrer&quot;&gt;Pac-Man&lt;/a&gt;. In this video game, Pac-Man can leave the screen on one edge, and come back onto the screen from the same position on the other side.&lt;/p&gt;&#xA;&lt;p&gt;But I want to abandon this analogy, and come up with something better, because:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If you haven't played Pac-Man, it's not very helpful - and how many people these days have? I think I've played Pac-Man on a TI calculator a total of once or twice in high school.&lt;/li&gt;&#xA;&lt;li&gt;If you &lt;em&gt;have&lt;/em&gt; played Pac-Man, it's not very helpful, because in a typical Pac-Man maze, the &amp;quot;tunnels&amp;quot; that allow this wrapping-around behavior only go one way: from left to right. So a Pac-Man level is more like a cylinder than a torus.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Are there better analogies?&lt;/p&gt;&#xA;" OwnerUserId="9609" LastActivityDate="2020-11-10T16:58:05.960" Title="Replacement for the Pac-Man grid analogy" Tags="|concept-motivation|intuition|topology|" AnswerCount="8" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19125" PostTypeId="2" ParentId="19124" CreationDate="2020-11-07T17:05:41.630" Score="19" Body="&lt;p&gt;A possibility is to show your students Google's implementation of the game Snake.  If you enter the term &lt;a href=&quot;https://www.google.com/search?q=snake&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;code&gt;snake&lt;/code&gt;&lt;/a&gt; into Google's search engine, the there is a box at the top showing&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KBL4q.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KBL4q.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If your particular Google bubble doesn't show this result, I &lt;em&gt;think&lt;/em&gt; that there is a &lt;a href=&quot;https://www.google.com/fbx?fbx=snake_arcade&quot; rel=&quot;nofollow noreferrer&quot;&gt;direct link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you hit &amp;quot;Play&amp;quot;, you can play a game of Snake, using the standard rules.  Play a couple of short rounds, and demonstrate what happens if you run into a wall (you lose).  Then, after losing a couple of rounds, hit the gear to open up a settings menu.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oWy4H.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oWy4H.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;On the second line of settings, choose the infinity symbol.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lpnvE.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lpnvE.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This version of the game is played on a torus.  If you eat enough apples, you can see how the wrapping around works.  Alternatively, instead of selecting the infinity symbol, the peace symbol also allows you to play on a torus, but without the centering (i.e. the game will behave like Asteroids or Pac-Man, where you leave one side of the screen, and reappear on the other).&lt;/p&gt;&#xA;&lt;p&gt;What is handy about this is that you get a very quick demonstration of a game being played on a torus, and this demonstration is available to any person who has a clever device (e.g. a desktop or laptop computer, a clever phone, etc).  Moreover, it has a rather visceral, concrete quality to it which, to my mind, is more convincing than a bunch of hand drawn pictures (though maybe not as convincing as actually making a torus out of paper or cloth).&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2020-11-10T14:37:23.573" LastActivityDate="2020-11-10T14:37:23.573" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19126" PostTypeId="2" ParentId="19085" CreationDate="2020-11-07T17:22:21.253" Score="2" Body="&lt;p&gt;&amp;quot;The function &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; raises &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; to the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the input&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The function &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; raises &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; to the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;sup&gt;th&lt;/sup&gt; power, where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the input&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;both ape the structure of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; and, despite containing a clause, sound less awkward/contrived than the alternatives.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2020-11-08T01:30:42.893" LastActivityDate="2020-11-08T01:30:42.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19127" PostTypeId="2" ParentId="19124" CreationDate="2020-11-07T18:59:08.107" Score="6" Body="&lt;p&gt;I don't think it's necessary or a good idea to invoke a specific game, because any given game will be one that only a small fraction of your students have played. Just say, &amp;quot;You know how in some video games, if you go off one side of the screen, you reappear on the other side? Say you have a game where this happens both left/right and top/bottom.&amp;quot;&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-11-07T18:59:08.107" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19128" PostTypeId="2" ParentId="19124" CreationDate="2020-11-08T14:47:43.813" Score="1" Body="&lt;p&gt;I'm not sure you need an analogy, I think you could explain it nicely just stating how to construct it from a square, but if you do, you could buy a couple of bicycle tubes (preferable for a children’s bike, which are smaller and wider). Then cut it open so you have a rectangle. You can then show the intact and the cut version side by side.&lt;/p&gt;&#xA;&lt;p&gt;(It might not be a nice rectangle, but I think it will work anyway)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/w1PTZ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/w1PTZ.jpg&quot; alt=&quot;bicycle tube&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13330" LastActivityDate="2020-11-08T14:47:43.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19129" PostTypeId="2" ParentId="19124" CreationDate="2020-11-08T15:19:46.753" Score="7" Body="&lt;p&gt;One thing that I like about the Pacman analogy is that, if you draw Pacman's eye (or Ms. Pacman's bow, which is my usual choice), then Pacman is not mirror symmetric. This means you can talk about nonorientable surfaces and draw how Pacman goes around a Mobius strip/Klein bottle/projective plane and comes back reversed.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2020-11-08T15:19:46.753" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19130" PostTypeId="2" ParentId="19088" CreationDate="2020-11-08T23:28:21.503" Score="0" Body="&lt;p&gt;It's not free (yet), but it is the only true &amp;quot;on the fly&amp;quot; WYSIWYG Windows LaTeX compiler that I know: BaKoMa TeX (&lt;a href=&quot;http://www.bakoma-tex.com/menu/about.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.bakoma-tex.com/menu/about.php&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The project is gathering $$$ to make the software public domain.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2020-11-08T23:28:21.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19131" PostTypeId="2" ParentId="19124" CreationDate="2020-11-09T07:28:56.290" Score="0" Body="&lt;p&gt;I have always found the best way to ingrain toroidals is by showing them in unexpected places. Drinking straws, tires, a hollow question mark, a mug... Figuring out how many holes real world objects have is in my opinion the best way to build intuition for their properties.&lt;/p&gt;&#xA;" OwnerUserId="7922" LastActivityDate="2020-11-09T07:28:56.290" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19132" PostTypeId="2" ParentId="19124" CreationDate="2020-11-09T14:55:39.607" Score="3" Body="&lt;p&gt;When you're picking out a movie on Netflix, there's a grid of movies &amp;amp; TV available.  If you keep scrolling to the right in any one of the rows, you eventually come back to the first movie in the row.  And if you keep scrolling down through the categories, eventually you come back to the first category you started out with.&lt;/p&gt;&#xA;&lt;p&gt;(At least, that's how the Netflix catalog interface works on my TV.  Streaming UI being what it is, I wouldn't be surprised if it's different on other devices, or if it changes next week.)&lt;/p&gt;&#xA;" OwnerUserId="13965" LastActivityDate="2020-11-09T14:55:39.607" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19134" PostTypeId="2" ParentId="19124" CreationDate="2020-11-09T21:17:26.913" Score="2" Body="&lt;p&gt;I have (once) used a square of felt with two strips of velcro glued to the felt.  Then just &amp;quot;identify&amp;quot; one opposing pair of edges in an orientation preserving way via velcro.  Demonstrate that one obtains a finite cylinder.  Then &amp;quot;identify&amp;quot; the two circle edges in an orientation preserving way.&lt;/p&gt;&#xA;&lt;p&gt;This also gave some room to talk about nonorientation preserving gluing, a Moebius strip, and the real projective plane (and its cross-cap).  You can't actually perform the identifications for all of these (since you &lt;em&gt;will&lt;/em&gt; obtain an embedding in &lt;span class=&quot;math-container&quot;&gt;$\Bbb{R}^3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$S^3$&lt;/span&gt;) which allows discussion of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-manifolds that don't embed in &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-space.&lt;/p&gt;&#xA;" OwnerUserId="5800" LastActivityDate="2020-11-09T21:17:26.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19136" PostTypeId="1" CreationDate="2020-11-10T09:00:59.567" Score="5" ViewCount="269" Body="&lt;p&gt;Two students in my game development course would like, as a final project, to develop a video game for teaching mathematics. In contrast to the &lt;a href=&quot;https://matheducators.stackexchange.com/q/4208/13986&quot;&gt;many other games for teaching maths&lt;/a&gt;, which focus mainly on problem solving, they want to focus on the concept of a mathematical proof, in particular: proofs of classic theorems in number theory, such as Euclid's theorem or the fundamental theorem of arithmetics. Students often find such proofs hard to learn due to their &amp;quot;infinite&amp;quot; nature: while any finite number of examples is not sufficient, there is a finitely-written proof that covers &lt;em&gt;all&lt;/em&gt; infinitely-many cases. The difficulty is both in understanding the proofs and in writing their own proofs.&lt;/p&gt;&#xA;&lt;p&gt;As an example in-game idea that might be used to illustrate these abstract concepts, we thought of a hero that must defeat a monster. The hero can shoot at the monster some special cases of the theorem (e.g. prime numbers), but this only slows the monster down temporarily. In order to defeat it completely, the hero must build an &amp;quot;induction machine&amp;quot; that can construct cases endlessly. There are many other proof concepts that should be considered.&lt;/p&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Are there video games that aim to teach the concepts of mathematical proofs (rather than just problem solving)?&lt;/li&gt;&#xA;&lt;li&gt;What are some research papers regarding the difficulties encountered by students in learning mathematical proofs?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13986" LastActivityDate="2020-11-10T09:00:59.567" Title="A video game for teaching the concept of a mathematical proof" Tags="|reference-request|proofs|games|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19137" PostTypeId="1" AcceptedAnswerId="19138" CreationDate="2020-11-10T14:39:42.077" Score="1" ViewCount="160" Body="&lt;p&gt;I am teaching a Calculus class with around 30 students. The classroom is about 100 m^2.&lt;/p&gt;&#xA;&lt;p&gt;So far I have been using myopenmath.com to assign homework problems to my students. Now I would like to deliver the Mid-term examination via myopenmath.com.&lt;/p&gt;&#xA;&lt;p&gt;The set-up in my mind is as follows: the students will bring their own devices into class and access myopenmath.com to get the exam problems. The exam is an open-book exam, but the students are not allowed to receive support from anyone else. I’ll wander around the class and supervise the exam.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My question&lt;/strong&gt;: do we have any kind of technology to support this exam so that it minimises the chance of trying to cheat from the students?&lt;/p&gt;&#xA;&lt;p&gt;My problem is that I have to run the exam on my own, without any support from the university. Of course, the traditional examination form (written exam) might help to prevent academic misconduct. But I love the advantage of myopenmath.com in providing immediate feedback and creating randomized questions.&lt;/p&gt;&#xA;" OwnerDisplayName="user14865" LastActivityDate="2020-11-10T14:49:22.793" Title="How to create an online examination in a small class that ensures certain academic integrity" Tags="|exams|online-instruction|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19138" PostTypeId="2" ParentId="19137" CreationDate="2020-11-10T14:49:22.793" Score="3" Body="&lt;p&gt;The best technology in those circumstances is the &amp;quot;print&amp;quot; feature of MyOpenMath. Could you have each student print out their copy of the exam, take it on paper, and then have access to the solutions immediately after the time runs out?&lt;/p&gt;&#xA;&lt;p&gt;No internet + randomized questions =&amp;gt; probably no cheating&lt;/p&gt;&#xA;&lt;p&gt;[In the &amp;quot;create print version&amp;quot;, you can select &amp;quot;create key&amp;quot; in case you want to have the full solutions available on paper, too.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2020-11-10T14:49:22.793" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19139" PostTypeId="2" ParentId="19124" CreationDate="2020-11-10T16:58:05.960" Score="1" Body="&lt;h2&gt;East to west on a world map&lt;/h2&gt;&#xA;&lt;p&gt;The world map is usually what I use to explain toroidal periodic boundary conditions. &lt;strong&gt;The analogy isn't perfect because the Earth isn't a torus, but it gets the important ideas across in a way everyone can understand.&lt;/strong&gt; Unfortunately, it doesn't work north-to-south. I used to use &lt;i&gt;Asteroids&lt;/i&gt;, but few students are familiar with the game. Everyone is familiar with the world map.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6RDSH.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6RDSH.jpg&quot; alt=&quot;World map modified from Mercator projection by Wikipedia user Strebe ( https://en.wikipedia.org/wiki/World_map#/media/File:Mercator_projection_SW.jpg) &quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;World map modified from &lt;a href=&quot;https://en.wikipedia.org/wiki/World_map#/media/File:Mercator_projection_SW.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mercator projection by Wikipedia user Strebe&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you take the map as is, then it looks like you have to fly east over Africa or Europe to get from California to China. However, using the minimum image convention, you see that a flight from California to China should go west over the periodic boundary to land in the left periodic image of China.&lt;/p&gt;&#xA;" OwnerUserId="12740" LastActivityDate="2020-11-10T16:58:05.960" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19140" PostTypeId="1" CreationDate="2020-11-10T19:43:26.817" Score="12" ViewCount="972" Body="&lt;p&gt;I get this question a lot from new students who are taking their first proof-based math class.  They are struggling because they don't have that fluency with proofs, to begin with.  They don't know what constitutes valid proof or they have trouble approaching questions that ask them to prove something*.&lt;/p&gt;&#xA;&lt;p&gt;*Note this is &lt;em&gt;different&lt;/em&gt; from asking an experienced student to prove a hard theorem, where the difficulty arises in the actual content of the proof, not the various methods of how to go about proving it.&lt;/p&gt;&#xA;&lt;p&gt;When I think back on how I learned proofs, I grew up spending many years reading proofs and solving hard problems, and then eventually I started constructing my own proofs.  But I would say it happened organically. It would be the same way a native speaker grows up in their home country, hearing the language being spoken around them and gradually assimilating it themselves. This is NOT useful to a student taking their proof-based class who has to immediately learn to be 'fluent' in the language of proofs.&lt;/p&gt;&#xA;&lt;p&gt;What do you tell a student in this position? Or what would you recommend to be the most efficient way to get that understanding?&lt;/p&gt;&#xA;&lt;p&gt;Edit: I realized there's really two main questions going on here:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How do you go about teaching proofs in general?&lt;/li&gt;&#xA;&lt;li&gt;Let's just say they're in an introductory class that already assumes some basic fluency of proofs.  (Like a first-year course in real analysis or something). How can they go about acquiring this fluency on their own?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14497" LastEditorUserId="14497" LastEditDate="2020-11-11T16:35:58.907" LastActivityDate="2020-12-28T21:08:47.457" Title="How do you teach students about the concept of a proof?" Tags="|mathematical-pedagogy|proofs|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19141" PostTypeId="2" ParentId="19140" CreationDate="2020-11-10T23:23:36.723" Score="5" Body="&lt;p&gt;Note: I'm currently learning proofs myself, but I wonder if a good book on mathematical proofs might be useful towards this goal. That along with a 1:1 experienced mathematics tutor. Books in this genre include How to Prove it by Velleman, How to Think Like a Mathematician by Houston, the Book of Proof by Hammack, and Mathematical Proofs a Transition to Advanced Mathematics by Pearson press.&lt;/p&gt;&#xA;" OwnerDisplayName="user13234" LastActivityDate="2020-11-10T23:23:36.723" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19142" PostTypeId="2" ParentId="19140" CreationDate="2020-11-11T15:20:59.680" Score="4" Body="&lt;p&gt;I'd probably introduce the idea of proof with arithmetic questions that very much seem true based on simple specific examples but would need to be proven. Usually the classic even-odd proofs like the following:&lt;/p&gt;&#xA;&lt;p&gt;If we let &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; be an integer...&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; is odd, then &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is odd.&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; is even, then &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is even.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; is odd if and only if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is odd.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And before even trying to do these questions, one would need to define &amp;quot;integer&amp;quot;, &amp;quot;even&amp;quot;, &amp;quot;odd&amp;quot; and what &amp;quot;if... then&amp;quot; means, as well as &amp;quot;if and only if&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But is the question &amp;quot;how to teach proofs?&amp;quot; or &amp;quot;how to teach that proofs are necessary?&amp;quot; If it's the latter, it might be useful to bring up things that seem true based upon &amp;quot;common sense&amp;quot; but are, in fact, false. If they know calculus, harmonic series could be a good example of something counterintuitive. Birthday problem could be another one.&lt;/p&gt;&#xA;&lt;p&gt;Another good proof would be the ones without words, like that of Pythagorean Theorem.&lt;/p&gt;&#xA;" OwnerDisplayName="user13544" LastEditorDisplayName="user13544" LastEditDate="2020-11-11T23:15:15.803" LastActivityDate="2020-11-11T23:15:15.803" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19144" PostTypeId="1" CreationDate="2020-11-14T01:22:06.227" Score="3" ViewCount="844" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt; What is a succinct, clear and purely &lt;em&gt;algebraic&lt;/em&gt; explanation of why&#xA;the product of the slopes of perpendicular lines is &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here I am aiming for high-school students (in the U.S.).&#xA;I have a purely geometric explanation (below), but I would like to supplement it with a purely algebraic explanation.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/wPnqn.jpg&quot; /&gt;&#xA;&lt;p&gt;Rotating a slope triangle by &lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="470" LastEditDate="2020-11-15T14:29:03.883" LastActivityDate="2020-11-15T14:29:03.883" Title="What is an algebraic explanation of why the product of the slopes of perpendicular lines is $-1$?" Tags="|secondary-education|geometry|linear-algebra|" AnswerCount="4" CommentCount="11" ClosedDate="2020-11-14T19:10:28.587" ContentLicense="CC BY-SA 4.0" />
  <row Id="19145" PostTypeId="2" ParentId="19144" CreationDate="2020-11-14T03:17:57.260" Score="4" Body="&lt;p&gt;Let us &lt;em&gt;define&lt;/em&gt; two lines &lt;span class=&quot;math-container&quot;&gt;$L_1: y=m_1x+b_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L_2: y = m_2x+b_2$&lt;/span&gt; to be &lt;strong&gt;perpendicular&lt;/strong&gt; if their intersection exists and forms a right angle. Clearly &lt;span class=&quot;math-container&quot;&gt;$m_1 \neq m_2$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$P=(x_o,y_o)$&lt;/span&gt; be the point of intersection. Then,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ m_1x_o+b_1 = y_o =  m_2x_o+b_2 $$&lt;/span&gt;&#xA;Observe &lt;span class=&quot;math-container&quot;&gt;$b_2-b_1 = (m_1-m_2)x_o$&lt;/span&gt; this will be important later. Furthermore, select &lt;span class=&quot;math-container&quot;&gt;$x_2 &amp;gt; x_o$&lt;/span&gt; and notice &lt;span class=&quot;math-container&quot;&gt;$Q = (x_2, m_1x_2+b_1) \in L_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$R =(x_2, m_2x_2+b_2) \in L_2$&lt;/span&gt;. The line-segments &lt;span class=&quot;math-container&quot;&gt;$PQ$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$PR$&lt;/span&gt; form adjacent legs of a right triangle with hypotenuse &lt;span class=&quot;math-container&quot;&gt;$QR$&lt;/span&gt; hence by the Pythogorean Theorem we have:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \| PQ \|^2+\|PR \|^2 = \| QR \|^2 $$&lt;/span&gt;&#xA;Notice,&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align} \| PQ \| &amp;amp;= \sqrt{(x_2-x_o)^2+(m_1x_2+b_1-y_o)^2} \\&#xA;&amp;amp;= \sqrt{(x_2-x_o)^2+(m_1x_2+b_1-m_1x_o-b_1)^2} \\&#xA;&amp;amp;= \sqrt{(x_2-x_o)^2+m_1^2(x_2-x_o)^2} \\&#xA;&amp;amp;= |x_2-x_o|\sqrt{1+m_1^2} \\&#xA;\end{align}&lt;/span&gt;&#xA;Also,&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align} \| PR \| &amp;amp;= \sqrt{(x_2-x_o)^2+(m_2x_2+b_2-y_o)^2} \\&#xA;&amp;amp;= \sqrt{(x_2-x_o)^2+(m_2x_2+b_2-m_2x_o-b_2)^2} \\&#xA;&amp;amp;= \sqrt{(x_2-x_o)^2+m_2^2(x_2-x_o)^2} \\&#xA;&amp;amp;= |x_2-x_o|\sqrt{1+m_2^2} \\&#xA;\end{align}&lt;/span&gt;&#xA;Finally, recall &lt;span class=&quot;math-container&quot;&gt;$b_2-b_1 = (m_1-m_2)x_o$&lt;/span&gt; and calculate&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align} \| QR \| &amp;amp;= \sqrt{(x_2-x_2)^2+(m_2x_2+b_2-(m_1x_2+b_1))^2} \\&#xA;&amp;amp;= |m_2x_2+b_2-m_1x_2-b_1| \\&#xA;&amp;amp;= |(m_2-m_1)x_2+(m_1-m_2)x_o| \\&#xA;&amp;amp;= |x_2-x_o||m_2-m_1| &#xA;\end{align}&lt;/span&gt;&#xA;Hence, by Pythagorean Theorem,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ |x_2-x_o|^2(1+m_1^2)+|x_2-x_o|^2(1+m_2^2) =|x_2-x_o|^2|m_2-m_1|^2$$&lt;/span&gt;&#xA;Thus, as &lt;span class=&quot;math-container&quot;&gt;$|x_2-x_o| \neq 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|a|^2 = a^2$&lt;/span&gt; we find&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 2+m_1^2+m_2^2 = (m_2-m_1)^2 = m_2^2-2m_1m_2+m_1^2 $$&lt;/span&gt;&#xA;Therefore,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \boxed{1 = -m_1m_2}. $$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-11-14T03:17:57.260" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19146" PostTypeId="2" ParentId="19144" CreationDate="2020-11-14T04:10:10.740" Score="4" Body="&lt;p&gt;Since you used &amp;quot; &lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt; &amp;quot;, let me suggest something motivated by trigonometry.&lt;/p&gt;&#xA;&lt;p&gt;Define the relative-slope between two lines by &lt;span class=&quot;math-container&quot;&gt;$$m_{rel}=\frac{m_2-m_1}{1+m_2m_1},$$&lt;/span&gt;&#xA;from the trigonometric identity &lt;span class=&quot;math-container&quot;&gt;$$\tan(\theta_2-\theta_1)=\frac{\tan\theta_2-\tan\theta_1}{1+\tan\theta_2\tan\theta_1}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If the lines are parallel [i.e. &lt;span class=&quot;math-container&quot;&gt;$\theta_2-\theta_1=0^\circ$&lt;/span&gt;], then &lt;span class=&quot;math-container&quot;&gt;$m_{rel}=0$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$m_2=m_1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If the lines are perpendicular [i.e. &lt;span class=&quot;math-container&quot;&gt;$\theta_2-\theta_1=90^\circ$&lt;/span&gt;], then &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\frac{1}{m_{rel}}=0$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$m_2m_1=-1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(The analogous argument for special relativity implies that the product of the slopes equals 1 for the spacelike-axis Minkowski-orthogonal to a timelike-axis.)&lt;/p&gt;&#xA;" OwnerUserId="10537" LastActivityDate="2020-11-14T04:10:10.740" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19147" PostTypeId="2" ParentId="19144" CreationDate="2020-11-14T05:50:55.917" Score="5" Body="&lt;p&gt;Here's the algebra-based proof I've used in a college algebra class. Perpendicular lines are defined as meeting at a right angle. Assume that we know the Pythagorean and distance formulas.&lt;/p&gt;&#xA;&lt;p&gt;A possible lemma is that slope of a line indicates how much &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; increases for a 1-unit increase in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; on that line. Given that &lt;span class=&quot;math-container&quot;&gt;$m = \Delta y / \Delta x$&lt;/span&gt;, when &lt;span class=&quot;math-container&quot;&gt;$\Delta x = 1$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$m = \Delta y / \Delta x = \Delta y / 1 = \Delta y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Assume we have two perpendicular lines of defined slope &lt;span class=&quot;math-container&quot;&gt;$m _1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m_2$&lt;/span&gt;. Call the point of intersection &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt;. Step to the right by &lt;span class=&quot;math-container&quot;&gt;$\Delta x = 1$&lt;/span&gt; unit. On one line you'll be at the point &lt;span class=&quot;math-container&quot;&gt;$(x+1, y + m_1)$&lt;/span&gt;, while on the other you'll be at &lt;span class=&quot;math-container&quot;&gt;$(x + 1, y + m_2)$&lt;/span&gt;. The three points form a right triangle and we can use the aforementioned formulas.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kLlGT.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kLlGT.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Call the lengths of the sides of the triangle &lt;span class=&quot;math-container&quot;&gt;$a, b, c$&lt;/span&gt;. By the distance formula, these lengths are:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a = \sqrt{ 1^2 + m_1 ^2}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$b = \sqrt{ 1^2 + m_2 ^2}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$c = \sqrt{(m_1 - m_2)^2}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then by the Pythagorean formula (and the binomial square formula) we get:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\implies 1^2 + m_1^2 + 1^2 + m_2^2 = (m_1 - m_2)^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\implies m_1^2 + m_2^2 + 2 = m_1^2 - 2 m_1 m_2 + m_2^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\implies 2 = -2 m_1 m_2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\implies -1 = m_1 m_2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm pretty fond of this proof, because it provides an added opportunity to get more practice and experience with the Pythagorean, distance, and binomial-square formulas, which are key topics in this course. (Perhaps the kernel of the idea here is equivalent to James' proof, but more brief?)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-11-14T05:50:55.917" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19148" PostTypeId="2" ParentId="19144" CreationDate="2020-11-14T13:51:04.857" Score="5" Body="&lt;p&gt;Have these students had trigonometry? If so, they may have seen the formulas for rotating a point around the unit circle.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x' = x\cos(\theta) - y\sin(\theta) $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ y' = x\sin(\theta) + y\cos(\theta) $$&lt;/span&gt;&#xA;If we think of the slope &lt;span class=&quot;math-container&quot;&gt;$m=\frac{y}{x}$&lt;/span&gt; relative to an origin of &lt;span class=&quot;math-container&quot;&gt;$\langle0,0\rangle$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; can be thought of as just a point that can be rotated around the origin. Now, a perpendicular line can be thought of as a rotation &lt;span class=&quot;math-container&quot;&gt;$\theta=90$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\theta=-90$&lt;/span&gt; degrees. In either case, &lt;span class=&quot;math-container&quot;&gt;$\cos(\theta)$&lt;/span&gt; will be &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. So now, our above formulas become:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x' = -y\sin(\theta) $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ y' = x\sin(\theta) $$&lt;/span&gt;&#xA;Calculate the slope of our new line:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ m' = \frac{y'}{x'} $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ m' = \frac{x\sin(\theta)}{-y\sin(\theta)} $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ m' = \frac{x}{-y} $$&lt;/span&gt;&#xA;Now multiply your slopes:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ mm' $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{y}{x}\frac{x}{-y} $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ -1 $$&lt;/span&gt;&#xA;Note, I'm not a mathematician, so this might not fly as a proof. But this type of trigonometry is fundamental in computer graphics applications like video games, so it has a really cool application that might help to engage students.&lt;/p&gt;&#xA;" OwnerUserId="13207" LastActivityDate="2020-11-14T13:51:04.857" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19150" PostTypeId="2" ParentId="12377" CreationDate="2020-11-15T00:20:05.043" Score="3" Body="&lt;p&gt;I have just been exploring numeracy (and literacy), which I find quite fascinating. Numeracy seems to be the application of mathematics in a range of different contexts, not just in mathematics classes and maths scenarios. The Australian education literature and formal definitions seems to have gone that way. However, number sense is a specific subsection or part of the overall numeracy umbrella concept.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.australiancurriculum.edu.au/f-10-curriculum/general-capabilities/numeracy/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.australiancurriculum.edu.au/f-10-curriculum/general-capabilities/numeracy/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;While numeracy focuses a broad application, number sense is a cognitive ability or internal familiarity with estimations and quantities. There is an inherent sense some people are great at eyeballing and getting a sense of numbers but there is a learnt part of number sense too. Guessing the number of candies in a jar would be a number sense example maybe? Number sense seems a more arithmetic concept, more concrete and definite.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.understood.org/en/learning-thinking-differences/child-learning-disabilities/math-issues/number-sense-what-you-need-to-know&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.understood.org/en/learning-thinking-differences/child-learning-disabilities/math-issues/number-sense-what-you-need-to-know&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The flip side of innumeracy includes people with difficulty with number sense, dyscalculia, different levels of mathematical competency (learnt) and also universal finding that humans are basically innumerate with large numbers and probabilities.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Innumeracy_(book)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Innumeracy_(book)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14947" LastEditorUserId="14947" LastEditDate="2020-11-16T01:43:31.400" LastActivityDate="2020-11-16T01:43:31.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19151" PostTypeId="2" ParentId="19017" CreationDate="2020-11-15T02:38:13.583" Score="1" Body="&lt;p&gt;50/50 is a common default whenever there is a sense of uncertainty. I like the generalisation by &lt;a href=&quot;https://builtin.com/data-science/probability&quot; rel=&quot;nofollow noreferrer&quot;&gt;data science guy Rob Arthur &lt;/a&gt; that humans tend to have only three settings -&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Impossible&lt;/li&gt;&#xA;&lt;li&gt;50/50&lt;/li&gt;&#xA;&lt;li&gt;Certain&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“I think human beings are just sort of hardwired to do view in only those three categories&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The other aspect to consider is that the advertising and the intensity of the discussion around the lottery may also have contributed to &lt;a href=&quot;https://www.verywellmind.com/availability-heuristic-2794824#:%7E:text=They%20suggested%20that%20the%20availability,reflections%20of%20the%20real%20world.&quot; rel=&quot;nofollow noreferrer&quot;&gt;an availability bias&lt;/a&gt;. When we know and hear about events, they become familiar and intensity of the discussion cements the event into our memory more intensely. Events that are familiar are assumed to be more common. Humans are biased to the familiar and exciting, so we assume that it is common too. Winning the lottery is far more exciting than the concept of wasting money on the ticket.&lt;/p&gt;&#xA;&lt;p&gt;Also, probability is a really hard concept for humans to understand and comprehend, especially when it comes to massive numbers such as 90 million. I really like&lt;a href=&quot;https://nrich.maths.org/7326&quot; rel=&quot;nofollow noreferrer&quot;&gt; Professor's Spiegelhalter's argument &lt;/a&gt;that probabilities are unintuitive. Humans are guided by intuition and emotions for most of our lives, but when it comes to probablities, we really need to manage and ignore our lifelong bias to intuition and emotion, something that is very difficult to do. Especially for your 13 year old.&lt;/p&gt;&#xA;&lt;p&gt;Learning how to manage these intense feelings is probably a useful part of the discussion and education about probabilities.&lt;/p&gt;&#xA;" OwnerUserId="14947" LastActivityDate="2020-11-15T02:38:13.583" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19153" PostTypeId="2" ParentId="14919" CreationDate="2020-11-15T05:36:02.787" Score="2" Body="&lt;p&gt;I suggest following the convention in your country or institution while mentioning others. ISO 80000 is not a widely accepted standard in every fields. Many mathematicians strongly disagree with some of the decisions of the ISO standard (See section 4 of &lt;a href=&quot;https://tug.org/TUGboat/tb41-1/tb127gregorio-math.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://tug.org/TUGboat/tb41-1/tb127gregorio-math.pdf&lt;/a&gt;). In practice, publications from AMS and ACM do not follow the ISO standard as well.&lt;/p&gt;&#xA;&lt;p&gt;The Chinese standard you mentioned was a mandatory standard which was written based on ISO 31-11, the predecessor of ISO 80000. It was turned to a recommendatory standard in 2017.&lt;/p&gt;&#xA;" OwnerUserId="14953" LastActivityDate="2020-11-15T05:36:02.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19154" PostTypeId="1" CreationDate="2020-11-15T16:19:06.813" Score="9" ViewCount="392" Body="&lt;p&gt;In the textbook I am using to teach mathematics to high school students I found the following illustration about composition of functions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/14J94.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/14J94.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I do not agree with this illustration. For me &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is the slicer, &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; is the sliced potato, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is the fryer and &lt;span class=&quot;math-container&quot;&gt;$f(g(x))$&lt;/span&gt; is the bowl of french fries. I would introduce the notation (not on the diagram though) of &lt;span class=&quot;math-container&quot;&gt;$f\circ g$&lt;/span&gt; for the process of slicing and then frying.&lt;/p&gt;&#xA;&lt;p&gt;Am I being overly pedantic?&lt;/p&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2020-11-18T16:44:33.710" Title="Question about function notation" Tags="|notation|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19156" PostTypeId="2" ParentId="7281" CreationDate="2020-11-15T18:17:49.377" Score="1" Body="&lt;p&gt;Mathematically there really is no difference. The words &amp;quot;fraction&amp;quot; and &amp;quot;ratio&amp;quot; simply express the concept of dividing one number by another, and are used pretty much interchangeably in mathematics. They need not refer to comparing a part to a whole. The words &amp;quot;fraction&amp;quot; and &amp;quot;ratio&amp;quot; are also used with reference to division of complex, and hypercomplex numbers, where the concept of a part to a whole is completely meaningless.&lt;/p&gt;&#xA;&lt;p&gt;One difference in how the words are used is that the word &amp;quot;fraction&amp;quot; is often used to represent a specific representation of a ratio. For example 3/4 and 6/8 might be considered different fractions which represent the same ratio. You might also run across phrases like &amp;quot;when dividing two complex numbers, first write the ratio as a fraction with a real denominator.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="14954" LastEditorUserId="14954" LastEditDate="2020-11-15T20:06:39.107" LastActivityDate="2020-11-15T20:06:39.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19157" PostTypeId="2" ParentId="19154" CreationDate="2020-11-15T20:12:46.913" Score="6" Body="&lt;p&gt;You are not being pedantic. The name of the process that slices is &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, and the result of slicing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the textbook presentation seems to be for students who are just encountering function notation. In the language of the Dubinsky school of constructivism, students at this stage are not ready to distinguish between &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;. It will take some time and &amp;quot;action&amp;quot; before functions are encapsulated by the students as primary mathematical objects:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Action-&amp;gt;Process-&amp;gt;Object-&amp;gt;Schema&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;So although you appreciate the difference, it might be too much to expect students at this stage to do the same.&lt;/p&gt;&#xA;&lt;p&gt;But still, if the seeds are not planted, the development might never get started.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2020-11-15T20:12:46.913" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19158" PostTypeId="1" CreationDate="2020-11-15T23:31:36.367" Score="2" ViewCount="341" Body="&lt;p&gt;Do any textbooks or (somewhat?) standard curricula introduce logarithms and their applications in arithmetic without assuming the students know any algebra?&lt;/p&gt;&#xA;&lt;p&gt;(I do not mean just the use of logarithms for doing arithmetic, but rather, understanding of logarithms.)&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2020-11-23T16:53:44.610" LastActivityDate="2020-12-03T01:58:20.447" Title="Logarithms chronologically before algebra" Tags="|arithmetic|logarithm|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19159" PostTypeId="2" ParentId="19044" CreationDate="2020-11-16T10:54:56.933" Score="5" Body="&lt;p&gt;Sharing the impressions of a person who earned 2 IMO bronze medals in his youth, but whose dreams of a successful research career were never truly fulfilled :-)&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Mathematics is, indeed, not only about problem solving, but it isn't only about building theory either. Individual mathematicians may place themselves near one of the end points of this &amp;quot;spectrum&amp;quot;, but the entire line segment is somewhat continuously populated.&lt;/li&gt;&#xA;&lt;li&gt;The end points feed (and need) each other. The need to solve problems drives theory. Think of all the theory Newton and Leibniz developed to solve a few physics problems. Think of all the theory developed by people trying to solve Fermat's Last. On the other hand, a new theory usually spawns new problems (calculus again).&lt;/li&gt;&#xA;&lt;li&gt;In that process, whenever a new kind of a problem emerges, the first attacks on it are chiefly &lt;em&gt;taking the existing tools to their limits&lt;/em&gt;. This is what, IMHO, learning to look for elementary solutions prepares the students for. If those attacks fail &lt;em&gt;then&lt;/em&gt; we may need some new theory.&lt;/li&gt;&#xA;&lt;li&gt;On the other hand, the theories we teach to the students also open up new ways of thinking. I'm not sure how to best phrase this, so I poke randomly with: A) theory allows us to put a problem into a new framework, B) conceptualizing a problem helps. Somewhat unsatisfactory word-dropping here, sorry.&lt;/li&gt;&#xA;&lt;li&gt;I somewhat equate using elementary tools only to getting some actual dirt on my hands. This is necessary. Students need to see a lot of that. Too many kids coming out of high school think that math is all about learning which theory/tool applies to which problem. This is fine for an engineer, may be even a physicist, not sure about a teacher. If we let them go through an entire undergrad program without unlearning that, I think of that as a failure. I mean, when facing a new problem the first reaction should not be to search for an entirely new method.&lt;/li&gt;&#xA;&lt;li&gt;Giving the students problems that require taking their (necessarily elementary) tools to the limit is also essential for motivating the gifted ones (and sieving them out of the rest of the pack).&lt;/li&gt;&#xA;&lt;li&gt;Consider an analogy with sports. The professional athletes don't come out of a vacuum. They competed in college, they competed in high school, they competed in junior high, probably earlier. We need to do the same.&lt;/li&gt;&#xA;&lt;li&gt;I do concede that the type of problems/exercises you seem to be concerned about &lt;em&gt;are not for everybody&lt;/em&gt;. A student can become a very competent teacher or engineer without ever needing to strain themselves with elementary tools only (though the engineer may still benefit from developing a similar mindset).&lt;/li&gt;&#xA;&lt;li&gt;If your concern is that some of the contest problems have become &lt;em&gt;a craft&lt;/em&gt; as opposed to &lt;em&gt;an art&lt;/em&gt; that is something I can associate with. Such problems are not the ones the contestants will remember though :-)&lt;/li&gt;&#xA;&lt;li&gt;I cannot reach a conclusion here. My brain is not strong enough. May be it is just that the elementary tools have their own appeal? &lt;a href=&quot;https://en.wikipedia.org/wiki/Atle_Selberg&quot; rel=&quot;noreferrer&quot;&gt;Selberg&lt;/a&gt; got a Fields medal and a position at IAS (Princeton) for an &lt;em&gt;elementary&lt;/em&gt; proof of the prime number theorem (among other things).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="282" LastActivityDate="2020-11-16T10:54:56.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19160" PostTypeId="2" ParentId="9689" CreationDate="2020-11-16T13:38:53.757" Score="0" Body="&lt;p&gt;We could approach defining perpendicularity by the notion of minimizing the distance from a point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; to a line &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then we will have an optimization problem with a constraint.&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$P=(x_0,y_0)$&lt;/span&gt; and if &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; is given by &lt;span class=&quot;math-container&quot;&gt;$y=mx+b$&lt;/span&gt;, then we must minimize &lt;span class=&quot;math-container&quot;&gt;$f(x,y)=(x-x_0)^2+(y-y_0)^2$&lt;/span&gt; subject to the constraint &lt;span class=&quot;math-container&quot;&gt;$mx-y+b=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Setting this up using the formalism of Lagrange multipliers, we work with &lt;span class=&quot;math-container&quot;&gt;$$F(x,y,\lambda)=(x-x_0)^2+(y-y_0)^2+\lambda(mx-y+b).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Next, &lt;span class=&quot;math-container&quot;&gt;$$\begin{array}{tcl}&#xA;\dfrac{\partial F}{\partial x}&amp;amp;=&amp;amp;2(x-x_0)+\lambda m=0\\&#xA;\dfrac{\partial F}{\partial y}&amp;amp;=&amp;amp;2(y-y_0)-\lambda =0\\&#xA;\end{array}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;leads to &lt;span class=&quot;math-container&quot;&gt;$$\lambda=2(y-y_0)=-\dfrac{2(x-x_0)}{m}$$&lt;/span&gt;&#xA;and hence &lt;span class=&quot;math-container&quot;&gt;$$-\dfrac{1}{m}=\dfrac{y-y_0}{x-x_0}.$$&lt;/span&gt;&#xA;Summarizing, if point &lt;span class=&quot;math-container&quot;&gt;$Q=(x,y)$&lt;/span&gt; on the line &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; minimizes &lt;span class=&quot;math-container&quot;&gt;$d(P,Q)$&lt;/span&gt;, then the slope of &lt;span class=&quot;math-container&quot;&gt;$\overline{PQ}$&lt;/span&gt;, which  is equal to &lt;span class=&quot;math-container&quot;&gt;$\dfrac{y-y_0}{x-x_0}$&lt;/span&gt;, must in turn be equal to &lt;span class=&quot;math-container&quot;&gt;$\dfrac{-1}{m}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Evidently, this approach would not be appropriate in a precalculus class. But as an instructor thinking about how to present this idea, it is heartening to know that there is in fact an approach that relies on neither trigonometry nor diagrams.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2020-11-16T13:38:53.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19161" PostTypeId="2" ParentId="19158" CreationDate="2020-11-17T01:42:20.257" Score="-1" Body="&lt;p&gt;One example where you might expose students to the idea of logarithms prior to algebra might be through log-scaled graphs, such as this simple example below:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://www.graphpad.com/guides/prism/7/user-guide/images/embim7.png&quot; alt=&quot;Example of Log-scale graphs&quot; /&gt; &lt;a href=&quot;https://www.graphpad.com/guides/prism/7/user-guide/when_to_use_a_logarithmic_axis.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;image credit&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Students might not have the technical language to express what is going on in the graph to the right, but I think that it is within the realm of their comprehension.  Especially with the advent of COVID, it might be useful to expose pre-algebra students to this kind of growth and model so that they might have a better understanding of why disease transmission is something that needs to be taken seriously.&lt;/p&gt;&#xA;&lt;p&gt;The comments to your post also had some excellent answers, though I thought this interp&lt;/p&gt;&#xA;" OwnerUserId="14961" LastActivityDate="2020-11-17T01:42:20.257" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19162" PostTypeId="2" ParentId="19117" CreationDate="2020-11-18T05:44:21.360" Score="3" Body="&lt;p&gt;What part of differential equations is included in calculus may simply reflect the need of engineering or other majors at your institution. For example, when I taught Calculus II at NCSU in Raleigh, North Carolina USA, we covered how to solve:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ ay''+by'+cy = 0 $$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$a,b,c$&lt;/span&gt; are constants. If you look in the major calculus texts, you'll likely find an appendix on &lt;em&gt;constant coefficient second order ODEs&lt;/em&gt;. Certainly it is within the grasp of students in Calculus II. To be totally honest, it's way easier than a lot of the integration and integral-calculus-based problem solving which is typical of Calculus II. In short, if you can solve a quadratic equation and follow a recipe based on those solutions then you can just write down the general solution &lt;span class=&quot;math-container&quot;&gt;$y = c_1y_1+c_2y_2$&lt;/span&gt;. Frankly, you could put the method in highschool algebra books if you really wanted to, the method involves pretty much zero calculus (modulo it's derivation).&lt;/p&gt;&#xA;&lt;p&gt;When facing the corresponding &lt;em&gt;non-homogeneous&lt;/em&gt; problem:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ ay''+by'+cy = g $$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$a,b,c$&lt;/span&gt; are constants and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is the &lt;em&gt;forcing function&lt;/em&gt;, the solution has the form &lt;span class=&quot;math-container&quot;&gt;$y = c_1y_1+c_2y_2+y_p$&lt;/span&gt;. We cover two methods to find &lt;span class=&quot;math-container&quot;&gt;$y_p$&lt;/span&gt; (the particular solution), the &lt;em&gt;method of undetermined coefficients&lt;/em&gt; and &lt;em&gt;variation of parameters&lt;/em&gt;. Undetermined coefficients is pretty much just educated guessing with a little differentiation and a lot of algebra. Variation of parameters, in-contrast, is a general method which solves all inhomogeneous problems up to an integral you may or may not be able to hack in practice.&lt;/p&gt;&#xA;&lt;p&gt;Before all this, we also covered a little about visualizing differential equations with direction fields, how to solve separable first order ODEs and the integrating factor method for solving linear first order problems.&lt;/p&gt;&#xA;&lt;p&gt;In short, these are the major calculational tools that solve probably 90 percent of the basic applied problems you'll run across in basic engineering. A mass on a spring with friction, or an RLC-circuit, both are described by constant coefficient second order ODEs, so naturally engineering wants their students to know this math before the course so they can go deeper into the intuition and analysis of such problems. The first order ODEs are also applicable to a vast bank of examples across a wide variety of majors.&lt;/p&gt;&#xA;&lt;p&gt;I would also point out, in Calculus III we typically discuss conservative vector fields as &lt;span class=&quot;math-container&quot;&gt;$\vec{F}$&lt;/span&gt; for which there exists &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; with&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \vec{F} = \nabla f = \left\langle \frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z} \right \rangle $$&lt;/span&gt;&#xA;For example, if &lt;span class=&quot;math-container&quot;&gt;$\vec{F} = \langle x,2,z^2 \rangle$&lt;/span&gt; then we need to find &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; for which&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{\partial f}{\partial x} = x, \qquad&#xA;\frac{\partial f}{\partial y} = 2, \qquad&#xA;\frac{\partial f}{\partial z} = z^2 $$&lt;/span&gt;&#xA;This is a &lt;em&gt;system of partial differential equations&lt;/em&gt;. It is standard to learn how to solve this in Calculus III. In this sense, we can't isolate the study of PDEs from the natural discussion of vector calculus. Btw, the solution is easily seen by guessing to be &lt;span class=&quot;math-container&quot;&gt;$f = \frac{1}{2}x^2+2\frac{1}{3}z^3+c$&lt;/span&gt; where you can choose &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; however you like. Furthermore, there are calculus texts which cover other ODEs later in Calculus III. For example, the exact equation &lt;span class=&quot;math-container&quot;&gt;$Mdx+Ndy=0$&lt;/span&gt; is nice to analyze around the same time that Green's Theorem is discussed.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, last paragraph aside, it's not the math major that drives DEqns into calculus, it's these applications.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, if we are to envision Calculus for math majors, I think the answer would be radically different than the adhoc collection of inertially propelled topics which forms our current canon.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2020-11-18T05:44:21.360" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19163" PostTypeId="2" ParentId="19154" CreationDate="2020-11-18T14:31:41.530" Score="4" Body="&lt;p&gt;I would argue that there is an issue with the diagram as labeled regardless of whether you are or are not being pedantic.  For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Consider the case where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; can be used interchangeably, as is done in many US high school courses.  In this situation, the labeling on the slicer and fryer are fine, because they denote the functions in question.  Since &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is being used as an abstract placeholder, though, it should not be used as the label for the potato; the potato is a specific instance of a value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is common in high school calculus courses, but &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; might be more appropriate for &lt;em&gt;potato&lt;/em&gt;).  In this scenario, the labels would be (in order): &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g(p)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(g(p))$&lt;/span&gt;.  You could then spend time discussing that &lt;span class=&quot;math-container&quot;&gt;$f(g(x))$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f\circ g$&lt;/span&gt; are a process that was used to make the fries, rather than the fries themselves.&lt;/li&gt;&#xA;&lt;li&gt;Consider the case where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; have separate distinct meanings, as some users have suggested.  In this case, we are using &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as an input, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; as a function, and &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; as the output to that function.  For this scenario, the potato can happily be &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and our labels in order would be &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(g(x))$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In either case, it seems reasonable to say that regardless of one's views on the finer points of function notation, the diagram uses inconsistent &amp;quot;mixed metaphor&amp;quot; labeling.&lt;/p&gt;&#xA;" OwnerUserId="14966" LastEditorUserId="14966" LastEditDate="2020-11-18T16:44:33.710" LastActivityDate="2020-11-18T16:44:33.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19164" PostTypeId="1" AcceptedAnswerId="19174" CreationDate="2020-11-18T22:00:36.977" Score="41" ViewCount="6709" Body="&lt;p&gt;What are some good answers to questions e.g. &amp;quot;why do we need to study square roots&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Of course the answer depends highly on who is asking. For the scope of my question, I have a student in mind, who is - given their interests for further education - likely never going to use square roots in adult life, but still has to learn them. (Of course my question is not about square roots specifically. It can be any topic which, realistically, will not be used &amp;quot;in real life&amp;quot; by the student.)&lt;/p&gt;&#xA;&lt;p&gt;I have some answers, but I feel that they all fall short of being a satisfactory answer:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;Because you have to know it for your grades, so later you will have more choice of what you want to do in life&amp;quot; - this is my honest answer, but I fear it might be a source of resentment for the student against the education system, which I don't want.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bring some forced example from &amp;quot;real life&amp;quot;, e.g., &amp;quot;John has a son whose age is the square root of his age. John is 49 years old. How old is his son?&amp;quot; - I want to avoid these at any cost, as any student will see through these and will be enforced in his view, that &amp;quot;math is nonsense&amp;quot;. Even if the example is better and &lt;em&gt;I think&lt;/em&gt; more relevant &amp;quot;in real life&amp;quot;, if it fails to resonate deeply with the student, then it will backfire as an example.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;It teaches you to think abstractly&amp;quot; - I am deeply skeptical of some generic problem solving ability. I think we are good at solving problems that we practice solving, and it's not obvious how one transfers to another problem. And even if generic problem solving activity is a thing, it is hard to see why practicing square roots is the best thing to improve it, instead of something else that the student is more interested in. I think, hearing this answer, most students will nod along and still think to themselves that math is nonsense.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;Distract from the question&amp;quot;. The &amp;quot;why learn this&amp;quot; question usually arises from a thought process, that is just distracting the student. He finds square roots hard to understand, so he is looking for an excuse to think about something else. He never seems to ask the &amp;quot;why do this&amp;quot; question for activities that he otherwise just enjoys, e.g., reading a novel, playing music, drawing etc. and if asked &amp;quot;why&amp;quot;, he will rationalize why it is a good thing to do. So anything that distracts from the &amp;quot;why question&amp;quot; and manages to bring focus back to thinking about the problem itself is good. (If you have any concrete strategies for this, I appreciate if you share it.) However: I still think, that the &amp;quot;why learn this&amp;quot; question is important and I would not like to ignore it completely.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am an applied mathematician myself, so I can bring real life examples where math (beyond basic arithmetic) is useful. But the student will likely not have the knowledge to fully appreciate it, so while he might be impressed momentarily, I don't think I can expect any consistent success. It certainly makes sense to try connecting the applications of math with school curriculum every once in a while, but honestly these are often very far away from each other.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Is there any good strategy to answer to the &amp;quot;why learn this&amp;quot; question, that I missed?&lt;/p&gt;&#xA;&lt;p&gt;Did I make a conclusion above, which is absolutely wrong?&lt;/p&gt;&#xA;&lt;p&gt;If possible, I am looking for answers which are based on at least some empirical evidence, but I am also interested in concrete examples where something has worked really well.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; thanks for all the contributions. Some further clarification about my question:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I look for direct answers and not analogies. Math is not sports, art, or music. To me, such answers are essentially distraction (see point 4 above).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Several answers are in the direction of &amp;quot;it's part of general knowledge&amp;quot;. I see this as appeal to authority and as such, the nicer version of appealing to grades (point 1 above).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some answers are quite specific, e.g., lots of suggestions are about finance (which btw I personally find really boring). Some topics in math connect better with finance than others and having one &amp;quot;go-to example&amp;quot; isn't a really useful mindset when dealing with a concrete student and a concrete topic.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14091" LastEditorUserId="851" LastEditDate="2020-11-27T06:39:36.497" LastActivityDate="2024-01-08T07:52:36.673" Title="How should I answer questions about the purpose of learning math?" Tags="|student-motivation|" AnswerCount="23" CommentCount="27" ContentLicense="CC BY-SA 4.0" />
  <row Id="19165" PostTypeId="2" ParentId="7281" CreationDate="2020-11-18T22:34:36.870" Score="0" Body="&lt;p&gt;Focusing on the &lt;em&gt;difference&lt;/em&gt; between the abstract concept of a ratio and a fraction is likely to confuse students who are just learning these concepts.&lt;/p&gt;&#xA;&lt;p&gt;The goal should be that they eventually understand and become skilled with the abstraction a &lt;strong&gt;fraction&lt;/strong&gt;. The concept being abstract means, that it can be applied in a wide variety situations and can be used as a model for many concrete examples.&#xA;The key to understanding is eventually seeing what is common between seemingly different things.&lt;/p&gt;&#xA;&lt;p&gt;In grade 5, it suffices to say, that the difference is in the notation:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;code&gt;a/b&lt;/code&gt; is meant to denote that &lt;code&gt;a&lt;/code&gt; is the part, &lt;code&gt;b&lt;/code&gt; is the whole,&lt;/li&gt;&#xA;&lt;li&gt;&lt;code&gt;a:b&lt;/code&gt; is meant to denote, that &lt;code&gt;a&lt;/code&gt; and &lt;code&gt;b&lt;/code&gt; are two complementary parts of the same whole.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The notation &lt;code&gt;a/b&lt;/code&gt; is useful and is a good basis to build further abstractions and calculations later on. The notation &lt;code&gt;a:b&lt;/code&gt; is less useful, and is mostly only used in noting down concrete problems, and is not developed further for more abstractions that students will use later as often as with the fraction &lt;code&gt;a/b&lt;/code&gt;.&lt;/p&gt;&#xA;&lt;p&gt;That is why it is not so useful to go into abstract explanations of what is the difference between the &amp;quot;definition&amp;quot; of a fraction and a ratio. Just explain it in the minimalistic sense of what the notation means, and then focus on developing their understanding of the concept of a fraction.&lt;/p&gt;&#xA;" OwnerUserId="14091" LastEditorUserId="14091" LastEditDate="2020-11-20T14:54:17.867" LastActivityDate="2020-11-20T14:54:17.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19166" PostTypeId="2" ParentId="19164" CreationDate="2020-11-18T23:42:05.647" Score="26" Body="&lt;p&gt;The closest I came to getting fired for something I said to a student. The student asked &amp;quot;When will I ever use this math in the future?&amp;quot; I responded, &amp;quot;Well, you won't, but the smart kids might.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, the right answer, even at the high school level, is to ask what they might wish to do in the future. The well rounded education keeps all their options open. When I graduated High School, I chose electrical engineering, and that was my major as a college freshman. Now, I've seen curriculum that allows a student to not declare until junior year in college.&lt;/p&gt;&#xA;&lt;p&gt;For those that are certain they'll not go into a STEM career and are willing to make that decision. I'd go down the path that there is a mathematical equivalent of illiteracy, called &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1572/examples-of-innumeracy&quot;&gt;innumeracy&lt;/a&gt;. Similar to how one needs to read, directions on a medicine bottle, a simple passage in a magazine, there's a level of mathematical understanding that helps us make sound investment decisions. Helps us adjust a recipe when we have 75% or 150% of the required ingredients. I've made a habit that when a student says they hate math and don't get it, to talk money. Who doesn't like money? And who would be happy to get ripped off due to their own bad math? Granted, they may not need to use square roots, but we are graduating kids who can't calculate compound interest or balance a checkbook.&lt;/p&gt;&#xA;&lt;p&gt;The made up word problems (your sec 2) usually seem contrived and uninteresting, students are eager to get problems they are likely to actually face in real life.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="14091" LastEditDate="2020-11-19T02:39:29.897" LastActivityDate="2020-11-19T02:39:29.897" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19167" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T00:10:47.960" Score="10" Body="&lt;p&gt;Find questions which are actually interesting to the student. Here is just one idea:&lt;/p&gt;&#xA;&lt;p&gt;Ask the students to make several rectangles with an area of 50.  They might find &lt;span class=&quot;math-container&quot;&gt;$1 \times 50$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2 \times 25$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$5 \times 10$&lt;/span&gt;, etc.  Then challenge them to come up with some examples with non-integer side lengths.  What are the perimeters of all of these rectangles?  Can they figure out which one has the smallest perimeter?  You might not be able to find it exactly, but see if you can make the side lengths as close as possible.  What kind of rectangle is it?&lt;/p&gt;&#xA;&lt;p&gt;I won't say that every student will find this interesting, but it is relatively undirected open play, and students can work with it at a variety of different levels of abstraction.  Some students will write down algebraic expressions to help them, and some will just play numerically.  At the end of the day, you will have many students with many different approximations for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{50}$&lt;/span&gt;.  Then you can introduce the concept to them, when it already has a natural application.&lt;/p&gt;&#xA;&lt;p&gt;Of course, you can take any natural instance of the square root and try to devise a similar investigation.  Another example would be the fact that you only need to check up to &lt;span class=&quot;math-container&quot;&gt;$\lfloor \sqrt{N} \rfloor$&lt;/span&gt; numbers when checking if &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is prime via trial division (this is related, of course, to the first example I gave).&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-11-19T02:36:12.673" LastActivityDate="2020-11-19T02:36:12.673" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19168" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T01:36:56.830" Score="7" Body="&lt;p&gt;This is among the oldest questions about learning that has been asked.&lt;/p&gt;&#xA;&lt;p&gt;When a young person started learning geometry with Euclid and asked him why he should learn geometry, Euclid replied, &amp;quot;Give him threepence, since he must make a gain out of what he learns.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I think you are overlooking some aspects related to your answer #3 about abstract thinking. Abstract thinking is not about developing problem solving skills &lt;em&gt;per se&lt;/em&gt; in the sense of George Polya. It is about developing and maintaining plasticity of the brain, in order to reason with new information in unfamiliar contexts. &lt;em&gt;Practicing&lt;/em&gt; square roots does not develop abstract thinking. But learning about them, becoming familiar with them, and then moving on to a fresh unfamiliar context such as inverse functions, perhaps eventually to the notion branch cuts exemplified by Riemann surfaces for &lt;span class=&quot;math-container&quot;&gt;$w=\sqrt{z}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$w=\ln{z}$&lt;/span&gt;, is a reward of abstract thinking--not the threepence Euclid gave to his student.&lt;/p&gt;&#xA;&lt;p&gt;There is another reason why we have a requirement that all high school students will have an opportunity to learn some mathematics. In the USA, compulsory education became the norm in the late 19th century, with Mississippi becoming the last state to enact a compulsory attendance law in 1918. I think there are many good reasons for this expectation, including equity, equal access, and societal need. It is a scary thought exercise to imagine what would happen if we decided to make mathematics in high school be an elective on the par of personal finance or woodworking. If we allowed high school students to opt out of learning mathematics and science, we might not have enough people in educational pipelines to sustain societal need for STEM. Maintaining such pipelines is a reason why society invests so much in public education. Of course such a reason is not likely to satisfy a reluctant learner. But having to teach reluctant learners is a bargain we make with compulsory education policy.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2020-11-19T03:21:04.977" LastActivityDate="2020-11-19T03:21:04.977" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19170" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T06:23:52.613" Score="7" Body="&lt;p&gt;I disagree with the idea that any student can be assumed to “never need” square roots, at the point where they are learning them. At that point the students are simply too young for us to make assumptions about their career paths.&lt;/p&gt;&#xA;&lt;p&gt;And while we can go back and forth about the particular utility of square roots, they’re part of a basic canon of math. It’s the kind of math you will always be assumed to know when studying anything at all advanced, and therefore has utility if you use math in any major capacity in your life. (And in order to teach someone a subject at all you have to believe there is some use in it...)&lt;/p&gt;&#xA;&lt;p&gt;I’m not sure how exactly how to package that for a student, but perhaps that is a start towards justifying it.&lt;/p&gt;&#xA;" OwnerUserId="14972" LastActivityDate="2020-11-19T06:23:52.613" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="19171" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T06:50:01.807" Score="15" Body="&lt;h1&gt;Money&lt;/h1&gt;&#xA;&lt;p&gt;Most people will not need to compute the trajectory of a ballistic projectile, but everyone will need to deal with money to live in any advanced society.  Furthermore, while many people can get by without doing any advanced math, it is pretty easy to introduce real-world examples which get hairy very quickly.  Compound interest on credit cards are perhaps a simple example.  Computing simple interest on a mortgage is also relevant to many people.  I would go so far as to assume that the student will be sufficiently wealthy at some point that they will want to invest in the stock market, at which point you can introduce any number of concepts related to expected returns on various investments, options, etc.&lt;/p&gt;&#xA;&lt;h1&gt;Make it Personal&lt;/h1&gt;&#xA;&lt;p&gt;Just to drive home the point that being numerically literate is important, you could offer the student a small financial incentive.  Tell them:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I have $5 (or a dollar or whatever) that I'll give you, if you can solve this math problem for me right here, with just pencil and paper, in 10 minutes or less.  I promise it is a problem you will likely face one day.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If the student agrees, then you follow up with:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose you are buying a house.  A mortgage lender offers you a mortgage at 5% interest.  You need $200,000 to buy the house.  At the end of a 30 year mortgage, how much money will you have paid out?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Obviously, this is a pretty easy problem, mathematically speaking, but likely well beyond the skill of the typical &amp;quot;Why do I have to learn this?&amp;quot; student.  Hopefully, they will get the point that math is pretty important.  If they say: &amp;quot;I don't need to know that.  There's calculators for that&amp;quot; then you can say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How do you know you are using the calculator correctly?  Most calculators allow you to solve the problem from multiple angles, by changing the interest rate, the monthly payment, the total cost, etc.  How will you know the answer you are getting is even in the right ballpark?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Gamify&lt;/h1&gt;&#xA;&lt;p&gt;There's lots of directions you can take this, including estimating taxes on wages, payments for a car, etc.  If you want to engage the whole class, then you can make a game where they have to compute some interest or such to get the best loan for a car, etc.  Give them multiple loans for high-ticket items, multiple jobs that pay different amounts on different schedules (weekly, biweekly, monthly) and have different tax rates, and challenge them to &amp;quot;spend&amp;quot; their allocated &amp;quot;working hours&amp;quot; maximizing their income and buying the most things given your mini-market.  Hopefully they will see that it's far more than a game, and the answers have real-world implications.&lt;/p&gt;&#xA;" OwnerUserId="13468" LastActivityDate="2020-11-19T06:50:01.807" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19172" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T10:02:41.077" Score="4" Body="&lt;p&gt;Following on from Lawnmower Man's answer, I think &lt;strong&gt;money is a very good example&lt;/strong&gt;: one that people are familiar with and understand the importance of.  But you don't need to go as far as mortgages to see the need for powers and roots.  Perhaps an example that students are likely to encounter much earlier is &lt;strong&gt;credit card interest&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example: &lt;em&gt;Credit card A charges 50% APR, while credit card B charges 4%/month.  Which is more expensive per month?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;(A naïve calculation might be: ‘50 / 12 ≈ 4.1667 &amp;gt; 4, so A is more expensive.’  Which is of course wrong.  To get the right answer, you need to understand percentages well enough to be thinking of factors of 1.04 and 1.50 — and then understand exponentiation well enough to take the twelfth root of the latter.  Giving it a monthly interest of ~3.4%.  This may come as a surprise to people unfamiliar with compound interest…)&lt;/p&gt;&#xA;" OwnerUserId="12762" LastEditorUserId="12762" LastEditDate="2020-11-19T14:18:36.417" LastActivityDate="2020-11-19T14:18:36.417" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19173" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T12:59:36.940" Score="3" Body="&lt;p&gt;Something I heard a few years ago.  Add this to your arsenal.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Will I ever need to use algebra?&amp;quot;  You may not go into STEM.  You may still get a white-collar job, as an office worker.  (Most people never become an NBA player or a movie star.) The office may use spreadsheets to keep track of things.&lt;/p&gt;&#xA;&lt;p&gt;Each office has one person who knows enough algebra to program their spreadsheets.  The others just enter the numbers, and the spreadsheet computes everything automatically.  So it may be true that the workers in that office do not need to know algebra.&lt;/p&gt;&#xA;&lt;p&gt;Now imagine they have a job opening.  After interviews, they end up with some qualified candidates.  They all have similar credentials and experience, except that one can program spreadsheets.  Which one will they hire?&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2020-11-19T12:59:36.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19174" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T13:38:59.310" Score="8" Body="&lt;p&gt;As you said, it depends on the type of student you're dealing with. Some of them will have an interest in x, and you just have to make a connection to x. On the other hand, some students won't have enough perspective/experience to appreciate most answers you might give. I've had younger students argue with me about learning fractions and decimals. You'd think that bringing up money would quickly end the conversation, but you'd be surprised.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes the question is driven by psychology instead of curiosity/a need for motivation. You have to figure out where the question is coming from. Challenging the student with undirected open play will get you nowhere if the student is looking for a distraction as you wrote in #4. You don't always have an ideal student.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, there are lots of things that we learn that can't be appreciated in the moment. That has to be the perspective you should communicate. Why do you learn x, y, z in this other class? Is there a purpose to that? You're not always going to get instant gratification from everything that you learn.&lt;/p&gt;&#xA;&lt;p&gt;Ultimately the student has to be ok with accepting &lt;em&gt;something&lt;/em&gt; on trust. You can tell them &amp;quot;it teaches you think abstractly,&amp;quot; but if they're already critical of the process, then is that going to be convincing?&lt;/p&gt;&#xA;" OwnerUserId="13694" LastEditorUserId="13694" LastEditDate="2020-11-21T02:13:16.660" LastActivityDate="2020-11-21T02:13:16.660" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19175" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T14:44:55.300" Score="43" Body="&lt;p&gt;I found that my former students (low achieving ninth graders in the U.S.) always responded best when I answered with this:&lt;/p&gt;&#xA;&lt;p&gt;You enjoy watching sports right? Whether it's Football, Basketball, the Olympics, or whatever. Or maybe not sports, but you like watching a rock concert or orchestra. You pay money to see the famous athletes or famous musicians play in the championship or to play their concert. You don't pay money to see them lift weights in the gym or to practice their scales and arpeggios. But we all recognize how important practicing those things are: Practicing scales and arpeggios helps your ability to sight-read music because you pick up on similar patterns. Practicing sports plays over and over helps you recognize when it will be useful against what kind of defense.&lt;/p&gt;&#xA;&lt;p&gt;Well, all of us sitting in the room solving for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; or graphing a linear equation, this is our version of lifting weights, practicing sports plays or arpeggios. This is not the useful stuff. We won't get paid money for doing this. But this skill we are doing will help us exercise our mind so that it can perform any future tasks on the fly. We are lifting weights in the weight room right now, so no one is trying to pay us for this. But this will help prepare you for different unexpected situations in the future.&lt;/p&gt;&#xA;&lt;p&gt;I do also mention what Alex Gramatikov brought up: Why learn anything else in any other class? Each class has a set of key concepts or overall general learning goals that apply to life on the larger scale: History is about cause and effect and compromise, composition/literature is about writing and argumentation, science is about looking for evidence and testing hypotheses, etc. Each class has something useful to learn on the larger scale, but they use incredibly specific things (Like reading The Scarlet Letter, or learning about the structure of DNA ,or learning about the War of 1812) to convey those lessons.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastEditorUserId="8527" LastEditDate="2020-11-19T21:44:57.923" LastActivityDate="2020-11-19T21:44:57.923" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19176" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T14:47:56.000" Score="5" Body="&lt;p&gt;Many/most of the comments and responses have assumed mathematics to be entirely in the field of &lt;em&gt;hard&lt;/em&gt; sciences. I think that there is an argument to be had that mathematics as a field is also an &lt;em&gt;art&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Mathematics is beautiful. It is frequently located in the Arts and Sciences departments of colleges and universities. Other specific art subjects (e.g., painting, sculpting, sketching) are hardly ever criticized with the phrase &amp;quot;when will I ever use this?&amp;quot;. There are elements of every part of life that are more tedious and banal. Then there are parts that are invigorating and exciting to explore. These feelings are largely subjective to each person.&lt;/p&gt;&#xA;&lt;p&gt;I think that part of being a &lt;em&gt;human being&lt;/em&gt; is being able to find the beauty and enjoyment in the tasks that are otherwise tedious and banal. There is beauty in square roots in the same way there is beauty in paintings. Whether they're displayed in a museum or on your refrigerator.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2020-11-19T14:47:56.000" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19177" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T15:43:44.267" Score="10" Body="&lt;p&gt;I think there is also a case to be made for &amp;quot;passive&amp;quot; uses.  For any problem you encounter that you &lt;em&gt;know&lt;/em&gt; is a math problem, you can always research and figure out how to do it. But there are situations that you won't necessarily realize are math related, but knowing math might help you.  Getting scammed in pyramid and similar schemes is one such instance. Spotting lies is another.&lt;/p&gt;&#xA;&lt;p&gt;For example, the movie Bloodsport is purportedly based on real events in the life of Frank Dux who claims, among other things, to have 56 consecutive knockouts in a single-elimination bracketed tournament.  To someone without &amp;quot;advanced&amp;quot; math skills like square roots and exponents, this may seem plausible.  To those with such skills, it quickly becomes evident why such a claim is, to put it extremely mildly, a bit unlikely&lt;/p&gt;&#xA;" OwnerUserId="7522" LastActivityDate="2020-11-19T15:43:44.267" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19178" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T17:12:19.717" Score="5" Body="&lt;p&gt;Things taught at school can be divided into three rough categories:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Key skills. Examples: reading, basic arithmetic.&lt;/li&gt;&#xA;&lt;li&gt;Knowledge that should leave a trace, causing someone to ask questions they wouldn’t otherwise think of. Examples: “If I’m thirsty during my camping trip, will I be able to drink water from the river? I vaguely remember something about microorganisms, I better look this up.” “My shoelaces constantly come undone, I think someone told me it matters which loop goes on top?” “Can I get pregnant before marriage?”&lt;/li&gt;&#xA;&lt;li&gt;Specialized knowledge that’s really best acquired at places other than secondary school, and only by those who are interested in such topics. Examples: accurate names of organs of a cockroach, exact dates of historical events, coding skills.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Square roots in particular are somewhere on the boundary of 1 and 2. It’s easy to imagine quite practical questions which require this knowledge, e. g. what is the longest rod that will fit into this box, or the relationship between the weight of a pizza and its diameter etc.&lt;/p&gt;&#xA;&lt;p&gt;However, I have a feeling that if you’re confronted with this question, you’ve already lost and your students already consider mathematics a grudge, looking for excuses to avoid working on it as much as possible. Therefore, the best way to address the question is to prevent it from arising. Great amounts of literature have been written on how to teach mathematics in a way that it feels like playing and not like performing tedious calculations—not that many schools seem to implement any of it.&lt;/p&gt;&#xA;" OwnerUserId="10394" LastActivityDate="2020-11-19T17:12:19.717" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19179" PostTypeId="1" CreationDate="2020-11-19T17:26:12.480" Score="5" ViewCount="463" Body="&lt;p&gt;This question came up while teaching ~16 year olds binary numbers. Why do place values increase to the left and not the other way round?&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2020-12-13T09:29:23.690" Title="Why do we write numbers with decreasing place values?" Tags="|number-theory|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19180" PostTypeId="2" ParentId="19179" CreationDate="2020-11-19T17:39:33.523" Score="4" Body="&lt;p&gt;This is probably better suited to the History of Science and Mathematics SE, but I'll take a semi-informed stab at it.&lt;/p&gt;&#xA;&lt;p&gt;Since we read from left to right, placing the most significant digits in that same order give us the best opportunity to quickly comprehend the magnitude of the number.  For instance, the speed of light is 299,792,458 meters per second.  I think your brain is much more equipped to jump to &amp;quot;Oh, around 300 million&amp;quot; by seeing the most significant digits and the number of commas than if the number were given as 854,297,992.&lt;/p&gt;&#xA;&lt;p&gt;We obviously had a lot of history before the Hindu-Arabic numbers came into existence.  But even things like Roman numerals would give you the ability to just think about the degree of precision you wanted by ignoring the &amp;quot;digits&amp;quot; like I and V and X and L that you felt like ignoring.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2020-11-19T17:51:21.417" LastActivityDate="2020-11-19T17:51:21.417" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19182" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T19:39:24.090" Score="11" Body="&lt;p&gt;(This answer is specific to square roots, not to the broader question except as an example.)&lt;/p&gt;&#xA;&lt;p&gt;I occasionally use square roots to make sense of the news.  For example, suppose a fire has been reported to have burned 100,000 acres.  How big an area is that, in terms that make sense to me?  There are 640 acres per square mile, so divide by 640 and take the square root -- 100,000 acres is the area of a square 12.5 miles on a side.  Now that's a description that gives me a sense of scale in terms of the world I live in.&lt;/p&gt;&#xA;" OwnerUserId="14982" LastActivityDate="2020-11-19T19:39:24.090" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19183" PostTypeId="1" AcceptedAnswerId="19200" CreationDate="2020-11-19T21:24:56.797" Score="4" ViewCount="2680" Body="&lt;p&gt;After explaining some basic trigonometry to my kid, such as &lt;span class=&quot;math-container&quot;&gt;$\sin (\alpha+\beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$&lt;/span&gt;, Law of sines, Law of cosines, I wonder if there are some interesting problems for him to work on? Even better if it's a book.&lt;/p&gt;&#xA;&lt;p&gt;Example for &amp;quot;interesting&amp;quot;, &lt;span class=&quot;math-container&quot;&gt;$\sin 0 = \frac{\sqrt{0}}{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sin \frac\pi 6 = \frac{\sqrt{1}}{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sin \frac\pi 4 = \frac{\sqrt{2}}{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sin \frac \pi 3 = \frac{\sqrt{3}}{2}$&lt;/span&gt;, while &lt;span class=&quot;math-container&quot;&gt;$\sin \frac\pi 2 = \frac{\sqrt{4}}{2}$&lt;/span&gt;. Or more details in wiki page &lt;a href=&quot;https://en.wikipedia.org/wiki/Trigonometric_constants_expressed_in_real_radicals&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trigonometric constants expressed in real radicals&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Explanation in searching of interesting problems -- I believe exercises are necessary for one to really get the ideas and tricks of trigonometry, but they might also be boring, so if there are a bunch of interesting problems, that'll be perfect.&lt;/p&gt;&#xA;" OwnerUserId="9059" LastEditorUserId="3453" LastEditDate="2021-09-16T23:18:32.723" LastActivityDate="2023-12-23T15:20:36.163" Title="Interesting Trigonometry problems" Tags="|trigonometry|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19184" PostTypeId="2" ParentId="19164" CreationDate="2020-11-19T21:46:06.193" Score="4" Body="&lt;p&gt;I, my kids, and my grand children have all had toys where shaped blocks had to be placed inside a box with various shaped holes. None of the blocks would fit through a hole of a different size.&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/PK0d5.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PK0d5.jpg&quot; alt=&quot;Block Shape Toy&quot; /&gt;&lt;/a&gt;&#xA;W also had large colored buttons with holes in them. And colored string to feed through the holes.&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/rcwuX.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rcwuX.jpg&quot; alt=&quot;Colored Buttons&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;AS an adult, I can say that I have &lt;em&gt;NEVER&lt;/em&gt; had the need to perform either of these tasks as an adult. Not have my children. Why, then do these toys exist?&lt;/p&gt;&#xA;&lt;p&gt;Because they help develop basic skills that we will need later in our life. Babies have no idea why it is important to learn to place a square block in a square hole, and not a round hole. They cannot possibly understand the value of this activity. Their elders do.&lt;br /&gt;&#xA;Math is another form of skill development. It is a mental skill, not a physical skill.&lt;br /&gt;&#xA;Driving an automobile requires math skills, even though we don't use a calculator. We still have to estimate distance, speed, etc. Knowing if the cashier at the store is ripping you off is a math skill. After I pay my bills, how much money do I have left. If I invest money at x% interest, how much will I have .&lt;br /&gt;&#xA;&lt;br&gt;&#xA;That is why we learn math -- all type of math, in school.&lt;br /&gt;&#xA;$0.02&lt;/p&gt;&#xA;" OwnerUserId="12766" LastActivityDate="2020-11-19T21:46:06.193" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19185" PostTypeId="2" ParentId="19183" CreationDate="2020-11-19T21:48:20.437" Score="4" Body="&lt;p&gt;I remember enjoying simplifying trig expressions.&#xA;You can find exercises all over the web. One&#xA;source: &lt;a href=&quot;https://www.math-exercises.com/planimetry/trigonometry-and-trigonometric-expressions&quot; rel=&quot;nofollow noreferrer&quot;&gt;math-exercises.com&lt;/a&gt;.&#xA;Here's a snippet:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CVTrU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CVTrU.png&quot; alt=&quot;Trig&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;Two more sources: &#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_206_Precalculus/7%3A_Trigonometric_Identities_and_Equations/7.1%3A_Simplifying_Trigonometric_Expressions_with_Identities&quot; rel=&quot;nofollow noreferrer&quot;&gt;math.libretexts.org&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.onlinemathlearning.com/trigonometric-identities.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;onlinemathlearning.com&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-11-19T21:48:20.437" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19186" PostTypeId="2" ParentId="19164" CreationDate="2020-11-20T04:02:26.417" Score="4" Body="&lt;h2&gt;Because many of the cool jobs need maths.&lt;/h2&gt;&#xA;&lt;p&gt;Want to build robots? You need maths.&lt;/p&gt;&#xA;&lt;p&gt;Want to become a demolition expert? You need maths.&lt;/p&gt;&#xA;&lt;p&gt;Want to design fast cars or jet planes? You need maths.&lt;/p&gt;&#xA;&lt;p&gt;Want to design rockets that will take us to Mars? You need maths.&lt;/p&gt;&#xA;&lt;p&gt;Want to build computer games? You need maths.&lt;/p&gt;&#xA;&lt;p&gt;Want to make lots of money on the stock exchange or sports betting? You need maths.&lt;/p&gt;&#xA;&lt;p&gt;Want to research black holes and supernovas? You need maths.&lt;/p&gt;&#xA;&lt;p&gt;Want to design buildings and bridges? You need maths.&lt;/p&gt;&#xA;&lt;p&gt;Want to make your own AI-based human-to-animal face filter for Tiktok? You need maths.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Any answer to this question has to be &lt;em&gt;exciting and inspirational&lt;/em&gt;.&lt;/strong&gt; You're not going to motivate anyone with reasons like 'to allow you to get a job as a spreadsheet drone', even if that's the most likely actual reason for needing to learn maths. High-school kids haven't yet cottoned on to the fact that they'll most likely live a mediocre existence as adults. At that age, they're absolutely sure they'll become a rock star, the next Picasso, a professional athlete or the Prime Minister. Arguments towards pragmatism won't stick.&lt;/p&gt;&#xA;&lt;p&gt;It's important to give kids heroes of a sort. Not just the intellectual giants of the past, but people who, with just a little bit of maths and a lot of gumption were able to do something cool. Introduce youtube channels like &lt;a href=&quot;https://www.youtube.com/user/destinws2&quot; rel=&quot;nofollow noreferrer&quot;&gt;SmarterEveryDay&lt;/a&gt; and games like &lt;a href=&quot;https://en.wikipedia.org/wiki/Kerbal_Space_Program&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kerbal Space Program&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Poly_Bridge&quot; rel=&quot;nofollow noreferrer&quot;&gt;Poly Bridge&lt;/a&gt; if they're not already aware of them.&lt;/p&gt;&#xA;&lt;p&gt;There are plenty of really cool vocations out there that need maths that these kids will not even be aware of. It's also important to note that they don't need to be &lt;em&gt;Mega Maths Geniuses&lt;/em&gt; to have even a chance of getting these jobs, that even a little bit of basic maths knowledge will help them out down the line.&lt;/p&gt;&#xA;" OwnerUserId="14988" LastActivityDate="2020-11-20T04:02:26.417" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19187" PostTypeId="2" ParentId="19164" CreationDate="2020-11-20T08:57:56.787" Score="3" Body="&lt;p&gt;Most answers here (with the exception of @AndrewSanfratello's) focus on finding a practical &lt;em&gt;purpose&lt;/em&gt; of mathematics (money, brain training, problem resolution, finding a job, keeping all doors open, etc.).&lt;/p&gt;&#xA;&lt;p&gt;These are all a posteriori justifications of mathematics. Mathematics have not been &amp;quot;created&amp;quot; for money or to find a good job.&#xA;Mathematics, history, arts, physics, philosophy, etc. is what make mankind.&lt;br /&gt;&#xA;It could be a definition of the difference between people and animals: We do practice them because we want, because we're curious, because we ask questions, because we imagine, because we observe, because we don't understand, because we have something to voice.&#xA;What's the purpose of learning &amp;quot;History&amp;quot;? &amp;quot;Latin&amp;quot;? &amp;quot;Music&amp;quot;? &amp;quot;Physics&amp;quot;? At the end of the scholarship, only a very few part of everything you learned is &amp;quot;needed&amp;quot; in your work and day-to-day life.&lt;/p&gt;&#xA;&lt;p&gt;You could live without knowing anything about the history of mankind, or without any hint of imaginary numbers, sure. The point is to show that this would be a poor living, and there's something bigger that can give more meaning to our lives, because &lt;em&gt;we can think and create&lt;/em&gt;.&lt;br /&gt;&#xA;Obviously, no one will be an expert of all fields, and most will forget &lt;em&gt;almost&lt;/em&gt; everything about imaginary numbers, or the art of Dostoevsky, or the replication of a human cell. But anyone will know that it exists, anyone can focus on a given topic if they want. Anyone could freely think of it later in their life. Anyone has thoughts in common with the rest of the mankind.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The truth is: we (try to) teach children with basics of everything important for mankind. We try to educate them as intelligent people, not as animals who just try to fill their instinctive needs. We want to show them the beauty of the thought, and the greatness of eminent people who came before us.&lt;/strong&gt;&lt;br /&gt;&#xA;&lt;strong&gt;My suggestion is: you could simply explain this bigger-than-us truth instead of looking for low-brow explanations that will give them ground to easily counter-argue.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="14991" LastEditorUserId="14991" LastEditDate="2020-11-20T11:06:59.730" LastActivityDate="2020-11-20T11:06:59.730" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19188" PostTypeId="2" ParentId="19164" CreationDate="2020-11-20T12:45:51.760" Score="3" Body="&lt;p&gt;Although there are many good answers, I want to respond specifically to point 4 that you made there, that you believe the question &amp;quot;why learn this?&amp;quot; is distracting the student from actually learning.&lt;/p&gt;&#xA;&lt;p&gt;I want to argue that learning &amp;quot;why learn this?&amp;quot; is actually more important than learning the topic.&lt;/p&gt;&#xA;&lt;p&gt;I was always terrible at maths when I was young. I got a U in my Mathematics GCSE (that is, I did so badly I was &amp;quot;unmarked&amp;quot;, I basically sat an exam and was so bad at it, I might as well have handed in a blank paper).&lt;/p&gt;&#xA;&lt;p&gt;It was only when I went to University and really knuckled down to learn maths for my dream career (programming) that I really started to understand it. I was learning things that were so much harder than I started with, because &lt;strong&gt;I wanted desperately to learn it.&lt;/strong&gt; Sadly, I was still as good at math as a sack of potatoes.&lt;/p&gt;&#xA;&lt;p&gt;It was only when I begged my professor to sit down and explain to me &lt;strong&gt;why I needed to know these things&lt;/strong&gt; that I finally started to learn.&#xA;I started learning from the top down, not the bottom up.&#xA;My professor started by explaining that finding the Hypotenuse of a Triangle is why you cut across the grass of a park rather than walk down one side of it and then turn. How knowing the square root of a number lets you write up a grid for that thing with equal rows and columns. How I can use maths to draw shapes on a computer.&#xA;&lt;strong&gt;He explained to me in real terms what it means to use this particular parts of mathematics&lt;/strong&gt;, then I understood &lt;strong&gt;why my answers were wrong&lt;/strong&gt;, instead of just being told my numbers weren't right and to try harder next time.&#xA;Thankfully, programming gave me context on how things could be used. Programmers use a lot of maths. Other people's lives won't need that.&lt;/p&gt;&#xA;&lt;p&gt;All of that is to say: the &amp;quot;why learn this?&amp;quot; question can come from at least two places. It can be an excuse to not put in the effort as a &amp;quot;I won't need this anyway&amp;quot; response, or it can be the exact motivator to learn the skill in the first place. In the last 8 years since I started University, I haven't once learnt a mathematical technique I didn't understand the &amp;quot;why&amp;quot; of. Not one single time.&lt;/p&gt;&#xA;" OwnerUserId="14995" LastActivityDate="2020-11-20T12:45:51.760" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19189" PostTypeId="2" ParentId="19164" CreationDate="2020-11-20T16:05:51.850" Score="4" Body="&lt;p&gt;You have 2 packs of 34 paving slabs.  You want to lay a square patio.  How many slabs wide and long can your patio be?&lt;/p&gt;&#xA;&lt;p&gt;You are building a shed.  The wall is 6 feet tall, and 8 feet long.  If you need a diagonal brace for reinforcement, how long does it need to be?&lt;/p&gt;&#xA;&lt;p&gt;You have a cupboard that is 42&amp;quot; wide, and 25&amp;quot; tall.  Can you fit a 48&amp;quot; TV in it?&lt;/p&gt;&#xA;&lt;p&gt;Do you have enough ribbon to tie around that present?  Is your ladder tall enough to clean the gutters?  Will that new wardrobe fit up the stairs to your bedroom?  Will the party bunting reach between the ceiling beams?  What diameter of cake tin do I need, to ensure that it isn't too thick or too thin?&lt;/p&gt;&#xA;&lt;p&gt;Moving away from just square roots: If 1 18&amp;quot; pizza costs the same as 2 12&amp;quot; pizzas, which gives you more pizza?  How much tax do you need to pay, on purchases or income?  Which savings account or mortgage is better value?&lt;/p&gt;&#xA;&lt;p&gt;Which equipment or attacks are better in their video games?  Fixed or percentage buffs?  Big hits, or Damage-over-time?  Guaranteed small numbers, or a chance for a big number?  Should they increase Critical Chance, or Critical Damage?&lt;/p&gt;&#xA;&lt;p&gt;Finally: there are a lot of people and companies out there who make a &lt;strong&gt;lot&lt;/strong&gt; of money out of other people being bad at maths: do your students want to be their suckers and victims?&lt;/p&gt;&#xA;" OwnerUserId="10016" LastActivityDate="2020-11-20T16:05:51.850" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19190" PostTypeId="2" ParentId="19164" CreationDate="2020-11-20T22:46:58.480" Score="2" Body="&lt;h2&gt;You learn math to build on top of those before you&lt;/h2&gt;&#xA;&lt;p&gt;We only get so many years in our life.  The best of us barely scrape together a hundred of them.  So there is something to be said for building on top of the work of those before us.&lt;/p&gt;&#xA;&lt;p&gt;If you want to dance, you can always create your own dance.  Indeed, some would argue that is the truest way.  But typically we build on those before us -- those people who built and shaped ballet, tap dance, ballroom dancing, hip hop, and more.  Every one of these has a language associated with it, and every one will point out that it's not reasonable to get good at their art without practicing so that you understand what is meant when people use terms like a pirouette.&lt;/p&gt;&#xA;&lt;p&gt;If you want to play a sport, you can always pioneer your own way.  Indeed, the games have been improved by this thinking.  But typically we build on those before us -- those people who built strategies and practice regimes.  We turn to a good coach.  Every sport has a language associated with it, and every one will point out that it's not not reasonable to get good at their sport without practicing first so that you understand what people mean when they use terms like zone defense.&lt;/p&gt;&#xA;&lt;p&gt;Likewise, there are many things where, if you want to get better at them, you turn to a physicist or an engineer.  We have our own languages, and they happen to be heavily steeped in mathematics.  If you want to get good at things they do, or leverage their words to make your life better, you will need to practice their art, just as one would practice a sport or dance.&lt;/p&gt;&#xA;&lt;p&gt;So if a direct answer is what you seek, the direct value of learning mathematics stems from one's ability to tap into the work done before us which is written in that language.  What work you find of value is really a person-by-person thing.  Some might find value in seeing the equations behind mortgages rather than just signing the papers for one.  Another might appreciate the ability work on the undercarriage of their car safely because they understand geometry.  But the common thread in all of them is the need to be able to take advantage of the works of literally thousands of years of thinkers.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2020-11-20T22:46:58.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19191" PostTypeId="2" ParentId="17161" CreationDate="2020-11-20T23:56:20.093" Score="0" Body="&lt;p&gt;I would challenge the premise of your question, that nice quotes are particularly useful.&lt;/p&gt;&#xA;&lt;p&gt;At best, they are neutral, that is, they are on the wall somewhere and no one ever notices them. At worst, they are downright misleading and misrepresenting reality.&lt;/p&gt;&#xA;&lt;p&gt;Take your quote as an example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Struggling is a good thing… it’s where learning happens, it’s what we professors are always doing in our research… the struggle is the most interesting place to be.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Sounds nice, but it's dubious at best. Struggling isn't always conductive to learning. Students often struggle but don't learn anything. To the contrary, much learning happens when students are given tasks they can be successful with.&lt;/p&gt;&#xA;&lt;p&gt;If you want to get your students stop worrying about making mistakes, then the most important thing you have to do is make sure, they are not punished for making mistakes.&lt;/p&gt;&#xA;&lt;p&gt;Avoid even the most subtle things, that make mistakes look like a personality defect. Try to prevent situations where students are likely to make mistakes publicly, but when it exceptionally happens, make sure, that students do are not mean to each other, and enforce this rule consistently.&lt;/p&gt;&#xA;&lt;p&gt;The second thing is to teach them well and set them up for success, so even if they make mistakes occasionally, it reduces over time, so they start having a reason to believe in themselves.&lt;/p&gt;&#xA;&lt;p&gt;This will be difficult, as most students have prior negative experiences in math class, so you have to undo previous pedagogical mistakes.&lt;/p&gt;&#xA;&lt;p&gt;Decoration is superficial, behavior is what matters. And I argue, that quotes and posters have much less of an impact on behavior than people think. I would even go so far as to remove most unnecessary decoration from the classroom, to reduce possible distractions.&lt;/p&gt;&#xA;" OwnerUserId="14091" LastEditorUserId="14091" LastEditDate="2020-11-21T00:01:21.587" LastActivityDate="2020-11-21T00:01:21.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19192" PostTypeId="2" ParentId="19164" CreationDate="2020-11-21T08:10:11.733" Score="2" Body="&lt;p&gt;There has alreadby been several good and thoughtful answers, but I think what it boils down to is that maths is always useful, and most maths up to highschool level has many practical, everyday uses that others have already mentioned.&lt;/p&gt;&#xA;&lt;p&gt;It is also intensely boring to a teenager (unless you are a nerd, like me). At that age you are always questioning authority, and to make it worse, you have already had years of boredom imposed on you in the form of learning how to spell, add, multiply and so on, so it is no wonder that you are tired of it; but I don't think it is likely that you would find any teenager, who regrets being able to read and write (thus being able to play computer games, for example).&lt;/p&gt;&#xA;&lt;p&gt;The trick to learn boring things is always the same: keep at it, don't think about why, pace yourself with small, achievable targets in a daily routine, and celebrate once you done your duty for the day. It's like washing dishes: Not a bundle of joy, but you get through it.&lt;/p&gt;&#xA;" OwnerUserId="13424" LastActivityDate="2020-11-21T08:10:11.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19193" PostTypeId="2" ParentId="19158" CreationDate="2020-11-22T01:53:47.063" Score="1" Body="&lt;p&gt;Chemistry books through the 80s, at least, taught the use of log tables for doing arithmetic calculations, without stressing the teaching of logs as inverse power functions.  Purely as a sort of table based slide rule.  You can make hs physics problems easy with round numbers but it's pretty much impossible with chemistry, given molar masses, gas laws, moles, etc. etc.&lt;/p&gt;&#xA;&lt;p&gt;The need for this went away after mid to late 70s when hand calculators had become cheap and ubiquitous in chemistry classes.  But I still remember working some calculations with log tables in the late 70s, early 80s in chem classes. The AP Chem exam had log tables, as aids, into the 90s, but it was just inertia.  Scientific calculators were the norm.&lt;/p&gt;&#xA;&lt;p&gt;Not precisely same, but similar is the use of tables for future and present time value of money in engineering economics, rather than calculating the details algebraically. See for example the chapter within the EIT reference manual by Lindberg.&lt;/p&gt;&#xA;" OwnerUserId="15011" LastActivityDate="2020-11-22T01:53:47.063" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19195" PostTypeId="1" AcceptedAnswerId="19196" CreationDate="2020-11-22T09:40:15.590" Score="4" ViewCount="120" Body="&lt;p&gt;I want to give to my students instructions/advises for drawing (directed) graphs (with few nodes, less than 15) in draft mode (before drawing final one on the official test paper).&lt;/p&gt;&#xA;&lt;p&gt;But I really have no idea what to say that would be relevant... Myself, I do it quite randomly before finalize.&lt;/p&gt;&#xA;&lt;p&gt;Is there some relevant instructions that could be given?&lt;/p&gt;&#xA;" OwnerUserId="13795" LastActivityDate="2020-11-22T12:59:12.710" Title="Start drawing a graph" Tags="|advising|graph-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19196" PostTypeId="2" ParentId="19195" CreationDate="2020-11-22T12:59:12.710" Score="2" Body="&lt;p&gt;There is quite a bit of literature on this topic in the &amp;quot;Graph Drawing&amp;quot; community.&#xA;Although the goal is algorithms for automatically drawing graphs,&#xA;several papers offer criteria which could be followed by hand. I'll mention two.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1) Eades, Peter, and Lin Xuemin. &amp;quot;How to draw a directed graph.&amp;quot; In 1989 &lt;em&gt;IEEE Workshop on Visual Languages&lt;/em&gt;, pp. 13-14. IEEE Computer Society, 1989.&#xA;&lt;a href=&quot;https://doi.org/10.1109/WVL.1989.77035&quot; rel=&quot;nofollow noreferrer&quot;&gt;DOI&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here their 1st criterion aims for a downward flow, but the preferred direction&#xA;could be, say, rightward.&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/IiYLr.png&quot; width=&quot;500&quot; /&gt;&#xA;&lt;p&gt;Then one applies the criteria in order: C1, C2, C3, each step refining the&#xA;previous.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(2) Gansner, Emden R., Eleftherios Koutsofios, Stephen C. North, and K-P. Vo. &amp;quot;A technique for drawing directed graphs.&amp;quot; &lt;em&gt;IEEE Transactions on Software Engineering&lt;/em&gt; 19, no. 3 (1993): 214-230.&#xA;&lt;a href=&quot;https://doi.org/10.1109/32.221135&quot; rel=&quot;nofollow noreferrer&quot;&gt;DOI&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IYpCN.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IYpCN.png&quot; alt=&quot;Gasner&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iyCk3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iyCk3.png&quot; alt=&quot;csh&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2020-11-22T12:59:12.710" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19197" PostTypeId="2" ParentId="19183" CreationDate="2020-11-22T16:13:22.163" Score="13" Body="&lt;p&gt;When introducing the sum of angle equations, I practiced presenting this. Starting with introducing a right triangle inside a rectangle. (I misplaced my notes, in which I had every step clearly laid out.) In the end, it was a great proof of both equations.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NvoGa.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NvoGa.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-11-22T16:13:22.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19200" PostTypeId="2" ParentId="19183" CreationDate="2020-11-24T18:15:20.507" Score="6" Body="&lt;p&gt;There are many interesting trigonometry problems in &amp;quot;Trigonometry for the Practical Man&amp;quot; by James Edgar Thompson. Here are some from Chapter 5:&lt;/p&gt;&#xA;&lt;p&gt;Problem 1&lt;/p&gt;&#xA;&lt;p&gt;From the top of a mountain three miles above sea level, the angle of depression of the&#xA;ocean horizon is found to be 2° 13' 50&amp;quot;. Find the radius of the earth.&lt;/p&gt;&#xA;&lt;p&gt;Answer: 3960 miles&lt;/p&gt;&#xA;&lt;p&gt;Problem 2&lt;/p&gt;&#xA;&lt;p&gt;An observer sights a telescope on the sun, then turns the telescope straight up to the zenith, measuring an angle of 30° 0' 13.2&amp;quot;. At this same moment (by previous agreement), another observer 8280 miles away reports the sun is just setting. Find the distance from the center of the earth to the center of the sun. (You will need the earth’s radius found in problem 1.)&lt;/p&gt;&#xA;&lt;p&gt;Answer: 92,800,000 miles&lt;/p&gt;&#xA;&lt;p&gt;Problem 3&lt;/p&gt;&#xA;&lt;p&gt;From a point on the surface of the earth, the sun is sighted with a transit, first on one edge and then on the other, and the angle between the lines of sight is found to be 32' 4&amp;quot;. Find the radius of the sun. (You will need the answers to problems 1 and 2.)&lt;/p&gt;&#xA;&lt;p&gt;Answer: 433,000 miles&lt;/p&gt;&#xA;&lt;p&gt;Problem 4&lt;/p&gt;&#xA;&lt;p&gt;On January 1, an astronomer measures the sun-earth-star angle of the star Proxima Centauri, and finds it to be 95° 27' 45&amp;quot;, and six months later measures the angle and finds it has changed to 84° 32' 13.6&amp;quot;. Find the distance from the earth to the star on January 1. (You will need to use the earth-sun distance found in problem 2.)&lt;/p&gt;&#xA;&lt;p&gt;Answer: 4.63 light years&lt;/p&gt;&#xA;" OwnerUserId="306" LastActivityDate="2020-11-24T18:15:20.507" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19201" PostTypeId="2" ParentId="19164" CreationDate="2020-11-25T02:23:05.670" Score="0" Body="&lt;p&gt;You'll need square roots for high school chemistry and physics.  If you never take those classes, yes you dont need the roots. If you do, you will. This answer is pretty generalizable. Math is a tool of science and engineering. You can't even be a nurse without some of this stuff.&lt;/p&gt;&#xA;&lt;p&gt;And by tool, I mean a tool of the classes, not the practice. I worked as a mechanical engineer and it was all drafting and selecting pumps and valves out of catalogs on the job.  But the courses required math.&lt;/p&gt;&#xA;&lt;p&gt;If you are going to do a trade, then you don't need that stuff.  And dont need about half of high school.  Let alone college.  Then again our society way overemphasizes school. Yale or jail fallacy.&lt;/p&gt;&#xA;" OwnerUserId="15011" LastActivityDate="2020-11-25T02:23:05.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19202" PostTypeId="2" ParentId="19088" CreationDate="2020-11-25T07:51:54.117" Score="0" Body="&lt;p&gt;I was using Overleaf for my classes, and switched over to Notion (for other reasons as well). One of Notion's amazing abilities is to render LaTeX in real-time, both in-line or as a block equation (which you can set through Notion's properties and switch back if necessary).&lt;/p&gt;&#xA;&lt;p&gt;You can also just paste images and drag-and-drop every block everywhere. I heartily recommend it.&lt;/p&gt;&#xA;&lt;p&gt;Bonus: Notion has awesome database capabilities which can also help you track your problem-sets and all sorts of course-related materials.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2020-11-25T07:51:54.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19203" PostTypeId="2" ParentId="7509" CreationDate="2020-11-25T12:55:27.693" Score="3" Body="&lt;p&gt;3 blue 1 brown has an outstanding video on Fourier series:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=r6sGWTCMz2k&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=r6sGWTCMz2k&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thinking of a periodic complex valued map as a parametric curve in the complex plane, and thinking of the complex exponentials as &amp;quot;clocks&amp;quot; running at different speeds, the problem becomes:  how do we draw anything using a bunch of revolving clocks stacked center to tip?&lt;/p&gt;&#xA;&lt;p&gt;He develops the formula for the Fourier transform while keeping a tight correspondence between the symbolic and the graphical.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-11-25T12:55:27.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19204" PostTypeId="1" AcceptedAnswerId="19207" CreationDate="2020-11-26T12:20:10.807" Score="21" ViewCount="4665" Body="&lt;p&gt;I was aware that students find the definition of function too abstract and thus find it difficult.  However, I thought, once you understand functions, the concept of injective and surjective functions are easy.  True to my belief students were able to grasp the concept of surjective functions very easily.  On the other hand, they are really struggling with injective functions.  Even after spending a lot of time, they often say &amp;quot;a function is one-one if every element in the domain has a unique image&amp;quot;.  I am unable to help my students as I do not understand the difficulty they are facing.  And this makes me feel helpless and frustrated.  Do students in general find the concept difficult?  If yes, what makes the concept difficult?&lt;/p&gt;&#xA;&lt;p&gt;I could think of a possible explanation, but I was not so sure.  The definition contains a &lt;span class=&quot;math-container&quot;&gt;$p\implies q$&lt;/span&gt; statement and I know many students find logic difficult.  So, the students might be finding injective functions difficult for similar reasons.&lt;/p&gt;&#xA;&lt;p&gt;Finally, if the concept is not generally difficult for students, then I might be doing something wrong.  What are the common mistakes teachers make while teaching the concept?  Knowing this would help me steer off the common pitfalls.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance for any help you can provide.&lt;/p&gt;&#xA;&lt;p&gt;Edit 1:  These students are mathematics major students fresh out of school.&lt;/p&gt;&#xA;&lt;p&gt;Edit 2: I accepted an answer as the answer really flushed out the line of thought I had in mind. However, I would still love to see more answers and suggestions on how to make the concept easier for the student.&lt;/p&gt;&#xA;" OwnerUserId="13830" LastEditorUserId="13830" LastEditDate="2020-11-27T06:50:12.370" LastActivityDate="2020-12-01T19:55:45.973" Title="Why is the concept of injective functions difficult for my students?" Tags="|undergraduate-education|functions|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19205" PostTypeId="2" ParentId="19204" CreationDate="2020-11-26T13:12:03.517" Score="8" Body="&lt;p&gt;The only example you give of what's going wrong is students' definition using the words &amp;quot;a function is one-one if every element in the domain has a unique image.&amp;quot; I suppose it literally translates into a statement that for every x in the domain, there exists a unique y such that f(x)=y. This is really more like a description of the fact that in our standard way of defining functions, functions are always single-valued (although some people do still describe things like the inverse tangent as multiple-valued functions).&lt;/p&gt;&#xA;&lt;p&gt;If a student offered this definition, and I was concerned that it represented a conceptual misunderstanding, I would probe more to ask what they meant. For example, you could ask them whether the sine is one-to-one by their definition.&lt;/p&gt;&#xA;&lt;p&gt;If they respond with a correct conceptual explanation, but you're concerned about their ability to express themselves precisely in standard mathematical language, then that would be a separate issue. You could explain why the word &amp;quot;unique&amp;quot; in their phrasing is misleading. This would be a concern for math majors at the university level. If they're not going to be mathematicians, this seems to me like an issue of mathematical writing style that is not worth worrying about.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-11-26T13:21:20.410" LastActivityDate="2020-11-26T13:21:20.410" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19206" PostTypeId="2" ParentId="19204" CreationDate="2020-11-26T16:32:15.433" Score="8" Body="&lt;p&gt;In his Abstract Algebra book, John Fraleigh mentions that this is a common mistake beginning students make. Exercise 37 in Section 0 (7th edition) asks the reader to make a pedagogical case for using the terminology &amp;quot;two-to-two&amp;quot; instead of one-to-one. He doesn't expect the new terminology to actually take hold, of course, but just discussing it might make things clearer to students.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2020-11-26T16:32:15.433" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19207" PostTypeId="2" ParentId="19204" CreationDate="2020-11-26T16:47:24.790" Score="31" Body="&lt;p&gt;I think you will find that almost everyone has this problem when they first starting to learn rigorous mathematics, and many students will never overcome this difficulty.&lt;/p&gt;&#xA;&lt;p&gt;The following three statements are logically equivalent to each other, but students will almost all find the first to be easier to understand than the second, and will not be mature enough to really understand the third (although they may like it for its brevity).  However, mathematicians almost universally prefer the second definition.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f:X \to Y$&lt;/span&gt; is not injective if and only if there exist &lt;span class=&quot;math-container&quot;&gt;$x_1, x_2 \in X$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x_1 \neq x_2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x_1) = f(x_2)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f: X \to Y$&lt;/span&gt; is injective if and only if &lt;span class=&quot;math-container&quot;&gt;$f(x_1)=f(x_2)$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$x_1=x_2$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x_1,x_2 \in X$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f: X \to Y$&lt;/span&gt; is injective if and only if each element in the image of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has a unique preimage.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It is easy to show a function is not injective:  you just find two distinct inputs with the same output.  Students can look at a graph or arrow diagram and do this easily.  If given a function they will look for two distinct inputs with the same output, and if they fail to find any, they will declare that the function is injective.  This conception fits extremely well with the first definition.&lt;/p&gt;&#xA;&lt;p&gt;The second definition is equivalent, but this equivalence is not at all obvious to students.  They are not accustomed to performing logical transformations as complex as this &amp;quot;on the fly&amp;quot;.  However, mathematicians almost universally prefer this definition (and for good reason:  it leads to a much simpler proof structure when you actually want to prove that a function is injective, and it is much easier to use when you know a function is injective.)&lt;/p&gt;&#xA;&lt;p&gt;Here is the symbolic proof of equivalence:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;&amp;amp;\neg [ \exists x_1 \in X \exists x_2 \in X (x_1 \neq x_2) \wedge (f(x_1) = f(x_2))] \\&#xA;&amp;amp;\equiv \forall x_1 \forall x_2 \neg [(x_1 \neq x_2) \wedge (f(x_1) = f(x_2))]\\&#xA;&amp;amp; \equiv \forall x_1 \forall x_2 (x_1 = x_2) \vee \neg (f(x_1) = f(x_2))\\&#xA;&amp;amp; \equiv \forall x_1 \forall x_2 (f(x_1) = f(x_2)) \implies (x_1 = x_2)&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think students could have conceptual trouble with any line of this reasoning (even subconsciously).  For instance, they could have trouble negating universally quantified statements, they could have trouble with de Morgan's laws, although I expect that most of them will get hung up on the last step (realizing that &lt;span class=&quot;math-container&quot;&gt;$p \implies q$&lt;/span&gt; is equivalent to &lt;span class=&quot;math-container&quot;&gt;$\neg p \vee q$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The second definition is hard for students to swallow for a number of reasons:  we state that &lt;span class=&quot;math-container&quot;&gt;$f(x_1) = f(x_2)$&lt;/span&gt; in the hypothesis.  For many students, if we have given a different name to two variables, it is because the values are not equal to each other.  Then in the conclusion, we say that they are equal!  Then the student might say &amp;quot;why did I give them two different names in the first place!?!&amp;quot;.  We did so to imagine a world where &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt; are not equal, and yet their function values agree.  We want to rule that world out, which is why we have the implication.  Students are used to only thinking about the &lt;span class=&quot;math-container&quot;&gt;$T \implies T$&lt;/span&gt; part of the implication truth table, but here we are using that &lt;span class=&quot;math-container&quot;&gt;$T \implies F$&lt;/span&gt; is false:  we are ruling out the hypothesis being true while the conclusion is false by declaring that the implication is true.&lt;/p&gt;&#xA;&lt;p&gt;The third definition adds an extra complication.  We might say something like 3. as a clarifying remark.  Since it is mostly devoid of logical complication (these are swept under the rug with the words &amp;quot;image&amp;quot;, &amp;quot;preimage&amp;quot;, and &amp;quot;unique&amp;quot;), this feels easier for the student.  However, they are in danger because they do not actually have a rigorous understanding of the meaning of these words.  This is the definition I think they are trying to mimic when they say &amp;quot;a function is one-one if every element in the domain has a unique image&amp;quot;.  Some students may also make this error because of a faulty linguistic assumption:  thinking at &amp;quot;one to one&amp;quot; means &amp;quot;one element goes to one element&amp;quot;.  I love @Thierry's suggestion in another answer to replace &amp;quot;one to one&amp;quot; with &amp;quot;two to two&amp;quot; to address this linguistic trap.&lt;/p&gt;&#xA;&lt;p&gt;This is my diagnosis of the problem, but I do not have great solutions for you.  I see it in all of my upper level math classes:  discrete math (where they are first introduced to injectivity), linear algebra, number theory, abstract algebra, etc.  I do not have a good solution.  Occasionally I will meet with a student intensively in office hours and discuss these points for hours.  Some of these encounters do seem to fix the difficulty, but I cannot pinpoint what made it &amp;quot;click&amp;quot; for the student.  It seems like an extremely persistent problem, and is probably indicative of some deep barriers to advancing in mathematics which I do not currently understand or know how to help people overcome.&lt;/p&gt;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&lt;p&gt;One thing which has helped a bit (I think) is giving explicit names for the following distinct concepts:&lt;/p&gt;&#xA;&lt;p&gt;A relation &lt;span class=&quot;math-container&quot;&gt;$R \subset X \times Y$&lt;/span&gt; is said to be&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;single valued&lt;/strong&gt; iff &lt;span class=&quot;math-container&quot;&gt;$(x,y_1) \in R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x,y_2) \in R \implies (y_1 = y_2)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;total&lt;/strong&gt; iff for all &lt;span class=&quot;math-container&quot;&gt;$x \in X$&lt;/span&gt; there exist &lt;span class=&quot;math-container&quot;&gt;$y \in Y$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$(x,y) \in R$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;injective&lt;/strong&gt; iff &lt;span class=&quot;math-container&quot;&gt;$(x_1,y) \in R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x_2,y) \in R$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$x_1 = x_2$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;surjective&lt;/strong&gt; iff for all &lt;span class=&quot;math-container&quot;&gt;$y \in Y$&lt;/span&gt; there exist &lt;span class=&quot;math-container&quot;&gt;$x \in X$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$(x,y) \in R$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This way I can at least verbally discuss student errors:  when students give the false definition you mention I can say &amp;quot;No, you are just saying that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is single valued, which is part of the definition of a function&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-11-29T14:30:50.160" LastActivityDate="2020-11-29T14:30:50.160" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19208" PostTypeId="1" CreationDate="2020-11-27T05:24:35.300" Score="5" ViewCount="340" Body="&lt;p&gt;I am about to teach a second course in analysis for advanced undergraduate students. The students have already studied roughly the first eight chapters of Rudin's &lt;em&gt;Principles of mathematical analysis&lt;/em&gt;. They have also had a course in Linear Algebra.&lt;/p&gt;&#xA;&lt;p&gt;I get approximately 36 hours of lectures and the course culminates in the classical theorems of vector analysis. I plan to spend 9-12 hours on differential calculus in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; and 12-15 hours on vector analysis. This leaves me with around 12 hours of lectures for integration. The question is which integral to cover? It seems impossible to cover the Lebesgue integral in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; in just twelve hours given that the students have not seen the Lebesgue integral on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. On the other hand, looking at chapter 7 of Trench's textbook (&lt;a href=&quot;http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF&quot; rel=&quot;noreferrer&quot;&gt;http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF&lt;/a&gt;), the Riemann integral in several variables seems just as technical as the Lebesgue integral and I am not sure whether the effort has any payoff as the target audience consists of students who are definitely going to pursue mathematics in the future.&lt;/p&gt;&#xA;&lt;p&gt;My question is whether there is any exposition of the Lebesgue integral in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; that can be covered in twelve lecture hours. This exposition should contain proofs of both Fubini's theorem and the change of variables. I would also be happy if someone points out the advantages of just sticking to the Riemann integral. It seems to me that it is possible to give a more-or-less complete presentation of the Riemann integral in several variables within twelve lecture hours.&lt;/p&gt;&#xA;" OwnerUserId="15034" LastActivityDate="2020-11-30T20:29:52.717" Title="The Riemann integral vs Lebesgue integral in several variables for advanced undergraduates" Tags="|mathematical-analysis|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19209" PostTypeId="2" ParentId="19204" CreationDate="2020-11-27T07:48:11.573" Score="20" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;On the other hand, they are really struggling with injective functions. Even after spending a lot of time, they often say &amp;quot;a function is one-one if every element in the domain has a unique image&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Have you asked your students what they mean by &amp;quot;unique&amp;quot; when they say that?&#xA;The reason I'm asking is that, by some commonly used definitions of the word, this statement is perfectly correct!&lt;/p&gt;&#xA;&lt;p&gt;In particular, &amp;quot;unique&amp;quot; in English can have either of the following meanings (among others):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;single, without alternative, only one (as in &amp;quot;the equation has a unique solution&amp;quot;), or&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;distinct from all others of its kind, not the same as any other (as in &amp;quot;ammonia has a unique smell&amp;quot; or &amp;quot;each genuine dollar bill has a unique serial number&amp;quot;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Indeed, I would even go so far as to suggest that, outside certain branches of mathematics, the second meaning is probably more common.  And if that's the meaning your students are using, then they &lt;em&gt;do&lt;/em&gt; have the right mental concept, even if they're expressing it in an unfortunately ambiguous way.&lt;/p&gt;&#xA;&lt;p&gt;(Of course, the different meanings also have a lot of overlap in practice — for example, most serial and ID numbers tend to be unique in both senses of the word.)&lt;/p&gt;&#xA;&lt;p&gt;Your students may not realize this potential ambiguity or, even if they do, may not be able to come up with a good alternative way of saying what they mean.  Next time you encounter this phrasing, you might want to try simply prompting your students with a question like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;When you say that 'every element has a unique image', do you mean that no element may have two different images, or that two different elements may not have the same image?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That should at least give you a clearer idea of what your students are thinking, and whether their problem lies in understanding the concept or merely in explaining it.  And, at the same time, you're also subtly pointing out the ambiguity and suggesting alternative ways of phrasing similar statements in the future.&lt;/p&gt;&#xA;" OwnerUserId="526" LastEditorUserId="526" LastEditDate="2020-11-27T08:45:37.197" LastActivityDate="2020-11-27T08:45:37.197" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19210" PostTypeId="1" CreationDate="2020-11-27T14:33:25.557" Score="6" ViewCount="246" Body="&lt;p&gt;I've been recently thinking a lot about what would be the &amp;quot;best possible instruction method&amp;quot;, and in regards to that, I've been drawn the question of what would be possible if a student had unlimited access to a knowledgeable teacher as their main method of instruction (as opposed to supplementary tutoring). In particular, I'm thinking about how this would enable the student to take a very active role in their education, while having just-in-time answers/feedback from an expert, and maybe aspects of pull-based learning. This train of thought has led me to articles on home schooling, special education and music education, but nothing of a more academic nature and almost nothing quantitative.&lt;/p&gt;&#xA;&lt;p&gt;If I were to try to study this myself, I'd choose high-school or undergrad mathematics (e.g. calculus) as something where assessment is relatively straightforward. I'd then possibly take a cohort of reasonably competent and motivated students and attempt to measure the outcomes of a course of studies where each student in the experimental group has unlimited (e.g. up to 8 hours a day) access to one-on-one instruction from a knowledgeable and patient teacher. I'd then try to measure various aspects of the instruction and attempt to correlate them with learner outcomes. In particular, I'd be very interested in the effectiveness of different ways of structuring that learning time.&lt;/p&gt;&#xA;&lt;p&gt;As a side note, I'm well aware that this educational approach is not scalable, but would like to just get some reference measurement as to what is possible.&lt;/p&gt;&#xA;&lt;p&gt;If you are familiar with any studies like that, either in mathematics or any other field, I would very much appreciate your insights and/or links to relevant literature.&lt;/p&gt;&#xA;" OwnerUserId="688" LastEditorUserId="2083" LastEditDate="2020-11-27T14:52:22.253" LastActivityDate="2021-11-17T23:40:46.030" Title="Measuring the effectiveness of one-on-one instruction?" Tags="|reference-request|education-research|tutoring|general-pedagogy|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19211" PostTypeId="2" ParentId="19208" CreationDate="2020-11-28T10:52:33.440" Score="2" Body="&lt;p&gt;The word &lt;em&gt;covered&lt;/em&gt; is unproductive, as it is so teacher-oriented. Any amount of material can be &lt;em&gt;covered&lt;/em&gt; in 12 hours, provided that you can speak and write fast enough, and when you reach your limit, you prepare slides full with formulas and flip them faster than anyone can even read. The material is &lt;em&gt;covered&lt;/em&gt;, right?&lt;/p&gt;&#xA;&lt;p&gt;If you want a streamlined presentation, take a look at at Folland's &amp;quot;Real analysis: modern techniques and applications&amp;quot;. It &lt;em&gt;covers&lt;/em&gt; the required material on pages from 19-81, so 62 pages in total. So, you'd need to &lt;em&gt;cover&lt;/em&gt; about 5 pages per lecture hour. In my experience, anything faster than 3 pages/hour is unreasonable. And the text is terse and it doesn't get any more streamlined than that. But there are really not so many innovations in Folland, I feel that the material is standard and other sources cover it in essentially the same way, just add more details.&lt;/p&gt;&#xA;&lt;p&gt;To put it differently, in my university there is a separate 5cr. course (28 hours of lectures) called &amp;quot;Measure and integration&amp;quot;, that is supposed to &lt;em&gt;cover&lt;/em&gt; approximately first 60 pages of Folland (not including Fubini). And it does not have a reputation of the easiest of courses. Admittedly, course does feel slow to &lt;em&gt;me&lt;/em&gt;, and our students are not very strong, but then again, it is impossible to answer your question without knowing how strong your students are.&lt;/p&gt;&#xA;&lt;p&gt;I do share the sentiment that teaching Riemann integration instead of Lebesgue is feels like a waste of time. But it is a fact that Lebesgue theory takes a non-trivial time to learn.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2020-11-28T10:52:33.440" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19212" PostTypeId="2" ParentId="19204" CreationDate="2020-11-28T13:14:20.770" Score="1" Body="&lt;p&gt;You mention that these students are fresh out of high school. I'm not an educator but I can tell a little about how I first experienced this. What is hard in the beginning is to translate dense math language to an intuitive picture in your mind. You learn how to do this after some practice but in the beginning this can be daunting. You can guide your students in this proces by providing drawings and intuition and motivate the definition of a surjection.&lt;/p&gt;&#xA;&lt;p&gt;Examples of this could be&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Explain the word origin. Injection comes from &amp;quot;throw/hurl in/onto&amp;quot;. We ofcourse know injection as putting something in something else. Surjection is defined as “a throwing over or on; an exaggeration, a hyperbole”. Does this make sense for the definition?&lt;/li&gt;&#xA;&lt;li&gt;draw pictures (which you probably already do)&lt;/li&gt;&#xA;&lt;li&gt;give concrete examples and counterexamples and explain why they are injective/not injective&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Once you have solid intuition about injective functions the definition should become a natural statement. Like you mentioned these students don't have a solid background in proofs/logic so they may understand &lt;span class=&quot;math-container&quot;&gt;$f(a)=f(b)\implies a=b$&lt;/span&gt; on a loose level but fail to see what this implies &lt;em&gt;in practice&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6749" LastActivityDate="2020-11-28T13:14:20.770" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19213" PostTypeId="2" ParentId="19210" CreationDate="2020-11-28T14:05:55.150" Score="2" Body="&lt;p&gt;My answer is anecdotal, certainly not a reference to a study or anything of the sort.&lt;/p&gt;&#xA;&lt;p&gt;Background - I work in a high school, as an in-house tutor. I've also subbed, and co-taught a sophomore algebra class.&lt;/p&gt;&#xA;&lt;p&gt;Our sophomore math is offered at 4 different levels, presumably so an honors student on track for AP calculus (AP = Advanced placement, the student is able to take an exam that might earn them college credit. Some high school students graduate with as many as 5 classes of college credit) isn't in the same class as one struggling with middle school math they haven't mastered.&lt;/p&gt;&#xA;&lt;p&gt;The classes I am co-teaching this year have a total of 48 students in 2 sections. They are at the second level from the highest. In theory, in the great bell curve of understanding, they should seem to be sliced out rather sharply. In reality, they seem to be a normal distribution of their own. The top 1/6, often completing the week's assignment, (which I load Saturday morning, and mean for it to be done over the course of the entire week) before we meet for class Monday. The bottom 1/6, not looking at the assignments at all, or claiming to do them 'on paper' but never offering proof. Yet, still complaining the quiz was either too tough, or its formatting difficult to follow. (the week's assignment was exactly the same material with different numbers.) Note, my 1/6 is a reference to the bell curve, and my observation that teachers target their lessons to the 1st standard deviation, i.e. the middle 2/3. Yes, I believe that individual (or really, small groups) attention, can keep those students from falling behind.&lt;/p&gt;&#xA;&lt;p&gt;The above is my attempt to paint a picture showing that even with the attempted stratification, the range from low to high is still too wide. That top 1/6 should have been placed higher, the bottom, lower. Either way, my experience with students that visit me for tutoring, is that the one on one lets me focus on exactly where they are in their understanding, and help get them to the next level. I hear, on a daily basis, &amp;quot;Now I understand it, why doesn't my teacher do it this way?&amp;quot; In return, I offer that the teacher has so much time to get through so much material. There are ways of explaining any math topic a second or third way, if the time were available.&lt;/p&gt;&#xA;&lt;p&gt;I'd welcome a better system with what we have available. i.e. in my case, 500 students in their sophomore year, divided by level, into 20 classes. In the perfect world scenario, the range of understanding from low to high wouldn't feel so wide. The benefit of one on one or smaller groups than the 20+ we have comes from being able to focus at a certain level. And, while I agree that one on one might be very effective, it's impractical. Separating 500 students into 20 levels? A pipedream, perhaps, but the system we have now is broken, and any move towards this would be an improvement.&lt;/p&gt;&#xA;&lt;p&gt;Op asked - &amp;quot;do you have any experience/familiarity with this one-on-one approach used as the main method of instruction?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Again, my experience is anecdotal - 4 years ago, a sophomore came into my tutoring room. I knew him from the class I frequently subbed. He didn't need any help. He told me he wanted to skip the next year's math, and planned to visit me for my help. (One class session twice per week). He was very motivated, needed very little actual attention, except when hitting a bit of trouble. In the end, he passed the exam to skip Junior math. He gave me credit for helping. I think he'd have passed without me.&lt;/p&gt;&#xA;&lt;p&gt;Second anecdote - This June, after school ended, I heard from 3 students who wanted to summer study to go up a level (not quite a jump of a full year, but the result is to be able to take calculus as a senior vs the precalc course.) This was the summer we would mostly avoid going out due to Covid, and I welcomed the challenge. In this case, I gave them 8 weeks of 2 hours, twice per week. This material was what they'd have taken as seniors. At the end of the summer all three were accepted into the calculus course.&lt;/p&gt;&#xA;&lt;p&gt;Both anecdotes support the benefit of one on one, or very small groups, but in both cases, these were advanced students to begin with. I have no experience of offering one on one to a student as a main class.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-11-29T12:49:02.907" LastActivityDate="2020-11-29T12:49:02.907" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19214" PostTypeId="2" ParentId="14765" CreationDate="2020-11-28T17:29:36.070" Score="1" Body="&lt;p&gt;I am not a fan of the so-called &amp;quot;Cauchy induction&amp;quot; mentioned in the other answer, for the reasons mentioned in my comment there. So here is my own offered answer (taken from &lt;a href=&quot;https://matheducators.stackexchange.com/a/7984/1550&quot;&gt;an earlier post of mine&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{Z}{\to}\mathbb{R}$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f(0) = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(1) = 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x{+}1) + 6 f(x{-}1) = 5 f(x)$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$x {\in} \mathbb{Z}$&lt;/span&gt;, prove that &lt;span class=&quot;math-container&quot;&gt;$f(x) = 3^x - 2^x$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$x {\in} \mathbb{Z}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Any student that can give a correct proof of this statement has at least an intermediate level of understanding of induction. Actually, difficulty in understanding induction has nothing to do with &amp;quot;backward-forward&amp;quot;, but simply has to do with not being taught how to do rigorous reasoning. It definitely is impossible for anyone who is familiar with an FOL deductive system to be unable to understand induction properly, as it is completely obvious what &lt;em&gt;exactly&lt;/em&gt; induction means from the &lt;em&gt;nature&lt;/em&gt; of formal deduction itself, &lt;a href=&quot;http://matheducators.stackexchange.com/a/10033/1550&quot;&gt;as explained here&lt;/a&gt;. Furthermore, it is impossible for anyone who knows how to construct formal FOL proofs to use induction wrongly in any way whatsoever.&lt;/p&gt;&#xA;&lt;p&gt;So why is it that people do not want to teach students formal FOL deduction? I have no idea. In my teaching experience, it takes &lt;em&gt;much less time&lt;/em&gt; for students to learn to use a reasonable Fitch-style system (such as &lt;a href=&quot;http://math.stackexchange.com/a/1684204/21820&quot;&gt;this variant&lt;/a&gt;) than to use a common programming language, and once they learn it they will never again be unclear about what is illogical about their arguments, because it is simply an open-and-shut case whether or not they can produce a formal proof.&lt;/p&gt;&#xA;&lt;p&gt;In other words, the phenomenon of students not understanding induction is merely a &lt;em&gt;symptom&lt;/em&gt; of the real problem of not understanding &lt;em&gt;basic logic&lt;/em&gt;. Solve that true malady and all bad symptoms (including induction issues, quantifier swapping, choice confusion, and so on) disappear automatically.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2020-11-28T17:29:36.070" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19215" PostTypeId="1" CreationDate="2020-11-29T18:59:33.507" Score="-2" ViewCount="120" Body="&lt;p&gt;There must be an error in this partial integration but I do not see it. Do you see it?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/udtkG.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/udtkG.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10261" LastActivityDate="2020-11-29T20:17:42.757" Title="Finding an error in a partial integration" Tags="|integration|" AnswerCount="1" CommentCount="1" ClosedDate="2020-11-29T23:47:45.637" ContentLicense="CC BY-SA 4.0" />
  <row Id="19216" PostTypeId="1" CreationDate="2020-11-29T19:10:01.413" Score="2" ViewCount="715" Body="&lt;p&gt;The &lt;a href=&quot;https://www.maa.org/math-competitions/amc-1012&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMC 10/12 test&lt;/a&gt; is a test used in a math competition for high school students.&lt;/p&gt;&#xA;&lt;p&gt;I have a few students that know LaTeX who are very young and are extremely advanced for their age in high school. As a teacher, I feel it is my job to get into some of the top tier schools for math when they graduate high school.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt; When I was in college, I became obsessed with graph theory and loved it! I became a high school math teacher to give back to the subject.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As a high school math teacher, I quickly realized that I am awful when it comes to the AMC 10/12 tests. In my opinion, every question is really cute. I feel the test gauges one's ability to memorize tricks in a methodically roundabout way. I want to teach students proofs by induction, more about LaTeX, and how to conjecture and prove beautiful theorems in math. Not to focus on having them think a certain way in order to memorize a set of tricks to tackle a set of questions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\textbf{Question:}$&lt;/span&gt; So, what should I do to help students get into these top schools? How much do the AMC 10/12 tests help students to get into top tier schools? If I show them a ton of theoretical math in LaTeX that they keep track of, would that be enough to get them into a school like MIT, etc.? Or should I keep on learning about this test for the sake of their educational future and not knowledge?&lt;/p&gt;&#xA;" OwnerUserId="12366" LastEditorDisplayName="user507" LastEditDate="2020-11-30T18:03:34.887" LastActivityDate="2020-12-01T12:59:26.233" Title="Is the AMC 10/12 Test the Difference Maker for Top Schools? What do Colleges Look for?" Tags="|undergraduate-education|secondary-education|usa|math-contests|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19217" PostTypeId="2" ParentId="19215" CreationDate="2020-11-29T20:17:42.757" Score="4" Body="&lt;p&gt;The term in parentheses in the middle expression should be the change in &lt;span class=&quot;math-container&quot;&gt;$x\frac{1}{x}$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$x = 1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x = 2$&lt;/span&gt; (which evaluates to zero), not just &lt;span class=&quot;math-container&quot;&gt;$x\frac{1}{x}$&lt;/span&gt; (which evaluates to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;But as @SueVanHattum says, we are here to defend mathematics education, not to practise it.&lt;/p&gt;&#xA;" OwnerUserId="14522" LastActivityDate="2020-11-29T20:17:42.757" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19218" PostTypeId="2" ParentId="19216" CreationDate="2020-11-30T02:50:49.517" Score="5" Body="&lt;p&gt;Those tests are not key to admissions to undergrad, even at top schools. That will depend more on overall grades and standardized test scores.  What schools used to call the whole man.&lt;/p&gt;&#xA;&lt;p&gt;I would also be careful about only pushing what appeals to you.  I see that way too much from the proofy types. But other aspects may appeal to other people.  Realize also, that it's probably too early to predict the kids will even be math majors in undergrad. Just because they are bright and following advanced topics you like to show them doesn't mean they will follow your path.  Could go into physics or some other STEM.&lt;/p&gt;&#xA;&lt;p&gt;And the time to worry about top schools is grad school. At a different stage.  The top school undergrad thing is way too emphasized in the us.  Many kids in undergrad are better off with a small liberal arts school or an affordable public university than Harvard. At a minimum, the relative benefit of Harvard is marginal. What is more crucial is the ability of the student.&lt;/p&gt;&#xA;" OwnerUserId="15011" LastEditorUserId="2083" LastEditDate="2020-12-01T12:59:26.233" LastActivityDate="2020-12-01T12:59:26.233" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19219" PostTypeId="2" ParentId="19208" CreationDate="2020-11-30T19:01:08.343" Score="1" Body="&lt;p&gt;The discussion seems to overlook this simple fact: the Lebesgue integral is not a generalization of and cannot serve as a substitute for the Riemann integral.&lt;/p&gt;&#xA;" OwnerUserId="10168" LastActivityDate="2020-11-30T19:01:08.343" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19220" PostTypeId="2" ParentId="19208" CreationDate="2020-11-30T20:29:52.717" Score="3" Body="&lt;p&gt;Another option (I have never attempted this) would be to claim the existence of a thing called &amp;quot;The Lebesgue integral&amp;quot;, list some carefully chosen theorems about it as axioms (maybe just linearity, some inequalities, and dominated convergence), and treat it as a black box for your course.  You could give an extremely high level sketch of how this would be defined, and promise them that they could learn the construction in another course.  Give them the course number of the course at this university where measure theory is developed in detail for further questions.  Whether such an option would be suitable depends on your goals for this course.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-11-30T20:29:52.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19221" PostTypeId="2" ParentId="19204" CreationDate="2020-12-01T19:55:45.973" Score="1" Body="&lt;p&gt;I find the view of solving equations very helpful, when I explain injective and surjective functions:&lt;/p&gt;&#xA;&lt;p&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f:X\to Y$&lt;/span&gt; is&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;injective&lt;/em&gt; if the equation &lt;span class=&quot;math-container&quot;&gt;$f(x)=y$&lt;/span&gt; has always &lt;em&gt;at most&lt;/em&gt; one solution,&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;surjective&lt;/em&gt; if the equation &lt;span class=&quot;math-container&quot;&gt;$f(x)=y$&lt;/span&gt; has always &lt;em&gt;at least&lt;/em&gt; one solution, and&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;bijective&lt;/em&gt; if the equation &lt;span class=&quot;math-container&quot;&gt;$f(x)=y$&lt;/span&gt; has always &lt;em&gt;exactly&lt;/em&gt; one solution.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3398" LastActivityDate="2020-12-01T19:55:45.973" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19222" PostTypeId="1" AcceptedAnswerId="19231" CreationDate="2020-12-01T20:58:51.127" Score="7" ViewCount="391" Body="&lt;p&gt;Is there any single variable complex analysis textbook which uses &lt;span class=&quot;math-container&quot;&gt;$\textrm{d}\bar{z}$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Every single variable text I have found defines what a complex line integral with respect to &lt;span class=&quot;math-container&quot;&gt;$\textrm{d}z$&lt;/span&gt; means, and also &lt;span class=&quot;math-container&quot;&gt;$\textrm{d}s$&lt;/span&gt;, but none of them defines an integral with respect to &lt;span class=&quot;math-container&quot;&gt;$\textrm{d}\bar{z}$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$\textrm{d}\bar{z} \wedge \textrm{d} z$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I want to be able to write computations like&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;\int_{d\Omega} \frac{f(z)}{z-\zeta} \textrm{d}z &amp;amp;= \iint_\Omega d\left( \frac{f(z)}{z-\zeta}\textrm{d}z \right)\\&#xA;&amp;amp;= \iint_\Omega \frac{\partial f}{\partial \bar{z}} \frac{1}{z-\zeta} \textrm{d}\bar{z} \wedge\textrm{d}z \\&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;without needing to convert everything back into &lt;span class=&quot;math-container&quot;&gt;$\textrm{d}x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\textrm{d}y$&lt;/span&gt; terms. No textbook I have uses these symbols at all, much less develops how to do this kind of calculus.&lt;/p&gt;&#xA;&lt;p&gt;Something I find to be completely crazy is that I have never seen the following computation in print anywhere that I can remember, even though it is surely one of the most fundamental computations in all of complex analysis:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;\int_{S^1} \frac{1}{z} \textrm{d}z &amp;amp;= \int_{S^1} \bar{z} \textrm{d}z \textrm{,  since $\frac{1}{z} = \bar{z}$ on $S^1$}\\&#xA;&amp;amp;= \iint_D \textrm{d}\bar{z} \wedge \textrm{d}z\\&#xA;&amp;amp;= \iint_D 2i\textrm{d}x \wedge \textrm{d}y \textrm{, since $\textrm{d}\bar{z} \wedge \textrm{d}z = 2i\textrm{d}x \wedge \textrm{d}y$}\\&#xA;&amp;amp;=2\pi i \textrm{, since the area of the unit circle is $\pi$}&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think this is also because the textbooks essentially ignore non-holomorphic functions and their exterior derivatives entirely.&lt;/p&gt;&#xA;&lt;p&gt;I never felt like I really understood single variable complex analysis until I learned how to do these kinds of computations while learning several complex variables.&lt;/p&gt;&#xA;&lt;p&gt;I am looking for such a book because I cannot really imagine teaching the subject without these tools.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-12-02T00:49:08.793" LastActivityDate="2020-12-04T17:43:00.417" Title="Single variable complex analysis textbook which uses differential forms" Tags="|textbooks|complex-numbers|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19223" PostTypeId="2" ParentId="7406" CreationDate="2020-12-02T13:08:49.483" Score="0" Body="&lt;p&gt;Some Stanford lecturers are switching to &lt;a href=&quot;https://us.edstem.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ed discussion as an alternative&lt;/a&gt; based on this recent &lt;a href=&quot;https://www.stanforddaily.com/2020/10/04/concerned-with-piazzas-data-privacy-management-some-professors-look-to-alternative-discussion-forums/&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; from Stanford daily.&lt;/p&gt;&#xA;" OwnerUserId="15056" LastEditorUserId="15056" LastEditDate="2020-12-02T13:13:53.883" LastActivityDate="2020-12-02T13:13:53.883" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19224" PostTypeId="1" CreationDate="2020-12-02T13:22:49.697" Score="3" ViewCount="546" Body="&lt;p&gt;Tl;dr: I am trying to teach my sister how the LCM algorithm works but I just can't figure out how to explain it intuitively. The best explanation I can give is that you're trying to construct a number that contains the common factors of each number we are taking LCM of as once, but I'm not sure how to explicitly show the connection of this idea with the steps in the algorithm.&lt;/p&gt;&#xA;&lt;p&gt;The algorithm is discussed under the section &amp;quot;Using the table-method&amp;quot; in this Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Least_common_multiple&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2020-12-02T19:43:57.850" Title="How to explain the LCM algorithm to an 11 year old?" Tags="|secondary-education|teaching|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19225" PostTypeId="1" CreationDate="2020-12-02T13:52:03.010" Score="3" ViewCount="441" Body="&lt;p&gt;My son is in 5th grade (US) and since he is doing remote learning, we have been doing a lot of topics in pre-algebra just using worksheets.  I'd like to start him on a formal middle school curriculum, preferably one that has rigorous fundamentals, similar to the Spivak calculus book I used way back when.&lt;/p&gt;&#xA;" OwnerUserId="2154" LastActivityDate="2020-12-03T21:27:24.863" Title="Looking for a rigorous middle school self-study math course" Tags="|algebra|self-learning|curriculum|middle-school|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19226" PostTypeId="2" ParentId="19224" CreationDate="2020-12-02T16:00:24.723" Score="7" Body="&lt;p&gt;EDIT&lt;/p&gt;&#xA;&lt;p&gt;Here are the CCSS for grade 6.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.corestandards.org/Math/Content/6/NS/B/4/&quot; rel=&quot;noreferrer&quot;&gt;http://www.corestandards.org/Math/Content/6/NS/B/4/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;At this age, perhaps she is just expected to &amp;quot;find the LCM of two numbers less than or equal to 12&amp;quot;.  They do not specify a method, but I would still advocate using the definition (as presented below).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I would start by making sure she knows, and can compute with, the definition of the LCM.&lt;/p&gt;&#xA;&lt;p&gt;For instance, to find the least common multiple of 60 and 28.&lt;/p&gt;&#xA;&lt;p&gt;First list all the multiples of 60 and all the multiples of 28:&lt;/p&gt;&#xA;&lt;p&gt;60, 120, 180, 240, 300, 360, 420, 480, ...&lt;/p&gt;&#xA;&lt;p&gt;28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 208, 336, 364, 392, 420, 448, ...&lt;/p&gt;&#xA;&lt;p&gt;The least number which is a multiple of both 60 and 28 is 420.  Interestingly, if you continue the list, the next common multiple would be 840, and the next one after that would be 1260.&lt;/p&gt;&#xA;&lt;p&gt;Work with this definition exclusively for a while.&lt;/p&gt;&#xA;&lt;p&gt;For the small numbers your sister is working with, this method might be sufficient.&lt;/p&gt;&#xA;&lt;p&gt;However, you might want to develop some ideas which could &amp;quot;speed things up&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;For instance, in this example you might notice that since &lt;span class=&quot;math-container&quot;&gt;$60 = 4*3*5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$28 = 4*7$&lt;/span&gt;, that there is no chance of the multiple &lt;span class=&quot;math-container&quot;&gt;$60*1, 60*2, 60*3, 60*4, 60*5$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$60*6$&lt;/span&gt; being a common multiple, since none of these have a factor of seven (why?), and ANY multiple of 28 will have a factor of 7.  We could have skipped all of those, and the first number we check (in this case) would be 60*7, which actually worked.&lt;/p&gt;&#xA;&lt;p&gt;Developing this circle of ideas further can lead to the method that Dave L Renfro mentions in the comments.  Understand, however, that we are secretly invoking uniqueness of prime factorization when we make such arguments, and that theorem is not obvious.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-12-02T19:43:57.850" LastActivityDate="2020-12-02T19:43:57.850" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="19228" PostTypeId="1" CreationDate="2020-12-03T01:05:30.297" Score="2" ViewCount="103" Body="&lt;p&gt;Where could I find a list, and ideally discussion, of primary school mathematics schemes (textbook series) in common use in Australia and New Zealand in the period immediately following decimalization of their currencies in 1966, 1967 (say the next 10 or 15 years)? These may be with or without Imperial units of length, weight, etc.&lt;/p&gt;&#xA;" OwnerUserId="15058" LastActivityDate="2020-12-03T01:05:30.297" Title="Reference request for 1960s-1970s Australia/NZ primary-level textbooks" Tags="|reference-request|textbooks|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19229" PostTypeId="2" ParentId="19158" CreationDate="2020-12-03T01:58:20.447" Score="0" Body="&lt;p&gt;Generally, it's presumed students know about directed numbers (which may or may not be considered algebra), because otherwise they can't use logarithms of numbers less than 1.&lt;/p&gt;&#xA;&lt;p&gt;But apart from that proviso, the answer is that this was absolutely the case in some countries, before calculators replaced logarithm tables. For students aiming for O-levels in England (taken at age 16 by roughly the top 25% in overall academic ability), logarithms were typically introduced at age 12-14, with an emphasis on practical calculations but little on theory. While it is also true that students were acquainted with some algebra at this age, logarithms were generally presented only numerically at this stage.&lt;/p&gt;&#xA;&lt;p&gt;For example, if you look at &lt;a href=&quot;https://www.cambridgeassessment.org.uk/our-research/archives-and-heritage/past-exam-material/maths/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the old O-level syllabi of one examining board,&lt;/a&gt; you'll see that logarithm tables were used in exams as late as 1984, and they were explicitly part of the arithmetic syllabus as late as 1974.&lt;/p&gt;&#xA;&lt;p&gt;For examples of textbooks with early introduction of logarithms, you could have a look at &lt;em&gt;General Arithmetic for Schools&lt;/em&gt; by Durell, or &lt;em&gt;Arithmetic, Part II&lt;/em&gt; by Siddons, Snell and Lockwood.&lt;/p&gt;&#xA;" OwnerUserId="15058" LastActivityDate="2020-12-03T01:58:20.447" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19231" PostTypeId="2" ParentId="19222" CreationDate="2020-12-03T06:41:12.340" Score="5" Body="&lt;p&gt;The book you are looking for is:&lt;/p&gt;&#xA;&lt;p&gt;Complex Variables&#xA;An Introduction&lt;/p&gt;&#xA;&lt;p&gt;Authors: Berenstein, Carlos A., Gay, Roger&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, there is no way this book is an introduction except for truly ambitious students. I would classify it as an advanced graduate course. The book is very well-written and I think this is the only book in which the Weierstrass factorization theorem is proved using the solution to the &lt;span class=&quot;math-container&quot;&gt;$\overline{\partial}$&lt;/span&gt;-equation.&lt;/p&gt;&#xA;" OwnerUserId="15034" LastActivityDate="2020-12-03T06:41:12.340" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19232" PostTypeId="2" ParentId="19225" CreationDate="2020-12-03T19:24:53.230" Score="3" Body="&lt;p&gt;I recommend starting him with &lt;a href=&quot;https://beastacademy.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beast Academy&lt;/a&gt; (maybe even go down to their level 4 or below). Beast Academy is very fun; the rigor may not be obvious, but they are building thinking skills.  When he's done with BA, then go to &lt;a href=&quot;https://artofproblemsolving.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Art of Problem Solving&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-12-03T19:24:53.230" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19233" PostTypeId="2" ParentId="19225" CreationDate="2020-12-03T20:13:11.183" Score="3" Body="&lt;p&gt;You could consider the series distributed in the United States by singaporemath.com:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Primary Mathematics&lt;/em&gt; for grades 5 and 6&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;New Elementary Mathematics&lt;/em&gt; for grades 7 and 8.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;You can also find later volumes of &lt;em&gt;New Elementary Mathematics&lt;/em&gt; for grades 9 and 10 for sale on the internet. &lt;em&gt;Primary Mathematics&lt;/em&gt; has three versions adapted for the U.S., while &lt;em&gt;New Elementary Mathematics&lt;/em&gt; is not adapted (hence uses metric measures only).&lt;/p&gt;&#xA;&lt;p&gt;Secondly, for grades 5 to 10, you could consider the recent Australian series &lt;em&gt;ICE-EM Mathematics&lt;/em&gt;, now in its third edition, whose authors include a number of PhD mathematicians. It is published by the Cambridge University Press and comes recommended by Fields medalist Terence Tao.&lt;/p&gt;&#xA;&lt;p&gt;Unlike the &lt;em&gt;Art of Problem Solving&lt;/em&gt; series mentioned in the comments, the Singaporean and Australian series are intended for a wider range of abilities (although &lt;em&gt;New Elementary Mathematics&lt;/em&gt; is probably for the top 20 to 30% in Singapore schools, which would equate to a smaller percentage in the U.S.)&lt;/p&gt;&#xA;&lt;p&gt;An alternative to these would be various mid-twentieth century British textbooks aimed at higher achievers, starting from about grade 7 level. One would have to tolerate the fact that they are obviously so old-fashioned, but the level of competence developed in algebra, trigonometry and particularly geometry (in students of a mathematical bent) is without comparison to more recent textbooks.&lt;/p&gt;&#xA;" OwnerUserId="15058" LastEditorUserId="15058" LastEditDate="2020-12-03T21:27:24.863" LastActivityDate="2020-12-03T21:27:24.863" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19234" PostTypeId="1" AcceptedAnswerId="19494" CreationDate="2020-12-04T00:46:59.080" Score="10" ViewCount="471" Body="&lt;p&gt;I have seen (by some of my former instructors) the following strategy applied as an alternative to traditional &amp;quot;midterms and final&amp;quot; assessment in a math course:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Students take a quiz weekly. Each quiz question is tied to a specific standard. On its own, this could serve as a mix of formative/summative assessment, depending on whether the grading scheme includes midterms as well and what weight these quizzes are assigned.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The quizzes are graded and handed back fairly quickly, with some amount of feedback.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;On each week's quiz, there are additional questions which cover the previous week's &amp;quot;new standards&amp;quot;, though they might be different questions. Students who did not perform as well as they wanted the first time can also attempt these questions and replace their grade for the previous week's question.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I know that this is &lt;em&gt;a type of&lt;/em&gt; to &amp;quot;standards based grading.&amp;quot; However, I'd like to know whether there is a name for this specific assessment scheme other than &amp;quot;standards based grading&amp;quot; (since this assessment scheme refers to a specific implementation of standards based grading) and whether there is any research regarding whether this scheme is effective in a college math classroom -- both in instructor time and student learning. A Google search has proved fruitless, because I do not know what terms to search for.&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="2083" LastEditDate="2020-12-04T08:39:45.027" LastActivityDate="2021-02-17T22:14:10.517" Title="Weekly quizzes as an alternative for midterms? What is this called?" Tags="|undergraduate-education|terminology|assessment|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19237" PostTypeId="2" ParentId="19222" CreationDate="2020-12-04T17:43:00.417" Score="2" Body="&lt;p&gt;I do appreciate the idea of using &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial}{\partial\bar{z}}$&lt;/span&gt; so that one can bring non-holomorphic functions into the course, and applications to physics. Your question brings the Poisson kernel to my mind.&lt;/p&gt;&#xA;&lt;p&gt;But let's assume we are talking about a first course on functions of a single complex variable &lt;span class=&quot;math-container&quot;&gt;$w=f(z)$&lt;/span&gt;. A philosophy of the course is to proceed as one does in a course such as Calculus I, where the focus is a function of a single real variable &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt;. One learns the definition of &lt;span class=&quot;math-container&quot;&gt;$f'(z)$&lt;/span&gt;, about &lt;span class=&quot;math-container&quot;&gt;$\int_Cf(z)\, dz$&lt;/span&gt;, Taylor series &lt;span class=&quot;math-container&quot;&gt;$\Sigma c_n z^n$&lt;/span&gt;, etc. One comes to understand the similarities &lt;em&gt;and surprising differences&lt;/em&gt; between the two &amp;quot;categories&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$w=f(z)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So from this perspective, we should hesitate to focus too much on &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial f}{\partial\bar{z}}$&lt;/span&gt; because this suggests we are working in the framework of &lt;span class=&quot;math-container&quot;&gt;$w=f(z,\bar z)$&lt;/span&gt;, not the framework of a function of a single complex variable &lt;span class=&quot;math-container&quot;&gt;$w=f(z)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of course we can think of the function &lt;span class=&quot;math-container&quot;&gt;$g(z)=\bar z$&lt;/span&gt; in the framework of &lt;span class=&quot;math-container&quot;&gt;$w=f(z)$&lt;/span&gt;. But we quickly move past this example after discovering that &lt;span class=&quot;math-container&quot;&gt;$g'(z)$&lt;/span&gt; does not exist by the limit definition of a function of a &lt;strong&gt;single&lt;/strong&gt; variable &lt;span class=&quot;math-container&quot;&gt;$g'(z)=\lim_{h\to 0}\dfrac{g(z+h)-g(z)}{h}$&lt;/span&gt;. As instructor of such an introductory course, I would prefer to not be forced to dance around the difference (or abuse of notation?) between &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(z,\bar z)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When we get to a course in several complex variables with creatures like &lt;span class=&quot;math-container&quot;&gt;$f(z,w)$&lt;/span&gt;, which is akin to multivariable Calculus III with creatures like &lt;span class=&quot;math-container&quot;&gt;$f(x,y)$&lt;/span&gt;, we are forced to confront partial derivatives &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial}{\partial{z}},\dfrac{\partial}{\partial{w}}$&lt;/span&gt; from the very beginning. In this course, we are ready to open the flood gates to &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial}{\partial\bar{z}}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$  \dfrac{\partial}{\partial\bar{w}}$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2020-12-04T17:43:00.417" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19238" PostTypeId="2" ParentId="19234" CreationDate="2020-12-05T20:12:55.073" Score="3" Body="&lt;p&gt;This sounds like a method described in the following paper as &lt;strong&gt;second chance grading&lt;/strong&gt;. The logistics are slightly different (less frequent exams with groups of topics, as opposed to weekly quizzes) but the main idea is similar.&lt;/p&gt;&#xA;&lt;p&gt;An internet search for &amp;quot;second chance grading&amp;quot; didn't reveal much beyond this paper, but perhaps the term will start to be used for things like this. &lt;a href=&quot;https://matheducators.stackexchange.com/questions/19234/weekly-quizzes-as-an-alternative-for-midterms-what-is-this-called#comment53273_19234&quot;&gt;As one commenter said&lt;/a&gt;, &amp;quot;Not every practice in college teaching has a formal identity or name.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Source: Oscar E. Fernandez (2020) Second Chance Grading: An Equitable, Meaningful, and Easy-to-Implement Grading System that Synergizes the Research on Testing for Learning, Mastery Grading, and Growth Mindsets, PRIMUS, DOI: 10.1080/10511970.2020.1772915&lt;/p&gt;&#xA;&lt;p&gt;Link: &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/10511970.2020.1772915&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tandfonline.com/doi/full/10.1080/10511970.2020.1772915&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The image shown here is from that paper. (It's paywalled but FYI: MAA members have access to all &lt;em&gt;PRIMUS&lt;/em&gt; issues.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WuxT4.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WuxT4.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2020-12-05T20:12:55.073" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19239" PostTypeId="2" ParentId="19140" CreationDate="2020-12-08T00:31:18.623" Score="6" Body="&lt;p&gt;I am someone who had a lot of trouble with this, having no formal education in higher mathematics until very recently, and thus having to figure out what is meant by a &amp;quot;correct&amp;quot; proof all on my own.  This is a topic that is close to my heart. Specifically your quote here&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But I would say it happened organically. It would be the same way a native speaker grows up in their home country, hearing the language being spoken around them and gradually assimilating it themselves. This is NOT useful to a student taking their proof-based class who has to immediately learn to be 'fluent' in the language of proofs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is very uplifting.  I find that a teacher/writer &lt;em&gt;not&lt;/em&gt; taking this difference into account was the primary barrier, so just by realizing this I think you are 90% of the way there.&lt;/p&gt;&#xA;&lt;p&gt;For me, the process of learning how to write proofs took an enormously long time, and was extremely frustrating. I tried books like How to Solve it and other similar books, but they did not help me at all.&lt;/p&gt;&#xA;&lt;p&gt;The issue for me turned out to be two fold.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;These books focused on proof &lt;em&gt;strategies&lt;/em&gt; (i.e. induction, contradiction, breaking down a problem into smaller cases, etc)  and not mathematical &lt;em&gt;foundations&lt;/em&gt; (i.e. formal systems, formalizing common number systems, consistency and &amp;quot;well defined&amp;quot;-ness, etc). Jumping into the former without the latter, to me at least, is like learning to swim without a pool to swim in. That is,  without having seen a branch of mathematics built &lt;em&gt;completely from scratch&lt;/em&gt; either through set theory, type theory, or any other mathematical foundation of your choice, your students will have no way of &amp;quot;hooking&amp;quot; their knowledge on to something tangible, and everything will feel extremely floaty and ad-hoc. Like you are just making up the rules as you go along.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;These books did not touch on the practical and philosophical issues associated with proof. That is, they don't answer the important questions &amp;quot;How can we know that these formal symbols on a page actually represent what we want them to represent intuitively?&amp;quot;. Another way of putting it is: &amp;quot;How do you &lt;em&gt;formalize&lt;/em&gt; something that hasn't been formalized before? How do you go from an informal fuzzy ideas about laws of physics being the same for everyone to General relativity? How do you go from fuzzy ideas about a transformation being 'natural' to category theory?&amp;quot;. This may not seem important for basic understanding, but if you are expecting them to learn proof techniques in a &amp;quot;non organic&amp;quot; way, then it is essential that they have these questions answered. The answers to these questions are what are often assumed as given; they are the essence, I think, of what is gained &amp;quot;organically&amp;quot; in the manner you describe. Without the answer to these questions, formalization will feel cold and unsatisfying.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I think there are several ways to mitigate these problems. But for me what finally made it all click was seeing two things, corresponding to the two points above&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I saw the basics of Real Analysis built up entirely formally from the axioms of ZFC in Tao's introductory book on the subject. That is, he defines sets, then natural numbers, then uses natural numbers to define rational numbers, and then uses those to define real numbers. I can't overstate how helpful this was. I also saw Peano Arithmetic and set theory built up  completely from the ground up in &lt;em&gt;Type Theory and Formal Proof&lt;/em&gt; by Nederpelt and Guevers. This book indirectly taught me good formal &amp;quot;strategies&amp;quot; of proof.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I saw the historical evolution of basic topology from its&#xA;informal roots to its modern form, in &lt;em&gt;Proofs and Refutations&lt;/em&gt;. This allowed me to see how we got to the level of rigor we have today, and more practically, how real research works in the real world.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I would look into these books if you haven't seem them for ideas about how to get this &amp;quot;organically&amp;quot; gained knowledge out of your head and into your student's head. I am not exactly sure how this could all be fit into a one semester course, but it really didn't click for me until I read all three books.&lt;/p&gt;&#xA;&lt;p&gt;But the essence I think, is making sure that they have an example of the right way to make some mathematical theory completely rigorous, defining every term, proving every theorem, etc. And also making sure that they have a clear understanding of why this rigor is there, preferably through reference to the historical evolution of the theory and not artificially constructed examples. (e.g. Once upon a time there was no formal definition of a polyhedron, then chaos ensued, now we have topology, once upon a time there was no formal definition of a limit, then chaos ensued, now we have real analysis, once upon a time there was no formal definition of volume, then chaos ensued, now we have measure theory, the list goes on and on...)&lt;/p&gt;&#xA;" OwnerUserId="6708" LastEditorUserId="6708" LastEditDate="2020-12-28T21:08:47.457" LastActivityDate="2020-12-28T21:08:47.457" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19240" PostTypeId="2" ParentId="19140" CreationDate="2020-12-08T01:21:53.220" Score="3" Body="&lt;p&gt;For your second question (how to rapidly review basic proof skills in courses which require these skills):&lt;/p&gt;&#xA;&lt;p&gt;I give all students in my junior and senior level courses a link to Michael Hutchings' &amp;quot;&lt;a href=&quot;https://math.berkeley.edu/%7Ehutching/teach/proofs.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Mathematical Arguments&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I give a short quiz on basic proof skills the first day of class.  If you do well on the quiz, you can skip several assignments which are meant as &amp;quot;proof skill remediation&amp;quot;, and follow the material in the link.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2020-12-08T01:21:53.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19242" PostTypeId="2" ParentId="18965" CreationDate="2020-12-08T14:46:00.243" Score="3" Body="&lt;p&gt;My recommendation for a usable, inexpensive drawing tablet would be the XP-Pen StarG40. It's what I've been using this year (in conjunction with Blackboard/Collaborate) and costs about $40 as of this writing. It's the lowest-cost tablet I've seen, and it works reasonably well (there are encoding issues in Collaborate that make it always a bit jittery, but it's perfectly smooth in MS Paint, for example).&lt;/p&gt;&#xA;&lt;p&gt;At the start of 2020, I got this, and also a somewhat higher-end model from the same brand for my graphical artist partner. To our surprise, the lower-end model actually works quite a bit better. She's used much higher-end Wacom models in the past and certainly think that those give a better experience (at much higher cost). On my recommendation, we got these XP-Pen models for everyone in our mathematics department.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1R97k.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1R97k.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B078YR2MTF&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/dp/B078YR2MTF&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-12-08T14:46:00.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19245" PostTypeId="1" CreationDate="2020-12-09T19:41:17.497" Score="8" ViewCount="1577" Body="&lt;p&gt;&lt;strong&gt;NOTE:&lt;/strong&gt;&#xA;This question will soon be duplicated, as I didn't make clear that I was a high school sophmore in the beginning. At first I thought it didn't matter, and somewhat arrogant to mention, but in the comments of HFBrowning's answer, we see:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;@JohnClever KCd's observation is spot on. The fact that you are currently a student changes the advice everyone would be giving you; I am certain that all the answers you have received were geared around you being older. I would either heavily modify your question, or consider starting a different one, since I'm not sure this one (with all the answers given) can be salvaged in its current state. – HFBrowning 2 hours ago&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm planning on creating a 'math club' that is, in truth, a semester's course covering set theory, logic, and abstract and linear algebra.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Curriculum:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The bulk of the course is the algebra. The set theory and logic act as preliminaries and are covered in the first pair of months, where abstract and linear algebra are covered in the next two pairs of months respectively. That being said, although I can see myself replacing the algebra with combinatorics, analysis, or number theory, I see the foundations as essential.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Pacing:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;We'll do 4-6 problems a day, usually about a third to a half of each section's problems, and working around 2-5 hours each day is expected (a little under an hour for each problem).&lt;/p&gt;&#xA;&lt;p&gt;This gives about 6 sections a week, assuming you work on Saturdays. This is about a 1.5 chapters a week, and because there are 4.2 weeks in a month, that's 6.3 chapters a month, usually enough to move on to another book. I know this is rather fast, so I half the time to be 6.3 chapters every &lt;em&gt;two&lt;/em&gt; months, or one book every two months. This allows us to do 2.75 books by the end of the school year, and assuming that we leave out some stuff, it seems perfectly reasonable.&lt;/p&gt;&#xA;&lt;p&gt;School starts back up in January, and ends in June, leaving 5.5 months. Since the logic and set theory is rather short compared to either the linear or abstract algebra, we won't need 6 months.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Miscellaneous Things About the Course:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I have no teaching experience, being a high school sophomore myself.&lt;/li&gt;&#xA;&lt;li&gt;The course will be taught from Mendelson, (for logic and set theory) Lang (for abstract Algebra), and Axler (for linear Algebra), the books I, myself, used to learn these subjects.&lt;/li&gt;&#xA;&lt;li&gt;It is assumed that anyone who joins the club has a great interest in math, and is sufficiently hardworking.&lt;/li&gt;&#xA;&lt;li&gt;Although the previous condition is obvious insofar as it is &lt;em&gt;necessary&lt;/em&gt;, I have tried as much as possible to make it true insofar as it is &lt;em&gt;sufficient&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Thus, I have tried to make the curriculum require as few prerequisites as possible, and may even end up replacing linear algebra with something else to that end, considering the amount of calculus-based examples it often uses.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Has anyone else already done this? I would presume yes, but finding a specific example seems difficult.&lt;/p&gt;&#xA;&lt;p&gt;How do I recruit students? I suppose I could leverage the concept of a 'super-hard math class covering concepts you've never heard of,' but it doesn't really seem &lt;em&gt;true&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Does the curriculum make sense? In my experience, enumerative combinatorics would be the standard 'higher-math' example, but combinatorics is also the bane of my existence, and I hate it with every fiber of my being. If I had to replace the algebra, I could probably have abstract algebra be some point-set topology and a bit of measure theory, which I could use later on to teach fractal geometry, so that they finally understand what's on the posters to every math class in America.&lt;/p&gt;&#xA;&lt;p&gt;How do I keep the students interested? I would assume it would be easier to do than a required curriculum course, especially considering how much more interesting I would suppose pure math would be than the unmotivated drudgery that is, for example, pre-calculus, but I also think that the sort of people who love math in high school would be the kind of people that sign up, and they might be expecting something more computational compared to the abstract stuff we're doing.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Clarifications:&lt;/strong&gt;&#xA;I now realize that people thought I was referring to Lang's graduate textbook, which I am not. I'm &amp;quot;ambitious&amp;quot;, but not evil. I am instead referring to his undergraduate textbook.&lt;/p&gt;&#xA;&lt;p&gt;At our school, I know that that basic computational linear algebra is taught in precalculus, which is why I thought Axler was reasonable, perhaps supplemented by some 3blue1brown for good measure.&lt;/p&gt;&#xA;&lt;p&gt;Mendelson is required reading for many philosophy majors as an introduction to formal logic, so I hardly think it too harsh for a math enthusiast.&lt;/p&gt;&#xA;" OwnerUserId="13911" LastEditorUserId="13911" LastEditDate="2020-12-11T17:41:50.970" LastActivityDate="2020-12-11T17:41:50.970" Title="Advice on teaching abstract algebra and logic to high-school students" Tags="|undergraduate-education|secondary-education|mathematical-pedagogy|course-design|tutoring|" AnswerCount="7" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="19247" PostTypeId="2" ParentId="19245" CreationDate="2020-12-10T16:04:15.927" Score="18" Body="&lt;p&gt;Here's my advice.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I have no teaching experience.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Remedy that first before you lay out plans for a 6-month course of study. Find some way where you can teach just for a single day in some way at the high school that you're targeting. After that, find some way where you can teach for a week (i.e., say, 4 interlinked days). I'd suggest that you restart your math-club planning after that.&lt;/p&gt;&#xA;&lt;p&gt;I think most of us find that the very first day of teaching forces a radical shift in perspective. I had ideas about teaching and student capacities that were shattered into a million pieces based on that first day. You should get on the other side of that experience first.&lt;/p&gt;&#xA;&lt;p&gt;Frankly, the outline as given in the OP is galactically infeasible for high school students, and you'll find yourself very frustrated if you try to pursue anything like it.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-12-10T16:04:15.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19248" PostTypeId="2" ParentId="19245" CreationDate="2020-12-10T18:57:28.590" Score="8" Body="&lt;p&gt;I’d like to give you some ideas, based on my own musings about starting a math club, of what to teach instead of Lang and Axler. You say you want to attract motivated, hardworking students, but that’s not necessarily equivalent to “mathematically mature” students. They will have gaps in their knowledge, and as Daniel Collins said--and I can attest to--you may be surprised at how big and numerous these gaps can be! Things that are old hat to you will be &lt;em&gt;new and exciting to them&lt;/em&gt;. Seemingly simple things such as drawing a diagram to illustrate completing the square can be very helpful. Or drawing dots for even/odd numbers and seeing how the rules for combining them work out by pairing up leftover dots or seeing how a dot is still left over. Or going over complex numbers in a more geometric way, perhaps starting from the fact that multiplication by &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; rotates by &lt;span class=&quot;math-container&quot;&gt;$180^{\circ}$&lt;/span&gt; and then seeking a rotation by &lt;span class=&quot;math-container&quot;&gt;$90^{\circ}$&lt;/span&gt; to get negative squares. Don’t assume that they’ve seen &lt;em&gt;any&lt;/em&gt; of this stuff. (I’m stressing visualization a lot in this answer partly because I’m a visual learner but mostly because I think it’s neglected in the usual high school curriculum.)&lt;/p&gt;&#xA;&lt;p&gt;As for number theory, how about starting with “clock arithmetic” (no need for fancy, intimidating terminology yet) and literally drawing a clock and counting steps, just as they were introduced to arithmetic on a number line? You can then “skip count” and see the patterns that emerge, maybe even creating nice shapes in the process by connecting the numbers you hit with lines. When does skip counting make a shape with a vertex at every number? What’s the pattern between the number of “hours” on the clock, the number you skip count with, and the number of vertices? This already getting into pretty serious math.&lt;/p&gt;&#xA;&lt;p&gt;How about drawing Pascal’s (Khayyam’s? Yang Hui’s?) Triangle with just the dots, no numbers. Ask for the number of paths to a lower point starting from the top and going down exactly one level at each step and only one step left or right. Gets pretty messy. But the only way to get to a point is to first get to one of the two points directly above it, so there’s our recursion formula! Now how about the number of teams we can pick from a given number of players? Well, let’s put one player at each level of the triangle and go left if they’re on the team, right if not. Now there’s a one-to-one correspondence between teams and paths. If they can also solve this by basic counting, then there’s a proof that the numbers in the rows add up to powers of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;. And if they don’t know counting, then you can still get the result by arguing inductively (yet still informally), noting that the zeroth row sums to &lt;span class=&quot;math-container&quot;&gt;$2^0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Some final suggestions: do something like the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; naturals (not by induction at first, but perhaps come back to it later if you teach induction) then ask how you can use this to easily get the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; even numbers. From these two results, can you get the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; odd numbers? Now, start from scratch and find the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; odd numbers by adding L shapes to squares. Can you now use this to get the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; numbers? What about the formula for the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; squares? If they know the formula for the volume of a pyramid, can they get it by taking a pyramid and then cutting off parts of it to get a step pattern?&lt;/p&gt;&#xA;&lt;p&gt;Of course, you can introduce plenty of basic linear algebra as well, which I didn’t even attempt to outline here. Even if you do none of this, hopefully it gives you an idea of what (I think) may appeal to bright high schoolers. Good luck!&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2020-12-10T18:57:28.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19249" PostTypeId="2" ParentId="19245" CreationDate="2020-12-10T19:05:54.687" Score="11" Body="&lt;p&gt;Regarding &amp;quot;How do I recruit students:&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;You should &lt;strong&gt;start here&lt;/strong&gt; -- you have started with this cool thing you want to do, and are wondering how to do it. But you should instead try to find some of these students first. What do they want? Where are they?&lt;/p&gt;&#xA;&lt;p&gt;If you have a core of 1-2 excited parents and 2-3 excited students, &lt;strong&gt;then&lt;/strong&gt; you can start planning.&lt;/p&gt;&#xA;&lt;p&gt;I have run events with almost no students before. It is embarrassing and depressing. Don't make the same mistake.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2020-12-10T19:05:54.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19250" PostTypeId="2" ParentId="19245" CreationDate="2020-12-10T23:06:55.460" Score="5" Body="&lt;p&gt;The idea as you've outlined just doesn't sound like a math club. I would start by first asking yourself what it is that you want to achieve. Just some possibilities, are you looking to:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Encourage more local high school students to pursue math/see the&#xA;beauty in it?&lt;/li&gt;&#xA;&lt;li&gt;Give kids who are already into math a rigorous outlet?&lt;/li&gt;&#xA;&lt;li&gt;Promote logical thinking in general?&lt;/li&gt;&#xA;&lt;li&gt;Or is it that you want the experience of teaching, and these are the topics &lt;strong&gt;you&lt;/strong&gt; are interested in? (By the way this answer is really not a great one)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The answer to this question will probably be illuminating.&lt;/p&gt;&#xA;&lt;p&gt;If the answer lies closer to #1, that could be a good fit for a math club, if you are willing to change your approach to be (1) simpler/gentler, and (2) more focused on exploration and letting the students' interests guide the topics. You could end up with some students who think fractals are pretty, and want to know how they &amp;quot;work&amp;quot;. Or pre-engineering students who would like to explore math because of an interest in robotics. But at the age you're targeting, I really doubt any student (unless you're surrounded by prodigies) is going to have an interest in abstract algebra.&lt;/p&gt;&#xA;" OwnerUserId="15088" LastEditorUserId="15088" LastEditDate="2020-12-10T23:16:45.863" LastActivityDate="2020-12-10T23:16:45.863" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19251" PostTypeId="2" ParentId="19245" CreationDate="2020-12-11T01:03:25.303" Score="4" Body="&lt;p&gt;I'm sorry, but like many commenters I do think your plan is far far too ambitious. Going over that material (I only know about Lang and Axler personally) will not give students any motivation for the extremely abstract concepts, and I think motivation is crucial, especially at the high school level. At best, you will teach them this abstract math and they will follow what you tell them, but have no idea why you are considering the questions you are considering/why you are using the definitions you are using/what the concepts mean on an intuitive level.&lt;/p&gt;&#xA;&lt;p&gt;This summer, I started and taught an abstract algebra class for high schoolers (also my first), but I used the book &amp;quot;Abstract Algebra An Introduction, 3rd Edition by Thomas W. Hungerford&amp;quot; instead of say Lang or Dummit&amp;amp;Foote. This introductory text by Hungerford starts with super concrete examples like &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z$&lt;/span&gt; and modular arithmetic, allowing me to introduce fields and then rings. Then, we moved onto polynomial rings &lt;span class=&quot;math-container&quot;&gt;$F[x]$&lt;/span&gt; and began talking about quotients. Then we moved onto more abstract but basic ring theory (isomorphism theorems, ideals, basic facts like these), and then onto field theory. Unfortunately the summer ended before I could get to basic group and Galois theory (which I really wanted to get to because it's really cool and deep -- and so makes students feel good about themselves; and it's good advertising to potential students -- i.e. &amp;quot;wanna know why there's no quintic formula? Come find out with some awesome college-level math!&amp;quot;), but that's when I would have introduced groups -- after rings and fields.&lt;/p&gt;&#xA;&lt;p&gt;I think that introducing groups first is too abstract for first-time students, and only after gaining intuition about more general abstract algebraic structures like rings/fields, can I more easily generalize to groups (by saying &amp;quot;rings are designed to be 'like' &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$F[x]$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cdot$&lt;/span&gt; are defined, and in fact because of this we can make rings out of functions &lt;span class=&quot;math-container&quot;&gt;$f: X\to \mathbb Z$&lt;/span&gt;, but beyond addition and multiplication functions have a very important operation: composition. Thus we define groups based off of this idea of function composition&amp;quot; (thus shedding light for students on why associativity but not commutativity is included in the axioms).&lt;/p&gt;&#xA;&lt;p&gt;And then during Galois theory ideas like series, solvability, and simplicity fall out more naturally. If I had even more time in my summer class, after Galois theory, I would go back to develop more theory about rings (PID, domains, whatever) and/or groups (orbits/stabilizers, class formula, Sylow, more series, free groups, whatever).&lt;/p&gt;&#xA;&lt;p&gt;To summarize, my advice for you is the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Find easier textbooks, i.e. ones with more concrete examples/intuition/motivation. I've already recommended an algebra book that I like for these high school classes; I'm sure there are many linear algebra ones out there.&lt;/li&gt;&#xA;&lt;li&gt;Do not set expectations so high, for yourself or students. First times are always challenging, so be patient and keep and open mind. If you do this more years, it will only get better as you get more experience.&lt;/li&gt;&#xA;&lt;li&gt;If after going through these more concrete books you still have time, go ahead and use Lang or Axler or whatever. But I doubt that you will have time. But again, to reemphasize: DO NOT undervalue good motivation. I promise you if you take the time to guide your students to really get a tangible feel for the subjects through lots of examples/pictures/analogies, they will internalize it much better then if you talk about the snake lemma on week 2.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;As I do have a bit of experience on teaching an abstract algebra class for high schoolers, I'd be happy to give you more details (like what my lesson plans were, or how I ran the class) if you reach out by email (which you can find by digging around my profile). I do think my class turned out decently; although the students weren't many, those that stayed around were just normal high schoolers from my old high school, and they grasped the material decently well, though I did get the impression that even my &amp;quot;slower&amp;quot; curriculum was a bit too fast for them. Best of luck to whatever plan of action you will take!&lt;/p&gt;&#xA;" OwnerUserId="13084" LastEditorUserId="13084" LastEditDate="2020-12-11T01:37:06.907" LastActivityDate="2020-12-11T01:37:06.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19252" PostTypeId="2" ParentId="19245" CreationDate="2020-12-11T03:09:25.980" Score="3" Body="&lt;p&gt;This sounds like a super interesting course - but I'm inclined to agree with the other answers here, it's a &lt;em&gt;lot&lt;/em&gt;. As Daniel Collins and Thierry pointed out, there are some things about how students think that are hard to anticipate before you've spent some time teaching. I've had some similar difficult experiences myself - the first time I tried to teach a proper logic class was awful! I had this idea that we were going to try to build arithmetic up from foundations over the course of a five-week intensive class. That plan went out the window on day three.&lt;/p&gt;&#xA;&lt;p&gt;Here's some observations that I think are relevant to your plan as it currently stands.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Abstraction is &lt;em&gt;hard&lt;/em&gt;. I mean, really hard. Think about how students usually make the transition from arithmetic to basic algebra - it takes a year! And that's peanuts compared to the transition from ordinary algebra to abstract algebra. For that reason, it would actually make more sense to focus on linear algebra first - there's more opportunity to be concrete.&lt;/li&gt;&#xA;&lt;li&gt;In the same vein: Just because one topic is philosophically prior to another doesn't mean that it should be &lt;em&gt;taught&lt;/em&gt; first. Just like we don't try to teach Peano Arithmetic (a formalized system laying out the fundamental definitions of addition, multiplication, and so on) before teaching arithmetic, and no one introduces Dedekind cuts as a first presentation of irrational numbers, set theory and logic don't need to be covered in order to adequately prepare a student for abstract algebra. For a formal treatment, you need some basics; but it's fully possible to teach finite group theory with no understanding of set theory at all. I worked with a teacher once who successfully taught the basics of finite group theory to fifth-graders!&lt;/li&gt;&#xA;&lt;li&gt;It sounds like your planned class isn't a prerequisite for anything and doesn't have a standardized curriculum. That could be an opportunity - try an &lt;em&gt;open-ended&lt;/em&gt; class. Introduce some fundamental topics, and then choose what to do next based on the abilities and interests of the students. That'll help you adapt your expectations, too. I've only rarely had the opportunity to do this, but it can be a lot of fun, for you and the students. If you choose to do this, aim to teach two- or three-week &amp;quot;modules&amp;quot;, introducing a new topic or delving further into a topic you've previously covered.&lt;/li&gt;&#xA;&lt;li&gt;Students are slow. Obviously I don't mean that in an insulting way - but every single time I've taught a new topic, I've been surprised at how long it takes. And the younger the students (no matter how clever and motivated they are!) the longer it takes. For perspective: A typical high-school calculus class covers in about a year the amount of content covered in a quarter at a university (in my part of the US, at least). Combining this with the fact that a typical undergraduate abstract algebra class covers &lt;em&gt;part&lt;/em&gt; of one textbook, 3 in a year is asking high-school students to pack about six years of learning into one!&lt;/li&gt;&#xA;&lt;li&gt;Don't focus so much on textbooks. If the textbook was the most important thing, these highly motivated students you're planning to teach would just get the textbook and learn it themselves. Focus on what &lt;em&gt;you&lt;/em&gt; can offer - what can you explain well? What are your favorite examples of key ideas? What can be &lt;em&gt;made&lt;/em&gt; with the ideas you want to teach?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2020-12-11T03:09:25.980" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19253" PostTypeId="2" ParentId="19245" CreationDate="2020-12-11T15:15:28.663" Score="2" Body="&lt;p&gt;I agree with many others here that your plan in its current form is overly ambitious. I will not repeat the excellent advice given so far, but I will add that one way to attract interest and bring things down to earth would be to use or even focus on concrete applications of abstract algebra. Students may enjoy learning about secret sharing, secret codes and the like. See &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18977/applications-of-abstract-algebra-outside-of-mathematics-and-suitable-textbook&quot;&gt;this question&lt;/a&gt; for some ideas. (Full disclosure: I asked the question and posted one of the answers.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2020-12-11T15:15:28.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19255" PostTypeId="1" CreationDate="2020-12-11T18:17:01.917" Score="9" ViewCount="1283" Body="&lt;p&gt;I don't know where I can find a rigorous statistics course or textbook. The closest thing I can think of is measure-theoretic probability theory, but I wouldn't really call that &amp;quot;statistics&amp;quot;. By 'close to statistics', I mean something that, although as mathematically rigorous as probability theory, can be reasonably substituted for a statistics course with minimal study.&lt;/p&gt;&#xA;&lt;p&gt;Does such a thing exist? Perhaps not, as the point of many statistics courses is in their applications.&lt;/p&gt;&#xA;" OwnerUserId="13911" LastActivityDate="2022-06-19T06:47:30.460" Title="Statistics, for the mathematically rigorous" Tags="|reference-request|textbooks|self-learning|statistics|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19256" PostTypeId="1" CreationDate="2020-12-11T18:21:42.137" Score="1" ViewCount="452" Body="&lt;p&gt;For a project I am considering on geometry in US high schools, I need a list of all the reasons that are usually allowed in two column geometry proofs there. I have a bachelor's degree in math and education.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;&lt;p&gt;Dan&lt;/p&gt;&#xA;" OwnerUserId="6103" LastActivityDate="2020-12-11T18:40:57.063" Title="US High School Geometry: What are all the &quot;reasons&quot; allowed in two column proofs?" Tags="|geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19257" PostTypeId="2" ParentId="19256" CreationDate="2020-12-11T18:40:57.063" Score="5" Body="&lt;p&gt;We don't have any standards about that.  The Common Core only specifies that students learn to do proofs about lines, triangles, quadrilaterals, and circles.  Using two-column proofs isn't mandated, and neither are a list of which kinds of proofs you need to be able to demonstrate.  Individual states can set that for themselves, but they may even push it down to individual districts or even individual teachers.&lt;/p&gt;&#xA;&lt;p&gt;It's kind of a pain in the neck, to be honest.  I live in New York state which has had a uniform final exam in geometry for the past 150 years or so.  But I've never gotten a straight answer even in our state system of how many intermediate steps you need between knowing that two lines are parallel to knowing that their adjacent angles are congruent.  Some people argue that you always need to say that the angles are right angles by &amp;quot;definition of perpendicular&amp;quot; and then conclude that they are congruent because &amp;quot;all right angles are congruent&amp;quot;, but others say that's pedantry and you can say that all right angles are congruent in a single step.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2020-12-11T18:40:57.063" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19258" PostTypeId="1" CreationDate="2020-12-11T22:23:50.667" Score="8" ViewCount="218" Body="&lt;p&gt;One of the biggest revolutions in education in the last few years seems to be the use of artificial intelligence to offer adaptive individualized learning environments as done by companies like &lt;a href=&quot;http://squirrelai.com/&quot; rel=&quot;noreferrer&quot;&gt;squirrel&lt;/a&gt;, &lt;a href=&quot;https://www.knewton.com/&quot; rel=&quot;noreferrer&quot;&gt;knewton&lt;/a&gt; or &lt;a href=&quot;https://www.aleks.com/&quot; rel=&quot;noreferrer&quot;&gt;aleks&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Are the &lt;a href=&quot;https://www.oecd.org/pisa/&quot; rel=&quot;noreferrer&quot;&gt;PISA&lt;/a&gt; data detailed enough to measure effects by such AI based approaches in mathematics? If so, was this point investigated and what are the results?&lt;/p&gt;&#xA;" OwnerUserId="15093" LastEditorUserId="15362" LastEditDate="2021-01-27T11:26:14.313" LastActivityDate="2021-02-14T19:23:53.167" Title="Effects of AI based education in mathematics in PISA like studies" Tags="|reference-request|technology-in-education|pisa|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19259" PostTypeId="2" ParentId="19255" CreationDate="2020-12-12T00:07:27.167" Score="4" Body="&lt;p&gt;I would suggest &amp;quot;All of Statistics&amp;quot; by Wasserman. It is reasonably concise and moderate in its demands on background, but much more mathematically serious, also covering a much wider range of material, than a typical first course in statistics.&lt;/p&gt;&#xA;" OwnerUserId="5233" LastActivityDate="2020-12-12T00:07:27.167" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19260" PostTypeId="2" ParentId="19255" CreationDate="2020-12-12T02:31:28.260" Score="6" Body="&lt;p&gt;I am not familiar with this book, but the title alone suggests it might&#xA;be worth examining for your purposes.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Statistics for Mathematicians: A Rigorous First Course&lt;/em&gt;.&#xA;Victor M. Panaretos.&#xA;Compact Textbook in Mathematics. Birkhäuser/Springer 142 (2016).&#xA;ISBN-10 : 9783319283395.&#xA;&lt;a href=&quot;https://link.springer.com/book/10.1007%2F978-3-319-28341-8&quot; rel=&quot;noreferrer&quot;&gt;Springer link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Intended for students of Mathematics taking their first course in Statistics, the focus is on Statistics for Mathematicians rather than on Mathematical Statistics.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;      &lt;img src=&quot;https://i.stack.imgur.com/J7EET.jpg&quot; height=&quot;240&quot; /&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-12-12T02:41:02.597" LastActivityDate="2020-12-12T02:41:02.597" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19261" PostTypeId="1" CreationDate="2020-12-12T05:58:59.480" Score="0" ViewCount="114" Body="&lt;p&gt;(YES) - included;&#xA;(NO) - explicitly excluded;&lt;/p&gt;&#xA;&lt;p&gt;as stated from the IOI syllabus(2019);&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;5.2 Discrete Structures (DS)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;DS1. Functions, relations, and sets&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;(YES)Functions (surjections, injections, inverses, composition)&lt;/p&gt;&#xA;&lt;p&gt;(YES) Relations (reflexivity, symmetry, transitivity, equivalence&#xA;relations, total/linear order relations, lexicographic order)&lt;/p&gt;&#xA;&lt;p&gt;(YES) Sets (inclusion/exclusion, complements, Cartesian products, power sets)&lt;/p&gt;&#xA;&lt;p&gt;(NO) Cardinality and countability (of infinite sets)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;DS2. Basic logic&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;(YES) First-order logic&lt;/p&gt;&#xA;&lt;p&gt;(YES) Logical connectives (incl. their basic properties)&lt;/p&gt;&#xA;&lt;p&gt;(YES) Truth tables&lt;/p&gt;&#xA;&lt;p&gt;(YES)Universal and existential quantification(Note: statements should avoid definitions with nested quantifiers whenever possible.)&lt;/p&gt;&#xA;&lt;p&gt;(YES) Modus ponens and modus tollens&lt;/p&gt;&#xA;&lt;p&gt;(YES)Normal forms&lt;/p&gt;&#xA;&lt;p&gt;(NO) Validity&lt;/p&gt;&#xA;&lt;p&gt;(NO) Limitations of predicate logic&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;DS3. Proof techniques&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;(YES) Notions of implication, converse, inverse, contrapositive, negation, and contradiction&#xA;Direct proofs, proofs by: counterexample, contraposition, contradiction&lt;/p&gt;&#xA;&lt;p&gt;(YES) Mathematical induction&lt;/p&gt;&#xA;&lt;p&gt;(YES)Strong induction (also known as complete induction)&lt;/p&gt;&#xA;&lt;p&gt;(YES)Recursive mathematical definitions (incl. mutually recursive definitions)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;DS4. Basics of counting&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;(YES) Counting arguments (sum and product rule, arithmetic and geometric progressions, Fibonacci numbers)&lt;/p&gt;&#xA;&lt;p&gt;(YES) Permutations and combinations (basic definitions)&lt;/p&gt;&#xA;&lt;p&gt;(YES)Factorial function, binomial coefficients&lt;/p&gt;&#xA;&lt;p&gt;(YES) Inclusion-exclusion principle&lt;/p&gt;&#xA;&lt;p&gt;(YES) Pigeonhole principle&lt;/p&gt;&#xA;&lt;p&gt;(YES) Pascal’s identity, Binomial theorem&lt;/p&gt;&#xA;&lt;p&gt;(NO)Solving of recurrence relations&lt;/p&gt;&#xA;&lt;p&gt;(NO)Burnside lemma&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;DS5. Graphs and trees&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;(YES)Trees and their basic properties, rooted trees&lt;/p&gt;&#xA;&lt;p&gt;(YES)Undirected graphs (degree, path, cycle, connectedness,&lt;/p&gt;&#xA;&lt;p&gt;Euler/Hamilton path/cycle, handshaking lemma)&lt;/p&gt;&#xA;&lt;p&gt;(YES) Directed graphs (in-degree, out-degree, directed path/cycle, Euler/Hamilton path/cycle)&#xA;(YES) Spanning trees&lt;/p&gt;&#xA;&lt;p&gt;(YES) Traversal strategies&lt;/p&gt;&#xA;&lt;p&gt;(YES) ‘Decorated’ graphs with edge/node labels, weights, colors&lt;/p&gt;&#xA;&lt;p&gt;(YES) Multigraphs, graphs with self-loops&lt;/p&gt;&#xA;&lt;p&gt;(YES) Bipartite graphs&lt;/p&gt;&#xA;&lt;p&gt;(YES) Planar graphs&lt;/p&gt;&#xA;&lt;p&gt;(NO)Hypergraphs&lt;/p&gt;&#xA;&lt;p&gt;(NO)Specific graph classes such as perfect graphs&lt;/p&gt;&#xA;&lt;p&gt;(NO)Structural parameters such as treewidth and expansion&lt;/p&gt;&#xA;&lt;p&gt;(NO)Planarity testing&lt;/p&gt;&#xA;&lt;p&gt;(NO)Finding separators for planar graphs&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;DS6. Discrete probability&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Applications where everything is finite (and thus arguments about probability can be easily turned into combinatorial arguments)&#xA;are (YES), everything more complicated is (NO)&lt;/p&gt;&#xA;&lt;p&gt;I am preparing for IOI and want a good discrete structure book/resource to learn the above mentioned subjects.&lt;/p&gt;&#xA;&lt;p&gt;I am currently taking Pre-Calculus relevant Mathematics in my country. I am in 10th grade, India.&lt;/p&gt;&#xA;&lt;p&gt;I have searched for courses and books but I don't how to find a good one.&lt;/p&gt;&#xA;&lt;p&gt;Any recommendations and edits are welcome.&lt;/p&gt;&#xA;" OwnerUserId="15092" LastActivityDate="2020-12-12T05:58:59.480" Title="Which resource(s) to use to learn the following syllabus of discrete structures?" Tags="|discrete-math|resource-request|computer-science|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19262" PostTypeId="1" CreationDate="2020-12-12T13:41:02.137" Score="38" ViewCount="3870" Body="&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt; To cope with online education, I taught &lt;a href=&quot;https://en.wikipedia.org/wiki/Linear_algebra&quot; rel=&quot;noreferrer&quot;&gt;linear algebra&lt;/a&gt; using a variant of the flipped classroom.  I recorded videos and put them up on YouTube and students presented the content in these videos during class - sometimes in pairs, sometimes individually.  During the presentation, I used to point out gaps or lack of understanding.&lt;/p&gt;&#xA;&lt;p&gt;Long story short, the students did not gain much from the course - especially towards the end of the course.  My students did terribly in their final exam.  As they all did terribly, I feel I failed them.  I had a conversation with them and they said that me pointing out their mistakes or lack of understanding demotivated them.  Sending them to a vicious cycle and the result of the final exam is a consequence.&lt;/p&gt;&#xA;&lt;p&gt;I also taught an &lt;a href=&quot;https://en.wikipedia.org/wiki/Inquiry-based_learning&quot; rel=&quot;noreferrer&quot;&gt;IBL&lt;/a&gt; course and a colleague attended this course.  He too told me that I shouldn't say &amp;quot;You are wrong&amp;quot; - it demotivates the students.  At the time I had argued, &amp;quot;how else will they learn?&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is it wrong to say &amp;quot;you are wrong&amp;quot;?  If yes, how do we correct mistakes in a way that is not demotivating?  My policy has been, give elaborate feedback, but grade leniently.  But, feedback often meant I point out their mistakes.  Of course, I was also generous with praise.  How can I give feedback that is not demotivating?&lt;/p&gt;&#xA;&lt;p&gt;I didn't use the exact phrase &amp;quot;you are wrong&amp;quot;.  I used different phrases in different situations.  I mostly said the argument was wrong.  I do, however, say &amp;quot;I don't think you have understood this concept&amp;quot;.  Sometimes they reply I understand; I am just unable to explain.  So, I generally add the disclaimer - &amp;quot;I do not mean you did not understand anything, but there is some lack of understanding.  If you cannot explain, that means you have not understood the concept well enough&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="13830" LastEditorUserId="851" LastEditDate="2020-12-13T14:07:32.890" LastActivityDate="2020-12-16T17:42:47.257" Title="How can I give feedback that is not demotivating?" Tags="|undergraduate-education|student-motivation|" AnswerCount="6" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="19263" PostTypeId="2" ParentId="19262" CreationDate="2020-12-12T16:02:04.883" Score="27" Body="&lt;p&gt;I don't think your students are responding negatively to being told they're wrong. They're responding negatively to being told they're wrong &lt;em&gt;in this specific context&lt;/em&gt;. You've set them up in a very high-stress situation where they have to watch your lecture video and then immediately act like little professors. Since the video isn't interactive, they have no opportunity to ask questions if they're confused. They know that their understanding is weak at this stage, so they're being set up for failure when you ask them to do the presentations.&lt;/p&gt;&#xA;&lt;p&gt;A side note: -- I use the flipped-classroom technique myself, but I continue to be baffled by the custom of implementing this using videos of lectures rather than having students read the book. This seems to me like a ridiculous practice with no sane justification.&lt;/p&gt;&#xA;&lt;p&gt;If you're going to do the flipped classroom, then you should start with some reason to do that. That reason should be based on evidence, and the practices you implement should be ones that have evidence to show that they work. I think most people who use the technique successfully make sure that once the students are in person with you, they start off with easy tasks and have plenty of opportunities to get help and ask questions. Only later should they be asked to do more difficult things. In linear algebra, an easy task would be something like finding the dot product &lt;span class=&quot;math-container&quot;&gt;$\langle 1,0,0 \rangle \cdot \langle 0,1,0 \rangle$&lt;/span&gt;. A student who attempts this but makes a mistake will not feel that they have been set up for failure.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-12-12T16:02:04.883" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="19264" PostTypeId="2" ParentId="19262" CreationDate="2020-12-12T17:40:13.427" Score="12" Body="&lt;p&gt;An effective educator should not only point out a student's mistake, but also provide the student with the opportunity to apply their newfound knowledge so that it is more likely to stick with them.&lt;/p&gt;&#xA;&lt;p&gt;The example I'm about to provide relates to high school algebra, but I'm sure it can be applied to college level math such as Linear Algebra.&lt;/p&gt;&#xA;&lt;p&gt;The other day I was helping a student with solving radical equations. We were working on solving &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x^2-4x}=x-4$&lt;/span&gt;. I let the student work on solving this independently until they arrived at a solution. Early on, when the student tried to square both sides of the equation, they incorrectly expanded &lt;span class=&quot;math-container&quot;&gt;$(x-4)^2$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x^2-16$&lt;/span&gt; (this common mistake is known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Freshman%27s_dream&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Freshman's Dream&amp;quot;&lt;/a&gt; if anyone is curious). I informed the student of their mistake, but I did not fix their work for them. Instead, my goal was to remind the student how to square binomials (I assumed this was a prerequisite for being able to solve radical equations).&lt;/p&gt;&#xA;&lt;p&gt;Rather than letting them continue solving the original problem, we put it on hold and I computed &lt;span class=&quot;math-container&quot;&gt;$(x+3)^2$&lt;/span&gt; for the student, drawing colorful arrows that represented FOIL (that is my username after all &lt;span class=&quot;math-container&quot;&gt;$\ddot\smile$&lt;/span&gt;). Then I asked the student to expand &lt;span class=&quot;math-container&quot;&gt;$(2x+5)^2$&lt;/span&gt; to boost their confidence. Afterwards, the student said they were ready to return to the original problem (if this was not the case, then I would have provided more exercises until they felt ready). With their newfound confidence, the student was able to correctly expand &lt;span class=&quot;math-container&quot;&gt;$(x-4)^2$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x^2-8x+16$&lt;/span&gt; in the original problem. From there, the student had no trouble finding the solution.&lt;/p&gt;&#xA;&lt;p&gt;I hope this example shows that constructive feedback is insufficient unless the student is, if time permits, &lt;strong&gt;shown&lt;/strong&gt; what their mistake was. I personally don't mind making up my own problem on the spot and working it out for the student. I also want to mention that if this path is chosen, &lt;strong&gt;it's important to not focus on the problem that the student was stuck on.&lt;/strong&gt; Choose a different problem to solve for the student. Then if they can go back and correctly modify their calculations for the problem they were struggling with, they will not only be learning, but they will also feel much better about their ability to succeed in math.&lt;/p&gt;&#xA;" OwnerUserId="13657" LastActivityDate="2020-12-12T17:40:13.427" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19265" PostTypeId="2" ParentId="19262" CreationDate="2020-12-12T20:32:53.707" Score="16" Body="&lt;p&gt;Here are some rhetorical devices I use to deliver corrections in class that are very gentle to students' egos:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;You have a slice of something correct there, but maybe we should look more closely at...&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;I think you're definitely on the right path here, and what should we do about this part?&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;That would definitely be the right answer in [this other situation]. But notice that we have [this current situation]. What should we do with this?&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;Hmmm, is that really the definition of that object? Perhaps we should go back and review the exact definition.&amp;quot; [Side note: One reason why it's good to have digital slides is to allow a quick jump-back to prior materials like this.]&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;Interesting. Now, it's always good to double-check our work, so let's check the result by [some validation method]...&amp;quot; [delivered in a case when I know on sight that the answer is incorrect]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;That said: I don't think that any of this will be a silver-bullet for the challenges faced by your students. I think that almost all instructors are facing great difficulties in the current emergency-distance-learning environment, a steady dropoff in participation rates, etc. If students start falling behind in any STEM course, and need increased personal responsibility in distance learning, then the falling-behind will snowball and you'll get these kinds of results at the end. Students saying they're discouraged by corrections may be a convenient ego-saving scapegoat, but it's not necessarily the full story.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2020-12-12T20:32:53.707" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19266" PostTypeId="2" ParentId="19262" CreationDate="2020-12-12T21:59:56.153" Score="7" Body="&lt;p&gt;I'm not sure how well this suggestion would fit into your specific course set up, but I want to share something that has been helpful for me in other contexts:&lt;/p&gt;&#xA;&lt;p&gt;I like to set a general theme/expectation that submitted work from students is supposed to &lt;em&gt;be helpful for a classmate reading along&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, when I see something on a problem set that is not quite right, or correct but not detailed enough, I get to say something like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;&amp;quot;I think that &lt;em&gt;I&lt;/em&gt; see what you're getting at here, but I worry that a classmate would be confused.&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It acknowledges that they have something reasonable, that they're &amp;quot;on the right track&amp;quot;. But it also points out that their work could be improved in some way. I think this may help point out errors without &amp;quot;demoralizing&amp;quot; the student.&lt;/p&gt;&#xA;&lt;p&gt;It also may push them to think this way in the future and in other contexts. In my experience, it tends to lead to conversations about the &lt;em&gt;mathematical ideas&lt;/em&gt; themselves and &lt;em&gt;how best to present them&lt;/em&gt;, as opposed to how many &amp;quot;points&amp;quot; it's worth.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2020-12-12T21:59:56.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19267" PostTypeId="2" ParentId="19179" CreationDate="2020-12-13T01:39:57.710" Score="0" Body="&lt;p&gt;It makes perfect sense if you consider the full range of real numbers instead of just integers. In any base (&lt;em&gt;decimal, binary, hex, whatever&lt;/em&gt;) the &lt;strong&gt;radix point&lt;/strong&gt; (&lt;em&gt;decimal point in base 10&lt;/em&gt;) is in the middle with exponential powers increasing positively to the left and negatively to the right.&lt;/p&gt;&#xA;&lt;p&gt;For any base ß :&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;ßⁿ + ßⁿ⁻¹ + ... + ß¹ + ß⁰ . ß⁻¹ + ß⁻² + ... + ß⁻ⁿ&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;This works in binary as well as any other number base. In fact it used to be common to operate on scaled binary like this in lieu of floating point.&lt;/p&gt;&#xA;" OwnerUserId="10376" LastActivityDate="2020-12-13T01:39:57.710" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19268" PostTypeId="2" ParentId="19179" CreationDate="2020-12-13T09:29:23.690" Score="1" Body="&lt;p&gt;Our numbering system comes from copying the Hindu-Arabic numbering system.&#xA;They write from right to left, so in their system, they start from the least significant digit. However, taking over the ordering as-is was easier, otherwise operations would need to have been adapted. So we write numerals in the same order even if we generally write text in the opposite direction.&lt;/p&gt;&#xA;&lt;p&gt;Note, that some operations are more &amp;quot;natural&amp;quot; in one order, others in the opposite. Comparison and division are naturally done from the most significant digit, addition and multiplication from the least significant digit.&lt;/p&gt;&#xA;&lt;p&gt;Also note, that this is connected to &lt;a href=&quot;https://en.wikipedia.org/wiki/Endianness&quot; rel=&quot;nofollow noreferrer&quot;&gt;endianness&lt;/a&gt; which is relevant for computer systems.&lt;/p&gt;&#xA;" OwnerUserId="14091" LastActivityDate="2020-12-13T09:29:23.690" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19269" PostTypeId="2" ParentId="19164" CreationDate="2020-12-13T17:09:30.360" Score="0" Body="&lt;p&gt;You'll probably only use ten percent of the things you study, so of course it would be much more efficient to concentrate on just that ten percent.&lt;/p&gt;&#xA;&lt;p&gt;Pick now and limit your options forever.&lt;/p&gt;&#xA;" OwnerUserId="10376" LastActivityDate="2020-12-13T17:09:30.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19272" PostTypeId="1" AcceptedAnswerId="19279" CreationDate="2020-12-15T15:59:18.527" Score="6" ViewCount="174" Body="&lt;p&gt;I'm now semi-retired after a career teaching physics (and occasionally math) at a community college, and am looking for opportunities to volunteer with disadvantaged youth. During my college career, I organized and ran a walk-in tutoring center for the physics department, with a combination of faculty and students as tutors. Doing this on a volunteer basis at a high school seems like it would be an effective use of my skills and enjoyable to do.&lt;/p&gt;&#xA;&lt;p&gt;Compared to the community college program I supervised, it seems like what I'm trying to do now would be much less work for a high school to support, since it wouldn't require funding or space. (During COVID, this would be over zoom.) I'm treating this as a real job search, and have worked up a resume and job letter, and lined up references. I've researched high schools in my area to find ones that have a lot of students qualifying for free lunches, and have made attempts to establish contact, although so far I have not had any immediate nibbles.&lt;/p&gt;&#xA;&lt;p&gt;I suspect a barrier here is that most high schools do not have any preexisting setup of this kind, so it's not like volunteering for an organization that already has &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; volunteers and can just plug in volunteer &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt;. Can anyone describe a successful program of this type at a high school in the US that they're familiar with? I'm interested in things like where the tutoring happens physically (in non-COVID times), who hires and supervises tutors (assistant principal? department head?), and how students are made aware of the availability of tutoring (maybe by their teachers, with announcements via the Schoology courseware?).&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorUserId="2083" LastEditDate="2020-12-16T08:03:32.033" LastActivityDate="2020-12-16T08:03:32.033" Title="Typical structure for walk-in math tutoring in US high schools?" Tags="|secondary-education|tutoring|usa|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19273" PostTypeId="1" CreationDate="2020-12-15T17:36:59.637" Score="9" ViewCount="104" Body="&lt;p&gt;I am curious about the experiences of people who had no formal mathematical training in their youth, but learned mathematics as adults.&lt;/p&gt;&#xA;&lt;p&gt;I imagine that there are a fair number of people in the US who were homeschooled in such a way that mathematics was almost entirely omitted.  There is also the phenomenon of &amp;quot;Sudbury schools&amp;quot;, in which students are given complete freedom to choose how they spend their time.  Many of these students could experience an education which is completely free of instruction in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I am curious about how people with this background adapt to mathematics as adults.  How long does it take them to &amp;quot;catch up&amp;quot; to their peers who were trained in the normal system?  Are there any long term mathematical disadvantages or advantages in this population?&lt;/p&gt;&#xA;&lt;p&gt;I would like to be made aware of research studies tackling these kinds of questions.  If they do not exist, I am serious about trying to undertake such research.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2020-12-15T19:51:49.207" LastActivityDate="2020-12-15T19:51:49.207" Title="Research into the mathematical education of adults with no prior formal mathematical experience" Tags="|education-research|undergraduate-research|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19274" PostTypeId="2" ParentId="19272" CreationDate="2020-12-15T18:51:35.910" Score="4" Body="&lt;p&gt;One factor that makes this endeavor more difficult has to do with background checks and liability. The school district will have policies and procedures that make it more difficult to allow volunteers to be present in the building. Oversight of non-employees becomes yet another ball for an administrator to juggle. So accept that your work might have to be taken &amp;quot;off site&amp;quot; and not be officially sanctioned by the school.&lt;/p&gt;&#xA;&lt;p&gt;Some ideas:&lt;/p&gt;&#xA;&lt;p&gt;Make inquiries with people on the local school board, and not just high school teachers and principals.&lt;/p&gt;&#xA;&lt;p&gt;Look at United Way projects, or propose one yourself. Often a collaboration with a community organization such as ones for Hmong and Southeast Asian communities can be productive.&lt;/p&gt;&#xA;&lt;p&gt;Investigate the MAA Tensor SUMMA program&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/programs-and-communities/outreach-initiatives/tensor-summa-grants&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/programs-and-communities/outreach-initiatives/tensor-summa-grants&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is a summary:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The Tensor Foundation has provided funding for the MAA to award Tensor SUMMA (Strengthening Underrepresented Minority Mathematics Achievement) grants for projects designed to encourage the pursuit and enjoyment of mathematics by students who are members of groups historically underrepresented in the field of mathematics. These include students who are African-American, Native American, Hispanic, or Pacific Islander. If your project will be targeted toward another group that you believe would qualify the project, you may wish to consult with us before submitting your proposal. Projects may be designed for middle school students, high school students, or college/university students.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Partner with a regional college or university that has a secondary-education program to find undergraduates interested in service-learning, or resume development. Also see if the regional college or university has a continuing education unit that might facilitate a program.&lt;/p&gt;&#xA;&lt;p&gt;I have seen many such models in my career. They all come with frustration and challenges, but with patience and willingness to adapt, they are rewarding and serve good social purpose.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2020-12-15T19:24:58.753" LastActivityDate="2020-12-15T19:24:58.753" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19276" PostTypeId="2" ParentId="19262" CreationDate="2020-12-15T21:11:01.907" Score="2" Body="&lt;p&gt;If you meet 3 times weekly, allocate 2 of the 3 days to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/19262/how-can-i-give-feedback-that-is-not-demotivating#comment53372_19264&quot;&gt;doing justice to the syllabus&lt;/a&gt; and the third to an open, but directed solving of &lt;em&gt;homework problems&lt;/em&gt; (beware of undirected, open discussion, which will simply waste time).&lt;/p&gt;&#xA;&lt;p&gt;Ideally you'll be able to&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;more rapidly cover the material&lt;/li&gt;&#xA;&lt;li&gt;discover what students are missing&lt;/li&gt;&#xA;&lt;li&gt;provide a better opportunity for students to teach in an inverted classroom&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The discussion day is a terrific opportunity for your &lt;em&gt;positive&lt;/em&gt; feedback; students will listen in a better light, ask better, and try to understand because it's worth &amp;quot;free&amp;quot; points on their homework.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://academia.stackexchange.com/a/116463/68971&quot;&gt;My answer to a similar trouble on Academia SE&lt;/a&gt; covers the process and structure in detail&lt;/p&gt;&#xA;" OwnerUserId="9273" LastEditorUserId="9273" LastEditDate="2020-12-15T21:31:45.463" LastActivityDate="2020-12-15T21:31:45.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19277" PostTypeId="2" ParentId="186" CreationDate="2020-12-15T22:07:49.390" Score="0" Body="&lt;p&gt;I will add to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/186/good-definition-for-introducing-real-numbers/199#199&quot;&gt;this answer&lt;/a&gt;. I think only the second way or something like it is something everyone can learn in elementary school with no struggle as long as they still teach fractions their current ways. The third one was Dedekind cuts but they probably meant of rational numbers which is very confusing to introduce from scratch using fractions. According to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14831/how-to-explain-fractions-to-7-year-old-kid/14833#14833&quot;&gt;this answer&lt;/a&gt;, people 14-18 struggle with fractions.&lt;/p&gt;&#xA;&lt;p&gt;I read the body of the question &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13150/how-to-teach-sum-of-fractions-to-students&quot;&gt;How to teach sum of fractions to students?&lt;/a&gt;. It was very confusing what they were saying. I thinking giving students a long list of direct instruction on how to add fractions is why they can't learn it. I think the task teachers gave them was to learn how to show their work and learn what the direct instruction on how to add them is. I think not all students can learn how to do that when they're in elementary school. I think the whole system on everything should be changed to not need them to learn it then anymore. Instead, teachers should do something like demonstrate the following example. They could ask &amp;quot;What is &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4} + \frac{5}{6}$&lt;/span&gt;? Then they will let the students think a little bit. Then they will demonstrate that it can be rewritten as &lt;span class=&quot;math-container&quot;&gt;$\frac{9}{12} + \frac{10}{12}$&lt;/span&gt;. Then they will give homework problems like that but only ask them to give the answer and not ask them to explain it. Naturally, they will ask themselves &amp;quot;I remember there was a way to do it. What was it?&amp;quot; without realizing it. They will find their own method in their head to do it. Maybe they will get the students to challenge themselves to get the answer with them. Then afterwards, they could tell them that what they were doing was taking the lowest common denominator. I'm not sure they will even figure out that that means lowest common multiple of the denominators or what lowest common multiple means. So they will tell them that the new denominator was the lowest common multiple of those numbers and be more direct and not even say &amp;quot;lowest common denominator&amp;quot; yet.&lt;/p&gt;&#xA;&lt;p&gt;Later on, I'm not sure the best way to introduce irrational numbers if they teach them fractions first. I suppose they could then introduce a discrete graph for each rational number that has the slope of that rational number, that is the function that assigns to each integer the floor function of that integer times that rational number For example, this would be the graph for the rational number &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{7}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NMBNj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NMBNj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then they could demonstrate the following example. Take the following partial graphs whos entirety are the images.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jtZy2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jtZy2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt; &amp;amp; &lt;a href=&quot;https://i.stack.imgur.com/Lkdr0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Lkdr0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now take another 2 images. The first takes 1 copy of the first followed by 2 copies of the second and the second takes 1 copy of the first followed by 3 copies of the second to get&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oyRzl.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oyRzl.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt; &amp;amp; &lt;a href=&quot;https://i.stack.imgur.com/gCdvj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gCdvj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now we can create another 2 images. The first takes 1 copy of the first followed by 2 copies of the second and the second takes 1 copy of the first followed by 3 copies of the second to get&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0GQBe.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0GQBe.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt; &amp;amp; &lt;a href=&quot;https://i.stack.imgur.com/JXK10.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JXK10.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This demonstrates how irrational numbers are possible. Then afterwards, maybe the real numbers could be constructed from the Dedekind cuts of the rational numbers.&lt;/p&gt;&#xA;&lt;p&gt;However, I think it's easier to deal just with halves quarters, eights and so on at first. Then it will be very easy to define &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; as a Dedekind cut of those numbers and then introduce all Dedekind cuts of those numbers. Then they will explain that not all of them can be gotten by dividing an integer by a nonzero integer if negative numbers were introduce before non whole numbers. Then it will make sense. They will be like &amp;quot;I don't see why they have to all be able to be gotten that way.&amp;quot; Then if they already know the associativity and distributivity of real number multiplication, maybe they will simplify things and won't introduce fraction notation at all and will instead say &amp;quot;Take the numbers that can be expressed as &lt;span class=&quot;math-container&quot;&gt;$x \div y$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are integers and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is not zero.&amp;quot; Given 2 numbers expressed that way like &lt;span class=&quot;math-container&quot;&gt;$(2 \div 3) \div (5 \div 7)$&lt;/span&gt;, they will then ask &amp;quot;How do you write it as an integer in its normal form divided by an integer in its normal form?&amp;quot; The answer will be &lt;span class=&quot;math-container&quot;&gt;$14 \div 15$&lt;/span&gt;. They might also later give another question of &amp;quot;How do you write &lt;span class=&quot;math-container&quot;&gt;$(3 \div 4) + (5 \div 6)$&lt;/span&gt; in that way?&amp;quot; The answer will be &lt;span class=&quot;math-container&quot;&gt;$19 \div 12$&lt;/span&gt;. That's right. They will not use improper fractions and get them to write &lt;span class=&quot;math-container&quot;&gt;$1 + (7 \div 12)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://math.stackexchange.com/questions/895076/irrational-numbers-in-reality&quot;&gt;https://math.stackexchange.com/questions/895076/irrational-numbers-in-reality&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11718" LastActivityDate="2020-12-15T22:07:49.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19278" PostTypeId="1" CreationDate="2020-12-15T23:21:45.783" Score="2" ViewCount="76" Body="&lt;p&gt;I want to know which topics should be covered if one wants to create an easygoing mathematical course that covers basic concepts that are not in main focus of standard courses i.e. what those concepts should be? Course should be on undergraduate level.&lt;/p&gt;&#xA;" OwnerUserId="15121" LastActivityDate="2020-12-15T23:21:45.783" Title="creating an easygoing mathematical course that covers basic concepts that are not in main focus of standard courses" Tags="|undergraduate-education|" AnswerCount="0" CommentCount="4" ClosedDate="2020-12-21T11:55:13.777" ContentLicense="CC BY-SA 4.0" />
  <row Id="19279" PostTypeId="2" ParentId="19272" CreationDate="2020-12-16T02:27:23.230" Score="2" Body="&lt;p&gt;I have served as a math tutor through my employer, a software firm that sponsors a local college prep Jesuit school serving &amp;quot;students of limited economic resources.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The tutoring was an organized affair arranged through the head of schools, enabled by our pre-existing sponsorship relationship. I think how it happened was that one of our employees who was interested in tutoring first gauged interest among the other employees to get a quorum of tutors, and then reached out to the school through our designated contact.&lt;/p&gt;&#xA;&lt;p&gt;Before being allowed to tutor, prospective tutors had to undergo a background check and take mandatory training through &lt;a href=&quot;https://www.virtusonline.org/virtus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;VIRTUS&lt;/a&gt;, a Catholic-specific program to teach adults how to recognize and report child abuse. I think the background check and the training were valid for three or five years.&lt;/p&gt;&#xA;&lt;p&gt;Each visit, tutors had to sign in at the school office with a government ID, which was checked by the &lt;a href=&quot;https://raptortech.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Raptor&lt;/a&gt; visitor management system against known lists of criminals and sex offenders.&lt;/p&gt;&#xA;&lt;p&gt;The tutoring was at a set time on the same day every week, once per week. Aside from the scheduling, it could be considered drop-in in that tutors could choose which weeks to go, and students could choose to attend tutoring or not. We had a different set of tutors and students every week. Subjects ranged from pre-algebra to calculus. It is a small school, with fewer than 600 students, so the students are known to the faculty and staff. Often the students who showed up did so because a teacher had either required or recommended it.&lt;/p&gt;&#xA;&lt;p&gt;Summary:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You're looking for an organized program. Check with your professional and fraternal organizations to see if something like this exists already.&lt;/li&gt;&#xA;&lt;li&gt;A one-time background check will be required, and possibly some training.&lt;/li&gt;&#xA;&lt;li&gt;You'll check in every time you visit the school.&lt;/li&gt;&#xA;&lt;li&gt;You may not get the same kids every time, so you may not be able to count on developing rapport.&lt;/li&gt;&#xA;&lt;li&gt;You may not get the same subject matter every time. It helps to be flexible and willing to tutor any type of high school math or science.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3593" LastActivityDate="2020-12-16T02:27:23.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19280" PostTypeId="2" ParentId="19262" CreationDate="2020-12-16T05:58:29.770" Score="4" Body="&lt;p&gt;Quote, &lt;em&gt;emphasis&lt;/em&gt; [and implications] added:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is it &lt;em&gt;wrong&lt;/em&gt; [for the teacher] to say &amp;quot;you are wrong&amp;quot;? If yes, how do we [the teacher] correct mistakes in a way that is not demotivating? My policy has been, [teachers] give elaborate feedback... But, feedback often meant &lt;em&gt;I&lt;/em&gt; point out their mistakes. Of course, I [the teacher] was also generous with praise. How can I [the teacher] give feedback that is not demotivating?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Perhaps your students responded productively to elaborate feedback, but most research and casual empiricism say they don't; students overwhelmingly pay attention to grades. [One can like or dislike this, but I see almost no room to debate its veracity.] In this case, they interpreted your feedback as &amp;quot;I'm going to get a low grade whether or not I try hard, so why put time into this class?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;EDIT: OP said that many students received good grades for the presentations and may have inferred that effort was not necessary. Overall, the comments and grading strike me as sending an inconsistent message.&lt;/p&gt;&#xA;&lt;p&gt;More importantly, though, this quote is all about &lt;em&gt;the teacher&lt;/em&gt; doing a lot of mathematical thinking while the students - presenters and audience - doing very little.&lt;/p&gt;&#xA;&lt;p&gt;I humbly suggest that what you're really looking for is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;How do I get students to think and persevere mathematically?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How can a student learn to assess the quality of their own math and that of others?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How can I give students better odds/opportunities to &lt;em&gt;shine&lt;/em&gt; when presenting in front of the class?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What motivates students to respond productively to feedback? [Hint: GRADES]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The types of comments from @FoiledIt24 and @Daniel R. Collins are quite good for dealing with richer and advanced content. That's definitely part of the solution.&lt;/p&gt;&#xA;&lt;p&gt;Here, I suggest an approach that can tackle all four of those questions at once: &lt;strong&gt;&lt;a href=&quot;http://etec.ctlt.ubc.ca/510wiki/Assessment_as_Learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;assessment as learning&lt;/a&gt;&lt;/strong&gt;. [This link is OK. If anybody has a better link please paste it in the comments!]&lt;/p&gt;&#xA;&lt;p&gt;You could implement assessment as learning as follows:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Discuss the learning outcomes with the students and create criteria for the various concepts and skills to master. Create one big rubric for the course plus smaller rubrics for other critical learning outcomes. Involve students in this as much as possible. Ideally, within a week of the course and/or unit(s) starting, you can say &amp;quot;Here are the scoreboards we will use to track grades and learning.&amp;quot; This is a large but worthy investment because without knowing goals and criteria of assessment, nothing below will work. Also, their involvement in creating rubrics will increase intrinsic motivation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Provide examples &lt;em&gt;of varying quality&lt;/em&gt; and have students assess them according to the criteria you've established together, then have students slap an A+ or C- or some other letter grade on them. Discuss as a class and ensure everyone knows the difference between A-quality work vs B-quality work vs C-quality work etc. These varying examples are arguably the single best place to tackle the most common mistakes and misconceptions. You might say something like: &amp;quot;The fictitious students Sue, Al, Bob, and Zach have all attempted their own proofs of the associativity of matrix multiplication. Pretend to be the teacher and mark their work according to the rubric we agreed upon earlier. Give each student specific suggestions. End by giving each a letter grade and prepare to justify it all to other members of your group. Try to reach a consensus with them, then we'll discuss it as a class.&amp;quot; And you can do so, knowing it's a safe place to share candid assessments, whether positive or negative.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As students work on proofs and problems on their own, give them feedback in the form of:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Reminding them of the criteria. &amp;quot;This is a word problem and one of the criteria is correct interpretation of numbers in context.&amp;quot; This is the best type of assistance.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;Keep thinking&amp;quot; questions and comments such as those mentioned elsewhere in this thread&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.understood.org/en/school-learning/for-educators/classroom-management/the-power-of-effective-praise-a-guide-for-teachers&quot; rel=&quot;nofollow noreferrer&quot;&gt;Descriptive praise&lt;/a&gt; [EDIT: I realize I gave this short thrift originally. I should have said that among pedagogical techniques that are &lt;em&gt;easy&lt;/em&gt; to implement, descriptive praise is IMHO the most powerful. Other suggestions in this post are much harder.]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hints that reference success but require deep thinking to utilize. &amp;quot;I see you're stuck on exercise 38, but you aced last week's problem set and its last problem is quite relevant here.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Highlighting specific progress and excellence in front of the class and celebrating it. Even as little as a short applause or a cheer or giving them a token prize. &amp;quot;Last week, Joe was unhappy with his proof by induction. This week his induction proof got an A- and the grand prize of a $0.25 chocolate bar. Look at both proofs and tell me what improvements he made. &lt;em&gt;Reference the criteria we discussed earlier.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;As students become more proficient in assessing their own work, you can start calling on them to present in front of the class. &amp;quot;Who has thoughts on this they'd like to share?&amp;quot; Or, more privately, you can speak to one student and say &amp;quot;Hey, listen, your work here is really elegant even though it's not quite complete. I know you don't like performing, but can I share a screenshot of that with the class?&amp;quot; After this, have the class discuss &lt;em&gt;in terms of the pre-established criteria&lt;/em&gt; what makes these such shining examples. When they're done commenting, as the teacher, say &amp;quot;I'd give this a B+. Not bad for 5 minutes work! Here's a similar problem. Now, everyone, gimme some A+ quality work!&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have students write in PENCIL a practice midterm as if it were the real thing, but do NOT count it for grades. Then, have them mark their own practice midterm in BRIGHT RED PEN, you guessed it, according to the criteria. (They might need a solution key, too.) Then, still in RED, each student writes a letter grade at the top of their test along with a bunch of specific comments to themselves on how to improve for the real midterm. Another student then approves the grade and comments, then submits everything to you and you give them written feedback on the whole thing - but mainly the RED stuff - ideally at least a few days before the real midterm. Trust me - they &lt;strong&gt;will&lt;/strong&gt; respond productively to &lt;strong&gt;this&lt;/strong&gt; feedback!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I hope this provides some useful ideas on how to ensure assessment and feedback are productive and motivating. It really just boils down to clarifying expectations, building on success, aligning incentives and feedback, and profound involvement of students.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2020-12-16T17:42:47.257" LastActivityDate="2020-12-16T17:42:47.257" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19281" PostTypeId="1" CreationDate="2020-12-16T06:27:08.917" Score="5" ViewCount="332" Body="&lt;p&gt;I've recently taken a new position within a math department at a large university. The department has an official policy that in most lower-level undergraduate classes (let's say anything in the calculus sequence and below), roughly half of the students should receive A's and B's, and the other half should receive C's and below.&lt;/p&gt;&#xA;&lt;p&gt;I tend to disagree with this policy, but being new to the department and rather inexperienced with the student population, I'd like to honor the policy for the time being. So for the first time in my teaching career, I plan to curve my class to achieve a desired distribution.&lt;/p&gt;&#xA;&lt;p&gt;One tenet of my teaching philosophy is that I try to be as transparent with students as possible about how they will be assessed, and how their performance will translate into a letter grade. This means that if I am to institute a curve next semester, I want to be able to tell students exactly how the curve will work. Of course, I'm very worried that telling them only half of them can get A's and B's will create a very competitive environment, so students will not want to study with and help each other, so learning will suffer (in fact, I'm almost certain this will happen). This leads to my question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How can I simultaneously&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;tell the students how I will assign grades,&lt;/li&gt;&#xA;&lt;li&gt;achieve the distribution desired by the department, and&lt;/li&gt;&#xA;&lt;li&gt;still get students to buy into working together?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Seems impossible to me - maybe it is...&lt;/p&gt;&#xA;&lt;p&gt;One quasi-solution (which potentially strays a bit from condition 2) is to have guaranteed grade cutoffs. For example, if a student achieves an 80% in the class, she is guaranteed at least a B-, even if 75% of the class scored higher. In this scenario, the perceived effect of any curve is that it &amp;quot;can only help and not harm.&amp;quot; This could give students at least some sense of being in control of their own destiny and that they are not only being assessed in comparison to their peers.&lt;/p&gt;&#xA;&lt;p&gt;The problem with this solution is that determining those grade cutoffs can be tricky at the outset. If I make them too high, then at some point in the semester it will become clear that the curve will supersede the cutoffs, and competition is likely to creep back in. If the cutoffs are too low, then I could be locked into awarding an overly generous distribution that would be frowned upon by my department.&lt;/p&gt;&#xA;&lt;p&gt;What can I do?&lt;/p&gt;&#xA;" OwnerUserId="21" LastActivityDate="2020-12-17T00:46:06.087" Title="Curving grades without creating competition among students" Tags="|undergraduate-education|student-motivation|grading|" AnswerCount="2" CommentCount="7" ClosedDate="2020-12-18T05:18:05.443" ContentLicense="CC BY-SA 4.0" />
  <row Id="19284" PostTypeId="2" ParentId="19281" CreationDate="2020-12-16T17:55:26.340" Score="4" Body="&lt;p&gt;I teach at a community college, where our goal is to help all students succeed. But they have other commitments, and many are not used to being A students, and don't expect that. So I have seldom managed to get half of them getting As and Bs. (Linear Algebra and Math for Elementary Teachers are the only courses I can remember that probably achieved that.)&lt;/p&gt;&#xA;&lt;p&gt;I am guessing that you're at an elite university, where most students will work very hard for those top grades, so your natural grade distribution is likely to be higher than mine. It would be great if there were a way to get data on the likely distribution without that sucky policy. It if were close, then your plan would be good.&lt;/p&gt;&#xA;&lt;p&gt;Also, if your overall grade distribution among multiple courses fits their nasty criteria, is it ok with them if some of your classes do better?&lt;/p&gt;&#xA;&lt;p&gt;I would definitely recommend fighting this policy once you have tenure.&lt;/p&gt;&#xA;&lt;p&gt;I was an undergrad at U of Michigan. I remember feeling like I did pretty well in my French class, and being shocked to get a C. It was that sort of grading system that did it. (I already had my language requirement covered. I was taking that course for fun! Grr. Can you tell I have strong feelings about this issue?)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2020-12-16T17:55:26.340" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19285" PostTypeId="2" ParentId="19281" CreationDate="2020-12-17T00:46:06.087" Score="3" Body="&lt;p&gt;Your question is pretty broad: &amp;quot;what can I do&amp;quot;.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;To start with, I think you are overestimating the danger of students outperforming some external &amp;quot;bar&amp;quot; on competence (e.g. AP test).  My experience has been that generally curving inflates the grades versus an a priori standard.  Note, this is a generalization.  I'm sure we could construct an example where this is not the case.  E.g. a required course in shop or mechanical drawing at a competitive engineering school might be one where we expect well over 50% to score a B or higher.  I don't think this is the case in lower div math weeder courses at a big stat university though.  Nor is it in the majority of cases...as a student, what I found was more of a relaxation knowing others at the low end will carry one, versus a competition to extreme...after all, even with a priori standards many students work well short of capacity.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Furthermore, very little knowledge is gained in a course like this from collaboration.  Students learn by drilling homework.  Those who drill more, learn more.  Collaboration/help is way overrated.  Individual practice is underrated.  (I also find most people who would help will still help even if there's a small disincentive.  After all the single person you help is in front of you...you're not lifting the entire curve...just the person who is in front of you, socially near.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In addition, I would counter that your own lack of confidence in setting a standard for A, B, C work and for designing such tests shows that you might even be better off going with the crude idea of a curve.  After all, you don't have a feel for how to set the bars without one!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You're new to the school.  And maybe a newer instructor in general (based on the hesitancy to gauge tests).  Why not try it out their way first?   You can always do something different later, when you have more basis to strike out on your own.  There's a gazillion things to work on in your teaching.  Why not concentrate on some of them and just roll with testing similar to how the rest of the department does it (at least this semester)?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P.s.  None of this is to say I'm pro curve.  The opposite, actually.  Just for different reasons than yours.&lt;/p&gt;&#xA;" OwnerUserId="15125" LastActivityDate="2020-12-17T00:46:06.087" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19286" PostTypeId="1" CreationDate="2020-12-18T02:31:53.643" Score="6" ViewCount="423" Body="&lt;p&gt;I am teaching complex analysis for undergraduates using the textbook Complex Variables and Applications, Brown and Churchill. I am looking for resources that I can use to find good problems for homework sets and take-home exams.&lt;/p&gt;&#xA;&lt;p&gt;The problems in this book are a poor choice for these assessments because there are complete solution manuals available online. I would appreciate any recommendations on where to find decent problems whose solutions are not readily available (or at least harder to find).&lt;/p&gt;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;" OwnerUserId="15127" LastActivityDate="2020-12-29T17:52:55.333" Title="Complex analysis books/resources where solutions are difficult to find" Tags="|undergraduate-education|reference-books|" AnswerCount="1" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="19287" PostTypeId="1" AcceptedAnswerId="19290" CreationDate="2020-12-19T01:05:20.873" Score="12" ViewCount="512" Body="&lt;p&gt;My students, when presented with an integral (&lt;a href=&quot;https://www.math.ucdavis.edu/%7Ekouba/CalcTwoDIRECTORY/usubdirectory/USubstitution.html&quot; rel=&quot;noreferrer&quot;&gt;source&lt;/a&gt;) like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int (2x+2)e^{x^2+2x+3} \ dx$$&lt;/span&gt;&#xA;are apt to recognize derivative patterns like &lt;span class=&quot;math-container&quot;&gt;$u' e^{u}$&lt;/span&gt; and reverse-engineer anti-derivatives rather than to utilize &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-substitution.&lt;/p&gt;&#xA;&lt;p&gt;With something like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int x \cdot \cos(5x^2) \ dx$$&lt;/span&gt;&#xA;they would rewrite to get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{1}{10}\int 10x \cdot \cos(5x^2) \ dx$$&lt;/span&gt;&#xA;and then unwind the chain-rule.&lt;/p&gt;&#xA;&lt;p&gt;So I wonder if they are missing out by avoiding &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-substitution. Are there applications later on that will be more challenging if they don't get more practice with it now, or is &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-substitution occasionally convenient but unnecessary?&lt;/p&gt;&#xA;" OwnerUserId="219" LastActivityDate="2020-12-24T08:09:24.447" Title="Usefulness of $u$-substitution in and beyond early Calculus?" Tags="|calculus|integration|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19288" PostTypeId="2" ParentId="19287" CreationDate="2020-12-19T03:46:42.890" Score="4" Body="&lt;p&gt;There are sometimes reasons to avoid &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-substitution if you can see the antiderivative without it: in a definite integral, you don't have to reparameterize the interval's endpoints to &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-values/units if you don't use &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-substitution. Of course, this can also be a downside in some circumstances.&lt;/p&gt;&#xA;&lt;p&gt;The fundamental thing is, though, that &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-substitution is really a reparameterization of the function and is intimately tied to the chain rule by the fact that we do reparameterization through composition.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2020-12-19T21:34:37.727" LastActivityDate="2020-12-19T21:34:37.727" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19289" PostTypeId="1" CreationDate="2020-12-19T14:49:35.193" Score="1" ViewCount="286" Body="&lt;p&gt;The definitions from Kenneth Rosen textbook are as :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A relation &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; on a set &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; such that for all &lt;span class=&quot;math-container&quot;&gt;$a,b ∈ A$&lt;/span&gt; ,if &lt;span class=&quot;math-container&quot;&gt;$(a,b) ∈ R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(b,a) ∈ R$&lt;/span&gt;,then &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt; is called &lt;strong&gt;antisymmetric&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A relation &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; on a set &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; such that for all &lt;span class=&quot;math-container&quot;&gt;$a,b ∈ A$&lt;/span&gt;  if &lt;span class=&quot;math-container&quot;&gt;$(a, b) ∈ R$&lt;/span&gt; implies that &lt;span class=&quot;math-container&quot;&gt;$(b, a) \notin R$&lt;/span&gt; is called &lt;strong&gt;asymmetric&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The problem which I face is that the terms &amp;quot;antisymmetric&amp;quot; and &amp;quot;asymmetric&amp;quot; seems to be same to me (something which is not symmetric) and as such, without revision after few days, I forget which definition corresponds to what. [I am bad at memorizing things]&lt;/p&gt;&#xA;&lt;p&gt;Could anyone suggest me a suitable mnemonics to remember the same.&lt;/p&gt;&#xA;" OwnerUserId="14166" LastActivityDate="2020-12-20T05:49:11.263" Title="Mnemonics to correlate the definition of &quot;asymmetric relation&quot; and &quot;antisymmetric relation&quot; with the terms" Tags="|self-learning|discrete-math|resource-request|set-theory|" AnswerCount="1" CommentCount="7" ClosedDate="2020-12-21T11:56:03.690" ContentLicense="CC BY-SA 4.0" />
  <row Id="19290" PostTypeId="2" ParentId="19287" CreationDate="2020-12-19T15:42:32.463" Score="14" Body="&lt;p&gt;A substitution is a very general procedure in mathematics. Your students will have experienced it many times already in algebra before they got to calculus, e.g., when solving two equations in two unknowns. More generally, it's an example of a change of variables, in the case where there is only one variable. An example in two variables would be changing from coordinates &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; to new coordinates &lt;span class=&quot;math-container&quot;&gt;$(u,v)=(y,-x)$&lt;/span&gt;, i.e., a 90-degree rotation. As a fancy example from an advanced subject, you could look at the transformation from Schwarzschild coordinates to &lt;a href=&quot;https://en.wikipedia.org/wiki/Kruskal%E2%80%93Szekeres_coordinates&quot; rel=&quot;noreferrer&quot;&gt;Kruskal-Szekeres coordinates&lt;/a&gt; for a black hole.&lt;/p&gt;&#xA;&lt;p&gt;It's often convenient to do a change of variables simply for convenience of notation. For example, if I was going to do a lengthy calculation involving a cylindrical tank, I might find it useful to switch from the radius to the cross-sectional area, just to keep the writing simpler.&lt;/p&gt;&#xA;&lt;p&gt;So the attitude should be simply that substitutions are things that we do naturally whenever it's convenient. They make life better by keeping things simpler. Who wants to make things hard?&lt;/p&gt;&#xA;&lt;p&gt;Even within the context of freshman calculus, there are plenty of cases where your students' pattern-recognition strategy won't work, e.g., &lt;span class=&quot;math-container&quot;&gt;$\int dx/\sqrt{1-x^2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As an example where you really, really want to do some substitutions, consider the problem of finding the magnetic field inside a solenoid of finite length, by integrating over loops of current. This gives an integral of the form &lt;span class=&quot;math-container&quot;&gt;$B=\int_p^q a^2(a^2+z^2)^{-3/2} dz$&lt;/span&gt;. Students should be taught that the very first thing to do here is the substitution &lt;span class=&quot;math-container&quot;&gt;$u=z/a$&lt;/span&gt;, simply because it cleans up the integral. After that, the substitution &lt;span class=&quot;math-container&quot;&gt;$\tan\theta=1/u$&lt;/span&gt; makes the integral into a trivial one. Moreover, the final result is &lt;em&gt;simpler&lt;/em&gt; when expressed in terms of &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; than in terms of &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;, so it should just be left that way. And &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; also has a direct geometrical interpretation.&lt;/p&gt;&#xA;&lt;p&gt;A similar example would be something like &lt;span class=&quot;math-container&quot;&gt;$\int_{-\infty}^\infty a^2(a^2+z^2)^{-3/2} dz$&lt;/span&gt;. It should be explained to students that this is an indefinite integral, not a definite integral, and that the first step should be to make it into a definite integral with the substitution &lt;span class=&quot;math-container&quot;&gt;$u=z/a$&lt;/span&gt;. In real-life applications, this type of substitution is almost always obvious because we want to change from a variable that has units to one that is unitless.&lt;/p&gt;&#xA;&lt;p&gt;IMO the term &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-substitution&amp;quot; should be dropped. We should just call this a substitution or a change of variables. That would make it more clear that this is just a normal and familiar mathematical practice. Furthermore, it would disabuse students of the notion that they will never have to make up their own notation for a change of variables -- something they are incredibly reluctant to do, because in general, they are almost never asked in a math course to make up notation for something.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-12-19T15:47:48.270" LastActivityDate="2020-12-19T15:47:48.270" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19291" PostTypeId="1" CreationDate="2020-12-19T16:23:39.407" Score="4" ViewCount="380" Body="&lt;p&gt;I plan to record lectures for a MOOC on Calculus sometime next year. The MOOC is targeted at an undergraduate audience that comprises engineers, math majors as well as majors in the sciences, etc.  They are all first year students. All students would have seen one variable calculus in quite some detail in high school. See here to get an idea of what students learn in school:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://ncert.nic.in/textbook.php?lemh1=ps-6&quot; rel=&quot;noreferrer&quot;&gt;https://ncert.nic.in/textbook.php?lemh1=ps-6&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://ncert.nic.in/textbook.php?lemh1=ps-7&quot; rel=&quot;noreferrer&quot;&gt;https://ncert.nic.in/textbook.php?lemh1=ps-7&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The course I am planning is a one year, two semester course (roughly 70-80 hours of recorded lectures) covering the standard topics of one variable calculus, sequences and series, multivariable calculus and vector analysis. Note that I have full control over the syllabus. I am unsatisfied with the presentation in the commonly used Calculus texts like Thomas's Calculus. I am looking for the following features in a textbook:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The books should be of reasonable length and not be 1000+ pages. Something around 600-700 pages would be preferable.&lt;/li&gt;&#xA;&lt;li&gt;The book should not be an analysis textbook! The books need only&#xA;give heuristic and physical or geometric arguments and proofs are&#xA;not necessary except for the easier theorems.&lt;/li&gt;&#xA;&lt;li&gt;The books should not have 50 exercises for each section. I believe that solving 5 well-chosen exercises per section is better than solving 50 exercises most of which are minor variants of each other.&lt;/li&gt;&#xA;&lt;li&gt;It is okay if the book does not have many applications. All the students who take the course will be doing courses in physics and chemistry simultaneously so they will see plenty of applications anyway.&lt;/li&gt;&#xA;&lt;li&gt;The book should be relatively stable in the sense that there should not be a new edition in the next 4-5 years. The videos will be recorded only once whereas the same recorded lectures will be rerun in several iterations of the course. It would be helpful if the recommended book for the course remains in print for at least the next 5 years.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The books that I have found that satisfy all my requirements are the two calculus books by Lang (Short Calculus and Calculus of Several Variables). There is one more: Single and Multivariable Calculus by David Guichard et al. which has a bonus of being open source.&lt;/p&gt;&#xA;&lt;p&gt;Are there more books that fit my requirements? If not, I will choose from the books above.&lt;/p&gt;&#xA;" OwnerUserId="15034" LastActivityDate="2020-12-30T10:07:51.400" Title="Looking for a calculus books with very specific requirements" Tags="|calculus|textbooks|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19292" PostTypeId="2" ParentId="12118" CreationDate="2020-12-19T17:11:00.477" Score="7" Body="&lt;p&gt;There are both intellectual and interpersonal issues here. It seems that, in an effort to be professional, you have ignored the interpersonal issues and have tried to focus only on the intellectual ones in the things you say to this student. This has not worked.&lt;/p&gt;&#xA;&lt;p&gt;Don't try to diagnose the student. He could have autism, he could have ADD, he could have some other invisible disability, or he could just be a person with an unpleasant personality. Instead, just discuss his behavior with him explicitly, using language that you won't feel bad about afterward if you find out he did have a disability.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Fred, I've been using social cues to let you know that there is a problem with your behavior, but it seems like you haven't understood those cues, so I'm going to be more explicit now.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Go on and tell him what the behaviors are and how he needs to change them. No shouting, no facebook, etc.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2020-12-19T17:11:00.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19293" PostTypeId="2" ParentId="19289" CreationDate="2020-12-20T04:55:21.170" Score="6" Body="&lt;p&gt;First, let's note that the terms as used by Rosen are standard definitions, as we can see on Wikipedia (&lt;a href=&quot;https://en.wikipedia.org/wiki/Antisymmetric_relation&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Asymmetric_relation&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;), as well as other resource sites. There was some question about this in the comments, so I thought to clarify this first. Perhaps reading those articles will give an added perspective for the OP.&lt;/p&gt;&#xA;&lt;p&gt;Now, I'm not going to offer a mnemonic -- I don't think it's a good practice. I almost always find there is some deeper meaning to mathematical structures, which when understood makes the relationships much clearer and makes a mnemonic unnecessary baggage. Usually I find that students reliant on mnemonic devices use them as a crutch, barely succeed in the current course of study, and fail to succeed at a later step.&lt;/p&gt;&#xA;&lt;p&gt;That said, here are some comments looking at the Rosen text (speaking of Kenneth Rosen, &lt;em&gt;Discrete Mathematics and its Applications&lt;/em&gt;, Seventh Edition) that may be clarifying.&lt;/p&gt;&#xA;&lt;p&gt;In Section 9.1, the definition of &lt;em&gt;antisymmetric&lt;/em&gt; appears in the main text, whereas &lt;em&gt;asymmetric&lt;/em&gt; only appears within the exercises of that section. I would highly encourage you to work through each of Exercises 18-24 in that section, as the sequence walks you through clarifying exactly what the difference is. (The term &lt;em&gt;asymmetric&lt;/em&gt; is only referenced in 4 other exercises throughout the rest of the text; it is largely only presented to walk through the sequence reflecting on the exact meaning of &lt;em&gt;antisymmetric&lt;/em&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Spoilers follow for those exercises.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;For what it's worth, I don't think that the phrasing of the &lt;em&gt;antisymmetric&lt;/em&gt; definition there is the best it could possibly be. When I present it in my class using that book, I rephrase it as the following (and this may be the only time I rephrase any definition in other than the exact way the book does):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Antisymmetric&lt;/strong&gt;: A relation &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; on a set &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; such that whenever &lt;span class=&quot;math-container&quot;&gt;$(a, b)&#xA; \in R$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$(b, a) \not \in R$&lt;/span&gt; (except when &lt;span class=&quot;math-container&quot;&gt;$a = b$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note that while the Wikipedia article gives two equivalent ways of phrasing this, my version shows a third way that I think is a bit clearer. Note that this is now obviously saying the same thing as the definition for asymmetry, with the added clause &amp;quot;except when &lt;span class=&quot;math-container&quot;&gt;$a = b$&lt;/span&gt;&amp;quot; at the end.&lt;/p&gt;&#xA;&lt;p&gt;So this highlights that antisymmetry and asymmetry are &lt;em&gt;almost&lt;/em&gt; the same thing, through which Exercises 9.1 walk the student. Any asymmetric relation is necessarily antisymmetric; but the converse does not hold. Specifically, the definition of antisymmetry permits a relation element of the form &lt;span class=&quot;math-container&quot;&gt;$(a, a)$&lt;/span&gt;, whereas asymmetry forbids that. So an asymmetric relation is necessarily irreflexive.&lt;/p&gt;&#xA;&lt;p&gt;The property of antisymmetry is fundamentally of more interest, because it is used in the definition of a &lt;em&gt;partial order&lt;/em&gt;, that is, a &lt;em&gt;poset&lt;/em&gt;. See the start of Section 9.6: &amp;quot;A relation &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; on a set &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is called a &lt;em&gt;partial ordering&lt;/em&gt; or &lt;em&gt;partial order&lt;/em&gt; if it is reflexive, antisymmetric, and transitive.&amp;quot; See also Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Partially_ordered_set&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The property of asymmetry is not as interesting in this sense, because it's not independent of the reflexive property; as noted above, any asymmetric relation is automatically irreflexive.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps working through Section 9.6 on your own, and seeing the major use-case for antisymmetry (as opposed to asymmetry), will also be a memorable exercise.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2020-12-20T05:49:11.263" LastActivityDate="2020-12-20T05:49:11.263" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19295" PostTypeId="2" ParentId="3858" CreationDate="2020-12-20T13:52:38.040" Score="1" Body="&lt;p&gt;I am a law student and realised i could use my knowledge of mathematics to study law. Since my brain is use to seeing formulas, I derived a technic in memorizing long chapters and quotes into equations&lt;/p&gt;&#xA;&lt;p&gt;BP &amp;gt; CC &amp;gt;  P/RPB.    Is my definition of morality.&lt;/p&gt;&#xA;&lt;p&gt;Body of principles derived from a particular code of conduct through a particular religion, philosophy or belief.&lt;/p&gt;&#xA;&lt;p&gt;My brain easily sees the letters and during my law essay writings I can expand from these formulas.&#xA;I have also summarized long essays with dates to very short formulas.&lt;/p&gt;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="15147" LastEditorUserId="205" LastEditDate="2023-10-07T06:01:12.347" LastActivityDate="2023-10-07T06:01:12.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19297" PostTypeId="1" CreationDate="2020-12-23T20:01:36.087" Score="2" ViewCount="246" Body="&lt;p&gt;I was trying to figure out the distinction of a &lt;strong&gt;root&lt;/strong&gt; and a &lt;strong&gt;zero&lt;/strong&gt; and found people in such discussions make distinctions between &lt;strong&gt;equation&lt;/strong&gt; vs. &lt;strong&gt;expression&lt;/strong&gt; vs. &lt;strong&gt;polynomial&lt;/strong&gt; without defining them.  What is the difference exactly?&lt;/p&gt;&#xA;&lt;p&gt;I'm especially confused by function.  Likely I've forgotten many details from my high school math such as this definition.  I can tell you about functions in several programming languages, but that's not the same thing at all.&lt;/p&gt;&#xA;" OwnerUserId="15162" LastActivityDate="2020-12-24T22:23:08.677" Title="Definition of equation vs. expression vs. polynomial" Tags="|functions|polynomials|roots|" AnswerCount="1" CommentCount="0" ClosedDate="2020-12-27T18:09:48.530" ContentLicense="CC BY-SA 4.0" />
  <row Id="19298" PostTypeId="2" ParentId="19297" CreationDate="2020-12-23T22:57:32.070" Score="6" Body="&lt;p&gt;Here are informal definitions of the terms that seem confusing to you:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A &lt;em&gt;&lt;strong&gt;function&lt;/strong&gt;&lt;/em&gt; is a relation between two sets, usually sets of numbers. It maps elements of the first set to elements of the second set.&lt;/li&gt;&#xA;&lt;li&gt;An &lt;em&gt;&lt;strong&gt;expression&lt;/strong&gt;&lt;/em&gt; is a combination of symbols representing a calculation, ultimately a number.&lt;/li&gt;&#xA;&lt;li&gt;An &lt;em&gt;&lt;strong&gt;equation&lt;/strong&gt;&lt;/em&gt; describes that two expressions are identical (numerically).&lt;/li&gt;&#xA;&lt;li&gt;A &lt;em&gt;&lt;strong&gt;polynomial&lt;/strong&gt;&lt;/em&gt; is a specific type of expression of one variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, a sum of powers of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; multiplied by numbers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I'm avoiding complexities and precision in the hope that this may start toward&#xA;untangling.&lt;/p&gt;&#xA;&lt;p&gt;A computer-language &lt;em&gt;function&lt;/em&gt; takes input and produces output, which is&#xA;analogous to a function mapping elements of the 1st set (input) to elements of the 2nd set (output).&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2020-12-24T22:23:08.677" LastActivityDate="2020-12-24T22:23:08.677" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19299" PostTypeId="2" ParentId="19287" CreationDate="2020-12-24T08:09:24.447" Score="4" Body="&lt;p&gt;Differentiation is a science; integration is an art.&lt;/p&gt;&#xA;&lt;p&gt;When you integrate, you have to guess what function differentiated to give the one you are looking at. Substitution is one of the methods available for re-writing the function in the hope of finding something more recognisable.&lt;/p&gt;&#xA;&lt;p&gt;Students who skip the substitution are actually showing more understanding than those who don't. It shows that they understand the chain rule well enough to recognise its effects without needing to explicitly write the substitution out (which just takes up more time). Doing this should be encouraged.&lt;/p&gt;&#xA;&lt;p&gt;You could argue that they are missing out on practice of carrying out the substitution method that they might need with harder cases. If the students are able to read and understand the mathematical notation properly, they won't need so much routine practice to be able to carry out the mechanical part. What they do need is practice at identifying situations where a substitution is useful (ie. where it's much harder to spot the anti-derivative). If they are not getting this practice it is because the questions they have been set are not suitable.&lt;/p&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2020-12-24T08:09:24.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19300" PostTypeId="2" ParentId="19291" CreationDate="2020-12-25T18:18:23.573" Score="1" Body="&lt;p&gt;I would recommend Balmohan Limaye textbooks as it fits all your demands. The books are &lt;a href=&quot;https://www.amazon.in/Multivariable-Calculus-Analysis-Undergraduate-Mathematics/dp/1493970844/ref=pd_lpo_14_t_0/258-6961501-1717468?_encoding=UTF8&amp;amp;pd_rd_i=1493970844&amp;amp;pd_rd_r=7f5c8742-1679-472b-9473-d39c4aa2506e&amp;amp;pd_rd_w=fnGtA&amp;amp;pd_rd_wg=7Oh4l&amp;amp;pf_rd_p=5a903e39-3cff-40f0-9a69-33552e242181&amp;amp;pf_rd_r=5KV3YC37Z79AMZJKD63S&amp;amp;psc=1&amp;amp;refRID=5KV3YC37Z79AMZJKD63S&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Course in Multivariable Calculus and Analysis&lt;/a&gt;&#xA;and &lt;a href=&quot;https://www.amazon.in/Course-Calculus-Analysis-Undergraduate-Mathematics-dp-3030013995/dp/3030013995/ref=dp_ob_title_bk&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Course in Calculus and Real Analysis&lt;/a&gt;, both by Ghorpade &amp;amp; Limaye.&lt;/p&gt;&#xA;" OwnerUserId="14662" LastEditorUserId="376" LastEditDate="2020-12-30T10:07:51.400" LastActivityDate="2020-12-30T10:07:51.400" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19301" PostTypeId="2" ParentId="17984" CreationDate="2020-12-26T08:29:30.207" Score="0" Body="&lt;p&gt;Consider the equation in its two different arrangement of variables: &lt;span class=&quot;math-container&quot;&gt;$y=x+5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=y-5$&lt;/span&gt;;&lt;/p&gt;&#xA;&lt;p&gt;for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (independent variable value) as input, I am getting a set of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; (values dependent on input '&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;' variable values).&#xA;Plotting them on a graph(where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is independent and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is the dependent variable) I get:(Line in red)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mWQ5M.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mWQ5M.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; (independent variable value) as input, I am getting a set of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (values dependent on input '&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;' variable values).&#xA;Plotting them on a graph (where &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is independent and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the dependent variable) I get:(Line in blue(overlapping the red line made by the first equation)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5QT1b.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5QT1b.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;which is the same thing (since it's the same equation), but in this case the axes aren't the same, they are in a way &amp;quot;interchanged&amp;quot;, so in order to represent the the inverse function (which is a function, lets say, &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; i.e a function where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is dependent and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is independent) we swap the variables so they can be plotted to get the inverse function in terms of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as independent variable.&lt;/p&gt;&#xA;&lt;p&gt;The inverse function is the line in green (with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as dependent variable)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5Cew9.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5Cew9.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, in conclusion, this is how I understand it and how I explain the &amp;quot;interchanging of x and y&amp;quot; to myself. It ultimately translates to undoing of a function that is done on independent variable x. Maybe you could explain it in this way while teaching.&lt;/p&gt;&#xA;" OwnerUserId="15167" LastEditorUserId="15167" LastEditDate="2020-12-27T05:28:39.773" LastActivityDate="2020-12-27T05:28:39.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19303" PostTypeId="1" CreationDate="2020-12-27T08:23:21.697" Score="6" ViewCount="453" Body="&lt;h2&gt;Background&lt;/h2&gt;&#xA;&lt;p&gt;I know the question in the title is very broad so I will try to explain it as succinctly as I can. Half my time is spent on as a researcher on didactics, while the other half is devoted to teaching. The course I discuss below is an introductory course to mathematics for those who only need some mathematics in their further studies, some brands of chemistry students, biology, CS students and so forth (whether they only need a bit is up for discussion). The curriculumn is a litle bit of set theory, algebra, functions, graphs, limits, derivation, integration and the most simple differential equations.&lt;/p&gt;&#xA;&lt;p&gt;We have done some surveys amongst the students where they have been given a series of simple problems within algebra. Factorize this fraction, simplify this expression, alongside some more involved problems such as evaluating 1 # (2 # 3) when they are given a # b = a/b + b/a.&#xA;This test is conducted in a closed environment, individually with no aids other than a pen and pencil. According to what we would expect university students to score, the results are abysmal.&lt;/p&gt;&#xA;&lt;p&gt;However, I also give them big hand ins during the semester, about 3 of them give or take. Each project takes about a month for them to solve these problems in a group, and they are fairly involved compared to the survey we did. These projects or exercises uses &amp;quot;real life&amp;quot; problems to motivate the students.&lt;/p&gt;&#xA;&lt;p&gt;We also did a post-test (not representative due to covid)&lt;/p&gt;&#xA;&lt;h2&gt;Results&lt;/h2&gt;&#xA;&lt;p&gt;According to what we would expect university students to score, the results are abysmal&#xA;in the pre and post-test.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand just looking at the what the students handed in, the results where fantastic.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Possible solutions and explanations&lt;/h2&gt;&#xA;&lt;p&gt;It was clear that several groups did use online CAS solutions to perform the algebra part of the projects. Note that the algebra part in these projects was just a small part of the problem. This was clear by the replies from the teaching assistants and a very small percentage of students even screenshoted WolframAlpha step by step in their hand ins!&lt;/p&gt;&#xA;&lt;p&gt;The projects were new as of this year, so they could not have obtained solutions from older students. In addition the numbers on every groups project varied, so they could not copy directly another groups solution.&lt;/p&gt;&#xA;&lt;p&gt;The projects counted indirectly count towards their grade as the plan was to have a individual pass / no pass grading system with a group oral exam at the end of the semester. The oral exam would be based upon the projects, and start by asking how they went about solving some of the problems. With some follow up questions to see if they understood what they did. Again not possible this semester.&lt;/p&gt;&#xA;&lt;h2&gt;Question&lt;/h2&gt;&#xA;&lt;p&gt;I realize that the pre and post-test are conducted in a very unnatural environment to the students. Yet, based on the results from these tests I would not have expected any of the students to be able to solve the complex projects.&lt;/p&gt;&#xA;&lt;p&gt;If I asked a student at point blank today if he would be able to find the derivative of (1-x)/(1+x) I doubt he or she could do it. However, seemingly in a group, with every resource available and enough time, they are. I think I am just a tad confused what it means to understand algebra.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Do my students know algebra / how do I know whether I have taught them anything?&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;My gut feeling is that algebra is more than symbolic manipulation, but I do not have sources to back me up on this.&lt;/p&gt;&#xA;" OwnerUserId="596" LastEditorUserId="596" LastEditDate="2020-12-27T18:17:22.307" LastActivityDate="2022-03-07T19:52:39.683" Title="Do my students know elementary algebra; do they just use online calculators or external help; and is this ok?" Tags="|mathematical-pedagogy|algebra|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19304" PostTypeId="2" ParentId="17984" CreationDate="2020-12-27T08:47:17.110" Score="4" Body="&lt;p&gt;I can't spot in any of the other answers what I think is the main point.&lt;/p&gt;&#xA;&lt;p&gt;Neither teaching students to swap variables or teaching them not to switch variables is really the solution, as both of these simply train students to carry out a mechanical process without understanding what is going on.&lt;/p&gt;&#xA;&lt;p&gt;Either way, for the student the 'inverse' of a function remains 'the formula I get at the end of carrying out these steps'.&lt;/p&gt;&#xA;&lt;p&gt;What is lacking is the more basic idea of what a function is. While a function is a formula written on a page, the concept of an inverse doesn't really mean much. I'd say what needs adding is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a function is a rule for taking an input number and giving an output number (this definition should be sufficient in the circumstances);&lt;/li&gt;&#xA;&lt;li&gt;and the same function can be written in more than one way.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I was really hit by how little understanding students have of functions when I wrote down &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2+2$&lt;/span&gt; and asked them to write down what &lt;span class=&quot;math-container&quot;&gt;$f(y)$&lt;/span&gt; was. Most of them got there after some thought, but really weren't sure they had the right answer (which explains why they were finding linear functions so hard).&lt;/p&gt;&#xA;&lt;p&gt;Once you have those ideas, you are in a better position to teach that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sometimes is it right to change the variables. Sometimes it's not. You have to make a decision based on the meaning you are attaching to the symbols you have written down.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4746" LastActivityDate="2020-12-27T08:47:17.110" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19306" PostTypeId="2" ParentId="19303" CreationDate="2020-12-27T19:52:55.843" Score="7" Body="&lt;p&gt;There are two levels at which someone can understand algebra. (1) They can do stylized tasks using a set algorithm, such as multiplying out &lt;span class=&quot;math-container&quot;&gt;$(a+b)(c+d)$&lt;/span&gt;. (2) They understand what it means and can apply it to real-life problems.&lt;/p&gt;&#xA;&lt;p&gt;In principle it is possible for someone to master #2 while still not being competent at #1. The reality is that this never happens.&lt;/p&gt;&#xA;&lt;p&gt;A CAS will help with #1, but not with #2. It's great that your class is designed around projects, which test level #2. But you seem to be assuming that a CAS is the only method your students have for cheating on these projects. They can instead use online cheating services such as chegg. If you see students who can't do level #1 on the pretest but turn in projects where they have suddenly mastered the more difficult level #2, then this is probably what is happening.&lt;/p&gt;&#xA;&lt;p&gt;You use an example in which students are asked to evaluate &amp;quot;1 # (2 # 3) when they are given a # b = a/b + b/a.&amp;quot; To me this clouds the waters. This is not a test of algorithmic knowledge of algebra, nor is it a test of whether students can apply algebra to real life. This is a test of their ability at abstraction, which is a completely different mental ability.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2020-12-27T19:58:32.170" LastActivityDate="2020-12-27T19:58:32.170" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19308" PostTypeId="2" ParentId="19286" CreationDate="2020-12-29T17:52:55.333" Score="2" Body="&lt;p&gt;To synthesize some of the remarks given in chat, you can look for very old (or very recent) texts in the hope that there are few or no solutions already available. However, with the rise of online services often providing quick answers to mathematics questions, this may be a lost cause. Effective ways of dealing with this problem might involve different approaches to assessment than the usual assignments and long exams. Ben Crowell suggests frequent short exams as a countermeasure. (Other countermeasures might include using heavily proctored environments or more individualized project-style assessment.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2020-12-29T17:52:55.333" CommentCount="1" CommunityOwnedDate="2020-12-29T17:52:55.333" ContentLicense="CC BY-SA 4.0" />
  <row Id="19309" PostTypeId="2" ParentId="2679" CreationDate="2020-12-30T20:22:38.920" Score="2" Body="&lt;p&gt;How about encapsulating the entirety of the explanation in a tree diagram, which is visual, accessible, and relevant to any Intro Probability classroom?&#xA;&lt;a href=&quot;https://i.stack.imgur.com/a7wbf.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/a7wbf.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;We drew this today while going over the Monty Hall problem to cap off the topic (Probability) by considering interesting / counter-intuitive problems like the Birthday Problem, medical tests and the Sally Clark case and using tree diagrams to tackle them.&lt;/p&gt;&#xA;&lt;p&gt;Here, without loss of generality, the contestant has chosen Door &lt;span class=&quot;math-container&quot;&gt;$1.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Extending the traditional probability tree by framing the contestant's &lt;strong&gt;winning decision&lt;/strong&gt; as a third trial makes this a self-contained visual summary of the game (probability experiment).&lt;/p&gt;&#xA;&lt;p&gt;P.S. On less mathy fronts, I find these two videos (&lt;a href=&quot;https://www.youtube.com/watch?v=7u6kFlWZOWg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brady's&lt;/a&gt;, &lt;a href=&quot;https://youtu.be/4Lb-6rxZxx0?t=167&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lisa Goldberg's&lt;/a&gt;) both cogent, and the OP's own explanation cleverly succinct!&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-09-15T09:31:20.190" LastActivityDate="2021-09-15T09:31:20.190" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19311" PostTypeId="2" ParentId="2679" CreationDate="2020-12-31T19:31:04.233" Score="1" Body="&lt;p&gt;I usually explain this by simulating, and then defusing, the mis-understanding using a simple table. You can set up any of the three cases of the problem, as they are obviously all the same, I'll use [1:Goat] [2:Goat] [3:Car] here:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;My choice        Monty Shows       Reward for Stick     Reward for Switch&#xA; 1 (goat)          2 (goat)               0                     1 &#xA; 2 (goat)          1 (goat)               0                     1 &#xA; 3 (car)           1 (goat)               1                     0 &#xA; 3 (car)           2 (goat)               1                     0 &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Point out to the student that that in the case where I chose 3, Monty can open either door number 1 or number 2, but in the other case, he has only one choice. However, so far all that we've done is to replicate the confusion because if you count up the stick vs switch rewards, it's 50/50 (2/2).&lt;/p&gt;&#xA;&lt;p&gt;The student will be, like, &amp;quot;See!&amp;quot; (and this is, indeed, the confusion that even many mathematicians who wrote into MVS's original column had!)&lt;/p&gt;&#xA;&lt;p&gt;But let's look at the probabilities of each row. Recall that the total has to be 1. In the first column, start by assigning 1/3rd to each row. But that's not right, because it adds to 4/3rds. Okay, maybe 1/4 each? But that makes no sense because where does the 4 come from; there are only 3 doors!? The only obvious remaining assignment is 1/3rd, 1/3rd, and then there's 1/3rd that the personal will choose the car -- that works out, so obviously we need to split this, so it's 1/6th for each of the last two rows.&lt;/p&gt;&#xA;&lt;p&gt;Aha! So there's an actual traceable imbalance in the rewards! Each of the +1 for switching rewards has 1/3rd, so you get 2/3rds for switching, but the +1s for sticking &lt;em&gt;together&lt;/em&gt; only add up to 1/3rd (1/6th each)!&lt;/p&gt;&#xA;&lt;p&gt;There are two advantages to this approach. First, the table is obvious (more obvious to most people than a tree). Second, you start by replicating the learner's understanding, then point out the mis-understanding, and lead them, using the same representation, to understanding. I find that I generally don't have to go father than pointing out the 1/6th for the latter two rows, and they figure it out themselves.&lt;/p&gt;&#xA;&lt;p&gt;(I want to make a comment about another common approach to explaining this, which I actually think is not a great way to do it. Specifically,, sometimes explainers will say something like: imagine that there were 100 boxes, 99 with goats, and when you choose one, Monty reveals 98 of them. There's several reasons why, although correct, isn't a great way to explain the problem. The main issue is that it doesn't map uniquely to the 3-door case; it's not clear, when Monty is revealing the 98 other goats, that he couldn't, at that point, have just revealed them of them, or one-at-a-time, which is an equally natural analog to the 3-door case, and is very confusing because you'd have to mentally run the simulation backwards to the 3-door case before being able to answer! That is, although the explanation is correct, it's not obvious that the correct analog of the 3-door case is to reveal 98, as opposed to 1, wrong doors.)&lt;/p&gt;&#xA;" OwnerUserId="14430" LastActivityDate="2020-12-31T19:31:04.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19312" PostTypeId="2" ParentId="18030" CreationDate="2021-01-01T22:46:42.363" Score="2" Body="&lt;p&gt;maybe this whiteboard:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.mathcha.io/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathcha.io/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://tableaunoir.github.io/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://tableaunoir.github.io/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Regards,&lt;/p&gt;&#xA;&lt;p&gt;Enrique&lt;/p&gt;&#xA;" OwnerUserId="15184" LastActivityDate="2021-01-01T22:46:42.363" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19313" PostTypeId="2" ParentId="3858" CreationDate="2021-01-02T22:52:07.397" Score="1" Body="&lt;p&gt;Your client has been fired from his job for failing the Company comprehensive drug screening program of all one thousand employees.&lt;/p&gt;&#xA;&lt;p&gt;The drug test is 99% accurate.&lt;/p&gt;&#xA;&lt;p&gt;How might you proceed counselor?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;OK, let's add some more specifics.&lt;/p&gt;&#xA;&lt;p&gt;Assume that there are 10 employees that are actually using illegal drugs. What is the probability of guilt for your client?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;It's 50/50, no better than a coin toss.&lt;/p&gt;&#xA;" OwnerUserId="10376" LastEditorUserId="10376" LastEditDate="2021-01-08T04:00:59.193" LastActivityDate="2021-01-08T04:00:59.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19314" PostTypeId="2" ParentId="16807" CreationDate="2021-01-03T15:05:15.603" Score="1" Body="&lt;p&gt;The observation that a rhombus's diagonals bisect its internal angles gives rise to an efficient method to find an angle bisector.&lt;/p&gt;&#xA;&lt;p&gt;Here, a British national-exam question hints this method to the candidate:&lt;a href=&quot;https://i.stack.imgur.com/MErAC.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MErAC.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-01-09T20:54:07.330" LastActivityDate="2021-01-09T20:54:07.330" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19315" PostTypeId="2" ParentId="16807" CreationDate="2021-01-03T18:12:05.947" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;OP: &amp;quot;I am fairly sure that rhombuses and kites are pretty useless.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Understanding parallelograms and rhombi is quite useful in pop-up card design:&lt;/p&gt;&#xA;&lt;p&gt;     &lt;a href=&quot;https://i.stack.imgur.com/wXqZC.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wXqZC.jpg&quot; alt=&quot;AgolXmas&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Constructed by Ian Agol.&#xA;&lt;a href=&quot;https://twitter.com/agolian/status/1340216414200680448&quot; rel=&quot;nofollow noreferrer&quot;&gt;Twitter link&lt;/a&gt;;&#xA;&amp;quot;inspired by @divbyzero's tweet.&amp;quot; 19Dec2020.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-01-04T00:35:07.287" LastActivityDate="2021-01-04T00:35:07.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19316" PostTypeId="2" ParentId="16807" CreationDate="2021-01-03T20:01:33.127" Score="1" Body="&lt;p&gt;It would be hard to say what you could get rid of without knowing what is on your state test.  In my state (NY), they push quads quite a bit.  I'll grant you that kites aren't interesting or important, but rhombuses are both.&lt;/p&gt;&#xA;&lt;p&gt;One of the usual capstone problems is being given the coordinates of four points and demonstrating that the quadrilateral formed from the vertices is a rhombus but not a square.  That's a pretty strong review question; it measures skill on the slope and distance formulas, interpreting those results in the context of congruent and parallel/perpendicular line segments, and then interpreting THOSE results in quadrilateral language.&lt;/p&gt;&#xA;&lt;p&gt;Another sample problem: if a rhombus has diagonals of 10 and 24 inches, what is its perimeter?  Again, a lot of facts that you'd need to work that out: the diagonals of any parallelogram bisect each other, all rhombuses are parallelograms, the diagonals of a rhombus are perpendicular, so you can use the Pythagorean theorem to solve for the side length, and then all sides of a rhombus are congruent so you can find the perimeter.&lt;/p&gt;&#xA;&lt;p&gt;Obviously, as other responses demonstrate, even STEM-based professionals lead fulfilling and successful careers without remembering these facts.  But I don't begrudge the Common Core and New York State for expecting my students to master them.  Understanding that there are conditions for determining that a quadrilateral is of a special type and there are implications for knowing that a quadrilateral is of a special type is the sort of applied logic that is so often used in the real world and so infrequently nurtured in students.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-01-03T20:01:33.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19317" PostTypeId="2" ParentId="18030" CreationDate="2021-01-04T05:37:46.953" Score="1" Body="&lt;p&gt;My suggestion is to use a Jupyter notebook or Google Colab notebook together with a browser marker plugin.  You can't quite type anywhere; but you can inline or center the latex per usual.  I use chrome with a free app called &amp;quot;page marker&amp;quot; that lets me draw on any window in chrome, whether that's my Jupyter notebook or a Colab one.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2021-01-04T05:37:46.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19318" PostTypeId="2" ParentId="19069" CreationDate="2021-01-04T13:34:02.180" Score="2" Body="&lt;p&gt;I think &lt;a href=&quot;https://www.robocompass.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.robocompass.com/&lt;/a&gt; fits the bill... if you are ok with something code-driven, which for me is just an amazing bonus for my students and for exposition.&lt;/p&gt;&#xA;&lt;p&gt;So, the way in which robocompass deviates from rusty compass is directly related to a major potential benefit. Worth a look, anyway!&lt;/p&gt;&#xA;" OwnerUserId="15192" LastActivityDate="2021-01-04T13:34:02.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19319" PostTypeId="2" ParentId="13032" CreationDate="2021-01-04T19:55:55.507" Score="0" Body="&lt;p&gt;It is obvious that the algorithm given in the accepted answer (in the Mathematics Stack Exchange Site) to the question “Divide a plane with 2n points into two equal halves” (at: &lt;a href=&quot;https://math.stackexchange.com/questions/3083344/divide-a-plane-with-2n-points-into-two-equal-halves/3083372#comment8100957_3083372&quot;&gt;https://math.stackexchange.com/questions/3083344/divide-a-plane-with-2n-points-into-two-equal-halves/3083372#comment8100957_3083372&lt;/a&gt;) can be slightly modified to fit the situation you ask about.&lt;/p&gt;&#xA;" OwnerDisplayName="user10552" LastActivityDate="2021-01-04T19:55:55.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19321" PostTypeId="1" CreationDate="2021-01-05T14:13:07.927" Score="6" ViewCount="438" Body="&lt;p&gt;I am looking for a simple solution to the following problem:&lt;/p&gt;&#xA;&lt;p&gt;I would like to create a math quiz with n questions via LaTex (printable, in pdf) where question 1 is chosen randomly from question bank #1, question 2 is chosen randomly form question bank #2, etc.&lt;/p&gt;&#xA;&lt;p&gt;Is there a simple way to do so? Thanks in advance.&lt;/p&gt;&#xA;&lt;p&gt;Andrea&lt;/p&gt;&#xA;" OwnerUserId="15199" LastActivityDate="2021-01-15T08:25:40.827" Title="Create a math quiz with LaTex by randomly choosing questions from a bank" Tags="|exams|testing|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19323" PostTypeId="2" ParentId="16807" CreationDate="2021-01-06T00:36:20.560" Score="4" Body="&lt;p&gt;I am quite surprised by the suggestion (and apparent consensus?) that kites and rhombi are not very important for high school geometry.  Here are three entirely different arguments for why they should, in fact, play a central role in the curriculum.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Argument 1: Understanding hierarchical relationships&lt;/strong&gt;.&#xA;One of the goals of secondary Geometry is to help students move from a basic conception of &amp;quot;shapes&amp;quot; as individual objects to a more sophisticated notion of &amp;quot;shapes&amp;quot; as generic representatives of categories of objects, and a corresponding focus on the relationships among those categories.  (For those who are familiar with the old &lt;a href=&quot;https://en.wikipedia.org/wiki/Van_Hiele_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;Van Hiele language for describing geometric understanding&lt;/a&gt;, I am talking about the transition from Level 1 to Level 2 and Level 3.).&lt;/p&gt;&#xA;&lt;p&gt;Students who do not make this transition may fail to understand, for example, that a proof of a property of isosceles trapezoids applies not only to the &lt;em&gt;specific&lt;/em&gt; trapezoid represented in a particular diagram, but to &lt;em&gt;any&lt;/em&gt; isosceles trapezoid satisfying the hypotheses of the theorem.  (See for example &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01273371&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chazan, 1993&lt;/a&gt;, particularly at p. 372.)&lt;/p&gt;&#xA;&lt;p&gt;At a basic level, the goal here is to move students from simpler statements about individual figures (&amp;quot;This is a rectangle because it has four right angles&amp;quot;) to claims about nesting of one category of figures inside another category (&amp;quot;All squares are rectangles, but not all rectangles are squares&amp;quot;), with deductive proof making it possible to generalize from a generic case to the category of which it is a representative. At a more advanced level, students are expected to develop an understanding of a hierarchical relationship among types of figures, as shown below. (&lt;a href=&quot;https://www.expii.com/t/the-quadrilateral-hierarchy-10524&quot; rel=&quot;nofollow noreferrer&quot;&gt;Image source&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kvpKT.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kvpKT.gif&quot; alt=&quot;hierarchical model of quadrilaterals&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus for example students should know (and be able to prove!) conditional statements like the following examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$ABCD$&lt;/span&gt; is both a rectangle and a rhombus, then it is a square (and conversely if &lt;span class=&quot;math-container&quot;&gt;$ABCD$&lt;/span&gt; is a square then it is both a rectangle and a rhombus)&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$ABCD$&lt;/span&gt; is both a parallelogram and a kite, then it is a rhombus (and conversely if it is a rhombus then it is both a parallelogram and a kite).&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$ABCD$&lt;/span&gt; is both an isosceles trapezoid and a parallelogram, then it is a rectangle (and conversely...)&lt;/li&gt;&#xA;&lt;li&gt;Every rectangle is a parallelogram (but not every parallelogram is a rectangle); every rhombus is a parallelogram (but not every parallelogram is a rhombus); every parallelogram is a trapezoid (but not every trapezoid is a parallelogram); every trapezoid is a quadrilateral (but not every quadrilateral is a trapezoid).&lt;/li&gt;&#xA;&lt;li&gt;etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Note by the way that in the classification scheme shown above, different types of quadrilaterals are defined &lt;em&gt;inclusively&lt;/em&gt;, rather than &lt;em&gt;exclusively&lt;/em&gt;.  Not all curricula follow this convention consistently; for a historical perspective on quadrilateral definitions and classification, &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13700/in-what-curricula-are-rectangles-defined-so-as-to-exclude-squares/13766#13766&quot;&gt;see my answer to a related question&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now if we were to stipulate, for the sake of discussion, that understanding these hierarchical relationships is one of the goals of the curriculum, consider what would happen if the categories of &amp;quot;kite&amp;quot; and &amp;quot;rhombus&amp;quot; were removed or omitted. Roughly 1/3 of the diagram above, and a corresponding number of hierarchical relationships would simply disappear -- there would be no way to even talk about them!  We &lt;em&gt;need&lt;/em&gt; kites and rhombi in the lexicon so that we can have a fully-developed theory of quadrilaterals.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Argument 2: Analyzing how quadrilaterals are like, and unlike, triangles.&lt;/strong&gt;  In the standard sequence of the Geometry course, students first study triangles, which they learn to classify into types according to two different schemes:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When classifying by &lt;em&gt;side lengths&lt;/em&gt;, every triangle is either scalene, isosceles, or equilateral.&lt;/li&gt;&#xA;&lt;li&gt;When classifying by &lt;em&gt;angle measures&lt;/em&gt;, every triangle is either acute, obtuse, or right.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These two different classification schemes interact in complicated ways: a right triangle can be scalene or isosceles; an isosceles triangle can be acute, obtuse, or right; and so on. However, some combinations are impossible.  For example, an equilateral triangle &lt;em&gt;cannot&lt;/em&gt; be right or obtuse, because of the following theorem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Theorem&lt;/em&gt;.  All equilateral triangles are equiangular, and conversely all equiangular triangles are equilateral.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This theorem is a corollary of the Base Angles Theorem and the Triangle Sum Theorem, the proofs of which rely on fundamental properties of parallels and triangle congruence; it is therefore a kind of capstone result for the &amp;quot;triangles unit&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;When students move from triangles to quadrilaterals, the situation is suddenly very different.  A quadrilateral may be equilateral without being equiangular, or conversely may be equiangular without being equilateral! This is an extremely important idea, one that is very counterintuitive for may students; anecdotally, I can report that many of the high school students I have tutored believe that if you know a quadrilateral has four congruent sides, it must be a square.  Informally, the reason this is not true is because quadrilaterals, unlike triangles, are not rigid; there is no &amp;quot;SSSS Congruence&amp;quot; property for quadrilaterals. (See the question, and my answer, at &lt;a href=&quot;https://math.stackexchange.com/questions/490242/what-is-the-precise-definition-of-a-rigid-shape/1577419#1577419&quot;&gt;&amp;quot;What is the precise definition of a rigid shape?&amp;quot;&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;So we need names for quadrilaterals that are equiangular (but not necessarily equilateral), and for quadrilaterals that are equilateral (but not necessarily equiangular).  We call the first type a &amp;quot;rectangle&amp;quot;, and we call the second type a &amp;quot;rhombus&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Similarly if we try to generalize the notion of &amp;quot;isosceles triangle&amp;quot; to the quadrilateral case, we might say (as a first attempt) that we are interested in figures in which some, but not all, of the sides are congruent.  On further reflection we realize that there are two worthwhile possibilities to consider here:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If two pairs of &lt;em&gt;opposite sides&lt;/em&gt; are congruent, we have a parallelogram.&lt;/li&gt;&#xA;&lt;li&gt;If two (disjoint) pairs of &lt;em&gt;adjacent sides&lt;/em&gt; are congruent, we have a kite.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Argument 3:  Transformation-based approaches to Geometry&lt;/strong&gt;.  I'll be briefer about this one, because transformation-based approaches to Geometry, while common internationally, are still somewhat rare in U.S. curricula (see p. 204 &lt;a href=&quot;https://www.routledge.com/Secondary-Mathematics-for-Mathematicians-and-Educators-A-View-from-Above/Weiss/p/book/9781138294677&quot; rel=&quot;nofollow noreferrer&quot;&gt;of this book&lt;/a&gt; for a brief discussion of the history of such approaches in the secondary curriculum), despite the fact that the Common Core State Standards &lt;a href=&quot;http://www.corestandards.org/Math/Content/HSG/introduction/&quot; rel=&quot;nofollow noreferrer&quot;&gt;emphasize them as fundamental&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In a transformation-based approach, figures are classified according to the types of symmetry they have.  For example, isosceles triangles have exactly one line of symmetry; equilateral triangles have three.  With quadrilaterals, again, things are more complicated:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A quadrilateral with two lines of symmetry that pass through the midpoints of its sides is a &lt;em&gt;rectangle&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;A quadrilateral with two lines of symmetry that pass through its vertices is a &lt;em&gt;rhombus&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;A quadrilateral with one line of symmetry that passes through the midpoints of (two of) its sides is an &lt;em&gt;isosceles trapezoid&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;A quadrilateral with one line of symmetry that passes through (two of) its vertices is a &lt;em&gt;kite&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;There is an important analogy lurking in the above observations:  kites and rhombi are, in an important way, &lt;em&gt;dual to&lt;/em&gt; isosceles trapezoids and rectangles.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Concluding thoughts:&lt;/strong&gt;. There is more that could be said on this; for example, one very fruitful avenue of student exploration is to study what kind of figure is produced when one joins together the midpoints of the sides of a quadrilateral.  (See &lt;a href=&quot;https://www.jstor.org/stable/10.5951/mathteacher.106.4.0269?seq=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://www.geogebra.org/m/azp4q3T4&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Varignon%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; for some examples.) This is variously referred to as the &amp;quot;dual&amp;quot; or &amp;quot;midpoint polygon&amp;quot; construction.  Some very interesting results can be discovered and proved by students; for example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The dual of any quadrilateral is a parallelogram&lt;/li&gt;&#xA;&lt;li&gt;The dual of any rectangle is a rhombus&lt;/li&gt;&#xA;&lt;li&gt;The dual of any rhombus is a rectangle&lt;/li&gt;&#xA;&lt;li&gt;Therefore the dual of a square is a square&lt;/li&gt;&#xA;&lt;li&gt;The double-dual of a rhombus is another rhombus; the double-dual of a rectangle is another rectangle&lt;/li&gt;&#xA;&lt;li&gt;More generally the double-dual of a parallelogram is another parallelogram similar to the original&lt;/li&gt;&#xA;&lt;li&gt;The dual of an isosceles trapezoid is also a rhombus&lt;/li&gt;&#xA;&lt;li&gt;The dual of a kite is also a rectangle&lt;/li&gt;&#xA;&lt;li&gt;etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Without rhombi and kites these kinds of investigations become essentially impossible.&lt;/p&gt;&#xA;&lt;p&gt;TL;DR:  if rhombi and kites did not exist, we would need to invent them in order to answer basic questions about the relationships among the other types of quadrilaterals.  Trying to do geometry without them would be like trying to teach algebra without the words &amp;quot;slope&amp;quot; or &amp;quot;quadratic&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-01-06T00:36:20.560" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19325" PostTypeId="1" CreationDate="2021-01-06T11:43:26.730" Score="0" ViewCount="380" Body="&lt;p&gt;I'm looking for a simple way to define mathematics to primary/elementary school teachers and explain some of the confusion children have.&lt;/p&gt;&#xA;&lt;p&gt;I'm hoping some Algebraist could help me properly state the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A number in and of itself has no true meaning. True in the sense that it relates to an existing object within our world. The question we need to ask is how do we teach children meaning if that meaning is not at first grounded within something concrete.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Numbers in and of themselves represent abstract notions and in the pure study of mathematics we study mostly patterns: the various patterns that emerge from these abstract notions and the various means through which some relation is developed or expressed between them. Meaning that between the value 1 and 2 for instance there is no relation except when explicitly defined for example as some additive operation, in general an addition of multiples of the unit element.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="15207" LastEditorUserId="2083" LastEditDate="2021-01-07T15:11:35.347" LastActivityDate="2021-02-22T03:36:41.340" Title="Defining mathematics to primary/elementary school teachers" Tags="|undergraduate-education|primary-education|teacher-preparation|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19326" PostTypeId="2" ParentId="19325" CreationDate="2021-01-06T17:22:09.390" Score="1" Body="&lt;p&gt;I highly recommend the book &lt;em&gt;The Number Sense: How the Mind Creates Mathematics&lt;/em&gt;, by  Stanislas Dehaene and published by Oxford University Press. Another book that comes to mind is &lt;em&gt;The Language Instinct: How The Mind Creates Language&lt;/em&gt;, by Steven Pinker. Both books have had broad impact on scholarship related to your question.&lt;/p&gt;&#xA;&lt;p&gt;As you try to refine your question and find an answer that satisfies you, it might help to think about how artificial intelligence &amp;quot;learns.&amp;quot; Let's take the sentence from your OP &amp;quot;&lt;em&gt;The question we need to ask is how do we teach children meaning if that meaning is not at first grounded within something concrete.&lt;/em&gt;&amp;quot; and replace &amp;quot;&lt;em&gt;children&lt;/em&gt;&amp;quot; by &amp;quot;&lt;em&gt;machines&lt;/em&gt;&amp;quot; or &amp;quot;&lt;em&gt;artificial intelligence.&lt;/em&gt;&amp;quot; To be sure, there is much controversy about all of this, related to meaning and understanding.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-01-06T17:22:09.390" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19327" PostTypeId="2" ParentId="19321" CreationDate="2021-01-06T18:44:53.077" Score="7" Body="&lt;p&gt;As Brandon mentioned in a comment, I'm working to develop an online platform for this use-case, where the banks are generated using SageMath code and &lt;a href=&quot;https://pretextbook.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;PreTeXt XML&lt;/a&gt;. Problem sets for linear algebra and differential equations are available to the public at &lt;a href=&quot;https://checkit.clontz.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://checkit.clontz.org/&lt;/a&gt; and I'll be working to add better support for new authors in the near future. The community for the CheckIt platform is on the #checkit-app channel of the &lt;a href=&quot;https://www.masterygrading.com/resources&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mastery Grading Slack&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="15209" LastEditorUserId="1608" LastEditDate="2021-01-13T13:38:26.880" LastActivityDate="2021-01-13T13:38:26.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19328" PostTypeId="2" ParentId="19325" CreationDate="2021-01-07T05:21:17.690" Score="0" Body="&lt;p&gt;Forget New Math, it does now work with young kids, and it is not needed for learning arithmetics in elementary school.&lt;/p&gt;&#xA;&lt;p&gt;Instead, first teach the concepts of &amp;quot;same as&amp;quot;, &amp;quot;more than&amp;quot;, &amp;quot;less than&amp;quot; by lining up real objects like apples or backpacks or horses one against another, making pairs. Then you ask, how do we figure out &amp;quot;more&amp;quot; or &amp;quot;less&amp;quot; or &amp;quot;same&amp;quot; without bringing my horses and your horses to the river? We can abstract horses into counters and line up counters mine against yours and compare them. Then you ask, can we abstract other things like cows or women or children with counters? Do they have to be the same counters? Can we use different ones? Can we use sticks instead of stones? Can we compare without lining up counters? Well, we write counters on paper instead of actually carrying stones or sticks. Pictures of counters is an abstraction of physical counters. When we get too many things, it becomes unwieldy, can we do better? Well, we can &amp;quot;count&amp;quot; the counters - a whole new idea and a process, and come up with - ta-da! - numbers. Comparing 3 and 5 is the same as comparing ||| and |||||, just different representation. How is it better? Without introducing place value, not much. So you explain how place value works, and how with limited number of digits we can now represent any countable number, another level of abstraction.&lt;/p&gt;&#xA;&lt;p&gt;But please, don't tell kids that numbers don't exist, that they represent the abstract notion blah-blah-blah. They tried it sixty years ago, and it did not work.&lt;/p&gt;&#xA;&lt;p&gt;Let me quote &lt;em&gt;Why Johnny Can't Add&lt;/em&gt; by Morris Kline:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Is 7 a number,&amp;quot; asks a teacher. The students, taken aback by the&#xA;simplicity of the question, hardly deem it necessary to answer; but&#xA;the sheer habit of obedience causes them to reply affirmatively. The&#xA;teacher is aghast. &amp;quot;If I asked you who you are, what would you say?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The students are now wary of replying, but one more courageous&#xA;youngster does do so: &amp;quot;I am Robert Smith.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The teacher looks incredulous and says chidingly, &amp;quot;You mean that you&#xA;are the name Robert Smith? Of course not. You are a person and your&#xA;name is Robert Smith. Now let us get back to my original question: Is&#xA;7 a number? 0f course not!  It is the name of a number. 5 + 2, 6 + 1,&#xA;and 8 - 1 are names for the same number. The symbol 7 is a numeral for&#xA;the number.&lt;/p&gt;&#xA;&lt;p&gt;The teacher sees that the students do not appreciate the distinction&#xA;and so she tries another tack. &amp;quot;Is the number 3 half of the number 8?&amp;quot;&#xA;she asks. Then she answers her own question: &amp;quot;Of course not! But the&#xA;numeral 3 is half of the numeral 8, the right half.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The students are now bursting to ask, &amp;quot;What then is a number?&amp;quot;&#xA;However, they are so discouraged by the wrong answers they have given&#xA;that they no longer have the heart to voice the question. This is&#xA;extremely fortunate for the teacher, because to explain what a number&#xA;really is would be beyond her capacity and certainly beyond the&#xA;capacity of the students to understand it. And so thereafter the&#xA;students are careful to say that 7 is a numeral, not a number. Just&#xA;what a number is they never find out.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2021-01-07T22:56:27.047" LastActivityDate="2021-01-07T22:56:27.047" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="19329" PostTypeId="1" AcceptedAnswerId="19330" CreationDate="2021-01-07T15:27:57.513" Score="2" ViewCount="230" Body="&lt;p&gt;I talked with my friend on December 29 2020. Then I talked with him again on January 03, 2021.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt; What was the year when you last talked with your friend?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A:&lt;/strong&gt; 2021.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt; And what was the year the previous time you talked?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Α:&lt;/strong&gt; 2020.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Comment:&lt;/strong&gt; So dutifully doing the arithmetic, &lt;span class=&quot;math-container&quot;&gt;$2021\, (-) \,2020 = 1$&lt;/span&gt;.&#xA;This means that 1 year has passed right?&lt;/p&gt;&#xA;&lt;p&gt;This is an almost universal joke around the change of the year (&amp;quot;it's been a year since we last talked&amp;quot;, etc). But how to explain this to a little child?&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorDisplayName="user507" LastEditDate="2021-01-07T23:33:43.727" LastActivityDate="2021-01-08T05:16:34.613" Title="How do we explain to a little child that a date in 2020 and a date in 2021 are not necessarily a year apart?" Tags="|mathematical-pedagogy|intuition|arithmetic|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19330" PostTypeId="2" ParentId="19329" CreationDate="2021-01-07T18:11:11.797" Score="8" Body="&lt;p&gt;I would use a number line.  This is the most straight forward way to explain being &amp;quot;in between&amp;quot; integers while giving some intuition with a visual.  It is possible that a school-age child would be familiar with it, too.&lt;/p&gt;&#xA;&lt;p&gt;The number line is particularly useful because you can demonstrate that the distance between a point near the end of 2020 and a point near the beginning of 2021 has length less than one.&lt;/p&gt;&#xA;&lt;p&gt;You might want to start with examples of smaller numbers first, or introduce the line before moving all the way to 2020.  You could even do this in context, saying &amp;quot;think about a change from the year 5 to the year 6&amp;quot;.  Be sure to talk about what it means to be &amp;quot;in&amp;quot; a year as being between the starting point of each year.&lt;/p&gt;&#xA;" OwnerUserId="219" LastEditorUserId="219" LastEditDate="2021-01-07T19:10:53.533" LastActivityDate="2021-01-07T19:10:53.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19331" PostTypeId="2" ParentId="19325" CreationDate="2021-01-07T20:20:37.287" Score="1" Body="&lt;p&gt;I would suggest taking it easy to start but maybe something like base numeration is a way in here.  There is no reason we have to use base 10 and students use of different bases can be very important to understanding operations that require regroupings, and that weird alignment under the typical presentation of multiplication algorithms.&lt;/p&gt;&#xA;&lt;p&gt;One example to check out is Roger Howe's essay &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/pmet/resources/PVHoweEpp-Nov2008.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  I found activities where teachers count, add, subtract, and multiply using base 4 blocks to be very effective in my elementary mathematics teaching courses.&lt;/p&gt;&#xA;" OwnerUserId="7003" LastActivityDate="2021-01-07T20:20:37.287" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19332" PostTypeId="2" ParentId="19329" CreationDate="2021-01-08T05:16:34.613" Score="3" Body="&lt;p&gt;My approach might be to try to use smaller units of time to draw comparisons. I would look at a clock make sure the child knew about seconds or minutes. We could talk about how long one second is, and then how long one minute is, and how on a standard digital clock, the time is usually displayed up to the minute. We could even watch the time change as it goes from one minute to the next.&lt;/p&gt;&#xA;&lt;p&gt;Then I would probably steer the conversation towards how long ago something was. Whether it was one second ago, 10 seconds ago, or 60 seconds ago, and what the displayed clock might look like. Then advancing the conversations from seconds and minutes, to minutes and hours, and increase the time units on up from there.&lt;/p&gt;&#xA;&lt;p&gt;This is all hypothetical, but since your question seems to be only hypothetical, this may satisfy your question. If you do try it, let me know how it turns out!&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2021-01-08T05:16:34.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19333" PostTypeId="2" ParentId="19321" CreationDate="2021-01-08T15:39:00.710" Score="7" Body="&lt;p&gt;So in theory different students could get different questions?  The software I know of that does this is WeBWorK.  It will even score student-entered answers for you.  See &lt;a href=&quot;https://webwork.maa.org&quot; rel=&quot;noreferrer&quot;&gt;LINK&lt;/a&gt; . Supported by the Mathematical Association of America.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2021-01-08T15:39:00.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19334" PostTypeId="2" ParentId="5805" CreationDate="2021-01-09T12:35:16.393" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;How can I utilize a spaced repetition system to help me learn Mathematics most effectively?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I use the novel Spaced Repetition/ Note-taking Software &amp;quot;RemNote&amp;quot; to do so.&#xA;(I am  &lt;strong&gt;not&lt;/strong&gt; affiliated with RemNote! I just find the program amazing.)&lt;/p&gt;&#xA;&lt;p&gt;RemNote allows me to connect my flashcards. Thus, I can alawys see the &amp;quot;bigger picture&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;My workflow basically consists of five steps:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Continually repeating the maths cards in my database&lt;/li&gt;&#xA;&lt;li&gt;Understanding new concepts by describing them myself&lt;/li&gt;&#xA;&lt;li&gt;Making Exercises and turn my mistakes into &amp;quot;feedback flashcards&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Exact planning for the next day&lt;/li&gt;&#xA;&lt;li&gt;Repeating the process&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If you are interested in my workflow, you can read more about it here: &lt;a href=&quot;https://tobiasbru.medium.com/how-to-study-maths-with-remnote-fae890d3c20d&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to study Maths with Spaced Repetition&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What are the best ways of making strong connections between mathematical knowledge you've learned?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In RemNote, bidirectional links allow connecting certain concepts. This way you can see new connections and also manifest them, as you have discovered them.&lt;/p&gt;&#xA;" OwnerUserId="15219" LastEditorUserId="2083" LastEditDate="2021-01-14T08:12:40.973" LastActivityDate="2021-01-14T08:12:40.973" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19335" PostTypeId="2" ParentId="369" CreationDate="2021-01-09T18:11:18.893" Score="1" Body="&lt;p&gt;Dolciani eventually went on to write an explicit 'Algebra 2' book, used in the 1970's and 1980's commonly.  That Algebra 2 book has an exact replica of the material in the SMSG Intermediate Mathematics.  So I think that should resolve it.  'Algebra 2' is the SMSG 'Intermediate Mathematics.'&lt;/p&gt;&#xA;" OwnerUserId="15220" LastActivityDate="2021-01-09T18:11:18.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19336" PostTypeId="2" ParentId="11384" CreationDate="2021-01-09T19:12:41.057" Score="0" Body="&lt;p&gt;Use Excel Online (Google Sheets).. And install &lt;a href=&quot;https://workspace.google.com/marketplace/app/power_tools/1058867473888&quot; rel=&quot;nofollow noreferrer&quot;&gt;Power Tools&lt;/a&gt; for Google Sheets.. Then in Google Sheets go to Addons tab and &lt;strong&gt;start&lt;/strong&gt; Power Tools. Then choose &lt;code&gt;Randomize&lt;/code&gt; from Power Tools menu. Select &lt;code&gt;Shuffle&lt;/code&gt;. Then select choices of your test in excel sheet. Then select &lt;code&gt;Cells in each row&lt;/code&gt; and click &lt;code&gt;Shuffle&lt;/code&gt; from Power Tools menu. This will shuffle each row's selected cells independently from one another.&lt;/p&gt;&#xA;" OwnerUserId="15221" LastActivityDate="2021-01-09T19:12:41.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19337" PostTypeId="2" ParentId="19255" CreationDate="2021-01-10T17:41:57.493" Score="1" Body="&lt;p&gt;As a student of mathematics taking a statistics course next semester, I have been plagued with this problem. I have found a solution in &amp;quot;Lectures on Probability Theory and Mathematical Statistics&amp;quot; by Marco Taboga, which is a collection of many lectures and exercises from the website &lt;a href=&quot;https://www.statlect.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.statlect.com/&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="15226" LastActivityDate="2021-01-10T17:41:57.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19338" PostTypeId="1" CreationDate="2021-01-10T18:28:46.577" Score="-1" ViewCount="116" Body="&lt;p&gt;I’m a master student in Turkey. I’m researching math textbooks from different counties to compare them for my thesis. However, it is really difficult to find them. I need your suggestions. Or is there anyone who can share &lt;strong&gt;elementary math book&lt;/strong&gt; with me? I’m so greatful if you can.&#xA;Thanks&lt;/p&gt;&#xA;" OwnerUserId="15227" LastActivityDate="2021-01-11T07:55:42.547" Title="Math elementary textbooks" Tags="|textbooks|mathematical-analysis|" AnswerCount="1" CommentCount="4" ClosedDate="2021-01-11T08:44:18.490" ContentLicense="CC BY-SA 4.0" />
  <row Id="19339" PostTypeId="2" ParentId="19338" CreationDate="2021-01-10T21:28:50.933" Score="1" Body="&lt;p&gt;Gunaydan.&lt;/p&gt;&#xA;&lt;p&gt;Here is a lit review of US math elementary textbooks evolution:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/41103881?seq=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/41103881?seq=1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Realistically your question is so broad that I think you need to learn more before you can even ask the right questions.  But maybe this will get you thinking.&lt;/p&gt;&#xA;&lt;p&gt;For example, I would think about restricting your scope to two to three countries and one topic.  Maybe...decimals and Turkey, US, France.*  Questions to consider could be at what age is the material taught, how consistent is it within the country (I imagine France is more uniform than the US), and what methodology is emphasized (mnemonics, drill, etc.)  [An alternate approach might be more around business aspects: cost, selection, publishers.]&lt;/p&gt;&#xA;&lt;p&gt;*Just an example.  Pick something that interests you.  I would also look at the approach in Turkey first, to orient yourself, prior to looking at other countries.&lt;/p&gt;&#xA;&lt;p&gt;FWIW, I would also not lose sight of the similarities.  There's a scene in the movie Mean Girls where the heroine says math is her favorite class because it's the same in all languages.  This is not 100% true.  But it's MUCH more true than history or language.  (Based on my personal experience in Turkey.)  So in the midst of identifying differences, don't lose sight of similarities, either.&lt;/p&gt;&#xA;" OwnerUserId="15228" LastEditorUserId="2083" LastEditDate="2021-01-11T07:55:42.547" LastActivityDate="2021-01-11T07:55:42.547" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19340" PostTypeId="2" ParentId="19255" CreationDate="2021-01-11T04:36:47.280" Score="1" Body="&lt;p&gt;Kevin Arlin already gave my &amp;quot;best&amp;quot; answer in &lt;em&gt;All of Statistics&lt;/em&gt;. That said, statistics is a large topic. If you're looking specifically at inference and statistical learning and want a rigorous exposition of most of the different methodologies used I would recommend &lt;em&gt;Elements of Statistical Learning&lt;/em&gt;, although it assumes a heavier mathematics background than you might expect (in particular, matrix differentiation and probability).&lt;/p&gt;&#xA;" OwnerUserId="9019" LastActivityDate="2021-01-11T04:36:47.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19341" PostTypeId="2" ParentId="220" CreationDate="2021-01-11T12:38:15.640" Score="4" Body="&lt;p&gt;I often teach a short unit on induction in elementary calculus classes—it comes up very naturally when trying to prove a version of the power rule (i.e. the version pertaining to functions of the form &lt;span class=&quot;math-container&quot;&gt;$x \mapsto x^n$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n\in\mathbb{N}$&lt;/span&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Lemma 1:&lt;/strong&gt; (Product Rule)  Let &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; be real valued functions on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;.  At any point &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; where both &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; are differentiable,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (fg)'(x) = f'(x) g(x) + f(x) g'(x). $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As it is not really germane to this discussion, I'll assume that the product rule has already been proved.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem 2:&lt;/strong&gt; (Power Rule, version 1) For any natural number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm{d}}{\mathrm{d}x} x^n = nx^{n-1}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Proof of Theorem 2 (by induction):&lt;/em&gt;  As a basis for induction, note that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{\mathrm{d}}{\mathrm{d}x} x^1&#xA;= \lim_{h\to 0} \frac{(x+h)^1 - x^1}{h}&#xA;= \lim_{h\to 0} \frac{h}{h}&#xA;= 1&#xA;= x^0. $$&lt;/span&gt;&#xA;Suppose that there is some natural number &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{\mathrm{d}}{\mathrm{d}x} x^k = k x^{k-1}. $$&lt;/span&gt;&#xA;Then&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\frac{\mathrm{d}}{\mathrm{d}x} x^{k+1}&#xA;&amp;amp;= \frac{\mathrm{d}}{\mathrm{d}x} \left[ x \cdot x^k \right] \\&#xA;&amp;amp;= \left[  \frac{\mathrm{d}}{\mathrm{d}x} x \right] x^k + x \left[ \frac{\mathrm{d}}{\mathrm{d}x} x^k \right] &amp;amp;&amp;amp; \text{(Product Rule)} \\&#xA;&amp;amp;= 1 \cdot x^k + k x \cdot x^{k-1} &amp;amp;&amp;amp; \text{(Induction Hypothesis)} \\&#xA;&amp;amp;= (k+1) x^k,&#xA;\end{align}&lt;/span&gt;&#xA;which is the desired result. &lt;span class=&quot;math-container&quot;&gt;$\square$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;It is worth noting, perhaps, that most elementary calculus texts prove this version of the power rule by appealing to the binomial theorem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem 3:&lt;/strong&gt; (Binomial Theorem)  Let &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; be positive real numbers and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; a natural number.  Then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k}. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Proof of Theorem 2 (using the binomial theorem):&lt;/em&gt;  Let &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; be a natural number.  Then&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\frac{\mathrm{d}}{\mathrm{d}x} x^n&#xA;&amp;amp;= \lim_{h\to 0} \frac{(x-h)^n - x^n}{h} \\&#xA;&amp;amp;= \lim_{h\to 0} \frac{ \left[ \sum_{k=0}^{n} \binom{n}{k} x^k h^{n-k} \right] - x^n}{ h} &amp;amp;&amp;amp; \text{(Binomial Theorem)} \\&#xA;&amp;amp;= \lim_{h\to 0} \sum_{k=0}^{n-1} \binom{n}{k} x^k h^{n-k-1} \\&#xA;&amp;amp;= \binom{n}{n-1} x^{n-1} &amp;amp;&amp;amp; \text{(all other terms have a factor of $h$)} \\&#xA;&amp;amp;= n x^{n-1},&#xA;\end{align}&lt;/span&gt;&#xA;which is the claimed result. &lt;span class=&quot;math-container&quot;&gt;$\square$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have several complaints about this approach:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Many&lt;/em&gt; elementary calculus students struggle with even basic arithmetic operations.  Throwing binomial coefficients, factorials, and sigma notation at them is at this point in the class is a little unfair, unless one is willing to move the sequences and series material to the beginning of the class (this material is usually taught near the end of the second semester of calculus at American institutions).  The sigma notation &lt;em&gt;can&lt;/em&gt; be avoided, but it gets replaced with ellipses, which, in my opinion, is less rigorous and a bit hand-wavy.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The binomial theorem, itself, requires proof, and the classical proof is by induction:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Proof of Theorem 3:&lt;/em&gt;  As a basis for induction, note that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(a+b)^1 = a+b = \binom{1}{0} a^0 b^1 + \binom{1}{1} a^1 b^0 = \sum_{k=0}^{1} \binom{1}{k} a^k b^{n-k}. $$&lt;/span&gt;&#xA;Suppose that there is some natural number &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (a+b)^k = \sum_{k=0}^{n} \binom{n}{k} a^k b^{n-k}. $$&lt;/span&gt;&#xA;Then&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;(a+b)^{k+1}&#xA;&amp;amp;= (a+b) (a+b)^k \\&#xA;&amp;amp;= (a+b) \sum_{k=0}^{n} \binom{n}{k} a^k b^{k-n} \\&#xA;&amp;amp;= \big[\kern-2pt\big[ \text{several tedious steps involving combinatorial identities} \big]\kern-2pt\big] \\&#xA;&amp;amp;= \sum_{k=0}^{n+1} \binom{n+1}{k} a^k b^{n+1-k},&#xA;\end{align}&lt;/span&gt;&#xA;as desired. &lt;span class=&quot;math-container&quot;&gt;$\square$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that I've skipped much of the computational detail here—a lot of those details are, themselves, rather off-topic for a calculus class.  In order to use the binomial to prove the product rule, I also have to discuss with students the intermediate results that I've skipped here.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I am not going to argue that proving the power rule in calculus &lt;em&gt;requires&lt;/em&gt; induction (I can't think of an alternative proof off the top of my head, but that doesn't mean anything).  On the other hand, I think that induction is the &lt;em&gt;best&lt;/em&gt; way to obtain the product rule (even if you choose to invoke the binomial theorem, there is an induction hiding behind the scenes), and it provides a good excuse to introduce induction.&lt;/p&gt;&#xA;&lt;p&gt;In an American &amp;quot;Introduction to Proofs&amp;quot; or elementary &amp;quot;Discrete Mathematics&amp;quot; course, either of which is usually offered &lt;em&gt;after&lt;/em&gt; students have taken calculus, I think that the first proof offers a nice call-back to calculus.  Students should already know the power rule, and may even have seen some kind of proof (probably a mysterious proof involving the binomial theorem, which they probably won't have seen a proof of).  Thus providing this as a first or second example gives some context for induction, and helps to connect earlier mathematical knowledge to new material.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2022-08-13T21:37:56.287" LastActivityDate="2022-08-13T21:37:56.287" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19342" PostTypeId="1" AcceptedAnswerId="19345" CreationDate="2021-01-11T20:08:55.157" Score="13" ViewCount="2082" Body="&lt;p&gt;As we try to work and teach in the midst of this pandemic, some problems arise when making online math exams. My question is simple: What could be an interesting basic differentiation question such that students doing the online exam still have to work a bit to get it, rather than simply Googling it or using WolframAlpha/Mathematica to solve it for them?&lt;/p&gt;&#xA;&lt;p&gt;Everything I can think of seems to be either too easy (meaning they can easily solve it online) or too demanding. I'm struggling to find a good balance, if there is any. This is a first-year math module I'm teaching (non-math students).&lt;/p&gt;&#xA;&lt;p&gt;Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="12603" LastEditorUserId="12603" LastEditDate="2021-01-13T21:46:14.443" LastActivityDate="2021-01-15T08:09:28.877" Title="Finding the Balance in a Math Question (Teaching)" Tags="|undergraduate-education|mathematical-pedagogy|derivative|cheating|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19343" PostTypeId="2" ParentId="19342" CreationDate="2021-01-11T22:44:29.463" Score="2" Body="&lt;p&gt;Related rate word problem question with some geometry needed as part of the problem.&lt;/p&gt;&#xA;&lt;p&gt;But.  I would caution you that if you are teaching weaker students, there is a problem in assigning tough problems in that you don't well evaluate their ability in basic methods.    I would instead suggest alternatives like short duration (fixed time) remote exams.  Also honor statements.  None of the methods are perfect, but they help, especially the former.&lt;/p&gt;&#xA;" OwnerUserId="15236" LastActivityDate="2021-01-11T22:44:29.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19344" PostTypeId="2" ParentId="19342" CreationDate="2021-01-12T02:57:51.660" Score="9" Body="&lt;p&gt;&lt;em&gt;Note&lt;/em&gt;:  It's possible that in the future, Wolfram Alpha will improve and be able to answer the questions in this answer, so it's best to actually try them in Wolfram Alpha first.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Use questions that involve conditional statements and generic expressions.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to\infty} f(x)=1$&lt;/span&gt;, then what is &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to\infty}\frac{f(x)}{x}$&lt;/span&gt; equal to?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Typing &lt;code&gt;limit of (f(x)/x) as x goes to infinity if limit of f(x) as x goes to infinity is equal to 1&lt;/code&gt; or &lt;code&gt;limit of f(x) as x goes to infinity is equal to 1, what is limit of f(x)/x as x goes to infinity?&lt;/code&gt; or &lt;code&gt;if limit of f(x) as x goes to infinity is equal to 1, then what is limit of f(x)/x as x goes to infinity?&lt;/code&gt; into Wolfram Alpha doesn't give the answer.&lt;/p&gt;&#xA;&lt;p&gt;A more difficult one:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to\infty} f(x)=1$&lt;/span&gt;, then what is &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to\infty}\frac{x-f(x)}{x+f(x)}$&lt;/span&gt; equal to?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One involving derivatives:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f'(x)=\sqrt{x}\cos(\pi x^2)$&lt;/span&gt;, then what is &lt;span class=&quot;math-container&quot;&gt;$f'(4)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Intentionally choose &lt;span class=&quot;math-container&quot;&gt;$f(x)=\int\sqrt{x}\cos(\pi x^2)\,\mathrm{d}x$&lt;/span&gt; to be ridiculously complicated.)  Typing &lt;code&gt;If f'(x)=x^0.5cos(pi x^2), then what is f'(4)?&lt;/code&gt; into Wolfram Alpha doesn't work.&lt;/p&gt;&#xA;&lt;p&gt;Make it clear to the students that &lt;span class=&quot;math-container&quot;&gt;$f'(4)$&lt;/span&gt; (which is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;), is not the same as &lt;span class=&quot;math-container&quot;&gt;$\frac{\mathrm{d}}{\mathrm{d}x}\left(f(4)\right)$&lt;/span&gt; (which is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;You can then ask (after discussing the chain rule) the more difficult question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f'(x)=\sqrt{x}\cos(\pi x^2)$&lt;/span&gt;, then what is &lt;span class=&quot;math-container&quot;&gt;$\frac{\mathrm{d}}{\mathrm{d}x}\left(f(x^2)\right)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2021-01-12T03:43:16.700" LastActivityDate="2021-01-12T03:43:16.700" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19345" PostTypeId="2" ParentId="19342" CreationDate="2021-01-12T13:15:39.943" Score="26" Body="&lt;ul&gt;&#xA;&lt;li&gt;I think it is helpful to let students know that you are looking for their thinking while problem-solving, and not just answers.  Then you can ask questions like:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Find all of the points on a circle of radius &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; that have a slope of &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;so that they can explain their problem-solving.  Even if they do end up using advanced calculators, they will have to manually break the problem down into steps, which demonstrates some understanding.  Alternatively, you can also give them a problem that you &lt;em&gt;know&lt;/em&gt; is too challenging, and turn the question into something like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Do not solve, but describe the approaches you would try in order to find &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$f(x)=$&lt;/span&gt; [function that's too hard]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You can also ask for their thinking directly.  I'm not sure of the complexity you are looking for, but for example:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The derivative of &lt;span class=&quot;math-container&quot;&gt;$\frac{x^2+4}{\sqrt{x}}$&lt;/span&gt; can be found using the quotient rule or the product rule. Show that both give the same result.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Problems based on pictures are harder to Google.  This one is based on a Khan Academy problem:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5WHE5.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5WHE5.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There are also &amp;quot;find-the-mistake&amp;quot; problems, like:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Carser took the following steps when finding the derivative of &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2 \cdot \sin(x)$&lt;/span&gt;.  Describe his mistake and explain how to correct it.  &lt;span class=&quot;math-container&quot;&gt;$$f(x)=x^2 \cdot \sin(x)$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$f'(x)=2x \cdot \cos(x)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="219" LastEditorUserId="219" LastEditDate="2021-01-12T15:13:47.740" LastActivityDate="2021-01-12T15:13:47.740" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19346" PostTypeId="2" ParentId="5805" CreationDate="2021-01-13T03:37:36.070" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How can I utilise a spaced repetition system to help me learn Mathematics most effectively?&lt;/li&gt;&#xA;&lt;li&gt;What are the best ways of making strong&#xA;connections between mathematical knowledge you've learned?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't think spaced-repetition flashcard systems lend themselves to abstractions, such as the logic and connections among different facts in math.&lt;/p&gt;&#xA;&lt;p&gt;I would suggest using &lt;a href=&quot;https://www.youtube.com/watch?v=b7o09a7t4RA&amp;amp;t=739s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ali Abdaal's Revision Timetable&lt;/a&gt; approach instead. Instead of &lt;em&gt;predicting&lt;/em&gt; what you'll need to review in the future, you can simply &lt;em&gt;track&lt;/em&gt; what you need to study next.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2021-01-13T03:37:36.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19348" PostTypeId="2" ParentId="19321" CreationDate="2021-01-15T08:25:40.827" Score="0" Body="&lt;p&gt;This can be done, although I haven't worked out a model to show how. Latex/Tex is a programming language in the sense that one can declare and initialize variables (\newcounter, \setcounter), iterate (there are while loops), and use conditionals (\ifthenelse). Additionally, one caan call a random number generator (I don't habitually use this, but I think it is \rand). Combined with \input, \include or some similar tool these ingredients are sufficient to write a latex file that generates an exercise sheet with problems drawn randomly from a bank.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2021-01-15T08:25:40.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19349" PostTypeId="1" CreationDate="2021-01-15T08:43:12.033" Score="25" ViewCount="5415" Body="&lt;p&gt;I know someone I really like, but sadly, that person has absolutely no experience in math or mathematical thinking above third grade mathematics (+, - are fine, but division already makes problems). This person is an adult and not stupid, but he never had any opportunity to get more educated and it worked for all his life like this.&lt;/p&gt;&#xA;&lt;p&gt;Now, he asked me why you'd need proofs for anything ever, since it's enough to see that an equation works after testing it for a few values.&lt;/p&gt;&#xA;&lt;p&gt;What very practical example could I use where an equation works for like the first 100 numbers in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt; or something like that, but fails afterwards? Something that doesn't need a long introduction into equations (since this is above his head), except for things like &amp;quot;? + 5 = 10, what is ?&amp;quot; (&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s needlessly complicate this since it &amp;quot;looks too mathy&amp;quot;).&lt;/p&gt;&#xA;" OwnerUserId="15257" LastEditorUserId="851" LastEditDate="2021-01-18T06:32:34.483" LastActivityDate="2021-02-26T21:30:11.823" Title="How can I explain why we need proofs to someone who has no experience in mathematical thinking?" Tags="|proofs|primary-education|" AnswerCount="23" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="19350" PostTypeId="2" ParentId="19349" CreationDate="2021-01-15T15:06:29.037" Score="13" Body="&lt;p&gt;Theorem: Suppose you start with the number 1, and you're allowed to multiply it by 2, 3, and/or 5, as many times as you like. In this way, it's possible to get any  whole number.&lt;/p&gt;&#xA;&lt;p&gt;Proof: Check this for 1, 2, 3, 4, 5, and 6. Also true for other random examples I think of, such as 10, 100, and 96. It's clearly true.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-01-15T15:06:29.037" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19351" PostTypeId="2" ParentId="19349" CreationDate="2021-01-15T16:56:19.270" Score="33" Body="&lt;p&gt;More fun than equations are patterns that seem to hold. Put a dot on a circle, connect it to all the other dots (none yet), there is 1 region. Second dot connects to first, two regions. Third dot connects to first two, there are now 4 regions. It sure looks like the regions are doubling. In fact, they are not. It's a fun problem, and takes you by surprise. After you find a number pattern that works better, you will want to make sure it always works (ie prove it).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example:&lt;/strong&gt; Put five dots on a circle and connect them all through straight lines. The number of regions is 16.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/c5YVU.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/c5YVU.png&quot; alt=&quot;Example&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;My paper on this: &lt;a href=&quot;https://scholarship.claremont.edu/jhm/vol1/iss2/7/&quot; rel=&quot;noreferrer&quot;&gt;https://scholarship.claremont.edu/jhm/vol1/iss2/7/&lt;/a&gt; (download is free)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2021-01-18T01:37:15.793" LastActivityDate="2021-01-18T01:37:15.793" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19352" PostTypeId="2" ParentId="19349" CreationDate="2021-01-15T17:39:45.600" Score="5" Body="&lt;p&gt;From a practical point of view, your friend is right.  One just has to be careful to test enough numbers.  Engineers don't believe calculus because it has proofs based on analysis which is based on set theory; engineers believe calculus because lots of engineers before them used calculus to build bridges and very few of the bridges fell down.  We have analysis in the first place, not primarily because of logical doubts about calculus, but because there were methods in summing Fourier series that were giving nonsensical results and people wanted to know why they were happening and what not to do to avoid them.&lt;/p&gt;&#xA;&lt;p&gt;An alien civilization could treat mathematics entirely empirically, as a kind of empirical science of quantities (or even as a branch of physics), and, while this treatment has some disadvantages, they could have all the technology we have.&lt;/p&gt;&#xA;&lt;p&gt;On Earth, Euclid decided that mathematics would be a good subject with which to teach and demonstrate the power of Aristotelian logic, and people ever since have found this approach to mathematics beautiful and attractive.  It is one of the crowning glories of our civilization.  Some of it may be hard for some people to understand, but no more so than Joyce's &lt;em&gt;Ulysses&lt;/em&gt; or Schoenberg's &lt;em&gt;Pierrot Lunaire&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2021-01-15T17:39:45.600" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19353" PostTypeId="2" ParentId="19349" CreationDate="2021-01-15T18:17:45.877" Score="14" Body="&lt;p&gt;One thing that can't be shown without proof techniques is the impossibility of something. One reasonably concrete example of this is the impossibility of constructing a square and a circle with the same area using a straight edge and a compass. Individual cases can be shown to not work and to demonstrate that it's a &lt;em&gt;hard&lt;/em&gt; problem, but there's a world of difference between hard and impossible.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, there's a big difference between knowing &lt;em&gt;that&lt;/em&gt; something works, and knowing &lt;em&gt;why&lt;/em&gt; something works. A good proof not only demonstrates that a given property or equation works but provides an understanding of why it works.&lt;/p&gt;&#xA;" OwnerUserId="15259" LastActivityDate="2021-01-15T18:17:45.877" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19354" PostTypeId="2" ParentId="19349" CreationDate="2021-01-15T18:30:04.960" Score="9" Body="&lt;p&gt;Because mathematicians want to be really, really certain.&lt;/p&gt;&#xA;&lt;p&gt;There are lots of ways you can be convinced that something is true, and be wrong.  By requiring proofs before you are convinced, you can make that amazingly less likely.&lt;/p&gt;&#xA;&lt;p&gt;A proof requires that someone lay bare all of the weaknesses in their claim, and show how those weaknesses are not a problem.  Someone checking a proof can look at it, and even without understanding it fully, can determine if the person writing the proof cheated or made a mistake.&lt;/p&gt;&#xA;&lt;p&gt;Imagine if you had a laser pointer, and when you pointed it at someone telling a lie, it was red, and if they told the truth, it was green.&lt;/p&gt;&#xA;&lt;p&gt;And it wasn't only &amp;quot;the truth as they see it&amp;quot;, but the absolute truth.  If they think the moon is made of green cheese, and you point the laser at them, and they say &amp;quot;the moon is made of green cheese&amp;quot; it will show &amp;quot;that is a lie&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;That superpower would be very useful.  Proofs in mathematics give other mathematicians that super power.  They can even point the laser at themselves, and see if their own statements are true or not!&lt;/p&gt;&#xA;&lt;p&gt;Now mathematical proofs isn't quite as good as that laser, because the super power only works on proofs, not on everything you say, and checking a proof takes work (it isn't instant).  Plus, finding proofs for something can be extremely hard.&lt;/p&gt;&#xA;&lt;p&gt;Mathematics relies on this to do some of its amazing things.  Mathematics builds these huge complex castles of thought, where each brick, each beam, each blob of cement is more mathematics.  It is only because we can &lt;strong&gt;rely&lt;/strong&gt; on the quality of the building-blocks that mathematics can build the complex stuff it can do.&lt;/p&gt;&#xA;&lt;p&gt;To see how this is useful, imagine that laser pointer again.  And you want to build the best castle.  So you walk around and say &amp;quot;this rock would be good for building that castle&amp;quot;, &amp;quot;I should build a wall here for my castle&amp;quot;, &amp;quot;this wall is strong enough to hold up my castle&amp;quot;, then check the laser pointer each time.  That super laser pointer can &lt;strong&gt;teach you&lt;/strong&gt; how to build the castle you want to build, even if you don't know how to.&lt;/p&gt;&#xA;&lt;p&gt;Those castles of thought let you do things like predict the weather (with imperfect accuracy, but way better than we could even 10 years ago), build skyscrapers, split the atom, fly to the moon, or invent a covid-19 vaccine.&lt;/p&gt;&#xA;&lt;p&gt;Now, not &lt;strong&gt;everything&lt;/strong&gt; in those tasks is done with mathematics.  But supporting all of that science are huge castles of mathematics that are relied upon to not be made of nonsense.&lt;/p&gt;&#xA;" OwnerUserId="9985" LastEditorUserId="15238" LastEditDate="2021-01-15T19:16:58.103" LastActivityDate="2021-01-15T19:16:58.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19355" PostTypeId="2" ParentId="19349" CreationDate="2021-01-15T19:16:52.167" Score="12" Body="&lt;p&gt;Maybe you can use some non-mathematical examples...&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Black_swan_theory&quot; rel=&quot;noreferrer&quot;&gt;Black swans&lt;/a&gt;. Europeans assumed that all swans were white, because all swans in Europe are white, to the extent that a black swan was considered impossible--like a &amp;quot;flying pig&amp;quot;. When they finally got to Australia and discovered that there are black swans, it was a big shock.&lt;/p&gt;&#xA;&lt;p&gt;Suppose someone tells you that they leave their car door unlocked every day and nothing bad has ever happened. How confident do you feel in leaving your car door unlocked? Do you think it's &lt;em&gt;impossible&lt;/em&gt; for your car to be burgled, just because your friend's car hasn't been up until now?&lt;/p&gt;&#xA;&lt;p&gt;Even better would be coming up with some examples relating to some area that your friend is interested in.&lt;/p&gt;&#xA;&lt;p&gt;Then you can show some concrete things that can be proven... Drawing four aces in a row from a normal deck is unlikely, drawing five is impossible.&lt;/p&gt;&#xA;" OwnerUserId="1908" LastActivityDate="2021-01-15T19:16:52.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19356" PostTypeId="2" ParentId="19349" CreationDate="2021-01-15T19:18:29.513" Score="5" Body="&lt;p&gt;Try solving &lt;a href=&quot;https://en.wikipedia.org/wiki/Pell%27s_equation&quot; rel=&quot;noreferrer&quot;&gt;Pell's equation&lt;/a&gt; by trial and error. For example:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2-61y^2=1$&lt;/span&gt; has a whole-number solution &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt;. Are there any others?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Try a few values for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, and it will appear not. Is that a proof that there are no more solutions?&lt;/p&gt;&#xA;&lt;p&gt;It turns out that the next smallest solution is &lt;span class=&quot;math-container&quot;&gt;$x=1766319049$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=226153980$&lt;/span&gt;, which would take quite a while to find by searching.&lt;/p&gt;&#xA;&lt;p&gt;It's a relatively simple problem to explain - a square of tiles can be rearranged into &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; smaller identical squares, with one left over - but for some values of the multiplier &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; it leads to very big numbers. For other values (squares) there is no other solution - you would search forever.&lt;/p&gt;&#xA;" OwnerUserId="15261" LastEditorUserId="15261" LastEditDate="2021-01-15T19:24:53.713" LastActivityDate="2021-01-15T19:24:53.713" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19357" PostTypeId="2" ParentId="19349" CreationDate="2021-01-15T19:24:55.613" Score="14" Body="&lt;p&gt;Most answers describe proof as an improved correctness checking tool.&#xA;But, I don't think this gets at the core of the issue. The goal of math isn't to check off theorems, but to understand them and consequently nature (broadest possible meaning of the word).&lt;/p&gt;&#xA;&lt;p&gt;Proofs are a way to describe insights into a subject in a clear manner.  Impossibility proofs tell you &amp;quot;why&amp;quot; something can't work. Constructive proofs show you &amp;quot;how&amp;quot; to make the thing you want. In general, good proofs are those with explanatory power and insight, they expose what's going on. Consequently, writing a proof forces you to understand the problem, making it a fantastic learning process.&lt;/p&gt;&#xA;&lt;p&gt;The fact that mathematicians continuously revisit the same subjects and theorems (lines, circles, etc) to develop new ways of looking at them supports this view.&lt;/p&gt;&#xA;&lt;p&gt;Even simple problems require a lot of understanding.&#xA;How many subsets are there of a given set? A natural thing to try is writing them for a small set like &lt;span class=&quot;math-container&quot;&gt;$\{1, 2, 3\}$&lt;/span&gt;. Notice that for the number 1, there are subsets that contain it, and subsets that don't. Those that don't include 1, are the subsets of &lt;span class=&quot;math-container&quot;&gt;$\{2, 3\}$&lt;/span&gt;, those that contain 1 are not.&#xA;So if we remove one element &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; from a set &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, and find &lt;span class=&quot;math-container&quot;&gt;$P(A - \{ x \})$&lt;/span&gt;, all of those sets are still subsets of A and none contain x. Adding x into each one creates all the subsets to complete P(A). Each additional element in a set thus double the number of subsets. After checking the empty set (base case), you can conclude &lt;span class=&quot;math-container&quot;&gt;$|P(A)| = 2^{|A|}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Writing out a table would tell you the same answer, but it wouldn't give the same insight into the subject (or certainty). As a a bonus, once we learn why it's &lt;span class=&quot;math-container&quot;&gt;$2^{|A|}$&lt;/span&gt; we also know how to construct subsets with an algorithm.&lt;/p&gt;&#xA;" OwnerUserId="15260" LastEditorUserId="15260" LastEditDate="2021-01-28T21:08:36.763" LastActivityDate="2021-01-28T21:08:36.763" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19358" PostTypeId="2" ParentId="19349" CreationDate="2021-01-15T22:11:57.860" Score="7" Body="&lt;p&gt;An example that directly contradicts the idea of &amp;quot;seeing that an equation works after testing it for a few values:&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem (not): The polynomial &lt;span class=&quot;math-container&quot;&gt;$x^2-x+41$&lt;/span&gt; is prime for every integer value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This works after testing it for every integer &lt;span class=&quot;math-container&quot;&gt;$x = 0, 1, 2, \dots, 40$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So &amp;quot;obviously&amp;quot; (according to your friend!) its value is a prime number for any integer value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Just don't try &lt;span class=&quot;math-container&quot;&gt;$x = 41$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Historical note: this was originally discovered by Euler, and mentioned in a letter to Bernoulli in 1771.&lt;/p&gt;&#xA;" OwnerUserId="7677" LastActivityDate="2021-01-15T22:11:57.860" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19359" PostTypeId="2" ParentId="19349" CreationDate="2021-01-15T22:18:46.253" Score="1" Body="&lt;p&gt;To some extent, your friend is totally right. If you want to check if an equation always holds, then just checking several values is often a nice and practical way to do that.&lt;/p&gt;&#xA;&lt;p&gt;But proofs can certainly come in handy sometimes. Here's a simplified version of a problem I encountered at work this week.&lt;/p&gt;&#xA;&lt;p&gt;We had 8 cables of various lengths and 8 conduits of various lengths. We needed to put one cable in each conduit. Also, each cable needed to be longer than the conduit it was in. If a cable was too short, it wouldn't connect the two components on the ends of the conduit, so that would be bad. On the other hand, if a cable was too long, we could simply tuck the excess away, and that would be perfectly fine.&lt;/p&gt;&#xA;&lt;p&gt;Here were the lengths of the conduits: 50 feet, 60 feet, 65 feet, 70 feet, 80 feet, 80 feet, 100 feet, 120 feet.&lt;/p&gt;&#xA;&lt;p&gt;And here were the lengths of the cables: 55 feet, 60 feet, 60 feet, 70 feet, 75 feet, 85 feet, 110 feet, 130 feet.&lt;/p&gt;&#xA;&lt;p&gt;If you simply start trying different ways of arranging the cables and conduits, you'll find that none of the ways you're trying work. But there are over 40,000 possible assignments of cables to conduits. Even if you try 1,000 arrangements and none of them work, how do you know that no solution exists?&lt;/p&gt;&#xA;&lt;p&gt;Fortunately, we don't have to rely on trial and error. If you know the right &amp;quot;trick,&amp;quot; then it's easy to prove that there really is no way of making these cables fit in these conduits. That's a lot easier.&lt;/p&gt;&#xA;&lt;p&gt;(I sent an email to the project manager stating that the cables we had weren't long enough, and that we needed some longer cables delivered.)&lt;/p&gt;&#xA;" OwnerUserId="930" LastActivityDate="2021-01-15T22:18:46.253" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19360" PostTypeId="2" ParentId="19349" CreationDate="2021-01-16T00:09:37.173" Score="4" Body="&lt;p&gt;get him to draw a triangle (maybe even freehand), and measure the internal angles.&#xA;point out all the things in the world built with triangles - electricity pylons, eiffel tower, maps etc - triangles are important, and its important in order to build our world to understand triangles completely.&lt;/p&gt;&#xA;&lt;p&gt;what was the sum of the internal angles? (in my experience it won't be 180degrees due to inaccurate readings)  Repeat it again with a few more triangles.  Then explain that internal angles always add up to 180degrees - its been &lt;strong&gt;proved&lt;/strong&gt;, so engineers dont need to guess or measure every triangle - they can just use the proved fact.&lt;/p&gt;&#xA;" OwnerUserId="15262" LastActivityDate="2021-01-16T00:09:37.173" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19361" PostTypeId="2" ParentId="19349" CreationDate="2021-01-16T01:32:50.907" Score="2" Body="&lt;p&gt;There is quite an amazing collection in response to the&#xA;11 yr-old MathOverflow&#xA;question, &lt;a href=&quot;https://mathoverflow.net/q/15444/6094&quot;&gt;Examples of eventual counterexamples&lt;/a&gt;.&#xA;(This a precursor to the already cited 9 yr-old MSE question,&#xA;&lt;a href=&quot;https://math.stackexchange.com/q/111440/237&quot;&gt;Examples of patterns that eventually fail&lt;/a&gt;.)&#xA;Here's one (among many) I like (&lt;a href=&quot;https://mathoverflow.net/a/52382/6094&quot;&gt;posted by Gerry Myerson&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The numbers &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$121$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$1211$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$12111$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$121111$&lt;/span&gt;, etc., are all composite - until you get to the one with &lt;span class=&quot;math-container&quot;&gt;$138$&lt;/span&gt; digits—that's a prime.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For example,&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;121 &amp;amp; = 11^2 \\&#xA;1211 &amp;amp; = 7 \cdot 173 \\&#xA;12111 &amp;amp; = 3 \cdot 11 \cdot 367 \\&#xA;121111 &amp;amp; = 281 \cdot 431 \\&#xA;1211111 &amp;amp; = 11 \cdot 23 \cdot 4787 &#xA;\end{align}&lt;/span&gt;&#xA;Here is the number with &lt;span class=&quot;math-container&quot;&gt;$137$&lt;/span&gt; digits (parsed with commas):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;12,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111&#xA;$$&lt;/span&gt;&#xA;It is divisible by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; (and &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt;, and ...).&#xA;Whereas the number with &lt;span class=&quot;math-container&quot;&gt;$138$&lt;/span&gt; digits is prime:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;121,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-01-16T02:07:50.573" LastActivityDate="2021-01-16T02:07:50.573" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19362" PostTypeId="2" ParentId="19349" CreationDate="2021-01-16T02:54:33.343" Score="0" Body="&lt;p&gt;Does your friend get bothered by multiplication?  You say &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt; are okay but division is problematic, so you skipped over multiplication.&lt;/p&gt;&#xA;&lt;p&gt;If your friend is okay with multiplication, and knows about breaking up numbers into prime factors, here's something to claim: every number has at most three different prime factors (&lt;span class=&quot;math-container&quot;&gt;$45 = 3 \cdot 3 \cdot 5$&lt;/span&gt; has just two different prime factors, for instance). This is true for all numbers up to &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt;, but of course it is ultimately wrong and the first number with four different prime factors is &lt;span class=&quot;math-container&quot;&gt;$2 \cdot 3 \cdot 5 \cdot 7 = 210$&lt;/span&gt;. So a pattern that holds for &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt;, or even &lt;span class=&quot;math-container&quot;&gt;$200$&lt;/span&gt;, examples need not hold all the time.&lt;/p&gt;&#xA;&lt;p&gt;Or how about this: if &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is prime then &lt;span class=&quot;math-container&quot;&gt;$2^p-1$&lt;/span&gt; is prime:&lt;/p&gt;&#xA;&lt;p&gt;(1) &lt;span class=&quot;math-container&quot;&gt;$2^2 - 1 = 3$&lt;/span&gt; is prime&lt;/p&gt;&#xA;&lt;p&gt;(2) &lt;span class=&quot;math-container&quot;&gt;$2^3 - 1 = 7$&lt;/span&gt; is prime&lt;/p&gt;&#xA;&lt;p&gt;(3) &lt;span class=&quot;math-container&quot;&gt;$2^5 - 1 = 31$&lt;/span&gt; is prime&lt;/p&gt;&#xA;&lt;p&gt;(4) &lt;span class=&quot;math-container&quot;&gt;$2^7 - 1 = 127$&lt;/span&gt; is prime&lt;/p&gt;&#xA;&lt;p&gt;(5) &lt;span class=&quot;math-container&quot;&gt;$2^{11} - 1 = 2047$&lt;/span&gt; is prime... just kidding: it is &lt;span class=&quot;math-container&quot;&gt;$23 \cdot 89$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So the pattern broke after five tries. Now ask your friend: &lt;em&gt;is some definite number of examples always good enough&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Claim: A number &lt;span class=&quot;math-container&quot;&gt;$n &amp;gt; 1$&lt;/span&gt; is prime exactly when &lt;span class=&quot;math-container&quot;&gt;$2^{n-1}-1$&lt;/span&gt; is divisible by &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.  This is correct for &lt;span class=&quot;math-container&quot;&gt;$n \leq 300$&lt;/span&gt;. It breaks for the first time at &lt;span class=&quot;math-container&quot;&gt;$n = 341$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$2^{340}-1$&lt;/span&gt; is divisible by &lt;span class=&quot;math-container&quot;&gt;$341$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$341$&lt;/span&gt; is composite: it is &lt;span class=&quot;math-container&quot;&gt;$11 \cdot 31$&lt;/span&gt;. The next counterexample is &lt;span class=&quot;math-container&quot;&gt;$561$&lt;/span&gt;, which is &lt;span class=&quot;math-container&quot;&gt;$3 \cdot 11 \cdot 17$&lt;/span&gt;. There are infinitely many further counterexamples.&lt;/p&gt;&#xA;&lt;p&gt;So again we have a pattern that works initially but eventually doesn't.  The lesson here is that in math, merely believing something because it works for a bunch of examples is not a viable method of knowing &lt;em&gt;what is correct&lt;/em&gt; and &lt;em&gt;what is not correct&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2021-01-16T03:52:51.433" LastActivityDate="2021-01-16T03:52:51.433" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19363" PostTypeId="2" ParentId="19349" CreationDate="2021-01-16T03:01:40.030" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;since it's enough to see that an equation works after testing it for a few values.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Would they be willing to bet their life on that if that equation was being used for an autopilot in an airplane or some other piece of equipment? Or if the equation takes millions of possible combinations of values?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose I make an engine. I test it and it works. Great. Now what? I can make that same engine again, but without understanding how it really works I will have a lot of trouble improving it and building upon it. Proofs are the mathematical equivalent of picking something apart to see &lt;em&gt;if&lt;/em&gt; and &lt;em&gt;why&lt;/em&gt; it is true.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Related to number #2, math is not really about the investigation&#xA;equations. It is about the investigation of concepts and ideas of&#xA;which equations are a very tiny part. Imagine a biologist comes up&#xA;with a theory: &amp;quot;All birds fly.&amp;quot; So he finds a few birds and they all&#xA;fly. Does that mean he is correct? Were his few tests good enough? We know that they were obviously not because a few odd cases were missed.&lt;/p&gt;&#xA;&lt;p&gt;Those few odd cases that were missed can cause big problems if&#xA;another mathematician assumed it was true and used it in their own&#xA;work for something else, or can cause literal disaster if it is&#xA;being used to calculate the mechanical stresses of a bridge or used&#xA;in a computer program that controls heavy machinery.&lt;/p&gt;&#xA;&lt;p&gt;For a biologist to really prove that he would need to find every&#xA;single bird in existence. That's obviously impractical, and even&#xA;when talking about simple equations, there are &lt;em&gt;far far&lt;/em&gt; more&#xA;numbers, let alone combinations of numbers that can go into an&#xA;equation. But unlike the biologist, in math you have the luxury&#xA;that it follows logic so you can actually verify whether something always works or whether it only works under certain conditions and figure out what those conditions are. Those are proofs.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="13250" LastEditorUserId="13250" LastEditDate="2021-01-16T20:58:31.647" LastActivityDate="2021-01-16T20:58:31.647" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19364" PostTypeId="2" ParentId="19349" CreationDate="2021-01-16T03:06:21.323" Score="2" Body="&lt;p&gt;Here's a surprising fact I've learned today on this very site:&lt;/p&gt;&#xA;&lt;p&gt;Have your friend raise with a calculator&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1.5^{1.5^{...}} $&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;At some point it goes to infinity.&lt;/p&gt;&#xA;&lt;p&gt;Now let him do the same with&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1.4^{1.4^{...}} $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Funnily it stop growing. That raises some questions about the previous one : did it actually stop growing? Or was just the calculator screen to small and it stops eventually? And how can we ever be sure that's the case? And what about 1.49?&lt;/p&gt;&#xA;&lt;p&gt;I guess you could do the same with the geometric series.&lt;/p&gt;&#xA;" OwnerUserId="15264" LastActivityDate="2021-01-16T03:06:21.323" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19365" PostTypeId="2" ParentId="19349" CreationDate="2021-01-16T11:30:12.847" Score="-1" Body="&lt;p&gt;&lt;strong&gt;Money, even imaginary money is a good motivator&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Much mathematical thinking historically, came from gambling and card playing.  If you want to avoid losing your shirt, you need to understand the mathematics. It is a very practical use of the subject.&lt;/p&gt;&#xA;&lt;p&gt;A simple experiment is to throw two dice.  Ask your friend to bet on which sum will come up most often (use a box of matches each). Make sure you choose first and choose seven.  Repeatedly throw the dice. You will win in the long term and your friend will become bankrupt. Ask if the dice are loaded. See if the other person can come up with an explanation of why certain sums are more likely than others. This will get them thinking.&lt;/p&gt;&#xA;&lt;p&gt;They have observed that seven is the most likely sum so now ask them to &lt;strong&gt;prove&lt;/strong&gt; that seven is the most likely result. This will help them to start constructing proofs.&lt;/p&gt;&#xA;&lt;p&gt;Eventually you may nudge them towards drawing a diagram like the following.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nhY5p.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nhY5p.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Forget precise probabilites, just point out that seven is more likely because there are more ways to achieve it with two dice.&lt;/p&gt;&#xA;&lt;p&gt;There are some good stories to be found online regarding gamblers being cheated by con-artists.&lt;/p&gt;&#xA;&lt;p&gt;From here you can point out that people with no knowledge of maths are vulnerable to being manipulated with respect to money. Collect a  few examples.**&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;** E.g. Pyramid schemes etc.&lt;/p&gt;&#xA;" OwnerUserId="5798" LastEditorUserId="5798" LastEditDate="2021-01-16T11:35:54.233" LastActivityDate="2021-01-16T11:35:54.233" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19366" PostTypeId="2" ParentId="19349" CreationDate="2021-01-16T11:40:12.043" Score="6" Body="&lt;p&gt;Put 100 red marbles in a box and 1 white marble. When you blindly pick a few marbles and they are all red, it does not mean all marbles in the box are red.&lt;/p&gt;&#xA;" OwnerUserId="15271" LastActivityDate="2021-01-16T11:40:12.043" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19367" PostTypeId="1" AcceptedAnswerId="19368" CreationDate="2021-01-16T13:35:55.043" Score="1" ViewCount="301" Body="&lt;p&gt;I am giving tutoring to a high school student since more than one year and half. He is about 17 years old. He has excellent results in mathematics, being best student in his class for the last year and half and he would finally understand every concept I want to teach up to now.&lt;/p&gt;&#xA;&lt;p&gt;His parents told me the first time we met that they'd like me to teach math, maybe in a more recreational way than school, even though it is up to me to decide.&lt;/p&gt;&#xA;&lt;p&gt;It has been several days I'm wondering myself if it is appropriate to play Chess with him. I like chess since a long time, I know it has a lot of relations with the mathematics, with the logic, many applications in computer science, game theory and so on. So would it be appropriate playing chess with him?&lt;/p&gt;&#xA;&lt;p&gt;During the one hour and half I spend teaching to him every week, I was thinking of proposing him a 15 minutes per player game. Maybe proposing him to start 15 minutes before the end of his lesson and possibly allowing the game to last after his lesson, which is find for me because I like playing chess and I like spending time doing mathematics with him.&lt;/p&gt;&#xA;&lt;p&gt;The cons I see in this idea is that Chess could be seen as a game and will not directly help him pursue his excellent grades in mathematics. Another con is that, although I like playing Chess, I'm not living thanks to it, nor playing at an excellent level. I only have 1500 elo. So maybe it is not so good playing with me for a student learning Chess?&lt;/p&gt;&#xA;&lt;p&gt;I think when I was younger and went to optionnal math classes, I would have loved playing Chess with one of my math teacher but that logically didn't happen.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I did try something similar last week, spending the last 15 minutes of the class showing him the very first LaTeX commands. He told me he liked it!&lt;/p&gt;&#xA;" OwnerUserId="15273" LastActivityDate="2021-01-16T17:25:43.727" Title="Is playing and teaching chess appropriate in private lessons?" Tags="|secondary-education|teaching|tutoring|discrete-math|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19368" PostTypeId="2" ParentId="19367" CreationDate="2021-01-16T14:17:34.427" Score="9" Body="&lt;p&gt;It's a bad idea.  (1) It's not that special.  He can get a game from chess very easily from anyone.  An individual session with you is not high value, not best use of time.  (2) It will raise hackles.&lt;/p&gt;&#xA;&lt;p&gt;I would suggest instead introducing some recreations that are less familiar instead. Playing Hex for instance is an idea.  The value is much higher than chess, given he has (likely) never played it before.  Whereas, he probably has chess exposure.&lt;/p&gt;&#xA;&lt;p&gt;In addition, it allows you to segue to John Nash and the like (suggest the kid look it up in Wiki afterwards).  Then maybe you can awaken some human interest in math, given the romanticism surrounding Nash.&lt;/p&gt;&#xA;&lt;p&gt;And I'm sure there are many other things you could do that would be new/interesting to the kid, and look new to his parents.  More than just pulling out the chess board.  Another idea is experimental derivation of pi by dropping a needle (has the advantage of being interactive) versus a lecture.&lt;/p&gt;&#xA;" OwnerUserId="15274" LastEditorUserId="15274" LastEditDate="2021-01-16T14:23:00.107" LastActivityDate="2021-01-16T14:23:00.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19369" PostTypeId="1" CreationDate="2021-01-16T15:48:43.423" Score="1" ViewCount="439" Body="&lt;p&gt;We’re precalculus honors teachers. In this year of Covid and reduced instructional time, what topics can we cut (Demana textbook) that would not hurt our kids in either calc AB or BC?&lt;/p&gt;&#xA;" OwnerUserId="15275" LastActivityDate="2021-01-18T14:03:08.447" Title="Honors Precalculus: what topics to cut?" Tags="|calculus|precalculus|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19370" PostTypeId="2" ParentId="19367" CreationDate="2021-01-16T17:25:43.727" Score="4" Body="&lt;p&gt;“ I know it has a lot of relations with the mathematics, with the logic, many applications in computer science, game theory and so on.“&lt;/p&gt;&#xA;&lt;p&gt;Do &lt;em&gt;you&lt;/em&gt; know these specific relationships and are you prepared to introduce/explain these things in the context of the game? If so, that sounds like it should take quite a bit of preparation on your part to teach a complex game and the science it touches on. Might be worth it if you can do all of that. But if you are only merely aware that “one can apply computer science principles to chess”, then I don’t think doing this on the clock is the right move. Maybe just see if this student wants to hang out after math time is done and learn the game.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2021-01-16T17:25:43.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19371" PostTypeId="2" ParentId="19349" CreationDate="2021-01-16T18:42:50.333" Score="1" Body="&lt;p&gt;It is possible to tile (tessellate) a plane with any triangle.&lt;br /&gt;&#xA;It is possible to tile a plane with any quadrilateral.&lt;br /&gt;&#xA;It is possible to tile a plane with some pentagons (such as 'houses' where the sides all have equal length, the base is a square, and the top is an equilateral triangle).&lt;br /&gt;&#xA;It is possible to tile a plane with some hexagons (such as regular hexagons).&lt;br /&gt;&#xA;It is impossible to tile a plane with any convex polygon with 7 or more sides. &lt;a href=&quot;https://mathstat.slu.edu/escher/index.php/Tessellations_by_Squares,_Rectangles_and_other_Polygons&quot; rel=&quot;nofollow noreferrer&quot;&gt;[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Depending on how open to being convinced your friend is, you could present a very contrived example like the question 'is n+100 &amp;gt; n+n for all integer n?'&lt;/p&gt;&#xA;&lt;p&gt;As a side-note, you can argue against induction in general with a parable like that of the inductive turkeys. &lt;a href=&quot;https://blog.timeneye.com/inductivist-turkey-thanksgiving-experience-time-tracking/&quot; rel=&quot;nofollow noreferrer&quot;&gt;full version&lt;/a&gt; *I believe the parable originated from Betrand Russell, but I don't know the original source&lt;br /&gt;&#xA;The short version is that for the first few years of a farm animal's life, the farmer showing up means food. But then one day without warning it means the farmer has come to kill them.&lt;/p&gt;&#xA;" OwnerUserId="15278" LastActivityDate="2021-01-16T18:42:50.333" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19373" PostTypeId="1" AcceptedAnswerId="19374" CreationDate="2021-01-17T18:07:20.477" Score="32" ViewCount="4339" Body="&lt;p&gt;I posted this question earlier today on the Mathematics site (&lt;a href=&quot;https://math.stackexchange.com/q/3988907/96384&quot;&gt;https://math.stackexchange.com/q/3988907/96384&lt;/a&gt;), but was advised it would be better here.&lt;/p&gt;&#xA;&lt;p&gt;I had a heated argument with someone online who claimed to be a school mathematics teacher of many years standing. The question which spurred this discussion was something along the lines of:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;A horseman was travelling from (location A) along a path through a forest to (location B) during the American War of Independence. The journey was of 22 miles. How far was it in kilometres?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;To my mind, the answer is trivially obtained by multiplying 22 by 1.6 to get 35.2 km, which can be rounded appropriately to 35 km.&lt;/p&gt;&#xA;&lt;p&gt;I was roundly scolded by this ancient mathematics teacher for a) not using the official conversion factor of 1.60934 km per mile and b) not reporting the correct value as 35.405598 km.&lt;/p&gt;&#xA;&lt;p&gt;Now I have serious difficulties with this analysis. My argument is: this is a man riding on horseback through a forest in a pre-industrial age. It would be impractical and impossible to measure such a distance to any greater precision than (at best) to the nearest 20 metres or so, even in this day and age. Yet the answer demanded was accurate to the nearest millimetre.&lt;/p&gt;&#xA;&lt;p&gt;But when I argued this, I was told that it was not my business to round the numbers. I was to perform the conversion task given the numbers I was quoted, and report the result for the person asking the question to decide how accurately the numbers are to be interpreted.&lt;/p&gt;&#xA;&lt;p&gt;Is that the way of things in school? As a trained engineer, my attitude is that it is part of the purview of anybody studying mathematics to be able to estimate and report appropriate limits of accuracy, otherwise you get laughably ridiculous results like this one.&lt;/p&gt;&#xA;&lt;p&gt;I confess I have never had a good relationship with teachers, apart from my A-level physics teacher whom I adored, so I expect I will be given a hard time over my inability to understand the basics of what I have failed to learn during the course of the above.&lt;/p&gt;&#xA;" OwnerUserId="15290" LastEditorUserId="8931" LastEditDate="2021-01-20T05:54:24.627" LastActivityDate="2021-01-21T13:16:34.870" Title="What is the current school of thought concerning accuracy of numeric conversions of measurements?" Tags="|secondary-education|real-numbers|" AnswerCount="11" CommentCount="35" ContentLicense="CC BY-SA 4.0" />
  <row Id="19374" PostTypeId="2" ParentId="19373" CreationDate="2021-01-17T19:16:48.470" Score="43" Body="&lt;p&gt;You're right. The random, anonymous person you met online is not competent. This is basic mathematical literacy, as taught in every freshman chemistry and physics class.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-01-17T19:16:48.470" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19375" PostTypeId="2" ParentId="19349" CreationDate="2021-01-18T01:15:00.847" Score="1" Body="&lt;p&gt;Keep this person as a friend and admit -- if asked -- that you have no good explanation for someone unfamiliar with mathematics. (In general, spouses of mathematicians never ask!) If your friend plays bridge or chess, discuss the strategies of the game. You could then say that proofs give mathematicians the &lt;em&gt;guarantee&lt;/em&gt; that their strategies will work. Engineers have methods that (almost) guarantee that their structures will not collapse.&lt;/p&gt;&#xA;" OwnerUserId="15294" LastActivityDate="2021-01-18T01:15:00.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19376" PostTypeId="2" ParentId="19373" CreationDate="2021-01-18T01:30:07.843" Score="15" Body="&lt;p&gt;When a tutoring student asks me about rounding, I tell them that absent specific instructions from a teacher, common sense should apply.&lt;/p&gt;&#xA;&lt;p&gt;For a conversion, 22 miles isn’t 22.0000 miles, there’s the assumption it’s been rounded. You can’t convert and find yourself with 6 digits of accuracy beyond the decimal. As you note, there’s a number of digits that result to be the nearest meter, millimeter, etc. which is absurd. Before GPS, I’d give directions accurate to 1/10 mile, as that’s what a car odometer reflects. Even that was often called a bit obsessive.&lt;/p&gt;&#xA;&lt;p&gt;My home scale gives me my weight to .1 lbs. Would it really be of value to have an extra digit of accuracy?&lt;/p&gt;&#xA;&lt;p&gt;A person’s height? The nearest inch will do.&lt;/p&gt;&#xA;&lt;p&gt;The one thing I warn about - don’t round while doing interim steps. This is a sure way to find that the final result may be off by enough to be graded as wrong. This issue commonly presents itself with trig functions which ask for a triangle side to the nearest 1/100. Rounding should be done as the final step.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-01-18T12:56:27.527" LastActivityDate="2021-01-18T12:56:27.527" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19377" PostTypeId="2" ParentId="19373" CreationDate="2021-01-18T06:27:40.880" Score="40" Body="&lt;p&gt;The product of two numbers should be given with as many significant digits as the least precise of the numbers multiplied (see &lt;a href=&quot;https://www.nku.edu/%7Eintsci/sci110/worksheets/rules_for_significant_figures.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.nku.edu/~intsci/sci110/worksheets/rules_for_significant_figures.html&lt;/a&gt;). 1.60934 km/mile has six significant digits (or, if a mile is defined to be an exact number of km, then the conversion factor has an infinite number of significant digits). 22 miles has two significant figures. We take the smaller of these two, which is two significant figures from the 22 miles. This means that rounding to 35 km is correct.&lt;/p&gt;&#xA;&lt;p&gt;It is a good idea to use, during one's work, at least one significant digit more than the final quantity needed, so it would have been good practice to use the conversion factor of 1.61 if this were a test, but for a casual online conversation, 1.6 is fine.&lt;/p&gt;&#xA;&lt;p&gt;The importance of getting significant figures correctly pales in comparison of basic decency. Even if this person had been correct, scolding you would not be. If you believe that someone is in error, you should express that view politely. It appears that this person's civility may have atrophied from having a captive audience with such a power differential that they have been able to dispense with basic politeness.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2021-01-18T06:27:40.880" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19378" PostTypeId="2" ParentId="19349" CreationDate="2021-01-18T09:49:36.237" Score="3" Body="&lt;p&gt;If you've got a sum you're keeping track of (e.g. you're adding up your purchases in a shop), you can quickly check whether your final answer is right by just repeatedly adding up the digits, ignoring any 9's, until you get a one-digit number, and then checking whether the sum of the digits of your sum is equal to the sum of the digits of the &lt;em&gt;summands&lt;/em&gt; after doing the same thing to &lt;em&gt;them&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, {£1.72, £5.99, £2.13}:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The sum is £9.84 on the one hand; ignoring nines, we get a digit sum of 8+4=12. The digit sum of 12 is 1+2 = 3, which is a single digit so we stop.&lt;/li&gt;&#xA;&lt;li&gt;The digit sum of the components - ignoring nines - is {10, 5, 6}, which adds up to 21, whose digits add up to 3, which is a single digit so we stop.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So our test says &amp;quot;you didn't make a mistake as far as I'm concerned&amp;quot;, because we got the same number 3 both times.&#xA;If I'd got my sum wrong and I'd made £9.45, the digit sum of £9.45 (ignoring nines) would be 9, which (ignoring nines) has a sum of 0, and we'd know we had made a mistake either in the sum or in our digit-sums check.&lt;/p&gt;&#xA;&lt;p&gt;This check is quicker and easier than just re-doing the sum, and as the list gets longer and the numbers get bigger, the technique gets cheaper and cheaper; and you can run the check incrementally, after every addition you perform.&lt;/p&gt;&#xA;&lt;p&gt;In real life, this technique has a very long history in accounting folklore, so we can be quite sure it works; it's called &amp;quot;casting out nines&amp;quot;. But what if we'd just come up with it ourselves? How could we know it actually works? It's &lt;em&gt;extremely&lt;/em&gt; unobvious that the procedure gives you any meaningful answers. It's so unclear &lt;em&gt;why&lt;/em&gt; it works that if I'd just discovered the technique for myself, I would be quite uncomfortable to use it. Only with a &lt;em&gt;proof&lt;/em&gt; would I be happy.&lt;/p&gt;&#xA;" OwnerUserId="7101" LastEditorUserId="7101" LastEditDate="2021-01-19T12:41:27.667" LastActivityDate="2021-01-19T12:41:27.667" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19379" PostTypeId="2" ParentId="19349" CreationDate="2021-01-18T13:47:24.120" Score="2" Body="&lt;p&gt;Let me be provocative.&lt;/p&gt;&#xA;&lt;p&gt;Your friend is right. Nobody needs proof except mathematicians. The reason you want absolute proof is that &lt;em&gt;you have a mathematical mindset.&lt;/em&gt; For all other people (farmers, carpenters, engineers, physicists) overwhelming evidence is sufficient.&lt;/p&gt;&#xA;&lt;p&gt;It is telling that most non-trivial statements about actual nature cannot be proven to be correct, and don't need to be. For example all &amp;quot;knowledge&amp;quot; we have about our surroundings is fundamentally, utterly wrong. There is no solid matter, there is no absolute time and space. Newton and Leibniz were entirely mistaken (there is no &lt;em&gt;Fernwirkung).&lt;/em&gt; A ball you throw does emphatically &lt;em&gt;not&lt;/em&gt; follow a parabola, even in a vacuum; another categorical error. Of course it does not even follow an ellipse. In fact, nobody in the world can tell you exactly where it goes because the three body problem is unsolvable. That is quite typical for real-world problems: We can only provide numerical approximations. But our assumptions and approximations are often &amp;quot;good enough&amp;quot; (Malcolm Gladwell).&lt;/p&gt;&#xA;&lt;p&gt;Proving mathematical statements is &lt;em&gt;art pour l'art&lt;/em&gt; for people who like puzzles. Nobody really needs it. (This being said, if your proof provides an insight it has value. But the value is in the additional insight.)&lt;/p&gt;&#xA;" OwnerUserId="11894" LastEditorUserId="11894" LastEditDate="2021-01-18T17:27:14.853" LastActivityDate="2021-01-18T17:27:14.853" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19380" PostTypeId="2" ParentId="19369" CreationDate="2021-01-18T14:03:08.447" Score="3" Body="&lt;p&gt;I'm not sure of your details, like how much you have already covered, how often your classes meet, etc., but here is how I would adjust this textbook for specifically preparing students for AP Calculus. I happen to use this textbook for precalc and I use Larson for AP Calculus AB.&lt;/p&gt;&#xA;&lt;h2&gt;Comments on Table of Contents&lt;/h2&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Functions and Graphs&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Polynomial, Power, and Rational Functions&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Exponential, Logistic, and Logarithmic Functions&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Trigonometric Functions&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Analytic Trigonometry&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Chapter 4 trig is necessary for AP Calc, but chapter 5 is not. I think it is important that students see some trig identities, but it is incredibly unlikely that any students will lose AP Calc exam points for not knowing a trig identity.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;&lt;strong&gt;Applications of Trigonometry&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Chapter 6 is useful for Calc BC but unnecessary for Calc AB. If your students will be going into Calc AB, and you are pressed for time, I would skip Chapter 6.  They will still need to understand particle motion for Calc AB, but they do not need this chapter to develop that understanding.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;7&quot;&gt;&#xA;&lt;li&gt;&lt;strong&gt;Systems and Matrices&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Linear algebra is an incredibly important topic, but one that just isn't necessary for AP Calc.  I would drop this chapter if really pressed for time.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;8&quot;&gt;&#xA;&lt;li&gt;&lt;strong&gt;Analytic Geometry in Two and Three Dimensions&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It is debatable how useful chapter 8 is for Calc BC, but I would say it is &lt;em&gt;not&lt;/em&gt; useful for Calc AB.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;9&quot;&gt;&#xA;&lt;li&gt;&lt;strong&gt;Discrete Mathematics&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I cover 9.1 through 9.5 but skip everything beyond that in chapter 9. Sequences and series are the most important for AP Calc from this chapter.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;10&quot;&gt;&#xA;&lt;li&gt;&lt;strong&gt;An Introduction to Calculus: Limits, Derivatives, and Integrals&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I would leave as much time for this chapter as possible, and especially focus on the concept of limits.  That way, students will be entering AP Calc next year ahead of the game in some respects.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="219" LastActivityDate="2021-01-18T14:03:08.447" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19381" PostTypeId="2" ParentId="19373" CreationDate="2021-01-18T14:46:56.530" Score="9" Body="&lt;p&gt;When I was in school, I once got an answer marked as error for having too many digits. IIRC it was in trigonometry and I had just written down as many digits as the calculator displayed. (I was able to discuss it away, but was told to avoid unreasonable amounts of digits in the future)&lt;/p&gt;&#xA;&lt;p&gt;That was in the 1990s in continental Europe, but I think it is still good enough for s counterexample: Not all teachers are like that.&lt;/p&gt;&#xA;" OwnerUserId="15298" LastActivityDate="2021-01-18T14:46:56.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19382" PostTypeId="2" ParentId="13464" CreationDate="2021-01-18T16:55:19.783" Score="1" Body="&lt;p&gt;There are really several distinct questions tangled up in this, so let me try to tease them apart.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The first question is, &lt;em&gt;Are there people who are&lt;/em&gt; &lt;em&gt;&lt;strong&gt;intrinsically&lt;/strong&gt;&lt;/em&gt; &lt;em&gt;incapable of creative work in mathematics&lt;/em&gt;?&lt;/li&gt;&#xA;&lt;li&gt;The second question is, &lt;em&gt;Are there people who&lt;/em&gt; &lt;em&gt;&lt;strong&gt;eventually become&lt;/strong&gt;&lt;/em&gt; &lt;em&gt;incapable of creative work in mathematics&lt;/em&gt;?&lt;/li&gt;&#xA;&lt;li&gt;The third question is, &lt;em&gt;What evidence exists for the answers to the above questions?&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The reason I separate out the first and second question is because the anecdotal case mentioned in the OP was not a blank slate, but rather someone who had already gone through a full sequence of primary and secondary education before the OP met him.  These initial years of education play a major role in shaping not only what mathematics people know, but also how they approach the subject.  There is an abundance of research that demonstrates that primary and secondary education tends to emphasize memorization and speed, and pays little attention (if any) to opportunities to reason creatively in novel contexts.  It seems extremely plausible to me that a student who is successful in such a context may find that whatever innate mathematical creative potential they may have once had becomes calcified, or atrophies (or whatever metaphor you think is best).&lt;/p&gt;&#xA;&lt;p&gt;To take a somewhat clichéd analogy, it is widely acknowledged that it is very difficult to become a proficient musician if you do not start taking lessons until adulthood.  It's not impossible, of course, and with hard work one can learn to play with some passable level of skill -- but to become a truly great musician may be beyond the reach of someone who starts too late.  Does that mean that such a person could &lt;em&gt;never&lt;/em&gt; have become a great musician?  Not necessarily -- but it does suggest that there is a window of opportunity that eventually closes.&lt;/p&gt;&#xA;&lt;p&gt;Now mathematics is not music, and mathematical creativity is not the same as musical talent, but I think there is something fruitful in the analogy nevertheless.  The story in the OP was about someone who, in adulthood, found that he was not able to do creative work in math.  It's noteworthy that this individual was &amp;quot;about 5 years older&amp;quot; than the narrator of the story, and had been away from mathematics instruction for that period of time.  It seems quite reasonable to me that whatever potential for creative work in mathematics this person once possessed may have simply dried up by the time he returned to university study.&lt;/p&gt;&#xA;&lt;p&gt;Anecdotes are not data (as the OP knows, which is why he asks for evidence), so my own anecdotal experience is not worth much here, but I don't think very many people will be surprised to learn that students who return to university study after a number of years in the workforce tend to struggle more so than their younger peers.  They have a lot of stresses on them; they are more likely to have spouses and children, they are less likely to have direct financial and emotional support from their parents, and so on.  The fact that the person in the OP's story had a nervous breakdown suggests, to me, that there was probably a lot going on that had nothing to do with his mathematical coursework.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-01-18T16:55:19.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19383" PostTypeId="2" ParentId="19373" CreationDate="2021-01-18T17:14:46.477" Score="2" Body="&lt;p&gt;&lt;em&gt;IF&lt;/em&gt; the horse ride were 22.00000000 miles then the other person would be right.&lt;/p&gt;&#xA;&lt;p&gt;Else if it were 22 miles then you should round the answer to zero decimal places.&lt;/p&gt;&#xA;&lt;p&gt;Some people are illogically pedantic without any rational reason for what they promulgate.&lt;/p&gt;&#xA;" OwnerUserId="15302" LastActivityDate="2021-01-18T17:14:46.477" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19384" PostTypeId="2" ParentId="19373" CreationDate="2021-01-18T17:27:33.170" Score="17" Body="&lt;p&gt;Just to play the &lt;del&gt;devil's&lt;/del&gt; teacher's advocate here: one &lt;em&gt;can&lt;/em&gt; make a point that rounding should be generally avoided but measurement uncertainty instead be expressed explicitly. Specifically, rounding errors should always be &lt;em&gt;much smaller&lt;/em&gt; than measurement errors. Now, if you have a figure of 22 miles, I'd interpret this as &lt;span class=&quot;math-container&quot;&gt;$(22\pm0.5)\mathrm{mi} = (35.4\pm0.8)\mathrm{km}$&lt;/span&gt;. I specified one more digit, but not only did I represent the center value better (which in your rounding adds a whopping 50% error), I also captured that the inaccuracy of that result is even &lt;em&gt;bigger&lt;/em&gt; than simply &lt;span class=&quot;math-container&quot;&gt;$35\:\mathrm{km}$&lt;/span&gt; would suggest. In particular, &lt;span class=&quot;math-container&quot;&gt;$36\:\mathrm{km}$&lt;/span&gt; is also within the range!&lt;/p&gt;&#xA;&lt;p&gt;How many digits to write out is then uncritical; in physics convention is to write two non-significant digits in both the value and uncertainty figure. One is usually enough, but when &lt;em&gt;completely&lt;/em&gt; omitting non-significant digits you do introduce excessive extra error. If the numbers are just stored in a computer, you should typically keep &lt;em&gt;all&lt;/em&gt; the digits of the binary number representation – with double precision that means you keep a rather absurd 16 decimals! It doesn't really increase the precision, but it also doesn't really cost anything or suggest too high precision (because uncertainty is stored separately), and it makes sure that rounding really will have no contribution to the error of the final result.&lt;/p&gt;&#xA;" OwnerUserId="6902" LastActivityDate="2021-01-18T17:27:33.170" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19385" PostTypeId="2" ParentId="19349" CreationDate="2021-01-18T19:26:19.910" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Now, he asked me why you'd need proofs for anything ever, since it's&#xA;enough to see that an equation works after testing it for a few values.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So let me give my take on a practical example which does not involve numbers or formulas, so it should be relative simple to understand.&lt;/p&gt;&#xA;&lt;p&gt;Having a mathematical proof for some calculation is like having a map.&lt;/p&gt;&#xA;&lt;p&gt;Let's say that you live in Village A and you want to visit Village&#xA;B. There is a large forest in between and there has been no contact&#xA;between the villages before so all you have to begin with is an&#xA;approximate direction. Well, you will then have to walk out in the forest&#xA;and try and try to find Village B. Assuming you find it and then put&#xA;up marks along the way, that's like a formula discovered that other&#xA;people can use. And if all you ever want is to travel back and forth&#xA;between Village A and Village B, then by all means you do not need to&#xA;do anything more.&lt;/p&gt;&#xA;&lt;p&gt;But is it the fastest way? Is there an alternative route that does not&#xA;have that many ups and downs over hills? What if part of the path becomes&#xA;inaccessible due to landslides or floods?&lt;/p&gt;&#xA;&lt;p&gt;If you have a map then you will be able to reason about alternatives&#xA;outside the already existing path without actually having to resort&#xA;to trial and error. Without a map you will just have to test out&#xA;alternatives and see if they fit. So maybe you found a new path that is&#xA;a bit shorter. Great, but is it the shortest? When can you stop testing&#xA;to conclude you have found the shortest path?&lt;/p&gt;&#xA;&lt;p&gt;Then maybe you want to visit Village C. If you have a map you are able&#xA;to find out already before you start that it is impossible to walk to&#xA;Village C because it is located far out on an island, whereas if you&#xA;just set out walking trying to find it you might be caught up in an&#xA;never ending quest doomed to fail.&lt;/p&gt;&#xA;&lt;p&gt;Now the above village scenario is perhaps a bit contrived,&#xA;but it is not extremely far away from the &lt;a href=&quot;https://en.wikipedia.org/wiki/Travelling_salesman_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;travelling salesman&#xA;problem&lt;/a&gt;&#xA;which is a real mathematical problem which has had a lot of research applied to it.&lt;/p&gt;&#xA;&lt;p&gt;For this problem, theory/knowledge/proofs outside an actual&#xA;formula/algorithm/implementation to find the shortest path tells us that there exists&#xA;some upper limit where it is impractically impossible to find the&#xA;optimal solution.&lt;/p&gt;&#xA;&lt;p&gt;Thus the approach of just testing a formula with a few values to&#xA;see that it works is not enough.&lt;/p&gt;&#xA;&lt;p&gt;Q.E.D.&lt;/p&gt;&#xA;" OwnerUserId="468" LastEditorUserId="468" LastEditDate="2021-01-18T19:34:27.193" LastActivityDate="2021-01-18T19:34:27.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19386" PostTypeId="2" ParentId="19373" CreationDate="2021-01-18T19:33:12.183" Score="2" Body="&lt;p&gt;You're both right, depending on the domain of discourse and the rules of engagement.&lt;/p&gt;&#xA;&lt;p&gt;In pure math, the traditional expectation is that the numbers given are exact unless stated otherwise, and answers are also to be exact unless stated otherwise. So when the mathematician read &amp;quot;22 miles,&amp;quot; he's using a tradition that means &amp;quot;exactly 22 miles.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But in the physical sciences, all measurements are understood to be inexact and approximations and rounding are either &amp;quot;allowed&amp;quot; or &amp;quot;expected&amp;quot; (depending on the logical rigor applied).&lt;/p&gt;&#xA;&lt;p&gt;In this case, the correct answer boils down to a question of semantics and assumptions.  What about this question:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;If a man traveled 22 miles, how far did he travel in kilometers?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;How would you answer that?  The &amp;quot;If&amp;quot; complicates things.  Some would say that it turns the question into a hypothetical that ignores the physical difficulties in measuring exactly 22 miles and turns it into a &amp;quot;given.&amp;quot; It's not a stretch to read the original question as a hypothetical, even without an explicit &amp;quot;If&amp;quot; at the beginning.&lt;/p&gt;&#xA;&lt;p&gt;Some traditions say that integers are always expected to be exact and that the question should have used &amp;quot;approximately 22 miles,&amp;quot; &amp;quot;22.0,&amp;quot; or a bar on the last significant digit to show it's a real number instead of an integer.&lt;/p&gt;&#xA;&lt;p&gt;Even in the physical sciences, scenarios used for pedagogical purposes are sometimes idealized in order to remove confounding factors that might distract from the main point being taught.  I don't think we know enough about the source of this question to know about what assumptions or simplifications are being made.&lt;/p&gt;&#xA;&lt;p&gt;You may argue that the use of &amp;quot;a man riding on horseback through a forest in a pre-industrial age&amp;quot; implies a real situation and an actual, inexact measurement. A counterargument is that the use of abstract identifiers &amp;quot;A&amp;quot; and &amp;quot;B&amp;quot; to to designate the starting and stopping point suggest an idealized situation.&lt;/p&gt;&#xA;&lt;p&gt;I would agree this is a good question for Mathematics Educators Stack Exchange.  It emphasizes that in the classroom (as well as life) it's important to be explicit about assumptions and expectations and to lay out the ground rules.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Adding a summary, based on comments, that tries to be more direct:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Use of significant digits only applies to inexact numbers such as measurements.&lt;/li&gt;&#xA;&lt;li&gt;In a problem like &amp;quot;Convert 22 miles to kilometers,&amp;quot; there is no reason to think 22 miles is a measurement.  Rather, it is a &amp;quot;given&amp;quot;: Something that is to be assumed or taken for granted for the sake of the problem.&lt;/li&gt;&#xA;&lt;li&gt;I think this question boils down to this: In the original question, is &amp;quot;22 miles&amp;quot; to be taken as a given or a measurement?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I don't think we can tell. (At least not without more context about where the question came from and why it was asked.)  The original question could merely be &amp;quot;Convert 22 miles to kilometers,&amp;quot; dressed up in a story to make it engaging or interesting.&lt;/p&gt;&#xA;&lt;p&gt;My reading of some of the comments suggests a point of view of &amp;quot;If the problem resembles a real-world situation, then it must be interpreted as a real-world situation.&amp;quot;  Or more succinctly: If 22 miles &lt;em&gt;could be&lt;/em&gt; interpreted as a measurement, then it &lt;em&gt;must be&lt;/em&gt; interpreted as a measurement. Or that by phrasing the question in a historical, real-world context, that somehow forces the measurement interpretation.  I don't follow that.  It ignores the way real-world people write, talk, and teach.&lt;/p&gt;&#xA;" OwnerUserId="5465" LastEditorUserId="5465" LastEditDate="2021-01-19T18:22:47.400" LastActivityDate="2021-01-19T18:22:47.400" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="19387" PostTypeId="2" ParentId="19349" CreationDate="2021-01-18T22:47:00.937" Score="7" Body="&lt;p&gt;Most of the answers assume, as does the OP, that the purpose of proof is &lt;em&gt;verification&lt;/em&gt;.  In fact &amp;quot;establishing the truth of a proposition&amp;quot; is only one of the reasons that mathematicians prove things.&lt;/p&gt;&#xA;&lt;p&gt;The most-cited paper on this subject is probably Michael de Villiers' &amp;quot;The Role and Function of Proof in Mathematics&amp;quot; (&lt;em&gt;Pythagoras&lt;/em&gt;, 1990, pp. 17-24).  De Villiers argues that, in addition to verification, proof serves all of the following functions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Proof is a means of explanation:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although it is possible to achieve quite a high level of confidence in the validity of a conjecture by means of quasi-empirical verification (e.g. accurate constructions and measurement; numerical substitution, etc.), this generally provides no satisfactory &lt;em&gt;explanation&lt;/em&gt; &lt;strong&gt;why&lt;/strong&gt; &lt;em&gt;it may be true&lt;/em&gt;. (emphasis in original)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Proof is a means of systematization:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... proof which manages to expose the underlying logical relationships between statements...  is an indispensable tool in the systematisation of various known results into a deductive system of axioms, definitions and theorems...  Some of the most important functions of a deductive systematisation are given as follows...&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;it helps with the &lt;strong&gt;identification&lt;/strong&gt; of inconsistencies, circular arguments and hidden or not explicitly stated assumptions&lt;/li&gt;&#xA;&lt;li&gt;it &lt;strong&gt;unifies&lt;/strong&gt; and &lt;strong&gt;simplifies&lt;/strong&gt; mathematical theories...&lt;/li&gt;&#xA;&lt;li&gt;it provides a useful &lt;strong&gt;global perspective&lt;/strong&gt; or &lt;strong&gt;bird's eye view&lt;/strong&gt; of a topic...&lt;/li&gt;&#xA;&lt;li&gt;it is helpful for &lt;strong&gt;applications&lt;/strong&gt; both within and outside mathematics, since it aids checking the applicability of a whole complex structure or theory simply by evaluating the suitability of its axioms and definitions&lt;/li&gt;&#xA;&lt;li&gt;it often leads to &lt;strong&gt;alternative&lt;/strong&gt; deductive systems which provide new perspectives and/or are more economical, elegant and powerful than existing ones&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Proof as a means of discovery&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is often popularly said by critics of the amount of deductive rigour at school level, that deduction in general (and proof in particular) is not a particularly useful heuristic device in the actual &lt;strong&gt;discovery&lt;/strong&gt; of new mathematical results. Such people naively seem to believe that theorems are virtually always first discovered by means of intuition and/or quasi-empirical methods, before they are verified by the production of proofs...  Perhaps this perception is due also in part to the stereotyped way in which proof is normally taught (first the result is presented &lt;em&gt;followed&lt;/em&gt; by the proof).  This view however is completely false, as there are numerous examples in the history of mathematics where new results were discovered/invented in a purely deductive manner...  To the working mathematician proof is therefore not merely a means of &lt;em&gt;a posteriori&lt;/em&gt; verification, but often also a means of exploration, analysis, discovery and invention.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Proof is a means of communication&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;According to this view proof is a unique way of communicating mathematical results between professional mathematicians, between lecturers and students, between teachers and pupils, and among students and pupils themselves...  Proof as a form of social interaction therefore also involves the &lt;strong&gt;subjective negotiation&lt;/strong&gt; of not only the meanings of concepts concerned, but implicitly also of the criteria for an acceptable argument.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In case your friend is not persuaded by de Villiers' analysis, Yehuda Rav, in the aptly-titled article &amp;quot;Why do we Prove Theorems?&amp;quot; (Philosophia Mathematica, 1999, 5-41), considers the following thought experiment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let us fancifully pretend that the general decision problem did after&#xA;all have a positive solution, that every axiomatisable theory was&#xA;decidable, and that a universal decision algorithm was invented and&#xA;implemented on oracular computers, marketed under the trade name of&#xA;PYTHIAGORA (Pythia + Pythagoras). There she is, sitting on the desktop&#xA;of every mathematician. Not only can PYTHIAGORA answer all our&#xA;questions, but she does so at the speed of light, having no complexes&#xA;about complexity of computation. Think how wonderful it all would be.&#xA;You want to know whether the Riemann hypothesis is true or not, just&#xA;type it in (using some appropriate computer language), and in a split&#xA;second, the answer is flashed back on your screen: 'true' or&#xA;'false'... We mathematicians would only have to produce conjectures,&#xA;and let PYTHIAGORA weed out the false from the true. What a paradise!&#xA;What a boon!&lt;/p&gt;&#xA;&lt;p&gt;Did I say boon? Nay, I say, no boon but doom! A universal decision&#xA;method would have dealt a death blow to mathematics, for we would&#xA;cease having ideas and candidates for conjectures. The main thesis to&#xA;be developed here is that the essence of mathematics resides in&#xA;inventing methods, tools, strategies and concepts for solving problems&#xA;which happen to be on the current internal research agenda or&#xA;suggested by some external application.  But conceptual and&#xA;methodological innovations are inextricably bound to the search for&#xA;and the discovery of proofs, thereby establishing links between&#xA;theories, systematising knowledge, and spurring further developments.&#xA;&lt;strong&gt;Proofs, I maintain, are the heart of mathematics, the royal road to creating analytic tools and catalysing growth&lt;/strong&gt;. (&lt;em&gt;Emphasis added&lt;/em&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In other words: the activity of searching for and finding proofs gives rise to new ideas, new questions, and new connections between previously unrelated fields of research.&lt;/p&gt;&#xA;&lt;p&gt;Seven years after Rav's paper, John Dawson wrote, in the same journal, a thematic sequel:  &amp;quot;Why Do Mathematicians Re-Prove Theorems?&amp;quot; (Philosophia Mathematica, 2006, 269-286). As the title suggests, mathematicians do indeed devote time and effort to finding new proofs of already proved results, something that would make no sense if the sole purpose of a proof were verification!  Indeed such proofs are sometimes highly regarded by the mathematical community; Dawson points out that &amp;quot;in 1950 a Fields Medal was awarded to Atle Selberg, in part for his elementary proof of the prime-number theorem.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Dawson identifies several reasons (and sub-reasons) for why mathematicians re-prove theorems.  Here is an abbreviated version of his list:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To remedy perceived gaps or deficiencies in earlier arguments.  (Subreasons:  to replace a non-constructive argument with a&#xA;constructive demonstration; to provide a more efficient algorithm for&#xA;performing a calculation; to eliminate controversial hypotheses, such&#xA;as the parallel postulate, the Axiom of Choice, the Riemann&#xA;Hypothesis, or the Continuum Hypothesis.)&lt;/li&gt;&#xA;&lt;li&gt;To employ reasoning that is simpler, or more perspicuous, than earlier proofs.  (Examples: it may reduce the number of computations,&#xA;or the number of cases to consider; it may be significantly shorter;&#xA;it may require fewer hypotheses or fewer conceptual prerequisites.)&lt;/li&gt;&#xA;&lt;li&gt;To demonstrate the power of different methodologies (for research, pedagogical, or ideological purposes).&lt;/li&gt;&#xA;&lt;li&gt;To provide a rational reconstruction (or justification) of historical practices.&lt;/li&gt;&#xA;&lt;li&gt;To extend a result, or to generalize it to other contexts.&lt;/li&gt;&#xA;&lt;li&gt;To discover a new route.&lt;/li&gt;&#xA;&lt;li&gt;Concern for methodological purity.&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;Finally, the existence of multiple proofs of theorems serves an overarching purpose that is often overlooked, one that is analogous to&#xA;the role of &lt;em&gt;confirmation&lt;/em&gt; in the natural sciences....&amp;quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Obviously, given the limitations of your friend's mathematical background, not all of these explanations are going to be comprehensible or persuasive.  But some of the main ideas should be clear regardless.  In particular the fact that proof has an explanatory power, that it can be used to discover new things, and that it can be used to reveal the underlying assumptions we may be unconsciously making, should be persuasive reasons why proof has value beyond its power to verify.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="123" LastEditDate="2021-02-26T21:30:11.823" LastActivityDate="2021-02-26T21:30:11.823" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19388" PostTypeId="2" ParentId="19373" CreationDate="2021-01-18T23:26:49.117" Score="28" Body="&lt;p&gt;Here's a joke I like to tell when people could use a reminder about precision vs accuracy:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A tour guide at Giza was explaining how the Pyramids were 4507 years&#xA;old. Someone in the crowd asked: &amp;quot;That's oddly specific. How do we know this?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Well. I was told they were 4500 years old when I&#xA;started working here 7 years ago.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm not sure the grumpy teacher you mentioned would be amused, though.&lt;/p&gt;&#xA;" OwnerUserId="9380" LastActivityDate="2021-01-18T23:26:49.117" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19389" PostTypeId="2" ParentId="19373" CreationDate="2021-01-20T00:10:06.280" Score="2" Body="&lt;p&gt;&lt;strong&gt;It depends&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I agree with just about everyone that the answer is 35, or perhaps 35.4 (a number I like better, see below). An answer of 35.405598 km is precise to the millimeter.  I've ridden horses; they don't work in millimeters.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Update:&lt;/em&gt; For what it's worth, after all this discussion, I think that the right number is &amp;quot;about 35 and a half&amp;quot; (not 35.5) kilometers.  Thirty five and half has about the same uncertainty as &amp;quot;22 miles&amp;quot; (maybe even more), and is within &amp;quot;horseshoes and hand-grenades&amp;quot; of the exact answer of &amp;quot;just about 35.4 exactly&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;As you acknowledge, the intermediate answer you came up with (using an approximate conversion factor) of 35.2 km is wrong; 35 km is a correct answer, but 35.2 km is just plain wrong.  It makes sense to consider that a distance of &amp;quot;22 miles&amp;quot; is likely more precise than &amp;quot;something between 21.5 and 22.5&amp;quot; which is what considering 22 as having only two significant figures means.  It's more like 22.0 miles (i.e., between 21.95 and 22.05 (which gives you an uncertainty of about 500 feet (about 160 m)).&lt;/p&gt;&#xA;&lt;p&gt;But, when you multiply 22.0 by 1.6, then your answer should definitely only have 2 significant figures (not because of the 22, but because of the 1.6).  You can tell that your 3 significant figure result is off, the &amp;quot;completely precise&amp;quot; number is off by 0.2 km (200 m) from your figure.  Horses are more accurate than hundreds of meters.&lt;/p&gt;&#xA;&lt;p&gt;What you want to do working with numbers is to get an understanding of both the precision and the accuracy of the measurement.  Saying something is about 22 miles, give or take 500 feet makes 22.0 about the right number to use.&lt;/p&gt;&#xA;&lt;p&gt;When doing a conversion, it's &lt;strong&gt;&lt;em&gt;always&lt;/em&gt;&lt;/strong&gt; best to use the most precise number you have for all intermediate work, and &lt;em&gt;only round back to the correct number of significant figures at the end of the calculation&lt;/em&gt;.  When doing distance calculations, I always use the fact that one inch is &lt;em&gt;exactly&lt;/em&gt; 2.54 centimeters (i.e. 2.54000000000, as many zeros as you want).  If I've got a calculator (or a slide-rule) handy, I'd do this:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;22 miles * 5280 ft/mile * 12 in/ft * 2.54 cm/in / 100 cm/m / 1000 m/km&#xA;= 35.405568 km&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Note that that number is off by 30 millimeters from what you quote.  My number is correct.  Also note that I carried the units through the calculation.  That way, I can do some &lt;em&gt;dimensional analysis&lt;/em&gt; and see that I get an answer in km, and that it's what I expect: (miles * (ft/mile) * (in/ft) * (cm/in) / (cm/m) / (m/km) works out to km).&lt;/p&gt;&#xA;&lt;p&gt;I'd look at that number and say &amp;quot;yes, it's 35.4 km.&amp;quot;  Also note that all those intermediate conversion constants are exact (the number of inches in a foot is exactly 12 - so you can treat 12 like 2.54, it has as many zeros as you want).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;But then again&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Way back when I was a student, I had a math prof who'd get upset at us engineers for saying the answer is about 35.4km.  He's say that two numbers can be equal, but &amp;quot;about equal&amp;quot; or &amp;quot;approximately equal&amp;quot; have no mathematical meaning.  Then he'd point out that it would be pretty easy to figure out that one was about equal to zero - and at the point, everything breaks.&lt;/p&gt;&#xA;&lt;p&gt;So, if you are in a math class and the teacher says &amp;quot;The relationship between miles and kilometers is 1.609344 km/mile, how many kilometers are there in 22 miles?&amp;quot;, then the answer is 35.405568 km, not 35.4 km.&lt;/p&gt;&#xA;&lt;p&gt;Note the absence of the horse in this phrasing of the question.&lt;/p&gt;&#xA;" OwnerUserId="13120" LastEditorUserId="13120" LastEditDate="2021-01-21T01:13:43.230" LastActivityDate="2021-01-21T01:13:43.230" CommentCount="27" ContentLicense="CC BY-SA 4.0" />
  <row Id="19390" PostTypeId="2" ParentId="19373" CreationDate="2021-01-20T05:21:26.360" Score="0" Body="&lt;p&gt;I would say that based on the words of the question, the answer is 35.&lt;/p&gt;&#xA;&lt;p&gt;It is not a distance or measurement of 22 miles between points A &amp;amp; B.  It is a journey between locations A &amp;amp; B.&lt;/p&gt;&#xA;&lt;p&gt;Next time you see the guy who scolded you, ask him how one should answer if asked, “What is the numerical value of pi minus e?”&lt;/p&gt;&#xA;" OwnerUserId="15317" LastActivityDate="2021-01-20T05:21:26.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19391" PostTypeId="1" CreationDate="2021-01-20T15:50:22.640" Score="4" ViewCount="250" Body="&lt;p&gt;In conceptually motivating the &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; definition and proof of a limit, I realized a new way of choosing the &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, consider &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to 4}\sqrt{x}=2$&lt;/span&gt;. In the &amp;quot;standard method&amp;quot;, you use the back-of-the-envelope calculation that leads to &lt;span class=&quot;math-container&quot;&gt;$$\frac{|x-4|}{\sqrt{x}+2} &amp;lt; \frac{|x-4|}{2}$$&lt;/span&gt; and so choose &lt;span class=&quot;math-container&quot;&gt;$\delta = 2 \epsilon$&lt;/span&gt;. But this leap (replacing &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}+2$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;) seems so unnatural to students who are encountering this for the first time.&lt;/p&gt;&#xA;&lt;p&gt;Instead, if you draw the conceptual picture of what's going on (which many of us I assume start with), then we get that for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt; to be between &lt;span class=&quot;math-container&quot;&gt;$2-\epsilon$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2+\epsilon$&lt;/span&gt;, it must be that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is between &lt;span class=&quot;math-container&quot;&gt;$(2-\epsilon)^2 = 4-4\epsilon+\epsilon^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(2+\epsilon)^2 = 4+4\epsilon+\epsilon^2$&lt;/span&gt;. Then, like we do with all the preceding numerical examples, we take the smaller of the two differences of these with &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; and get &lt;span class=&quot;math-container&quot;&gt;$\delta = 4\epsilon - \epsilon^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The resulting proof looks like this:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\epsilon&amp;gt;0$&lt;/span&gt; and set &lt;span class=&quot;math-container&quot;&gt;$\delta = 4\epsilon - \epsilon^2$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$|x-4| &amp;lt; \delta$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$$4-4\epsilon + \epsilon^2= 4-\delta &amp;lt; x &amp;lt; 4+\delta = 4+ 4\epsilon - \epsilon^2 &amp;lt; 4+ 4\epsilon + \epsilon^2.$$&lt;/span&gt;&#xA;Hence &lt;span class=&quot;math-container&quot;&gt;$$(2-\epsilon)^2 &amp;lt; x &amp;lt; (2+\epsilon)^2$$&lt;/span&gt; and thus &lt;span class=&quot;math-container&quot;&gt;$2-\epsilon &amp;lt; \sqrt{x} &amp;lt; 2+ \epsilon$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$|\sqrt{x}-2| &amp;lt; \epsilon.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is totally analogous to the argument we expect them to make with numerical examples. So here's my question: &lt;strong&gt;Why don't we teach it this way?&lt;/strong&gt; Why isn't this the way it's done/introduced in textbooks?&lt;/p&gt;&#xA;&lt;p&gt;Some possibilities:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;This only works when the function has an obvious inverse. So it wouldn't work for, say, &lt;span class=&quot;math-container&quot;&gt;$\lim_{x \to 1} x^5-3x+4 = 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Technically if &lt;span class=&quot;math-container&quot;&gt;$\epsilon \ge 4$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\delta = 4\epsilon - \epsilon^2 \le 0$&lt;/span&gt;, and so really we have to break it up into pedantic cases.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So maybe I've already answered my own question, but then the question becomes: &lt;strong&gt;Is there any educational benefit to using/showing this type of reasoning as opposed to the usual?&lt;/strong&gt; Has anyone else done this in class and did it help or hinder your students understanding/proving?&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorDisplayName="user507" LastEditDate="2021-01-20T21:53:41.057" LastActivityDate="2021-01-21T19:01:34.547" Title="Differing Choices of $\delta$ in a Limit" Tags="|proofs|limits|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19392" PostTypeId="2" ParentId="19373" CreationDate="2021-01-20T19:50:03.653" Score="1" Body="&lt;p&gt;It depends on the level of the class.&lt;/p&gt;&#xA;&lt;p&gt;I would expect someone who has a recent undergraduate degree in mathematics to have experienced significant figures at at least some point in their life, either in high school or in college. I would also expect common sense to kick in and say that the level of accuracy proposed is unreasonable.&lt;/p&gt;&#xA;&lt;p&gt;But it is reasonable to dodge the topic when teaching arithmetic, algebra, etc., because the students usually have a hard enough time as it is. Sometimes you can arrange for numbers that come out evenly anyway, but if you're stuck trying to teach an awkward conversion (miles to km) or if the task is to teach something about decimals or fractions then you may be unable to avoid it.&lt;/p&gt;&#xA;&lt;p&gt;For example, &amp;quot;Alex had five cookies and split them evenly with Blake. How many did each of them get?&amp;quot; Two and a half, and we aren't going to quibble over how precisely half of a cookie was achieved.&lt;/p&gt;&#xA;&lt;p&gt;If your students are advanced enough to be working with more precise numbers (and, presumably, starting to question what level of accuracy is acceptable) then the best way to dodge it is to simply specify what rounding you want in the question: &amp;quot;Answer to x decimal places.&amp;quot; That way you can specify the correct precision without the students having to understand how to calculate what the correct precision should be.&lt;/p&gt;&#xA;&lt;p&gt;That's much simpler for the student to understand than the official way, which is &lt;a href=&quot;https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors&quot; rel=&quot;nofollow noreferrer&quot;&gt;according to NIST&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;The precision of your conversion should be based on &lt;em&gt;relative error&lt;/em&gt;. If error isn't specified, then you can infer it from the number of digits in the values given. Use a conversion factor with equal or more precision to that to preform the calculation. Then you round the result to produce a relative error that is of the order of the original.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$22$&lt;/span&gt; miles implies an error range of plus or minus &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; mile which is &lt;span class=&quot;math-container&quot;&gt;$2.\overline{27}\%$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Using a conversion factor of &lt;span class=&quot;math-container&quot;&gt;$1.61$&lt;/span&gt; kilometers per mile (which has better than &lt;span class=&quot;math-container&quot;&gt;$2.\overline{27}\%$&lt;/span&gt; accuracy, note that &lt;span class=&quot;math-container&quot;&gt;$1.6$&lt;/span&gt; is not accurate enough)... &lt;span class=&quot;math-container&quot;&gt;$22*1.61=35.42$&lt;/span&gt; km. We could also use &lt;span class=&quot;math-container&quot;&gt;$1.609$&lt;/span&gt; or any more precise conversion, it &lt;em&gt;does not matter&lt;/em&gt; because we will be rounding. (For example, in this case, &lt;span class=&quot;math-container&quot;&gt;$22*1.609 = 35.398$&lt;/span&gt; km.)&lt;/p&gt;&#xA;&lt;p&gt;Now we round... &lt;span class=&quot;math-container&quot;&gt;$40$&lt;/span&gt; km would have a relative error &lt;span class=&quot;math-container&quot;&gt;$5\div40\approx12.5\%$&lt;/span&gt; which is too much, &lt;span class=&quot;math-container&quot;&gt;$35.4$&lt;/span&gt; would have a relative error of &lt;span class=&quot;math-container&quot;&gt;$0.05\div35.4\approx0.141%$&lt;/span&gt; which is too little. &lt;span class=&quot;math-container&quot;&gt;$35$&lt;/span&gt; km has a relative error of &lt;span class=&quot;math-container&quot;&gt;$0.5\div35\approx 1.43\%$&lt;/span&gt; which is just right. Note that we get the same (rounded) answer regardless of how much precision we used in the conversion factor, as long as the conversion factor met a minimum level precision.&lt;/p&gt;&#xA;&lt;p&gt;Question: Why do we assume plus or minus one-half mile? Wouldn't a distance of 22 miles be measured more accurately than that?&lt;/p&gt;&#xA;&lt;p&gt;Answer: No. If anything it is likely to be much worse. (Disclaimer, I'm not doing sigfigs in this section, I just can't be bothered.)&lt;/p&gt;&#xA;&lt;p&gt;In American revolutionary war era from the &lt;a href=&quot;https://www.nypl.org/blog/2015/08/12/traveling-with-jefferson&quot; rel=&quot;nofollow noreferrer&quot;&gt;New York Public Library&lt;/a&gt;, Thomas Jefferson measuring exactly 22 miles would have actually gone over 22.3 miles (and he was a bit obsessive about measurements):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Before he left on the trip, Jefferson bought from a Philadelphia watchmaker an odometer that counted the revolution’s of his carriage’s wheel. He had measured distance based “on the belief that the wheel of the Phaeton [his carriage] made exactly 360. revoln’s in a mile.” After the trip, though, he re-measured circumference of the wheel and found that it made only 354.95 revolutions in a mile. So for every seventy-one miles Jefferson thought he traveled, he had actually traveled seventy-two.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But I use my car odometer, not a carriage! It's much more reliable! ...nope. From &lt;a href=&quot;https://www.motus.com/variations-in-odometer-readings/&quot; rel=&quot;nofollow noreferrer&quot;&gt;motus.com&lt;/a&gt;, if your car odometer says 22 miles then it could be anywhere between 21.12 to 22.88 miles:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Surprisingly, there is no federal law that regulates odometer accuracy. The Society of Automotive Engineers set guidelines that allow for a margin of error of plus or minus four percent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Actually I use GPS, that's very accurate! ....nope again. GPS has a margin of error on every position measurement made, plus error from the distance between measurements. Essentially your path is like a coastline and the GPS can suffer from the &lt;a href=&quot;https://en.wikipedia.org/wiki/Coastline_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;coastline paradox&lt;/a&gt;. From &lt;a href=&quot;https://www.singletracks.com/mtb-gear/do-gps-units-always-overestimate-trail-distances-no/&quot; rel=&quot;nofollow noreferrer&quot;&gt;singletracks.com&lt;/a&gt; (with pictures and a good explanation: In a very, very bad case (steep trail, lots of switchbacks) your GPS may report 22 miles when you've actually gone 44 miles! Holy guacamole.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...]GPS reports the full loop is right at 1 mile long. In fact, everyone else who rides this trail gets roughly the same measurement. But the trail always “feels” much longer than that.&lt;/p&gt;&#xA;&lt;p&gt;Recently I started riding with a wheel-based cyclocomputer, which I calibrated and verified during one of our track tests. Measuring this particular trail with the cyclocomputer reveals the trail is actually closers to 2 miles long, meaning our GPS units are underestimating distance by half!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm not going to find sources for inaccuracy of distance calculated by counting steps, the time it took to travel, etc. It's pretty obvious that no one (and no horse) actually moves at that even of a pace for 22 miles.&lt;/p&gt;&#xA;&lt;p&gt;So accepting 22 miles on a trail as being between 21.5 and 22.5 is actually pretty generous. Better to just call it a day and say it was &amp;quot;some distance&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="1908" LastEditorUserId="1908" LastEditDate="2021-01-21T13:16:34.870" LastActivityDate="2021-01-21T13:16:34.870" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19393" PostTypeId="2" ParentId="19391" CreationDate="2021-01-21T05:31:37.540" Score="4" Body="&lt;p&gt;You are answering your own question. But also keep in mind a bigger picture. The purpose of introducing the &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; definition of continuity is &lt;em&gt;not&lt;/em&gt; to make people proficient in finding a &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;  for a particular situation like &lt;span class=&quot;math-container&quot;&gt;$f(x)=\sqrt{x}$&lt;/span&gt;. Rather, the purpose is to develop appreciation of the need for a formal definition of continuity. The examples where we have students find &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; in specific situations are the necessary exercises that build this appreciation for the abstract definition.&lt;/p&gt;&#xA;&lt;p&gt;These exercises of finding &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; for simple, explicit functions are the scaffolding that enables a student to make sense of how continuity is invoked in the proof of the intermediate value theorem or the fundamental theorem of calculus, and to appreciate examples of functions that are continuous but nowhere differentiable, or discontinuous everywhere except at a point, for example.&lt;/p&gt;&#xA;&lt;p&gt;If one's understanding of continuity is to ever develop beyond the &amp;quot;trace the curve without lifting the pencil&amp;quot; level, then exercises of finding &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; in simple concrete situations are necessary. But proficiency in these exercises is not the goal. The goal is to provide scaffolding resulting in an understanding of the abstract definition, so one can make sense of continuity in theoretical situations.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-01-21T05:31:37.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19394" PostTypeId="2" ParentId="19391" CreationDate="2021-01-21T06:21:47.717" Score="1" Body="&lt;p&gt;I don't know whether this is a comment or an answer.&lt;/p&gt;&#xA;&lt;p&gt;There is a bigger picture to keep in mind, which is that, beyond developing the students' understanding of the epsilon-delta definition and their ability to find delta, we also want to develop the students' cleverness.  There's a certain art to finding the unnatural leaps that enable someone to make the delicate estimates that are necessary in more advanced (and difficult) pieces of analysis.  Students need to learn that math (and life!) require cleverness and have opportunities to develop it.&lt;/p&gt;&#xA;&lt;p&gt;Of course there is a need to balance competing goals.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2021-01-21T06:21:47.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19395" PostTypeId="2" ParentId="5614" CreationDate="2021-01-21T15:41:16.450" Score="2" Body="&lt;p&gt;Here is what has worked for me, and believe me this is something I wrestle with every year. I love that they ask this question and I don't want to discourage curiosity, I want to empower those kiddos.&lt;/p&gt;&#xA;&lt;p&gt;I always say long story short, your x-value always wants to get back to 0, the parent function. So if a function is f(x-2), +2 is what will make it equal 0, which seems to go well after learning ZPP.&lt;/p&gt;&#xA;&lt;p&gt;However, they always follow up  with &amp;quot;well why doesn't it do the opposite for the 'y-axis'&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;That is because the y value doesn't directly affect x, ultimately it is added or subtracted on after the fact. So it will just in crease or decrease the value as written. usually works.&lt;/p&gt;&#xA;&lt;p&gt;However if they still persist (Yay!) generating a table of values tends to make them think twice now and remember how functions shift.&lt;/p&gt;&#xA;" OwnerUserId="15330" LastEditorUserId="9019" LastEditDate="2021-01-21T20:37:34.190" LastActivityDate="2021-01-21T20:37:34.190" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19396" PostTypeId="2" ParentId="19391" CreationDate="2021-01-21T19:01:34.547" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;But this leap (replacing &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}+2$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;) seems so unnatural to students who are encountering this for the first time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Students will encounter it eventually, for example in their next calculus course someone will ask them&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Does this series converge?&#xA;&lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty \frac{1}{\sqrt{n} + 2}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and a solution might start with the fact that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$n \geq 4, \frac{1}{\sqrt{n}+2} \geq \frac{1}{\sqrt{n} + \sqrt{n}} = \frac12 \frac{1}{\sqrt{n}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So you might as well show them how to estimate expressions with other expressions now. It's just a useful tool.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-01-21T19:01:34.547" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19397" PostTypeId="1" CreationDate="2021-01-21T21:20:20.303" Score="14" ViewCount="1012" Body="&lt;p&gt;I've been pretty dissatisfied with the way the Pythagorean theorem is usually taught, mainly for two reasons:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The chosen proof feels like magic and I don't feel like I have a better understanding of why the theorem is true after looking at it. I feel this way about all rearrangement proofs, the &amp;quot;Behold!&amp;quot; proof, and Euclid's I.47 for example.&lt;/li&gt;&#xA;&lt;li&gt;Even when a good proof is chosen, the theorem statement itself remains mysterious. How would one have guessed that &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2=c^2$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I am thus interested in having a &amp;quot;&lt;a href=&quot;http://cognitivemedium.com/srs-mathematics&quot; rel=&quot;noreferrer&quot;&gt;discovery fiction&lt;/a&gt;&amp;quot; for the Pythagorean theorem, where one goes all the way from not even knowing the theorem statement to discovering an intuitive proof. I'm not necessarily looking for something that is historically accurate, just a compelling story of how someone might discover things on their own.&lt;/p&gt;&#xA;&lt;p&gt;I wrote a &lt;a href=&quot;https://www.lesswrong.com/posts/rTC8MgPuYfXEw3WLp/discovery-fiction-for-the-pythagorean-theorem&quot; rel=&quot;noreferrer&quot;&gt;blog post&lt;/a&gt; giving one approach, but I am worried my audience (high school students) might find it too difficult to follow.&lt;/p&gt;&#xA;&lt;p&gt;I'm wondering how to improve on what I have, or how to come up with some other &amp;quot;discovery path&amp;quot;. I'm also interested in hearing more generally about how to teach the Pythagorean theorem to high school students (and which trade-offs one should make between, e.g. theoretical satisfyingness vs comprehensibility).&lt;/p&gt;&#xA;" OwnerUserId="9917" LastActivityDate="2021-02-10T20:58:23.710" Title="How to teach the Pythagorean theorem in a satisfying way to high school students?" Tags="|secondary-education|mathematical-pedagogy|geometry|proofs|concept-motivation|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19399" PostTypeId="1" CreationDate="2021-01-22T01:11:29.300" Score="3" ViewCount="352" Body="&lt;p&gt;I am to give the following for an interview:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;a short 7–10-minute teaching demonstration on logarithms. Please consider this as your first 10 minutes of introducing logarithms as if you have not previously mentioned the word to this class. Treat this as closely to what you would do during an in-person classroom course. Assume that you are teaching to a typical audience of community college students enrolled in a College Algebra class, who may not have previously encountered logarithms.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Now the caveat. In my previous jobs I use a bit of flipped classroom- having the students read a bit or watch a short video and answer several questions- prior to coming into class. This introduces the topic if only vaguely. Once in class, after a short icebreaker with their peers, they jump into group worksheets which (hopefully) motivate the topic and bring the students through the topic with a series of (inquiry-based learning) questions. The active worksheet is the focus of the class. When I call attention to the front of the room (when we reconvene for full class discussion), which is done intermittingly, it is to give some answers to problems, summarize, prompt students to do work on the board or ask additional questions. In other words I try to minimize the old fashioned &amp;quot;talking at them&amp;quot; and I've eliminated powerpoint slides for example (this is based on student feedback.)&lt;/p&gt;&#xA;&lt;p&gt;Can anyone give me advice on how to reconcile this with the interview as mentioned above? Logistically do I try to tell the committee ahead of time? Do I prepare a 7-10 minute presentation the old fashioned way and then tell them how it'll actually go down in my active learning class?&lt;/p&gt;&#xA;&lt;p&gt;Right now I am planning to prepare the active learning inquiry-based worksheet as I would for this class topic, and I will explain to the committee that students will preferably have already seen a bit of an intro before physically coming into class. But I will give the 7-10 minute introduction, with the active learning caveat in mind, as if logs require this special 7-10 minute introduction before students jump into the worksheet. Can anyone comment on this strategy?&lt;/p&gt;&#xA;&lt;p&gt;To further complicate all this of course, I am assuming my job would start Fall semester and that the expectation is for in-person classes. If that happens, and if students are still distanced within the class, this leads into a whole other problem about how to do active work under these conditions.&lt;/p&gt;&#xA;&lt;p&gt;So more sub-questions: can I ask the committee outright what the expectation is by Fall and whether there will always be an online component regardless? Do I approach all this as if I need to have separate teaching strategies for both in-person (distanced or not) and online, permanently?&lt;/p&gt;&#xA;" OwnerUserId="11880" LastActivityDate="2021-01-22T09:25:07.097" Title="short teaching demo on logs; but by someone who uses active learning" Tags="|online-instruction|interactive-teaching|inquiry-based-learning|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19400" PostTypeId="2" ParentId="19399" CreationDate="2021-01-22T01:27:59.073" Score="2" Body="&lt;p&gt;I introduce logarithms as answering the question &amp;quot;to what power do I raise this base to get this result?&amp;quot; Ideally I would review exponents beforehand, but 7 minutes is no time at all.  Presumably these students will be familiar with exponents, so you can pose problems like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$2^{x}=32$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$3^{x}=27$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$4^{x}=16$$&lt;/span&gt;&#xA;Then you can introduce the &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; notation. It is always nice when you can build the concept first and then offer notation to support the concept. I would introduce the notation by translating the first examples , i.e.,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$2^{x}=32 \iff \log_{2}(32)=x$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$3^{x}=27 \iff \log_{3}(27)=x$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$4^{x}=16 \iff \log_{4}(16)=x$$&lt;/span&gt;&#xA;Obviously there is more to cover, and you should go over vocabulary as you introduce the notation, but this is how I teach logarithms in 7 minutes and require the students to do most of the explaining. The only thing you would be explicitly telling them is the notation.&lt;/p&gt;&#xA;" OwnerUserId="219" LastEditorUserId="219" LastEditDate="2021-01-22T01:34:21.877" LastActivityDate="2021-01-22T01:34:21.877" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19401" PostTypeId="2" ParentId="19397" CreationDate="2021-01-22T03:08:07.563" Score="13" Body="&lt;p&gt;I really like the idea of a &amp;quot;discovery fiction&amp;quot; -- it gives a name to something I often try to use when teaching.  Here is one suggestion.&lt;/p&gt;&#xA;&lt;p&gt;I will try to come back and write a more elaborated version of this answer later, with diagrams and proper notation, but briefly:&lt;/p&gt;&#xA;&lt;p&gt;(a) Don't let on that you are going to prove the Pythagorean Theorem -- don't even mention it.  (Your students probably know the statement already, as it is commonly taught [without proof] in middle school, at least in the United States.)  Instead, start with the fact that an altitude drawn to the hypotenuse of a right triangle divides the triangle into two smaller, similar triangles.  This is a good application of similarity arguments.&lt;/p&gt;&#xA;&lt;p&gt;(b) So we have three similar triangles (which we can informally call &amp;quot;big&amp;quot;, &amp;quot;medium&amp;quot;, and &amp;quot;small&amp;quot;).  Using proportionality, we discover that each leg of the &amp;quot;big&amp;quot; triangle is the geometric mean of two lengths, (hypotenuse of big triangle) and (portion of hypotenuse of big triangle).  Write those two geometric mean relationships down, both with and without square roots.&lt;/p&gt;&#xA;&lt;p&gt;(c) Now &amp;quot;suddenly notice&amp;quot; that if you add the two GM equations without square roots together, you find an equation of the form (leg)^2 + (leg)^2 = (hyp)^2.  Amazing!&lt;/p&gt;&#xA;&lt;p&gt;(d) Oh, wait...  That sounds familiar.  Has anybody seen this before?  (Wait for one of the students to say &amp;quot;Isn't that the Pythagorean Theorem?&amp;quot;)&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-01-22T03:08:07.563" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19402" PostTypeId="2" ParentId="19399" CreationDate="2021-01-22T05:25:38.733" Score="4" Body="&lt;p&gt;Explain what you explained here to start the interview. Do a 3-5 minute intro that introduces logarithms the way your video would introduce it. Then spend the rest of the time doing whatever else you would do in class.&lt;/p&gt;&#xA;&lt;p&gt;Bring the activity with you. Hand it out to the committee.&lt;/p&gt;&#xA;&lt;p&gt;Basically I don't see any reason to do anything other than &lt;strong&gt;give a faithful representation of what you would actually do in the job&lt;/strong&gt;, just like for any interview.&lt;/p&gt;&#xA;&lt;p&gt;If the place you are interviewing is any good, then the committee is made up of mostly educators, and this will be a good task for an interview: it tests your ability to follow instructions from administration, keep track of time, and concisely explain tricky topics.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-01-22T05:25:38.733" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19404" PostTypeId="2" ParentId="19399" CreationDate="2021-01-22T09:25:07.097" Score="-2" Body="&lt;p&gt;I introduce logarithms via a number guessing game. I wrote a run-of-the-mill Python program to do this. It asks for an integer, &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;1$&lt;/span&gt;, and the class has to guess the random integer between 1 and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. Each time they do not get the answer right, a message lets them know if the number is higher or lower.&lt;/p&gt;&#xA;&lt;p&gt;After a couple of rounds, I ask them what they think the maximum number of needed guesses should be. By this time, many students pick up on how they can cut the number of options in half at each step. However, it often isn't easy for them to articulate that into a mathematical answer.&lt;/p&gt;&#xA;&lt;p&gt;Before I tell them the answer, I introduce logarithms as the inverse of the exponential function and how they are used to find unknown exponent values. After a couple of examples of rewriting exponential form to logarithmic form, we revisit the guessing game. By this time, students suspect the maximum number of guesses has something to do with logarithms.&lt;/p&gt;&#xA;&lt;p&gt;We discuss the &amp;quot;cut in half&amp;quot; pattern and many students jump to &lt;span class=&quot;math-container&quot;&gt;$\log_2(n)$&lt;/span&gt;, although I do hear a few base one-halfs too. Luckily it doesn't take long for students to see that the logarithm can not have a base of 1/2 in this game.&lt;/p&gt;&#xA;&lt;p&gt;After looking at a few more examples, including when &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt; and when &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a power of 2, the class eventually comes to the conclusion that the answer must be &lt;span class=&quot;math-container&quot;&gt;$\log_2(n)+1$&lt;/span&gt;, rounded down to the nearest integer (I don't cover floor and ceiling functions when talking about families of functions; however, I mention them here).&lt;/p&gt;&#xA;" OwnerUserId="625" LastActivityDate="2021-01-22T09:25:07.097" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19405" PostTypeId="2" ParentId="19397" CreationDate="2021-01-22T22:04:32.213" Score="6" Body="&lt;p&gt;This is a great question. Love it love it love it!!&lt;/p&gt;&#xA;&lt;p&gt;The following is just what occurs to me off the top of my head.&lt;/p&gt;&#xA;&lt;p&gt;Show a graph paper grid with a dot at the origin and a dot at (3,4). Say we want to find the distance between the dots. We could guess that it would be 3+4=7. Well, that would be right if these were city blocks, but it's not the right answer as the crow flies. Roughly what do you think the distance is as the crow flies?&lt;/p&gt;&#xA;&lt;p&gt;OK, let's measure it. Could someone draw it on the board at a scale where the coordinates are 30 cm and 40 cm? Thanks. What do you get when you measure it? 49 cm? Can you estimate the millimeters? I know it's rough. OK, you got 49.3 cm.&lt;/p&gt;&#xA;&lt;p&gt;So now we know that the formula x+y isn't right. Actually, our answer looks really close to 50 cm. We don't know who the first person was to notice this, but supposedly it was already known to the ancient Egyptians. (Picture of rope with knots.) Yes, the conjecture is right, the true answer really is 5, but the Egyptians never knew why.&lt;/p&gt;&#xA;&lt;p&gt;Here's one thing we can tell for sure about the mystery formula that gives the distance in terms of x and y. Suppose I draw a circle with a radius of 5, centered on the origin. It passes through our point from before. Now look at the top of the circle. It looks really flat there. If you were to zoom in on that point under an infinitely powerful microscope, it would look &lt;em&gt;perfectly&lt;/em&gt; flat. You wouldn't even be able to see that it was curved, just like you can't tell that the earth is round when you're standing on the earth.&lt;/p&gt;&#xA;&lt;p&gt;This is a hint that the relationship is nonlinear, meaning that it isn't just about ordinary addition and subtraction, like x+y or 2x+3y or something like that. I'm going to give you a preview of the actual answer, and then we'll come back and prove it. People probably did this originally as a guess and check (or guess and measure), and may never have been able to give an actual proof until hundreds of years later.&lt;/p&gt;&#xA;&lt;p&gt;So the actual expression relating the distance d to x and y is &lt;span class=&quot;math-container&quot;&gt;$d^2=x^2+y^2$&lt;/span&gt;. This is called nonlinear because it has the exponents in it. When you square a big number, it gets really big. When you square a small number, it gets really small by comparison. What do you get when you square 1000? What do you get when you square 0.1?&lt;/p&gt;&#xA;&lt;p&gt;This makes sense in terms of the close-up flatness idea. At a point like this (point to somewhere close to the top of the circle), x is really small. The square of a small number is really small, so &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; is so small, we hardly care about it. So &lt;span class=&quot;math-container&quot;&gt;$d^2\approx y^2$&lt;/span&gt;. Can anyone solve this equation for y?&lt;/p&gt;&#xA;&lt;p&gt;[...continue from here, lead in to formal proof...]&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-01-22T22:04:32.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19407" PostTypeId="1" CreationDate="2021-01-24T00:33:19.640" Score="9" ViewCount="290" Body="&lt;p&gt;I am currently advising a student at his final project (it is a graduation course for people who will become math teachers). We've chosen to pick some basic mathematical rules which are (or at least should be) known by students (like &lt;span class=&quot;math-container&quot;&gt;$a^{-1} = \frac{1}{a}, \forall a \in \mathbb{R}^*$&lt;/span&gt;) and explain the reasons why those rules actually make sense.&lt;/p&gt;&#xA;&lt;p&gt;I am totally convinced that, in most cases, it is good for students to understand why mathematical rules work. I could write a very long text justifying this belief but, from an academic point of view, my personal opinion has no value. I need academic articles/books written by researchers of mathematical education supporting this idea. Could you please suggest me some articles/books along these lines?&lt;/p&gt;&#xA;&lt;p&gt;From now, all I've got are some lines from G. Polya which resemble those ideas, but they are still not exactly what I want.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="15337" LastEditorUserId="376" LastEditDate="2021-01-24T09:40:27.270" LastActivityDate="2021-01-25T19:45:09.420" Title="Seeking references for why it is good that students understand why mathematical rules work" Tags="|undergraduate-education|reference-request|undergraduate-research|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19408" PostTypeId="2" ParentId="19407" CreationDate="2021-01-24T01:54:50.013" Score="7" Body="&lt;p&gt;The key term you are interested in is &amp;quot;conceptual knowledge&amp;quot; (more specifically, &amp;quot;conceptual understanding&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://www.itws.org/NCTM-ContentProcessCoreStandards.pdf&quot; rel=&quot;noreferrer&quot;&gt;this document&lt;/a&gt; from the National Council of Teachers of Mathematics,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Students who lack understanding of a topic may rely on procedures too heavily. Without&#xA;a flexible base from which to work, they may be less likely to consider analogous&#xA;problems, represent problems coherently, justify conclusions, apply the mathematics&#xA;to practical situations, use technology mindfully to work with the mathematics,&#xA;explain the mathematics accurately to other students, step back for an overview, or&#xA;deviate from a known procedure to find a shortcut. In short, a lack of understanding&#xA;effectively prevents a student from engaging in the mathematical practices.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I do not have a copy of the book &amp;quot;Concept-Rich Mathematics Instruction&amp;quot; by Meir Ben-Hur (Association for Supervision and Curriculum Development, 2006) (and I have not read it), but I suspect you might find what you're looking for there.  You can read its first chapter (&amp;quot;Conceptual Understanding&amp;quot;) &lt;a href=&quot;http://www.ascd.org/publications/books/106008/chapters/Conceptual-Understanding.aspx&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2021-01-24T02:58:29.007" LastActivityDate="2021-01-24T02:58:29.007" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19409" PostTypeId="2" ParentId="19407" CreationDate="2021-01-24T13:19:06.200" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I am totally convinced that, in most cases, it is good for students to understand why mathematical rules work. I could write a very long text justifying this belief but, from an academic point of view, my personal opinion has no value. I need academic articles/books written by researchers of mathematical education supporting this idea.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is there an agreed upon goal of mathematics education?  It seems that you are assuming (correct me if I am wrong) that there is a &amp;quot;real goal&amp;quot; of mathematics education, and that &amp;quot;understanding the reasoning&amp;quot; is not one of those goals.  Then your question is for evidence &amp;quot;understanding the reasoning&amp;quot; supports the &amp;quot;real goal&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I do not view things this way. I think that everyone has their own individual motivations for how they choose to engage (or not engage) with mathematics, and they are all equally valid.  I have personal beliefs that engaging with mathematics in particular ways can enrich someones life, so I have built a career around sharing that perspective.  I know that one's life &lt;em&gt;can&lt;/em&gt; be enriched in this way because my own life has been so enriched. No academic justification needed.&lt;/p&gt;&#xA;&lt;p&gt;For an analogy, I find it very enriching to forage for wild foods.  When I am hiking and eating a pawpaw, and someone asks me what that is, I do not need a research study to justify sharing my enthusiasm for my hobby.&lt;/p&gt;&#xA;&lt;p&gt;Now there are interesting math education research questions you could ask which are related to this question.  It would be interesting to know the answer to questions like &amp;quot;Does having an explanation of a rule increase the chance that the rule will be applied correctly?&amp;quot;.  This is a question about objective reality, and hence has a chance at being answered by research.  The value judgement that this would be a &amp;quot;good thing&amp;quot; is not answerable by research.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-01-24T13:19:06.200" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19410" PostTypeId="1" AcceptedAnswerId="19412" CreationDate="2021-01-24T13:46:02.220" Score="9" ViewCount="1637" Body="&lt;p&gt;From one textbook we use in our High School -&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4OZ4X.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4OZ4X.jpg&quot; alt=&quot;Some text (see transcription below) with a drawing of a trapezoid to the right. The trapezoid is labeled &amp;quot;Trapezoid&amp;quot;.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Transcription:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A &lt;em&gt;&lt;strong&gt;trapezoid&lt;/strong&gt;&lt;/em&gt; is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called &lt;em&gt;bases&lt;/em&gt; of the trapezoid.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And from Wikipedia -&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In Euclidean geometry, a convex quadrilateral with at least one pair&#xA;of parallel sides is referred to as a trapezium (/trəˈpiːziəm/) in&#xA;English outside North America, but as a trapezoid[1][2] (/ˈtræpəzɔɪd/)&#xA;in American and Canadian English.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One other textbook in my school follows the Wikipedia definition.&lt;/p&gt;&#xA;&lt;p&gt;The former definition excludes parallelograms and rectangles. The latter, defines both to be a subset of trapezoids.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;How do we address this with students?&lt;/strong&gt; I'm starting to get objections to the textbook image I posted, with the student either recalling having learned it differently in a prior class, or searching and declaring another source as the contradiction to our textbook. I can offer a response of &amp;quot;This is one of those math naming properties that doesn't have 100% agreement.&amp;quot; Although this doesn't seem satisfying.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2021-01-26T04:10:09.707" LastActivityDate="2021-01-26T04:10:09.707" Title="Definition of Trapezoid" Tags="|geometry|terminology|definitions|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="19411" PostTypeId="2" ParentId="19410" CreationDate="2021-01-24T14:03:43.357" Score="8" Body="&lt;p&gt;Unfortunately, we don't have a set of universally agreed upon definitions in mathematics. It might seem like we do (or should), especially in Geometry with its long history and so much agreement, but the truth is that we use different definitions frequently. One proof of this is the differing definitions in your textbooks! That is just the nature of a subject that evolved over millennia across a world of cultures. This is why it is so important to rigorously define your meaning in whatever context in which you're working.&lt;/p&gt;&#xA;&lt;p&gt;In the case of trapezoids, I have heard passionate arguments for both sides, but that debate really is not important.  The important piece is: &lt;strong&gt;define your terms.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;h3&gt;How to address this for students.&lt;/h3&gt;&#xA;&lt;p&gt;Two thoughts:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;This is an excellent opportunity to demonstrate to students that they can use whichever definition they need as long as they clearly state which one they're using! It is also a great way to engage students in formalizing logical arguments. I understand it might not seem satisfying to just state that there is disagreement, but maybe you could turn that into a productive discussion.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Focus on whichever definition will best serve them. For example, use the definition that is used for SAT, ACT, AP Exams, etc. Not because those should be authoritative organizations in mathematics, but because it will simplify things for students by minimizing discrepancies in definitions when they sit for high-stakes exams.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="219" LastActivityDate="2021-01-24T14:03:43.357" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="19412" PostTypeId="2" ParentId="19410" CreationDate="2021-01-24T14:48:42.440" Score="21" Body="&lt;p&gt;I would use this to help students understand three &amp;quot;meta&amp;quot; ideas:&lt;/p&gt;&#xA;&lt;p&gt;(1) Math is not about memorizing lots of random trivia. In the real world, if you go up to a mathematician and ask them which definition of a trapezoid is right, they will just smile indulgently. They don't know or care.&lt;/p&gt;&#xA;&lt;p&gt;(2) There is not always a consensus about definitions. Get over it, boys and girls! In STEM, it's very common that when you read something, you need to check which definitions they're using.&lt;/p&gt;&#xA;&lt;p&gt;(3) In general, in math, we prefer to make our definitions in such a way that theorems come out tidy and with a minimum of special-casing. For this purpose, it's usually good to have the things that fit definition A be a subset of the things that fit definition B. By this rule of thumb, it's preferable to define a parallelogram as being a trapezoid. If not, then any time you want to prove a theorem whose conclusion is &amp;quot;X is a trapezoid,&amp;quot; you will probably have to uglify it by making the conclusion &amp;quot;X is either a trapezoid or a parallelogram.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Often, a reason why books will sometimes choose exclusive definitions (so that a square is not a rectangle, and a parallelogram is not a trapezoid) is that they have a low estimate of their students' intelligence. Students operating at lower intellectual levels (as well as very young kids) have trouble understanding how these definitions can be inclusive.&lt;/p&gt;&#xA;&lt;p&gt;In this particular example, there is a possible advantage of choosing the exclusive definition, which is that then we have two sides that we can pick out as the &amp;quot;bases.&amp;quot; It's a trade-off.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-01-25T22:40:02.117" LastActivityDate="2021-01-25T22:40:02.117" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19413" PostTypeId="2" ParentId="19407" CreationDate="2021-01-25T09:48:48.167" Score="0" Body="&lt;p&gt;A while ago I was asking myself the same question: why &lt;span class=&quot;math-container&quot;&gt;$a^{-1} = \frac{1}{a}$&lt;/span&gt;. I haven't read anything on the topic. I just figured that this was engineered, based on existing and obvious definitions of exponentiation for positive integers, like &lt;span class=&quot;math-container&quot;&gt;$a^{2} = a \times a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a^{1} = a$&lt;/span&gt; and then you want to do some operations with them, like multiplying and dividing. So, if &lt;span class=&quot;math-container&quot;&gt;$a^{1} \times a^{2} = a \times a \times a = a^{3}$&lt;/span&gt;, that is, the exponents are added up, then it would be nice if the same happened when we go into the zero and then the negatives: &lt;span class=&quot;math-container&quot;&gt;$a^{0} \times a^{2}$&lt;/span&gt; should be &lt;span class=&quot;math-container&quot;&gt;$a^{2}$&lt;/span&gt;, which means that &lt;span class=&quot;math-container&quot;&gt;$a^{0}$&lt;/span&gt; should be 1. Likewise, &lt;span class=&quot;math-container&quot;&gt;$a^{-1} \times a^{3}$&lt;/span&gt; should be &lt;span class=&quot;math-container&quot;&gt;$a^{2}$&lt;/span&gt;, which means that &lt;span class=&quot;math-container&quot;&gt;$a^{-1}$&lt;/span&gt; should be &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{a}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, this rule works because it has been designed to work, not because it is a divine law.&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2021-01-25T19:45:09.420" LastActivityDate="2021-01-25T19:45:09.420" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19415" PostTypeId="2" ParentId="19410" CreationDate="2021-01-26T03:27:36.247" Score="2" Body="&lt;p&gt;The wikipedia definition is the right one.&lt;/p&gt;&#xA;&lt;p&gt;A square is a rectangle. A rectangle is a trapezoid.&lt;/p&gt;&#xA;&lt;p&gt;Yeah, at times you can have specific/different definitions. But, the trapezoid one is pretty clear cut. You'd be hard pressed to find a (non contrived) theorem that applies to trapezoid that suddenly stops working because the shape is also a rectangle.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me like the &amp;quot;exactly one pair&amp;quot; definition is only there to be less confusing to students (but, is it a rectangle or a trapezoid?). To me, that's not a good reason. Rather using this weird definition is a missed opportunity to discuss interesting concept: A rectangle is a trapezoid, but a trapezoid is not a rectangle.&lt;/p&gt;&#xA;" OwnerUserId="14023" LastActivityDate="2021-01-26T03:27:36.247" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19416" PostTypeId="1" CreationDate="2021-01-26T22:13:18.007" Score="5" ViewCount="239" Body="&lt;p&gt;We are now in lock-down, so while homeschooling my son I get to se exactly what he does for math. He has been getting a huge amount of repetitive practicing of really simple math, despite being quite good at math.&#xA;My question is, given his skill level and his current enjoyment of math is there any point to this endless repetition, or should I go head to head with the teacher about it?&#xA;Is there such a thing as the recommended number of repetitions?&lt;/p&gt;&#xA;&lt;p&gt;By really huge amount I mean he will have to do between 100 and 140 additions of numbers under 10, mostly +1 and +2 additions, in one take, at least once a week. This takes 10 minutes without any breaks.&lt;/p&gt;&#xA;&lt;p&gt;He has been doing additions of numbers under 10 for at least 1.5 year now almost without errors, and additions of numbers up to 100 for about a year. He does this for fun, he likes it when we randomly ask him to sum two numbers. Last summer (6 months ago) he liked showing off adding up much larger numbers, eg 367 + 58.&#xA;This is outside of school. I think the started with additions in school about 1.5 years ago.&lt;/p&gt;&#xA;&lt;p&gt;In addition to the above repetitive task his math tasks consists of more or less useful tasks. Each day of the week he does 1 or 2 tasks, the other tasks can be any of the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;practice telling the time from a clock, or setting a clock to some time&lt;/li&gt;&#xA;&lt;li&gt;(learning to do) multiplication&lt;/li&gt;&#xA;&lt;li&gt;adding and subtracting from 10-20&lt;/li&gt;&#xA;&lt;li&gt;adding and subtracting numbers &amp;lt; 100&lt;/li&gt;&#xA;&lt;li&gt;proportionality (one ice-cream costs 2 euros, how much are 3 ice-creams)&lt;/li&gt;&#xA;&lt;li&gt;estimating on an unmarked axis (eg placing 45 and 75 on a line marked only with 0 and 100)&lt;/li&gt;&#xA;&lt;li&gt;various arithmetic 'games'. For example he has to write an addition as a substraction by moving numbers around, but also quite difficult ones where he has to find all the rows and columns that add up to 50 in a 3x3 square of numbers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I cannot square the level of that last task with doing 100 sums of adding 1 or 2 to a number under 10.&lt;/p&gt;&#xA;&lt;p&gt;It has been a long time since I was in school, but I don't think we ever had to do more then 50 sums in a row, probably not more then 25. Because we were writing down the answers we were slower, he does his tasks on a computer. Still, he could do less repetitions and do something else with the time saved by typing instead of writing.&#xA;(Maybe practice writing).&lt;/p&gt;&#xA;&lt;p&gt;The tasks are 'adaptive' and 'individual', so different children get different tasks, and are decided on by a computer program. Being a software developer myself, I worry that something has gone wrong here.&#xA;Even if the computer program performs as specified, I worry that all this repetition of tasks my child already mastered will eventually rob him of the fun he has doing math.&lt;/p&gt;&#xA;&lt;p&gt;When I ask him if he finds these tasks boring, he says he does. But he is also proud to get so many answers in a row correct. My guess is it is psychologically rewarding the same way playing minesweeper is.&lt;/p&gt;&#xA;" OwnerUserId="15362" LastEditorUserId="15362" LastEditDate="2021-01-27T15:34:43.343" LastActivityDate="2021-01-27T18:02:41.730" Title="Sensible amount of repetition 7 year old" Tags="|reference-request|primary-education|teaching|arithmetic|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19418" PostTypeId="1" CreationDate="2021-01-27T03:55:42.527" Score="2" ViewCount="241" Body="&lt;p&gt;Some students have sniveled that most examples of Bayes' Theorem use non-integer numbers. I want to try a Bayes' Theorem chart that uses just single digit Natural Numbers &lt;span class=&quot;math-container&quot;&gt;$\le 9$&lt;/span&gt;. To complete the table below most comfortably for teenagers,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;what are the simplest stories?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;what natural numbers ≤ 9 contrast the base rate fallacy the most? Please don't repeat a number.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/q/16977/15364&quot;&gt;The biggest number in this similar question&lt;/a&gt; still uses two digits, and rehashes the common example of letting D be be a disease and &lt;span class=&quot;math-container&quot;&gt;$H_0$&lt;/span&gt; be a negative (diagnostic) test result. What other &lt;span class=&quot;math-container&quot;&gt;$H_0, D$&lt;/span&gt; are more intuitive?   Green denotes true positive and negative, red false positive and negative.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{array}{r|cc|c}&#xA;\text{Number of occurrences}&amp;amp;D &amp;amp;\lnot D  &amp;amp;\text{Total}\\ \hline&#xA;H_a &amp;amp;\color{green}{\Pr(D)\Pr(+|D)}&amp;amp;\color{red}{\Pr(D^C)\Pr(+|D^C)}&amp;amp;\text{add the 2 left entries}\\&#xA;H_0 &amp;amp;\color{red}{\Pr(D)\Pr(-|D)}&amp;amp;\color{green}{\Pr(D^C)\Pr(-|D^C)}&amp;amp;\text{add the 2 left entries}\\ \hline&#xA;\text{Total}&amp;amp;\text{add the 2 above entries}&amp;amp;\text{add the 2 above entries}&amp;amp;\text{single digit integer}&#xA;\end{array}$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user15364" LastEditorDisplayName="user15364" LastEditDate="2021-01-29T04:02:56.693" LastActivityDate="2021-01-29T04:02:56.693" Title="What story and one-digit Natural Numbers best fit Bayes' Theorem chart?" Tags="|probability|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19419" PostTypeId="1" CreationDate="2021-01-27T04:40:49.207" Score="2" ViewCount="363" Body="&lt;p&gt;This question applies to solely books already accompanied by an ISM (&amp;quot;Instructor's Solution Manual) — access is restricted to instructors. Some don't even sell a SSM (Students' Solution Manual) that provide solutions for just half of the exercises.&lt;/p&gt;&#xA;&lt;p&gt;How can I persuade an author by emailing her to release the ISM to students? Has anyone succeeded in this?&lt;/p&gt;&#xA;&lt;p&gt;Here are my three rebuttals to most authors' defense — &lt;a href=&quot;https://old.reddit.com/r/changemyview/comments/9sa9c4/cmv_textbooks_should_not_offer_practice_problems/e8nc6j4/&quot; rel=&quot;nofollow noreferrer&quot;&gt;[t]eachers often use textbooks to assign homework problems. If they give a key to all problems, the teachers will have to use a different resource which will be a hassle for students as well.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Instructors can assign exercises from another book, without divulging their choice of textbook to students. Multiple textbooks cover most topics in undergraduate math and statistics.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some authors deliberately make the half of exercises without solutions (even numbered)  harder. Even if a SSM provides solutions for the easier other half, then students get to practice just easier questions to their disservice.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even if a SSM accompanies a book, students have to buy another textbook or Schaum's problem book for more practice. Students and libraries have to spend more money buying this second book. cond book.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user15364" LastEditorDisplayName="user15364" LastEditDate="2021-01-28T07:00:42.770" LastActivityDate="2021-01-29T00:09:16.507" Title="How can I convince authors to publish the Instructor's Solution Manual to students?" Tags="|textbooks|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19420" PostTypeId="2" ParentId="11433" CreationDate="2021-01-27T15:04:59.210" Score="0" Body="&lt;p&gt;I'm not sure whether this is useful or not, but if the students are used to the notation &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and you wish to allow them mental continuity, you could always try the technique of saying:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; be a variable whose domain is the set of real numbers. Then we define the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; to be [whatever formula].&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In that way you kill two birds with one stone: you keep the familiar notation without which a student can often get (at least temporarily) lost, while at the same time adding some small aspect to the rigor which is required when you get deeper into the subject.&lt;/p&gt;&#xA;&lt;p&gt;At that stage you can then start talking about &amp;quot;domain&amp;quot; and &amp;quot;codomain&amp;quot; and &amp;quot;image&amp;quot; (but steer clear of &amp;quot;range&amp;quot; as the term is dismayingly ambiguous), and at &lt;em&gt;that&lt;/em&gt; point you can then introduce the idea that a function is &lt;em&gt;actually&lt;/em&gt; a triple consisting of domain, codomain and the subset of the cartesian product of the latter -- by which time they are also now probably ready to consider the concept of the general relation.&lt;/p&gt;&#xA;&lt;p&gt;Sooner or later, many students may well find themselves on a course which explores the abstract concepts themselves: injections, surjections, bijections, etc. and will need to understand the subtleties like the image of a union equals the union of the images, while intersections are more complicated than that ... for physics students it is probably not appropriate to explore this, at least at this stage, but having the proper language for the various &amp;quot;parts&amp;quot; of a function is always useful to have in one's toolkit.&lt;/p&gt;&#xA;" OwnerUserId="15290" LastActivityDate="2021-01-27T15:04:59.210" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19421" PostTypeId="2" ParentId="19416" CreationDate="2021-01-27T18:02:41.730" Score="7" Body="&lt;p&gt;A once per week ten minute session is objectively not using up that much of the kid's time.  In addition, he's not even as much turned off on it as you are.  Also, I think you will find that speed increases with volume, so it becomes nonlinear.&lt;/p&gt;&#xA;&lt;p&gt;None of this is to say that I love the particular drill.  Sounds too easy for a smart 7yo.  Just that it's not so bad to do something non-optimal.  I mean, heck Ralph Maccio had to spend a whole day painting fences and buffing cars.  This ain't anywhere as grasshopper as that. ;-)&lt;/p&gt;&#xA;&lt;p&gt;The bigger issue might be that he is ready for more challenging material.  But even here, it sounds like you are getting 50-75% of a loaf.  And that side issue is bigger than the ten minute weekly sessions.&lt;/p&gt;&#xA;&lt;p&gt;In any case, there are limits to how much a group class can be individualized.  I would avoid complaining about something that is this minor (10 mins per week).  Life isn't perfect.  Part of growing up is learning that, soldiering on, and doing good things regardless, rather than getting too tied up with minor issues.&lt;/p&gt;&#xA;" OwnerUserId="15369" LastActivityDate="2021-01-27T18:02:41.730" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19422" PostTypeId="1" AcceptedAnswerId="19423" CreationDate="2021-01-27T20:09:17.377" Score="4" ViewCount="187" Body="&lt;p&gt;Judea Pearl, in his book &amp;quot;Probabilistic reasoning in intelligent systems&amp;quot; uses a handful of stories over and over again, each time to demonstrate a different point. (His &amp;quot;Alarm&amp;quot; story comes to mind)&lt;/p&gt;&#xA;&lt;p&gt;I assume that the perceived advantage of such a strategy is that students gain familiarity with the example and so no time and mental effort is wasted in understanding the story each time, making it easier to focus on the actual point. But surely there's also something to be gained from going over different examples, showing, for example, the applicability of the taught methods in different situations.&lt;/p&gt;&#xA;&lt;p&gt;What are some other advantages/disadvantages of this approach? When is it recommended?&lt;/p&gt;&#xA;&lt;p&gt;(I'm mainly interested in undergrad level courses)&lt;/p&gt;&#xA;" OwnerUserId="15371" LastActivityDate="2021-02-19T05:46:15.837" Title="Pros/Cons of using a single story in multiple examples to demonstrate different points" Tags="|undergraduate-education|examples|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19423" PostTypeId="2" ParentId="19422" CreationDate="2021-01-27T20:49:52.807" Score="5" Body="&lt;h3&gt;Cognitive Load&lt;/h3&gt;&#xA;&lt;p&gt;I agree that the advantage of a repeated story is a familiarity with the context so that cognitive load is minimized. &lt;a href=&quot;https://en.wikipedia.org/wiki/Cognitive_load&quot; rel=&quot;noreferrer&quot;&gt;Cognitive load theory&lt;/a&gt; describes different types of load, and while you cannot minimize some types of cognitive load, for example the &lt;em&gt;intrinsic&lt;/em&gt; difficulty of the topic that you are teaching, you do have control over other types, such as &lt;em&gt;extraneous&lt;/em&gt; load of instructional details like problem contexts and material choice/design.  One way to do that, as you point out, is to reuse similar contexts.&lt;/p&gt;&#xA;&lt;h3&gt;Abstraction &amp;amp; Transfer Learning&lt;/h3&gt;&#xA;&lt;p&gt;An advantage in changing-up the context is in helping students to abstract by de-contextualizing problems.  A significant goal in education is &lt;em&gt;transfer learning&lt;/em&gt;, or the ability of students to re-apply knowledge in new but similar contexts. This is also an area of research in machine learning. When we write our curricular units we are required to identify &amp;quot;transfer knowledge&amp;quot;, or what it is that students will be able to use across multiple domains.&lt;/p&gt;&#xA;&lt;p&gt;It is helpful for students to understand concepts by examples, but also by non-examples. We want them to &amp;quot;know&amp;quot; a concept both by what it &lt;em&gt;is&lt;/em&gt; and by what it &lt;em&gt;is not&lt;/em&gt;. To me, that is a significant advantage of seeing a concept in a variety of contexts.  Seeing only the same context repeatedly can be risky if students might attribute an unrelated piece of the context to the concept or vice versa. For example, I once created an awful lesson on Law of Sines where I kept using similar side ratios for all of the examples, so students ended up learning the ratios of that particular triangle very well without really understanding Law of Sines.&lt;/p&gt;&#xA;&lt;p&gt;Really, I don't think &amp;quot;transfer learning&amp;quot; exists! I think we either develop new schema or we map/project new contexts onto existing schema. Ok maybe that's exactly what transfer learning is... Brains are weird.&lt;/p&gt;&#xA;&lt;h3&gt;Caveat of Contexts&lt;/h3&gt;&#xA;&lt;p&gt;Contexts often are not distinct, and there tend to be subcontexts. So it is possible to see different contexts within the same story. I'm not familiar with &amp;quot;Probabilistic Reasoning in Intelligent Systems&amp;quot;, but it may be that Pearl repeats the same stories while at the same time creating new contexts of understanding. Maybe it's similar to how a show can have different problems in different episodes while maintaining characters and settings.&lt;/p&gt;&#xA;" OwnerUserId="219" LastEditorUserId="219" LastEditDate="2021-01-27T20:55:35.147" LastActivityDate="2021-01-27T20:55:35.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19424" PostTypeId="2" ParentId="19419" CreationDate="2021-01-28T13:00:34.447" Score="2" Body="&lt;p&gt;I don't think you can convince them.  The practice has evolved for marketing reasons.  (I have seen a couple recent books where it was explicitly said in the preface that the author cut answers for 50% of the book in a later edition, to appeal to the market.)  And you won't change this dynamic.  It appeals to gatekeeper instructors (even though they rationalize the practice with other excuses...which you won't get them to change those either).  Of course this sucks for students really trying to learn the topic by drill and check.  In other words, I agree with you.  But you won't change the practice.  There's many ways that college is a ripoff (look at the whole textbook new edition and defeat the resale market racket also).&lt;/p&gt;&#xA;&lt;p&gt;Just buy (or assign, but I get the impression you are a student) a Schaum's or similar drill book.&lt;/p&gt;&#xA;&lt;p&gt;I have seen occasional books that do have all/most of the answers:  Thomas Finney calculus (at least the old editions), Tenanbaum ODE, Weinberger PDE.&lt;/p&gt;&#xA;&lt;p&gt;I have also occasionally seen texts that had a purchasable solution guide with full answers and even hand written, reproduced, full solutions.  Lindeberg EIT/PE books (obviously mostly for the self study market, thus not needing to market to gatekeepers).  But also I had an organic chem test (Streitweiser and Heathcock, I think) that had this.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  It is interesting to me that this is a &amp;quot;how do I convince&amp;quot; or &amp;quot;what are arguments for position A&amp;quot; question.  Have seen a fair amount of these.  You will get more help when the forum community agrees with the proposition.  (E.g. why proofs are dreamy and calculations suck.)  You will get less when asking &amp;quot;not A&amp;quot; in terms of community norms.&lt;/p&gt;&#xA;" OwnerUserId="15369" LastEditorUserId="15369" LastEditDate="2021-01-28T13:25:28.883" LastActivityDate="2021-01-28T13:25:28.883" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19425" PostTypeId="2" ParentId="19418" CreationDate="2021-01-28T18:18:34.033" Score="6" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Two professional athletes and six fans are eating at a restaurant table.  Both of the professional athletes are wearing their jerseys, while only half of the fans are wearing jerseys.  Given a person at the table wearing a jersey, what is the probability that they are a professional athlete?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't know why you think that students understand BT better with single-digit numbers.  Seems to me that with a hundred fans it becomes far clearer how unlikely it is that a jersey-wearing person is an athlete even though all of the athletes are wearing jerseys.  But you do you.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-01-28T18:18:34.033" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19426" PostTypeId="2" ParentId="19419" CreationDate="2021-01-29T00:02:52.483" Score="4" Body="&lt;p&gt;Welcome to matheducators.SE! It's great to have an undergraduate student participating in the site.&lt;/p&gt;&#xA;&lt;p&gt;The question is basically written as if this were the year 1996, when I started my teaching career. Today, we have the internet.&lt;/p&gt;&#xA;&lt;p&gt;Your question includes a lot of money-related material. There is no fundamental reason why students need to keep getting financially exploited by the textbook publishers. For the undergrad math and stat courses you're talking about, there are generally multiple &lt;a href=&quot;http://theassayer.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;free options on the internet&lt;/a&gt;. Most instructors just don't care about the economic victimization of their students. They've gotten used to using a particular non-free textbook, and they're not going to change now, because it would be work.&lt;/p&gt;&#xA;&lt;p&gt;Your question is written as though the mechanism for economic victimization were still the cost of the books, but this is no longer true, because you can get your book for free on the internet at &lt;a href=&quot;http://libgen.rs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://libgen.rs/&lt;/a&gt; . For lower-division math, the mechanism for economic victimization is no longer the cost of accessing the book, it's the cost of an account on a system like mymathlab. Again, there are nonexploitative alternatives (e.g., WeBWorK), and again, most instructors don't use them because they're used to using the expensive thing already and changing over would be work.&lt;/p&gt;&#xA;&lt;p&gt;Your question is written as though instructors could still keep secrets from their students:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Instructors can assign exercises from another book, without divulging their choice of textbook to students.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This won't work, because the internet exists. If your instructor assigns you a problem, and you don't know what book it's from, cut and paste the text into a search engine. The first hit will be on chegg.com, your central resource for cheating and copyright violation.&lt;/p&gt;&#xA;&lt;p&gt;And finally, your question is written as though it still made sense for teachers to be assigning problems for points and making those points enough to have a big effect on their students' grades. This does not make sense today, because students can easily get solutions from chegg.&lt;/p&gt;&#xA;&lt;p&gt;I'm the author of some &lt;a href=&quot;http://lightandmatter.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;free physics texts&lt;/a&gt; (and also one free math text). About half of the problems have already been illegally copied onto chegg, and have solutions available there. Because of this, I've given up on using homework grades as more than a feedback and progress-tracking thing. (I count them for 1% of my students' grades.) My students check their machine-checkable answers using free sotfware, and I grade their non-machine-checkable answers by hand. After that, I email them solutions.&lt;/p&gt;&#xA;&lt;p&gt;Your question is somewhat helpful in that, although I don't agree with almost any of your logic, you made me rethink whether I want to simply put the solutions in the books at this point. I don't see any significant benefit to doing that, except perhaps for some small number of students who are using the books for self-study and can't get an instructor to comment on their work. The downside is that many of the people using my books don't agree with my opinions and therefore are still giving 10 or even 20% of their students' grades for homework. Those people would probably be very upset with me and stop using my book.&lt;/p&gt;&#xA;&lt;p&gt;For your own education, given the reality of how the courses you're taking are being run, I would suggest that you simply work hard to solve the problems without access to the solutions. Get going far enough in advance so that if you get stuck, you have time to get help. You can get help by going to your TA or instructor's office hours, or to an on-campus tutoring center. If you want examples with solutions, use the examples in your textbook -- most commercial texts have a huge number of these in every chapter.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-01-29T00:09:16.507" LastActivityDate="2021-01-29T00:09:16.507" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19427" PostTypeId="2" ParentId="18047" CreationDate="2021-01-29T04:13:45.283" Score="0" Body="&lt;p&gt;Have you tried the MAA's book reviews? You can filter by topic and rating. Because you prefer &amp;quot;a book that emphasizes theory over procedure&amp;quot;, try &lt;a href=&quot;https://www.maa.org/press/maa-reviews/browse?field_tags_tid=37267&amp;amp;field_bll_rating_tid=All&amp;amp;field_maa_review_value=All&quot; rel=&quot;nofollow noreferrer&quot;&gt;Probability Theory&lt;/a&gt; as well as &lt;a href=&quot;https://www.maa.org/press/maa-reviews/browse?field_tags_tid=37266&amp;amp;field_bll_rating_tid=All&amp;amp;field_maa_review_value=All&quot; rel=&quot;nofollow noreferrer&quot;&gt;Probability&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user15364" LastActivityDate="2021-01-29T04:13:45.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19428" PostTypeId="2" ParentId="18047" CreationDate="2021-01-30T15:44:13.637" Score="0" Body="&lt;p&gt;I have used this one. It's adequate and free!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://saylordotorg.github.io/text_introductory-statistics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://saylordotorg.github.io/text_introductory-statistics/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2021-01-30T15:44:13.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19429" PostTypeId="2" ParentId="14077" CreationDate="2021-01-31T12:07:30.383" Score="2" Body="&lt;p&gt;I think this is only a partial answer to your question.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Ideals, Varieties, and Algorithms&lt;/em&gt; by Cox, Little &amp;amp; O'Shea presents affine varieties and their parametrizations in Sections 2 and 3, respectively, of Chapter 1. This material could complement or supplement the coverage of curves and surfaces in a multivariable calculus course. Then Section 4 of Chapter 3 (Elimination Theory) introduces singular points (and envelopes), including how to find tangent lines to algebraic curves by looking for multiple roots. Perhaps it could be useful.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2021-01-31T12:07:30.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19430" PostTypeId="2" ParentId="14081" CreationDate="2021-02-01T19:33:30.720" Score="1" Body="&lt;p&gt;Some of my students love &lt;a href=&quot;https://global.oup.com/academic/product/number-theory-9780198846734?cc=ca&amp;amp;lang=en&amp;amp;&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Number Theory: Step by Step&lt;/em&gt; published in Dec. 2020 by Kuldeep Singh&lt;/a&gt;. The author provides solutions &lt;a href=&quot;https://maths-for-all.co.uk/linear-algebra/&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt; to ALL exercises. And the author uses color.&lt;/p&gt;&#xA;" OwnerDisplayName="user15364" LastActivityDate="2021-02-01T19:33:30.720" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19433" PostTypeId="2" ParentId="14081" CreationDate="2021-02-02T19:16:37.543" Score="0" Body="&lt;p&gt;Warwick's old &lt;a href=&quot;https://homepages.warwick.ac.uk/%7Emaskal/past_teaching/MA246/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Number Theory&lt;/a&gt; course is short, taught via workbooks (so is intended for independent study), regularly taken by first-year students, and available online. The new version (&lt;a href=&quot;http://homepages.warwick.ac.uk/staff/J.E.Cremona/courses/MA257/ma257.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF&lt;/a&gt;) is also available online, but is perhaps a little less gentle, and doesn't have the same form.&lt;/p&gt;&#xA;" OwnerUserId="11771" LastActivityDate="2021-02-02T19:16:37.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19434" PostTypeId="1" CreationDate="2021-02-03T00:34:15.643" Score="25" ViewCount="6060" Body="&lt;p&gt;The cross product is an important vector operation in in any serious multivariable calculus course.  In most textbooks that I'm aware of, right after the definition, we always introduce the determinant formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ &#xA;  \mathbf{u} \times \mathbf{v} = &#xA;  \det \begin{bmatrix}&#xA;    \mathbf{i} &amp;amp; \mathbf{j} &amp;amp; \mathbf{k} \\&#xA;    u_1 &amp;amp; u_2 &amp;amp; u_3 \\&#xA;    v_1 &amp;amp; v_2 &amp;amp; v_3&#xA;  \end{bmatrix}&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\mathbf{i}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbf{j}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbf{k}$&lt;/span&gt; are the unit vectors in the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; directions (in a right-handed coordinate system).&lt;/p&gt;&#xA;&lt;p&gt;This is pretty standard, and it is how I learned this. The first time I taught this, however, I had to stop myself in the middle of the lecture, because as I was explaining this to students I realized that this makes no sense.  I must have realized the nonsensical nature of this equation long ago, but it was not until I had to explain this to someone else that I realized how bad this is.&lt;/p&gt;&#xA;&lt;p&gt;That was four years ago, and I have taught this course several times since then. Now I still introduce this equation just because it is in the textbook, but I try to be extra careful not to mislead students.&lt;/p&gt;&#xA;&lt;p&gt;I see very little value of teaching this equation:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It makes no sense. ([&lt;strong&gt;Update&lt;/strong&gt;] I regret not stating this explicitly and hence misled many answers. Here I'm saying &amp;quot;it makes no sense&amp;quot; in the same way that &lt;span class=&quot;math-container&quot;&gt;$1+1=0$&lt;/span&gt; makes no sense. To disprove this claim, simply state this equation as a statement without introducing any new terms or redefine any existing terms in a 2nd year college level multivariable calculus course)&lt;/li&gt;&#xA;&lt;li&gt;Most students in my class don't know how to compute 3x3 determinants, so for this to make sense, I have to teaching that, which is not trivial. ([&lt;strong&gt;Update&lt;/strong&gt;] This statement is not completely true. I'm sure a good percentage of students have learned determinants, but when they enter my class, they still won't be able to compute one unless I spend a lot of class time to teach them)&lt;/li&gt;&#xA;&lt;li&gt;In order to teach this right, we have to teaching mathematics wrong --- any student who feels comfortable writing down this formula either has no idea what a matrix is or can happily ignore the meanings behind mathematical symbols. In either case, he/she will not deserve a degree in mathematics.&lt;/li&gt;&#xA;&lt;li&gt;It only marginally makes computation faster (even that is arguable).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Are my views reasonable? If yes, why do almost every textbook include it?&lt;/p&gt;&#xA;&lt;p&gt;P.S., of course I understand we could perform some serious mathematical yoga and inject meaning into the determinant formula: E.g. by passing to &lt;span class=&quot;math-container&quot;&gt;$\Lambda^*(\mathbb{R}^3)$&lt;/span&gt;. But I consider that to be more of a rationalization after the fact.&lt;/p&gt;&#xA;&lt;p&gt;[&lt;strong&gt;Update&lt;/strong&gt;] I guess I should have stated a few things explicitly:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I'm certainly not questioning the value of formal notations/expressions in undergraduate level classes. I teach Gram-Schmidt determinants in senior level numerical analysis, which is also meaningless.  I have no issue with that, because most students at this level had some exposure to mathematical nonsense that are useful, and it is thus not hard to make this distinction. My question is specifically about multivariable calculus in first or second year in college. It is my observation that students in this stage have poor understanding for the distinction between purely formal and actual meaningful expressions (probably nonexistence). Well, either that, or I'm teaching in the wrong university.&lt;/li&gt;&#xA;&lt;li&gt;This formal expression clearly works and I guess anyone teaching such a course knows exactly what it means. It is nothing but a more compact way of writing down&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\mathbf{u} \times \mathbf{v} = &#xA;[(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{i}] \, \mathbf{i} +&#xA;[(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{j}] \, \mathbf{j} +&#xA;[(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{k}] \, \mathbf{k}$$&lt;/span&gt;&#xA;This is just how cofactor expansion works. I don't think we have any question there. The question is about whether or not teach this as the primary way of computation in 1st/2nd year level multivariable course.&lt;/li&gt;&#xA;&lt;li&gt;Some comments/answers say this is a useful way to memorize the cross product formula. Sure. But is it much better than the alternatives? What about just write down&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(u_1 \mathbf{i} + u_2 \mathbf{j} + u_3 \mathbf{k}) \times (v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k})$$&lt;/span&gt;&#xA;and just ask students to multiply them out? There is no additional information to memorize in this case. Or what about the &amp;quot;Xyzzy&amp;quot; mnemonic? That's even faster. So we are certainly not comparing with just the definition here.&lt;/li&gt;&#xA;&lt;li&gt;I also have small scale data collected from 3 colleges last year showing the difference in how quickly average students can computing cross product using the formal determinantal formula vs just the naive multiplication technique is basically unmeasurably small. (Hopefully this year we can get proper IRB approval and perform a slightly more scientific experiment)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user13395" LastEditorDisplayName="user13395" LastEditDate="2021-02-05T19:39:46.853" LastActivityDate="2024-03-17T08:02:53.727" Title="Why do we still teach the determinant formula for cross product? And is it as bad as I think it is?" Tags="|undergraduate-education|calculus|notation|vector-calculus|vectors|" AnswerCount="11" CommentCount="21" ContentLicense="CC BY-SA 4.0" />
  <row Id="19435" PostTypeId="2" ParentId="19434" CreationDate="2021-02-03T08:00:55.927" Score="13" Body="&lt;p&gt;Sure, give up on the mixed-object determinant formula. Instead use tensor arithmetic. Let &lt;span class=&quot;math-container&quot;&gt;$\epsilon_{ijk}$&lt;/span&gt; be the completely antisymmetric symbol. Also, while we're at it, let's give up on quaternionic notation for unit-vectors and instead use &lt;span class=&quot;math-container&quot;&gt;$\hat{x},\hat{y},\hat{z}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\hat{x_i}$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$i=1,2,3$&lt;/span&gt; then you have an elegant formula:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \vec{A} \times \vec{B} = \sum_{i,j,k} \epsilon_{ijk} A_iB_j \hat{x_k} $$&lt;/span&gt;&#xA;Moreover, you then have a completely legit platform to share the formula for the determinant:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ det(A) = \sum_{i_1,i_2,\dots, i_n} \epsilon_{i_1i_2\dots i_n}A_{1i_1}A_{2i_2}\cdots A_{ni_n} $$&lt;/span&gt;&#xA;I wish this was in standard textbooks. It would have been helpful to me at a certain point in my mathematical youth. Also, the apparent definition of the determinant in terms of Laplace's expansion by minors is truly horrible. Of course, eventually, if we think longer on this we will be driven to wedge products.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the hypocritical formula can be naturally interpreted to simply mean the formula I write here with the dreaded &lt;span class=&quot;math-container&quot;&gt;$\epsilon_{ijk}$&lt;/span&gt;-type notation. Furthermore, the determinant trick extends to higher dimensional orthogonal complements:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x=\sum_{i_1,\dots, i_n} \epsilon_{i_1i_2\dots i_n}(v_1)_{i_1}(v_2)_{i_2} \cdots (v_{n-1})_{i_{n-1}}\hat{x}_{i_n} \in \{ v_1,v_2, \dots , v_{n-1} \}^{\perp} $$&lt;/span&gt;&#xA;and &lt;span class=&quot;math-container&quot;&gt;$\| x \| = 1$&lt;/span&gt;. Most students would know what you meant if you expressed the above as (for the sake of brevity I'll just do &lt;span class=&quot;math-container&quot;&gt;$n=4$&lt;/span&gt;)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \text{det} \left[ \begin{array}{c} &#xA;\hat{x}_1 \ \hat{x}_2 \ \hat{x}_3 \ \hat{x}_4 \\&#xA;\vec{A} \\&#xA;\vec{B} \\&#xA;\vec{C}&#xA;\end{array}\right]$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\hat{x}_1 = \langle 1,0,0,0 \rangle$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec{A},\vec{B},\vec{C} \in \mathbb{R}^4$&lt;/span&gt;. For example, &lt;span class=&quot;math-container&quot;&gt;$\vec{A} = \langle 0,1,0,0 \rangle$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\vec{B} = \langle 0,0,1,0,\rangle$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec{C} = \langle 0,0,0,1 \rangle$&lt;/span&gt; gives us:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \text{det} \left[ \begin{array}{cccc} &#xA;\hat{x}_1 &amp;amp; \hat{x}_2 &amp;amp; \hat{x}_3 &amp;amp; \hat{x}_4 \\&#xA;0 &amp;amp; 1 &amp;amp; 0 &amp;amp; 0 \\&#xA;0 &amp;amp; 0 &amp;amp; 1 &amp;amp; 0 \\&#xA;0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 1 \\&#xA;\end{array}\right] = \hat{x}_1.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-02-03T08:00:55.927" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="19436" PostTypeId="2" ParentId="19434" CreationDate="2021-02-03T14:12:57.463" Score="7" Body="&lt;p&gt;I agree with the OP: the determinant is only a mnemonic device, but (a) students&#xA;may not know how to evaluate a determinant, so it's useless as a mnemonic, and&#xA;(b) it is not a &amp;quot;legal&amp;quot; determinant and so mathematicallly misleading.&lt;/p&gt;&#xA;&lt;p&gt;I would rather start with &lt;span class=&quot;math-container&quot;&gt;$|a \times b| = |a| |b| \sin \theta$&lt;/span&gt; as a contrast to &lt;span class=&quot;math-container&quot;&gt;$a \cdot b = |a| |b| \cos \theta$&lt;/span&gt;, and derive the coordinate formula (if you feel a derivation is needed) by expanding &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$a_x \hat{i} + a_y \hat{j} + a_z \hat{k}$&lt;/span&gt; and similarly for &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. Writing out the terms is a bit of a mess, but after simplifications such as &lt;span class=&quot;math-container&quot;&gt;$\hat{i} \times \hat{i} = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\hat{i} \times \hat{j} = \hat{k}$&lt;/span&gt;, one eventually reaches the coordinate formula. That this requires distributivity and anticommutativity is a opportunity to explain those properties.&lt;/p&gt;&#xA;&lt;p&gt;This derivation could be a homework assignment rather than part of the lecture.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="-1" LastEditDate="2021-02-04T12:08:52.043" LastActivityDate="2021-02-04T12:08:52.043" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19437" PostTypeId="2" ParentId="19434" CreationDate="2021-02-03T21:20:52.053" Score="4" Body="&lt;p&gt;As a preliminary, it's important to realize the incredibly low level of most students who are learning this stuff. Low means a low intellectual level and previous exposure to low-intellectual-level instruction, with low expectations for intellectual understanding. Low means that students who have previously been exposed to &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{i}}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{j}}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{k}}$&lt;/span&gt; find it an insuperable barrier to adapt to &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{x}}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{y}}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{z}}$&lt;/span&gt;. Low means that after encountering the dot and cross products twice (in first-semester physics and multivariable calculus), they still can't tell you under what circumstances the dot product is zero, or under what circumstances the cross product is zero.&lt;/p&gt;&#xA;&lt;p&gt;Yes, the presentation using a determinant is awful and should be nuked from orbit.&lt;/p&gt;&#xA;&lt;p&gt;A good way to go is the following. First we introduce the dot product and build intuition for it through interesting applications, such as computing work, or finding the angle between two vectors. Then we point out that the dot product is rotationally invariant, so that although we may compute it using components that depend on our choice of a coordinate system, the result doesn't actually depend on how we rotate the axes. This is easy to see because the dot product can be defined as &lt;span class=&quot;math-container&quot;&gt;$\textbf{A}\cdot\textbf{B}=|\textbf{A}||\textbf{B}|\cos\theta$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is the angle between the vectors.&lt;/p&gt;&#xA;&lt;p&gt;Next we provide some motivation for why we would want to multiply two vectors to get a vector. In a physics class, this could come in the context of angular momentum. We would like this definition of multiplication to be rotationally invariant, like the dot product. Based on these considerations, we clearly must have &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{x}}\times\hat{\textbf{x}}=0$&lt;/span&gt;, because otherwise there is no rotationally invariant way to decide what direction to pick for the result. By similar arguments, we determine that &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{x}}\times\hat{\textbf{y}}$&lt;/span&gt; must lie along the &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; axis. We fix it to be &lt;span class=&quot;math-container&quot;&gt;$\hat{\textbf{z}}$&lt;/span&gt;, which is arbitrary up to a nonzero multiplicative constant. Choosing the sign of this constant is arbitrary, and amounts to choosing a right-hand rule. We practice the right-hand rule on some examples.&lt;/p&gt;&#xA;&lt;p&gt;Finally, we build up a 3x3 multiplication table for cross products of the unit vectors in a particular xyz coordinate system. Any cross product we want to evaluate, given the two vectors in component form, can then be evaluated by using the distributive property (bilinearity).&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-02-03T21:20:52.053" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19438" PostTypeId="2" ParentId="19434" CreationDate="2021-02-03T22:48:09.933" Score="21" Body="&lt;p&gt;My answer is probably not very useful when teaching in high school. I'll just mention here a few reasons why this definition is in fact a good one, and why it's a good idea to teach this formula at a university level mathematics course.&#xA;&lt;span class=&quot;math-container&quot;&gt;$\newcommand{\u}{\mathbf{u}}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\newcommand{\v}{\mathbf{v}}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\newcommand{\w}{\mathbf{w}}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\newcommand{\R}{\mathbb{R}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let me rephrase it a bit. Choosing a positively oriented orthonormal basis &lt;span class=&quot;math-container&quot;&gt;$\mathbf{e_1},\mathbf{e_2},\mathbf{e_3}$&lt;/span&gt; (e.g., the standard basis) and expressing &lt;span class=&quot;math-container&quot;&gt;$\u = (u_1,u_2,u_3)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\v = (v_1,v_2,v_3)$&lt;/span&gt; in this basis, we define the cross product as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\mathbf{u} \times \mathbf{v} = &#xA;\det \begin{bmatrix}&#xA;u_1 &amp;amp; v_1 &amp;amp; \mathbf{e_1} \\&#xA;u_2 &amp;amp; v_2 &amp;amp; \mathbf{e_2} \\&#xA;u_3 &amp;amp; v_3 &amp;amp; \mathbf{e_3} &#xA;\end{bmatrix}.&#xA;$$&lt;/span&gt;&#xA;To make this definition rigorous, we should add one more thing - by the above determinant we actually mean the Laplace expansion with respect to the last column. One could think of it as an exercise in the spirit of &lt;a href=&quot;https://youtu.be/XFDM1ip5HdU&quot; rel=&quot;noreferrer&quot;&gt;this 3Blue1Brown video&lt;/a&gt; - in mathematics, one is often led to make sense of some nonsensical notions, and this is a good example of this. There's more than one way of formalizing this definition, and all sensible solutions are probably equivalent.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;There are certain properties that are easy to see with this definition. For any &lt;span class=&quot;math-container&quot;&gt;$\w \in \R^3$&lt;/span&gt; the dot product with &lt;span class=&quot;math-container&quot;&gt;$\u \times \v$&lt;/span&gt; can be computed as&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;(\u \times \v) \cdot \w &#xA;&amp;amp; = \left( \det \begin{bmatrix}&#xA;u_1 &amp;amp; v_1 &amp;amp; \mathbf{e_1} \\&#xA;u_2 &amp;amp; v_2 &amp;amp; \mathbf{e_2} \\&#xA;u_3 &amp;amp; v_3 &amp;amp; \mathbf{e_3} &#xA;\end{bmatrix} \right)&#xA;\cdot \w \\&#xA;&amp;amp; = \det \begin{bmatrix}&#xA;u_1 &amp;amp; v_1 &amp;amp; \mathbf{e_1} \cdot \w \\&#xA;u_2 &amp;amp; v_2 &amp;amp; \mathbf{e_2} \cdot \w \\&#xA;u_3 &amp;amp; v_3 &amp;amp; \mathbf{e_3} \cdot \w&#xA;\end{bmatrix} \\&#xA;&amp;amp; = \det \begin{bmatrix}&#xA;u_1 &amp;amp; v_1 &amp;amp; w_1 \\&#xA;u_2 &amp;amp; v_2 &amp;amp; w_2 \\&#xA;u_3 &amp;amp; v_3 &amp;amp; w_3&#xA;\end{bmatrix}.&#xA;\end{align*}&lt;/span&gt;&#xA;In fact, the above can serve as an alternative definition. With the geometric interpretation of determinant, it can be made into a coordinate-free definition (which I was taught as the basic definition at university). Moreover, some other things are evident:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$(\u \times \v) \cdot \u = 0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(\u \times \v) \cdot \v = 0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\u \times \v$&lt;/span&gt; is bilinear in &lt;span class=&quot;math-container&quot;&gt;$\u,\v$&lt;/span&gt;, and also alternating: &lt;span class=&quot;math-container&quot;&gt;$\v \times \u = - \u \times \v$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;if &lt;span class=&quot;math-container&quot;&gt;$\u$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\v$&lt;/span&gt; are linearly independent, then &lt;span class=&quot;math-container&quot;&gt;$\u \times \v$&lt;/span&gt; is nonzero, and &lt;span class=&quot;math-container&quot;&gt;$\u$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\v$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\u \times \v$&lt;/span&gt; form a positively oriented basis.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Another advantage of this definition is that it easily generalizes. There's nothing special in &lt;span class=&quot;math-container&quot;&gt;$\R^3$&lt;/span&gt; - a similar formula defines the cross product &lt;span class=&quot;math-container&quot;&gt;$\v_1 \times \ldots \times \v_{n-1}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt; vectors in an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-dimensional space (with dot product), with the same basic properties. The easiest example of this is in &lt;span class=&quot;math-container&quot;&gt;$\R^2$&lt;/span&gt;: the &lt;em&gt;cross product&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$\u = (u_1,u_2)$&lt;/span&gt; is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\det \begin{bmatrix}&#xA;u_1 &amp;amp; \mathbf{e_1} \\&#xA;u_2 &amp;amp; \mathbf{e_2} &#xA;\end{bmatrix} &#xA;= (-u_2,u_1).&#xA;$$&lt;/span&gt;&#xA;This is exactly the vector &lt;span class=&quot;math-container&quot;&gt;$\u$&lt;/span&gt; rotated by &lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt;, as one may expect.&lt;/p&gt;&#xA;" OwnerUserId="15367" LastEditorUserId="15367" LastEditDate="2021-02-03T23:33:22.383" LastActivityDate="2021-02-03T23:33:22.383" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19439" PostTypeId="2" ParentId="19434" CreationDate="2021-02-04T04:34:43.697" Score="16" Body="&lt;p&gt;I think you're correct, strictly speaking, but overreacting. I just checked two calculus books, Rogawski and Strang. Rogawski defines the cross product as the &amp;quot;symbolic&amp;quot; determinant (his words) in question but then defines exactly what he means by this determinant in the same line, so I don't see a problem there. Strang mentions the determinant mnemonic later in his presentation and claims that it's &amp;quot;probably illegal...but it works&amp;quot;. Again, that seems reasonable to me (and humorous, as Strang often is).&lt;/p&gt;&#xA;&lt;p&gt;I figure that anyone who hasn't seen 3x3 determinants before won't know enough to even be confused in the first place, while anyone who has can easily see that it's just a formal determinant that spits out whatever &amp;quot;official&amp;quot; definition they've already seen.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2021-02-04T04:34:43.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19440" PostTypeId="2" ParentId="19434" CreationDate="2021-02-04T13:08:07.680" Score="11" Body="&lt;p&gt;I had a lecturer who thought the same as you, so suggested we just learnt the definition of the cross product (which he derived along the same lines as James S Cook's answer). Nobody liked his lectures because he pitched them way too high for our first-year brains.&lt;/p&gt;&#xA;&lt;p&gt;The point is, it's a very useful mnemonic device, and when you teach it, make that clear, but make sure you teach it, because nobody will like you if you don't.&lt;/p&gt;&#xA;" OwnerUserId="15406" LastEditorUserId="15406" LastEditDate="2023-03-04T12:58:49.333" LastActivityDate="2023-03-04T12:58:49.333" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19441" PostTypeId="1" CreationDate="2021-02-04T20:54:45.960" Score="0" ViewCount="82" Body="&lt;p&gt;I am looking for a book or any resources related to this book. It is called Pattern and Structure Mathematics Awareness Program (there is a book one and book two).&lt;/p&gt;&#xA;&lt;p&gt;I have tried to order it but it is out of stock in the three places that I ordered from.&lt;/p&gt;&#xA;&lt;p&gt;Can anyone shed light on where I can get this from?&lt;/p&gt;&#xA;&lt;p&gt;Thanks and have a good one!&lt;/p&gt;&#xA;" OwnerUserId="14905" LastEditorUserId="376" LastEditDate="2021-02-05T08:54:52.573" LastActivityDate="2021-02-05T08:54:52.573" Title="Pattern and Structure Mathematics Awareness Program (pasmap)" Tags="|reference-books|resource-request|book|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19442" PostTypeId="2" ParentId="19434" CreationDate="2021-02-04T23:42:01.510" Score="4" Body="&lt;p&gt;&lt;strong&gt;Just go with the Cartesian formula.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The first time I saw a cross product was in computer code and I always felt like the typical way it's coded is the easiest way to compute &lt;span class=&quot;math-container&quot;&gt;$\mathbf{c} = \textbf{a} \times \mathbf{b}$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$c_x = a_y b_z - a_z b_y$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$c_y = a_z b_x - a_x b_z$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$c_z = a_x b_y - a_y b_x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The pattern is obvious. Each component of &lt;span class=&quot;math-container&quot;&gt;$\mathbf{c}$&lt;/span&gt; is the difference between the products of the other two different-direction components of &lt;span class=&quot;math-container&quot;&gt;$\mathbf{a}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{b}$&lt;/span&gt;. The only tricky part is which product gets the minus sign. You just have to remember that the component of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; (or the first multiplicand) in the first term is the next letter in the xyz series from the component of the cross product. &lt;span class=&quot;math-container&quot;&gt;$c_x$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$a_y$&lt;/span&gt; in the first term. &lt;span class=&quot;math-container&quot;&gt;$c_y$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$a_z$&lt;/span&gt; in the first term. For &lt;span class=&quot;math-container&quot;&gt;$c_z$&lt;/span&gt;, we hit the end of series and start back at &lt;span class=&quot;math-container&quot;&gt;$a_x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;No need to think about matrices or tensors.&lt;/p&gt;&#xA;&lt;p&gt;You can also pretty easily see the properties of the cross product from this form. It's pretty clear that &lt;span class=&quot;math-container&quot;&gt;$\textbf{a} \times \mathbf{a} = 0$&lt;/span&gt;, since &lt;span class=&quot;math-container&quot;&gt;$a_i a_j = a_j a_i$&lt;/span&gt;. It's also pretty clear that &lt;span class=&quot;math-container&quot;&gt;$\textbf{b} \times \mathbf{a} = -\textbf{a} \times \mathbf{b}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12740" LastActivityDate="2021-02-04T23:42:01.510" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19443" PostTypeId="2" ParentId="19441" CreationDate="2021-02-05T03:08:57.327" Score="2" Body="&lt;p&gt;I would borrow it from a library. Use InterLibrary Loan (ILL) if in the US.  This means you can get the volumes from the distant library sent to yours.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.worldcat.org/search?q=Pattern+and+Structure+Mathematics+Awareness+Program&amp;amp;qt=owc_search&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.worldcat.org/search?q=Pattern+and+Structure+Mathematics+Awareness+Program&amp;amp;qt=owc_search&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15413" LastActivityDate="2021-02-05T03:08:57.327" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19445" PostTypeId="2" ParentId="19434" CreationDate="2021-02-05T16:08:05.487" Score="8" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;this makes no sense&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;But it does.&lt;/strong&gt; A determinant exists for any &lt;span class=&quot;math-container&quot;&gt;$n\times n$&lt;/span&gt; matrix for which the Leibniz formula&lt;span class=&quot;math-container&quot;&gt;$$\det A:=\sum_{\sigma\in S_n}\epsilon_\sigma\prod_{i=1}^nA_{i\sigma_i}$$&lt;/span&gt;is fully antisymmetric. This is why, for example, &lt;a href=&quot;https://en.wikipedia.org/wiki/Quaternionic_matrix#Determinants&quot; rel=&quot;nofollow noreferrer&quot;&gt;quaternionic matrices lack determinants&lt;/a&gt;: their elements don't commute. But if we apply it to the case at hand, we don't have that problem (as long as the entries of &lt;span class=&quot;math-container&quot;&gt;$u,\,v$&lt;/span&gt; are &lt;a href=&quot;https://en.wikipedia.org/wiki/C-number&quot; rel=&quot;nofollow noreferrer&quot;&gt;c-numbers&lt;/a&gt;), because vectors commute with scalars. The Gram-Schmidt determinants you've also taught admit the same defence.&lt;/p&gt;&#xA;&lt;p&gt;If you object to this, perhaps you have a narrower definition of when determinants are defined. From such a perspective, my definition is not &amp;quot;meaningless&amp;quot;, it's a different meaning. But we typically go with the most general workable definition in mathematics, as it pays dividends. Nor is being comfortable writing it evidence of misunderstanding or ignoring certain things. At &lt;em&gt;worst&lt;/em&gt;, it's an &lt;a href=&quot;https://en.wikipedia.org/wiki/Abuse_of_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;abuse of notation&lt;/a&gt;, which is not an insult; it's a reference to utility.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It only marginally makes computation faster... But is it much better than the alternatives [for memorization]?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The main reason to learn this expression is to understand it, not to compute with it or have a mnemonic. Its geometric significance has been discussed in multiple answers already, so I won't repeat those points. What I'd add is it facilitates a &lt;a href=&quot;https://en.wikipedia.org/wiki/Sanity_check&quot; rel=&quot;nofollow noreferrer&quot;&gt;sanity check&lt;/a&gt; of such details as the cross product's antisymmetry, the scalar triple product's full antisymmetry, and each being multilinear.&lt;/p&gt;&#xA;" OwnerUserId="5794" LastEditorUserId="5794" LastEditDate="2024-03-17T08:02:53.727" LastActivityDate="2024-03-17T08:02:53.727" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19446" PostTypeId="2" ParentId="19434" CreationDate="2021-02-05T18:58:31.697" Score="1" Body="&lt;p&gt;The question of whether formal relationships &amp;quot;make sense&amp;quot; is an old one in mathematics. People asked the question about whether negative numbers &amp;quot;made sense&amp;quot; (how can you start with three pebbles and take away six of them?) or complex numbers (what number when squared gives minus 1?). But we found that putting them in algebraic expressions &lt;em&gt;as if they were sensible numbers&lt;/em&gt; proved useful, and so eventually they got accepted.&lt;/p&gt;&#xA;&lt;p&gt;A matrix may be initially defined as a grid of numbers - integer, real, complex - and that makes sense. We have some intuitive idea of what it means. But formally we can put other objects inside a matrix, and so long as we have some operations analogous to addition and multiplication (i.e. the objects are elements of a ring or something similar), we can reproduce a lot of the matrix theory we've already built up for reals in the new arena. So integers modulo k, or quaternions, or matrices could just as easily be used. If the new entity doesn't have quite the same properties as the reals or integers, not all the theory gets reproduced. Matrices of integers often don't have inverses when the corresponding matrix of reals does. Real and complex matrices have significant differences in the theories that arise, so you do have to be careful with this sort of generalisation. It is important to emphasise to students that this usage isn't standard, and you can't apply any real-number matrix theory you know without going back to basics. But that doesn't make it invalid, and it is frequently the case that what are initially merely formal relationships and parallels can reflect deeper relationships. (As with the progression from natural numbers to integers to rationals to reals to complex numbers...)&lt;/p&gt;&#xA;&lt;p&gt;I'm guessing that the reason you consider the formula to &amp;quot;make no sense&amp;quot; is that it mixes vectors and scalars in the same algebra, and while the subset of operations needed for a determinant are allowed (addition of vectors, multiplication of a vector by a scalar), there are other operations that are not (multiplication of two vectors, addition of a vector to a scalar). There is in fact a more general algebra where these operations &lt;em&gt;are&lt;/em&gt; allowed: Geometric Algebra, (aka Clifford algebra) &lt;em&gt;does&lt;/em&gt; allow these operations. It generalises vector algebra to incorporate scalars, vectors, bivectors, pseudovectors, spinors, complex numbers, and quaternions. You can add a vector to a scalar, in the same sort of way as adding a real number to an imaginary one, and the multiplication of vectors in Geometric Algebra incorporates both the dot product and cross product of vector algebra into a single, usually invertible operation.&lt;/p&gt;&#xA;&lt;p&gt;If instead of defining matrices over the reals, you define them over the multivectors of geometric algebra, the determinant now makes algebraic sense. Whether it helps intuitively is another matter. Geometric Algebra has a far simpler way of expressing cross products: as the dual to the antisymmetric part of the product. In 3D geometric algebra &lt;span class=&quot;math-container&quot;&gt;$ab=(ab+ba)/2+(ab-ba)/2=a\cdot b+a\wedge b=a\cdot b+I(a \times b)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is the unit trivector and acts like an imaginary unit. This dual is only needed to turn the valid-in-all-dimensions bivector &lt;span class=&quot;math-container&quot;&gt;$(ab-ba)/2=a\wedge b$&lt;/span&gt; (which vector algebra can't cope with) into a pseudovector - something looking a lot like a vector but which has different transformation properties under reflection.&lt;/p&gt;&#xA;&lt;p&gt;The determinant formula probably reflects some deeper geometric relationship in Geometric Algebra, but I agree that exploring something like that is not suitable for a beginner/introductory class. But as for &amp;quot;not making sense&amp;quot;, there are plenty of things in vector algebra that don't make sense. Why can dot products be done in any dimension but cross products only work in three dimensions? (Or non-uniquely and without some of the most useful properties, seven.) Why are there two different products, both of them producing objects of a different type? (Dot product produces scalars, cross product produces pseudovectors.) Why can't you invert vector multiplication, as with real, rational, complex, and matrix multiplication, to get a 'vector division' operation? What is a pseudovector anyway? What do you get if you add a vector to a pseudovector? Why are the products non-associative? [&lt;span class=&quot;math-container&quot;&gt;$(a\cdot b)\cdot c$&lt;/span&gt; is undefined, &lt;span class=&quot;math-container&quot;&gt;$(a\times b)\times c\neq a\times (b\times c)$&lt;/span&gt;.] You have entities to represent directed length (i.e. vectors) but not directed areas or volumes. You have entities to represent translations but not the other isometries like rotations, reflections, screws, etc. Translation is the limit of rotation as the centre-of-rotation is moved towards infinity, so why aren't they represented by the same type of object? Angular velocity - treated as a vector - can't be integrated directly to yield angular displacement. Angular velocities add like vectors, but angular displacements (i.e. rotations) don't. It's messy and ugly. But as you say, most teachers don't notice, as vector algebra is now the conventional way it has 'always' been taught.&lt;/p&gt;&#xA;" OwnerUserId="15421" LastActivityDate="2021-02-05T18:58:31.697" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19447" PostTypeId="2" ParentId="19434" CreationDate="2021-02-06T21:42:18.377" Score="1" Body="&lt;p&gt;In my opinion &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; matrices, particularly &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; real matrices are fairly easy to understand as mathematical objects, even to folks who haven't been exposed to linear algebra. &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; real matrices can be motivated by linear functions in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt;, (equivalently) as scaled rotations of the plane that fix the origin, as a convenient way to represent complex numbers, or not motivated at all.&lt;/p&gt;&#xA;&lt;p&gt;The determinant formula for &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; matrices specifically is compact, easy to remember, and can be justified if necessary by its geometric interpretation as the signed area of a parallelogram.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \left| \begin{matrix} x &amp;amp; y \\ z &amp;amp; w \end{matrix} \right| = xw - yz $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I will introduce the nonstandard notion of a &lt;em&gt;cyclic index&lt;/em&gt;. If &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a vector in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x_{(0)}$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$x_{3}$&lt;/span&gt; and is &lt;span class=&quot;math-container&quot;&gt;$x_{(100)}$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_{(2)}$&lt;/span&gt; is trivially &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt;. The componentwise entries of the cross product can be written as follows.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ s = a \times b $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ s_{(i+0)} = \left| \begin{matrix} a_{(i+1)} &amp;amp; b_{(i+1)} \\ a_{(i+2)} &amp;amp; b_{(i+2)} \end{matrix} \right| $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;By defining the &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;th component of the result vector &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; in terms of the &lt;em&gt;next&lt;/em&gt; dimension mod &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$i+1$&lt;/span&gt; symbolically) and the &lt;em&gt;next next&lt;/em&gt; dimension mod &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$i+2$&lt;/span&gt; symbolically), we can remove the need to remember whether the determinant associated with &lt;span class=&quot;math-container&quot;&gt;$\hat{i}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\hat{j}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\hat{k}$&lt;/span&gt; is supposed to be negated or not.&lt;/p&gt;&#xA;" OwnerUserId="13521" LastEditorUserId="13521" LastEditDate="2021-02-07T04:47:02.177" LastActivityDate="2021-02-07T04:47:02.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19448" PostTypeId="1" CreationDate="2021-02-07T07:49:11.670" Score="6" ViewCount="78" Body="&lt;p&gt;I plan to work with my students on solving real-world problems through modelling them. Now it is my idea to follow the modelling cycle below.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5DAMv.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5DAMv.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The idea is to find with the help of two values that have been investigated by an institute for market research the point of maximum profit located on the inverse demand function. This function takes a certain quantity of a good and maps to it the price per unit the vendor can take.&lt;/p&gt;&#xA;&lt;p&gt;So far for the background. Do you see a way to differentiate working on this topic except of beeing open in which ways of modelling are accepted as adequate?&lt;/p&gt;&#xA;" OwnerUserId="10036" LastEditorUserId="376" LastEditDate="2021-02-07T16:51:41.327" LastActivityDate="2021-02-07T16:51:41.327" Title="Support modelling cycle through differentiated means" Tags="|secondary-education|applied-mathematics|applications|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19450" PostTypeId="2" ParentId="19434" CreationDate="2021-02-07T22:16:41.727" Score="1" Body="&lt;p&gt;It surprises me that nobody has brought up the &lt;a href=&quot;https://en.wikipedia.org/wiki/Rule_of_Sarrus&quot; rel=&quot;nofollow noreferrer&quot;&gt;rule of Sarrus&lt;/a&gt; yet:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For &lt;span class=&quot;math-container&quot;&gt;$\mathbf{u} \times \mathbf{v}$&lt;/span&gt;, make the table &lt;span class=&quot;math-container&quot;&gt;$\begin{array}{|ccc|cc} \mathbf{i}&amp;amp; \mathbf{j}&amp;amp; \mathbf{k}&amp;amp; \mathbf{i}&amp;amp; \mathbf{j} \\ u_1 &amp;amp; u_2 &amp;amp; u_3 &amp;amp; u_1 &amp;amp; u_2 \\ v_1 &amp;amp; v_2 &amp;amp; v_3 &amp;amp; v_1 &amp;amp; v_2 \end{array}$&lt;/span&gt; , then sum up products of diagonals (with signs, as usual).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(If we want Wikipedia to rather link to the cross product, there is also &lt;a href=&quot;https://en.wikipedia.org/wiki/Sarrus%27s_scheme&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sarrus's scheme&lt;/a&gt;, but what is presented there is different enough to not be immediately recognisable as the same as the determinant rule; I suspect they may have been more similar, but don't care enough to go examine the page history.)&lt;/p&gt;&#xA;&lt;p&gt;Using the rule of Sarrus has a number of advantages.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It is clearly a memnonic device, rather than something pretending to be a formula, so there is no problem with the rigor.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is &lt;em&gt;usable&lt;/em&gt; even for students who have not learnt to compute determinants in a previous course (and also for those who did learn that, but have now forgotten the details).&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Depending on what is covered in your multivariate calculus course, you may of course need to do determinants anyway to perform change of variables in multiple integrals, but that is far more likely to get deferred to a more advanced course than the cross product is. Even when you cover change of variables in multiple integrals, it tends to be dominated by the standard cartesian/polar/spherical changes where people anyway look up the factor in a table.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For those students who care, you can explain that the reason the &lt;em&gt;textbook&lt;/em&gt; instead writes that formula with the determinant is that &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; determinants can be calculated using the exact same rule, which have tempted some authors to bend the interpretations so that they could present the cross product as a determinant. Then point out that a determinant which could be applied to a matrix mixing scalar and vector entries must in general also cope with products of two vectors — in particular it must be known how to evaluate such products before one starts using it to evaluate that determinant…&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It's of course not impossible to define such a determinant. The easiest approach is probably to say that the matrix elements are quaternions, as it is perfectly possible to define &lt;span class=&quot;math-container&quot;&gt;$\det\colon \mathbb{H}^{n \times n} \longrightarrow \mathbb{H}$&lt;/span&gt; (there's even a method for approximating the &lt;a href=&quot;https://en.wikipedia.org/wiki/Permanent_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;permanent&lt;/a&gt; which makes use of such determinants), but then care is needed in the definition and algebraic manipulation — do &lt;em&gt;you&lt;/em&gt; know which of your favourite determinant properties were derived under the assumption that element multiplication is commutative? Probably not. Even something as basic as &lt;span class=&quot;math-container&quot;&gt;$\begin{vmatrix} 0 &amp;amp; \mathbf{i} \\ \mathbf{j} &amp;amp; 0 \end{vmatrix}$&lt;/span&gt; ends up having a different value if elements are multiplied left-to-right than it does if they're multiplied top-to-bottom.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It might feel as if not using a determinant in the definition weakens the link to the triple product formula&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \mathbf{u} \mathbin{\pmb{\cdot}} (\mathbf{v} \times \mathbf{w}) = \begin{vmatrix} u_1 &amp;amp; u_2 &amp;amp; u_3 \\ v_1 &amp;amp; v_2 &amp;amp; v_3 \\ w_1 &amp;amp; w_2 &amp;amp; w_3 \end{vmatrix} $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;but getting to this from the nonsense determinant formula for the cross product &lt;span class=&quot;math-container&quot;&gt;$\mathbf{v} \times \mathbf{w}$&lt;/span&gt; still requires doing and undoing cofactor expandsion along the first row, so the proof length remains the same for a definition using the rule of Sarrus.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Mind you, taking the triple product formula as definition of the cross product provides easy routes not only to getting explicit expressions for the elements of the cross product (just let &lt;span class=&quot;math-container&quot;&gt;$\mathbf{u}$&lt;/span&gt; range over the vectors in the standard basis), but also for identifying &lt;span class=&quot;math-container&quot;&gt;$\Vert \mathbf{v} \times \mathbf{w} \Vert$&lt;/span&gt; as the area of the parallelogram spanned by &lt;span class=&quot;math-container&quot;&gt;$\mathbf{v}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{w}$&lt;/span&gt; (which in turn leads to the &lt;span class=&quot;math-container&quot;&gt;$\left\Vert \mathbf{v} \right\Vert \left\Vert \mathbf{w} \right\Vert \sin\theta$&lt;/span&gt; formula for &lt;span class=&quot;math-container&quot;&gt;$\Vert \mathbf{v} \times \mathbf{w} \Vert$&lt;/span&gt;) as a consequence of that determinant being the (signed) volume of the parallelepiped spanned by &lt;span class=&quot;math-container&quot;&gt;$\mathbf{u}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbf{v}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{w}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's probably also worth pointing out that while this definition caters for the cross product as the &lt;span class=&quot;math-container&quot;&gt;$n=3$&lt;/span&gt; instance of the &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;-valued &lt;span class=&quot;math-container&quot;&gt;$(n{-}1)$&lt;/span&gt;-ary product of vectors in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;, it does not so much cater for the cross product as being the &lt;span class=&quot;math-container&quot;&gt;$n=3$&lt;/span&gt; exterior product of two &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-forms or bracket in &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{so}(n)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Still, even if being notationally rigorous in the definition of the cross product, teaching a multivariate calculus course probably means that you will also have to address the even larger degree of symbolic interpretation that is needed to write the curl as &lt;span class=&quot;math-container&quot;&gt;$\nabla \times \mathbf{F}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14052" LastActivityDate="2021-02-07T22:16:41.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19451" PostTypeId="1" AcceptedAnswerId="19453" CreationDate="2021-02-07T23:15:24.633" Score="5" ViewCount="251" Body="&lt;p&gt;I'm planning to teach a series of lessons on expanding and factoring binomials using a certain visual method for my edTPA assessment. Each lesson is pretty much going to be structured the same way in that the focus will be on providing as many practice problems as time allows (with some intermittent breaks to reduce cognitive demand and students can ask questions if they need clarification on something).&lt;/p&gt;&#xA;&lt;p&gt;Rather than directly stating the concepts and procedures, I will reveal these by providing examples for the students to work on during the lesson. I envision giving the students &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; problems at a time and then we will discuss the solutions as a class, pointing out key concepts that relate to the lesson goals (i.e. multiplying two binomials is like multiplying the length and width of a rectangle to get the area). The exercises will get slightly more advanced as the lesson progresses (going from working with whole numbers, such as &lt;span class=&quot;math-container&quot;&gt;$2(3+4)$&lt;/span&gt;, to polynomials like &lt;span class=&quot;math-container&quot;&gt;$2x(3x+4)$&lt;/span&gt;. Students have already learned how to do simplify this expression using the Distributive Property. In this lesson they will use the &lt;a href=&quot;http://youtube.com/watch?v=DnAIhwJw5aY&quot; rel=&quot;nofollow noreferrer&quot;&gt;Box Method&lt;/a&gt; instead.&lt;/p&gt;&#xA;&lt;p&gt;This approach embraces my philosophy that students learn math by doing math, but what is the exact name of this teaching style? Has there been any research on its advantages/disadvantages?&lt;/p&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="13657" LastEditDate="2021-02-10T22:30:28.917" LastActivityDate="2021-02-11T05:49:18.727" Title="What is the name of this teaching style?" Tags="|teaching|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19452" PostTypeId="1" CreationDate="2021-02-08T00:12:19.230" Score="5" ViewCount="182" Body="&lt;p&gt;Robert  Ash (1935-2015) touts in the &lt;a href=&quot;https://faculty.math.illinois.edu/%7Er-ash/RV/RVPreface.pdf&quot; rel=&quot;noreferrer&quot;&gt;Preface&lt;/a&gt; to &lt;a href=&quot;https://faculty.math.illinois.edu/%7Er-ash/RV.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Real Variables with Basic Metric Space Topology&lt;/em&gt;&lt;/a&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I rely especially on one of the most useful of all learning devices: the inclusion of detailed solutions to exercises. Solutions to problems are commonplace in ele-mentary texts but quite rare (although equally valuable) at the upper division undergraduate and graduate level. This feature makes the&#xA;book suitable for independent study, and further widens the audience.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Fortunately Ash publishes solutions to all exercises in his textbooks. Michael Spivak does for &lt;em&gt;Calculus&lt;/em&gt;. Who or what factors decide whether all exercises will have solutions published?&lt;/p&gt;&#xA;" OwnerDisplayName="user15364" LastActivityDate="2021-02-08T17:22:24.937" Title="What determines whether a textbook publishes all solutions to exercises?" Tags="|textbooks|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19453" PostTypeId="2" ParentId="19451" CreationDate="2021-02-08T03:00:35.967" Score="2" Body="&lt;p&gt;I think the spiral approach describes the part of the lesson design that you are focusing on here.  I won't deny you the opportunity to practice with Google Scholar, but you should have no trouble finding research exploring both its strengths and weaknesses in mathematics education.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-02-08T03:00:35.967" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19454" PostTypeId="2" ParentId="19452" CreationDate="2021-02-08T12:35:16.233" Score="2" Body="&lt;p&gt;I can see two things here.&lt;/p&gt;&#xA;&lt;p&gt;(1) As the quote remarks, including solutions makes the book better suited for independent study.&lt;/p&gt;&#xA;&lt;p&gt;(2) Including solutions makes the book less suited for unmotivated students.  For example, students taking the class because it is required, but trying as hard as they can to do as little work as possible.  Does anyone argue that copying solutions from the book (or from the internet, etc.) leads to less learning than actually thinking about the problems?&lt;/p&gt;&#xA;&lt;p&gt;So I think there is an argument for not including solutions, or including only selected solutions, in a textbook for use in a course with a substantial number of unmotivated students.  As some of us know, many lower-level math courses are of this type.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2021-02-08T12:35:16.233" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19455" PostTypeId="2" ParentId="19452" CreationDate="2021-02-08T16:11:01.660" Score="2" Body="&lt;p&gt;I don't think the answer is the same for lower-division texts, upper-division texts, and graduate texts. The situation isn't the same for all of these. Therefore in this answer, I'll focus only on lower-division texts, which are what I know about. (I teach at a community college.)&lt;/p&gt;&#xA;&lt;p&gt;For this type of class, systems like mymathlab are the publisher's way of maintaining profits when students can buy used books or download digital copies for free. They sell mymathlab to instructors as a way of not having to hand-grade homework. If solutions were in the book, or legally available, students would put the answers in mymathlab and get 100%, which would break this system. Chegg also breaks the system, but it gives plausible deniability.&lt;/p&gt;&#xA;&lt;p&gt;Before the internet, there was a concern that less motivated students would not have the intellectual maturity to solve the problems themselves if the full solution was in the book. This no longer makes sense, because such students have other options. For a computation-based math class such as freshman calculus, they can use software to do their problems. For word problems or mathematical problems in other subjects like physics and chemistry, they will pay for a chegg membership, which is a one-stop cheating solution for all their classes. It is true, however, that many instructors are in denial about these technological realities and don't want to change their practices, so they would &lt;em&gt;perceive&lt;/em&gt; this to be a problem with openly publishing solutions.&lt;/p&gt;&#xA;&lt;p&gt;In a comment, Joseph O'Rourke says he thinks this has to do with the cost of preparing solutions and with page costs.&lt;/p&gt;&#xA;&lt;p&gt;I doubt that the cost of preparing solutions is a major factor. People have generally written up solutions for their own problems as they use their own book, before it gets published (or they may get a grad student to write solutions). There is the cost of putting these in a nicer-looking format, but I think for this type of lower-division text, that has to happen anyway, because people looking to adopt the book want to know whether it has an instructor's solutions manual.&lt;/p&gt;&#xA;&lt;p&gt;Re production costs, the cost of physically producing paper textbooks stopped being a really significant fraction of their retail price about 20-30 years ago. There has been exponential growth in prices while production costs for paper books have stayed about the same. And in any case, within the last 5 years or so I've seen about 80-90% of STEM students shift to digital books. Often they rent them. They don't want to carry around a backpack with 30 pounds of textbooks inside, and their reading habits are digital. And in any case, the solutions don't have to be included in the printed object. They can just be in a PDF file. Publishers have actually given up at this point on making money on lower-division STEM books. Their revenue stream for these courses is now from software like mymathlab and Mastering Physics.&lt;/p&gt;&#xA;&lt;p&gt;In summary, it's not clear to me that there is a positive net value for anyone in making solutions to a lower-division STEM textbook openly available. Students have access to solutions anyway, so why bother? Having access to solutions is also probably a net negative educationally, since most students use them in an immature way. Making them openly available would increase the publisher's development costs slightly, and they can't sell the product because students would just redistribute it on the internet. It would save students the $15/month and moral degradation of using chegg, but the publishers don't care about these things.&lt;/p&gt;&#xA;&lt;p&gt;More broadly, I think this is a situation where technological change has brought about unforseen consequences, and created a system that is dysfunctional and yet still profitable for the publishers. The system is broken and needs to be fixed, but providing solutions openly won't fix what's wrong and will also destroy the publishers' revenue stream (because it would eliminate any remaining justification for systems like mymathlab).&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-02-08T17:22:24.937" LastActivityDate="2021-02-08T17:22:24.937" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19456" PostTypeId="1" AcceptedAnswerId="19457" CreationDate="2021-02-09T01:21:33.010" Score="7" ViewCount="277" Body="&lt;p&gt;I have been tutoring a wide range of math subjects for many years. Recently, I began tutoring a girl in high school geometry (in California, for context). This semester of the course is starting with a pretty standard introduction to geometric proof writing. Her class has learned several relevant concepts including the three properties of equivalence relations (reflexive, symmetric, and transitive), and they are now beginning the section on the triangle congruence theorems (Side-Side-Side, Side-Angle-Side, etc.).&lt;/p&gt;&#xA;&lt;p&gt;During proof-writing exercises, the teacher has been telling students that statements such as &lt;span class=&quot;math-container&quot;&gt;$\overline{AB} \cong \overline{BA}$&lt;/span&gt; are instances of the &lt;strong&gt;symmetric&lt;/strong&gt; property of congruence. I was taught that congruence was an equivalence relation between geometric objects, and therefore, the way that a geometric object is labelled is irrelevant (at least for geometric objects that have no directionality - like line segments). Additionally, I consider the symmetric property to be a conditional statement that requires a premise (e.g. &lt;span class=&quot;math-container&quot;&gt;$\overline{AB} \cong \overline{CD}$&lt;/span&gt;) to then draw a conclusion (viz. &lt;span class=&quot;math-container&quot;&gt;$\overline{CD} \cong \overline{AB}$&lt;/span&gt;). As a result, I would consider the statement &lt;span class=&quot;math-container&quot;&gt;$\overline{AB} \cong \overline{BA}$&lt;/span&gt; to be an instance of the reflexive property because (a) it relates a geometric object to itself and (b) it is a simple statement rather than a conditional.&lt;/p&gt;&#xA;&lt;p&gt;I approached the teacher about this since I was certain it must have been a misunderstanding by my student, but the teacher insisted that only statements like &lt;span class=&quot;math-container&quot;&gt;$\overline{AB} \cong \overline{AB}$&lt;/span&gt; should be considered reflexive. I understand where this idea might come from since statements like &lt;span class=&quot;math-container&quot;&gt;$\triangle{ABC} \cong \triangle{BCA}$&lt;/span&gt; are not true in general. However, such triangle congruences have an implied ordering or directionality that does not exist for a line segment.&lt;/p&gt;&#xA;&lt;p&gt;Obviously, I will tutor my student to follow her teacher's instructions (even though I disagree with them), but it makes me curious: &lt;strong&gt;Is this the standard way in which the symmetric property of congruence is taught in the US?&lt;/strong&gt; If not, would this be an example of the reflexive property or some other property?&lt;/p&gt;&#xA;" OwnerUserId="15436" LastEditorUserId="2083" LastEditDate="2021-02-09T11:30:08.203" LastActivityDate="2021-02-09T20:39:05.493" Title="Is $\overline{AB} \cong \overline{BA}$ usually taught as an instance of the symmetric property of congruence?" Tags="|secondary-education|geometry|proofs|usa|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19457" PostTypeId="2" ParentId="19456" CreationDate="2021-02-09T13:44:47.427" Score="4" Body="&lt;p&gt;For me, this is more of a foundational issue than one of secondary education. I would suggest that &lt;span class=&quot;math-container&quot;&gt;$\overline {AB}=\overline {BA}$&lt;/span&gt; is a matter of equality rather than congruence, given the definition of line segment as the locus of points between &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;.  (Strictly speaking, I suppose, it is the locus of points &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$AC+BC=AB$&lt;/span&gt;.)  Based on this, it would follow that &lt;span class=&quot;math-container&quot;&gt;$\overline {AB}\cong\overline {BA}$&lt;/span&gt; follows from the reflexive principle since they are the same line segment.&lt;/p&gt;&#xA;&lt;p&gt;I can appreciate the teacher's point to a degree.  For instance, if you are showing that &lt;span class=&quot;math-container&quot;&gt;$\triangle ABC\cong\triangle DCB$&lt;/span&gt;, you could make an argument that the reflexive property statement should be &lt;span class=&quot;math-container&quot;&gt;$\overline{BC}\cong\overline{CB}$&lt;/span&gt; even if you were going to go on to justify it with the reflexive property.  But I've never known anyone before who used or demanded that level of pedantry outside of maybe computer-based proof checkers.&lt;/p&gt;&#xA;&lt;p&gt;That said, justifying that statement by the symmetric property of congruence is not valid.  The symmetric property of congruence states that if &lt;span class=&quot;math-container&quot;&gt;$X\cong Y$&lt;/span&gt;, then you can infer &lt;span class=&quot;math-container&quot;&gt;$Y\cong X$&lt;/span&gt;.  The statement that two line segments are symmetric to one another is so nonsensical that &lt;a href=&quot;https://en.wikipedia.org/wiki/Not_even_wrong&quot; rel=&quot;nofollow noreferrer&quot;&gt;it isn't even wrong&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(Disclaimer: I studied and teach in New York state, so I don't know about California.  &lt;a href=&quot;https://matheducators.stackexchange.com/a/19257/12619&quot;&gt;As I have mentioned before&lt;/a&gt;, the CCSS don't have much to say about the details of exposing students to proofs, so California could arguably have struck out on its own here.)&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2021-02-09T17:27:18.650" LastActivityDate="2021-02-09T17:27:18.650" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19458" PostTypeId="1" CreationDate="2021-02-09T19:47:15.683" Score="2" ViewCount="448" Body="&lt;p&gt;Most differential calculus courses begin with the theory (and analysis) of differentiation, followed by computations, and likewise integral calculus courses. That's a lot for a three credit course, which is why most college courses of this sort carry four credits.&lt;/p&gt;&#xA;&lt;p&gt;I initially wondered if it were possible to cut back these courses to three credits by removing the analysis portions, but this question goes the other way. I am now wondering what would happen if colleges &amp;quot;supersized&amp;quot; these calculus courses by adding most of an intermediate analysis material and making them six credits. The advantage is that students could learn both theory and practice in one course. Also the average math program requires about 39 credits, so the student will have completed 12 credits by the end of the freshman year, and could complete the major in six or seven semesters instead of eight.&lt;/p&gt;&#xA;&lt;p&gt;Are there any university programs that offer this kind of supersized option in introductory mathematics? (Some years ago, Yale had an &amp;quot;early concentration&amp;quot; option in Economics, featuring six credits of principles and intermediate microeconomics in the freshman first term, and similarly for macroeconomics in the second term.)&lt;/p&gt;&#xA;" OwnerUserId="1333" LastEditorUserId="1333" LastEditDate="2021-02-10T17:40:03.667" LastActivityDate="2021-02-10T17:40:03.667" Title="Are there any university programs that &quot;supersize&quot; calculus courses?" Tags="|mathematical-pedagogy|calculus|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19459" PostTypeId="2" ParentId="19456" CreationDate="2021-02-09T20:39:05.493" Score="2" Body="&lt;p&gt;In axiomatizations of Euclidean plane geometry such as the ones by Hilbert or Tarski, the statement &lt;span class=&quot;math-container&quot;&gt;$\overline {AB}\cong\overline {BA}$&lt;/span&gt; is a postulate. In Tarski's system, this congruence axiom is explicitly called &amp;quot;Reflexivity of Congruence.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-02-09T20:39:05.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19460" PostTypeId="2" ParentId="19458" CreationDate="2021-02-09T21:11:05.187" Score="8" Body="&lt;p&gt;A.  CIT is (in)famous for having a freshman calculus course, with substantial analysis content.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.caltech.edu/%7E2014-15/1term/ma001a/&quot; rel=&quot;noreferrer&quot;&gt;http://www.math.caltech.edu/~2014-15/1term/ma001a/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The course is not supersized in terms of credit size though.  And it presupposes AP5 level of capability for the incoming students.  And the kids hate it.&lt;/p&gt;&#xA;&lt;p&gt;B.  For various reasons, I think the suggestion to mix more real analysis into freshman calculus is a bad idea.  (A recurring suggestion though.)&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Mixes in content that most students (science, engineering, econ) will never need.  This is the bulk of a typical class.  Most of these kids, if they validated calc 1/2, would move to calc 3 and ODEs, well, well before a theoretical calc course.  Which realistically most will never take...even at grad level.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Furthermore, realize that most math majors don't self designate, at least in the states, freshman year.  As a hedgie, you should understand &amp;quot;option value&amp;quot;, the whole trope of the Hollywood sequel option validating the negative NPV of a first movie.  Well...learning all that shit before, is forfeiting option value.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Moves away from gradual progressive learning and deepening insight to a &amp;quot;learn it all at once&amp;quot; approach, that is not pedagogically optimal.  Realize the human brain is imperfect and requires training, in a very different fashion from the way a computer can do everything at once, provided an instruction set.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Taking an extra couple credits has to come from somewhere.  If you go from 4 to 6, where are you getting these 2 credits from?  As a financial guy, think of this as an investment decision.  Not &amp;quot;money from the air&amp;quot;, but a trade-off.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15439" LastEditorUserId="15439" LastEditDate="2021-02-09T21:16:15.297" LastActivityDate="2021-02-09T21:16:15.297" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19461" PostTypeId="2" ParentId="14461" CreationDate="2021-02-09T22:20:07.063" Score="2" Body="&lt;p&gt;In the US, the norm has been (and still is) for there to be an INTRO to ODEs within the second semester of a stereotypical calculus course.  You can see this as long ago as the Granville text from over 100 years ago.  And it was almost identical in the Thomas Finney text I used in the 1980s, which was also AP aligned.  (AP calc now still requires an ODE intro, but has backed off on second order constant coefficient homo/nonhomo requirement, versus what was required back in the day.)&lt;/p&gt;&#xA;&lt;p&gt;I believe the rationale for this was that some students (then and still) only have two semesters of calculus.  And there is the capacity to allow a short intro to ODEs (and to series/sequences).  This gives the kids who never go further (bio and econ and chem majors and such), something.&lt;/p&gt;&#xA;&lt;p&gt;For the physics and engineering and math (and at the Academies, all the kids), they will get a calc 3 and ODE course.  The ODE course DOES have overlap with the intro from second semester calc.  However, such overlap is not the end of the world.  HS chem and bio and physics overlap with college (or AP) chem, bio, and physics.  For most of the kids, stuff is hard enough.  Seeing some parts of it twice is a feature, not a bug.  (We are closer to dogs than to silicon...we are meat.)&lt;/p&gt;&#xA;" OwnerUserId="15439" LastActivityDate="2021-02-09T22:20:07.063" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19462" PostTypeId="2" ParentId="19397" CreationDate="2021-02-10T20:58:23.710" Score="4" Body="&lt;p&gt;I really like the idea of &lt;a href=&quot;http://cognitivemedium.com/srs-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;discovery fictions&lt;/a&gt; because they capture an essential component of how mathematics is best understood and communicated.  Almost all ideas in mathematics follow simply and inexorably from previous ideas, and understanding any mathematical discipline consists almost entirely of figuring out how you could have developed these ideas on your own.&lt;/p&gt;&#xA;&lt;p&gt;That being said, &lt;em&gt;&lt;strong&gt;there are some statements in mathematics that are much simpler than any of their explanations&lt;/strong&gt;&lt;/em&gt;.  Such a statement is true because it's true, and no explanation will ever be entirely satisfying, because none of the explanations reflect the beauty and power of the statement itself.  These are the kinds of statements that people bring up when they argue that mathematics is &amp;quot;discovered&amp;quot; instead of invented.  They often become famous theorems, and sometimes many different proofs are known.&lt;/p&gt;&#xA;&lt;p&gt;The Pythagorean theorem is a paradigmatic example of such a statement.  Most statements in geometry are best understood using their explanations, but the Pythagorean theorem is more like a diamond that mathematicians unearthed when we started digging into the subject.  There is nothing expected about it, and no matter how many proofs of it we find there is no way to completely dispel the mystery of why it was there.&lt;/p&gt;&#xA;&lt;p&gt;The real history of the Pythagorean theorem agrees with this assessment.  The history is that the &lt;em&gt;statement itself&lt;/em&gt; was discovered several times by several different cultures.  The Egyptians seem to have been aware of the (3,4,5) triple in 2000 BCE (apparently used as a way of making right angles  for building construction), and by 1000 BCE both the Babylonians and the Chinese were aware of many different Pythagorean triples as well as the general rule.  However, it wasn't until 500 years later (around 500 BCE) that the first demonstration of the theorem was given by Pythagoras.  This is perhaps the first significant example in history of the now familiar mathematical process of forming a &lt;em&gt;conjecture&lt;/em&gt; based on empirical evidence, and then later finding a proof of the conjecture.&lt;/p&gt;&#xA;&lt;p&gt;So my argument is that, unlike almost every other statement in secondary-school mathematics, there isn't a good &lt;a href=&quot;http://cognitivemedium.com/srs-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;discovery fiction&lt;/a&gt; for the Pythagorean theorem, and in fact any such fiction would be highly misleading.  Instead, the Pythogorean theorem is shining example of mathematics behaving like a science, with a process of discovery based on first gathering empirical evidence, then guessing the general rule, and finally verifying it, with the difference between mathematics and science being that the verification comes in the form of a proof instead of an experiment.  This method of discovering mathematical truth is very powerful and important, and there are very few genuine examples of it in secondary mathematics, so I would advocate for presenting the Pythagorean theorem from that point of view rather than straining to make the theorem seem like an &amp;quot;intuitive&amp;quot; or &amp;quot;obvious&amp;quot; part of the development of geometry.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2021-02-10T20:58:23.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19463" PostTypeId="2" ParentId="19451" CreationDate="2021-02-11T05:49:18.727" Score="3" Body="&lt;p&gt;I'm not entirely sure if it fits the teaching style you described, but it at least partially resembles the &lt;em&gt;discovery method&lt;/em&gt;. Classrooms that adhere strictly to the discovery method typically have little direction provided by the teacher, whereas you seem to be facilitating the discussion more.&lt;/p&gt;&#xA;&lt;p&gt;I'm not familiar with any specific research on the discovery method, but what I would consider a more extreme version of the discovery method is the &lt;em&gt;Moore method&lt;/em&gt;, of which I am aware of some studies.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2021-02-11T05:49:18.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19465" PostTypeId="1" AcceptedAnswerId="20704" CreationDate="2021-02-12T06:52:15.393" Score="5" ViewCount="250" Body="&lt;p&gt;The critical points of a function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; are candidates for local extrema, i.e., if a function changes from increasing to decreasing, or vice versa, it must happen at a critical point.&lt;/p&gt;&#xA;&lt;p&gt;Is there an analogous term for candidates for inflection points? The mathematical definition is analogous - if a function changes from positive concavity to negative concavity at a point, or vice versa, it must happen at a point &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; in the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$f''(a)=0$&lt;/span&gt; or where &lt;span class=&quot;math-container&quot;&gt;$f''(a)$&lt;/span&gt; does not exist - but what do we call such a point (if we call it anything at all)?&lt;/p&gt;&#xA;" OwnerUserId="21" LastEditorUserId="21" LastEditDate="2021-02-12T07:22:11.190" LastActivityDate="2021-04-08T18:45:25.003" Title="Term for candidates for inflection points" Tags="|calculus|terminology|derivative|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19466" PostTypeId="2" ParentId="17083" CreationDate="2021-02-12T14:04:30.297" Score="1" Body="&lt;p&gt;I do this with Mathematica. Remember that each row operation is left multiplication by an elementary matrix.&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;(* This is the number of rows of the matrix. *)&#xA;n = 5;&#xA;&#xA;(* This is a matrix with 1 in position (i,j) and 0 everywhere else. *)&#xA;e[i_,j_]:=Table[If[{a,b}=={i,j}, 1, 0], {a,n}, {b,n}];&#xA;&#xA;(* This switches rows i and j *)&#xA;switch[i_, j_] := IdentityMatrix[n] + e[i, j] + e[j, i] - e[i, i] - e[j, j];&#xA;&#xA;(* This multiplies row i by t *)&#xA;mult[i_, t_] := IdentityMatrix[n] + (t - 1) e[i, i];&#xA;&#xA;(* This adds u times row i to row j *)&#xA;add[i_, j_, u_] := IdentityMatrix[n] + u e[j, i];&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Given a matrix &lt;code&gt;M&lt;/code&gt;, I can do one step of row reduction with a computation like &lt;code&gt;add[1,2,-3].M&lt;/code&gt; and can do nested steps by &lt;code&gt;add[1,3,5].add[1,2,-3].M&lt;/code&gt; etcetera.&lt;/p&gt;&#xA;&lt;p&gt;Of course, if you just want the final answer, that is &lt;code&gt;RowReduce[M]&lt;/code&gt;.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2021-02-13T02:08:42.297" LastActivityDate="2021-02-13T02:08:42.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19467" PostTypeId="2" ParentId="13846" CreationDate="2021-02-12T14:12:23.837" Score="4" Body="&lt;p&gt;Here are projects I have written and used. I assign these to student groups of 3-6 students, mostly Engineering majors, mostly sophomores. In class time is about 30 minutes per project to introduce the set up; then they work out of class for about 2 weeks per project (and come to office hours frequently during that time).&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.math.lsa.umich.edu/courses/214/project_files/LinearProgramming.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Linear programming&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.math.lsa.umich.edu/courses/214/project_files/Camera.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Camera matrices&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.math.lsa.umich.edu/courses/214/project_files/ECC.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Error correcting codes&lt;/a&gt; The first draft of this one was written by Martin Strauss.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.math.lsa.umich.edu/courses/214/project_files/LeastSquares.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Least squares data fitting&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.math.lsa.umich.edu/courses/214/project_files/Springs.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oscillations of mechanical systems&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I want to make another one on Fourier transforms.&lt;/p&gt;&#xA;&lt;p&gt;If anyone would like to use these, send me an e-mail and I'll send you LaTeX source and auxilliary files. Please give me credit. At some point, I'll get around to doing a formal Creative Commons release of these.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2021-02-12T14:12:23.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19468" PostTypeId="1" CreationDate="2021-02-12T22:15:26.357" Score="4" ViewCount="302" Body="&lt;p&gt;A natural way to reason about Euclidean geometry&#xA;using modern mathematical language&#xA;is to define Euclidean space&#xA;as an affine space &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;&#xA;directed by a finite-dimensional real vector space &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, a bare vector space structure doesn't provide us with enough language&#xA;to define things like lengths or perpendicularity,&#xA;so we're ought to endow &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; with some additional structure.&#xA;The well-known solution is to equip &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; with an inner product.&#xA;The length of &lt;span class=&quot;math-container&quot;&gt;$v \in V$&lt;/span&gt; is then &lt;em&gt;defined&lt;/em&gt; as &lt;span class=&quot;math-container&quot;&gt;$\sqrt{(v, v)}$&lt;/span&gt;,&#xA;and, &lt;em&gt;by definition&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$v^1 \in V$&lt;/span&gt; is perpendicular to &lt;span class=&quot;math-container&quot;&gt;$v^2 \in V$&lt;/span&gt;&#xA;iff &lt;span class=&quot;math-container&quot;&gt;$(v^1, v^2) = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, in order to convince someone that&#xA;the concepts we intuitively know as &amp;quot;length&amp;quot; and &amp;quot;perpendicularity&amp;quot;&#xA;are indeed expressible in terms of a&#xA;positive defined bilinear form,&#xA;we unavoidably(?) need to apply to the Pythagorean theorem.&#xA;Thus, in order to justify&#xA;that our model of Euclidean space is adequate,&#xA;we need to invoke some facts about the Euclidean space,&#xA;which is an awkward thing to do while we are &lt;em&gt;defining&lt;/em&gt; it!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Are there any (hopefully more intuitive and convincing) ways&#xA;of &lt;em&gt;defining&lt;/em&gt; length and orthogonality in Euclidean space,&#xA;other &lt;em&gt;defining&lt;/em&gt; them in terms of an inner product?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;UPD: Let me try to rephrase my question in a different way.&lt;/p&gt;&#xA;&lt;p&gt;Suppose you are giving someone an &lt;em&gt;introductory&lt;/em&gt; course on Euclidean geometry. Imagine that you decided to stick to a linear algebraic spirit. So you postulate the existence of the set of points and the set of translations, and introduce the axioms regarding the translations and their relationship with numbers and with points. Now you can talk as much as you want about affine geometry.&lt;/p&gt;&#xA;&lt;p&gt;Next you need to introduce some axioms relating to length and perpendicularity. You &lt;em&gt;could&lt;/em&gt; define an inner product and even provide an example of it, but defining the distance in terms of it won't go well, because in order to believe that such definition is reasonable, your audience should know the Pythagorean theorem, which they don't, because you are just giving them an introductory course right now:)&lt;/p&gt;&#xA;&lt;p&gt;Is there any satisfactory way out of this situation?&lt;/p&gt;&#xA;" OwnerUserId="15451" LastEditorUserId="15451" LastEditDate="2021-02-14T14:21:13.707" LastActivityDate="2021-02-14T21:49:34.080" Title="Can we define length and perpendicularity not via an inner product?" Tags="|geometry|linear-algebra|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19469" PostTypeId="2" ParentId="19468" CreationDate="2021-02-13T00:23:49.063" Score="3" Body="&lt;p&gt;I'm not quite sure if the question is the same as the question in the title.&lt;/p&gt;&#xA;&lt;p&gt;For the question in the title: Sure, we can certainly develop the concept of length independently from inner product. The result would be a normed vector space. We only need 4 properties for the &amp;quot;norm&amp;quot; function:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nonnegativity&lt;/li&gt;&#xA;&lt;li&gt;A vector has zero norm if and only if the vector is zero&lt;/li&gt;&#xA;&lt;li&gt;Linearity for positive scalars&lt;/li&gt;&#xA;&lt;li&gt;Triangle inequality&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;There is no reference to inner product. What's more interesting is that not all normed vector space structure we can put on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;, including some quite useful ones, are induced by inner products.&lt;/p&gt;&#xA;&lt;p&gt;For the other part, since perpendicularity means orthogonality, the reference to inner product is already implied.&lt;/p&gt;&#xA;&lt;p&gt;Footnote: in Riemannian geometry, &amp;quot;perpendicular curves&amp;quot; can be defined in a way that seems to be independent from inner product. Although I don't think that counts because it is impossible to define a Riemannian structure without inner product anyway.&lt;/p&gt;&#xA;" OwnerDisplayName="user13395" LastEditorDisplayName="user13395" LastEditDate="2021-02-13T00:30:49.613" LastActivityDate="2021-02-13T00:30:49.613" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19470" PostTypeId="2" ParentId="17083" CreationDate="2021-02-13T02:18:06.890" Score="2" Body="&lt;p&gt;Not a software exactly, but super useful for those in the process of learning row reduction is the website: &lt;a href=&quot;http://www.math.odu.edu/%7Ebogacki/cgi-bin/lat.cgi?c=rref&quot; rel=&quot;nofollow noreferrer&quot;&gt;Linear Algebra Toolkit, by P. Bogacki&lt;/a&gt;. You just choose the number of rows and columns and it will produce the reduced row echelon form, and show each step along the way with the indicated row-op clearly stated. I've used this for years.&lt;/p&gt;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/T7QRz.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/T7QRz.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2021-02-13T18:41:08.963" LastActivityDate="2021-02-13T18:41:08.963" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19471" PostTypeId="1" AcceptedAnswerId="19472" CreationDate="2021-02-13T12:39:32.750" Score="7" ViewCount="1164" Body="&lt;p&gt;To evaluate &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;  &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; we cannot use&#xA;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{A}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;  &lt;span class=&quot;math-container&quot;&gt;$\sqrt{B}$&lt;/span&gt; = &lt;span class=&quot;math-container&quot;&gt;$\sqrt{AB}$&lt;/span&gt; as the result would be 1.&lt;/p&gt;&#xA;&lt;p&gt;I know (?) that we must first respect that the initial numbers become imaginary, and of course &lt;span class=&quot;math-container&quot;&gt;$i^{2} $&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This came up in school (high school) yesterday, a teacher called me in as I walked by, and while I was confident of the answer, we are still looking to understand what rule, theorem, etc, makes this so.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-02-15T21:48:49.563" LastActivityDate="2021-02-15T21:48:49.563" Title="Multiplying the square roots of negative numbers before we calculate a result using $i$" Tags="|algebra|" AnswerCount="5" CommentCount="5" ClosedDate="2021-02-18T09:55:47.137" ContentLicense="CC BY-SA 4.0" />
  <row Id="19472" PostTypeId="2" ParentId="19471" CreationDate="2021-02-13T13:30:08.860" Score="12" Body="&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identities&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The identities &lt;span class=&quot;math-container&quot;&gt;$(bc)^x=b^xc^x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(b/c)^x=b^x/c^x$&lt;/span&gt; are valid when &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; are positive real numbers and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a real number.  But a calculation using principal branches shows that&#xA;&lt;span class=&quot;math-container&quot;&gt;${\displaystyle 1=(-1\cdot -1)^{\frac {1}{2}}\not =(-1)^{\frac {1}{2}}(-1)^{\frac{1}{2}}=-1}$&lt;/span&gt;&lt;br /&gt;&#xA;and&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;${\displaystyle i=(-1)^{\frac {1}{2}}=\left({\frac {1}{-1}}\right)^{\frac {1}{2}}\not ={\frac {1^{\frac {1}{2}}}{(-1)^{\frac {1}{2}}}}={\frac {1}{i}}=-i}$&lt;/span&gt;&lt;br /&gt;&#xA;On the other hand, when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is an integer, the identities are valid for all nonzero complex numbers.&lt;br /&gt;&#xA;If exponentiation is considered as a multivalued function then the possible values of &lt;span class=&quot;math-container&quot;&gt;$(−1 \cdot −1)^{1/2}$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$\{1,−1\}$&lt;/span&gt;. The identity holds, but saying &lt;span class=&quot;math-container&quot;&gt;$\{1\}=\{(−1\cdot −1)^{1/2}\}$&lt;/span&gt; is wrong.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I suppose there is no name for the &amp;quot;rule&amp;quot; that says the identity &lt;span class=&quot;math-container&quot;&gt;$\sqrt{A}\sqrt{B}=\sqrt{AB}$&lt;/span&gt; is wrong when &lt;span class=&quot;math-container&quot;&gt;$A,B&amp;lt;0$&lt;/span&gt;, just a rule saying that the identity is right if &lt;span class=&quot;math-container&quot;&gt;$A,B\ge 0$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2021-02-13T13:30:08.860" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19473" PostTypeId="2" ParentId="19468" CreationDate="2021-02-13T13:45:29.060" Score="2" Body="&lt;p&gt;Here is an approach which works only(?) in two dimensional case and seems reasonably satisfying to me. Hope it helps to clarify the question!&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; be a &lt;strong&gt;two-dimensional&lt;/strong&gt; real vector space.&#xA;Consider a structure on &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; consisting of:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A linear transformation &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$r^2 = -1$&lt;/span&gt; chosen up to sign.&lt;/li&gt;&#xA;&lt;li&gt;A nondegenerate skew-symmetric bilinear form &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; chosen up to sign.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;(&amp;quot;chosen up to sign&amp;quot; means that we consider both &lt;span class=&quot;math-container&quot;&gt;$(-r, s)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(r, -s)$&lt;/span&gt; to be the same structure as &lt;span class=&quot;math-container&quot;&gt;$(r, s)$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Given such a structure, one can make definitions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A &lt;em&gt;length&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$v \in V$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\sqrt{|s(v, r \cdot v)|}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$v^1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v^2$&lt;/span&gt; are &lt;em&gt;perpendicular&lt;/em&gt; iff &lt;span class=&quot;math-container&quot;&gt;$v^1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r \cdot v^2$&lt;/span&gt; are linearly dependent.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Arguably(!), those notions and definitions are better connected to intuitive geometric concepts. After all, if we think of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; as a right angle turn, and think of &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; as an oriented parallelogram area, the definitions of length and perpendicularity immediately agree with intuition!&lt;/p&gt;&#xA;&lt;p&gt;There is a natural bijective correspondence&#xA;between inner products&#xA;and structures of this kind. Indeed:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is equipped with an inner product,&#xA;there is exactly one up to sign orthogonal transformation&#xA;satisfying the &lt;span class=&quot;math-container&quot;&gt;$r^2 = -1$&lt;/span&gt; equation,&#xA;and exactly one up to sign skew-symmetric bilinear form &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;&#xA;which gives &lt;span class=&quot;math-container&quot;&gt;$±1$&lt;/span&gt; on any orthonormal base.&lt;/li&gt;&#xA;&lt;li&gt;If, conversely, &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is equipped with &lt;span class=&quot;math-container&quot;&gt;$(±r, ±s)$&lt;/span&gt; pair as above,&#xA;define &lt;span class=&quot;math-container&quot;&gt;$f(v^1, v^2) := s(v^1, r \cdot v^2)$&lt;/span&gt;.&#xA;&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is bilinear and symmetric (this easily follows from &lt;span class=&quot;math-container&quot;&gt;$r^2 = -1$&lt;/span&gt;).&#xA;Moreover, if &lt;span class=&quot;math-container&quot;&gt;$v \neq 0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r \cdot v$&lt;/span&gt;&#xA;are linearly independent (this again follows from &lt;span class=&quot;math-container&quot;&gt;$r^2 = -1$&lt;/span&gt;).&#xA;Because &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; is nondegenerate, we conclude that &lt;span class=&quot;math-container&quot;&gt;$\forall v \neq 0 : f(v, v) \neq 0$&lt;/span&gt;.&#xA;Thus, the quadriatic form associated with &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is definite,&#xA;so it's either positive definite or negative definite,&#xA;so either &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-f$&lt;/span&gt; is a valid inner product.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15451" LastActivityDate="2021-02-13T13:45:29.060" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19474" PostTypeId="2" ParentId="19471" CreationDate="2021-02-13T20:20:11.213" Score="14" Body="&lt;p&gt;In modern pedagogy, there has been a tendency to eradicate the notion of a multiple-valued function, but it's really quite a useful notion, and it's the way people always used to talk about these things. In this language, both 0 and &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; are inverse tangents of 0, and both 2 and &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; are square roots of 4. This is IMO the most natural way of dealing with the problem you describe. Then the identity &lt;span class=&quot;math-container&quot;&gt;$\sqrt{A}\sqrt{B}=\sqrt{AB}$&lt;/span&gt; is not really an identity, because all three of the square root symbols represent multi-valued operators. Similarly, the identity &lt;span class=&quot;math-container&quot;&gt;$\tan^{-1}\tan x=x$&lt;/span&gt; is not an identity; it's an identity modulo &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The educational and practical problem with the eradication of multiple-valued functions is that it plays into students' belief that &amp;quot;the answer&amp;quot; is whatever their calculator says it is. Therefore, if their calculator says that the inverse tangent of 0 is 0, it must be right. This causes them to get wrong answers to real-life problems.&lt;/p&gt;&#xA;&lt;p&gt;For students who have been carefully taught that all operations are single-valued, and who may not be able/willing to absorb this approach, you can also tackle this by asking them the following. If &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a real number, and &lt;span class=&quot;math-container&quot;&gt;$x^2=4$&lt;/span&gt;,  is it true that &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt;? For students operating at this kind of concrete level, the message that they should be remembering is that when you take square roots of both sides of an equation, you have to write &lt;span class=&quot;math-container&quot;&gt;$\pm \sqrt{\ }$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-02-13T23:45:52.323" LastActivityDate="2021-02-13T23:45:52.323" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19475" PostTypeId="2" ParentId="19471" CreationDate="2021-02-14T01:50:41.187" Score="2" Body="&lt;p&gt;The underlying idea is that &amp;quot;the&amp;quot; square root of a number is only well defined when that number is positive, as we then have a consistent way to discriminate both square roots by taking the positive one. So &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt; is the positive number that, when squared, gives the positive number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. This works because the product of two positive numbers is a positive number.&lt;/p&gt;&#xA;&lt;p&gt;The general statement is that the product of a root of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and of a root of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is a root of &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt;. This still works for &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; is a root of &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$i^2=-1$&lt;/span&gt; is a root of 1. It simply isn't &amp;quot;the&amp;quot; root of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, as it is not positive.&lt;/p&gt;&#xA;" OwnerUserId="15456" LastActivityDate="2021-02-14T01:50:41.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19476" PostTypeId="2" ParentId="19471" CreationDate="2021-02-14T03:38:23.677" Score="2" Body="&lt;p&gt;The first radical is negative i.  The second one is positive i. ;-)  Almost not trolling.  Similar point as with Ben.&lt;/p&gt;&#xA;&lt;p&gt;This question may not be 100% identical, but reminds me of a lot of the other questions we get that sort of key around the issue/confusion of radicals meaning &amp;quot;positive root&amp;quot; or meaning &amp;quot;either root&amp;quot;.  FWIW, i is neither positive nor negative.&lt;/p&gt;&#xA;&lt;p&gt;I think I have seen i defined as the number squared that gives you -1. (Not using a radical in the definition.)  Of course this makes it clear that either i or negative i satisfy.&lt;/p&gt;&#xA;&lt;p&gt;From a practical teaching standpoint, I would avoid getting into this unless the kids bring it up.  Sounds like you are confused on it. (If they do, just explain that the identify only holds for positive numbers and leave it at that.)  I guess you could similarly wonder if y=3x/x is 3 at x=0 or undefined.&lt;/p&gt;&#xA;" OwnerUserId="15459" LastActivityDate="2021-02-14T03:38:23.677" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19477" PostTypeId="2" ParentId="19468" CreationDate="2021-02-14T04:46:18.347" Score="1" Body="&lt;p&gt;In general, manifolds have not a norm, but a metric, which is a function that takes two points as input and gives their distance as output. A space with addition/subtraction and a norm has a metric (the distance between two points is the length of their difference), and metric that respects linearity is a norm (the length of a vector is the distance between it and the origin).&lt;/p&gt;&#xA;&lt;p&gt;When doing constructions in plane geometry, if we're given a line &lt;span class=&quot;math-container&quot;&gt;$L_1$&lt;/span&gt; and a point &lt;span class=&quot;math-container&quot;&gt;$p_0$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$L_0$&lt;/span&gt;, we can construct a line &lt;span class=&quot;math-container&quot;&gt;$L_2$&lt;/span&gt; perpendicular to &lt;span class=&quot;math-container&quot;&gt;$L_1$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$p_0$&lt;/span&gt; by first finding two points &lt;span class=&quot;math-container&quot;&gt;$p_1, p_2$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$L_1$&lt;/span&gt; that are the same distance from &lt;span class=&quot;math-container&quot;&gt;$p_0$&lt;/span&gt;, then finding points that are the same distance from &lt;span class=&quot;math-container&quot;&gt;$p_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p_2$&lt;/span&gt;. If we take this as our definition of &amp;quot;perpendicular&amp;quot;, we can generalize this to any metric space.&lt;/p&gt;&#xA;&lt;p&gt;Once a metric &lt;span class=&quot;math-container&quot;&gt;$d:X \times X \rightarrow \mathbb R$&lt;/span&gt; is defined on a set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, given a curve &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, a point &lt;span class=&quot;math-container&quot;&gt;$p_0$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, and a parameterization &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb R \rightarrow X$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f(0)=p_0$&lt;/span&gt;, we can define a set &lt;span class=&quot;math-container&quot;&gt;$S= \{p:\lim_{t \rightarrow 0}  \frac {d(f(t),p)d(f(-t),p_0)}{d(f(-t),p)d(f(t),p_0)}=1\}$&lt;/span&gt; (there's a few more details in making sure the denominator is non-zero that I won't get into).&lt;/p&gt;&#xA;&lt;p&gt;If we require that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; have a &amp;quot;speed&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, that is, &lt;span class=&quot;math-container&quot;&gt;$\lim_{t \rightarrow 0} \frac {d(f(t),f(0))}t=1$&lt;/span&gt;, then we can simplify the definition of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\{p:\lim_{t \rightarrow 0}  \frac {d(f(t),p)}{d(f(-t),p)}=1\}$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is also a line, then we can simplify further to &lt;span class=&quot;math-container&quot;&gt;$S = \{p:d(f(t),p)=d(f(-t),p) \} \text{for some $t$}$&lt;/span&gt; (for a line, the value of &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; doesn't matter).&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2021-02-14T04:46:18.347" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19478" PostTypeId="2" ParentId="19471" CreationDate="2021-02-14T16:52:14.997" Score="2" Body="&lt;p&gt;I think trying to invoke a &amp;quot;rule&amp;quot; handed down from above would not be too beneficial for students. It just reinforces the idea that math is a bunch of rules to memorize and not think about (I know that's not your intention of course). The reason the statement is false for imaginary numbers is pretty simple: there are two possible ways to define &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;, and neither one makes the statement true (at least using the standard definition of square root for nonnegative reals; see below). There's nothing more or less to it.&lt;/p&gt;&#xA;&lt;p&gt;I think what would be helpful is for the students to try different combinations of definitions (say, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt; is non&lt;em&gt;positive&lt;/em&gt; for nonnegative reals and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}=i$&lt;/span&gt;) and see whether the rule holds for reals and/or imaginary numbers then. This reinforces the idea that the &amp;quot;rules&amp;quot; of math depend on our definitions and objects of study, and may be true in some cases and false in others.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2021-02-14T16:52:14.997" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19479" PostTypeId="2" ParentId="19258" CreationDate="2021-02-14T19:23:53.167" Score="2" Body="&lt;p&gt;In an attempt to answer your question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Are the PISA data detailed enough to measure effects by such AI based approaches in mathematics?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I found three things that point to the answer being &lt;strong&gt;no&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I dug through the &lt;a href=&quot;https://www.oecd.org/pisa/data/2018database/CY7_201710_QST_MS_SCQ_NoNotes_final.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2018 PISA School Questionnaire&lt;/a&gt;, which is administered to school principals, and found no question on AI that would allow the data to be parsed in this way. I only searched this questionnaire, so it's possible the question comes up elsewhere, but I would be surprised if it was. (For reference, the full list of master questionnaires from 2018 can be found &lt;a href=&quot;https://www.oecd.org/pisa/data/2018database/#d.en.516012&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A draft of the &lt;a href=&quot;https://pisa2021-maths.oecd.org/files/PISA%202021%20Mathematics%20Framework%20Draft.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2021 PISA framework&lt;/a&gt; has lots of mentions of Computer-Based Assessments of Mathematics (CBAM) and computer simulations, but no reference to Artificial Intelligence (AI) that I can find. The draft I found was from November 2018 though, so its entirely possible that something has been added to address this, but (again) I would be surprised if it was.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;My third point is anecdotal, but perhaps valuable, and it echoes the early comments to the original post. In the not so recent past (but still less than 10 years ago) I was teaching a college remedial mathematics course at an institution in the New York City area that was using an &amp;quot;Artificial Intelligence&amp;quot;-esque system for students to complete modules of work. The company that we used sought feedback on the system and provided cash incentives for educators who found mistakes in their software (either mistakes, typos, or sequencing), which gave me the feeling that these billed &amp;quot;Artificial Intelligence&amp;quot; learning environments are in actuality just logic trees that software engineers and mathematics educators have built out by brute force. Now, it has been some time since I have taught with this software, but I have seen nothing in the interim to suggest that the AI software you're referring to is anything but &amp;quot;hype&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="267" LastActivityDate="2021-02-14T19:23:53.167" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19480" PostTypeId="2" ParentId="19468" CreationDate="2021-02-14T21:49:34.080" Score="4" Body="&lt;p&gt;The following comes from &lt;a href=&quot;https://projecteuclid.org/download/pdf_1/euclid.afm/1485893376&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://projecteuclid.org/download/pdf_1/euclid.afm/1485893376&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$(V,|\cdot|)$&lt;/span&gt; be a normed vector space.&lt;/p&gt;&#xA;&lt;p&gt;Define &lt;span class=&quot;math-container&quot;&gt;$v \perp w$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$|v-w|^2 = |v|^2+|w|^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Define a &amp;quot;normed perpendicularity space&amp;quot; as a normed vector space where the set of vectors orthogonal to a given vector is always a subspace.&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$(V,|\cdot|)$&lt;/span&gt; arises from an inner product.  I do not have time for a full write up, but the details are in the paper and associated references.&lt;/p&gt;&#xA;&lt;p&gt;These axioms are geometrically reasonable, and do not depend on the presence of an inner product.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-02-14T21:49:34.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19481" PostTypeId="1" AcceptedAnswerId="19482" CreationDate="2021-02-15T18:37:51.447" Score="28" ViewCount="2880" Body="&lt;p&gt;I've been working with a teacher, helping her with tech. One of the things I help with is to convert PDF formatted quizzes or tests to DeltaMath for the students to take online. The issue that I face is exemplified by the following image (the question text is simply, &amp;quot;solve for line segments A, B, and C.&amp;quot; -&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wEhDH.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wEhDH.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;On the left is the question as presented to me. On the right is my effort on Desmos to graph the segments to scale. Now, the question I have - Should figures be to scale? Or is there some pedagogical value to &amp;quot;Make no assumptions beyond straight lines are in fact, lines. Use only the information given.&amp;quot;&#xA;Knowing the class as well as is possible, remote, I'd bet there would be far fewer correct answers on the left image. This question is part of a second year high school introduction to trigonometry, where the 'special right triangles', 30-60-90, and 45-45-90 are introduced.&lt;/p&gt;&#xA;&lt;p&gt;If I were a student I would find the left image to either be a mean trick, meant to confuse, or a lazy teacher not willing to take the time to produce a scale image.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: I appreciate the answers here, especially for the fact that I learn by seeing an alternate approach to what my position is. For what it's worth, remote learning has taken our weekly class time from 240 minutes to 160. The answers here have me believe that &amp;quot;not to scale&amp;quot; is an important skill to navigate. I could spend a full 3 classes using published graphs in the media whose chosen X-Y scales obfuscate the conclusion one should reach. Real life examples to make that point. If my perfect scaling has a student get B=6 too easily, I'm ok with that. The 'A' value is what will tell me if they understood this lesson.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-02-18T15:15:19.607" LastActivityDate="2021-02-19T00:01:26.343" Title="Should figures be presented to scale?" Tags="|secondary-education|geometry|" AnswerCount="10" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="19482" PostTypeId="2" ParentId="19481" CreationDate="2021-02-15T23:46:02.653" Score="22" Body="&lt;p&gt;As an answer so that I can paste in a picture: I think that the problem with the one on the left is that it is possibly &amp;quot;good enough&amp;quot; that someone might think that it is to scale and that apparent regularities are real. (Well, maybe not one of those &amp;quot;right angles&amp;quot;.) In particular, someone might become under the impression that the length 10 line is supposed to be horizontal and immediately jump to thinking that there is a 30 degree angle between the b &amp;amp; 10 lines.&lt;/p&gt;&#xA;&lt;p&gt;I'd make the figure &amp;quot;worse&amp;quot; so that no one could imagine it is more than a sketch of the general idea:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zldFU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zldFU.png&quot; alt=&quot;irregular version of questioner's figure&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The answer sheet would get the precise and tidy figure.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="15479" LastEditDate="2021-02-18T18:06:16.857" LastActivityDate="2021-02-18T18:06:16.857" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19483" PostTypeId="2" ParentId="19481" CreationDate="2021-02-16T01:33:00.477" Score="11" Body="&lt;p&gt;I would split the difference by creating an accurate diagram in Desmos and then giving the students a hand drawing based on that diagram.  That way, students could estimate their answer before solving and check their answer against that original estimate.  But they couldn't conclude that &lt;span class=&quot;math-container&quot;&gt;$b=6$&lt;/span&gt; by measuring the length of the known side.&lt;/p&gt;&#xA;&lt;p&gt;Pedagogically, the two skills that students need to leave this course with are extracting the verifiable information from a diagram and creating a diagram based on a word problem.  By giving them a diagram that is neither helpful nor misleading, you are forcing them to develop the former skill while modeling the latter for them to do themselves later.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-02-16T01:33:00.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19484" PostTypeId="2" ParentId="19481" CreationDate="2021-02-16T04:45:26.347" Score="6" Body="&lt;p&gt;Since your job is helping the teacher with technology and the teacher should be making the pedagogical decisions, ask her her preference and do it that way. Maybe she meant for the pictures not to be to scale, or maybe they came out that way due to her poor technology skills. Just ask.&lt;/p&gt;&#xA;" OwnerUserId="450" LastActivityDate="2021-02-16T04:45:26.347" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19485" PostTypeId="2" ParentId="19481" CreationDate="2021-02-16T07:13:03.210" Score="3" Body="&lt;p&gt;Our SAT problems were frequently not to scale and I think it added to the challenge.  Such drawings were labeled clearly &amp;quot;not to scale&amp;quot;. If the students will be taking the SAT's then it is good practice.&lt;/p&gt;&#xA;&lt;p&gt;Whether it is appropriate for this particular quiz/test depends on where the students are in the studies. First they need scale drawings before they tackle non-scale drawings.  The teacher needs to show them how &amp;quot;not to scale&amp;quot; drawings can be misleading and how to solve despite what their eyes show them.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;not to scale&amp;quot; drawing in this question is unusually misleading.  Here it looks like b=c and a = 3.  I think it a mistake to present such a picture on a quiz or test to the students unless they are used to working with misleading diagrams.&lt;/p&gt;&#xA;&lt;p&gt;Of course the final decision is the teacher's and not the person doing the technology.  There is no reason the person doing the technology can't point out what s/he has gleaned from this post to the teacher.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-02-16T07:13:03.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19486" PostTypeId="2" ParentId="19481" CreationDate="2021-02-16T08:37:09.420" Score="7" Body="&lt;p&gt;A mathematics teacher once told me, &amp;quot;at a ratio of 4:5, a rectangle becomes a square.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;While this is obviously not true in a strict sense, it holds the deeper truth that a rectangle at 4:5 or worse is not a good example to represent a &lt;em&gt;generic&lt;/em&gt; rectangle to young students, unless the goal is to teach or test that squares are also rectangles.&lt;/p&gt;&#xA;&lt;p&gt;That means illustrations should be either to scale, or &lt;em&gt;obviously&lt;/em&gt; not to scale, or scale-ness needs to be talked about.&lt;/p&gt;&#xA;&lt;p&gt;DKNguyen mentioned the ellipse in his comment. The Wikipedia article is a nice example. It uses one &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Ellipse-def0.svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;niclely elliptical ellipse&lt;/a&gt; for general purposes and three &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Ellipse-var.svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;differently shaped ellipses&lt;/a&gt; to explain edge cases.&lt;/p&gt;&#xA;" OwnerUserId="8201" LastEditorUserId="8201" LastEditDate="2021-02-17T06:19:27.260" LastActivityDate="2021-02-17T06:19:27.260" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19487" PostTypeId="1" CreationDate="2021-02-16T10:55:59.893" Score="3" ViewCount="83" Body="&lt;p&gt;I have a quite broad questions. I'm a math teacher in Switzerland (high school level, school year 9-12). We teach following topics (among others): quadratic equations, quadratic functions, system of linear equations, exponential funcion and logarithm, combinatorics, probability theory, calculus.&lt;/p&gt;&#xA;&lt;p&gt;We start to teach certain courses in English (but our studends are German speaking), therefore I'm looking for suitable textbooks, exercises material, suitable magazines (or links to such material) for CLIL (Content and Language Integrated Learning) learners in English.&lt;/p&gt;&#xA;&lt;p&gt;I already purchased Hease's Mathematics for the international student (HL core) which is quite good and I'm looking for similar material.&lt;/p&gt;&#xA;&lt;p&gt;Any help and good hints are appreciated.&lt;/p&gt;&#xA;" OwnerUserId="15200" LastActivityDate="2021-02-16T10:55:59.893" Title="Material and textbooks for CLIL learners" Tags="|secondary-education|textbooks|english-language-learners|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19488" PostTypeId="2" ParentId="19481" CreationDate="2021-02-16T13:51:34.897" Score="2" Body="&lt;p&gt;This question relates to an interesting feature of ancient geometric proofs: The ancient geometers often drew deliberately distorted figures in order to force the prover to NOT depend on any accidental features of the figures in formulating their proofs.  For example, when drawing a chord of a small arc of a circle, which is in fact a straight line segment, they would actually draw the chord &lt;em&gt;as an arc&lt;/em&gt; (enclosing a thick lens-like area between the chord and the circle) in order to force the prover to not confuse the chord with the arc.  You might consider drawing your own figures accordingly [with explanatory statements so the student understands what you are doing].&lt;/p&gt;&#xA;" OwnerUserId="15468" LastActivityDate="2021-02-16T13:51:34.897" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19489" PostTypeId="2" ParentId="19481" CreationDate="2021-02-16T13:59:45.760" Score="19" Body="&lt;p&gt;There is no value in drawing the figure &lt;em&gt;exactly&lt;/em&gt; to scale, but the left-hand figure is inaccurate to the point where it is positively misleading.&lt;/p&gt;&#xA;&lt;p&gt;Since the angle marked 30 degrees is actually drawn greater than 45 degrees, it gives the impression that a is less than 3 (or maybe equal to it, if the student takes the &amp;quot;30 degree&amp;quot; angle as being shown at exactly 45). Of course a is actually greater than 3.&lt;/p&gt;&#xA;&lt;p&gt;The ratio of the lines with lengths 3 and 10 is &amp;quot;obviously wrong&amp;quot; as well.&lt;/p&gt;&#xA;&lt;p&gt;Students should learn to draw their own diagrams that are &amp;quot;near enough right&amp;quot; to be useful as a sanity check on their calculations. The left hand diagram one is not.&lt;/p&gt;&#xA;&lt;p&gt;I still remember some advice from my high-school math teacher: &lt;em&gt;never&lt;/em&gt; draw a diagram showing an &amp;quot;arbitrary&amp;quot; angle that is close to 45 of 90 degrees, to avoid making incorrect assumptions. For example, students should know that for angles less than 45 degrees, the sine of the angle is smaller than the cosine (and vice versa for angles between 45 and 90). A &amp;quot;correctly&amp;quot; drawn figure is a good sanity check against using the wrong function.&lt;/p&gt;&#xA;&lt;p&gt;There are a some well known &amp;quot;nonsense proofs&amp;quot; which are based on drawing an incorrect figure, for example a proof that all triangles are isosceles (see &lt;a href=&quot;http://jdh.hamkins.org/all-triangles-are-isosceles/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jdh.hamkins.org/all-triangles-are-isosceles/&lt;/a&gt;) which depends on drawing a figure where a point is &lt;em&gt;inside&lt;/em&gt; the triangle, when in fact it is always &lt;em&gt;outside&lt;/em&gt; (unless the triangle really is isosceles, in which case it the point is on the base of the triangle). Don't go down that type of rabbit hole by accident!&lt;/p&gt;&#xA;" OwnerUserId="7677" LastEditorUserId="11" LastEditDate="2021-02-18T19:56:09.840" LastActivityDate="2021-02-18T19:56:09.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19490" PostTypeId="2" ParentId="19481" CreationDate="2021-02-16T17:29:28.360" Score="4" Body="&lt;p&gt;&lt;strong&gt;Consider the possibility that the exams will do this&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I don't know where you are, or how practices have changed, but when I was sitting my exams at 16 in the UK, the standard practice for the national exams was drawings that were wildly misleading in this manner. I assume the point is to make students concentrate on what is &lt;em&gt;actually&lt;/em&gt; known as opposed to assuming based on the drawing, but from the point of view of a secondary teacher this is something that they need to prepare students for &lt;em&gt;regardless&lt;/em&gt; of whether they, personally, think it is good practice.&lt;/p&gt;&#xA;" OwnerUserId="6399" LastActivityDate="2021-02-16T17:29:28.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19491" PostTypeId="1" CreationDate="2021-02-17T09:39:34.950" Score="3" ViewCount="157" Body="&lt;p&gt;We can spend a lot of time thinking of material which we have basics for without ever studying the original ideas from a textbook, for example, once one has finished regular derivatives, it is pretty motivated to jump to partial derivatives and other ideas related to it.&lt;/p&gt;&#xA;&lt;p&gt;Then a person self studying is faced with a choice, either he could think of the material more without opening the book or he could see how it was done before, so, in this dilemma, what is the correct 'choice' to make for maximizing our competence long term?&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorDisplayName="user507" LastEditDate="2021-02-17T14:17:13.387" LastActivityDate="2021-02-17T17:51:02.433" Title="Self-study: how much should we try to figure out material on our own before studying from a textbook?" Tags="|self-learning|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19492" PostTypeId="2" ParentId="19491" CreationDate="2021-02-17T17:51:02.433" Score="9" Body="&lt;p&gt;My advice (IN GENERAL) is to follow the book VERY explicitly and closely. You still should be doing lots of active things, not just &amp;quot;reading&amp;quot;.  But use the book as a scaffold.  In self instruction, the dangers of flailing around (to include quitting) are way higher than the dangers of missing something book did not cover or not thinking for yourself.  So just pick a book and follow it and spend almost all your time there.&lt;/p&gt;&#xA;&lt;ol start=&quot;0&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Ideally use a &amp;quot;programmed learning&amp;quot; text, such as by Stroud.  &lt;a href=&quot;https://en.wikipedia.org/wiki/Ken_Stroud&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ken_Stroud&lt;/a&gt;  If not able to find that, try to pick a text that is accessible/pleasant.  It should have good problems.  Good means including easy drill work not just &amp;quot;prove what we said&amp;quot; or &amp;quot;interesting hard&amp;quot;, having answers for checking, significant amount of them.  Avoid texts that are overly difficult or that require significant amounts of instruction/tutoring.  This includes many of the premier hard books you will see recommended on Q&amp;amp;A sites.  The biggest danger in self study is giving up.  NOT doing something too easy.  Doing a ho-hum program perfectly will get you way more than doing a ballbuster program sporadically.  Note, you can always go back and do an extra book or selected sections afterwards.  (This is the same principle as in music and sports training.  It is a practical pedagogy point, not a math point.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Work each example, on paper, as you get to it.  Try on your own, than look at how book did it, if needed. Then repeat on own, if you had to look.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Work the homework problems after each section.  Check your answers.  Rework any problems missed.  Do the whole missed problem over, writing it out from beginning to end, even if you &amp;quot;only had a dumb mistake&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At the end of your notebook, working backwards, keep a list of questions.  This is a parking lot for questions from reading or problems.  If the program is appropriately scoped (easy enough), you should definitely be able to keep moving through sections of the book.  NOT having to get on the computer and ask a SE question or call a tutor/friend.  I.e.  keep moving, if at all possible.  Finish the section studied in text/problems.  BUT have a parking lot for questions (something you didn't understand, something new you wonder, out of scope areas, etc.)  Then follow up occasionally to answer the questions. (Leave space in your notebook for the answers.  Personally, I find that often as you finish the section or perhaps go to the next one, you will be able to answer your own questions (and then write in your own answer in the space you left).  But having this organized list, means you can very efficiently target small amounts of tutor time (or ask on Q&amp;amp;A sites...but tutor more efficient).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15459" LastActivityDate="2021-02-17T17:51:02.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19493" PostTypeId="1" AcceptedAnswerId="19515" CreationDate="2021-02-17T19:50:10.707" Score="5" ViewCount="709" Body="&lt;p&gt;I am interested in learning graph theory, and from many resources I came to know that Douglas West's &lt;em&gt;Introduction to Graph Theory&lt;/em&gt; is a good textbook. But since I am doing self-study, it is at times hard to understand the material without proper guidance.&lt;/p&gt;&#xA;&lt;p&gt;So if anyone knows about any video lecture series on graph theory which uses the said text for reference, then it will be quite helpful to me. It would be like taking an actual course, getting input from the instructor, and then improvising on the learning thereafter by reading the text.&lt;/p&gt;&#xA;&lt;p&gt;Being a self-learner, I am having a bit of difficulty in understanding a few concepts.&lt;/p&gt;&#xA;&lt;p&gt;Any help would be appreciated. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="14166" LastEditorUserId="376" LastEditDate="2021-02-23T08:35:44.947" LastActivityDate="2021-02-23T21:52:20.563" Title="Is there any video lecture series on Graph Theory which uses &quot;Introduction to Graph Theory&quot; by Douglas West? as the text" Tags="|reference-request|self-learning|resource-request|graph-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19494" PostTypeId="2" ParentId="19234" CreationDate="2021-02-17T22:14:10.517" Score="4" Body="&lt;p&gt;Opal: Your proposed scheme does indeed sound like the &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/10511970.2020.1772915&quot; rel=&quot;nofollow noreferrer&quot;&gt;Second Chance Grading scheme&lt;/a&gt; I created and wrote about in PRIMUS (see the link and figure Brendan referenced). I'm happy to chat further with you -- and anyone else interested -- about the system, how it has fared since I wrote the article -- including during the pandemic (spoiler alert: it's still awesome!) -- and its connections to other grading systems.&lt;/p&gt;&#xA;&lt;p&gt;As others have already pointed out, the scheme you (Opal) envision has elements of mastery grading in it. It also attempts to operationalize and synthesize the latest research on growth mindsets and aspects of the scholarship of teaching and learning. I highly recommend reading my article, as well as the references I cite, for more details. Send me and email (ofernand@wellesley.edu) if you can't find the article for free through your institution or library and I'll send it to you.&lt;/p&gt;&#xA;" OwnerUserId="15474" LastActivityDate="2021-02-17T22:14:10.517" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19495" PostTypeId="2" ParentId="19481" CreationDate="2021-02-18T13:01:46.150" Score="3" Body="&lt;p&gt;I noticed in France a trend that did not exist before: there are many exercises (usually at the equivalent of US grade 7-9 = French 5&lt;sup&gt;ème&lt;/sup&gt;-3&lt;sup&gt;ème&lt;/sup&gt;) where the pupils are asked either to work on purposedly incorrect figures (a sketchy version of the left on in your question) or are asked to make quick handwritten sketches (no ruler or compass).&lt;/p&gt;&#xA;&lt;p&gt;This is to force them to not assume anything from the sketch.&lt;/p&gt;&#xA;&lt;p&gt;I find it a good idea when working on such exercises with my children.&lt;/p&gt;&#xA;" OwnerUserId="8076" LastEditorUserId="8076" LastEditDate="2021-02-18T16:10:42.197" LastActivityDate="2021-02-18T16:10:42.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19496" PostTypeId="2" ParentId="19422" CreationDate="2021-02-18T17:06:13.617" Score="2" Body="&lt;p&gt;Didactic stories should be simple, clear, and short. Think Aesop’s fables.  If they become more complex, at some point they become case studies.  (Case studies are good to use, but it’s a separate question.)&lt;/p&gt;&#xA;&lt;p&gt;Why repeat them? I don’t know Pearl’s book, so I am not familiar with his purposes in repeating stories. So I’ll offer reasons why I sometimes repeat them.&lt;/p&gt;&#xA;&lt;p&gt;First let’s note that short stories and fables themselves often repeat a story within the tale, usually the magic number of three times, each time with variation.  Repetition makes the story (or the lesson) easier to remember.  The variations advance our understanding of the problem presented in the story, usually with the third time being the charm.  And repetition makes the lesson easier to remember.&lt;/p&gt;&#xA;&lt;p&gt;A story that teaches a fundamental principle might be repeated as often as the fundamental principle is applied.  It being a fundamental principle, it will probably be applied relatively often in different contexts.  Learning to apply it in such contexts must be an important goal in the course.  Richard Hamming in “The Art of Doing Science and Engineering: Learning How to Learn” tells a story of two people, one who makes a random walk and one whose directions of motion somewhat line up.  The first makes progress at an average rate proportional to &lt;span class=&quot;math-container&quot;&gt;$\sqrt{n}$&lt;/span&gt; and the progress of the second will be proportional to &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.  Over time, the second person will outstrip the first.  He revisits this story several times in examining examples of engineers who had successful careers and those who had stunted careers. It is one of the fundamental points of his course: Making decisions according to your career goals will advance your career more successfully than making them without any direction in mind.  It must be effective because I read that book over fifteen years ago.  It’s also a geeky parable, which for his audience and for me makes it easier to remember.&lt;/p&gt;&#xA;&lt;p&gt;The following reason for repetition probably is not what Pearl does.  Here variation is used in the repetitions to illustrate significant differences when elements are changed.  I don’t really know a “story” per se that does this, but I can think of calculus examples.  For instance &lt;span class=&quot;math-container&quot;&gt;$(\sin x)/x$&lt;/span&gt; has three types of limits, but some students memorize, literally, “the limit of &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.”  And they may very well write that no matter whether &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is approaching zero, infinity, or a nonzero number.  @Carter’s answers also points out the importance of variation to help keep students from associating irrelevant elements of context to an example and from ignoring important elements.&lt;/p&gt;&#xA;&lt;p&gt;Now for a reason not to repeat a story.  A story that can be repeated multiple times, each time to illustrate a different fundamental principle, is probably too complicated to tell even once.  Without a specific example, I might be going out on a limb.  They apparently retold the &lt;em&gt;Iliad&lt;/em&gt; many, many times in ancient Greece, and Aristotle draws several different lessons from it.  It would be hard to do that in a semester-long class, though.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2021-02-19T05:46:15.837" LastActivityDate="2021-02-19T05:46:15.837" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19497" PostTypeId="1" CreationDate="2021-02-18T22:55:13.123" Score="2" ViewCount="948" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;There are &lt;span class=&quot;math-container&quot;&gt;$50$&lt;/span&gt; bacteria at 10:00 and each divides simultaneously into 2 every 20 minutes. The process of dividing into two is known as binary fission. Let &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; be the number of fissions that have occurred since 10:00. At what time does the tenth fission occur?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Expected answer is as follows.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The tenth fission occur &lt;span class=&quot;math-container&quot;&gt;$20 \times 10=200$&lt;/span&gt; minutes after 10:00. As 200 minutes equal to 3 hours and 20 minutes, the tenth fission occurs at 13:20.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unexpected answer is as follows.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As there are 50 bacteria that simultaneously divide themselves so all the first 50 fissions occur at 10:20. Thus the tenth fission, of course, occurs at 10:20 because all the 50 bacteria divide into two simultaneously at 10:20.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Question&lt;/h1&gt;&#xA;&lt;p&gt;How should I reword the question to avoid the second answer above?&lt;/p&gt;&#xA;&lt;p&gt;(I am not a native English speaker.)&lt;/p&gt;&#xA;" OwnerUserId="5696" LastEditorUserId="5696" LastEditDate="2022-08-08T15:22:01.793" LastActivityDate="2022-08-08T15:22:01.793" Title="How to reduce ambiguity in the following question?" Tags="|language-use|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19498" PostTypeId="2" ParentId="19497" CreationDate="2021-02-18T23:12:53.897" Score="13" Body="&lt;p&gt;The ambiguous answer is relatively correct (actually you need to know how old they are at the start of the problem).  But each fission is an individual splitting, not a generation.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps this:&lt;/p&gt;&#xA;&lt;p&gt;A certain species of bacteria splits into two cells ( a process called fission) every 20 minutes (the &amp;quot;generation&amp;quot; length), assuming proper growth conditions.  Assume we have a sample of 50 new bacteria at 10 AM at the experiment start.  The cells are in a container with proper growth conditions throughout the experiment.&lt;/p&gt;&#xA;&lt;p&gt;A.  Ten generations later, from the experiment start, what time is it?&lt;/p&gt;&#xA;&lt;p&gt;B.  How many bacteria are there in the container?&lt;/p&gt;&#xA;" OwnerUserId="15486" LastActivityDate="2021-02-18T23:12:53.897" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19499" PostTypeId="2" ParentId="19481" CreationDate="2021-02-19T00:01:26.343" Score="2" Body="&lt;p&gt;Here are the set of rules I would adhere to when producing 'not to scale' diagrams:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;straight lines on the diagram represent straight lines&lt;/li&gt;&#xA;&lt;li&gt;acute, right, obtuse, reflex angles on the diagram represent acute, right, obtuse, reflex angles respectively&lt;/li&gt;&#xA;&lt;li&gt;the relative length order of lines on the diagram is accurate (i.e. if AB is longer than BC on the diagram, then AB is really longer than BC)&lt;/li&gt;&#xA;&lt;li&gt;equal lengths or parallel sides might be indicated with marks/arrows along the lines, but the absence of such marks does not imply the lines are unequal/non-parallel&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I find that students are often expected to infer these rules from experience but it is valuable to be explicit about them in discussion to help students understand.&lt;/p&gt;&#xA;&lt;p&gt;For some classes, especially with more able students, it can be worth taking extra time to emphasise the point that they should not make assumptions based on the diagram (for example, give them a set of questions with identical triangles labelled differently). But for students getting used to trigonometry or who struggle with it, work on the principle of avoiding anything in a diagram that is likely to mislead them.&lt;/p&gt;&#xA;" OwnerUserId="1295" LastActivityDate="2021-02-19T00:01:26.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19501" PostTypeId="2" ParentId="19497" CreationDate="2021-02-19T04:29:13.763" Score="2" Body="&lt;p&gt;Use the word &amp;quot;generation&amp;quot; instead of &amp;quot;fission&amp;quot; as also suggested in @guest answer above. Also I couldn't understand the use of this sentence fragment, &amp;quot;&lt;strong&gt;Let n be the number of fissions that have occurred&lt;/strong&gt; since 10:00.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2021-02-19T04:29:13.763" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19503" PostTypeId="1" AcceptedAnswerId="19506" CreationDate="2021-02-19T17:05:08.957" Score="8" ViewCount="892" Body="&lt;p&gt;I know that undergrad math books use colors, like Gilbert Strang in his undergrad Linear Algebra textbooks and &lt;em&gt;Measure, Integration &amp;amp; Real Analysis&lt;/em&gt; by Sheldon Axler. Many first year calculus textbooks do also, but many aren't that reputable.&lt;/p&gt;&#xA;&lt;p&gt;If I had to convince an older, more stubborn author to use color, what reputable textbooks by more reputable authors will make the best case?&lt;/p&gt;&#xA;" OwnerDisplayName="user15364" LastEditorUserId="376" LastEditDate="2021-02-20T09:40:31.703" LastActivityDate="2021-02-20T09:40:31.703" Title="What famous graduate math textbooks use color?" Tags="|textbooks|graduate-education|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19504" PostTypeId="2" ParentId="19497" CreationDate="2021-02-19T18:40:53.593" Score="4" Body="&lt;p&gt;Neither of your sample answers are correct.&lt;/p&gt;&#xA;&lt;p&gt;You ask :&amp;quot;At what time does the tenth fission occur?&amp;quot;&#xA;The problem is that &amp;quot;Fission&amp;quot; is the incorrect word, as that would relate to individual fissions of individual bacteria.&lt;br /&gt;&#xA;If 5 bacteria reproduce at the same instant, then in that instant 5 fissions occurred.&lt;/p&gt;&#xA;&lt;p&gt;I believe you intend to ask when the tenth &lt;em&gt;&lt;strong&gt;generation&lt;/strong&gt;&lt;/em&gt; has occurred. Or possibly the more precise but verbose &amp;quot;At what time will a given cell in the sample undergo the tenth consecutive fission?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Secondly:&lt;br /&gt;&#xA;As there is no information given on the age distribution of the individual cells in the sample,&lt;br /&gt;&#xA;the first fission could occur at 10:00 + 0 seconds(very unlikely!). In which case that bacteria's descendants will undergo their 10th generation of fission at 13:00.&#xA;The first fission could also occur at 10:00 + 20 minutes(also very unlikely!). In which case that bacteria's descendants will undergo their 10th generation of fission at 13:20.&lt;/p&gt;&#xA;&lt;p&gt;The correct answer is a statistical distribution between 13:00 and 13:20&lt;/p&gt;&#xA;&lt;p&gt;You need to add information that the bacteria are at the beginning of their lifecycle.&lt;/p&gt;&#xA;&lt;p&gt;Also: Why bother complicating the question by starting with a count of 50? It has no relevance to &lt;em&gt;this&lt;/em&gt; question, because the answer you want is the same whether the starting colony is 1 or 50 or 1 billion.&lt;/p&gt;&#xA;&lt;p&gt;The students are likely to stumble on this question because they know the chance of 50 bacteria having the exact same age, deciding to fission in perfect synchrony, is zero. Making &lt;em&gt;this&lt;/em&gt; part of the lesson about a single starting cell removes this invisible and unneeded stumbling block.&lt;/p&gt;&#xA;" OwnerUserId="15493" LastActivityDate="2021-02-19T18:40:53.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19505" PostTypeId="1" CreationDate="2021-02-19T21:19:39.317" Score="11" ViewCount="290" Body="&lt;p&gt;I am teaching Linear Algebra this semester, with pre-recorded classes and weekly meetings, and the students have a few short and simple exercises to solve every week, which count a bit (20%) to their final course grade.&lt;/p&gt;&#xA;&lt;p&gt;This was designed in order to make sure that the students study a bit every week, and do not postpone studying the subject to the very last minute, as experience tells me and my colleagues that this is extremelly inefficient, specially in a (relatively) abstract course as Linear Algebra (this course focuses on abstract finite-dimensional Linear Algebra, with reduced use of matrices and linear systems, which are studied in a previous course).&lt;/p&gt;&#xA;&lt;p&gt;However, I noticed that most student just leave their weekly assignments to the very last minute. Out of 51 students, only 2 have even opened this week's assignment assignments, which are due on Sunday, as of Friday.&lt;/p&gt;&#xA;&lt;p&gt;As expected, the previous assignments grades are low, bordering on 60% mark. I should also add that they can take the weekly assignments as many times as they want to improve their grades, and the assignments are quite easy.&lt;/p&gt;&#xA;&lt;p&gt;So I am looking for references (articles published in reputable peer-reviewed journals) which discuss the different practices of studying; e.g. comparing studying a bit every day vs accumulating a lot of material to study all at once, and how those different approaches affect mathematics learning.&lt;/p&gt;&#xA;&lt;p&gt;I'm not even sure of what terms I should be looking for, as this is not my are of expertise, but I found that &amp;quot;cramming&amp;quot; seems to be related to the second approach mentioned above. All I could find were a few articles mentioning that cramming can be effective for students who already have good study habits, or for areas other than mathematics.&lt;/p&gt;&#xA;&lt;p&gt;tl;dr:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I'm looking for reputable articles comparing different ways of studying mathematics (spacing out the material or condensing it).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14410" LastActivityDate="2021-02-28T02:15:13.027" Title="Reference on study habits; cramming vs studying a bit every day" Tags="|reference-request|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19506" PostTypeId="2" ParentId="19503" CreationDate="2021-02-20T07:21:13.093" Score="18" Body="&lt;p&gt;One of the comments above mentions &amp;quot;the huge increase in cost&amp;quot; for using color in a book. The large cost increase for using color in a book was true twenty years ago. However, now the cost differential is quite modest. My book &lt;em&gt;Measure, Integration &amp;amp; Real Analysis&lt;/em&gt;, which was mentioned in the question as an example of a math textbook that uses color, was published by Springer in its Graduate Texts in Mathematics series. Although the book uses color, its list price is the same or lower than other books is this series, considerably less than the list price of its (black-and-white) main competitors. Furthermore, the electronic version of this book is legally free (at &lt;a href=&quot;https://measure.axler.net/&quot; rel=&quot;noreferrer&quot;&gt;https://measure.axler.net/&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The main use of color in &lt;em&gt;Measure, Integration &amp;amp; Real Analysis&lt;/em&gt; is to distinguish definitions (in yellow boxes) from theorems (in blue boxes). I think this use of color to give these visual clues and the occasional mathematically significant color photos help make the book more student-friendly.&lt;/p&gt;&#xA;" OwnerUserId="14775" LastActivityDate="2021-02-20T07:21:13.093" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19507" PostTypeId="2" ParentId="19325" CreationDate="2021-02-22T03:36:41.340" Score="0" Body="&lt;p&gt;I teach kids, small one 5-6 years to 11-12 years. So I have realised that a kid's life itself is not bound to anything concrete. They are best receptacles of Knowledge. But they are not good at reproducing it. See kids are capable of learning anything however abstract, but the trick is that teacher should be visualising it within. As long as it is &amp;quot;real&amp;quot; in the teacher's imagination, kids will learn it. If it is obscure, abstract, they will not be able to describe what they saw/learned but they learned. They do look into your soul. Personal experience.&lt;/p&gt;&#xA;&lt;p&gt;As far as numbers is concerned I don't know whether this is the answer but this is what I did. I realised that there is a path to Knowledge and one can choose any, but then there is &amp;quot;THE KNOWLEDGE&amp;quot; ITSELF without ANYTHING.&#xA;So the usual path is from &amp;quot;things&amp;quot; to &amp;quot;Numbers&amp;quot;. &amp;quot;What is this?&amp;quot;, &amp;quot;An apple Teacher!&amp;quot;, &amp;quot;No, it's ONE APPLE!&amp;quot;. I thought I'll take kids from Numbers to things. So I had to Explain NUMBER ITSELF without any &amp;quot;thing&amp;quot;.&#xA;I told them that a number is nothing but a series of empty places, each place has a name(One's, Two's) etc. and each place has a value(1, 10) etc. So what does each place contains? A NUMBER. &amp;lt;&amp;lt; Right here I explain to them difference between number and numerals. So each place contains ONLY ONE numeral. So Numeral 2 at 10's place means that this Number contains 2 tens in it. When this series of numerals is attached to a &amp;quot;thing&amp;quot; or property of a thing, then it is known as a NUMBER.&#xA;So Numerals tell about a &amp;quot;number&amp;quot; and a &amp;quot;number&amp;quot; tells us about things.&lt;/p&gt;&#xA;&lt;p&gt;And thankfully kids get it...&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2021-02-22T03:36:41.340" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19509" PostTypeId="2" ParentId="19505" CreationDate="2021-02-23T17:45:33.147" Score="4" Body="&lt;p&gt;If you can ignore the specific UBC context, this might be the type of paper&#xA;the OP seeks:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Eric Eich.&#xA;&amp;quot;The Cognitive Science of Learning Enhancement: Optimizing Long-Term Retention.&amp;quot; (2011).&#xA;Univ. British Columbia.&#xA;&lt;a href=&quot;https://ctlt.ubc.ca/resources/isotl/resources-archives/the-cognitive-science-of-learning-enhancement-optimizing-long-term-retention/&quot; rel=&quot;nofollow noreferrer&quot;&gt;HTML link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here's an excerpt comparing &amp;quot;massed&amp;quot; vs. &amp;quot;spaced&amp;quot; learning:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/xfmvR.png&quot; width=&quot;400&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&amp;quot;The advantage of spacing over massing in inductive learning seems even more surprising in view of Figure 3 [...].  As is apparent in the figure, most participants believed they learned more from massed than from spaced presentations, even though their performance on the name-selection test proved otherwise.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Massed&amp;quot; and &amp;quot;spaced&amp;quot; learning/practice are key search terms in this&#xA;literature.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-02-23T18:39:30.730" LastActivityDate="2021-02-23T18:39:30.730" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19510" PostTypeId="1" CreationDate="2021-02-23T18:51:05.760" Score="23" ViewCount="6551" Body="&lt;p&gt;Main question:  Does anyone have any good/interesting applications of the rational root theorem or ways to teach it that don't involve conveniently ignoring computer-based tools in order to avoid rote checking  of long root candidates lists?&lt;/p&gt;&#xA;&lt;p&gt;Longer rationale:  The rational root theorem feels like a redundant waste of time.&lt;/p&gt;&#xA;&lt;p&gt;For example, the very first guided problem  in my school's textbook (Holt, Algebra 2) demonstrates how to generate a candidate list, but then it immediately advises the student to use a graphing calculator to narrow down that list to the most likely roots they see on the graph in order to not have to plug and chug 10+ candidates.  Why go through the trouble of generating that list if you're just going to look at the graph to identify the root?  I get the point about verification, but why bother with the list at all if you trust the computer enough to believe its graph?&lt;/p&gt;&#xA;&lt;p&gt;My students and I go through a few examples and practice problems and then check them on wolframalpha, but my students respond: &amp;quot;Why did just do all that work at the beginning?&amp;quot;  I'm at a bit of a loss at what to say.  This isn't like the quadratic formula, which not only is quick, but also there's clear motivation for not totally relying on computer-generated solutions (e.g., works for all quad polys; gives exact form of irrational and complex roots, not rounded).  Moreover, it seems like an artificial chore to construct example polynomials with crazy enough rational roots (e.g., 12345/654321) to call into question the precision of computer-generated solutions.  In other words, problems with numbers difficult enough to make us suspicious of calculator rounding are highly artificial and time-consuming to produce, as well as likely to require a computer to solve in a reasonable amount of time anyway without extreme drudgework.&lt;/p&gt;&#xA;&lt;p&gt;I do see some drill-based skill-value in the repetitive process (factoring the constant term and leading coefficients and listing p/q for each, practice with evaluating a polynomial a various values), and I certainly think that discussions of what the theorem can and cannot tell you are important (e.g., if no rational roots, then the list doesn't apply).  But more than many other topics of Algebra 2, this topic feels like its problems are far more artificially constructed, its solution process and methods are far less broadly illuminating, and the students are just going through the motions to get information that is more easily, more quickly, and more practically obtained in other ways.&lt;/p&gt;&#xA;&lt;p&gt;Thoughts?&lt;/p&gt;&#xA;&lt;p&gt;[Update: 2/24]&#xA;I generally agree with the responses saying that the RRT is a useful logical tool to help step the student up towards Calculus-level processes that require multiple systematic steps and the observation/juggling of multiple pieces of info in a given math expression.&lt;/p&gt;&#xA;&lt;p&gt;I guess I'm just frustrated with the textbook presentation and style of problem I've seen up to now, which not only focuses on obtaining root values (rather than logically ruling things out), but also seems blandly circular in that you're often specifically constructing polynomials to have or not have specific rational roots (and for any appreciably high-degree polynomial example (10+), using a computer program to multiply it out).  As noted in some response below, many polynomials that actually come up in practice are unlikely to have nice rational roots -- hence my concern about wasting time plugging and chugging after the candidate list has been generated.&lt;/p&gt;&#xA;&lt;p&gt;I think I'm going to spend a bit of time writing problems more like the algebra puzzle in a comment below, i.e., provide the student with a finite set of options along with ancillary info which, when combined with the RRT, will rule out all but 1 or 2 of the options.  This &amp;quot;tool for logic puzzle&amp;quot; approach seems much more fruitful and useful for stimulating abstract thinking than the &amp;quot;generate a list and try them all&amp;quot; presentation that practically begs you to use a computer to grind through rote calculations.&lt;/p&gt;&#xA;&lt;p&gt;Thanks for all your comments!&lt;/p&gt;&#xA;&lt;p&gt;[Update 2: 2/25]&lt;br /&gt;&#xA;Related thread:  Should Eisenstein's Criterion be taught in high school?&#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/1044/should-eisenstein-s-criterion-be-taught-in-high-school&quot;&gt;Should Eisenstein’s criterion be taught in high-school?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10572" LastEditorUserId="10572" LastEditDate="2021-02-25T16:20:59.750" LastActivityDate="2022-12-21T15:29:14.023" Title="Why do we teach the Rational Root Theorem? (high school algebra 2)" Tags="|secondary-education|technology-in-education|solving-polynomials|rational-numbers|roots|" AnswerCount="9" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="19511" PostTypeId="2" ParentId="19510" CreationDate="2021-02-23T19:25:18.780" Score="13" Body="&lt;p&gt;Agreed...it's one of the less useful parts of high school algebra. But not because &amp;quot;you could use a computer&amp;quot;--you could say that about almost everything.  And then we get some of the same people who push the &amp;quot;use a computer&amp;quot; who are surprised when their kids flounder because of lack of manipulational ability in calculus. ;)&lt;/p&gt;&#xA;&lt;p&gt;The reason this topic is less useful is because you really don't end up using it to any reasonable extent (I can't ever recall using it!) in a conventional calculus, physics, engineering, or chemistry course.   Full stop.  Regardless if you &amp;quot;check answers with WA&amp;quot; (I wouldn't) or not.  So, if you're time-limited, just don't bother with it, or skimp on the amount of time, don't test it much, etc.  And do more with power laws or the like.&lt;/p&gt;&#xA;&lt;p&gt;If you do have time and student capacity (e.g. upper track kids in an enriched course), I would go ahead and cover it.  There is something about being able to move from guessing how to factor to some sort of systematic method, that is appealing after all the time spent factoring, especially for stronger kids.  [And do cubics and quartics also--IF you have top kids.  Something kind of beautiful about these discoveries.] And the work done is useful practice to be able to handle longer chains of reasoning and manipulation.&lt;/p&gt;&#xA;&lt;p&gt;Maybe a tenous application would be that there is something similar in trying the different possibilities as compared to combinatrics, truth tables, full factorials, issue/logic trees, etc.  Not saying THE SAME, but saying some similar &amp;quot;muscles&amp;quot; as for example Turing's bombe trying combinations of rotors and wiring. Again, very vague and loose.  But a connection.&lt;/p&gt;&#xA;&lt;p&gt;Also, of course, you're only spending PART of the year on this stuff.  So it's not the end of world to do a small amount.  But of course, priorities.  Don't push lower utility, harder topics if you aren't getting more core topics mastered by the students.&lt;/p&gt;&#xA;" OwnerUserId="15514" LastEditorUserId="15514" LastEditDate="2021-02-23T21:05:05.623" LastActivityDate="2021-02-23T21:05:05.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19512" PostTypeId="2" ParentId="19510" CreationDate="2021-02-23T19:25:30.290" Score="33" Body="&lt;p&gt;I would say the standard implementations of the Rational Root Theorem (make a huge list for the sake of making the huge list) indeed feel like a complete waste of time. However, the theorem can sometimes combine nicely with technology tools. I'll provide an example of this.&lt;/p&gt;&#xA;&lt;p&gt;Consider the following algebra puzzle:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$7x^3 -39x^2+52x+30 = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you are allowing a graphing calculator but not a computer algebra system, then the rational root theorem is pretty helpful.&lt;/p&gt;&#xA;&lt;p&gt;Start by realizing that it is a cubic polynomial, so it has at least one real root. We can go look at the graph, hoping to find a nice root:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RedNM.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RedNM.png&quot; alt=&quot;Desmos plot of the real root of the polynomial&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Uhoh.. 0.429. My graphing calculator doesn't know the exact value of that root. It could be any number of things. But &lt;strong&gt;because of the rational root theorem&lt;/strong&gt;, I know that if it is a rational root, it is some number of sevenths. This drastically decreases the amount of effort needed to find that indeed our real solution is &lt;span class=&quot;math-container&quot;&gt;$x = -\frac37$&lt;/span&gt;. If you need all three values of x, you can then proceed to divide by &lt;span class=&quot;math-container&quot;&gt;$7x+3$&lt;/span&gt; to factor the polynomial and find the two complex zeros.&lt;/p&gt;&#xA;&lt;p&gt;In summary, the rational root theorem can be useful as a trial-and-error shortener: it severely restricts the number of &amp;quot;trials&amp;quot; you have to make, and in a situation with a graphing calculator but no computer algebra system, that can still have some value.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-02-23T19:25:30.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19513" PostTypeId="2" ParentId="19510" CreationDate="2021-02-23T20:15:56.477" Score="12" Body="&lt;p&gt;I think an important learning outcome of this part of high school Algebra 2 (theory of polynomials) is developing mathematical capacity to juggle several disparate abstract tools all at once, similar to how in calculus, one uses first derivative test, second derivative test, concavity, asymptotes, intercepts, end behavior, etc., all at once, in some mysterious mash up in the brain, to understand the graph a function. The analogous abstract tools juggled in high school Algebra 2 are rational zero test, Descartes' rule of signs, degree and parity of degree, sign of leading coefficient, factor theorem for intercepts, synthetic division, bound theorem for roots, conjugate pair theorem, etc.  So these are like &amp;quot;training wheels&amp;quot; for what is done in calculus.&lt;/p&gt;&#xA;&lt;p&gt;The point is not to develop fluency with one specific tool such as the rational zero test. Rather, the point it to train the mind to use many tools simultaneously in a mathematical context.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-02-23T20:47:47.943" LastActivityDate="2021-02-23T20:47:47.943" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19515" PostTypeId="2" ParentId="19493" CreationDate="2021-02-23T21:52:20.563" Score="4" Body="&lt;p&gt;The Canvas class for Dartmouth's Spring 2020 course in Graph Theory, Math 38, seems to be mostly open. According to the &lt;a href=&quot;https://canvas.dartmouth.edu/courses/39351/assignments/syllabus&quot; rel=&quot;nofollow noreferrer&quot;&gt;syllabus&lt;/a&gt;, the course uses the 2nd edition of West's &lt;em&gt;Introduction to Graph Theory&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Course Description&lt;br /&gt;&#xA;This course will cover the fundamental concepts of graph theory: simple graphs, digraphs, Eulerian and Hamiltonian graphs, trees, matchings, networks, paths and cycles, graph colorings, and planar graphs. Famous problems in graph theory include: the Minimum Connector Problem (building roads at minimum cost), the Marriage Problem (matching men and women into compatible pairs), the Assignment Problem (filling jobs with applicants), the Network Flow Problem (maximizing flow in a network), the Committee Scheduling Problem (using the fewest time slots), the Four Color Problem (coloring maps with four colors so that adjacent regions have different colors), and the Traveling Salesman Problem (visiting a list of cities minimizing the traveled distance).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you navigate to the &lt;a href=&quot;https://canvas.dartmouth.edu/courses/39351/pages&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pages&lt;/a&gt;, you'll see Week 1, Week 2, etc. Behind each of those links is a wrap-up of the week's lectures.&lt;/p&gt;&#xA;&lt;p&gt;The course ran for 10 weeks, 3 sessions per week. Each week seemed to have 8-12 prerecorded video lectures, most hosted on the &lt;a href=&quot;https://math.dartmouth.edu/%7Enadia/math38/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dartmouth Mathematics Department&lt;/a&gt; server, but a handful linked from YouTube. In addition, there was a live Zoom session that was recorded. The Zoom recordings appear to be inaccessible without a login, but notes taken during the live sessions are available. Also, each session has a PDF of the professor's lecture notes. Finally, if you want to work the weekly problem sets, those are available, too, but the solutions have been removed.&lt;/p&gt;&#xA;&lt;p&gt;Available:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;About 100 prerecorded video lectures (10-15 minutes each)&lt;/li&gt;&#xA;&lt;li&gt;About 8 problem sets&lt;/li&gt;&#xA;&lt;li&gt;About 20 live Zoom notes&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Not available:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Recordings of live Zoom sessions&lt;/li&gt;&#xA;&lt;li&gt;Solutions to problem sets&lt;/li&gt;&#xA;&lt;li&gt;Quizzes, midterm, and final exam&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3593" LastActivityDate="2021-02-23T21:52:20.563" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19516" PostTypeId="2" ParentId="13331" CreationDate="2021-02-24T02:57:13.097" Score="1" Body="&lt;p&gt;Note that set theory when used to present ideas in mathematics makes the work of reading and intuitively verifying these harder. Several &amp;quot;such thats&amp;quot; or &amp;quot;conditions thats&amp;quot; in any  work may well enable and inform the work of subsequent metamathematics  , but intuitions are needed to make maths  intelligeable and applicable  rather than formalisms , which  still require further work to understand them!&lt;/p&gt;&#xA;&lt;p&gt;This will perhaps be why  in particular set theory is preferred  as a TOOL In books  where study of fundamental maths ideas is the Intention, rather than  application or origination of math. A guess would be that Wikipedia's  theoretically inclined approach to giving definitions in terms of sets has distorted perceptions of this?&lt;/p&gt;&#xA;" OwnerUserId="15517" LastActivityDate="2021-02-24T02:57:13.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19517" PostTypeId="2" ParentId="19510" CreationDate="2021-02-24T16:46:49.580" Score="7" Body="&lt;p&gt;I would reluctantly agree that it's not a particularly powerful tool if you have electronics at your disposal.  But I might double down and say that you should be teaching synthetic substitution as well so that students can factor cubic and quartic polynomials without their calculators.  About a year ago, I went down a YouTube rabbit hole of watching videos about factoring cubics and quartics by hand in Hindi (a language I do not speak).  I am given to believe that they do not teach RRT in India, and it is clear from the large number of &amp;quot;intuitive&amp;quot; hacks that they showcase to factor cubic and quartic polynomials that would have fallen almost instantly to RRT.&lt;/p&gt;&#xA;&lt;p&gt;One demonstration I like for giving a nice survey of pre-calculus algebra and trig tools is proving that &lt;span class=&quot;math-container&quot;&gt;$x:=\sin 18^\circ=\frac{-1+\sqrt5}4$&lt;/span&gt;.  Let's take &lt;span class=&quot;math-container&quot;&gt;$\theta=18^\circ$&lt;/span&gt; for the sake of clarity.  Then it is obvious from the cofunction identity that &lt;span class=&quot;math-container&quot;&gt;$\cos2\theta=\sin3\theta$&lt;/span&gt;.  Using the double and triple angle formulas, we can turn that into &lt;span class=&quot;math-container&quot;&gt;$1-2\sin^2\theta=3\sin\theta-4\sin^3\theta$&lt;/span&gt;, or more simply as &lt;span class=&quot;math-container&quot;&gt;$1-2x^2=3x-4x^3$&lt;/span&gt;.  So &lt;span class=&quot;math-container&quot;&gt;$\sin 18^\circ$&lt;/span&gt; is a root of &lt;span class=&quot;math-container&quot;&gt;$4x^3-2x^2-3x+1$&lt;/span&gt;.  Here is where we can use the rational root theorem to quickly observe that &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is a root of that polynomial (and &lt;span class=&quot;math-container&quot;&gt;$x\neq 1$&lt;/span&gt;), obviously, so by the Remainder Theorem we can then divide the polynomial by &lt;span class=&quot;math-container&quot;&gt;$x-1$&lt;/span&gt; and use the Quadratic formula to get the desired answer (again, since we can foresee that the root we want satisfies &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;x&amp;lt;1$&lt;/span&gt; by the definition of sine).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I'm not sure if it goes without saying or not, but the Remainder theorem &lt;strong&gt;is&lt;/strong&gt; essential for helping students to understand why the difference quotient &lt;span class=&quot;math-container&quot;&gt;$\frac{f(x)-f(a)}{x-a}$&lt;/span&gt; always has a removable discontinuity at &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a polynomial function.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-02-24T16:46:49.580" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19518" PostTypeId="2" ParentId="19510" CreationDate="2021-02-24T21:34:31.353" Score="13" Body="&lt;p&gt;The RRT is not taught in isolation.  It is taught as a collection of tools for (partially) factoring polynomials.  It should be taught with &lt;a href=&quot;https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs&quot; rel=&quot;nofollow noreferrer&quot;&gt;Descartes' rule of signs&lt;/a&gt; and some form of &lt;a href=&quot;https://en.wikipedia.org/wiki/Polynomial_long_division&quot; rel=&quot;nofollow noreferrer&quot;&gt;polynomial division&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is direct preparation for understanding the proof of &lt;a href=&quot;https://en.wikipedia.org/wiki/Eisenstein%27s_criterion&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eisenstein's criterion&lt;/a&gt; for irreducibility.  The &amp;quot;shadow&amp;quot; of that application is via &lt;a href=&quot;https://en.wikipedia.org/wiki/Gauss%27s_lemma_(polynomial)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss's lemma&lt;/a&gt;: factorization over the rationals and factorization over the integers are essentially equivalent.  (At the cited link, the statement that is directly to this point is &amp;quot;A non-constant polynomial in &lt;span class=&quot;math-container&quot;&gt;$\Bbb{Z}[x]$&lt;/span&gt; is irreducible in &lt;span class=&quot;math-container&quot;&gt;$\Bbb{Z}[x]$&lt;/span&gt; if and only if it is both irreducible in &lt;span class=&quot;math-container&quot;&gt;$\Bbb{Q}[x]$&lt;/span&gt; and primitive in &lt;span class=&quot;math-container&quot;&gt;$\Bbb{Z}[x]$&lt;/span&gt;.&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;So why do we have these three tools?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Descartes's rule of signs: are there any roots in the ray &lt;span class=&quot;math-container&quot;&gt;$[a,\infty)$&lt;/span&gt;?  (Obtained by looking at the signs of the coefficients of &lt;span class=&quot;math-container&quot;&gt;$p(x-a)$&lt;/span&gt;.)  Likewise, in the ray &lt;span class=&quot;math-container&quot;&gt;$(-\infty, a)$&lt;/span&gt;.  This can be used to find intervals for each of which there is either an isolated root, a multiple root, or one or more complex pairs of roots.&lt;/li&gt;&#xA;&lt;li&gt;For each such interval, can there be a rational root?  This frequently collapses the list of rational candidates by discarding those outside of the intervals.  These first two steps can also be done in the reverse order or iteratively: the RRT suggests a set of partitions to binary search through the list of RRT candidates, letting lack of change in the sign of coefficients indicate that the set of roots in the ray has not changed.&lt;/li&gt;&#xA;&lt;li&gt;Synthetic (or long) division of found rational roots gives an entirely different set of coefficients.  So any yet not-yet-found rational roots have to be on the lists for both the dividend and the quotient.  One can also take the intersections of the rule of signs intervals for both the dividend and the quotient.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;What does this practice?  You have a bunch of tools, some will give you partial information and some will not.  Learn how to select the correct tool, how to apply that tool, and how to update the information you have about the roots from the result of that application.  This is the same pattern one needs for statistical hypothesis testing and (very directly) solving differential equations.  It is a form of Bayesian inference (with all the equivalences that go with that).&lt;/p&gt;&#xA;&lt;p&gt;Students always imagine they will have immediate access to technology.  This is false.  In my own direct experience, I was standing in a test minefield, outside of range of cell towers, and unable to use a phone or other poorly RF-shielded devices for the obvious reason.  And I still had to make useful calculations in that setting.  What was I calculating?  The zero-crossing of an inverse function inferred from gathered discrete data.  So as recently as a couple of years ago, real, live people had to find real, live roots of real, live polynomial models of functions without any access to technology.&lt;/p&gt;&#xA;" OwnerUserId="5800" LastEditorUserId="5800" LastEditDate="2021-03-22T20:09:57.200" LastActivityDate="2021-03-22T20:09:57.200" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19519" PostTypeId="2" ParentId="19510" CreationDate="2021-02-24T23:20:05.463" Score="12" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$x = \sqrt 2 + \sqrt 7$&lt;/span&gt; prove that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is irrational.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;   x - \sqrt 2 &amp;amp;= \sqrt 7 \\&#xA;   x^2 - 2\sqrt 2x + 2 &amp;amp;= 7 \\&#xA;   x^2 -5 &amp;amp;= 2\sqrt 2x \\&#xA;   x^4 -10x^2 +25 &amp;amp;= 8x^2 \\&#xA;   x^4 -18x^2 + 25 &amp;amp;= 0 &#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2 + \sqrt 7$&lt;/span&gt; is a root of &lt;span class=&quot;math-container&quot;&gt;$x^4 -18x^2 + 25$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;According to RRT, if  &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is rational, then &lt;span class=&quot;math-container&quot;&gt;$x = \pm 1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x=\pm 5$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$ x=\pm 25$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = \pm 1 \implies x^4 -18x^2 + 25 = 8 \ne 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = \pm 5 \implies x^4 -18x^2 + 25 = 200 \ne 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = \pm 25 \implies x^4 -18x^2 + 25 = 379400 \ne 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2 + \sqrt 7$&lt;/span&gt; must be irrational.&lt;/p&gt;&#xA;" OwnerUserId="5144" LastActivityDate="2021-02-24T23:20:05.463" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19520" PostTypeId="2" ParentId="19510" CreationDate="2021-02-25T01:15:40.847" Score="5" Body="&lt;p&gt;Theorem: For every integer &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, the polynomial &lt;span class=&quot;math-container&quot;&gt;$x^3 - mx^2 - (m+1)x - 1$&lt;/span&gt; is irreducible among polynomials with rational coefficients.&lt;/p&gt;&#xA;&lt;p&gt;Proof:  This polynomial has degree &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, so if it is a product of lower-degree polynomials, the decomposition is (linear)(quadratic) or (linear)(linear)(linear). Either way, there is a linear factor and thus a rational root (&lt;span class=&quot;math-container&quot;&gt;$ax + b$&lt;/span&gt; with rational &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a \not= 0$&lt;/span&gt; has root &lt;span class=&quot;math-container&quot;&gt;$-b/a$&lt;/span&gt;). By the rational roots theorem, if &lt;span class=&quot;math-container&quot;&gt;$x^3 - mx^2 - (m+1)x - 1$&lt;/span&gt; has a rational root that root must be of the form &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is a factor of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is a factor of &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\pm 1$&lt;/span&gt;.  Let's try each of them:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x) = x^3 - mx^2 - (m+1)x - 1 \Rightarrow f(1) = 1 - m - (m+1) - 1 = -2m -1 \not= 0&#xA;$$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x) = x^3 - mx^2 - (m+1)x - 1 \Rightarrow f(-1) = -1 - m  + m + 1 - 1= -1 \not= 0.&#xA;$$&lt;/span&gt;&#xA;Therefore &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; doesn't have &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; as a root, so it is irreducible as a polynomial with rational coefficients. QED&lt;/p&gt;&#xA;&lt;p&gt;Good luck trying to prove every polynomial in an infinite family of examples is irreducible by computer.  Do you only ever illustrate results in math in your class with individual examples, instead of an infinite family of examples?&lt;/p&gt;&#xA;&lt;p&gt;Homework problem: each student needs to come up with their own infinite family of cubic polynomials and use the rational root theorem to prove every polynomial in the family is irreducible (in &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q[x]$&lt;/span&gt;).  In contrast, there is no simple nonconstant polynomial formula whose values are all prime numbers: &lt;span class=&quot;math-container&quot;&gt;$5n+2$&lt;/span&gt; is prime for infinitely many &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, but it's also composite for infinitely many &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. We think &lt;span class=&quot;math-container&quot;&gt;$n^2 + 1$&lt;/span&gt; is prime for infinitely many integers &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, but this is still an unsolved problem, and in any case it is composite for sure infinitely often (let &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; be an odd number bigger than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2021-02-25T01:15:40.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19522" PostTypeId="2" ParentId="13331" CreationDate="2021-02-25T03:29:42.400" Score="1" Body="&lt;p&gt;Set theory, as well as being a subject in its own right, is a convenient language in which most of mathematics can be expressed.&lt;/p&gt;&#xA;&lt;p&gt;English, as well as being a subject in its own right, is also a convenient language in which most of mathematics can be expressed, but nobody would ask &amp;quot;Why isn't English taught at the outset of math education?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;When learning a new subject, it's useful to go somewhat in order of historical development. That's one reason why we learn whole numbers before trigonometry, for example.&lt;/p&gt;&#xA;&lt;p&gt;Expressing the fundamentals of every part of mathematics in terms of sets is something that people have only really been doing for the last century or so. It's quite a sophisticated point of view which doesn't fit very well with understanding what mathematical objects actually are.&lt;/p&gt;&#xA;&lt;p&gt;For example, the concept of ordered pair is much easier to understand if it's presented in the usual way (&lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; is a thing with the property that &lt;span class=&quot;math-container&quot;&gt;$(a,b) = (c,d)$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$a=c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b=d$&lt;/span&gt;) rather than defining &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; as the set &lt;span class=&quot;math-container&quot;&gt;$\{\{a\}, \{a, b\}\}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On a personal note, I can remember in primary school doing some set theory with strings and objects.&lt;/p&gt;&#xA;&lt;p&gt;You put the objects on the floor and had to put loops of string around some of the objects to create sets. It all seemed rather boring and pointless. It wasn't until I started university, or maybe the final year of high school, that I realised what sets had to do with mathematics at all.&lt;/p&gt;&#xA;" OwnerUserId="814" LastActivityDate="2021-02-25T03:29:42.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19523" PostTypeId="1" CreationDate="2021-02-25T10:48:52.500" Score="2" ViewCount="182" Body="&lt;p&gt;This question concerns teaching a proof of the theorem that if a polynomial &lt;span class=&quot;math-container&quot;&gt;$f \in k[x]$&lt;/span&gt; over an &lt;em&gt;infinite&lt;/em&gt; field &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is the zero function (i.e. &lt;span class=&quot;math-container&quot;&gt;$f(a) = 0$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$a \in k$&lt;/span&gt;) then it is also the zero polynomial (i.e. the coefficients of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; are all zero). (Note that over a &lt;em&gt;finite&lt;/em&gt; field, one can have &lt;em&gt;nonzero&lt;/em&gt; polynomials that are &lt;em&gt;zero&lt;/em&gt; functions.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof&lt;/strong&gt;. We will use the fact that a nonzero polynomial in &lt;span class=&quot;math-container&quot;&gt;$k[x]$&lt;/span&gt; of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; has at most &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; distinct roots, that is, a &lt;em&gt;finite&lt;/em&gt; number of roots. Since &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is the zero function (condition), &lt;span class=&quot;math-container&quot;&gt;$f(a) = 0$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$a \in k$&lt;/span&gt; (definition). However, since &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is infinite (condition), &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has infinitely many roots and thus is the zero polynomial. Were any of the polynomial coefficients nonzero, this would lead to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; having a finite number of roots. However, this would contradict &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; having infinitely many roots and hence there cannot be any nonzero coefficients.&lt;/p&gt;&#xA;&lt;p&gt;What could be done to improve the clarity of the exposition so that the logic of the proof is easier to follow?&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2021-02-25T12:40:13.017" Title="Improving exposition of a proof about polynomials over infinite fields" Tags="|undergraduate-education|proofs|abstract-algebra|polynomials|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19524" PostTypeId="2" ParentId="19523" CreationDate="2021-02-25T12:40:13.017" Score="5" Body="&lt;p&gt;How about this:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; be an infinite field, and let &lt;span class=&quot;math-container&quot;&gt;$f \in k[x]$&lt;/span&gt;.  Assume &lt;span class=&quot;math-container&quot;&gt;$f(t) = 0$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$t \in k$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Assume to the contrary that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not the zero polynomial.  Then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a polynomial of degree &lt;span class=&quot;math-container&quot;&gt;$n \geq 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Choose distinct elements &lt;span class=&quot;math-container&quot;&gt;$z_1, z_2, z_3, \dots, z_{n+1}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;By repeated use of the linear factor theorem, we know that &lt;span class=&quot;math-container&quot;&gt;$f(x) = (x-z_1)(x-z_2)(x-z_3) \dots (x-z_{n+1})g(x)$&lt;/span&gt; for some nonzero polynomial &lt;span class=&quot;math-container&quot;&gt;$g \in k[x]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This show that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has degree at least &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt;, which contradicts the supposition that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; had degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thus our supposition was false, and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; must be the zero polynomial.&lt;/p&gt;&#xA;&lt;p&gt;A few subtle points:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;There is actually a small inductive argument hidden in the &amp;quot;repeated use of linear factors theorem&amp;quot;, which you may or may not want to glide over.&lt;/li&gt;&#xA;&lt;li&gt;One must be careful to explain why &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; cannot be the zero polynomial.  If it were, then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; would be the zero polynomial contradicting the hypothesis.  Understanding the distinction between formal polynomials and polynomial functions is a likely stumbling block in this proof.&lt;/li&gt;&#xA;&lt;li&gt;It is worth pointing out that the linear factor theorem does not fail for the zero polynomial, since &lt;span class=&quot;math-container&quot;&gt;$(x-z)$&lt;/span&gt; divides the zero polynomial for any &lt;span class=&quot;math-container&quot;&gt;$z \in k$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;It is also worth showing that this theorem has content by displaying a polynomial over a finite field which is nonzero as a polynomial, but whose associated function is the constant zero function.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-02-25T12:40:13.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19525" PostTypeId="2" ParentId="19510" CreationDate="2021-02-25T15:25:26.370" Score="5" Body="&lt;p&gt;I really think this questions gets to a bigger issue. This is part of a general problem with our approach to math education.&lt;/p&gt;&#xA;&lt;p&gt;The point of high school math class, from any practical standpoint, is &lt;em&gt;not&lt;/em&gt; to teach students things that they will constantly be using all the time (or even ever). Empirically this is verified in myriad examples, not just the rational root theorem.&lt;/p&gt;&#xA;&lt;p&gt;So I think it's better to not even pretend like we're trying to teach people things that are immediately useful. Absolutely 100% of word problems that students encounter before reaching vector calculus or maybe even differential equations are completely and utterly contrived, and usually made trivial with computer assistance. Students see through this, and by my observation they are always complaining about how non-useful the math they're learning is. It's interesting to observe that they don't make that same complaint nearly as often about other classes, yet I can honestly say (as far as I'm aware) I have never used a single thing I learned in high school outside of math, foreign-language, and web design classes. I certainly share your frustration with textbooks for not realizing this; but you are the teacher, and even if you have to follow the curriculum of the book, you do not have to adopt its attitudes.&lt;/p&gt;&#xA;&lt;p&gt;So then what is the point of teaching math in high school?&lt;/p&gt;&#xA;&lt;p&gt;There's a couple of reasons I can think of that are not related to direct applicability:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It forces students to think abstractly and to grasp new concepts. The rational root theorem is an example here. Simply put, it forces you to think about polynomials in a different way than simply finding the roots with WolframAlpha or by the quadratic formula. From my experience as both a mathematician and as a math educator, it is impossible to be good at math unless you can understand every concept in multiple ways. High school math, if nothing else, is an exercise in being able to think abstractly and think in new ways. If done correctly, it should help to keep student's minds flexible and indirectly fosters creativity. High school math is to the mind what PE is to the body.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is foundational for more advanced mathematics. Analogies to phonics classes in elementary school, you need to learn the basics of English orthography if you want to be able to read and write effectively. It's not directly useful for the purposes of reading comprehension to need to be able to read words out loud, but this is a necessary prerequisite step in learning English. In the same way, all the basic concepts you learn in high school aren't broadly applicable, but it's necessary to go through them in order to get to the parts of math that are applicable. Perhaps not 100% of the things you learn are necessary foundations, so this probably doesn't directly relate to the rational root theorem (it'd be hard to argue that's foundational knowledge for any higher branch of math). But it is another reason to teach the basic, non-applicable concepts.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Math is something that we as a society believe to be valuable and it's culturally and historically enriching to learn. To me this is a weak argument but it is the entire reason we teach history and literature in high school. Thus learning theorems that have historical significance is beneficial just like reading Shakespeare or learning about the War of 1812 is beneficial.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user13995" LastEditorUserId="5794" LastEditDate="2022-12-21T15:29:14.023" LastActivityDate="2022-12-21T15:29:14.023" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19526" PostTypeId="1" CreationDate="2021-02-26T09:28:55.147" Score="7" ViewCount="368" Body="&lt;p&gt;In a &lt;a href=&quot;https://matheducators.stackexchange.com/q/10478/376&quot;&gt;previous question&lt;/a&gt;, I asked about the pros and cons of teaching rings &lt;em&gt;before&lt;/em&gt; groups in abstract algebra. Recently, it has come to my attention that there is a third approach - a unified approach - as exemplified by Matej Brešar in his &lt;em&gt;Undergraduate Algebra: A Unified Approach&lt;/em&gt; in which he introduces the various algebraic structures in parallel. See &lt;a href=&quot;https://www.maa.org/press/maa-reviews/undergraduate-algebra-a-unified-approach&quot; rel=&quot;noreferrer&quot;&gt;https://www.maa.org/press/maa-reviews/undergraduate-algebra-a-unified-approach&lt;/a&gt; for a review by Louisa Catalano, from which I quote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The key novelty and unique feature of the book is that, in this first part, analogous topics on different algebraic structures are considered simultaneously (for example, the section on substructures introduces subgroups, subrings, subfields, etc., subsequently; or another example, the section on normal subgroups and quotient groups is followed by the section on ideals and quotient rings).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What are the pros and cons of taking a parallel approach to algebraic structures instead of teaching them sequentially?&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2021-02-26T17:26:14.273" LastActivityDate="2021-02-27T15:14:16.007" Title="Rings in parallel with groups in abstract algebra" Tags="|undergraduate-education|course-design|abstract-algebra|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19527" PostTypeId="2" ParentId="19526" CreationDate="2021-02-26T16:34:41.390" Score="3" Body="&lt;p&gt;I think this certainly makes a lot more sense than teaching rings &lt;em&gt;before&lt;/em&gt; groups.  I think it might be a wash whether you teach all of the parts of groups and then all of the parts of rings and note the similarities along the way or if you collate them like this book does, but at least this seems pedagogically viable.&lt;/p&gt;&#xA;&lt;p&gt;I think my larger concern is how the definition of these structures are spiraled on top of each other.  I've seen definitions of a group as a set endowed with a binary operation that satisfies these seven axioms and a ring is a set endowed with two binary operations that satisfies twelve axioms (where the number of axioms might vary).  I think it's much stronger to start by defining a monoid as a structure and then say that a group is a monoid where each element has an inverse and a ring is a commutative group with respect to addition and a monoid with respect to multiplication that satisfies the distributive properties.  If you do that, then notions like subrings and ideals will make more sense to the students whether you teach groups and rings separately or concurrently.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2021-02-26T17:39:18.000" LastActivityDate="2021-02-26T17:39:18.000" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19528" PostTypeId="2" ParentId="19526" CreationDate="2021-02-26T22:40:00.893" Score="6" Body="&lt;p&gt;I think this is an interesting question. In the US undergraduate mathematics curriculum, one often finds a sequence of courses &amp;quot;Abstract Algebra I&amp;quot; and &amp;quot;Abstract Algebra II.&amp;quot; I think there is lots of variation. Typically groups are in the first course, but Sylow theorems might be in the second course, along with rings. And the second course is often &amp;quot;topical&amp;quot; depending on the instructor's mathematical interests and perspectives.&lt;/p&gt;&#xA;&lt;p&gt;As I think about this question of teaching rings in parallel with groups, I wonder how this would interface with a two-semester sequence Abstract I/II. At many institutions, only the first course is required for majors, especially for secondary education mathematics majors. The second course is then offered as a elective. Therefore, I think the question must be considered in terms of the local audience and requirements for the major and its various emphases.&lt;/p&gt;&#xA;&lt;p&gt;Another factor, which is not too significant in this case, is consideration of transfer-credit equivalency. By the time a student is taking abstract algebra, they are probably settled into the institution, the department, and emphasis for their mathematics major and so they will not be transferring in or out. But if this question were transplanted to teaching Calculus in an innovative approach--say introduce integration very early in Calculus 1 (Apostol's classic textbook starts with integration!), then caution is needed. At the lower division level, transferring from one institution to another is common and so &amp;quot;transfer-credit currency&amp;quot; becomes an important consideration if we strive to be transfer student friendly.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-02-26T22:40:00.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19529" PostTypeId="1" CreationDate="2021-02-27T11:20:52.457" Score="14" ViewCount="1422" Body="&lt;p&gt;In several online high school teaching resources (I do not want to single out any) I see that &lt;span class=&quot;math-container&quot;&gt;$a-b$&lt;/span&gt; is referred to as the conjugate of &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt; with no restriction on &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I can understand that &lt;span class=&quot;math-container&quot;&gt;$a-bi$&lt;/span&gt; is the conjugate of &lt;span class=&quot;math-container&quot;&gt;$a+bi$&lt;/span&gt; when we consider complex numbers.This is a standard terminology.&lt;/p&gt;&#xA;&lt;p&gt;After &lt;a href=&quot;https://math.stackexchange.com/questions/3478221/is-the-conjugate-of-irrational-numbers-standard-terminology&quot;&gt;asking a question&lt;/a&gt;, I was directed to &lt;a href=&quot;https://en.wikipedia.org/wiki/Conjugate_element_(field_theory)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galois extensions&lt;/a&gt;, so now I understand why we call &lt;span class=&quot;math-container&quot;&gt;$a-\sqrt{b}$&lt;/span&gt; the conjugate of &lt;span class=&quot;math-container&quot;&gt;$a+\sqrt{b}$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are rational and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{b}$&lt;/span&gt; is irrational).&lt;/p&gt;&#xA;&lt;p&gt;However, I still don't understand why it is justified to use the term &amp;quot;conjugate&amp;quot; when we simply change the sign in a binomial. Is it really necessary to give names to everything? Is a formal similarity enough to use the same expression as a name? In the previous two examples the similarity is more than formal. The two type of conjugation have similar properties (since they are both special cases of a general theory). However, I do not see how changing the sign in a binomial without any restriction on the terms in the binomial fits in this general theory.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/rD6e7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rD6e7.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14957" LastEditorUserId="14957" LastEditDate="2021-03-01T06:57:40.277" LastActivityDate="2021-03-01T06:57:40.277" Title="Is &quot;conjugate of a binomial&quot; a standard terminology?" Tags="|terminology|" AnswerCount="5" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="19530" PostTypeId="2" ParentId="19529" CreationDate="2021-02-27T13:59:16.700" Score="-3" Body="&lt;p&gt;A Google search shows that this is common terminology.&lt;/p&gt;&#xA;&lt;p&gt;Just because it's strange to you doesn't mean it is strange to the world.&lt;/p&gt;&#xA;&lt;p&gt;Also a minus bi is often called the COMPLEX conjugate.  That is to differentiate from the usual conjugate, I guess.&lt;/p&gt;&#xA;&lt;p&gt;A comment asks if any books use this terminology.  Well Google books shows several.  Barron's pre calculus is a recent one.  But also several algebra books from the late nineteenth and early twentieth century.  Note that I'm excluding usage in terms of complex conjugated or binomial surds.  But even with the very frequent usage of binomial surd, it would seem that the qualification surd differentiates from normal binomial conjugates.&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure the usage is most common. But it is certainly not uncommon.  Really, what we are getting is a sounds strange to me from questioner.  And Google is a helpful tool to show frequent usage. And I would caution questioners and answers to generalize from own personal experience.  Sounds strange to me doesn't mean strange to the world.&lt;/p&gt;&#xA;&lt;p&gt;For FB, well would a rose by any other name smell as sweet? We are not even kvetching about divided by zero.  We are discussing terminology.  And if people use the term such, how can that be wrong.  Especially if common.  It's just language.&lt;/p&gt;&#xA;&lt;p&gt;Heck.  My dictionary, lists one of the definitions as pairs having features in common but opposite in some particular.  Further, the terms COMPLEX conjugate or binomial conjugate SURD each imply some need to further define the term conjugate...that is referring to binomials with imaginary and real or rational and sqrt irrational parts.  I guess...as opposed to other sorts of conjugates.&lt;/p&gt;&#xA;&lt;p&gt;So certainly a binomial conjugate, used as Sue and many other English speaking math teachers, for over a hundred years, is reasonable usage.  It's not bastsrdizing the language.  It fits fine with other uses if the word, is well defined in class, is supported in dictionaries, is common enough to be frequent, at least.  Not saying it is majority though.&lt;/p&gt;&#xA;&lt;p&gt;So anyhow if the usage is common, what is your problem with it.  A rose is a rose by any other name.  Oh...and a spade is a shovel. &lt;/p&gt;&#xA;&lt;p&gt;And as far as the opposite binomial surd, for rationalizing, so what? Standard method works as is and is easy to learn.  Why complicate?&lt;/p&gt;&#xA;" OwnerUserId="15011" LastEditorUserId="15011" LastEditDate="2021-02-28T17:27:59.173" LastActivityDate="2021-02-28T17:27:59.173" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19531" PostTypeId="2" ParentId="19526" CreationDate="2021-02-27T15:14:16.007" Score="4" Body="&lt;p&gt;Group theory has lots of applications in the sciences, such as to crystallography and quantum mechanics. Rings and fields basically don't, except in trivial ways like doing linear algebra over the complex numbers rather than the reals. I assume that's the reason why the curriculum is normally arranged the way it is, so that STEM students have the option of taking only a first-semester algebra class that focuses on groups.&lt;/p&gt;&#xA;&lt;p&gt;I double-majored in math and physics as an undergrad, then went on to become a physicist. I never used the material from my second-semester algebra class. My first-semester algebra class ended up by developing some of the theory of the classification of finite groups, which from what I remember was intellectual stimulating but had no applications for me. From my point of view, that time would have been better spent on introducing Lie groups and representations.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-02-27T15:14:16.007" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19532" PostTypeId="1" CreationDate="2021-02-27T17:33:16.753" Score="10" ViewCount="2791" Body="&lt;p&gt;I had an idea to improve education in my country with the help of textbooks by popular authors. I thought to publish videos on YouTube with comments and practical explanations about what the author wanted to say. However, the problem is that any textbook is copyrighted and cannot be quoted. This annoys me wildly, these rules just limit the creative ways of teaching the subject, especially mathematics. What do you think about that? Is it wise to put up such barriers?&lt;/p&gt;&#xA;" OwnerUserId="15528" LastActivityDate="2021-03-03T08:59:12.817" Title="Why are all educational textbooks copyrighted? ©" Tags="|textbooks|technology-in-education|teaching|reference-books|book|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="19533" PostTypeId="2" ParentId="19529" CreationDate="2021-02-27T18:21:05.653" Score="8" Body="&lt;p&gt;Calling both &lt;span class=&quot;math-container&quot;&gt;$a+ib \mapsto a-ib$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a+b \sqrt{2} \mapsto a-b \sqrt{2}$&lt;/span&gt; conjugations is reasonable because of the general understanding given by Galois Theory (which has been known much longer than our lifetimes even if the educational establishment can't see fit to require it in our education). Using the same name is in part reasonable beyond the abstract union of the concepts in that similar calculational strategies follow for both:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{1}{2+3i} = \frac{1}{2+3i} \cdot \frac{2-3i}{2-3i} = \frac{2-3i}{13} $$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{1}{2+3\sqrt{2}} = \frac{1}{2+3\sqrt{2}}\cdot \frac{2-3\sqrt{2}}{2-3\sqrt{2}} = \frac{2-3\sqrt{2}}{-14} $$&lt;/span&gt;&#xA;However, there is danger in using something without understanding. If we try to naively extend this idea of conjugation to other numbers such as &lt;span class=&quot;math-container&quot;&gt;$1+\sqrt[3]{7} \mapsto 1-\sqrt[3]{7}$&lt;/span&gt; then we'll find the &lt;em&gt;rationalization scheme&lt;/em&gt; above falls flat:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{1}{1+\sqrt[3]{17}} = \frac{1}{1+\sqrt[3]{17}}\cdot \frac{1-\sqrt[3]{17}}{1-\sqrt[3]{17}} = \frac{1-\sqrt[3]{17}}{1+(\sqrt[3]{17})^2} $$&lt;/span&gt;&#xA;the denominator is still ugly. To find the magic number to rationalize &lt;span class=&quot;math-container&quot;&gt;$1+\sqrt[3]{17}$&lt;/span&gt; we can consider the polynomial &lt;span class=&quot;math-container&quot;&gt;$x^3-17$&lt;/span&gt; which clearly takes &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{17}$&lt;/span&gt; as a root. I'll use my &lt;a href=&quot;https://billcookmath.com/index.html?page=./sage/algebra&quot; rel=&quot;noreferrer&quot;&gt;brother's handy-dandy Sage-based polynomial Euclidean algorithm calculator&lt;/a&gt;  to see that the &lt;span class=&quot;math-container&quot;&gt;$gcd(x^3-17,x+1) = 1$&lt;/span&gt; and in fact:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/mSTNV.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mSTNV.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{-1}{18}(x^3-17)+\left(\frac{1}{18}x^2-\frac{1}{18}x+\frac{1}{18} \right)(x+1) = 1. $$&lt;/span&gt;&#xA;Now, set &lt;span class=&quot;math-container&quot;&gt;$x = \sqrt[3]{17}$&lt;/span&gt; in the identity above to see that:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \left(\frac{1}{18}(\sqrt[3]{17})^2-\frac{1}{18}\sqrt[3]{17}+\frac{1}{18} \right)(1+\sqrt[3]{17}) = 1. $$&lt;/span&gt;&#xA;Which goes to show:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{1}{1+\sqrt[3]{17}} = \frac{(\sqrt[3]{17})^2-\sqrt[3]{17}+1}{18} $$&lt;/span&gt;&#xA;To calculate the same using Galois conjugates is more difficult since the splitting field of &lt;span class=&quot;math-container&quot;&gt;$x^3-17$&lt;/span&gt; has more than one conjugation and essentially you have to multiply by the product of all the Galois conjugates to produce an integer.&lt;/p&gt;&#xA;&lt;p&gt;My apologies if this is slightly vague, I am trying to avoid getting further into the story of Galois theory here. You can look at Section 14.1, Example 5 in Dummit and Foote's 3rd edition where the Galois group of &lt;span class=&quot;math-container&quot;&gt;$x^3-2$&lt;/span&gt; is shown to be &lt;span class=&quot;math-container&quot;&gt;$S_3$&lt;/span&gt; which means there are five nontrivial automorphisms which we would call &lt;em&gt;Galois conjugates&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;All of this said, I would not personally say &lt;span class=&quot;math-container&quot;&gt;$a-b$&lt;/span&gt; is the conjugate of &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt;. I suspect such terminology is made because of similarity in the pattern discussed here for &lt;span class=&quot;math-container&quot;&gt;$a+bi$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$a+b\sqrt{2}$&lt;/span&gt;. If I am going to do something with both &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a-b$&lt;/span&gt; I'll likely say something about remembering the difference of squares formula &lt;span class=&quot;math-container&quot;&gt;$a^2-b^2 = (a+b)(a-b)$&lt;/span&gt;. But, that's just my personal preference.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-02-27T18:21:05.653" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19534" PostTypeId="2" ParentId="19529" CreationDate="2021-02-27T18:34:42.230" Score="3" Body="&lt;p&gt;I have used this terminology for years. It helps students to see that being able to factor a square minus a square is powerful. There is one more context for this that I know of. When a or b is a trig function, multiplying by the conjugate gives us something that fits a Pythagorean identity. (I am now going to test whether that is always true, or just usually true.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-02-27T18:34:42.230" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19535" PostTypeId="2" ParentId="19529" CreationDate="2021-02-27T18:59:53.593" Score="8" Body="&lt;p&gt;As James mentioned, to fully appreciate the term &amp;quot;conjugate&amp;quot; here we need to know a little about Galois theory.&lt;/p&gt;&#xA;&lt;p&gt;The field &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}[\sqrt{2}] =\{a+b\sqrt{2} \;|\; a,b \in \mathbb{Q}\}$&lt;/span&gt; contains the field of rational numbers &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;. If we look for all automorphisms of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}[\sqrt{2}]$&lt;/span&gt; that leave &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt; fixed pointwise, we get the Galois group: &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Gal}(\mathbb{Q}[\sqrt{2}]/\mathbb{Q})$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In particular, &lt;span class=&quot;math-container&quot;&gt;$\varphi \in \mathrm{Gal}(\mathbb{Q}[\sqrt{2}]/\mathbb{Q})$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$\varphi$&lt;/span&gt; is an invertible mapping from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}[\sqrt{2}]$&lt;/span&gt; to itself and preserves addition and multiplication. It turns out that there are only two such mappings. The identity map: &lt;span class=&quot;math-container&quot;&gt;$\mathrm{id}(a+b\sqrt{2})=a+b\sqrt{2}$&lt;/span&gt; and the conjugate map: &lt;span class=&quot;math-container&quot;&gt;$\varphi(a+b\sqrt{2})=a-b\sqrt{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What is going on here is that &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}[\sqrt{2}]$&lt;/span&gt; is obtained by attaching all of the roots of the irreducible (over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;) polynomial &lt;span class=&quot;math-container&quot;&gt;$x^2-2$&lt;/span&gt;. These Galois automorphisms arise from the two ways we can permute the roots of &lt;span class=&quot;math-container&quot;&gt;$x^2-2$&lt;/span&gt;. Namely: &lt;span class=&quot;math-container&quot;&gt;$\pm\sqrt{2} \mapsto \pm\sqrt{2}$&lt;/span&gt; (the identity) or &lt;span class=&quot;math-container&quot;&gt;$\pm\sqrt{2} \mapsto \mp\sqrt{2}$&lt;/span&gt; (the conjugate).&lt;/p&gt;&#xA;&lt;p&gt;We can do the same thing for the complex numbers &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; extending the reals &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. Again, the Galois group only has two elements (called the identity map and the conjugate map). Again, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; is generated over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; by the roots of an irreducible (over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;) polynomial &lt;span class=&quot;math-container&quot;&gt;$x^2+1$&lt;/span&gt; and these maps come from permuting the roots of this polynomial. Namely: &lt;span class=&quot;math-container&quot;&gt;$\pm i \mapsto \pm i$&lt;/span&gt; (the identity) and &lt;span class=&quot;math-container&quot;&gt;$\pm i \mapsto \mp i$&lt;/span&gt; (the conjugate).&lt;/p&gt;&#xA;&lt;p&gt;Anytime (in characteristic 0) we have a quadratic extension (two dimensional extension) of one field over another, we'll get a Galois group with exactly two elements: an identity and a conjugate map.&lt;/p&gt;&#xA;&lt;p&gt;The image of an element under a Galois automorphism is called a conjugate of that element. But in the context of a quadratic extension, if we say &amp;quot;conjugate&amp;quot; we usually mean the non-trivial one (not using the identity map and just getting our element itself right back).&lt;/p&gt;&#xA;&lt;p&gt;Much more generally, if we take any Galois extension &lt;span class=&quot;math-container&quot;&gt;$\mathbb{K}/\mathbb{F}$&lt;/span&gt; and consider some Galois automorphism &lt;span class=&quot;math-container&quot;&gt;$\varphi \in \mathrm{Gal}(\mathbb{K}/\mathbb{F})$&lt;/span&gt; then given &lt;span class=&quot;math-container&quot;&gt;$a,b \in \mathbb{K}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$b=\varphi(a)$&lt;/span&gt;, we call &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; conjugates (relative to that Galois group).&lt;/p&gt;&#xA;" OwnerUserId="453" LastEditorUserId="453" LastEditDate="2021-02-28T05:05:56.707" LastActivityDate="2021-02-28T05:05:56.707" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19536" PostTypeId="2" ParentId="19532" CreationDate="2021-02-28T01:05:21.090" Score="31" Body="&lt;p&gt;Under the laws in most countries, a work such as a book is automatically copyrighted as soon as it is written down. Putting the copyright notice in the book isn't what makes it copyrighted. The notice just makes it easier to sue for copyright violation, because it's harder for the person who copied to claim they didn't know the work was copyrighted.&lt;/p&gt;&#xA;&lt;p&gt;For this reason, it's pretty much true that all textbooks published within the last century are still in copyright. However, many people do write textbooks and intentionally make them free to access. I run a web site, &lt;a href=&quot;http://theassayer.org/&quot; rel=&quot;noreferrer&quot;&gt;http://theassayer.org/&lt;/a&gt; , that catalogs such books. People refer to these as open educational resources, OERs.&lt;/p&gt;&#xA;&lt;p&gt;You can probably produce a commentary or work of criticism on a copyrighted work without violating copyright, and likewise for quoting short sections. This varies from country to country, e.g., US fair use is very permissive.&lt;/p&gt;&#xA;&lt;p&gt;If you want to make a derivative work from an OER, you can do so. However, you should read the license carefully. Some licenses, such as the CC-By-SA license used by wikipedia, require that your derivative work be released under the same license.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-02-28T01:05:21.090" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19537" PostTypeId="2" ParentId="19505" CreationDate="2021-02-28T02:15:13.027" Score="3" Body="&lt;p&gt;To add to the excellent suggestions already made, I intersperse findings from this article (&lt;a href=&quot;https://www.aft.org/sites/default/files/periodicals/dunlosky.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.aft.org/sites/default/files/periodicals/dunlosky.pdf&lt;/a&gt;) throughout the semester in all my classes. The article reviews the results of a 2013 deep dive by educational psychologists into the effectiveness of 10 of the most popular study strategies students use (e.g, flash cards, highlighting the textbook). The psychologists found that practice testing and distributed practice were the two most effective of the 10 techniques they looked at. But that's only a tiny part of the story. I highly recommend taking a look at Table 4 in their main article (&lt;a href=&quot;https://journals.sagepub.com/doi/pdf/10.1177/1529100612453266&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://journals.sagepub.com/doi/pdf/10.1177/1529100612453266&lt;/a&gt;), which details the effectiveness of study strategies based on their intended purpose (e.g., memorization). &lt;br&gt; &lt;br&gt; Another suggestion for Luiz -- since I use flipped classrooms too -- that follows up on Daniel's comment (wondering if there's a way to tell that your students are watching your videos) is to have your students submit written reflections on your videos. My reflections are Google Forms that ask them to: (1) summarize the main mathematical takeaways from the lesson; (2) identify the most interesting part of the lesson; (3) tell me what, if anything, they still find confusing; (4) share anything else that's on their mind; and (5) let me know if there are any issues happening now or on the horizon that might impact their learning. I read over these reflections before each class, use them to customize the class sessions, and construct lesson debriefs using them that answer all their individual questions in red text. I then upload this document (appropriately anonymized) to the LMS. Oh, and I encourage them to send themselves a copy of their Form submission. I sell this by telling them that the more care they take now with their reflections the easier their studying will be later when they go back and review their excellent past reflections; works every time and helps them see the value and payoff of good work now. I'm happy to share my Form with anyone interested.&lt;/p&gt;&#xA;" OwnerUserId="15474" LastActivityDate="2021-02-28T02:15:13.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19538" PostTypeId="2" ParentId="19529" CreationDate="2021-02-28T15:34:09.457" Score="2" Body="&lt;p&gt;I don't think that this usage is standard, although it might be common. Still, I would argue that it is a reasonable enough usage that it is worth trying to fit it into the more standard theory.&lt;/p&gt;&#xA;&lt;p&gt;As observed in the comments, the &lt;span class=&quot;math-container&quot;&gt;$a+b\to a-b$&lt;/span&gt; rule doesn't make much sense if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are concretely realized as numbers. However, it does make sense if we are looking at an expression which lives in the multivariate polynomial ring &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}[a,b]$&lt;/span&gt;. This ring also supports several other forms of conjugacy (i.e. ring automorphisms) such as changing the sign of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and swapping &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;'s and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;'s.&lt;/p&gt;&#xA;&lt;p&gt;tl;dr: It is fine as long as you realize that what's being conjugated are formal expressions, not the numbers themselves.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2021-02-28T15:34:09.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19539" PostTypeId="1" CreationDate="2021-02-28T18:26:38.090" Score="5" ViewCount="312" Body="&lt;p&gt;I have a college student who I've been tutoring over the past two semesters. He's a hard worker and quite bright. Unfortunately, I do not feel like I am having much success in helping him learn mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I work with him on a weekly basis, usually somewhere between 2 and 4 hours. He is currently taking college algebra, and took a remedial algebra course last semester.&lt;/p&gt;&#xA;&lt;p&gt;There are two issues I notice while he is working:&lt;/p&gt;&#xA;&lt;p&gt;1.) Whatever I teach him, he can't seem to retain on a long-term basis. For instance, last semester I'd taught him about factoring and how it relates to distribution, and he had gotten the hang of doing both. This semester, he seems to have almost completely forgotten all of it. He has forgotten his exponent rules, rules for working with fractions, and so on as well. I feel like he does not really understand what a fraction is either, as he cannot easily see the relationship between fractions and decimals, or percentages, or the relationship between fractions and division. So fractions are a point where we trip up. I've tried many strategies that will help him, including going over the forgotten material, and making cheat sheets to help him look things up easily when he's forgotten, but even then it's like he's forgotten the procedure for working with those aforementioned subjects. Needless to say, he's struggling with his college algebra course because it picks up assuming you know all of those things.&lt;/p&gt;&#xA;&lt;p&gt;2.) He seems to be a bit of a perfectionist. He will spend hours on a single problem that he does not understand how to solve. I've suggested that he try a timer, and spend no more than maybe 15-20 minutes on a problem (these are problems I think are probably meant to be done in closer to 5 minutes -- his tests are 30 questions over two hours), but that suggestion does not seem to have helped.&lt;/p&gt;&#xA;&lt;p&gt;To be clear, I think this student is bright. When I build things up for him from the ground up, he follows the argument. I ask questions while I make the argument to ensure that he understands the logic between one step and the next, and to the best of my ability he does. But when it comes time to do the work, I feel like either he did not follow the argument or simply cannot remember it. This student works hard, and I feel like it is more my fault that he is not processing this material than his, but I cannot figure out which approach I should take.&lt;/p&gt;&#xA;&lt;p&gt;Has anyone had any experiences with a student like this? A motivated, hard-working student that just is not getting it. If there are any suggestions, they would be greatly appreciated.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt; I have a hard-working, motivated student who just seems like he is not understanding the material from his math courses, and I need help figuring out strategies on how to move forward with him.&lt;/p&gt;&#xA;" OwnerUserId="15533" LastActivityDate="2021-03-01T17:24:56.667" Title="I need strategies for tutoring a bright, motivated student currently taking college algebra who just doesn't seem to retain the things I teach" Tags="|undergraduate-education|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="19540" PostTypeId="2" ParentId="19539" CreationDate="2021-02-28T19:09:15.427" Score="4" Body="&lt;p&gt;You make the (common) error of emphasizing explanation/understanding as the KEY aspect of pedagogy.  However, the key is PRACTICE, not preaching.  Humans are not computers, to which an instruction set, given once, can be run indefinitely and perfectly, once scanned properly.&lt;/p&gt;&#xA;&lt;p&gt;Now, of course, nothing wrong with an explanation.  And it can help.  But if the trainee is not performing, you need more DRILL.  For instance, you don't even mention how he comes to you with a notebook full of solved problems or the like.  He needs more practice.  He just does.&lt;/p&gt;&#xA;&lt;p&gt;And if the procedures seem long/complex to him (not by words, but by demonstrated performance), then you need to drill simpler problems to start.  And a more progressive (SLOWER) rate of increasing the complexity over time.&lt;/p&gt;&#xA;&lt;p&gt;That's the actionable advice.  Other than that, I would not assume he is as bright as you think...some students appear so conversationally, maybe based on being poised, but are not, in terms of brute intellect.  &amp;quot;College algebra&amp;quot; is a remdial course in the US and has been so for last 50 years at least.  The expectation for a stereotypical freshman is to be taking first semester calculus.  If you're in &amp;quot;college algebra&amp;quot;, you're behind.  This is not 1940 any more, where some high schools didn't even teach trigonometry.  I'm not expecting the kid to be in calculus in high school.  But if he hits college and is still taking algebra he's behind a normal track.&lt;/p&gt;&#xA;&lt;p&gt;Also, remember, &amp;quot;you can't win them all&amp;quot;.  Some students, you just won't help.  (This isn't meant to beg the question...just that people are not roots of equations.  You will not solve every one.)&lt;/p&gt;&#xA;" OwnerUserId="15534" LastEditorUserId="15534" LastEditDate="2021-02-28T19:17:21.793" LastActivityDate="2021-02-28T19:17:21.793" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19541" PostTypeId="2" ParentId="19532" CreationDate="2021-03-01T10:05:52.890" Score="4" Body="&lt;p&gt;Copyright is related to works, not ideas or facts.&lt;/p&gt;&#xA;&lt;p&gt;To quote the &lt;a href=&quot;https://www.oxfordlearnersdictionaries.com/definition/english/copyright_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oxford Learners Dictionaries&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;if a person or an organization holds the copyright on a piece of&#xA;writing, music, etc., they are the only people who have the legal&#xA;right to publish, broadcast, perform it, etc., and other people must&#xA;ask their permission to use it or any part of it&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The important part is that copyright is always related to &amp;quot;works&amp;quot;; which means anything tangible in some form or fashion. The details on this vary between different legislatures, but copyright never covers &lt;em&gt;ideas&lt;/em&gt;, only ever the expression of those.  The source of the term is literally the right to &lt;em&gt;copy&lt;/em&gt; something. Ideas cannot be copied, hence the category simply does not apply.&lt;/p&gt;&#xA;&lt;p&gt;As an example: if you look at &lt;a href=&quot;https://paleoflourish.com/recipe-copyright/&quot; rel=&quot;nofollow noreferrer&quot;&gt;copyrights on cooking recipes&lt;/a&gt;, it is fine to copy and re-use the actual recipe content (i.e., the &amp;quot;idea&amp;quot; - the list of ingredients and the instructions of how to combine them) as long as you do not copy that verbatim. If there is accompanying text, copyright regulations certainly apply to that. The itemized list (&amp;quot;5 apples&amp;quot;, &amp;quot;3 eggs&amp;quot;) is usually not copyrighted because it's just too trivial - there are only so many ways to represent this. Of course, if you wrap your recipe in stories about the country where the recipe is from, or add history lessons or some kind of story, then &lt;em&gt;those&lt;/em&gt; bits are definitely copyright-worthy material.&lt;/p&gt;&#xA;&lt;p&gt;The same applies to any other field. Nobody can copyright a mathematical proof, a computer algorithm, an instruction of how to build a house, or any other &amp;quot;content&amp;quot;. You can only ever copyright the &amp;quot;work&amp;quot; which explains the ideas (in the case of the computer algorithm, that would be the actual program implementing it, for example). While there are plenty of discussions about where the exact borders lie, you can always take the information content and wrap it in your own words / videos / audio representation.&lt;/p&gt;&#xA;&lt;p&gt;Finally, be sure not to mix this up with patents - patents are basically the opposite and &lt;em&gt;can&lt;/em&gt; be used to protect ideas; the requirements to receive a patent are therefore also &lt;em&gt;much&lt;/em&gt; more strict. The copyright is owned by default, in many jurisdictions, or with trivial effort (e.g., just by writing it somewhere near your work).&lt;/p&gt;&#xA;&lt;p&gt;So to wrap it up: you can use all &lt;em&gt;information&lt;/em&gt; from the textbooks freely (as long as you create your own material to express it), but you usually can never just copy stuff verbatim. Exceptions exist (&amp;quot;fair use&amp;quot;) but I think that's not what you're asking about.&lt;/p&gt;&#xA;&lt;p&gt;US-specific sources for reference &lt;a href=&quot;https://en.wikipedia.org/wiki/Baker_v._Selden&quot; rel=&quot;nofollow noreferrer&quot;&gt;Baker vs Selden&lt;/a&gt;, &lt;a href=&quot;https://www.law.cornell.edu/uscode/text/17/102&quot; rel=&quot;nofollow noreferrer&quot;&gt;17 U.S. Code § 102&lt;/a&gt;, similar regulations would be applicable in other legislations.&lt;/p&gt;&#xA;" OwnerUserId="8146" LastEditorUserId="8146" LastEditDate="2021-03-03T08:59:12.817" LastActivityDate="2021-03-03T08:59:12.817" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19542" PostTypeId="2" ParentId="19539" CreationDate="2021-03-01T15:07:56.537" Score="5" Body="&lt;p&gt;The title says the student doesn't &amp;quot;retain,&amp;quot; but your final summary at the bottom that he doesn't &amp;quot;understand.&amp;quot; These seem like totally different things to me. From your description, it seems like the problem is that he doesn't understand, and therefore when he learns an algorithm, it's a bunch of arbitrary mumbo-jumbo, like being asked to memorize a poem in Greek if you don't speak Greek.&lt;/p&gt;&#xA;&lt;p&gt;I think you're being unrealistic when you say that he's bright. If he had normal educational opportunities in the US in K-12, then the fact that he landed in remedial algebra in college means that he has always been a poor math student. You describe him as being able to follow your reasoning when you lay it out for him step by step, and in a comment you say that his online homework system offers him models, which he follows step by step. This is parroting, which is not a high level of intellectual activity.&lt;/p&gt;&#xA;&lt;p&gt;More drill is not the answer. He has had plenty of drill in the past and it hasn't worked. This student does not seem like he's going to succeed unless he gains a better fundamental understanding of things like the conceptual meaning of division, and changes his mental habits. He needs to change his mental habits so that he's no longer using algorithmic crutches to get through problems that he doesn't understand conceptually. So for example, say he doesn't understand how to simplify &lt;span class=&quot;math-container&quot;&gt;$(a^9)^7$&lt;/span&gt;. You can remind him of the identity or make a cheat sheet for him, and that's probably what he wants, but that isn't going to help. So in this situation, you say something like, OK, let's do an example with simpler numbers. Say it was &lt;span class=&quot;math-container&quot;&gt;$(a^2)^3$&lt;/span&gt;. Now exponentiation is repeated multiplication, so can you write the part inside the parens that way? OK, now do you see what will happen when you do the outside exponentation the same way? How many factors of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; did you get? Why did it turn out to be 6?&lt;/p&gt;&#xA;&lt;p&gt;The problems you describe with not understanding division and ratios conceptually are extremely common. Kids take a wrong turn in 4th grade, and they end up in college still not understanding these concepts. This has to be approached conceptually from a variety of angles. For example, in a word problem with some context, you can talk about trends, such as what would happen if the denominator got bigger. If he says that a 7% change is a factor of 0.07, talk him through an example with 7% tax on a \$100 TV and get him to come up with the fact that they're not selling him the TV for $7. Show him dimensional analysis so that he can see whether it makes sense that meters divided by meters/second gives units of time. Conceptual understanding of multiplication and division comes through a long process of seeing things like this from all these different angles, and developing mental habits of sense-making.&lt;/p&gt;&#xA;&lt;p&gt;The idea of you making cheat sheets for him is really bad IMO. Encourage him to make cheat sheets for himself if he needs them. He needs to be independent and take responsibility for organizing his own knowledge. That's part of the process of gaining conceptual understanding.&lt;/p&gt;&#xA;&lt;p&gt;So we're talking about getting this student to change his mental habits and develop better conceptual understanding of the last decade's worth of math instruction -- not the last semester. Realistically, most students at this age are not willing to put in the kind of sustained effort that would be required to completely retrain their brains. And making this kind of change takes years, so if they're already in a college STEM major, then they don't have enough time to make the change. You don't say what the student's major is or his intended career. If he's an engineering major, then it's essentially impossible to accomplish these things this late in the game. If he wants to be a pharmacist, then it's realistic to set a goal like getting him to the level where he can convert milligrams to grams without &lt;em&gt;ever&lt;/em&gt; getting the conversion factor wrong.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-03-01T15:29:03.060" LastActivityDate="2021-03-01T15:29:03.060" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19543" PostTypeId="1" CreationDate="2021-03-01T16:24:50.127" Score="2" ViewCount="244" Body="&lt;p&gt;My child in grade 3 often needs a reminder because in a task where he needs to do addition he would subtract instead of adding  or v.v ignoring the signs given in a task. He would also forget adding carry.&lt;/p&gt;&#xA;&lt;p&gt;Why this happens and what can be done to ensure that he does not commit this mistake in future?&lt;/p&gt;&#xA;&lt;p&gt;He understands the procedure as well as concept but he often commits this silly mistake. What kind of task should be given to him so that he keeps the sign in mind before and during solving the task?&lt;/p&gt;&#xA;" OwnerUserId="13103" LastActivityDate="2021-03-01T16:24:50.127" Title="Why students do not notice sign of plus/minus?" Tags="|primary-education|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19544" PostTypeId="2" ParentId="19539" CreationDate="2021-03-01T17:24:56.667" Score="0" Body="&lt;p&gt;tl;dr Does he need to know algebra, or just to pass the course? Rant follows.&lt;/p&gt;&#xA;&lt;p&gt;The kinds of struggles you describe suggest that he won't/can't major in a science. Perhaps  college algebra is simply a hurdle he must jump in order to get on with his literature or history or psychology, where he may indeed be a good student.&lt;/p&gt;&#xA;&lt;p&gt;If that's the case, what you need to do for him is help him pass the exams. Find some samples, then do lots of problems just like those.&lt;/p&gt;&#xA;&lt;p&gt;The sad fact is that although algebra is an important step on the path to more real mathematics, &amp;quot;college algebra&amp;quot; is often a conspiracy. The problems on the exams are just like the ones on the book with different numbers (perhaps different letters, but usually just x and y). When a student passes the exam the teacher can pretend to have taught something the student can pretend to have learned. But it all vanishes right after the exam.&lt;/p&gt;&#xA;&lt;p&gt;Final note. Fractions are conceptually difficult. In elementary school they occur as divisions of pies, points on the number line, and in formal manipulations. Understanding how those ideas fit together is hard for teachers, let alone for students. See &lt;a href=&quot;https://math.stackexchange.com/questions/1127483/how-to-make-sense-of-fractions/1127776#1127776&quot;&gt;https://math.stackexchange.com/questions/1127483/how-to-make-sense-of-fractions/1127776#1127776&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4984" LastActivityDate="2021-03-01T17:24:56.667" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19545" PostTypeId="1" CreationDate="2021-03-01T18:25:54.697" Score="6" ViewCount="219" Body="&lt;p&gt;I am running into problems explaining to my students in high school what exactly is sample space in probabilities, especially with identical objects.&lt;/p&gt;&#xA;&lt;p&gt;For example, according to Q6.2.3 of &lt;a href=&quot;http://homepages.math.uic.edu/%7Ebpower6/math160/math160-6-2.pdf&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; UIC website, the sample space for a letter is chosen at random from the word “MISSISSIPPI” is S={M, I, S, P}. The reason given was the letters, for example S, are indistinguishable.&lt;/p&gt;&#xA;&lt;p&gt;However, I have also seen solution to question such as:&lt;/p&gt;&#xA;&lt;p&gt;“Write down the sample space of the experiment of drawing a marble at random from a bag containing 5 identical blue (B) marbles and 4 identical red (R) marbles.”&lt;/p&gt;&#xA;&lt;p&gt;would be S={B1,B2,B3,B4,B5,R1,R2,R3,R4} but &lt;strong&gt;not&lt;/strong&gt; S={B,R} like in the example above.&lt;/p&gt;&#xA;&lt;p&gt;When a student asked me why is the marbles in the second example distinct if they are identical and I couldn’t really give a convincing answer. I explained that in the first example, we are not drawing a letter from a bag but instead we are picking a letter from the word right in front of us. So we either choose a M, I, S or P. But in the second example, the marbles are in a bag  so we can’t see the balls. And when we pick, we don’t know what we will be drawing. But when a student asked “then wouldn’t the second example be S={B,R} since we either pick up a blue or red marble.&lt;/p&gt;&#xA;&lt;p&gt;A second student then asked then why can’t we distinguish the letters in the first problem such that S={M,I1,S1,S2,I2,S3,S4,I3,P1,P2,I4}&lt;/p&gt;&#xA;&lt;p&gt;I was a bit dumbfounded and could not give a convincing explanation. Any suggestions how should I better explain it?&lt;/p&gt;&#xA;" OwnerUserId="15538" LastEditorUserId="15538" LastEditDate="2021-03-01T18:31:36.440" LastActivityDate="2021-03-02T09:59:33.027" Title="Difficulty in explaining sample space" Tags="|secondary-education|probability|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19546" PostTypeId="1" CreationDate="2021-03-01T19:02:54.627" Score="65" ViewCount="8732" Body="&lt;p&gt;I have been professionally tutoring a wide range of students (from elementary school through graduate school) for many years. Most of them are from the United States. I generally focus on helping my students develop procedural fluency built on a conceptual foundation (i.e. &amp;quot;now I can do the thing and I won't forget how to do the thing because I understand what it is and why it's useful&amp;quot;). I think that a conceptual understanding is very important but only because it serves a utilitarian purpose of helping students remember methods and feel empowered as they work. With this in mind, I happen to think that knowing how to add, subtract, multiply, and divide by hand is important. However, I am beginning to think that perhaps I am simply biased because of my upbringing, and that such things really aren't valuable anymore.&lt;/p&gt;&#xA;&lt;p&gt;The vast majority of my students are younger than 20 years old, and the way they've been taught math is almost unrecognizable to me. Most of them have (for as long as I've known them) never taken a math or science test during which a calculator wasn't explicitly allowed. As a predictable result, they rely on their calculators to do all calculations. I suspect that most of them could rattle off some of the times tables if their life actually depended on it, but I've seen all of them type single digit multiplication and addition problems into their calculators. I know for a fact that many of them never learned the old trick for multiplying by 10 (you know, putting a zero at the end), and almost none of them know how to check for divisibility above 2. On top of that, my grade school and middle school students generally don't remember any method for conveniently adding, subtracting, or multiplying by hand. Some of them can multiply by drawing boxes or number lines, but the ones who can don't understand what they are doing.&lt;/p&gt;&#xA;&lt;p&gt;For older students, we generally have bigger problems and more abstract topics to tackle, but for my youngest students (4th - 8th grade), I try to encourage them to develop a familiarity with the simplest forms of by-hand addition, subtraction, and multiplication (you know: one number on top of the other, carry the 2, etc.). While the methods are usually familiar to them, this sort of encouragement generally goes nowhere because their teachers don't care and because they always have either a dedicated calculator or their phone handy.&lt;/p&gt;&#xA;&lt;p&gt;I tell students that calculators aren't always allowed (maybe that's not true anymore). I tell them that knowing is faster than typing it in (the difference is admittedly minimal). I remind them that different calculators have different syntaxes, so it's important to have a way to do it by hand if only to check the calculator (true, but not persuasive). I even joke that the world might end, and they'll need math when the electrical grid goes down (they remind me that their calculator has a little solar panel on it).&lt;/p&gt;&#xA;&lt;p&gt;Truthfully, it's depressing, and what's worse, I'm not sure I'm convinced anymore either. I mean, I don't know any convenient algorithms for computing square roots or logarithms by hand but that never held me back.&lt;/p&gt;&#xA;&lt;p&gt;So, to restate the question, is there actually any demonstrable, practical value in learning to add, subtract, multiply, and divide by hand (even if the reason might not be persuasive to a 10-year-old)?&lt;/p&gt;&#xA;" OwnerUserId="15436" LastEditorDisplayName="user18187" LastEditDate="2023-01-13T05:31:31.923" LastActivityDate="2023-10-23T14:43:50.890" Title="Is there a virtue to learning how to compute by hand?" Tags="|reference-request|primary-education|arithmetic-operations|arithmetic|middle-school|" AnswerCount="17" CommentCount="33" ContentLicense="CC BY-SA 4.0" />
  <row Id="19547" PostTypeId="2" ParentId="19546" CreationDate="2021-03-02T01:06:30.850" Score="9" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Brian D. Rude, &amp;quot;The Case For Long Division.&amp;quot; 2004. &lt;a href=&quot;http://www.brianrude.com/calndv.htm&quot; rel=&quot;noreferrer&quot;&gt;HTML link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is a somewhat long (unpublished) article (which I haven't studied carefully),&#xA;but maybe the excerpt below suffices to give the gist&#xA;of it. Before this excerpt, among his closing sentences are: &amp;quot;But a calculator should be more than a paperweight. Let’s teach for understanding.&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;My argument is that long division is just one procedure, among many, that promotes the usefulness of arithmetic, that helps to make arithmetic a life skill rather than just one more subject forgotten, one more door opened but then allowed to close again by lack of relevance to everyday life. I will make my argument for long division purely on the utilitarian aim of education. My argument is that even the person who has no intention of learning anything more than perhaps ninth grade algebra should learn long division. [...] But my argument has very little to do with being able to divide one number by another when an actual division problem arises in everyday life. We’ll use a calculator for that. My argument is that we should not expect anyone to really make sense out of arithmetic if they can’t do long division. If you can’t really make sense out of arithmetic a calculator is nothing more than a paperweight.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm not evaluating his argument; just posting as someone's strongly held opinion.&#xA;Although I certainly agree that we should &amp;quot;teach for understanding&amp;quot;!&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-03-02T01:06:30.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19548" PostTypeId="2" ParentId="19545" CreationDate="2021-03-02T04:51:25.250" Score="7" Body="&lt;p&gt;The (correct) sample space depends on how the probability problem has been framed:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The set of letters in “&lt;span class=&quot;math-container&quot;&gt;$MISSISSIPPI$&lt;/span&gt;” is indeed &lt;span class=&quot;math-container&quot;&gt;$S_1=\{M,I,S,P\}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In a probability experiment with sample space &lt;span class=&quot;math-container&quot;&gt;$S_1,$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$P(M)=\frac1{11};P(I)=P(S)=\frac4{11};P(P)=\frac2{11}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;On the other hand, we might prefer a classical probability experiment&#xA;where all outcomes are equally likely. Distinguishing among different&#xA;occurrences of each letter, our sample space is now&#xA;&lt;span class=&quot;math-container&quot;&gt;$S_2=\{M,I_1,S_1,S_2,I_2,S_3,S_4,I_3,P_1,P_2,I_4\}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here, the &lt;em&gt;event&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\{I_1,I_2,I_3,I_4\}$&lt;/span&gt; of choosing letter &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; has probability &lt;span class=&quot;math-container&quot;&gt;$\frac4{11},$&lt;/span&gt;&#xA;whereas the &lt;em&gt;outcome&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$I_2$&lt;/span&gt; has probability &lt;span class=&quot;math-container&quot;&gt;$\frac1{11}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-03-02T05:06:54.307" LastActivityDate="2021-03-02T05:06:54.307" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19549" PostTypeId="2" ParentId="19546" CreationDate="2021-03-02T04:52:10.033" Score="27" Body="&lt;p&gt;I find the ability to estimate calculations quite useful and I think you need to be able do do calculations to estimate them.  If you are keeping a grocery budget, I would suggest you should know what your groceries will cost within &lt;span class=&quot;math-container&quot;&gt;$10\%$&lt;/span&gt; before they are rung up.  To know that, you need to be able to add and multiply in your head.  You don't need many digits of precision, but you need to be able to do it.  I was in engineering during my career and it helps a lot to know what is important and what is not.&lt;/p&gt;&#xA;" OwnerUserId="12340" LastActivityDate="2021-03-02T04:52:10.033" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19550" PostTypeId="2" ParentId="19546" CreationDate="2021-03-02T06:22:56.293" Score="58" Body="&lt;p&gt;The following response is written with elementary-to-high-school mathematics in mind.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A lack of a decent number sense really does encumber making sense of&#xA;and parsing word problems, as well as the process of exploring&#xA;solution strategies. It is akin to interpreting a passage written in a&#xA;not-so-familiar dialect: attention to the actual substantive problem&#xA;at hand is being competed for.&lt;/li&gt;&#xA;&lt;li&gt;Performing a sequence of algebraic manipulation involves&#xA;micro-decisions along the way that don't require a calculator, but definitely do require facility/flexibility with arithmetic and fractions, to be achieved efficiently &lt;em&gt;and&lt;/em&gt; effectively.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Using the calculator as a crutch significantly atrophies the student's numerical fluency and impairs the development of their intuition and mental dexterity. With a weak number sense, even basic procedures consume valuable working-memory space—sometimes causing a bottleneck—interfering with higher levels of competence (interpreting problems, drawing connections, devising strategies, reasoning correctly, etc.).&lt;/p&gt;&#xA;&lt;p&gt;The student who is comfortable computing by hand avoids these pitfalls. In contrast, a key side effect of being over-reliant on the calculator is the subconscious tendency to reach for and jab at it as a distraction from actually trying to make sense of problems.&lt;/p&gt;&#xA;&lt;p&gt;To be sure, mathematics is certainly not just computation/arithmetic; however, being adept at the latter enables us to focus on—and &lt;a href=&quot;https://matheducators.stackexchange.com/a/2004/372&quot;&gt;informs&lt;/a&gt;—the more creative aspects of the former: recognising patterns, discerning structure, comprehending &amp;amp; modelling &amp;amp; solving problems, and communicating with accuracy &amp;amp; unambiguity &amp;amp; conciseness &amp;amp; persuasiveness &amp;amp; wit.&lt;/p&gt;&#xA;&lt;p&gt;Although I agree that long or non-simple computations should be outsourced to or &lt;em&gt;aided&lt;/em&gt; by a computer/calculator, I believe that, especially in elementary school, students ought to be given ample opportunities to cultivate number sense and numerical fluency, and the attendant self-confidence. Calculator-prohibited tests aren't a bad idea to prevent calculator “addiction” down the road.&lt;/p&gt;&#xA;&lt;p&gt;Students who learn to depend on a calculator for trivial computations will have a curtailed risk comprehension and statistical and scientific literacy. More fundamentally, they will lack the invaluable virtue of numeracy, which facilitates independent sanity checking, estimation and error-checking—not just at the end of a problem solution, but throughout the process—not just on paper, but also in routine day-to-day situations.&lt;/p&gt;&#xA;&lt;p&gt;P.S. &lt;a href=&quot;https://matheducators.stackexchange.com/a/26369/372&quot;&gt;Jochen's examples&lt;/a&gt; illustrate how a paucity of basic estimation skills, or low competence with ratios and percentages, impoverishes routine everyday situations and is in some ways as limiting as illiteracy.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-10-08T07:37:31.263" LastActivityDate="2023-10-08T07:37:31.263" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19551" PostTypeId="2" ParentId="19546" CreationDate="2021-03-02T09:08:30.537" Score="3" Body="&lt;p&gt;I think if we can introduce a variety of algorithm's for teaching addition and multiplication, it'd be beneficial for the student. No matter what mathematics level you are, algorithmic thinking is always important. When I say algorithmic thinking, I mean to say that we have a set of rules to tackle a problem such that applying the rules in the correct order always solves the problem.&lt;/p&gt;&#xA;&lt;p&gt;But, other than accuracy, I don't think we should try to 'test' the computational limit of students like asking them to multiply / add large digit numbers, we should stick to working with small numbers fast and quick because that is the skill which is important in real life for example for buying something.&lt;/p&gt;&#xA;&lt;p&gt;At a higher level, I think for applied science subjects, we should just stop putting it as an exercise for students to calculate as it simply check if student understands the topic or not. If a question in a test comes in such a way, then it tests the students ability to execute a standard algorithm rather than their grasp of the topic (the second being the important thing when dealing with an applied sciences).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Truthfully, it's depressing, and what's worse, I'm not sure I'm convinced anymore either. I mean, I don't know any convenient algorithms for computing square roots or logarithms by hand but that never held me back.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As for this, I think introducing some higher level mathematics help a lot. For example, introduce differential calculus and Taylor series as a method to overcome this. I am a believer that we can introduce the concept to a person who simply has done basic differentiation and grasped polynomials (I've written such an &lt;a href=&quot;https://brianbabu890.medium.com/a-brief-introduction-to-taylor-series-47416e7b9123&quot; rel=&quot;nofollow noreferrer&quot;&gt;introduction here&lt;/a&gt;) though I'm not sure how easy it is to explain differentiation to a ten year old. There was some discussion on a similar problem in this &lt;a href=&quot;https://math.stackexchange.com/questions/4042364/why-does-an-exponential-function-eventually-get-bigger-than-a-quadratic&quot;&gt;MSE post&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Other ways include introducing how that topic could be applied into other fields, for example consider this theorem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For any compose number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, one of it's prime factor must be less than&lt;span class=&quot;math-container&quot;&gt;$ \sqrt{x}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And other number theory / modular arthimetic &lt;a href=&quot;https://www.youtube.com/watch?v=lJ3CD9M3nEQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;results&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;To be honest,  I wouldn't say this answer is completely original, these ideas had been implemented by professor Greist in his online MOOC course on &lt;a href=&quot;https://www.youtube.com/watch?v=LYCwHcHRhCg&amp;amp;list=PLKc2XOQp0dMwj9zAXD5LlWpriIXIrGaNb&quot; rel=&quot;nofollow noreferrer&quot;&gt;calculus&lt;/a&gt; where he assumed knowledge knowledge of basic differentiation and then began with Taylor's theorem. If you see through the playlist, he brings in the applications of calculus in various fields such as economics, thermodynamics , life sciences and more&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2021-03-02T09:08:30.537" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19552" PostTypeId="2" ParentId="19545" CreationDate="2021-03-02T09:59:33.027" Score="2" Body="&lt;p&gt;I don't agree with your explanation that there is any inherent difference in the bag of balls and the letters of Mississippi.  I do think that the difference is how you define the sample space, ie as a space with equally like probabilities, or a set of distinguishable outcomes that do not have eqully likely probabilities.&lt;/p&gt;&#xA;&lt;p&gt;If a sample space assumes that all probabilities are equally likely (which many treatments of probabilities do), then you would have to use&#xA;S(1) ={M,I1,S1,S2,I2,S3,S4,I3,P1,P2,I4}&lt;br /&gt;&#xA;as your student suggested.&lt;/p&gt;&#xA;&lt;p&gt;Some treatments of probability do not assume that all probabilities are equally likely and they might allow you to call the set of letters,&#xA;S(2)={M,I,S,P}&#xA;the sample space. However I would call it a set of distinguishable outcomes that do not have equally likely probabilities.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-03-02T09:59:33.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19553" PostTypeId="2" ParentId="19546" CreationDate="2021-03-02T11:43:52.913" Score="25" Body="&lt;p&gt;Yes!  But the virtue doesn't lie in being able to do the calculation but in gaining a feel for numbers as well as algorithmic thinking.&lt;/p&gt;&#xA;&lt;p&gt;I teach Computer Science freshmen and one of the first things we need to do is introducing base 2 as well as number systems working with modulo (two's complement).  We also introduce basic circuitry to do addition/subtraction and multiplication/division.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In order to understand why &amp;quot;shifting&amp;quot; left/right in binary is multiplication/division by 2, it's helpful to have seen the same concept with 10&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;understanding the, very frequent, check for even/odd in binary is much easier when you have a basic grasp of how number systems work&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;binary subtraction is a hard one for many, I frequently need to re-teach them decimal subtraction first.  If I give them a subtraction &lt;span class=&quot;math-container&quot;&gt;$a-b$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;b$&lt;/span&gt; and ask them to do it by hand, they don't manage it (since we create infinitely leading 9s/1s).  This is an important insight when implementing algorithmic subtraction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Understanding algebraic groups like the one set up with two's complement (equivalence classes using modulo) is much easier.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Long division and written multiplication gives easy insights on why, e.g., multiplication of two &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; digit numbers requires at most &lt;span class=&quot;math-container&quot;&gt;$2p$&lt;/span&gt; digits.  Similarly, why addition can only increase the number of digits by 1.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, half/full/carry-ripple adders are intuitive if you have worked with written addition (carry digits, etc.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, rest assured, all of this is useful.  However, it's a whole different ordeal to convince your students.  Depending on their level you can try to ask them to &amp;quot;algorithmicize&amp;quot; addition (and even program it!) or ask questions like &amp;quot;whats the maximum number I can have if I multiply two &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; digit numbers?&amp;quot;, with emphasis on &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, i.e., not a fixed number.  They also need to explain &lt;em&gt;why&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you can find exercises that actually make them need to use written calculations, they will be motivated to learn these.&lt;/p&gt;&#xA;" OwnerUserId="13956" LastActivityDate="2021-03-02T11:43:52.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19554" PostTypeId="2" ParentId="19546" CreationDate="2021-03-02T14:27:44.030" Score="13" Body="&lt;p&gt;I attended the Computer based math educational summit back in 2016 and found their ideas interesting. I agree with some of their points and disagree with others, but it is certainly interesting to look at the following diagram from their website.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/71f3b.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/71f3b.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;They argue, that if we let computers (proper ones, not handheld calculators) do the repetitive calculations, then we can focus more on setting up problems and interpreting results. Moreover, we can tackle more interesting problems. This is a different type of thinking, but logic, reasoning, algorithmic thinking is certainly part of it.&#xA;Their website is &lt;a href=&quot;https://www.computerbasedmath.org/&quot; rel=&quot;noreferrer&quot;&gt;https://www.computerbasedmath.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2021-03-02T14:27:44.030" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19555" PostTypeId="2" ParentId="19546" CreationDate="2021-03-02T15:12:18.943" Score="9" Body="&lt;p&gt;Beyond having worked as a programming teacher I have no experience with math education, but  this is a topic I &lt;a href=&quot;https://matheducators.stackexchange.com/q/2617/860&quot;&gt;have been fascinated with for years&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Arguments in favor of mental/manual arithmetic can be typically categorized as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;h3&gt;It's important in daily life&lt;/h3&gt;&#xA;&lt;p&gt;The argument typically goes that you need to be able to do arithmetic a lot in daily life with the most common example being buying items in the shop. It's good to be aware that physical shopping has been on the decline for decades and even in physical shops &lt;a href=&quot;https://digitalcontent.api.tesco.com/v1/media/clubcard-mktg/scan+as+you+shop/cc-scan%2Bas%2Byou%2Bshop-stamp-3.jpeg&quot; rel=&quot;noreferrer&quot;&gt;handheld scanners that tell you the sum on the go&lt;/a&gt; are on the rise.&lt;/p&gt;&#xA;&lt;p&gt;As someone who has relatively bad arithmetic skills and seeing how most people function with even worse arithmetic skills in daily life I dare say this is just not true nowadays.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;h3&gt;It's important to be able to do arithmetic for maths&lt;/h3&gt;&#xA;&lt;p&gt;Let me very clearly state that this might be true for students who will be studying pure maths at university level. For everybody else the most important life skill is typically being able to reduce a problem into steps that allow coming to a conclusion. The typical math education example of this is word problems. The ability to express a real problem as a maths problem is typically the hardest part, but any time spend teaching students arithmetic takes away from the time spend on teaching &amp;quot;problem solving&amp;quot;. Understanding for example what integration means and when it's useful is far more valuable than the months we spend on actually doing integrations by hand.&lt;/p&gt;&#xA;&lt;p&gt;To this day I value the ability to express something in mathematical terms, as the next step is going to a calculator or wolfram alpha to actually get to a conclusion/the next step. In my experience most mathematical educations are incredibly focused on the mathematical computation and the 'solve actual problems' is barely touched upon.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;h3&gt;It's useful to catch mistakes&lt;/h3&gt;&#xA;&lt;p&gt;The argument goes that being able to do mental arithmetic makes it easier to notice mistakes. This is without a doubt completely true. At the same time I have noticed that nearly the same benefit is gained when you learn to just do 'order of magnitude estimates' which I do myself.&lt;/p&gt;&#xA;&lt;p&gt;At the end of the day it boils down to teaching an intuitive understanding of the mathematics itself, rather than teaching a student to become a human calculator. Being able to look at &amp;quot;find &lt;code&gt;∫[x sin(x)] dx&lt;/code&gt;&amp;quot; and being able to imagine how &lt;code&gt;x sin(x)&lt;/code&gt; would look, and thus how the integral of it would kind of look is incredibly valuable. That means that regardless of how you calculate &lt;strong&gt;and use&lt;/strong&gt; that integral, you will be able to tell whether the result is complete bullshit or not.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The current highest voted answer calls using a calculator a crutch, but I would really want to ask everybody to consider that using a CAS calculator or WA is just offloading the 'stupid, but error prone' part of mathematics and leaving the user with the truly hard parts. Computers and calculators aren't a crutch, they are - if properly taught - a stronger foundation. In my own high school I did the highest level of maths and every single student in my class was better than me in actually &amp;quot;doing&amp;quot; traditional mathematics, but 10% of our final grade came from two essays where we had to mathematically model, analyze and interpret simplified real problems and I comparatively aced those essays as my programming interests had always created a greater focus on the bigger picture (where and how can I use what I am taught). That disparity where I was helping students who were far better at mathematics than me fascinated me for years which led me 7 years ago to computerbasedmath.org which posits that far too little time is spend on &lt;a href=&quot;https://matheducators.stackexchange.com/a/19554/860&quot;&gt;definition, abstraction and interpretation, because all time in maths education is spend on computation&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="860" LastActivityDate="2021-03-02T15:12:18.943" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19557" PostTypeId="2" ParentId="19546" CreationDate="2021-03-02T19:24:53.953" Score="10" Body="&lt;p&gt;I have thought a lot about this question since posting it, and having read the other answers and the many comments, I want to add a perspective that no one else seems to have given.&lt;/p&gt;&#xA;&lt;p&gt;Most of the real work in doing math is understanding and conceptualizing the problem rather than in computing an answer. This will be apparent to anyone who has read a mathematical journal article which included no numerals anywhere in its text. This will also be apparent to a high school student in algebra or calculus who spends most of their time &amp;quot;setting up&amp;quot; their problem rather than &amp;quot;solving it.&amp;quot; The value of a calculator is to make the final step of a problem (i.e. solving it) easier; the value of pen and paper is to make the conceptualizing part (i.e setting it up) easier. To forgo learning pencil-and-paper computations early on is to build habits that make conceptualization more difficult later.&lt;/p&gt;&#xA;&lt;p&gt;In learning arithmetic, conceptualizing a problem is often as simple as determining whether to multiply or divide. Writing this down seems like a trivial formality even to (perhaps, especially to) children. As a result, it can appear to a student that mathematics is merely the process of figuring out what operation to perform and then doing it. To an extent this may be true, but figuring out what to do is generally very hard. In fact, it is so hard that we emphasize word problems at all stages of math education precisely to train this skill. In later courses, it becomes so challenging that pages of hand-written work are often necessary if for no other reason than to keep a record of what you were thinking in case your answer is later determined to be incorrect. However, because of the seeming triviality of arithmetic, it can seem that the conceptualization is trivial too and that we may as well skip the formality of writing it down.&lt;/p&gt;&#xA;&lt;p&gt;This is exactly where the rubber meets the road. Figuring out whether to multiply or divide two numbers is anything but trivial - especially for children just beginning their mathematical education. And the skills and habits they learn at this stage will inform their habits later on. Do we really belive that a child who never learns to bear the yoke of long division will have the patience to sort through integration by parts? Or worse yet, can we really believe that children who are not instilled with the virtue of showing their work will suddenly learn to keep meticulous notes when it really matters to themselves or their colleagues?&lt;/p&gt;&#xA;&lt;p&gt;By-hand computations are useful for many reasons: building familiarity with useful ideas, improving estimation skills, and creating a sense of independence and self-reliance. As others have pointed out, all of these things are important and valuable in daily adult life. However, one virtue stands out to me as most valuable: learning to show your work. What will a student who has never learned to show their work do when they encounter a problem (mathematical or otherwise) which can't be solved in one step? I think many will throw up their hands in frustration. I think many will believe that if they can't do it in their heads or on their phone then it is too challenging to be done at all.&lt;/p&gt;&#xA;&lt;p&gt;There will come a day in a student's life when writing out the details of multiplication will cease to be important, but careful and well-documented thought will never lose its value. And if we're not teaching children to be careful, accurate, and useful thinkers, then what are we doing?&lt;/p&gt;&#xA;" OwnerUserId="15436" LastEditorUserId="15436" LastEditDate="2021-03-02T20:48:33.357" LastActivityDate="2021-03-02T20:48:33.357" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19558" PostTypeId="2" ParentId="19546" CreationDate="2021-03-02T19:26:28.273" Score="21" Body="&lt;p&gt;I taught at the elementary and high school levels.  At times we used calculators and at times we didn't. Students benefit from experience both ways. Students need to learn that calculators are only a tool and they still have to think.  Students also need to learn that having a calculator doesn't guarantee that their computation will be correct.  Finally those that can't compute have trouble with algebra.&lt;/p&gt;&#xA;&lt;p&gt;I usually allowed my 5th grade students to use a calculator for our chapter on mean, median, and mode. I taught them that if you are adding 10 numbers to find the mean, you should add more than once because there was no guarantee that you would get an accurate answer the first time.  After a test using calculators the students were always surprised that even with the calculator the test was still hard.&lt;/p&gt;&#xA;&lt;p&gt;In 9th grade we introduced graphing calculators. Students who didn't understand the order of operations had trouble figuring out when to put in parentheses, resulting in lots of wrong answers.  Students who didn't understand that 0.33333333 is an estimate of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt;   couldn't understand why their answer wasn't exactly right.  Similar mistakes were made with pi.&lt;/p&gt;&#xA;&lt;p&gt;When I taught algebra, I discovered how important it is understand computation with fractions.  A student won't be able to add &lt;span class=&quot;math-container&quot;&gt;$\frac{x}{3x+2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{x+1}{3x-2}$&lt;/span&gt; unless he already knows how to add   &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{5}$&lt;/span&gt;.  As mentioned above a student who doesn't understand order of operations will have trouble simplifying expressions. A student who doesn't understand that &lt;span class=&quot;math-container&quot;&gt;$\frac{x}{3}$&lt;/span&gt; is the same as &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}{x}$&lt;/span&gt; will have trouble solving equations.&lt;/p&gt;&#xA;&lt;p&gt;There is a place for calculators, but it is only a tool and students need to be able to calculate, estimate, and solve problems.  Students need to learn to use the calculator efficiently and effectively.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-03-02T19:26:28.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19559" PostTypeId="2" ParentId="18047" CreationDate="2021-03-02T20:56:23.743" Score="1" Body="&lt;p&gt;I found &lt;a href=&quot;http://www.randomservices.org/random/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.randomservices.org/random/&lt;/a&gt; useful. It is written in a formal language, but it has examples, exercises and also interactive demonstrations/illustrations.&lt;/p&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2021-03-02T20:56:23.743" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19560" PostTypeId="2" ParentId="19546" CreationDate="2021-03-02T23:23:25.267" Score="2" Body="&lt;p&gt;You're entirely correct and there is huge value to being able to compute by hand.&lt;/p&gt;&#xA;&lt;p&gt;For a most basic instance from the real world, here in the UK I once interviewed a candidate for office manager who had spent about 10 years working in a tax office, and been promoted three times. To me, that should have meant he ate, breathed and slept numbers.&lt;/p&gt;&#xA;&lt;p&gt;I asked him what 7x8 was, why it was special and what other number grouping had similar characteristics. To me, anyone happy with what US Americans would call junior high school arithmetic should have known all three answers.&lt;/p&gt;&#xA;&lt;p&gt;My candidate, however, spent several seconds looking for a calculator and failing to find one, resorted to pen and paper - remember, this was for 7x8.&lt;/p&gt;&#xA;&lt;p&gt;As it happened, this was in an open-plan office and everyone else was both ear-wigging and either more my age, or from South Africa; theoretically a generation or so &amp;quot;behind&amp;quot; the UK.&lt;/p&gt;&#xA;&lt;p&gt;Even with pencil and paper, my candidate was so clearly uncomfortable with 7x8 that I asked him - slightly less bluntly - &amp;quot;Do you realise, everyone else in the room had all the answers before you even realised you might need a calculator?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;He replied: &amp;quot;Oh, my God! Now I see why older people are always complaining about calculators. I just wish someone had explained that before.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="13952" LastActivityDate="2021-03-02T23:23:25.267" CommentCount="26" ContentLicense="CC BY-SA 4.0" />
  <row Id="19561" PostTypeId="2" ParentId="19546" CreationDate="2021-03-03T00:24:09.977" Score="2" Body="&lt;p&gt;You can't be overly reliant on technology.&lt;/p&gt;&#xA;&lt;p&gt;I once bought an item from a shop during a power outage.  Cash register was non-functional.  Not enough light to power a solar calculator.  The cashier had to compute a 10% discount by hand.&lt;/p&gt;&#xA;&lt;p&gt;She did it.... the long way.&lt;/p&gt;&#xA;" OwnerUserId="15549" LastActivityDate="2021-03-03T00:24:09.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19563" PostTypeId="2" ParentId="19546" CreationDate="2021-03-03T13:18:17.160" Score="5" Body="&lt;p&gt;&lt;strong&gt;By hand&lt;/strong&gt; =/= in your head.  Having some faculty for mental arithmetic is good.  Having a fluency for deconstructing a 'problem' into basic mathematical operations is pretty essential.  However many people learn, and continue to benefit, by &lt;strong&gt;writing out&lt;/strong&gt; the 'sums'.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If Fred drives at 45 miles per hour, how far will he go in 45 minutes?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I expect most people here have done 3/4 of 45 in their heads before reaching the end of the question.  (And never had to worry if the units were correct.)  But if you can't jump to the easy/obvious method then what do you do?  One way is&#xA;&amp;quot;Obviously not as far...  Aha!  45 60ths... Now that's 45 over 60&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt; 45&#xA; ----&#xA; 60&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;and times the speed&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt; 45&#xA; ----  X  45  =&#xA; 60&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;So now I have a calculation plan.&amp;quot;  What happens next is anybody's guess:  Personally I would use a calculator to check my guess of 34-ish with 4 5 ÷ 6 0 × 4 5 =.  I'm getting old and the 45/60 = 3/4 = .25 mental bit (see the error!) is not worth the strain.  Some people retain information by reading it. (Others by doing or hearing.)  Now that 'template' is available for the next time Fred drives for 1.3 hours at 47 mph.&lt;/p&gt;&#xA;&lt;p&gt;This approach means that students need a pencil and paper.  &lt;em&gt;Eeek!&lt;/em&gt;  A good foundation for pencil and paper is sums in an exercise book.  Fractions and calculators don't mix.  Fractions are a key part of our mental grasp of worldly things.&lt;/p&gt;&#xA;" OwnerUserId="15556" LastActivityDate="2021-03-03T13:18:17.160" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19564" PostTypeId="1" AcceptedAnswerId="19565" CreationDate="2021-03-03T14:39:48.563" Score="31" ViewCount="5054" Body="&lt;p&gt;&lt;sup&gt;(I was inspired by the comments in &lt;a href=&quot;https://matheducators.stackexchange.com/a/19560/77&quot;&gt;this answer&lt;/a&gt; to ask this question.)&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have some multiplication table cards from &lt;a href=&quot;https://en.wikipedia.org/wiki/Kumon&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kumon&lt;/a&gt; that have a list of commonly mistaken multiplications: &lt;span class=&quot;math-container&quot;&gt;$7\times 8, 4\times 8, 11\times 12, 7\times 9, 6\times 7, 12\times 8, 4\times 7, 6\times 8, 9\times 12, 8\times 9, 11\times 11$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$6\times 9$&lt;/span&gt; (in this order).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lGEMwm.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lGEMwm.jpg&quot; alt=&quot;Kumon multiplication table cards&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I assume that this is based on data they obtained from the numerous children who have answered their worksheets.&lt;/p&gt;&#xA;&lt;p&gt;Is there some other source (a study, perhaps) that lists the products of single digits that children usually get wrong?&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="12255" LastEditDate="2021-04-12T15:37:30.390" LastActivityDate="2021-04-12T15:37:30.390" Title="Which product of single digits do children usually get wrong?" Tags="|primary-education|arithmetic-operations|multiplication-table|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="19565" PostTypeId="2" ParentId="19564" CreationDate="2021-03-03T15:42:38.510" Score="46" Body="&lt;p&gt;&lt;a href=&quot;https://www.theguardian.com/news/datablog/2013/may/31/times-tables-hardest-easiest-children&quot; rel=&quot;noreferrer&quot;&gt;https://www.theguardian.com/news/datablog/2013/may/31/times-tables-hardest-easiest-children&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are links to a dataset in the article. As far as I can tell, this isn't a formal study:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But some new data generated by pupils at Caddington Village School in&#xA;Bedford sheds light on which multiplications are actually the hardest&#xA;– and how kids do overall.&lt;/p&gt;&#xA;&lt;p&gt;The data is generated by an app produced by an app developed by&#xA;education tech firm Flurrish, and in total the 232 children who&#xA;participated produced more than 60,000 answers. Here's how they did&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the data is of unknown quality, but the graph is both pretty and pretty believable. (Except that I'd probably label the graph below &lt;em&gt;in&lt;/em&gt;accuracy rather than accuracy.)&lt;/p&gt;&#xA;&lt;p&gt;It's notable that the data is slightly asymmetric but I'm unsure if that is statistically significant. i.e. Do kids use commutativity?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Gy7ZT.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Gy7ZT.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2021-03-04T23:41:39.453" LastActivityDate="2021-03-04T23:41:39.453" CommentCount="27" ContentLicense="CC BY-SA 4.0" />
  <row Id="19566" PostTypeId="1" CreationDate="2021-03-03T15:57:07.250" Score="9" ViewCount="792" Body="&lt;p&gt;In the context of a standard undergraduate Calculus sequence, I've noticed there is a big emphasis on teaching the algebra part of Calculus. What I mean by this is that a student may feel more comfortable finding, say, a limit by computing a bunch of values approaching some number, or by going from the graph. But yet, it seems we spend an inordinate amount of time on finding certain limits by factoring, or rationalizing, etc.&lt;/p&gt;&#xA;&lt;p&gt;Same goes for being able to find a derivative from the definition, or with finding indefinite integrals by hand. Integration is an algorithmic task which we can only perform for hand picked cases at great pains, and there is a huge class of functions for which there is no antiderivative in elementary functions. Given that CAS can do it much more easily, why exactly do we require students to learn this?&lt;/p&gt;&#xA;&lt;p&gt;While I understand some of these things are interesting in their own right to a mathematician (for example, partial fractions brings to a discussion of what polynomials are irreducible over the reals), I'm not sure who in the real world actually goes around trying to find an indefinite integral by hand, or calculating derivatives from their definition. Again, I do understand these are important skills, but I frequently hear the argument that these are the most important skills, as opposed to having an intuitive &amp;quot;feel&amp;quot; for limits, or for when derivatives come into models, for example.&lt;/p&gt;&#xA;&lt;p&gt;So my question is:&lt;/p&gt;&#xA;&lt;p&gt;Do engineers/scientists,etc. actually need to know the algebraic part of Calculus? How?&lt;/p&gt;&#xA;&lt;p&gt;Concrete examples would be greatly appreciated. Thanks!&lt;/p&gt;&#xA;" OwnerUserId="15558" LastEditorDisplayName="user507" LastEditDate="2021-03-04T13:22:52.743" LastActivityDate="2021-12-29T18:57:25.100" Title="Why teach the algebraic Calculus?" Tags="|calculus|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="19567" PostTypeId="2" ParentId="19566" CreationDate="2021-03-03T18:48:38.150" Score="6" Body="&lt;p&gt;I find that the algebraic manipulations are similar to those in physics and engineering (and to a lesser extent chemistry).  Much more so than looking at a graph on a calculator or the like.  Both the specific calculus tricks as well as the work similarity (multi-step rearrangements) needed in HW problems in physics/engineering.&lt;/p&gt;&#xA;&lt;p&gt;Note that engineers often don't do calculus after they leave school.  Maybe someone working for NASA, but that's a rarity.  Lots of people doing HVAC and the like.  And that is really just algebra and arithmetic and drafting and keeping track of details.&lt;/p&gt;&#xA;&lt;p&gt;As a business consultant, there is a lot of manipulation of numbers and formulas and the like in Excel (or even estimates in meetings in your head/discussion).  Basically word problems (which is simple applied algebra...making sure the ratios are right side up).  But you'd be amazed how many people struggle with this.&lt;/p&gt;&#xA;&lt;p&gt;So, if I hire an engineer or a consultant, I would prefer him to have had a classical algebra calculus background.  It's kind of overkill for what he does...but it makes sure he has what he needs.  Would be leery of kids that did a bunch of concept and graph looking stuff, but couldn't do multi-step calculations.  I would not underestimate the value (and mental load) of being able to keep track of multi-step problems for math majors either.&lt;/p&gt;&#xA;&lt;p&gt;For a nursing student it's definitely overkill that they can handle partial fractions integration.  Arguably, it is overkill for doctors as well.  Although, I might just appreciate the &amp;quot;weeder&amp;quot; aspect of filtering out the less bright pre-meds.  At a weaker school, I can also buy the value for business/econ/social science students of stripping out a lot of the intricate tricks.  But some basic algebra calculus still of benefit.  But just less volume of it.  (At better, say top of state school and up, I would expect all the kids to handle a typical Thomas Finney calculus course.)&lt;/p&gt;&#xA;" OwnerUserId="15563" LastActivityDate="2021-03-03T18:48:38.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19568" PostTypeId="1" CreationDate="2021-03-03T23:13:01.197" Score="11" ViewCount="607" Body="&lt;p&gt;I noticed many seemingly reputable online sources have &amp;quot;incorrect&amp;quot; description of&#xA;directional derivatives for real-valued functions in several variables.&#xA;Here, by &amp;quot;incorrect&amp;quot; I mean it disagree with the definitions in the textbooks I'm familiar with.&#xA;&lt;s&gt;Of course, I'm open the possibility that all the textbooks I have been using are wrong.&lt;/s&gt; (Update: well, that's probably impossible).&lt;/p&gt;&#xA;&lt;p&gt;For a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; variables &lt;span class=&quot;math-container&quot;&gt;$\mathbf{x}=(x_1,\dots,x_n)$&lt;/span&gt;&#xA;and a unit vector &lt;span class=&quot;math-container&quot;&gt;$\mathbf{v}$&lt;/span&gt;,&#xA;almost all sources defines directional derivatives as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ D_{\mathbf{v}} f(\mathbf{x}) = \lim_{h \to 0} \frac{f(\mathbf{x} + h \mathbf{v}) - f(\mathbf{x})}{h}. $$&lt;/span&gt;&#xA;So far so good.&#xA;Then many online sources state that&#xA;if the partial derivatives exist, then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ &#xA;  D_{\mathbf{v}} f(\mathbf{x}) = &#xA;  \frac{\partial f}{\partial x_1} v_1 &#xA;  + \dots +&#xA;  \frac{\partial f}{\partial x_n} v_n.&#xA;$$&lt;/span&gt;&#xA;This is &lt;strong&gt;not true&lt;/strong&gt;.&#xA;At least this disagrees with all the textbook my department is using (Ron Larson).&#xA;We can also easily come up with counterexamples&#xA;where the left hand side is &lt;strong&gt;undefined&lt;/strong&gt; while the right hand is &lt;strong&gt;defined&lt;/strong&gt;&#xA;(Update: or even better/worse counterexamples where&#xA;one side is &lt;strong&gt;nonzero&lt;/strong&gt; and the other side is &lt;strong&gt;zero&lt;/strong&gt;).&#xA;Indeed, if students took the above statement literally,&#xA;then the existence of partials implies the existence of all directional derivatives,&#xA;which renders many exam questions meaningless.&lt;/p&gt;&#xA;&lt;p&gt;Some major reputable sources that has this problem:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://openstax.org/books/calculus-volume-3/pages/4-6-directional-derivatives-and-the-gradient&quot; rel=&quot;noreferrer&quot;&gt;OpenSTAX Calculus volume 3&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathworld.wolfram.com/DirectionalDerivative.html&quot; rel=&quot;noreferrer&quot;&gt;Wolfram MathWorld&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://tutorial.math.lamar.edu/classes/calciii/directionalderiv.aspx&quot; rel=&quot;noreferrer&quot;&gt;Paul's online notes&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Just to name few well known ones.&#xA;These three also happen to be the first 3 search results in my Google search results&#xA;for &amp;quot;directional derivative&amp;quot;.&#xA;So these are what students are most likely to see.&lt;/p&gt;&#xA;&lt;p&gt;YouTube sources are not much better,&#xA;the 3 videos with the highest views that I can see are&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=tDPp5uWSIiU&amp;amp;t&quot; rel=&quot;noreferrer&quot;&gt;Professor Leonard&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=N_ZRcLheNv0&quot; rel=&quot;noreferrer&quot;&gt;Khan Academy&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=uCY0XYXgYQo&quot; rel=&quot;noreferrer&quot;&gt;patricJMT&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Only the last one makes this distinction explicit.&lt;/p&gt;&#xA;&lt;p&gt;This disagreement causes real issues in classroom.&#xA;In addition to confusing students,&#xA;students very often justifies incorrect answers on exams&#xA;by referencing these sources.&#xA;It is not easy to convince students that somehow I am to be trusted&#xA;more than these reputable sources.&#xA;(no one is going to care about counterexamples).&lt;/p&gt;&#xA;&lt;p&gt;My question is why are so many seemingly reputable sources wrong about this rather important distinction? (Update: this is an honest question. Why are they all wrong about same thing in exact same way? Carelessness in the exact same way? Yet, none of them confuse the existence of partials and differentiability. Why not be careless there too?)&lt;/p&gt;&#xA;" OwnerDisplayName="user13395" LastEditorDisplayName="user13395" LastEditDate="2021-03-04T04:15:58.617" LastActivityDate="2021-03-18T21:24:20.480" Title="Why are so many online sources &quot;wrong&quot; about directional derivatives?" Tags="|calculus|vector-calculus|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="19569" PostTypeId="2" ParentId="19568" CreationDate="2021-03-04T03:32:05.343" Score="5" Body="&lt;p&gt;It is worth noting that your definition of directional derivative allows for non-continuous functions to be differentiable (in all directions). It isn't enough to look at just straight-line paths for the definition. See page 117 of &lt;em&gt;Counterexamples&lt;/em&gt; &lt;em&gt;in&lt;/em&gt; &lt;em&gt;Analysis&lt;/em&gt; for two interesting counterexamples. One of them is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x,y)=\frac{x^2 y}{x^4+y^2}, \quad f(0,0)=0$$&lt;/span&gt;&#xA;which is differentiable along each line through the origin, but is not continuous there. (Check it on parabolas which go through the origin. e.g. &lt;span class=&quot;math-container&quot;&gt;$y=\pm x^2$&lt;/span&gt;) Once you know that the function is actually differentiable, the limit you provide will actually give the correct answer.&lt;/p&gt;&#xA;&lt;p&gt;To answer the direct question: because it is hard to get this stuff totally correct without losing the main point and making it look like a real analysis textbook.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2021-03-04T03:40:43.807" LastActivityDate="2021-03-04T03:40:43.807" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19570" PostTypeId="2" ParentId="19566" CreationDate="2021-03-04T13:40:22.773" Score="9" Body="&lt;p&gt;It's certainly not possible to omit instruction in these topics for &lt;em&gt;all&lt;/em&gt; students. After all, someone has to be able to write the code for the computer algebra system. Someone who wants to be a mathematician or physicist needs these skills, and using a CAS is not a substitute. For example, they will encounter multivariable versions of some of the single-variable techniques and theorems, and they can't be prepared to deal with those unless they have gotten their hands dirty with them in single-variable calculus.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, it is ridiculous that people who want to be pharmacists, dentists, and physical therapists are being forced to learn how to integrate using hyperbolic trig substitutions. Here in California, calculus requirements have been gradually becoming universal for students on these tracks. It started with the UC system, and now it's gotten to the point where AFAICT all such students are being told by their counselors to take calculus and calculus-based physics.&lt;/p&gt;&#xA;&lt;p&gt;So this is not an issue that math departments created, and it's not one that they can solve. Math departments are acting as service departments here.&lt;/p&gt;&#xA;&lt;p&gt;The reason that these requirements have been inflated in life sciences and allied health fields is simply supply and demand. There are too many students who want to do these majors and go into these professional programs. The market has responded in two ways. First, there is the phenomenon of Caribbean med schools and vet schools, and also the less selective DPT programs. I've had some very good students go to those schools and have successful careers, but lower-performing students from Caribbean med schools end up not getting internships, which is a horrible outcome. The second way that the market has responded is by simply making students jump through hoops in order to get into the impacted majors at better schools, and the better professional programs. There, forcing students to learn hyperbolic trig substitutions is simply a way of filtering them by diligence and intellectual ability. They could be required to learn Homeric Greek instead, and it would serve the same purpose.&lt;/p&gt;&#xA;&lt;p&gt;Engineering majors are a middle ground. I teach physics to engineering majors, and my main complaint is that they are force-fed the random set of topics of second-semester calculus without gaining any clear picture of the structure of the subject or being able to separate what's important from what's not. A typical symptom of this is that they can do an integral in 5 minutes that would take me 30 minutes (because the techniques are freshly practiced for them), but when I ask them to do a Taylor series, their reaction is something like, &amp;quot;I don't remember that.&amp;quot; To me, this is the wrong way around. Taylor series are massively important and practical, whereas forgetting one's integration tricks is not a problem in an era of computer algebra systems.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-03-04T13:47:13.267" LastActivityDate="2021-03-04T13:47:13.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19571" PostTypeId="2" ParentId="19566" CreationDate="2021-03-04T16:29:12.313" Score="8" Body="&lt;p&gt;Since you asked about engineers and scientists, the answer is easy; yes they need to know about algebraic aspects of calculus. There is no way to fully understand physics without having a complete and functioning toolset of algebra. Every trick we do in calculus and more are needed to derive theorems and solve problems in physics. Do engineers and scientists need to understand Physics ? That depends, do you want them to understand things from basic principles or are you content that they adopt a blackbox mentality that anything too physical they have to leave to experts. I would hope we want them to gain a mastery of physics that allows them to reasonably create their own models from applying appropriate physical law etc. to the problem at hand. Algebraic skill is a base level skill towards that goal. Furthermore, mastery of power series and comprehension of special functions far beyond what we even dream of teaching in the usual DEqns course would be helpful. For practicing physicists, if they actually care about understanding the mathematics used in their discipline, so much more than this is needed.&lt;/p&gt;&#xA;&lt;p&gt;Like everything else nowadays, the question is do we want to put in the effort to do it correctly, or do we just want to do the bare minimum. Maybe I'm just getting old :)&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-03-04T16:29:12.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19574" PostTypeId="2" ParentId="19546" CreationDate="2021-03-05T03:26:45.827" Score="6" Body="&lt;p&gt;Calculus student had a final result of&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2\pi}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Which she told me was 1.57. I immediately realized that she had calculated&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac12\pi$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;from keying in 1/2&lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There’s no going back on calculator use, I realize. What I strive for is to have the student who performed a series of calculations (for, in this case a related rates problem in calculus) to manage to recognize that final calculation should result in something less than .166 or really at least less than 1. The 1.57 should have struck her as an incorrect answer.&lt;/p&gt;&#xA;&lt;p&gt;The issue is also one of calculator skills. I try to maintain an approach that in life, any tool needs to be used carefully. A knife is important in the kitchen, but sloppy usage will cut your hand. Calculator use requires mindfulness as well. Using parenthesis properly is part of the process as is an understanding of the expected result as in my example.&lt;/p&gt;&#xA;&lt;p&gt;Similar to this simple parentheses issue, I’ve seen trig exams showing multiple steps indicating an understanding of all the rules the student was expected to know. The final answers? All wrong. The student’s calculator was in radian mode while the questions were all in degrees. I think that the calculate by hand skill would build the math sense that would lead to the recognition that a series of answers make little sense.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-03-05T03:26:45.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19575" PostTypeId="2" ParentId="369" CreationDate="2021-03-05T07:30:18.463" Score="3" Body="&lt;p&gt;The SMSG textbook for algebra II and trig was entitled &amp;quot;Modern Algebra and Trigonometry, Book 2&amp;quot; by Dolciani, Berman, and Wooton, published by Houghton-Mifflin. We used this during my junior year in high school (1965-66). I wish I'd kept the book instead of returning it to our bookstore for credit, because I recently had to purchase a used (paperback) edition for around $20 from Amazon. I use this as a reference, along with &amp;quot;Modern Algebra, Book 1&amp;quot; and &amp;quot;Modern Introductory Analysis&amp;quot;, for the precalculus course I'm now teaching. All three books, published in the early-to-mid 60s, receive high accolades from Amazon customers, including myself.&lt;/p&gt;&#xA;" OwnerUserId="15592" LastActivityDate="2021-03-05T07:30:18.463" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19579" PostTypeId="2" ParentId="19546" CreationDate="2021-03-05T19:30:44.763" Score="7" Body="&lt;p&gt;You really need to be able to do sums in your head when debating or negotiating. To prove this, show your students these famous car-crash interviews: (1) &lt;a href=&quot;https://www.youtube.com/watch?v=kAEixRNPQmE&quot; rel=&quot;noreferrer&quot;&gt;Diane Abbott (Labour) floundering horrifically&lt;/a&gt; over numbers she could neither remember nor even estimate in her head. The coolness and speed with which the interviewer questioned her numbers must itself have put her in a mental tailspin, leaving her completely unable to think straight. (2) &lt;a href=&quot;https://www.youtube.com/watch?v=6FYjG3ix-bw&quot; rel=&quot;noreferrer&quot;&gt;Natalie Bennett, Green Party leader in 2015, doing similarly appallingly&lt;/a&gt; over the cost of building half a million homes. Both these ladies probably still find the memory of these moments keeps them awake at night.&lt;/p&gt;&#xA;" OwnerUserId="15596" LastActivityDate="2021-03-05T19:30:44.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19581" PostTypeId="2" ParentId="4351" CreationDate="2021-03-06T16:06:52.153" Score="8" Body="&lt;p&gt;Given the equation &lt;span class=&quot;math-container&quot;&gt;$$\lvert2x\rvert=x-1,$$&lt;/span&gt; rewriting the left-hand side as &lt;span class=&quot;math-container&quot;&gt;$\pm2x$&lt;/span&gt; results in the &lt;strong&gt;unique candidate&lt;/strong&gt; solution set &lt;span class=&quot;math-container&quot;&gt;$\left\{-1,\frac13\right\}$&lt;/span&gt; &lt;strong&gt;that nonetheless fails&lt;/strong&gt; to satisfy the equation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}\forall x\in\mathbb R \Bigg[\quad\quad\quad&amp;amp;x\in\emptyset\\\iff{}&amp;amp;\bigg(x&amp;lt;0 \,\text{ and}\,-2x=x-1\bigg) \:\text{ or }\: \bigg(x\geq0 \,\text{ and }\, 2x=x-1\bigg)\\\iff{}&amp;amp;\lvert2x\rvert=x-1\\\implies{}&amp;amp;\pm2x=x-1\\\iff {}&amp;amp;x\in\left\{-1,\frac13\right\}\quad\Bigg].\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Principle_of_explosion&quot; rel=&quot;nofollow noreferrer&quot;&gt;deductive explosion&lt;/a&gt;, performing a valid operation on an inconsistent equation, which has no solution, has given rise to an &lt;a href=&quot;https://math.stackexchange.com/a/3927547/21813&quot;&gt;extraneous&lt;/a&gt; solution set.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-07-25T15:16:06.040" LastActivityDate="2022-07-25T15:16:06.040" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19582" PostTypeId="1" CreationDate="2021-03-06T18:15:34.630" Score="4" ViewCount="326" Body="&lt;p&gt;From reviews on Amazon of the various high school math texts by Mary Dolciani et al of the SMSG, I assume that there might be a successor to the approach (referred to as “the new math”) taken by the SMSG. Does anyone have any insight into this, and whether there really were “math wars” when the SMSG series was introduced in the 1960s in response to the Soviets’ launching of Sputnick?&lt;/p&gt;&#xA;" OwnerUserId="15592" LastEditorUserId="2083" LastEditDate="2021-03-07T17:59:34.317" LastActivityDate="2021-09-03T18:13:37.143" Title="Successor to School Mathematics Study Group (SMSG)" Tags="|mathematical-pedagogy|usa|history|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19584" PostTypeId="1" CreationDate="2021-03-09T19:13:41.237" Score="5" ViewCount="389" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Tilt originated as a poker term for a state of mental or emotional confusion or frustration in which a player adopts a less than optimal strategy, usually resulting in the player becoming over-aggressive. Tilting is closely associated with another poker term, &amp;quot;steam&amp;quot;. - &lt;a href=&quot;https://en.wikipedia.org/wiki/Tilt_(poker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I encounter a similar pattern when I solve 'difficult' math problems in a drill (a practice sheet with a lot of questions ), once I get a few and struggle with a few that I can't solve a question after setting up the basic idea for it/ Can't get up the set up at all, then I got into this tilt mode where I do worse and worse skipping questions through the drill.&lt;/p&gt;&#xA;&lt;p&gt;So,  how do we reduce the amount of 'stress' / 'mental load' that comes when we solve math problems?&lt;/p&gt;&#xA;&lt;p&gt;One strategy that I adopted early on is going for a few questions instead of a large number, this helped me develop my mathematics abilities a lot , but I feel it is mostly useless when I try to do a large number of problems similar to those we receive in standardized tests.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorUserId="2083" LastEditDate="2021-03-10T07:04:40.707" LastActivityDate="2021-03-13T00:23:51.610" Title="How to reduce tilting when going over drills?" Tags="|self-learning|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19585" PostTypeId="2" ParentId="19584" CreationDate="2021-03-09T21:09:30.980" Score="3" Body="&lt;p&gt;A few things that I find help.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Make sure you have some easy drill problems at the start of a homework session.  Most books are actually well set up for this approach.  Having I, II, and III problems (or A, B, C).  The first category are plug and chug.  Last category require a little bit of derivation or more detailed word problems.  Intermediate are usually multi-step calculations.  I find that doing some of the &amp;quot;easy&amp;quot; problems sort of gets you in the zone.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Make sure you get through the whole session.  Even if you skip some, come back and do all.  You'll often find that your subconscious has done some work on the problems while moved on.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Go back to the text, if you are just missing a lot.  REad the section.  Work the example problems.  (You should be doing this anyways, before drill!)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you have to peak at answer or hint or if you worked it and got it wrong, you need to REPEAT the whole problem.  Actually rework all the cheated or missed problems as a &amp;quot;section&amp;quot; after the end of your drill. Even if you had a &amp;quot;dumb mistake&amp;quot; or say &amp;quot;got it&amp;quot; after checking.  You still have to rework it.  And in your mind treat it as a &amp;quot;fresh problem&amp;quot;.  This sounds hokey, but it gets you more into a slight discovery mode.  OVerall, at the end of the day, you need to treat this stuff as a bit more like gymnastics routines or music performance.  You need to acquire &amp;quot;automaticity&amp;quot; with basic courses like calculus or linear algebra.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Avoid music or any distractions (e.g. net).  It can be a little arduously monastic.   But you will toughen up the willpower muscles over time.  Like lifting weights or running cross country.  You're not just working on the work, you're working on your CAPACITY for work.  &amp;quot;How do you walk around the world?&amp;quot;  &amp;quot;One step at a time, but your stride gets better as you walk.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P.S. It's a little hard to advise when so general.  Not clear what course, your ability, motivation, etc.  Since this is a human behavior problem, not a math problem, there is no precise answer.  And a single person might try different approaches.  Or different people might have different things that click for them.  And...please take this as &amp;quot;fellow sinner&amp;quot; advice.  Not as holier than thou.&lt;/p&gt;&#xA;" OwnerUserId="15618" LastEditorUserId="8571" LastEditDate="2021-03-13T00:23:51.610" LastActivityDate="2021-03-13T00:23:51.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19586" PostTypeId="1" CreationDate="2021-03-10T01:43:14.333" Score="4" ViewCount="156" Body="&lt;p&gt;I am looking for resources for teaching math modeling to high school teachers with rusty math background. It will be a 6-week course. Some tips/directions on simple projects would be helpful. I would want to introduce coding for numerically solving systems of ODEs.&lt;/p&gt;&#xA;" OwnerUserId="15619" LastActivityDate="2021-03-15T00:30:47.127" Title="Resources for teaching mathematical modeling to high school teachers" Tags="|teaching|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19587" PostTypeId="2" ParentId="19586" CreationDate="2021-03-10T03:39:44.383" Score="2" Body="&lt;p&gt;The information and handbooks at &lt;a href=&quot;https://m3challenge.siam.org/resources&quot; rel=&quot;nofollow noreferrer&quot;&gt;SIAM Mathworks Modelling Challenge&lt;/a&gt; pretty helpful.  One handbook introduces the reader to the modeling process and the second one introduces the reader to the communication of computation related to modeling (including ODEs via spreadsheets--you could probably generalize what they discuss to solve the logistic equation to systems).  There are also have links to sample problems from previous competitions.&lt;/p&gt;&#xA;" OwnerUserId="1537" LastActivityDate="2021-03-10T03:39:44.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19588" PostTypeId="1" CreationDate="2021-03-10T07:51:52.167" Score="2" ViewCount="321" Body="&lt;p&gt;I am not sure which StackExchange to ask this question but I am interested in the relative underperformance of Western countries such as Australia, US, UK compared to other higher performing countries in the world. More specifically whether there are specific areas of mathematics of underperformance that have been discussed or highlighted before.&lt;/p&gt;&#xA;&lt;p&gt;The PISA testing results that I have found online are very broad and generic but does not break down into the usual specific areas of maths for example, like &lt;a href=&quot;https://www.oecd-ilibrary.org/docserver/9789264266490-9-en.pdf?expires=1615355002&amp;amp;id=id&amp;amp;accname=guest&amp;amp;checksum=7680A40DC65D2E99DBFCF7587658EC5C&quot; rel=&quot;nofollow noreferrer&quot;&gt;this report from the OECD.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am aware that another test TIMSS is problematic when comparing the &lt;a href=&quot;https://www.newscientist.com/article/mg21728985-800-west-vs-asia-education-rankings-are-misleading/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;West&amp;quot; versus the rest&lt;/a&gt; but I wonder if there are specific subjects areas that are linked to underperformance rather than a general pedagogical approach issue.&lt;/p&gt;&#xA;" OwnerUserId="14947" LastEditorDisplayName="user507" LastEditDate="2021-03-10T13:13:30.437" LastActivityDate="2021-03-10T13:13:30.437" Title="Which areas of maths should underperforming western countries focus on?" Tags="|mathematical-pedagogy|exams|assessment|" AnswerCount="0" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="19589" PostTypeId="1" CreationDate="2021-03-11T10:53:17.217" Score="9" ViewCount="420" Body="&lt;p&gt;On Mathematics StackExchange for a particular instance, it is highly recommended that questioners mark up their questions using &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt;. A surprising number of mathematicians and student mathematicians are perfectly happy to admit that they don't know &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; and are not planning on learning it.&lt;/p&gt;&#xA;&lt;p&gt;To me, that is an attitude which is beginning to make less and less sense as time marches on. Perhaps 12 or 15 years ago &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; was not necessarily the best option, but now we have such convenient and (fairly) high-quality tools like MathJax, it appears to me that for a mathematician &lt;em&gt;not&lt;/em&gt; to want to learn &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; is akin to an English student not prepared to learn how to use a word processor.&lt;/p&gt;&#xA;&lt;p&gt;What is the general opinion of people in the educational and academic sphere: is it feasible, desirable and / or possible to make it a general requirement for a degree in mathematics, or physics, or even engineering, to attend a short module on developing fluency in basic &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; of some description?&lt;/p&gt;&#xA;&lt;p&gt;I understand that certain word commercial processors may have extensions which allow the generation of inline mathematics, which themselves use a compiler that builds some interim &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; code, but these tend to be cumbersome and unwieldy, and can cause unnecessary bloat, and have considerable limitations.&lt;/p&gt;&#xA;" OwnerUserId="15290" LastEditorUserId="15290" LastEditDate="2021-03-11T14:26:32.450" LastActivityDate="2021-03-27T23:54:54.797" Title="Worth introducing a mandatory short module on $\LaTeX$ into a mathematics degree?" Tags="|mathematical-pedagogy|technology-in-education|specific-software|" AnswerCount="6" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19591" PostTypeId="2" ParentId="19589" CreationDate="2021-03-11T14:15:23.870" Score="8" Body="&lt;p&gt;I do think that having an introduction and primer to LaTeX in the course of a college math/computing degree is appropriate and beneficial.&lt;/p&gt;&#xA;&lt;p&gt;For example, I work this into my discrete math course, and require that weekly submissions to the Discussions board be in that format (the Blackboard LMS supports LaTeX entry). My community-college students often have notable skill deficiencies, but even here, the students who will pass the course can usually pick it up in a day, from a one-paqe summary handout.&lt;/p&gt;&#xA;&lt;p&gt;I agree with Ben Crowell that an entire dedicated course on the subject would be overkill.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-03-13T04:21:18.090" LastActivityDate="2021-03-13T04:21:18.090" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19592" PostTypeId="2" ParentId="19589" CreationDate="2021-03-11T18:34:48.013" Score="2" Body="&lt;p&gt;One clear objection to teaching Latex to all undergrads that would be raised in a faculty meeting would be: Latex indeed is pervasive in mathematical and theoretical physics publications (it is not that common in experimental physics, I don't know about engineering), but the majority of undergrads will not stay in academia, but will enter the job market once they graduate.&lt;/p&gt;&#xA;&lt;p&gt;Now, one may argue that Latex is such a powerful tool that even outside academia one could profit from learning it. But if other people in the company do not know Latex, then its usefulness declines seriously, since you cannot share a project.&lt;/p&gt;&#xA;&lt;p&gt;A more subtle objection would be about portability. Recall that Latex was created as a typesetting tool that would generate the same document in any computer. One chemist noted to me that, in the age of smartphones, Latex-generated documents lose their edge because although high-quality they are difficult to read in small screens.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: To clarify the portability issue, this is relevant for the majority of people who will end up outside academia and will be producing documents which are expected/advantageous to be read in small devices. For writing papers or student examinations portability is irrelevant and arguably Latex is by far the best option.&lt;/p&gt;&#xA;&lt;p&gt;It is important to consider that Latex is not just a tool to write equations, which is the part that mathjax implements, but a full typesetting application. It is easy to learn how to produce a standard document, but extremely complicated if you want to be able to change the diagrammatics, not to mention that Latex is infamous for having one of the worst error messages of all time. I would wager that the average user of Latex is happy to send a document whose output contains errors as long as it compiles and &amp;quot;looks fine&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Wrapping up, while I still love latex and continue to spread its gospel, it seems unlikely that the average student would benefit much from learning Latex to justify one extra module that will consume a faculty member, specially considering that while everybody in a math department uses latex, the majority of mathematicians don't know enough of latex to teach it in a course.&lt;/p&gt;&#xA;" OwnerUserId="15277" LastEditorUserId="15277" LastEditDate="2021-03-12T02:50:10.530" LastActivityDate="2021-03-12T02:50:10.530" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19593" PostTypeId="2" ParentId="19589" CreationDate="2021-03-11T20:31:20.090" Score="3" Body="&lt;p&gt;(feel free to change to comment):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are going to ADD something, then you need to make an argument over what to cut.  Or say camels have it easy and can carry more straw (they don't, but at least adresse the issue).  I see this mental lapse all the time.  Life is an optimization/choice problem.  Not an &amp;quot;add&amp;quot; problem.  By the way, starting to think about this would have made even your forum-y post a little more thoughtful.  You would have prompted yourself to consider duration for instance, addressing the Dan Collins object.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would beware of assuming that the reception you recieve on Stack Exchange (a computer-y platform) is the same as what the bulk of teachers or students feel.  I've noted a huge pro LATEX bias here...even to the extent of thinking it's expected.  But the rest of the world seems to get by surprisingly fine without it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(related to 1) &amp;quot;Required&amp;quot; is a big stick to swing.  After all, DC mentions how he does it ad hoc without a requirement.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why are you sure this needs to be covered in a class. I've actually never used it (not a math major granted).  If I ever needed it, I think I could get it done WHEN I NEEDED IT.  I mean we routinely get the calculator/WA pushers saying we should spend time on that.  But a strong pen and paper student can completely add that stuff, very easily, when he needs it.  (And the weak pen and paper, spent time on gom-puter crutch, not so much in reverse.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15621" LastEditorUserId="15621" LastEditDate="2021-03-11T20:51:51.657" LastActivityDate="2021-03-11T20:51:51.657" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19594" PostTypeId="2" ParentId="19584" CreationDate="2021-03-12T05:26:52.693" Score="6" Body="&lt;p&gt;The American comedian Henny Youngman had this joke:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The patient says, &amp;quot;Doctor, it hurts when I do this.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The doctor says, &amp;quot;Then don't do that!”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately, some people have the incorrect idea that mathematics is about &amp;quot;solving problems&amp;quot;, and lots of them, but it's not. The truth is that math since Euclid is about establishing and understanding basic relationships through carefully-defined concepts, theorems, and proofs.&lt;/p&gt;&#xA;&lt;p&gt;The exercises are just a way to ascertain whether the student understands the principles instead of cheating their way through the material. If you emphasize a giant mass of exercises to the detriment of the concepts, then you've got the tail wagging the dog. Unfortunately, as societies try to present a facade that more of their citizens understand math, science, and engineering, this kabuki-theater becomes more prevalent, and the robotic (i.e., mechanically computable) exercise lists grow, absurdly, without bound.&lt;/p&gt;&#xA;&lt;p&gt;Older textbooks from around a century ago had &lt;em&gt;much&lt;/em&gt; shorter exercise lists, carefully curated, to highlight the essential discoveries in the presentations. Books now have longer exercise lists mostly to try throw chaff at cheaters, but it gives a bad impression about where the emphasis should be.&lt;/p&gt;&#xA;&lt;p&gt;If you understand the essential concepts, then you don't need to do endless drill exercises. You will know what's at the heart of them all.&lt;/p&gt;&#xA;&lt;p&gt;The OP already has their own answer, of course, writing:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One strategy that I adopted early on is going for a few questions&#xA;instead of a large number, this helped me develop my mathematics&#xA;abilities a lot&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, I suspect that they've been given a confusing message by their country's high-stakes and overly-intensive testing regime (again, a response to widespread cheating). Unfortunately there's nothing we as students or instructors can do to solve that. But we can share confidence that if we understand the essential principles, then any standardized-format problems will fall before us. The power of that kind of abstraction is, after all, &lt;em&gt;the whole point of mathematics&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2021-03-12T05:26:52.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19595" PostTypeId="1" CreationDate="2021-03-12T16:42:39.360" Score="7" ViewCount="987" Body="&lt;p&gt;In the optimization section of Calculus 1 a common problem is to find the minimum distance between a curve and a point. I'd like to extend this idea and be able to compute the minimum distance between two (smooth and non-intersecting) curves. However I can't seem to justify that it should occur when the slopes of the two curves are parallel. This fact is visually intuitive, but I'd like to establish it more algebraically.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is there a Calculus-1-level justification that the&#xA;minimum distance between curves occurs when the slopes are equal?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The justifications I've been able to find so far rely on Vector and/or Multivariable Calculus. And perhaps that's the true nature of this type of problem and so it just can't be done in Calculus 1.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Alternatively,&lt;/strong&gt; feel free to solve the following problem using only Calculus 1 optimization techniques that would generalize to slightly more difficult curves (i.e. no tricks allowed that take advantage of the symmetry/simplicity of these specific curves):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Find the minimum distance between the curves &lt;span class=&quot;math-container&quot;&gt;$xy=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=-x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(This, of course, has the solution of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; between &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2021-03-12T18:32:12.097" LastActivityDate="2021-11-28T18:50:08.300" Title="Distance Between Curves in First-Semester Calculus" Tags="|calculus|optimization|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19596" PostTypeId="2" ParentId="19595" CreationDate="2021-03-12T17:28:05.213" Score="12" Body="&lt;p&gt;How about this? Say the two curves are &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, with the min distance achieved&#xA;at point &lt;span class=&quot;math-container&quot;&gt;$a \in A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b \in B$&lt;/span&gt;. Let me assume both curves are smooth,&#xA;or at least have 1st derivatives everywhere.&lt;/p&gt;&#xA;&lt;p&gt;Claim: the segment &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt; is orthogonal to &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&#xA;Suppose to the contrary that &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt; is not orthogonal to &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&#xA;Then one can move &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; slightly &lt;span class=&quot;math-container&quot;&gt;$\pm$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;&#xA;(toward the smaller angle formed by &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt; and the tangent to &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;)&#xA;and shorten &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Because &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt; is orthogonal to both curves, the slopes at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and at &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; must&#xA;be parallel.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;     &lt;a href=&quot;https://i.stack.imgur.com/tgp0g.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tgp0g.jpg&quot; alt=&quot;MinDisk&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The figure illustrates local shortening by moving to the smaller-angle side.&#xA;(But it does not illustrate the actual min distance, which is&#xA;achieved elsewhere.)&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-03-12T18:57:29.800" LastActivityDate="2021-03-12T18:57:29.800" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19597" PostTypeId="2" ParentId="19595" CreationDate="2021-03-14T08:06:36.500" Score="3" Body="&lt;p&gt;I would be more careful with the phrasing of the question. Here are some points to explain why. The first point is an answer under certain conditions.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When looking for the distance between a point and a curve (in two dimensions), do you prove that for a point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; not on the curve when &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is the closest point on the curve, then &lt;span class=&quot;math-container&quot;&gt;$\overleftrightarrow{PB}$&lt;/span&gt; is normal to the curve, or do you accept an intuitive argument? If this can be assumed, then Joseph O'Rourke's answer extended with user2913's comment is a solution: If &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; is the shortest distance between the two curves, then &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is closest to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; (on curve 1), so the tangent at &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; for curve 1 is perpendicular to &lt;span class=&quot;math-container&quot;&gt;$\overleftrightarrow{AB}$&lt;/span&gt;. Similarly, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; (on curve 2) is optimal for &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, so the tangent at &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; for curve 2 is perpendicular to &lt;span class=&quot;math-container&quot;&gt;$\overleftrightarrow{AB}$&lt;/span&gt;. Hence the two tangents must be parallel.&lt;/li&gt;&#xA;&lt;li&gt;However, the closest point that is assumed in the previous bullet only exists under certain conditions. Here is an example. The distance from the origin to the parametric spiral &lt;span class=&quot;math-container&quot;&gt;$((1+1/t)\times\cos t,(1+1/t)\times\sin t)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, but there is no closest point on the spiral to the origin.&lt;/li&gt;&#xA;&lt;li&gt;As for two curves, here is an example. The distance between the graphs of &lt;span class=&quot;math-container&quot;&gt;$y=1+1/(1+x^2)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, yet there are no points on the curves with distance &lt;span class=&quot;math-container&quot;&gt;$1.$&lt;/span&gt; On the graph of &lt;span class=&quot;math-container&quot;&gt;$y=1+1/(1+x^2)$&lt;/span&gt; the point &lt;span class=&quot;math-container&quot;&gt;$(0,2)$&lt;/span&gt; has the property that the tangent is parallel to the line &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt;, but this still does not give the minimal distance.&lt;/li&gt;&#xA;&lt;li&gt;There are also examples on a bounded region: The distance between the spiral &lt;span class=&quot;math-container&quot;&gt;$((1+1/t)\times\cos t,(1+1/t)\times\sin t)$&lt;/span&gt; and the circle &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2=0.25$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$0.5$&lt;/span&gt;, but there are no points on the two curves with distance &lt;span class=&quot;math-container&quot;&gt;$0.5$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;All in all, the point I am trying to make here is that the first intuitive assumption we make is that there are points realising the minimal distance (either between point and curve or between two curves). This assumption is not always true, but if there are such points, then the condition about parallel tangent lines can be proved within Calculus-1.&lt;/p&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2021-03-14T08:06:36.500" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19598" PostTypeId="2" ParentId="19586" CreationDate="2021-03-15T00:30:47.127" Score="1" Body="&lt;p&gt;I would recommend checking in with the &lt;a href=&quot;https://www.mtsu.edu/jstrayer/modules/modules2.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;MODULE&lt;/em&gt;&lt;span class=&quot;math-container&quot;&gt;$(S^2)$&lt;/span&gt;&lt;/a&gt; project (Mathematics Of Doing, Understanding, Learning and Educating for Secondary Schools).  A few notes:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This project is currently developing materials for Geometry, Algebra, Statistics, and Modeling.  Obviously, it is the last of these that would be of interest for you.  You can find links to some sample materials at the the bottom of the page I linked to above.&lt;/li&gt;&#xA;&lt;li&gt;The program materials are designed for use with preservice teachers, but might also work very well with the &amp;quot;rusty teacher&amp;quot; you have in mind.&lt;/li&gt;&#xA;&lt;li&gt;The materials are still under development, and are therefore something of a work in progress.  I am not certain, but I believe the Modeling team is currently seeking partners to pilot their materials.&lt;/li&gt;&#xA;&lt;li&gt;Full disclosure: I am a member of the Geometry writing team, and have piloted the Algebra materials.  However, I don't have any direct involvement with the Modeling group.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Another resource you might find helpful is the recently-published book &lt;em&gt;&lt;a href=&quot;https://www.routledge.com/The-Learning-and-Teaching-of-Mathematical-Modelling/Niss-Blum/p/book/9781138730700&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Learning and Teaching of Mathematical Modelling&lt;/a&gt;&lt;/em&gt; by Mogens Niss and Werner Blum (Routledge, 2020), part of the &lt;a href=&quot;https://www.routledge.com/IMPACT-Interweaving-Mathematics-Pedagogy-and-Content-for-Teaching/book-series/IMPACT&quot; rel=&quot;nofollow noreferrer&quot;&gt;IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching) book series&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-03-15T00:30:47.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19599" PostTypeId="2" ParentId="19582" CreationDate="2021-03-15T00:54:32.480" Score="2" Body="&lt;p&gt;There was indeed a great deal of pushback to the New Math.  You can find a brief summary of this on pp. 12-16 of &lt;a href=&quot;https://www.routledge.com/Secondary-Mathematics-for-Mathematicians-and-Educators-A-View-from-Above/Weiss/p/book/9781138294677&quot; rel=&quot;nofollow noreferrer&quot;&gt;my textbook&lt;/a&gt;, from which I will quote liberally:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To say that these curricula were ambitious would be a dramatic&#xA;understatement. Inspired by the mathematicians of the Bourbaki&#xA;movement, New Math curricula stressed mathematical structures as a&#xA;key topic. The language and notation of sets was introduced as early&#xA;as 1st grade; students were told to distinguish the word number&#xA;(referring to a quantity) from numeral (referring to the&#xA;representation of a number in written form), and to focus on properties such as order and equivalence. In upper elementary grades,&#xA;students wrote numbers in different base systems (i.e. base 6 and base&#xA;2 in addition to the familiar base 10). At the middle-grades level,&#xA;students were taught to distinguish between an &amp;quot;open sentence&amp;quot; (like&#xA;&lt;span class=&quot;math-container&quot;&gt;$3x + 5 = 17$&lt;/span&gt;) and an &amp;quot;equation&amp;quot; (such as &lt;span class=&quot;math-container&quot;&gt;$2 (3 + 5) = 6 + 10$&lt;/span&gt;). So strong&#xA;was the emphasis on set theory and precise language that some felt the&#xA;New Math went too far. Richard Feynman, who served on the California&#xA;State Curriculum Commission in 1964, objected to what he saw as the&#xA;excesses of the New Math's emphasis on language:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In regard to this question of words, there is also in the new mathematics books a great deal of talk about the value of precise language — such things as that one must be very careful to distinguish a number from a numeral and, in general, a symbol from the object that it represents... For example, one of the books pedantically insists on pointing out that a picture of a ball and a ball are not the same thing. I doubt that any child would make an error in this particular direction. It is therefore unnecessary to be precise in the language and to say in each case, &amp;quot;Color the picture of the ball red,&amp;quot; whereas the ordinary book would say, &amp;quot;Color the ball red.&amp;quot;... Although this sounds like a trivial example, this disease of increased precision rises in many of the textbooks to such a pitch that there are almost incomprehensibly complex sentences to say the very simplest thing. In a first-grade book (a primer, in fact) I find a sentence of the type: &amp;quot;Find out if the set of the lollypops is equal in number to the set of girls&amp;quot; — whereas what is meant is: &amp;quot;Find out if there are just enough lollypops for the girls.&amp;quot;...&lt;/p&gt;&#xA;&lt;p&gt;If we would like to, we can and do say, &amp;quot;The answer is a whole number less than 9 and bigger than 6,&amp;quot; but we do not have to say, &amp;quot;The answer is a member of the set which is the intersection of the set of those numbers which is larger than 6 and the set of numbers which are smaller than 9.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;It will perhaps surprise most people who have studied these textbooks to discover that the symbol &lt;span class=&quot;math-container&quot;&gt;$\cup$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\cap$&lt;/span&gt; representing union and intersection of sets, and the special use of the brackets &lt;span class=&quot;math-container&quot;&gt;$\{\}$&lt;/span&gt; and so forth, all the elaborate notation for sets that is given in these books, almost never appear in any writings in theoretical physics, in engineering, in business arithmetic, computer design, or other places where mathematics is being used. I see no need or reason for this all to be explained or to be taught in school. It is not a useful way to express one’s self. It is not a cogent and simple way. It is claimed to be precise, but precise for what purpose?...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It will come as little surprise to anyone who has lived through the more recent waves of mathematics (and other curriculum) reforms over the past several years, and the sometimes intense political controversies that have accompanied them, to learn that the New Math met with an intense and prolonged backlash from parents, teachers, politicians and other stakeholders. Critiques ran the gamut, from the gentle mockery of &lt;a href=&quot;https://www.youtube.com/watch?v=UIKGV2cTgqA&amp;amp;feature=youtu.be&quot; rel=&quot;nofollow noreferrer&quot;&gt;comedian Tom Lehrer’s song New Math&lt;/a&gt; and &lt;a href=&quot;https://assets.amuniversal.com/88014a50f891013014ff001dd8b71c47&quot; rel=&quot;nofollow noreferrer&quot;&gt;whimsical references in Charles Schulz’s Peanuts cartoon&lt;/a&gt;, to the no-holds-barred attack of the best-selling 1973 book &lt;a href=&quot;https://en.wikipedia.org/wiki/Why_Johnny_Can%27t_Add&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Why Johnny Can’t Add: The Failure of the New Math&lt;/em&gt;&lt;/a&gt;, written by New York University mathematics professor Morris Kline...&lt;/p&gt;&#xA;&lt;p&gt;It would probably be giving Kline's book too much credit to claim that it single-handedly ended the New Math, but it is certainly correct that it served as a punctuation mark at the end of the era. By the late 1970s the New Math had virtually disappeared from classroom use. The era of the New Math had given way to what came to be known as the Back to Basics movement.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That last sentence contains the answer to the question in the title:  Yes, there was a successor to the New Math movement, and it was generally known as the Back to Basics movement.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-03-15T00:54:32.480" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19601" PostTypeId="2" ParentId="19595" CreationDate="2021-03-17T07:32:20.667" Score="1" Body="&lt;p&gt;So, you want to find the minimum distance between two curves in the most general form using calculus and optimization.&#xA;The curves you are interested at this particular example are:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;xy=1 ~~and~~ y=-x&#xA;\tag{01}&#xA;\end{equation}&lt;/span&gt;&#xA;The very first step is to write them in parametric form. i.e,&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;xy=1  \Rightarrow &#xA;\begin{cases}&#xA;x_1 &amp;amp;=t  \\&#xA;y_1 &amp;amp;=\frac{1}{t} &#xA;\end{cases}&#xA;\tag{02}&#xA;\end{equation}&lt;/span&gt;&#xA;Similarly,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;y=-x  \Rightarrow &#xA;\begin{cases}&#xA;x_2 &amp;amp;=s  \\&#xA;y_2 &amp;amp;=-s &#xA;\end{cases}&#xA;\tag{03}&#xA;\end{equation}&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; are the parameter. You can define their range based on the domain of defination of the curves. e.g, for first curve &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; can take any value &lt;span class=&quot;math-container&quot;&gt;$(-\infty,\infty)$&lt;/span&gt; except 0.&#xA;Now our objective is to find the pair of &lt;span class=&quot;math-container&quot;&gt;$(t,s)$&lt;/span&gt; for which the distance is minimum. Let's define the distance between any two points located on these two curves as &lt;span class=&quot;math-container&quot;&gt;$D_{12}$&lt;/span&gt;. i.e,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{eqnarray}&#xA;D_{12}^2 &amp;amp;=(x_1-x_2)^2+(y_1-y_2)^2 \\&#xA;&amp;amp;=(t-s)^2+(\frac{1}{t}+s)^2&#xA;\tag{06}&#xA;\end{eqnarray}&lt;/span&gt;&#xA;We need to find the &lt;span class=&quot;math-container&quot;&gt;$(t,s)$&lt;/span&gt; corresponding to the minimum value of the objective function  &lt;span class=&quot;math-container&quot;&gt;$D_{12}^2$&lt;/span&gt;. Note that, since &lt;span class=&quot;math-container&quot;&gt;$D_{12}$&lt;/span&gt; is a positive scalar-valued function, hence minimizer of &lt;span class=&quot;math-container&quot;&gt;$D_{12}$&lt;/span&gt; will also be a minimizer of &lt;span class=&quot;math-container&quot;&gt;$D_{12}^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For extremum value of the objective function,&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp;\frac{d}{dt}{D_{12}^2} =0 \\&#xA;&amp;amp;\Rightarrow 2(t-s)+2(\frac{1}{t}+s)(-t^2)=0 \\&#xA;&amp;amp;\Rightarrow (t-s)-\frac{1}{t^2}(\frac{1}{t}+s)=0 \tag{09}\\&#xA;&amp;amp;\frac{d}{ds}{D_{12}^2} =0 \\&#xA;&amp;amp;\Rightarrow 2(t-s)(-1)+2(\frac{1}{t}+s)=0 \\&#xA;&amp;amp;\Rightarrow (t-s)=(\frac{1}{t}+s) \tag{10}&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Solve those two equations in a smart way. Substitute the second equation(\ref{10}) in the first one(\ref{09}) (sorry! equation \label-ing is not working). You will get,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp; (\frac{1}{t}+s)-\frac{1}{t^2}(\frac{1}{t}+s)=0 \\&#xA;&amp;amp; (1-\frac{1}{t^2})(t-s)=0 &#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hence, either &lt;span class=&quot;math-container&quot;&gt;$(1-\frac{1}{t^2})=0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(t-s)=0$&lt;/span&gt;.&#xA;Now, &lt;span class=&quot;math-container&quot;&gt;$(t-s)=0$&lt;/span&gt; is not possible. Because if you substitute this to the second equation(\ref{10}) you will end up having &lt;span class=&quot;math-container&quot;&gt;${1+s^2}=0$&lt;/span&gt;. Which is not possible as &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; is a real valued parameter.&#xA;Then, only option left is,&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{eqnarray}&#xA;(1-\frac{1}{t^2})=0 \\&#xA;t = \pm 1&#xA;\end{eqnarray}&lt;/span&gt;&#xA;Substitute &lt;span class=&quot;math-container&quot;&gt;$t=1$&lt;/span&gt; in second equation to get &lt;span class=&quot;math-container&quot;&gt;$s=0$&lt;/span&gt;. Similarly &lt;span class=&quot;math-container&quot;&gt;$t=-1$&lt;/span&gt; will give &lt;span class=&quot;math-container&quot;&gt;$s=0$&lt;/span&gt;.&#xA;The closest point on parametric form are &lt;span class=&quot;math-container&quot;&gt;$(t_1=1,s_1=0)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(t_2=-1,s_2=0)$&lt;/span&gt;.&#xA;The corresponding pints on curves in cartesian co-ordinate are &lt;span class=&quot;math-container&quot;&gt;$[(1,1),(0,0)]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$[(-1,-1),(0,0)]$&lt;/span&gt; on respective curves. The minimum distance is &lt;span class=&quot;math-container&quot;&gt;$\sqrt(2)$&lt;/span&gt;. See the image below:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/WucZU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WucZU.png&quot; alt=&quot;The solution points&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;See, even though you missed one solution, mathematics guided us to both the solutions.&lt;/p&gt;&#xA;&lt;p&gt;To generalize this by stating that 'at minimum distance the slope is parallel', this might work for 2D but not in higher dimensions or between a curve and surface. Think of a 3D line and a parabola when both are not in the same plane. The concept of 'slope' is very vague when you go dimensions higher than 2D.&lt;/p&gt;&#xA;" OwnerUserId="15188" LastActivityDate="2021-03-17T07:32:20.667" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19602" PostTypeId="2" ParentId="7237" CreationDate="2021-03-17T13:04:53.037" Score="2" Body="&lt;p&gt;This is more of a long comment than an answer.&lt;/p&gt;&#xA;&lt;p&gt;Some of the other answers here are advocating for factoring rather than case analysis.  I just want to point out that proving the theorem that for any two real numbers &lt;span class=&quot;math-container&quot;&gt;$ab = 0$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$a=0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$b=0$&lt;/span&gt; requires the same sort of reasoning as @Aeryk advocates in the accepted answer:&lt;/p&gt;&#xA;&lt;p&gt;In one direction, if we assume that &lt;span class=&quot;math-container&quot;&gt;$ab=0$&lt;/span&gt;, then we consider two cases:  if &lt;span class=&quot;math-container&quot;&gt;$a=0$&lt;/span&gt;, we are done.  If &lt;span class=&quot;math-container&quot;&gt;$a \neq 0$&lt;/span&gt;, then we can divide both sides by &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to conclude that &lt;span class=&quot;math-container&quot;&gt;$b=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the other direction, assuming either &lt;span class=&quot;math-container&quot;&gt;$a=0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$b=0$&lt;/span&gt; quickly leads to &lt;span class=&quot;math-container&quot;&gt;$ab=0$&lt;/span&gt; by multiplying both sides by the appropriate factor.&lt;/p&gt;&#xA;&lt;p&gt;All of this is just to say that we cannot truly avoid case analysis through factoring:  we can only postpone it.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-03-17T13:04:53.037" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19603" PostTypeId="2" ParentId="18338" CreationDate="2021-03-18T00:38:33.637" Score="1" Body="&lt;p&gt;The text, &amp;quot;Linear Algebra Done Right&amp;quot; Springer(1997) by Sheldon Axler explicitly addressed the same problem. Matrices first appear in the last chapter (10).&lt;/p&gt;&#xA;" OwnerUserId="6202" LastActivityDate="2021-03-18T00:38:33.637" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19604" PostTypeId="1" CreationDate="2021-03-18T16:16:25.657" Score="5" ViewCount="343" Body="&lt;p&gt;I had some mathematics education during my high school and Electrical Engineering studies, but I never used any of them during my career as a software professional. Now I am again coming across Linear Algebra, complex numbers and Calculus while learning certain topics in computer science such as Quantum Computing and Machine Learning.&lt;/p&gt;&#xA;&lt;p&gt;Where I did my studies, maths education was pathetic as we were taught in a way that barely helps to develop intuition and real appreciation for the subject. I had to do rote learning all of the matrix operations and formulas and differentiation rules. Same with Fourier transformation etc which I can vaguely remember. Then we repeatedly solved problems using an outrageous number of sample problems so that you can score in the exam.&lt;/p&gt;&#xA;&lt;p&gt;That is why my maths knowledge is very poor for the level I was formally educated and I realized that in-order to be able to do good quality work in the future using mathematics as a tool, I need to reeducate myself.&lt;/p&gt;&#xA;&lt;p&gt;After doing scores of hours of curriculum and text book research and course auditing, I compiled a list of courses and text books suitable for self study at my level.&lt;/p&gt;&#xA;&lt;p&gt;My curriculum is below:&lt;/p&gt;&#xA;&lt;p&gt;Semester 1:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Single Variable Calculus - MIT OCW and textbook by James Stewart, 8th ed&lt;/li&gt;&#xA;&lt;li&gt;Discrete Mathematics - MIT OCW and corresponding textbook from&#xA;them&lt;/li&gt;&#xA;&lt;li&gt;Linear Algebra - MIT OCW and textbook by Gilbert Strang&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Semester 2:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Multi Variable Calculus - MIT OCW and James Stewart, 8th ed&lt;/li&gt;&#xA;&lt;li&gt;Differential Equation - edX (textbook not decided yet, probably Stewart is enough)&lt;/li&gt;&#xA;&lt;li&gt;Matrix Methods in Data Analysis, Signal Processing, and Machine Learning - MIT OCW and a new textbook from Strang&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Semester 3:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Linear/Non linear/Convex Optimization - edX&lt;/li&gt;&#xA;&lt;li&gt;Introduction to Complex Analysis - coursera&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Semester 4:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Probabilistic Systems Analysis and Applied Probability - MIT OCW and textbook by Bertsekas&lt;/li&gt;&#xA;&lt;li&gt;Statistics for Applications - MIT OCW and textbook &lt;em&gt;The elements of statistical learning&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I made this &amp;quot;semester&amp;quot; thing based on the idea that all of these single courses are covered in a single semester in a college. There are also some rough estimation from course creators on effort needed to self study the topic - which is usually around 150 hours.&lt;/p&gt;&#xA;&lt;p&gt;I am 2-3 months into this process and now I am realizing that I won't be able to cover a given course in 150-200 hours which I had allotted. The problem seem to be doing exercises, especially what I consider as difficult ones. As an example, Discrete Mathematics book had an innocent looking problem &lt;em&gt;&amp;quot;Prove that &lt;span class=&quot;math-container&quot;&gt;$\sqrt2 ^ \sqrt2$&lt;/span&gt; may be rational using Proof by cases&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is one of the 50 or so problems given and this alone took me a few hours and a whole lot of new insights such as transcendental numbers, Gilfond't constant etc. That's great, but I would be done in...may be 4 years instead of 2 I had planned.&lt;/p&gt;&#xA;&lt;p&gt;So my question is - where do I draw lines when it comes to exercises and how do students/professors in good Western universities manage to finish a course in a single semester? Do they just do only part of the textbook? What about problems? Attempting only some of the similar problems sound reasonable, but what if they are very diverse and interesting questions? As I am detached from formalized education for too long, it would be appreciated to get some insights into these issues.&lt;/p&gt;&#xA;" OwnerUserId="15437" LastEditorUserId="15437" LastEditDate="2021-04-27T14:29:05.243" LastActivityDate="2021-04-27T14:29:05.243" Title="Self study curriculum for a working professional who is enthusiastic about mathematics" Tags="|self-learning|exercises|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="19605" PostTypeId="2" ParentId="19568" CreationDate="2021-03-18T21:24:20.480" Score="1" Body="&lt;p&gt;Well, I guess the answer is rather obvious - this relation is true if we additionally assume that the partial derivatives are continuous, in which case the function is differentiable. (The actual proof of this is deemed too technical for many such courses.) This condition is almost always met in practice, and a large fraction of courses (say, those for physicists, engineers etc.) are teaching calculus-style, computational aspect, not even bothering with rigorous definitions and proofs.&lt;/p&gt;&#xA;&lt;p&gt;Of the sources your mention, &amp;quot;Paul's online notes&amp;quot; is clearly like that. Lots of &amp;quot;worked examples&amp;quot;, and as far as I can see, it does not even give a definition of a differentiable function. It says that your formula follows from chain rule, but the conditions for the latter to be applicable are never discussed. So, I wouldn't really say it even makes a mistake here - it just doesn't discuss the class of functions to which the formula applies, and &lt;em&gt;that is true for everything there,&lt;/em&gt;  not just this formula. Something similar may apply to Wolfram MathWorld.&lt;/p&gt;&#xA;&lt;p&gt;In some cases, the condition of continuity of partial derivatives is simply forgotten. I feel it is unlikely that a properly trained mathematician would make such a mistake, so that may be a sign of the overall qualification of people involved, in which case there will be more mistakes. But I guess the popularity and &amp;quot;reputation&amp;quot; of these online sources hinges more on how well they explain things than on being correct on every detail.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2021-03-18T21:24:20.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19606" PostTypeId="2" ParentId="19589" CreationDate="2021-03-18T21:41:46.060" Score="3" Body="&lt;p&gt;At the University where I'm teaching, there is such a course mandatory for all students (not only in math, but also physicists, chemists and computed scientists) and recommended in the first year of studies.&lt;/p&gt;&#xA;&lt;p&gt;It's a 1 credit course (meaning the it &amp;quot;officially&amp;quot; assumes about 25 hours of work), and it's offered online - the student is supposed to read tutorials, practice, then take a practice test, then the exam; they can also ask questions from the teacher.&lt;/p&gt;&#xA;&lt;p&gt;I would say it's a very reasonable arrangement. The students will, at the very least, be required to write a bachelor thesis, then possibly a master thesis. At which point, if you don't offer such a course, you would just assume that they are somehow supposed to have learnt to typeset math on their own. Requiring something they haven't been explicitly taught may be frustrating (although that is culture-dependent, of course). And quick learners will just easily pass the exam.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2021-03-18T21:41:46.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19607" PostTypeId="1" AcceptedAnswerId="19609" CreationDate="2021-03-19T00:16:52.107" Score="17" ViewCount="2164" Body="&lt;p&gt;Are there some &lt;em&gt;natural&lt;/em&gt; contexts in which a double exponential occurs,&#xA;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to the (&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; to the &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x ^ {(y ^ z)} \;, \textrm{or} \;, a ^ {(b ^ c)} \; \textrm{?}&#xA;$$&lt;/span&gt;&#xA;Of course one can contrive many problems that ask computational questions concerning towers of exponents,&#xA;possibly challenging problems.&#xA;And in my own area of research, &lt;span class=&quot;math-container&quot;&gt;$2^{n^2}$&lt;/span&gt; is not uncommon;&#xA;EXP is exponential-time &lt;span class=&quot;math-container&quot;&gt;$O( 2^ {n^k} )$&lt;/span&gt;.&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Generalized_game&quot; rel=&quot;noreferrer&quot;&gt;Generalized chess&lt;/a&gt; falls in this class.&lt;/p&gt;&#xA;&lt;p&gt;I am more seeking some context in which such a (limited) tower&#xA;occurs naturally and would be understood&#xA;and appreciated by high-school or early-college students.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-03-19T00:41:48.917" LastActivityDate="2021-03-22T06:18:19.397" Title="Natural occurrences of a to the (b to the c)?" Tags="|functions|exponential|exponent|" AnswerCount="11" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19608" PostTypeId="2" ParentId="19607" CreationDate="2021-03-19T02:21:11.880" Score="2" Body="&lt;p&gt;Inspired by your example, I like the number of matrices, tensors, binary or n-ary relations over a specified set (where the domain and range of such a relation need not match), such as asking about colorings using &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; colors of a &lt;span class=&quot;math-container&quot;&gt;$n \times n$&lt;/span&gt; board. Continuous or &amp;quot;non-combinatorial&amp;quot; applications feel hard to come up with, perhaps due to dimensionless exponential constraints, but might also exist?&lt;/p&gt;&#xA;" OwnerUserId="4679" LastActivityDate="2021-03-19T02:21:11.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19609" PostTypeId="2" ParentId="19607" CreationDate="2021-03-19T03:40:39.303" Score="27" Body="&lt;p&gt;If your students have learned some statistics, then you could point out that the normal distribution's probability density function uses this double exponential.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2021-03-19T03:40:39.303" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19610" PostTypeId="2" ParentId="19607" CreationDate="2021-03-19T12:44:37.613" Score="12" Body="&lt;p&gt;Count all possible functions mapping &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; input bits to &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt; output bits: For &lt;em&gt;each&lt;/em&gt; of the &lt;span class=&quot;math-container&quot;&gt;$2^i$&lt;/span&gt; input combinations, each output has two possible outputs (&lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;), i.e. you have &lt;span class=&quot;math-container&quot;&gt;$2^{2^i}$&lt;/span&gt;, leading to a total of &lt;span class=&quot;math-container&quot;&gt;$2^{o\cdot 2^i}$&lt;/span&gt; binary functions. And course if you're working with &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; different values per input/output instead of binary you end up with &lt;span class=&quot;math-container&quot;&gt;$n^{o\cdot n^i}$&lt;/span&gt; functions.&lt;/p&gt;&#xA;" OwnerUserId="1120" LastActivityDate="2021-03-19T12:44:37.613" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19611" PostTypeId="1" AcceptedAnswerId="19613" CreationDate="2021-03-19T13:41:46.283" Score="7" ViewCount="457" Body="&lt;p&gt;Context: I am an assistant professor of mathematics at a small school in the US.&lt;/p&gt;&#xA;&lt;p&gt;Many of my courses have homework assignments administered by WeBWorK.  A small minority of students struggle mightily (mostly with parenthesis) when assembling inline text expressions like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\texttt{[(e^(3x^2-7)(6x))(5x+2)-(e^(3x^2-7))(5)][(5x+2)^2]}&#xA;$$&lt;/span&gt;&#xA;in order to represent mathematical expressions like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{(e^{3x^2-7}(6x))(5x+2)-(e^{3x^2-7})(5)}{(5x+2)^2}.&#xA;$$&lt;/span&gt;&#xA;This brings me to my question: Are there any online tools that could allow a student to assemble something in &amp;quot;pretty print&amp;quot; and then decompose the given expression into inline text?   Wolfram Alpha and LaTeX both go the wrong way--they transform inline text into beautiful expressions.&lt;/p&gt;&#xA;&lt;p&gt;Note: I believe that the students learn something significant from learning how to carefully type these inline expressions, but for those who really struggle, I don't want that to be what prevents them from completing an assignment.&lt;/p&gt;&#xA;" OwnerUserId="5988" LastEditorUserId="5988" LastEditDate="2021-03-20T00:10:19.597" LastActivityDate="2021-03-22T03:21:08.677" Title="Do any tools exist that convert &quot;pretty print&quot; equations into &quot;inline text&quot; equations?" Tags="|homework|online-instruction|software|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19612" PostTypeId="2" ParentId="19607" CreationDate="2021-03-19T15:22:31.717" Score="4" Body="&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; represents the set of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; students of a school, then &lt;span class=&quot;math-container&quot;&gt;$\mathscr P(S)$&lt;/span&gt; is the set of rosters for all possible clubs that that school could host (if we assume that any two clubs with exactly the same set of students merged their interests into a single club charter).  In that same scenario, the yearbook that printed pictures of all of the clubs would be a member of &lt;span class=&quot;math-container&quot;&gt;$\mathscr P(\mathscr P(S))$&lt;/span&gt;.  Thus, there would be &lt;span class=&quot;math-container&quot;&gt;$2^{2^n}$&lt;/span&gt; possible yearbooks based on the different numbers and arrangements of clubs the students could form.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-03-19T15:22:31.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19613" PostTypeId="2" ParentId="19611" CreationDate="2021-03-19T15:39:52.620" Score="5" Body="&lt;p&gt;Here's an answer (but you might not like it*): &lt;a href=&quot;https://www.mathway.com/Algebra&quot; rel=&quot;noreferrer&quot;&gt;Mathway&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You can intuitively create expressions in their rendered chat box. Then highlight the expression and copy and paste into a regular text box. When you do, it gives it inline.&lt;/p&gt;&#xA;&lt;p&gt;For example, entering into Mathway the expression you mentioned:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/AKxw8.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AKxw8.png&quot; alt=&quot;Mathway expression entry&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And then highlighting, copying (I had to do a Ctrl+A to highlight it and a Ctrl+C to copy), and pasting gives: &lt;code&gt;((e^(x^(2)+7)(6x))(5x+2)-(e^((x^(2)+7)))(5))/((5x+2)^(2))&lt;/code&gt;&lt;/p&gt;&#xA;&lt;p&gt;* I say you might not like it because the program is designed to help the students solve problems/cheat and you might not want your students interacting with such a site.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2021-03-19T15:39:52.620" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19614" PostTypeId="2" ParentId="19607" CreationDate="2021-03-19T16:56:27.960" Score="3" Body="&lt;p&gt;It is easy to square a number. So instead of computing &lt;span class=&quot;math-container&quot;&gt;$\exp(z)$&lt;/span&gt; you can compute &lt;span class=&quot;math-container&quot;&gt;$\exp(z/2^k)^{2^k}$&lt;/span&gt; for some suitably large positive integer &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;. This is a very simple way of accelerating the convergence of the Taylor series for &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt;, also known as &amp;quot;argument reduction&amp;quot;. The double-exponential is important here because it allows you to get the &lt;span class=&quot;math-container&quot;&gt;$2^k$&lt;/span&gt;-th power using just &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; squarings; the technique will not work without perfect powers.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-03-19T16:56:27.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19615" PostTypeId="2" ParentId="19607" CreationDate="2021-03-19T22:19:42.183" Score="2" Body="&lt;p&gt;I think there are really two different ways that this can occur. (1) You can have people writing math in real-world contexts where it makes sense to use a notation that is this type of tower of exponents. (2) You can have a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that blows up like &lt;span class=&quot;math-container&quot;&gt;$a^{b^x}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$a^{x^b}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x\rightarrow\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I'm a physicist, and the contexts where I see #1 are generally things having to do with statistical physics. For example, I can have &lt;span class=&quot;math-container&quot;&gt;$10^{23}$&lt;/span&gt; atoms in a box, and I can ask what is the probability that if I check at a certain time, all the atoms will be on the left side. This will be &lt;span class=&quot;math-container&quot;&gt;$2^{-10^{23}}$&lt;/span&gt;. Similar examples are things like the time scale for a white dwarf to spontaneously decay into a black hole by quantum tunneling, or probabilities involving Boltzmann brains.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-03-19T22:19:42.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19616" PostTypeId="2" ParentId="19607" CreationDate="2021-03-19T23:39:32.770" Score="3" Body="&lt;p&gt;What a nice collection of answers!&lt;/p&gt;&#xA;&lt;p&gt;I am inclined to use &lt;span class=&quot;math-container&quot;&gt;$\lfloor A^{3^n} \rfloor$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mills%27_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mill's constant&lt;/a&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A \approx 1.3063778838630806904686144926 \;,&#xA;$$&lt;/span&gt;&#xA;just because it is astounding that this evaluates to a prime for every &lt;span class=&quot;math-container&quot;&gt;$n \in \mathbb{N}$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2, 11, 1361, 2521008887, \ldots \;.&#xA;$$&lt;/span&gt;&#xA;See &lt;a href=&quot;https://oeis.org/A051254&quot; rel=&quot;nofollow noreferrer&quot;&gt;A051254&lt;/a&gt;.&#xA;But really, @JoelReyesNoche's answer is best.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-03-19T23:39:32.770" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19617" PostTypeId="2" ParentId="19607" CreationDate="2021-03-20T01:15:36.553" Score="8" Body="&lt;p&gt;The number of undirected graphs of order &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$2^{n \choose 2} = 2^{(n^2-n)/2}$&lt;/span&gt; (e.g. consider the adjacency vector as a binary-encoded number). You can describe this in terms of the number of different arrangements in which &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; classmates can be friends. This also has the benefit of being easy to enumerate for small numbers.&lt;/p&gt;&#xA;" OwnerUserId="14054" LastActivityDate="2021-03-20T01:15:36.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19618" PostTypeId="2" ParentId="4367" CreationDate="2021-03-20T03:27:18.187" Score="0" Body="&lt;p&gt;A neglected aspect of teaching binary is the question of how to pronounce it. I will deal with this question.&lt;/p&gt;&#xA;&lt;p&gt;In computer science it often makes sense to pronounce binary numbers like telephone numbers; a mere series of numerals. But in a math class, quantities are being considered, and a comparison with base ten counting and arithmetic is implicit if not explicit.&#xA;Current pronunciation systems for binary are inadequate. That is why binary numbers when being taught to beginners, for example fifth graders, should be pronounced in a way that is analogous to how the teacher has been pronouncing base ten numbers when doing counting and arithmetic, and not analogous to how the teacher recites a telephone number.&lt;/p&gt;&#xA;&lt;p&gt;Let’s dive in using fourteen as an example.&lt;/p&gt;&#xA;&lt;p&gt;Fourteen, when converted from our everyday base ten to a binary number is written ‘1110'. Sometimes a subscript ‘two’ or subscript ‘2’ is appended making it ‘1110two’ or ‘1110₂’. Note that this seems to be best that Medium.com can do to display a subscript ‘two’ or ‘2’. I will use ‘1110two’ as being the safer, less ambiguous format.&lt;/p&gt;&#xA;&lt;p&gt;Most math teachers pronounce 1110two ‘one one one oh base two’, or ‘one one one zero base two’, or if the context makes it unambiguous, just ‘one one one oh’, or ‘one one one zero’. This works, but it is not ideal because one is simply reading out the digits. It is like pronouncing ‘27’ as ‘two seven’ rather than ‘twenty-seven’. Or pronouncing 7² as ‘seven superscript two’ rather than ‘seven squared’. Or pronouncing French ‘chat’ (which is pronounced in French roughly as ‘shah’ and means ‘cat’ — the meowing animal) as ‘see aitch ay tee’.&lt;/p&gt;&#xA;&lt;p&gt;Not many math teachers, thank goodness, would pronounce 14 ÷ 2 as ‘one four divided by two’. But when teaching arithmetic in binary, the teacher probably would pronounce the exact same problem, 1110two ÷ 10two as ‘one one one zero divided by one zero’. This seems to be a great pity. The reason for it may be the lack of a well-known way to pronounce binary numbers that is analogous to the way we pronounce base ten numbers when doing arithmetic or counting.&lt;/p&gt;&#xA;&lt;p&gt;I say ‘when doing arithmetic or counting’ because when the number is merely a phone number or other code or data, and therefore not a number that that represents a quantity, and not a number whose magnitude needs to be understood, and not a number that will be used in any calculation, then it is fine and very efficient to recite the string of numerals.&lt;/p&gt;&#xA;&lt;p&gt;You could probably find a math teacher who ‘pronounces’ 1110two as ‘fourteen’. This also works, but it is not ideal because it is not so much pronouncing the binary number as calculating the decimal representation and pronouncing that. It is like ‘pronouncing’ 7² as ‘forty-nine’, or ‘pronouncing’ French ‘chat’ as ‘cat’, which would be calculating what 7² is and pronouncing that, and translating French ‘chat’ into English ‘cat’ and pronouncing that.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps half of all people that have an opinion on it, if the Internet ‘answers’ sites are to go by, think that 1110two should be pronounced ‘one thousand one hundred ten base two’. This is a terrible method, because that way you can never safely shorten it to just ‘one thousand one hundred ten’ because of the danger of it being understood as being the familiar one thousand one hundred ten (of base ten). Also, there will always be the danger that you will be understood to mean 10001010110two which is 1110ten converted to a binary format. There is no way to make the listener understand that you don’t mean the latter, no matter how you phrase it. Whether you say ‘one thousand one hundred in/written in/read as/base two’ it will never be clear that you don’t mean 10001010110two. It is like pronouncing French ‘chat’ as English ‘chat’ (as ‘chit-chat’).&lt;/p&gt;&#xA;&lt;p&gt;If you demonstrate or teach counting, arithmetic, and measuring in binary while reciting the numbers like phone numbers, you will have a false comparison with decimal unless you pronounce the decimal numbers in the same way.&lt;/p&gt;&#xA;&lt;p&gt;It is for the above reasons that I propose that math teachers, and indeed anyone who wishes to appreciate arithmetic, counting, measurement, and other mathematical thinking using binary numbers should pronounce binary numbers in a way that is analogous to how we pronounce our everyday base ten numbers when not reciting a phone number or a room number or other code or data.&lt;/p&gt;&#xA;&lt;p&gt;If you demonstrate or teach counting, arithmetic, and measuring in binary while reciting the numbers like phone numbers, you will have a false comparison with decimal unless you pronounce the decimal numbers in the same way.&lt;/p&gt;&#xA;&lt;p&gt;The student has little chance of appreciating the full beauty and power of binary when all numbers are reduced to strings of numerals pronounced like binary telephone numbers. And with decimal counting using normal good decimal pronunciation being compared with binary counting using mere recitation of numerals the student cannot make an accurate or enlightened comparison between the two bases.&#xA;Counting aloud in binary using my pronunciation system.&lt;/p&gt;&#xA;&lt;p&gt;Try counting from one to sixteen in binary, pronouncing the binary numbers in a way that is roughly analogous to how you’d do it in base ten:&lt;/p&gt;&#xA;&lt;p&gt;1two, pronounced ‘one’. Note that 1two is analogous to 1ten because each is the smallest number and the smallest numeral in its base, but also to 9ten because each is the biggest single digit number and the biggest numeral in its base.&lt;/p&gt;&#xA;&lt;p&gt;10two, pronounced ‘two’. Note that 10two is analogous to 10ten because each is the smallest two digit number.&lt;/p&gt;&#xA;&lt;p&gt;11two, pronounced ‘two one’. Note that 11two is analogous 11ten because it is one more than 10two but also to 19ten because each is the largest two digit number that starts with a ‘1’ and also to 99ten because each is the largest two digit number.&lt;/p&gt;&#xA;&lt;p&gt;100two, pronounced ‘four’. Note that 100two is analogous to 100ten because each is the smallest three digit number but also analogous to 900ten because each is the biggest three digit number that ends in ‘00’.&lt;/p&gt;&#xA;&lt;p&gt;101two, pronounced ‘four one’. Note that 101two is analogous to 101ten because each is one bigger than the smallest three digit number in its base. This is not the only analogy here.&lt;/p&gt;&#xA;&lt;p&gt;110two, pronounced ‘four two’.&lt;/p&gt;&#xA;&lt;p&gt;‘111two, pronounced ‘four two one’. Note that 111two is analogous to 999ten because each is the largest three digit number in its base.&lt;/p&gt;&#xA;&lt;p&gt;1001two, pronounced ‘eight one’.&lt;/p&gt;&#xA;&lt;p&gt;1010two, pronounced ‘eight two’.&lt;/p&gt;&#xA;&lt;p&gt;1011two, pronounced ‘eight two one’.&lt;/p&gt;&#xA;&lt;p&gt;1100two, pronounced ‘eight four’.&lt;/p&gt;&#xA;&lt;p&gt;1101two, pronounced ‘eight four one’.&lt;/p&gt;&#xA;&lt;p&gt;1110two, pronounced ‘eight four two’.&lt;/p&gt;&#xA;&lt;p&gt;1111two, pronounced ‘eight four two one’.&lt;/p&gt;&#xA;&lt;p&gt;10000two, pronounced…what? ‘Sixteen’ would seem to be the logically implied name. But ‘sixteen’ is a base ten number. It’s short for ‘six and ten’. That’s confusing. And it will be worse when we get to bigger numbers. ‘Thirty-two’ (‘three tens and two’), ‘sixty-four’ (‘six tens and four’), and ‘one hundred twenty eight’(ten tens and two tens and eight’, and so on, getting longer and longer and more and more difficult and confusing and more and more obviously being base ten numbers.&lt;/p&gt;&#xA;&lt;p&gt;The solution is to find short, easily pronounced and easily recalled names to replace ‘sixteen’ for 10000two, ‘thirty-two’ for 100000two, and so on. But there are an awful lot of names needed, because binary numbers are so long. Compare 1000two with 8ten. It’s four digits compared with one. So these names should be super easy to recall, because so many are needed and no one will be able to face learning so many names unless they are super duper easy to learn.&lt;/p&gt;&#xA;&lt;p&gt;The easiest to recall names would be names that can be deduced from the number. This way, no names need to be memorized, only the way to deduce the name from the number needs be memorized.&lt;/p&gt;&#xA;&lt;p&gt;But for now, to keep things simple, I’ll just tell you that my new name for 10000two is ‘ri’ and my new name for 100000two is ‘li’, and my new name for 1000000two is ‘shi’. 10000000two is ‘ki’. 100000000two is ‘fi’. 1000000000two is ‘pi’.&lt;/p&gt;&#xA;" OwnerUserId="15662" LastActivityDate="2021-03-20T03:27:18.187" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19619" PostTypeId="2" ParentId="19595" CreationDate="2021-03-20T04:53:45.180" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;[T]o justify that it should occur when the slopes of the two curves are parallel:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/yRRti.png&quot; width=&quot;450&quot;&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt; is a minimum, then the curve through &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; has no points in the interior of the circle centered at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; with radius &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt;.  (Any point inside is closer to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; than &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.)  Likewise, the curve through &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; has no points in the interior of the circle centered at &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; with radius &lt;span class=&quot;math-container&quot;&gt;$ba$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Since it is presumed that the curves are smooth and have well defined tangents, the tangents to the curves at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; must coincide with the tangents to the corresponding circles at the same points.&lt;sup&gt;*&lt;/sup&gt;  Therefore the tangents at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are perpendicular to the segment &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt; and hence parallel.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;*&lt;/sup&gt;I've found students don't usually hold with Steiner's dictum, &amp;quot;Geometry stimulates, while calculating replaces, thinking&amp;quot; (per Dirk Struik), so you might feel obliged to haul out some &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;'s. OTOH, students love the idea that things are tangent if they don't cross and touch at only one point.  But if necessary, here goes:  Let &lt;span class=&quot;math-container&quot;&gt;$y = f(x)$&lt;/span&gt; be the equation of the curve through &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y = g(x)$&lt;/span&gt; be the equation of the (semi)circle through &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and centered at &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.  Then &lt;span class=&quot;math-container&quot;&gt;$f(x)-g(x)$&lt;/span&gt; is zero at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.  If the derivative of &lt;span class=&quot;math-container&quot;&gt;$f(x)-g(x)$&lt;/span&gt; is nonzero at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f(x)-g(x)$&lt;/span&gt; changes sign at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&lt;sup&gt;**&lt;/sup&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; enters the circle, contrary to hypothesis.  Therefore &lt;span class=&quot;math-container&quot;&gt;$f'(x)=g'(x)$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, and the tangent lines have the same slope and pass through the same point, which is what needed to be shown.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;**&lt;/sup&gt;If this isn't obvious, one can prove, in the same way Fermat's Critical Point Theorem is proved, that if &lt;span class=&quot;math-container&quot;&gt;$h(c)=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h'(c)\ne0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$h(x)$&lt;/span&gt; changes sign at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;. Or one can prove it directly from the limit definition. If &lt;span class=&quot;math-container&quot;&gt;$[h(x)-h(c)]/(x-c)=h(x)/(x-c)$&lt;/span&gt; approaches a nonzero value &lt;span class=&quot;math-container&quot;&gt;$h'(c)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; close enough to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;,&#xA;then &lt;span class=&quot;math-container&quot;&gt;$h(x)$&lt;/span&gt; must change sign when &lt;span class=&quot;math-container&quot;&gt;$x-c$&lt;/span&gt; changes sign. Etc.  This can also be proved geometrically using &lt;a href=&quot;https://doi.org/10.1080/00029890.1964.11992395&quot; rel=&quot;nofollow noreferrer&quot;&gt;H.A. Thurston's definition of the tangent line&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2021-03-21T12:53:31.880" LastActivityDate="2021-03-21T12:53:31.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19620" PostTypeId="1" CreationDate="2021-03-20T10:49:00.730" Score="1" ViewCount="72" Body="&lt;p&gt;I will be teaching a combination of business calculus and business linear algebra, two classes that have been around awhile at my school. I’m assuming people are familiar with these types of classes. I’m wondering if anyone can tell me a couple topics that were classically in these classes that were probably discarded over time, and a couple topics that either have been added to more modern forms of the classes, or SHOULD be added.&lt;/p&gt;&#xA;" OwnerUserId="11880" LastActivityDate="2021-03-20T10:49:00.730" Title="History of business calculus/linear algebra curriculum" Tags="|calculus|linear-algebra|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19621" PostTypeId="1" AcceptedAnswerId="19622" CreationDate="2021-03-20T11:31:29.047" Score="2" ViewCount="276" Body="&lt;p&gt;I will be teaching discrete mathematics. I’m wondering if anyone can tell me a couple topics that were classically in this class (in any undergraduate version) that were probably discarded over time, and a couple topics that either have been added to more modern forms of this class, or SHOULD be added.&lt;/p&gt;&#xA;&lt;p&gt;Even better if there’s something more recent you can think of that was deleted in favor of something else.&lt;/p&gt;&#xA;" OwnerUserId="11880" LastEditorUserId="2083" LastEditDate="2021-03-20T12:10:26.240" LastActivityDate="2021-04-01T05:39:58.003" Title="History of discrete math curriculum" Tags="|undergraduate-education|discrete-math|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19622" PostTypeId="2" ParentId="19621" CreationDate="2021-03-20T13:08:16.643" Score="5" Body="&lt;p&gt;This is not a question that can be easily answered without careful historical research. Here are some guesses in the absence of such research.&lt;/p&gt;&#xA;&lt;p&gt;What's been added?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Algorithm analysis: big-&lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; notation. Dijkstra's algorithm.&lt;/li&gt;&#xA;&lt;li&gt;Aspects of complexity theory: polynomial-time vs. beyond polynomial.&lt;/li&gt;&#xA;&lt;li&gt;Discussion of the TSP problem, Christofides' heuristic.&lt;/li&gt;&#xA;&lt;li&gt;Cryptography, RSA encryption.&lt;/li&gt;&#xA;&lt;li&gt;Graph Laplacian, possibly even touching eigenvalues.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;What's been removed?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Some topics from discrete probability, e.g., Birthday paradox.&lt;/li&gt;&#xA;&lt;li&gt;Some topics from linear algebra, e.g., Cramer's rule.&lt;/li&gt;&#xA;&lt;li&gt;Some advanced material on solving recurrence relations, e.g., second-order recurrences, multiple roots.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-03-20T13:08:16.643" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="19623" PostTypeId="1" CreationDate="2021-03-21T20:33:59.983" Score="3" ViewCount="156" Body="&lt;p&gt;&lt;strong&gt;Circumstance&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hello,&lt;/p&gt;&#xA;&lt;p&gt;I am looking for books that encourage creative problem-solving at the high school level. I attended secondary school in the United States and I have noticed that there are techniques that I was not taught thoroughly.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Request&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Please suggest books that address the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Substitution/change of variable for single variable equations&lt;/li&gt;&#xA;&lt;li&gt;Factoring non-algebraic expressions&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;smart&amp;quot; manipulation of expressions&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I have listed YouTube videos depicting the &lt;em&gt;kinds&lt;/em&gt; of problems I would like to be able to solve. The videos are in Russian, but they all involve solving equations.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=SrBg1BkW24g&quot; rel=&quot;nofollow noreferrer&quot;&gt;Problem 1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=Kipr842uj7c&quot; rel=&quot;nofollow noreferrer&quot;&gt;Problem 2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=2E5J6x1nDog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Problem 3&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=JRg1pKXvaDk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Problem 4&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=0ZZRmgXXlHs&quot; rel=&quot;nofollow noreferrer&quot;&gt;Problem 5&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am aware of The Art of Problem Solving but I feel it addresses contest math.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="15670" LastActivityDate="2021-03-21T20:53:20.240" Title="High school algebra books that encourage creative/novel problem solving" Tags="|secondary-education|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19624" PostTypeId="2" ParentId="19623" CreationDate="2021-03-21T20:53:20.240" Score="3" Body="&lt;p&gt;I like &lt;em&gt;&lt;strong&gt;The Art and Craft of Problem Solving&lt;/strong&gt;&lt;/em&gt;, by Paul Zeitz. But you probably already know of it, since it's mentioned on the aops site.&lt;/p&gt;&#xA;&lt;p&gt;I think it addresses some of what you list above. But it goes so much deeper too.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-03-21T20:53:20.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19625" PostTypeId="2" ParentId="19607" CreationDate="2021-03-21T22:44:09.510" Score="1" Body="&lt;p&gt;Let &lt;em&gt;n&lt;/em&gt; be given and consider the set of logical formulas with &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; variables.  Two formulas are &lt;em&gt;equivalent&lt;/em&gt;  if they are both true, or both false, for any assignment of truth values to the variables.  For example, the formulas &lt;span class=&quot;math-container&quot;&gt;$\lnot a\land b$&lt;/span&gt;  and &lt;span class=&quot;math-container&quot;&gt;$\lnot(a\lor\lnot b)$&lt;/span&gt; are equivalent, because both are true only when &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is false and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is true.  This naturally partitions the family of formulas into classes, with two formulas in the same class if they are equivalent.&lt;/p&gt;&#xA;&lt;p&gt;How many classes are there?  The answer is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2^{2^n}.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="259" LastActivityDate="2021-03-21T22:44:09.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19626" PostTypeId="2" ParentId="19607" CreationDate="2021-03-22T02:03:47.780" Score="4" Body="&lt;p&gt;Gompertz model was created in 1825 to study human mortality curves. From the 1920's, it was used in economic fields, and from there it was also used in Biology to study cells and microorganisms, such as microbes, growth of tumours, and survival of cancer patients. The model is described by the differential equation &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{dN}{dt} = rN \ln \left(\dfrac{K}{N} \right),$$&lt;/span&gt; and in the context of growth of tumours, &lt;span class=&quot;math-container&quot;&gt;$N=N(t)$&lt;/span&gt; is the number of cells at time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is a positive constant related to the maximum possible size of the tumour.&lt;/p&gt;&#xA;&lt;p&gt;The general solution for the model reads &lt;span class=&quot;math-container&quot;&gt;$N(t)= Ke^{-Ce^{rt}}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is a constant related to the initial condition &lt;span class=&quot;math-container&quot;&gt;$N(0)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;References:&#xA;&lt;a href=&quot;https://doi.org/10.1371/journal.pone.0178691&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1371/journal.pone.0178691&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3380" LastActivityDate="2021-03-22T02:03:47.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19627" PostTypeId="2" ParentId="19611" CreationDate="2021-03-22T03:21:08.677" Score="2" Body="&lt;p&gt;As mentioned in the comments, Mathquill (&lt;a href=&quot;http://docs.mathquill.com/en/latest/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://docs.mathquill.com/en/latest/&lt;/a&gt;) allows pretty print entry (and copy/paste from other mathquill environments) and outputs latex. Khan Academy's KAS.js (&lt;a href=&quot;https://github.com/Khan/KAS&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://github.com/Khan/KAS&lt;/a&gt;) can parse latex and output inline math. I've stuck the two together here: &lt;a href=&quot;https://trkern.github.io/inliner.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://trkern.github.io/inliner.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The output isn't optimal, since KAS makes changes to an equation's structure at the parsing stage.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-03-22T03:21:08.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19628" PostTypeId="2" ParentId="19607" CreationDate="2021-03-22T06:18:19.397" Score="3" Body="&lt;p&gt;The maximum of a large number of independent, identically distributed random variables -- e.g., the highest flood observed over a long period -- has an &lt;a href=&quot;https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution&quot; rel=&quot;nofollow noreferrer&quot;&gt;extreme value distribution&lt;/a&gt;. One common case is the Gumbel distribution, whose cdf has the double-exponential form &lt;span class=&quot;math-container&quot;&gt;$e^{-e^{-x}}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14524" LastActivityDate="2021-03-22T06:18:19.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19629" PostTypeId="1" CreationDate="2021-03-22T20:54:31.557" Score="3" ViewCount="292" Body="&lt;p&gt;This semester in first-semester Calculus I've been trying to focus on how to do Calculus calculations when given a table of data since this seems to be of importance to science majors. There are pretty standard methods for computing tabular derivatives and tabular antiderivatives (integrals). It appears, though, that doing one operation and then doing the other does not give back the data you started with.&lt;/p&gt;&#xA;&lt;p&gt;Before I get into my long example below, the main question is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Most Important Question&lt;/strong&gt;: Is there a way, using tables of data, to demonstrate the Fundamental Theorem of Calculus?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;A Long Example&lt;/h1&gt;&#xA;&lt;p&gt;Suppose we have the data&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;1&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;5&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;9&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;13&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;17&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;7&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;15&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;31&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;For data in the middle of the table, we use the average of the slopes to the right and left to approximate the derivative at that point, e.g. &lt;span class=&quot;math-container&quot;&gt;$$f'(5) = \text{Average}\left(\frac{3-1}{5-1},\frac{7-3}{9-5}\right) = \frac{3}{4}.$$&lt;/span&gt; For endpoints, you just use the only slope you can compute, e.g. &lt;span class=&quot;math-container&quot;&gt;$f'(1) = \frac{3-1}{5-1} = \frac{1}{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Doing so gives you the table&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;1&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;5&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;9&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;13&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;17&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;7&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;15&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;31&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1/2&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3/4&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3/2&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;4&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;Going in reverse, suppose we know &lt;span class=&quot;math-container&quot;&gt;$f(1)=1$&lt;/span&gt; and, given the derivatives we just computed, now compute &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at the other points. The standard way to do this is using the average of a left-hand and right-hand sum (aka trapezoid rule) to approximate the integral, e.g.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(9) = f(1) + \int_1^9 f'(x)dx = 1 + \frac{4}{2}\left(\frac{1}{2}+2\cdot\frac{3}{4}+\frac{3}{2}\right) = 8 \neq 7$$&lt;/span&gt;&#xA;The full table looks like&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;     &lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;1&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;5&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;9&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;13&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;17&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;7&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;15&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;31&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1/2&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3/4&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3/2&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;4&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$f(1)+\int_1^x f'(t)dt$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3.5&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;8&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;17&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;31&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;First Question&lt;/strong&gt;: How do we &amp;quot;fix&amp;quot; the inner data points so that &lt;span class=&quot;math-container&quot;&gt;$f(x) = f(1)+\int_1^x f'(t)dt$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Going the other direction is worse. Suppose now we had just the last two rows of the previous table and then compute a derivative from the last row. Then it looks like&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;           &lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;1&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;5&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;9&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;13&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;17&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1/2&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3/4&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3/2&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;4&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$F(x) = 1+\int_1^x f'(t)dt$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3.5&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;8&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;17&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;31&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$F'(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;5/8&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;7/8&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;27/16&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;23/8&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3.5&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Second Question&lt;/strong&gt;: How do we &amp;quot;correct&amp;quot; the process so that &lt;span class=&quot;math-container&quot;&gt;$f'(x)=F'(x)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Third Question&lt;/strong&gt;: Why doesn't this work? (I'm guessing it has to do with the essence of an approximation and the weird edge effects.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Bonus Question&lt;/strong&gt;: Do those who actually do applied mathematics with data in a research setting have to worry about this?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="401" LastEditorUserId="117" LastEditDate="2021-03-22T23:20:12.310" LastActivityDate="2021-03-23T23:55:55.027" Title="Tabular Fundamental Theorem of Calculus" Tags="|calculus|applied-mathematics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="19630" PostTypeId="2" ParentId="19629" CreationDate="2021-03-22T21:43:27.350" Score="3" Body="&lt;p&gt;If&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;1&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;5&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;9&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;13&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;17&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;3&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;7&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;15&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;31&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;Then we can approximate the slopes at the midpoint of each subinterval:&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;3&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;7&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;11&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;15&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;0.5&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;2&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;4&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;Going in reverse, if given that &lt;span class=&quot;math-container&quot;&gt;$f(1)=1$&lt;/span&gt; and the above table of values, we can approximate&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(5) = f(1)+ \displaystyle \int_1^5 f'(t) \textrm{d}t \approx 1+0.5(4) = 3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(9) = f(1) + \displaystyle \int_1^9 f'(t) \textrm{d}t \approx 1+0.5(4)+1(4) = 7$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;etc.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-03-22T21:43:27.350" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19631" PostTypeId="2" ParentId="19629" CreationDate="2021-03-23T17:31:03.343" Score="3" Body="&lt;p&gt;The key term to search for this is &amp;quot;calculus of finite differences&amp;quot;. (&lt;a href=&quot;https://en.wikipedia.org/wiki/Finite_difference&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Finite_difference&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;It looks like a common practice to solve this is to look at &amp;quot;forward differences&amp;quot;, essentially letting &amp;quot;f'(x)&amp;quot; = f(x+1)-f(x). This solves the &amp;quot;fencepost error&amp;quot; by associating the &amp;quot;fence segment&amp;quot; between x and x+1 to the &amp;quot;fencepost&amp;quot; at x. This gives one a really nice fundamental theorem as in &lt;a href=&quot;https://people.math.sc.edu/howard/Classes/555c/finiteDiff.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://people.math.sc.edu/howard/Classes/555c/finiteDiff.pdf&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f(x) = F(x+1)-F(x)$&lt;/span&gt;, then:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sum\limits_{k=a}^b f(k) = F(b+1)-F(a)$$&lt;/span&gt;&#xA;Which holds by a basic telescoping sum.&lt;/p&gt;&#xA;&lt;p&gt;If you want to use your notion of derivative, we can see what's happening if we make a table for a general function f: (I'm using sampling point spacing of 1 to make the calculations easier and show the general idea)&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;x&lt;/th&gt;&#xA;&lt;th&gt;1&lt;/th&gt;&#xA;&lt;th&gt;2&lt;/th&gt;&#xA;&lt;th&gt;3&lt;/th&gt;&#xA;&lt;th&gt;4&lt;/th&gt;&#xA;&lt;th&gt;5&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&amp;quot;f'(x)&amp;quot;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$f(2)-f(1)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{f(2)-f(1)}2+\frac{f(3)-f(2)}2$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{f(4)-f(3)}2+\frac{f(3)-f(2)}2$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{f(5)-f(4)}2+\frac{f(4)-f(3)}2$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$f(5)-f(4)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;If we add up all of these, we get:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac32 [f(2)-f(1)] + [f(3)-f(2)] + [f(4)-f(3)] + \frac32 [f(5)-f(4)]$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Which is very close to what we want:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$[f(2)-f(1)] + [f(3)-f(2)] + [f(4)-f(3)] + [f(5)-f(4)] = f(5)-f(1),$$&lt;/span&gt; but the end differences are being counted an extra .5 times. This is why your trapezoid rule method gives the correct answers: each middle difference counts fully since it shows up in both the left-hand and right-hand sums, but each end difference is only counted half since it is only counted in one of the two sums.&lt;/p&gt;&#xA;&lt;p&gt;If we take the trapezoid rule from 1 to 4 (instead of 1 to 5), however, we get:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$[f(2)-f(1)]+[f(3)-f(2)]+.75[f(4)-f(3)]+.25[f(5)-f(4)]$$&lt;/span&gt; Which is nearly what we want:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$[f(2)-f(1)]+[f(4)-f(2)]+[f(4)-f(3)] = f(4)-f(1)$$&lt;/span&gt;&#xA;To correct this we need to add in a term of &lt;span class=&quot;math-container&quot;&gt;$$-.25f(3) + .5f(4) - .25f(5)$$&lt;/span&gt;&#xA;In general we need an analogous error correction term for any integral with internal upper bound, and if we want our &amp;quot;integral&amp;quot; to start at an internal point, we need a correction term for that too. Note that these error terms vanish as the distance between sampling points approaches 0 for nice functions, and only depend on what's happening at the endpoints of the &amp;quot;integral&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;My advice is to use forward differences and left-hand sums to make things cleaner for a demonstration for students. Note that this pairing works even with different spacings for sample points.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastEditorUserId="15671" LastEditDate="2021-03-23T23:55:55.027" LastActivityDate="2021-03-23T23:55:55.027" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="19632" PostTypeId="2" ParentId="16587" CreationDate="2021-03-23T19:18:23.410" Score="3" Body="&lt;p&gt;A good approach for engineers might be to connect the &amp;quot;slope of a graph&amp;quot; to sensitivity: the factor by which a small change in input gets multiplied to produce a change in the output. You can connect this to both understanding how changes propagate through a system and error analysis.&lt;/p&gt;&#xA;&lt;p&gt;You can either do this purely symbolically to derive the definition of the derivative (with h as your error), or start by analyzing the problem of sensitivity for linear functions, and then show that tangent lines are good approximations for a graph up close, and then derive the definition through secant lines approaching tangent lines.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-03-23T19:18:23.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19633" PostTypeId="2" ParentId="16587" CreationDate="2021-03-23T20:46:07.130" Score="0" Body="&lt;p&gt;I would go with a graphical interpretation to start with.  I remember learning this sort of thing in the USN sub force for power transients on reactors (rods, reactivity, power).  It was especially useful for the enlisted men since they had less math.  But even for officers, I felt that it gave huge insight to have to draw curves (even by eye, phenomenological, directional) of derivative or integral.  I had a 5 on AP BC (and a rocking 5, didn't encounter a single problem I couldn't solve).  But still, I felt the visual interpretation was new, cool, useful.  You actually draw the curves &amp;quot;above each other&amp;quot; (so you can connect the inferences).  Definitely made me think more about the actual physics, than just some algevra-laced calculus.&lt;/p&gt;&#xA;&lt;p&gt;And then after that, show a bit of math.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  In addition to actually helping the kids, I think if you show something like this, visually, it will give the hiring committee that feeling that you actually care about the kids.  And are a cool fish, pedagogically.  Two homicided birds.  ;-)&lt;/p&gt;&#xA;&lt;p&gt;P.s.s. Don't do rod height...too geeky submarine.  But maybe, pushing down the accelerator pedal on a car.&lt;/p&gt;&#xA;&lt;p&gt;P.s.s.s.  The hiring committee figures you have the props on algebra-based calculus.  But show them a little creatitivity and outreach.  Cha-ching, bay-bee@  Buy me a beer, if you get the gig.&lt;/p&gt;&#xA;" OwnerUserId="15678" LastEditorUserId="15678" LastEditDate="2021-03-24T00:29:27.833" LastActivityDate="2021-03-24T00:29:27.833" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19634" PostTypeId="1" CreationDate="2021-03-24T07:09:55.043" Score="2" ViewCount="290" Body="&lt;p&gt;A standard pre-calculus curriculum consists of the study of elementary functions: Polynomials, rational functions, (circular and hyperbolic) trigonometric functions, exponential functions, their inverses, and sums, products and composites of these. Moreover, when it comes to understanding transformations, we teach only affine transformations: dilations, reflections (about the axes, or &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;), and translations.&lt;/p&gt;&#xA;&lt;p&gt;I originally answered this question to myself as: &lt;em&gt;Well, affine transformations preserve the elementary functions&lt;/em&gt;. For example, &amp;quot;all&amp;quot; sine functions are of the form &lt;span class=&quot;math-container&quot;&gt;$$f(x) = a \sin(bx + c) +d,$$&lt;/span&gt; which means that all sine functions are described by affine transformations of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and affine transformations of &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But since we allow composites in the definition of elementary functions (or, at least wikipedia does), my previous answer has no meaning.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This leads me to ask the following (potentially meaningless and uninteresting question):&#xA;&lt;strong&gt;Is there a mathematically interesting answer to why affine transformations are distinguished when studying graphical transformations?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; While the question is motivated by pedagogy, I'm not interested in &amp;quot;affine is the easiest, so that's why we teach it&amp;quot; -- This answer is unenlightening and no fun.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; In the comments, Matt F. asked &lt;em&gt;Are the affine transformations the largest possible Lie group of (everywhere-defined) transformations of the plane?&lt;/em&gt; While this is not my question, it is this flavour of question that I am asking.&lt;/p&gt;&#xA;" OwnerUserId="15680" LastActivityDate="2021-03-26T03:53:31.400" Title="Why do we typically only teach high-school students affine transformations of elementary functions?" Tags="|mathematical-pedagogy|mathematical-analysis|linear-algebra|graphing|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="19635" PostTypeId="1" CreationDate="2021-03-25T02:03:11.970" Score="13" ViewCount="465" Body="&lt;p&gt;I'm teaching Calculus I. It's time for the derivative of &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt;. In previous semesters, I've told&#xA;students we mainly do this just for closure, so that we know that we can find derivatives of every function possible.&lt;/p&gt;&#xA;&lt;p&gt;Are there any applications of &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Does anyone actually use logarithmic differentiation?&lt;/p&gt;&#xA;&lt;p&gt;Edited to add: Are there any applications of this that come up in multi-variable calculus?&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2021-03-25T20:19:30.657" LastActivityDate="2021-03-27T00:06:17.687" Title="Are there any applications of $x^x$?" Tags="|calculus|applications|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="19636" PostTypeId="2" ParentId="19635" CreationDate="2021-03-25T04:19:57.950" Score="3" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$n^n$&lt;/span&gt; shows up in combinatorics (number of lists of numbers from 1-n of length n), but I doubt it will have applications for the physical world: &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; cannot have units, since it doesn't make sense to raise something to a power with units.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-03-25T04:19:57.950" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19637" PostTypeId="2" ParentId="15168" CreationDate="2021-03-25T04:43:06.660" Score="1" Body="&lt;p&gt;For high-dimensional multivariable functions it's hard to imagine a geometric tangent, but the notion of linear approximation makes perfect sense. The derivative (usually called the Jacobian matrix &lt;a href=&quot;https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant&lt;/a&gt;) still satisfies (for &lt;span class=&quot;math-container&quot;&gt;$\vec{x} \approx \vec{a}$&lt;/span&gt;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\vec{f}(\vec{x}) \approx J (\vec{x}-\vec{a}) + \vec{f}(a)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt; is the Jacobian matrix of &lt;span class=&quot;math-container&quot;&gt;$\vec{f}$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$\vec{a}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Linear approximations are also useful in solving differential equations. Solving for the position of a pendulum as a function of time requires approximating &lt;span class=&quot;math-container&quot;&gt;$\sin(x) \approx x$&lt;/span&gt; for small values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; if one wants an analytic solution (&lt;a href=&quot;https://en.wikipedia.org/wiki/Pendulum_(mathematics)#Small-angle_approximation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Pendulum_(mathematics)#Small-angle_approximation&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-03-25T04:43:06.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19639" PostTypeId="2" ParentId="15168" CreationDate="2021-03-25T13:35:17.513" Score="0" Body="&lt;p&gt;Linear approximation is taught as standard method for interpolating tabular data in many engineering areas.  For instance steam tables.  Sort of a crutch to give you more value than the resolution of the table.&lt;/p&gt;&#xA;" OwnerUserId="15688" LastActivityDate="2021-03-25T13:35:17.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19640" PostTypeId="2" ParentId="19634" CreationDate="2021-03-25T16:00:08.263" Score="2" Body="&lt;p&gt;We can think of these transformations from two perspectives:  as mappings of the plane or as graphs of pre/post composition of the function.&lt;/p&gt;&#xA;&lt;p&gt;For instance, the map &lt;span class=&quot;math-container&quot;&gt;$T: \mathbb{R}^2 \to \mathbb{R}^2$&lt;/span&gt; given by &lt;span class=&quot;math-container&quot;&gt;$(a,b) \mapsto (a+2,b)$&lt;/span&gt; is the translation of the plane two units to the right.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\phi: \mathbb{R} \to \mathbb{R}$&lt;/span&gt; be the function &lt;span class=&quot;math-container&quot;&gt;$\phi(x) = x-2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{R} \to \mathbb{R}$&lt;/span&gt; be an arbitrary function, and let &lt;span class=&quot;math-container&quot;&gt;$\Gamma_f$&lt;/span&gt; be the graph of this function.&lt;/p&gt;&#xA;&lt;p&gt;Then it is a theorem that &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; maps &lt;span class=&quot;math-container&quot;&gt;$\Gamma_f$&lt;/span&gt; bijectively onto &lt;span class=&quot;math-container&quot;&gt;$\Gamma_{f \circ \phi}$&lt;/span&gt;.  This theorem is usually stated in much less awesome language in precalculus textbooks.&lt;/p&gt;&#xA;&lt;p&gt;In precalculus, we almost entirely restrict ourself to the second perspective of function pre/post composition.  The first notion is almost entirely absent:  at this level, we only treat real valued functions &amp;quot;of one variable&amp;quot;, so mappings like &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; are not defined or even thought of as functions.&lt;/p&gt;&#xA;&lt;p&gt;In precalculus we &lt;strong&gt;do&lt;/strong&gt; think about pre/post composition with functions other than affine functions.  This happens every time we graph a composition of two functions like &lt;span class=&quot;math-container&quot;&gt;$\sin(x^2)$&lt;/span&gt;.  It also happens with  &amp;quot;log/log&amp;quot; plots.  We just do not generally think of it as a &amp;quot;transformation&amp;quot; of the graph, because we do not have a notion of a mapping of the plane to itself.  Even with the affine functions this is only done informally.&lt;/p&gt;&#xA;&lt;p&gt;Pre/post composition with affine functions is an important special case with important properties.  In particular, we can reason about these using the theorems and techniques of high school geometry (while we cannot similarly reason about more general mappings of the plane, requiring either linear algebra for more general affine mappings, or multivariable calculus for smooth mappings).&lt;/p&gt;&#xA;&lt;p&gt;We do not generally consider other affine transformations because (as noted) we do not consider functions &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2 \to \mathbb{R}^2$&lt;/span&gt; at all, if we did they do not correspond to function pre/post composition, they do not take graphs of functions to graphs of functions, and they require too much linear algebra to be understandable to students at this level.  The one exception (as you note) is reflection in &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;, which is important because it corresponds to taking the inverse of a function.&lt;/p&gt;&#xA;&lt;p&gt;Moreover there are extremely practical reasons to be concerned with pre/post composition with affine functions:  they represent &amp;quot;affine coordinate&amp;quot; changes, which come up all the time in real life!&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$F(t)$&lt;/span&gt; represents the temperature in Fahrenheit &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; hours after noon, and &lt;span class=&quot;math-container&quot;&gt;$C(t)$&lt;/span&gt; represents the temperature in Celsius &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; minutes after 8:00am, then&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$F(t) = \frac{9}{5}C(60 \cdot (t+4)) + 32$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;or in other words&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$F = \phi_2 \circ C \circ \phi_1$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$\phi_1(t) = 60 \cdot (t+4)$&lt;/span&gt; is the conversion from &amp;quot;hours after noon&amp;quot; to &amp;quot;minutes after 8:00am&amp;quot; and &lt;span class=&quot;math-container&quot;&gt;$\phi_2(x) = \frac{9}{5}x+32$&lt;/span&gt; is the conversion from Celsius to Fahrenheit.&lt;/p&gt;&#xA;&lt;p&gt;Since affine changes of coordinates are so ubiquitous in the universe (and are indeed the basic building blocks of all other coordinate changes:  this is called linear algebra and  multivariable  differential calculus), it makes sense to attend to the basics in high school!&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-03-25T18:03:57.227" LastActivityDate="2021-03-25T18:03:57.227" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="19642" PostTypeId="2" ParentId="19634" CreationDate="2021-03-25T20:36:16.810" Score="1" Body="&lt;p&gt;I'll give a less elegant answer:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;because the dilations and translations are easy to understand and&#xA;implement.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If we were interested in preserving shape under the transformations then surely rigid motions or orthogonal transformations would be a better scope for the discussion. Sadly, highschool students (in the USA) do not typically learn linear algebra, much less use of eigencooordinates to orthonormally diagonalize quadratic forms. This material is not particularly difficult, for example Lay's Linear Algebra text is fairly elementary and includes a section. Some of the better Calculus texts also include rotated forms of conic sections (Anton does at least in the version I looked at last)&lt;/p&gt;&#xA;&lt;p&gt;If I wanted to make the graphical transformations interesting to the type of graph considered I might try asking:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;what class of transformations maintains the graph type for a given type of graph&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For example, which class takes lines to lines, or parabolas to parabolas, or rational functions to rational functions. There is a lot of flexibility in these questions and I suspect they would be rather challenging to curious highschool students (which exist).&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-03-25T20:36:16.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19643" PostTypeId="2" ParentId="19635" CreationDate="2021-03-25T21:18:38.673" Score="6" Body="&lt;p&gt;For the expression &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt; we could focus on finding occurrences of &lt;span class=&quot;math-container&quot;&gt;$x\ln(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;One direction is Stirling's approximation &lt;span class=&quot;math-container&quot;&gt;$\ln(N!)\sim N\ln(N)$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$N!$&lt;/span&gt; is like &lt;span class=&quot;math-container&quot;&gt;$N^N$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Another direction is that the prime number theorem gives an estimate for the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th prime &lt;span class=&quot;math-container&quot;&gt;$p_n$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$p_n=n\ln(n)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Yet another direction involves entropy. Entropy might be given by an expression like &lt;span class=&quot;math-container&quot;&gt;$S=-\Sigma_i p_i\ln(p_i)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$p_i$&lt;/span&gt; are probabilities. Then &lt;span class=&quot;math-container&quot;&gt;$e^S$&lt;/span&gt; is like the number of micro-states of a system, and is often expressed combinatorially with factorials, bringing us back to Stirling!&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-03-26T01:21:06.603" LastActivityDate="2021-03-26T01:21:06.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19644" PostTypeId="2" ParentId="19634" CreationDate="2021-03-26T03:53:31.400" Score="0" Body="&lt;p&gt;I think there's a lot of variation in content of precalculus.  I had a semester of theory of functions and semester of analytic geometry, both including some aspects of calculus itself, in a strong public school. But in some ways you could have skipped that stuff and moved right into calculus after algebra two trig.&lt;/p&gt;&#xA;&lt;p&gt;Rotations were definitely included.   And looking in analytics books, are still a normal topic. Didn't kill me to have them, but wouldn't have missed them either.  Not a technique I remember using much in standard stem undergrad courses. And lot of long formulas to write and remember. But arguably easier to look up now and use than if you looked them up for first time with no working experience.&lt;/p&gt;&#xA;&lt;p&gt;But big picture I would be wary of, one, assuming your experience is typical or should be.  And two, there are always competing priorities of time and difficulty versus content.&lt;/p&gt;&#xA;&lt;p&gt;P.s. I find the term affine transformation a little theoretical.  Not sure if that's how discussed by normal students or teachers of precalc.  But has been a long time so apologies if that is how topic is discussed in high school.&lt;/p&gt;&#xA;" OwnerUserId="15688" LastActivityDate="2021-03-26T03:53:31.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19645" PostTypeId="2" ParentId="13649" CreationDate="2021-03-26T16:29:46.767" Score="3" Body="&lt;p&gt;Building off the circle example, you can actually work out the centripetal acceleration formula by implicitly differentiating twice. If your students aren't familiar with vectors you can just plug in x = 0 and y = 1:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x^2+y^2=r^2$$&lt;/span&gt;&#xA;Differentiate with respect to t:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$2x\cdot x' + 2y \cdot y' = 0$$&lt;/span&gt;&#xA;Plugging in x = 0, y = r, you can solve &lt;span class=&quot;math-container&quot;&gt;$y' = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Differentiating again, we get:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$2x'\cdot x' + 2x\cdot x'' + 2y' \cdot y' +2 y \cdot y'' = 0$$&lt;/span&gt;&#xA;Plugging in everything we know (&lt;span class=&quot;math-container&quot;&gt;$x=0, y=r, y'=0$&lt;/span&gt;):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$2(x')^2 + 2 r y'' = 0$$&lt;/span&gt;&#xA;Which we can solve to get:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y'' = -\frac{(x')^2}{r}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I also throw in the example of the squircle (&lt;span class=&quot;math-container&quot;&gt;$x^4+y^4 = 1$&lt;/span&gt;) and an elliptic curve (&lt;a href=&quot;https://en.wikipedia.org/wiki/Elliptic_curve&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Elliptic_curve&lt;/a&gt;) for fun. A key fact about elliptic curves that is used in applications is that any tangent line at a point with intersects the curve at exactly one other point, and if the point of tangency has rational coordinates, so does the intersection point.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-03-26T16:29:46.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19646" PostTypeId="1" CreationDate="2021-03-26T17:16:46.027" Score="1" ViewCount="88" Body="&lt;p&gt;Previously I had asked a &lt;a href=&quot;https://matheducators.stackexchange.com/q/19493/14166&quot;&gt;question&lt;/a&gt; about something similar, but more constrained. But now I ask something more general. I just got hold of the Linear Algebra by Prof Gilbert of MIT and they are just awesome. Especially the use of chalk board and explanations.&lt;/p&gt;&#xA;&lt;p&gt;I am used to explanation in that type of environment. Can anyone provide me with such type of material on graph theory. Does MIT have one ?&lt;/p&gt;&#xA;" OwnerUserId="14166" LastEditorUserId="14166" LastEditDate="2021-03-29T08:03:30.030" LastActivityDate="2021-03-29T08:03:30.030" Title="Is there any video lecture series on (UG level) graph theory (might not be specific on any books) but the video/content quality is like that of MIT?" Tags="|self-learning|resource-request|graph-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="19647" PostTypeId="1" CreationDate="2021-03-26T17:33:43.097" Score="2" ViewCount="58" Body="&lt;p&gt;Typically universities ask students to evaluate each course at the end of semester. &amp;quot;On the scale of 1 to 5 how well did X do Y&amp;quot;-type questions.&lt;/p&gt;&#xA;&lt;p&gt;I rather do a &lt;em&gt;continuous&lt;/em&gt; survey, from the first day to the last. While the topics are still hot in each person's mind &amp;quot;what is working and what is not working&amp;quot;-type questionnaire.&#xA;More of an open format and less of 1 through 5 format.&#xA;I think, especially for online classes, this will help us keep things running smoother.&lt;/p&gt;&#xA;&lt;p&gt;The question is what is a good format or software for doing such a thing.&#xA;Typically we want it to be anonymous but limited to students who are taking the class.&lt;/p&gt;&#xA;&lt;p&gt;Is there a version of Blackboard that allows anonymous comments by students?&#xA;Have you used surveying sites for this purpose?&#xA;If you have you done such a thing, what were pros and cons, what software or site did you use?&lt;/p&gt;&#xA;" OwnerUserId="1472" LastActivityDate="2021-03-27T08:45:48.610" Title="How to do continuous course evaluation and feedback" Tags="|teacher-evaluations|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="19648" PostTypeId="2" ParentId="19635" CreationDate="2021-03-26T23:57:49.820" Score="4" Body="&lt;p&gt;Logarithmic differentiation is used in &lt;em&gt;mathematics&lt;/em&gt; when dealing with infinite products in complex analysis.  Note the construction of &lt;span class=&quot;math-container&quot;&gt;$f’/f$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; does not require logarithms of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; as a middle step (except intuitively), so it is perfectly fine to use even when &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; takes non-positive values. The prime number theorem and the link between prime numbers and the Riemann hypothesis involve the logarithmic derivative of the Riemann zeta-function. That is all too advanced for your students. The topic of logarithmic differentiation feels like an archaic topic in a calculus course, since there are almost no worthwhile problems to use it on. Feynman liked to use it to differentiate rational functions with numerator and denominator in factored form. I do not think your students would find this interesting at all.&lt;/p&gt;&#xA;&lt;p&gt;I think time spent in a calculus course on &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt; or the “most general” case of &lt;span class=&quot;math-container&quot;&gt;$f(x)^{g(x)}$&lt;/span&gt; is not time well spent. These are worthless things for students to spend time on in calculus when there are so many more important things to discuss.  The exponential-type functions that matter are &lt;span class=&quot;math-container&quot;&gt;$x^c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c^x$&lt;/span&gt; for constant &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;. You can branch out and use the chain rule on &lt;span class=&quot;math-container&quot;&gt;$e^{-x^2}$&lt;/span&gt;, which matters in probability, but &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt; is a time sink.  That &lt;span class=&quot;math-container&quot;&gt;$x\log x$&lt;/span&gt; shows up in math is &lt;em&gt;not&lt;/em&gt; a compelling argument to care about &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt; for a real variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;: a function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is not important just because its logarithm is important. Otherwise we’d make a big deal about &lt;span class=&quot;math-container&quot;&gt;$e^{\sin x}$&lt;/span&gt; and in basic courses that never shows up in practical situations. The function of a real variable &lt;span class=&quot;math-container&quot;&gt;$x^x$&lt;/span&gt; is a curiosity and that is all.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2021-03-27T00:06:17.687" LastActivityDate="2021-03-27T00:06:17.687" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20648" PostTypeId="2" ParentId="19647" CreationDate="2021-03-27T08:45:48.610" Score="2" Body="&lt;h1&gt;Consider a less technological solution&lt;/h1&gt;&#xA;&lt;p&gt;Students, especially in modern times, are already using a big bunch of different online platforms and their functionalities. Adding one more is not a good solution, and, in particular, since it is voluntary, it would likely not get used that much anyway.&lt;/p&gt;&#xA;&lt;p&gt;At NTNU each course has a group of student representatives (referansegruppe), maybe four to seven of them per course (one per study program, or maybe everyone if it is a small course). They acts as the official representatives of the rest of the students and also write a short report at the end of the course. In my experience (with a large course intended for example for biologists and geologists) I got reasonable feedback from the representatives. Note that they can speak for anonymous students or the course student body as a whole, so it solves the issue of anonymity. It seems that at least some other Norwegian universities have similar arrangements, though the details and terminology vary.&lt;/p&gt;&#xA;&lt;p&gt;But if you choose to burden the students with yet more systems or forms to fill, one option is to use the questionares offered by Microsoft, Google and no doubt several other actors; some of these are certainly free when it comes to price and your university might have sold their soul to Microsoft or Google, or just otherwise have some kind of arrangement with some firm that allows making surveys. Typically the surveys can be anonymous. If such a system is not integrated with your university website, then you can still distribute the link by the usual means. It is quite unlikely for anyone outside the course to sabotage the survey, even if it is technically possible.&lt;/p&gt;&#xA;&lt;p&gt;I did use some such surveys and faced no problems. They were not associated with course feedback, so I can't comment on if they are fit for that in particular.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-03-27T08:45:48.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20649" PostTypeId="1" AcceptedAnswerId="20650" CreationDate="2021-03-27T11:14:42.643" Score="8" ViewCount="1625" Body="&lt;p&gt;I love mathematics and I know pretty much everyone in this community does too! That feeling of being stuck on a mathematical problem for days as you try to unearth the complexity in front of you is painful but that final emotion of solving it, of discovering something new is certainly magical. I can only imagine the pure bliss Sir Andrew Wiles felt when he cracked Fermat's Last Theorem or when Isaac Newton and Gottfried Leibniz discovered the essence of Calculus. Mathematics is, without doubt, beautiful and can help us understand the most counterintuitive ideas. But while so many are deeply passionate about math, the subject often carries a negative connotation to the general public. In this post, I seek to explore and understand why - why is something so spectacular as math hated by so many?&lt;/p&gt;&#xA;&lt;p&gt;In my experience the answer is a result of the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The education system puts greater importance on getting done with the syllabus rather than getting done with the syllabus &lt;strong&gt;well&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;For example, take the quadratic formula &lt;span class=&quot;math-container&quot;&gt;$x=\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In almost all cases, teachers resort to simply writing this on a whiteboard and asking students to memorize it for their examinations. Apart from a few, no one sits down and allows students to discover it themselves, to think about the equation &lt;span class=&quot;math-container&quot;&gt;$ax^{2} + bx + c = 0$&lt;/span&gt; in the most creative ways. This, in my opinion, kills a child's imagination and paints this hard, boring picture of mathematics in their head while they have gained no intuition for the formula whatsoever. As a result, they end up disliking mathematics.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Students argue that stuff like the quadratic equation is most likely never going to be useful in their life. This is a fair point. As a future doctor, you may never have to go beyond some basic arithmetic and statistics in your career so learning about this quadratic equation seems fairly useless. What students often fail to realize, however, in my experience, is that the journey of discovering and understanding the quadratic formula is itself a worthy experience - that the critical thinking and creative skills it develops can help them everywhere in life. As a result of this oblivion, they end up disliking maths.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;While these are just &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; observations, I am sure there are many more that you must have encountered in your mathematical journey. What are those other reasons and more importantly, how do we fix them?&lt;/p&gt;&#xA;&lt;p&gt;Edit: This post had been previously closed because it was &amp;quot;opinion-based&amp;quot;. I understand that, but I am completely willing to hear other people's opinions, their refutations to my observations, and their own observations. After all, if I don't present my opinion, how will other people comment on it, how can one have a discussion? I sincerely request this post to not be closed again because its only intention is to spark discussion, not to indoctrinate people with personal ideas.&lt;/p&gt;&#xA;" OwnerDisplayName="user15697" LastActivityDate="2023-11-24T22:41:52.153" Title="Why do so many people hate math?" Tags="|secondary-education|" AnswerCount="9" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="20650" PostTypeId="2" ParentId="20649" CreationDate="2021-03-27T13:12:54.797" Score="12" Body="&lt;p&gt;You asked a big question. Maybe some of these ideas are worth investigating more deeply. The two first explanations are completely generic, while the last two are more specific to mathematics; I would advise to not forget the power of the generic explanations. But I don't know to what extent these are good explanations or what other good ones there might be. This is something mathematics education has grappled with for a long time and is grappling with right now.&lt;/p&gt;&#xA;&lt;h1&gt;It is forced on people&lt;/h1&gt;&#xA;&lt;p&gt;Some people don't like music or playing instruments, yet might have had to play or sing at school. Some have bad memories of it.&lt;/p&gt;&#xA;&lt;p&gt;Some people don't like physical activity while at school, yet might be forced to do it. A whole lot of bad memories is associated to it.&lt;/p&gt;&#xA;&lt;p&gt;The same with foreign languages, philosophy and really whatever else is being forced on people at school. But there is a lot of mathematics and most people don't see the beautiful mathematics outside the hated school context.&lt;/p&gt;&#xA;&lt;h1&gt;It is difficult&lt;/h1&gt;&#xA;&lt;p&gt;Many people don't like doing difficult things. It seems mathematics is difficult. Thus many people don't like it.&lt;/p&gt;&#xA;&lt;h1&gt;Rote versus creative?&lt;/h1&gt;&#xA;&lt;p&gt;Is doing lots of rote calculation something that makes people like mathematics less, when compared to the creative type of mathematics? Maybe; at least there are serious trends in teacher education, at least in Finland and Norway, aimed towards reducing the amount of rote learning. See for example &amp;quot;problem-based learning&amp;quot; and similar keywords for more.&lt;/p&gt;&#xA;&lt;h1&gt;Abstract versus applied?&lt;/h1&gt;&#xA;&lt;p&gt;Is abstract mathematics more or less motivating than applied mathematics? ATD, the anthropological theory of the didactic by Yves Chevallard, is an example of a serious theory that advocates for mathematics applications first, and also in a creative way. There are doubtless others.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;new mathematics&amp;quot; was an example of going strongly abstraction-first, but presumably it could be done in a more effective manner, too.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-03-27T13:12:54.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20651" PostTypeId="2" ParentId="19589" CreationDate="2021-03-27T18:45:00.903" Score="2" Body="&lt;p&gt;I think it is not a good idea to oblige students to take a course on latex. Students do need to learn to write, and they can be taught to use latex as part of being taught to write, if need be, but the basics of latex are &lt;em&gt;easily&lt;/em&gt; learned independently, and students need to learn to learn things independently. What might not be obvious to students is that it is useful to learn latex, but this is not hard to communicate.&lt;/p&gt;&#xA;&lt;p&gt;My experience with directing undergraduate final projects written by engineering students is that some learn latex to write their projects, others get by with word/open office, and it's not necessarily a good use of time for some students to be learning latex, although it certainly makes sense for a student whose written project will incorporate a large number of mathematical expressions or equations. On the other hand, in my experience a student whose project is focused on coding (who could learn latex in a day with no difficulty) generally has little need for what latex offers.&lt;/p&gt;&#xA;&lt;p&gt;One should also remember that while learning enough latex to write a math paper is very easy, learning enough latex to format nicely a document full of images and tables can be quite time consuming, and, there are alternatives both easier to learn and easier to use well.&lt;/p&gt;&#xA;&lt;p&gt;There are lots of tools that students can learn - AutoCad, C++, Matlab, Rust, Innovator, Latex, Maple, Sage, R, etc. and there should be some selection because no one can learn everything nor does anyone need to learn everything.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2021-03-27T18:45:00.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20652" PostTypeId="2" ParentId="19589" CreationDate="2021-03-27T21:42:26.440" Score="4" Body="&lt;p&gt;It is true that &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; is a language,&#xA;but I lived through the transition from &lt;a href=&quot;https://www.wordperfect.com/en/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WordPerfect&lt;/a&gt;&#xA;and &lt;a href=&quot;https://en.wikipedia.org/wiki/WordStar&quot; rel=&quot;nofollow noreferrer&quot;&gt;WordStar&lt;/a&gt; to Microsoft Word,&#xA;and I lived through the transition from &lt;a href=&quot;https://www.troff.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;troff&lt;/a&gt; to &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt;.&#xA;Already &lt;a href=&quot;https://www.overleaf.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Overleaf&lt;/a&gt;,&#xA;&lt;a href=&quot;https://www.mathjax.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathJax&lt;/a&gt; and other similar software developments&#xA;are lowering the TeX entrance-threshold.&lt;/p&gt;&#xA;&lt;p&gt;In the future, &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; will likely move &amp;quot;under the hood&amp;quot; as have&#xA;other typesetting languages.&#xA;This argues for supplemental instruction, rather than new courses and new requirements.&lt;/p&gt;&#xA;&lt;p&gt;Cf. Asimov's &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Feeling_of_Power&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Feeling of Power&lt;/a&gt;.&#xA;Could I be the last troff expert? :-)&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-03-27T23:54:54.797" LastActivityDate="2021-03-27T23:54:54.797" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20653" PostTypeId="2" ParentId="10758" CreationDate="2021-03-28T00:45:57.513" Score="1" Body="&lt;p&gt;A common error I see when teaching function composition is students seeing it as multiplication. Many factors contribute to this, but examples with multiplication in them don't help:&lt;/p&gt;&#xA;&lt;p&gt;Q: Let f(x) = 2x and g(x) = x+1. What is f(g(x))?&lt;/p&gt;&#xA;&lt;p&gt;A: f(g(x)) = 2(x+1)&lt;/p&gt;&#xA;&lt;p&gt;Some students will see this and think that the answer somehow involves multiplying f(x) by g(x).&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-03-28T00:45:57.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20654" PostTypeId="2" ParentId="5986" CreationDate="2021-03-28T02:25:36.423" Score="2" Body="&lt;p&gt;Found this after debating with my 8th grader son trying to bounce players of diagonal walls in a canvas js based game of his own making:&lt;/p&gt;&#xA;&lt;p&gt;I agree with many posts above that we have it all backwards. Vectors don't come into our lives through physics anymore, but through video games and early coding.&lt;/p&gt;&#xA;&lt;p&gt;Kids do translation of stuff using scratch in pre-k. They bounce things that move off walls using scratch in pre-k. Then we tell them to do a little python and canvas js, until they show up with all the vector calculus problems:&lt;/p&gt;&#xA;&lt;p&gt;Intersection - that's english for clash detection, i.e. don't run into a wall.&lt;/p&gt;&#xA;&lt;p&gt;Reflection - pong is easy because the wall is vertical. How do you do it when the walls float in 2-D space?&lt;/p&gt;&#xA;&lt;p&gt;X-product - guy is on one side off the wall and then on the other - deal with it. That's a slam dunk case of how to explain the x-product. And the triple product for that matter (if you happen to operate a turret with a cannon).&lt;/p&gt;&#xA;&lt;p&gt;Dot-product - they learn y= mx+n and its totally useless when you want to program snake. You need vectors. I've seen 100's of lines of code simply trying to determine which quadrant we're in.&lt;/p&gt;&#xA;&lt;p&gt;The answer is before trig. And definitely before physics.&lt;/p&gt;&#xA;" OwnerUserId="15701" LastEditorUserId="15701" LastEditDate="2021-03-28T02:45:24.913" LastActivityDate="2021-03-28T02:45:24.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20655" PostTypeId="2" ParentId="5986" CreationDate="2021-03-28T16:18:49.390" Score="0" Body="&lt;p&gt;As with Ben C, I agree that kids differ in their abilities.  Usually lower, slower than the people who frequent this site.  Case in point, the casually name-dropped Tom Apostol calculus, is not typical.  So, &amp;quot;I have met kids&amp;quot; is not a good indicator for general ability.  You are dealing with a population of non-identical units.  People are not congruent shapes.&lt;/p&gt;&#xA;&lt;p&gt;In a moderately accelerated G/T (i.e. upper track) setting in the States, I saw vectors just touched on a little bit in Algebra 2 (10th grade).  Not much done with them, but quick exposure.  And I'm old enough to have gone through a curriculum that was still a bit touched by the ambitious notions post-Sputnick.  Vectors were more heavily covered in precalculus (especially in analytic geometry) which was after trigonometry.  And they were heavily used in calculus BC and in HS (non calc-based) physics.  Most of the kids in the G/T were a year ahead (basically expected).&lt;/p&gt;&#xA;&lt;p&gt;So you had:&lt;/p&gt;&#xA;&lt;p&gt;Normal track:&#xA;first, light exposure: 11th grade (Alg 2 or Alg 2/trig).  Full exposure 12th grade (precalc).  And often HS physics (in some cases requiring some direct teaching of vectors or baby trig by the instructor to students that hadn't had it yet...which is unfortunate, but seemed to be managed OK, by a good instructor.)&lt;/p&gt;&#xA;&lt;p&gt;G/T track: a year earlier.&lt;/p&gt;&#xA;&lt;p&gt;I don't think the short exposure in Algebra 2 was critical.  And it all clicked more when you really covered them properly in pre-calc.  And especially in physics, where so many problems in mechanics involved separating downward gravitational accceration and sideways movement into independent components.  Bomb-dropping for instance.  But you do end up needing trig, for many problems.  So in general, I think any heavy work in vectors, should be post trigonometry.  Pre-trig light exposure to vectors was (opinion, based on self and classmates) not strongly needed, but not daunting either.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, as with many questions here about 'why don't we do more, or do more earlier', I would remind that there are a fixed number of hours, t, for instruction.  If you do more of Y in a given year, you need to consider the impact on X.&lt;/p&gt;&#xA;" OwnerUserId="15710" LastActivityDate="2021-03-28T16:18:49.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20656" PostTypeId="2" ParentId="11203" CreationDate="2021-03-29T16:51:01.353" Score="0" Body="&lt;p&gt;As pointed out by @AmyB and @DanielRCollins, an &lt;strong&gt;equation&lt;/strong&gt; is simply a statement that two expressions are equal.&lt;/p&gt;&#xA;&lt;p&gt;@Adam remarks that the &lt;a href=&quot;https://math.stackexchange.com/a/4254850/21813&quot;&gt;equation-identity distinction&lt;/a&gt; doesn't seem useful; on the contrary, it &lt;em&gt;is&lt;/em&gt; instructive to distinguish an identity from a “contingent” equation:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$z\in\mathbb C,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\;C_n$&lt;/span&gt; denote &lt;span class=&quot;math-container&quot;&gt;$\left(z^2-2z\cos\left(\frac{n\pi}9\right)+1\right),$&lt;/span&gt; and consider the three separate arguments &lt;span class=&quot;math-container&quot;&gt;$$\left(z^3-1\right)\left(z^6+z^3+1\right)=\left(z^3-1\right)C_2C_4C_8 \\ \therefore \left(z^6+z^3+1\right)=C_2C_4C_8; \tag1$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$z^2-1=\left(z+1\right)\left(z-1\right) \\ \therefore z-1=z-1; \tag2$$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$$z\left(z^3-1\right)=7\left(z^3-1\right) \\ \therefore z=7\tag3.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt; Why are inferences (&lt;a href=&quot;https://www.youtube.com/watch?v=tmasA1j67t8&amp;amp;lc=UgzgZ3APS-ySblZJNN14AaABAg&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;,2) correct, but inference (3) incorrect?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A:&lt;/strong&gt; Premises (1,2) are &lt;strong&gt;identities&lt;/strong&gt; (which hold on the domain of the &lt;em&gt;variable&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;), whereas premise (3) is an &lt;strong&gt;equation&lt;/strong&gt; that's contingent on &lt;em&gt;unknown&lt;/em&gt; (but particular) values of &lt;span class=&quot;math-container&quot;&gt;$z.$&lt;/span&gt; A separate case &lt;span class=&quot;math-container&quot;&gt;$(z^3-1)=0$&lt;/span&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7237/rationale-for-not-dividing-both-sides-of-an-equation-by-x-ex-6x2-12x/19602#comment54810_19602&quot;&gt;&lt;em&gt;necessarily&lt;/em&gt;&lt;/a&gt; branches out when dividing &lt;span class=&quot;math-container&quot;&gt;$(z^3-1)$&lt;/span&gt; out of premise (3)—but not so when a similar division is performed on premises (1,2): conclusions (1,2) remain identities.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-09-19T19:11:57.957" LastActivityDate="2021-09-19T19:11:57.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20657" PostTypeId="1" CreationDate="2021-03-29T17:37:26.133" Score="0" ViewCount="155" Body="&lt;p&gt;Which academic subjects examine what the advantages and disadvantages of the various number bases are, e.g. besides base ten: base twelve, base sixteen, base eight, base two and the ways that they can be written and pronounced.&lt;/p&gt;&#xA;&lt;p&gt;For example, base eight 10001 can be pronounced &amp;quot;one zero zero zero one base eight&amp;quot;, &amp;quot;four thousand ninety-six one&amp;quot;, &amp;quot;two to the power twelve one&amp;quot;, and some other ways, and there is no consensus about which way is best. Would this be a mathematical question, or a cognitive science question, a mathematical education question or some other kind of question?&lt;/p&gt;&#xA;&lt;p&gt;For another example, to what academic subject does the question of how to write thirty-six in base thirty-seven belong?&lt;/p&gt;&#xA;" OwnerUserId="15662" LastEditorUserId="15662" LastEditDate="2021-03-29T20:10:59.777" LastActivityDate="2021-04-01T11:50:35.203" Title="Which academic subjects examine what the advantages and disadvantages of the various number bases are?" Tags="|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20658" PostTypeId="2" ParentId="352" CreationDate="2021-03-29T17:59:49.290" Score="-1" Body="&lt;p&gt;I think the multiplication table should be learned, but only up to ten times ten. As a student I would appreciate a teacher that made the task as easy as possible, and part of that is teaching only what is needed. The teacher can tell students that they can be glad they don't need to learn it past ten times ten.&lt;/p&gt;&#xA;&lt;p&gt;There is a naturalness of stopping at ten times ten that you don't have with stopping at twelve times twelve. If it's good to know twelve times twelve, then why not thirteen times thirteen, and ninety-nine times ninety-nine? Thus the student feels less sense of satisfaction in a complete job done, paradoxically.&lt;/p&gt;&#xA;&lt;p&gt;Dozens don't matter much in math class. The original reason for teaching to twelve times twelve presumably had something&lt;/p&gt;&#xA;&lt;p&gt;It also seems to me that learning or teaching to twelve times twelve is partly based on a sort of greed, where one thinks how easy it would be to do that, and then makes the mistake of thinking that that is a reason to do it.&lt;/p&gt;&#xA;&lt;p&gt;There is a danger that the table will be learned imperfectly as a result of trying to learn the bigger table.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit: I am starting to doubt whether the whole table should be memorized. Maybe that should be optional. If the student seems not to like it, perhaps only part of the table should be memorized, up to five times five perhaps. When the student needs to multiply seven times eight, it can be turned into (5 + 2) * (4 + 4) = 20 + 20 + 8 + 8 = 56. When greater speed is needed a calculator can be used or a multiplication table on paper could be used. Or how about having the student create a times table for his or her own future use, using pencil and paper calculations only?&lt;/p&gt;&#xA;&lt;p&gt;If a student on his or her own comes up with the idea that he or she wants to memorize the table, then great. I am worried about the effects of any extrinsic motivation to memorize it being present.&lt;/p&gt;&#xA;&lt;p&gt;If the table it to be memorized, at least by stages ask the student to do it in bite size chunks and find a way to make it fun. Failure here can have life-long consequences.&lt;/p&gt;&#xA;" OwnerUserId="15662" LastEditorUserId="15662" LastEditDate="2021-04-16T13:45:00.277" LastActivityDate="2021-04-16T13:45:00.277" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20659" PostTypeId="1" CreationDate="2021-03-29T18:21:27.023" Score="1" ViewCount="353" Body="&lt;p&gt;When memorizing and recalling the times table, I learned to say &amp;quot;six sevens are forty-two&amp;quot;, and always wondered what it would be like to learn to say &amp;quot;six times seven equals forty-two&amp;quot; and whether it would be harder. Likewise, of course, with all the other ones e.g. &amp;quot;seven eights are fifty-six&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="15662" LastEditorUserId="15662" LastEditDate="2021-03-29T20:15:58.520" LastActivityDate="2021-03-31T06:39:58.417" Title="Which works better for learning 6 * 7 = 42: saying &quot;six sevens are forty-two&quot;, or &quot;six times seven equals forty-two&quot;?" Tags="|mathematical-pedagogy|primary-education|education-research|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="20660" PostTypeId="2" ParentId="20659" CreationDate="2021-03-29T18:34:54.640" Score="-3" Body="&lt;p&gt;I think for first learning, it is easier to think of the former, not the latter.  So two twos make a four.  After all, what is &amp;quot;times&amp;quot;, for someone new to it.&lt;/p&gt;&#xA;&lt;p&gt;But by the time you are doing the whole multiplication table (and six sevens is pretty high up into it), you've got more familiarity with multiplication and can just start saying &amp;quot;times&amp;quot;, which is more efficient (to say or to think about).&lt;/p&gt;&#xA;" OwnerUserId="15718" LastActivityDate="2021-03-29T18:34:54.640" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20661" PostTypeId="2" ParentId="20657" CreationDate="2021-03-29T19:28:55.840" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Which academic subjects examine what the advantages and disadvantages of the various number bases are, e.g. besides base ten: base twelve, base sixteen, base eight, base two and the ways that they can be written and pronounced.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't know whether this has been investigated academically. You should use Semantic scholar, Google scholar and other academic search engines for this purpose.&lt;/p&gt;&#xA;&lt;p&gt;But: It is not so that any given question necessarily is part of a fixed academic subject, or any of them. Academia does not work this way. If someone has researched, or will research, this question, then they would search for an academic journal that would accept the result and write the paper for it. The particular would depend on the precise research question, but also the background of the author.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, this could also be something like bachelor's or master's thesis, which would be nominally under some academic field, but the connection might be very nominal. Or it might be a side issue in a PhD thesis or some book etc.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For another example, to what academic subject does the question of how to write thirty-six in base thirty-seven?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From a mathematical perspective, the list of number symbols could be extended in an arbitrary way. Consider hexadecimals for an example of how this has been done. Since this does not have clear applications (as far as I know), and all the ways of doing this are mathematically equivalent anyway, I doubt people have bothered to think about this. The relevant academic subject depends on who needs such notation; developing it would presumably be a side effect of whatever it is being developed for.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2021-04-01T11:50:35.203" LastActivityDate="2021-04-01T11:50:35.203" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20665" PostTypeId="2" ParentId="20659" CreationDate="2021-03-31T06:39:58.417" Score="1" Body="&lt;p&gt;&lt;em&gt;Not a proper acceptable answer, just an expansion of my original comment:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;A literal/direct/mechanistic recitation probably involves a lighter cognitive load than a quirkier sentence-translated recitation. So:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;one times five equals five&lt;/li&gt;&#xA;&lt;li&gt;six times seven equals forty-two&lt;/li&gt;&#xA;&lt;li&gt;eleven times twelve equals one-hundred-and-thirty-two;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;versus&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;one (copy of) five is five&lt;/li&gt;&#xA;&lt;li&gt;six (lots of) sevens are forty-two&lt;/li&gt;&#xA;&lt;li&gt;eleven twelves are one-hundred-and-thirty-two.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Native English-language speakers may prefer the latter, while non-native English-language speakers may prefer the former.&lt;/p&gt;&#xA;&lt;p&gt;@DanielRCollins: &lt;em&gt;I would certainly not want to hear something like &amp;quot;twenty fives is one hundred&amp;quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Yes, in the sentence-translation (as opposed to mechanistic recitation),&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the fact that &amp;quot;twenty fives&amp;quot; and &amp;quot;twenty-five&amp;quot; are almost&#xA;homophonous,&lt;/li&gt;&#xA;&lt;li&gt;the asymmetrical adverb(&lt;em&gt;twenty&lt;/em&gt;)-noun(&lt;em&gt;fives&lt;/em&gt;) structure,&lt;/li&gt;&#xA;&lt;li&gt;the pluralisation,&lt;/li&gt;&#xA;&lt;li&gt;the fact that the noun but not the adverb is pluralised,&lt;/li&gt;&#xA;&lt;li&gt;the option between &amp;quot;is&amp;quot; &amp;amp; &amp;quot;are&amp;quot; (&amp;quot;20 fives &lt;em&gt;is&lt;/em&gt; 100&amp;quot; and &amp;quot;20 fives &lt;em&gt;are&lt;/em&gt;&#xA;100&amp;quot; are differently meaningful),&lt;/li&gt;&#xA;&lt;li&gt;the natural-language ambiguity of &lt;em&gt;in what sense&lt;/em&gt; 20 fives&#xA;is/are/becomes 100,&lt;/li&gt;&#xA;&lt;li&gt;etc.,&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;all impose cognitive burdens. The process is more akin to a narration than a streamlined recitation, and furthermore detracts from the commutativity of the multiplication operation.&lt;/p&gt;&#xA;" OwnerUserId="372" LastActivityDate="2021-03-31T06:39:58.417" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20666" PostTypeId="1" AcceptedAnswerId="20675" CreationDate="2021-03-31T07:09:55.070" Score="8" ViewCount="633" Body="&lt;p&gt;In an Advanced Calculus course, students were asked to prove &lt;span class=&quot;math-container&quot;&gt;$$|a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$$&lt;/span&gt;&#xA;for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; real numbers &lt;span class=&quot;math-container&quot;&gt;$a_1,a_2,...a_n$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am teaching assistant for this course, and one of my students replied like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$|a_1+a_2|\le |a_1|+|a_2|$&lt;/span&gt; by triangular inequality.&lt;/p&gt;&#xA;&lt;p&gt;Then, &lt;span class=&quot;math-container&quot;&gt;$|a_1+(a_2+...+a_n)| \le |a_1|+|a_2+...+a_n|$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$|a_1|+|a_2+(a_3+...+a_n)| \le |a_1|+|a_2|+ |a_3+...+a_n|$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Repeating this ,&lt;/p&gt;&#xA;&lt;p&gt;|&lt;span class=&quot;math-container&quot;&gt;$a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I gave him a bad grade (5 from 10), so he complained. It is my turn to reply.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is the student's proof correct?&lt;/li&gt;&#xA;&lt;li&gt;Is the point deduction justified? The professor and I expected the students to use mathematical induction for this proof.&lt;/li&gt;&#xA;&lt;li&gt;How can I most clearly communicate the grading rationale to the student in our upcoming conversation? How can I defend the grading policy against student objections?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I posted the question on mathstackexchange too:&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/4083896/grade-on-proving-a-1-a-2-a-n-le-a-1a-2-a-n?noredirect=1#4083901&quot;&gt;https://math.stackexchange.com/questions/4083896/grade-on-proving-a-1-a-2-a-n-le-a-1a-2-a-n?noredirect=1#4083901&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15725" LastEditorUserId="1230" LastEditDate="2021-04-01T14:52:10.757" LastActivityDate="2021-04-02T16:57:11.030" Title="Grade on proving |$a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$" Tags="|calculus|mathematical-analysis|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="20667" PostTypeId="2" ParentId="19621" CreationDate="2021-03-31T21:34:15.593" Score="2" Body="&lt;p&gt;You should look at the stated contents of the course you are being asked to teach. Then take a look at what they already have learned when taking this course (perhaps add a bit of refresher, or give them a summary text). Next look at the use they'll have of the material included in the course in later courses, or perhaps in their later life.&lt;/p&gt;&#xA;&lt;p&gt;The above should give a decent skeleton, fill in gaps, check how deep to dig into e.g. proofs (generally useful for mathematics, required ---but not really stated like that--- for programming and other areas) or techniques in combinatorics, graph theory, ...&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2021-03-31T21:34:15.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20668" PostTypeId="2" ParentId="19621" CreationDate="2021-04-01T05:39:58.003" Score="4" Body="&lt;p&gt;If you are teaching to an audience that will largely be computer science majors (which is often the case), then you should look at the ACM guidelines for an undergraduate computer science curriculum, which includes guidelines for what should be in a discrete math course.  Looking at older versions of these guidelines will also give you some sense of how this course has changed.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2021-04-01T05:39:58.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20669" PostTypeId="2" ParentId="12308" CreationDate="2021-04-01T16:33:16.983" Score="0" Body="&lt;p&gt;It is absolutely a valuable skill. HOWEVER, students should also learn to be aware of the difference between finding the true answer and approximations. Approximations involve errors no matter how small it is unless it is zero.&lt;/p&gt;&#xA;" OwnerUserId="6370" LastActivityDate="2021-04-01T16:33:16.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20670" PostTypeId="1" AcceptedAnswerId="20673" CreationDate="2021-04-02T02:50:14.697" Score="25" ViewCount="3030" Body="&lt;p&gt;Some students are prone to making small calculation errors. Not errors in understanding, but errors like adding or multiplying integers incorrectly, or dropping a negative sign. &lt;em&gt;Unsystematic errors&lt;/em&gt; in the terminology &lt;a href=&quot;https://matheducators.stackexchange.com/a/10879/5220&quot;&gt;referenced here&lt;/a&gt;. Sometimes there's an obvious cause to this, like if the student writes sloppily and literally loses negative signs in their calculations or can't read their own handwriting, or if they're rushing through the calculation.&lt;/p&gt;&#xA;&lt;p&gt;But suppose I've got a student who writes neatly and performs calculations carefully and methodically, but still makes a ton of small errors. How do I coach this student? &amp;quot;You need more practice&amp;quot; is probably true, but can't I be more helpful than that?&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2021-04-05T17:56:54.003" Title="How do you coach students who often make small errors?" Tags="|reference-request|students-mistakes|struggling-students|" AnswerCount="7" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="20671" PostTypeId="1" CreationDate="2021-04-02T12:56:38.240" Score="18" ViewCount="299" Body="&lt;p&gt;Many definitions in mathematics are &amp;quot;fully crystalized&amp;quot;.  Sometimes the form of these definitions might be somewhat baffling to the uninitiated.&lt;/p&gt;&#xA;&lt;p&gt;For example, the definition of a relation from &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; as a subset of &lt;span class=&quot;math-container&quot;&gt;$X \times Y$&lt;/span&gt; is extremely economical, and makes formulating and proving theorems about relations very efficient.&lt;/p&gt;&#xA;&lt;p&gt;This same economy is a disadvantage when teaching the concept.  I want my students to understand that &amp;quot;x is a brother of y&amp;quot; is an example of a relation on the set of people.  In this context, saying that the relation &amp;quot;is a brother of&amp;quot; actually &lt;strong&gt;is&lt;/strong&gt; a set of ordered pairs of people seems philosophically disturbing.  I would rather speak of this set of ordered pairs as a kind of ledger sheet which records all of the brother relationships between people.  For instance (Steve, Kayla) will be on that list because Steve (me) is the brother of Kayla (my sister), but (Kayla, Steve) will not be on this ledger sheet.  If I accurately record all of the information about all pairs of people in this list, then I can replace the relation &amp;quot;x is a brother of y&amp;quot; with the equivalent relation &amp;quot;(x,y) is on the 'brother of' ledger sheet&amp;quot;.  Since any relation can be so translated, we skirt the issue of &amp;quot;actually&amp;quot; defining what a relation is by instead using this construction as a proxy.&lt;/p&gt;&#xA;&lt;p&gt;I think that this kind of definition is quite common.  We often use an easily available &amp;quot;proxy&amp;quot; for a concept to provide an economical definition.  We then need to mentally translate each time we use the concept.  This skill at conceptual translation is important to develop, but seems difficult to assess or help students to attain.&lt;/p&gt;&#xA;&lt;p&gt;Questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Has this conceptual skill of translating between concepts and their proxies been identified and studied in the literature?&lt;/li&gt;&#xA;&lt;li&gt;What are the advantages and disadvantages of straying from the &amp;quot;official&amp;quot; definitions to allow easier assimilation?  For instance, defining a relation instead as a two variable predicate might be &amp;quot;closer&amp;quot; to the students native conception of the concept.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-04-03T15:55:56.817" Title="Tension between the most intuitive definition vs. the most common definition of a concept" Tags="|undergraduate-education|definitions|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="20672" PostTypeId="2" ParentId="20670" CreationDate="2021-04-02T14:49:34.687" Score="1" Body="&lt;h3&gt;Allow and promote using a calculator&lt;/h3&gt;&#xA;&lt;p&gt;This is one of the recommendations listed for accommodating students with &lt;a href=&quot;https://en.wikipedia.org/wiki/Dyscalculia&quot; rel=&quot;nofollow noreferrer&quot;&gt;dyscalculia&lt;/a&gt;, but this could help whether or not a student has dyscalculia. It's not really &lt;em&gt;coaching&lt;/em&gt; and it doesn't address the underlying source of errors, but at the very least it will keep those arithmetic errors from cropping up and getting in the way of conceptual understanding. On the assessment side of things too, allowing for calculators will keep those small errors from giving a skewed measure of a student's ability, especially in the world of online computer-graded assessments.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2021-04-02T14:49:34.687" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="20673" PostTypeId="2" ParentId="20670" CreationDate="2021-04-02T14:56:36.693" Score="22" Body="&lt;h3&gt;Ask the student to &amp;quot;talk through&amp;quot; their calculations&lt;/h3&gt;&#xA;&lt;p&gt;Having a student verbalize their calculation may force them to pay more attention (or a different kind of attention) to their work that causes them to catch the errors as they make them. This feels very related to &lt;a href=&quot;https://en.wikipedia.org/wiki/Rubber_duck_debugging&quot; rel=&quot;noreferrer&quot;&gt;rubber duck debugging&lt;/a&gt;. At the very least, if you're working with a student one-on-one, if they don't catch their own error then listening to them verbalize their calculation can help you catch their error as they make it, and maybe give you some insight into what thought led to that error.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2021-04-02T14:56:36.693" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20674" PostTypeId="1" CreationDate="2021-04-02T16:31:56.357" Score="5" ViewCount="518" Body="&lt;p&gt;Why are higher level math textbooks almost completely black and white? I can't think of any math textbooks on a subject more advanced than calculus that uses colors. &lt;em&gt;Edited to add that the comments by Dave L Renfro and J W include examples of books that do use color.&lt;/em&gt; I have to imagine the use of colors has some educational purpose or psychological function, but I can't seem to find studies about it applying to age groups past K-12 (I'd be happy to read if you can point any out to me).&lt;/p&gt;&#xA;&lt;p&gt;An exception to the presentation of advanced math content with color may be in video format, so there must be some visual appeal to the use of color. I just can't come up with the reason why this is not applied in books, even when printing is not an issue (as with ebooks, for example).&lt;/p&gt;&#xA;&lt;p&gt;Historically, I imagine it was simply cheaper to print in all black and white and maybe people just prefer what they're used to. But it doesn't explain why even books offered in electronic format would stick to this rule.&lt;/p&gt;&#xA;&lt;p&gt;From a practical point of view I understand that mathematical concepts don't have associated colors and don't require them (outside, maybe, of coloring results in graph theory and knot theory, where often enough colors are just represented by symbols).&lt;/p&gt;&#xA;&lt;p&gt;I've heard the argument that colors make work look &amp;quot;unprofessional&amp;quot; or &amp;quot;inelegant&amp;quot; but I'm also questioning why that is, since it can be a largely subjective assessment.&lt;/p&gt;&#xA;&lt;p&gt;I'm biased because I like using colors. When I take notes or write on the board, I like color coding, highlighting, and in general having quick visual &amp;quot;shortcuts&amp;quot; to find information on the page. I understand this can be distracting for others and I'd like to better understand why.&lt;/p&gt;&#xA;&lt;p&gt;Any thoughts?&lt;/p&gt;&#xA;&lt;p&gt;Edits: adjusted the question to say &amp;quot;most&amp;quot; as it is clear there are some counterexamples.&lt;/p&gt;&#xA;" OwnerUserId="15735" LastEditorUserId="15735" LastEditDate="2021-04-02T18:09:10.760" LastActivityDate="2021-04-04T15:53:20.050" Title="Why are most college level math textbooks black and white only?" Tags="|mathematical-pedagogy|textbooks|teaching|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20675" PostTypeId="2" ParentId="20666" CreationDate="2021-04-02T16:57:11.030" Score="2" Body="&lt;p&gt;In my opinion whether this proof is correct or not depends on the actual wording of the question. This recursive argument shows that the statement is true for 100 numbers (since it is true for 2, 3, 4, so we eventually reach 100) It also shows that the statement is true for 1000 numbers (we eventually reach 1000). For &lt;strong&gt;any&lt;/strong&gt; fixed natural number this is a correct argument.&#xA;However, if the students were asked to show that the statement is true for &lt;strong&gt;all&lt;/strong&gt; positive integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, then an extra step is needed. This is where quoting the principle of mathematical induction comes in. It is important for students to understand that the principle of mathematical induction is not just a shortcut to shorten a proof. It is a necessary axiom to move from &amp;quot;true for any fixed &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&amp;quot; to &amp;quot;true for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2021-04-02T16:57:11.030" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20676" PostTypeId="2" ParentId="11203" CreationDate="2021-04-02T17:19:54.577" Score="0" Body="&lt;p&gt;When I was thinking about a similar question, I found the article &amp;quot;What is an equation&amp;quot; by S. Marcus and S.M. Watt ( &lt;a href=&quot;https://www.academia.edu/3287674/What_is_an_Equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.academia.edu/3287674/What_is_an_Equation&lt;/a&gt; ) As far as I understand they discuss the difference between &amp;quot;equation&amp;quot; and &amp;quot;equality&amp;quot; in some languages. In some language (like english) there is no difference, but in others (like french) these two concepts are different.&lt;/p&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2021-04-02T17:19:54.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20677" PostTypeId="2" ParentId="20674" CreationDate="2021-04-03T00:35:53.777" Score="4" Body="&lt;p&gt;Part of the cost of a printed book is the cost of paper, printing, and binding (PPB). These costs can be lowered with economies of scale when you are printing a lot of copies. Many of the costs are basically costs to set up for a run, while the incremental cost of printing one more book is small. (This is for traditional printing, not print-on-demand.) Upper-division and graduate books don't sell a lot of copies, so their PPB costs will be much higher for the same design. Many of these books, especially graduate texts, are probably unprofitable for the publisher and author, or the margin may be very tight.&lt;/p&gt;&#xA;&lt;p&gt;Part of PPB costs is printing, and the cost of printing is proportional to the number of colors of ink used, which is four for a full-color book. That's a lot more money.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Historically, I imagine it was simply cheaper to print in all black and white and maybe people just prefer what they're used to. But it doesn't explain why even books offered in electronic format would stick to this rule.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Most books these days are offered both in print and electronically, so this actually makes printing costs &lt;em&gt;more&lt;/em&gt; of an issue. Let's say in the past you would have produced 3000 copies of a printed graduate text in one printing, at cost X. Today, if 2/3 of the students are either buying an electronic text or getting a copy from Library Genesis, then maybe you print 1000 copies, but the production cost is still roughly X, because most of the costs are setup costs. So now production costs are almost three times higher in proportion to the sticker price.&lt;/p&gt;&#xA;&lt;p&gt;Producing a book with a fancier layout also incurs more costs for design, and if you want to include photos then there are royalties that the publisher has to pay to the photographer for each copy of the book. For a low-selling, unprofitable upper-division book, these costs just make it even more unprofitable.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-04-03T00:44:14.180" LastActivityDate="2021-04-03T00:44:14.180" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20678" PostTypeId="2" ParentId="20671" CreationDate="2021-04-03T00:38:01.023" Score="5" Body="&lt;p&gt;Your question makes me think of the definition of linear (in)dependence used in Lay's Linear Algebra. It does not feel natural, and yet it's the simplest definition to work with.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1Au2s.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1Au2s.png&quot; alt=&quot;definition of linear (in)dependence&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I start with a more natural definition, that if one vector can be made as a combination of others in the set (ie it depends on those others), then the set is linearly dependent. And then I show that the two definitions are equivalent. It's still confusing for students.&lt;/p&gt;&#xA;&lt;p&gt;I believe it always helps to connect with what students already know. So I believe that this is helpful. But I do not have solid evidence to back up that belief.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-04-03T00:38:01.023" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20679" PostTypeId="2" ParentId="20670" CreationDate="2021-04-03T00:50:19.890" Score="14" Body="&lt;p&gt;I used to have this problem.  What helps me more than anything is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Solve it two different ways if you can and make sure they agree&lt;/li&gt;&#xA;&lt;li&gt;If you are finding a general formula, test it on some examples&lt;/li&gt;&#xA;&lt;li&gt;If neither of the above are possible, re-read every step of your work with an attitude like it's trying to sell you a used car.&lt;/li&gt;&#xA;&lt;li&gt;Adopt the useful exaggeration that any unchecked step is probably wrong.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15739" LastEditorUserId="15739" LastEditDate="2021-04-03T00:58:18.487" LastActivityDate="2021-04-03T00:58:18.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20680" PostTypeId="2" ParentId="20671" CreationDate="2021-04-03T15:55:56.817" Score="5" Body="&lt;p&gt;Another example of a crystalized definition is Dedekind's approach to defining &lt;em&gt;finite&lt;/em&gt;. First we define a set &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; be infinite if it is equivalent to a proper subset, i.e., if there is an injection &lt;span class=&quot;math-container&quot;&gt;$f:S\to S$&lt;/span&gt; that is not a surjection. Only then do we define what is means for a set to be &lt;em&gt;finite&lt;/em&gt;. A set is finite if it is not infinite. &amp;quot;Hilbert's Hotel&amp;quot; is a meme for this approach.&lt;/p&gt;&#xA;&lt;p&gt;Obviously, people think only about finite sets in the early stages of their cognitive development. Later they learn about big numbers (the &amp;quot;number of grains of sand found in every region whether inhabited or uninhabited&amp;quot;--Archimedes, &lt;em&gt;The Sand Reckoner&lt;/em&gt;), and infinite sets.&lt;/p&gt;&#xA;&lt;p&gt;Only after this development occurs, either individually or in mathematical culture, do we see the brilliance of Dedekind's approach to defining finite, which is to first characterize infinite sets.&lt;/p&gt;&#xA;&lt;p&gt;Hans Freudenthal, a Dutch mathematician who took an interest in mathematics education in the 1970s and founded the journal &lt;strong&gt;Educational Studies in Mathematics&lt;/strong&gt;, wrote about this definition of Dedekind in his book &lt;em&gt;Didactical Phenomenology of Mathematical Structures&lt;/em&gt;. He says that this definition &amp;quot;presupposes a well-stuffed arsenal of mathematical strategies.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-04-03T15:55:56.817" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20681" PostTypeId="1" AcceptedAnswerId="20682" CreationDate="2021-04-03T17:44:52.217" Score="7" ViewCount="976" Body="&lt;p&gt;I am thinking of taking up a position at a liberal art college. I have taught mathematics at large public universities but I have no idea what is it like to work at a liberal art colleges. So what are things I should know about teaching mathematics at a liberal art college? Can you recommend some resources to read so I may be better prepared for this new job?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorUserId="2083" LastEditDate="2021-04-03T19:05:06.333" LastActivityDate="2021-04-04T00:45:57.350" Title="How to be a good math teacher at a liberal art college?" Tags="|undergraduate-education|usa|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20682" PostTypeId="2" ParentId="20681" CreationDate="2021-04-03T23:46:30.870" Score="17" Body="&lt;p&gt;Let me reply as someone who left a university in favor of a Liberal Arts College (LAC).&#xA;Although @AlexanderWoo's point is valid, my experience supports the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;At a LAC, the walls between the departmental silos are lower,&#xA;enabling relatively easy cross-disciplanary collaborations.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At a LAC with a strong faculty-governance tradition, you can play an&#xA;all-college role that is almost impossible to achieve at a&#xA;university.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are passionate about teaching, it is uplifting to work at an institution&#xA;that prioritizes and rewards teaching.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I've found the engagement with students much more direct: There is less&#xA;separation between students and faculty.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The best students at a LAC are comparable to the best students at&#xA;a university. So one can still research-collaborate with the top students.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The scholarship that counts for tenure is more varied than at a university.&#xA;For example, writing a textbook could gain you tenure at a LAC, but&#xA;rarely at a university.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To counter-balance the above with two negative points:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It rarely possible to remain active at the research-frontier of a popular math&#xA;subdiscipline. You might need to adjust your research agenda toward less-traveled&#xA;paths, and less-dependent on graduate students.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Univ&lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt;LAC transition is nearly&#xA;irreversible. It would not be easy to move from a LAC to&#xA;a university later in your career.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-04-04T00:45:57.350" LastActivityDate="2021-04-04T00:45:57.350" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20683" PostTypeId="2" ParentId="20670" CreationDate="2021-04-03T23:59:21.840" Score="2" Body="&lt;p&gt;I believe that the majority of students with this problem are doing what is called &amp;quot;end-gaining&amp;quot; in the Alexander Technique:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.hilaryking.net/glossary/end-gaining&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.hilaryking.net/glossary/end-gaining&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm not an expert, but I think that in this case, it means that they are overly focused on getting to the end of the exercise, and they have a mistaken belief that the means for getting there are less important than the end result.&lt;/p&gt;&#xA;&lt;p&gt;They might also be unconsciously reflecting a disdain for &amp;quot;easy&amp;quot; maths, such as arithmetic, that they have picked up from their educators. Or worse, consciously cultivating an &amp;quot;Absent minded Professor&amp;quot; act, by which they hope to demonstrate their brilliance ! (&amp;quot;I don't have time or patience for that middle-school stuff&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Apprentices in trades can't afford to neglect the basics, and are not allowed to, by their mentors. I believe that we also should (kindly and patiently) enforce proper attention to the &amp;quot;boring details&amp;quot; also. Often this can be achieved just by example.&lt;/p&gt;&#xA;" OwnerUserId="6893" LastActivityDate="2021-04-03T23:59:21.840" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20684" PostTypeId="1" CreationDate="2021-04-04T06:56:11.847" Score="0" ViewCount="367" Body="&lt;p&gt;Tom Lehrer claims and the audience seems to agree with him that the 'old way', before new math to do subtraction was to say, for example, 'three from two is nine carry the one'. I never heard of this method. Does anyone still use it? What is it called? &lt;a href=&quot;https://www.youtube.com/watch?v=UIKGV2cTgqA&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=UIKGV2cTgqA&lt;/a&gt; is the video showing Tom Lehrer's bit. Does it really not involve understanding what you are doing as the new math activists claimed (I presume).&lt;/p&gt;&#xA;&lt;p&gt;Is it known what Lehrer really thought about this part of new math?&lt;/p&gt;&#xA;&lt;p&gt;Lehrer's example is 342 - 173.&lt;/p&gt;&#xA;&lt;p&gt;Edit: Also, when was the borrowing method of teaching subtraction invented?&lt;/p&gt;&#xA;" OwnerUserId="15662" LastEditorUserId="15662" LastEditDate="2021-04-06T06:30:22.907" LastActivityDate="2022-08-09T15:32:16.230" Title="Did Americans, before new math came in to schools, really say, 'three from two is nine carry the one&quot;, instead of borrowing ten from the tens column?" Tags="|arithmetic-operations|new-math|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20685" PostTypeId="2" ParentId="20670" CreationDate="2021-04-04T08:52:31.827" Score="4" Body="&lt;p&gt;&lt;strong&gt;Teach methods for verifying results.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Often once you have an answer, there are simple calculations to check its correctness. For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Numerical calculations: round the results and compute backwards. &lt;strong&gt;Example:&lt;/strong&gt; if you get 1234 * 56 / (789 - 987) = 349, compute e.g. 300 * (700 - 900) / 50 = -1200 which is not close to 1234, because of the dropped minus sign.&lt;/li&gt;&#xA;&lt;li&gt;Solving equations: substitute the result back to original equation, both sides should have same value.&lt;/li&gt;&#xA;&lt;li&gt;Simplifying equations: test it on a few values.&lt;/li&gt;&#xA;&lt;li&gt;Integrating a function: take the derivative of the result.&lt;/li&gt;&#xA;&lt;li&gt;Verbal problems: rephrase the problem backwards. &lt;strong&gt;Example:&lt;/strong&gt; &lt;em&gt;&amp;quot;Cities are 100 km apart. Car 1 travels 80 km/h, car 2 travels 30 km/h towards each other. How long until they meet?&amp;quot;&lt;/em&gt; If you get answer of 1 hour, rephrase as &lt;em&gt;&amp;quot;Where are the cars after 1 hour?&amp;quot;&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Geometry problems: check basic identities. For example if the diagram has triangles, verify the sum of corners is 180 degrees.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;It's worth noting that the same methods will also work for testing individual parts of a long problem, in order to find where exactly the mistake occurred.&lt;/p&gt;&#xA;" OwnerUserId="14647" LastActivityDate="2021-04-04T08:52:31.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20686" PostTypeId="2" ParentId="20684" CreationDate="2021-04-04T08:55:12.053" Score="10" Body="&lt;p&gt;It sounds like a variation of subtracting that I learned in high school (1968). My instructor called it European subtraction.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{ccc}&#xA;  &amp;amp; 3 &amp;amp; 4 &amp;amp; 2 \\&#xA;- &amp;amp; 1 &amp;amp; 7 &amp;amp; 3 \\&#xA;\hline&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You start by saying &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; plus &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt;.&#xA;You write the &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; as shown and &amp;quot;carry&amp;quot; the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, of the &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt;, as a subscript of the &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; as is shown below.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{ccc}&#xA;  &amp;amp; 3 &amp;amp; 4 &amp;amp; 2 \\&#xA;- &amp;amp; 1 &amp;amp; 7_{\color{red} 1} &amp;amp; 3 \\&#xA;\hline&#xA;  &amp;amp;   &amp;amp;   &amp;amp; \color{red} 9 \\&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; plus &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$14$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{ccc}&#xA;  &amp;amp; 3 &amp;amp; 4   &amp;amp; 2 \\&#xA;- &amp;amp; 1_{\color{red}1} &amp;amp; 7_1 &amp;amp; 3 \\&#xA;\hline&#xA;  &amp;amp;   &amp;amp; \color{red}6   &amp;amp; 9 \\&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Finally, &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; plus &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{ccc}&#xA;  &amp;amp; 3 &amp;amp; 4   &amp;amp; 2 \\&#xA;- &amp;amp; 1_1 &amp;amp; 7_1 &amp;amp; 3 \\&#xA;\hline&#xA;  &amp;amp; \color{red}1   &amp;amp; 6   &amp;amp; 9 \\&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The big advantage of this method is that you can do many subtractions at once.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{ccc}&#xA;  &amp;amp; 5 &amp;amp; 3 &amp;amp; 4 &amp;amp; 2 \\&#xA;- &amp;amp; 1 &amp;amp; 9 &amp;amp; 2 &amp;amp; 3 \\&#xA;- &amp;amp; 2 &amp;amp; 9 &amp;amp; 4 &amp;amp; 5 \\&#xA;\hline&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;, plus &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;, plus &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$14$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$19$&lt;/span&gt;, plus &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$23$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;, plus &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{llll}&#xA;  &amp;amp; 5   &amp;amp; 3   &amp;amp; 4   &amp;amp; 2 \\&#xA;- &amp;amp; 1   &amp;amp; 9   &amp;amp; 2   &amp;amp; 3 \\&#xA;- &amp;amp; 2_2 &amp;amp; 9_1 &amp;amp; 4_1 &amp;amp; 5 \\&#xA;\hline&#xA;  &amp;amp;   &amp;amp; 4   &amp;amp; 7   &amp;amp; 4 \\&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think this is a wonderful way to subtract, but, I would not teach it to grade school students.&lt;/p&gt;&#xA;" OwnerUserId="5144" LastEditorUserId="5144" LastEditDate="2021-04-04T09:19:15.760" LastActivityDate="2021-04-04T09:19:15.760" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="20687" PostTypeId="1" CreationDate="2021-04-04T15:35:02.507" Score="7" ViewCount="1038" Body="&lt;p&gt;I have been looking at questions on Math Stack Exchange and I am frequently coming across topics that sound as if they could have been optional chapters in a high school geometry class, but I have never heard of them.  &amp;quot;Radical circles&amp;quot;, &amp;quot;Euler line&amp;quot;, and so forth.  It feels almost as if there were a Geometry II class that I somehow skipped, leaving a hole in my knowledge.&lt;/p&gt;&#xA;&lt;p&gt;Note I am &lt;em&gt;not&lt;/em&gt; looking for advanced courses like differential geometry or projective geometry, but rather things that look like challenge problems in ordinary high school geometry.  In particular I would like to find a textbook for self-study.  I own several high school &amp;quot;Geometry I&amp;quot; textbooks already.  I would like a textbook for this hypothetical &amp;quot;Geometry II&amp;quot; course.  Any suggestions are appreciated.&lt;/p&gt;&#xA;" OwnerUserId="15750" LastEditorUserId="4791" LastEditDate="2021-04-09T15:34:04.323" LastActivityDate="2023-10-23T12:41:17.987" Title="What is a good second book in high school geometry?" Tags="|secondary-education|textbooks|geometry|" AnswerCount="8" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20688" PostTypeId="2" ParentId="20674" CreationDate="2021-04-04T15:53:20.050" Score="2" Body="&lt;p&gt;Use of color definitely can be helpful in learning mathematics.  Presentation matters.  I have given up on texts using an awful font: if learning the idea requires 95% of my brainpower, and deciphering the bad presentation uses up 10% of my brainpower, it makes the difference between learning and not learning.&lt;/p&gt;&#xA;&lt;p&gt;That said, using color intelligently requires work and careful thought.  Visual design is a different skill from straight pedagogy.  I imagine some authors simply do not want to put in the extra work of deciding how to arrange colors.  Certainly just formatting math for my workbook was so exhausting I wouldn't have wanted to add the extra labor of formatting for colors too.&lt;/p&gt;&#xA;&lt;p&gt;When I self-published a physics workbook, the cost of printing color was much higher than black and white.  That will sometimes make a difference; I wanted to use color but only in a few spots, so I sacrificed that to keep the books cheaper.  As for ebooks, they could use color, but again, you have to put in extra effort to have two versions, one black and white for printing and one color for e-publishing.&lt;/p&gt;&#xA;&lt;p&gt;An important change that is probably in the works is the ability to display a e-text with the user's choice of foreground/text and background colors.  This will make a difference for those with vision disabilities.  (One of my favorite inventions that I have heard about in recent years is the font designed for dyslexic people!). I had a student with dyspraxia who had to use yellow paper to be able to read their own work.&lt;/p&gt;&#xA;&lt;p&gt;When working difficult proofs with students, I have sometimes written each line on the whiteboard in a different color.  So they could say, &amp;quot;At the start of the green line, why is the first term positive?&amp;quot; and so forth.  It was one more timesaver in communication.&lt;/p&gt;&#xA;&lt;p&gt;Use of color can certainly help, but it requires more work and expense in general.&lt;/p&gt;&#xA;" OwnerUserId="15750" LastActivityDate="2021-04-04T15:53:20.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20690" PostTypeId="2" ParentId="20687" CreationDate="2021-04-04T17:00:16.400" Score="3" Body="&lt;p&gt;A quick Google search returned the following texts based on key words from your posting: rational circle, Euler line, geometry textbook.  Based on skimming the contents and prefaces, I think they are good chance to match your desires:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A Beautiful Journey through Olympiad Geometry, Stefan Lozanovski, 2016&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.olympiadgeometry.com/the-book.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.olympiadgeometry.com/the-book.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Euclidean Geometry in Math Olympiads, Evan Chen, 2016&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.evanchen.cc/geombook.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://web.evanchen.cc/geombook.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;At least the latter link seems to have additional resources to the named book.&lt;/p&gt;&#xA;" OwnerUserId="15752" LastActivityDate="2021-04-04T17:00:16.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20691" PostTypeId="2" ParentId="20687" CreationDate="2021-04-04T19:24:03.447" Score="9" Body="&lt;p&gt;I recommend either &lt;em&gt;Excursions in Geometry&lt;/em&gt; by Ogilvy or &lt;em&gt;Geometry Revisited&lt;/em&gt; by Coxeter and Greitzer. Both are cheap too.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2021-04-04T19:24:03.447" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="20692" PostTypeId="2" ParentId="20687" CreationDate="2021-04-05T01:57:33.770" Score="2" Body="&lt;p&gt;I wonder if you'd enjoy an older textbook on Solid Geometry. (The only ones I've ever seen are quite old. Maybe there's newer.) But that may not address the particular topics you mentioned.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-04-05T01:57:33.770" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20693" PostTypeId="2" ParentId="19546" CreationDate="2021-04-05T05:35:08.667" Score="3" Body="&lt;p&gt;I remember getting my first 18 inch slide ruler.  I had worked hard, poison oak filled, hours making fire breaks in the Oakland hills to make enough money to get it. It was aluminum, had a spring loaded cursor that slid like it was greased, and the C and D scales (I think) lined up perfectly. It was huge and had scales I was never going to use. I loved it.&lt;/p&gt;&#xA;&lt;p&gt;So what do I do first? I was in the kitchen an looked down at the kitchen table. I would measure the width of the table, the angle to the opposite corner, and calculate the length of the table. I was trying to figure out how to set up my protractor to measure the angle when my dad walked in. My pop was a very successful travel agent but he didn't know or care what 8 times 8 was. That's what adding machines are for. He stood there watching me. I'm sure he thought I was insane.  &amp;quot;Why don't you just measure it&amp;quot;? &amp;quot;Well&amp;quot;, i said. &amp;quot;Suppose this is a plot of land and there is a river going through here.&amp;quot; &amp;quot;OK&amp;quot;. He replied, and continued standing there. I got the angle, to what I hoped was the nearest half a degree, picked up the slide rule, moved the scale, moved the cursor and read off a number. Knowing I couldn't be that accurate, I picked up the slide rule again and converted the fractional part of the answer into sixteenths of an inch. &amp;quot;51 and three quarter inches&amp;quot;, I proclaimed. I held the end of the measuring tape on one end while my dad pulled it over to the other end. &amp;quot;54 inches&amp;quot; he said. &amp;quot;Well,&amp;quot; I said. &amp;quot;With the tools I was using, &amp;quot;That's not too bad.&amp;quot; To my surprise, my dad was amazed. How did you do that? I knew better than to go into more detail than saying this slide rule has scales on it that convert angles into numbers that you can use it to calculate distances and multiply and divide numbers.&lt;/p&gt;&#xA;&lt;p&gt;I almost never used that slide rule to calculate an answer. First, I calculated the answer myself, then I used the slide rule to check my answer. Yet I treated it with the same respect that a marine treats his rifle. It was always clean, properly lubricated, and the cursor was always lint free.&lt;/p&gt;&#xA;&lt;p&gt;Eventually my slide ruler went the way of land lines, record players, and tv sets that you used pliers to change the channel with. It's a shame. I learned a lot about mathematics by figuring out how slide rules worked. Maybe now we need to show them how spreadsheets work. And hand-held plotting calculators.&lt;/p&gt;&#xA;" OwnerUserId="5144" LastActivityDate="2021-04-05T05:35:08.667" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20695" PostTypeId="2" ParentId="20670" CreationDate="2021-04-05T15:19:03.527" Score="5" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;When there is an obvious way to check the final result, don't give partial credit to students who make a mistake and fail to catch it. If they're solving a system of equations, they should be checking by substituting back in. If they're integrating, they should be checking by differentiation. Do give partial credit if they do the check on an exam, but run out of time and write on their paper that they know it failed the check but didn't have time to track down the mistake.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Assign problems in which dimensional analysis is relevant, and make this a required part of the problem. Do an example early in the semester where you demonstrate the technique, and refer your students to a written presentation of it. (College STEM majors should have seen it multiple times already, but could probably use the review.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Require sense-making.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Encourage the use of computer algebra systems on homework, when appropriate, but discourage its use in inappropriate situations where an expert would not resort to a CAS. The open-source CAS Maxima is free.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Example: A sample of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; atoms of a radioactive isotope decays over time according to &lt;span class=&quot;math-container&quot;&gt;$N=Ae^{-\lambda t}$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is large enough that we can treat it as a continuous variable. (a) Infer the units of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;. (b) Find the rate of decay, in units of decays per second. (c) Check that the units of your result make sense. (d) Check that your answer makes sense in the limit &lt;span class=&quot;math-container&quot;&gt;$t\rightarrow\infty$&lt;/span&gt;. What is the interpretation of this limit? (e) Suppose that we do such an experiment using a larger value of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. Interpreting your answer from part b, what is the mathematical effect of increasing &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;? Does it increase the rate of decay, or decrease it? Now compare this mathematical trend with what is expected physically. Does it make sense?&lt;/p&gt;&#xA;&lt;p&gt;I have a bunch of problems in this style included in &lt;a href=&quot;http://lightandmatter.com/fund/&quot; rel=&quot;noreferrer&quot;&gt;my OER calculus text&lt;/a&gt;. The book describes dimensional analysis in sections 1.7.3 and 1.9.&lt;/p&gt;&#xA;&lt;p&gt;Dimensional analysis is particularly powerful because if you make a habit of doing it reflexively at every line, it tends to catch mistakes immediately rather than at the end of the problem.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-04-05T15:33:25.663" LastActivityDate="2021-04-05T15:33:25.663" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20696" PostTypeId="2" ParentId="20670" CreationDate="2021-04-05T17:56:54.003" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;As far as &amp;quot;just practice&amp;quot;, you shouldn't trivialize this or think of it as so simple as yes, no.  Your job is to train the trainee.  Many errors are common and repeated.  So?  This is not so different from sports or music.  Develop a set of drills, games, etc.  Try different things with different people.&lt;/p&gt;&#xA;&lt;p&gt;The problem here is not so Boolean as math itself.  People are not OR gates.  I don't mean to come down too hard on you, but I just want to emphasize that this is a big topic and is central to being a TEACHER as opposed to a creator of math.  You need to wallow around in there and try different things (worksheets, books, Kahn Academy, purchased games, competitive drill, etc., and see what seems to work better with different students.&lt;/p&gt;&#xA;&lt;p&gt;And no, there is no simple square peg for the hole style answer from YA or SE or whatever.  It's a big topic. And yes, it may take time and yes some students will struggle more than others. Teaching people to ski the bumps parallel is not instant gratification either.  Math is tricky to master, for us organic brains.  So are languages, music, sports, and work.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have your selection of problems include some easier ones (less steps).  This is especially important at the beginning of a session and with subjects that are more error prone.  By having a decent fraction of the problems that are simpler, the trainee can get some experience at solving at least some problems correctly and not become discouraged.  Perhaps he also gets some confidence from the simpler successes and incorporates them in longer problems.  Of course, there should be some selection of problems that test the trainee with the multistep calculations with which he struggles.  But don't make the mistake of doing 100% like that.  (Note also that this is in the OPPOSITE direction of teachers who want to emphasize &amp;quot;project style&amp;quot; homework assignments.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Whenever a problem is missed for a dumb mistake, make the student repeat the entire problem (after finishing it) with a fresh sheet of paper.  It gives free drill on the concept itself and is a light &amp;quot;stick&amp;quot; to encourage doing it right.  It is also more likely to be easy for the student (i.e. reasonable progression) rather than an all new problem or the same problem after a long duration away.  This may seem hokey, but it really works.  Near term feedback.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do NOT verbally minimize the importance of calculation errors.  I.e. &amp;quot;Oh it was just a dumb mistake.&amp;quot;  Treat it more like tennis where the ball is in/out and it doesn't matter how pretty the stroke.  And I'm not saying you in particular do this, just a general precaution.  I would be like this all the time.  But in this case especially (where the problem is errors), one should not accept the pattern which needs to be trained away.  And I'm not saying to crucify the kids either.  Just eschew saying &amp;quot;it's OK&amp;quot;.  Since it's not.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I was concerned in comments that you didn't want to bother estimating the error frequency.  You are looking at a practical phenomenon.  One having to do with human performance.  Not Euclid.  One of the ways to get your hands wrapped around a soft topic like this is to start by making simple estimates (e.g. frequency) and to segment the errors into different types.  And yes, I do think knowing this will help with devising solutions as well.  For one thing it gets you involved in studying the phenomenon more.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15752" LastActivityDate="2021-04-05T17:56:54.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20697" PostTypeId="1" CreationDate="2021-04-06T02:13:28.877" Score="0" ViewCount="204" Body="&lt;p&gt;I just looked at &lt;a href=&quot;https://en.wikipedia.org/wiki/Matriculation_exam_(Finland)&quot; rel=&quot;nofollow noreferrer&quot;&gt;matriculation exams from Finland&lt;/a&gt;. They have both basic and advanced level exams. Most US high school seniors could not pass the basic exam. If each US state were to create its own math matriculation exam, with basic and advanced options, the level of math among college freshmen would increase tremendously. Why there is no attempt to emulate Finland, whose education is considered the best in the world?&lt;/p&gt;&#xA;" OwnerUserId="9777" LastEditorDisplayName="user507" LastEditDate="2021-04-06T13:01:50.837" LastActivityDate="2021-04-06T14:01:45.893" Title="Matriculation exams like in Europe" Tags="|geometry|precalculus|usa|finland|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="20699" PostTypeId="2" ParentId="20697" CreationDate="2021-04-06T13:34:46.043" Score="6" Body="&lt;p&gt;Finland is a small country with a centralized political system and a university system that is essentially public and free, and highly centralized on a couple of big schools. The US is a big country with a federal system of government and a higher education system that is a mix of public and private, spread across an entire continent. In the US, the federal government has essentially no role in education -- it doesn't fund it and only regulates it in certain very limited ways.&lt;/p&gt;&#xA;&lt;p&gt;So the first barrier in your plan is that it can't be instituted politically. The federal government is not going to expropriate the University of Chicago from its private owners, take over Cal State LA from the California state government, etc. Without that kind of centralized control, you can't institute national standards for college admissions.&lt;/p&gt;&#xA;&lt;p&gt;The second barrier to your plan is that the electorate in the US doesn't want it. The voters have decided that they want higher education in the US to be very broadly available. In Finland, it looks like about 3% of the population is enrolled in higher education at any given time. In California (whose population is 7 times greater than that of Finland), the figure is about 6%, of which about half is community college students. (Counts of community college students are somewhat meaningless, since many CC students have only tenuous ongoing ties with the schools where they are theoretically students.) Community colleges have nonselective admissions. By your plan, they would close up shop, since the four-year schools already have enough capacity for the number of students who quality under any kind of selective admissions standards.&lt;/p&gt;&#xA;&lt;p&gt;You seem to imagine that in Finland, the examination requirement has made everyone graduate from high school with a high level of academic preparation for college. That doesn't seem to be the case, since their total enrollment in schools with selective admissions is about the same as ours, at 3% of the total population.&lt;/p&gt;&#xA;&lt;p&gt;Standardized testing is great. It's one of the few things that prevents the total dumbing down of education to some kind of horribly low level dictated by popular demand. It's also great because it's one of the strongest anti-racist and anti-classist institutions in our society. It's a barrier to the kind of racism and classism that says that it's OK for a high school with brown or working class students to provide inferior educational opportunities and pretend that those opportunities are not inferior. But a standardized test is sort of like a COVID test. It doesn't cure the problem, it just makes it harder for policy makers to hide their heads in the sand and pretend that nothing is wrong.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-04-06T14:01:45.893" LastActivityDate="2021-04-06T14:01:45.893" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20701" PostTypeId="2" ParentId="20687" CreationDate="2021-04-07T15:32:40.943" Score="4" Body="&lt;p&gt;In my (lack of) experience, the second class in Euclidean geometry is actually an undergraduate course that seems to be often called College Geometry.  And, yeah, there are so many fascinating topics there that are accessible to gifted HS students that don't really measure up to other topics when it comes to career and college readiness, but they were the pinnacle of mathematical and logical  knowledge before calculus was developed.&lt;/p&gt;&#xA;&lt;p&gt;Something that is more readable than a textbook but just as informative that I really loved was &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1616145870&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Secrets of Triangles&lt;/a&gt; by Posamentier.  There aren't formal exercises, but there are plenty of claims in the narrative that you can verify for yourself on a sheet of scrap paper.  I also read a college geometry textbook by Posamentier that I really appreciated if you are particular about a less casual narrative style.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="511" LastEditDate="2021-04-07T16:18:23.013" LastActivityDate="2021-04-07T16:18:23.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20702" PostTypeId="2" ParentId="20687" CreationDate="2021-04-07T16:17:06.050" Score="1" Body="&lt;p&gt;@MatthewDaly's mention of &lt;em&gt;The Secrets of Triangles&lt;/em&gt; reminded me of&#xA;the just released &lt;em&gt;A Cornucopia of Quadrilaterals&lt;/em&gt;, which I've been reading.&#xA;(Alsina, Claudi, and Roger B. Nelsen. Vol. 55. American Mathematical Soc., 2020.)&lt;/p&gt;&#xA;&lt;p&gt;For example, for a bicentric quadrilateral (cyclic and tangential) of side lengths &lt;span class=&quot;math-container&quot;&gt;$a,b,c,d$&lt;/span&gt; and angles &lt;span class=&quot;math-container&quot;&gt;$A,B,C,D$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;a + c &amp;amp;= b + d \\&#xA;A + C &amp;amp;= B + D&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7L9SN.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7L9SN.jpg&quot; alt=&quot;CoverImage&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And now a sequel of sorts,&#xA;&lt;em&gt;A Panoply of Polygons&lt;/em&gt;&#xA;(&lt;a href=&quot;https://bookstore.ams.org/view?ProductCode=DOL/58&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMS link&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/aSobC.jpg&quot; width=&quot;300&quot; /&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2023-10-23T12:41:17.987" LastActivityDate="2023-10-23T12:41:17.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20703" PostTypeId="2" ParentId="20687" CreationDate="2021-04-07T17:19:08.417" Score="3" Body="&lt;p&gt;I took an undergraduate course in &amp;quot;advanced planar geometry&amp;quot; in preparation for secondary teaching. It was from the text by Isaacs, &amp;quot;Geometry for College Students (Pure and Applied Undergraduate Texts)&amp;quot; and is available for purchase &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0821847945&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. There are also less upstanding ways to obtain this text.&lt;/p&gt;&#xA;&lt;p&gt;It appears to cover all the topics you've mentioned: the Euler line is the topic of 2.C and radical circles in 3.D.&lt;/p&gt;&#xA;&lt;p&gt;Other topics of note that I enjoyed: the nine-point circle, discussion of Simson lines, the butterfly theorem, and Ceva's theorem, and finally the discussion of &amp;quot;constructible numbers&amp;quot; at the end.&lt;/p&gt;&#xA;&lt;p&gt;The text should be accessible to anyone with a basic mathematical maturity with proof, ideally at least introductory proof at the college level, but it re-introduces &amp;quot;Geometry I&amp;quot; in its first chapter, slightly more formally.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2021-04-07T17:19:08.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20704" PostTypeId="2" ParentId="19465" CreationDate="2021-04-08T02:47:32.437" Score="2" Body="&lt;p&gt;The calculus textbook I'm currently using (Burzynski, Applied Calculus for Business, Life and Social Sciences, XYZ Textbooks) uses the term &amp;quot;hypercritical point&amp;quot;. A quick web search indicates that a few other sources use it as well.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastEditorUserId="15671" LastEditDate="2021-04-08T18:45:25.003" LastActivityDate="2021-04-08T18:45:25.003" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20705" PostTypeId="1" CreationDate="2021-04-08T05:41:09.807" Score="6" ViewCount="460" Body="&lt;p&gt;I was reading this book:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Dynamics of Particles and Rigid Bodies: A Self-Learning Approach&amp;quot; by Mohammed F. Daqaq&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the preface, the author explains of his &amp;quot;flipped classroom exercises&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I have presented a set of exercises, which I call &amp;quot;Flipped&#xA;classroom exercises&amp;quot;. Flipping the classroom is a modern teaching technique which has been shown to be very effective in teaching topics requiring mathematical derivations. In a typical flipped classroom, the instructor will briefly discuss the material for ten minutes, do some of the important derivations, and solve one example. Subsequently, the instructor hands the&#xA;students one or more problems to solve on their own, but provides some guidelines that help the students by dividing the problem into several sub-problems. The student will then spend the rest of the class trying to solve as many of these problems as possible. The problems are carefully designed to have an increasing difficulty level. At the end of the class, the full solution of the problems is given to the student.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;An example from the text:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Flipped Classroom Exercise 1.1&#xA;Find the rotation matrix necessary to take you from a certain frame, N , to another frame B by performing a successive 2-1-3 rotation using angles (...)&#xA;To answer this exercise, follow the following steps:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Which rotation takes place first? What is the rotation matrix associated with it?&lt;/li&gt;&#xA;&lt;li&gt;Which rotation takes place second? What is the rotation matrix associated with it?&lt;/li&gt;&#xA;&lt;li&gt;Which rotation takes place third? What is the rotation matrix associated with it?&lt;/li&gt;&#xA;&lt;li&gt;Multiply the rotation matrices obtained in steps 1, 2, and 3. Since the 2-rotation occurs first, the matrix obtained in step 1 must be on the far left. Show that the transformation matrix from N to B can be written as ....&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Are there any guides to creating such exercises so I can use in teaching?&lt;/li&gt;&#xA;&lt;li&gt;Are there any other textbooks that uses this approach?&lt;/li&gt;&#xA;&lt;li&gt;Is this supposed to be similar to inquiry based learning?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15777" LastEditorUserId="12255" LastEditDate="2021-04-09T05:16:14.190" LastActivityDate="2021-04-09T05:16:14.190" Title="&quot;Flipped classroom exercises&quot; resources" Tags="|undergraduate-education|inquiry-based-learning|resources|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20706" PostTypeId="2" ParentId="20705" CreationDate="2021-04-08T11:39:42.833" Score="9" Body="&lt;p&gt;This is not really a complete answer to your question (which is pretty broad), but would not have fit in a comment.&lt;/p&gt;&#xA;&lt;p&gt;What is described in the quoted text is one person's particular way of implemented a flipped classroom. One thing that I find sad about it is that it never mentions the role of a textbook in presenting the material -- which is impossible to do adequately in a 10-minute presentation. An alternative model, which I have used throughout my career, is closer to what is described by Mazur, Peer Instruction, which emphasizes reading. Yet another model is to record a lecture on video and have students watch it before class -- I suppose on the theory that students are all illiterate.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is this supposed to be similar to inquiry based learning?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As I understand it, inquiry-based learning is a different thing. It's a model where the students to some extent construct the subject for themselves. For instance, rather than presenting students with the definition of the derivative, you could have them do examples where they make numerical approximations to the slope of &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^3$&lt;/span&gt;, etc., form conjectures, and eventually are led to constructing the definition of the derivative in terms of the limit for themselves. I've done this technique, and I think can be made to work well in gen ed classes, but it's not well suited to classes for STEM majors. It gets awkward because you're at least temporarily hiding information from the students, and then at some point they have to be encouraged to synthesize things into some more organized form. In my experience, it can randomly work better or worse depending on the group of students you get and whether there are students in the group who are natural leaders.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Are there any other textbooks that uses this approach?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some ways of implementing the flipped classroom don't require any special features in the textbook. If you want to implement it using group activities or whole-class active learning exercises rather than individual activities, then you need a source of tested and well-designed activities. For example, my &lt;a href=&quot;http://lightandmatter.com/fund/&quot; rel=&quot;noreferrer&quot;&gt;OER first-semester calculus text&lt;/a&gt; has discussion questions sprinkled throughout the text.&lt;/p&gt;&#xA;&lt;p&gt;Before diving in to a technique like this, I would suggest that you carefully study the pedagogical literature and make sure that whatever practices you follow have empirical support. I mainly teach physics, not math, and the impression I get is that there is overwhelming empirical support for active learning in physics, but not as much in math. And active learning does not necessarily mean a flipped classroom. In the empirical studies in freshman physics, the common denominator for success seems to be that you get the students to talk, and the words coming out of their mouths are about concepts, not plugging numbers into formulas.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-04-08T20:02:56.157" LastActivityDate="2021-04-08T20:02:56.157" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="20707" PostTypeId="2" ParentId="19465" CreationDate="2021-04-08T14:34:17.497" Score="5" Body="&lt;p&gt;The term &lt;em&gt;Hergert Numbers&lt;/em&gt; is sometimes used in my specific region of the US for the values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$f''(x) = 0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$f''(x)$&lt;/span&gt; is undefined. This is in reference to Rodger Hergert, an Illinois community college professor who sometime in the 1990s became very frustrated with the fact that these numbers had no good name.&lt;/p&gt;&#xA;&lt;p&gt;But it doesn't &lt;em&gt;really&lt;/em&gt; matter what you call them as long as you are precise about it and distinguish them from inflection points, so I recommend the whimsical option where you name them after a current math educator -- call the points you are talking about &lt;em&gt;&lt;strong&gt;Hergert points&lt;/strong&gt;&lt;/em&gt; and the x-values &lt;em&gt;&lt;strong&gt;Hergert numbers&lt;/strong&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If a student asks who Hergert is, you can grin and truthfully say &amp;quot;some guy like me who was annoyed that these didn't have a name.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If you do adopt the terminology, he'll send you a shirt, too. As of the time of this writing, the directions to get the shirt are at &lt;a href=&quot;http://www.hergertnumbers.org/&quot; rel=&quot;noreferrer&quot;&gt;hergertnumbers.org&lt;/a&gt; under &amp;quot;If you would like your own Hergert Numbers t-shirt.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-04-08T14:34:17.497" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20708" PostTypeId="2" ParentId="20687" CreationDate="2021-04-08T17:12:08.953" Score="1" Body="&lt;p&gt;So some great books on geometry and written for middle/high schoolers are those written by Kiselev and translated to English by Alexander Givental. The textbooks are &amp;quot;Planimetry&amp;quot; and &amp;quot;Stereometry&amp;quot; respectively. Maybe try taking a look in either of those books and see if there is something that suits you?&lt;/p&gt;&#xA;&lt;p&gt;Here's a link to the first.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0977985202&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5746" LastActivityDate="2021-04-08T17:12:08.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20709" PostTypeId="1" CreationDate="2021-04-09T02:26:42.163" Score="7" ViewCount="515" Body="&lt;p&gt;&lt;strong&gt;TL;DR version:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It seems to me that high school curricula no longer distinguish between &amp;quot;horizontal shift&amp;quot; and &amp;quot;phase shift&amp;quot;, or between &amp;quot;frequency&amp;quot; and &amp;quot;angular frequency&amp;quot;, when teaching sinusoidal functions.  Is this perception accurate, and if so is this a recent change in usage?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Long version:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;When not at my day job, I tutor a few high school students, and I recently I have noticed that some of them seem to be learning to describe sinusoidal functions in a way that differs from how I understand important words.  In particular, the distinction between &amp;quot;horizontal shift&amp;quot; and &amp;quot;phase shift&amp;quot; seems to be elided, as is the distinction between &amp;quot;frequency&amp;quot; and &amp;quot;angular frequency&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Let me use an example to illustrate what I mean.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$H(t) = 8 \sin \left( \frac{2\pi}{5}(t - 2) \right) + 9$$&lt;/span&gt;&#xA;which can be written in the equivalent form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$H(t) = 8 \sin \left( \frac{2\pi}{5}t - \frac{4\pi}5 \right) + 9$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here let's assume that &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is time, measured in seconds, and &lt;span class=&quot;math-container&quot;&gt;$H(t)$&lt;/span&gt; is the height of something, measured in meters.  My understanding is that the correct way to describe the features of this function is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The &lt;em&gt;amplitude&lt;/em&gt; is &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; meters.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;midline&lt;/em&gt; is &lt;span class=&quot;math-container&quot;&gt;$y = 9$&lt;/span&gt; meters (this describes the equilibrium position of the oscillating object).&lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;period&lt;/em&gt; is &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; seconds.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;frequency&lt;/em&gt; is &lt;span class=&quot;math-container&quot;&gt;$\frac 15$&lt;/span&gt; cycles/second, i.e. &lt;span class=&quot;math-container&quot;&gt;$\frac 15$&lt;/span&gt; Hz.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;angular frequency&lt;/em&gt; is &lt;span class=&quot;math-container&quot;&gt;$\frac{2\pi}5$&lt;/span&gt; radians/second; note that this is the same as the coefficient of &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; inside the argument of the sine function.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;horizontal shift&lt;/em&gt; is &lt;span class=&quot;math-container&quot;&gt;$t = 2$&lt;/span&gt; seconds; this means that the graph has been shifted &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; seconds to the right, and therefore, when graphing the ``first'' period of the wave, we would begin at &lt;span class=&quot;math-container&quot;&gt;$t = 2$&lt;/span&gt;, rather than at &lt;span class=&quot;math-container&quot;&gt;$t = 0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;phase shift&lt;/em&gt; is &lt;span class=&quot;math-container&quot;&gt;$t = \frac{4\pi}5$&lt;/span&gt; radians; the interpretation of this is that the graph has been shifted &lt;span class=&quot;math-container&quot;&gt;$\frac 25$&lt;/span&gt; of a cycle to the right (because a full cycle is &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt; radians).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;What I am seeing:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Students are being taught that in an expression of the form &lt;span class=&quot;math-container&quot;&gt;$A \sin (bt)$&lt;/span&gt;, the constant &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is called the &amp;quot;frequency&amp;quot;, and that the period is computed as &lt;span class=&quot;math-container&quot;&gt;$T = \frac{2\pi}{b}$&lt;/span&gt;.  I agree with the latter formula, but not the word attached to the constant &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.  Frequency is measured in Hz, so it &lt;em&gt;must&lt;/em&gt; be the reciprocal of the period -- no?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students are also being taught that in an expression of the form &lt;span class=&quot;math-container&quot;&gt;$A \sin (b(t-h))$&lt;/span&gt;, the quantity &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is called &lt;em&gt;both&lt;/em&gt; &amp;quot;horizontal shift&amp;quot; and &amp;quot;phase shift&amp;quot; -- apparently these are synonyms now? The quantity &lt;span class=&quot;math-container&quot;&gt;$bh$&lt;/span&gt;, which is what &lt;em&gt;I&lt;/em&gt; would call the &amp;quot;phase shift&amp;quot;, does not seem to have a name, and expressions like &lt;span class=&quot;math-container&quot;&gt;$\sin(3t - \pi)$&lt;/span&gt; must first be rewritten as &lt;span class=&quot;math-container&quot;&gt;$\sin\left( 3 \left(t - \frac{\pi}3 \right) \right)$&lt;/span&gt;, so that the &amp;quot;phase shift&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$\pi/3$&lt;/span&gt; can be identified.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I would chalk this up to careless teachers, but I'm finding this in published (online) materials as well.  For example, &lt;a href=&quot;https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/5.6/primary/lesson/phase-shift-of-sinusoidal-functions-pcalc&quot; rel=&quot;nofollow noreferrer&quot;&gt;here is a CK-12 textbook that explicitly says &amp;quot;Phase shift is the horizontal shift left or right for periodic functions&amp;quot;&lt;/a&gt;, and here is &lt;a href=&quot;https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/5.5/primary/lesson/frequency-and-period-of-sinusoidal-functions-pcalc/#&quot; rel=&quot;nofollow noreferrer&quot;&gt;the same book defining frequency as &amp;quot;the number of cycles that occur in &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt;&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now I am not by any means a language prescriptivist; it's fine with me if the meaning of words changes, or if words are used in different ways by different communities. (All definitions are conventions, and all conventions are local.)  But I am curious where the boundaries of these communities are, and in &lt;em&gt;who&lt;/em&gt; uses words in specific ways.&lt;/p&gt;&#xA;&lt;p&gt;So now, my questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In what curricula are the distinctions &amp;quot;frequency&amp;quot;/&amp;quot;angular frequency&amp;quot; and &amp;quot;horizontal shift&amp;quot;/&amp;quot;phase shift&amp;quot; preserved, and in which are they elided?  (Answers might start with: &amp;quot;I teach X in country Y, and the convention here is...&amp;quot;)&lt;/li&gt;&#xA;&lt;li&gt;Have these usages changed over time -- and in particular, is my sense that this is a recent shift accurate?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To a lesser extent I am interested in discussion of the pros and cons of both approaches, but primarily I am looking not for opinions, but for factual information about how these words are used, and by whom.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="851" LastEditDate="2021-04-12T15:37:00.847" LastActivityDate="2021-04-12T15:37:00.847" Title="Phase shift vs. horizontal shift, and frequency vs. angular frequency in sinusoidal functions" Tags="|secondary-education|trigonometry|vocabulary|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20710" PostTypeId="1" CreationDate="2021-04-09T04:27:07.947" Score="6" ViewCount="140" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Evidence suggests that even instructional approaches that produce conceptual&#xA;gains may leave students reliant and expecting to be reliant on guidance from instructors (Redish et al., 1998). Students do not expect to be able to address situations they have not encountered before or to judge for themselves when an answer makes sense. &lt;strong&gt;Instead, students' principal method for assessing their understanding is to check that their answers to exercises align with the published solutions.&lt;/strong&gt;&#xA;(National Research Council. 2013. Adapting to a Changing World: Challenges and Opportunities in Undergraduate Physics Education. Washington, DC: The National Academies Press. &lt;a href=&quot;https://doi.org/10.17226/18312&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.17226/18312&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is there any method to avoid the issue highlighted in bold? One approach might be to teach students to check whether their answers make sense (if it is to derive an equation, then check special cases.). But this begs the question of how to teach this technique systematically. Any suggestions?&lt;/li&gt;&#xA;&lt;li&gt;Would you recommend further reading on this topic? I understand I am coming from a Physics education standpoint, but I believe it is equally applicable to Mathematics as well.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Some might be tempted to just not give answers at all, like in many textbooks (especially Math), but this assumes students have already mastered the aforementioned technique.&lt;/p&gt;&#xA;" OwnerUserId="15777" LastEditorDisplayName="user507" LastEditDate="2021-04-09T14:05:29.053" LastActivityDate="2021-04-09T14:12:18.213" Title="Students reliant on answers provided, but not their own reasoning?" Tags="|undergraduate-education|homework|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20711" PostTypeId="2" ParentId="20709" CreationDate="2021-04-09T13:58:53.663" Score="8" Body="&lt;p&gt;These books are simply reflecting the longstanding and universal usage in physics and engineering, which is that these words can have either meaning, and any ambiguity is normally either resolved by context or unimportant.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-04-09T13:58:53.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20712" PostTypeId="2" ParentId="20710" CreationDate="2021-04-09T14:12:18.213" Score="3" Body="&lt;p&gt;Like you, I'm a physicist. Yes, you teach students this by teaching them to check their own answers. They way you do this is by repeatedly assigning them work in which they're explicitly required to do so. &lt;a href=&quot;https://matheducators.stackexchange.com/a/20695/507&quot;&gt;Example&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Would you recommend further reading on this topic? I understand I am coming from a Physics education standpoint, but I believe it is equally applicable to Mathematics as well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The best review papers I know of on physics education research are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;von Korff, &lt;a href=&quot;https://arxiv.org/pdf/1603.00516.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/pdf/1603.00516.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hake, 2001, &amp;quot;Lessons From the Physics-Education Reform Effort,&amp;quot; &lt;a href=&quot;https://arxiv.org/abs/physics/0106087&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/physics/0106087&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Sadly, I think research-based reform in physics education has been killed off by a combination of technological factors including MasteringPhysics and Chegg.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Evidence suggests that even instructional approaches that produce conceptual gains may leave students reliant and expecting to be reliant on guidance from instructors (Redish et al., 1998).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This would be a wonderful problem to have, since practices that produce conceptual gains are essentially never implemented.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-04-09T14:12:18.213" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20713" PostTypeId="2" ParentId="20709" CreationDate="2021-04-09T19:05:33.217" Score="4" Body="&lt;p&gt;In engineering and physics, in the English literature, &lt;em&gt;angular frequency&lt;/em&gt; is oftentimes abbreviated in &lt;em&gt;frequency&lt;/em&gt;. Sometimes this is explicitly stated at the beginning, but mostly it is given implicitly. However, the intended audience is expected to not make any confusion. The usage is certainly not recent, and I think I've seen it already about 40 years ago in even older books.&lt;/p&gt;&#xA;&lt;p&gt;It's worth noting that other languages employ instead two very distinct terms for the two concepts, and there's less risk of confusing abbreviations. For instance, in Italian and  French the angular frequency is called &lt;em&gt;pulsazione&lt;/em&gt; and &lt;em&gt;pulsation&lt;/em&gt;, respectively. Similar terms are probably used also in Spanish and German. Actually, English has a similarly distinct term too, &lt;a href=&quot;https://www.merriam-webster.com/dictionary/pulsatance&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;pulsatance&lt;/em&gt;&lt;/a&gt;, but this has never become widespread, and it'd probably understood by very few people. For a list of these terms in various languages see the entry at the &lt;a href=&quot;https://www.electropedia.org/iev/iev.nsf/display?openform&amp;amp;ievref=103-07-03&quot; rel=&quot;nofollow noreferrer&quot;&gt;The World's Online Electrotechnical Vocabulary&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For what concerns &lt;em&gt;phase shift&lt;/em&gt; and &lt;em&gt;horizontal shift&lt;/em&gt;, I've never found authors who would  not distinguish between the two, even though what you call &lt;em&gt;horizontal shift&lt;/em&gt; is more commonly called &lt;em&gt;time shift&lt;/em&gt;, or similarly for other quantities. I'd say that your example can be an idiosyncrasy limited to few people.&lt;/p&gt;&#xA;&lt;p&gt;Finally, in engineering, what you call &lt;em&gt;midline&lt;/em&gt; is more commonly called &lt;em&gt;DC component&lt;/em&gt;, &lt;em&gt;DC offset&lt;/em&gt; or &lt;em&gt;vertical offset&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5367" LastActivityDate="2021-04-09T19:05:33.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20714" PostTypeId="1" CreationDate="2021-04-09T21:37:45.330" Score="4" ViewCount="621" Body="&lt;p&gt;Learning mathematics and learning &lt;a href=&quot;https://www.youtube.com/watch?v=zv1Fuqhxia8&quot; rel=&quot;nofollow noreferrer&quot;&gt;parkour&lt;/a&gt; seem to have a lot in common. Both can be done on varying levels, but to progress in either one needs to overlearn and build basic skills so that these skills are automatic and accurate. Practitioners of both claim that anyone can do them, although to the beginner each of the disciplines looks daunting, if not impossible, to do at any more than a rudimentary level. Although for mathematics this is more of a byproduct, both of these disciplines also require practitioners to overcome and manage their fears.&lt;/p&gt;&#xA;&lt;p&gt;Now, parkour is something that naturally fascinates young people. The practice of mathematics, at some level, is a mental analogue of parkour. Try as I might, I have not been able to find any videos or blogposts that discuss this analogy, or how one might use or combine the teaching of one of these things with the other in order to develop a healthy &amp;quot;growth mindset&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;My question is: What are some resources for this?&lt;/p&gt;&#xA;&lt;p&gt;Here and there I have found some hoaky things online trying to connect parkour to math in the most unsatisfying way (to me)...namely to try to use math or physics to study parkour movements. This is not what I am looking for, here. I am looking for something that discusses the similarities of how mathematicians approach problem solving and how traceurs do...I'm talking about habits.&lt;/p&gt;&#xA;&lt;p&gt;I should address what might seem to be a massive oversight or mistake embedded in this question: being stuck certainly doesn't seem to fit into the idea of being able to fluidly overcome obstacles...and we know being stuck is very important to learning and doing mathematics. This said, the similarity I propose here is based on the (realistic, I think) assumption that mathematicians do not view being &lt;em&gt;stuck&lt;/em&gt; as a condition of inactivity, and this is precisely the point I'd like the analogy to make for students. A stuck mathematician doesn't generally look stuck, in the traditional sense.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2021-04-09T21:55:04.897" LastActivityDate="2023-12-24T12:08:00.940" Title="Parkour and Mathematical Practice?" Tags="|student-motivation|problem-solving|" AnswerCount="2" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="20715" PostTypeId="1" CreationDate="2021-04-10T18:18:39.517" Score="4" ViewCount="347" Body="&lt;p&gt;I want to stress to my students that we should be careful with how we treat the equals sign and that we should always make sure that the units match. However, sometimes I write&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;11:40 - 15 \text{ min} =  11:25 &#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and I am wondering whether I should stop doing it. One problem is that I would have trouble justifying the notation to a student.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Should I avoid writing it in this manner, and what are good alternatives?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Remark:&lt;/strong&gt; From an abstract mathematical standpoint it seems like the notation really means that we have an action of the group &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}\cdot \min$&lt;/span&gt; acting on the set of military hours &lt;span class=&quot;math-container&quot;&gt;$hh:mm$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Clarification:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;By &lt;span class=&quot;math-container&quot;&gt;$11:40$&lt;/span&gt; I mean that it has passed &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt; hours and &lt;span class=&quot;math-container&quot;&gt;$40$&lt;/span&gt; minutes since midnight, i.e. &lt;span class=&quot;math-container&quot;&gt;$11:40 \text{ a.m.}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;One of my issues is that the operation &lt;span class=&quot;math-container&quot;&gt;$20 \min - 9 \text{ a.m.}$&lt;/span&gt; does not seem meaningful, but &lt;span class=&quot;math-container&quot;&gt;$9 \text{ a.m.}-20 \min $&lt;/span&gt; does.&lt;/li&gt;&#xA;&lt;li&gt;There might be some additional issues with the cases &lt;span class=&quot;math-container&quot;&gt;$00:15 - 45 \min = 23:30$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6527" LastEditorUserId="6527" LastEditDate="2021-04-11T09:00:25.610" LastActivityDate="2021-04-12T15:30:42.770" Title="Should I avoid writing: $ 11:40 - 15 \text{ min} = 11:25$, and what are alternatives to this way of writing?" Tags="|notation|" AnswerCount="7" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20716" PostTypeId="1" AcceptedAnswerId="20719" CreationDate="2021-04-10T19:32:01.490" Score="1" ViewCount="406" Body="&lt;p&gt;Like the title says. I am self studying intro to proofs(How to prove it by velleman) so I can start an introduction to analysis. I am wondering if I should complete all the exercises in the textbook(about 25 proofs a chapter), or I should just roughly skim through and complete, for instance, just odds. Doing all the problems takes a lot of time and I want to get started on Spivak Calculus as soon as possible. However, doing every problem also give me more practice. Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="15802" LastActivityDate="2021-04-10T23:20:17.160" Title="Should I do all the proof practice problems in How to Prove It, an intro to proofs book?" Tags="|textbooks|proofs|self-learning|homework|problem-solving|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20717" PostTypeId="2" ParentId="20715" CreationDate="2021-04-10T21:57:03.993" Score="7" Body="&lt;p&gt;Absolute time is an example of what is called an affine space; a mathematical structure where you can consider the difference of two points (which in this case would be duration), and that second space is linear.  The affine space doesn't have a linear structure itself in the most abstract case, but it gets an interesting structure of its own when you think about all the different facets of the linearity of the translation space.  Another common example of an affine space is the Euclidean plane, where you can't add two points but you can add a vector to a point to yield the resultant point of that translation.&lt;/p&gt;&#xA;&lt;p&gt;What you're doing is perfectly cromulent from a notational standpoint.  You don't mention the age of the students you are teaching.  But if they call you out on it and you don't want to explain all of this, you could demur and pretend that &amp;quot;11:40&amp;quot; is a different notation for the duration since midnight and you can combine them just like you might say &amp;quot;5'8&amp;quot; + 3 inches&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-04-10T21:57:03.993" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20718" PostTypeId="2" ParentId="20716" CreationDate="2021-04-10T23:06:52.990" Score="2" Body="&lt;p&gt;Firstly, congratulations for taking on this task. Of course it is very important to actually complete problems, rather than just read the notes.&lt;/p&gt;&#xA;&lt;p&gt;I would say that completing the odd numbers is a sensible compromise - you can go back and complete the even questions at a later date for more practice or revision.&lt;/p&gt;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="15746" LastActivityDate="2021-04-10T23:06:52.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20719" PostTypeId="2" ParentId="20716" CreationDate="2021-04-10T23:20:17.160" Score="3" Body="&lt;p&gt;I have this book.  A lot of the problems are pretty similar and there are a lot in there.&lt;/p&gt;&#xA;&lt;p&gt;Normally, I'm in favor of doing all the exercises (since you get faster as you go).  But I really don't think that quantity of drill is required for a strong student, especially since this is a course that some people still eschew (not taking any proofs class).  Maybe do all the Spivak problems though.&lt;/p&gt;&#xA;&lt;p&gt;The other thing is the book only has answers for the odds.  So unless you have an answer guide, you can't check the evens.&lt;/p&gt;&#xA;" OwnerUserId="15803" LastActivityDate="2021-04-10T23:20:17.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20720" PostTypeId="2" ParentId="20715" CreationDate="2021-04-11T00:18:40.207" Score="1" Body="&lt;p&gt;I  think the concern about careful use of units here is a bit of a red herring because the representation of time is not numeric.&lt;/p&gt;&#xA;&lt;p&gt;A good framework to resolve this conundrum might be that of datatypes. Perhaps there are some future computer scientists in your class.&lt;/p&gt;&#xA;&lt;p&gt;Time represented in the format HH:MM can be thought of as a special datatype, similar to number, text, float, integer, percent. Subtraction would be done by converting first to numeric, and then converting back to HH:MM&lt;/p&gt;&#xA;&lt;p&gt;In this case, 11:40 would convert to 0.486111111111111, in the sense that noon which is halfway through the day,  would convert to 0.50&lt;/p&gt;&#xA;&lt;p&gt;Fifteen minutes has numeric value 0.0104166666666667. We (the computer) does the subtraction to get 0.4756944444444.&#xA;Upon converting back to HH:MM datatype we get 11:25.&lt;/p&gt;&#xA;&lt;p&gt;I would not worry about what feels like sloppy use of units because HH:MM is not numeric. If a student asks, give them a developmentally appropriate &amp;quot;datatype&amp;quot; answer.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-04-11T00:18:40.207" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20721" PostTypeId="2" ParentId="20715" CreationDate="2021-04-11T01:30:29.490" Score="2" Body="&lt;p&gt;Consider the following statement:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{8}{5} - 1 = \frac{3}{5}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You could argue that the data types don't match: &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{5}$&lt;/span&gt; is an ordered pair; why would you be able to subtract an integer from it?&lt;/p&gt;&#xA;&lt;p&gt;The reason is that you can define the way subtraction works for these objects even though they are not the same data type. You need to prove that your operation is well-defined, but once you do so, you are fine.&lt;/p&gt;&#xA;&lt;p&gt;You can also cohesively define how subtraction works with a time and a time interval, so... do it. &lt;strong&gt;It's fine.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-04-11T01:30:29.490" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="20722" PostTypeId="2" ParentId="20715" CreationDate="2021-04-11T06:30:54.430" Score="5" Body="&lt;p&gt;I read the left-hand side as 11 minutes and 40 seconds minus 15 minutes. (Analogous to 11.4 - 15). So I expected the right-hand side to be negative.&lt;/p&gt;&#xA;&lt;p&gt;I would not write this, especially as a teacher.&lt;/p&gt;&#xA;&lt;p&gt;I would prefer &lt;span class=&quot;math-container&quot;&gt;$11:40-00:15=11:25$&lt;/span&gt; hours, with the understanding that the units refer to the leftmost grouping&lt;/p&gt;&#xA;" OwnerUserId="5884" LastActivityDate="2021-04-11T06:30:54.430" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20723" PostTypeId="2" ParentId="20715" CreationDate="2021-04-11T11:37:09.240" Score="5" Body="&lt;p&gt;I taught elementary school. I think the clearest way to write this for elementary school children is:&lt;/p&gt;&#xA;&lt;p&gt;11 hours 40 min - 15 min =  11 hours 25 min&lt;/p&gt;&#xA;&lt;p&gt;You could then conclude from the above that 11:25 is 15 minutes earlier than 11:40.&#xA;You could also regroup as necessary.&lt;/p&gt;&#xA;&lt;p&gt;11 hours 15 min - 40 min = ?&lt;/p&gt;&#xA;&lt;p&gt;This should be regrouped as 10 hours 75 min - 40 min = 10 hours 35 min.&#xA;We can now conclude that 40 minutes earlier than 11:15 is 10:35.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-04-11T11:37:09.240" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20724" PostTypeId="2" ParentId="20715" CreationDate="2021-04-11T17:22:39.297" Score="0" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It is as cromulent (thanks, MatthewDaly!) to write &lt;span class=&quot;math-container&quot;&gt;$$11.40\text{&#xA;   a.m.} \color{#00F}- 15\text{ min} = 11.25\text{ a.m.}$$&lt;/span&gt; as it is to&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/3500269/what-happens-when-a-vector-is-subtracted-from-a-given-point#comment7198833_3500283&quot;&gt;write&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$$(4,3) \color{#00F}-\begin{pmatrix}1&#xA;   \cr1\end{pmatrix}=(3,2),$$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$“\color{#00F}-”$&lt;/span&gt; here can be&#xA;thought of (is implicitly defined) as a variant subtraction&#xA;operation: one whose first argument is a time/Cartesian -point, second&#xA;argument is a duration/vector, and output is a time/Cartesian -point.&lt;/p&gt;&#xA;&lt;p&gt;When using the 24-hour clock, I write &lt;span class=&quot;math-container&quot;&gt;$$11\;40 \color{#00F}- 15\text{&#xA;   min} = 11\;25.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I avoid writing &lt;span class=&quot;math-container&quot;&gt;$$11:40 \color{#00F}- 15\text{ min} = 11:25$$&lt;/span&gt; though,&#xA;as it is ambiguous whether it is nearing noon or midnight.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-04-11T18:59:44.167" LastActivityDate="2021-04-11T18:59:44.167" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20725" PostTypeId="1" CreationDate="2021-04-11T17:36:45.457" Score="6" ViewCount="413" Body="&lt;p&gt;I want to start by sharing my teaching philosophy: I believe in teaching in a way that is primarily student-centered. That is, I prefer to do less talking to give the student(s) more time to practice working on math problems, which I believe is the best way for them to build conceptual understanding and procedural fluency. Conversely, the more modeling I do for the student(s), I feel that they risk tricking themselves into thinking they understand the mathematics, which is detrimental to their learning.&lt;/p&gt;&#xA;&lt;p&gt;Now on to the main point: I work for an online tutoring company that offers instant tutoring sessions where I meet with students &amp;quot;on the spot&amp;quot; (there is no ongoing relationship between me and the tutee). A lot of my tutoring experience has come from hosting these types of sessions, mainly because it fits better into my daily schedule and it's less time-consuming than applying for tutoring opportunities that are more competitive. During instant tutoring sessions, I try to stick to my teaching philosophy as described above, but sometimes doing so does not work out. Practically all students ask for Homework Help during an instant tutoring session. Most students begin by saying something along the lines of &amp;quot;How would you solve this problem?&amp;quot; (which to me is like saying &amp;quot;What is the answer to this question?&amp;quot;). My response is usually &amp;quot;Let me show you a similar example&amp;quot; and I will change the numbers around so I can model the steps that the student must take, or I will use the Socratic Method to determine what they are specifically having trouble understanding and ask probing questions to help them understand how to at least start the solution to the math problem.&lt;/p&gt;&#xA;&lt;p&gt;I've always viewed Homework Help as the lowest form of tutoring because from my experience, the student is usually focused on getting it done rather than learning. I still try to make these situations work because I feel guilty if I tell the student that I can't help them, and I don't want them to give me a bad rating either. Unfortunately, when I agree to help the student with their homework while sticking to my teaching philosophy (that is, making it clear that I won't just give them the answer), sometimes the student abruptly leaves the online session. Needless to say, it's frustrating to do the right thing from an educational standpoint and essentially get the door slammed in my face.&lt;/p&gt;&#xA;&lt;p&gt;(Side note: the Covid-19 pandemic may also have an affect on this behavior since students are tired of staring at a computer screen all day, but I am ignoring that here. I have observed this behavior even before the pandemic started.)&lt;/p&gt;&#xA;&lt;p&gt;Thankfully, it's not always this bad. During a recent instant tutoring session, one student asked &amp;quot;Can we work on this problem together?&amp;quot; (it was on graphing a rational function). I liked hearing this because the student understood that my responsibility as the tutor was to provide &lt;em&gt;guidance&lt;/em&gt;, not &lt;em&gt;answers&lt;/em&gt;. As the student worked through the problem, the student asked &lt;em&gt;good&lt;/em&gt; questions when he was unsure how to proceed, such as &amp;quot;Is the vertical asymptote where the function is undefined?&amp;quot;. This behavior is more desirable because the student was showing effort instead of trying to get me to say the answer. Unfortunately, not all students are like this one when it comes to instant tutoring.&lt;/p&gt;&#xA;&lt;p&gt;Recently, I began to realize that perhaps instant tutoring is meant to allow students to get a quick answer to their HW. I recently came across this &lt;a href=&quot;https://www.prnewswire.com/news-releases/varsity-tutors-announces-on-demand-tutoring-now-available-in-every-us-middle-and-high-school-subject-300702991.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; where it is described that a student's grade improved from (presumably) one instant tutoring session. To me, it sounds like the tutor is the one who earned the improved grade! I don't believe that as the tutor, I should be doing the student's work because it's &lt;em&gt;their&lt;/em&gt; responsibility. Otherwise, how else will they learn?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My question is:&lt;/strong&gt; What is the true purpose of instant tutoring? Is it meant to be a Q and A where the tutor just gives the answer? Or should I stick to my teaching philosophy during instant tutoring sessions and focus on improving how I interact with students?&lt;/p&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="13657" LastEditDate="2021-04-12T20:03:53.953" LastActivityDate="2021-04-14T18:09:04.383" Title="What is the purpose of &quot;instant&quot; tutoring?" Tags="|tutoring|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20726" PostTypeId="1" CreationDate="2021-04-11T22:05:02.613" Score="3" ViewCount="74" Body="&lt;p&gt;Say I have the problem: I roll a die three times and I am interested in the probability of ending up with two 1's.&lt;/p&gt;&#xA;&lt;p&gt;My impression is that a single roll is called a trial.&lt;/p&gt;&#xA;&lt;p&gt;What is the full 3-roll action called? I thought it was called an experiment, but I've seen experiment used synonymous with trial.&lt;/p&gt;&#xA;&lt;p&gt;Using the binomial distribution, we call getting a 1 here a &amp;quot;success&amp;quot; and getting anything else a &amp;quot;failure.&amp;quot; This might confuse students who think it's a &amp;quot;failure&amp;quot; to get three 1's. Is there an alternative/better way to say this acquiring of a 1 vs. not a 1?&lt;/p&gt;&#xA;" OwnerUserId="11880" LastEditorUserId="2083" LastEditDate="2021-04-12T05:30:01.970" LastActivityDate="2021-04-12T10:51:52.767" Title="simple statistics (binomial) terminology" Tags="|terminology|statistics|discrete-math|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20728" PostTypeId="2" ParentId="20726" CreationDate="2021-04-12T05:20:14.583" Score="1" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;You might point out that the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-trial experiment has &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; possible&#xA;outcomes and &lt;span class=&quot;math-container&quot;&gt;$256$&lt;/span&gt; possible events, which are subsets of the sample&#xA;space; whereas each trial has &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; possible outcomes, which can be&#xA;contextually characterised as a success and a failure.&lt;/p&gt;&#xA;&lt;p&gt;To distinguish between trial outcomes and experiment outcomes (or to&#xA;remind about their distinction), I sometimes call the former&#xA;“sub-outcomes” and the latter “elementary events”. When introducing&#xA;the terminology, I indicate that “sub-outcome” is a nonstandard word&#xA;(actually, I coined it).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To directly address your question:&lt;/p&gt;&#xA;&lt;p&gt;I would point out that the assignments ‘success’ &amp;amp; ‘failure’&#xA;naturally arise out of Bernoulli trials having exactly two outcomes&#xA;of fixed probabilities.&lt;/p&gt;&#xA;&lt;p&gt;In contrast, it is not so useful to call the&#xA;event &lt;span class=&quot;math-container&quot;&gt;$\{n11,1n1,11n\}$&lt;/span&gt; or its outcomes “successes”; moreover, we might be&#xA;interested in multiple events in the experiment, in which case the notion of a&#xA;“successful” event (or elementary event) will have to keep varying according to the event being considered.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-04-12T10:51:52.767" LastActivityDate="2021-04-12T10:51:52.767" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="20729" PostTypeId="1" CreationDate="2021-04-12T12:35:02.663" Score="7" ViewCount="4315" Body="&lt;p&gt;I am teaching 12-16 year olds.&lt;/p&gt;&#xA;&lt;p&gt;How should I write down the Pythagorean theorem equation?&lt;/p&gt;&#xA;&lt;p&gt;Some alternatives:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\text{leg}_1^2 + \text{leg}_2^2 = \text{hypotenuse}^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$b_1^2 + b_2^2 = h^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2 + y^2 = z^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$AB^2 + AC^2 = BC^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\text{(longest side length)}^2 = \text{(shortest side length)}^2 + \text{(remaining side length)}^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I know it is context dependent, but say I want to calculate the length of the diagonal in a rectangle, or say I just want to refer to the Pythagorean theorem in general.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&#xA;Based on the comments I would also like to suggest&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\text{(First leg)}^2 + \text{(Second leg)}^2 = \text{(Hypotenuse)}^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;2nd edit:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I assume that the students have already seen and worked with the statement of the Pythagorean theorem in some variation before.&lt;/p&gt;&#xA;" OwnerUserId="6527" LastEditorUserId="2083" LastEditDate="2021-04-13T10:46:00.173" LastActivityDate="2021-04-15T14:54:29.130" Title="What is the preferred way to denote the Pythagorean theorem equation?" Tags="|secondary-education|geometry|notation|middle-school|" AnswerCount="7" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="20730" PostTypeId="2" ParentId="20729" CreationDate="2021-04-12T14:04:14.490" Score="6" Body="&lt;p&gt;The only one of these that looks objectionable to me is the one that calls the hypotenuse &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, since in a triangle the letter &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; usually refers to the triangle's height (which could be either one of the legs but could not be the hypotenuse).&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-04-12T14:04:14.490" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20731" PostTypeId="2" ParentId="20729" CreationDate="2021-04-12T14:39:32.813" Score="19" Body="&lt;h1&gt;Common knowledge&lt;/h1&gt;&#xA;&lt;p&gt;The formula &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2 = c^2$&lt;/span&gt; is common knowledge and the words for hypotenuse and leg (is &amp;quot;cathetus&amp;quot; not used in English?) are basic mathematical vocabulary. Including these seems a good idea.&lt;/p&gt;&#xA;&lt;h1&gt;Connections to other mathematics&lt;/h1&gt;&#xA;&lt;p&gt;The notation with AB, CA and BC might be something the students have used or will use in less analytical geometry. Maybe you have an opportunity to mention that other context or tie these together, now or in a geometrical context.&lt;/p&gt;&#xA;&lt;h1&gt;Jargon&lt;/h1&gt;&#xA;&lt;p&gt;Using some formulation without too much jargon is recommended; all the variables might be as much nonsense for some pupils, so this speaks for including some formulation that uses more natural language. This would also give you the opportunity to discuss why we use letters as variables in place of words (note that this is commonly not done in programming, for example; mathematics is peculiar here and an explanation might be order).&lt;/p&gt;&#xA;&lt;p&gt;This also suggests avoiding needlessly difficult notation like subscripts, unless you feel that the students could use practice there and are ready for it, and would not have too much difficulty with Pythagoras. All extra cognitive load makes learning the main subject harder.&lt;/p&gt;&#xA;&lt;h1&gt;Conflicts in notation&lt;/h1&gt;&#xA;&lt;p&gt;As mentioned Chris in his answer, &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; already has a different meaning in the same context, so you might want to avoid this. Think backwards and forwards to see if there are other unfortunate notational conflicts.&lt;/p&gt;&#xA;&lt;h1&gt;Curriculum&lt;/h1&gt;&#xA;&lt;p&gt;What does the curriculum that binds you say, if anything? Or maybe some other relevant guidelines. They probably don't go into this much detail.&lt;/p&gt;&#xA;&lt;h1&gt;What does research say?&lt;/h1&gt;&#xA;&lt;p&gt;You might want to a quick literature search on for example Semantic scholar (or other academic search engine of your choice), search for something like &amp;quot;pythagorean theorem didactic&amp;quot;, pick the first few articles that are accessible and seem relevant, skim through them and check if they discuss the matter of notation or link to something that discusses it. It is very possible to not find relevant stuff right away, but worth a few minutes, at least.&lt;/p&gt;&#xA;&lt;p&gt;You can spend as much or as little time as you want on this, but at least skimming an article or two might also give other ideas on your teaching, so why not?&lt;/p&gt;&#xA;&lt;h1&gt;In the end...&lt;/h1&gt;&#xA;&lt;p&gt;You have to check how much time you have and what the most important concerns for your students in the situation you are in. You can't do everything.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-04-12T14:39:32.813" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="20732" PostTypeId="2" ParentId="20715" CreationDate="2021-04-12T15:30:42.770" Score="1" Body="&lt;p&gt;In statistics this type of data is called &lt;strong&gt;interval data&lt;/strong&gt; - data where differences (&amp;quot;intervals&amp;quot;) make sense but not necessarily other operations. This is the data-science way to say what Matthew Daly is saying in his answer.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Dates and times are a standard example used in statistics classes. For example, you can find the difference between two years:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;2021 - 1999 = 22 years.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This means that if you like, you can rearrange the equation to say things like&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;1999 + 22 years = 2021.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;2021 - 22 years = 1999.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;but similar expressions make no sense, for example&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;22 years - 2021 ??&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;1999 + 2021 ??&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Other common examples of interval data, like temperatures, have the same issue but the issues are hidden by a subtle trick in the way we speak. For example, you can find the difference in temperature between an object that is 75° and another that is 60°:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;75° - 60° = 15°&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But the &amp;quot;15°&amp;quot; is a different kind of thing; it is an &lt;strong&gt;interval&lt;/strong&gt;, not the temperature of an object.&lt;/p&gt;&#xA;&lt;p&gt;If I tell you to add this 15° (interval) to the 75° (a temperature), that works just fine.&lt;/p&gt;&#xA;&lt;p&gt;But if I instead tell you I have two objects, where one is 75° (a temperature) and the other is 60° (a temperature), and then ask you to find the &lt;em&gt;sum of their temperatures&lt;/em&gt;, you should feel some cognitive issue here -- can you really add two temperatures? It's never a useful or meaningful thing to do. You can't really add those two temperatures together in any meaningful way, just like you can't add &amp;quot;noon&amp;quot; plus &amp;quot;2 o'clock.&amp;quot; This is &lt;strong&gt;interval data&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As far as I know, times and dates are the best example of interval data where this issue manifests clearly and does not have any linguistic tricks to obscure the issue.&lt;/p&gt;&#xA;&lt;p&gt;So -- for interval data, you have two types of quantities: &amp;quot;measurements&amp;quot; and &amp;quot;intervals&amp;quot; (in Matthew Daly's answer he refers to these as two spaces). You can do the following things:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Measurement - Measurement = Interval&lt;/li&gt;&#xA;&lt;li&gt;Measurement &lt;span class=&quot;math-container&quot;&gt;$\pm$&lt;/span&gt; Interval = Measurement&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But other operations don't make sense.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-04-12T15:30:42.770" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20733" PostTypeId="2" ParentId="20725" CreationDate="2021-04-12T16:32:23.530" Score="2" Body="&lt;p&gt;Stick to your philosophy IMO. In fact, I would double down on it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I've always viewed Homework Help as the lowest form of tutoring&#xA;because from my experience, the student is usually focused on getting&#xA;it done rather than learning. I still try to make these situations&#xA;work because I feel guilty if I tell the student that I can't help&#xA;them, and I don't want them to give me a bad rating either.&#xA;Unfortunately, when I agree to help the student with their homework&#xA;while sticking to my teaching philosophy (that is, making it clear&#xA;that I won't just give them the answer), sometimes the student&#xA;abruptly leaves the online session. Needless to say, it's frustrating&#xA;to do the right thing from an educational standpoint and essentially&#xA;get the door slammed in my face.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If at the start of the lesson they are like, &amp;quot;Oh, I have to do my bloody maths homework... here it is I guess... how do I do it?&amp;quot; and then you explain it to them via a similar example, and then there is a back and forth game between you and them (either they &lt;em&gt;say&lt;/em&gt; they &amp;quot;get it&amp;quot; but really they don't, or they say they don't get it) and they become more and more impatient until they decide to &amp;quot;slam the door in your face&amp;quot;, then screw them! That's their loss. If the student repeatedly asks you to do their homework for them, then tell them, &amp;quot;no, I'm not a homework service&amp;quot;. It's not worth getting a bad reputation by not properly tutoring students. And you will get a bad reputation if you do students' homework for them because that's a form of cheating and everyone will know you as the &amp;quot;cheater tutor&amp;quot;. Do you really want that reputation, or do you want a strong reputation among parents of being a... helpful tutor? Everyone knows that doing someone's homework for them doesn't help them to learn anyway...&lt;/p&gt;&#xA;&lt;p&gt;Usually on these websites, the parents write a message before the first lesson starts. If they write, &amp;quot;Wants help with homework&amp;quot;, then IMO you should make crystal clear to the student's parent(s) &lt;em&gt;before the first lesson even starts&lt;/em&gt; that you're not a homework completion service.&lt;/p&gt;&#xA;&lt;p&gt;In terms of &amp;quot;how you should tutor&amp;quot;, there are many books on this, but firstly I think it is rarely useful to just hand the student the answer on the plate without any effort. This is mainly because there are correct and incorrect methods for arriving at the right answer. If you just tell them the right answer, they may well be using an incorrect method, and so will convince themselves in a sort of &amp;quot;post-rationalisation&amp;quot; that their &lt;em&gt;method&lt;/em&gt; is sound, which it might not have been. You need them to explain their &lt;em&gt;method&lt;/em&gt; to you.&lt;/p&gt;&#xA;&lt;p&gt;This ties in with my final point: I think as a general rule, you want the student to get to a point where they are capable of explaining the concept to you or someone else. Sometimes my students do this and when doing so their wording or English is poor, because they are putting all their energy into &lt;em&gt;focusing on expressing the newly-learned concept,&lt;/em&gt; but this is fine and all part of the learning process: they are not going to be able to explain concepts perfectly first time.&lt;/p&gt;&#xA;&lt;p&gt;So I would double down on your tutoring philosophy. If students in these &amp;quot;instant tutoring sessions&amp;quot; keep probing you to &lt;em&gt;just tell them the answers to questions&lt;/em&gt;, then all you can do is tell them and their parents that you didn't sign up to be a homework completion service, and explain to them that it won't benefit &lt;em&gt;anyone&lt;/em&gt; if you did their homework for them. In fact it is detrimental to you, the tutor, because it damages the purity of your reputation. It is also probably mentally and emotionally taxing having to aid someone else in cheating on their homework, and this taxation can weigh on your mind and ultimately waste your time outside of lessons. It damages the student by getting them into cheating habits. It also gives them a false sense of security that they can do the work when really they can't, which means they will do worse in exams than they expect to do.&lt;/p&gt;&#xA;&lt;p&gt;It's not worth it: don't do it.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2021-04-12T16:38:51.017" LastActivityDate="2021-04-12T16:38:51.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20735" PostTypeId="1" CreationDate="2021-04-12T19:55:48.070" Score="1" ViewCount="256" Body="&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/4099618/why-is-finite-group-theory-often-included-under-algebra&quot;&gt;[cross posted from mse]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;the class of all finite groups is not closed under produtcs - example: the product over all finite cyclic groups - thus it is not a &lt;a href=&quot;https://en.wikipedia.org/wiki/Variety_(universal_algebra)#Birkhoff%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;variety of algebras&lt;/a&gt;, ie, it's not axiomatizable by universal sentences.&lt;/p&gt;&#xA;&lt;p&gt;in fact more is true: it's not axiomatizable at all, &lt;a href=&quot;https://en.wikipedia.org/wiki/Compactness_theorem#Applications&quot; rel=&quot;nofollow noreferrer&quot;&gt;by a consequence of the compactness theorem for first order theories&lt;/a&gt;, namely, that a theory having models of arbitrarily large finite size - again, take the cyclic groups - must have an infinite model.&lt;/p&gt;&#xA;&lt;p&gt;proofs of theorems pertaining to finite but not to arbitrary groups have more of an 'arithmetical flavor' than a '&lt;a href=&quot;https://en.wikipedia.org/wiki/Abstract_nonsense&quot; rel=&quot;nofollow noreferrer&quot;&gt;general nonsensical&lt;/a&gt;' one, and it's not hard to envision the subject being developed inside first-order number theory.&lt;/p&gt;&#xA;&lt;p&gt;so what is it that makes the subject of finite groups 'algebraic'? why does it appear on, say, syllabi on undergrad and graduate algebra?&lt;/p&gt;&#xA;" OwnerUserId="15843" LastEditorUserId="15843" LastEditDate="2021-04-15T11:23:34.090" LastActivityDate="2021-04-15T11:23:34.090" Title="in what sense is the subject of finite group theory 'algebraic'?" Tags="|abstract-algebra|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="20736" PostTypeId="2" ParentId="20735" CreationDate="2021-04-12T20:42:25.777" Score="7" Body="&lt;p&gt;Typically, students learn about finite groups long before they think of &amp;quot;variety of algebras&amp;quot; or &amp;quot;compactness theorem&amp;quot;.  I suppose you also want to exclude fields from &amp;quot;algebra&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;I would say: topics for an algebra syllabus are chosen according to how useful they are; not according to how they fit into some general framework.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2021-04-12T20:42:25.777" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20737" PostTypeId="2" ParentId="20729" CreationDate="2021-04-12T21:17:08.087" Score="-2" Body="&lt;p&gt;I prefer to state it the traditional way, not involving multiplying a length by itself, but involving areas of those polygons known as squares.&lt;/p&gt;&#xA;&lt;p&gt;Draw two lines at right angles to the hypotenuse at its two endpoints to get two sides of the square other than the hypotenuse itself, and draw the fourth side, and there you have a square, whose area is the sum of the areas of two squares having the two legs as one side.&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2021-04-12T21:17:08.087" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20738" PostTypeId="2" ParentId="20725" CreationDate="2021-04-12T23:17:05.577" Score="4" Body="&lt;p&gt;Answering your question seems to require filling in a little context, since the notion of tutoring without an ongoing tutor-tutee relationship is pretty general. Clicking through to the PR piece that you linked to, it seems that&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;This is a for-profit company.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They're online.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They cater to middle-school and high-school students.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This was announced before COVID.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is the purpose of “instant” tutoring?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It looks like its purpose is to make money for the company. As to the reason for picking this model over one in which there is a continuing relationship, most likely that reason is efficiency and labor costs. The efficiency seems pretty clear. As far as labor costs, this model makes you, the tutor, an instantly replaceable widget. If you ask for a raise or get a better-paying opportunity, they don't have a problem replacing you.&lt;/p&gt;&#xA;&lt;p&gt;I would be more sympathetic to their model if this had been a response to the educational meltdown caused by COVID.&lt;/p&gt;&#xA;&lt;p&gt;Your educational philosophy seems sound. If you feel like you can make it fit within this system, then there is no reason to change it.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-04-12T23:17:05.577" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20739" PostTypeId="2" ParentId="20729" CreationDate="2021-04-13T06:41:31.687" Score="10" Body="&lt;p&gt;In Olympiad geometry, &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the so-called &lt;em&gt;standard notation&lt;/em&gt; for the sides of a triangle, so it makes sense to use it consistently when referring to  a triangle (in isolation). However, in any case the general principle is &lt;strong&gt;introducing all your notation&lt;/strong&gt;. Writing&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pythagoras' theorem states that &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2=c^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pythagoras' theorem states that &lt;span class=&quot;math-container&quot;&gt;$leg^2+leg^2=hyp^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is sloppy and should be avoided. Try to force yourself to take the high road and write&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For a right triangle with legs &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; and hypothenuse &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, Pythagoras' theorem states that &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2=c^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you are using the formula in a longer problem involving a rectangle, usually there is already notation that has been introduced earlier:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let a rectangle be given with length &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; and width &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; be its diagonal.&#xA;[...]&#xA;By Pythagoras' theorem, &lt;span class=&quot;math-container&quot;&gt;$l^2+w^2 = d^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4930" LastActivityDate="2021-04-13T06:41:31.687" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20742" PostTypeId="2" ParentId="20729" CreationDate="2021-04-13T13:30:45.067" Score="9" Body="&lt;p&gt;Two alternatives I have seen used (am not necessarily recommending them, but will list some pros and cons) which don't seem to have been mentioned yet.&lt;/p&gt;&#xA;&lt;h2&gt;Don't denote it algebraically at all! Draw a picture instead&lt;/h2&gt;&#xA;&lt;p&gt;For the lower end of the 12-16 age range, I've seen this work really well. You literally draw the squares sticking out from the triangle. Write the areas of the squares &lt;em&gt;inside&lt;/em&gt; the squares. The two smaller squares add together to make the big square.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/migwk.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/migwk.png&quot; alt=&quot;Drawing squares on the sides of a 3-4-5 triangle to find the missing leg&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Advantages&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Even students with weak algebra skills can excel at this topic, at least at the three most common questions for beginners (use two legs to find hypotenuse; use leg and hypotenuse to find remaining leg; use three sides to verify if triangle is right-angled).&lt;/li&gt;&#xA;&lt;li&gt;Kids seem to enjoy drawing it out! It makes the topic much less intimidating and emphasises the shape/geometry aspect rather than the potentially off-putting algebra.&lt;/li&gt;&#xA;&lt;li&gt;Two of the most common mistakes with Pythagoras are (a) mixing up +/- depending on whether we should be finding hypotenuse/remaining leg; (b) forgetting to square-root at the end. Even among very weak students, drawing a picture seems to almost completely eliminate error (a) - it's obvious which the two smaller squares are, which should add up to make the big one - and substantially reduces error (b).&lt;/li&gt;&#xA;&lt;li&gt;Conceptually, makes clear the symmetry of the Pythagorean theorem with respect to the triangle's legs, with neither such finickity notation as &lt;span class=&quot;math-container&quot;&gt;$\text{leg}_1^2 + \text{leg}_2^2$&lt;/span&gt;, nor the potential to mislead of &lt;span class=&quot;math-container&quot;&gt;$\text{leg}^2 + \text{leg}^2$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Reinforces to students the geometric meaning of squaring and square-rooting. This is pretty useful among students around 12 years old, who by this stage are usually conceptually familiar with addition, subtraction, multiplication and division, and who generally see squaring as a special case of multiplication, but who often struggle with &amp;quot;what does square-rooting mean?&amp;quot; Those who try to see it as a special case of division often make mistakes like treating square-rooting as equivalent to dividing by two. Doing lots of drawing of squares while working through exercise questions on Pythagoras is an excellent want to build familiarity with what's at this age a rather alien operation.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Drawbacks&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Miss, or rather postpone, a chance to practise algebra and show how different areas of maths (algebra/geometry) are interlinked.&lt;/li&gt;&#xA;&lt;li&gt;Not so good for more abstract rather than numerical questions, e.g. when the side lengths are given only as algebraic expressions or you're applying Pythagoras as part of a mathematical proof e.g. of the law of the cosines.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Clearly if you want students to deal with more advanced material later, you can't stick to non-algebraic approaches forever. But I think &amp;quot;just draw a picture&amp;quot; is at least worth considering as an introductory teaching strategy for younger or less confident groups, for the first year that they see it. A potential time to switch to an algebraic notation might be when trigonometric ratios are introduced, since by this stage you probably want to be applying algebra to triangle problems. I say &amp;quot;probably&amp;quot; rather than &amp;quot;certainly&amp;quot; because there are, in fact, advocates of using non-algebraic approaches even to sin, cos, tan, e.g. by using &amp;quot;formula triangles&amp;quot;. Again, this has pros and cons. But if you are going to label the sides of a right-angled triangle as the adjacent, opposite and hypotenuse, this gives you an alternative way to write the Pythagorean theorem.&lt;/p&gt;&#xA;&lt;h2&gt;Denote the two legs as the adjacent and opposite&lt;/h2&gt;&#xA;&lt;p&gt;For example, if you use three-letter abbreviations for the sides, you can put:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\text{opp}^2 + \text{adj}^2 = \text{hyp}^2 $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Advantages&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Consistent with how you teach trigonometry.&lt;/li&gt;&#xA;&lt;li&gt;Makes it unambiguously clear that the hypotenuse is the side that goes on its own side, which e.g. &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2$&lt;/span&gt; doesn't.&lt;/li&gt;&#xA;&lt;li&gt;A good starting point for teaching trigonometric identities. For example, dividing by &lt;span class=&quot;math-container&quot;&gt;$\text{hyp}^2$&lt;/span&gt; produces easily recognised fractions like &lt;span class=&quot;math-container&quot;&gt;$\frac{\text{opp}^2}{\text{hyp}^2}$&lt;/span&gt; and so we get &lt;span class=&quot;math-container&quot;&gt;$\sin^2 \theta + \cos^2 \theta = 1$&lt;/span&gt;. You could use a notation like &lt;span class=&quot;math-container&quot;&gt;$\frac {a^2}{c^2}$&lt;/span&gt; instead, but it doesn't tell the story quite so clearly.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Drawbacks&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In principle, the Pythagorean theorem doesn't distinguish between the two legs, and it shouldn't be necessary to label an angle of interest so that the opposite and adjacent can be distinguished. In practice, once trig is taught, many schools drill students to mark on their angle of interest and label the three sides as opp, adj, hyp as soon as they see a right-angled triangle, so this may not be a serious flaw. (Arguably it's harmful conceptually to hide the symmetry with respect to the legs, but I doubt students will be hampered by this in their final exams. If they learned Pythagoras by the &amp;quot;just draw it&amp;quot; method beforehand, they may already be well aware of this symmetry anyway.)&lt;/li&gt;&#xA;&lt;li&gt;Depending on the structure of your curriculum, trigonometric ratios might be taught several years after the Pythagorean theorem, so it's a long wait until this notation becomes available to you.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5640" LastEditorUserId="5640" LastEditDate="2021-04-13T15:42:48.463" LastActivityDate="2021-04-13T15:42:48.463" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="20743" PostTypeId="2" ParentId="20729" CreationDate="2021-04-14T09:04:38.487" Score="3" Body="&lt;p&gt;Whatever you choose, make sure to follow some basic rules:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Clearly state the preconditions and make sure they are understood (right triangle in your case).&lt;/li&gt;&#xA;&lt;li&gt;Clearly state the meaning of the symbols (e.g. which symbols stand for the sides adjacent to the right angle, and which for the third one).&lt;/li&gt;&#xA;&lt;li&gt;Use the symbols consistently (don't make e.g. the same symbol &amp;quot;leg&amp;quot; stand for two different sides with two different lengthes).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Being sloppy at these topics creates lots of problems later on (students just guessing when to apply some formula, students not understanding the notion of a variable, ...).&lt;/p&gt;&#xA;" OwnerUserId="15862" LastActivityDate="2021-04-14T09:04:38.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20744" PostTypeId="2" ParentId="20725" CreationDate="2021-04-14T17:15:37.067" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;My question is: What is the true purpose of instant tutoring? Is it&#xA;meant to be a Q and A where the tutor just gives the answer? Or should&#xA;I stick to my teaching philosophy during instant tutoring sessions and&#xA;focus on improving how I interact with students?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Let me pile on the pessimism here.&lt;/p&gt;&#xA;&lt;p&gt;For the company, instant tutoring is about making money, not learning. If you don't believe me, ask them what they measure and how those measures correspond to payments. Really! Send them an email and ask. I'd love to hear if they even &lt;em&gt;try&lt;/em&gt; to measure learning beyond subjective reviews along the lines of:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;With the tutor Cheating Charlie, all my homework got done super fast so I'm really happy. The teacher's off my back, my parents aren't nagging, and all those right answers made my grades go up! Five stars for Cheating Charlie!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;With the tutor Mastery Mandy, I really truly learned that 'multiply' does not always mean 'increase', and we had SO MUCH FUN with games, pictures, hands-on experiences, practical applications, etc. Boy do I understand and appreciate multiplication now! But my homework on rational number calculations didn't get done and my grade went down and my teacher and parents got angry with me and I can't get into my school's accelerated math program. Now I can't get into AP Calculus which makes it harder to get into university and I can't even apply for the scholarships I wanted, so Mastery Mandy probably ruined my life. She sucks. I'd give her 0 stars if I could. My parents have committed to pestering customer service until we get the refund and apology we deserve.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One of the central problems of education is that virtually &lt;strong&gt;everything&lt;/strong&gt; revolves around the incentive of grades - NOT learning, NOT work habits, NOT character development, NOT attitude, NOT long-term opportunities - and grades are not always an accurate or insightful measure of learning. They are, however, like other incentives, &lt;a href=&quot;https://fs.blog/2017/10/bias-incentives-reinforcement/&quot; rel=&quot;nofollow noreferrer&quot;&gt;vastly underestimated&lt;/a&gt; in their power to motivate.&lt;/p&gt;&#xA;&lt;p&gt;To most students, the purpose of instant tutoring (and a lot of other tutoring) is &lt;strong&gt;not&lt;/strong&gt; learning but to get answers on the page that will please other people in the short-run.&lt;/p&gt;&#xA;&lt;p&gt;As for the dishonesty and upcoming pain (bad habits, inability to enjoy or learn further math, likely terrible performance on an upcoming test), delusions are extremely widespread. 98% of chain smokers believe they'll quit within 5 years, but only 1% or 2% actually do. Some believe that chain smoking is actually a justifiable net positive, &amp;quot;...at least for now.&amp;quot; Likewise, 98% of students believe they'll successfully deal with those other problems down the road, but virtually none will because math is cumulative, habits are hard to break, and most importantly, the incentives do not change.&lt;/p&gt;&#xA;&lt;p&gt;This may all seem utterly terrible.&lt;/p&gt;&#xA;&lt;p&gt;That's because it is.&lt;/p&gt;&#xA;&lt;p&gt;But there is hope.&lt;/p&gt;&#xA;&lt;p&gt;My advice: Stop taking students who only want help with an imminent deadline. Only work with students who do not have imminent deadlines (typically younger students, but not always) and give them delightful experiences with rigorous math. Then, your reputation and record will actually correspond, you will do far more good for the students, you will be able to teach according to your philosophy, and word will spread so quickly that you won't need to give a cut of revenue to a matching agency for finding you students. Furthermore, if you give students experiences of mastery and deep understanding from a young age, they are more likely to seek that as they go through subsequent math courses with all their perverse incentives.&lt;/p&gt;&#xA;&lt;p&gt;I spent 8 years agonizing over the kind of pain you expressed in the original post. I've tried &lt;em&gt;everything&lt;/em&gt; to move the needle from &amp;quot;grade grabbing&amp;quot; to &amp;quot;real learning&amp;quot;. Let me assure you, with rising university admission standards, the needle has gone exclusively in the wrong direction. All my time and effort was wasted because I underestimated the power of incentives. If anything, my students (and their teachers, parents, counsellors, school administrators, etc.) tended to demonstrate the &amp;quot;backfire effect&amp;quot;, i.e. their beliefs that they should prioritize &amp;quot;grade grabbing&amp;quot; over real learning got &lt;em&gt;stronger&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A few years ago, I took the path described in my advice above, and I only wish I had taken that path from Day 1. I would have saved myself a lot of headache, earned more, and done more good.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2021-04-14T18:09:04.383" LastActivityDate="2021-04-14T18:09:04.383" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20746" PostTypeId="1" CreationDate="2021-04-15T07:11:21.417" Score="7" ViewCount="3562" Body="&lt;p&gt;A math student may write very long and detailed answers, just because he or she does not know what to look for, for example in Geometry proofs.&lt;/p&gt;&#xA;&lt;p&gt;Or - a student may just write an arbitrary step without explaining.&lt;/p&gt;&#xA;&lt;p&gt;Overall, it can become tedious and even frustrating to grade math homework or exams.&lt;/p&gt;&#xA;&lt;p&gt;Do you share the same frustration, and how do you handle it in the best possible way?&lt;/p&gt;&#xA;" OwnerUserId="7960" LastActivityDate="2022-07-04T05:02:39.243" Title="How do you handle the frustration of having to GRADE student exams / homework?" Tags="|mathematical-pedagogy|exams|grading|" AnswerCount="7" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20747" PostTypeId="2" ParentId="11813" CreationDate="2021-04-15T08:54:06.687" Score="0" Body="&lt;p&gt;That's a good but very tough question. It seems to me that by only explaining &lt;em&gt;what&lt;/em&gt; the student is required to do, or by correctly explaining what and incorrectly explaining &lt;em&gt;why&lt;/em&gt; it is that way, the student is stimulated to figure out for himself or herself the 'why' bit.&lt;/p&gt;&#xA;&lt;p&gt;Spike Milligan's father told him lies quite a bit when he was a child, and he turned out a genius who was very good at figuring things out for himself.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;You cannot teach a man anything, you can only help him find it within himself.&amp;quot; -'Galileo&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;If you teach a man anything, he will never learn.&amp;quot; ― George Bernard Shaw&lt;/p&gt;&#xA;&lt;p&gt;I suspect that a very narrow view of understanding is being taken when you say that someone is able to solve an equation correctly but doesn't understand what he or she is doing. As I see it, some sort of understanding is needed, otherwise the answer will not be correct, though it might not be the type of understanding we think is desirable.&lt;/p&gt;&#xA;&lt;p&gt;Maybe the teacher should make the student figure out the meaning of what he or she is doing by putting challenging problems to the student that can't be solved without the supposedly superior/desired type of understanding.&lt;/p&gt;&#xA;" OwnerUserId="15662" LastEditorUserId="15662" LastEditDate="2021-04-15T09:02:44.023" LastActivityDate="2021-04-15T09:02:44.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20748" PostTypeId="2" ParentId="20746" CreationDate="2021-04-15T10:03:42.723" Score="2" Body="&lt;p&gt;I grit my teeth and do it. Skimming long answers for the relevant parts is a skill.&lt;/p&gt;&#xA;&lt;p&gt;One way of handling the frustration is talking about it with someone. If that someone is also grading the same exams or homework, or was/is involved in teaching the course, this can even be genuinely useful.&lt;/p&gt;&#xA;&lt;p&gt;But really, on the scale of frustrations, this is not a big deal. At least to me.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-04-15T10:03:42.723" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20749" PostTypeId="2" ParentId="20746" CreationDate="2021-04-15T14:52:48.713" Score="3" Body="&lt;p&gt;&lt;em&gt;Overall, it can become tedious and even frustrating to grade math homework or exams. Do you share the same frustration, and &lt;strong&gt;how do you handle it in the best possible way&lt;/strong&gt;?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some random ideas I have heard others use:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Choose a small subset of proof questions to grade in detail. You're right that proofs can be long to write and difficult to parse for the grader. Maybe just grade the &amp;quot;most important&amp;quot; one from a homework set and then post solutions to the others after they are turned in.&lt;/li&gt;&#xA;&lt;li&gt;Develop (early) a set of expectations that include the things you want to see and the things you won't tolerate. [e.g. &amp;quot;Making any claim, such as &amp;quot;angle &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$25^{\circ}$&lt;/span&gt;, without supplying written evidence or justification, will mean a deduction in points&amp;quot;] Then use a short-hand for marking these various things so you're not writing a ton to point out the same kinds of issues on every paper. This can help with your grading time, and it may reduce the number of occurrences of these things in your students' written work.&lt;/li&gt;&#xA;&lt;li&gt;Do something else while you're grading exams. I knew someone who said they graded one student's proof, then did 5 push-ups, graded another student's proof, then did 5 sit-ups, etc. [This was during the pandemic lock-down when they weren't getting enough exercise, and apparently this kept them motivated to get the grading done.]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="470" LastActivityDate="2021-04-15T14:52:48.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20750" PostTypeId="2" ParentId="20729" CreationDate="2021-04-15T14:54:29.130" Score="3" Body="&lt;p&gt;I really wish people would &lt;em&gt;stop&lt;/em&gt; teaching the Pythagorean Theorem as &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2$&lt;/span&gt;, for the following reason:  Give your students the diagram below, and ask them to solve for &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.  At least 1/3 of a typical high school class will write &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2$&lt;/span&gt; and report back to you that &lt;span class=&quot;math-container&quot;&gt;$c = 5$&lt;/span&gt;.  The problem is that the equation &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2$&lt;/span&gt; is so memorable, so ingrained in them, that automaticity kicks in and overrides any consideration about what the letters actually are supposed to refer to.  I much prefer a semantic description such as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(\textrm{first leg})^2 + (\textrm{second leg})^2 = (\textrm{hypotenuse})^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kspKv.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kspKv.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-04-15T14:54:29.130" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20751" PostTypeId="2" ParentId="20746" CreationDate="2021-04-15T15:40:11.450" Score="5" Body="&lt;p&gt;I actually get excited to grade my students exams. I see the test/exams as the whole point of the class: the chance for the student to demonstrate what they know or don't know about certain problems. It's like the chance to get inside of my students head and see &amp;quot;oh, that's what they were thinking.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I also feel that grading is a responsibility that teachers owe to their students, ideally as soon as possible. I once heard a high school teacher say &amp;quot;I am giving them the unit 5 test today...but I haven't even started grading their unit 4 test.&amp;quot; Can you imagine being a student in that class? Being held accountable for new information when you don't even know how you did on the older information?&lt;/p&gt;&#xA;&lt;p&gt;Two things help me look forward to grading:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I started to put a rather difficult question the students had not seen before as the last question of a test. This was always worth 7-10 points, so I explained that it was the &amp;quot;A student question.&amp;quot; If you could mimic the formulas and regurgitate some memorized steps, you could get a B. But if you wanted an A, you'll need to earn points on the last question. This made me excited to see what solutions the students came up with, especially if it was different than mine. That gave me something to look forward to when grading.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I started to collect data on &lt;em&gt;every&lt;/em&gt; test question. I think most teachers unofficially get a feel for how the whole class did after grading a test or perhaps they might even keep track of everyone's final grades so they could report the class average and median. But I took this a step further and collected data on every student for every question. Then I could keep track of what the hardest questions were and which students were getting them right (or if the whole class missed what I deemed an 'easy' question!). If all I did was keep track of the final grades, then seeing the class average of 75% didn't feel like it meant anything to me. But if I see that on problem #4, the average score was 3 out of 8 points, then I can go back and find ways to re-iterate this concept. So I saw grading not as &amp;quot;what scores should I put in the computer&amp;quot; but more as &amp;quot;what didn't I teach so well and need to go back over.&amp;quot; Adopting this mindset made it easier to begin grading a large stack of papers because it was ultimately about me doing my job better, not just attaching numbers next to student names.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I haven't done this myself, but I have seen teachers that do this:&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;You could not hand write specific feed back in detail on each test (maybe just underline or circle where the student lost points). Then write up a document that shows what the most common mistakes were on each problem. That way, you're correcting everyone's mistakes in one document instead of over and over again multiple times on different tests. If the students aren't sure why they lost points or if they lost points for reasons that aren't on the &amp;quot;common mistakes document,&amp;quot; then you can meet with them individually to discuss it.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2021-04-15T15:40:11.450" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20752" PostTypeId="2" ParentId="20746" CreationDate="2021-04-15T15:55:26.893" Score="13" Body="&lt;p&gt;I have found that, for myself, implementing standards based grading has eliminated this frustration entirely.  I now find grading to be enjoyable.&lt;/p&gt;&#xA;&lt;p&gt;I have a collection of standards for my course like&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Evaluate the truth value of an expression such as &lt;span class=&quot;math-container&quot;&gt;$(T \Leftrightarrow F) \implies (F \wedge T)$&lt;/span&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Give a Fitch style proof outline for a theorem such as &lt;span class=&quot;math-container&quot;&gt;$\forall x \exists y (A(x,y) \wedge (\neg B(x)))$&lt;/span&gt; &amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Prove simple statements about parity and divisibility such as 'The sum of any two odd numbers is even'.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The students are first given &amp;quot;rehearsal sets&amp;quot;.  These are like &amp;quot;homework&amp;quot;.  They try their best, submit their attempt to me by a certain date, then gain access to the solutions, make corrections, and resubmit.  Grading these is basically for completion and takes no time at all.&lt;/p&gt;&#xA;&lt;p&gt;They are then given a &amp;quot;performance set&amp;quot; for the standard.  These are collections of problems which I believe demonstrate mastery of the standard.  They are grouped into &amp;quot;attempts&amp;quot;.  I might have 15 problems grouped into 5 attempts of 3 problems each.  So each student will submit the same 3 problems as their first attempt.  If they demonstrate mastery, then they are done.  If not, they should get help, read my feedback, and then attempt the second set.  They continue this until they either give up on the standard, or master it.&lt;/p&gt;&#xA;&lt;p&gt;The grade in the course is 15% homework, 10% &amp;quot;honest attempts&amp;quot; at mastering each standards, and the remaining 75% of the grade is derived from mastering standards.  There is no such thing as partial credit:  either I believe you really understand what you are doing, or I do not, on each standard.&lt;/p&gt;&#xA;&lt;p&gt;I learned that a lot of my frustration with grading stemmed from the feeling that it wasn't fair, and that my feedback didn't matter.  If I deduct 3 points on a problem, and write a paragraph explaining why there was a problem, the student has little incentive to correct the issue:  maybe this kind of question will not come up again.  The whole course is a guessing game: what kind of question will the instructor ask?&lt;/p&gt;&#xA;&lt;p&gt;SBG eliminates all of that.  I lay out my expectations in extraordinary detail.  There is no guessing what I want the student to know:  it is all extremely explicit.  My feedback is not wasted.  When I give feedback, I learn about the present state of knowledge of the student, and the student learns what they need to fix on their next attempt.  The student has to persist in learning the content until I deem their understanding suitable.  There is no &amp;quot;good enough&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-04-15T15:55:26.893" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="20754" PostTypeId="2" ParentId="20746" CreationDate="2021-04-16T14:26:11.347" Score="2" Body="&lt;p&gt;Grading may be tedious.  But overly long answers can be averted.  A student may write a lot, hoping that the mostly irrelevant text will contain whatever the teacher is looking for. But in doing so, the student demonstrates a lack of knowledge – and won't get full credit. No matter what truths are hidden in the sea of irrelevant wrapping. Students write this way only because teachers let them.&lt;/p&gt;&#xA;&lt;p&gt;Telling the students that you grade this way, should help even more. &amp;quot;If you are unsure, don't write down every irrelevant fact you know of; this will be punished.&amp;quot;  The point of an exam is to have the student demonstrate their knowledge. Including lots of irrelevant stuff, means they don't know what is relevant for the question. Such an answer gets a bad grade; even if it also includes a correct answer to the question.&lt;/p&gt;&#xA;" OwnerUserId="15883" LastActivityDate="2021-04-16T14:26:11.347" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20755" PostTypeId="1" CreationDate="2021-04-16T14:30:41.423" Score="-1" ViewCount="174" Body="&lt;p&gt;What research has been done on the effects of requiring students to learn to count and do some easy arithmetic in an alternative number base, for example binary, base four, base six, base eight, base twelve, base sixteen, or base twenty?&lt;/p&gt;&#xA;" OwnerUserId="15662" LastEditorUserId="15662" LastEditDate="2021-04-19T02:02:45.767" LastActivityDate="2021-04-19T02:02:45.767" Title="What research has been done on the effects of requiring students to learn to count in an alternative number base such a binary or base eight?" Tags="|undergraduate-education|secondary-education|primary-education|" AnswerCount="2" CommentCount="7" ClosedDate="2021-04-18T18:48:48.583" ContentLicense="CC BY-SA 4.0" />
  <row Id="20756" PostTypeId="2" ParentId="20755" CreationDate="2021-04-16T20:03:31.247" Score="4" Body="&lt;p&gt;I can't answer the OP's questions, but I'll just mention that&#xA;a local 6th-grade teacher (in the U.S.) has a successful unit on base-&lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;.&#xA;It is mentioned in the recent article below. Sometimes he called it &amp;quot;star-fish math.&amp;quot;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;James Henle.&#xA;&amp;quot;Math for Grades 1 to 5 Should Be Art.&amp;quot;&#xA;&lt;em&gt;Mathematical Intelligencer&lt;/em&gt;. &lt;strong&gt;42&lt;/strong&gt;, pages 64–69, Dec. 2020.&#xA;&lt;a href=&quot;https://link.springer.com/journal/283/volumes-and-issues/42-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer link&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;And then there is numeration. Kids can invent a method of naming numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[This teacher] used to tell his sixth-graders every year the tale of the mysterious mathematician Professor Étoile, who invented a new way of notating numbers using the letters &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt;. His method was lost, [the teacher] explains, when he fell off a boat and was eaten by an octopus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[The teacher] then divided the class into groups. Each group was tasked with creating its own system of numeration using &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt;. After a week, they presented their methods in a mini-math conference. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[He] did this every year with great success. His students came up with many creative systems.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course &lt;em&gt;étoile&lt;/em&gt; is French for: star!&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-04-17T00:35:49.047" LastActivityDate="2021-04-17T00:35:49.047" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20757" PostTypeId="2" ParentId="20755" CreationDate="2021-04-16T20:17:08.913" Score="3" Body="&lt;p&gt;I'd say yes, and I'd go with binary if you had to do any one alternative base simply because it's so relevant to computing and technology, and in my experience teaching discrete math, once you understand binary, related bases like octal and hex are pretty simple to pick up. But I don't think the converse is necessarily true.&lt;/p&gt;&#xA;&lt;p&gt;Ideally I'd like math and CS students to learn base 2, 8, and 16, and then learn how base 10 notation is related. From there students can do whatever they want; I sometimes teach student base 36 and use 0-9 plus A-Z. After they learn the general concept of base-&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; representation, it's all the same.&lt;/p&gt;&#xA;&lt;p&gt;Using some familiar everyday objects to represent bits can be fun for teaching binary. For example, asking students how they might build a binary clock like the one below can generate some good ideas:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MViVi.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MViVi.jpg&quot; alt=&quot;Binary clock&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Or, ask students how they would do this if all they had was a single flag that they could either hold up or put down. Or my personal favorite, counting to 31 using only the fingers of one hand (literally, &amp;quot;digits&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;For hexadecimals, &lt;a href=&quot;https://www.youtube.com/watch?v=k-GH3mbvUro&quot; rel=&quot;nofollow noreferrer&quot;&gt;this scene from &lt;em&gt;The Martian&lt;/em&gt;&lt;/a&gt; is really good.&lt;/p&gt;&#xA;&lt;p&gt;Finally, should language departments teach this? Heck no! Why should they get all the fun?&lt;/p&gt;&#xA;" OwnerUserId="942" LastActivityDate="2021-04-16T20:17:08.913" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20758" PostTypeId="1" CreationDate="2021-04-17T07:19:58.530" Score="1" ViewCount="218" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hello, I am an undergraduate in CS. I would like to study Graph Theory on my own (self-study) for a competitive examination (named GATE). It is an examination for undergraduates and as such, I would like to concentrate on the topics which are there normally in a standard UG graph theory course. The &lt;a href=&quot;https://drive.google.com/file/d/15m1czm6LIqRhskbqLPNvcfqY0ot_8ZdT/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;syllabus&lt;/a&gt; for the examination.&lt;/p&gt;&#xA;&lt;p&gt;The pattern is as follows, 65 questions, time 3 hours.&#xA;So for each question, I can afford at most 2.76 mins, which is quite a constrained amount of time, considering the level of the questions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://pyq.ravindrababuravula.com/subjects/&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; is to give an idea about the level of the questions&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;As per the given syllabus, the exam asks quite involving questions on Graph Theory, with questions based on properties of the graph after applying certain algorithms, tree formed by traversals, or coloring, or other important but at times unseen (by the candidates) properties or concepts. Mostly they tend to ask questions from unseen topics, which probably one has not handled previously unless he is quite lucky. In such cases, they definite the topics and then asks about them.&lt;/p&gt;&#xA;&lt;p&gt;Usually, questions are like select the most appropriate options or marking the true statements.&lt;/p&gt;&#xA;&lt;p&gt;Now it is quite difficult (for me at least) to understand in 2.76 mins the new concept (defined in the question), read the statements, find counterexamples for them to prove them wrong. At worse for some statements, I might be unable to find the counterexample in that 2.76 mins, but a more difficult counterexample might exist.&lt;/p&gt;&#xA;&lt;p&gt;One can have a look at the type of question in graph theory &lt;a href=&quot;https://pyq.ravindrababuravula.com/topic/?tp=Graph-Theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://pyq.ravindrababuravula.com/subject/?cs=Algorithms&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://pyq.ravindrababuravula.com/subject/?cs=Data-Structures&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Situation/Request&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have gone through &lt;em&gt;CLRS&lt;/em&gt; and I am acquainted with the terminologies of graph theory algorithms and properties of trees produced by traversals, etc. But the thing is that all these I have learned, by going through the text many times and I have been quite slow at grasping those concepts given in &lt;em&gt;CLRS&lt;/em&gt;. I have even gone through &lt;em&gt;Kenneth Rosen's Discrete Mathematics&lt;/em&gt; texts.&lt;/p&gt;&#xA;&lt;p&gt;[I have solved most of the exercise questions of CLRS (except for the &lt;span class=&quot;math-container&quot;&gt;$\star$&lt;/span&gt; questions) mostly on my own but only for once, long ago, while reading the text for the first time. Some questions even took a single day to think (too slow indeed). Rosen has a huge no of questions at the end of each text, that too I have solved not all but selectively as I found few problems repetitive there, and that too I have done only once. But I have read the text many more times (that too selectively) so that I do not forget the things which I have learned...]&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Could anyone recommend me some good text that would help me to build &lt;em&gt;intuition&lt;/em&gt; behind graph theory properties or statements unknown to me. A text which does not explain the concept with rigorous mathematical proofs. [I tried to read Douglas West's book, but it was so rigorous, that after completing the proof, one has to go through it once again to see, what he was actually proving]. I would like to read a text which is easy and explains the concepts using intuition and examples and logical reasoning, rather than going into rigorous mathematical proofs, assuming dangerous notational conventions, with 100s of greek symbols, and then proving something, which only experts can probably understand. I hope can express myself clearly.&#xA;A text with many examples/illustrations of how every theorem which it states could be applied to problems. Probably one that also includes properties of trees and graphs, after some algorithms are applied to it, like, questions about articulation point in a DFS tree, properties of cross-edges, back-edges, etc.&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The texts which I have already read are classic and fine, but I find it difficult at times to answer questions in the GATE, (might be due to the short time or might be that I am not that sharp) but still I want to sharpen my skills in dealing with these problems of graph theory. Even the weightage of this subject is quite high every year.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="14166" LastActivityDate="2021-04-20T19:56:27.047" Title="An intuitive (non rigorous) text book on graph theory which is student friendly with vivid illustrations" Tags="|undergraduate-education|reference-request|textbooks|self-learning|graph-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20759" PostTypeId="1" AcceptedAnswerId="20771" CreationDate="2021-04-17T14:27:55.627" Score="13" ViewCount="2629" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Milton_Gregory#The_Seven_Laws_of_Teaching&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;  shows that in 1886 John Milton Gregory outlined his &amp;quot;The Seven Laws of Teaching&amp;quot;;  asserting  that a teacher should:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Know thoroughly and familiarly the lesson you wish to teach; or, in other words, teach from a full mind and a clear understanding.&lt;/li&gt;&#xA;&lt;li&gt;Gain and keep the attention and interest of the pupils upon the lesson. Refuse to teach without attention.&lt;/li&gt;&#xA;&lt;li&gt;Use words understood by both teacher and pupil in the same sense—language clear and vivid alike to both.&lt;/li&gt;&#xA;&lt;li&gt;Begin with what is already well known to the pupil in the lesson or upon the subject, and proceed to the unknown by single, easy, and natural steps, letting the known explain the unknown.&lt;/li&gt;&#xA;&lt;li&gt;Use the pupil's own mind, exciting his self-activities. keep his thoughts as much as possible ahead of your expression, making him a discoverer of truth.&lt;/li&gt;&#xA;&lt;li&gt;Require the pupil to reproduce in thought the lesson he is learning—thinking it out in its parts, proofs, connections, and applications til he can express it in his own language.&lt;/li&gt;&#xA;&lt;li&gt;Review, review, REVIEW, reproducing correctly the old, deepening its impression with new thought, correcting false views, and completing the true.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question is in today's on-line high stakes testing environment, how many of his &amp;quot;Laws of Teaching&amp;quot; are still relevant; in particular, to the teaching of mathematics at the K-12 grade school level. &lt;a href=&quot;https://en.m.wikipedia.org/wiki/How_People_Learn&quot; rel=&quot;noreferrer&quot;&gt;(a modern reference on How People Learn)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="216" LastEditorUserId="216" LastEditDate="2021-04-19T02:47:26.527" LastActivityDate="2021-04-19T03:46:51.027" Title="How many of &quot;The Seven Laws of Teaching&quot; are still relevant for teaching maths today?" Tags="|secondary-education|mathematical-pedagogy|primary-education|online-instruction|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="20760" PostTypeId="2" ParentId="20758" CreationDate="2021-04-17T14:42:14.317" Score="1" Body="&lt;p&gt;That book gets ripped pretty hard on Amazon.  The Dover texts by Trudeau and Chartrand are supposedly easier and friendlier, per reviews.  And will be cheap, since Dover.&lt;/p&gt;&#xA;&lt;p&gt;If you want to develop familiarity and speed, I would certainly not eschew (i.e. I would do) problems that are repetitive.  You'll get more practiced at the concept.  Also more practiced at speed itself.  You are a sharp fellow and this is an optimization problem.  But if you're struggling with speed/familiarity/recall, I think more problems is a better answer than multiple rereadings (that is unusual...I find that I master a text, than can use it as a desk reference...very easily since it is &amp;quot;mastered&amp;quot;...if I really need super speed/skill again, it's best developed by selective drill...but that is a refresher from having done all the problems long ago).&lt;/p&gt;&#xA;&lt;p&gt;Here is a &amp;quot;for dummies&amp;quot; article:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.freecodecamp.org/news/i-dont-understand-graph-theory-1c96572a1401/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.freecodecamp.org/news/i-dont-understand-graph-theory-1c96572a1401/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, I realize you are pretty sharp and knowledgeable given the work on the West book so far.  Still, maybe this helps to get back to more pictorial intuition.  It also has a bit more of a CS/industry slant, so perhaps that motivates you to remember concepts.  Humans are not silicon-based math logic systems, we are overtrained hunter/gatherers.  We remember things better with &amp;quot;import&amp;quot; and with multiple frames of reference, e.g. applied and theory both, pictorial and algebraic, etc.&lt;/p&gt;&#xA;&lt;p&gt;P.s. I'm not a math theory graph expert or student at all--just little bit in practical scheduling/PM work.  Essentially I just Googled for stuff that might help you.  And applied some critical thinking, looking at textbook prefaces and the like.  I know Amazon reviews are not perfect, but suggest looking at them.  Also at prefaces and even skimming texts themselves before investing in deep effort.  Of course at some point you have to jump in.  But be wary of single point advice on texts.  Especially on SE and from professors there can be a tendency to advise the books that are hardest versus most pedagogically effective.  Or to ignore the questioner's stipulations, like telling someone who is a weak student to self study Rudin.  (I sort of did too in your case...because the links were &amp;quot;hard&amp;quot; for me, but at least I knew I did..and I did click them at least...kudos for the detail by the way!) Also, for some reason Reddit is often more reasonable, helpful.&lt;/p&gt;&#xA;" OwnerUserId="15891" LastActivityDate="2021-04-17T14:42:14.317" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20761" PostTypeId="2" ParentId="20759" CreationDate="2021-04-17T14:49:45.053" Score="12" Body="&lt;p&gt;They are basic, friendly pieces of pedagogical advice.  Most pre-college teaching is very much STILL in this mold.&lt;/p&gt;&#xA;&lt;p&gt;Where we fall down is in high-end universities and graduate schools, where pedagogy is less emphasized in the paradoxical belief that harder material should be learned with worse training methods.  Or that smart students don't need/benefit from gradual training methods.  In contrast, for high stakes professional training (like jet pilots and NFL football players) very careful attention is paid to using efficient methods of pedagogy.&lt;/p&gt;&#xA;&lt;p&gt;I remember irking the hell out of the dad of a shipmate of mine (he was a CSU-level chem prof) when I told him that what was great about the Naval Academy was that it was &amp;quot;taught like high school&amp;quot;.  But I meant it.  And I saw how it was way more efficient than what a relative got at a top named school.  (Probably more expensive for the school to deliver though.)&lt;/p&gt;&#xA;" OwnerUserId="15891" LastEditorUserId="15891" LastEditDate="2021-04-17T14:58:37.393" LastActivityDate="2021-04-17T14:58:37.393" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20762" PostTypeId="2" ParentId="20746" CreationDate="2021-04-17T15:28:02.283" Score="3" Body="&lt;p&gt;Here's a short grab-bag of ideas; things I've found useful.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Design the test for gradeability&lt;/strong&gt;. As the instructor, you have the power (hopefully) to design a test that you look forward to grading. Some of us get locked into legacy traditions from examples we've seen, or things mentors dictated. We don't need to necessarily follow those same habits if they're depressing to us. Think about what you can manage to reasonably grade, that makes for a quality assessment, and take responsibility for something you're eager to grade and get results from.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Use a reasonable point scale.&lt;/strong&gt; Joel David Hamkins suggested that one use a &lt;a href=&quot;https://matheducators.stackexchange.com/a/14412/5563&quot;&gt;5-point qualitative scale&lt;/a&gt; to grade any mathematical question. I started using that and it made my process much easier. For some reason I can't understand, a lot of people get stuck on a 10-point scale for questions of any type. That's unnecessarily granular and you wind up wasting time debating over &amp;quot;6 points or 7?&amp;quot;. Consider also the magical number &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Magical_Number_Seven,_Plus_or_Minus_Two&quot; rel=&quot;nofollow noreferrer&quot;&gt;7±2&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Minimize writing feedback.&lt;/strong&gt; From recollection, studies show that written feedback on graded tests is mostly ignored by students. It seems really hard to avoid (I still have not really succeeded to date), but of course the written feedback is the most time-consuming part, and arguably should be avoided. Provide a common solution sheet and refer students to that. (I do that, but still to date can't avoid giving individual feedback. Maybe someday.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Consider digital testing.&lt;/strong&gt; This is really an adjunct to the item above. I've learned in the last year, re: feedback, that I can save a lot of time with an online test, and a text FAQ/common errors file to the side, from which I copy-and-paste common feedback observations. At least for some of my classes I'm going to continue having students, in the classroom under my proctoring, log on to the computer system and take digital tests in the future, partly for this purpose.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(In addition, I use &lt;em&gt;some&lt;/em&gt; multiple-choice questions on definitions and terms, and fill-in-the-blank numerical calculations, which can be digitally auto-graded. Forbid the thought of an &lt;em&gt;all&lt;/em&gt;-multiple-choice test, though.)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="6053" LastEditDate="2022-07-04T05:02:39.243" LastActivityDate="2022-07-04T05:02:39.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20763" PostTypeId="1" AcceptedAnswerId="20768" CreationDate="2021-04-17T15:29:20.330" Score="13" ViewCount="567" Body="&lt;p&gt;&lt;strong&gt;Case:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Exam Problem given to student at university: Give a problem/context illustrating the operation &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{5} + \frac{2}{3}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Answer by student: Anna and Beatrice buy flowers for grandpa for his birthday. Anna buys &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; flowers , &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; of which are daffodils. Beatrice buys &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; flowers, &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; of which are daffodils. What fraction of the flowers are daffodils?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Remarks:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The student seems to confuse the operation &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b} + \frac{c}{d}$&lt;/span&gt; with the some other operation I am not sure how to denote, but my attempt is: &lt;span class=&quot;math-container&quot;&gt;$(a,b) \star (c,d) = \frac{a+c}{b+d}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The student seems to think that &amp;quot;4 out of 5&amp;quot; could be related to &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{5}$&lt;/span&gt;. However, in this case there actually is a difference between &amp;quot;6 out of 8&amp;quot; and &amp;quot;3 out of 4&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am not sure that I am asking the right question, but I observed this case in real life and I would like to analyze it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;What property of fractions or addition of fractions could they be misunderstanding, and how would you explain to the student where they have gone wrong so that they don't repeat this in the future?&lt;/p&gt;&#xA;" OwnerUserId="6527" LastEditorUserId="6527" LastEditDate="2021-04-17T15:39:38.033" LastActivityDate="2021-04-18T16:48:09.930" Title="Analyzing an answer to the following problem: Give meaning to $\frac{4}{5} + \frac{2}{3}$" Tags="|students-mistakes|arithmetic-operations|fractions|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20765" PostTypeId="2" ParentId="20763" CreationDate="2021-04-17T16:06:18.813" Score="8" Body="&lt;p&gt;The word you are looking for is &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mediant_(mathematics)&quot; rel=&quot;noreferrer&quot;&gt;mediant&lt;/a&gt;&lt;/em&gt;.&#xA;The mediant of two fractions &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{c}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{b}{d}$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\frac{a+b}{c+d}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;According to Wikipedia,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is sometimes called the &lt;strong&gt;freshman sum&lt;/strong&gt;, as it is a common mistake in the early stages of learning about addition of fractions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Teachers often do this in grading papers.  For example, a test has two parts: the first has 10 items, the second has 20 items.  The student got 8 items in the first part correctly (8/10), and 12 items in the second part correctly (12/20).  The &amp;quot;total score&amp;quot; of the student is (8+12)/(10+20) = 20/30.  Note that the &amp;quot;total score&amp;quot; is &lt;em&gt;not&lt;/em&gt; 2/3.&lt;/p&gt;&#xA;&lt;p&gt;I haven't tried this yet, but perhaps instead of just &amp;quot;explaining to the student where they have gone wrong,&amp;quot; you might also want to show the student that this operation (mediant) is sometimes the correct thing to do (as in my example).&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2021-04-17T16:13:30.930" LastActivityDate="2021-04-17T16:13:30.930" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="20766" PostTypeId="2" ParentId="20763" CreationDate="2021-04-17T16:23:30.573" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;What &lt;strong&gt;property of fractions or addition of fractions&lt;/strong&gt; could they be&#xA;misunderstanding, and how would you explain to the student where they&#xA;have gone wrong so that they don't repeat this in the future?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[Emphasis added.]&lt;/p&gt;&#xA;&lt;h1&gt;Short Answer&lt;/h1&gt;&#xA;&lt;p&gt;Fractions are numbers and they behave like numbers when we do operations on them. Many students never learn this.&lt;/p&gt;&#xA;&lt;p&gt;You (and the student) may be focusing on things like common denominators and equivalent fractions and why we add numerators but not denominators and then simplifying at the and maybe converting to a mixed or whole number and...&lt;/p&gt;&#xA;&lt;p&gt;These are all procedures and rules. Nothing in there emphasizes that a fraction is a number.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps you could ask your student to draw similar pictures based on similar context for the following two equations. See if they can determine answers by inspection. Tell them that you can.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$13 - 2 =$&lt;/span&gt; _____&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$13 - \frac{1}{6} = $&lt;/span&gt; _____&lt;/p&gt;&#xA;&lt;p&gt;If they're struggling to think of a context, tell them to start with 13 loaves of bread.&lt;/p&gt;&#xA;&lt;p&gt;This has a good chance of revealing if they think of fractions as numbers.&lt;/p&gt;&#xA;&lt;p&gt;Generally speaking, don't do &lt;em&gt;any exercise ever&lt;/em&gt; involving two fractions until the student demonstrates understanding by doing that exercise with one fraction (new knowledge) and one whole number (prior knowledge).&lt;/p&gt;&#xA;&lt;h1&gt;Long Answers&lt;/h1&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/15284/7151&quot;&gt;Mine&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.berkeley.edu/%7Ewu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Best answers from Hung Hsi-Wu&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2021-04-17T16:23:30.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20767" PostTypeId="2" ParentId="20746" CreationDate="2021-04-17T16:29:38.717" Score="3" Body="&lt;p&gt;In addition to all the other GREAT advice, I would just advise to do it ASAP and do it fast (quick and dirty).  And give yourself a beer afterwards.  Students really appreciate the prompt feedback.  It is HUGE to them.  Tests are big events.  And they want the info ASAP.  I would always do it same day and post the results (clear names for students in the military, code names for...ugh..snowflake civilians).  In addition to the benefit to the students, you don't have that awful hanging weight on your soul.  Just done.&lt;/p&gt;&#xA;&lt;p&gt;Um.  And don't grade homeworks (assign it, but don't collect/grade it).  And don't assign projects.&lt;/p&gt;&#xA;&lt;p&gt;The other thing that helps you (and your kids) is regularity.  Do tests on the same day (ever FRI for HS, every second FRI for college).  It seems so arbitrary, but routine is helpful for managing the animals, to include yourself.  Plus it makes that FRI beer taste even better.&lt;/p&gt;&#xA;&lt;p&gt;Don't do (much) written feedback on test papers.  And develop an acronym set--you can publish it.  Things like SF-1 (sig figs minus 1).  Or NWNC (no work, no credit).  Or ACF (answered carried forwards).  Learned that at nuke school in the military.  The kids will learn it and get a little fun out of it.  Like that badge of experience from Mr. Hardoof's class.&lt;/p&gt;&#xA;&lt;p&gt;Take the test prior to giving it.  Then take it again during the exam, while proctoring the students.  You will be fast and it will be easy.  But it gives some empathy, some brothers in arms, with the participants.  And one of the repetitions can be used as your published solution.&lt;/p&gt;&#xA;" OwnerDisplayName="user15892" LastEditorDisplayName="user15892" LastEditDate="2021-04-17T16:40:20.793" LastActivityDate="2021-04-17T16:40:20.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20768" PostTypeId="2" ParentId="20763" CreationDate="2021-04-17T19:11:01.427" Score="9" Body="&lt;p&gt;The student who designed this problem wasn't thinking about the different wholes.&lt;/p&gt;&#xA;&lt;p&gt;IN your students problem, there are 3 different wholes.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Anna's flowers - The whole is 5 flowers and &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{5}$&lt;/span&gt; are daffodils&lt;/li&gt;&#xA;&lt;li&gt;Beatrice's flowers The whole is 3 flowers and &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{3}$&lt;/span&gt; are daffodils&lt;/li&gt;&#xA;&lt;li&gt;The flowers of Anna and Beatrice combined. The whold is 8 flowers and &lt;span class=&quot;math-container&quot;&gt;$\frac{6}{8}$&lt;/span&gt; are daffodils.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;When we take a fraction we have to be aware of the whole. You can't compare fractions of different wholes.  This is why &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; of an apple is less than &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; of a watermelon even though: &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2} = \frac{1}{2}$&lt;/span&gt;.  Furthermore, you can't add the two halves (&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}+\frac{1}{2}=1$&lt;/span&gt;) because they are halves of different things. If we have &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; an apple and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; a watermelon, we do not have 1 whole fruit.&lt;/p&gt;&#xA;&lt;p&gt;Your student has shown a lack of awareness of the importance of the whole, by adding the fractions that are fractions of different wholes and by not specifying the whole in the question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What fraction of the flowers are daffodils?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To correctly give a context for adding two fractions the 2 fractions must be of the same whole.&#xA;.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2021-04-18T05:22:34.560" LastActivityDate="2021-04-18T05:22:34.560" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20771" PostTypeId="2" ParentId="20759" CreationDate="2021-04-17T22:14:16.537" Score="14" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;OP: &amp;quot;Refuse to teach without attention.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In my role as chair, I attended an instructor's class where he &lt;em&gt;really&lt;/em&gt;&#xA;refused to advance until he was certain the students were all with him, via detailed verbal feedback. The students responded, stopped the presentations and asked questions.&lt;/p&gt;&#xA;&lt;p&gt;I've changed my own teaching as a result of watching how this can work.&lt;/p&gt;&#xA;&lt;p&gt;In response to questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The topic was confidence intervals via bootstrap sampling.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$37$&lt;/span&gt; students. &lt;span class=&quot;math-container&quot;&gt;$75$&lt;/span&gt;-minute class.&lt;/li&gt;&#xA;&lt;li&gt;He kept asking &amp;quot;Does that make sense?&amp;quot; He asked many simple questions,&#xA;almost like call &amp;amp; response in jazz. For more complicated questions, he&#xA;formed in-place groups for them to discuss and report out.&lt;/li&gt;&#xA;&lt;li&gt;I didn't notice any student totally zoned-out. But my presence may have altered the atmosphere a bit.&lt;/li&gt;&#xA;&lt;li&gt;The magic was not so much his techniques in the observed class. Rather, it was the rapport he earlier established so that the students felt&#xA;safe in acknowledging they didn't get it, to assume they all &lt;em&gt;can&lt;/em&gt; get it, to ask without embarrassment for an explanation to be repeated.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-04-18T13:14:04.150" LastActivityDate="2021-04-18T13:14:04.150" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="20772" PostTypeId="2" ParentId="20763" CreationDate="2021-04-18T09:46:24.447" Score="4" Body="&lt;ul&gt;&#xA;&lt;li&gt;The student was conflating the &lt;strong&gt;sum&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$$\frac45+\frac23$$&lt;/span&gt; and the&#xA;&lt;strong&gt;weighted average&lt;/strong&gt; (where the weights account for group-size differences)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left(\frac{\color{#00F}5}{\color{#180}{5+3}}\right)\frac45+\left(\frac{\color{#00F}3}{\color{#180}{5+3}}\right)\frac23\\=\frac{4+2}{5+3}$$&lt;/span&gt;&#xA;of the two given fractions/rates.&lt;/li&gt;&#xA;&lt;li&gt;Clearly, the &lt;em&gt;sum&lt;/em&gt; of two positive fractions is necessarily greater&#xA;than each of them, whereas their &lt;em&gt;weighted average&lt;/em&gt; lies somewhere&#xA;between them.&lt;/li&gt;&#xA;&lt;li&gt;The student's misinterpretation (“What fraction of the flowers are&#xA;daffodils?”) is akin to claiming that the overall performance on an&#xA;exam comprising several papers and in which every mark is worth the same value equals the &lt;span class=&quot;math-container&quot;&gt;$\color{#C00}{\text{sum}}$&lt;/span&gt;&#xA;of the individual scaled scores.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{60+80}{100+100}=\frac12\left(\frac{60}{100}+\frac{80}{100}\right)\\\color{#C00}=\frac{60}{100}+\frac{80}{100}.$$&lt;/span&gt; In this example, the student would've been likelier to catch the discrepancy, since the averaging and summing are easier to contrast, since the given fractions share a common denominator.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-04-18T16:48:09.930" LastActivityDate="2021-04-18T16:48:09.930" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20773" PostTypeId="1" AcceptedAnswerId="20777" CreationDate="2021-04-18T22:04:20.810" Score="12" ViewCount="9179" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The students in my country are supposed to be able to work with and answer questions about functions at the age of around 15. This is asserted in the standard mathematics curriculum for middle school students.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I think the definition of a function is extremely abstract and technical. Of course, I understand its importance in higher mathematics, but we are not only preparing kids to become mathematicians. Additionally, the definition of a function is rather cumbersome, and I have trouble stating it clearly at any level.&lt;/p&gt;&#xA;&lt;p&gt;I question why we put so much emphasis on the concept of a &amp;quot;function&amp;quot; while we could be broadening our scope a bit to all relations between variables. As an example, the equation of a circle does not fall directly under the &amp;quot;function&amp;quot; category. Why does the term &amp;quot;function&amp;quot; even need to be taught to most children?&lt;/p&gt;&#xA;&lt;p&gt;Most of the text books only deal with linear functions, quadratic functions, and 1/x.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Why should or shouldn't we teach functions to 15 year olds?&lt;/p&gt;&#xA;" OwnerUserId="6527" LastActivityDate="2021-05-08T08:43:17.593" Title="Why should or shouldn't we teach functions to 15 year olds?" Tags="|secondary-education|functions|definitions|" AnswerCount="7" CommentCount="25" ContentLicense="CC BY-SA 4.0" />
  <row Id="20774" PostTypeId="2" ParentId="20773" CreationDate="2021-04-19T01:12:50.313" Score="27" Body="&lt;p&gt;I think &amp;quot;function&amp;quot; is one of those notions that can be presented in different ways to people at different ages and who have different levels of ambition in math. This is similar to the notion of a &amp;quot;set,&amp;quot; which I was taught in school ca. age 5 or 6 in an age-appropriate way, but then learned about at a different level in college.&lt;/p&gt;&#xA;&lt;p&gt;At a basic level, there is not much to say about functions that is very difficult or complicated:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A function takes an input and gives an output.&lt;/li&gt;&#xA;&lt;li&gt;We usually write the function to the left of its input: &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;. But sometimes we use other types of notation, such as &lt;span class=&quot;math-container&quot;&gt;$|x|$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Strictly speaking, a function is supposed to have one well-defined output, which makes it like a function on a calculator. By this definition, the square root is not a function, but we can make it into a function by specifying the positive root, or we can relax our notion of &amp;quot;function&amp;quot; to include multi-valued functions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;When students get to the point where they're drawing graphs of functions, they can be told that rule 3 is the vertical line test. Notions like inverting a function, composition of functions, one-to-one functions, and so on do not have to be introduced early on. I don't see the point of introducing non-obvious terminology at this stage, such as domain, range, and so on. I would not take a typical class of 15 year olds and start teaching them how to notate the composition of functions, nor would I belabor the fact that a function cannot always be written down as a formula.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-04-19T01:12:50.313" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="20776" PostTypeId="2" ParentId="20759" CreationDate="2021-04-19T03:46:51.027" Score="1" Body="&lt;p&gt;Good list.&lt;/p&gt;&#xA;&lt;p&gt;One of the above comments is true. In the modern classroom, if you wait to get everyone’s attention, you won’t be able to teach anything. That is especially true in today’s math classrooms. Many students are not motivated to learn. Math is objective and cumulative. It is easy to see where someone is on the spectrum of math understanding. However, with social promotion, many students move on (are moved on) with D- grades, and the teacher is blamed for their lack of motivation and understanding. Picture 11th graders coming into a Trig class on Day One and declaring to the teacher that they don’t do fractions.&lt;/p&gt;&#xA;&lt;p&gt;We should have two math options for high school graduation: Applied Math (calculators and spreadsheets), and Theoretical Math (axioms and proofs, development of the math systems of algebra, geometry, trig, calculus, stats, etc.). The first group would go straight into the work force or attend community college; the 2nd to 4-year colleges and universities.&lt;/p&gt;&#xA;&lt;p&gt;Oh, and in elementary school, teach the basic arithmetic facts: addition, subtraction, multiplication, and division of natural numbers, integers, and rational numbers (fractions, decimals, and per cents). Worksheet based. You must get this before you move on. Don’t fear the Drill and Kill method of teaching elementary facts.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Tom&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10517" LastActivityDate="2021-04-19T03:46:51.027" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20777" PostTypeId="2" ParentId="20773" CreationDate="2021-04-19T04:06:26.413" Score="37" Body="&lt;p&gt;In the U.S. Common Core standards, functions are supposed to be introduced in the 8th grade, i.e., around age 13-14. So arguably age 15 is a year or two behind where they ought to be.&lt;/p&gt;&#xA;&lt;p&gt;The standard for the 8th grade says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Understand that a function is a rule that assigns to each input&#xA;exactly one output.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So honestly that really doesn't seem like a hugely complicated idea to introduce.&lt;/p&gt;&#xA;&lt;p&gt;Consider that things called &amp;quot;functions&amp;quot; will, at this point or very soon, be all around the students. They are intrinsic to:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mathematics&lt;/li&gt;&#xA;&lt;li&gt;Computer programming&lt;/li&gt;&#xA;&lt;li&gt;Standard office software like Excel&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Therefore it seems highly reasonable to give them the general concept at this time or earlier.&lt;/p&gt;&#xA;&lt;p&gt;If motivation for why it's important is required, then you should provide that. As a first stab I can usually say something like: It's easier to deal with a process when we know it has just one result, rather than possibly many. (In the back of my head I'm thinking about things like 1-to-1 vs. 1-to-many relationships in a database, and the very different ways we have to handle that.)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-04-20T12:59:33.213" LastActivityDate="2021-04-20T12:59:33.213" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="20778" PostTypeId="2" ParentId="20773" CreationDate="2021-04-19T13:41:53.033" Score="16" Body="&lt;p&gt;Functions are far broader and more applicable than you give them credit for.  Consider the following:&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;Country or state&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;Capital&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;Elevation (in meters)&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Bolivia&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Sucre&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;2783&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Ecuador&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Quito&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;2763&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Colombia&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Bogata&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;2619&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Eritrea&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Asmara&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;2363&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Ethiopia&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Addis Ababa&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;2362&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Mexico&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Ciudad de Mexico&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;2216&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;New Mexico&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Santa Fe&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;2152&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Wyoming&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Cheyenne&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1856&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Colorado&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Denver&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1613&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Nevada&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Carson City&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1462&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Utah&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Salt Lake City&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1308&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Montana&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Helena&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;1262&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;Your students would be able to determine the elevation of Quito or the capital of Wyoming by interpreting that table far earlier than 15, and they're evaluating functions in order to do it.  The only other thing you need to do is show them an example of where a table fails to be useful.&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;January&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;31&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;February&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;28&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;February&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;29&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;March&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;31&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;April&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;30&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;Again, much younger students would be able to describe why that table is deficient.  So a function is a &amp;quot;math thing&amp;quot; that unambiguously associates exactly one output to any input.  From there, you go on to talking about how we typically use it in mathematics, to evaluate an expression with one or more variables.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="11" LastEditDate="2021-05-06T15:49:23.283" LastActivityDate="2021-05-06T15:49:23.283" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="20779" PostTypeId="1" AcceptedAnswerId="20784" CreationDate="2021-04-19T16:51:51.927" Score="7" ViewCount="487" Body="&lt;p&gt;Let us assume that I want to graph any of the functions below.&lt;/p&gt;&#xA;&lt;p&gt;A) A can of soda costs &lt;span class=&quot;math-container&quot;&gt;$\$1$&lt;/span&gt;. Draw a graph depicting the total cost as a function of the number of cans you buy.&lt;/p&gt;&#xA;&lt;p&gt;Comment: One cannot actually buy &lt;span class=&quot;math-container&quot;&gt;$4.6$&lt;/span&gt; cans of soda.&lt;/p&gt;&#xA;&lt;p&gt;B) Ellen is a member at a gym. She pays &lt;span class=&quot;math-container&quot;&gt;$\$30$&lt;/span&gt; at the start of every month and she paid an initial fee of &lt;span class=&quot;math-container&quot;&gt;$\$10$&lt;/span&gt; for a membership card. Draw a graph depicting the amount of money Ellen has paid after &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; months.&lt;/p&gt;&#xA;&lt;p&gt;Comment: The graph is in my opinion something like a step-function if interpreted as a function on the real numbers.&lt;/p&gt;&#xA;&lt;p&gt;C) You bring a large cake into the classroom, intending to share it with the other students. Draw a graph depicting the amount of cake each student gets.&lt;/p&gt;&#xA;&lt;p&gt;Comment: One cannot have &lt;span class=&quot;math-container&quot;&gt;$7.3$&lt;/span&gt; students.&lt;/p&gt;&#xA;&lt;p&gt;It is in my opinion rather hard to say exactly what the domains are in the examples above. However, I want to give some justification in the case that I should &amp;quot;approximate&amp;quot; the functions as being &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \to \mathbb{R}$&lt;/span&gt;. Should I avoid giving word problems like these? Should I avoid mentioning this issue if no student points it out? Should I rather ask them to make a table and not to attempt to graph the functions at all?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;When working with 12-16 year olds, how should I depict graphs of functions that are technically not &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \to \mathbb{R}$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="6527" LastEditorUserId="6527" LastEditDate="2021-04-19T16:58:25.530" LastActivityDate="2021-04-19T21:20:46.543" Title="When working with 12-16 year olds, how should I graph functions when the domain technically isn't $\mathbb{R}$?" Tags="|secondary-education|functions|middle-school|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20780" PostTypeId="1" CreationDate="2021-04-19T17:05:00.713" Score="3" ViewCount="249" Body="&lt;p&gt;I am going to teach this probability and statistics course in a couple of weeks. The probability part can be made very interesting, in my opinion, easily. But I am a little worried that I might make the statistics part a boring collection of information and algorithmic manipulations. So my question is, how do I make the statistics part of the course interesting?&lt;/p&gt;&#xA;&lt;p&gt;The syllabus for the stat part is roughly the contents of this &lt;a href=&quot;https://open.umn.edu/opentextbooks/textbooks/459&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5453" LastEditorUserId="238" LastEditDate="2021-04-20T17:38:12.040" LastActivityDate="2021-04-24T01:19:02.230" Title="How to make an introductory course on Statistics interesting" Tags="|undergraduate-education|statistics|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="20781" PostTypeId="2" ParentId="20773" CreationDate="2021-04-19T17:19:40.560" Score="9" Body="&lt;p&gt;Some abstraction is perfect for children. Mitsumasa Anno was a Japanese writer of children's books and he had many innovative approaches to introduction functions. This is from Anno's Math Games II. You can even see the introduction of function inverse.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QcDlR.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QcDlR.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-04-19T17:19:40.560" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="20782" PostTypeId="2" ParentId="20779" CreationDate="2021-04-19T20:54:59.357" Score="1" Body="&lt;p&gt;I would do points, rather than a connected curve for A.  Except if you are looking at large amounts, I wouldn't be so fussy and just draw a line.  This actually has some use if you consider say different rates sold by a monopolist practicing price descrimination, for instance.  (I think this is beyond your audience though.)&lt;/p&gt;&#xA;&lt;p&gt;For B, if you are interested in total spend, a connected stepwise curve is good because it emphasizes the time aspect (i.e. you've still spent $40 halfway through the first month.  If you are looking at the individual spend points, I quite like the &amp;quot;arrow&amp;quot; graphic that is used in engineering economics for project spend points (important for eventual discounting to NPV).&lt;/p&gt;&#xA;&lt;p&gt;See here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=arrow%20graphic%20discounted%20cash%20flow&amp;amp;tbm=isch&amp;amp;tbs=rimg:CSyBEnHYY2UPYUhr0s4VTtuo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=arrow%20graphic%20discounted%20cash%20flow&amp;amp;tbm=isch&amp;amp;tbs=rimg:CSyBEnHYY2UPYUhr0s4VTtuo&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But again, this is really a bit beyond your audience.  Really, I would not emphasize that stuff so much.  Even if you are doing practical problems in commercial arithmetic, you can just solve the problems with linear (or whatever) approximations and then go in afterwards and either pick a set point one higher or lower (based on the logic of the situation).  E.g. Number of men to dig a ditch by a deadline.  If the answer comes out 7.3, than &amp;quot;8&amp;quot; is the right response.  In other words, I would worry about other stuff more. But anyhow...here's a response to the question.&lt;/p&gt;&#xA;" OwnerUserId="15904" LastActivityDate="2021-04-19T20:54:59.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20783" PostTypeId="2" ParentId="20780" CreationDate="2021-04-19T21:13:28.720" Score="1" Body="&lt;p&gt;That's pretty normal to have a lot of ground to cover.  Even with the more friendly books, it still ends up being a lot of concepts and formulas.  Given this, I think you sort of have to make your peace with the idea that kids will not master everything, especially in the long term. I probably wouldn't try some fundamental change to improve things since it may make things worse (like adding a big project or the like).  Just engage with the kids, be animated and sort of cheerlead through the thing.  Maybe tell a sea story or two.&lt;/p&gt;&#xA;&lt;p&gt;Whereas I have an iconic recall of the quadratic formula, I don't think I've ever used a chi test in a class or business/engineering situation.  And I don't remember the formula.  Or what is different about it versus a t test.  But at least I sorta &amp;quot;heard of it&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In terms of basic lasting concepts, I think they should have lasting knowledge that normal distribution is a &amp;quot;bell curve&amp;quot; and is a decent guess when you have no clue.  Along with the very basic concepts of confidence intervals (not every tricky caveat or logic issue but just the idea of 90% CI).&lt;/p&gt;&#xA;&lt;p&gt;After that, just the info that there are a lot of concepts, formulas, and things can be tricky.  So if someone quotes a statistic know that it may have issues.  And if you are working on something statistical, bring in a professional/watch out for errors.  [But again, I know &amp;quot;rigor people&amp;quot; hate this, but I think you have to set the bar lower at &amp;quot;exposure&amp;quot; rather than &amp;quot;mastery&amp;quot;.  Pound smart, not penny foolish.]&lt;/p&gt;&#xA;" OwnerUserId="15904" LastActivityDate="2021-04-19T21:13:28.720" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20784" PostTypeId="2" ParentId="20779" CreationDate="2021-04-19T21:20:46.543" Score="9" Body="&lt;p&gt;In all of these examples, I would draw a line unless that idealization led to missing something essential in the problem. These idealizations happen all the time, and they tend to be very useful. E.g., for the flow of water, we normally don't need to consider that water consists of discrete molecules. Ditto for examples like immigration, fluctuations in the population of a certain species of flower, or photons hitting a solar cell.&lt;/p&gt;&#xA;&lt;p&gt;On a philosophical level, number systems like the rational numbers, real numbers, and hyperreal numbers are all just ways of modeling reality. The model never corresponds perfectly with reality. Nobody has ever measured a number in real life that could be definitively classified as irrational or irrational, for example.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-04-19T21:20:46.543" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20785" PostTypeId="2" ParentId="1760" CreationDate="2021-04-20T15:00:54.490" Score="0" Body="&lt;p&gt;There's a recent book by Nicholas A. Scoville: &lt;em&gt;Discrete Morse Theory&lt;/em&gt;, AMS Student Mathematical Library, 2019.&lt;/p&gt;&#xA;&lt;p&gt;Publisher's page: &lt;a href=&quot;https://bookstore.ams.org/stml-90&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://bookstore.ams.org/stml-90&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;MAA review: &lt;a href=&quot;https://www.maa.org/press/maa-reviews/discrete-morse-theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/maa-reviews/discrete-morse-theory&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It looks very accessible to the undergraduate - some background in proof writing is recommended as is some linear algebra, but the book is largely self-contained. It's full of diagrams and examples. Could be a good place for engineering and computer science students, among others, to get a first look at simplicial complexes, Betti numbers, persistent homology and related topics in a discrete Morse theory setting. The relative concreteness of the examples helps keep the wall of abstraction from looming up too high.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2021-04-20T15:00:54.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20786" PostTypeId="2" ParentId="20773" CreationDate="2021-04-20T20:13:04.953" Score="7" Body="&lt;p&gt;If 10-11 year olds can learn programming (which some have done even before Scratch was a thing), then it's hardly a leap at all to suggest that 15 year olds can learn functions since they are a common element of programming languages.&lt;/p&gt;&#xA;&lt;p&gt;Every student is going to be different and some are going to be better at math than others, but I think this notion that there is some predefined linear order of teaching math is a little misguided. Even the most intimidating concepts, such as quaternions, can be taught with the right approach, something that surprises me every time with e.g. 3Blue1Brown's videos.&lt;/p&gt;&#xA;&lt;p&gt;Just because any given subject in math has all sorts of crazy depth to it and endless connections to other mathematical concepts, there is still a ton of value in just the surface level concepts. You don't need to teach in-depth category theory to teach functions when a basic understanding of domain and range is enough to solve the problems at hand.&lt;/p&gt;&#xA;" OwnerUserId="15559" LastActivityDate="2021-04-20T20:13:04.953" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20787" PostTypeId="1" CreationDate="2021-04-20T20:53:21.517" Score="1" ViewCount="183" Body="&lt;p&gt;I'm teaching a physics class that has a year of calculus as a prerequisite, but as so often happens, many of my students have forgotten a lot of the much more basic math from earlier classes. In particular, many don't seem to know how to manipulate logs and exponentials or how to do things like changing &lt;span class=&quot;math-container&quot;&gt;$2^x$&lt;/span&gt; to a base-e exponential.&lt;/p&gt;&#xA;&lt;p&gt;Question: Can anyone recommend a 3-4 page brief summary and review of these topics? I'm looking for something that my students can access for free online.&lt;/p&gt;&#xA;&lt;p&gt;I found this, which seemed too brief for my purposes: &lt;a href=&quot;https://wou.edu/mathcenter/files/2015/09/Exponents-and-Logarithms.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://wou.edu/mathcenter/files/2015/09/Exponents-and-Logarithms.pdf&lt;/a&gt; For example, it doesn't show what the graphs look like, it doesn't seem to explicitly show how to change the base of an exponential, and it doesn't make contact with calculus.&lt;/p&gt;&#xA;&lt;p&gt;OpenStax College Algebra has a 100-page chapter that is much too lengthy and contains too much cruft for my purposes: &lt;a href=&quot;https://assets.openstax.org/oscms-prodcms/media/documents/CollegeAlgebra-OP.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://assets.openstax.org/oscms-prodcms/media/documents/CollegeAlgebra-OP.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The Wikipedia articles &amp;quot;Exponential&amp;quot; and &amp;quot;Exponentiation&amp;quot; have too little of the basic material and wander off into topics that I see as not very relevant, such as combinatorics.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-04-20T21:13:47.387" LastActivityDate="2021-04-21T15:19:27.203" Title="Resource request: 3-4 page review of exponentials and logs" Tags="|resource-request|exponential|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20791" PostTypeId="2" ParentId="20773" CreationDate="2021-04-22T05:37:03.207" Score="3" Body="&lt;p&gt;We shouldn't need to teach functions to 15 year olds, because ideally they should have already learned programming since primary school, including mathematical and general functions and inverse functions. Programming, including demos, games and robotics, is the best motivator to learn math in my opinion.&lt;/p&gt;&#xA;" OwnerUserId="3287" LastEditorUserId="3287" LastEditDate="2021-05-03T02:03:38.130" LastActivityDate="2021-05-03T02:03:38.130" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20792" PostTypeId="1" CreationDate="2021-04-22T10:45:32.040" Score="2" ViewCount="436" Body="&lt;p&gt;I've taken Calculus 1 and it's time to relearn because I've forgotten some of it. But it's been a couple months since I've done any solid mathematics. I was hoping for a book that would include comprehensive &amp;amp; rigorous text on precalculus, probability &amp;amp; combinatorics, and analysis. I am aware it's too much to ask for a single book and that supplementary material is available through other books, but my problem is that these modern books aren't rigorous enough in their text. They oversimplify everything and bloat the book with unnecessary figures explaining basic things, and their countless excercises also provide no real value as all they require is rote memorization of the given techniques.&lt;/p&gt;&#xA;&lt;p&gt;My geometry &amp;amp; trig has also gotten a bit rusty over the months.&lt;/p&gt;&#xA;&lt;p&gt;Where should I begin? What book(s) should I go with? Is there a list?&lt;/p&gt;&#xA;&lt;p&gt;EDIT: Clarification on my use of the word &amp;quot;rigor&amp;quot;:&#xA;I meant a book that is challenging, a bit formal in nature, and includes useful and fruitful topics not usually found in traditional maths textbooks.&lt;/p&gt;&#xA;&lt;p&gt;SECOND EDIT: I've found the book I was looking for, after browsing a bunch of similar threads, I've found the precious jewel: &lt;a href=&quot;https://i.stack.imgur.com/3F1fS.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://i.stack.imgur.com/3F1fS.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It was published in 1891, about the time of my grand-grand father! It contains a plethora of information!&lt;/p&gt;&#xA;" OwnerUserId="15922" LastEditorUserId="15922" LastEditDate="2021-04-22T18:23:59.400" LastActivityDate="2021-05-04T23:13:49.363" Title="Looking for rigorous books to review geometry, trig, and precalculus" Tags="|textbooks|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20793" PostTypeId="2" ParentId="20792" CreationDate="2021-04-22T12:41:50.333" Score="3" Body="&lt;p&gt;Frank Ayres First Year College Math (Schaum's Outline).  I have/like the original 1958 edition (easy to get used), but the newer edition with co-author is probably OK, also.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0007DPVM2&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Theory-problems-first-college-mathematics/dp/B0007DPVM2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since it is a review, it is written economically and clearly.  Directed at the student as the customer, not a committee of teachers.  Not a ganglion-basher.&lt;/p&gt;&#xA;&lt;p&gt;Covers first and second year HS algebra, plane and solid geometry, plane trig, analytical geometry, function concept, and very short/easy intro to calculus (pre-calc style).&lt;/p&gt;&#xA;&lt;p&gt;It does not cover prob/stats.  (You'll need something else, preferably not a doorstop but something of a review nature or &amp;quot;for business students&amp;quot; or the like.)&lt;/p&gt;&#xA;&lt;p&gt;Has example problems and exercises, with answers.  But not an annoying amount.  And you can decide how much to read versus practice. (But you really DO need to practice.  One of the reasons you are having to review is likely lack of drill earlier.)&lt;/p&gt;&#xA;&lt;p&gt;Other than that, if you have access to your old calculus 1 text, I recommend looking at it.  Even if it's not perfect, at least you are familiar with it.&lt;/p&gt;&#xA;&lt;p&gt;P.s. I am trying to address the real question, I think you are asking.  I would just be careful about using the word &amp;quot;rigor&amp;quot;.  That is probably not what you really need.  Or at least not the way many people on MSE or MESE use it (highly abstract, very difficult).  Given, the totality of what you said (taken only through Calc 1, need to review pre-calc), you're not the candidate for stuff that is more difficult.  You need to work on the basics...need to acquire automaticity in manipulations involving trig, power laws, logs, etc.  Any standard review (or even omnibus) text will be &amp;quot;correct enough&amp;quot; for you to get value.  What you need to do is disciplined review and practice of the basics.  Seat of pants to seat of chair.  Door closed.  Walkman (or ipod or whatever) off.&lt;/p&gt;&#xA;" OwnerUserId="15923" LastEditorUserId="15923" LastEditDate="2021-04-22T13:47:13.613" LastActivityDate="2021-04-22T13:47:13.613" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="20794" PostTypeId="2" ParentId="20780" CreationDate="2021-04-24T01:19:02.230" Score="3" Body="&lt;p&gt;Expanding a bit on my comment, there is (1) a new textbook available for&#xA;project-based intro stats, (2) an online syllabus describing a course based&#xA;on community projects, and (3) an academic paper concluding that&#xA;&amp;quot;the project-based course ... provides a promising model for getting students hooked on the power and excitement of applied statistics.&amp;quot;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;(1) Chelsea Myers. &lt;em&gt;Project-Based R Companion to Introductory Statistics&lt;/em&gt;.&#xA;&lt;a href=&quot;https://www.routledge.com/Project-Based-R-Companion-to-Introductory-Statistics/Myers/p/book/9780367262082&quot; rel=&quot;nofollow noreferrer&quot;&gt;Routledge link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(2) Cindy Kaus, &lt;em&gt;Introductory Statistics with Community-Based Projects&lt;/em&gt;.&#xA;&lt;a href=&quot;https://ncsce.net/introductory-statistics-with-community-based-projects/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Metropolitan link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(3) Dierker, Lisa, Jane Robertson Evia, Karen Singer-Freeman, Kristin Woods, Janet Zupkus, Alan Arnholt, Elizabeth G. Moliski, Natalie Delia Deckard, Kristel Gallagher, and Jennifer Rose. &amp;quot;Project-based learning in introductory statistics: Comparing course experiences and predicting positive outcomes for students from diverse educational settings.&amp;quot; &lt;em&gt;International Journal of Educational Technology and Learning&lt;/em&gt; 3, no. 2 (2018): 52-64.&#xA;&lt;a href=&quot;http://scipg.com/index.php/101/article/view/26&quot; rel=&quot;nofollow noreferrer&quot;&gt;Journal link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-04-24T01:19:02.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20795" PostTypeId="1" AcceptedAnswerId="20797" CreationDate="2021-04-24T16:36:07.943" Score="5" ViewCount="298" Body="&lt;p&gt;I just read an interesting &lt;a href=&quot;https://betterexplained.com/articles/quadratic-formula/&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; that helps to understand completing the square, and prove the quadratic equation from a geomterical perspective.&lt;/p&gt;&#xA;&lt;p&gt;My question is how do I understand the graphical perspective&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/330px-Polynomialdeg2.svg.png&quot; alt=&quot;Text&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;in terms of this new geometric one&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://betterexplained.com/wp-content/webp-express/webp-images/uploads/quadratic/double-b.png.webp&quot; alt=&quot;Text&quot; /&gt;&lt;/p&gt;&#xA;" OwnerUserId="15918" LastActivityDate="2021-04-27T14:17:02.297" Title="Geometric and Graphical perspective on completing the square" Tags="|algebra|geometry|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20796" PostTypeId="1" CreationDate="2021-04-25T01:48:29.920" Score="2" ViewCount="689" Body="&lt;p&gt;I have always wondered whether it is necessary or not.    For me, it seems that it is enough to teach them the intuitive idea, that is, limit is just an approximation of a certain process.&lt;/p&gt;&#xA;&lt;p&gt;what do you think about it? I am basically interested in knowing if there is pedagogical reasons based on the nature of their training.&lt;/p&gt;&#xA;&lt;p&gt;I hope you get my concern because my english is not very good looking.&lt;/p&gt;&#xA;" OwnerUserId="15933" LastEditorUserId="2083" LastEditDate="2021-04-25T06:15:49.757" LastActivityDate="2021-04-25T20:40:56.900" Title="Is it necessary to teach the definition of a limit for engineering majors?" Tags="|undergraduate-education|calculus|engineering-mathematics|syllabus|" AnswerCount="4" CommentCount="2" ClosedDate="2021-04-25T20:41:46.793" ContentLicense="CC BY-SA 4.0" />
  <row Id="20797" PostTypeId="2" ParentId="20795" CreationDate="2021-04-25T01:51:11.380" Score="9" Body="&lt;p&gt;Note: as in the originally referenced article, I'll take &lt;span class=&quot;math-container&quot;&gt;$a=1$&lt;/span&gt; to keep it less complicated.&lt;/p&gt;&#xA;&lt;p&gt;I think it's important to first establish two key issues about the graphical setting:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What does it mean to &amp;quot;be a square&amp;quot; in a graph.&lt;/li&gt;&#xA;&lt;li&gt;What information are we assuming we can &amp;quot;read&amp;quot; from the graph.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;For (1), in the geometric setting this notion is obvious (e.g. rectangle with all equal sides or the area of such) and algebraically it's something of the form &lt;span class=&quot;math-container&quot;&gt;$(\text{stuff})^2$&lt;/span&gt;. Motivated by these ideas, it seems that graphically to &amp;quot;be a square&amp;quot; should be to be parabolic with vertex on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis. (Or perhaps even vertex at the origin.)&lt;/p&gt;&#xA;&lt;p&gt;For (2), let's assume we can obtain the coordinates of the vertex, &lt;span class=&quot;math-container&quot;&gt;$(h,k)$&lt;/span&gt;. Or, if you prefer, the focus and the directrix that define the parabola initially, but from them we can find the vertex as the midpoint between the focus and the directrix.&lt;/p&gt;&#xA;&lt;p&gt;Putting (1) and (2) together, we can &amp;quot;complete the square&amp;quot; by a vertical translation of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; units which results in a parabola with vertex on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis.&lt;/p&gt;&#xA;&lt;p&gt;For example, given the graph of &lt;span class=&quot;math-container&quot;&gt;$y=x^2-x-2$&lt;/span&gt; we find that the vertex is at &lt;span class=&quot;math-container&quot;&gt;$(1/2,-9/4)$&lt;/span&gt;. So we subtract &lt;span class=&quot;math-container&quot;&gt;$-9/4$&lt;/span&gt; from both sides to get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y-\left(-\frac{9}{4}\right) = x^2-x-2-\left(-\frac{9}{4}\right)$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y+\frac{9}{4} = \left(x-\frac{1}{2}\right)^2$$&lt;/span&gt;&#xA;Notice, of course, that what is being squared is &lt;span class=&quot;math-container&quot;&gt;$x-h$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Looking at this I would probably agree with the criticism that it seems a little too trivial and it's not nearly as cool or demonstrative as the geometric scenario. However, I think for students it could be a good example of how physicists/mathematicians/scientist often change coordinate systems to make equations easier. What once was &lt;span class=&quot;math-container&quot;&gt;$y=x^2-x-2$&lt;/span&gt; becomes &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt; after a vertical and horizontal shift. So we were just measuring from the &amp;quot;wrong&amp;quot; origin to begin with.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2021-04-25T01:51:11.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20798" PostTypeId="2" ParentId="20796" CreationDate="2021-04-25T04:45:54.050" Score="4" Body="&lt;p&gt;Well it doesn't really feel right to get degrees in engineering and gain years of engineering experience without even knowing what a limit actually is. And even though many engineers will do just fine without having been exposed to the rigorous definition of a limit, some engineers will need to be familiar with rigorous definitions/proofs if they ever pursue a career in academia or if they ever have to read textbooks/papers that haven't been specifically written for engineers.&lt;/p&gt;&#xA;" OwnerUserId="15934" LastActivityDate="2021-04-25T04:45:54.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20799" PostTypeId="2" ParentId="20796" CreationDate="2021-04-25T12:49:52.090" Score="1" Body="&lt;p&gt;No, it's definitely not &amp;quot;necessary&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I'm not an engineering major, but roomed with one, did a general engineering minor, and worked in/around mechanical, nuclear, mining and chemical engineering (had electrical on staff too).  Passed my EIT and was at one time, about to take the PE (mechanical) exam.  Most engineers in the workplace don't even use calculus, let alone epsilon-delta.  For that matter I've worked with a (small number of, but still violates a Euclidean &amp;quot;necessary&amp;quot;) licensed PEs who didn't even have a college degree.&lt;/p&gt;&#xA;&lt;p&gt;In their normal undergrad training (fluids, thermo, etc.), engineers will often use calculus.  So you need calculus to get through a BS in E.  But you don't need or use epsilon-delta in those courses, either in the derivations or the homework drill or the projects.&lt;/p&gt;&#xA;&lt;p&gt;Even if we use a less picky interpretation of &amp;quot;necessary&amp;quot; (very helpful), I would not put epsilon delta up there.  For one thing, the current stereotypical STEM training in the US, gives an exposure to epsilon delta, with a few problems, at the beginning of a calc course.  But doesn't come back to it, doesn't use it.  So it's not even that important for undergrad calculus or diffyQs.&lt;/p&gt;&#xA;&lt;p&gt;Now, does it hurt to have something like this?  I don't think so.  Especially given the limited time spent on in a typical calculus course based on a text like Thomas.  I mean at least you've seen it so someone mentions it, it's not a mythical creature.  And it's really just a bunch of detailed algebra pushing symbols and equations around (my same feeling about series)--maybe builds the algebra muscles in a manner helpful for and similar to multistep homework problems in fluid hydraulics or power systems.  But it's not firmly connected--just general muscle building.  For the vainshingly small but not converging to zero population of undergrad engineers that go on to some theoretical Ph.D. (and even within that a small set of them) and turn out to need this sort of stuff, they at least were exposed to it.  And can pick it up more as needed, for their work.  (Not to say they couldn't even with no previous exposure.)&lt;/p&gt;&#xA;&lt;p&gt;Being practical, we can always come up with things that might in some circumstance be helpful (learning Latin, say).  But life is finite. It is practical.  It's an engineering problem, with constraints, costs. etc.  So I definitely would not use the strong word necessary (even if we use it colloquially to mean strongly important) to describe formal limits.  There are a gazillion things useful to engineers and a lot on their slate already.  There is limited time in the day and limited brains in the skull.  So, I wouldn't go beyond the week or two at beginning of a calc course, as done now.&lt;/p&gt;&#xA;" OwnerUserId="15935" LastEditorUserId="15935" LastEditDate="2021-04-25T12:55:29.050" LastActivityDate="2021-04-25T12:55:29.050" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20800" PostTypeId="1" CreationDate="2021-04-25T14:09:28.473" Score="5" ViewCount="99" Body="&lt;p&gt;Whether we are preparing undergraduates for research in industry or academia effective collaboration is an important higher skill. I think there are two aspects to this in mathematics - thinking together about problems and working, where the latter often allows differentiation of roles.&lt;/p&gt;&#xA;&lt;p&gt;In subjects such as engineering it can be clearer how one might set up group projects which model how research happens in this field. How can we incorporate this into a course which is very heavy on individual ability to understand and apply proof techniques to the exclusion of most other skills (such as undergraduate mathematics degrees in the UK, where students study maths specifically and are able to specialise into pure/applied/etc. in the second or third year)?&lt;/p&gt;&#xA;&lt;p&gt;Some students will study together and some will not, while this is not discouraged can it be encouraged or made into a formal objective? Does studying together on lecture notes or individual problem sheets actually teach collaborative problem-solving, or would it be better to design specific tasks or resources to encourage group work?&lt;/p&gt;&#xA;" OwnerUserId="15936" LastActivityDate="2021-04-25T14:09:28.473" Title="Objectives for group work in undergraduate pure maths" Tags="|undergraduate-education|mathematical-pedagogy|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20801" PostTypeId="2" ParentId="20796" CreationDate="2021-04-25T14:45:00.560" Score="4" Body="&lt;p&gt;It is probably not the place of mathematics educators to decide what mathematics courses engineering majors should take. But a good reference point is ABET accreditation. Over 600 universities in the US have ABET accredited engineering programs. We should defer to the professionals who set these standard and assess outcomes. Here is a description of their current criteria:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.abet.org/accreditation/accreditation-criteria/criteria-for-accrediting-engineering-programs-2021-2022/&quot; rel=&quot;nofollow noreferrer&quot;&gt;ABET 2021-22 Criteria&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;They refer to college level mathematics, and say &amp;quot;For illustrative purposes, some examples of college-level mathematics include calculus, differential equations, probability, statistics, linear algebra, and discrete mathematics.&amp;quot; Accredited undergraduate programs require a minimum of 30 semester credit hours of college level mathematics and basic sciences. This document gives more specific expectations for different types of engineering programs. For example, Civil Engineering requires mathematics through differential equations.&lt;/p&gt;&#xA;&lt;p&gt;I think from this it is clear that a &amp;quot;survey course&amp;quot; in calculus would be inappropriate for engineering students. Students in engineering programs are expected to take the same sort of single variable calculus course taken by other STEM majors, including math majors.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-04-25T14:45:00.560" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20802" PostTypeId="1" CreationDate="2021-04-25T19:08:42.210" Score="19" ViewCount="4900" Body="&lt;p&gt;Look at the following example:&lt;/p&gt;&#xA;&lt;p&gt;Which picture has four apples?&lt;/p&gt;&#xA;&lt;p&gt;A&lt;a href=&quot;https://i.stack.imgur.com/Tpm46.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Tpm46.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;B &lt;a href=&quot;https://i.stack.imgur.com/AOv29.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AOv29.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;C &lt;a href=&quot;https://i.stack.imgur.com/lZNmQ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lZNmQ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;D &lt;a href=&quot;https://i.stack.imgur.com/BWqpH.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BWqpH.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;B is the expected answer but should not the correct answer be BCD? Technically if a set has &lt;strong&gt;exactly&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; elements, then it has &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; elements if &lt;span class=&quot;math-container&quot;&gt;$k\leq m$&lt;/span&gt;. This is also how we talk in everyday language:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Do you have three dollars?&amp;quot;&#xA;&amp;quot;Yes.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The second speaker is not indicating he has exactly three dollars. He simply indicates that he has &lt;strong&gt;at least&lt;/strong&gt; three dollars.&lt;/p&gt;&#xA;&lt;p&gt;So I am wondering if we are teaching children correct logic here. Shouldn't the original question be rephrased as &amp;quot;which picture has &lt;strong&gt;exactly&lt;/strong&gt; four apples&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="-1" LastEditDate="2021-04-26T17:40:35.623" LastActivityDate="2021-10-10T18:45:02.247" Title="Children's counting problems: Is this question phrased correctly?" Tags="|logic|preschool-education|" AnswerCount="9" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="20803" PostTypeId="2" ParentId="20796" CreationDate="2021-04-25T20:40:56.900" Score="2" Body="&lt;p&gt;The concepts behind limits are actually very important to engineering (in the form of error/precision analysis), but are rarely phrased that way.&lt;/p&gt;&#xA;&lt;p&gt;Given a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, we can imagine an engineering situation where there is some desired range of outputs from the function, but the engineer has control over the value of the inputs of the function. If &lt;span class=&quot;math-container&quot;&gt;$f(c) = L$&lt;/span&gt;, one might want to set the value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; close to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is within some error &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;. But since this is a real-world engineering situation, one cannot set the value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; precisely, only aim for a target value and specify a level of precision. That is, we should aim to make &lt;span class=&quot;math-container&quot;&gt;$x = c$&lt;/span&gt; and hopefully there is some level of precision &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; we can set on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; so that if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is within &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is within the desired range.&lt;/p&gt;&#xA;&lt;p&gt;One can then ask about the relationship between the precision necessary on the input to produce a desired precision on the output. For differentiable functions, this relationship is linear (in the limit to 0), and the multiplier is the derivative.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-04-25T20:40:56.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20804" PostTypeId="2" ParentId="20802" CreationDate="2021-04-25T20:42:48.403" Score="33" Body="&lt;p&gt;I don't think there's anything wrong with the wording; it's clear what is being asked. Your example with the three dollars is also not always the way we speak in everyday language. If you ask someone with three children if they have two children, they're unlikely to say &amp;quot;yes&amp;quot; and leave it at that.&lt;/p&gt;&#xA;&lt;p&gt;Getting more silly, a bicycle isn't a unicycle despite the fact that bicycles have at least one wheel. The root words for unicycle are &amp;quot;one&amp;quot; and &amp;quot;wheel,&amp;quot; but a unicycle is defined to have exactly one wheel even though no root word for &amp;quot;exactly&amp;quot; appears. Exactly one wheel is just the more natural interpretation, just as exactly four apples is the more natural interpretation.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastEditorUserId="12671" LastEditDate="2021-04-26T16:26:21.417" LastActivityDate="2021-04-26T16:26:21.417" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="20805" PostTypeId="2" ParentId="20802" CreationDate="2021-04-25T21:51:57.170" Score="16" Body="&lt;p&gt;Perhaps &amp;quot;shows&amp;quot; instead of &amp;quot;has&amp;quot;. If you asked me to show you 4 apples, I can't think of a logical argument in favor of me grabbing 5 apples and smiling smugly.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2021-04-25T21:51:57.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20806" PostTypeId="2" ParentId="20802" CreationDate="2021-04-26T03:36:08.743" Score="2" Body="&lt;p&gt;I think a lot of it has to do with the age of the child, and what the goal of the question is.  If this is for children just learning their numbers, like say 4 or 5 years old, then I think B is the correct answer as thy are not being asked to stretch their logical capabilities, but to simply recognize and call out the difference between 4 of something, or 5 of something, etc.&lt;/p&gt;&#xA;&lt;p&gt;However if this was for older children who already have comfort with basic numbers concepts and the goal was to demonstrate more creative thinking, then I believe B, C, and D would be the correct answer.&lt;/p&gt;&#xA;" OwnerUserId="13160" LastEditorUserId="13160" LastEditDate="2021-04-26T21:21:15.637" LastActivityDate="2021-04-26T21:21:15.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20807" PostTypeId="2" ParentId="20802" CreationDate="2021-04-26T12:50:54.927" Score="2" Body="&lt;h3&gt;If you allowed this, how do you grade answers?&lt;/h3&gt;&#xA;&lt;p&gt;Suppose the same maths test said &amp;quot;John has 2 apples and Lucy has 3 apples.  How many apples do they have in total?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;By your logic, the child could say &amp;quot;1&amp;quot; and be entirely correct.  If you have 5 apples and someone asks you &amp;quot;do you have an apple?&amp;quot;, the answer of course is &amp;quot;yes&amp;quot;.  So clearly your proposal fails, because your logic is not internally consistent.&lt;/p&gt;&#xA;&lt;p&gt;It is important to remember that what we say casually in everyday use is &lt;em&gt;&lt;strong&gt;NOT&lt;/strong&gt;&lt;/em&gt; mathematically sound, nor even necessarily logically or factually sound.  Technically, if we ask &amp;quot;which set has 4 members?&amp;quot; (or perhaps more accurately, &amp;quot;which set has an ordinal of 4?&amp;quot;), then a set which has 5 members does &lt;em&gt;&lt;strong&gt;NOT&lt;/strong&gt;&lt;/em&gt; meet the question.  For an even better example of how everyday usage does not match mathematical usage, look up the differences between Boolean AND/OR and how we might actually use &amp;quot;and&amp;quot; and &amp;quot;or&amp;quot; in conversation.&lt;/p&gt;&#xA;" OwnerUserId="6456" LastActivityDate="2021-04-26T12:50:54.927" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20808" PostTypeId="2" ParentId="20802" CreationDate="2021-04-26T14:27:22.277" Score="20" Body="&lt;p&gt;When we describe counts in natural language, there's almost always an implicit &amp;quot;exactly&amp;quot; when phrasing like this. We use phrases like &amp;quot;at least 4&amp;quot; when we want a more general description.  Most children who have reached a development level where this quiz would be reasonable will probably already have learned this.&lt;/p&gt;&#xA;&lt;p&gt;In fact, this is why there's a common joke:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Q: How many months have 28 days?&lt;br /&gt;&#xA;A: Just one, February.&lt;br /&gt;&#xA;Q: No, they &lt;em&gt;all&lt;/em&gt; do.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The punch line works because we normally don't treat &amp;quot;have 28 days&amp;quot; as meaning &amp;quot;have 28 days, and possibly others&amp;quot;, but when someone points it out we can see the potential ambiguity. But some people probably still won't get the joke, because the implicit &amp;quot;exactly&amp;quot; is so pervasive.&lt;/p&gt;&#xA;" OwnerUserId="10551" LastActivityDate="2021-04-26T14:27:22.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20809" PostTypeId="1" CreationDate="2021-04-26T15:06:41.110" Score="4" ViewCount="442" Body="&lt;p&gt;So I'm wrapping up my second year teaching high school calculus, and my first as an AP course.  In addition to review, I'd like to end the year by giving the students a taste of the kinds of real-world applications that they can now start to appreciate now that they have a conceptual understanding of the fundamentals of calculus.  Obviously, we've seen plenty of word problems that show how optimation, related rates, and integrals can lead to effective decision-making, but I'd like to go a bit deeper.&lt;/p&gt;&#xA;&lt;p&gt;Here are a few ideas I have of topics to show them.  Can you suggest other topics in the same vein?  My list is currently heavy on modeling with differential equations that are easy to conceptualize even if they are hard to solve.  Other topics across the calculus curriculum that might engage students would be very welcome.&lt;/p&gt;&#xA;&lt;p&gt;In rough order of difficulty to comprehend:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The equation for continuously compounded interest basically comes from realizing that interest is the derivative of a saving account's balance and is proportional to the size of the balance.&lt;/li&gt;&#xA;&lt;li&gt;Same thing for radioactive decay being an exponential function, since the change in the number of radioactive ions at any instant is proportional to the total number of radioactive ions.&lt;/li&gt;&#xA;&lt;li&gt;Without getting too deep into physics, knowing that the acceleration of a moving frictionless spring is proportional and opposite to its distance from the equilibrium point suggest a sinusoidal motion of the spring since sine and cosine are equal to the negative of their second derivatives.&lt;/li&gt;&#xA;&lt;li&gt;The now-familiar shape of time vs. infected people in an epidemic can be inferred from the S-I-R model, which is a nonlinear system of differential equations but still easy to understand where the equations come from.&lt;/li&gt;&#xA;&lt;li&gt;Similarly, it is relatively intuitive to understand the Lotka-Volterra equations to model population sizes in a predator-prey dynamic.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-04-28T21:41:09.313" Title="Showing applications of calculus to intro students" Tags="|calculus|examples|applications|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20810" PostTypeId="2" ParentId="20809" CreationDate="2021-04-26T15:20:32.360" Score="2" Body="&lt;p&gt;While you're talking about physics, you might want to mention generally the way physics is computed in video games. The positions and velocities of various objects are updated every fraction of a second, with the new velocity being the old velocity plus acceleration (due to forces) times time and the new position being the old position plus velocity times time. You'll have to gloss over the vector bits though.&lt;/p&gt;&#xA;&lt;p&gt;You can also talk about Newton's Method for numerically solving equations. This is actually how many computer algebra systems handle numerically solving equations.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-04-26T15:20:32.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20811" PostTypeId="2" ParentId="20809" CreationDate="2021-04-26T15:41:51.063" Score="3" Body="&lt;p&gt;I worry about throwing a bunch of application/modeling that relies on diffyQs beyond the AP curriculum.  You are combining tougher aspect (applications are harder than straight math) along with new math content.  This is sort of what I hate about how PDEs are taught (relatively unfamiliar application, like string tension, plus the new math needed to solve it).  In contrast, calculus and ODEs (and lower) seem to take an approach of teaching the equation solving first, before practicing/complicating with applications (i.e. word problems).&lt;/p&gt;&#xA;&lt;p&gt;I do really like predator-prey though from your list.   I would take an approach that emphasizes graphing, observing (maybe a bit of Excel plugging) rather than the terminology of Lotke Volterra.  Independent of the math, what I like about this application is the gutty vitality and intuitive understanding of it. Animals eating each other.  Thing that's more easy to think about than mechanical or financial topics (and definitely than electrical or chemicals, which are always more &amp;quot;invisible&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;An alternate suggestion might be to cover 2nd order constant coefficient ODEs.  Used to be part of the standard Calc BC in the 80s.  They slightly watered down the class a few years ago.&lt;/p&gt;&#xA;" OwnerUserId="15947" LastEditorUserId="15947" LastEditDate="2021-04-26T15:47:14.867" LastActivityDate="2021-04-26T15:47:14.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20812" PostTypeId="2" ParentId="20802" CreationDate="2021-04-26T18:27:13.697" Score="11" Body="&lt;p&gt;Nearly every test like this includes instructions to choose the &amp;quot;best answer&amp;quot; to cover exactly this scenario. This looks like it's part of a test of basic counting skills, and in that context, the best answer is B. While one could make an argument for either C or D, I can't imagine an argument for either of those being the &lt;em&gt;best&lt;/em&gt; answer when B is present.&lt;/p&gt;&#xA;" OwnerUserId="15949" LastActivityDate="2021-04-26T18:27:13.697" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20813" PostTypeId="1" CreationDate="2021-04-26T22:44:28.043" Score="1" ViewCount="931" Body="&lt;p&gt;I'm trying to explain some basic concept to my kid (he just started learning basic algebra following &lt;a href=&quot;https://www.amazon.co.uk/Discovering-Algebra-Investigative-Approach-Mathematics/dp/1559533404&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discovering Algebra: An Investigative Approach&#xA;by Jerald Murdock&lt;/a&gt; ). For example homeomorphism means a donut is (in a way) the same as a coffee cup.&lt;/p&gt;&#xA;&lt;p&gt;Coming to metric spaces, it's easy to explain. The basic concept is that there's a distance between any two points. This can be shown as&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Taxicab_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;taxi distance&lt;/a&gt; vs Euclidean distance (which I coined as &amp;quot;pigeon distance&amp;quot;)&lt;/li&gt;&#xA;&lt;li&gt;a hiking map measuring distances by how&#xA;many minutes it will take, due to different types of landforms: sand, swamp, clay, meadow, etc&lt;/li&gt;&#xA;&lt;li&gt;given two cities on earth, the distance in between is the same but looks quite different in different &lt;a href=&quot;https://en.wikipedia.org/wiki/Map_projection&quot; rel=&quot;nofollow noreferrer&quot;&gt;map projection&lt;/a&gt;, so conversely, on the same map, different distances could be defined&lt;/li&gt;&#xA;&lt;li&gt;a sphere shape &lt;a href=&quot;https://en.wikipedia.org/wiki/Lantern#/media/File:Chinese_lantern_night_sky_lijiang_yunnan_china.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;lantern&lt;/a&gt; with an ant wandering on the surface, then projected to floor, ceiling and walls of the room, measure the distance of the internal part of the room by the distance on the surface of the lantern.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;However I couldn't find simple examples in real life for normed vector space.&lt;/p&gt;&#xA;&lt;p&gt;I thought of a flat area with a strong wind where walking vertical to the wind is  different to walking along the direction of the wind. However the effort is also due to if one walks along the wind direction or against it, so it's not a nice example.&lt;/p&gt;&#xA;&lt;p&gt;Can you suggest some good real life examples of normed vector spaces?&lt;/p&gt;&#xA;" OwnerUserId="9059" LastEditorUserId="9059" LastEditDate="2021-04-27T09:58:31.550" LastActivityDate="2021-04-27T12:54:39.253" Title="Are there real life examples of normed vector spaces?" Tags="|topology|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="20814" PostTypeId="2" ParentId="20795" CreationDate="2021-04-27T00:18:31.437" Score="4" Body="&lt;p&gt;I say that the story of quadratics -- their algebra and their graphing -- is a beautiful story of the of power of symmetry. If I have a rectangle of area 36, you know little about that rectangle: it was area 36. But if I add I have a (maximally) symmetrical rectangle of area 36, then you know everything about it: it must be a 6-by-6 square. The mention of symmetry collapses uncertainty into certainty here. That's the power of the &amp;quot;quadrus&amp;quot; method (the &amp;quot;square&amp;quot; method) that allows you to solve equations of the form &lt;span class=&quot;math-container&quot;&gt;$ax^2 + bx + c = q$&lt;/span&gt;, aka quadratic equations. (Get the name?)&lt;/p&gt;&#xA;&lt;p&gt;This algebraic method leads you to see that any graph of an equation &lt;span class=&quot;math-container&quot;&gt;$y = ax^2 +bx + c$&lt;/span&gt; is going to be the same beautiful symmetric shape as the graph of &lt;span class=&quot;math-container&quot;&gt;$y = ax^2$&lt;/span&gt;. Ahh ... symmetry again! Let's use it to our advantage!&lt;/p&gt;&#xA;&lt;p&gt;Graph: &lt;span class=&quot;math-container&quot;&gt;$y = (x-2)(x-6)+7$&lt;/span&gt;.&#xA;Okay. I see &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=6$&lt;/span&gt; staring me in the face. They each give &lt;span class=&quot;math-container&quot;&gt;$y = 7$&lt;/span&gt;. Oh! I have two symmetrical points on a symmetrical graph. The line of symmetry must be at &lt;span class=&quot;math-container&quot;&gt;$x=4$&lt;/span&gt;, halfway between. At &lt;span class=&quot;math-container&quot;&gt;$x=4$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=3$&lt;/span&gt;. Oh! The graph is now clear.&lt;/p&gt;&#xA;&lt;p&gt;Graph: &lt;span class=&quot;math-container&quot;&gt;$y = 2x^2 + 8x +5$&lt;/span&gt;&#xA;Okay. &lt;span class=&quot;math-container&quot;&gt;$y =2x(x+4) + 5$&lt;/span&gt;.&#xA;I see &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=-4$&lt;/span&gt; both give &lt;span class=&quot;math-container&quot;&gt;$y=5$&lt;/span&gt;. Ahh. Line of symmetry is at &lt;span class=&quot;math-container&quot;&gt;$x=-2$&lt;/span&gt;. Put in this value and the graph of the equation becomes clear.&#xA;The power of symmetry!&lt;/p&gt;&#xA;&lt;p&gt;Yes &lt;a href=&quot;https://www.youtube.com/channel/UCib_J32VI8rQI_LCFXn1XAA&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/channel/UCib_J32VI8rQI_LCFXn1XAA&lt;/a&gt; has the videos and here is a PDF of everything, in case you are interested: &lt;a href=&quot;http://www.jamestanton.com/?p=2296&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jamestanton.com/?p=2296&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;p.s. The graph of y = x^2 is a symmetrical &amp;quot;U&amp;quot;-shaped graph. If you add something non-symmetrical to it, like 2x+1, and so graph y = x^2+2x+1, it should be a shock to see the same symmetrical U -shape graph appear! Seriously, whoa! But the algebra of completing the square shows this is really the graph of y= (x+1)^2 with x=-1 behaving like zero. In general, as people are showing, the algebra of completing the square gives, for example:&lt;/p&gt;&#xA;&lt;p&gt;y = 2x^2 - 4x + 1&lt;/p&gt;&#xA;&lt;p&gt;2y = 4x^2 - 8x + 2 (I am following my nose after drawing a picture of a square and wishing for easy numbers)&lt;/p&gt;&#xA;&lt;p&gt;2y+2 = 4x^2 - 8x + 4&lt;/p&gt;&#xA;&lt;p&gt;2y+2 = (2x-2)^2 = 4(x-1)^2  (looking at my picture)&lt;/p&gt;&#xA;&lt;p&gt;y = 2(x-1)^2 - 1&lt;/p&gt;&#xA;&lt;p&gt;Yep... it is just y=x^2 again, made steeper, with x=1 behaving like zero, and everything shifted down one.&lt;/p&gt;&#xA;&lt;p&gt;This stuff is really about teaching kids the confidence to follow their wits and commonsense. (If I really needed a graph of y = 2x^2 - 4x + 1 I would just type it into Desmos ... So what is this material teaching really in the 21st century?)&lt;/p&gt;&#xA;" OwnerUserId="13256" LastEditorUserId="13256" LastEditDate="2021-04-27T14:17:02.297" LastActivityDate="2021-04-27T14:17:02.297" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20815" PostTypeId="2" ParentId="20813" CreationDate="2021-04-27T00:51:09.593" Score="5" Body="&lt;p&gt;I assume the question is based on your desire to show your students some examples of real-world motivation for the context. If so, then good for you, but it would be nice if you could edit the question to spell that out a little more, including an explanation of the level of the course and your students' background.&lt;/p&gt;&#xA;&lt;p&gt;[EDIT] The OP edited the question to say that this is for a young kid starting algebra, not a class of college students, so a lot of the examples below will be at too high a level.&lt;/p&gt;&#xA;&lt;p&gt;Every type of vector you encounter in freshman physics lives in some normed vector space. Force vectors are a normed vector space, as are momenta, velocities, displacements, magnetic fields, and so on. These examples historically were the ones that led to the creation of the vector-scalar system by Gibbs and Heaviside around 1880, so they are the prototypical examples of the word &amp;quot;vector.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;For infinite-dimensional examples, I think the earliest important examples were also from physics -- wavefunctions in quantum mechanics. In this application, the norm has an interpretation relating to probability. This norm arises from a complex-valued inner product, which is a measure of the similarity of one wavefunction to another.&lt;/p&gt;&#xA;&lt;p&gt;Other types of waves, such as sound waves, also live in infinite-dimensional vector spaces, and there the norm typically is a measure of energy. One can define a real-valued inner product between sound waves and such, but it's not particularly physically interesting to my knowledge.&lt;/p&gt;&#xA;&lt;p&gt;It's actually not easy to find a student-friendly physics example of a vector space that is &lt;em&gt;not&lt;/em&gt; normed. Relativity uses vectors with what physicists call a metric, but it isn't positive-definite. So that's a good non-example, but most students probably won't have the background knowledge to be able to understand it. One of the foundational problems in attempts to create a theory of quantum gravity is that it doesn't seem possible to define a norm, and yet we can't make sense of quantum mechanics without a norm. Maybe a somewhat less far-out example of a non-normed vector space in physics would be that sometimes we want to talk about idealized waves such as plane waves, whose norm would diverge.&lt;/p&gt;&#xA;&lt;p&gt;From outside the field of physics, the best examples I know of are from statistics, where things like correlations can be expressed as inner products, and again norms are probabilities.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-04-27T12:54:39.253" LastActivityDate="2021-04-27T12:54:39.253" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20816" PostTypeId="2" ParentId="20802" CreationDate="2021-04-27T06:47:37.780" Score="8" Body="&lt;p&gt;I showed this question to my three-year old son. His response - because he counted the apples one by one in each picture, passing &amp;quot;4&amp;quot; each time - was B, C and D. Hence, we need to take into account how children arrive at their conclusion, since they do not apply formal logic. The thought process is very different from the abstract approach a programmer might take when deciding whether his boolean expression is correctly formulated. I have worked with kids taking ability score tests, and sometimes even the very strictly standardised question sets with pictures, where the question itself was worded correctly, elicit unusual answers. But many children, if prompted, give a very compelling answer. &amp;quot;Odd one out&amp;quot; questions are especially problematic, and the question here is a variation of that.&#xA;It does not matter if the formal logic is the correct one, linguistically or mathematically, if children can not apply the rules in a conscious way. But of course we want a specific answer - at least for grading - and not find out how the child's analytical capabilities work exactly. To limit the risk of such unusual reasoning, best remove ambiguities, albeit without introducing logical loops that frame other questions in a more complicated way.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Which picture shows exactly 4 apples&amp;quot; is the most accurate phrasing, and it avoids having to put &amp;quot;at least&amp;quot; into other questions to keep the logic consistent (see the &amp;quot;how many children do you have&amp;quot; example given in another reply).&lt;/p&gt;&#xA;" OwnerUserId="15952" LastActivityDate="2021-04-27T06:47:37.780" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20818" PostTypeId="2" ParentId="20802" CreationDate="2021-04-27T23:22:28.357" Score="0" Body="&lt;p&gt;This was not the question:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Which pictures have four apples?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This test is a multiple choice question that does not appear to have other than the 4 unique choices: a, b, c, d.&lt;/p&gt;&#xA;&lt;p&gt;There is no all of the above or b-d choice which can be selected.&lt;/p&gt;&#xA;&lt;p&gt;If the test was a fill in the blank write in, there is a bit of ambiquity.&lt;/p&gt;&#xA;&lt;p&gt;More importantly though, examining the response with the student would be far better than pushing a black and white mindset.&lt;/p&gt;&#xA;" OwnerUserId="15960" LastActivityDate="2021-04-27T23:22:28.357" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20820" PostTypeId="2" ParentId="20809" CreationDate="2021-04-28T14:55:12.723" Score="1" Body="&lt;p&gt;More of a comment than an answer but I don't have the rep to post comments. When my AP calc professor was introducing derivatives, we dropped Barbie dolls out of a window and timed their descent lol. We found average speeds that way, and then he asked us if we could determine the speed it was falling at exactly 1 second. There's not a good way to do this without derivatives, which was a fun segue into the concept.&lt;/p&gt;&#xA;&lt;p&gt;Generalizing more, the relationship of acceleration to velocity to position is always good for explaining derivatives and integrals, and even an intuitive proof for the fundamental theorem of calculus (if you're driving from one location to another, by the time you reach your destination all the tiny velocity * delta time increments summed up end up equaling the distance you traveled, or final position minus initial position [ F(b) - F(a) ]. And the units work out too, since m/s * s = m).&lt;/p&gt;&#xA;" OwnerUserId="15966" LastActivityDate="2021-04-28T14:55:12.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20822" PostTypeId="1" CreationDate="2021-04-28T20:42:02.587" Score="6" ViewCount="102" Body="&lt;p&gt;As I wrap up my Spring semester online courses, I'm starting to think about next Fall when (hopefully) our university will return to full in-person classes. Because of COVID, over the last year I have more-or-less turned 3 of my in-person courses into fully on-line courses. My main question here essentially is: &lt;strong&gt;what's the best way to change them back into in-person courses?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;For one of these courses (call it &amp;quot;Course 1&amp;quot;), I was already half way through transitioning to a flipped-classroom and so being forced to record a bunch of video lectures just moved that along quicker. When we're back in-person, class time will consist of a short intro or review, mostly focus on students working on problems/homework in groups and presenting solutions, and finish with a quick summary.&lt;/p&gt;&#xA;&lt;p&gt;For &amp;quot;Course 2&amp;quot; I can mostly do the same as Course 1, but I have very few in-class problem sets created. In-class group work was nonexistent in this course before and during COVID, but I can imagine working in that direction. Both Course 1 and 2 are courses that I've realized lecture alone isn't working as well as one would like, and so I'm happy to work toward giving students more time-on-task.&lt;/p&gt;&#xA;&lt;p&gt;The main issue is &amp;quot;Course 3&amp;quot; which is a bit higher level than the other two. First of all, the thought of creating suitable problem sets for group work for even half of the class time is overwhelming. Pre-COVID the students in this class had a lot of success with a lecture-only format, so I feel less pressure to change the mode. Yet I now possess the whole course in video form, so certainly there's an advantage to be had &lt;em&gt;somehow&lt;/em&gt;. It seems foolish to just return to what I was doing pre-COVID and keep all my digital materials secret from the in-person students.&lt;/p&gt;&#xA;&lt;p&gt;I'm sure all of us are in a similar situation with our own individual nuances. So I'm looking for &lt;strong&gt;advice from anyone who has successfully turned an online course into an in-person course.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What did you keep and make available for your in-person students?&lt;/li&gt;&#xA;&lt;li&gt;Was there anything pre-existing that you decided against using in order to re-create it live in the classroom (e.g. certain lectures)?&lt;/li&gt;&#xA;&lt;li&gt;We all know now that it takes a ton of writing/creating to go from in-person to online. How much new material did you have to write/create to move from online to in-person?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(Not that it really matters, but just in case: Course 1 and 2 are first and second semester calculus. Course 3 is an introduction to Discrete Math.)&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2021-04-28T20:42:02.587" Title="Converting an Online Course Back to In-Person" Tags="|teaching|online-instruction|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20823" PostTypeId="2" ParentId="20809" CreationDate="2021-04-28T21:41:09.313" Score="0" Body="&lt;p&gt;Here are several real-world problems that are mostly accessible to first-year students.  Some of them require a little bit of differential equation manipulation, but a minimum amount:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Geometry (where do equations of volumes of cones / spheres / etc come from)&lt;/li&gt;&#xA;&lt;li&gt;Deriving the Interest Rate Equation&lt;/li&gt;&#xA;&lt;li&gt;Deriving the Logistic Curve&lt;/li&gt;&#xA;&lt;li&gt;Deriving the Kelly Criterion&lt;/li&gt;&#xA;&lt;li&gt;Lotka-Volterra (predator/prey) models&lt;/li&gt;&#xA;&lt;li&gt;Physics:&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Deriving Kinematic equations&lt;/li&gt;&#xA;&lt;li&gt;Deriving kinetic energy equation&lt;/li&gt;&#xA;&lt;li&gt;Deriving escape velocity equation&lt;/li&gt;&#xA;&lt;li&gt;Deriving the rocket equation&lt;/li&gt;&#xA;&lt;li&gt;Deriving Newton's Law of cooling&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;Electronics&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Calculating RMS voltage&lt;/li&gt;&#xA;&lt;li&gt;Calculating capacitor filling times&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;Chemistry&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Several rate equations are based on calculus&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2021-04-28T21:41:09.313" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20824" PostTypeId="1" AcceptedAnswerId="20826" CreationDate="2021-04-29T00:35:57.143" Score="7" ViewCount="1557" Body="&lt;p&gt;I am currently student teaching for an Integrated Math 1 class (which is similar to Algebra 1) that consists of 9th graders. I have been teaching my students how to solve linear systems using Elimination.&lt;/p&gt;&#xA;&lt;p&gt;Here is one example we did in class:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{cases} x-y=10 \\ 2x+y=5 \end{cases}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The students recognized that they should add the second equation from the first. However, a common mistake I noticed was that some students did not combine like-terms correctly (which they have previously learned). For example, one student (who has an &lt;a href=&quot;https://en.wikipedia.org/wiki/Individualized_Education_Program&quot; rel=&quot;nofollow noreferrer&quot;&gt;IEP&lt;/a&gt;) added the equations and wrote &lt;span class=&quot;math-container&quot;&gt;$$2x+y=15$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Clearly, this student did not recall how to add &lt;span class=&quot;math-container&quot;&gt;$x+2x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-y+y$&lt;/span&gt;, but he did correctly add &lt;span class=&quot;math-container&quot;&gt;$10+5=15$&lt;/span&gt; (presumably because the &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; and the &lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt; were &amp;quot;visible&amp;quot;). After discussing with my cooperating educator (CE), she suggested that I rewrite the original system as &lt;span class=&quot;math-container&quot;&gt;$$\begin{cases} \color{red}{1}x-\color{red}{1}y=10 \\ 2x+\color{red}{1}y=5 \end{cases}$$&lt;/span&gt; to make the coefficients stand out.&lt;/p&gt;&#xA;&lt;p&gt;But then it occurred to me: why is that in math, we do not normally write, for instance, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Now the obvious reason may be that any number times &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is itself, so it is more efficient to write &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, and consequently, it is more efficient to write &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt;. But why explain this to a 9th grader when they can just simply get in the habit of always writing &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt;? If anything, this will help the students &lt;em&gt;see&lt;/em&gt; the coefficient and make less mistakes when adding or subtracting like-terms. Otherwise, I find myself explaining to the students &amp;quot;Don't forget that there's an invisible &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; next to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;, which doesn't really stick with them. This issue would be avoided if the students were accustomed to writing &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt; instead of just &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Recognizing that my students have a variety of skill levels (and the fact that I am young and naive), it might be overkill for some of them to always write &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; if they already know what they are doing, so I understand that my proposal will depend the students' mathematical abilities. Hence, my questions are: &lt;strong&gt;For the purpose of teaching, should we normalize writing &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;? Or is this approach better suited for students with special learning needs instead of forcing &lt;em&gt;all&lt;/em&gt; students to write this way?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="13657" LastEditDate="2021-04-30T20:33:05.320" LastActivityDate="2021-05-03T23:29:41.750" Title="Why do we write $x$ instead of $1x$?" Tags="|secondary-education|algebra|notation|" AnswerCount="6" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="20825" PostTypeId="2" ParentId="20824" CreationDate="2021-04-29T01:02:26.953" Score="7" Body="&lt;p&gt;Part of our job is to teach students to read and write mathematics properly. We shouldn't be turning out students who say &amp;quot;derive&amp;quot; when they mean &amp;quot;differentiate,&amp;quot; say &amp;quot;cancel&amp;quot; when they mean &amp;quot;eliminate,&amp;quot; write &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;, or who don't know that &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt; is a notation for the exponential function. Therefore it would be a disservice to any student to encourage them to write weird stuff like &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt; as a routine thing instead of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. By the same token, you should not be teaching the student to write &lt;span class=&quot;math-container&quot;&gt;$1x^1+0$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I also don't understand why anyone would hypothesize that misunderstanding something like this would lead to the particular answer that the student wrote. Now if a student wrote &lt;span class=&quot;math-container&quot;&gt;$3x+x=3x$&lt;/span&gt;, and explained that they reasoned there &amp;quot;was nothing in front of the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;,&amp;quot; then you would have reason to explain to them that &lt;span class=&quot;math-container&quot;&gt;$3x+x$&lt;/span&gt; is the same as &lt;span class=&quot;math-container&quot;&gt;$3x+1x$&lt;/span&gt;, but only as an explanation, not as something they should always make a habit of writing.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-04-29T01:02:26.953" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="20826" PostTypeId="2" ParentId="20824" CreationDate="2021-04-29T03:43:58.773" Score="27" Body="&lt;p&gt;I think the basic answer is that there are all sorts of things that we could write, but don't.  We usually leave off things that are redundant, but we can add them back in when convenient.&lt;/p&gt;&#xA;&lt;p&gt;For instance, instead of &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt; we could also write &lt;span class=&quot;math-container&quot;&gt;$1\times 1\times 1\times x$&lt;/span&gt;.  Or we could write &lt;span class=&quot;math-container&quot;&gt;$x + 0$&lt;/span&gt; or even &lt;span class=&quot;math-container&quot;&gt;$0x^3 + 0x^2 + x + 0$&lt;/span&gt;.  We could also write &lt;span class=&quot;math-container&quot;&gt;$0yz + 0y^2 + 0x^2 + 1\times 1\times x + 23 \times 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes it is even &lt;em&gt;helpful&lt;/em&gt; to add these things in.  For instance, sometimes it is helpful to write &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$2\times\frac{1}{2}\times x$&lt;/span&gt;.  The point is that there are a &lt;em&gt;lot&lt;/em&gt; of equivalent ways to write things, and we choose the ones that are more helpful for us or help us think about the question more clearly.&lt;/p&gt;&#xA;&lt;p&gt;However, in absence of other considerations, we like to write things in the shortest way possible, which is why we leave off the 1.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2021-04-29T03:43:58.773" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20828" PostTypeId="2" ParentId="134" CreationDate="2021-04-29T18:17:54.950" Score="1" Body="&lt;p&gt;This is simple, but effective.  Consider doing a weekly, period-long test, every Friday.  I remember having that in HS for pre-calc and calc.  At first was surprised by the frequency.  But ended up admiring the pedagogic effect.  Tests are some of the most effective drill (because of the drive for prep and the higher stakes...and the learning from the results).  In addition, I find the &amp;quot;midterm plus a final&amp;quot; practice of many college courses is fraught with issues (cramming, etc.)  and ends up being too high stakes, too limited in feedback.  Weekly is a better frequency.&lt;/p&gt;&#xA;&lt;p&gt;And keep it on FRI.  Weekend free for the students.  And regularity is a powerful social tool.  I've even used it to drive behavior by multi billion dollar CEOs.  Pavlov did good shit...don't dismiss it.&lt;/p&gt;&#xA;&lt;p&gt;This needn't keep you from introducing more drill, less dulcet tones, into other parts of the classes.  But right away, it is replacing several hours of sage time with practice.&lt;/p&gt;&#xA;" OwnerUserId="15977" LastEditorUserId="15977" LastEditDate="2021-04-29T18:22:55.030" LastActivityDate="2021-04-29T18:22:55.030" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20829" PostTypeId="1" CreationDate="2021-04-30T14:09:05.710" Score="3" ViewCount="206" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;tl;dr: Why do so many students have poor intuition of numbers, and&#xA;what can be done about it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I've always been good with numbers. As a maths tutor, one of the things I notice is how poor the &lt;em&gt;intuition&lt;/em&gt; and &lt;em&gt;techniques&lt;/em&gt; that some of my students have when it comes to approximating numbers (amount of things) and fractions of things.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Quickly approximate how many lines are on the following page, just by looking at it:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YJLIO.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YJLIO.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;When I ask this question to my students, they will either count the number of lines &amp;quot;One, two, three, four...&amp;quot;, which takes ages and moreover is tedious, or will just stare at the page and guess a random number. Some will guess like &amp;quot;seventeen&amp;quot; or &amp;quot;fifty&amp;quot;, or &amp;quot;no idea&amp;quot;, which worries me, as it shows their subitising skills aren't up to par. That said, not all are bad at this and some students will make a reasonable approximation of twenty to forty.&lt;/p&gt;&#xA;&lt;p&gt;One method I use is to approximate &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; lines by: Scrolling my eyes from line &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to line &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; while counting &amp;quot;One, two, three&amp;quot; in my head. Then keep counting in my head at the same speed, scrolling my eyes at the same speed, and then stop scrolling my eyes once I get to the number &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; in my head. This will take all of about &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; seconds for me to do. Now I know that I won't have counted exactly &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; lines, but that's besides the point: I'm trying to get an approximation/estimation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Bk4gD.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Bk4gD.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Once I have this &amp;quot;length of &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;&amp;quot;, I'll then make copies of the &amp;quot;length of &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; lines&amp;quot;, using my imagination. Basically, doing this in my mind's eye:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EoteA.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EoteA.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Although in reality what I think I do is I try to look at a line in line the bottom of the first arrow, then look at a line in line with the bottom of the second arrow, then look at a line in line with the bottom of the third arrow, and notice that the bottom of the third arrow is very near the line closest to the bottom of the page.&lt;/p&gt;&#xA;&lt;p&gt;This technique of &amp;quot;close-ish&amp;quot; approximation is &lt;em&gt;useful&lt;/em&gt; because now I can try and do things with the information I have, without having spent &lt;span class=&quot;math-container&quot;&gt;$30$&lt;/span&gt; seconds figuring out how many lines there actually are, which I don't care about. I can do the above process very quickly (in less than &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; seconds) due to using it so often. This shortcut saves lots of time and therefore I think if students did this, they would lose less focus in maths because using these estimations help get close to the right answer quickly. They should use these shortcuts when approximating to see if their answers make sense.&lt;/p&gt;&#xA;&lt;p&gt;Now let's say I'm writing an essay with the above paper and I know I can write on average &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; words per line. The teacher requires the essay be minimum 600 words. I have 10 pages left in my paper pad. Do I have to buy a new one on the way home from school or not? Now a lot of my students will struggle with this question, spend &lt;span class=&quot;math-container&quot;&gt;$5-10$&lt;/span&gt; minutes on it, as if it's a really difficult problem-solving exercise, and then some will give up, some will give the wrong answer, and some will give the right answer, assuming the right answer is &amp;quot;yes there is enough paper left because &lt;span class=&quot;math-container&quot;&gt;$8\times 30\times 10 = 2400 &amp;gt; 600.$&lt;/span&gt; &amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But here is my thought process: &lt;span class=&quot;math-container&quot;&gt;$8\times 30\times 10 = 2400,\ $&lt;/span&gt; so  you only need &lt;span class=&quot;math-container&quot;&gt;$\frac{600}{2400} = \frac{1}{4}$&lt;/span&gt; of your &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; pages (i.e. &lt;span class=&quot;math-container&quot;&gt;$2\frac{1}{2}$&lt;/span&gt; pages), but since you're going to be making notes and drafts on your essay, you might need to use &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; pages total. Plus there's the other homework and I need more paper for school tomorrow and I don't want to buy it from the shop tomorrow morning so, hmmm... so yes I really do need to buy another book pad now. If I had &lt;span class=&quot;math-container&quot;&gt;$30$&lt;/span&gt; pages rather than &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; pages left in my pad, I would then think, &amp;quot;hmmm... how much paper do I need for homework tonight? Maybe like 15 pages total. And I'm pretty sure I can get away with &lt;span class=&quot;math-container&quot;&gt;$30-15=15$&lt;/span&gt; pages at school tomorrow, so I'm not going to bother going to the stationary shop after school today&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So I think it's this fluent usage of maths within a self-conversation that improves my intuition. It is realistic that I would have actually thought these things on the way home from school, whereas I imagine students give up on this thought process and just think, &amp;quot;I'm going to get a new pad from the stationary shop now&amp;quot;, or &amp;quot;I'll borrow paper from James at school tomorrow&amp;quot;, or most likely they won't think about it at all. In order to have a thought process similar to my one, one must have enough confidence in their own thought process to begin with.&lt;/p&gt;&#xA;&lt;p&gt;But I think it's exactly this confident application of arithmetic in self-conversations assisting in everyday life things that helped people including myself get really good at doing arithmetic quickly and well. So we maybe have a kind of a chicken-and-egg thing: students won't do this unless they have confidence, but they won't ever be confident unless they start doing it.&lt;/p&gt;&#xA;&lt;p&gt;Now, my thought processes outlined above will not be a revelation to anyone here (I would hope). But I think lots of students don't use this and other similar methods to estimation-check their working, or solve everyday mini-problems. So I guess my over-arching questions are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Q1) what can be done to help students improve these important quick&#xA;estimation skills?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Q2) How can we be encouraging students who are not naturally good at maths and arithmetic and fractions to do this on a day-to-day basis as part of their self-conversation? Or is it unethical or unrealistic to expect to be able to influence their self-conversation?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Originally my question also included &amp;quot;estimating fractions&amp;quot; (which &lt;em&gt;is&lt;/em&gt; closely related to this question) to help improve intuition with fractions, but this question is already overly-long and a ramble (apologies for that), so maybe I will ask that as a separate question in the future.&lt;/p&gt;&#xA;&lt;p&gt;Possible answer to my own question: I need to convey to my students the methods to help improve intuition of numbers outlined above (and various other methods) in a clear way. I think I already do this though. But maybe I need to do it &lt;em&gt;more&lt;/em&gt; slowly and clearly? I'm not sure.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2023-06-12T16:58:20.130" LastActivityDate="2023-06-14T11:28:16.653" Title="&quot;Rough subitising / estimation&quot; for better intuition and ability to apply arithmetic" Tags="|intuition|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="20830" PostTypeId="1" AcceptedAnswerId="20832" CreationDate="2021-04-30T15:40:12.277" Score="10" ViewCount="2387" Body="&lt;p&gt;I am going to teach 2nd-year undergraduate students in applied math or computer science a course called &amp;quot;Discrete Mathematics for Computer Science&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Most students who take this course plan to apply for &lt;strong&gt;graduate schools&lt;/strong&gt;.&#xA;This is a new course, so I will decide the syllabus myself.&lt;/p&gt;&#xA;&lt;p&gt;I studied computer science for my PhD and my research is mostly in probability and combinatorics. So for most part of the course, I plan to teach these two topics.&lt;/p&gt;&#xA;&lt;p&gt;However, I have also seen that mathematical logic is included in similar courses in some other university. This means&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Propositional logic:  Combining propositions, truth table, axiomatic rule&lt;/li&gt;&#xA;&lt;li&gt;Predicate Logic: Negating quantifiers, combining quantifiers, and order of quantifiers&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am not familiar with this topic, so it's hard for me to judge if including it will be beneficial for students when they go to graduate schools. Is mathematical logic still an active domain of research in computer science?&lt;/p&gt;&#xA;&lt;p&gt;Should I include it in my course?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorUserId="851" LastEditDate="2021-05-04T03:09:50.097" LastActivityDate="2021-05-04T10:08:15.153" Title="Should mathematical logic be included in a discrete mathematics course for computer science?" Tags="|computer-science|" AnswerCount="10" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="20831" PostTypeId="2" ParentId="20824" CreationDate="2021-04-30T15:58:10.860" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Clearly, this student did not recall how to add &lt;span class=&quot;math-container&quot;&gt;$x+2x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-y+y$&lt;/span&gt;, but&#xA;he did correctly add &lt;span class=&quot;math-container&quot;&gt;$10+5=15$&lt;/span&gt; (presumably because the &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; and the&#xA;&lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt; were &amp;quot;visible&amp;quot;). After discussing with my cooperating educator,&#xA;she suggested that I rewrite the original system as &lt;span class=&quot;math-container&quot;&gt;$$\begin{cases}&#xA; \color{red}{1}x-\color{red}{1}y=10 \\ 2x+\color{red}{1}y=5&#xA; \end{cases}$$&lt;/span&gt; to make the coefficients stand out.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't think it makes sense to make everyone in the class write &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1y$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, the main reason being that generally speaking, people and textbooks don't use &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Did you get to the bottom of &lt;em&gt;how&lt;/em&gt; the student got to: &lt;span class=&quot;math-container&quot;&gt;$2x+y = 15$&lt;/span&gt; ? Then you can simply explain to the student why their specific working out was incorrect and help them do it the right way. You can also explain to him/her that you &lt;em&gt;can&lt;/em&gt; write &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt;, but in mathematical textbooks and literature, it is written as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for brevity, so it is better to stick to the most common way to write it, for communication's sake.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastActivityDate="2021-04-30T15:58:10.860" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20832" PostTypeId="2" ParentId="20830" CreationDate="2021-04-30T16:53:32.160" Score="15" Body="&lt;p&gt;Certainly the rudiments of logic are needed in computer science:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Enough to understand Boolean expressions in &lt;strong&gt;if&lt;/strong&gt;-statements. &lt;code&gt;if (a &amp;lt; pi) and (not finished)&lt;/code&gt;, etc.&lt;/li&gt;&#xA;&lt;li&gt;Enough to understand proofs by contradiction, and induction proofs.&lt;/li&gt;&#xA;&lt;li&gt;Enough to understand satisfiability, 3SAT.&lt;/li&gt;&#xA;&lt;li&gt;Enough to understand the undecidability of the halting problem for Turing machines.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Not needed: First-order predicate calculus. Resolution. Model theory. The Löwenheim–Skolem theorem. Etc.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-04-30T16:53:32.160" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="20833" PostTypeId="2" ParentId="20830" CreationDate="2021-04-30T17:33:24.843" Score="3" Body="&lt;p&gt;Practical suggestion: I would doublecheck the rest of the standard curriculum and see if this stuff is covered elsewhere (or a good portion of it).  For instance, would be surprised if they are not seeing Boolean algebra somewhere.  (Heck, I wasn't even CS and got that in a survey EE course.)&lt;/p&gt;&#xA;" OwnerUserId="15979" LastActivityDate="2021-04-30T17:33:24.843" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20834" PostTypeId="2" ParentId="8052" CreationDate="2021-04-30T20:39:28.733" Score="2" Body="&lt;p&gt;Mathematics is often thought of as a bunch of clever people doing weird things with symbols for reasons that are difficult to relate to reality. Whilst this is not entirely false, it is for the most part (almost humorously so in my opinion), the antithesis of the actual role of mathematics in today's world.&lt;/p&gt;&#xA;&lt;p&gt;I challenge anyone to seriously tackle the world's major problems without first being an expert in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Our planet is on the verge of human extinction due to climate change. There will be wars over water. There is currently a global pandemic. There are billions of people living in poverty. There is famine, disease, etc...&lt;/p&gt;&#xA;&lt;p&gt;Has any of the progress made (towards solving these problems) in the past been  done without rigorous problem-solving? Without collecting and analysing data (statistics)?&lt;/p&gt;&#xA;&lt;p&gt;Ah, &lt;em&gt;rigorous problem solving&lt;/em&gt;. This is, in my opinion, the most accurate definition of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone seriously think we can tackle any of these problems without greatly advancing technology for example machine learning or &lt;a href=&quot;https://en.wikipedia.org/wiki/Computer_simulation&quot; rel=&quot;nofollow noreferrer&quot;&gt;computer simulation&lt;/a&gt;, in which being good at mathematics is a prerequisite to even understanding to what is going on, let alone being able to work towards solutions?&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Why would / how can you not learn mathematics?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Even if you don't care about the world's serious problems, but just want to further your own career, you may want to advertise your product or service to make it arrive at the top of a google search. In which case, it may help to understand how &lt;a href=&quot;https://en.wikipedia.org/wiki/PageRank&quot; rel=&quot;nofollow noreferrer&quot;&gt;google's pagerank algorithm&lt;/a&gt; works and how you can exploit it (although there are already people who are specialist in this type of work- but clearly &lt;em&gt;they&lt;/em&gt; need to be good at maths.)&lt;/p&gt;&#xA;&lt;p&gt;Then there are Scientists who must use maths to both gather and analyse data and also to help solve problems e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical modelling&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What about trying to come up to solutions to economic problems. If you take a look at &lt;a href=&quot;https://economics.stackexchange.com/&quot;&gt;economics stackexchange&lt;/a&gt;. Take a look at some of the questions on there and notice the high proportion of questions with some reasonably heavy mathematical machinery in them.&lt;/p&gt;&#xA;&lt;p&gt;I could easily give another thousand examples, but you can find lists of the applications of mathematics if you do a quick google search.&lt;/p&gt;&#xA;&lt;p&gt;Again, in response to the original question of, &amp;quot;Why do I need to learn mathematics?&amp;quot;. I don't see how you can do without mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Not to the extent of air and water, I'm not delusional. For example, you &lt;em&gt;can avoid&lt;/em&gt; mathematics and pursue a career of art or music or entertainment or &amp;quot;social media influencer&amp;quot; and whilst I don't think these people are entirely useless to society, the fact remains that they are not the problem-solvers.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2021-04-30T21:11:55.230" LastActivityDate="2021-04-30T21:11:55.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20835" PostTypeId="2" ParentId="57" CreationDate="2021-04-30T21:02:41.883" Score="1" Body="&lt;p&gt;I have been a TA for 6 courses, and I have observed the following general pattern of the weekly 55-minute tutorial sessions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The instructor assigns problem sheets a few days in advance. The problem sheets contain:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;routine exercises&lt;/li&gt;&#xA;&lt;li&gt;problems that can be solved by an application of an important theorem taught that week&lt;/li&gt;&#xA;&lt;li&gt;problems that need to be broken down into multiple steps by fitting together two or more previously known results&lt;/li&gt;&#xA;&lt;li&gt;multi-step problems that cover ideas beyond what was covered in class&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The instructor emphasises that everyone should do their best to attempt every problem in the tutorial sheet before attending the tutorial. During the tutorial, I solve some selected problems on the board in detail.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;First, I pick one routine exercise of each type and display all the steps in my solution, and point out how the template works for similar problems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then, I pick the problems that demonstrate the application of a theorem, and again solve one problem of each type in detail and point out the signs that a particular theorem is to be applied, and also how to adjust the solution in similar situations.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This takes around 20 minutes. Sometimes the students offer alternative solutions or ask questions to clarify some steps, so I spend additional time as necessary.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For the problems that require multiple steps, I go over each one slowly, explaining each part of the breakdown carefully. In this part, I specifically ask the students for their ideas. Their suggestions are usually on the right track, so in such cases I invite them to elaborate further, and ask them to help me by suggesting what I should write next on the board. Even when a suggestion is incorrect or unfruitful, it offers a nice teaching moment to try and explore why that method won't work. The final step in such problems is usually some computation, either numerical or symbolic, that I leave for them to verify on their own. I have found the students to be confident of their computational skills, so they are comfortable with this: I do give them the final answer or expression for them to verify that they have got it right.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;By this time, there is usually very little time left to cover the difficult multi-step problems, so I open the floor for discussion. Students usually ask questions related to previous problems, or about the theory discussed in class that week, and that closes the tutorial session.&lt;/p&gt;&#xA;&lt;p&gt;The final exam never tests the students based on the difficult multi-step problems. So, those who were unable to attempt them are not at a disadvantage. At the same time, the bright students enjoy the challenge posed by those problems. I always stay back after the tutorial to answer any further questions, and I usually receive a couple of questions about those problems, which I'm happy to discuss.&lt;/p&gt;&#xA;&lt;p&gt;This method of running the tutorials seems to bring good results. In my experience, the weakest students need clear worked-out solutions to the simpler problems, which they can refer to later on. So they remain attentive for the first 20–30 minutes, and start zoning out a little after 40 minutes.&lt;/p&gt;&#xA;&lt;p&gt;Asking the bright students to solve the routine problems instead does not seem to help, because they can't express their work in a clear manner right off the bat, which is quite understandable. I also find that these students are able to find sufficient internal motivation to get through the first 20–30 minutes of easy problem-solving without losing much focus. So, I have a fully attentive class for at least 30 minutes, which is great. Then, when I engage the bright students in solving the more difficult problems, they are quite eager to offer their insights when attacking these problems, which is great because these are meant to develop their intuition and their understanding of the course material.&lt;/p&gt;&#xA;&lt;p&gt;Somewhat contrary to your description of your class, I find that the students resist engaging in discussion groups, and that any attempt to force them to do so results in a dissatisfying tutorial experience for everyone involved.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;To summarise:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I give detailed solutions to each &amp;quot;type&amp;quot; of routine exercise and application of a theorem. This usually takes 20–30 minutes.&lt;/li&gt;&#xA;&lt;li&gt;I invite discussion on the more conceptual problems, taking suggestions from the students on how to tackle them. This takes another 15–25 minutes.&lt;/li&gt;&#xA;&lt;li&gt;If any time is remaining, I spend that answering further questions on the tutorial problems or the theory covered in class.&lt;/li&gt;&#xA;&lt;li&gt;I also choose to stay back after the tutorial is officially done to answer deeper questions, and any other questions that could not be taken up during the tutorial due to lack of time.&lt;/li&gt;&#xA;&lt;li&gt;I specifically avoid making the students work in groups, and also avoid making the bright students explain the solutions to the routine exercises to the rest of the students.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15884" LastActivityDate="2021-04-30T21:02:41.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20836" PostTypeId="2" ParentId="20830" CreationDate="2021-05-01T02:10:07.113" Score="6" Body="&lt;p&gt;Senior computer science student here (graduating in December). I took a similar course titled &amp;quot;Discrete Math for Computing&amp;quot; in my freshman year. It's been a few years so my memory is hazy (and having just this morning finished the most stressful semester I've ever had doesn't help with that), but going through my homework files, it seems the topics were roughly:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;propositional and predicate logic, including identities, inference, and proving arguments&lt;/li&gt;&#xA;&lt;li&gt;mathematical induction&lt;/li&gt;&#xA;&lt;li&gt;recursive sequences (e.g. Fibonacci)&lt;/li&gt;&#xA;&lt;li&gt;set theory&lt;/li&gt;&#xA;&lt;li&gt;combinatorics (counting, combinations, permutations, but not probability)&lt;/li&gt;&#xA;&lt;li&gt;graph theory and trees, including tree traversal (preorder, inorder, postorder) and how those traversal methods relate to arithmetic expressions (e.g. reverse Polish notation)&lt;/li&gt;&#xA;&lt;li&gt;Boolean algebra, linking back to the logic and set theory sections, and including Karnaugh maps&lt;/li&gt;&#xA;&lt;li&gt;base-&lt;em&gt;n&lt;/em&gt; arithmetic (including a variety of &lt;em&gt;n&lt;/em&gt; from 2 to 16, but with a focus on binary and hexadecimal)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As I mentioned, this was a course designed for an incoming freshman's first semester (requiring only a math placement test result qualifying them for pre-calc or calculus), so it was more of a broad survey, without many deep dives into a single topic. It was also a mandatory course for all computer science students, not one intended only for students going to graduate school.&lt;/p&gt;&#xA;&lt;p&gt;Some future courses in the major expanded on some of the topics from this course; e.g. our statistics course expanded on combinatorics and really dived into probability, and we have a separate &amp;quot;Logic for Computer Science&amp;quot; course (also required) typically taken as a junior or senior that takes the propositional and predicate logic section from the discrete math course and expands on that topic over a full semester. Things like finite state machines and the halting problem are covered in various other mandatory courses in the major.&lt;/p&gt;&#xA;&lt;p&gt;Since your course is for sophomores and apparently an elective, this obviously wouldn't perfectly apply to you, but hopefully it's helpful to see what an existing similar course elsewhere teaches.&lt;/p&gt;&#xA;" OwnerUserId="15983" LastActivityDate="2021-05-01T02:10:07.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20837" PostTypeId="2" ParentId="20830" CreationDate="2021-05-01T03:57:33.527" Score="5" Body="&lt;p&gt;One very important point to realize is that &lt;strong&gt;using FOL&lt;/strong&gt; is completely different from &lt;strong&gt;studying FOL&lt;/strong&gt;. The first is necessary for all logical reasoning, and hence you should include it in your course, if not how will those students ever learn to use FOL? The second is not; it is a mathematical investigation of FOL itself, and is carried out in a suitable meta-system MS, and is what the field of &amp;quot;mathematical logic&amp;quot; is all about. While mathematical logic is very interesting, it is &lt;strong&gt;not&lt;/strong&gt; a good idea to attempt to teach it to beginners who do not yet know how to use FOL, not the least because working in MS requires using FOL already!&lt;/p&gt;&#xA;&lt;p&gt;So yes the topics you mentioned (PL and then FOL including basic stuff like quantifier order) ought to be taught to students as early as possible. In fact, almost all courses with the title &amp;quot;mathematical logic&amp;quot; will &lt;em&gt;not&lt;/em&gt; teach those stuff since it is already assumed, and they instead cover theorems of MS including semantic completeness of FOL and syntactic incompleteness of certain formal systems.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-05-01T03:57:33.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20838" PostTypeId="2" ParentId="20830" CreationDate="2021-05-01T04:00:55.877" Score="16" Body="&lt;p&gt;The Association for Computing Machinery (which is the professional association for computer scientists) puts out a curriculum guideline for computer science.  While the guideline is for what an entire program should include and does not specify how it should be broken into courses, the 41 hours of instruction it recommends under the heading of &amp;quot;Discrete Structures&amp;quot; is frequently taught as one course under a title like &amp;quot;Discrete Mathematics for Computer Science&amp;quot;.  They recommend:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;4 hours on sets, relations, and functions&lt;/li&gt;&#xA;&lt;li&gt;9 hours on basic logic&lt;/li&gt;&#xA;&lt;li&gt;11 hours on proof techniques&lt;/li&gt;&#xA;&lt;li&gt;5 hours on basic counting&lt;/li&gt;&#xA;&lt;li&gt;4 hours on graphs and trees&lt;/li&gt;&#xA;&lt;li&gt;8 hours on discrete probability&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For more details, see p. 76-81 in &lt;a href=&quot;https://www.acm.org/binaries/content/assets/education/cs2013_web_final.pdf&quot; rel=&quot;noreferrer&quot;&gt;ACM 2013 computer science curriculum&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2021-05-01T04:00:55.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20839" PostTypeId="1" CreationDate="2021-05-01T12:34:31.567" Score="2" ViewCount="106" Body="&lt;p&gt;Suppose you have a carton that has the form of a square with sides of length a. If we want to produce a box out of it whose height is x we might deduce the following formula:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$V_a(x)= x(a-2x)^2=a^2 x - 4 a x^2 + 4 x^3$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I want to explore this formula with my students because it is very interesting to see that the specified points in the title of this thread are both located on two special curves. Since I don't want to say to my students here is your exercise and start working, I am reflecting on creating a good introduction that is also exciting for the students.&lt;/p&gt;&#xA;&lt;p&gt;Since the locus has no use in an application related to this exercise I decided that I don't want to use a problemoriented introduction. The point is that every problemoriented introduction to this topic or discovery oriented process leads to application contexts which only need the y-coordinates of the specified points. So I hope that you might have an idea for an introduction to my students that is a little bit funny or exciting.&lt;/p&gt;&#xA;&lt;p&gt;Remark: My students are able to examine functions based on their term by using the concept of the first, second and third derivative.&lt;/p&gt;&#xA;" OwnerUserId="10036" LastActivityDate="2021-05-01T12:34:31.567" Title="Locus of the maximal turning point and the point of inflection" Tags="|mathematical-analysis|derivative|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20840" PostTypeId="2" ParentId="20830" CreationDate="2021-05-01T13:52:53.370" Score="11" Body="&lt;p&gt;This is standard in the Discrete Mathematics course. I teach it, and all of the published textbooks I've looked at start similarly with the unit on logic. A few points on this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I mean, you need that stuff.&lt;/li&gt;&#xA;&lt;li&gt;Math majors take the course, not just CS majors.&lt;/li&gt;&#xA;&lt;li&gt;Frequently the whole &lt;em&gt;raison d'être&lt;/em&gt; of the discrete math course is a replacement in lieu of a dedicated introduction to proofs course.&lt;/li&gt;&#xA;&lt;li&gt;The CS majors need that platform in order in order to engage with proofs about algorithms, re: halting, correctness, tractability.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;My first instinct would just be to pick a good textbook and follow the same sequence; if you're mostly focused on research, then the textbook author has thought about the scaffolding issues much more deeply than you have.&lt;/p&gt;&#xA;&lt;p&gt;My second suggestion, if that's not compelling, is to look at the rest of the sequence of your program. Has logic already been taught formally in another course? Then you can skip it. Has it not? Then the discrete math course is the intended delivery mechanism.&lt;/p&gt;&#xA;&lt;p&gt;Finally: Not everything in the undergraduate program needs to link immediately to a trendy current research topic. Your students need the basics first, including (perhaps first and foremost: see the &lt;a href=&quot;https://en.wikipedia.org/wiki/Trivium&quot; rel=&quot;noreferrer&quot;&gt;Trivium&lt;/a&gt;) classical logical.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2021-05-01T13:52:53.370" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20841" PostTypeId="2" ParentId="20824" CreationDate="2021-05-01T19:34:14.537" Score="1" Body="&lt;p&gt;I wouldn't introduce the notation 1x, which they won't see on any standardized test or in any higher level class, unless it is necessary for student with an IEP.  Instead I suggest the following solution (to be proposed to your CE).&lt;/p&gt;&#xA;&lt;p&gt;You say in the comments that you hadn't taught combining like terms to the class, but they had learned it before.  It would seem to me that they have forgotten what they learned and need a review of the material and problems to help drill the concepts.&lt;/p&gt;&#xA;&lt;p&gt;Tell the class that many have forgotten how to combine like terms which is essential for all the work you will be doing so you are going to review.  When reviewing working with like terms:  6x - x,  I would say that since x is the same as one x, then 6x - x = 5x. You've explained it without writing.&#xA;If they need a concrete example -  have a 6 pack of soda and take away one can of soda, leaving 5 cans.&lt;/p&gt;&#xA;&lt;p&gt;Once you have reviewed it, give them some problems for drill. Walk around and check that they are doing it correctly.  If needed give a few drill problems at the start of each class, until the class has gotten it.&#xA;I would think this would be enough for mastery.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-05-01T19:34:14.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20842" PostTypeId="1" CreationDate="2021-05-01T22:26:59.093" Score="6" ViewCount="856" Body="&lt;p&gt;So, I realize this can be a broad question, so I'll narrow it down. I have lived in Spain and own several Math textbooks from that country (the equivalent of 8th grade and high school Math). Has anyone here been able to note major differences in the way Limits are taught as well as Introductory Calculus in general? I have noticed that in Spain, they always tend to begin with Sequences before introducing the Limit. I have not seen that approach here in the US. Am I wrong? If so, what book would be closest to the way Calculus/High School level maths are taught in Spain? Also, my impression is (and again, I could be wrong) but just comparing textbooks, the Math level seems to be much more demanding/rigorous in Spain than it is here in the US.&lt;/p&gt;&#xA;" OwnerUserId="15987" LastEditorUserId="2083" LastEditDate="2021-05-02T06:03:30.477" LastActivityDate="2021-05-03T12:57:29.693" Title="Calculus limits taught in the US vs Spain?" Tags="|calculus|usa|limits|spain|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20843" PostTypeId="2" ParentId="20842" CreationDate="2021-05-01T23:35:34.217" Score="4" Body="&lt;p&gt;In my experience, we don't really expect students to have a strong understanding of limits in Calc 1 in the US. I've always been told to skip the epsilon-delta chapter, and sequences are never brought up until Calc 2.&lt;/p&gt;&#xA;&lt;p&gt;When I took a real analysis class, we covered limits of sequences first. It seems like a good approach if you want to get a strong understanding for limits.&lt;/p&gt;&#xA;&lt;p&gt;Lax &amp;amp; Terrell's Calculus with Applications (&lt;a href=&quot;https://link.springer.com/book/10.1007%2F978-1-4614-7946-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/book/10.1007%2F978-1-4614-7946-8&lt;/a&gt;) Starts with:&lt;/p&gt;&#xA;&lt;p&gt;Chapter 1: Numbers and Limits&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1.1: Inequalities&lt;/li&gt;&#xA;&lt;li&gt;1.2: The Least Upper Bound Theorem&lt;/li&gt;&#xA;&lt;li&gt;1.3: Sequences and their Limits&lt;/li&gt;&#xA;&lt;li&gt;1.4: The Number e&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-05-01T23:35:34.217" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20844" PostTypeId="2" ParentId="20830" CreationDate="2021-05-02T07:14:46.257" Score="3" Body="&lt;p&gt;The prerequisites of the course are important in recognizing your students' degree of exposure and penetration into these topics. If they've already completed curricula on formal logic and fundamental proofs, then repeating these in depth across too many lectures might be suboptimal and too easy/boring; on the other hand, providing a review at the beginning of the semester might be helpful. Summarily, it depends on the coursework that ALL students need to take during freshman year.&lt;/p&gt;&#xA;&lt;p&gt;If they haven't taken any course in formal logic or math proofs, then I'd recommend:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Week 1: intro formal logic, truth tables, propositional logic, common logical fallacies&lt;/li&gt;&#xA;&lt;li&gt;Week 2: algebraic proofs, contrapositives, proof by contradiction&lt;/li&gt;&#xA;&lt;li&gt;Week 3: (strong) induction&lt;/li&gt;&#xA;&lt;li&gt;Week 4: minterm/maxterm, logic statement simplification, kmaps, finite state machines / automata (&amp;quot;digital electronics week&amp;quot;)&lt;/li&gt;&#xA;&lt;li&gt;Week 5: set theory basics and notation; countable and uncountable infinite sets (cardinality); subsets / partitioning&lt;/li&gt;&#xA;&lt;li&gt;Week 6: the definition of functions; relations; bijectivity&lt;/li&gt;&#xA;&lt;li&gt;Week 7: conditional probability, Bayes, Bernoulli processes&lt;/li&gt;&#xA;&lt;li&gt;Week 8: intro for combinatorics and permutations; counting two ways; binomial theorem; bijection counting&lt;/li&gt;&#xA;&lt;li&gt;Week 9: combinatorial proofs with inclusion/exclusion; counting subsets&lt;/li&gt;&#xA;&lt;li&gt;Week 10: trees and traversal&lt;/li&gt;&#xA;&lt;li&gt;Week 11: basic graphs, isomorphism; coloring; min paths and optimization techniques&lt;/li&gt;&#xA;&lt;li&gt;Week 12: GCDs; Euclidean algorithm; modular arithmetic, Fermat's, Diophantine equations&lt;/li&gt;&#xA;&lt;li&gt;Week 13+: applications in efficient algorithm design; algorithmic complexity; ensuring code correctness and making assertions; identifying and dealing with edge cases; statistical computing; recursion; network optimization; cryptography.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15990" LastActivityDate="2021-05-02T07:14:46.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20845" PostTypeId="2" ParentId="18567" CreationDate="2021-05-02T08:39:49.397" Score="2" Body="&lt;p&gt;One way to remember the mathematics that you learn is to create a narrative that explains how the concepts are threaded together. This narrative need not have any historical relevance (and it usually cannot, since historical developments are usually tangled and messy). The goal is to lay down a road from start to finish, passing through all the important checkpoints (definitions, theorems, etc.) in the material.&lt;/p&gt;&#xA;&lt;p&gt;Repeat this narrative to yourself as you develop it, continuously refining it as you learn more of the material. Of course, you may not necessarily have a linear path through all the topics you need to cover, so create bylanes wherever necessary in the same manner. Ideally, every proof of a theorem should be a mini-narrative on its own. Repeat these narratives to yourself over and over as you create them and as you learn the material.&lt;/p&gt;&#xA;&lt;p&gt;This should give you a strong base which you can use to recall the mathematics you learned even after a long time. Perhaps the roads you created become dusty and unused, perhaps the small paths become overgrown and slightly hidden from sight, but the broad directions will still lie there waiting for you to traverse them again. With just a bit of effort the next time, you can clear away the paths and speed along confidently from start to finish, or wherever you wish to go.&lt;/p&gt;&#xA;&lt;p&gt;This is related to what Timothy Chow calls &amp;quot;an exposition problem&amp;quot;&lt;sup&gt;1&lt;/sup&gt; (also popularized by Tim Gowers in his blog posts&lt;sup&gt;2,3&lt;/sup&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;All mathematicians are familiar with the concept of an &lt;em&gt;open research problem&lt;/em&gt;. I propose the less familiar concept of an &lt;em&gt;open exposition problem&lt;/em&gt;. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves. The proofs should be &amp;quot;natural&amp;quot; in Donald Newman’s sense&lt;sup&gt;4&lt;/sup&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This term … is introduced to mean not having any ad hoc constructions or &lt;em&gt;brilliancies&lt;/em&gt;. A &amp;quot;natural&amp;quot; proof, then, is one which proves itself, one available to the &amp;quot;common mathematician in the streets.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have found this method to be very useful in not only retaining in my memory the mathematics that I learn, but also in explaining the theory to others on the fly.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;Footnotes&lt;/h3&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;T. Y. Chow (2008). &amp;quot;A beginner's guide to forcing.&amp;quot; &lt;a href=&quot;https://arxiv.org/abs/0712.1320&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv:0712.1320&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;W. T. Gowers (2009, May 20). &amp;quot;A solution to an exposition problem.&amp;quot; &lt;em&gt;Gowers's Weblog&lt;/em&gt;.  &lt;a href=&quot;https://gowers.wordpress.com/2009/05/20/a-solution-to-an-exposition-problem/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gowers.wordpress.com/2009/05/20/a-solution-to-an-exposition-problem/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;W. T. Gowers (2011, Nov 20). &amp;quot;Normal subgroups and quotient groups.&amp;quot; &lt;em&gt;Gowers's Weblog&lt;/em&gt;. &lt;a href=&quot;https://gowers.wordpress.com/2011/11/20/normal-subgroups-and-quotient-groups/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gowers.wordpress.com/2011/11/20/normal-subgroups-and-quotient-groups/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;D. J. Newman (1998). &amp;quot;Analytic Number Theory.&amp;quot; Springer–Verlag.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15884" LastActivityDate="2021-05-02T08:39:49.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20847" PostTypeId="2" ParentId="10680" CreationDate="2021-05-03T04:23:10.180" Score="1" Body="&lt;p&gt;i'd like to recommend&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.amazon.co.uk/Isaac-Newton-Apple-Dead-Famous/dp/0590114069/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Isaac Newton and his Apple&lt;/a&gt; and &lt;a href=&quot;https://www.amazon.co.uk/Murderous-Maths-10-Books-Slip/dp/1407131931/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Murderous maths&lt;/a&gt; series by Poskitt Kjartan (Author) and Philip Reeve (Illustrator). Such books explained many basic math concepts in lots of jokes. My son really loves it (10 out of 10, same as Roblox and icecream).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prof Richard Evan Schwartz's &lt;a href=&quot;http://www.richardevanschwartz.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;picture books&lt;/a&gt; especially Gallery of the Infinite, which explains Cantor diagonalization argument,  Cantor-Bernstein Theorem etc in a quite attractive way.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.amazon.co.uk/dp/0691149151/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Prime Suspects: The Anatomy of Integers and Permutations: A Mathematical Sciences Investigation&lt;/a&gt; by Andrew Granville (Author) and Jennifer Granville (Illustrator), it discussed number theory in the format of comic book.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9059" LastActivityDate="2021-05-03T04:23:10.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20848" PostTypeId="2" ParentId="10680" CreationDate="2021-05-03T04:57:41.397" Score="4" Body="&lt;p&gt;The ultimate for this, which is an entire curriculum, is &amp;quot;&lt;a href=&quot;https://www.rainbowresource.com/category/9813/Life-of-Fred.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Life of Fred&lt;/a&gt;&amp;quot; book series. It is fantastic, and you can get an entire math education, elementary to undergrad, just through Fred books.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastEditorUserId="9594" LastEditDate="2021-05-03T11:44:47.837" LastActivityDate="2021-05-03T11:44:47.837" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20849" PostTypeId="2" ParentId="20842" CreationDate="2021-05-03T12:57:29.693" Score="5" Body="&lt;p&gt;IME, it's a generational thing.  When I was in HS in the US in the early 80's, we studied convergent sequences and Cauchy sequences in the year before AP Calculus.  (We didn't get to the punchline that the real numbers were the completion of the rationals, so it was not clear at the time what Cauchy had to do with anything.)&lt;/p&gt;&#xA;&lt;p&gt;Nowadays in the US, it seems like the goal is to expose a broad range of majors to the concepts of calculus at the expense of rigor.  Limits are arguably the biggest thing that got pitched.  In our defense, recognizing continuity is highly intuitive and then you can retroactively describe the limit as the behavior of a function &amp;quot;near&amp;quot; a member of its domain instead of at the value (which can be gathered by reading a graph or building a table of values).  Students majoring in math or engineering then take Real Analysis later in their studies which goes into all of the pedantic details.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-05-03T12:57:29.693" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20850" PostTypeId="2" ParentId="20824" CreationDate="2021-05-03T17:35:05.633" Score="8" Body="&lt;p&gt;I would like to comment on the question: &lt;strong&gt;For the purpose of teaching, should we normalize writing 1x instead of x?&lt;/strong&gt; not for attempting an answer but to ask why one should consider it a (not so) reasonable idea.&lt;/p&gt;&#xA;&lt;p&gt;I think that behind such question there is the idea that if I write &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; then the algorithm for sum would be clearer. I think what's wrong in this position is the idea of solving teaching problems always via algorithms: &lt;em&gt;if I give my students the universal working algorithm they'll just need applying it&lt;/em&gt;. Which is however the opposite of understanding.&lt;/p&gt;&#xA;&lt;p&gt;It's a teaching direction that does not work, in my opinion, for two reasons:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;it is practically impossible to write decent math without such notational abuses (then shouldn't we more completely write &lt;span class=&quot;math-container&quot;&gt;$1\cdot x$&lt;/span&gt;)? Attempts to eliminate them all are a path to failure.&lt;/li&gt;&#xA;&lt;li&gt;learning how to deal with such abuses is a good way to improve your understanding. And one should not free the path of a student from learning opportunities.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It is however important to develop awareness: a teacher should be aware that for someone such ambiguities are source of misunderstandings so should be ready to unravel and explain. Students should grow in awareness that at a number of points notational shortcuts can be dangerous ans should reflect on them.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2021-05-03T17:35:05.633" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20851" PostTypeId="2" ParentId="20830" CreationDate="2021-05-03T19:40:29.893" Score="3" Body="&lt;p&gt;&amp;quot;Discrete Math&amp;quot; is a pretty broad way of categorizing topics in modern mathematics. It was required in my second year of undergrad CS, and mathematical logic was the central theme. We used &amp;quot;Discrete Mathematics and Its Applications&amp;quot; by Kenneth Rosen, and the syllabus followed the same progression as the chapters in the text. We covered every chapter except the last one (&amp;quot;Modeling Computation&amp;quot; which was covered extensively in an upper division course dedicated to that topic). It more or less went like this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;propositional logic, predicates, quantifiers, and proof techniques&lt;/li&gt;&#xA;&lt;li&gt;sets, functions, sequences, series, cardinality, countable sets&lt;/li&gt;&#xA;&lt;li&gt;algorithms and complexity&lt;/li&gt;&#xA;&lt;li&gt;basic number theory (divisibility and congruence, modular arithmetic, residue classes, primality)&lt;/li&gt;&#xA;&lt;li&gt;induction, recursive relations, solving linear recurrence relations, inclusion/exclusion&lt;/li&gt;&#xA;&lt;li&gt;counting, generalized permutations and combinations&lt;/li&gt;&#xA;&lt;li&gt;basic probability, Bayes, expected value&lt;/li&gt;&#xA;&lt;li&gt;brief introduction to graphs/trees, their properties and common operations&lt;/li&gt;&#xA;&lt;li&gt;boolean algebra and functions (covered in greater detail in another required course)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;We took several detours to topics outside of the text that our instructor thought would be beneficial, if not interesting (we proved Euler's identity and that Q is dense in R). It was challenging, but I liked that class. I still use the book as a reference occasionally. This was ~15 years ago at one of the larger midwestern universities.&lt;/p&gt;&#xA;" OwnerUserId="15293" LastEditorUserId="15293" LastEditDate="2021-05-03T19:53:20.057" LastActivityDate="2021-05-03T19:53:20.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20852" PostTypeId="2" ParentId="20830" CreationDate="2021-05-03T22:48:54.253" Score="5" Body="&lt;p&gt;I speak from the perspective of a someone who got a bachelors in CS who is currently teaching a coworker programming. I have no formal training in education. I believe Propositional Logic is essential to a CS education and programming in general, so if your students aren't getting this knowledge elsewhere (which Discrete Math for a CS student atleast in my case was the only oppurtunity I had to learn this stuff in depth) make sure they are getting it in your class. Predicate Logic feels essential as well. There is a great amount of simplicity that can come into a program by not overcomplicating your propositional logic. Also if you don't know the equivalence of different propositional statements when you get in the work force it would be hard to read others code if they do it differently then you might generally.&lt;/p&gt;&#xA;&lt;p&gt;As a very simple and overly trivial example &lt;code&gt;!(a == b)&lt;/code&gt; can be written as &lt;code&gt;a != b&lt;/code&gt;. And &lt;code&gt;!(a &amp;amp;&amp;amp; b)&lt;/code&gt; can sometimes be written as &lt;code&gt;!a || !b&lt;/code&gt; by De Morgan's law. Not knowing these logical equivalences can continue to confuse a CS student.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Predicate Logic: Negating Quantifiers, Combining Quantifiers, Order of Quantifiers I am not familiar with this topic, so it's hard for me to judge if including it will be beneficial for students when they go to graduate schools. Is mathematical logic still an active domain of research in computer science? Should I include it in my course?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Predicate logic is important to young students, because it provides a building block to help them reason about correctness of Computer Programs and how to refactor (and possibly simplify or even improve efficiency of) them and keep the same outcome just as it was with propositional logic. For example one such example I gave to my coworker recently was about negating quantifiers. The two basic quantifiers are the existential quantifier (usually stated as a there exists statement. Such as &amp;quot;there exists a P such that it is Q&amp;quot;), and the universal quantifier (usually stated as an all statement, such as &amp;quot;all Ps are Qs&amp;quot;). We thought through a simple exercise with my coworker that said something to the effect of: &amp;quot;If not all of the numbers are greater than 3 return true otherwise false&amp;quot;. Now it might be obvious to some, but the secret that predicate logic teaches you in the &amp;quot;NOT ALL Ps are Qs&amp;quot; part of the statement is that it's true &amp;quot;if and only if&amp;quot; (a.k.a. IFF, and in other less formal words only as long as) &amp;quot;THERE EXISTS atleast one P such that it is NOT Q&amp;quot;. If and only if is another way to state the two statements are either true or false together.&lt;/p&gt;&#xA;&lt;p&gt;So, given the above problem statement &amp;quot;If not all of the numbers are greater than 3 return true otherwise false&amp;quot;, do you need to always check ALL of the n numbers? The answer is no not always, because as soon as you find THERE EXISTS one counterexample such that the statement is NOT the case you can break the loop. Yes this is a very trivial and simple example in this case only gave minor efficiency gains, but reasoning and creating logical proofs by induction and knowing things like loop invariants (such as ALL &lt;span class=&quot;math-container&quot;&gt;$i - 1$&lt;/span&gt; elements up to the point of iteration are sorted) which both almost always use some form of quantifier will help your CS students so much in their Graduate level CS classes.&lt;/p&gt;&#xA;&lt;p&gt;At some point they will probably have to prove correctness of algorithms in their graduate and hopefully their undergraduate studies. I only took one graduate level class in CS (during my bachelor's degree) and that was Computational Geometry which my teacher described as very similar to his Advanced Algorithms class he gave to undergraduate students. This computational geometry class I had to use both propositional logic as well as predicate logic on basically every single question and for sure for every assignment. That graduate class was basically pure theory with no code, but it was absolutely enlightening and logical proofs were absolutely a corner stone of that class.&lt;/p&gt;&#xA;&lt;p&gt;Unless you are absolutely certain the CS students are learning these logical proof methods as a prereq to your class and to their other theory heavy CS classes, I would highly encourage you to teach them it. If you want help with learning the basics of Propositional and Predicate Logic I would check out the first and second chapters of the great &lt;a href=&quot;https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/readings/&quot; rel=&quot;noreferrer&quot;&gt;course notes from the Mathematics for Computer Science course&lt;/a&gt; taught at MIT (basically that course is their version of Discrete Math, so feel free to borrow ideas from them on what should be taught to CS students). I learned all I know about propositional and predicate logic from those notes and my Discrete Math course in college, and it was never repeated to that same extent anywhere else in my CS education.&lt;/p&gt;&#xA;" OwnerUserId="16002" LastActivityDate="2021-05-03T22:48:54.253" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20853" PostTypeId="2" ParentId="20824" CreationDate="2021-05-03T23:29:41.750" Score="3" Body="&lt;p&gt;It is important to become comfortable with disambiguation. For example, in listening we must use context to distinguish between the homophones “reed” and “read” as in “I will read a book tonight.” In reading, we must use context to know when to pronounce “read” with a &lt;em&gt;long e&lt;/em&gt; sound as in “reed” or with a &lt;em&gt;short e&lt;/em&gt; sound as in “red.” The brain is an amazing disambiguation machine: it is  &lt;strong&gt;much&lt;/strong&gt; better than a search engine such as Google.&lt;/p&gt;&#xA;&lt;p&gt;We do students a disservice by trying to insulate them from exercising disambiguation skills. Maybe instead of trying to change the common practice of writing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$1x$&lt;/span&gt;, you could approach this issue by taking &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; and writing it as &lt;span class=&quot;math-container&quot;&gt;$x+x$&lt;/span&gt;. If one is having difficulty seeing that  &lt;span class=&quot;math-container&quot;&gt;$2x+x$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$3x$&lt;/span&gt;, then the teacher could write &lt;span class=&quot;math-container&quot;&gt;$(x+x)+x$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$2x+1x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When students are learning rules of exponents and getting confused with &lt;span class=&quot;math-container&quot;&gt;$x^nx^m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x^n)^m$&lt;/span&gt; it can help to trot out the examples &lt;span class=&quot;math-container&quot;&gt;$a^2a^3$&lt;/span&gt;=&lt;span class=&quot;math-container&quot;&gt;$(aa)(aaa)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(a^2)^3=(aa)(aa)(aa)$&lt;/span&gt;. Eventually the brain becomes quick at sorting out &lt;span class=&quot;math-container&quot;&gt;$a^{n+m}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a^{nm}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We cannot avoid disambiguation. In fact we need to embrace it, because it underlies important skills.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-05-03T23:29:41.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20854" PostTypeId="2" ParentId="20830" CreationDate="2021-05-04T10:08:15.153" Score="3" Body="&lt;p&gt;&lt;strong&gt;TL;DR: FOL is the Foundation of almost all Information Representation&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'd recommend to include FOL for one more reason:&lt;/p&gt;&#xA;&lt;p&gt;It's the foundation of nearly all information representation we do in software engineering, e.g. the direct basis of relational database systems.&lt;/p&gt;&#xA;&lt;p&gt;Knowing about FOL makes clear what the limitations of such representations are, e.g. the virtually non-existent support for negating statements.&lt;/p&gt;&#xA;&lt;p&gt;And personally, I find the interpretation of predicates as natural-language sentences with placeholders very useful, as a documentation what the bits and bytes in our data storages mean.&lt;/p&gt;&#xA;" OwnerUserId="15862" LastActivityDate="2021-05-04T10:08:15.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20855" PostTypeId="2" ParentId="20792" CreationDate="2021-05-04T23:13:49.363" Score="3" Body="&lt;p&gt;I like &lt;a href=&quot;http://www.williamchen-mathematics.info/ln.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Chen's lecture notes&lt;/a&gt; a lot. Clear, nicely structured. He wrote a lot, on a variety of topics ranging from first-year topics to rather advanced stuff.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2021-05-04T23:13:49.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20856" PostTypeId="1" CreationDate="2021-05-05T06:33:00.107" Score="0" ViewCount="1003" Body="&lt;p&gt;Where can I find manual solution for textbooks like Advanced Linear Algebra by Rotman or Introduction to Smooth manifolds by Lee? any help would be appreciated&lt;/p&gt;&#xA;" OwnerDisplayName="user16012" LastActivityDate="2021-05-05T06:59:35.590" Title="manual solutions to graduate textbooks" Tags="|textbooks|" AnswerCount="2" CommentCount="6" ClosedDate="2021-05-05T15:04:00.250" ContentLicense="CC BY-SA 4.0" />
  <row Id="20857" PostTypeId="2" ParentId="20856" CreationDate="2021-05-05T06:47:38.300" Score="1" Body="&lt;p&gt;Sometimes, publishers will sell (or send for free) solutions manuals to problems in their textbooks if you are a teacher that is using the textbook in class.  They might ask for proof of this (for example, a class webpage listing the textbook as the reference with your name as the teacher).  (This happened to me a few decades ago---a probability textbook by Peebles from McGraw-Hill.)  Visit the publisher's website or contact the publisher directly to find out if they have such a manual available.&lt;/p&gt;&#xA;&lt;p&gt;Of course, some authors do not create solutions manuals for their textbooks.  There is a chance that someone else has created such a manual (for example, I know a faculty member who did this).  You might find these by using an internet search engine.  But be careful in this case because you would not be sure if the solutions are correct.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2021-05-05T06:47:38.300" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20858" PostTypeId="2" ParentId="20856" CreationDate="2021-05-05T06:59:35.590" Score="-2" Body="&lt;p&gt;I assume you mean roman not rotman.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;lee smooth manifolds. look up 'SOLUTIONS TO INTRODUCTION TO SMOOTH MANIFOLDS BY JOHN M. LEE, 2012, SPRINGER' by ernest yeung&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;lee smooth manifolds again. try here: &lt;a href=&quot;https://wj32.org/wp/mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://wj32.org/wp/mathematics/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Steven Roman’s Advanced Linear Algebra. try here: &lt;a href=&quot;https://wj32.org/wp/mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://wj32.org/wp/mathematics/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5987" LastActivityDate="2021-05-05T06:59:35.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20859" PostTypeId="2" ParentId="2121" CreationDate="2021-05-07T02:53:20.357" Score="3" Body="&lt;p&gt;Answer 1: Ease of use.&#xA;Some things are easier to use if they are symmetrical. Eg a box lid, bricks, shape sorter toy, device connections such as HDMI. (Think of the frustration using a USB device because of the standards limited symmetry).&lt;/p&gt;&#xA;&lt;p&gt;Answer 2: Safety of use.&#xA;Some things are safer to use if they have limited or no symmetrical. Eg 3 pin plug, SIM cards, PC memory cards&lt;/p&gt;&#xA;" OwnerUserId="16030" LastActivityDate="2021-05-07T02:53:20.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20860" PostTypeId="2" ParentId="2121" CreationDate="2021-05-07T16:19:33.957" Score="2" Body="&lt;p&gt;It's a key aspect of chemical structure determination.  I once saw someone prove that a posited isomer was the same molecule (there was a subtle C2 axis), to the flummoxing of a seminar speaker.  For similar reasons it is a critical aspect of NMR and IR spectrum analysis (affects peak numbers).&lt;/p&gt;&#xA;&lt;p&gt;Cotton's Group Theory Chemical Applications is the classic reference.  But there are many competing texts or just shorter sections within inorganic chem texts.  Or articles on the Web.  (Google away.  For starters, the Wiki article on Molecular Symmetry is decent and accessible.)&lt;/p&gt;&#xA;&lt;p&gt;Cotton is a (relatively easy) grad chem text.  But there are many lab manuals or parts of chem, even at the HS level that discuss symmetry.  It's probably not the most important aspect of chemistry (would put stoichiometry, gas laws, reaction balancing, periodic trends, etc. earlier).  But it is a fun, insightful, easy aspect, especially if done in a lab  (with clay, toothpicks and foam, etc.)&lt;/p&gt;&#xA;&lt;p&gt;Also, within HS biology it is normal to say that humans (and many creatures) have bilateral symmetry.  The mirror plane that runs through you.  (Relatively speaking, of course, there are off-axis internal organs.)  Starfish have a C5 axis and the five mirror planes (again, within reason).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Also, in terms of math, I think this is one of the easiest, yet still abstract, areas to learn about.  If you want a special topic, it's an easier fit than pushing matrices or projective geometry or induction or the like.&lt;/p&gt;&#xA;" OwnerUserId="16033" LastActivityDate="2021-05-07T16:19:33.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20861" PostTypeId="2" ParentId="2121" CreationDate="2021-05-07T18:37:56.777" Score="0" Body="&lt;p&gt;Some very simple examples:&lt;/p&gt;&#xA;&lt;p&gt;The commutative property of multiplication is a symmetry. When you learn your times table in grade school, this symmetry cuts the amount of memorization roughly in half.&lt;/p&gt;&#xA;&lt;p&gt;In STEM, students learn lots of equations that have squares in them. Almost always, there is a nice concept that can be harvested by observing that the square gets rid of the effect of any negative sign. As a random example, the heat output by a resistor is given by &lt;span class=&quot;math-container&quot;&gt;$P=I^2R$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is the current. Flipping the sign of &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; makes no difference, and that makes sense because a resistor can't be made into a refrigerator just by reversing the current. STEM students should be coached to go through this simple thought process every time they encounter a square.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes we need to break a symmetry for practical reasons, such as when two people are trying to pass through a door, and they do the dance where each person is trying to let the other through, but they keep blocking each other inadvertently. Another example is the arbitrary choice that for English-language books, the title is printed on the spine in such a way that when you tilt your head to the right, you can read all the titles of the books on a shelf.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/fznGs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fznGs.png&quot; alt=&quot;climbing not&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The photos show another example where asymmetry is an advantage. This is a piece of rock climbing equipment known as a &lt;a href=&quot;https://en.wikipedia.org/wiki/Nut_(climbing)&quot; rel=&quot;nofollow noreferrer&quot;&gt;nut&lt;/a&gt;. It's used for the same purpose as a piton, but it doesn't damage the rock. You stick it in a crack, tug to seat it firmly, and then clip the rope to it. The first nuts were actually literal mechanical nuts, as in nuts and screws, that British climbers scavenged off of railroad tracks and put nylon cord through. Later people started manufacturing purpose-made nuts like this one, and they started intentionally making them with asymmetric shapes like this. The asymmetry means that any given nut can be placed in four different ways, which means that you have a much better chance of finding a good fit to a given crack.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-05-07T18:37:56.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20862" PostTypeId="1" CreationDate="2021-05-07T23:03:02.013" Score="7" ViewCount="180" Body="&lt;p&gt;Preparing for a class I teach this summer and stumbled upon a technology problem.&lt;/p&gt;&#xA;&lt;p&gt;I want to show my students a graph, ask students to annotate it privately (eg mark mean on a histogram) and then overlay all of their images with the true mean to show the distribution of their guesses. Is there an implemented solution?&lt;/p&gt;&#xA;" OwnerUserId="16036" LastActivityDate="2021-05-09T19:48:04.230" Title="Looking for a collaborative drawing solution" Tags="|online-instruction|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20864" PostTypeId="2" ParentId="2121" CreationDate="2021-05-08T01:01:35.070" Score="0" Body="&lt;p&gt;I recently used symmetry to work out the coordinates of the vertices of an archimedian-ish solid. This significantly reduced the number of variables I needed when setting up and solving the equations. A simpler example might be using symmetry to work out the coordinates of vertices of a hexagon or octogon. Once you've determined the x and y coordinates of a nontrivial point, you can use symmetry to work out all the other coordinates, so you only need two variables, and two equations for distances.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-05-08T01:01:35.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20865" PostTypeId="2" ParentId="2121" CreationDate="2021-05-08T02:19:22.733" Score="1" Body="&lt;p&gt;I am not sure whether to give this as a comment or an answer, but sometimes symmetry can give us clever simple solutions to things. Understanding symmetry together with angles, for example, gives this nifty technique for building a &lt;a href=&quot;https://www.youtube.com/watch?v=RKoOFApig-A&quot; rel=&quot;nofollow noreferrer&quot;&gt;really cool-looking cabinet&lt;/a&gt;. (Video: a method using complementary/supplementary angles to get a cabinet that fits together nicely, using the symmetry of an isosceles trapezoid, in particular, and its angle relations).&lt;/p&gt;&#xA;&lt;p&gt;I particularly like this motivation for learning symmetry  because it gives an application for students who can't imagine themselves studying advanced math or science -- but who could imagine themselves as craftspeople.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2021-05-08T02:19:22.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20866" PostTypeId="2" ParentId="20773" CreationDate="2021-05-08T08:43:17.593" Score="3" Body="&lt;p&gt;Most of pre 1900 mathematics can be done without the modern function concept. Hints that this was actually the case can be found in this hsm question &lt;a href=&quot;https://hsm.stackexchange.com/q/6104/3462&quot;&gt;Who first considered the &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; as an object in itself, and who decided to call it a function?&lt;/a&gt; or if you skim through Leonard Eulers books on differentiation and integration (you'll have to look very hard to ever find an &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;So for the mathematics most people will ever learn in a modern curriculum, instead of using functions, one could express almost anything of interest just using variable quantities and equations between them. For instance, instead of writing &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt; you could just write &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt; and continue from there. That's implicitly what's still happening when students are taught the modern function concept at the age of 15: teachers introduce it but rarely use it in isolation later on. Just like in many programming languages we can use &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; but we'll never need (or be able) to use &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; as an object itself. This indirect use of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; probably makes it harder for students to understand what the fuss is all about.&lt;/p&gt;&#xA;&lt;p&gt;Also mathematicians pre 1900 where much more lax about the idea of a unique value of the dependent variable corresponding to a given value of the independent one, which is why for instance we still talk of an &amp;quot;implicit function&amp;quot; when we discuss the equation of the circle. So they had the broader perspective you mention.&lt;/p&gt;&#xA;&lt;p&gt;In summary I doubt that people need to learn the modern definition with 15.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2021-05-08T08:43:17.593" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20867" PostTypeId="1" CreationDate="2021-05-08T22:02:00.557" Score="4" ViewCount="572" Body="&lt;p&gt;I want to learn probability theory and discrete math. However, I also need to brush up on computational calculus and linear algebra. Would you recommend only studying one subject at a more intense pace for a shorter period of time or would you recommend taking on multiple subjects over a longer period of time- e.g. learning probability and discrete math whilst also reviewing calculus but like taking in less information about any given subject each time i learn, or doing subjects on alternating days.&lt;/p&gt;&#xA;&lt;p&gt;Example of one subject- learning probability and only probability every day for 3 hours for one month until i finish a series of lectures&lt;/p&gt;&#xA;&lt;p&gt;example of multiple- learning probability 3 days a week for 3 hours and reviewing for 3 hours the other 3 days&lt;/p&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;p&gt;another example of multiple - learning probability for 90 minutes and reviewing for 90 minutes every day.&lt;/p&gt;&#xA;&lt;p&gt;My goal is to maximize retention and understanding while still keeping a good pace. If I don't use it within a month I usually lose it.&lt;/p&gt;&#xA;&lt;p&gt;What do you think?&lt;/p&gt;&#xA;" OwnerUserId="15802" LastActivityDate="2021-05-09T16:58:36.640" Title="Study multiple subjects at the same time or deep dive into one?" Tags="|self-learning|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20868" PostTypeId="2" ParentId="20867" CreationDate="2021-05-08T22:35:47.680" Score="2" Body="&lt;p&gt;I advise one, not two.  The biggest danger with self study is quitting.  Keep it simple and get an earlier victory.  Will reduce quitting danger.  Retention can be kept by periodic review.  And/or really just brushing up, WHEN YOU NEED IT later in life.  Nobody expects you to remember every trick, if you are not continuing to use the stuff and/or doing a follow-on course to use it.  But it sure comes back quick when needed.  Especially if you MASTERED (look up the concept) it at the time.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  Note, I'm concerned about hearing you talk about getting through lectures.  Don't.  Get through a book and do lots of problems.  Same thing with review.  Review means a retest of your abilities, followed by corrective action to do some more drill and retest again.  It does not mean more passive viewing/reading.  See:  &lt;a href=&quot;https://www.scotthyoung.com/blog/2012/11/13/why-lectures/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.scotthyoung.com/blog/2012/11/13/why-lectures/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16038" LastActivityDate="2021-05-08T22:35:47.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20870" PostTypeId="2" ParentId="20862" CreationDate="2021-05-09T16:04:47.157" Score="3" Body="&lt;p&gt;I'm the only person in my department who hasn't gone deep into Nearpod, but it seemed in training that this was part of their appeal.  If you couldn't overlay all of their estimates simultaneously, you could certainly display each one in rapid succession.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-05-09T16:04:47.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20871" PostTypeId="2" ParentId="20867" CreationDate="2021-05-09T16:10:56.883" Score="2" Body="&lt;p&gt;Terms to search for include &amp;quot;blocked practice&amp;quot; and &amp;quot;interleaving&amp;quot;; the latter is also known as &amp;quot;mixed practice&amp;quot; or &amp;quot;varied practice&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;For instance, see &lt;a href=&quot;https://effectiviology.com/interleaving/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://effectiviology.com/interleaving/&lt;/a&gt; (excerpt below)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Interleaving is a learning technique that involves mixing together different topics or forms of practice, in order to facilitate learning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Interleaving helps people retain new information, acquire new skills, and improve existing abilities in a wide range of domains, such as math, music, and sports.&#xA;When deciding what kind of material to interleave, you should use logical criteria given the reason that you want to interleave material in the first place, and make sure that the items that you interleave aren’t too similar or too different.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The effectiveness of interleaving varies based on factors such as the type of material involved and the environment in which the material is learned, so you should assess the situation when deciding whether and how to interleave; if possible, you should also assess the effectiveness of the interleaving over time, and experiment with different approaches to it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Interleaving can be hard, and people sometimes underestimate its effectiveness; this is important to keep in mind both when it comes to interleaving in your own learning and when it comes to interleaving while teaching others.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In your specific situation, you might want to look into any overlap between your courses that could reinforce your retention and understanding of concepts. Perhaps discrete probability comes up in discrete mathematics (some courses/books include it). Certainly, permutations and combinations are likely to be covered in discrete mathematics; these topics are also used in probability. Integrals (calculus) may also come up when studying probability. Linearity from linear algebra also shows up in the linearity of operations of differentiation and integration. Vector spaces in linear algebra can have &amp;quot;vectors&amp;quot; that are sequences or functions. Matrices can arise in the graph theory part of discrete mathematics. More generally, standard proof techniques might be needed in both linear algebra and discrete mathematics.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2021-05-09T16:58:36.640" LastActivityDate="2021-05-09T16:58:36.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20872" PostTypeId="2" ParentId="20862" CreationDate="2021-05-09T18:55:48.803" Score="3" Body="&lt;p&gt;If you're using zoom, I think you might be able to display your graph on the whiteboard and have them all annotate. If not, there are many online whiteboards available for collaboration. A colleague recently recommended app.ziteboard.com. (I haven't tried it out yet.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-05-09T18:55:48.803" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20873" PostTypeId="2" ParentId="20862" CreationDate="2021-05-09T19:48:04.230" Score="3" Body="&lt;p&gt;Desmos classroom activities are capable of doing this sort of thing. &lt;a href=&quot;https://teacher.desmos.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://teacher.desmos.com&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are lots of existing activities you can copy that might do the exact thing you want.&lt;/p&gt;&#xA;&lt;p&gt;(The front page mentions setting up “classrooms”. You don’t have to create classrooms to use Desmos activities as far as I know. Pretty sure you can still have a stand-alone activity the students sign into just with their names.)&lt;/p&gt;&#xA;" OwnerUserId="1853" LastActivityDate="2021-05-09T19:48:04.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20874" PostTypeId="1" CreationDate="2021-05-10T13:19:18.380" Score="5" ViewCount="369" Body="&lt;p&gt;Teaching discrete mathematics, we pose (from the textbook) questions on finding compositions of relations, notably, relations on very small finite sets with only 3 or 4 elements (as an introductory and tested exercise).&lt;/p&gt;&#xA;&lt;p&gt;I'm searching for some way of being very specific and concrete in pointing out the individual &amp;quot;linkages&amp;quot; that are producing elements in the composition relation. I've been frustrated that none of the textbooks I've checked ever shows an explicit example being worked out like this. It's always definition, boom, final composition relation in its entirety.&lt;/p&gt;&#xA;&lt;p&gt;For example, here's an exercise from the Rosen Discrete Math book:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; be the relation &lt;span class=&quot;math-container&quot;&gt;$\{(1,2),(1,3),(2,3),(2,4),(3,1)\}$&lt;/span&gt;, and let&#xA;&lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; be the relation &lt;span class=&quot;math-container&quot;&gt;$\{(2,1),(3,1),(3,2),(4,2)\}$&lt;/span&gt;. Find &lt;span class=&quot;math-container&quot;&gt;$S \circ R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;(*) Here's where I want some scratch/explanatory work, resulting in the answer:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\{(1,1),(1,2),(2,1),(2,2)\}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, my instinct is to start writing something like: &lt;span class=&quot;math-container&quot;&gt;$(1,2) + (2,1) \implies (1, 1)$&lt;/span&gt;, etc., but that's a multifold abuse of notation -- which my students are already greatly struggling with, so I want to set a good example.&lt;/p&gt;&#xA;&lt;p&gt;I really want something that can be written briefly in one line of text per element in the composition  (e.g., not converting to a digraph and saying &amp;quot;look at this, it's easy&amp;quot;, or any other trick to make the problem &amp;quot;easier&amp;quot; -- the point is to document production of each individual element).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What's the best way to show work in finding elements of a relational composition (at point (*) above)?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-05-11T20:26:25.597" LastActivityDate="2021-05-13T11:23:26.367" Title="What's a good notation to show elements of relation composition?" Tags="|notation|discrete-math|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="20876" PostTypeId="2" ParentId="20874" CreationDate="2021-05-10T22:45:57.170" Score="3" Body="&lt;p&gt;Thanks for your feedback on my previous answer, which contained a misunderstanding. Here's a new try. I believe the following is the way to express the thought that you were trying to express as &lt;span class=&quot;math-container&quot;&gt;$(1,2) + (2,1) \implies (1, 1)$&lt;/span&gt;, using only the notation your book seems to be using:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\{(1,2)\}\circ\{(2,1)\}=\{(1,1)\}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;That is, your book defines a relation as a set of ordered pairs. We restrict our attention to one element of S and one element of R, forming singleton sets which represent relations that connect only two things (i.e., each of their graphs would be a single dot). We compose these two singleton relations and get one element of the composition of S with R. The complete composition of S and R would be the union of all such compositions of singletons with singletons.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-05-10T22:51:31.250" LastActivityDate="2021-05-10T22:51:31.250" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20877" PostTypeId="1" CreationDate="2021-05-12T04:07:19.757" Score="7" ViewCount="851" Body="&lt;p&gt;There's a nice example of why people dislike proof by exhaustion on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_by_exhaustion&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia page&lt;/a&gt;. The problem statement is &amp;quot;prove that all years in which the Modern Olympics are held are divisible by 4&amp;quot;. One can prove this by induction by showing that the first year, 1896, is divisible by 4, and applying the fact that all Olympic games are 4 years after the previous one. Alternatively one can prove this by exhaustion by listing out all the years in which Olympic games were held, and checking that they are all divisible by 4. Naturally the first method is preferable, not only because it's simpler, but also because it proves the statement indefinitely into the future.&lt;/p&gt;&#xA;&lt;p&gt;Only problem with this is that there's going to be an Olympic Games in 2021 thanks to COVID, and 2021 is not divisible by 4.&lt;/p&gt;&#xA;&lt;p&gt;What is a good replacement for this example? The obvious thing is to choose some well-known, reliably repeating event, but it's not easy to think of something that isn't annual. There is Halley's Comet, but the period of that is not a nice round number (75.32 years).&lt;/p&gt;&#xA;&lt;p&gt;NB: I notice the Wikipedia article has changed to specifically exclude the 2021 Olympics, but the more exceptions are listed (the cancelled Olympics during WW1 and WW2 are already listed) the less elegant the example becomes, too.&lt;/p&gt;&#xA;" OwnerUserId="16009" LastActivityDate="2021-06-12T15:13:24.203" Title="Example of why proof by exhaustion is inelegant" Tags="|proofs|examples|" AnswerCount="5" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="20878" PostTypeId="2" ParentId="20874" CreationDate="2021-05-12T06:06:18.817" Score="6" Body="&lt;p&gt;To explain why any particular ordered pair is in &lt;span class=&quot;math-container&quot;&gt;$S \circ R$&lt;/span&gt;, you can just show that it satisfies the definition, which says that &lt;span class=&quot;math-container&quot;&gt;$(a,c)$&lt;/span&gt; is in &lt;span class=&quot;math-container&quot;&gt;$S \circ R$&lt;/span&gt; if there exists &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$(a,b) \in R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(b,c) \in S$&lt;/span&gt;.  To show this is true, you can just give an example of such &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and observe that &lt;span class=&quot;math-container&quot;&gt;$(a,b) \in R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(b,c) \in S$&lt;/span&gt;.  The two things here are connected with the word &amp;quot;and&amp;quot;; if you want to use a symbol for that, it should be &lt;span class=&quot;math-container&quot;&gt;$\wedge$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\And$&lt;/span&gt;, not &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, you could write:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(1,3) \in R \quad \And \quad (3,2) \in S, \quad\text{so} \quad (1,2) \in S \circ R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is already quite short, and I don't think you should write anything further removed from the definition of &lt;span class=&quot;math-container&quot;&gt;$S \circ R$&lt;/span&gt; just to shorten it further.&lt;/p&gt;&#xA;" OwnerUserId="870" LastActivityDate="2021-05-12T06:06:18.817" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="20879" PostTypeId="2" ParentId="20877" CreationDate="2021-05-12T18:34:07.530" Score="4" Body="&lt;p&gt;A classical example of proof by exhaustion is to establish an integer &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is prime by trial divisions. One uses trial division by candidates for divisors up to &lt;span class=&quot;math-container&quot;&gt;$\sqrt{N}$&lt;/span&gt;. This can be made slightly less exhausting by using the method of the Sieve of Eratosthenes.&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is large, say 250 digits, then this approach to establishing primality is not practical, even with a computer. There are algorithms for primality testing that are much more efficient but they do require some theory beyond the brute force &amp;quot;method of exhaustion&amp;quot; approach. For this sort of reason, the method of exhaustion is sometimes deemed &amp;quot;inelegant.&amp;quot; &amp;quot;Elegance&amp;quot; would be to develop or learn some theory to approach the problem smartly.&lt;/p&gt;&#xA;&lt;p&gt;The method of exhaustion takes an interesting turn when the number of possible cases to check is  infinite. For example, we might try to prove there is an odd perfect number by starting with 3 and working until we find an odd perfect number &lt;span class=&quot;math-container&quot;&gt;$2n+1$&lt;/span&gt;. People have tried this approach to no avail. Perhaps there will be an elegant theoretical advancement some day that proves there are no odd perfect numbers. Or perhaps a quantum computer will find an odd perfect number some day. Or perhaps the problem will be unsolved forever!&lt;/p&gt;&#xA;&lt;p&gt;There are arithmetic statements &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; which are false for every &lt;span class=&quot;math-container&quot;&gt;$n\in{\bf N}$&lt;/span&gt; but which are independent of standard systems of axioms. The statement  &lt;span class=&quot;math-container&quot;&gt;$[\hbox{for all }n: P(n)]$&lt;/span&gt; would have neither a proof nor a counterexample. Therefore to establish such a statement &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; is always false would require the method of exhaustion to check for every case &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, which is not possible in finite time. I guess these arithmetic statements get the last laugh!&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-05-12T18:39:21.243" LastActivityDate="2021-05-12T18:39:21.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20880" PostTypeId="2" ParentId="17900" CreationDate="2021-05-12T23:47:24.350" Score="5" Body="&lt;p&gt;My friend, who is currently teaching an abstract algebra course, recently provided me with an example of an incorrect proof that I think showcases a few types of faulty reasoning that are common among students. I'm sharing it here with his permission.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proposition:&lt;/strong&gt; Suppose &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is a ring and &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is a prime ideal in &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;. For all ideals &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$JK \subseteq I$&lt;/span&gt; then either &lt;span class=&quot;math-container&quot;&gt;$J \subseteq I$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$K \subseteq I$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Proof.&lt;/em&gt; Suppose &lt;span class=&quot;math-container&quot;&gt;$JK \subseteq I$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$\sum_{n, m} a_nb_m \in I$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is an ideal, it is closed under addition so each &lt;span class=&quot;math-container&quot;&gt;$a_nb_m$&lt;/span&gt; is in &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is prime, either &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$b_m$&lt;/span&gt; is in &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;. Therefore either &lt;span class=&quot;math-container&quot;&gt;$J \subseteq I$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$K \subseteq I$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There are a few obvious problems with this proof.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First, it is not clear what &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b_m$&lt;/span&gt; are. Presumably the &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt;'s are some finite set of elements in &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt; and the &lt;span class=&quot;math-container&quot;&gt;$b_m$&lt;/span&gt;'s are some finite set of elements in &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, but this should be stated explicitly. This is an instance of &lt;strong&gt;The Mysterious Stranger.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;Next, saying that &amp;quot;since &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is closed under addition, each &lt;span class=&quot;math-container&quot;&gt;$a_nb_m$&lt;/span&gt; is in &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;&amp;quot; is not correct. It is mixing up &amp;quot;closed under addition&amp;quot; with its converse, so this is an instance of &lt;strong&gt;Implication Confusion.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;The conclusion of the proof is more or less a non-sequitor. All that the proof has really established is that either &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; contains some element which is in &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;, which is not enough to show that either &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is entirely contained in &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;. I would say this is an example of &lt;strong&gt;Wishful Thinking.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;Finally, there is an element of &lt;strong&gt;The Tangle&lt;/strong&gt; to the whole proof. It is not really clear what the strategy of the entire proof is, why the sum &lt;span class=&quot;math-container&quot;&gt;$\sum_{n, m} a_nb_m$&lt;/span&gt; is being introduced or what the goal of each step is. It's likely that such a proof is mostly the result of pattern-matching the appearance of proofs given in class in the hopes that this will lead to an approximation of a correct proof (and thus partial credit).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13482" LastActivityDate="2021-05-12T23:47:24.350" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20881" PostTypeId="2" ParentId="20874" CreationDate="2021-05-13T09:00:42.870" Score="6" Body="&lt;p&gt;I'd avoid introducing a new notation for such a limited scope: you won't probably use it elsewhere during the class, and it wouldn't be used elsewhere in the literature.&lt;/p&gt;&#xA;&lt;p&gt;I have three suggestions. The first is a simple variation of &lt;a href=&quot;https://matheducators.stackexchange.com/a/20878/5367&quot;&gt;Trevor Wilson's answer&lt;/a&gt;, which mirrors the first relation sign to keep the bridging elements together:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$R\ni(1,3) \And (3,2) \in S\Rightarrow (1,2) \in S \circ R$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The second is to use a table, structured as follows:&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;&lt;span class=&quot;math-container&quot;&gt;$(a,b) \in R$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;th&gt;&lt;span class=&quot;math-container&quot;&gt;$(b,c) \in S$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;th&gt;&lt;span class=&quot;math-container&quot;&gt;$(a,c) \in S \circ R$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(1,3)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(3,1)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(1,3)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(3,2)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(1,2)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;...&lt;/td&gt;&#xA;&lt;td&gt;...&lt;/td&gt;&#xA;&lt;td&gt;...&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;The third is another table, where the first column lists the pairs from &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, and the first row lists the pairs from &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;. Then, you mark the intersections which have an element in common (this might possibly reduce mistakes from the students):&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;&lt;/th&gt;&#xA;&lt;th&gt;&lt;span class=&quot;math-container&quot;&gt;$(2,1)$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;th&gt;&lt;span class=&quot;math-container&quot;&gt;$(3,1)$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;th&gt;&lt;span class=&quot;math-container&quot;&gt;$(3,2)$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;th&gt;&lt;span class=&quot;math-container&quot;&gt;$(4,2)$&lt;/span&gt;&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(1,2)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(1,3)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(1,2)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(2,3)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(2,1)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$(2,2)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;...&lt;/td&gt;&#xA;&lt;td&gt;...&lt;/td&gt;&#xA;&lt;td&gt;...&lt;/td&gt;&#xA;&lt;td&gt;...&lt;/td&gt;&#xA;&lt;td&gt;...&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;" OwnerUserId="5367" LastEditorUserId="5367" LastEditDate="2021-05-13T11:23:26.367" LastActivityDate="2021-05-13T11:23:26.367" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20882" PostTypeId="2" ParentId="20877" CreationDate="2021-05-13T21:32:47.807" Score="9" Body="&lt;p&gt;Again, the more standard label would be &amp;quot;by brute force&amp;quot; or &amp;quot;case by case&amp;quot;, rather than &amp;quot;by exhaustion&amp;quot;, since the latter refers to a proto-calculus method dating back at least to Archimedes.&lt;/p&gt;&#xA;&lt;p&gt;My own reaction to lengthy case-by-case treatment is that such a discussion seems to give no &lt;em&gt;reason&lt;/em&gt; for the phenomenon. That is, a &amp;quot;good explanation&amp;quot; hopefully/presumably encompasses the various seemingly-special reasons that work in the possibly myriad &amp;quot;cases&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes the trade-off is dubious: some results about semi-simple real Lie groups are just basic linear algebra for the classical groups, but intrinsic (=case-independent) arguments often require much more, and are not necessarily illuminating ... especially with regard to classical groups (such as &lt;span class=&quot;math-container&quot;&gt;$GL_n(\mathbb R)$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2021-05-13T21:32:47.807" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20883" PostTypeId="2" ParentId="20877" CreationDate="2021-05-13T23:26:51.787" Score="8" Body="&lt;p&gt;Perhaps equally interesting are theorems that can only be proved&#xA;&amp;quot;by exhaustion&amp;quot;---by a long case analysis---as far as we know.&#xA;For example, that &lt;a href=&quot;https://math.stackexchange.com/q/2427/237&quot;&gt;there are only a finite number of sporadic simple groups&lt;/a&gt;&#xA;seems to be such a theorem.&#xA;According to &lt;a href=&quot;https://math.stackexchange.com/a/2498/237&quot;&gt;Jack Schmidt&lt;/a&gt;, Gerhard Michler studied the issue&#xA;and thinks that&#xA;&amp;quot;it should not be taken for granted that there are only finitely many sporadic groups&amp;quot;: there is a natural procedure to possibly generate an infinite number. But&#xA;it doesn't.&lt;/p&gt;&#xA;&lt;p&gt;Another example, closer to home, suggested by&#xA;@user52817 in a comment.&#xA;In &lt;a href=&quot;https://mathoverflow.net/q/199097/6094&quot;&gt;an MO question&lt;/a&gt;, I asked which unfoldings of the hypercube tile &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-space.&#xA;My question was recently &lt;a href=&quot;https://mathoverflow.net/a/392828/6094&quot;&gt;answered by Moritz Firsching&lt;/a&gt;, using integer programming:&#xA;All &lt;span class=&quot;math-container&quot;&gt;$261$&lt;/span&gt; unfoldings of the hypercube tile &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-space!&#xA;But the proof is by exhaustion of the &lt;span class=&quot;math-container&quot;&gt;$261$&lt;/span&gt; cases; no overarching&#xA;theorem is known.&#xA;This raises the question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Q&lt;/strong&gt;&lt;/em&gt;. Is it true that, for every &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, each of the&#xA;unfoldings of the &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;-dimensional cube&#xA;tiles &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^{d-1}$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We now know this is true for &lt;span class=&quot;math-container&quot;&gt;$d=3,4$&lt;/span&gt;:&#xA;The &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt; unfoldings of the &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; cube tile &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt;,&#xA;and the &lt;span class=&quot;math-container&quot;&gt;$261$&lt;/span&gt; unfoldings of the &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^4$&lt;/span&gt; hypercube tile &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-05-15T22:26:22.603" LastActivityDate="2021-05-15T22:26:22.603" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20884" PostTypeId="2" ParentId="17900" CreationDate="2021-05-14T02:14:26.480" Score="3" Body="&lt;p&gt;Three categories of error I often encounter are &amp;quot;Incorrect negation&amp;quot;, &amp;quot;trouble with quantifiers&amp;quot; and &amp;quot;Confusing general parameters with specific objects&amp;quot;.  Everybody knows that students have trouble dealing with negating quantifiers in the context of, for example, an &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; proof in an Analysis course.  We also see these issues arise in Linear Algebra, where student work like the following is rampant:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\vec{v}$&lt;/span&gt; be any nonzero vector; prove that &lt;span class=&quot;math-container&quot;&gt;$\{\vec{v}, \vec{0}\}$&lt;/span&gt; is linearly dependent&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solution&lt;/strong&gt;.  Let &lt;span class=&quot;math-container&quot;&gt;$a\vec{v} + b\vec{0} = \vec{0}$&lt;/span&gt;.  If &lt;span class=&quot;math-container&quot;&gt;$a = b = 0$&lt;/span&gt; then this is a trivial relation; therefore the vectors are linearly&#xA;dependent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What's happening in this muddle is something like this:  Students know that &amp;quot;linearly independent&amp;quot; means &amp;quot;there is no nontrivial linear relation on the vectors&amp;quot;; they try to negate it, but get tangled up in the triple negative (&amp;quot;it is not true that there is no nontrivial...&amp;quot;), and end up thinking that &amp;quot;linearly dependent&amp;quot; means &amp;quot;there is a trivial linear relation on the vectors&amp;quot;. The use of the letters &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, which &lt;em&gt;seem&lt;/em&gt; like general parameters in the first sentence, but are subsequently revealed in the very next sentence to have been very specific constants all along, helps to obscure the obvious error.&lt;/p&gt;&#xA;&lt;p&gt;In fact on our most recent final exam we included, as a true/false question,the statement&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$S = \{v_1 \, \dots, v_m \}$&lt;/span&gt; is a set of vectors, and &lt;span class=&quot;math-container&quot;&gt;$a_1v_1 + \cdots a_nv_n = 0$&lt;/span&gt; is a trivial relation, then &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is linearly independent.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A distressingly large fraction of our students marked this as true.  They seem to have trouble understanding that &amp;quot;There are &lt;em&gt;no nontrivial relations&lt;/em&gt;...&amp;quot; means something different from &amp;quot;There &lt;em&gt;is a trivial relation&lt;/em&gt;...&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-05-14T02:14:26.480" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20885" PostTypeId="2" ParentId="17900" CreationDate="2021-05-14T03:12:37.417" Score="9" Body="&lt;p&gt;The Q&amp;amp;A so far has gone into a lot of relatively sophisticated stuff. The students I teach are mostly engineering majors at a community college, so their mistakes are a lot simpler. They have generally been exposed to proof writing in high school geometry. However, my experience is that this does not translate well into a more general skill. Many of them learned &amp;quot;two-column proofs,&amp;quot; which are pretty artificial and constrained.&lt;/p&gt;&#xA;&lt;p&gt;Often they will &amp;quot;prove&amp;quot; something by giving an example, e.g., they'll prove that the dot product is commutative by making up a couple of vectors and doing the computations. This also comes up in physics labs, where they claim that their lab results prove Newton's second law. They don't want to believe that a counterexample can disprove a general statement, but one piece of evidence can't prove it.&lt;/p&gt;&#xA;&lt;p&gt;They write something that they claim is a proof, but they never make use of one of the assumptions, and this assumption is necessary.&lt;/p&gt;&#xA;&lt;p&gt;In my experience students in elementary classes like first-semester calculus need to get some experience writing really simple, easy proofs so that they can get past these initial stages. Unfortunately, I don't think most instructors assign any such work in these courses. I would imagine that this is a big reason for the meltdowns I hear about when math majors start their upper-division coursework.&lt;/p&gt;&#xA;&lt;p&gt;Here's an example of what I would like them to be able to do. Suppose they know enough about the dot product to know how to compute it from cartesian components. Then if I ask them to prove that the dot product is commutative, ideally they would be able to write something like this: &amp;quot;The dot product can be expressed in terms of the products of the two vectors' components. Since the components are real numbers, and multiplication of real numbers is commutative, it follows that the dot product is commutative.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But even this extremely simple-looking argument contains some pretty sophisticated mental activity. Students are trying to relate everything to their previous education, and their previous education contains a couple of possible models of this. One model is concrete computation involving numbers, so they produce an example involving numbers and call that a proof. Another model is two-column proofs in high school geometry, but that model doesn't carry over here, because it doesn't help them to focus on the essentials, ignore inessential elements, and write a sentence meant for a human to read.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-05-14T15:24:00.503" LastActivityDate="2021-05-14T15:24:00.503" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20887" PostTypeId="1" AcceptedAnswerId="20889" CreationDate="2021-05-14T17:46:41.483" Score="5" ViewCount="1153" Body="&lt;p&gt;I am going to teach an undergraduate statistics course next year.&#xA;My plan is to spend less time on theory and a more time on teaching students to doing some statistics on computer. However, this raises the questions of how to evaluate students fairly and discourage cheating.&#xA;If you have taught courses involving some programming, how did you test students?&#xA;What are the pitfalls to avoid?&lt;/p&gt;&#xA;&lt;p&gt;Here are some ideas which comes to my mind, each with some drawbacks --&lt;/p&gt;&#xA;&lt;h2&gt;Randomized assignments&lt;/h2&gt;&#xA;&lt;p&gt;Gave each student a slightly different assignment with some randomized parameters or data.&lt;/p&gt;&#xA;&lt;p&gt;The drawback of this is that it is going to make grading more difficult.&lt;/p&gt;&#xA;&lt;h2&gt;Group project&lt;/h2&gt;&#xA;&lt;p&gt;Divide students in groups and ask them to propose a project. Each group then chooses a project proposed by other students, collect and analyze the data, and then present their result to the class.&lt;/p&gt;&#xA;&lt;p&gt;From my own experience as a student, usually the stronger students in a group do most of the work, and the weak ones almost do nothing.&lt;/p&gt;&#xA;&lt;h2&gt;In-class closed-book exam&lt;/h2&gt;&#xA;&lt;p&gt;Give some students some problems to solve on their own laptops within limited time, in class, and without Internet access.&lt;/p&gt;&#xA;&lt;p&gt;The problem with this approach is that it is difficult to enforce no Internet access (we don't have computer labs). And there is a bit danger that if a student's suddenly have problem with his coding environment, he/she won't be able to do the exam.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2021-05-25T15:10:03.927" Title="How to test student's skills in programming or using software?" Tags="|exams|computer-programs|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="20888" PostTypeId="1" CreationDate="2021-05-14T17:55:11.183" Score="3" ViewCount="195" Body="&lt;p&gt;OK so this is really old school. Back in the 1980s, as a latchkey kid, I played on the computers in the library in elementary and middle school. There was this one bizarre educational game that I'd love to find more about.&lt;/p&gt;&#xA;&lt;p&gt;It involved connecting a bunch of nodes with wires, creating a big rube goldberg contraption that did things to words or numbers. Press the play button and stuff pops out of some of the nodes, travels along all the little conveyer belts, interacts in some other nodes, and pops out in various places. It was a very &amp;quot;just play with it and see what it does&amp;quot; kind of game. I forget if there was even a purpose, like being required to get a certain output into a bucket or something.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The Incredible Machine&amp;quot; is close but no cigar. It's not falling objects in a literal rube goldberg machine, it's squares connected by wires, like some modern day audio-processing pro software:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9WuCX.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9WuCX.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But this was a kiddie game. Mid-80s.&lt;/p&gt;&#xA;&lt;p&gt;Cross-posted on Retrogaming: &lt;a href=&quot;https://retrocomputing.stackexchange.com/questions/19837/looking-for-an-educational-game-from-long-ago&quot;&gt;https://retrocomputing.stackexchange.com/questions/19837/looking-for-an-educational-game-from-long-ago&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16072" LastEditorUserId="16072" LastEditDate="2021-05-15T15:43:36.517" LastActivityDate="2021-05-20T03:58:11.827" Title="Looking for an educational game from long ago, possibly called Mother Goose" Tags="|secondary-education|primary-education|" AnswerCount="2" CommentCount="6" ClosedDate="2021-05-21T16:17:51.937" ContentLicense="CC BY-SA 4.0" />
  <row Id="20889" PostTypeId="2" ParentId="20887" CreationDate="2021-05-14T19:57:32.793" Score="6" Body="&lt;p&gt;I teach intro to Comp Sci and my main methodology is to flip the classroom, give the lecture in a video to be watched before class, make class a lab, give them homework-like assignments that they work on in front of me to practice and break and fail and ask questions, and then homework is a project that's either the same thing with a slight variation, or a combination of several things practised during the lab. I don't give tests because I'd rather them spend time on projects than study to an exam.&lt;/p&gt;&#xA;&lt;p&gt;I generally don't worry about randomizing their projects too much, but I do try to randomize the INPUT. I design them so I can write a test harnass to run all their programs with some test cases, check the output, and print if the output was correct. So to that end, it can be randomized, just to make sure they're not hardcoding output.&lt;/p&gt;&#xA;&lt;p&gt;Example 1: labs one day include reading 1 filename on the command line, opening the file, and counting lines and characters. Homework: also count words, and have it work for N files on the command line. (With ample additional lecture notes on what goes into counting words, and leeway if they're just close, as there are easy and hard algos for doing this.)&lt;/p&gt;&#xA;&lt;p&gt;Example 2: labs include printing out looped data in html, including tables. homework is to take a previous homework that printed matrices (and their cross product) just in text, and to add all the tags inbetween so that it formats properly.&lt;/p&gt;&#xA;&lt;p&gt;I can feed in my own files to test their output for example 1. And for the other example, their matrices already have randomized data in them, so I wrote an output reader that reads their output and checks to see if the math was correct, also guaranteeing that their output isn't hardcoded.&lt;/p&gt;&#xA;&lt;p&gt;Beyond that, I just hand check to see there's no direct copying going on.&lt;/p&gt;&#xA;" OwnerUserId="16072" LastEditorUserId="16072" LastEditDate="2021-05-16T14:27:50.740" LastActivityDate="2021-05-16T14:27:50.740" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20890" PostTypeId="2" ParentId="20887" CreationDate="2021-05-15T00:54:12.567" Score="4" Body="&lt;p&gt;Among other courses, I teach &lt;em&gt;Introduction to CompSci via Programming&lt;/em&gt;.&#xA;I use group projects because every post-grad job will involve programming in&#xA;groups.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I've had success with elementary 2D graphics, allowing the students&#xA;considerable artistic latitude: spirographs, fish swimming in an acquarium, etc.&#xA;Various requirements, i.e., use of classes. Difficult to grade, but cheating not an issue.&lt;/li&gt;&#xA;&lt;li&gt;Less innovative but still effectively engaging: Design and implement a simple game, some variety of breakout or PacMan. Again with only the most primitive graphics.&lt;/li&gt;&#xA;&lt;li&gt;Among my most successful projects was to capture a sampled stream of all Twitter tweets within a specific time period (quite a firehose), and ask the students to parse and then analyze the tweets for, say, emotion, negativity, political leaning, English vs. Spanish, etc. I spent ~&lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; hrs figuring out how to capture a Twitter stream, which I then made available to them. Again, difficult to grade but cheating not an issue. They felt they were doing something &amp;quot;real&amp;quot;&#xA;(including running out of memory), and could potentially make sociological discoveries.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-05-15T01:19:19.610" LastActivityDate="2021-05-15T01:19:19.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20891" PostTypeId="2" ParentId="20888" CreationDate="2021-05-15T18:10:31.787" Score="1" Body="&lt;p&gt;Was it Widget Workshop? &lt;a href=&quot;https://en.wikipedia.org/wiki/Widget_Workshop&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Widget_Workshop&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some screenshots here: &lt;a href=&quot;https://www.mobygames.com/game/widget-workshop-the-mad-scientists-laboratory/screenshots&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mobygames.com/game/widget-workshop-the-mad-scientists-laboratory/screenshots&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can play it in an emulator here: &lt;a href=&quot;https://archive.org/details/widgetworkshop&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/widgetworkshop&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-05-15T18:10:31.787" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20892" PostTypeId="1" CreationDate="2021-05-18T00:21:26.523" Score="17" ViewCount="507" Body="&lt;p&gt;About 15 years ago, I heard a math education professor give a talk about how computer algebra systems would change the kinds of questions teachers would ask high school and first year college students. Exercises in factoring and polynomial long division would become de-emphasized just as those asking students to extract square roots by hand to the third decimal place were deemphasized after calculators became widespread. New types of questions exploring patterns would be possible.&lt;/p&gt;&#xA;&lt;p&gt;The need for remote teaching and assessment rising this past year has renewed my interest in this topic. Many instructors have had to make all their assessments open-book, open-internet.&lt;/p&gt;&#xA;&lt;p&gt;So my question has two parts. One is about a curriculum, particularly at the precalculus level: Is there a curriculum that emphasizes topics and skills that CAS would not render &amp;quot;irrelevant&amp;quot; (for lack of a better term) and that utilizes CAS to enrich understanding?&#xA;The other part is about assessments/problem sets: is a there a body of wolfram alpha-proof questions out there for assessing students?&lt;/p&gt;&#xA;&lt;p&gt;Edit: I regret implying that factoring will be irrelevant with CAS. In that particular case, I am looking for questions that would build or assess this skill even if the student answering the questions has access to CAS -perhaps something like &lt;a href=&quot;https://www.openmiddle.com/factoring-quadratics-2-integer-solutions/&quot; rel=&quot;noreferrer&quot;&gt;this Open Middle problem&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="690" LastEditorUserId="690" LastEditDate="2021-05-22T02:28:49.917" LastActivityDate="2021-06-02T10:03:07.557" Title="Is there a math curriculum that is aware of CAS and the internet?" Tags="|secondary-education|technology-in-education|assessment|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="20894" PostTypeId="2" ParentId="17864" CreationDate="2021-05-19T19:52:13.680" Score="1" Body="&lt;p&gt;You can't. Mathematics is hard, and its beauty must be earned by hard work.&lt;/p&gt;&#xA;&lt;p&gt;Think about how you might have realized the beauty of maths. You most probably have thought very hard about a problem, and when you finally put the pieces together, your reward was a feeling of beauty. This reward then made you more likely to put in effort and get more rewards, which became a positive feedback loop.&lt;/p&gt;&#xA;&lt;p&gt;The point is, you can't just show the beauty of maths. Maths is not a nice painting or a piece of music where beauty can be enjoyed effortlessly. Without the effort put in to understand the concepts, your students won't recognize it.&lt;/p&gt;&#xA;&lt;p&gt;The best you can do is to present the material in a straightforward, mathematically most honest way, without try-hard ways to &amp;quot;make it interesting&amp;quot;. Students recognize immediately if the teacher is trying too hard to &amp;quot;sell&amp;quot; the curriculum, or tries to trick them into being interested, and they resist instinctively. Get them put in the work, by being strict, if necessary, be honest to the mathematics you want to present, and earn their respect. Then some of them, who put in the work, might get hooked by the positive feedback loop.&lt;/p&gt;&#xA;" OwnerUserId="14091" LastActivityDate="2021-05-19T19:52:13.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20895" PostTypeId="2" ParentId="20888" CreationDate="2021-05-20T03:58:11.827" Score="1" Body="&lt;p&gt;This sounds a bit like &lt;a href=&quot;https://www.myabandonware.com/game/rockys-boots-cp/play-cp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rocky's Boots&lt;/a&gt;, a game in which you connected logic gates with wires to build machines that would kick objects with specific properties.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-05-20T03:58:11.827" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20896" PostTypeId="1" AcceptedAnswerId="20901" CreationDate="2021-05-20T08:41:25.030" Score="8" ViewCount="719" Body="&lt;p&gt;It is observed many times that students do not ask the right questions in the classroom. They just attend the lectures passively. Rather than asking the questions o get their doubt cleared, they assume something else and then get their concepts wrong. Should the students in a math class be encouraged to ask useful questions?&#xA;Is it worth the effort in persuading and forcing the students to ask genuine questions?&lt;/p&gt;&#xA;&lt;p&gt;Here are a few reasons which I found useful-&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Asking questions makes the class more interactive rather than just passive learning.&lt;/li&gt;&#xA;&lt;li&gt;Asking proper questions makes the students engage more deeply with math, which results in more critical thinking.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Are there any more aspects apart from the above listed, to the benefits of asking questions?&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance for the help :-)&lt;/p&gt;&#xA;" OwnerUserId="15858" LastActivityDate="2021-05-20T21:19:45.023" Title="Importance of asking questions in a mathematics class" Tags="|secondary-education|advising|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20897" PostTypeId="1" CreationDate="2021-05-20T08:57:44.813" Score="1" ViewCount="114" Body="&lt;p&gt;You can play chess online anytime you want to, even against a human opponent of your level, even in hourly competitions.&#xA;It would be nice to have the same thing in math, at least for kids.&#xA;Everyone could compete on their own level, learn new things by solving problems, maintain some ELO like score.&#xA;I see two issues that make this harder than chess: new problems need to be generated for each contest and the correctness of the solutions needs to be verified.&lt;/p&gt;&#xA;&lt;p&gt;I think that having new problems is manageable by working with a huge enough database, such things already &lt;a href=&quot;https://cemc2.math.uwaterloo.ca/contest/PSG/school/index.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;exist&lt;/a&gt;, even if for a different level/difficulty. But enthusiastic people could upload their own problems. Also, by the time someone meets all the problems from a given level, probably they have mastered them well enough so that they are ready to move on to the next level.&lt;/p&gt;&#xA;&lt;p&gt;About verifying solutions, a simple solution is to use multiple choice tests, or fields where contestants can enter numerical values.&#xA;This might be adequate for young children, but probably they are not the target audience for online contests.&lt;br /&gt;&#xA;Another solution could be contestants checking each other's arguments and receiving some points for this as well, but this can lead to extended debates and opinion-based decisions.&lt;br /&gt;&#xA;Finally, for old enough students (from high school) one could require that the solutions are entered in some theorem prover, like &lt;a href=&quot;https://en.wikipedia.org/wiki/Lean_(proof_assistant)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lean&lt;/a&gt;, which automatically checks for correctness. This would make the competition similar to coding contests.&#xA;Another advantage of this is that the correctness of proposed competition problems can also be verified automatically (though not their level).&lt;/p&gt;&#xA;&lt;p&gt;Are there any other obstacles for online math competitions?&lt;/p&gt;&#xA;" OwnerUserId="7002" LastActivityDate="2021-05-20T13:19:44.280" Title="Hourly math contests online" Tags="|online-instruction|math-contests|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20898" PostTypeId="2" ParentId="20897" CreationDate="2021-05-20T13:19:44.280" Score="4" Body="&lt;p&gt;Art of Problem Solving has a game called &amp;quot;&lt;a href=&quot;https://artofproblemsolving.com/ftw&quot; rel=&quot;nofollow noreferrer&quot;&gt;For the Win&lt;/a&gt;.&amp;quot; I don't have any experience with it myself, but from what I've heard of it, it is essentially what you described. A player can host a room with a set of rules (how many questions and how much time per question). It also keeps a rating system, although I don't know if there's a way to separate different levels (I think an 800 ELO player could join a room with a 2400 ELO player for example). The game draws on a large pool of AMC or AIME style problems and the students have to type in their answers (no multiple choice).&lt;/p&gt;&#xA;&lt;p&gt;The major draw back (again, only from what I've heard) is that students end up playing it so much, that they memorize the answers. The points you get per question are based on how quickly you answered, so memorizing answers and getting a question right in 1 second maximizes your points. The site says they have 15,000+ problems curated, so perhaps the &amp;quot;memorizing answers based on the first sentence&amp;quot; problem will go away as they add more and more questions to their database.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2021-05-20T13:19:44.280" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20899" PostTypeId="1" CreationDate="2021-05-20T16:48:01.030" Score="1" ViewCount="65" Body="&lt;p&gt;Activities are a great way to engage students with Mathematics. And that also helps them to visualise mathematical facts. For example, Pi in its decimal form appears to have no pattern (we used to say). But, if we ask students to colour code digits (upto some 500 or 600), they will automatically be able to see that.&lt;/p&gt;&#xA;&lt;p&gt;If we allow students to play with periodicity of repeating decimals after a while they will figure out Midy's property. &lt;a href=&quot;https://web.iitd.ac.in/%7Eatripath/publications/Midy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://web.iitd.ac.in/~atripath/publications/Midy.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, what all other concepts can help students to get engaged while having fun.&lt;/p&gt;&#xA;" OwnerUserId="16098" LastEditorUserId="16098" LastEditDate="2021-05-20T17:32:31.527" LastActivityDate="2021-05-20T17:32:31.527" Title="What are different Mathematical concepts which can be introduced to 12-16 year students in an activity based way" Tags="|secondary-education|mathematics-in-daily-life|recreational-learning|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20900" PostTypeId="2" ParentId="20896" CreationDate="2021-05-20T16:57:44.730" Score="2" Body="&lt;p&gt;From our experience we have seen that most of the times they do not understand what to ask. And this is basically because students usually feel disconnected with the subject. To change that, we think that some fun way or activity based introduction of concepts is necessary. That boosts their capability to engage - and along the way learns to think and question.&lt;/p&gt;&#xA;" OwnerUserId="16098" LastActivityDate="2021-05-20T16:57:44.730" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20901" PostTypeId="2" ParentId="20896" CreationDate="2021-05-20T21:19:45.023" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Should the students in a math class be encouraged to ask useful questions?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is it worth the effort in persuading and forcing the students to ask genuine questions?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No, that's the wrong approach.  You can't &lt;em&gt;force&lt;/em&gt; someone to ask genuine questions; that won't aid their learning.  And persuasion isn't really relevant.  Encouragement, yes, but that's different from persuasion.&lt;/p&gt;&#xA;&lt;p&gt;The best approach is to coax into action the student's mind to ask questions, to derive material from the basic principles presented, to discover things for themselves.&lt;/p&gt;&#xA;&lt;p&gt;As to HOW to do that, it's not easy to put into a can.  This is actually, in my opinion, the key attribute that makes for an excellent teacher: they can prompt the student to ask questions for themselves.&lt;/p&gt;&#xA;&lt;p&gt;It's not that the questions themselves are beneficial.  It's that an inquisitive attitude toward the subject, and mental participation with the learning process, are beneficial, and these attributes will of course manifest in the form of questions (among other ways).  &lt;em&gt;Forcing&lt;/em&gt; the student to ask questions for the sake of asking questions misses this point.&lt;/p&gt;&#xA;" OwnerUserId="5751" LastActivityDate="2021-05-20T21:19:45.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20902" PostTypeId="1" CreationDate="2021-05-21T02:42:45.473" Score="2" ViewCount="233" Body="&lt;p&gt;This question may be just too broad in scope, but some form of it has been on my mind during this year of remote learning as I imagine a future cycle of educational upheaval.&lt;/p&gt;&#xA;&lt;p&gt;Much of what we teach in mathematics involves algorithms: ways to calculate certain things, methods for solving equations, etc.&lt;/p&gt;&#xA;&lt;p&gt;I am curious what (and possibly how) you would teach if it was assumed that everyone had access to calculating devices (e.g. Symbolab, Wolfram|Alpha) and could use them at any time, and that teaching students methods for calculating, simplifying and solving numerical and algebraic expressions/equations was decided to be outside the scope of what you could teach. [Note: an introduction to how a calculation works could be fine, but it would be pointless&lt;span class=&quot;math-container&quot;&gt;${}^\star$&lt;/span&gt; to put these types of questions on an exam as the answers are just a few button clicks away.]&lt;/p&gt;&#xA;&lt;p&gt;(&lt;span class=&quot;math-container&quot;&gt;${}^{\star}$&lt;/span&gt; Unless you were merely testing whether someone brought their calculator that day)&lt;/p&gt;&#xA;&lt;p&gt;I ask this to determine what things (skills, understandings, habits) educators think remain at (or should be brought into) the core of this discipline in the age of technology that can perform all of the arithmetic and algebra we teach. Note that I am &lt;em&gt;not&lt;/em&gt; suggesting that it would be the best scenario to disallow certain arithmetic or algebraic methods, but rather I am asking what, in their absence, we would choose to teach to students. Just what survives this hypothetical scenario?&lt;/p&gt;&#xA;&lt;p&gt;I am interested mainly in first- or second-year topics in college, such as precalculus, trigonometry, calculus, differential equations and linear algebra.&lt;/p&gt;&#xA;&lt;p&gt;Are there colleges that have officially adopted such a program, or do you imagine a curriculum or specific set of outcomes for any of these courses that eschews teaching/focusing-on/assessing calculation skills or algebraic manipulation? Would any of these courses cease to exist or might some be combined?&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2021-05-21T11:30:47.523" Title="How to teach if calculations and algebraic manipulations are off limits" Tags="|undergraduate-education|calculations|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20903" PostTypeId="1" AcceptedAnswerId="20907" CreationDate="2021-05-21T07:31:47.787" Score="2" ViewCount="210" Body="&lt;p&gt;The questions that I want ask are the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What are the most important and effective topics to conduct a workshop?&lt;/li&gt;&#xA;&lt;li&gt;What fraction of workshop should include lectures, activity, problem solving, etc.&lt;/li&gt;&#xA;&lt;li&gt;What could be the optimal number (roughly) of students in a workshop?&lt;/li&gt;&#xA;&lt;li&gt;What all other parameters one should think while designing a workshop?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="16098" LastActivityDate="2021-05-23T12:33:18.307" Title="If one wants to conduct a 1/2 day workshop in Mathematics for 12-16 year old students - how one should go about preparing the workshop" Tags="|undergraduate-education|secondary-education|applied-mathematics|seminar|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20904" PostTypeId="2" ParentId="20902" CreationDate="2021-05-21T11:30:47.523" Score="10" Body="&lt;p&gt;I highly recommend looking at the Calculus exams from the University of Michigan.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://dhsp.math.lsa.umich.edu/examshops.html&quot; rel=&quot;noreferrer&quot;&gt;https://dhsp.math.lsa.umich.edu/examshops.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;These problems tend to focus on extracting relevant information for solving problems from a variety of representations (symbolic, graphical, numerical, verbal), and on translating between these representations.&lt;/p&gt;&#xA;&lt;p&gt;Example:  Here is a problem where the student needs to translate from a verbal description of some quantity to a symbolic expression.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://dhsp.math.lsa.umich.edu/exams/115exam1/f18/p5.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://dhsp.math.lsa.umich.edu/exams/115exam1/f18/p5.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example:  Here is a problem where the student needs to extract information about a function from its graph.  Problem 4.b.ii is very nice, since it really requires the student to understand composition and one sided limits (even though one sided limits are not obviously indicated).  Problem 4.b.iii requires the student to have really understood the graphical meaning of the standard difference quotient.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://dhsp.math.lsa.umich.edu/exams/115exam1/f17/p4.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://dhsp.math.lsa.umich.edu/exams/115exam1/f17/p4.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The exams are full of superbly crafted questions of this form.  Symbolic computation is kept to a minimum:  a &amp;quot;gateway&amp;quot; exam which is mastery based ensures that students are developing calculation skill.  The essential outcomes of the course center on reasoning and conceptual understanding.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-05-21T11:30:47.523" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20905" PostTypeId="2" ParentId="520" CreationDate="2021-05-21T12:58:30.460" Score="0" Body="&lt;p&gt;One very good application of Eigen Value and Vectors is Principal Component Analysis in Machine Learning.&#xA;This application can actually help to visualise the mathematical concept better.&lt;/p&gt;&#xA;" OwnerUserId="16098" LastActivityDate="2021-05-21T12:58:30.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20906" PostTypeId="2" ParentId="520" CreationDate="2021-05-21T15:35:54.557" Score="2" Body="&lt;p&gt;This may be too advanced for some students in a first-semester linear algebra course, but those who have had some Physics may be impressed by the diagonalization of a moment of inertia tensor.  Take some irregularly-shaped 3-dimensional object, and form a &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; matrix describing all of the components of the moments of inertia around various axes; the equations of motion for rotating the object around an arbitrary axis are extremely complicated.  Now diagonalize the system, finding the principal axes of the object: suddenly they are simple (if you rotate around one of those axes).&lt;/p&gt;&#xA;&lt;p&gt;Demonstration:  Bring in a rock or some other irregularly-shaped object and spin it like a top.  We &lt;em&gt;instinctively&lt;/em&gt; align it with one of its principal axes, because we can &amp;quot;feel&amp;quot; them in our hands.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-05-21T15:35:54.557" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20907" PostTypeId="2" ParentId="20903" CreationDate="2021-05-21T15:54:36.090" Score="3" Body="&lt;p&gt;Math Circles are one type of &amp;quot;workshop&amp;quot; I'd recommend. Math Circles can take many forms: More students or fewer, more lecture or less. But the idea is that the students do most of the work, and the leaders are there to give helpful nudges. (Perhaps 10 students with one leader, 15 with two, 7 or 8 for each leader if more.)&lt;/p&gt;&#xA;&lt;p&gt;Good problems are &amp;quot;low floor, high ceiling&amp;quot;, easy to get started, and full of rich content. They also can be approached from many directions. You need problems that will intrigue students. Resources include: &lt;a href=&quot;http://www.geometer.org/mathcircles/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tom Davis's page&lt;/a&gt;, &lt;a href=&quot;https://mathcircles.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathcircles site&lt;/a&gt;, &lt;a href=&quot;https://www.nationalmathfestival.org/people/national-association-math-circles-namc&quot; rel=&quot;nofollow noreferrer&quot;&gt;National Association of Math Circles&lt;/a&gt;, and a few posts on my blog, including &lt;a href=&quot;https://mathmamawrites.blogspot.com/2012/03/oakland-math-circle-does-spot-it.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one about doing a math circle with the game Spot It&lt;/a&gt;. There are no &lt;em&gt;important&lt;/em&gt; topics, only good ones and not so good.&lt;/p&gt;&#xA;&lt;p&gt;Instead of having all participants working on one problem, you can have a festival of mathematics, like the &lt;a href=&quot;https://www.jrmf.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Julia Robinson Mathematics Festival&lt;/a&gt;, with a different problem sheet or puzzle collection at each table (along with a table leader).&lt;/p&gt;&#xA;&lt;p&gt;Since most math leaders have not been trained in this style of work, it's important that the leaders get experience playing with problems together before leading.&lt;/p&gt;&#xA;&lt;p&gt;(I see that you are in India. I just now googled &amp;quot;math circles India&amp;quot; and I see a number of promising looking sites. Maybe the people behind those will have some advice for you too.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2021-05-21T16:07:26.007" LastActivityDate="2021-05-21T16:07:26.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20908" PostTypeId="1" CreationDate="2021-05-21T18:01:32.750" Score="4" ViewCount="267" Body="&lt;p&gt;Mathematical (geometric) constructions are an interesting way to engage students. It also helps in better understanding of different geometrical properties.&#xA;For example, Sierpinski triangle or square, Steiner chains.&#xA;While constructing, it immensely helped students to have a better grasp on geometrical properties.&lt;/p&gt;&#xA;&lt;p&gt;Are there any list one can refer to? Or propose whatever you think could be tried.&lt;/p&gt;&#xA;" OwnerUserId="16098" LastEditorDisplayName="user507" LastEditDate="2021-05-31T22:56:17.843" LastActivityDate="2021-06-02T14:53:50.280" Title="Are there any list of mathematical constructions which can challenge 12-16 year old students?" Tags="|undergraduate-education|secondary-education|geometry|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20909" PostTypeId="2" ParentId="20903" CreationDate="2021-05-21T20:57:58.250" Score="1" Body="&lt;p&gt;One tactical suggestion is to do a hands on statistical workshop.  Stats is still math, right?  I mean, even if lower class, back of the bus math.  ;)&lt;/p&gt;&#xA;&lt;p&gt;I know of one with miniature catapults, that is used for corporate SiX Sigma efforts.  But you could look for others that are simpler or just make your own (tossing paper airplanes or the like).  It leads to several interesting topics (average, std deviation, sample size, factors, etc.)  And it involves some fun moving and doing.&lt;/p&gt;&#xA;&lt;p&gt;And really, you're not doing any serious investigation of physics/aerodynamics.  But using them to explore mathematics.  If you will allow me to creep the ghetto of stats into math.  Plus just the time duration and the like will fit well into a workshop.  And the KEY for workshops is to keep the animals BUSY (working, producing, trying, etc.) not viewing slides and getting unhappy.&lt;/p&gt;&#xA;" OwnerUserId="16105" LastActivityDate="2021-05-21T20:57:58.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20910" PostTypeId="1" CreationDate="2021-05-21T23:10:42.803" Score="9" ViewCount="2192" Body="&lt;p&gt;I see a push toward having undergraduate curriculums built around 8-week classes. This is mostly in the online education in the USA. Recently I have seen a number of these in sophomore or junior-level math classes.  For example you have 8 weeks to cover the entire standard calculus III, and this is not a summer class. Essentially there would be four short semesters in the usual academic year, i.e. September-May. In each of these four accelerated semesters the regular material of a course is to be covered at double the usual speed.&lt;/p&gt;&#xA;&lt;p&gt;Is there a study that looks at the result or justifies the approach?&lt;/p&gt;&#xA;&lt;p&gt;Edit 1: Early Spring 2023&lt;/p&gt;&#xA;&lt;p&gt;Thankfully, the university I work at mostly stopped this practice. After an 180 turn we are going back to the old format. Among the odd events was that the length of a semester was not really 8 weeks either, it could be as short as 6.5 weeks!&lt;/p&gt;&#xA;&lt;p&gt;Edit 2: Late Spring 2023&lt;/p&gt;&#xA;&lt;p&gt;I believe I  finally understood the reason behind offering short courses. Businesses are willing to support their employees taking one course at a time. So this creates a loophole. Make a course run on an &amp;quot;8-week&amp;quot; schedule. Then student can take two courses during a semester with employer support. But if businesses change their language and say you can take  9 or 12 credit hours per calendar year, and no more than 3 or 4 hours at a time, then this loophole will be closed.&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2023-04-21T06:04:19.980" LastActivityDate="2023-04-21T06:04:19.980" Title="Is there a study that compares 8-week vs 16-week math classes?" Tags="|undergraduate-education|vector-calculus|usa|online|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="20911" PostTypeId="2" ParentId="20910" CreationDate="2021-05-22T01:29:35.047" Score="10" Body="&lt;p&gt;My school, a community college in California, looked at this sort of thing about 10 years ago when the union was pushing to shorten the semester from 18 weeks to 16 weeks. We have also historically had 6-week or 8-week summer classes, and many faculty have always been skeptical about the quality of instruction in those classes. Often when I see a student who doesn't seem to have the knowledge covered by the prerequisite for my class, I check their transcript and see the same pattern. They took the class once or twice and failed it, then took it a third time in a short summer class and passed.&lt;/p&gt;&#xA;&lt;p&gt;At the time when we were discussing the shortening of the semester by two weeks, the supporters of the proposal went around saying that &amp;quot;studies&amp;quot; showed that success was actually higher with the shorter semester. Later it turned out that these studies never actually existed.&lt;/p&gt;&#xA;&lt;p&gt;The problem with trying to do a study of this is that you can't control for all the variables.&lt;/p&gt;&#xA;&lt;p&gt;The student population taking an 8-week summer course is a different population than the one taking a 16-week course during a regular semester. For example, we would get a lot of students from UC who would show up during the summer at our community college in order to take care of their one remaining requirement.&lt;/p&gt;&#xA;&lt;p&gt;My school finally did shorten the semester by 2 weeks. You would think we could look at success rates before and after, and see if there was a change. But most faculty responded to the change by cutting material and assigning less work, so again there is an uncontrolled variable.&lt;/p&gt;&#xA;&lt;p&gt;For a lot of faculty who don't care about their jobs, an 8-week online course is the greatest deal ever. You've already recorded your canned videos. The homework is in the textbook publisher's online system, so you don't have any grading. You were always doing zero work, and now you're just doing zero work twice as fast. Then in November you take that trip to Costa Rica.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-05-22T01:29:35.047" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20912" PostTypeId="1" CreationDate="2021-05-22T06:17:45.150" Score="5" ViewCount="190" Body="&lt;p&gt;So, what we essentially mean is students will apply their knowledge on divisibility, factorization, prime numbers, lcm, gcf, decimals, fractions, etc to play the game. A somewhat different approach to practice maths and to help them through with these concepts.&#xA;Depending on age there will be advanced levels with more other topics involved in designing the game.&lt;/p&gt;&#xA;&lt;p&gt;Are there any such game app?&lt;/p&gt;&#xA;" OwnerUserId="16098" LastEditorDisplayName="user507" LastEditDate="2021-05-30T15:26:59.193" LastActivityDate="2021-06-01T16:37:18.327" Title="Are there any mathematics based game apps which require students (between 10 - 16 years) to apply their maths knowledge to play the game" Tags="|secondary-education|number-theory|mathematics-in-daily-life|recreational-learning|rational-numbers|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20916" PostTypeId="2" ParentId="20903" CreationDate="2021-05-23T12:33:18.307" Score="5" Body="&lt;p&gt;Glad you are seeking peer support for this. Leaders' connections to good math friends make their mathematics events shine!&lt;/p&gt;&#xA;&lt;p&gt;Some suggestions, based on my experiences organizing math events:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Know your audience.&lt;/li&gt;&#xA;&lt;li&gt;KNOW YOUR AUDIENCE. Are your students there for goal-oriented work, such as an academic credit or winning a competition, or are they playing and exploring? How does their life experience and identity intersect with their math work? Are the families supportive? What's their past math experience, and how do you know? Are the students well-fed, well-rested, and well-supplied when you meet, or do you need to provide general well-being support and supplies as well? Who finances your endeavor (families, a non-profit, a school system...) and what are their expectations of you? What struggles in the relevant communities are likely to affect your event (poverty, political tensions...)? You may want to include some of that info in your next question about organizing events.&lt;/li&gt;&#xA;&lt;li&gt;What's your setting: physical or virtual venue and what kind, during or after school, how many people (students, teachers, volunteers), what supplies, etc.?&lt;/li&gt;&#xA;&lt;li&gt;Who are you? Workshops are built on personal trust towards people leading them. So are peer networks. You may want to show your face(s) and name(s) to your prospective audience and to the peers whose support you seek.&lt;/li&gt;&#xA;&lt;li&gt;What are hopes and dreams of the members of your organization in regard to your event? If all goes well, what will change for you and yours?&lt;/li&gt;&#xA;&lt;li&gt;What is your organization's big goal, philosophy, manifesto? What change are you trying to cause in mathematics education overall? What is it all about for you? The more specific you can be here, the better it will help to plan your event.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If you do not know your audience and setting, start there. Find a community that needs a math workshop, and adjust your work to THEIR needs. Become community-responsive.&lt;/p&gt;&#xA;&lt;p&gt;Thank you for sharing your journey!&lt;/p&gt;&#xA;" OwnerUserId="118" LastActivityDate="2021-05-23T12:33:18.307" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20917" PostTypeId="2" ParentId="20910" CreationDate="2021-05-24T18:00:27.747" Score="10" Body="&lt;p&gt;In all scenarios, unless the school judges the success of the course by some external metric it is very difficult to objectively compare different methods of instruction. The plain fact is that instructors adjust the difficulty to fit the term.&lt;/p&gt;&#xA;&lt;p&gt;Summer school in-person instruction can be very effective if the students are focused on just one or two classes. However, if they are working and don't have the time it is disastrous since the lack of elasticity in their schedule means they can't recover if they fall behind.&lt;/p&gt;&#xA;&lt;p&gt;Online 8 week courses are not a good format for serious math in my experience. The selling point for online work is that school is something you do in the margin of your life. Like the old nights and weekends plans we used to get with cell phones. The thing is, calculus III or real analysis or abstract algebra etc... these are not things you do on the edge of your day. Especially when you've not really absorbed the prerequisites.&lt;/p&gt;&#xA;&lt;p&gt;As an instructor, the students who are so underprepared which come to online 8 week courses present an impossible problem. It's just about impossible to teach math well without face to face dialogue, ok, perhaps that's too strong, but it is much harder to assess if you are reaching your target audience with your help. The systematic lack of feedback by most online students is simply maddening.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2021-05-25T05:21:59.847" LastActivityDate="2021-05-25T05:21:59.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20918" PostTypeId="2" ParentId="20892" CreationDate="2021-05-25T02:09:35.730" Score="7" Body="&lt;p&gt;You could look at &lt;a href=&quot;https://neilstrickland.github.io/maths_with_maple/&quot; rel=&quot;noreferrer&quot;&gt;https://neilstrickland.github.io/maths_with_maple/&lt;/a&gt;.  This is a full set of material for a CAS-based course which is mostly calculus but has some precalculus.  It was originally taught using Maple, but I have recently added files that do most of the same work using Maxima instead.  (Maxima is open source, but Maple is not.)  Everything is released under a Creative Commons license.  The idea of the course was to deemphasise routine calculation and use CAS to enhance understanding; you can judge for yourself how well I succeeded.&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2021-05-25T02:09:35.730" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20919" PostTypeId="2" ParentId="20887" CreationDate="2021-05-25T15:10:03.927" Score="2" Body="&lt;p&gt;If you have access to a Moodle server, you could look at CodeRunner (&lt;a href=&quot;https://coderunner.org.nz/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://coderunner.org.nz/&lt;/a&gt;).  This is a Moodle question type that can be used for automated assessment of programming exercises in a variety of languages.  The documentation says the following about supported languages:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;CodeRunner currently natively upports Python2 (considered obsolescent), Python3, C, C++, Java, PHP, Pascal, JavaScript (NodeJS), Octave and Matlab. However, other languages are easily supported without altering the source code of either CodeRunner just by scripting the execution of the new language within a Python-based question.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Personally I have set up a server with CodeRunner that one of my colleagues is using for teaching Python.  He seems happy with it, but I have not had much involvement since the initial setup.)&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2021-05-25T15:10:03.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20920" PostTypeId="1" CreationDate="2021-05-28T17:58:53.780" Score="-6" ViewCount="246" Body="&lt;p&gt;I love pure mathematics, so much. However, I recently did probability and statistics and applied mathematics courses, and I'm confused with statistics. There's no proofs and logic in applied math courses. I've always wondered why &lt;span class=&quot;math-container&quot;&gt;$P(A \cup B)=P(A)+P(B)- P(A \cap B)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16126" LastEditorUserId="13657" LastEditDate="2021-05-28T19:11:28.757" LastActivityDate="2021-05-28T19:17:07.327" Title="Why is Pure Mathematics so much easier than Applied Mathematics?" Tags="|applied-mathematics|" AnswerCount="1" CommentCount="1" ClosedDate="2021-05-30T22:03:27.953" ContentLicense="CC BY-SA 4.0" />
  <row Id="20921" PostTypeId="2" ParentId="20920" CreationDate="2021-05-28T19:17:07.327" Score="3" Body="&lt;p&gt;Applied mathematics can mean both proof-based approach to an applied subject, or to pure numerical and heuristic approach to problems.&lt;/p&gt;&#xA;&lt;p&gt;So: both pure and applied mathematics can be proof-based. However, applied issues tend to be messier, as real world is messy and applications tend to come from there. Not that pure mathematics does not also get messy sometimes...&lt;/p&gt;&#xA;&lt;p&gt;Whether you consider proof-based or more heuristic stuff easier is for you to decide, but it is a matter of personality and prior learning. Neither is inherently easier or harder. This seems to be your issue.&lt;/p&gt;&#xA;&lt;p&gt;For the stochastics issue, you might want to search &amp;quot;probability of union&amp;quot;, for example &lt;a href=&quot;https://duckduckgo.com/?q=probability+of+union&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://duckduckgo.com/?q=probability+of+union&lt;/a&gt; . If the answers do not satisfy you, you might want to find some lecture notes or a book that covers the basics of stochastics. It is sure to be discussed.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-05-28T19:17:07.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20923" PostTypeId="1" CreationDate="2021-05-30T23:18:30.360" Score="22" ViewCount="1990" Body="&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;My friend told me that when she was in elementary school, the teacher would ask questions like &amp;quot;If you have &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; apples and eat &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; of them, how many apples do you have left?&amp;quot; A kid in the class answered &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, to which the teacher replied &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; what? &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; elephants?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Seemingly the teacher wanted to emphasize the importance of using units.&lt;/p&gt;&#xA;&lt;p&gt;My friend told me that this led her to believe for some time that unitless numbers were meaningless, e.g. that calculating the area of a rectangle with side lengths &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; was meaningless.&lt;/p&gt;&#xA;&lt;p&gt;I have also been wondering whether one should write calculations as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$6 \text{ apples} - 2 \text{ apples} = 4 \text{ apples}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$20 \text{ cakes} / 5 \text{ cakes per person} = 4 \text{ persons}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;or as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$6-4 = 2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$20/5 = 4$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;What does research indicate about how one should treat units in elementary school? Hopefully some of the concerns above have been addressed quite directly in some research paper.&lt;/p&gt;&#xA;" OwnerUserId="6527" LastEditorUserId="20719" LastEditDate="2023-08-13T10:09:37.257" LastActivityDate="2023-08-13T10:09:37.257" Title="What does research indicate about how one should treat units in elementary school?" Tags="|primary-education|units|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20924" PostTypeId="2" ParentId="20923" CreationDate="2021-05-31T00:12:42.230" Score="16" Body="&lt;p&gt;There is a distinction to be made between numbers that depend for their meaning on an arbitrary standard of measurement and those that don't. If I ask how many states there are in the US, the answer is 50. If I ask how far it is to your dentist, and you say 15, then that's just wrong. 15 miles? 15 blocks? 15 minutes? This is pretty standardized as usage in science and engineering, which matches usage in ordinary speech.&lt;/p&gt;&#xA;&lt;p&gt;I don't think this is an issue that can be settled by research. Research can't tell us whether poetry should rhyme, whether men should wear skirts, or whether it's OK to put a 30 amp fuse on a circuit that's only wired to be able to handle 20 amps. These are matters of taste, convention, and utility. The standard usage for units in science and engineering is based on a combination of taste, convention, and utility.&lt;/p&gt;&#xA;&lt;p&gt;Students at the elementary school level should of course be taught to write numbers with units when it's normal and correct to do so. In addition, kids at this age need to absorb a conceptual understanding of the meaning of multiplication and division. Writing fake units like &amp;quot;5 apples&amp;quot; or &amp;quot;5 cakes per person&amp;quot; is a nice way to help them develop competence at reasoning about what operations make sense, because they can check that the result of their calculation has units that make sense, as in your cakes-people example. However, it should be taught to them as a method of checking, keeping track, and deciding what makes sense -- not as an arbitrary rule.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-05-31T00:19:06.693" LastActivityDate="2021-05-31T00:19:06.693" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="20925" PostTypeId="1" CreationDate="2021-05-31T18:00:58.730" Score="3" ViewCount="237" Body="&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am working with 10-13 year olds.&lt;/p&gt;&#xA;&lt;p&gt;I saw a problem which requires the students to convert &lt;span class=&quot;math-container&quot;&gt;$6/27$&lt;/span&gt; into decimal notation. I can think of some methods, but I have not tested using any of the following methods in a classroom setting:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Method 1:&lt;/strong&gt; One can do the calculation&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{c c c c c c c c}&#xA;6 \hspace{5mm} 0 &amp;amp; 0 &amp;amp; : &amp;amp; 27 &amp;amp; = &amp;amp; 0 &amp;amp; 2 \\ &#xA;0 \hspace{5mm}  0 &amp;amp; \\&#xA;\overline{6 \hspace{5mm} 0} &amp;amp;\\&#xA;5 \hspace{5mm} 4 &amp;amp;\\&#xA;\overline{\hspace{7mm} 6} \\&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;with appropriately placed commas, to arrive at&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;6/27 = 0.\overline{2}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Method 2:&lt;/strong&gt; If one recalls that &lt;span class=&quot;math-container&quot;&gt;$1/9 = 0.\overline{1}$&lt;/span&gt;, then one can compute&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\frac{6}{27} = \frac{2}{9} = 2 \cdot \frac{1}{9} = 2 \cdot 0.\overline{1} = 0.\overline{2}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Method 3:&lt;/strong&gt; If one recalls that &lt;span class=&quot;math-container&quot;&gt;$1/3 = 0.\overline{3}$&lt;/span&gt;, then one can compute&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\frac{6}{27} = \frac{2}{9} = \frac{2}{3} \cdot \frac{1}{3} = 0.\overline{6} \cdot 0.\overline{3}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and then one can perhaps proceed with a multiplication strategy.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;What are strategies that a 10-13 year old could use to convert &lt;span class=&quot;math-container&quot;&gt;$6/27$&lt;/span&gt; into decimal notation?&lt;/p&gt;&#xA;" OwnerUserId="6527" LastEditorUserId="6527" LastEditDate="2021-05-31T18:06:22.497" LastActivityDate="2021-06-19T01:33:09.457" Title="What are strategies that a 10-13 year old could use to convert 6/27 into decimal notation?" Tags="|primary-education|division|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20926" PostTypeId="2" ParentId="20925" CreationDate="2021-05-31T20:02:14.297" Score="6" Body="&lt;p&gt;I prefer method 2, which has the student simplifying first.  Even if the student doesn't know that &lt;span class=&quot;math-container&quot;&gt;$&#xA; \frac{1}{9} =  0.\overline{1} &#xA;$&lt;/span&gt;,  the student can still divide 2.00 by 9.&lt;/p&gt;&#xA;&lt;p&gt;If the student uses method 1, s/he misses the chance to see the pattern in ninths and is more likely to make a mistake because s/he is dividing by 27 (a 2 digit number) instead of a one digit number.  On the other hand, it is a good method because it works for all fractions.&lt;/p&gt;&#xA;&lt;p&gt;I would never use method 3,  since the student is multiplying 2 repeating decimals and they have no understanding of how that will work.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-05-31T20:02:14.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20927" PostTypeId="2" ParentId="20908" CreationDate="2021-05-31T20:02:29.383" Score="3" Body="&lt;p&gt;Assuming that by &amp;quot;Geometric constructions&amp;quot;, you mean &amp;quot;ruler and compass constructions&amp;quot;, here is a sequence of progressively-more-difficult constructions that begin at the trivial and end at the impossible:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The &amp;quot;standard&amp;quot; constructions:&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Perpendicular bisector of a line segment.&lt;/li&gt;&#xA;&lt;li&gt;Angle bisector.&lt;/li&gt;&#xA;&lt;li&gt;Perpendicular to a line through a point.&lt;/li&gt;&#xA;&lt;li&gt;Duplicated line segment.&lt;/li&gt;&#xA;&lt;li&gt;Line parallel to a line through a point.&lt;/li&gt;&#xA;&lt;li&gt;Mirroring a point in a line.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;Some easy extensions:&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Line disection (into arbitrarily many pieces)&lt;/li&gt;&#xA;&lt;li&gt;Angles of 30, 45, 60, and 90 degrees.&lt;/li&gt;&#xA;&lt;li&gt;An equilateral triangle.&lt;/li&gt;&#xA;&lt;li&gt;Triangles given:&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A triangle to copy.&lt;/li&gt;&#xA;&lt;li&gt;Two angles and a side&lt;/li&gt;&#xA;&lt;li&gt;Two sides and the angle between.&lt;/li&gt;&#xA;&lt;li&gt;Three sides&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;The incircle and circumcircle of a triangle.&lt;/li&gt;&#xA;&lt;li&gt;The diameter of a circle.&lt;/li&gt;&#xA;&lt;li&gt;A circle through three points.&lt;/li&gt;&#xA;&lt;li&gt;The tangents to a circle passing through a given point on or outside the circle.&lt;/li&gt;&#xA;&lt;li&gt;A regular square, pentagon, hexagon, octagon, dodecagon (inscribed in a given circle or from one side), or a regular polygon with any (small) number of sides in &lt;a href=&quot;https://oeis.org/A003401&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; list.&lt;/li&gt;&#xA;&lt;li&gt;Any results from &lt;em&gt;Elements&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;Significantly harder problems:&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Some of the Wernick List (construct a triangle given three of: the vertices, the circumcenter, the side midpoints, the centroids, the altitude feet, the orthocentre, the feet of the internal angle bisectors, and the incenter - obviously not the unsolvable cases!)&lt;/li&gt;&#xA;&lt;li&gt;The Mohr-Mascheroni constructions.&lt;/li&gt;&#xA;&lt;li&gt;The Poncelet–Steiner constructions.&lt;/li&gt;&#xA;&lt;li&gt;Calculate the best approximation to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; that you can using a ruler and compass.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/1811360/a-challenging-straightedge-and-compass-construction&quot;&gt;This&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/questions/2298228/difficult-geometric-construction&quot;&gt;this&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;Trick questions&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Bisect an angle using only a ruler (not impossible, but the fact that I said &amp;quot;ruler&amp;quot; (as in the physical object) not &amp;quot;straightedge&amp;quot; is important, and it's nothing to do with the markings).&lt;/li&gt;&#xA;&lt;li&gt;Angle trisection, squaring the circle, constructing a regular heptagon.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="11771" LastActivityDate="2021-05-31T20:02:29.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20928" PostTypeId="2" ParentId="20923" CreationDate="2021-06-01T01:07:36.770" Score="11" Body="&lt;p&gt;The &lt;a href=&quot;https://journals.sagepub.com/doi/full/10.1177/0731948718790089&quot; rel=&quot;noreferrer&quot;&gt;CRA approach&lt;/a&gt; says to move from concrete to pictorial to abstract concepts. The idea of numbers without units or identifiers is treating numbers as an abstraction, and a positive goal. You start with the concrete examples - 3 apples plus 2 apples is 5 apples, 3 pebbles plus two pebbles is 5 pebbles, 3 people plus 2 people is 5 people, and spot the pattern that lets you say 3 plus 2 is 5. What is &amp;quot;3&amp;quot; on its own? Three whats? The point is &amp;quot;3&amp;quot; represents a pattern with an empty slot for the unit/type of object into which you can insert anything.&lt;/p&gt;&#xA;&lt;p&gt;When there is only one entity under discussion being counted, it is perfectly legitimate (and a positive achievement) to use the abstraction and let context fill it in. The answer to &amp;quot;If you have 6 apples and eat 2 of them, how many apples do you have left?&amp;quot; is actually 4, and not 4 apples, because the units to use are specified in the question. If you had asked &amp;quot;If you have 6 apples and eat 2 of them, what do you have left?&amp;quot;, then the correct answer would be 4 apples. There is a subtle distinction that kids pick up automatically from usage - in this case, it sounds like the kid who answered the question understood the rules better than the teacher did.&lt;/p&gt;&#xA;&lt;p&gt;What I think the teacher is trying to address is the tendency to overdo the abstraction, and abstract in situations where it is not safe to do so. The most common situations are where there are several different units in use, or where you are writing numbers for someone who might not know the context.&lt;/p&gt;&#xA;&lt;p&gt;If you have six boxes of eggs (with six eggs to a box) and you drop two of them, how many do you have left? We have two different units: eggs, and boxes. Did you drop two eggs or two boxes? Are you asking how many eggs or how many boxes are left? You can no longer assume a unit, and must specify at every step.&lt;/p&gt;&#xA;&lt;p&gt;The rules over when units are or are not required are complicated. Sometimes you can set the formal rules out explicitly - in which case you need to get them right, and explain all the exceptions and corner cases too, or cause more confusion when usage conflicts with the rules you've just given. Or you can teach them by example, using units when needed, and not when you don't, and let children's natural language acquisition skills deduce the rules subconsciously. That's usually easier and more accurate, but  often results in knowledge that people can't explain - they just know it &amp;quot;feels wrong&amp;quot; to say it a certain way.&lt;/p&gt;&#xA;&lt;p&gt;As mathematics gets more formal and precise, all this intuitive encoding of knowledge starts to get in the way, and people lose track of the complicated manipulations being applied subconsciously and get confused about what concepts they're using. Being careful about units (and more generally, about &lt;em&gt;types&lt;/em&gt;) is one way to resolve this confusion. But it is a stage of abstraction beyond the abstraction of pure numbers.&lt;/p&gt;&#xA;&lt;p&gt;You have to teach it in stages. First you count concrete objects - 3 apples, 3 pebbles, 3 people. Then you recognise the pattern, and abstract them all as a &amp;quot;3&amp;quot;. &lt;em&gt;Only when the children are able to do that reliably&lt;/em&gt;, do you start to explore the multi-unit ambiguous situations requiring more precision, and put the units back in. And at a much, much later stage of education, you can start to pull apart the different types and the rules governing them more formally (cardinal numbers, ordinal numbers, rationals, real numbers, vectors, torsors, groups, dimensional analysis, etc.). It may help that later learning if you give some &lt;em&gt;indication&lt;/em&gt; early on that these are really different types of things that we are tacitly mixing up together, so it doesn't come as such a shock later, but it just confuses things to try to explain everything at once too early, or to skip over important intermediate stages.&lt;/p&gt;&#xA;&lt;p&gt;I can't tell from what you write whether the teacher is trying to prevent students using the abstraction at all, or whether having taught that stage, they are moving on to the next step of explaining the exceptions where you do need to keep the concrete types. The &amp;quot;how many apples do you have left?&amp;quot; question is not a good example, but I don't know if that's what the teacher actually used, or is your friend's paraphrase.&lt;/p&gt;&#xA;&lt;p&gt;But I would say that children need to learn &lt;em&gt;both&lt;/em&gt; approaches. They need to be able to abstract pure numbers from a concrete situation, and they also need to be able to keep and manipulate units and types in more complex multi-unit situations. The latter is a significantly more advanced stage of understanding, that requires a firm grasp of the abstraction process first. This isn't based on any pedagogical research - it's based on the structure of the mathematical concepts being taught. It's not that one approach is better than the other and should be the only one taught, it's that you need to teach them &lt;em&gt;both&lt;/em&gt;, one after the other.&lt;/p&gt;&#xA;" OwnerUserId="16138" LastActivityDate="2021-06-01T01:07:36.770" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="20929" PostTypeId="1" CreationDate="2021-06-01T03:40:09.933" Score="2" ViewCount="190" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/12185/how-to-respond-to-solve-this-equation-in-a-basic-algebra-class&quot;&gt;A past question&lt;/a&gt; described a school where many teachers insisted that answers to algebra problems had to be phrased in set-theoretic language or notation. For example, when asked to solve &lt;span class=&quot;math-container&quot;&gt;$2x+3=6−x$&lt;/span&gt;, students were expected to say that the solution set was &lt;span class=&quot;math-container&quot;&gt;$\{1\}$&lt;/span&gt;, not that &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt;. In a comment, the OP wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I think that there is a history in my department of using a particular textbook that stresses this distinction, and that over the past several decades, it has crept into the culture here to the point where it is codified in course descriptions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now I'm wondering who the authors of such textbooks were. It feels like old math education, in the vein of New Math... Presumably the makers were American? Unless a school was importing their textbooks from overseas?&lt;/p&gt;&#xA;" OwnerUserId="5597" LastEditorDisplayName="user507" LastEditDate="2021-06-01T12:33:57.660" LastActivityDate="2021-06-07T00:16:51.550" Title="When and where were textbooks that use set notation for basic algebra solutions?" Tags="|textbooks|history|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="20930" PostTypeId="2" ParentId="20912" CreationDate="2021-06-01T16:37:18.327" Score="2" Body="&lt;p&gt;While this is not quite an app, I believe it fits within the spirit of the question: Desmos activities.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://teacher.desmos.com/collections/featured&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://teacher.desmos.com/collections/featured&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;While some Desmos activities are very math-forward, others follow Dan Myers' &amp;quot;Three Acts&amp;quot; format, &lt;a href=&quot;https://teacher.desmos.com/activitybuilder/custom/57e04d6fc8c899190bdbb450?collections=5d9d12cd9bb4713ab3b0d267&quot; rel=&quot;nofollow noreferrer&quot;&gt;giving students an interesting scenario and asking them to solve it with math,&lt;/a&gt; and showing the students what their computations give them in real time. Others ask students to get &lt;a href=&quot;https://teacher.desmos.com/activitybuilder/custom/566b317b4e38e1e21a10aafb?collections=5da6462c8b305273be677729&quot; rel=&quot;nofollow noreferrer&quot;&gt;marbles to slide through certain points&lt;/a&gt;. Many are teacher-paced, but most can be student-paced, and all can be modified by instructors.&lt;/p&gt;&#xA;&lt;p&gt;One other (probably popular) answer that I do not yet see is the &lt;a href=&quot;https://dragonbox.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dragonbox&lt;/a&gt; series. In particular, algebra 12+ and Elements give puzzles that rely on algebra and on geometry to solve!&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2021-06-01T16:37:18.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20931" PostTypeId="1" CreationDate="2021-06-01T21:55:35.833" Score="17" ViewCount="6711" Body="&lt;p&gt;The imagined students are in elementary school, say around 9-13 years old.&lt;/p&gt;&#xA;&lt;p&gt;I want to use rather precise terminology when talking to my students. However, it seems like we typically use the same terminology for a geometric figure and for its boundary.&lt;/p&gt;&#xA;&lt;p&gt;Does the word &amp;quot;triangle&amp;quot; refer to the &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-dimensional geometric figure or the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-dimensional geometric figure? It seems to me like we kind of treat &amp;quot;triangle&amp;quot; to mean either, or even both at the same time.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QesHG.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QesHG.png&quot; alt=&quot;two things called triangle&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are similar issues with other terms, like &amp;quot;square&amp;quot;, &amp;quot;circle&amp;quot; and &amp;quot;cylinder&amp;quot;. Here I can imagine the word &amp;quot;cylinder&amp;quot; also being used for cylinders open at the bottom, or a cylinder open in both ends. There seems to be some distinction when talking about &amp;quot;balls&amp;quot; and &amp;quot;spheres&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I fear that when the terminology is not precise enough that this will lead to unexpected problems when defining and working with other concepts later on. For instance, if one defines a triangle as some union of edges and vertices, then one needs to be a bit careful when defining the area of the triangle.&lt;/p&gt;&#xA;&lt;p&gt;Many of the geometric figures are so elementary that they are deeply rooted in daily language, and there seems to be no great solution.&lt;/p&gt;&#xA;&lt;p&gt;I have played with the idea of calling things something like &amp;quot;1D-triangle&amp;quot; and &amp;quot;2D-triangle&amp;quot;, but this seems to make things worse and I have not come up with good alternative terminology.&lt;/p&gt;&#xA;&lt;p&gt;My current go-to would be to tell my students something like &amp;quot;We call this a triangle, but we also call this a triangle. Sometimes the distinction matters, but most often it is clear from context.&amp;quot; Another possibility would be to just let the students realize the ambiguity themselves. I don't really have any evidence that suggests that this actually is an issue.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;What can (and should) an educator do about ambiguous terms like &amp;quot;triangle&amp;quot;, &amp;quot;square&amp;quot;, etc?&lt;/p&gt;&#xA;" OwnerUserId="6527" LastEditorUserId="6527" LastEditDate="2021-06-03T09:34:46.617" LastActivityDate="2023-10-04T13:17:02.383" Title="What can (and should) an educator do about ambiguous terms like &quot;triangle&quot;, &quot;square&quot;, etc?" Tags="|geometry|primary-education|terminology|definitions|language-use|" AnswerCount="12" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="20932" PostTypeId="2" ParentId="20931" CreationDate="2021-06-02T03:57:03.320" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Many of the geometric figures are so elementary that they are deeply rooted in daily language, and there seems to be no great solution.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I agree with you here, and I think this is the key point. To me they are clearly well-defined: &amp;quot;Triangle&amp;quot;, &amp;quot;square&amp;quot;, and polygons in general, are bounded regions on the Euclidean plane, i.e., 2D figures.&lt;/p&gt;&#xA;&lt;p&gt;In your pictures, the left one is a triangle. When we say &amp;quot;the triangle ABC&amp;quot;, it is a (very) economic way to describe a 2d region enclosed by a &amp;quot;three-non-colinear-points-and-three-lines&amp;quot; system.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the intention of your right picture is to show the edges of a triangle. A more technical definition would be a &amp;quot;closed polygonal chain&amp;quot;, roughly speaking, just a set of points connected by lines.  (&lt;a href=&quot;https://en.wikipedia.org/wiki/Polygonal_chain&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Polygonal_chain&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Maybe the most remarkable difference between those objects is the area. In Euclidean Geometry, the area of DEF is 0 (the area of 1 line is zero, so the area of 3 lines is just 0+0+0). Nevertheless, the area &lt;em&gt;enclosed by&lt;/em&gt; DEF is the same as the area of the &amp;quot;triangle ABC&amp;quot;, and it is nonzero.&lt;/p&gt;&#xA;" OwnerUserId="3380" LastEditorUserId="14023" LastEditDate="2021-06-02T12:16:31.837" LastActivityDate="2021-06-02T12:16:31.837" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20933" PostTypeId="2" ParentId="20931" CreationDate="2021-06-02T04:31:34.070" Score="3" Body="&lt;p&gt;A triangle is born from three non-collinear points and the axiom that two points determine a line. In the context of neutral geometry, a triangle has no structure other than three lines and three points. In particular, there is no notion of the interior of a triangle without more axioms.&lt;/p&gt;&#xA;&lt;p&gt;In the real projective plane, one cannot define the &amp;quot;interior&amp;quot; of a triangle. Similarly, the notion of the interior of a triangle is problematic in finite geometries.&lt;/p&gt;&#xA;&lt;p&gt;In Euclidean geometry, one uses Hilbert's betweenness axiom (plane separation) and Pasch's theorem to make sense of the interior of a triangle.&lt;/p&gt;&#xA;&lt;p&gt;As educators, when teaching Euclidean geometry, we should try to use vocabulary that makes a distinction between a triangle and its interior.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-06-02T04:31:34.070" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20934" PostTypeId="2" ParentId="20892" CreationDate="2021-06-02T09:35:35.430" Score="1" Body="&lt;p&gt;Some brief resources-&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://ComputerBasedMath.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;ComputerBasedMath.org&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.wolfram.com/education/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.wolfram.com/education/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.wolfram.com/books/?source=nav&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.wolfram.com/books/?source=nav&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In particular, &lt;a href=&quot;https://www.wolfram.com/books/profile.cgi?id=9764&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.wolfram.com/books/profile.cgi?id=9764&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/2617/is-there-a-calculator-centered-secondary-school-curriculum-somewhere/2633#2633&quot;&gt;Is there a calculator centered secondary school curriculum somewhere?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2071" LastEditorUserId="2071" LastEditDate="2021-06-02T10:03:07.557" LastActivityDate="2021-06-02T10:03:07.557" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20936" PostTypeId="2" ParentId="20931" CreationDate="2021-06-02T12:51:47.290" Score="26" Body="&lt;p&gt;One encounters exactly the same issue teaching multivariable calculus when one treats integrals over three-dimensional regions and integrals over the surfaces that are their boundaries. In particular the word &lt;em&gt;sphere&lt;/em&gt; is particularly confusing in this context. (Mathematicians use &lt;em&gt;sphere&lt;/em&gt; to mean the the two-dimensional surface; colloquial speech and some physicists and engineers use the same word to describe the three-dimensional ball.)&lt;/p&gt;&#xA;&lt;p&gt;Except when teaching students studying mathematics per se, my sense is that the best practice is simply to say clearly what interpretation is intended. So one can say &lt;em&gt;the triangle formed by the edges connecting vertices A, B, and C&lt;/em&gt; or &lt;em&gt;the triangular region with vertices A, B, and C&lt;/em&gt;. Writing this I betray that I tend to use &lt;em&gt;triangle&lt;/em&gt; to mean the one-dimensional polygonal curve and say something like &lt;em&gt;triangular region&lt;/em&gt; to refer to the region it bounds and would usually simply speak of the &lt;em&gt;the triangle with vertices A, B, and C&lt;/em&gt; and the &lt;em&gt;triangular region&lt;/em&gt; that it bounds. In calculus I might speak of a &lt;em&gt;the surface of the sphere&lt;/em&gt; and the &lt;em&gt;spherical solid&lt;/em&gt;. In my private usage I would regard the first phrase as redundant (a sphere is a surface), but in the classroom I am talking to students and (hopefully) not to myself (I think of &lt;em&gt;area&lt;/em&gt; as &lt;em&gt;two-dimensional volume&lt;/em&gt; but I know better than to ever call an area a volume in calculus class).&lt;/p&gt;&#xA;&lt;p&gt;At any rate, in the classroom what matters is being clear and consistent. One does not want to make (fussy) fine distinctions with an audience unprepared to hear fine distinctions, and one does not want to use the same word(s) to refer to a region and its boundary. The second goal is more fundamental, but meeting it should not entail cumbersome and/or subtle terminology, particularly with students, such as typical high school students, who are simply unprepared to hear mathematical and terminological subtleties. We want to help them learn to listen more carefully, but this process is gradual, and should not come at the cost of more basic matters.&lt;/p&gt;&#xA;&lt;p&gt;One should also try to be aware of what other teachers the students might have might say in regards to the same figures. So when I teach multivariable calculus I try to figure out what the professor teaching the same students electromagnetism says in regards to spheres/balls. I don't necessarily follows his/her practice, but if I can I make clear where mine differs from it.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2021-06-02T12:51:47.290" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20937" PostTypeId="2" ParentId="20908" CreationDate="2021-06-02T14:53:50.280" Score="3" Body="&lt;p&gt;I would like to add a few of my favourites:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Apollonian constructions (construct a circle tangent to three objects, where the objects can be points, lines or circles). There are 10 of these problems, some are standard, some are tricky.&lt;/li&gt;&#xA;&lt;li&gt;Construct the golden ratio (this is related to the regular pentagon mentioned above).&lt;/li&gt;&#xA;&lt;li&gt;Solve a quadratic equation using straightedge and compass (the coefficients and the unit is given as line segments).&lt;/li&gt;&#xA;&lt;li&gt;Trisect an angle using paper folding (this cannot be done in general using straightedge and compass).&lt;/li&gt;&#xA;&lt;li&gt;Using a piece of string and two pins, one can draw an ellipse. &lt;a href=&quot;https://www.math.lsu.edu/%7Esmolinsk/GeometricConstructions_5_2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;This paper&lt;/a&gt; investigates constructions using a tool like this.&lt;/li&gt;&#xA;&lt;li&gt;The impossibility of constructing the centre of a circle using only a straightedge.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2021-06-02T14:53:50.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20938" PostTypeId="2" ParentId="20931" CreationDate="2021-06-02T15:35:27.300" Score="1" Body="&lt;p&gt;I don't really know what's the big problem. Terms need not be ambiguous; it's up to you to define and use them in clear and unambiguous manner. For example you can use &amp;quot;triangular region&amp;quot; or &amp;quot;cylindrical volume&amp;quot; to clearly differentiate from &amp;quot;triangle&amp;quot; and &amp;quot;cylindrical surface&amp;quot;, and of course you have to define whether your region/volume includes the boundary. I also don't see why you say you have to be careful when defining area of a triangle. Just define the area of a polygon via the shoelace formula and you're done (it is better than unsigned area).&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-06-02T15:35:27.300" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="20939" PostTypeId="2" ParentId="20931" CreationDate="2021-06-02T15:48:24.013" Score="41" Body="&lt;p&gt;I'm confused.  Are you really going to try to make this sort of distinction when teaching geometric figures to students &lt;em&gt;&amp;quot;around 9-13 years old&amp;quot;&lt;/em&gt;?  Students that age (and engineers my age -- much, much older) think that a triangle is a triangle.  It's a polygon formed by three non-colinear points.&lt;/p&gt;&#xA;&lt;p&gt;A triangle has many ways you can think about it.  Three points, and the connecting line segments.  The three angles.  The area of the triangle. The three ways you can measure the height of a triangle (used in calculating that area). Whether the triangle is equilateral, isosceles, acute, scalene.  Etc. But, it's a triangle.&lt;/p&gt;&#xA;&lt;p&gt;I finished at or near the top of my math classes all the way through K-12 and junior college (the top got a lot more crowded in my engineering math classes) I've never bothered to think of it any differently, and I can't imagine anything but confusion if you tried to make a distinction.&lt;/p&gt;&#xA;&lt;p&gt;Take a circle.  If I cared about the &lt;em&gt;1-dimensional&lt;/em&gt; aspect of it (your term, not mine), I'd speak of the circumference of the circle.  If I cared about the region it bounds, I might talk of the area.  If I care about it's trigonometric features, I'd discuss them separately.  But, a circle is a circle is a circle.&lt;/p&gt;&#xA;" OwnerUserId="13120" LastEditorUserId="13120" LastEditDate="2021-06-02T15:56:10.487" LastActivityDate="2021-06-02T15:56:10.487" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="20940" PostTypeId="2" ParentId="20931" CreationDate="2021-06-02T15:58:13.193" Score="23" Body="&lt;p&gt;I think the distinction you are raising is not natural to students at this age. I teach undergraduates and graduate students, not elementary schoolers, but I find that it is not natural for undergraduates who have not had a theoretical math course. &lt;b&gt;In my experience, students do not naturally think of geometric figures as sets of points.&lt;/b&gt;&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$P = (-1,-1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$Q = (1,0)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$R = (0,1)$&lt;/span&gt;, then students understand just fine the idea that &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; is in the interior of the triangle &lt;span class=&quot;math-container&quot;&gt;$PQR$&lt;/span&gt;, and is not on the boundary of the triangle &lt;span class=&quot;math-container&quot;&gt;$PQR$&lt;/span&gt;, but they don't think of the triangle &lt;span class=&quot;math-container&quot;&gt;$PQR$&lt;/span&gt; as a set of points, so the question of whether or not &lt;span class=&quot;math-container&quot;&gt;$(0,0) \in PQR$&lt;/span&gt; is not a natural one to them. (They also don't distinguish naturally between the open triangle, which does not contain its boundary, and the closed triangle which does.)&lt;/p&gt;&#xA;&lt;p&gt;When you get to point set toplogy or to multivariate integration, you need to raise this issue. Until then, though, I think it is just fine to use the words &amp;quot;interior of the triangle&amp;quot; or &amp;quot;boundary of the triangle&amp;quot; when you really need to specify, and ignore the issue otherwise. (Again, I'm not an elementary teacher, though.)&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;If you do want to teach this distinction, you might want this cartoon:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kK2R8.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kK2R8.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2021-06-27T18:07:44.557" LastActivityDate="2021-06-27T18:07:44.557" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20942" PostTypeId="2" ParentId="20931" CreationDate="2021-06-02T17:51:01.807" Score="9" Body="&lt;p&gt;I confess, when you start talking about 1-D triangles, my own first thought is &amp;quot;how can you have non-colinear points in 1-D?&amp;quot;. So, I imagine most students that age will have a far more difficult time with that.&lt;/p&gt;&#xA;&lt;p&gt;Keep in mind age appropriateness. For 9-12 year old children, you are generally looking at a level of psychological development characterized by concrete reasoning (Piaget as the primary theorist here). For that level, a triangle is commonly imagined as something physical, having the attribute of three edges/corners. As a shape, it is understood (once the concepts are introduced) to have the additional properties of area and perimeter. The formal definition of these from an advanced math perspective is probably beyond what most students in that age range will understand.&lt;/p&gt;&#xA;&lt;p&gt;Instead, stick with simple physical properties they should be familiar with, commonly described as the &amp;quot;enclosed area&amp;quot;, &amp;quot;filled area&amp;quot;, or simply &amp;quot;area&amp;quot;, versus the &amp;quot;perimeter&amp;quot;, &amp;quot;boundary&amp;quot;, or &amp;quot;outline&amp;quot;, as needed. Save the advanced definitions for advanced students - eg, those taking various Euclidian and non-Euclidian geometry classes, multivariate calculus, etc. If it is easier for you, you can suggest that, unless you say otherwise, when you name a shape, you mean the boundary and not the enclosed area, but keep in mind that the majority of your students will need the reminder whenever the question comes up.&lt;/p&gt;&#xA;&lt;p&gt;You could also give them a heads up that there is a formal definition, but for your purposes you will be going with the common usage as above. (I like this approach when discussing color, for example, since you will get art students talking about primary and secondary color, which is useful for some purposes, but not the same as specific frequencies of light as discussed in science and physics classes.) A brief heads up of &amp;quot;these work for reasons we aren't going to get into this year&amp;quot;, with a later &amp;quot;offline&amp;quot; discussion for those who are interested enough to ask, is probably the best approach.&lt;/p&gt;&#xA;" OwnerUserId="16151" LastActivityDate="2021-06-02T17:51:01.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20943" PostTypeId="1" CreationDate="2021-06-03T15:13:30.807" Score="10" ViewCount="1130" Body="&lt;p&gt;I was recently looking through job listings in my local area (US high school, ages 14-18) and I found two boarding schools with vacancies, one was male-only and the other was female-only. This got me thinking:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;QUESTION&lt;/strong&gt;:&#xA;is there any research or evidence that suggests different approaches to teaching math / different assignments are better suited for different genders?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Should the teaching style of the math teachers at these two schools be very different? For example, would an all male classroom do better with direct instruction? Open problem solving? Socratic Method? What about an all female class, would they do better with Socratic Method? End of unit projects vs end of unit tests?&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2021-06-09T21:08:10.063" Title="Different Teaching Styles for Different Genders?" Tags="|secondary-education|education-research|gender|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="20944" PostTypeId="2" ParentId="20943" CreationDate="2021-06-03T15:40:06.290" Score="17" Body="&lt;p&gt;I can't cite research to back me up on this, and you will want answers that reference research. But.&lt;/p&gt;&#xA;&lt;p&gt;Most of the differences between groups of boys and groups of girls come from how they've been socialized. Our culture pushes boys one way and girls another. Because of this (and not any inherent differences) I imagine that the girls will enjoy cooperation more, and the boys will enjoy competition more. (I personally enjoy competition, but most women in math are outliers to some extent.)&lt;/p&gt;&#xA;&lt;p&gt;Research does show that timed tests harm female students more than male students. But I don't know how that information might affect best teaching practices.&lt;/p&gt;&#xA;&lt;p&gt;The most important effect of classes for girls only is that the girls have more space to shine and get more attention from their teachers. A good book on how much more attention the boys get in mixed-gender classrooms is &lt;a href=&quot;https://www.simonandschuster.com/books/Still-Failing-at-Fairness/David-Sadker/9781416552475&quot; rel=&quot;noreferrer&quot;&gt;Failing at Fairness&lt;/a&gt; (updated as Still Failing at Fairness), by Myra and David Sadker. (Myra died right after the first edition came out.) This does not suggest a need for different practices in single gender classrooms.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-06-03T15:40:06.290" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="20945" PostTypeId="2" ParentId="20931" CreationDate="2021-06-03T15:56:02.720" Score="7" Body="&lt;p&gt;I agree with the point that a lot of the answers here are making - some distinctions, while correct and important, are not accessible to the age group you're talking about - I also want to point out an important benefit of &lt;em&gt;not&lt;/em&gt; making the distinction for them. While it is crucial in higher mathematics to be able to be extremely precise, it's &lt;em&gt;also&lt;/em&gt; important to be able to understand the &amp;quot;sense&amp;quot; of what's going on, even when it's not made explicit.&lt;/p&gt;&#xA;&lt;p&gt;For example, consider the well-known fact that &amp;quot;the volume of a sphere is &lt;span class=&quot;math-container&quot;&gt;$\frac43\pi r^3$&lt;/span&gt;&amp;quot;. I doubt anyone here would disagree with that, but a student who's highly aware of &lt;em&gt;all&lt;/em&gt; of the important differences in math might raise any or all of the following concerns:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;Sphere&amp;quot; may refer to a figure in any number of dimensions, so the statement claims that spheres have a fixed volume no matter the dimension!&lt;/li&gt;&#xA;&lt;li&gt;The only sphere that actually has volume (volume being explicitly three-dimensional) ought to be the one in &lt;em&gt;four&lt;/em&gt; dimensions, but then this formula is incorrect.&lt;/li&gt;&#xA;&lt;li&gt;The English term &amp;quot;a&amp;quot; is ambiguous; do we mean that there &lt;em&gt;is&lt;/em&gt; a sphere with this volume, or that &lt;em&gt;every&lt;/em&gt; sphere has this volume?&lt;/li&gt;&#xA;&lt;li&gt;Strictly speaking, the term &amp;quot;sphere&amp;quot; makes sense in any metric space, but the term &amp;quot;volume&amp;quot; does not.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To make that statement really, truly, mathematically precise, in a way that will be sure to never set a student up for misunderstanding in the future, we would have to say the following:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The three-dimensional Lebesgue measure of every ball in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; of radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, under the standard Euclidean metric, is &lt;span class=&quot;math-container&quot;&gt;$\frac43\pi r^3$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Now, certainly, a student who prefers this second formulation over the first &lt;em&gt;knows more math&lt;/em&gt;; they must, at minimum, be aware of the terminology used in measure theory, metric spaces, and higher-dimensional geometry. But the student who prefers the first formulation &lt;em&gt;understands the concept better&lt;/em&gt;, because the formula really isn't about &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-dimensional geometry or Lebesgue measures!&lt;/p&gt;&#xA;&lt;p&gt;An important skill for students to build is navigating the connection between the &amp;quot;fuzzy but intuitive&amp;quot; nature of natural language and the &amp;quot;precise but confusing&amp;quot; nature of mathematical language. If you jump directly to the fully precise version, they lose the opportunity to develop that skill!&lt;/p&gt;&#xA;" OwnerUserId="10081" LastEditorUserId="893" LastEditDate="2021-06-06T14:23:03.127" LastActivityDate="2021-06-06T14:23:03.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20946" PostTypeId="1" CreationDate="2021-06-03T19:39:28.357" Score="24" ViewCount="4581" Body="&lt;p&gt;The assumed age of the students is 10-15 years old.&lt;/p&gt;&#xA;&lt;p&gt;What is the danger in saying that two triangles are &amp;quot;the same&amp;quot; or &amp;quot;equal&amp;quot; instead of saying that they are congruent? It seems to me like the term &amp;quot;congruent&amp;quot; is an unnecessarily rigorous term to introduce to children at that age when we already have vocabulary that seems to work just fine.&lt;/p&gt;&#xA;&lt;p&gt;Intuitively, at least for me, what the concept of &amp;quot;congruency&amp;quot; captures, that &amp;quot;sameness&amp;quot; might not, is that the former includes the notion of triangles being mirror images of each other. However, for triangles this distinction does not seem to matter fully, as you can still move (in 3-space) any two congruent triangles so that they overlap.&lt;/p&gt;&#xA;&lt;p&gt;So why do we really introduce the notion that two triangles are &amp;quot;congruent&amp;quot; instead of just saying that they are &amp;quot;the same&amp;quot; or &amp;quot;equal&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="6527" LastEditorUserId="6527" LastEditDate="2021-06-03T21:30:48.037" LastActivityDate="2021-07-22T03:06:07.190" Title="Why do we introduce the notion that triangles are &quot;congruent&quot; instead of just saying that they are &quot;the same&quot; or &quot;equal&quot;?" Tags="|geometry|primary-education|terminology|middle-school|" AnswerCount="9" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="20947" PostTypeId="2" ParentId="20946" CreationDate="2021-06-03T19:52:58.987" Score="56" Body="&lt;p&gt;Colloquially, there's a lot of conceptual overlap between all of these terms, but &amp;quot;sameness&amp;quot; is not a well-defined mathematical property. Congruent shapes need not be &amp;quot;the same&amp;quot; or &amp;quot;equal&amp;quot; in all respects - they can be rotated differently, or be in different positions, or be different colors, or have different names, or differ in any other characteristic that's not size and shape.&lt;/p&gt;&#xA;&lt;p&gt;Congruency defines &lt;em&gt;exactly which properties&lt;/em&gt; need to be &amp;quot;the same&amp;quot; for two shapes to be considered as such.  &amp;quot;Sameness&amp;quot; is an ill-defined concept that doesn't describe which properties should or should not be the same in order to qualify, and &amp;quot;equality&amp;quot; somewhat implies that &lt;em&gt;all&lt;/em&gt; properties must be the same, which also isn't quite true of congruency. Congruency is a precise term that indicates equality of side lengths and angles and their arrangement, but nothing more or less. If you want to describe congruency as &amp;quot;sameness&amp;quot; you'd need to elaborate on &lt;em&gt;exactly what characteristics&lt;/em&gt; need to be &amp;quot;the same&amp;quot; in order to qualify, in which case you're just back to the definition of congruency.&lt;/p&gt;&#xA;&lt;p&gt;One could reasonably argue that two congruent triangles drawn in different colors are not &amp;quot;the same&amp;quot;. You could also argue that two congruent triangles drawn on different parts of the page and rotated differently are not &amp;quot;equal&amp;quot; to one another (as &amp;quot;equality&amp;quot; of shapes isn't defined in the first place). Congruency is the right term to describe exactly what properties you're considering. &amp;quot;Sameness&amp;quot; or &amp;quot;equality&amp;quot; are reasonable terms to introduce the general notion of congruency in a non-rigorous way, but they are too imprecise and open to interpretation in a way that &amp;quot;congruency&amp;quot; is not.&lt;/p&gt;&#xA;" OwnerUserId="5959" LastEditorUserId="5959" LastEditDate="2021-06-04T19:02:44.563" LastActivityDate="2021-06-04T19:02:44.563" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="20948" PostTypeId="1" CreationDate="2021-06-03T20:38:47.643" Score="2" ViewCount="144" Body="&lt;p&gt;I recently came across a very old Algebra textbook from the 1860s, and on the chapter discussing &amp;quot;arithmetical progression&amp;quot;, it says there are &amp;quot;20 cases for arithmetical progression&amp;quot;. The book then shows a table with 20 different formulae for arithmetic progressions. My question is this: I have looked in many other new textbooks and have only seen 1 or 2 basic formulae for arithmetic progression but no where near as many as in the 1860s textbook. Am I missing something or were 19th century textbooks in Math much more rigorous and detailed?  I'll give the first 4 formulae listed:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Given: a, d, n      l = a + (n - 1)d&lt;/li&gt;&#xA;&lt;li&gt;Given: a, d, S      l = -1/2d±&lt;span class=&quot;math-container&quot;&gt;$\sqrt{2dS + (a - 1/2d)^2}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Given: a, n, S      l = &lt;span class=&quot;math-container&quot;&gt;$\frac{2S}{N}$&lt;/span&gt; - a&lt;/li&gt;&#xA;&lt;li&gt;Given: d, n, S      l = &lt;span class=&quot;math-container&quot;&gt;$\frac{S}{n}$&lt;/span&gt; + &lt;span class=&quot;math-container&quot;&gt;$\frac{(n-1)d}{2}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I have never seen it set up this way. Any thoughts? They go on to list 15 more forms!&lt;/p&gt;&#xA;" OwnerUserId="15988" LastEditorUserId="2083" LastEditDate="2021-06-04T07:38:22.387" LastActivityDate="2021-06-07T01:15:08.787" Title="Arithmetical Progression" Tags="|algebra|textbooks|history|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="20949" PostTypeId="2" ParentId="20931" CreationDate="2021-06-03T21:06:58.120" Score="14" Body="&lt;p&gt;Educator here who has worked with many students in the aforementioned age range (9-13) on triangles and squares. In my experience, it has never come up that a confusion between the boundary and the interior of a plane region was relevant to problem solving at that grade level. For these types of elementary shapes, the boundary and the interior completely determine one another anyway, so which one you pick to be the True Meaning Of  A Triangle wouldn't make much of a difference.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me that the precision gained by saying &amp;quot;The triangle is the set of boundary points&amp;quot; is a small payoff compared to the drawback of having to pedantically reword every statement to be unambiguous (and the resulting increase in prolixity/verbosity). Is a ten-year-old going to understand the area of a triangle formula better if I tell him, &amp;quot;Here is the formula for the area of &lt;em&gt;the interior region of&lt;/em&gt; a triangle&amp;quot;, vs. &amp;quot;Here's the formula for area of a triangle&amp;quot;? They're going to know what I mean either way. Having worked with students that struggle on math (and in some cases, may be apathetic towards the concept of gaining a deeper understanding), none of them would have benefited from splitting hairs over the distinction. Conversely, there's significant gain in conciseness using a word like &amp;quot;triangle&amp;quot; as an umbrella concept that can refer to one or more of:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;any three non-collinear points&lt;/li&gt;&#xA;&lt;li&gt;the line segments connecting same&lt;/li&gt;&#xA;&lt;li&gt;the 2D region inside those line segments in the plane passing through all three&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;and letting the context clarify which one is meant. The umbrella concept also reinforces for the student that these are all interrelated, rather than balkanizing them artificially into &amp;quot;this one is about the 2D region, this one is about the 1D perimeter, and this one is about the 3 zero-dimensional corner points&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="14424" LastActivityDate="2021-06-03T21:06:58.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20950" PostTypeId="1" CreationDate="2021-06-03T21:15:19.703" Score="4" ViewCount="469" Body="&lt;p&gt;&lt;strong&gt;Let them discover &lt;span class=&quot;math-container&quot;&gt;$\;\ell=2\pi R\;$&lt;/span&gt;  by their own, at least the invariance of &lt;span class=&quot;math-container&quot;&gt;$\ell/R$&lt;/span&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Do you think that the following method works well in a mathematics class of  elementary school? What do you think about the idea that math education system would remove the  dictation of the math formulas, instead, it would encourage the students to observe some part or the whole formula themeselves? In the following post I mainly focuse on &amp;quot;The perimeter of circle&amp;quot; in elementary schools.&lt;/p&gt;&#xA;&lt;p&gt;As a generalization of this question, at the last lines of this post, I will  extend this question to high school geometry, for example sphere area.&lt;/p&gt;&#xA;&lt;p&gt;Do you have any ideas or similar suggestions or some situations you already experienced? When you were student, how did you feel about memorizing and be dictated of mathematical formula?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Here is my suggestion about discovering processess of circle perimeter formula in elementary school:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Let's distribute a few circles, with different radii, among some different groups of  students of elementary school. (Circles that are made of flexible wires and easily form a line segment. Or we can use some pulley and  thread or-string. The radius of pulley is easilly measurable. The perimeter of circle is the length of thread ) Then let the students calculate the ratio of the perimeter of the circle to the length of the radius and they  compare the result of each other.&#xA;We do this in a situation where they are not yet familiar with the formula for the circumference of a circle. But they would observe that the ratio of perimeter by radius  is independent of our chosen circle.They would observe that the ratio is the same for measurments of all groups. &lt;strong&gt;Perhaps they would be surprised.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We do not expect them to discover the number&lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; but hopefully they observe and discover that &lt;span class=&quot;math-container&quot;&gt;$\ell/R$&lt;/span&gt; does not depend on datas so is a universal constant.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Generalization to high school geometric formulas:&lt;/strong&gt;&#xA;How can one extend this idea to high school problems for example observing that the surface area of sphere  divided by &lt;span class=&quot;math-container&quot;&gt;$R^2$&lt;/span&gt; is independent of chosen sphere?&lt;/p&gt;&#xA;&lt;p&gt;Your answers and  comments are very appreciated.&lt;/p&gt;&#xA;" OwnerUserId="6154" LastEditorUserId="6154" LastEditDate="2021-06-04T11:30:36.750" LastActivityDate="2021-06-13T19:01:03.017" Title="Is it a good idea for elementary school students to observe and discover the &quot;circle perimeter formula&quot; themselves without being dictated to?" Tags="|secondary-education|geometry|primary-education|technology-in-education|education-research|" AnswerCount="5" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="20951" PostTypeId="2" ParentId="20946" CreationDate="2021-06-03T21:24:57.327" Score="30" Body="&lt;p&gt;The smart-aleck answer is that most congruent triangles, or congruent figures more generally, aren't actually &amp;quot;the same&amp;quot; or &amp;quot;equal&amp;quot;. Usually when we say two things are &amp;quot;the same&amp;quot;, we mean that they are not just indistinguishable, but that they are literally the same exact thing. &amp;quot;Equality&amp;quot; means two numbers are the same, or two sets have all the same elements as each other. So &amp;quot;sameness/equality&amp;quot; for geometric figures means that every point of one figure is also a point of the other figure. But this definition is obviously too restrictive if we instead want to talk about two figures that have the same &lt;em&gt;size and shape,&lt;/em&gt; but might be different considered as sets of points, as is typically the case in geometry.&lt;/p&gt;&#xA;&lt;p&gt;You can imagine the confusion for a beginner student in geometry if they were presented with constantly moving goalposts for when two figures are &amp;quot;the same&amp;quot;, where sometimes we literally meant they have all the same points, and sometimes we only meant that their measurements are equal in every respect (or equivalently, that a sequence of rigid motions exists to put one atop the other). Or on the teacher end, having to deal with a smart-aleck student going, &amp;quot;Those triangles are obviously different triangles, one's over here and the other's over there; why are you saying they're the same?&amp;quot; So it's convenient to have a shorthand word for &amp;quot;same size and shape, but not necessarily same points,&amp;quot; when that's actually what we mean.&lt;/p&gt;&#xA;" OwnerUserId="14424" LastActivityDate="2021-06-03T21:24:57.327" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="20952" PostTypeId="2" ParentId="20931" CreationDate="2021-06-04T10:37:19.103" Score="1" Body="&lt;p&gt;What you are trying to do is way too complicated for this age group. You have been exposed to years of math but all this stuff is new for the students. What might seem easy/trivial for you might be very complicated for them. If you try to explain it this way you run the risk of confusing your students thoroughly and possibly even creating an aversion for math in general.&lt;/p&gt;&#xA;&lt;p&gt;The solution is simple. If there's ever a risk of confusion just spend one or two sentences to explain exactly what you mean. Maybe add even a picture.&lt;/p&gt;&#xA;" OwnerUserId="6749" LastActivityDate="2021-06-04T10:37:19.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20953" PostTypeId="1" AcceptedAnswerId="20964" CreationDate="2021-06-04T14:41:08.483" Score="6" ViewCount="308" Body="&lt;p&gt;I am writing a syllabus for a statistics course which I will teach next year. My plan includes a group project for students. To reflect the contributions of each students, I want to grade the project as a whole and let students decides how to split the points. But  students who read my syllabus pointed out that this make them feel awkward.&lt;/p&gt;&#xA;&lt;p&gt;Is there a better way for grading group/collaborative project/work?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorDisplayName="user11702" LastEditDate="2021-06-05T13:29:53.760" LastActivityDate="2021-06-19T19:38:08.020" Title="How to grade group project or group assignment?" Tags="|grading|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20954" PostTypeId="2" ParentId="20950" CreationDate="2021-06-04T16:18:11.903" Score="2" Body="&lt;p&gt;Yes, it is always good when students independently discover mathematical facts, in contrast to being told that some old Greek dude figured it all out 2,500 years ago.  It makes students feel useful and participatory, it prepares them for a future where they will be investigating phenomena where formulas aren't as well established, and the experience forms mental pathways that you don't get from reading a formula in a box on a worksheet.&lt;/p&gt;&#xA;&lt;p&gt;If anything, I think your exercise is rushing the students towards quantification.  Have the students take a piece of string that is the length of the circumference of their circle.  (Dental floss works well as an inexpensive, disposable, and easily cuttable string.)  Then, using their floss, they can measure how many times they could go straight end-to-end across the middle of the circle.  Then ask the students to compare their results with their neighbor and they will find that they all came up with &amp;quot;a little more than three&amp;quot;.  From there, you can dive into calculations, but that will verify the students' conjecture instead of forming it.&lt;/p&gt;&#xA;&lt;p&gt;As far as the surface area of a sphere goes, it's a formula that you never forget after doing &lt;a href=&quot;https://youtu.be/jaL8Kuv6YHo&quot; rel=&quot;nofollow noreferrer&quot;&gt;the orange peeling activity&lt;/a&gt;.  My tip is to have a knife to cut the oranges in half so you can have a good (albeit slightly runny) template for your circles, and to make 6 circles instead of 4 to add some uncertainty of what the final result will be.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-06-04T16:18:11.903" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20955" PostTypeId="2" ParentId="20950" CreationDate="2021-06-04T18:35:38.073" Score="2" Body="&lt;p&gt;I'm NOT so sure that one should emphasize discovery learning for the FIRST learning of a topic.  Humans are imperfect computing machines.  You have to consider the ma...person in the loop, as the military would say.&lt;/p&gt;&#xA;&lt;p&gt;Also, young children have high capability and even enjoyment of recognition and memorization.  Consider the game &amp;quot;Go Fish&amp;quot; for instance, or even the decades long affinity they have for memorizing state capitals. There is time.  When kids are in calculus, the will learn a different reason, intuition for geometric formulas.&lt;/p&gt;&#xA;&lt;p&gt;Note:  I am not 100% opposed to discovery learning, even at young ages.  But I would just be cautious about thinking this is the only path to Heaven.&lt;/p&gt;&#xA;" OwnerDisplayName="user16150" LastActivityDate="2021-06-04T18:35:38.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20956" PostTypeId="2" ParentId="20953" CreationDate="2021-06-04T18:40:22.823" Score="2" Body="&lt;p&gt;Don't do it.  You are inviting drama.  If you are going to do group work (debate-able and hopefully not whole hog), just split it like Solomon.  Chop the baby!&lt;/p&gt;&#xA;&lt;p&gt;P.s. If you LIKE dramahz, there is still the dynamic of group selection to give you that enjoyment.  (Unless you force the assignments...killjoy.)&lt;/p&gt;&#xA;" OwnerDisplayName="user16150" LastActivityDate="2021-06-04T18:40:22.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20957" PostTypeId="2" ParentId="20948" CreationDate="2021-06-04T19:25:27.660" Score="5" Body="&lt;p&gt;These are the first term, last term, difference between successive terms, number of terms being counted from first, and sum from first to last term of an arithmetic sequence.&lt;/p&gt;&#xA;&lt;p&gt;The formulae allow for finding each given the others (there are five things to find and four ways to pick three of the rest as required). They are all rearrangements of the basic formulae for the progression and the sum.&lt;/p&gt;&#xA;&lt;p&gt;At least in theory, modern algebra classes try to teach the understanding of what creates the rules, not just memorise the rules themselves, so that students today need only two formulae and can reinvent whatever form they need.&lt;/p&gt;&#xA;" OwnerUserId="6724" LastActivityDate="2021-06-04T19:25:27.660" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20958" PostTypeId="2" ParentId="20931" CreationDate="2021-06-04T21:52:54.683" Score="6" Body="&lt;p&gt;Programmers consider the naming of things to be one of the three leading problems in our field.&lt;/p&gt;&#xA;&lt;p&gt;For the cases you describe, we do already have a well-established and widely-used set of terms that even non-computer users should recognize and be familiar with.&lt;/p&gt;&#xA;&lt;p&gt;A circle can be called a &lt;strong&gt;solid&lt;/strong&gt; or &lt;strong&gt;filled&lt;/strong&gt; circle, contrasted with &lt;strong&gt;wireframe&lt;/strong&gt; or &lt;strong&gt;outlined&lt;/strong&gt; circle.&lt;/p&gt;&#xA;&lt;p&gt;Generally &amp;quot;filled/outlined&amp;quot; suggest 2D, while &amp;quot;solid/wireframe&amp;quot; suggest 3D, but that's not a hard and fast rule. You could pick one pair and use it universally, if you wanted. Or even use both pairs synonymously.&lt;/p&gt;&#xA;&lt;p&gt;In the third dimension, there's an ambiguity again: what is the term for a shape formed from planes? Maybe &lt;strong&gt;box&lt;/strong&gt; or &lt;strong&gt;container&lt;/strong&gt;?&lt;/p&gt;&#xA;&lt;p&gt;For the cylinder problem, &amp;quot;open-ended&amp;quot; feels a little ambiguous: I'd tend to be more explicit about which specific faces are missing, so &amp;quot;open-topped&amp;quot; or &amp;quot;open both ends&amp;quot;.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Filled or solid cube, cylinder, prism, sphere, tetrahedron.&lt;/li&gt;&#xA;&lt;li&gt;Open-topped cubic box.&lt;/li&gt;&#xA;&lt;li&gt;Open-topped square outline.&lt;/li&gt;&#xA;&lt;li&gt;Cylindrical box, open both ends; aka &lt;strong&gt;tube&lt;/strong&gt;/&lt;strong&gt;pipe&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Wireframe cube.&lt;/li&gt;&#xA;&lt;li&gt;A filled circle; aka &lt;strong&gt;disk&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;There are areas of uncertainty, still: geometrically speaking, is it meaningful to speak of a wireframe sphere, torus, or other curved shape? I'd argue they'd be ambiguous, as you can't say how the curves are subdivided. And in theory, a wireframe cylinder should just be the two wireframe circles of the ends, but that's not what people would imagine: they'd picture connecting lines subdividing the curve, too. But these are edge cases unlikely to be visited by your target audience.&lt;/p&gt;&#xA;&lt;p&gt;So for kids and teens, and indeed for almost all cases, I think you'd do well with &lt;strong&gt;wireframe/outlined, box/container, filled/solid&lt;/strong&gt;, with explicit specification of any missing edges or faces.&lt;/p&gt;&#xA;" OwnerUserId="16182" LastActivityDate="2021-06-04T21:52:54.683" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20959" PostTypeId="1" AcceptedAnswerId="20960" CreationDate="2021-06-05T01:27:20.050" Score="3" ViewCount="356" Body="&lt;p&gt;My 10 year old daughter is trying to read this book — please explain in Simple English that she'll grasp.  Kindly see the embolded phrases below. The author doesn't expound why the 3D &amp;quot;mental pictures&amp;quot; are &amp;quot;usually enough&amp;quot;. Scilicet, why doesn't &amp;quot;this impoverished vision&amp;quot; hinder high-dimensional geometry, or at least deprive or forestall you from learning all about it?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;      In the same way, a point in three-dimensional space is described by a list&#xA;of three coordinates (x,y,z). And nothing except habit and craven fear keeps us from pushing this further. A list of four numbers can be thought of as a point&#xA;in four-dimensional space, and a list of ten numbers, like the California&#xA;temperatures in our table, is a point in ten-dimensional space. Better yet, think&#xA;of it as a ten-dimensional vector.&lt;br /&gt;&#xA;      Wait, you may rightfully ask: How am I supposed to think about that?&#xA;What does a ten-dimensional vector look like?&lt;br /&gt;&#xA;      It looks like this:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ijnrz.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ijnrz.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;      That’s the dirty little secret of advanced geometry. It may sound&#xA;impressive that we can do geometry in ten dimensions (or a hundred, or a&#xA;million . . .), &lt;strong&gt;but the mental pictures we keep in our mind are two- or at most&#xA;three-dimensional. That’s all our brains can handle. Fortunately, this&#xA;impoverished vision is usually enough.&lt;/strong&gt;&lt;br /&gt;&#xA;      High-dimensional geometry can seem a little arcane, especially since the&#xA;world we live in is three-dimensional (or four-dimensional, if you count time,&#xA;or maybe twenty-six-dimensional, if you’re a certain kind of string theorist,&#xA;but even then, you think the universe doesn’t extend very far along most of&#xA;those dimensions). Why study geometry that isn’t realized in the universe?&lt;br /&gt;&#xA;      One answer comes from the study of data, currently in extreme vogue.&#xA;Remember the digital photo from the four-megapixel camera: it’s described&#xA;by 4 million numbers, one for each pixel. (And that’s before we take color&#xA;into account!) So that image is a 4-million-dimensional vector; or, if you like,&#xA;a point in 4-million-dimensional space. And an image that changes with time&#xA;is represented by a point that’s &lt;em&gt;moving around&lt;/em&gt; in a 4-million-dimensional&#xA;space, which traces out a curve in 4-million-dimensional space, and before&#xA;you know it you’re doing 4-million-dimensional calculus, and then the fun can&#xA;really start.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ellenberg, &lt;em&gt;How Not to Be Wrong&lt;/em&gt; (2014), pages 338-9.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2021-07-03T15:18:06.437" Title="Explain to 10 year old — Why do 3D mental pictures usually suffice for high-dimensional geometry?" Tags="|geometry|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20960" PostTypeId="2" ParentId="20959" CreationDate="2021-06-05T02:22:54.453" Score="3" Body="&lt;p&gt;Well, to start, &amp;quot;usually enough&amp;quot; is ambiguous enough to allow a bit of wiggle room. That said, there are plenty of times where knowing how dimensions 1, 2, and 3 act to start to see the special cases break down and see the general pattern. For example, Jordan is hinting at vector spaces in the cited section. Dimension 1 is a special case where vector addition is just regular addition, but when you move to dimension 2 you start to see that there is something different going on. e.g. You can't do things like multiply 2D vectors like you could in 1D. (Or at least, there isn't just &lt;em&gt;one&lt;/em&gt; way of doing it. You can multiply component-wise, or view them as complex numbers, or use the 3D cross product or inner product to get a scalar result, etc.) Once you keep on going to 3D, some more of the stuff particular to 1 and 2 dimensions falls away. 3D has its own peculiarities, but there is a continuous core set of facts/axioms that keeps on going &amp;quot;all the way up&amp;quot; forming the concept of a vector space. Some parts of this core even continue to infinite dimensional spaces.&lt;/p&gt;&#xA;&lt;p&gt;Now, you can't expect this core set of vector space axioms to tell you everything. Those California temperatures are probably a time series and the images have a 2D spatial structure and these structures aren't captured by viewing them as raw vectors. But that's all structure that they have &lt;em&gt;in&lt;/em&gt; &lt;em&gt;addition&lt;/em&gt; to their vector structure, not &lt;em&gt;instead&lt;/em&gt; &lt;em&gt;of&lt;/em&gt;. i.e. We have some sort of foundation on which to build.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2021-06-05T02:22:54.453" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20961" PostTypeId="2" ParentId="20953" CreationDate="2021-06-05T03:14:41.343" Score="5" Body="&lt;p&gt;When I taught Math for Elementary Teachers, I had some group projects that I believed would help them learn problem-solving. They had to work together. They had to each write it up individually. In their write-ups, they had to credit group members who had helped with the solution process. They each got an individual grade for their individual write-ups (which they couldn't do well on if the group hadn't figured things out together).&lt;/p&gt;&#xA;&lt;p&gt;I have not felt a need for graded group work recently, but this seemed like a good way to both require them to work together and grade them individually.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-06-05T03:14:41.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20962" PostTypeId="1" CreationDate="2021-06-05T13:44:46.213" Score="43" ViewCount="12187" Body="&lt;p&gt;Lately, my students keep telling me why what we are learning is not important. They ask me when will we use this in the real world?&lt;/p&gt;&#xA;&lt;p&gt;I explain how math is important in gambling, cooking, finance, engineering, programming, electronics, philosophy, and logic. I state how we live in a digital world, and there is a ton of application out there. As a pure mathematician at heart, I point out how important proofs are in research and how difficult theorems are to come up with. I explain how pure mathematicians don't even need to find appication, and their work is incredibly important, e.g. Euclid and his discoveries in number theory and geometry. I explain how math is beautiful and similar to art, patterns, and design. I have played movies where mathematicians had huge roles and impacts on history. I explain the importance of education and study in order to find happiness and get out of poverty. I explain how education is important to get what you want in life.&lt;/p&gt;&#xA;&lt;p&gt;Still... students comment how little math they use and others use generally in their daily lives. It's like if I am not using the math I am using tonight, then why do I bother learning it?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What is a good response to get students out of this mindset? How do you dare them to try to do these things? I am really into psychology and how to persuade students to change their mindset.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="12366" LastEditorUserId="470" LastEditDate="2021-06-06T20:45:47.893" LastActivityDate="2024-01-04T08:03:53.127" Title="Response to Students Who Say &quot;This Is Not Important&quot;" Tags="|secondary-education|applications|psychology|" AnswerCount="24" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="20963" PostTypeId="2" ParentId="20962" CreationDate="2021-06-05T14:19:57.383" Score="16" Body="&lt;p&gt;The American Mathematical Society provides posters promoting awareness of mathematics, its beauty, and applications.  Here's a quote from the &lt;a href=&quot;https://www.ams.org/posters&quot; rel=&quot;noreferrer&quot;&gt;AMS Posters&lt;/a&gt; website:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Students frequently ask when they will use the math we learn in real life, and your posters provide great visuals to support the answers to this question.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The AMS also have &amp;quot;&lt;a href=&quot;https://www.ams.org/publicoutreach/mathmoments/mathmoments&quot; rel=&quot;noreferrer&quot;&gt;Mathematical Moments&lt;/a&gt;,&amp;quot; a &amp;quot;series of posters that promote appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture.&amp;quot;  These are available in different languages.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2021-06-05T14:19:57.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20964" PostTypeId="2" ParentId="20953" CreationDate="2021-06-05T16:37:04.907" Score="8" Body="&lt;p&gt;My approach (that has worked successfully with my students for over a decade) has been to give equal grades to all students in the group unless I uncover evidence that someone has been free riding.&lt;/p&gt;&#xA;&lt;p&gt;To that end, I survey the teams at the end of the project and ask them to evaluate the contributions of all of the team members.  I also require an explicit statement of contributions in the project report.&lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2021-06-05T16:37:04.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20965" PostTypeId="2" ParentId="20962" CreationDate="2021-06-05T19:11:29.903" Score="4" Body="&lt;p&gt;W.G.,&lt;/p&gt;&#xA;&lt;p&gt;I think you have to engage based on the situation.  If it is someone wanting an argument, or to disrupt the class, you can engage a little, but at a certain  (quick) point, need to move on and just teach.  It's not a winnable argument, per se.&lt;/p&gt;&#xA;&lt;p&gt;That's not to say that no motivation should ever be given.  It is good if you can do so at times along the way (without long discussions, just little asides like &amp;quot;this helped crack the Nazi's code, or this is important in biology, or whatever).  And of course, this is not the only motivation.  You should be motivating by modulating your voice, having energy, being excited, showing JOY when you solve a problem.  In other words, the personal dynamic.&lt;/p&gt;&#xA;&lt;p&gt;As far as pure math and beauty and all that:  Even crusty old applied people like I were thrilled when Wiles solved FLT.  So there is some thrill to pure discoveries.  But really in a sense students get that since integration by parts is new TO THEM.  But you've gotta realize 95%+ of the population sees math as a tool of science (and business and the like).  So, mix in a little beauty at times.  But if you think everyone is heading to being like you, thinking like you, you're ignoring reality.&lt;/p&gt;&#xA;&lt;p&gt;Really, the major driver for kids to learn math is to be able to handle undergrad science/engineering classes.  And it's not a stupid rationale.  It's a very real thing in front of them.  Calculus as a co/pre-req for entry level physics for instance.&lt;/p&gt;&#xA;&lt;p&gt;If they want to be doctors, they need to take chem, physics, bio and organic chem. That's just to deal with the MCAT.  You could abstractly argue about how much of that is needed for doctors, but we live in the real world.  You want a big house and a trophy wife, gotta pass those MCATs.  Let's deal with what we're actually facing.&lt;/p&gt;&#xA;&lt;p&gt;Probably for your own good, it would be nice to learn at least a little more about how math IS used in college homework problems in chem, physics, engineering, econ, etc.  You don't need to run off an get a masters.  But you could learn a lot from skimming a few textbooks.  And I say this with humility, but I've seen a huge pattern of pure types lacking even the start of awareness of these areas.  There aware that there's some stuff called applications.  But they have zero feel for it.  Do things like citing an incredibly minor and high level diffyQ for one random Ph.D. thesis in biophysics modeling, but don't know that imaginary numbers are key in all of alternating current EE.  But the good thing about being very ignorant, is that can very quickly move from zero to something. Large payoff for minor investment.&lt;/p&gt;&#xA;&lt;p&gt;P.s. Based on your questions, I'm assuming you're dealing with college-bound HS students.  But of course you should have told us this and told it to yourself.  The audience is a key variable in all these pedagogy questions.  8th graders are different than 12th than college.  Top track different from average.  Etc.&lt;/p&gt;&#xA;" OwnerUserId="16186" LastActivityDate="2021-06-05T19:11:29.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20966" PostTypeId="2" ParentId="20962" CreationDate="2021-06-05T23:26:36.143" Score="18" Body="&lt;p&gt;I've never had success with giving a list of applications to such students - because, realistically, we &lt;em&gt;don't&lt;/em&gt; use most of the math we teach. For example, I teach early equations in one of my classes, things like &amp;quot;solve the equation &lt;span class=&quot;math-container&quot;&gt;$5x = 15$&lt;/span&gt;&amp;quot;. If asked why this question is &amp;quot;useful&amp;quot;, I could give this answer:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Well, suppose you were planning a barbeque with four friends, and you had 15 hamburger patties in the freezer, and you wanted to know how many hamburgers each person could have.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Except &lt;em&gt;we don't use math like that.&lt;/em&gt; Nobody, no matter how steeped in math, would solve the above problem by defining a variable, setting up an equation, and solving that equation. Even I would just solve this problem by dividing 15 by 5.&lt;/p&gt;&#xA;&lt;p&gt;I could, instead, give this answer:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Solving equations like &lt;span class=&quot;math-container&quot;&gt;$5x = 15$&lt;/span&gt; is the first step in learning to solve more complicated equations, which you'll need to do for higher-level math classes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But this isn't terribly convincing either; if they aren't planning on taking later math classes, what's the point? And even if they are, it just pushes the problem back one step - why are those more complicated equations useful? If I'm being totally honest, even I hardly ever use an equation for anything that isn't math.&lt;/p&gt;&#xA;&lt;p&gt;Instead, my response is always about the &lt;em&gt;modes of thought.&lt;/em&gt; Solving an equation like &lt;span class=&quot;math-container&quot;&gt;$5x = 15$&lt;/span&gt; is an exercise in precise thought; writing equations, defining variables, and so on are exercises in precise communication. And thinking and communicating precisely are crucial skills in literally every field. If you haven't developed the level of organized thinking required to handle the exponent laws, it's going to be very hard for you to make sense of all the complicated details of a loan; if you aren't able to connect the numbers in an expression to their underlying meaning, it's going to be hard to tell whether two for \$3 is a better or worse deal than one for \$1.&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2021-06-05T23:26:36.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20967" PostTypeId="2" ParentId="20962" CreationDate="2021-06-06T00:41:33.613" Score="46" Body="&lt;p&gt;&lt;em&gt;&amp;quot;Lately, my students keep telling me why what we are learning is not important. They ask me when will we use this in the real world?&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;There's a quick reply to this that I think people won't like, and you would have to be very careful using, but actually makes a more serious than it first appears. One possible answer is: &amp;quot;You're quite right. You don't need maths to mop the toilets out for a living. Is that your plan?&amp;quot; Many people do end up in basic jobs. There's nothing wrong with that, if that's genuinely what you want out of life. And many people who do it are happy and fulfilled with their lot. It's &lt;em&gt;their&lt;/em&gt; life. You just need to be sure it's an informed choice, that they're going into with their eyes open to the consequences.&lt;/p&gt;&#xA;&lt;p&gt;But really you need to find out why they're asking. It could be that they genuinely find it hard to connect what they're doing to real world applications, or that they &lt;em&gt;do&lt;/em&gt; see a need for maths in daily life but not the stuff you're teaching. (People need to do things like fill in tax returns, pay bills, plan finances, calculate interest for loans, run a business, calculate fuel usage, buy supplies in the right amounts, understand economics, etc. Maths lessons often don't teach any of that.) In this case, find out what they want to do with their lives, or what they do do, and then give examples built around their particular applications.&lt;/p&gt;&#xA;&lt;p&gt;Or it might be that they don't enjoy it, feel they're not good at it, and are seeking some excuse not to do it, or a reason not to feel bad about not being able to do it. Dismissing it as &amp;quot;unimportant&amp;quot; is one way to do that. In which case, it's a waste of time giving reasons why it really &lt;em&gt;is&lt;/em&gt; essential - that just makes them feel &lt;em&gt;even worse&lt;/em&gt;. There you need to build their confidence, help them catch up.&lt;/p&gt;&#xA;&lt;p&gt;Or it may be a cultural issue. Sometimes people belong to a cultural identity that has negative views about maths and studying. Girls may think it's a &amp;quot;boy's subject&amp;quot;. Boys may think it's &amp;quot;unmanly&amp;quot; or &amp;quot;geeky/nerdy&amp;quot; to be studious/academic/clever (as opposed to being physically tough, good at sports, etc.). Teens are very sensitive to what characteristics are seen as attractive or unattractive to the opposite sex. Sometimes you get racial, religious, or class stereotypes, where studiousness is associated with particular groups that are negatively regarded, or where there are strong religiously-based expectations that oppose, for example, girls studying, or the adoption of Western values, or non-religious values generally. Where gangs and criminality are a problem, respecting authority and conforming to the expectations of mainstream society are often seen very negatively by their peers. Friends, families, and communities all apply social pressures. Cultural conflict is a much harder problem to solve. There may be a way to fit mathematics positively into their worldview, but it may be that the only way is to tackle it head on and &lt;em&gt;change&lt;/em&gt; their worldview, and that obviously has some very serious ethical complications that need to be considered first. Is is &lt;em&gt;right&lt;/em&gt; to destroy elements of their culture, even if it is &amp;quot;for their own good&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Without knowing what their issue is, trying to come up with solutions is a very broad question. Ask more questions about what &lt;em&gt;they&lt;/em&gt; want from life, what they value, and what they think they have to offer. And be open to the possibility that maybe they're right, and maths is &lt;em&gt;not&lt;/em&gt; what they need.&lt;/p&gt;&#xA;" OwnerUserId="16188" LastActivityDate="2021-06-06T00:41:33.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20968" PostTypeId="2" ParentId="20946" CreationDate="2021-06-06T08:51:12.117" Score="3" Body="&lt;p&gt;The problem is that there are different kinds of 'samenesses' (equivalence relations). One of these is congruency, but another is similarity. If you don't teach your students the word &lt;em&gt;congruent&lt;/em&gt;, and use the word &lt;em&gt;same&lt;/em&gt; instead, what word are you going to use when introducing similarity, since similar triangles can also be thought of as being the same, but in a different way than congruent triangles. It is important for the students to understand that the concept of being the same is captured by multiple different notions in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="16192" LastActivityDate="2021-06-06T08:51:12.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20969" PostTypeId="2" ParentId="20946" CreationDate="2021-06-06T11:34:10.513" Score="-4" Body="&lt;p&gt;If a triangle has the same three angles as another angle they are said to be congruent. The lengths of their sides need not be equal though. So two congruent triangles need not be the same.&lt;/p&gt;&#xA;&lt;p&gt;Even if the lengths of the sides are the same, the triangles need not be the same. They can be mirror images of one another.&lt;/p&gt;&#xA;&lt;p&gt;So congruence of triangles is a broader vision of triangles as the same triangles. Congruent triangles can have different side length (though the ratio of corresponding sides is the same), they can be mirror images, or they can be rotated wrt to each other.&lt;/p&gt;&#xA;" OwnerUserId="16194" LastActivityDate="2021-06-06T11:34:10.513" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20970" PostTypeId="1" CreationDate="2021-06-06T12:49:57.917" Score="0" ViewCount="603" Body="&lt;p&gt;Here is a question I gave on an exam last year. Please let me know what you think of the question (eg: if it is a fair question, easy, difficult, etc). A lot of students were upset for questions like this, but apparently what I am noticing is that they are woefully unprepared for Math coming into the college level, even in so called &amp;quot;higher&amp;quot; Math classes. I can't speak for all colleges, obviously, but that is my impression in several.&lt;/p&gt;&#xA;&lt;p&gt;The two sides of an isosceles triangle have a length of &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;, each side, and the angle between them is the value of a random variable called &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; with a function of proportional density to &lt;span class=&quot;math-container&quot;&gt;$x(\pi-x)$&lt;/span&gt; in each point &lt;span class=&quot;math-container&quot;&gt;$x \in \left(0, \dfrac{\pi}{2}\right)$&lt;/span&gt;. Calculate the function of density of the area of the triangle and the mean (expected value) of the random variable.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This is for a Calculus III with Analytic Geometry course (and the requirement is that a student has grade of C or better in the previous course). It is an honors course. Most of the students are majoring in Math education or simply Math. Their expectation is to then go on to Probability I and II, so obviously we cover a range of topics, including Probability and analysis topics.&lt;/p&gt;&#xA;" OwnerUserId="15988" LastEditorUserId="15988" LastEditDate="2021-06-08T12:07:04.977" LastActivityDate="2021-06-08T23:11:19.413" Title="Proportional density function question" Tags="|functions|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="20971" PostTypeId="1" AcceptedAnswerId="20984" CreationDate="2021-06-06T13:15:03.390" Score="2" ViewCount="258" Body="&lt;p&gt;I would say continuity is the idea that points numerically close to the input point of a function agree with the value of the function at that point. Now, suppose I introduce the idea of a derivative, it takes two close by points of the function and computes the slope.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \lim_{x \to a} f(x) = f(a)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, if a function is continuous i.e: really close by points numerically agree, how can there be a variation in the function to define the derivative?&lt;/p&gt;&#xA;&lt;p&gt;Excuse me if the question is poorly phrased.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2021-06-09T08:25:35.943" Title="How to explain continuity and differentiability to highschool students?" Tags="|calculus|" AnswerCount="2" CommentCount="3" ClosedDate="2021-06-10T04:45:58.997" ContentLicense="CC BY-SA 4.0" />
  <row Id="20972" PostTypeId="2" ParentId="20962" CreationDate="2021-06-06T13:35:15.600" Score="2" Body="&lt;p&gt;Give real world examples of how maths applies to everyday job functions, go beyond cash, tax and cost planning and apply it to physical tasks.&lt;/p&gt;&#xA;&lt;p&gt;It is important in almost any real world building project to check for square and for setout. An understanding of triangulation, pythagoras and the use of 3/4/5 triangle rule in measurement is the only thing that will guarantee success when you are trying to construct either horizontal or vertical elements.&lt;/p&gt;&#xA;&lt;p&gt;Painters, carpet layers, dress makers or anybody that buys things in liquid form or on rolls need to analyse and calculate coverage of that material to simplify the task and minimise wastage. How much paint/cloth one needs to coat/wrap a spherical object requires a mathematical solution. So is the amount of Helium required to inflate 10 party balloons, cook steaks on a BBQ grill from a propane tank etc.&lt;/p&gt;&#xA;&lt;p&gt;Anybody who handles loose materials will need to calculate and consider bulk density, pile mass, slump angle, friction and surface area for storage and handling. How much space do you need for the dirt you are digging out of the hole and how far away should the spoil pile be so it doesn't fall back in on you? This requires an understanding of geometry, angle of repose and base shape. Show your students that some animals dig conical pits in sand to catch prey and how the hole created is on the verge of slumping until it's disturbed by the prey and collapses aka the Great Pit of Carkoon!&lt;/p&gt;&#xA;" OwnerUserId="16191" LastActivityDate="2021-06-06T13:35:15.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20973" PostTypeId="2" ParentId="20946" CreationDate="2021-06-06T13:39:04.530" Score="0" Body="&lt;p&gt;The same triangles can always be put on top of one another, while congruent triangles can't, generally. If we confine ourselves to the 2d plane, that is. Putting on top requires the third dimension though, so let's state that they can be put on top of one another without rotating them around an axis in the plane.&lt;/p&gt;&#xA;" OwnerUserId="16194" LastActivityDate="2021-06-06T13:39:04.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20974" PostTypeId="2" ParentId="20970" CreationDate="2021-06-06T13:50:46.543" Score="6" Body="&lt;p&gt;You are asking a lot of them, even though each little bit of the problem isn't especially difficult and was probably covered in a class that they have taken. You need to remember&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a bit of trig/geometry to calculate the area from an angle,&lt;/li&gt;&#xA;&lt;li&gt;a bit of statistics/calculus to get the normalization constant and a formula for the expected value, and&lt;/li&gt;&#xA;&lt;li&gt;a bit of calculus to do the integration of trig functions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Each of these topics has been taught to them in a disconnected manner, so simply combining the topics from different courses is a challenge in its own right.&lt;/p&gt;&#xA;&lt;p&gt;As always, difficulty depends on what you have been teaching them and how much time they had to do it.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2021-06-06T13:50:46.543" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="20975" PostTypeId="2" ParentId="20970" CreationDate="2021-06-06T19:09:19.387" Score="11" Body="&lt;p&gt;People respond well to perceived effort and investment on the part of others. When I get an email that is worded poorly from a colleague, or start reading a paper that has not been carefully revised, I am more likely to tune out and abandon the situation.&lt;/p&gt;&#xA;&lt;p&gt;It is possible that you put significant effort into creating this question, but when I read it, it &lt;strong&gt;feels&lt;/strong&gt; like a question that was thrown out to me in first draft form. If a professor gave me this question, I would &lt;strong&gt;feel&lt;/strong&gt; that it is unlikely that the question is even possible, let alone good. Additionally, your opinion of the students is clearly low as shown by your comments, and the students are going to be able to detect that.&lt;/p&gt;&#xA;&lt;p&gt;I would say the question is basically impossible as written, not because of the math, but &lt;strong&gt;because of the social cues&lt;/strong&gt; above.&lt;/p&gt;&#xA;&lt;p&gt;Here is what happens when I try to read this question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The two sides of an isosceles triangle have a length of l, each side,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What is meant by &amp;quot;&lt;em&gt;the two sides&lt;/em&gt; of a triangle?&amp;quot; Triangles have three sides. &amp;quot;The&amp;quot; here indicates that these two sides are the only two sides. That's not true. From context I guess you don't care which two sides, but the word &amp;quot;the&amp;quot; starts off the sentence with confusion. Then, what is &amp;quot;each side&amp;quot; doing in this question? It doesn't clarify anything. Maybe you were worried I would think that two of the sides had a total length of &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;? Then you would have used the word &amp;quot;total.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I am fourteen words into the question and I have already had to stop three times to figure out what is being asked. As a student, I am somewhat hostile already: the person writing the question did not take much effort to communicate with me. All of this effort could have been avoided if the instructor had drawn a picture of a triangle, or revised the first draft of the question. I do not have high hopes.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the angle between them is the value of a random variable called &amp;quot;x&amp;quot; with a function of proportional density to x(π - x) in each point x ∈ (0, π/2).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What? At this point, in a timed test, I am looking for a different question. This question is either here to try to get me to waste time, or it is here for some other student to get correct -- it's certainly not for me.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Calculate the function of density of the area of the triangle&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I do not know what this is and I don't know any student that would know what this is.&lt;/p&gt;&#xA;&lt;p&gt;If you want to ask students this question, you need to work hard on revising it to communicate what you are asking. Here is a second draft of the first couple of sentences, which at least communicate the triangle you have in mind without requiring significant mental effort on the part of students who already know you don't expect them to get the question correct:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Two sides of an isosceles triangle each have length &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;. The angle between these two sides is given by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, which is between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\pi/2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-06-06T19:09:19.387" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="20976" PostTypeId="2" ParentId="20962" CreationDate="2021-06-06T19:36:59.627" Score="7" Body="&lt;p&gt;(not complete)&lt;/p&gt;&#xA;&lt;p&gt;You should have mentioned which were some of the topics where your student's noise pollution rates increased drastically. We could have been able to tailor our responses specific to your curriculum.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, I remember one of my friends arguing why the chapter on polynomials was a silly chapter with no real-life application [ &lt;em&gt;questions like&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$p\left(x\right)=4x^{3}+10x^{2}+5x\ +4$&lt;/span&gt; find the zeroes, etc ]. I must confess I really believed in his argument that what we were learning was not useful. But then came high school-level physics and chemistry which was essentially piggybacking here and there based on the &lt;strong&gt;&lt;em&gt;core principles&lt;/em&gt;&lt;/strong&gt; of the chapter on polynomials, such as finding zeros. I would also argue that there could have been more applications on the chapter that I just can't recollect, but if I remember changing my mind afterwards then that chapter must have been really useful.&lt;/p&gt;&#xA;&lt;p&gt;This may sound like a silly anectode, but math has always been like that. You learn something at an elementary level realises its application only later. This is true at a scientific level too. There are many times where abstract ideas in the field of mathematics were given real-world application (courtesy of physics) only several decades after the discovery.&lt;/p&gt;&#xA;&lt;p&gt;I suggest maybe going to the maths and science history stack exchange and see if you can ask them for an example of mathematical abstract which was given use only many years later. The only example coming to my mind is the heavy reliance of the General theory of relativity on Topological space as opposed to Euclidean space (which to be frank must have had 0 application) and of course, the Euler number making its way into Electrical engineering and Quantum mechanics. Oh yes along with iota. If you can find any formulas that knock them out, then they may understand how even the square root of negative 1 can be useful (ill edit my answer better tomorrow as its late-night)&lt;/p&gt;&#xA;&lt;p&gt;Now assuming you have found a great example, ask your students how much of them want to become scientists, engineers&lt;/p&gt;&#xA;&lt;p&gt;In the words of Richard Feynmann&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mathematicians prepare abstract reasoning that’s ready ‘to be used’ even though they don’t know what it’s being used for&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(I must also confess i am not a teacher but just a student who likes maths)&lt;/p&gt;&#xA;" OwnerUserId="16196" LastActivityDate="2021-06-06T19:36:59.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20977" PostTypeId="2" ParentId="20962" CreationDate="2021-06-06T22:32:34.670" Score="43" Body="&lt;h2&gt;Math is just as useless as almost any other subject&lt;/h2&gt;&#xA;&lt;p&gt;As a math tutor, I've thought about this &lt;strong&gt;a lot&lt;/strong&gt; over the last 15 years or so. Aside from tutoring, I don't use my math education in &amp;quot;the real world&amp;quot;. Here's a list of educational requirements that I have &lt;em&gt;also&lt;/em&gt; never used in &amp;quot;the real world&amp;quot;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;American literature (Hemingway, Flannery O'Connor, Langston Hughes)&lt;/li&gt;&#xA;&lt;li&gt;British literature (Shakespeare, ummm... I'm sure I studied some other British authors)&lt;/li&gt;&#xA;&lt;li&gt;Gym class (volleyball, crab soccer, archery, etc.)&lt;/li&gt;&#xA;&lt;li&gt;Latin&lt;/li&gt;&#xA;&lt;li&gt;Biology&lt;/li&gt;&#xA;&lt;li&gt;Chemistry&lt;/li&gt;&#xA;&lt;li&gt;Physics&lt;/li&gt;&#xA;&lt;li&gt;95% of History (e.g., &amp;quot;the age of explorers!&amp;quot;) - besides most of what I learned is either completely false, presented poorly, or heavily biased in favor of the myth of American exceptionalism&lt;/li&gt;&#xA;&lt;li&gt;Finding library books using a card catalog and the Dewey Decimal system&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For some reason, math seems to be the only subject students are forced to learn that causes them to ask &amp;quot;why do we have to learn this?&amp;quot; I don't know why. But outside of a very short list of actually applicable subjects (expository essays in English class, health class, driver's education, and shop class, and art (!!) for me), &lt;strong&gt;the vast majority of primary and secondary education is just as &amp;quot;useless&amp;quot; as mathematics&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;h2&gt;So why do we teach &lt;em&gt;any&lt;/em&gt; of this stuff?&lt;/h2&gt;&#xA;&lt;p&gt;There is one thing that I said to a student one time about why we learn math and he &lt;em&gt;said&lt;/em&gt; it made sense to him. It was similar to this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you want to excel in certain sports, you might do some strength training. So you might do bench presses, curls, dead lifts, or other esoteric muscle isolation exercises. None of those actions, like curling a barbell, are thing you would actually do in a sport like baseball or football (American) or football (Soccer). But manly athletes do strength training to help them be more effective in their sport. The point is, the things that you do to &lt;strong&gt;strengthen&lt;/strong&gt; muscles does not have to be the same thing you are doing when you &lt;strong&gt;use&lt;/strong&gt; your stronger muscles. The muscles don't care how they got strong - they are strong and humans with strong muscles can do things that humans with weaker muscles can't.&lt;/p&gt;&#xA;&lt;p&gt;School is like strength training for your brain muscles. We tell you to do all of these arbitrary mental tasks, like the mental equivalent of curling a barbell, not because you'll be doing mental barbell curls later on in life, but simply to strengthen that part of your brain, so you can do &lt;em&gt;whatever you want&lt;/em&gt; with that part of your brain later in life. The same reasoning applies to learning British literature and Physics. Your brain doesn't care how it got strong. And humans with stronger brains can do things that humans with weaker brains can't.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Note: only &lt;em&gt;after&lt;/em&gt; typing all of this up did I discover that this question is almost certainly a duplicate of a question where the highest voted answer is quite similar to my answer here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/19164/how-should-i-answer-questions-about-the-purpose-of-learning-math/19175#19175&quot;&gt;How should I answer questions about the purpose of learning math?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5176" LastEditorUserId="5176" LastEditDate="2021-06-06T22:48:34.580" LastActivityDate="2021-06-06T22:48:34.580" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20978" PostTypeId="2" ParentId="20929" CreationDate="2021-06-07T00:16:51.550" Score="1" Body="&lt;p&gt;It really depends entirely on what is being asked. I have often required set notation from my students (but I teach at the college level). Either is fine for me, but if the instruction is to define the answer within a particular form, then I do believe it is necessary. If the student did not follow the instruction, then I would most certainly deduct points. I have even deducted points for things like &amp;quot;put your first initial and surname&amp;quot; if a student puts the full name. This is to train them to be precise when asked to be precise. This is important because what if, for example, you have an absolute value equation?&lt;/p&gt;&#xA;&lt;p&gt;Now, in response to your question about the book: Mostly in former Soviet textbooks will you see this as well as continental European texts. In the case of the US, I can definitely recommend you look at Dolciani's Modern School Mathematics (1960s or 70s version).&lt;/p&gt;&#xA;" OwnerUserId="15988" LastActivityDate="2021-06-07T00:16:51.550" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20979" PostTypeId="2" ParentId="20962" CreationDate="2021-06-07T00:42:03.883" Score="0" Body="&lt;p&gt;I point them to GH Hardy's essay: A Mathematician's Apology. Aside from any &amp;quot;applications&amp;quot;, I got into Math for the beauty of it, not because of any &amp;quot;applications&amp;quot;, so I try to show that to my students. There are a lot of things we do in life that may not have a utility of purpose, but are nonetheless enjoyable or beautiful.&lt;/p&gt;&#xA;&lt;p&gt;As Hardy said: &amp;quot;Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="15988" LastActivityDate="2021-06-07T00:42:03.883" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="20980" PostTypeId="2" ParentId="20948" CreationDate="2021-06-07T01:15:08.787" Score="2" Body="&lt;p&gt;I'm not that crazy about the long list on that one topic.  The one potential benefit would be to have kids derive each from the definition of arithmetic progression.  Sort of just getting practice in manipulation.&lt;/p&gt;&#xA;&lt;p&gt;Maybe a little similar to how playing with different trig identities is useful.  Nobody needs to memorize all of them.  But knowing a few and deriving others is a good muscle-building activity. And I'm not saying it has to be the bare minimum.  But probably much more towards the minimum than the converse and emphasis on deriving others as needed.  For sure, the practice in manipulation is helpful later in puzzle solving activities of anti-differentiation.&lt;/p&gt;&#xA;" OwnerUserId="16199" LastActivityDate="2021-06-07T01:15:08.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20981" PostTypeId="2" ParentId="20962" CreationDate="2021-06-07T01:33:15.263" Score="3" Body="&lt;p&gt;Because the sum of all the subjects helps to make for a well rounded education. This keeps all opportunities in life open to you.&lt;/p&gt;&#xA;&lt;p&gt;The math learned by sophomore year of high school may actually be enough math for the average person. But even though a lawyer may not ever use the law of cosines, the law school they want to get into expects a HS transcript with good grades across the board.&lt;/p&gt;&#xA;&lt;p&gt;At another level, I was tasked to teach an MCAS (our local state standardized test for sophomore HS) class, teaching to the skills required. To that class, I asked if they were comfortable reading street signs, or the articles in a paper or magazine. They said they were. I told them there was a level of math understanding that was similar. Enough to not get ripped off in a store, to adjust recipes ingredients up or down, and so on.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-06-07T01:33:15.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20982" PostTypeId="2" ParentId="20946" CreationDate="2021-06-07T02:08:16.727" Score="9" Body="&lt;p&gt;Historically (and by historically I mean &amp;quot;in Euclid's &lt;em&gt;Elements&lt;/em&gt;&amp;quot;) the word &amp;quot;equal&amp;quot;, when applied to geometric figures, meant &amp;quot;equal in magnitude&amp;quot;.  So for example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Euclid refers to two segments as &lt;em&gt;equal&lt;/em&gt; if they have the same length&lt;/li&gt;&#xA;&lt;li&gt;Two triangles are &lt;em&gt;equal&lt;/em&gt; if they have the same area&lt;/li&gt;&#xA;&lt;li&gt;Two solids are &lt;em&gt;equal&lt;/em&gt; if they have the same volume&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For example, &lt;em&gt;Elements&lt;/em&gt; Book I, Proposition 35 says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Parallelograms which are on the same base and between the same parallels are equal to one another.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(See figure below, from Fitzpatrick's English translation of the &lt;em&gt;Elements&lt;/em&gt;, based on the Greek text of Heiberg.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XQ8ON.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XQ8ON.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Notice that &amp;quot;equal&amp;quot;, in this usage, does &lt;em&gt;not&lt;/em&gt; mean &amp;quot;congruent&amp;quot;; nor does it mean &amp;quot;the same, identical&amp;quot;.  With this idea as background, the word &amp;quot;congruent&amp;quot; plays an important role: it allows us to say that two figures are not only equal in area but are precisely the same shape, i.e. can be superimposed onto one another.&lt;/p&gt;&#xA;&lt;p&gt;Language, of course, is constantly evolving.  At some point people stopped using the word &amp;quot;equal&amp;quot; to mean &amp;quot;equal in area&amp;quot;, and began using it to mean &amp;quot;the same mathematical object&amp;quot;.  I am not sure exactly when this switch happened, but I suspect it was in the latter part of the 19th century, as set theoretical ideas began permeating all aspects of mathematics.  The greatest sea change in Geometry during this time was &lt;a href=&quot;https://en.wikipedia.org/wiki/Erlangen_program&quot; rel=&quot;noreferrer&quot;&gt;Felix Klein's Erlangen Programme&lt;/a&gt;, which sought to reframe Euclidean Geometry as the study of properties that are invariant under isometries of the plane.  In this context it is important to distinguish between the statements &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\triangle ABC = \triangle XYZ$&lt;/span&gt;&amp;quot; (which means that the two triangles are the same mathematical object) and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\triangle ABC \cong \triangle XYZ$&lt;/span&gt;&amp;quot; (which means that there exists an isometry mapping one triangle onto the other).&lt;/p&gt;&#xA;&lt;p&gt;(This is also when &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13700/in-what-curricula-are-rectangles-defined-so-as-to-exclude-squares/13766#13766&quot;&gt;the partitional classification of quadrilaterals found in Euclid began to gave way to the hierarchical classification scheme most of us are more familiar with&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;The effect of this linguistic switch was that the relative valences of the words &amp;quot;equal&amp;quot; and &amp;quot;congruent&amp;quot; became reversed:  whereas in Euclid's work the word &amp;quot;equal&amp;quot; is a &lt;em&gt;weaker&lt;/em&gt; relation than the word &amp;quot;congruent&amp;quot; (in that congruent figures are always equal, but the converse is not true), nowadays the word &amp;quot;equal&amp;quot; describes a relationship &lt;em&gt;stronger&lt;/em&gt; than congruence (in that &amp;quot;equal figures&amp;quot; are (trivially) congruent, but the converse is false.)&lt;/p&gt;&#xA;&lt;p&gt;So there is one reason why it is important to have different words: to distinguish between different notions of &amp;quot;same&amp;quot; (&amp;quot;same area&amp;quot; vs. &amp;quot;same figure&amp;quot; vs. &amp;quot;same size and shape&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;I would go even further, and say that a lot of the vocabulary we use in Geometry is introduced to disentangle notions that, to a naive student, seem equivalent.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For example:  to many students, a statement like &amp;quot;rectangle &lt;span class=&quot;math-container&quot;&gt;$ABCD$&lt;/span&gt; is bigger than rectangle &lt;span class=&quot;math-container&quot;&gt;$PQRS$&lt;/span&gt;&amp;quot; seems perfectly sensible. That is because they have an unexamined notion of &amp;quot;relative size&amp;quot;.  But it is possible for one rectangle to have a larger &lt;em&gt;diameter&lt;/em&gt; than another, and simultaneously a smaller &lt;em&gt;area&lt;/em&gt;, and equal perimeter!  (Which is larger:  a &lt;span class=&quot;math-container&quot;&gt;$6 \times 8$&lt;/span&gt; rectangle, or a &lt;span class=&quot;math-container&quot;&gt;$7 \times 7$&lt;/span&gt; rectangle?)  So we introduce vocabulary (area, perimeter, diameter) in order to unpack these distinct notions of &amp;quot;size&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;A second, more sophisticated example: if you ask students to find a point in the interior of a triangle that is equidistant from all three vertices, they will say &amp;quot;in the center&amp;quot;.  If you ask students to find a point in the interior of a triangle that is equidistant from all three sides, they will also say &amp;quot;in the center&amp;quot;.  It seems both obvious and unproblematic to them that &amp;quot;center of a triangle&amp;quot; means something -- and that it means &lt;em&gt;one&lt;/em&gt; thing.  But in fact there are multiple notions of &amp;quot;center&amp;quot;, all of which are worth naming, and therefore we need different names (circumcenter, incenter, orthocenter, centroid) to distinguish them.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2021-06-09T01:02:20.293" LastActivityDate="2021-06-09T01:02:20.293" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20983" PostTypeId="2" ParentId="20970" CreationDate="2021-06-07T02:32:26.573" Score="8" Body="&lt;p&gt;Like most of the respondents here, I had to read this problem several times before I could understand what you are asking.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The phrase &amp;quot;a random variable called &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with a function of proportional density to &lt;span class=&quot;math-container&quot;&gt;$x(\pi - x)$&lt;/span&gt;&amp;quot; is both nonstandard usage and potentially misleading.  On my initial read of the problem I thought you were trying to describe the mass density of a triangular lamina, &lt;a href=&quot;https://openstax.org/books/calculus-volume-3/pages/5-6-calculating-centers-of-mass-and-moments-of-inertia&quot; rel=&quot;noreferrer&quot;&gt;a common problem type in multivariable calculus&lt;/a&gt;.  But under that interpretation the rest of the problem didn't make any sense (or at least I couldn't make sense out of it).&lt;/li&gt;&#xA;&lt;li&gt;Only after reading the problem a few times did I finally realize that you meant that &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a random variable with &lt;strong&gt;probability density function proportional to&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x(\pi- x)$&lt;/span&gt;&amp;quot;, and that by &amp;quot;function of density of the area of the triangle&amp;quot; you were asking for the corresponding probability density function of the area.&lt;/li&gt;&#xA;&lt;li&gt;In particular notice that the adjective &amp;quot;proportional&amp;quot; in your sentence is misplaced; it belongs alongside the preposition &amp;quot;to&amp;quot;, as in &amp;quot;proportional to&amp;quot;, not modifying the word &amp;quot;density&amp;quot; as you have it.&lt;/li&gt;&#xA;&lt;li&gt;Some authors prefer the phrase &amp;quot;probability distribution function&amp;quot; over &amp;quot;probability density function&amp;quot; precisely to avoid the possible linguistic confusion I experienced between the two meanings of the word &amp;quot;density&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;Who are your students?  In your question you refer both to &amp;quot;Math coming into the college level&amp;quot; and &amp;quot;even in so called 'higher' Math classes&amp;quot;.  But these are not at all the same populations of students.  It is not at all reasonable to make inferences about what one group can or should know based on the performance of the other group.&lt;/li&gt;&#xA;&lt;li&gt;Specifically, there is no reason whatsoever to expect students &amp;quot;coming into the college level&amp;quot; to have ever seen probability density functions (pdfs) before.  They are not part of the K-12 curriculum and are not commonly taught in lower-level undergraduate mathematics courses.  Some textbooks (e.g. Hughes-Hallett) include them in a section on applications of integration in Calc 2, but many universities -- including my own -- routinely skip that application, as there simply isn't enough time in the semester to cover everything in the book.&lt;/li&gt;&#xA;&lt;li&gt;You have not said what your class is on, only that it is a class for Math education majors.  Is it a class on probability?  Have you done other examples of probability density functions?  Or is perhaps a class on probability a prerequisite for students in your class, so that it is reasonable to presume that this is prior knowledge?  The wording of your question seems to suggest that you expect students to &lt;em&gt;already know&lt;/em&gt; this language, which makes me think that you have not used it before.  If this is the case, I would suggest you take a look at your own department's course offerings, especially those typically taken in a sequence prior to the course you teach, and ask yourself where a student would have been likely to encounter this before.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;One final thought:  various bits and pieces of your phrasing combine to suggest to me that your own undergraduate education was not in the United States, and that English is not your first language.  (If I am wrong, I sincerely apologize for the inference.)  If that is the case, it might explain a number of related phenomena: your use of nonstandard language, your students' difficulty understanding what you mean, and your surprise at what students do and don't know.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDITED TO ADD:&lt;/strong&gt;&#xA;Incidentally, I was curious about the problem so decided to solve it myself, and found that the probability distribution function of the area is given by the function&#xA;&lt;span class=&quot;math-container&quot;&gt;$$g(A) = \frac{24 \arcsin\left(\frac{2A}{l^2}\right) \left(\pi - \arcsin\left(\frac{2A}{l^2}\right) \right)}{\pi^3 \sqrt{l^4 - 4A^2}}$$&lt;/span&gt;&#xA;It's possible that some clever identity could be used to simplify this, but if there is, I don't see it. Of course it's also possible that I have made an error in my solution, in which case I hope someone (perhaps the OP?) will post the correct answer.  But if I am right, the ugliness of the answer is itself (in my opinion) strong evidence that this is not a reasonable question for an exam.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2021-06-07T03:49:27.960" LastActivityDate="2021-06-07T03:49:27.960" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="20984" PostTypeId="2" ParentId="20971" CreationDate="2021-06-07T04:57:17.283" Score="4" Body="&lt;h1&gt;Continuity&lt;/h1&gt;&#xA;&lt;h2&gt;Method 1&lt;/h2&gt;&#xA;&lt;p&gt;One of the most intuitive ways that I like to think about what it means for a function to be &amp;quot;continuous&amp;quot; is by the &amp;quot;drawing test&amp;quot;: can you draw the function's graph on a piece of paper without ever removing your pen/pencil from the paper? A continuous function is a function whose graph is an &lt;em&gt;unbroken curve&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;h2&gt;Method 2&lt;/h2&gt;&#xA;&lt;p&gt;Another way that might be helpful in understanding what it means for a function to be continuous is to actually look at the types of &lt;em&gt;discontinuities&lt;/em&gt; for a function. You can read Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Classification_of_discontinuities&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; on the types of discontinuities for more information. In particular, there are three types of discontinuities:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A &lt;strong&gt;removable&lt;/strong&gt; discontinuity:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/qWGr7.png&quot; width=&quot;200&quot;/&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;A &lt;strong&gt;jump&lt;/strong&gt; discontinuity:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/fM2Za.png&quot; width=&quot;200&quot;/&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;An &lt;strong&gt;infinite&lt;/strong&gt; (or essential) discontinuity:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/nPB4r.png&quot; width=&quot;200&quot;/&gt;&#xA;&lt;p&gt;As you can see, in neither of these scenarios can we draw the function's graph without removing our pencil from the page. Hence, the function is not continuous.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;Here's a nice visual summary of each discontinuity:&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/CR0T4.jpg&quot; width=&quot;700&quot;/&gt;&#xA;&lt;h1&gt;Differentiability&lt;/h1&gt;&#xA;&lt;p&gt;Differentiability is a more technical and lengthier topic than continuity, as differentiability brings about discussions regarding smoothness and multi-variability. I'll leave it up to you to teach/introduce differentiability to your students as per the specified curriculum and course outline. The topic is rich, as even in a typical Calculus 3 course, differentiability is studied in depth.&lt;/p&gt;&#xA;&lt;p&gt;That being said, sticking to functions with a single variable, and proceeding in a fashion as we did above, we know intuitively now that a continuous function is one that has no &amp;quot;breaks&amp;quot; in its graph. Similarly, a differentiable function from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; is a function that not only has no breaks in its graph, but also has a well-defined plane that is tangent to the graph at each point. Hence, we expect a differentiable function to not have any &lt;em&gt;sharp&lt;/em&gt; folds, corners, or &amp;quot;peaks&amp;quot; in the graph; the graph must be smooth.&lt;/p&gt;&#xA;" OwnerUserId="16204" LastActivityDate="2021-06-07T04:57:17.283" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20985" PostTypeId="2" ParentId="20962" CreationDate="2021-06-07T05:51:05.087" Score="2" Body="&lt;p&gt;My kids have asked me this and i just responded &amp;quot;You are wrong. History, geography and most subjects i learnt at school were pretty much useless to me, and i have forgotten most of it, but maths is the only subject I have needed on a daily basis. I need it for all kinds of things.&amp;quot; From time to time, i involve them in the maths i need. When i am working out materials i need for DIY. When i was calculating flow for draining the garden. Budgeting. Etc. My young kids now fully accept and understand that they need it.&lt;/p&gt;&#xA;&lt;p&gt;Part of the problem is that many of the subjects ARE useless and i dont understand why they are taught. There is too much academia. Kids should learn gardening, DIY, electrics, money management and dealing with government offices, sports, diet, cooking, typing, web development, image processing, and a plethora of other topics far more important than the 18th century textbook based crap schools offer today. Perhaps if schools offered more topics where kids had to work and think for themselves then they would see when and why they need to calculate things and come to you for guidance with the context of real projects.&lt;/p&gt;&#xA;" OwnerUserId="13491" LastActivityDate="2021-06-07T05:51:05.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20986" PostTypeId="2" ParentId="20962" CreationDate="2021-06-07T10:27:53.843" Score="1" Body="&lt;p&gt;There is an extra dimension, because they might not be using mathematics directly. I am a big fan of knowing how and why something works, and there are many applications of mathematics they use every day but don't think about.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The Adds they see when theybrowse the net are based on statistical algorithms, using many factors&lt;/li&gt;&#xA;&lt;li&gt;The phones they use to communicate work internally using binary numbers, so if we didn't have some of the mathematical concepts we do, they would have to still use paper strips and cards to use them&lt;/li&gt;&#xA;&lt;li&gt;The production facilities producing their clothes and gadgets most likely use CAM and CAD to function - also in essence based on counting.&lt;/li&gt;&#xA;&lt;li&gt;the games and such they play use maths to calculate all sorts of aspects - view angle, effects, etc - there is a reason that graphics cards are being hoarded for bitcoin mining - they crunch a lot of numbers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I would play some games - like &amp;quot;find the maths in your hobby&amp;quot; - or &amp;quot;go looking for the mathematical aspects to every piece in your bag&amp;quot;, or if should be more direclty available &amp;quot;how close can you get to the sum in your head when you buy groceries&amp;quot; or &amp;quot;where can we get to with this amount of gas/level of charge on our battery&amp;quot; - and &amp;quot;can we get home again&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;It could also be stuff like the ratio of the sides of &lt;code&gt;Ax&lt;/code&gt; paper being &lt;code&gt;sqrt(2)&lt;/code&gt;and the possibilities that gives - if mathematic wasn't involved - we'd have a lot of different not related sizes maybe...&lt;/p&gt;&#xA;&lt;p&gt;Personally I am a big fan of Presh Talwaker and NumberPhile on Youtube, and i like to be aware of the potentials, but i rarely get to apply it. appart from recently when I was building CAD models of a woodworking project. parameterization of angle calculations and side lengths is directly transferable. The issue is of course,that the students don't necessarily have that sort of hobby - yet. And when they get there later in lfe I hope they remember some of what they did learn in school.&lt;/p&gt;&#xA;&lt;p&gt;You could also ask them to try to come to class only bringing stuff that has NO relation to mathematics at all - challenge them to prove to you that mathematics isnt' everywhere.&lt;/p&gt;&#xA;&lt;h2&gt;TL:DR&lt;/h2&gt;&#xA;&lt;p&gt;use analogies and real world examples of how the students' life would be without mathematics.&lt;/p&gt;&#xA;" OwnerUserId="16208" LastActivityDate="2021-06-07T10:27:53.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20987" PostTypeId="2" ParentId="20962" CreationDate="2021-06-07T10:34:49.953" Score="12" Body="&lt;p&gt;Trigger warning: math enthusiasts do not like this answer.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;They ask me when will we use this in the real world?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When students ask when they will use something in &amp;quot;real life&amp;quot;, they are rarely expecting an actual answer to how something is used.&lt;/p&gt;&#xA;&lt;p&gt;They are telling you: &amp;quot;I don't understand this.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Rather than coming up with try-hard ways to directly answer their question, focus your energy on teaching effectively, so that they understand the math that is taught to them.&lt;/p&gt;&#xA;&lt;p&gt;It is not to say that it is not important to highlight connections of math to other areas of life, &lt;em&gt;when they arise naturally&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I recommend that you read &amp;quot;How I wish I'd taught maths&amp;quot; from Craig Barton. Chapter 2.3 discusses this topic at some length.&lt;/p&gt;&#xA;" OwnerUserId="14091" LastEditorUserId="14091" LastEditDate="2021-06-07T16:27:57.850" LastActivityDate="2021-06-07T16:27:57.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20988" PostTypeId="2" ParentId="20962" CreationDate="2021-06-07T12:03:04.007" Score="4" Body="&lt;p&gt;A simple answer would be: &amp;quot;You will date more than one person in your life but you won't know in advance who is the one, unless you give them all a substantial try. If you never do that you'll miss an important part of a full life.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="11894" LastActivityDate="2021-06-07T12:03:04.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20989" PostTypeId="1" CreationDate="2021-06-07T14:37:14.147" Score="3" ViewCount="189" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0iox8.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0iox8.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is the solution I presented:&lt;/p&gt;&#xA;&lt;p&gt;(a) P(G|B) = (5/24) * (.125) = .026. Since you select ONE senior manager, then P(B) is 4/23. Now, if we are dealing with a second employee who is also a senior manager, then (4/23) * (.125) = .022. So, these are the criteria. Therefore, the TOTAL probability under these conditions becomes&#xA;.026 * .022 = .056%&lt;/p&gt;&#xA;&lt;p&gt;(b) Let's deal with the partner now. (6/24) * (.375) = .094. Since you select one employee who is a junior partner, then (3/23) * (.500) = .065. Taking the data together, then: (.094) * (.065) = .611%&lt;/p&gt;&#xA;&lt;p&gt;(c) (10/24) * (.125) = .052. If you choose one senior level employee, you can use it to find the probability you are looking for in this question.&lt;/p&gt;&#xA;&lt;p&gt;(9/23) * (.500) = .196. Then 8/22 * .500 = .182. So, the total probability has to be .19%&lt;/p&gt;&#xA;&lt;p&gt;The student is stuck and I am not sure where I am failing to be more clear: The student asks&#xA;&amp;quot;Why do you calculate P(G|B)? I know that it is the probability that an employee is senior-level given that they are a manager. Why do you calculate this?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Student subsequently asks:&lt;/p&gt;&#xA;&lt;p&gt;Shouldn't &lt;span class=&quot;math-container&quot;&gt;$P(G|B) =\frac{P(GandB)}{P(B)}$&lt;/span&gt; ? I think you meant &lt;span class=&quot;math-container&quot;&gt;$P(G and B) = \frac{5}{24} * 0.125$&lt;/span&gt;&#xA;Also you are assuming that the &lt;span class=&quot;math-container&quot;&gt;$P(G)$&lt;/span&gt; is constant?&#xA;Initially there are &lt;span class=&quot;math-container&quot;&gt;$0.125*24 = 3$&lt;/span&gt; seniors&#xA;After the first senior manager has been chosen, the &lt;span class=&quot;math-container&quot;&gt;$P(G)$&lt;/span&gt; changes to &lt;span class=&quot;math-container&quot;&gt;$2/23$&lt;/span&gt; right? Or am I misunderstanding something?&lt;/p&gt;&#xA;&lt;p&gt;I explained by getting into a discussion about the intersection of events. I also clarified it in a different way:&lt;/p&gt;&#xA;&lt;p&gt;You have 24 total employees. 10/24 are associates, 6/24 are partners, 5/24 are managers, and 3/24 are entrepreneurs.&lt;/p&gt;&#xA;&lt;p&gt;They can either be entry, junior or senior according to your question. Also, you provide the data: your question says that entry level is 50%, junior 37.5% and senior 12.5%.&lt;/p&gt;&#xA;&lt;p&gt;Think of it then this way:&lt;/p&gt;&#xA;&lt;p&gt;50% you can write it out as 1/2. So 1/2 * 10/24 = 5/24 ----&amp;gt; entry level associate.&lt;/p&gt;&#xA;&lt;p&gt;3/8 * 10/24 = 5/32 &amp;gt; there's your junior level associate.&lt;/p&gt;&#xA;&lt;p&gt;1/8 * 10/24 = 5/96 &amp;gt; there's your senior level associate.&lt;/p&gt;&#xA;&lt;p&gt;Now look again at your question part A. You can break it down with fractions:&lt;/p&gt;&#xA;&lt;p&gt;P(E) = 5/192 + 5/192 =10/192 = 0.0520&lt;/p&gt;&#xA;&lt;p&gt;I can go on, but can anyone let me know their thoughts? I am unsure how to get this across to the student and I've already tried 3 ways, but he is still stuck.&lt;/p&gt;&#xA;" OwnerUserId="15988" LastActivityDate="2021-06-07T21:18:58.503" Title="Not sure what a student is misunderstanding on this Stat question" Tags="|probability|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="20990" PostTypeId="2" ParentId="20962" CreationDate="2021-06-07T14:43:02.033" Score="3" Body="&lt;h3&gt;Give them practical examples&lt;/h3&gt;&#xA;&lt;p&gt;Engineering is a good start.  But equally well, you could relate it to simple DIY.&lt;/p&gt;&#xA;&lt;p&gt;The most obvious example is Pythagoras.  You're building a covered area next to your house.  The house wall is 3m before you get to the gutters; and you want the posts to be 2.5m tall.  You've got 3m of patio.  How long is the roof? because you need to make sure you buy enough tiles for it.&lt;/p&gt;&#xA;&lt;p&gt;Frighteningly, I have had a conservatory fitted by someone who couldn't do this calculation to work out how big the roof needed to be.  He needed to get someone to hold a tape measure for him to measure that distance.  I told him beforehand what the answer would be, and it was. :)  To be fair, he did a decent job of putting it together, but he was completely out of his depth on any kind of design work.&lt;/p&gt;&#xA;&lt;p&gt;Or even more complex trigonometry.  Last year I built raised flower beds on a square area, using wooden sleepers.  To make it look nice and give me access, I decided to put paths along the diagonals.  Then I decided that it'd look better to make the outside walls curve around.  To make that look good, I had the path running at 45 degrees, then three sections of wall to make the curve.  How long did they need to be, and what angle did I need to cut to make them join properly?  And then since I was building with overlapping joints with 10cm of overlap to hold everything together, how long did each section have to be in total, and how many 2.4m sleepers did I need to buy to build it?&lt;/p&gt;&#xA;&lt;p&gt;This isn't an imaginary example, by the way.  I have physically built this, and I literally did draw the diagram and use the Sine Rule to work out lengths of each section.  Sure, I could have chopped things roughly to fit and hacked it around a bit, but this way I could guarantee it would fit (within the bounds of final fine adjustment.)&lt;/p&gt;&#xA;" OwnerUserId="6456" LastActivityDate="2021-06-07T14:43:02.033" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20991" PostTypeId="2" ParentId="20962" CreationDate="2021-06-07T17:15:59.613" Score="2" Body="&lt;p&gt;Here a lot of subjects at university require quite a bit of math. Students who didnt pay a lot of attention during high school have a hard time and a lot fail. And it is usually a surprise for the students when studying majors like psychology, or archeology that they have to pass one or two math modules (mostly aiming at statistics, but you need also a bit of analysis when talking about statistical functions)&lt;/p&gt;&#xA;" OwnerUserId="11794" LastActivityDate="2021-06-07T17:15:59.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20992" PostTypeId="2" ParentId="20962" CreationDate="2021-06-07T18:21:09.327" Score="2" Body="&lt;p&gt;Why do people go to the gym to work out or take spin classes? Is lifting heavy weights or spinning a wheel on a stationary bike a useful skill they need to practice to succeed in their ordinary life? For most people, no. They do it, because going through those motions helps train their body to endure physical stress without wearing out. That helps them everyday, whether it's making a shopping trip to the mall, playing with their kids, going on a fun mountain bike trek, or just climbing the stairs to get to their  office for work. They do hard physical things to make all the other physical things in their life easier.&lt;/p&gt;&#xA;&lt;p&gt;Math education is the same way. Most people don't need to know how to derive a function, or calculate the cosign of an angle, or even know what those things mean. I'm a software engineer by profession with a Computer Science degree, so I had to take a lot of math in college. I can pretty honestly say I have not done any hard math other than some light algebra since I graduated almost 20 years ago. I have forgotten far more than I remember from school. But I am ever so thankful for the math education I got, because I know its there every time I do my taxes, or try to figure out if that discount at the store is really a good deal, or have to solve a hard logic problem at work. I don't use most of what I learned in my math classes, but because I went through the hard work of learning it, my brain is better trained to handle the everyday problems I do encounter. It makes everything in my life easier.&lt;/p&gt;&#xA;" OwnerUserId="13470" LastActivityDate="2021-06-07T18:21:09.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20993" PostTypeId="2" ParentId="20962" CreationDate="2021-06-07T19:15:11.437" Score="7" Body="&lt;p&gt;&lt;strong&gt;You do math to get better at logical thinking, not to get better at math.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Similar to how world-class athletes train in the gym by doing many exercises that you'd never see them stop and do mid-match, learning and doing math trains your brain to make you better at other every day tasks.&lt;/p&gt;&#xA;&lt;p&gt;A lot of math (most of it, in my opinion) is learning how to analyze a problem and break it down into steps you know how to solve. As you advance in your education the problems get more complex and you learn to identify which variables are relevant and what they relate to from context. That's a skill that's applicable in nearly everyone's life, from choosing whether you want an electric vs gas car based on purchase price and operating cost over 5 years, to planning a barbecue with friends. They might not think that those are very complicated problems, but that's the point: thanks to their their mathematical foundations they can easily analyze the problem.&lt;/p&gt;&#xA;" OwnerUserId="10023" LastActivityDate="2021-06-07T19:15:11.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="20994" PostTypeId="2" ParentId="20989" CreationDate="2021-06-07T21:18:58.503" Score="6" Body="&lt;p&gt;I see a few things going on here that may be confusing your student.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The problem itself is a problematic one; it assumes but does not state that seniority is independent of position, and that the employee of Company II chooses randomly who to meet with. The first assumption might be considered realistic, but the second definitely is not! As a general rule, students get very confused when presented with a problem that's asking them to make unreasonable assumptions.&lt;/li&gt;&#xA;&lt;li&gt;Worse, the problem presents a situation which cannot happen. If we are to believe that seniority is independent of position, that 1/8 of the employees are senior, and that five employees are managers, then there is only 5/8 of a senior manager! That means that the correct answer to problem (a) should be zero, because you don't have two entire senior managers to select.&lt;/li&gt;&#xA;&lt;li&gt;In your solution: What are the events &amp;quot;G&amp;quot; and &amp;quot;B&amp;quot;? From context, using the numbers you're giving, it looks like &amp;quot;B&amp;quot; is &amp;quot;a person is a manager&amp;quot; and &amp;quot;G&amp;quot; is &amp;quot;a person is senior&amp;quot;. But &lt;em&gt;which&lt;/em&gt; person are you talking about, since four are being selected? And why these variable names? This is contributing to some of your student's confusion; when they say P(G) should go from 5/24 to 2/23, that's because they're misinterpreting the &amp;quot;5/24&amp;quot; as the probability of being senior, and then calculating &amp;quot;the number of seniors remaining divided by the number of people remaining&amp;quot;. The best way I know to teach using this kind of notation is to be extremely explicit about what each event is, and to have the variable used in some way connected to the word; for example, maybe &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$S_1$&lt;/span&gt;&amp;quot; would be &amp;quot;the first person chosen is senior&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$M_1$&lt;/span&gt;&amp;quot; would be &amp;quot;the first person chosen is a manager&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;Using conditional probability here is an interesting choice. If &amp;quot;B&amp;quot; does mean &amp;quot;a person is a manager&amp;quot; and &amp;quot;G&amp;quot; means &amp;quot;a person is senior&amp;quot;, then &amp;quot;P(G|B)&amp;quot; would mean &amp;quot;the probability that a person is senior, given that they are a manager&amp;quot; or in more ordinary English &amp;quot;the probability that a manager selected at random is senior&amp;quot;, which here would be 1/8. Your student is correct to say that the calculation you're doing would be more accurately notated as &amp;quot;P(G and B)&amp;quot;, because &amp;quot;and&amp;quot; is the connective that generally represents intersectional probability.&lt;/li&gt;&#xA;&lt;li&gt;Finally - your solution has an error! Your calculation in (a), for example, gives the probability that two employees selected in a row are both senior managers (discarding for the moment the inconsistencies in the problem noted above). But that isn't what the question is asking; &lt;em&gt;four&lt;/em&gt; employees are selected, and only two must be senior managers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I would write out a recommended solution here, to try showing to your student, but after a few minutes of thinking I'm forced to realize that the issues with the problem that I noted earlier make that impossible. If we try the careful book-keeping you were doing, keeping track of how many managers are left and so on, then we rapidly run into the &amp;quot;5/8-of-a-manager&amp;quot; issue; if we don't, then your student's complaint about assuming a probability is constant when it isn't comes up. So, &lt;strong&gt;my recommendation would be to walk through the problem with the student and point out what's wrong with it.&lt;/strong&gt; A student can benefit a lot from seeing how a problem can be &amp;quot;wrong&amp;quot;!&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2021-06-07T21:18:58.503" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="20997" PostTypeId="2" ParentId="20970" CreationDate="2021-06-07T23:27:16.973" Score="3" Body="&lt;p&gt;To me, this question seems poorly worded.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Not sure what &amp;quot;the&amp;quot; two sides of a triangle even means.  Do you mean the two equal sides?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;After that, seems like we get some info about an angle and have to use that to predict info about the area.  [Probably using the formula for area.  Donno.  Guessing.]  OK.  I'd say, this is medium hardish.  Not end of the world.  But certainly, not easy drill.  I hope you are covering easier drill.  Or your students are such brainiacs they don't need that checked.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In some ways, this question feels very 19th century to me.  Lot of language to translate into math.  Then execute the math.  Then translate the math back into language.  Not saying this is the end of the world, of course.  Heck, look how people struggle with basic chem stoichiometry!  All that said, I would be leery of pushing this with weaker students or even with strong students who haven't been trained for the game.  And I even think there is some benefit to these translations (life is word problems).  But...don't trivialize the challenge.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's a mistake that you don't tell us more about the audience (age and &amp;quot;track&amp;quot;).  These are key aspects in pedagogy.  What's right/helpful for Susan Randall in Harvard is not what's right for  a lower track kid in Brooklyn.  This is a PART OF THE EQUATION (in pedagogical questions).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cris C. is a sweety.  If he's ragging on you, you're hurting.  I'm the mean bridge troll.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="16217" LastActivityDate="2021-06-07T23:27:16.973" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="20998" PostTypeId="2" ParentId="20962" CreationDate="2021-06-08T16:39:42.567" Score="2" Body="&lt;h1&gt;It Depends, do they want a &lt;em&gt;Good&lt;/em&gt; job?&lt;/h1&gt;&#xA;&lt;p&gt;Consider this list made by US News and World Report of the &lt;a href=&quot;https://money.usnews.com/careers/best-jobs/rankings/the-100-best-jobs&quot; rel=&quot;nofollow noreferrer&quot;&gt;100 Best Jobs&lt;/a&gt;.  Almost all of the jobs on this list fall into one of two groups:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Medical/Health Careers, and ...&lt;/li&gt;&#xA;&lt;li&gt;Jobs that require &lt;em&gt;A Lot of Math Skills&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Go ahead, go through the list: &lt;em&gt;software developer, statistician, financial analyst&lt;/em&gt;, etc.  What do they all have in common?  That's right, those all require lots of math knowledge and skills.  And it doesn't matter if you use a different source or list, they all pretty much look like this one.  Lots of medicine and lots of math.  Take your pick (but you're not going to get into med school with bad grades in math either).&lt;/p&gt;&#xA;" OwnerUserId="556" LastEditorUserId="556" LastEditDate="2021-06-08T16:46:38.237" LastActivityDate="2021-06-08T16:46:38.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21000" PostTypeId="2" ParentId="20962" CreationDate="2021-06-08T22:07:33.870" Score="2" Body="&lt;p&gt;To be honest, if we're speaking of secondary school mathematics, most people don't need them in real life. Most people won't ever find a situation, where they would need them.&lt;/p&gt;&#xA;&lt;p&gt;Most of the occupations that need math need nothing that isn't in the elementary school syllabus. And still, elementary school syllabus has things that have no practical applications for the very majority of people. Who needs trigonometry? For sure, you have needed them to setup the trebuchet, but even then, you've just used the tables. And for finance analysis, you don't even need to understand the concept of function to make the calculations, or type the formula in the excel.&lt;/p&gt;&#xA;&lt;p&gt;So don't pretend to find out abstract examples that show how useful the stuff is in the real life. Pupils are not stupid, they have a good filling if something is reality near or alienated. Concentrate on the real positives.&lt;/p&gt;&#xA;&lt;p&gt;First of all, &lt;strong&gt;studies&lt;/strong&gt;. Math is a requirement for technical studies. Can a secondary school pupil know for sure he/she will never want to study on the technical university? Most likely not. Not learning math will take that opportunity away from them. It doesn't matter if math is really needed. I've studied information sciences, I needed to master math to pass the entrance exams and to pass first 2 years, I've never ever used stuff I've learned there in the 'real' life, but I would never get to that 'real' life without that.&lt;/p&gt;&#xA;&lt;p&gt;Second, &lt;strong&gt;brain training&lt;/strong&gt;. Mathematics are just the training for the abstract thinking. It's a great training, probably one of the best training that you can do in school. It's very hard to overestimate how hardly we need it in the occupations that doesn't directly need higher mathematics.&lt;/p&gt;&#xA;&lt;p&gt;So if you're asked, why do they need to learn that stuff, ask them, are they sure, they want to loose the opportunity to choose every uni they want in the future, and to train their brain now?&lt;/p&gt;&#xA;" OwnerDisplayName="user16237" LastActivityDate="2021-06-08T22:07:33.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21001" PostTypeId="2" ParentId="20970" CreationDate="2021-06-08T23:11:19.413" Score="0" Body="&lt;p&gt;I sketched out a partial solution to this problem on a piece of scratch paper. I could be missing a simplification or getting something wrong, but the following is what I came up with. It took me about a page of math to get the pdf of A, expressed in terms of x. To get the pdf of A, one would then have to invert the function A(x) and substitute, which would be doable but looks messy. After that, one would need to do an integral in order to get the expectation value. I would be pleasantly surprised if that integral came out to be something that could be done in closed form without a huge mess.&lt;/p&gt;&#xA;&lt;p&gt;(1) This problem seems much too long and messy to use as a single problem on a timed exam.&lt;/p&gt;&#xA;&lt;p&gt;(2) It requires putting together a lot of knowledge: probability theory, integration, differentiation, and trig manipulation. Of the topics on this list, the relevant knowledge of probability is not something that would be a prerequisite for third-semester calculus at the school where I taught before I retired. You say that you covered it during the your own class, but one shouldn't expect students to master topics like this that, from your description, are more like a preview of later coursework.&lt;/p&gt;&#xA;&lt;p&gt;(3) The English is not very good and requires a great deal of additional effort to parse.&lt;/p&gt;&#xA;&lt;p&gt;You describe the class as an honors class in which most of the students are math education majors. This sounds like a little bit of a jackalope. Math education majors are typically extremely poor math students. Perhaps the honors designation led you to overestimate the level of the students. At my school, &amp;quot;honors&amp;quot; is extremely watered down and pretty meaningless.&lt;/p&gt;&#xA;&lt;p&gt;I would expect students' response to this question to depend on factors such as how much partial credit you gave, how difficult the other questions on the test were, and what grading scale you're using.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-06-08T23:11:19.413" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="21002" PostTypeId="2" ParentId="20943" CreationDate="2021-06-09T03:01:09.540" Score="7" Body="&lt;p&gt;I think the cognitive science is quite clear on this: Teachers should focus on learner similarities, not differences.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ascd.org/publications/educational-leadership/feb12/vol69/num05/Teaching-to-What-Students-Have-in-Common.aspx&quot; rel=&quot;noreferrer&quot;&gt;Teaching to What Students Have in Common&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The corollary to the author's first &amp;quot;Must Have&amp;quot; (factual knowledge) is that there is one area in which differences matter tremendously: Prior Knowledge. That should determine the majority of the differentiation in your classroom.&lt;/p&gt;&#xA;&lt;p&gt;My rough guess is:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;90% of students would be better off if their teachers took prior knowledge and universal properties of cognitive architecture more seriously and largely ignored other differences.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The 10% of students who need other types of differentiation are obvious exceptions, such as students with medical issues, social anxiety, disability, troubled home environments, poverty, culture shock, enormous age differences, enormous time demands such as a job or child, etc. and other non-cognitive and non-academic factors.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;~0% of students need differentiation according to &amp;quot;learning styles&amp;quot; (they don't exist), gender differences, etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2021-06-09T03:01:09.540" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21003" PostTypeId="2" ParentId="20962" CreationDate="2021-06-09T06:00:12.893" Score="2" Body="&lt;p&gt;&lt;strong&gt;You need &lt;em&gt;applied&lt;/em&gt; mathematics to make reasonable life choices.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Talk with them about compound interest and &lt;a href=&quot;https://en.wikipedia.org/wiki/Payday_loan&quot; rel=&quot;nofollow noreferrer&quot;&gt;payday loans&lt;/a&gt;. (And make sure that the example also includes some fees.) Talk about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Basic_reproduction_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;reproduction number&lt;/a&gt; of a virus and what increasing or decreasing it by a couple of percent means, some months down the line. Talk through different mobile phone contracts, comparing prepaid with fixed costs. How many minutes do you have to use the phone per day/week/month to make contract X better than contract Y? How much will it cost you? Talk to them about the meaning of pre-election polls, exit polls, and election results, and how one would &lt;em&gt;expect&lt;/em&gt; them to differ.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;You need &lt;em&gt;pure&lt;/em&gt; mathematics to understand and practice &lt;em&gt;applied&lt;/em&gt; mathematics.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Using real-world examples might enhance the learning experience, but each example takes a lot of time and effort. Also, too many real-world terms might obscure the underlying pattern. Both the loan example and the pandemic example are exponential growth. But a real-world loan also has fees, and a real-world pandemic happens in a finite population, so neither of them is easy to calculate.&lt;/p&gt;&#xA;&lt;p&gt;So show them the basic formula, and ask them if it helps them to understand the practical application better.&lt;/p&gt;&#xA;" OwnerUserId="8201" LastEditorUserId="8201" LastEditDate="2021-06-09T17:50:41.337" LastActivityDate="2021-06-09T17:50:41.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21004" PostTypeId="2" ParentId="20971" CreationDate="2021-06-09T08:25:35.943" Score="1" Body="&lt;p&gt;For continuity, I would generally follow one or both methods suggested by CuriosityCalls' answer (modulo my reservations about essential discontinuities, which I expressed in my comment to that answer).&lt;/p&gt;&#xA;&lt;p&gt;For differentiability, in one sense this is straightforward for students who have already encountered the definition of the derivative as &lt;span class=&quot;math-container&quot;&gt;$\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$&lt;/span&gt;. In this case, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is differentiable just in case that limit exists.&lt;/p&gt;&#xA;&lt;p&gt;Of course in practice, at the high-school level, matters are not so simple, since we often want to avoid having to compute that limit to determine whether the derivative exists. At this level, it is typically sufficient for students to know that derivatives have the intermediate value property, and hence a derivative can not have a jump discontinuity. This is enough to justify simple cases like &lt;span class=&quot;math-container&quot;&gt;$f(x)= |x|$&lt;/span&gt; not having a derivative at &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;, and the intuitive idea that a function is not differentiable at a point if its graph has a &amp;quot;sharp corner&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;However, to avoid incorrect generalisation, I would also consider showing students the function&#xA;&lt;span class=&quot;math-container&quot;&gt;$g(x) = \left\{ \begin{array}{ll} x\sin\left(\frac{1}{x}\right) &amp;amp; x \neq 0 \\ 0 &amp;amp; x = 0 \end{array} \right. $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and observe that while the derivative does not have a jump discontinuity at 0, the derivative does not exist at 0. This is just to emphasise that what matters for whether a function is differentiable at &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; is whether the limit of the difference quotient exists; showing that a (hypothetical) derivative is not continuous does NOT prove that the derivative does not exist. The test described above is just a handy shortcut method that can be used when the functions are suitably nice.&lt;/p&gt;&#xA;" OwnerUserId="9455" LastActivityDate="2021-06-09T08:25:35.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21005" PostTypeId="2" ParentId="20943" CreationDate="2021-06-09T12:18:45.133" Score="0" Body="&lt;p&gt;High school boys and girls obviously tend to differ from each other in their behavior, which is probably mostly due to social factors (as opposed to hormones). But if you base your teaching methods on these differences, then you risk perpetuating the same sexism in society that created those differences. It would be different if we knew there were certain &lt;em&gt;inherent&lt;/em&gt; differences in how boys and girls learn math.&lt;/p&gt;&#xA;&lt;p&gt;The tools of science are in principle capable of detecting certain types of inherent sex-linked differences. For example, if a certain trait is caused solely by a gene on the Y chromosome, then that will cause certain well-defined statistical correlations between related people of various sexes and types of genetic relatedness. For example, this is how we know that the most common type of color blindness is due to a gene on the Y chromosome. There can also be inherent sex-linked differences in things like lateralization of the two hemispheres of the brain, which are I believe caused by hormones, but there is no scientific magic bullet for showing that these directly cause intellectual differences.&lt;/p&gt;&#xA;&lt;p&gt;In reality, when you try to measure whether there are inherent &lt;em&gt;intellectual&lt;/em&gt; differences between groups, you tend to get swamped by random and systematic errors. On any test, males and females will have overlapping bell curves, and although there will be some difference &lt;span class=&quot;math-container&quot;&gt;$\Delta\mu$&lt;/span&gt; between the means, that difference will almost always be very small compared to the width &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; of each bell curve. The one case I've heard of where it's &lt;em&gt;not&lt;/em&gt; true that &lt;span class=&quot;math-container&quot;&gt;$\Delta\mu \ll \sigma$&lt;/span&gt; for sex-linked mathematical ability is that males tend to be better at visualizing three-dimensional rotations. I don't know of any case where such a difference is both large and demonstrably inherent.&lt;/p&gt;&#xA;&lt;p&gt;IMO this makes it pointless to try to teach one group differently than another.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-06-09T12:35:42.347" LastActivityDate="2021-06-09T12:35:42.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21007" PostTypeId="2" ParentId="20962" CreationDate="2021-06-09T20:08:58.323" Score="3" Body="&lt;p&gt;Would it not be interesting to ask students who voice such opinions to substantiate their answers with examples of situations where one would think that mathematics would be of importance but is not?&lt;/p&gt;&#xA;&lt;p&gt;One could ask a student to give a presentation of why number theory is of no importance in cryptography or why calculus is of no importance for optimisation problems. The goal should be that the student can substantiate his/her opinion but this will then require him/her to get enough of an understanding of mathematics to be able to explain convincingly why it is actually of no importance. And then the student will discover that his/her objection did not make sense.&lt;/p&gt;&#xA;" OwnerUserId="7148" LastActivityDate="2021-06-09T20:08:58.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21008" PostTypeId="2" ParentId="20943" CreationDate="2021-06-09T21:08:10.063" Score="0" Body="&lt;p&gt;I find in sports instruction, that girls are more willing to listen to coaching.  Boys are more bone-headed and want to do techniques how they think they should, instead of how advised.  Also, boys are more tolerant or even responsive to challenges (tough football coach or bootcamp), whereas girls do better with encouragement, positive versus negative feedback.&lt;/p&gt;&#xA;&lt;p&gt;Overall, I find girls more pleasant to deal with.  ;-)&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't say these are huge differences and they are more based on personality and response to coaching (education) in general, not to math.  But you'll notice it.  Try teaching a boy to surf and he won't listen to you and will get his ass kicked.  A girl will and will be standing first day.  (Yeah, they have better flotation, but they also paddle weaker.  And it's just night and day how they take coaching.)&lt;/p&gt;&#xA;&lt;p&gt;Maybe less of an impact in a non-physical task, but I think the personality still comes through.  I saw it when TAing chemistry.  Girls more willing to ask for, and accept instruction.  (Interesting to get the Smith prof's take.)&lt;/p&gt;&#xA;&lt;p&gt;[Generalizations of course, about populations.]&lt;/p&gt;&#xA;" OwnerUserId="16253" LastActivityDate="2021-06-09T21:08:10.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21009" PostTypeId="2" ParentId="20946" CreationDate="2021-06-09T21:19:52.357" Score="1" Body="&lt;p&gt;I can tell about how it went in Russia, presumably, that might have gone the same way in other countries as well.&lt;/p&gt;&#xA;&lt;p&gt;Traditionally, congruent shapes were called &amp;quot;equal&amp;quot;. Then, in the 1960's, it was felt that some set-theoretic notions should be introduced into the school geometry, a charge led by leading mathematicians such as Kolmogorov. Since there's a notion of equality for sets, and &amp;quot;a triangle &lt;em&gt;is&lt;/em&gt; a set of points, but two triangles being equal &lt;em&gt;does not&lt;/em&gt; imply they are equal as sets&amp;quot; sounds contradictory, a notion of congruence was introduced.&lt;/p&gt;&#xA;&lt;p&gt;The whole project was, however, a complete disaster,  in part because of genuine failures in the implementation, but mostly because of lack of qualification or desire in the teacher's community to switch. There was a huge backlash, with the word &amp;quot;congruent&amp;quot; one of the targets of mockery. So after some time, it was rolled back, and textbooks reverted to &amp;quot;equal triangles&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But the problem is, if you want to give any definitions at all, then a modern definition of e. g. a circle would likely contain words like &amp;quot;a set of points&amp;quot;. And it is hard to justify &amp;quot;definitions&amp;quot; like &amp;quot;line has only length but no width&amp;quot; when rigorous definitions are nearly as easy to state. So, some definitions, and some set-theoretic language, and hence also &lt;em&gt;congruent triangles&lt;/em&gt; crept into some textbooks. It helped that new generations of teachers were raised for which the notion of a set is not as alien as for the old guard.&lt;/p&gt;&#xA;&lt;p&gt;So, as it happened, in the middle school we studied &lt;em&gt;equal triangles&lt;/em&gt;, then, in the high school, we were told that it is more correct to call them &lt;em&gt;congruent&lt;/em&gt; (without much explanation), and later on we were introduced to some set-theoretic language that &lt;em&gt;did&lt;/em&gt; explain the reason. But I understand that some textbook authors prefer to avoid that switch and introduce the proper terminology from the beginning.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2021-06-09T21:19:52.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21010" PostTypeId="1" AcceptedAnswerId="21011" CreationDate="2021-06-10T02:37:36.337" Score="3" ViewCount="2009" Body="&lt;p&gt;How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to symbolically integrate definite integrals, when their graphs look so unremarkable and straightforward?&lt;/p&gt;&#xA;&lt;p&gt;I used &lt;a href=&quot;https://www.desmos.com/calculator/gwqjdpjruv&quot; rel=&quot;nofollow noreferrer&quot;&gt;Desmos&lt;/a&gt; to graph &lt;span class=&quot;math-container&quot;&gt;$\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \, dx$&lt;/span&gt; below. &lt;a href=&quot;https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/&quot; rel=&quot;nofollow noreferrer&quot;&gt;u/camelCaseCondition's comment&lt;/a&gt; reaffirms that &lt;a href=&quot;https://math.stackexchange.com/q/1142705&quot;&gt;the  solution&lt;/a&gt; is knotty and effortful, but not why:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale) is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least &lt;em&gt;be understandable&lt;/em&gt; had you taken Calc I-III and Complex Analysis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/PI35D.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;My question is more specific than &lt;a href=&quot;https://math.stackexchange.com/q/5303&quot;&gt;Why do statements which appear elementary have complicated proofs?&lt;/a&gt; and &lt;a href=&quot;https://cs.stackexchange.com/q/128980&quot;&gt;Why are mathematical proofs so hard?&lt;/a&gt;, because I'm asking especially about definite integrals with uneventful graphs, but that high schoolers can effortlessly graph without software.&lt;/p&gt;&#xA;" OwnerDisplayName="user15364" LastEditorUserId="470" LastEditDate="2021-06-11T14:11:45.543" LastActivityDate="2021-06-19T14:51:18.950" Title="Why can an easily graphable definite integral, be labyrinthine to evaluate?" Tags="|integration|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="21011" PostTypeId="2" ParentId="21010" CreationDate="2021-06-10T03:25:57.587" Score="5" Body="&lt;p&gt;Integration is tricky in general because it is like trying to find the question given an answer. For differentiation, we have a set of rules that work, and if we are systematic and careful enough, we can symbolically compute any derivative using those rules. Integration is tougher because you are trying to undo those rules to arrive back at the starting point. Furthermore, since many functions have equivalent forms (before and after simplification for instance), it can be very tricky to determine which rules were applied as two functions may look similar but have very different integrals.&lt;/p&gt;&#xA;&lt;p&gt;Going back to the question-answer analogy, think of this way. Suppose you know the answer to a question is five, and you are interested in determining the question. From this information alone, it is pretty much impossible to determine the original question. Why? We're missing a lot of context. Maybe the five represents dollars and is the amount we spent on lunch today. Maybe the five represents shoes and the question is how many shoes do we have. Now, suppose we knew the answer is five years. We would have a better chance of knowing what the question is here and can rule out some of the previous options, but again, the lack of context makes it hard to know precisely what that question is.&lt;/p&gt;&#xA;&lt;p&gt;The same thing happens in integration. You are given an answer (a function to integrate), and you are asked to determine the question (the integral) that the answer corresponds to. This integrand is usually given in a simplified form meaning it's akin to knowing the answer is five; the simplifications make us lose contextual information useful for solving the integral. As you apply your integration rules, you regain more and more of the context for the answer until you know enough to arrive at the question.&lt;/p&gt;&#xA;&lt;p&gt;Last note: the graphing is easier because ultimately, the graph does not have to be perfect. It can be a very good approximation, but due to human error or finiteness (of computer arithmetic), it is never a perfect. Making the graph is like being in the right ballpark; you are super close, but you might miss the mark by a smidge. Symbolic computations on the other hand are perfect; they do not discard any information; therefore, they are harder to arrive at.&lt;/p&gt;&#xA;" OwnerUserId="14024" LastEditorUserId="14024" LastEditDate="2021-06-10T03:33:24.243" LastActivityDate="2021-06-10T03:33:24.243" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21012" PostTypeId="2" ParentId="21010" CreationDate="2021-06-10T04:25:53.680" Score="3" Body="&lt;p&gt;It may help to connect this with the general problem of finding formulas for datasets/graphs. After all, numerically computing the function value of the antiderivative is easy, it's finding a formula to connect those values that's hard.&lt;/p&gt;&#xA;&lt;p&gt;We have lots of real world problems like this: if you want to figure out the trajectory of a thrown object, you can start with a parabolic model then gradually refine the model to account for more and more details. But at every stage the model is incomplete.&lt;/p&gt;&#xA;&lt;p&gt;You could also go more into detail about the process of computing a definite integral using Riemann sums, and show them how there's a tricky summation/limit they have to compute to get the exact answer. Maybe show them some limits where it's hard to tell what number they converge to just looking at them numerically. Done well, this could be a good motivation for a several key ideas (limits, Riemann sums) often not given the proper level of motivation in a standard calc class.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-06-10T04:25:53.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21013" PostTypeId="2" ParentId="20946" CreationDate="2021-06-10T06:48:36.263" Score="2" Body="&lt;p&gt;I agree with the point in the other answers, that the difference is in the approach, how rigorous we would like to be.&lt;/p&gt;&#xA;&lt;p&gt;However, I would like to also mention that this rigour in discussing elementary geometry is not extended to some other parts of teaching mathematics in high school. I do not think that too many high school teachers discuss vectors in a rigorous way (for example as equivalence classes of directed line segments). I think it is pretty common to refer to directed line segments of opposite sides of a parallelogram as equal vectors or same vectors.&lt;/p&gt;&#xA;&lt;p&gt;I guess, the difference is the goal. Elementary geometry is a good topic where axiomatic, rigorous approach can be introduced in a way that is accessible to high school students, so it is natural to insist on formal language.&lt;/p&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2021-06-10T06:48:36.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21014" PostTypeId="2" ParentId="21010" CreationDate="2021-06-10T07:42:33.693" Score="8" Body="&lt;p&gt;Calculating the definite integral is just as easy as graphing. You just plot your graph on a millimeter grid paper and count the number of squares below the graph.&lt;/p&gt;&#xA;&lt;p&gt;Of course, it does not give you an exact answer, only an approximation, but so is graphing. If it is easy to discern some approximate or qualitative features of the problem, it does not mean that it is equally easy to get a precise, quantitative answer.&lt;/p&gt;&#xA;&lt;p&gt;Answering concretely why the &lt;em&gt;indefinite&lt;/em&gt; integration is hard, it is useful to compare it to differentiation. What you are integrating in calculus courses are &lt;em&gt;elementary&lt;/em&gt; functions, that is, those that can be obtained from a finite set of basic functions (constants, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\arcsin x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\log x$&lt;/span&gt;) by a finite number of applications of a finite set of operations (&lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cdot$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$/$&lt;/span&gt;, composition). We know the derivatives of all the basic functions, and for each operation, there's a rule the derivative in terms of the operands and their derivatives (e. g. &lt;span class=&quot;math-container&quot;&gt;$(fg)'=f'g+fg'$&lt;/span&gt;). This makes the differentiation straightforward: you just follow how your function was constructed from the basic functions and apply the rules. This also proves that a derivative of an elementary function is an elementary function.&lt;/p&gt;&#xA;&lt;p&gt;In integration, there are no rules for &lt;span class=&quot;math-container&quot;&gt;$\int fg$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\int f^a$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\int f(g)$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;. In fact, one can sometimes &lt;em&gt;prove&lt;/em&gt; that &lt;span class=&quot;math-container&quot;&gt;$\int fg$&lt;/span&gt; is not an elementary function even though &lt;span class=&quot;math-container&quot;&gt;$\int f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\int g$&lt;/span&gt; are elementary, for example, this is the case for &lt;span class=&quot;math-container&quot;&gt;$f(x)=\log x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)=\frac{1}{x-1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now, why the definite integration is hard is simply because the go-to method for that is to do indefinite integration first and then plug in the limits. If your indefinite integral is not an elementary function, you may try to apply other methods (such as the complex-analytic contour integration), but they only work in very special cases. This is the case in your example - no elementary anti-derivative, but other ad hoc methods work.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastEditorUserId="10349" LastEditDate="2021-06-11T11:32:31.397" LastActivityDate="2021-06-11T11:32:31.397" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21015" PostTypeId="2" ParentId="21010" CreationDate="2021-06-10T12:31:13.553" Score="3" Body="&lt;p&gt;This is really more of an intrinsic math question than an ed question.  And it comes down to &amp;quot;going backwards is hard&amp;quot;.  There are just not always analytical solutions (or easy ones) to definite (or indefinite) integrals.  But the same thing applies with diffyQs.  Or quintic and higher polynomials.  Or many algebraic expressions with a mix of logs, trig, polynomial.&lt;/p&gt;&#xA;&lt;p&gt;And the graphability is kind of irrelevant.  And whether that's through a computer or even just manually on paper.&lt;/p&gt;&#xA;" OwnerUserId="16263" LastActivityDate="2021-06-10T12:31:13.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21016" PostTypeId="2" ParentId="21010" CreationDate="2021-06-10T17:32:54.750" Score="5" Body="&lt;p&gt;I think a good analogy would be: the equations &lt;span class=&quot;math-container&quot;&gt;$x = x^2 - 6$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x = e^x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x = \cos x$&lt;/span&gt; all &lt;em&gt;look&lt;/em&gt; equally complicated; but algebraic techniques are well suited to the first one and not the other two. (The second one can be solved easily but not by algebraic techniques; the third one basically can't be solved at all symbolically.) Moreover, the equation &lt;span class=&quot;math-container&quot;&gt;$x = x^{2.001} - 6$&lt;/span&gt; is also hopeless to solve using algebra.&lt;/p&gt;&#xA;&lt;p&gt;Math is an incredibly precise subject; differences that seem tiny to us can make a huge difference to the techniques available to us.&lt;/p&gt;&#xA;" OwnerUserId="13919" LastActivityDate="2021-06-10T17:32:54.750" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21017" PostTypeId="2" ParentId="20950" CreationDate="2021-06-10T17:51:26.773" Score="1" Body="&lt;p&gt;A good think is to students pick up several rounded objects (wheels, cans, glasses, etc.) and mesure with rule the length and the diameter and simply divide. And observe if something happens. You can see this idea &lt;a href=&quot;http://canterano.somenxavier.xyz/activitats/fitxes-anton-aubanell/www.xtec.cat/%7Eaaubanel/Fitxes/F13.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; (in Catalan - &lt;a href=&quot;https://translate.google.com/translate?sl=ca&amp;amp;tl=en&amp;amp;u=http://canterano.somenxavier.xyz/activitats/fitxes-anton-aubanell/www.xtec.cat/%257Eaaubanel/Fitxes/F13.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;automatic English translation&lt;/a&gt;). It is original from &lt;a href=&quot;https://twitter.com/antonaubanell?lang=es&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anton Aubanell&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Just a note: for measuring the length I think the better is to pick up a rope and surround the object you want to mesure to. Put a mark in the rope and mesure from the beginning to the mark of the rope.&lt;/p&gt;&#xA;" OwnerUserId="12352" LastEditorUserId="12352" LastEditDate="2021-06-10T17:57:56.917" LastActivityDate="2021-06-10T17:57:56.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21018" PostTypeId="2" ParentId="18793" CreationDate="2021-06-10T18:12:27.703" Score="0" Body="&lt;p&gt;Why simply let students whatever they want. Make groups of students based on what they want: equations, fractions, geometry, etc. Some students need more patience and some students are fast. The last group of students could switch to other topic. Why we have to teach the same thing at the same time to everything?&lt;/p&gt;&#xA;&lt;p&gt;The idea is picked from the &lt;a href=&quot;https://www.amazon.es/escuela-del-mundo-revoluci%C3%B3n-educativa/dp/8434431351/ref=sr_1_1?dchild=1&amp;amp;keywords=Salman%20Khan&amp;amp;qid=1623348705&amp;amp;s=books&amp;amp;sr=1-1&amp;amp;srs=11454515031&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt; of Salman Khan.&lt;/p&gt;&#xA;" OwnerUserId="12352" LastActivityDate="2021-06-10T18:12:27.703" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21019" PostTypeId="2" ParentId="858" CreationDate="2021-06-10T20:17:19.760" Score="1" Body="&lt;p&gt;My own assessment is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Teaching training: do training, a lot:&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Enroll courses (better outside your area)&lt;/li&gt;&#xA;&lt;li&gt;Read blogs from teachers&lt;/li&gt;&#xA;&lt;li&gt;Read books&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;Non-local knowledge:&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do you know latest teaching theories&lt;/li&gt;&#xA;&lt;li&gt;Do you know what other countries do?&lt;/li&gt;&#xA;&lt;li&gt;Do you know any rellevant teacher outside your country?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;Non isolament:&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do you know any other collegues outside your school?&lt;/li&gt;&#xA;&lt;li&gt;Do you collaborate with them?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If you answer 'yes' to many, then you are a good teacher.&lt;/p&gt;&#xA;&lt;p&gt;It works for me.&lt;/p&gt;&#xA;" OwnerUserId="12352" LastActivityDate="2021-06-10T20:17:19.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21020" PostTypeId="2" ParentId="21010" CreationDate="2021-06-11T04:53:46.823" Score="3" Body="&lt;p&gt;If I were a teacher, this would be my response to the question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why are some integrals easy to solve while others are tricky, complicated or downright impossible?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;That is a very good question, and one I've asked many times and never gotten a satisfying answer for. The best I've seen is that differentiation is almost always possible because the information you're after - i.e. the slope on the graph - is only dependent on what the function is doing locally, whereas integration is like asking about the entire life history of the function. You have to find an answer that takes into account every part of the initial function. In a way you're adding new information, while differentiation is easier because it is taking information away.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If we take a step back from calculus, another way of thinking about it, the function is like a set of rules in a game or a simulation, and integration is like playing it out to the end. Imagine you're a complete beginner watching a grandmaster chess game. It's pretty easy to figure out the rules about how the different pieces are allowed to move just by watching. That's like differentiation. Instead of that, imagine you're told all the rules about how the pieces move and you now have to beat a grandmaster. That's integration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now I still don't find that fully satisfying as an answer. As a high school student I was surprised to hear that there were integrals that cannot be solved. Surely if mathematicians were smarter, or they tried harder, they ought to be able to solve them?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I'm afraid this is a misconception, and the way we teach maths is partly to blame. Every problem we have given you is a problem that has a solution. In this course we won't ever give you a problem that cannot be solved using the techniques that we have taught you. But in real life there are many problems that can't be solved exactly. Some of them you might have even stumbled across while goofing around with pen and paper. A couple examples are &lt;span class=&quot;math-container&quot;&gt;$ \int e^{sin(x)} dx$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\int log(log(x)) dx$&lt;/span&gt;. You can try these now if you want, but there is no trick, substitution or way of rearranging them that will give you an integral that anyone can solve. In fact it's been mathematically proven that these integrals, and others like them actually cannot be solved.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But wait a minute I hear you say. I can draw these equations, which means I can shade in the area underneath them. Surely that means a solution exists? I mean, yes it does, but it does not guarantee that you'll be able to find a combination of &lt;span class=&quot;math-container&quot;&gt;$sin(x)$&lt;/span&gt;,&lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; or any of these other functions you know that equals the solution.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you go on to do maths or physics at university, you'll learn ways around this. You learn how to approximate these integrals and find solutions that are 'good enough'. For just about every real world problem this is indeed good enough. You already do this when you find the circumference of a circle. Pi just keeps going on forever but that doesn't stop us from using 3.14 or pressing the pi button on our calculators that only goes up to 3.1415926.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You'll also learn all about special functions. These are all the integrals that we couldn't solve, so we just gave them names so that they are their own thing. You can still do useful stuff with them. So going back to the equation above we can just decide one day that &lt;span class=&quot;math-container&quot;&gt;$\int log(log(x)) dx = Fred(x)$&lt;/span&gt;. Later, if we come across the integral &lt;span class=&quot;math-container&quot;&gt;$\int cos(x) + 2log(log(x)) dx$&lt;/span&gt; we know that the solution will be &lt;span class=&quot;math-container&quot;&gt;$sin(x) + 2Fred(x) + c$&lt;/span&gt;, so then we use the techniques we learn at Uni to approximate what Fred(x) is, and then we have an answer that's close enough.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Finally, I'd like to point out that you see these sorts of mysteries all through mathematics. There are lots of places where &lt;em&gt;we don't know why things are the way they are&lt;/em&gt;. For example, and this is going a bit off topic here, there is this object that lives in 196,883 dimensional space called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Monster_group&quot; rel=&quot;nofollow noreferrer&quot;&gt;Monster Group&lt;/a&gt; that is of astounding symmetry. No one knows why it's there, or why it's the biggest object of it's kind. We also don't have a formula for finding every prime number. There was a mathematician called John Von Neumann, perhaps the smartest mathematician ever, who once said 'Young man, in mathematics you don't understand things. You just get used to them'. Sometimes, when you're dealing with complex systems, you can't necessarily expect them to make sense, they're just the way they are. Just look at the world we live in now and tell me &lt;em&gt;that&lt;/em&gt; makes sense. Which makes all the times when you &lt;em&gt;can&lt;/em&gt; make sense of stuff all the more special. Maybe some of you guys will go on to do maths at uni and figure out how to make sense of some of this stuff.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But yeah, I'm not a teacher, nor am I a mathematician, and I suspect the above answer wouldn't survive their attention spans.&lt;/p&gt;&#xA;" OwnerUserId="14988" LastActivityDate="2021-06-11T04:53:46.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21021" PostTypeId="2" ParentId="20877" CreationDate="2021-06-11T16:30:59.870" Score="4" Body="&lt;p&gt;Nearly every classical or recreational math result about a finite set is an example.&lt;/p&gt;&#xA;&lt;p&gt;Can we have a walk in Königsberg so as to cross each of its &lt;a href=&quot;https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg&quot; rel=&quot;nofollow noreferrer&quot;&gt;seven bridges&lt;/a&gt; once? Well, we can attempt starting at southeast point then cross northward then cross back then go east... Never mind, Euler had a better approach.&lt;/p&gt;&#xA;&lt;p&gt;Can we tile a chessboard with A1 and H8 removed by &lt;span class=&quot;math-container&quot;&gt;$1\times 2$&lt;/span&gt; tiles? Well, we can try placing the first domino to A2A3 then second to A4A5 ... But it is easier to notice that we have removed two black squares.&lt;/p&gt;&#xA;&lt;p&gt;Can we decompose 1087 into a sum of two squares? We can try 1+1086 or 4+1083 or 9+1078 etc. and check that none of the second terms is a square, or we can notice that an odd sum of two squares equals 1 modulo 4, and 1087 equals 3.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2021-06-11T16:30:59.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21022" PostTypeId="2" ParentId="858" CreationDate="2021-06-11T23:07:52.513" Score="1" Body="&lt;p&gt;There only one reliable way to do this, and it's also very straightforward and easy to do. Give your students a well-constructed, research-based test that measures their knowledge of the subject. Do this once at the beginning of the semester and once at the end. Subtract.&lt;/p&gt;&#xA;&lt;p&gt;This gives you a number, which has interest only if you have something else to compare with. You can try to compare with your own past results, or with the results of your colleagues at your school, but generally the random and systematic errors make this difficult, and it also won't alert you to the case where you and your colleagues are both doing something wrong. It's better to compare with published results that have large sample sizes.&lt;/p&gt;&#xA;&lt;p&gt;It's easy to do this. What's harder is to identify ways of improving your results (if they appear sub-optimal) and sustaining any such improvements over time against countervailing forces.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-06-11T23:07:52.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21023" PostTypeId="1" AcceptedAnswerId="21024" CreationDate="2021-06-12T09:42:38.433" Score="15" ViewCount="2629" Body="&lt;p&gt;I (a student) am doing a presentation on Taylor series in my class (12th grade, in Germany if this is relevant). I am looking for a good example where you can see when Taylor series might be useful. Something like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Consider this problem with this function. [Some information on how the problem is hard to solve with its current function]. Wouldn't it be practical to have a &amp;quot;polynomial version&amp;quot; of this function to solve the problem? [Some information on why this would make the problem easier]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course, I would be open to other examples/applications that an average 12th-grade student would understand.&lt;/p&gt;&#xA;&lt;p&gt;I looked at &lt;a href=&quot;https://math.stackexchange.com/q/218421/&quot;&gt;What are the practical applications of the Taylor Series?&lt;/a&gt; on Math SE, but none of the applications/examples there seem to be showing the practicability of Taylor series. For example, we (unfortunately) haven't done differential equations.&lt;/p&gt;&#xA;&lt;p&gt;In his video &lt;a href=&quot;https://youtu.be/3d6DsjIBzJ4&quot; rel=&quot;noreferrer&quot;&gt;Taylor series | Chapter 11, Essence of calculus&lt;/a&gt; (which basically is how I learnt Taylor series), 3blue1brown (Grant Sanderson) tells how he was first confronted with Taylor series:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] this cosine function made the problem awkward and unwieldly. But by approximating [using Taylor series], everything fell into place much more easily.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It isn't explained how exactly the approximation made things easier, but I think this is the sort of example/application I'm looking for for my introduction – some problem, for example, where you would easily see that a polynomial approximation is a lot better to work with.&lt;/p&gt;&#xA;" OwnerUserId="16285" LastActivityDate="2021-09-26T13:02:52.007" Title="An introductory example for Taylor series (12th grade)" Tags="|secondary-education|calculus|examples|concept-motivation|" AnswerCount="7" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="21024" PostTypeId="2" ParentId="21023" CreationDate="2021-06-12T13:34:31.710" Score="25" Body="&lt;p&gt;One practical reason for choosing a Taylor Series approximation of a function over the function itself is if you are able to compute using only the four arithmetic operations.  For example, if you are asked to find the cosine of an angle and the only computing device you have is a four-function calculator, then you can get a good approximation of the cosine of the angle using the first few terms of the Taylor Series of the cosine function.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2021-06-12T13:34:31.710" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21025" PostTypeId="2" ParentId="20877" CreationDate="2021-06-12T15:13:24.203" Score="1" Body="&lt;p&gt;Prove that all popes had less than 150 years when they died.&lt;/p&gt;&#xA;&lt;p&gt;You could look at each of 260 of them and verify that this holds.&lt;/p&gt;&#xA;&lt;p&gt;Or you could take the fact that the oldest known person lived up to 122 years, which can be used to automatically conclude that all popes can only be less than this.&lt;/p&gt;&#xA;" OwnerUserId="10413" LastActivityDate="2021-06-12T15:13:24.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21026" PostTypeId="2" ParentId="20950" CreationDate="2021-06-12T15:32:56.397" Score="3" Body="&lt;p&gt;Over all I think the idea, of hands on for something like C/D is an invariant is laudable, but there are a few caveats.&lt;/p&gt;&#xA;&lt;p&gt;First, students at that age are very aware of precision. The answer &amp;quot;the ratio is 3 and a bit&amp;quot; is fine, but they will not accept it is a constant if one group measures 3.1, the second group 3.2, a third group 3.15. Indeed, in that case you end up asking them to put aside their concrete observations and accept the teacher's assertion &amp;quot;it's your measurements that weren't accurate enough&amp;quot;, which will be counterproductive - it says &amp;quot;you're not good enough&amp;quot; and &amp;quot;were you a better student you would have measured this right.&amp;quot;  Both false, and counterproductive to encouraging an interest in the mathematical arts.&lt;/p&gt;&#xA;&lt;p&gt;You also need circles over several orders of magnitudes. If all students have circles about the same size, say 1 to 5 inches, that is not ideal; again the variability in measurements will make them skeptical that what you state as a truth is in fact a truth they have observed. Better to also go to the playground and measure some HUGE circles too, say some with 10 or 20 foot diameters. Switching to metric: a centimeter, 10 centimeter, 1 meter, 10 meter diameters. After seeing that circles of very different sizes all have SIMILAR (and with the sloppiness of elementary student measurement, NOT IDENTICAL) C/D ratios, you can PROPOSE that the C/D ratio is a constant/invariant. If you are pursuasive, you MIGHT convince them that the ratio COULD BE invariant. If you are lucky the students will begin (and just begin) to believe you that the ratio of C/D is invariant.&#xA;So then what?&lt;/p&gt;&#xA;&lt;p&gt;Trying to figure out the exact number, pi - 3.14159... is beyond elementary school. Honestly, beyond high school. But you can get them to believe it is a number that is bigger than 3, bigger than 3 1/4, smaller than 3 1/2 or 3 1/7. And you can use it to tease, foreshadow, anticipate ideas in middle school and high school math (the existence of irrationals, transcendentals, etc)&lt;/p&gt;&#xA;&lt;p&gt;And then you CAN bring up that it's hard question, and that very very clever people, like Archimedes, Zu Chongzhi, Liu Hu, Madhava, and Aryabhata did a huge amount of work to get the value more accurate - and importantly, between upper and lower bounds. And that getting the value of pi to better than 1% (two digits) precision took on the order of a thousand years.&lt;/p&gt;&#xA;&lt;p&gt;(note, one of my pet peeves in teaching is how often ideas that took decades to millennia for the brightest minds our earth has ever produced to come to grips with are insultingly and demeaningly presented to students as ideas so simple and obvious that the student must be &amp;quot;stupid&amp;quot; if they don't grasp the concept as the instant it comes from the teacher's lips.)&lt;/p&gt;&#xA;&lt;p&gt;All this can tie into conversations about how good is good enough a measurement. The idea of precision. And the difference between an exact answer, which in some ways can't be had because the number is &amp;quot;irrational&amp;quot; - and conversations on what that means, and &amp;quot;good enough&amp;quot; for - building something, and that different precisions are needed for different tasks. And that a mathematician (as opposed to a carpenter, baker, engineer) wants to know it EXACTLY.&lt;/p&gt;&#xA;&lt;p&gt;Anyways, just some quick notes about, sorry not more coherent.&lt;/p&gt;&#xA;" OwnerUserId="15606" LastActivityDate="2021-06-12T15:32:56.397" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="21027" PostTypeId="2" ParentId="21023" CreationDate="2021-06-13T05:16:38.153" Score="5" Body="&lt;p&gt;Following the link you have in your post I found an answer mentioning combinatorics. Formal power series (generating functions) are used often in probability and combinatorics. After a bit of search on the internet I found an &lt;a href=&quot;https://en.wikipedia.org/wiki/Examples_of_generating_functions&quot; rel=&quot;noreferrer&quot;&gt;interesting (for me) example&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The coefficients of the expansion of &lt;span class=&quot;math-container&quot;&gt;$f(x)=\dfrac{x}{1-x-x^2}$&lt;/span&gt; are the numbers in the Fibonacci series.This is not really along the lines you mention in your question (it is not a polynomial approximation of a function), rather, it is a function, which in some sense has all information about the Fibonacci series.&lt;/p&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2021-06-13T05:16:38.153" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21028" PostTypeId="2" ParentId="21023" CreationDate="2021-06-13T08:04:01.617" Score="19" Body="&lt;p&gt;An excellent introductory example would be exponential function &lt;span class=&quot;math-container&quot;&gt;$\exp(x) = e^x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;By definition, this is the function that is its own derivative, i.e. &lt;span class=&quot;math-container&quot;&gt;$\exp'(x) = \exp(x)$&lt;/span&gt;. That's all nice and swell from a mathematical stand point, and it makes it easy to prove interesting properties of the function. &lt;strong&gt;But how do you actually compute it?&lt;/strong&gt; The definition above is totally useless for computing concrete points on this function like &lt;span class=&quot;math-container&quot;&gt;$\exp(1)$&lt;/span&gt;. Again, mathematicians will frequently simply call this result &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, but how do you even compute its numerical value of &lt;span class=&quot;math-container&quot;&gt;$e = 2.71828182845904523536$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;That's where the Taylor expansion comes to the rescue: Since we require &lt;span class=&quot;math-container&quot;&gt;$\exp(0) = 1$&lt;/span&gt;, and since we know that &lt;span class=&quot;math-container&quot;&gt;$\exp'(x) = \exp(x)$&lt;/span&gt;, we can easily conclude that all derivitives have the value &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x = 0$&lt;/span&gt;. As such, it's trivial to write down the tailor expansion of &lt;span class=&quot;math-container&quot;&gt;$\exp()$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\exp(x) \approx \sum_{n = 0}^∞\frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \dots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;And this we can compute to any precision we like&lt;/strong&gt;. In fact, mathematicians have shown that this expansion actually tends to the &lt;span class=&quot;math-container&quot;&gt;$\exp()$&lt;/span&gt; function, so the &lt;span class=&quot;math-container&quot;&gt;$\approx$&lt;/span&gt; sign above can be replaced with an &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; sign. For instance, we can now easily give an approximation&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$e = \exp(1) \approx 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} \approx 2.71$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="16293" LastActivityDate="2021-06-13T08:04:01.617" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="21029" PostTypeId="2" ParentId="21023" CreationDate="2021-06-13T17:07:11.677" Score="6" Body="&lt;p&gt;I suggest looking at the deduction for the equation of a simple pendulum. While a differential equation is involved, it is very simple. And it is made simple by the fact that one can safely replace, for small enough &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;, the sine &lt;span class=&quot;math-container&quot;&gt;$\sin\theta$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2021-06-13T17:07:11.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21030" PostTypeId="2" ParentId="20950" CreationDate="2021-06-13T18:51:49.437" Score="2" Body="&lt;p&gt;The question of when and why less guidance works or doesn't work is far from obvious. Instead of providing my personal opinion, as many have already done in other answers, I will try to collect some pointers to research that has already tried to answer this question, which you can use to critically review your own approach.&lt;/p&gt;&#xA;&lt;p&gt;Andrew Blair in his article &lt;a href=&quot;https://www.inquirymaths.org/posts/inquiryisnotdiscoverylearning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inquiry is not discovery learning&lt;/a&gt; distinguishes between discovery and inquiry learning, as a response to critics of discovery learning. He makes the following distinction:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;discovery learning&lt;/em&gt;, which he defines as when &amp;quot;students are expected to derive a procedure or concept from an activity devised by the teacher&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;inquiry&lt;/em&gt;, where the procedure or concept that appears at the end of the discovery process is incorporated into the course of an inquiry&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In discovery learning, the teacher attempts to preserve the pretence of discovery, even to the extent of withholding knowledge; in inquiry, the teacher, as a participant in the classroom activity, aims to introduce subject-specific knowledge when it is most relevant and meaningful to her students.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Clark et. al on the other hand, in &lt;a href=&quot;https://www.aft.org/sites/default/files/periodicals/Clark.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Putting Students on the Path to Learning&lt;/a&gt; advocates for fully guided instruction:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Decades of research clearly demonstrate that for novices (comprising virtually all students), direct, explicit instruction is more effective and more efficient than partial guidance. So, when teaching new content and skills to novices, teachers are more effective when they provide explicit guidance accompanied by practice and feedback, not when they require students to discover many aspects of what they must learn.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Clark et. al. does not advocate using explicit instruction all the time, and recognize the need for independent inquiry, but suggest to use it to practice newly gained knowledge rather than to discover it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Minimally guided instruction can increase the achievement gap. [..] more-skilled learners tend to learn more with less-guided instruction, but less-skilled learners tend to learn more with more-guided instruction. For these relatively weak students, the failure to provide strong instructional support produced a measurable loss of learning. The implication of these results is that teachers should provide explicit instruction when introducing a new topic, but gradually fade it out as knowledge and skill increase.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;TL;DR the details of how you design such activity are important. You have to carefully consider how and when you provide explanation and instruction along the way of the inquiry.&lt;/p&gt;&#xA;" OwnerUserId="14091" LastEditorUserId="14091" LastEditDate="2021-06-13T19:01:03.017" LastActivityDate="2021-06-13T19:01:03.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21032" PostTypeId="2" ParentId="21023" CreationDate="2021-06-13T19:37:22.340" Score="12" Body="&lt;p&gt;I feel it a bit strange that no one mentioned it, but a famous example is how Newton quickly estimated &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; by the Taylor series.&lt;/p&gt;&#xA;&lt;p&gt;Here is a quick sketch:&lt;/p&gt;&#xA;&lt;p&gt;First note that the area of a quarter circle is given by the integral:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \int_0^1 \sqrt{1-x^2} dx = \frac{\pi}{4} \tag{1}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now look at the left side.. &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;1$&lt;/span&gt; .... which means... BINOMIAL EXPANSION TIME!&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \int_0^1 \sqrt{1-x^2} dx = \int_0^1 ( 1- \frac12 x^2  - \frac18 x^4... \text{higher order}) dx= \left[1 -\frac{1^3}{12} - \frac{1^5}{40} + \text{higher order terms} \right] \tag{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now equate (1) and (2), and multiply both side by (4)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \pi = \left[  4 - \frac13 -\frac{1}{10}.. \right]$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Adding up the sums in the infinite series on the right, we get a fast approximation for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. However, Newton went further and tweaked this technique to get even faster convergence i.e: closer to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; with less terms. Veritasium made a &lt;a href=&quot;https://www.youtube.com/watch?v=gMlf1ELvRzc&quot; rel=&quot;nofollow noreferrer&quot;&gt;lovely video&lt;/a&gt; on this.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Note: Newton figured out binomial series by 'experiment' , so it is inaccurate to say newton used Taylor series but now that we understand binomial series is just the same as Taylor series, I supppose it is fine. &lt;a href=&quot;https://www.youtube.com/watch?v=xv_PWwdDWDk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ref&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Even before the Veritasium video, a friend had showed me this proof in their calculus book. I am grateful for them. ;D&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorUserId="14627" LastEditDate="2021-06-15T08:59:07.830" LastActivityDate="2021-06-15T08:59:07.830" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21033" PostTypeId="2" ParentId="21023" CreationDate="2021-06-13T19:58:11.610" Score="2" Body="&lt;p&gt;Another Newton example:&lt;/p&gt;&#xA;&lt;p&gt;Physics is full of differential equations, and one may ask how Newton dealt with them when most of the tips and tricks we know in the modern days were not yet discovered. Well, with his mighty series methods of course.&lt;/p&gt;&#xA;&lt;p&gt;In most introductory Physics classes, simple harmonic motion is introduced as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{d^2 x}{dt^2 } = -k x$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For simplicity sake, let's assume &lt;span class=&quot;math-container&quot;&gt;$k=1$&lt;/span&gt;, then, to illustrate series in the above, assume &lt;span class=&quot;math-container&quot;&gt;$x= a_o  + a_1 t + a_2t^2....= \sum_i a_i t^i$&lt;/span&gt;, then plug into the DE:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ (2a_2 + 3 \cdot 2 a_3 t+ 4 \cdot 3 a_4 t^2...) = -(a_o + a_1 t + a_2 t^2...)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Rearrange like terms:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ (2a_2 + a_o) +  ( 3 \cdot 2 a_3+ a_1) t+ (4 \cdot 3 a_4 + a_2)t^2...=0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Considering the initial condition &lt;span class=&quot;math-container&quot;&gt;$x(0)=0$&lt;/span&gt; [physically imagine giving a kick from the equilibrium position], only odd terms survive:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ (3 \cdot 2 a_3 + a_1)t + ( 5 \cdot 4 a_5 +a_3 )t^3...=0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can write the above as a summation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \sum_{k=1} \left[ (2k+1)(2k) a_{2k+1} + a_{2k-1}  \right] t = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For the whole polynomial to be zero  always, the coefficient has to be always zero, we get a recurrence from that:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ a_{2k+1} = -\frac{a_{2k-1} }{(2k+1)(2k)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The solution to this recurrence is given as &lt;span class=&quot;math-container&quot;&gt;$a_{2k+1} = (-1)^{k} \frac{1}{(2k+1)!}$&lt;/span&gt;, plug this back into our initial series expression for &lt;span class=&quot;math-container&quot;&gt;$x(t)$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x(t) =  \sum_i (-1)^i \frac{t^{2i+1} }{(2i+1)!}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorUserId="8023" LastEditDate="2021-06-14T12:48:37.340" LastActivityDate="2021-06-14T12:48:37.340" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21034" PostTypeId="1" CreationDate="2021-06-14T06:58:03.823" Score="4" ViewCount="150" Body="&lt;p&gt;&lt;em&gt;&lt;strong&gt;Questions&lt;/strong&gt;&lt;/em&gt;: Are we really taking students into account FULLY when writing textbooks for various areas? Also, are we being unintentionally elitist or dismissive when neglecting to take a more humble approach in writing explanations/proofs and making decisions about providing solutions to exercises?&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Let me explain&lt;/em&gt;: (using geometry as an example I'm familiar with)&lt;/p&gt;&#xA;&lt;p&gt;Many textbooks (for (a well known) example, the Topological, Smooth and Riemannian Manifolds texts by Jack Lee) are written with exercises and no solutions to better encourage learning and reduce the undesired yielding to temptation by even more conscientious students etc etc.&lt;/p&gt;&#xA;&lt;p&gt;My issue with this approach in Math education is that it really stands in the way of self-learners. There is also a tendency to write some books with super concise notation (a common one being little more than a passing mention of 'oh these are Christoffel symbols') in the name of brevity and with the far too common reference to 'mathematical maturity'. My personal feeling about this last bit is that it often gives an escape for simply bad pedagogy. But putting aside personal feelings, I would again say this is a great hindrance to self learners!&lt;/p&gt;&#xA;&lt;p&gt;Consider how students often progress to become knowledgeable and move on to serious graduate work for instance. A lot of it depends on the school itself. Take a large school like Waterloo or UChicago that offers every course under the sun and the programs are geared towards a more thorough treatment of prerequisites leading to tougher courses (like Symplectic geometry, say). A student in these elite places has access to Profs and fellow students and even seminars that are instrumental to their progress but are not available at smaller schools or certain entire countries. As a result, they have two choices if they hope to try for certain more demanding areas of Math: Move or Chalk it up to impossible.&lt;/p&gt;&#xA;&lt;p&gt;Isn't it a duty to write in ways that don't (even unintentionally) assume you're talking to either an extraordinary student or one that is smart and has access to the right instructors?&lt;/p&gt;&#xA;&lt;p&gt;Now I know we can't write amazing textbooks that are self-contained to cater to every student in every circumstance but I believe some obvious steps like opting for full solutions and fleshing out more details even in some 'higher level' texts will have somewhat of a democratizing effect on math education.&lt;/p&gt;&#xA;" OwnerUserId="16296" LastActivityDate="2021-06-14T13:04:32.647" Title="Textbooks with solutions and catering to different circumstances" Tags="|undergraduate-education|mathematical-pedagogy|textbooks|course-design|struggling-students|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21035" PostTypeId="2" ParentId="21034" CreationDate="2021-06-14T12:52:19.160" Score="0" Body="&lt;p&gt;I believe it is a commercial choice.  Universities and public schools prefer to specify such books because it gives the teacher a gatekeeper role in the learning process.  In addition, some teachers like to assign graded homework (rather than using tests for all evaluation) and just use the book.  A book marketed to self studiers would not keep the answers (not worked solutions, per se) secret.  For example my father's WW2 math textbooks (&amp;quot;War Department Education Manuals&amp;quot;) all had answers included.&lt;/p&gt;&#xA;&lt;p&gt;Also, personally, I am a rather undisciplined creature.  But the times I have self-studied courses, I had zero problem treating the problems as drill, doing the drill, and using the answers for feedback.  Felt no temptation to peek--are you tempted to cheat on reps during a workout on your own? Feedback is important to find mistakes (even silly ones) and also for the psychological &amp;quot;gamification&amp;quot; aspect of seeing how you did.  Would play games like &amp;quot;try to get 100%&amp;quot; in my head.  In the rare cases that I couldn't solve a drill problem, a glance at the solution is actually kind of a useful hint to then go and try to work the problem.  [And of course, since it was drill, I made sure to work every problem, to re-work any missed ones, even if &amp;quot;silly mistake&amp;quot;, and to work, without consulting text (while reworking) any problems that I had to [very rarely] sneak a peak at the answer or the text lesson.]  Realize that self-study is rather monastic and difficult to stay committed to.  So some aspect of feedback is helpful psychologically (and this is an important aspect, do not underestimate it).  Furthermore, consider the whole concept of programmed learning, which involves interaction/feedback in a very step by step manner.&lt;/p&gt;&#xA;&lt;p&gt;I would also add that extremely advanced textbooks (e.g. topology) tend to have small markets and are written as labors of love by people who are rather smart at math (but not at teaching).  Often the writers are more interested in explaining to themselves and in not being caught in a mistake of math (similar to Wiki math articles) than they are in the hard slog of actual communication, actual training of the unwashed.  And there is not a sufficient market to push the pedants for stronger pedagogy (in contrast to more common/easy subjects).&lt;/p&gt;&#xA;" OwnerUserId="16298" LastEditorUserId="16298" LastEditDate="2021-06-14T12:59:08.637" LastActivityDate="2021-06-14T12:59:08.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21036" PostTypeId="2" ParentId="21023" CreationDate="2021-06-14T12:55:57.653" Score="3" Body="&lt;p&gt;Here is one more example: A student if often taught to memorize the l'hopital rule : Hawken  you see a &lt;span class=&quot;math-container&quot;&gt;$\frac{0}{0}$&lt;/span&gt;, you plug in the numbers for the limit then algorithmically like a computer you must take derivative of numerator and denominator and then you have the true value of the limit.&lt;/p&gt;&#xA;&lt;p&gt;OK, but why does that work? A rigorous answer is difficult but Taylor's theorem can shed light:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \lim_{x \to a} \frac{P(x)}{Q(x)} = \lim_{x \to a} \frac{ P(a) + \frac{P'(a)}{1!} (x-a) + \frac{P''(a) }{2!} (x-a)^2... +O\big((x-a)^3) \big)}{ Q(a) + \frac{Q'(a) }{1!}(x-a) + \frac{Q''(a)}{2!} (x-a)^2.. +O\big( (x-a)^3 \big)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the case of &lt;span class=&quot;math-container&quot;&gt;$\frac{0}{0}$&lt;/span&gt; from directly plugging in numbers that just means &lt;span class=&quot;math-container&quot;&gt;$P(a)=Q(a)=0$&lt;/span&gt; and we have the ratio between first derivatives.. ahhh so that was what it meant all along: L'hopital is just saying to find the limit of a ratio going to &lt;span class=&quot;math-container&quot;&gt;$\frac{0}{0}$&lt;/span&gt; on direct substitution, take the ratio of a point close to &lt;span class=&quot;math-container&quot;&gt;$'a'$&lt;/span&gt; instead.. but what's the value at the point controlled by? Answer to that: The derivatives.&lt;/p&gt;&#xA;&lt;p&gt;Also, if someone wasn't paying attention in algebra class and never figured out how to use remainder theorem, then Taylor also gives an easy proof for it as well. &lt;a href=&quot;https://math.stackexchange.com/questions/3769138/generalized-remainder-theorem-for-powers-of-linear-factors&quot;&gt;Ref&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorUserId="14627" LastEditDate="2021-06-14T23:20:42.803" LastActivityDate="2021-06-14T23:20:42.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21037" PostTypeId="2" ParentId="21034" CreationDate="2021-06-14T13:04:32.647" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Isn't it a duty to write in ways that don't (even unintentionally) assume you're talking to either an extraordinary student or one that is smart and has access to the right instructors?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No. It isn't a duty to write textbooks in a particular way, or to write textbooks at all.&lt;/p&gt;&#xA;&lt;p&gt;The most common way that advanced textbooks get written is that people teach a course for a while and use that as a basis for a textbook. Unsurprisingly, the primary audience of such a textbook is &lt;em&gt;their own students&lt;/em&gt;, and more generally, students who are similarly situated.&lt;/p&gt;&#xA;&lt;p&gt;It's already the case that there's not a lot of incentive to write textbooks. If it were an expectation that, as part of writing a textbook, one should put in an additional substantial amount of work (like writing full solutions) to make the book &lt;em&gt;less useful&lt;/em&gt; to the primary audience, the result would mostly be that people would write fewer textbooks.&lt;/p&gt;&#xA;&lt;p&gt;It's certainly a good thing when people write for a broader audience, but if you want that to happen, the place to look is at the systemic incentives, not trying to put more obligations on individual people who are already trying to make their subject more accessible.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2021-06-14T13:04:32.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21038" PostTypeId="1" AcceptedAnswerId="21041" CreationDate="2021-06-14T19:56:05.710" Score="6" ViewCount="1278" Body="&lt;p&gt;I will describe what I mean by the above with an example.&lt;/p&gt;&#xA;&lt;p&gt;Suppose you are a professor, about to teach a first Calculus course in a university. There are dozens, if not hundreds, of calculus books out there. How do you choose which one to use as a reference for your course? You may narrow it down to a few books, but how do you actually read the ones in the final list to make a choice?&lt;/p&gt;&#xA;&lt;p&gt;In another example, continuing from the preceding one, maybe you just choose the one you used as a student, but next month a new Calculus book comes about and attracts a lot of attention. How do you read it to know if, next semester, you could use it instead of the original one?&lt;/p&gt;&#xA;&lt;p&gt;I cannot possibly believe a math professor has the time to study them in depth to know what they contain, what are their pros/cons, etc, as, at least in my experience, it takes from weeks to months to read a book in depth, doing some exercises, even if you already know most of the contents.&lt;/p&gt;&#xA;&lt;p&gt;So do they just read the books like a novel? Do a quick en-passant through each one to have some idea of the differences?&lt;/p&gt;&#xA;&lt;p&gt;I would really like to know the answer, since, as a student, every good book I know has been recommended by someone else - but, at some point, I would like to be able to make my own judgement.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;&lt;p&gt;Just to clarify, I know there is no &amp;quot;correct&amp;quot; answer, but I would really like people who have been through this' opinions/techniques. Also, I used calculus only as an example, so the same question can be asked regarding any type of math book.&lt;/p&gt;&#xA;&lt;p&gt;PS: This is a port of my original (now closed) question on MSE, which was asked to be ported over to this community&lt;/p&gt;&#xA;" OwnerUserId="16299" LastEditorUserId="5563" LastEditDate="2021-06-16T18:56:17.853" LastActivityDate="2021-06-17T04:01:15.297" Title="How do math professors select textbooks?" Tags="|undergraduate-education|textbooks|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="21040" PostTypeId="2" ParentId="21038" CreationDate="2021-06-14T23:08:17.917" Score="9" Body="&lt;p&gt;I think this answer depends completely on the professor and the department.&lt;/p&gt;&#xA;&lt;p&gt;You ask specifically about a freshman calculus text. At many schools, there are ten or twenty sections of this class every semester, so there is a tendency to want to agree on the same book for the entire department. This becomes an especially important issue if the book is non-free and sold as a single volume that covers two semesters. In this situation, the individual professor has zero choice, so there is no issue. There may be a textbook selection committee or a specific faculty member who does administrative work for all sections of the class.&lt;/p&gt;&#xA;&lt;p&gt;The majority of faculty are adjunct. They typically have zero choice about books, especially because they're often hired at the last minute.&lt;/p&gt;&#xA;&lt;p&gt;If someone is fortunate enough to have the choice, then I think the criteria are entirely individual. I would personally be looking for a free book. Many professors are locked into a certain book because they're used to using the ancillaries for that book. Some people have pet topics, and if their topic isn't included, they won't consider the book.&lt;/p&gt;&#xA;&lt;p&gt;90% of commercial texts for a subject like freshman calc are extremely similar in terms of the exposition, artwork, and problems. A professor may have preferences on certain things, but they can judge those things without reading a 1000-page book from cover to cover. For instance, they may want a book that's structured so that they can skip some of the material on limits.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-06-14T23:08:17.917" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21041" PostTypeId="2" ParentId="21038" CreationDate="2021-06-15T05:34:15.907" Score="6" Body="&lt;p&gt;One of my favorite mathematicians, when it comes to teaching, once wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Can one learn mathematics by reading it?&#xA;I am inclined to say no. Reading has an edge over listening because reading is more active - but not much. Reading with pencil and paper on the side is very much better - it is a big step in the right direction.&#xA;The very best way to read a book, however, with, to be sure, pencil and paper on the side, is to keep the pencil busy on the paper and throw the book away. (Halmos, R.P., 1975).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, supplementary to other answers, if I had a choice, how do I decide what book to use for teaching? Here are some thoughts:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Usually I look at the problems: Are they providing the opportunities to learn what is required?&lt;/li&gt;&#xA;&lt;li&gt;Is chapter organization in a way that allows me to structure learning processes as intended?&lt;/li&gt;&#xA;&lt;li&gt;I look at the exposition of some critical instances and examples. how are they explained. E.g. in calculus, how is the notion of a derivative explained?&lt;/li&gt;&#xA;&lt;li&gt;Not to neglect: The &amp;quot;look and feel&amp;quot; of a book is important. Is language respectful and accessible and figures and graphics of high quality?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Actually, one could possibly write a book about 'what are quality books in math' and not come to an end. But to me, a main question is: Does the book help me to engage students to actively do mathematics? Put another way:&#xA;What mathematics does the book provide in order to keep the pencil of my students busy?&#xA;And no, I don't throw the book away :-).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Halmos, P.R. (1975). &lt;a href=&quot;https://doi.org/10.1080/00029890.1975.11993856&quot; rel=&quot;noreferrer&quot;&gt;The Teaching of Problem Solving&lt;/a&gt;, the first section in:&#xA;Halmos, P. R., Moise, E. E., &amp;amp; Piranian, G. The Problem of Learning to Teach. The American Mathematical Monthly, 82(5), 466–476.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2021-06-15T05:34:15.907" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="21043" PostTypeId="2" ParentId="20959" CreationDate="2021-06-15T14:23:37.727" Score="4" Body="&lt;p&gt;Part of learning to do higher dimensional geometry is learning which aspects of your low dimensional intuition are good or not. I can't visualize two smooth &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-planes meeting transversely at a point, which I know happens in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^4$&lt;/span&gt;. But I am experienced enough to know that, when thinking about dimensions and transversality, I need to compute, because my two and three dimensional experience will lead me astray.&lt;/p&gt;&#xA;&lt;p&gt;Similarly, my visual intuition tells me that the diagonal of a unit cube is only a bit longer than the side; certainly not more than twice as long: But in 1,000,000 dimensional space, the diagonal is 1,000 times the side length! Again, I have spent enough time on high dimensional computations to know that I need to compute with the Pythagorean theorem when I have a high dimensional vector with many nonzero entries, because all those small squares can add up to something big.&lt;/p&gt;&#xA;&lt;p&gt;I think part of Jordan's statement is based on the fact that he is experienced enough to know where the traps are, so what remains seems clear and trouble free.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2021-06-15T14:23:37.727" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21044" PostTypeId="1" CreationDate="2021-06-15T18:02:43.887" Score="3" ViewCount="632" Body="&lt;p&gt;We want to teach about words that have multiple meanings, in mathematics and&#xA;English for middle school children (e.g., function, root, or, volume, angle, constant).  This is for students who are learning  English as a second language. That is, how do you teach words that have different connotations or meanings in mathematics compared to English as taught in literature class.&lt;/p&gt;&#xA;&lt;p&gt;Other conversation on this forum have identified these words, creating a nice reference list. &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10832/examples-of-vocabulary-that-have-different-meanings-in-mathematics-compared-to&quot;&gt;Examples of vocabulary that have different meanings in Mathematics compared to &amp;quot;everyday&amp;quot; English&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now I am asking how to teach these words with multiple meanings.&lt;/p&gt;&#xA;" OwnerUserId="16303" LastEditorUserId="2083" LastEditDate="2021-06-15T19:15:07.470" LastActivityDate="2021-07-16T00:19:45.387" Title="Teaching words with multiple meanings in Mathematics and English" Tags="|secondary-education|primary-education|middle-school|english-language-learners|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21046" PostTypeId="2" ParentId="21044" CreationDate="2021-06-15T20:09:00.303" Score="0" Body="&lt;p&gt;One very practical approach is just to teach the math in close to 100 percent English. Math is actually easier than literature or history to learn in a foreign language.  So don't do it bilingually.  Explanations, sure.  Fine.  But nominally 100 percent immersion in math class.  This also helps to make clear just subliminally that you are using terms in their math context such as line or area...no differently than for English language only students who can tell by context when a word is used for a mathematical purpose as opposed to essay writing usage.&lt;/p&gt;&#xA;&lt;p&gt;For instruction in English itself, this is a normal second language class. History may be bilingual, if students are weak.  And if at all possible, avoid assigning classes on a third language.  My sister did immersion year in German high school. Math was easiest.  Well after Englush.  Then science was next...easy except for occasional vocabulary.  German lit and history were hardest because of the verbal content. Well...except for Latin.  That was just a bridge too far and they eventually acceded to letting her drop it.  Just too hard to learn a third language in a second language.&lt;/p&gt;&#xA;" OwnerUserId="16305" LastActivityDate="2021-06-15T20:09:00.303" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="21048" PostTypeId="1" AcceptedAnswerId="21055" CreationDate="2021-06-16T14:47:04.923" Score="48" ViewCount="4261" Body="&lt;p&gt;When introducing an abstraction it is important (in my opinion) to have a wide variety of examples of this abstraction.&lt;/p&gt;&#xA;&lt;p&gt;Since finite dimensional real vector spaces are classified up to isomorphism by their dimension, it is a little difficult to find examples of abstract vector spaces which &amp;quot;feel&amp;quot; very different from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;.  It can be a little difficult to justify the extra work involved in making the abstraction because of this.&lt;/p&gt;&#xA;&lt;p&gt;I have a short list of examples which I like to use:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The space of polynomials of degree less than or equal to &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$n \in \mathbb{N}$&lt;/span&gt;.  Interesting applications include interpolation and splines.&lt;/li&gt;&#xA;&lt;li&gt;The space of rational functions with a given denominator.  An interesting application is partial fraction decomposition.&lt;/li&gt;&#xA;&lt;li&gt;The space of &lt;span class=&quot;math-container&quot;&gt;$n \times m$&lt;/span&gt; real matrices.  One application here is to image analysis, if you view the entries of the matrix as indicating brightness of a pixel located on a screen.  Projecting into a lower dimensional subspace is a form of image compression.  One super interesting application is &lt;a href=&quot;https://www.sciencemag.org/news/2018/01/mind-reading-algorithm-can-decode-pictures-your-head&quot; rel=&quot;noreferrer&quot;&gt;mind reading&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The space of sequences satisfying a linear recurrence relation.  The shift operator is a linear operator from this space to itself.  Finding the eigenbasis of the shift operator allows one to extract an explicit formula for the &lt;span class=&quot;math-container&quot;&gt;$n^{th}$&lt;/span&gt; term of such a sequence.  One notable example is an explicit formula for the Fibonacci sequence.&lt;/li&gt;&#xA;&lt;li&gt;Real Homology and cohomology of a simplicial complex. The vector spaces are the spaces of chains/cochains and the linear maps are the boundary maps. I have never used these as examples in an intro linear algebra course, but it seems like one could assign a student project about this with significant scaffolding.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I am hoping to generate a &amp;quot;big list&amp;quot; which would be useful to teachers of linear algebra.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-06-22T11:00:55.100" LastActivityDate="2024-03-15T23:00:38.727" Title="Big list of &quot;interesting&quot; abstract vector spaces" Tags="|examples|linear-algebra|big-list|" AnswerCount="14" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="21049" PostTypeId="2" ParentId="21048" CreationDate="2021-06-16T17:32:50.083" Score="8" Body="&lt;p&gt;Below are two elementary/expository papers that I know for generating strange examples, although how interesting they are will vary from person to person (however, a connection with the special theory of relativity does exist), and unless some additional conditions are imposed I don't believe they're finite-dimensional. Nonetheless, I think it's worth giving these references because it's likely that people who google-land on this question in the future would be interested.&lt;/p&gt;&#xA;&lt;p&gt;The first paper is freely available (see also &lt;a href=&quot;https://www.google.com/search?q=%22Generating+exotic-looking+vector+spaces%22&amp;amp;filter=0&quot; rel=&quot;noreferrer&quot;&gt;this google search&lt;/a&gt;) and the second paper is behind a paywall.&lt;/p&gt;&#xA;&lt;p&gt;Michael A. Carchidi, &lt;a href=&quot;https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/generating-exotic-looking-vector-spaces&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Generating exotic-looking vector spaces&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;College Mathematics Journal&lt;/strong&gt; 29 #4 (September 1998), pp. 304−308.&lt;/p&gt;&#xA;&lt;p&gt;Daniel S. Kalman and Prescott K. Turner, &lt;a href=&quot;https://www.tandfonline.com/doi/pdf/10.1080/0020739790100203&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Algebraic structures with exotic structures&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;International Journal of Mathematical Education in Science and Technology&lt;/strong&gt; 10 #2 (April−June 1979), pp. 173−174.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2021-06-16T17:38:05.443" LastActivityDate="2021-06-16T17:38:05.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21050" PostTypeId="2" ParentId="21038" CreationDate="2021-06-16T18:20:50.087" Score="3" Body="&lt;p&gt;I know you asked about mathematicians and college texts, but perhaps my experience in elementary school would be worthwhile. Several times I was asked to evaluate new texts for our curriculum. Since I taught only math to grades 1-6, my input was considered worthwhile.&lt;/p&gt;&#xA;&lt;p&gt;When looking at new books:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I looked to see what topics were covered. For example, our coverage of geometry such as perimeter/area was poor, and when we replaced the books I looked to make sure this deficiency would be taken care of.&lt;/li&gt;&#xA;&lt;li&gt;I looked to see if there were a range of problems for weak, average, and strong students. I also checked to see if these were clearly marked, as the average elementary school teacher wouldn't be able to make a judgement easily.&lt;/li&gt;&#xA;&lt;li&gt;I looked to see if the plan for the curriculum was realistic in terms of time.&lt;/li&gt;&#xA;&lt;li&gt;I checked how problem solving was taught.&lt;/li&gt;&#xA;&lt;li&gt;I looked to see if there were new approaches for teaching things and considered if the time spent training the teachers would be a worthwhile tradeoff for the new approach.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-06-16T18:20:50.087" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21051" PostTypeId="1" CreationDate="2021-06-16T19:52:31.620" Score="-2" ViewCount="195" Body="&lt;p&gt;Here is an example of a lesson I did on linear equations where my objective was to show that they are equations of first degree. The reason I do it this way is because I tend to find that students only see first degree equations as being of the form: mx + b, meaning that they can ONLY look like, say:&lt;/p&gt;&#xA;&lt;p&gt;2x + 3 = 5.&lt;/p&gt;&#xA;&lt;p&gt;An equation of the first degree (aka linear equation) with a single variable is defined by the following statement:&lt;/p&gt;&#xA;&lt;p&gt;mx + b = 0    where &lt;span class=&quot;math-container&quot;&gt;$m \neq 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m, b \in R$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;or any other equation that is equal to the above statement.&lt;/p&gt;&#xA;&lt;p&gt;Example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(x + 1)(x - 1)  = x^2 + 2x + 1$&lt;/span&gt; is an linear equation.&lt;/p&gt;&#xA;&lt;p&gt;Reasoning:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2 - 1 = x^2 + 2x + 1$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\leftrightarrow -1=2x+1$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\leftrightarrow 2x + 2 = 0$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus: x + 1 = 0&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Since I was chided in a previous post for not providing context, I will do so now: I shared this approach with a College Algebra class (first year college students) and it is NOT an honors class. What do you think about my approach? I welcome any comments (positive or negative).&lt;/p&gt;&#xA;" OwnerUserId="15988" LastEditorUserId="2083" LastEditDate="2021-06-17T05:20:36.927" LastActivityDate="2021-06-17T05:20:36.927" Title="A linear equation -- my approach" Tags="|algebra|terminology|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="21052" PostTypeId="2" ParentId="21038" CreationDate="2021-06-16T20:25:46.947" Score="1" Body="&lt;p&gt;I am fortunate enough that I can choose my own. I typically just recommend certain classics and use my own book in class. It has always worked quite well since I do not use the typical textbooks used in most average American colleges/college Math courses. I usually tell them to get my book (since I draw heavily from the exercises there) and then I share with them an entire bibliography of books that are optional for them to buy or borrow from wherever they find them.&lt;/p&gt;&#xA;&lt;p&gt;How do I select them? I tend to select classics over the newer texts that do not really give an emphasis on the things I want covered. So, for example, I always recommend Apostol or Spivak. I give things from a more theoretical, less &amp;quot;applied&amp;quot; approach.&lt;/p&gt;&#xA;" OwnerUserId="15988" LastActivityDate="2021-06-16T20:25:46.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21053" PostTypeId="2" ParentId="21051" CreationDate="2021-06-16T20:36:21.350" Score="5" Body="&lt;p&gt;I don't approve of teaching students unusual meanings for standard terms. If I correctly understand your example, &lt;span class=&quot;math-container&quot;&gt;$\sin x=x$&lt;/span&gt; would be a linear equation. And whether &lt;span class=&quot;math-container&quot;&gt;$ax^2+bx+c$&lt;/span&gt; is linear depends on whether &lt;span class=&quot;math-container&quot;&gt;$b^2=4ac$&lt;/span&gt;. The issue becomes worse with polynomials of higher degree.&lt;/p&gt;&#xA;&lt;p&gt;It's not clear to me what you mean by two equations being equal (in &amp;quot;or any other equation that is equal to the above statement&amp;quot;). From your example, it appears that you just mean &amp;quot;equivalent&amp;quot;. Or maybe &amp;quot;equivalent via very elementary algebraic manipulations&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2021-06-16T20:36:21.350" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21054" PostTypeId="2" ParentId="21051" CreationDate="2021-06-16T21:10:04.240" Score="5" Body="&lt;p&gt;I think I would tend to disagree with your terminology's bent. The label of the equation speaks to the type of the expression which forms the equation. The type of solution set does not label the equation. Quadratic equations are not differently labeled when they have different solution sets. Furthermore, since there are literally infinitely many equivalent expressions which maintain the same solution set, I think anchoring terminology on the solution set is not a good choice. Your example is quadratic because it is formed with a quadratic expression. Simple as that.&lt;/p&gt;&#xA;&lt;p&gt;To give some more absurd examples,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x^{11} = 0, \qquad x^{8}=0 \qquad x=0, \qquad sinh(x)= 0 \qquad x(\cos^2 x + \sin^2 x+ln(e^x)-x )=0$$&lt;/span&gt;&#xA;are not all the same equation. Same solution set ? Sure. Same equation, no.&lt;/p&gt;&#xA;&lt;p&gt;Or, more like your example, is &lt;span class=&quot;math-container&quot;&gt;$x^{10}+x = x^{10}-x+2$&lt;/span&gt; a linear equation ? I would say no, however, it allows a reduction to the linear equation &lt;span class=&quot;math-container&quot;&gt;$x=-x+2$&lt;/span&gt; hence &lt;span class=&quot;math-container&quot;&gt;$2x=2$&lt;/span&gt; and we conclude &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt;. I want some term other than &amp;quot;same&amp;quot; to communicate that steps were require to bridge the gap from the given equation to the linear equation.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-06-16T21:10:04.240" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21055" PostTypeId="2" ParentId="21048" CreationDate="2021-06-16T21:14:52.203" Score="24" Body="&lt;p&gt;Here are some more examples:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$C[a,b]$&lt;/span&gt;, the set of continuous real-valued functions on an interval &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt;. This abstract vector space has some very nice properties that make it very good for a first-semester linear algebra course:&lt;/p&gt;&#xA;&lt;p&gt;a. It has a natural inner product on it, given by &lt;span class=&quot;math-container&quot;&gt;$\langle f, g \rangle = \int_a^b f(t)g(t) \, dt$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;b. It contains the (infinite-dimensional) vector space &lt;span class=&quot;math-container&quot;&gt;$\mathcal P$&lt;/span&gt; of all polynomial functions (restricted to &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt;) as a subspace, as well as every (finite-dimensional) vector space &lt;span class=&quot;math-container&quot;&gt;$\mathcal P_n$&lt;/span&gt; of polynomial functions of degree &lt;span class=&quot;math-container&quot;&gt;$\le n$&lt;/span&gt; for given &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;c.  Considered as an inner product space with respect to the inner product in (a), orthogonal projection onto &lt;span class=&quot;math-container&quot;&gt;$\mathcal P_n$&lt;/span&gt; produces the best possible polynomial approximation of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;d.  It also contains the (infinite-dimensional) vector space &lt;span class=&quot;math-container&quot;&gt;$\mathcal F$&lt;/span&gt; consisting of all convergent Fourier series on &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt;, as well as every (finite-dimensional) vector space &lt;span class=&quot;math-container&quot;&gt;$\mathcal F_n$&lt;/span&gt; of Fourier series of order &lt;span class=&quot;math-container&quot;&gt;$\le n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;e. Considered as an inner product space with respect to the inner product in (a), orthogonal projection onto &lt;span class=&quot;math-container&quot;&gt;$\mathcal F_n$&lt;/span&gt; produces the best possible  approximation as a Fourier series of order &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Subspaces of the examples in the OP:&lt;/p&gt;&#xA;&lt;p&gt;a. The vector space of upper-triangular &lt;span class=&quot;math-container&quot;&gt;$n\times n$&lt;/span&gt; matrices&lt;/p&gt;&#xA;&lt;p&gt;b. The vector space of symmetric (or skew-symmetric) &lt;span class=&quot;math-container&quot;&gt;$n \times n$&lt;/span&gt; matrices&lt;/p&gt;&#xA;&lt;p&gt;c.  The vector space &lt;span class=&quot;math-container&quot;&gt;$\mathcal P_{\textrm{even}}$&lt;/span&gt; of polynomials with only even-degree terms, and the corresponding vector space &lt;span class=&quot;math-container&quot;&gt;$\mathcal P_{\textrm{odd}}$&lt;/span&gt;.  Notice that &lt;span class=&quot;math-container&quot;&gt;$\mathcal P = \mathcal P_{\textrm{even}} + P_{\textrm{odd}}$&lt;/span&gt;, a nice example of subspace addition, and moreover the differentiation operator &lt;span class=&quot;math-container&quot;&gt;$D:\mathcal P \to \mathcal P$&lt;/span&gt; is a linear transformation that maps these two subspaces onto each other.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Vector spaces of functions of a particular form.  These can be somewhat artificial but nevertheless have nice pedagogical properties.  For example, for any fixed real numbers &lt;span class=&quot;math-container&quot;&gt;$k, r$&lt;/span&gt;, the set &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; consisting of functions of the form &lt;span class=&quot;math-container&quot;&gt;$f(x) = A\sin kx + B\cos kx + Ce^{rx}$&lt;/span&gt; is a subspace of &lt;span class=&quot;math-container&quot;&gt;$C[a,b]$&lt;/span&gt;. The differentiation operator &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; maps &lt;span class=&quot;math-container&quot;&gt;$V \to V$&lt;/span&gt;, and if you choose the natural basis for &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, then the matrix of &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; relative to that basis has a nice geometric interpretation as a &lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt; rotation in the plane spanned by &lt;span class=&quot;math-container&quot;&gt;$\sin kx$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos kx$&lt;/span&gt;, followed by a dilation by a factor &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; in that plane and a dilation by a factor &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; in the direction of the subspace spanned by &lt;span class=&quot;math-container&quot;&gt;$e^{rx}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, let's not forget:  &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt;, considered as a 2-dimensional real vector space.  The isomorphism with &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^2$&lt;/span&gt; is so obvious that sometimes we forget these are not &amp;quot;actually&amp;quot; the same set.  (Yes, I know &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt; is sometimes defined as &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^2$&lt;/span&gt; with additional structure, but it need not be defined that way, and students do not see it that way unless they are taught to.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2023-12-24T21:00:15.817" LastActivityDate="2023-12-24T21:00:15.817" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="21056" PostTypeId="2" ParentId="21048" CreationDate="2021-06-17T00:29:11.710" Score="10" Body="&lt;p&gt;The set of solutions to a system of linear homogeneous ODEs is a vector space, and the dimension of this vector space is equal to the total order of the system.  The idea that every solution is the linear combination of some &amp;quot;special&amp;quot; set of solutions is just a statement that we can always find a basis of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; elements in an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-dimensional vector space.&lt;/p&gt;&#xA;&lt;p&gt;This allows for an obvious extension, if you want to get into it:  the set of solutions to an inhomogeneous system of linear ODEs is an affine space.&lt;/p&gt;&#xA;" OwnerUserId="13965" LastActivityDate="2021-06-17T00:29:11.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21057" PostTypeId="2" ParentId="21051" CreationDate="2021-06-17T00:50:23.667" Score="6" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Example:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$(x+1)(x−1)=x^2+2x+1$&lt;/span&gt; is an linear equation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I agree with all above answers. I wouldn't say that &lt;em&gt;it is&lt;/em&gt; a linear equation, but &lt;em&gt;it reduces to&lt;/em&gt;, or &lt;em&gt;it is solved by&lt;/em&gt; or any other phrase you want to use instead. Because, I mean, that's the idea right?&lt;/p&gt;&#xA;&lt;p&gt;I feel that the relaxed terminology is not quite accurate, and it might generate confusion when they see these ideas in the Cartesian plane. Recall that &lt;span class=&quot;math-container&quot;&gt;$ax+b=c$&lt;/span&gt; represents the intersection of two &amp;quot;lines&amp;quot; in the plane (students might associate &amp;quot;lines&amp;quot; to &amp;quot;linear&amp;quot;, and that's ok). But &lt;span class=&quot;math-container&quot;&gt;$(x+1)(x−1)=x^2+2x+1$&lt;/span&gt; is the intersection of two parabolas.&lt;/p&gt;&#xA;&lt;p&gt;Finally, if you treat &lt;span class=&quot;math-container&quot;&gt;$$(x+1)(x−1)=x^2+2x+1$$&lt;/span&gt; as a linear equation, it is very likely that you have problems when they say that &lt;span class=&quot;math-container&quot;&gt;$$(x+1)(x−1)=\color{red}{2x^2}+2x+1$$&lt;/span&gt; is a linear equation as well.&lt;/p&gt;&#xA;" OwnerUserId="3380" LastEditorUserId="3380" LastEditDate="2021-06-17T01:07:23.197" LastActivityDate="2021-06-17T01:07:23.197" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21058" PostTypeId="2" ParentId="21048" CreationDate="2021-06-17T00:58:22.627" Score="6" Body="&lt;p&gt;Some more (rather general) examples of vector spaces over any field &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;For any set &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;, the space &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}^S$&lt;/span&gt; of mappings from &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}$&lt;/span&gt;, with pointwise addition and scalar multiplication. (Actually, every vector space is isomorphic to a subspace of such a space, due to the existence of a basis.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;More generally, if &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is a set and &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is a vector space over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}$&lt;/span&gt;, then so is &lt;span class=&quot;math-container&quot;&gt;$V^S$&lt;/span&gt;, again with pointwise addition and scalar multiplication.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One of my favourites (in case that we want to consider fields different from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt;, too): The power set &lt;span class=&quot;math-container&quot;&gt;$2^S$&lt;/span&gt; of a given set &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is a vector space over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}_2 = \{0,1\}$&lt;/span&gt;, where addition is defined to be the symmetric difference, and &lt;span class=&quot;math-container&quot;&gt;$0 \cdot A := \emptyset$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1 \cdot A := A$&lt;/span&gt; for each &lt;span class=&quot;math-container&quot;&gt;$A \subseteq S$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Actually, this space is isomorphic to &lt;span class=&quot;math-container&quot;&gt;$(\mathbb{F}_2)^S$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A note on example 1. in the question:&lt;/p&gt;&#xA;&lt;p&gt;Again, if you consider general fields (and thus also fields of non-zero characteristic), polynomials even yield two classes of vectors spaces: polynomials in the &amp;quot;algebraic&amp;quot; sense (which are, strictly speaking, just lists of their coefficients) and polynomial functions.&lt;/p&gt;&#xA;&lt;p&gt;The mapping which maps each polynomial to its corresponding polynomial function (which is injective if and only if the field has characteristic zero) is a nice example of a linear map which is, maybe, not so easy to understand immediately in terms of matrices.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;On a more general note, I'd say that one of the major motivations for making the abstraction from spaces such as &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}^n$&lt;/span&gt; to general vector spaces is that - even when we restrict ourselves to the finite-dimensional case - general vector spaces quite are flexible in terms of constructing new spaces from given ones. So let me continue the list of examples by giving examples of &amp;quot;standard constructions&amp;quot;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;As mentioned implicitly in previous answers, a vector subspace of a vector space is again a vector space in its own right. Thus, in particular, the kernel and the range of a linear map is a vector space (and hence, so is the set of solutions to a homogeneous linear equation).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$V_1, \dots, V_n$&lt;/span&gt; are vector spaces, then so is &lt;span class=&quot;math-container&quot;&gt;$V_1 \times \dots \times V_n$&lt;/span&gt; (with componentswise addition and scalar multiplication.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; is a vector subspace of a vector space &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, then the quotient space &lt;span class=&quot;math-container&quot;&gt;$V/U$&lt;/span&gt; is a vector space.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The dual space &lt;span class=&quot;math-container&quot;&gt;$V'$&lt;/span&gt; of a vector space is a vector space (this is a special case of the space of all linear mappings between two given vector spaces - an example that was mentioned in a comment by James S. Cook).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2021-06-17T00:58:22.627" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="21059" PostTypeId="2" ParentId="21048" CreationDate="2021-06-17T01:26:16.213" Score="6" Body="&lt;p&gt;The spin states of an electron form a two-dimensional vector space over the complex numbers. Designate &amp;quot;spin up&amp;quot; and &amp;quot;spin down&amp;quot; for a basis. The vector space structure is a consequence of the linearity of the Schrodinger equation.&lt;/p&gt;&#xA;&lt;p&gt;The computer science slant on this situation uses the word &amp;quot;qubits.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-06-17T01:33:32.190" LastActivityDate="2021-06-17T01:33:32.190" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21060" PostTypeId="2" ParentId="21038" CreationDate="2021-06-17T04:01:15.297" Score="4" Body="&lt;p&gt;I will address the related question of how to &lt;em&gt;find&lt;/em&gt; textbooks.  You have to do this before you can select one, which the other answers address nicely.&lt;/p&gt;&#xA;&lt;p&gt;A good place to start is with the &lt;a href=&quot;https://aimath.org/textbooks/&quot; rel=&quot;nofollow noreferrer&quot;&gt;AIM Open Textbook Initiative&lt;/a&gt;.  From their site:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The American Institute of Mathematics (AIM) seeks to encourage the adoption of open source and open access mathematics textbooks. The AIM Editorial Board has developed evaluation criteria to identify the books that are suitable for use in traditional university courses. The Editorial Board maintains a list of Approved Textbooks which have been judged to meet these criteria.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I count 19 broad topics, most with multiple texts.  My experience with a few of these has been overwhelmingly positive, with a very high average production value.&lt;/p&gt;&#xA;" OwnerUserId="3443" LastActivityDate="2021-06-17T04:01:15.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21062" PostTypeId="1" CreationDate="2021-06-17T07:43:18.573" Score="6" ViewCount="727" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;How and what mathematics must be taught for training engineering students with the mathematics required for Machine Learning?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How can one conduct training of mathematics required for application-based topics like ML? The students are basically engineering students having covered basic Engineering Mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Can anyone please give any tips for an intuitive way of teaching ML Maths?&#xA;Also, recommendations on any good books that the students will find easy to comprehend are welcome...&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;- the scope of the course is to make the students industry-ready with principles of mathematics used in ML topics. It is not as deep as research Mathematics but certainly more rigorous and focused than the general Engineering Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="15858" LastEditorUserId="8023" LastEditDate="2021-06-18T09:13:34.413" LastActivityDate="2021-06-18T09:13:34.413" Title="Teaching Mathematics to a Machine Learning Class" Tags="|undergraduate-education|engineering-mathematics|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="21063" PostTypeId="2" ParentId="21062" CreationDate="2021-06-17T12:05:45.257" Score="-1" Body="&lt;p&gt;If you are talking about a short exposure (hard to tell but guessing), I think running an in class lab on multiple regression, would be good.  There's a lot of hype around &amp;quot;machine learning&amp;quot; and &amp;quot;AI&amp;quot;, but in practice, it often has elements of regressions, DOE, etc. In any case, those tools are very close and have uses of their own.  Plus if you do an in class lab with Stata or R, where they can play around with degrees of freedom, fitting, second order effects (squared terms and interactions), it will not feel like &amp;quot;math class&amp;quot;, but something sort of computer-ish and useful.  And stats IS math. (Not) sorry, Rudin luvverz!&lt;/p&gt;&#xA;&lt;p&gt;It's very hard to answer this, because we don't know the context.  Are you talking about all the foundational mathematics for these students (so basic calculus and the like)?  Or sooper high level research? (Doubt that, think you are talking bachelor's students trying to cash in on the buzzword.  But of course, some MIT Ph.D. might be using some hairy stuff and doing theory papers, needing all the high end gadgets of optimization.)&lt;/p&gt;&#xA;&lt;p&gt;If we take for granted, these are CS BS students, and they will get the normal slew of math courses for CS (calculus, linear algebra).  They are not going to want to be forced through a bunch of hairy stuff that doesn't directly apply.  There is a reason why (most of) these students are doing CS instead of physics.  ;-)&lt;/p&gt;&#xA;" OwnerUserId="16311" LastActivityDate="2021-06-17T12:05:45.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21064" PostTypeId="2" ParentId="21062" CreationDate="2021-06-17T12:49:26.383" Score="6" Body="&lt;p&gt;This may help, the labs and associated materials for a course&#xA;&lt;em&gt;CSC 294: Computational Machine Learning&lt;/em&gt;:&#xA;&lt;a href=&quot;https://github.com/comp-machine-learning-general/course-materials&quot; rel=&quot;nofollow noreferrer&quot;&gt;github link&lt;/a&gt;. See &lt;code&gt;course-materials/Labs/&lt;/code&gt;, Jupyter Notebooks:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;k-means&lt;/li&gt;&#xA;&lt;li&gt;PCA&lt;/li&gt;&#xA;&lt;li&gt;SVD&lt;/li&gt;&#xA;&lt;li&gt;k-NN&lt;/li&gt;&#xA;&lt;li&gt;Gradient descent&lt;/li&gt;&#xA;&lt;li&gt;SVM&lt;/li&gt;&#xA;&lt;li&gt;Decision trees&lt;/li&gt;&#xA;&lt;li&gt;Deep learning&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-06-17T13:33:36.160" LastActivityDate="2021-06-17T13:33:36.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21065" PostTypeId="2" ParentId="21048" CreationDate="2021-06-17T12:55:53.130" Score="12" Body="&lt;p&gt;The vector space &lt;span class=&quot;math-container&quot;&gt;$V = C^{\infty}(\mathbb{R},\mathbb{R})/\mathbb{R}[x]$&lt;/span&gt; of smooth functions modulo polynomials. Note that &lt;span class=&quot;math-container&quot;&gt;$ d/dx \colon V\to V $&lt;/span&gt; is an isomorphism, so that we have a nice inverse &lt;span class=&quot;math-container&quot;&gt;$\int \colon V \to V $&lt;/span&gt;, taking the class of a function to the class of an antiderivative. So suddenly, the indefinite integral operation is well-defined. Note that it is also an &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}[x]$&lt;/span&gt;-module, and that composition with polynomials from the right is a well-defined operation.&lt;/p&gt;&#xA;" OwnerUserId="16312" LastEditorUserId="16312" LastEditDate="2021-06-18T09:38:13.263" LastActivityDate="2021-06-18T09:38:13.263" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21067" PostTypeId="2" ParentId="21062" CreationDate="2021-06-18T01:12:57.877" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://dustingmixon.wordpress.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dustin Mixon&lt;/a&gt; at The Ohio State University has written &lt;a href=&quot;https://dustingmixon.wordpress.com/2019/01/07/math-8610-mathematics-of-data-science/&quot; rel=&quot;nofollow noreferrer&quot;&gt;rigorous notes on the Mathematics of Data Science&lt;/a&gt; that cover both &amp;quot;fundamentals&amp;quot; (matrix analysis, convex optimization, probability) and &amp;quot;applications&amp;quot; (dimensionality reduction, clustering, compressed sensing).&lt;/p&gt;&#xA;&lt;p&gt;These notes are pitched at a reasonably high graduate level, but they contain plenty of approachable material worth sharing with an undergraduate audience.&lt;/p&gt;&#xA;" OwnerUserId="3443" LastActivityDate="2021-06-18T01:12:57.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21068" PostTypeId="2" ParentId="21062" CreationDate="2021-06-18T07:08:45.883" Score="1" Body="&lt;p&gt;I am sure you will enjoy this &lt;a href=&quot;https://www.coursera.org/specializations/mathematics-machine-learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics for Machine Learning specialization&lt;/a&gt; by UCL&lt;/p&gt;&#xA;" OwnerUserId="8023" LastActivityDate="2021-06-18T07:08:45.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21069" PostTypeId="2" ParentId="978" CreationDate="2021-06-18T10:39:37.567" Score="-3" Body="&lt;p&gt;I am not a  teacher, I am a student but  according to my experience firstly we must try to create his interest in mathematics.&#xA;Every concept of mathematics is relate to daily life things and explain the concept by giving daily life example. Clear the basic concept of mathematics of students.&lt;/p&gt;&#xA;" OwnerUserId="16319" LastActivityDate="2021-06-18T10:39:37.567" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21070" PostTypeId="1" AcceptedAnswerId="21074" CreationDate="2021-06-18T15:18:08.110" Score="7" ViewCount="468" Body="&lt;p&gt;I would like to start out next year by creating a classroom community. This year I noticed a lot of burnout towards the second half of the year. Basically, I want to see what kinds of games/activities worked for people. I would like my students to develop agency because many of them come into the math classroom resigned to fail or barely pass. As an example, the first class the students could answer some basic questions about themselves and then pass the focus to the next student of their choice (rather than the instructor passing the focus). I know another teacher who starts her class by giving a 2-minute simple riddle, which allows all students to participate and be able to contribute.&lt;/p&gt;&#xA;" OwnerUserId="16321" LastActivityDate="2021-06-29T15:21:09.163" Title="Creating an Engaging Class Atmosphere" Tags="|secondary-education|mathematical-pedagogy|student-motivation|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="21071" PostTypeId="2" ParentId="978" CreationDate="2021-06-18T16:45:14.297" Score="2" Body="&lt;p&gt;But, &lt;span class=&quot;math-container&quot;&gt;$\frac{64}{16} = \frac{4}{1} = 4$&lt;/span&gt;. So...&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-06-18T16:45:14.297" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21072" PostTypeId="2" ParentId="21070" CreationDate="2021-06-18T17:40:01.113" Score="2" Body="&lt;p&gt;Rather than trying to figure out a trick to use, you might want to see what are the core activities in the classroom and what, if anything, you can do about those. The family life etc. of the pupils can also have a huge effect, especially in corona times, but there is less you can do about those.&lt;/p&gt;&#xA;&lt;p&gt;The main ideas in the literature I have met are that activities should allow the pupils to investigate and discover things and be open-ended. For stuff in English you might want to read Boaler's book on Mathematical mindsets. It is very American, but if you can read it in spite of that, it has many good ideas to this effect. Otherwise you might want to look at any recent books in the curriculums of your country's teacher training institutes.&lt;/p&gt;&#xA;&lt;p&gt;You might want to de as little explaining and lecturing as possible, while have the pupils work, alone and in groups, as much as possible. The problems should be accessible to everyone, but contain depth at the same time. Avoid routine calculations and closed problems with only a single right answer.&lt;/p&gt;&#xA;&lt;p&gt;Of course, you might already be doing all of this; in that case, please edit your question with a description of what you are doing at the moment!&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-06-18T17:40:01.113" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21074" PostTypeId="2" ParentId="21070" CreationDate="2021-06-18T19:28:18.623" Score="14" Body="&lt;p&gt;I agree with @Tommi that creating community is bigger than one activity. I do a number of things at the start of semester, but building community is also in the way I teach, every day of class. (I teach college, currently online via zoom.)&lt;/p&gt;&#xA;&lt;p&gt;You might check out &lt;a href=&quot;https://www.francissu.com/post/100-questions-about-mathematics&quot; rel=&quot;noreferrer&quot;&gt;Francis Su's blog&lt;/a&gt;, which is related to his book, &lt;em&gt;Mathematics for Human Flourishing&lt;/em&gt;. I also found the article &amp;quot;&lt;a href=&quot;https://arxiv.org/pdf/2008.02835.pdf&quot; rel=&quot;noreferrer&quot;&gt;Todxs cuentan:  building community and welcoming humanity from the first day of class&lt;/a&gt;&amp;quot;, by Federico Ardila–Mantilla, inspiring, especially his use of music. (I still haven't done that, but I definitely want to.)&lt;/p&gt;&#xA;&lt;p&gt;I think helping students understand the beauty of math is helpful, and I ask my students to read these articles:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://grantwiggins.wordpress.com/2012/12/07/math-as-artistry-an-interview-with-steve-strogatz-mathematician/&quot; rel=&quot;noreferrer&quot;&gt;Conversation between Steven Strogatz and Grant Wiggin&lt;/a&gt;s&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.newstatesman.com/sci-tech/2013/05/third-culture-power-and-glory-mathematics&quot; rel=&quot;noreferrer&quot;&gt;Article by Ian Stewart&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.maa.org/external_archive/devlin/LockhartsLament.pdf&quot; rel=&quot;noreferrer&quot;&gt;Lockhart's Lament&lt;/a&gt; (by Paul Lockhart)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In terms of equity, I recommend &lt;em&gt;Whistling Vivaldi&lt;/em&gt;, by Claude Steele. It explains the concept of stereotype threat and the research behind it. One intervention that seems small but helps a lot is to ask students to write about their values at the beginning of a course.&lt;/p&gt;&#xA;&lt;p&gt;I also think it helps when they think about how to learn math, and I ask them to read a number of articles about that, including &lt;a href=&quot;https://www.mathvalues.org/masterblog/lift-evry-voice-supporting-dvc-umoja-students-in-math&quot; rel=&quot;noreferrer&quot;&gt;this one by Jamylle Carter which emphasizes building community&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As we work together in class:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I pause often to check for understanding,&lt;/li&gt;&#xA;&lt;li&gt;I notice the spark of right thinking in wrong answers,&lt;/li&gt;&#xA;&lt;li&gt;I accept them where they are and thank them for asking questions,&lt;/li&gt;&#xA;&lt;li&gt;If they can't answer a question I've asked, I ask an easier one, and provide scaffolding via questions, to get them to where they can answer the original question.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Do these practices build community? I'm not sure. But they do build safety, which is a first step toward building community.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-06-18T19:28:18.623" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21075" PostTypeId="2" ParentId="21070" CreationDate="2021-06-18T20:07:33.457" Score="5" Body="&lt;p&gt;In the age before covid, back when we were still human, I found breaking the class into groups and having them solve about a half dozen problems on the board was moderately successful in getting them involved. You need a lot of board space and a few minutes before class to write up problems. I would give them 10-20 minutes to attempt, then I'd spend the rest of the class either affirming or denying their answers. I pepper in criticisms of notation etc. as appropriate, ideally striking a friendly posture. I don't do these quizzes all the time, and I don't do them at all in the upper level classes. Sometimes lecture is the most efficient manner to competently cover material.&lt;/p&gt;&#xA;&lt;p&gt;Another approach, keep a stack of 3x5 cards with their names and randomly select them for questions as you go. This doesn't work as well in my estimation, many kids these days are pretty introverted w.r.t. math class. It's not always wise to antagonize their shyness directly. In contrast, with the groups, everybody is getting out of their comfort zone so it's less personal.&lt;/p&gt;&#xA;&lt;p&gt;All of this said, every class is different. There is no substitute for just talking to each of them individually to try to forge some small friendship and/or understanding of why they are in your class and what are their aims in seeking higher education. ( I don't recommend this last paragraph for large lecture sections !)&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-06-18T20:07:33.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21076" PostTypeId="2" ParentId="21048" CreationDate="2021-06-18T20:24:32.463" Score="10" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; be a set, and let &lt;span class=&quot;math-container&quot;&gt;$\mathcal A$&lt;/span&gt; be an algebra of subsets of &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;.  Then &lt;span class=&quot;math-container&quot;&gt;$\mathcal A$&lt;/span&gt; is a vector space over the field &lt;span class=&quot;math-container&quot;&gt;$\mathbb F_2 = \{0,1\}$&lt;/span&gt;, with the operation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;E \mathbin{\Delta} F = (E \cup F)\setminus (E \cap F)&#xA;$$&lt;/span&gt;&#xA;as addition, and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;0E = \varnothing,\qquad 1E=E&#xA;$$&lt;/span&gt;&#xA;as scalar multiple.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Any finite-dimensional vector space over &lt;span class=&quot;math-container&quot;&gt;$\mathbb F_2$&lt;/span&gt; has cardinal &lt;span class=&quot;math-container&quot;&gt;$2^k$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is the dimension.  So we conclude any finite algebra of sets has cardinal &lt;span class=&quot;math-container&quot;&gt;$2^k$&lt;/span&gt;.  In particular, without using this try to prove that there is no algebra of sets with &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; elements.&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="205" LastEditDate="2024-03-13T21:48:44.007" LastActivityDate="2024-03-13T21:48:44.007" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21077" PostTypeId="1" AcceptedAnswerId="21081" CreationDate="2021-06-18T20:58:49.073" Score="5" ViewCount="273" Body="&lt;p&gt;This question is targeted at teachers who taught both low and high level mathematics. I have a group of students that I'm currently teaching precalculus and they seem to be doing really well in all computational areas. For example, they can all do a problem like &amp;quot;Find the partial fraction decomposition of &lt;span class=&quot;math-container&quot;&gt;$\frac{3x}{x^2-4}$&lt;/span&gt;&amp;quot; or &amp;quot;Compute &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{x\to\infty}\frac{x^3+2e^x}{3-e^x}$&lt;/span&gt;&amp;quot;. However, as soon as we hit abstract notions, they fall apart. We just reached Intermediate Value Theorem and they just can't seem to make the leap into abstraction. Consider a multiple choice question:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; be a continuous function and let &lt;span class=&quot;math-container&quot;&gt;$f(1) = 3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(3)=11$&lt;/span&gt;. Which of the below statements must be true?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is increasing on the interval &lt;span class=&quot;math-container&quot;&gt;$(1,3)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(2)=7$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x) = 5$&lt;/span&gt; for some value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; has a local maximum in the interval &lt;span class=&quot;math-container&quot;&gt;$(1,3)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The majority of the students chose 1 and only a few chose the correct answer 3. We did 3 or 4 similar exercises and I explained each one (or so I thought). By the time we did a similar question #4, there was only a moderate improvement. Can someone suggest how to help students make a leap into the abstraction? How do you teach abstract concepts to students who have only encountered computational problems in their math careers?&lt;/p&gt;&#xA;" OwnerUserId="16321" LastActivityDate="2021-06-18T23:11:05.427" Title="Making the leap from Pre-Calculus to Calculus" Tags="|secondary-education|mathematical-pedagogy|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21078" PostTypeId="2" ParentId="20910" CreationDate="2021-06-18T21:31:34.273" Score="-1" Body="&lt;p&gt;A good area to look at would be colleges that do &amp;quot;one course at a time&amp;quot;.  (Also, at summer schools.)&lt;/p&gt;&#xA;&lt;p&gt;See for example, this link:  &lt;a href=&quot;https://www.cornellcollege.edu/one-course-at-a-time/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cornellcollege.edu/one-course-at-a-time/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16311" LastEditorUserId="15884" LastEditDate="2023-01-30T13:37:45.343" LastActivityDate="2023-01-30T13:37:45.343" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21079" PostTypeId="2" ParentId="20925" CreationDate="2021-06-18T21:37:06.643" Score="3" Body="&lt;p&gt;Simplify first.  Then do the problem by long division.  With a number of such problems, the student will quickly realize that 2/9 gives .222... and 4/9 gives .4444 and the like.  It's actually a good introduction to such stuff.  But make them grind the long division first, several times, to see where it all comes from.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;times .111...&amp;quot; seems cute and appeals to the mathy sorts here, but I don't think it is as foundation building (for a neophyte) as cranking the long division a few times and then realizing all the x/9 (x from 1 to 8), end up being .xxx.  (This is a pedagical, psychological issue, not a math concept.)&lt;/p&gt;&#xA;" OwnerUserId="16311" LastActivityDate="2021-06-18T21:37:06.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21081" PostTypeId="2" ParentId="21077" CreationDate="2021-06-18T23:11:05.427" Score="11" Body="&lt;p&gt;Here's one quick idea of an exercise that might shake students out of their narrow thinking.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Give the students three coordinate grids with (1,3) and (3,11) filled in with closed circles.  (I'll use the numbers from your example, but I'd probably choose better numbers if I were doing this for real.)&lt;/li&gt;&#xA;&lt;li&gt;In the first grid, ask the students to graph a continuous function on [1,3] that goes through the two points.&lt;/li&gt;&#xA;&lt;li&gt;Then, in the second grid, ask them to graph a DIFFERENT continuous function that goes through the two points.&lt;/li&gt;&#xA;&lt;li&gt;Then have them look over the four multiple choice options and see how many of the options are satisfied by both of their graphs.&lt;/li&gt;&#xA;&lt;li&gt;In the third grid, graph a continuous function that goes through the two points such that only one of the four conditions is satisfied by all three of their graphs.&lt;/li&gt;&#xA;&lt;li&gt;Have the students pair up, describe what they did, and then hold a class-wide conversation where people shared what their pairs talked about.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-06-18T23:11:05.427" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21082" PostTypeId="2" ParentId="20925" CreationDate="2021-06-19T01:33:09.457" Score="0" Body="&lt;p&gt;If the decimal expansion of &lt;span class=&quot;math-container&quot;&gt;$\frac13$&lt;/span&gt; is known (as in Method 2), then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac6{27}=\frac29=\left(\frac23\right)/3=(0.\overline{66})/3=0.\overline{22}.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2021-06-19T01:33:09.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21083" PostTypeId="2" ParentId="21048" CreationDate="2021-06-19T08:23:10.270" Score="3" Body="&lt;p&gt;Expanding on the answer by James Cook, the whole zoo of spaces in poly-linear algebra:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the dual space &lt;span class=&quot;math-container&quot;&gt;$V^\star$&lt;/span&gt; of a given space;&lt;/li&gt;&#xA;&lt;li&gt;the space &lt;span class=&quot;math-container&quot;&gt;$V^\star\otimes V^\star$&lt;/span&gt; of all bilinear forms;&lt;/li&gt;&#xA;&lt;li&gt;the space &lt;span class=&quot;math-container&quot;&gt;$V^\star\otimes W$&lt;/span&gt; of all linear operators from &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt;;&lt;/li&gt;&#xA;&lt;li&gt;for &lt;span class=&quot;math-container&quot;&gt;$V=W$&lt;/span&gt;, the subspaces of symmetric and anti-symmetric operators;&lt;/li&gt;&#xA;&lt;li&gt;the spaces &lt;span class=&quot;math-container&quot;&gt;$\Lambda^k(V)$&lt;/span&gt; of anti-symmetric bilinear &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-forms on &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The advantage of these examples is functoriality; i. e., a linear operator on &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; gives rise to linear operators on &lt;span class=&quot;math-container&quot;&gt;$V^\star,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\Lambda^k(V)$&lt;/span&gt;, etc., and this respects composition. From this observation, we readily get a coordinate-free definition of determinant, with the property that the determinant of a product is a product of determinants being automatic.&lt;/p&gt;&#xA;&lt;p&gt;On a more elementary note, it is good to at some point introduce the students to the idea that the rows and columns of a matrix need not be indexed by integers (and in particular, that the rows and columns of an &lt;span class=&quot;math-container&quot;&gt;$n\times n$&lt;/span&gt; matrix may be indexed by &lt;em&gt;different&lt;/em&gt; sets.) This conveys some idea of usefulness of abstraction: while we can enumerate any finite set, this enumeration is arbitrary, and so is identification to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;. For an example, take an adjacency matrix of a graph, or a transition matrix of a Markov chain.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2021-06-19T08:23:10.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21084" PostTypeId="1" AcceptedAnswerId="21086" CreationDate="2021-06-19T09:01:30.950" Score="4" ViewCount="512" Body="&lt;p&gt;I'm preparing a holiday class in computer graphics. The class will be held in English. I'm a French speaker and I'm fighting with some words which have lots of meanings to find the right one in the context of maths or physics.&lt;/p&gt;&#xA;&lt;p&gt;The current one is &amp;quot;&lt;em&gt;repère&lt;/em&gt;&amp;quot; (an origin, and in my case 2 or 3 basis vectors) in the context of &amp;quot;&lt;em&gt;changement de repère&lt;/em&gt;&amp;quot;. The translation for &amp;quot;&lt;em&gt;repère&lt;/em&gt;&amp;quot; that sounds to be the best so far is &amp;quot;&lt;em&gt;frame&lt;/em&gt;&amp;quot;, but is that right? Can we  then speak about &amp;quot;&lt;em&gt;frame shift&lt;/em&gt;&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="16328" LastEditorUserId="376" LastEditDate="2021-06-19T10:21:44.653" LastActivityDate="2021-06-19T23:26:17.940" Title="What is the English word for the French &quot;repère&quot;?" Tags="|terminology|linear-algebra|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21085" PostTypeId="2" ParentId="21084" CreationDate="2021-06-19T10:11:44.760" Score="5" Body="&lt;p&gt;The set of basis vectors is just called the basis in English, and so your topic is usually called &lt;a href=&quot;https://en.wikipedia.org/wiki/Change_of_basis&quot; rel=&quot;noreferrer&quot;&gt;change of basis&lt;/a&gt;. Google is telling me that &amp;quot;repère&amp;quot; is translated as &amp;quot;coordinates&amp;quot;, which would be the family of scalar variables that are used to define each point in the space in terms of the basis vectors.&lt;/p&gt;&#xA;&lt;p&gt;Bonne chance!&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-06-19T10:11:44.760" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21086" PostTypeId="2" ParentId="21084" CreationDate="2021-06-19T16:24:18.143" Score="5" Body="&lt;p&gt;Elie Cartan wrote a paper in 1935 that was foundational for differential geometry. It is titled&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;La Méthode de Repère Mobile, La Théorie des Groupes Continus, et Les Espaces Généralisés&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In English, we often refer to the &amp;quot;method of moving &lt;em&gt;frames&lt;/em&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-41/issue-11/Review--Elie-Cartan-La-M%C3%A9thode-de-Rep%C3%A8re-Mobile-La/bams/1183498480.full?tab=ArticleFirstPage&quot; rel=&quot;noreferrer&quot;&gt;Here is a link to a review of the paper&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-06-19T16:24:18.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21087" PostTypeId="2" ParentId="20959" CreationDate="2021-06-19T18:52:17.040" Score="1" Body="&lt;p&gt;Because the 2/3-dimensional visualisations are equally unnecessary. Some mathematicians who naturally think along visual lines find them a useful way to structure their thoughts, but (for mathematicians not still in the &lt;a href=&quot;https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;pre-formal&lt;/a&gt; stage of their development), it's just a tool to help them reason about some more formal system. Other mathematicians do not use such tools (for example, those of us with aphantasia certainly do not), and so the visualisation cannot be essential to the process.&lt;/p&gt;&#xA;" OwnerUserId="11771" LastActivityDate="2021-06-19T18:52:17.040" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="21088" PostTypeId="2" ParentId="21048" CreationDate="2021-06-19T19:31:08.307" Score="6" Body="&lt;p&gt;This is the example that made me realize the importance of abstract linear algebra as an undergraduate: the edge spaces and the vertex space of a graph (&lt;a href=&quot;https://en.wikipedia.org/wiki/Edge_and_vertex_spaces&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Edge_and_vertex_spaces&lt;/a&gt;).  The edge graph of a graph &lt;span class=&quot;math-container&quot;&gt;$(V, E)$&lt;/span&gt; is the vector space of functions from &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; to the 2-element field.  The vertex space of a graph is defined likewise.  These spaces actually have applications in graph theory. A purely graph-theoretic notion such as existence of a cycle has a corresponding linear algebraic notion in terms of linear endomorphisms, and sometimes useful in solving graph-theoretic problems.  (Of course, these may be regarded as a special case of other examples.)&lt;/p&gt;&#xA;" OwnerUserId="16332" LastEditorUserId="376" LastEditDate="2024-03-09T06:52:16.737" LastActivityDate="2024-03-09T06:52:16.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21089" PostTypeId="2" ParentId="20953" CreationDate="2021-06-19T19:38:08.020" Score="1" Body="&lt;p&gt;If you wish to assess individual capabilities, use an individual assignment. Group projects are fundamentally and unavoidably terrible at doing so. Use group assignments, instead, to encourage collaboration - this is, you may notice, completely incompatible with your suggested marking system, which actively encourages antagonistic behaviours towards other group members.&lt;/p&gt;&#xA;&lt;p&gt;As far as actual methods of assessing group projects goes, I'd suggest just... not. If for some reason you actually must have a number next to it in some system, grade on completion. If not, just don't write any numbers down. Give verbal feedback to the group, of course, and I suppose you could give them some sort of numerical or letter grade if you really wanted, but I'm not convinced that there's any particular benefit to doing so, and I certainly wouldn't write that number down anywhere (at least not anywhere that matters).&lt;/p&gt;&#xA;" OwnerUserId="11771" LastActivityDate="2021-06-19T19:38:08.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21090" PostTypeId="2" ParentId="21084" CreationDate="2021-06-19T23:26:17.940" Score="1" Body="&lt;p&gt;Yes, (ignoring diacritical marks) Cartan's &amp;quot;repere mobile&amp;quot; is &amp;quot;moving frame&amp;quot;, but that refers to a more complicated idea than I think will come up in basic linear algebra. In basic linear algebra, it would make sense to just say &amp;quot;frame&amp;quot;, rather than &amp;quot;moving frame&amp;quot;, but/and the plain &amp;quot;frame&amp;quot; is better translated to contemporary English as &amp;quot;coordinate system&amp;quot;, I think&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2021-06-19T23:26:17.940" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21091" PostTypeId="2" ParentId="176" CreationDate="2021-06-19T23:32:31.910" Score="0" Body="&lt;p&gt;Because knowing mathematics is mostly about gls, whereas teaching mathematics is mostly about mls.&lt;/p&gt;&#xA;&lt;p&gt;The terms ‘general life situation’ (gls) and ‘momentary life situation’ (mls) were introduced by Kurt Lewin, founder of Social Psychology, in his classic treatise ‘Principles of Topological Psychology’.&lt;/p&gt;&#xA;" OwnerDisplayName="user10552" LastActivityDate="2021-06-19T23:32:31.910" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="21092" PostTypeId="2" ParentId="14601" CreationDate="2021-06-19T23:48:18.830" Score="3" Body="&lt;p&gt;The Jordan Curve Theorem is the canonical example of this.&lt;/p&gt;&#xA;&lt;p&gt;Also, the fact that the sum of the first n odd numbers is n squared is obvious from the well-known ‘wrapping’ diagram, but the actual proof is via mathematical induction, which is a challenge for many students.&lt;/p&gt;&#xA;&lt;p&gt;Also: The theorem on sphere-packing, long time in coming, merely confirmed what what obvious to everyone. (“While the Kepler conjecture is intuitively obvious, the proof remained surprisingly elusive.” – Wolfram MathWorld)&lt;/p&gt;&#xA;&lt;p&gt;Also: The maximum possible area bounded by a curve of fixed length is a circle. (Dido’s problem)&lt;/p&gt;&#xA;&lt;p&gt;Also: There are 3 regular tessellations.&lt;/p&gt;&#xA;&lt;p&gt;Also: The base angles of an isosceles triangle are equal.&lt;/p&gt;&#xA;&lt;p&gt;Also: A cannon ball travels farthest when fired at a 45-degree angle (ignoring air resistance).&lt;/p&gt;&#xA;&lt;p&gt;and also perhaps:&#xA;Every bounded infinite set of numbers has an accumulation point. (Bolzano-Weierstrass theorem)&lt;/p&gt;&#xA;&lt;p&gt;and also perhaps:&#xA;If there is an injective mapping from A to B, and an injective mapping from B to A, then there is a bijective mapping between them. (Schrӧder-Bernstein theorem)&lt;/p&gt;&#xA;&lt;p&gt;and also perhaps:&#xA;A continuous function defined on a closed interval attains a maximum value. (high-point theorem)&lt;/p&gt;&#xA;" OwnerDisplayName="user10552" LastEditorDisplayName="user10552" LastEditDate="2021-06-20T18:10:30.780" LastActivityDate="2021-06-20T18:10:30.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21093" PostTypeId="2" ParentId="14601" CreationDate="2021-06-20T00:16:06.000" Score="0" Body="&lt;p&gt;Belatedly:&lt;/p&gt;&#xA;&lt;p&gt;It is my considered opinion that a reasonably refined &amp;quot;common sense&amp;quot; is rarely &lt;em&gt;truly&lt;/em&gt; wrong, with regard to &lt;em&gt;meaningful&lt;/em&gt; mathematical questions. Perhaps wrong on some technical disqualifications... that's easy enough.&lt;/p&gt;&#xA;&lt;p&gt;To my observation, a main reason that mathematics is useful, and clarifying about (human interaction with) the world, is that it is a good narrative for accurate narratives about the world...&lt;/p&gt;&#xA;&lt;p&gt;The absolutely amazing thing is that in many cases we can somehow see that, in the language of that narrative, no other outcome was possible. The language of the narrative constrained reality? Whoa. How often does that happen?&lt;/p&gt;&#xA;&lt;p&gt;:)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2021-06-20T00:16:06.000" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21094" PostTypeId="1" CreationDate="2021-06-20T20:51:59.810" Score="0" ViewCount="277" Body="&lt;p&gt;I came to know about Gilbert Strang's two books, &amp;quot;Introduction to Linear Algebra&amp;quot; and &amp;quot;Linear Algebra and its Applications&amp;quot;. The first is the one used as the text in the 18.06 linear algebra MIT course. I have almost all the videos of 18.06 which deal with basics linear algebra (like left out the portions of Markov matrices, differential equations, Fourier series, etc). Though Gilbert Strang's videos are awesome and highly intuitive, it does not define many things, from which I have encountered questions like: Block determinants, rank, nullity theorem, the general form of the characteristic equation (using the concept that rth co-efficient is the sum of the principal minors of order r), principal minors, and many other small pieces which can be used to answer MCQ question in competitive examinations.&lt;/p&gt;&#xA;&lt;p&gt;Though the second book, has concepts like rank, nullity, etc, but even there, many concepts which I need are missing, and it is more inclined towards application stuff.&lt;/p&gt;&#xA;&lt;p&gt;Please can anyone recommend me an intuitive yet comprehensive and easily readable (student-friendly) book on Linear Algebra which do not focus much on applications, just basics&lt;/p&gt;&#xA;" OwnerUserId="14166" LastEditorDisplayName="user507" LastEditDate="2021-06-21T14:16:25.197" LastActivityDate="2021-06-22T05:14:23.550" Title="Highly intuitive yet comprehensive and easily readable (student friendly) book on linear algebra which do not focus much on applications, just basics" Tags="|reference-request|textbooks|self-learning|linear-algebra|reference-books|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="21095" PostTypeId="1" CreationDate="2021-06-21T12:33:14.347" Score="7" ViewCount="339" Body="&lt;h3&gt;Background&lt;/h3&gt;&#xA;&lt;p&gt;I have recently started tutoring a 14-year-old student who is diagnosed with ADHD and was struggling with passing a basic math course (think basic linear equations systems, first and second degree equations, evaluating polynomials, Pythagoras theorem, etc.).&lt;/p&gt;&#xA;&lt;p&gt;For reasons not relevant here, the student is not under any medication, so it's hard for her to concentrate at times. I manage to keep our classes engaging, give her breaks when it's obvious she needs them, keep homework light, etc. I believe she suffers from low selfsteem when it comes to studies, so I am very encouraging, always frame criticism positively, highlight what she did right even if she got the answer wrong, etc.&lt;/p&gt;&#xA;&lt;p&gt;She understands what the goal of the exercises are and the rules to operate within the exercises, but will often &amp;quot;forget&amp;quot; through an exercise (e.g. get the sign wrong when multiplying two integers; or completely butcher the process of solving a system of linear equations when the day before she knew the method well).&lt;/p&gt;&#xA;&lt;p&gt;I have noticed that those errors are less frequent the more she practices one kind of exercise (for example, she would get signs wrong all the time when first working on linear equations, and barely gets them wrong now after a few weeks of practice). So I think practice and consistency is key for her, but it is obvious that she cannot &amp;quot;grind&amp;quot; them for hours (or even one hour) as a neurotypical person might be able to.&lt;/p&gt;&#xA;&lt;h3&gt;The situation&lt;/h3&gt;&#xA;&lt;p&gt;Her grades went up, but she still failed the course, so she needs to retake the exam in September.&lt;/p&gt;&#xA;&lt;p&gt;I have identified the following issues:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;She often skips steps when doing something, often leading to numerical errors she wouldn't make otherwise.&lt;/li&gt;&#xA;&lt;li&gt;She won't double check an exercise is correct before considering it &amp;quot;finished&amp;quot;, leading to errors which she would probably detect otherwise.&lt;/li&gt;&#xA;&lt;li&gt;She sometimes loses focus and considers an exercise &amp;quot;finished&amp;quot; before actually finishing it (for example, not calculating the value actually being asked but stopping one step before).&lt;/li&gt;&#xA;&lt;li&gt;She is lacking sufficient practice in many areas of the subject, so sometimes doubts what is to be done when it should be automatic at this stage.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I believe all of them can be improved with practice, by making it a habit to double check results (assisted by a calculator), writing down what is being asked and not considering the exercise done until one can draw a rectangle around the answer, etc. Furthermore, it is clear that practice cannot come from 4 days of intense study, as she simply lacks the capacity to sit down and work on something for extended periods of time.&lt;/p&gt;&#xA;&lt;h3&gt;My idea&lt;/h3&gt;&#xA;&lt;p&gt;I thought it might be good to have her do a couple of exercises a day under &amp;quot;exam conditions&amp;quot; (meaning she is expected to deliver her very best work, and particularly double check the answers). These exercises should not take more than 30 minutes in total, so she could fit them whenever she wants through her completely-free summer day. By the end of summer, she should have more than enough practice with each kind of exercise.&lt;/p&gt;&#xA;&lt;p&gt;My problem with this is that I have seen what happens when she is &amp;quot;forced&amp;quot; to do homework when she doesn't feel like it: she delivers crap (knowing it is crap), to get her parents off her back. That's a waste of time for everyone involved (particularly me, who need to write feedback which I know she doesn't really need, as she knows that's not how you approach the exercise anyway).&lt;/p&gt;&#xA;&lt;p&gt;I really want to change the focus from &amp;quot;Your task is to get this over with as soon as possible&amp;quot; to &amp;quot;Your task is to do this right&amp;quot;. I can't think of a way of doing that, though. Measures like &amp;quot;If you the exercise wrong I will have your parents make you re-do it&amp;quot; sort of defy the purpose of not having the exercises become a drag, plus I know she can't do good work even if she wants to after a while. Plus I'd really like to not have it become a situation in which she is monitored by her parents and feels more pressure, as I really want her to not hate the whole experience.&lt;/p&gt;&#xA;&lt;h3&gt;My question&lt;/h3&gt;&#xA;&lt;p&gt;What would be a good way of getting someone like the student I've described actually get work done and get into good habits when solving exercises (not skipping steps, making it clear what the answer to the problem is, etc.)? It's not so much a matter of motivation or not understanding the course work, just inability to actually do the required grinding the way other people might be able to.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;NOTE:&lt;/strong&gt; I am aware this might be offtopic, but it's literally an issue I encountered as a mathematics educator, and I couldn't find a more suitable community. I also think the answer does depend on the subject being math, as it allows for a different method of studying the syllabus compared to other subjects.&lt;/p&gt;&#xA;" OwnerUserId="16341" LastEditorUserId="16341" LastEditDate="2021-06-21T17:42:32.827" LastActivityDate="2021-10-05T21:10:27.123" Title="How to get student with ADHD to practice exercises consistently" Tags="|secondary-education|spain|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="21096" PostTypeId="2" ParentId="21094" CreationDate="2021-06-21T14:37:50.380" Score="1" Body="&lt;p&gt;Despite the last word in the name, you might consider Bernard Kolman's Linear Algebra with Applications.  Most of it is not applications and you can ignore the parts that are (tending to be at the ends of chapters or exercise sessions), and just get help with the basics.&lt;/p&gt;&#xA;&lt;p&gt;FWIW, I find it incredibly easy and gentle.  Almost entirely requires no calculus, for instance.  Will get you comfortable with the basic terms/concepts in &amp;quot;mattress theory&amp;quot; before moving on to those tougher books that all the smart people on the Internets push.&lt;/p&gt;&#xA;&lt;p&gt;It's an older book--I have 3rd edition, 1984.  You should be able to get a cheap copy from a used bookseller.&lt;/p&gt;&#xA;" OwnerUserId="16342" LastActivityDate="2021-06-21T14:37:50.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21097" PostTypeId="2" ParentId="21094" CreationDate="2021-06-21T23:54:33.603" Score="1" Body="&lt;p&gt;Perhaps take a look at &lt;a href=&quot;https://www.math.brown.edu/streil/papers/LADW/LADW.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Linear Algebra Done Wrong&amp;quot;&lt;/a&gt; by Treil. Free PDF download, so you can take a thorough look. Others swear by Axler's &lt;a href=&quot;https://linear.axler.net&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Linear Algebra Done Right&amp;quot;&lt;/a&gt;, to which the previous is a reaction, but this one isn't available for free.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2021-06-21T23:54:33.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21098" PostTypeId="2" ParentId="21094" CreationDate="2021-06-22T05:14:23.550" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://archive.org/details/HalmosP.R.FiniteDimensionalVectorSpaces.SpringerVerlag205s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Finite Dimensional Vector Spaces&lt;/a&gt; by &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Halmos/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Halmos&lt;/a&gt; is worth the read.  This was picked up by &lt;a href=&quot;https://store.doverpublications.com/by-subject-mathematics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dover&lt;/a&gt; and a physical copy should be relatively inexpensive.&lt;/p&gt;&#xA;" OwnerUserId="3443" LastActivityDate="2021-06-22T05:14:23.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21099" PostTypeId="2" ParentId="21048" CreationDate="2021-06-22T07:37:10.583" Score="1" Body="&lt;p&gt;Something else that I haven't seen explicitly:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; be a field. A &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-algebra &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; with ring homomorphism &lt;span class=&quot;math-container&quot;&gt;$\phi:k\to A$&lt;/span&gt; is a &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-vector space with scalar multiplication:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lambda \cdot a=\phi(\lambda)\cdot a$$&lt;/span&gt;&#xA;(where &lt;span class=&quot;math-container&quot;&gt;$\lambda\in k, a\in A$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;A quite cool example:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; be the following ring (known as the Weyl Algebra):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A=\mathbb{C}\langle x,y\rangle/\langle yx-xy-1\rangle$$&lt;/span&gt;&#xA;If you haven't seen this before, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}\langle x,y\rangle$&lt;/span&gt; is the free algebra with basis in letters &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt;. This is similiar-ish to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}[x,y]$&lt;/span&gt; except that the variables don't commute:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$xy\ne yx$$&lt;/span&gt;&#xA;You can have something like:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(3i)xy+\sqrt{2}y^2 x- yxy+\cdots$$&lt;/span&gt;&#xA;Anyway, by quotienting by the free algebra generated by &lt;span class=&quot;math-container&quot;&gt;$yx-xy-1$&lt;/span&gt; we have the relation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$yx=xy+1$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;There's some fun stuff in there.&lt;/p&gt;&#xA;&lt;p&gt;Algebras like these are closely related to differential operators, and differential equations, and some have shed light on certain partial differential equations. So what be seen as weird, has really important application.&lt;/p&gt;&#xA;" OwnerUserId="10662" LastActivityDate="2021-06-22T07:37:10.583" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21100" PostTypeId="1" CreationDate="2021-06-22T08:48:38.090" Score="2" ViewCount="362" Body="&lt;p&gt;Peter Liljedahl discusses a lot about &amp;quot;&lt;a href=&quot;https://www.edutopia.org/article/building-thinking-classroom-math&quot; rel=&quot;nofollow noreferrer&quot;&gt;thinking classrooms&lt;/a&gt;&amp;quot;. There is research and books, which show, that his methods transform classes from &amp;quot;non-thinking&amp;quot; to &amp;quot;thinking&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;If that is true, there should be evidence in grade improvement, like, higher final grades compared to control classes, or higher STEM participation.&lt;/p&gt;&#xA;&lt;p&gt;What is the evidence for this?&lt;/p&gt;&#xA;" OwnerUserId="14709" LastEditorDisplayName="user507" LastEditDate="2021-06-22T12:46:23.760" LastActivityDate="2021-06-22T12:46:23.760" Title="Peter Liljedahl &quot;Thinking Classroom&quot; pedagogy: what is evidence for its effectiveness?" Tags="|grading|general-pedagogy|classroom-management|classroom-interaction|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="21101" PostTypeId="2" ParentId="21048" CreationDate="2021-06-22T12:14:06.257" Score="4" Body="&lt;p&gt;Probably not considered mathematics but I think still useful for students:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The space of possible velocities a particle in space can have&lt;/li&gt;&#xA;&lt;li&gt;The space of possible angular momenta a rigid body in space can have&lt;/li&gt;&#xA;&lt;li&gt;The space of forces that can act on a particle in space&lt;/li&gt;&#xA;&lt;li&gt;...&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These are abstract vector spaces in the sense that they are not supplied with a canonical basis and hence not canonically isomorphic to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;. These spaces are are also not canonically isomorphic to each other (unless someone believes that SI units are canonical.)&lt;/p&gt;&#xA;&lt;p&gt;In general physical quantities deliver many more examples and I find it instructive to also mention one dimensional examples to students, like electrical charge.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2021-06-22T12:14:06.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21102" PostTypeId="1" AcceptedAnswerId="21104" CreationDate="2021-06-23T01:06:42.290" Score="5" ViewCount="650" Body="&lt;p&gt;When I was an undergrad studying abstract algebra, we used the second edition of Artin and covered groups first and then rings. Fields, vector spaces, and algebras came later, I think.&lt;/p&gt;&#xA;&lt;p&gt;I remember finding group theory difficult because of the number of concepts (cosets, group actions, &amp;amp;c) and the large amounts of only-slightly-different notations for things like orbits and stabilizers. Another thing I remember pretty clearly is being confused by the different meanings of juxtaposition in different contexts (e.g. &lt;span class=&quot;math-container&quot;&gt;$gHg^{-1}$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$ghg^{-1}$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$gX$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is a set).&lt;/p&gt;&#xA;&lt;p&gt;I'm wondering whether there are any books or perhaps courses that don't start with either groups or rings and instead pick something more unusual. I'm not a teacher; my interest in this is entirely theoretical.&lt;/p&gt;&#xA;&lt;p&gt;Based on &lt;a href=&quot;https://matheducators.stackexchange.com/questions/10478/rings-before-groups-in-abstract-algebra&quot;&gt;this question&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/19526/rings-in-parallel-with-groups-in-abstract-algebra&quot;&gt;this question&lt;/a&gt;, it seems like teaching groups or rings first is popular, with rings being firmly in second place.&lt;/p&gt;&#xA;&lt;p&gt;There are other things though that are perhaps reasonable candidates for an introductory algebraic structure.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Fields -- there are a few ready examples, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt;, and the finite fields. And there's cool stuff like &lt;span class=&quot;math-container&quot;&gt;$\mathbb{H}$&lt;/span&gt; if we're willing to consider skew fields.&lt;/li&gt;&#xA;&lt;li&gt;Vector spaces -- row/column vectors with matrices as linear maps and some other examples should be familiar from linear algebra, which is usually taught first.&lt;/li&gt;&#xA;&lt;li&gt;Lattices -- lattices are reasonably straightfoward (in my opinion)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And there are some other things that are weird, but also perhaps reasonable choices.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Monoids or semigroups&lt;/li&gt;&#xA;&lt;li&gt;Semirings -- rings without negation.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13521" LastEditorUserId="13521" LastEditDate="2021-06-23T18:22:10.180" LastActivityDate="2021-06-23T18:22:10.180" Title="Undergraduate-level abstract algebra books or courses that don't start with groups or rings" Tags="|undergraduate-education|abstract-algebra|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21104" PostTypeId="2" ParentId="21102" CreationDate="2021-06-23T06:13:33.030" Score="8" Body="&lt;p&gt;Not sure if this counts for the OP's purpose, but I started with Hungerford's &lt;em&gt;Abstract Algebra: An Introduction&lt;/em&gt;. The first 3 chapters go like this:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Arithmetic in &lt;span class=&quot;math-container&quot;&gt;$\mathbb  Z$&lt;/span&gt; revisited&lt;/li&gt;&#xA;&lt;li&gt;Congruence in &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z$&lt;/span&gt; and modular arithmetic (i.e., &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z_n$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Rings&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Etc. The motivating thesis is explicitly laid out in the Preface and a Thematic Table of Contents -- that it's better to go from well-known concepts to more abstract in fairly small gradual steps (i.e., the reverse of OP's experience with Artin).&lt;/p&gt;&#xA;&lt;p&gt;Ironically, I now look at a book that starts with groups enviously for its brevity and elegance (rings as a combination of two groups plus distribution), but it seems pretty easy to believe that Hungerford's approach is less of a shock-treatment to students.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-06-23T16:28:56.437" LastActivityDate="2021-06-23T16:28:56.437" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21105" PostTypeId="2" ParentId="21102" CreationDate="2021-06-23T12:32:09.277" Score="4" Body="&lt;p&gt;A nice example is Paolo Aluffi's &lt;a href=&quot;http://dx.doi.org/10.1090/gsm/104&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Algebra: Chapter 0&lt;/em&gt;&lt;/a&gt;. The first chapter (40 pages) is &lt;em&gt;Preliminaries: Set theory and categories&lt;/em&gt;. Groups and rings follow in the next three chapters.&lt;/p&gt;&#xA;&lt;p&gt;This book is special as it follows a &lt;em&gt;cathegorically minded viewpoint&lt;/em&gt; (in the author's words), which I like very much. According to the introduction, the book is suitable for upper-level undergraduate or beginning graduate courses.&lt;/p&gt;&#xA;" OwnerUserId="15367" LastActivityDate="2021-06-23T12:32:09.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21108" PostTypeId="1" CreationDate="2021-06-24T13:08:36.623" Score="1" ViewCount="289" Body="&lt;p&gt;I have been examining sites that illustrate the use of the block model for solving problems. I am an adult with a degree in math and I have an interest in seeing how math can be taught.&lt;/p&gt;&#xA;&lt;p&gt;From what I have seen, block modeling can be a useful introduction to algebra. The blocks provide a visual way of working with unknown variables. However, there are cases where the illustrated use of block models requires more ingenuity and abstract reasoning than simple use of algebra.&lt;/p&gt;&#xA;&lt;p&gt;I saw one problem given as follows: The ratio of money between A and B is &lt;span class=&quot;math-container&quot;&gt;$5:4.$&lt;/span&gt; If each of A and B spends &lt;span class=&quot;math-container&quot;&gt;$\$45,$&lt;/span&gt; then the ratio of the money they have left is &lt;span class=&quot;math-container&quot;&gt;$13:10.$&lt;/span&gt; How much money did each have originally?&lt;/p&gt;&#xA;&lt;p&gt;This is the solution that was given. To solve the problem, observe that &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; is one greater than &lt;span class=&quot;math-container&quot;&gt;$4.$&lt;/span&gt; If we multiply both &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$3,$&lt;/span&gt; giving &lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$12,$&lt;/span&gt; we now have the same ratio but now we have a difference of &lt;span class=&quot;math-container&quot;&gt;$3.$&lt;/span&gt; This allows us to form &lt;span class=&quot;math-container&quot;&gt;$13$&lt;/span&gt; out of the &lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; out of the &lt;span class=&quot;math-container&quot;&gt;$12,$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; left over for each. This common difference of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; blocks represent &lt;span class=&quot;math-container&quot;&gt;$\$45,$&lt;/span&gt; so each block is &lt;span class=&quot;math-container&quot;&gt;$\$22.50,$&lt;/span&gt; so A and B started respectively with &lt;span class=&quot;math-container&quot;&gt;$15 \times \$22.50$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$12 \times \$22.50.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Anyone able to think of that would have no problem parameterizing the original values as \$5x and \$4x and solving (5x - 45)/(4x - 45) = 13/10.&lt;/p&gt;&#xA;&lt;p&gt;Here is an example of the Singapore block model that I think works well. A person spends 1/6 of the money in their wallet and has \$15 dollars left over.  How much did they start with?  Create 6 blocks. Color one to represent the 1/6 that was spent.  The remaining 5 blocks must add to \$15, so each block is \$3. Add the \$3 in the colored block to the \$15 to get \$18. Someone learning algebra may have a little trouble setting up the equation even though it is a relatively simple problem.  The block model makes it easy to see how everything relates.&lt;/p&gt;&#xA;" OwnerUserId="8590" LastEditorUserId="8590" LastEditDate="2021-06-25T17:07:29.400" LastActivityDate="2021-07-06T10:48:27.893" Title="Is there a book or Web site that explains how to use and teach the Singapore math block model?" Tags="|mathematical-pedagogy|" AnswerCount="2" CommentCount="4" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21110" PostTypeId="2" ParentId="21070" CreationDate="2021-06-25T18:36:31.010" Score="3" Body="&lt;p&gt;There's a lot of good answers here - just to add one tactic that's worked well for me, consider adding low-stakes assignments that allow students to use skills they don't ordinarily think of as part of a math class. In one of my classes this last year, I included a weekly discussion post as part of the class, usually centered around a philosophical question related to what we were doing in class that week; one of the earliest discussions was about the question of whether math is something that is &amp;quot;discovered&amp;quot; or &amp;quot;invented&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Since this wasn't a philosophy class, the assignment wasn't graded in any strict way - if a student wrote a post that showed engagement, and replied to someone else's post in a way that showed engagement, they received full points. And the discussions were, overall, a very tiny portion of the course grade. But what I noticed was that, apparently as a consequence, I had many more of the less-mathematically-comfortable students bringing their whole selves to class - students who were used to thinking &amp;quot;oh, I'm really a literature person, there's nothing for me to offer in a math class&amp;quot; (or similar) had started to feel like there really was something for them to contribute. The result was a much more energetic and engaged atmosphere than I usually see - even though we were meeting over Zoom!&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2021-06-25T18:36:31.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21112" PostTypeId="1" AcceptedAnswerId="21113" CreationDate="2021-06-27T20:27:50.653" Score="9" ViewCount="785" Body="&lt;p&gt;From &lt;a href=&quot;http://lchc.ucsd.edu/MCA/Mail/xmcamail.2011_06.dir/pdfn8bJa6mo4c.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Development of Algebra in the Elementary Mathematics Curriculum of V.V. Davydov&lt;/a&gt; (by&lt;br /&gt;&#xA;Jean Schmittau, State University of New York at Binghamton&#xA;, and Anne Morris , University of Delaware ) one briefly understands about the mathematics curriculum Davydov prepared for Russian school:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The  curriculum itself (which Schmittau used as written in the&#xA;implementation with US  students) consists of nothing but a carefully&#xA;developed sequence of problems, which  children are expected to solve.&#xA;The problems are not broken down into steps for the  children, they&#xA;are not given hints, and there is no didactic presentation of the&#xA;material. There is nothing to read but one problem after another. The&#xA;third grade  curriculum, for example, consists of more than 900&#xA;problems. Teachers, in turn,  present the children with these&#xA;problems, and they do not affirm the correctness of  solutions; rather&#xA;the children must come to these conclusions from the mathematics&#xA;itself. The children learn to argue their points of view without,&#xA;however, becoming  argumentative.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How peculiar ! I would like to have a look at the text books.&lt;/p&gt;&#xA;&lt;p&gt;However, seems nowhere could I find them, namely:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Davydov, V. V., Gorbov, S. F., Mikulina, G. G., Saveleva, O. V.&#xA;(1999a).  Mathematics Class 1. J. Schmittau (Ed.). Binghamton, NY:&#xA;State University of  New York.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Davydov, V. V., Gorbov, S. F., Mikulina, G. G., Saveleva, O. V.&#xA;(2000a).  Mathematics Class 2. J. Schmittau (Ed.). Binghamton, NY:&#xA;State University of  New York.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Davydov, V. V., Gorbov, S. F., Mikulina, G. G., Saveleva, O. V., &amp;amp;&#xA;Tabachnikova,  N. L. (2001). Mathematics 3rd Grade. J. Schmittau&#xA;(Ed.). Binghamton, NY:  State University of New York.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;May I ask, where could I find them?&lt;/p&gt;&#xA;" OwnerUserId="9059" LastEditorDisplayName="user507" LastEditDate="2021-06-27T22:50:42.493" LastActivityDate="2021-06-28T18:52:48.073" Title="Looking for copies of Vasily Vasilovich Davydov's mathematics curriculum" Tags="|textbooks|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21113" PostTypeId="2" ParentId="21112" CreationDate="2021-06-28T06:27:36.730" Score="8" Body="&lt;p&gt;This is not an answer, but more like a long comment.&lt;/p&gt;&#xA;&lt;p&gt;I could not easily find a free PDF of these textbooks - the measures to prevent book piracy work better now in Russia than a decade or two ago. Anyway, I found several screenshots, which I post below. The method has New Math-y feel to me, that is, instead of comparing and counting things it starts with comparing abstract quantities, and one has to make a leap to realize that letters mean quantities.&lt;/p&gt;&#xA;&lt;p&gt;While I like some of the ideas in these textbooks, there are enough things that irk me, similar to the stuff that Common Core-aligned programs push, like the exercise 149 below for 2nd grade: two-dimensional arrangement of things, letters, arrows, and I really cannot make sense of the K -&amp;gt; _ -&amp;gt; M thing. Maybe if it were explained by a teacher, I would get it, but why use non-standard notation when one can use a standard one, or just write in plain words?&lt;/p&gt;&#xA;&lt;p&gt;Same with exercises 110 and 113 for the 3rd grade, here the idea of what the arrows depict is clearer, but then again, why not use standard notation? Taking exercise 113 as an example, the arrows on the left mean multiplication: &lt;span class=&quot;math-container&quot;&gt;$ 8 \times 5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$8 \times 4$&lt;/span&gt;, but the arrows on the right mean addition: 32 + 40.&lt;/p&gt;&#xA;&lt;p&gt;So, before anyone starts hunting for these textbooks (and there are a whole bunch of them: teacher's edition, student textbook, student exercise book, student exercise book with solutions, etc) one should take a second look at this method. I am not very impressed.&lt;/p&gt;&#xA;&lt;p&gt;At the same time I want to point out, that usage of segments (&amp;quot;rods&amp;quot;), boxes, counters is traditional for Russian/Soviet elementary math education. Also, traditionally pupils did not use worksheets, and there were very few &amp;quot;fill the box for answer&amp;quot; exercises, mostly in the 1st grade. The screenshots below show the placeholders for answers, something that is more traditional for American textbooks.&lt;/p&gt;&#xA;&lt;p&gt;Word problems are the staple of Russian/Soviet math curriculum, so whatever method you take, you'll see problems about trains, ships, pipes that fill a pool, workers that work in parallel on the same big assignment, etc. In this regard Russian programs have many common elements. I personally prefer a more traditional approach that starts from abstracting counters from real things (cows, cars), then comparing counters, then counting counters, then writing these tick marks as digits, etc. This approach does not exclude algebraic-like exercises.&lt;/p&gt;&#xA;&lt;p&gt;For example, here is a 5th grade word problem: a child care center purchased two kinds of candies of the same price: 9 kg of the first kind and 13 kg of the second kind. The cost of the second kind of candies is $16 more than of the first kind. What is the total cost of the candies?&lt;/p&gt;&#xA;&lt;p&gt;Another very important thing to keep in mind: Russian school curricula is very structured with tight relationships between subjects. There is more freedom nowadays in choosing this or that program to teach this or that subject, but still there are only three to five different programs that are popular, and they must cover specific topics in specific sequence. Algebra and geometry start from 7th grade and continue until the final grade, in parallel. Physics starts from 7th grade and goes for five years. Chemistry goes for at least two years. All these word problems with ships and trains use uniform motion first, but when physics starts, uniformly accelerated motion is used too, plus basic trigonometry - in 7th grade! So, I think it is not very productive to take just a single subject like mathematics/algebra from Russian program and use it elsewhere.&lt;/p&gt;&#xA;&lt;p&gt;I don't like how American school program is structured, it basically does not exist. There are some vague conventions that algebra and geometry start in high school, that geometry, which is only one year course, must follow Algebra 1, that physics is usually not taken in 9th grade, etc. Different districts, schools and even individual teachers choose whatever courses they feel like, and it is impossible to plan, say, learning about right triangle and Pythagorean theorem a couple of weeks before solving a problem in a physics class about a boat, crossing a river. I see that some courses like Core-Plus try to incorporate basics of physics into math course only to fail miserably.&lt;/p&gt;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&lt;p&gt;1st grade:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ThQDG.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ThQDG.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;2nd grade:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tKxEr.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tKxEr.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;3rd grade:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TBIW7.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TBIW7.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7930" LastEditorUserId="7930" LastEditDate="2021-06-28T18:52:48.073" LastActivityDate="2021-06-28T18:52:48.073" CommentCount="24" ContentLicense="CC BY-SA 4.0" />
  <row Id="21114" PostTypeId="1" CreationDate="2021-06-28T15:10:15.227" Score="5" ViewCount="439" Body="&lt;p&gt;I am currently assisting a course for future teachers at university level for a joint education in maths and physics in Germany and I have a question regarding the use and possible application of the following result.&lt;/p&gt;&#xA;&lt;p&gt;We defined the annihilator for an element A of a K-algebra &lt;span class=&quot;math-container&quot;&gt;$\mathcal A$&lt;/span&gt; as the set of polynomials over the field K, that return zero, i.e.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$Ann(A):=\{f\in K[x]: \tilde f(A)=0\}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The tilda indicates the evaluation-homomorphism. We proved that any such f can be decomposed into a unique normed polynomial h and some rest, such that every &lt;span class=&quot;math-container&quot;&gt;$f = g\cdot h$&lt;/span&gt;. This can be used to show that a nilpotent matrix of order k, has &lt;span class=&quot;math-container&quot;&gt;$M_A(x)=x^k$&lt;/span&gt; as minimal polynomial.&lt;/p&gt;&#xA;&lt;p&gt;My problem: it is an interesting tool to find the minimal polynomial of say a real valued matrix. But I do not see the relevance for future teachers as this seems to be &amp;quot;just&amp;quot; a technical result without any applications that one could explain to someone without extensive mathematical education.&#xA;I do know, that the subspace of linear functions with the same property is a subspace of the dual space.&lt;/p&gt;&#xA;&lt;p&gt;So, are there any nice applications for this method in the realm of euclidean geometry? or physics? or maths?&lt;/p&gt;&#xA;&lt;p&gt;If you feel this is not the right place to ask such an involved question, feel free to suggest an alternative. I have posted this very question to math.stackexchange, too.&lt;/p&gt;&#xA;" OwnerUserId="16382" LastActivityDate="2021-06-28T20:48:29.120" Title="Applications of the annihilator from linear algebra" Tags="|secondary-education|linear-algebra|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21115" PostTypeId="2" ParentId="20959" CreationDate="2021-06-28T17:11:15.623" Score="0" Body="&lt;p&gt;If you want to show the detailed displacement pattern of a cantilever beam you can use the mode shape vector. In 1D, the mode shape vector of 6 degree of freedom (DOF) lumped mass cantilever beam (which is basically a line with 6 mass points dots) will contain 6 entries. Each entry suggests the position of a lumped mass point (DOF). If you use only 2 mass points (DOF), you get just two values and can show an inclined line. But that would be way too erroneous. You could use 3 to 4 dots and get a very good approximation though. In this case, the structure is 1D. But the deformation vector goes up to 6 Dimension? So I think, vectors only contain refined information in a mathematical sense.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/u0m90.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/u0m90.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The photo example is similar. Lets say there are 10000 pixels in the image. Assume a red apple and so each of these pixels are filled with red color and its variants. If you want to show a green apple you fill those pixels with green and its variants. But there will still be only 10000 pixels. And the pixel positions are still the same.&#xA;So if you want to explain this through vectors, each pixel would need 3 entries: &lt;span class=&quot;math-container&quot;&gt;$$\{\textrm{x coordinate},\textrm{y coordinate},\textrm{intensity of red color}\}$$&lt;/span&gt; In this case, the position of pixels determine the shape of the object. Once the positions are fixed, we need colors to differentiate. Colors are only adding more information. Similarly higher dimensions add only more information.&lt;/p&gt;&#xA;&lt;p&gt;Physical dimensions are only 3. And that's it. Period. There are no higher dimensions in the physical realm. It's a dangerous lie when someone says they are able to visualize 10-dimensional physical space. The extra dimensions can be seen only as additional details/information such as: details/time/temperature/color/etc.&lt;/p&gt;&#xA;" OwnerUserId="16380" LastEditorUserId="16380" LastEditDate="2021-07-03T15:18:06.437" LastActivityDate="2021-07-03T15:18:06.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21116" PostTypeId="2" ParentId="21114" CreationDate="2021-06-28T17:47:01.223" Score="4" Body="&lt;p&gt;The method of annihilators from constant coefficient ODEs is interesting. If &lt;span class=&quot;math-container&quot;&gt;$L = P(D) \in \mathbb{R}[D]$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$D = d/dt$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$deg(P)=n$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$L[y]=0$&lt;/span&gt; defines an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th order &lt;strong&gt;homogeneous&lt;/strong&gt; ODE. In this case, &lt;span class=&quot;math-container&quot;&gt;$Ann(L)$&lt;/span&gt; is a finite dimensional subspace of the infinite dimensional space of smooth functions on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. So, one interesting thing is just that annihilators give a formalism to characterize solution sets as they relate to the structure of an operator. This alone is not terribly interesting, the &lt;em&gt;method of annihilators&lt;/em&gt; typically refers to the following mathematical slight of hand: given a &lt;strong&gt;nonhomogeneous&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th order ODE &lt;span class=&quot;math-container&quot;&gt;$L[y] = g$&lt;/span&gt; we can apply the method of annihilators if there exists &lt;span class=&quot;math-container&quot;&gt;$A \in \mathbb{R}[D]$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$deg(A)=k$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$A[g]=0$&lt;/span&gt;. Then, &lt;span class=&quot;math-container&quot;&gt;$AL[y] = A[g] = 0$&lt;/span&gt;. By the theory of ODEs, there exists a fundamental solution set for &lt;span class=&quot;math-container&quot;&gt;$AL[y]=0$&lt;/span&gt; say &lt;span class=&quot;math-container&quot;&gt;$y_1,y_2,\dots, y_n, y_{n+1}, \dots , y_{n+k}$&lt;/span&gt;. Moreover, without loss of generality we may suppose &lt;span class=&quot;math-container&quot;&gt;$y_1,y_2,\dots, y_n$&lt;/span&gt; is a fundamental solution set of &lt;span class=&quot;math-container&quot;&gt;$L[y]=0$&lt;/span&gt; since solutions of &lt;span class=&quot;math-container&quot;&gt;$L[y]=0$&lt;/span&gt; are also solutions of &lt;span class=&quot;math-container&quot;&gt;$AL[y]=0$&lt;/span&gt;. In fact, we can argue &lt;span class=&quot;math-container&quot;&gt;$y=\displaystyle\sum_{j=1}^{n+k} c_jy_j$&lt;/span&gt; serves as the general solution of &lt;span class=&quot;math-container&quot;&gt;$L[y]=g$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, &lt;span class=&quot;math-container&quot;&gt;$y''-y = e^t$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$L = D^2-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A=D-1$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$A[e^t]=0$&lt;/span&gt;. Note, &lt;span class=&quot;math-container&quot;&gt;$AL = (D-1)(D^2-1) = (D+1)(D-1)^2$&lt;/span&gt; thus &lt;span class=&quot;math-container&quot;&gt;$AL[y]=0$&lt;/span&gt; has solution &lt;span class=&quot;math-container&quot;&gt;$y = c_1e^{-t}+c_2e^t+c_3te^t$&lt;/span&gt;. Observe &lt;span class=&quot;math-container&quot;&gt;$L[y] = e^t$&lt;/span&gt; gives&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (D^2-1)(c_1e^{-t}+c_2e^t+c_3te^t) = e^t $$&lt;/span&gt;&#xA;hence&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ c_3D^2[te^t] - c_3te^t = e^t $$&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ c_3(te^t+2e^t) - c_3te^t = e^t $$&lt;/span&gt;&#xA;consequently,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 2c_3e^t = e^t $$&lt;/span&gt;&#xA;and we find &lt;span class=&quot;math-container&quot;&gt;$c_3 = \frac{1}{2}$&lt;/span&gt; thus &lt;span class=&quot;math-container&quot;&gt;$y = c_1e^{-t}+c_2e^t+ \frac{1}{2}te^t$&lt;/span&gt; is the general solution of &lt;span class=&quot;math-container&quot;&gt;$y''-y=e^t$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Notice the nonhomogeneous problem can be viewed as a finite dimensional linear algebra problem since &lt;span class=&quot;math-container&quot;&gt;$ann(AL)$&lt;/span&gt; is an &lt;span class=&quot;math-container&quot;&gt;$(n+k)$&lt;/span&gt;-dimensional vector space by the existence and uniqueness theory for constant coefficient ODEs. I think this is fairly removed from direct matrix math. Of course, pick a basis, and we can be right back to the matrix.&lt;/p&gt;&#xA;&lt;p&gt;Beyond this, the concept of annihilators play a large role in the study of differential forms and integral submanifolds. But, you might be looking for a very different sort of example, so I'll stop here.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-06-28T17:47:01.223" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21117" PostTypeId="2" ParentId="21114" CreationDate="2021-06-28T20:48:29.120" Score="1" Body="&lt;p&gt;I'm not a theoretical math type, more of a technical guy.  But I'm used to seeing the annihilator method and operator discussed in ODE courses.  For example pages 193-195 of Spiegel, third edition Applied Differential Equations.&lt;/p&gt;&#xA;&lt;p&gt;If there are other knuckle-draggers like I in your course (since the physics slant), perhaps they will find such a discussion (having to do with characteristic equation of familiar second order ODE with constant coefficients) easier to grok than the holomorphism and set notation above.&lt;/p&gt;&#xA;" OwnerUserId="16385" LastActivityDate="2021-06-28T20:48:29.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21118" PostTypeId="2" ParentId="21070" CreationDate="2021-06-29T15:21:09.163" Score="1" Body="&lt;p&gt;My experience is at university level so it may not fit your requests.&lt;/p&gt;&#xA;&lt;p&gt;During my online course I planned and did the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;opened each lecture with one musical piece that was chosen by students. They all agreed it helped them gaining concentration and momentum&lt;/li&gt;&#xA;&lt;li&gt;asked them, with a small reward on grades, to write a log journal - one page at least per week - with personal comments on what they liked/disliked, found interesting, and in general reflecting on their learning experience. This had to be sent privately to me, with the agreement that (obscuring names) I could have used  and commented  this material in classroom as well.&lt;/li&gt;&#xA;&lt;li&gt;gave the opportunity to jointly work on a small project in groups of 3. Projects were chosen by a list. My course was on differentiable curves and each project should produce as outcome computations of the curve invariants as well as comments on why the chosen curve was interesting/connected to other parts of math/historically relevant. Again small leverage on grades.&lt;/li&gt;&#xA;&lt;li&gt;asked them to jointly produce, again in small groups, commented set of notes that I will use in next year class. So produce material to be used by junior students in future classes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;All this was quite effective and (together with a couple of video-parties at moments in which I felt they were losing momentum) helped in creating an active participation in (virtual) classroom.&lt;/p&gt;&#xA;&lt;p&gt;One thing that I did in the past (not virtual) was the following: I would open each lecture with 5 minutes recalling an argument of their choice with the agreement that they were supposed to choose the argument and agree on it before me entering the classroom. Again this somewhat worked.&lt;/p&gt;&#xA;&lt;p&gt;I hope I'll be able to reproduce some of this in presence next year. But it has been a hell of work, to say it clear.&lt;/p&gt;&#xA;" OwnerUserId="465" LastActivityDate="2021-06-29T15:21:09.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21121" PostTypeId="1" AcceptedAnswerId="21123" CreationDate="2021-07-01T20:20:53.580" Score="8" ViewCount="999" Body="&lt;p&gt;Here are some ideas and a few questions I've been pondering lately related to the teaching of functions in college algebra and precalculus:&lt;/p&gt;&#xA;&lt;p&gt;Based on my experience, the teaching of functions usually starts with the discussion of relations, and leads to eventually defining a function from &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; as a relation that associates each element of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; (the domain) with exactly one element of &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; (the codomain). From here there tends to be a lot of focus on questions that involve determining whether an equation &amp;quot;defines &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;, or &amp;quot;is a function&amp;quot;. The focus on these types of questions seems to often lead students (including me when I was taking college algebra) to think of a function as &amp;quot;an equation that passes the vertical line test&amp;quot;, or &amp;quot;an equation where each input has exactly one output&amp;quot;. This seemed like a pretty productive way of thinking about functions, until I got into more advanced math courses and they started describing a function as &amp;quot;a set of ordered pairs&amp;quot;. Also, since I had the understanding that a function is &amp;quot;an equation where each input has exactly one output&amp;quot;, I often thought of the rule &lt;span class=&quot;math-container&quot;&gt;$f(x)=y$&lt;/span&gt; itself as a function (this idea is reinforced throughout college algebra textbooks. They repeatedly refer to the rule &lt;span class=&quot;math-container&quot;&gt;$f(x)=y$&lt;/span&gt; itself as a function), when this technically isn't true. In order to properly define a function, we need to explicitly state the domain and codomain, correct?&lt;/p&gt;&#xA;&lt;p&gt;This led to even more confusion as I looked back on the type of questions I was asked in college algebra and precalculus. The formal notation that we see in courses like real analysis that usually looks something like &amp;quot;Define &lt;span class=&quot;math-container&quot;&gt;$f:X \to Y$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$f(x)=y$&lt;/span&gt;&amp;quot;. This notation was essentially non existent in college algebra, precalculus, and most of calculus. I never once thought of a function as a set, nor did I think of the domain and codomain as being part of the formal definition of a function.&lt;/p&gt;&#xA;&lt;p&gt;As I continued to think back, more questions came up. For example, does it really make sense to ask a student to &amp;quot;find the domain of the following function?&amp;quot; Isn't the domain something that should be included in the defining of the function itself? Also, what does &amp;quot;the domain&amp;quot; even mean? This seems to imply that each rule &lt;span class=&quot;math-container&quot;&gt;$f(x)=y$&lt;/span&gt; has one single domain set associated with it. I guess the convention is to find the &amp;quot;largest&amp;quot; one with the &amp;quot;least&amp;quot; number of restrictions? (I know I'm not being precise when I say &amp;quot;largest&amp;quot; and &amp;quot;least&amp;quot; here, but hopefully y'all understand what I'm trying to say)&lt;/p&gt;&#xA;&lt;p&gt;Another issue is finding the inverse of a function, or determining whether two functions are inverses of each other. Does it even make sense to ask this without explicitly stating the domain and codomain? I mean, &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt; has an inverse function if we restrict the domain to the non-negative reals, but if I were to just ask &amp;quot;Does the function &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt; have an inverse function?&amp;quot; (textbooks often ask these kinds of questions), I would expect almost everyone to answer &amp;quot;no&amp;quot;, because it is not injective. Then again, whether a function is injective or surjective also depends entirely on the domain and codomain, which in this case are not explicitly stated.&lt;/p&gt;&#xA;&lt;p&gt;Many of the textbooks I've referred to seem to mirror the issues I'm pointing out, but it is still possible that my experience is the exception, not the rule. Either way, this all leads me to wonder if there are ways to introduce and build on the concept of a function that are more productive, and I'm curious to hear some other perspectives from people with more experience.&lt;/p&gt;&#xA;&lt;p&gt;TLDR: Understanding the formal definition of a function is extremely important for understanding concepts like one to one, onto, inverse functions, and perhaps many more, but it seems as though we often don't put much (or any) emphasis on it in college algebra, precalculus, or calculus. Perhaps this is something we should make note of and try to improve so that our students are leaving these courses with a productive and accurate way of thinking about functions.&lt;/p&gt;&#xA;" OwnerUserId="13742" LastEditorUserId="470" LastEditDate="2021-07-02T06:22:28.193" LastActivityDate="2021-07-08T16:16:40.847" Title="Does the way we often introduce the concept of a function make sense?" Tags="|undergraduate-education|functions|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21122" PostTypeId="2" ParentId="21121" CreationDate="2021-07-02T03:27:16.077" Score="1" Body="&lt;p&gt;I appreciate your concern and have felt similar despair when teaching about functions and the awkward game of &amp;quot;finding their domains&amp;quot; in precalculus courses.&lt;/p&gt;&#xA;&lt;p&gt;But then I lift myself up by thinking about analytic continuation. A real variable construct like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt; somehow just knows, without human intervention, what its domain should be. In this case, analytic continuation takes &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt; to have a domain that is a two-sheeted Riemann surface that is a double covering of &lt;span class=&quot;math-container&quot;&gt;${\bf C}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;More generally, an analytic function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, whose domain and values are defined initially only infinitesimally, just &amp;quot;knows&amp;quot; how to expand and find its maximal domain without meddling by brains. Think about the spiraling Riemann surface that serves as the maximal domain for the logarithm, or the analytic continuation of &lt;span class=&quot;math-container&quot;&gt;$f(p)=\displaystyle{\sum_{n=1}^{\infty} \dfrac{1}{n^p}}$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$p&amp;gt;1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;${\bf C}\setminus\{1\}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-07-02T03:35:26.877" LastActivityDate="2021-07-02T03:35:26.877" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21123" PostTypeId="2" ParentId="21121" CreationDate="2021-07-02T08:30:24.807" Score="9" Body="&lt;p&gt;The question is probably too opinion-based to allow for a definite answer, but let me offer a few reasons why I think that we should rather &lt;em&gt;not&lt;/em&gt; ignore domains and codomains when we teach functions to, say, highschool students (yes, this means I'm suggesting to do it differently from how it is now done in many places):&lt;/p&gt;&#xA;&lt;p&gt;(1) The OP is of course right that, from a purely mathematical point of view, it does not make sense to talk about a function without specifying its domain and codomain. Domain and codomain are just part of the &lt;em&gt;definition&lt;/em&gt; of a function.&#xA;One might claim (as done, for instance, in the comments under the question) that this were merely &amp;quot;technical&amp;quot;, but I disagree: Changing the domain of a function can change it's properties &lt;em&gt;entirely&lt;/em&gt;, which indicates that it is not a mere technicality or subtlety.&lt;/p&gt;&#xA;&lt;p&gt;It even has a close connection to very practical questions which we see in the newspaper every day: if we consider the evolution of some quantity (say, economical, or social, or meteorological, or, well, epidemiological...) over a period of time, then the properties of this function (of time) do often depend very strongly on the precise period of time that we consider. This is a very relevant insight, and it is closely related to the domain of functions and to the operation of restricting a function to a smaller domain. And while it's certainly absolutely possbile to understand this without knowing what the domain of a function is, it at least shows that things like &amp;quot;domains&amp;quot; are by no means something abstract or technical which were completely removed from reality.&lt;/p&gt;&#xA;&lt;p&gt;(2) One might make a point that talking domains or codomains were to complicated or &amp;quot;advanced&amp;quot; for student to grasp too early in their education. I don't think that's accurate, though:&lt;/p&gt;&#xA;&lt;p&gt;When I went to highschool (in Germany) we were taught &amp;quot;functions&amp;quot; in precisely the same imprecise manner as described by the OP. However, we had to learn a lot of things in mathematics which are much more involved than the simple question what the domain and codomain of a function are (for instance, I don't buy that anybody finds derivatives, integrals, or applications of the central limit theorem - even when done without any proofs - easier than the fact that a function needs a domain and a codomain).&lt;/p&gt;&#xA;&lt;p&gt;(3) While I believe that teaching things in a less rigorous manner does sometimes (or even often) make them easier for students (in particular, in highschool) to grasp, I am far from convinced that this is always the case - and the concept at hand seems to be a prototypical counterexample:&lt;/p&gt;&#xA;&lt;p&gt;For instance, I vaguely remember that when I went to school a lot of students had a hard time to understand the concept of inverse functions. I have severe doubts that this were due to some intrinsic difficulty of the concept - in fact, inverse functions are probably one of the simplest concepts in mathematics. However, when students are never taught what a domain and a codomain of a function is and are instead made to believe that a function is just some kind of evaluation rule between numbers, I'm not surprised that they don't understand what an inverse function is and that they find computing an inverse function a completely pointless alogrithmic endeavour.&lt;/p&gt;&#xA;&lt;p&gt;(4) Here's the part of the story that I'm personally most familiar with:&lt;/p&gt;&#xA;&lt;p&gt;Whenever I'm involved in teaching first year students who major in mathematics at a German university, their first mathematics courses are typically, &amp;quot;Linear Algebra&amp;quot; and &amp;quot;Real Analysis in one variable&amp;quot;. One of the (various) things that many students really struggle with is to understand what a function really is. In particular, many students don't understand that, say, for a function&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; f: \mathbb{R} \to \mathbb{R}, \quad x \mapsto f(x)&#xA;$$&lt;/span&gt;&#xA;it is &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; - and not &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; - which is the function. It doesn't fit how they were taught to think about functions in school: there, a function was not a mapping with a domain and a codomain, but it was somekind of &amp;quot;formula&amp;quot; which they had to manipulate in some way.&lt;/p&gt;&#xA;&lt;p&gt;While this seems like a subtlety at first glance, it makes it very difficult for them to grasp many concepts in Linear Algebra: for instance, understanding and using linear maps between to function spaces is very difficult if one doesn't see the difference between &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So the fact that functions are taught as &amp;quot;formulas&amp;quot; rather than as mappings with domains and codomains in school, has the very unpleasent consequence that we have to &amp;quot;unteach&amp;quot; the students this idea when they enter university to study mathematics or a closely related subject. (Of course, the real problem here is not that it makes life a bit harder for faculty and for teaching assistants, but that it makes life much harder for the students.)&lt;/p&gt;&#xA;" OwnerUserId="16250" LastEditorUserId="16250" LastEditDate="2021-07-03T03:06:58.420" LastActivityDate="2021-07-03T03:06:58.420" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="21124" PostTypeId="1" AcceptedAnswerId="21125" CreationDate="2021-07-03T08:46:01.023" Score="-3" ViewCount="1226" Body="&lt;p&gt;OUP published &lt;a href=&quot;https://global.oup.com/academic/product/visual-complex-analysis-9780198534464?cc=ca&amp;amp;lang=en&amp;amp;&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Visual Complex Analysis&lt;/em&gt;&lt;/a&gt; in 1999. MAA published &lt;a href=&quot;https://www.maa.org/press/maa-reviews/visual-group-theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Visual Group Theory&lt;/em&gt;&lt;/a&gt; in 2009. But no visualization counterparts have been published for other  subjects like (Commutative) Algebra, Combinatorics, (Algebraic, Differential) Geometry, Harmonic Analysis, Matrix Theory, Measure or Ergodic Theory, (Advanced) (Real)  Analysis, Rings Fields Modules, Topology. These subjects are abstruser than univariate calculus, but are the mathematicians listed below incapable of writing books on them?&lt;/p&gt;&#xA;&lt;p&gt;Why don't these tyro authors  publish, and wouldn't they profit more from, the aforementioned &lt;em&gt;Visual&lt;/em&gt; equivalents to assist students visualizing other subjects, or problem books with full solutions to each problem?&#xA;Why do these tyros persist in publishing  univariate calculus textbooks that remain untrodden? These tyros must know that the market for univariate calculus textbooks is saturated and has little chance of being disrupted.  In actuality, these tyros failed, because most instructors are unfamiliar with their untrodden textbooks probably because they don't say anything new or reform calculus education.&lt;/p&gt;&#xA;&lt;p&gt;Instructors don't have time to read and compare the glut of univariable calculus textbooks. I read this review of Peter Mercer's &lt;a href=&quot;https://www.maa.org/press/maa-reviews/more-calculus-of-a-single-variable&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;More Calculus of a Single Variable&lt;/em&gt; (2014)&lt;/a&gt;, but it doesn't distinguish it from other allegedly revolutionary books like &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-a-modern-approach&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Karl Menger's Calculus: A Modern Approach&lt;/em&gt; (2007)&lt;/a&gt;, &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-a-rigorous-first-course&quot; rel=&quot;nofollow noreferrer&quot;&gt;Daniel Velleman's &lt;em&gt;Calculus: A Rigorous First Course&lt;/em&gt; (2017)&lt;/a&gt; or &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-from-approximation-to-theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dan Sloughter's &lt;em&gt;Calculus From Approximation to Theory&lt;/em&gt; (2020)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Alex Himonas, &lt;a href=&quot;https://www.maa.org/publications/maa-reviews/calculus-ideas-applications&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus: Ideas &amp;amp; Applications&lt;/em&gt;&lt;/a&gt; (2003).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since there are so many calculus books and the coverage in them is so similar, I cannot say that this book is outstanding.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Dale Varberg, Edwin Purcell, and Steven Rigdon, &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-6&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus&lt;/em&gt;&lt;/a&gt; (2006).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So: a mostly traditional book, with the real virtue of brevity but otherwise not too imaginative.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Robert Smith and Roland Minton, &lt;a href=&quot;https://www.maa.org/publications/maa-reviews/calculus-early-transcendental-functions&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus: Early Transcendental Functions&lt;/em&gt;&lt;/a&gt; (2007).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There's a numbing sameness to most calculus books. In part, that's inevitable: how many paths are there through this particular garden? Given that one must visit the usual tourist spots, there is little space for being really original. [...] Beyond that, this strikes me as plain-vanilla.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Jon Rogawski, &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus&lt;/em&gt;&lt;/a&gt; (2008).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But now we come to the key question: Given that there are already thousands of calculus books in print, is it valuable to have a new calculus book that is just like nearly all of them, except that it is has clearer explanations and fewer errors? It's hard to make this sound exciting, and in fact I am not excited by it. Most calculus books are deadly dull, and even though this one has lots of pretty pictures and interesting sidebars I still had a hard time getting through it. I would have liked it much better if it had addressed some of the issues raised by the reform movement.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Arnold Ostebee, &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-from-graphical-numerical-and-symbolic-points-of-view-volume-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus: From Graphical, Numerical, and Symbolic Points of View&lt;/em&gt;&lt;/a&gt; (2008).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When reviewing a general calculus text, the main question the reviewer needs to answer is “how is this book different from the existing mainstream choices?” The answer, in this case, is that the book is more ambitious and moves at a faster pace than most competing textbooks. It also makes a few unusual choices in the order in which the topics are covered.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Laura Taalman and Peter Kohn, &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus&lt;/em&gt;&lt;/a&gt; (2013).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Calculus textbook market is crowded, and the books on that market are very similar to one another. It is completely normal for two books in that category to overlap by ninety percent or more in their coverage of topics&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Peter Lax, Maria Terrell, &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-with-applications&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus with Applications&lt;/em&gt;&lt;/a&gt; (2013).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My criticisms and suggestions aside, this is an altogether excellent text.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Michael P. Sullivan and Kathleen Miranda, &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-early-transcendentals&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus: Early Transcendentals&lt;/em&gt;&lt;/a&gt; (2014).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Other than these aspects, there is nothing to differentiate this book from other books designed for the three-semesters of calculus market. &lt;strong&gt;In many ways I find myself wishing that there would be a moratorium on new calculus books because the field seems to have hit a wall regarding anything new in textbooks. [Embolding mine]&lt;/strong&gt; I could have used this book to teach calculus 20 years ago and I could use my book of twenty years ago to teach it now and the outcomes would not be significantly different.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Alexander Hahn, &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-in-context&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus in Context&lt;/em&gt;&lt;/a&gt; (2017).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Overall, this reviewer would not choose to use this text as the main text for a calculus course, but would (and has) utilized the myriad of applications as starting points for student projects.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Deborah Hughes-Hallett, Andrew Gleason, &lt;em&gt;Calculus: Single and Multivariable&lt;/em&gt;, &lt;a href=&quot;https://www.wiley.com/en-us/Calculus%3A+Single+and+Multivariable%2C+8th+Edition-p-9781119696551&quot; rel=&quot;nofollow noreferrer&quot;&gt;8th ed. (2018)&lt;/a&gt;. But the MAA hasn't reviewed it since the &lt;a href=&quot;https://www.maa.org/press/maa-reviews/calculus-single-and-multivariable&quot; rel=&quot;nofollow noreferrer&quot;&gt;2005 4th ed.&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user155" LastEditDate="2021-07-05T02:35:27.470" LastActivityDate="2021-07-05T17:22:45.910" Title="Why so many single-variable calculus textbooks? But still no equivalents to Visual Group Theory or Complex Analysis for other subjects?" Tags="|textbooks|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="21125" PostTypeId="2" ParentId="21124" CreationDate="2021-07-03T13:58:22.760" Score="22" Body="&lt;p&gt;Single variable calculus is in a very different situation from the other topics you mention.  The number of students in the US who take single variable calculus is at least two orders of magnitude up from the number who study any of those other subjects, because it's a requirement at many schools for a wide variety of majors.&lt;/p&gt;&#xA;&lt;p&gt;This leads to several effects&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The market for selling such books is huge, so an unsuccessful calculus book may sell substantially more copies than a highly successful measure theory book. (Even if we assume the people writing these books aren't particularly motivated by the royalties, those sales are also one plausible measure of the impact of the book.)&lt;/li&gt;&#xA;&lt;li&gt;Professors have to teach the subject a lot, which motivates them to want to teach it in a way they like.&lt;/li&gt;&#xA;&lt;li&gt;Because the topic is taught to younger, less mathematically mature students, issues with the textbook are harder to smooth over.&lt;/li&gt;&#xA;&lt;li&gt;Because the course has obligations to many departments outside math, it's relatively hard to innovate. Someone who wants to significantly change what specific topics, say, a combinatorics course covers can often just do that unilaterally, or at least negotiate it with just a couple other people. But even curricular changes to the calculus course may have to be discussed, not only at the departmental level, but possibly negotiated with other departments as well.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Put together, I think it's clear why that would motivate some people to write books which, from their perspective, improve the way calculus is taught while still being compatible with the basic strictures of the standard American calculus course.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2021-07-03T13:58:22.760" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21126" PostTypeId="2" ParentId="21124" CreationDate="2021-07-04T07:04:35.857" Score="6" Body="&lt;p&gt;I agree with Henry Towsner's answer, but I also believe the premise of the question is false.&lt;/p&gt;&#xA;&lt;p&gt;For example, my college adopted Sullivan &amp;amp; Miranda 2e (one of the &amp;quot;glut&amp;quot; of calculus textbooks listed in the question) in our most recent calculus textbook search, in part because it seemed more readable than the usual options. Every example in the exposition directly indicates to the student which exercises they should try next, before they read further in the text. This supports the instructors in my department who focus on teaching students how to read textbooks (a skill many students have not learned before reaching us).&lt;/p&gt;&#xA;&lt;p&gt;If a reviewer is &lt;em&gt;not&lt;/em&gt; focused on this aspect of teaching mathematics, perhaps looking at the Sullivan &amp;amp; Miranda text would cause the reviewer to despair that &amp;quot;the field seems to have hit a wall regarding anything new in textbooks.&amp;quot; To my committee, it looked great. Perhaps many of the textbooks listed have a non-trivial following like this one.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-07-04T07:04:35.857" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21127" PostTypeId="2" ParentId="21121" CreationDate="2021-07-04T10:14:47.673" Score="1" Body="&lt;p&gt;The definition of function as a binary relation on two sets is related to the set theoretic foundations of mathematics. Under this foundations, we consider only sets as the primitive building blocks for all mathematics. You have mentioned functions, but other mathematical objects are defined (sometimes awkwardly) as some set. For example, tuples of numbers (a,b) are defined as the set {a, {a,b}}. Natural numbers are usually defined in set theory either using the Zermelo ordinals (0={}, 1={{}}, 2={{{}}}, ...) or the von Neumann ordinals (0=∅, 1={∅}, 2={∅, {∅}}, ...).&lt;/p&gt;&#xA;&lt;p&gt;Some mathematicians are unsatisfied with the foundations. Many of the proposed alternative foundations consider functions as a primitive concept on the same level as the concept of set. These ideas usually come from category theorists or type theorists. For example, Lawvere's Elementary Theory of the Category of Sets (ETCS) is basically an axiomatization of an elementary topos with a natural numbers object. On his original paper, Lawvere leaves the concepts of function, domain, codomain and composition undefined on purpose, he just states on the axioms the rules that shall relate these concepts. In other words, he's not interested on the &lt;em&gt;substance&lt;/em&gt; of these concepts, but on the &lt;em&gt;structure&lt;/em&gt; that arise given the axioms on these concepts.&lt;/p&gt;&#xA;&lt;p&gt;My point is that many mathematicians share your concern with the foundations of mathematics and feel that the way they think about mathematics is inconsistent with the formalism. Many have proposed alternative, more intuitive foundations.&lt;/p&gt;&#xA;&lt;p&gt;On the exercise about &amp;quot;finding the domain of a function&amp;quot;, I'm on the same boat of the other replies, a function on ECTS must be related to a unique domain and codomain from the moment it's defined. I would rephrase such exercises from:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Find the domain of f(x)=√(x^2-1)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;to&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Define a function f : X -&amp;gt; R, such that f(x)=√(x^2-1). For the answer to be correct, no other function g : Y -&amp;gt; R must exist such that f = g∘i for some proper inclusion i: X -&amp;gt; Y.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="16400" LastActivityDate="2021-07-04T10:14:47.673" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21128" PostTypeId="2" ParentId="21121" CreationDate="2021-07-04T22:57:23.410" Score="2" Body="&lt;p&gt;First, let's keep in mind who the students are who are taking these courses. They are either high school students or college students who are taking high-school math for remediation. A typical student in such a class is thinking of becoming a dentist or selling used cars. Your concerns will make no sense to such a student and will be of no educational benefit to them.&lt;/p&gt;&#xA;&lt;p&gt;There is nothing wrong with writing down an equation and taking that as the definition of a certain function, including its domain. There are some conventions for how to interpret such a thing. To the extent that these conventions are sometimes nontrivial, the nontrivial cases are of no interest in applications. For example, if I use &lt;span class=&quot;math-container&quot;&gt;$y=(\sqrt x)^2$&lt;/span&gt; to define a function, then we have to worry about whether this is defined for negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Your textbook probably defines a convention according to which the domain doesn't include negative values, but in a real-life application we wouldn't decide that based on the convention, we would decide it based on real-world context. So just skip the &amp;quot;find the domain&amp;quot; exercises and instead do some applications and word problems, and your students will be better off.&lt;/p&gt;&#xA;&lt;p&gt;There is nothing special or holy about the definition of a function as a set of ordered pairs. The world's greatest mathematicians prior to ca. 1900 did fine without such a definition. Set theory is not the only possible foundation for mathematics, and working it out as a such a foundation is of no interest to students at this level. If 0.1% of them will become math majors in college, then they'll learn that in their upper-division courses.&lt;/p&gt;&#xA;&lt;p&gt;There is actually no real need to define a function. It can be a primitive notion, like lines and points in Euclidean geometry. You can point out that the graph of a function passes the vertical line test, but that isn't a definition of a function, it's just a theorem about graphs. You can point out that some functions have inverses and some don't.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-07-04T22:57:23.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21129" PostTypeId="2" ParentId="21108" CreationDate="2021-07-05T03:07:33.267" Score="1" Body="&lt;p&gt;It's been many years since I looked at it, and I no longer have my copy available, but I &lt;em&gt;believe&lt;/em&gt; that &lt;em&gt;Elementary Mathematics for Teachers&lt;/em&gt; by Scott Baldridge and Thomas Parker was written to be used alongside the Singapore Math Curriculum.  (In fact, a quick Google search shows that Baldridge and Parker's book is &lt;a href=&quot;https://shop.singaporemath.com/index.php/product/elementary-mathematics-for-teachers-parker-and-baldridge/&quot; rel=&quot;nofollow noreferrer&quot;&gt;available for sale from Singapore Math, Inc&lt;/a&gt;, which would seem to support my memory.)  I think it almost certainly discusses how to use and teach all of the various models that are used in Singapore Math, at some length.&lt;/p&gt;&#xA;&lt;p&gt;(Just to add a brief, opinionated disclaimer: this is not an endorsement of that particular text, which I think is problematic in many respects.  But it is almost certainly the correct answer to the OP's question.)&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-07-05T03:07:33.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21130" PostTypeId="2" ParentId="21124" CreationDate="2021-07-05T17:22:45.910" Score="1" Body="&lt;p&gt;I think the question as asked here is just misinformed. There are numerous books on pretty much all advanced mathematical topics which contain both answers and solutions. If you wish to read on intuition there is so much available on Mathoverflow etc. that this question and it's presuppositions must surely be an affront to those who labored to gain insight in previous eras where the internet did not allow us to see such vistas.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps this is in part an answer, why write a book when most new students are too busy searching the net for information. If you seek an audience, is a book really the format of choice now ?&lt;/p&gt;&#xA;&lt;p&gt;Of course, I value books and I'd like to finish a few of my own, but I will admit this is more about quality control and finishing some of my own thoughts. If I wanted to influence students and push the edge of understanding, I think You Tube video tutorials are probably more where it's at.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-07-05T17:22:45.910" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21131" PostTypeId="1" AcceptedAnswerId="21144" CreationDate="2021-07-06T04:28:27.267" Score="18" ViewCount="7334" Body="&lt;p&gt;Why is differential calculus often presented before integral calculus?&lt;/p&gt;&#xA;&lt;p&gt;Note: I'm still learning calculus at the moment.&lt;/p&gt;&#xA;&lt;p&gt;It seems that many elementary calculus texts describe differential calculus before integral calculus. They start with an informal intuition into the concept of a limit and how to calculate various limits. They then go on to describe the derivative via physical applications and/or the tangent/secant line approach. To me it seems that the integral would be more intuitive to understand first, and then limits and derivatives. To the best of my knowledge, this is also how calculus was historically developed. Is there a particular reason for presenting the derivative before the integral in elementary calculus?&lt;/p&gt;&#xA;" OwnerDisplayName="user13234" LastActivityDate="2023-08-26T15:07:45.940" Title="Why is differential calculus often presented before integral calculus?" Tags="|calculus|" AnswerCount="9" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="21132" PostTypeId="2" ParentId="21108" CreationDate="2021-07-06T10:48:27.893" Score="1" Body="&lt;p&gt;This is a book I read years ago. It is quite general though, maybe you need something more practical &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0033HXD1K&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Singapore-Model-Method-Learning-Mathematics/dp/B0033HXD1K#customerReviews&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I don't personally endorse the method, nor the book. I think it is very specific and probably such detailed method wouldn't fit for students when they go on to secondary schools. I had myself this issue with students coming from Steiner's school.&lt;/p&gt;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="16379" LastActivityDate="2021-07-06T10:48:27.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21133" PostTypeId="1" CreationDate="2021-07-06T11:56:11.903" Score="1" ViewCount="52" Body="&lt;p&gt;Often, I find it useful to teach some Python to students. In-person classes are easy in this regard, I can just go to &lt;a href=&quot;https://colab.research.google.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;colab.research.google.com&lt;/a&gt; and show them the concepts there. (It requires no technical knowledge to use it, that's why I prefer that over locally-run Jupyter notebooks.)&lt;/p&gt;&#xA;&lt;p&gt;Despite the claim that it is good for collaborative working, it does &lt;em&gt;not&lt;/em&gt; update live as I write content in those notebooks, as Google Jamboard would, for example.&lt;/p&gt;&#xA;&lt;p&gt;Are there websites which provide Jupyter notebooks (with Python 3 backend) which I could use to work with students real-time? By real-time, I mean that I want my students to see what I have written in a cell within a second at least.&lt;/p&gt;&#xA;" OwnerUserId="16410" LastActivityDate="2021-07-06T11:56:11.903" Title="Which websites provide real-time updated Jupyter notebooks for Python teaching?" Tags="|interactive-teaching|" AnswerCount="0" CommentCount="6" ClosedDate="2021-07-07T09:27:54.817" ContentLicense="CC BY-SA 4.0" />
  <row Id="21134" PostTypeId="2" ParentId="21131" CreationDate="2021-07-06T14:00:35.883" Score="37" Body="&lt;p&gt;While there are no theoretical difficulties with developing integration first (&amp;quot;from scratch&amp;quot; measure theory books demonstrate this), it presents some pedagogical challenges.&lt;/p&gt;&#xA;&lt;p&gt;Most problems in existing Calculus courses have &amp;quot;closed form&amp;quot; solutions.  It is rare to just set up a problem and then numerically evaluate.  The primary tool available for exact calculation of definite integrals is the Fundamental Theorem of Calculus.  Finding &lt;span class=&quot;math-container&quot;&gt;$$\int_0^1 x^5 \textrm{d}x$$&lt;/span&gt; can be done numerically or exactly using excruciating Riemann sum calculations.  We like to know that &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx} x^6 = 6x^5$&lt;/span&gt; before covering integration so that this integral can be evaluated exactly using the fundamental theorem.&lt;/p&gt;&#xA;&lt;p&gt;I personally think it would be a better story if we taught integration first, got practice setting up all kinds of integrals to solve practical problems, and used numerical approximation as our only evaluation tool.  This would drive home what the definite integral IS.  Then do differential calculus in all of its glory.  Then the FTOC would actually be surprising and useful (Wow!  Now we can calculate these integrals exactly instead of just approximating!).  As it is, most students think the FTOC is the definition of the definite integral, which is unfortunate.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-07-06T14:00:35.883" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="21135" PostTypeId="2" ParentId="21131" CreationDate="2021-07-06T14:16:45.467" Score="18" Body="&lt;p&gt;The text I know of that does integration first is: Apostol, &lt;em&gt;Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;An advantage of using this for top-rate incoming university students is that they do not think &amp;quot;I already know this&amp;quot; and tune out [as they may do when the course begins with differentiation].&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2021-07-06T14:16:45.467" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21136" PostTypeId="2" ParentId="21131" CreationDate="2021-07-06T18:57:20.147" Score="8" Body="&lt;p&gt;I think one of the main reasons for the ordering, limits-&amp;gt; derivatives -&amp;gt; integrals is that it follows in order of difficulty in some sense of the word.&lt;/p&gt;&#xA;&lt;p&gt;In reality the situation is more complex, but if you start with the minimal (read &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;) definition of a limit it's fairly straightforward. It's also almost essential for even a semi rigorous treatment of both differential and integral calculus.&lt;/p&gt;&#xA;&lt;p&gt;Differentiation comes pretty much next in complexity. In reality the basic version (assuming stuff you have is almost everywhere &lt;span class=&quot;math-container&quot;&gt;$C^\infty$&lt;/span&gt;) is actually quite a bit easier than limits, but since (it looks as if) you're using limits to calculate it, it's more complex. At the same time derivatives are important for a lot of other things you want the students to be learning, mainly basic physics, where you get differential equations at every corner.&lt;/p&gt;&#xA;&lt;p&gt;Integration is by far the hardest both in terms of students understanding and actual computation. Unlike differentiation where (mostly) anything that can be written neatly has a neat (read made up of elementary functions) derivative, most integrals don't actually have a closed form solution. With &amp;quot;most&amp;quot; here meaning many of the ones students can think of.&lt;/p&gt;&#xA;&lt;p&gt;All in all, this order is in many ways natural, progressing from easier to harder and building on previously learned material. In reality integration both 2d and 3d is a lot more complex than derivatives. Both in terms of the limits employed (you have to take a limit over partitions, so you need a new notion of limit) and the idea that's it's &amp;quot;finding the area under the curve&amp;quot; is actually surprisingly complex since you don't actually know what &amp;quot;area&amp;quot; is until you come up some version of integration.&lt;/p&gt;&#xA;" OwnerUserId="5896" LastActivityDate="2021-07-06T18:57:20.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21137" PostTypeId="2" ParentId="21131" CreationDate="2021-07-06T18:59:24.663" Score="19" Body="&lt;p&gt;One issue is that for differentiation, you can find the derivatives of so-called &amp;quot;elementary functions&amp;quot; (e.g. powers, exponentials, logarithms, trig functions) directly from the definition of a limit, and there are then standard ways to find the derivatives of an expression knowing the derivatives of its component parts (e.g. the product and quotient rules, function-of-a-function, etc.)&lt;/p&gt;&#xA;&lt;p&gt;Therefore it is straightforward to generate a large number of &amp;quot;textbook exercises&amp;quot; of graded difficulty.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, most &amp;quot;simple-looking functions&amp;quot; do &lt;em&gt;not&lt;/em&gt; have closed-form integrals, and the easiest way to find a set of basic functions that &lt;em&gt;do&lt;/em&gt; have closed form integrals is to recognize that &amp;quot;the function you want to integrate looks like a derivative that you already know&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In fact many of the so-called &amp;quot;special functions&amp;quot; in mathematics are &lt;em&gt;defined&lt;/em&gt; as the integral of a simple-looking function (e.g. the gamma function, which generalizes the idea of a factorial for non-integer values).&lt;/p&gt;&#xA;&lt;p&gt;Of course as other answers have said, if you taught integration using only &lt;em&gt;numerical&lt;/em&gt; methods as an initial way to evaluate integrals, that problem does not exist, but (at high school or university level) most students will not have much if any understanding of numerical methods, and therefore there are too many possibilities for &amp;quot;garbage-in garbage out&amp;quot; exercises where students have no way to check their work.&lt;/p&gt;&#xA;&lt;p&gt;Typing formulas into Wolfram's integral evaluator isn't &amp;quot;learning math&amp;quot; - and it isn't even an interesting task, unless you are going to &lt;em&gt;do&lt;/em&gt; something interesting with the output.&lt;/p&gt;&#xA;" OwnerUserId="7677" LastEditorUserId="376" LastEditDate="2023-08-26T15:07:45.940" LastActivityDate="2023-08-26T15:07:45.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21138" PostTypeId="2" ParentId="21131" CreationDate="2021-07-06T22:48:18.557" Score="8" Body="&lt;p&gt;I'd like to comment on why, indeed, it would be reasonable to present the subject with integrals first... (which is not what is done, nearly universally, I understand).&lt;/p&gt;&#xA;&lt;p&gt;In particular, I'd argue that the notion of &amp;quot;derivative&amp;quot; is more sophisticated than &amp;quot;area under a curve&amp;quot;. This includes the subtlety of &amp;quot;instantaneous rate of change&amp;quot;. Yes, people'd been thinking about this a long time (cf. Zeno's paradoxes).&lt;/p&gt;&#xA;&lt;p&gt;But/and, yes, a fundamental reason for the success of &amp;quot;calculus&amp;quot; as a very basic mathematical gadget is the fundamental theorem of calculus, giving explicit and elementary (indefinite) integrals for many interesting functions. Explicit (elementary) expressions obviously allow many more manipulations and experimentation than subtler &amp;quot;estimates&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;For that matter, the first really rigorous results about differential equations used conversion of differential equations to integral equations, and invocation of theorems about compact operators. (See Volterra, Hilbert, Schmidt.)&lt;/p&gt;&#xA;&lt;p&gt;In my own experience, I did think it was charming that slopes to polynomial curves could be exactly determined, but it was &lt;em&gt;really&lt;/em&gt; amazing that areas under such graphs could be &lt;em&gt;precisely&lt;/em&gt; (not just numerically) determined, and in forms that were astonishingly simple.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2021-07-06T22:48:18.557" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21139" PostTypeId="1" CreationDate="2021-07-07T04:19:26.187" Score="0" ViewCount="454" Body="&lt;p&gt;I used to learn Real Analysis before Complex Analysis in my bachelor study, but now the order is reversed in my university.&lt;br /&gt;&#xA;I would like to ask which order is better to learn the subjects, and which order is better to teach the subjects?&lt;/p&gt;&#xA;" OwnerUserId="16416" LastEditorUserId="376" LastEditDate="2021-07-07T07:02:56.887" LastActivityDate="2021-07-22T16:36:47.883" Title="Real before complex analysis or vice versa?" Tags="|undergraduate-education|mathematical-analysis|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="21140" PostTypeId="2" ParentId="21131" CreationDate="2021-07-07T11:23:26.697" Score="4" Body="&lt;p&gt;As with a huge number of the &amp;quot;why do we do this order&amp;quot; questions, or the &amp;quot;why don't we teach real analysis before calculus&amp;quot; questions, they seem to tacitly assume that the order of teaching is based on philosophical explanation or even formal mathematical proof.  WRONG!&lt;/p&gt;&#xA;&lt;p&gt;This is NOT the reason why instruction is structured the way it is.  It is structured based on imperfect beings that learn harder subjects after easier ones, because that works better pedagogically.  Instruction is a maximization problem of time, brains, etc.  To get the most sense pounded into skulls within the practical limits that pertain in the practical world.  And in the case of integration, it is harder than differentiation (going backwards) and makes more sense afterwards.  So.  Practical pedagogy uber theoretical explication.&lt;/p&gt;&#xA;" OwnerUserId="16422" LastActivityDate="2021-07-07T11:23:26.697" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="21142" PostTypeId="2" ParentId="21131" CreationDate="2021-07-07T14:39:19.953" Score="19" Body="&lt;p&gt;Because:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://xkcd.com/2117/&quot; rel=&quot;noreferrer&quot;&gt;https://xkcd.com/2117/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And now I'm padding my answer with 21 characters.&lt;/p&gt;&#xA;" OwnerUserId="16425" LastActivityDate="2021-07-07T14:39:19.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21144" PostTypeId="2" ParentId="21131" CreationDate="2021-07-07T21:36:17.123" Score="7" Body="&lt;p&gt;David Bressoud has a 2019 book that argues for teaching integration before derivatives, and only later teaching limits, partly because that matches the way that the ideas developed historically: &amp;quot;Calculus Reordered: A History of the Big Ideas&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="14443" LastActivityDate="2021-07-07T21:36:17.123" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21145" PostTypeId="1" AcceptedAnswerId="21146" CreationDate="2021-07-07T23:42:56.650" Score="8" ViewCount="1210" Body="&lt;p&gt;Is there a place where I can buy well made calculus 2 and calculus 3 lesson plans and power points?&lt;/p&gt;&#xA;&lt;p&gt;There is a good chance I'll be having my first semester of full time teaching next semester. If so, I'll be teaching four different classes, which would obviously be a ton of prep work. I was thinking if I could find a place to buy good power points to guide my lectures that would be great.&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="16437" LastActivityDate="2021-07-09T01:15:45.507" Title="Is there a place where I can buy well made calculus 2 and calculus 3 lesson plans and power points?" Tags="|teacher-preparation|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="21146" PostTypeId="2" ParentId="21145" CreationDate="2021-07-08T00:05:53.527" Score="20" Body="&lt;p&gt;Assuming you are newly part of a department, &lt;strong&gt;ask someone in the department if they have anything you can use&lt;/strong&gt;. They may direct you to publisher-provided materials or just give you a big pile of things.&lt;/p&gt;&#xA;&lt;p&gt;Many instructors are worried that this kind of question will be seen as a sign of &lt;em&gt;weakness&lt;/em&gt;, but actually seeking out expert materials well in advance of the semester is often seen as a sign of &lt;em&gt;strength&lt;/em&gt;. This is also a good excuse to talk to some people in your department.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-07-08T00:05:53.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21147" PostTypeId="2" ParentId="21121" CreationDate="2021-07-08T11:13:33.573" Score="0" Body="&lt;p&gt;In mathematics, at least, two concepts of function are used. We will distinguish them by using uppercase and lowercase letters:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Function is a univalent relation f, i.e. (x, y), (x, y ') ∈ f -&amp;gt; y = y'.&lt;/li&gt;&#xA;&lt;li&gt;FUNCTION is a triple (f, D, K), which we usually write f: D -&amp;gt; K, which consists of sets D and K and a univalent relation f such that the set of its first components is D, and the set of its second components is contained in K, i.e. f (D) ⊆ K.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In both cases, the set of the first components of f is called the domain of f, symbolically Dom f, and the set of second components of f is called the image of f, symbolically Im f. We define y = f (x) as (x, y) ∈ f, f (S) as {y: (∃x∈S) y = f (x)} and f-1 (S) as {x: (∃y∈S) y = f (x)}.&lt;/p&gt;&#xA;&lt;p&gt;Note that for a function (i.e. the univalent relation) it makes sense to ask someone to determine its domain, in the same way as it makes sense to ask someone to determine its image. For a FUNCTION it makes sense to ask someone to determine its image, but it does not make sense to ask someone to determine its domain because it is already given.&lt;/p&gt;&#xA;&lt;p&gt;Of course, you can understand each FUNCTION as a function (because the graph of a FUNCTION is a function), but you cannot understand a function as a FUNCTION (because a function determines domain D but does not determine codomain K).&lt;/p&gt;&#xA;&lt;p&gt;Obviously, function is a simpler concept. Then why is FUNCTION used in school? What does the FUNCTION offer that the function does not offer?&lt;/p&gt;&#xA;&lt;p&gt;The concept of injection is the same in both cases. The function f is an injection from A iff&lt;/p&gt;&#xA;&lt;p&gt;(∀x, x'∈ A) (x ≠ x' -&amp;gt; f (x) ≠ f (x ')).&lt;/p&gt;&#xA;&lt;p&gt;If we omit &amp;quot;from A&amp;quot; we mean &amp;quot;from the domain of f&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The concept of composition is simpler for functions. The composition of the functions f and g is the function f◦g defined by&lt;/p&gt;&#xA;&lt;p&gt;f◦g (x)  = f (g (x))        i.e.       (∃u) (u = g (x) &amp;amp; y = f (u)).&lt;/p&gt;&#xA;&lt;p&gt;Of course, it is possible to prove that Dom f◦g = g-1 (Dom f) and Im f◦g = f (Im g).&lt;/p&gt;&#xA;&lt;p&gt;For FUNCTIONS, the notion of composition is more complex. Composition of the FUNCTIONS f: D -&amp;gt; K and g: D1 -&amp;gt; K1 is the FUNCTION&lt;/p&gt;&#xA;&lt;p&gt;f◦g : g-1 (Im g ∩ Dom f) -&amp;gt; K1&lt;/p&gt;&#xA;&lt;p&gt;defined by&lt;/p&gt;&#xA;&lt;p&gt;f◦g (x) = f (g (x)).&lt;/p&gt;&#xA;&lt;p&gt;The concept of inverse function is simpler for functions. The function f-1 is the inverse function of the function f iff&lt;/p&gt;&#xA;&lt;p&gt;y = f (x) ↔ x = f-1 (y).&lt;/p&gt;&#xA;&lt;p&gt;For FUNCTIONS, the concept of inverse is more complex. THE FUNCTION f-1: D’ -&amp;gt; K' is the inverse FUNCTION of the FUNCTION f: D -&amp;gt; K iff&lt;/p&gt;&#xA;&lt;p&gt;y = f (x) ↔ x = f-1 (y)   and   D '= f (D) &amp;amp; K '= D&lt;/p&gt;&#xA;&lt;p&gt;(or if you insist that only bijections have inverses, D' = K and K '= D).&lt;/p&gt;&#xA;&lt;p&gt;A FUNCTION is a surjection iff f (D) = K. The concept in this form has no meaning for functions (but see below).&lt;/p&gt;&#xA;&lt;p&gt;A FUNCTION is a bijection iff it is surjection and injection. The concept in this form has no meaning for functions (but see below).&lt;/p&gt;&#xA;&lt;p&gt;Obviously function is a simpler and easier concept. Then why do we insist on FUNCTIONS at school?&lt;/p&gt;&#xA;&lt;p&gt;I guess because of the important concepts of surjection and bijection that have no meaning for functions. But functions doesn't really have any problems with these concepts.&lt;/p&gt;&#xA;&lt;p&gt;A function is a surjection on B if its image is B.&lt;/p&gt;&#xA;&lt;p&gt;A function is a bijection from A on B if it is an injection from A and a surjection on B.&lt;/p&gt;&#xA;&lt;p&gt;Using these concepts, we can easily say, for example, that sin x is a bijection from [-π, π] on [-1, 1] and that therefore there is an inverse bijection sin-1 x from [-1, 1] to [-π, π].&lt;/p&gt;&#xA;&lt;p&gt;In terms of FUNCTIONS you have to talk about the restriction of the FUNCTION sin x: R -&amp;gt; R to a different FUNCTION sinx: [-π,π]  -&amp;gt; R, if you allow inversions of injections or to the sin x: [-π,π]  -&amp;gt; [-1,1], if you only allow inversions of bijections (but then you get involved in the problem, does the notion of restriction allow you to change the codomain).&lt;/p&gt;&#xA;&lt;p&gt;I think it is evident what is simpler and clearer.&lt;/p&gt;&#xA;&lt;p&gt;In addition, the more complex concept of FUNCTION is easily defined if we need it (we need it, for example, when we start talking about the space of all functions from A to B, etc.). A function from A to B (this is what we have so far called FUNCTION) is a function with domain A and the image contained in B.&lt;/p&gt;&#xA;" OwnerUserId="10261" LastEditorUserId="10261" LastEditDate="2021-07-08T16:16:40.847" LastActivityDate="2021-07-08T16:16:40.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21148" PostTypeId="2" ParentId="21139" CreationDate="2021-07-08T17:10:32.613" Score="3" Body="&lt;p&gt;While, yes, precise advise would depend considerably on context, I do think there are some features that should/could influence curriculum design/choices.&lt;/p&gt;&#xA;&lt;p&gt;Basic complex analysis (Cauchy's theorem and corollaries, power series and Laurent expansions, residues, ...) functions very well, answers questions, and can feel like a fulfillment or happy continuation of the positive aspects of calculus. That is, we &lt;em&gt;can&lt;/em&gt; compute derivatives of lots of things, and compute some integrals, in lucky cases. Amazing. And, in complex analysis, &amp;quot;functions&amp;quot; &lt;em&gt;are&lt;/em&gt; (representable as) power series, which is about as good as it gets.&lt;/p&gt;&#xA;&lt;p&gt;Basic real analysis can be perceived as a lot of bad news and reports of danger. Yes, &amp;quot;rigor&amp;quot; is being imparted to calculus, but can easily be perceived as expensive, without much operational benefit besides the rigor itself.&lt;/p&gt;&#xA;&lt;p&gt;But, ok, it's good to have a bit of that basic real analysis before proceeding too much further with complex analysis, since some aspects really do depend on finicky estimates, issues of uniformity-or-not, auxiliary functions not as nice as holomorphic, as opposed to the algebraic ideas in the compact Riemann surfaces direct.&lt;/p&gt;&#xA;&lt;p&gt;Beyond the basic real analysis stuff, for applications (to complex analysis and to other things), the kind of things that have seemed most persistently useful to me involve distribution-theory (generalized functions), Fourier series and Fourier transforms, and things sometimes labelled &amp;quot;functional analysis&amp;quot;, rather than subtleties about pointwise behavior and measure theory.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2021-07-08T18:24:43.690" LastActivityDate="2021-07-08T18:24:43.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21149" PostTypeId="2" ParentId="21145" CreationDate="2021-07-08T22:51:12.370" Score="6" Body="&lt;p&gt;As noted in a comment, most major calculus textbooks at this time (or more generally: almost anything in the standard undergraduate track, at least freshman-sophomore years) comes with slideshow presentations for the instructor. It's a very common use-case, and one of the principal value-add selling points for textbook publishers these days. It's likely you'll need an instructor account at the publisher's website; ask your publisher textbook sales contact.&lt;/p&gt;&#xA;&lt;p&gt;That said, to date, I haven't personally seen any that were grade-A &amp;quot;well made&amp;quot;. In my experience, I have to commit to revising/editing/upgrading/rewriting them on an ongoing basis, to tune them to my specific course and stylistic focus. Maybe all instructors are too idiosyncratic to fully accept another person's slide organization; but it can be a strategically smart way to get started (esp. if you've received a lot of big new preps at once, and/or you need to focus on your research).&lt;/p&gt;&#xA;&lt;p&gt;Chris Cunningham's suggestion to ask around at your institution is a good one, of course. I'm the point person at my department for resources like that, and I consider it a good sign of planning and professionalism when people ask. It's probably less likely for slides to be in use/available by traditional math faculty -- but I have friends in other disciplines for whom using department-provided slides is &lt;em&gt;mandatory&lt;/em&gt;, so you should check to be sure.&lt;/p&gt;&#xA;&lt;p&gt;Finally, as one example, the free OpenStax calculus textbooks likewise have instructor slide decks available &lt;em&gt;for free&lt;/em&gt;. Again, you'll need to make an instructor account to access them. See the page below and search for &amp;quot;PowerPoint Slides&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://openstax.org/details/books/calculus-volume-2?Instructor%20resources&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax Calculus 2 Instructor Resources&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-07-09T01:15:45.507" LastActivityDate="2021-07-09T01:15:45.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21151" PostTypeId="1" AcceptedAnswerId="21152" CreationDate="2021-07-09T03:57:40.007" Score="-2" ViewCount="428" Body="&lt;p&gt;Someone asked me this question today. Why do you honestly believe that high school math test scores in America (particularly the United States) are low compared to other countries? I thought about this question a lot, and I honestly think it has to do with another question and that is the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why has there been a rapid decline of responsibility for students in the United States of America? What inspired this movement? Was it parent-driven? Was it student-driven? Was it stress-related and done for emotional reasons? Was it about equity? There has to be a reason for it, and I can't put my finger on it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There has been this gradual decline of responsibility for students to do anything now which has led to this deficit. There is no demand to do problems 1-30 and 40-55* (odds) anymore. Turn your homework a few days late. No worries or responsibility there! We now allow retakes on all assessments and don't hold students responsible to study and do well the first time. There was this big retake your test to understand the material movement when I was in school, and there was no responsibility for students to study the first time to do well. Now, there are no more timed tests for multiplication tests; you can count on your fingers and understand (6)(3)=18 because you have to understand six groups of three is eighteen. There's no responsibility to memorize that! Other countries have stepped up their homework, timed tests, and responsibility. In America, we don't hold students responsible for retaining any common sense math skills based on pop quiz assessments like we used to. I find continually that students don't know what basic terms are such as mean, volume, and really trivial math skills as there is no responsibility for students to memorize and retain these facts at a young age via our assessments. Instead, we just assess one common core standard a day and call it good at that. You also can get a 4.0 with proficiencies and get 4 questions wrong. So, again my question is why has there been this shift away from this responsibility?&lt;/p&gt;&#xA;" OwnerUserId="12366" LastEditorUserId="12366" LastEditDate="2021-07-09T15:51:56.517" LastActivityDate="2021-07-10T15:00:22.427" Title="Why are math test scores dropping in America? Lack of student responsibility movement" Tags="|secondary-education|mathematical-pedagogy|usa|" AnswerCount="1" CommentCount="16" ClosedDate="2021-07-11T16:14:05.040" ContentLicense="CC BY-SA 4.0" />
  <row Id="21152" PostTypeId="2" ParentId="21151" CreationDate="2021-07-09T21:54:19.887" Score="14" Body="&lt;p&gt;Some of us would point to political pressures to evidence higher &amp;quot;success&amp;quot; in terms of increased graduation rates, which wind up pressuring institutions to reduce standards and pass students regardless of whether they've mastered material skills or not.&lt;/p&gt;&#xA;&lt;p&gt;Note that U.S. high school graduation rates have been spiking upwards in recent years, which provides opportunities for systems and states to congratulate and defend themselves:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1mizM.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1mizM.png&quot; alt=&quot;Increasing U.S. high school graduation rates 1990-2015&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Decreasing_graduation_completion_rates_in_the_United_States&quot; rel=&quot;noreferrer&quot;&gt;Image from Wikipedia&lt;/a&gt;. Note this is in the context of an article discussing how college graduation rates are &lt;em&gt;failing&lt;/em&gt; to increase significantly in the same time frame.&lt;/p&gt;&#xA;&lt;p&gt;Here's an article on the &lt;a href=&quot;https://www1.cuny.edu/sites/cunyufs/2017/02/13/graduationratesrisingpreparednessfallingnocauseforcunytocelebrate/&quot; rel=&quot;noreferrer&quot;&gt;CUNY (City of New York) University Faculty Senate blog in 2017&lt;/a&gt; summarizing this trend towards higher-graduation-from-high-school, but lower-preparation-for-college rates, with a number of interesting links for further reference:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Both the State and the City of New York have recently reported a rise&#xA;in high school graduation rates.  But college preparedness still&#xA;leaves much to be desired.  It presents a major challenge for colleges&#xA;that admit students whose reading, writing and mathematical skills are&#xA;not at the level needed for academic success.&lt;/p&gt;&#xA;&lt;p&gt;Close to 80 percent of the Department of Education (DOE) graduates&#xA;entering CUNY in the Fall of 2015 needed remediation of some kind.&#xA;While the NYC DOE cites a 79 percent  graduation rate as evidence that&#xA;schools are improving, only an average 37 percent of students graduate&#xA;college-ready...&lt;/p&gt;&#xA;&lt;p&gt;Both the State and City have been altering their standards for&#xA;graduation to boost those numbers. Now CUNY has changed its&#xA;requirements and methods for placing people into remedial classes.&lt;/p&gt;&#xA;&lt;p&gt;What this portends remains to be seen for students just entering into&#xA;credit bearing courses with skills that are even lower than in the&#xA;recent past. One strategy is to provide a non-algebra path to a&#xA;degree. Such intellectual skills and mental habits as might be&#xA;inculcated by the mastery of algebra are thus being set aside in the&#xA;hopes that those no longer burdened by graduation requirements&#xA;including algebra will graduate and find some sort of employment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Indeed, since that blog post was written, CUNY has waived the requirement for an &lt;a href=&quot;http://www.helpyourmath.com/cunyExam.html&quot; rel=&quot;noreferrer&quot;&gt;elementary algebra skills test&lt;/a&gt; (at roughly the 9th grade level) that was previously required for graduation -- precisely because it was found to be impossible to structure such an exam in a way that everyone would pass it. Moreover, they've eliminated all entry-placement tests and remedial courses (leading the way among other large institutions in the U.S.); this disposes of the embarrassment around the &amp;quot;percent [who] needed remediation&amp;quot; number, as it simply doesn't exist anymore. And they've declared by fiat that all students will be entered to college-credit bearing classes regardless of skill level. For example, from a 2019 university memo on the new &amp;quot;corequisite&amp;quot; course model (emphasis as in original):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The college must allow enrollment of students who are &lt;em&gt;not&lt;/em&gt; skills proficient.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Scherer &amp;amp; Anson in their book &lt;em&gt;Community Colleges and the Access Effect&lt;/em&gt; make the point that high schools practically guaranteeing graduation, and community colleges guaranteeing acceptance via open admissions, set up students to not perceive hard work as a requirement (Chapter 8; quoting from Skelly &amp;amp; Laurence, &amp;quot;Tracking College Readiness&amp;quot;, &lt;em&gt;The School Administrator&lt;/em&gt; 2011):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Community colleges’ open enrollment policies have a negative effect on&#xA;student motivation during high school particularly during the senior&#xA;year. Seniors going to a “JC” (junior college) know their admission is&#xA;guaranteed, so they often slack off and avoid challenging course work,&#xA;particularly during their senior year. The bad habits formed in high&#xA;school are not easily shaken.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In light of this perceived end-game to the political pressure for increased graduation rates (previously at high schools, and now at least at lower-level, public-funded colleges), we might reflect more generally on what is known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Campbell%27s_law&quot; rel=&quot;noreferrer&quot;&gt;Campbell's Law&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The more any quantitative social indicator is used for social&#xA;decision-making, the more subject it will be to corruption pressures&#xA;and the more apt it will be to distort and corrupt the social&#xA;processes it is intended to monitor.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-07-10T15:00:22.427" LastActivityDate="2021-07-10T15:00:22.427" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="21153" PostTypeId="2" ParentId="17080" CreationDate="2021-07-11T17:33:57.180" Score="2" Body="&lt;p&gt;In Mandarin Chinese:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;number&lt;/strong&gt;: 号数&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;numeral&lt;/strong&gt;: 数字&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum&lt;/strong&gt; (with assist from Justin Hancock)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is the distinction between those words? Are &amp;quot;numbers&amp;quot; concepts related to quantity and &amp;quot;numerals&amp;quot; symbols used to represent numbers? Some sources online are suggesting that distinction, while others are suggesting they're synonymous.   &lt;em&gt;-Joe&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It isn't my native tongue, but to my understanding,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{blue}号\color\red数$&lt;/span&gt; (&lt;strong&gt;number&lt;/strong&gt;) refers to a natural number or its value or count, whereas &lt;span class=&quot;math-container&quot;&gt;$\color\red数量$&lt;/span&gt; means quantity. The character &lt;span class=&quot;math-container&quot;&gt;$\color\red数$&lt;/span&gt; actually means ‘numerical/quantitative’. And, depending on the context, &lt;span class=&quot;math-container&quot;&gt;$\color\red数目$&lt;/span&gt; means either number or quantity. Finally, &lt;span class=&quot;math-container&quot;&gt;$\color{blue}号码$&lt;/span&gt; refers to a numerical code/identification/label, rather than a number in the mathematical sense.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\color\red数\color{violet}字$&lt;/span&gt; (&lt;strong&gt;numeral&lt;/strong&gt;; synonym: &lt;span class=&quot;math-container&quot;&gt;$\color\red数目\color{violet}字$&lt;/span&gt;) refers to a natural number's symbolic representation. The character &lt;span class=&quot;math-container&quot;&gt;$\color{violet}字$&lt;/span&gt; literally means the written word.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Hopefully more informative than confusing! Chinese is a rather fuzzy/flexible/non-absolute language indeed.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-10-26T10:32:36.380" LastActivityDate="2023-10-26T10:32:36.380" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="21154" PostTypeId="2" ParentId="20962" CreationDate="2021-07-11T18:38:51.910" Score="3" Body="&lt;p&gt;Perhaps it's the mathematical way of thinking about and approaching problems that is useful, and this way of mathematical thinking can be carried over into many domains of study and work. So while there might not immediately be direct real world applications to parts of it, it's the way of mathematical thinking and approaching problems that is useful. That being said, mathematics can be such a rewarding and interesting subject of study. Perhaps there is room for improvement in this area. How can mathematics be approached to where students can genuinely gain more of an interest in it?&lt;/p&gt;&#xA;" OwnerDisplayName="user13234" LastEditorDisplayName="user13234" LastEditDate="2021-07-11T19:04:38.877" LastActivityDate="2021-07-11T19:04:38.877" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21155" PostTypeId="1" AcceptedAnswerId="21156" CreationDate="2021-07-11T22:21:37.883" Score="2" ViewCount="118" Body="&lt;p&gt;There are dozens (maybe thousands) of websites that explain what triangular numbers, square numbers, etc. are. I'm searching for a &lt;em&gt;printed book&lt;/em&gt; that includes this material, preferably at a level that would be appropriate for an elementary school-aged student (i.e., someone 11 or younger).  It's not typically found in elementary school textbooks or other curricula, so this would probably need to be a &amp;quot;Cool Math&amp;quot;-type of extracurricular book.  Does anybody know of such a source?&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2023-02-21T19:49:10.313" Title="Reference request: an introduction to triangular, square, and other figurate numbers" Tags="|reference-request|primary-education|number-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="21156" PostTypeId="2" ParentId="21155" CreationDate="2021-07-12T01:46:46.940" Score="4" Body="&lt;p&gt;There is a publication in the ERIC archive called &lt;a href=&quot;https://archive.org/details/ERIC_ED035550/page/n161/mode/2up?q=%22triangular+numbers%22+&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Recreational Mathematics&lt;/em&gt;&lt;/a&gt;*, authored by William L. Schaaf, that is a bibliography of books and journal articles about recreational mathematics, organized by topic. A search in this document for &amp;quot;triangular numbers&amp;quot; yields quite a few results, for example, a page of books about &amp;quot;number curiosities&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MvvoX.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MvvoX.png&quot; alt=&quot;triangular numbers&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A specific book that treats triangular numbers is Martin Gardner's &lt;a href=&quot;https://archive.org/details/mathematicalcarn00gard/page/196/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mathematical Carnival&lt;/em&gt;&lt;/a&gt;, which starts from Pascal's Triangle and shows the different sorts of numbers that can be extracted from it. As Gardner puts it, &amp;quot;the pattern is so simple that a 10-year-old can write it down.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2y1O0.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2y1O0.jpg&quot; alt=&quot;Gardner&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I suggest a search through ERIC for some more recent references, or a search through the Internet Archive for some possibly older ones, may result in a list of suitable books. Some of the &amp;quot;out of print&amp;quot; books may have been revived in an affordable Dover printing.&lt;/p&gt;&#xA;&lt;p&gt;*Not to be confused with the now-defunct &lt;a href=&quot;https://en.wikipedia.org/wiki/Journal_of_Recreational_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Journal of Recreational Mathematics&lt;/em&gt;&lt;/a&gt; or the newer journal called &lt;a href=&quot;https://sciendo.com/journal/RMM&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Recreational Mathematics Magazine&lt;/em&gt;&lt;/a&gt;!&lt;/p&gt;&#xA;" OwnerUserId="3593" LastEditorUserId="3593" LastEditDate="2023-02-21T19:49:10.313" LastActivityDate="2023-02-21T19:49:10.313" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21157" PostTypeId="1" CreationDate="2021-07-12T08:49:29.907" Score="-1" ViewCount="83" Body="&lt;p&gt;Ideally, these videos should talk about a real life problem which student can relate to and then showing logically how the problem can be solved. Later bringing in the right mathematical tool to solve the problem. This will have lasting impact on a student's mind.&#xA;Slowly, for many of them, the question that always many possess &amp;quot;what is the application of maths&amp;quot; will be erased from their mind.&lt;/p&gt;&#xA;" OwnerUserId="16098" LastEditorUserId="16098" LastEditDate="2021-07-12T08:55:04.077" LastActivityDate="2021-07-18T19:34:16.457" Title="Can anyone help give pointers of Mathematics Story Telling Animated Videos for Age Group 10 - 15?" Tags="|secondary-education|mathematics-in-daily-life|videos|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21158" PostTypeId="1" AcceptedAnswerId="21160" CreationDate="2021-07-12T17:30:52.657" Score="2" ViewCount="190" Body="&lt;p&gt;I am a maths graduate student.&lt;/p&gt;&#xA;&lt;p&gt;A little background: In the coming semester, I will be assisting a prof whom I admire and whom I also want to thank a lot, and I actually will have a lot of free time. Honestly though, I myself love doing those assistant tasks, and in general love managing teaching as well since I am bound to teach in college later on. Hence, I would love to use this opportunity to make his teaching run even more smoothly as much as I can. Hopefully this post can collect some ideas for me and all who might need.&lt;/p&gt;&#xA;&lt;p&gt;I will start by some restriction: I am not going to grade papers. Also, I obviously cannot and must not replace the prof during his teaching haha. I will however conduct tutorials, but I am pretty sure I am allowed and would love to offer more help to the prof, be proactive, and initiate and suggest ideas to him to &lt;em&gt;ease his teaching job&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My question: For those of you who are (perhaps college) maths teachers, or who are assistants yourselves, what do you think are many things that a maths assistant can do &lt;strong&gt;more&lt;/strong&gt; to help professor in teaching?&lt;/p&gt;&#xA;&lt;p&gt;In other words, for those who are teaching, &amp;quot;It will be amazing if my assistant can &lt;strong&gt;also&lt;/strong&gt; ...(blank)...&amp;quot;, or for those who are assistants, &amp;quot;If I have all the reasons to &lt;strong&gt;further&lt;/strong&gt; improve the professor's teaching, I will ...(blank)...&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Notice the emboldened words, because I am looking for giving additional help outside of what I am obliged to do, and because I have done some of those additional help and will keep doing them, such as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Off-tutorial, I additionally provide timely feedback to students who are emailing me. It is actually not a must (off-tutorial), but for me, it is.&lt;/li&gt;&#xA;&lt;li&gt;I also report to profs instances of students having difficulty on some concepts when I see them during tutorials.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The above are tutorial-related stuff. I can google how to be a good (maths) tutor for many more. What I plan to add are (feel free to comment on these too):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I like to attend his lectures live (if it is not pre-recorded).&lt;/li&gt;&#xA;&lt;li&gt;I like to help making his lecture notes more complete (if not yet), like adding what he said during lectures or correcting the notes. Especially, if his proofs leave missing details (which I can obviously expect), I can try adding them to the notes.&lt;/li&gt;&#xA;&lt;li&gt;I like to provide more examples + exercises, as the lack of such is not uncommon, right? xD&lt;/li&gt;&#xA;&lt;li&gt;I like to initiate a discussion in our uni learning platform (whether directly related to the materials or related to applications instead) to introduce perhaps some exciting stuff.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I feel like there can be more, or something that is related to the class being offline/online, but I am currently out of idea. What else do you think can be done?&lt;/p&gt;&#xA;&lt;p&gt;Thank you very much. Apologize for the long text.&lt;/p&gt;&#xA;&lt;p&gt;Remark: The course will be about pure math (algebra-related stuff).&lt;/p&gt;&#xA;&lt;p&gt;EDITS:&#xA;Thanks for the replies. The course is a little bit higher level course for undergraduate, and it will involve some linear algebra and abstract algebra. It is an elective course.&lt;/p&gt;&#xA;" OwnerUserId="13576" LastEditorUserId="143" LastEditDate="2021-07-14T11:04:34.897" LastActivityDate="2021-07-14T11:04:34.897" Title="How can maths assistants at a college be extra helpful for professors?" Tags="|undergraduate-education|algebra|teaching|lecture-notes|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="21160" PostTypeId="2" ParentId="21158" CreationDate="2021-07-13T11:02:03.200" Score="4" Body="&lt;p&gt;I suggest holding &lt;strong&gt;office hours&lt;/strong&gt;, both one-on-one, and possibly in small groups, to help students with the assignments, or just to clarify the material presented in the lectures. Of course the prof may hold such office hours,&#xA;but since you are giving tutorials, it seems natural, and might serve as&#xA;a lower-stakes alternative for those afraid to visit the prof.&lt;/p&gt;&#xA;&lt;p&gt;By &amp;quot;office hours&amp;quot; I mean you are available somewhere during regular, specified hours for walk-in help, no appointment necessary. This can be more effective than email consultations.&lt;/p&gt;&#xA;&lt;p&gt;(I'm assuming that this differs from &amp;quot;I will however conduct tutorials.&amp;quot;)&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-07-13T11:02:03.200" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21161" PostTypeId="1" CreationDate="2021-07-13T03:34:46.443" Score="11" ViewCount="255" Body="&lt;p&gt;I am parent to a 4-year-old son who is mathematically precocious. An example of what I mean (since I'm sure guys like Gauss were proving theorems at 4):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;He multiplies and divides small numbers easily, like knowing that if there are 3 kids and 12 Halloween treats then everyone gets 4&lt;/li&gt;&#xA;&lt;li&gt;He understands the idea of prime&#xA;numbers as those numbers of candies which can't be equally divided&#xA;among his cousins&lt;/li&gt;&#xA;&lt;li&gt;He has figured out by himself the prime numbers up&#xA;to ~20&lt;/li&gt;&#xA;&lt;li&gt;He is a LEGO fanatic and uses math as necessary - he can&#xA;calculate that he needs nine 2x4 LEGO bricks to cover a&#xA;6x12 plate before he starts.&lt;/li&gt;&#xA;&lt;li&gt;He asked me what 12 times 12 is, then he announced that the carpet in the doctor's office has 144 squares.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I'm wondering how to help him engage more with math since he seems to enjoy it and is good at it. He's not ready yet for &amp;quot;Martin Gardner junior&amp;quot; style questions like &amp;quot;How many balance weighings would it take to identify the one low-weight gem among 9?&amp;quot; And I'm not sure I want to start him on things like memorizing addition and multiplication tables since that's not very relatable to the real world.&lt;/p&gt;&#xA;&lt;p&gt;What kinds of concepts, questions, techniques, books, activities, etc. should I be introducing him to?&lt;/p&gt;&#xA;" OwnerUserId="16497" OwnerDisplayName="Akdinv" LastEditorUserId="15884" LastEditDate="2021-07-15T13:59:37.410" LastActivityDate="2021-07-16T06:59:59.343" Title="What is a good math &quot;curriculum&quot; for a mathematically precocious 4-year old?" Tags="|arithmetic-operations|resource-request|gifted-students|preschool-education|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="21163" PostTypeId="2" ParentId="21161" CreationDate="2021-07-14T15:27:08.543" Score="5" Body="&lt;p&gt;Here are some random toys, games, and things to make:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Flexagon#Hexaflexagons&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Flexagon#Hexaflexagons&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sprouts_(game)&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Sprouts_(game)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Nim&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Nim&lt;/a&gt; -- It's not that exciting as a game, but it's susceptible to solving by recursion, which is kind of cool. Martin Gardner wrote an article on a version called Wythoff's Nim, which is isomorphic to a game on a chess board. The solution on the chess board is easy to visualize.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Racetrack_(game)&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Racetrack_(game)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Penrose_tiling&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Penrose_tiling&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ultimate_tic-tac-toe&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ultimate_tic-tac-toe&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you don't mind risking having him become a physicist rather than a mathematician, you could encourage him to branch out a little bit from legos and build other types of things. I remember building stuff like a paddlewheel boat that ran on a rubber band, a home-made parachute so I could drop toy soldiers out the window of our third-floor apartment. There are books on origami and paper airplanes.&lt;/p&gt;&#xA;&lt;p&gt;There are children's books on things like codes and ciphers, Egyptian hieroglyphs.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-07-14T15:27:08.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21164" PostTypeId="1" CreationDate="2021-07-14T15:39:11.917" Score="9" ViewCount="664" Body="&lt;p&gt;I am looking for resources (books, websites, etc.) for mathematics relevant to or in the context of &lt;a href=&quot;https://en.wikipedia.org/wiki/Sustainability&quot; rel=&quot;nofollow noreferrer&quot;&gt;sustainability&lt;/a&gt;, broadly construed, at upper secondary or early undergraduate level - so not necessarily assuming calculus or differential equations as background, although possibly introducing some notions of these subjects.&lt;/p&gt;&#xA;&lt;p&gt;So far, I have found the book &lt;a href=&quot;https://www.springer.com/gp/book/9783319766591&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics for Sustainability&lt;/a&gt; by Roe, deForest &amp;amp; Jamshidi; Springer 2018, in the Texts for Quantitative Critical Thinking series (Book website: &lt;a href=&quot;https://math-for-sustainability.com/;&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://math-for-sustainability.com/;&lt;/a&gt; review at &lt;a href=&quot;https://www.ams.org/journals/notices/202011/rnoti-p1784.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ams.org/journals/notices/202011/rnoti-p1784.pdf&lt;/a&gt;). It would appear to be an excellent book, although those outside the US will perhaps prefer a version with more use of metric units. It is organized around the  concepts of measurement, flows, networks, change, risk and decisions, along with relevant case studies.&lt;/p&gt;&#xA;&lt;p&gt;For a dynamical systems approach, there is &lt;a href=&quot;https://press.princeton.edu/books/paperback/9780691122045/critical-transitions-in-nature-and-society&quot; rel=&quot;nofollow noreferrer&quot;&gt;Critical Transitions in Nature and Society&lt;/a&gt; by Scheffer, Princeton University Press 2009. The dynamical systems concepts are explained with the assistance of diagrams and graphs, with relevant equations in an appendix. Well worth reading and a useful supplementary resource.&lt;/p&gt;&#xA;&lt;p&gt;What are other good resources for teaching sustainability topics using mathematics, or teaching mathematics applied to sustainability?&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2021-07-16T06:17:48.970" LastActivityDate="2022-10-18T12:58:05.847" Title="Resources for mathematics for sustainability" Tags="|undergraduate-education|secondary-education|applications|resource-request|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="21165" PostTypeId="2" ParentId="21164" CreationDate="2021-07-14T17:39:09.437" Score="5" Body="&lt;p&gt;Possibly worth looking at is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Earth Algebra: College Algebra With Applications to Environmental Issues&lt;/strong&gt; by Christopher Schaufele and Nancy Zumoff (&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0065008863&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;1995 1st edition&lt;/a&gt; and &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0321015975&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;1998 2nd edition&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://tribalcollegejournal.org/earth-algebra-college-algebra-applications-environmental-issues/&quot; rel=&quot;noreferrer&quot;&gt;this review&lt;/a&gt; and &lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED417937.pdf&quot; rel=&quot;noreferrer&quot;&gt;this project/funding document&lt;/a&gt;. For what it's worth, in Spring 1993 I interviewed (on campus) for a tenure-track position at the university this was developed and saw a copy of the &lt;a href=&quot;https://www.worldcat.org/title/earth-algebra-college-algebra-with-applications-to-environmental-issues/oclc/26934973&quot; rel=&quot;noreferrer&quot;&gt;1993 preliminary version&lt;/a&gt;. I don't know how well this experimental approach to the traditional college algebra course went (I was not offered the position, and back then you couldn't google on the internet for information like this), but the faculty were certainly excited about it when I was there. Although there were a lot of innovations/criticisms around this time (in the U.S. at least) about &amp;quot;rainforest math&amp;quot; &lt;strong&gt;[1]&lt;/strong&gt;, up to this time I believe the innovations were limited to various extended exercises, tangential text discussions, etc. in otherwise traditional type textbooks. I think (but I'm not entirely sure) their book was the first college algebra textbook in which the algebra was framed and developed in order to analyze a single theme of topical interest. As I understand it, their idea was to pick a single theme of topical interest that algebra could be developed around, rather than sprinkle the text with a large number of disjointed and unrelated non-traditional applications, and for this theme they chose environmental issues.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; A pejoratively used term for what some considered to be ill-fitting/forced applications of school-level mathematics in order to capture students' interest. I don't know if the term &amp;quot;rainforest math&amp;quot; was widely used in 1993, but it was definitely widely used by the late 1990s.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2021-07-14T17:39:09.437" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21166" PostTypeId="1" CreationDate="2021-07-14T21:30:54.600" Score="9" ViewCount="468" Body="&lt;p&gt;This is my first semester being an instructor of record for a college algebra course. One of the sections we cover is &amp;quot;methods of solving quadratic equations&amp;quot;, where we discuss factoring and using the zero product property, the square root method, completing the square, and using the quadratic formula.&lt;/p&gt;&#xA;&lt;p&gt;One of the questions I pondered while creating their exam was: &amp;quot;Should I assess their ability to use a particular method to solve a quadratic equation, or should I assess their ability to solve a quadratic equation in general?&amp;quot; In other words, should I give the quadratic equation and specify which method they need to use in order to get full credit, or should I give a quadratic equation and allow them to use any method they choose to find the solution set?&lt;/p&gt;&#xA;&lt;p&gt;It seems that the benefit of the former is that it will encourage them to study all of the methods we've learned and practice using them. If I allow them to use any method they choose, they may decide to just learn the quadratic formula and disregard the other three altogether.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, I think that being able to successfully figure out which method to use to solve a particular problem is a skill that is worthy of assessing. This is the point that makes me lean toward giving them a quadratic equation and allowing them to use any method they choose. It also seems like posing a question in this way is more &amp;quot;authentic&amp;quot; in a sense. If they get to calculus and attempt to find the critical points of a polynomial function, they aren't going to be told whether they should use the zero product property or complete the square to solve the resulting quadratic.&lt;/p&gt;&#xA;&lt;p&gt;So I am torn, which is why I thought it would be beneficial to get some feedback from some more experienced instructors. Do you assess your students ability to use a particular problem solving method, or do you assess your students general ability to solve problems?&lt;/p&gt;&#xA;&lt;p&gt;Another context that this could come up is with series convergence.&lt;/p&gt;&#xA;" OwnerUserId="13742" LastEditorUserId="13742" LastEditDate="2021-07-14T21:36:20.157" LastActivityDate="2021-07-22T16:16:07.827" Title="Should we assess ability to use specific problem solving methods, or general ability to solve problems?" Tags="|undergraduate-education|exams|assessment|testing|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="21167" PostTypeId="2" ParentId="21166" CreationDate="2021-07-15T01:51:43.817" Score="8" Body="&lt;p&gt;As a fellow college algebra (et. al.) instructor, I think that &lt;strong&gt;knowing specific methods&lt;/strong&gt; is a necessary thing to confirm. That's what I do on my tests. If you want and have time to add yet another general question (so as to assess &lt;strong&gt;both&lt;/strong&gt;), then that's commendable.&lt;/p&gt;&#xA;&lt;p&gt;The point is that the specific methods will pop up in future courses in different contexts, and your students need to be proficient enough with them to follow the development. This could be in the context of more advanced definitions, proofs, algorithms, etc. Here are some use-cases off the top of my head (maybe others can come up with better ones):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Method of factoring&lt;/strong&gt;: You don't mention that the Fundamental Theorem of Algebra is part of your course. At some point math students should digest this, and factoring is critical to their awareness. Ultimately it leads to the fact that any polynomial of degree &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; can be factored into exactly &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; factors, a wonderfully clear observation. Factoring will be used to reduce lots of stuff in calculus, so getting more practice there is good.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Solve by square roots&lt;/strong&gt;: Students should be aware that solving an equation by square roots gives &lt;em&gt;two&lt;/em&gt; solutions. This prepares them for higher radicals in the complex plane; cube-roots of unity have three values, etc. And it's needed as part of completing the square. (Also personally I feel it's by far the simplest method, and if a technical person can't do this mentally they'll seem slow.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Completing the square&lt;/strong&gt;: The essential thing is that this is the method by which you'll prove the validity of the quadratic formula. (You do that in class, right?) With this in hand, I find it's actually a fun class moment to see how much the students can help drive that symbolic proof. This activity paves the way for more proof-based courses in the future; you should verify that students have the tools to follow that proof, say. Another application: manipulating general circle equations so as to easily graph them.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In some sense, the quadratic formula is actually the most limited of all the techniques you've listed, because -- while formulas exist for 3rd and 4th degree equations -- a major achievement in algebra was proving that &lt;em&gt;no such formula can exist&lt;/em&gt; for equations of the 5th degree in higher. So solving by a formula is ultimately a dead-end. In contrast, at least hypothetically, any such equation still has a predictable number of roots, and can always be factored.&lt;/p&gt;&#xA;&lt;p&gt;IMO: It's easy to lose sight of the fact that almost everything we do in a math course like this is an &lt;em&gt;exercise&lt;/em&gt;, not really a &lt;em&gt;problem&lt;/em&gt;. (And I'm careful to use that former terminology.) Does anyone in the real world need to literally solve a quadratic equation by hand? Well, no: you can just use some technology (Wolfram Alpha comes to mind.)&lt;/p&gt;&#xA;&lt;p&gt;The real point of our math classes is to &lt;em&gt;understand concepts&lt;/em&gt;; like, any nonlinear polynomial is factorable (in complex numbers), that &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th roots have (you know) &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; roots, and that we need to verify our truths through proof (such as the techniques of completing the square vis-a-vis the quadratic formula). The test questions we present are merely somewhat-regrettable verifications that the students have indeed absorbed the mathematical facts.&lt;/p&gt;&#xA;&lt;p&gt;So again on my test at this point I ask specific questions directing that students show me they know how factoring, completing the square, and the quadratic formula work. Separately, a homework quiz has questions on solving by roots, the key step in completing the square, and recreating one step in the proof of the quadratic formula.&lt;/p&gt;&#xA;&lt;p&gt;If you add another general question so that you've also assessed their general ability, great -- but you will probably find they just use the quadratic formula, so you're not getting much extra information from that. Personally I might lean more towards a written question a la, &amp;quot;Given the following equation, which method would be simplest and why?&amp;quot;, but you might get a bunch of pushback on that (and I don't have space on my own tests for it).&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-07-15T08:49:02.317" LastActivityDate="2021-07-15T08:49:02.317" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21168" PostTypeId="2" ParentId="21164" CreationDate="2021-07-15T09:09:45.147" Score="3" Body="&lt;p&gt;Since asking the question, I've come across Thomas J. Pfaff's website, &lt;a href=&quot;https://sustainabilitymath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sustainability Math&lt;/a&gt;. Plenty of material there, including projects and links to further resources. Contains a blog that refers to current/recent events.&lt;/p&gt;&#xA;&lt;p&gt;I found Pfaff's website when perusing &lt;a href=&quot;http://mpe.dimacs.rutgers.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics of Planet Earth&lt;/a&gt;, specifically the &lt;a href=&quot;http://mpe.dimacs.rutgers.edu/curriculum-material/&quot; rel=&quot;nofollow noreferrer&quot;&gt;list of curriculum materials&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(Made this community wiki - feel free to upvote (or not) based on intrinsic usefulness, without me gaining rep for an answer to my own question.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2021-07-15T14:44:05.953" LastActivityDate="2021-07-15T14:44:05.953" CommentCount="0" CommunityOwnedDate="2021-07-15T09:09:45.147" ContentLicense="CC BY-SA 4.0" />
  <row Id="21169" PostTypeId="1" CreationDate="2021-07-15T13:50:02.257" Score="7" ViewCount="191" Body="&lt;p&gt;I have taught some SBG courses in the past, and I am planning on converting all of my courses to SBG.  I have enjoyed teaching these courses, I believe that students have learned more, and students also seem to enjoy the courses more (judging from comments and student evaluation scores).&lt;/p&gt;&#xA;&lt;p&gt;One issue I have is keeping the number of standards down when I am designing a course.  I am designing a discrete math course now, and I have 16 standards just for the logic portion!  I still need standards on sets,  relations and functions, combinatorics, and graph theory.  If I end up with ~15 standards for each of these, I will be looking at a total of ~75 standards, which seems kind of silly.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have resources on SBG design principles which could help me?&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorDisplayName="user507" LastEditDate="2021-07-15T21:24:24.007" LastActivityDate="2021-07-15T21:24:24.007" Title="Standards based grading design principles" Tags="|course-design|standards-based-grading|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21171" PostTypeId="2" ParentId="21169" CreationDate="2021-07-15T17:27:09.440" Score="5" Body="&lt;p&gt;I don't have any specific resources, but here's what I did recently to convert my Business Calc class to SBG: Go back over a whole semester's batch of quizzes and exams and on each question ask yourself what skill/concept you were trying to assess. Doing this I ended up with about 33 standards (I say &amp;quot;about&amp;quot; because 21 of them come in 3 different &amp;quot;flavors&amp;quot; depending on the type of info they're given, e.g. formulas, graphs, or tables.) It helped that I had multiple but synonymous versions of each quiz/exam. It allowed me to see the broader idea rather than the specific calculation.&lt;/p&gt;&#xA;&lt;p&gt;Of course, this can also lead to a realization that some of the things one might have asked on quizzes/exams in the past are maybe not as important as you thought. That's a beneficial part of SBG.&lt;/p&gt;&#xA;&lt;p&gt;I used a similar technique last summer when transferring my Discrete Math class to SBG, too. In that case I ended up with 32 standards. So IMHO I think the 30-35 range is what's optimal.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2021-07-15T17:27:09.440" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21174" PostTypeId="2" ParentId="21044" CreationDate="2021-07-16T00:19:45.387" Score="3" Body="&lt;p&gt;Personally, the perspective I recommend is kind of the reverse. Despite the linked thread, I find that &lt;em&gt;most&lt;/em&gt; of the time, the mathematical term has &lt;em&gt;some&lt;/em&gt; reason why it was picked for the mathematical concept in question. It wasn't picked in an arbitrary or malicious fashion. At some point when the practitioner first used it, it seemed like the best English word for the concept in question.&lt;/p&gt;&#xA;&lt;p&gt;Of course, natural language has ambiguity and multiple meaning to most words: but if you dig into the English dictionary, usually &lt;em&gt;one&lt;/em&gt; of the several senses for a word is directly applicable to the math concept. As someone who does a lot of reading/writing as well as math instruction (and usually wants to dig into the history of math concepts before teaching them), the English connotation is clear, and I get a bit frustrated that my students' English isn't strong enough for that to be immediately evident -- as well as that there's no space in normal books for that connection, nor time in class sessions.&lt;/p&gt;&#xA;&lt;p&gt;So: On reading the OP's question my heart kind of leaped for joy a tiny a bit at having the opportunity to make these connections clear. I already spend time in some cases reflecting explicitly on the English meaning of some more difficult terms as I introduce them in my college courses -- e.g.: predicate, relation, equivalence -- but I often wish wistfully I had time to do that more comprehensively.&lt;/p&gt;&#xA;&lt;p&gt;Let me pick 3 semi-random examples from the linked thread of &amp;quot;different meanings&amp;quot;. For convenience I'm using &lt;a href=&quot;https://www.dictionary.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;dictionary.com&lt;/a&gt; as my reference.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.dictionary.com/browse/even&quot; rel=&quot;nofollow noreferrer&quot;&gt;Even&lt;/a&gt;: The primary definition in English is &amp;quot;level; flat; without surface irregularities; smooth&amp;quot;. Now, this same term has been used since ancient Greece (written ζυγός αριθμός). The point is that the number is &lt;em&gt;evenly divisible by 2&lt;/em&gt;, that is, when halved, there are no &amp;quot;irregularities&amp;quot; in the way of any awkward remainder or fraction in the result.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.dictionary.com/browse/series&quot; rel=&quot;nofollow noreferrer&quot;&gt;Series&lt;/a&gt;: Primary definition here is &amp;quot;a group or a number of related or similar things, events, etc., arranged or occurring in temporal, spatial, or other order or succession; sequence&amp;quot;. Clearly that's appropriate for both a run of TV show and an ordered list of values being summed. (The only &lt;em&gt;nuance&lt;/em&gt; being that in English series and sequence are totally synonymous, where in math one does need to remember they're a bit more specific.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.dictionary.com/browse/volume&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volume&lt;/a&gt;: In this case the relevant sense is a bit further down the list: &amp;quot;a mass or quantity, especially a large quantity, of something: [e.g.] &lt;em&gt;a volume of mail&lt;/em&gt;&amp;quot;. So this sense of &amp;quot;filling an amount of space&amp;quot; is definitely present in English usage. Consider the English word &lt;em&gt;voluminous&lt;/em&gt;, &amp;quot;forming, filling, or writing a large volume or many volumes&amp;quot; which has the same sense of filling-space, but is not (to my knowledge) used in mathematics.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;A few other things that I find my students respond quite well to when I take a momentary historical detour: (1) the fact that the radical symbol comes from an over-time transmogrified &amp;quot;r&amp;quot;, being an abbreviation for radix/root, (2) the fact that our summation symbol &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; (Sigma) is the Greek equivalent to our capital &amp;quot;S&amp;quot;, the obvious choice to abbreviate &amp;quot;summation&amp;quot; (and you can easily draw out the graphical evolution from one to the other; mostly, drop the lower bar), (3) likewise for capital &lt;span class=&quot;math-container&quot;&gt;$\Pi$&lt;/span&gt; (Pi), which is the analog to our capital &amp;quot;P&amp;quot;, the obvious choice to abbreviate &amp;quot;product&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So in short, you should look at this class as a huge &lt;em&gt;opportunity&lt;/em&gt; -- not to highlight the differences, but to underscore the &lt;em&gt;similarities&lt;/em&gt; of the meanings between the English and mathematical terms, that would in many cases by cryptic to people with weak English vocabulary. The more research and connections you can make like that, the stronger it should be in your students' minds, and you get to serve two goals at once with this project. In some sense I'm even a tiny bit envious of your opportunity. Good luck!&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2021-07-16T00:19:45.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21175" PostTypeId="2" ParentId="21161" CreationDate="2021-07-16T06:45:45.853" Score="2" Body="&lt;p&gt;Some additional ideas and links:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Strategical thinking:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/D%C5%8Dbutsu_sh%C5%8Dgi&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/D%C5%8Dbutsu_sh%C5%8Dgi&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Programming:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.turingtumble.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.turingtumble.com/&lt;/a&gt; (Technically 8+ but the concept should be understandable for younger kids as well. Note: it involves small marbles so use caution whether it's appropriate to be used by a smaller child.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Any kind of board game with a linear progression, eg. &lt;a href=&quot;http://jakevdp.github.io/blog/2017/12/18/simulating-chutes-and-ladders/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chutes and Ladders&lt;/a&gt;. It has been &lt;a href=&quot;https://siegler.tc.columbia.edu/wp-content/uploads/2019/02/sieg-ram08.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;shown&lt;/a&gt;, that playing linear numerical board games promotes children's numerical development.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://mathlesstraveled.files.wordpress.com/2020/03/figurate-1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Forward difference and counting dots in different shapes&lt;/a&gt;  is a natural extension to the things he seems to have already discovered&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14091" LastEditorUserId="14091" LastEditDate="2021-07-16T06:59:59.343" LastActivityDate="2021-07-16T06:59:59.343" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21177" PostTypeId="2" ParentId="21164" CreationDate="2021-07-16T20:56:38.527" Score="4" Body="&lt;p&gt;Before listing 'resources', let me start with some official documents (e.g. to clarify the difference of sustainable math and math for sustainability, see &lt;a href=&quot;https://xkcd.com/1007/&quot; rel=&quot;nofollow noreferrer&quot;&gt;xkcd:Sustainable&lt;/a&gt; :-).&lt;/p&gt;&#xA;&lt;p&gt;The official UN Site lists the 17 Goals for sustainability; especially &lt;a href=&quot;https://www.un.org/sustainabledevelopment/education/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Goal 4: Education&lt;/a&gt;. Still debating what this all means, I find the paper by Brundiers et. al. (2021, see below for reference) gives a nice overview. Although the article is behind a paywall, the relevant figures are available on &lt;a href=&quot;https://www.researchgate.net/publication/343265417_Key_competencies_in_sustainability_in_higher_education-toward_an_agreed-upon_reference_framework&quot; rel=&quot;nofollow noreferrer&quot;&gt;researchgate&lt;/a&gt;, put short: different 'competencies' intertwined yield what is called 'key competencies for sustainable development'.&lt;/p&gt;&#xA;&lt;p&gt;Math relevant: problem-solving, strategic-thinking, ...&#xA;So to speak: All math is relevant! To me, what's important is not to think of the modelling aspect only. In math we  need to sort this out a little more!&lt;/p&gt;&#xA;&lt;p&gt;In the UN-model, there are different aspects of 'eduacation for sustainable development' (ESD) mentioned: Relations to environmental, economic, social and political&#xA;questions. I like the 'Math of Planet Earth' Series exactly for this reason: It shows more that just applications of differential equations to environmental questions.&lt;/p&gt;&#xA;&lt;p&gt;Let me first add: UNESCO &amp;amp; MGIEP have a wonderful text about &lt;a href=&quot;https://mgiep.unesco.org/article/textbooks-for-sustainable-development-a-guide-to-embedding&quot; rel=&quot;nofollow noreferrer&quot;&gt;general aspects for textbooks&lt;/a&gt; in relation to ESDs and explicate that in examples. What I find interesting is that they don't start in high school. It starts in Kindergarten: Kids getting their grips to quantities, measurement and numbers. That's where maths starts relating to ESD: Discovering you in the world around. They list different affordances that are also met by resources that are not specifically mentioned in relation to 'sustainability' (see p. 50), such as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;use real world contexts that relates to the life of students&lt;/li&gt;&#xA;&lt;li&gt;use real data in your teaching, not given numbers (or discuss with students realistic guesses)&lt;/li&gt;&#xA;&lt;li&gt;don't oversimplify, but enable students to grapple with complexity ('corse at the appropriate level)&lt;/li&gt;&#xA;&lt;li&gt;respect, courage and agency in teaching are important; thus it is not only about the material but also how you enact with them when teaching.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In this broader sense, I would like to add different resources, trying not to repeat the ones already mentioned in other answers.&#xA;Also, I didn't check the list for the age group appropriate for the materials. And of course this is far from complete...&lt;/p&gt;&#xA;&lt;h2 id=&quot;general&quot;&gt;General:&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Kaslik (2016) &lt;a href=&quot;https://sites.google.com/site/offthebeatenmathpath/math-in-a-vibrant-and-sustainable-society&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math in a Vibrant and Sustainable Society&lt;/a&gt;. Resources on different topics: I like it because it (now) drives on a positive path showing opportunities; many other models are quite pessimistic; this book startet like that and switched...&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://scotdec.org.uk/resources/issue-to-action-maths/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Issue to Action: Maths&lt;/a&gt; Nice examples related to climate, gender and migration.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://cbmp.gse.upenn.edu&quot; rel=&quot;nofollow noreferrer&quot;&gt;Community Based Math Project&lt;/a&gt; from Philly shows ways how to connect mathematical questions to your local environment and see the world around you through 'math lens'.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2 id=&quot;political&quot;&gt;Political:&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Saari (2001). &lt;a href=&quot;https://bookstore.ams.org/elect/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chaotic elections, a mathematician looks at voting&lt;/a&gt;.&#xA;I think Saari's book is a classic showing the different affordances of election systems and showing mathematically that there isn't a perfect one (&lt;a href=&quot;https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arrow's theorem&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2 id=&quot;social&quot;&gt;Social:&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Steen (2001) &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/QL/MathAndDemocracy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics and Democracy&lt;/a&gt; is a classic as well, showing the importance to be able to think about quantities in everyday life and different aspects of it.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.radstats.org.uk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Radical Statistics&lt;/a&gt;. Talking about the use of 'real data' and where to find it; e.g. on&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.gapminder.org/tools/#$chart-type=bubbles&amp;amp;url=v1&quot; rel=&quot;nofollow noreferrer&quot;&gt;GapMinder&lt;/a&gt;. Still one of the best tools to explore real world data!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2 id=&quot;economical&quot;&gt;Economical:&lt;/h2&gt;&#xA;&lt;p&gt;Yes, just about every book on Econometrics or aiming about 'financial literacy' (many others know any good resources on that...)&lt;/p&gt;&#xA;&lt;h2 id=&quot;environmental&quot;&gt;Environmental:&lt;/h2&gt;&#xA;&lt;p&gt;Most of the resources I know... mentioned in other answers and :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://maths4sustainability.wordpress.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maths for sustainability&lt;/a&gt; Many examples mainly related to energy usage.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://usemathanywhere.myfreesites.net&quot; rel=&quot;nofollow noreferrer&quot;&gt;USE*Math&lt;/a&gt; Nice examples of real world math in nature&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Brundiers, K., Barth, M., Cebrián, G. et al. Key competencies in sustainability in higher education—toward an agreed-upon reference framework. Sustain Sci 16, 13–29 (2021). &lt;a href=&quot;https://doi.org/10.1007/s11625-020-00838-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/s11625-020-00838-2&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2021-07-16T20:56:38.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21178" PostTypeId="2" ParentId="8258" CreationDate="2021-07-17T08:47:04.610" Score="0" Body="&lt;p&gt;Crosslink to a similar question on &lt;a href=&quot;https://math.stackexchange.com/questions/1078059/pre-algebra-book-recommendation&quot;&gt;Mathexchange:preAlgebra-books&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Recommended there; and emphasized here: I like the book by Gelfand and Shen as a &amp;quot;plain math / no thrill&amp;quot; book:&lt;/p&gt;&#xA;&lt;p&gt;Gelfand, Shen (2003). &lt;a href=&quot;https://books.google.ch/books/about/Algebra.html?id=Z9z7iliyFD0C&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra&lt;/a&gt;. Boston: Birkhäuser&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2021-07-17T08:47:04.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21179" PostTypeId="1" CreationDate="2021-07-18T17:31:11.010" Score="0" ViewCount="81" Body="&lt;p&gt;I've designed a few applied problems with business analytics/money calculations. However, I'm not sure which age (or rather, which grade) are those best to introduce.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;on one hand, solution uses only + - * / and percents&lt;/li&gt;&#xA;&lt;li&gt;on the other hand, cognitive load is high, because too many word concepts are laid out together&lt;/li&gt;&#xA;&lt;li&gt;they also contain uncertainty, there is not enough information for precise solution, so it requires introducing variables/parameters/assumptions&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I really tried to model business and money stuff.&lt;/p&gt;&#xA;&lt;p&gt;So which grade do those exercises best belong to?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The company XXX spent 300_000 UAH and developed an online MMO game. All the users are predicted to pay cumulatively $10_000 each month. How many months is required for game to pay off?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The company XXX doesn't get all the money users pay it. 30% takes the market place (Apple Store or Google Play), 18% are taxed by govt, it also has to compensate 70_000 UAH/month to each of 2 programmers. Now, when would game pay off?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;30_000 users in-game cumulatively pay \$10000 per month. How many users should you buy additionally, so you have enough money for Lexus RX 350 credit (600_000 UAH)?  You can buy users at Instagram for \$1 per user.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14709" LastActivityDate="2021-07-18T20:49:26.960" Title="Which grade students should be able to solve applied economy / business / profit prediction exercises?" Tags="|secondary-education|applications|exercises|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21180" PostTypeId="1" AcceptedAnswerId="21187" CreationDate="2021-07-18T17:56:34.557" Score="1" ViewCount="311" Body="&lt;p&gt;I am trying to qualify for AIME this year and get a score of at least &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;. (If anyone doesn't know, AIME is the American Invitational Mathematics Examination. You need to get a decent score on the AMC (American Mathematical Contest) to qualify for taking the AIME. I was wondering if there were any books or resources that could be very helpful for the AMC.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="16472" LastEditorUserId="2083" LastEditDate="2021-07-19T06:23:06.967" LastActivityDate="2021-07-20T20:16:33.183" Title="Resources to qualify for AIME" Tags="|usa|math-contests|resources|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="21181" PostTypeId="2" ParentId="21157" CreationDate="2021-07-18T18:05:20.390" Score="1" Body="&lt;p&gt;Not quite &amp;quot;animated&amp;quot;, but still very nice: James Tanton Youtube&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/c/JamesTanton_SquineCosquineTanq/videos&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/c/JamesTanton_SquineCosquineTanq/videos&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;He starts with a kind of problem, sometimes applied, but often just non-standard, and then shows mathematician viewpoint on problem solution.&lt;/p&gt;&#xA;&lt;p&gt;You may start discovering with his &amp;quot;Why I became mathematician&amp;quot; &lt;a href=&quot;https://www.youtube.com/watch?v=b86QcxdrNF4&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=b86QcxdrNF4&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14709" LastActivityDate="2021-07-18T18:05:20.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21182" PostTypeId="2" ParentId="21157" CreationDate="2021-07-18T19:10:10.937" Score="0" Body="&lt;p&gt;A guess animation and real life are opposite? On the real life side, &lt;a href=&quot;https://docs.google.com/spreadsheets/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsfqouLWNNC6Z4/edit#gid=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dan Mayer's collection&lt;/a&gt; of 3-Act Tasks provide many examples with video for the first act. You might need to look for the ones that are appropriate to your age group.&#xA;Note: If you're not familiar with the format of '3-act tasks', you might look at Dan Mayer's TED-talk &lt;a href=&quot;https://www.ted.com/talks/dan_meyer_math_class_needs_a_makeover?utm_campaign=tedspread&amp;amp;utm_medium=referral&amp;amp;utm_source=tedcomshare&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathclass needs a makeover&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Not exactly real life, but animated and real math: I like &lt;a href=&quot;http://www.flatlandthemovie.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Flatland&lt;/a&gt; a lot. The ed-version also comes with activity sheets, plenty of activities to this old &lt;a href=&quot;https://en.wikipedia.org/wiki/Flatland&quot; rel=&quot;nofollow noreferrer&quot;&gt;book from Edwin Abbott&lt;/a&gt; can be found online.&#xA;Actually there's math accessible on many grade level. from just the definition of 'dimension' to the recursive formulas counting k-dim Elements of an n-dim Cube. Very fun! There's a second part introducing the idea of different geometries using the fact, that the angle sum of a triangle is not 180° (thus - wow - we are not living on a plane but on a sphere...). But - don't want to spoil ;-).&lt;/p&gt;&#xA;&lt;p&gt;You might already have encountered Oliver Knill's site on &lt;a href=&quot;http://people.math.harvard.edu/%7Eknill/mathmovies/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math in the Movies&lt;/a&gt;. The site is general, but there are also animated examples appropriate for the age group you're asking for.&#xA;Actually quite the opposite, but I used the episode looking at &lt;a href=&quot;http://people.math.harvard.edu/%7Eknill/mathmovies/swf/bartproblem.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bart Simpson solving word problems&lt;/a&gt; in classes, stimulating good discussions, also on 'real life situations'.&lt;/p&gt;&#xA;&lt;p&gt;hope that's interesting to you.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastEditorUserId="9004" LastEditDate="2021-07-18T19:34:16.457" LastActivityDate="2021-07-18T19:34:16.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21183" PostTypeId="2" ParentId="21179" CreationDate="2021-07-18T20:49:26.960" Score="2" Body="&lt;p&gt;For gifted students, you could do them from algebra on.  Say starting at grade 9 (1age 14).  In some ways, they are very similar to the sorts of applied algebra that are involved in chemistry (but still seem to challenge university students...word problems are hard!)&lt;/p&gt;&#xA;&lt;p&gt;Conceptually even earlier, you could use them, from grad 6 (11).  I would just watch out, that the kids are pretty sharp and have the time.  And maybe make it fun somehow, competition or something.&lt;/p&gt;&#xA;&lt;p&gt;I would not use these with middle or lower track kids, though (ever).&lt;/p&gt;&#xA;&lt;p&gt;And I would use them more as an enrichment, challenge. Not replacing more regular and immediate drill work.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  Just my gut feel, not a teacher.&lt;/p&gt;&#xA;" OwnerUserId="16474" LastActivityDate="2021-07-18T20:49:26.960" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21184" PostTypeId="1" CreationDate="2021-07-19T18:38:28.287" Score="6" ViewCount="332" Body="&lt;p&gt;I am mildly interested in research on mathematics education in the Waldorf/Steiner educational context, primarily in the upper secondary school setting, but not exclusively.&lt;/p&gt;&#xA;&lt;p&gt;Information about Waldorf/Steiner:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Waldorf_education&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Waldorf_education&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.waldorfeducation.org/waldorf-education/in-our-schools/awsna-principles-for-waldorf-schools&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.waldorfeducation.org/waldorf-education/in-our-schools/awsna-principles-for-waldorf-schools&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Even if this is a niche, given the breadth of research in mathematics education, I am somewhat surprised by how little can be found on this via google search. I presume one just needs to know where to look. My questions:&lt;/p&gt;&#xA;&lt;p&gt;What is the state of the art?&#xA;What exactly has been studied?&#xA;How active is the research?&#xA;What are the relevant journals?&#xA;etc.&lt;/p&gt;&#xA;" OwnerUserId="5345" LastEditorUserId="5345" LastEditorDisplayName="user507" LastEditDate="2022-03-25T20:22:42.743" LastActivityDate="2022-03-25T20:22:42.743" Title="Research on Waldorf/Steiner mathematics education" Tags="|secondary-education|reference-request|education-research|big-list|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="21185" PostTypeId="1" AcceptedAnswerId="23253" CreationDate="2021-07-20T03:29:30.250" Score="3" ViewCount="109" Body="&lt;p&gt;This year, I will be teaching a Math Modeling course to a group of motivated high school students. One topic we will discuss is providing good search results from an online query. Here's the problem.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Imagine you run a large online retailer. A potential customer searches for &amp;quot;toaster oven&amp;quot;. You can show them several different offers from different vendors. Vendor 1 has rating 4.3/5.0, price of \$20, and delivery time of 1 week. Vendor 2 has rating 4.7/5.0, price of \$25, and a delivery time of 5 days. Vendor 3 has rating 3.9/5.0, price of \$18, and delivery time of 2 weeks. In what order should you show the results to the customer on your website?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After discussing this specific problem, I will ask students to come up with a formula with variables RATING, PRICE, and DELIVERY TIME that provides reliable results in a variety of situations.&lt;/p&gt;&#xA;&lt;p&gt;One thing I'm lacking is a data set that students can work with. Do you know where to find a data set of several hundred retail items along with a rating, price, and delivery time for each?&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="16484" LastActivityDate="2021-08-09T21:09:28.880" Title="Math Modeling search results data set" Tags="|secondary-education|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21187" PostTypeId="2" ParentId="21180" CreationDate="2021-07-20T20:16:33.183" Score="2" Body="&lt;p&gt;Since no one has yet answered, as suggested in a comment I'm including the links I gave in a comment. Others are welcome to add to this (make sure they're not already included at the first three links), and maybe this should be a &lt;a href=&quot;https://meta.stackexchange.com/questions/11740/what-are-community-wiki-posts&quot;&gt;Community Wiki answer&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://artofproblemsolving.com/store/list/all-products&quot; rel=&quot;nofollow noreferrer&quot;&gt;Art of Problem Solving books&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/press/ebooks/problem-books&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA problem solving books&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.imomath.com/index.php?options=347&quot; rel=&quot;nofollow noreferrer&quot;&gt;Recommend Books at the IMO website&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;After exhausting all of these, a google search for &lt;a href=&quot;https://www.google.com/search?q=math+%22contest%22+%22books%22&amp;amp;filter=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;math + &amp;quot;contest&amp;quot; + &amp;quot;books&amp;quot;&lt;/a&gt; and a google-books search for &lt;a href=&quot;https://www.google.com/search?q=math+%22contest%22&amp;amp;tbm=bks&amp;amp;filter=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;math &amp;quot;contest&amp;quot;&lt;/a&gt; will lead you to additional books.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2021-07-20T20:16:33.183" CommentCount="0" CommunityOwnedDate="2021-07-21T15:40:12.743" ContentLicense="CC BY-SA 4.0" />
  <row Id="21188" PostTypeId="2" ParentId="21185" CreationDate="2021-07-20T21:23:30.540" Score="0" Body="&lt;p&gt;You may have better luck finding data in a different area.&lt;/p&gt;&#xA;&lt;p&gt;Many forms of GEO analytics have rich sets of data for things like: income, political, home prices, etc that may be more practical. The very concept of average, median, whatever can be a challenge by State, County, population vice geographic area. If you compute averages by County, what does that say about the State?&lt;/p&gt;&#xA;&lt;p&gt;Even using recent election data can be an interesting challenge figuring how to break it down and present it.&lt;/p&gt;&#xA;&lt;p&gt;Another potentially interesting data source and analysis is historical Stock Trades incorporating Time, Volume, and Price. It can be quite interesting to see sudden high Volume jumps &lt;strong&gt;preceding&lt;/strong&gt; sudden price changes. It makes you go &lt;em&gt;Hmmmm&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10376" LastActivityDate="2021-07-20T21:23:30.540" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21189" PostTypeId="1" CreationDate="2021-07-20T23:24:12.830" Score="6" ViewCount="214" Body="&lt;p&gt;These days I got curious about something: Are there measurements of how much mathematics people remember after high-school?&lt;/p&gt;&#xA;" OwnerUserId="272" LastActivityDate="2021-07-20T23:24:12.830" Title="Are there measurements of how much mathematics people remember after high-school?" Tags="|secondary-education|reference-request|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21190" PostTypeId="2" ParentId="21184" CreationDate="2021-07-20T23:49:23.657" Score="2" Body="&lt;p&gt;Three references.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia has an article,&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Studies_of_Waldorf_education#2012_US_Study&quot; rel=&quot;nofollow noreferrer&quot;&gt;Studies of Waldorf education&lt;/a&gt;,&#xA;which cites various publications, not, however, specifically focused on mathematics education.&#xA;[Not the same Wikipedia article to which @JoelReyesNoche refers.]&lt;/p&gt;&#xA;&lt;p&gt;For example,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1) A 2012 study compared the reading and math standardized test scores obtained in public Waldorf schools in the United States with scores for their districts as a whole, as well as scores in matched comparison schools. Public Waldorf school scores were lower than those of regular public schools in earlier grades (second and third), but higher in seventh and eighth grade.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Abigail L. Larrison; Alan J. Daly; Carol VanVooren (October 5, 2012). &amp;quot;Twenty Years and Counting: A Look at Waldorf in the Public Sector Using Online Sources&amp;quot;. &lt;em&gt;Current Issues in Education&lt;/em&gt;. 15 (3).&lt;/p&gt;&#xA;&lt;p&gt;Another reference from the same Wikipedia article:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(2) A 2009 PISA study found that, compared to state school students, European Waldorf students are significantly more capable in the sciences, slightly less capable in mathematics; and comparable in reading ability.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(PISA =  Programme for International Student Assessment)&lt;/p&gt;&#xA;&lt;p&gt;And here is statement from &lt;a href=&quot;https://thewaldorfschool.org/mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;a particular US Waldorf school&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(3) WSL’s math curriculum largely mirrors the Massachusetts state standards; however, the range of teaching techniques is broader.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-07-20T23:49:23.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21191" PostTypeId="2" ParentId="20946" CreationDate="2021-07-21T18:27:23.723" Score="1" Body="&lt;p&gt;This is much less rigorous, but I taught my students that numbers or measures can be equal, but objects--like segments, polygons, and angles--cannot be equal, they must be congruent. It was a distinction I gave particularly to help with proofs, where they often need to say two angle measures are equal (and use the equal sign) before saying the angles themselves are congruent (with the congruent symbol).&lt;/p&gt;&#xA;" OwnerUserId="16486" LastActivityDate="2021-07-21T18:27:23.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21192" PostTypeId="2" ParentId="21166" CreationDate="2021-07-22T16:16:07.827" Score="0" Body="&lt;p&gt;Assessment is the measure of progress and decides a student's capability or competence o get into the next level.&lt;/p&gt;&#xA;&lt;p&gt;General ability to solve problems is a broad term.&lt;/p&gt;&#xA;&lt;p&gt;But, as you said -   &amp;quot;Should I assess their ability to use a particular method to solve a quadratic equation, or should I assess their ability to solve a quadratic equation in general?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The term general in the above statement is only allowing flexibility to a student in whatever way he/she can solve a problem to score high. But, does it really assess anything as regards to &amp;quot;general ability to solve problems&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In our experience and opinion, &amp;quot; General ability to solve problems&amp;quot; gets enhanced when a student understands and appreciates the need of solving problem using different techniques. Only then, he truly finds beauty and essence of the different methods and automatically generalized in his/her mind the right applicability of a method to solve problems. When he becomes capable in generalizing different methods or finds connection  between them, their problem solving ability gets enhanced.&lt;/p&gt;&#xA;" OwnerUserId="16098" LastActivityDate="2021-07-22T16:16:07.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21193" PostTypeId="2" ParentId="21139" CreationDate="2021-07-22T16:36:47.883" Score="0" Body="&lt;p&gt;Our take is Real Analysis before Complex Analysis.&lt;/p&gt;&#xA;&lt;p&gt;Real Numbers when couldn't accommodate square root of negative number - complex numbers were introduced. If you consider, complex number as an ordered pair - basically elements of &lt;span class=&quot;math-container&quot;&gt;$R^{2}$&lt;/span&gt; - can that be better conceived without understanding the properties on R.&lt;/p&gt;&#xA;&lt;p&gt;Mathematics teaches us ways to generalize abstract structures and as such Real Analysis comes as a particular case of Complex Analysis. Without having understood the properties on real numbers how can we appreciate the generalization mechanism to higher dimensions.&lt;/p&gt;&#xA;&lt;p&gt;There can be other schools of thought - but we prefer Real Analysis before Complex Analysis.&lt;/p&gt;&#xA;" OwnerUserId="16098" LastActivityDate="2021-07-22T16:36:47.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="21197" PostTypeId="1" AcceptedAnswerId="21199" CreationDate="2021-07-25T05:43:48.227" Score="11" ViewCount="3241" Body="&lt;p&gt;I'm teaching a linear algebra class and I'm considering presenting eigenvectors and eigenvalues without using determinants, as in Axler's book Linear Algebra Done Right. (See also Axler's paper &amp;quot;&lt;a href=&quot;https://www.axler.net/DwD.html&quot; rel=&quot;noreferrer&quot;&gt;Down with determinants!&lt;/a&gt;&amp;quot;.)&lt;/p&gt;&#xA;&lt;p&gt;However, I need students to be able to compute eigenvalues and eigenvectors by hand on homework and quizzes. I am considering having them do it like this: To find the eigenvalues of a matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, we need to find the scalars &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$A - \lambda I$&lt;/span&gt; has a nonzero null vector. We can find those values of &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; just by doing row reduction on &lt;span class=&quot;math-container&quot;&gt;$A - \lambda I$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone foresee any problems with this approach? Is there a better way to handle this?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; The course I'm teaching is a 5 week review of linear algebra &amp;quot;boot camp&amp;quot; for Master's of Science in Data Science students at the University of San Francisco. The textbook is Introduction to Linear Algebra by Strang. The course is not very abstract -- we work mainly in &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt; and do a lot of computations by hand -- but the students tend to be strong. The course could be described as an accelerated, 5 week review of a sophomore level undergraduate introduction to linear algebra course, covering topics such as least squares, eigenvalues and eigenvectors, and the SVD (and PCA). The students come from various technical backgrounds, but most have taken such a sophomore level course (at least) while they were in college.&lt;/p&gt;&#xA;&lt;p&gt;I am also interested in the idea of taking this approach to computing eigenvalues by hand (avoiding determinants) in a sophomore level undergrad introduction to linear algebra course.&lt;/p&gt;&#xA;" OwnerUserId="2049" LastEditorUserId="2049" LastEditDate="2021-07-25T23:31:31.870" LastActivityDate="2021-07-27T04:41:29.150" Title="Computing eigenvalues by hand without determinants" Tags="|linear-algebra|eigenvalues|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="21198" PostTypeId="2" ParentId="21197" CreationDate="2021-07-25T17:24:32.900" Score="4" Body="&lt;p&gt;Presumably, on a homework or quiz, the matrices involved with be 2x2. Let's look at the bookkeeping labor involved for this example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$A=\pmatrix{\ \ 3&amp;amp;1\\ -4&amp;amp;1}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We need to row reduce &lt;span class=&quot;math-container&quot;&gt;$$\pmatrix{3-\lambda&amp;amp;1\\ -4&amp;amp; 1-\lambda}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The main step is &lt;span class=&quot;math-container&quot;&gt;$R_2\to \frac{4}{3-\lambda}R_1+R_2$&lt;/span&gt;&#xA;(Side note, remember to investigate &lt;span class=&quot;math-container&quot;&gt;$\lambda=3$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;p&gt;This results in &lt;span class=&quot;math-container&quot;&gt;$$\pmatrix{3-\lambda&amp;amp; 1\\ 0&amp;amp;\frac{4+(3-\lambda)(1-\lambda)}{3-\lambda}}=\pmatrix{3-\lambda&amp;amp; 1\\ 0&amp;amp;\frac{\lambda^2-4\lambda+7}{3-\lambda}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So the condition we arrive at is that &lt;span class=&quot;math-container&quot;&gt;$\lambda^2-4\lambda+7=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Think about the bookkeeping overhead required, only to arrive at the same quadratic equation which could have been found much quicker with &lt;span class=&quot;math-container&quot;&gt;$\det(A-\lambda I)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On a quiz, is it realistic to ask students to go through this method of row reduction  which has more room for clerical errors, just to find the eigenvalues of a modest 2x2 matrix? Why not point out that in end, we are going to get a quadratic which can be more easily arrived at by using a determinant?&lt;/p&gt;&#xA;&lt;p&gt;If the choice to avoid all mention of determinants is made, then I would either not ask students to find eigenvalues without software, or I would make the pedagogical choice to teach QR, which is among the most important algorithms in numerical linear algebra.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-07-25T18:05:04.570" LastActivityDate="2021-07-25T18:05:04.570" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="21199" PostTypeId="2" ParentId="21197" CreationDate="2021-07-26T13:42:13.140" Score="9" Body="&lt;p&gt;I sympathize with Axler's &amp;quot;Down with Determinants!&amp;quot; because when I was originally taught determinants the definition was offered without any motivation. Later, in an upper division course, I learned that a determinant could be interpreted as the product of the eigenvalues, and it was only at that point that I felt that I understood what the determinant was. This is the definition that Axler proposes using.&lt;/p&gt;&#xA;&lt;p&gt;There is nothing wrong with this approach in principle, but it only makes sense when you're doing linear algebra over the complex numbers. That means that it's incompatible with your chosen text, since Strang does determinants fairly early and gets into complex numbers only at the very end of the book. I don't think it's a good idea to depart so radically from the sequence of topics in your chosen book. You describe two courses, a standard sophomore review class and a 5-week rapid review for students who have already had that class. For the standard intro class, I would predict massive failure. Most of your students will be engineering majors, and although you're at a school with highly selective admissions, I think you'd find many students really struggling to figure out what was going on if you departed this much from the presentation in the text. For the review class, you might be able to get away with more, because they've already seen the subject, but the rapid pace of a 5-week class also leaves less room for error.&lt;/p&gt;&#xA;&lt;p&gt;Axler's main complaint, echoed by my experience, was the lack of proper motivation for the determinant. I looked at the motivation for this topic in the fourth edition of Strang, and IMO it is indeed pretty bad. He introduces the determinant casually in problem set 2.3, #8, with essentially no motivation (or none that a real-world student would be likely to pick up on). Then he never mentions determinants again until ch. 5, where he abruptly starts talking about them in a tone that suggests that students already know all about them.&lt;/p&gt;&#xA;&lt;p&gt;My suggestion would be either to use a better book, or just to use your lectures to patch up these defects in Strang. As an alternative book, &lt;a href=&quot;https://joshua.smcvt.edu/linearalgebra/&quot; rel=&quot;noreferrer&quot;&gt;Hefferon&lt;/a&gt; is really nice, it's free, and the motivation for the determinant seems much more acceptable to me than what Strang does.&lt;/p&gt;&#xA;&lt;p&gt;If I was constrained to teach from Strang or wanted to use it despite the bad presentation of the determinant, then I think my &amp;quot;patch&amp;quot; presentation would go something like the following.&lt;/p&gt;&#xA;&lt;p&gt;We've seen that a matrix can be interpreted as a kind of transformation, a function that takes a point in R^n to some other point in R^n. As with many things in life, these operations can be undoable or not undoable. If you make a typo in your English paper, you use the undo function in your word processor. But if you launch the nuclear missiles, there is no undo.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, it's not always obvious from looking at a matrix whether it's invertible or not. This is mainly because a matrix can be expressed in more than one basis. For example, diag(1,0) is obviously not undoable -- geometrically, it's a projection onto the x axis. But if we rotate our basis vectors by 45 degrees, we get this matrix ..., and in this form it's not at all obvious that the matrix isn't invertible.&lt;/p&gt;&#xA;&lt;p&gt;A nice way to see what's going on here is to take a unit square in the Cartesian plane and see what happens to it when we apply the matrix. Diag(1,0) takes this square and squashes it flat, making its area become zero. Clearly if you eliminate the area of a geometrical figure, you're losing some information, so that's not an invertible operation. Every linear transformation in R^2 has some number associated with it, which is the factor by which it changes areas. That number is called the determinant. Here are some other examples, interpreted geometrically. (Show a rotation and diag(1,2) as examples.) You can see that for a diagonal 2x2 matrix with positive elements, the determinant is the product of the diagonal elements.&lt;/p&gt;&#xA;&lt;p&gt;Our definitions will all actually come out simpler if we reinterpret &amp;quot;area&amp;quot; to be a signed quantity. You've already seen this kind of redefinition in high school geometry and trig, where at some point you learn about negative angles. With this redefinition, the determinant of a diagonal 2x2 matrix is always the product of its diagonal elements. When the determinant is negative, we're mirror-imaging things.&lt;/p&gt;&#xA;&lt;p&gt;Two sweet things about this definition: (1) It's independent of what basis we choose. (2) It's actually not hard to generalize to R^n. In R^1 we would just change &amp;quot;area&amp;quot; to &amp;quot;length&amp;quot;, in R^3 we'd use volume, and so on.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastEditorDisplayName="user507" LastEditDate="2021-07-26T13:53:13.523" LastActivityDate="2021-07-26T13:53:13.523" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21200" PostTypeId="2" ParentId="21197" CreationDate="2021-07-26T14:10:14.180" Score="6" Body="&lt;p&gt;I am going to just play here. Take&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;A = \begin{bmatrix} 1 &amp;amp; 2 \\ 3 &amp;amp; -4 \end{bmatrix}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;Linear Algebra Done Right&amp;quot; idea for establishing existence of eigenvectors is that an eigenvector is invariant after multiplication by &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; (up to scaling).  So it makes sense to iterate &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; in our hunt for an eigenvector.&lt;/p&gt;&#xA;&lt;p&gt;The list of vectors &lt;span class=&quot;math-container&quot;&gt;$\vec{e}_1, A(\vec{e_1}), A^2(\vec{e_1})$&lt;/span&gt; must be linearly dependent since it is a collection of 3 vectors in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\textrm{RREF}\left(&#xA;\begin{bmatrix} 1 &amp;amp; 1 &amp;amp; 7\\ 0 &amp;amp; 3 &amp;amp; -9\end{bmatrix} \right) = \begin{bmatrix} 1 &amp;amp; 0 &amp;amp; 10\\ 0 &amp;amp; 1 &amp;amp; -3\end{bmatrix}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus we have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$A^2(\vec{e}_1) = 10\vec{e_1}+-3A(\vec{e}_1)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Bringing all the terms to one side we have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(A^2+3A-10I)\vec{e}_1 = \vec{0}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This did give us the characteristic polynomial without determinants!  It will not always (such as in the case your first guess actually was an eigenvector).  I am not sure about the relationship in general, aside the the fact that the polynomial you get must be a factor of the characteristic polynomial.&lt;/p&gt;&#xA;&lt;p&gt;Factoring we have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(A+5I)(A-2I)\vec{e}_1 = \vec{0}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So either &lt;span class=&quot;math-container&quot;&gt;$\vec{e_1}$&lt;/span&gt; is an eigenvector with eigenvalue &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; (which it is clearly not) or  &lt;span class=&quot;math-container&quot;&gt;$(A-2I)\vec{e}_1$&lt;/span&gt; is an eigenvector with eigenvalue &lt;span class=&quot;math-container&quot;&gt;$-5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So we have determined that &lt;span class=&quot;math-container&quot;&gt;$(A-2I)(\vec{e_1}) = \begin{bmatrix} -1 \\ 3\end{bmatrix}$&lt;/span&gt; must be an eigenvector with eigenvalue &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;, which we can easily check.&lt;/p&gt;&#xA;&lt;p&gt;This does give us one eigenvector/eigenvalue pair (as it necessarily must), but I think it is no coincidence that we got the eigenvector with greatest magnitude.  Iterating this way would tend to select for this eigenvector.  I am not sure if there is a useful way to extract the other eigenvector for &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; this way.&lt;/p&gt;&#xA;&lt;p&gt;Sorry if this was not much help!&lt;/p&gt;&#xA;&lt;p&gt;EDIT:  Playing again.  What if we factor in the other order?&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(A-2I)(A+5I)\vec{e}_1 = \vec{0}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then we are saying either &lt;span class=&quot;math-container&quot;&gt;$\vec{e_1}$&lt;/span&gt; is an eigenvector with eigenvalue &lt;span class=&quot;math-container&quot;&gt;$-5$&lt;/span&gt; (which it clearly is not), or &lt;span class=&quot;math-container&quot;&gt;$(A+5I)(\vec{e}_1)$&lt;/span&gt; is an eigenvector with eigenvalue &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So we have in fact determined that &lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix} 6 \\ 3\end{bmatrix}$&lt;/span&gt; is an eigenvector with eigenvalue 2!&lt;/p&gt;&#xA;&lt;p&gt;This gives us our basis of eigenvectors.&lt;/p&gt;&#xA;&lt;p&gt;This actually seems pretty satisfying.  I would like to play more with this (especially in the case of higher dimensional eigenspaces, higher dimensions, etc).  Maybe this method is &amp;quot;workable&amp;quot; enough for most purposes of an intro linear algebra class.  I also really like that it combines finding the eigenvalue/eigenvector pair into one piece of reasoning, instead of first finding the eigenvalues and then finding the corresponding eigenvector.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-07-26T14:16:17.667" LastActivityDate="2021-07-26T14:16:17.667" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="21201" PostTypeId="2" ParentId="21197" CreationDate="2021-07-27T04:41:29.150" Score="4" Body="&lt;p&gt;Students tend to remember what they &lt;em&gt;do&lt;/em&gt; much better than what they're told. If this is a student's first introduction to linear algebra, what's more important for them to remember about eigenvalues?&lt;/p&gt;&#xA;&lt;p&gt;I'm personally fond of setting up the equation &lt;span class=&quot;math-container&quot;&gt;$A\pmatrix{x\\y} = \lambda \pmatrix{x\\y}$&lt;/span&gt; and solving the system of equations itself (using row reduction techniques but not writing it as a matrix). This got me really far in some physics classes where we needed to solve for eigenvalues and the corresponding eigenvectors. This approach helps reinforce the definition of eigenvector/value and the relationship between row reduction and solving systems of equations, but you risk students not remembering the existence of the &lt;span class=&quot;math-container&quot;&gt;$\det(A-\lambda I)=0$&lt;/span&gt; equation if they don't use it.&lt;/p&gt;&#xA;&lt;p&gt;Building on user52817's answer, we can solve&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\pmatrix{3&amp;amp;1\\-4&amp;amp;1}\pmatrix{x\\y} = \lambda \pmatrix{x\\y}$$&lt;/span&gt;&#xA;By setting up the equations&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{eqnarray}3x+y=\lambda x\\ -4x+y = \lambda y\end{eqnarray}$$&lt;/span&gt;&#xA;Bring everything to one side:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{eqnarray}(3-\lambda)x+y=0\\ -4x+(1-\lambda)y = 0\end{eqnarray}$$&lt;/span&gt;&#xA;Multiply the top by &lt;span class=&quot;math-container&quot;&gt;$-4$&lt;/span&gt; and the bottom by &lt;span class=&quot;math-container&quot;&gt;$(3-\lambda)$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{eqnarray}-4(3-\lambda)x-4y=0\\ -4(3-\lambda)x+(3-\lambda)(1-\lambda)y = 0\end{eqnarray}$$&lt;/span&gt;&#xA;Subtract top from bottom:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$((3-\lambda)(1-\lambda)-4)y=0$$&lt;/span&gt;&#xA;So &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt; (an important case, leading to the zero eigensystem) or &lt;span class=&quot;math-container&quot;&gt;$\lambda^2-4\lambda+7=0$&lt;/span&gt;&#xA;You can stop here if you just want to solve for &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; or go one step further and plug in to solve for the relationship between &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. This system is underconstrained, reinforcing the idea that you really have an eigenspace, and any multiple of an eigenvector is an eigenvector.&lt;/p&gt;&#xA;&lt;p&gt;This seems to be a modification of what you're doing. I encourage you to give it a try! The only risk is that your students might not remember the existence of the &lt;span class=&quot;math-container&quot;&gt;$\det(A-\lambda I)=0$&lt;/span&gt; trick.&lt;/p&gt;&#xA;&lt;p&gt;Having students solve &lt;span class=&quot;math-container&quot;&gt;$\det(A-\lambda I)=0$&lt;/span&gt; reinforces the skills for calculating determinants, and your students are likely to remember the existence of this formula. But you risk students remembering this equation next semester as just a computation trick, and not hanging on to the theory behind why it works (since this isn't used in applying the formula). The formula does produce a nice aha moment when you do remember why it works, though.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-07-27T04:41:29.150" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="21203" PostTypeId="1" AcceptedAnswerId="21206" CreationDate="2021-07-28T09:53:50.620" Score="19" ViewCount="7549" Body="&lt;p&gt;EDIT (two years later): I was saddened to realise that no-one seems to care at the school level. Everything I thought might be a problem ended up as a non-issue because no-one challenged anything. The word limit is used in class but no one stops to ask what is actually meant by it.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Background: I am new to this site, but have 1500 reputation on the main Maths Stack. I am (age-wise) a secondary student of maths, but for a very long time have been informally learning at home and been studying linear algebra, calculus and formalism that goes far beyond the A-level and Further Maths curriculums. “A-level”, for non-UK readers, means people from 17-18 years old, just starting to learn calculus but without any real formal background - I’ve seen the textbooks and they really just state facts and don’t explain them! I have been helping my friends with their maths for a long time, and have acted as an informal tutor to them. I’ve recently taken up tutoring work for work experience, and I hope to start tutoring A-level students online soon. That may look very pretentious and arrogant, but for context my teachers said I didn’t even need to turn up to A-level class when I formally “begin” the A-level next year.&lt;/p&gt;&#xA;&lt;p&gt;Question:&lt;/p&gt;&#xA;&lt;p&gt;Anyway, I am not a veteran teacher by any means. After the summer, when I’m tutoring for money or helping my friends, inevitably the courses will get to derivative calculus.&lt;/p&gt;&#xA;&lt;p&gt;But when I was studying introductory calculus, I remember being extremely aggravated by not being told the reasons for things. In particular, the thing that really got me and held back my understanding for at least a month or so was: “but aren’t these calculus mathematicians just constantly dividing by zero? How can anyone call this formal! It’s sloppy maths!!” And while that is obviously a false statement that only a naive student would make, I couldn’t get over it - until I did. I can’t remember exactly what helped me: the idea that works for me now is the picture of the derivative as a limit, and the epsilon-delta definition is my friend here, but importantly it will not be taught to my friends or tutees during their time at secondary school!&lt;/p&gt;&#xA;&lt;p&gt;So, I am dreading the day that one of my observant friends or observant tutees asks: “but aren’t we just dividing by zero?” because I can’t give a formal response, and I have zero proper educational background so I don’t know what the &lt;em&gt;intuitive&lt;/em&gt; response should be, that actually works for pupils. I could point them to 3 blue 1 brown, but he does step into formalism here and there which is excellent for the students' learning and general appreciation but it is potentially scary for students just being introduced to calculus. Does anyone have any ideas, experience-backed, for how to take a curious student past the “derivative paradox”? It stumped my learning for a long time and for an A-level student with hardly any time at all that could be disastrous, and I don’t have the heart to tell them to just bottle up their curiosity and plough on with the course - that isn’t maths, that’s oppressive textbook education!&lt;/p&gt;&#xA;&lt;p&gt;As a concrete example (but I invite the answerer to not focus on solely this example):&lt;/p&gt;&#xA;&lt;p&gt;The power law for integer powers (and for non-integers too, but the base A-level doesn't cover the Taylor expansion of the binomial theorem) is very easily proven like this (this is a proof I thought of myself a while ago, I hope it's correct!)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align}\lim_{\delta x\to0}\frac{(x+\delta x)^n-x^n}{\delta x}&amp;amp;=\lim_{\delta x\to0}\frac{\sum_{k=0}^n{n\choose k}(\delta x)^k\cdot x^{n-k}-x^n}{\delta x}\\&amp;amp;=\lim_{\delta x\to0}\frac{(x^n-x^n)+n\cdot\delta x\cdot x^{n-1}+o(\delta x)}{\delta x}\\&amp;amp;=nx^{n-1}+\lim_{\delta x\to0}\frac{o(\delta x)}{\delta x}\\&amp;amp;=nx^{n-1}\end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now my younger self would have had several quibbles with this: &amp;quot;why is &lt;span class=&quot;math-container&quot;&gt;$(\delta x)^2$&lt;/span&gt; &amp;quot;smaller&amp;quot; than &lt;span class=&quot;math-container&quot;&gt;$\delta x$&lt;/span&gt;? &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is just &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, right? And why are we doing maths where there are zeros on both the numerator, and the denominator, that's undefined,... right? What do you mean by &lt;em&gt;vanishingly&lt;/em&gt; small?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;And I am personally a great believer in trying to prove things for oneself when learning; whenever I hear a Wikipedia article or a YouTube video state some fact, I always try to prove it for myself before continuing with the article/video. This is very rewarding, as it hones my formalism and understanding, and I would just love to ask a friend or tutee to prove the power law of derivatives using what they know about the binomial expansion and walk them through it but still leave the burden of proof with them... but I don't think I can, for A-level, judging by what I've seen of the A-level textbooks and so on. There are just too many unanswered quibbles!&lt;/p&gt;&#xA;&lt;p&gt;Any advice is appreciated here. I want the best for my friends/tutees, and unanswerable quibbles due to the necessary evils of standardised, institutional education could damage their motivation/confidence that maths actually works.&lt;/p&gt;&#xA;&lt;p&gt;To be very very precise: my question is about how to specifically soothe a curious student quibbling about the derivative “paradox” in the same way that I once (detrimentally, with no one to guide me) did. I’m not interested in general ideas of whether we should or shouldn’t push rigour into introductory calculus.&lt;/p&gt;&#xA;" OwnerUserId="16527" LastEditorUserId="16527" LastEditDate="2023-05-10T18:48:44.513" LastActivityDate="2023-05-22T04:54:05.267" Title="How does one tutor an A-level student past the derivative paradox?" Tags="|secondary-education|calculus|teaching|derivative|" AnswerCount="20" CommentCount="28" ContentLicense="CC BY-SA 4.0" />
  <row Id="21205" PostTypeId="1" CreationDate="2021-07-28T14:55:22.823" Score="3" ViewCount="53" Body="&lt;p&gt;I use the CCSS(M) and proficiency education to assess students. Is it okay to repeat the standards &lt;a href=&quot;http://www.corestandards.org/Math/Practice/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; when assessing students for proficiency-based education? The idea of these standards in my opinion is to make them repeatable.&lt;/p&gt;&#xA;&lt;p&gt;Proficiency-based education is typically assessed on a scale 0, 1, 2, 3, 4 where a 3 is proficient for any given standard. The idea of proficiency typically means once a student is proficient then they move on. In my opinion, the standards I listed above are repeatable and need to be integrated on their own. I am curious to see what others do who use proficiency-based education.&lt;/p&gt;&#xA;" OwnerUserId="12366" LastActivityDate="2021-07-28T14:55:22.823" Title="Implementing Standards for Mathematical Practice in CCSS(M) (Proficiency Education)" Tags="|common-core|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="21206" PostTypeId="2" ParentId="21203" CreationDate="2021-07-28T15:30:39.193" Score="31" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;There is no royal road to geometry. - Euclid&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Nor calculus.&lt;/p&gt;&#xA;&lt;p&gt;The essence of calculus thinking is really the limit concept. One needs to wrap one's mind around that. Formally: it's the core technique that defines derivatives and integrals. Poetically: it's the eye-of-the-needle through which you must pass to get to the next level of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;There are many educators nowadays who argue that epsilon-delta definitions should be left aside in starting calculus courses, and I'm left perpetually agog at that trend. All of the (rather basic) calculus texts I've ever seen include the (absolutely essential) definition in their introductory chapter on derivatives. To skip over that is just shooting oneself in the foot.&lt;/p&gt;&#xA;&lt;p&gt;So your options are basically just these: (a) teach limits and epsilon-delta proofs so as to digest the real definition/explanation, or (b) hand-wave it and leave it in perpetual ineffable mystery. There's really no other path forward.&lt;/p&gt;&#xA;&lt;p&gt;(I like the Stein/Barcellos text &lt;a href=&quot;https://openlibrary.org/books/OL1533671M/Calculus_and_analytic_geometry.&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus and Analytic Geometry&lt;/em&gt;&lt;/a&gt; (1992) that I taught from in college; the sequence highlights this emphasis, and everything seems clearly laid out -- but I guess it's not one of the &amp;quot;big names&amp;quot; and maybe not radical enough to make a huge splash.)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-05-19T00:10:24.593" LastActivityDate="2023-05-19T00:10:24.593" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="21207" PostTypeId="2" ParentId="21203" CreationDate="2021-07-28T19:20:55.503" Score="19" Body="&lt;p&gt;We routinely get questions related to pushing more rigor in early calculus.  Usually from outstanding students and based on sample of one I like it that way logic.&lt;/p&gt;&#xA;&lt;p&gt;There's a reason why things are the way they are. And that's because most students would get the opposite of a benefit pedagogically by emphasizing increased rigor in early calculus. It's not rationales holding people back.  But lack of drill, algebra and trig working ability, etc.&lt;/p&gt;&#xA;&lt;p&gt;For a weak student, you would be better off to help them by just doing more drill. That builds familiarity.  To the extent you discuss concepts, analogies like tangents and the like will help the more.&lt;/p&gt;&#xA;&lt;p&gt;Really truly. It is not lack of epsilon delta holding back someone from integration by parts. It us lack of practice.  Of something new.&lt;/p&gt;&#xA;&lt;p&gt;Humans are not rule based computers. They are imperfect systems that need practice.&lt;/p&gt;&#xA;" OwnerUserId="16535" LastActivityDate="2021-07-28T19:20:55.503" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="21208" PostTypeId="2" ParentId="21203" CreationDate="2021-07-28T22:04:28.157" Score="12" Body="&lt;p&gt;Start with a numerical example. Say you want to find the gradient of the tangent to &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt;. Obviously the point itself is &lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt;. Pick a nearby point, say &lt;span class=&quot;math-container&quot;&gt;$x=1.1$&lt;/span&gt;. A moment with a calculator shows &lt;span class=&quot;math-container&quot;&gt;$y=1.21$&lt;/span&gt; and the gradient of this chord is &lt;span class=&quot;math-container&quot;&gt;$0.21/0.1=2.1$&lt;/span&gt;. Now pick a closer point, &lt;span class=&quot;math-container&quot;&gt;$x=1.01$&lt;/span&gt;. We again use the calculator to find &lt;span class=&quot;math-container&quot;&gt;$y=1.0201$&lt;/span&gt; and the gradient of this chord is &lt;span class=&quot;math-container&quot;&gt;$0.0201/0.01=2.01$&lt;/span&gt;. And so on. The subsequent terms are &lt;span class=&quot;math-container&quot;&gt;$2.001$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2.0001$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2.00001$&lt;/span&gt;, ... which are clearly approaching &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Once you have built up a bit of numerical intuition, so that they can &lt;em&gt;see&lt;/em&gt; that as the other point gets closer and closer to the tangent point, the gradient gets closer and closer to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, then introduce algebra, with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\delta x$&lt;/span&gt; in place of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$0.000...1$&lt;/span&gt;. Match each algebraic term to the corresponding numerical series. That way, they can see some terms stay the same and others rapidly vanish into tinier and tinier corrections. Then state that there is some mathematical machinery needed to make this &amp;quot;limit&amp;quot; process formal and rigorous, which they will be introduced to later, but for the moment they should accept the intuitive proposal that as the two points get closer together, the chord gets closer to the tangent line, and you can safely assume that those terms shrinking towards zero can be dropped. Indeed, you might need to be careful that they don't &lt;em&gt;go too far the other way&lt;/em&gt; and regard it as &amp;quot;just obvious&amp;quot;. Mention the possibility of sharp corners, and they can probably see how that would go wrong without you having to go into details.&lt;/p&gt;&#xA;&lt;p&gt;Once you have established it for a simple case like &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt;, then do a few more simple polynomial cases using the same numerical-then-algebraic approach, you should be able to go through the steps more quickly, then check to see if they're happy to move to a purely algebraic approach. At any time later if they feel unsure, they can always just stick in a tiny number like &lt;span class=&quot;math-container&quot;&gt;$0.000001$&lt;/span&gt; and see what happens.&lt;/p&gt;&#xA;" OwnerUserId="16536" LastActivityDate="2021-07-28T22:04:28.157" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="21209" PostTypeId="2" ParentId="21203" CreationDate="2021-07-29T00:04:27.673" Score="4" Body="&lt;p&gt;This might be opening up another can of works but have you considered introducing infinitesimals? Like imaginary numbers, they are created by appending a new element to the reals, but instead of root -1 the element added is smaller than any positive real number but greater than 0, effectively the &amp;quot;dx&amp;quot; from the integral. Calculus can be consistently derived from this as well as from limits, and to some people it might seem more intuitive.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand to others it might just be an additional source of confusion, but it might at least help you feel better about handwaving justifications&lt;/p&gt;&#xA;" OwnerUserId="16537" LastActivityDate="2021-07-29T00:04:27.673" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="22209" PostTypeId="2" ParentId="21203" CreationDate="2021-07-29T01:42:36.023" Score="31" Body="&lt;p&gt;I teach calculus at a community college in the U.S. (2 year college, from which many students transfer to a university). I explain limits from about day two in an informal way (&amp;quot;h gets infinitely close to 0&amp;quot;), and talk about the problems with saying infinitely close (but I keep saying it...). I tell students that our learning journey will match the journey made by mathematicians. The 150 years it took between Newton/Leibniz and the precise definition of limits will take us about 6 weeks. I keep saying infinitely close (but not equal), and keep wondering with them how that can be.&lt;/p&gt;&#xA;&lt;p&gt;You said your example wasn't the important part, but I see a few problems with it pedagogically. Using two letters for one variable () can be confusing for students. I have most often seen h used for this, and since I'm not from the U.K., I thought I'd check. I googled something like 'derivative from definition' and found &lt;a href=&quot;https://www.youtube.com/watch?v=IGTcIUriGBs&quot; rel=&quot;noreferrer&quot;&gt;this video&lt;/a&gt;. It looks like it might help you think about this.&lt;/p&gt;&#xA;&lt;p&gt;Also, I hope you would not use the combinatorics notation with students. It looks pretty scary. It helps to keep to one scary new thing at a time. When I teach this, we multiply out (x+h)^2, (x+h)^3, and (x+h)^4. I ask students to look for patterns, and after those three examples they generally see that we always start with x^n + n*x^(n-1). That's all you need. The rest has an h in it that can be factored out. And since h &lt;em&gt;does not equal&lt;/em&gt; 0, the common factor can be canceled. This makes sense, even without the epsilon-delta definition.&lt;/p&gt;&#xA;&lt;p&gt;For your own interest, you might want to check out non-standard analysis. (Berkeley was wrong, Newton's infinitesimals could find a firm foundation. Instead of using limits, we expand our number system.) It doesn't help students to understand the basics, but it's kind of mind-bending.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-07-29T01:42:36.023" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="23209" PostTypeId="2" ParentId="21203" CreationDate="2021-07-29T04:27:04.460" Score="4" Body="&lt;p&gt;The only way I found to explain this is with infinitesimals.  It's not that &lt;span class=&quot;math-container&quot;&gt;$dX$&lt;/span&gt; is vanishingly small... its that it is &amp;quot;infintessimally small.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;My usual approach to explaining all of this is to start with Zeno's paradoxes.  His most famous is good enough -- the idea that you can't run to the end of a football field without first running to the half way point, and you can't get there without running to the quarter way point, and so on.  One might argue that this shows motion is impossible, for it would require infinitely many steps!  Really all it does is show that this way of thinking is insufficient to match our physical reality.&lt;/p&gt;&#xA;&lt;p&gt;Calculus was originally called &amp;quot;the calculus of infintessimals.&amp;quot;  Calculus simply means &amp;quot;a procedural way of calculating something,&amp;quot; but this particular calculus was so astonishingly valuable that it croweded out the term, and now we just refer to it as &amp;quot;calculus.&amp;quot;  It turns out that is mighty difficult to come up with a system of mathematics which handles infintessimals and is &lt;em&gt;consistent&lt;/em&gt;.  Many have tried.  Newton and Liebnitz were the first to come up with a formal way of dealing with infintessimals which was consistent.&lt;/p&gt;&#xA;&lt;p&gt;We see quickly that such a calculus is a resolution to Zeno's paradox, defining a meaningful way to add up an infinite number of infintessimally small segments.  And that, I find, is key to understanding &lt;em&gt;why&lt;/em&gt; we do what we do in calculus.  The purpose of calculus was to solve equations that could only be solved if we could have an infinite number of steps.  To do that, we created limits, derivatives, and integrals, all of which are consistent ways of allowing infintessimals to coexist meaningfully with normal numbers.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2021-07-29T04:27:04.460" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23210" PostTypeId="2" ParentId="21203" CreationDate="2021-07-29T06:37:21.193" Score="8" Body="&lt;p&gt;I admit that I'm unable to follow the proof you give as an example in your question, but am I correct when I assume that your question simply wonders how to reconcile &lt;span class=&quot;math-container&quot;&gt;$dy/dx$&lt;/span&gt; with the fact that &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; approaches zero — and hence is &lt;em&gt;considered zero&lt;/em&gt; by your students?&lt;/p&gt;&#xA;&lt;p&gt;Then I'd simply explain the limit operation visually by exploring a curve. This can be done on a blackboard. The limit operation is essentially a deep zoom into the curve at a given x, much like a zoom into the Mandelbrot set. I'd visually present this as a succession of increasing detailed looks through a symbolic &amp;quot;magnification glass&amp;quot;. And all differentiable curves have one thing in common: When you zoom in deeper and deeper, the detail becomes a straighter and straighter line. The inclination of the emerging straight line at that point is the derivative. That the result becomes more and more exact the deeper you zoom in is obvious, and the mental jump to saying &amp;quot;the virtual end point of this zoom is the result we are looking for&amp;quot; is quite intuitive.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Then&lt;/em&gt; you have a wonderful motivation and opening to present the mathematical tools which perform exactly that process in a rigorous and proven fashion. The rigorous mathematical toolkit of calculus is necessary because intuition leads easily astray with infinite values. Establishing firm theoretical ground is one of the great achievements of humanity.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The crucial point is to understand a limit operation as a process, not a static value.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;When, for a differentiable function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, we write&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f'(x) = \lim \limits_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} = \frac{dy}{dx}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;we state that, when we make &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; smaller and smaller, the fraction gets closer and closer to a finite value. The &lt;em&gt;limit&lt;/em&gt; notation describes the &lt;em&gt;dynamic behavior&lt;/em&gt; of the formula when the variable &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; becomes smaller and smaller. The limit its value approaches is not the result of setting &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; 0; it is the vanishing point of an operation that does not have an end. We can &lt;em&gt;prove&lt;/em&gt; that this vanishing point exists and which value it has, but it is not reached for &lt;em&gt;any&lt;/em&gt; value of &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt;, least of all for 0. The meaning of the formula is that we can get &lt;em&gt;as close as we like&lt;/em&gt; when we make &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; smaller again — a process.&lt;/p&gt;&#xA;" OwnerUserId="11894" LastEditorUserId="11894" LastEditDate="2021-07-30T22:19:57.067" LastActivityDate="2021-07-30T22:19:57.067" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="23211" PostTypeId="2" ParentId="21203" CreationDate="2021-07-29T07:42:37.763" Score="4" Body="&lt;p&gt;I do not know your textbooks, but when I first learned about limits in school (I believe in my school system that happened when we were about 15/16 years old, but it has been an awful long time ago, so I'm not sure), the &lt;span class=&quot;math-container&quot;&gt;$\delta-\epsilon$&lt;/span&gt; approach was used, with a strong geometric aspect. I cannot remember every having any inclination that there were any divisions by zero or anything along these lines.&lt;/p&gt;&#xA;&lt;p&gt;I.e., &lt;span class=&quot;math-container&quot;&gt;$\lim$&lt;/span&gt; was introduced right from the start like in this picture:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Limit_(mathematics)#/media/File:L%C3%ADmite_01.svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Limit_(mathematics)#/media/File:L%C3%ADmite_01.svg&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The formalism &amp;quot;Whenever a point x is within a distance δ of c, the value f(x) is within a distance ε of L.&amp;quot; translates directly to simply drawing lines into that image. Don't start out by drawing a tangent at &lt;span class=&quot;math-container&quot;&gt;$(x,f(x))$&lt;/span&gt;, but draw a line from &lt;span class=&quot;math-container&quot;&gt;$(x-\delta,f(x-\delta)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(x+\delta,f(x+\delta)$&lt;/span&gt;. Then pick a smaller &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; and draw more lines. It is intuitively visible that they converge to the tangent.&lt;/p&gt;&#xA;&lt;p&gt;At this point, you can point out the relationship of the quotient &lt;span class=&quot;math-container&quot;&gt;$\epsilon/\delta$&lt;/span&gt; to the inclination/angle of the line.&lt;/p&gt;&#xA;&lt;p&gt;Obviously you should pick a beneficial example at first, nothing too weird. &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt;, and don't pick &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; as your &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The point of the exercise is not primarily to show that there is a tangent here. Of course that is useful too for an intuitive understanding in the case that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a school-level &amp;quot;simple&amp;quot; function. But the main point is that it is obvious to see that even though &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; gets ever smaller, and &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; does so too, there is no danger of ever dividing anything by zero.&lt;/p&gt;&#xA;&lt;p&gt;Take as long as they need to understand this very simple beginner concept. Then pick one of the other aspects, for example, the formal &lt;span class=&quot;math-container&quot;&gt;$\delta-\epsilon$&lt;/span&gt; definition. Or you can do a hard switch and talk about series limits, just to drive home the point that &amp;quot;limit&amp;quot; is never something that happens at a particular point (be it at some &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; where you want to calculate &lt;span class=&quot;math-container&quot;&gt;$f'(c)$&lt;/span&gt;, or in the case of a limit towards infinity), but that saying &amp;quot;limit&amp;quot; means a short hand for a &amp;quot;moving target&amp;quot; when going towards some value (or going on forever in the case of infinity).&lt;/p&gt;&#xA;&lt;p&gt;If I recall correctly, we actually &lt;em&gt;did&lt;/em&gt; derive the usual common limits in school by plugging things into the formal definition; i.e. the &amp;quot;rules&amp;quot; were not simply written down and then followed by rote. You definitely can try that as well, you do not need uni level education for that.&lt;/p&gt;&#xA;" OwnerUserId="8146" LastEditorUserId="8146" LastEditDate="2021-08-02T06:48:53.917" LastActivityDate="2021-08-02T06:48:53.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23212" PostTypeId="2" ParentId="21203" CreationDate="2021-07-29T09:33:08.407" Score="1" Body="&lt;p&gt;Personally I found the calculus is done in two parallel tracks: 1) on how essentially the derivatives and integrals are, and 2) the delta-epsilon language to be rigorous. Sometimes there needs a &amp;quot;translation&amp;quot; in the mind when one tries to understand or compose a proof.&lt;/p&gt;&#xA;&lt;p&gt;I would like introduce &amp;quot;Calculus made easy&amp;quot; by Silvanus P. Thompson, even better the version with Martin Gardener's notes, it established the essential ideas of calculus with the original Newton's languages.&lt;/p&gt;&#xA;&lt;p&gt;For &amp;quot;divided by 0&amp;quot; paradox, I would recommend &lt;a href=&quot;https://math.stackexchange.com/questions/12906/the-staircase-paradox-or-why-pi-ne4&quot;&gt;https://math.stackexchange.com/questions/12906/the-staircase-paradox-or-why-pi-ne4&lt;/a&gt;  as an example to show that what matters here is the ratio of two infinitesimal amounts&lt;/p&gt;&#xA;" OwnerUserId="9059" LastActivityDate="2021-07-29T09:33:08.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23213" PostTypeId="2" ParentId="21203" CreationDate="2021-07-29T12:58:29.120" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a reason why the binomial theorem and calculus were the subject of study by Newton---they &lt;em&gt;are&lt;/em&gt; closely related!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There are computational methods that work consistently and then there are &lt;em&gt;proofs&lt;/em&gt; that these methods are &amp;quot;correct&amp;quot;. This represents two &lt;em&gt;different&lt;/em&gt; approaches to mathematics. Many working mathematicians were (in their youth) enchanted by proofs and so that has become the dominant approach. It does not &lt;em&gt;have&lt;/em&gt; to be so!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A lot of early mathematics is more about computational methods that are consistent. One &lt;em&gt;does not&lt;/em&gt; fully justify them with proofs. Practical and heuristic examples are used instead. Counter-examples are used to limit their applicability.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For example, we say that multiplication is repeated addition. To what extent is this &lt;em&gt;proved&lt;/em&gt; in school? Proving basic arithmetical laws for integers requires induction, let alone for rational numbers and don't even bring up decimal or real numbers. We talk about angles and distances being measured even though proving that such measurements have the right properties requires limits; in the case of angles it is even more tricky than for distances since they can only be bisected. So, from the point of view of &amp;quot;proof&amp;quot; a lot is missing in school mathematics. Why are we balking at calculus?!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Coming to your question, I believe it is &lt;em&gt;reasonable&lt;/em&gt; to introduce calculus via an algebraic method like: (1) &lt;span class=&quot;math-container&quot;&gt;$a(x+h)-ax = ah$&lt;/span&gt; to justify the linear case followed by (2) &lt;span class=&quot;math-container&quot;&gt;$(x+h)(y+g)=xy+(xg+yh)+\dots$&lt;/span&gt; as an introduction to Leibniz rule. When &lt;span class=&quot;math-container&quot;&gt;$g,h&amp;lt;1$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$gh&amp;lt;\min(g,h)$&lt;/span&gt;. This can be used to explain why we are &lt;em&gt;uninterested&lt;/em&gt; in the remaining terms. One can use similar ideas to justify the integral &lt;span class=&quot;math-container&quot;&gt;$\int_a^b x dx = (b^2-a^2)/2$&lt;/span&gt; and use this heuristic to justify its extension to &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;TL;DR: The Leibniz rule is &amp;quot;king&amp;quot; as it is a useful computational method. Justifying it on the basis of &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proofs is an afterthought.&lt;/p&gt;&#xA;" OwnerUserId="13498" LastEditorUserId="13498" LastEditDate="2021-07-29T13:04:02.880" LastActivityDate="2021-07-29T13:04:02.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23214" PostTypeId="2" ParentId="21203" CreationDate="2021-07-29T20:11:12.283" Score="4" Body="&lt;p&gt;I am a father of 14 and 17 yo children in France. I have a rudimentary understanding of math through my studies (PhD in physics) and I always used math as a useful toolbox.&lt;/p&gt;&#xA;&lt;p&gt;When my older kid had derivatives (during COVID, which in France was an educatory disaster), he was disappointed because he could not understand what this was after reading his book.&lt;/p&gt;&#xA;&lt;p&gt;It was very much understandable: the first phrase is about R that maps to R and then comes a fraction with a lim in front.&lt;/p&gt;&#xA;&lt;p&gt;To me, this is the worst way to teach something - physics was guilty of the same thing by dropping a formula and good luck with that.&lt;/p&gt;&#xA;&lt;p&gt;I started with a physical example (velocity, then heat), showing how the derivative is built. The limit was quite easily digested, and then he had a eureka moment when he understood that &lt;strong&gt;a derivative is a function factory&lt;/strong&gt;: a function comes in, and a completely different function comes out, carrying some interesting features of the input one.&lt;/p&gt;&#xA;&lt;p&gt;All this was purposedly discussed with a lot of hand waving (as we do in physics), sacrifying the precision on the altar of understanding.&lt;/p&gt;&#xA;&lt;p&gt;He got everything, came back to the initial formula which then was clear, and easily went through the topic.&lt;/p&gt;&#xA;" OwnerUserId="8076" LastEditorUserId="8076" LastEditDate="2022-05-31T09:11:51.293" LastActivityDate="2022-05-31T09:11:51.293" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23215" PostTypeId="2" ParentId="21203" CreationDate="2021-07-29T21:11:16.283" Score="-2" Body="&lt;p&gt;As an intuitive aid to the calculus I like to explain that infinity is a number so big that it wouldn't make any difference if the you made it bigger. This is really an epsilon-delta argument in a disguise that makes it look friendlier. It can adapted to any limit situation.&lt;/p&gt;&#xA;" OwnerUserId="17545" LastActivityDate="2021-07-29T21:11:16.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23216" PostTypeId="2" ParentId="21203" CreationDate="2021-07-30T04:25:27.150" Score="7" Body="&lt;p&gt;First off - I 100% agree with Collins here that there's no shortcuts. To &lt;em&gt;really&lt;/em&gt; understand how derivatives work, you need to learn the epsilon-delta definition. But it's my sense that you're not really shooting for that here - it sounds like what you want is a way to &lt;em&gt;convince&lt;/em&gt; a student, not necessarily a way to &lt;em&gt;prove&lt;/em&gt; it to them. The difference can be hard for someone advanced in math to spot, because you're so used to proving things that you've forgotten that you once didn't know how!&lt;/p&gt;&#xA;&lt;p&gt;When it comes to derivatives: the funny thing is, most students (I speak from the perspective of the US math curriculum here) are actually &lt;em&gt;already comfortable&lt;/em&gt; with one example of dividing by zero - removable discontinuities, or &amp;quot;holes&amp;quot;. For example, the function&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{x(x + 2)}{x}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;has a &amp;quot;hole&amp;quot; in it at &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt;, because &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt; is undefined. But this is probably very much a &amp;quot;well, &lt;em&gt;technically&lt;/em&gt;&amp;quot; sort of situation for them - &lt;em&gt;technically&lt;/em&gt; it's not defined at &lt;span class=&quot;math-container&quot;&gt;$x = 0$&lt;/span&gt;, but look, you can just cancel the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s and it works just fine, filling the hole. Thus, in this case, it makes sense to say that what was &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt; turns out to be &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You can move from there into a convincing explanation of why derivatives work: we're not dividing by zero, we're filling a hole! The trick is to find a way to write the function that admits the cancellation you need. Of course, that isn't always easy or even possible with pre-calculus techniques; but in my experience most students are prepared to accept &amp;quot;it makes sense in this case, and it will make sense in these other cases once you learn a bit more&amp;quot; as a convincing argument.&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2021-07-30T04:25:27.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23217" PostTypeId="2" ParentId="21203" CreationDate="2021-07-30T12:50:53.370" Score="2" Body="&lt;p&gt;Hopefully, they know some basic physics, e. g., that if you release a heavy object, its velocity grows uniformly, and after &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; seconds it will have velocity &lt;span class=&quot;math-container&quot;&gt;$gt$&lt;/span&gt;. Now, ask them, if they had to prove this experimentally, how would they go about it? They probably will come up with some experimental design involving stopwatch, then you point out that they are in fact measuring average velocities and not momentary velocities. Then you may discuss how you get better and better validation of the uniform acceleration law by taking finer instruments and smaller intervals between measurements, but ultimately there's some kind of &amp;quot;right answer&amp;quot; that you are approaching with these improvements.&lt;/p&gt;&#xA;&lt;p&gt;The obvious advantage here is that they have already an internalized notion of momentary velocity and have no intuitive doubts that such a thing exists.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2021-07-30T12:50:53.370" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23219" PostTypeId="2" ParentId="21203" CreationDate="2021-07-30T18:48:00.263" Score="4" Body="&lt;p&gt;So many answers, but not yet pointed out is that your proposed proof of the derivative of &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is wrong because it uses the wrong notation. There is a difference between Big-O and little-o (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Landau_notation#Family_of_Bachmann%E2%80%93Landau_notations&quot; rel=&quot;nofollow noreferrer&quot;&gt;the formal definitions of Landau notation&lt;/a&gt; for details).&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\def\lfrac#1#2{{\large\frac{#1}{#2}}}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;That said, there is a way to make things rigorous without sacrificing on the intuition. See, suppose you have variables &lt;sup&gt;[1]&lt;/sup&gt; &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; that are varying with respect to &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; (e.g. time) such that &lt;span class=&quot;math-container&quot;&gt;$\lfrac{dx}{dt}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\lfrac{dy}{dt}$&lt;/span&gt; are both defined at any point. We can focus our attention on any specific point, and consider any other point (with a different value for &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;). Let &lt;span class=&quot;math-container&quot;&gt;$Δx,Δy,Δt$&lt;/span&gt; be the changes in &lt;span class=&quot;math-container&quot;&gt;$x,y,t$&lt;/span&gt; when you move from the first to the second point. By definition, &lt;span class=&quot;math-container&quot;&gt;$\lfrac{Δx}{Δt} ≈ \lfrac{dx}{dt}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$Δt → 0$&lt;/span&gt;, which can be intuitively expressed by saying that, if &lt;span class=&quot;math-container&quot;&gt;$Δt$&lt;/span&gt; eventually gets close enough but not equal to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\lfrac{Δx}{Δt}$&lt;/span&gt; eventually stays close enough to &lt;span class=&quot;math-container&quot;&gt;$\lfrac{dx}{dt}$&lt;/span&gt;. Similarly &lt;span class=&quot;math-container&quot;&gt;$\lfrac{Δy}{Δt} ≈ \lfrac{dy}{dt}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$Δt → 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(x·y)}{dt}$&lt;/span&gt;? &lt;span class=&quot;math-container&quot;&gt;$Δ(x·y) = (x+Δx)·(y+Δy)-x·y$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$= x·Δy+Δx·y+Δx·Δy$&lt;/span&gt;, so we have &lt;span class=&quot;math-container&quot;&gt;$\lfrac{Δ(x·y)}{Δt} = x·\lfrac{Δy}{Δt}+\lfrac{Δx}{Δt}·y+\lfrac{Δx}{Δt}·\lfrac{Δy}{Δt}·Δt$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$≈ x·\lfrac{dy}{dt}+\lfrac{dx}{dt}·y+\lfrac{dx}{dt}·\lfrac{dy}{dt}·Δt$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$≈ x·\lfrac{dy}{dt}+\lfrac{dx}{dt}·y$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$Δt → 0$&lt;/span&gt;. Why can we make these approximations? Because &lt;span class=&quot;math-container&quot;&gt;$x,y,\lfrac{dx}{dt},\lfrac{dy}{dt}$&lt;/span&gt; are all finite, and approximate equalities are closed under addition and multiplication by a finite factor, and because &lt;span class=&quot;math-container&quot;&gt;$Δt ≈ 0$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$Δt → 0$&lt;/span&gt;. Therefore &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(x·y)}{dt} = x·\lfrac{dy}{dt}+\lfrac{dx}{dt}·y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So we have rigorously proven the product rule for derivatives while keeping the algebraic benefits of Leibniz notation. We can easily obtain &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^k)}{dt}$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$k∈ℕ^+$&lt;/span&gt; using the product rule and induction. Trivially &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^1)}{dt} = \lfrac{dt}{dt} = 1 = 1·t^0$&lt;/span&gt;. Given any &lt;span class=&quot;math-container&quot;&gt;$k∈ℕ^+$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^k)}{dt} = k·t^{k-1}$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^{k+1})}{dt} = \lfrac{d(t^k·t)}{dt}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$= t^k·\lfrac{dt}{dt}+\lfrac{d(t^k)}{dt}·t$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$= t^k·1+k·t^{k-1}·t = (k+1)·t^k$&lt;/span&gt;, by the product rule. Therefore by induction we get the desired theorem.&lt;/p&gt;&#xA;&lt;p&gt;Again, completely rigorous, and yet without any need to have any unclear bound on many terms.&lt;/p&gt;&#xA;&lt;p&gt;We can extend this to negative integers and even rational powers. Consider any point at which &lt;span class=&quot;math-container&quot;&gt;$t ≠ 0$&lt;/span&gt;. Take any &lt;span class=&quot;math-container&quot;&gt;$k∈ℕ^+$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^k·t^{-k})}{dt} = \lfrac{d(1)}{dt} = 0$&lt;/span&gt;. Also &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^k·t^{-k})}{dt} = t^k·\lfrac{d(t^{-k})}{dt}+\lfrac{d(t^k)}{dt}·t^{-k}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$= t^k·\lfrac{d(t^{-k})}{dt}+k·t^{k-1}·t^{-k} = t^k·\lfrac{d(t^{-k})}{dt}+k·t^{-1}$&lt;/span&gt; by the product rule. Thus we have &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^{-k})}{dt} = -t^{-k}·(k·t^{-1}) = -k·t^{-k-1}$&lt;/span&gt;, as desired.&lt;/p&gt;&#xA;&lt;p&gt;I leave it as an exercise for you to extend it to rational powers, which can be done via the same tools. Hint: Take any &lt;span class=&quot;math-container&quot;&gt;$p,q∈ℤ$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$q &amp;gt; 0$&lt;/span&gt;. Consider any point at which &lt;span class=&quot;math-container&quot;&gt;$t &amp;gt; 0$&lt;/span&gt;. Then prove by induction that &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^{k/q})}{dt} = k·t^{(k-1)/q}·\lfrac{d(t^{1/q})}{dt}$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$k∈ℕ$&lt;/span&gt;. Thus &lt;span class=&quot;math-container&quot;&gt;$1 = \lfrac{d(t^{q/q})}{dt} = q·t^{(q-1)/q}·\lfrac{d(t^{1/q})}{dt}$&lt;/span&gt;, and so &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^{1/q})}{dt} = 1/q·t^{1/q-1}$&lt;/span&gt;, yielding &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^{k/q})}{dt} = k/q·t^{k/q-1}$&lt;/span&gt; as desired.&lt;/p&gt;&#xA;&lt;p&gt;This formalization of derivatives can easily extend to implicit differentiation. We simply define &lt;span class=&quot;math-container&quot;&gt;$\lfrac{dy}{dx} = r$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is a constant and &lt;span class=&quot;math-container&quot;&gt;$Δx → 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\lfrac{Δy}{Δx} ≈ r$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$Δt → 0$&lt;/span&gt;. Intuitively, &lt;span class=&quot;math-container&quot;&gt;$\lfrac{dy}{dx}$&lt;/span&gt; is defined to be &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is a fixed constant and moving the other point close enough causes the value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to also move close enough and the ratio of &lt;span class=&quot;math-container&quot;&gt;$Δy$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$Δx$&lt;/span&gt; to stay close enough to &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now we can easily prove the chain rule &lt;span class=&quot;math-container&quot;&gt;$\lfrac{dz}{dx} = \lfrac{dz}{dy}·\lfrac{dy}{dx}$&lt;/span&gt; at any point where &lt;span class=&quot;math-container&quot;&gt;$\lfrac{dz}{dy},\lfrac{dy}{dx}$&lt;/span&gt; are both defined. I leave that as an exercise.&lt;/p&gt;&#xA;&lt;p&gt;Implicit differentiation is of course useful in practical applications of derivatives to many fields, and &lt;a href=&quot;http://math.stackexchange.com/a/2176223/21820&quot;&gt;here&lt;/a&gt; is one example.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, it is actually mathematically difficult to prove &lt;span class=&quot;math-container&quot;&gt;$\lfrac{d(t^r)}{dt} = r·t^{r-1}$&lt;/span&gt; for arbitrary &lt;span class=&quot;math-container&quot;&gt;$r∈ℝ$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t &amp;gt; 0$&lt;/span&gt;, in the sense that the easiest way is probably via the exponential function and natural logarithm and via the definition of &lt;span class=&quot;math-container&quot;&gt;$t^r = \exp(r·\ln(t))$&lt;/span&gt;. None of these are easy, so I would say that we cannot handle this at the high-school level.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;sup&gt;[1]&lt;/sup&gt; Here &amp;quot;variable&amp;quot; is in the older sense of a symbol used to denote a varying quantity, and not in the modern sense of a symbol in a logical formalism of mathematics.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-07-30T18:48:00.263" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="23220" PostTypeId="2" ParentId="21203" CreationDate="2021-07-30T19:49:33.847" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;my question is about how to specifically soothe a curious student quibbling about the derivative “paradox” in the same way that I once (detrimentally, with no one to guide me) did. I’m not interested in general ideas of whether we should or shouldn’t push rigour into introductory calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Do not soothe them. Encourage them. Cheer them on as they fight this fight.&lt;/p&gt;&#xA;&lt;p&gt;No one will challenge your own understanding of a subject like a curious student. But rather than fear this challenge embrace it because if you listen well you're about to discover whole different ways to think about this. This doesn't mean you're way of thinking about it is wrong. It's simply not the only correct way.&lt;/p&gt;&#xA;&lt;p&gt;What the student is searching for is a way to think about this that they can fit into their own way of thinking about math. Now sure, they need to think about it correctly but they need a correct way that works for them. The problem is, you likely have no idea what that is. But that's fine. Just listen to them. When you spot misconceptions correct them gently. Maybe with your own penetrating questions, thought experiments, and stories. Be prepared to be wrong. Take it with grace. Don't dictate how they think. Judge their methods by the results. Hinting at your way of thinking may help but don't insist on it. Make it fun.&lt;/p&gt;&#xA;&lt;p&gt;As an example let me tell you I've successfully tutored a Summa Cum Laude student in mathematics. I've tutored, in a university mathlab, calculus, linear algebra, differential equations, and statistics. And I still can't do times table flash cards.&lt;/p&gt;&#xA;&lt;p&gt;Yeah my brain just doesn't work that way. I've found ways around that problem. I know a lot of patterns that let me work out the answer but some cards still leave me stammering.&lt;/p&gt;&#xA;&lt;p&gt;Yet after working in the lab I can factor fairly large numbers in my head.&lt;/p&gt;&#xA;&lt;p&gt;So yeah, I'm weird. And it makes it hard to teach me. Because I come at the typical problems in very different ways. And some teachers and tutors can't handle that.&lt;/p&gt;&#xA;&lt;p&gt;But what I've learned over the years is that I'm not that weird. Many people have their own styles that play to their strengths rather than their weaknesses.&lt;/p&gt;&#xA;&lt;p&gt;Because of that my advice is to listen to them. Learn what you can about how they think and how they model this problem in their heads. Test what they can relate to and use that.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“but aren’t we just dividing by zero?”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is a voice calling out from many previous lessons about this being undefined. You are about to change how they look at the whole universe so don't trivialize this question. Take them seriously.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Yes we are. Kinda. But we're telling the story of how you get to it. Without that story it's still undefined. We're trying to keep zero from destroying that information.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;What that answer is attempting to do is let them know that their previous lessons weren't useless lies. They were lies leading to a deeper more useful lie because that's what all models are. If they weren't they'd be the real thing.&lt;/p&gt;&#xA;&lt;p&gt;That answer might not land with them. But it is trying to speak to them at the same level. They want a simple explanation of how this fits with what they learned before and what's changed since then. Find some way to explain that to them that will stand up to the lessons yet to come.&lt;/p&gt;&#xA;" OwnerUserId="3305" LastEditorUserId="3305" LastEditDate="2021-07-30T22:35:07.790" LastActivityDate="2021-07-30T22:35:07.790" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23222" PostTypeId="1" CreationDate="2021-08-01T13:57:18.630" Score="-4" ViewCount="180" Body="&lt;p&gt;In one of my programs I have a function I call &lt;code&gt;reduce(n)&lt;/code&gt; which associates to &lt;code&gt;n&lt;/code&gt; the recursive sum of &lt;code&gt;n&lt;/code&gt;'s digits until this sum is one digit long. For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;12 -&amp;gt; [1, 2] =&amp;gt; 3&lt;/li&gt;&#xA;&lt;li&gt;149 -&amp;gt; [1, 4, 9} -&amp;gt; 14 -&amp;gt; [1, 4] =&amp;gt; 5&lt;/li&gt;&#xA;&lt;li&gt;and so on&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Is there an official name for this operation ?&lt;/p&gt;&#xA;" OwnerUserId="16328" LastActivityDate="2021-08-08T14:37:57.410" Title="Is there an **official** name for the following &quot;digit reduction&quot; operation?" Tags="|terminology|arithmetic|" AnswerCount="1" CommentCount="3" ClosedDate="2021-08-02T00:56:57.267" ContentLicense="CC BY-SA 4.0" />
  <row Id="23223" PostTypeId="2" ParentId="23222" CreationDate="2021-08-01T14:00:47.623" Score="4" Body="&lt;p&gt;A 5 second Google (is your friend) shows the operation referred to as digit sum or sum of digits.  WA even has an operator that performs this.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?&amp;amp;q=sum+of+digits&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?&amp;amp;q=sum+of+digits&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Tangentially related, you might also look at the concept of the checksum.  I'm familiar with that from military navigation/targeting, but the wiki article discusses a lot of other uses.  It's tangential, I guess since, per wiki, the checksum is not forced down recursively to a single digit.  (Although I've seen some military use where it was, because the field had a defined digit size of 1.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Checksum&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Checksum&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17560" LastEditorUserId="-1" LastEditDate="2021-08-08T14:37:57.410" LastActivityDate="2021-08-08T14:37:57.410" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23225" PostTypeId="1" CreationDate="2021-08-02T12:13:54.137" Score="2" ViewCount="598" Body="&lt;p&gt;So, in more advanced courses and fields in math like undergraduate and more advanced, how does one make a proof? He makes trials of everything he knows about the problem? What are the instructions to make proofs? When someone reads theorems and other proofs should he make preparations to answer the problems he will try to answer?&lt;/p&gt;&#xA;&lt;p&gt;When i try i mostly fail solving the problems. Especially in courses like introduction to metric spaces or general topology. What should i do to correct this? When i read the answers to the problems the process of proving them confuses me. Could you help in this?&lt;/p&gt;&#xA;&lt;p&gt;Also, because i have failed in exams sometimes, and i found them difficult, i find difficulty trying to read and solve problems because of fear.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastActivityDate="2021-08-03T17:23:37.333" Title="Trials and instructions on proofs" Tags="|proofs|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="23226" PostTypeId="2" ParentId="23225" CreationDate="2021-08-02T12:55:23.880" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;OP: &amp;quot;how does one make a proof?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This (or its equivalent) could help:&lt;/p&gt;&#xA;&lt;p&gt;Velleman, Daniel J. &lt;em&gt;How to prove it: A structured approach&lt;/em&gt;. Cambridge University Press, 2019. 2nd Edition.&#xA;&lt;a href=&quot;https://www.cambridge.org/core/books/how-to-prove-it/6D2965D625C6836CD4A785A2C843B3DA&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cambridge link&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;     &lt;a href=&quot;https://i.stack.imgur.com/G6jmc.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/G6jmc.jpg&quot; alt=&quot;cover&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-08-02T12:55:23.880" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23227" PostTypeId="2" ParentId="23225" CreationDate="2021-08-02T13:47:34.057" Score="10" Body="&lt;p&gt;There's no simple switch to turn on to rectify this.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You need to work.  And keep working.  Not trying from fear is not going to get you anywhere.  If you have worse skill/capability, you probably need to work even MORE than those with higher skill.  And &amp;quot;giving up&amp;quot; or &amp;quot;not doing homework&amp;quot; are the most common reasons for people not progressing in a course...and especially with self study (one of your interests).  &amp;quot;Gotta work&amp;quot; may sound trivial but it is not. It's a common failing.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You need to do partial work even on problems that intimidate you.  Rewrite the statement.  Draw a diagram.  Do a trivial manipulation.  At least it gets you into the topic a little bit more actively, versus passive looking at it.  &lt;strong&gt;How you just came here for an answer (versus doing a Google search and sharing the insights) &lt;em&gt;likely reflects a failing you have in your proof work as well&lt;/em&gt;.&lt;/strong&gt;  You need to do AND show some scratch.  Not just expect a deus ex machina.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to find/use books appropriate for your skill level.  There is a very bad tendency on the web (SE especially) of people recommending only the toughest books, even for students that ask for an easy one.  If you know you are struggling with benchpressing 225#, back off and try 135#. Or 115# (or whatever).  You can do the Rudin and Spivak and all that jazz later (if you choose).  But be very wary of following the recommendations for the most skilled, advanced (or even already know the topic!) students.  There's a reason why Feynman lectures failed at Cal Tech.  And why students (even there!) benefited more from a standard approach, at least at first.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Make sure that you have (or at least firm up as you go), manipulation ability and prerequisite course skills.  For instance, I was weak at partial fractions (in algebra class), but in the course of integral calculus and ODEs (Laplace transforms), I corrected that issue.  But if you come on something like that (e.g. some aspect of set theory, or of series convergence testing), don't dismiss the error as from an earlier course.  Look at it as a &amp;quot;free&amp;quot; opportunity to practice, upgrade, or familiarize yourself with that skill also.  I would not get so diverted that you are unwilling to try any new material.  But at least firm up old things that were not quite mastered.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="17566" LastActivityDate="2021-08-02T13:47:34.057" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23228" PostTypeId="1" CreationDate="2021-08-02T18:43:50.010" Score="5" ViewCount="3209" Body="&lt;p&gt;Specifically the one updated by Martin Gardner. I'm not studying as part of a high school or college course (I, in the near future, will though) just as a personal project.&lt;/p&gt;&#xA;" OwnerUserId="17571" LastActivityDate="2021-08-06T18:55:43.737" Title="Is Calculus Made Easy by Silvanus P. Thompson a good book for first-time calculus learners?" Tags="|secondary-education|calculus|textbooks|self-learning|recreational-learning|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="23229" PostTypeId="2" ParentId="23228" CreationDate="2021-08-02T19:21:15.883" Score="9" Body="&lt;p&gt;Yes. The book is well written and fun. It cuts out a ton of the ridiculous cruft that has clogged up the current crop of commercial freshman calculus texts. IIRC Gardner moderates some of the dated and sexist language.&lt;/p&gt;&#xA;&lt;p&gt;Be aware of how this book's approach relates to the history and to currently fashionable ways of presenting calculus. It uses infinitesimals rather than limits. Both infinitesimals and limits are perfectly fine ways of introducing calculus. Historically, there were some concerns that infinitesimals were inherently inconsistent, but it turns out that that's not the case.&lt;/p&gt;&#xA;&lt;p&gt;When teaching yourself something intellectually challenging, I would always suggest that you find multiple sources of information rather than just focusing on one book. There is a text by Keisler that is free online and introduces freshman calculus using infinitesimals, but with a bit more of the modern machinery. Compared to Thompson, it's boring and pedantic and slow and lacking in personality, but it's worth having on hand while working through Thompson. You may also be interested in my own textbook &lt;a href=&quot;http://lightandmatter.com/fund/&quot; rel=&quot;noreferrer&quot;&gt;Fundamentals of Calculus&lt;/a&gt;, which introduces both infinitesimals and limits while focusing mainly on techniques and applications rather than foundational issues.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-08-02T19:21:15.883" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="23230" PostTypeId="2" ParentId="23225" CreationDate="2021-08-03T00:28:27.133" Score="3" Body="&lt;p&gt;Another text on this...&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;How to Read and Do Proofs&lt;/em&gt;, Daniel Solow&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/01MiN.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/01MiN.jpg&quot; alt=&quot;picture&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2021-08-03T00:28:27.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23231" PostTypeId="2" ParentId="23228" CreationDate="2021-08-03T02:24:01.920" Score="4" Body="&lt;p&gt;After giving the book another look I have to expand on my comment above and make it stronger: &lt;strong&gt;no, this isn't a good book for a first-time learner&lt;/strong&gt;, and in fact I think it's a terrible choice. I looked at the second edition (not the Gardner update), but unless Gardner practically rewrote the book I don't think this matters much.&lt;/p&gt;&#xA;&lt;p&gt;At the very beginning, derivatives are explained by simply stating that &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; is small and so &lt;span class=&quot;math-container&quot;&gt;$(dx)^2$&lt;/span&gt; and higher powers are, like, really really small and they don't matter so we ignore them. I'm not being unfair either, this is the actual essence of his argument. It leads to wonderful chains of reasoning such as &lt;span class=&quot;math-container&quot;&gt;$y+dy=x^2+2x\cdot dx+(dx)^2$&lt;/span&gt; but let's drop the small term (no, not that one...the &lt;em&gt;really&lt;/em&gt; small one dummy) and actually &lt;span class=&quot;math-container&quot;&gt;$y+dy=x^2+2x\cdot dx$&lt;/span&gt;. As a real world example, we're told to think of &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; as a flea on an ox (which would bother the ox) and &lt;span class=&quot;math-container&quot;&gt;$(dx)^2$&lt;/span&gt; as a flea on the flea (which wouldn't bother the ox). But wait, we can make that flea &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; as small as we wish, and a flea as small as we wish certainly wouldn't bother an ox. What's a beginning student supposed to make of this?&lt;/p&gt;&#xA;&lt;p&gt;Thompson and anyone who has already learned calculus can make sense of it in hindsight, but how can it be viewed as anything but nonsense by a first-time reader? Some of the other books listed in comments are far better imo. I already recommended Hughes Hallet, and I can also highly recommend &lt;em&gt;What is Calculus About&lt;/em&gt; by Sawyer and &lt;em&gt;Calculus: An Intuitive and Physical Approach&lt;/em&gt; by Kline (in Dave L. Renfro's link).&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2021-08-03T02:24:01.920" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="23232" PostTypeId="2" ParentId="23228" CreationDate="2021-08-03T13:16:19.407" Score="3" Body="&lt;p&gt;In a sense it is OK.  In the sense that almost any book (except for one that is very difficult, and assigned to a weaker student and/or one without strong motivation to prevail) is OK.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;It's really more about your sticktoitiveness than anything else, what you get out of it.&lt;/strong&gt;  I'm working on a language study right now.  I have several different texts available on my shelf.  And there are plusses and minuses.  But really, the key thing that will determine if I prevail is if I stick to it or not.  (I know several times I have not!)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I would probably advise a more standard book.  But particularly one that has answers to the exercises (not worked solutions per se, just answers).&lt;/strong&gt;  You need to do your own drill and to check yourself.  The intermediate review grammar I am self studying from has all the exercise answers.  It's designed for usage.  Not for acceptance by teacher committees that pan books with drill answers.  And it sure as heck improves my understanding to see that I missed something and then check why (often figuring out my own mistake, but in a few cases, prompting a question of someone, which I keep a &amp;quot;parking lot&amp;quot; for.)&lt;/p&gt;&#xA;&lt;p&gt;I like Granville although it is a tiny bit dry.  But it uses simple words.  And has short sections with minimal text (not in a proofy &amp;quot;intuitive to most casual observer&amp;quot; way but more in a KISS way).  You can get one of the older editions free online in a pdf.  Although really cost should not be a discriminating factor.  Think of the value of your time invested.  There is a huge cheap market for old books on Amazon, so just buying an old edition is a no brainer.&lt;/p&gt;&#xA;&lt;p&gt;I also like Thomas Finney prior to about 1982 (when Thomas retired).  Studied with their AP version in early 80s.  [You can see the blue kaleidoscope cover of the book in the movie Stand and Deliver.]  It is slightly more theoretical than Granville though, without being some baby real analysis craziness.  I probably benefited some from partially working Granville first.&lt;/p&gt;&#xA;&lt;p&gt;I also recommend to look at the Schaum's Outline.  In general this sort of series is very useful for adult learners looking to review an old topic (minimal text for one thing, having answers, etc.)  More of a businessman's &amp;quot;80-20&amp;quot; attitude versus the fussy complete-ism that you get from high end types on SE.  Some posters here pan the series, but I think they are much more user friendly for a self studier.  (And, like dieting, giving up is the key danger, so picking something user friendly is a good way to go.  In that sense, the book you listed may be nice just because it is a little more amusing and so you may stick with it.  You could even try an older politically incorrect version to provoke your laughs--either at or with, your choice.  ;)&lt;/p&gt;&#xA;&lt;p&gt;I have not used it, but one text I hear referred to is &amp;quot;Calculus for the Practical Man&amp;quot;.  Maybe also similar in being pretty gentle and marketed at self studiers and especially those who are a bit more dilettantish.&lt;/p&gt;&#xA;" OwnerUserId="17576" LastActivityDate="2021-08-03T13:16:19.407" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="23233" PostTypeId="2" ParentId="23225" CreationDate="2021-08-03T17:23:37.333" Score="2" Body="&lt;p&gt;Perhaps Beck and Geoghegan's &lt;a href=&quot;http://math.sfsu.edu/beck/papers/aop.noprint.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The art of proof&amp;quot;&lt;/a&gt;, Martin's &lt;a href=&quot;https://personal.math.vt.edu/daymv/ProofsBook/IPaMV.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;An Introduction to Proof and Mathematical Vernacular&amp;quot;&lt;/a&gt; or Hammack's &lt;a href=&quot;http://www.people.vcu.edu/%7Erhammack/BookOfProof/BookOfProof.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The Book of Proof&amp;quot;&lt;/a&gt; are relevant. All are free PDF downloads, you can peruse them and see if they are a match for you.&lt;/p&gt;&#xA;&lt;p&gt;Due to it's placement in most computer science curricula, most introductory books on discrete mathematics include a hefty introduction to proofs. You'll also find loads of lecture notes for such classes around.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2021-08-03T17:23:37.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23234" PostTypeId="2" ParentId="23228" CreationDate="2021-08-04T01:18:01.870" Score="4" Body="&lt;p&gt;The fact that the book uses infinitesimals (presented informally) rather than limits is definitely a big deal, and I think you will have to read the book for yourself to see if it works for you. When I was exposed to the idea of &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; as some kind of infinitely small quantity, my philosophical confusion completely blocked me. But the idea of a limit, even with its complicated &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; definition, made sense. But lots of other people quickly see how to manipulate and interpret &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; viewed as infinitesimals, and you might be like them.&lt;/p&gt;&#xA;&lt;p&gt;Just be aware that, when you take calculus, you will likely be taught using limits.&lt;/p&gt;&#xA;" OwnerUserId="13126" LastActivityDate="2021-08-04T01:18:01.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23235" PostTypeId="1" AcceptedAnswerId="23252" CreationDate="2021-08-04T06:00:27.173" Score="5" ViewCount="464" Body="&lt;p&gt;As I have myself struggled a bit with this concept, I would like to present my own explanation of it.&#xA;Context:&#xA;Loss of significance is a loss of precision, not necessarily accuracy.&lt;/p&gt;&#xA;&lt;p&gt;And for a long time, I didn't know the difference between the two concepts.&lt;/p&gt;&#xA;&lt;p&gt;Explanation:&#xA;Now first, I show them a calculation of subtracting two numbers close to each other in floating point. We then try to calculate the initial and final relative and absolute errors.&lt;/p&gt;&#xA;&lt;p&gt;We then observe that the relative error has become much larger, and in addition to that, we have lost many significant figures.&lt;/p&gt;&#xA;&lt;p&gt;Context: when I was presented this in my first course of numerical analysis, I said: &amp;quot;so? Who cares? We are still close to the number!&amp;quot;,&amp;quot;the absolute error is still bounded by the addition of the errors!&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But then when I realized the difference between precision and accuracy, I convinced myself that this was a loss of precision.&lt;/p&gt;&#xA;&lt;p&gt;Explanation: the relativity of precision:&lt;/p&gt;&#xA;&lt;p&gt;Intuitively, we can observe that as we increase the size of the Actual number to be estimated A, relative to the size of the absolute error, we see that the cluster of numbers&#xA;(Whose average is a measure of precision) gets &amp;quot;closer together&amp;quot;, and that might give us sufficient precision.&lt;/p&gt;&#xA;&lt;p&gt;Example:&#xA;Like trying to hit a ship with rockets, an error area of 1 meter is amazing.&lt;/p&gt;&#xA;&lt;p&gt;But hitting an apple with a bow and arrow , and 1 meter is too imprecise, too big to hit the apple consistently.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Last part of explanation:&#xA;I then say,&amp;quot; we want to be consistent with our data, and a loss of relative error takes that away from us. So we &amp;quot;care&amp;quot; about loss of significance.&lt;/p&gt;&#xA;&lt;p&gt;I myself am only partially satisfied with this explanation, and find that there is something missing, or perhaps it is in part incorrect.&lt;/p&gt;&#xA;&lt;p&gt;I would appreciate your comments or any guidance from your part.&lt;/p&gt;&#xA;" OwnerUserId="17580" LastActivityDate="2022-08-22T20:44:31.220" Title="How to explain loss of significance in numerical analysis?" Tags="|concept-motivation|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23237" PostTypeId="2" ParentId="23235" CreationDate="2021-08-04T19:14:55.390" Score="4" Body="&lt;p&gt;&lt;strong&gt;Significance&lt;/strong&gt; is precision which is reproducability.&lt;/p&gt;&#xA;&lt;p&gt;Accuracy is a degree of correctness.&lt;/p&gt;&#xA;&lt;p&gt;Consider two 1 foot rulers lying on a table. Both are marked in nominal inches:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;One ruler is made of Invar-42 low expansion nickel-iron alloy but is obviously only about 10 inches in length despite being numbered to 12.&lt;/li&gt;&#xA;&lt;li&gt;The other looks to be 12 inches in length but is made of stretchy rubber.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Ruler #1 is very precise (albeit inaccurate) and reading off many significant figures from it has value and reproducability.&lt;/p&gt;&#xA;&lt;p&gt;Ruler #2 is more accurate but measuring the same thing multiple times with it will give wildly varying precision, as in the significance is low.&lt;/p&gt;&#xA;&lt;p&gt;Another thought approach is weighing a known high precision, to many significant figures, mass. If you are using a &lt;em&gt;bathroom scale&lt;/em&gt;, anything more precise than a tenth pound would have no significance. &lt;em&gt;On my bathroom scale it's more like 2 pounds.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="10376" LastActivityDate="2021-08-04T19:14:55.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23238" PostTypeId="2" ParentId="23235" CreationDate="2021-08-05T12:31:10.753" Score="3" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In statistics, business and science, I keep hearing about how &lt;strong&gt;precision&lt;/strong&gt; = consistency = low variability = closeness of the data points = reproducibility &amp;amp; repeatability.&lt;/p&gt;&#xA;&lt;p&gt;In this case, precision and accuracy are orthogonal concepts, and &lt;strong&gt;imprecision is not thought to lead to inaccuracy&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here, we are discussing ‘precision’ with reference to a set of repeated measurements.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In physics and &lt;a href=&quot;https://academic.oup.com/nsr/article/7/12/1795/6017311&quot; rel=&quot;nofollow noreferrer&quot;&gt;numerical analysis&lt;/a&gt;, however, I think of &lt;strong&gt;precision&lt;/strong&gt; not as a measure of data dispersion, but as a measure of the resolution of the instrument or the recordings or the calculations.&lt;/p&gt;&#xA;&lt;p&gt;In this sense, &lt;strong&gt;imprecision&lt;/strong&gt; is about uncertainty more than about inconsistency, and &lt;strong&gt;can lead to inaccuracy&lt;/strong&gt; (loss of correctness), due to the approximation (rounding and truncation) errors being cumulatively propagated by an algorithm, its numerical instability, or a problem's ill condition.&lt;/p&gt;&#xA;&lt;p&gt;Here, we are discussing ‘precision’ with reference to a single or set of non-repeated measurement(s).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://physics.stackexchange.com/questions/657598/what-is-meant-by-precision#comment1474589_657605&quot;&gt;This comment&lt;/a&gt; nicely links these two aspects/interpretations of ‘precision’: “The closer the repeated results are to each other, the more confident you can be about expressing the result with a greater number of digits.”&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-08-22T20:44:31.220" LastActivityDate="2022-08-22T20:44:31.220" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="23240" PostTypeId="2" ParentId="23228" CreationDate="2021-08-06T18:55:43.737" Score="1" Body="&lt;p&gt;Also, get Steven Strogatz's &amp;quot;Infinite Powers&amp;quot; Book.  It doesn't have any work problems.  He very carefully explains many of the fundamental ideas of calculus.  I was fuzzy on some of the ideas, and his book really cleared some items up.  Calculus has a &amp;quot;tortoise and a hare&amp;quot;, aspect to it.  If you take it slowly you will learn more than if you try to race through it.&lt;/p&gt;&#xA;" OwnerUserId="17597" LastActivityDate="2021-08-06T18:55:43.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23241" PostTypeId="1" CreationDate="2021-08-07T02:57:41.593" Score="-2" ViewCount="114" Body="&lt;p&gt;I am building a math game, the Problem I am not a math teacher. :)&lt;/p&gt;&#xA;&lt;p&gt;So I am trying to find the best way to create questions for different ages and different levels, I tried to google it but no luck.&lt;/p&gt;&#xA;&lt;p&gt;In more details I want to know for each age what type of questions I should add what is the complexity and best way to move for next level for the kids. So its more math and education question.&lt;/p&gt;&#xA;&lt;p&gt;Example that comes in mind: Age 5 should have addition between 0 and 10 and player can go to next level if he passes first 3 levels (30 question). etc....&lt;/p&gt;&#xA;&lt;p&gt;Kindly if you know any article about this will be great&lt;/p&gt;&#xA;" OwnerUserId="17598" LastActivityDate="2021-08-07T04:42:44.860" Title="Math questions for math game" Tags="|games|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23242" PostTypeId="2" ParentId="23241" CreationDate="2021-08-07T04:06:49.580" Score="8" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Go research it.  This is not suitable for small answer Q&amp;amp;A.  It's too big a topic--what topics to cover for each grade of math!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Go look at some of the existing products in this genre.  Buy a few, play them, read the Amazon reviews on their plus, minus, etc.  You do realize your game will not be the first creature of this species to walk the Earth, right?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Build some alphas and test them.  That is what you want to do first, anyways, right?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P.s.  You'd do a lot better by drilling down on one particular area, not looking at all ages.  And no, don't ask me, which one.  Go wallow around until you find one that looks interesting to you.&lt;/p&gt;&#xA;&lt;p&gt;P.s.s. Although this board covers all math teaching in scope in theory, in practice, it has very few pre-college teachers, and those few are in high schools.  You need to look for other areas for input for lower level math teacher input.  There is sort of an ecosystem of teacher blogs for instance.  Or just talk to local teachers.&lt;/p&gt;&#xA;" OwnerUserId="17599" LastEditorUserId="17599" LastEditDate="2021-08-07T04:12:05.410" LastActivityDate="2021-08-07T04:12:05.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23243" PostTypeId="2" ParentId="23241" CreationDate="2021-08-07T04:42:44.860" Score="3" Body="&lt;p&gt;Welcome! Primary math standards vary by location, but you can see some examples from the US (I just googled &amp;quot;k12 math standards&amp;quot;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.michigan.gov/documents/mde/K-12_MI_Math_Standards_REV_470033_7_550413_7.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.michigan.gov/documents/mde/K-12_MI_Math_Standards_REV_470033_7_550413_7.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://education.ky.gov/curriculum/standards/kyacadstand/Documents/Kentucky_Academic_Standards_Mathematics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://education.ky.gov/curriculum/standards/kyacadstand/Documents/Kentucky_Academic_Standards_Mathematics.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nd.gov/dpi/sites/www/files/documents/Academic%20Support/v3.Mathematics%20Standards%20Final%208.14.17.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nd.gov/dpi/sites/www/files/documents/Academic%20Support/v3.Mathematics%20Standards%20Final%208.14.17.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;These documents are quite long. Khan Academy has a more succinct list of skills, and comes with some example auto-generated problems: &lt;a href=&quot;https://www.khanacademy.org/math/k-8-grades&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.khanacademy.org/math/k-8-grades&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In terms of how to move students on from one level to the next, I suggest reading into Leitner Boxes: &lt;a href=&quot;https://ncase.me/remember/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ncase.me/remember/&lt;/a&gt; .&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-08-07T04:42:44.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23244" PostTypeId="1" AcceptedAnswerId="23249" CreationDate="2021-08-07T10:53:37.483" Score="3" ViewCount="1328" Body="&lt;p&gt;I have recently started some summer tuition for a student who has failed in mathematics at UK &lt;a href=&quot;https://en.wikipedia.org/wiki/Key_Stage&quot; rel=&quot;nofollow noreferrer&quot;&gt;key stage&lt;/a&gt; 3 (age 14), and trying to plumb the depths of his knowledge. Having engaged him in some simple exercises in simultaneous equations in 2 variables, with a view to presenting these equations as lines in a Cartesian plane and showing him where they cross, I have identified a particular problem which concerns me.&lt;/p&gt;&#xA;&lt;p&gt;He can (just about) understand the concept of &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; being points on a plane, and how an equation &lt;span class=&quot;math-container&quot;&gt;$a x + b y = k$&lt;/span&gt; can be converted into an equation of the form &lt;span class=&quot;math-container&quot;&gt;$y = m x + c$&lt;/span&gt;, but when I came to giving him some simple examples I pulled off the top of my head, I found that the inevitable fractions that came his way as a result were completely beyond his ken.&lt;/p&gt;&#xA;&lt;p&gt;I found myself having to explain how to add fractions, and how to multiply fractions, and just about how to divide fractions, most of which he vaguely remembers from many years ago; however, the concept of the reciprocal, and the technique of turning the fraction upside down, was completely alien to him, and it frightened him.&lt;/p&gt;&#xA;&lt;p&gt;The question I have: is it &lt;em&gt;expected&lt;/em&gt; that he ought to be fluent with manipulating fractions? He keeps converting to decimal everywhere he goes, particularly in applications where he has to work out, for example, how far he can go on 18 litres if 15 litres takes him n miles -- which is all very well till first thing he does is he divides by 15 and chops off the recurring 1/3 at the first digit.&lt;/p&gt;&#xA;&lt;p&gt;I am putting a set of exercises in his way to allow him to manipulate algebraic expressions. I believe it may be necessary to insist he does some algebraic exercises every day. Does anyone know of any good works which I may use to support this student? Most of my library is either way too advanced or positively antiquarian.&lt;/p&gt;&#xA;" OwnerUserId="15290" LastEditorUserId="2083" LastEditDate="2021-08-07T18:01:24.690" LastActivityDate="2021-08-07T18:01:24.690" Title="What is the expected fluency with fractions at UK key stage 3?" Tags="|algebra|fractions|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="23247" PostTypeId="2" ParentId="23244" CreationDate="2021-08-07T15:40:24.917" Score="7" Body="&lt;p&gt;We DON'T need to know the intricacies of UK standards to say yes students are expected to know fractions before algebra.  However, your puzzlement over it (like a new concept to you that a kid who fails algebra might be weak on arithmetic!) is concerning.  And similarly, you are not the first, by any means, &amp;quot;good at math, but weak at teaching it&amp;quot; person to be flummoxed by such a common occurrence.  But, there have been many discussions on this forum where people note the difficulty in teaching weaker kids at lower level colleges, with fractions being a notable hurdle.  So it is something that experienced TEACHERS as opposed to mathematicians are not surprised by.&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure that starting with two equations, two unknowns is the right spot either.  I don't know for sure, but would Bayesian bet that this is one of the later topic in the kid's year of math. They are one of the harder parts of algebra 1.  A little more looking at the kid's overall ability and deficits is also in order.  Also, you ought to get some texts and skim them to understand what topics are taught when.  Not just plunge ahead.  Even a very short perusal would bring your knowledge (of the customary teaching of the topic, not of math itself!) up hugely.&lt;/p&gt;&#xA;&lt;p&gt;In terms of practical work, I wouldn't give up.  And wouldn't just go all the way back to arithmetic either.  But mix in a little bit of that stuff along the way (but separated from the algebra or at least with easier topics, like single equation in x).  Oh...and try to assign problems with simpler arithmetic.  It is just some extra work by you to design such problems that don't have the annoying fractions.  (Or if they do, start with 1/2--look up the training concept of &amp;quot;progression&amp;quot;.)  The problem comes when you assign what (for the kid) are harder topics (systems of linear equations) along with hardest arithmetic. For some reason us meat computers slip a gear when you combine a couple things that are medium hard.  Unlike silicon where either it knows the subroutine or doesn't.  And you can just pile as many on as you want.&lt;/p&gt;&#xA;" OwnerUserId="17602" LastEditorUserId="17602" LastEditDate="2021-08-07T15:48:33.993" LastActivityDate="2021-08-07T15:48:33.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23248" PostTypeId="2" ParentId="23244" CreationDate="2021-08-07T16:28:35.643" Score="7" Body="&lt;p&gt;Yes, he will have learnt how to do arithmetic with fractions, equivalent fractions, fractional/decimal/percentage equivalence, reciprocals, and how to solve problems involving fractions in Key Stage 3 - it is part of the National Curriculum, which can be found &lt;a href=&quot;https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Whether he ought to be fluent in manipulating fractions really depends on whether he will be taking Higher Tier or Foundation Tier GCSE. A Foundation Tier student will likely spend some time consolidating previous work on fractions, whereas higher tier students should be fluent in working with fractions by the end of Key Stage 3. It is not expected that students should know anything about algebraic fractions yet - that is usually covered in Key Stage 4.&lt;/p&gt;&#xA;&lt;p&gt;For resources, &lt;a href=&quot;https://www.mathsgenie.co.uk/resources/27_fractions-adding-subtracting-multiplying-dividing.pdf&quot; rel=&quot;noreferrer&quot;&gt;Maths Genie&lt;/a&gt; and &lt;a href=&quot;https://corbettmaths.com/2019/08/29/equations-involving-fractions-practice-questions/&quot; rel=&quot;noreferrer&quot;&gt;Corbett Maths&lt;/a&gt; both have plenty of practice questions in the style of UK exams. You might also like to look at GCSE papers for more difficult questions - almost everything that is in the Key Stage 3 curriculum can also be tested at GCSE.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2021-08-07T16:28:35.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23249" PostTypeId="2" ParentId="23244" CreationDate="2021-08-07T17:32:30.573" Score="8" Body="&lt;p&gt;I tutor maths as a full-time job, and many of the students I have tutored over the past few years are not fluent in their times tables up to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; or arithmetic with fractions, or both. And it's not like I select students with these low abilities on purpose.&lt;/p&gt;&#xA;&lt;p&gt;And when I say they are not fluent, I mean that they are nowhere near as good as me at them (arithmetic with fractions). I mean, not so humble brag: I'm quite quick with arithmetic. But this isn't an excuse for students to be really bad or really slow. Some of my students are fine, but a lot of them are too slow or uncertain of their calculations when I begin tutoring them. With regards to fraction arithmetic in particular, this can be anything from:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The student having &amp;quot;forgot&amp;quot; how to add, subtract, multiply and divide by fractions&lt;/li&gt;&#xA;&lt;li&gt;OR The student adding fractions incorrectly, like 1/2 + 1/3 = 2/5 because you add the numerator and denominator&lt;/li&gt;&#xA;&lt;li&gt;OR The student is good at fractions in general, but struggles with mixed fractions&lt;/li&gt;&#xA;&lt;li&gt;OR The student is too slow because they find calculating stressful or they are simply uncertain that their calculation methods are correct- for example, it takes them about 5 minutes to calculate 3/4 - 2/3, when it should take at most 30 seconds in my opinion.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;OR something else...&lt;/p&gt;&#xA;&lt;p&gt;It is my belief that students &lt;em&gt;must&lt;/em&gt; become fluent at their &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; times tables first, &lt;em&gt;then&lt;/em&gt; get good at fraction arithmetic as well as fraction problems, &lt;em&gt;then&lt;/em&gt; you can move on to ratio/proportion and then algebra and so forth. Of course, you should not just be teaching in terms of numbers; use pictorial diagrams like pie charts and rectangles to help visual learners learn fractions. And number patterns e.g. &lt;span class=&quot;math-container&quot;&gt;$10 $&lt;/span&gt; times something means you just add a &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; on the end of the number...&lt;/p&gt;&#xA;&lt;p&gt;But yeah, if you try to teach someone algebra when they are poor at fractions, then well... that will only get them so far in my experience. Perhaps other educators have experienced otherwise- but I don't see how this is possible.&lt;/p&gt;&#xA;&lt;p&gt;This is not to say that arithmetic is the &lt;em&gt;only&lt;/em&gt; thing you should be focusing on. If your student wants to go through what they are doing at school or are interested in another topic, then by all means spend a small amount of time going through that topic with them. But until they are just good enough at arithmetic to progress onto harder topics, you don't have much choice but to focus on arithmetic. Also, it should be mentioned that this process of getting them up to scratch at arithmetic &lt;em&gt;shouldn't&lt;/em&gt; take a huge amount of time &lt;em&gt;if&lt;/em&gt; the student puts the effort to improve in. If they don't put in the effort, then try to motivate them. If the student is trying hard to learn but simply cannot, and you have tried everything to help them learn the arithmetic, try to determine if they have learning difficulties.&lt;/p&gt;&#xA;&lt;p&gt;To this end, &lt;a href=&quot;https://www.amazon.co.uk/New-Maths-Catch-Up-Workbook-Answers/dp/1789080584/ref=sr_1_1?dchild=1&amp;amp;keywords=ks3%20catch%20up&amp;amp;qid=1628354494&amp;amp;sr=8-1&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; are some good &amp;quot;catch-up books&amp;quot; focusing on fractions, decimals and percentages for KS3. There are &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; books in the series and the arithmetic gets progressively harder. I have only tried this with one student who particularly struggled with fractions. Previously he stagnated on fraction arithmetic for over a year. I would say by working his way through these books, he improved massively on fractions. I must say though, that it was through his own desire to improve and realisation that he &lt;em&gt;must pass GCSE maths&lt;/em&gt; that motivated him to actually put some work in and improve. But it certainly is handy that those books exist, and they're relatively cheap, so if it's arithmetic you want your student to work on then buy the books and get them to work through them. One book per week or two weeks should be a fair pace for most students who are weak at arithmetic. During this process of them completing this homework, you can identify their weaknesses and help them improve.&lt;/p&gt;&#xA;&lt;p&gt;Now it sounds like you're also looking for algebra books. But both CGP and Collins have lots of KS3/ GCSE level books. So try fish around on amazon to find the most relevant one to suit your student's needs.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.amazon.co.uk/IGCSE%C2%AE-Textbook-Higher-Course-Revision/dp/1782944370/ref=sr_1_13?crid=180HK1INGFG9F&amp;amp;dchild=1&amp;amp;keywords=igcse%20maths%20edexcel%209-1%20cgp&amp;amp;qid=1628356677&amp;amp;sprefix=igcse%20maths%2Caps%2C162&amp;amp;sr=8-13&quot; rel=&quot;noreferrer&quot;&gt;This&lt;/a&gt; book is my go-to book for helping KS3/GCSE students who are good enough at the fundamentals to progress, as it has a lot of exercises for them to practice.&lt;/p&gt;&#xA;&lt;p&gt;Someone else suggested mathsgenie.co.uk, and I second this website.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2021-08-07T17:38:55.633" LastActivityDate="2021-08-07T17:38:55.633" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="23250" PostTypeId="1" CreationDate="2021-08-08T00:57:40.230" Score="11" ViewCount="442" Body="&lt;p&gt;I teach this course from David Lay's &lt;em&gt;&lt;strong&gt;Linear Algebra and Its Applications&lt;/strong&gt;&lt;/em&gt;, which on the whole is a great textbook and explains things well. It does not explain the steps of LU factorization well, so I started exploring online to see some other explanations. I noticed that most sources describe it only for square matrices. That would make an explanation of the steps much easier.&lt;/p&gt;&#xA;&lt;p&gt;My question here is whether there is any need (in a sophomore level course) to include non-square matrices (in the LU factorization topic)?&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-03-06T20:34:03.437" LastActivityDate="2022-03-06T20:34:03.437" Title="Teaching LU Factorization in a sophomore-level Linear Algebra course" Tags="|undergraduate-education|linear-algebra|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="23251" PostTypeId="2" ParentId="23250" CreationDate="2021-08-08T01:56:27.580" Score="3" Body="&lt;p&gt;This is just a data-point from my own background. I double-majored in math and physics as an undergrad and got my PhD in physics. I did a postdoc and then became a community college teacher for 25 years. I taught 98% physics and 2% math.&lt;/p&gt;&#xA;&lt;p&gt;I'm pretty sure we never saw LU decomposition at all in my sophomore course at Berkeley. I think I learned it for the first time in an upper-division course. (I think we may have seen it both in upper-division linear algebra and in numerical analysis.) If you asked me today, I would not remember what it was for or how to do it, just that it's the name of some way of factoring a matrix. The fact that it can even be defined for a non-square matrix is a complete surprise to me.&lt;/p&gt;&#xA;&lt;p&gt;Speaking purely as a physicist, we basically never use non-square matrices. The most frequent application is to operators in quantum mechanics, which are always maps from a given space back to the same space, so in the finite-dimensional case they're always square matrices. In fact, we usually only care about operators that are either unitary (for time-evolution) or Hermitian (for observables).&lt;/p&gt;&#xA;&lt;p&gt;I don't know, maybe an economist, for example, would have a different point of view, but my impression is that doing LU factorization at all is kind of esoteric for a sophomore class, and the fact that it can be done for non-square matrices is super esoteric.&lt;/p&gt;&#xA;" OwnerDisplayName="user507" LastActivityDate="2021-08-08T01:56:27.580" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="23252" PostTypeId="2" ParentId="23235" CreationDate="2021-08-09T05:42:20.163" Score="7" Body="&lt;p&gt;If they're not impressed by small errors, point out &lt;a href=&quot;https://en.wikipedia.org/wiki/Catastrophic_cancellation&quot; rel=&quot;noreferrer&quot;&gt;catastrophic cancellation&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As a computer programmer, these issues are the kind of thing that we pay great attention to.  &lt;a href=&quot;https://www.cs.utexas.edu/users/flame/laff/alaff/a2appendix-catastrophic-cancellation.html&quot; rel=&quot;noreferrer&quot;&gt;We even modify the quadratic formula to avoid these problems&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Where you really get in trouble is when you use equations that are built around exact assumptions, like &amp;quot;energy is conserved.&amp;quot;  When you use equations where energy isn't conserved, you can get spectacularly absurd results because you were deponent on energy being the same before and after.&lt;/p&gt;&#xA;&lt;p&gt;You may have to show them non-linear responses first.  Linear responses to errors can be as boring as they think they are.  Non-linear can be spectacular.  Guidance and Control systems can be particularly unfriendly.  You can mathematically show that they are stable, but due to numerical imprecision, the simulation version of the algorithms can be wildly unstable.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2021-08-09T05:42:20.163" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23253" PostTypeId="2" ParentId="21185" CreationDate="2021-08-09T21:09:28.880" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://www.kaggle.com/PromptCloudHQ/toy-products-on-amazon&quot; rel=&quot;nofollow noreferrer&quot;&gt;This dataset&lt;/a&gt; seems pretty darn close to what you want if you swap in the number of items in stock for delivery time and trim it down a little.&lt;/p&gt;&#xA;&lt;p&gt;It's a dataset of toy products on Amazon that includes close to 10,000 data points. I'd specifically restrict it to five of the columns: &lt;code&gt;product_name&lt;/code&gt;, &lt;code&gt;price&lt;/code&gt;, &lt;code&gt;number_available_in_stock&lt;/code&gt;, &lt;code&gt;number_of_reviews&lt;/code&gt; (could be an interesting additional variable to have them consider - i.e. a 4.7 average rating with 20 reviews is probably worse than a 4.5 average rating with 200 reviews), and &lt;code&gt;average_review_rating&lt;/code&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you're very in favor of the delivery time variable, you could probably just add in some randomly generated delivery times when you do the rest of the data set formatting? But as others have pointed out, delivery time is highly dependent on where the user is, so number in stock might be a reasonable substitute that gets at the same idea (and is already in the data set).&lt;/p&gt;&#xA;&lt;p&gt;For future reference, kaggle is a pretty good site to check for free, large data sets.&lt;/p&gt;&#xA;" OwnerUserId="8366" LastActivityDate="2021-08-09T21:09:28.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23254" PostTypeId="1" CreationDate="2021-08-10T09:38:50.020" Score="-2" ViewCount="128" Body="&lt;p&gt;A company report its net profit 1,000,000 in 2000,and earned 1,100,000 in 2001,so the earning increase rate from 2000 to 2001 is as below:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;1,100,000-1,000,000)/1,000,000 = 0.1 =10% &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;A company report its net loss 1,000,000 in 2000,and earned 1,100,000 in 2001,so what is the the earning increase rate from 2000 to 2001 ?&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;(1,100,000-(-1,000,000))/(-1,000,000) = -2.1 = -210% &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Should i say that the earning increase rate from 2000 to 2001 is  &lt;code&gt;-210%&lt;/code&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Let suppose a extrem case:&lt;/p&gt;&#xA;&lt;p&gt;A company report a zero profit  in 2000,and earned 1,100,000 in 2001,what is the the earning increase rate from 2000 to 2001 ?&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;(1,100,000 - 0)/ 0 # no meaning!&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="12422" LastActivityDate="2021-08-10T12:33:35.867" Title="How to solve the simple elementary school math issue--calculate increase rate" Tags="|teacher-preparation|division|" AnswerCount="1" CommentCount="1" ClosedDate="2021-08-10T16:14:13.880" ContentLicense="CC BY-SA 4.0" />
  <row Id="23255" PostTypeId="2" ParentId="23254" CreationDate="2021-08-10T12:33:35.867" Score="2" Body="&lt;p&gt;A company will use NM indicating, as you noted, &amp;quot;No Meaning&amp;quot;.&#xA;This is less of a right/wrong problem as it is a problem that opens up a discussion. The actual meaning of these +/- percents and what then mean for the company.&lt;/p&gt;&#xA;&lt;p&gt;I'll draw a comparison. Without explicit instructions, students often have no idea how many digits to round to in their final answer. When a population problem results in a decimal, I ask them to use common math sense. The answer is an integer, of course, but if the problem continued, use the decimal, lest a series of rounding errors accumulates through the calculations.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-08-10T12:33:35.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23258" PostTypeId="2" ParentId="23250" CreationDate="2021-08-11T08:59:50.200" Score="13" Body="&lt;p&gt;Poole's &lt;em&gt;Linear Algebra: A Modern Introduction&lt;/em&gt;, 2nd edition, relegates the non-square case of the LU factorization to an exercise. Strang's &lt;em&gt;Introduction to Linear Algebra&lt;/em&gt;, 5th edition, does square systems only for LU, unless I've missed something tucked away somewhere. When covering LU in a first linear algebra course, I think it's fine to keep to the square case for this topic. Non-square linear systems can receive more attention later in the course when covering linear independence, subspaces, dimension, rank and the like.&lt;/p&gt;&#xA;&lt;p&gt;In a &lt;a href=&quot;https://matheducators.stackexchange.com/questions/23250/teaching-lu-factorization-in-a-sophomore-level-linear-algebra-course#comment60898_23251&quot;&gt;comment&lt;/a&gt;, you also wonder whether to skip LU entirely. I think it has a place as a first introduction to matrix factorizations. As Steven Gubkin &lt;a href=&quot;https://matheducators.stackexchange.com/questions/23250/teaching-lu-factorization-in-a-sophomore-level-linear-algebra-course#comment60896_23250&quot;&gt;writes&lt;/a&gt;, it uses only &amp;quot;row reduction steps in the Gaussian elimination algorithm, together with an understanding of how to achieve row operations using matrix multiplication.&amp;quot; It can also be used as a springboard to numerical linear algebra, teaching the algorithms used to solve problems in practice alongside the theory, but as Ben Crowell &lt;a href=&quot;https://matheducators.stackexchange.com/questions/23250/teaching-lu-factorization-in-a-sophomore-level-linear-algebra-course#comment60905_23251&quot;&gt;points out&lt;/a&gt;, this could be delayed until a first course in numerical analysis. My concern here is that some students may never take such a course and should arguably be given some exposure to numerics.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2021-08-11T08:59:50.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23259" PostTypeId="1" CreationDate="2021-08-14T01:41:14.247" Score="10" ViewCount="2057" Body="&lt;p&gt;Migrated from &lt;a href=&quot;https://mathoverflow.net/questions/401664/remote-work-in-math-academia&quot;&gt;Mathoverflow.net&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is a questions about teaching/research in math academia.&lt;/p&gt;&#xA;&lt;p&gt;During the pandemic, many things have been moved online: courses, seminars, informal gatherings, etc. As a student, I've had a positive experience with online courses: they allowed me to work at my own schedule (because lectures were recorded), and I was not tied down to a geographic location.&lt;/p&gt;&#xA;&lt;p&gt;There seems to be a strong argument to be made in favor of continuing this post-pandemic which may bring a host of benefits to all levels of math academia. (I've mentioned some of the benefits on the student side.)&lt;/p&gt;&#xA;&lt;p&gt;On the teaching side, I could see how remote teaching can eliminate many of the inefficiencies that were present pre-pandemic. For instance, why do we need to give three identical lectures on Calculus I, three times a week, every year (many of which are taught very poorly or unenthusiastically)? I think everybody would benefit if one records a set of lectures taught by a very good teacher, once every few years. (Imagine how much time this would free up for everyone.)&lt;/p&gt;&#xA;&lt;p&gt;I could see how remote work can be beneficial for math academics outside of teaching. For instance, I hear about a lot of people leaving math academia because they cannot get an academic position in a city they want to live in (or, similar problems tied to geographic location). Can we not resolve this remote work? Of course, there are benefits to interacting with people face-to-face, but I still don't see why all academic positions have to be in-person.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My Question&lt;/strong&gt;. Is there any movement within math academia (either in the U.S. or elsewhere) to make the following changes:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;permanently move some (or all) teaching online (either undergraduate&#xA;or graduate level)?&lt;/li&gt;&#xA;&lt;li&gt;Have &amp;quot;remote&amp;quot; academic positions? (i.e. be&lt;br /&gt;&#xA;affiliated with a research institution, but not be required to be&lt;br /&gt;&#xA;physically present at a certain location)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Subquestion&lt;/strong&gt;. What are arguments against implementing either of the above (if any)? (So far, I haven't heard any convincing arguments about the above issues from anyone. My conclusion is that things are the way they are largely because of inertia. If people have strong arguments in favor of doing in-person work all the time, I would be interested in hearing about them.)&lt;/p&gt;&#xA;&lt;p&gt;Note: I considered posting this on academia.stackexchange, but I realized a lot of the question was math-specific. (For instance, the content of Freshman Calculus or Linear Algebra will not change 20 years from now, whereas in some other field of natural science, there might be a groundbreaking discovery that forces people to reevaluate the fundamentals of the field.)&lt;/p&gt;&#xA;&lt;p&gt;Note 2: On MathOverflow, the question faced many oppositions. (In short, people thought a.) most people preferred in person instruction to remote, and b.) there is enough variation in presentation that warrants the in person instruction. Someone also implied that in person instruction was the justification for their salary. Please see the link above for details and for the exact phrasing.) I still am not convinced with the above arguments. Is there a definitive argument in favor of in person instructions?&lt;/p&gt;&#xA;" OwnerUserId="17622" LastEditorUserId="276" LastEditDate="2021-08-17T15:12:30.503" LastActivityDate="2021-08-17T15:12:30.503" Title="Remote work in math academia" Tags="|online-instruction|online|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="23260" PostTypeId="2" ParentId="23259" CreationDate="2021-08-14T04:45:27.843" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;On the teaching side, I could see how remote teaching can eliminate many of the inefficiencies that were present pre-pandemic. For instance, why do we need to give three identical lectures on Calculus I, three times a week, every year (many of which are taught very poorly or unenthusiastically)? I think everybody would benefit if one records a set of lectures taught by a very good teacher, once every few years. (Imagine how much time this would free up for everyone.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Personally, I think there's a huge difference between &amp;quot;giving a lecture&amp;quot; and &amp;quot;teaching&amp;quot;. The first one focuses on the speaker; the second one, on the listener/student. I agree with you that pre-recorded sessions will save hours for the speakers, so they will have more time to research. But, if you want pre-recorded sessions, why don't we just use Youtube? There are thounsand or millions of youtubers doing that for a long time ago, at a really high level, in all languages, and using the best tools to catch their audience. The reason is quite simple, and it's because the main ingredient in the teaching process are the students, and the feedback they provide in class. In a fancy language, the evolution of a lecture is uniquely determined by the initial conditions of the system, which are distinct provided two different group of students.&lt;/p&gt;&#xA;&lt;p&gt;I have always believed that teaching requires more social skills than academic ones, and it's so weird because we were forced to develop digital skills, and to be constantly using new tools and softwares. So, academically speaking, we are now giving better lectures than before. Nevertheless, this social side is the one we miss the most due to pandemic.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Regarding the research issue, actually I don't see any &amp;quot;good&amp;quot; reason to be forced to do an exclusive research job at one place or another. Maybe, it could be argued some financial issues, such as fraud, taxes or whatever (maybe too much Netflix at this point). However, and to the best of my knowledge, there is no University that will hire you exclusively for research.&lt;/p&gt;&#xA;" OwnerUserId="3380" LastActivityDate="2021-08-14T04:45:27.843" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="23261" PostTypeId="2" ParentId="23259" CreationDate="2021-08-14T06:40:50.037" Score="9" Body="&lt;p&gt;To start with, look up the well-written-about phenomenon of MOOC attrition rates.&lt;/p&gt;&#xA;&lt;p&gt;Other than that, this is a very long &amp;quot;question&amp;quot; based on extrapolating a personal (sample of one) experience/belief, with little/no basic research by yourself, and asking for a broad discussion ('list all the counter reasons'), rather than a focused question.&lt;/p&gt;&#xA;&lt;p&gt;Other than that, you emphasize the lecture too much (video or in person) as a part of the learning process.  What trains you is working problems, getting tested on them, and asking questions about the parts you don't get. NOT listening to lectures.&lt;/p&gt;&#xA;" OwnerUserId="17623" LastEditorUserId="17623" LastEditDate="2021-08-14T06:48:11.850" LastActivityDate="2021-08-14T06:48:11.850" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23262" PostTypeId="2" ParentId="23259" CreationDate="2021-08-14T07:40:03.260" Score="10" Body="&lt;p&gt;The question presents teaching as lecturing; that is, as presenting content in an organized manner. This is not without value; it is easier to understand well-organized content than non-organized content. Hopefully the level of the presentation is at least somewhat tailored to the audience, too.&lt;/p&gt;&#xA;&lt;p&gt;This is essentially a behaviourist view of learning and knowledge: the teacher presents the knowledge at a suitable pace and suitable bits, which the students absorb. Thus there is learning.&lt;/p&gt;&#xA;&lt;p&gt;We might contrast this with social-constructive view of learning. Here, learning happens when people make sense of the material, first together and then in the process move to personal understanding. This is a lot more challenging to do via a pre-recorded video lecture.&lt;/p&gt;&#xA;&lt;p&gt;This also happens in any kind of learning situation where students are discovering something and the teacher is taking the role of helping them along.&lt;/p&gt;&#xA;&lt;p&gt;One might say that these approaches are very challenging in the context of traditional lectures. This is a problem with traditional lectures.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-08-14T07:40:03.260" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23265" PostTypeId="2" ParentId="23259" CreationDate="2021-08-15T00:57:23.377" Score="24" Body="&lt;p&gt;&lt;em&gt;Context: This answer is based an experiences from maths programs at German universities. Things might be considerably different in other countries.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is mainly an answer to the teaching part of your subquestion, but first a few preliminary comments:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It is definitely possible to set up an entire maths program at a university in a purely remote way, and this has already been done long before Covid 19 occurred.&lt;/p&gt;&#xA;&lt;p&gt;For instance, the largest university in Germany (in terms of students) is (and was, also before the pandemic) &lt;a href=&quot;https://www.fernuni-hagen.de/english/&quot; rel=&quot;noreferrer&quot;&gt;Fernuniversität Hagen&lt;/a&gt;, which offers exclusively remote programs. Thy offer both a Bachelor's and a Master's program in maths. You don't have to (and I guess, you probably can't) show up in person, except for the exams at the end of each semester.&lt;/p&gt;&#xA;&lt;p&gt;A similar concept is offered by &lt;a href=&quot;https://www.openuniversity.edu/&quot; rel=&quot;noreferrer&quot;&gt;Open University&lt;/a&gt; in the UK.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You won't get any argument from me if you claim that there is a lot of &amp;quot;inertia&amp;quot; in how mathematics is taught at universities, and that many things should and could be improved.&lt;/p&gt;&#xA;&lt;p&gt;I am also under the impression that the disruptions in teaching caused by the pandemic took off some of the inertia and opened various options for improvements, which at least some people are happy to use.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;However, I do not see any way how such improvements could or should result in a large fraction of the teaching being done online, let alone by means of mostly standardised online teaching materials.&lt;/p&gt;&#xA;&lt;p&gt;You ask for concrete arguments against implementation of online teaching on a large scale - so here we go.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Disclaimer:&lt;/em&gt; The following comments are exclusively about mathematics; I do not make any claims about different academic fields.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Remote teaching&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;While remote teaching can be done and has been done for a long time by some institutions, it can have some severe negative effects which make it, from my point of view, hardly suitable for a large fraction of students:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Learning mathematics becomes much easier if students work together.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;When I began studying mathematics (in 2007 - damn, I'm getting old...), one of the very first things we were told by various lecturers and tutors was that it is extremely hard to be successful if you only work on your own. We were thus encourage to work on the homework problems together with our peers. We were also encouraged (and sometimes even required) to write down and submit the solutions to all our homework problems in pairs. To most students this was extremely helpful. You learn and understand much more if you discuss the problems you're trying to solve with other students. You also learn to communicate about mathematics, which is an extremely important skill.&lt;/p&gt;&#xA;&lt;p&gt;Of course, remote teaching does not make cooperation between students impossible, but it makes it much harder. In particular, if remote teaching becomes somekind of standard, many students probably won't move any longer to the city where their university is located. Thus, they can't simply meet their peers in person, unless they are lucky and a few of their peers happen to live close to them (and they happen to get along well with precisely those peers). So in many cases, the only available option will then be to work with their peers remotely, too. This might work for a few students, but for most of them it will most likely come close to a disaster: it's more than difficult enough to get used to understand and discuss mathematics as a freshman. Doing this remotely will add considerably to these difficulties.&lt;/p&gt;&#xA;&lt;p&gt;Technical tools such as video calls and tablets are useful for some purposes, but there is simply no way to make an online meeting between, say, three first year students who try to solve their homework in Linear Algebra nearly as efficient as the same meeting when they all sit at the same table.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Students are social beings.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Of course, you can go to a bar at your hometown, with people completely unrelated to your maths program. However, when it comes to motivation for their studies, many students will feel much more motivated if they have a close and positive relationship to some students from the same program (with whom they can, for instance, discuss the course materials, discuss their homework, learn for exams, and so on.) Positive social relationships are much (much, much) easier to establish and to maintain if people meet in person and in high frequency.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Physical presence provides daily routine.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Yes, I know, students are adults, and all this. But that's besides the point. Most adults work more efficiently with daily routine, too. It is, obviously, possible to establish daily routine without physical presence, too. But it is usually harder.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Asking questions is much easier when you are physically present.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This refers to lectures, tutorials, and also office hours. If a student without much mathematical experience yet, asks me to explain something, this will be much easier if we are both in the same room. During the last few weeks, infection numbers in Germany have been quite low, so I was able to meet some students in office instead of remotely. Whenever I suggested this to a student, they were quite happy (or even relieved) to do this - apparently, they were all under the impression that things are much easier to discuss when everybody is in the same room.&lt;/p&gt;&#xA;&lt;p&gt;Again, it is far from impossible to ask and answer questions remotely. It is just harder and less efficient in most cases.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Lecturers are social beings.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Personally, I like teaching quite a lot. I love mathematics, and I like the idea to give students the opportunity to become passionate about mathematics, too. Well, most of them don't become really passionate about it, of course, but I still like it to see how they learn and understand something that as new to them.&lt;/p&gt;&#xA;&lt;p&gt;In order to experience this, I need to &lt;em&gt;meet&lt;/em&gt; my students - in real life, not just  on some computer screen. Not giving me this opportunity will, in the long run, most likely drag down my motivation. Dragging down the motivation of people who are passionate about teaching is not a particularly wise course of action if you're interested in high quality teaching.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Re-usage of materials results in lower quality, unless the materials are standardized.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;One of the main advantageous of teaching a course several times is that it gives the lecturer the option to improve each time. But this is only possible if the lecturer does not re-use, for instance, the same videos every year. At first glance, one might argue that changing only parts of the materials still saves times, but in my experience this is unlikely to work.&lt;/p&gt;&#xA;&lt;p&gt;For instance, a year ago I taught a course in point set topology. During the course I took a lot of notes about things that I would like to improve in case that I'm going to teach the course again. These note will most likely result in re-organizing all the contents of the course. I did the class as &amp;quot;inverted classroom&amp;quot;, with a manuscript and videos. But I will be essentially unable to reuse any video at all when I teach the course again.&lt;/p&gt;&#xA;&lt;p&gt;Hence, online course will, in general, only save time if most of the materials are somehow standardized, and the same materials are used by many lecturers. This brings us to the next point.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Usage of standardized materials&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;A preliminary remark about textbooks.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;There seems to be a misconception in the original post, where it is claimed that materials are already highly standardized for undergraduate mathematics. It is worthwhile mentioning that the idea to base an undergraduate course on a textbook seems to be very much a US thing. (When I started to regularly read posts here and on Academia StackExchange I was first quite confused how naturally many people spoke about &amp;quot;the&amp;quot; textbook in their course.)&lt;/p&gt;&#xA;&lt;p&gt;In fact, I can't remember a single course or lecture I've attended which was based on a textbook. Instead, we were typically given a list of books, along with the remark &amp;quot;These books treat the same topic as the course, but of course there are various differences. We advise you to have a look at these books, choose one that suites your preferences, and get it from the library so that you have a reference which gives you a second perspective on the topic.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The lecturer would then give the lecture based on their own course design. (Obviously, many lecturers use textbooks as a source of reference when designing and preparing a course; but in Germany - and I think the same is true for various further European countries - they typically don't follow one single textbook. Exceptions do exist, obviously.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So using standardized materials is far from universal. But also in places where undergraduate course are often based on textbooks, the suggestion in the original posts amounts to more, namely to also use standardized videos for online teaching.&lt;/p&gt;&#xA;&lt;p&gt;So, to continue the list, here are some objections against using standardized materials in order to &amp;quot;save time&amp;quot;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Mathematics is done by mathematicians.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is in extremely important point, I think (and I find it a bit surprising how rarely it is mentioned in discussions about this topic).&lt;/p&gt;&#xA;&lt;p&gt;It might be somewhat related to the academic system - but in my experience, first year students in mathematics find themselves in a very strange situation: they have been dealing with all these nice computations of derivates and integrals, and minima and maxima of functions, in school but have hardly had any significant exposure to what mathematics really means. Then they enter university and are essentially blown away both by the level of abstraction and by the pace of the courses.&lt;/p&gt;&#xA;&lt;p&gt;(A nice example: In my very first week aa a university student, I attended my first Linear Algebra lecture ever. The lecturer talked a few minutes about organizational matters; then the mathematics started like this: &amp;quot;Alright, you also attend a course on analysis, and there you'll learn some basics about propositional logics and elementary set theory during the next two weeks; you will also learn there what a field is. That's important, because we need this now. Definition 1.1: Vector spaces.&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;Now, this is tough for most students (of course, one might start on a less heavy note than this Linear Algebra course, but this will only briefly delay the shock), and there is a high risk that many students will very quickly come to the conclusion that mathematics at universities is just abstract non-sense which mortal beings like them cannot understand.&lt;/p&gt;&#xA;&lt;p&gt;Now if all this stuff is presented to them in videos that have been recorded by a different person they never met, and their lecturer will only answer their questions on the material rather than teaching the material, the &amp;quot;not comprehensible by mortal beings&amp;quot;-effect will probably even amplify by an order of magnitude or so.&lt;/p&gt;&#xA;&lt;p&gt;So to put it differently: I strongly believe that most students need to see a real and alive human being in front of them who presents and explains the contents of the course. They need this in order to understand that mathematics is not some completely remote mumbo jumbo done by weird professors who are sitting somewhere in basement offices without windows, in cities at the other end of the country, and recording videos - but by mathematicians who exist in the same universe as the students do, and are standing there right in front of the classroom and communicate with the students in person.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Lecturers learn a lot by designing a course.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is an argument not only against standardized videos, but also a reason why I don't mainly following a single textbook when designing a course. Whenever I prepare a course, it turns out that I learn a lot (this is an understatement, actually) about the things I'm going to teach. This is true both at the level of single results (I might, e.g., include a theorem in a lecture that I haven't been aware of before, or might find a proof of the theorem somewhere that I haven't known before and use it in the course, or I might already know the theorem but have never bothered to study its proof before) and on the level of the entire course (which definitions and results do I need to treat first; which organization of the material keeps the proofs as smooth and efficient as possible; which organization of the material is likely to provide proper motivation for the students, and so on).&lt;/p&gt;&#xA;&lt;p&gt;The understanding of such things that I get from designing the entire course on my own is much higher than the understanding I would gain from just following a textbook. Consider for instance the following line of reasoning:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;In Chapter 2, I include a Lemma 2.4 which is instrumental in the proof of Theorem 2.5; in fact, there would be a shorter and more direct proof of Theorem 2.5, but Lemma 2.4 combined with Example 4.9 (which we will treat later) will allow me to give a very efficient and easy-to-follow proof of a major result later on in Chapter 5.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;These dependencies between various parts of a theory are extremely difficult to grasp of you don't organize the material on your own. The major advantage of a good textbook - that the material is already very well-organized - is also a major drawback here: In a really well-thought book, the material will be so smoothly organized that you often don't even notice how much thought went into it and why the organization is precisely done the way it is.&lt;/p&gt;&#xA;&lt;p&gt;So, to sum up: I learn much more of I design my course materials on my own.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Lecturers who give the lecture themselves will be better prepared for questions.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Our human minds tend to be lazy. If somebody provides my with a video and a chapter of the lecture notes for a course, and I'm supposed to answer students' questions on this material, I will most likely briefly go through the lecture notes before class, and then see what happens.&lt;/p&gt;&#xA;&lt;p&gt;Of course, I will be able to answer many questions of the students with this kind of preparation, and I'm confident that my answers will be of reasonably high quality on average.&lt;/p&gt;&#xA;&lt;p&gt;However, there is no chance that my answers will be as good as they would be if I had prepared and given the lecture myself. When I have to give a lecture, I will take a sheet of paper before the lecture, and write down most of the content of the lecture for me, once again. In particular, I will write down the proofs that I'm going to discuss. Obviously this will make me much more familiar with the contents than just reading briefly through the lecture notes.&lt;/p&gt;&#xA;&lt;p&gt;Of course, I &amp;quot;could&amp;quot; do them same thorough preparation even if there's the pre-recorded video and I'm just supposed to answer questions. But I'm quite sure that I won't, and I'm also quite sure that many colleagues won't, either. You could argue that this is stupid behaviour, that we should overcome the underlying psychological bias, and that we should thus prepare in the same way as if we were to give the lecture ourselves. But this is besides the point: Good management decisions are not about what people should do, they are about what people will do.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;It is important to keep the contents diverse.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This point is not so much about the students' or faculties' perspective, but rather about society (or at least, mathematics) as a whole. Where I live, university are almost exclusively funded by taxpayers' money - so there's a legitimate requirement that teaching at universities should not only benefit the students, but also society on a larger scale.&lt;/p&gt;&#xA;&lt;p&gt;I argue that it is beneficial to cover a very diverse range of ways to teach undergraduate maths courses. It is true that, for instance, the basics of Linear Algebra are unlikely to change in the next few decades, and it is also true that the contents of a course on, say, Linear Algebra are to some extent pre-defined by the syllabus (which will, in most cases, be designed by the math department).&lt;/p&gt;&#xA;&lt;p&gt;But even under these boundary conditions, it is possible to vary the approach to a course on Linear Algebra considerably. If we were to use a set of pre-defined videos to teach this course, numerous students would learn Linear Algebra in precisely the same way, in many other perspectives on Linear Algebra would likely be hardly recognized. Such a loss of diversity is not a particular good foundation for scientific progress (to put it mildly).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Lecturers can be somewhat idiosyncratic.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is one more point about the motivation of the lecturer. Many mathematicians  I know (including myself) tend to have quite a few preferences (not too say: strong opinions) on how certain things should be done in teaching in general, and in certain courses in particular. Urging (or even forcing them) to use pre-defined material, with quite restricted options to adjust it to their own preference, is quite likely to have a negative impact on their motivation.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I try to follow the leitmotiv &amp;quot;If I do something, I try to do it right.&amp;quot; So if you gave my three different textbooks with associated video courses and asked me to choose one of the three and to give a course by providing the students with these materials and answering questions, chances are that I would find that none of the three sets of materials &amp;quot;does it right&amp;quot;, and consequentially I would care much less about the course. (Important note: Obviously, &amp;quot;do it right&amp;quot; is completely subjective here, and there will, in general, not be any objective reason to back up my feeling that the materials should be presented differently. However, this doesn't change the fact that I would be much less motivated to teach from materials which do not fit my preferences.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastEditorUserId="276" LastEditDate="2021-08-16T23:58:59.307" LastActivityDate="2021-08-16T23:58:59.307" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23267" PostTypeId="1" CreationDate="2021-08-17T20:53:09.183" Score="-4" ViewCount="112" Body="&lt;p&gt;My First Job Interview presentation topic for a permanent assistant lecturer position in Mathematics and Data science is below:&lt;/p&gt;&#xA;&lt;p&gt;''You have been asked by your Head of Department to design an elective module for the final year of the honours undergraduate Mathematics and Data Science programme. You are required to give a five-minute presentation pitching your module to a group of students.''&lt;/p&gt;&#xA;&lt;p&gt;Could anyone help/suggest to me how or what to prepare for slides. What are the modules I should offer? Below is the whole syllabus for them. Thanks for any help and suggestions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ayUgV.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ayUgV.png&quot; alt=&quot;Year 4&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kH2GK.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kH2GK.png&quot; alt=&quot;Year 3&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AhCyh.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AhCyh.png&quot; alt=&quot;Year 2&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mfZW0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mfZW0.png&quot; alt=&quot;Year 1&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15570" LastEditorUserId="15570" LastEditDate="2021-08-17T23:22:41.120" LastActivityDate="2021-08-17T23:22:41.120" Title="Help me to prepare for my first job interview presentation" Tags="|course-design|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23268" PostTypeId="2" ParentId="23267" CreationDate="2021-08-17T22:50:07.093" Score="1" Body="&lt;p&gt;I'm assuming a &amp;quot;module&amp;quot; is a course.  But perhaps you could clarify what that means in terms of time extent.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You have 5 minutes, so that is 3 slides (max).  Might even just do one!  Yes, realio trulio.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I suggest picking something that INCLUDES statistics.  Math and data science basically says stats.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;My advice is to pick something topical that will interest students.  Something that helps them get jobs or laid or the like.  Internet crap, maybe.  Or pharma stuff (if at a school with strong bio).  Or Covid.  Or global warming.  Or oil (if at UT).  Etc.  If you can find something that interests YOU as well, great.  But try to look for the win/win.  NOT what interests you, but not others.  Something that interests both.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Market research or surveying are sort of easy things.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Consider some course with a modicum of practical stats teaching (to include survey design, DOE, etc.  NOT theoretical stats crapola.)  And working the kids to do projects.  (which will probably include data gathering*, analysis, modeling**).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;*Use Survey Monkey.&lt;/p&gt;&#xA;&lt;p&gt;**Many schools have student licenses for Stata or the like.  But there's also R.&lt;/p&gt;&#xA;&lt;p&gt;Bottom line.  Make it sexy. Make it sizzle. BE ENTHUSIASTIC.&lt;/p&gt;&#xA;&lt;p&gt;Other than that, practice it several times.  And BE ENTHUSIASTIC.  Even phony enthusiasm is catching.  BE OUTGOING.&lt;/p&gt;&#xA;" OwnerUserId="17637" LastActivityDate="2021-08-17T22:50:07.093" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23272" PostTypeId="1" AcceptedAnswerId="23273" CreationDate="2021-08-19T21:06:27.390" Score="14" ViewCount="5166" Body="&lt;p&gt;I am tutoring a grade 8 student that failed his math exam, but surprisingly it seems that he is more than smart enough: he understands everything I say really quickly, and he knows most of the ideas from class before I even say them.&lt;/p&gt;&#xA;&lt;p&gt;When he solves some exercises beside me he does very well, so I was thinking why he failed the exam, and I couldn't find a reason.&#xA;Also when I give him some homework and he does some parts wrong, then when I just point them out without saying anything more, he knows what is wrong and how to correct it.&lt;/p&gt;&#xA;&lt;p&gt;So my question is, how do I help such a student?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;He understands all the ideas in his lessons (parms, right triangles, first degree equations, identities and real number operations)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;He finishes his exams on time (he is not slow in solving)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;3-He said that he does not have exam anxiety, and he does not stress out in his exams.&lt;/p&gt;&#xA;&lt;p&gt;So what is the problem?&lt;/p&gt;&#xA;&lt;p&gt;I think he is just reluctant, but how do I solve that?&lt;/p&gt;&#xA;" OwnerUserId="12486" LastEditorUserId="12486" LastEditDate="2021-08-21T08:30:33.850" LastActivityDate="2022-07-12T07:08:22.177" Title="A smart student that struggles in exams" Tags="|exams|tutoring|struggling-students|psychology|anxiety|" AnswerCount="7" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="23273" PostTypeId="2" ParentId="23272" CreationDate="2021-08-19T21:32:18.940" Score="25" Body="&lt;p&gt;There are several possible explanations. Without much more information, it is impossible to give a clear-cut answer.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Perhaps your student is able to &amp;quot;read&amp;quot; your unconcisious reactions very well, and you are telegraphing the solution.&lt;/li&gt;&#xA;&lt;li&gt;Some people just go blank in stressful situations, like an exam (you say it's not the case, but it might be anyway).&lt;/li&gt;&#xA;&lt;li&gt;Some people are just overconfident, and don't read exam questions carefully (solving the wrong problems, making stupid mistakes by not reviewing their work).&lt;/li&gt;&#xA;&lt;li&gt;Care only for the &amp;quot;big picture&amp;quot;, understand what it is all about and not slogging through doing detailed work to really get proficient in applications.&lt;/li&gt;&#xA;&lt;li&gt;The problems you solve while studying are just different than the exam. E.g., you review how to solve equations, and the exam asks to set up the equations (and solve them).&lt;/li&gt;&#xA;&lt;li&gt;Perhaps there are language problems in understanding the exams.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I'd try to set up some exam-like study sessions (leave the student to solve a sequence of problems by themselves, and then go over the result together with a fine toothed comb). That would help train for exams, and highlight any problem areas/origins). Also review the exams given to see where the problems lie.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2021-08-20T15:18:46.680" LastActivityDate="2021-08-20T15:18:46.680" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="23274" PostTypeId="2" ParentId="2763" CreationDate="2021-08-19T23:57:06.533" Score="0" Body="&lt;p&gt;No, the highway of multiplication is not repeated addition, but its on-ramp is, just as the highway of the product of topological spaces is not simply their Cartesian product, but its on-ramp consists of studying the Cartesian product of finitely-many topological spaces. These can be considered to be caterpillar-versus-butterfly situations, or as logarithmic (inverted 'L') situations, the rapid-rise portion corresponding to the initial state, and the slow-rise portion corresponding to the final state. Artifact-evolution follows the same inverted 'L' pattern. For example, the changeover from the biplane to the monoplane follows this pattern.&lt;/p&gt;&#xA;&lt;p&gt;So, no, don't avoid teaching multiplication as repeated addition, but point out, when the time is right, that multiplication eventually takes on a life of its own.&lt;/p&gt;&#xA;" OwnerDisplayName="user10552" LastActivityDate="2021-08-19T23:57:06.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23275" PostTypeId="2" ParentId="23272" CreationDate="2021-08-20T09:17:39.467" Score="8" Body="&lt;p&gt;I had a similar student, these 2 things helped.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Take mock tests with different criteria, like a test where questions are easy but paper is too lengthy to complete in time or sometimes very few questions but too difficult to solve and you`ll be able to filter out the exact problem.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask your student to explain concepts to you or take a theoretical test.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="17644" LastActivityDate="2021-08-20T09:17:39.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23276" PostTypeId="2" ParentId="23272" CreationDate="2021-08-20T12:45:40.453" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;When he solves some exercises beside me he does very well, so I was thinking why he failed the exam, and I couldn't find a reason. Also when I give him some homework he do some parts wrong, but when I just point them out without saying anything he knows what is wrong and how to correct it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is the problem. Carelessness.&lt;/p&gt;&#xA;&lt;p&gt;I too have this problem a lot, I think the thing is being unable to focus the attention into the task at hand (there are sometimes where there is just lack of knowledge as well). It may be due to the following issues : psychological issues, stress, attention disorders, anxiety disorders.&lt;/p&gt;&#xA;&lt;p&gt;A diagnostic test one could do is to do is to observe the student when they solve and see how their body language/ posture changes. Another is to simply ask them to speak out their thought process as they work through the exercises. Funny enough, I find myself to make the lesser mistakes when I am speaking aloud what I am doing.&lt;/p&gt;&#xA;&lt;p&gt;Another thing is to see where the mistakes are made and try to find practical solutions for it, for example I remember making a lot of mistakes when copying information from the problem onto the paper. I would get the digits all messed up.&lt;/p&gt;&#xA;&lt;p&gt;A way I fixed this was to know actively tell myself to look at the digits in the question paper when copying them. Yes,  this sounds simple but this is something I had to train myself to do.. my eyes would fly all over the place&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorUserId="14627" LastEditDate="2021-08-21T02:57:58.347" LastActivityDate="2021-08-21T02:57:58.347" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="23277" PostTypeId="1" CreationDate="2021-08-20T16:26:55.747" Score="1" ViewCount="129" Body="&lt;p&gt;I completed a BSc in Computer Science recently and am going on to do an MSc in Data Science. However, the only focussed math module I had during CS was in the first year and I didn't do too well. I really want to ace it this year so I'm trying to get spend some time getting it. I've got a 'Content outline' which I've retrieved from the UK University's website so I have a list of things to do. Issue is, I know I'm gonna struggle and more sure that I'll forget what I learned if I ever learn it. How can I concrete the newly found knowledge in my brain. Is there a (fun) or repetitive method that anyone here have had success with when learning math?&lt;/p&gt;&#xA;&lt;p&gt;Statistics is new territory to me too (I think). Any suggestions, recommendations or advice is greatly appreciated.&lt;/p&gt;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="17646" LastActivityDate="2021-08-20T18:59:35.780" Title="How can I effectively learn and master Math and Statistics for Data Science?" Tags="|calculus|statistics|probability|applied-mathematics|computer-science|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23278" PostTypeId="2" ParentId="23277" CreationDate="2021-08-20T18:45:21.527" Score="2" Body="&lt;p&gt;You could try writing careful and detailed notes for yourself, and by &amp;quot;careful and detailed&amp;quot;, I mean something you think even your former teachers would think is good and something you would feel comfortable in lending to someone else who needed to review the material. Maybe use two or three different colors of pens, if the notes are handwritten -- one color for problem statements, another color for problem solutions, another color for facts/theorems to remember, etc.&lt;/p&gt;&#xA;&lt;p&gt;The reason I am emphasizing the quality of the notes is that this will force you to spend time thinking carefully about the material, and thus you will likely retain it better. Also, the notes will be something you can refer back to if &amp;quot;I'll forget what I learned if I ever learn it&amp;quot; comes to pass, but my guess is that the mental calm you'll get from preparing the notes will reduce your fears, thus allowing your memory to function better.&lt;/p&gt;&#xA;&lt;p&gt;Also, depending on the subject matter and how well you know it, you could write problems on index cards, with solutions on the reverse side. This would work well, I think, for probability and combinatorics problems, since most of that involves practice in identifying how to solve the problems, and not with learning a lot of theory stuff. The index cards can then be ordered or grouped in various ways, in case your initial way of ordering/grouping them changes as you become more familiar with the material, something that is difficult to do with pages and pages of ordinary handwritten notes (because you'll have to decide in advance whether topic A comes before topic B, etc. when writing). Of course, for digitally-typed notes, you can rearrange sections and problems even after writing them.&lt;/p&gt;&#xA;&lt;p&gt;By the way, nothing about this is specific to statistics or data science. I used the notecards method in Fall 2011 when I spent a few months studying Hilbert style propositional logic systems (e.g. see &lt;a href=&quot;https://math.stackexchange.com/a/94483/13130&quot;&gt;this answer&lt;/a&gt;), and I wanted to keep track of those results that could be proved using only modus ponens and the deduction theorem (e.g. the positive implication fragment of intuitionistic propositional logic), and then those results one could additionally prove in the slightly larger system of Johansson's minimal negation implication logic, and then those results one could additionally prove in intuitionistic implication logic, and finally those results one could additionally prove in ordinary implication logic. Regarding written notes and such for myself on various topics, I've been doing this since middle school (ages 12-14, 1971-73).&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2021-08-20T18:45:21.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23279" PostTypeId="2" ParentId="23277" CreationDate="2021-08-20T18:59:35.780" Score="2" Body="&lt;p&gt;First, I would try to emphasize accessible texts, exercises, etc.  If you are weak, you don't load 225 on the bar and start benching.  You absolutly SHOULD get in the gym.  And will have faster gains than someone who is already very strong.  But you need to be progressive.  Going too heavy will hurt/discourage you.  Similarly with studying something you're weak in. Be progressive (not politically, but &amp;quot;bit by bit&amp;quot;).  In particular, be wary of many people on MSE who are sooper smart in math and weak on training weak students, who prescribe books/courses that are too tough and/or theoretical for weak/applied students.&lt;/p&gt;&#xA;&lt;p&gt;Second, take a look at the course descriptions, syllabi, and texts and see what math is required.  Emphasize the first semester.  See where you think you might have issues/not.&lt;/p&gt;&#xA;&lt;p&gt;Try to stay on top of the courses overall.  Pre-read the lessons in the book (actually do this) and (try to) work the problems before the lectures.  Don't get behind, don't cram.  Go to office hours.  Etc. A stronger student might be able to survive with a non-optimal study regime, but you are already fighting with one arm behind your back.&lt;/p&gt;&#xA;&lt;p&gt;Write down your steps during any derivations or homeworks.  If you miss something because of a &amp;quot;stupid mistake&amp;quot; rework the problem all over again and do it correctly.&lt;/p&gt;&#xA;&lt;p&gt;I would not despair on the math.  Realistically stats is a blizzard of formulas.  But at the end of the day, if you have intuitions about some basic concepts, you can do most of it (for example least squares regression is &amp;quot;like drawing a line through the data&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Hang in there and have a little fun.  Hard work pays off.  Really, it does.  You can do this.&lt;/p&gt;&#xA;" OwnerUserId="17648" LastActivityDate="2021-08-20T18:59:35.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23280" PostTypeId="2" ParentId="23272" CreationDate="2021-08-20T21:24:43.527" Score="8" Body="&lt;p&gt;I was this student throughout school and University. Professors would comment that I was smart, that I knew the material, but I'd still do poorly on exams. I made careless mistakes: I would sometimes forget to finish questions half-way through, clearly make simple arithmetic errors and just otherwise not give the exam the attention it deserved.&lt;/p&gt;&#xA;&lt;p&gt;Studying was also a struggle, it was the worst part of school for me. It'd sap time and energy, and I'd get frustrated that I was spending so much time studying and getting nowhere.&lt;/p&gt;&#xA;&lt;p&gt;And then I was diagnosed with ADHD as an adult, and it all suddenly made sense.&lt;/p&gt;&#xA;&lt;p&gt;The forgetting stuff on the exam, the difficulties studying and the overall carelessness. I'm not saying your student has ADHD, but it's something that could be worth looking in to. From experience, I can tell you that none of the usual strategies for studying worked for me, and I tried a lot over my 10-year undergrad career.&lt;/p&gt;&#xA;&lt;p&gt;Apologies if this is an off-topic answer, but I couldn't post a comment.&lt;/p&gt;&#xA;" OwnerUserId="17649" LastActivityDate="2021-08-20T21:24:43.527" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23281" PostTypeId="2" ParentId="23272" CreationDate="2021-08-21T10:38:20.457" Score="3" Body="&lt;p&gt;That same thing happened to me when I was in school.  Even in University it still happened to me.  Even now as a programmer I still see the same pattern happening to me.&lt;/p&gt;&#xA;&lt;p&gt;I was conscious of the problem, but had no solution.  In fact, I don't think of it as a problem, but as an advantage (I'll discuss about this below).&lt;/p&gt;&#xA;&lt;p&gt;When I do maths exams, I like so much maths problems that I always remember (like in video) all of the exam, including the questions and the answers I gave, and even remember all of my mental process to derive such answers.&lt;/p&gt;&#xA;&lt;p&gt;But I don't like so much reading the exam itself.  And when I don't like something, I'm not very interested in it, to the point I don't care what I do with it or what results/consequences it has.&lt;/p&gt;&#xA;&lt;p&gt;One of the first examples of the above that I remember is the following:  I remember doing an exam when I was 11 when after exiting the exam room I remembered the whole exam and I did it perfect... I did, but didn't read one question that was very short between two long questions, and therefore didn't answer that one.  It simply hadn't reached my attention.  The rest of the exam was perfect.&lt;/p&gt;&#xA;&lt;p&gt;Another example I remember very well is the following:  I did another maths exam when I was 17, and even though I like them, I don't like to lose 2h of my time in an exam; it simply bores me, and my performance decreases considerably.  And at that age, exams start being 2h in Spain (and at university I had 4h exams, in which more like this happened).  In those kinds of exams, either I nail it in a very short time (&amp;lt;&amp;lt;1h) and I get out of the exam room immediately, or I'm doing it really bad.  In that exam, when I was 17, I almost nailed it in maybe 30 minutes or so.  But that day I preferred sleeping in the exam chair rather than going to the corridor, so after giving the teacher the exam paper I simply slept on the exam chair (my teacher was OK with that).  After some long sleep, I suddenly woke up and told my teacher that there was a mistake in my deduction and I told her the exact place on the exam paper, and asked her if I could fix it.  As she knew for sure that I hadn't talked to anyone or cheated at all, and I was still inside the exam time, she allowed me to fix it.  She was very amused that I had been reviewing all of my exam while sleeping.&lt;/p&gt;&#xA;&lt;p&gt;Other times I may forget part of the question while I'm answering it.  I may have already answered the &amp;quot;interesting&amp;quot; (to me) part of the question, and then I (unconsciously) lose interest on the rest of the question, maybe because the rest of it is very obvious, or some other reasons...&lt;/p&gt;&#xA;&lt;p&gt;The conclusion I got for myself after so many years of repeating these things, is that it's not a problem. I simply can't concentrate my attention on things that don't really interest me, but I can concentrate exceptionally on things that I like.  So, OK, I can live with that, I'll simply accept it, and try to continue working on things that attract me, and avoid those that don't.  I can make a very good profit of that ability of mine to live.  And in the end, not everyone can do what everyone else does; trying hard to &amp;quot;fix&amp;quot; it is probably only going to bring unhappiness, because there's nothing to be fixed.&lt;/p&gt;&#xA;&lt;p&gt;What I'd do with this kid if I were a teacher, I'd make as much of an exception as I can (as much as the laws allow), and grade him according to the understanding of maths that he shows in reality, and not just in the exams.&lt;/p&gt;&#xA;&lt;p&gt;As per myself, I had to change schools when I was 12 due to this. Later on I continued getting much lower grades in high school and university than I could have gotten (still had quite good ones), and even in the end I didn't finish my degree because of 1.5 ECTS (out of 240 ECTS) because I lost interest in it.  And have I had any serious problems due to that? Nope; I'm very happy, and have had very good jobs.  Even if he may have the same problems in the future, I'd just not try so hard to fix them, or at least not so hard that it demotivates him to do what he likes.  Specializing is not bad; in fact it's a good thing, IMO.&lt;/p&gt;&#xA;" OwnerUserId="17651" LastEditorUserId="17651" LastEditDate="2021-08-21T22:59:34.703" LastActivityDate="2021-08-21T22:59:34.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23282" PostTypeId="1" AcceptedAnswerId="23283" CreationDate="2021-08-21T14:43:40.630" Score="7" ViewCount="1383" Body="&lt;p&gt;I currently work as a substitute teacher at a local high school and the topic in one of the classes is sequences, series and limits. Because I always disliked learning about a topic without having some sense of what problem it addresses or what new possibilities it opens, I wanted to present a motivating example in the first lesson. What I had in mind was an application of sequences, series or limits that is easy enough to grasp without already knowing the topic but still reliant on its ideas to motivate its study.&lt;/p&gt;&#xA;&lt;p&gt;To clarify some of the terms I am using: By sequences, series and limits I mean the basic theory of sequences (explicit formulas, recursive formulas, arithmetic sequence, geometric sequence, divergent and convergent sequences), the simple series (arithmetic series, geometric series, divergence of the harmonic series) and the definition of convergence using an epsilon bound.&lt;/p&gt;&#xA;&lt;p&gt;After doing some research on the web I went with Zenos paradox of &lt;a href=&quot;https://plato.stanford.edu/entries/paradox-zeno/#AchTor&quot; rel=&quot;noreferrer&quot;&gt;Achilles and the Tortoise&lt;/a&gt; and after presenting it, letting the students discuss it among themselves and arguing Zenos position, I showed them how the paradox can be resolved with an infinite series. In the last part however, I had to simply state that this infinite summation yields a finite value and that proving this fact and similar things would be the content of the following lectures.&#xA;Afterwards I felt like I had presented a solution that wasn't really understandable and that left people rather disappointed than hungry for more.&#xA;So my question is if any of you know motivating examples for the study of sequences, series and limits to get high school students interested in the topic?&#xA;Ideally I would like these examples to highlight the importance of dealing with infinities. I know there are plenty of cool problems involving sequences and series that connect to geometry, physics or finance ( for example here: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/11106/examples-of-arithmetic-and-geometric-sequences-and-series-in-daily-life&quot;&gt;Examples of arithmetic and geometric sequences and series in daily life&lt;/a&gt; ) but I haven't found ones that met my criteria of demonstrating the need for sequences, series and dealing with infinity in an accessible way.&lt;/p&gt;&#xA;" OwnerUserId="17653" LastActivityDate="2021-10-20T08:54:56.317" Title="Motivating example for sequences, sums and limits in high school" Tags="|examples|sequences|series|infinity|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23283" PostTypeId="2" ParentId="23282" CreationDate="2021-08-21T15:56:42.430" Score="11" Body="&lt;p&gt;This application is known as &amp;quot;&lt;a href=&quot;https://www.investopedia.com/terms/g/gross-up.asp&quot; rel=&quot;noreferrer&quot;&gt;gross-up&lt;/a&gt;&amp;quot; in accounting.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You run the finances for a small business. The boss would like to give an employee a \$100 bonus for their hard work. However, the employee's earnings are taxed at 20%. The boss says &amp;quot;okay fine; pay them \$120 and then they will get \$100 out of it.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;a) If you pay the employee \$120, how much money do they actually take&#xA;home?&lt;/p&gt;&#xA;&lt;p&gt;b) Can you cover the difference by adding a few dollars to the&#xA;payment?&lt;/p&gt;&#xA;&lt;p&gt;c) How much money should you actually pay the employee for&#xA;them to take home \$100?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The final answer of \$125 can be found by either&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$(1-0.2)x = 100$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x = 100 + 20 + 4 + 0.80 + \ldots$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-08-21T15:56:42.430" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23284" PostTypeId="2" ParentId="23282" CreationDate="2021-08-21T16:29:05.100" Score="3" Body="&lt;p&gt;The common puzzle of giving a few terms and asking for the next are examples of (generating) sequences by some particular rule.&lt;/p&gt;&#xA;&lt;p&gt;A series is just a sequence, summed together. Ask e.g. for the sum &lt;span class=&quot;math-container&quot;&gt;$1 + 2 + \dotsb + n$&lt;/span&gt; (a finite series).&lt;/p&gt;&#xA;&lt;p&gt;For infinite series/limits, ask for &lt;span class=&quot;math-container&quot;&gt;$0.1 + 0.01 + \dotsb$&lt;/span&gt;, you can plot the sucsesive terms and show they get ever nearer &lt;span class=&quot;math-container&quot;&gt;$1 / 9$&lt;/span&gt;. Or use &lt;span class=&quot;math-container&quot;&gt;$1 + 1/2 + 1/4 + \dotsm$&lt;/span&gt;. Bonus point is to point out that e.g &lt;span class=&quot;math-container&quot;&gt;$0.1 + 0.1 + \dotsm$&lt;/span&gt; doesn't have a limit, so that the terms tend to zero is a requisite for convergence, then show that &lt;span class=&quot;math-container&quot;&gt;$1 + 1/2 + 1/3 + \dotsm$&lt;/span&gt; doesn't converge either (simplest proof is to compare with &lt;span class=&quot;math-container&quot;&gt;$1 + (1/2) + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + \dotsm$&lt;/span&gt; and note that each term is less than the respective term of the series and each parentesis adds up to &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2021-08-22T01:40:11.950" LastActivityDate="2021-08-22T01:40:11.950" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23285" PostTypeId="2" ParentId="23282" CreationDate="2021-08-21T16:50:38.300" Score="6" Body="&lt;p&gt;I'm not sure that starting with an applied motivation (derivation, word problem) is the best way to introduce this topic.  Look at how your experiment failed.  This is because &amp;quot;word problems are hard&amp;quot;.  Really, they are. One of my big criticisms of most PDE texts is introducing the topic as an engineering problem.  And I say this as an applied guy!&lt;/p&gt;&#xA;&lt;p&gt;Had an ODE teacher who always introduced new techniques with a motivating applied problem. But I found this much harder than learning the technique first.  Then, perhaps practicing it later, after mastery, on some applied problems (probably well after sufficient drill on simple non-applied exercises).  Had a sub for that course, who just stressed for us the main techniques and exercises and the like that were &amp;quot;just algebra&amp;quot; and it was really refreshing for a class that was getting annoying to me.&lt;/p&gt;&#xA;&lt;p&gt;Now, this doesn't mean zero motivation.  But I think it's really sufficient to just mention that these are techniques useful in such and such fields very quickly and without proving how or deriving it.  Like if I were teaching imaginary numbers the first time, I might mention that they're useful in electronics.  But I'm not getting into ELI the ICE monster.  It's just a throwaway remark to say that the topic has uses.&lt;/p&gt;&#xA;&lt;p&gt;This doesn't mean you shouldn't know the applications.  Yes, you should, because you should have a deeper knowledge of the topic than your students.  It enriches you.  But you don't need to show that.  Remember these topics are new to the students.  Every fresh crop of kids brings fresh ignorance.  So, try to introduce things gently and progressively.&lt;/p&gt;&#xA;" OwnerUserId="17654" LastActivityDate="2021-08-21T16:50:38.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23286" PostTypeId="2" ParentId="23282" CreationDate="2021-08-21T17:47:42.303" Score="3" Body="&lt;p&gt;There are lots of good physics examples involving equilibrium. For example, you can set up a pendulum and show how the amplitude forms a sequence that decays exponentially toward zero, or describe putting a hot object and a cold object in thermal contact and measuring the temperature at some fixed time interval.&lt;/p&gt;&#xA;&lt;p&gt;When you ask students questions like, &amp;quot;How could we give a numerical description of how strongly friction affects the pendulum?,&amp;quot; they will often come up with answers like &amp;quot;Count how many swings it takes until it stops moving.&amp;quot; They usually find it pretty intellectually interesting to consider the possibility that there is never a discrete point at which the motion stops (although eventually the motion will certainly become too weak to be detectable given background perturbations such as air currents and vibration from passing trucks).&lt;/p&gt;&#xA;&lt;p&gt;Epidemiological or biological models might provide a motivation similar to Zeno's paradox but without the silly philosophical cruft. When you consider the population of a certain species over time, there can be cases of positive feedback, which cause exponential growth or decay, and also negative feedback, which causes oscillations. The exponential decay cases are mathematically the same as Zeno's paradox. E.g., we expect the number of people who have been exposed to covid to exponentially approach the entire population. To make this a discrete sum, simply pose the model as a discrete one, by talking about the total cases per year or something.&lt;/p&gt;&#xA;" OwnerUserId="17655" LastEditorUserId="17655" LastEditDate="2021-08-21T17:52:45.810" LastActivityDate="2021-08-21T17:52:45.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23287" PostTypeId="2" ParentId="23272" CreationDate="2021-08-22T01:03:52.960" Score="4" Body="&lt;p&gt;•Have him create a short 10-question exam for you and/or for some younger students. (just to give him different perspective - from that of the test maker) Explain why some missed some questions and some missed others.&lt;/p&gt;&#xA;&lt;p&gt;•Give him the challenge of creating a math test by giving him the answers first and he has to create a question to get that answer.&lt;/p&gt;&#xA;&lt;p&gt;•Discuss with him what it means to be indifferent.  Then discuss his indifference about completing tests.  Explain how the test is REALLY a way to give the teacher a grade.  It is a tool teachers use to determine how well they are able to convey the information to the students.  Isn't that the truth? :)&lt;/p&gt;&#xA;&lt;p&gt;•Have him learn about the skill of test-MAKING. Maybe that will help with his interest.&lt;/p&gt;&#xA;&lt;p&gt;•Create a game of creating a word problem by going back and forth between you and him. You start off by writing &amp;quot;find x&amp;quot;  then he adds a variable or a number, then your turn to add a symbol or number, etc... back and forth... This will turn into a random string of statements, variables, numbers etc....  This will illustrate how test questions are hard to make and some DO NOT have an answer.&lt;/p&gt;&#xA;&lt;p&gt;•Create a test for him where ONE of the 5 questions is not solvable.  If he can spot the UNSOLVABLE one then he doesn't have to do the other 4.  But he does have to show on paper how that unsolvable question is unsolvable.   This might help him to look at tests as ways to show the teacher they wrote good questions and all of them were solvable.&lt;/p&gt;&#xA;&lt;p&gt;Just some random thoughts.. I really hope this helps or inspires.&#xA;You are precious for asking and wanting to help this young person.&lt;/p&gt;&#xA;&lt;p&gt;Blessings.&lt;/p&gt;&#xA;" OwnerUserId="17658" LastActivityDate="2021-08-22T01:03:52.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23288" PostTypeId="2" ParentId="23272" CreationDate="2021-08-22T07:16:51.377" Score="4" Body="&lt;p&gt;Khan Academy has great problems for drill.  I have often used it for careless students. Advantages are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;there are no clues from you&lt;/li&gt;&#xA;&lt;li&gt;if it's wrong the computer will say so  and he doesn't get credit for understanding, so he will learn to focus on accuracy&lt;/li&gt;&#xA;&lt;li&gt;he will have a chance to correct it which will reinforce his understanding and accuracy.&lt;/li&gt;&#xA;&lt;li&gt;he can do this on his own.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Bottom line is many motivated kids often outgrow their carelessness.  I was very careless at 12, but by 14 I was no longer making careless mistakes. Of course this doesn't help the present, so I suggest trying Khan academy.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-08-22T07:16:51.377" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23289" PostTypeId="2" ParentId="23282" CreationDate="2021-08-22T22:39:01.737" Score="0" Body="&lt;p&gt;Taylor series are really useful tools for solving a number of calc 2 problems without analytic solutions. It's a bit more advanced than some of these other examples, but might still be viable as a way of tying a number of topics together.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-08-22T22:39:01.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23290" PostTypeId="1" AcceptedAnswerId="23291" CreationDate="2021-08-23T15:30:17.637" Score="1" ViewCount="658" Body="&lt;p&gt;I am an highschool student (I'm starting soon the italian equivalent of 12th grade) and I have many problems with the way hs math education works. I don't understand why everything is explained only at an intuitive level, and there are no clear and precise definitions/proofs.&lt;/p&gt;&#xA;&lt;p&gt;The biggest problem is the approach to real numbers: we always use them even though we have never defined what they are, which is understandable because it's not really easy to explain Dedekind Cuts/Cauchy sequences to highschool students, but we also have never touched upon on why the real numbers are different from the rational ones, that is also the reason why everything we do works: the least upper bound/completeness property of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. As a consequence of this, we get a lot of definitions that are not really definitions but are more like &amp;quot;defining things into existence&amp;quot;, for example this year we learned about the square root function and this was the definition: &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$x \geq 0$&lt;/span&gt;. The square root of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the unique number &lt;span class=&quot;math-container&quot;&gt;$r \geq 0$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$r^2 = x$&lt;/span&gt;&amp;quot;. And the same thing happened for exponentials and logarithms (by the way it's not so hard to define these functions using only elementary techniques: you can use some simple inequalities + the completeness property and you're done).&lt;/p&gt;&#xA;&lt;p&gt;The other big problem is the approach to mathematics, in highschool math most of the stuff you do is use some result you never really understood to simplify an arbitrary expression or solve an arbitrary equation, instead of doing abstract thinking/problem solving or analyzing some key examples in the theory to really understand the results.&lt;/p&gt;&#xA;&lt;p&gt;Why is that? Why doesn't anyone try to change this?&lt;/p&gt;&#xA;" OwnerUserId="17662" LastEditorUserId="2083" LastEditDate="2021-08-23T17:53:56.837" LastActivityDate="2021-08-29T11:20:38.763" Title="Why is highschool math so unrigorous?" Tags="|secondary-education|algebra|real-numbers|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="23291" PostTypeId="2" ParentId="23290" CreationDate="2021-08-23T16:40:24.440" Score="8" Body="&lt;p&gt;For the same reason that elementary counting numbers of more than a single digit are explained as &lt;em&gt;ones&lt;/em&gt;, &lt;em&gt;tens&lt;/em&gt;, &lt;em&gt;hundreds&lt;/em&gt;, &lt;em&gt;etc.&lt;/em&gt; . The concept of powers and exponents has not yet been developed. Later it is re-cast in terms of powers of &lt;strong&gt;ten&lt;/strong&gt; with a &lt;strong&gt;decimal&lt;/strong&gt; point delineating the positive/negative powers. Even then, it's often never further defined as powers of a &lt;strong&gt;Base&lt;/strong&gt; with a &lt;strong&gt;radix&lt;/strong&gt; point delineation.&lt;/p&gt;&#xA;&lt;p&gt;The ability to successfully use the mechanics does not necessarily require understanding the abstract concepts, just as the ability to use a smart phone or computer does not require understanding how they work.&lt;/p&gt;&#xA;&lt;p&gt;It's simply more practical to teach functionality first. Abstract understanding can come later, or not at all in many cases.&lt;/p&gt;&#xA;" OwnerUserId="10376" LastActivityDate="2021-08-23T16:40:24.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23292" PostTypeId="1" CreationDate="2021-08-23T18:04:22.000" Score="13" ViewCount="194" Body="&lt;p&gt;I've been tutoring mathematics at university level for over 10 years, and one of the more common requests from students is worked solutions for sheets of exercises. Most educators I've worked with seem to think that &lt;em&gt;not&lt;/em&gt; including worked solutions, and instead encouraging students to ask tutors questions, is the superior method for educating mathematics students. I also subscribe to this belief. I was wondering if there are any studies that either confirm or refute this belief.&lt;/p&gt;&#xA;" OwnerUserId="10407" LastActivityDate="2021-08-23T18:04:22.000" Title="What studies exist, comparing the efficacy of exercise sheets with or without worked solutions?" Tags="|undergraduate-education|mathematical-pedagogy|reference-request|education-research|exercises|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="23293" PostTypeId="2" ParentId="23290" CreationDate="2021-08-24T00:23:54.650" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;As a consequence of this, we get a lot of definitions that are not really definitions but are more like &amp;quot;defining things into existence&amp;quot;, for example this year we learned about the square root function and this was the definition: &amp;quot;Let x≥0. The square root of x is the unique number r≥0 such that r2=x&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You're just mistaken about what rigor consists of. It's very common throughout all of mathematics simply to start from a set of axioms and work forward. This is how Euclid worked, and it's also how people created ZFC set theory, which is the most common foundational framework for mathematics today.&lt;/p&gt;&#xA;&lt;p&gt;More valid criticisms of this definition might be: (1) It hides a claim of existence inside a definition. (2) It makes it unclear whether this existence is intended to be a theorem, or an axiom. (3) If you want to talk about third roots and so forth, then you may need a proliferation of such axioms, which is kind of ugly. But these are all more like stylistic issues, and they could be easily remedied without having to do an explicit construction of the reals from set theory.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I don't understand why everything is explained only at an intuitive level, and there are no clear and precise definitions/proofs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The only example you give in the question is a definition that is precise. You haven't given any examples of what you think is an unclear or imprecise proof.&lt;/p&gt;&#xA;" OwnerUserId="17668" LastEditorUserId="17668" LastEditDate="2021-08-24T00:34:13.290" LastActivityDate="2021-08-24T00:34:13.290" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23294" PostTypeId="1" CreationDate="2021-08-24T07:41:15.297" Score="7" ViewCount="370" Body="&lt;p&gt;When I was in grade 11, I was fortunate enough to attend a high school that offered an optional course in vector geometry. The course was taught out of the book &lt;em&gt;Analytic Geometry with an Introduction to Vectors and Matrices&lt;/em&gt; by Murdoch. The high school course covered the first &lt;span class=&quot;math-container&quot;&gt;$2/3$&lt;/span&gt; of the book, so it didn't really deal with matrices.&lt;/p&gt;&#xA;&lt;p&gt;After that, my next contact with linear algebra was in a highly abstract form as an undergraduate in Europe. (To give an example, the proofs of the theorems on kernels and images of linear mappings appealed to the analogous facts about groups. Matrices were introduced &lt;em&gt;after&lt;/em&gt; linear mappings, quotient and dual spaces, etc.) This was made possible in part by the fact that my European classmates had all studied vector geometry the way I had as part of their normal school curriculum, and perhaps even more in depth.&lt;/p&gt;&#xA;&lt;p&gt;In any case, I never really experienced the typical beginning linear algebra course most Americans do, so when I eventually had to teach one a number of years back, it was something foreign to me. I taught a college linear algebra course out of &lt;em&gt;Elementary Linear Algebra&lt;/em&gt; by Anton, the compulsory textbook. I was shocked that my students' first contact with linear algebra could possibly be so ungeometric. The chapters on matrices and determinants came before the chapter on vectors. Had I not made significant changes to the presentation of the material, much of the work on linear equations, matrices and determinants would have been unsupported by what I thought were indispensable geometric interpretations.&lt;/p&gt;&#xA;&lt;p&gt;My questions are as follows.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Do college teachers agree with my feeling (a hunch, really) that the typical approach to linear algebra in the U.S. that deemphasizes vectors until a late stage is likely to compromise non-specialist students' ability to use linear algebra in later studies and in real-world applications?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If so, what is the impediment in the U.S. to shifting vector geometry to an earlier stage of the high school or college curriculum? Could there ever be an AP Vectors?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Could it be that the study of mechanics acts as a kind of back door to facility with vectors for science students, but students going on to economics or social science are handicapped because they never get an equivalent opportunity?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="17669" LastActivityDate="2021-08-24T14:00:04.517" Title="Vector geometry as a prelude to linear algebra" Tags="|linear-algebra|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="23295" PostTypeId="2" ParentId="23294" CreationDate="2021-08-24T14:00:04.517" Score="5" Body="&lt;p&gt;&amp;quot;You fight with the army you brought.&amp;quot;  Nobody is going to rearrange the standard HS prep courses so that you can run a non-standard (in the US) linear algebra course.  You'd be better off teaching a standard course and concentrating on the actual student interactions--do a good job within your constraints--rather than dreaming about alternate scenarios.  And for God's sake, please don't derail things with trying to remediate [what you see as] a gap at the beginning of the course, thus abandoning the scaffolding of the text!&lt;/p&gt;&#xA;&lt;p&gt;Is it possible that some economists end up weak on vectors?  Sure.  Especially the lower ability, back of the class, not heading to grad school, tranche.  But so what?  They need vectors a hell of a lot less for their standard undergrad courses than the naval architects need vectors for their standard undergrad courses.  And the Narchs will get what they need in HS/college physics class as well as statics, dynamics, strength of materials, etc.  And heck, I run into econ professors (especially the chartist, time series obsessed, macro types) that make sunk cost or basic supply and demand logic errors.  So, there's like just the econ itself that they need to learn!&lt;/p&gt;&#xA;&lt;p&gt;Life is a linear programming optimization problem.  With real constraints.  Time, money, IQ, etc.  Yes, there will be some gaps in people's knowledge.  Try to go for the easy wins and the quickest, biggest improvements.  Not obsessing on &amp;quot;oh there's a gap&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I would also be a little bit wary of having to have geometric intuition first.  Yes, it is nice to have intuitions, when they can be QUICKLY and EASILY given (&amp;quot;derivative is like a tangent line&amp;quot;).  But if it takes too much time to give--and asking for an AP course, or changing the first few weeks of course order, IS too much time--then you are actually increasing the cognitive load, not decreasing it.  You might be better off teaching some symbol pushing FIRST and then developing the insights LATER.&lt;/p&gt;&#xA;&lt;p&gt;One of the big reasons why Feynman Lectures suck versus Halliday and Resnick is that the they make more sense as a second exposure than the initial one.  But the mechanical cannonball questions are a more powerful, time-efficient and cognitive load efficient way to &amp;quot;build your physics muscles&amp;quot; first.  (Even Feynman, who was very intuition driven and liked thinking things out on first principles, realized this at times.  For example, when he needed to understand E&amp;amp;M better for some work he was doing, he got a standard book and mechanically worked every exercise.)&lt;/p&gt;&#xA;&lt;p&gt;I would also be wary of extrapolating what worked for you (clearly by profession an outlier in terms of aptitude) or even Europe (with traditionally a more elite/smaller college prep &amp;quot;track&amp;quot;), with what is best for the population in question here (US students).  Perhaps you're even right.  But consider that there's a reason why different materials are used with different tracks.  At least be uncertain.&lt;/p&gt;&#xA;" OwnerUserId="17671" LastActivityDate="2021-08-24T14:00:04.517" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="23296" PostTypeId="1" CreationDate="2021-08-24T18:12:21.397" Score="3" ViewCount="261" Body="&lt;p&gt;A math textbook gives the following problem:&#xA;What is the solution of 3(2x-1) – 2(3x+4) = 11x.&#xA;The solution is an integer.&lt;/p&gt;&#xA;&lt;p&gt;Using Excel to choose some numbers at random and compute the remaining numbers, I want to create problems like that one, with coefficients and solutions that are integers. Rightly or wrongly, I’ve conceptualized my task like this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a(bx + c) + d(ex + f) = gx&lt;/li&gt;&#xA;&lt;li&gt;abx + ac + dex + df = gx&lt;/li&gt;&#xA;&lt;li&gt;ac + df = - abx - dex + gx&lt;/li&gt;&#xA;&lt;li&gt;ac + df = (-ab - de + g)x&lt;/li&gt;&#xA;&lt;li&gt;(ac + df) / (-ab - de + g) = x&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So if we arbitrarily choose values of a, b, d, e, g and x, we can solve for the numerator. For example, if a=2, b=3, d=5, e=7, and g=11, then the denominator is -30. If we choose x=3, then the numerator is 3*(-30) = -90.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;(ac + df) = -90&lt;/li&gt;&#xA;&lt;li&gt;2c + 5f = -90&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Somehow I’m stuck on that last step. How do I pick values of c and f so that this works? Of course I don’t want just an example of values of c and f that solve here, but a general approach that will give values that are integers.&lt;/p&gt;&#xA;&lt;p&gt;Or can you suggest a better approach to the original problem?&lt;/p&gt;&#xA;" OwnerUserId="8258" LastActivityDate="2021-08-27T01:20:43.543" Title="How to create an equation of a given form with integer coefficients and variables" Tags="|algebra|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="23297" PostTypeId="2" ParentId="23296" CreationDate="2021-08-24T22:04:42.153" Score="0" Body="&lt;p&gt;Your approach to automating the product of such problems seems workable, but it does require you take care to remember all the details of your set-up. Let me suggest a far less clever, but very portable method.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Stand in front of board.&lt;/li&gt;&#xA;&lt;li&gt;Write the solution you want, let's say &lt;span class=&quot;math-container&quot;&gt;$x= 13$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Write an arithmetic sentence which is true off the top of your head like:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 3(13+1) = 42 = 26+16 = 2(13) + 16 $$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;You don't let the class see what I wrote above, you then write:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 3(x+1) = 2x+16 $$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Erase the evidence and turn around, ask them to solve it.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If you didn't make a mistake the answer is &lt;span class=&quot;math-container&quot;&gt;$x=13$&lt;/span&gt;. If you did make a mistake, bluff, it's still a legit algebra problem to solve.&lt;/p&gt;&#xA;&lt;p&gt;Added: or if you must have an equation of the particular form the OP desired,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 3(-2\cdot 13+10)+4(3\cdot 13-1) = 8 \cdot 13 $$&lt;/span&gt;&#xA;hence the equation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ 3(-2x+10)+4(3x-1) = 8x $$&lt;/span&gt;&#xA;The idea can be used to create equations of most any form we desire. Simply begin with the answer and work backwards.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2021-08-25T04:17:05.510" LastActivityDate="2021-08-25T04:17:05.510" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23298" PostTypeId="2" ParentId="23296" CreationDate="2021-08-24T23:17:11.330" Score="1" Body="&lt;p&gt;Your &lt;span class=&quot;math-container&quot;&gt;$2 c + 5 f = -90$&lt;/span&gt; can be solved using &lt;a href=&quot;https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bézout's identity&lt;/a&gt;: As &lt;span class=&quot;math-container&quot;&gt;$\gcd(2, 5) = 1$&lt;/span&gt;, you know that there are &lt;span class=&quot;math-container&quot;&gt;$2 u + 5 v = 1$&lt;/span&gt; (for example, &lt;span class=&quot;math-container&quot;&gt;$u = 3, v = -1$&lt;/span&gt;), and manufacturing your &lt;span class=&quot;math-container&quot;&gt;$c, f$&lt;/span&gt; if trivial, &lt;span class=&quot;math-container&quot;&gt;$c = u \cdot (-90) = -180, f = v \cdot (-90) = 90$&lt;/span&gt;. Replacing in the first equation this gives you a similar equation for &lt;span class=&quot;math-container&quot;&gt;$a, d$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2021-08-25T00:13:39.740" LastActivityDate="2021-08-25T00:13:39.740" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23300" PostTypeId="1" AcceptedAnswerId="23301" CreationDate="2021-08-27T23:22:16.080" Score="2" ViewCount="96" Body="&lt;p&gt;The title basically says it all. I need to prepare material for a whole classroom of elementary statistics students, so if anyone wants to help me out in the name of math education, that'd be great!&lt;/p&gt;&#xA;" OwnerUserId="16437" LastEditorUserId="12255" LastEditDate="2021-08-29T16:20:57.017" LastActivityDate="2021-08-29T16:20:57.017" Title="Looking for good examples/explanations of maximum likelihood estimators for discrete random variables" Tags="|undergraduate-education|examples|statistics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="23301" PostTypeId="2" ParentId="23300" CreationDate="2021-08-28T00:40:56.863" Score="3" Body="&lt;p&gt;This might be helpful:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pishro-Nik, Hossein. &lt;em&gt;Introduction to probability, statistics, and random processes.&lt;/em&gt; (2016). &lt;a href=&quot;https://www.probabilitycourse.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Open access textbook&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/mybvr.png&quot; width=&quot;400&quot; /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.probabilitycourse.com/chapter8/8_2_3_max_likelihood_estimation.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Section 8.2.3&lt;/a&gt;: Maximum Likelihood Estimation.&lt;/p&gt;&#xA;&lt;p&gt;His exposition includes five examples, several discrete, with solutions.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;I have a bag that contains 3 balls. Each ball is either red or blue.&amp;quot; Etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-08-28T00:40:56.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23302" PostTypeId="1" CreationDate="2021-08-28T12:04:45.843" Score="5" ViewCount="1589" Body="&lt;p&gt;What do you call a fraction that has one fraction in the numerator and also one in the denominator? I mean (a/b)/(c/d). The word by word translation from my native language would be: composite fraction. But there is no entry on Google search page for that.&lt;/p&gt;&#xA;" OwnerUserId="17682" LastEditorUserId="12255" LastEditDate="2021-08-29T10:36:23.030" LastActivityDate="2021-08-31T16:47:52.753" Title="Composite fraction?" Tags="|terminology|fractions|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="23303" PostTypeId="2" ParentId="23302" CreationDate="2021-08-28T12:48:47.887" Score="1" Body="&lt;p&gt;I googled fraction over fraction and was led to the term &amp;quot;complex fraction&amp;quot;.  Note that this includes more different cases than the one you identified, but does include fraction over fraction.  Lots of hits on that term and it is commonly used, for example at Khan Academy.&lt;/p&gt;&#xA;&lt;p&gt;See:  &lt;a href=&quot;https://www.merriam-webster.com/dictionary/complex%20fraction&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.merriam-webster.com/dictionary/complex%20fraction&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17683" LastActivityDate="2021-08-28T12:48:47.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23304" PostTypeId="1" CreationDate="2021-08-28T12:51:33.023" Score="6" ViewCount="330" Body="&lt;p&gt;In Norway a widely used concept is that of &amp;quot;tiervenner&amp;quot;, &amp;quot;ten friends&amp;quot; (my translation to Finnish is &amp;quot;kymppikaverit&amp;quot;). This simply means numbers (implicitly positive integers) whose sum is ten. There are some resources that name this in Danish, too.&lt;/p&gt;&#xA;&lt;p&gt;Is a similar term used in other languages? Or, in the absence of such a term, is the concept in use when introducing numbers and simple arithmetic to children?&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-08-28T23:15:59.347" Title="Is there an equivalent for &quot;tiervenner&quot; in English or other languages?" Tags="|primary-education|terminology|language-use|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23305" PostTypeId="2" ParentId="23302" CreationDate="2021-08-28T14:27:51.980" Score="6" Body="&lt;p&gt;The term &lt;strong&gt;complex rational expression&lt;/strong&gt; (or &lt;strong&gt;complex fraction&lt;/strong&gt;) is commonly used, in U.S. algebra/college algebra texts, to refer to rational expressions where the numerator and/or denominator contain sums or differences of other rational expressions.&lt;/p&gt;&#xA;&lt;p&gt;Note that technically this would not apply to an expression &lt;span class=&quot;math-container&quot;&gt;$\frac{a/b}{c/d}$&lt;/span&gt;, which has no sums or differences in it; but perhaps by extension you could use the term, and others would know what you're talking about. Here's the presentation from Sullivan, &lt;em&gt;Algebra &amp;amp; Trigonometry&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ue0NZ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ue0NZ.png&quot; alt=&quot;Sullivan definition of complex rational expressions&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2021-08-28T14:27:51.980" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="23306" PostTypeId="2" ParentId="23302" CreationDate="2021-08-28T15:08:14.440" Score="11" Body="&lt;p&gt;I like your term. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Fraction&quot; rel=&quot;noreferrer&quot;&gt;wikipedia article on fractions&lt;/a&gt; also mentions they are called &lt;em&gt;complex fractions&lt;/em&gt; or &lt;em&gt;compound fractions&lt;/em&gt;. Personally, I dislike the term complex fraction as it is obviously going to be interpreted as things like &lt;span class=&quot;math-container&quot;&gt;$\frac{3+i}{2-i}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I think I call them &amp;quot;fractions of fractions&amp;quot; which is really in tune with your term &lt;em&gt;composite&lt;/em&gt; or the synonymous term &lt;em&gt;compound&lt;/em&gt;. Furthermore, I tend to use the following language for&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{ \ \frac{a}{b} \ }{\frac{c}{d}} $$&lt;/span&gt;&#xA;I say&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is the numerator of the numerator&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is the denominator of the numerator&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the numerator of the denominator&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is the denominator of the denominator&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For some variety I also speak of the numerator as &amp;quot;upstairs&amp;quot; and the denominator as &amp;quot;downstairs&amp;quot; when I'm talking through the math I'm writing.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-08-28T15:08:14.440" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23307" PostTypeId="2" ParentId="23304" CreationDate="2021-08-28T16:04:38.963" Score="5" Body="&lt;p&gt;I have seen the phrase &amp;quot;make a ten&amp;quot; in wide circulation.  Example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.mathcoachscorner.com/2020/11/make-a-ten-strategy-for-addition/&quot; rel=&quot;noreferrer&quot;&gt;https://www.mathcoachscorner.com/2020/11/make-a-ten-strategy-for-addition/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-08-28T16:04:38.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23308" PostTypeId="2" ParentId="23304" CreationDate="2021-08-28T23:15:59.347" Score="1" Body="&lt;p&gt;In German, the word is &amp;quot;Zehnerzerlegung&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="16304" LastActivityDate="2021-08-28T23:15:59.347" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23309" PostTypeId="1" CreationDate="2021-08-29T08:33:49.257" Score="1" ViewCount="126" Body="&lt;p&gt;Can anything in math ultimately be analysed into symbols, equations, formulas, with as exeption perhaps the Euclidean geometry we know with triangles,straight lines etc? Can also proofs, definitions and theorems be analysed in this way? for example, when we say function, continuous, matrix, tensor, metric tensor, continous, symmetric, integrable, bounded all these words can be translated into symbols and formulas.&lt;/p&gt;&#xA;&lt;p&gt;Can this be done with anything in math except Euclidean geometry and use anything in terms of symbols and formulas when reading and proving and understanding in math? Do mathematicians do this when reading , proving and understanding math?&lt;/p&gt;&#xA;&lt;p&gt;Should i try to translate any words in math into symbols and formulas? Does this help? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastActivityDate="2021-08-29T16:08:38.197" Title="Analysis of math and its elements" Tags="|self-learning|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="23310" PostTypeId="2" ParentId="23290" CreationDate="2021-08-29T11:20:38.763" Score="4" Body="&lt;p&gt;&lt;strong&gt;Do not confuse rigorous with foundations-first.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Highschool math is neither, but it is perfectly possible to do rigorous math without starting with the foundations. A key aspect here is being utterly clear on what is taken for granted as a prerequisite. Developing mathematics foundations-first is challenging enough when addressing an audience of mathematicians who already know where the journey is leading. Keeping a novice motivated to follow along will be near-impossible, as the investment required of them before stuff starts to make sense is just too much.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Many teachers would not be qualified to rigorously do math.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;As future teachers will not only need to take subject-specific courses, but also courses about how to teach the subject to children. In many systems, a teacher isn't single-subject either. Thus, a math teacher will often have taken (far) fewer math courses at university then someone with a BSc in Mathematics. They'll often have seen some level of rigour in math during their lectures, but being able to rigorously do math themselves is a different story.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Foundations-first school teaching was attempted once. It didn't go well.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/New_Math&quot; rel=&quot;nofollow noreferrer&quot;&gt;New Math&lt;/a&gt; movement lead to an attempt for math teaching based on set theory around the 1960s, in a couple of countries. The experiment was ultimately deeemed a failure. Of course, one can debate whether this is because the attempt was handled in an incompetent way (see the previous item) or whether it was a bad idea in the first place. But this failure will be an obstacle for any similar attempt to reform school teaching.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Probably few pupils could cope.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Mathematics is a special mode of thought, and a lot of people really struggle with the kind of abstract reasoning required. Already basic stuff such as processing a definition doesn't fit in with how most people think - they'd expect to learn a concept by generalizing from examples instead. When proving a statement mathematicians essentially think about ways how the statement could be wrong -- non-mathematicians often look for confirmation instead.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Those that would benefit might be better off elsewhere.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Those rare pupils who really want a rigorous math education are probably better off looking for it elsewhere than at highschool. I, for example, stopped attending math at school in the 10th grade and instead took math courses at university. This was a bit tricky to set up, but that would be the route I'd recommend to any pupil with opinions on whether Dedekind cuts or the Cauchy-completion are a better way introduce the reals!&lt;/p&gt;&#xA;&lt;p&gt;Highschool math teaching definitely needs to be overhauled though, but not with the OP or me-20-years-ago as examples in mind.&lt;/p&gt;&#xA;" OwnerUserId="13471" LastActivityDate="2021-08-29T11:20:38.763" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="23311" PostTypeId="2" ParentId="23309" CreationDate="2021-08-29T16:08:38.197" Score="5" Body="&lt;p&gt;This perspective is known as Formalism &lt;a href=&quot;https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)&lt;/a&gt;. It is sometimes the case that working out the details of a symbolic argument is tedious and unenlightening, in which case it is easier to simply convince someone that you could in theory write out the details in a purely formal way. It is my perspective that most mathematical arguments are this: attempts to convince the audience that a completely formal symbolic proof exists.&lt;/p&gt;&#xA;&lt;p&gt;If you're going to make an exception for Euclidean geometry, you might also need to make an exception for graph theory or category theory, however I don't think you need to: geometric figures can be encoded in a linear symbolic language (just think about how you would encode these diagrams in a computer programming language).&lt;/p&gt;&#xA;&lt;p&gt;Translating math into symbols is excellent practice, both as practice working with symbolic mathematics, and with the topic at hand (they say that a good way to learn is to explain something in different words!).&lt;/p&gt;&#xA;&lt;p&gt;Just be aware that there's a reason mathematicians don't communicate exclusively in symbols: the key ideas and intuitions behind an argument are usually best communicated in words.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-08-29T16:08:38.197" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23312" PostTypeId="1" CreationDate="2021-08-29T21:59:17.497" Score="1" ViewCount="285" Body="&lt;p&gt;How does &lt;a href=&quot;https://www.macmillanlearning.com/college/us/product/The-Basic-Practice-of-Statistics/p/1319244378&quot; rel=&quot;nofollow noreferrer&quot;&gt;David S. Moore; William I. Notz; Michael Fligner's 2021 9 edn. &lt;em&gt;The Basic Practice of Statistics&lt;/em&gt;&lt;/a&gt; differ from &lt;a href=&quot;https://www.macmillanlearning.com/college/us/product/Introduction-to-the-Practice-of-Statistics/p/1319244440&quot; rel=&quot;nofollow noreferrer&quot;&gt;David S. Moore; George P. McCabe; Bruce Craig's 2021 10 edn &lt;em&gt;Introduction to the Practice of Statistics&lt;/em&gt;&lt;/a&gt;? Publisher's the same. Please don't discuss or recommend other books in this thread.  Out of these two, I prefer a book that&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;uses color, esp. for diagrams and pictures.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;offers a free, accessible solutions manual to all exercises in the book.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;David S. Moore is credited as the first author, but he's &lt;a href=&quot;https://www.stat.purdue.edu/%7Edsmoore/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Long retired. Please be aware that all of my texts are now other hands, even though they continue to bear my name. Please direct text questions to the current authors.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2021-09-04T17:52:42.560" Title="The Basic Practice of Statistics vs. Introduction to the Practice of Statistics" Tags="|textbooks|statistics|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23313" PostTypeId="1" CreationDate="2021-08-30T19:58:02.063" Score="2" ViewCount="138" Body="&lt;p&gt;(a) How to enhance learners interest in learning equal sharing with fractions in grade 3?&#xA;(b) Which effective activities will be used to help learners in learning fractions?&#xA;(c) What can be created to teach equal sharing to grade 3?&lt;/p&gt;&#xA;" OwnerUserId="17691" LastActivityDate="2021-08-31T12:44:07.063" Title="Creating best hand on activities on how to teach equal sharing with fractions in grade 3" Tags="|primary-education|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="23314" PostTypeId="2" ParentId="10636" CreationDate="2021-08-30T22:55:58.417" Score="2" Body="&lt;p&gt;If anyone is still interested...&lt;/p&gt;&#xA;&lt;p&gt;Calculus Video Projects have some resources that contain conceptual problems under their 'Resources' tab. They also have conceptual videos for Calculus 1 topics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://calcvids.org/videos/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://calcvids.org/videos/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17692" LastActivityDate="2021-08-30T22:55:58.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23315" PostTypeId="2" ParentId="23313" CreationDate="2021-08-31T12:44:07.063" Score="4" Body="&lt;p&gt;In my experience, the best manipulative for sharing is food, especially if the children are going to get to eat it afterwards.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I have used saltines for fourths. One saltine is a whole and the four sections are fourths.&lt;/li&gt;&#xA;&lt;li&gt;I have used juice boxes.  A package of 8 juice boxes is one whole and each juice box is one eighth.&lt;/li&gt;&#xA;&lt;li&gt;Cereal is very good to use because you can make a group of any number of pieces be the whole and so you have control over the size of the group and the fraction. Fruit loops have the added benefit of being different colors.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If you are going to use food, remember to take into account any allergies, parent preferences (some schools limit sugar at the parent's request), and hygiene.  Are the children going to eat what they handle?  In that case they need to have clean hands when they start and instructions not to touch their faces or shoes while they are handling the food.&lt;/p&gt;&#xA;&lt;p&gt;If you want to avoid food, you can use sheets of small stickers and cut them into groups of whatever size you'd like.&lt;/p&gt;&#xA;&lt;p&gt;As for activities.  Divide children into pairs, threes, or different sized groups. Decide what is the whole. Give the children an amount which is a multiple of the group size.&lt;/p&gt;&#xA;&lt;p&gt;Here is one activity:  Have the whole be 24 pieces of cereal.  Divide children into groups of 2, 3, 4, and 6. Demonstrate with a group of 3.  Figure out what fraction of the group each child is - for example 3 children - each child is 1/3.  Then give them 1 whole (24 pieces of cereal) and tell them to share equally.  Have them write 1/3 of 24 is 8 (since they will each get 8).  Have each group share their cereal and write a similar sentence.  When they have the write sentence - add cereal so that each child has 10.  Emphasize that the bigger denominators have less of a share because there are more children to share with, so 1/2 is greater than 1/3.&lt;/p&gt;&#xA;&lt;p&gt;Here is another activity.  Divide children into pairs.  Have them divide different amounts of cereal.  The children will see halves of different numbers.&lt;/p&gt;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-08-31T12:44:07.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="23316" PostTypeId="2" ParentId="23302" CreationDate="2021-08-31T16:47:52.753" Score="3" Body="&lt;p&gt;In addition to &amp;quot;complex fraction&amp;quot; and &amp;quot;compound fraction&amp;quot;, they are also sometimes called &amp;quot;nested fractions&amp;quot;, as can be shown (for example) &lt;a href=&quot;https://www.khanacademy.org/math/algebra-basics/alg-basics-algebraic-expressions/alg-basics-nested-fractions/v/algebraic-expressions-with-fraction-division&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-08-31T16:47:52.753" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24317" PostTypeId="1" CreationDate="2021-09-02T09:59:27.220" Score="-3" ViewCount="88" Body="&lt;p&gt;What is abstract math about? I think we can not visualise probably what we read. Or can we? I am talking about the theorems, definitions and proofs in areas of math like Riemannian geometry, differentiable manifolds, group theory, functional analysis with vector spaces, spaces with inner product,.&lt;/p&gt;&#xA;&lt;p&gt;So, what remains to do? Think of the logic behind math, remember what you read and when given the chance to prove an exercise or problem, use logic? But what can give me the directions and instructions to prove what i want to prove? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastActivityDate="2021-09-02T11:08:51.007" Title="Abstract math and making proofs" Tags="|proofs|self-learning|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24318" PostTypeId="2" ParentId="24317" CreationDate="2021-09-02T11:08:51.007" Score="1" Body="&lt;p&gt;When learning advanced (or simple, too) mathematics what I see is a process of sense-making. This is a combination of visualization, thinking in terms of processes, building intuition by calculating a lot, knowing key examples, etc.&lt;/p&gt;&#xA;&lt;p&gt;When learning one develops these thinking skills that allow them, personally, to process mathematics. Some of them are learned from others, others built by oneself.&lt;/p&gt;&#xA;&lt;p&gt;When asked to prove something, there are several strategies. There is a famous book of Pólya György (in English: George Pólya) about this, for example. One key step I find myself taking is using the representations I have at hand to figure out if the claim is true or what happens if I try to contradict it, and by this mean understand the problem better.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-09-02T11:08:51.007" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24319" PostTypeId="1" AcceptedAnswerId="24321" CreationDate="2021-09-02T13:44:13.420" Score="4" ViewCount="533" Body="&lt;p&gt;This semester, I am teaching discrete math for computer science students. Today I taught solving linear recurrence equations. The way I did it was not rigors. Instead I used the method of advanced operators described &lt;a href=&quot;https://www.rellek.net/book/s_recurrence_solving.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. I feel this is a bit easier to understand for CS students.&lt;/p&gt;&#xA;&lt;p&gt;However, there were two top students coming to me after class, suggesting that it is much clearer to do it from a vector space perspective, which they have already learned in some linear algebra courses before.&lt;/p&gt;&#xA;&lt;p&gt;There are three options for me.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Change how I teach this part next time.&lt;/li&gt;&#xA;&lt;li&gt;Give a make-up lecture to explain everything from a vector-space perspective.&lt;/li&gt;&#xA;&lt;li&gt;Do a survey and see what the majority of students wants.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;All three options are feasible for me.&lt;/p&gt;&#xA;&lt;p&gt;I am inclined to choose the last one. Maybe the majority have a different feeling about the lecture. What do you think?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2021-09-03T20:21:00.460" Title="Should I gave a make up lecture if some students found what I taught is a bit unclear?" Tags="|undergraduate-education|linear-algebra|recursion|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24321" PostTypeId="2" ParentId="24319" CreationDate="2021-09-02T18:56:56.653" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I am inclined to choose the last one. Maybe the majority have a different feeling about the lecture.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you run a survey and the majority of students &amp;quot;thought the last lecture was too difficult&amp;quot;, that won't necessarily be because they already know linear algebra and would have found a vector space perspective easier to understand. It is possible that some of your students don't have a strong understanding of vector spaces, so they probably wouldn't understand what your offer entails (e.g. (re)learning a topic so they can learn a brand new topic).&lt;/p&gt;&#xA;&lt;p&gt;However, if &lt;em&gt;you&lt;/em&gt; think students knowing both perspectives is important, then by all means add another lecture on the topic. If this is mainly because your two top students already know how to do something from their linear algebra course (which it sounds like they do), and you deliberately chose to teach the method from the linked textbook, then I wouldn't burden the rest of the class with a change of direction on a &amp;quot;what if&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;If you're really curious about what is a &amp;quot;better&amp;quot; way to teach the topic, experiment with the different approach next term.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2021-09-02T18:56:56.653" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24323" PostTypeId="1" CreationDate="2021-09-03T00:04:31.760" Score="6" ViewCount="224" Body="&lt;p&gt;I am teaching at the remedial level in college and I am trying to put together an argument on why we should explain the notations for mixed fractions and the terms proper/improper fractions before we test to see if students can skip the class.   My theory of the case is that it has been some time since they have ever used these terms/concepts and we should give a brief reminder of the concepts before we test them (That way students who are capable should be able to do it).&lt;/p&gt;&#xA;&lt;p&gt;However, I'm having trouble with my google-fu figuring out just when do students stop using these terms.  Answers would ideally be in order of preference:  Florida,  United States,  International.&lt;/p&gt;&#xA;" OwnerUserId="2201" LastActivityDate="2021-09-04T13:02:36.420" Title="At what grade/age levels do students in Florida/USA stop using mixed fractions?" Tags="|fractions|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24324" PostTypeId="2" ParentId="24319" CreationDate="2021-09-03T14:38:09.010" Score="5" Body="&lt;p&gt;My philosophy is &lt;em&gt;schedule über alles&lt;/em&gt;. As an academic, time is your most constrained resource. This goes for balancing the hours in your work vs. research, teaching, and service requirements. And in this case it goes for the limited number of class sessions in your semester.&lt;/p&gt;&#xA;&lt;p&gt;Do you not have a day-by-day schedule for your term? You should. For me, it's the &lt;em&gt;very first thing&lt;/em&gt; I map out when preparing a course. What you really don't want (and I've seen it happen to other instructors) is to slide loosely through the term and then realize you need to somehow squeeze 3 chapters of content into the last week of class. The awkwardness you now feel is nothing compared to that.&lt;/p&gt;&#xA;&lt;p&gt;So keep in mind that you must proceed forward to the following areas of content. If not, be clear-headed: exactly what will get cut from the course if you give a makeup lecture? Make a deliberate choice. And what will get cut from your other work this week as you prepare the new unplanned lecture?&lt;/p&gt;&#xA;&lt;p&gt;In a college course, students should be expected to fill in gaps outside of class on their own. Actually, that's where the majority of the work should be happening. In particular if it's your top students, and they've found an alternative method that satisfies them, at your option, just let them use either method. But for other students I would lean towards sticking with the book presentation which they can consult and reference more easily.&lt;/p&gt;&#xA;&lt;p&gt;I recommend the Steven Krantz book &lt;em&gt;How to Teach Mathematics&lt;/em&gt; by the AMS as the go-to for general math-teaching philosophy. Section 2.7 on &amp;quot;Handouts&amp;quot; seems relevant here:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you give a class hour on Stoke's theorem and feel that you have not&#xA;made matters clear, then you might be inclined to draw up a handout to&#xA;help students along. You also might suspect that this extra effort on&#xA;your part will improve your teaching evaluations and, in particular,&#xA;that students will appreciate all the additional work that you have&#xA;put in. Well, it won't and they don't...&lt;/p&gt;&#xA;&lt;p&gt;What I can do is examine my own conscience and tell you what I see. If&#xA;I give a lesson that is not up to snuff, or if I do a poor job&#xA;explaining what curvature is, or if I goof up a proof in class, then I&#xA;can salve my conscience by writing up a handout. It takes about an&#xA;hour, it is a way of doing penance, and it is a way of working past&#xA;the guilt of having screwed up in class. In my heart of hearts,&#xA;however, I know that what I &lt;em&gt;should&lt;/em&gt; do is strive to give better&#xA;classes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In summary: Of your listed options, (1) is the most on-target. Krantz goes on to suggest that giving a handout online as an &lt;em&gt;optional&lt;/em&gt; added resource to students can be okay. I would recommend in the main sticking to the chosen text presentation, regardless if some advanced students have found another method. Spending more class sessions on the topic should be totally off the table, as that way lies a cascade of further problems as other topics get squeezed out of the semester.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="470" LastEditDate="2021-09-03T20:21:00.460" LastActivityDate="2021-09-03T20:21:00.460" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24325" PostTypeId="1" AcceptedAnswerId="24327" CreationDate="2021-09-03T17:18:21.580" Score="4" ViewCount="85" Body="&lt;p&gt;I was wondering if someone knows what is the name of the app used in the live quiz made in min 4:30 here &lt;a href=&quot;https://www.youtube.com/watch?v=pq9LcwC7CoY&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=pq9LcwC7CoY&lt;/a&gt;, or if someone knows a similar app.&lt;/p&gt;&#xA;&lt;p&gt;The key thing that I'm looking for is that the bar chart doesn't show the answers, while it does show the number of people who have answered.&lt;/p&gt;&#xA;" OwnerUserId="17701" LastActivityDate="2021-09-03T20:46:50.083" Title="Poll software used for live quiz" Tags="|software|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24327" PostTypeId="2" ParentId="24325" CreationDate="2021-09-03T20:46:50.083" Score="5" Body="&lt;p&gt;Earlier in the video (&lt;a href=&quot;https://youtu.be/pq9LcwC7CoY?t=82&quot; rel=&quot;noreferrer&quot;&gt;at this point&lt;/a&gt;), Grant tells us that the poll is done through &lt;a href=&quot;https://itempool.com/&quot; rel=&quot;noreferrer&quot;&gt;Itempool&lt;/a&gt;. You can create an account there and create something called a &amp;quot;live challenge&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2021-09-03T20:46:50.083" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24328" PostTypeId="1" CreationDate="2021-09-04T05:08:41.260" Score="0" ViewCount="575" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How does &lt;a href=&quot;https://www.cengage.com/mathematical-statistics-with-applications-44-7th-edition-7e-wackerly-mendenhall-scheaffer/9780495110811/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dennis Wackerly, Richard L. Scheaffer's &lt;em&gt;Mathematical Statistics with Applications&lt;/em&gt;, 7th Edition  published in 2007&lt;/a&gt; MATERIALLY differ from &lt;a href=&quot;https://www.pearson.com/us/higher-education/program/Larsen-Introduction-to-Mathematical-Statistics-and-Its-Applications-An-6th-Edition/PGM276291.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Richard J. Larsen, Morris Marx's &lt;em&gt;Introduction to Mathematical Statistics and Its Applications&lt;/em&gt;, 6th Edition published in 2018&lt;/a&gt;? &lt;a href=&quot;https://www.legacy.com/us/obituaries/gainesville/name/dennis-wackerly-obituary?pid=192704250&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wackerly died in 2019&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Does the 11 year difference matter?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I riffled them at the library. They both offer a Student's Solution Manual to half the exercises, and a password protected Instructor Solutions Manual to all exercises. But I PREFER students to be able to freely access all solutions to all exercises!&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;I loathe they're both grayscale! It's blindingly — pun intended — obvious that color can ameliorate graphs and charts! See below.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qTIPe.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qTIPe.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/KCp14.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KCp14.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt;In the &lt;a href=&quot;https://faculty.math.illinois.edu/%7Er-ash/RV/RVPreface.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Preface&lt;/a&gt; to &lt;a href=&quot;https://faculty.math.illinois.edu/%7Er-ash/RV.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Real Variables with Basic Metric Space Topology&lt;/em&gt;&lt;/a&gt;, Robert  Ash (1935-2015) touts&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I rely especially on one of the most useful of all learning devices: the inclusion of detailed solutions to exercises. Solutions to problems are commonplace in ele-mentary texts but quite rare (although equally valuable) at the upper division undergraduate and graduate level. This feature makes the&#xA;book suitable for independent study, and further widens the audience.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user155" LastEditDate="2021-09-04T21:43:58.077" LastActivityDate="2021-09-04T21:43:58.077" Title="Wackerly (2007) vs. Larsen (2018) — Introduction to Mathematical Statistics" Tags="|textbooks|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24329" PostTypeId="2" ParentId="24323" CreationDate="2021-09-04T11:54:49.017" Score="1" Body="&lt;p&gt;In parts of the UK and Commonwealth countries, mixed numbers (aka mixed fractions) are commonly used till about 16 years old. So, mark schemes typically specify &lt;span class=&quot;math-container&quot;&gt;$5\frac14$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\frac{21}4.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For example, do a keyword search for “mixed”:&lt;br&gt;&#xA;     &lt;a href=&quot;https://www.cambridgeinternational.org/Images/597037-2023-2024-syllabus.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cambridgeinternational.org/Images/597037-2023-2024-syllabus.pdf&lt;/a&gt;&lt;br&gt;&#xA;    &lt;a href=&quot;https://www.cambridgeinternational.org/Images/557161-2022-2024-syllabus.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cambridgeinternational.org/Images/557161-2022-2024-syllabus.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-09-04T13:02:36.420" LastActivityDate="2021-09-04T13:02:36.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24330" PostTypeId="2" ParentId="24328" CreationDate="2021-09-04T17:42:25.997" Score="1" Body="&lt;p&gt;I used the Wackerly text in the typical two-semester math stats course that I taught for many years. Mendenhall was the original author.&lt;/p&gt;&#xA;&lt;p&gt;I have not used the Larsen text.&lt;/p&gt;&#xA;&lt;p&gt;The main reason I chose the Wackerly text is because of its nice treatment of linear models using matrices. See section 11.10 Fitting the Linear Model by Using Matrices, and section 13.13 Analysis of Variance Using Linear Models. And it is accessible even though we do not have a linear algebra prerequisite. Larsen does not appear to have anything similar.&lt;/p&gt;&#xA;&lt;p&gt;I have no problem with grayscale only, especially in an upper level statistics course. And in my opinion the 11-year difference does not matter.&lt;/p&gt;&#xA;&lt;p&gt;And in my experience students can find every instructor manual on the internet, whether or not it is password protected.&lt;/p&gt;&#xA;" OwnerUserId="306" LastActivityDate="2021-09-04T17:42:25.997" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24331" PostTypeId="2" ParentId="23312" CreationDate="2021-09-04T17:52:42.560" Score="1" Body="&lt;p&gt;Basic Practice is a lower level book than Intro Practice.&lt;/p&gt;&#xA;&lt;p&gt;I used Basic Practice for a no-prerequisite data analysis course, and I used Intro Practice for an algebra-prerequisite elementary stats course.&lt;/p&gt;&#xA;&lt;p&gt;Both are in color and have free instructor manuals that students can easily find on the internet.&lt;/p&gt;&#xA;" OwnerUserId="306" LastActivityDate="2021-09-04T17:52:42.560" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24332" PostTypeId="1" AcceptedAnswerId="24333" CreationDate="2021-09-05T16:57:30.517" Score="12" ViewCount="5147" Body="&lt;p&gt;I act as a tutor sometimes for students who are self-studying undergraduate-level math.  Most of the students have already earned an undergraduate degree in something and some of the students are PhDs from other STEM fields.  How do I deal with students who seem to not really be doing the work but seem to be shifting that onto me like &amp;quot;No one ever showed me that?&amp;quot; or &amp;quot;Why do I need to do this anyway?&amp;quot; even though they have proclaimed to wanting to learn?&lt;/p&gt;&#xA;&lt;p&gt;Here is an actual case study.  A student claims to have a PhD in a STEM field and to be a published faculty member at an R1 institution, and to have taught there for 20 years.  He wants to learn calculus, differential equations, linear algebra, probability, stochastic processes, and &amp;quot;the equivalent of a math degree&amp;quot; in order to aid him in a research idea he has.  He does not want to enroll in school again for some reason.  He claims to want to learn how to read and write proofs and make mathematical arguments.  He is happy spending several years or so self-studying.&lt;/p&gt;&#xA;&lt;p&gt;At the start the student said he knew some calculus - up to basic integration.  After an assessment it turns out that he did not know any trigonometry, for example, how to solve equations involving logarithms, and that he could not differentiate &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sqrt{x^2 + 1}$&lt;/span&gt;, for example.  He considered writing &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sqrt{x^2 + 1}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$f(x) = (x^2 + 1)^{1/2}$&lt;/span&gt; as something &amp;quot;not obvious&amp;quot;.  I think he knew some ideas of math or even advanced math, but could not actually solve any problems or something.&lt;/p&gt;&#xA;&lt;p&gt;So we spent weeks/months reviewing precalculus.  The setup was that he reads sections from a textbook (he picked the book), I assign problems from that textbook, and then we review a selection of those problems.  What would happen is that when reviewing the supposedly already attempted problems he would just say &amp;quot;I got stuck&amp;quot; or &amp;quot;no one showed me how to do that&amp;quot; and not have even attempted half of them.  At first I was patient and emphasized the gain that could be realized by practicing reading (which he said he knew that fact), but after a while it seemed that he was not really reading closely at all or not reading and working on problems regularly and just started being disagreeable about the problems for some reason.&lt;/p&gt;&#xA;&lt;p&gt;We finally moved into calculus using a well-known text.  For some of the basic problems he did fine.  But for some of the true/false style questions or prove-using-the-definition questions he would get stuck and ask &amp;quot;what do I need this for&amp;quot; and be argumentative.  I reviewed how to do the questions anyway, but I feel like I am being bs'ed.  I mean at the outset he said he wanted to learn to prove things, but I do not sense a genuine interest at all.&lt;/p&gt;&#xA;&lt;p&gt;The money is good, but this feels wrong to me, like I have to accept some level of bs from him.  My feeling is that your education is your responsibility; self-studying is not compulsory.  Am I nuts?  How do I handle such students professionally?  Should I care?  It is hard to believe that this was really even a faculty member somewhere.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt; After some consideration and after reading the comments here I have decided to drop the student.  After some digging around online I see that the student has an MD, not a PhD, and the institution shows only a brief affiliation at a lab for several years in the early 2000s.  At that time he was listed along with six others on a publication.  He turned out to have a personal website too where he lists himself as a current assistant professor at the institution, but the institution only shows the old medical lab affiliation.&lt;/p&gt;&#xA;&lt;p&gt;I have also resolved as per the answers to state upfront what I offer, where it might fit with the student's goals, and my expectations, mathematical and behavioral.  I think a detailed mathematical plan at the outset is important as well as periodic check-ins.&lt;/p&gt;&#xA;" OwnerUserId="9816" LastEditorUserId="9816" LastEditDate="2021-09-09T18:14:52.683" LastActivityDate="2021-09-09T18:14:52.683" Title="Dealing with disagreeable students and not compromising" Tags="|self-learning|tutoring|" AnswerCount="6" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="24333" PostTypeId="2" ParentId="24332" CreationDate="2021-09-05T20:22:12.027" Score="20" Body="&lt;p&gt;It sounds like your students are not getting what they wanted from your tutelage; since they are not getting a formal credential from their work with you, their likeliest motivation is that they think additional math knowledge will aid them in their research projects or careers. As you said, the case-study student's objective is to learn the equivalent of a math degree to &amp;quot;aid him in a research idea he has&amp;quot;, but he questions the relevance of the material. It sounds like he is slowly realizing that most of the material in a 4-year math degree will not actually be helpful in efficiently advancing his research project. Might you suggest that he does not need a math degree himself but rather a collaborator with math background who can more easily help him separate the wheat from the chaff on his particular research idea?&lt;/p&gt;&#xA;&lt;p&gt;Here are a few different choices for dealing with your students going forward:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Be up front with the fact that you are offering to tutor a standard course of a given subject and that much of the material &lt;strong&gt;will not&lt;/strong&gt; be directly relevant to their research project / career advancement and only provide unquantifiable 2nd-order benefits such as &amp;quot;general problem-solving skills&amp;quot;. This approach is bad for business but cuts to the chase and deters students who will give you a lot of grief later. The students who are still interested might be more likely to actually do the work.&lt;/li&gt;&#xA;&lt;li&gt;Offer to create a custom curriculum for each student depending on their specific goals. Though highly labor-intensive, this approach would be less like tutoring them and more like collaborating with the students on their own projects (and thus possibly more rewarding than walking everyone through the same trite problem sets?). It could also be difficult to execute well without lots of experience and could frustrate the students if the topic sequencing is not optimal.&lt;/li&gt;&#xA;&lt;li&gt;Keep doing everything the same way but work on dissociating your personal response from the work you're being asked to do. This is the best option if you are trying to maximize your revenue and ROI from your tutoring activities.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2021-09-05T20:22:12.027" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24334" PostTypeId="1" CreationDate="2021-09-06T05:39:15.050" Score="2" ViewCount="197" Body="&lt;p&gt;Context: University studies in Norway, master level, to become a teacher at either primary education (barneskole, classes 1-7) or mostly middle school level (ungdomsskole, classes 8-10 and pupils of age about 13-16, but can also teach classes 5-7 from barneskole). I am interested in the first couple of years of mathematics education courses.&lt;/p&gt;&#xA;&lt;p&gt;The law allows giving the students either letter grades, which amount to numbers 1-5 but use letters for whatever reason, or simple pass/fail as grades.&lt;/p&gt;&#xA;&lt;p&gt;The contents of the courses are a mix of fairly simple mathematics and a fairly big dose of didactics. Exams are sometimes written, sometimes oral.&lt;/p&gt;&#xA;&lt;p&gt;What are the effects of having pass/fail or more explicit grades on student learning, stress levels, etc.? My intuition suggests that grades encourage towards short-term instrumental learning to get a good grade and increase stress and anxiety related to the exams, so I would be inclined to think that pass/fail would give more room for learning to master the subject matter and maybe reduce the negative emotional effects of having exams by a bit. But maybe there are studies in a context similar to this, so I don't have to trust my intuition only.&lt;/p&gt;&#xA;&lt;p&gt;This potentially related question seems to happen in a different context and discusses things like graded homework, which we do not have in any case: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18544/would-it-make-sense-for-math-courses-to-be-pass-fail&quot;&gt;Would it make sense for math courses to be pass/fail?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-09-06T16:35:41.937" Title="Effect of grades versus pass/fail on teacher education students" Tags="|undergraduate-education|exams|teacher-training|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24335" PostTypeId="2" ParentId="24334" CreationDate="2021-09-06T16:35:41.937" Score="4" Body="&lt;p&gt;Some papers you might be interested in:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;Consequences of school grading systems on adolescent health: evidence from a Swedish school reform.&amp;quot; (2019), focused on the switch from informal review to grades in years 6-7.&#xA;&lt;a href=&quot;https://www.researchgate.net/publication/336989196_Consequences_of_school_grading_systems_on_adolescent_health_evidence_from_a_Swedish_school_reform&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.researchgate.net/publication/336989196_Consequences_of_school_grading_systems_on_adolescent_health_evidence_from_a_Swedish_school_reform&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;A Change to Pass/Fail Grading in the First Two Years at One Medical School Results in Improved Psychological Well-Being&amp;quot; (2009)&#xA;&lt;a href=&quot;https://journals.lww.com/academicmedicine/Fulltext/2009/05000/A_Change_to_Pass_Fail_Grading_in_the_First_Two.28.aspx/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://journals.lww.com/academicmedicine/Fulltext/2009/05000/A_Change_to_Pass_Fail_Grading_in_the_First_Two.28.aspx/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-09-06T16:35:41.937" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24336" PostTypeId="1" CreationDate="2021-09-06T17:53:51.763" Score="3" ViewCount="208" Body="&lt;p&gt;I have a hypothesis that I have not yet implemented, and am seeking guidance before I do. I have always taught algebra (and above) in such a way that the students understand the derivations. I check their understanding through discussion. The problem is that I have found that these students are unable to recreate the derivations, nor do anything on their own. My hypothesis is that if, in addition, I require them to copy my on-board derivations exactly in their notebooks, that, although this might distract from understanding somewhat, it will give them a good sense of the mechanics, and so might better support their being able to create, or at least recreate, their own. I resist this for obvious reasons, but I’m at whit’s end with students that clearly understand, but cannot actually do derivations.&lt;/p&gt;&#xA;" OwnerUserId="14430" LastEditorUserId="2083" LastEditDate="2021-09-07T05:44:53.870" LastActivityDate="2021-09-11T15:47:47.780" Title="The value of making students copy your derivations exactly" Tags="|secondary-education|mathematical-pedagogy|middle-school|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24337" PostTypeId="2" ParentId="24332" CreationDate="2021-09-06T21:28:33.373" Score="16" Body="&lt;p&gt;I think the real issue here is that you thought you were essentially doing undergraduate tutoring, and you weren't.  You were doing &lt;em&gt;adult education&lt;/em&gt;, and that is not the same thing.&lt;/p&gt;&#xA;&lt;p&gt;When someone is in their 40's and has not worked - daily - with math since college...they no longer know any math.  They have probably even lost much of their &lt;em&gt;high school&lt;/em&gt; math, as you discovered when your student could no longer do basic trigonometry.&lt;/p&gt;&#xA;&lt;p&gt;If you talk to other people who do adult education for returning students in their 40's, you will discover that they have the same experience(s) you do - odd incongruities between the student's status in life and their knowledge level; often inexplicable gaps in what seems to you to be obvious knowledge or necessary sequences of knowledge; seemingly random patterns of retained and lost knowledge.  And, most importantly, &amp;quot;grumpiness&amp;quot; on the part of the student, who is at least as confused as you are by the amount of skill they have lost in this area.&lt;/p&gt;&#xA;&lt;p&gt;Unless you are prepared to rebuild the student's fundamentals before proceeding and unless you are prepared to tolerate a little curmudgeonliness, you shouldn't take on students of this type.&lt;/p&gt;&#xA;" OwnerUserId="17714" LastActivityDate="2021-09-06T21:28:33.373" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="24338" PostTypeId="1" AcceptedAnswerId="24340" CreationDate="2021-09-06T23:29:00.190" Score="7" ViewCount="2936" Body="&lt;p&gt;I taught the proof inclusion-exclusion principle to CS students yesterday. While the proof is not too long, it does involves quite a bit notations. I could feel that most students lost interests a few minutes into the proof.&lt;/p&gt;&#xA;&lt;p&gt;When we teach a proof that takes longer than 5 minutes, how can we help students to follow? Or maybe we should just ask students to read proofs by themselves?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorUserId="17719" LastEditDate="2021-09-07T21:42:45.377" LastActivityDate="2021-09-08T13:35:33.487" Title="How to keep students' attention while teaching a proof?" Tags="|undergraduate-education|proofs|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24339" PostTypeId="2" ParentId="24332" CreationDate="2021-09-07T00:11:21.500" Score="5" Body="&lt;p&gt;Organic chemists are well known for &amp;quot;only needing to count to 4&amp;quot;.  Sure, they have some math in their studies, but it's really not used for natural products synthetic chemists.&lt;/p&gt;&#xA;&lt;p&gt;I disagree with the commenter who said to confront the fellow, demanding proof of his professorship.  If you really care, look it up yourself.  But it's inappropriate to make the lesson an interrogation of him.&lt;/p&gt;&#xA;&lt;p&gt;I think this person is pretty much a rare case.  That said, if you are doing individualized tutoring, you should realize that it's not just the material gaps of your students that will vary, but also motivation, personality, etc.  For one thing, he's truly an &amp;quot;adult learner&amp;quot;.  There will be some differences in how you handle him versus how you handle what is essentially a delayed adolescent (undergrad, not working full time).  That doesn't mean you don't want to move him along.  But just have a little feel for how people can be different.  And still try to bring him along.  The &amp;quot;not compromising&amp;quot; gave me an impression you expect everyone to be the same, respond to the same sort of plan.&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't think of it as &amp;quot;compromising&amp;quot;, but just try to help the trainee, given his tendencies.  It's not that strange to have a client in physical training, music, etc. who pays for help but does not always practice rigorously.&lt;/p&gt;&#xA;&lt;p&gt;Rather than getting into an argument (the direct approach) see if you can gently tease/encourage the fellow to do more.  Like...&amp;quot;sure, I agree it's boring to do drill problems...and after all you wouldn't want to master things.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Try to come up with some plan that gets the fellow interests and that is not 100% boring drill.  Maybe some chemical applications or the like.  Or have some aspect of the lesson that is more gamified.  Like a 5 minute quiz at beginning.  Donno, exactly.  Try some different things.&lt;/p&gt;&#xA;&lt;p&gt;I doubt proofs are a good use of time.  He has more basic skill deficits.  And it's not well connected to his needs in chemistry (maybe being able to read mathy articles in J. Phys. Chem.)&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I would also encourage you to encourage the student to buy Frank Ayres First Year College Math.  It's at about the right level (precalc, with a VERY gentle &amp;quot;intro to calc&amp;quot;).  Having something shorter and friendlier versus a bunch of logorrheic doorstops may be helpful for him to refer to.  Also, there is a strong emphasis on drill.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0007DPVM2&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/dp/B0007DPVM2&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17716" LastEditorUserId="17716" LastEditDate="2021-09-07T00:33:32.573" LastActivityDate="2021-09-07T00:33:32.573" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24340" PostTypeId="2" ParentId="24338" CreationDate="2021-09-07T05:44:08.020" Score="12" Body="&lt;p&gt;You want the students to work, not be an audience, because being an audience is highly ineffective. (Some of the best students do know how to work even when they are treated as an audience.)&lt;/p&gt;&#xA;&lt;p&gt;Can the proof be broken into smaller pieces, some of which the students could do? Are there exercises that are the same as parts of the proof? Is the entire proof something that they could manage if working on it as a class, perhaps after you have given a high level idea of the proof?&lt;/p&gt;&#xA;&lt;p&gt;Just showing or covering material does not lead to that much learning, so you might just as well not do it and instead spend that time on some activity that makes the students think. (Some exceptions might exist; some clever tricks, for example, might well be illustrated by a short and to the point proof, but hopefully with activity, too.)&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-09-07T05:44:08.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24341" PostTypeId="2" ParentId="24338" CreationDate="2021-09-07T05:50:17.557" Score="6" Body="&lt;p&gt;I was taught to follow the CRA model:  Concrete, Representational, Abstract.   That's the order you should present things.&lt;/p&gt;&#xA;&lt;p&gt;So for instance,  you could start the concept off with actual physical objects that have multiple qualities (Say,  stickers on them, shape, color, etc.),  and show that if you count all the qualities you overcount by everything that's in 2 categories,  so you subtract those but now you have undercounted by anything that was in 3, etc.&lt;/p&gt;&#xA;&lt;p&gt;From that you can take it to the representational on the board,  with a Venn Diagram of 3 sets, pointing out the overcount,undercount/corrections.&lt;/p&gt;&#xA;&lt;p&gt;Only at the end do you go the abstract with the terminology and the general case.&lt;/p&gt;&#xA;" OwnerUserId="2201" LastActivityDate="2021-09-07T05:50:17.557" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24342" PostTypeId="2" ParentId="24332" CreationDate="2021-09-07T06:19:56.233" Score="1" Body="&lt;p&gt;You could consider using a less &amp;quot;pure maths&amp;quot; textbook in your tutoring. Instead you could use a maths-heavy textbook more related to their field of study and making them work through the proofs in detail. For example, if your student has a chemistry background you could use a &lt;a href=&quot;https://www.wiley.com/en-us/Chemical%2C+Biochemical%2C+and+Engineering+Thermodynamics%2C+5th+Edition-p-9781119321286&quot; rel=&quot;nofollow noreferrer&quot;&gt;book on thermodynamics&lt;/a&gt; and work through the proofs of thermodynamics and equations of state.&lt;/p&gt;&#xA;&lt;p&gt;Where they get stuck on those proofs you can then turn back to your pure maths books to provide the necessary theorems or techniques.&lt;/p&gt;&#xA;" OwnerUserId="17717" LastActivityDate="2021-09-07T06:19:56.233" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24343" PostTypeId="2" ParentId="24336" CreationDate="2021-09-07T08:00:14.023" Score="1" Body="&lt;p&gt;In general, trying to learn formal language or algorithms leads to many pupils not understanding what is going on and thus making bizarre mistakes, because they have a rough sense of what the result should look like, but little sense as to why.&lt;/p&gt;&#xA;&lt;p&gt;This might be counteracted by the understanding created through discussion, but that is hard to say without further details.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-09-07T08:00:14.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24344" PostTypeId="2" ParentId="24338" CreationDate="2021-09-07T10:06:54.270" Score="6" Body="&lt;p&gt;I always had fun following maths proofs, both in school and uni, but that was mostly through self-motivation/interest. I wager that most of my co-students more or less slept through the same lessons or were only kept awake because at that time (pre-internet/pre-digital) we were forced to write down whatever the teacher/prof wrote on the chalkboard.&lt;/p&gt;&#xA;&lt;p&gt;So you're not alone and it does not seem to be a trivial endeavour.&lt;/p&gt;&#xA;&lt;p&gt;As you wrote in a comment, you do not need your students to repeat the proofs, or come up with their own; you just want them to be more engaged in their time they sit in your classroom.&lt;/p&gt;&#xA;&lt;p&gt;So I would suggest for you to do exactly that: engage them. Don't just present the proof, but at every single step, ask them what they think comes next. Don't just grab the one pupil that you know &amp;quot;groks&amp;quot; that stuff, but try getting a discussion going, or ask those more at risk of nodding off...&lt;/p&gt;&#xA;&lt;p&gt;You might or might not get through the whole proof using this method, but at least you will engage them and force/encourage them to think about the individual steps. You'd need to tailor your proof so that they have a running chance to come up with good ideas of course; maybe give a few more hints when it comes to solve the crux of the problem.&lt;/p&gt;&#xA;" OwnerUserId="8146" LastEditorUserId="8146" LastEditDate="2021-09-07T11:57:46.753" LastActivityDate="2021-09-07T11:57:46.753" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24348" PostTypeId="2" ParentId="24332" CreationDate="2021-09-07T21:50:26.043" Score="2" Body="&lt;p&gt;I enjoy teaching math, I get paid for dealing with the b.s.. If it turns out that there is more b.s. to deal with that I initially expected, my price goes up.&lt;/p&gt;&#xA;&lt;p&gt;I don't explicitly tell the client that that's the reason why the price is now higher, but I set my price to the b.s. level. So either the client pays me enough to put up with it all, or decides that they don't want to pay my price, and they are free to find some other tutor.&lt;/p&gt;&#xA;&lt;p&gt;(Oh, and if it's not obvious, my price and willingness to take a particular job vary with my fiscal needs and current work load - Just like everywhere.)&lt;/p&gt;&#xA;" OwnerUserId="14507" LastActivityDate="2021-09-07T21:50:26.043" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24349" PostTypeId="2" ParentId="24332" CreationDate="2021-09-07T22:53:34.483" Score="3" Body="&lt;p&gt;It may be that his style of learning differs from yours. I have an MA in math, and when my son was in high school helped tutor him in math and physics. I always like the &amp;quot;aha moment&amp;quot; in proofs, where the purpose of previous obscure statements becomes plain. My son hates that - when I delivered the punch line he would get upset and say &amp;quot;where did that come from? How did he know to do that?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;He ended up going to college as a film major, changing to physics, and is now completing his PhD in quantum physics. He aced every math course he took in college and grad school (including differential equations, which I hated). He sees math as a tool, and my aesthetic approach didn't appeal to him at all. A more pragmatic approach, where he learned math as necessary to support concrete physics problems he was studying, made all the difference.&lt;/p&gt;&#xA;" OwnerUserId="17724" LastActivityDate="2021-09-07T22:53:34.483" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24350" PostTypeId="2" ParentId="1056" CreationDate="2021-09-08T07:06:16.173" Score="2" Body="&lt;p&gt;Expanding my comment into an answer:&lt;/p&gt;&#xA;&lt;p&gt;So you want the umbrella term for a predicate (propositional function), like the triangle or Cauchy–Schwarz inequality, that's always true, but not necessarily so regardless of interpretation. So, ‘validity’ and ‘tautology’ are ruled out.&lt;/p&gt;&#xA;&lt;p&gt;If the object is an &lt;a href=&quot;https://math.stackexchange.com/a/4254850/21813&quot;&gt;equality&lt;/a&gt; instead of inequality, I'd have suggested ‘identity’ (a universally-quantified equation)...&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;@RyanG Yes, that's what I want. Although, as mentioned in a recent comment, I'm skittish around the word &amp;quot;predicate&amp;quot;. I like 'identity' because it's a word that students already know, e.g. from trig. The question is whether it's too much of a reach to extend the word to boolean expressions that aren't equalities.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;...For inequalities, we might similarly distinguish between “&lt;strong&gt;identity inequalities&lt;/strong&gt;” and non-identity inequalities? This is clearly not standard terminology either, but a google search did return &lt;a href=&quot;https://virtualnerd.com/algebra-1/linear-inequalities/multi-step-examples/variables-both-sides/identity-definition&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; definition as the first result:&lt;/p&gt;&#xA;&lt;p&gt;“An &lt;strong&gt;identity inequality&lt;/strong&gt; is an inequality that is true no matter what values we plug in for the variable.”&lt;/p&gt;&#xA;&lt;p&gt;P.S. I agree that &amp;quot;predicate&amp;quot; is a bad choice for the purpose: not only is it distractingly formal-logical, it even has two (related but) distinct definitions within the field of logic.&lt;/p&gt;&#xA;&lt;p&gt;P.P.S. Possibly &lt;a href=&quot;https://math.stackexchange.com/a/4231999/21813&quot;&gt;related&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-09-26T16:34:51.840" LastActivityDate="2021-09-26T16:34:51.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24351" PostTypeId="1" CreationDate="2021-09-08T09:53:19.040" Score="30" ViewCount="4773" Body="&lt;p&gt;I'm teaching a second year &amp;quot;Introduction to Theoretical Computer Science&amp;quot; course, and one of the skills/habits I've tried to instill in the students is to actually read definitions, take them seriously and go back to consult them when appropriate. This has been less successful than I would like. As I am very frequently seeing theory questions over at cs.stackexchange where &amp;quot;look at the definition!&amp;quot; is the natural answer, I don't seem to be the only one struggling with this.&lt;/p&gt;&#xA;&lt;p&gt;Some example for situations I have in mind are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A) I'm introducing right-linear grammars (where rules are of the form &lt;span class=&quot;math-container&quot;&gt;$T \rightarrow aU$&lt;/span&gt;) and context-free grammars (where the constraint is that rules have a single non-terminal on the left). Then I'm asking whether all right-linear grammars are context-free, expecting that students look at the two definitions are arrive at &amp;quot;obviously, yes&amp;quot;. But a lot of them seem to instead try to remember whether I ever told them.&lt;/li&gt;&#xA;&lt;li&gt;B) Students trying to learn all kinds of facts regarding the empty set by heart (e.g. &lt;span class=&quot;math-container&quot;&gt;$\emptyset \subseteq A$&lt;/span&gt; is always true) rather than recognizing that these are just consequences of the general definitions.&lt;/li&gt;&#xA;&lt;li&gt;C) Students feeling stuck questions like &amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\mathrm{NP}$&lt;/span&gt;-complete and &lt;span class=&quot;math-container&quot;&gt;$A \leq_p B$&lt;/span&gt;, is &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; necessarily &lt;span class=&quot;math-container&quot;&gt;$\mathrm{NP}$&lt;/span&gt;-complete?&amp;quot; while being a bit hazy about what &lt;span class=&quot;math-container&quot;&gt;$\mathrm{NP}$&lt;/span&gt;-complete means. I'd want them to look up the definitions of terms they are uncertain about before throwing their hands in the air.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;What I've tried is just straight-on telling them that and how definitions are central, and that if in doubt, they should look at them. I reminded them of &amp;quot;look at the definition&amp;quot; when they asked questions of this nature. I promised them they could avoid having to learn plenty of stuff by heart if they'd only started at what the definitions say.&lt;/p&gt;&#xA;&lt;p&gt;Are there any better techniques to get this point across? Or maybe some insight why this seems to be so difficult?&lt;/p&gt;&#xA;" OwnerUserId="13471" LastEditorUserId="-1" LastEditDate="2021-09-10T09:01:11.177" LastActivityDate="2021-09-12T13:48:22.793" Title="Getting students to actually read definitions" Tags="|undergraduate-education|definitions|" AnswerCount="10" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="24352" PostTypeId="1" AcceptedAnswerId="24385" CreationDate="2021-09-08T12:57:45.620" Score="2" ViewCount="222" Body="&lt;p&gt;Suppose I want to subtract 46 from 52. Instead of the borrowing method, I can use this method:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{r}&#xA;&amp;amp; 5 &amp;amp; 2\\&#xA;-\!\!\!\!\!\!&amp;amp; 4 &amp;amp; 6 \\&#xA;\hline&#xA;&amp;amp; &amp;amp; -4 \\&#xA;+\!\!\!\!\!\!&amp;amp; 1 &amp;amp; 0 \\&#xA;\hline&#xA;&amp;amp; &amp;amp; 6&#xA;\end{array}&lt;/span&gt;&#xA;where I am essentially adding the difference in the ones place (-4) to the difference in the tens place (10). Similarly, suppose that I want to add 79 to 52. I can write&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{r}&#xA;&amp;amp;  &amp;amp; 5 &amp;amp; 2\\&#xA;+\!\!\!\!\!\!&amp;amp; &amp;amp; 7 &amp;amp; 9 \\&#xA;\hline&#xA;&amp;amp;  &amp;amp; 1 &amp;amp; 1 \\&#xA;+\!\!\!\!\!\!&amp;amp; 1 &amp;amp; 2 &amp;amp; 0 \\&#xA;\hline&#xA;&amp;amp; 1 &amp;amp; 3 &amp;amp; 1&#xA;\end{array}&lt;/span&gt;&#xA;where I am adding the numbers in the ones place together to get 11 and the numbers in the tens place together to get 120. I then add these two results together to get the final answer. Is there a name for this method of column addition and subtraction?&lt;/p&gt;&#xA;" OwnerUserId="16438" LastActivityDate="2021-09-12T05:41:27.947" Title="Is there a name for this method of column addition and subtraction?" Tags="|arithmetic-operations|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24353" PostTypeId="2" ParentId="24352" CreationDate="2021-09-08T13:24:27.923" Score="4" Body="&lt;p&gt;I don't know if your idea has a name, but it feels weird when you try to apply it to something like 10-4:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{r}&#xA;&amp;amp; 1 &amp;amp; 0\\&#xA;-\!\!\!\!\!\!&amp;amp; \! &amp;amp; 4 \\&#xA;\hline&#xA;&amp;amp; &amp;amp; -4 \\&#xA;+\!\!\!\!\!\!&amp;amp; 1 &amp;amp; 0 \\&#xA;\hline&#xA;&amp;amp; {\color{red}1} &amp;amp; {\color{red}-}{\color{red}4}&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since the sum of the ones place (-4+0 = -4) and the sum of the tens place (0+1=1), the &amp;quot;6&amp;quot; you put on your example seems to be forced, at least to me. I mean, if you want to teach this to kids, you will have to give a second thought on the rules.&lt;/p&gt;&#xA;" OwnerUserId="3380" LastActivityDate="2021-09-08T13:24:27.923" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24354" PostTypeId="2" ParentId="24338" CreationDate="2021-09-08T13:35:33.487" Score="3" Body="&lt;p&gt;Several tricks:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Surprise&lt;/strong&gt;: present your theorem as surprising (i.e. nonobvious). How would I go about showing this is actually true? Then develop the line of attack to show them that this can actually be done; ideally, but not necessarily with a trick. I got more than one &amp;quot;wow/awesome&amp;quot; using this method. This is actually a good sign, because this means that students made the transition from treating math as accounting to seeing that it's actually an exciting endeavour.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Intuition&lt;/strong&gt;: this is exactly the opposite of 1., but sometimes more suitable. Explain what the theorem says and show how this is actually intuitive. Now show how to cast intuition into formal language.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Processual&lt;/strong&gt;: sometimes neither 1 nor 2 are possible. Show them the technical steps. Repeat and rinse. This needs practice. This is very local and again complementary to 1 and 2, but as students see and take similar steps again and again, they gain familiarity and thus lose fear of approaching proofs themselves, by having the &amp;quot;muscle memory&amp;quot; already entrained. This is essentially implementing von Neumann's dictum: &amp;quot;Young man, in mathematics you don't understand things. You just get used to them.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14579" LastActivityDate="2021-09-08T13:35:33.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24356" PostTypeId="2" ParentId="24351" CreationDate="2021-09-08T16:55:52.850" Score="3" Body="&lt;p&gt;I try to stress that definitions are just useful shortcuts, ways to put several disparate phenomena under the same roof (like context free grammars and the others in the classification), and &lt;em&gt;show&lt;/em&gt; the similarities/usefulness of said definitions. One of the basic &amp;quot;proof techniques&amp;quot; taught in the discrete math course (first one that for our students means real proofs, not mostly &amp;quot;learn when/how to apply formula XYZ&amp;quot;) is &amp;quot;If nothing looks useful, try unraveling the definitions&amp;quot;. It is widely useful, for example when dealing with congruences modulo &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; (often rewriting &lt;span class=&quot;math-container&quot;&gt;$a \equiv b \pmod{m}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$a = b + k m$&lt;/span&gt; allows to see patterns/simplify).&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2021-09-08T16:55:52.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24357" PostTypeId="2" ParentId="24351" CreationDate="2021-09-08T17:13:01.477" Score="2" Body="&lt;p&gt;Maybe do an in-class exercise with 1 simple (fast) problem, where the students don't have access to the definition.  For ease of grading, use a multiple choice answer format.  Collect papers and set aside.&lt;/p&gt;&#xA;&lt;p&gt;Then follow up with similar exercise, but with the appropriate definition included the handout.  Have students pass the papers and grade them in class, both sets.  Then write the performance down on the board, for each set.&lt;/p&gt;&#xA;&lt;p&gt;This might seem childishly simple, but I think you need to make it simpler, to drive different behavior.  If you give them a lecture on &amp;quot;read the definitions&amp;quot;, they will tune out.  Or keep it for infrequent high scale tests (mid-term/final).  Or project style homeworks. Instead of all that, do something simple.&lt;/p&gt;&#xA;" OwnerUserId="17730" LastActivityDate="2021-09-08T17:13:01.477" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24358" PostTypeId="2" ParentId="24351" CreationDate="2021-09-09T02:39:02.707" Score="17" Body="&lt;p&gt;First of all, you should test them on remembering the definitions.&lt;/p&gt;&#xA;&lt;p&gt;Second, there are probably a significant number of your students who do not understand the definitions.  Suppose you gave them an example of a grammar that was not right-linear and the definition of a right-linear grammar, and asked them why, according to the definition, the example grammar was not right-linear.  Many of them would not be able to accomplish this task.  You may have to teach them to do this task for various definitions and hope that they figure out how to do this for definitions where they have not been shown how to do it for that definition.  In my experience, there are a few students who do not learn this skill in general.  (In other words, suppose you give examples for various definitions throughout the course and, at the end, give the definition for a context-sensitive grammar and an example of a non-context-sensitive grammar, and ask why the grammar is not context-sensitive.  There may be some students who simply still cannot point to the production that violates the rule.)&lt;/p&gt;&#xA;&lt;p&gt;Third, many of your students probably have some trouble with basic logic.  Probably not at the level where they could not compare the definitions of right regular grammar and context free grammar, but that might be at the edge of their logical capacities.  In some students, this could be caused by a working memory capacity issue, where they really cannot remember both definitions simultaneously well enough to think about both of them at the same time.  If you cannot do logic, it's much less useful to remember the definitions, because you cannot use them to conclude anything (since you also need logic to do that).&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2021-09-09T02:39:02.707" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24359" PostTypeId="1" AcceptedAnswerId="24362" CreationDate="2021-09-09T07:37:27.860" Score="12" ViewCount="955" Body="&lt;p&gt;I am teaching in a hybrid mode (zoom+classroom) this semester.&lt;/p&gt;&#xA;&lt;p&gt;I would like to encourage collaboration between students.&#xA;So I divided them into pairs and asked them to establish a communication channel between pairs, using whatever apps they like.&lt;/p&gt;&#xA;&lt;p&gt;Then I give them exercises in class, and ask them to discuss their answer with their peers. After a few minutes, I choose a random group to answer the question.&lt;/p&gt;&#xA;&lt;p&gt;However, it seems that the students simply try to solve the problem themselves, and never really talk with their remote pairs.&lt;/p&gt;&#xA;&lt;p&gt;I wonder if it there is any way to change this?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2021-09-09T16:48:09.750" Title="How to encourage students to discuss with themselves in a remote teaching environment?" Tags="|remote-teaching|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24360" PostTypeId="2" ParentId="24351" CreationDate="2021-09-09T08:25:34.930" Score="5" Body="&lt;p&gt;This is something I know very well from the other side, as an amateur student of maths for the last 40 years or so. In my experience, reading a definition and actually understanding it are two very different things. Once you understand the definition at a deeper level, you very often forget how hard it was to get to that point. It simply takes a certain amount of tedious grind, using what you know, to actually understand.&lt;/p&gt;&#xA;&lt;p&gt;I blame to a very large extent the way maths is taught in school, where you too often focus on learning a handful of useful results, without really going into the fundamental understanding of the basics. This is what I call the 'accountant's mindset': you learn to use a number of ready-made tools that enable you to carry out routine tasks, that require no deeper understanding of what makes the tools work. I think the only cure is to keep going back to the definitions and emphasising how you use the definition and why they must be what they are.&lt;/p&gt;&#xA;&lt;p&gt;I think it was Steve Awodey who once said that too much teaching focuses on 'what' when it was equally important, if not more, to talk about 'why': a definition tells what a concept is, but it doesn't excplain why it must be that way; very often, if you know the practical motivations behind the definition, you can easily construct it yourself - and use it where appropriate.&lt;/p&gt;&#xA;&lt;p&gt;In the ideal world, one should always be able to draw a direct line from first principles to whatever you are working at; or rather, the opposite way: you should always be able to answer 'Why' with a reference back to definitions, and then on to 'what is the reason for that definition', etc all the way down to something like first order logic. If the reasoning stops at 'Theorem x.y.z', then you are an accountant in my book :-)&lt;/p&gt;&#xA;" OwnerUserId="13424" LastActivityDate="2021-09-09T08:25:34.930" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24361" PostTypeId="2" ParentId="24351" CreationDate="2021-09-09T08:39:04.263" Score="1" Body="&lt;p&gt;I’ve been able to inculcate students with the awareness that definitions are not &lt;em&gt;just&lt;/em&gt; legal agreements—which they basically are—or boring fine print, by &lt;em&gt;persistently demonstrating&lt;/em&gt;&lt;sup&gt;҂&lt;/sup&gt; using definitions to successfully deal with problems: for example, in modelling scenarios and translating texts, manipulating expressions, and crafting explanations.&lt;/p&gt;&#xA;&lt;p&gt;Frequently, applying definitions is not just the most direct way to accomplish the task at hand, but the only way; in such cases, I specifically emphasise this point.&lt;/p&gt;&#xA;&lt;p&gt;However, the process really does require time and trust, and is part of the larger process of &lt;em&gt;encouraging sense-making&lt;/em&gt;&lt;sup&gt;҂&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Eventually, students even become comfortable with mathematical vocabulary having multiple meanings depending on context and even the exam board—which reinforces the importance of referring to the definition!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addednum&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;҂&lt;/sup&gt; I couldn't have expanded my points better than how Reese has said it in &lt;a href=&quot;https://matheducators.stackexchange.com/a/24365/372&quot;&gt;this answer&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;acculturating students to reflexively turn to definitions rather&#xA;than to regard them parenthetically, and&lt;/li&gt;&#xA;&lt;li&gt;getting hands dirty in demystifying definitions and unravelling meaning, perhaps even thinking about their whys and hows.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-09-12T05:41:33.590" LastActivityDate="2021-09-12T05:41:33.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24362" PostTypeId="2" ParentId="24359" CreationDate="2021-09-09T09:10:24.840" Score="11" Body="&lt;p&gt;I have been using small groups in Zoom, typically 3-4 students per group. Typically random groups, but also more open and longer term tasks. Sometimes with a few minutes of individual work first, sometimes without. This has worked out fine, with people even using cameras a lot of the time when they are in those small groups.&lt;/p&gt;&#xA;&lt;p&gt;It is quite a lot of burden to ask them to figure out their own means of establishing a workspace. You should teach them how to do this or you should make it as automatic as possible (which Zoom makes possible). The more they have to think and choose and mess around, the easier it is just to not bother and instead work by oneself.&lt;/p&gt;&#xA;&lt;p&gt;Also, if the exercises are very short (a few minutes), the time it takes to set up an external communication is long when compared to the length of the task. Lots of friction there. And with only two people, they both have to be awake and active enough before communication can happen.&lt;/p&gt;&#xA;&lt;p&gt;My concrete suggestions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Use Zoom to set up the groups.&lt;/li&gt;&#xA;&lt;li&gt;Have groups of three to five people, so that even if some of them are not active, the others can still collaborate.&lt;/li&gt;&#xA;&lt;li&gt;Give clear but open-ended tasks or tasks that allow several approaches. This helps in creating mathematical dialogue. If the tasks are simple calculations there is a big risk that someone solves and shows, or that several people work in parallel without much discussion.&lt;/li&gt;&#xA;&lt;li&gt;Have tasks that are long enough for the students to figure out that they should be doing group work, figure out what they should be doing, get started doing it, and have some time to do it, too.&lt;/li&gt;&#xA;&lt;li&gt;If you use Zoom functionality, you can cycle through the groups, seeing what people are doing there, whether they understood the activity or not, how far they have come, and there is also the opportunity to have a discussion with them. I usually start with the smallest group, the group I guess might need the most help, or a random group, and visit them briefly starting from there. After a few groups are finished or have accomplished the main points (always good to have some extra activity for the fastest groups) I typically end the small group sessions. Then I select a random person or a random group and ask them to report what they have accomplished, asking if others have comments or commenting myself. This keeps the students accountable.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-09-09T09:10:24.840" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24363" PostTypeId="2" ParentId="24351" CreationDate="2021-09-09T09:14:34.557" Score="8" Body="&lt;p&gt;You asked:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Students trying to learn all kinds of facts [...] by heart rather than recognizing that these are just consequences of the general definitions. [...] Are there any better techniques to get this point across? Or maybe some insight why this seems to be so difficult?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The answer is simply that the students are unable to do &lt;strong&gt;basic logical reasoning&lt;/strong&gt;. They don't even understand that the statement &amp;quot;Every object of type &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is also an object of type &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;&amp;quot; is of the form &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$∀x{∈}S\ ( \ x∈T \ )$&lt;/span&gt;&amp;quot;, nor how to even start proving it, even though it is &lt;strong&gt;obvious&lt;/strong&gt; to anyone with a rudimentary grasp of basic (first-order) logic that one can prove it using the following &lt;a href=&quot;http://math.stackexchange.com/a/1684204/21820&quot;&gt;Fitch-style&lt;/a&gt; outline:&lt;/p&gt;&#xA;&lt;p&gt;  Given any &lt;span class=&quot;math-container&quot;&gt;$x∈S$&lt;/span&gt;:&lt;br /&gt;&#xA;    [Possibly expand definition of &lt;span class=&quot;math-container&quot;&gt;$x∈S$&lt;/span&gt; otherwise you know &lt;strong&gt;nothing&lt;/strong&gt; about &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.]&lt;br /&gt;&#xA;    ...&lt;br /&gt;&#xA;    [Possibly deduce definition of &lt;span class=&quot;math-container&quot;&gt;$x∈T$&lt;/span&gt; otherwise you &lt;strong&gt;usually cannot&lt;/strong&gt; conclude...]&lt;br /&gt;&#xA;    &lt;span class=&quot;math-container&quot;&gt;$x∈T$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It should be obvious that you cannot possibly do anything if you didn't use any of the definitions of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;! But do students know that? And do teachers know that?&lt;/p&gt;&#xA;&lt;p&gt;Ultimately, the only solution is for teachers of any mathematical subject to require every proof to be written in a logically structured format (whether or not Fitch-style). If students understand that we cannot consider something to be a proof unless it follows permitted logical deductive rules, then all your problems will vanish.&lt;/p&gt;&#xA;&lt;p&gt;Also, students should be taught &lt;a href=&quot;https://math.stackexchange.com/a/1399880/21820&quot;&gt;what definitions really mean&lt;/a&gt;, and observe that the simple example there is a theorem about odd naturals that cannot be proven without using the definition of &amp;quot;odd&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastEditorUserId="1550" LastEditDate="2021-09-09T10:25:54.197" LastActivityDate="2021-09-09T10:25:54.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24364" PostTypeId="2" ParentId="24351" CreationDate="2021-09-09T14:18:10.027" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;insight why this seems to be so difficult?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have studied CS with a side-dish of math (some decades ago) and I was surprised by (but very much enjoyed) the actual theoretical aspects of CS. I generally didn't know what to expect from a CS curriculum except that in my case it was abundantly clear that I wanted to study whatever there is to know about computers etc.&lt;/p&gt;&#xA;&lt;p&gt;Therein might lie your problem. Many of your students probably had no idea what CS entails, didn't expect it so heavily related to maths (i.e., logic, proofs, formulae etc.) and are quite unable to cope with languages, grammars, automata, lambda calculus, O-notation and so on. Also I'd suggest that a healthy percentage couldn't care less and have zero intrinsic interest. At least that was the case in the cohort I studied back then.&lt;/p&gt;&#xA;&lt;p&gt;In my case, at least 50% of all students bowed out within the first 4 semesters, and without having any statistics, I assume that was because of maths and theoretical CS, mostly.&lt;/p&gt;&#xA;&lt;p&gt;Another point is this: since the beginning of my time in the workforce, first as programmer and then as still-programming/developing/architecting/coaching teamlead, I over and over find that young programmers (who, in my company, usually have studied CS or something very close at uni levels) love to solve problems by cut&amp;amp;paste; by going to Stack Overflow; by reading tutorials and so on. Having worked closely with dozens if not more people over the decades, I think I met exactly one person who (like me) actually went to the reference documention of whatever tool or programming language we used, first, when trying to understand it or solve a deep problem. Almost everybody else seems to have a deep aversion against really &amp;quot;grokking&amp;quot; some topic from first principles.&lt;/p&gt;&#xA;&lt;p&gt;Sure, just going to a tutorial and copying some lines of code is usually quicker, and often the software kind of works at the end; but for me personally, I get my satisfaction from deeply understanding what I am doing; I am fascinated by digging deep. Meanwhile, I am very aware that this is &lt;em&gt;not&lt;/em&gt; normal for everybody, even CS students. This is a bit of a parallel to your definitions - the core documentation of whatever you're using is the definition, and a tutorial would be something like a practical example.&lt;/p&gt;&#xA;&lt;p&gt;As to &amp;quot;why&amp;quot; that is, I assume that CS (especially theoretical, but basically all of it) is by its very nature not &amp;quot;natural&amp;quot; for humans to learn and do. This is probably also the reason why we have such a shortage, worldwide, of people really excelling at it. If it were easy and came to most of us as a matter of course, we would have boatloads of people easily chugging out software and advancing the field...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Are there any better techniques to get this point across?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately I find it very hard to suggest anything except repeating your point over and over again. At least in my country, alas, due to changes to the school system in the last decade (EU -&amp;gt; Pisa...) I am afraid that schools do not have the time or capacity anymore to really bring the pupils to uni-level logical thinking, primarily cramming facts and forgetting them as quick as possible to make space for the next test. This is only partially cynical - that opinion comes from talking with and experiencing the education of some of them, including one of my adult children.&lt;/p&gt;&#xA;&lt;p&gt;Maybe in your next class, take out the first 2-3 sessions to talk about nothing else except how to approach these things (basically, what you explained in your post). Make them aware what they have to expect, and give them techniques to actually do the work. Maybe, in the first time, after actually teaching your first few definitions, take another hour or two just for Q&amp;amp;A where you test them on it and make it routine for them to go to the definitions.&lt;/p&gt;&#xA;&lt;p&gt;Maybe, on your test sheets, write a message &amp;quot;go back to your definitions whenever you don't know what to do&amp;quot; in big letters.&lt;/p&gt;&#xA;" OwnerUserId="8146" LastActivityDate="2021-09-09T14:18:10.027" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24365" PostTypeId="2" ParentId="24351" CreationDate="2021-09-09T15:56:11.873" Score="28" Body="&lt;p&gt;In my experience, students are often predisposed to &amp;quot;learn&amp;quot; by memorizing facts; that's how much of their early education worked, so that's what they're used to. When you give them a definition and then a bunch of consequences of the definition, they don't think &amp;quot;okay, I have a definition and then 87 examples of using the definition&amp;quot;, they think &amp;quot;okay, I have 88 facts about this thing, and one of them is called 'the definition'.&amp;quot; The definition is often the most abstract and complicated fact - which means that even if you tell them &amp;quot;the definition is the most important one&amp;quot;, they'll instinctively avoid it if they possibly can. To give an example in a different context: say you're trying to write a program in Python, and you want to make a particular thing happen. Obviously, you &lt;em&gt;could&lt;/em&gt; go back to fundamentals - if you're prepared to write machine code, you can make the computer do whatever the heck you want - but the way you're &lt;em&gt;actually&lt;/em&gt; going to approach it is to Google the thing you want to do and find out which Python command does it.&lt;/p&gt;&#xA;&lt;p&gt;To get students to make real use of a definition, you need to do two things. One is what it sounds like you're already doing - &lt;strong&gt;raise the importance of the definition above the other facts.&lt;/strong&gt; I'd recommend, if you aren't doing it already, taking time to demonstrate solving problems in class where you explicitly demonstrate going back to the definition. Don't just pick out the relevant part of the definition, either: restate the whole definition, to demonstrate &amp;quot;this is what I reflexively do to solve the problem&amp;quot;, and then identify the parts you actually need.&lt;/p&gt;&#xA;&lt;p&gt;The second thing you need to do is &lt;strong&gt;demystify the definition.&lt;/strong&gt; Don't let it be just another fact. For example, the formal definition of &amp;quot;NP-complete&amp;quot; might be phrased as (partially lifted from Wikipedia):&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;A decision problem C is NP-complete if it can be solved in polynomial time by a non-deterministic Turing machine, and furthermore every problem which can be solved in polynomial time by a non-deterministic Turing machine can be converted in polynomial time by a deterministic Turing machine into an instance of C.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;But no one's going to use that definition. I'm no computer science researcher, but I doubt even the best in the field think about it this way except when they're actively writing a formal proof. For working with the ideas, it's better to phrase it like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;A decision problem C is NP-complete if the solution can be checked easily, and every problem with easily-checked solutions can be easily phrased as an example of C.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is a terrible definition in the literal sense, because it's very vague; but it conveys the idea well enough to make it easy to use. The example problem you gave, for example (if &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is NP-complete and &lt;span class=&quot;math-container&quot;&gt;$A \leq_p B$&lt;/span&gt;, is &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; necessarily NP-complete) is much more approachable by this definition.&lt;/p&gt;&#xA;&lt;p&gt;The way that I generally encourage this demystification is by asking not for the definition, but for what the term &lt;strong&gt;means.&lt;/strong&gt; To many students, a &amp;quot;definition&amp;quot; is a thing they have to memorize. &amp;quot;What does X mean&amp;quot; is a thing they have to understand. It takes some work to get them there - you have to model paraphrasing things in a way that is loose but useable, and you have to demonstrate switching back to the precise definition when needed - but if you can get them thinking about what things mean, you'll see a lot of progress.&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2021-09-09T15:56:11.873" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24366" PostTypeId="2" ParentId="24359" CreationDate="2021-09-09T16:48:09.750" Score="7" Body="&lt;p&gt;Tommi's answer here is excellent. Just to add some additional suggestions - in my experience, a main issue with online learning is that it requires more structure than in-person does.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Try adapting the Think-Pair-Share model - that is, give them some time to think about the problems &lt;em&gt;before&lt;/em&gt; asking them to group up. It's a natural impulse to start the problems on their own, so give them a little room to do that before making them talk to each other.&lt;/li&gt;&#xA;&lt;li&gt;Have group-work as an explicit part of the requirements (and the goals of the assignment). I.e., instead of &amp;quot;you need to solve these problems and the tools you have include group work&amp;quot;, present it as &amp;quot;you need to learn to work with a group, and the way we're going to practice that is by trying to solve these problems&amp;quot;. If you can, having part of your rubric (for the assignment or for the whole class) be something like &amp;quot;collaborates well with classmates&amp;quot; can help.&lt;/li&gt;&#xA;&lt;li&gt;You can try assigning specific roles within the group (usually with groups of three to five, as Tommi suggested). Giving each student a specific role to play that isn't just &amp;quot;problem-solver&amp;quot; makes them feel more part of the group. Generic versions include having a &amp;quot;scribe&amp;quot; (who writes down the work) and a &amp;quot;reporter&amp;quot; (who's responsible for sharing back to the class later). If you know your students well, you might also try roles that play to their strengths. I had some success with one group when I had one student be the &amp;quot;idea-generator&amp;quot;, one the &amp;quot;idea-refiner&amp;quot;, and one the &amp;quot;accountability checker&amp;quot; - the first student was really good at coming up with ideas but had trouble critically evaluating whether they made sense, the second was really good at detail work once they had an idea to work with, and the third tended to struggle with the math but was excellent at spotting when someone was making an unfounded assumption or taking an unreasonable step.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2021-09-09T16:48:09.750" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24368" PostTypeId="1" AcceptedAnswerId="24406" CreationDate="2021-09-10T03:17:30.967" Score="6" ViewCount="298" Body="&lt;p&gt;I am going to do an online exam next week.&#xA;Since this will be done by students at home,&#xA;there is no way to guarantee that they do not cheat.&#xA;I am thinking of asking for a proof of work, i.e., photos of their sketch papers.&#xA;It certainly cannot stop anyone from cheating, but maybe it can make it a bit harder.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, this is an elective course.&#xA;So students may not have too much incentives to cheat.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Update: I have decided to not to ask for this. Ultimately, there is no way in preventing cheating in home exams. It is better to save honest student the time.&lt;/p&gt;&#xA;&lt;p&gt;As for the exam, I am teaching discrete mathematics, and I would like to ask students to apply theorems to solve problems. I don't really test computations because I allow them to use WolframAlpha to do the computation.&lt;/p&gt;&#xA;&lt;p&gt;I also want to test their ability to discover mathematics, i.e., ask them to do a bit of counting, such as the number of binary trees, and discover a formula for these numbers, and prove it.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorDisplayName="user11702" LastEditDate="2021-09-19T23:06:03.423" LastActivityDate="2021-09-25T23:37:49.047" Title="Should I ask for a proof-of-work for online exam?" Tags="|undergraduate-education|cheating|remote-teaching|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24369" PostTypeId="2" ParentId="353" CreationDate="2021-09-10T20:21:04.467" Score="1" Body="&lt;p&gt;In my opinion, the truth-table definition of material implication is disturbing because &amp;quot;if ..., then ...&amp;quot; is used in mathematics in two distinct ways (and no similar distinction exists for the other connectives like NOT, AND, and OR):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;&amp;quot; means that you can start out with &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, make some deductions, and end up with &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;. Or more precisely, it's a sentence you can interact with in the following two ways: (a) You can conclude &amp;quot;if &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;&amp;quot; by starting out with &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and then deriving &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;. (This is often called a conditional proof or direct proof.) (b) You can make use of a sentence &amp;quot;if &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;&amp;quot; if in addition you know &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, to conclude that &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is true. (This is called &lt;em&gt;modus ponens&lt;/em&gt;.)&lt;/li&gt;&#xA;&lt;li&gt;As the truth table definition.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These end up being the same given the rest of classical logic as background, of course, but they are &lt;em&gt;conceptually distinct&lt;/em&gt;. I think what is necessary is to connect these two concepts. In other words, for the truth table definition to make sense, and for a logical connective to be called an &amp;quot;implication&amp;quot; or &amp;quot;conditional&amp;quot;, it has to be shown equivalent to the deductive sense (1) above. In mathematical logic we have the distinction between &lt;span class=&quot;math-container&quot;&gt;$\{p\} \vdash q$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vdash p\to q$&lt;/span&gt;, which are shown to be equivalent by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Deduction_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;deduction theorem&lt;/a&gt; (and its converse), but I believe it is possible to show this same kind of thing without bringing in all of the scary formal notation from mathematical logic.&lt;/p&gt;&#xA;&lt;p&gt;I have a &lt;a href=&quot;https://riceissa.github.io/material-conditional/&quot; rel=&quot;nofollow noreferrer&quot;&gt;longer essay&lt;/a&gt; that implements the above strategy for explaining the material implication.&lt;/p&gt;&#xA;" OwnerUserId="9917" LastEditorUserId="9917" LastEditDate="2023-06-12T19:54:28.150" LastActivityDate="2023-06-12T19:54:28.150" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24370" PostTypeId="1" CreationDate="2021-09-10T23:34:12.750" Score="5" ViewCount="229" Body="&lt;p&gt;I taught propositional logic a few weeks ago using &lt;a href=&quot;http://discrete.openmathbooks.org/dmoi3/sec_propositional.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discrete Mathematics: An Open Introduction, 3rd edition&lt;/a&gt;.&#xA;See chapter 1 and 3.1.&lt;/p&gt;&#xA;&lt;p&gt;The topics includes&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;logical connectives&lt;/li&gt;&#xA;&lt;li&gt;implications&lt;/li&gt;&#xA;&lt;li&gt;converse and contrapositive&lt;/li&gt;&#xA;&lt;li&gt;universal and existential predicate&lt;/li&gt;&#xA;&lt;li&gt;truth table&lt;/li&gt;&#xA;&lt;li&gt;valid logic argument&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The students found the exercises in the book a little bit too easy for them.&#xA;So I gave some &lt;a href=&quot;https://puzzlewocky.com/brain-teasers/knights-and-knaves/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Knight-and-Knaves&lt;/a&gt; type logic puzzles. Then students ask what does this has to do with what they learned in class.&lt;/p&gt;&#xA;&lt;p&gt;I wonder how can I give some logic problems, which are hard, but not like puzzles.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorDisplayName="user11702" LastEditDate="2021-09-12T00:48:55.067" LastActivityDate="2021-12-17T22:36:18.850" Title="Where can I find hard exercises for logic?" Tags="|undergraduate-education|logic|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24372" PostTypeId="2" ParentId="1252" CreationDate="2021-09-11T04:59:46.363" Score="0" Body="&lt;p&gt;I have realized that explaining WHAT Number is and how it is formed explains many of the doubts of the students like borrow, carry and doubts like the one mentioned above. Once you explain properly how those digits after point are formed and what they mean, then simply telling them that there is no 100th part left that's why there is a zero at hundredth place and so on. BUT before that you'll have to properly explain how a number comes to hundredth place and what does it mean.&#xA;I usually explain Numbers as a series of Empty Spaces, each Space has:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A Name (e.g. Units).&lt;/li&gt;&#xA;&lt;li&gt;To Understand: We say it's 1's place.&lt;/li&gt;&#xA;&lt;li&gt;To calculate: Value of Numeral at that place we multiply with place value in this case 1.&lt;/li&gt;&#xA;&lt;li&gt;And can hold JUST ONE Numeral.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Then I go on to explain how starting from 0 a Number is made. Starting with 0 at 1's place we keep increasing the value(usually an example of cows leaving the barn and counting) till we reach 10 and in that case we move 1 to 10's place and put 0 at Unit's and then we again start increasing value at Unit's place.&#xA;As added info, I say Numerals tell about Number and Number tells us about this world. Numeral will just tell how many 1a or 10s or 100s are there in the number whereas a Number tells us about something in this world, say number of people in Village.&lt;/p&gt;&#xA;&lt;p&gt;So what I have realized is that if we PROPERLY introduce Kids to Numbers then many of the doubts about +, -, numbers etc. can be taken care of....&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2021-09-11T04:59:46.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24373" PostTypeId="1" CreationDate="2021-09-11T05:03:18.930" Score="2" ViewCount="438" Body="&lt;p&gt;A student (kid) of mine asked this question to me. I am not sure what to make of it or how do I answer it.&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;How many zeros do I need to add to get a non-zero value?&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Could someone help? If I were to explain it slightly mathematically, what approach I can take?&lt;/p&gt;&#xA;" OwnerUserId="12610" LastEditorUserId="372" LastEditDate="2021-09-13T18:14:29.943" LastActivityDate="2021-09-20T00:17:38.983" Title="How many zeros do we need to add to get a nonzero value?" Tags="|arithmetic|real-numbers|" AnswerCount="3" CommentCount="12" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24374" PostTypeId="2" ParentId="24373" CreationDate="2021-09-11T07:44:54.250" Score="2" Body="&lt;p&gt;✔ &lt;strong&gt;explaining by contrapositive&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Adding up any number of zeros gives zero; so, it is impossible to obtain a nonzero value by adding up just zeros.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;explaining by deriving contradiction&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; be a nonzero value.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Assume&lt;/em&gt; that &lt;span class=&quot;math-container&quot;&gt;$n\in\mathbb Z_0^+$&lt;/span&gt; zeros are required to additively obtain &lt;span class=&quot;math-container&quot;&gt;$c,$&lt;/span&gt; i.e.,  &lt;span class=&quot;math-container&quot;&gt;$\underbrace{0+0+\ldots+0}=c.\\\quad\;n\text{ times}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$n\times 0=c;$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$c=0.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hence, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is both zero and nonzero—which is absurd; our assumption must have been false; i.e., no countable number of zeros can sum to a nonzero value.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum (expansion of my first comment below)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Consider a target-archery competition in which a contestant has scored &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; points. Then the probability that they have hit the circular target is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; yet, given any point on it, the probability that they have hit that point is exactly &lt;span class=&quot;math-container&quot;&gt;$0.$&lt;/span&gt; Here, the sum of &lt;em&gt;uncountably&lt;/em&gt; many zeros turns out to be nonzero. (&lt;a href=&quot;https://en.wikipedia.org/wiki/Measure_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Measures&lt;/a&gt; only need to be countably additive.)&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-09-15T14:43:47.060" LastActivityDate="2021-09-15T14:43:47.060" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24375" PostTypeId="2" ParentId="24351" CreationDate="2021-09-11T12:51:31.170" Score="3" Body="&lt;p&gt;Give them the definitions they should learn. Then, various exercises that can be solved using the definitions. Occationally, an exercise may invoke older definitions for repetition.&lt;/p&gt;&#xA;&lt;p&gt;When we learned the Pythagoras theorem, lots of triangle exercises followed.&lt;/p&gt;&#xA;&lt;p&gt;Also, tell them early on: &amp;quot;Memorizing everything won't get you an A here. Perhaps not even a C. This is math, and you must be able to reason, and combine several facts &amp;amp; techniques to arrive at an answer.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Also: &amp;quot;In math, you don't learn all the answers, you learn how to work out the answers. If I ask, what is 987*365, you don't complain you never learned that particular product.  You learned a method for working out ALL products. CS works the same way, to some extent.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="15883" LastActivityDate="2021-09-11T12:51:31.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24376" PostTypeId="2" ParentId="24351" CreationDate="2021-09-11T13:02:09.000" Score="4" Body="&lt;p&gt;I've said in the past that the math discipline has a problem of jumping into the higher-levels of conceptual difficulty with exercises too fast. Let's say we take &lt;a href=&quot;https://facultyedsolutions.org/the-six-levels-of-learning/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bloom's Taxonomy&lt;/a&gt; as a model. Non-STEM teachers constantly complain that they spend all their time at Level 1, memorizing facts, and can't move past that. Math teachers, on the other hand, give a definition and then taking that for granted, launch immediately to at least Level 3, applications.&lt;/p&gt;&#xA;&lt;p&gt;Note in the OP's example they've given a definition or two and then immediately expect logical inferences based on those definitions.&lt;/p&gt;&#xA;&lt;p&gt;I understand, because I don't have time to do any better in my math courses (e.g., discrete mathematics for CS majors) -- there's too much dense content in the curriculum to do any differently. But if I had more time I would:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Test remembering definitions directly with multiple-choice or fill-in-the-blank quizzes&lt;/li&gt;&#xA;&lt;li&gt;Test basic understanding of definitions with true/false quiz questions of whether different cases qualify for the definition at hand. (E.g.: Is 5 an integer? Is 1/2? Is -8? Is 0? Is pi?)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I manage to work in a little bit of that in lower-level algebra classes, but as I say, time is very constrained in our math classes.&lt;/p&gt;&#xA;&lt;p&gt;At one point another SE poster once wrote something like, &amp;quot;People don't understand the importance of definitions until they have to use them in anger (i.e., prove something)&amp;quot;. That's sort of true, and if students get to the point of a proof-writing course, and they're still intransigent about not reading definitions, then they probably just fail.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-09-12T13:48:22.793" LastActivityDate="2021-09-12T13:48:22.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24377" PostTypeId="2" ParentId="24373" CreationDate="2021-09-11T13:23:50.387" Score="2" Body="&lt;p&gt;Suppose that you want to know that &lt;span class=&quot;math-container&quot;&gt;$\underbrace{0+\cdots+ 0}_{N} = 0$&lt;/span&gt;. We know that &lt;span class=&quot;math-container&quot;&gt;$0+0=0$&lt;/span&gt; and also have that &lt;span class=&quot;math-container&quot;&gt;$\underbrace{0+\cdots+ 0}_{n+1} = (\underbrace{0+\cdots+ 0}_{n}) +0$&lt;/span&gt; by associativity. Then proceed by mathematical induction, reducing &lt;span class=&quot;math-container&quot;&gt;$(\underbrace{0+\cdots+ 0}_{n}) +0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$0+0=0$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2021-09-11T13:23:50.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24378" PostTypeId="1" AcceptedAnswerId="24383" CreationDate="2021-09-11T14:21:23.177" Score="2" ViewCount="747" Body="&lt;p&gt;I'm teaching a class called &amp;quot;Math for Elementary Teachers.&amp;quot; The main goal of the course is to prepare the prospective teachers for an exam that I believe is called &amp;quot;Praxis&amp;quot; (some sort of standardized math test that people who want to teach elementary education have to take). I'm teaching in the state of Iowa if that matters.&lt;/p&gt;&#xA;&lt;p&gt;The course uses the book: &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/013475882X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&#xA;Mathematical Reasoning for Elementary Teachers - Media Update 7th Edition&#xA;by Calvin Long (Author), Duane DeTemple (Author), Richard Millman (Author) &lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In two semesters of the class, we are scheduled to cover every section of the above book. Does anyone know enough about this incoming standardized test and this textbook to know if this an effective way to prepare them? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="16437" LastEditorUserId="5597" LastEditDate="2021-09-12T17:22:36.107" LastActivityDate="2021-09-12T17:22:36.107" Title="Preparing elementary teachers for the praxis exam" Tags="|undergraduate-education|primary-education|usa|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24379" PostTypeId="2" ParentId="1056" CreationDate="2021-09-11T15:15:46.250" Score="1" Body="&lt;p&gt;Your self-answer is wrong, because a &lt;strong&gt;tautology&lt;/strong&gt; must be true under &lt;strong&gt;every&lt;/strong&gt; interpretation, but your example is &lt;strong&gt;certainly not so&lt;/strong&gt;. There is no single word for what you want, but the standard terminology is that your formula is &lt;strong&gt;true under standard/intended interpretation&lt;/strong&gt;. And you can note that it is conventional to say just &amp;quot;true&amp;quot; to mean that when dealing with arithmetical formulae. The failure to understand this conventional meaning of &amp;quot;true arithmetical sentence&amp;quot; underlies a large fraction of misunderstandings of Godel's incompleteness theorems due to the phrase &amp;quot;&lt;a href=&quot;https://math.stackexchange.com/a/1873544/21820&quot;&gt;true but unprovable statements&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Nor is &amp;quot;universally true&amp;quot; correct for the same reason.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-09-11T15:15:46.250" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24380" PostTypeId="2" ParentId="24336" CreationDate="2021-09-11T15:47:47.780" Score="2" Body="&lt;p&gt;Mere discussion is useless. You need to actually force students to produce proofs, if you want them to ever be able to prove anything. In particular, from my experience I can be quite sure that most students &lt;strong&gt;cannot&lt;/strong&gt; produce perfectly correct mathematical proofs of any newly encountered theorems (say at the undergraduate level) unless they have spent at least roughly 1000 hours practicing proving. Of course you need to in the first place provide them some kind of deductive system, for which the only practical style seems to be Fitch-style (e.g. &lt;a href=&quot;http://math.stackexchange.com/a/1684204/21820&quot;&gt;this one&lt;/a&gt; or the one in &lt;a href=&quot;http://homepages.uc.edu/%7Emartinj/Symbolic_Logic/341%20Syllabus,%20Textbook,%20Handouts,%20Notes/LPL%20textbook.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;LPL&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;So what you need is literally to have them do &lt;strong&gt;many&lt;/strong&gt; proof exercises, &lt;strong&gt;many&lt;/strong&gt; times. To give a very simple analogy, you can discuss all the details you know about building a steam engine with a student, but if that student never tries to build a steam engine, you can be very sure that the student will utterly fail on an actual attempt, no matter how much they 'understood you through discussion'.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-09-11T15:47:47.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24381" PostTypeId="2" ParentId="24352" CreationDate="2021-09-11T16:31:55.183" Score="1" Body="&lt;p&gt;Your method is circular. Try:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;1000000&#xA;−     1&#xA;−−−−−−−&#xA;100000&#xA;     −1&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Now what??&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-09-11T16:31:55.183" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24382" PostTypeId="2" ParentId="24378" CreationDate="2021-09-11T17:16:18.543" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The main goal of the course is to prepare the prospective teachers for an exam that I believe is called &amp;quot;Praxis&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think the best thing would be to get a copy of a test prep book for the Praxis (the specific version that applies in your case), or the Praxis's &amp;quot;Official Guide&amp;quot; (perhaps named differently) and incorporate into your class as many questions from it as seems reasonable during the semester. By &amp;quot;incorporate into your class&amp;quot;, I mean lecture examples, in-class small-groups work (if possible, given COVID issues), homework problems, short quiz and major test problems, etc.&lt;/p&gt;&#xA;&lt;p&gt;I don't know anything about the book you cited, but any book that claims to help prepare students for the Praxis test should be fine as long as your mastery of the material is reasonably good and you have a good supply of Praxis-like problems to use.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2021-09-11T17:16:18.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24383" PostTypeId="2" ParentId="24378" CreationDate="2021-09-11T19:47:46.897" Score="18" Body="&lt;p&gt;This is not an answer, but I think it's too important to leave as a comment.&lt;/p&gt;&#xA;&lt;p&gt;I strongly urge you to rethink your goals for this course.&lt;/p&gt;&#xA;&lt;p&gt;Your goals should be to prepare your students to teach mathematics in elementary schools.&lt;/p&gt;&#xA;&lt;p&gt;The Praxis test is a not terrible minimum measure of this ability, but, frankly, it's not that good a measure, and many people who pass the Praxis are still pretty terrible teachers of elementary school math, particularly if they have been taught to the test without really understanding what they are learning.&lt;/p&gt;&#xA;&lt;p&gt;My impression is that half of US university students don't understand multiplication of integers, in the sense that they don't understand that if you repeat a process that takes m steps n times, you end up having to do m*n total steps.  In many cases (and this is corroborated by what many elementary teachers say about themselves and each other), this is because they had 3rd and 4th grade teachers who don't understand this either.  (Never mind addition of fractions.  Basically no one understands that.)&lt;/p&gt;&#xA;&lt;p&gt;Your job is to help stop this cycle of ignorance, not prepare people for a standardized test.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2021-09-11T19:47:46.897" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24384" PostTypeId="1" CreationDate="2021-09-12T05:05:50.080" Score="0" ViewCount="348" Body="&lt;p&gt;I teach kids in a village and I was wondering if there is a list of Facts, Procedures, things, a kid 6-15 years of age should know to be able to easily understand things at his/her level of Math. Basics like What are Numbers? Operations etc.&lt;/p&gt;&#xA;" OwnerUserId="12610" LastEditorUserId="2083" LastEditDate="2021-09-13T05:09:42.637" LastActivityDate="2021-09-13T05:09:42.637" Title="What do kids(6-15) need to know in Math to be able to understand Math easily and effectively?" Tags="|primary-education|india|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24385" PostTypeId="2" ParentId="24352" CreationDate="2021-09-12T05:35:39.990" Score="2" Body="&lt;p&gt;The method of addition and subtraction that you mention is not new.  For now, I provide one reference, but I'm sure there are others.  Note that your method of subtraction makes one big assumption that is not needed for the traditional method: you assume that students are familiar with negative integers.&lt;/p&gt;&#xA;&lt;p&gt;The website &lt;em&gt;Knowledge Over Grades&lt;/em&gt; uses a slight variant of your subtraction method: instead of &amp;quot;subtracting from right to left&amp;quot; it does the subtraction of larger powers of 10 first.  It calls the method &amp;quot;&lt;a href=&quot;https://www.knowledgeovergrades.net/blog/subtract-multi-digit-numbers-without-regrouping-or-carrying&quot; rel=&quot;nofollow noreferrer&quot;&gt;subtraction of multi-digit numbers without regrouping or borrowing&lt;/a&gt;.&amp;quot;  (It assumes that the student is familiar with place values and negative integers.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/0PRry.png&quot; alt=&quot;subtract multi-digits without regrouping or borrowing&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that there seems to be a typo in the image.  (It uses the word &lt;em&gt;carrying&lt;/em&gt; instead of &lt;em&gt;borrowing&lt;/em&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;The website calls the addition method you mention &amp;quot;&lt;a href=&quot;https://www.knowledgeovergrades.net/blog/adding-multi-digit-numbers-by-place-value-without-regrouping-or-carrying&quot; rel=&quot;nofollow noreferrer&quot;&gt;addition of multi-digit numbers without regrouping or carrying&lt;/a&gt;.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2021-09-12T05:41:27.947" LastActivityDate="2021-09-12T05:41:27.947" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24386" PostTypeId="2" ParentId="24384" CreationDate="2021-09-12T05:41:36.210" Score="1" Body="&lt;p&gt;I have written before about &lt;a href=&quot;https://matheducators.stackexchange.com/a/16705/1550&quot;&gt;what should be taught at the high-school level&lt;/a&gt;, and anyone who knows basic logic as described there will &amp;quot;&lt;em&gt;be able to easily understand things at his/her level of Math&lt;/em&gt;&amp;quot; and beyond to every area of mathematics in the future as well. Besides that, you should also let students (especially but not just children) &lt;a href=&quot;https://matheducators.stackexchange.com/a/18990/1550&quot;&gt;explore concrete interesting mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I think it cannot be denied that attempting to teach or do mathematics without a solid foundation of basic logic is just like writing software by copy-pasting from StackOverflow. So basic logic has to be taught well. On the other hand, you need to motivate learning the basics, and that is what the games, puzzles and other recreational mathematics are for.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-09-12T05:41:36.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24387" PostTypeId="2" ParentId="16886" CreationDate="2021-09-12T05:56:20.813" Score="0" Body="&lt;p&gt;Although this question was asked 2 years back but the Knowledge I acquired recently so here it goes.&lt;/p&gt;&#xA;&lt;p&gt;The objective is not to &amp;quot;understand&amp;quot; but to Fall in Love. Falling in Love is different, so you'll have to have Relation at a Personal Level with Math and the Elements in it. Open a Math Book NOT TO UNDERSTAND but to &amp;quot;get to know&amp;quot; better, date sort of and let your curiosity guide you. That's how I learned Math. Opened a topic in Wikipedia and what ever I didn't understand I clicked on that link and then another, sometimes it took a week-10 days to return back to original topic. That's when I realized Numbers is what people don't understand properly and Numbers are the ONLY REALLY ABSTRACT entity in Mathematics. I studied about Numbers a lot and finally what I learned is that usual way is for a Teacher to teach Math using things like 1 Apple, 2 Apples, 3 Apples. I thought let's try and understand Numbers without using &amp;quot;things&amp;quot;; JUST THE NUMBER ITSELF. Chasing that line of thought I arrived at this&amp;gt;&amp;gt;&amp;gt; Numbers are nothing but a SERIES OF EMPTY SPACES. Each Space has a name and a value associated with it. Each Space can contain ONLY ONE Numeral. So Numerals don't tell us anything about the World, they simple tell us about the Number as to how many units, tens or hundreds it contains. So this Series of Numerals forms a number and then Number tells us something about the world. So this explanation happened to me, I didn't read it like this anywhere. And it happened because I had that personal relation with Math.&lt;/p&gt;&#xA;&lt;p&gt;PISA also says that &amp;quot;Experience&amp;quot; of/while Learning plays a very important part in whole Learning process. Approach it with lot of Heart and Be Attentive....&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2021-09-12T05:56:20.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24389" PostTypeId="1" AcceptedAnswerId="24394" CreationDate="2021-09-15T20:33:39.370" Score="2" ViewCount="84" Body="&lt;p&gt;Logo is a programming language that has, as main (and probably only) purpose the teaching of geometry and introductory computer science&lt;/p&gt;&#xA;&lt;p&gt;I am looking for an interpreter for logo that runs online and accepts commands in portuguese.&lt;/p&gt;&#xA;&lt;p&gt;For reference, here is one in english: &lt;a href=&quot;https://www.calormen.com/jslogo/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.calormen.com/jslogo/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have looked online and found no such site.&lt;/p&gt;&#xA;&lt;p&gt;If there does not exist a online interpreter, the next best thing would be one that runs on windows and needs no instalation. (the third best thing, one that runs on windows and needs instalation)&lt;/p&gt;&#xA;&lt;p&gt;I am aware of only one implementation of logo that allows for commands in portuguese, kturtle, and it seems to require linux&lt;/p&gt;&#xA;" OwnerUserId="1662" LastEditorUserId="1662" LastEditDate="2021-09-17T16:53:22.057" LastActivityDate="2021-09-18T16:44:02.640" Title="running the logo language online, in portuguese" Tags="|software|specific-software|" AnswerCount="1" CommentCount="5" ClosedDate="2021-10-04T19:40:41.163" ContentLicense="CC BY-SA 4.0" />
  <row Id="24391" PostTypeId="2" ParentId="19183" CreationDate="2021-09-17T16:57:50.790" Score="2" Body="&lt;p&gt;You can use the double angle formulas (together with the small angle approximations &lt;span class=&quot;math-container&quot;&gt;$\sin(\theta) \approx \theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos(\theta) \approx 1$&lt;/span&gt;) to approximate sine and cosine of any number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by using the small angle approximations for &lt;span class=&quot;math-container&quot;&gt;$\frac{x}{2^n}$&lt;/span&gt; and doubling until you get &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  This is a pretty cool &amp;quot;proto-calculus&amp;quot; way to compute sine and cosine using only hand calculations.  Makes a nice coding project too!&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-09-17T17:03:45.500" LastActivityDate="2021-09-17T17:03:45.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24392" PostTypeId="2" ParentId="14919" CreationDate="2021-09-17T22:08:31.780" Score="2" Body="&lt;p&gt;In particular, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt; is used to mean either &lt;span class=&quot;math-container&quot;&gt;$\{0, 1, \dotsc\}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\{1, 2, \dotsc\}$&lt;/span&gt;, depending on the exact area/author. Some write &lt;span class=&quot;math-container&quot;&gt;$\mathbf{N}$&lt;/span&gt; for this. It is important for your students to know there are several meanings in use for this particular notation, even if you pick one and stick to it religiously. Personally I stick to the second (most often used), and write &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}_0$&lt;/span&gt; for the first when needed.&lt;/p&gt;&#xA;&lt;p&gt;More generally, for some notations there are several conflicting definitions, while some notions have a lot of different notations (e.g. &lt;span class=&quot;math-container&quot;&gt;$f(x) = O(g(x))$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$f(x) \ll g(x)$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Always&lt;/em&gt; check any (perhaps just local) definitions. Mathematics notation and terminology is a living, changing language, with it's own dialects and local quirks.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2021-09-21T16:09:35.847" LastActivityDate="2021-09-21T16:09:35.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24393" PostTypeId="2" ParentId="353" CreationDate="2021-09-18T02:56:20.437" Score="1" Body="&lt;p&gt;If you will eventually be teaching the basic methods of proof (conditional proof, proof by contradiction, etc.) in your course for math majors, you might consider starting with the truth tables for NOT, AND and OR and postpone the truth table for IMPLIES until they understand some of those basic methods of proof. Then they should be able to understand a proof of ~A =&amp;gt; [A =&amp;gt; B] (see below) which corresponds to lines 3 and 4 of the usual truth table for IMPLIES, and IIUC is usually where problems seem to arise. (Screenshot from my proof checker.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PKaDn.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PKaDn.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2021-09-18T03:01:48.783" LastActivityDate="2021-09-18T03:01:48.783" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24394" PostTypeId="2" ParentId="24389" CreationDate="2021-09-18T16:44:02.640" Score="5" Body="&lt;p&gt;I think &lt;a href=&quot;https://scratch.mit.edu/&quot; rel=&quot;noreferrer&quot;&gt;Scratch&lt;/a&gt; might meet your needs well.  It isn't formally Logo (although there is no formal Logo standard TBH), but it has the same turtle graphics capabilities.  It's a far more visual language where you kind of snap together &amp;quot;lego blocks&amp;quot; to create code.  It's a neat system that avoids syntax errors entirely.  If you're using it to teach programming more than geometry, it also prepares students for 21st century languages more than Logo does, with lots of multimedia support and teaching an event-driven paradigm.  It's a completely web-based side with a large global userbase uploading and sharing their projects.  Taking a quick glance at the language support, they seem to support both Portuguese and Brazilian Portuguese.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-09-18T16:44:02.640" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24395" PostTypeId="1" CreationDate="2021-09-19T09:44:09.273" Score="2" ViewCount="171" Body="&lt;p&gt;My son is in Class or Grade 5 in India. he really likes mathematics. Apart from school work, he does ixl.com, website dedicated for math exercises of his class.&#xA;he does those exercises relatively easily.&#xA;i want to encourage him in mathematics,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;how should i encourage my son, so that he is engaged and curiosity remains ?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;can anyone suggest math books for him to continue to learn problem solving ?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15090" LastEditorUserId="2083" LastEditDate="2021-09-19T11:33:13.900" LastActivityDate="2021-09-20T22:58:54.480" Title="How to encourage grade 5 kid very interested in mathematics?" Tags="|primary-education|reference-books|gifted-students|book|india|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24396" PostTypeId="2" ParentId="24373" CreationDate="2021-09-20T00:17:38.983" Score="-1" Body="&lt;p&gt;You don't say what age the student is or what their mathematical background is. Your question is also phrased as a mathematics question, not an education question. Both of these things make your question, in its present form, not very well suited to this site.&lt;/p&gt;&#xA;&lt;p&gt;If the student is in high school and hasn't had calculus, then a reasonable thing to do here would be to explain that there are various number systems that extend the reals to include infinite quantities. Examples:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Extended_real_number_line&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Extended_real_number_line&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hyperreal_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Hyperreal_number&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Surreal_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Surreal_number&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The student's question can only be answered in such an extended number system, not in the real number system, since the reals don't include infinity. The student's question is equivalent to asking 1/0, and actually none of these systems make that a well-defined thing. But, e.g., the hyperreals do allow you to divide 1 by an infinitely small number and get an infinite result.&lt;/p&gt;&#xA;&lt;p&gt;If the student has had calculus, then you can tell them that they're basically reinventing indeterminate forms.&lt;/p&gt;&#xA;" OwnerUserId="17778" LastActivityDate="2021-09-20T00:17:38.983" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24397" PostTypeId="2" ParentId="24395" CreationDate="2021-09-20T19:28:50.450" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://beastacademy.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beast Academy&lt;/a&gt; is wonderful. It goes from a level 2 to a level 5, but those are higher than U.S. grade levels. Also, I have some of my favorite books listed &lt;a href=&quot;https://mathmamawrites.blogspot.com/p/math-books.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;at my blog&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-09-20T19:28:50.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24398" PostTypeId="2" ParentId="24395" CreationDate="2021-09-20T22:58:54.480" Score="2" Body="&lt;p&gt;The &lt;a href=&quot;https://www.khanacademy.org/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kahn Academy&lt;/a&gt; is quite extensive, and in my experience of high quality. It is partitioned by grades (kindergarten, 1st, .2nd, .., 8th), and by topic (arithmetic, geometry, pre-algebra, etc.) It has a nice interactive interface, practice questions and levels. A student can have a personalized dashboard. They've now branched out to Economics and AP-Biology among other topics. A non-profit organization.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-09-20T22:58:54.480" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24399" PostTypeId="1" CreationDate="2021-09-21T15:25:44.403" Score="-4" ViewCount="234" Body="&lt;p&gt;Note: Updated based on &lt;a href=&quot;https://math.meta.stackexchange.com/questions/34125&quot;&gt;this&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In my course, my instructor posed the following exercise:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; be the subset  of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$S=\{(a_1,a_2,a_3...a_n) | a_2 = \pm a_1, a_3=...=a_n=0 \}$&lt;/span&gt;. Define addition and multiplication as follows: For &lt;span class=&quot;math-container&quot;&gt;$(a_1,a_2, ..., a_n),(b_1,b_2, ..., b_n) \in S$&lt;/span&gt;, define &lt;span class=&quot;math-container&quot;&gt;$(a_1,a_2, ..., a_n)+(b_1,b_2, ..., b_n)=(a_1+b_1,a_2+b_2, ..., a_n+b_n)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(a_1,a_2, ..., a_n)(b_1,b_2, ..., b_n)=(a_1 \times b_1,a_2 \times b_2, ..., a_n \times b_n)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; are the usual addition and multiplication in &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;. Which axioms in the definition of a field are satisfied by &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;? Is &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; a field and why?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, I notice that the given operation &lt;span class=&quot;math-container&quot;&gt;$+: S^2 \to \mathbb R^n$&lt;/span&gt; is not closed i.e. its image is not a subset of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;. Now, I notice that the given operation &lt;span class=&quot;math-container&quot;&gt;$+: S^2 \to \mathbb R^n$&lt;/span&gt; is not closed i.e. its image is not a subset of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;. In particular it really just says 'define' instead of like giving explicitly the domain and range as &lt;span class=&quot;math-container&quot;&gt;$+: S^2 \to \mathbb R^n$&lt;/span&gt;. I think this is much like: &lt;a href=&quot;https://math.stackexchange.com/questions/856226/how-do-you-prove-the-domain-of-a-function&quot;&gt;https://math.stackexchange.com/questions/856226/how-do-you-prove-the-domain-of-a-function&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I asked my instructor about this and they responded as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(It) is part of the assignment (as to whether or not the operations are closed). If an operation is not closed, which of the rest of the axioms are satisfied?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Question 1: So basically this is a trick for students like someone studying this for the 1st time may think of 'define' to really mean &lt;span class=&quot;math-container&quot;&gt;$+: S^2 \to S$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$+: S^2 \to \mathbb R^n$&lt;/span&gt; and thus not realise the operation is not closed ?&lt;/p&gt;&#xA;&lt;p&gt;Question 2:&lt;/p&gt;&#xA;&lt;p&gt;If not, then what?&lt;/p&gt;&#xA;&lt;p&gt;If so, then is this a fair trick? I mean is the question umambiguous, 'reasonable' and 'of considerable pedagogical value' (terms used in comments below) ? Or is this question perhaps unclear and an unfair way to trick students?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Cross-posted on maths se: &lt;a href=&quot;https://math.stackexchange.com/questions/4256472/can-you-talk-about-the-rest-of-the-field-axioms-when-the-operations-are-not-cl&quot;&gt;https://math.stackexchange.com/questions/4256472/can-you-talk-about-the-rest-of-the-field-axioms-when-the-operations-are-not-cl&lt;/a&gt; --&amp;gt; I justify the cross-post in that maths education se might respond more to the trick of questions while maths se might respond more to the talking about field axioms when assumptions are violated.&lt;/p&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2021-09-22T20:13:50.860" LastActivityDate="2021-09-22T20:13:50.860" Title="Can you talk about (the rest of the) field axioms when the operations are not closed?" Tags="|mathematical-pedagogy|terminology|abstract-algebra|definitions|" AnswerCount="1" CommentCount="17" ClosedDate="2021-09-22T09:50:34.753" ContentLicense="CC BY-SA 4.0" />
  <row Id="24400" PostTypeId="2" ParentId="24399" CreationDate="2021-09-22T02:19:36.630" Score="2" Body="&lt;p&gt;This is a little long for a comment so it will have to be an answer. First, you should talk to your professor; I'm pretty sure that they will be able to definitively clear your issue up with a small amount of back-and-forth discussion.&lt;/p&gt;&#xA;&lt;p&gt;You seem to have trouble starting around your encounter with the word &amp;quot;define&amp;quot;. I'm not really sure what we are supposed to get out of your link. (Is that a game? an anime? it certainly isn't a universal enough piece of culture to pass unexplained.) At an informal level, when we say &amp;quot;define...&amp;quot; it merely sets up some common terminology. In this case, an extension of &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; to vectors. Someone else could propose a different &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; on the same set at a later time, but this is the one you will be working with in this problem. Maybe it satisfies the field axioms and maybe it doesn't; calling them &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; doesn't mean much.&lt;/p&gt;&#xA;&lt;p&gt;Secondly you get tripped up around the words &amp;quot;binary operation&amp;quot;. A binary operation on &lt;span class=&quot;math-container&quot;&gt;$\mathcal{S}$&lt;/span&gt; is simply a function &lt;span class=&quot;math-container&quot;&gt;$\mathcal{S}\times \mathcal{S} \to \mathcal{S}$&lt;/span&gt;. i.e. it takes two elements of &lt;span class=&quot;math-container&quot;&gt;$\mathcal{S}$&lt;/span&gt; to one element of &lt;span class=&quot;math-container&quot;&gt;$\mathcal{S}$&lt;/span&gt;. Aside from verifying that the operations are well-defined and have the proper (co-)domains, there is nothing to do.&lt;/p&gt;&#xA;&lt;p&gt;But, &lt;em&gt;trust me&lt;/em&gt;, your professor wanted you to do something &lt;em&gt;so much simpler&lt;/em&gt;. You have a direct sum of commutative rings and hence a commutative ring. What else do you need for a field?&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2021-09-22T02:19:36.630" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="24401" PostTypeId="1" CreationDate="2021-09-22T15:42:15.300" Score="8" ViewCount="742" Body="&lt;p&gt;So recently I was teaching high school calculus to a high school class and I was wondering about the pedagogically best way to make students actually understand why the derivatives of the exponential &amp;amp; logarithm functions are what they are (instead of just being told what they are). The students already knew some basic calculus including derivatives of power functions and rules of differentiation. The following is the approach that I came up with and I am curious to learn &lt;em&gt;how it compares to alternative approaches and whether people think that those alternative approaches are of more pedagogical value, and why.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Start by tabulating values and making precise drawings of &lt;span class=&quot;math-container&quot;&gt;$2^x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3^x$&lt;/span&gt;, then draw tangents at &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; and work out their slope with a ruler and calculator. You get approx. &lt;span class=&quot;math-container&quot;&gt;$0.7$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1.1$&lt;/span&gt;, so arguably there must be a value &lt;span class=&quot;math-container&quot;&gt;$2&amp;lt;a&amp;lt;3$&lt;/span&gt; for which at &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; the function &lt;span class=&quot;math-container&quot;&gt;$a^x$&lt;/span&gt; has a tangent with slope exactly &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. Call this value Euler's number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; (thereby establishing a working definition of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;) and consider the corresponding exponential function &lt;span class=&quot;math-container&quot;&gt;$f(x)=e^x$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; as a base.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;From that definition of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; we know that &lt;span class=&quot;math-container&quot;&gt;$f'(0)=1$&lt;/span&gt;, which translates to &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{h \to 0}\frac{e^h-1}{h}=1$&lt;/span&gt; by first principles. Rearranging this for &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$(1+h)^{\frac{1}{h}}\to e$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$h\to 0$&lt;/span&gt;, from which students can work out the value of &lt;span class=&quot;math-container&quot;&gt;$e=2,718...$&lt;/span&gt; with a calculator.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Now work out the derivative of &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; from first principles (differential quotient). This is easy enough, because &lt;span class=&quot;math-container&quot;&gt;$e^{x+h}=e^x e^h$&lt;/span&gt; and the limit &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{h\to 0}\frac{e^h-1}{h}=1$&lt;/span&gt; is known from the chosen definition of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;. One finds that &lt;span class=&quot;math-container&quot;&gt;$f'(x)=e^x$&lt;/span&gt;, which is a cool result.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Define the natural logarithm &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt; as the inverse function of &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; (logarithms and inverse functions are known). In order to work out the derivative of &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt; start with the ansatz &lt;span class=&quot;math-container&quot;&gt;$e^{\ln x} = x$&lt;/span&gt;, which is clear from the chosen definition of &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt;. Differentiate both sides by applying the chain rule and using the now known fact that &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; is its own derivative. This way one immediately obtains the result that &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt; has derivative &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given this, it is now easy to work out the derivative of general &lt;span class=&quot;math-container&quot;&gt;$a^x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\log_a x$&lt;/span&gt; by chain-ruling the expression &lt;span class=&quot;math-container&quot;&gt;$a^x = (e^{\ln a})^x = e^{\ln a\times x}$&lt;/span&gt; and differentiating &lt;span class=&quot;math-container&quot;&gt;$\log_a x=\frac{\ln x}{\ln a}$&lt;/span&gt;. From this it becomes clear that the original exact slope values were &lt;span class=&quot;math-container&quot;&gt;$\ln 2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\ln 3$&lt;/span&gt; and could have been worked out by pressing a calculator button, which students found amusing.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In fact, this definition of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; can be made rigorous (assuming familiarity with uniform convergence) and thereby may be of some pedagogical value at undergraduate university level, too.&lt;/p&gt;&#xA;" OwnerUserId="5345" LastEditorUserId="3344" LastEditDate="2022-06-09T20:15:39.113" LastActivityDate="2022-06-09T20:15:39.113" Title="Exponential &amp; logarithm in a high school calculus class" Tags="|mathematical-pedagogy|calculus|derivative|exponential|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24402" PostTypeId="2" ParentId="5614" CreationDate="2021-09-23T18:32:57.377" Score="0" Body="&lt;p&gt;I'm a student who has just come across this exact problem.&lt;/p&gt;&#xA;&lt;p&gt;I've thought about it for a while and the greatest explanation I could come up with is this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The value of x in an equation is dependant on the value of y, not dependant on the numbers around the x; e.g. it's coefficients and the constants on its side of the equation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The value of y in an equation(for a given point) is dependant on the values around the x in inverse, since it's on the other side of the equation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Since the value of x is dependant on y which is dependant on the opposites of the coefficients and constants around x, x is dependant on the opposites of the coefficients and constants around x, meaning that when they're changed, the equation gets shifted by the opposite of what's been done(as an equation is just a series of points that this idea can be though of applying to each individually).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This probably won't help, but it's the one that helps me understand it the clearest. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="17798" LastActivityDate="2021-09-23T18:32:57.377" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24404" PostTypeId="1" CreationDate="2021-09-25T02:32:49.213" Score="7" ViewCount="210" Body="&lt;p&gt;I am looking for input/advice regarding whether a student I just began tutoring may have dyscalculia - and, if so, how to go about broaching the subject / assisting them as best as possible.&lt;/p&gt;&#xA;&lt;p&gt;I'd rather not get into what cause me to become worried of this for privacy reasons, except that a student I just began tutoring in introductory university physics (calculus based) demonstrated an extremely worrying difficulty with basic mathematical reasoning, despite giving me the impression that they were otherwise quite intelligent. Whenever I presented them with a question, I felt they would answer seemingly randomly, almost as if they were picking a word or phrase I had used earlier in the hopes it was correct. In short, I have worked with others in the past who have found math quite difficult, but this felt like something else entirely.&lt;/p&gt;&#xA;&lt;p&gt;If you have worked with students suffering from dyscalculia in the past, I would appreciate any input you have on how to possibly identify it and differentiate it from simply finding math difficult.&lt;/p&gt;&#xA;" OwnerUserId="17808" LastActivityDate="2021-09-25T02:32:49.213" Title="How to recognize possible dyscalculia in a student?" Tags="|calculus|tutoring|physical-sciences|dyscalculia|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24406" PostTypeId="2" ParentId="24368" CreationDate="2021-09-25T23:37:49.047" Score="2" Body="&lt;p&gt;I also teach discrete mathematics, and do so online, including exams, in the last two years.&lt;/p&gt;&#xA;&lt;p&gt;All of my test questions in that course are open-response, using the &amp;quot;essay&amp;quot; type of question on the Blackboard learning management system. Students are required to use either LaTeX or the built-in formula editor for any symbolic math statements.&lt;/p&gt;&#xA;&lt;p&gt;To my understanding, learning logic, rules of inferences, and writing proofs (likely for the first time) is the chief goal of such a course. I don't see how justifying and explaining reasoning can be disentangled from that essential work. I'm not in favor of saying that answers are one thing, and justifications are another (both for conceptual and logistical purposes).&lt;/p&gt;&#xA;&lt;p&gt;On the issue of cheating-deterence (for which I'm the head faculty expert on the issue), my understanding is that the best practice is to make test questions distinguished between different students. For me, this might mean programmatically altering some of the questions a bit so (a) answers might be a tiny bit different, and/or (b) someone posting a question to Chegg (et. al.) can be immediately identified.&lt;/p&gt;&#xA;&lt;p&gt;For more on the anti-cheating issue, see this question:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/18477/what-are-the-best-practices-for-giving-online-tests&quot;&gt;What are the best practices for giving online tests?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2021-09-25T23:37:49.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24407" PostTypeId="1" CreationDate="2021-09-26T18:34:56.877" Score="0" ViewCount="378" Body="&lt;p&gt;I need to understand &amp;quot;implicit differentiation&amp;quot; and after that I need to be able to explain it to a student.&lt;/p&gt;&#xA;&lt;p&gt;Here is an example:&lt;/p&gt;&#xA;&lt;p&gt;Find the formula of a tangent line to the following curve at the given point using implicit differentiation.&lt;/p&gt;&#xA;&lt;p&gt;x+xy+y^2=7 at a point (1,2)&lt;/p&gt;&#xA;&lt;p&gt;What is the best way of explaining  that?&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="13628" LastActivityDate="2021-10-03T20:16:20.390" Title="implicit differentiation, formula of a tangent line" Tags="|calculus|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24408" PostTypeId="2" ParentId="24407" CreationDate="2021-09-26T23:14:13.453" Score="3" Body="&lt;p&gt;Without being rigourous, the equation &lt;span class=&quot;math-container&quot;&gt;$F(x,y)=0$&lt;/span&gt; defines a relation between &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. To promote this relation to a function, some restrictions in the &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;-plane has to be imposed. Assuming that such conditions exists, you can say that the equation &lt;span class=&quot;math-container&quot;&gt;$F(x,y)=0$&lt;/span&gt; is defining &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; implicitly as a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In some cases, we will be able to manipulate the equation &lt;span class=&quot;math-container&quot;&gt;$F(x,y)=0$&lt;/span&gt; to find an explicit representation of the desired function, by solving the expression for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. In other cases, it won't be able to solve, but in both cases the derivate &lt;span class=&quot;math-container&quot;&gt;$y' = \dfrac{dy}{dx}$&lt;/span&gt; should exists as a function.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A simple example:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$y^3 - x = 0$&lt;/span&gt; defines &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; implicitly as a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. But in this case, we can solve the expression for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, which leads to &lt;span class=&quot;math-container&quot;&gt;$y= \sqrt[3]{x}$&lt;/span&gt;, &lt;strong&gt;and then&lt;/strong&gt;, &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{dy}{dx} = \dfrac{1}{3\sqrt[3]{x^2}}$$&lt;/span&gt; Implicit differentiation will help to compute the derivative without the solving-for-y process. This requires the chain rule, because in general: &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{dL}{dx} = \dfrac{dL}{dy}\cdot \dfrac{dy}{dx}$$&lt;/span&gt; Thus, using properties of derivatives,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$y^3 - x = 0 \implies \dfrac{d(y^3)}{dx} - \dfrac{d(x)}{dx} = \dfrac{d(0)}{dx}$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{d(y^3)}{dy}\cdot \dfrac{dy}{dx} - 1 = 0$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$3y^2\cdot \dfrac{dy}{dx} - 1 = 0 \implies \dfrac{dy}{dx} = \dfrac{1}{3y^2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It should be noted that &lt;span class=&quot;math-container&quot;&gt;$\dfrac{1}{3y^2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\dfrac{1}{3\sqrt[3]{x^2}}$&lt;/span&gt; are indeed the same expression, provided the explicit representation &lt;span class=&quot;math-container&quot;&gt;$y=\sqrt[3]{x}$&lt;/span&gt; of the function.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Another example&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$e^{y}-y=x$&lt;/span&gt; is an expression where there's no way to find an explicit representation. Nevertheless, you can assume that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and use implicit differentiation to find the derivative of the function:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{d(e^{y})}{dx} - \dfrac{d(y)}{dx} = \dfrac{d(x)}{x}$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{d(e^y)}{dy}\cdot \dfrac{dy}{dx} - \dfrac{dy}{dx} = 1 $$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{dy}{dx} \left( e^y - 1\right) = 1 \implies \dfrac{dy}{dx} = \dfrac{1}{e^y-1}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/K7bfP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/K7bfP.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As you can see, in the relation &lt;span class=&quot;math-container&quot;&gt;$e^{y}-y=x$&lt;/span&gt; there are two ways (or &amp;quot;branches&amp;quot;) to define &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. The domain of the function is &lt;span class=&quot;math-container&quot;&gt;$[1,+\infty[$&lt;/span&gt; and the range for the possitive branch is &lt;span class=&quot;math-container&quot;&gt;$[0,+\infty[$&lt;/span&gt;. This function is differentiable in &lt;span class=&quot;math-container&quot;&gt;$]1,+\infty[$&lt;/span&gt;, because for &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt; and therefore, the derivative is ill-defined. You can also deduce some other properties of the function. For example, since &lt;span class=&quot;math-container&quot;&gt;$y&amp;gt;0 \implies e^y&amp;gt;1$&lt;/span&gt;, the derivative is always possitive and therefore, the possitive branch is a stricly increasing function.&lt;/p&gt;&#xA;&lt;p&gt;I hope this helps for the explanation.&lt;/p&gt;&#xA;" OwnerUserId="3380" LastActivityDate="2021-09-26T23:14:13.453" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24409" PostTypeId="2" ParentId="24407" CreationDate="2021-09-27T01:18:09.377" Score="2" Body="&lt;p&gt;I haven't taught calculus in a while and looking back, implicit differentiation is one of the (many) topics that I think I could have taught better. If I was teaching it now I would emphasize that the derivative is a local property, and show how by zooming in on a point on the graph we can find a region where &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (except when we can't, and maybe the students can see when that happens). Let's call this function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then, using your example, I would rewrite &lt;span class=&quot;math-container&quot;&gt;$x+xy+y^2=7$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x+xg(x)+g(x)^2=7$&lt;/span&gt;, and emphasize that it holds &lt;em&gt;locally&lt;/em&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$x+xg(x)+g(x)^2$&lt;/span&gt; is a constant function in our little region, its derivative is zero there. Now we just use basic differentiation rules to get an expression for &lt;span class=&quot;math-container&quot;&gt;$g'$&lt;/span&gt;, and thus the tangent line.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastEditorUserId="12671" LastEditDate="2021-10-03T20:16:20.390" LastActivityDate="2021-10-03T20:16:20.390" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24410" PostTypeId="1" CreationDate="2021-09-27T07:01:29.893" Score="4" ViewCount="283" Body="&lt;p&gt;After students learn how to use probabilistic simulations, what strategies can one use to encourage them to understand analytical results anyway? For example, I'm struggling to find a compelling reason/example to teach about the maths behind expected values when they can just run a simple simulation. I'm talking about students interested in applying probability, not students who appreciate pure maths.&lt;/p&gt;&#xA;" OwnerUserId="7936" LastActivityDate="2022-04-12T15:26:16.893" Title="Probability — analytical results instead of simulations" Tags="|examples|concept-motivation|probability|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24411" PostTypeId="2" ParentId="24407" CreationDate="2021-09-27T13:42:58.507" Score="0" Body="&lt;p&gt;I introduce implicit differentiation in two parts:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How to differentiate dependent variables (i.e. &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx} y = \frac{dy}{dx}$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;li&gt;If two functions are equal not just at a point, their derivatives are equal&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And then go through a bunch of examples including the squircle &lt;span class=&quot;math-container&quot;&gt;$x^4+y^4=1$&lt;/span&gt; and plenty of just &amp;quot;find &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx} \left(xy + x^2 + y^2\right)$&lt;/span&gt;&amp;quot;-type problems.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-09-27T13:42:58.507" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24412" PostTypeId="2" ParentId="24407" CreationDate="2021-09-27T16:46:46.740" Score="2" Body="&lt;p&gt;I think it is important to constantly connect implicit differentiation with the chain rule. At first, there might be examples where a third variable is the independent variable, and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are dependent variables. For example, find &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$x(t)^2+y(t)^2=1$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$&lt;/span&gt;. In this case, the coordinate variables are of the same status: dependent variables. Somehow, putting the coordinate variables on this same status makes it more clear that the chain rule is in play. To finish, we do have to employ the chain rule again, in the form &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then the next step is the situation were &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the independent variable, and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; depends on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. So in effect, &lt;span class=&quot;math-container&quot;&gt;$x=t$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=y(t)$&lt;/span&gt; but we might as well drop &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; and think &lt;span class=&quot;math-container&quot;&gt;$x=x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y(x)$&lt;/span&gt;, and consider finding &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2=1$&lt;/span&gt;, but reinforce the notation &lt;span class=&quot;math-container&quot;&gt;$x^2+y(x)^2=1$&lt;/span&gt;. Students will be annoyed at having to write &lt;span class=&quot;math-container&quot;&gt;$x^2+y(x)^2=1$&lt;/span&gt; but this is good medicine to enforce at first.&lt;/p&gt;&#xA;&lt;p&gt;Eventually, students learn to &lt;em&gt;first&lt;/em&gt; think about what variables are dependent and which is the independent variable, before jumping into the mechanics of differentiation rules. Only then can they be freed from the instructor's requirement of writing the explicit dependency such as &lt;span class=&quot;math-container&quot;&gt;$x(t)^2+y(t)^2=1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x^2+y(x)^2=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When I grade a problem like this on an exam, students know that they will at least get partial credit if they write the equation with the correct explicit dependency.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-09-28T15:00:19.397" LastActivityDate="2021-09-28T15:00:19.397" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24413" PostTypeId="2" ParentId="24410" CreationDate="2021-09-29T16:10:57.747" Score="7" Body="&lt;p&gt;Your question could apply generally to why should anyone learn the math &amp;quot;behind&amp;quot; anything, if they can easily compute the answer on a computer. I don't think they ALWAYS should. There should be a reason. For example, I had an old professor who said that when calculators first became common, some professors insisted that students should still learn to take square roots (of non square integers) by hand, in case all of the batteries died. I doubt that would be a popular opinion today.&lt;/p&gt;&#xA;&lt;p&gt;The main reasons (not exhaustive) that I think that someone should learn the math &amp;quot;behind&amp;quot; anything is if:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;That helps them to understand the concept better&lt;/li&gt;&#xA;&lt;li&gt;That helps them build toward more advanced material that they will study later&lt;/li&gt;&#xA;&lt;li&gt;They will face situations in which they cannot use a computer&lt;/li&gt;&#xA;&lt;li&gt;They will face situations in which doing it without a computer is more efficient (e.g. faster)&lt;/li&gt;&#xA;&lt;li&gt;That helps them apply a sanity check on the results returned by the computer&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If someone is already in their chosen profession, and will not be studying anything else later, perhaps #2 doesn't apply. But, if they want to eventually model disease transmission using stochastic differential equations, then that could entail computing the expected value of quantities based on equations (not specific values of the probabilities), and hence a computer simulation probably cannot be used to determine the general solution (only solutions for particular values of the variables). That may not be the quality of analysis that could be published in medical journals.&lt;/p&gt;&#xA;&lt;p&gt;I've had students in college algebra classes make mistakes on a calculator resulting in answers that are obviously incorrect, and when I ask, &amp;quot;Does that answer make sense?&amp;quot;, they just hold up the calculator toward me. In general it can help to have some understanding of what you are doing, rather than having complete faith in the computer/calculator (and your ability to enter the correct input).&lt;/p&gt;&#xA;&lt;p&gt;But if someone is paying to take a college course for a particular purpose, and that purpose is satisfied by learning to use simulations to compute quantities, great.&lt;/p&gt;&#xA;" OwnerUserId="12506" LastActivityDate="2021-09-29T16:10:57.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24414" PostTypeId="2" ParentId="24410" CreationDate="2021-09-29T20:24:04.857" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;they can just run a simple simulation&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Simulations outside the classroom are often anything but simple. Brute force simulations are often a lot of work to set up, take a long time to calculate, and take even more computational resources to make sufficiently accurate. It is therefore necessary to have more advance tools in one's toolbox.&lt;/p&gt;&#xA;&lt;p&gt;Here is a very nice worked example of &lt;a href=&quot;http://jakevdp.github.io/blog/2017/12/18/simulating-chutes-and-ladders/&quot; rel=&quot;noreferrer&quot;&gt;Chutes and Ladders&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14091" LastActivityDate="2021-09-29T20:24:04.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24415" PostTypeId="2" ParentId="24410" CreationDate="2021-09-29T21:39:16.210" Score="2" Body="&lt;p&gt;Your question prompted me to reread a comment that John Tukey wrote in the March 1979 edition of the &lt;em&gt;Journal of the American Statistical Association&lt;/em&gt; (pp. 121-122).&lt;/p&gt;&#xA;&lt;p&gt;He referred to what he calls &amp;quot;the whole data analyst-statistician.&amp;quot; Such a person is able to &amp;quot;take quite different views and adopt quite different styles as the needs change.&amp;quot; Evidently, you want your students who are working adults to grow in these terms.&lt;/p&gt;&#xA;&lt;p&gt;Tukey goes on to describe probability modelers as wanting to believe that their models are entirely correct, and data analysts as regarding their models as &amp;quot;benchmarks in a wilderness, and expecting little truth.&amp;quot; These are two opposite ends of a spectrum. He refers to the development of &amp;quot;robust/resistant techniques&amp;quot; that will bring these two together. His &amp;quot;whole data analyst-statistician&amp;quot; is an interdisciplinary thinker who can bring these perspectives of models together. A person who can go back and forth between these two ends of the spectrum is someone who appreciates the meaning of the famous statement by George Box, who was a colleague of John Tukey: &amp;quot;All models are wrong, but some are useful.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Your question about bringing students to appreciate both analytic results and simulations leads us naturally to the philosophical thinkings of these two giants of statisticians (Tukey and Box). One idea for inspiring motivated undergraduates is to have them read the first seven pages of Tukey's 1962 paper &amp;quot;The Future of Data Analysis.&amp;quot; I find Tukey's opening sentence to be both chilling and relevant to your predicament: &lt;em&gt;For a long time I have thought I was a statistician, interested in inferences from the particular to the general. But as I have watched mathematical statistics evolve, I have had cause to wonder and to doubt.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="12506" LastEditDate="2022-04-12T15:26:16.893" LastActivityDate="2022-04-12T15:26:16.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24416" PostTypeId="1" CreationDate="2021-10-01T12:15:05.493" Score="2" ViewCount="4032" Body="&lt;p&gt;Could anyone familiar with Singapore please tell me what is the name of a common textbook used for Math in Singapore's secondary schools? I am looking for topics covered for ages 14-18. I am very interested in what I have heard about Singapore's math education, so it's mostly to see their approach since I also do not like the vast majority of American math textbooks.&lt;/p&gt;&#xA;" OwnerUserId="15988" LastEditorUserId="2083" LastEditDate="2021-10-01T13:30:39.647" LastActivityDate="2022-03-11T17:37:30.407" Title="Singapore Math textbooks for secondary/&quot;high school&quot;?" Tags="|textbooks|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24417" PostTypeId="2" ParentId="24416" CreationDate="2021-10-01T17:47:56.903" Score="8" Body="&lt;p&gt;I'm only moderately familiar with Singapore, but here is what I know.&lt;/p&gt;&#xA;&lt;p&gt;There was an older textbook series for ages 12 to 16 called &amp;quot;New Elementary Mathematics: Syllabus D&amp;quot; Books 1 to 4. I doubt all schools used it, but I think it was from the same publishing company that put out the elementary-level books that were eventually sold in the U.S. I'm not sure what they use now.&lt;/p&gt;&#xA;&lt;p&gt;The series was formerly sold in the United States. Here is a review by, I assume, a homeschooling U.S. parent: &lt;a href=&quot;https://cathyduffyreviews.com/homeschool-reviews-core-curricula/math/math-grades-7-8/new-elementary-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cathyduffyreviews.com/homeschool-reviews-core-curricula/math/math-grades-7-8/new-elementary-mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you do a search for &amp;quot;Ordinary Level Mathematics&amp;quot; (ages 12-16), &amp;quot;Additional Mathematics&amp;quot; (ages 14-16) and &amp;quot;Advanced Level Mathematics&amp;quot; (ages 16-18), you'll turn up some materials used now.&lt;/p&gt;&#xA;&lt;p&gt;See here for the official curricula at O-Level: &lt;a href=&quot;https://en.wikipedia.org/wiki/Singapore-Cambridge_GCE_Ordinary_Level&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Singapore-Cambridge_GCE_Ordinary_Level&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For A-Level, you ought to be able to find syllabi by searching for H1 Mathematics, H2 Mathematics, H2 Further Mathematics and H3 Mathematics. According to this thread, textbooks aren't widely used at this level, but there are textbooks for similar exams offered by &amp;quot;Cambridge International&amp;quot; up to the &amp;quot;Further Maths&amp;quot; level: &lt;a href=&quot;https://www.reddit.com/r/SGExams/comments/jzz91o/a_levels_h2_maths_and_further_maths_textbook/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.reddit.com/r/SGExams/comments/jzz91o/a_levels_h2_maths_and_further_maths_textbook/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The entire Singapore curriculum at the secondary level is modelled on older curricula in use in England (probably about pre-1985). So if you want more textbooks like that, you can look at the books in use in England in that period. They were very different from the ones used now because at that time the top 25% had their own curricula and were sometimes educated in separate schools. Beware of pre-1970 books, though, as they use Imperial units and non-decimal (!) currency.&lt;/p&gt;&#xA;&lt;p&gt;If you want to see what was typical in the early 1980s for ages 11 to 15 or 16 in England (in the top 25%), there are some books on the Internet Archive called &amp;quot;New General Mathematics for West Africa&amp;quot; by Channon and McLeish Smith. These are basically the same as the British textbooks but with &amp;quot;London&amp;quot; replaced with &amp;quot;Lagos&amp;quot; and pounds with local currencies, etc. (I'm not saying these are the best textbooks, but at least you can look at them.) For &amp;quot;Additional Mathematics&amp;quot; for advanced students at ages 15-16, see &amp;quot;Pure Mathematics: A First Course&amp;quot; by Backhouse, also on the Internet Archive. This was also a first-year book at Advanced Level (age 16-17) for those who hadn't taken Additional Mathematics. I believe all of this is quite similar in approach to the Singapore curriculum.&lt;/p&gt;&#xA;&lt;p&gt;The main difference between the Singapore syllabus and most English syllabi about pre-1980 appears to be that some pure geometry was taken out and some vectors and transformations added in.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: Here are some links to official syllabi and additional information provided to me in the comments.&lt;/p&gt;&#xA;&lt;p&gt;O-Level:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/olevel/2023syllabus/4052_y23_sy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2023 syllabus for O-Level Mathematics&lt;/a&gt; (Pure Mathematics, Probability &amp;amp; Statistics)&lt;br&gt;&#xA;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/olevel/2023syllabus/4052_y23_sp_1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Specimen Paper 1 for O-Level Mathematics&lt;/a&gt;&lt;br&gt;&#xA;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/olevel/2023syllabus/4052_y23_sp_2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Specimen Paper 2 for O-Level Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/olevel/2023syllabus/4049_y23_sy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2023 syllabus for O-Level Additional Mathematics&lt;/a&gt; (Pure Mathematics)&lt;br&gt;&#xA;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/olevel/2022syllabus/4049_y21_sp_1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Specimen Paper 1 for O-Level Additional Mathematics&lt;/a&gt;&lt;br&gt;&#xA;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/olevel/2022syllabus/4049_y21_sp_2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Specimen Paper 2 for O-Level Additional Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A-Level:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/alevel/2023syllabus/9758_y23_sy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2023 syllabus for H2 Mathematics&lt;/a&gt; (Pure Mathematics : Probability &amp;amp; Statistics = 7:3)&lt;br&gt;&#xA;with Additional Mathematics as prerequisite&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/alevel/2023syllabus/9649_y23_sy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2023 syllabus for H2 Further Mathematics&lt;/a&gt; (Pure Mathematics : Probability &amp;amp; Statistics = 3:1)&lt;br&gt;&#xA;with H2 Mathematics as corequisite&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/alevel/2023syllabus/9820_y23_sy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2023 syllabus for H3 Mathematics&lt;/a&gt; (Pure Mathematics)&lt;br&gt;&#xA;with H2 Mathematics as corequisite&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/alevel/2023syllabus/List_MF26_y23_sy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;List of formulae for A-Level Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/alevel/2023syllabus/8865_y23_sy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2023 syllabus for H1 Mathematics&lt;/a&gt; (Pure Mathematics : Probability &amp;amp; Statistics = 4:6)&lt;br&gt;&#xA;restricted to non-H2 candidates; a proper subset of Additional Mathematics and H2 Mathematics&lt;/p&gt;&#xA;&lt;p&gt;The O-level (Elementary) Mathematics course takes place over four years (13-16yo), and the Additional Mathematics alongside it over the final 2 years (15-16yo); they are designed to occur in parallel, so the texts don't overlap, although some pre-requisite stuff from EM does get reviewed at the beginning of the relevant AM chapters, and complementary topics across the two subjects occur alongside each other.&lt;/p&gt;&#xA;&lt;p&gt;About half of all 16-year-olds take Additional Mathematics, a minority which will go on (17-18yo) to take H1 Mathematics instead of H2 Mathematics, which will have been taken by half of all the college students (most of the STEM ones and some of the non-STEM ones). H2 Further Mathematics is a very unpopular course; almost nobody takes H3 Mathematics.&lt;/p&gt;&#xA;&lt;p&gt;About 30% of H2 Mathematics overlaps (at different depths) with Additional Mathematics, and about 60% of the latter overlaps with the former.&lt;/p&gt;&#xA;" OwnerUserId="17833" LastEditorUserId="372" LastEditDate="2022-03-11T17:37:30.407" LastActivityDate="2022-03-11T17:37:30.407" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24418" PostTypeId="1" AcceptedAnswerId="24419" CreationDate="2021-10-01T18:17:59.517" Score="13" ViewCount="1289" Body="&lt;p&gt;A &amp;quot;growth mindset&amp;quot; can be defined this way, see &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3514770/&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In a fixed mindset students believe their basic abilities, their intelligence, their talents, are just fixed traits. They have a certain amount and that's that, and then their goal becomes to look smart all the time and never look dumb.&lt;/p&gt;&#xA;&lt;p&gt;In a growth mindset students understand that their talents and abilities can be developed through effort, good teaching and persistence. They don't necessarily think everyone's the same or anyone can be Einstein, but they believe everyone can get smarter if they work at it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Are there any studies that attempt to show a measurable effect of a student possessing a growth mindset? What are the implications of these studies for math educators?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2022-06-15T02:14:22.037" Title="Are there research studies that attempt to determine the value of a &quot;Growth Mindset?&quot;" Tags="|reference-request|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24419" PostTypeId="2" ParentId="24418" CreationDate="2021-10-02T04:36:55.997" Score="22" Body="&lt;p&gt;Psychologist Carol Dweck's &amp;quot;growth mindset&amp;quot; theory has become a popular solution and intervention technique in (mostly American) schools of all ages. We might say that it's become the new version of the &amp;quot;self-esteem&amp;quot; movement seen in the 80's. While Dweck first developed the theory in the 90's, it's really taken hold of popular consciousness from the 2010's on.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, we should remember that psychology has an ongoing &lt;a href=&quot;https://en.wikipedia.org/wiki/Replication_crisis&quot; rel=&quot;noreferrer&quot;&gt;replication crisis&lt;/a&gt; in many of its landmark findings. Many of the &amp;quot;easy&amp;quot; ideas for transformative effects have not borne fruit over the years, and been later found to have tainted methods by core researchers. Sure enough, in recent years many or most of the large-scale, high-quality attempts at replicating the claims of growth mindset have failed to so. Here are a few examples:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Li, Y., &amp;amp; Bates, T. C., Ph.D. (2017)&lt;/strong&gt;. &lt;em&gt;Does growth mindset improve children’s IQ, educational attainment or response to setbacks? Active-control interventions and data on children’s own mindsets.&lt;/em&gt; &lt;a href=&quot;https://doi.org/10.31235/osf.io/tsdwy&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.31235/osf.io/tsdwy&lt;/a&gt; (Study done in China, students aged 9-13 years, N = 624).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;No effect of the classic growth mindset manipulation was found for&#xA;either moderate or more difficult material... children’s mindsets were&#xA;unrelated to resilience to failure for either outcome measure...&#xA;Finally, in 2 studies relating mindset to grades across a semester in&#xA;school, the predicted association of growth mindset with improved&#xA;grades was not supported. Neither was there any association of&#xA;children’s mindsets with their grades at the start of the semester.&#xA;Beliefs about the malleability of basic ability may not be related to&#xA;resilience to failure or progress in school.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Bahník, Štěpán, and Marek A. Vranka (2017).&lt;/strong&gt; &lt;em&gt;Growth mindset is not associated with scholastic aptitude in a large sample of university applicants.&lt;/em&gt; Personality and Individual Differences 117: 139-143. &lt;a href=&quot;https://doi.org/10.1016/j.paid.2017.05.046&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1016/j.paid.2017.05.046&lt;/a&gt; (Study of university students taking an admissions test in the Czech Republic, N = 5653).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We found that results in the test were slightly negatively associated&#xA;with growth mindset (r = −0.03). Mindset showed no relationship with&#xA;the number of test administrations participants signed up for and it&#xA;did not predict change in the test results. The results show that the&#xA;strength of the association between academic achievement and mindset&#xA;might be weaker than previously thought.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Foliano, F., Rolfe, H., Buzzeo, J., Runge, J., &amp;amp; Wilkinson, D. (2019).&lt;/strong&gt; &lt;em&gt;Changing mindsets: Effectiveness trial.&lt;/em&gt; National Institute of Economic and Social Research. &lt;a href=&quot;http://psychbrief.com/growth-mindset-fails/&quot; rel=&quot;noreferrer&quot;&gt;Summary at PsychBrief&lt;/a&gt;. (Study in England, Year 6 students, N = 4584).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The difference between the control group and the intervention group on&#xA;all 3 primary outcomes [math, reading, GPS] was 0... The difference between the groups for all 4 secondary&#xA;outcomes was also 0... This RCT was a highly powered test of the&#xA;efficacy of growth mindset in a real-world environment across a wide&#xA;range of schools in the England. The fact none of the primary or&#xA;secondary outcomes were distinguishable from 0 raises serious&#xA;questions as to the efficacy of growth mindset for Year 6 students...&#xA;Given the evidence so far, it is unrealistic to expect growth mindset&#xA;to have large and/or wide-scale impact.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Caitlin Brez, Eric M. Hampton, Linda Behrendt, Liz Brown &amp;amp; Josh Powers (2020)&lt;/strong&gt; &lt;em&gt;Failure to Replicate: Testing a Growth Mindset Intervention for College Student Success&lt;/em&gt;, Basic and Applied Social Psychology, 42:6, 460-468, &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/01973533.2020.1806845&quot; rel=&quot;noreferrer&quot;&gt;DOI: 10.1080/01973533.2020.1806845&lt;/a&gt; (U.S. study, university math &amp;amp; psychology students, N = 2607).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The pattern of findings is clear that the intervention had little&#xA;impact on students’ academic success even among sub-samples of&#xA;students who are traditionally assumed to benefit from this type of&#xA;intervention (e.g., minority, low income, and first-generation&#xA;students)... These findings support some of the emerging literature that&#xA;demonstrates that growth mindset interventions may not be as effective&#xA;as once thought... The proposition that a one-time intervention at the&#xA;postsecondary level will result in long-term measurable student&#xA;outcomes was not supported in the present study.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Now, a not-uncommon defense in a number of these cases in psychology is that the attempts to replicate didn't properly recreate the conditions or variables for a true test. The counter-argument here would be the &lt;a href=&quot;https://en.wikipedia.org/wiki/Observer-expectancy_effect&quot; rel=&quot;noreferrer&quot;&gt;observer-expectancy effect&lt;/a&gt; -- in some cases a primary researcher has even argued that &lt;em&gt;only they&lt;/em&gt; have the necessary knowledge to ever do so. Indeed, Dweck has made the &amp;quot;not anyone can do a replication&amp;quot; argument (&lt;a href=&quot;https://www.buzzfeed.com/tomchivers/what-is-your-mindset&quot; rel=&quot;noreferrer&quot;&gt;BuzzFeed News interview&lt;/a&gt;). In response, Nick Brown, who developed the &lt;a href=&quot;https://en.wikipedia.org/wiki/GRIM_test&quot; rel=&quot;noreferrer&quot;&gt;GRIM&lt;/a&gt; (Granularity-Related Inconsistency of Means) test and found several errors in Dweck's seminal paper, said this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The question I have is: If your effect is so fragile that it can only&#xA;be reproduced [under strictly controlled conditions], then why do you&#xA;think it can be reproduced by schoolteachers?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Finally, psychologist Russell Warne &lt;a href=&quot;https://russellwarne.com/2020/01/03/the-one-variable-that-makes-growth-mindset-interventions-work/&quot; rel=&quot;noreferrer&quot;&gt;wrote on his blog&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I discovered the one characteristic that the studies that support&#xA;mindset theory share and that all the studies that contradict the&#xA;theory lack: Carol Dweck... So, there you go! Growth mindsets can&#xA;improve academic performance  –if you have Carol Dweck in charge of&#xA;your intervention.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is somewhat hyperbolic, but clarifies the issue at stake. Growth mindset theory fits fairly snugly into the basket of psychological &amp;quot;quick fixes&amp;quot; that make up the replication crisis, broadly cuts against long-standing findings from neuroscience on intelligence, and is racking up more failures-to-replicate as it garners more attention. Like other similar principles that came before, it's probably a bad bet that institutional interventions based on the theory will be worth the resources spent on them.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2022-06-15T02:14:22.037" LastActivityDate="2022-06-15T02:14:22.037" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24420" PostTypeId="1" CreationDate="2021-10-02T09:18:46.407" Score="19" ViewCount="3282" Body="&lt;p&gt;I cannot recall ever hearing the terms &amp;quot;improper fraction&amp;quot; and &amp;quot;proper fraction&amp;quot; outside of an elementary and middle school setting. At some point in my mathematics education people began to simply say &amp;quot;fraction&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Has there been any research into the benefits of differentiating between these two concepts?&lt;/p&gt;&#xA;&lt;p&gt;What is the rationale for this differentiation?&lt;/p&gt;&#xA;" OwnerUserId="6527" LastEditorUserId="2083" LastEditDate="2021-10-02T09:56:52.597" LastActivityDate="2023-03-26T13:12:06.130" Title="What is the rationale for distinguishing between proper and improper fractions?" Tags="|secondary-education|reference-request|primary-education|fractions|" AnswerCount="9" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="24421" PostTypeId="2" ParentId="24420" CreationDate="2021-10-02T12:41:10.833" Score="25" Body="&lt;p&gt;&lt;strong&gt;added Oct 6&lt;/strong&gt;&lt;br /&gt;&#xA;The reason mixed numbers are found in US education is that mixed numbers are found outside of school in the US, so the children need to learn to understand them.  Mixed numbers are found in road signs, cooking recipes, length measurements, and so on.  (Denis Nardin commented that mixed numbers are never seen in Italy.  Meters, centimeters, and millimeters easily become decimals; miles, yards, feet, and inches do not.) (But see the Hong Kong roadsign below, with ½ km.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Also no research, just an explanation.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In order for the general public to understand you, you cannot say&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\text{Add }\frac{4}{3}\text{ liters of water.}&#xA;\tag{1}$$&lt;/span&gt;&#xA;instead, you must say&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\text{Add }\; 1\,\frac{1}{3}\;\text{ liters of water.}&#xA;\tag2$$&lt;/span&gt;&#xA;Thus, gradeschool kids need to learn how to get their answer in the form &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt;.  To teach this, there must be terminology for these two forms of the answer, such as &amp;quot;improper fraction&amp;quot; and &amp;quot;mixed number&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GEjkf.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GEjkf.jpg&quot; alt=&quot;measuring cups&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/lJzr9.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lJzr9.jpg&quot; alt=&quot;banana bread&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/vVkVE.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vVkVE.jpg&quot; alt=&quot;St Ives&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/cvlOG.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cvlOG.jpg&quot; alt=&quot;Hong Kong&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/h57SO.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/h57SO.jpg&quot; alt=&quot;speed&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Can you imagine a road sign &amp;quot;Fenstanton &lt;span class=&quot;math-container&quot;&gt;$\frac{9}{4}$&lt;/span&gt;&amp;quot; ?&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2023-03-26T13:12:06.130" LastActivityDate="2023-03-26T13:12:06.130" CommentCount="21" ContentLicense="CC BY-SA 4.0" />
  <row Id="24422" PostTypeId="2" ParentId="24420" CreationDate="2021-10-02T12:48:34.077" Score="11" Body="&lt;p&gt;I do not know of any relevant research.&lt;/p&gt;&#xA;&lt;p&gt;Here are my own not-research-informed ideas.&lt;/p&gt;&#xA;&lt;p&gt;Most people refer to fractions as parts of a whole.  If someone says &amp;quot;I lost a fraction of a pound on my diet&amp;quot;, you can be fairly certain that they didn't lose &lt;span class=&quot;math-container&quot;&gt;$\frac{23}{1}$&lt;/span&gt; pounds.&lt;/p&gt;&#xA;&lt;p&gt;Since the common usage of the word and the mathematical usage differ, it is useful to draw attention to the fact that fractions can be larger than one by giving them a special name.  I think the word &amp;quot;improper fraction&amp;quot; is a bit unfortunate because it carries the implication that such fractions are undesirable.&lt;/p&gt;&#xA;&lt;p&gt;This is a little bit similar to the word &amp;quot;or&amp;quot;.  In English the word can be used in at least two ways.  In mathematics, when we need to be precise, we invent the terms &amp;quot;inclusive or&amp;quot; and &amp;quot;exclusive or&amp;quot; to make the distinction.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="5597" LastEditDate="2021-10-02T19:24:02.270" LastActivityDate="2021-10-02T19:24:02.270" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24423" PostTypeId="2" ParentId="24418" CreationDate="2021-10-02T12:57:10.637" Score="4" Body="&lt;p&gt;Yeager and Walton's Social-Psychological Interventions in Education: They're Not Magic &lt;a href=&quot;https://www.researchgate.net/publication/258182797_Social-Psychological_Interventions_in_Education_They%27re_Not_Magic&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.researchgate.net/publication/258182797_Social-Psychological_Interventions_in_Education_They%27re_Not_Magic&lt;/a&gt; is a good &amp;quot;meta-study&amp;quot; for growth mindset and similar interventions. It discusses the psychology behind these types of interventions: how students who struggle academically may feel and the feedback loops that keep them struggling academically. It also explains ways in which social-psychological interventions may backfire, which can explain the replication issue in Daniel R. Collins' answer.&lt;/p&gt;&#xA;&lt;p&gt;A specific study to add to your list is Niiya, Crocker, and Bartmess' &amp;quot;From vulnerability to resilience: learning orientations buffer contingent self-esteem from failure&amp;quot; &lt;a href=&quot;https://pubmed.ncbi.nlm.nih.gov/15563324/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://pubmed.ncbi.nlm.nih.gov/15563324/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In terms of implications, Joe Feldman's Grading for Equity describes how a traditional grading system reinforces a fixed mindset and a retakes-based system can support a growth mindset.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-10-02T12:57:10.637" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24424" PostTypeId="2" ParentId="24420" CreationDate="2021-10-02T17:14:38.050" Score="8" Body="&lt;p&gt;Student are introduced to fractions as part of a whole. They are then taught that improper fractions can be more than a whole - this is not ideal terminology or helpful for understanding. Improper fraction is a terrible name since it implies that there is something wrong with the fraction.&lt;/p&gt;&#xA;&lt;p&gt;Once student start to do calculations with fractions greater than one, the distinction helps make calculations easier.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When multiplying and dividing you would convert mixed numbers to &amp;quot;improper fractions&amp;quot; to make the calculation easier.&lt;/li&gt;&#xA;&lt;li&gt;When adding and subtracting you would work with mixed numbers and not improper fractions.&lt;/li&gt;&#xA;&lt;li&gt;Converting between mixed numbers and improper fractions (which is necessary in many different problems) can strengthen understanding of fractions if taught properly.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The distinction is important at the elementary school level.  From the comment below I see it is important at the high school level too.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2021-10-02T17:58:12.990" LastActivityDate="2021-10-02T17:58:12.990" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24425" PostTypeId="1" AcceptedAnswerId="24428" CreationDate="2021-10-04T01:47:46.413" Score="1" ViewCount="5946" Body="&lt;p&gt;A basic terminology question for a foreign speaker.&lt;/p&gt;&#xA;&lt;p&gt;Please correct me if wrong. (Let's ignore the commutative property of + and * here)&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;5 add by 10 is 5 + 10&#xA;5 add 10 is 10 + 5&#xA;5 subtract by 10 is 5 - 10&#xA;5 subtract 10 is 10 - 5 &#xA;5 multiply by 10 is 5 * 10&#xA;5 multiply 10 is 10 * 5&#xA;5 times 10 is 5 * 10&#xA;5 divide by 10 is 5 / 10&#xA;5 divide 10 is 10 / 5&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Also, do we only say 'plus' and 'minus', never 'plus by' or 'minus by'?&lt;/p&gt;&#xA;" OwnerUserId="17837" LastActivityDate="2021-10-04T06:50:29.573" Title="Is there a difference between 'subtract' and 'subtract by'?" Tags="|terminology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24426" PostTypeId="2" ParentId="24425" CreationDate="2021-10-04T02:28:18.437" Score="4" Body="&lt;p&gt;This should probably be moved to math.SE, since I don't see anything here about teaching. But almost every line you wrote is incorrect English.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Add by&amp;quot; is gramatically incorrect. It should be &amp;quot;plus.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;5 add 10&amp;quot; should be &amp;quot;5 plus 10&amp;quot; (or &amp;quot;5 and 10&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;subtract by -&amp;gt; minus&lt;/p&gt;&#xA;&lt;p&gt;5 subtract 10 -&amp;gt; 5 subtracted from 10 (if you want it to mean 10-5)&lt;/p&gt;&#xA;&lt;p&gt;5 multiply by 10 -&amp;gt; 5 multiplied by 10&lt;/p&gt;&#xA;&lt;p&gt;5 divide by 10 -&amp;gt; 5 divided by 10&lt;/p&gt;&#xA;&lt;p&gt;5 divide 10 is incorrect English.&lt;/p&gt;&#xA;" OwnerUserId="17838" LastActivityDate="2021-10-04T02:28:18.437" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24427" PostTypeId="1" CreationDate="2021-10-04T04:21:54.810" Score="0" ViewCount="103" Body="&lt;p&gt;This is an example of the AC Method:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ x^2 + 16x +63 $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(1) &lt;span class=&quot;math-container&quot;&gt;$x² + 7x$&lt;/span&gt; (2) &lt;span class=&quot;math-container&quot;&gt;$9x + 63$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(1) &lt;span class=&quot;math-container&quot;&gt;$x(x + 7)$&lt;/span&gt; (2) &lt;span class=&quot;math-container&quot;&gt;$9(x + 7)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;so we have:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$x(x + 7)+ 9(x + 7)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(1) with (2) The Result is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ (x+9)(x+7) $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have more examples in letters:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/waCvR.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/waCvR.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Comparing for example with this list:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Factoring out the GCF.&lt;/li&gt;&#xA;&lt;li&gt;The sum-product pattern.&lt;/li&gt;&#xA;&lt;li&gt;The grouping method.&lt;/li&gt;&#xA;&lt;li&gt;The perfect square trinomial pattern.&lt;/li&gt;&#xA;&lt;li&gt;The difference of squares pattern.&lt;/li&gt;&#xA;&lt;li&gt;Newton-Girard formulae&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I don't know this Methods and is because I am not a native speaker maybe I know all of them just the language barrier...&lt;/p&gt;&#xA;" OwnerUserId="13238" LastEditorUserId="13238" LastEditDate="2021-10-04T07:08:40.367" LastActivityDate="2021-10-04T07:08:40.367" Title="Is the AC Method of Factoring polynomials more popular and used by teachers than others methods of factoring polynomials?" Tags="|algebra|curriculum|tutoring|precalculus|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24428" PostTypeId="2" ParentId="24425" CreationDate="2021-10-04T06:50:29.573" Score="8" Body="&lt;p&gt;We don't say &amp;quot;subtract by&amp;quot; or &amp;quot;add by&amp;quot;.  'by' is used with the verbs &lt;em&gt;multiply&lt;/em&gt; and &lt;em&gt;divide&lt;/em&gt; in sentences like:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Divide 6 by 3 -&amp;gt; 6/3&lt;/li&gt;&#xA;&lt;li&gt;6 divided by 3 -&amp;gt; 6/3&lt;/li&gt;&#xA;&lt;li&gt;multiply 5 by 4 -&amp;gt; 5*4&lt;/li&gt;&#xA;&lt;li&gt;5 multiplied by 4 -&amp;gt; 5*4&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;On the other hand, we can use 'to' and 'from' with the verbs &lt;em&gt;add&lt;/em&gt; and &lt;em&gt;subtract&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Subtract 5 from 10 -&amp;gt; 10-5&lt;/li&gt;&#xA;&lt;li&gt;Add 5 to 10 -&amp;gt; 10+5&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The above ways of giving the expressions are verb-based.  We can also use other words to make them 'operator-based', where there is no verb, but an English word to replace the operator:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;6 divided by 3 -&amp;gt; 6/3&lt;/li&gt;&#xA;&lt;li&gt;5 times 4 -&amp;gt; 5*4&lt;/li&gt;&#xA;&lt;li&gt;10 minus 5, or 10 take away 5 -&amp;gt; 10-5&lt;/li&gt;&#xA;&lt;li&gt;10 plus 5 -&amp;gt; 10+5&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12571" LastActivityDate="2021-10-04T06:50:29.573" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24429" PostTypeId="2" ParentId="24420" CreationDate="2021-10-04T07:33:47.703" Score="7" Body="&lt;p&gt;The use of rulers with fractional inches is the first thing that springs to mind.&lt;/p&gt;&#xA;&lt;p&gt;Like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zIboC.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zIboC.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The four keys have a width of &lt;span class=&quot;math-container&quot;&gt;$2 \frac{11}{16}$&lt;/span&gt; inches at the tops of the key caps.&lt;/p&gt;&#xA;&lt;p&gt;If I calculated a length, and got &lt;span class=&quot;math-container&quot;&gt;$\frac{43}{16}$&lt;/span&gt; inches, I'd have to convert it to &lt;span class=&quot;math-container&quot;&gt;$2 \frac{11}{16}$&lt;/span&gt; to actually measure it.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The &amp;quot;improper fraction&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$\frac{43}{16}$&lt;/span&gt; is easier to use for further calculations.&lt;/li&gt;&#xA;&lt;li&gt;The &amp;quot;proper fraction&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$2 \frac{11}{16}$&lt;/span&gt; is easier to use in the real, physical world.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4800" LastEditorUserId="4800" LastEditDate="2021-10-04T08:30:16.397" LastActivityDate="2021-10-04T08:30:16.397" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24431" PostTypeId="1" CreationDate="2021-10-04T09:16:05.713" Score="0" ViewCount="123" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BZcg4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BZcg4.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are two theorems:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;All three angles AC_1B, AC_2B, AC_3B are equal. In general: All angles above a chord are equal.&lt;/li&gt;&#xA;&lt;li&gt;The size of any angle above a chord AB is half the size of angle AOB where O is the center of the circle.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I am Czech. In my native language there is a name of both these two statements separately.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;věta o obvodovém úhlu&lt;/li&gt;&#xA;&lt;li&gt;věta o středovém úhlu&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;What are these two theorems called in English?&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17840" LastEditorUserId="29" LastEditDate="2021-10-05T19:18:17.883" LastActivityDate="2021-10-05T19:18:17.883" Title="Names of two circle theorems in English" Tags="|geometry|" AnswerCount="1" CommentCount="0" ClosedDate="2021-10-10T19:55:22.797" ContentLicense="CC BY-SA 4.0" />
  <row Id="24432" PostTypeId="2" ParentId="24431" CreationDate="2021-10-04T10:47:34.163" Score="2" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Inscribed_angle#Theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;inscribed angle theorem&lt;/a&gt;.&#xA;Sometimes known as the &lt;a href=&quot;https://www.youtube.com/watch?v=WxxqPXXbDx8&amp;amp;ab_channel=JustinRyan&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Star Trek Lemma&lt;/em&gt;&lt;/a&gt;.&#xA;A generalization of &lt;a href=&quot;https://en.wikipedia.org/wiki/Thales%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thale's Theorem&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt; &lt;a href=&quot;https://i.stack.imgur.com/pNfS2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pNfS2.png&quot; alt=&quot;InscAngle&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;Wikipedia image&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-10-04T10:47:34.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24434" PostTypeId="2" ParentId="24418" CreationDate="2021-10-04T12:25:42.533" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8299535/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yeagar and Dweck 2021&lt;/a&gt; take a look at various criticisms (&amp;quot;controversies&amp;quot;) of mindset theory and argue for why they are flawed. Dweck is a central actor behind the theory, so one can doubt the neutrality of the arguments. I am not quite inside enough of this research to comment on whether the arguments in the article hold up to critical scrutiny.&lt;/p&gt;&#xA;&lt;p&gt;The article also provides an answer to the question posed here; it cites several positive results on mindset theory and discusses how to understand them and what limitations they have.&lt;/p&gt;&#xA;&lt;p&gt;The central issues or arguments in the paper are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Predictive power of mindsets on student outcomes. The authors claim that there is heterogeneity (different students respond to mindset in different ways), but overall these is a weak positive correlation between mindset and achievement in national-level studies and often stronger effects with students who struggle academically.&lt;/li&gt;&#xA;&lt;li&gt;Student interventions work, provided they are proper ones. There is some work on what a proper intervention is like. There is a pre-registered USA-level experiment, also analyzed by external people, that verified that for students of lower achievements there was significant improvement. (But what about higher achievement?) There is more reference to heterogeneity.&lt;/li&gt;&#xA;&lt;li&gt;Effect sizes of around 0,2 standard deviations in weaker schools and 0,1 among weaker students have been observed in pre-registered studies, according to the article. There is also an argument for why such effect sizes are significant.&lt;/li&gt;&#xA;&lt;li&gt;Teacher-targeted mindset interventions have not been successful thus far. The authors argue that this is because they are an emerging field.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Summarizing, the article points out several pre-registered studies or studies based on national datasets that suggest effects at least among weaker students. The article claims heterogeneity as the main culprit for failed results in some studies. I did not check the sample groups of the pre-registered studies.&lt;/p&gt;&#xA;&lt;p&gt;Someone with a greater understanding of the methodology and the research field will have to comment further on the quality of the arguments in the article.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-10-04T12:25:42.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24436" PostTypeId="2" ParentId="24420" CreationDate="2021-10-04T20:16:49.277" Score="8" Body="&lt;p&gt;Interestingly enough, we do not have such a distinction in France. From the French Wikipedia &lt;a href=&quot;https://fr.wikipedia.org/wiki/Fraction_(math%C3%A9matiques)&quot; rel=&quot;nofollow noreferrer&quot;&gt;article about fractions&lt;/a&gt; (emphasis mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dans l'enseignement français depuis la fin du xixe siècle, la fraction est définie comme le quotient de deux nombres entiers sans contrainte sur la taille du numérateur et du dénominateur (...)&lt;/p&gt;&#xA;&lt;p&gt;In French education, since the end of the 19th century, a fraction is defined as a division of two whole numbers, &lt;strong&gt;without constraints on the size of the numerator and denominator&lt;/strong&gt; (...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think I recall that at the very, very first introduction of fractions my children saw the version &lt;span class=&quot;math-container&quot;&gt;$1\frac{3}{4}$&lt;/span&gt; but it was quickly replaced by &lt;span class=&quot;math-container&quot;&gt;$\frac{7}{4}$&lt;/span&gt; and never came back. I do not remember how it was when I was learning fractions, but I do not remember ever having used the &lt;span class=&quot;math-container&quot;&gt;$1\frac{3}{4}$&lt;/span&gt; version.&lt;/p&gt;&#xA;&lt;p&gt;On a related note, I find the &lt;span class=&quot;math-container&quot;&gt;$1\frac{3}{4}$&lt;/span&gt; version particularly unintuitive, it suspiciously looks like &lt;span class=&quot;math-container&quot;&gt;$1\times \frac{3}{4}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;: Now that I think of it, it may be that the US use of proper vs improper fractions come from their very heavy use in measurements.&lt;/p&gt;&#xA;&lt;p&gt;In Europe, we would never say that something is &lt;span class=&quot;math-container&quot;&gt;$1\frac{2}{11}$&lt;/span&gt; meters. We would say it is 1,18 m (with a comma :)) - because of the decimal nature of the metric system.&lt;/p&gt;&#xA;&lt;p&gt;We basically never have a need to use fractions in everyday life, only in calculations (where the &amp;quot;improper&amp;quot; form is easier to use)&lt;/p&gt;&#xA;" OwnerUserId="8076" LastEditorUserId="8076" LastEditDate="2021-10-06T07:23:08.787" LastActivityDate="2021-10-06T07:23:08.787" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24437" PostTypeId="2" ParentId="24420" CreationDate="2021-10-05T00:56:05.297" Score="5" Body="&lt;p&gt;I did some research, which you can follow &lt;a href=&quot;https://www.themathdoctors.org/improper-fractions-what-are-they-and-why/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, that explains the why of the nomenclature we use. Basically, something is &lt;em&gt;proper&lt;/em&gt; if it is contained in something else, and &lt;em&gt;improper&lt;/em&gt; otherwise. For example, the &lt;em&gt;proper divisors&lt;/em&gt; of 12 are 1, 2, 3, 4, and 6. This set excludes 12, because that represents the entire portion, e.g. no fraction occurs. Similarly, in set theory, a proper subset is a set that contains no elements not in the parent set, and is missing at least one value from the parent set. Similarly, an improper fraction contains &lt;em&gt;at least&lt;/em&gt; the whole.&lt;/p&gt;&#xA;&lt;p&gt;The reason why we learn improper fractions briefly is to (a) introduce a useful form for operating with rational numbers (e.g. &lt;span class=&quot;math-container&quot;&gt;$\frac32\times\frac34$&lt;/span&gt; is slightly easier to math than &lt;span class=&quot;math-container&quot;&gt;$1\frac12\times\frac34$&lt;/span&gt;), and (b) also to introduce the concept of proper and improper sets. In addition, in order to have a &lt;em&gt;proper solution&lt;/em&gt;, one cannot use solutions which are not irreducible, as those are &lt;em&gt;improper solutions&lt;/em&gt;. For example, &lt;span class=&quot;math-container&quot;&gt;$1+1=\frac42$&lt;/span&gt; would be an improper solution, for hopefully obvious reasons.&lt;/p&gt;&#xA;&lt;p&gt;This is obviously not really discussed in higher maths, by the time this topic is revisited, you're now talking about rational numbers and rational expressions, rather than simply fractions. This could be the reason why it's not talked about using that exact nomenclature in higher maths.&lt;/p&gt;&#xA;" OwnerUserId="17846" LastEditorUserId="17846" LastEditDate="2021-10-05T01:06:53.700" LastActivityDate="2021-10-05T01:06:53.700" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24438" PostTypeId="2" ParentId="21095" CreationDate="2021-10-05T09:56:51.110" Score="2" Body="&lt;p&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt; I'm not a psychologist and I'm not trained to deal with &amp;quot;complicated&amp;quot; students such as this. It's perfectly possible the method I used was counterproductive, I wouldn't know. However, we eventually were successful and the student managed to master the concepts and passed the exam comfortably.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Method used&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I provided the student with a sheet detailing how to solve each type of problem step by step, with examples; another sheet about how to verify the results they got were correct; and another sheet with tips and tricks to detect errors as soon as possible, to avoid working needlessly.&lt;/li&gt;&#xA;&lt;li&gt;Each day I would send the student 30 minutes worth of exercises.&lt;/li&gt;&#xA;&lt;li&gt;The exercise sheet looked very similar day after day: solve a couple linear equations systems using two different methods, solve a problem requiring Pythagoras theorem with a small twist (do something with the result of applying Pythagoras), and solving a problem which requires setting your own equations system. This would go on for 8 weeks.&lt;/li&gt;&#xA;&lt;li&gt;The student would tackle the exercises whenever she wanted through the day, and split it in chunks if they so desired; they would send them back to me for feedback.&lt;/li&gt;&#xA;&lt;li&gt;I would not demand proof that she had followed any of the advice I gave in the sheets, but made sure to let her know if she made mistakes which would have been caught had she followed my advice.&lt;/li&gt;&#xA;&lt;li&gt;If an exercise wasn't properly solved I would hint at what kind of mistake it was, and tell her that she had to send me a corrected the exercise before I send any more exercises. My aim here was for her to be able to detect mistakes in her own work.&lt;/li&gt;&#xA;&lt;li&gt;The student would be in charge of their own workload: I would not push them or pester them. If she had to redo some exercises, I would be happy for her to do it the next day. At most, if she had failed to send the answers the previous day, I would let her know I would not send any more exercises until the previous ones were handed in, to not overwhelm her.&lt;/li&gt;&#xA;&lt;li&gt;Whenever possible, I would not involve the parents. It had to be clear that it was something she had to do for her own sake and out of her own volition, not to please anyone (her parents or me).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;How it went&lt;/h2&gt;&#xA;&lt;p&gt;I noticed some serious improvement the first couple of weeks. Exercises went from being impossible to mark (as it wasn't clear what she was trying to accomplish) to being understandable, and in many cases correct.&lt;/p&gt;&#xA;&lt;p&gt;At one point she asked me for a few days off, as she knew she was delivering at a good pace and was a bit overloaded. I gladly let her take a week off, and it was really hard to reel her in afterwards (the quality declined significantly from before the vacation).&lt;/p&gt;&#xA;&lt;p&gt;However, she was motivated and managed to put in the hours (despite my advice not to push herself too hard to avoid frustration/burn out). She ended up realising that it was actually less work to verify the exercises before handing them in and trying to detect mistakes than having to redo the same exercise over and over again. It was still difficult for her, and she would often forget despite me reminding her daily that it was a step she should do, for her own sake, before handing things in.&lt;/p&gt;&#xA;&lt;p&gt;In the end she was comfortable with the exercises. She knew where to look for mistakes when things didn't add up, but it was very mentally demanding for her to do so (she'd rather redo the exercise completely without looking at her first attempt than re-read her own work to find a minor mistake), so it wasn't possible to do away with the silly mistakes completely. I learnt to first not show my frustration about this and then just not be frustrated by it.&lt;/p&gt;&#xA;&lt;p&gt;Still the result was good enough, and honestly I think close to the best that could be achieved given the circumstances. I had to switch my aim from &amp;quot;Have her do the exercises perfectly, following every single step, verifying things, etc.&amp;quot; to &amp;quot;Have her do exercises well enough&amp;quot;, as attempts of achieving the first resulted in great frustration for both of us.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Particularities&lt;/h2&gt;&#xA;&lt;p&gt;I think something which played in my favour was the fact that I was doing it for free, as she's somewhat close to me. Hence, when she tried to fool me (she tried a few times, by solving the exercises online) or I saw she made absolutely no effort and was wasting my time, I could play the card of &amp;quot;I'm doing this for you, not for me. I know you have places you'd rather be, but I guarantee I do too. I'm glad to be able to help you, but I really don't want to waste my time or yours, so if you're not willing to take this seriously we can just call it quits&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I &amp;quot;only&amp;quot; had to do this twice: once when she tried to fool me by copying her answers from somewhere two days in a row, after I let it slide with some complicit talk the previous day; and the second time when she delivered utterly terrible work for three days in a row.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Personal conclusions&lt;/h2&gt;&#xA;&lt;p&gt;It became clear to me that there's a limit to how well she can do under her current circumstances (untreated severe ADHD). I adjusted my expectations to that, reassured her when she got frustrated with herself and had an honest talk with the parents about my impressions. Anything else is outside my control.&lt;/p&gt;&#xA;" OwnerUserId="16341" LastEditorUserId="16341" LastEditDate="2021-10-05T10:57:50.123" LastActivityDate="2021-10-05T10:57:50.123" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24439" PostTypeId="2" ParentId="21095" CreationDate="2021-10-05T21:10:27.123" Score="0" Body="&lt;p&gt;I think you are doing a good job, engaging with the kid and moving things along.  There's a lot we don't know about the brain and about pedagogy, even for normal learners.  Then for syndromes like ADHD, there's a lot unknown about what exactly that means.  Surely not the same thing for each person so diagnosed.  And then there are confounding variables to ADHDness (IQ, others).  So good job just establishing a program and seeing what works.&lt;/p&gt;&#xA;&lt;p&gt;Drill and repetition help all students.  Some need more than others, but even very original thinkers like Richard Feynman benefited from it.&lt;/p&gt;&#xA;&lt;p&gt;Few minor additions to consider:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Force entire rework of missed problems.  Even if (or especially if) it was a &amp;quot;dumb mistake&amp;quot;.  I don't mean rework every single problem.  But all the missed ones.  So if she does five problems and misses two, rework those.  And then when reworking them, she (or you or I or any student) should do it from a &amp;quot;blank sheet&amp;quot;.  I.e. forced to write the whole problem down with any steps or the like.  (It becomes a chance to do free practice of the known parts of the problem and gives more correction than just pen and ink fixing one component.)  This may not be as applicable if you are doing single step arithmetic.  But it applies to any multi-step process (long division, solving for x, fraction multiplication, etc.).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to create some small gamification of her work.  So if she does a set of 10 (or 5 or 3) problems, make it a goal to see what percent she gets right.  Can even track this from day to day and give her some little feeling of trying to beat the day before.  Personally I would give a small sweet for a perfect set.  (But commenters will disagree.)  Could do gold stars if you prefer.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to build towards doing some set of problems at a time.  If she can't grind 20 in a row, make it 10 or 5 or 3.  But something more than just working each problem and immediately checking the answer at the back of the book.  Doing several of them and then checking the set.  I do think you can start to build up her capacity to do longer sets over time.  Especially with material where she is relatively strong.  This will benefit her in school with tests and really is something needed for the work world as well.  She may have a lower concentration capacity than the average, but I would not assume that it cannot be improved over time.  Or that the average student can't improve this over time.  Especially with material he/she becomes deeply invested in.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="17849" LastActivityDate="2021-10-05T21:10:27.123" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24440" PostTypeId="2" ParentId="24420" CreationDate="2021-10-07T00:26:31.640" Score="2" Body="&lt;p&gt;The appearance of &amp;quot;mixed numbers&amp;quot; is inherent to the Euclidean algorithm. For example,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\eqalign{\scriptsize{\frac{355}{113}}\rightarrow&amp;amp; 335=\color{red}{3}\cdot113+16,\ \scriptsize{\frac{355}{113}}=3\scriptsize{\frac{16}{113}}\\&#xA;\scriptsize{\frac{113}{16}}\rightarrow&amp;amp;113=\color{red}{7}\cdot16+1,\phantom{xx}\ \scriptsize{\frac{113}{16}}=7\scriptsize{\frac{1}{16}}\\&#xA;\scriptsize{\frac{16}{1}}\rightarrow&amp;amp;\ 16=\color{red}{16}\cdot1+0,\phantom{xxx} \scriptsize\frac{16}{1}=16\frac{0}{1}&amp;amp;}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and then we encode this as the continued fraction&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{355}{113}=\color{red}{3}+\cfrac{1}{\color{red}{7}+\cfrac{1}{\color{red}{16}}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Notice the natural appearance of mixed numbers &lt;span class=&quot;math-container&quot;&gt;$7\frac{1}{16}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3\frac{16}{113}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The Euclidean algorithm, which is one of the oldest known algorithms, involves repeated conversions of positive rational numbers &lt;span class=&quot;math-container&quot;&gt;$p/q$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$p&amp;gt;q$&lt;/span&gt; to mixed numbers:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{a}{b}=n_0+\frac{r_0}{b}\rightarrow\frac{b}{r_0}=n_1+\frac{r_1}{r_0}\rightarrow\cdots $$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-10-15T14:43:32.277" LastActivityDate="2021-10-15T14:43:32.277" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24444" PostTypeId="2" ParentId="267" CreationDate="2021-10-08T15:40:18.847" Score="6" Body="&lt;p&gt;I would like to use this as an opportunity to make an important distinction.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof by contradiction&lt;/strong&gt; is an argument of the form:&lt;/p&gt;&#xA;&lt;p&gt;Assume &lt;span class=&quot;math-container&quot;&gt;$\neg p$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Argue a contradiction under this assumption.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Conclude &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof of negation&lt;/strong&gt; is an argument of the form&lt;/p&gt;&#xA;&lt;p&gt;Assume &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Argue a contradiction under this assumption.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Conclude &lt;span class=&quot;math-container&quot;&gt;$\neg p$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I learned about the difference from Andrej Bauer here:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/&quot; rel=&quot;noreferrer&quot;&gt;http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A classical mathematician might not distinguish these proofs, because they think &lt;span class=&quot;math-container&quot;&gt;$\neg (\neg p) \equiv p$&lt;/span&gt;, but a constructive mathematician will make this distinction.&lt;/p&gt;&#xA;&lt;p&gt;I would like to go through Jim Belk's list to illustrate the difference:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Infinitude of primes.  As noted, this &lt;em&gt;can&lt;/em&gt; be phrased as a proof by contradiction, but it can also be viewed as a completely constructive result:  given a list of prime numbers, it gives an algorithm for constructing a new prime which is not in the list.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Irrationality of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;.  This is proof of negation.  The definition of &amp;quot;irrational&amp;quot; is &amp;quot;not rational&amp;quot;.  You prove the negation of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is rational&amp;quot; by assuming it is and obtaining a contradiction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sum of a rational and irrational is irrational.  Again, this is proof of negation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cantor's diagonalization argument:  To prove there is no bijection, you assume there is one and obtain a contradiction.  This is proof of negation, not proof by contradiction.  I will point out that, similar to the infinitude of primes example, this can be rephrased more constructively. Given an injection &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N} \to \mathbb{R}$&lt;/span&gt;, this argument explicitly produces an element of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; which is not in the image.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;No bijection from &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}(X)$&lt;/span&gt;:  similar remarks to Cantor's argument apply.  As usually phrased it is proof of negation, and it can be rephrased more constructively as a recipe which takes an injection from &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}(X)$&lt;/span&gt; and produces an element of &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}(x)$&lt;/span&gt; which is not in the image.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Russel's proof that there is no set of all sets.  I am not sure this counts as a formal argument.  It is more of a metamathematical theorem, that we should not attempt to construct a formal system which allows unbounded set formation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\vdots$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I think all of the other examples are also proof of negations, rather than proof by contradiction.  Many of the theorems can be rephrased more powerfully without using negation in the theorem statement at all, and they have direct proofs in this case.  None of them, that I can see, can both be rephrased without negation in the statement and require proof by contradiction, rather than direct proof, to demonstrate them.&lt;/p&gt;&#xA;&lt;p&gt;Usually proof by contradiction can be avoided, and doing so creates direct proofs which are more constructive.  Even for the non-constructive mathematician this is good mathematical hygiene:  the intermediate results proven during a proof by contradiction are useless to your later work (they only hold under a false premise), while the intermediate results obtained in the direct proof are all immediately useful in real circumstances.&lt;/p&gt;&#xA;&lt;p&gt;Proof of negation is unavoidable though.   We can actually define &lt;span class=&quot;math-container&quot;&gt;$\neg p$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$p \implies F$&lt;/span&gt;. If you are trying to prove that a certain number is &lt;em&gt;not&lt;/em&gt; rational, we must show that assuming it is rational leads to contradiction.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-10-08T15:40:18.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24445" PostTypeId="1" CreationDate="2021-10-08T19:26:23.470" Score="15" ViewCount="4873" Body="&lt;p&gt;When teaching proofs by contradiction of an implication &lt;span class=&quot;math-container&quot;&gt;$P \Rightarrow Q$&lt;/span&gt;, one starts by assuming both &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and (not &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;), and then reaches a contradiction. The problem is, most elementary proofs of this type are &amp;quot;fake,&amp;quot; in the sense that the assumption &lt;span class=&quot;math-container&quot;&gt;$``P&quot;$&lt;/span&gt; is never used.&#xA;A typical example is proving the proposition&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;if &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; is even then &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is even&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;by contradiction. One assumes both &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; is even&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is odd&amp;quot;. Then one shows that if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is odd then &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; is also odd, reaching a contradiction with the assumption &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; is even.&amp;quot;&#xA;In reality, however, this assumption was never used in the argument, so this is what I call a &amp;quot;fake&amp;quot; proof by contradiction, since one can rephrase it into a contrapositive proof.&lt;/p&gt;&#xA;&lt;p&gt;My question, then, is, can anyone provide a relatively simple (for educational purposes) &amp;quot;true&amp;quot; proof by contradiction of a proposition &lt;span class=&quot;math-container&quot;&gt;$``P \Rightarrow Q&quot;$&lt;/span&gt; in which one would really need BOTH the assumption (not &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;) and the assumption (&lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;) to actually reach some kind of contradiction?&lt;/p&gt;&#xA;" OwnerUserId="17857" LastEditorUserId="13585" LastEditDate="2024-01-14T14:45:41.983" LastActivityDate="2024-01-14T14:45:41.983" Title="I want a &quot;true&quot; proof by contradiction of an implication $P \Rightarrow Q$" Tags="|proofs|logic|" AnswerCount="6" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="24446" PostTypeId="2" ParentId="24445" CreationDate="2021-10-08T20:47:25.927" Score="10" Body="&lt;p&gt;No, I suspect this situation never occurs. Here is why:&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; really implies &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, then we know logically that &lt;span class=&quot;math-container&quot;&gt;$\neg Q$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$\neg P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thus if you assume &lt;span class=&quot;math-container&quot;&gt;$\neg Q$&lt;/span&gt;, you will be able to deduce &lt;span class=&quot;math-container&quot;&gt;$\neg P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In this way, you never &amp;quot;need&amp;quot; the assumption &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-10-08T20:47:25.927" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24448" PostTypeId="2" ParentId="24445" CreationDate="2021-10-09T01:02:55.217" Score="9" Body="&lt;p&gt;I think you are overlooking the fact that proof by contradiction must invoke the tautology  &lt;span class=&quot;math-container&quot;&gt;$(P\ \hbox{or}\  \neg P)$&lt;/span&gt;, called the &lt;em&gt;law of excluded middle&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To prove &lt;span class=&quot;math-container&quot;&gt;$P\Rightarrow Q$&lt;/span&gt; by contradiction, we show that &lt;span class=&quot;math-container&quot;&gt;$(\neg Q\Rightarrow\neg P)$&lt;/span&gt;.  The next step, which is where we deduce the conclusion &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, is where we &lt;em&gt;must&lt;/em&gt; invoke the assumption that &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is &amp;quot;true.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So in proof by contradiction, we do not &lt;em&gt;actively&lt;/em&gt; use &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. Instead, we &lt;em&gt;passively&lt;/em&gt; use the assumption &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; in the last step of the proof when we invoke the law of excluded middle. In effect, proof by contradiction swaps the active use of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; that we would use in a direct proof with active use of &lt;span class=&quot;math-container&quot;&gt;$\neg Q$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-10-09T02:31:43.190" LastActivityDate="2021-10-09T02:31:43.190" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24451" PostTypeId="2" ParentId="24420" CreationDate="2021-10-09T03:45:57.110" Score="0" Body="&lt;p&gt;For me it is simply a difference between diving 1 unit vs. dividing a group(more than 1 unit). I teach very small kids and sometimes I explain them a difficult topic for example fractions and try to find out WHAT EXACTLY are they not understanding, why they don't understand. So I have found that difference of diving/splitting 1 vs dividing/splitting group is important...&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2021-10-09T03:45:57.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24452" PostTypeId="2" ParentId="24445" CreationDate="2021-10-09T04:51:56.030" Score="32" Body="&lt;p&gt;As you've noticed, there are (at least) three potential ways of proving an implication &lt;span class=&quot;math-container&quot;&gt;$p \Rightarrow q$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Assume &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, and conclude &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Assume &lt;span class=&quot;math-container&quot;&gt;$\neg q$&lt;/span&gt;, and conclude &lt;span class=&quot;math-container&quot;&gt;$\neg p$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Assume both &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\neg q$&lt;/span&gt;, and derive a contradiction.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If I understand you right, you're asking for a proof which is of the third kind. Moreover, you're asking for a proof which is &lt;em&gt;essentially&lt;/em&gt; of the third kind, rather than being a proof which is essentially a proof of the first or second kind, but which has been dressed up as a proof of the third kind.&lt;/p&gt;&#xA;&lt;p&gt;In other words, you're asking for a proof which takes &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\neg q$&lt;/span&gt; and &amp;quot;meets in the middle,&amp;quot; rather than starting with one premise and reasoning all the way to the negation of the other premise.&lt;/p&gt;&#xA;&lt;p&gt;My suggestion is below.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; If a number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is odd, then &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is not even. &lt;em&gt;(A number is defined as&lt;/em&gt; even &lt;em&gt;if it is of the form &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; for integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, and&lt;/em&gt; odd &lt;em&gt;if it is of the form &lt;span class=&quot;math-container&quot;&gt;$2n + 1$&lt;/span&gt; for integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof.&lt;/strong&gt; We will prove this by contradiction. Suppose that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is odd and even. Then, for some integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and some integer &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = 2 n$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = 2 p + 1.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This means that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;2 n &amp;amp;= 2 p + 1\\&#xA;2 n - 2 p &amp;amp;= 1\\&#xA;n - p &amp;amp;= \tfrac12.&#xA;\end{align*}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, since &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; are integers, &lt;span class=&quot;math-container&quot;&gt;$n - p$&lt;/span&gt; is an integer, whereas &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt; is not an integer. This is a contradiction, and so our proof is complete.&lt;/p&gt;&#xA;" OwnerUserId="930" LastActivityDate="2021-10-09T04:51:56.030" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24454" PostTypeId="1" AcceptedAnswerId="24459" CreationDate="2021-10-09T20:12:35.637" Score="3" ViewCount="677" Body="&lt;p&gt;Question in title. I'd like to draw a perfect circle on the whiteboard using &amp;quot;expo&amp;quot; dry erase markers. Is there a store I could buy such a compass at? Thank you&lt;/p&gt;&#xA;" OwnerUserId="16437" LastActivityDate="2021-10-11T13:24:25.373" Title="Where can I buy a compass that can hold an &quot;Expo&quot; dry erase whiteboard marker?" Tags="|geometry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24455" PostTypeId="2" ParentId="24454" CreationDate="2021-10-09T20:25:14.727" Score="5" Body="&lt;p&gt;This looks like it should work with any (regularly-sized) dry erase marker, since just the drawing tip fits through the holes:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/m2yge.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/m2yge.png&quot; alt=&quot;Compass set for a whiteboard.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[Note: I don't want to link directly to the ebay post since those change over time, and I'm not suggesting this seller. It is currently the first item that Google returns when I search &amp;quot;suction cup draw circle whiteboard&amp;quot;.]&lt;/p&gt;&#xA;&lt;p&gt;Alternatively, what about just getting a regular suction cup that will stick to the whiteboard, tying one end of a string to the metal hook (if it has one), and tying the other end of the string around your dry erase marker?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qN8tNs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qN8tNs.png&quot; alt=&quot;A suction cup with metal hook on end&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="77" LastEditDate="2021-10-10T02:22:22.823" LastActivityDate="2021-10-10T02:22:22.823" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24457" PostTypeId="2" ParentId="20802" CreationDate="2021-10-10T18:45:02.247" Score="0" Body="&lt;p&gt;It is wrong exactly for the reasons you have pointed out. There shouldn't be an issue with including the word 'exactly' as in 'exactly 4 apples'.&lt;/p&gt;&#xA;" OwnerUserId="5987" LastActivityDate="2021-10-10T18:45:02.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24459" PostTypeId="2" ParentId="24454" CreationDate="2021-10-11T02:27:13.850" Score="4" Body="&lt;p&gt;I bought the product pictured below from Amazon, around four years ago, for about $10. That exact product is no longer available, but if you're willing to search Amazon for &amp;quot;whiteboard compass&amp;quot;, then you'll see a number of similar products.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/t1kC9.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/t1kC9.jpg&quot; alt=&quot;Whiteboard compass next to foot-long ruler&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's worked satisfactorily for me, although I don't put it through heavy use (frankly, just one day a year in a college algebra class).&lt;/p&gt;&#xA;&lt;p&gt;I'd previously tried the &amp;quot;jury rigged on a string&amp;quot; method and that did not work well enough for me. First, there was nothing holding the angle of the marker fixed, so it would wiggle around the circle and not match up properly at the end. Secondly, the little bit of winding in the string at the center would also throw off the consistent radius. Third, the pen would tend to slip out of the string.&lt;/p&gt;&#xA;&lt;p&gt;The compass I got is large (bordering on unwieldy), but I like the fact that it illustrates/resembles a &amp;quot;traditional&amp;quot; compass, so students have a visual sense of what we mean by that term normally. Granted the limited number of times I use it, I'm very happy with that little bit of tradeoff.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-10-11T13:24:25.373" LastActivityDate="2021-10-11T13:24:25.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24460" PostTypeId="2" ParentId="24445" CreationDate="2021-10-11T22:30:31.940" Score="2" Body="&lt;p&gt;I think you may be looking at this the wrong way.&lt;/p&gt;&#xA;&lt;p&gt;There are three distinct logical rules which are all equivalent. Each rule holds for all propositions &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Rule 1: &lt;span class=&quot;math-container&quot;&gt;$\neg \neg P \implies P$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is the classic &amp;quot;proof by contradiction&amp;quot; rule.&lt;/p&gt;&#xA;&lt;p&gt;Rule 2: &lt;span class=&quot;math-container&quot;&gt;$(\neg P \implies \neg Q) \implies (Q \implies P)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is &amp;quot;proof by contrapositive&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Rule 3: &lt;span class=&quot;math-container&quot;&gt;$P \lor \neg P$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is the so-called &amp;quot;law of excluded middle&amp;quot; (a name which I don't like, but that's another story).&lt;/p&gt;&#xA;&lt;p&gt;Each of these logical rules can be proved using the others (and the other rules of constructive logic, also known as intuitionist logic).&lt;/p&gt;&#xA;&lt;p&gt;(2) &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; (1): Take &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; to be true. Then &lt;span class=&quot;math-container&quot;&gt;$\neg P \implies \neg Q$&lt;/span&gt; is the statement that &lt;span class=&quot;math-container&quot;&gt;$\neg P$&lt;/span&gt; implies false, which is the same as saying &lt;span class=&quot;math-container&quot;&gt;$\neg \neg P$&lt;/span&gt;. Conversely, saying &lt;span class=&quot;math-container&quot;&gt;$Q \implies P$&lt;/span&gt; is the same as saying that true implies &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, which is equivalent to &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$(\neg P \implies \neg Q) \implies (Q \implies P)$&lt;/span&gt; is equivalent to &lt;span class=&quot;math-container&quot;&gt;$\neg \neg P \implies P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(3) &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; (2): Suppose &lt;span class=&quot;math-container&quot;&gt;$\neg P \implies \neg Q$&lt;/span&gt;. Suppose &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;. Now we have &lt;span class=&quot;math-container&quot;&gt;$P \lor \neg P$&lt;/span&gt;. In the case that &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is true, we have &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. And in the case that &lt;span class=&quot;math-container&quot;&gt;$\neg P$&lt;/span&gt; holds, we have &lt;span class=&quot;math-container&quot;&gt;$\neg Q$&lt;/span&gt;, which contradicts &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;. This proves that &lt;span class=&quot;math-container&quot;&gt;$(\neg P \implies \neg Q) \implies (Q \implies P)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(1) &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; (3): We first prove the statement &lt;span class=&quot;math-container&quot;&gt;$\neg \neg (P \lor \neg P)$&lt;/span&gt;. Suppose that &lt;span class=&quot;math-container&quot;&gt;$\neg (P \lor \neg P)$&lt;/span&gt;. Now suppose &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$P \lor \neg P$&lt;/span&gt;. But this contradicts &lt;span class=&quot;math-container&quot;&gt;$\neg (P \lor \neg P)$&lt;/span&gt;. Therefore, &lt;span class=&quot;math-container&quot;&gt;$\neg P$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$P \lor \neg P$&lt;/span&gt;. But this contradicts &lt;span class=&quot;math-container&quot;&gt;$\neg (P \lor \neg P)$&lt;/span&gt;. Therefore, &lt;span class=&quot;math-container&quot;&gt;$\neg \neg (P \lor \neg P)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We now apply proof by contradiction to conclude that &lt;span class=&quot;math-container&quot;&gt;$\neg \neg (P \lor \neg P) \implies P \lor \neg P$&lt;/span&gt;, and we therefore conclude that &lt;span class=&quot;math-container&quot;&gt;$P \lor \neg P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So the moral of the story here is that any proof which is done using any one of these three rules can always be rephrased to use another of the 3 rules. In particular, any proof using rule 1 can always be rephrased to use only rule 2.&lt;/p&gt;&#xA;&lt;p&gt;There is another way to look at this which illustrates why it almost always seems more natural to you to use proof by contrapositive as opposed to proof by contradiction.&lt;/p&gt;&#xA;&lt;p&gt;Consider the fact that whenever we're proving just about any fact whatsoever, we're proving this fact in the context of some assumptions that we've made.&lt;/p&gt;&#xA;&lt;p&gt;For example, suppose I know that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a positive real number, and I wish to prove &lt;span class=&quot;math-container&quot;&gt;$\exists y \in \mathbb{R} (y \cdot x = 1)$&lt;/span&gt;. Then one can also view this as proving the implication &lt;span class=&quot;math-container&quot;&gt;$(x &amp;gt; 0) \implies \exists y \in \mathbb{R} (y \cdot x = 1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In particular, let's suppose we've made an assumption &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, and then we have proved &lt;span class=&quot;math-container&quot;&gt;$\neg \neg P$&lt;/span&gt; on our way to proving &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; by contradiction.&lt;/p&gt;&#xA;&lt;p&gt;Then we can also view this proof as a proof of &lt;span class=&quot;math-container&quot;&gt;$Q \implies \neg \neg P$&lt;/span&gt;. This is equivalent to proving &lt;span class=&quot;math-container&quot;&gt;$\neg (Q \land \neg P)$&lt;/span&gt;, which is in turn equivalent to proving that &lt;span class=&quot;math-container&quot;&gt;$\neg P \implies \neg Q$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So our proof, in the context of &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, that &lt;span class=&quot;math-container&quot;&gt;$\neg \neg P$&lt;/span&gt; holds can also be viewed as a proof of &lt;span class=&quot;math-container&quot;&gt;$\neg P \implies \neg Q$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From there, we wish to conclude, in the context of the assumption &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, that &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; holds. That is, we wish to conclude &lt;span class=&quot;math-container&quot;&gt;$Q \implies P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So in fact, what we're &amp;quot;really doing&amp;quot; is proving &lt;span class=&quot;math-container&quot;&gt;$\neg P \implies \neg Q$&lt;/span&gt; and then concluding from this fact that &lt;span class=&quot;math-container&quot;&gt;$Q \implies P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So in this sense, all proofs by contradiction with any assumptions whatsoever are actually proofs by contrapositive. We collect all the assumptions into a single proposition &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, prove &lt;span class=&quot;math-container&quot;&gt;$\neg P \implies \neg Q$&lt;/span&gt;, and then conclude that &lt;span class=&quot;math-container&quot;&gt;$Q \implies P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This illustrates the fact that proof by contradiction is really just proof by contrapositive where no assumptions are made. Of course, making no assumptions is equivalent to only assuming &lt;span class=&quot;math-container&quot;&gt;$\top$&lt;/span&gt;. Put another way, proof by contradiction is just proof by contrapositive in the case where &lt;span class=&quot;math-container&quot;&gt;$Q = \top$&lt;/span&gt;. And this is indeed exactly how we proof that proof by contradiction is a valid technique using proof by contrapositive (exactly (2) &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; (1) above).&lt;/p&gt;&#xA;" OwnerUserId="17883" LastEditorUserId="17883" LastEditDate="2021-10-12T20:37:10.467" LastActivityDate="2021-10-12T20:37:10.467" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="24461" PostTypeId="2" ParentId="20649" CreationDate="2021-10-12T01:45:17.017" Score="-1" Body="&lt;p&gt;The poet Samuel Taylor Coleridge had an answer (namely, that imagination is underutilized). You can read his full critique of mathematics education here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://allpoetry.com/A-Mathematical-Problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://allpoetry.com/A-Mathematical-Problem&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user10552" LastActivityDate="2021-10-12T01:45:17.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24462" PostTypeId="2" ParentId="20649" CreationDate="2021-10-12T02:23:12.763" Score="8" Body="&lt;p&gt;Do people hate math, or do people hate that they have to do math in school ?&#xA;I suppose every so often we meet a true hater, someone who reviles math in every avenue. But, I think that is anomalous. In reality, most people hate math because it is harder than their other subjects by in large.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, that is not a product of math teachers being &amp;quot;mean&amp;quot;, it is a product of every earlier math course that cut a corner and just let kids slide by. Unfortunately, prerequisites matter so when an 18 year old gets to college and is not comfortable with arithmetic, it's a battle of overcoming years of neglect. The same is true for highschool if the highschool course holds the standard it ought.&lt;/p&gt;&#xA;&lt;p&gt;All of this said, one of my principle aims in any lower math class is to try to win some over to see math is not all bad. In fact, math can be fun when we try to understand it rather than just do it. I do things like show them more than one way to solve a problem and I talk a bit about the history which led to us being able to compactly do algebra without awkward lengthy sentences of old-english prose. Still, there are always those who I fail to reach. I can only hope the next brave soul who teaches them breaks down their hate.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-10-12T02:23:12.763" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24463" PostTypeId="2" ParentId="24445" CreationDate="2021-10-12T11:29:18.817" Score="2" Body="&lt;p&gt;&lt;strong&gt;Claim:&lt;/strong&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Compact_space&quot; rel=&quot;nofollow noreferrer&quot;&gt;Compact&lt;/a&gt; (&lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;) metric space implies &lt;a href=&quot;https://en.wikipedia.org/wiki/Sequentially_compact_space&quot; rel=&quot;nofollow noreferrer&quot;&gt;sequentially compact&lt;/a&gt; (&lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Proof.&lt;/em&gt; Assume &lt;span class=&quot;math-container&quot;&gt;$P\land \lnot Q$&lt;/span&gt;. We use &lt;span class=&quot;math-container&quot;&gt;$\lnot Q$&lt;/span&gt; to construct an infinite sequence with no limit points. Because there are no limit points, each point in our space has an open neighbourhood containing only finitely many points of the sequence. These neighbourhoods form a cover and we use &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; to pick out a finite subcover. But a finite subcover cannot contain the infinite sequence. □&lt;/p&gt;&#xA;&lt;p&gt;This was the simplest example I could find. You can also consider proofs using &lt;a href=&quot;https://en.wikipedia.org/wiki/Strategy-stealing_argument&quot; rel=&quot;nofollow noreferrer&quot;&gt;strategy stealing arguments&lt;/a&gt;, which can be simpler to explain informally.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As to how this relates to other answers saying proof by contradiction is logically equivalent to proof by contrapositive. You can view this as a proof of &lt;span class=&quot;math-container&quot;&gt;$\lnot Q \Rightarrow \lnot P$&lt;/span&gt;. The contradiction arises because it is the easiest way to prove &lt;span class=&quot;math-container&quot;&gt;$\lnot P \Leftrightarrow P\to\bot$&lt;/span&gt;, which some like to call a &lt;a href=&quot;http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/&quot; rel=&quot;nofollow noreferrer&quot;&gt;proof of negation&lt;/a&gt;. A contradiction is in some sense unavoidable when proving a negative, e.g. &lt;a href=&quot;https://mathoverflow.net/q/32011/164831#comment71928_32011&quot;&gt;prove &lt;span class=&quot;math-container&quot;&gt;$\sqrt2$&lt;/span&gt; is not rational&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="17869" LastEditorUserId="17869" LastEditDate="2021-10-12T11:48:37.887" LastActivityDate="2021-10-12T11:48:37.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24464" PostTypeId="1" CreationDate="2021-10-12T15:14:14.157" Score="1" ViewCount="312" Body="&lt;p&gt;I am a high school student really into algebra and algebraic geometry.&lt;/p&gt;&#xA;&lt;p&gt;I want to expose this sort of math to other high school students that have the motivation and ability.&lt;/p&gt;&#xA;&lt;p&gt;For a long time, I had no idea how to learn more math after learning calculus, and I feel my middle school and early high school years were wasted as a result. One day, I coincidentally found a post on r/math that had a link to a bunch of Springer PDFs that were being given out for free. This experience was incredibly eye-opening, and I was exposed to all sorts of math that I had never seen before.&lt;/p&gt;&#xA;&lt;p&gt;I want to find similar students to me that never got the chance to find an online copy of Lang, or their father's calculus textbook. Furthermore, I want to give a mathematical experience better than picking up a random book and struggling through it while you're supposed to be doing homework.&lt;/p&gt;&#xA;&lt;p&gt;In short, I want to teach advanced mathematics (leaning towards abstract algebra/foundations) to high school (or even middle school) students, with no prerequisites other than demonstrated commitment to learning and calculus.&lt;/p&gt;&#xA;&lt;p&gt;In order to accomplish this goal, I have a few questions.&lt;/p&gt;&#xA;&lt;p&gt;(1) When I first asked this question, I thought that we were going to go through parts I and II of Lang's &lt;em&gt;Algebra&lt;/em&gt;, all of Mendelson's logic book, and Axler's linear algebra book. I based the curriculum and pacing heavily on Math 55 at Harvard. But then the answers made me realize that the majority of students are not willing to neglect their grades and social life to study math for 60 hours a week. Should I postpone the project to the summer looking for the students who can, or lower my standards?&lt;/p&gt;&#xA;&lt;p&gt;(2) How do I find and recruit students? Nobody at my school is interested, and even if they were, I doubt most of them would put in the effort.&lt;/p&gt;&#xA;&lt;p&gt;(3) What general methods for teaching should I know of? How do I write a lesson plan, schedule a curriculum, and grade proofs?&lt;/p&gt;&#xA;" OwnerUserId="13911" LastEditorUserId="13911" LastEditDate="2021-10-15T12:18:51.130" LastActivityDate="2021-10-15T14:17:57.067" Title="Advice on teaching advanced mathematics to high school students" Tags="|secondary-education|abstract-algebra|logic|gifted-students|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24465" PostTypeId="1" CreationDate="2021-10-12T16:20:14.590" Score="2" ViewCount="97" Body="&lt;p&gt;In recent times, I have been studying mathematics (topics related to mathematical logic, more specifically) independently and I feel a great deal of passion about it. However, although I have made peace with the fact that it's a lonely endeavor and my motivation is enough to carry on, I have also become preoccupied with the veracity and quality of my learning; basically, I'm worried about misinterpreting things I read and/or that my proofs are fallacious.&lt;/p&gt;&#xA;&lt;p&gt;For example, currently I'm reading [1] and working on the exercises with great pleasure (&lt;a href=&quot;https://github.com/ale64bit/logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;these&lt;/a&gt; are some of my notes). I also worked through some of [2] (again, &lt;a href=&quot;https://github.com/ale64bit/kaktusat&quot; rel=&quot;nofollow noreferrer&quot;&gt;some&lt;/a&gt; of my notes). I tried to contact several professors in my area asking for advice and available programs, but I never got a reply from them. Unfortunately I do not have a university degree and my current life situation doesn't allow me enrolling in one.&lt;/p&gt;&#xA;&lt;p&gt;How can one make progress as an independent student and gain the additional confidence/reassurance that could be attained, perhaps, with the help of an advisor, a teacher or fellow students?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;[1] Shoenfield, J.R. Mathematical Logic. Reading, Mass., Addison-Wesley Pub. Co (1967)&lt;/p&gt;&#xA;&lt;p&gt;[2] Knuth, D. E.: The Art of Computer Programming, Volume 4, Fascicle 6: Satisfiability. Pearson Education (US) (2015)&lt;/p&gt;&#xA;" OwnerUserId="17891" LastActivityDate="2021-10-12T16:20:14.590" Title="How to make progress as an independent student?" Tags="|self-learning|logic|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24466" PostTypeId="2" ParentId="24464" CreationDate="2021-10-12T22:38:30.313" Score="1" Body="&lt;p&gt;For (2), I'd suggest creating the lessons first as blog posts or youtube videos and then sharing them. I think the math reddit might be a good place, but I'm not active there. These can then in turn serve as advertisements for any active curricula you'd like to implement.&lt;/p&gt;&#xA;&lt;p&gt;For (3):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Start with an overall structure of the course. Make a list of course-level goals.&lt;/li&gt;&#xA;&lt;li&gt;Create assessments (the equivalent of a midterm/final) based on these course-level goals. Even if you don't wind up using these assessments, it's useful to have them to structure the course.&lt;/li&gt;&#xA;&lt;li&gt;Expand your list of goals to a list of goals for each week/module/lesson. If your target audience is really no prerequisites, pay close attention to everything someone needs to know to be able to answer all of the questions you created in step 2.&lt;/li&gt;&#xA;&lt;li&gt;Create assessments based on each week's goals.&lt;/li&gt;&#xA;&lt;li&gt;Create lessons in preparation for those assessments.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-10-12T22:38:30.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24467" PostTypeId="1" CreationDate="2021-10-13T06:10:36.330" Score="3" ViewCount="257" Body="&lt;p&gt;I have a student (age 14) who is valiantly striving to improve his algebra. He knows the &amp;quot;rules&amp;quot;, and he more-or-less manages to apply them correctly, and hence is generally able to tackle the &amp;quot;simplify this&amp;quot; which his curriculum routinely throws at him.&lt;/p&gt;&#xA;&lt;p&gt;But then he encounters something like:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$-14 x + 21 = 7 x^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and he completely fails to notice that he can divide every term by &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When I pointed out to him that cancelling big numbers out of his expressions (and in this context, &amp;quot;big&amp;quot; seems to be anything greater than or equal to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;) his reaction was: &amp;quot;How can I tell what number to cancel by?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;And at that point it occurred to me that was unable to recognise that, in the expression above, he could not tell that &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; was a divisor of each of &lt;span class=&quot;math-container&quot;&gt;$14$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$21$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Back in my day, we learned our &amp;quot;times tables&amp;quot;, and hence recognising by sight every &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-digit number's prime factors is instant, so I never even thought that would be a problem.&lt;/p&gt;&#xA;&lt;p&gt;Now I realise that because pupils are no longer expected to learn arithmetic, they are also unable to &amp;quot;recognise&amp;quot; numbers in this way.&lt;/p&gt;&#xA;&lt;p&gt;Apart from giving this dude lots of practice doing basic arithmetic (which doesn't work because he doesn't do it), what strategies can I suggest for him so he can gain facility at &amp;quot;cancelling down&amp;quot; composite numbers like this?&lt;/p&gt;&#xA;" OwnerUserId="15290" LastActivityDate="2021-10-18T15:13:12.330" Title="How to train facility with numbers?" Tags="|algebra|struggling-students|arithmetic|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24468" PostTypeId="2" ParentId="24467" CreationDate="2021-10-13T21:49:45.153" Score="5" Body="&lt;p&gt;I am guessing that you were hired by his parents to tutor him. If that's the case, he isn't necessarily on board. That's a tough situation to be in.&lt;/p&gt;&#xA;&lt;p&gt;You made a few errors in your post&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;Back in my day, we learned our &amp;quot;times tables&amp;quot;, and hence recognising by sight every 2-digit number's prime factors is instant&amp;quot; That's not true. There are many people who have &amp;quot;learned their times tables&amp;quot; but still have problems with factoring. The times table is one direction, and factoring is another.&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;Now I realise that because pupils are no longer expected to learn arithmetic...&amp;quot; I'm curious why you think this is so. I do not see any evidence of this, and am in touch with lots of teachers (and a few students).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;What I hear is the basic problem is that you know he needs to practice some basics, and he's not into it, and won't. I recommend fun. There are plenty of educational video games for multiplication and some for factoring. There's a game you could play with him (not online), called either  Divisor Miser or Tax Collector. (I can't believe I've never blogged about that. &lt;a href=&quot;https://www.filepicker.io/api/file/GI9qktuJQFucOpNwKEXS&quot; rel=&quot;noreferrer&quot;&gt;Here's a reasonable write-up&lt;/a&gt; of it.) There's a lovely board game called Prime Climb. I'm sure there's lots more.&lt;/p&gt;&#xA;&lt;p&gt;I will add more to this answer if I think of more.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-10-13T21:49:45.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24469" PostTypeId="2" ParentId="24467" CreationDate="2021-10-13T22:04:36.363" Score="-1" Body="&lt;p&gt;Why is it a priority to give him the ability to do these simplifications?  I can do them and naturally go that direction.  But, like you, am to the right side of the bell curve on math topics.&lt;/p&gt;&#xA;&lt;p&gt;Seems to me that the priority should be to get him to do the algorithms first (add like quantities to each side, isolate Xs on left and numbers on the right, etc.).  Once he has total facility with that, sure then (THEN!) work on the recognition of numbers and the like. But first things first.&lt;/p&gt;&#xA;" OwnerUserId="17898" LastActivityDate="2021-10-13T22:04:36.363" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24470" PostTypeId="2" ParentId="24420" CreationDate="2021-10-14T02:11:51.067" Score="6" Body="&lt;p&gt;Here's another use-case that came up in my college remedial algebra class tonight (and again, this boils down to translations to mixed numbers): &lt;strong&gt;Finding fractions in a graph&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So the specific example that presented itself tonight was a book exercise: &amp;quot;Graph the equation: &lt;span class=&quot;math-container&quot;&gt;$y = - \frac 5 3$&lt;/span&gt;&amp;quot;. At that point, my students could tell me that this would be a horizontal line in the Cartesian plane, and that it should go through &lt;span class=&quot;math-container&quot;&gt;$-\frac 5 3$&lt;/span&gt; on the y-axis. But where is that? Nobody could find it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;ME: Can anyone tell me what two integers &lt;span class=&quot;math-container&quot;&gt;$-\frac 5 3$&lt;/span&gt; is between?&lt;/p&gt;&#xA;&lt;p&gt;STUDENT: Between 3 and 5.&lt;/p&gt;&#xA;&lt;p&gt;ME: No.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So given that none of the students could answer it, I suggested: it's probably helpful if we convert to a mixed number. What is that? Well, now I'm talking about taking this improper fraction and writing as a whole number plus a proper fraction (with little underlined blanks with those words underneath). But no one can accomplish that, so I reviewed the long division algorithm, and came up with &lt;span class=&quot;math-container&quot;&gt;$-\frac 5 3 = -1 \frac 1 3$&lt;/span&gt;. This at least lets us answer the question above: &amp;quot;This value is between &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;&amp;quot; (and then finish the exercise by drawing a horizontal line at that height).&lt;/p&gt;&#xA;&lt;p&gt;Then my students asked for another one like that (presumably because it seemed opaque to them). The next exercise in the book was, &amp;quot;Graph the equation: &lt;span class=&quot;math-container&quot;&gt;$y = - \frac {15} 4$&lt;/span&gt;&amp;quot;. Again, none of my students could find that location, i.e., say what two integers straddle it, because none of them could convert to a mixed number. I went through a long division again, etc. One student said he had no idea how I was coming up with &amp;quot;the fraction part&amp;quot;. So for the purposes of this class I had to refer him to tutoring -- where I assume they'll be using the terms &amp;quot;proper fraction&amp;quot; and &amp;quot;improper fraction&amp;quot; to talk about those parts of the translation-to-mixed-number process, and so actually &lt;em&gt;find&lt;/em&gt; where the value is on a number line.&lt;/p&gt;&#xA;&lt;p&gt;You can see an identical problem in finding other points, intercepts, slopes, etc. that have fractional components, if people are unable to recognize the need and convert to the mixed number format.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-10-14T19:35:50.340" LastActivityDate="2021-10-14T19:35:50.340" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24471" PostTypeId="1" CreationDate="2021-10-14T06:57:16.673" Score="5" ViewCount="275" Body="&lt;p&gt;I am doing a maths PhD and naturally that involves leading exercise classes for undergraduate students. The idea is that the students just show up and work through the problem set and I'm there to discuss with them if need be. This is the third time I've done this and the first time was for a first year class which many students regularly attended, but the last 2 have been for third year classes and (as I'm sure is common) I am now lucky if 3 people show up.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have any experience with how to encourage the students to engage with the classes? Or as a side question, is it even important that they do? They are given the solutions to the problems a week later anyway but I have no idea if they are actually doing them or not.&lt;/p&gt;&#xA;&lt;p&gt;(This of course raises the possibility of never giving them the solutions but I am not very comfortable with that idea if there are students who for whatever reason can't show up. Maybe we could give them the solutions later in the term?)&lt;/p&gt;&#xA;&lt;p&gt;In the first year class that I taught one problem each week was marked and counted towards the final grade. I don't like this so much as, especially being first years, they spent a ridiculous amount of time on the question and didn't focus on the others and were generally very stressed out by it. I wonder if it would be worth having a question marked that doesn't count towards the grade? In my undergraduate we had to hand in a whole set of problems every week but they didn't count to anything so it kept you engaged and you got real feedback but you also had the space to make mistakes. However this was an unusual situation because I went to Cambridge where the teaching is in much smaller groups (1 or 2 people) so you would show up anyway.&lt;/p&gt;&#xA;" OwnerUserId="17901" LastActivityDate="2021-10-27T17:03:07.497" Title="How can I encourage students to show up for exercise classes?" Tags="|undergraduate-education|mathematical-pedagogy|student-motivation|exercises|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24472" PostTypeId="2" ParentId="24467" CreationDate="2021-10-14T08:07:08.777" Score="9" Body="&lt;p&gt;If he won't practice on his own, then I suggest the following.  At the beginning of each tutoring session, give him a blank 10 by 10 times table and have him fill it in.  He can then refer to it while he is working.  If he needs to know the factors of 63, then he can find them.&lt;/p&gt;&#xA;&lt;p&gt;I have done this with students who had difficulty learning the times tables.  Every time they fill out the table they get faster.  If he doesn't know something, he will have to figure it out and you can give him different tips that may help him in the future.  For example if he is doing the 8 times table, he can add 8 to each previous number, or double the 4 times table.  When doing 9 times a number, he can subtract that number from 10 times that number or notice the pattern in the table.&lt;/p&gt;&#xA;&lt;p&gt;If he balks at doing this, explain that his choice is to learn the facts, or to do this each time he sits down to work with you. Do NOT accept one that was filled in before, because you have to see him fill it in each time.&lt;/p&gt;&#xA;&lt;p&gt;Hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2021-10-18T15:13:12.330" LastActivityDate="2021-10-18T15:13:12.330" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24473" PostTypeId="2" ParentId="24471" CreationDate="2021-10-15T06:56:44.253" Score="2" Body="&lt;ol start=&quot;0&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;It would be helpful if you told us more about the class and the demographic.  Is this freshman calculus at Enormous State University?  Or abstract algebra at Princeton?  Kind of changes the answer based on the type of drill the peeps are doing and what their capabilities are.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I recommend sharing written solution (or at least answer) keys at the sessions.  Maybe that will drive attendance. But I think drill problem solutions (or at least answers) should be available immediately, even pre-emptively, in the book.  When I self-study math or language or whatever, I do the drill first.  Then turn to the back and see how I did.  Then rework the problem if I missed it.  Moving the corrective feedback even further back in the term seems like going the wrong direction.  Heck, even having it a week later seems like an overly long feedback loop.  How am I supposed to advance my knowledge if I have to wait a week to find my mistakes?  And yes if someone wants to show up and take the key and leave, let them.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To the extent you know typical questions, mistakes on graded tests, within their course, and can share that with students, that can be beneficial.  But since you are not a long term professor, it's unlikely you know that in detail.  But at least you have your own experience from doing the courses.  (Although maybe not if you are some Oxbridge person and are at Cal now.)  If schools really cared, they would pay (higher cost, higher experience) tenured profs to do these sessions and incentivize pay for teaching versus research...but clearly they don't and that is just the world we live in.  All that said, the more you can help the kids to do better on their tests, the more benefit for them.  But see my postscript first and don't spend extra time on it outside of the sessions themselves.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P.s.  In terms of &amp;quot;is it important&amp;quot;?  No.  What's important is that you work on your own Ph.D., in particular the original research part of it.  There are a lot of things wrong with the teaching methods at R1 schools, and a massive amount of money (including the opportunity time of the students) being spent poorly for the results achieved.  But it's not your priority to fix them (nor can you).  Just show up and collect your stipend and do your research.  That's the most important thing...your research.  You are a student too.  And not getting paid much.&lt;/p&gt;&#xA;" OwnerUserId="17903" LastActivityDate="2021-10-15T06:56:44.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24474" PostTypeId="2" ParentId="24464" CreationDate="2021-10-15T14:17:57.067" Score="1" Body="&lt;p&gt;Re (1), you should lower your standards.  As previously discussed, the population for Math55 in high school is tiny.  (Add onto that the difficulty of finding them and it just makes little sense to try to develop this.)  In addition, you really lack the math knowledge OR the practical pedagogical experience to develop Math55 for high school (even the time, given you are in high school).&lt;/p&gt;&#xA;&lt;p&gt;But I think it would be better to abandon this project and concentrate instead on your own learning.  If you need interaction (natural human need, you talked before about wanting a club), I would look into math teams, Olympiads, etc.  Yes, I know the math team at your school didn't satisfy you, but typically if you do well there are playoffs, recognition, etc. and in that case you'll be drawn into a circle of bright mathy youngsters at other schools.&lt;/p&gt;&#xA;&lt;p&gt;Re (2), this is a problem people all over the world struggle with this, in various topics.  In some ways, it's really more of a practical marketing question (but non-trivial).  Honestly rather than trying to build something, you're better off just concentrating on your own development.  If you want, you could blog about it.  But don't be surprised how hard it is to get traffic.  And again, you really lack the chops or the time to develop even an interesting blog on the topic.  But if you treat it more as journaling than maybe OK.&lt;/p&gt;&#xA;&lt;p&gt;Re (3), you need to develop, try, and read about several practical approaches.  This is a huge topic.&lt;/p&gt;&#xA;&lt;p&gt;Overall, lots of interesting things here that you could learn about (Fermi problem estimation of a population, how to build interest across the Internet, and pedagogy/engagement in high end math).  But I would just learn about them in the course of trying to create this thing.  Which you won't do successfully. It's like starting a mail order computer kit company.  You're unlikely to become Dell, but you'll learn some interesting things along the way.  (And if you do become Dell, go you.  But still, you need to learn by doing.)  Really, I would just get through high school and do Olympiads.  And learn Japanese.&lt;/p&gt;&#xA;" OwnerUserId="17903" LastActivityDate="2021-10-15T14:17:57.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24475" PostTypeId="2" ParentId="7526" CreationDate="2021-10-15T14:52:30.083" Score="1" Body="&lt;p&gt;It depends on your goals for the examination. Typically the goal is to assess students' understanding, or &lt;em&gt;measure&lt;/em&gt; them in some similar way. Will having some prompts on your exam that are identical to prompts some students have answered before throw off the &lt;em&gt;accuracy&lt;/em&gt; of this measurement? I don't think so. Especially since there's a positive correlation between how well a student does on an exam and how much time they spend studying (looking over past materials) anyways.&lt;/p&gt;&#xA;&lt;p&gt;A relevant anecdote: a past instructor of mine gave us access to every midterm and final exam he wrote in his decades-long career. In that week before each exam &lt;em&gt;I learned so much&lt;/em&gt; working through those past exams. Granted, it was a CS course and the exam was assessing &amp;quot;can I &lt;em&gt;do&lt;/em&gt; this&amp;quot; more than &amp;quot;do I &lt;em&gt;know&lt;/em&gt; this.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2021-10-15T14:52:30.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24476" PostTypeId="1" CreationDate="2021-10-15T15:04:47.140" Score="13" ViewCount="1265" Body="&lt;p&gt;I'm looking for public repositories/banks of past undergraduate math exams. No solutions necessary. I just want inspiration for writing my own exams and practice exams. It'd be nice to collect links to these repositories in one place.&lt;/p&gt;&#xA;&lt;p&gt;Side-quest: I'm also interested in &lt;em&gt;old&lt;/em&gt; exams just to compare their questions/styles with modern exams. It'd be cool to know the class scores on old exams too, but that's a separate query :)&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2021-10-15T15:15:50.013" LastActivityDate="2022-08-05T16:52:41.327" Title="Where can I find public repositories of past math exams?" Tags="|undergraduate-education|exams|resource-request|big-list|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24477" PostTypeId="2" ParentId="24476" CreationDate="2021-10-15T15:06:49.340" Score="12" Body="&lt;p&gt;Quite a few universities publicly post the math exams their faculty write:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;UC Berkeley &lt;a href=&quot;https://math.berkeley.edu/courses/archives/exams&quot; rel=&quot;nofollow noreferrer&quot;&gt;hosts an archive of their past exams&lt;/a&gt;, sorted by &lt;a href=&quot;http://guide.berkeley.edu/courses/math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;course&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;University of Michigan hosts past exams for some classes. There's not a consolidated repository though: past exams are hosted under individual course webpages. For example here are the past exams for &lt;a href=&quot;http://www.math.lsa.umich.edu/courses/215/17exampractice/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math215 Mutlivariable Calculus&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similarly the University of Washington hosts past exams under the webpage for each course. For example here is the &lt;a href=&quot;https://sites.math.washington.edu/%7Em120/testindex.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;exam archive for their Math120 Precalulus class&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.uab.edu/cas/mathematics/calculus-testbank&quot; rel=&quot;nofollow noreferrer&quot;&gt;UAB hosts their past &lt;em&gt;calculus&lt;/em&gt; exams&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;University of Colorado Boulder &lt;a href=&quot;https://www.colorado.edu/amath/academics/exam-archives&quot; rel=&quot;nofollow noreferrer&quot;&gt;hosts some of their past undergraduate and graduate exams&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2022-08-05T16:52:41.327" LastActivityDate="2022-08-05T16:52:41.327" CommentCount="2" CommunityOwnedDate="2021-10-15T15:06:49.340" ContentLicense="CC BY-SA 4.0" />
  <row Id="24478" PostTypeId="2" ParentId="24476" CreationDate="2021-10-15T15:12:22.620" Score="0" Body="&lt;p&gt;Quite a few websites host PDFs of academic materials for the sake of student &lt;strike&gt;cheating&lt;/strike&gt; studying. They can be tough to navigate and find specific things, but they do host tons of content.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.chegg.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chegg&lt;/a&gt; comes to mind, although it's easier to search their site for particular questions rather than find exams. Through their &lt;a href=&quot;https://www.chegg.com/uversity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chegg Uversity&lt;/a&gt; push for academic materials though, I expect they'll have an large bank of materials available (for an access fee?) soon.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="15884" LastEditDate="2021-10-17T13:40:57.403" LastActivityDate="2021-10-17T13:40:57.403" CommentCount="0" CommunityOwnedDate="2021-10-15T15:12:22.620" ContentLicense="CC BY-SA 4.0" />
  <row Id="24479" PostTypeId="2" ParentId="24476" CreationDate="2021-10-15T17:00:19.410" Score="3" Body="&lt;p&gt;USNA has a good site with old math exams and some other course materials (but mostly old tests, solutions).  Well organized by course, and even listed by common course name versus cryptic course numbers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.usna.edu/MathDept/resources/index.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.usna.edu/MathDept/resources/index.php&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you want pre-Internet materials, my advice is to search ad hoc by course name, including reaching out to old professors directly (but also Internet and Google Scholar).&lt;/p&gt;&#xA;&lt;p&gt;Although really my advice is to give questions similar to the drill homework, especially at schools that are not MITish in difficulty.  If the kids master that, then they've mastered what needs to be tested.  You should be able to write these easily from looking at the drill book or competing texts (not saying to plagiarize, can modify details).&lt;/p&gt;&#xA;" OwnerUserId="17903" LastEditorUserId="17903" LastEditDate="2021-10-17T01:09:56.613" LastActivityDate="2021-10-17T01:09:56.613" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24480" PostTypeId="2" ParentId="7526" CreationDate="2021-10-15T17:14:35.190" Score="1" Body="&lt;p&gt;I think it's a great idea.  The point is to drive learning, not trickiness. If they've mastered the typical problem set (including the workings), that's fine.  You can't know how to work 50 tests without having learned the topic.&lt;/p&gt;&#xA;&lt;p&gt;I would try to avoid tests that are all or nothing (single question).  And avoid infrequent exams.  I.e. biweekly is better than &amp;quot;final/midterm only&amp;quot;.  This will reduce the luck component sufficiently.  (Also, many other reasons why you should not give infrequent super high stakes problems, regardless.)&lt;/p&gt;&#xA;" OwnerUserId="17903" LastActivityDate="2021-10-15T17:14:35.190" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24481" PostTypeId="2" ParentId="24476" CreationDate="2021-10-16T19:55:10.770" Score="2" Body="&lt;p&gt;The University of Cambridge makes past undergraduate exams available &lt;a href=&quot;https://www.maths.cam.ac.uk/undergrad/pastpapers/past-ia-ib-and-ii-examination-papers&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In addition, old example sheets for some undergraduate courses can be found &lt;a href=&quot;https://www.maths.cam.ac.uk/undergrad/examplesheets&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2021-10-16T19:55:10.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24482" PostTypeId="1" CreationDate="2021-10-19T03:37:44.523" Score="-4" ViewCount="192" Body="&lt;p&gt;What would be to teach in most countries and education systems?&lt;/p&gt;&#xA;&lt;p&gt;(Taught before &amp;quot;high education frames&amp;quot;, i.e. before doing bachelor of arts in mathematics).&lt;/p&gt;&#xA;" OwnerUserId="17917" LastEditorUserId="17917" LastEditDate="2021-10-19T05:49:30.560" LastActivityDate="2021-10-20T04:54:41.057" Title="What is the most fundamental continuous function in calculus after a constant (totally straight) line?" Tags="|calculus|functions|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24483" PostTypeId="2" ParentId="24482" CreationDate="2021-10-19T05:03:23.877" Score="2" Body="&lt;p&gt;The linear function is a pretty good first example: they are very fundamental and everything is simple.&#xA;But rather than think in terms of the most fundamental thing to come next, I would encourage you to think in terms of typical examples that show the breadth of the concept.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Linear function&lt;/li&gt;&#xA;&lt;li&gt;Polynomial of maybe second degree (not linear, but still fairly familiar)&lt;/li&gt;&#xA;&lt;li&gt;Constant function (too elementary a case for students to understand easily, and valuable due to that)&lt;/li&gt;&#xA;&lt;li&gt;Absolute value (or some other piecewise defined function)&lt;/li&gt;&#xA;&lt;li&gt;Square root (not defined everywhere)&lt;/li&gt;&#xA;&lt;li&gt;Maybe &lt;span class=&quot;math-container&quot;&gt;$x^{-1}$&lt;/span&gt; (if you want to get tricky)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For signal processing you might go more in the direction of sine and cosine as simple examples and in general use more periodic examples.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2021-10-19T18:51:35.220" LastActivityDate="2021-10-19T18:51:35.220" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24484" PostTypeId="1" AcceptedAnswerId="24486" CreationDate="2021-10-19T05:55:26.520" Score="0" ViewCount="805" Body="&lt;p&gt;Both function types contain a straight line on a plane, so if a student asks for the difference between them what would be a standard or recommended way to explain it?&lt;/p&gt;&#xA;" OwnerUserId="17917" LastEditorUserId="376" LastEditDate="2021-10-19T07:55:36.813" LastActivityDate="2021-10-21T14:11:05.957" Title="How to explain the difference between a constant function and a linear function?" Tags="|calculus|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24485" PostTypeId="2" ParentId="24484" CreationDate="2021-10-19T08:06:02.977" Score="2" Body="&lt;p&gt;A constant function does not vary with the independent variable: its value remains the same no matter what &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is (assuming &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the independent variable).&lt;/p&gt;&#xA;&lt;p&gt;As a formula, a line can be described by &lt;span class=&quot;math-container&quot;&gt;$y = ax + b$&lt;/span&gt;. The special case of a constant function has &lt;span class=&quot;math-container&quot;&gt;$a = 0$&lt;/span&gt;, so the formula can be written succinctly as &lt;span class=&quot;math-container&quot;&gt;$y = b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you want to bring in the idea of slope, note that a constant function has slope zero and that the graph is a horizontal line.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2021-10-19T08:06:02.977" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24486" PostTypeId="2" ParentId="24484" CreationDate="2021-10-19T08:08:39.627" Score="2" Body="&lt;p&gt;Mathematically: &lt;span class=&quot;math-container&quot;&gt;$y = mx +b$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$m = 0$&lt;/span&gt; (constant function), or &lt;span class=&quot;math-container&quot;&gt;$m \neq 0$&lt;/span&gt; (linear).&lt;/p&gt;&#xA;&lt;p&gt;Intuitively:  discuss something that is responsive or not.  Perhaps adding weight to a scale, before (or after) it hits the highest level on the indicator.&lt;/p&gt;&#xA;" OwnerUserId="17903" LastEditorUserId="3380" LastEditDate="2021-10-21T14:11:05.957" LastActivityDate="2021-10-21T14:11:05.957" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24487" PostTypeId="1" CreationDate="2021-10-19T15:49:05.563" Score="9" ViewCount="1084" Body="&lt;p&gt;I've been using wordpress.com, but it renders LaTeX terribly! There must be a blogging platform that does a good job of this. Can someone recommend one?&lt;/p&gt;&#xA;" OwnerUserId="14430" LastEditorUserId="6053" LastEditDate="2021-10-20T17:24:12.683" LastActivityDate="2021-10-20T17:24:12.683" Title="Is there a blog platform that renders LaTeX well?" Tags="|latex|" AnswerCount="1" CommentCount="10" ClosedDate="2021-10-23T22:16:01.680" ContentLicense="CC BY-SA 4.0" />
  <row Id="24488" PostTypeId="2" ParentId="24487" CreationDate="2021-10-19T22:52:18.390" Score="15" Body="&lt;p&gt;It is common to use &lt;a href=&quot;https://www.mathjax.org/&quot; rel=&quot;noreferrer&quot;&gt;MathJax&lt;/a&gt;.&#xA;This is used on this StackExchange site for LaTeX-like input&#xA;(not full LaTeX of course).&#xA;MathJax is available for WordPress via plugins,&#xA;such as &lt;a href=&quot;https://wordpress.org/plugins/jetpack/&quot; rel=&quot;noreferrer&quot;&gt;JetPack&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;See &lt;a href=&quot;https://docs.mathjax.org/en/v1.0/platforms/wordpress.html&quot; rel=&quot;noreferrer&quot;&gt;Installing MathJax in&#xA;WordPress&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;There is also the &lt;a href=&quot;https://wordpress.org/plugins/simple-mathjax/&quot; rel=&quot;noreferrer&quot;&gt;Simple&#xA;MathJax&lt;/a&gt; plugin for WordPress.&lt;/li&gt;&#xA;&lt;li&gt;Another how-to: &lt;a href=&quot;https://yutsumura.com/mathjax-in-wordpress/&quot; rel=&quot;noreferrer&quot;&gt;How to Use MathJax on WordPress&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastActivityDate="2021-10-19T22:52:18.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24489" PostTypeId="2" ParentId="24482" CreationDate="2021-10-20T04:54:41.057" Score="1" Body="&lt;p&gt;The category-theorist in me would argue the most fundamental function is the identity function, so &lt;span class=&quot;math-container&quot;&gt;$f(x)=x$&lt;/span&gt; might be a good one to go to next.&lt;/p&gt;&#xA;&lt;p&gt;I say this one because, when you really start diving down into what &amp;quot;fundamental&amp;quot; is, the existence of an identity mapping is really key to a lot of structures.  In particular, categories.  And, of course, its a dirt simple function, which is probably what you're going for.&lt;/p&gt;&#xA;&lt;p&gt;Of course this being said, the constant function and identity function can also be tricky first functions because they are &lt;em&gt;so&lt;/em&gt; simple that it's hard to get any feel for functions at all with them.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2021-10-20T04:54:41.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24490" PostTypeId="2" ParentId="23282" CreationDate="2021-10-20T08:54:56.317" Score="0" Body="&lt;p&gt;I found another way to present Zenos paradox in a more approachable fashion on the &lt;a href=&quot;https://www.mathteacherscircle.org/news/mtc-magazine/ws2017/freeing-the-clones/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Teachers Circle&lt;/a&gt;. They pose the problem as follows,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose that Homer would like to walk home. In order to do so, he must walk halfway. After he does this, he must walk half of the remaining distance, and then half that distance, etc. Since he will always have half the remaining distance to go, he will never reach home. Logically, we know that Homer can reach home. So how can we resolve this paradoxical situation?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this case one can write down the series of distances that Homer has to walk easily and then prove that it is finite by the drawing the following diagramm.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qQAz9.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qQAz9.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17653" LastActivityDate="2021-10-20T08:54:56.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24491" PostTypeId="1" CreationDate="2021-10-20T10:10:05.393" Score="2" ViewCount="389" Body="&lt;p&gt;It is common for students to come unstuck by there inability to manipulate fractions (eg calculating gradients, algebraic fractions, etc) yet these same students can handle decimal numbers competently. How is this possible?&lt;/p&gt;&#xA;" OwnerUserId="998" LastActivityDate="2021-10-28T09:03:21.843" Title="Is it possible to understand decimals without understanding fractions?" Tags="|fractions|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24492" PostTypeId="1" CreationDate="2021-10-20T10:38:39.183" Score="2" ViewCount="369" Body="&lt;p&gt;Should i solve math problems by writing the answers to papers or notebooks with pencils or should i try solving them in my head at undergraduate studies at university?&lt;/p&gt;&#xA;&lt;p&gt;Also, sometimes after learning theorems, definitions or proofs should i write them on papers or notebooks to confirm than i know them well?&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastActivityDate="2021-10-20T15:04:02.823" Title="Solving math problems and learning" Tags="|proofs|self-learning|writing|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24493" PostTypeId="2" ParentId="24492" CreationDate="2021-10-20T14:46:57.233" Score="7" Body="&lt;p&gt;Write stuff down.  If you only do it in your head, you'll fake yourself into thinking you know more than you do.  When you force yourself to write things down, you show the steps and ensure you really know them all.  Clear writing is clear thinking.&lt;/p&gt;&#xA;&lt;p&gt;In addition, it gives you more investment in the answer, especially when you check your drill homework and mark a check or an ex next to written work.&lt;/p&gt;&#xA;&lt;p&gt;Also it mimics what you have to do for exams.&lt;/p&gt;&#xA;&lt;p&gt;P.s. It is fine to write the steps quickly.  In fact speed--not spastic, but with some pace--is desirable.  ( &lt;a href=&quot;https://www.youtube.com/watch?v=BqWqzvnbnjU&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=BqWqzvnbnjU&lt;/a&gt; )  But get the whole explication down.  Yes.  In writing.  Own it!&lt;/p&gt;&#xA;" OwnerUserId="17926" LastEditorUserId="17926" LastEditDate="2021-10-20T15:04:02.823" LastActivityDate="2021-10-20T15:04:02.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24494" PostTypeId="1" AcceptedAnswerId="24495" CreationDate="2021-10-20T16:01:08.340" Score="2" ViewCount="639" Body="&lt;p&gt;What are the best type of answer choices for multiple choice questions?  Numerical only like the probability and financial math actuarial exams?&lt;/p&gt;&#xA;&lt;p&gt;I have an opportunity to utilize a multiple choice computerized system to let students practice and prepare for exams for a first calculus course at a university in the USA. This opportunity would be an optional self-learning/self-assessment supplement; not a requirement.&lt;/p&gt;&#xA;&lt;p&gt;Course grades are based on three in-person, hand-written, free-response exams and on weekly quizzes.  Students are not allowed calculators or notes for them.  They are to use analytic-type reasoning and are awarded partial credit.  HW is never collected.&lt;/p&gt;&#xA;&lt;p&gt;Since the only option is multiple choice I need to decide if the answer choices should be all only numerical or not.  My concern is that revealing non-numerical answer choices gives away too much.  But then numerical only options limits what types of questions I can ask I think.&lt;/p&gt;&#xA;&lt;p&gt;I mean suppose I want to ask about the MVT.  Numerically, I could ask to find a value in the interval guaranteed by the theorem.  But I also think it's important for them to verify that the theorem even applies which seems to me to be non-numerical.&lt;/p&gt;&#xA;&lt;p&gt;Example of a numerical question that does not reveal too much in my opinion:&lt;/p&gt;&#xA;&lt;p&gt;Q. Suppose there exists whole numbers &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; in the interval &lt;span class=&quot;math-container&quot;&gt;$[1,9]$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f(x) = c\sin(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f^{\prime}(\frac{d}{e}\pi) = 4$&lt;/span&gt;.  Then &lt;span class=&quot;math-container&quot;&gt;$c + d$&lt;/span&gt; equals&lt;/p&gt;&#xA;&lt;p&gt;A. 5 .... B. 6 .... C. 7 .... D. 8 .... E. 9&lt;/p&gt;&#xA;" OwnerUserId="17928" LastActivityDate="2021-10-20T18:16:37.337" Title="Write calculus multiple choice questions with only numerical answer choices?" Tags="|assessment|test-design|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24495" PostTypeId="2" ParentId="24494" CreationDate="2021-10-20T18:05:06.313" Score="7" Body="&lt;p&gt;I think it is good to pose both numerical and non-numerical problems. While the numerical problems are generally the focus in a Calculus class, a strong conceptual understanding helps make everything easier. You specifically mention theorems, so you could have problems like the following:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;1) Which of the following conditions are required to use the Mean Value Theorem?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I: &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous on &lt;span class=&quot;math-container&quot;&gt;$[a, b]$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;II: &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous on &lt;span class=&quot;math-container&quot;&gt;$(a, b)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;III: &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differentiable on &lt;span class=&quot;math-container&quot;&gt;$(a, b)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;IV: &lt;span class=&quot;math-container&quot;&gt;$f(a) = f(b)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Answer Choices:&lt;/p&gt;&#xA;&lt;p&gt;a) I and III only.&lt;/p&gt;&#xA;&lt;p&gt;b) II and III only.&lt;/p&gt;&#xA;&lt;p&gt;c) I, III, and IV only.&lt;/p&gt;&#xA;&lt;p&gt;d) II, III, and IV only.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;2) Which of the following is true for the function &lt;span class=&quot;math-container&quot;&gt;$f(x)=|x|$&lt;/span&gt;?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;a) There exists some &lt;span class=&quot;math-container&quot;&gt;$c\in(-1, 1)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f'(c)=\frac{|1|-|-1|}{1-(-1)}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;b) The Intermediate Value Theorem applies on the interval &lt;span class=&quot;math-container&quot;&gt;$(-1, 1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;c) All of the above.&lt;/p&gt;&#xA;&lt;p&gt;d) None of the above.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;3) Suppose &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differentiable on &lt;span class=&quot;math-container&quot;&gt;$[a, b]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(a)&amp;lt;f(b)$&lt;/span&gt;. Which of the following is true?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I: You can apply the Mean Value Theorem on the interval &lt;span class=&quot;math-container&quot;&gt;$[a, b]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;II: You can apply the Intermediate Value Theorem on the interval &lt;span class=&quot;math-container&quot;&gt;$[a, b]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;III: You can apply Rolle's Theorem on the interval &lt;span class=&quot;math-container&quot;&gt;$[a, b]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;IV: You can apply the Extreme Value Theorem on the interval &lt;span class=&quot;math-container&quot;&gt;$[a, b]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;a) I and IV only.&lt;/p&gt;&#xA;&lt;p&gt;b) II and IV only.&lt;/p&gt;&#xA;&lt;p&gt;c) I, III, and IV only.&lt;/p&gt;&#xA;&lt;p&gt;d) I, II, and IV only.&lt;/p&gt;&#xA;&lt;p&gt;e) II only.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;4) The graph of the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is displayed below. Which of the following is true about &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;?&lt;/strong&gt; (Please pretend I added a graph here.)&lt;/p&gt;&#xA;&lt;p&gt;I: You can apply the Mean Value Theorem on the interval &lt;span class=&quot;math-container&quot;&gt;$[0, 1]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;II: You can apply the Intermediate Value Theorem on the interval &lt;span class=&quot;math-container&quot;&gt;$[0, 1]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;III: &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differentiable on &lt;span class=&quot;math-container&quot;&gt;$(0, 1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;IV: &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous on &lt;span class=&quot;math-container&quot;&gt;$(0, 1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;a) III only&lt;/p&gt;&#xA;&lt;p&gt;b) IV only&lt;/p&gt;&#xA;&lt;p&gt;c) II and III only&lt;/p&gt;&#xA;&lt;p&gt;d) I, II, III, and IV&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;5) Suppose &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt;. Find the value of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f'(c)=\frac{9-4}{3-2}$&lt;/span&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;a) 2.5&lt;/p&gt;&#xA;&lt;p&gt;b) 6.25&lt;/p&gt;&#xA;&lt;p&gt;c) No such &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; exists because &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not continuous on &lt;span class=&quot;math-container&quot;&gt;$[2, 3]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;d) No such &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; exists because &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not differentiable on &lt;span class=&quot;math-container&quot;&gt;$[2, 3]$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14024" LastEditorUserId="14024" LastEditDate="2021-10-20T18:16:37.337" LastActivityDate="2021-10-20T18:16:37.337" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24496" PostTypeId="1" AcceptedAnswerId="24519" CreationDate="2021-10-21T03:24:54.217" Score="4" ViewCount="325" Body="&lt;p&gt;Is it good idea to encourage students to look up more about particular topics that interest them? The idea is that I don't think they will understand how to read math literature.&lt;/p&gt;&#xA;&lt;p&gt;for example, if a student in a higher level algebra course asks &amp;quot;can every polynomial be solved?&amp;quot; Is it beneficial to point them to the Abel-Ruffini theorem? They can clearly understand that the result is no, at least not written in radicals, without proof. In order for them to actually understand the depth of the result requires a whole course on abstract algebra which is way beyond their level of comprehension.&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="10335" LastEditDate="2021-10-22T22:44:13.970" LastActivityDate="2021-10-28T22:12:46.740" Title="is it beneficial to encourage high school students to conduct their own 'research' in mathematics?" Tags="|secondary-education|education-research|general-pedagogy|abstract-algebra|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24497" PostTypeId="2" ParentId="7526" CreationDate="2021-10-21T15:57:06.077" Score="0" Body="&lt;p&gt;In general there is no good reason to repeat questions as such (even if they are not made available, one should assume they &lt;em&gt;are&lt;/em&gt; available, being circulated online or via messaging). If students know that questions might be repeated and solutions to past exams are available, many students will simply memorize the solutions.&lt;/p&gt;&#xA;&lt;p&gt;It's reasonable to put similar questions from year to year, as one generally wants to test similar competencies from year to year. However, excessive similarity from year to year generates a situation where students learn how to solve particular problem typologies rather than how to do mathematics. Being able to solve problems of a particular form and structure is a particularly &lt;em&gt;useless&lt;/em&gt; skill, and it is a bad idea to encourage students in this direction because they already receive a lot of push in this direction, from family, friends, themselves, and paid tutors.&lt;/p&gt;&#xA;&lt;p&gt;It will work if there are variations in the problem statements (what appear to an instructor minor variations can be far less minor from a student's point of view - I give an example below) and if the subset of problem typologies used on any given exam is a small subset of all those available (that is to say, over the years there are 15-20 different kinds of problems, in any given year maybe 4 or 5 appear). This makes the exam less predictable but retains the desired effect of guiding students in their preparation.&lt;/p&gt;&#xA;&lt;p&gt;What I mean by minor variation is the following. I can ask&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Find all &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$Ax = rx$&lt;/span&gt; has a nontrivial solution.&lt;/li&gt;&#xA;&lt;li&gt;Find all &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$(A - rI)x = 0$&lt;/span&gt; has a nontrivial solution.&lt;/li&gt;&#xA;&lt;li&gt;Find the eigenvalues of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These are mathematically equivalent problems, but their statements are not equivalent in difficulty (they are listed in order of decreasing difficulty). In particular, many students are unable to solve 1 because they can't see that it is the same as 2; many students learn a procedure for finding eigenvalues (calculate some determinant) and don't relate eigenvalues to the equation appearing in 2, so 3 is easier than 2. One might put some variant of this question some years (varying A, and not every year), varying the difficulty, by changing the phrasing as above, depending on where one wants to center the exam's difficulty and what other problems appear on the exam (maybe form 3 is ok this year because other questions are more conceptual).&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2021-10-21T15:57:06.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24498" PostTypeId="2" ParentId="24491" CreationDate="2021-10-22T01:09:36.557" Score="6" Body="&lt;p&gt;Normally I avoid semantics, but in this case, they're quite important.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is it possible to &lt;strong&gt;understand&lt;/strong&gt; decimals without &lt;strong&gt;understanding&lt;/strong&gt; fractions?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h2&gt;No. Such an understanding is literally impossible.&lt;/h2&gt;&#xA;&lt;p&gt;With a strict and literal sense of the word &amp;quot;&lt;strong&gt;understanding&lt;/strong&gt;&amp;quot;, no. If you do not understand fractions - and tenths in particular - then the conventional meaning of &lt;span class=&quot;math-container&quot;&gt;$3.7$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$3\frac{7}{10}$&lt;/span&gt; cannot make sense. Without this basic factual knowledge, a &lt;em&gt;precise&lt;/em&gt; understanding of &lt;span class=&quot;math-container&quot;&gt;$10.1\div3.7$&lt;/span&gt; is literally impossible.&lt;/p&gt;&#xA;&lt;p&gt;And make no mistake: By the hundreds of millions, &lt;a href=&quot;https://matheducators.stackexchange.com/a/15284/7151&quot;&gt;students have no idea WTF is going on with fractions due to the way they were taught&lt;/a&gt;. So their understanding of decimals such as &lt;span class=&quot;math-container&quot;&gt;$3.7$&lt;/span&gt; will range from non-existent to reasonable, but never strong nor precise.&lt;/p&gt;&#xA;&lt;h2&gt;Numbers Written in Decimal Format Activate Prior Knowledge of Whole Numbers &amp;amp; Money&lt;/h2&gt;&#xA;&lt;p&gt;When calculating &lt;span class=&quot;math-container&quot;&gt;$10.1\div3.7=$&lt;/span&gt; ____ , it &lt;em&gt;is&lt;/em&gt; possible for fractionally weak students to have reasonable&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;intuitions&lt;/li&gt;&#xA;&lt;li&gt;estimates (&amp;quot;It's kinda like &lt;span class=&quot;math-container&quot;&gt;$\$10\div\$4$&lt;/span&gt;&amp;quot;)&lt;/li&gt;&#xA;&lt;li&gt;written procedures and&lt;/li&gt;&#xA;&lt;li&gt;quotients in simplest decimal form&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;One reason that fractionally deficient students find &lt;span class=&quot;math-container&quot;&gt;$10.1\div3.7=$&lt;/span&gt; ____ somewhat intuitive is that it &lt;em&gt;looks and acts&lt;/em&gt; much more like whole number knowledge and money. Reasoning that &lt;span class=&quot;math-container&quot;&gt;$3.7$&lt;/span&gt; is like &lt;span class=&quot;math-container&quot;&gt;$\$3.70$&lt;/span&gt; which is nearly &lt;span class=&quot;math-container&quot;&gt;$\$4$&lt;/span&gt; is basically correct, even if students have no understanding of tenths or hundredths or how those correspond to dimes and pennies. The algorithm for dividing &lt;span class=&quot;math-container&quot;&gt;$10.1$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$3.7$&lt;/span&gt; is just a couple of dots different from the algorithm for dividing &lt;span class=&quot;math-container&quot;&gt;$101$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$37$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;h2&gt;Numbers Written in Fraction Format Activate Dread &amp;amp; Confusion&lt;/h2&gt;&#xA;&lt;p&gt;In contrast, a student could look at an equivalent expression with fractions, &lt;span class=&quot;math-container&quot;&gt;$\frac{101}{10}\div\frac{37}{10}$&lt;/span&gt;, and think: &amp;quot;Can I cancel the zeroes? Can I cancel the tens? What do I divide first? There's so much division! Do I multiply? Do I flip something? Then do I multiply after? What are the rules for denominators? Oh, I know, maybe I &lt;em&gt;cross-multiply&lt;/em&gt; and divide after? Can I just multiply through by 10 and have no more fractions? That would make this question &lt;em&gt;way&lt;/em&gt; easier. Wait... You want me to estimate &lt;em&gt;this&lt;/em&gt;? How the heck do I do that? Oh, you want me to &lt;em&gt;draw&lt;/em&gt; it? You can't &lt;em&gt;draw&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{101}{10}$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$101$&lt;/span&gt; out of &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; is impossible. I can't show you &lt;span class=&quot;math-container&quot;&gt;$101$&lt;/span&gt; out of my &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; fingers. God, I f^%@&amp;amp;ing hate fractions. I &lt;em&gt;hate&lt;/em&gt; them. I hate them &lt;em&gt;so much&lt;/em&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;As linked above, many students actually are that confused when it comes to fractions.&lt;/p&gt;&#xA;&lt;h2&gt;Conclusion&lt;/h2&gt;&#xA;&lt;p&gt;Numbers written in decimal format tend to activate prior knowledge of whole numbers while the exact same numbers written in fractional format will often activate dreadful confused memories of weird procedures. It's no wonder why students instinctively switch towards decimal numbers - those are familiar.&lt;/p&gt;&#xA;&lt;p&gt;And students, by the millions, will, of course, soon desire the impossible conversion of &lt;span class=&quot;math-container&quot;&gt;$\frac{x+3}{x^2-9}$&lt;/span&gt; to decimal format at which point weeder math classes will&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Yank them from any academic or professional future involving algebra or higher-level math and&lt;/li&gt;&#xA;&lt;li&gt;Leave a terrible impression of math and their own abilities forever.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Until fractions are taught as numbers and decimals are taught as fractions with base 10 denominators, the phenomena described in the original post will continue.&lt;/p&gt;&#xA;" OwnerUserId="7151" LastActivityDate="2021-10-22T01:09:36.557" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24499" PostTypeId="2" ParentId="24401" CreationDate="2021-10-22T05:50:37.340" Score="6" Body="&lt;p&gt;I normally go other way around.&lt;/p&gt;&#xA;&lt;p&gt;Start with a logarithm, which by definition maps multiplication to addition: &lt;span class=&quot;math-container&quot;&gt;$\ln{ab} = \ln{a} + \ln{b}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;A detour into a history could be also useful: multiplication is hard, addition is easy; this mapping, they say, prolonged the lives of astronomers&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Prove, by a sheer geometrical argument, that the &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \int_1^x\dfrac {1}{t}dt$&lt;/span&gt; satisfy the above definition. You may shamelessly use the idea of an area of a curvilinear trapezoid. They will understand.&lt;/p&gt;&#xA;&lt;p&gt;Use a fundamental theorem to demonstrate that &lt;span class=&quot;math-container&quot;&gt;$(\ln{x})' = \dfrac{1}{x}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Introduce the inverse function, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. Show that it maps addition to multiplication. Show how it naturally extends the idea of power. Use the the inverse function derivative theorem to demonstrate that &lt;span class=&quot;math-container&quot;&gt;$f' = f$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Make sure to spend time to discuss how &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; emerges, how other bases come to play, and what makes &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; special.&lt;/p&gt;&#xA;" OwnerUserId="17936" LastEditorUserId="17936" LastEditDate="2021-10-22T06:11:23.283" LastActivityDate="2021-10-22T06:11:23.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24500" PostTypeId="2" ParentId="14552" CreationDate="2021-10-22T12:10:31.360" Score="6" Body="&lt;p&gt;I am going to try the following activity as a first introduction to Mathematical Induction on Monday next week.  I will let you know how it goes.&lt;/p&gt;&#xA;&lt;p&gt;The implication &lt;span class=&quot;math-container&quot;&gt;$P(k) \implies P(k+1)$&lt;/span&gt; let's you &amp;quot;hop around&amp;quot; the natural numbers, deciding the proof of new statements using your knowledge of the truth value of old statements.  However, it is a bit too straightforward to see what all the fuss is about.  The underwhelming and boring nature of the hopping (just put one foot in front of the other) doesn't really permit any play.  Without play there can be no learning.  So here, I first pose two more interesting rules for &amp;quot;hopping&amp;quot; which give interesting play opportunities (they are actually a puzzle).  The final example gives the &amp;quot;obvious&amp;quot; induction rule, which should now feel truly obvious to the student.  At this point we will formalize what we have learned as the principle of mathematical induction.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Alice, Bob, and Chelle are three mathematicians.  As mathematicians, they have a love of certain numbers.  Also, as mathematicians, their love is quite idiosyncratic.&lt;/p&gt;&#xA;&lt;p&gt;Alice loves the natural numbers 1 and 2.  Also, if she loves the natural number &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, then she also loves the natural number &lt;span class=&quot;math-container&quot;&gt;$k+5$&lt;/span&gt;.  Which natural numbers can you be certain that Alice loves?  Are there any natural numbers you can be sure she does not love?  Are there any natural numbers you just do not have enough information about to decide this question?&lt;/p&gt;&#xA;&lt;p&gt;Bob loves the natural number 5.  If he loves the natural number &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, then he also loves the natural number &lt;span class=&quot;math-container&quot;&gt;$2k$&lt;/span&gt;.  Also, if he loves the natural number &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;, he also loves &lt;span class=&quot;math-container&quot;&gt;$j-2$&lt;/span&gt;. Which natural numbers can you be certain that Bob loves?  Are there any natural numbers you can be sure he does not love?  Are there any natural numbers you just do not have enough information about to decide this question?&lt;/p&gt;&#xA;&lt;p&gt;Chelle loves the natural number 1.  If she loves the natural number &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, then she also loves the natural number &lt;span class=&quot;math-container&quot;&gt;$k+1$&lt;/span&gt;.  Which natural numbers can you be certain that Chelle loves?  Are there any natural numbers you can be sure she does not love?  Are there any natural numbers you just do not have enough information about to decide this question?&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-10-22T12:10:31.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24501" PostTypeId="2" ParentId="24491" CreationDate="2021-10-23T00:48:05.360" Score="3" Body="&lt;p&gt;So much depends upon what you mean by &amp;quot;understand&amp;quot; decimals. Think about this in terms of &lt;a href=&quot;https://en.wikipedia.org/wiki/Chinese_room&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Searle's Chinese Room&lt;/strong&gt;&lt;/a&gt;. Also see this &lt;a href=&quot;https://www.youtube.com/watch?v=D0MD4sRHj1M&quot; rel=&quot;nofollow noreferrer&quot;&gt;BBC Studios video&lt;/a&gt; with Marcus du Sautoy. In this thought experiment, we imagine writing a computer program that gives person A, locked in a room and who knows nothing about Chinese characters, enough instructions to mindlessly manipulate characters to trick person B outside the room and who understands Chinese, into thinking that person A &amp;quot;understands&amp;quot; Chinese.&lt;/p&gt;&#xA;&lt;p&gt;Replace Chinese characters in the Searle Chinese Room thought experiment with decimals, and then with rational numbers (fractions). My sense is that it would be much easier to write the AI instructions for manipulation of decimals than it would be for fractions. So in this sense, I think the answer to your question is &amp;quot;Yes, it is possible to understand decimals without understanding fractions.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I do agree that reliance on digital calculators inhibits a person's ability to understand fractions. Think about the wild discussions about whether &lt;span class=&quot;math-container&quot;&gt;$0.9\overline{9}$&lt;/span&gt; equals 1, or if it is a tiny bit less than 1. Clearly, people have a difficult time with this concept and this points to a subtle relationship between decimals and rational numbers. Yet if you ask people if &lt;span class=&quot;math-container&quot;&gt;$0.3\overline{3}$&lt;/span&gt; is equal to &lt;span class=&quot;math-container&quot;&gt;$\frac13$&lt;/span&gt; or slightly less, I think there is no misunderstanding:  &lt;span class=&quot;math-container&quot;&gt;$99.\overline9\%$&lt;/span&gt; of people will easily agree that &lt;span class=&quot;math-container&quot;&gt;$0.3\overline{3}$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$\frac13$&lt;/span&gt;. Why is this? I hypothesize that it is because the input &lt;span class=&quot;math-container&quot;&gt;$1\div3$&lt;/span&gt; on a digital calculator results in &lt;span class=&quot;math-container&quot;&gt;$0.3333333333$&lt;/span&gt;. The brain parses the finite string as &lt;span class=&quot;math-container&quot;&gt;$0.3\overline3$&lt;/span&gt; as so of course this equals &lt;span class=&quot;math-container&quot;&gt;$\frac13$&lt;/span&gt;. I have often thought how diabolically fun it would be to design a calculator that upon entering 1 from the keypad, the displayed result is &lt;span class=&quot;math-container&quot;&gt;$0.9999999999$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2021-10-23T00:48:05.360" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24502" PostTypeId="1" CreationDate="2021-10-23T04:52:38.893" Score="0" ViewCount="165" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Linear_function&amp;amp;oldid=1027148144&quot; rel=&quot;nofollow noreferrer&quot;&gt;It appears in the current version of the Wikipedian article Linear function&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In mathematics, the term linear function refers to two distinct but&#xA;related notions:&lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Linear_function&amp;amp;oldid=1027148144&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In calculus and related areas, a linear function is a function whose&#xA;graph is a straight line, that is, a polynomial function of degree&#xA;zero or one.[2] For distinguishing such a linear function from the&#xA;other concept, the term affine function is often used.[3]&lt;/p&gt;&#xA;&lt;p&gt;In linear&#xA;algebra, mathematical analysis,[4] and functional analysis, a linear&#xA;function is a linear map.[5]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Which context (&amp;quot;notion&amp;quot;) of the two contexts of the &lt;em&gt;linear function&lt;/em&gt; concept should be taught first?&#xA;&lt;br&gt;Which one typically contain more basic math?&#xA;&lt;br&gt;The Calculus context or the linear algebra context?&lt;/p&gt;&#xA;" OwnerUserId="17940" LastActivityDate="2021-10-23T22:17:06.383" Title="Which context of the two contexts of the linear function concept should be taught first?" Tags="|linear-algebra|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24503" PostTypeId="2" ParentId="24476" CreationDate="2021-10-23T06:16:08.743" Score="1" Body="&lt;p&gt;Indian stastical instituite also has a archive of past exams and notes here for&#xA;&lt;a href=&quot;https://www.isibang.ac.in/%7Eadean/infsys/database/&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathematics&lt;/a&gt; and some basic courses in statistics and computer science&lt;/p&gt;&#xA;" OwnerUserId="14662" LastActivityDate="2021-10-23T06:16:08.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24504" PostTypeId="2" ParentId="24502" CreationDate="2021-10-23T15:02:05.323" Score="2" Body="&lt;p&gt;The concepts are unrelated (at least in the context of mathematical teaching), and students are at least roughly familiar with both before they get to calculus or linear algebra.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In calculus, a linear function from &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; to itself is of the form &lt;span class=&quot;math-container&quot;&gt;$f(x)=ax+b$&lt;/span&gt;, because it has a constant derivative and its graph is a straight line in the plane.&lt;/li&gt;&#xA;&lt;li&gt;In linear algebra, the only linear automorphism from &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; to itself is of the form &lt;span class=&quot;math-container&quot;&gt;$f(x)=ax$&lt;/span&gt;, because only that carries the linear structure of the domain into the range of the function.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In practice (at least in any linear algebra courses and textbooks I've read), there is no confusion between the two concepts.  Those treatments refer to linear transformations or linear operators (or linear forms in more specific cases where the codomain is &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;) instead of linear maps, so it's never struck me until just now that anyone would call it a linear function.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-10-23T15:02:05.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24505" PostTypeId="2" ParentId="12690" CreationDate="2021-10-23T15:53:22.957" Score="2" Body="&lt;p&gt;I have been searching the corect wording for a while in the same context, that is, implementing a function that represents the decimal expansion of &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt;. But then, I landed in a word desert when I needed that function to handle base 2 to 10.&lt;/p&gt;&#xA;&lt;p&gt;I naturally came to google &lt;code&gt;&amp;quot;n-imal&amp;quot; fractions&lt;/code&gt; and found that the term &amp;quot;n-imal&amp;quot; is used here and there (@see ref. below) to designate a fraction expressed in base n, or the fractional part of a rational expressed in base n, but it seems no strict convention exists.&lt;/p&gt;&#xA;&lt;p&gt;By fractional part I mean the non-integer part represented by the digits after the radix point (I'm french to complicate the thing so I'm not sure if it's clear).&lt;/p&gt;&#xA;&lt;p&gt;Using &amp;quot;n-ary&amp;quot; is ambiguous, for example we already use &amp;quot;binary&amp;quot; to mean that something is &amp;quot;binary&amp;quot;, but how do we name the fractional part of a binary number ? We're left with &amp;quot;binary&amp;quot; or &amp;quot;bit&amp;quot; (binary digit), something is definitely missing.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, &amp;quot;n-imal&amp;quot; allows for words like &amp;quot;binimal&amp;quot; to be used, which can't be confused with &amp;quot;binary&amp;quot; (ie. for a binary system, or the binary representation of a number).&lt;/p&gt;&#xA;&lt;p&gt;The problem is also that most of the time, we wrongly refer to base-10 numbers as &amp;quot;decimal&amp;quot; numbers, but we should call them &amp;quot;decanary&amp;quot; (or &amp;quot;denary&amp;quot; ?) numbers so that &amp;quot;decimals&amp;quot; can only refer to the fractional part of a decanary number, or by extension, to decanary numbers having a fractional part. The same logic should apply for &amp;quot;hexadecanary&amp;quot; vs &amp;quot;hexadecimal&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;It seems most people don't care about this, but one of the most important thing in mathematics and computer science is to avoid ambiguity.&lt;/p&gt;&#xA;&lt;p&gt;For consistency, I would recommend using something like :&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;base&lt;/th&gt;&#xA;&lt;th&gt;n-imal&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;2&lt;/td&gt;&#xA;&lt;td&gt;binimal&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;3&lt;/td&gt;&#xA;&lt;td&gt;trinimal&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;4&lt;/td&gt;&#xA;&lt;td&gt;quadrimal&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;5&lt;/td&gt;&#xA;&lt;td&gt;pentimal&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;6&lt;/td&gt;&#xA;&lt;td&gt;heximal&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;7&lt;/td&gt;&#xA;&lt;td&gt;septimal&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;8&lt;/td&gt;&#xA;&lt;td&gt;octimal&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;9&lt;/td&gt;&#xA;&lt;td&gt;9-imal (?)&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;10&lt;/td&gt;&#xA;&lt;td&gt;decimal&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;...&lt;/td&gt;&#xA;&lt;td&gt;???&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;So to answer the question, I would say that :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; is to fraction as &lt;span class=&quot;math-container&quot;&gt;$.c$&lt;/span&gt; is to &lt;strong&gt;n-imal&lt;/strong&gt; (with &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;b$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; represents the fractional part of &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; in base-n).&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; is to fraction as &lt;span class=&quot;math-container&quot;&gt;$.5$&lt;/span&gt; is to &lt;strong&gt;decimal&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{01}{10}_{2}$&lt;/span&gt; is to fraction as &lt;span class=&quot;math-container&quot;&gt;$.1_{2}$&lt;/span&gt; is to &lt;strong&gt;binimal&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; = &lt;span class=&quot;math-container&quot;&gt;$.5_{10}$&lt;/span&gt; : one half equals zero point five decimals, or zero decimal five.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; = &lt;span class=&quot;math-container&quot;&gt;$.1_{2}$&lt;/span&gt; : one half equals zero point one binimal, or zero binimal one.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Here some references using n-imal/n-mal wording :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.fr/books?id=XuXDrtthXHcC&amp;amp;pg=PA196&amp;amp;lpg=PA196&amp;amp;dq=%22n-imal%22#v=onepage&amp;amp;q=%22n-imal%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Topology of Chaos: Alice in Stretch and Squeezeland&lt;/a&gt; (&lt;a href=&quot;https://www.physics.drexel.edu/%7Ebob/PHYS750_NLD/branched_man.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Branched Manifolds&lt;/a&gt;, p.195-196)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.fr/books?id=92C-DwAAQBAJ&amp;amp;pg=PA243&amp;amp;lpg=PA243&amp;amp;dq=%22binimal%22#v=onepage&amp;amp;q=%22binimal%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;ARM 64-Bit Assembly Language&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.exploringbinary.com/bicimals/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rick Regan advocating for the use of &amp;quot;bicimal&amp;quot;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.fr/books?id=5WjDDwAAQBAJ&amp;amp;pg=PA140&amp;amp;lpg=PA140#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Knowledge and the Philosophy of Number: What Numbers Are and How They Are Known&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.tapatalk.com/groups/dozensonline/idea-for-pronouncing-digital-fractions-in-multiple-t2227-s24.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Idea for pronouncing digital fractions in multiple bases: &amp;quot;N-imals&amp;quot;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ijpam.eu/contents/2013-84-3/11/11.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;International Journal of Pure and Applied Mathematics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.fr/books?id=gks1CgAAQBAJ&amp;amp;pg=PA224&amp;amp;lpg=PA224&amp;amp;dq=%22binimal%22#v=onepage&amp;amp;q=%22binimal%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Modern Assembly Language Programming with the ARM Processor&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;University of Illinois : &lt;a href=&quot;http://new.math.uiuc.edu/public348/cantor.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cantor Cardinality&lt;/a&gt;, &lt;a href=&quot;http://new.math.uiuc.edu/public348/analysis/bolzano.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bolzano-Weierstrass&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="17945" LastEditorUserId="17945" LastEditDate="2021-10-27T10:54:49.833" LastActivityDate="2021-10-27T10:54:49.833" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24506" PostTypeId="2" ParentId="24502" CreationDate="2021-10-23T22:17:06.383" Score="4" Body="&lt;p&gt;The first is taught first.&lt;/p&gt;&#xA;&lt;p&gt;(In the US, typically) lines are taught in 9th grade Algebra 1.  You learn to graph and write equations for lines.  It's a huge part of the year.&lt;/p&gt;&#xA;&lt;p&gt;The more advanced (but different) concept is from college/grad school classes.  Linear algebra is a college class, sophomore year.  Real analysis is taught junior year of college.  Functional analysis is a graduate course.&lt;/p&gt;&#xA;&lt;p&gt;I would add also that the term &amp;quot;linear&amp;quot; is used in differential equations for specific sorts of diffyQs. (Which are typically a required sophomore level college course, ODEs, and an optional junior level course, PDEs.)&lt;/p&gt;&#xA;&lt;p&gt;I agree with Matt.  The concepts are not confused by students.  For one thing the kids who are in 9th grade, don't know all that other stuff coming later (which many will never have).  And then the kids who do get these harder classes, later, with the use of the term linear, will have plenty of time/explanation to realize these are not y=mx+b beasts they are dealing with.&lt;/p&gt;&#xA;&lt;p&gt;This isn't the only case where words may mean different things in different courses.  Makes me think of this review:  &lt;a href=&quot;https://www.econjobrumors.com/topic/review-of-abstract-algebra-dummit-foote&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.econjobrumors.com/topic/review-of-abstract-algebra-dummit-foote&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17947" LastActivityDate="2021-10-23T22:17:06.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24507" PostTypeId="2" ParentId="19103" CreationDate="2021-10-25T02:53:21.073" Score="-1" Body="&lt;p&gt;With others, I think the mechanical aspects of adding etc. to sides of the equation is easier at first.  And super helpful in really driving the concept of the equality idea.  When you do it in detail, you have a set of steps where you add, divide, etc. on both sides of the equation.  NOT &amp;quot;flipping over the equals and reversing sign&amp;quot;--that is sort of a quick way to think of it later.&lt;/p&gt;&#xA;&lt;p&gt;So 2x + 5 = 6 - 7x&lt;/p&gt;&#xA;&lt;p&gt;[Add 7x to both sides.]&lt;/p&gt;&#xA;&lt;p&gt;2x + 5 +7x = 6 - 7x +7x&lt;/p&gt;&#xA;&lt;p&gt;[Combine terms in x.]&lt;/p&gt;&#xA;&lt;p&gt;9x + 5 =  6&lt;/p&gt;&#xA;&lt;p&gt;[subtract 5 from each side.]&lt;/p&gt;&#xA;&lt;p&gt;9x + 5 -6 = 6 - 6&lt;/p&gt;&#xA;&lt;p&gt;[combine the numbers]&lt;/p&gt;&#xA;&lt;p&gt;9x - 1 = 0&lt;/p&gt;&#xA;&lt;p&gt;[add one to each side]&lt;/p&gt;&#xA;&lt;p&gt;9x -1 + 1 = 0 + 1&lt;/p&gt;&#xA;&lt;p&gt;[combine the numbers]&lt;/p&gt;&#xA;&lt;p&gt;9x = 1&lt;/p&gt;&#xA;&lt;p&gt;[Divide both sides by 9.]&lt;/p&gt;&#xA;&lt;p&gt;9x/9 = 1/9&lt;/p&gt;&#xA;&lt;p&gt;[Do the division.]&lt;/p&gt;&#xA;&lt;p&gt;x = 1/9&lt;/p&gt;&#xA;&lt;p&gt;Kids should actually write this whole thing down (not the bracketed comments, but the step by step solution).  It's very powerful for them to learn this process. Remember that &amp;quot;x&amp;quot; is a scary thing.  We're doing math with letters in it now!!!  So doing this repeated, clear manipulation is useful for them to start understanding equations and how to manipulate them.&lt;/p&gt;&#xA;&lt;p&gt;Factors are kind of a short cut.  And not always easy to identify by eye.  So teach them the methodical method first.  Then when having a general comfort with letters in math and with manipulations, they can move to factoring.&lt;/p&gt;&#xA;&lt;p&gt;Plus if you have exes and numbers on both sides of the equals, you're going to need to do some manipulation anyway, even if you factor later.  But I would still teach the brute force step by step method first.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  The &amp;quot;by inspection&amp;quot; stuff is fine and I do it and the sharper kids jump to it also.  (What Feynman called solving it with arithmetic instead of algebra.)  But it's not always easy to do this with a complicated expression.  And there's nothing to stop going over this stuff later.  But one thing at a time.  Especially with weaker students.&lt;/p&gt;&#xA;" OwnerUserId="17947" LastActivityDate="2021-10-25T02:53:21.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24508" PostTypeId="1" CreationDate="2021-10-26T00:44:14.490" Score="-6" ViewCount="156" Body="&lt;p&gt;I am not a mathematics teacher, I just want to know which function is more elementary and is more reasonable to be learnt first:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Constant function (&lt;code&gt;f(x)=1&lt;/code&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Identity function (&lt;code&gt;f(x)=x&lt;/code&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Thank you,&lt;/p&gt;&#xA;" OwnerUserId="17940" LastActivityDate="2021-10-26T19:06:53.737" Title="Which function is more elementary: Constant function or identity function?" Tags="|mathematical-pedagogy|" AnswerCount="1" CommentCount="7" ClosedDate="2021-10-27T00:33:06.780" ContentLicense="CC BY-SA 4.0" />
  <row Id="24509" PostTypeId="2" ParentId="24508" CreationDate="2021-10-26T13:49:22.593" Score="1" Body="&lt;p&gt;In algebra 1, kids learn y =mx+b.  Then, soon after and during same year m=0 or b=0 are special cases of the general.  Not sure which of the subcases first between each other (doesn't matter much) and in any cases, they're learned as types of lines.  (Of course the one other line they learn is x = k, which of course is not a function--is a relation--and is not a subset of y = mx +b.)&lt;/p&gt;&#xA;&lt;p&gt;Given your previous question about real analysis and linear algebra content, I don't really get the question here.  Seems so simple for someone so advanced.  If it's not a troll (which it, Bayesian bet, probably is), then I'm confused what you're asking and how to help/engage.&lt;/p&gt;&#xA;&lt;p&gt;(Stereotypically, in the US) you'll have algebra 1 in 9th, geometry in 10th, algebra2 in 11th and get function concept as part of 12th precalc.  So in learning the function concept, you have some background of knowledge to use.&lt;/p&gt;&#xA;&lt;p&gt;Bottom line is if you start working on the function CONCEPT, you're generally doing that after already having some exposure to equations, graphing, etc. so you have a baseline of experience to match against the function concept itself.  And then you work on function concept (in/out machine, pairs of numbers, certain equations, graphically).  In particular, understanding functions (no more than one y for an x) versus non-function relations.  Like y=mx+b versus xsq+ysq=rsq.  And the domain and range and blabla.&lt;/p&gt;&#xA;&lt;p&gt;But when you learn the function CONCEPT, you're concentrating on the concept itself.  You don't have to learn it fresh for different polynomials.  You already have a background on those erquations from algebra 1 and 2.  That hasn't disappeared.  In fact, it's a resource to use.  Maybe you'll learn a few functions that are new (like excluded end point, drawn with open circle, or like top half of a circle).  But you don't have to relearn y=mx+b.&lt;/p&gt;&#xA;" OwnerUserId="17957" LastEditorUserId="17957" LastEditDate="2021-10-26T19:06:53.737" LastActivityDate="2021-10-26T19:06:53.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24510" PostTypeId="1" CreationDate="2021-10-26T20:58:03.143" Score="0" ViewCount="296" Body="&lt;p&gt;Here is a question on a test I asked recently in a Calculus I class I teach on Saturdays. These are students who are planning to major in Math, and they are already taking AP Calculus BC in their high schools, but some want to take it a step further and practice more (eg: go deeper) on the weekends. Anyway, here is the question and, with permission of the student, I have posted his solution. I have my own solution too, but it is different in scope, so I do not want to post it yet. Any other approaches to this problem? As a hint, I will say that my solution uses the Weierstrass Theorem (the extreme value theorem). I tried to add more tags to the post, but apparently more points are required.&lt;/p&gt;&#xA;&lt;p&gt;A sign, posted on a wall, has its upper and lower edges represented by heights m and n, respectively, with reference to a reader's vision. How far from the wall should a person reading the sign be located so that the visual angle determined by the pupil and the edges is at a maximum?&lt;/p&gt;&#xA;&lt;p&gt;A student solved it as follows:&lt;/p&gt;&#xA;&lt;p&gt;Let us suppose &lt;span class=&quot;math-container&quot;&gt;$m&amp;gt;n$&lt;/span&gt; (in other words, m represents the height of the top edge and r represents the height of the bottom edge, so &lt;span class=&quot;math-container&quot;&gt;$m&amp;gt;n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thus, &lt;span class=&quot;math-container&quot;&gt;$$\theta_1=\arctan\left(\frac{m}{d}\right), \theta_2=\arctan\left(\frac{n}{d}\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We are trying to maximize the difference between &lt;span class=&quot;math-container&quot;&gt;$\theta_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\theta_2$&lt;/span&gt;. Essentially we need to maximize &lt;span class=&quot;math-container&quot;&gt;$\arctan\left(\frac{m}{d}\right)-\arctan\left(\frac{n}{d}\right)$&lt;/span&gt;.&#xA;We take the derivative with respect to &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; to get:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{n}{n^2+d^2}-\frac{m}{m^2+d^2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Set this equal to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and solve for &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; to get:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$d=\sqrt{mn}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="15988" LastActivityDate="2022-03-26T04:02:20.217" Title="Distance from the wall when reading a sign? A Calc I/II problem" Tags="|calculus|" AnswerCount="1" CommentCount="3" ClosedDate="2022-04-01T01:22:38.410" ContentLicense="CC BY-SA 4.0" />
  <row Id="24511" PostTypeId="2" ParentId="24510" CreationDate="2021-10-27T02:21:53.120" Score="1" Body="&lt;p&gt;I always find it fun to try to avoid calculus in problems like this, so if you want to show your students some applications of geometry and trig, how about something like this: use geometric reasoning to see that the angle is maximized when your eye is at the bottom of the circle containing your eye and the poster (since, looking at the angle at the center of the circle, you want &lt;span class=&quot;math-container&quot;&gt;$mn$&lt;/span&gt; to be a chord of the smallest such circle).&lt;/p&gt;&#xA;&lt;p&gt;Now we get &lt;span class=&quot;math-container&quot;&gt;$\tan\phi=n/d$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\tan\theta=\frac{m-n}{2d}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\tan(\theta+\phi)=m/d$&lt;/span&gt;. The angle sum formula and algebra take care of the rest. (I tried to resize the picture but sticking on an &amp;quot;s&amp;quot; was too small and an &amp;quot;m&amp;quot; previewed as almost all black! If I figure it out I'll edit.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OfQ6M.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OfQ6M.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2021-10-27T02:21:53.120" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24512" PostTypeId="1" CreationDate="2021-10-27T04:29:46.397" Score="15" ViewCount="4701" Body="&lt;p&gt;Why do people think long division of polynomials is complicated ?&lt;/p&gt;&#xA;&lt;p&gt;I heard this expressed recently and it seems like an odd sentiment. For me, synthetic division is complicated and totally adhoc because it looks nothing like the division of integers I remember from my childhood. In contrast, long division of polynomials is just the same process I know and love from the arithmetic of my youth. To me, long division of polynomials is just a natural extension of long division of numbers.&lt;/p&gt;&#xA;&lt;p&gt;I generally refuse to cover synthetic division because it's not significantly faster and it fails to cover division by nonlinear polynomials. Why teach something which is weird and not general. Just to make that particular calculation faster ?&lt;/p&gt;&#xA;&lt;p&gt;In addition, long division of polynomials requires multiplication of polynomials in the calculation. This is a skill which needs reinforcement. In contrast, synthetic division is a one-way street of shifting numbers around without any particular attachment to the larger discussion of polynomial algebra.&lt;/p&gt;&#xA;&lt;p&gt;What am I missing ?&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-10-30T09:44:04.907" Title="Why the fear of polynomial long division?" Tags="|undergraduate-education|mathematical-pedagogy|algebra|" AnswerCount="5" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="24513" PostTypeId="2" ParentId="24512" CreationDate="2021-10-27T10:33:32.067" Score="4" Body="&lt;p&gt;I think the fear of poly long division is based on the amount of symbol manipulation and writing.  So, I get how the students feel.  But completely agree with you that this needs reinforcement.&lt;/p&gt;&#xA;&lt;p&gt;For synthetic division, I don't remember ever using synthetic division in any later math or science classes. (To the extent, I don't recall learning it.  Probably did, went to very good school...but maybe even didn't.)  Maybe synthetic division is a little like the tic-tac-toe (tabular integration) shown in Stand and Deliver.  Kind of a cool flourish and worthwhile if you do a gazillion problems.  And does take some of the pain of writing and symbol manipulation away.  But still more important to learn/understand the long way as a base.&lt;/p&gt;&#xA;&lt;p&gt;In a time constrained environment and with weaker kids, I would pitch the synthetic division.  It's pretty optional.  Like I said, don't recall using it later in any other course.  I would keep the poly long division since it's kind of analogous to manipulation done later in partial fractions and that gets used in lots of other courses.&lt;/p&gt;&#xA;" OwnerUserId="17958" LastActivityDate="2021-10-27T10:33:32.067" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24514" PostTypeId="2" ParentId="24512" CreationDate="2021-10-27T13:03:23.033" Score="11" Body="&lt;p&gt;I like to include synthetic division as a topic in a college algebra or precalculus course. It is an opportunity to take a 20 minute digression to talk about &lt;a href=&quot;https://en.wikipedia.org/wiki/Horner%27s_method&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Horner's method&lt;/strong&gt;&lt;/a&gt;, which is used in several computer science algorithms.&lt;/p&gt;&#xA;&lt;p&gt;It is a clever way to evaluate a polynomial that improves the naive approach of evaluating all the terms which might be large numbers (appearances of &lt;span class=&quot;math-container&quot;&gt;$c_k 10^k$&lt;/span&gt;), and then adding all these evaluated terms, which might include subtractions of large integers. This sort of perspective can begin opening the eyes of some STEM students to issues related to numerical algorithms. It might put some to sleep. But it excites me, so they wake up! It is the unique day of the semester when I demonstrate using Matlab.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-10-27T13:18:15.527" LastActivityDate="2021-10-27T13:18:15.527" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="24515" PostTypeId="2" ParentId="24471" CreationDate="2021-10-27T17:03:07.497" Score="0" Body="&lt;p&gt;How about offering participation in the classes for extra credit? Or, perhaps participation can be in place of a homework assignment, or in place of a quiz grade? This might take off the pressure of graded assessments, and encourage students to show up and engage with the material without worrying about a grade.&lt;/p&gt;&#xA;" OwnerUserId="17960" LastActivityDate="2021-10-27T17:03:07.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24516" PostTypeId="2" ParentId="24512" CreationDate="2021-10-27T19:29:47.420" Score="9" Body="&lt;p&gt;(This answer is me speaking as a student.  I taught both polynomial long division and synthetic to my first calculus class, but realized by the end of the unit that it was really unneeded for AP questions and didn't teach it again.)&lt;/p&gt;&#xA;&lt;p&gt;So as a person who factors polynomials every once in a while, I don't like the large number of sign flips polynomial division requires for the subtraction phase and the fact that you only need to misread one to get the wrong answer.  I also prefer the reduced writing space that synthetic uses.  (When I learned both techniques back in the day, we were asked to factor cubics and quartics with the rational root test, so there was plenty of scratch paper used up for trial and error.)&lt;/p&gt;&#xA;&lt;p&gt;I get that synthetic looks artificial in the context of numeric long division.  But, TBH, that's a pretty artificial manipulation of figures itself.  I wonder if synthetic would look more attractive to you if your teachers had switched you to short division once you were ready for it....&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2021-10-27T19:29:47.420" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24517" PostTypeId="2" ParentId="24491" CreationDate="2021-10-27T19:30:24.617" Score="1" Body="&lt;p&gt;If some student has grown up in a country where the metric system is used in everyday life (such as continental Europe), I would &lt;strong&gt;definitely say that this is possible&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;p&gt;A student not being familiar with fractions will not know how much &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{12}h$&lt;/span&gt; is; however, he is aware how much 5 minutes is.&lt;/p&gt;&#xA;&lt;p&gt;Such a student will have no idea how much &lt;span class=&quot;math-container&quot;&gt;$\frac{7}{360}h$&lt;/span&gt; is; however, he will have an idea how much 70 seconds is.&lt;/p&gt;&#xA;&lt;p&gt;In the metric system, most &amp;quot;larger units&amp;quot; are &lt;span class=&quot;math-container&quot;&gt;$10^n$&lt;/span&gt; times a &amp;quot;smaller unit&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$1\text{km}=10^3\text{m}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$1\text{m}=10^2\text{cm}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$1\text{cm}=10\text{mm}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A child growing up in a country where the metric system is used in everyday life learns quite early that nearly every unit (kilometers, kilograms ...) works like this.&lt;/p&gt;&#xA;&lt;p&gt;If you tell the student that &lt;span class=&quot;math-container&quot;&gt;$x=123.456789$&lt;/span&gt;, he will not be aware that &lt;span class=&quot;math-container&quot;&gt;$123.456789=123+\frac{456789}{1000000}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, he might think that numbers work the same way as units and think of a pseudo-unit (&lt;span class=&quot;math-container&quot;&gt;$\triangle$&lt;/span&gt;) which is defined as: &lt;span class=&quot;math-container&quot;&gt;$\triangle:=\frac{1}{1000}$&lt;/span&gt;. However, not knowning about fractions, the student would not be aware that &lt;span class=&quot;math-container&quot;&gt;$\triangle$&lt;/span&gt; is defined that way.&lt;/p&gt;&#xA;&lt;p&gt;Now the student would think of &lt;span class=&quot;math-container&quot;&gt;$123.45678905$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$123\text{k}\triangle + 456\triangle + 789\text{m}\triangle + 50\mu\triangle$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I see no reason why it should not be possible that some students think about decimal numbers this way.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;By the way&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you were told: &amp;quot;&lt;em&gt;The time required is 0.345 days&lt;/em&gt;&amp;quot;, most people would also have no idea how much &lt;span class=&quot;math-container&quot;&gt;$\frac{69}{200}\text{d}$&lt;/span&gt; is, but they would think about &lt;span class=&quot;math-container&quot;&gt;$8\text{h} + 16\text{min} + 48\text{s}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7627" LastEditorUserId="7627" LastEditDate="2021-10-28T09:03:21.843" LastActivityDate="2021-10-28T09:03:21.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24518" PostTypeId="2" ParentId="24512" CreationDate="2021-10-27T20:29:07.420" Score="3" Body="&lt;p&gt;How did you learn to multiply 2 or more digit numbers?&lt;/p&gt;&#xA;&lt;p&gt;If you learned &amp;quot;long multiplication&amp;quot;, then so be it.&lt;/p&gt;&#xA;&lt;p&gt;But if you learned the grid method, then synthetic division links right back to that.  In synthetic division, you write out the grid and fill in the gaps to find &amp;quot;what do you need to multiply by the divisor to get the polynomial in the dividend.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This means if you understand why grid multiplication works (which is easier to explain than long multiplication), you can understand why &amp;quot;synthetic division&amp;quot; works.&lt;/p&gt;&#xA;&lt;p&gt;You can teach this division method as just shifting numbers around.  But the strength in the method is not in how quickly students can find an answer, but in the connection that they can make to grid multiplication and the notion of division as the inverse of multiplication.&lt;/p&gt;&#xA;" OwnerUserId="5361" LastEditorUserId="6053" LastEditDate="2021-10-28T21:04:10.717" LastActivityDate="2021-10-28T21:04:10.717" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="24519" PostTypeId="2" ParentId="24496" CreationDate="2021-10-28T22:12:46.740" Score="2" Body="&lt;p&gt;In general, no.  First of all, definitely don't send them to original (old or cutting edge) literature.  It's just too hard.  But really sending them to advanced textbooks is too tricky also.  Kids in Algebra 2 need to learn that stuff first, not Artin.&lt;/p&gt;&#xA;&lt;p&gt;Second, when I say &amp;quot;in general&amp;quot;, probably the majority of your kids don't have A/A+ level capability in what they are doing now.  Throwing more on them is not kind, in that case.&lt;/p&gt;&#xA;&lt;p&gt;I do think there is a way to do enrichment.  But it's maybe a bit more of a tease.  &amp;quot;If you ever take abstract algebra, they'll show you why quintic general solution doesn't exist.&amp;quot;  I say this as someone who had it explained to me thus.  AND who still has only this level of understanding.  (I've never taken abstract algebra, so that's my level of understanding of it.)&lt;/p&gt;&#xA;&lt;p&gt;For the sharper, more curious kids, you can maybe direct them to contests.  It won't just be the problems then, but some interaction with other kids in an after class setting.  Also Numberphile or Mathologer videos are good stuff.  Or the Andrew Wiles video (ripped copy has been up on Vimeo for years).  There are some good books also, in particular Flatland.&lt;/p&gt;&#xA;&lt;p&gt;Avoid any push for doing original research.  It's just not feasible for high school kids in math.   If you have to, just do some statistics things or descriptives (like the thread we had on pictures/sculptures of shapes).  I.e. take a &amp;quot;big tent&amp;quot; view of what passes for math.  Your administrators won't know the difference and will probably think you are complying.  (Maybe even you are, if you think a little sideways.)&lt;/p&gt;&#xA;" OwnerUserId="17965" LastActivityDate="2021-10-28T22:12:46.740" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24520" PostTypeId="2" ParentId="24512" CreationDate="2021-10-30T09:44:04.907" Score="2" Body="&lt;p&gt;This is a perspective from an adult who self-studies math. I felt this way about synthetic division up until recently, when I sat down to actually learn synthetic division. But then, once I got it, it made perfect sense why people like it. It's less mentally taxing once you get a handle on it.&lt;/p&gt;&#xA;&lt;p&gt;If you say &amp;quot;why should they care about that?&amp;quot;, then I direct you to the quadratic formula. Why do you teach it? You can always complete the square. But we don't. Why teach the power rule in calculus? You can always derive it just as fast from the definition of a derivative rather quickly. But we teach it.&lt;/p&gt;&#xA;&lt;p&gt;Students generally feel like, once they get beyond a certain part of math, they just want to be able to get it done quickly without a lot of mental effort or space. The division isn't the main point of what they're doing, so they want to get it over with and back to the fun part of math.&lt;/p&gt;&#xA;&lt;p&gt;My grandpa was a math teacher, and he would teach me a lot of shortcuts, including this weird way of multiplying by multiple of 5. I know how to do square roots longhand, even though Newton's method is right there. Mathematicians love their shortcuts, too.&lt;/p&gt;&#xA;" OwnerUserId="17969" LastActivityDate="2021-10-30T09:44:04.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24521" PostTypeId="1" CreationDate="2021-10-30T21:26:34.277" Score="8" ViewCount="2097" Body="&lt;p&gt;What would you guys say the requirements for teaching &amp;quot;Math for Elementary Teachers&amp;quot; should be? Is this generally agree'd upon? Would you recommend it be taught by somebody who has a background or is knowledgeable in Elementary education? Is there any reason that a run-of-the-mill math PhD with little background in elementary education should not teach it?&lt;/p&gt;&#xA;" OwnerUserId="16437" LastActivityDate="2021-10-31T08:23:13.937" Title="Teaching Math for Elementary Teachers" Tags="|primary-education|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24523" PostTypeId="2" ParentId="24521" CreationDate="2021-10-31T04:20:19.013" Score="16" Body="&lt;p&gt;The essential problem with U.S. K-6 education is that the teachers don't actually understand the math they're teaching. And they also don't have any vision about why it's important later on. (I've had a number of community college students say they were going into elementary education precisely because they thought it was the least math-oriented profession they could think of.)&lt;/p&gt;&#xA;&lt;p&gt;In that view, the more we can get the non-math education people out of a closed loop, the better. These prospective K-6 teachers should have at least some exposure to how actual math professionals approach the subject. So having someone in a math department teach the course seems like a good idea. Value-add if they're &lt;em&gt;not&lt;/em&gt; indoctrinated in the soft-arts education ideology &lt;em&gt;du jour&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Better than that, of course, would be if we could pull the plug on the whole broken tactic of poorly-trained education majors teaching the math subject, and instead get disciplinary experts in at the K-6 level.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-10-31T08:23:13.937" LastActivityDate="2021-10-31T08:23:13.937" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="24524" PostTypeId="2" ParentId="24521" CreationDate="2021-10-31T06:37:16.857" Score="22" Body="&lt;p&gt;I don't know where you're from, so I'll address this from a North American perspective.&lt;/p&gt;&#xA;&lt;p&gt;If you're thinking research-level mathematics is useful to learning how to deal with 5-year-olds, you might just need a reality check.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is there any reason that a run-of-the-mill math PhD with little background in elementary education should not teach it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;School teachers are my &lt;strong&gt;heroes&lt;/strong&gt;. Their jobs are &lt;a href=&quot;https://slate.com/human-interest/2013/12/teaching-in-americas-highest-need-communities-isnt-rocket-science-its-harder.html&quot; rel=&quot;noreferrer&quot;&gt;radically harder than they appear&lt;/a&gt;, and if you (any reader) think that's being too generous, then go get 30+ underprivileged 12-year-olds - who have virtually no understanding of fractions, decimals, base 10, or division - excited about non-terminating decimals.&lt;/p&gt;&#xA;&lt;p&gt;Administer a lesson that not only teaches the math in logical detail and completeness, but also inspires wonder, better work habits, and confidence.&lt;/p&gt;&#xA;&lt;p&gt;Go on now.&lt;/p&gt;&#xA;&lt;p&gt;Give it a shot.&lt;/p&gt;&#xA;&lt;p&gt;It will only take you an hour or so.&lt;/p&gt;&#xA;&lt;p&gt;Report back.&lt;/p&gt;&#xA;&lt;p&gt;We'll follow up in a few years to see what impact your lesson had.&lt;/p&gt;&#xA;&lt;p&gt;Go now!&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;School teachers work in a field that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Provides virtually zero long-term feedback. In mid-April some decades ago, your kindergarten teacher's lesson had some kind of impact on your life. What was it? Who measures it? Who informed the kindergarten teacher of the results?&lt;/li&gt;&#xA;&lt;li&gt;Provides &lt;a href=&quot;http://www.danielwillingham.com/uploads/5/0/0/7/5007325/willingham_final_with_links.docx&quot; rel=&quot;noreferrer&quot;&gt;very little useful training&lt;/a&gt; and even &lt;a href=&quot;https://www.nctq.org/dmsView/Learning_About_Learning_Report&quot; rel=&quot;noreferrer&quot;&gt;less scientific insight&lt;/a&gt; into learning. Practical issues such as behavior management, learning English, health and socioeconomic problems, time-efficient lesson prep and marking, dealing with parents, cognition of learning, etc. tend to get little attention in teacher schools yet this constitutes the vast majority of what teachers have to deal with.&lt;/li&gt;&#xA;&lt;li&gt;Immerses them in &lt;a href=&quot;https://www.youtube.com/watch?v=oxZzoVp5jmI&quot; rel=&quot;noreferrer&quot;&gt;intuition-deceiving scenarios&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Requires them to work alone then provides little time to reflect deeply - as you are continuously managing children - and even less time to collaborate with colleagues. It's one reason why teachers one room apart can vary in quality so radically.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;An elementary teacher trainer needs to address &lt;em&gt;those&lt;/em&gt; problems.&lt;/p&gt;&#xA;&lt;p&gt;What % of &amp;quot;run-of-the-mill math PhD[s]&amp;quot; have the expertise in both cognition &lt;em&gt;and&lt;/em&gt; 8-year-olds to do so?&lt;/p&gt;&#xA;&lt;p&gt;And then there's &lt;em&gt;the math&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;&amp;quot;They think a remainder of 3 is the same as decimal 3&amp;quot;&lt;/h3&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://www.thestar.com/yourtoronto/education/2016/05/13/for-many-teachers-math-just-doesnt-add-up.html&quot; rel=&quot;noreferrer&quot;&gt;Elementary teachers’ weak math skills spark mandatory crash courses&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;That a disastrously large share of elementary teachers had outrageously harmful and stupid pseudomath educations is surprising to some outside of the world of education, but is easliy verifiable within that world.&lt;/p&gt;&#xA;&lt;p&gt;I concur with the other poster who said some people become teachers at least partly because it allows them to hide their lack of math skills. This is the self-perpetuation of a broken math education system.&lt;/p&gt;&#xA;&lt;p&gt;So, when it comes to math, teacher trainees often arrive fearful of, resentful of, and resistant to any math or math pedagogy courses.&lt;/p&gt;&#xA;&lt;p&gt;What % of &amp;quot;run-of-the-mill PhD[s]&amp;quot; will have even the faintest idea what to do during the &lt;em&gt;second&lt;/em&gt; hour with an adult who can't stop crying over a protractor and pie chart assignment?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;[Purely Speculative] Top 4 List of Requirements for Elementary Math Teacher Trainers&lt;/h3&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Lots of experience in treating math phobia with warmth, empathy, patience, etc. and converting that phobia into wonder and joy and mastery. Many teacher trainees need to learn math for real this time. The way they're taught in an elementary math methods course is likely to be one of the only antidotes to the dog %#&amp;amp;@ pseudomath they've experienced for the past 20+ years. Teacher trainees need to experience that there is more to math than weird rules and rote drills. Math appears in games, puzzles, stories, magic tricks, history, role playing, art, engineering, economics, medicine, etc... and of course, I'm still a passionate supporter of well-designed paper-and-pencil work, such as &lt;a href=&quot;https://jumpmath.org/&quot; rel=&quot;noreferrer&quot;&gt;JUMP Math&lt;/a&gt;. [&amp;quot;Well-designed&amp;quot; meaning they gently but firmly lead to deep insight.] This antidote needs to be proven, potent, and unforgettable. A PhD in math is largely irrelevant.&lt;/li&gt;&#xA;&lt;li&gt;Deep pedagogical content knowledge of math, especially along the treacherous path of Whole Numbers → Whole Number Arithmetic → Rational Numbers → Rational Number Arithmetic → Algebra. Most students fall off this path only to never recover. Many elementary teacher trainees will be among them, yet will need to go through all of it &lt;em&gt;again&lt;/em&gt; - within a semester! - and then be ready to pass along those teachings to their own students. The future grade 3 teachers needs to learn X, Y, and Z, possiby for the first time, but deeply enough to know &amp;quot;I have to ask 8-year-olds these tough questions about X now so that in grade 6 they'll be ready for Y which prepares them for Z in grade 8 and success in high school math.&amp;quot; This is a tall order that has little to do with a PhD in math and a lot to do with working with kids of varied ages.&lt;/li&gt;&#xA;&lt;li&gt;A treasure trove of video clips of real children studying math in real classrooms.  Trainees need to see children cry with terror during a lesson that appears to be going well. They need to see students be triumphantly amazed and beg for more after activities that were &amp;quot;obviously&amp;quot; a boring waste of time. Trainees need to see every student repeat &amp;quot;Addition and subtraction are opposites&amp;quot; every morning from September to May, then, in June, see 60% of the class tackle &lt;span class=&quot;math-container&quot;&gt;$90-?=70$&lt;/span&gt; with their fingers and 30% feel totally confident solving it with &lt;span class=&quot;math-container&quot;&gt;$90+70=160$&lt;/span&gt;. PhD: Not required. Lots of classroom time: Definitely required.&lt;/li&gt;&#xA;&lt;li&gt;Lots of connections to nearby elementary school teachers who can support the trainer with both social proof and provide first-hand experience of the above to the trainees. Again, a PhD is irrelevant while experience within school systems is invaluable.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;An elementary math teacher trainer does not need a PhD in math, but they need to be extraordinary in so many other ways, from being a math therapist to being a documentary filmmaker. They need saintly patience. They need, likely, decades of impressive experience. They probably need to be leaders in the math education community.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps I'm just dreaming of &lt;strong&gt;super heroes&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;All of the above above is a &lt;a href=&quot;https://www.saffo.com/02008/07/26/strong-opinions-weakly-held/&quot; rel=&quot;noreferrer&quot;&gt;strong opinion, weakly held&lt;/a&gt; by someone with no formal expertise in this area. I look forward to anyone's feedback. :-)&lt;/p&gt;&#xA;" OwnerUserId="7151" LastEditorUserId="7151" LastEditDate="2021-10-31T06:45:07.387" LastActivityDate="2021-10-31T06:45:07.387" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="24525" PostTypeId="1" CreationDate="2021-11-02T08:49:59.443" Score="6" ViewCount="1394" Body="&lt;p&gt;Consider the following problems:&lt;/p&gt;&#xA;&lt;p&gt;A) You have 20 problems for your math homework this week, and you want to do 1/5 of them today. How many problems do you need to do today?&lt;/p&gt;&#xA;&lt;p&gt;B) You need to read a 20 page book this week, and you want to read the first 1/5 of the book today. How many pages do you need to read today?&lt;/p&gt;&#xA;&lt;p&gt;C) You need to run a total of 20 kilometers this week, and you want to run 1/5 of that today. How many kilometers do you need to run today?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Are these problems conceptually different enough that an educator needs to carefully deal with each of these scenarios, or will a child that can (actually) solve one of these problems also be able to solve the others?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;To explain why I consider these problems different:&lt;/p&gt;&#xA;&lt;p&gt;A) You have 20 problems. You can create 5 groups with 4 problems in each.&lt;/p&gt;&#xA;&lt;p&gt;B) You have 20 pages. You can create 5 groups with 4 pages in each. However, in this case there is an implied order of the pages. This feels more like a scaling of a discrete set, than placing objects into groups of equal size.&lt;/p&gt;&#xA;&lt;p&gt;C) The kilometers are not really objects and so it doesn't really make sense to group them. One could of course imagine them as &amp;quot;objects&amp;quot;, but arguably we are really performing some kind of scaling operation.&lt;/p&gt;&#xA;" OwnerUserId="6527" LastEditorUserId="6527" LastEditDate="2021-11-02T10:57:29.530" LastActivityDate="2021-11-04T15:46:18.133" Title="Are these fraction problems different enough to warrant individual consideration?" Tags="|secondary-education|primary-education|middle-school|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24526" PostTypeId="2" ParentId="24525" CreationDate="2021-11-02T11:18:46.460" Score="8" Body="&lt;p&gt;i don’t think they are.&#xA;In fractions there are (at least) 3 analogies: set (discrete objects), area (or volume), and length.&lt;/p&gt;&#xA;&lt;p&gt;Your 1st is set, 2nd maybe area (more like length) and 3rd is length.&lt;/p&gt;&#xA;&lt;p&gt;You could re-write  #2 so that your filling a glass or jug or jar with something (or using 1/5 of an amount)&lt;/p&gt;&#xA;&lt;p&gt;But,  i think they are all conceptually the same. a fraction of a whole.&lt;/p&gt;&#xA;&lt;p&gt;Why not ask about missing parts or what is the whole, Then these would be conceptually different thus modifying /  increasing  the rigor and assess a students understanding of fractions (here they may just be applying an algorithm / process):&lt;/p&gt;&#xA;&lt;p&gt;why not  ask that after the 1st day you left 4/5 of the problems left. If there are 16 problems left, how many problems in total are there (or you completed 4 or 1/5 of the problems, how many problems are there in total). You could also ask if you completed 4 problems out of 20, what is the fraction of the problems you completed or have not completed.&lt;/p&gt;&#xA;&lt;p&gt;You could even ask an open question. You’ve completed 1/5 of the problems. How many problems are there in total? How many problems did you complete? (would be challenging because student would wonder: where are the numbers to work with , and you’ll have to tell them. make them up!)&lt;/p&gt;&#xA;&lt;p&gt;Each one of these suggestions asks a different aspect of the problem (generate a fraction of a part , a missing part, or the whole).&lt;/p&gt;&#xA;&lt;p&gt;These are just some  possibilities (and analogies) that  can really modify what you are asking for in these problems.&lt;/p&gt;&#xA;&lt;p&gt;You could even work with fractions greater than 1 also ;)&lt;/p&gt;&#xA;" OwnerUserId="17986" LastActivityDate="2021-11-02T11:18:46.460" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24527" PostTypeId="2" ParentId="24525" CreationDate="2021-11-02T13:09:54.477" Score="6" Body="&lt;p&gt;This depends on when in the &amp;quot;fractions curriculum&amp;quot; this happens.&lt;/p&gt;&#xA;&lt;p&gt;If the children know that &amp;quot;a fraction of&amp;quot; really means to multiply by this fraction, then all problems are equal enough that one single student will be able to solve either all or none of them.&lt;/p&gt;&#xA;&lt;p&gt;If the problems are used to teach &amp;quot;fraction of is multiplication by&amp;quot;, it will be useful to consider them separately.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2021-11-02T13:09:54.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24528" PostTypeId="2" ParentId="24525" CreationDate="2021-11-02T14:38:29.550" Score="1" Body="&lt;p&gt;Yes, the second and third have an implied order to them that the first doesn't, but it doesn't impact the basic operation (look at the words and translate them to 20 times 1/5.)  I think myself (and most students) would sense something a little different, but still perform the operation.  After all you're not asking a question that requires some different use of this additional information.  Maybe, to be fair to you, we could say the latter problems are slightly harder since there's a slight distraction in the order.&lt;/p&gt;&#xA;&lt;p&gt;My advice is to avoid worrying about what confusions might occur and deal with them more as they occur.  Lots of theoretical people want to anticipate every eventuality and then end up teaching with masses of caveats, explanations, etc.  But math is hard enough.  It's easier to get a simple explanation than an elaborate one.  (Kids are not computers, they have limited processing capacity.)  I actually don't think the order thing will flummox the kids but even if it does, I would deal with it afterwards rather than ahead of time.&lt;/p&gt;&#xA;&lt;p&gt;Also, can't you just observe and report your experience?  Why the need/desire to consider the problem sans kids, rather than looking at what actually happens?  Surely the actual results in teaching setting would inform our answers more than just looking at the problems without recourse to that information.&lt;/p&gt;&#xA;" OwnerUserId="17989" LastActivityDate="2021-11-02T14:38:29.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24530" PostTypeId="2" ParentId="24525" CreationDate="2021-11-03T15:05:54.493" Score="0" Body="&lt;p&gt;It would depend on the student.  If this is the first time they are seeing fractions of a set, then I don't think one problem would be enough for them to get it unless they are very quick. If it is the first time they are seeing fractions of a set, it would be worthwhile for them to see more problems.  Of course it might be better to change the numbers in the problems in that case.  On the other hand some children benefit from seeing the application of the same numbers in different contexts.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-11-03T15:05:54.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24531" PostTypeId="2" ParentId="24525" CreationDate="2021-11-03T15:41:04.163" Score="3" Body="&lt;p&gt;One point where C is different from A and B is that a distance can scaled (or split) by any fraction. So 20km can be split into 6 equal parts, even if the result cannot (yet) be represented by the numbers the students know.&lt;/p&gt;&#xA;&lt;p&gt;But 20 problems or 20 pages cannot be split into 6 equal parts. At least nothing simple comes to mind.&lt;/p&gt;&#xA;&lt;p&gt;In other words, C is a problem from a somewhat different family than A and B, in that they generalise in different ways (in the sense of varying the problem &amp;quot;parameters&amp;quot;/constants/numbers).&lt;/p&gt;&#xA;" OwnerUserId="12521" LastActivityDate="2021-11-03T15:41:04.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24532" PostTypeId="2" ParentId="23235" CreationDate="2021-11-03T19:58:03.977" Score="0" Body="&lt;p&gt;This is pretty much a scientific methods question (I guess math can concern itself also).  But in any case, if you look at any first chapter of a college (or even high school) chemistry or physics text, you are very likely to get a great discussion of this topic.&lt;/p&gt;&#xA;&lt;p&gt;FWIW, the analogy that &amp;quot;sings&amp;quot; to me is riflery or archery (or I guess darts).  A marksmen will hit a &amp;quot;tight pattern&amp;quot;.  But if the sights are off, it will be displaced.  The visual of the bullseye is very powerful here.  A video example, from minute 1:00 to 1:45:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=nXrsB2RMo2w&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=nXrsB2RMo2w&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Of course the same thing applies with artillery or NGFS.&lt;/p&gt;&#xA;" OwnerUserId="17994" LastActivityDate="2021-11-03T19:58:03.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24533" PostTypeId="2" ParentId="24525" CreationDate="2021-11-03T22:44:16.493" Score="0" Body="&lt;p&gt;My state has a standardized test for math given in the second year of high school. It has a low bar for passing, about 30%, but it's a requirement for graduation.&lt;/p&gt;&#xA;&lt;p&gt;I teach a class to help the students who are struggling to pass. In this class, I use a similar strategy as I use for the SAT or ACT (college entrance exams). That is, to read a question and be able to articulate the math problem being asked. I've encountered students who can calculate the area of a circle, but have trouble when the problem is posed as, say, a radio tower with a certain transmission distance, asking how many square miles of coverage the radio station has.&lt;/p&gt;&#xA;&lt;p&gt;Often, I'll project a problem onto the board and cross out the extra details that don't really matter. In many cases, the students are English language learners, and they may be fine math students, but have difficulty with long word problems where the vocabulary really has nothing to do with the math. In this case, the minor differences in your 3 problems get crossed out and &amp;quot;1/5 of 20&amp;quot; is all that remains.&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, there are slight changes you can make ala jmg's suggestion and others, which would change it up enough to not feel redundant to the typical student. The difference among the 3 feels too slight to matter.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-11-04T10:24:42.563" LastActivityDate="2021-11-04T10:24:42.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24534" PostTypeId="2" ParentId="24525" CreationDate="2021-11-04T15:46:18.133" Score="1" Body="&lt;p&gt;I'll point out that the OP's question really contains two independent questions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Are these problems conceptually different enough that an educator&#xA;needs to carefully deal with each of these scenarios...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It would be a bad idea for an educator to spend time presenting each of the given questions independently and analyzing them. First, we don't have that much time available. Second, trying to identify and clarify some subtle distinction is more likely to further confuse students on those nuances than anything else. Third, it seems to be just the tip of the iceberg to how many iterations could be made to this and any other problem, and I don't see why the discussion would end with just these 3 variations for this, or any other problem.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... or will a child that can (actually) solve one of these problems&#xA;also be able to solve the others?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Well, that's a separate issue, and I wouldn't guarantee that all students actually will be able to perfectly transfer their abstract knowledge from case to another. In fact, it's pretty well established with numerous examples that that's &lt;em&gt;not&lt;/em&gt;, completely generally, the case. Maybe a student just gets distracted by a word or phrasing they don't know in one instance.&lt;/p&gt;&#xA;&lt;p&gt;My understanding is that it's been pretty well established that the key to student reading proficiency is a legacy of reading wide and &lt;em&gt;deep subject matter&lt;/em&gt;.* If students have read on the subject in question, they'll be more comfortable and proficient dealing with the word problem. If not, they'll likely to be distracted and unable to parse the sentence. We can't know in general which camp they'll fall in when they get to these problems, so: no guarantees.&lt;/p&gt;&#xA;&lt;p&gt;But nonetheless, it's not a good use of time trying to analyze and distinguish the nuances of these problems; that's not really where any difficulty will reside, and would increase confusion rather than decrease it. Time in the math class is best spent on the abstracted math itself, not other issues; if subject-matter difficulties arise, then you might have to assist with that on a case-by-case basis.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;(*) Can't find a good link at this writing; if anyone can share one in comments I'd edit it in.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2021-11-04T15:46:18.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24536" PostTypeId="1" CreationDate="2021-11-04T19:55:40.373" Score="4" ViewCount="633" Body="&lt;p&gt;When teaching addition and multiplication of fractions, I seem to recall some advice on this site that one should first treat the cases&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a \cdot \frac{c}{d}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a + \frac{c}{d}$&lt;/span&gt; before moving on to the more general &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b} \cdot \frac{c}{d}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b} + \frac{c}{d}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This made think about the case &lt;span class=&quot;math-container&quot;&gt;$a \cdot \frac{1}{b}$&lt;/span&gt; first. I would like to demonstrate in some way, and preferably (but not necessarily) geometrically, that &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b} = a \cdot \frac{1}{b}$&lt;/span&gt;, but I am having trouble finding an actual difference between &lt;span class=&quot;math-container&quot;&gt;$a \cdot \frac{1}{b}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, the answer might just be that &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a \cdot \frac{1}{b}$&lt;/span&gt; have the same representation, and therefore they are equal.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Is there an agreed upon difference between how we represent &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a \cdot \frac{1}{b}$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="6527" LastActivityDate="2022-06-25T08:19:56.003" Title="Is there an agreed upon difference between how we represent $\frac{a}{b}$ and $a \cdot \frac{1}{b}$?" Tags="|primary-education|fractions|middle-school|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24537" PostTypeId="2" ParentId="24536" CreationDate="2021-11-04T21:32:35.197" Score="11" Body="&lt;p&gt;The common core state standards definition of the fraction &lt;span class=&quot;math-container&quot;&gt;$\frac{N}{D}$&lt;/span&gt; of a unit is to subdivide the unit into &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; equal sized pieces.  Each of these pieces is defined to be &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{D}$&lt;/span&gt; of the unit.  Then &lt;span class=&quot;math-container&quot;&gt;$\frac{N}{D}$&lt;/span&gt; is defined to be &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; of these pieces.&lt;/p&gt;&#xA;&lt;p&gt;Under these definitions, I think there is no difference between &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{D}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{N}{D}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-11-05T00:27:15.600" LastActivityDate="2021-11-05T00:27:15.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24538" PostTypeId="2" ParentId="24536" CreationDate="2021-11-05T14:48:21.850" Score="3" Body="&lt;p&gt;As an abstract concept, I don't think they have different meanings.  As with many situations, though, different notations can represent different situations in stories.  As an example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$N \cdot \frac{1}{D}$&lt;/span&gt; could represent having &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{D}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; different things, while &lt;span class=&quot;math-container&quot;&gt;$\frac{N}{D}$&lt;/span&gt; could represent having &lt;span class=&quot;math-container&quot;&gt;$\frac{N}{D}$&lt;/span&gt; of the same thing.  We're discussing the same quantity, but the organization is different.&lt;/p&gt;&#xA;&lt;p&gt;For example, let our story problem be about pizzas with &lt;span class=&quot;math-container&quot;&gt;$N=3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$D=8$&lt;/span&gt;.  The amount &lt;span class=&quot;math-container&quot;&gt;$N \cdot \frac{1}{D}$&lt;/span&gt; suggests to me that I might have had one slice (&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{8}$&lt;/span&gt; of the pizza) of 3 different pizzas, while &lt;span class=&quot;math-container&quot;&gt;$\frac{N}{D}$&lt;/span&gt; might suggest to me that I had 3 slices of the same pizza.&lt;/p&gt;&#xA;&lt;p&gt;This is certainly not a requirement to think in these terms, and some people may view them reversed -- but if you are trying to communicate to students who can't conceptualize them as being the same, this sort of example might bridge the gap for them.  In both cases, regardless of what else happened, I had three slices of pizza, whether &lt;span class=&quot;math-container&quot;&gt;$3\cdot\frac{1}{8}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{8}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(For a more abstract geometric representation, you could use the standard rectangles and circles common for discussing fractions; the comparison would still be one fraction from many shapes vs multiple fractions from the same shape.)&lt;/p&gt;&#xA;" OwnerUserId="14966" LastActivityDate="2021-11-05T14:48:21.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24539" PostTypeId="2" ParentId="24536" CreationDate="2021-11-05T15:39:09.890" Score="-2" Body="&lt;p&gt;I think students need to become used to thinking, reading and computing with various representations of fractions: a/b, a 1/b's, a:b, etc.&lt;/p&gt;&#xA;&lt;p&gt;Mathematically, your two examples are the same and even not that hard to resolve, conceptually.  But, sort of as you discuss, a/b versus a*(1/b) is a simpler example of what could be more complex representations and calculations, like those involved in multiplication, division (the whole cross multiplication thingie), addition (the whole common terms thingie), etc.&lt;/p&gt;&#xA;&lt;p&gt;However, pedagogically, I think you are better off just teaching and building familiarity, conventionally.  Rather than trying to anticipate and preclude every edge case and conceptual hurdle before it occurs.  For instance, I think (as with most research mathematicians), you show a false belief that some sort of proof will build comprehension for very young learners.  It's not like learning to count is enhanced by learning the Peano axioms.&lt;/p&gt;&#xA;&lt;p&gt;Also, I think the effort to brainstorm possible hurdles and then analyze them and come up with correctives is a much weaker approach to improving pedagogy than working with students, trying different things, and observing them.  Educational methodology is a natural science, like birdwatching or chemistry or the like.  It's not a Euclidean derivation.&lt;/p&gt;&#xA;" OwnerUserId="18003" LastActivityDate="2021-11-05T15:39:09.890" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24540" PostTypeId="2" ParentId="24536" CreationDate="2021-11-05T17:38:42.870" Score="4" Body="&lt;p&gt;Straightedge and compass construction is an interesting way to thing about this as an educator. This perspective would &lt;em&gt;not&lt;/em&gt; be something you actually take into an elementary school classroom. Rather it is for the teacher to think about this from a higher viewpoint. Thinking about what we are teaching from higher viewpoints makes teaching more fun, and so it makes us better teachers.&lt;/p&gt;&#xA;&lt;p&gt;There are standard constructions for &lt;span class=&quot;math-container&quot;&gt;$a\cdot b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jIIuZ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jIIuZ.png&quot; alt=&quot;multiply&quot; /&gt;&lt;/a&gt; &lt;a href=&quot;https://i.stack.imgur.com/9Nw3o.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9Nw3o.png&quot; alt=&quot;divide&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now let's use notation in the answer posted by Steven Gubkin and compare &lt;span class=&quot;math-container&quot;&gt;$\frac{N}{D}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N\cdot\frac{1}{D}$&lt;/span&gt;, which strongly suggests an assumption that &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; are positive integers.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;For &lt;span class=&quot;math-container&quot;&gt;$\frac{N}{D}$&lt;/span&gt;, we start with a unit length &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and construct a segment of length &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;. Then we use the division construction to construct &lt;span class=&quot;math-container&quot;&gt;$\frac{N}{D}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;For &lt;span class=&quot;math-container&quot;&gt;$N\cdot\frac{1}{D}$&lt;/span&gt; we use the division construction to construct &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{D}$&lt;/span&gt; and then use the multiplication construction to construct &lt;span class=&quot;math-container&quot;&gt;$N\cdot\frac{1}{D}$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is assumed to be an integer, we can do the multiplication by marking off &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; copies of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{D}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It is a proverbial &amp;quot;exercise for the reader&amp;quot; to reconcile that (1) and (2) result in the same length.&lt;/p&gt;&#xA;&lt;p&gt;More generally, we can start with segments of length &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and a unit, where we do not assume &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are integers. We use the division and multiplication constructions to construct segments of length &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a\cdot\frac{1}{b}$&lt;/span&gt;. Then we step back and try to see that the two segments are equal in length.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-11-05T17:44:39.720" LastActivityDate="2021-11-05T17:44:39.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24541" PostTypeId="2" ParentId="24536" CreationDate="2021-11-06T15:21:04.780" Score="6" Body="&lt;p&gt;You should be aware that, in standard developments, the two expressions being considered here are true &lt;em&gt;by definition&lt;/em&gt;. The quotient &lt;span class=&quot;math-container&quot;&gt;$\frac a b$&lt;/span&gt; is just a shorthand (&amp;quot;syntactic sugar&amp;quot;) way of writing &lt;span class=&quot;math-container&quot;&gt;$a \cdot \frac 1 b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, here it in Sullivan's &lt;em&gt;Algebra and Trigonometry&lt;/em&gt;, Review Section R.1:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UGWck.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UGWck.png&quot; alt=&quot;Definition of quotient, per Sullivan&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You'll see that in any standard textbook on algebra or precalculus. Just looking at other books on my shelf: It's the same in Rosen, &lt;em&gt;Discrete Mathematics&lt;/em&gt;, Appendix 1 (Axioms for the Real Numbers, in Exercises). Etc.&lt;/p&gt;&#xA;&lt;p&gt;Of course, such a definition follows soon after the axiom for real numbers that observes any nonzero real number &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; has a multiplicative inverse that we denote &lt;span class=&quot;math-container&quot;&gt;$\frac 1 b$&lt;/span&gt;. This is completely standard, and likewise follows the definitions in abstract algebra that a ring is an object with additive inverses, a division ring is an object with multiplicative inverses, etc. (Such objects including real numbers, obv.)&lt;/p&gt;&#xA;&lt;p&gt;In short, in the standard development, this equality is not something you can prove, justify, or demonstrate; it's simply an invention of a new shorthand piece of writing. It's founded directly in the assumed multiplicative-inverse axiom for real numbers, and reflects the development at the level of abstract algebra.&lt;/p&gt;&#xA;&lt;p&gt;However, the &lt;em&gt;interesting&lt;/em&gt; thing (and possibly what the OP meant) is that there is a difference in the definition and meaning of &lt;span class=&quot;math-container&quot;&gt;$a \div b$&lt;/span&gt; versus &lt;span class=&quot;math-container&quot;&gt;$\frac a b $&lt;/span&gt;, whose equivalence does indeed take a short proof. Steven Gubkin expanded more on this in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14114/how-to-resolve-the-new-definition-of-subtraction-and-division-seen-in-college-al&quot;&gt;his answer to my related question here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-11-08T15:16:43.693" LastActivityDate="2021-11-08T15:16:43.693" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24542" PostTypeId="2" ParentId="267" CreationDate="2021-11-07T09:58:09.823" Score="1" Body="&lt;p&gt;Something more trivial could be the following one:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$a^2=0\Rightarrow a=0.$$&lt;/span&gt;&#xA;Assume that &lt;span class=&quot;math-container&quot;&gt;$a^2=0$&lt;/span&gt; and at the same time &lt;span class=&quot;math-container&quot;&gt;$a\neq0$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$a\in\mathbb{R}.$&lt;/span&gt; Then, there exists &lt;span class=&quot;math-container&quot;&gt;$a^{-1}\in\mathbb{R}$&lt;/span&gt; so:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$a^2=0\Rightarrow a^{-1}a^2=0\Rightarrow a=0,$$&lt;/span&gt;&#xA;so you indeed need both &lt;span class=&quot;math-container&quot;&gt;$a^2=0$&lt;/span&gt; as well as &lt;span class=&quot;math-container&quot;&gt;$a\neq0$&lt;/span&gt; to arrive to a contradiction.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2021-11-07T09:58:09.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24543" PostTypeId="1" AcceptedAnswerId="24547" CreationDate="2021-11-07T10:07:07.183" Score="3" ViewCount="195" Body="&lt;p&gt;My son graduates from secondary school in 2022. But he shall attend university in Australia or UK —  where you must apply for  &lt;em&gt;ONE major&lt;/em&gt; — not liberal arts in the USA. He wants to pick &lt;a href=&quot;https://www.imperial.ac.uk/study/ug/courses/2021/mathematics-department/mathematics-statistics-finance-bsc/&quot; rel=&quot;nofollow noreferrer&quot;&gt;BSc Math with Statistics for Finance&lt;/a&gt;, or &lt;a href=&quot;https://www.sydney.edu.au/courses/subject-areas/major/financial-mathematics-and-statistics2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Financial Math and Statistics&lt;/a&gt;. His secondary school &lt;em&gt;MATH&lt;/em&gt; teacher advised us that statisticians must  work with software and programming.&lt;/p&gt;&#xA;&lt;p&gt;But son is legally blind, and suffers from &lt;a href=&quot;https://en.wikipedia.org/wiki/Computer-induced_medical_problems&quot; rel=&quot;nofollow noreferrer&quot;&gt;Computer-induced medical problems&lt;/a&gt;. We have taken him to consult two ophthalmologists and three optometrists. They all advise the same thing  — he must try to minimize using computer and screen devices. None of them took university statistics, and they cannot comment whether son should study statistics at university.&lt;/p&gt;&#xA;&lt;p&gt;What can you advise please? &lt;strong&gt;Will his legal blindness and computer induced medical problems, thwart or stymie him from BSc statistics?&lt;/strong&gt; &lt;em&gt;&lt;strong&gt;What about PhD statistics?&lt;/strong&gt;&lt;/em&gt;  Thank you.&lt;/p&gt;&#xA;" OwnerUserId="18007" LastEditorUserId="18007" LastEditDate="2021-11-07T21:53:24.560" LastActivityDate="2021-11-08T02:42:13.383" Title="To what degree can the legally blind, with computer induced medical problems, study statistics?" Tags="|undergraduate-education|graduate-education|statistics|special-needs-education|disability|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24544" PostTypeId="2" ParentId="14114" CreationDate="2021-11-07T15:22:26.153" Score="4" Body="&lt;p&gt;Here is what the CCSS have to say when division and fractions are being introduced in grade 3:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.corestandards.org/Math/Content/3/introduction/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.corestandards.org/Math/Content/3/introduction/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In other words, division is (as you suggest) being defined as an inverse to multiplication.  Since the CCSS defines multiplication in a way which distinguishes the two factors (&lt;span class=&quot;math-container&quot;&gt;$A \times B$&lt;/span&gt; is the number of units &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; groups, is each group has &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; units), there are actually TWO distinct definitions of division which must be unified through the commutative property of multiplication:&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;How many groups&amp;quot; definition of division:&lt;/p&gt;&#xA;&lt;p&gt;We define &lt;span class=&quot;math-container&quot;&gt;$A \div B$&lt;/span&gt; to be the number &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; which solves the multiplication problem &lt;span class=&quot;math-container&quot;&gt;$C \times B = A$&lt;/span&gt;.  For example &lt;span class=&quot;math-container&quot;&gt;$12 \div 4 = 3$&lt;/span&gt; since if want a certain number of groups, each containing 4 units, to give us 12 units total, then we must have 3 of those groups.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;How many units in each group&amp;quot; definition of division:&lt;/p&gt;&#xA;&lt;p&gt;We alternatively define &lt;span class=&quot;math-container&quot;&gt;$A \div B$&lt;/span&gt; to be the number &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; which solves the multiplication problem &lt;span class=&quot;math-container&quot;&gt;$B \times C = A$&lt;/span&gt;.  For example &lt;span class=&quot;math-container&quot;&gt;$12 \div 4 = 3$&lt;/span&gt; since if want 4 groups, each containing a certain number of units, to give us 12 units total, then we must have 3 units in each group.&lt;/p&gt;&#xA;&lt;p&gt;The definition of &lt;span class=&quot;math-container&quot;&gt;$\frac{A}{B}$&lt;/span&gt; of a unit is to take the unit amount, split it into &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; equal sized parts to obtain the unit fraction &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{B}$&lt;/span&gt;.  Then &lt;span class=&quot;math-container&quot;&gt;$\frac{A}{B}$&lt;/span&gt; is defined to be equal to &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; of these parts of size &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{B}$&lt;/span&gt;.  For instance, the definition of &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{3}$&lt;/span&gt; of a pound would be to take &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; pound, split it into three equal sized pieces each called &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; pound.  Then four of these pieces is &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{3}$&lt;/span&gt; pound.&lt;/p&gt;&#xA;&lt;p&gt;According to these definitions, there is no direct link between &lt;span class=&quot;math-container&quot;&gt;$A \div B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{A}{B}$&lt;/span&gt;.  However, we can argue their equality using the definitions, both &amp;quot;intuitively&amp;quot; and more formally.&lt;/p&gt;&#xA;&lt;p&gt;From an intuitive &amp;quot;how many units in each group&amp;quot; perspective &lt;span class=&quot;math-container&quot;&gt;$A \div B$&lt;/span&gt; could be thought of as the answer to the question:  &amp;quot;I have A cupcakes, and B people to share them with.  What fraction of a cupcake will each person receive?&amp;quot;.  One way to answer this is to split each cupcake into &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; parts.  Now I can give &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{B}$&lt;/span&gt; to each person from &lt;strong&gt;each&lt;/strong&gt; of the &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; cupcakes, yielding &lt;span class=&quot;math-container&quot;&gt;$\frac{A}{B}$&lt;/span&gt; cupcakes for each person.  Thus &lt;span class=&quot;math-container&quot;&gt;$A \div B = \frac{A}{B}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From an intuitive &amp;quot;how many groups&amp;quot; perspective, &lt;span class=&quot;math-container&quot;&gt;$A \div B$&lt;/span&gt; could be thought of as the answer to the question : &amp;quot;I have A pounds of flour.  It takes B pounds to make one recipe.  How many recipes can I make?&amp;quot;.  One way to answer this question is to split each recipe into &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; equal parts.  Then it takes &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; pound of flour to make &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{B}$&lt;/span&gt; recipes.  Since I have &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; pounds of flour, I can make &lt;span class=&quot;math-container&quot;&gt;$\frac{A}{B}$&lt;/span&gt; recipes.&lt;/p&gt;&#xA;&lt;p&gt;From a more formal/algebraic perspective we might make the following analogous definitions and proofs:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$A,B \in \mathbb{R}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$B \neq 0$&lt;/span&gt;.  We define &lt;span class=&quot;math-container&quot;&gt;$A \div B$&lt;/span&gt; as the real number &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$B \times C = A$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If we were being really formal, I supposed existence and uniqueness of this number would need to be addressed.  I have never seen any elementary text address the uniqueness part.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$A, B \in \mathbb{R}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$B \neq 0$&lt;/span&gt;.  Define &lt;span class=&quot;math-container&quot;&gt;$\frac{A}{B} = AB^{-1}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$B^{-1}$&lt;/span&gt; is the multiplicative inverse of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Note that this corresponds to our intuitive treatment of fractions.  &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{B}$&lt;/span&gt; was defined as the number such that &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; of them yields &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;:  aka it was defined as the multiplicative inverse of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now to check that &lt;span class=&quot;math-container&quot;&gt;$A \div B = \frac{A}{B}$&lt;/span&gt;, we just need to check that &lt;span class=&quot;math-container&quot;&gt;$\frac{A}{B}$&lt;/span&gt; satisfies the definition of the quotient:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\begin{align*}&#xA;B \times \frac{A}{B} &amp;amp;= B \times (A \times \frac{1}{B})\\&#xA; &amp;amp;= A \times (B \times \frac{1}{B})\\&#xA;&amp;amp;= A \times 1\\&#xA;&amp;amp;= A&#xA;\end{align*}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This whole discussion will fly way over the head of almost any student though.  Ideally the teacher can be aware of these issues, so that they can design tasks which target the intuitive development of these ideas.&lt;/p&gt;&#xA;&lt;p&gt;There is a tension between the logical development of the ideas, which has a strict progression from the definitions, and the desired end state, in which the intuitions, understandings, and equivalences are so strong that it is easy to forget what originally implied what.  You want to build number and operation sense which is so strong that these ideas are all applied intuitively at the subconscious level.  I am not sure how to resolve this tension.  We want a really big &amp;quot;The following are all equivalent&amp;quot; statements to be living (implicitly) in the mind of each student, but we cannot get there without a logical progression.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-11-07T15:22:26.153" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24545" PostTypeId="2" ParentId="14114" CreationDate="2021-11-07T15:50:33.770" Score="-2" Body="&lt;p&gt;This isn't every possibility, but a rough sketch of approaches, rank ordered by what I think will help the (weak, given they are taking this course in college) students the most.  Which I believe is the more important objective.  Not fussy details of math or of what is most interest to the (more sophisticated) instructors.  But helping the kids.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Avoid dwelling on the theoretical justifications (at all) and pursue a course of familiarization via practice.  Yes, the kids need to learn the ability to manipulate symbols (like polynomial long division) as they do numbers, but this is best accomplished by light practice with the numbers, followed by extensive practice with x-containing expressions.  Perhaps even juxtaposed (when they get to the symbols at least, OK if you do some all arithmetic drill earlier, but doing a long division number problem, itself, followed by a polynomial one, is an effective way to build confidence, willingness.  If you instead make them question things they already are familiar with (subtraction of numbers), or at least think they are, you will derail them.  As other have pointed out, one could spend immense detail on 1+1, but this is not the right time for that.  I would even wager that it is hard to learn that level of abstract material without some base in the topic first (why it is easier not to have to learn Rudin before conventional calculus.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teach the definition you (and they) are comfortable with, the conventional one. But then move to significant calculational practice to build familiarity.  (The way to learn quantum mechanics is to do particle in the box problems, not axiom-dwelling.  In some cases, you really need a sort of familiarity with the materials, before dwelling on definitions or different axioms or notations [like bra ket] are pedagogically manageable.  (There's a 1.5, where you glide over it and move to practice very fast.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teach the book's definition.  Then practice.  (There's probably a 2.5 where you glide over it and emphasize practice, though.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do both, then practice.  This has the disadvantage of spending way too much time on fussy details and bogging the kids down, when they need familiarization with manipulating x-expressions.  Note, I'm not saying little edge cases are irrelevent (like stuff the &amp;quot;divided by zero&amp;quot; hawks watch for), but even there they are better dealt with after or during the course of building basic familiarity/competence.  Not pre-emptively.  That is too much cognitive load and not progressive enough.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="18010" LastEditorUserId="18010" LastEditDate="2021-11-07T15:57:11.613" LastActivityDate="2021-11-07T15:57:11.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24546" PostTypeId="1" CreationDate="2021-11-07T21:44:40.090" Score="0" ViewCount="104" Body="&lt;p&gt;My sister received her scores for &amp;quot;End-of-Course Examination Program&amp;quot; test results. There is a section &amp;quot;Algebra 1&amp;quot; and underneath a table with rows:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Algebra&lt;/li&gt;&#xA;&lt;li&gt;Functions&lt;/li&gt;&#xA;&lt;li&gt;Numbers and Quantity; Interpreting Data&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Can somebody explain and show examples of problems that belong to that &amp;quot;Numbers and Quantity; Interpreting Data&amp;quot; category?&lt;/p&gt;&#xA;" OwnerUserId="18011" LastActivityDate="2021-11-08T03:00:10.093" Title="What is Algebra 1 &quot;Number and quantity; Interpreting Data&quot;?" Tags="|algebra|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24547" PostTypeId="2" ParentId="24543" CreationDate="2021-11-08T02:16:41.893" Score="2" Body="&lt;p&gt;I suspect the avenue to answering this revolves much more around his condition and his reaction to it than it does around statistics as a field.  For one thing &amp;quot;legal blindness&amp;quot; is a pretty different thing than complete blindness.  I also admit to ignorance of &amp;quot;computer induced medical problems&amp;quot; syndrome (for example how well is it understood, how much does it vary, etc.--just having a wikilink is not like Euclidean certainty.)  Also, of course there have been completely blind people who still made contributions in science/methods.  Bottom line is I would get some more second opinions (including having him examined) as well as checking out some forums or the like for the blind community.  As well as just working with him and seeing how he does.&lt;/p&gt;&#xA;&lt;p&gt;On a simpler level, I do think working in statistics involves a significant amount of reading, writing (typing nowadays), and coding.  If he has a hard time with those activities, it may hold him back.  For example (statistic of one), I had a (since recovered) low brain blood flow several years ago and found that it interfered with detail work.  Still pretty bright, could converse, even do PPT and Excel in a business setting.  But Word footnotes (never an issue previously) were extremely hard on me.  Something about detailed reading was trickier for me.&lt;/p&gt;&#xA;&lt;p&gt;So I really think attention to him and trial of different activities is the key.  More so than theoretical ideas of what is required for statistics.&lt;/p&gt;&#xA;&lt;p&gt;On a practical level, if it's something you/he want to do, I would not let the first contrary answer* turn you off.  At the same time, from a Bayesian standpoint, it's not promising that the first answer was negative.  And we don't know how much of an examination they did, etc.  So...good look into it some more, since...why not?  But don't assume that the initial advice was invalid either.  Just check some more.&lt;/p&gt;&#xA;&lt;p&gt;*Rereading, I see it is 5 people that you talked to, not just 1.  That doesn't sound good.  Still, if it's something you want to do, maybe check in with some other people, such as a neurologist or psychologist who specializes in treatment of the blind.  I guess also talking to someone in the blind community (or an occupational therapist) might be helpful as well.  For example, an avenue to explore would be how well can Braille substitute for computer text, especially in the work of an applied field. (I don't know, would check to learn.)&lt;/p&gt;&#xA;" OwnerUserId="18010" LastEditorUserId="18010" LastEditDate="2021-11-08T02:42:13.383" LastActivityDate="2021-11-08T02:42:13.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24548" PostTypeId="2" ParentId="24546" CreationDate="2021-11-08T03:00:10.093" Score="1" Body="&lt;p&gt;I suggest to look at her textbook.  If that does not make it clear, ask the teacher.  We can't guess what the words mean in the specific context of your sis's course.  It's not like there's only one right answer, one course.  And we have no idea what they were pushing in that specific course.&lt;/p&gt;&#xA;" OwnerUserId="18010" LastActivityDate="2021-11-08T03:00:10.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24549" PostTypeId="1" AcceptedAnswerId="24555" CreationDate="2021-11-08T18:45:53.780" Score="37" ViewCount="8937" Body="&lt;p&gt;When teaching my first discrete math class recently, I found a subset of about 5 out of 35 of my primarily computer science students who I struggled to reach. If these students simply struggled with the material this would be one thing, but they ended up disruptive to the rest of the classroom environment, so I would like suggestions on how to better handle this subset of students in the future.&lt;/p&gt;&#xA;&lt;p&gt;I will try to characterize them through example behavior, though of course this characterization is bound to be inaccurate and no one student met all bullet points below. My question is, how to most effectively teach proof-based content to students who exhibit subsets of this behavior, or perhaps how to most effectively handle their impact on a classroom's attitude towards the class, and proof in general (since these students are often very vocal and self-assured, as they are perhaps accustomed to having better grades in computation-based courses). Some example behavior:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Argues against the notion of doing problems on homework that are not exactly identical to the problems in class (this is common in lower classes, but it is a key feature of the students I am describing)&lt;/li&gt;&#xA;&lt;li&gt;Refuses to acknowledge differences in if-then statements in natural language versus propositional logic, even when those differences are explicitly addressed in one-on-one conversation (e.g. argues against the professor's statement that &amp;quot;if 3&amp;gt;5 then 3&amp;gt;4&amp;quot; is a true proposition in class and in office hours, and will argue that the instructor does not know anything rather than concede that they could be wrong.)&lt;/li&gt;&#xA;&lt;li&gt;Insists that the model proofs provided by the instructor contain superfluous information, which is in fact necessary to include at their level.&lt;/li&gt;&#xA;&lt;li&gt;Calls inductive proofs &amp;quot;circular&amp;quot; in complete confidence, not realizing that they do not understand induction.&lt;/li&gt;&#xA;&lt;li&gt;Makes negative comments about how nothing in the class makes sense during weekly group work, derailing their groups.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I thought that, for example, explicitly discussing the purpose of homework in the class, explicitly addressing the differences between natural language and propositional logic in class, providing 3 model proofs for the students to grade and rank before midterms with my own grade and rank provided later, and going over the logic of induction during every example inductive proof we did for a week would at least dissuade these students. But unfortunately my class had the most negative attitude I have ever experienced, and this was reflected in my student evaluations that term, despite always receiving high evaluations before.&lt;/p&gt;&#xA;&lt;p&gt;Can these conflicts be avoided? Is this something that is bound to appear in any early-proofs course, and if so, how can these students be better reached -- by instruction or simple &amp;quot;classroom management&amp;quot; ?&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="11" LastEditDate="2021-11-15T16:26:50.613" LastActivityDate="2021-11-22T08:33:31.257" Title="Combative students in proofs classes" Tags="|undergraduate-education|student-motivation|classroom-management|classroom-interaction|" AnswerCount="10" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="24551" PostTypeId="2" ParentId="24549" CreationDate="2021-11-08T22:51:16.847" Score="17" Body="&lt;p&gt;EDIT: I would like to clarify that my response below is not intended to be definitive.  This is an extremely difficult problem to have.  It is perhaps the most difficult problem one can have as a teacher:  a complete breakdown of the trust in each other which is needed to make communication possible.  The idea below is only a stab in the dark, which has many obvious potential drawbacks.  Very open to hearing other approaches which are less problematic.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Unfortunately, it seems that these students do not trust you as a subject matter expert. This is a difficult place to be.  If they are convinced that your proofs are invalid, that your understanding of logic is faulty, and that your understanding of mathematical induction is circular, it will be very difficult to regain any sort of healthy classroom atmosphere.&lt;/p&gt;&#xA;&lt;p&gt;Put yourself in their shoes:  if you were taking a class and the instructor was clearly spouting nonsense (for instance, a biology class where the instructor clearly had a Lamarckian rather than Darwinian understanding of evolution), you too might act disruptively.  I certainly would!&lt;/p&gt;&#xA;&lt;p&gt;This is an extremely serious problem and it calls for serious solutions.  Here are my suggestions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Hold a meeting with the disruptive students and see if I am right:  do they consider you to be mathematically or logically incompetent?&lt;/li&gt;&#xA;&lt;li&gt;If so, you need to be prepared with a plan to address their concerns. Since the students themselves (clearly) do not have the logical prowess necessary to judge you, all you are left with is an appeal to authority.  See if there are any instructors at your institution who they trust as mathematical authorities.&lt;/li&gt;&#xA;&lt;li&gt;Invite these students, and their chosen authority, to a meeting together.  Discuss these issues together (vacuous implications, mathematical induction, the standards you are holding them to in their logical arguments, etc.).  When their authority figure confirms that they are wrong, and that you are right, this might have some impact.  Hopefully, once they realize that &lt;em&gt;they&lt;/em&gt; are wrong, they can begin really listening to your instruction and trying to learn from it.&lt;/li&gt;&#xA;&lt;li&gt;Make it very clear to them that you will not tolerate disruptive conversations and behaviors during class.  Make a clear distinction between genuine questions (in which both the asker and the receiver carry a curiosity about each other, and a humbleness in their responses), to questions which are designed only to inflame or to dismantle authority.  Let the students know that the later sort of question will be identified as inappropriate and that you will ask a student who persists in these attacks to leave the classroom.  Same comments apply to derailing groupwork:  let them know that they are derailing their group, and that if they cannot stay on task they will be asked to leave.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-11-10T15:59:16.633" LastActivityDate="2021-11-10T15:59:16.633" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24552" PostTypeId="2" ParentId="24549" CreationDate="2021-11-08T23:48:22.930" Score="9" Body="&lt;p&gt;Some helpful feelings of mine about teaching:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;You can't force someone to learn. As an undergraduate professor, I am responsible for being a resource to provide my students with the information they need and some motivation and structure. However, my students are welcome to pass up those resources.&lt;/li&gt;&#xA;&lt;li&gt;I am responsible for providing a classroom environment conducive to learning. If your troublesome students are calling out and disrupting class or disrupting groupwork, provide them with clear warnings and then ask them to leave the classroom if it continues. This is much easier said than done: it is very unpleasant to be confrontational.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-11-08T23:48:22.930" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24553" PostTypeId="2" ParentId="24549" CreationDate="2021-11-09T02:35:50.763" Score="29" Body="&lt;p&gt;You don't say in the question what kind of school this is. It must be a four-year school rather than a community college, but there is no indication of what its admissions standards are like. If this is a state school that's easy to get into, say Cal State Fullerton, and a lot of these students are math education students, then you're extremely lucky that only 10% of your students are behaving in this way. On the other hand, if this is Berkeley or the University of Chicago, then what you describe would seem very anomalous.&lt;/p&gt;&#xA;&lt;p&gt;Your students are probably going through a very drastic transition right now. If your student population is anything like the one at the school that I recently retired from, then about half of them have probably passed their two years of lower-division math under COVID conditions where the tests were online and they cheated on them. They may have learned little or nothing. The test asks them to differentiate &lt;span class=&quot;math-container&quot;&gt;$\sin(x^2)$&lt;/span&gt;, so they use Wolfram Alpha to find out the answer. If the test asks them something that Wolfram Alpha can't answer, they get on their Discord server during the exam and see if anyone knows. Now, they're in a class where someone is actually reading their work carefully. So not only is it their first time actually having their learning evaluated, but they also completely lack the background knowledge needed in order to tackle the material in your class.&lt;/p&gt;&#xA;&lt;p&gt;From your username and some comments above, we know that you're female. This doesn't help. Many students will discount your expertise.&lt;/p&gt;&#xA;&lt;p&gt;I'm male, but when I started teaching I was fairly young, and although for the most part that made it easier to connect with students, it did also tend to make some of them treat me with less respect. It helped somewhat when I started dressing a little more formally. On the first day of class, you might also want to give a brief professional bio. This could be stuff like &amp;quot;Gee whiz, I work on topology, and here's why that's super interesting -- here's a picture of a coffee cup turning into a doughnut!,&amp;quot; but it can also be an occasion when you briefly mention your educational background.&lt;/p&gt;&#xA;&lt;p&gt;As far as disruption of class, you have to strike a balance. If you shut them down completely, then it wrecks the atmosphere of intellectual inquiry, fails to model normal collegial behavior in academia, and risks worsening their lack of confidence in you, since authoritarianism is often a refuge of the incompetent. But if you let them revisit the same issues over and over, take too much time, or behave disrespectfully, then it diminishes the education that can happen in the room. If they're taking too much time on their bogus objection, say that: &amp;quot;Sorry, but we've spent quite a bit of time on your question, and now it's time to move on.&amp;quot; You will probably see other students smiling and nodding when you say this. If they're revisiting something you've already addressed, say that: &amp;quot;This is the same issue you raised last Wednesday, and we discussed it then and I suggested that you take a look at section  16.7 in the book on induction. Unfortunately it's not appropriate to revisit this further at this class, but please do feel free to come by my office hours if you're having trouble with this topic.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Over the years I have visited many people's classes to evaluate their teaching. When there was inappropriate behavior by students, I almost never saw a response from the instructor that seemed firm enough to me. Most people hate conflict and public confrontation, so they err on the side of softness. A typical example would be that the instructor hands back a pile of homework papers, then puts the stack of unclaimed papers on the desk at the front of the room and begins class. Then a student comes in 15 minutes late, walks over to the stack of papers, and stands between the teacher and the rest of the class digging through the pile. Most instructors will simply ignore this, which IMO is wrong because it's extremely distracting and disrespectful. All it takes is: &amp;quot;Ah, John, what you're doing is inappropriate. You're blocking people's view of the board and it's very distracting. Please wait until after class to pick up your graded paper.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="18014" LastEditorUserId="11" LastEditDate="2021-11-15T16:29:36.920" LastActivityDate="2021-11-15T16:29:36.920" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24554" PostTypeId="2" ParentId="24549" CreationDate="2021-11-09T10:22:25.100" Score="-1" Body="&lt;p&gt;Well, there are interesting philosophical questions that remain in logic.  Regarding entailment, for example.  Why not talk to them about it?  eg. Relevance Logic.  And induction too, might be seen as rather intemperate.  Even, perhaps, as exhibiting a debauched lack of discipline.  Maybe have a side talk about axioms vs. axiom schemes, and eg. Predicative Arithmetic.  It's not like the topic of logic is closed.  That might make them feel more comfortable about learning one particular model of rational, deductive thinking.  Some of us instinctually rebel, when we're taught something that doesn't feel like it makes natural sense to us.  Showing that it's one model, might help put a certain distance between them and the thing they are studying, and make it easier for them to swallow.&lt;/p&gt;&#xA;" OwnerUserId="18016" LastEditorUserId="18016" LastEditDate="2021-11-12T21:38:01.880" LastActivityDate="2021-11-12T21:38:01.880" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="24555" PostTypeId="2" ParentId="24549" CreationDate="2021-11-09T14:03:47.187" Score="25" Body="&lt;p&gt;There's probably no silver bullet.&lt;/p&gt;&#xA;&lt;p&gt;But one tool I use is in these situations (e.g., I teach discrete mathematics etc. at a U.S. community college) is to &lt;em&gt;very closely align&lt;/em&gt; with a good textbook. In fact, here's my personal note to myself in my checklist for preparing a new course:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The most important thing is to &lt;strong&gt;TEACH FROM A GOOD BOOK&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My motivation here is precisely that aligning with the book gives voice to a &amp;quot;second opinion&amp;quot; in the course, that I as the instructor have not gone off the rails or become delusional in what I'm presenting or asking students to do (in terms of exercises and test questions). If someone does push back, I can point to a particular section or wording in the book for backup. If they claim to know everything in the class, I ask if they can complete a certain exercise in the book.&lt;/p&gt;&#xA;&lt;p&gt;I'll confess that I wish this made more difference than it does in practice. A vanishingly small number of my students are actually consulting the textbook at all; the vast majority accept being fully dependent on what I personally say. But philosophically it gives me confidence (which might help the OP), and in certain circumstances it's been helpful for me to be hyper-aware of the book sequencing so I could refer students there, or quote some definition or exercise as a backup.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2021-11-09T14:03:47.187" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24556" PostTypeId="1" CreationDate="2021-11-09T19:16:30.427" Score="6" ViewCount="359" Body="&lt;p&gt;I'm not entirely how best to pose this question, so that it fits within the guidelines (so edits / suggestions for modification are warmly welcome).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I'm interested in exploring effective strategies for helping students solidify their basic ability to recognize nonsensical expressions and was curious to know what has worked for others or what others have tried.&lt;/strong&gt; I'll illustrate the kind of &amp;quot;nonsense&amp;quot; I mean with two examples.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;e.g. 1: nonsense commonly recognized by students.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In my experience, most HS and university students working only within the set of real numbers have no trouble recognizing something like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-2}$&lt;/span&gt; as being &amp;quot;meaningless&amp;quot; or &amp;quot;nonsensical&amp;quot; or simply undefined.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;e.g. 2: nonsense not always recognized by students.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Again, drawing from experience, I find a surprisingly large number of students who would recognize the above are less likely to notice that &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$y = \sqrt{3-x}$&lt;/span&gt;  for &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 1$&lt;/span&gt;&amp;quot; doesn't make sense (i.e. assuming one is not using a domain convention in which this would be understood to actually mean &amp;quot;for &lt;span class=&quot;math-container&quot;&gt;$1 &amp;lt; x \leq 3$&lt;/span&gt;&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;N.B.: I'm not interested in focusing on the particular issues with this specific example&lt;/strong&gt;, I'm interested in any effective strategies others have used for helping students develop the instinct to reflect &amp;amp; question what they are reading, to make sure it makes some sort of sense; i.e. to get them in the habit of thinking about the math they are working with and not just &amp;quot;going through the motions.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Again, I know this is quite fuzzy and subjective; however, it seems like it falls under the acceptable type of subjective questions; happy to edit or remove, if I'm mistaken.&lt;/p&gt;&#xA;" OwnerUserId="5797" LastActivityDate="2021-11-19T18:18:24.843" Title="Effective Strategies for Helping Students Recognize Nonsensical Expressions? (HS - Undergrad Level)" Tags="|undergraduate-education|secondary-education|notation|exercises|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24557" PostTypeId="1" CreationDate="2021-11-09T21:29:32.027" Score="4" ViewCount="106" Body="&lt;p&gt;Google Forms enables locked mode (where students cannot move to other tabs or apps without submitting the form) only if students use certain laptops called Chromebooks.&lt;/p&gt;&#xA;&lt;p&gt;I'm looking for any platform to create Math tests (thus, possibly compatible with Equatio or any other friendly way of displaying formulas and plots) that enables a locked mode for other kind of laptops and mobile devices, too.&lt;/p&gt;&#xA;&lt;p&gt;I tried with Microsoft Forms, but I wasn't able to set the locked mode. Any suggestions?&lt;/p&gt;&#xA;" OwnerUserId="18024" LastActivityDate="2021-11-09T21:29:32.027" Title="Quiz in locked mode without managed Chromebooks?" Tags="|testing|test-design|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24558" PostTypeId="2" ParentId="24549" CreationDate="2021-11-09T21:54:30.717" Score="1" Body="&lt;h1&gt;Take Them Seriously&lt;/h1&gt;&#xA;&lt;p&gt;Authority cannot be demanded or mandated; it must be earned.  Frankly, I think you can and must earn it.  You obviously take your job seriously, because you took the time to think carefully about these students, their disruptive behavior, take notes, and solicit advice.  That's great!  When a student challenges you in class, you should buy yourself time by saying: &amp;quot;That's a great point.  I'm going to take a note and we will discuss that later.&amp;quot;  Then write down their objection and move on.  Within the next day or so, bring up the objection in class, and delay the rest of your content to do so, so that the rest of the class can see that 1) you are taking all the students seriously, and 2) frivolous or unfounded interruptions cost the entire class.&lt;/p&gt;&#xA;&lt;h1&gt;Example Responses&lt;/h1&gt;&#xA;&lt;p&gt;Since you mentioned these are mainly CS students, I will provide responses tailored for that community.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Argues against the notion of doing problems on homework that are not exactly identical to the problems in class&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We're going to do a mock interview.  You three have worked at Twitter for the last 5 years as back-end server engineers.  I am the hiring manager at Google where you are looking for your next career transition.&lt;/p&gt;&#xA;&lt;p&gt;Bobby, tell me a bit about the work you've done at Twitter.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Let him BS an answer for a few seconds to warm them up and build up their confidence.  Nod and smile and give small affirmations (&amp;quot;Very good!  Impressive!&amp;quot;).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ok, now why do you think you would be a good fit for Google?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;More rambling BS...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ok, that's all well and good, but based on your performance in MATH-243, I am led to believe that you only work on projects that are identical to work you've already done.  You are aware that Google does not have any micro-blogging products, are you not?  You see, we need engineers to design and build things &lt;em&gt;that have never been done before&lt;/em&gt;.  There are no examples for them to look at and study.  They need to apply the principles they learned elsewhere, and generalize them to completely new areas.  Which part of your MATH-243 history demonstrates this skill?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Refuses to acknowledge differences in if-then statements in natural language versus propositional logic, even when those differences are explicitly addressed in one-on-one conversation (e.g. argues against the professor's statement that &amp;quot;if 3&amp;gt;5 then 3&amp;gt;4&amp;quot; is a true proposition in class and in office hours, and will argue that the instructor does not know anything rather than concede that they could be wrong.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This one is actually very easy to address.  You just need to learn to speak in their language.  Ask them this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let's try a different proposition:&lt;/p&gt;&#xA;&lt;p&gt;if rocket_launchers &amp;gt; rifles then rocket_launchers &amp;gt; pistols&lt;/p&gt;&#xA;&lt;p&gt;Is this proposition true or false?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Any self-respecting gamer will instantly see that this is a trap.  Because the natural-language version is true for some games, false for others, and both true and false for yet other games, depending on the map, the team, the enemies, etc.  And they know that their friends in the same class will vehemently object to any possible answer they give with embarrassing counter-examples.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But let's not debate this amongst ourselves.  Let's ask a computer to decide!&lt;/p&gt;&#xA;&lt;p&gt;let a = 3&lt;/p&gt;&#xA;&lt;p&gt;let b = 5&lt;/p&gt;&#xA;&lt;p&gt;let c = b - 1&lt;/p&gt;&#xA;&lt;p&gt;if a &amp;gt; b then a &amp;gt; c&lt;/p&gt;&#xA;&lt;p&gt;Is &lt;em&gt;&lt;strong&gt;this&lt;/strong&gt;&lt;/em&gt; proposition true or false?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This instantly teleports the problem directly into the center of their world, and challenges them with an idea that they &lt;em&gt;should&lt;/em&gt; have already encountered if they have taken any CS classes at all yet.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Insists that the model proofs provided by the instructor contain superfluous information, which is in fact necessary to include at their level.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The basic theme here is that you have a core of students with logic bugs in their programming.  Unfortunately, you have to put on your hacker hat and debug their wetware.  This means coming up with a similar proof which fails without the &amp;quot;superfluous information&amp;quot;.  That can be quite challenging, but if demonstrated clearly and effectively in class, should also be a real eye-opener for everyone.  And I guarantee you it would earn you a tremendous amount of authority, as well.  It isn't easy to come up with these on the fly, so don't.  Make sure you are well-prepared with whatever background research you need to do before responding to these challenges.&lt;/p&gt;&#xA;&lt;p&gt;Most importantly, whenever possible, get the students themselves to lead the class down the path of the answer until they get stuck.  They and the class should ideally &amp;quot;discover&amp;quot; the bug on their own.  Try to avoid saying: &amp;quot;That doesn't work&amp;quot; and replace it with: &amp;quot;What happens if?&amp;quot;  &amp;quot;What about this?&amp;quot;  Whatever statements you want to put in the students heads will work best if the students themselves say it out loud and believe they are the originators of the statement.  Try to work out the &amp;quot;failure modes&amp;quot; of the flawed proofs beforehand so you can readily point them out in class.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Calls inductive proofs &amp;quot;circular&amp;quot; in complete confidence, not realizing that they do not understand induction.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is a &amp;quot;circular proof&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Well, it's a thing where you assume the thing you are trying to prove!&lt;/p&gt;&#xA;&lt;p&gt;Ok, let's try one out.  Everyone come to the front and draw a number from this bowl.  Now one of you will be the Prover, and another will be an Oracle.  The rest will be the numbers.  Now line up, in positions decided by the Oracle.  Ok, the Prover will stand over here with his back to the rest of the class.&lt;/p&gt;&#xA;&lt;p&gt;My theorem is this: the list is in ascending order from left to right iff, for every student, the student to their left has a smaller number.&#xA;Now, Prover, you may ask the Oracle to interrogate any student and the one to their left to see which is larger.  When you are done, you should announce whether the list is in ascending order or not.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is not really an &amp;quot;inductive proof&amp;quot; in the conventional sense, but has the same elements as one, and should help programmer types visualize what is actually happening in the abstract mathematical space.  The special condition where a student does not have a partner to their left is obviously the base case, and you should leave it to the class to discover it.&lt;/p&gt;&#xA;&lt;p&gt;The Oracle can decide whether to put the students in order or not, and you should do both ways without announcing it to the Prover.&lt;/p&gt;&#xA;&lt;p&gt;After the Prover announces their results, you can challenge them: &amp;quot;But you just assumed that to begin with, right?  All the rest of the work was unnecessary, because this is circular, right?&amp;quot;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Makes negative comments about how nothing in the class makes sense during weekly group work, derailing their groups.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This is more difficult, but indicates that some students may require tutoring.  Ideally, your dept. has volunteer or paid tutors available to assist.  If other students are succeeding in the class, you can simply point out that their sentiment is clearly not universal, but you would like to help them catch up, and here are some tutoring resources we can look at.&lt;/p&gt;&#xA;" OwnerUserId="13468" LastActivityDate="2021-11-09T21:54:30.717" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24559" PostTypeId="1" CreationDate="2021-11-10T01:54:02.997" Score="5" ViewCount="367" Body="&lt;p&gt;I am a first-year Pre-Calculus teacher at a public high school in the United States. The curriculum I am using often contains &amp;quot;&lt;a href=&quot;https://www.nctm.org/noticeandwonder/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Notice and Wonder&lt;/a&gt;&amp;quot; activities whenever a new concept is about to be introduced. The purpose of this type of activity, at least from my perspective, is to get the students to make sense of something that is unfamiliar to them. Students also get to hear their classmates' ideas on how to make sense of what is presented, which encourages a classroom environment where finding multiple ways to approach a problem is valued and appreciated.&lt;/p&gt;&#xA;&lt;p&gt;I have tried to implement a few Notice and Wonder activities but I do not think I carried them out successfully. What I usually do is give students about 1 minute to observe the task and then I ask for volunteers to share what they noticed and wondered. For whatever reason, this is when I get a lot of blank stares; not many students have been willing to share what they noticed. I tell my students that there is no wrong answer when it comes to sharing what they noticed, but this has had little effect. Since I teach older students (11th and 12th graders), perhaps they are less enamored with the idea of Notice and Wonder.&lt;/p&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Is my understanding of the purpose of a Notice and Wonder activity valid?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;What is the best way to implement a Notice and Wonder activity?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Some ideas:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do a &lt;a href=&quot;https://www.readingrockets.org/strategies/think-pair-share&quot; rel=&quot;nofollow noreferrer&quot;&gt;Think-Pair-Share&lt;/a&gt; so that more students are confident enough to volunteer.&lt;/li&gt;&#xA;&lt;li&gt;Call on students regardless of whether or not they raise their hand (i.e. use a lottery system for randomness).&lt;/li&gt;&#xA;&lt;li&gt;Continue to maintain a classroom environment where the product (i.e. correct answer) is far less important than the process.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="13657" LastEditDate="2021-11-14T01:10:25.930" LastActivityDate="2021-11-14T01:10:25.930" Title="How to Effectively Implement a &quot;Notice and Wonder&quot; Activity?" Tags="|secondary-education|problem-design|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24560" PostTypeId="2" ParentId="24549" CreationDate="2021-11-10T05:50:53.137" Score="5" Body="&lt;p&gt;The problem with induction is common (I'm sure you are aware of that). With an audience of CS majors I would try and utilize the connection between induction and recursion. Basically driving home the point that induction is the tool for proving that a function defined by recursion gives the predicted outcome.&lt;/p&gt;&#xA;&lt;p&gt;I would go for the throat and use &lt;a href=&quot;https://en.wikipedia.org/wiki/Ackermann_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ackermann's function&lt;/a&gt;. We define the function &lt;span class=&quot;math-container&quot;&gt;$f(x,y)$&lt;/span&gt; with natural number inputs &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; by the following innocent looking rules:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(0,y)=y+1$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x+1,0)=f(x,1)$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x+1,y+1)=f(x,f(x+1,y))$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The students may have the desire to code this in whichever programming language they choose. But the last one is a real killer, and makes the most powerful computer scream for mercy.&lt;/p&gt;&#xA;&lt;p&gt;A sequence of exercises around the theme could be:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Why/how do these relations define the function for all inputs &lt;span class=&quot;math-container&quot;&gt;$x,y\in\Bbb{N}$&lt;/span&gt;? (this is actually somewhat delicate, you may want to put it last)&lt;/li&gt;&#xA;&lt;li&gt;Calculate (or run) a few test values &lt;span class=&quot;math-container&quot;&gt;$f(1,y)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$y=0,1,2,\ldots,5$&lt;/span&gt;. Form a conjecture as to what &lt;span class=&quot;math-container&quot;&gt;$f(1,y)$&lt;/span&gt; is in simpler terms. Prove it by induction on &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The same with &lt;span class=&quot;math-container&quot;&gt;$f(2,y)$&lt;/span&gt;. Calculate a few values, form a conjecture, prove it by induction.&lt;/li&gt;&#xA;&lt;li&gt;The same with &lt;span class=&quot;math-container&quot;&gt;$f(3,y)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Then &lt;span class=&quot;math-container&quot;&gt;$f(4,2)$&lt;/span&gt; (or make that &lt;span class=&quot;math-container&quot;&gt;$f(4,2021)$&lt;/span&gt; is you feel particularly cruel).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Anyone who tries with a few lines of code alone will run into a limit on recursion depth. Guaranteed :-)&lt;/p&gt;&#xA;&lt;p&gt;Anyway, using the connection that when proving anything about recursion &lt;strong&gt;induction is the only tool&lt;/strong&gt; may make the penny drop.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I only had to teach a related class to CS majors once. I gave them the answer for &lt;span class=&quot;math-container&quot;&gt;$f(1,y)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(2,y)$&lt;/span&gt; and had them prove it for general &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; by induction. &lt;span class=&quot;math-container&quot;&gt;$f(3,y)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(4,2)$&lt;/span&gt; where extra assignments. I may have given the formula for &lt;span class=&quot;math-container&quot;&gt;$f(3,y)$&lt;/span&gt; as well. It did generate a lot of discussion. Like one student asked whether I'm at all worried about recursion depth :-) Which is really the point for computer scientists. You can actually do &lt;span class=&quot;math-container&quot;&gt;$f(1,y)$&lt;/span&gt; in class.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, it may be worth your while to use a (simpler) recursively defined function when explaining induction. Fibonacci may be enough. If Fibonacci is too much, start with integer multiplication &lt;span class=&quot;math-container&quot;&gt;$0*y=0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(x+1)*y=x*y+y$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2021-11-10T06:11:36.170" LastActivityDate="2021-11-10T06:11:36.170" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24561" PostTypeId="2" ParentId="24559" CreationDate="2021-11-10T15:53:56.870" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Yes, I think your general understanding of the purpose is correct.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In my view, it has four goals.  For the weaker student, it gives them a chance to find the footing to start asking questions.  For the average student, it gives them a chance to start making connections to prior knowledge.  For the gifted student, it gives them a chance to hypothesize and see if their intuition is valid.  For you, it allows you to gauge where students are in this continuum on this topic.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;strong&gt;There is no best answer to this question; it will depend on you and your students.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Do your best to understand why your particular students may not be engaging -- if you have a good rapport with them, you can even try asking one or two apart from the crowd.  If the problem is authenticity (as was mentioned in the comments), try creating your own similar activities to increase relevance.  If the problem is anxiety at being wrong, given them low risk activities that are harder to opt out of, like Vote With Your Feet/Valley of Values until they start feeling comfortable with sharing.  If the problem is not understanding the expectations, modeling a question or two for them will help quite a bit.  Sometimes, though, especially as a new teacher, it's that we simply aren't waiting long enough for a response.  As an AP Calculus teacher, my students are very mathematically inclined, and yet sometimes I still get the blank stares when I ask certain questions.  As long as you've created a classroom environment where it's safe to be &amp;quot;wrong&amp;quot;, there's no harm in letting those quiet moments stretch out while you're waiting for an answer to be offered&lt;/p&gt;&#xA;" OwnerUserId="14966" LastActivityDate="2021-11-10T15:53:56.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24562" PostTypeId="1" CreationDate="2021-11-10T17:21:41.847" Score="2" ViewCount="137" Body="&lt;p&gt;What are the advantages and wins of solving already solved problems? Does it help me know how to become better at problem solving and test my abilities on it? Does it help me become better and have more possibilities of solving open problems?&lt;/p&gt;&#xA;&lt;p&gt;When i find already solved problems difficult to solve, should i look at the answer? Should i try to solve them by reading the theory from books or something else like leave them for some time like one, two or three days and come back to solve them?&lt;/p&gt;&#xA;&lt;p&gt;I am an undergraduate student who tries to get his degree in math and i am interested in solving open problems.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastActivityDate="2021-11-10T17:21:41.847" Title="Solved , open problems and levels of difficulty" Tags="|problem-solving|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24564" PostTypeId="2" ParentId="24549" CreationDate="2021-11-11T03:09:16.640" Score="5" Body="&lt;p&gt;This is a tricky situation. Here are some strategies that I have followed in my Discrete Math for CS class, perhaps they will be of some use to you.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I make it clear that the purpose of the class is to understand how to read mathematical definitions and what constitutes a proof and that this understanding is to be inculcated in a hands-on manner, i.e., by reading and understanding definitions and learning how to write proofs of theorems based on those definitions. This goes some way in answering your first question: homework questions &lt;em&gt;must&lt;/em&gt; be different because it is not the specific topic that is important but the process.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I begin with logic and present it as an abstraction, i.e., I actively discourage the students (who are at the sophomore level) from using their &amp;quot;real world&amp;quot; ideas while trying to understand logic. A good way to do this is to begin with the law of the excluded middle which doesn't hold in the real world. Essentially I present logic as a formal system with its own rules. Individual propositions like &amp;quot;3 &amp;gt; 4&amp;quot; are thought of as statements that bring their own truth value with them from some other place and, while we are studying logic, we do not dispute the truth value of a particular proposition, but just look at what truth values emerge when we put propositions together.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At the beginning of the semester I share with them guidelines on how to write a proof based on a discussion provided in Sec 2.7 of the text by Lehmann, Leighton and Meyer (MIT OCW Fall 2010).&#xA;With this as a basis it can become clear what is essential in a proof and what is superfluous. Perhaps having such a benchmark will give you a stronger platform for arguing for the necessity of including extra material in proofs. However, my suggestion to you would be: don't include anything superfluous in a proof. You can always consider having an extra discussion before or after. You can also consider having versions of the proof from the baggiest to the leanest. In fact such a discussion might benefit the students immensely.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am hopeful that if you present Induction after propositional logic and proofs by deduction, you can present it as iterative deduction. This presentation in LLM (see ref above) changed the way I approached Induction. In line with my comment #1 above, I find that students come in with certain wrongheaded ideas that need to be unlearned. LLM's text provides some examples of how proof techniques like Induction, when wrongly applied, lead to preposterous conclusions (&amp;quot;All horses are black&amp;quot;). Such examples can be used to try and convince the students to follow things in a more programmatic way so that they get things right at the sophomore level. As they proceed to later years the intuition will also come.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Your 5th point about negative comments is something I have not encountered, possibly because I teach in India where the students tend to be deferential to professors.&lt;/p&gt;&#xA;" OwnerUserId="18039" LastEditorUserId="18039" LastEditDate="2021-11-11T03:15:51.673" LastActivityDate="2021-11-11T03:15:51.673" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24565" PostTypeId="2" ParentId="24549" CreationDate="2021-11-11T16:57:39.247" Score="0" Body="&lt;p&gt;You do not specify how many of these students have a background in cs, but you indicate some of them does. Computer science stress to a high degree concrete and clear logic used to solve concrete real world problems as effectively as possible. Therefore connecting whatever proof you are trying to teach to concrete, real world problems and demonstrating how you can use a specific type of proof to resolve such a problem would be helpful in making computer science students grasp that concept.&lt;/p&gt;&#xA;&lt;p&gt;Proof based content is very close to the heart of cs. The subject matter will matter to those students at a very personal level.&lt;/p&gt;&#xA;&lt;p&gt;Another alternative: Instead of trying to immediately teach a student that does not understand what you are talking about at a fundamental level, instead listen intently to him and try to pinpoint his underlying assumptions and why he reasons the way he does. It would be good to do this on a one to one basis initially, or at least in a small group. Once you understand the underlying assumptions and reasoning of the student you should have an easier time reaching not just that student, but most students with a similar psychology.&lt;/p&gt;&#xA;&lt;p&gt;Another alternative: If you have the option to bring in a good guest lecturer  this can be a good way to reach students you have trouble with. Sometimes you simply need a different approach to reach people. If the guest lecturer does manage to explain things to these combative students, you can learn from him.&lt;/p&gt;&#xA;&lt;p&gt;Note that all of the above assume your combative students have a similar background or personality. Good luck.&lt;/p&gt;&#xA;" OwnerUserId="18046" LastActivityDate="2021-11-11T16:57:39.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24568" PostTypeId="1" AcceptedAnswerId="24571" CreationDate="2021-11-12T19:24:54.330" Score="22" ViewCount="12595" Body="&lt;p&gt;I was just tutoring someone and we went through some sort of diagnostic test thing, when the following question came up.&lt;/p&gt;&#xA;&lt;p&gt;Question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Here is some information about &lt;span class=&quot;math-container&quot;&gt;$50$&lt;/span&gt; people who took the driving test:&#xA;&lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt; of the &lt;span class=&quot;math-container&quot;&gt;$50$&lt;/span&gt; people are teenagers. One quarter of the adults&#xA;failed the driving test. The number of adults who passed the driving&#xA;test was &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; more than the number of teenagers who passed the test.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The final part of the multi-part question was:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A person is chosen at random. What is the probability that they passed the driving test? [1 mark]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The student answered: &lt;span class=&quot;math-container&quot;&gt;$\ \frac{40}{50}.\ $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I said, &amp;quot;that's correct, but you should simplify the answer because it's always good to simplify fractions unless there is a good reason not to.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;They replied, &amp;quot;but it's the last part of the question (and so there isn't another part of the question that relies on simplification of this fraction), simplifying takes up time that they could be spending doing other questions in the test.&amp;quot; So basically, they disagreed with me.&lt;/p&gt;&#xA;&lt;p&gt;I then stopped for pause and admitted, maybe they're right, but I'm not sure if they would get the mark if they do not fully simplify the fraction or give the answer as a decimal. But I'm not sure I am correct about this. Are they correct and I am mistaken? What is the correct response to my student?&lt;/p&gt;&#xA;&lt;p&gt;Source of the test, for anyone interested: &lt;a href=&quot;https://qualifications.pearson.com/content/dam/pdf/GCSE/mathematics/GCSE%20Maths%20Online%20Study%20Course_Diagnostic%20Assessment.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://qualifications.pearson.com/content/dam/pdf/GCSE/mathematics/GCSE%20Maths%20Online%20Study%20Course_Diagnostic%20Assessment.pdf&lt;/a&gt;&#xA;Note that the test doesn't give instructions to what form the answer should be given.&lt;/p&gt;&#xA;&lt;p&gt;And the context is that this is a GCSE student I have just begun tutoring, and I basically gave them this assessment to broadly test their ability of fundamentals (arithmetic, shapes, problem solving...) just to get a feel for the level they are at, because students often over- or occasionally under-estimate their own abilities, as well as testing their speed and their accuracy in test conditions. As already stated, I &lt;em&gt;expect&lt;/em&gt; students to simplify fractions where possible unless there is a good reason not to do so because that is what I have been taught to do and it has fared me well. This entire question is basically me calling into question whether or not simplifying fractions is a reasonable expectation in this context.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2021-11-13T16:00:33.310" LastActivityDate="2023-03-20T14:43:47.933" Title="Student asked me if it is necessary to simplify fractions at the end of answering a question. I'm not sure how to respond" Tags="|secondary-education|exams|fractions|" AnswerCount="14" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24569" PostTypeId="2" ParentId="24568" CreationDate="2021-11-12T19:47:38.877" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I then stopped for pause and admitted, maybe they're right, but I'm&#xA;not sure if they would get the mark if they do not fully simplify the&#xA;fraction or give the answer as a decimal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Life is not about getting points. A person who doesn't automatically simplify 40/50 to 4/5 (or 0.8) is being silly and annoying, or showing a lack of competence. This person may be unlucky enough to have a teacher who doesn't enforce this expectation. If so, then they're lucky to have you to provide the guidance that they should be getting.&lt;/p&gt;&#xA;" OwnerUserId="18054" LastActivityDate="2021-11-12T19:47:38.877" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24570" PostTypeId="1" CreationDate="2021-11-12T19:58:22.910" Score="7" ViewCount="313" Body="&lt;p&gt;Context: I teach at a small liberal arts school in the US.  The students of interest in this question are generally in their first year.&lt;/p&gt;&#xA;&lt;p&gt;In courses like precalculus and calculus I like to use functions involving fractional powers like &lt;span class=&quot;math-container&quot;&gt;$x^\frac{2}{3}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x^\frac{7}{5}$&lt;/span&gt;.  Occasionally that leads to the evaluation of expressions like &lt;span class=&quot;math-container&quot;&gt;$(-8)^\frac{2}{3}$&lt;/span&gt;.  Of course, this is equal to &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, but not all calculators are able to resolve this expression, unfortunately.&lt;/p&gt;&#xA;&lt;p&gt;Graphing calculators like the TI-83 and TI-84 have no issues, and the TI-89 does not complain, provided that you have its complex format set to &amp;quot;real&amp;quot; instead of &amp;quot;rectangular&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;However, I have found that basic calculators like the TI-30XIIS and TI-36 respond to this expression with domain errors.&lt;/p&gt;&#xA;&lt;p&gt;Most curiously, these basic calculators &lt;em&gt;can&lt;/em&gt; evaluate expressions with composite powers like &lt;span class=&quot;math-container&quot;&gt;$((-8)^\frac{1}{3})^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$((-8)^2)^\frac{1}{3}$&lt;/span&gt;.  In fact, they seem to be able to handle negative numbers to the power of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{n}$&lt;/span&gt; (of course, provided that &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is odd).&lt;/p&gt;&#xA;&lt;p&gt;This leaves me with a few questions...&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there any workaround for this other than telling me students to (a) understand rational exponents more thoroughly, (b) buy a better calculator, or (c) break every fractional exponent into repeated powers?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is it about the TI-30XIIS that allows it to evaluate the decomposed expressions but not the original?  I've heard it said that it uses logarithms to handle rational exponents, so that might explain why the first expression ran into issues--but that does not explain why the decomposed expressions were ok!  (Maybe they're coded differently?)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Any insight that anyone can provide would be much appreciated!  (Also, I'm not sure if this question is appropriate here, or if it should be migrated to math.stackexchange.)&lt;/p&gt;&#xA;" OwnerUserId="5988" LastEditorUserId="5988" LastEditDate="2021-11-13T00:18:24.253" LastActivityDate="2021-11-13T03:24:25.583" Title="Rational Powers of Negative Numbers on Basic Calculators" Tags="|calculators|exponent|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="24571" PostTypeId="2" ParentId="24568" CreationDate="2021-11-12T20:35:17.157" Score="19" Body="&lt;p&gt;The answer to your question depends on the pedagogical goal of the exercise, and what learning outcomes you have identified.  It basically comes down to the following question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is manipulating fractions one of the skills which you are emphasizing in this class?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If one of the goals of your class is to get students to more handily work with fractions, then yes, you should require them to simplify fractions.  On the other hand, if working with fractions is not one of the identified learning outcomes for the class, then you probably shouldn't insist upon it.&lt;/p&gt;&#xA;&lt;p&gt;However, if you are not the one setting the marking guidelines, then the only possible answer is &amp;quot;Ask whoever is in charge.&amp;quot;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As an example from my own teaching:  in my precalculus classes, we spend some time talking about various equations for a line.  Starting from very basic Greek geometry, we can conclude that if &lt;span class=&quot;math-container&quot;&gt;$(x_1,y_1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x_2, y_2)$&lt;/span&gt; are any two points in the plane, then &lt;em&gt;an&lt;/em&gt; equation for the line is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (y-y_1)(x_1 - x_2) = (x - x_1)(y_1 - y_2). $$&lt;/span&gt;&#xA;After a little bit of manipulation, this can be written as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ y - y_1 = \left( \frac{y_1 - y_2}{x_1 - x_2} \right) (x-x_1) $$&lt;/span&gt;&#xA;(assuming that &lt;span class=&quot;math-container&quot;&gt;$x_1 \ne x_2$&lt;/span&gt;) which is the &lt;em&gt;point-slope&lt;/em&gt; equation for a line.  For example, the line through the points &lt;span class=&quot;math-container&quot;&gt;$(2,5)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(1, 3)$&lt;/span&gt; is given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ y-3 = \left( \frac{3-5}{1-2} \right)(x-1). $$&lt;/span&gt;&#xA;I am perfectly happy to give full credit to this answer on an exam, as I expect my students to be capable of basic arithmetic, and the goal of the exercise is to demonstrate recall of this equation.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, I would &lt;em&gt;expect&lt;/em&gt; them to simplify this equation in the context of a question like &amp;quot;Find an equation for a line which is perpendicular to the line through &lt;span class=&quot;math-container&quot;&gt;$(2,5)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(1,3)$&lt;/span&gt; which passes through the point &lt;span class=&quot;math-container&quot;&gt;$(-4,5)$&lt;/span&gt;.  Certainly, if they wrote&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ y - 5 = -\left( \frac{1-2}{3-5} \right)(x+4), $$&lt;/span&gt;&#xA;I would probably give them full marks (as it demonstrates understanding of the key concept), but I would encourage them to simplify the fraction, as this will (1) make it easier to see the relation between the two slopes (&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{2}$&lt;/span&gt;) and (2) simplify further computation.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2021-11-12T20:42:07.357" LastActivityDate="2021-11-12T20:42:07.357" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24572" PostTypeId="2" ParentId="12410" CreationDate="2021-11-12T20:46:05.743" Score="0" Body="&lt;p&gt;I remember getting the math bug in school from reading Martin Gardner's Scientific American column &amp;quot;Mathematical Games&amp;quot;.  While I didn't understand everything I read, I was fascinated. These columns have been collected into several books, the first of which is &amp;quot;Hexaflexagons and Other Mathematical Diversions&amp;quot;. The entire collection is available in CD format as &amp;quot;Martin Gardner's Mathematical Games Cdr Edition&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="5005" LastActivityDate="2021-11-12T20:46:05.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24573" PostTypeId="2" ParentId="24568" CreationDate="2021-11-12T20:55:50.707" Score="13" Body="&lt;p&gt;I tell my students this story when this issue comes up:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Imagine you are answering the phone at the local pizza place. Someone on the other end says &amp;quot;Yes, I'd like to place an order. I'd like twelve thirds pizzas.&amp;quot; What would you think?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They often come up with the following explanations:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It is a prank call.&lt;/li&gt;&#xA;&lt;li&gt;Maybe I misheard them?&lt;/li&gt;&#xA;&lt;li&gt;Maybe they don't know what thirds are?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;When someone gives a final answer like &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{40}{50}$&lt;/span&gt;,&amp;quot; the possible explanations are the same.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-11-12T20:55:50.707" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24574" PostTypeId="2" ParentId="24568" CreationDate="2021-11-12T21:51:46.407" Score="43" Body="&lt;p&gt;The short answer to your question is: &lt;strong&gt;everyone is right&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I agree with people here that in many contexts, &lt;span class=&quot;math-container&quot;&gt;$0.75$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4}$&lt;/span&gt; would be a more desirable answer than &lt;span class=&quot;math-container&quot;&gt;$\frac{45}{60}$&lt;/span&gt;.  I also agree with several here that when the context is &amp;quot;out of 50 people&amp;quot;, an unreduced fraction like &lt;span class=&quot;math-container&quot;&gt;$\frac{40}{50}$&lt;/span&gt; makes perfect sense.  Certainly, when I write \$13.25 cents as &amp;quot;Thirteen Dollars and &lt;span class=&quot;math-container&quot;&gt;$\frac{25}{100}$&lt;/span&gt;&amp;quot; on my checks, the bank would be concerned about me if I wrote &amp;quot;Thirteen Dollars and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;.&amp;quot;  What matters more than anything else is the &lt;strong&gt;context&lt;/strong&gt; of the question and the &lt;strong&gt;audience&lt;/strong&gt; for whom the audience is intended.&lt;/p&gt;&#xA;&lt;p&gt;Consider simplifying radicals -- this is an important skill in Geometry class in US High Schools, especially when dealing with fractions with radical denominators.  Leaving a fraction of the form &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\sqrt{8}}$&lt;/span&gt; is unheard of, &lt;em&gt;because dealing with radicals, and rationalizing denominators, is a skill taught in the course&lt;/em&gt;.  Now consider the same situation in an AP Calculus class.  Students in this course are not only allowed, &lt;em&gt;but encouraged by the format of the test&lt;/em&gt; to leave this fraction alone in the free-response section.  Why?  Because verifying that the student performed the Calculus correctly is more important than seeing if they make a computational mistake in the simplifying algebra afterward.&lt;/p&gt;&#xA;&lt;p&gt;Most multiple choice standardized tests, including the AP Calculus test, will show simplified fractions most of the time -- but not all of the time.&lt;/p&gt;&#xA;&lt;p&gt;As a result, in my courses, I typically explicitly vary the context, and make it obvious when I do so.  In one chapter, we might be emphasizing simplification, so I will do all practice problems with a simplified answer, and note in the test instructions in bold that answers must be simplified to receive full credit.  In another chapter, I will state that simplification isn't as important, practice problems will be done with non-simplified answers, and the test instructions will either say that simplification is unnecessary or omit discussing simplification at all.  (I do the same thing with calculator use, and the same thing with being able to look up or reference formulas.).&lt;/p&gt;&#xA;&lt;p&gt;In my mind, this helps them be prepared for varying contexts they will encounter in the future.&lt;/p&gt;&#xA;" OwnerUserId="14966" LastActivityDate="2021-11-12T21:51:46.407" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="24575" PostTypeId="2" ParentId="24570" CreationDate="2021-11-12T23:34:13.523" Score="7" Body="&lt;p&gt;&lt;strong&gt;Answer:&lt;/strong&gt; The calculators are fine, it is the question's premise &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$(-8)^{2/3} = 4$&lt;/span&gt;&amp;quot; that needs to be fixed.&lt;/p&gt;&#xA;&lt;p&gt;Now this was a bit polemic, of course. Here's a more detailed description of what I mean:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;About definitions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since results of computations are not dictated to us be a higher power, but are consequences of the definitions of the involved operations, the question is: What definition of fractional powers of negative numbers yields the equality &lt;span class=&quot;math-container&quot;&gt;$(-8)^{2/3} = 4$&lt;/span&gt;? One can certainly give a definition which yields this result (see below), but in my experience such a definition is hardly ever used, for the following reason:&lt;/p&gt;&#xA;&lt;p&gt;Whenever we define a mathematical concept, there's a number of issues that we should typically consider to evaluate whether the definition - let's call it &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; - that we are putting forth is useful (or &amp;quot;good&amp;quot;):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; should be well motivated.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; should be useful in some sense.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; should have (to a certain extent) nice mathematical properties. (At least if we intend to build rich theory about it.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; should be consistent with previous notions that are generalized by &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In case that even more general concepts than &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; are available (or even common) in the literature, it is desirable that &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; be consistent with these more general concepts.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It's not always possible to satisfy all these points, and &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; might well turn out to be fruitful if only a subset of them are satisfied. But the smaller this subset is, the less likely it become that it is a good choice to use &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Fractional powers &lt;span class=&quot;math-container&quot;&gt;$x^{\alpha}$&lt;/span&gt; for positive numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Here, &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is a rational number.)&lt;/p&gt;&#xA;&lt;p&gt;I think it is useful to first point out why things work so well for &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 0$&lt;/span&gt;.&#xA;There are, of course, various different approaches to define fractional powers of positive numbers - but they all turned out to be equivalent (in the sense that they all yield the same result). Here's what happens to the points 1)-5) above:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Depending on the approach one uses to define &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 0$&lt;/span&gt;, there are various ways to motivate the definition. (I guess there's no point in listing some of them here.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The (equivalent) definitions of &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; are, for instance, useful because they tell use something about solutions of equations: For a non-zero integer &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; and a real number &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 0$&lt;/span&gt;, the number &lt;span class=&quot;math-container&quot;&gt;$y = x^{1/q}$&lt;/span&gt; is the only positive real number that satisfies &lt;span class=&quot;math-container&quot;&gt;$y^q = x$&lt;/span&gt;. This shows that positive numbers to the power &lt;span class=&quot;math-container&quot;&gt;$1/q$&lt;/span&gt; are a useful concepts. This idea that &amp;quot;fractional powers solve equations&amp;quot; is consistent with more general fractional exponents: For non-zero rational &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and for positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, the number &lt;span class=&quot;math-container&quot;&gt;$y = x^{1/\alpha}$&lt;/span&gt; is the only positive solution of &lt;span class=&quot;math-container&quot;&gt;$y^\alpha = x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For fixed &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 0$&lt;/span&gt;, there are various nice properties of the mapping &lt;span class=&quot;math-container&quot;&gt;$\alpha \mapsto x^\alpha$&lt;/span&gt;. For instance, it is a group homomorphism from the additive group of rational numbers into the multiplicitive group of positive real numbers. Moreoever, it is continuous, and it's continuous extension to &lt;span class=&quot;math-container&quot;&gt;$\alpha \in \mathbb{R}$&lt;/span&gt; is even differentiable, and in fact even real analytic.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The definition is consistent with the definition of &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; for integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The definition is also consistent with the definition of &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; for real &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 0$&lt;/span&gt; (that one obtains either by continuous continuation of fractional powers, or by using the real exponential function and logarithm). It is even consistent with the definition of &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; that one obtains for &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{C} \setminus (-\infty,0]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\alpha \in \mathbb{C}$&lt;/span&gt; by using the principal branch of the complex logarithm.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Fractional powers &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; for negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, let &lt;span class=&quot;math-container&quot;&gt;$x &amp;lt; 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\alpha = p/q \in \mathbb{Q}$&lt;/span&gt; for integers &lt;span class=&quot;math-container&quot;&gt;$q \not= 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;. There are serious problems to give a general ad hoc definition of &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; since, for even &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;, there is no real number &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; which satisfies &lt;span class=&quot;math-container&quot;&gt;$y^q = x$&lt;/span&gt;. So an ad hoc definition of &lt;span class=&quot;math-container&quot;&gt;$x^{p/q}$&lt;/span&gt; can apparently only work for odd &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; - in which case we can define &lt;span class=&quot;math-container&quot;&gt;$x^{p/q}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$(x^{1/q})^p$&lt;/span&gt;, where we define &lt;span class=&quot;math-container&quot;&gt;$x^{1/q}$&lt;/span&gt; to be the only real number whose &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;'s power is &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. This ad hoc definition does indeed yield &lt;span class=&quot;math-container&quot;&gt;$(-8)^{2/3} = 4$&lt;/span&gt;. So let's check what happens to points 1-5 with this definition:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;One can certainly find a reasonable motivation for such a definition (for instance, that one whises to have a similar concept as for positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;). But even here there's a first problem: Since we have, for instance, also &lt;span class=&quot;math-container&quot;&gt;$\alpha = \frac{2p}{2q}$&lt;/span&gt;, this definition only works if we first cancel all common factors of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; - so the definition of &lt;span class=&quot;math-container&quot;&gt;$x^{\alpha}$&lt;/span&gt; now relies on a certain representation of the number &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; (instead of giving the definition for every representation and just showing that this is consistent). Strictly speaking, that's not a problem, but it makes the definition of &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; at least a bit less &amp;quot;natural&amp;quot; than for positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$\alpha = 1/q$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is odd, one can say as in the case of positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that it makes sense to define &lt;span class=&quot;math-container&quot;&gt;$x^{1/q}$&lt;/span&gt; to be the unique real number whose &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;'s power is &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. However, in contrast to the positive case, this does not generalise to fractional exponents whose numerator is even: Indeed, if we use the definition described above - and thus obtain &lt;span class=&quot;math-container&quot;&gt;$(-8)^{2/3} = 4$&lt;/span&gt; -, then &lt;span class=&quot;math-container&quot;&gt;$y = (-8)^{2/3} = 4$&lt;/span&gt; is &lt;em&gt;not&lt;/em&gt; a solution of the equation &lt;span class=&quot;math-container&quot;&gt;$y^{3/2} = -8$&lt;/span&gt;. So the idea that &lt;span class=&quot;math-container&quot;&gt;$y =  x^{1/\alpha}$&lt;/span&gt; solves &lt;span class=&quot;math-container&quot;&gt;$y^\alpha = x$&lt;/span&gt; breaks down with this definition - and thus one of the major reasons why fractional powers are useful is lost.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This is where it becomes really bad. From an algebraic point of view we can at least still say that &lt;span class=&quot;math-container&quot;&gt;$\alpha \mapsto x^\alpha$&lt;/span&gt; is a group homomorphism from the additive group &lt;span class=&quot;math-container&quot;&gt;$G := \{p/q: \, p,q \in \mathbb{Z}, \; q \text{ odd}\}$&lt;/span&gt; into the multiplicative group &lt;span class=&quot;math-container&quot;&gt;$(0,\infty)$&lt;/span&gt; (although the domain of the group homomorphism is, well, a bit special). But from an analytic point of view, everything breaks down: When we endow the set &lt;span class=&quot;math-container&quot;&gt;$G \subseteq \mathbb{R}$&lt;/span&gt; with the topology induced by &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, the mapping &lt;span class=&quot;math-container&quot;&gt;$\alpha \mapsto x^\alpha$&lt;/span&gt; is far from being continuous: If &lt;span class=&quot;math-container&quot;&gt;$p,q$&lt;/span&gt; are both odd, then &lt;span class=&quot;math-container&quot;&gt;$x^{p/q}$&lt;/span&gt; will be negative. But if &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is even and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is odd, &lt;span class=&quot;math-container&quot;&gt;$x^{p/q}$&lt;/span&gt; will be positive. Since both sets &lt;span class=&quot;math-container&quot;&gt;$\{p/q: \; p,q \text{ odd}\}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\{p/q: \; p \text{ even}, q \text{ odd}\}$&lt;/span&gt; are dense in &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, the mapping &lt;span class=&quot;math-container&quot;&gt;$\alpha \mapsto x^\alpha$&lt;/span&gt; is nowhere continuous.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This point seems to be ok. The definition is still consistent with the definition of &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; for integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This doesn't go well, either. Since &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; does not depend continuously on &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, we can't even extend the definition to real &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; by a continuity argument. Our definition of &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; is also completely at odds with complex analysis, for two reasons: (i) The principal branch of the complex logarithms gives us a definition of &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; for all complex numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; except for those in &lt;span class=&quot;math-container&quot;&gt;$(-\infty,0]$&lt;/span&gt;. So we chose to define &lt;span class=&quot;math-container&quot;&gt;$x^{\alpha}$&lt;/span&gt; for precisely those numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for which the principal branch of the logarithm doesn't yield a result. (ii) If we choose another branch of the logarithm, we will get a definition of &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; for negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; - but with this definition, &lt;span class=&quot;math-container&quot;&gt;$x^\alpha$&lt;/span&gt; will depend continuously on &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and hence, it cannot be consistent with the definition that we gave.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Summary:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;For the reasons pointed out above, it is in my experience very uncommon to define &lt;span class=&quot;math-container&quot;&gt;$x^{p/q}$&lt;/span&gt; for negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, odd &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; and general integers &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Therefore:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Please let the basic calculators live :-), they are apparently doing a good job (at least with respect to this question).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Please don't tell your students that &lt;span class=&quot;math-container&quot;&gt;$(-8)^{2/3}$&lt;/span&gt; were &amp;quot;of course&amp;quot; equal to &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; - this does them a disservice because the underlying definition is not useful with respect to solving equations, very badly behaved from an analytic point of view (continuity!), and incosistent with common concepts from complex analysis.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Additional remarks:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;As I mentioned above, some people certainly define &lt;span class=&quot;math-container&quot;&gt;$x^{1/q}$&lt;/span&gt; for negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and odd &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; to be the only real number whose &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;'s power is &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. This has some of the disadvantages mentioned above (in particular, the inconistency with the complex logarithm), but it also has the advantage that it scores highly on point 2: Solving equations is one of the most fundamental things in maths, so a notation which helps us to discuss solutions to certain equations can often be useful. Moreover, this definition is not so badly behaved with respect to 3: Since the set &lt;span class=&quot;math-container&quot;&gt;$\{1/q: \, q \text{ odd}\}$&lt;/span&gt; has no accumulation points in &lt;span class=&quot;math-container&quot;&gt;$(0,\infty)$&lt;/span&gt;, there are no continuity problems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is quite illuminating to check how the common definition &lt;span class=&quot;math-container&quot;&gt;$0^0 := 1$&lt;/span&gt; scores on points 1-5. It might be somewhat ambivalent with regard to 3 (in particular, with respect to continuity issues) but it is absolutely unbeatable with regard to 2: All typical formulas from basic algebra or analysis that include sums over powers (for instance: the binomial expansion, the geometric sum, Taylor expansions) would immediately break down - or at least require extremly bizzare exception handling - if we didn't define &lt;span class=&quot;math-container&quot;&gt;$0^0$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. So this is an nice example where certainly not all of the points 1-5 are satisfied, but where some of the points outrank the others so stronlgy that the definition is completely common.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastEditorUserId="16250" LastEditDate="2021-11-13T03:24:25.583" LastActivityDate="2021-11-13T03:24:25.583" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="24576" PostTypeId="2" ParentId="397" CreationDate="2021-11-13T11:37:32.023" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Most of them just aren't used to having to present their thoughts/work in a way beyond &amp;quot;here's a bunch of equations in a row and here's the final answer in a box&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I find students emulate very well the style of how they've been instructed. Their presentation style reflects the lectures, tutorials, explanations, and texts they have been exposed to.&lt;/p&gt;&#xA;&lt;p&gt;If we want students to adopt a clearer, less terse method of explanation, then we must not subjugate them to a style opposite of these objectives.&lt;/p&gt;&#xA;" OwnerUserId="18057" LastActivityDate="2021-11-13T11:37:32.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24577" PostTypeId="2" ParentId="24568" CreationDate="2021-11-13T15:26:35.600" Score="1" Body="&lt;p&gt;The probability is 4/5’s or 80% not 40/50. The number of people that passed could be described as 40/50 (aka 40 out of 50), but that would require a different question.&lt;/p&gt;&#xA;&lt;p&gt;Whether the answer should or should not be simplified is going to depend upon the question. It’s fundamentally no different from whether the answer should be 1/3, .3, .33, .3333, etc: the question determines the form and precision.  The question should be worded in such a way that the form and precision is obvious.&lt;/p&gt;&#xA;" OwnerUserId="18060" LastActivityDate="2021-11-13T15:26:35.600" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24578" PostTypeId="2" ParentId="24568" CreationDate="2021-11-13T16:34:12.933" Score="8" Body="&lt;p&gt;Í am a physics teacher, not a mathematics teacher, but I would reward the mark.&lt;/p&gt;&#xA;&lt;p&gt;It was a multipart question and you did not show us the other parts. But when I formulate tests I try to have all parts independently of each other, so a mistake in a previous part does not impact the latter parts. Assuming that the parts are independent, the student does have to do a lot of work for 1 mark. First calculate the number of adults, the number of adults that fail, the number of adults that pass, the number of teens that pass and lastly the total number of persons that pass. And this is the shortest path (other solutions require more steps). So what is the mark for: for the students to show he has the insight to make all steps above, or to show he can simplify fractions?&lt;/p&gt;&#xA;" OwnerUserId="18061" LastActivityDate="2021-11-13T16:34:12.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24579" PostTypeId="2" ParentId="24568" CreationDate="2021-11-13T17:22:24.537" Score="6" Body="&lt;p&gt;&lt;strong&gt;tl;dr:&lt;/strong&gt; the answer should be given in the reduced form.&lt;/p&gt;&#xA;&lt;p&gt;Say there are min. 40 students who are taking the exam. Would you want to find the reduced version of the fraction &lt;span class=&quot;math-container&quot;&gt;$394.784176044 /  493.480220054$&lt;/span&gt; 40 times? (assuming every time the answer will be given differently but of same complexity)&lt;/p&gt;&#xA;&lt;p&gt;Another issue is that the rational number has infinitely many fractional representation, hence the fact that student gave the correct answer doesn't mean that you can verify the correctness of the answer in a finite amount of time. That is why, to make the answer unique, you have to stick with a unique representation (or a small subset of possible representations) to make sure that a correct answer can easily be recognised!&lt;/p&gt;&#xA;" OwnerUserId="9513" LastActivityDate="2021-11-13T17:22:24.537" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24580" PostTypeId="2" ParentId="24556" CreationDate="2021-11-13T17:23:52.293" Score="1" Body="&lt;p&gt;I don't have experience teaching this skill directly, but in my calculus teaching I always tried to develop in students the ability to visualize the graph of a function, given its symbolic expression. With your example I think you want to get students to the point where they immediately picture shape of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3-x}$&lt;/span&gt;, which instantly tells them where it doesn't make sense. Exercises in which they are required to sketch a graph without explicit plotting or use of software should develop this skill. This would happen after teaching the shapes of a variety of standard functions and how these shapes are affected by basic algebraic transformations.&lt;/p&gt;&#xA;&lt;p&gt;In solving equations, getting students in the habit of checking their solutions, and giving them carefully selected examples where standard manipulations (such as squaring both side) lead to spurious solutions, should alert them to the fact that not all manipulations preserve truth value for all values of the variables. In such examples you can then ask them to pinpoint the step where the spurious solution entered, and have them identify why that happened. These are habits, and can only be developed by practice.&lt;/p&gt;&#xA;&lt;p&gt;As for Dave Renfro's example in the comments, you want students to reach the point where they immediately become uneasy on seeing such a problem. The graph is more complicated than in your example, but students should be able to notice that &lt;span class=&quot;math-container&quot;&gt;$x^2+\frac{1}{x^2}$&lt;/span&gt; is large both when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is close to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is far from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. So if a solution to &lt;span class=&quot;math-container&quot;&gt;$x^2+\frac{1}{x^2}=\frac{1}{4}$&lt;/span&gt; exists, it can only lie in some delicately chosen sweet spot. But they should quickly come to suspect that no such sweet spot exists. You could try guiding their thinking as follows: ask if they can use &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; real, &lt;span class=&quot;math-container&quot;&gt;$x^2+\frac{1}{x^2}=\frac{1}{4}$&lt;/span&gt;, and basic knowledge of algebra to put bounds on the two terms separately, with the goal being to get them to see that both terms have to lie between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;. Also try to get them to notice that the two terms are reciprocals of each other. So if one term is less than &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;, what can they say about the other?&lt;/p&gt;&#xA;&lt;p&gt;I think that the common element in these two examples is that potential problems are recognized when students  see an algebraic expression as representing a computational process. In &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3-x}$&lt;/span&gt;, a square root is going to be taken, and students know that there will be a problem if the radicand is negative. So what is the radicand? Well, &lt;span class=&quot;math-container&quot;&gt;$3-x$&lt;/span&gt;. So when is that negative? The equation &lt;span class=&quot;math-container&quot;&gt;$x^2+\frac{1}{x^2}=\frac{1}{4}$&lt;/span&gt; tells me that two (clearly positive) numbers add to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;. So both numbers are &lt;span class=&quot;math-container&quot;&gt;$\le\frac{1}{4}$&lt;/span&gt;. But they are reciprocals of each other. Can two reciprocals both be less than &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="1737" LastEditorUserId="1737" LastEditDate="2021-11-16T17:33:17.460" LastActivityDate="2021-11-16T17:33:17.460" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24581" PostTypeId="2" ParentId="24568" CreationDate="2021-11-13T20:51:03.927" Score="22" Body="&lt;p&gt;I am a GCSE Maths examiner. For a question like this, any correct equivalent decimal, percentage or fraction, whether simplified or not, would receive full marks. It is only specifically if it says in the question that the answer should be simplified that a simplified fraction is required to obtain the final accuracy mark. The reason for this is that the question is not testing ability to simplify fractions unless this is mentioned in the question.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastEditorUserId="8900" LastEditDate="2021-11-14T14:19:32.810" LastActivityDate="2021-11-14T14:19:32.810" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="24582" PostTypeId="2" ParentId="24568" CreationDate="2021-11-14T07:21:01.637" Score="5" Body="&lt;p&gt;Let me start with an &lt;a href=&quot;https://philosophy.stackexchange.com/questions/77369/why-is-it-wrong-to-answer-a-question-with-a-tautology-isnt-22-correct-when&quot;&gt;analogous question&lt;/a&gt; on phil-SE:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Teacher:  What is 2+2?&lt;br /&gt;&#xA;Student: 2+2 is 2+2&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;What is wrong with this answer?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now it may &lt;em&gt;seem&lt;/em&gt; to be an issue with using tautologies for communication. However as I point out&#xA;law-of-identity/tautology is a red-herring to address this issue.  To see that, let's change the exchange slightly:&lt;/p&gt;&#xA;&lt;p&gt;Q : What is 2+3&lt;br /&gt;&#xA;A : 2+3 is 3+2&lt;/p&gt;&#xA;&lt;p&gt;Most of us would admit this is correct but &amp;quot;uh-uh&amp;quot;.&#xA;Yet the answer is sufficiently different from the question that pure logic — aka tautologies/law-of-identity etc does not help us.&lt;/p&gt;&#xA;&lt;h1&gt;We need a math answer not a logic one&lt;/h1&gt;&#xA;&lt;p&gt;Note: Ironically some overzealous member there misleadingly edited the question-title adding the word &amp;quot;tautology&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;As I commented there&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What you are looking for is the idea behind the technical term &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Ground_expression&quot; rel=&quot;nofollow noreferrer&quot;&gt;ground term&lt;/a&gt;&#xA;of Rewrite systems.&#xA;Alternatively &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Beta_normal_form&quot; rel=&quot;nofollow noreferrer&quot;&gt;normal form&lt;/a&gt;&#xA;in the lambda calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Likewise here: sometimes 40/50 could be a better answer. Sometimes not.&lt;/p&gt;&#xA;&lt;p&gt;If we back off from this specific question it's clearly a question about...&lt;/p&gt;&#xA;&lt;h1&gt;The definition of &amp;quot;Simplification&amp;quot;&lt;/h1&gt;&#xA;&lt;p&gt;Now much of school-math, at least arithmetic and algebra are just about simplification.&lt;/p&gt;&#xA;&lt;p&gt;But even just going from school arithmetic to algebra &lt;em&gt;the rules change&lt;/em&gt;:&#xA;So in arithmetic &lt;span class=&quot;math-container&quot;&gt;$2+3$&lt;/span&gt; unequivocally simplifies to &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But in algebra between &lt;span class=&quot;math-container&quot;&gt;$x^2 - 5x +6$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x-3) (x-2)$&lt;/span&gt; there is no a priori reason why one is simpler than the other.&#xA;(And calling the first &amp;quot;simplified&amp;quot; and the second &amp;quot;factorized&amp;quot; only adds to the basic problem viz that simplification is context defined)&lt;/p&gt;&#xA;&lt;p&gt;In short therefore, at least for math teachers, if not the general math-literate public, it would be a good idea to go from a fuzzy notion of simplification to...&lt;/p&gt;&#xA;&lt;h1&gt;A reified concept of simplification&lt;/h1&gt;&#xA;&lt;p&gt;This question (and many such) suggests that it's increasingly imperative to teach basics of &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Lambda_calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;lambda-calculus&lt;/a&gt; (LC) and/or &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Rewriting&quot; rel=&quot;nofollow noreferrer&quot;&gt;rewriting-systems&lt;/a&gt;&#xA;(RS) as fundamental (and not &amp;quot;advanced&amp;quot;) math. That is: put something like LC/RS on par with at least calculus, if not algebra/geometry, in the math curriculum.&lt;/p&gt;&#xA;&lt;p&gt;Also this &lt;a href=&quot;https://philosophy.stackexchange.com/questions/77369/why-is-it-wrong-to-answer-a-question-with-a-tautology-isnt-22-correct-when#comment215164_77369&quot;&gt;conversation&lt;/a&gt; is worth reproducing.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now I show you a circle and ask: what is the ratio of the perimeter and the diameter? The answer is Pi. Now I ask: what is Pi? The answer is: it is the ratio of the perimeter and diameter of any circle. That's circular. You could also say that Pi is approximately 3.14, but it's not exactly 3.14, so there really is no better answer than the circular &amp;quot;Pi is the ratio of the perimeter and the diameter, and the ratio of the perimeter and the diameter is Pi.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My response :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Yes... Sometimes one writes &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; as 3.14, sometimes as the more correct  &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80#Formulae_yielding_%CF%80&quot; rel=&quot;nofollow noreferrer&quot;&gt;sophisticated formulae&lt;/a&gt;.   And sometimes we leave &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. (not to mention sometimes making pi into &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; as I've done or &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; into pi as you've ). IOW people answering condescendingly are only succeeding in displaying their ignorance of the non-triviality hiding behind the issue of simplification.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Summary&lt;/h1&gt;&#xA;&lt;p&gt;The notion of simplification is not simple! Nor unique.&#xA;It should at the least be made rigorous.&lt;/p&gt;&#xA;&lt;p&gt;And preferably mechanically computable. Though thats a dispensable frill.&lt;/p&gt;&#xA;" OwnerUserId="17672" LastEditorUserId="17672" LastEditDate="2021-11-16T06:01:19.743" LastActivityDate="2021-11-16T06:01:19.743" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24583" PostTypeId="2" ParentId="24568" CreationDate="2021-11-14T13:35:53.803" Score="4" Body="&lt;p&gt;The general purpose of simplification or writing an expression in some &amp;quot;standard form&amp;quot; is to be able to have a unique canonical answer, and thereby easily compare student work to an official answer for grading, to spot-check against an answer at the back of a book, to compare between two cooperating students, etc.&lt;/p&gt;&#xA;&lt;p&gt;So generally the protocol in a classroom or testing situation will be &amp;quot;yes&amp;quot;, it should be written in the canonical unique form to support that kind of efficient check.&lt;/p&gt;&#xA;&lt;p&gt;I also point out that in the linked sample test, some questions are marked with a &amp;quot;calculator&amp;quot; symbol, while the present question is not. We can deduce the final answer is not meant to be given as a decimal (which would change this advice: admittedly no need in the &lt;em&gt;middle&lt;/em&gt; of a work product), but rather to be given as a fraction. And given the above protocol, it should be simplified, as that's almost surely the form on the answer sheet being used for grading.&lt;/p&gt;&#xA;&lt;p&gt;(That said, the answer from A. Goodier claims from experience that actual GCSE Maths examiners &lt;em&gt;don't&lt;/em&gt; require that; whereas I agree with the commenter about that not being fully coherent.)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-03-17T15:23:04.207" LastActivityDate="2023-03-17T15:23:04.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24584" PostTypeId="1" CreationDate="2021-11-14T17:34:48.690" Score="8" ViewCount="622" Body="&lt;p&gt;I would like to introduce counting principles associated with each of the 4 basic operations (addition, subtraction, multiplication, and division) before introducing permutations and combinations in my Discrete Math course.&lt;/p&gt;&#xA;&lt;p&gt;It is easy for me to come up with lots of simple problems involving the first three operations, but coming up with natural division problems is proving difficult for me.&lt;/p&gt;&#xA;&lt;p&gt;Here are a few I have come up with:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Count the number of edges and vertices of the icosahedron.  If you overcount the number of vertices and edges by treating each of the 20 triangles as distinct, we get &lt;span class=&quot;math-container&quot;&gt;$60$&lt;/span&gt; vertices and &lt;span class=&quot;math-container&quot;&gt;$60$&lt;/span&gt; edges.  However, each edge of the icosahedron corresponds to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; of the edges we have counted:  hence there are really &lt;span class=&quot;math-container&quot;&gt;$60 \div 2 = 30$&lt;/span&gt; edges.  Similarly , each vertex of the icosahedron corresponds to &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; of the vertices we overcounted.  Hence there are &lt;span class=&quot;math-container&quot;&gt;$60 \div 5  =12$&lt;/span&gt; vertices.&lt;/li&gt;&#xA;&lt;li&gt;5 people sit around a round table.  We do not care about their exact position:  only their clockwise arrangement around the table.  How many seating arrangements are possible?  First over count the number of arrangements by keeping track of their exact position in 5 chairs.  There are &lt;span class=&quot;math-container&quot;&gt;$5!$&lt;/span&gt; such possibilities. However, &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; exact seating assignments correspond to each clockwise arrangement.  So there are &lt;span class=&quot;math-container&quot;&gt;$5! \div 5 = 4!$&lt;/span&gt; such clockwise arrangements.&lt;/li&gt;&#xA;&lt;li&gt;The &amp;quot;handshake&amp;quot; problem is another good one:  we need to divide by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; to address overcounting.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In general, we use division when we want to count the number of elements of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;, and we can recognize that &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is bijective with &lt;span class=&quot;math-container&quot;&gt;$X/ \sim$&lt;/span&gt; for some easier to count set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and an equivalence relation &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; whose equivalence classes all have the same cardinality &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.  Then we count &lt;span class=&quot;math-container&quot;&gt;$|S| = |X| \div c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I think this is an important counting principle.  In particular, I want students to have exposure to this idea before we get the formula for combinations, which involves first counting the number of permutations and then dividing by the number of permutations which correspond to the same combination.&lt;/p&gt;&#xA;&lt;p&gt;Please contribute as many natural elementary &amp;quot;division&amp;quot; combinatorics problems as you can invent!&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-11-16T14:05:51.463" Title="Simple combinatorics problems using division" Tags="|discrete-math|exercises|combinatorics|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24585" PostTypeId="2" ParentId="24584" CreationDate="2021-11-14T18:49:10.213" Score="9" Body="&lt;p&gt;You can do necklaces composed of colored beads. For simplicity the number of beads needs to be prime. If there are to be &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; beads with &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; choices of color, then there are &lt;span class=&quot;math-container&quot;&gt;$a^p$&lt;/span&gt; ways to arrange the colors if one bead is identified as the origin. Since &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is prime, the only necklaces with repeating color arrangements are the monochromatic necklaces, of which there are &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. So the number of non-monochromatic arrangements is &lt;span class=&quot;math-container&quot;&gt;$a^p-a$&lt;/span&gt; and the total number of necklaces where rotated arrangements are considered equivalent is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;a+\frac{a^p-a}{p}.&#xA;$$&lt;/span&gt;&#xA;This nicely proves the non-trivial fact that &lt;span class=&quot;math-container&quot;&gt;$a^p-a$&lt;/span&gt; is divisible by &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, which is Fermat's little theorem.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2021-11-14T18:49:10.213" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24586" PostTypeId="2" ParentId="24568" CreationDate="2021-11-14T20:29:25.143" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The test should indicate in what format the answer is expected to be given. This should be clear both to you and to the students. Otherwise grading becomes subjective. If you accept 40/50, would you accept another student giving 120/150, and another 468/585?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The test covers basic math skills. Simplifying fractions seems to fit the overall material covered by the test. It seems appropriate to require students to simplify the fractions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If the student doesn't want to spend their time simplifying the fraction, you need to do it. How much time are you willing to spend? Consider you spent time writing a SE question about it. If the student knew they were asked to simplify the fractions and they leave it to you, they are disrespectful.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="18074" LastActivityDate="2021-11-14T20:29:25.143" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24587" PostTypeId="2" ParentId="24584" CreationDate="2021-11-15T08:01:26.373" Score="1" Body="&lt;p&gt;Just in case this is not obvious, obtaining the general formula for number of ways to choose &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; distinct items can be done by this technique. First choose a sequence of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; distinct items from those, and then apply the &lt;span class=&quot;math-container&quot;&gt;$k!$&lt;/span&gt;-to-&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; mapping from those sequences to sets of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; items, erasing their ordering.&lt;span class=&quot;math-container&quot;&gt;$\def\lfrac#1#2{{\large\frac{#1}{#2}}}$&lt;/span&gt; Since there are &lt;span class=&quot;math-container&quot;&gt;$\lfrac{n!}{(n-k)!}$&lt;/span&gt; sequences, the desired number of ways is &lt;span class=&quot;math-container&quot;&gt;$\lfrac{n!}{k!·(n-k)!}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There is also a more symmetric proof of the same formula: The desired number of ways is simply the number of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-digit binary strings with exactly &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; ones. Well, just label all the zeros &lt;span class=&quot;math-container&quot;&gt;$0_1,0_2,...0_{n-k}$&lt;/span&gt; and all the ones &lt;span class=&quot;math-container&quot;&gt;$1_1,1_2,...,1_k$&lt;/span&gt;, and the number of permutations of these is just &lt;span class=&quot;math-container&quot;&gt;$n!$&lt;/span&gt;. Now erase the labels, which is a &lt;span class=&quot;math-container&quot;&gt;$(n-k)!·k!$&lt;/span&gt;-to-&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; mapping onto the desired set that you want to count.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-11-15T08:01:26.373" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24588" PostTypeId="2" ParentId="24568" CreationDate="2021-11-15T09:14:06.867" Score="4" Body="&lt;p&gt;(Adding this because I want to - the first version of this answer is near the end after a horizontal line.)&lt;/p&gt;&#xA;&lt;p&gt;I think that the acceptable answers should succeed in &lt;em&gt;communicating&lt;/em&gt; with whoever is reading the answer. And the context (who is reading, what the answer is supposed to tell to the reader,...) should be taken into account.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Here &lt;span class=&quot;math-container&quot;&gt;$40/50$&lt;/span&gt; is ok, unsimplified, because the entire population is a nice round number &lt;span class=&quot;math-container&quot;&gt;$50$&lt;/span&gt;, and this answer conveys the desired piece of information in a useful manner.&lt;/li&gt;&#xA;&lt;li&gt;Similarly, I don't care whether a student choose to write &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\sqrt2/2$&lt;/span&gt;. Or &lt;span class=&quot;math-container&quot;&gt;$\dfrac{22}7$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$3\dfrac17$&lt;/span&gt;. True, with more difficult fractions the latter form may give a better about the size of number. I mean, if I'm applying for a grant, and the formulas tell me that the budget total is &lt;span class=&quot;math-container&quot;&gt;$10345712/171$&lt;/span&gt; euros, I still should not use that form of the number in my application.&lt;/li&gt;&#xA;&lt;li&gt;To reiterate the point &lt;a href=&quot;https://matheducators.stackexchange.com/a/1965/282&quot;&gt;from my old answer&lt;/a&gt;. Simplification should perhaps be seen as &lt;strong&gt;a tool&lt;/strong&gt; for converting an answer into a form that is better suited at communicating with the reader as opposed to as &lt;strong&gt;an end&lt;/strong&gt; in itself. In addition to learning simplification tricks, the students could (should?) be tested (or at least trained) in exhibiting good judgement about which forms of an answer are useful for the reader.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Not really answering the title question, and possibly this should be a comment only.&lt;/p&gt;&#xA;&lt;p&gt;But because your pain is my pain, may be we can work around this as follows.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Whenever the question has a numerical answer, and we really want the kids to simplify the end result, may be a way to achieve this is to turn the question into a multiple choice one? Give a list of alternatives, and make the students pick: A) 37/50, B) 4/5, C) 12/25, D) 3/4?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course, this may lead to different kinds of games. Like students complaining if the only numerically correct answer is not fully simplified.&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2021-11-17T09:33:37.587" LastActivityDate="2021-11-17T09:33:37.587" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24589" PostTypeId="2" ParentId="24584" CreationDate="2021-11-15T09:23:52.050" Score="2" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\def\lfrac#1#2{{\large\frac{#1}{#2}}}$&lt;/span&gt;The number of permutations of &lt;span class=&quot;math-container&quot;&gt;$[1..n]$&lt;/span&gt; in which &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; comes before &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is just &lt;span class=&quot;math-container&quot;&gt;$\lfrac12n!$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$n ≥ 2$&lt;/span&gt;. Clearly the equivalence relation is to consider reversals to be the same, though I would not quite say that the desired configurations are the equivalence classes, since each is rather a representative of a class. This fact, however, is useful for the following harder problem.&lt;/p&gt;&#xA;&lt;p&gt;The number of bracelets made from &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; distinct beads, considering it to be invariant under rotation or flipping, is &lt;span class=&quot;math-container&quot;&gt;$\lfrac12(n-1)!$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$n ≥ 3$&lt;/span&gt;. This is a good example that demonstrates that boundary cases are important. Also, it is quite impossible to define what a bracelet even is without using the equivalence relation. Here we have &lt;span class=&quot;math-container&quot;&gt;$n!$&lt;/span&gt; permutations of the beads, and we consider two of them to be equivalent iff one can be obtained from the other by rotations and/or flips. To count the equivalence classes, we can use the powerful technique of &lt;strong&gt;canonicalization&lt;/strong&gt;, which is the idea of finding a useful &lt;strong&gt;canonical form&lt;/strong&gt;. For this problem, given any permutation, its canonical form is obtained by rotating it to make &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; at the front and then reversing the rest if necessary to make &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; come before &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. Each equivalence class has exactly one canonical form, because every sequence of rotations/flips is equivalent to a sequence of rotations and then at most one flip, so any two canonical forms in the same class must be identical. Also, canonical forms in different classes must be different by definition of the classes. Thus we just have to count canonical forms, which yields the answer by the earlier fact.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-11-15T09:23:52.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24591" PostTypeId="2" ParentId="19210" CreationDate="2021-11-16T17:14:48.560" Score="3" Body="&lt;p&gt;Coming back to my question a year later, I'm happy to say that I recently had a breakthrough, by stumbling upon &lt;a href=&quot;https://en.wikipedia.org/wiki/Bloom%27s_2_sigma_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bloom's 2 Sigma Problem&lt;/a&gt;. In &lt;a href=&quot;http://web.mit.edu/5.95/readings/bloom-two-sigma.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;his 1984 paper&lt;/a&gt;, Bloom discusses findings replicated across subjects and age groups showing that when an &amp;quot;average student&amp;quot; was taught by a &amp;quot;good tutor&amp;quot; via Mastery Learning, over a period of time, they perform approximately 2 full standard deviations better than those in the control group, who received conventional classroom instruction, putting the tutored students at the 98th percentile in their class.&lt;/p&gt;&#xA;&lt;p&gt;I also discovered a more recent &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00461520.2011.611369&quot; rel=&quot;nofollow noreferrer&quot;&gt;2011 paper by VanLehn&lt;/a&gt; which found human tutoring to only provide an effect size of 0.79 sigma, but it's locked behind a paywall and I haven't had a chance to go over it yet.&lt;/p&gt;&#xA;&lt;p&gt;So I have a tangible starting point here, which I hope would be useful to others as well, and I'd appreciate any comments regarding additional studies or further detail in these studies that would better answer the question.&lt;/p&gt;&#xA;" OwnerUserId="688" LastEditorUserId="688" LastEditDate="2021-11-17T23:40:46.030" LastActivityDate="2021-11-17T23:40:46.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24593" PostTypeId="1" CreationDate="2021-11-17T23:26:02.760" Score="3" ViewCount="347" Body="&lt;p&gt;Traditionally, I have always taught evaluating expressions before teaching linear equations. But, I was recently given a remedial class of students that have to cover the bare minimums (and we have until mid-December to finish). Luckily, I have great flexibility with what I can do to the syllabus, so for the first time ever, I have completely cut out evaluating expressions since they won´t even be tested on this on the final exam.&lt;/p&gt;&#xA;&lt;p&gt;My question is more if anyone else has done this, or thinks this is not a good way to go. Most of my students in that particular class have ZERO to little formal math background, a lot of them did not even finish high school, and they barely get by with mean, median, mode, rounding, etc. I started equations with them today, and they seemed ¨fine&amp;quot; for the most part. Of course, I also have spent the past week emphasizing positive and negative integer operations, so they are pretty OK with that so far. The textbook itself does  not cover linear equations until after the section on evaluating expressions.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: Upon request for &amp;quot;non-native&amp;quot; English speakers:&lt;/p&gt;&#xA;&lt;p&gt;Evaluating expressions simply means in the US to plug in numbers for the given variable values of the algebraic expression. Thus, for example, an exercise would be:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Evaluate a + b + c&amp;quot; if a = 1, b = 2, c = 3.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The expressions can be as simple as that, or very much more complicated/interesting/beautiful. But, you get the picture.&lt;/p&gt;&#xA;&lt;p&gt;Linear equations simply mean basic equations where you solve for an unknown variable.&lt;/p&gt;&#xA;&lt;p&gt;Example: x + 5 = 10. What is x? Or 2x + 20 = 40, what is x? Etc.&lt;/p&gt;&#xA;" OwnerUserId="15988" LastEditorUserId="2083" LastEditDate="2021-11-18T19:53:42.330" LastActivityDate="2021-11-18T19:53:42.330" Title="Teaching Solving Linear Equations before teaching evaluating expressions" Tags="|undergraduate-education|mathematical-pedagogy|algebra|remedial-courses|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24594" PostTypeId="2" ParentId="24536" CreationDate="2021-11-18T01:15:40.360" Score="3" Body="&lt;p&gt;For most everybody here, the expressions &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a \div b$&lt;/span&gt; look identical, but to elementary students they are not.  I mention this because I think the more salient distinction here is not between &lt;span class=&quot;math-container&quot;&gt;$a \cdot \frac 1b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac ab$&lt;/span&gt;, but rather between &lt;span class=&quot;math-container&quot;&gt;$a \div b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a \cdot \frac 1b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a \div b$&lt;/span&gt; is usually understood as the answer to one of the two following questions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A segment (or other quantity) of length (or other magnitude) &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is divided into &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; pieces of equal size.  What is the size of each piece?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A segment (or other quantity) of length (or other magnitude) &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is divided into pieces, each of size &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.  How many pieces are there?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The first question is called a &lt;em&gt;partitive model&lt;/em&gt; for division, and the second question is called a &lt;em&gt;quotative model&lt;/em&gt; for division.&lt;/p&gt;&#xA;&lt;p&gt;In contrast, &lt;span class=&quot;math-container&quot;&gt;$a \cdot \frac 1b$&lt;/span&gt; is usually represented as the answer to the question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A segment (or other quantity) of length (or other magnitude) &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is divided into &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; equal pieces.  If we take &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; of those pieces together, what is their combined size?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Understanding why &lt;span class=&quot;math-container&quot;&gt;$a \div b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a \cdot \frac 1b$&lt;/span&gt; are equal seems to me a non-trivial question.  It amounts to an understanding that it does not matter whether you first multiply and then divide, or whether you first divide and then multiply.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2021-11-18T01:15:40.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24595" PostTypeId="2" ParentId="17706" CreationDate="2021-11-18T08:00:46.560" Score="1" Body="&lt;p&gt;I fully agree with the self-accepted Answer, and am adding an orthogonal point:&lt;/p&gt;&#xA;&lt;p&gt;in mathematics, &lt;span class=&quot;math-container&quot;&gt;$$p \Leftrightarrow q \Leftrightarrow r$$&lt;/span&gt; is commonly understood as &lt;span class=&quot;math-container&quot;&gt;$$(p \Leftrightarrow q) \;\text{ and }\; (q \Leftrightarrow r),$$&lt;/span&gt; while in formal logic, &lt;span class=&quot;math-container&quot;&gt;$$p \leftrightarrow q \leftrightarrow r\tag{*}$$&lt;/span&gt; is commonly understood as &lt;span class=&quot;math-container&quot;&gt;$$p \leftrightarrow (q \leftrightarrow r),\tag1$$&lt;/span&gt; which is &lt;strong&gt;not&lt;/strong&gt; logically equivalent to &lt;span class=&quot;math-container&quot;&gt;$$(p \leftrightarrow q) \;\text{ and }\; (q \leftrightarrow r)\tag2.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Notice that for the assignment &lt;span class=&quot;math-container&quot;&gt;$(p,q,r)=$&lt;/span&gt;  (false, false, true), &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; is true but &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; is false.)&lt;/p&gt;&#xA;&lt;p&gt;So, in fact, for &lt;span class=&quot;math-container&quot;&gt;$(*),$&lt;/span&gt; the purely symbolic (albeit grammatically unfortunate) verbalisation “&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;”, is actually is more accurate than the grammatically-correct “&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$q,$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;”, as the latter is a misinterpretation!&lt;/p&gt;&#xA;&lt;p&gt;Here, any grammatically-correct verbalisation necessarily involves &lt;em&gt;some&lt;/em&gt; interpretation; in fact, since &lt;span class=&quot;math-container&quot;&gt;$(*)$&lt;/span&gt; is in some quarters understood as &lt;span class=&quot;math-container&quot;&gt;$(p \leftrightarrow q) \leftrightarrow r\tag3,$&lt;/span&gt; disambiguation is even necessary in the proess of avoiding a purely-symbolic reading.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-11-18T14:05:25.260" LastActivityDate="2021-11-18T14:05:25.260" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24596" PostTypeId="2" ParentId="24593" CreationDate="2021-11-18T17:13:51.823" Score="9" Body="&lt;p&gt;I hope this does not come across as overly harsh:  I do not think that thinking of teaching as &amp;quot;covering material&amp;quot; in a particular order is a useful framework.&lt;/p&gt;&#xA;&lt;p&gt;If your students are solving equations like &lt;span class=&quot;math-container&quot;&gt;$3x+4 = 19$&lt;/span&gt;, but are unable to evaluate &lt;span class=&quot;math-container&quot;&gt;$3x+4$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x = 5$&lt;/span&gt; to check to see if they are correct (or, if they get an incorrect answer, to see for themselves that their answer is incorrect), then they are not doing anything of intellectual value.  It is impossible to &amp;quot;solve&amp;quot; the equation without understanding how to evaluate the expression on the lefthand side.&lt;/p&gt;&#xA;&lt;p&gt;One can manipulate the equation according to some rules, obtain &lt;span class=&quot;math-container&quot;&gt;$x=5$&lt;/span&gt;, and circle it, but this would just be mimicry of mathematics, not the genuine article.&lt;/p&gt;&#xA;&lt;p&gt;So my answer is:  you do not need to specifically devote a day to &amp;quot;evaluating expressions&amp;quot;, but you had better be sure that the students do achieve this outcome in tandem with solving equations.  If you do not, then all of the work you have done will be meaningless.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-11-18T17:13:51.823" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24598" PostTypeId="2" ParentId="24593" CreationDate="2021-11-18T18:25:04.967" Score="2" Body="&lt;p&gt;Your context is students who have not learned much from the mathematics education they have been exposed to thus far. I guess, but can not be sure, that they have already been exposed to someone showing them algorithms and asking them to repeat. Because they are performing badly now, likely the teachers have tried to take this into account by teaching these things to them very slowly and thoroughly.&lt;/p&gt;&#xA;&lt;p&gt;How has it been working thus far?&lt;/p&gt;&#xA;&lt;p&gt;Given this, it is a tempting idea that you should do something, anything, else. What exactly this something else is seems to vary a bit, but the literature I have been exposed to recently has been all about open questions and questions with low threshold to doing something but with lots of room to expand.&lt;/p&gt;&#xA;&lt;p&gt;Some that might be relevant for your students are (and this is speculation on my part, so a truckful of salt etc.)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Figurate_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Figurate_number&lt;/a&gt; : have a pattern, ask them to continue it, figure out a rule, evaluate to check. This should practice evaluation of expressions, though mostly with positive integers. But then you could ask if we can make sense of the two-and-halft figure or zeroth figure or minus tenth figure; what might such mean?&lt;/li&gt;&#xA;&lt;li&gt;how complex an equation can you make that has a given answer?&lt;/li&gt;&#xA;&lt;li&gt;how much do different telephone contracts cost in terms of fixed costs and use of data, calls etc.? Which ones are cheapest for particular amounts of use? At which point do the costs of different contracts break even? Maybe use the contracts the students have. Or take something more relevant for their everyday life. Obviously, make sure to explicitly draw the connection to the more abstract mathematics, or the skills won't translate outside the context. Note that this problem trains both of the skills you mention.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This perspective would suggest approaching the issue more in terms of interesting and open questions and less from the perspective of a staircase of simple skills.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-11-18T18:25:04.967" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24599" PostTypeId="1" AcceptedAnswerId="24640" CreationDate="2021-11-18T18:50:29.323" Score="2" ViewCount="806" Body="&lt;p&gt;I have some background from completing Silvanus Thompson's book, but I didn't fully grasp the later chapters in it. I'm going to use khan academy and maybe other resources as a supplement.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;How useful would Stewart's book be for a student with my background?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How about a student with no calculus background preparing for the AP?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How does it compare for students who are taking AP calculus AB exam and students taking AP calculus BC exam?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Edit: My book's version is &lt;em&gt;Stewart Calculus early transcendentals ninth edition metric version&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="17571" LastEditorUserId="17571" LastEditDate="2021-11-25T18:37:15.757" LastActivityDate="2022-11-24T18:39:27.037" Title="Is Stewart Calculus a good book for AP calculus exam prep?" Tags="|secondary-education|calculus|textbooks|student-question|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24600" PostTypeId="1" AcceptedAnswerId="24645" CreationDate="2021-11-18T22:16:27.233" Score="4" ViewCount="356" Body="&lt;p&gt;In my Business Calculus class (U.S. college-level), we discuss three aspects of cost: Total Cost &lt;span class=&quot;math-container&quot;&gt;$C(q)$&lt;/span&gt;, Marginal Cost &lt;span class=&quot;math-container&quot;&gt;$MC(q)$&lt;/span&gt;, and Average Cost &lt;span class=&quot;math-container&quot;&gt;$A(q)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is quantity produced. The defining relationships are:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{dC(q)}{dq} = MC(q)$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$A(q) = \frac{C(q)}{q}$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$C(q) = C(0) + \int_0^q MC(q)\ dq$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$C(0)$&lt;/span&gt; is referred to as the Fixed Costs.&lt;/p&gt;&#xA;&lt;p&gt;A question on a recent quiz was:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given the cost function &lt;span class=&quot;math-container&quot;&gt;$C(q) = q^2+50$&lt;/span&gt;, what is the average cost when&#xA;producing 10 units?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is a pretty straightforward calculation: &lt;span class=&quot;math-container&quot;&gt;$\frac{10^2+50}{10} =15$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For the exam I'd like to ask (likely a bonus question now since I'm obviously struggling enough to post here) essentially the same question but in a position-velocity scenario instead.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The main problem I'm running into is how to ask the question since I can't figure out the position-analogous thing to Average Cost.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;At first, considering units in each scenario, I thought it should be something like &amp;quot;Average Velocity&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given the position function &lt;span class=&quot;math-container&quot;&gt;$s(t) = t^2+50$&lt;/span&gt;, what is the average velocity after 10 seconds?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But that doesn't match up quite right because the solution to this is &lt;span class=&quot;math-container&quot;&gt;$\frac{(10^2+50)-(0^2+50)}{10} = 10$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It seems like the obstruction to forming the concept I want is that, in the Cost scenario, the Fixed Costs get incorporated into the averaging (whereas in the average velocity they get subtracted out). I'm not sure how the Position question should be phrased to treat the initial position the same as Fixed Costs.&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2021-11-18T23:10:07.963" LastActivityDate="2023-04-28T16:20:03.300" Title="Average Cost to Velocity Analogy" Tags="|calculus|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24601" PostTypeId="2" ParentId="24600" CreationDate="2021-11-18T23:50:32.340" Score="-1" Body="&lt;p&gt;I'm a physicist by training. What struck me immediately as awkward or ill-defined about your physics question is that it assumes there's a natural starting point at &lt;span class=&quot;math-container&quot;&gt;$t=0$&lt;/span&gt;. This is a totally wrong/alien way of thinking to a physicist. There is nothing special about &lt;span class=&quot;math-container&quot;&gt;$t=0$&lt;/span&gt;. So to phrase the physics question in a way that is a natural fit to the subject and the way the subject is conceptualized by workers in the field, it should really be something like &amp;quot;Given ..., what is the average velocity over the time interval from 0 to 10 seconds?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This just isn't perfectly analogous to your business example, where producing zero units really is special in some sense. I suppose in a businessy context, C(0) represents the fact that you have fixed costs such as property taxes, which accrue regardless of whether you produce anything. Then I suppose C(q)-C(0) might be referred to as something like an incremental cost, relative to keeping your factory in mothballs.&lt;/p&gt;&#xA;&lt;p&gt;Please also keep in mind that average velocity simply isn't of any interest in physics. The only reason it's discussed in the first chapter of freshman physics texts is to disabuse students of the notion that its definition, which is what they learned in grade school, would be of interest.&lt;/p&gt;&#xA;" OwnerUserId="18102" LastActivityDate="2021-11-18T23:50:32.340" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24602" PostTypeId="2" ParentId="24600" CreationDate="2021-11-19T06:07:20.590" Score="3" Body="&lt;p&gt;These exercises are indeed not analogous to one another:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;“Given the &lt;strong&gt;position&lt;/strong&gt; function &lt;span class=&quot;math-container&quot;&gt;$s(t) = t^2+50$&lt;/span&gt;, what is the &lt;strong&gt;mean position&lt;/strong&gt; over the first 10 seconds?”&lt;/li&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given the &lt;strong&gt;position&lt;/strong&gt; function &lt;span class=&quot;math-container&quot;&gt;$s(t) = t^2+50$&lt;/span&gt;, what is the &lt;strong&gt;average velocity&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$\color{brown}{\text{over the first}}$&lt;/span&gt; 10 seconds?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given the &lt;strong&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{brown}{\textbf{total}}$&lt;/span&gt; cost&lt;/strong&gt; function &lt;span class=&quot;math-container&quot;&gt;$\color{brown}{T}C(q) = q^2+50$&lt;/span&gt;, what is the &lt;strong&gt;average &lt;span class=&quot;math-container&quot;&gt;$\color{brown}{\textbf{unit}}$&lt;/span&gt; cost&lt;/strong&gt; when producing 10 units?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Exercise 1 wants the &lt;strong&gt;mean value thus far of a function&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$$\text{mean position}=\frac{\int_0^{t}s(t)\,\mathrm dt}t.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Exercise 2 wants the &lt;strong&gt;mean value thus far of the derivative of a smooth function&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$$\text{mean velocity}=\frac{s(t)-s(0)}t.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Exercise 3 wants the &lt;strong&gt;mean incremental value thus far of a &lt;em&gt;running total&lt;/em&gt;&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$$\text{mean unit cost}=\frac{TC(q)}q\\ \color\red{\ne}\frac{TC(q)-TC(0)}q\;\text{ or }\;\frac{\displaystyle\sum^q_{i=0} TC(i)}q\;\text{ or }\;\frac{\displaystyle\sum^q_{i=1} TC(i)}q.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This exercise fits your requirement, since it is analogous to Exercise 3 (and, noting that &lt;span class=&quot;math-container&quot;&gt;$d(0)\equiv0,$&lt;/span&gt; a special case of Exercise 2's concept):&lt;/p&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Given the &lt;strong&gt;distance-travelled&lt;/strong&gt; function &lt;span class=&quot;math-container&quot;&gt;$d(t) = t^2+50t$&lt;/span&gt;, what is the &lt;strong&gt;average speed thus far&lt;/strong&gt;? &lt;span class=&quot;math-container&quot;&gt;$$\left(\frac{d(t)-d(0)}t=\right)\text{average speed}\boldsymbol=\frac{d(t)}t.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Both exercises 3 and 4 start from a &lt;em&gt;cumulative&lt;/em&gt; function instead of a &lt;em&gt;state&lt;/em&gt; function, and want to derive the average &lt;em&gt;slope&lt;/em&gt; since the beginning. The former is why their numerators involve no subtraction.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Reply to comments&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Rereading this, it's worth noting that &lt;span class=&quot;math-container&quot;&gt;$\color{brown}{\text T}$&lt;/span&gt;C(10) = &lt;span class=&quot;math-container&quot;&gt;$\color{brown}{\text T}$&lt;/span&gt;C(0) + MC(1)+MC(2)+...+MC(10). So MC plays the role of instantaneous velocity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes, and the analogous summation is &lt;span class=&quot;math-container&quot;&gt;$$\text{distance travelled } d(t)=\int_0^t\left|\frac{\mathrm ds}{\mathrm dt}\right|\,\mathrm dt,$$&lt;/span&gt; noting that &lt;strong&gt;distance travelled&lt;/strong&gt; passes through the origin and &lt;span class=&quot;math-container&quot;&gt;$\left|\frac{\mathrm ds}{\mathrm dt}\right|$&lt;/span&gt; means instantaneous speed.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So it's still unclear to me what the motion measurement analogous to Average &lt;span class=&quot;math-container&quot;&gt;$\color{brown}{\text{Unit}}$&lt;/span&gt; Cost would be.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\text{average unit cost}=\frac{\text{total cost}}{\text{quantity}}=\frac{TC(q)}q$$&lt;/span&gt; is analogous to &lt;span class=&quot;math-container&quot;&gt;$$\text{average speed}=\frac{\text{distance travelled}}{\text{time}}=\frac{d(t)}t.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;N.B. &lt;strong&gt;average unit cost&lt;/strong&gt; = &lt;strong&gt;average marginal cost&lt;/strong&gt; iff fixed cost = $0.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-04-28T16:20:03.300" LastActivityDate="2023-04-28T16:20:03.300" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24603" PostTypeId="1" AcceptedAnswerId="24633" CreationDate="2021-11-19T17:19:04.217" Score="3" ViewCount="150" Body="&lt;p&gt;To make a polygon on a traditional geoboard, one usually stretches a rubber band around the vertices. No problem there.&lt;/p&gt;&#xA;&lt;p&gt;When making a simple line segment, however, a rubber band is typically stretched between the two end points. This creates a “segment” that looks like two closely spaced parallel lines.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Is there a way to make an actual line segment on a geoboard?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I looked for loop-less rubber bands with little holes on each end to fit over the pegs, but I had no luck.&lt;/p&gt;&#xA;&lt;p&gt;Computer-based/virtual geoboards tend not have this problem, and seeing these made me wonder how it could be done on physical boards.&lt;/p&gt;&#xA;" OwnerUserId="13229" LastActivityDate="2021-11-26T06:12:18.817" Title="Line segments on a geoboard" Tags="|geometry|manipulatives|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24604" PostTypeId="1" CreationDate="2021-11-19T17:20:06.797" Score="16" ViewCount="569" Body="&lt;p&gt;For people teaching high school standardized curriculums such as AP, IB, or A-Levels, how do you find the balance between preparing students for the standardized test compared to ensuring they understand the material?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In an ideal world, the two are the same. And whether standardized testing is good or not is another question. I would like to highlight that in these three programs, the goal of the course is typically to prepare the students for the respective examinations. This is contrasted with ACT/SAT preparation taking only a small part of class time.&lt;/p&gt;&#xA;&lt;p&gt;I want to know more about i) structuring the class to target a relatively wide range in students background and goals (including differences in motivation, ability, plans, and outcomes); ii) selecting questions that are good for learning vs that are common on the exam; iii) the role that past exam questions play in instruction and assessment.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: to allow responses from people who do not teach/research standardized curriculums, the alternate question is &amp;quot;In an environment where the teacher can predict but not change the scope of a math course, how to teach high-level problem solving skills? How big of a role should problem solving exercises play, comparison to theory or other exercises?&amp;quot; The context is important since I am not interested in the teacher changing the course coverage to allow more time spent on harder problems.&lt;/p&gt;&#xA;" OwnerUserId="18050" LastEditorUserId="18050" LastEditDate="2021-12-07T04:27:35.677" LastActivityDate="2021-12-07T04:27:35.677" Title="How to balance between teaching to the standardized test vs understanding?" Tags="|secondary-education|mathematical-pedagogy|assessment|classroom-interaction|standards-based-grading|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24605" PostTypeId="2" ParentId="24603" CreationDate="2021-11-19T17:40:20.623" Score="2" Body="&lt;p&gt;You could cut open the rubber bands and tie loops/knots at the ends. Downside of course is that you'd have to manually tie the knots for however many you need.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: It occurs to me that it's possible to tie knots without cutting them in half, then just make sure the rubber band go on one side of the pegs. Probably eaiser this way, too.&lt;/p&gt;&#xA;" OwnerUserId="18050" LastEditorUserId="18050" LastEditDate="2021-11-23T05:46:57.997" LastActivityDate="2021-11-23T05:46:57.997" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24606" PostTypeId="2" ParentId="24556" CreationDate="2021-11-19T18:18:24.843" Score="0" Body="&lt;p&gt;Students at the high school level expect to see examples somewhat similar to test questions. This is also true to some extent at the first/second year undergrad level as well. Therefore, you should clearly give examples that highlight domain restrictions or extraneous roots. These should be accompanied with a graph of the function.&lt;/p&gt;&#xA;&lt;p&gt;It's also a good idea to always do a &amp;quot;reality-check&amp;quot; at the end of questions you do as teaching examples. For example if the question asks for number of people, ask the class if &amp;quot;23.6 persons&amp;quot; or $17.82343 is the final answer. When solving an equation quickly plug the answer back into the equation to show that it works.&lt;/p&gt;&#xA;&lt;p&gt;Then when you get to one with an extraneous solution, plug answer back as usual but pause right before attempting to divide by 0 or evaluate square root of negative numbers and ask class &amp;quot;I think something's wrong. Can someone help me out?&amp;quot;. A &lt;em&gt;little&lt;/em&gt; bit of drama and overreaction may help keep students engaged thus more likely for them to remember to spot &amp;quot;nonsense&amp;quot;. As usual homework questions should be curated to reinforce this habit of verifying the solutions.&lt;/p&gt;&#xA;" OwnerUserId="18050" LastActivityDate="2021-11-19T18:18:24.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24607" PostTypeId="2" ParentId="24604" CreationDate="2021-11-19T18:45:14.120" Score="1" Body="&lt;p&gt;You have zero responsibility to teach to the test. A kid who actually understands calculus should get a 5 on the AP exam. Problem solved.&lt;/p&gt;&#xA;&lt;p&gt;Kids who want test prep can simply download sample exams and familiarize themselves with the types of questions on the exams.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I want to know more about i) structuring the class to target a relatively wide range in students background and goals;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you have a student with too poor a background to do well in AP calculus, then realistically that student doesn't belong in AP calculus. If that student takes AP calculus anyway, achieves a poor understanding of calculus, and therefore doesn't pass the AP exam, then that's not some horrible tragedy. They can take calculus again in college, and they'll probably pass because of the previous exposure. That's a good outcome.&lt;/p&gt;&#xA;" OwnerUserId="18105" LastActivityDate="2021-11-19T18:45:14.120" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24608" PostTypeId="1" CreationDate="2021-11-20T13:52:18.840" Score="2" ViewCount="71" Body="&lt;p&gt;I am looking for articles in French preferably (or in English otherwise, but my english is bad..) on the following topics:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;study of assessments made in class after a problem;&lt;/li&gt;&#xA;&lt;li&gt;phase of conclusion of a problem;&lt;/li&gt;&#xA;&lt;li&gt;institutionalization of problems.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am already familiar with the IREM site (French site), but I did not find articles there specific to the phases of conclusion of the problems. This is part of a thesis, and I am currently a young teacher in high school.&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance for your suggestions, good day.&lt;/p&gt;&#xA;" OwnerUserId="18107" LastActivityDate="2021-11-20T13:52:18.840" Title="Search for educational articles" Tags="|secondary-education|education-research|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24609" PostTypeId="2" ParentId="397" CreationDate="2021-11-21T06:44:49.730" Score="1" Body="&lt;p&gt;&lt;em&gt;Inspired by Paul Lockhart's &amp;quot;A Mathematician's Lament&amp;quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I would argue that what students actually need is play. Play is essential for learning especially at the earliest stages of education. Given that the central element of mathematics is proof, this means that students should form their own conjectures from an early age. They should also attempt to justify them using a verbal explanation (ideally without much mathematical jargon which the child does not understand yet).&lt;/p&gt;&#xA;&lt;p&gt;Contrast this with how we currently teach students: start by assuming these facts as a given, and then force students to memorise them through rote-and-drill repetition. This is the complete opposite of what we should be doing. Rather, students should firstly ask the right questions, for example by drawing a semicircle, and then wondering what happens to the angle inscribed in it as the point moves. Then they will ponder, struggle, try and then try again, before they might make another copy and discover a rectangle hidden within the circle:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/prhlj.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/prhlj.gif&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Aha! Since we made a copy of the triangle, each pair of opposite lengths must be the same, so we have a rectangle! Therefore the angle inscribed in a semicircle is 90º!&lt;/p&gt;&#xA;&lt;p&gt;This is how we make students engage with mathematics, discovering the intrinsic value of truth and beauty hidden within it.&lt;/p&gt;&#xA;&lt;p&gt;Students need real experience with mathematics through inquiry and through play before they can progress with applying the same methods to harder problems, or manipulating different methods of proof such as proof by induction. In other words, students need to have lots and lots of fun with mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Even in high school curricula where mathematical investigation and reasoning are given significant attention such as the IB Diploma, used in many international schools around the world, investigation is treated as a subject in isolation, divorced from the rest of the course. Students are guided through a bewildering assortment of topics throughout the course where mathematical concepts are seen as disparate and disconnected. Investigation is limited to a month or two where students start from scratch, particularly those that have come from more traditional curricula. After 12 years of rote memorisation, of seeing mathematics as a game where symbols are manipulated as if by magic, no wonder students come to struggle with mathematics later. If all they have seen are grammar and conjugation drills in their English class, how will they be equipped to write and read stories, plays, novels and poems, just as they will engage with the art of proof?&lt;/p&gt;&#xA;&lt;p&gt;In mathematics, communication is key and notation is secondary. All of those pesky symbols: &lt;span class=&quot;math-container&quot;&gt;$\alpha, \beta, \gamma, \Sigma, \Pi$&lt;/span&gt; can come later. And so students need to get in the right frame of mind as viewing mathematics as a narrative, with an inherent narrative structure of exposition, discovery and conclusion, not as a meaningless bash of symbols. We do the exact same thing with English, or students' native languages.&lt;/p&gt;&#xA;&lt;p&gt;So for students who have been completely deprived of this throughout their &amp;quot;education&amp;quot;, we need to start small. Students will naturally be horrified at the sight of a research paper as it will be shockingly new. Start by using very simple yet perplexing examples: why does a pyramid take up a third the space of the cuboid bounding it? They will very often not know why. Spend as much time with them as possible, no matter how long it takes for them to understand. Tell them that mathematics is like any work of art or music in its beauty and its lack of relevance in the real world. First give them as much space and time to make mistakes, to ask &amp;quot;what if?&amp;quot;. Gently prompt them in the right direction when they get lost, which will involve the destruction of many assumed truths that they may hold sacred. You will need to hold their hand (yes, even in undergrad) and then let go when they are ready.&lt;/p&gt;&#xA;&lt;p&gt;This will mean forgetting what you already know about teaching. Get them to chat to each other, present their work on the whiteboards, get up from your lectern and wander around the room (if you haven't already). And you will give them the power of independent thought and autonomy, divorced from any pressure about passing grades or graduation requirements. Because that is what it means to be a great teacher and a friend, a great mentor and a learner, and what it means to play. Make your classroom as unbounded as the imagination and curiosity of students. And in time they will learn to write.&lt;/p&gt;&#xA;" OwnerUserId="13864" LastActivityDate="2021-11-21T06:44:49.730" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24610" PostTypeId="1" CreationDate="2021-11-21T14:07:58.053" Score="19" ViewCount="7195" Body="&lt;p&gt;I have some very basic questions about how to teach the logarithm to high school students:&lt;/p&gt;&#xA;&lt;p&gt;First of all, is it better to introduce it as a function with a graph or is it better to treat it like a black box with certain properties at first with which to solve exponential equations?&lt;/p&gt;&#xA;&lt;p&gt;And once the best starting point has been used, how can I lead my students to a deeper understanding of logarithms? Are there any particularly conducive methods? Like e.g. have them solve twenty problems for each rule of logarithm?&lt;/p&gt;&#xA;&lt;p&gt;One of my problems with logarithms is that after I tell them that the logarithm is the inverse function of the exponential then there is nothing more to think about: If they are supposed to solve an equation then they can just apply the logarithm to both sides of the equation. That's boring.&lt;/p&gt;&#xA;&lt;p&gt;Ideally, I'd like to pose interesting problems only.&lt;/p&gt;&#xA;" OwnerUserId="377" LastEditorUserId="851" LastEditDate="2021-11-22T20:44:04.520" LastActivityDate="2021-11-26T04:50:44.900" Title="How should I teach logarithms to high school students?" Tags="|secondary-education|logarithm|" AnswerCount="10" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="24611" PostTypeId="2" ParentId="24610" CreationDate="2021-11-21T15:24:05.247" Score="28" Body="&lt;p&gt;Coming from the perspective of someone who reteaches this material at the college level, neither the graph perspective nor the list of properties perspective really translate into a deep understanding later. For the former, all you get is an understanding that the graph goes up, but not a lot, and is too vague to really be useful. For the latter, they don't remember the rules and the &amp;quot;big picture&amp;quot; that the logarithm transforms multiplication into addition (i.e. is a group homomorphism from &lt;span class=&quot;math-container&quot;&gt;$(\mathbb{R}^+,\times)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(\mathbb{R},+)$&lt;/span&gt;) gets spotty understanding.&lt;/p&gt;&#xA;&lt;p&gt;So, I'd like to introduce another interpretation for the logarithm, analogous to an interpretation for division. That is, you can interpret the integer part of &lt;span class=&quot;math-container&quot;&gt;$a / b$&lt;/span&gt; as the number of times that you can subtract &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and stay above &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. Similarly, the integer part of &lt;span class=&quot;math-container&quot;&gt;$\log_b(a)$&lt;/span&gt; is the number of times that you can divide &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and stay above &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. There are several things you can get from that which might aid understanding:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$1+\lfloor \log_{10}(n) \rfloor $&lt;/span&gt; is the number of decimal digits of an integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;20&amp;quot; questions: If you have &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; things and want to pick out individuals by asking yes/no questions, &lt;span class=&quot;math-container&quot;&gt;$\lceil \log_{2}(n) \rceil$&lt;/span&gt; is the minimum number you have to ask to distinguish every one of them. You can tie this into questions about how much personal information is required to distinguish every person in a country.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2021-11-22T14:38:18.013" LastActivityDate="2021-11-22T14:38:18.013" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="24612" PostTypeId="2" ParentId="24610" CreationDate="2021-11-21T16:21:32.383" Score="0" Body="&lt;p&gt;This is a pretty broad question, but here are some suggestions.&lt;/p&gt;&#xA;&lt;p&gt;Start with some remediation of the notion of a proportionality. In the K-12 curriculum, most students only see direct proportionalities, not proportionalities of the form &lt;span class=&quot;math-container&quot;&gt;$y \propto x^n$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$n\ne 1$&lt;/span&gt;. Do an example like a comparison of the number of jelly beans that can fit in one jar compared to another, when the second jar has the same shape but double the linear dimensions. If they don't understand this sort of thing, then it's basically a crippling issue when they get to logarithms. Convince them to reason about things like &lt;span class=&quot;math-container&quot;&gt;$y_1/y_2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_1/x_2$&lt;/span&gt;, using an example where using these ratios makes it easy to solve the problem in your head, while not using ratios makes it a mess. E.g., for the jelly bean problem, show how awful it gets if they insist on explicitly writing down the formula for the volume of a cylinder and then just cranking out the algebra. Point out that ratios are unitless, which is a big win. Point out that the factors like pi are destined to cancel out. This sort of thing is necessary, or else they will never talk or think about ratios.&lt;/p&gt;&#xA;&lt;p&gt;Give them tasks in which they translate the identities into the simplest and most transparent possible example using arithmetic. For example, &lt;span class=&quot;math-container&quot;&gt;$\log(a^b)=b\log a$&lt;/span&gt; can be exemplified using &lt;span class=&quot;math-container&quot;&gt;$10^3$&lt;/span&gt; and log base 10.&lt;/p&gt;&#xA;&lt;p&gt;Connect to STEM examples such as pH, radioactive half-life, or the number of bits needed to represent an address on a computer with &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; bytes of memory. Another computer-sciency example is the number of steps needed in order to look up a word in the dictionary, if you follow an algorithm where you repeatedly bisect.&lt;/p&gt;&#xA;&lt;p&gt;Do the technique where you extract an unknown exponent from a log-log plot of data points. This is most transparent if you start with artificial data and use log base 10, e.g., the data points are (1,5), (10,500), (100,50000). After showing a simple example like this, show an interesting real one. For example, people have measured the amount of effort required for a team of computer programmers to write n lines of code.&lt;/p&gt;&#xA;&lt;p&gt;Print out a simplified slide rule and hand out scissors, have them use it to do multiplication.&lt;/p&gt;&#xA;&lt;p&gt;Pose this question to them. Hey, suppose your friend says, &amp;quot;I learned how to manipulate logs in high school, but I never understood what it was about.&amp;quot; Explain to your friend in 10 words or less. My own answer would be something like &amp;quot;Logs turn multiplication into addition.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="18111" LastActivityDate="2021-11-21T16:21:32.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24613" PostTypeId="2" ParentId="24610" CreationDate="2021-11-21T20:21:38.847" Score="16" Body="&lt;p&gt;How do you teach students about the operator &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{}$&lt;/span&gt;? It's a similar operator in many ways; when dealing with cube roots I try to show them these things:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{x}$&lt;/span&gt; asks &amp;quot;&lt;strong&gt;which base&lt;/strong&gt; to the third power gives us &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;?&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x) = \sqrt[3]{x}$&lt;/span&gt; is a one-to-one function, and &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; has a graph of a certain shape.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{}$&lt;/span&gt; is the inverse operation to the cube, so it's useful for undoing cubes.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{}$&lt;/span&gt; does not distribute over addition, but it does over multiplication.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So, for logarithms I do the corresponding things:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log_3{x}$&lt;/span&gt; asks &amp;quot;&lt;strong&gt;which exponent&lt;/strong&gt; with base 3 gives us &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;?&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x) = \log_3(x)$&lt;/span&gt; is a one-to-one function, and &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; has a graph of a certain shape.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log_3$&lt;/span&gt; is the inverse operation to exponentiation with base &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, so it's useful for undoing exponentiation with base &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log_3$&lt;/span&gt; does not distribute over addition, and whoa, look at what it does with multiplication.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;A good exercise following this presentation would be:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Consider these two equations: &lt;span class=&quot;math-container&quot;&gt;$x^3 = 729$&lt;/span&gt;; &lt;span class=&quot;math-container&quot;&gt;$3^x = 729$&lt;/span&gt;. How do they look similar? How are they different? Solve each of them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-11-21T20:21:38.847" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24614" PostTypeId="2" ParentId="24610" CreationDate="2021-11-22T00:24:19.527" Score="7" Body="&lt;p&gt;I would start with explaining exponentials as repeated multiplication. For example, we look at the sequence &lt;span class=&quot;math-container&quot;&gt;$2^1=2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2^2=2\times 2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2^3=2\times 2\times 2$&lt;/span&gt;... and call it one 'two', two 'twos', three 'twos'. Then show the rule for multiplication goes like &lt;span class=&quot;math-container&quot;&gt;$(2\times 2)\times (2\times 2\times 2)$&lt;/span&gt; and say that we find the total number of twos by adding up the number of twos in each bracket. &lt;span class=&quot;math-container&quot;&gt;$2^2\times 2^3=2^5$&lt;/span&gt;. So we can do &lt;em&gt;multiplication&lt;/em&gt; of blocks of twos by &lt;em&gt;adding&lt;/em&gt; the number of twos in each block. We have turned multiplying into adding. And then the 'logarithm to base 2' is &lt;em&gt;the number of twos in a block&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Repeat for several different numbers, like powers of 3, 4, 10 to establish the concept for whole numbers. 10 should already be familiar to them, so it links it back to existing knowledge.&lt;/p&gt;&#xA;&lt;p&gt;Then we 'fill in the gaps' between the whole numbers. We show that square roots act like they are half way between the whole numbers. &lt;span class=&quot;math-container&quot;&gt;$2^{1/2}=\sqrt{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2^1=\sqrt{2}\times \sqrt{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2^{3/2}=\sqrt{2}\times \sqrt{2}\times \sqrt{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2^2=\sqrt{2}\times \sqrt{2}\times \sqrt{2}\times \sqrt{2}$&lt;/span&gt;. The logarithms go up in steps of a half. And if we plot the points on a graph, we can gradually fill in the gaps to get a smooth curve.&lt;/p&gt;&#xA;&lt;p&gt;Then for any number, we can use the smooth curve to describe it as being very close to some number of twos (possibly fractional) multiplied together, and use the previous trick to turn any multiplication into addition. Slide rules are a very good example to demonstrate how the knowledge can be applied. Students can make a simple one themselves, and you can show them a professional one and maybe talk a little about the history of calculation before calculators. And for something more up to date, compound interest on loans is very important to anyone who wants to buy something expensive like a house. '3% per annum compound for 20 years' means a block of twenty 1.03's multiplied together. If you want to borrow \$1000 now and pay back no more than \$1100 in six months time, what's the maximum annual interest rate you can afford? Gambling is another fun example - how many times do you have to toss a coin before the odds of winning every time drop below one in a million?&lt;/p&gt;&#xA;&lt;p&gt;As a rule, any new concept should be first introduced with concrete examples, then with pictures or visual metaphors, it should ideally be connected to a real world problem they can relate to, and finally turned into an abstract definition.  Revisit the whole concrete-pictorial-abstract build-up several times over a period of a few days, with gaps in between. People remember things they find they repeatedly need to know - the act of recalling it affixes it more firmly as a long-term memory.&lt;/p&gt;&#xA;" OwnerUserId="18112" LastActivityDate="2021-11-22T00:24:19.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24615" PostTypeId="2" ParentId="24610" CreationDate="2021-11-22T05:30:47.940" Score="1" Body="&lt;p&gt;Log-linear and log-log graph paper can help visualization. (If paper is passé, you can find templates on the web.) Plot the same function on the different scales.&lt;/p&gt;&#xA;&lt;p&gt;Examples help. Familiar examples.&lt;/p&gt;&#xA;&lt;p&gt;E.g., the signal strength bars on mobile phones are decibels and are logarithmic.&lt;/p&gt;&#xA;&lt;p&gt;E.g., the &lt;a href=&quot;https://en.wikipedia.org/wiki/Richter_magnitude_scale&quot; rel=&quot;nofollow noreferrer&quot;&gt;Richter scale&lt;/a&gt;. It may not be so familiar unless you live in California, but cool, and in newspapers.&lt;/p&gt;&#xA;" OwnerUserId="17835" LastEditorUserId="851" LastEditDate="2021-11-26T04:50:44.900" LastActivityDate="2021-11-26T04:50:44.900" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24616" PostTypeId="2" ParentId="24549" CreationDate="2021-11-22T08:33:31.257" Score="1" Body="&lt;p&gt;Their behaviour is classic &lt;a href=&quot;https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dunning-Kruger&lt;/a&gt;. The way to change their behaviour is to give them more knowledge about the subject (in this case e.g. mathematical induction).&lt;/p&gt;&#xA;&lt;p&gt;I know this seems really difficult or perhaps even impossible: At the moment they seem to be working against you and seem to be actively refusing to acquire more skill.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps you could give them extra easy questions to work on to start with and gradually increase the level of complexity from there while making sure that you don't lose them on the way. That's how I would go about this. As long as they are having small experiences of success they have no reason to act up because it keeps them happy and motivated. You need to eliminate all sources of frustration for them.&lt;/p&gt;&#xA;&lt;p&gt;It might also help to tell them, although in less direct words, that, just because they don't immediately understand something it does not mean that they are stupid. Maybe use 5 minutes to introduce them to the idea of mindsets and Carol Dweck.&lt;/p&gt;&#xA;" OwnerUserId="377" LastActivityDate="2021-11-22T08:33:31.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24617" PostTypeId="2" ParentId="24610" CreationDate="2021-11-22T09:27:55.173" Score="4" Body="&lt;p&gt;Although I am not a teacher, I would recommend going with the historical perspective. Logarithms were invented by John Napier (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Logarithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Logarithm&lt;/a&gt;) basically to make multiplication easier. Instead of having to multiply several large numbers, you looked up their logarithms in the table, then only had to add them and consult the table again in reverse direction with the sum to find the product.&lt;/p&gt;&#xA;&lt;p&gt;Bring a book with the logarithmic tables and demonstrate the process.&lt;/p&gt;&#xA;&lt;p&gt;This was how I learned about logarithms from my father and also how I taught them to my children. The process seemed utterly magical to me and really stuck in my memory.&lt;/p&gt;&#xA;&lt;p&gt;Students today will also be amazed at the concept of doing manual calculations and how much work they were but also how important.&lt;/p&gt;&#xA;&lt;p&gt;Once the properties of logarithms as aid to multiplication have been established, you can start deriving what properties they need to have, e.g. that base 10 logarithms increase by 1 for each factor of 10, etc..&lt;/p&gt;&#xA;&lt;p&gt;That builds the bridge to the relationship of logarithms and exponentiation, and from there you can introduce logarithms of different bases and finally the natural logarithms.&lt;/p&gt;&#xA;" OwnerUserId="18116" LastActivityDate="2021-11-22T09:27:55.173" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24618" PostTypeId="2" ParentId="24610" CreationDate="2021-11-22T11:10:12.507" Score="5" Body="&lt;p&gt;Instant intuition giver: &lt;strong&gt;Logarithm computes the number of digits.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;One could start by letting the students think how such a function should behave.&lt;/p&gt;&#xA;&lt;p&gt;One possible idea: That's just for whole numbers. Is there a way to interpolate? (Turns out yes, there is -&amp;gt; give the formal definition for decadic logarithm.)&lt;/p&gt;&#xA;&lt;p&gt;Another one: Different number systems. Binary numbers are longer than their standard power-of-ten equivalent, shouldn't the logarithm be different. (Answer: depends on the number system but it's only a constant factor, generalize to base-N logarithm.)&lt;/p&gt;&#xA;&lt;p&gt;Once these two questions have been answered, the full definition is there and all the cool properties like &amp;quot;is the inverse of exponentiation&amp;quot; and that it lowers multiplication to addition can be introduced.&lt;br /&gt;&#xA;The students can then use the number-of-digits intuition to validate that all these logarithm laws do, in fact, make sense to them.&lt;/p&gt;&#xA;&lt;p&gt;Caveat: This would have been the best approach for my own personal intuition, which may or may not align with that of others.&lt;br /&gt;&#xA;However, I do know that starting with the raw definition &amp;quot;it's the inverse of exponentiation&amp;quot; is very hard to grasp in high school, I was not the only one who was struggling with it. I suspect that's because in high school, the only examples of invertible functions are addition and multiplication, where the inverse function can be conceptually split into two operations, &amp;quot;take the inverse of the second operand&amp;quot; and &amp;quot;apply the original operation using that inverse&amp;quot;; this tactics will fail for logarithms, so people will be left with no way to build an intuition.&lt;br /&gt;&#xA;That number-of-digits-plus-sensible-interpolation intuition has been carrying me through decades, so I am mildly confident maybe it is useful to other students as well.&lt;/p&gt;&#xA;" OwnerUserId="15740" LastActivityDate="2021-11-22T11:10:12.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24619" PostTypeId="2" ParentId="24610" CreationDate="2021-11-22T15:07:35.967" Score="4" Body="&lt;p&gt;I think it would be more useful to teach logarithms as the inverse of exponents&lt;sup&gt;1&lt;/sup&gt;, and compare most of the properties of logarithms with the properties of exponents, because most of the properties of logarithms are related to those of exponents, or those of other logarithmic properties.&lt;/p&gt;&#xA;&lt;p&gt;Here’s a list of common (as in often talked about, and not that complicated) logarithmic properties and how they relate to properties of exponents.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Logarithms are the inverse of exponents. &lt;span class=&quot;math-container&quot;&gt;$log_b(b^x)=x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ b^{log_b(x)}=x$&lt;/span&gt; is probably the simplest way to describe this in relation to exponents, and if the students have already been thought inverse functions, they shouldn’t have much trouble with this because they’ll already understand that &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(f(x))=x$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;The fact that &lt;span class=&quot;math-container&quot;&gt;$log(a)+log(b)=log(ab)$&lt;/span&gt; can be described by comparing it to a similar law for exponents. In exponents, &lt;span class=&quot;math-container&quot;&gt;$10^a*10^b=10^{a+b}$&lt;/span&gt; is the related law.&lt;/li&gt;&#xA;&lt;li&gt;The logarithm rule saying &lt;span class=&quot;math-container&quot;&gt;$b*log(a)=log(a^b)$&lt;/span&gt; can be compared to the exponent rule &lt;span class=&quot;math-container&quot;&gt;$(a^b)^c=a^{bc}$&lt;/span&gt;, or alternatively it can be considered as an extension of &lt;span class=&quot;math-container&quot;&gt;$log(a)+log(b)=log(ab)$&lt;/span&gt; because multiplication is repeated addition (excluding unusual cases like imaginary numbers, which normally are not thought with basic level logarithms).&lt;/li&gt;&#xA;&lt;li&gt;The logarithm rule saying &lt;span class=&quot;math-container&quot;&gt;$log(a)-log(b)=log(a/b)$&lt;/span&gt; is related to the exponent rule &lt;span class=&quot;math-container&quot;&gt;$\frac{10^a}{10^b}=10^{a-b}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;The rules &lt;span class=&quot;math-container&quot;&gt;$log_x(x)=1$&lt;/span&gt; is an extension of the rule that &lt;span class=&quot;math-container&quot;&gt;$log_x(x^b)=b$&lt;/span&gt; and b is just equal to one.&lt;/li&gt;&#xA;&lt;li&gt;The rule that &lt;span class=&quot;math-container&quot;&gt;$log_m(1)=0$&lt;/span&gt; is related to the exponent rule that &lt;span class=&quot;math-container&quot;&gt;$m^0=1$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;The logarithm rule that &lt;span class=&quot;math-container&quot;&gt;$\frac{log_a(b)}{log_a(c)}=log_c(b)$&lt;/span&gt; cannot be compared to any exponential rule that I know of, so you can either make your students simply memorize it, or if you don’t plan on your students getting tested on this rule (since from what I can tell, it isn’t as popular as the other rules), just not teach it to them.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Important note:&lt;/strong&gt; this idea falls apart if you haven’t thought your students all of the exponent rules and what inverse functions are.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt;I think this because &lt;a href=&quot;https://www.nbcnews.com/better/health/how-get-better-remembering-things-according-neuroscience-ncna882426&quot; rel=&quot;nofollow noreferrer&quot;&gt;this NBC article&lt;/a&gt; says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;We remember things because&lt;/strong&gt; they either stand out, &lt;strong&gt;they relate to and can easily be integrated in our existing knowledge base&lt;/strong&gt;, or it’s something we retrieve, recount or use repeatedly over time&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Emphasis mine)&lt;/p&gt;&#xA;&lt;p&gt;So relating logarithms to exponents will make them easier to remember.&lt;/p&gt;&#xA;" OwnerUserId="17999" LastEditorUserId="17999" LastEditDate="2021-11-22T18:01:20.907" LastActivityDate="2021-11-22T18:01:20.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24620" PostTypeId="2" ParentId="24610" CreationDate="2021-11-22T19:10:59.220" Score="5" Body="&lt;p&gt;All of the math explanations here are very good. And they all use math to explain math. Sometimes it is helpful to understand the rationale; concept or the &amp;quot;Why was this invented / Why does someone need this;&amp;quot; or even the 'real world' explanation behand the math.&lt;/p&gt;&#xA;&lt;p&gt;I had a story I used to tell. It went something like this:&lt;br /&gt;&#xA;Suppose you need to count to a very large number. Let's say [sic] &amp;quot;A Jillion.&amp;quot;&lt;br /&gt;&#xA;You start out counting 1, 2, 3, 4, etc. up to 10.&lt;br /&gt;&#xA;Now, the ones digit repeats: 11, 12, 13, and so on. But this is now the 'uninteresting' part. We know this already. So, this time, you count by Tens: 10, 20, 30, 40, etc. We are not truly ignoring the numbers in between, we know they are there, but they don't necessarily add anything to the count.&lt;br /&gt;&#xA;When we got to 100, we again change our scale and count by hundreds: 100, 200, 300, 400, etc.&lt;br /&gt;&#xA;At 1,000, we re-scale and count by thousands: 1,000; 2,000; 3,000; 4,000 etc.&lt;br /&gt;&#xA;We do this until we reach &amp;quot;A Jillion.&amp;quot;&lt;br /&gt;&#xA;&lt;br&gt;&#xA;This is the concept of Base 10 Logarithms.&lt;br /&gt;&#xA;&lt;br&gt;&#xA;Why do this?&lt;br /&gt;&#xA;If I am poor, and someone asks me how much money I have, I will tell them in dollars and cents: \$4.60, because that is the scale I need to describe the number.&lt;br /&gt;&#xA;If, on the other hand, I am a multi-billionaire, and someone asks me how much money I have, dollars and cents don't really matter: $4.6 Billion, because Billions is the scale I need to neatly describe the number.&lt;br /&gt;&#xA;&lt;br&gt;&#xA;Why do I need to do that? It is convenient and efficient, but it is also useful in other areas.&#xA;Bill Nye has a great explanation using the Richter scale (which is a logarithmic scale.)&lt;br /&gt;&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=1qbg7orb1lc&quot; rel=&quot;noreferrer&quot;&gt;Bill Nye the Science Guy - Earthquakes (Richter scale)&lt;/a&gt;&lt;br /&gt;&#xA;&lt;br&gt;&#xA;So, basically, logarithms allow me to adjust the scale for very large or very small numbers.&lt;/p&gt;&#xA;&lt;p&gt;Now, introducing &amp;quot;The Math&amp;quot; to work out these scenarios will, hopefully, make more sense.&lt;/p&gt;&#xA;&lt;p&gt;$0.02&lt;/p&gt;&#xA;" OwnerUserId="12766" LastActivityDate="2021-11-22T19:10:59.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24622" PostTypeId="1" CreationDate="2021-11-22T23:55:17.920" Score="2" ViewCount="74" Body="&lt;p&gt;I am planning to teach a course on Statistics using &lt;a href=&quot;https://store.doverpublications.com/0486492710.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beginning Statistics with Data Analysis by Mosteller, Fienberg and Rourke&lt;/a&gt;. On page xii of the book, the authors said &amp;quot;A solutions manual is available to the instructor ... &amp;quot;. May I know where I can purchase/request/download this solution manual? My search for this solution manual was unsuccessful.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2021-11-22T23:55:17.920" Title="Where can I find the solutions manual to Beginning Statistics with Data Analysis by Mosteller, Fienberg and Rourke?" Tags="|reference-request|statistics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24623" PostTypeId="1" CreationDate="2021-11-23T14:32:01.783" Score="1" ViewCount="437" Body="&lt;p&gt;I have a master in math, teaching experience as a graduate student, TA experience and tons of in-person tutoring. I feel like a pretty prime candidate to be an online tutor. Can anyone give me some advice on how to get started? Thanks!&lt;/p&gt;&#xA;" OwnerUserId="16437" LastActivityDate="2023-07-14T16:46:32.977" Title="How do I get started as an online math tutor?" Tags="|tutoring|online|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24624" PostTypeId="2" ParentId="24610" CreationDate="2021-11-23T21:17:53.443" Score="2" Body="&lt;p&gt;Slide rules are a good tactile method of exploring logs. You can use them to calculate the &lt;span class=&quot;math-container&quot;&gt;$log_ba$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b^a$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$a\cdot b$&lt;/span&gt;. They also visually illustrate the properties of logs, such as &lt;span class=&quot;math-container&quot;&gt;$\log(xy)=\log(x)+\log(y), \log(x^y)=y\log(x), \log_yx=\frac{\log_bx}{\log_by}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2021-11-23T21:17:53.443" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24625" PostTypeId="2" ParentId="41" CreationDate="2021-11-24T15:37:37.733" Score="0" Body="&lt;p&gt;When doing density calculations students sometimes mistake my &amp;quot;Rho&amp;quot; for &amp;quot;P&amp;quot;. Need to curl the stem a lot more to avoid this happening. Also when teaching hydrocarbon chemistry need to always make sure that my double and triple bonds are clear as sometimes the lines are not far enough apart and students mistake it for a single bond.&lt;/p&gt;&#xA;" OwnerUserId="18136" LastActivityDate="2021-11-24T15:37:37.733" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24626" PostTypeId="1" CreationDate="2021-11-24T16:44:16.753" Score="18" ViewCount="3676" Body="&lt;p&gt;In high school (in the US, at least), it is common to define the domain of a function as the set of real numbers for which the function is well-defined and returns a real result. Then students are asked questions like, &amp;quot;What is the domain of &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt;?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In most university-level classes, the domain and codomain of a function are simply part of the definition of a function. When I define a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, I write &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$f : A \to B$&lt;/span&gt; be the function given by &lt;span class=&quot;math-container&quot;&gt;$f(x) = \ldots$&lt;/span&gt;&amp;quot; In this context, the question of what the domain of a function becomes trivial.&lt;/p&gt;&#xA;&lt;p&gt;It's my opinion that the second approach is superior, as we don't always want the domain to be the largest set possible, nor do we always deal with only functions of real numbers. And of course changing the domain and codomain can change properties of the function (e.g., injectivity).&lt;/p&gt;&#xA;&lt;p&gt;My question is: why, historically and pedagogically, do we teach domain this way in high school?&lt;/p&gt;&#xA;" OwnerUserId="14206" LastActivityDate="2021-11-29T00:55:14.247" Title="Why is there a disconnect in the usage of &quot;domain&quot; between high school and higher mathematics, and where does it come from?" Tags="|secondary-education|functions|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24627" PostTypeId="2" ParentId="24626" CreationDate="2021-11-24T19:53:51.063" Score="23" Body="&lt;p&gt;In real-world applications, the typical case is that the domain is neither implicit in an expression we write down, nor explicitly stated along with the expression. Rather, one uses knowledge of the real world to decide what numbers make sense as inputs to the function. So for example, if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; represents an investment, and &lt;span class=&quot;math-container&quot;&gt;$f(x)=x-3$&lt;/span&gt; represents the profit on that investment, then we need to look at the business context to decide what values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; make sense. If we're selling short in the stock market, then &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;0$&lt;/span&gt; could make sense. If we're buying a fast-food franchise, then the domain is something like &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;a$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is some fairly large, positive amount of money.&lt;/p&gt;&#xA;&lt;p&gt;In applications, it's sort of a modern mathematical lie-told-to-children that one variable is a function of the other. In the era of Gauss and Euler, there was no such notion of a function, just relations between variables. In physics, there are typically no cause-and-effect relationships. For example, in Newton's second law &lt;span class=&quot;math-container&quot;&gt;$a=F/m$&lt;/span&gt; the acceleration is often thought of as an effect due to the force, but in reality, there is no such cause-and-effect relationship mathematically inherent in Newton's laws. Laws of physics are typically just differential equations, and they have time-reversal symmetry. Often we solve an entire problem and find a relationship between &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, and only then do we worry about what's a function of what or what parts of the &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; plane are meaningful in the real world. So for example, if a weight &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; is hung from the middle of a cord, which is slanted at an angle &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;, it's not true that &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; causes &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; or that &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; causes &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt;. And only at the end of solving the problem would we be likely to be able to clarify for ourselves that &lt;span class=&quot;math-container&quot;&gt;$\theta=0$&lt;/span&gt; is not physically possible.&lt;/p&gt;&#xA;&lt;p&gt;An obvious disadvantage of always stating domains explicitly is that it would be a pain. For instance, if a calculus textbook has 100 exercises in which students differentiate given functions, we just want to be able to write something like &lt;span class=&quot;math-container&quot;&gt;$\sin(x^2)$&lt;/span&gt;, without having to write a lot of irrelevant words. What would be the point here of restricting the values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;? When we calculate the derivative, we get an expression that is some function that, even if restricted to some smaller domain, could be analytically extended in a unique and natural way to the whole real line. And of course, we're doing freshman calculus, so the domain of discourse is the real line. It's very normal to do mathematics with an implied domain of discourse, e.g., every axiomatic presentation of Euclidean plane geometry is going to have an implied domain of discourse which is the plane.&lt;/p&gt;&#xA;" OwnerUserId="18137" LastActivityDate="2021-11-24T19:53:51.063" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24628" PostTypeId="2" ParentId="24626" CreationDate="2021-11-24T21:49:19.900" Score="13" Body="&lt;p&gt;We give high-schoolers many different explanations of the word &amp;quot;function.&amp;quot; Here are a few that are either implied or outright stated at various points in a student's education:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A function is an expression in terms of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. This is pretty unusual all by itself, but it may appear in conjunction with (2) or (3) (i.e. students may reason that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is the function, and since &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; equals some expression, therefore the expression is the function too). Students may become uncomfortable if the variable is not &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;A function is an equation of the form &lt;span class=&quot;math-container&quot;&gt;$f(x) = \mathrm{[something]}$&lt;/span&gt;, where [something] is in terms of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Students may believe that the function can be subsequently &amp;quot;abbreviated&amp;quot; as &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;. If there is also a &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; and you're teaching function compositions, then there will often be confusion about how &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; can have &amp;quot;multiple different values&amp;quot; (i.e. when you write &lt;span class=&quot;math-container&quot;&gt;$f(g(x))$&lt;/span&gt;, the value that is given to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is &amp;quot;not the same&amp;quot; as the value that is given to &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, but some students may believe that both values are &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; because that's how &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is written); students may reconcile this by blindly applying the correct substitution without understanding what they are doing. But it gets them points on the exam or homework, so the student thinks they are learning.&lt;/li&gt;&#xA;&lt;li&gt;A function is an equation of the form &lt;span class=&quot;math-container&quot;&gt;$y = \mathrm{[something]}$&lt;/span&gt;, where [something] is in terms of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Students may believe that it specifically has to be &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, in the right order, rather than any other pair of variables.&lt;/li&gt;&#xA;&lt;li&gt;A function is a set of ordered pairs &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt;, such that no two (distinct) pairs have the same &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. The word &amp;quot;distinct&amp;quot; is often omitted from this definition, but it probably doesn't matter in practice because students are generally going to assume that's what it means anyway.&lt;/li&gt;&#xA;&lt;li&gt;As in (2), but allowing for piecewise notation. Students may believe that piecewise functions are somehow &amp;quot;less valid&amp;quot; than &amp;quot;regular&amp;quot; functions, and/or try to rewrite piecewise functions as non-piecewise functions. For that matter, students believe that there is a such thing as &amp;quot;a piecewise function&amp;quot; in the first place.&lt;/li&gt;&#xA;&lt;li&gt;As in (3), but allowing for implicit functions. Again, students may resist implicit functions, but solving an implicit function for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is at least a useful activity some of the time.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;All of these definitions are at best incomplete and at worst outright wrong.&lt;/strong&gt; (4) is somewhat close to being right, but like all of these definitions, it does not supply a codomain and therefore cannot explain the concept of a surjective function (for example, if you take the floor function over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \rightarrow \mathbb{R}$&lt;/span&gt;, it is not surjective, but if you take it over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \rightarrow \mathbb{Z}$&lt;/span&gt;, it is surjective, yet both of these functions contain exactly the same set of ordered pairs in their respective graphs). Unlike the other definitions, (4) &lt;em&gt;does&lt;/em&gt; imply a specific domain - the domain is just the set of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; coordinates. However, students might not notice this subtlety, and the existence of the other (wronger) definitions tends to deemphasize the idea that the domain can be chosen arbitrarily. (4) is also incompatible with algebraic and (for the most part) analytic manipulation, so students will have to work with the other definitions regardless of what they think of (4).&lt;/p&gt;&#xA;&lt;p&gt;There is one exercise at the high-school level that does describe functions more or less correctly, but it tends to get neglected in favor of the definitions given above. It usually looks like this: Students are shown a diagram with two adjacent ellipses or circles, each containing a small number of labeled points, and arrows from points in the left ellipse to points in the right ellipse. Students are then (correctly) told that these diagrams show (binary) relations, and are typically asked to identify which relations are functions, one-to-one, etc. These diagrams contain all of the required elements of a function, including the codomain. However, this visualization is usually dropped in favor of the so-called &amp;quot;vertical line test&amp;quot; (i.e. draw vertical lines through the graph of a function, and make sure that each line intersects the graph at most once) and &amp;quot;horizontal line test&amp;quot; (the same, but with horizontal lines), which can be used on infinite relations/functions (unlike drawing out an explicit relation diagram one point at a time). I suspect that many students prefer the vertical/horizontal line tests, because they are specific &amp;quot;recipes&amp;quot; that students can use to &amp;quot;get the right answer&amp;quot; and do well on exams or homework (whereas trying to visualize infinitely many arrows between points is going to be difficult at best).&lt;/p&gt;&#xA;&lt;p&gt;The other serious problem at the high-school level is that most students don't properly appreciate the distinction between &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;. Of course, mathematicians often ignore this distinction as well, because it's a rather fine and technical distinction which only sometimes matters, but many students are not even aware that there is a distinction to be drawn in the first place. If you cannot even write &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; by itself without confusing half the class, how do you expect them to understand &lt;span class=&quot;math-container&quot;&gt;$f: A \rightarrow B$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;TL;DR:&lt;/strong&gt; We don't give students a proper explanation of &amp;quot;domain&amp;quot; because we don't give them a proper explanation of &amp;quot;function,&amp;quot; and that in turn is because of high-school's emphasis on algebra, graphing, and other specific ways of applying the math, rather than on a broader conceptual foundation.&lt;/p&gt;&#xA;" OwnerUserId="8474" LastEditorUserId="8474" LastEditDate="2021-11-29T00:55:14.247" LastActivityDate="2021-11-29T00:55:14.247" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="24629" PostTypeId="2" ParentId="24626" CreationDate="2021-11-24T22:27:21.520" Score="9" Body="&lt;p&gt;In my high school days (1970s), the set of all numbers for which an expression is defined (and, implicitly, real) was called the “natural domain” of the function defined by the expression. The domain of a function with an unspecified domain was to be understood to be its natural domain, a convenient assumption. I’m not sure this is an accurate answer to why pedagogically it was done, but it seemed important that we be able to determine when an expression is defined. We also had to determine the domain of a composition, a somewhat complicated process that required patience and care and was easy (for me) to blunder through.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2021-11-24T22:27:21.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24631" PostTypeId="2" ParentId="24626" CreationDate="2021-11-25T22:52:29.910" Score="3" Body="&lt;p&gt;At the secondary level, students tend to think of &amp;quot;real numbers&amp;quot; as being literal: any number that isn't in the real number system isn't a &amp;quot;real number&amp;quot;, and any time you're given a variable, the default is that it can be any real number, and the default is that a function applies to all real numbers. The idea of a function that takes positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;, and simply isn't defined on nonpositive numbers, is hard to grasp; why would it also take negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;? Unless there's some reason why it &lt;em&gt;can't&lt;/em&gt; be defined in a region, they will take it as defined in that region. And the idea of, say, a function from the equivalence classes of &lt;span class=&quot;math-container&quot;&gt;$(x-y \in \mathbb N)\rightarrow x \sim y$&lt;/span&gt; to points on a circle is going to make them very confused.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2021-11-25T22:52:29.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24632" PostTypeId="2" ParentId="24604" CreationDate="2021-11-26T05:56:07.750" Score="1" Body="&lt;p&gt;I acknowledge the subjective nature of the question and hereby provide an answer from my own experiences as a tutor, hoping to &lt;strong&gt;inspire better answers from more qualified teachers&lt;/strong&gt;. I am in the midst of completing my teaching certificate so I'm somewhat familiar with general teaching principles and the different approaches. I don't intend to turn this into a discussion since I have no intentions of persuading anyone or responding to others' claims.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The three programs for the most part give access or charge a small fee to past exam papers; this is in part to uphold a transparency and consistency in grading and expectations.&lt;/p&gt;&#xA;&lt;p&gt;Generally in my tutoring sessions, I first emphasize a firm grasp of the background and central concepts, and strong algebraic and error-finding skills. This is more than enough to achieve an above-average grade in such programs. But to achieve the top marks, I rely upon the abundance of past papers. Some reoccuring problem solving skills and motifs are identified and assigned to tutees. When I was in high school, I was taught to just drill past papers questions which was not only time-consuming but possibly also inefficient. I am keen to see if others have the opposite issue of having too much theory and not enough practice.&lt;/p&gt;&#xA;&lt;p&gt;For weak or average students I assign relatively common questions and for stronger one I give an even spread of difficulties, not emphasizing the easy nor the hard questions. This is mainly to give students a realistic expectation of the exam, and to not overwhelm them. This is easy to accomplish in a private tutoring environment, so I was curious as to how teachers manage this balance in the classroom. In a classroom there are more differences than just skills or math experience. In my experiences as a student and my observations, the most practical and common approach is to give every one the same material and instruction, so to prioritize timely and adequate coverage of the syllabus.&lt;/p&gt;&#xA;&lt;p&gt;The overemphasis on past papers is more apparent in students, especially if their teacher is subpar and they are looking to &amp;quot;be able to do past papers&amp;quot;. I say this despite my high school teacher giving us loads of past paper questions and dedicate about half of in class time on such problems. It is better for the teacher to overemphasize on past papers, because a student has a much tougher time navigating the resources if they were to buy past papers without guidance.&lt;/p&gt;&#xA;&lt;p&gt;Therefore I believe the primary role of the teacher is to make sure the students understand the material, via using a variety of resources and technology, and to highlight sufficient past paper problems so the students understand specific expectations and reoccuring themes. This should pave a good starting place for if they want to do additional problems on their own.&lt;/p&gt;&#xA;" OwnerUserId="18050" LastActivityDate="2021-11-26T05:56:07.750" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24633" PostTypeId="2" ParentId="24603" CreationDate="2021-11-26T06:12:18.817" Score="2" Body="&lt;p&gt;One practical way to do this might be to loop the rubber band over one end, and to then twist it before looping it over the other end. In this way the parallel lines combine to become one.&lt;/p&gt;&#xA;&lt;p&gt;This would not work well with &amp;quot;fat&amp;quot; rubber bands, but should be fine with regular rubber bands.&lt;/p&gt;&#xA;" OwnerUserId="9521" LastActivityDate="2021-11-26T06:12:18.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24634" PostTypeId="2" ParentId="24623" CreationDate="2021-11-26T06:13:17.693" Score="3" Body="&lt;p&gt;I have much fewer qualifications and I see myself as a fine tutor, so no doubt you are even more justified in your belief.&lt;/p&gt;&#xA;&lt;p&gt;There are various websites that allow tutors to set up a profile, post details and rates, but allow the tutors to interact directly with the clients. Many these sites either charge to client or far more commonly charge the tutors, typically for a first-contact. I don't want to give specific names as the one I used, and such sites in general, was susceptible to false information or even scams. I joined one with a low bar of entry and I'm sure with a Masters you can join more selective ones.&lt;/p&gt;&#xA;&lt;p&gt;If you join an online community of students you are looking to tutor, you may be occasionally allowed to advertise your services. This depends on the nature and type of the student community. This is more to build reputation than to gain leads per se.&lt;/p&gt;&#xA;&lt;p&gt;This is more true for high school, but you also want to contact the parent or whoever is paying for the services, and high schoolers and undergrads aren't willing to pay as much. Unless you want to tutor some international program or competition, most parents want a local tutor so you should seek to advertise locally, even if you meet online. Along these lines, see if you can be allowed to advertise on campus or even be listed as a tutor.&lt;/p&gt;&#xA;&lt;p&gt;For international programs or competitions, it's best to open a website and do some SEO. The technical details should refer to stackexchange or the alike.&lt;/p&gt;&#xA;" OwnerUserId="18050" LastActivityDate="2021-11-26T06:13:17.693" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24635" PostTypeId="1" AcceptedAnswerId="24636" CreationDate="2021-11-26T22:41:23.160" Score="4" ViewCount="862" Body="&lt;p&gt;To prove, e.g., the identity &lt;span class=&quot;math-container&quot;&gt;$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$&lt;/span&gt;, I remembered working, in high school, in the following way. Expanding the LHS gives&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;(a^2+b^2)(c^2+d^2)=a^2c^2+a^2d^2+b^2c^2+b^2d^2,\qquad(1)&#xA;\end{equation}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;while, expanding the RHS gives&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;(ac-bd)^2+(ad+bc)^2&amp;amp;=a^2c^2+b^2d^2+2acbd+a^2d^2+b^2c^2-2abcd\\&#xA;&amp;amp;=a^2c^2+a^2d^2+b^2c^2+b^2d^2,    \qquad(2)&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Comparing the RHS of (1) and (2) one deduces that &lt;span class=&quot;math-container&quot;&gt;$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;First question:&lt;/strong&gt; Is this considered an acceptable proof at a post-secondary level? Or should one work differently, like, for instance, by completing the square&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;(a^2+b^2)(c^2+d^2)&amp;amp;=a^2c^2+a^2d^2+b^2c^2+b^2d^2=a^2c^2+a^2d^2-2abcd+b^2c^2+b^2d^2+2abcd\\&#xA;&amp;amp;=(ac-bd)^2+(ad+bc)^2&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As another example consider the proof that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}+\sqrt{6}&amp;lt;\sqrt{15}$&lt;/span&gt;. Teachers in secondary school (and even some lecturers in engineering schools I know) usually work like this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Taking the square of each side of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}+\sqrt{6}&amp;lt;\sqrt{15}$&lt;/span&gt;  gives  &lt;span class=&quot;math-container&quot;&gt;$2+6+2\sqrt{2}\sqrt{6}&amp;lt;15$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Rearranging gives &lt;span class=&quot;math-container&quot;&gt;$2\sqrt{2}\sqrt{6}&amp;lt;7$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Squaring again gives &lt;span class=&quot;math-container&quot;&gt;$48&amp;lt;49$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Since &lt;span class=&quot;math-container&quot;&gt;$48&amp;lt;49$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}+\sqrt{6}&amp;lt;\sqrt{15}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Nevertheless, according to A Concise Introduction to Pure Mathematics of Martin Liebeck, previous argument is not a proof. Indeed, citing the author &lt;em&gt;We have shown that if P is the statement we want to prove, and Q is the statement that 48 &amp;lt; 49, then P⇒Q; but this tells us nothing about the truth or otherwise of P.&lt;/em&gt;&#xA;The proper proof starts by supposing the veracity of the contrary&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;\sqrt{2}+\sqrt{6}\geq\sqrt{15}&#xA;\end{equation}&lt;/span&gt;&#xA;We have then&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\sqrt{2}+\sqrt{6}\geq\sqrt{15}&amp;amp;\Rightarrow \left(\sqrt{2}+\sqrt{6}\right)^2\geq\left(\sqrt{15}\right)^2\Rightarrow 2+6+2\sqrt{2}\sqrt{6}\geq 15\\&#xA;&amp;amp;\Rightarrow 2\sqrt{12}\geq 7\Rightarrow 4\times 12\geq 49\Rightarrow 47\geq 48&#xA;\end{align*}&lt;/span&gt;&#xA;that is a contradiction.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Second question:&lt;/strong&gt; Is this lack of rigor on the part of teachers justified?&lt;/p&gt;&#xA;" OwnerUserId="18147" LastActivityDate="2022-02-15T06:15:47.797" Title="Questions about proofs" Tags="|proofs|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24636" PostTypeId="2" ParentId="24635" CreationDate="2021-11-26T23:29:43.453" Score="8" Body="&lt;p&gt;In general, it is very difficult to answer the question about whether or not the lack of rigor is &amp;quot;justified&amp;quot;, as the ultimate goal is competence and literacy in math and logical reasoning, so what is justified for one student may not be justified for another, as each student's situation and current understanding will be different. What constitutes something being justified on the part of the teacher is whether the teacher can correctly identify the gaps and poor habits on the side of the student, and then choose the best method of explaining and fixing them.&lt;/p&gt;&#xA;&lt;p&gt;That's the ideal case anyways. In the case of teaching a class containing many students, often times the teacher has a goal that they need to meet and cannot cater to every single student at every single step to make sure they don't fall behind. The teachers make their best effort, but their time and resources are limited. There is always some push and pull there. Putting these generalities aside, I'll respond to the specific questions now.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The first question of proving &lt;span class=&quot;math-container&quot;&gt;$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$&lt;/span&gt; is pretty straightforward. There's not that much more you can do to make it more rigorous without getting into some very technical machinery that is not expected of students until they reach formal proof-writing courses in university. This is mostly because everything involves equality and basic arithmetic properties like distributive and commutative.&lt;/p&gt;&#xA;&lt;p&gt;The second question is significantly trickier. The proof by contradiction method certainly is rigorous and one way of approaching the problem, but the other method shown can also be made rigorous if one carefully constructs the reasoning used to justify each statement.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Taking the square of each side of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}+\sqrt{6}&amp;lt;\sqrt{15}$&lt;/span&gt;  gives  &lt;span class=&quot;math-container&quot;&gt;$2+6+2\sqrt{2}\sqrt{6}&amp;lt;15$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;One can prove that if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are both positive, and &lt;span class=&quot;math-container&quot;&gt;$a &amp;lt; b$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\sqrt{a} &amp;lt; \sqrt{b}$&lt;/span&gt;. If this statement is proved, then it follows that as long as &lt;span class=&quot;math-container&quot;&gt;$2+6+2\sqrt{2}\sqrt{6}&amp;lt;15$&lt;/span&gt; is true, then &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}+\sqrt{6}&amp;lt;\sqrt{15}$&lt;/span&gt; must also be true.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Rearranging gives &lt;span class=&quot;math-container&quot;&gt;$2\sqrt{2}\sqrt{6}&amp;lt;7$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;One can also show that if &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;b$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$a+c&amp;lt;b+c$&lt;/span&gt;, where all the variables are real numbers. This would allow for the &amp;quot;rearranging&amp;quot; of values being done here.&lt;/p&gt;&#xA;&lt;p&gt;Once these intermediate concepts are proven, they can be used as the building blocks of proving this statement in this way, but as you can see, this involves a lot more steps if one chooses to take this approach in order to be rigorous. Most of the time, when teachers ignore these steps to make this rigorous, they are assuming that the students already understand the truth of the concepts being applied and are able to intuitively understand the logic behind it.&lt;/p&gt;&#xA;&lt;p&gt;The reason why rigor is important for problems like these is that students need to be careful not to build bad habits and apply reasoning in fallacious ways. For instance, just because &lt;span class=&quot;math-container&quot;&gt;$a &amp;gt; b$&lt;/span&gt; does not imply that &lt;span class=&quot;math-container&quot;&gt;$a^2 &amp;gt; b^2$&lt;/span&gt; (assuming real values). Just because &lt;span class=&quot;math-container&quot;&gt;$a^2 &amp;gt; b^2$&lt;/span&gt; does not imply that &lt;span class=&quot;math-container&quot;&gt;$a &amp;gt; b$&lt;/span&gt;. There are important restrictions to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; in order for these statements to hold true, specifically that they both must be positive.&lt;/p&gt;&#xA;&lt;p&gt;But even if we understood that given positive real values, &lt;span class=&quot;math-container&quot;&gt;$a &amp;gt; b$&lt;/span&gt; implies that &lt;span class=&quot;math-container&quot;&gt;$a^2 &amp;gt; b^2$&lt;/span&gt;, that doesn't mean that this is the reason why we can square both sides of an inequality. Instead of proving &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}+\sqrt{6}&amp;lt;\sqrt{15}$&lt;/span&gt; by showing that &lt;span class=&quot;math-container&quot;&gt;$2+6+2\sqrt{2}\sqrt{6}&amp;lt;15$&lt;/span&gt; is true, we would be assuming &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}+\sqrt{6}&amp;lt;\sqrt{15}$&lt;/span&gt; to prove that &lt;span class=&quot;math-container&quot;&gt;$2+6+2\sqrt{2}\sqrt{6}&amp;lt;15$&lt;/span&gt; is true, the opposite of what we are trying to do.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As you can see, it is very easy to slip and make wrong arguments when writing proofs. The point of rigor is making sure that we aren't just loosely applying things that feel right but aren't actually valid. Whether it is justified or not depends on if the teacher is conveying these concepts properly, and if it is within the purview of the teacher's goals.&lt;/p&gt;&#xA;" OwnerUserId="9383" LastEditorUserId="9383" LastEditDate="2021-11-29T16:32:21.120" LastActivityDate="2021-11-29T16:32:21.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24637" PostTypeId="1" CreationDate="2021-11-26T23:31:00.343" Score="4" ViewCount="168" Body="&lt;p&gt;I’ve seen at least two phrases to describe a fixed difference between two numbers, i.e., “constant difference” and “common difference.”&lt;/p&gt;&#xA;&lt;p&gt;For example, if Sibling A is 10 years older than Sibling B today, you can expect that the difference in their ages will &lt;em&gt;always&lt;/em&gt; be 10 years.&lt;/p&gt;&#xA;&lt;p&gt;Some questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Are these two phrases always interchangeable, or is there some nuance that would be good for an educator or student to understand?&lt;/li&gt;&#xA;&lt;li&gt;Does this principle go by any other names?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13229" LastActivityDate="2021-11-27T17:22:33.577" Title="Are ‘constant difference’ and ‘common difference’ synonymous?" Tags="|terminology|arithmetic|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24638" PostTypeId="2" ParentId="24637" CreationDate="2021-11-27T13:34:24.080" Score="4" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a &lt;strong&gt;constant difference&lt;/strong&gt; between A's and B's ages.&lt;/p&gt;&#xA;&lt;p&gt;Here, the value of interest is constant as the input, time, varies.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a &lt;strong&gt;common difference&lt;/strong&gt; between successive terms of sequence S.&lt;/p&gt;&#xA;&lt;p&gt;Here, the value of interest is common across varying objects of comparison.&lt;/p&gt;&#xA;&lt;p&gt;But I could equally say there is a &lt;em&gt;constant difference&lt;/em&gt; between successive terms (as I vary the index). Here, the disparity between &lt;em&gt;constant difference&lt;/em&gt; and &lt;em&gt;common difference&lt;/em&gt; is just a matter of framing and emphasis, and isn't important.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2021-11-27T17:22:33.577" LastActivityDate="2021-11-27T17:22:33.577" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24639" PostTypeId="2" ParentId="24635" CreationDate="2021-11-27T14:47:59.507" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Comparing the RHS of (1) and (2) one deduces that &lt;span class=&quot;math-container&quot;&gt;$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;First question:&lt;/strong&gt; Is this considered an acceptable proof at a post-secondary level?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It's perfectly valid to meet in the middle like this.&lt;/p&gt;&#xA;&lt;p&gt;It's even valid to start with the desired result then work towards &lt;span class=&quot;math-container&quot;&gt;$1=1$&lt;/span&gt; or some established identity, as long as the logical connective between each step is justifiably and explicitly displayed as &lt;span class=&quot;math-container&quot;&gt;$'{\iff}'.$&lt;/span&gt; (Typically, the implicit punctuation between lines is “therefore”, in which case—since anything is a consequence of pigs flying—it is invalid to start with the desired result.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Second question:&lt;/strong&gt; Is this lack of rigor on the part of teachers justified?  Teachers in secondary school (and even some lecturers in&#xA;engineering schools I know) usually work like this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Taking the square of each side of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}+\sqrt{6}&amp;lt;\sqrt{15}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Rearranging gives &lt;span class=&quot;math-container&quot;&gt;$2\sqrt{2}\sqrt{6}&amp;lt;7$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Squaring again gives &lt;span class=&quot;math-container&quot;&gt;$48&amp;lt;49$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Since &lt;span class=&quot;math-container&quot;&gt;$48&amp;lt;49$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}+\sqrt{6}&amp;lt;\sqrt{15}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The proper proof starts by supposing the veracity of the contrary&#xA;.....&#xA;that is a contradiction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The proof-by-contradiction comparison is immaterial, since the above proof attempt is related to it in neither intent nor substance.&lt;/p&gt;&#xA;&lt;p&gt;The above proof is actually fine and valid if&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;it is explicitly noted that both values of each inequality being&#xA;squared are nonnegative, and&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$'{\iff}'$&lt;/span&gt; is explicitly displayed between steps.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(|p|&amp;lt;|q|\iff p^2&amp;lt;q^2.)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;While these may not be stressed in writing, are they perhaps at least being communicated orally? It is also unclear whether their omission is deliberate or due to ignorance.&lt;/p&gt;&#xA;&lt;p&gt;Echoing CosmoVibe, it's hard to categorically say aye or nay to your question, since the desirable level of rigour muchly depends on context and circumstances.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum&lt;/strong&gt; (moving my comments up into the post)&lt;/p&gt;&#xA;&lt;p&gt;@Daniel Well, I did say that this practice is valid as long as the 'iff' between each step is (1) justifiably and (2) explicitly displayed: it is critical that (1) our chain of reasoning must &lt;em&gt;genuinely&lt;/em&gt; be flowing in both directions, and (2) our presentation make this abundantly clear.&lt;/p&gt;&#xA;&lt;p&gt;In practice, because trying to ensure such “reversibility” at every step is prone to carelessness (people aren't used to having to keep checking the reverse flow while driving forward) or, more commonly, misunderstanding &amp;amp; disregard, I can understand why teachers discourage this practice or lie or believe that it's invalid.&lt;/p&gt;&#xA;&lt;p&gt;It's anyway easy and always valid to present &lt;em&gt;just&lt;/em&gt; the required direction of argument after having used the 'cheat' direction to devise the proof's meat. This—just like filtering out extraneous solutions at the end of equation-solving—also sidesteps having to continually ensure that the &lt;span class=&quot;math-container&quot;&gt;$'{\iff}'$&lt;/span&gt;s are genuine.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-02-15T06:15:47.797" LastActivityDate="2022-02-15T06:15:47.797" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24640" PostTypeId="2" ParentId="24599" CreationDate="2021-11-27T17:50:52.197" Score="2" Body="&lt;p&gt;Your question is somewhat broad, and answerable, but this answer should not be considered as providing advice.  Your question may have gone unanswered for two reasons.  First, people may not know of the available resources for obtaining a copy of your book of interest for review.  And second, nobody wants you to have a bad experience based on what happened after following a recommendation that was given. Of course, nobody wants you to have a bad outcome. You have to take responsibility for making the effort to achieve success.&lt;/p&gt;&#xA;&lt;p&gt;You have indicated you have no essential background in calculus, and that this is a somewhat new beginning in trying to learn the material.  One caution, though... Take your time.  If you do not have a pre-calculus background of knowledge, you may do well to review all of that material first.  Are your skills ready for calculus? The usual prerequisites are algebra I and II, geometry, trigonometry, and precalculus or math analysis.  Having background knowledge in analytic geometry will also be helpful and this is usually part of the precalculus series of topics.  Make sure you know these prerequisites first.  The usual requirements for entry into calculus are successful grades in prerequisite courses, first.  If you are uncertain about your readiness or skills, ask for some guidance from the folks at your guided course of instruction.  Or ask for help from other professional resources, such as your school math department, or counselor.  You will find that having at the ready reference texts for each of the prerequisite subject areas, will be helpful in refreshing your memory and retaining your skills when you have a question about something basic that you have previously learned.  There is an old saying that &lt;em&gt;the mark of a good education is not what you have learned, but what you have forgotten.&lt;/em&gt; Consequently, for those courses and your studies in calculus, &lt;em&gt;keep all of your notes and worked problems!&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;A suggestion would be that you review the College Board requirements of the AP course, &lt;a href=&quot;https://apcentral.collegeboard.org/courses/ap-calculus-ab/course-audit&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; as the information is complete regarding supporting texts and material covered in the tests.  The text list they give does not include the Stewart 9th edition text of interest, but nevertheless, does mention several preceding editions, including Stewart, &lt;em&gt;Calculus for AP,&lt;/em&gt; which may be of interest.  A complete sample syllabus of AP required material may be found &lt;a href=&quot;https://apcentral.collegeboard.org/pdf/ap-calculus-ab-sample-syllabus-1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. This AP syllabus is one of four various syllabi using different texts, however this syllabus does not use the Stewart text.  Nevertheless, the covered material will be the similar, syllabus to syllabus, and this syllabus is an &lt;em&gt;excellent&lt;/em&gt; guide.&lt;/p&gt;&#xA;&lt;p&gt;The Stewart 9th edition text was given a cursory review, and was found to comprehensively cover material in calculus up to, and &lt;em&gt;including&lt;/em&gt; an introduction to differential equations.  Of particular note is that the book has many fine illustrations and apparently very complete examples that support the subject material of each chapter.  Problem sets are numerous and odd-numbered problems have answers provided at the back of the text.  The index is complete. Each chapter has sub-topic study guides and overviews to assist the student in reviewing and keeping in mind essential background information.&lt;/p&gt;&#xA;&lt;p&gt;Given that you can obtain and review the book at no charge, and that you can access a syllabus for the AP course, and further that you may be using a guided course of instruction, you may be ahead to download the Stewart text and begin a review of the material.&lt;/p&gt;&#xA;&lt;p&gt;One final thought: If your guided course of study uses a different text, then that text may be best. Your guided course of study has your interests at heart in providing you with the material necessary to pass the AP exam. Asking for their thoughts may be helpful regarding the Stewart text.&lt;/p&gt;&#xA;" OwnerUserId="18149" LastEditorUserId="-1" LastEditDate="2021-12-01T13:29:24.287" LastActivityDate="2021-12-01T13:29:24.287" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24641" PostTypeId="2" ParentId="24626" CreationDate="2021-11-27T20:30:25.273" Score="5" Body="&lt;p&gt;At A-level (16+ in the UK) students are never asked questions such as &amp;quot;what is the domain of &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt; in formal examination questions.  In my teaching I mention that such a question is meaningless, the domain is part of the definition of the function and the function isn't fully defined until a domain is given.&lt;/p&gt;&#xA;&lt;p&gt;Question papers are careful always to state the domain - a matter of confusion when students are answering questions in which the domain is not very relevant:  &amp;quot;What are we supposed to &lt;strong&gt;do&lt;/strong&gt; with the &lt;span class=&quot;math-container&quot;&gt;$x\in \mathbb{R}$&lt;/span&gt;?&amp;quot; is a fairly common question (and in many cases the answer is &amp;quot;nothing, its just there ensure the function is fully defined&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The only context in which students are asked for a domain is when they are defining a function themselves, generally as an inverse or composition of functions.  So it would be a valid question to state&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x +1, x\in\mathbb{R}\ x&amp;gt;2$&lt;/span&gt;. Find the inverse function &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Students would then only be awarded full marks if they gave both a formula for the inverse and its domain.  The domain is required, even when it is not explicitly asked for.&lt;/p&gt;&#xA;&lt;p&gt;The disconnect with higher maths is that a co-domain is never mentioned, so questions of a function being &amp;quot;onto&amp;quot; can't be considered.&lt;/p&gt;&#xA;" OwnerUserId="5361" LastActivityDate="2021-11-27T20:30:25.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24642" PostTypeId="2" ParentId="19595" CreationDate="2021-11-27T22:26:55.427" Score="1" Body="&lt;p&gt;I like the geometric approaches of Joseph O’Rourke and my other answer, but here is an algebraic-calculus approach that is Calculus-I level except for the number of variables. It also assumes differentials are covered in Calculus I.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let the two curves be given by &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=g(x)$&lt;/span&gt;. Having them in this explicit form isn’t critical, but we will assume &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is locally a differentiable function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&#xA;Let &lt;span class=&quot;math-container&quot;&gt;$P_1(x_1,y_1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P_2(x_2,y_2)$&lt;/span&gt; be arbitrary points on the first and second curves respectively.&#xA;Thus&#xA;&lt;span class=&quot;math-container&quot;&gt;$$dy_1 = m_1\,dx_1, \qquad dy_2 = m_2\, dx_2 \,,$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$m_1=f’(x_1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m_2=g’(x_2)$&lt;/span&gt; are the slopes of the tangent lines at &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P_2$&lt;/span&gt; respectively.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; be the distance between &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P_2$&lt;/span&gt;.&#xA;Then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$L^2 = (x_1-x_2)^2 + (y_1-y_2)^2 \,.$$&lt;/span&gt;&#xA;At an extremum of &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$dL=0$&lt;/span&gt;, which implies&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\eqalign{&#xA;0&amp;amp;=(x_1-x_2) (dx_1-dx_2) + (y_1-y_2) (dy_1-dy_2) \cr&#xA; &amp;amp;=(x_1-x_2) (dx_1-dx_2) + (y_1-y_2) (m_1\,dx_1-m_2\,dx_2) \,. \tag{1}\cr&#xA;}$$&lt;/span&gt;&#xA;Now &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt; are independent, so &lt;span class=&quot;math-container&quot;&gt;$dL=0$&lt;/span&gt; no matter what relationship between &lt;span class=&quot;math-container&quot;&gt;$dx_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dx_2$&lt;/span&gt; we consider.&#xA;We can hold &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt; constant and vary &lt;span class=&quot;math-container&quot;&gt;$P_2$&lt;/span&gt; or vice versa; or we can vary them together.&#xA;Some results deduced from (1):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\eqalign{&#xA;dx_2=0\hfil &amp;amp;\Longrightarrow P_1P_2 \perp \text{tangent line at}\ P_1 \cr&#xA;dx_1=0\hfil &amp;amp;\Longrightarrow P_1P_2 \perp \text{tangent line at}\ P_2 \cr &#xA;dx_1=dx_2 &amp;amp;\Longrightarrow m_1=m_2 \Longrightarrow \text{tangent lines at}\ P_1,\, P_2\ \text{are parallel} \cr&#xA;}$$&lt;/span&gt;&#xA;The last example yields the desired result.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2021-11-28T18:14:20.470" LastActivityDate="2021-11-28T18:14:20.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24643" PostTypeId="2" ParentId="19595" CreationDate="2021-11-28T16:54:41.783" Score="2" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$h: \mathbb{R} \to \mathbb{R}$&lt;/span&gt; be a function.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; be an arbitrary point in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt;.  We want to minimize the distance from the point &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; to the graph of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is a single variable calculus problem.&lt;/p&gt;&#xA;&lt;p&gt;We are minimizing &lt;span class=&quot;math-container&quot;&gt;$(x-a)^2 + (h(x) - b)^2$&lt;/span&gt;.  Let &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; be the abscissa where the absolute minimum occurs.  This occurs when &lt;span class=&quot;math-container&quot;&gt;$2(x_0-a)+2(h(x_0)-b)h'(x_0) = 0$&lt;/span&gt;.  In other words, it occurs when &lt;span class=&quot;math-container&quot;&gt;$h'(x_0) = -\frac{x_0-a}{h(x_0)-b}$&lt;/span&gt;.  The slope of the line segment connecting &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(x_0, h(x_0))$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\frac{h(x_0)-b}{x_0-a}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This shows that the slope of the line segment connecting &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(x_0,h(x_0))$&lt;/span&gt; is perpendicular to the tangent line to &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$(x_0,h(x_0))$&lt;/span&gt;, since the slopes are opposite and reciprocal.&lt;/p&gt;&#xA;&lt;p&gt;We have shown, using only single variable calculus, that the tangent line to the point &lt;span class=&quot;math-container&quot;&gt;$(x_0,h(x_0))$&lt;/span&gt; on one function (&lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;) which is closest to a given point &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; must be perpendicular to the line connecting these two points.&lt;/p&gt;&#xA;&lt;p&gt;Now let &lt;span class=&quot;math-container&quot;&gt;$p_1 = (a_1,f(a_1))$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p_2 = (a_2,g(a_2))$&lt;/span&gt; be the pair of points on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; which minimize the distance between the graphs of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$p_2$&lt;/span&gt; is the closest point on the graph of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; to the point &lt;span class=&quot;math-container&quot;&gt;$p_1$&lt;/span&gt;, so the segment &lt;span class=&quot;math-container&quot;&gt;$\overline{p_1p_2}$&lt;/span&gt; is perpendicular to the tangent line to &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$p_2$&lt;/span&gt;.  Symmetrically, the line segment &lt;span class=&quot;math-container&quot;&gt;$\overline{p_1p_2}$&lt;/span&gt; must be perpendicular to the tangent line to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$p_1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Hence, the tangent lines at &lt;span class=&quot;math-container&quot;&gt;$p_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p_2$&lt;/span&gt; must be parallel.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-11-28T18:50:08.300" LastActivityDate="2021-11-28T18:50:08.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24644" PostTypeId="2" ParentId="24600" CreationDate="2021-11-29T18:59:06.723" Score="2" Body="&lt;p&gt;I think the key thing making these examples feel different is the &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Level_of_measurement&quot; rel=&quot;nofollow noreferrer&quot;&gt;level of measurement&lt;/a&gt;&lt;/strong&gt; of the two quantities. The short version of the story is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If there is a well-defined, meaningful zero point for your data, it might be &lt;strong&gt;ratio data&lt;/strong&gt;, so it could make sense to divide such quantities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If there is no well-defined, meaningful zero-point, then the data is at best called &lt;strong&gt;interval data;&lt;/strong&gt; dividing these does not make any mathematical sense.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Quantities produced are &lt;strong&gt;ratio data&lt;/strong&gt;. Zero is zero; special and well-defined.&lt;/p&gt;&#xA;&lt;p&gt;Positions are &lt;strong&gt;interval data&lt;/strong&gt;. Zero is arbitrary; it can be moved around.&lt;/p&gt;&#xA;&lt;p&gt;So when you divided the total cost by 10 in your economics example, you got a meaningful answer, while you noticed that you couldn't reasonably divide your position by 10 -- the only way to convert your positions into ratio data was to convert them into displacement (which is &lt;strong&gt;ratio&lt;/strong&gt; level!) by finding &lt;span class=&quot;math-container&quot;&gt;$(10^2+50)−(0^2+50)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For a physics example that parallels the one you want, you will need to use something that has a meaningful physical zero value so that you have ratio data;  degrees Kelvin, barometric pressure, molarity of a solution, and so on. Position in meters, temperature in degrees Celsius, and other interval data will give you the same awkwardness you found in your question.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2021-11-29T18:59:06.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24645" PostTypeId="2" ParentId="24600" CreationDate="2021-11-30T19:40:18.663" Score="2" Body="&lt;p&gt;After reading the responses and thinking about this a lot since posting, I have decided to answer my own post (certainly not my intent when I first posted).&lt;/p&gt;&#xA;&lt;p&gt;IMHO the analogous motion question would be the following&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose the position of Car C at time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; seconds is given by &lt;span class=&quot;math-container&quot;&gt;$C(t) = t^2+50$&lt;/span&gt; (meters). Car B, starting at the origin at &lt;span class=&quot;math-container&quot;&gt;$t=0$&lt;/span&gt;, wants to travel at a constant speed such that, after 10 seconds, it has caught up to Car C. What is that constant speed?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Answer: Car B needs to travel at a constant speed of &lt;span class=&quot;math-container&quot;&gt;$\frac{C(10)}{10} = 15 \text{ m/s}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There's a couple of reasons why this is what I was originally looking for:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The solution has the correct form, &lt;span class=&quot;math-container&quot;&gt;$\frac{C(T)}{T}$&lt;/span&gt;, without introducing any subtractions. This means that, like Average Cost, it approaches &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$T \to 0^+$&lt;/span&gt;. This is often a key conceptual point in distinguishing between Average Cost and Marginal Cost.&lt;/li&gt;&#xA;&lt;li&gt;There is no calculus required to understand and/or find the solution. You do not have to understand or consider instantaneous velocity. Again, this mirrors Average Cost which students usually understand quicker precisely because it doesn't really require any calculus knowledge. (In fact, all the extra calculus surrounding the topic of Average Cost usually works against them as they try to shoe-horn it into some sort of Calculus idea.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="401" LastActivityDate="2021-11-30T19:40:18.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24646" PostTypeId="1" CreationDate="2021-12-01T11:01:00.663" Score="2" ViewCount="193" Body="&lt;p&gt;There is a distinction between substantive and syntactic knowledge particularly in mathematics. There are several definitions for both knowledge. However, what are the indicators and the examples of substantive knowledge and syntactic knowledge such that we can explain the constructs operationally?&lt;/p&gt;&#xA;" OwnerUserId="18153" LastActivityDate="2021-12-01T11:01:00.663" Title="What are differences and the examples of syntactic knowlede and substantive knowledge in mathematics?" Tags="|education-research|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24647" PostTypeId="1" CreationDate="2021-12-01T18:53:34.743" Score="6" ViewCount="878" Body="&lt;p&gt;I have a new math pupil. She is an emotional 12 year old girl who starts to cry when she doesn't understand something.&lt;/p&gt;&#xA;&lt;p&gt;Her father said she is sensitive. I was asked to do more praising and encouraging from her parents.&lt;/p&gt;&#xA;&lt;p&gt;I am looking for ways how to provide a better service.&lt;/p&gt;&#xA;&lt;p&gt;Have you dealt with a similar situation before? What is the best thing to do?&lt;/p&gt;&#xA;" OwnerUserId="18154" LastEditorUserId="143" LastEditDate="2021-12-26T13:48:17.800" LastActivityDate="2021-12-26T13:48:17.800" Title="Advice on how to tutor a twelve year old that starts to cry?" Tags="|tutoring|struggling-students|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24648" PostTypeId="2" ParentId="24647" CreationDate="2021-12-02T06:41:53.860" Score="5" Body="&lt;p&gt;Her parents say she is sensitive and needs praise and encouragement, but you say she has gaps in her knowledge.&lt;/p&gt;&#xA;&lt;p&gt;Some suggestions:&lt;/p&gt;&#xA;&lt;p&gt;Assess what she knows and doesn't know. Give her an oral written assessment of the skills she needs. Make sure she understands that you need to know what to teach her.&lt;/p&gt;&#xA;&lt;p&gt;Make the parents your partners. Tell them you are happy to encourage and praise her, but she also has to learn the material, which is hard because she has gaps.  Tell them about your plan to assess her and ask them to encourage her to approach the assessment without worry so that you can best figure out how to teach her.  Ask them to encourage her to be comfortable saying I don't know and also to be comfortable when making mistakes.&lt;/p&gt;&#xA;&lt;p&gt;Give her opportunities to succeed. This will be easier to do once you have assessed her. Ask her easy questions that she can answer.&lt;/p&gt;&#xA;&lt;p&gt;Praise and encouragement should be for effort not for success.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2021-12-23T08:45:04.973" LastActivityDate="2021-12-23T08:45:04.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24649" PostTypeId="2" ParentId="24647" CreationDate="2021-12-02T10:48:34.780" Score="9" Body="&lt;h1&gt;Empty praise is no good&lt;/h1&gt;&#xA;&lt;p&gt;Do not give praise when it is not deserved. It is not healthy and does not lead to stronger character or anything else good.&lt;/p&gt;&#xA;&lt;h1&gt;Tasks with no wrong answer&lt;/h1&gt;&#xA;&lt;p&gt;Maybe, and I am guessing here, the pupil reacts badly to being shown inadequate. Maybe mathematics has always been a subject full of right/wrong -type exercises: you either get it or you do not, and it is right or wrong.&lt;/p&gt;&#xA;&lt;p&gt;How about trying tasks that are not of this type? There are many sources for these types of questions, but one typical thing is to ask the pupil to make something. Maybe you give an answer and them to create an exercise. Or maybe it is an open exercise with free parameters the pupil can themselves choose.&lt;/p&gt;&#xA;&lt;p&gt;The point here is to reduce the emotional intensity and the matter of winning and losing, showing you are smart or stupid.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2021-12-02T10:48:34.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24651" PostTypeId="1" CreationDate="2021-12-03T04:39:23.340" Score="6" ViewCount="341" Body="&lt;p&gt;I teach vector calculus. I love both Math3D and Geogebra. But I have reached a limit in terms of what these programs can do. Some examples of features that I wish Math3D had:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Draw vector fields with color or line weight representing magnitude, rather than length.&lt;/li&gt;&#xA;&lt;li&gt;Draw a slice of a vector field with vectors anchored just along a surface.&lt;/li&gt;&#xA;&lt;li&gt;Copy/paste objects with the same settings (e.g. making a new vector with all the same formatting of a vector I'm already using.)&lt;/li&gt;&#xA;&lt;li&gt;More robust operations (e.g. curl, div, etc.)&lt;/li&gt;&#xA;&lt;li&gt;Parameterize piecewise curves (or logic like &amp;quot;show point P if parameter T &amp;gt; 1, else suppress,&amp;quot; etc.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I could keep going! So I am just looking to upgrade to the next level of software sophistication. I've looked around some, but the lists of visualization software I've found are overwhelming and it's hard to know where to start. Any recommendations?&lt;/p&gt;&#xA;" OwnerUserId="18159" LastActivityDate="2021-12-07T00:03:37.020" Title="More advanced (free) alternatives to Geogebra and Math3D?" Tags="|calculus|vector-calculus|graphing|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24652" PostTypeId="1" CreationDate="2021-12-03T19:40:32.583" Score="8" ViewCount="332" Body="&lt;p&gt;Base 10 sets are readily available from educational suppliers and retail sites, as are the largest element in these sets, sometimes known as &amp;quot;decimeter cubes&amp;quot; (10 x 10 x 10 cm).&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;unit cubes&amp;quot; (1 x 1 x 1 cm) in such sets are the basis for the larger cousins, including &amp;quot;rods&amp;quot; and &amp;quot;flats.&amp;quot; (See &lt;a href=&quot;http://mathsmaterials.com/base-ten-blocks/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for more info.)&lt;/p&gt;&#xA;&lt;p&gt;The unit cubes that I use, made by SI Manufacturing, weigh exactly one gram each so they can be utilized for a range of STEM activities. I think this kind of cube is fairly common.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;However, I can't locate any decimeter cubes that have a mass of one kilogram, i.e., the equivalent of 1,000 my unit cubes. The models that I can find seemed to be designed to be lighter weight, presumably to reduce costs. Has anyone heard of a 1 kg decimeter cube? Can you recommend a source?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vPOS3.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vPOS3.jpg&quot; alt=&quot;Image of Base-Ten Set&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13229" LastActivityDate="2023-02-22T20:02:55.310" Title="Is there such thing as a 1 kg &quot;decimeter cube?&quot;" Tags="|manipulatives|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="24653" PostTypeId="1" CreationDate="2021-12-04T21:50:56.400" Score="2" ViewCount="200" Body="&lt;p&gt;Any references for core textbooks aimed at English language learners at the university level?&lt;/p&gt;&#xA;&lt;p&gt;I have need of them for:&#xA;Calculus I-II-III,&#xA;Linear Algebra, and&#xA;Differential Equations.&lt;/p&gt;&#xA;&lt;p&gt;Most of my students are in a unique situation where the only language they can sort of read is English, but not well enough for the standard textbooks to be accessible to them. Currently, I am essentially wasting my students' money since their books are unreadable to them.&lt;/p&gt;&#xA;" OwnerUserId="18165" LastActivityDate="2021-12-07T14:47:16.020" Title="ELL textbooks for core university math?" Tags="|undergraduate-education|textbooks|reference-books|english-language-learners|" AnswerCount="2" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="24654" PostTypeId="2" ParentId="24653" CreationDate="2021-12-05T19:57:24.343" Score="2" Body="&lt;p&gt;You've said that your students are at a &amp;quot;university&amp;quot; (i.e., not a community college), and have clarified that they're in the US. Your characterization is that &amp;quot;most of&amp;quot; them find a standard textbook not &amp;quot;accessible,&amp;quot; or &amp;quot;unreadable.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;It's unclear what you mean when you say that you want to cater to &amp;quot;English language learners&amp;quot; but  &amp;quot;most of the students are Americans.&amp;quot; ELL is a term used to mean people whose native language is not English. Given your student population, it sounds like probably the majority of them are &lt;em&gt;not&lt;/em&gt; people whose native language is not English.&lt;/p&gt;&#xA;&lt;p&gt;Terms like &amp;quot;unreadable&amp;quot; are absolute, whereas reading comprehension is relative.&lt;/p&gt;&#xA;&lt;p&gt;You cannot guarantee your students the ability to benefit from a college education. All you can guarantee them is college-level instruction that is appropriate for students who have the preparation and motivation to succeed in college-level work. You can also, if you choose to do so, guarantee that none of them will be limited to a second-class education by the structure of your class, or held back from improving their academic skills and critical thinking ability because the class is taught at an inappropriate pre-college level.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Currently, I am essentially wasting my students' money since their books are unreadable to them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If money is the issue, then choose an OER text. There are many such texts to choose from at the lower-division level, especially for freshman calc.&lt;/p&gt;&#xA;" OwnerUserId="18168" LastActivityDate="2021-12-05T19:57:24.343" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24655" PostTypeId="2" ParentId="24653" CreationDate="2021-12-06T14:52:31.593" Score="2" Body="&lt;p&gt;I think it's broadly unlikely that there are any calculus-track textbooks aimed at English Language Learners. But that's just an educated guess.&lt;/p&gt;&#xA;&lt;p&gt;If you use OER textbooks, then you won't be wasting any of your students' money. I recommend the ones from &lt;a href=&quot;https://openstax.org/subjects/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax&lt;/a&gt;, which in my opinion are fairly high quality. That includes the full calculus sequence, but not linear algebra or differential equations.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2021-12-07T14:47:16.020" LastActivityDate="2021-12-07T14:47:16.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24656" PostTypeId="1" AcceptedAnswerId="24667" CreationDate="2021-12-06T15:44:56.810" Score="6" ViewCount="3235" Body="&lt;p&gt;I am tutoring a student who doesn't really understand how multiplication works or what distributivity is, I want to explain him how distributivity of multiplication over addition on an abstract level works like:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ (a+b)(c+d) = a(c+d) + b(c+d) = ac+ad+bc+bd$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The student seems to struggle in understanding on an intuitive level why the expanding of brackets work in multiplication. Would there be a way to explain the above sequence of manipulation without using any reference to area formulas?&lt;/p&gt;&#xA;&lt;p&gt;Please keep it as simple as possible.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorUserId="14627" LastEditDate="2021-12-07T02:57:28.920" LastActivityDate="2021-12-13T10:42:06.573" Title="How do I explain the distributivity of multiplication to a student without using the analogy of areas?" Tags="|multiplication-table|" AnswerCount="10" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="24657" PostTypeId="2" ParentId="24656" CreationDate="2021-12-06T16:45:09.033" Score="4" Body="&lt;p&gt;If you want to avoid area, then the next closest thing is arrays.&lt;/p&gt;&#xA;&lt;p&gt;See:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.khanacademy.org/math/7th-grade-foundations-engageny/7th-m2-engage-ny-foundations/7th-m2-tc-foundations/a/distributive-intro&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.khanacademy.org/math/7th-grade-foundations-engageny/7th-m2-engage-ny-foundations/7th-m2-tc-foundations/a/distributive-intro&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://makemathmoments.com/progression-of-multiplication/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://makemathmoments.com/progression-of-multiplication/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If arrays are still too close to area, then you'll have to think of another &amp;quot;good&amp;quot; way to visualize multiplication in general. If it truly is a good way, then the distributive property should come naturally within it.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2021-12-06T16:45:09.033" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24658" PostTypeId="2" ParentId="24656" CreationDate="2021-12-06T17:20:17.347" Score="1" Body="&lt;p&gt;Despite your requirement, I recommend using array and area models.&lt;/p&gt;&#xA;&lt;p&gt;I suggest engaging them in tasks such as the following:&lt;/p&gt;&#xA;&lt;p&gt;Write as many algebraic expressions as you can (using integers and the 4 operations) to count the number of x's in the following diagram:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;xxx&#xA;xxxxx&#xA;xxxxx&#xA;   xx&#xA;   xx&#xA;   xx&#xA;xxxxx&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Possible answers include &lt;span class=&quot;math-container&quot;&gt;$3\cdot 4+2 \cdot 6$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$7\cdot 5 - 2 - 3 \cdot 3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Also engage them in the other direction:&lt;/p&gt;&#xA;&lt;p&gt;Can you draw a picture of x's which has &lt;span class=&quot;math-container&quot;&gt;$5\cdot (2 + 4) - 3\cdot 3$&lt;/span&gt; dots in it?&lt;/p&gt;&#xA;&lt;p&gt;Work your way up to expressions like the one you are struggling with.  Be sure to pair the geometric work with symbolic translations as much as possible.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-12-06T17:20:17.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24659" PostTypeId="2" ParentId="24604" CreationDate="2021-12-06T17:59:22.150" Score="2" Body="&lt;p&gt;Upon the request of the OP I'll elaborate on my comment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If the AP and other test makers were doing their job the test would be vastly different from year to year precisely to discourage teaching to the test.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have a variety of different data which informs my disdain for the AP test as it is currently practiced as well as the &amp;quot;SOL's&amp;quot; (Standard Of Learning) tests which shape elementary math teaching in Virginia (where I used to live).&lt;/p&gt;&#xA;&lt;p&gt;I'll begin with the SOL's. I had a neighbor who taught math to middle school kids, he described how he taught them &amp;quot;tricks&amp;quot; to get the problem right on the SOL. From what I heard, the focus of the learning was on the tricks to help pass the SOLs rather than on building a unified mathematical framework infused with both concept and calculation.&lt;/p&gt;&#xA;&lt;p&gt;With the AP test, I think they're allowed a graphing calculator now ? Yep:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://apstudents.collegeboard.org/exam-policies-guidelines/calculator-policies&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://apstudents.collegeboard.org/exam-policies-guidelines/calculator-policies&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, that's not a good sign. I can't see a reason you would need a graphing calculator for a well-designed calculus test. Unless you're testing ability to use a graphing calculator (why on earth encourage that ? there are far better alternatives online and... again, really inappropriate to a calculus placement test). So, I don't have a lot of firsthand testimony from students who took AP calculus and described their course, but I know that as a department we saw reason to be less and less trusting of anything but a 4 or 5 on the exam. At that point, it's more an indication of intelligence and study habit than actual knowledge of calculus.&lt;/p&gt;&#xA;&lt;p&gt;Why ? Because the test is largely unchanged, so if you study the standard problem types you'll be able to make a good score if you are intelligent. Your inability to add fractions etc. can even be hidden by the ever present calculator.&lt;/p&gt;&#xA;&lt;p&gt;Or, if you're really clever and a cheat, you can program the answers of the test into your graphing calculator after a scout student takes the test and publishes the questions and answers on the appropriate &amp;quot;help&amp;quot; site. Here I conjecture, but anyone who has experience with the Chegg homework feedback loop knows my speculation is not an idle one.&lt;/p&gt;&#xA;&lt;p&gt;Beyond the specific situations I list above, I think the general response we receive in teaching students speaks to the abuse of math in schools. I have had countless students (good students) come to me mid-semester the first time they have my course and say something along the lines of &amp;quot;your teaching is so different&amp;quot;. Why ? Well, I think there are a couple things I do which are never going to be high on the priority list of a teacher who teaches to the test.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;emphasis is on definitions&lt;/li&gt;&#xA;&lt;li&gt;proofs of theorems which involve the methods of the course are given&lt;/li&gt;&#xA;&lt;li&gt;compare and contrast definitions&lt;/li&gt;&#xA;&lt;li&gt;ask inverse questions&lt;/li&gt;&#xA;&lt;li&gt;contrast graphical verses analytical methods&lt;/li&gt;&#xA;&lt;li&gt;warn about pathological cases&lt;/li&gt;&#xA;&lt;li&gt;discuss historical evolution of math concepts&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In contrast, a person who teaches to the test has two main questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;which questions are allowed on the test ?&lt;/li&gt;&#xA;&lt;li&gt;how can we reliably do the questions as quick as possible ?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In the above mode of teaching, definitions may not even be stated, theorems are seldom proved and the bulk of time is spent solving problems. I am guilty of such teaching in my absolute lowest level teaching, but to teach calculus this way is to rob students of beauty.&lt;/p&gt;&#xA;&lt;p&gt;This danger is not isolated to calculus. In fact, any suitably standardized test or curriculum has the danger of squelching creativity and the best teaching for the sake of the pragmatism of passing the test.&lt;/p&gt;&#xA;&lt;p&gt;But, the existence of a standard test is not the automatic death of teaching. For example, my wife took advanced math in highschool in Hong Kong. They have to take a test which is quite serious just to get into the math track if I understand correctly. That test didn't discourage good teaching, rather it promoted it because the test was a bit unpredictable and their were many different kinds of problems you might face. That test was administered in such a way that the teachers who taught the students were not the teachers who proctored the test (a much more serious business than what we find at American institutions).&lt;/p&gt;&#xA;&lt;p&gt;I'm torn on the merit of standardized testing. In principle, I see great merit in replacing course credits with passing certification tests because many students are trapped in universities whose standards are so low that professors are held hostage by the weakest students.( I can envision replacing a math degree with holding certifications in different subjects. That would free students from the burden of liberal arts dogma) In practice, we cannot teach the course as it ought to be because, well, the students. The existence of an external test which created without direct regard to the weakness of students at a particular institution gives serious students a way to prove their skill. That said, such a test once it becomes too familiar after years of creation and the perhaps inevitable recycling of questions, becomes gamified and discourages good teaching of the topic.&lt;/p&gt;&#xA;&lt;p&gt;This is why I made my comment, in my view the creators of standardized tests must engage in a certain randomness to fight against the gamification.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-12-06T17:59:22.150" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24660" PostTypeId="1" CreationDate="2021-12-06T18:38:53.217" Score="1" ViewCount="779" Body="&lt;p&gt;Pro's and cons of number line model vs color counter model&lt;/p&gt;&#xA;&lt;p&gt;When teaching multiplication to elementary schoolers, the &amp;quot;number line model&amp;quot; and &amp;quot;color counter model&amp;quot; are both widely used techniques. Can somebody help me to understand some of the pro's and cons for either model? Thank you&lt;/p&gt;&#xA;" OwnerUserId="16437" LastActivityDate="2021-12-08T18:27:46.163" Title="Pro's and cons of number line model vs color counter model" Tags="|primary-education|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24661" PostTypeId="2" ParentId="24660" CreationDate="2021-12-06T20:43:03.267" Score="2" Body="&lt;p&gt;As someone who never heard of the color counter model, I find it overly complicated.&lt;/p&gt;&#xA;&lt;p&gt;I assume that the question is asked in the context of signs when multiplying integers. While both methods arrive at the correct results, I can't see any advantages of the color counter approach.&lt;/p&gt;&#xA;&lt;p&gt;I'd rather fear that this method hides the fact - and more importantly, the reason - why products of integers with equal sign are positive.&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2021-12-08T18:27:46.163" LastActivityDate="2021-12-08T18:27:46.163" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24663" PostTypeId="2" ParentId="24656" CreationDate="2021-12-06T23:24:19.820" Score="17" Body="&lt;p&gt;This is based on a framing device I used in a Khan Academy comment helping students get an intuitive appreciation of the distributive rule.  It is just a description of &lt;span class=&quot;math-container&quot;&gt;$a(b+c)=ab+ac$&lt;/span&gt;, but it might be enough to jump start a student's intuition.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Every day at work, I eat a lunch of two sandwiches and three cookies.  I like to be organized by packing five lunch boxes on the weekend for the upcoming work week.  That way, I can just grab a lunch box out of my refrigerator each day on the way to work.  At the beginning of the work week, how many sandwiches and cookies do I have prepared?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Students with a solid understanding of multiplication word problems don't seem to have trouble correctly calculating that there are ten sandwiches and fifteen cookies.  When prompted, they get that they can treat counting the sandwiches and counting the cookies as two separate problems.&lt;/p&gt;&#xA;&lt;p&gt;All that remains is showing how algebra just gives us a slightly different notation that allows us to describe the situation in a single expression.  In this example, &lt;span class=&quot;math-container&quot;&gt;$2S+3C$&lt;/span&gt; is an expression that models the contents of a lunch box (which can't be simplified because sandwiches and cookies are not &amp;quot;like terms&amp;quot;) and 5(2S+3C) is an expression that describes the contents of five lunch boxes.  Once you understand that, the distributive rule just verifies the students' own calculation that &lt;span class=&quot;math-container&quot;&gt;$$5(2S+3C)=(5\cdot2S)+(5\cdot3C)=10S+15C$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Keep in mind that I came up with this inquiry for an online community.  If I had the opportunity to sit at a table with an individual student, using manipulatives would be much more powerful.  For instance, draw four circles on a piece of paper and put three pennies and two paper clips on each circle.  Then sweep all the pennies and paper clips into a single pile and ask how many of each there are.  Again, the key is helping students understand that they already knew the answer.  The thing that makes the distributive property cool is not that it is a new fact, but rather that algebraic notation lets us talk about subconsciously familiar numeric relationships that were previously much harder to express.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2021-12-08T02:03:10.570" LastActivityDate="2021-12-08T02:03:10.570" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24664" PostTypeId="2" ParentId="24651" CreationDate="2021-12-07T00:03:37.020" Score="3" Body="&lt;p&gt;If you have familiarity with Python programming, you might want to consider the Manim library. It is frequently updated and, looking at your list of requirements, I believe Manim will check all of the boxes.&lt;/p&gt;&#xA;&lt;p&gt;There are a lot of resources for learning and working with Manim; particularly on YouTube.&lt;/p&gt;&#xA;" OwnerUserId="625" LastActivityDate="2021-12-07T00:03:37.020" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24665" PostTypeId="2" ParentId="15371" CreationDate="2021-12-07T05:05:15.147" Score="2" Body="&lt;p&gt;You may use the division symbol. This &lt;strong&gt;division by a base&lt;/strong&gt; has an important meaning: &lt;strong&gt;the duality&lt;/strong&gt;. This notation has a lot of benefits, including reducing the amount of computations and explicitly showing the logic of linear algebra.&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$\alpha = (\alpha_1 ,\ \alpha_2 ,\ \cdots ,\ \alpha_n)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\beta = ( \beta_1 ,\ \beta_2,\ \cdots ,\ \beta_n)$&lt;/span&gt; are bases, which will be regarded formally as rows, then the matrix representing the change-of-base from &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; is denoted by &lt;span class=&quot;math-container&quot;&gt;$$\frac \alpha \beta \ \ \ \ \text{or}\ \ \ \ \beta \backslash \alpha \in K^{n\times n},$$&lt;/span&gt;&#xA;(but &lt;strong&gt;not&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$\alpha / \beta$&lt;/span&gt;) where &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is the base field. As I mentioned below, This notation is still valid even if &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is not a basis, but just an ordered tuple of vectors. In this case &lt;span class=&quot;math-container&quot;&gt;$\beta\backslash \alpha$&lt;/span&gt; may not be a change-of-base matrix, though.&lt;/p&gt;&#xA;&lt;p&gt;This notation has some benefits: for example we may write &lt;span class=&quot;math-container&quot;&gt;$v = \alpha x$&lt;/span&gt; to say that &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; is the linear combination &lt;span class=&quot;math-container&quot;&gt;$x_1 \alpha_1 + x_2 \alpha_2 + \cdots + x_n \alpha_n$&lt;/span&gt; of the base &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; by the coefficient (column) vector &lt;span class=&quot;math-container&quot;&gt;$x = [x_1 \ x_2 \ \cdots \ x_n]^T$&lt;/span&gt;. Then it is natural to use symbols like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x = \frac v \alpha \ \ \ \ \text{or} \ \ \ \ \alpha \backslash v. $$&lt;/span&gt;&#xA;Now suppose &lt;span class=&quot;math-container&quot;&gt;$y = [y_1 \ \cdots \ y_n]^T$&lt;/span&gt; is the coordinate column vector of &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; w.r.t. the base &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;. Then now we can write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ y = \frac \alpha \beta x, \ \ \ \ \text{that is,} \ \ \ \ \frac v \beta = \frac \alpha \beta \frac v \alpha \ \ \ \ \text{or} \ \ \ \ \beta \backslash v = (\beta \backslash \alpha)(\alpha \backslash v). $$&lt;/span&gt;&#xA;Be careful: perform cancellations only along the line &lt;span class=&quot;math-container&quot;&gt;$\backslash$&lt;/span&gt;, not &lt;span class=&quot;math-container&quot;&gt;$/$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The above notaions &lt;span class=&quot;math-container&quot;&gt;$\beta \backslash \alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\beta \backslash v$&lt;/span&gt; are compatible in the sense that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac \alpha \beta = \frac {(\alpha_1,\ \cdots,\ \alpha_n)} \beta = \left[ \frac {\alpha_1} \beta \ \frac {\alpha_2} \beta \  \cdots \  \frac {\alpha_n} \beta \right]. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We may define several similar symbols to make things involving change-of-base easy. Let &lt;span class=&quot;math-container&quot;&gt;$T: V \to V$&lt;/span&gt; be a linear map. Denote by &lt;span class=&quot;math-container&quot;&gt;$T\alpha$&lt;/span&gt; the ordered tuple &lt;span class=&quot;math-container&quot;&gt;$(T\alpha_1 ,\ \cdots, \ T\alpha_n) \in V^n$&lt;/span&gt; of vectors. Then the matrix representation of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\alpha\backslash T\alpha$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;With some more work, one may easily verify the formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac {T\beta} \beta = \frac \alpha \beta \frac {T\alpha} \alpha \frac \beta \alpha, $$&lt;/span&gt;&#xA;which means that the matrix of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; w.r.t. &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; is the conjugate of  the matrix of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; w.r.t &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\beta\backslash\alpha$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Even more benefit:&lt;/strong&gt; If the space &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is just &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt;, then the tuples &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; are merely the matrices, and the three notations below denotes completely the same thing:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac \alpha \beta = \beta \backslash \alpha = \beta ^{-1} \alpha, $$&lt;/span&gt;&#xA;and this is the reason why I recommended you not to use the symbol &lt;span class=&quot;math-container&quot;&gt;$\alpha / \beta$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Again more benefit:&lt;/strong&gt;  We may denote the coordinate map &lt;span class=&quot;math-container&quot;&gt;$V \to \mathbb R^n$&lt;/span&gt; w.r.t the base &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; by just&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac 1 \alpha : v \mapsto \frac v \alpha, $$&lt;/span&gt;&#xA;and its inverse by just&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \alpha: x = [x_1 \ \cdots \ x_n]^T \mapsto \alpha x = \sum x_i \alpha_i. $$&lt;/span&gt;&#xA;In this viewpoint we find the relations:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \alpha \frac 1 \alpha \ \ \ \text{ equals the identity map } V \to V, $$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac 1 \alpha \alpha \ \ \ \text{ equals the identity matrix } I. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that the notations in this answer enjoys the associative law.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Here is another benefit.&lt;/strong&gt; For a base &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, we may promise to denote the dual base &lt;span class=&quot;math-container&quot;&gt;$(\alpha_i^*)$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$V^*$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\alpha \backslash 1$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac 1 \alpha := [ \alpha_1^* ,\ \alpha_2^* ,\ \cdots ,\ \alpha_n^* ]^T, $$&lt;/span&gt;&#xA;where the dual base is regarded as a column. This makes no conflict with `Again more benefit'. And our first notation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac \beta \alpha $$&lt;/span&gt;&#xA;now may be regarded as a multiplication of column and a row, producing a square matrix:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac \beta \alpha = \frac 1 \alpha \beta = \begin{bmatrix} \alpha_1^* \\ \alpha_2^* \\ \vdots \\ \alpha_n^* \end{bmatrix} [ \beta_1 \ \ \beta_2 \ \ \cdots \ \ \beta_n ], $$&lt;/span&gt;&#xA;which is exactly equal to the change-of-base matrix. This is the genuine meaning of our &lt;strong&gt;division by a basis&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So let's take a look at an example. How this notation may be used? Let me prove a famous theorem in linear algebra:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; Let &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; be a fin dim vector space over a field &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$V^*$&lt;/span&gt; be the dual space of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$V^{**}$&lt;/span&gt; the double dual. Then there is a linear isomorphism of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; onto &lt;span class=&quot;math-container&quot;&gt;$V^{**}$&lt;/span&gt; that is independent of the choice of base of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof.&lt;/strong&gt; Let &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; be the dimension of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$\alpha = (\alpha_1 ,\ \cdots,\ \alpha_n) \in V^n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\beta \in V^n$&lt;/span&gt; be bases of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, which are regarded as a row form. Then they induces the dual bases &lt;span class=&quot;math-container&quot;&gt;$\alpha^* = \alpha \backslash 1 = [\alpha_1^* ,\ \cdots,\ \alpha_n^*]^T$&lt;/span&gt; and  &lt;span class=&quot;math-container&quot;&gt;$\beta^* = \beta \backslash 1$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$V^*$&lt;/span&gt;, which are regarded as columns. We use a lemma:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Lemma.&lt;/strong&gt; The change-of-base matrix&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac {\alpha^*} {\beta^*} $$&lt;/span&gt;&#xA;from &lt;span class=&quot;math-container&quot;&gt;$\alpha^*$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\beta^*$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$\alpha \backslash \beta$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac 1 \alpha = \frac \beta \alpha \frac 1 \beta. $$&lt;/span&gt;&#xA;&lt;strong&gt;Remark:&lt;/strong&gt; It is well-known that the resulting matrix must be the &lt;em&gt;transposed inverse&lt;/em&gt;, but the reason why we got just the inverse is that we are regarding the dual bases as columns, not rows. So the lemma does not defy your background knowledge; if you regard the dual bases as rows, then you'll get the familiar &lt;em&gt;transposed inverse&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So let's continue to working with the theorem. If we apply the lemma twice, then we know that the changeofbase matrix from &lt;span class=&quot;math-container&quot;&gt;$\alpha^{**}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\beta^{**}$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$\beta \backslash \alpha$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now define a linear isomorphism &lt;span class=&quot;math-container&quot;&gt;$T:V \to V^{**}$&lt;/span&gt; by the linear extension of the assignment &lt;span class=&quot;math-container&quot;&gt;$\alpha_i \mapsto \alpha_i^{**}$&lt;/span&gt;. We want to show that &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is independent of choice of the base; that is, &lt;span class=&quot;math-container&quot;&gt;$T\beta_i = \beta_i^{**}$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;. This is equivalent to showing that the matrix representation &lt;span class=&quot;math-container&quot;&gt;$\beta^{**} \backslash T\beta$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; equals the identity matrix. This is easily done by our notations:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac {T\beta} {\beta^{**}} = \frac{\alpha^{**}}{\beta^{**}} \frac {T\alpha}{\alpha^{**}} \frac {\beta} {\alpha} = \frac \alpha \beta \cdot I \cdot \frac \beta \alpha = I. $$&lt;/span&gt;&#xA;This completes the proof. //&lt;/p&gt;&#xA;&lt;p&gt;So our division notation allows us not to do enormous scalar computations. This is &lt;strong&gt;the other great benefit&lt;/strong&gt;, I think.&lt;/p&gt;&#xA;" OwnerUserId="18173" LastEditorUserId="18173" LastEditDate="2021-12-10T12:09:27.070" LastActivityDate="2021-12-10T12:09:27.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24666" PostTypeId="2" ParentId="24656" CreationDate="2021-12-07T06:11:44.753" Score="4" Body="&lt;p&gt;Reteach multiplication of multi-digit numbers (eg &lt;span class=&quot;math-container&quot;&gt;$12 \times 34$&lt;/span&gt;) by pen and paper in terms of &lt;span class=&quot;math-container&quot;&gt;$(10+2)(30+4)$&lt;/span&gt;. You should reconcile their early memories of arithmetics with this somewhat new concept of distribution.&lt;/p&gt;&#xA;&lt;p&gt;Then redo the same problem but break the factors into different numbers, eg &lt;span class=&quot;math-container&quot;&gt;$(10+2)(30+4) = (5+7)(28+6)$&lt;/span&gt; to allow them to see it has nothing to do with the number 10 or decimal representations.&lt;/p&gt;&#xA;" OwnerUserId="18050" LastEditorUserId="18050" LastEditDate="2021-12-10T02:52:05.903" LastActivityDate="2021-12-10T02:52:05.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24667" PostTypeId="2" ParentId="24656" CreationDate="2021-12-07T11:50:56.927" Score="17" Body="&lt;p&gt;It depends on how the student conceptualises multiplication. First, &lt;em&gt;find that out&lt;/em&gt;. &lt;em&gt;Then&lt;/em&gt; design an approach that uses it.&lt;/p&gt;&#xA;&lt;p&gt;For example, if they think of multiplication as repeated addition, then you might try examples like:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3\times(4+5)=(4+5)+(4+5)+(4+5)=(4+4+4)+(5+5+5)=(3\times 4)+(3\times 5)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This approach gets messy trying to multiply out more than one bracket, so has to be taken slowly. Always do the brackets one at a time with this approach.&lt;/p&gt;&#xA;&lt;p&gt;If they think of it as counting objects arranged in a rectangle, set out some objects on the table, or draw some pictures. How many red diamonds are there  here? How does this picture represent the statement: &lt;span class=&quot;math-container&quot;&gt;$3\times(4+5)=3\times 4+3\times 5$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3pb4S.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3pb4S.png&quot; alt=&quot;Rectangle of squares&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Counting objects arranged in a rectangle is &lt;em&gt;visually&lt;/em&gt; intuitive, which helps a lot, and makes an easy starting point to move on to interpreting it as an area. I personally think it's an easier approach than rearranging symbols as above. But the important thing is to &lt;em&gt;find out where they are now&lt;/em&gt;, and then chart a course from there to where you want to go.&lt;/p&gt;&#xA;&lt;p&gt;Also, start with concrete examples. Then illustrate the concept with pictures and diagrams. Finally, present the abstract/symbolic, once the basic intuition is there for it to connect to. &amp;quot;Multiplying out brackets&amp;quot; is the symbolic/abstract end stage. If they have been only shown the symbolic manipulation, that would explain why they have no intuitive grasp of why following the procedure works, and you won't get far without going back and introducing the supporting concepts.&lt;/p&gt;&#xA;&lt;p&gt;If they haven't come across the area analogy, then I agree it is going to make it more difficult to introduce it right in the middle of explaining multiplying out brackets. But this ought to be telling you that you're trying to do things too fast, and out of order. &lt;em&gt;Don't even think&lt;/em&gt; about teaching 'multiplying out brackets' until you have first taught 'multiplying', with an intuitive, visual analogy like areas or counting in rectangles or fixed-size boxes. (Like 'Widgets come in boxes of three. If we have four boxes and add five more boxes, how many widgets?') Then when you teach each abstract method, you have a visual/intuitive picture to link it back to.&lt;/p&gt;&#xA;&lt;p&gt;If you want to use several such analogies/pictures, then introduce them &lt;em&gt;all&lt;/em&gt; up front when teaching what multiplication is. Then later, when you run into difficulties with one approach, you have alternatives to fall back on, and you don't have to be introducing new topics in the middle of an explanation.&lt;/p&gt;&#xA;" OwnerUserId="18174" LastActivityDate="2021-12-07T11:50:56.927" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="24668" PostTypeId="2" ParentId="24656" CreationDate="2021-12-07T12:31:32.420" Score="1" Body="&lt;p&gt;My highschool teacher  would say&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you have two times three apples and five plums, you have six apples and ten plums. That is,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2(3a+5p)=2\times3\,a+2\times5\,p&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="475" LastActivityDate="2021-12-07T12:31:32.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24670" PostTypeId="2" ParentId="24656" CreationDate="2021-12-08T01:23:50.680" Score="0" Body="&lt;p&gt;2 apples + 3 apples = 5 apples&lt;/p&gt;&#xA;&lt;p&gt;Why?&lt;/p&gt;&#xA;&lt;p&gt;2 apples + 3 apples =  (2+3) apples = 5 apples&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;=&amp;quot; on the left is distributivity.&lt;/p&gt;&#xA;" OwnerUserId="11741" LastActivityDate="2021-12-08T01:23:50.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24671" PostTypeId="2" ParentId="24660" CreationDate="2021-12-08T01:53:07.283" Score="4" Body="&lt;p&gt;The color counter model is as follows:  model an integer as a collection of positive and negative particles. I will use an ordered pair &lt;span class=&quot;math-container&quot;&gt;$(P,N)$&lt;/span&gt; to denote the number of positive and negative chips present.&lt;/p&gt;&#xA;&lt;p&gt;The interesting thing here is that different pairs can represent the same integer. &lt;span class=&quot;math-container&quot;&gt;$(3,0) = (4,1) = (5,2)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(0,2) = (1,3) = (2,4)$&lt;/span&gt; for instance. &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; is equivalent to &lt;span class=&quot;math-container&quot;&gt;$(c,d)$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$a+d = b+c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is actually the official &lt;em&gt;definition&lt;/em&gt; of the integers when you build them up from scratch.  As such it is logically prior to the number line from a rigorous development.  It also rhymes with the definition of equivalent fraction:  in the foundations of mathematics, we actually &lt;em&gt;define&lt;/em&gt; the rational numbers as equivalence classes of pairs of integers (ordinate not being equal to zero) with &lt;span class=&quot;math-container&quot;&gt;$(a,b) ~ (c,d)$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$a*d=c*d$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is all far too abstract for school children, but they can learn to &amp;quot;cancel&amp;quot; pos/neg pairs and &amp;quot;create&amp;quot; pos/neg pairs when it suits them.  This is valuable work, and can support learning the rules for symbolic manipulation of signed numbers.&lt;/p&gt;&#xA;&lt;p&gt;I think that both models are valuable.  Negative numbers are strange beasts, and it takes a lot of work to understand their internal consistency, how their behavior is the &amp;quot;only sensible&amp;quot; way to make the properties of arithmetic work out, how they can sensibly model real world phenomena (temperature, debt, etc).&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2021-12-08T01:53:07.283" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24672" PostTypeId="2" ParentId="24656" CreationDate="2021-12-08T06:36:25.793" Score="0" Body="&lt;p&gt;Assuming you get them through the first part, about how a(b+c) distributes, the second part is just distribution again. It's just symbol manipulation. Think of (a+b), for now, as a single value. Then (a+b)(c+d) gives (a+b)c + (a+b)d by simple distribution.&lt;/p&gt;&#xA;&lt;p&gt;I feel like many students learn FOIL so they can go straight from (a+b)(c+d) to the full ac+ad+bc+bd expansion. But FOIL is just a derived shortcut. You're showing the actual process, where (a+b)(b+c) is distribution twice.&lt;/p&gt;&#xA;&lt;p&gt;I think NulliusInVerba really has it for the first part. Arrays are merely a visual way to show &amp;quot;multiplication is a shortcut for repeated addition&amp;quot;. When we sketch 3x6 as a 3 by 6 rectangle, that's just a pretty way to say &amp;quot;3x6 means 6+6+6, which we can think of as 3 strips of 6's, side-by-side&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;With &amp;quot;repeated addition&amp;quot;, x(y+z) is defined as (y+z)+(y+z) ... x times. Commutativity and association then let us regroup them as x y's and x z's. Reversing the rule (multiple additions can be rewritten as multiplication&amp;quot; gives us the distributed xy+xz.&lt;/p&gt;&#xA;" OwnerUserId="13766" LastEditorUserId="13766" LastEditDate="2021-12-08T15:28:35.033" LastActivityDate="2021-12-08T15:28:35.033" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24673" PostTypeId="2" ParentId="24656" CreationDate="2021-12-08T09:00:35.493" Score="1" Body="&lt;p&gt;I recommend &lt;a href=&quot;https://en.wikipedia.org/wiki/Cuisenaire_rods&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cuisenair Rods&lt;/a&gt;. While one could argue that their use essentially breaks down to area it's very intuitive and accessible. As always, a physical involvement like manually handling the rods aids &amp;quot;grasping&amp;quot; an issue.&lt;/p&gt;&#xA;&lt;p&gt;If you arrange, say, 6 rows of 5+2 rods you can split them in a pile of 2-rods and a pile of 5-rods before adding the rows, or leave them connected and count the entire rows (whose length is 5+2 each); there is no easier explanation than that.&lt;/p&gt;&#xA;" OwnerUserId="11894" LastActivityDate="2021-12-08T09:00:35.493" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24674" PostTypeId="2" ParentId="24416" CreationDate="2021-12-09T07:24:01.273" Score="5" Body="&lt;p&gt;There are no common textbook(s) widely used across schools.&lt;/p&gt;&#xA;&lt;p&gt;Instead, each individual school/teacher will usually have their own notes (that they guard jealously but are usually leaked out).&lt;/p&gt;&#xA;&lt;p&gt;Check out this &amp;quot;Holy Grail&amp;quot; &lt;a href=&quot;https://drive.google.com/drive/folders/1gC6GQLgcuoHzwDXtEGzTwvzCz0YsYuwg&quot; rel=&quot;noreferrer&quot;&gt;collection&lt;/a&gt; of resources (mostly exam papers but also has some notes).&lt;/p&gt;&#xA;" OwnerUserId="18188" LastActivityDate="2021-12-09T07:24:01.273" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24675" PostTypeId="1" CreationDate="2021-12-09T23:37:14.030" Score="5" ViewCount="265" Body="&lt;p&gt;A good model for algebra are Algebra Tiles.&lt;/p&gt;&#xA;&lt;p&gt;See: &lt;a href=&quot;https://calculate.org.au/wp-content/uploads/sites/15/2019/03/lesson-sequence-for-algebra-tiles.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://calculate.org.au/wp-content/uploads/sites/15/2019/03/lesson-sequence-for-algebra-tiles.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this presentation they recommend an aspect ratio of 3:14 for the &amp;quot;x&amp;quot; tile. This implicitly defines x = 4.666, though, of course x is a variable not a particular number! &lt;strong&gt;Why do they recommend this?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Another choice would be 1:10, though this could confuse algebra with place value.&lt;/p&gt;&#xA;&lt;p&gt;Other numbers/ratios could be 2, 1.5, 2, 3 or even &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="998" LastActivityDate="2021-12-11T14:59:39.573" Title="What's the best ratio for algebra tiles?" Tags="|algebra|manipulatives|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24676" PostTypeId="2" ParentId="24675" CreationDate="2021-12-10T10:46:54.330" Score="2" Body="&lt;p&gt;The makers of algebra tiles don't want students concluding that x = 2 or 3.  To avoid that kinds of confusion is the reason that they don't have an integer ratio such as 1:2 or 1:3.&lt;/p&gt;&#xA;&lt;p&gt;Futhermore when modeling a quadratic, the student will naturally, line up the tiles so that the x edges touch.  Imagine if the ratio was 1:3 and the student had 3x, the student could create an alternate model and leap to incorrect conclusions.&lt;/p&gt;&#xA;&lt;p&gt;I don't know what advantages you see of having a ratio of &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2021-12-10T10:46:54.330" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24677" PostTypeId="2" ParentId="24675" CreationDate="2021-12-10T12:45:35.100" Score="8" Body="&lt;p&gt;They must have a mathematician on their team!  There is a very good reason to recommend &lt;span class=&quot;math-container&quot;&gt;$x=4.666$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Consider a given length &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;.  We want to compare to &lt;span class=&quot;math-container&quot;&gt;$ax_0 $&lt;/span&gt;  to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$a,b \in \{1,2,3,...,10\}$&lt;/span&gt;.  The minimum difference &lt;span class=&quot;math-container&quot;&gt;$\textrm{min}(|ax_0 -b|)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$a,b \in \{1,2,3,..., 10\}$&lt;/span&gt; will represent the &amp;quot;closest&amp;quot; one can make some number &lt;span class=&quot;math-container&quot;&gt;$ax_0$&lt;/span&gt; to the number &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; using some number of &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; tiles and unit tiles (restricting yourself to between 1 and 10 of each type of tile).&lt;/p&gt;&#xA;&lt;p&gt;Here is a graph of &lt;span class=&quot;math-container&quot;&gt;$f(x) = \textrm{min}(|ax -b|)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$a,b \in \{1,2,3,..., 10\}$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UoSnH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UoSnH.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/aazy484fzp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Desmos link&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As you can see, &lt;span class=&quot;math-container&quot;&gt;$4.\bar{6}$&lt;/span&gt; is one local maximum of this function, with an output of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt;.  This means that if you select a tile length of &lt;span class=&quot;math-container&quot;&gt;$x=4.\bar{6}$&lt;/span&gt;, the closest a student will be able to get any number of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; inch tiles and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; inch tiles is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; of an inch, which should be highly visible!&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-12-11T14:59:39.573" LastActivityDate="2021-12-11T14:59:39.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24678" PostTypeId="2" ParentId="24416" CreationDate="2021-12-10T16:55:39.840" Score="3" Body="&lt;p&gt;Anonymous' answer is excellent and comprehensive. To add on to that, as a Singaporean, I can share my experience with the education system here.&lt;/p&gt;&#xA;&lt;p&gt;For ages 13-16 (secondary school) different teachers tend to have their own teaching materials, whereas for ages 17-18 (junior college) notes tend to be issued on a per-school basis, referred to as &amp;quot;lecture notes&amp;quot;. To my knowledge, there is no standard textbook widely used in the country for any secondary school / junior college level math education.&lt;/p&gt;&#xA;&lt;p&gt;However, it should also be noted that Singapore's education system has different &amp;quot;tracks&amp;quot; for students of varying academic levels / interests, most notably for ages 13-16 there are the Normal (Technical), Normal (Academic), and Express tracks, whereas for ages 17-18 there are Junior Colleges and then there are Polytechnics. Details may vary widely based on the different academic tracks, or based on the school, or even based on the individual teacher. It may be the case that a particular teacher chooses to work with a certain textbook for example, though most in my experience do not.&lt;/p&gt;&#xA;&lt;p&gt;To answer your underlying question of what is taught in Singapore's math education system, I suggest you to Google &amp;quot;Singapore O Level math syllabus&amp;quot; and &amp;quot;Singapore A Level math syllabus&amp;quot;. (Edit: I realised these were already mentioned in Anonymous' answer.) O Levels are a national exam generally taken at the end of the year for 16 year olds, and the A levels are generally taken for 18 year olds. The syllabus documents are made publicly available by the national exam board, the SEAB, online. All teachers in Singapore teach based on these exam syllabi and many have an exam-centric approach, so this is a very good guide to what is actually taught in classrooms.&lt;/p&gt;&#xA;&lt;p&gt;If you have further questions, feel free to ask in the comments.&lt;/p&gt;&#xA;" OwnerUserId="10206" LastEditorUserId="10206" LastEditDate="2021-12-10T17:03:23.613" LastActivityDate="2021-12-10T17:03:23.613" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24679" PostTypeId="1" AcceptedAnswerId="24686" CreationDate="2021-12-11T12:19:17.500" Score="24" ViewCount="4074" Body="&lt;p&gt;I tutor a young girl aged 11 (grade 4).&lt;/p&gt;&#xA;&lt;p&gt;She is doing OK for her age, but I have observed that she has a tendency for rigid ways of thinking. She is usually more inclined to follow rules and stick to known ways of doing things which I think limits her learning.&lt;/p&gt;&#xA;&lt;p&gt;Examples:&lt;/p&gt;&#xA;&lt;p&gt;Calculating 2×7×3×5: she understands that she can multiply 2 and 5 first to get 10, and easily get the answer. However, when faced with similar tasks, she still prefers going from left to right and &amp;quot;does not like&amp;quot; to consider if there might be an easier way.&lt;/p&gt;&#xA;&lt;p&gt;Or, consider the following exercise:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let's add 3400 and 5700. Adam adds 34 hundred and 57 hundred and gets 9100. Ben adds 3400 and 5000 to get 8400, and adding the remaining 700 he gets 9100 as well. John adds 3400 and 700 to get 4100, then adds 5000 and also gets 9100. Explain how each of them did it. How would you do it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When faced with such tasks, she always &amp;quot;likes&amp;quot; the way to split the numbers because she thinks working with hundreds is &amp;quot;strange&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I think it is normal to understand one way easier than others. But I observed that she has a tendency to put up resistance when I try to give her exercises that are more easily solved in the ways that challenges the &amp;quot;order&amp;quot; that she thinks things should be done. I think this limits her learning, because eg. to understand why certain procedures work, or to be able to check results etc. all depend on understanding things from different angles.&lt;/p&gt;&#xA;&lt;p&gt;How can I encourage her to think in ways that she might at first find &amp;quot;strange&amp;quot; or seemingly defying &amp;quot;order&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="14091" LastEditorUserId="14091" LastEditDate="2021-12-12T20:20:30.890" LastActivityDate="2021-12-19T20:06:56.870" Title="How to encourage young student to think in unusual ways?" Tags="|primary-education|" AnswerCount="10" CommentCount="20" ContentLicense="CC BY-SA 4.0" />
  <row Id="24680" PostTypeId="2" ParentId="24679" CreationDate="2021-12-11T13:21:43.047" Score="1" Body="&lt;p&gt;games and puzzles?&#xA;logic games? math based games?&lt;/p&gt;&#xA;&lt;p&gt;could you model using anchor numbers (sums of 5 and 10s?)&#xA;de/recomposing numbers into place value?&lt;/p&gt;&#xA;&lt;p&gt;multiply in this manner?&#xA;add in this manner?&lt;/p&gt;&#xA;&lt;p&gt;when doing multiplication reason (double or triple or half)&lt;/p&gt;&#xA;&lt;p&gt;i wonder how much she thinks of math as rigid rules versus conceptual / relational (how math / numbers are related to each other?)&lt;/p&gt;&#xA;&lt;p&gt;even empty number lines and showing jumps (again, decomposing numbers by groups of 10)&lt;/p&gt;&#xA;&lt;p&gt;MODEL this way of thinking?&lt;/p&gt;&#xA;" OwnerUserId="18197" LastActivityDate="2021-12-11T13:21:43.047" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24682" PostTypeId="2" ParentId="24679" CreationDate="2021-12-11T14:02:14.343" Score="5" Body="&lt;p&gt;My feeling here is that she might not be much interested in math, so simply doesn't enjoy looking for alternate ways of solving problems.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand - at least according to your examples - there is not much benefit of finding alternate ways.&lt;/p&gt;&#xA;&lt;p&gt;I'd look for problems being hard solved the traditional way and very easy after some reorganisation.&lt;/p&gt;&#xA;&lt;p&gt;E.g.&lt;br /&gt;&#xA;5 / 9 * 4 * 9&lt;br /&gt;&#xA;37 * 67 + 63 * 67 {= 100 * 67}&lt;/p&gt;&#xA;&lt;p&gt;Maybe also give limited time to even more highlight difference.&lt;/p&gt;&#xA;&lt;p&gt;Also, what helped me to look for alternative ways is trying calculate in my head as there I really felt forced to look for the easiest way.&lt;/p&gt;&#xA;&lt;p&gt;Once she experience benefit of finding alternative ways, she probably will try to apply it also on her own.&lt;/p&gt;&#xA;" OwnerUserId="14808" LastEditorUserId="14808" LastEditDate="2021-12-11T16:37:16.910" LastActivityDate="2021-12-11T16:37:16.910" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24683" PostTypeId="2" ParentId="12118" CreationDate="2021-12-11T17:00:19.307" Score="0" Body="&lt;p&gt;First of all, I would never put up with such rudeness from anyone, let alone a student. I would certainly speak to the student about that (politely, of course!), and then if that fails, take it up with the administration or whoever is in charge of your tutoring department.&lt;/p&gt;&#xA;&lt;p&gt;Second, WRONG on you to give the student the answer! I understand the frustration, but giving him the answer means he was able to bully you into something that is not appropriate. We have a tutoring office in my college, and I make it explicit to tutors to NOT give the answers to the students. They can tell them about their steps (not GIVE steps, but confirm if the steps are right, or guide them through steps, but never feed the answers or the methods). Also, you are taking away time from other students who have an equal right to be there and get through their lessons/work.  That's not fair at all.&lt;/p&gt;&#xA;&lt;p&gt;If the student is indeed autistic as alleged, or has some kind of IEP, then you need your college to give you extra support on this if you are not a trained professional in this matter. For example, I am not (and have no desire to be) a trained professional in special ed matters. So, when I have had students in my class who need an IEP or extra help, the college assigns an assistant for this matter, or gives me the resources I need for that particular student.&lt;/p&gt;&#xA;&lt;p&gt;You mention the student has a medical bracelet, as well as hygiene issues. Again, that definitely needs to be communicated to the administration to see how they can help you. I once had a female student with very poor personal hygiene, and it would have been very uncomfortable for me (a male) to address this alone with her. Instead, I had her go see the Dean of students (another female) and they addressed this issue. Other students were complaining about her BO as well. Thankfully, the issue was resolved efficiently.&lt;/p&gt;&#xA;" OwnerUserId="15988" LastActivityDate="2021-12-11T17:00:19.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24684" PostTypeId="2" ParentId="24679" CreationDate="2021-12-12T12:44:53.430" Score="15" Body="&lt;h3&gt;Caveat&lt;/h3&gt;&#xA;&lt;p&gt;Having spent a decade in education, and many of those years working with a body of students only 10% were at or above grade level (and I'm pretty sure that the metric didn't indicate any of the latter), I've seen the sort of rigidity you mention. Anyone who thinks education isn't primarily an exercise in psychology hasn't spent much time in the classroom. Given the poor way math is taught in the US, math is more psychological than most. The challenge to inculcate creative thinking in math is perhaps &lt;em&gt;the&lt;/em&gt; challenge, &lt;a href=&quot;https://www.mathvalues.org/masterblog/2019/1/26/what-is-mathematical-creativity-how-do-we-develop-it-and-should-we-try-to-measure-it-part-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;so make sure you understand it yourself&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;h3&gt;Why Math Students Are Rigid&lt;/h3&gt;&#xA;&lt;p&gt;In mathematics education, there has been a tension between fundamentalists (let's stick to the arithmetic/geometric fundamentals which generally means lots of rote memorization) and math education philosophies that look to enrich a child's conceptual understanding, like &lt;a href=&quot;https://en.wikipedia.org/wiki/New_Math&quot; rel=&quot;nofollow noreferrer&quot;&gt;New Math&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Common_Core_State_Standards_Initiative&quot; rel=&quot;nofollow noreferrer&quot;&gt;Common Core&lt;/a&gt;. I've found students often have psychological preferences much like the former, &amp;quot;fundamentalists&amp;quot; and &amp;quot;imaginative&amp;quot; thinkers. A &amp;quot;fundamentalist&amp;quot; tends to believe there's one way to do things, and that getting correct answers is important, whereas &amp;quot;imaginative&amp;quot; students are creative in their approaches (I had a secondary student essentially reinvent math pictographs) to do problems. The latter group of students likes to get right answers, but tends to prefer finding a way that makes sense to them.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/National_Council_of_Teachers_of_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;NCTM&lt;/a&gt; has criticized (rightfully IMNSHO) math education as a being a mile wide and an inch deep. Given the sequential nature of math, I think the rigid thinking of children (barring the occurrence in psychological disorders like &lt;a href=&quot;https://en.wikipedia.org/wiki/Obsessive%E2%80%93compulsive_personality_disorder&quot; rel=&quot;nofollow noreferrer&quot;&gt;OCPD&lt;/a&gt;) often hinges upon an emotional experience of math that might be described as a brutal series of rejections for students who show little talent for it or are deprived of a quality year of education, often at no fault of the student. A bad 3rd-grade math teacher may result in a struggle in math for the next 9 years, and for a child sensitive to failure, that's a lot of rejection. Standardized tests tend to exacerbate that rejection.&lt;/p&gt;&#xA;&lt;h3&gt;Changing Math Thinking&lt;/h3&gt;&#xA;&lt;p&gt;When I earned my math degree, I saw that the attitudes about math didn't change much from K to graduate classes. There's an absolute lack of wonder about mathematics. It's tough to bring about wonder, and some students just aren't gifted in math and fewer want to learn. Gauss on math education:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I have a true aversion to teaching. The perennial business of a professor of mathematics is only to teach the ABC of his science; most of the few pupils who go a step further, and usually to keep the metaphor, remain in the process of gathering information, become only Halbwisser [one who has superficial knowledge of the subject], for the rarer talents do not want to have themselves educated by lecture courses, but train themselves. And with this thankless work the professor loses his precious time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Depending on the age of the student, &lt;strong&gt;you need to get into the student's head&lt;/strong&gt;. With a senior in high school, you can just have the conversation about how why they do what they do, how they feel when they get wrong answers, etc. On the other hand, with a younger student, you have to be a little more indirect and infer.  For most students who are rigid in their approach, it's simply a combination of poor math skills and a dislike-to-fear of failure. A rigid student is just clinging to an algorithm to avoid appearing completely ignorant, more often than not, and sadly, they've often been rewarded for it in the past. That's why so many people hate the Common Core: it doesn't focus on answers so much as reasoning, and well, critical thinking is hard work.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Once you have a sense of why it's so important for a student to stick to an algorithm, then, if you have good rapport with a student, &lt;strong&gt;persuade them to broaden their thinking&lt;/strong&gt;. Math understanding comes from having a dialog, even with yourself when working on a problem, and students are conditioned to execute an &lt;a href=&quot;https://en.wikipedia.org/wiki/Algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;algorithm&lt;/a&gt; than applying a series of &lt;a href=&quot;https://en.wikipedia.org/wiki/Heuristic&quot; rel=&quot;nofollow noreferrer&quot;&gt;heuristics&lt;/a&gt;. But, one thing that has been made clear is that mathematical problem solving involves &lt;a href=&quot;https://en.wikipedia.org/wiki/Defeasible_reasoning&quot; rel=&quot;nofollow noreferrer&quot;&gt;defeasible reasoning&lt;/a&gt;. Math solutions are very much based on intuitions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To develop a student's inner dialog to engage in problem-solving, it often helps to &lt;strong&gt;teach problem-solving strategies&lt;/strong&gt;. &lt;a href=&quot;https://www.google.com/books/edition/How_to_Solve_It/X3xsgXjTGgoC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;George Polya's work on the matter&lt;/a&gt; is a good start. Some students (those with ADHD or impoverished grammars and vocabularies) struggle to solve math because they have a hard time reasoning through it. But I suspect, since the matter is an empirical question, that most students simply haven't had practice at it. When I taught, in my math department, of the 25 teachers, only 2 of us actually had math degrees, and new teachers often had little insight of the material. I suspect that's the norm in the US. You have to make use of &lt;a href=&quot;https://en.wikipedia.org/wiki/Teachable_moment&quot; rel=&quot;nofollow noreferrer&quot;&gt;teachable moments&lt;/a&gt; and role model math problem-solving.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;As you may have heard, &lt;a href=&quot;https://en.wikipedia.org/wiki/Royal_Road#A_metaphorical_%22Royal_Road%22_in_famous_quotations&quot; rel=&quot;nofollow noreferrer&quot;&gt;there is no royal road to geometry&lt;/a&gt;, but there are things you can do:&lt;/p&gt;&#xA;&lt;p&gt;I had three practices in the classroom to encourage a child to understand that making mistakes in math is the norm (not the exception), and my students would complain. First, I used to make my students &lt;strong&gt;use pens instead of pencils&lt;/strong&gt; because and they would protest that I'd see all of their mistakes; second, I would &lt;strong&gt;give vocabulary and writing assignments&lt;/strong&gt; and assessments devoid of calculation. Third, I would &lt;strong&gt;give problems AND the answer and ask for work and rationale&lt;/strong&gt;. You would think I was pulling teeth without analgesics. The usual rejoinder was &amp;quot;this isn't how you do math!&amp;quot; I ultimately had to &lt;a href=&quot;https://en.wikipedia.org/wiki/Flipped_classroom&quot; rel=&quot;nofollow noreferrer&quot;&gt;flip the classroom&lt;/a&gt;, and I encountered this strategy in the university as well.&lt;/p&gt;&#xA;&lt;p&gt;More directly,&lt;/p&gt;&#xA;&lt;p&gt;I'd say a good first start is to redesign &lt;strong&gt;your lesson and materials around the answer of getting from the givens to the conclusion&lt;/strong&gt;. For instance, if your student is struggling with associativity in addition, give worksheets that ask for two ways to go from 3 + 6 + 9 to the answer 18. This way you're focusing on the path instead of the destination. &lt;strong&gt;Use graphic organizers&lt;/strong&gt;. At the elementary level, if you have steps, you almost invariably have &lt;strong&gt;arrows and boxes&lt;/strong&gt; meant to suggest what gets combined. Don't do a lot of problems, &lt;strong&gt;do a few problems repeatedly and well to encourage understanding&lt;/strong&gt;. Teach &lt;strong&gt;non-standard algorithms&lt;/strong&gt;. I can add three-digit numbers in my head, but that's because I've had a lot of practice with non-standard methods. &lt;strong&gt;Use lots of vocabulary&lt;/strong&gt; including Q&amp;amp;A; when my child started addition, we talked about addends, adders, sums, and doubles as well as ten frames, groups, addition, combination, etc. Most of my at-risk high schoolers knew less math vocabulary than my 1st-grade daughter. Multiplication can be talked about as a product of factors, or more simply &lt;em&gt;m&lt;/em&gt; groups of &lt;em&gt;n&lt;/em&gt;. So, 3 x 2 = 6 is also three groups of two things is six things that are more concrete. Mathematics is terse, and it helps if you develop semantics to accompany that syntax.&lt;/p&gt;&#xA;&lt;p&gt;I could go on, but there are lots of tricks to the trade, and most of them are specific to the type of math involved. Make problems concrete. For commutativity or transformations, I would talk about putting on socks and shoes (non-commutative). Methods like FOIL for products of binomials, mnemonic devices for memorization, games, etc. Teaching math is no different than repairing cars in that anyone can do it, but few can do it well. Ultimately, keep on developing your skills is the best long-term strategy. Being a math teacher is a lot like being a wizard, and you have to have a spell book of materials and techniques.&lt;/p&gt;&#xA;" OwnerUserId="18201" LastEditorUserId="18201" LastEditDate="2021-12-19T20:06:56.870" LastActivityDate="2021-12-19T20:06:56.870" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24685" PostTypeId="2" ParentId="24679" CreationDate="2021-12-12T14:39:55.410" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Let's add 3400 and 5700. Adam adds 34 hundred and 57 hundred and gets 9100. Ben adds 3400 and 5000 to get 8400, and adding the remaining 700 he gets 9100 as well. John adds 3400 and 700 to get 4100, then adds 5000 and also gets 9100. Explain how each of them did it. How would you do it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When faced with such tasks at age 9, I would be bored out of my wits. As others have suggested, the problem may simply be that what you are giving her is just downright uninteresting. &lt;strong&gt;Why&lt;/strong&gt; should I care what is the fastest way of adding ten numbers? If I can't be made to care, can a child?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How can I encourage her to think in ways that first seem &amp;quot;strange&amp;quot; or seemingly defying &amp;quot;order&amp;quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Frankly, this is likely to be an unproductive way of teaching. Mathematics is &lt;strong&gt;not&lt;/strong&gt; about trying to find strange or disordered ways of solving problems! Rather, it is about appreciating and understanding the underlying &lt;strong&gt;beauty&lt;/strong&gt; of mathematics. Have you shown her anything in mathematics that &lt;em&gt;she&lt;/em&gt; finds beautiful? If not, then maybe you should reconsider what your intended goal is.&lt;/p&gt;&#xA;&lt;p&gt;Note that even if your goal is for students to solve arithmetic problems fast, you should not forbid your students from using any methods that they like, and instead reward the speed of solving. More generally, mathematics olympiad problems at the appropriate level (e.g. &lt;a href=&quot;https://artofproblemsolving.com/wiki/index.php/AMC_8_Problems_and_Solutions&quot; rel=&quot;noreferrer&quot;&gt;AMC8&lt;/a&gt;) are a good source of questions that encourage thinking &lt;strong&gt;without forcing&lt;/strong&gt; the student to use any particular method. All that matters is how many questions they can solve within the time limit!&lt;/p&gt;&#xA;&lt;p&gt;But if your goal is to convey the beauty of mathematics, then you need to actually demonstrate that. Ad-hoc tricks to speed up arithmetic calculations are only very slightly interesting, unless you can show &lt;strong&gt;significant&lt;/strong&gt; benefit or beauty in the 'non-standard' methods. For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(1+2+3+\cdots+99) + (99+98+97+\cdots+1) = 100×99$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac1{1×2}+\frac1{2×3}+\frac1{3×4}+\cdots\frac1{9×10} = (\frac11-\frac12)+(\frac12-\frac13)+(\frac13-\frac14)+\cdots+(\frac19-\frac1{10})$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$S = \frac13+\frac1{3^2}+\frac1{3^3}+\cdots+\frac1{3^9}$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$S×3 = 1+\frac13+\frac1{3^2}+\cdots+\frac1{3^8} = 1+S-\frac1{3^9}$&lt;/span&gt;. Thus ...&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;These are the &lt;em&gt;real tricks&lt;/em&gt; that students should be shown. Paltry tricks like &lt;span class=&quot;math-container&quot;&gt;$2×7×3×5 = 21×10$&lt;/span&gt; are not convincingly a big deal, unless your goal is merely to be able to finish a list of arithmetic exercises faster than her.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-12-12T14:39:55.410" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="24686" PostTypeId="2" ParentId="24679" CreationDate="2021-12-12T15:19:27.217" Score="25" Body="&lt;p&gt;A lot of the answers here are suggesting giving examples where the alternatives you're interested in are &lt;em&gt;necessary&lt;/em&gt; (or at least wildly easier than the &amp;quot;rigid&amp;quot; approach). That's a good option, but in my experience - depending on the student - it might have the opposite of the desired effect, because these examples will always feel &amp;quot;forced&amp;quot;. She may feel that the examples you're giving are examples cooked up specifically to poke at her weaknesses, which can cause a student to retreat.&lt;/p&gt;&#xA;&lt;p&gt;As an alternative suggestion: when I teach creative mathematical thinking, I present it as a &lt;strong&gt;freedom&lt;/strong&gt;, not a &lt;strong&gt;task&lt;/strong&gt;. The fact that we have many ways to solve the same problem means that we have the freedom to choose which approach to use; this is a good thing! Importantly, though, in order to get a student to a point where they actually do think of it as a freedom, I have to really give them the freedom to choose - and they may choose the &amp;quot;rigid&amp;quot; option. My priority, then, is to make sure that they're making an &lt;strong&gt;informed choice&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, for example, looking at the problem &lt;span class=&quot;math-container&quot;&gt;$2 \times 7 \times 3 \times 5$&lt;/span&gt;, I might start by asking the student to write down an approach - not an answer, just an approach, like &amp;quot;multiply 2 by 7, then by 3, then by 5&amp;quot;. Then ask her to write &lt;em&gt;another&lt;/em&gt; approach, and then to decide which approach she wants to use and &lt;em&gt;why&lt;/em&gt;. If the answer is &amp;quot;I want to use the 'rigid' approach because I like straightforward things better&amp;quot;, then fine - the goal is to instill the habit of considering her options, not to get her to use the option I think is easiest. When I've used this strategy with students, they tend to gradually get better at spotting alternatives and more likely to spot those alternatives without prompting; and this then makes those alternatives feel &amp;quot;easier&amp;quot; to them, because they aren't so hard to think about.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Relatedly, I've also had some success simply changing my language around students. It's natural to ask &amp;quot;how do we solve this problem&amp;quot; or &amp;quot;how do we start this problem&amp;quot;; but that language implicitly suggests that there's only one correct way. I tend to phrase it as &amp;quot;how can we solve this problem&amp;quot; or &amp;quot;how can we start this problem&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2021-12-12T15:19:27.217" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24687" PostTypeId="2" ParentId="24679" CreationDate="2021-12-12T22:18:36.677" Score="3" Body="&lt;p&gt;In a comment I wrote &amp;quot;Perhaps her need for autonomy is part of this.&amp;quot; I was asked to elaborate in an answer. I'll try.&lt;/p&gt;&#xA;&lt;p&gt;People learn what they want to learn. Real motivation cannot come from the outside. (It has been shown that extrinsic motivation doesn't last and boomerangs.) If a student has little interest in math, then a tutor can make their time together more enjoyable (or at least less painful), but cannot make the student care about mathematics.&lt;/p&gt;&#xA;&lt;p&gt;If we look at how young children learn before they encounter schooling, they pick what they learn, and throw themselves into it with gusto. Learning to walk and talk takes amazing dedication, and they all do it. There are not kids who just didn't learn that.&lt;/p&gt;&#xA;&lt;p&gt;Schooling teaches kids to work by the clock and to obey authority far more effectively than it teaches content matter.&lt;/p&gt;&#xA;&lt;p&gt;So a student comes to a tutor, most likely because their parents required it. The student is not going to be happy about losing some of the little bit of free time they have. They are not going to come into it feeling competent. And they are not going to have burning questions about the mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I have tutored students like this. I would say I had almost no success in 2 of the 3 most recent cases. In the 3rd, I asked the student if there was anything he liked that was at all related to math. (Not sure now how I framed that question.) He liked logic puzzles. So that was a big part of how we spent our time. And I &amp;quot;taught&amp;quot; very little there. If I had &amp;quot;taught&amp;quot; more, I would have stolen his sense of competence, and possibly his love of logic puzzles. I did tell him that logic was my first love in math, and that I had studied logic in grad school.&lt;/p&gt;&#xA;&lt;p&gt;Humans have deep needs. After food, water, and shelter, one of our greatest needs is autonomy. Another is learning/discovery/making sense. I believe that autonomy is required for deep learning to take place.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-12-12T22:18:36.677" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24688" PostTypeId="2" ParentId="24679" CreationDate="2021-12-12T22:28:24.250" Score="2" Body="&lt;h1&gt;Geometry&lt;/h1&gt;&#xA;&lt;p&gt;The more different ways one has of looking at a problem, the better the data for training the internal model. Instead of giving a symbolic multiplication problem, try a geometric one.  Get some small blocks, form a rectangular solid with it, and ask the student how many blocks are in the solid.  Rearrange the solid in different ways so that the &amp;quot;easy&amp;quot; multiplications occur on different axes, and ask the student each time why they chose the strategy they did.&lt;/p&gt;&#xA;&lt;p&gt;After she has solved the blocks problem, write down the exact same problem on paper.  Show multiple permutations of each problem, so she can see how they are equivalent, but that the &amp;quot;best&amp;quot; strategy that she chose depended not on following a prescribed rigid rule, but rather by making a value judgment about what might be the path of least resistance (the &amp;quot;easiest&amp;quot; route).  Hopefully, she will recognize that the &amp;quot;rule&amp;quot; she ought to follow is one that she discovers herself, and that the rules she is given can be improved upon by looking at the problems differently.&lt;/p&gt;&#xA;&lt;h1&gt;Fermi Estimation&lt;/h1&gt;&#xA;&lt;p&gt;This might seem a premature for an 11 year old, but I think not.  Learning that math can be done approximately and still be useful should be a universal lesson that all humans learn explicitly (especially since most of us use this fact implicitly anyway).  Get several jars and marbles (or beads, or whatever has a consistent size).  Put different numbers in them, and ask her how many marbles are in each one.  Then, collect a few jars and ask her how many are in both together.&lt;/p&gt;&#xA;&lt;p&gt;She will realize that she can guess there are 50 marbles in one jar, but she won't really know if there are 55 or 60 or 77 marbles in it.  And when she goes to add the jars together, she will be forced to give up on precision in the lower digits and just do the math on the higher ones.  She won't spell this out explicitly (unless she displays a sudden jump in precocity), but you can explain it to her.&lt;/p&gt;&#xA;&lt;p&gt;If she insists on trying to make precise estimates, put time bounds on the problem.  Tell her you are going to show her jars, and she has to add them in 5 seconds or whatever puts her on the edge of her calculating/estimating abilities.  This will force her to estimate to only the highest digit of precision.  After the 5 seconds are up, cover the jars with a towel so she can't see them any more.&lt;/p&gt;&#xA;&lt;p&gt;Then write down the same marble problems on paper, with the real numbers, and show her how close she got by using different strategies.&lt;/p&gt;&#xA;&lt;h1&gt;Toys&lt;/h1&gt;&#xA;&lt;p&gt;If you know something about the toys the student plays with, you can connect your math lessons to her world even more directly.  For instance, LEGO bricks are perfect for forming geometric solids.  Most likely, if she plays with them at all, she will already have some intuition for how to do math with them.  If she does any kind of crafts involving beads, buttons, sticker sheets, etc., you can use these objects as the things-to-be-counted.  She probably already does some basic arithmetic on them, and if she is faster at solving problems using these objects, then you can ask what rules/strategies she is using to show that she has already invented her own rules which might be different from the textbook strategies.&lt;/p&gt;&#xA;&lt;p&gt;If she doesn't play with LEGOs, you can introduce them as a good exercise for learning math while also being a fun way to relax or exercise creativity.&lt;/p&gt;&#xA;" OwnerUserId="13468" LastActivityDate="2021-12-12T22:28:24.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24689" PostTypeId="2" ParentId="24679" CreationDate="2021-12-13T08:40:40.843" Score="1" Body="&lt;p&gt;I have often observed my daughter doing the same thing.&lt;/p&gt;&#xA;&lt;p&gt;I sometimes play a game with her, where I will give her a problem like yours 2×7×3×5 (albeit with addition because she is young). Then I ask her how fast we can solve a bunch of these problems. Gameifying it works for her.&lt;/p&gt;&#xA;&lt;p&gt;A bunch or all of the problems are just permutations on the original. This reinforces the idea with her that she has already solved the subproblem. Maybe such a strategy will help you.&lt;/p&gt;&#xA;" OwnerUserId="12100" LastActivityDate="2021-12-13T08:40:40.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24690" PostTypeId="2" ParentId="24656" CreationDate="2021-12-13T09:38:38.050" Score="1" Body="&lt;p&gt;If you do not want to use area, here is another way:&lt;/p&gt;&#xA;&lt;p&gt;You have 9 bracelets. Each bracelet has 7 beads on it. Now you put 4 bracelets on your left hand and 5 on your right.&lt;/p&gt;&#xA;&lt;p&gt;How many beads you have in total? &lt;br&gt;&#xA;How many beads you have on your left hand? How many on your right?&lt;/p&gt;&#xA;&lt;p&gt;Points of discussion:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;With any number of bracelets, you can count beads by removing them from the bracelets one by one. On or off the bracelets, you have the same number of beads.&lt;/li&gt;&#xA;&lt;li&gt;To get the total in both hands, you can count the beads on the left and the right and then add, or, take all bracelets down and count. Either way the number of the beads is the same.&lt;/li&gt;&#xA;&lt;li&gt;If you have some number of a thing in one hand and some number in another, then you have the sum of counts of the thing in both hands.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The likely reaction to this, once they get it, is &amp;quot;meh, of course&amp;quot; because it is so trivial, that it does not even seem to be something of use.&lt;/p&gt;&#xA;&lt;p&gt;So the main task is to connect the idea to the symbolic identity &lt;code&gt;a(b+c)=ab+ac&lt;/code&gt;, so that they understand that it serves as a model for the reasoning above.&lt;/p&gt;&#xA;&lt;p&gt;When both factors are sums, of course demonstrating with areas is much simpler and usually more satisfying. But if they understand the above you can also convince them, that the result is the same if the distributive property is applied twice.&lt;/p&gt;&#xA;" OwnerUserId="14091" LastEditorUserId="14091" LastEditDate="2021-12-13T10:42:06.573" LastActivityDate="2021-12-13T10:42:06.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24691" PostTypeId="2" ParentId="24679" CreationDate="2021-12-13T17:22:04.177" Score="3" Body="&lt;p&gt;(Disclaimer: I'm not an educator, though I did study maths to university level. But this draws on a memory from the pupil's point of view, so I hope it may still be helpful.)&lt;/p&gt;&#xA;&lt;p&gt;When I was in junior school, aged around 9 or 10, my teacher gave me an open-ended project to look at over the holiday.&lt;/p&gt;&#xA;&lt;p&gt;She drew a very simple grid of numbers:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;1  2  3  4  5  6  …&#xA;2  3  4  5  6  7  …&#xA;3  4  5  6  7  8  …&#xA;4  5  6  7  8  9  …&#xA;5  6  7  8  9 10  …&#xA;…  …  …  …  …  …&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;And she indicated a square of numbers within the grid. I can't remember which one; say for example it was:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    3  4&#xA;    4  5&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;She pointed out that if you add up the top-left and bottom-right numbers (3 + 5) you get the same result as if you add up the top-right and bottom-left numbers (4 + 4).&lt;/p&gt;&#xA;&lt;p&gt;And I think she told me to investigate that — no further direction or clarification; she just left me to look at that over the holiday and see what I could come up with.&lt;/p&gt;&#xA;&lt;p&gt;So I went on to prove that the same is true for every possible square of 4 adjacent numbers, anywhere on the grid. And that it's true even if the numbers aren't adjacent, as long as they form the corners of any square, or any rectangle. I also found and proved a result or two from multiplying the numbers instead of adding. (I've forgotten how much more I did; this was a long time ago!)&lt;/p&gt;&#xA;&lt;p&gt;I found this exercise really interesting at that age because it was open-ended. The algebra is of course trivial, but I was probably fairly new to the idea of proving general results, and of investigating and discovering for myself — so this was far more about imagination and curiosity than anything else I'd done before, and far less about trying to identify the solutions that the teacher expected.&lt;/p&gt;&#xA;&lt;p&gt;Maybe a similarly open-ended task, pregnant with interesting results to be discovered, might help students like that in the question look at something mathematical without any rules to follow or previous examples to imitate — and without any pressure to find predetermined ‘right answers’?&lt;/p&gt;&#xA;" OwnerUserId="12762" LastActivityDate="2021-12-13T17:22:04.177" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24692" PostTypeId="1" CreationDate="2021-12-13T21:29:54.217" Score="0" ViewCount="77" Body="&lt;p&gt;I am interested in code that generates an urn with various colored balls for an economics experiment. I came across this thread: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14302/urn-containing-colored-balls-generator&quot;&gt;Urn (containing colored balls) generator?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, the server that the code was on has been taken down, and I can't seem to find a way to use this site to contact the user that originally coded it.&lt;/p&gt;&#xA;&lt;p&gt;Are there ways to either (a) contact the user who wrote the code or (b) see if code like this has been done in the past?&lt;/p&gt;&#xA;&lt;p&gt;Thanks,&#xA;Keaton&lt;/p&gt;&#xA;" OwnerUserId="18209" LastActivityDate="2021-12-13T21:29:54.217" Title="Code for Urn (containing colored balls) generator?" Tags="|technology-in-education|online-instruction|probability|" AnswerCount="0" CommentCount="2" ClosedDate="2021-12-20T22:43:32.653" ContentLicense="CC BY-SA 4.0" />
  <row Id="24695" PostTypeId="2" ParentId="24679" CreationDate="2021-12-14T17:05:23.573" Score="0" Body="&lt;p&gt;I'll comment on a point of view, perhaps it's useful.&lt;/p&gt;&#xA;&lt;p&gt;You want to induce and encourage flexible ways of thinking.&lt;/p&gt;&#xA;&lt;p&gt;The value of that, IMHO, lies in that flexible ways of thinking increase the chances of understanding (real) problems more deeply and finding solutions to (real) problems, and/or finding new solutions, and/or better in some metric.&lt;/p&gt;&#xA;&lt;p&gt;I don't think your end goal is for her to always use the easiest way to do multiplication. Or addition.&lt;/p&gt;&#xA;&lt;p&gt;Because, doing arithmetic by a fixed method is in itself easy: you don't have to &lt;em&gt;think&lt;/em&gt; (which is &amp;quot;expensive&amp;quot;), just do. A fixed method gives you a tool to use easily to solve larger problems, so that you don't have to think about every little detail.&lt;/p&gt;&#xA;&lt;p&gt;So, perhaps, it might help you if you tell her something like this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Look, I know you like/prefer doing it that way. I'm not trying to make you change your way. I'm just asking you to find alternative ways to help you learn that: &lt;em&gt;the ability to find alternative ways&lt;/em&gt;, thinking of options, of possibilities. This is just an example.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12521" LastActivityDate="2021-12-14T17:05:23.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24696" PostTypeId="1" AcceptedAnswerId="24698" CreationDate="2021-12-14T20:21:19.580" Score="6" ViewCount="419" Body="&lt;p&gt;I'm engineer and I love linear-algebra. I finished a couple of days the Linear-Algebra on OpenCourseMIT to refresh my memory and I have been doing ton of exercises. I would like to teach to engineer students, so I think it would be a linear-algebra more on the &amp;quot;practical&amp;quot; side.&lt;/p&gt;&#xA;&lt;p&gt;But I would like to ask you for any tips, suggestions or books that may help me in this process.&lt;/p&gt;&#xA;&lt;p&gt;For those who teach it, how would you approach it? What are you methods and way?&lt;/p&gt;&#xA;&lt;p&gt;Anything will be more than welcome.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="18214" LastActivityDate="2022-02-18T00:28:06.080" Title="Preparing to be a linear-algebra teacher: Any tips/suggestions?" Tags="|undergraduate-education|linear-algebra|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24697" PostTypeId="2" ParentId="24696" CreationDate="2021-12-15T00:59:28.130" Score="6" Body="&lt;p&gt;You ask for sources &amp;quot;more on the 'practical' side.&amp;quot;&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/0fHbC.jpg&quot; width=&quot;200&quot; /&gt;&#xA;&lt;p&gt;May I suggest consulting &lt;em&gt;Linear Algebra Through Geometry&lt;/em&gt; by Thomas Banchoff and John Wermer (&lt;a href=&quot;https://link.springer.com/book/10.1007/978-1-4612-4390-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer link&lt;/a&gt;.)&#xA;You can download each chapter for free.&lt;/p&gt;&#xA;&lt;p&gt;The figures are dated but still geometric intuition is developed&#xA;throughout.&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/ebMcb.png&quot; width=&quot;400&quot; /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Also I see that MIT Open Courseware has a&#xA;&lt;strike&gt;course&lt;/strike&gt; video entitled&#xA;&lt;em&gt;Geometry of Linear Algebra&lt;/em&gt;:&#xA;&lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/ax-b-and-the-four-subspaces/the-geometry-of-linear-equations/geometry-of-linear-algebra/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&#xA;I haven't looked at it, but from the title it likely similarly emphasizes&#xA;geometric intuition.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2021-12-15T16:02:53.313" LastActivityDate="2021-12-15T16:02:53.313" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24698" PostTypeId="2" ParentId="24696" CreationDate="2021-12-15T14:22:17.723" Score="7" Body="&lt;p&gt;I have been teaching linear algebra for six years the best book that covers the concepts, mainly for engineers,  is &lt;strong&gt;Linear Algebra and Its Applications&lt;/strong&gt; for &lt;em&gt;David C. Lay.&lt;/em&gt; And if you want a more slightly abstract book (useful for mathematicians) you can check &lt;strong&gt;LINEAR ALGEBRA&lt;/strong&gt; for Jim &lt;em&gt;Hefferon&lt;/em&gt;. The last book I would like to recommend is Linear Algebra Done Right&amp;quot; by Sheldon Axler is an excellent book but I have never used but there is a huge vote for it.&lt;/p&gt;&#xA;" OwnerUserId="18218" LastActivityDate="2021-12-15T14:22:17.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24699" PostTypeId="1" AcceptedAnswerId="24700" CreationDate="2021-12-15T22:16:45.597" Score="1" ViewCount="305" Body="&lt;p&gt;Is it possible to write a problem statement as follows:&lt;/p&gt;&#xA;&lt;p&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is defined on &lt;span class=&quot;math-container&quot;&gt;$]0,1[$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$f(x)=x$&lt;/span&gt;. Determine &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to 1}f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Or should one write always as:&lt;/p&gt;&#xA;&lt;p&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is defined on &lt;span class=&quot;math-container&quot;&gt;$]0,1[$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$f(x)=x$&lt;/span&gt;. Determine &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to 1-}f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If only one one these is correct, how to teach that the other one is incorrect?&lt;/p&gt;&#xA;" OwnerUserId="18221" LastActivityDate="2021-12-16T07:39:34.713" Title="Limit from both sides or from left?" Tags="|notation|limits|" AnswerCount="1" CommentCount="1" ClosedDate="2021-12-16T12:07:40.037" ContentLicense="CC BY-SA 4.0" />
  <row Id="24700" PostTypeId="2" ParentId="24699" CreationDate="2021-12-16T07:29:59.670" Score="-1" Body="&lt;p&gt;Context is everything. If you are talking about limits from left and right, above and below, then it's a good idea to indicate using a superscript of + or -. On the other hand, generally when you are talking about a limit without reference to the left or right, above or below, then you are talking about a limit from both directions. What's the conceptual difference?&lt;/p&gt;&#xA;&lt;p&gt;Limits from one side are about the idea of getting closer and closer in an infinite way. This idea is at the heart of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Infinitesimal&quot; rel=&quot;nofollow noreferrer&quot;&gt;infinitesimal&lt;/a&gt; in calculus (and of supreme importance in &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;analysis&lt;/a&gt; further on). A good way of thinking about this is that an infinitesimal is a non-zero number that is so small, that's it's not important in a calculation. Consider how when you take a half of a half of a half, and so on, the distance in question becomes so tiny that it might be better just to round in real life.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, limits from two sides of a line, the ones without the superscript are about the idea that if you have an infinitesimal on the left and and an infinitesimal on the right, what you really have is &lt;a href=&quot;https://en.wikipedia.org/wiki/Continuous_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;continuity&lt;/a&gt;. Continuity is an important idea in calculus, because &lt;a href=&quot;https://en.wikipedia.org/wiki/Difference_quotient&quot; rel=&quot;nofollow noreferrer&quot;&gt;derivatives are really averages in disguise&lt;/a&gt;. And to have an average, you need to have a line determined by two secants.&lt;/p&gt;&#xA;&lt;p&gt;So you understand, two points can be near to each other, or they can be a specific distance apart. The first is called a 'neighborhood', and the latter is called a 'measure'. The limit of a difference quotient (the derivative) calculates a measure and then attributes the measure to the point, justified by nearness. THAT'S at the heart of the derivative, and the most important lesson from calculus, as well as the hardest part to learn since it's hidden behind so much calcluation and symbols.&lt;/p&gt;&#xA;&lt;p&gt;So, if you have a point, and you take a point infinitesimally to the left and infinitesimally to the right, you run a line through the points; since the distance between the secant line and the center point is also infinitesimally close, you can pretend that the line parallel to the secant line is the same as the one that is tangent to the curve at the point. That is, the two lines have the same slope because we can &amp;quot;round away&amp;quot; the difference given by the infinitesimal. Thus, a derivative is actually an approximation.&lt;/p&gt;&#xA;&lt;p&gt;So, whether or not you use the +/- (and it's always a good idea to use them in pairs and not drop the +) is a question of whether you are trying to teach the idea of an infinitesimal, the discrete amount &amp;quot;lower delta x&amp;quot;, or trying to teach the idea of continuity, which expresses a geometric intuition about curvature.&lt;/p&gt;&#xA;&lt;p&gt;Once the relation of the infinitesimal and continuity are understood (the points are infinitely near but not measurable), THEN a student is ready to understand the &lt;a href=&quot;https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%CE%B4)-definition_of_limit&quot; rel=&quot;nofollow noreferrer&quot;&gt;epison-delta definition&lt;/a&gt; of continuity which grounds 'geometric curvature' in 'algebraic implication', logically speaking.&lt;/p&gt;&#xA;" OwnerUserId="18201" LastEditorUserId="18201" LastEditDate="2021-12-16T07:39:34.713" LastActivityDate="2021-12-16T07:39:34.713" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="24701" PostTypeId="2" ParentId="9990" CreationDate="2021-12-17T11:07:21.900" Score="2" Body="&lt;p&gt;This is a subtle issue. It goes to the heart of the difference between math and physics.&lt;/p&gt;&#xA;&lt;p&gt;That A implies B does not entail that B implies A. One encounters frequently the errant reasoning that it does even among engineering students in the university (yesterday a student told me that because a matrix was diagonalizable it must be symmetric).&lt;/p&gt;&#xA;&lt;p&gt;However, that A implies B and one has observed B provides evidence for believing A. This statement is somehow the basis for the scientific method.&lt;/p&gt;&#xA;&lt;p&gt;In the example given in the original post, that Maria bought ice-cream today &lt;em&gt;does&lt;/em&gt; provide some evidence that she might have gone to the beach today.&lt;/p&gt;&#xA;&lt;p&gt;The difference is that between deductive reasoning and inference.&lt;/p&gt;&#xA;&lt;p&gt;Inferential reasoning is more common, more natural, and more powerful (why does anyone believe Euclid's axioms?).&lt;/p&gt;&#xA;&lt;p&gt;I quote from V. I. Arnold (&lt;a href=&quot;https://doi.org/10.1134/S1560354714060100&quot; rel=&quot;nofollow noreferrer&quot;&gt;Translation of the V. I. Arnold paper “From Superpositions to KAM Theory” (Vladimir Igorevich Arnold. Selected — 60, Moscow: PHASIS, 1997, pp. 727–740)&lt;/a&gt;), who in his inimitably polemical manner explains the issue better than I can anyway:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now it became possible to apply the techniques developed in the&#xA;problem of adiabatic invariants. As soon as I accomplished that,&#xA;Kolmogorov suggested that I should submit the paper on perpetual&#xA;adiabatic invariance to ZhETF, the main physical journal in the&#xA;USSR. A few weeks later, M. A. Leontovich (who was, as far as I&#xA;remember, a deputy to the editor-in-chief of ZhETF) invited me to&#xA;his home (near the Atomic Energy Institute of the USSR Academy of&#xA;Sciences) to discuss the manuscript. Having fed me, as usual, by&#xA;boiled buckwheat and calling me, as usual, “Dimka” (M. A. called me&#xA;in such a way until his death), Mikhail Aleksandrovich explained to me&#xA;that the paper could not be published in ZhETF due to the following&#xA;reasons.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The manuscript contained the words “theorem” and “proof” forbidden in ZhETF.&lt;/li&gt;&#xA;&lt;li&gt;The manuscript claimed that “A implies B” while every physicist knew examples showing that B does not imply A.&lt;/li&gt;&#xA;&lt;li&gt;The manuscript used the unintelligible terms “Lebesgue measure”, “invariant tori”, “Diophantine conditions”. Mikhail Aleksandrovich&#xA;therefore proposed that I should rewrite the paper.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Now I realize how&#xA;right he was in defending a physical journal from the Bourbaki-like&#xA;mathematical jargon. For instance, indeed, while claiming that “A&#xA;implies B” the author must point out explicitly whether the converse&#xA;holds, otherwise any reader not spoiled by the mathematical slang&#xA;would understand the claim as “A is equivalent to B”.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The issue is that mathematical deductive reasoning is rarely if ever applicable outside of mathematics where it should be replaced by the Bayesian paradigm of what Polya called plausible inference, of which it is an extreme case.&lt;/p&gt;&#xA;&lt;p&gt;The moral for teaching is that the difference between necessary and sufficient conditions is not something to be passed over lightly and that confusion in regards to it is not necessarily evidence of stupidity. When one teaches that all symmetric matrices are diagonalizable one must remind students that not all diagonalizable matrices are symmetric. Moreover, all orthogonally diagonalizable matrices are symmetric ...&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2021-12-17T14:58:00.310" LastActivityDate="2021-12-17T14:58:00.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24702" PostTypeId="2" ParentId="24647" CreationDate="2021-12-17T19:48:31.770" Score="9" Body="&lt;p&gt;Intense emotions surrounding math are not uncommon. Importantly, they tend to &amp;quot;feed back&amp;quot; on themselves - the memory of a bad experience with math makes doing math more stressful later, which then creates new bad experiences. So, taking action sooner rather than later will help immensely. Here are some suggestions.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There's no such thing as being just &amp;quot;sensitive&amp;quot;. Emotions have a reason; the more you can find out about the reason, the better you can help. What makes this student distressed? Is it when learning something new, or when trying to use something they think they should already know? Are there particular topics that cause more stress than others? Is it &lt;em&gt;starting&lt;/em&gt; a problem that causes stress, or does the stress kick in somewhere in the middle? Are word problems a greater stressor? Answers to any of these questions will help you figure out how to help the student.&lt;/li&gt;&#xA;&lt;li&gt;As another answer here pointed out, &lt;strong&gt;empty praise is worse than useless.&lt;/strong&gt; If a student picks up on you praising them for nothing, they won't trust you anymore, and they'll rely on their own judgment of how they're doing - which will always mean feeling bad about their math ability. That said, &lt;strong&gt;there is always something to praise honestly.&lt;/strong&gt; When I'm working with a student in this situation, I'll often praise the process (&amp;quot;see, setting it up like this was great, it was only later that things went wrong&amp;quot;), inventiveness (&amp;quot;well, that didn't turn out right, but it was a really cool way to approach it!&amp;quot;), and even awareness (&amp;quot;that's great, being able to spot when you don't know something is a super important skill&amp;quot;).&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Lower the stakes.&lt;/strong&gt; The perception of math as a &amp;quot;right-answer-or-bust&amp;quot; field is toxic. In between the right-or-wrong questions, ask questions that don't have clear-cut answers. I'm a fan of process-suggestion questions (&amp;quot;which of our tools do you guess might help here?&amp;quot;) and even emotion questions (&amp;quot;what's your gut feeling about this problem? What does that tell you?&amp;quot;).&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Model error-positive thinking.&lt;/strong&gt; Many students get stressed about math because they feel (often by observation of instructors, stronger students, and popular media) that the &amp;quot;right&amp;quot; way to do math is to always be sure of exactly what to do next, never make mistakes, and never run into dead ends. But &lt;em&gt;no one does that.&lt;/em&gt; The math I teach is math I've been practicing almost daily for more than a decade, and even I mess up sometimes. Even when I look perfect, it's just because I'm good at noticing and fixing my mistakes in my head before anyone sees them. I've found that it helps to turn off that reflex in myself: let the student see me make mistakes, puzzle over what to do next, or try something that doesn't end up working. In doing so, I can show them the right way to emotionally deal with an error: you welcome it as an opportunity to practice spotting and fixing mistakes.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Give control, where possible.&lt;/strong&gt; In my experience, many students think of math as something that happens &lt;em&gt;to&lt;/em&gt; them, not something they &lt;em&gt;do&lt;/em&gt;. And of course that's stressful - when you feel like you're fending off an attack, why wouldn't you get upset? It can help to provide opportunities for the student to take control of the situation. That can be as simple as giving them a choice about which problem to attempt next, or suggesting multiple ways to solve a problem (e.g., to find &lt;span class=&quot;math-container&quot;&gt;$2 \times 3 \times 5$&lt;/span&gt;, you might ask &amp;quot;would you rather multiply &lt;span class=&quot;math-container&quot;&gt;$2 \times 3$&lt;/span&gt; first, or start with &lt;span class=&quot;math-container&quot;&gt;$2 \times 5$&lt;/span&gt;?&amp;quot;).&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Self-care is a skill.&lt;/strong&gt; At 12, many (most?) students are still working on developing their coping mechanisms. Older students might be expected to recognize when they're feeling stressed out and frustrated, and to know what steps to take to get back on-track. But a 12-year-old probably needs help with that. Practice naming emotions, and offering simple solutions. For example: &amp;quot;It seems like you're getting frustrated. Do you want to take a break for a few minutes?&amp;quot; Or, &amp;quot;You seem upset about this problem. Do you want to talk about why?&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2021-12-17T19:48:31.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24703" PostTypeId="2" ParentId="24370" CreationDate="2021-12-17T20:42:05.187" Score="1" Body="&lt;p&gt;You might want to consider mining Ramsey Theory for that purpose. In particular, the theorem on friends and strangers has a proof with beautiful logic. (For that matter, so does Euler’s proof of the necessity of the form of even perfect numbers.)&lt;/p&gt;&#xA;" OwnerDisplayName="user10552" LastActivityDate="2021-12-17T20:42:05.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24704" PostTypeId="1" AcceptedAnswerId="24732" CreationDate="2021-12-17T21:16:44.620" Score="8" ViewCount="502" Body="&lt;p&gt;I'm currently teaching a mini-seminar to high school students, most of whom have at most a background in Algebra/Algebra II (in the US high school system) about finding roots of polynomials.&lt;/p&gt;&#xA;&lt;p&gt;In general, doing so is rather hard, and by the Abel-Ruffini theorem, there doesn't exist a radical (and field operation) based formula for solving general degree five polynomials and up.&lt;/p&gt;&#xA;&lt;p&gt;I'd like to convey the intuition for why polynomials are so hard to crack, and here's my idea so far.&lt;/p&gt;&#xA;&lt;p&gt;When I learned the Abel-Ruffini theorem, my intuition was that for &amp;quot;most&amp;quot; polynomials (with full Galois groups), the roots were so symmetric that they almost &amp;quot;protect&amp;quot; each other: i.e., knowing one will unlock the rest.&lt;/p&gt;&#xA;&lt;p&gt;As an analogy, a symmetric knot on your earbuds is far harder to untangle than an asymmetric knot as it's hard to know where to start. Or maybe Sudoku is a good analogy, as the unknown cells almost protect each other, and symmetric Sudokus (by which I mean ones in which unknown cells are all tied together) are much harder to solve than ones with &amp;quot;gimme&amp;quot; cells.&lt;/p&gt;&#xA;&lt;p&gt;As a final analogy, I was thinking about systems of linear equations, and how systems with a few variables as far easier to solve than the rest makes for a much easier system in general.&lt;/p&gt;&#xA;&lt;p&gt;However, all of the above is handwaving nonsense. Is this intuition meaningful? Is there a way of presenting these ideas without confusing the kids?&lt;/p&gt;&#xA;" OwnerUserId="19228" LastActivityDate="2021-12-31T09:21:30.563" Title="Explaining the intuition for why finding roots of polynomials is hard" Tags="|solving-polynomials|roots|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24705" PostTypeId="2" ParentId="24370" CreationDate="2021-12-17T22:27:55.807" Score="2" Body="&lt;p&gt;Here are all of the textbook chapters and assignments related to logic which I wrote for my Discrete Mathematics course last semester.  We spent about half of the course time (8 weeks) on this content.&lt;/p&gt;&#xA;&lt;p&gt;It contains a full treatment of the connectives and quantifiers, and includes problems from a &amp;quot;boolean calculus&amp;quot; perspective, from a &amp;quot;natural deduction&amp;quot; perspective, and interweaves connections to real mathematical usages by using and proving theorems about equalities, inequalities, parity statements, and divisibility statements.&lt;/p&gt;&#xA;&lt;p&gt;All of these are still fairly rough and incomplete (especially chapter 4 of the textbook).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://drive.google.com/drive/folders/19pXwLTowCwz1frNnVFNk0_v7KknxqXyJ?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://drive.google.com/drive/folders/19pXwLTowCwz1frNnVFNk0_v7KknxqXyJ?usp=sharing&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-12-17T22:36:18.850" LastActivityDate="2021-12-17T22:36:18.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24706" PostTypeId="1" AcceptedAnswerId="24715" CreationDate="2021-12-19T21:44:11.067" Score="10" ViewCount="9237" Body="&lt;p&gt;Most students find math unfathomable, labyrinthine by the time of univariate &lt;a href=&quot;https://math.stackexchange.com/q/20578&quot;&gt;integration&lt;/a&gt; (&lt;a href=&quot;https://redd.it/a8oh6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reddit&lt;/a&gt;).  Even overachievers – who ace undergraduate math without studying – will eventually be convoluted by math, like in post-doctoral &lt;a href=&quot;https://old.reddit.com/r/math/comments/9p4bf0/when_does_math_get_really_hard/e7yz49o/&quot; rel=&quot;nofollow noreferrer&quot;&gt;abstract algebra&lt;/a&gt; or number theory. &lt;strong&gt;But exactly why’s (&lt;a href=&quot;https://old.reddit.com/r/explainlikeimfive/comments/5y8gqs/eli5why_do_some_people_find_mathematics_hard_no/deotgy0/&quot; rel=&quot;nofollow noreferrer&quot;&gt;abstract&lt;/a&gt;) math way knottier and thornier than law or medicine?&lt;/strong&gt; I’m comparing math with law and medicine as they’re highly sought non-math subjects.&lt;/p&gt;&#xA;&lt;h3&gt;&lt;strong&gt;Rule out these  non-answers because they also appertain to law and medicine.&lt;/strong&gt;&lt;/h3&gt;&#xA;&lt;p&gt;Pre-suppose no dyscalculia. Pre-suppose no &lt;a href=&quot;https://redd.it/3fm9ab&quot; rel=&quot;nofollow noreferrer&quot;&gt;ineffective teaching&lt;/a&gt; that cannot be the decisive reason, because math still bewilders and overwhelms students even when taught skillfully by teachers who have PhDs in math.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Information overload and quantity&lt;/strong&gt;. Information changes more swiftly in law and medicine. Laws are daily created, amended, repealed by administrative authorities, civil servants, or politicians. Old medications stop working (e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Drug_resistance&quot; rel=&quot;nofollow noreferrer&quot;&gt;drug resistance&lt;/a&gt;), new medications are patented, and new diseases crop up.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://old.reddit.com/r/math/comments/8a66ai/why_is_math_so_complicated_and_hard/dwwbgog/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Incrementalism&lt;/a&gt;.&lt;/strong&gt; Law and medicine “also build on each other in a way that most topics don’t, especially before college”. Law schools make contract law a prerequisite for commercial or maritime law.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="75" LastEditDate="2021-12-21T12:21:21.243" LastActivityDate="2021-12-23T08:46:22.783" Title="Why's math way more puzzling, abstruse than law and medicine?" Tags="|mathematical-pedagogy|" AnswerCount="14" CommentCount="20" ClosedDate="2021-12-24T11:16:19.460" ContentLicense="CC BY-SA 4.0" />
  <row Id="24708" PostTypeId="2" ParentId="24706" CreationDate="2021-12-20T00:05:11.550" Score="17" Body="&lt;p&gt;Ineffective teaching is absolutely part of it. Math is about understanding and problem-solving. Problem-solving is hard. And students who aren't already into math want it easier. So the teacher &amp;quot;helps&amp;quot; by showing them steps (me included!), and then they aren't really learning as much.&lt;/p&gt;&#xA;&lt;p&gt;K-6 teachers teach elementary school because they love kids (most, anyway), but that doesn't mean they like math, or understand it at all. Their mathphobia gets passed along to the students.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2021-12-20T00:05:11.550" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24709" PostTypeId="2" ParentId="24706" CreationDate="2021-12-20T01:16:45.033" Score="16" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Even overachievers — who ace high school calculus without studying — will eventually be puzzled or convoluted by math, like in undergraduate or graduate Analysis or Abstract Algebra. But why exactly is (abstract) math way knottier and thornier than law or medicine?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You have used the term &amp;quot;overachievers&amp;quot; to describe people who ace high school calculus without studying, but within them, there's some who even ace undergraduate and graduate-level analysis and abstract algebra courses. &lt;em&gt;&lt;strong&gt;They will still run into challenges eventually, if for example they go on to become a researcher who wishes to publish new findings or solutions to unsolved problems in those areas.&lt;/strong&gt;&lt;/em&gt;  But that is also true for medicine and law students at that level (i.e. people who are actually practicing the subject professionally for a living, will find that there's a lot of challenging unanswered questions, like why is this patient not responding to this medication like the textbooks would suggest, and what legal reasoning can I use to acquit the defendant I'm representing?).&lt;/p&gt;&#xA;&lt;p&gt;For the most elite of the &amp;quot;overachievers&amp;quot; (to use your term!), it will often not be hard to get through medical school, law school, and graduate-level analysis and abstract algebra courses, and the biggest challenges will come when trying to advance the cutting edge of the subject (proving unsolved conjectures in analysis, discovering a cure for cancer, reforming the legal system).&lt;/p&gt;&#xA;&lt;p&gt;Perhaps it's the &amp;quot;mid-range overachievers&amp;quot; where the question becomes a lot more interesting. these are people who aced high school but struggle in analysis and abstract algebra courses. I could probably write forever on the subject (and since this is my first time posting on this site, I'll leave space for others to weigh in too!) but I do agree with you that the &amp;quot;mid-range overachievers&amp;quot; in mathematics will struggle more than the &amp;quot;mid-range overachievers&amp;quot; in medicine or law. Out of the many things I can say, perhaps one stands out: I've always been able to pick up a law or medicine textbook and understand what I'm reading, and learn whatever new terms I come across by looking them up further (&lt;em&gt;&lt;strong&gt;as long as it's written in standard English)&lt;/strong&gt;&lt;/em&gt;, but a random math textbook will rarely read like a novel, and will have tonnes of symbols and notation &lt;em&gt;&lt;strong&gt;that vary from author to author&lt;/strong&gt;&lt;/em&gt;, and &lt;em&gt;&lt;strong&gt;most of all,&lt;/strong&gt;&lt;/em&gt; I find myself needing to read &lt;em&gt;&lt;strong&gt;extremely slowly and carefully&lt;/strong&gt;&lt;/em&gt; with &lt;em&gt;&lt;strong&gt;many hours of thinking in between&lt;/strong&gt;&lt;/em&gt; to understand the proofs, whereas in medicine and law I am typically just reading a regurgitation of facts &lt;em&gt;&lt;strong&gt;without having to try to reproduce the proofs of those facts on my own&lt;/strong&gt;&lt;/em&gt;. In analysis and abstract algebra courses, even at the lowest of undergraduate levels, &lt;em&gt;&lt;strong&gt;we want students to actually do the proofs on their own, not just to know what the theorems/facts are&lt;/strong&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13687" LastEditorUserId="13687" LastEditDate="2021-12-20T08:56:38.130" LastActivityDate="2021-12-20T08:56:38.130" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24710" PostTypeId="2" ParentId="18801" CreationDate="2021-12-20T02:02:15.820" Score="0" Body="&lt;p&gt;Equivalence classes provide a tool for describing information. If you have an object &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and an equivalence relation &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, one piece of information about &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &amp;quot;which &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;-equivalence class is it in&amp;quot;. What's more, any piece of information can be described in this way by constructing an equivalence relation where two things are equal if this attribute of theirs is the same.&lt;/p&gt;&#xA;&lt;p&gt;Usually we have the notion of information first, and can construct an equivalence relation based on that, but sometimes it's reversed: the equivalence relation allows us to capture an important piece of information that's hard to describe. For instance, the asymptotic behavior of a function is a really important piece of information, but to get at that information you really need to start with the notion of asymptotic equivalence.&lt;/p&gt;&#xA;&lt;p&gt;This is close to one of my favorite equivalence relations: two sequences being eventually equal, which doesn't actually capture any information about the particular value of the sequence at any point, but still captures important information about it.&lt;/p&gt;&#xA;&lt;p&gt;The best example of this is Myhill-Nerode equivalence. We say that two computers in particular states are Myhill-Nerode equivalent if no matter what inputs we feed into both computers, we get the same outputs. This equivalence relation captures exactly the information we need to keep track of to determine the future behavior of a computer. Any other information we're keeping track of is unnecessary.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2021-12-20T02:02:15.820" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24711" PostTypeId="2" ParentId="24706" CreationDate="2021-12-20T02:25:05.233" Score="37" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Univariate calculus — e.g. integration (see also Reddit) — is when most students find math unfathomable and labyrinthine.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Well, not really. Actually most students never reach this level of math, and most students who have difficulty with math have difficulty with much more basic math than this.&lt;/p&gt;&#xA;&lt;p&gt;Suppose, for example, that I tell you that a bedroom community called Havensleep has an area of 17,000 acres, and I also tell you that the average housing lot in Havensleep is 0.43 acres. Can you figure out how many lots there are in Havensleep?&lt;/p&gt;&#xA;&lt;p&gt;In my experience, most college students are unable to solve this, even with the aid of a calculator. They're given the numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, but they don't know what arithmetic operation to do with them. Should they do the product &lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt;? The sum &lt;span class=&quot;math-container&quot;&gt;$x+y$&lt;/span&gt;? Should they find &lt;span class=&quot;math-container&quot;&gt;$x/y$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$y/x$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;This is supposed to have been grade school math, but the way grade school math was taught, one was always told what operations to do.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why is math way more difficult, puzzling, abstruse compared to other subjects like law and medicine?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't think this is true at all. I would have a terrible time with premed classes such as o-chem, for example, because I'm bad at memorization. Ditto for a lot of the graduate curriculum in law school. Law can be very difficult and abstract. Can you explain what a tort is? Can you explain why Roe v Wade is important constitutional law even though Casey has completely replaced its criteria for regulation of abortion? Do you know which constitutional amendments Roe invokes and how this makes it different from previous case law? Can you make a coherent argument as to whether it does this in a valid way or not?&lt;/p&gt;&#xA;&lt;p&gt;In general, people just tend to be bad at things that they're never given practice doing. The Havensleep example is one such thing that students are not given practice with. But in most public K-12 schools in the US most students get very little exposure to things like recognizing parts of speech, using the metric system, recognizing the sounds of different musical instruments, or breaking up their writing into paragraphs.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ineffective teaching cannot be a reason, because students are still bewildered, overwhelmed by math even when taught well by teachers who have PhDs in math.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Subject matter knowledge is necessary but not sufficient to make someone a good teacher.&lt;/p&gt;&#xA;" OwnerUserId="19242" LastActivityDate="2021-12-20T02:25:05.233" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="24712" PostTypeId="2" ParentId="24706" CreationDate="2021-12-20T03:35:12.877" Score="12" Body="&lt;p&gt;The standards in math are way higher because of supply and demand.&lt;/p&gt;&#xA;&lt;p&gt;Lawyers and doctors directly help people.  We need lots of them.  Mathematicians do some difficult to understand work that might possibly be useful to someone somewhere sometime.  We need some mediocre lawyers and doctors just so that all the necessary legal and medical work in our society can get done.  We don't need mediocre mathematicians (though some people who might have been mediocre mathematicians end up being engineers (or lawyers!)).&lt;/p&gt;&#xA;&lt;p&gt;Is mathematics easier than creative writing or philosophy, which are fields with a similar level of demand?  I seriously doubt it.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2021-12-20T03:35:12.877" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="24714" PostTypeId="2" ParentId="24706" CreationDate="2021-12-20T07:54:52.877" Score="17" Body="&lt;p&gt;Many good answers already, but here's one more thought: the bar in math is set much higher.&lt;/p&gt;&#xA;&lt;p&gt;When you're doing law or medicine or languages or whatever, there's a fairly broad spectrum of &amp;quot;OK&amp;quot;. You don't need to know stuff perfectly, you can also arrive at acceptable solutions which are not ideal. You can also rely on understanding some things only partially or intuitively.&lt;/p&gt;&#xA;&lt;p&gt;In contrast, maths is very strict. It has to be, because it's completely abstract and talks about absolute truths and falsehoods. You cannot &amp;quot;sorta kinda&amp;quot; prove a theorem, it has to be an ideal, watertight proof, or else it doesn't count at all. There's still plenty of room for intuition, but intuition alone is not enough - you need to follow it up with precise, concrete logic.&lt;/p&gt;&#xA;&lt;p&gt;And human mind is not very well suited for that. It prefers to categorize the world in vague, fuzzy patterns, and this razor-sharp ultra-precise thinking does not come naturally to it. You need more training to get your brain to accept and memorize and utilize these kinds of patterns.&lt;/p&gt;&#xA;" OwnerUserId="14011" LastActivityDate="2021-12-20T07:54:52.877" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="24715" PostTypeId="2" ParentId="24706" CreationDate="2021-12-20T09:34:49.690" Score="28" Body="&lt;p&gt;The perceived difficulty of abstract math is due to two factors:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You learn math at school, but it is actually very different from what you do at university. In school you are applying rules to get some numerical result. And the rules are easily mapped to concepts you encounter in real life. This changes in university. Suddenly you work almost exclusively with proofs, and real world applications are often only an after thought.&lt;/p&gt;&#xA;&lt;p&gt;This change of pace causes pupils that are good at math in school to struggle at university, because they got into something different then they originally thought.&lt;/p&gt;&#xA;&lt;p&gt;The common idea of what studying law or medicine is, is much closer to the real thing.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Abstract math is abstract.&#xA;If you pick up some law text you can understand the basic ideas directly. You will certainly miss some finer points, but most of the words used will have a meaning similar to the one they have in every day usage.&lt;/p&gt;&#xA;&lt;p&gt;Medicine is a little more difficult. There are many special terms that you have never heard before – like Latin names for every organ, tissue and part of your body. But most of these terms can be  easily explained with a single paragraph of text in common in English and an image. So again you can read and understand the gist of it as long as you can access Wikipedia. Of course, this changes once you get into things like e.g. biochemistry.&lt;/p&gt;&#xA;&lt;p&gt;Compare that to a university level text book on math. After a few pages of introduction, you'll be confronted with pages of pages with very little of actual English. Most will be special symbols which have absolutely zero meaning in real life. In order to understand it, you cannot just look up certain terms, you have to look up every single symbol.&lt;/p&gt;&#xA;&lt;p&gt;Take for example Euler's identity: &lt;span class=&quot;math-container&quot;&gt;$e^{i\pi }+1=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Every symbol has a very specific meaning, and even the positioning of terms above and to the right, or right next to each other have meaning. And this is a short and simple example. And at first glance, this has nothing to do with your real life. There is no experience from your life to which you can tie this new concept. And tying new information to existing information is something very important for learning. So studying math easily ends up being like learning new concepts – while at the same time – being like learning a new language to express these concepts in a language, using a very different alphabet and grammar than the one you are used to.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19243" LastEditorUserId="75" LastEditDate="2021-12-21T08:15:15.740" LastActivityDate="2021-12-21T08:15:15.740" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24716" PostTypeId="2" ParentId="24706" CreationDate="2021-12-20T10:16:06.617" Score="8" Body="&lt;p&gt;I think a large part of this is that the media bombard us from an early age with the message that maths is boring and difficult.  Neither of these things are true, the point is that the media industry is dominated by people with arts and humanities degrees, so they are just telling us that &lt;em&gt;they&lt;/em&gt; found maths difficult and boring, because &lt;em&gt;their&lt;/em&gt; interests lay elsewhere.&lt;/p&gt;&#xA;&lt;p&gt;Another part of the problem is that maths, unlike e.g. art, has a stage where we have to do a lot of groundwork with no immediately useful benefit to us.  However we can't get to the more interesting and useful aspects of maths until we have got past that stage.  This is a reason why more children play football than cricket.  All you need to be able to play football is some basic level of fitness and co-ordination.  You can play without technique or strategy.  That doesn't work in cricket, if you can't play a solid defensive shot, you won't bat long against a bowler that knows what they are doing.  This means that it can take you a lot longer to get to the point in cricket where you really understand what is going on and enjoying it fully.&lt;/p&gt;&#xA;&lt;p&gt;For students to learn &lt;em&gt;any&lt;/em&gt; subject effectively you need them to be interested in learning, so if the majority of them have been told for years that maths is difficult and boring, and they can't see the point in e.g. algebra, because it isn't useful in their daily lives, as a teacher you have a much more difficult job on your hands than e.g. an art teacher.&lt;/p&gt;&#xA;&lt;p&gt;On the flip side, you only need to be a little bit good at maths to gave a genuinely useful skill (as it is relatively uncommon), where as you have to be extremely good at English to have a similarly useful level of skill.&lt;/p&gt;&#xA;&lt;p&gt;BTW, I am not an overachiever.  I got a D at A-level (not good), but these days I am happy writing/reading about reproducing kernel Hilbert spaces.  People learn things at different rates at different times in their lives, and a lot of it has to do with how much they want to learn at the time.&lt;/p&gt;&#xA;" OwnerUserId="802" LastEditorUserId="802" LastEditDate="2021-12-20T10:21:31.123" LastActivityDate="2021-12-20T10:21:31.123" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24717" PostTypeId="2" ParentId="24706" CreationDate="2021-12-20T12:57:48.637" Score="10" Body="&lt;p&gt;Using only my personal experience: I think it is indeed that mathematics as a science is entirely abstract, as Jens said in another answer.&lt;/p&gt;&#xA;&lt;p&gt;Both law and medicine are sciences that have a strong connection to real world events or items that everybody can relate to. In effect they deal with failures of humans and failures of their everyday interactions. That is unknown to exactly nobody; solving the problems of either field is of interest to everybody. That's why everybody talks at parties about their medical problems or their divorce proceedings, but very few people outside of department Christmas parties have small talk about Abelian groups.&lt;/p&gt;&#xA;&lt;p&gt;The case is very different with the part of math that is directly related to real-world issues. You &lt;em&gt;can&lt;/em&gt; have party talk about folding paper to get to the moon, or about an exponentially growing number of infections, or the chances to draw a flush on the river. Few people have trouble understanding math up to, say, 10th grade where you count things, divide amounts, have parabolic trajectories of balls etc. That's the kind of math that is at the level of law and medicine. You have a real-world problem and a systematic approach to solve it. Medicine and law then continue to heap on more of the same; what is difficult for students is the sheer amount of stuff to cram into their heads, not the increasing abstraction.&lt;/p&gt;&#xA;&lt;p&gt;That's different with math. You don't have another 42 similar laws after the commutative and distributive one. You also don't need to learn multiplication tables for the first 52 natural numbers — but that's what law students must do in the United States, and medical students with the bones and blood vessels in the body. That is why those subjects are hard. That is not why math is hard. In math you see that there is a general pattern in the laws governing the concrete operations (addition) and sets (apples) you started with. Math then goes on defining and making statements about &lt;em&gt;these abstract patterns.&lt;/em&gt; It's as if M.D.s, after looking at three human specimens, would start making generalized assumptions about the self organization of matter in a space-time matrix governed by certain natural constants where the matrix loci are only allowed certain parametrized interactions with their neighbors. That's not what doctors do: They try to save their patients' lives. That is as if math professors would be concerned with counting change in the supermarket or engineer bridges. That is not what mathematicians do.&lt;/p&gt;&#xA;&lt;p&gt;The secret to understanding abstract math is, I think, to become acquainted with the abstract building blocks so that they take on the role of things the student can relate to, as if they were part of the real world. Series and sequences, limits, operations and relations and so on become almost tangible objects about which calculus and abstract algebra make statements. I think it's a bit like a muscle memory of the brain: You make certain concepts your own and use them with ease so that they can form the basis for the next abstraction. This &amp;quot;mental muscle memory&amp;quot; is built with practice. To some, of course, this practice doesn't feel like a chore, very much like a gifted musician builds muscle memory on the side by doing what they like best, while others loathe their piano practice, or their math homework.&lt;/p&gt;&#xA;&lt;p&gt;But however you get there: You need to be familiar with the concept of a function and of derivation to understand the concept of a differential equation, and so on.&lt;/p&gt;&#xA;" OwnerUserId="11894" LastEditorUserId="11894" LastEditDate="2021-12-20T14:19:45.177" LastActivityDate="2021-12-20T14:19:45.177" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24718" PostTypeId="2" ParentId="24706" CreationDate="2021-12-20T13:01:36.387" Score="6" Body="&lt;p&gt;Sorry for one more answer...&lt;/p&gt;&#xA;&lt;p&gt;Math needs a special way of thinking, different from our everyday practice (and, to some degree from law and medicine).&lt;/p&gt;&#xA;&lt;p&gt;In everyday life, we often use similarities and analogies. If something is similar enough to an established truth, we accept it. To my understanding, the whole system of precedents in law is based on that kind of thinking. And in medicine, a close match in symptoms often is enough to correctly diagnose an illness.&lt;/p&gt;&#xA;&lt;p&gt;Our brain is hard-wired by evolution to excel at &amp;quot;pattern matching&amp;quot;, at classifying things by similarities.&lt;/p&gt;&#xA;&lt;p&gt;In math, there is no such thing as &amp;quot;similar to truth&amp;quot;. Something is either correct or incorrect, with nothing in-between.&lt;/p&gt;&#xA;&lt;p&gt;And here comes my criticism on (some approaches to) math education. Students can achieve high math grades by doing things the &amp;quot;pattern matching&amp;quot; way, by just finding the &amp;quot;most similar formula&amp;quot; and blindly applying it. Give them a quadratic equation using &lt;code&gt;c&lt;/code&gt; as the variable instead of &lt;code&gt;x&lt;/code&gt;, and they're lost. But we don't do that, we always use &lt;code&gt;x&lt;/code&gt;, and so we support the non-math pattern matching approach instead of the exact thinking process needed in math.&lt;/p&gt;&#xA;&lt;p&gt;And then, later on, when exact thinking &amp;quot;the math way&amp;quot; is needed, it has not been trained enough.&lt;/p&gt;&#xA;" OwnerUserId="15862" LastActivityDate="2021-12-20T13:01:36.387" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24719" PostTypeId="2" ParentId="24706" CreationDate="2021-12-20T22:56:40.690" Score="3" Body="&lt;p&gt;Aspects of this answer may be specific to the US.  I treat this as an empirical question about perceptions (i.e., what people &lt;em&gt;think&lt;/em&gt; is more puzzling) than a question about what subject actually &amp;quot;is.&amp;quot;  I think three factors are particularly relevant.&lt;/p&gt;&#xA;&lt;p&gt;The first:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A lot of people see math relatively early in life, while most people do not see law or medicine, or at least not until relatively later in life. This causes difficulties in math to be more readily apparent than other difficulties.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The question refers to &amp;quot;univariate integration&amp;quot; and this is a good illustration of the point.  Integration is taught in most US high schools and almost every US college or university, where it is taken by many students who are not planning careers that directly involve math.  There is no law or medicine counterpart to this!  (Many students entering law or medical school in the US have no pre-existing legal or medical education, and &amp;quot;popular&amp;quot; treatments of law and medicine often smooth over details in the interest of storytelling in a way that a calculus instructor is under no obligation to do.)&lt;/p&gt;&#xA;&lt;p&gt;People from all sorts of backgrounds might have a common experience of once having taken something like calculus.  Calculus thus becomes a frame of reference for &amp;quot;something abstruse or difficult.&amp;quot;  There is no similar legal or medical touchstone.  And there's no reason why the touchstone &amp;quot;has&amp;quot; to be math.  Maybe 100 or 200 years ago, a better reference point for puzzling and abstruse might have been learning Latin or ancient Greek.  That may have been more common reference point.&lt;/p&gt;&#xA;&lt;p&gt;(It might be observed that calculus is not reflective of most professional uses of mathematics.  This is true, but there is a closer parallel than there is between a history class and legal education/practice, or between a biology class and medical education/practice.)&lt;/p&gt;&#xA;&lt;p&gt;The second:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Legal and medical education are accompanied by significant &amp;quot;buy-in&amp;quot; from students in the form of voluntary commitment of time and money.  This creates selection effects (reducing the number of people who would find something abstruse), psychological effects (having committed to something, one may be less inclined to perceive difficulties as insurmountable), and social effects (having publicly committed to something, one may be less likely to complain about difficulties).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These effects are not present when people are forced to take a subject for a purpose other than pursuing a career that uses it (which is, on a person-hours basis, the vast majority of all mathematics education).&lt;/p&gt;&#xA;&lt;p&gt;The third:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Formalized mathematics seems at least potentially amenable to being &amp;quot;figured out,&amp;quot; and less of a matter of opinion, than many legal and medical issues.  People therefore have higher expectations about how much sense math ought to make, and these higher expectations are violated when one struggles with math in a way that expectations might not be violated when struggling with difficult issues in other areas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="15759" LastActivityDate="2021-12-20T22:56:40.690" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24720" PostTypeId="2" ParentId="24647" CreationDate="2021-12-21T02:45:59.897" Score="2" Body="&lt;p&gt;Both &lt;a href=&quot;https://matheducators.stackexchange.com/a/24648&quot;&gt;Amy B&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/a/24702&quot;&gt;Reese Johnston&lt;/a&gt; have given some excellent suggestions but make sure you pay attention so that the student doesn't use tears (even unconsciously) to avoid work or get spoonfed answers.  Without observation or more details, I cannot tell if the parents are simply accommodating a more sensitive child or have fallen into the habit of removing anything that frustrates the child.&lt;/p&gt;&#xA;" OwnerUserId="8163" LastEditorUserId="15884" LastEditDate="2021-12-23T11:02:33.513" LastActivityDate="2021-12-23T11:02:33.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24721" PostTypeId="2" ParentId="24706" CreationDate="2021-12-21T10:30:54.310" Score="4" Body="&lt;p&gt;&lt;strong&gt;TL;DR:&lt;/strong&gt; The problem with math is that it is layers of abstraction. You need some familiarity (not just bare understanding) with one layer before you can progress. Most students don’t realise this and thus become frustrated when they fail to understand something even though it is clearly defined.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Many things have already been said, but I need to challenge your second assumption:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Rule out these  non-answers because they also appertain to law and medicine.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;[…]&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Incrementalism.&lt;/strong&gt; Law and medicine &lt;a href=&quot;https://old.reddit.com/r/math/comments/8a66ai/why_is_math_so_complicated_and_hard/dwwbgog/&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;also build on each other in a way that most topics don't, especially before college&amp;quot;&lt;/a&gt;. Law schools make contract law a prerequisite for commercial or maritime law.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Incrementalism is no binary thing: All fields have some of it, but there is a wide spectrum from deep fields (with lot of increment) to broad fields (with a lot of content).&#xA;&lt;strong&gt;(Pure) mathematics is clearly on the deep side of this spectrum:&lt;/strong&gt;&#xA;For example, you can be a number theorist by only being good at this one thing and knowing about a few related mathematical subfields. The number theory that doesn’t require depth has already been discovered millennia ago. By contrast, a general medical practitioner needs to know about a lot of common ailments and a bit about biology, chemistry, physics, pharmacy, psychology, bureaucracy, and mathematics.&lt;/p&gt;&#xA;&lt;p&gt;This is exacerbated by the abstractness of mathematics. In fact, I would argue that the core of mathematics is not proofs, but finding the right abstraction to be able to formulate useful theorems and facilitate proving them. And this abstraction comes in layers on layers. &lt;strong&gt;The problem with this layered abstraction is that you have to be somewhat fluent in one layer to grasp the next one.&lt;/strong&gt; Otherwise you quickly have too many things you need to mentally juggle at the same time.&lt;/p&gt;&#xA;&lt;p&gt;Consider your example of integration: To understand what integration is, you need to have a clear understanding of functions (amongst other things). To understand functions, you need to understand variables. To understand variables, you need to understand numbers and elementary arithmetics. And the last things are already something on which most children spend the entirety of primary school. (Yes, we also teach them stuff like multiplying with pen and paper, but that’s only a means to the end of understanding numbers and multiplication, since pocket calculators can do this better than any human.) And university-level mathematics puts layers of layers on top of that.&lt;/p&gt;&#xA;&lt;p&gt;By contrast, consider MRNA vaccines (i.e., some top-notch medicine). To understand how these work, you need to first understand what cells are and how DNA works. On this basis, you can continue to understand what MRNA does, how viruses work, and how the immune system works. There are some layers here, but only some and they are not that abstract. For example, you can see cells under a microscope and you can explain DNA by analogy to blueprints. I can understand these things without any university-level education on these matters, and I can even explain them to a child in primary school.&#xA;(Also see &lt;a href=&quot;https://xkcd.com/2425/&quot; rel=&quot;noreferrer&quot;&gt;this XKCD&lt;/a&gt;.) Mind that I am far from being able to develop an MRNA vaccine myself, because I considerably lack breadth in medicine, pharmacy, molecular genetics, etc.&lt;/p&gt;&#xA;&lt;p&gt;Now back to your question: Since mathematics requires you to wrap your mind around layers of abstraction, what determines your learning speed is how quickly you can understand abstract concepts and automatise them (so you can apply them without occupying too much of your brain). If you go too fast, you will inevitably hit a barrier where you run into something that is clearly spelt out to you, but you cannot understand because you just lack experience with the previous layer of abstraction. &lt;strong&gt;This is naturally frustrating, in particular if you do not understand why you are struggling&lt;/strong&gt; (which is quite common because most professors do not realise or tell you these things; at least nobody told me). Something similar also happens in other deep fields, such as physics, philosophy, or linguistics.&lt;/p&gt;&#xA;&lt;p&gt;By contrast, the limiting factor of learning speed in a broad field is the amount of stuff you can learn. You will hit barriers, but they are of a different and more obvious nature like bad explanations, being applications of another field, or you being tired from cramming too much knowledge into your brain since you last slept.&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2021-12-21T10:30:54.310" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24722" PostTypeId="2" ParentId="24706" CreationDate="2021-12-21T16:59:34.210" Score="3" Body="&lt;p&gt;Because math is about truth rather than consensus.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2021-12-21T16:59:34.210" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24724" PostTypeId="2" ParentId="24647" CreationDate="2021-12-22T09:16:00.987" Score="0" Body="&lt;p&gt;Let me clarify the situation: you are her teacher, not her parent. This makes it your job to explain matter to her, and it's the job of the parents to prepare her for that.&lt;/p&gt;&#xA;&lt;p&gt;As you have stated:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Her father said she is sensitive. I was asked to do more praising and encouraging from her parents.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is not your job! Not at all!&lt;/p&gt;&#xA;&lt;p&gt;Let's face it: crying is an enormous time and focus waster: when a person starts crying, he or she looses his or her focus for quite a long time, so it should happen as less as possible.&lt;br /&gt;&#xA;I remember that, also at the age of 12, during one lesson, I've been crying too. As far as I know, the teacher did not even speak to me, he just let it happen, and I never had any bad feelings towards that teacher for that.&lt;/p&gt;&#xA;&lt;p&gt;I have the impression that the parents are not dealing well with their sensitive child and let the burden on you, and that's not the way education is to be done: it's your job to deliver the matter and it's their job to make her ready for dealing with that matter.&lt;/p&gt;&#xA;&lt;p&gt;Oh, if you think I'm being harsh here, just think about this situation: I imagine you have a regular class of about 20 students, and you are planning of adapting yourself to the &amp;quot;special&amp;quot; needs of one single student. Another student might notice that and invent a &amp;quot;special&amp;quot; need for him- or herself too, do you see the snowball effect coming? ... at the end you'll be dealing with &amp;quot;special&amp;quot; needs that much that you won't be able to teach your class anymore.&lt;/p&gt;&#xA;&lt;p&gt;Does this mean there is nothing you can do? Well, the easiest you can do is increase praising of every student in your class. Like this, her need to praising might be fulfilled (or at least reduced) and you don't make an exception for one student. But don't go further than that.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2021-12-22T09:16:00.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24725" PostTypeId="2" ParentId="24706" CreationDate="2021-12-22T09:54:08.410" Score="0" Body="&lt;p&gt;I correspond somewhat to the profile invoked by the OP, having experienced difficulty with a part of the maths (and physics) in my bachelor's in Electronic Engineering.&lt;/p&gt;&#xA;&lt;p&gt;I think the problem lies not in the abstraction of mathematics, but in its expression.&lt;/p&gt;&#xA;&lt;p&gt;I reason and memorize verbally. At school, maths concepts are verbalized, which permits understanding by a wider audience. High school calculus is relatively easy to verbalize.&lt;/p&gt;&#xA;&lt;p&gt;Higher maths (and physics) are much more symbolic. I find it difficult to self-educate because I can't verbalize the equations that I read. I find Physics Stack Exchange interesting, but my eyes glaze over when I arrive at the equations. On the other hand, I find accounting easy because numbers can be verbalized.&lt;/p&gt;&#xA;&lt;p&gt;So I would say that the problem the OP describes arises when the verbalizers can no longer verbalize the equations they are required to understand and memorize. They need to develop a symbolic understanding in high school, or understand (as I did) that higher maths is not for them.&lt;/p&gt;&#xA;" OwnerUserId="19268" LastEditorUserId="376" LastEditDate="2021-12-22T15:31:51.473" LastActivityDate="2021-12-22T15:31:51.473" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24727" PostTypeId="2" ParentId="24706" CreationDate="2021-12-22T20:44:27.617" Score="2" Body="&lt;p&gt;My point of view on this is that it's all about whether progression in the subject gets more abstract or more concrete.&lt;/p&gt;&#xA;&lt;p&gt;Law and medicine are both concentrations of much larger and more broad fields. Medicine is a particular corner of biology and law is a particular corner of the intersection between logic and political science. The further one goes in these fields, the more concrete and specific things get.&lt;/p&gt;&#xA;&lt;p&gt;Math, unlike professional subjects, is the inverse. Math starts teaching you things that seem concrete, and then begins to generalize those relationships and algorithms.&lt;/p&gt;&#xA;&lt;p&gt;Think about it this way: Arithmetic is algorithms for numbers, algebra is a generalization of arithmetic for that applies to large subsets of all numbers. Linear algebra is a generalization of algebra to multiple equations at once. Calculus is bridging the gap between discrete and continuous functions using algebra. And it just gets more general from there.&lt;/p&gt;&#xA;&lt;p&gt;At some point, everyone runs out of the intellectual skill needed to think in ever more abstract terms (Some sooner than others).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;TLDR:&lt;/strong&gt; Professional subjects are concrete applications of general concepts. Math is generalizing from concrete observable use cases. Thinking in ever more abstract terms is a different problem with fewer systems to guide students, versus thinking in ever more concrete ones.&lt;/p&gt;&#xA;" OwnerUserId="19272" LastEditorDisplayName="user155" LastEditDate="2021-12-23T08:46:22.783" LastActivityDate="2021-12-23T08:46:22.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24728" PostTypeId="2" ParentId="24704" CreationDate="2021-12-23T04:17:52.050" Score="2" Body="&lt;p&gt;I advocate for an explanation of why the strategies for quadratics don't immediately work for higher orders. When finding the roots of a quadratic, there are two predominant strategies: factoring, and somehow taking its square root.&lt;/p&gt;&#xA;&lt;p&gt;Using Viete's formulas, we can get sum and product of roots. This is fine for quadratic since we have two roots and two equations, but for higher orders this more or less results in guessing and checking. It's also entirely possible for a real polynomial to have zero rational roots, which evades the rational root theorem. See &lt;a href=&quot;https://math.stackexchange.com/questions/181857/factoring-a-polynomial-having-no-rational-roots&quot;&gt;example&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The other strategy of taking the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th root requires cleverly manipulating the formula into a perfect power. All quadratics can be written as &lt;span class=&quot;math-container&quot;&gt;$0 = \text{multiplier} (\text{power}) + \text{constant}$&lt;/span&gt;, but it's not as simple for higher powers because instead of a constant it's another function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, which is not very useful if you are trying to take its &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th root.&lt;/p&gt;&#xA;&lt;p&gt;Not sure if you absolutely need an &amp;quot;analogy&amp;quot; to explain the intuition, but I'll edit my answer once I think of an appropriate one.&lt;/p&gt;&#xA;" OwnerUserId="18050" LastActivityDate="2021-12-23T04:17:52.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24729" PostTypeId="1" CreationDate="2021-12-23T15:04:55.400" Score="7" ViewCount="4538" Body="&lt;p&gt;I have to teach limits to infinity of real functions of one variable. I would like to start my course with a beautiful example, not simply a basic function like &lt;span class=&quot;math-container&quot;&gt;$1/x$&lt;/span&gt;. For instance, I thought of using the functions linked to the propagation of covid-19 and show that, under the basic model, the number of contaminations will go to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; when time goes to &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt;. However, this is a bad idea because the model is not so easy to explain and moreover students are sick of covid-subjects.&lt;/p&gt;&#xA;&lt;p&gt;Hence, I ask you some help to find interesting examples from physics, geography, etc ... I suppose that an example with &amp;quot;time&amp;quot; going to &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt; would be nice.&lt;/p&gt;&#xA;&lt;p&gt;NB : This a crossed-question with MSE : &lt;a href=&quot;https://math.stackexchange.com/questions/4340560/nice-examples-of-limits-to-infinity-in-real-life&quot;&gt;https://math.stackexchange.com/questions/4340560/nice-examples-of-limits-to-infinity-in-real-life&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19276" LastActivityDate="2021-12-27T17:16:35.373" Title="Nice examples of limits to infinity in real life" Tags="|calculus|limits|mathematics-in-daily-life|infinity|" AnswerCount="8" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24730" PostTypeId="2" ParentId="24729" CreationDate="2021-12-23T16:57:36.303" Score="14" Body="&lt;p&gt;If you are willing to take some time to explain the model and do some simulation, I &lt;em&gt;really&lt;/em&gt; like to show students a logistic growth model (in discrete time).  The basic setup is something like the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Assume that there is a population of rabbits living in some area.  We are going to make a few assumptions about this population:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the rabbits breed once every month (say, at the end of the month);&lt;/li&gt;&#xA;&lt;li&gt;the number of baby rabbits born at the beginning of each month is proportional to the total number of rabbits (essentially, we can assume that every rabbit has some number of babies during the month); and&lt;/li&gt;&#xA;&lt;li&gt;there are limited resources in this area, which means that if there are too many rabbits, rabbits will starve at a faster rate, and fewer rabbits will survive to the beginning of the next month in order to breed.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After some work, we can get a discrete time model, which looks something like the following:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ P(t_{n+1}) = P(t_n)\left( 1 + r\left( 1- \frac{P(t_n)}{K}\right) \right),$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$P(t_n)$&lt;/span&gt; is the rabbit population at the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th timestep (i.e. at the beginning of month &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;), &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is some measure of intrinsic growth (essentially, how many baby rabbits every rabbit would have if the environment did not constrain the growth of the rabbit population), and &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is the carrying capacity of the environment (i.e. the maximum sustainable population size).&lt;/p&gt;&#xA;&lt;p&gt;This model can be simulated pretty quickly with a &lt;a href=&quot;https://docs.google.com/spreadsheets/d/13KTvEvXIhs5MxClbkl6ge8GyhWAJ4O9qIQYagssM99A/edit?usp=sharing&quot; rel=&quot;noreferrer&quot;&gt;spreadsheet&lt;/a&gt;.  I typically don't introduce this example until the second semester of calculus, when we talk (briefly) about differential equations.  But I do think that this would be doable in a first lecture on limits, and I think it pings some intuition (sequential limits are more intuitive than continuous limits, I think).&lt;/p&gt;&#xA;&lt;p&gt;As an added bonus, you can tweak the parameters to see different kinds of behaviour in one model.  For example,  take &lt;span class=&quot;math-container&quot;&gt;$K=1000$&lt;/span&gt; (I'll keep that constant), start with &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; rabbits, and suppose that the intrinsic growth rate is &lt;span class=&quot;math-container&quot;&gt;$r=0.1$&lt;/span&gt;.  Then the population increases monotonically to the carrying capacity:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RV7G6.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RV7G6.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, if &lt;span class=&quot;math-container&quot;&gt;$r = 1.96$&lt;/span&gt;, we can start with an initial population of &lt;span class=&quot;math-container&quot;&gt;$900$&lt;/span&gt; or so, and the population will ping-pong back and forth across the carrying capacity, but eventually die down to a limit:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xIfdh.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xIfdh.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In both of these cases, we can see a sequence which has a limit, but the limit is approached differently.  Once sequence is monotonic, while the other oscillates, but damps down.  Make &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; a bit bigger (say, &lt;span class=&quot;math-container&quot;&gt;$r=2.2$&lt;/span&gt;), and you get something really interesting:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Dx1Qk.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Dx1Qk.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this case, the sequence does not have a limit, but it does oscillate between two different values (more or less).  So the limit does not exist, but it fails to exist in an interesting way.  And then you can see something really crazy by increasing the intrinsic growth rate to something like &lt;span class=&quot;math-container&quot;&gt;$r = 2.8$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kvnqZ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kvnqZ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this case, the behaviour of the system is chaotic.  Again, the limit fails to exist, but we get to see something interesting happen.&lt;/p&gt;&#xA;&lt;p&gt;And, if you really want, you  can also just give them the function which models continuous logistic growth:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ P(t) = \frac{K P(0)}{P(0) + (K-P(0))\mathrm{e}^{-rt}}. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is a limit which can be computed by hand, but one can also graph the behaviour of the function and see how it behaves, e.g. using &lt;a href=&quot;https://www.geogebra.org/calculator/qhjg7wcf&quot; rel=&quot;noreferrer&quot;&gt;GeoGebra&lt;/a&gt;.&lt;a href=&quot;https://i.stack.imgur.com/I9B5t.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/I9B5t.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2021-12-23T16:57:36.303" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24731" PostTypeId="2" ParentId="24704" CreationDate="2021-12-23T21:41:20.220" Score="2" Body="&lt;p&gt;We can rescale a polynomial so its leading coefficient is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; without changing the roots.  So let's focus on polynomials with leading coefficient &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.  If its degree is &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; then it has &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; coefficients after the leading term. Writing down that polynomial amounts to saying what the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; non-leading coefficients are.&lt;/p&gt;&#xA;&lt;p&gt;When you are given a polynomial of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; (with leading coefficient &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;) and want to find its roots, you are essentially told the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; non-leading coefficients and want to get the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; roots from that information. Let's see what system of equations this amounts to for some small values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, starting with &lt;span class=&quot;math-container&quot;&gt;$n = 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$n = 2$&lt;/span&gt;: when&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x^2 + bx + c = (x-r)(x-s) = x^2 - (r+s)x + rs,&#xA;$$&lt;/span&gt;&#xA;we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;b = -r-s, \ \ c = rs.&#xA;$$&lt;/span&gt;&#xA;We are told&#xA;&lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and want to figure out &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;.  So we have to&#xA;find &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;r + s = -b, \ \ rs = c.&#xA;$$&lt;/span&gt;&#xA;This is not a system of linear equations in &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, but with a little algebraic trickery we can &amp;quot;break the symmetry&amp;quot; in &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; to extract &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(r-s)^2 = r^2 - 2rs + s^2 = (r+s)^2 - 4rs = b^2 - 4c \Longrightarrow &#xA;r-s = \pm \sqrt{b^2-4c}.&#xA;$$&lt;/span&gt;&#xA;Thus we have the system of &lt;em&gt;linear&lt;/em&gt; equations&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;r + s = -b, \ \ r - s = \pm\sqrt{b^2-4c}.&#xA;$$&lt;/span&gt;&#xA;This can be solved for &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; by standard methods of algebra.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$n = 3$&lt;/span&gt;: when&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x^3 + bx^2 + cx + d = (x-r)(x-s)(x-t) = x^2 - (r+s+t)x + (rs+rt+st)x - rst,&#xA;$$&lt;/span&gt;&#xA;we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;b = -r-s-t, \ \ c = rs+rt+st, \ \ d = -rst.&#xA;$$&lt;/span&gt;&#xA;We are told&#xA;&lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; and want to figure out &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;.  So we have to&#xA;find &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;r+s+t = -b, \ \ rs +rt+st = c, \ \ rst = -d.&#xA;$$&lt;/span&gt;&#xA;As in degree &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, we are faced with a nonlinear system of equations in the unknowns &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;.  With a &lt;em&gt;lot&lt;/em&gt; more algebraic work, this can be converted into a system of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; linear equations in &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; using cube roots of unity.  But already we see the level of algebra looks more  difficult than in degree &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I am going to stop here, because I think it is already clear the math is getting hard and we're already in degree &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2021-12-23T21:41:20.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24732" PostTypeId="2" ParentId="24704" CreationDate="2021-12-23T23:25:30.420" Score="1" Body="&lt;p&gt;There is a closed form expression for the roots of a quintic, it is just that the solutions do not always simplify to expressions involving field operations and radicals.&lt;/p&gt;&#xA;&lt;p&gt;For example, using Wolfram Alpha, one root of the equation &lt;span class=&quot;math-container&quot;&gt;$x^5-x+a=0$&lt;/span&gt; is given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x_1 =a\phantom{.}{}_4F_3\left(\frac15, \frac25, \frac35, \frac45;\frac12, \frac34,\frac54;\frac{3125a^4}{256}\right)$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$_4F_3$&lt;/span&gt; is a &lt;a href=&quot;https://en.wikipedia.org/wiki/Generalized_hypergeometric_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;generalized hypergeometric function&lt;/a&gt;. The other four roots have similar expressions. This approach involves the Bring radical and the reversion of a Taylor series, e.g., finding the Taylor series for &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(a)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$f(x)=x-x^5$&lt;/span&gt;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Bring_radical#Series_representation&quot; rel=&quot;nofollow noreferrer&quot;&gt;See this reference&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The exact form solutions for a quintic in terms of hypergeometric functions simplify to expressions in radials only in special circumstances, which can be quantified by algebraic properties of the Galois group of the polynomial. In order for the solutions to simplify in terms of radicals, the Galois group must deviate from the generic case of the full symmetric group &lt;span class=&quot;math-container&quot;&gt;$S_5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It turns out that by substitutions, polynomials of degree &lt;span class=&quot;math-container&quot;&gt;$n=2,3,4,5$&lt;/span&gt; can be reduced to the form &lt;span class=&quot;math-container&quot;&gt;$x^n+a_1x+a_0$&lt;/span&gt;. For &lt;span class=&quot;math-container&quot;&gt;$n=2$&lt;/span&gt; this is called &lt;em&gt;completing the square.&lt;/em&gt;  This reduction is not possible in general for &lt;span class=&quot;math-container&quot;&gt;$n\ge6.$&lt;/span&gt; However, there are formulas for the roots of a general polynomial in terms of theta-functions. &lt;a href=&quot;https://en.wikipedia.org/wiki/Thomae%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;You can see these formulas here.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2021-12-29T19:26:21.620" LastActivityDate="2021-12-29T19:26:21.620" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24733" PostTypeId="2" ParentId="24729" CreationDate="2021-12-23T23:53:50.037" Score="7" Body="&lt;p&gt;Deeba and Rushkady Go to Town: A Fanciful Real-Life Story&lt;/p&gt;&#xA;&lt;p&gt;A festival was just starting in town, and Deeba and Rushkady walked toward the square headed to the Infinite Pancake Eating Contest.  The contest would end only when all participants agreed it was over.  Deeba said, “Rushkady, since you always eat pancakes twice as a fast as me, no matter how fast or slow I eat them, I think you should give me a head start.“&lt;/p&gt;&#xA;&lt;p&gt;“How much of a head start do you think would make a fair contest?” asked Rushkady.&lt;/p&gt;&#xA;&lt;p&gt;“One hundred.”&lt;/p&gt;&#xA;&lt;p&gt;“Not enough. I would overtake you.”&lt;/p&gt;&#xA;&lt;p&gt;“One thousand.”&lt;/p&gt;&#xA;&lt;p&gt;“Still not enough.”&lt;/p&gt;&#xA;&lt;p&gt;“No way!” exclaimed Deeba.&lt;/p&gt;&#xA;&lt;p&gt;“Look, if I eat twice as fast as you, then eventually I will eat twice as much as you, whatever head start I give you” replied Rushkady.&lt;/p&gt;&#xA;&lt;p&gt;“Not if I had a head start of a million!”&lt;/p&gt;&#xA;&lt;p&gt;“Yes, I would, or nearly so, if the contest went on long enough.”&lt;/p&gt;&#xA;&lt;p&gt;Who was right?&lt;/p&gt;&#xA;&lt;p&gt;[It also gives an intuitive explanation of one of L’Hôpital’s rules.]&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2021-12-23T23:53:50.037" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24734" PostTypeId="2" ParentId="24729" CreationDate="2021-12-24T00:27:39.913" Score="1" Body="&lt;p&gt;The stability of the solar system is a classic real-life problem about time going to &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt;.  It’s not an easy problem in which to understand the mathematics, not even close to the introductory level of &lt;span class=&quot;math-container&quot;&gt;$1/x$&lt;/span&gt;.  Its pedagogical value lies in the problem statement being easy to understand and to relate to, and it is a problem to which people have devoted a lot of energy over the centuries.  In other words, it’s more inspiring than explanatory.&lt;/p&gt;&#xA;&lt;p&gt;Scott Tremaine traces the stability problem back to Isaac Newton in &lt;a href=&quot;https://www.ias.edu/ideas/2011/tremaine-solar-system&quot; rel=&quot;nofollow noreferrer&quot;&gt;Is the Solar System Stable?&lt;/a&gt;, which calculus students should be able to read, and forward through many famous mathematicians, Lagrange, Laplace,…,Poincaré,….  I think one could argue that the problem goes back further, to Aristotle and probably earlier.  Once you realize that there’s a universe and a solar system that changes, you are bound to wonder how long you or the human race will be safe.  However, one could also argue that it became a problem in mathematical analysis and conceivable in terms of limits at infinity only after Newton developed a mathematical theory of gravitation.&lt;/p&gt;&#xA;&lt;p&gt;According to Tremaine, the orbits will remain stable for as long as the sun is expected to live, about 8 billion years.  That is, in our current understanding, we do not expect time to be able to approach &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt; for those of us in the solar system, not as a real-life problem.  Such limitations of reality afford the teacher an opportunity to discuss why people (and not just calculus students) model phenomena with &lt;span class=&quot;math-container&quot;&gt;$t \rightarrow +\infty$&lt;/span&gt;, when in practice &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; just being large enough is sufficient.&lt;/p&gt;&#xA;&lt;p&gt;A more detailed treatment of the problem may be found in Jacques Laskar, &lt;a href=&quot;https://arxiv.org/pdf/1209.5996.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Is the Solar System Stable?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2021-12-24T00:27:39.913" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24735" PostTypeId="2" ParentId="18911" CreationDate="2021-12-24T12:54:36.227" Score="2" Body="&lt;p&gt;I recently encountered -&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3^{x-1}+2^{x}=5$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I spent too much time trying to use logs to isolate X and got nowhere. I then rearranged to look at it as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$y=3^{x-1}+2^{x}-5$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and that's when I realized that this was a classic case of the need for Newton's Method. I'd say that I'd strongly advise against underestimating the value of this technique.&#xA;(FWIW - After using this method for the problem I shared, I searched here to find Q&amp;amp;A on  this. Ironic to see this as the question that resulted)&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-12-24T12:54:36.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24736" PostTypeId="2" ParentId="24729" CreationDate="2021-12-24T13:41:04.290" Score="3" Body="&lt;p&gt;Maybe these are too easy but...&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;If you drop a bouncy ball, eventually you will stop hearing it bounce!&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Well, we can model the ball by accounting for the predominant mode of energy loss via collision with the floor but ignoring air resistance and assuming constant gravitational acceleration. Then on each bounce with impact speed &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, its rebound speed is &lt;span class=&quot;math-container&quot;&gt;$v·r$&lt;/span&gt; for some constant &lt;span class=&quot;math-container&quot;&gt;$r∈(0,1)$&lt;/span&gt;, which is also called &lt;a href=&quot;https://en.wikipedia.org/wiki/Coefficient_of_restitution&quot; rel=&quot;nofollow noreferrer&quot;&gt;the coefficient of restitution&lt;/a&gt;, where each bounce cuts the ball's kinetic energy down to &lt;span class=&quot;math-container&quot;&gt;$r^2$&lt;/span&gt; times the previous amount.&lt;/p&gt;&#xA;&lt;p&gt;Then by energy conservation the ball's height on each bounce is &lt;span class=&quot;math-container&quot;&gt;$r^2$&lt;/span&gt; times the height of the previous bounce. Hence after &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; bounces the height the ball reaches would be &lt;span class=&quot;math-container&quot;&gt;$r^{2k}$&lt;/span&gt; times the initial dropping height. Taking &lt;span class=&quot;math-container&quot;&gt;$k→∞$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$r^{2k}→0$&lt;/span&gt;, and so for any &lt;span class=&quot;math-container&quot;&gt;$ε &amp;gt; 0$&lt;/span&gt; the ball height eventually no longer exceeds &lt;span class=&quot;math-container&quot;&gt;$ε$&lt;/span&gt;. Since the sound comes from the ball's energy, past that point the bounces will have too little energy for you to hear it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Given any bouncy ball, there is some constant &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; such that no matter how high you drop it, if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; seconds elapse between the first two bounces, then it stops bouncing within &lt;span class=&quot;math-container&quot;&gt;$c·x$&lt;/span&gt; seconds!&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;We use the same assumptions as before. Then the time taken between consecutive bounces forms a geometric progression with ratio &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;. Thus the total time taken by all bounces after the first bounce is &lt;span class=&quot;math-container&quot;&gt;$x·\sum_{k=0}^∞ r^k$&lt;/span&gt;. How is that for a real-life limit to infinity?&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2021-12-24T13:41:04.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24737" PostTypeId="2" ParentId="24729" CreationDate="2021-12-24T22:48:19.530" Score="3" Body="&lt;p&gt;I have actually just finished teaching a first semester of calculus, and when I started it I was concerned about examples to start with. And as far as I can see, first two semesters of differential and integral calculus are about approximations: finite difference is an approximation of derivative (and vice versa!), Riemann sum is an approximation of Riemann integral (and vice versa), member of sequence is an approximation of the limit (and vice versa), etc. Problem is that (in Russia) school students are not trained in approximate calculations at all. They do have a hands-on experience with calculators or numerical software, so they have an IDEA of approximate calculations, but they are not formally trained.&lt;/p&gt;&#xA;&lt;p&gt;So I started with a question: &amp;quot;What is &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;?&amp;quot; Of course, somebody inevitably answered that it is a length of diagonal of the square, and it's not a rational fraction. Then we considered a number of ways to approximate &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; with rationals:&lt;/p&gt;&#xA;&lt;ol start=&quot;0&quot;&gt;&#xA;&lt;li&gt;Initial bounds: &lt;span class=&quot;math-container&quot;&gt;$1&amp;lt; \sqrt{2} &amp;lt; 2$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Finding the next decimal digit by trial and error (much work, boring).&lt;/li&gt;&#xA;&lt;li&gt;Bisection (a bit less work, also boring). Connection to binary system.&lt;/li&gt;&#xA;&lt;li&gt;Continued fractions.&lt;/li&gt;&#xA;&lt;li&gt;I didn't do that, but Taylor expansion might be a good idea. Or not. It's tricky to get a sequence that can be proven to converge to &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Of course, little to no proofs are given at this time. The point is to acquaint students with ideas of:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;quality of approximation (a good approximation lies near the approximated value);&lt;/li&gt;&#xA;&lt;li&gt;convergence (broadly speaking, approximations get better over time);&lt;/li&gt;&#xA;&lt;li&gt;speed of convergence (some types of approximations get better much faster or much slower than the others).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And then you can drop the definition of the limit (I used &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; variant). The point is that a converging sequence might lose some quality of approximation locally, but over the time such losses (dropping below the certain quality threshold &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;) must become impossible. In other words, you can lose some quality of approximation, but if the sequence converges, the quality of approximation will get as good as requested (and stay that good after that).&lt;/p&gt;&#xA;&lt;p&gt;This is actually what I did, and I think the students got a better grasp of &amp;quot;what in god's name is happening?&amp;quot; than usual.&lt;/p&gt;&#xA;&lt;p&gt;Hope this helps :)&lt;/p&gt;&#xA;" OwnerUserId="19281" LastActivityDate="2021-12-24T22:48:19.530" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24738" PostTypeId="2" ParentId="24729" CreationDate="2021-12-24T23:42:56.713" Score="1" Body="&lt;p&gt;A nice easy limit that can be seen in everyday life.&lt;/p&gt;&#xA;&lt;p&gt;Suppose your student looks along a perfectly level road, with perfect visibility. They look down at an angle that allows them to see the road 10m ahead of them. Then 20m ahead. Then 30m ahead. And so on, for any amount of distance ahead.&lt;/p&gt;&#xA;&lt;p&gt;You can easily show, draw, find an equation, even tabulate, and intuitively perceive, that the angle they look at tends ever closer to the horizontal (0 or 90, however you choose to define it), but never quiiiiiiite reaches it.&lt;/p&gt;&#xA;&lt;p&gt;You can also show how limits work in mathematics - for any angle (0-or-90) + ε close to the horizontal you can name, however tiny, the value will get closer to 0 or 90 than that, if you look far enough ahead. (And you can demonstrate/calculate when it will, for 1/100, 1/1000000 and 10&lt;sup&gt;-100&lt;/sup&gt; of a degree if needed, in this case, to make it concrete,and show it never reaches 0-or-90 even so: infinity isn't a &amp;quot;number&amp;quot;.) Classic  &amp;quot;≤ ε&amp;quot; stuff, ties nicely into how and why we define a limit that way.&lt;/p&gt;&#xA;&lt;p&gt;Its one of the easiest examples I can think of, for introducing the concepts or a limit in an everyday context.&lt;/p&gt;&#xA;" OwnerUserId="16176" LastEditorUserId="16176" LastEditDate="2021-12-24T23:48:21.573" LastActivityDate="2021-12-24T23:48:21.573" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24739" PostTypeId="2" ParentId="24729" CreationDate="2021-12-26T06:09:30.070" Score="1" Body="&lt;p&gt;As an amateur photographer, my first thought was the lens position for focus on an object.  It should be easy to intuitively explain that the closer the object the further the image because the lens only bends the rays so much.  As the object gets very far away, the location of the image moves toward a limit, which is the focal length of the lens.  Like the 1/x example, it can show that you don't have to be &amp;quot;close to infinity&amp;quot; to be near the limit.&lt;/p&gt;&#xA;" OwnerUserId="12340" LastActivityDate="2021-12-26T06:09:30.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24740" PostTypeId="2" ParentId="13705" CreationDate="2021-12-26T22:56:47.503" Score="-1" Body="&lt;p&gt;Agents and missions are a good choice for maximum pairing problems for bipartite graphs. Edge linking agent to a mission means that agent can carry out the mission, so there are no edges linking mission to mission or agent to agent.&#xA;Agent can't do another agent, or we have an entirely different problem :)&lt;/p&gt;&#xA;" OwnerUserId="19281" LastActivityDate="2021-12-26T22:56:47.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24741" PostTypeId="2" ParentId="13705" CreationDate="2021-12-27T16:18:26.040" Score="1" Body="&lt;p&gt;It occurred to me in thinking about the &lt;a href=&quot;https://en.wikipedia.org/wiki/M%C3%A9nage_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;ménage problem&lt;/a&gt; that one could reformulate it as a story about students and advisors attending an awards dinner, with the requirement that advisors and students sit alternately, with no student sitting next to their own advisor. One must stipulate that no student has multiple advisors attending the dinner and that no advisor has multiple students attending the dinner.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2021-12-27T16:18:26.040" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24742" PostTypeId="2" ParentId="24729" CreationDate="2021-12-27T17:16:35.373" Score="2" Body="&lt;p&gt;&lt;strong&gt;Answer:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I feel like a lot of these suggestions are quite complicated.  Not complicated for us, but complicated for new-to-the-topic students.  If you must (but see below) go with a life example, I think doing something like Zeno's paradox of motion might be helpful.  first half of the trip, half of the second half of the trip, etc.  (1/2^x)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Too-long-for-comment comment:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I strongly disagree pedagogically with the impulse behind &amp;quot;I would like to start my course with a beautiful example, not simply a basic function like 1/x.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Every new crop of grain does not require a new sickle design.  For the students, these topics, themselves, are new and wonderful revelations, even with simple explanations.  And the theory of gradualism implies starting with the simplest examples first, not the hardest.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, introducing topics as derivations of word problems ADDS COGNITIVE LOAD to what is already a new and moderately difficult task (for the student). &amp;quot;Word problems are hard.&amp;quot;  Yes, verily, they are.&lt;/p&gt;&#xA;&lt;p&gt;One of my pet peeves on PDE texts is they want to introduce topics as derivations of engineering problems in statics or heat transfer.  ODE texts are better as they have a semi-standard method of first teaching, training, practicing the algebra-manipulation of the techniques, with significant homework.  Then, in next chapter, after success with the symbol manipulation, having the students do various word problems (from physics, econ, bio, etc.).  These then give the students practice in translating to/from the real world and symbol world, as well as giving extra practice in the given techniques.  But the cognitive load is easier as the students have already practiced the symbol manipulations, but are adding in some word problem complication on top.  But this is different than having a NEW (to the student) symbol manipulation taught along with all the distractions of an engineering problem (or Covid, or econ or whatever).  And I say all this as someone much more applied in orientation and career than the average college/HS math teacher.   Not opposed to applied topics, just opposed to a poor pedagogical approach.&lt;/p&gt;&#xA;&lt;p&gt;In general, there's a huge theme of questions here showing a desire of the teacher (often much smarter and more math experienced than his students, and usually with a background of grad school research) wanting to do something that INTERESTS THE TEACHER as opposed to what is BEST PEDOGICALLY for his students.&lt;/p&gt;&#xA;&lt;p&gt;For the student, doing his first limits to infinity, a simple function is a feature, not a bug. And while it may be boring to the teacher to use a classic example, used in many different texts, the student is someone seeing the concept itself for the very first time.  It's novel to him.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps there has even been some evolutionary process by which the texts converged on a particular approach, example, intro.  &amp;quot;Tried and true.&amp;quot;  This is not to say one should never innovate or that it is impossible to improve or experiment.  Or that practices in math education have never converged on wrong approaches.  But one should be a bit cynically skeptical of new teachers pushing new methods, to include self skepticism.&lt;/p&gt;&#xA;" OwnerUserId="19293" LastActivityDate="2021-12-27T17:16:35.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24744" PostTypeId="2" ParentId="14653" CreationDate="2021-12-27T20:07:13.593" Score="2" Body="&lt;p&gt;Below is an image that shows the net change in &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; over each interval as a (finite) &amp;quot;differential,&amp;quot; which is the same in this case as a term in a Riemann sum for the integral.  The net change is given by the change in &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; for the black, dashed secant line and is equal to the change in &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; along the purple, solid tangent line obtained via the Mean Value Theorem.  The sum of the net changes over all intervals gives &lt;span class=&quot;math-container&quot;&gt;$F(b)-F(a)$&lt;/span&gt; and is equal to the sum of the differentials, which is a Riemann sum. The appropriate limit is the integral of &lt;span class=&quot;math-container&quot;&gt;$F'=f$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hLGsw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hLGsw.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is an animation of the same idea, with different labeling:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/asLAp.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/asLAp.gif&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;One is accustomed to resort to area as an intuitive basis, since most or all textbooks do. I am.  But position and velocity make a good basis as well, taking &lt;span class=&quot;math-container&quot;&gt;$\Delta F$&lt;/span&gt; to be the change in position and &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; the change in time.  It is from that point of view I find the above figures intuitively convincing, rather than trying to think in terms of area being the height of the graph.  It reflects a typical application of integration, to break a motion from &lt;span class=&quot;math-container&quot;&gt;$F(a)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$F(b)$&lt;/span&gt; into indefinitely many parts, apply some formula, and integrate/sum to obtain that can be used to solve problems.  Such applications were a monthly if not weekly feature of the year-long mathematical methods physics course I took, and the above shows the fundamental first step.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2021-12-27T20:07:13.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24745" PostTypeId="2" ParentId="18911" CreationDate="2021-12-28T10:54:16.620" Score="2" Body="&lt;p&gt;Many traditional calculus classes completely omit all mention of numerical calculation and approximation (in any concrete sense). It is certainly viable to cut Newton's method and it might make sense if students can't properly manipulate logarithms and trigonometric functions.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, numerical calculation and approximation introduce lots of interesting and useful ideas (notice I did not say anything about their direct utility). Newton's method is instructive in this sense. Moreover it is one of the few methods accessible at the elementary level that is actually useful and used in practice (Simpson's rule and, later, Runge-Kutta are the other two that come to mind). Teaching it might help to explain how calculus is used for something other than passing exams.&lt;/p&gt;&#xA;&lt;p&gt;One way of motivating Newton's method is: follow the tangent line until it intersects the horizontal axis. This makes it an instructive application related to derivatives and the tangent line approximation. Often the tangent line approximation (the linear Taylor approximation) is presented as if it were important with little or no application that serves to justify in the student's mind the claimed importance.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps teaching approximation techniques can help eliminate a bit of the magic thinking associated with calculator use. Students generally have no idea how any digital approximation is actually calculated nor that what devices return are approximations. How in the world did anyone calculate square roots before there were handheld calculators? How do calculators do it now?&lt;/p&gt;&#xA;&lt;p&gt;It should not be that hard to teach Newton's method in a short, self-contained way. Students typically like algorithms. Both the bisection method and Newton's method are easy to teach and easy to learn. Both are easily presented in a step by step way. It should be easier to teach Newton's method to mediocre uninterested students than it is to teach the chain rule or integration by parts. What can be a bit complicated about teaching Newton's method is that sometimes it fails. However, it may be good for students to see this too.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2021-12-28T10:54:16.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24747" PostTypeId="1" AcceptedAnswerId="24749" CreationDate="2021-12-29T04:17:02.230" Score="0" ViewCount="408" Body="&lt;p&gt;How can we understand differential equations and Integration in real life so that we can understand calculus easily. All we do here, at university level is memorize calculus and get the answer. We cannot relate these beautiful equations to other physical phenomenon because we just memorize and didn't understand.&lt;/p&gt;&#xA;" OwnerUserId="19303" LastActivityDate="2022-01-14T00:21:48.247" Title="How can I visualize differential equations and Integration in real life?" Tags="|calculus|differential-equations|mathematics-in-daily-life|physical-sciences|engineering-mathematics|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24749" PostTypeId="2" ParentId="24747" CreationDate="2021-12-29T07:57:27.757" Score="4" Body="&lt;p&gt;You have asked two very different questions. I'll leave the differential equations for someone else. There is one particular application of integration which is my favorite last problem to do in Calc I. (We got behind this semester, and I was very sad not to have time for this. It feels like a perfect grand finale to me.)&lt;/p&gt;&#xA;&lt;p&gt;You probably learned the formula for volumes of cones before high school. It makes sense that the base (&lt;span class=&quot;math-container&quot;&gt;$πr^2$&lt;/span&gt;) and the height (h) would be involved. And that it would be smaller than a cylinder. But why &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt;?  There is no easy way to see where the  &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; comes from (that I know of) before you can do integration.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Finding Volume of a Cone&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let's say our cone has radius R and height H. Put the point of the cone at the origin, and run its axis along the x-axis, so that its height is an x-coordinate. Then you know that it will go through the point (H,R). The line through that point and the origin is &lt;span class=&quot;math-container&quot;&gt;$y = \frac{R}{H}x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now we imagine rotating that line around the x-axis, to create the cone. We also imagine slicing the cone vertically, so that the volume is made up of an infinite number of circular disks that are infinitely thin. Each disk has volume &lt;span class=&quot;math-container&quot;&gt;$πr^2h$&lt;/span&gt;, where r becomes &lt;span class=&quot;math-container&quot;&gt;$\frac{R}{H}x$&lt;/span&gt; and h becomes dx (representing an infinitely thin bit of the x-axis), so each slice has volume &lt;span class=&quot;math-container&quot;&gt;$π(\frac{R}{H}x)^2dx$&lt;/span&gt;. Now we add them up. But that is exactly what the definite integral means, adding up an infinite number of infinitely small function values:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{0}^{H}\pi \left ( \frac{R}{H}x \right )^{2}dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now the fundamental theorem of calculus tells you that you can find a value for this by using anti-derivatives:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{0}^{H}\pi \left ( \frac{R}{H}x \right )^{2}dx&#xA;= \pi \int_{0}^{H}  \frac{R^2}{H^2}x^2 dx&#xA;= \pi \frac{R^2}{H^2} \frac{x^3}{3} &#xA;\Big|_0^H&#xA;=\pi \frac{R^2}{H^2} \frac{H^3}{3}&#xA;=\frac{\pi}{3} R^2H $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Tada!&lt;/p&gt;&#xA;&lt;p&gt;If that makes sense, you can also find the volume of a sphere. It's just a little bit harder.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="11" LastEditDate="2021-12-31T22:42:29.973" LastActivityDate="2021-12-31T22:42:29.973" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24750" PostTypeId="2" ParentId="19566" CreationDate="2021-12-29T18:57:25.100" Score="1" Body="&lt;p&gt;To provide a different angle from the great replies already offered, I think other factors include:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the technology required for a computational approach has only (relatively) recently become readily accessible to a wide audience (&amp;amp; still isn't in many places);&lt;/li&gt;&#xA;&lt;li&gt;depending on the approach, the use of software can hijack the course, making it more about learning to use a specific software tool, as opposed to learning the underlying ideas;&lt;span class=&quot;math-container&quot;&gt;$^{\color{red}1}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;numerical approaches have non-trivial limitations -- not just issues related to floating point (&lt;em&gt;e.g.&lt;/em&gt; a naive choice of &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; values can easily lead to erroneous conclusions);&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Of course, there are a bunch of other reasons for the current emphasis on an algebraic approach -- probably enough to fill several volumes! Personally, I've find that students benefit from a mix of approaches.&lt;/p&gt;&#xA;&lt;p&gt;As to your explicit question, I do think physicists and engineers benefit from being comfortable with algebraic manipulations, but certainly not at the expense of a broader understanding of the core concepts of calculs (which, unfortunately, does seem to happen, a lot, although I'm not sure it's fair to place all the fault on algebra...).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^{\color{red}1}$&lt;/span&gt;: In a sense, you could argue this is exactly what happens with the algebraic approach, except the tool that hijacks the course is algebra; this points to Dave Renfro's comment about &amp;quot;students learn algebra in calculus and learn calculus in differential equations.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="5797" LastActivityDate="2021-12-29T18:57:25.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24751" PostTypeId="1" CreationDate="2021-12-29T19:01:00.617" Score="1" ViewCount="202" Body="&lt;p&gt;I'm revisiting the materials I've put together for &lt;strong&gt;students taking a non-proof-based intro to calculus, and my goal is for them to have a clear but rough sense of a limit as a bound&lt;/strong&gt; (basically enough of a genuine understanding for the limit definition of the derivative to have actual meaning to them; but the focus of the course will very much be using calculus as a tool).&lt;/p&gt;&#xA;&lt;p&gt;I've started off by introducing limits via tables of values (using simple functions, to establish the idea of &amp;quot;approaching-but-not-equal&amp;quot;) and would like to next move to direct substitution (and then gradually illustrate the limitations of direct substitution); however, I'm a bit at a loss as to how to introduce direct substitution.&lt;/p&gt;&#xA;&lt;p&gt;The texts I've consulted all introduce direct substitution by invoking continuity in one way or another (&lt;em&gt;i.e.&lt;/em&gt; directly or indirectly), but this is a bit too ouroboros for my liking (insofar as it just muddies the water to invoke continuity to justify limits, when the only definition of continuity they'll encounter uses a limit -- and only later on, at that).&lt;/p&gt;&#xA;&lt;p&gt;At this point, &lt;strong&gt;I'm leaning towards something as vague as &amp;quot;for many functions we can ... &lt;em&gt;introduce substitution&lt;/em&gt;&amp;quot;, -- any suggestions for a cleaner introduction / justification?&lt;/strong&gt; or is this just not possible in the scope of an intro course for &amp;quot;applied calculus&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="5797" LastEditorUserId="5797" LastEditDate="2021-12-30T00:15:14.967" LastActivityDate="2021-12-30T14:25:47.390" Title="Introducing direct substitution in an intro calculus course" Tags="|calculus|concept-motivation|applied-mathematics|limits|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24752" PostTypeId="2" ParentId="24747" CreationDate="2021-12-29T19:15:47.957" Score="0" Body="&lt;p&gt;Ibrahim:&lt;/p&gt;&#xA;&lt;p&gt;I would not think of it as either/or.  Really, you should be able to understand and use the concepts in various ways.  Algebraic symbol manipulation, &amp;quot;word problems&amp;quot; (to include not just physics, but chem, econ, business, bio, etc.), as well as graphical views.  Perhaps even other conceptions (e.g. 500+ years ago, algebra was less developed and even basic algebra problems were &amp;quot;word problems&amp;quot; (but not practical situations, just verbose description of equations).&lt;/p&gt;&#xA;&lt;p&gt;All of this said, it's very important not to exclude the symbol-pushing in a desire to get graphical or engineering approaches.  Symbol-pushing was developed because it is easy and efficient and fast.  Now, if you choose to add some graphical methods or practical word problems, fine.  But still, make sure the kids are strong on the basic x-pushing around stuff.  For one thing, many of these problems are solved by translation to/from the abstracted symbol world.  Also, really the abstracted symbols are usually easier.&lt;/p&gt;&#xA;&lt;p&gt;For Integration:&lt;/p&gt;&#xA;&lt;p&gt;All of this said, my recommendation would be to add moderate amounts of simpler word problems at the end of lessons or chapters, once the students are practiced in the techniques.  Of course some topics are more prone to this than others (e.g. related rates).  Also some topics are inherently more geometrical (e.g. volumes of rotated surfaces).  Or even just curve visualization (the whole blabla about finding extremum, roots, aspymptotes, etc.)  But don't feel the need to make every topic have a geometric or word-problem approach.  For instance, partial fractions are just a storm of algebra pushing symbols...roll with it and just treat it as what it is (a grunch of algebra).  But of course Archimedes method is a very geometric topic (at least in its explication).  So...be eclectic.&lt;/p&gt;&#xA;&lt;p&gt;Also, one practical uses of integration are just &amp;quot;box counting&amp;quot; on graph paper (estimating the integral).  For instance, King Hubbert made the point that with a fixed reserve of oil to deplete, the exact shape, timing, or even height of a peak (or multiple ones) in the production curve is irrelevant to the idea that in the end, the total integral under the curve (boxes on graph paper) is fixed.  [His ideas were flawed as his estimates were low, plus total resource is actually affected by future prices and technology...but the basic explanation in his 1956 paper is a beautiful description of the concept of integral calculus, even for a non-ideal functional curve.]&lt;/p&gt;&#xA;&lt;p&gt;Another knuckle-cruncher aspect of integration that any older scientist will be familiar with is weighing cut paper curves (e.g. NMR spectra) to get the area under the curve.  I probably wouldn't waste time in math class on that, let the chemistry teachers do it.  But if you want it, there's an activity.&lt;/p&gt;&#xA;&lt;p&gt;For Differentiation:&lt;/p&gt;&#xA;&lt;p&gt;Pedagogically, I love the approach that most ODE texts use to do most explication and practice FIRST in symbol pushing world and then later do some engineering type problems.  Just think this is easier on the students.  Word problems are hard, so it's just easier to get a new technique without a bunch of spinach about flow rates in and out of tanks or the like.  But then afterwards, you can do a host of problems to at least get some exposure and intuition about differential equations in the dynamics of kinematics, fluids, electronics, radioactivity, etc.  [And most STEM will get more in their physics and engineering classes, but this is a good start for them.]&lt;/p&gt;&#xA;&lt;p&gt;Two common topics in ODE texts that have some graphical aspects are phase plane and predator prey system of equations.  So if you want that aspect, well enjoy the time on those topics at least.  Personally, I felt like they were a bit extraneous to the hard core symbol pushing, like in the second order non-homo constant coefficient DE.  But I guess you could try to push students like me to spend time on them.  (But it will be tough, since most schools don't give adequate time to cover a whole ODE course and you may be under pressure to cut topics, not do enrichment.)&lt;/p&gt;&#xA;" OwnerUserId="19308" LastEditorUserId="19303" LastEditDate="2022-01-14T00:21:48.247" LastActivityDate="2022-01-14T00:21:48.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24753" PostTypeId="2" ParentId="2754" CreationDate="2021-12-29T19:34:28.930" Score="0" Body="&lt;p&gt;I think if you have a job/task where you need to do this conversion routinely (e.g. moving between two different data sets), than it makes sense to just have this small conversion factor memorized.  Derive it once and then practice with it a lot and just have it in ready access memory as a known quantity (like sin of 30 degrees).&lt;/p&gt;&#xA;&lt;p&gt;But in the likely case you are not doing very routine work emphasizing just these two units, I think it's actually better to just do it the &amp;quot;long way&amp;quot; each time.  After all, there will be a blizzard of units (grams, moles, mm Hg, BTUs, horsepower, tons of refrigeration, electon-volts, etc. etc. that need conversion).&lt;/p&gt;&#xA;&lt;p&gt;It is a normal and helpful muscle-building part of chemistry class to get practice with &amp;quot;railroad track&amp;quot; problems.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.katmarsoftware.com/articles/railroad-track-unit-conversion.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.katmarsoftware.com/articles/railroad-track-unit-conversion.htm&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Look, for example, at one of the later, longer, examples in this article.  Note also that one of the good things about this method is, because of the emphasis on unit cancellation, you are less likely to &amp;quot;use a ratio upside down&amp;quot; (something I have seen award-winning scientists, McKinsey/Goldman sharpies, and even dummies like myself do...it is a common error).&lt;/p&gt;&#xA;&lt;p&gt;So, yeah...make em line up the whole railroad track of conversions.  1000m/km, 60 s/min, 60min/hr.  And rattle them all down.  Yes, the answer is surprisingly small/simple (3.6).  But unless students are routinely using just that conversion, rather than having many to deal with, I think it's wrong move to memorize this specific conversion.&lt;/p&gt;&#xA;" OwnerUserId="19308" LastActivityDate="2021-12-29T19:34:28.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24754" PostTypeId="2" ParentId="9837" CreationDate="2021-12-29T19:59:35.653" Score="0" Body="&lt;p&gt;A small example:&lt;/p&gt;&#xA;&lt;p&gt;Calculators were allowed on the 1983 in USA (I took it) AP Calculus tests.  There had been a long semi-political push to &amp;quot;get technology in&amp;quot;.  However, students using calculators performed worse than those that did not.  Ha!  So they pulled it out of the AP Calc.  (And delayed putting it into SATs, GREs, etc., which was the real end game of the tech pushers.)&lt;/p&gt;&#xA;&lt;p&gt;The tech-pushing NCTM licked its wounds for a decade, but eventually the Empire Struck Back with TI-time-required AP calculus problems in the mid 90s.&lt;/p&gt;&#xA;&lt;p&gt;Comment:&lt;/p&gt;&#xA;&lt;p&gt;For me, the issue has always been about a fetishization of &amp;quot;technology&amp;quot; (gotta get them computers in the classroom so our kids can compete with the Japanese...but then they sit and get dust...and the kids don't learn the basics either).&lt;/p&gt;&#xA;&lt;p&gt;By the late 70s, early 80s, scientific calculators (Sharp or Casio or those crazy Polish HPs) were ubiquitous in high school physics and (especially) chemistry classes.  Cheap, easy to use, and allowed without any objection by teachers, to deal with the inconvenient final calculations of solved problems in stoichiometry, gas laws, equilibrium, etc.&lt;/p&gt;&#xA;&lt;p&gt;They really weren't (and still aren't) needed to deal with practice problems in calculus.  My point is that it shouldn't be about loving or hating the &amp;quot;tech&amp;quot;, but about the right tool for the right job.  The reality of tech.  Not the silly/stupid political will to dump money and appear to be doing something.  Not forcing things like the &amp;quot;squint at screen&amp;quot; TI-required questions.&lt;/p&gt;&#xA;&lt;p&gt;In general, I don't find strong-at-algebra-manipulation students to have a hard time when moving into doing Excel.  Or when using a calculator in chem class.  It's kind of easy compared to partial fractions or epsilon delta games.  So there's not really some crucial need for strong students to practice key-pushing.  If anything, it can be a crutch for weaker students.  I'm always amused by the yearly questions on College Confidential by students who are aghast that their college math class won't allow calculators (and the invariable, true, answer is &amp;quot;you don't need them&amp;quot;).&lt;/p&gt;&#xA;" OwnerUserId="19308" LastEditorUserId="2083" LastEditDate="2024-01-16T06:37:13.750" LastActivityDate="2024-01-16T06:37:13.750" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24755" PostTypeId="2" ParentId="18772" CreationDate="2021-12-29T20:22:07.600" Score="2" Body="&lt;p&gt;The following is not a foolproof solution, but one thing I've done for remote exams is have questions where you have students make up their own numbers, you can add some restrictions to make sure they don't make a problem too easy or find a problem done in the textbook.&lt;/p&gt;&#xA;&lt;p&gt;For example, I've asked things like&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;here is &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; matrix with two or three entries blank. Place some non-zero numbers in those positions and compute the determinant.&amp;quot; They have to show their work, and if they show the correct process, and I don't care about the answer and so that way I don't have to actually compute the determinant.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Find a 3 x 3 matrix with (1,1,1) in the nullspace. (When grading, I just have to sum the rows to make sure they add up to 0.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Make up a system of 3 equations in 5 variables and show how to solve it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Using this method, I have caught a few students using Chegg (their number choices were identical to the ones chosen by Chegg, in addition to rest of the work identical layout, etc). This doesn't circumvent someone from hiring a &amp;quot;tutor&amp;quot; (or going to a website with no searchable posts) to solve problems as then I do not have a paper trail.&lt;/p&gt;&#xA;&lt;p&gt;I haven't taught calculus in a while, but maybe something like &amp;quot;let f(x) be a trig function, g(x) be an exponential function. Compute the derivative of some non-constant function built out of f and g and involving the term f*f (or you could be more specific and ask compute f(3x)g(9x), f(g(3x)), etc&lt;/p&gt;&#xA;&lt;p&gt;Obviously things like WolframAlpha can still be used for this.&lt;/p&gt;&#xA;&lt;p&gt;Edit March 2022:&#xA;I’ve also in some classes giving up on exams by having students record a video of them explaining the material  as if they were teaching it to someone who doesn’t know it… this isn’t directly relevant to your question about online exams, but I only came to this idea after getting tired of cheating on online exams despite trying various things, that this assignment is harder (but not impossible) to cheat on, and in some good ways, possibly better for learning than an exam, but I don’t have data to back this assertion.&lt;/p&gt;&#xA;" OwnerUserId="10289" LastEditorUserId="10289" LastEditDate="2022-03-16T04:24:40.460" LastActivityDate="2022-03-16T04:24:40.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24756" PostTypeId="2" ParentId="24751" CreationDate="2021-12-30T14:17:48.110" Score="3" Body="&lt;p&gt;I think that it makes sense to introduce continuity in the same lesson that you introduce limits.&lt;/p&gt;&#xA;&lt;p&gt;Here is a sketch of a lesson plan:&lt;/p&gt;&#xA;&lt;p&gt;Give them this link&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/rlu2zgcjyf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.desmos.com/calculator/rlu2zgcjyf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Group work:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is &lt;span class=&quot;math-container&quot;&gt;$f(2)$&lt;/span&gt; defined or undefined?&lt;/li&gt;&#xA;&lt;li&gt;As &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, what are the values of &lt;span class=&quot;math-container&quot;&gt;$f(t)$&lt;/span&gt; approaching?&lt;/li&gt;&#xA;&lt;li&gt;Make a table of values for &lt;span class=&quot;math-container&quot;&gt;$t = 1.9, 1.99, 1.999$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t=2.1, 2.01, 2.001$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Summarize and introduce the words and notation for limits.  Stress the connection between the numerical and graphical approaches.&lt;/p&gt;&#xA;&lt;p&gt;Then show them this related graph:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/zxivpgtrfl&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.desmos.com/calculator/zxivpgtrfl&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Point out that, in this graph &lt;span class=&quot;math-container&quot;&gt;$g(2)$&lt;/span&gt; exists and is equal to the limit which had been calculated before.  Define continuity at a point.&lt;/p&gt;&#xA;&lt;p&gt;Now reveal the formulas:&lt;/p&gt;&#xA;&lt;p&gt;The formula for the first function is &lt;span class=&quot;math-container&quot;&gt;$f(x) = \frac{x-2}{x^2 - 3x + 2}$&lt;/span&gt;.  Plug in &lt;span class=&quot;math-container&quot;&gt;$f(1.99)$&lt;/span&gt; as a sanity check (and to reinforce the numerical/symbolic connection).&lt;/p&gt;&#xA;&lt;p&gt;Point out that attempting to plug in &lt;span class=&quot;math-container&quot;&gt;$x= 2$&lt;/span&gt; results in &lt;span class=&quot;math-container&quot;&gt;$\frac{0}{0}$&lt;/span&gt;, which is undefined.  If we make &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; close to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, but not equal to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, then we will get a number close to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; divided by a number close to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.  Spend some time here.  If you have clickers you could ask some multiple choice questions like &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{0.0000001}{7}$&lt;/span&gt; is (a) extremely small, (b) extremely large, (c) neither&amp;quot;.  Then repeat for &lt;span class=&quot;math-container&quot;&gt;$\frac{7}{0.0000001}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{0.0001}{0.0000001}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{0.0000001}{0.0001}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\frac{0.0002}{0.0001}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Summarize the findings:  when investigating comparing fractions making the numerator smaller makes the fraction smaller, but making the denominator smaller makes the fraction larger.  In the case we are looking at there is a &amp;quot;tug of war&amp;quot;:  both are approaching zero.  Which of these drives will win? As the last three numerical examples show in the last paragraph, the answer depends on the relatives sizes of the two quantities, which is hard to predict by just looking at the formula.&lt;/p&gt;&#xA;&lt;p&gt;Question:  can anyone simplify the formula &lt;span class=&quot;math-container&quot;&gt;$\frac{x-2}{x^2-3x+2}$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Answer:  Someone gets &lt;span class=&quot;math-container&quot;&gt;$g(x) = \frac{1}{x-1}$&lt;/span&gt;.  Point out that this function is identical to the one we started with, expect it &lt;strong&gt;is&lt;/strong&gt; defined at &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt;.  Plug in &lt;span class=&quot;math-container&quot;&gt;$g(1.99)$&lt;/span&gt; as a sanity check.  In other words, this is the formula for the second function we graphed.&lt;/p&gt;&#xA;&lt;p&gt;Can we tell the limit of this function as &lt;span class=&quot;math-container&quot;&gt;$x \to 2$&lt;/span&gt; easily?  Yes!  If &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is getting close to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, it makes sense that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x-1}$&lt;/span&gt; is getting close to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2-1} = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In other words, we have found &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x \to 2} f(x)$&lt;/span&gt; by using algebra to find another function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$f(x) = g(x)$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x \neq 2$&lt;/span&gt;, but &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt;.  So finding &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x \to 2} g(x)$&lt;/span&gt;  is easy:  just plug it in (direct substitution).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The work I wish I could require for these kinds of problems reflects this way of thinking.&lt;/p&gt;&#xA;&lt;p&gt;In my ideal world, here is what a student response to the following limit problem would look like:&lt;/p&gt;&#xA;&lt;p&gt;Find &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x \to 4} \frac{x-4}{\sqrt{x} - 2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The function &lt;span class=&quot;math-container&quot;&gt;$f(x) = \frac{x-4}{\sqrt{x} - 2}$&lt;/span&gt; has a domain of &lt;span class=&quot;math-container&quot;&gt;$x \in [0,4) \cup (4, \infty)$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not defined at &lt;span class=&quot;math-container&quot;&gt;$x=4$&lt;/span&gt;, it is not continuous at &lt;span class=&quot;math-container&quot;&gt;$x=4$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is close to &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, the formula for &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; gives me something of the form &lt;span class=&quot;math-container&quot;&gt;$\frac{\textrm{close to 0}}{\textrm{close to 0}}$&lt;/span&gt;, which is indeterminate.  So I will need to do more work to determine the limit.&lt;/p&gt;&#xA;&lt;p&gt;However,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;f(x) &amp;amp;= \frac{x-4}{\sqrt{x} - 2}\\&#xA; &amp;amp;= \frac{(\sqrt{x}-2)(\sqrt{x}+2)}{\sqrt{x} - 2}\\&#xA; &amp;amp;= \frac{1}{\sqrt{x}+2}&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus &lt;span class=&quot;math-container&quot;&gt;$g(x) = \frac{1}{\sqrt{x}+2}$&lt;/span&gt; is a new function which is continuous on &lt;span class=&quot;math-container&quot;&gt;$[0, \infty)$&lt;/span&gt; and agrees with &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; everywhere except for &lt;span class=&quot;math-container&quot;&gt;$x=4$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA; \lim_{x \to 4} f(x) &amp;amp;= \lim_{x  \to 4} g(x)\\&#xA; &amp;amp;= g(4)\\&#xA;   &amp;amp;= \frac{1}{\sqrt{4}+2}\\&#xA;   &amp;amp; = \frac{1}{2+2}\\&#xA;   &amp;amp;= \frac{1}{4}&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x \to 4} \frac{x-4}{\sqrt{x}-2} = \frac{1}{4}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I will also note that this kind to analysis pinpoints why so many instructors would be uncomfortable with the work:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\displaystyle \lim_{x \to 4} (x+3) = \displaystyle \lim_{x \to 4} 7 = 7$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;even though all of the equalities are correct.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2021-12-30T14:25:47.390" LastActivityDate="2021-12-30T14:25:47.390" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24758" PostTypeId="2" ParentId="10939" CreationDate="2021-12-31T12:31:59.420" Score="4" Body="&lt;p&gt;First, metacognition may sometimes actively interfere with task performance. Second, the costs of engaging in metacognitive strategies may under certain circumstances outweigh its benefits. Third, metacognitive judgments or feelings involving a negative self-evaluation may detract from psychological well-being.&lt;/p&gt;&#xA;" OwnerUserId="19314" LastActivityDate="2021-12-31T12:31:59.420" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24759" PostTypeId="2" ParentId="24747" CreationDate="2021-12-31T12:43:09.777" Score="0" Body="&lt;p&gt;Differentiation and integration considered by all scientists throughout the ages as one of the best sciences that guided the mind of man over all times&lt;/p&gt;&#xA;&lt;p&gt;The fields of the use of calculus are very wide. It enters into many fields and are not limited to specific people or to those who use it only. But to almost all human beings. Here are some examples of its benefits:&lt;/p&gt;&#xA;&lt;p&gt;1-What do we do if we are asked to calculate the amount of water required to fill a large swimming pool?&lt;/p&gt;&#xA;&lt;p&gt;The answer is to determine the shape of the swimming pool and find its size. Therefore, we find the size of the water that will fill it. If it is a cubic or parallel rectangle, or .. or .., finding its size is not difficult in any way because these geometric shapes are regular.any student can find their size..But ... what if the shape of the swimming pool is not a regular geometric shape !! it begins with a slight gradient and then the slope descends steeply. Then the sides of the pool become curved, or semi-elliptical. Then it tends to rise slightly. Is it (easy) to find the size of the water to fill this The pool ? Of course yes !! it is the science of calculus!&lt;/p&gt;&#xA;&lt;p&gt;2-Cars:&lt;/p&gt;&#xA;&lt;p&gt;A car doesnt leave from the factory without knowing where the center of its mass and weight and central axis, to determine the factors of security and safety on different roads and different speeds of the car ..&lt;/p&gt;&#xA;&lt;p&gt;This is done only by calculus.&lt;/p&gt;&#xA;&lt;p&gt;3-Design:&lt;/p&gt;&#xA;&lt;p&gt;The graphic engineer uses calculus to determine the difference and change of three-&#xA;dimensional models and how it will change when exposed to multiple conditions. This helps him to create a very realistic environment in 3D movies or video games.In games like the Need 4 speed or GTA, all the statistics that the player sees during the race on the screen include the speed of the car and the distance between him and the contestants and the time between his car and each car of the participants in the race in the second and tenth of the second !! these calculations of differentiation and integration are done in the moment and moment while enjoying the game !! This principle applies to many games ..&lt;/p&gt;&#xA;&lt;p&gt;4- Space&lt;/p&gt;&#xA;&lt;p&gt;Space engineers often use calculus when planning long missions to launch a probe because they need different speeds in the probe's orbit proportional to gravity and altitude. Calculus helps them to determine all these variables with infinite accuracy !!&lt;/p&gt;&#xA;&lt;p&gt;also uses calculus physicists, doctors, biologists, chemists, astronauts, economists, technologists and others ...&lt;/p&gt;&#xA;&lt;p&gt;Let me pass you a quote given by a great Physicist, Engineer, Inventor.&lt;/p&gt;&#xA;&lt;p&gt;The day science begins to study non-physical phenomena, it will make more progress in one decade than in all the previous centuries of its existence.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nikola Tesla.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Basically calculus is used in every single thing in our day and knowing it saves hundreds of years&lt;/p&gt;&#xA;&lt;p&gt;I hope I helped a little ..&#xA;HAVE A GREAT DAY&lt;/p&gt;&#xA;" OwnerUserId="19314" LastActivityDate="2021-12-31T12:43:09.777" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24760" PostTypeId="1" CreationDate="2022-01-01T08:23:33.880" Score="1" ViewCount="418" Body="&lt;p&gt;Too many students lack the luxuries of time and effort to mull exercises and problems. They must juggle MULTIPLE jobs to pay exorbitant tuition fees. Single parents or adult learners  must prioritize their children or full time job. &lt;em&gt;&lt;strong&gt;Presume these unquestionable harsh realities in the USA — please don't challenge these postulates herein.&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;They admit to me — after reading the textbook or watching lectures, they dive right into perusing full solutions, then asking questions about the full solution that bewilder them.&lt;/p&gt;&#xA;&lt;p&gt;But they reaffirm  this time saving method is craftier, and DOESN'T harm their learning or grades one tad! Rather than spending &lt;span class=&quot;math-container&quot;&gt;$t_1$&lt;/span&gt; on attempting the solution (&lt;strong&gt;which they completely lack in reality&lt;/strong&gt;), then &lt;span class=&quot;math-container&quot;&gt;$t_2$&lt;/span&gt; on understanding parts of the solution that befuddle them — they find it shrewder to skip &lt;span class=&quot;math-container&quot;&gt;$t_1$&lt;/span&gt; and start at &lt;span class=&quot;math-container&quot;&gt;$t_2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Are these students correct? Any peer reviewed evidence? Can commencing with, and tearing into, the full solutions — before and without attempting to solve exercises or problems at all — improve your learning and grades?&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="470" LastEditDate="2022-01-01T21:32:43.917" LastActivityDate="2022-01-01T21:32:43.917" Title="Is it more efficacious, productive to jump to perusing full solutions — before and without attempting to solve problems?" Tags="|mathematical-pedagogy|usa|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24761" PostTypeId="1" CreationDate="2022-01-01T14:42:03.480" Score="5" ViewCount="511" Body="&lt;p&gt;Okay. My questions are: How do some people do doctorates in mathematics and spend so much time like three to six years trying to answer one or two open problems? How do they have the patience, psychological strength, perseverance to solve those problems for so much time?&lt;/p&gt;&#xA;&lt;p&gt;Other questions: People like Perelman, Wiles who spend seven years each one of them for the Poincare conjecture, Fermat's last theorem to prove them respectively, how do they do it? Do they know beforehand some of the possible ways to solve those problems? What should someone study to ultimately solve problems like those? How will he know if he follows the correct paths to prove them?&lt;/p&gt;&#xA;&lt;p&gt;Should i try to solve problems that do not take so many years or months to solve at the beginning at least? Perhaps they take like two weeks or a month to solve or less than that? How could i find those open problems?&lt;/p&gt;&#xA;&lt;p&gt;I am an undergraduate student in math at university. I try to read some math and physics that are not included in the material covered in my undergraduate studies.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastActivityDate="2022-08-04T04:16:29.410" Title="Doctorate and examples of difficult solved problems" Tags="|proofs|self-learning|problem-solving|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="24765" PostTypeId="1" CreationDate="2022-01-01T23:26:26.413" Score="14" ViewCount="995" Body="&lt;p&gt;I believe that good math courses are structured around measurable learning goals. For example, &amp;quot;can correctly replace a line integral with an equal double integral using Green's Theorem&amp;quot; or &amp;quot;can use the Sylow Subgroup Theorems to prove the smallest non-abelian simple group has order 60.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I also believe that good math courses should give students a deeper understanding than &amp;quot;follow these rules&amp;quot;. In particular, at least the major level courses should certainly include proofs of main theorems, such as the Sylow Subgroup Theorems.&lt;/p&gt;&#xA;&lt;p&gt;But I cannot come up with a measurable learning goal (one that doesn't start with &amp;quot;understands&amp;quot;) that would motivate a class activity that includes proving important theorems.&lt;/p&gt;&#xA;&lt;p&gt;My example of using the Sylow Theorems is a good learning goal, but if that is the only goal, there is no reason to explain the proof of the Sylow Theorems. But I also feel that &amp;quot;Can reproduce a proof of the Sylow Theorems&amp;quot; is going too far.&lt;/p&gt;&#xA;&lt;p&gt;I hope you can understand the tension I'm getting at. Have you found any way to justify teaching well-known (beautiful!) proofs in a learning-goal motivated course design?&lt;/p&gt;&#xA;" OwnerUserId="19319" LastActivityDate="2022-01-03T17:57:17.310" Title="Is there a measurable learning goal related to understanding proofs of important theorems?" Tags="|proofs|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="24767" PostTypeId="2" ParentId="24765" CreationDate="2022-01-02T06:01:37.610" Score="5" Body="&lt;p&gt;Your quest for measurable learning outcomes suggests a framework rooted in Bloom’s Taxonomy and the higher-education assessment industry that has been built around it. As I think about your question, I think about inquiry-based learning [IBL], especially when implemented in upper-division proof-based mathematics classes. In this &lt;a href=&quot;https://www.ams.org/journals/notices/201706/rnoti-p570.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Notice of the American Mathematical Society Doceamus&lt;/a&gt; article, we find this description:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Students in IBL proof-based courses are asked to develop the fundamental concepts and to produce the proofs of the important theorems. This may require abandoning a traditional textbook in favor of a customized sequence of tasks that meets&#xA;the students where they are mathematically&#xA;and is designed to guide them on a journey&#xA;of mathematical discovery. &lt;em&gt;As opposed to&#xA;completing exercises after an instructor&#xA;has covered the relevant material,&lt;/em&gt; students&#xA;decipher definitions, explore examples,&#xA;make conjectures, and prove theorems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These &lt;a href=&quot;http://www.inquirybasedlearning.org/ibl-video&quot; rel=&quot;nofollow noreferrer&quot;&gt;videos&lt;/a&gt; from the Academy of Inquiry Based Learning show how IBL can look in a classroom.&lt;/p&gt;&#xA;&lt;p&gt;Evidently, the “learning outcomes” of such an IBL approach are not neatly measured by the so called Bloom taxonomy action verbs of the assessment industry. Of course it is not realistic to use IBL in every undergraduate mathematics course, and I am not suggesting that you need to adapt this approach. Rather, I am pointing out that it is OK to give yourself license and not be a slave to  measurable learning outcomes. It’s OK to design your course so that you can include proofs of important theorems in your class. Not everything you do in class as the instructor needs to be associated to an “action verb” for assessment.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="12619" LastEditDate="2022-01-02T21:20:40.757" LastActivityDate="2022-01-02T21:20:40.757" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24768" PostTypeId="2" ParentId="24765" CreationDate="2022-01-02T17:17:30.700" Score="6" Body="&lt;p&gt;I assign this reading early on in the kind of undergraduate classes where I expect students to understand proofs:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ma.rhul.ac.uk/%7Euvah099/Maths/HoddsAlcockInglisSelfExplanation.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.ma.rhul.ac.uk/~uvah099/Maths/HoddsAlcockInglisSelfExplanation.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This handout is based on the following research:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/10.5951/jresematheduc.45.1.0062#metadata_info_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;https://www.jstor.org/stable/10.5951/jresematheduc.45.1.0062#metadata_info_tab_contents&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Once students understand the difference between &amp;quot;self-explanation&amp;quot; vs. &amp;quot;paraphrasing&amp;quot; or &amp;quot;monitoring&amp;quot;, I can then make a measurable standard like:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;I can write self-explanations for various proofs of the Sylow Theorems&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;This does not require a student to reproduce a proof (not one of your goals), but it does require that the student can comprehend a proof that someone else has written and provide evidence.  I probably wouldn't put such an assessment as part of a timed examination, but I might assign it as part of a homework assignment, or the take home portion of an examination.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-01-02T17:17:30.700" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24769" PostTypeId="1" CreationDate="2022-01-03T00:25:05.757" Score="5" ViewCount="158" Body="&lt;p&gt;I am a sophomore math undergraduate and so far all of my university courses have been online due to the pandemic. I am really curious what you guys think about the efficiency of teaching mathematics via a drawing tablet compared to teaching it using the regular blackboards/whiteboards/whatever other types of boards exist that are found in most lecture halls.&lt;/p&gt;&#xA;&lt;p&gt;Let me give some motivation for this question since I think that it may seem odd for a student to ask this. Many of my professors claim that online math lecturing is not as good as in person math lecturing due to the limited space you have when you use a drawing tablet (Note: I am only going to focus on this point in this question. It is quite obvious that in person teaching is much better because we are social beings, but I want to discuss this particular aspect of online teaching). By this they mean that instead of having, say, 3 blackboards that you can write on, you just have as much space as your screen and that's basically it. Since I have only taken university math courses online, I can't really decide whether they are right about this and I don't want to judge based only on my high school experience. But the fact that this has been constantly brought up by many different people has inevitably got me thinking about whether my lectures would be much better (I have to mention that I believe that they are good in general) if classes were in person or if this is just something that stems from, say, a more canonical view towards lecturing. If you ask me, the biggest advantage of these drawing tablets is that the professor can save their notes and then send them to their students and in this way you basically get &amp;quot;free&amp;quot; lecture notes (by free I mean that the lecturer doesn't need to spend some extra time typesetting some notes, he just maybe has to polish a bit what he wrote). I find this to be superior to the traditional setting where you have to worry about getting the notes from someone in case you have some emergency and can't attent the lecture. In my view this advantage is pretty big and it seems to me that maybe after the pandemic is over and everyone resumes in person classes these drawing tablets could still be useful for some lectures if not all. But then again, my university education so far has been online, so I don't think that I am necessarily right. That's why I thought that this may be a good question to ask here.&lt;/p&gt;&#xA;" OwnerUserId="19326" LastActivityDate="2022-01-03T00:25:05.757" Title="What do you think, is teaching on an actual board more efficient than using an online board?" Tags="|undergraduate-education|technology-in-education|online-instruction|online|lecturing|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24770" PostTypeId="1" CreationDate="2022-01-03T14:17:59.977" Score="3" ViewCount="103" Body="&lt;p&gt;I am teaching a linear algebra class for math majors and non-majors out of the first 4 chapters of Lay's book. My plan is to have the students read a section prior to each class, have them answer a few questions about the reading (submitted online before class), and discuss their answers during class. (There will be a group work component too, but that's not why I am here.)&lt;/p&gt;&#xA;&lt;p&gt;I am trying to think of questions that would help them engage thoughtfully with the material. I feel good about asking some questions like &amp;quot;find the row echelon form of the following matrix,&amp;quot; but I think there are some more foundational questions that I might ask, like &amp;quot;What new results/definitions are introduced in this section&amp;quot; or &amp;quot;how does this section relate to the previous section,&amp;quot; or &amp;quot;what new problem is introduced in this section&amp;quot; etc. The kinds of question that experts ask themselves all the time (possibly without thinking about it).&lt;/p&gt;&#xA;&lt;p&gt;Can you suggest any such questions?&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2022-01-03T15:54:18.857" Title="Questions to help better understand the textbook" Tags="|undergraduate-education|linear-algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24771" PostTypeId="2" ParentId="24770" CreationDate="2022-01-03T15:54:18.857" Score="2" Body="&lt;p&gt;Lay is the textbook I use. (And I love it.) I wonder if you'd find the true-false questions valuable. There are usually 8 of them (2 problems near #23 and 24, usually with 4 parts each). Those seem like questions you could ask after the reading but before lecturing.&lt;/p&gt;&#xA;&lt;p&gt;[I don't do what you're doing. I do have them take notes on each section, but they turn those in with the homework, after lecture.]&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-01-03T15:54:18.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24772" PostTypeId="2" ParentId="24765" CreationDate="2022-01-03T17:57:17.310" Score="6" Body="&lt;p&gt;Several professors I've had over the years, when asked why they were showing us a proof well beyond our fabrication, justified it as follows.&lt;/p&gt;&#xA;&lt;p&gt;Mathematics essentially contains a finite number of &amp;quot;tricks&amp;quot;. As you develop your skill in mathematics, you accrue more of these tricks that you have seen employed in various contexts. It is rare that a proof requires an ingenious trick -- it is more common that a proof is a series of these tricks strung together with some mathematical thread. There is ingenuity in this stringing together, of course, but the tricks are not new. The justification given is then two-fold:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To show us more of these tricks so that we may go on to use them (or modifications) in subsequent proofs. Famous proofs are often famous because they used a new, ingenious trick.&lt;/li&gt;&#xA;&lt;li&gt;To show us how one goes about stringing together these tricks into a coherent proof.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I might add a third reason as well. Mathematics is beautiful, and I think it is important that students gain an appreciation for this. Showing them a proof you find beautiful, and explaining why it is so, is definitely working towards that goal.&lt;/p&gt;&#xA;" OwnerUserId="19332" LastActivityDate="2022-01-03T17:57:17.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24773" PostTypeId="2" ParentId="10349" CreationDate="2022-01-03T19:10:33.923" Score="1" Body="&lt;p&gt;When doing a proof by contradiction of &amp;quot;if A then B&amp;quot;, you get to assume A and you also get to assume NOT B. Proving it directly, you only get to assume A. So proof by contradiction gives you more to work with from the start. I think that's why students like it.&lt;/p&gt;&#xA;" OwnerUserId="19319" LastActivityDate="2022-01-03T19:10:33.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24774" PostTypeId="2" ParentId="22" CreationDate="2022-01-03T22:28:17.773" Score="2" Body="&lt;p&gt;I assume your students have seen Linear Algebra.&lt;/p&gt;&#xA;&lt;p&gt;Remember in Linear Algebra how you sometimes have to solve &lt;span class=&quot;math-container&quot;&gt;$Ax = b$&lt;/span&gt; for a matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; with more columns than rows? You usually get free variables, right? So after reducing to echelon form you have something like:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{pmatrix} 1 &amp;amp; 0 &amp;amp; 2 \\ 0 &amp;amp; 1 &amp;amp; -1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You know from Linear Algebra that there are infinitely many solutions. How do you find them? Well, you can freely choose &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; and solve for the remaining variables.&lt;/p&gt;&#xA;&lt;p&gt;For instance if &lt;span class=&quot;math-container&quot;&gt;$z=0$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$x = 3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y = 3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Or, if &lt;span class=&quot;math-container&quot;&gt;$z=1$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$x = 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y = 4$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Indeed, &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; can be any value and the system can still be solved (we call &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; a free variable). In other words, once &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; is nailed down, the other two variables are determined uniquely. And another way of saying that is to say there is a unique function &lt;span class=&quot;math-container&quot;&gt;$\vec{g}(z)$&lt;/span&gt; such that the equation&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{pmatrix} 1 &amp;amp; 0 &amp;amp; 2 \\ 0 &amp;amp; 1 &amp;amp; -1 \end{pmatrix}\begin{pmatrix} \vec{g}(z) \\ z \end{pmatrix} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;is satisfied for any value &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; (forgive the abuse of notation in &lt;span class=&quot;math-container&quot;&gt;$\vec{g}(z)$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Well the existence of this function &lt;span class=&quot;math-container&quot;&gt;$\vec{g}(z)$&lt;/span&gt; is the conclusion of the Implicit Function Theorem. Whenever you have a situation with more outputs than inputs, like you have a function &lt;span class=&quot;math-container&quot;&gt;$f : \mathbb{R}^n \to \mathbb{R}^m$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$m&amp;gt;n$&lt;/span&gt; and you're solving &lt;span class=&quot;math-container&quot;&gt;$f(\vec{x}) = \vec{b}$&lt;/span&gt;, there will be a unique function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; of the free variables such that the equation &lt;span class=&quot;math-container&quot;&gt;$$f(\; g(\text{free variables}), \text{free variables} \;) = \vec{b}$$&lt;/span&gt; is satisfied for any value of the free variables within reason (and &amp;quot;within reason&amp;quot; means within a small neighborhood of the base point where linear reasoning like this is going to be valid).&lt;/p&gt;&#xA;&lt;p&gt;Thinking back to our matrix example, we can see that the number of free variables is &lt;span class=&quot;math-container&quot;&gt;$\#\text{outputs} - \#\text{inputs}$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$m-n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Well, there is one catch which is there is some condition that the total derivative has to have rank &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; or something, but that makes sense because you only get free variables when your matrix has the right rank.&lt;/p&gt;&#xA;" OwnerUserId="19319" LastEditorUserId="19319" LastEditDate="2022-01-04T18:32:45.753" LastActivityDate="2022-01-04T18:32:45.753" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24775" PostTypeId="2" ParentId="17198" CreationDate="2022-01-03T22:55:07.610" Score="2" Body="&lt;p&gt;Many studies about active learning are summarized and linked to &lt;a href=&quot;https://profteacher.com/2017/08/07/evidence-connecting-teaching-practices-and-equity/&quot; rel=&quot;nofollow noreferrer&quot;&gt;at this link&lt;/a&gt;. I do agree with the commenters Dirk and David, who mention that there are pros and cons to any teaching method, and that studies are usually based around averages, so they might not be applicable to every situation.&lt;/p&gt;&#xA;" OwnerUserId="19319" LastActivityDate="2022-01-03T22:55:07.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24777" PostTypeId="1" AcceptedAnswerId="24778" CreationDate="2022-01-06T21:54:08.387" Score="7" ViewCount="265" Body="&lt;p&gt;In French language, arithmetic statements are often read, at the elementary school level, as , say, &amp;quot; deux et deux font quatre&amp;quot; , i.e. something like &amp;quot; two and two make four&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Out of this arises a belief according to which the &lt;span class=&quot;math-container&quot;&gt;$ \Large= $&lt;/span&gt; symbol expresses some sort of action , either an action performed by numbers themselves or by the person that operates the mental activity of computation which is supposed to be denoted by the &lt;span class=&quot;math-container&quot;&gt;$\Large +$&lt;/span&gt; sign.&lt;/p&gt;&#xA;&lt;p&gt;This first belief may, in the head of older students, be replaced by the idea that &lt;span class=&quot;math-container&quot;&gt;$\Large =$&lt;/span&gt; means &amp;quot; has the same magnitude &amp;quot; or &amp;quot; has the same value as&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I tried to show to high school students that the supposedly active meaning of &lt;span class=&quot;math-container&quot;&gt;$\Large =$&lt;/span&gt; does not work anymore when the equality is reversed : &lt;span class=&quot;math-container&quot;&gt;$2+2$&lt;/span&gt; may ( arguably)  &amp;quot; make&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; , but would one say that  &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; &amp;quot; makes &amp;quot; &lt;span class=&quot;math-container&quot;&gt;$2+2$&lt;/span&gt; ?&lt;/p&gt;&#xA;&lt;p&gt;But I did no manage to convince them that, at least in the case of arithmetic statements, the &amp;quot; has the same magnitude &amp;quot; interpretation is not correct.&lt;/p&gt;&#xA;&lt;p&gt;The identity meaning seems simply unbelievable  to students.&lt;/p&gt;&#xA;" OwnerUserId="19341" LastEditorUserId="2083" LastEditDate="2022-01-07T09:36:43.670" LastActivityDate="2022-01-08T02:23:27.090" Title="How to convince a high school student that the $=$ symbol denotes identity?" Tags="|arithmetic-operations|equal-sign|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="24778" PostTypeId="2" ParentId="24777" CreationDate="2022-01-07T07:45:59.517" Score="10" Body="&lt;p&gt;Although some of the trouble may be language dependent, there is also the issue (here in the U.S., too, in English) that young students get the feeling, perhaps from teachers, even, that = is like saying &amp;quot;and here's what comes next&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I have seen research on this issue. I can't find exactly what I was looking for, but did find &lt;a href=&quot;https://math.vanderbilt.edu/schectex/commerrs/#Consciousness&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;To put it simply: Some students ... use the equals sign (=) as a symbol for the word &amp;quot;then&amp;quot; or the phrase &amp;quot;the next step is.&amp;quot; For instance, when asked to find the third derivative of x4+7x2–5, some students will write &amp;quot;x4+7x2–5 = 4x3+14x = 12x2+14 = 24x.&amp;quot; Of course, those four expressions are not actually equal to one another.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;A slight variant of this error consists of connecting several different equations with equal signs, where the intermediate equals signs are intended to convey &amp;quot;equivalent to&amp;quot; --- for example, x = y – 3 = x+3 = y. This is very confusing and altogether wrong, because equality is transitive --- i.e., if a=b and b=c then a=c, but x certainly is not equal to x+3. It would be better to replace that middle equals sign with some other symbol. The most obvious symbol for this purpose is ≡, which means &amp;quot;is equivalent to,&amp;quot; but that symbol has the disadvantage of looking too much like an equals sign, and thus possibly leading to the same confusion. Thus, a better choice would be ↔ or ⇔, both of which mean &amp;quot;if and only if.&amp;quot; Thus, I would rewrite the example above as x = y – 3 ⇔ x+3 = y.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The author called this 'stream of consciousness' notation.&lt;/p&gt;&#xA;&lt;p&gt;I have seen both of these issues, and I just keep reminding students that they are writing something incorrect.&lt;/p&gt;&#xA;&lt;p&gt;Edited to add: Thanks to the comments, &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7964/issues-with-equals-where-does-this-come-from-and-how-do-i-combat-it&quot;&gt;here's a link to a similar question, in which the accepted answer&lt;/a&gt; includes links to research.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-01-08T02:23:27.090" LastActivityDate="2022-01-08T02:23:27.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24780" PostTypeId="2" ParentId="11651" CreationDate="2022-01-07T13:35:23.460" Score="2" Body="&lt;p&gt;I think a lot of results are &amp;quot;accessible&amp;quot; in terms of a pop science-y explanation, rather than following a detailed proof.  And since you're looking for emotional motivation, I actually think the former is the more important!&lt;/p&gt;&#xA;&lt;p&gt;For example, see this (quite engaging) video about moving sofas around corners.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=rXfKWIZQIo4&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=rXfKWIZQIo4&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Many of the mathologer and numberphile and such videos are very accessible, motivating, and quite well done (content and production and engaging presenters).&lt;/p&gt;&#xA;&lt;p&gt;Heck, I would even include the Andrew Wiles doc by Horizon.  It has some real connection to school age children (Wiles even says he was motivated at age 10).  It doesn't matter so much that I don't understand the details of the proof itself.  The problem is describable and a topical (literally naming &amp;quot;topics&amp;quot; in advanced math) explanation of the achievement is possible.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.dailymotion.com/video/x1btavd&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.dailymotion.com/video/x1btavd&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Probably a key thing is that the problem itself is describable (even if the details of the proof are not).  For instance the Poincare Conjecture as a problem is tricky to understand, just as a concept.  I mean I don't get it...so that must mean it's hard, right?  ;)&lt;/p&gt;&#xA;&lt;p&gt;You can maybe still mention that &amp;quot;they bagged a recent big problem&amp;quot; if someone solves the Riemann hypothesis tomorrow, but harder to really talk about the topic.  And also note that it was the less general, but more understandable FLT problem that really motivated Wiles to his seven-year attic-sweat, not the more math-important (because of generality) Taniyama conjecture.&lt;/p&gt;&#xA;&lt;p&gt;But I also would not underestimate the appeal of older &amp;quot;aha&amp;quot;s.  After all, they are new to the student and are thrilling achievements of Man.  I remember thinking Archimedes must have been a freaking time traveler when we learned his method for area under the curve of a parabola.  Sure smells like integral calculus to me!&lt;/p&gt;&#xA;&lt;p&gt;E.g. here is an engaging mathologer video on solving cubics:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=N-KXStupwsc&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=N-KXStupwsc&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Or this Veritasium  video on Newton's advances in calculating pi (I might wait for a BC calc class, not do it in geometry though as the methods are sort of calculus and &amp;quot;algebra two&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=gMlf1ELvRzc&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=gMlf1ELvRzc&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So I wonder if the desire to highlight research results is more a desire of the teacher than the students.  (After all the teacher is probably a math researcher or at least was one during grad school.)  I did chem research during grad school, but was a bit skeptical (as a TA) of big wheel tenured profs at an R1 who wanted to connect in recent research results to a service course in undergrad chem (mostly non-major students, vast swath of pre-meds).  Felt like they were trying to justify themselves and the whole concept of an R1 (which I do not agree is really beneficial for undergrad training).  And they were falsely deprecating the importance to the student of learning concepts, tricks that are not new research results, but are new to the student.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  But I would try to keep it light and motivational and not derail your course time with too much external content.  Unless the kids are very strong to start with (thus tolerating more enrichment as the basic content itself is easier on them).&lt;/p&gt;&#xA;&lt;p&gt;P.s.s.  Also take a look at the American Scientist columns (or books) by Brian Hayes.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.americanscientist.org/author/brian_hayes&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.americanscientist.org/author/brian_hayes&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;They are good popularizations.  I dated an R1 prof who was doing cutting edge research in one of the topics (it was graphs or queuing or Hamming cubes or something like that) and she was impressed that I basically had an overview of her area from such a Hayes column.  She took a look at the article in her area and said it was a lot better than she expected.&lt;/p&gt;&#xA;" OwnerUserId="19343" LastEditorUserId="19343" LastEditDate="2022-01-07T13:44:38.667" LastActivityDate="2022-01-07T13:44:38.667" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24782" PostTypeId="1" AcceptedAnswerId="24811" CreationDate="2022-01-08T16:50:05.440" Score="4" ViewCount="381" Body="&lt;p&gt;From the book&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The chapter on Second-Order Differential Equations, as well as the associated&#xA;appendix section on complex numbers, has been moved to the website.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It doesn't mention a reason in the book and I couldn't find an answer after searching the internet. Is there a reason related to pedagogy? Does the publisher profit from it?&lt;/p&gt;&#xA;" OwnerUserId="17571" LastActivityDate="2022-01-15T20:29:33.643" Title="Why has the chapter on second-order differential equations been moved to the website instead of being put in the book in Stewart Calculus 9th edition?" Tags="|textbooks|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24783" PostTypeId="1" AcceptedAnswerId="24785" CreationDate="2022-01-08T21:41:46.533" Score="5" ViewCount="1526" Body="&lt;p&gt;If someone never had any experience with mathematical proofs and had only classes like Calc I-III (which he passed, without paying any attention to the proofs present in the textbooks), how long would it take this person to master the Book of Proofs by Richard H. Hammack?&lt;/p&gt;&#xA;&lt;p&gt;Now a few clarifications: This person would learn this book completely alone, at home, so no other students, professors or any other type of math enthusiasts. Let's say this person would learn 2 hours a day. And by &amp;quot;mastering&amp;quot; I mean this person would solve 90% of all exercises without looking at the solutions and understand all theorems.&lt;/p&gt;&#xA;" OwnerUserId="19230" LastEditorUserId="19230" LastEditDate="2022-01-26T10:30:42.170" LastActivityDate="2022-01-26T10:30:42.170" Title="How long would it take someone to master the topics in the book &quot;Book of Proof&quot; by Hammack and similar?" Tags="|proofs|self-learning|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24784" PostTypeId="2" ParentId="24783" CreationDate="2022-01-08T22:36:06.003" Score="6" Body="&lt;p&gt;The text says it is designed for a 14-week semester.  With an instructor, we could guess that means 3 hours a week of class and 6 hours a week of additional work ( = 126 hours?).  I cannot say whether someone without an instructor could do it in that time, or could do it at all.  That will vary greatly depending on the student.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2022-01-08T22:36:06.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24785" PostTypeId="2" ParentId="24783" CreationDate="2022-01-08T23:33:00.547" Score="11" Body="&lt;p&gt;This is a text for an &amp;quot;introduction to proofs&amp;quot; course.&lt;/p&gt;&#xA;&lt;p&gt;It might not be well-known outside mathematical circles, because mathematics educators don't like to advertise this fact, but, outside of fairly selective universities, most students taking such courses fail to learn the material despite earning a passing grade.  The majority of students never learn to develop or organize on their own any but the simplest proofs, or understand any significantly complex proof.  This inability persists despite another year or two of coursework beyond the &amp;quot;introduction to proofs&amp;quot; course that develops and uses these skills.&lt;/p&gt;&#xA;&lt;p&gt;Hence the answer for the average student is forever.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, a student who has an excellent natural grasp of logic has essentially no need for this course and will be able to learn all the material in a week or two.  The most selective universities tend to not have a course based on a similar text and simply expect their students to intuitively pick up this material while taking more advanced math courses.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2022-01-08T23:33:00.547" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24786" PostTypeId="1" CreationDate="2022-01-08T23:40:01.867" Score="1" ViewCount="333" Body="&lt;p&gt;Some of &lt;em&gt;&lt;strong&gt;my 16 year old students&lt;/strong&gt;&lt;/em&gt; hanker after the formula  for the # of k-permutations of n objects, with x types, where &lt;span class=&quot;math-container&quot;&gt;$r_1, ⋯, r_x$&lt;/span&gt; = the number of each type of object. This is more generalized &lt;a href=&quot;https://matheducators.stackexchange.com/a/7765/19348&quot;&gt;than this question&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What books accessibly teach this formula? What books gently expound — fill in all gaps and steps in —  &lt;a href=&quot;https://mathoverflow.net/a/37218&quot;&gt;this answer&lt;/a&gt; by Prof. &lt;a href=&quot;https://mathoverflow.net/users/972/suresh-venkat&quot;&gt;Suresh Venkatasubramanian&lt;/a&gt;?&lt;/p&gt;&#xA;&lt;p&gt;The book doesn't have to be written for 16 year olds. You can recommend books for undergraduates, but they must be readable and  easygoing.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user155" LastEditDate="2022-01-09T07:46:21.600" LastActivityDate="2023-01-08T19:46:38.507" Title="What books teach the formula for the # of k-permutations of n objects, with x types, and $r_1,⋯,r_x$ = the number of each type of object?" Tags="|reference-books|combinatorics|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24787" PostTypeId="2" ParentId="24782" CreationDate="2022-01-09T00:58:34.377" Score="6" Body="&lt;p&gt;It's probably not just cost control--it's revenue capture.  There's definitely some MBA games going on in academic publishing.  Just look at the edition frequency (that's not to design a better book, it's to mess with the resale market).  So when you see these sort of alternate payment models always suspect a revenue scheme.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Sometimes you have to buy (or have your school buy) membership in a online HW-solver type of club to see these extras.  This is extra $$ for the publisher.  And also a means of control for colleges (i.e. making it harder for self studiers.  Gatekeeper stuff.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Furthermore any online structure is a hassle since it is not resalable.  It's also not enduring.  What if the provider goes defunct, for example?  It's like the trend in &amp;quot;software as a service&amp;quot; rather than having a CD.  (Same with movies from streaming services versus buying a CD.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm not sure about your specific example.  But I suspect some of the game with the online service (for extras or for HW) is related to edition control.  For example, they can STOP having the online Xth edition content when they come out with X+1 or X+2.  (see intro.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;More tenuous but still conceivable is something related to &amp;quot;price discrimination&amp;quot;.  &lt;a href=&quot;https://en.wikipedia.org/wiki/Price_discrimination&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Price_discrimination&lt;/a&gt;  Marketers try to come up with schemes to separate the cost sensitive from cost insensitive and then charge different prices (often with trivial differences in the product).  The point here is to compete on price only when you need to.  (With a total free market, transparency, and a commodity product this is impossible.  But there are many deviations from ideality.  So marketers try to exploit.  This is why discounting exists for instance, versus just across the board price changes like on oil, grain, etc.) I don't know exactly how they'd tie it in with a price discrimination scheme, but it's the sort of thing marketers think about.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Other than that, it's probably slightly related to the AP BC change  (slow and small step backwards in coverage, &amp;quot;dumbing down&amp;quot; by cutting content).  They cut 2nd order diffyQs a few years ago.  (Also shells for volumes of rotation.)  I think lots of colleges still have it.  And it's sort of a normal part of a traditional text (Granville, old Thomas, etc.)  So they probably need to keep it in for schools that haven't dumbed down yet like AP did.  But AP is probably a huge market.  And may be a trend that some colleges are also following, to water down.&lt;/p&gt;&#xA;&lt;p&gt;P.s. After writing this I did a quick Google search on 'use of online content to capture revenue in textbook publishing'.  Several articles came back, talking about this as a scheme for publishers to fight the used market.  For example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.reuters.com/article/usa-education-textbook/feature-textbook-publishers-revamp-ebooks-to-fight-used-market-idUSL1N0FN1TO20130723?edition-redirect=uk&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.reuters.com/article/usa-education-textbook/feature-textbook-publishers-revamp-ebooks-to-fight-used-market-idUSL1N0FN1TO20130723?edition-redirect=uk&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Note this is a 2013 article--not a new MBA game.)&lt;/p&gt;&#xA;&lt;p&gt;P.s.s.  One of the things that makes it possible for publishers to play these games is the partnership with schools (gatekeeper effect).  Obviously if you are a self studier, just buy a good copy of old style Thomas (even has all the answers in the back.)  If you check Amazon reviews, you can find the editions that are praised, for example.  In general, people still like many Dover texts that are 50+ years old, so recency is not very important in lower level mathematics.  (Be careful and read the reviews though).&lt;/p&gt;&#xA;" OwnerUserId="19349" LastEditorUserId="19349" LastEditDate="2022-01-09T14:25:04.997" LastActivityDate="2022-01-09T14:25:04.997" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24788" PostTypeId="2" ParentId="24783" CreationDate="2022-01-09T01:26:58.787" Score="4" Body="&lt;p&gt;I have that book.  It's actually a very easy gentle book.  I think easier than the medium difficulty calculus books or ODE books (not Spivak) that I have.  I don't think you'd have any issues self studying it.  Note, that it only has answers for 50% of the exercises, but the amount of exercises is vast (and like I said, the content is easy).  So I think you could get the gist just by working the odds.   For the occasional times you need help, can go online.  But probably don't need it that often.&lt;/p&gt;&#xA;&lt;p&gt;I do think you should also try to develop your ability to work through things though when studying calculus or ODEs (or other courses).  Often there are proofs or at least derivations, initial examples, etc.  It's a good skill to learn to pre-read such texts and be an active learner (working the material, doing the derivations as you go).  If the development is progressive enough, often you can do some of the derivations on your own, checking back as needed.  It will help you build up your problem solving muscles.&lt;/p&gt;&#xA;&lt;p&gt;(Added)  P.s.  Just saw the remarks by Woo and I agree.  If you think you can be a non-curious calculus/ODE student, go do this (very easy) book and then run into Rudin, you're crazy.  Again, I think following and solving/cosolving the proofs, derivations, etc. in Thomas Finney, Tennanbaum and Pollard, or the like will actually be more helpful.  And closer to the type of content you'll have in real analysis, if that's where you're heading.  If you can handle it, I would even just go straight to Spivak or a book like that now.  After all, you've had calculus, even if you didn't pay attention to the proofs and derivations.&lt;/p&gt;&#xA;" OwnerUserId="19349" LastEditorUserId="19349" LastEditDate="2022-01-09T01:32:45.303" LastActivityDate="2022-01-09T01:32:45.303" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24789" PostTypeId="2" ParentId="24783" CreationDate="2022-01-09T22:23:11.503" Score="9" Body="&lt;p&gt;(I had tried to add a clarifying comment to @AlexanderWoo's good answer, but it was mysteriously deleted.)&lt;/p&gt;&#xA;&lt;p&gt;My point was, and is, that it is not constructive to think of &amp;quot;proofs&amp;quot; as a thing separate from normal human discussion of things. Rather, proofs are really &lt;em&gt;explanations&lt;/em&gt;, or discussions that &lt;em&gt;persuade&lt;/em&gt;. There is no magic formula, and &amp;quot;proof&amp;quot; is not an alien thing.&lt;/p&gt;&#xA;&lt;p&gt;Yes, many math curricula (perhaps especially in the U.S., these days) in high school and the beginning of college or university have essentially no content that is about explanation or persuasion, but only about inexplicable rules.&lt;/p&gt;&#xA;&lt;p&gt;I do have some acquaintance with various &amp;quot;introduction to proof&amp;quot; textbooks and courses, and I am not a fan of any of them, for various reasons. Instead, as suggested in Alex W's answer, simply reading relatively easy proofs of &lt;em&gt;real&lt;/em&gt; things (rather than contrived) is the way to learn what the &lt;em&gt;point&lt;/em&gt; of &amp;quot;proofs&amp;quot; is. It's not conforming to rules in a ritual, it's about careful appraisal of causality and such...&lt;/p&gt;&#xA;&lt;p&gt;One particular problem in terms of understanding mathematics, is the notion that we should not trust our physical intuition. It is perverse to invalidate beginners' self-confidence and critical judgement by telling them things like this. Mathematics is not about inscrutable stuff behaving wildly contrary to our physical experience, and it is perverse to portray it as such. One iconic example is the intermediate value theorem in calculus. On one hand, it is &amp;quot;obviously true&amp;quot;... but it is non-trivial to prove, because a precise notion of &amp;quot;real number&amp;quot; is necessary, to begin. But that should not make us doubt the fact.&lt;/p&gt;&#xA;&lt;p&gt;Especially when I hear people say &amp;quot;what should I memorize?&amp;quot; it worries me. Of course, memorization can be a first step toward understanding, but I strongly prefer thinking of &amp;quot;talking about math&amp;quot; as more akin to throwing and catching a ball, or walking across a room. Maybe it can be axiomatized (and that may be of some interest), but it seems that our best understanding of it is achieved by viewing it as &amp;quot;real&amp;quot;, as opposed to disconnected from our reality.&lt;/p&gt;&#xA;&lt;p&gt;So, sure, look at &amp;quot;intro to proofs&amp;quot; books, if they are fun, and do exercises in them, if they're fun, but that's further from real math than &amp;quot;T-ball&amp;quot; (hitting a ball off a &amp;quot;tee&amp;quot;, for little kids) is from baseball.&lt;/p&gt;&#xA;&lt;p&gt;Better to look at relatively elementary texts that address some bit of mathematics itself, rather than claiming to talk about &amp;quot;proofs&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-01-09T22:23:11.503" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24790" PostTypeId="1" AcceptedAnswerId="24792" CreationDate="2022-01-10T00:31:26.500" Score="2" ViewCount="232" Body="&lt;p&gt;Every year, some 17 y.o. student makes the mistake of stopping &lt;span class=&quot;math-container&quot;&gt;$P(n, r)$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$\color{darkorange}{(n - r)}$&lt;/span&gt;, rather than &lt;span class=&quot;math-container&quot;&gt;$\color{forestgreen}{(n - (r - 1))}$&lt;/span&gt;. Because they are in their last year of high school, they do understand why. But they goof up because this is counterintuitive, because &lt;span class=&quot;math-container&quot;&gt;$P(n, r)$&lt;/span&gt; contains &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, but the last term contains &lt;span class=&quot;math-container&quot;&gt;$\color{forestgreen}{r - 1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As lined in red below, I want students to understand this, not just memorize. How can they intuit this?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0DZP7.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0DZP7.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;David Patrick, &lt;a href=&quot;https://artofproblemsolving.com/wiki/index.php/David_Patrick&quot; rel=&quot;nofollow noreferrer&quot;&gt;BS Math &amp;amp; Computer Science, MS Math (Carnegie Mellon),  PhD Math (MIT)&lt;/a&gt;. &lt;em&gt;Introduction to Counting &amp;amp; Probability&lt;/em&gt; (2005), p 20.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2022-01-10T15:27:01.997" Title="How can 17 y.o. high school students intuit that P(n, r) stops at $n - (r - 1)$, not $n - r$?" Tags="|combinatorics|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24791" PostTypeId="2" ParentId="24790" CreationDate="2022-01-10T02:28:22.977" Score="6" Body="&lt;p&gt;The way I think about it, and also the way I thought about it in high school, is &amp;quot;it's a product of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; consecutive numbers, sort of like &lt;span class=&quot;math-container&quot;&gt;$r \times (r-1) \times \cdots \times 2 \times 1$&lt;/span&gt;. So the difference between the first factor and the last factor is &lt;span class=&quot;math-container&quot;&gt;$r-1$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This is an issue that shows up in lots of places: In a list of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; consecutive integers, the difference between the first one and the last one is &lt;span class=&quot;math-container&quot;&gt;$r-1$&lt;/span&gt;. It even has a name: &lt;a href=&quot;https://en.wikipedia.org/wiki/Off-by-one_error#Fencepost_error&quot; rel=&quot;noreferrer&quot;&gt;Fence post error&lt;/a&gt;. So just learn that this is a standard place one can go wrong and get in the habit of subtracting the last term from the first term and comparing with the difference between &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; when you do computations like this.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2022-01-10T02:28:22.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24792" PostTypeId="2" ParentId="24790" CreationDate="2022-01-10T02:28:59.420" Score="5" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(n,r)$&lt;/span&gt; computes the number of permutations of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; objects (out of a choice of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;). So in the end there should only be &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; terms in the product. Since the 1st term is &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and the 2nd term is &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt;, then the &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;th term is &lt;span class=&quot;math-container&quot;&gt;$n-(r-1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What they are making is an &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Off-by-one_error&quot; rel=&quot;nofollow noreferrer&quot;&gt;off-by-one error&lt;/a&gt;. In a discrete topics course, they need to learn about this common mistake.&lt;/p&gt;&#xA;&lt;p&gt;Update: Another way to explain it is to ask how many items are you NOT using. The answer is &lt;span class=&quot;math-container&quot;&gt;$n-r$&lt;/span&gt;. So then &lt;span class=&quot;math-container&quot;&gt;$n-r$&lt;/span&gt; should NOT be a term in the product. So in your example of 3 officers out of 30. There are &lt;span class=&quot;math-container&quot;&gt;$30-3=27$&lt;/span&gt; UNused people. So the product counts down from 30 but does NOT use 27 (so stops at 28).&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2022-01-10T03:17:41.983" LastActivityDate="2022-01-10T03:17:41.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24793" PostTypeId="2" ParentId="24790" CreationDate="2022-01-10T04:21:35.357" Score="1" Body="&lt;p&gt;I think of these problems in terms of slots. I want to form a product whose factors are the number of ways of filling each of the &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; slots. When filling the &lt;span class=&quot;math-container&quot;&gt;$j^\text{th}$&lt;/span&gt; slot, the number of items that can go in that slot is &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; minus the number of items that are unavailable because they have already been placed in a slot. But this number of unavailable items is &lt;span class=&quot;math-container&quot;&gt;$j-1$&lt;/span&gt; because prior to filling the &lt;span class=&quot;math-container&quot;&gt;$j^\text{th}$&lt;/span&gt; slot &lt;span class=&quot;math-container&quot;&gt;$j-1$&lt;/span&gt; items will have been placed.&lt;/p&gt;&#xA;&lt;p&gt;So for the first slot, &lt;span class=&quot;math-container&quot;&gt;$n-0$&lt;/span&gt; items are available. (&lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; items have previously been placed.)&lt;/p&gt;&#xA;&lt;p&gt;For the second slot, &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt; items are available. (&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; item has previously been placed.)&lt;/p&gt;&#xA;&lt;p&gt;For the &lt;span class=&quot;math-container&quot;&gt;$j^\text{th}$&lt;/span&gt; slot, &lt;span class=&quot;math-container&quot;&gt;$n-(j-1)$&lt;/span&gt; items are available. (&lt;span class=&quot;math-container&quot;&gt;$j-1$&lt;/span&gt; items have previously been placed.)&lt;/p&gt;&#xA;&lt;p&gt;For the &lt;span class=&quot;math-container&quot;&gt;$r^\text{th}$&lt;/span&gt; slot, &lt;span class=&quot;math-container&quot;&gt;$n-(r-1)$&lt;/span&gt; items are available. (&lt;span class=&quot;math-container&quot;&gt;$r-1$&lt;/span&gt; items have previously been placed.)&lt;/p&gt;&#xA;&lt;p&gt;Here's another nice way of thinking about the final formula,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;P(n,r) = \frac{n!}{(n-r)!}.&#xA;$$&lt;/span&gt;&#xA;We want to fill only &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; slots, but suppose we start out by adding &lt;span class=&quot;math-container&quot;&gt;$n-r$&lt;/span&gt; additional slots so that we have a full complement of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; slots. These can, of course, be filled in &lt;span class=&quot;math-container&quot;&gt;$n!$&lt;/span&gt; ways. Now we hide the &lt;span class=&quot;math-container&quot;&gt;$n-r$&lt;/span&gt; added slots because we didn't want them anyway. Since there are &lt;span class=&quot;math-container&quot;&gt;$(n-r)!$&lt;/span&gt; ways the hidden slots can be filled, our &lt;span class=&quot;math-container&quot;&gt;$n!$&lt;/span&gt; configurations will include &lt;span class=&quot;math-container&quot;&gt;$(n-r)!$&lt;/span&gt; copies of each visible configuration. So we need to divide &lt;span class=&quot;math-container&quot;&gt;$n!$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$(n-r)!$&lt;/span&gt; to obtain the number of distinct visible configurations.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastEditorUserId="1737" LastEditDate="2022-01-10T07:25:37.440" LastActivityDate="2022-01-10T07:25:37.440" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24794" PostTypeId="2" ParentId="24790" CreationDate="2022-01-10T15:27:01.997" Score="5" Body="&lt;p&gt;This is somewhere between an answer and a comment. I think the other answers mentioning the fence post error get to the heart of it. In other words, is this really about permutations specifically? Because if their intuition says that &lt;span class=&quot;math-container&quot;&gt;$n-r$&lt;/span&gt; should appear in the formula, they're correct; &lt;span class=&quot;math-container&quot;&gt;$P(n,r)=n!/(n-r)!$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Their real problem doesn't seem to be faulty intuition (because who really has an intuitive feel for, say, the number of permutations of 36 objects out of 177?). It seems to me that they're just vaguely remembering a formula or a process that they don't have a complete understanding of and plugging in the wrong numbers, ie they're &lt;em&gt;hoping&lt;/em&gt; or &lt;em&gt;guessing&lt;/em&gt; that the answer takes a certain form, not intuiting. The fact that they do understand why they can stop at &lt;span class=&quot;math-container&quot;&gt;$n-r+1$&lt;/span&gt; when it's explained to them is a good sign.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2022-01-10T15:27:01.997" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24795" PostTypeId="1" CreationDate="2022-01-10T21:02:49.727" Score="1" ViewCount="523" Body="&lt;p&gt;I'm hoping to start a masters in mathematics in the fall, and am hoping to find a good book on mathematical statistics to study so that I'll be able to take graduate level mathematical statistics once I start my degree. For context, my undergrad was in music, and I'm in the midst of taking prerequisites to qualify for the masters program. I'm taking finals for Linear Algebra and Calc III, and after that I will be taking Differential Equations, Methods of Proof, Real Variables, and Abstract Algebra. Are there any books you recommend for self-study in mathematical statistics? (preferably not too expensive if possible -- trying to save for school) Thank you!&lt;/p&gt;&#xA;" OwnerUserId="19352" LastActivityDate="2022-01-11T19:58:58.170" Title="Best books for mathematical statistics self-study?" Tags="|self-learning|statistics|probability|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24796" PostTypeId="1" AcceptedAnswerId="24823" CreationDate="2022-01-10T22:49:22.083" Score="10" ViewCount="244" Body="&lt;p&gt;I used to post my slides before a class. But I noticed that many students simply read it while in class instead of listening . So I am thinking not doing it in the future. But they can still get it from students who have taken the class before.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2022-01-19T20:54:28.807" Title="Should I upload slides before or after a class" Tags="|undergraduate-education|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="24797" PostTypeId="2" ParentId="24795" CreationDate="2022-01-11T17:57:30.443" Score="2" Body="&lt;p&gt;Given you seem to have zero exposure to this material, I would just look for a reasonable undergrad text.&lt;/p&gt;&#xA;&lt;p&gt;I would avoid classes &amp;quot;for business&amp;quot; or the like (although really they are way better than nothing!)&lt;/p&gt;&#xA;&lt;p&gt;I would also be a little careful about asking for &amp;quot;mathematical statistics&amp;quot;.  Some people may interpret that as asking for a hyper theoretical course.&lt;/p&gt;&#xA;&lt;p&gt;[Edit:  I just checked your post on MSE and, as expected, one user has already commented in minutes, suggesting a very hard core already grad level stats book (Shao, Springer publ.) that emphasizes &amp;quot;measure-theoretic probability theory&amp;quot;.  This is insane for a person with zero previous prob/stats experience and who is a music major looking for a reasonable math masters.  But it's par for the course on SE to see these kind of recommend a too-hard book and ignore progression in the learning process.  And ignore the student's particulars.]&lt;/p&gt;&#xA;&lt;p&gt;You do want a standard middle of the road class that a math dept would teach.  It's fine if there is some calculus involved.  But you don't want something super theory heavy.  Not with what I read between the lines in your math background (not extensive) and ambitions (masters). Plus it's really beneficial to just get some insights in prob/stats, not just a blizzard of formulas.&lt;/p&gt;&#xA;&lt;p&gt;A good general, low-cost, easy-to-self-study option is the Schaum's Outline:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/007179557X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Schaums-Outline-Probability-Statistics-4th/dp/007179557X/ref=cm_cr_arp_d_product_top?ie=UTF8&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's calculus based, but very gentle.  Not encyclopedic, but covers the key areas.  And an omnibus can be daunting.  Better to make a first pass that is more prioritized.  And it is cheap.  And easy to read, has answer key, etc. (good option for self studiers.)&lt;/p&gt;&#xA;" OwnerUserId="19355" LastEditorUserId="19355" LastEditDate="2022-01-11T19:58:58.170" LastActivityDate="2022-01-11T19:58:58.170" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="24798" PostTypeId="1" CreationDate="2022-01-12T04:42:19.843" Score="5" ViewCount="2882" Body="&lt;p&gt;Proofs, or any mathematical derivation, appearing in any real setting, such as a book or textbook or talk, or even when we're teaching it in class, includes a great deal of surrounding explanation. But when we ask students to regurgitate proofs, we ask for what is merely the skeletal core of the proof in this rib-like two column format. At the very least, it seems to me, proofs ought to have three columns, including a new multi-row lefthand column that describes (at least) the approach being taken in that section of the proof. By &amp;quot;multi-row&amp;quot; I mean that the proposed new lefthand &amp;quot;row&amp;quot; can encompass multiple rows of the basic two-column proof. The resulting format would look like this:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    =================|===============|==============&#xA;      What we're     |   &amp;lt;c=&amp;lt;c       |  Reflective Property&#xA;       up to in      |  tABC~=tDEF   |  SAS&#xA;       this section  |    etc...     |  etc...&#xA;    =================|===============|==============&#xA;      What we're     |    etc...     |  etc...&#xA;       up to in      |    etc...     |  etc...&#xA;       this section  |    etc...     |  etc...&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;The context would, I think, largely reflect the reasoning and planning that went into (goes into) the proof, and would commonly, I think, end up representing lemmas that participate in the larger proof. One could say that these lemmas ought to be rolled off into prior proofs of their own, and I would agree. But we do not provide a way, in geometry, of naming and organizing proofs usefully so that prior short proofs (technically lemmas) can be looked up and referred to easily.&lt;/p&gt;&#xA;&lt;p&gt;Because we do not have a clear naming scheme for proofs, we cannot call upon them as one would functions in a programming language. Indeed, one might wonder why student of geometry aren't being taught geometry like one would teach a programming language: Here's a bunch of functions (lemmas) you can use, and here's how to use them. We do do this for some things, like the triangle congruence lemmas (SAS etc), and for some logical rationales (CPCTC, etc), but the dozen random theorems (lemmas) regarding parallelograms, mid segments, and so on aren't ready-to-hard functions with clear naming, so we end up re-deriving/proving them in the middle of other proofs, which makes the proofs into these long-winded, un-memorable and ultimately unwieldy things.&lt;/p&gt;&#xA;&lt;p&gt;The three-column format I'm proposing at least offers a way to internally organize proofs into logical segments so that even without addressing the problem of the previous paragraph, at least there is a a way of making the substructure explicit.&lt;/p&gt;&#xA;" OwnerUserId="14430" LastEditorUserId="376" LastEditDate="2022-01-15T12:39:17.473" LastActivityDate="2022-01-15T12:39:17.473" Title="Should proofs include a third “context” column?" Tags="|secondary-education|proofs|usa|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="24799" PostTypeId="1" CreationDate="2022-01-12T11:01:10.750" Score="1" ViewCount="204" Body="&lt;p&gt;I read sometimes mathematical works of others outside my undergraduate studies. I think i can not follow the understanding of the proofs of theorems sometimes. What should i do? Should i read other things connected to the proofs i read to understand the proofs?&lt;/p&gt;&#xA;&lt;p&gt;I want to make my own theorems and proofs but i can not solve the open problems i read, although they are considered difficult from others.What should i do to make theorems and proofs? At least for them to be new and accepted from the scientific community.&lt;/p&gt;&#xA;&lt;p&gt;Could i make my own open problems and conjectures or questions and try to prove them? If i do it, what should i read and what do most of the mathematicians do to solve them?&lt;/p&gt;&#xA;&lt;p&gt;When reading theorems without the proofs and learning them, how will i know if a problem needs those theorems to be solved? About the proofs, how should i use what i learn from them on other possible solutions of problems?&lt;/p&gt;&#xA;&lt;p&gt;What does someone learn from proofs?&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastActivityDate="2022-01-13T14:00:14.443" Title="Math outside of undergraduate studies and proofs" Tags="|proofs|self-learning|problem-solving|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24800" PostTypeId="2" ParentId="24799" CreationDate="2022-01-12T18:37:40.217" Score="4" Body="&lt;p&gt;It sounds to me like you need exposure to mathematical topics beyond what is covered&#xA;in an undergraduate math major. Here's one recommendation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Fuks, Dmitrij Borisovič, and Serge Tabachnikov. &lt;em&gt;Mathematical Omnibus: Thirty Lectures on Classic Mathematics&lt;/em&gt;. American Mathematical Soc., 2007.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;      &lt;a href=&quot;https://i.stack.imgur.com/UoPne.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UoPne.jpg&quot; alt=&quot;Omnibus&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are similar collections, but this one is both broad and quite accessible.&#xA;From a review by Harriet Pollatsek:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The thirty lectures in&#xA;Omnibus are organized into eight chapters (number of lectures in parentheses): Arithmetic and&#xA;Combinatorics (3), Equations (5), Envelopes and&#xA;Singularities (4), Developable Surfaces (3), Straight&#xA;Lines (4), Polyhedra (6), Surprising Topological&#xA;Constructions (2), and Ellipses and Ellipsoids (3).&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Added&lt;/strong&gt;&lt;/em&gt;. An example from the last chapter, Lecture 28 is entitled&#xA;&amp;quot;Billiards in Ellipses and Geodesics on Ellipsoids,&amp;quot;&#xA;drawing on material from&#xA;Tabachnikov's book &lt;em&gt;Geometry and Billiards&lt;/em&gt;. American Mathematical Society, 2005.&#xA;There is considerable current research&#xA;on this topic:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Reznik, Dan, Ronaldo Garcia, and Jair Koiller. &amp;quot;Can the elliptic billiard still surprise us?.&amp;quot; &lt;em&gt;The Mathematical Intelligencer&lt;/em&gt; 42, no. 1 (2020): 6-17.&#xA;&lt;a href=&quot;https://link.springer.com/article/10.1007/s00283-019-09951-2#Bib1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2022-01-13T14:00:14.443" LastActivityDate="2022-01-13T14:00:14.443" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24801" PostTypeId="1" AcceptedAnswerId="24803" CreationDate="2022-01-13T02:42:16.277" Score="3" ViewCount="658" Body="&lt;p&gt;I’m tutoring an 8th grade student in Algebra 1, and he showed me that their class learned how to find standard deviation and &lt;strong&gt;mean absolute deviation&lt;/strong&gt; using the following formulas:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$SD=\sqrt{\displaystyle\frac{\Sigma (x_i-\mu)^2}{n}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\textbf{Mean Absolute Deviation}=\displaystyle\frac{\Sigma |x_i-\mu|}{n}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I did not learn this when I was in Algebra 1 in 8th grade, and I was in the honors class. Is this because of Common Core? I know they’re trying to scatter more stats materials in the regular curriculum but I’m just shocked he was working on this at his level. He eventually understood it and was getting the correct answers but he definitely struggled a lot with it today before getting there. Just don’t understand why they’re covering this at this level — it seems a little advanced to me.&lt;/p&gt;&#xA;" OwnerUserId="19352" LastEditorUserId="2083" LastEditDate="2022-01-13T09:25:56.343" LastActivityDate="2022-01-13T09:25:56.343" Title="Why is my 8th grade Algebra 1 tutoring student learning mean absolute deviation and standard deviation?" Tags="|secondary-education|algebra|statistics|middle-school|common-core|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="24802" PostTypeId="2" ParentId="10092" CreationDate="2022-01-13T05:03:52.507" Score="3" Body="&lt;p&gt;Additionally, I would consider using &lt;a href=&quot;https://www.khanacademy.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Khan Academy&lt;/a&gt; as supplementary practice with the books that people have included on here. Khan Academy includes videos and many practice problems with detailed solutions given. That's how I reviewed high school math after not taking it for a while.&lt;/p&gt;&#xA;" OwnerUserId="19352" LastActivityDate="2022-01-13T05:03:52.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24803" PostTypeId="2" ParentId="24801" CreationDate="2022-01-13T08:06:44.963" Score="4" Body="&lt;p&gt;Short answer: Yes, the Common Core (correctly, IMO) believes that workers in the twenty-first century should have an elementary grounding in some topics of what you probably believe to be college-level statistics.&lt;/p&gt;&#xA;&lt;p&gt;Longer answer: Every state is free to interpret and implement the standards as they wish.  This is especially true at the high school level, where the standards are broken down by topic instead of by grade level.&lt;/p&gt;&#xA;&lt;p&gt;For instance, in New York, I don't believe students are ever directly exposed to the formula for variance and standard deviation.  But students in Algebra 2 are asked to calculate p-values and the usual confidence intervals for normal distributions using graphing calculators.  In Algebra I, by contrast, students need to be able to perform linear regressions and calculate correlation (again with a calculator) and be able to interpret bivariate data given a chart.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;If I were to criticize what your student is being asked to do, it isn't because of difficulty so much as authenticity.  Statistical analysis is something students should be well-practiced at, but they will be using tools like spreadsheets or statistical software to perform the calculations.  But either your state feels differently or your student's teacher feels like taking the time to supplement the standards to involve grunt work.  In the end, it's not a hill I would fight for on either side.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2022-01-13T08:06:44.963" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24804" PostTypeId="2" ParentId="24798" CreationDate="2022-01-13T14:11:56.753" Score="3" Body="&lt;p&gt;I think maybe the correct answer to this question is something simple like&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Sure, do something else if you like.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I also recommend reading &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2093/is-there-any-evidence-about-the-effectiveness-of-table-proofs-in-pre-college-m?rq=1&quot;&gt;this question&lt;/a&gt; because the good answer there may give you some insights into how imposing a structure on proofs can help and how it can hurt.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="14024" LastEditDate="2022-01-14T00:21:53.727" LastActivityDate="2022-01-14T00:21:53.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24805" PostTypeId="2" ParentId="24798" CreationDate="2022-01-13T19:09:15.343" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;OP: What seems to be missing as taught is an explicit indication of the plan.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even though this is far afield from your concentration on geometry, it&#xA;illustrates your point. Michael Sipser's text,&#xA;&lt;em&gt;Introduction to the Theory of Computation,&lt;/em&gt;&#xA;includes many &amp;quot;proof idea&amp;quot; sections prior to launching into each formal proof.&#xA;I've taught from this text and this really works.&#xA;Such &amp;quot;proof idea&amp;quot; thumbnails can be presented in any format (Sipser just&#xA;uses prose paragraphs), a 3rd column if you prefer.&lt;/p&gt;&#xA;&lt;p&gt;        &lt;a href=&quot;https://i.stack.imgur.com/A0Lyx.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/A0Lyx.jpg&quot; alt=&quot;Cover&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-01-13T19:09:15.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24807" PostTypeId="1" CreationDate="2022-01-14T01:13:46.733" Score="4" ViewCount="222" Body="&lt;p&gt;I'm an undergraduate who doesn't find analysis particularly interesting, but I'm taking a calculus on manifolds course next semester, so I'm reviewing measure and integration theory since my grasp on the subject should be stronger, but it's so difficult to actually sit down and learn the material when I'm uninterested in it. With subjects I like, I can sit down and read a textbook for hours every day without any problem.&lt;/p&gt;&#xA;&lt;p&gt;Does anybody have any advice? Topology was my favorite course so when I can think of problems topologically or have some topological motivation, it helps me want to learn these other subjects, but the textbook I use doesn't take a topological approach. If anybody has any advice (or textbook recommendations), I'd greatly appreciate it.&lt;/p&gt;&#xA;&lt;p&gt;Also, I can't just not learn this subject, either&lt;/p&gt;&#xA;" OwnerUserId="19362" LastActivityDate="2022-01-14T22:34:41.137" Title="How do you study subjects you're not that interested in" Tags="|undergraduate-education|textbooks|self-learning|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24808" PostTypeId="2" ParentId="24807" CreationDate="2022-01-14T12:36:38.320" Score="3" Body="&lt;p&gt;Do you think you will enjoy learning calculus on manifolds?  I think there is a lot to appreciate there, and it opens the doors to a lot of beautiful mathematics and physics(differential geometry, differential topology, general relativity, de Rham cohomology, Hodge theory, etc).&lt;/p&gt;&#xA;&lt;p&gt;You can mostly get by in calculus on manifolds with just Riemannian integration.  Measure theory isn't really needed (although it is certainly a nicer theory of integration, and makes a lot of things easier and more coherent).&lt;/p&gt;&#xA;&lt;p&gt;My personal advice would be to stop forcing yourself to study something which you are not currently excited about.  If you care about calculus on manifolds and measure theory crops up during your study, that might provide you with a more natural motivation to brush up on it.  Seeing it used for something you care about can be the key to starting to care.&lt;/p&gt;&#xA;&lt;p&gt;I thought my measure theory class was the most boring thing ever.  I could barely will myself to study, and then only for the grade.  It wasn't until I started learning some functional analysis that I gained an appreciation for measure theory and was able to return to its study with gusto.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-01-14T12:36:38.320" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24809" PostTypeId="2" ParentId="24807" CreationDate="2022-01-14T22:34:41.137" Score="4" Body="&lt;p&gt;Back in my student days I found that: Studying my favorite subjects could be done in almost any setting, regardless of distractions.  But to study less favorite subjects I had to be a a quiet place with no distractions.  This was in the Olden Days when I did not have distractions like an iPhone in my pocket that went with me everywhere.  Believe it or not: back then libraries were quiet places with books, not lively places with computers!&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2022-01-14T22:34:41.137" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24810" PostTypeId="2" ParentId="10092" CreationDate="2022-01-15T17:42:32.867" Score="3" Body="&lt;p&gt;There is an important genre of textbooks on high school mathematics written to be used by students who are preparing for careers in teaching mathematics at the high school level.  These textbooks are often written in a more rigorous fashion and cover the material in more depth than the textbooks used by high school students.&lt;/p&gt;&#xA;&lt;p&gt;A recent example of this is the series of three textbooks by Hung-Hsi Wu, &lt;em&gt;&lt;a href=&quot;https://bookstore.ams.org/cdn-1642150396209/mbk-132/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra and Geometry&lt;/a&gt;&lt;/em&gt;, &lt;em&gt;&lt;a href=&quot;https://bookstore.ams.org/cdn-1642150382041/mbk-131/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rational Numbers to Linear Equations&lt;/a&gt;&lt;/em&gt;, and &lt;em&gt;&lt;a href=&quot;https://bookstore.ams.org/mbk-133/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pre-Calculus, Calculus, and Beyond&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2022-01-15T17:42:32.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24811" PostTypeId="2" ParentId="24782" CreationDate="2022-01-15T20:29:33.643" Score="0" Body="&lt;p&gt;Nevertheless, things may not be as they seem.  The excised material is available in Stewart's 6th edition Calculus.  The material on complex numbers is within the scope of information covered algebra and trigonometry, or in a pre-calculus or math analysis course.  The material on second order differential equations is supplemental and suitable for inclusion in a general course on differential equations.  The publishing editor's decision to remove this material from the 9th edition text was apparently intended to focus the subject matter more succinctly on Calculus as the principal subject.  This is understandable, particularly since further study in the subjects comprising complex numbers and second order differential equations is more advanced and specific to study in those areas&lt;/p&gt;&#xA;" OwnerUserId="19369" LastActivityDate="2022-01-15T20:29:33.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24812" PostTypeId="2" ParentId="9847" CreationDate="2022-01-16T21:09:28.707" Score="2" Body="&lt;p&gt;If you want to convince someone that certain triples are the side lengths of a right triangle, you can exhibit pictures like the following, which show the &lt;span class=&quot;math-container&quot;&gt;$(4,3,5)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(12,5,13)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(8,15,17)$&lt;/span&gt; right triangles.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mPZfQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mPZfQ.png&quot; alt=&quot;three tilted rectangles with shaded right triangle&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The idea here is to let the given triple be &lt;span class=&quot;math-container&quot;&gt;$(a,b,c)$&lt;/span&gt;.  Then one of the diagonals of the tilted rectangle is vertical and has length &lt;span class=&quot;math-container&quot;&gt;$2c$&lt;/span&gt;, while half of the other diagonal, which is on a slant, coincides with the hypotenuse of the right triangle with legs &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. Since the diagonals of a rectangle are equal, this demonstrates that the hypotenuse of the right triangle has length &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Some details: the corners of the tilted rectangle are at &lt;span class=&quot;math-container&quot;&gt;$(0,c)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(0,-c)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(-a,-b)$&lt;/span&gt;. You know this is a rectangle because the slope of the line joining the first and second points is &lt;span class=&quot;math-container&quot;&gt;$-\frac{c-b}{a}$&lt;/span&gt;, while the slope of the line joining the second and third points is &lt;span class=&quot;math-container&quot;&gt;$\frac{c+b}{a}$&lt;/span&gt;. These are negative reciprocals of each other because the equality&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;-\left(-\frac{c-b}{a}\right)^{-1}=\frac{a}{c-b}=\frac{c+b}{a}&#xA;$$&lt;/span&gt;&#xA;is equivalent to the Pythagorean theorem. Although we are pretending we don't know the Pythagorean theorem, the given triple must satisfy it nonetheless. At this point, the negative reciprocal property is just be a property that happens to be satisfied by the given numbers. The corners of the right triangle are at &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(0,b)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is actually only a few steps shy of a proof of the Pythagorean theorem, as long as one is willing to countenance use of properties of similar triangles and a bit of algebra.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VaNgw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VaNgw.png&quot; alt=&quot;tilted rectangle with shaded right triangle and labeled vertices and edges&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Given &lt;span class=&quot;math-container&quot;&gt;$\triangle ACB$&lt;/span&gt; with opposite sides &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, extend the line through &lt;span class=&quot;math-container&quot;&gt;$\overline{AC}$&lt;/span&gt; and draw the circle with center &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and radius &lt;span class=&quot;math-container&quot;&gt;$\overline{AB}$&lt;/span&gt;, intersecting the line at &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;. The rectangle is now easily constructed. Since &lt;span class=&quot;math-container&quot;&gt;$\triangle DCB$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\triangle BCE$&lt;/span&gt; are similar, we have the result,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{c-b}{a}=\frac{a}{c+b}.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If one wishes to stick with purely geometric reasoning, construct three squares. Construct &lt;span class=&quot;math-container&quot;&gt;$ACJK$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\overline{AC}$&lt;/span&gt; (area is &lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt;). Construct &lt;span class=&quot;math-container&quot;&gt;$ADLM$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\overline{AD}$&lt;/span&gt; (area is &lt;span class=&quot;math-container&quot;&gt;$c^2$&lt;/span&gt;). Then, drawing the perpendicular to &lt;span class=&quot;math-container&quot;&gt;$\overline{CB}$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and the perpendicular to &lt;span class=&quot;math-container&quot;&gt;$\overline{DE}$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, the two perpendiculars meeting at &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, subdivide rectangle &lt;span class=&quot;math-container&quot;&gt;$BCEF$&lt;/span&gt; by a line &lt;span class=&quot;math-container&quot;&gt;$\overline{GH}$&lt;/span&gt; into a square &lt;span class=&quot;math-container&quot;&gt;$EFGH$&lt;/span&gt; (area is &lt;span class=&quot;math-container&quot;&gt;$a^2$&lt;/span&gt;) and a rectangle &lt;span class=&quot;math-container&quot;&gt;$BCHG$&lt;/span&gt;. The Pythagorean theorem is the statement that the gnomon (L-shaped region) &lt;span class=&quot;math-container&quot;&gt;$CDLMKJ$&lt;/span&gt; and the square &lt;span class=&quot;math-container&quot;&gt;$EFGH$&lt;/span&gt; have equal area.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/U5i5v.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/U5i5v.png&quot; alt=&quot;tilted rectangle with shaded triangle, labeled vertices, and added squares&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This can be proved by cut-and-paste geometry with the aid of the line perpendicular to &lt;span class=&quot;math-container&quot;&gt;$\overline{GH}$&lt;/span&gt; through the point &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\overline{GH}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\overline{BE}$&lt;/span&gt; intersect. Let this line intersect &lt;span class=&quot;math-container&quot;&gt;$\overline{CB}$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\overline{EF}$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. Then rectangle &lt;span class=&quot;math-container&quot;&gt;$CNPE$&lt;/span&gt; equals the gnomon in area. Since this rectangle's area equals the area of &lt;span class=&quot;math-container&quot;&gt;$\triangle BCE$&lt;/span&gt; plus the area of &lt;span class=&quot;math-container&quot;&gt;$\triangle EPI$&lt;/span&gt; minus the area of &lt;span class=&quot;math-container&quot;&gt;$\triangle BNI$&lt;/span&gt;, while the area of square &lt;span class=&quot;math-container&quot;&gt;$EFGH$&lt;/span&gt; equals  the area of &lt;span class=&quot;math-container&quot;&gt;$\triangle EFB$&lt;/span&gt; plus the area of &lt;span class=&quot;math-container&quot;&gt;$\triangle IHE$&lt;/span&gt; minus the area of &lt;span class=&quot;math-container&quot;&gt;$\triangle IGB$&lt;/span&gt;, and corresponding triangles in these calculations are congruent in pairs, the result is established.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/im5F5.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/im5F5.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1737" LastEditorUserId="1737" LastEditDate="2022-01-16T21:15:09.977" LastActivityDate="2022-01-16T21:15:09.977" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24813" PostTypeId="1" CreationDate="2022-01-17T10:36:55.220" Score="0" ViewCount="93" Body="&lt;p&gt;I try to have a problem solving approach to learning math. What i mean by this is if someone sets some questions or problems regarding the material i am reading how should i answer or what questions or problems should i make from the material i am reading to help me in learning,&lt;/p&gt;&#xA;&lt;p&gt;But, i confuse it with problem solving generally. Generally, i have tested my skills on problem solving and sometimes i fail to solve some problems. Probably i should not confuse these two ways or approaches?&lt;/p&gt;&#xA;&lt;p&gt;Also, what should i do to feel better about math and learning and not feel like having fear or sadness?I have read if i remember correctly that in the United States of America, many people have fear and sadness about learning and math and this affects me at least a little i think.&lt;/p&gt;&#xA;&lt;p&gt;Generally, i read undergraduate material in math and i try some papers which are uploaded on arxiv from researchers. Sometimes, i read free material uploaded from other sources.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastActivityDate="2022-01-17T10:36:55.220" Title="Problem solving approach to learning and psychology" Tags="|self-learning|problem-solving|psychology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24814" PostTypeId="1" AcceptedAnswerId="26537" CreationDate="2022-01-17T18:44:17.487" Score="4" ViewCount="454" Body="&lt;p&gt;I am not sure if my question is relative to this meta but I still want to put forth my thoughts and concerns and questions because I think its not just me but others too who have similar issues.&lt;/p&gt;&#xA;&lt;p&gt;My concerns mostly revolve around students with mental and learning disabilities in academia and specifically from undergraduate level and upwards.&lt;/p&gt;&#xA;&lt;p&gt;What kind of learning paths they should adopt?&lt;br /&gt;&#xA;How should teachers modify courses for them?&lt;br /&gt;&#xA;What would be criteria for their scholarships?&lt;/p&gt;&#xA;&lt;p&gt;For it to be more clear and explicit I am going to use myself as an example.&lt;/p&gt;&#xA;&lt;p&gt;I got admitted into this graduate programme for theoretical computer science two years ago. In both of my semesters from the first year, I would find myself completely lost in whatever I was doing even in the topics and subjects which I was confident about. I showed my concerns to my professors after a couple of discussions with them I decided to take relevant Bachelors courses which would improve my basics. So I did that but I had to go through same difficulties in them so again after introspection and discussions with fellow academics, I enrolled myself officially into Bachelors but same problems exist even now.&lt;/p&gt;&#xA;&lt;p&gt;Recently I(27) have been diagnosed with dyslexia and ADHD I don't know how I am going to efficiently continue my academic career. My university is going to be helpful in terms of time required to finish my degree, but having to do Bachelors again from 1st world country after moving from 3rd world country is not economical at all and getting scholarships is not something you can be surely definite about.  Most of the scholarships I have found they don't support my requirements (having to finish degree in longer time). So I want to know how to tackle such situations in European education systems because scholarships comes with responsibility of maintaining sufficient credits? So should there be a leniency in scholarships for such students?&lt;/p&gt;&#xA;&lt;p&gt;In my courses I have always felt that I need more time to finish assignments in order to have a good grasp of concepts but considering the disability I have and reports from my doctors I have only gotten to have extra time in the assignments not what I actually have proposed. So I want to ask that does my need justify my demand? Because I have found this in repetition that more time on exams seems not that of useful if I have not grasped concepts properly, since I would be spending time just to get bare minimum marks so that I have enough average before the exam for each course.&lt;/p&gt;&#xA;&lt;p&gt;Now I am at the point where I have passed two analysis(real) courses, one combinatorics course and I this loop of thinking is not stopping that as the courses get more advanced I don't know what to do because its just that I did not get to have a grasp of proofs properly. I have already revisited concepts and proofs from my combinatorics course twice with solving problems but whenever I look at same things after sometime I can't even recall definitions that I have to reread whole topic or sometimes chapter. I know I am still an amateur in this regard but blanking on definitions is a concern IMO after you have passed a course and repeated it. So such students should be given leniency according to their need? I ask this because a lot teachers put emphasis on time for homework when classes begin.&lt;/p&gt;&#xA;&lt;p&gt;I told and asked this because I do math for passion and I want to do it right(not perfect), perfect is what you strive for.&lt;/p&gt;&#xA;" OwnerUserId="19378" LastActivityDate="2023-06-14T10:25:33.530" Title="Mental Health in Mathematics" Tags="|undergraduate-education|exams|homework|graduate-education|special-needs-education|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24815" PostTypeId="1" CreationDate="2022-01-18T07:58:05.513" Score="5" ViewCount="260" Body="&lt;p&gt;Which books contain practice questions — preferably with full solutions — to assist 15 year olds with the &lt;a href=&quot;https://betterexplained.com/articles/learning-how-to-count-avoiding-the-fencepost-problem/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fence Post&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Off-by-one_error#Fencepost_error&quot; rel=&quot;nofollow noreferrer&quot;&gt;Off by One&lt;/a&gt; error?&lt;/p&gt;&#xA;&lt;p&gt;Most students at my institution have not heard of this name, though some recognized the error.  Some of them are hankering after more practice exercises, to assist them with preventing it! Even after I cover the concept in general, though without exercises, too many of them fail to spot and avoid it on exams and tests.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorDisplayName="user155" LastEditDate="2022-01-18T22:43:49.233" LastActivityDate="2022-01-31T19:40:04.650" Title="For 15 year olds, are there exercises — with full solutions — on the Fence Post or Off by One error?" Tags="|reference-request|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24816" PostTypeId="1" AcceptedAnswerId="24817" CreationDate="2022-01-18T10:12:50.397" Score="3" ViewCount="272" Body="&lt;p&gt;In my institution I have a friend who is part of this Math club where they write blog posts and share them online at medium.com. However, there is an issue with my friends post.After they shared it with me to go through, check for errors and edit, I looked online and found &lt;a href=&quot;https://www.cantorsparadise.com/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;. My issue is that their working and explanations are all copied from the person who wrote this, my friend has only changed a few words(5-10). I am worried that if they post this they will get into a lot of trouble since plagiarism is dealt with very harshly in my institution. However, they denied copying from there. I don't mean to sound rude, but I do not believe them. How should I go about this? Should I report them? FYI: There were no citations in their article.&lt;/p&gt;&#xA;" OwnerUserId="19265" LastEditorUserId="19265" LastEditDate="2022-05-15T13:37:00.913" LastActivityDate="2022-05-15T13:37:00.913" Title="Is copying working and explanation plagiarism in this context?" Tags="|secondary-education|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24817" PostTypeId="2" ParentId="24816" CreationDate="2022-01-18T18:46:00.050" Score="3" Body="&lt;p&gt;There is no way to know the provenance of the information found in the online blog that was the suspected source.  Several people have written about this odd mathematical series, including Wolfram.  Your friends may have been trying to efficiently summarize, as best they could, the considerable information they found in attempting to educate others and clarify this rather strange mathematical series which adds to -1/12.  Is there any way to claim that a math instructor's class lecture notes doing the same thing would be called plagiarism? Well, probably not.  Nevertheless, someone may wish to argue the issue depends on one's view of the situation.&lt;/p&gt;&#xA;" OwnerUserId="19382" LastActivityDate="2022-01-18T18:46:00.050" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24818" PostTypeId="2" ParentId="24816" CreationDate="2022-01-18T21:31:38.773" Score="1" Body="&lt;p&gt;You've warned him.  I would take zero further action.&lt;/p&gt;&#xA;&lt;p&gt;It's not even clear to me that your institution will punish someone for plagiarism that is not part of a class.  But in any case that's up to them.  If they find out.&lt;/p&gt;&#xA;" OwnerUserId="19383" LastActivityDate="2022-01-18T21:31:38.773" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24819" PostTypeId="1" CreationDate="2022-01-18T22:12:12.417" Score="2" ViewCount="108" Body="&lt;p&gt;Like the title says, I am looking for ways to measure the complexity of an algebra equation.  For now, I am focused on linear equations, but I would think any metrics could be generalized for quadratic, etc.  I hoping the metrics can be useful for isolating student trouble spots and measuring progress in a rigorous way, as well as have a way to make progressively more complex equations to solve (this is specifically for U.S. Algebra I students).&lt;/p&gt;&#xA;&lt;p&gt;Some examples to point out what I am looking for:&lt;/p&gt;&#xA;&lt;p&gt;(1) &lt;span class=&quot;math-container&quot;&gt;$3x + 2 = 11$&lt;/span&gt;.  In this simple equation there are two operations, the moving the 2 over, and then dividing by 3.  I would think these operations would have different measures?&lt;/p&gt;&#xA;&lt;p&gt;(2) &lt;span class=&quot;math-container&quot;&gt;$2(x+3)+1=13$&lt;/span&gt;.  Here there are two additional operations beyond (1), the distributive property and combining the numbers on the left.  Already (1) and (2) could contain 4 quantitative measures.&lt;/p&gt;&#xA;&lt;p&gt;(3) &lt;span class=&quot;math-container&quot;&gt;$3-2(4-2x)=11$&lt;/span&gt;.  Here the working of negative signs add a different complexity.&lt;/p&gt;&#xA;&lt;p&gt;(4) &lt;span class=&quot;math-container&quot;&gt;$2x+3(x+3) = 4x+10$&lt;/span&gt;.  Here there are &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s to combine, and an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to move across the equals sign.&lt;/p&gt;&#xA;&lt;p&gt;(5) &lt;span class=&quot;math-container&quot;&gt;$2x+1=6$&lt;/span&gt;.  I am distinguishing this from the 1st equation because the answer is a fraction.  Is this considered more complex?&lt;/p&gt;&#xA;&lt;p&gt;So I am hoping there can be a set of scores that I can assign to an equation to determine complexity.  I've seen a lot of people talk implicitly about complexity, but I have not seen any explicit calling out of what makes equations complex.&lt;/p&gt;&#xA;" OwnerUserId="5265" LastActivityDate="2022-01-18T22:12:12.417" Title="Measures to quantify complexity of algebra equation" Tags="|algebra|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24820" PostTypeId="1" AcceptedAnswerId="24821" CreationDate="2022-01-19T04:22:10.007" Score="13" ViewCount="2839" Body="&lt;p&gt;I'm a lousy artist. If I want my Gaussian curves to be accurately drawn when I use a whiteboard, or work with pen &amp;amp; paper, what are my options?&lt;/p&gt;&#xA;&lt;p&gt;Is there a way to use a straight edge, or compass, or some other trick to getting accurate curves from the Gaussian family?&lt;/p&gt;&#xA;&lt;p&gt;I want to make more symmetrical sketches where the error of any given area under the curve is minimized.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hnvdQ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hnvdQ.png&quot; alt=&quot;Normal distribution&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19385" LastEditorUserId="19385" LastEditDate="2022-01-19T08:36:57.810" LastActivityDate="2022-01-19T23:37:04.787" Title="How do I sketch a good gaussian curve freehanded, or by using only common sketching tools?" Tags="|calculus|geometry|statistics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24821" PostTypeId="2" ParentId="24820" CreationDate="2022-01-19T04:44:15.010" Score="21" Body="&lt;p&gt;I would put dots where I want 1 standard deviation to be, because I know that's where the inflection points are. (I just graphed &lt;span class=&quot;math-container&quot;&gt;$y=e^{-x^{2}/2}$&lt;/span&gt; on desmos, and I see that the inflection points are at a height of about 60% of the maximum, so that's about 3/5ths of the way up. But I didn't do that step back when I taught statistics.)&lt;/p&gt;&#xA;&lt;p&gt;And then the curve is almost down to 0 at 3 standard deviations. Draw the horizontal axis below, show tick marks from -3 to 3 standard deviations, put in the dots at one standard deviation and about 3/5ths  as high at the max, and then draw (thinking about the concavity as you go).&lt;/p&gt;&#xA;&lt;p&gt;Actually, I think I did practice a lot back then. Because my graphs were sometimes very much not symmetrical. Also, students' graphs are worse, so I gave them a sheet of empty normal curves (6 per side) for them to use. And told them they could have as many copies as they wanted.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BRttC.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BRttC.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-01-19T21:11:03.883" LastActivityDate="2022-01-19T21:11:03.883" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24822" PostTypeId="2" ParentId="24820" CreationDate="2022-01-19T19:59:33.577" Score="5" Body="&lt;p&gt;One trick for free-handing a more symmetrical bell curve is to draw it in two strokes, each starting from the center point (once right-to-left and once left-to-right).&lt;/p&gt;&#xA;&lt;p&gt;Perhaps you could make a stencil out of cardboard, which you could trace onto the board.&lt;/p&gt;&#xA;&lt;p&gt;However, I wonder about the pedagogical value of perfectly accurate drawings. There is an important difference between a &amp;quot;sketch&amp;quot; and a &amp;quot;plot&amp;quot;, and they each have their place. I would think that your students are probably not going to be able to reproduce any figures that rely on subtle scaling factors in their notes.&lt;/p&gt;&#xA;" OwnerUserId="19319" LastActivityDate="2022-01-19T19:59:33.577" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24823" PostTypeId="2" ParentId="24796" CreationDate="2022-01-19T20:54:28.807" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Then, what is it that I can add in a class? That is what confuses me.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That is the exciting part about teaching mathematics.  No matter how clear the slides (or book or ...) and no matter how complete the proofs, there is a big role for helping students understand.  I'll answer your comment rather than the question, as it seems that you would really like to upload slides ahead of time.&lt;/p&gt;&#xA;&lt;p&gt;Here are some examples of things you can do instead of read slides in class, in rough order of increasing distance from &amp;quot;reading the slides&amp;quot;.  This is by no means comprehensive.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do an example similar to one you've prepared, but with some different twist (not necessarily just different numbers).  For instance, if a slide shows a simple matrix multiplication, have one where the multiplication is commutative.  Or where the multiplication doesn't even exist in the other direction!&lt;/li&gt;&#xA;&lt;li&gt;Better: have the &lt;em&gt;students&lt;/em&gt; do an example that is similar, but different!  For instance, perhaps your slide has a row reduction to a degenerate matrix, but the students are given one (or several split up among groups!) that might or might not be nonsingular.  See what they get.&lt;/li&gt;&#xA;&lt;li&gt;Have a quiz based on a few of the slides at the beginning.  Then you aren't just posting the slides ahead of time, you're expecting they will look at them.  (I don't really recommend this with slides, but with a video this could work, and a lot of classroom management software now supports embedding basic types of quiz questions inside of narrated slideshows ...)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.inquirybasedlearning.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Help your students discover your theorems in class&lt;/a&gt; using inquiry-based (or one of the many other names for this) learning.  You can still do this with a &amp;quot;slides&amp;quot; as your main source, though in this case posting slides afterward might be a better bet ... and as you can tell, this is now pretty far from a slides-based lecture pedagogy.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I'm not sure what level (nor in what cultural milieu) you are teaching, but these can be translated to many different environs.  You may have to &amp;quot;sell&amp;quot; new pedagogies to students unfamiliar with them, but, in my view, this is worth the effort.  That, of course, is a big enough discussion for several other questions.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2022-01-19T20:54:28.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24824" PostTypeId="2" ParentId="24820" CreationDate="2022-01-19T23:37:04.787" Score="1" Body="&lt;p&gt;Consider using a tool such as a 'flexible curve' to draw your curves on a whiteboard or paper. It's like a stiff mouldable ruler - you bend it into your required shape and then draw along it.&lt;/p&gt;&#xA;" OwnerUserId="19386" LastActivityDate="2022-01-19T23:37:04.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24826" PostTypeId="1" CreationDate="2022-01-20T11:40:45.997" Score="0" ViewCount="221" Body="&lt;p&gt;After reading some papers about special kinds of algebras and rings like Gorenstein rings, Dickson algebras, Cayley-Dickson construction,  i want to ask do examples of general abstract objects in math like groups, rings, manifolds, abstract curvatures, modular forms that help us understand helps us know where the directions of math go to?&lt;/p&gt;&#xA;&lt;p&gt;For example, the n-manifold in n dimentions is defined and as special cases we have curves and surfaces, another example is the groups , where we have the set of real numbers equipped with addition and gives us the abelian group &lt;span class=&quot;math-container&quot;&gt;$(R,+)$&lt;/span&gt;, or with the complex numbers. Then we learn other examples of groups like the group of matrices equipped with addition or multiplication where the first is abelian group and the second nonabelian respectively.&lt;/p&gt;&#xA;&lt;p&gt;Questions: Does this way help us understand generally things in math?By considering special cases? Do you know cases where this perhaps is not applied? If it is not applied could it be applied by analogy?If i try to read math by considering special cases of objects to help me is this valid and okay?&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastEditorUserId="17553" LastEditDate="2022-01-20T11:59:10.700" LastActivityDate="2022-01-20T12:14:43.213" Title="Abstract math, examples and understanding or visualising" Tags="|self-learning|examples|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24827" PostTypeId="2" ParentId="24826" CreationDate="2022-01-20T12:14:43.213" Score="3" Body="&lt;p&gt;I think that constructing examples which satisfy and do not satisfy the definitions you are learning is essential.  In fact, I include this as an essential learning outcome tied to every definition in the courses I teach.&lt;/p&gt;&#xA;&lt;p&gt;You seem to be worried about whether you are learning math &amp;quot;the right way&amp;quot; (or perhaps &amp;quot;the most efficient way&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;There is no royal road to geometry.  Everyone must clear their own path.  I remember an interview with Terry Tao (iirc) where he describes wriggling in bed, thinking of his body as a 2 dimensional wave.  This helped him achieve a crucial insight in a problem he was working on which concerned Fourier analysis.&lt;/p&gt;&#xA;&lt;p&gt;This example just highlights that people are going to engage in all kinds of interesting physical, mental, and social behaviors when they are learning anything, or solving any problem.  As long as you are not hurting anyone, you are okay!&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-01-20T12:14:43.213" CommentCount="6" CommunityOwnedDate="2022-01-20T12:14:43.213" ContentLicense="CC BY-SA 4.0" />
  <row Id="24828" PostTypeId="2" ParentId="7199" CreationDate="2022-01-20T14:31:03.503" Score="4" Body="&lt;p&gt;In addition to the good answers already given, here are some additional ideas.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Joseph's answer predates (or is just at the beginning of?) the official start off the &lt;a href=&quot;https://nscoville.github.io/website/TRIUMPHS/OurStudentProjects/Projects/projects.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;TRIUMPHS project&lt;/a&gt;, which has many, many more such mini-projects.  They are intended for not history of math classes, but are very useful for those as well.  The &lt;a href=&quot;https://digitalcommons.ursinus.edu/triumphs_linear/1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chinese linear algebra one&lt;/a&gt; is a very good choice, for instance.&lt;/li&gt;&#xA;&lt;li&gt;In addition to Sue's excellent examples, I suggest these ones that my students seem to enjoy: Egyptian fractions (&lt;a href=&quot;https://tasks.illustrativemathematics.org/content-standards/tasks/839&quot; rel=&quot;nofollow noreferrer&quot;&gt;explicit computation&lt;/a&gt;); Greek numerals (the &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Greek_numbers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;learned system here&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Anything Euler: his amicable number formula, finding an Eulerian trail, Euler characteristic ... a lot of these are possible to compute pretty quickly in a classroom setting&lt;/li&gt;&#xA;&lt;li&gt;Doing a derivative using Newton's fluxion notation.  Any math history book should have this - but it's nice since his approach makes finding slopes of a variety of curves (not just a function graph) doable, analogous to implicit differentiation&lt;/li&gt;&#xA;&lt;li&gt;Solving a cubic or quartic polynomial using the various formulas of Cardano's era.  I don't personally like this, but students seem to.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2022-01-20T14:31:03.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24829" PostTypeId="1" CreationDate="2022-01-20T14:58:16.933" Score="6" ViewCount="464" Body="&lt;p&gt;I'm Italian and I've watched some videos from Americans and noticed a weird thing. Let's talk about a linear trigonometric equation like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sin x+\cos x+\sqrt3=0.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I've seen Americans solving it by just doing an&#xA;irrational equation, which takes more time and is pointless.&lt;/p&gt;&#xA;&lt;p&gt;Here we are taught that there are three ways to solve it:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;We let &lt;span class=&quot;math-container&quot;&gt;$Y = \sin x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$X = \cos x$&lt;/span&gt; and do a system with the equation of the trigonometric circle, &lt;span class=&quot;math-container&quot;&gt;$X^2 + Y^2 = 1$&lt;/span&gt;, then we find the values of the sine and cosine and find the related angles, we call this the graphic method.&lt;/li&gt;&#xA;&lt;li&gt;We let &lt;span class=&quot;math-container&quot;&gt;$a= \sin x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b=\cos x$&lt;/span&gt;, calculate &lt;span class=&quot;math-container&quot;&gt;$r=\sqrt{a^2+b^2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\alpha =\arctan \frac{b}{a}$&lt;/span&gt; and use the formula &lt;span class=&quot;math-container&quot;&gt;$r \cdot \sin(x+\alpha)=c$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; in this case is &lt;span class=&quot;math-container&quot;&gt;$-\sqrt 3$&lt;/span&gt;. We call this method of the auxiliary angle.&lt;/li&gt;&#xA;&lt;li&gt;We use the formulae &lt;span class=&quot;math-container&quot;&gt;$\sin x=\frac{2t}{1-t^2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos x=\frac{1-t^2}{1+t^2}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$t= \tan(x/2)$&lt;/span&gt; and solve the equation.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Isn't this taught in the US?&lt;/p&gt;&#xA;" OwnerUserId="19390" LastEditorUserId="376" LastEditDate="2022-01-21T16:07:38.287" LastActivityDate="2022-01-21T17:17:32.857" Title="Is Trigonometry done differently in the US?" Tags="|precalculus|trigonometry|usa|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24830" PostTypeId="2" ParentId="7199" CreationDate="2022-01-21T03:23:18.173" Score="3" Body="&lt;p&gt;Have them use an abacus.  (I prefer to teach using a &lt;a href=&quot;https://en.wikipedia.org/wiki/Suanpan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chinese abacus&lt;/a&gt;, which has 2+5 beads per column, and not using a &lt;a href=&quot;https://en.wikipedia.org/wiki/Soroban&quot; rel=&quot;nofollow noreferrer&quot;&gt;Japanese abacus&lt;/a&gt;, which has 1+4 beads per column.)&lt;/p&gt;&#xA;&lt;p&gt;I propose the following activity.  (I've only done this in undergraduate classes a handful of times, but the students seemed to enjoy it.)  Teach them how to use the abacus to add numbers.  Do &lt;em&gt;not&lt;/em&gt; teach them how to subtract.  Then have them add nines in succession, with you starting the process.  Start from 0, then add 9, but add 9 as +5+1+1+1+1.  Do this three or four times.  Then have volunteers do the addition by 9.  The students will find this quite difficult at the start.  Then (in my experience), a student will discover (&lt;em&gt;on their own&lt;/em&gt;) that it is much easier to do the addition by 9 as +10-1.  Once a student has done this, point it out to the class.  Praise the student for finding (without being prompted) a simpler procedure than the one taught by the teacher.  This ends the activity.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2022-02-09T11:10:21.687" LastActivityDate="2022-02-09T11:10:21.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24831" PostTypeId="2" ParentId="24829" CreationDate="2022-01-21T07:21:31.190" Score="9" Body="&lt;p&gt;As you've tagged this as precalculus, I'll say that there is no standard precalculus curriculum in the United States and very little &amp;quot;official&amp;quot; guidance about what such a curriculum should contain.  I think we've always struggled with the puzzle of what to teach in the fourth year of high school to students who mastered the &amp;quot;essential&amp;quot; math in three years.  Over the next generation, I predict we'll just toss up our hands and teach introductory Calculus to everyone.  As we stand at the moment, the class is a hodgepodge of interesting topics that don't require calculus (like matrices and conic sections) and preparation for calculus (like convergent sequences and the definition of the limit).&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;typical&amp;quot; trigonometry curriculum (and, again, there is no uniformity in the United States) is spread out over several years.  Typically, second-year students are learning plane geometry, and learn about using trigonometric functions on acute angles to solve right triangles.  Third year students then study the unit circle to extend the trig functions to have a domain of all real numbers.  This goes off in two directions: the development of the trig identities (specifically the Law of Sines and Law of Cosines to allow solving oblique triangles) and an analysis of the general sinusoidal function family .&lt;span class=&quot;math-container&quot;&gt;$y=a+b\sin(cx+d)$&lt;/span&gt;. In practice, there are too many useful trig identities and too many other topics to cover them all in a single year for a non-gifted class, so some of the topics are pushed into the precalculus stewpot or abandoned altogether.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Just for lulz, I tried solving this problem with my American education.  My first attempt was essentially your method 1, because the Girard-Newton identities are a bug that has bitten me recently.  But that led me to needing to find the roots of a quadratic equation with irrational coefficients and I ran out of patience with following that trail further.  Also, it's not how I would have solved this problem when I was a student anyways.&lt;/p&gt;&#xA;&lt;p&gt;So here is my second attempt, which is how I would teach students to solve equations of this sort if it were mandated that they learn it (and to be frank, I'm not sure why it should be). It uses the Pythagorean identity and the sine double angle identity that are part of the Common Core.  So while an Algebra 2 student may not be taught specifically how to solve equations of this sort, they should have little difficulty verifying my work.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sin x+\cos x=-\sqrt3\\(\sin x+\cos x)^2 =3\\\sin^2x+\cos^2x+2\sin x\cos x=3\\1+\sin 2x=3\\\sin 2x=2$$&lt;/span&gt; which has no solution (in the reals) because the values of sine only range from -1 to 1.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2022-01-21T17:17:32.857" LastActivityDate="2022-01-21T17:17:32.857" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24832" PostTypeId="2" ParentId="24829" CreationDate="2022-01-21T16:15:20.893" Score="-1" Body="&lt;p&gt;First, I think there is much more to trigonometry overall, than just this sort of problem (which is on a little bit the trickier side).  If you look at a book, there will be plenty of trig that is much more foundational.  So I think just saying &amp;quot;trig courses are different in 'murica...oooh la la&amp;quot; is not the right overall assessment, given the probable majority of content that's the same.&lt;/p&gt;&#xA;&lt;p&gt;My copy of Schaum's precalculus says there's no hard and fast rule on how to solve different trig equations, but advocates trying the following three approaches:&lt;/p&gt;&#xA;&lt;p&gt;A. Factoring (if already possible) into equation equal to zero.&lt;/p&gt;&#xA;&lt;p&gt;B. Converting into a single function (e.g. getting rid of some cossq to have all sinsq in an equation).&lt;/p&gt;&#xA;&lt;p&gt;C. Squaring both sides.  They have the specific example sinx + cosx = 1, which is almost exactly yours (but ends up easy to factor).  You &amp;quot;throw&amp;quot; cosx functions &amp;quot;onto the other side&amp;quot; and then square both sides, then replace sinsq with 1- cossq.  Then group terms and you have  a quadratic eqn in cosx.  Really, it's not that different than if you substituted the radical sq(1-sinsqx) immediately for cosx and then cleaned it up by squaring after.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I kind of get the impression method 1 ends up the same as what Schaum's talked about.  Or those (unlinked) videos you mentioned.  I do think the methods 2 and 3 you showed are interesting.  But unless you're deep in a strong trig course, you're not going to be jamming that sort of stuff all the time.  Yeah...good to have it at the time.  But as memory fades, not something you use all the time in higher courses.&lt;/p&gt;&#xA;&lt;p&gt;P.s.s.  I solved the problem using the Schaum's method and the quadratic method. Ans: x =arccos((-sqrt(3)+/-i)/2).  Whatever the heck that means.  I'm assuming the complex argument results for same reason as Daly not finding any answer.  But if you'd given a different constant and the solution was in the reals, the method would have gotten there fine.  And I didn't find it at all laborious to use.  Maybe seven lines (and I like to show all the steps).&lt;/p&gt;&#xA;" OwnerUserId="19396" LastActivityDate="2022-01-21T16:15:20.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24833" PostTypeId="1" CreationDate="2022-01-21T17:01:08.130" Score="4" ViewCount="466" Body="&lt;p&gt;I'm in my 3rd year of pure math bachelors, and a thorny issue keeps on reappearing for me. I understand the theory, for a day, maybe two. I've reached the point where our lecturers no longer detail every single step, but leave some details to be filled out myself. Most of the time I have no issue filling in the blanks myself, I understand everything, I can follow the line of thought. But after a few days I just forget everything, conditions for a theorem, etc. Even what I studied yesterday can be forgotten today. I feel like there is a finite box inside my head, and after its full if I want to put something in I have to push something out (involuntarily).&lt;/p&gt;&#xA;&lt;p&gt;Are there ways to solve my problem? It's very worrying for me I can debate with my friends about the theory when I have access to my notes, but keep on failing theoretical exams (written or oral).&lt;/p&gt;&#xA;" OwnerUserId="19397" LastActivityDate="2022-03-16T04:07:24.540" Title="How do people 'master' a subject? What is the difference with 'memorizing' (not purely memorizing, but rather having it drilled inside the brain) it?" Tags="|undergraduate-education|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24834" PostTypeId="2" ParentId="24833" CreationDate="2022-01-21T22:38:09.307" Score="5" Body="&lt;p&gt;Create lessons, blog posts or the like, explaining what you want to learn to an audience of an appropriate level - could be just yourself, or your friends, students of a similar level, etc.  You often learn what you teach, even when not literally teaching as a paid tutor or instructor.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2022-01-21T22:38:09.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24836" PostTypeId="2" ParentId="24833" CreationDate="2022-01-22T16:13:25.730" Score="1" Body="&lt;p&gt;I realize the work is harder and more abstract, but you need to try to use conventional methods of pedagogy, nonetheless.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Make a set of flashcards for key definitions.  Drill them. You can carry a pack and go over them when stuck in a line or waiting for class to start or the like (little lost time parts of the day). I know this sounds juvenile, but it will help.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Install a system of rolling review.  Note, rolling review does NOT mean re-reading stuff.  It means retesting yourself.  By redoing proofs, doing problems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to figure out a way to do some simpler drill problems for each lesson, not just super hard ones.  Look at alternate books, etc.  Or even just doing the proof several times.  Maybe even hokey things, like creating two examples and two counterexamples.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do all the homework problems for each lesson, when you get it.  You say you can handle the material at the time.  But don't recall it later.  Well, do all the problems then.  Not just the assignment.  But everything in the book.  [For rolling review, use other books with more problems, or just repeat the old ones...at least it proves you're not forgetting stuff!]  The point is that you need to &amp;quot;overtrain&amp;quot;.  Don't say &amp;quot;I get it&amp;quot;.  Drill the content into your brain like Umbridge's pen into Harry Potter's hand.  I see students in all kinds of coaching, testing, etc. (not pure math, but many subjects) who think &amp;quot;I got it&amp;quot;, but they really haven't firmed up the knowledge enough.  And it shows under pressure.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given you are failing, figure out some plan to allocate more time to the coursework.  Ask advice from an advisor, but make it your responsibility in the end.  Can you slim down the number of courses?  Eschew work or clubs?  In the end, if it's bad enough, you may want to drop to an easier track (applied?) if possible.  But try to make this work first with best efforts.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Meet regularly with your academic advisor for discussion of your situation.  It can be a little monastic to do so much math homework, so if you have some human connection understanding your effort, maybe that helps you put in the work.  Also, he may have advice for specific classes (yes, different/additional to your course teacher).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;See your teachers regularly with questions and to review homework issues.  For one thing it's also human connection.  And it may help on the content also.   Not to brown nose.  But with specific things (keep a list in your notebook...I use the back and go towards the front, since notes and homework drill goes front to back.)  A lot of teachers will see you outside of &amp;quot;office hours&amp;quot; if you just make an appointment (can try dropins also, but be prepared for a no...but then they'll usually specify something later).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This is controversial, but I have mixed feelings (mostly negative) on study groups.  The human interaction is nice.  But you can get distracted with socializing.  Also, you are not going to have the study group on the tests.  And then if others are much stronger, you may just sort of nod your head and make a few remarks.  But not really be training yourself to know the material.  I would probably moderate the use of study groups (lower time spent there) and try to do prework on your own and have specific questions for your colleagues, when you do go to them.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19404" LastActivityDate="2022-01-22T16:13:25.730" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24837" PostTypeId="2" ParentId="10970" CreationDate="2022-01-22T19:24:01.753" Score="1" Body="&lt;p&gt;The International Baccalaureate program has a sequence of Math courses.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/IB_Diploma_Programme&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.m.wikipedia.org/wiki/IB_Diploma_Programme&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The mathematics section covers topics up to integral calculus.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2022-01-22T19:24:01.753" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24838" PostTypeId="1" CreationDate="2022-01-23T16:43:15.413" Score="2" ViewCount="155" Body="&lt;p&gt;So far I've had three types of professors in my undergrad studies when it comes to choosing the main text for the course:&lt;br /&gt;&#xA;&lt;strong&gt;Type A:&lt;/strong&gt; these are the professors who pick some standard textbook(s) in English that is(are) world renowned and will mostly use this(these) as the main source for the material they teach and they will rigorously follow it. Examples of such textbooks would be Do Carmo's Differential Geometry text or Rudin's Principles of Mathematical Analysis aka &amp;quot;baby Rudin&amp;quot;.&lt;br /&gt;&#xA;&lt;strong&gt;Type B:&lt;/strong&gt; these are those who pick some textbook written in our native language that is more or less a traditional book for the subject in our country and follow it (or them if they choose more than one book, but this is rare in this typology) just like the people in Type A. Most of these professors pick such a textbook that is usually published by the publishing house of our university, so I would call these &amp;quot;local&amp;quot; texts. In case you are wondering, I live in southeast Europe, so we are not talking about books written in some international language, this is why I won't include examples.&lt;br /&gt;&#xA;&lt;strong&gt;Type C:&lt;/strong&gt; those who write their own lecture notes (following various textbooks that may also appear in Type A or Type B classes that I described above). By write I don't necessarily mean write them in LaTeX and provide us with some book-like PDF. Most of them actually just write their notes on the blackboard (well, the virtual board nowadays) and we copy them down.&lt;br /&gt;&#xA;I wonder which of these approaches is actually the best. I've had classes that fall in either of these types that were really good and I can say that I learnt a lot of things, but I am unsure which is the best (I am mostly wondering due to my interest in pedagogy). I've heard Type A or Type C professors claiming that the books used by Type B professors are bad simply because they are not well established texts around the world. But I think that anyone's lecture notes are subject to the same criticism because after all the textbooks mentioned in Type B (and I think most textbooks in the world) originated from some lecture notes. I am curious what your experience has been with taking/giving classes that fall in either of these types. I feel that Type A classes are the best becuse their content is standard and thorough. The textbooks I mentioned in Type B are also thorough most of the times, but they certainly do contain more typos and mistakes then the standard texts on the subject that is taught. The Type C approach usually feels like the most lightweight of all of them simply because those lecture notes will 90% of the times be exactly what is covered in class, so you don't really learn anything new by reading them, they come in handy only if you miss class and you want to know exactly what the lecturer discussed.&lt;/p&gt;&#xA;" OwnerUserId="19326" LastEditorUserId="19326" LastEditDate="2022-01-24T14:42:45.127" LastActivityDate="2022-01-24T14:42:45.127" Title="Worldwide standard textbooks vs textbooks from one's home country vs lecture notes by various people - pros and cons" Tags="|undergraduate-education|textbooks|general-pedagogy|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="24840" PostTypeId="1" AcceptedAnswerId="24845" CreationDate="2022-01-24T23:57:44.433" Score="-2" ViewCount="365" Body="&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/924203/reversing-the-order-of-summation-in-sum-i-1n-i#comment1907997_924214&quot;&gt;This comment doesn't fulfill my students or me&lt;/a&gt;, because it doesn't demystify this trick of writing &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt; forward, then backwards, then adding. &lt;strong&gt;What would spur students to action these unnatural steps? How would students vaticinate this trick?&lt;/strong&gt; &lt;em&gt;&lt;strong&gt;I'm hankering after an answer other than &amp;quot;You just have to memorize this trick after you see it&amp;quot;!&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Our textbook below also doesn't expatiate this trick!&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Problem 3.6: Find a formula for the sum of the first &lt;em&gt;n&lt;/em&gt; positive integers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://math.codidact.com/uploads/KVTZdiwCZCmmTb1kMFgwCas3&quot; alt=&quot;Image alt text&quot; /&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;David Patrick, &lt;em&gt;Introduction to Counting &amp;amp; Probability&lt;/em&gt; (2005), p 54.&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastActivityDate="2022-02-08T03:09:39.083" Title="How can students prognosticate to rewrite the same sum backwards, then add the same sum twice?" Tags="|proofs|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24841" PostTypeId="2" ParentId="18977" CreationDate="2022-01-25T04:16:02.047" Score="2" Body="&lt;p&gt;The answers above are good - crystallography and cryptography are huge.  Here are a couple more &amp;quot;smaller&amp;quot; applications that occur to me that show up in textbooks fairly frequently (both group theory, not rings/fields).  It would be tough to make a whole course out of either one of these, but definitely doable to spend a day to a week or more covering these sorts of applications if you didn't have to cover ground in preparation for fields/modules/Galois theory.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Puzzle solving, such as Rubik's Cube.  There is &lt;a href=&quot;https://jhupbooks.press.jhu.edu/title/adventures-group-theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;at least one textbook&lt;/a&gt; (of sorts) that focuses heavily on this (with Sage code too!) by David Joyner.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Pretty much any combinatorics book beyond a very first one will cover things like the (Cauchy-)Frobenius(-Burnside) Lemma for counting various patterns, which is a serious application of group theory (&lt;a href=&quot;https://www.mtosmt.org/issues/mto.07.13.4/mto.07.13.4.hook.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;even in music theory&lt;/a&gt;).  &lt;a href=&quot;https://www.appliedcombinatorics.org/book/ch_polya.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Keller and Trotter&lt;/a&gt; is one I have used bits and pieces of on occasion, and it's &lt;a href=&quot;https://github.com/mitchkeller/applied-combinatorics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;open source too&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2022-01-25T04:16:02.047" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24842" PostTypeId="2" ParentId="24840" CreationDate="2022-01-25T13:03:01.590" Score="4" Body="&lt;p&gt;I think the main message that you need to communicate to your students, since it is not coming from the book, is that genuine problem solving involves several stages. To quote Polya, these are &amp;quot;understanding the problem&amp;quot; (i.e., considering the sum from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;), &amp;quot;devising a plan&amp;quot; (forming a conjecture of what we believe the sum to equal) , &amp;quot;carrying out the plan&amp;quot; (writing a formal proof of the conjecture), and then &amp;quot;looking back&amp;quot; (understanding our solution and alternatives to take something away for future problems). Even though this is a simple problem, it can still be used to illustrate the stages and cultivate a more realistic and productive attitude toward problem-solving than the initial surprised reaction of &amp;quot;how did I not think of that trick?&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The explanation in the book is a concise path from the problem statement to the conclusion; it does not illustrate or offer much guidance on &lt;strong&gt;forming a conjecture&lt;/strong&gt;, which is a big part of the battle on this problem for a new student (if not so difficult for Gauss). I think with a conjecture in mind, it would be easier to come up with the &amp;quot;reverse the sum&amp;quot; justification.&lt;/p&gt;&#xA;&lt;p&gt;The natural next question: how does one form a conjecture? Just as commenter Rusty Core said -- by playing with the numbers. It's a little difficult for me to put myself into the shoes of a student who has never seen this identity before, but one approach that came to mind is to mock up some data in a spreadsheet (or calculate a few values by hand) and speculate on a pattern. With the spreadsheet, it should be easy to check that your guess is at least consistent with a large number of cases. From there, you could use a more structured approach like induction to actually prove your conjectured formula.&lt;/p&gt;&#xA;&lt;p&gt;Another way to form a conjecture is to draw a picture for inspiration... here is a doodle I drew up over coffee:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Qu7I9.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Qu7I9.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So I drew a few arrays of shaded circles and then &lt;strong&gt;conjectured&lt;/strong&gt; that it seems you can complete the arrays to be a square (empty circles).&lt;/p&gt;&#xA;&lt;p&gt;Translating that into symbols:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \sum_{i=1}^{n} i = n^2 - \sum_{i=1}^{n-1} i$$&lt;/span&gt;&#xA;which can be seen to be equivalent to the familiar identity, and from there you can use some more structured approach like induction for the formal proof.&lt;/p&gt;&#xA;&lt;p&gt;Alternatively, you could start by just considering the sum in isolation and then manipulate it into equivalent algebraic forms in the hopes of finding a pattern. This is the solution outlined in the book. Having seen this solution, the reader has the opportunity to &amp;quot;look back&amp;quot; and ask what they can take away from this to future problems? One thing they may not have considered is to not only think of the sum as having components but also to think of it as a whole (i.e., assign it a label, &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;). Then you might ask: is there a way to produce a simple formula in terms of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; performing &amp;quot;simple&amp;quot; operations on &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;? If we can limit the operations on &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; to operations with constants or functions of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, then we may be able to solve resulting equations for &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. I'm not sure how much total mileage you can get out of this line of thinking in discrete math, but it comes up at least one more time when deriving the formula for the sum of a finite geometric series.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2022-01-25T13:03:01.590" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24844" PostTypeId="1" CreationDate="2022-01-25T14:43:56.860" Score="8" ViewCount="453" Body="&lt;p&gt;I'd like to ask my students whether some real function is differentiable at a certain &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;. I prefer not telling them that they have to use the definition of the derivative, but to instead present a question that by its nature would require the usage of the definition. Usually I achieve this goal by giving them some piecewise defined function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; which would force them to use the definition.&lt;/p&gt;&#xA;&lt;p&gt;The problem is the following theorem which can easily be proved using the mean value theorem: if &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is continuous in a neighbourhood of &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;, and differentiable in that neighbourhood except possibly at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; itself, then in case &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to x_0} f'(x)$&lt;/span&gt; exists then &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is differentiable at that point with derivative equal to that limit, and the result is also similarly&#xA;correct for one sided limits and one sided derivatives.&lt;/p&gt;&#xA;&lt;p&gt;This means that students often use this theorem either because they have seen a TA use it, or worse, because they don't see why it would require proof in the first place. In either case, this does not check their understanding of the definition (and I am left with the dilemma of how to grade these correct solutions which implicitly use a theorem which was not proven in class).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My question is:&lt;/strong&gt; are there exercises which you must solve using the definition of the derivative and which do not easily allow a cop out such as the one above?&lt;/p&gt;&#xA;&lt;p&gt;My thoughts are that there are 3 options-&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A function which is only differentiable at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; but not in a neighbourhood of &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;. For example the function given in &lt;a href=&quot;https://math.stackexchange.com/questions/1314576/differentiability-at-a-point-does-not-imply-differentiability-on-a-neighbourhood&quot;&gt;this question&lt;/a&gt;. All generalizations of this example seem to be very similar to the original one, so it is not a good exam question. There are more examples in the answer to that question, but they all rely on infinite series which have not yet been covered at this point.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A function for which continuity at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; is hard to prove. This seems unlikely, because if it's hard to prove continuity, it will also be hard to differentiate the function at that point.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A function which is not differentiable at all, not at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; nor around it. This would disallow calculating &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; and then taking its limit, but then we are left with mostly boring examples and in any case the solution is always that the derivative does not exist.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I'd appreciate any ideas for such a function which forces you to differentiate &amp;quot;by hand&amp;quot;, or any thoughts whatsoever on how to tackle this problem.&lt;/p&gt;&#xA;" OwnerUserId="19415" LastActivityDate="2022-01-29T19:42:10.473" Title="Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?" Tags="|calculus|exams|definitions|problem-design|derivative|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24845" PostTypeId="2" ParentId="24840" CreationDate="2022-01-25T16:18:42.443" Score="10" Body="&lt;p&gt;This answer has already more or less appeared in some comments, but it's &lt;em&gt;not&lt;/em&gt; expected that every student should come up with this &amp;quot;trick&amp;quot; on their own. I know I didn't. The important thing is that they see that there's nothing especially tricky about it, and given the right circumstances it's something they &lt;em&gt;could have&lt;/em&gt; come up with. Most people have noticed the fact that &lt;span class=&quot;math-container&quot;&gt;$1+9=2+8=3+7=\cdots=9+1$&lt;/span&gt;, and from there it's not much of a leap to Gauss' method. Nobody should feel bad if they don't notice a seemingly tricky method right away. What's important is that they understand it (maybe they even think it's beautiful and inspires them to learn math!) and possibly use a similar method later. It doesn't have to be a case of memorizing some incomprehensible rule.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2022-01-25T16:18:42.443" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24847" PostTypeId="2" ParentId="24844" CreationDate="2022-01-27T01:57:57.280" Score="7" Body="&lt;p&gt;I'll argue that this will be likely not feasible for a test question.&lt;/p&gt;&#xA;&lt;p&gt;There's several points in the calculus progression where there's a &amp;quot;hard bottleneck&amp;quot; of some sort, but once you get past that, you can leverage the new fact to easily solve previously-hard problems. One example: finding the derivative of a monomial (from the definition + binomial theorem), plus the sum rule, and then all polynomials become trivial. Second example: finding the derivative of the first trigonometric function, and then all trig functions become trivial.&lt;/p&gt;&#xA;&lt;p&gt;Yes, ideally we want our students to understand those key proofs. No, after you've done it once, it's not really possible to put it back in the bottle and forget the new foundational fact.&lt;/p&gt;&#xA;&lt;p&gt;So while somewhat awkward, I'll propose that to confirm understanding of those critical proofs, you just have to give an exercise that says, &amp;quot;prove this from the definition&amp;quot;. Anything else will likely be far too complicated or unnatural in an exam situation (per tag in OP).&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2022-01-27T01:57:57.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24848" PostTypeId="2" ParentId="24833" CreationDate="2022-01-27T02:58:52.537" Score="5" Body="&lt;p&gt;This is purely anecdotal, but it helped for me so perhaps it's useful for you too.&lt;/p&gt;&#xA;&lt;p&gt;Instead of trying to memorise definitions, theorems, proofs etc. I think it is important for your brain to do some active exploring. By this I mean, try to understand why the definitions are the way they are, what &amp;quot;philosophical&amp;quot; idea they try to capture, and how they connect to other concepts. Try to see how the conditions imposed in the definitions allow for the key theorems to be proven, and what happens when you strengthen/weaken these conditions. Or, at the very least, try to come up with the proofs for a theorem or understand the definition on your own, independently of the textbook first, rather than reading through and then trying to recall it. This active discovery process is imo essential for doing mathematics.&lt;/p&gt;&#xA;&lt;p&gt;For example, when learning linear algebra, ask yourself: why do we need to consider vector spaces defined over a field? Why do we need the ability to divide scalars, what happens if we replace it by a commutative ring? Well, it turns out that you get modules and some key results about vector spaces don't carry over, e.g. uniqueness of the size of a basis, which makes vector spaces (especially those of finite dimension) significantly simpler. This gives you some deep understanding that is very difficult to forget; moreover, you more easily see the connections between different fields that will be helpful in the future.&lt;/p&gt;&#xA;&lt;p&gt;I think actively trying to memorise things in mathematics, including the definitions, is totally against the spirit of the subject. It won't even be effective anyway, especially not in the long run. Don't do that, you won't enjoy it anyway.&lt;/p&gt;&#xA;" OwnerUserId="10206" LastEditorUserId="10206" LastEditDate="2022-01-27T10:26:24.710" LastActivityDate="2022-01-27T10:26:24.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24850" PostTypeId="2" ParentId="24844" CreationDate="2022-01-27T19:38:46.773" Score="8" Body="&lt;p&gt;Here are some example questions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/TrMWjm.png&quot; alt=&quot;Graph&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;The graph of the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is given above. Evaluate the following limits.  If the limit is infinite, write &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-\infty$&lt;/span&gt; as appropriate.  If the limit does not exist write &amp;quot;DNE&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;&amp;amp;\lim_{h \to 0^+} \frac{f(4+h)-f(4)}{h}\\&#xA;&amp;amp;\lim_{h \to 0} \frac{f(4+h)-f(4)}{h}\\&#xA;&amp;amp;\lim_{x \to 4^-} \frac{f(4)-f(x)}{4-x}\\&#xA;&amp;amp;\lim_{h \to 0^-} \frac{f(2+h)-f(2)}{h}\\&#xA;&amp;amp;\lim_{x \to 2^+} \frac{f(x)-f(2)}{x-2}\\&#xA;&amp;amp;\lim_{h \to 0} \frac{f(3+h)-f(3)}{h}&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;To successfully answer such questions, the student must have a pretty strong grasp of one sided limits, and the interpretation of difference quotients as slopes.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="667" LastEditDate="2022-01-29T19:33:25.623" LastActivityDate="2022-01-29T19:33:25.623" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24851" PostTypeId="2" ParentId="24833" CreationDate="2022-01-27T22:48:42.460" Score="5" Body="&lt;p&gt;Perhaps as much an extended comment as an &amp;quot;answer&amp;quot;: yes, sheer memory inevitably plays a role in almost any endeavor. But the comparison I'd want to make is that we don't &amp;quot;memorize&amp;quot; how to walk across a room, or throw or catch a ball, or many other things-we-do. We do not &lt;em&gt;formalize&lt;/em&gt; them: we do not attempt to describe what's going on, nor the goals, by a list of rules. Rather, just as with a driver's license test in the U.S., the formal test itself is quite disconnected from one's skills as a driver.&lt;/p&gt;&#xA;&lt;p&gt;As @JochenGlueck commented, thinking to memorize things (with &amp;quot;flash cards&amp;quot;, etc.), is at best a shallow &amp;quot;fix&amp;quot; to your situation. Rather, I'd recommend thinking that (contrary to the tradition of many textbooks) the &amp;quot;definitions&amp;quot; are convenient naming conventions that (by this year) have proven very relevant to talking about real phenomena. So (good) &lt;em&gt;examples&lt;/em&gt; are the longer-lasting point... Those are the reasons the terminology was invented.&lt;/p&gt;&#xA;&lt;p&gt;Also, for me, at least, thinking that the objects and their properties are &lt;em&gt;real&lt;/em&gt;, rather than made-up formal abstract non-existent things, helps me take them seriously, and remember them.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-01-27T22:48:42.460" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24852" PostTypeId="2" ParentId="24844" CreationDate="2022-01-28T12:40:02.777" Score="3" Body="&lt;p&gt;A nice class of examples that can be used for this purpose are functions that are differentiable but not continuously differentiable (since the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\lim_{x \to x_0} f'(x)$&lt;/span&gt; trick&amp;quot; doesn't work for them). A typical example is the following exercise:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Exercise.&lt;/strong&gt; Let &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{R} \to \mathbb{R}$&lt;/span&gt; be given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  f(x) = &#xA;  \begin{cases}&#xA;    x^2\sin(\frac{1}{x}) \quad &amp;amp; \text{if } x \not= 0, \\ &#xA;    0 \quad &amp;amp; \text{if } x = 0.&#xA;  \end{cases}&#xA;$$&lt;/span&gt;&#xA;Show that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differentiable at &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; and compute &lt;span class=&quot;math-container&quot;&gt;$f'(0)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The differential quotient at &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is clearly &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$f'(0)$&lt;/span&gt; exists and is equal to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. However, for &lt;span class=&quot;math-container&quot;&gt;$x \not= 0$&lt;/span&gt; one has&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  f'(x) = 2x\sin(\frac{1}{x}) - \cos(\frac{1}{x}),&#xA;$$&lt;/span&gt;&#xA;which does not converge as &lt;span class=&quot;math-container&quot;&gt;$x \to 0$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2022-01-28T12:40:02.777" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24853" PostTypeId="1" CreationDate="2022-01-28T13:38:40.663" Score="-4" ViewCount="165" Body="&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: i didn't mean it like this programme should do this or that. i mean other people are accusing me like 'your programme should've had this or that' (actually they're saying that i should know this or that &lt;em&gt;anyway&lt;/em&gt; given what my programme taught me, but let's just pretend it's the accusations i sound outside these round brackets). so my hope is that it's not weird at all so i can defeat them in the argument.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Of course we knew what Bayes' theorem was and we have all the tools to answer this at either basic or advanced level, but we never learned explicitly  the following (and consequently anything about prior and posterior probability except for what I &lt;a href=&quot;https://stats.stackexchange.com/questions/173056/how-exactly-do-bayesians-define-or-interpret-probability&quot;&gt;had to self-study for a make-up project&lt;/a&gt; but even then I didn't yet know the ff)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://stats.stackexchange.com/questions/192179/why-does-a-probability-of-0-or-1-remain-unchanged-with-new-information-intuitiv&quot;&gt;no probability that is 0 or 1 should ever change.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;&lt;strong&gt;Basic&lt;/strong&gt;:&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$(\Omega, \text{sigma-algebras don't exist}, \mathbb P)$&lt;/span&gt; be a&#xA;probability space.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$A,B$&lt;/span&gt; be events with &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; non-zero probability. Then:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(A|B)=1$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$P(A)=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://stats.stackexchange.com/questions/186619/does-an-unconditional-probability-of-1-or-0-imply-a-conditional-probability-of-1&quot;&gt;https://stats.stackexchange.com/questions/186619/does-an-unconditional-probability-of-1-or-0-imply-a-conditional-probability-of-1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/1494682/is-it-correct-to-say-that-pa-1-to-pab-1-pa-1-to-pab-pa&quot;&gt;https://math.stackexchange.com/questions/1494682/is-it-correct-to-say-that-pa-1-to-pab-1-pa-1-to-pab-pa&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/1515756/is-a-probability-of-0-or-1-given-information-up-to-time-t-unchanged-by-informati&quot;&gt;https://math.stackexchange.com/questions/1515756/is-a-probability-of-0-or-1-given-information-up-to-time-t-unchanged-by-informati&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h1&gt;&lt;strong&gt;Advanced&lt;/strong&gt;:&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given a filtered probability space &lt;span class=&quot;math-container&quot;&gt;$(\Omega, \mathscr{F}, \{\mathscr{F}_n\}_{n \in \mathbb{N}}, \mathbb{P})$&lt;/span&gt;, let &lt;span class=&quot;math-container&quot;&gt;$A \in \mathscr{F}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$$\exists t \in \mathbb{N} \ \text{s.t.} \ E[1_A | \mathscr{F_t}] = 1 \ \text{a.s.}$$&lt;/span&gt; Does it follow that &lt;span class=&quot;math-container&quot;&gt;$$E[1_A | \mathscr{F_{s}}] = E[1_A | \mathscr{F_t}] \ \text{a.s.} \ \forall s &amp;gt; t \ ?$$&lt;/span&gt; What about &lt;span class=&quot;math-container&quot;&gt;$\forall s &amp;lt; t$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://stats.stackexchange.com/questions/192179/why-does-a-probability-of-0-or-1-remain-unchanged-with-new-information-intuitiv&quot;&gt;https://stats.stackexchange.com/questions/192179/why-does-a-probability-of-0-or-1-remain-unchanged-with-new-information-intuitiv&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://stats.stackexchange.com/questions/192242/is-e1-a-mathscrf-t-0-textor-1-rightarrow-e1-a-mathscrf&quot;&gt;https://stats.stackexchange.com/questions/192242/is-e1-a-mathscrf-t-0-textor-1-rightarrow-e1-a-mathscrf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://stats.stackexchange.com/questions/180073/prove-disprove-probability-of-0-or-1-almost-surely-will-never-change-and-has-n&quot;&gt;https://stats.stackexchange.com/questions/180073/prove-disprove-probability-of-0-or-1-almost-surely-will-never-change-and-has-n&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h1&gt;Question(s):&lt;/h1&gt;&#xA;&lt;p&gt;Is it weird if those never come up in discussions, homeworks/assignments, textbooks or exams in...&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;in a regular maths undergrad programme?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;in an applied maths undergrad programme?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;in an applied maths grad programme?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;in a quant undergrad programme?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;in a quant grad programme?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I ask 4 and 5 in case some other areas of applied maths really don't make use of probability or statistics (reeeeeeaaaaaally though? &lt;em&gt;applied&lt;/em&gt; maths?)&lt;/p&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2022-01-28T16:57:24.493" LastActivityDate="2022-01-28T16:57:24.493" Title="Is it weird for an undergrad or grad quant/applied maths(/even pure maths) programme to not teach that probabilities of 0 or 1 will never change?" Tags="|undergraduate-education|graduate-education|statistics|probability|" AnswerCount="1" CommentCount="18" ClosedDate="2022-01-28T20:07:17.063" ContentLicense="CC BY-SA 4.0" />
  <row Id="24855" PostTypeId="2" ParentId="24853" CreationDate="2022-01-28T16:54:49.670" Score="1" Body="&lt;p&gt;It's not weird.  The institutions are busy teaching the bulk of their topics.  Busy with conveying (what is at the end commercially important) knowledge to imperfect recipients, with limited time.  Obsessing on this one particular question (or the earlier quadrilateral wording minutia obsession) would be weird (by which I mean a poor use of time/resources).&lt;/p&gt;&#xA;&lt;p&gt;[If done by the institutions.  Whether weird for you is a separate question.]&lt;/p&gt;&#xA;&lt;p&gt;Now that doesn't mean nothing is to be gained by your obsession.  Might be, might not.  But that's for your own edificaton.  Expecting the rest of the world to have your agenda is silly though.  People are different, institutions are different. Not everyone has your priorities and interests.  This is an important thing for you to know as a teacher.  To have some empathy, thoughtfulness of others.&lt;/p&gt;&#xA;" OwnerUserId="19429" LastActivityDate="2022-01-28T16:54:49.670" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24856" PostTypeId="1" CreationDate="2022-01-29T08:31:13.873" Score="3" ViewCount="102" Body="&lt;p&gt;I am going to teach an undergraduate statistics course for the first time next semester. I would like students to do a project by applying some of the things they learned to real-world data. Is there any sources (books, websites) with suitable examples for something likes this?&lt;/p&gt;&#xA;&lt;p&gt;The topics covered in the course will be&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Prior and Posterior Distributions, Conjugate Prior Distributions, Bayes Estimators&lt;/li&gt;&#xA;&lt;li&gt;Maximum Likelihood Estimators&lt;/li&gt;&#xA;&lt;li&gt;The Sampling Distribution of a Statistic, The Chi-Square Distributions, The t-distributions, Confidence&#xA;Intervals, Fisher Information&lt;/li&gt;&#xA;&lt;li&gt;Testing Simple Hypotheses, Uniformly Most Powerful Tests, Uniformly Most Powerful Tests, The t Test, Comparing Two Normal Distributions&lt;/li&gt;&#xA;&lt;li&gt;Goodness-of-Fit, Contingency Tables, Tests of Homogeneity&lt;/li&gt;&#xA;&lt;li&gt;Linear Regression&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Update: One idea I have is to ask students to find a paper with published data and verify the statistics done in this paper is accurate.  Is this a good idea?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorDisplayName="user11702" LastEditDate="2022-02-01T04:09:07.490" LastActivityDate="2022-02-01T04:09:23.290" Title="Where can I find examples of statistics projects?" Tags="|reference-request|statistics|" AnswerCount="0" CommentCount="2" ClosedDate="2022-02-11T01:49:43.450" ContentLicense="CC BY-SA 4.0" />
  <row Id="24857" PostTypeId="2" ParentId="24844" CreationDate="2022-01-29T14:02:52.330" Score="6" Body="&lt;p&gt;Give a function together with some properties of the function, but do not give  a formula, and ask for the derivative.&lt;/p&gt;&#xA;&lt;p&gt;An example:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{R} \to \mathbb{R}$&lt;/span&gt; be a function which enjoys the following two properties:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(t+s) = f(t)f(s)$&lt;/span&gt; for all real numbers &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x \to 0} \frac{f(x) - 1}{x} = 3$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Prove that &lt;span class=&quot;math-container&quot;&gt;$f'(x) = 3f(x)$&lt;/span&gt; for all real &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;These kinds of problems used to appear on the AP Calculus BC &amp;quot;essay portion&amp;quot; in the 60s and early 70s.  I know because my AP calc teacher had us do every single AP problem going back to the inception of the AP test.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-01-29T14:02:52.330" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24858" PostTypeId="2" ParentId="24844" CreationDate="2022-01-29T19:42:10.473" Score="5" Body="&lt;p&gt;This might not be what you want, but sometimes a backwards problem appears more difficult than it seems and the difficulty is immediately removed when you know and apply the definition of the derivative.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem:&lt;/strong&gt; Calculate the following limit:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \lim_{x \rightarrow a} \frac{\cos^2(3x)-\cos^2(3a)}{x- a} $$&lt;/span&gt;&#xA;&lt;strong&gt;Solution:&lt;/strong&gt; By definition of derivative, the limit above is &lt;span class=&quot;math-container&quot;&gt;$f'(a)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$f(x) = \cos^2(3x)$&lt;/span&gt; hence &lt;span class=&quot;math-container&quot;&gt;$f'(a) = -6\sin(3a)\cos(3a)$&lt;/span&gt; and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \lim_{x \rightarrow a} \frac{\cos^2(3x)-\cos^2(3a)}{x- a} = -6\sin(3a)\cos(3a).$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2022-01-29T19:42:10.473" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24859" PostTypeId="2" ParentId="4427" CreationDate="2022-01-29T22:07:51.177" Score="3" Body="&lt;p&gt;My advice is to avoid the use of script or fancy fonts.  Even if you have the skills to pull it off--and I'm very impressed by the boardsmanship of people here, see some of the old threads--many of your students will struggle with it in notes.  The work is hard enough on its own, so try to avoid anything that adds mental hurdles (fancy notation).&lt;/p&gt;&#xA;&lt;p&gt;I have seen lots of books that show vectors, for example, in script or bold or the like.  But an arrow over the top is much easier to scribble in notes.  Especially for those like me, with poor handwriting.&lt;/p&gt;&#xA;&lt;p&gt;I would also avoid the use of Fraktur or of those Greek letters that are easily confused (the v looking thing, nu?)  The same point applies.  Even if you are good at it, some of the notetakers will not be.&lt;/p&gt;&#xA;&lt;p&gt;Like with the arrows on the top vectors, there are usually alternatives to show more variables (capital letters, underscores, etc.) that are simpler.&lt;/p&gt;&#xA;" OwnerUserId="19433" LastActivityDate="2022-01-29T22:07:51.177" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24860" PostTypeId="1" CreationDate="2022-01-30T22:11:44.597" Score="3" ViewCount="579" Body="&lt;p&gt;I am trying to write a very very basic trig primer, from scratch.&lt;/p&gt;&#xA;&lt;p&gt;Early on, I wish to discuss slopes (say, the slope of a line through the origin), and I would like to give the 'right' hints on how the slope of a line is related to the angle of the line with the x-axis.&lt;/p&gt;&#xA;&lt;p&gt;How would you go?  Are there families of cases (slope values) which provide easier to grasp insights (besides, say, slope=1)?&lt;/p&gt;&#xA;&lt;p&gt;Note, the examples should be used to introduce arctan(), and so a definition of arctan() should not be implied or preassumed.&lt;/p&gt;&#xA;&lt;p&gt;Sorry to be quite vague.&lt;/p&gt;&#xA;" OwnerUserId="7086" LastActivityDate="2024-03-02T17:03:47.927" Title="How to teach relationship between slopes and angles?" Tags="|trigonometry|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24861" PostTypeId="2" ParentId="24860" CreationDate="2022-01-30T23:51:21.633" Score="3" Body="&lt;p&gt;You want to convince students that an angle in a right triangle depends on the slope of the line (forming the hypotenuse).&lt;/p&gt;&#xA;&lt;p&gt;At this point, they are likely already familiar with calculating the slope of a line. I wouldn't worry about choosing a nice ratio of sides for the slope, but rather I would give students a protractor and have them draw (probably on graph paper) different right triangles whose hypotenuses have the same slope. I'd have them measure the corresponding interior angles to confirm that they are all equal in measure. If you did this with enough different slopes (i.e. each student gets a different slope to give their hypotenuses), then you could show them their compiled data in a table (or dots on a coordinate plane), announce that this must be a function, and then give them its proper name and show them that their calculator has it built-in.&lt;/p&gt;&#xA;&lt;p&gt;You could extend the idea to the other trig ratios, all using the pictures they already drew.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2022-01-30T23:51:21.633" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24862" PostTypeId="2" ParentId="24860" CreationDate="2022-01-31T09:58:49.743" Score="2" Body="&lt;p&gt;If you don't want to use a calculator, I'd adapt Nick C's answer as follows:&lt;/p&gt;&#xA;&lt;p&gt;Use &lt;span class=&quot;math-container&quot;&gt;$\frac y x$&lt;/span&gt; as as slope, with &lt;span class=&quot;math-container&quot;&gt;$x=10$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y = 1, 2, 3, 4, 5, 10, 20$&lt;/span&gt;; have the students draw the lines and measure the angle. I assume that the students are aware that the slope does not depend on the choice of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, but you could also vary this.&lt;/p&gt;&#xA;&lt;p&gt;If you're looking for ways to determine the angle without measuerment, the only useful values are probably&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;slope &lt;span class=&quot;math-container&quot;&gt;$\frac{1} {\sqrt{3}}$&lt;/span&gt; for an angle of &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}{6}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;slope &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; for an angle of &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}{4}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;slope &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3}$&lt;/span&gt; for an angle of &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}{3}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The plot of the angle vs. slope is easily obtained and shows the relation you're looking for.&lt;/p&gt;&#xA;&lt;p&gt;One could also discuss the behavior for &lt;span class=&quot;math-container&quot;&gt;$y \to \infty$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\alpha \to 90^\circ$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Depending on the setting, you could also use the relation to form pairs (or groups) of students to work together: hand out cards with either angle or slope (maybe different fractions for the same ratio) and have the students find their partners who have the &amp;quot;same line&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2022-01-31T12:03:53.550" LastActivityDate="2022-01-31T12:03:53.550" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24863" PostTypeId="1" CreationDate="2022-01-31T19:11:09.847" Score="6" ViewCount="114" Body="&lt;p&gt;I tutor high school math students. Students often struggle with a problem they had completed few months prior. Like any skill, it's natural to forget what you learned after a while.&lt;/p&gt;&#xA;&lt;p&gt;As high school instructors/teachers/tutors, &lt;strong&gt;how do you maximize retention of a skill and how to quickly get the student to recall/relearn what was forgotten? How often do you assess students' retention of previous topics outside of finals, eg if quadratics was introduced early in the semester, include a question requiring solving quadratics in the current unit assessment?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In many Asian and Eastern European countries, students drill through tons of (often similar) problems while in my experience this is not as emphasized in North America, where a stronger focus is on exploration. Do either muscle memory or understanding correlate with length and quality of retention?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I'm still new to this site. I try to phrase the same question in different ways.&lt;/p&gt;&#xA;" OwnerUserId="18050" LastActivityDate="2022-01-31T19:11:09.847" Title="Maximize retention" Tags="|secondary-education|mathematical-pedagogy|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24864" PostTypeId="2" ParentId="24815" CreationDate="2022-01-31T19:40:04.650" Score="4" Body="&lt;p&gt;You only really need to give them one type of question:&lt;/p&gt;&#xA;&lt;p&gt;Count the numbers in this list: [insert any finite easily indexed sequence]&lt;/p&gt;&#xA;&lt;p&gt;Examples:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$4,5,6, \cdots, 27$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$112, 115, 118, \cdots, 334$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$4, 2.5, 1, \cdots, -29$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$4, 8, 16, \cdots, 2048$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$9, 16, 25, 36, \cdots, 400$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And then teach them to solve ALL of these problems by method of bijections, forming one-to-one pairings of lists by explicit rules.&lt;/p&gt;&#xA;&lt;p&gt;For instance, take the second one: &lt;span class=&quot;math-container&quot;&gt;$112, 115, 118, \cdots, 334$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Subtract &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; to everything in the list: &lt;span class=&quot;math-container&quot;&gt;$12, 15, 18, \cdots, 234$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Divide by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$4, 5, 6, \cdots, 78$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Subtract &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$1,2,3, \cdots, 75$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So the answer is &lt;span class=&quot;math-container&quot;&gt;$75$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There are infinitely many ways to do this problem, and it doesn't matter how many bijections you make, you will always end up at the correct answer. This is superior to the formula for the number of items in a sequence for several reasons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the formula doesn't explain why the +1 is there&lt;/li&gt;&#xA;&lt;li&gt;the formula must be memorized&lt;/li&gt;&#xA;&lt;li&gt;this method is more general and doesn't require you memorize the +1&lt;/li&gt;&#xA;&lt;li&gt;this method also very easily explains why the +1 is there&lt;/li&gt;&#xA;&lt;li&gt;this method can be applied in way more creative ways&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;It is critically important to teach this method for some major reasons. Firstly, bijections are the foundation of combinatorics. It is how we define counting in addition to being an exceedingly powerful abstract problem solving tool. Secondly, this method gives students a way to guarantee they have the correct answer; it is the way in which we check our answer to counting problems in the same way that plugging values into an equation is how we check our answer in algebra and how proofs are how we check our answer in geometry. And finally, if they get used to applying this method and thinking about counting in this way, it builds excellent habits and avoids many common pitfalls.&lt;/p&gt;&#xA;" OwnerUserId="9383" LastActivityDate="2022-01-31T19:40:04.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24865" PostTypeId="2" ParentId="18977" CreationDate="2022-01-31T22:56:30.650" Score="2" Body="&lt;p&gt;A bit of a strange and abstract answer, but for variety:&lt;/p&gt;&#xA;&lt;p&gt;I have found personally that even just the basics of abstract algebra and group theory have helped me immensely pedagogically, as it has allowed me to understand abstractions better, and fundamentally altered how I think and process ideas and concepts.&lt;/p&gt;&#xA;&lt;p&gt;Firstly, group theory is essentially the study of symmetry. It helps you understand &lt;em&gt;how&lt;/em&gt; things can be symmetrical, and the different ways you can represent the same object. Anywhere that symmetry exists, you can think of it in terms of group theory. Let's say you and 3 other friends are playing a game together, and for variety sake, you want to switch up the player order each game. With a good understanding of the &lt;span class=&quot;math-container&quot;&gt;$S_4$&lt;/span&gt; group, you don't need to remember which orders you've tried, you can easily construct many simple ways to determine the next ordering. This example is a little contrived, but hopefully you can see where it could appear innocuously in every day life. Most people won't see &amp;quot;player order&amp;quot; as something having symmetry.&lt;/p&gt;&#xA;&lt;p&gt;Here is an example to make the next paragraph feel less abstract: Let &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; be a rotation 90 degrees clockwise around the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis. Let &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; be a rotation 90 degrees clockwise around the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis. Let &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; be a rotation 90 degrees counter-clockwise around the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis. I have a cube and I want to apply the &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; rotations in that order. What will the cube look like?&lt;/p&gt;&#xA;&lt;p&gt;First we represent the overall rotation as &lt;span class=&quot;math-container&quot;&gt;$a+b+c = a+c+b$&lt;/span&gt;. The &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; rotations cancel, so we are just left with &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. Grab a cube and try it yourself! You will find this does &lt;em&gt;not&lt;/em&gt; get you the right answer. Wait, what?&lt;/p&gt;&#xA;&lt;p&gt;When learning math, there are several major hierarchies of concepts students learn as they get more advanced, in the sense that they must be very comfortable manipulating and understanding them in order to reach the next step of mathematical maturity: numbers, arithmetic, expressions, functions, logic, properties. The &amp;quot;properties&amp;quot; level is examined when you study abstract algebra. What do algebras look like if the commutative and associative properties don't hold? There are a lot of implicit assumptions we make when writing mathematical statements that we often take for granted, that are now muscle memory (like when we solve an equation). But by examining the &lt;em&gt;properties&lt;/em&gt; of the algebra, we can catch our mistakes and justify our reasoning. In this problem, we assumed that 3D rotations are commutative, that &lt;span class=&quot;math-container&quot;&gt;$a+b = b+a$&lt;/span&gt;, but this is in fact &lt;em&gt;not true&lt;/em&gt; (this is a special case of matrix multiplication). Having a good grasp of abstract algebra makes it easier for you to challenge those assumptions.&lt;/p&gt;&#xA;&lt;p&gt;In general, the idea that you can basically do algebra with just about anything as long as your operations are defined properly, is a pretty cool and powerful idea. The idea that the algebra systems are (sort of) defined by their properties and not by what the elements represent means seemingly unrelated things are the same. This changes one's attitude towards math significantly, and you start to realize there is just as much invention in math as there is discovery. We aren't just proving theorems, we are also inventing definitions.&lt;/p&gt;&#xA;&lt;p&gt;Once you broaden your perspective in this way, you will have a healthier approach towards higher mathematics, a shift in philosophy, and an appreciation for abstraction. Then it doesn't seem too difficult to imagine that there could be plenty of applications of these concepts intuitively through art and design, programming, playing games, etc. even if it is difficult to identify specific ones. The more complex the math, the less explicit the applications and the more important they are in changing how we think and our philosophies.&lt;/p&gt;&#xA;" OwnerUserId="9383" LastActivityDate="2022-01-31T22:56:30.650" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24866" PostTypeId="2" ParentId="24860" CreationDate="2022-02-01T02:26:53.180" Score="1" Body="&lt;p&gt;I'd probably show them a picture that shows the relationship.  If &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is measured in radians, then in the unit circle figure below, the area of the two triangles is &lt;span class=&quot;math-container&quot;&gt;$\tan \theta = m$&lt;/span&gt; and the area of the intersection with the circle is &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;. Since they are areas, they are easy to compare.  It is easy to see that &lt;span class=&quot;math-container&quot;&gt;$\tan\theta &amp;gt; \theta$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;\theta&amp;lt;\pi/2$&lt;/span&gt;, and it is easy to see roughly how the difference grows as &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; increases.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/phVcZ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/phVcZ.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you want to compute &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; for given values of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, it is possible to do so approximately, but it takes a lot of work and I wouldn't advise it.&lt;/p&gt;&#xA;&lt;p&gt;However, it's not too hard to make a first approximation.  First, it is easy to see that &lt;span class=&quot;math-container&quot;&gt;$\tan\theta = \theta\ +$&lt;/span&gt; the two parts of the triangles outside the circle.  If we approximate these by triangles, replacing the circular arc by a straight line, then we can find the area of each little triangle to be&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;{1\over2}\left(1 - {1 \over \sqrt{1+m^2}}\right)\,m \,,&#xA;$$&lt;/span&gt;&#xA;by dropping a vertical line from the vertex on the circle and using similar triangles to derive the height from the base &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;.&#xA;From this, we get that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\theta = {m \over \sqrt{1+m^2}}$$&lt;/span&gt;&#xA;plus the amount cut off by straightening the arc.&lt;/p&gt;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$m = 0.5$&lt;/span&gt;, we get an approximation of &lt;span class=&quot;math-container&quot;&gt;$\theta \approx 0.447$&lt;/span&gt;&#xA;where actually &lt;span class=&quot;math-container&quot;&gt;$\theta = 0.464$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A slightly more complicated way we can calculate &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is to recursively bisect the angle and inscribe triangles inside the sector.  This will require the half-angle formulas to get the approximation in terms of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, so that might be an impediment for a motivating example at the beginning of trigonometry.&lt;/p&gt;&#xA;&lt;p&gt;After &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; bisections, the area of the inscribed triangles add up to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$2^k \sin (\theta/2^k) \,.$$&lt;/span&gt;&#xA;We then apply the half-angle formula for sine (once) and the half-angle formula for cosine &lt;span class=&quot;math-container&quot;&gt;$k-1$&lt;/span&gt; times, until we arrive at an argument of &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;.  Finally we substitute for &lt;span class=&quot;math-container&quot;&gt;$\cos \theta$&lt;/span&gt;, the formula in terms of &lt;span class=&quot;math-container&quot;&gt;$m=\tan\theta$&lt;/span&gt;, namely &lt;span class=&quot;math-container&quot;&gt;$\cos\theta = 1/\sqrt{1+m^2}$&lt;/span&gt;.&#xA;It makes the work easier if we write the half-angle formulas as substitutions in the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\eqalign{&#xA;\sin\alpha &amp;amp;\mapsto {1\over2}\sqrt{\mathstrut2-2\cos 2\alpha} \cr&#xA;2\cos\alpha &amp;amp;\mapsto \sqrt{\mathstrut2+2\cos 2\alpha} \cr&#xA;}$$&lt;/span&gt;&#xA;Then a nice recursive pattern emerges.&lt;/p&gt;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$k=4$&lt;/span&gt;, the steps are as follows:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\eqalign{&#xA;&amp;amp; 2^4 \sin \big(\theta/2^4\big) \cr&#xA;&amp;amp; 2^3 \sqrt{2 - 2 \cos \big(\theta/2^3\big)} \cr&#xA;&amp;amp; 2^3 \sqrt{2 - \sqrt{2 + 2 \cos \big(\theta/2^2\big)}} \cr&#xA;&amp;amp; 2^3 \sqrt{2 - \sqrt{2 + \sqrt{2 + 2 \cos \big(\theta/2\big)}}} \cr&#xA;&amp;amp; 2^3 \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{\mathstrut2 + 2 \cos \theta}}}}, \quad \cos\theta = 1/\sqrt{1+m^2} \cr&#xA;}$$&lt;/span&gt;&#xA;This gives an approximation of &lt;span class=&quot;math-container&quot;&gt;$\theta \approx 0.463583$&lt;/span&gt; compared to the correctly rounded value of &lt;span class=&quot;math-container&quot;&gt;$0.463648$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The general pattern in terms of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$2^k \sqrt{2 - \sqrt{2 + \sqrt{2 + \cdots \sqrt{2 + 2\big/\sqrt{1+m^2}}}}} \,,$$&lt;/span&gt;&#xA;where the total number of square-roots is &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;.&#xA;It has a linear convergence rate of &lt;span class=&quot;math-container&quot;&gt;$1/4$&lt;/span&gt;, that is,&#xA;the error asymptotically decreases as &lt;span class=&quot;math-container&quot;&gt;$1/4^k$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2022-02-02T15:32:58.637" LastActivityDate="2022-02-02T15:32:58.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24867" PostTypeId="2" ParentId="27" CreationDate="2022-02-01T16:04:23.570" Score="1" Body="&lt;p&gt;The proof by similarity (by finding scaled versions of the triangle within itself) is one option&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7tCVL.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7tCVL.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16420" LastActivityDate="2022-02-01T16:04:23.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24869" PostTypeId="1" CreationDate="2022-02-02T17:33:41.303" Score="12" ViewCount="2178" Body="&lt;p&gt;I love the game of &lt;a href=&quot;https://en.wikipedia.org/wiki/Sprouts_(game)&quot; rel=&quot;noreferrer&quot;&gt;Sprouts&lt;/a&gt; but unfortunately at the moment it seems to be a bit too complex for the young student I have introduced to it. She already knows &lt;a href=&quot;https://en.wikipedia.org/wiki/Tic-tac-toe&quot; rel=&quot;noreferrer&quot;&gt;noughts and crosses&lt;/a&gt; (also called Tictactoe). I need a game that is&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;more complex than noughts and crosses,&lt;/li&gt;&#xA;&lt;li&gt;less complex than Sprouts, and&lt;/li&gt;&#xA;&lt;li&gt;similar to both in having a small number of possible moves at each turn.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Either a boardgame or a pencil-and-paper game will be fine. The game of &lt;a href=&quot;https://en.wikipedia.org/wiki/Dots_and_Boxes&quot; rel=&quot;noreferrer&quot;&gt;Dots and Boxes&lt;/a&gt; using a small grid of squares is a possibility, but I'm hoping to find something that is better at getting a student into the frame of mind of &amp;quot;&lt;em&gt;if I go there, he has three possibilities; now if he chooses the first, then...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="19443" LastActivityDate="2022-02-04T22:00:29.220" Title="A game that is between Noughts and Crosses and Sprouts for complexity?" Tags="|games|" AnswerCount="9" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="24870" PostTypeId="2" ParentId="24869" CreationDate="2022-02-02T17:41:06.407" Score="7" Body="&lt;p&gt;Mill is the natural next step up.  You can draw the board yourself on the back of a checkers board (if not already printed there).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Nine_men%27s_morris&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Nine_men%27s_morris&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note, Wiki talks about flying being an option (see &amp;quot;phase three&amp;quot;), but I think this is the traditional method.  I've never seen else.  And is a fun phase.&lt;/p&gt;&#xA;&lt;p&gt;Supposedly the game is solved, but I find it much more than complex tic tac toe, so it will retain her interest (and yours) well.&lt;/p&gt;&#xA;" OwnerUserId="19444" LastActivityDate="2022-02-02T17:41:06.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24871" PostTypeId="2" ParentId="24869" CreationDate="2022-02-02T18:41:20.277" Score="4" Body="&lt;p&gt;Noughts and crosses on a larger board (we used school notebooks when school was boring) and you need five in a row to win. Also good use for partially used notebooks: all squares that have something in them are unavailable, all empty squares available.&lt;/p&gt;&#xA;&lt;p&gt;The game works in theory with more than two players, too, for extra flexibility, but you sometimes get kingmaker situations, which is no fun.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-02-02T18:41:20.277" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24872" PostTypeId="2" ParentId="24869" CreationDate="2022-02-03T00:43:27.137" Score="15" Body="&lt;p&gt;I'll suggest &lt;strong&gt;Ultimate Tic-Tac-Toe&lt;/strong&gt;. (I believe this is what &lt;a href=&quot;https://matheducators.stackexchange.com/questions/24869/a-game-that-is-between-noughts-and-crosses-and-sprouts-for-complexity#comment63142_24869&quot;&gt;Sue mentioned in her comment&lt;/a&gt;, as well.)&lt;/p&gt;&#xA;&lt;p&gt;The main ideas is to take a 3x3 tic-tac-toe board and replace each of the 9 squares with its own 3x3 board. When you make a move, you put an X or O in one of the 81 small squares. Whichever &lt;em&gt;small square&lt;/em&gt; you play in, that tells your opponent &lt;em&gt;which big square&lt;/em&gt; they must play in next. Check out the image for an example: I play the red X, which is in the top-right corner of that particular 3x3 board, so you must respond by playing &lt;em&gt;somewhere in the top-right corner of the &lt;strong&gt;big&lt;/strong&gt; board&lt;/em&gt; (the blue squiggly area).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OfA3V.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OfA3V.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;That image comes from a post on the &lt;strong&gt;Math With Bad Drawings&lt;/strong&gt; blog, and that post has a lot more information about the rules and strategy of the game: &lt;a href=&quot;https://mathwithbaddrawings.com/2013/06/16/ultimate-tic-tac-toe/&quot; rel=&quot;noreferrer&quot;&gt;https://mathwithbaddrawings.com/2013/06/16/ultimate-tic-tac-toe/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;An internet search for &amp;quot;ultimate tic tac toe&amp;quot; reveals several places to play the game online, including this one (which I like because you can play against an AI &lt;em&gt;or&lt;/em&gt; share a link to play online against a friend): &lt;a href=&quot;https://ultimate-t3.herokuapp.com/&quot; rel=&quot;noreferrer&quot;&gt;https://ultimate-t3.herokuapp.com/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's pretty easy to play with pencil and paper, of course, too!&lt;/p&gt;&#xA;&lt;p&gt;I think this fits what you're looking for because each move you make only has a few options (other than the very first move, you will always have at most 9 options, and you'll have fewer as the game goes on). It's also possible to look ahead a move or two (or more), and this can help your student develop some logical thinking skills along the lines of: &amp;quot;If I go here, you'll have to go &lt;em&gt;there&lt;/em&gt;, which means...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But most of all, this game is &lt;em&gt;fun!&lt;/em&gt; I've used it as an ice-breaker / just-for-fun activity in classes before and it always goes over well.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2022-02-03T00:43:27.137" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24873" PostTypeId="2" ParentId="24869" CreationDate="2022-02-03T07:26:18.560" Score="6" Body="&lt;p&gt;&amp;quot;Connect 4&amp;quot; and &amp;quot;Gomoku&amp;quot; are similar to tic-tac-toe, but more interesting. Gomuko is simply tic-tac-toe on a larger board, where one has to connect stones into lines of length 5 with a few extra rules to reduce the first player advantage.&lt;/p&gt;&#xA;&lt;p&gt;In connect 4, again the goal is to connect lines of a certain length, 4 this time, but now you can only place your disks at the lowest unoccupied position in a column.&lt;/p&gt;&#xA;&lt;p&gt;You can also play the games above on a hexagonal grid instead of a square grid, if you want some more variation.&lt;/p&gt;&#xA;&lt;p&gt;Something different and a bit more complex, but also nice is the game Hive. In Hive, each player either moves or places a hexagonal tile with an insect on it adjacent to the already placed tiles, according to specific placement and movement rules. The goal is to encircle the enemy Queen (bee). Thinking up to 4 moves deep is often required for good strategy, although good intuition can also get you pretty far. If you think Hive has too many rules, you can try removing some of the pieces from the game.&lt;/p&gt;&#xA;" OwnerUserId="9497" LastActivityDate="2022-02-03T07:26:18.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24874" PostTypeId="2" ParentId="24869" CreationDate="2022-02-03T09:18:38.807" Score="3" Body="&lt;p&gt;Sorry, another answer.  Was thinking of go and saw another person had answered that.&lt;/p&gt;&#xA;&lt;p&gt;Another similar game that many bright children enjoy is &lt;strong&gt;Othello&lt;/strong&gt;.  I find it a little more relaxing than chess or even checkers.  There's just something enjoyable about flipping the pieces.&lt;/p&gt;&#xA;&lt;p&gt;Probably want to buy a set, they are pretty attractive.    [Edit: But read the reviews, supposedly modern sets a little less rich in feel versus 70s versions.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B01MXWTYLF&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Othello-Classic-Game-2-Player/dp/B01MXWTYLF&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;and see also:  &lt;a href=&quot;https://en.wikipedia.org/wiki/Reversi&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Reversi&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19444" LastEditorUserId="19444" LastEditDate="2022-02-03T09:33:25.720" LastActivityDate="2022-02-03T09:33:25.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24876" PostTypeId="2" ParentId="24869" CreationDate="2022-02-04T07:46:12.050" Score="1" Body="&lt;p&gt;&lt;em&gt;&lt;strong&gt;Dots and Boxes&lt;/strong&gt;&lt;/em&gt; on boards of at least 5x5 boxes can be deep. (Elwyn Berlekamp, &lt;em&gt;The Dots-and-Boxes Game&lt;/em&gt;) If that's too big a game, try 4x4 boxes. That game isn't exactly trivial, but then a game  might end in an 8-8 draw.&lt;/p&gt;&#xA;&lt;p&gt;I see you've already considered it, but I argue in its favour that, after a while, long walls split the board into independent regions, and you need to think &amp;quot;shall I play in &lt;em&gt;this&lt;/em&gt; region or &lt;em&gt;that&lt;/em&gt; region?&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="6983" LastActivityDate="2022-02-04T07:46:12.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24877" PostTypeId="2" ParentId="24869" CreationDate="2022-02-04T08:06:05.203" Score="2" Body="&lt;p&gt;Games in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tafl_games&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tafl family&lt;/a&gt;. These are board games where pieces move (like rooks) from square to square, so there is that element of &amp;quot;if they move there, I move here&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;There are several versions of various sizes. The smallest I know is &lt;em&gt;&lt;strong&gt;Brandubh&lt;/strong&gt;&lt;/em&gt;, shown on the above Wikipedia page and at &lt;a href=&quot;https://boardgamegeek.com/boardgame/33692/brandubh&quot; rel=&quot;nofollow noreferrer&quot;&gt;BoardGameGeek&lt;/a&gt;. The board is 7x7; the defender has king and 4 men; the attacker has 8.&lt;/p&gt;&#xA;" OwnerUserId="6983" LastActivityDate="2022-02-04T08:06:05.203" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24878" PostTypeId="2" ParentId="24869" CreationDate="2022-02-04T16:15:21.053" Score="0" Body="&lt;p&gt;The game of Gale, described on this page, might be about right.  The flavor is similar to sprouts, but you can control the complexity by changing the grid size.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Shannon_switching_game&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Shannon_switching_game&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Variations are also described on the Wiki page.&lt;/p&gt;&#xA;" OwnerUserId="19450" LastActivityDate="2022-02-04T16:15:21.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24879" PostTypeId="2" ParentId="24869" CreationDate="2022-02-04T22:00:29.220" Score="0" Body="&lt;p&gt;You could try &lt;a href=&quot;https://en.wikipedia.org/wiki/Nim&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nim&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://withkoji.com/%7Eblackslate/nim&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's an online version of the game&lt;/a&gt;, where you can play either against another human (on the same device) or against the computer.&lt;/p&gt;&#xA;&lt;p&gt;If you click on the hamburger menu in the top left, you can choose a number of different hints that will allow you to work out the winning strategy. In particular, if you select the Step by Step checkbox, it will gradually work through all the possible combinations of pieces, starting with the easiest, so that, at each step, you can find a move that leaves the board in a combination that you already knows is winnable.&lt;/p&gt;&#xA;" OwnerUserId="3464" LastActivityDate="2022-02-04T22:00:29.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24881" PostTypeId="2" ParentId="24840" CreationDate="2022-02-07T20:01:45.577" Score="2" Body="&lt;p&gt;Adding to the other good answers, a very-slight variant of that trick is (to me) more understandable and more scale-able to other problems, &amp;quot;once the lesson is learned&amp;quot;. Namely, &lt;em&gt;grouping&lt;/em&gt; of terms in a large sum. In the example at hand, it's to group first-and-last, second-and-next-to-last, etc. Yes, when the total number of summands is &lt;em&gt;even&lt;/em&gt; this pairs everything, and then it's &lt;em&gt;odd&lt;/em&gt; the middle term is left over. With an even number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, we get &lt;span class=&quot;math-container&quot;&gt;${n\over 2}\cdot (n+1)$&lt;/span&gt;. With an odd number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; we get &lt;span class=&quot;math-container&quot;&gt;${n-1\over 2}\cdot (n+1)+{n\over 2}$&lt;/span&gt;. As we know, these give the same formula...&lt;/p&gt;&#xA;&lt;p&gt;The grouping idea does also apply to &amp;quot;telescoping&amp;quot; sums, for example.&lt;/p&gt;&#xA;&lt;p&gt;So, as in other answers, it's both learning re-usable ideas from seeing such things, as well as acquiring methodological approaches to come up with then on one's own.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-02-07T20:01:45.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24882" PostTypeId="2" ParentId="24840" CreationDate="2022-02-08T03:09:39.083" Score="1" Body="&lt;p&gt;&lt;strong&gt;Just draw it.&lt;/strong&gt; You will stumble on this method of proof immediately.&lt;/p&gt;&#xA;&lt;p&gt;Let's say you don't know the formula yet, so you draw &lt;span class=&quot;math-container&quot;&gt;$\sum_{i=1}^5 i$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bl1mV.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bl1mV.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You notice that this shape is a triangle, so maybe it is half of a square like in geometry.&lt;/p&gt;&#xA;&lt;p&gt;But when you draw another copy of the triangle:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/goCu2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/goCu2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;you notice that it didn't form a square after all; in fact it was half of a 5 x 6 rectangle. You conjecture immediately that &lt;span class=&quot;math-container&quot;&gt;$\sum_{i=1}^n i = \frac{n(n+1)}{2}$&lt;/span&gt; and start to look for an algebra trick to prove it.&lt;/p&gt;&#xA;&lt;p&gt;But the &amp;quot;trick&amp;quot; is just to look at the rows.&lt;/p&gt;&#xA;&lt;p&gt;The rows show that you should pair the red 5 with the blue 1, the red 4 with the blue 2, and so on, which is exactly the trick you have there in your book. No mystical insight needed.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2022-02-08T03:09:39.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24883" PostTypeId="1" CreationDate="2022-02-08T17:17:06.957" Score="19" ViewCount="4862" Body="&lt;p&gt;I will be teaching Calculus 1 soon, using Stewart's &lt;em&gt;Calculus: Early Transcendentals&lt;/em&gt; as a reference. I can't help but recall my time in high school AP Calculus and my first semester undergraduate calculus, and how much I (along with many other students) struggled with the topic of solids of revolution and using integrals to compute their volumes. My personal frustration: Are all these fancy tricks and methods of finding these volumes actually useful? I never got a satisfactory answer to that question through the remainder of my undergrad years, and to this day it seems to be little more than an &amp;quot;I teach it because it's the standard&amp;quot; kind of topic. Now that I'm going to be teaching undergraduate calculus myself, I have half a mind to omit the topic altogether and spend more time on topics that might be more useful to engineering and science students, such as basic numerical methods for approximating derivatives and integrals.&lt;/p&gt;&#xA;&lt;p&gt;Would I be doing my students a disservice if I skipped this topic altogether? Or am I right in thinking that the precious little time we have in a semester is better spent on other, more relevant topics?&lt;/p&gt;&#xA;" OwnerUserId="19467" LastActivityDate="2023-02-26T22:40:57.343" Title="Do I really need to cover solids of revolution in my Calculus I class?" Tags="|calculus|teaching|applications|" AnswerCount="6" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="24884" PostTypeId="2" ParentId="24883" CreationDate="2022-02-08T17:28:27.683" Score="8" Body="&lt;p&gt;It is usually covered in Calculus II, I believe. I do a little of it at the end of Calculus I, because it is so beautiful, and it feels like a perfect closing to the show.&lt;/p&gt;&#xA;&lt;p&gt;Useful? What in Calculus I is useful as it is? It's the concepts that are useful, and this is a new concept for the students.&lt;/p&gt;&#xA;&lt;p&gt;Do they know why the volume of the cone uses a 1/3? Probably not. And it is such a simple problem, once you have the tool of volumes of rotation. Beautiful.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-02-08T17:28:27.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24885" PostTypeId="2" ParentId="24883" CreationDate="2022-02-08T17:59:24.180" Score="12" Body="&lt;p&gt;At my institution, we teach out of &lt;em&gt;Thomas' Calculus&lt;/em&gt; (not the early transcendentals version, thank goodness).  Volumes and surfaces of revolution show up in a chapter titled &amp;quot;Applications of Definite Integrals&amp;quot;, which also includes arc length, some elementary fluids problems, and moments &amp;amp; centers of mass.  My impression is that the goal of the chapter is to give students a number of examples in which integrals give geometrically or physically useful / meaningful results.&lt;/p&gt;&#xA;&lt;p&gt;At my institution, this chapter is usually taught during the last two or three weeks of the semester.  Philosophically, I think that is the right place for it.  I don't expect students to &lt;em&gt;master&lt;/em&gt; the topics, but I do think that they benefit from exposure.  There are ideas in that chapter which recur, and there are some interesting problems which are resolved using elementary calculus techniques:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;computing volumes using cross-sections foreshadows the Fubini-Tonelli theorem, which comes back in multivariable calculus;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;computing volumes and surface areas of revolution exposes students to the idea of exploiting symmetries in order to simplify computations;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;using these techniques, it is possible to show that a cylinder is precisely a ball plus a cone---this is a fascinating result, if you ask me;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Y2SPk.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Y2SPk.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;computing moments and centers of mass gives insight into a physically meaningful process (one can &amp;quot;numerically integrate&amp;quot; by plotting a curve on a material of uniform density, cutting out the region, and balancing it on a fulcrum), but &lt;em&gt;also&lt;/em&gt; exposes students to one of the essential ideas of probability theory;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;and so on...&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I don't think that it is the end of the world if these examples are skipped, and I generally don't cover ever section in that chapter (because time is limited), but I do think that students benefit from seeing the material.  It gives them something more concrete to work with, and provides a number of examples which, in some sense, justify the previous 14 weeks of lecture.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2022-02-09T02:06:34.107" LastActivityDate="2022-02-09T02:06:34.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24886" PostTypeId="2" ParentId="24883" CreationDate="2022-02-08T18:20:12.927" Score="14" Body="&lt;p&gt;Pardon me ignoring your Calculus question,&#xA;but there is some beautiful mathematics here, e.g.,&#xA;Cavalieri’s principle. So there is an opportunity to connect&#xA;the calculus to these &amp;quot;fascinating results&amp;quot; (to quote&#xA;Xander Henderson).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bJcf9.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bJcf9.png&quot; alt=&quot;Bracelet&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Tom M. Apostol and Mamikon A. Mnatsakan.&#xA;&amp;quot;Volumes of Solids Swept Tangentially Around Cylinder.&amp;quot;&#xA;&lt;em&gt;Forum Geometricorum&lt;/em&gt;.&#xA;Volume 15 (2015) 13–44.&#xA;&lt;a href=&quot;https://forumgeom.fau.edu/FG2015volume15/FG201503.pdf&quot; rel=&quot;noreferrer&quot;&gt;PDF download&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-02-08T18:20:12.927" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24887" PostTypeId="1" CreationDate="2022-02-08T18:50:40.570" Score="2" ViewCount="170" Body="&lt;p&gt;I am teaching a class of 16 students aged around 15-16. They are all very social except for one older male student who has a severe kind of social anxiety. In the class there is also a gifted girl, she is really exceptional in that she is a self-learner and extremely curious. I would like to try to make these people work in small groups as research shows that it's a good way to boost question frequency and metacognition. The problem is how should I group these students taking into account these peculiarities? Ideally I would not like to make groups of three, rather couples. Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="6286" LastEditorUserId="6286" LastEditDate="2022-02-08T19:22:28.600" LastActivityDate="2022-02-13T22:11:09.463" Title="what is a good student grouping in this situation?" Tags="|gifted-students|anxiety|heterogeneous-groups|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24888" PostTypeId="2" ParentId="24883" CreationDate="2022-02-08T19:56:12.550" Score="41" Body="&lt;p&gt;An operation is born when we recognize the regularity in repeated reasoning.&lt;/p&gt;&#xA;&lt;p&gt;Take multiplication for example.  If we are living lives which involve even a modest amount of arithmetic thinking, we will encounter lots of situations where we need to calculate the total number of units present in some number of equal sized groups.  We might solve these problems in ad-hoc ways at first, but it becomes more and more convenient to &amp;quot;formalize&amp;quot; the process as a new abstract operation.  We can then develop algorithms for computing the product, etc.&lt;/p&gt;&#xA;&lt;p&gt;It is amazing that one operation allows us to solve all of the following problems:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I buy 34 valentines day cards (one for each of my classmates).  Each costs $2.45.  How much money did I spend?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A cake recipe requires &lt;span class=&quot;math-container&quot;&gt;$2\frac{1}{3}$&lt;/span&gt; sticks of butter.  I want to make &lt;span class=&quot;math-container&quot;&gt;$2.5$&lt;/span&gt; recipes.  How many sticks of butter do I need?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am riding my bike at &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt; miles per hour for 2.3 hours.  How many miles did I ride?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The same is true for integration.  Before we &amp;quot;invent&amp;quot; the definite integral, we should understand what kinds of situations it is abstracting.  Here are some:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I am riding my bike for 3 hours, but my speed is variable.  My speed can be described as &lt;span class=&quot;math-container&quot;&gt;$f(t)$&lt;/span&gt; miles per hour where &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is the time in hours.  How many miles did I travel?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am trying to find the area under a curve from &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x=3$&lt;/span&gt;, but the height of the curve is variable.  The height of the curve is given by &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; inches, where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the number of inches to the right of the origin.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am trying to find the work done by raising a bucket out of a well, but the bucket is leaking so its weight is variable.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am trying to find the volume of a solid, but the area of the cross sections is variable.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In each situation the &amp;quot;ad hoc&amp;quot; method to solve the problem has a commonality:  we chop the problem up into tiny pieces, approximate the answer for each tiny piece by assuming the variable quantity is &amp;quot;essentially constant&amp;quot; over the tiny pieces, and sum up the approximations.  To make the approximation better, make your pieces smaller (so that the &amp;quot;basically constant&amp;quot; approximation of each piece becomes more accurate).&lt;/p&gt;&#xA;&lt;p&gt;This process of chopping, approximating, and summing, and then taking a limit happens so frequently and in so many diverse situations that it makes sense to &amp;quot;formalize&amp;quot; this process as a new operation called integration.&lt;/p&gt;&#xA;&lt;p&gt;Teaching the operation without also teaching what kinds of problems it solves would be similar to teaching someone how to operate a circular saw without ever showing them a piece of wood.&lt;/p&gt;&#xA;&lt;p&gt;The volume of solids (and solids of revolution) are one example of the &amp;quot;integration idea&amp;quot;: chop up the solid into lots of thin slabs, and approximate that the cross sectional area is constant so that your slab is a cylinder (whose volume you can calculate).  Then sum up all of the volumes of the thin slabs.  Making the slabs thinner and thinner should give better and better approximations.&lt;/p&gt;&#xA;&lt;p&gt;While you can convey these ideas without using this particular example (there are millions of applications to choose from), volume is a classic.  In particular seeing where the formulas for the volumes of cones and spheres comes from is culturally important.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-02-08T19:56:12.550" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24889" PostTypeId="2" ParentId="24883" CreationDate="2022-02-08T21:59:59.573" Score="1" Body="&lt;p&gt;It's a forum style opinion, but I recommend to teach volumes, not your special topic.  Sorry, no Euclidean proof, but will at least blabla about why I feel that way.  Maybe it helps you get the heart to do it.&lt;/p&gt;&#xA;&lt;ol start=&quot;0&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;As far as location (calc 1 or 2), I think 2 makes more sense as it is a little trickier topic.  Also, the Naval Academy (very traditional) uses Stewart in order and has this topic in their second semester (their syllabus online, not sure how to link a PDF though).  Really, I would coordinate with department unless you are teaching 2.  If it's in a chapter of Stewart you are supposed to hit (&amp;quot;get through #X chapter&amp;quot;), than that sort of answers your question.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As far as just not doing it ever...and substituting a topic of your own interest, I wouldn't.  Teach the traditional course.  This is an entry, service course.  Things are hard enough for you, for the kids...might as well try driving down the middle of the road.  Spend your efforts, ingenuity on pedagogy, training (they are non trivial)...more than on changing the content.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;FWIW, I liked the topic (granted I was a strong student and my classmates strong also).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Would still prefer it to numerical topics.  Many students are/were like me also.  The special enriched sections of calculus with computers were reviled, for instance.  Was better to place out...or take the normal track.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think it's a most crucial topic ever.  In the sense of say max/min optimizations...or knowing that acceleration is the derivative of velocity.  Like if this was calc for business or for nurses or the like, sure cut it...and tricky integrations also.  But that's when you are generally watering down.  Not when you are going off the reservation to do a numerical topic.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think there is some general benefit in getting students to think about more complex problems (just in the sense of things to track, manipulations).  Also, I'm not even a visual person...but it was good for me to think a little about 3-d shapes.  And for engineers and the like, good for them also.  I mean Joseph's illustration is beautiful.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also sort of a &amp;quot;bridge to calc 3&amp;quot; type topic, maybe not directly...but in spirit.  Moving out of the plane.  Also physics 2 E&amp;amp;M involves some spheres and weird calc 3 style integrals.  Not that I recall ever solving a volume.  But still...kinda next door.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I do appreciate that you are reacting to your own experience.  And in the absence of anything, it is a Bayesian input.  But I would not overly bias a sample of one.  Also, you may get a different feel, love, comprehension of the topic after teaching it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19468" LastActivityDate="2022-02-08T21:59:59.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24890" PostTypeId="2" ParentId="24887" CreationDate="2022-02-08T22:59:58.167" Score="7" Body="&lt;p&gt;I don't have any solid suggestions regarding these two students.  I would have no concerns about the gifted student, and I might speak with the anxious student to put them with others they are most comfortable with.&lt;/p&gt;&#xA;&lt;p&gt;But I would recommend groups of 4, mainly because there is lots of information available on how to help them work well together. &lt;a href=&quot;http://mathmamawrites.blogspot.com/2011/07/complex-instruction.html&quot; rel=&quot;noreferrer&quot;&gt;I blogged about it after attending a workshop on &amp;quot;Complex Instruction&amp;quot;&lt;/a&gt;, and referenced the 4 roles they suggested you have students take in groups.&lt;/p&gt;&#xA;&lt;p&gt;I have found Henri Picciotto's work very helpful. &lt;a href=&quot;https://www.mathed.page/teaching/group-work.pdf&quot; rel=&quot;noreferrer&quot;&gt;Here's what he says about groups.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-02-08T22:59:58.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24891" PostTypeId="1" AcceptedAnswerId="24895" CreationDate="2022-02-09T16:43:16.603" Score="8" ViewCount="303" Body="&lt;p&gt;There are many many elegant algebraic (more generally, non-geometric) proofs, but fewer of them are both accessible and interesting to a pre-calculus student. Three nice examples of what I'm looking for are the proof of the irrationality of the square root of 2, Cantor's diagonal argument (although I don't know if that's considered an actual proof), and the proof of Euler's formula by algebraic manipulation of the expansions of exp, sin, and cos. I'm seeking a dozen or so more like these, i.e., mostly algebraic, elegant, fun (if you're a math geek :-), and short (you could write them down in ~1 page). Thanks in advance!&lt;/p&gt;&#xA;&lt;p&gt;(Clarification: If the proof has geometric aspects, like the diagonal argument, that's fair game. What I want to avoid is primarily geometric proofs, like those you do in your first geometry course. It's not that I'm against those, it's just that those are really easy to find as there are books and books of them -- like, specifically, 13 of them! :-) Also, I'm trying to avoid &amp;quot;proofs without words&amp;quot;, which I find personally annoying because, although they may not literally have words once you see how the proof works, getting there usually involves a LOT of explanation! So the label &amp;quot;proofs without words&amp;quot; is somewhat disingenuous to my mind.)&lt;/p&gt;&#xA;" OwnerUserId="14430" LastActivityDate="2022-02-13T18:26:49.177" Title="Seeking short algebraic proofs that an Algebra2 student can appreciate" Tags="|proofs|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24892" PostTypeId="2" ParentId="24883" CreationDate="2022-02-10T22:29:31.867" Score="19" Body="&lt;p&gt;A point to consider that has not been emphasized much in other answers so far: removing a topic from a syllabus in a service course should not be done before getting input from instructors of other courses that have yours as a prerequisite. People who teach Calculus III might &lt;em&gt;assume&lt;/em&gt; their students have experience computing volumes by the method you want to skip. They would not be happy to find out their students never saw the method in Calculus II first (a simpler setting).&#xA;Talk not just to the math department but other departments that have their majors take your course where you think volume integrals might matter: physics, engineering, maybe not economics.&lt;/p&gt;&#xA;&lt;p&gt;I will share a personal experience. Some years ago, a colleague and I decided unilaterally to cut the section on arc length from our Calculus II course: it was one lecture, we did not use it later in the course, students were just memorizing the formula, and it does not lend itself to concrete calculations outside of a few mostly artificial curves. So we cut it. All was fine… until the next semester when the Calculus III instructors wrote to us and asked if it was really true we had not taught arc length integrals. It turns out they were teaching arc length of space curves and told students it is similar to what they saw for arc length of plane curves in Calculus II. The students then said they had never seen arc length integrals in any form before. Oops! My department then created a policy that service courses could not cut topics from the syllabus without clearing it first with an undergraduate course committee.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2023-02-26T22:40:57.343" LastActivityDate="2023-02-26T22:40:57.343" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24895" PostTypeId="2" ParentId="24891" CreationDate="2022-02-11T16:27:09.130" Score="7" Body="&lt;p&gt;A lot of the &amp;quot;pre-calculus curriculum&amp;quot; (at least in my day) would seem to qualify.&lt;/p&gt;&#xA;&lt;p&gt;I remember my textbook containing proofs of:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The quotient-remainder theorem for real polynomials.&lt;/li&gt;&#xA;&lt;li&gt;The corollary that roots are in correspondence with linear factors.&lt;/li&gt;&#xA;&lt;li&gt;The corollary that a polynomial of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; can have at most &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; roots.&lt;/li&gt;&#xA;&lt;li&gt;The rational root theorem.&lt;/li&gt;&#xA;&lt;li&gt;The derivation of the quadratic formula, and the use of discriminants to distinguish reducible from irreducible quadratics.&lt;/li&gt;&#xA;&lt;li&gt;The binomial theorem.&lt;/li&gt;&#xA;&lt;li&gt;A lot of interesting results in analytic geometry concerning conic sections (how to complete the square with two variable quadratics, how to rotate a conic section into &amp;quot;standard form&amp;quot;, how to compute the location of foci and directrixes, how to compute eccentricities, etc).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The course I took did not expect me to learn or reproduce these proofs:  only utilize (some of) the results.  I have fond memories of learning this stuff anyways (I have always been a completionist:  I wanted to master the whole book).&lt;/p&gt;&#xA;&lt;p&gt;I think it is worth remembering that even if these &amp;quot;basic&amp;quot; things have lost their luster for you, a high school student is meeting these for the first time.  If introduced with care, to the right students, proving these results can be a profound mathematical experience.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-02-11T16:27:09.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24897" PostTypeId="2" ParentId="24891" CreationDate="2022-02-13T18:26:49.177" Score="3" Body="&lt;p&gt;My student the other day was delighted with my &amp;quot;proof&amp;quot; that the perpendicular to a line of gradient &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; has gradient &lt;span class=&quot;math-container&quot;&gt;$-1/m$&lt;/span&gt;, which I showed him by drawing the standard rise-over-run triangle for one line, drawing a perpendicular to that line, then turning the paper &lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt; and drawing the rise-over-run triangle for that, pointing out that it's the same triangle as the first one I drew, then turning it back the right way and showing that the rise-over-run of the second line had magically turned into minus the run over the rise of the first. He loved it.&lt;/p&gt;&#xA;" OwnerUserId="15290" LastActivityDate="2022-02-13T18:26:49.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24898" PostTypeId="2" ParentId="24887" CreationDate="2022-02-13T22:11:09.463" Score="1" Body="&lt;p&gt;I strongly urge you to make groups of three. The third student inproves the resources of the group, breaks deadlocks and generally, in my experience provides a better dynamic. This would also make it possible for the gifted student to be a resource for two other people: that is also to her benefit as explaining and challenging ideas deepens and embeds understanding of complex topics. The socially awkward student should have some say in how he is grouped and the possibility to temporarily withdraw if he becomes overwhelmed.&lt;/p&gt;&#xA;" OwnerUserId="3483" LastActivityDate="2022-02-13T22:11:09.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24901" PostTypeId="2" ParentId="7199" CreationDate="2022-02-15T05:07:09.697" Score="1" Body="&lt;p&gt;I saw a You Tube video recently which was shockingly clear on the topic of quadratic and cubic equations and their history. See: &lt;a href=&quot;https://www.youtube.com/watch?v=cUzklzVXJwo&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Imaginary Numbers were Invented&lt;/a&gt;. In particular, the video shows quite nicely how completing the square was literally completing a square. This could easily be adapted to an exercise for the classroom.&lt;/p&gt;&#xA;&lt;p&gt;1.) Study &lt;span class=&quot;math-container&quot;&gt;$x^2+26x = 27$&lt;/span&gt; by visualizing the &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; as the area of an &lt;span class=&quot;math-container&quot;&gt;$x \times x$&lt;/span&gt; square and &lt;span class=&quot;math-container&quot;&gt;$26x$&lt;/span&gt; as a &lt;span class=&quot;math-container&quot;&gt;$26 \times x$&lt;/span&gt; rectangle. Use colored paper to make it easy to track. Or, have them color a template if you want coloring as part of the exercise.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wPgm0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wPgm0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;2.) Cut the &lt;span class=&quot;math-container&quot;&gt;$x \times 13$&lt;/span&gt; rectangle in two and arrange one half horizontally with the &lt;span class=&quot;math-container&quot;&gt;$x \times x$&lt;/span&gt; piece and the &lt;span class=&quot;math-container&quot;&gt;$x \times 13$&lt;/span&gt; piece. Rotate the other half and set it vertically so the &lt;span class=&quot;math-container&quot;&gt;$x \times x$&lt;/span&gt; is over a &lt;span class=&quot;math-container&quot;&gt;$13 \times x$&lt;/span&gt; piece.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ej34y.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ej34y.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;3.) &lt;strong&gt;Complete the square&lt;/strong&gt; with a &lt;span class=&quot;math-container&quot;&gt;$13 \times 13$&lt;/span&gt; square.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/sbKMg.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sbKMg.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Notice to be fair we added an area of &lt;span class=&quot;math-container&quot;&gt;$169$&lt;/span&gt; to our picture so we must likewise add &lt;span class=&quot;math-container&quot;&gt;$169$&lt;/span&gt; to both sides:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x^2+13x+13x+169 = 27+169 = 196 $$&lt;/span&gt;&#xA;The equation above represents the areas of the parts of the square (left hand side) and the total area of &lt;span class=&quot;math-container&quot;&gt;$14^2 = 196$&lt;/span&gt; (right hand side). Since the big square pictured has side-length &lt;span class=&quot;math-container&quot;&gt;$14$&lt;/span&gt;. Thus &lt;span class=&quot;math-container&quot;&gt;$14 = x + 13$&lt;/span&gt; and we find &lt;span class=&quot;math-container&quot;&gt;$\boxed{x=1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I really like this demonstration for two reasons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It gives a reasonable origin story to the terminology &lt;strong&gt;completing the square&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;It makes the history of ignoring negative solutions to polynomial equations much more reasonable. The mathematicians of that era were not really studying polynomial algebra. Rather, they were asking particular geometric questions for which only geometrically reasonable answers were permitted.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Moving past this particular demonstration, I think it is interesting to note mathematicians were hung up on a &lt;em&gt;visualization&lt;/em&gt;. Often students seek visualization as a means of understanding. Sometimes, that is a wrong idea. To truly understand something, you might need to give up on visualization and instead embrace an &lt;em&gt;algebraic approach&lt;/em&gt;. For the problem visualized here,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x^2+26x = 27 \ \Rightarrow \ (x+13)^2-14^2=0 \ \Rightarrow (x+13-14)(x+13+14)=0 $$&lt;/span&gt;&#xA;hence &lt;span class=&quot;math-container&quot;&gt;$x = 1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x=-27$&lt;/span&gt;. A single line of algebra has replaced a page of cumbersome pictures.&lt;/p&gt;&#xA;&lt;p&gt;Of course, seeking an algebraic understanding is not a universal advice. Just like some screws require a flat-head and others require a star-head. Each problem has tools which are best. Knowing which tool to use, that is an art we're still learning.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2022-02-15T05:07:09.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24902" PostTypeId="1" CreationDate="2022-02-15T11:10:45.637" Score="2" ViewCount="200" Body="&lt;p&gt;In the late 80’s and early 90’s there was the idea of ‘calculus reform’ and some emphasis and syllabus changed. The order of doing things in calculus also changed with the advantage of technology.&#xA;Similarly in linear algebra, there was a linear algebra curriculum study group that produced some really good ways of teaching linear algebra and highlighted curriculum changes. This was produced in the January 1993 College Mathematics Journal.&#xA;Has any similar work been covered in (Further) Engineering Mathematics? I am looking for what are important topics to cover and any work or research on the teaching of Engineering Mathematics. I am looking for some sort of framework.&lt;/p&gt;&#xA;" OwnerUserId="1567" LastActivityDate="2022-02-15T14:07:27.207" Title="How and what to teach on a second year Engineering Mathematics?" Tags="|undergraduate-education|mathematical-pedagogy|applied-mathematics|mathematics-in-daily-life|engineering-mathematics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24903" PostTypeId="2" ParentId="24902" CreationDate="2022-02-15T14:07:27.207" Score="4" Body="&lt;p&gt;I'm unaware of a major reform in engine math.  If you look at the major selling textbooks, for instance, they remain pretty traditional.  The most popular, Kreyszig, is one that I had in the mid-80s (and fifth edition already then).  Granted there can be innovation (these author name texts become brands like Grey's Anatomy), but scanning the book I don't see much difference, now.&lt;/p&gt;&#xA;&lt;p&gt;I did a Google Scholar search and there's pretty little out there.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://scholar.google.com/scholar?hl=en&amp;amp;as_sdt=0%2C47&amp;amp;q=reform+%22engineering+mathematics%22&amp;amp;btnG=&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://scholar.google.com/scholar?hl=en&amp;amp;as_sdt=0%2C47&amp;amp;q=reform+%22engineering+mathematics%22&amp;amp;btnG=&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Several of the top hits are actually papers on reforming all math that engineers take, with an emphasis really on calculus.  See for instance the several paper on Wright State University, but if you read them, you see the issue was engineers failing first year calculus.  Probably reflecting a general issue of sending a lot of kids to college and then weeder courses inevitably (and easily) culling many of them.  But in any case, it's not really about &amp;quot;engine math&amp;quot; or what physicists call a &amp;quot;math methods&amp;quot; course.&lt;/p&gt;&#xA;&lt;p&gt;I would even caution you that in calculus, the reform movement (funded by NSF in the 90s) has pretty much crested, past, and increasingly been forgotten.  Can argue if this was the intransigence of the teachers or the drawbacks of the reform.  I grant the former, but think the latter applies also.  But in any case, even if it rocked, it didn't &amp;quot;win&amp;quot;, not like flat TVs killed CRTs.  Thomas sells more than Hughes Hallet.  Stewart is pretty commercial and tries to be all things, but I wouldn't really consider it a reform text.  My point is that even if you find some article (or text) on reformed engine math, I wouldn't assume that it was tested and successful in the market place. At least look at it carefully yourself, before pushing it, to see what you like...not just because it is labeled &amp;quot;reform&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="19496" LastActivityDate="2022-02-15T14:07:27.207" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24904" PostTypeId="1" AcceptedAnswerId="24907" CreationDate="2022-02-15T17:35:30.530" Score="25" ViewCount="3770" Body="&lt;p&gt;As a math teacher, I am always trying to self-assess my teaching methods. I am trying a lot of different methods but I would like to organize my study on the subject without weighing too much on the kids with a trial and error approach, so I'm looking for some scientific research on effective math teaching strategies and practices for high school students. There are a lot of ideas out there online but based on what I found either a) these ideas are really narrow-focused only on small children or b) the ideas are &amp;quot;common sense&amp;quot; ideas that &lt;em&gt;might&lt;/em&gt; work but are not really backed by any scientific study.&lt;/p&gt;&#xA;&lt;p&gt;Any help?&lt;/p&gt;&#xA;" OwnerUserId="6286" LastActivityDate="2022-02-18T14:41:24.583" Title="Are there science-backed effective teaching strategies?" Tags="|secondary-education|mathematical-pedagogy|teacher-development|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24907" PostTypeId="2" ParentId="24904" CreationDate="2022-02-15T19:21:05.333" Score="44" Body="&lt;p&gt;In terms of controlled experiments, then, yes. Note that most are opposite or orthogonal to virtually all pedagogical norms in math education.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Active recall. &amp;quot;Put away all your notes and textbooks and computers. Here is a blank piece of paper. Show me everything you've learned in the past couple of weeks.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Spaced repetition. You can certainly read about this elsewhere. :-)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Elaboration. &amp;quot;Don't just copy written steps. Tell me: How do you know this is true? How do you know it's useful? How can you check your work? What does it really mean?&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Interleaving. &amp;quot;Here are 12 word problems about a bakery. All the word problems involved the numbers 32 and 8, but you'll have to figure out whether it's division, multiplication, addition, subtraction, exponents, brackets, negatives, fractions, whatever. &lt;em&gt;You&lt;/em&gt; need to learn to make those distinctions.&amp;quot; More resources: &lt;a href=&quot;https://ssddproblems.com/&quot; rel=&quot;noreferrer&quot;&gt;SSDD Problems&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Multiple Representations. I assume you're already doing this, but in lower grades, it might be: &amp;quot;How can you represent the number of forks needed using those manipulatives? Can you draw it? Can you write the number in words? Can you write it using math symbols?&amp;quot; In upper grades, it's &amp;quot;What does this relationship look like as a graph? An equation? A story problem? Words? A table? A diagram? Can you tell me how they all represent the same information?&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Concrete examples. This has two implications. First, you should START with the concrete example. NOT &amp;quot;Here's two weeks of studying transformations and propreties of &lt;span class=&quot;math-container&quot;&gt;$f(x)=b^x$&lt;/span&gt;... and now here's a (magic) formula for compound interest. Now plug in these numbers.&amp;quot; But rather, &amp;quot;How do credit cards work? What happens if you don't pay over time? Is it good to just add 2%/month to calculate what our debt will be in 10 years?&amp;quot; See &lt;a href=&quot;https://www.math.ucsd.edu/%7Ejrabin/publications/ProblemFreeActivity.pdf&quot; rel=&quot;noreferrer&quot;&gt;Problem-Free Activity and Intellectual Need&lt;/a&gt; for more info. To this I'd add that students must be able to distinguish examples from superficially similar non-examples, plus they should be able to &lt;em&gt;generate&lt;/em&gt; examples and non-examples on their own. Note that virtually NO textbooks or math courses emphasize any of this.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Alternating worked examples and exercises. Note that virtually ALL math textbooks have an information &amp;amp; example section first, then exercises after. At most they sprinkle some &amp;quot;Can you regurgitate/mimick?&amp;quot; tasks in&#xA;the information sections. Most math lessons follow the same structure. [Edit: I am unaware of &lt;em&gt;scientific&lt;/em&gt; evidence on this, but it seems obvious that frequently walking around the class to see if your students are &lt;em&gt;actually&lt;/em&gt; following the lesson should be helpful. Frequent alternating is one structure that allows this.]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask probing questions. My take on this: Fight &lt;a href=&quot;https://matheducators.stackexchange.com/a/13215/19499&quot;&gt;confirmation bias&lt;/a&gt;. Do NOT ask &amp;quot;What can my students do?&amp;quot; [fertile ground for confirmation bias]. Instead, imagine a brilliant math teacher saying &amp;quot;I can prove &lt;em&gt;right now&lt;/em&gt; that the apparent learning you perceive is all an illusion. The students didn't learn anything.&amp;quot; How would that teacher prove their case? [And how often can most of us answer that question confidently?]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Read more at&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.learningscientists.org/&quot; rel=&quot;noreferrer&quot;&gt;Learning Scientists&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=oxZzoVp5jmI&quot; rel=&quot;noreferrer&quot;&gt;How We Learn vs How We Think We Learn&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.nctq.org/dmsView/Learning_About_Learning_Report&quot; rel=&quot;noreferrer&quot;&gt;Learning About Learning&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Make it Stick [name of book]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Problems with these recommendations include [edited list]&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They ignore the problem of many students arriving in class with grossly inadequate prior knowledge.&lt;/li&gt;&#xA;&lt;li&gt;They surface lots of bad news and discomfort. Students and teachers are often shocked at the low performance. They infer struggles that these teaching methods are ineffective - if not outright harmful - despite overwhelming evidence from controlled experiments. If you implement these things, prepare for tremendous pushback.&lt;/li&gt;&#xA;&lt;li&gt;They don't help much with tight deadlines or most standardized tests.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, I consider it a design problem to best implement those strategies into my pedagogy. And it's one helluva tough problem.&lt;/p&gt;&#xA;&lt;p&gt;Looking elsewhere, JUMP Math does an extraordinary job of breaking math into small steps. This approach has been &lt;a href=&quot;https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0223049&quot; rel=&quot;noreferrer&quot;&gt;tested in an RCT&lt;/a&gt; against my country's former mainstream math textbook series and JUMP came out on top, by far.&lt;/p&gt;&#xA;&lt;p&gt;IMHO, many of the problems generated by: (a) that mainstream math textbook series and (b) progressive approaches towards math education generally - are solved by the work of &lt;a href=&quot;https://peterliljedahl.com&quot; rel=&quot;noreferrer&quot;&gt;Peter Liljedahl&lt;/a&gt;. I am not personally familiar with the science or controlled experiments on this, but he has pretty vast experience and data.&lt;/p&gt;&#xA;&lt;p&gt;If I think of anything else, I'll add it. :-)&lt;/p&gt;&#xA;&lt;p&gt;Edit in response to @NotThatGuy 's comment. Here's more on &lt;a href=&quot;https://matheducators.stackexchange.com/a/13215/19499&quot;&gt;confirmation bias&lt;/a&gt;. (Please read the hypothetical example linked before moving on.) Often, teachers think &amp;quot;Can my students do these calculations correctly? If yes, then they must have mastered the concepts!&amp;quot; That inference is, in a huge share of cases, an example of confirmation bias that pervades math education from kindergarten to &lt;a href=&quot;https://blog.mrmeyer.com/2018/what-does-fluency-without-understanding-look-like/&quot; rel=&quot;noreferrer&quot;&gt;middle and high school&lt;/a&gt; to &lt;a href=&quot;https://newsroom.ucla.edu/stories/why-so-many-u-s-students-arent-learning-math&quot; rel=&quot;noreferrer&quot;&gt;community college&lt;/a&gt; up to &lt;a href=&quot;https://www.harvardmagazine.com/2012/03/twilight-of-the-lecture&quot; rel=&quot;noreferrer&quot;&gt;physics at Harvard&lt;/a&gt; to &lt;a href=&quot;https://hub.jhu.edu/2018/04/12/carl-wieman-cue2-science-learning/&quot; rel=&quot;noreferrer&quot;&gt;Nobel Prize laureate-picked graduate students&lt;/a&gt;. Confirmation bias is &lt;em&gt;practically everywhere in math education&lt;/em&gt;. Every math educator should be aware of this, they should stop inferring conceptual mastery from calculations, and prepare tasks and assessments that detect such illusions of learning.&lt;/p&gt;&#xA;" OwnerUserId="19499" LastEditorUserId="19499" LastEditDate="2022-02-17T22:47:42.710" LastActivityDate="2022-02-17T22:47:42.710" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="24908" PostTypeId="1" CreationDate="2022-02-15T21:30:46.570" Score="1" ViewCount="97" Body="&lt;p&gt;I'm looking for a PC software whiteboard application that allows for custom RGB background colors and custom RGB pen colors.  Most I've seen offer only a handful of color choices for the pen.&lt;/p&gt;&#xA;&lt;p&gt;Lucidspark, an online app, for example, offers custom RGB pen colors, but does not allow the canvas color to be changed.&lt;/p&gt;&#xA;&lt;p&gt;This is to be used for my lecture/chalk-talk recordings for undergrad higher-ed math.&lt;/p&gt;&#xA;&lt;p&gt;I prefer something local to my computer, and as cheap as possible, of course.  Besides custom color features I think only pen thickness options would be helpful.&lt;/p&gt;&#xA;" OwnerUserId="17928" LastActivityDate="2022-03-18T17:02:55.237" Title="Which whiteboard/canvas app with custom background and color options" Tags="|software|videos|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24909" PostTypeId="2" ParentId="24696" CreationDate="2022-02-15T22:06:54.127" Score="4" Body="&lt;p&gt;A major problem with most early linear algebra classes is that they emphasize solving problems by hand using rref, as these methods do not translate well to computer calculations since they are unstable and thus can give inaccurate results.  While engineering students may not understand the technical definition of stability, they should be taught that&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Stable algorithms are important for linear algebra on the computer&lt;/li&gt;&#xA;&lt;li&gt;Some simple algorithms (like rref) are good for human calculation, but not stable&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;With this in mind, when I teach a similar applied course, I plan my lectures around making sure I get as far as eigenvalue decomposition, QR factorization (if possible), and (perhaps most importantly) SVD and at least one good application of each.  This allows students to see 2-3 good natural bases for calculations besides just the standard basis, and is enough background for most basic applied linear algebra.  Getting as far as SVD is a doable challenge in a one semester course, so the necessary prerequisites determine pretty much everything else.&lt;/p&gt;&#xA;&lt;p&gt;Strang's linear algebra textbook is one of the best known textbooks that follows this trajectory, but in my opinion it's a bit too advance (with not enough concrete calculations) for a first textbook for engineering students.  (It would be an excellent resource for yourself as you're brushing up, as are &lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-1-the-geometry-of-linear-equations/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Strang's own lectures&lt;/a&gt;.) Thus I agree with @MrProof and his suggestion of Lay's textbook for students, if you want to follow a textbook closely.  It's concrete and has enough calculations to be more accessible for the population you are targeting.  It's good to supplement any textbook with &lt;a href=&quot;https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&quot; rel=&quot;nofollow noreferrer&quot;&gt;3Blue1Brown&lt;/a&gt;'s excellent series of videos on linear algebra, which give geometric intuition in a way that no static textbook page will be able to.&lt;/p&gt;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;" OwnerUserId="9542" LastEditorUserId="470" LastEditDate="2022-02-18T00:28:06.080" LastActivityDate="2022-02-18T00:28:06.080" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24910" PostTypeId="2" ParentId="24904" CreationDate="2022-02-16T11:10:28.713" Score="4" Body="&lt;p&gt;Maybe not so concrete on practical strategies, but here goes:&#xA;One key concept that has always resonated strongly with me is in David Tall's work where he talks about &lt;em&gt;compression&lt;/em&gt; being the prime mover in understanding mathematics, e.g. this one: &lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED489653.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://files.eric.ed.gov/fulltext/ED489653.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Mathematics is extremely hierarchical. To understand multiplication, you need to understand addition first, that's just common sense. But really you need more than that, you need to &lt;em&gt;compress&lt;/em&gt; the concept of addition such that you can do addition using only a small part of your intellectual capacity. Only then can you learn multiplication.&lt;/p&gt;&#xA;&lt;p&gt;The same concept goes all the way up. You have no hope of learning Itô calculus if you have not compressed the ordinary theory of Riemann integration.&lt;/p&gt;&#xA;&lt;p&gt;You can use this concept as a lens with which you view your teaching. Is the student struggling because it is difficult to understand the new thing, or because they have not yet achieved compression of the old thing?&lt;/p&gt;&#xA;" OwnerUserId="19505" LastActivityDate="2022-02-16T11:10:28.713" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24911" PostTypeId="1" CreationDate="2022-02-16T15:28:14.900" Score="0" ViewCount="178" Body="&lt;p&gt;What should a tutor do? I have students who even ask for homeworks, because they have desire for new knowledge in math. However, in my experience, most pupils hate mathematics. Should I ask their parent to be checking on that? I do private online classes for students 5th-13th Grade.&lt;/p&gt;&#xA;" OwnerUserId="19510" LastEditorUserId="19510" LastEditDate="2022-02-16T15:33:58.910" LastActivityDate="2022-02-22T19:41:39.993" Title="student doesn't bring a homework in a 1-to-1 class" Tags="|teaching|remote-teaching|" AnswerCount="2" CommentCount="2" ClosedDate="2022-02-25T21:58:07.120" ContentLicense="CC BY-SA 4.0" />
  <row Id="24912" PostTypeId="1" CreationDate="2022-02-16T16:08:10.023" Score="6" ViewCount="846" Body="&lt;p&gt;Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function.&#xA;How can Laplace transforms be introduced so that students are motivated to learn? It needs to have an impact.&#xA;What are the applications of Laplace transforms?&lt;/p&gt;&#xA;" OwnerUserId="1567" LastActivityDate="2022-02-17T18:21:32.777" Title="What is the best way to introduce Laplace transforms on an Engineering Mathematics course?" Tags="|undergraduate-education|mathematical-pedagogy|examples|engineering-mathematics|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24913" PostTypeId="2" ParentId="24908" CreationDate="2022-02-16T16:41:51.050" Score="1" Body="&lt;p&gt;The free version of myViewBoard works.  It allows for custom background and pen colors.  You can download it as a PC app or use it online as a web app.&lt;/p&gt;&#xA;" OwnerUserId="17928" LastActivityDate="2022-02-16T16:41:51.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24914" PostTypeId="2" ParentId="24912" CreationDate="2022-02-16T17:33:33.637" Score="5" Body="&lt;p&gt;In a &lt;a href=&quot;https://en.wikipedia.org/wiki/Classical_control_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;classical control theory&lt;/a&gt; course, Laplace transforms are used to calculate transfer functions. As suggested by @Isaiah, see what your colleagues are doing. Skipping material can be risky when other courses assume yours has covered it.&lt;/p&gt;&#xA;&lt;p&gt;@Graham's &lt;a href=&quot;https://matheducators.stackexchange.com/a/24919/376&quot;&gt;answer&lt;/a&gt; goes into more detail about applications. It's good to be familiar with them so you can mention them, but maybe only in passing as motivation. If you're teaching an engineering mathematics course, it may not necessarily be your role to cover the applications themselves, or at least not all of them - your colleagues will go over them in detail in their own courses. Again, to strike the right balance, have a chat about what their topics are and what they'd like you to do in your course as mathematical preparation.&lt;/p&gt;&#xA;&lt;p&gt;In addition, familiarity with Laplace transforms can pave the way to understanding other integral transforms, such as the z-transform and Fourier transform, which students will probably meet in courses on signal processing or partial differential equations.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="376" LastEditDate="2022-02-17T15:51:42.543" LastActivityDate="2022-02-17T15:51:42.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24915" PostTypeId="1" CreationDate="2022-02-16T17:42:44.997" Score="5" ViewCount="332" Body="&lt;p&gt;Does anyone have extensive classroom experience regarding the best verb tense to use when writing word problems at an elementary or middle school level? I have been writing some lessons recently and I never really thought about it until now. My M.Ed. program focused mainly on high school mathematics and placed much more emphasis on theory than classroom experience. I'd really like to know what works best in practice. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="1914" LastActivityDate="2022-02-16T17:42:44.997" Title="Word problems written in past tense, present tense, or future tense" Tags="|mathematical-pedagogy|primary-education|middle-school|language-use|word-problems|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24916" PostTypeId="2" ParentId="24911" CreationDate="2022-02-16T18:44:12.553" Score="1" Body="&lt;p&gt;Tell the student and the parents straight. They are paying to improve the student's maths skills. They are paying for extra lessons because the student isn't at the standard they would like. For this improvement to occur as much, if not more, must happen outside the lesson in the student's own time. That's what I tell them. I also say if they want to change tutor be wary of those who just turn up and take money. But then I am fortunate I have enough work to turn down students who don't study.&lt;/p&gt;&#xA;" OwnerUserId="7954" LastActivityDate="2022-02-16T18:44:12.553" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24917" PostTypeId="2" ParentId="24912" CreationDate="2022-02-17T03:57:40.543" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;I would not use&amp;quot; is the incorrect framework.  You need to teach your students, most efficiently, what they need as a service for their courses.  Not what you like/don't.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I actually think it's a gentler way (more progressive pedagogy) to learn the topic to just do some problems, do some calculations, etc. before dealing with applications.  Applications are word problems.  Word problems are hard.  And for LT, the applications are things like electrical circuits and mechanical controls that are not as easy to understand as a canon ball parabola or the like (not out of the world crazy, but still a little more abstract).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Take a look at the approach in Kreyszig (chapter five in the fifth edition).  It's a pretty standard approach, you'd get in any ODE standalone text.  Very tiny amount of motivation (not a physical problem driver like how PDE texts introduce methods).  Then a bunch of techniques, the use of the tables, the partial fractions, etc.  Few applications mixed in along the way and at the end.  But emphasis on developing skill in the manipulations first (sans physical problems) within most sections.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You should know/figure out the applications of LT.  They are classic ones and in almost all books.  The textbook should inform you more than we do.  FWIW, even though they like to list some mechanical applications, the honest thing for engineers is that that LT is really more important for EEs and Systems Engineers.  Not so much for mechanical (and all the mechanical related ones like aero, chem, etc.)  This is not to say the mechies don't have it required also.  Just they won't really see it a lot later on...just one of those things they have to do as part of their general engineering.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P.s.  See this related question:  &lt;a href=&quot;https://matheducators.stackexchange.com/questions/11126/should-i-teach-laplace-transforms-how-much?rq=1&quot;&gt;Should I teach Laplace Transforms? How much?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19516" LastActivityDate="2022-02-17T03:57:40.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24918" PostTypeId="2" ParentId="24912" CreationDate="2022-02-17T05:20:10.087" Score="3" Body="&lt;p&gt;Laplace Transforms treat discontinuous forces in a natural fashion. To solve problems with discontinuous forcing functions without the transform would require tedious fitting of solutions from one region to the next at each point of discontinuity. In contrast, the Laplace transform smooths all these into one algebraic problem which once solved gives the solution as a simple inverse transform. In addition, the method allows the introduction of the transfer function which is interesting in its own right. I challenge your assertion that it is easier to work problems without the method. Even continuous problems have nice solutions with the transform. For example, it allows the derivation of the multipliers in the repeated root solution.&lt;/p&gt;&#xA;&lt;p&gt;To me, the utility I sketch above motivates the use. But, there is a way to motivate it by an appeal to its similarity to an idea in signal processing. That is not something for which I currently have intuition. There is a well-known MIT DEqns professor who spends about 20 minutes explaining the meaning of the transform. If you search for it, you'll find it. Sorry I am lazy at the moment.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2022-02-17T05:20:10.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24919" PostTypeId="2" ParentId="24912" CreationDate="2022-02-17T14:06:24.290" Score="3" Body="&lt;h3&gt;Modelling physical or electrical behaviour&lt;/h3&gt;&#xA;&lt;p&gt;Mass-spring-damper, resistor-capacitor-inductor or pump-tank-pipe systems can all be modelled using the Laplace transform.  On its own this allows students to solve practical problems such as setting up a car suspension.  Setting up a damper for critical damping is the obvious first step, but you can also step between Laplace-domain and time-domain work to choose a spring rate which will limit the suspension excursion to x centimetres (to prevent bottoming-out the suspension).&lt;/p&gt;&#xA;&lt;p&gt;State-space notation allows you to also model hysteresis and time delays in systems, but state-space inherently builds on Laplace and cannot be understood before you have already grasped Laplace.&lt;/p&gt;&#xA;&lt;h3&gt;Classical control theory&lt;/h3&gt;&#xA;&lt;p&gt;Classical control theory uses PID controllers, phase lead-lag networks, and pole-zero calculations, in conjunction with models of the physical or electrical systems being controlled.  All of this is 100% tied into Laplace, and nothing else is a suitable alternative.  It is literally not possible to discuss classical control theory without Laplace.&lt;/p&gt;&#xA;&lt;h3&gt;Filter design&lt;/h3&gt;&#xA;&lt;p&gt;IIR filters are designed for particular frequency responses.  These always use Laplace, because substituting &amp;quot;s = jw&amp;quot; in the Laplace transfer function gives you the frequency response.  This is relevant for all engineering disciplines, as well as anything involving audio.  These filters are the building blocks for most practical filters, and understanding their behaviour is key to understanding the benefits (and drawbacks) of digital filtering with FIRs.&lt;/p&gt;&#xA;" OwnerUserId="6456" LastEditorUserId="6456" LastEditDate="2022-02-17T18:21:32.777" LastActivityDate="2022-02-17T18:21:32.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24920" PostTypeId="1" AcceptedAnswerId="24929" CreationDate="2022-02-17T22:12:59.373" Score="18" ViewCount="2338" Body="&lt;p&gt;Here are three &amp;quot;cancellations&amp;quot; seen during algebra simplification, two of which are invalid.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1) &lt;span class=&quot;math-container&quot;&gt;$\frac{x + 6}{6} = \frac{x+6\hspace-1.2ex\diagup}{6\hspace-1.2ex\diagup} = x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(2) &lt;span class=&quot;math-container&quot;&gt;$\frac{6x + 1}{6} = \frac{6\hspace-1.2ex\diagup x + 1}{6\hspace-1.2ex\diagup} = x + 1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(3) &lt;span class=&quot;math-container&quot;&gt;$\frac{6x}{6} = \frac{6\hspace-1.2ex\diagup x}{6\hspace-1.2ex\diagup}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What is the correct student-understandable feedback for the first two errors?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For (1), I have tried:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Adding 6 does not cancel with dividing by 6; those are not inverse operations.&lt;/li&gt;&#xA;&lt;li&gt;Adding 6 would cancel with subtracting 6: &lt;span class=&quot;math-container&quot;&gt;$x + 6 - 6 = x + 6\hspace-1.8ex\diagup - 6\hspace-1.8ex\diagup$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Dividing by 6 would cancel with multiplying by 6: &lt;span class=&quot;math-container&quot;&gt;$\frac{6x}{6} = \frac{6\hspace-1.2ex\diagup x}{6\hspace-1.2ex\diagup}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, this feedback implies that perhaps (2) should be allowed, since the 6 is certainly multiplied in (2), so why not divide by 6 to undo it?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For both (1) and (2), I have also tried:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The numerator contains two terms, both of which are divided by 6, and the incorrect cancellation unfairly deprives the other term of being divided by 6.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, this way of thinking implies that perhaps we are robbing the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; of its division by 6 in (3).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;What is the best feedback for error (1) and error (2)?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2022-02-22T00:18:53.897" Title="What is the right feedback for incorrect cancellation?" Tags="|algebra|grading|" AnswerCount="8" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="24921" PostTypeId="2" ParentId="24920" CreationDate="2022-02-17T22:50:40.443" Score="30" Body="&lt;p&gt;My feedback might be more like this: When you are not 100% sure that a step works, try it with numbers. If x=2, does your expression keep the same value before and after your cancellation?&lt;/p&gt;&#xA;&lt;p&gt;Also, you didn't say which level course. I see mistakes like this even among calculus students, so I have a worksheet I ask them to do in class, called &lt;a href=&quot;https://drive.google.com/file/d/1IaG-oW5N7gseXqSIJ0xvfP2tH7i0tSX7/view?usp=sharing&quot; rel=&quot;noreferrer&quot;&gt;Algebra Temptations&lt;/a&gt;. Step 1 is to mark each &amp;quot;identity&amp;quot; True or False. They are all false. Most students don't see this. We then discuss them.&lt;/p&gt;&#xA;&lt;p&gt;If you are teaching a course like beginning algebra, I'd recommend thinking about how to teach or reteach this topic. I do say things like 1*x=x, so if we can get our fraction in a form where we have a 1 (or n/n) times something, then the 1 can go away, but otherwise &amp;quot;canceling&amp;quot; changes the value and doesn't work.&lt;/p&gt;&#xA;&lt;p&gt;I do not expect most students to carefully read and think about my written feedback. So it seems better to ask them questions than to bother trying to explain perfectly why their work doesn't make sense.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-02-17T22:50:40.443" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24922" PostTypeId="2" ParentId="24920" CreationDate="2022-02-18T02:11:06.437" Score="11" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;What is &lt;strong&gt;the&lt;/strong&gt; correct student-understandable feedback for the first two errors? [emphasis added]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am not sure there exists a singular answer to this.&lt;/p&gt;&#xA;&lt;h3&gt;Ideally,&lt;/h3&gt;&#xA;&lt;p&gt;the educational trajectory should have students doing and saying things like this.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Grade 3 student: &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{14}{3}=4\frac{2}{3}$&lt;/span&gt;... this is more than 4 wholes but less than 5 wholes. Fractions are numbers which mostly reside between consecutive whole numbers.&amp;quot; [Student shows the &lt;span class=&quot;math-container&quot;&gt;$\frac{14}{3}=4\frac{2}{3}$&lt;/span&gt; with area diagrams, set diagrams, number lines, manipulatives, etc.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Grade 5 student: &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{12}{9}=\frac{12\times5}{9\times5}=\frac{10}{45}=\frac{12\times30}{9\times30}=\frac{12\div3}{9\div3}=\frac{4}{3} ...$&lt;/span&gt;. Every fraction has an infinite number of equivalent fractions. You generate them by dividing or multiplying both the top and bottom by the same number. *You do not generate them by adding or subtracting the same amount from the top and bottom.&amp;quot; [Student shows this by generating examples with varied representations of the idea.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Grade 10 student: &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{ab}{ac}=\frac{b}{c}$&lt;/span&gt;. Fractions can only be simplified by cancelling common factors &lt;em&gt;of the entire numerator and of the entire denominator&lt;/em&gt;. This generalizes the idea of equivalent fractions.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Based on that grade 10-level understanding, one can then start discussing, comparing, and contrasting structures.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Teacher: &amp;quot;Can you see the form &lt;span class=&quot;math-container&quot;&gt;$\frac{ab}{ac}$&lt;/span&gt; in these three rational expressions? If so, how? Show me if you can and let's start testing simplifications empirically.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'd consider this to be a mathematically coherent experience.&lt;/p&gt;&#xA;&lt;h3&gt;But the reality ...&lt;/h3&gt;&#xA;&lt;p&gt;is more like this.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Students of many ages: &amp;quot;I hate fractions. I HATE them. &lt;em&gt;HATE&lt;/em&gt;. They're nothin' but &lt;a href=&quot;https://matheducators.stackexchange.com/a/15284/19499&quot;&gt;weird crap stains on a page that have been hurting my brain for years&lt;/a&gt;. But I've noticed that if I follow even weirder rules, then teachers get off my back and I can return to more enjoyable aspects of life. So just tell me what the weird rules are, tell me if my answer's right, and I'll forget all of this when you walk away, but that's OK as long as I'm not working on fractions any more. And... OMG NO I DON'T WANT TO DO FRACTIONS WITH LETTERS IN THEM HOLY COW YOU MADE THEM EVEN WORSE.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In these cases, there are obviously no easy answers. The &lt;em&gt;only&lt;/em&gt; road to real understanding here is to meet students where they're at on that ideal trajectory then work up the ladder of abstraction one rung at a time. But that may require ignoring current coursework on which students are about to be graded. You want them to learn, but they want to get good grades NOW. And if they perceive rote cramming as the best way to get good grades, they'll do it, long-term consequences be damned.&lt;/p&gt;&#xA;&lt;p&gt;Let me know if you are able to solve &lt;em&gt;that&lt;/em&gt; one.&lt;/p&gt;&#xA;" OwnerUserId="19499" LastActivityDate="2022-02-18T02:11:06.437" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24923" PostTypeId="1" AcceptedAnswerId="24925" CreationDate="2022-02-18T03:07:23.060" Score="4" ViewCount="165" Body="&lt;p&gt;I have a class of 31 students. There are about 5 to 10 of them absent during each meeting. Right now I am not enforcing attendance because some of them may have legitimate reasons to be absent and it's difficult to keep track of this every class. Also, I think a few students may indeed be able to learn from reading the textbook by themselves.&lt;/p&gt;&#xA;&lt;p&gt;But on the other hand, some of the students who are absent performed quite poorly in the mid-term. So I wonder if my tolerance of absenteeism has an overall negative effect on students.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorUserId="12255" LastEditDate="2022-02-18T16:56:45.197" LastActivityDate="2022-02-18T16:56:45.197" Title="What to do with students who are often absent from class?" Tags="|undergraduate-education|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24924" PostTypeId="2" ParentId="24920" CreationDate="2022-02-18T03:38:26.313" Score="12" Body="&lt;p&gt;I tell my students that they can only cancel &lt;em&gt;factors&lt;/em&gt;, and tell them to factor the numerator and denominator (to remind them what a factor is).&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-02-18T03:38:26.313" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24925" PostTypeId="2" ParentId="24923" CreationDate="2022-02-18T06:27:01.227" Score="1" Body="&lt;p&gt;I do not have a solution, but two points of view. Disregarding either is simplistic.&lt;/p&gt;&#xA;&lt;h1&gt;Good teachers have good attendance&lt;/h1&gt;&#xA;&lt;p&gt;In this perspective, it is your duty as a teacher to make the education so good that students want to attend. Maybe you are just lecturing, not engaging them, teaching at too low or high level?&lt;/p&gt;&#xA;&lt;h1&gt;A student responsibility to be there&lt;/h1&gt;&#xA;&lt;p&gt;In this perspective the teacher offers their expertise and knowledge, and the student can choose to make use of it or not. Some students lack the skills or the motivation to get through and you must learn to accept this. You can not save everyone. So, if the students choose to not attend, that is their choice as an adult. Who are you to critize them?&lt;/p&gt;&#xA;&lt;h1&gt;Complicating factors&lt;/h1&gt;&#xA;&lt;p&gt;There are all kinds of issues beyond the two named above. Effective teaching puts the learners outside their comfort zone, so good teaching is not always purely pleasant; it is engaging and does not hurt, but requires energy to participate in and puts one to test. The levels of participation requires and stress that students accept varies from person to person, so most likely the level you choose will not suit everyone.&lt;/p&gt;&#xA;&lt;p&gt;There are also all kinds of habits and expectations in play. Students might see their position as one who sits in on lectures and absorbs knowledge, or as someone who is responsible for their own learning, or in a variety of other ways. These different positions interact with your teaching methods in various ways, leading to presence, absence, learning or not learning.&lt;/p&gt;&#xA;&lt;p&gt;So all I can say at this point is good luck and try out various teaching methods and ways of organizing things.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-02-18T06:27:01.227" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24926" PostTypeId="2" ParentId="24911" CreationDate="2022-02-18T08:45:48.183" Score="2" Body="&lt;p&gt;I tutor students on zoom.  I tell them to email me their homework by 5 PM the day before the next lesson. If they do, I have a chance to see how they did with the work BEFORE we meet.  If they don't, I get in touch with them and tell them that there is no point in meeting if they haven't done the work. Sometimes they will ask if they can send the work a little later the same day and I allow it and other times we cancel the lesson.  This works with college students.&lt;/p&gt;&#xA;&lt;p&gt;Since you tutor younger students I suggest that you soften this approach a little as well as involving the parents. Tell the parents the student hasn't been bringing the work.  Tell the parents you will be implementing a policy where they must email the homework in advance. If they do, it will be win -win as they will be keeping up with your work as well as giving you a chance to assess them before you meet.  If they don't explain to the parents that you will be cancelling the lessons until the homework is done.  Hope this works for you.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2022-02-22T19:41:39.993" LastActivityDate="2022-02-22T19:41:39.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24928" PostTypeId="1" CreationDate="2022-02-18T15:05:09.887" Score="3" ViewCount="189" Body="&lt;p&gt;I'm collecting additional enrichment content for my calculus students.  I'm looking for examples of equations that are used in various fields, but which can be derived at least somewhat straightforwardly from elementary calculus.  It's okay if there is a minor additional concept involved or requires some basic differential equation-like manipulation.&lt;/p&gt;&#xA;&lt;p&gt;Here's my current set, and I would love to hear what others could be presented!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Geometry&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Volume of a cone: &lt;span class=&quot;math-container&quot;&gt;$v = \frac{1}{3}\pi\, r^2h$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Volume of a sphere: &lt;span class=&quot;math-container&quot;&gt;$v = \frac{4}{3}\pi\,r^3$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Grazing goat problem&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Money&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Compound interest rate equation: &lt;span class=&quot;math-container&quot;&gt;$p = p_0\,e^{rt}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;The Kelly Criterion: &lt;span class=&quot;math-container&quot;&gt;$r = \frac{bp + p - 1}{p}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Biology&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The logistics curve: &lt;span class=&quot;math-container&quot;&gt;$p = \frac{Kp_0e^{rt}}{K + p_0\left(e^{rt} - 1\right)}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Collector's Curve: &lt;span class=&quot;math-container&quot;&gt;$y = -Ke^{-\frac{x}{K}} + K$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Physics&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Kinematic equations: &lt;span class=&quot;math-container&quot;&gt;$p = \frac{1}{2}At^2 + v_0t + p_0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Kinetic energy formula: &lt;span class=&quot;math-container&quot;&gt;$E_k = \frac{1}{2}mv^2$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Escape velocity formula: &lt;span class=&quot;math-container&quot;&gt;$v = \sqrt{\frac{2Gm}{r}}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Rocket equation: &lt;span class=&quot;math-container&quot;&gt;$v = v_0 + -v_e\ln\left(\frac{m_0}{m}\right)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Newton's law of cooling: &lt;span class=&quot;math-container&quot;&gt;$q = q_\text{env} - \left(q_\text{env} - q_0\right)e^{-Kt}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Electronics&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Capacitor-based timers: &lt;span class=&quot;math-container&quot;&gt;$v = v_\text{final}\left(1 - e^{-\frac{t}{RC}}\right)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Calculating RMS voltage: &lt;span class=&quot;math-container&quot;&gt;$v_\text{RMS} = \sqrt{\frac{1}{2}}A$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I updated this list to include the actual equations for most of them.  I'm not looking for a &amp;quot;type of problem&amp;quot; but a specific equation that people in that field would recognize that can be derived.  I would appreciate answers that included specific equations that could be derived through calculus.  Some fields missing above that would be helpful include economics, chemistry, meteorology/climatology, astronomy, etc.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastEditorUserId="9594" LastEditDate="2022-02-18T22:46:22.190" LastActivityDate="2022-02-21T11:22:08.493" Title="Good Examples of Equations Derived from Elementary Calculus" Tags="|calculus|examples|differential-equations|integration|derivative|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24929" PostTypeId="2" ParentId="24920" CreationDate="2022-02-18T16:01:57.167" Score="2" Body="&lt;p&gt;My thought process says the issue here is a misunderstanding of stylistic conventions leading to improper order of operations.&lt;/p&gt;&#xA;&lt;p&gt;A fraction like &lt;span class=&quot;math-container&quot;&gt;$\frac{a+b}{c+d}$&lt;/span&gt; has implied parentheses: &lt;span class=&quot;math-container&quot;&gt;$(a+b)\div(c+d)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\frac{(a+b)}{(c+d)}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;PEMDAS tells us we do parentheses first, then our multiplication/division step. If we're canceling a factor in the numerator and denominator, it has to be after we've taken care of the parentheses. The important issues in the implied parentheses are addition/subtraction operations.&lt;/p&gt;&#xA;&lt;p&gt;If we ignore the implied parentheses, it's conceivable that PEMDAS tells us to multiply, then divide, then add:&#xA;&lt;span class=&quot;math-container&quot;&gt;$\frac{6x+1}{6}$&lt;/span&gt;&lt;br /&gt;&#xA;multiply &lt;span class=&quot;math-container&quot;&gt;$\Rightarrow 6\cdot x = 6x$&lt;/span&gt;&lt;br /&gt;&#xA;divide &lt;span class=&quot;math-container&quot;&gt;$\Rightarrow 6x\div 6 = x$&lt;/span&gt;&lt;br /&gt;&#xA;add &lt;span class=&quot;math-container&quot;&gt;$\Rightarrow x + 1 = (x+1)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;A fraction like &lt;span class=&quot;math-container&quot;&gt;$\frac{6x}{6}$&lt;/span&gt; still has the implied parentheses: &lt;span class=&quot;math-container&quot;&gt;$(6\cdot x)\div 6$&lt;/span&gt;. But the associative and commutative properties of multiplication mean it doesn't really matter when we're just using them to group factors instead of addends.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\require{cancel}(6⋅x)÷6$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=(x⋅6)÷6$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=x⋅6÷6$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=x⋅\cancelto{1}{(6÷6)}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can still partially cancel sums using the distributive property, but it doesn't neatly cancel out the whole factor.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{6x+1}{6}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=\frac{1}{6}⋅(6x+1)$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=\frac{\cancel{6}x}{\cancel{6}}+\frac{1}{6}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=x+\frac{1}{6}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;There can also be problems with division and subtraction, since order of operations is trickier there, but the concept still works if we're careful. The rule here is that division and subtraction are commutative with other divisors and subtrahends, so if we group all the factors/addends on the left and all the divisors/subtrahends on the right, we can still manipulate the problem to find identities that cancel out.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{abc}{abc}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=(abc)÷(abc)$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=a⋅b⋅c÷a÷b÷c$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=a⋅b⋅c÷c÷b÷a$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=ab\cancelto{1}{(c÷c)}÷b÷a$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=a\cancelto{1}{(b÷b)}÷a$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=a÷a$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Because a fraction like &lt;span class=&quot;math-container&quot;&gt;$\frac{abc}{abc}$&lt;/span&gt; has already effectively grouped factors on the top and divisors on the bottom, we can immediately cancel them regardless of position, since we know they can always be re-arranged as above.&lt;/p&gt;&#xA;" OwnerUserId="5758" LastEditorUserId="5758" LastEditDate="2022-02-22T00:18:53.897" LastActivityDate="2022-02-22T00:18:53.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24930" PostTypeId="1" CreationDate="2022-02-18T16:31:11.673" Score="7" ViewCount="190" Body="&lt;p&gt;In my Intro to Proofs course I have an activity (more details below) where the goal is to write a collaborative proof of a statement. What makes it work as a whole class activity is that, to prove the statement, students must first prove 4 independent lemmas. These 4 lemmas can be put together to make the final proof. This means I can initially make 4 groups, each to work on one lemma. Then, after a while, reorganize into groups of 4 where each group member had a different lemma. And these groups are tasked with creating the proof of the statement.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately the example below is the only proof I know that has this kind of naturally independent pieces. Most proofs aren't so easily compartmentalized.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;So, what are some other statements/proofs that similarly can be naturally broken into separate (but equally difficult) lemmas?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Details of existing activity:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Main statement: The constant &lt;span class=&quot;math-container&quot;&gt;$e = \sum_{n=0}^\infty \frac{1}{n!}$&lt;/span&gt; is irrational.&lt;/li&gt;&#xA;&lt;li&gt;Lemma 1: For &lt;span class=&quot;math-container&quot;&gt;$a,b \in \mathbb{Z}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$$b!\left(\frac{a}{b} - \sum_{n=0}^b \frac{1}{n!}\right) \in \mathbb{Z}.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Lemma 2: For integer &lt;span class=&quot;math-container&quot;&gt;$b&amp;gt;0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$$b!\left(e - \sum_{n=0}^b \frac{1}{n!}\right) &amp;gt;0.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Lemma 3: For integers &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; b \le n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$$\frac{b!}{n!} \le \frac{1}{(b+1)^{n-b}}.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Lemma 4: For &lt;span class=&quot;math-container&quot;&gt;$b \ge 1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$$\sum_{k=b+1}^\infty \frac{1}{(b+1)^{k-b}} = \frac{1}{b}.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Note: Please do not post solutions and/or links to solutions for this activity.&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2022-02-18T16:43:56.023" LastActivityDate="2022-02-26T09:57:33.720" Title="Proofs with Independent Parts" Tags="|undergraduate-education|proofs|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24931" PostTypeId="2" ParentId="24928" CreationDate="2022-02-18T17:36:52.293" Score="0" Body="&lt;p&gt;I think there is a lot with rates versus cumulative.&lt;/p&gt;&#xA;&lt;p&gt;For example oil production.  But also any buildup or depletion. Drugs. PFAS. Radioactivity. Etc. Pressire equalization through an orifice. The concept and derivation of half life.&lt;/p&gt;&#xA;&lt;p&gt;Edit: responding to your edit.&lt;/p&gt;&#xA;&lt;p&gt;N = N0e−λt&lt;/p&gt;&#xA;&lt;p&gt;λ=ln(2)t1/2≈0.693t1/2&lt;/p&gt;&#xA;&lt;p&gt;See also:  &lt;a href=&quot;https://courses.lumenlearning.com/physics/chapter/31-5-half-life-and-activity/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://courses.lumenlearning.com/physics/chapter/31-5-half-life-and-activity/&lt;/a&gt; (but does not contain the derivation)&lt;/p&gt;&#xA;&lt;p&gt;Note that Beers Law for UV-tranmittance (or radiaoctive shielding) are interesting examples where the variable is distance rather than time but the same effect of attenuation occurs.  (People talk about &amp;quot;tenth thickness&amp;quot; instead of half life, but same concept.)&lt;/p&gt;&#xA;" OwnerUserId="19529" LastEditorUserId="19516" LastEditDate="2022-02-21T11:22:08.493" LastActivityDate="2022-02-21T11:22:08.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24932" PostTypeId="2" ParentId="24920" CreationDate="2022-02-18T17:47:23.540" Score="1" Body="&lt;p&gt;For these examples, you could also show some extended cancellation computations where they don't have to factor but just recognize that each fraction can be written as the sum of two pieces:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ &#xA;\begin{align}&#xA;\frac{x+6}{6} &amp;amp;= \frac{x}{6} + \frac{6}{6} \\&#xA;&amp;amp;= \frac{x}{6} + 6 (\frac{1}{6}) \\&#xA;&amp;amp;= \frac{x}{6} + 1&#xA;\end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;or for the second example since they already understand the third example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{6x+1}{6} = \frac{6x}{6} + \frac{1}{6} = x + \frac{1}{6}$$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This way it's clear that &amp;quot;cancellation&amp;quot; is not some new operation with its own rules -- it is just a consequence of simpler rules we already know about fraction addition and multiplying numbers.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastEditorUserId="1230" LastEditDate="2022-02-18T17:52:59.727" LastActivityDate="2022-02-18T17:52:59.727" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24933" PostTypeId="2" ParentId="24930" CreationDate="2022-02-18T20:25:23.750" Score="5" Body="&lt;p&gt;The example you have is great! I imagine it would be really hard to actually have all parts with similar difficulty. E.g. in your example, L2 is an immediate observation, while L1 and L3 (particularly the latter) require some manipulations with &lt;span class=&quot;math-container&quot;&gt;$n!$&lt;/span&gt;. It's still great, but I would expect (I might be wrong, of course) the group solving L2 to finish much faster than other groups.&lt;/p&gt;&#xA;&lt;p&gt;Here are two examples that I think have a similar spirit, although the division of work between different parts is far from equal.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Irrationality of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A proof of &lt;span class=&quot;math-container&quot;&gt;$\pi \notin \mathbb{Q}$&lt;/span&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational#Bourbaki%27s_proof&quot; rel=&quot;nofollow noreferrer&quot;&gt;found in Bourbaki's book&lt;/a&gt; (it might be due to someone else, see one of the comments) is naturally divided into parts, somewhat similarly to your proof of &lt;span class=&quot;math-container&quot;&gt;$e \notin \mathbb{Q}$&lt;/span&gt;. Take &lt;span class=&quot;math-container&quot;&gt;$f_n(x) = \frac{1}{n!} b^n x^n (\pi-x)^n$&lt;/span&gt;. Then:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;L1: &lt;span class=&quot;math-container&quot;&gt;$\int_0^\pi f_n(x) \sin(x) dx &amp;gt; 0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;L2: &lt;span class=&quot;math-container&quot;&gt;$\int_0^\pi f_n(x) \sin(x) dx \to 0$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$n \to \infty$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;L3: If &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is a fraction &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt;, then derivatives of any order of &lt;span class=&quot;math-container&quot;&gt;$f_n$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; have integer values. &lt;em&gt;Possible hint:&lt;/em&gt; write &lt;span class=&quot;math-container&quot;&gt;$b^n x^n (\pi-x)^n$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x^n (a-bx)^n$&lt;/span&gt;, and use the symmetry between &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=\pi$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;L4: If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is any polynomial of degree &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\int_0^\pi f(x) \sin(x) dx$&lt;/span&gt; is a sum of terms &lt;span class=&quot;math-container&quot;&gt;$\pm \left[ f^{(k)}(x) \sin x \right]_0^\pi$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\pm \left[ f^{(k)}(x) \cos x \right]_0^\pi$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$k=0,1,\ldots,2n$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Here L3 and L4 together imply &lt;span class=&quot;math-container&quot;&gt;$\int_0^\pi f_n(x) \sin(x) dx \in \mathbb{Z}$&lt;/span&gt;, which contradicts L1 and L2.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Liouville's theorem and a construction of a transcendental number&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Lioville's theorem states that irrational algebraic numbers don't have efficient rational approximations. More precisely: if &lt;span class=&quot;math-container&quot;&gt;$a \in \mathbb{R}$&lt;/span&gt; is an algebraic number of degree &lt;span class=&quot;math-container&quot;&gt;$d&amp;gt;1$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$|a-\frac{p}{q}| \ge \frac{c}{q^d}$&lt;/span&gt;. The proof consists of basically three steps, and in the fourth step one can use the theorem to show that a certain number is not algebraic.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;L1: Assume that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is a root of the polynomial &lt;span class=&quot;math-container&quot;&gt;$f(x) = a_0+a_1 x+\ldots+a_d x^d$&lt;/span&gt; with integer coefficients. Assume that &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; here is the smallest possible. Then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; doesn't have rational roots.&lt;/li&gt;&#xA;&lt;li&gt;L2: Assume that &lt;span class=&quot;math-container&quot;&gt;$f(x) = a_0+a_1 x+\ldots+a_d x^d$&lt;/span&gt; has integer coefficients and &lt;span class=&quot;math-container&quot;&gt;$f(p/q) \neq 0$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$p,q \in \mathbb{Z}$&lt;/span&gt;). Then &lt;span class=&quot;math-container&quot;&gt;$|f(p/q)| \ge q^{-d}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;L3: Let &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; be any polynomial. Justify that there's a constant &lt;span class=&quot;math-container&quot;&gt;$C &amp;gt; 0$&lt;/span&gt; (depending on &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;) such that &lt;span class=&quot;math-container&quot;&gt;$|f(x)-f(a)| \le C |x-a|$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \in [a-1,a+1]$&lt;/span&gt;. &lt;em&gt;Remark:&lt;/em&gt; one doesn't even need derivatives to check that.&lt;/li&gt;&#xA;&lt;li&gt;L4: Let &lt;span class=&quot;math-container&quot;&gt;$L := \sum_{n=1}^\infty 10^{-n!}$&lt;/span&gt; (Liouville's number). Then for any &lt;span class=&quot;math-container&quot;&gt;$d \in \mathbb{N}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c &amp;gt; 0$&lt;/span&gt; there is a rational approximation &lt;span class=&quot;math-container&quot;&gt;$p/q$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$q^d |a-\frac{p}{q}| &amp;lt; c$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Here L1, L2, L3 together imply Liouville's theorem mentioned before. If &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; were algebraic (i.e., a root of a polynomial with integer coefficients), it would contradict this law.&lt;/p&gt;&#xA;" OwnerUserId="15367" LastEditorUserId="15367" LastEditDate="2022-02-26T09:57:33.720" LastActivityDate="2022-02-26T09:57:33.720" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24934" PostTypeId="1" CreationDate="2022-02-19T00:19:41.603" Score="-2" ViewCount="184" Body="&lt;p&gt;Question: Is there such a thing as an Applied Maths profession?&lt;/p&gt;&#xA;&lt;p&gt;In my engineering experience we were often at a loss as to which mathematical approach to take, and there seemed no obvious way to find out except by an informed empirical approach, haphazard and inefficient. The mathematicians and professors we consulted seemed inward-looking, although each had their deep specialities which they were naturally keen to fit to the problem, sadly with poor results when we tried it.&#xA;What we needed was an Applied Maths guy, with a wide, perhaps shallow, education who could point us in the right direction, using experience, intuition, insider knowledge, a contacts network and good luck.&#xA;The advice we could get came mostly from fellow engineer CTOs, whose maths speciality had been crucial for success. However, if our product research stepped outside their experience, there seemed no-one to turn to.&lt;/p&gt;&#xA;&lt;p&gt;Real-world engineering problem-solving presents maths challenges in a wide-ranging, uncontrolled manner. The engineer needs an index to the world of maths, hoping for a handful of potentially useful approaches to try. However, the taxonomy of maths is apparently not organised by application. It does not say, 'If you have mostly straight lines, then use this (high-school) stuff; but if it's photogrammetric construction, you'll need post-grad spherical geometry at least. There is no user interface.&lt;/p&gt;&#xA;&lt;p&gt;The maths profession teaches and studies maths, but does not seem to provide a service that outsiders could use to access a few potentially fertile approaches. This makes applying maths inefficient - one either gets lucky, maybe having studied the right thing, or is in for a long haul. Hiring a mathematician is the obvious path to take, but which speciality do we advertise for?&lt;/p&gt;&#xA;&lt;p&gt;My intuition is that something analogous to the electronic industry's Application Engineer is needed. Without these guys being deployed by each chip and component maker, supported by endless application notes 'Use this chip to measure infra-red', working engineers would never have found chips etc to solve their problems.&lt;/p&gt;&#xA;&lt;p&gt;So, who is out there, actually promoting maths 'components' as solutions to users' questions, on your behalf, so that stuff gets applied in the real world?&lt;/p&gt;&#xA;" OwnerUserId="19530" LastActivityDate="2022-02-19T14:08:53.237" Title="Is applying maths a 'hard problem' and how does the profession help?" Tags="|problem-solving|applied-mathematics|inverse-problems|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="24935" PostTypeId="2" ParentId="24920" CreationDate="2022-02-19T10:24:51.400" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;What is the correct student-understandable feedback for the first two errors?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;Cancelling means dividing the numerator and the denominator by the same number. What do you get if you divide the whole term &lt;span class=&quot;math-container&quot;&gt;$6x+1$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;?&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Either the student realizes the mistake or you are now at the true core of the problem: the student doesn't remember how sums are divided and this topic must be covered again.&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2022-02-20T11:08:06.327" LastActivityDate="2022-02-20T11:08:06.327" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24937" PostTypeId="2" ParentId="24934" CreationDate="2022-02-19T13:11:47.337" Score="-1" Body="&lt;p&gt;There is a field called applied math (degrees given).  Sometimes it is just a &amp;quot;concentration&amp;quot; of a relevant math degree, more often undergrad.*  But even here, it just means the kids took a few more applied courses and a few less pure.  [But both pure and applied math are big topics, and kids have limited slots in their schedules, so it's not like either one covers the waterfront.]  And that they (more) try to get a job in industry.  But they don't turn into consulting mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;I don't think there are consulting mathematicians, the way there are consulting statisticians.  But even here, don't be surprised how much they specialize by field (e.g. health statistics).  Or that a lot of what they do, does not mean knowing the biggest catalogue of fancy techniques of stats, but more about how to avoid/correct typical problems in study design or interpretation.&lt;/p&gt;&#xA;&lt;p&gt;All that said, I think part of being a thoughtful knowledge worker or manager in the modern era is the ability to find the particular expertise you need.  I mean, I don't know details of market research, law, financial modeling.  But I know enough to recognize situations and access experts.  And even to figure out what subfield is relevant and make sure the expert can handle the subfield.  And even if I don't know the subfields, can educate myself to the extent of recognizing a situation and determining the right sort of expert needed.  And then tracking him down.  And paying him.  Or hiring him, full time.**&lt;/p&gt;&#xA;&lt;p&gt;For example, the problem you described here requires a projection geometry.  So fine, get someone who knows that.  But don't expect every math or applied math person to be skilled at it.  That's like expecting every random mechE to know a very particular issue (say expansion joints/anchors in piping).  They won't all know everything.  You should be capable of Googling or just talking to people (telephone game) to get to the right sort of expertise.  Heck, I did this before the Internet!&lt;/p&gt;&#xA;&lt;p&gt;I think a lot of your issue is that you (sound to be) in computer code writing, not traditional engineering.  And coders are even more mathematically weak than physical scientists and engineers.  Engineers in construction, oil production, aerospace, chip design, etc. etc. are capable of finding help.  They very often may not be very mathematical themselves (especially after graduation, in normal tasks) but they are at least aware of colleagues who are more mathematical and can advise on the correct approach, formula, etc. when they (very infrequently) deal with something mathematical.&lt;/p&gt;&#xA;&lt;p&gt;What I've seen in structural mechanical programs (finite element modeling) and in molecular modeling (e.g. Gaussian) is that software companies hire at least part of their development resources from appropriate math (or even just mathematically astute science/engineering).  That way, they don't just have a bunch of coders on staff who don't know the needed math/science for what the program does.&lt;/p&gt;&#xA;&lt;p&gt;I do agree with your issues talking to professors.  They all want to do &amp;quot;their stuff&amp;quot; and get grants to do it.  It can be very unlikely to find someone who has exactly what you need, for a consulting contract.  And even then, they are more used to writing papers and trying to get promoted to full professor and the like than in solving real problems for a company.  It wasn't always like this, but academia and academic math/science has become very specialized as fields have advanced.  And the high availability of mass research funding has allowed this to occur.  Not saying there are not some outliers, but that's not the norm.  And it's a common problem that companies have dealing with professors (not even wanting to hear your problem, just tell you their research).  It's why some universities set up industry outreach groups or FFRDCs, because they know their regular faculty doesn't work well with industry.&lt;/p&gt;&#xA;&lt;p&gt;Really, I think your best bet is just to hire a few smart, willing to work, wanting to get out of academia kids who can figure out the raytracing for you and learn enough of the code monkey stuff also.  (And you have to give them some time, too.  Don't expect manna from heaven.  They have to figure it out, also.)  It's not an immediate solution, but it's how you build a workforce that can take care of your needs.&lt;/p&gt;&#xA;&lt;p&gt;[Edit:  loved the comment about physicists...get some washed out of academia, but hungry one.  There are many of them.  Even if he/she/thing doesn't understand the projection geometry, he'll go learn it.  Your &amp;quot;didn't want to take ODEs&amp;quot; computer scientist grads won't.]&lt;/p&gt;&#xA;&lt;p&gt;Note* and Note**: That you didn't have at least a &amp;quot;Googled and read stuff&amp;quot; level of awareness of what universities offer that they call &amp;quot;applied math&amp;quot; before running to a SE with a forum question, worries me.  Shows similar lack of initiative as with finding appropriate consultants/employees to fix your code/business problems.&lt;/p&gt;&#xA;" OwnerUserId="19516" LastEditorUserId="19516" LastEditDate="2022-02-19T13:27:15.600" LastActivityDate="2022-02-19T13:27:15.600" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24938" PostTypeId="2" ParentId="24934" CreationDate="2022-02-19T14:08:53.237" Score="0" Body="&lt;p&gt;I don't think there is any existing service for connection mathematical generalists with people in need of applied mathematical assistance in the way you suggest.  Might be a good start up in and of itself!&lt;/p&gt;&#xA;&lt;p&gt;For your immediate problem I would suggest formulating a &amp;quot;minimum viable problem&amp;quot;:  strip away all of the non-essentials and formulate the easiest version of the mathematical problem you are trying to solve.&lt;/p&gt;&#xA;&lt;p&gt;Post this problem to math.stackexchange.  If you get interesting or insightful replies, you could reach out to those people to do some consulting work for you.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-02-19T14:08:53.237" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24939" PostTypeId="2" ParentId="24920" CreationDate="2022-02-19T18:42:38.187" Score="3" Body="&lt;p&gt;I disagree with all your options because it is not solving the &lt;strong&gt;real&lt;/strong&gt; issue, which is partly alluded to by &lt;em&gt;BravoMath&lt;/em&gt;. This error only arises with students who were never &lt;strong&gt;correctly taught&lt;/strong&gt; the true &lt;strong&gt;meaning of division&lt;/strong&gt; &lt;em&gt;and&lt;/em&gt; the meaning and purpose of &lt;strong&gt;precedence rules&lt;/strong&gt;, so that is what you will have to do first, to solve the issue properly, and permanently. And of course the student must be taught the true meaning of multiplication before that, but I shall omit that from this post otherwise it would be too long.&lt;/p&gt;&#xA;&lt;p&gt;~ ~ ~&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Meaning of division&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The meaning of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x ÷ y$&lt;/span&gt;&amp;quot; is the quantity &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$x = y × z$&lt;/span&gt;. If there is no such &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;, then &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x ÷ y$&lt;/span&gt;&amp;quot; is &lt;em&gt;meaningless&lt;/em&gt;. For example, &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$6 ÷ 2$&lt;/span&gt;&amp;quot; means &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$6 = 2 × 3$&lt;/span&gt;, and the meaning of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$1 ÷ 5$&lt;/span&gt;&amp;quot; is the quantity &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$1 = 5 × s$&lt;/span&gt;. For a concrete example of the latter, if you have &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; pizza and divide it by &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;, the result is exactly the amount of pizza such that &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; times that is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; pizza. Moreover, &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$1 ÷ 0$&lt;/span&gt;&amp;quot; is meaningless, because there is no &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$1 = 0 × z$&lt;/span&gt;. At this level (elementary school), division means &lt;strong&gt;nothing else&lt;/strong&gt;. (Do not even think about multiplicative inverses.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Precedence rules&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;We need to communicate what we mean accurately to another person. This involves rules that we &lt;em&gt;choose&lt;/em&gt; and &lt;em&gt;agree to use&lt;/em&gt;. It is like natural language, where we both agree that if I say &amp;quot;A and B came&amp;quot; I am telling you that &amp;quot;A came&amp;quot; and also that &amp;quot;B came&amp;quot;. Why? It's simply an agreement on a shared language, so that we know what each other is talking about. Mathematics is exactly the same; we agree on rules for what an arithmetic expression like &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$1+2×3$&lt;/span&gt;&amp;quot; means. The rules have to tell us exactly what operations to do, on what, in what order. It is &lt;strong&gt;very important to explain&lt;/strong&gt; all the &lt;em&gt;non-mathematical&lt;/em&gt; things about agreement that I have said here!&lt;/p&gt;&#xA;&lt;p&gt;(Note that you also need to teach what juxtaposition means and the rules for interpreting it. At this level, you can just say that you should insert multiplication symbols in-between juxtaposed numbers before reading it. But privately you should keep in mind that in actual mathematical writing it is not correct. Consider &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\cos 2x$&lt;/span&gt;&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\prod_{k=1}^n f(k) g(k)$&lt;/span&gt;&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\sum_{k=1}^n f(k) · \sum_{m=1}^n g(m)$&lt;/span&gt;&amp;quot;. In case you are interested, one rule that seems to work for almost all conventions is that juxtaposition has higher precedence than everything else except brackets and postfix operations.)&lt;/p&gt;&#xA;&lt;p&gt;~ ~ ~&lt;/p&gt;&#xA;&lt;p&gt;In my teaching experience, not one student who understood the above two basic concepts ever made the mistakes you mentioned in your post (apart from careless errors). Why? Because the meaning of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{x+6}{6}$&lt;/span&gt;&amp;quot; is &lt;strong&gt;by definition&lt;/strong&gt; the quantity &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$6 × z = x+6$&lt;/span&gt;. Certainly the student knows that &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$6 × x$&lt;/span&gt;&amp;quot; is not the same as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x+6$&lt;/span&gt;&amp;quot;, so they cannot &lt;em&gt;believe&lt;/em&gt; the wrong answer. Similarly, &lt;span class=&quot;math-container&quot;&gt;$\frac{6x+1}{6}$&lt;/span&gt; is the quantity &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$6 × w = 6x+1$&lt;/span&gt;. Again, the student knows (from proper teaching of multiplication) that &lt;span class=&quot;math-container&quot;&gt;$6 × (x+1) = ( 6 × x ) + 6$&lt;/span&gt;, which is obviously different from &lt;span class=&quot;math-container&quot;&gt;$6 × x + 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The point is, with the above definition of division, there is &lt;strong&gt;simply no cancellation&lt;/strong&gt;. &lt;span class=&quot;math-container&quot;&gt;$\frac{6x}{6} = x$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$6x = 6 × x$&lt;/span&gt;, and for &lt;strong&gt;no other reason&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2022-02-19T18:42:38.187" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24940" PostTypeId="2" ParentId="24920" CreationDate="2022-02-19T21:04:24.340" Score="1" Body="&lt;p&gt;Make them recall what &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; means: substitute values (they do it, not you) and see that it's wrong. But this has already been said.&lt;/p&gt;&#xA;&lt;p&gt;Another thing is use words. When students don't know how to read formulas, use plain, long, boring sentences: &amp;quot;take a number, add 6 to it and then divide the total by 6&amp;quot;. They must be able to convert from sentences to formulas and viceversa. If they can't do that easily everything that follows will only be great frustration.&lt;/p&gt;&#xA;&lt;p&gt;The motivation for this is: &lt;a href=&quot;https://blogs.ams.org/matheducation/2019/09/30/is-there-a-switch-for-making-sense/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://blogs.ams.org/matheducation/2019/09/30/is-there-a-switch-for-making-sense/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19533" LastActivityDate="2022-02-19T21:04:24.340" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24941" PostTypeId="2" ParentId="24928" CreationDate="2022-02-19T22:22:14.227" Score="1" Body="&lt;p&gt;Electric field due to a charged rod of length &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;. If the charged rod is placed from &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x=L$&lt;/span&gt; then we ask what the electric field due to the uniformly charged rod at &lt;span class=&quot;math-container&quot;&gt;$(x,0)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt;L$&lt;/span&gt;. Coulomb's Law for &lt;span class=&quot;math-container&quot;&gt;$dQ$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$(z,0)$&lt;/span&gt; gives&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ dE = \frac{1}{4\pi \epsilon_o}\frac{dQ}{(x-z)^2}$$&lt;/span&gt;&#xA;for &lt;span class=&quot;math-container&quot;&gt;$0 \leq z \leq L$&lt;/span&gt;. Assuming uniform charge density &lt;span class=&quot;math-container&quot;&gt;$dQ = \frac{Q}{L}dz$&lt;/span&gt; hence the total electric field directed in the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-direction at &lt;span class=&quot;math-container&quot;&gt;$(x,0)$&lt;/span&gt; is:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ E = \int dE = \frac{Q}{4\pi \epsilon_oL}\int_0^L\frac{dz}{(x-z)^2} = \frac{Q}{4\pi \epsilon_oL}\left[ \frac{1}{x-L}-\frac{1}{x}\right] = \frac{1}{4\pi \epsilon_o}\frac{Q}{x(x-L)}$$&lt;/span&gt;&#xA;When &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt;&amp;gt; L$&lt;/span&gt; notice &lt;span class=&quot;math-container&quot;&gt;$E \sim \frac{1}{4\pi \epsilon_o}\frac{Q}{x^2}$&lt;/span&gt;. There are many such examples to be gleaned from introductory electrostatics.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2022-02-19T22:22:14.227" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24942" PostTypeId="1" CreationDate="2022-02-20T04:32:32.067" Score="9" ViewCount="680" Body="&lt;p&gt;So, I was minding my own business and I thought I had defined inverse cotangent in the natural fashion. In particular, we define inverse tangent as the inverse of tangent restricted to &lt;span class=&quot;math-container&quot;&gt;$(-\pi/2, \pi/2)$&lt;/span&gt;. We all know this. So, I thought, &lt;span class=&quot;math-container&quot;&gt;$y = \cot(x)$&lt;/span&gt; has vertical asymptotes at &lt;span class=&quot;math-container&quot;&gt;$x = n \pi$&lt;/span&gt; for any integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; so surely the natural domain to restrict cotangent for inversion would be &lt;span class=&quot;math-container&quot;&gt;$(0, \pi)$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; \cot^{-1}(x) &amp;lt; \pi$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\lim_{ x\rightarrow \infty} \cot^{-1}(x) = 0$&lt;/span&gt; whereas &lt;span class=&quot;math-container&quot;&gt;$\lim_{x \rightarrow -\infty} \cot^{-1}(x) = \pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then I made the mistake of checking my work. I found that I have not followed the apparent proper path and inverted cotangent across its asymptote at &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;. Apparently, &lt;span class=&quot;math-container&quot;&gt;$cot^{-1}$&lt;/span&gt; is understood as the inverse for cotangent restricted to &lt;span class=&quot;math-container&quot;&gt;$(-\pi/2,0) \cup (0, \pi/2]$&lt;/span&gt; which leads to the graph of inverse cotangent having a hideous discontinuity as it jumps from &lt;span class=&quot;math-container&quot;&gt;$-\pi/2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\pi/2$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://mathworld.wolfram.com/InverseCotangent.html&quot; rel=&quot;noreferrer&quot;&gt;Mathword at Wolfram&lt;/a&gt; where the offensive (to my taste) definition is discussed at length. I guess this goes back to Euler. However, I also noticed I am not alone as the website above says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A different but common convention (e.g., Zwillinger 1995, p. 466; Bronshtein and Semendyayev, 1997, p. 70; Jeffrey 2000, p. 125) defines the range of &lt;span class=&quot;math-container&quot;&gt;$\cot^{-1}x$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$(0,\pi)$&lt;/span&gt;, thus giving a function that is continuous on the real line R. Extreme care should be taken where examining identities involving inverse trigonometric functions, since their range of applicability or precise form may differ depending on the convention being used.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question is simply this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Which definition of inverse cotangent do you favor in your teaching ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2022-02-20T19:30:27.043" LastActivityDate="2022-02-22T02:01:43.827" Title="What's the Deal with Inverse Cotangent?" Tags="|undergraduate-education|trigonometry|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24943" PostTypeId="2" ParentId="24942" CreationDate="2022-02-20T05:06:54.330" Score="-3" Body="&lt;p&gt;Not 100% sure if I'm following you, but if I am, the former.  All it is is the cot function rotated 90 degrees.  Graph it and you'll see.  Just the arccot has horizontal asymptotes and the cot has vertical ones.&lt;/p&gt;&#xA;&lt;p&gt;And I'm fine with allowing the definition to include multiples of 2pi for either of them.  So what if it's a relation instead of a function?&lt;/p&gt;&#xA;&lt;p&gt;Don't have to restrict it to the primary interval.  If you do, sure, have a discontinuity.  Who cares... I mean regular cot and tan do funky things when sin or cos become zero.  So what if their inverses do also.&lt;/p&gt;&#xA;&lt;p&gt;Just graph it manually and I think you'll feel better about it.   Never bothered me.  And this is how both books I have show the graphs. Lyman Kells et al Plane and Spherical Trig.  And Frank Ayres Schaum's Outline for First Year College Math.&lt;/p&gt;&#xA;" OwnerUserId="19535" LastEditorUserId="19535" LastEditDate="2022-02-20T05:16:38.903" LastActivityDate="2022-02-20T05:16:38.903" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24944" PostTypeId="1" AcceptedAnswerId="25010" CreationDate="2022-02-20T12:55:02.410" Score="4" ViewCount="278" Body="&lt;p&gt;I am a 17-year-old secondary school student from India studying math, physics, and other subjects. I lag behind other classmates only because I work slowly when doing addition, subtraction, multiplication, and division. Others are adept at working with numbers, adding, subtracting, multiplying, and dividing in a matter of seconds in their heads. But I have to follow the traditional way as I lack the mental ability. How do I become faster?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I am a 17-year-old secondary school student from India studying math, physics, and other subjects. I am currently in grade 11. I am preparing for competitive exams for engineering colleges.&lt;/li&gt;&#xA;&lt;li&gt;I am math dyslexic or dyscalculic. It is a big problem for me. I have to spend a huge amount of my time on the basic arithmetic operations involved in physics and mathematics courses. I lag behind other classmates only because I work slowly when doing addition, subtraction, multiplication, and division.&lt;/li&gt;&#xA;&lt;li&gt;Calculators are strictly prohibited here. Solving maths becomes tedious and consumes a huge amount of time. I lost all my energy doing the basic operations alone.&lt;/li&gt;&#xA;&lt;li&gt;My other classmates can easily and quickly handle the numbers.&lt;/li&gt;&#xA;&lt;li&gt;I get less scores in timed tests and examinations in school despite I know all topics in physics and mathematics.&#xA;I don't find the time sufficient to solve the sums during the tests. I'm finding it tougher to complete my studies and am losing interest because of this. This is highly demotivating.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I have to count the numbers like 6+5 with my fingers to add them up to 11.&lt;/li&gt;&#xA;&lt;li&gt;I spend almost a minute solving something like 965×532.&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{aligned}965\\ \dfrac{\times 532}{1930}\\ 28950\\ \dfrac{482500}{513380}\end{aligned}&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;I also have trouble remembering multiplication tables. I once again have to follow the traditional method of multiplication like a 7-year-old kid.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user19536" LastEditorDisplayName="user19536" LastEditDate="2022-02-22T04:33:42.507" LastActivityDate="2022-03-07T22:44:04.043" Title="Math dyslexia is a big problem for me. I lag behind my classmates" Tags="|secondary-education|arithmetic-operations|mental-math|time-management|india|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24945" PostTypeId="1" CreationDate="2022-02-21T03:11:52.497" Score="3" ViewCount="89" Body="&lt;p&gt;I have been trying the &lt;a href=&quot;https://matheducators.stackexchange.com/a/3/11702&quot;&gt;think-pair-share&lt;/a&gt; technique in my undergraduate linear algebra class of about 30 students. Some students quite enjoy the experience. But I noticed that some other students simply use the time to check their cellphones or do what-ever they want.&lt;/p&gt;&#xA;&lt;p&gt;I am picking a random student to share the answer now. But since we have a large class, it is unlikely that one is chosen. So many of them just take the chance and ignore the exercise.&lt;/p&gt;&#xA;&lt;p&gt;Should I pick the students who clearly do not work on the problem instead?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorUserId="11" LastEditDate="2022-02-21T17:51:31.337" LastActivityDate="2022-02-21T17:51:31.337" Title="How to make student work in think-pair-share?" Tags="|undergraduate-education|classroom-management|classroom-interaction|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24946" PostTypeId="1" CreationDate="2022-02-21T12:41:38.393" Score="3" ViewCount="173" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I want to introduce my students to some big names in mathematics, and one of the names I want to bring up is Richard Borcherds, known for his contributions to the fields of number theory, group theory, and abstract algebra.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;EDIT: I should have mentioned. This is a group of students who have expressed an interest in pursuing math as their major or minor in higher education.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;As a part of this introduction, I want to introduce the students to these three fields as well. Number theory and group theory I have covered. But I'm looking for some nice ways to introduce abstract algebra to someone who hasn't gotten that far in their education yet.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Are there any ways I can relate abstract algebra to the pre-calc algebra they already know?&lt;/p&gt;&#xA;&lt;p&gt;What are some good examples of real-life problems that can be solved with abstract algebra?&lt;/p&gt;&#xA;&lt;p&gt;Even with the phasing out of visualizing things, in favor of axioms and abstractions, are there any examples of things that &lt;em&gt;can&lt;/em&gt; be visualized?&lt;/p&gt;&#xA;" OwnerUserId="796" LastEditorUserId="796" LastEditDate="2022-02-21T14:17:36.837" LastActivityDate="2022-03-01T06:39:24.453" Title="What is a good way to broach the subject of abstract algebra, to a student in Calc 1, or pre-calc?" Tags="|abstract-algebra|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24947" PostTypeId="2" ParentId="24946" CreationDate="2022-02-21T13:33:41.920" Score="-4" Body="&lt;p&gt;I think it's nice to be able to give some knowledge of further courses in math, of a very &amp;quot;who's who in the zoo&amp;quot; sort.  By which I mean descriptive, not axiomatic, and not even a detailed or working description.  Just some extremely light and vague concept to attach to a word.  And then let it go.&lt;/p&gt;&#xA;&lt;p&gt;Not sure why you are pushing this guy, Richard Borcherds.  I've never heard of him.  I trust you that he's a big wheel in some way and interesting to people further advanced in their studies.  But I think it's &amp;quot;A Bridge Too Far&amp;quot; for high school.  Be a little wary of sharing your passion, what interests YOU, with what is good for your students.  I think you're better off showing them the Horizon video on Andrew Wiles.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.dailymotion.com/video/x223gx8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.dailymotion.com/video/x223gx8&lt;/a&gt; (probably copyright violating, but you can either buy a physical copy of the video or license or...just decide to steal it.)&lt;/p&gt;&#xA;&lt;p&gt;In terms of awareness of math topics fields, I actually think what's closer to interest to them is something that talks about the terms and how they connect to school courses, further down the line.  Not so much, where's active research going on.  (Yeah, I get that you probably are interested in coal face of research...but remember what I said about pushing your interests versus thinking about audience.)  We get questions here all the time about &amp;quot;what comes after Calculus BC&amp;quot; (US).  Showing a lack of understanding even of 3rd semester calculus and ODEs, that most engineers go on to take a year later.&lt;/p&gt;&#xA;&lt;p&gt;I think for AA, just knowing that it's like high school algebra but more...uh...abstract is kind of enough.  Tell them that it's a core course that math majors take in undergrad, but engineers and scientists almost always don't...don't need it.  And that it's related to how they proved that 5th level and higher polynomials don't have a &amp;quot;quadratic formula&amp;quot;-style solution. [This connects to high school algebra and teases that the topic actually did something.]  I include GT within AA.  You really don't need to get complicated enough to differentiate.&lt;/p&gt;&#xA;&lt;p&gt;And then, let it go.  Teach them the pre-calc they need to know.  Honest, even just knowing the (names of, not really even the content) basic future math courses in undergrad for science/engineering (who are much more numerous than math majors) and math majors, is an above and beyond enrichment.  Then get back to teaching the actual topic you're supposed to.&lt;/p&gt;&#xA;" OwnerUserId="19544" LastActivityDate="2022-02-21T13:33:41.920" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24948" PostTypeId="2" ParentId="24942" CreationDate="2022-02-21T18:24:38.927" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SqMbR.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SqMbR.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is from Algebra and Trigonometry by Paul Foerster. His definition has Cot following Cos, which to me is perfectly logical, and what we use in my HS.&lt;/p&gt;&#xA;&lt;p&gt;Respectfully, Wolfram is no more an authority than Foerster. It seems we agree, and if a student approached me regarding this, I'll stick with &amp;quot;apparently, there are multiple conventions. you need to use what your teacher has stated in their class.&amp;quot; Add this to the list. (Deprecating septagon and Pluto were the first two on my long list of changes between college and working in a HS nearly 30 years later.)&lt;/p&gt;&#xA;&lt;p&gt;Edit: when I engaged in a related conversation, I was led to &lt;a href=&quot;https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;This Document&lt;/a&gt; which offers -&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wDrYz.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wDrYz.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And at least was written by a body with some authority. If they disagreed with us, I'd have to rethink my strongly held belief. ISO is a well recognized organization offering agree upon standards.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-02-21T18:36:53.093" LastActivityDate="2022-02-21T18:36:53.093" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24949" PostTypeId="2" ParentId="24942" CreationDate="2022-02-22T01:48:47.650" Score="8" Body="&lt;p&gt;For teaching, I think this situation gives a good excuse to talk about how in mathematics, sometimes we get to embrace ambiguity.&lt;/p&gt;&#xA;&lt;p&gt;One thing that differentiates the two definitions is that they satisfy different &amp;quot;functional equations.&amp;quot; For one, we have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\hbox{arccot}(x)=\displaystyle{\frac{\pi}{2}}-\arctan(x)$$&lt;/span&gt; while for the other, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\hbox{arccot}(x)=\arctan\left(\frac{1}{x}\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Both are attractive. The second is an odd function, the first is not.&lt;/p&gt;&#xA;&lt;p&gt;The second is not continuous at the origin, which many people find disturbing (&amp;quot;offensive&amp;quot; in the OP), while the first is not continuous at infinity.&lt;/p&gt;&#xA;&lt;p&gt;One nice thing about the second one is that the branch cut for  &lt;span class=&quot;math-container&quot;&gt;$f(z)=\hbox{arccot}(z)$&lt;/span&gt; is compact and confined: it goes along the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis from &lt;span class=&quot;math-container&quot;&gt;$-i$&lt;/span&gt; to  &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;. The branch cut for the first wanders out to infinity. If I were to be offended by one or the other, I would be offended by the ugly branch cut for the first. I think that the compact branch cut is more aesthetic, but we have to trade this for continuity at the origin, caused by the fact that the origin is right in the middle of the compact branch cut.&lt;/p&gt;&#xA;&lt;p&gt;When teaching this in calculus, I refer to the &amp;quot;debate&amp;quot; and suggest to students that this is one of the things that makes mathematics interesting. I mention briefly how it connects to complex numbers. I use different computation engines to evaluate &lt;span class=&quot;math-container&quot;&gt;$\hbox{arccot}(-1)$&lt;/span&gt;. Some will give &lt;span class=&quot;math-container&quot;&gt;$3\pi/4$&lt;/span&gt; and others &lt;span class=&quot;math-container&quot;&gt;$-\pi/4$&lt;/span&gt;. You can show how both make sense by looking at the unit circle. When teaching complex variables, much more time can be spent on this issue, and it's really fun!&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2022-02-22T02:01:43.827" LastActivityDate="2022-02-22T02:01:43.827" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24950" PostTypeId="1" AcceptedAnswerId="24955" CreationDate="2022-02-22T02:08:26.153" Score="15" ViewCount="2269" Body="&lt;p&gt;Function composition means, roughly, taking the output of a function and applying it to the input of another function. If we define an object C to represent this operation, we could say &lt;span class=&quot;math-container&quot;&gt;$C(f,g) = f∘g$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$C(f,g)(x) = f(g(x))$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Say we had a similar but slightly different operation &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; (for “followed by”) such that &lt;span class=&quot;math-container&quot;&gt;$F(f,g) = g∘f$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$F(f,g)(x) = g(f(x))$&lt;/span&gt;. We could make a symbol for this operation, maybe &lt;span class=&quot;math-container&quot;&gt;$→$&lt;/span&gt;, so that what we wrote before reads as &lt;span class=&quot;math-container&quot;&gt;$F(f,g) = f → g$&lt;/span&gt;. This could read as “&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; followed by &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;” or “&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and then &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;”.&lt;/p&gt;&#xA;&lt;p&gt;My suspicion is that &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is a much more natural and intuitive operation than &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, since (in English) we read from left to right, so seeing &lt;span class=&quot;math-container&quot;&gt;$f*g$&lt;/span&gt; feels more like &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;-then-&lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; than &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;-then-&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. My second suspicion is that &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; are entirely equivalent and translating between use of one and the other is trivial. To my knowledge and experience, composition is a widely used operation, but followed-by is not. Why?&lt;/p&gt;&#xA;&lt;p&gt;Are there certain branches of maths, specific proofs, or particular applications that benefit by using &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;? Is anything known about the origins of this notation? Is followed-by actually used in some context I might not be familiar with? As followed-by is much more intuitive to me and makes certain problems a lot more obvious, might it be a good idea, every time I see &lt;span class=&quot;math-container&quot;&gt;$f∘g$&lt;/span&gt; to replace it with &lt;span class=&quot;math-container&quot;&gt;$g→f$&lt;/span&gt; or are there cases where that could lead me astray or make things more complicated?&lt;/p&gt;&#xA;" OwnerUserId="6771" LastEditorUserId="75" LastEditDate="2022-02-23T11:23:48.057" LastActivityDate="2022-02-24T21:04:32.777" Title="Why do we use functional composition in the order we do?" Tags="|mathematical-pedagogy|notation|functions|" AnswerCount="4" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="24951" PostTypeId="2" ParentId="24950" CreationDate="2022-02-22T04:33:07.440" Score="14" Body="&lt;p&gt;In general applications, defining the evaluation rule of &lt;span class=&quot;math-container&quot;&gt;$f\circ g$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$(f\circ g)(x)=f(g(x))$&lt;/span&gt;  has a lower &lt;a href=&quot;https://en.wikipedia.org/wiki/Cognitive_load#Extraneous&quot; rel=&quot;noreferrer&quot;&gt;extraneous cognitive load&lt;/a&gt; than the way you suggest.  As a sample of how inconveniencing that can be, go ahead and proofread the two following sequences of equations to see if there is a mistake in my exposition.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$((f\circ g)\circ h)(x)=h((f\circ g)(x))=h(g(f(x)))=(g\circ h)(f(x))=(f\circ(g\circ h))(x)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$((f\circ g)\circ h)(x)=(f\circ g)(h(x))=f(g(h(x)))=f((g\circ h)(x))=(f\circ(g\circ h))(x)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Esoteric rules of symbolic manipulation aren't bad in and of themselves, especially if it ultimately leads to a clearer understanding of the underlying mathematical structures.  But I don't think that's the case here: as far as I can see, all you get is something that is easier to translate into English step-by-step.  But IMO, the strength of mathematical symbolic language is that it offers efficiency and clarity away from the vagaries of natural language.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;That being said, there are subfields of mathematics where that cognitive load is more intrinsic than extraneous and therefore.  For instance, when studying permutation groups, it is unclear whether the product &lt;span class=&quot;math-container&quot;&gt;$fg$&lt;/span&gt; of two permutations is most naturally understood as &amp;quot;do &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; first and then &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;&amp;quot; or symbolically as &lt;span class=&quot;math-container&quot;&gt;$(fg)(x)=f(g(x))$&lt;/span&gt;.  As the comments suggest, I am given to believe that some old-school authors in abstract and linear algebra used postfix functional notation to establish the clarity.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2022-02-22T04:33:07.440" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="24952" PostTypeId="2" ParentId="7963" CreationDate="2022-02-22T06:31:17.090" Score="0" Body="&lt;h2&gt;Preface&lt;/h2&gt;&#xA;&lt;p&gt;This question is a bit old, so maybe I should just leave it be, but it intrigued me as I'd never seen points assigned like this. As I've never seen points assigned like this, I can't comment on common scoring systems, but I can come up with simple systems as you've emphasized in the question.&lt;/p&gt;&#xA;&lt;p&gt;Of possible note, I'm unable to access the Harvard scoring system linked in the question. It gives me an error that I'm not authenticated, which likely means I need a student or faculty account at Harvard to access it, which I don't have.&lt;/p&gt;&#xA;&lt;h2&gt;The Simple Function&lt;/h2&gt;&#xA;&lt;p&gt;Let:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; be the normal score for the question,&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$max$&lt;/span&gt; be the maximum score a hard question can be worth,&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; be the multiplier for the maximum score, so &lt;span class=&quot;math-container&quot;&gt;$M=\frac{max}{n}$&lt;/span&gt;,&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; be the additive portion of the multiplier, so &lt;span class=&quot;math-container&quot;&gt;$m=M-1$&lt;/span&gt;,&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; be the p-value as a percent of students who got the question right,&lt;br /&gt;&#xA;and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; be the actual value of the question after p-values are considered.&lt;/p&gt;&#xA;&lt;p&gt;Then use:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n(1+m-mp)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which gives a range of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$M\cdot n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then you can set &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; to something reasonable, like 1.5, and a student who just answers the hardest questions can only get 50% more points than a student who just answers the easiest questions.&lt;/p&gt;&#xA;&lt;h2&gt;Existing Scoring Methods&lt;/h2&gt;&#xA;&lt;p&gt;I've seen points assigned by subjective difficulty, or by length of time taken (which is still somewhat subjective, but can be measured for an &amp;quot;average&amp;quot; case). The idea being that you get a fairly constant number of points based on the effort or time spent, regardless of which questions you worked on.&lt;/p&gt;&#xA;&lt;p&gt;For example, if questions 1-10 take one minute each, and questions 11-15 take five minutes each, we could assign 3 points each for the first 10, and 15 points each for the last 5. A student who answers all questions should spend about 35 minutes and could get up to 105 points (which could be capped, allowing a mistake or two while still getting a perfect score, or not). That gives 20 extra minutes for the slow students, etc.&lt;/p&gt;&#xA;&lt;p&gt;A person who only spends 20 minutes on the test could answer all the short questions and two long questions for 60 points, or they could answer four long questions for the same 60 points. Etc.&lt;/p&gt;&#xA;&lt;h2&gt;Points by Percent Answered Correctly&lt;/h2&gt;&#xA;&lt;p&gt;If the goal is to objectively assign difficulty using the percentage of students who answered the question correctly, the trick is to find a function that assigns the range 0-1 in a way you find fair.&lt;/p&gt;&#xA;&lt;p&gt;Your point system says the value, v, of an answer is given by &lt;span class=&quot;math-container&quot;&gt;$v=\frac{n}{p}$&lt;/span&gt;, where n is the number of points normally graded, and p is the percent of students who got the question right.&lt;/p&gt;&#xA;&lt;p&gt;This puts a lot of extra weight on the questions only a few students get right. The person who gets all five questions right gets 100%, someone who only gets question 5 right gets 68%, and someone who only misses question 5 (which is 95% of the class) gets a maximum of 32%. That's clearly a bad scoring system.&lt;/p&gt;&#xA;&lt;h2&gt;Scaling the P-Factor&lt;/h2&gt;&#xA;&lt;p&gt;We can use scaling and offsets to tweak that system. Currently, since the denominator, p, can scale down to 0, we can get arbitrarily large score values. If we put p through a function such that the denominator has a limited range, we can limit the score of any particular answer.&lt;/p&gt;&#xA;&lt;p&gt;For example, &lt;span class=&quot;math-container&quot;&gt;$v=\frac{n}{0.9 + \frac{p}{10}}$&lt;/span&gt; gives us a minimum score of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$p=1$&lt;/span&gt;, and a maximum score of &lt;span class=&quot;math-container&quot;&gt;$\frac{n}{0.9}\approx 1.1n$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$p=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I prefer to think in terms of multipliers, instead of divisors, so I'd modify it a bit to &lt;span class=&quot;math-container&quot;&gt;$v=n\cdot(1.1-\frac{p}{10})$&lt;/span&gt;, giving a score range of exactly &lt;span class=&quot;math-container&quot;&gt;$\{n, 1.1n\}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We can further extend that to any arbitrary maximum relative score. Let &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; be the maximum relative score we can give, so &lt;span class=&quot;math-container&quot;&gt;$M=2$&lt;/span&gt; would give a final score of &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt;. Then let &lt;span class=&quot;math-container&quot;&gt;$m=M-1$&lt;/span&gt;, so it's the additional relative score. So for &lt;span class=&quot;math-container&quot;&gt;$M=2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$m=1$&lt;/span&gt;, giving 100% extra score for the question nobody answered.&lt;/p&gt;&#xA;&lt;p&gt;Now, for any given &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, we can write &lt;span class=&quot;math-container&quot;&gt;$v=n(1+m-mp)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Examples:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$M=2$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$m=1$&lt;/span&gt;:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n(1+1-1p)$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n(2-\{1,0\})$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=\{1,2\}n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$M=1.5$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$m=0.5$&lt;/span&gt;:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n(1+0.5-0.5p)$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n(1.5-\{0.5, 0\})$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=\{1, 1.5\}n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;h2&gt;Negative Offsets&lt;/h2&gt;&#xA;&lt;p&gt;We can offset the result so easy questions are worth less than &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and hard questions are worth more than &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, but we have to be careful that we're not assigning negative points for large values of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$m&amp;lt;2$&lt;/span&gt;, we can do &lt;span class=&quot;math-container&quot;&gt;$v=n(1+m-mp-\frac{m}{2})$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=1+\frac{m}{2}-mp$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$m=1$&lt;/span&gt;, that gives&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n(\{1,2\} - 0.5)$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=\{0.5, 1.5\}n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$m=0.5$&lt;/span&gt;, it's&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n(\{1,1.5\} - 0.25)$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=\{0.75, 1.25\}n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Setting &lt;span class=&quot;math-container&quot;&gt;$m=0.4$&lt;/span&gt;, gives&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=\{0.8, 1.2\}n$&lt;/span&gt;&lt;br /&gt;&#xA;which gives the hardest questions 50% more points than the easiest.&lt;/p&gt;&#xA;&lt;h2&gt;Piecewise Offsets&lt;/h2&gt;&#xA;&lt;p&gt;For large values of m, it's probably better to have offsets greater than 1 be multiplicatively proportional to offsets less than 1. So if the highest score is &lt;span class=&quot;math-container&quot;&gt;$n\cdot M$&lt;/span&gt;, the lowest score is &lt;span class=&quot;math-container&quot;&gt;$\frac{n}{M}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, now we're getting into the same issue as your original function: large discrepancies make hard questions too profitable. But the math still works, so we'll do it for the sake of discussion. Plus it might be useful for something like Jeopardy, where hard questions are supposed to be worth a lot. Setting M to 2.45 gives hard answers a 50% bonus over easy answers like we did above, but at that point you might as well just save yourself some headache and use the easy formula.&lt;/p&gt;&#xA;&lt;p&gt;Modify the above function (aka &amp;quot;create a new function from scratch and don't bother writing down the derivation&amp;quot;) to get&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\require{amsmath}v=\begin{cases} &#xA;      n\cdot[\frac{m+1}{2} - p\cdot(m-1)] &amp;amp; p\leq 0.5 \\&#xA;      n\cdot[1-(2\cdot p - 1)\cdot(1-\frac{2}{m+1})] &amp;amp; p\gt 0.5 &#xA;   \end{cases}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Example, &lt;span class=&quot;math-container&quot;&gt;$M=5$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$m=4$&lt;/span&gt;, we want our maximum score to be &lt;span class=&quot;math-container&quot;&gt;$5n$&lt;/span&gt; and our minimum score to be &lt;span class=&quot;math-container&quot;&gt;$\frac{n}{5}$&lt;/span&gt;. Because we're offsetting, actual max is &lt;span class=&quot;math-container&quot;&gt;$\frac{5n}{2}=2.5n$&lt;/span&gt; and actual min is &lt;span class=&quot;math-container&quot;&gt;$2\cdot\frac{n}{5}=\frac{n}{2.5}$&lt;/span&gt;. &lt;sup&gt;1&lt;/sup&gt; The system works best for &lt;span class=&quot;math-container&quot;&gt;$M&amp;gt;2$&lt;/span&gt;; otherwise, you get lower scores for hard questions.&lt;/p&gt;&#xA;&lt;p&gt;When &lt;span class=&quot;math-container&quot;&gt;$p=0$&lt;/span&gt;, we use&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n\cdot[\frac{m+1}{2} - p\cdot(m-1)]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[\frac{4+1}{2} - 0\cdot 3]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[2.5-0]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=2.5\cdot n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When &lt;span class=&quot;math-container&quot;&gt;$p=0.25$&lt;/span&gt;, we use&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n\cdot[\frac{m+1}{2} - p\cdot(m-1)]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[\frac{4+1}{2} - 0.25\cdot 3]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[2.5-0.75]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=1.75\cdot n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When &lt;span class=&quot;math-container&quot;&gt;$p=0.5$&lt;/span&gt;, we use&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n\cdot[\frac{m+1}{2} - p\cdot(m-1)]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[\frac{4+1}{2} - 0.5\cdot 3]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[2.5-1.5]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=1\cdot n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When &lt;span class=&quot;math-container&quot;&gt;$p=0.5$&lt;/span&gt;, we could use&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n\cdot[1-(2\cdot p - 1)\cdot(1-\frac{2}{m+1})]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[1-(2\cdot 0.5 - 1)\cdot(1-\frac{2}{4+1})]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[1-(0)\cdot(1-0.4)]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[1 - 0]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=1\cdot n$&lt;/span&gt;&lt;br /&gt;&#xA;showing there's continuity at the break.&lt;/p&gt;&#xA;&lt;p&gt;When &lt;span class=&quot;math-container&quot;&gt;$p=0.75$&lt;/span&gt;, we use&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n\cdot[1-(2\cdot p - 1)\cdot(1-\frac{2}{m+1})]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[1-(2\cdot 0.75 - 1)\cdot(1-\frac{2}{4+1})]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[1-(0.5)\cdot(1-0.4)]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[1 - 0.5\cdot 0.6]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[1-0.3]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=0.7\cdot n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When &lt;span class=&quot;math-container&quot;&gt;$p=1$&lt;/span&gt;, we use&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$v=n\cdot[1-(2\cdot p - 1)\cdot(1-\frac{2}{m+1})]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[1-(2\cdot 1 - 1)\cdot(1-\frac{2}{4+1})]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[1-(1)\cdot(1-0.4)]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=n\cdot[1 - 0.6]$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$=0.4\cdot n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; If you prefer, you can define &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; as actual max, then &lt;span class=&quot;math-container&quot;&gt;$m=2\cdot M-1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$M&amp;gt;1$&lt;/span&gt; gives good results, and the rest follows properly.&lt;/p&gt;&#xA;&lt;h2&gt;Selecting Sample Data&lt;/h2&gt;&#xA;&lt;p&gt;There are a few different ways to assign the actual p-values. First, values can be measured against the current class, previous class(es), or some standard.&lt;/p&gt;&#xA;&lt;p&gt;The easiest way is to give the test questions to some volunteers and see how many they get right. Record the p-values and assign scores to each question as above. Now, each time you grade future tests, the score multiplier for that question is already known.&lt;/p&gt;&#xA;&lt;p&gt;To get a larger sample set, you can aggregate the scores over each class. After a few semesters, you've got hundreds of samples for each question instead of the four samples from TAs you coerced into helping you.&lt;/p&gt;&#xA;&lt;p&gt;At some point, it might be good to drop your earliest classes, so you have a rolling average over 5 years or something. This keeps scores current, while also smoothing out anomalies.&lt;/p&gt;&#xA;&lt;p&gt;You can also assign the p-value based on how many students got it right on this exact test. This is probably fine for a competition, but leads to a big problem for classwork: if my score is based on a p-value from other people in the class, it's in my best interest to not help them, or to even mislead them into scoring poorly on certain topics I put more effort into.&lt;/p&gt;&#xA;&lt;p&gt;By scoring this test based on previous classes, I have no reason to deliberately sabotage my current classmates, which is generally a good thing.&lt;/p&gt;&#xA;&lt;h2&gt;Scoring Partial Credit&lt;/h2&gt;&#xA;&lt;p&gt;The second thing to consider when assigning p-values is partial credit. The above scoring assumes questions are pass/fail. In the real world, we often assign some points to answers that are incomplete or partially wrong, but got some parts right.&lt;/p&gt;&#xA;&lt;p&gt;In this case, instead of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; being the percent of right answers, we simply make it the average percent of maximum score attained. If a question is normally worth 10 points, and the average score is 6.2, then &lt;span class=&quot;math-container&quot;&gt;$p=62\%$&lt;/span&gt; for that question.&lt;/p&gt;&#xA;&lt;h2&gt;The Pendulum Effect&lt;/h2&gt;&#xA;&lt;p&gt;When we set pressures on a population based on previous performance, we can introduce a pendulum effect where the population bounces between different extremes.&lt;/p&gt;&#xA;&lt;p&gt;Classes one year might do well in one metric, then we change the scoring to motivate the next year's class to do better in areas the first class scored poorly. Then the second class performs poorly in the area the first class did well on. So we change the scoring again and we're back where we started.&lt;/p&gt;&#xA;&lt;p&gt;Instead of radically changing scoring each semester, it's better to slowly change scoring. So we only slightly incentivize year 2 in the areas year 1 was bad at. If year 2 still does badly in that area, we raise the incentive a bit more for year 3, etc.&lt;/p&gt;&#xA;&lt;p&gt;The idea is to get to a point where no topic has particularly higher p-values than another topic, implying that students are getting a well-rounded experience from the class.&lt;/p&gt;&#xA;&lt;p&gt;By aggregating p-values over several years, it helps smooth these jumps so we tend towards an equilibrium in the middle, instead of ping-ponging between extremes.&lt;/p&gt;&#xA;&lt;p&gt;Of course, this requires that students are aware of the scoring, and which topics are worth more points. Otherwise they won't know to spend a bit more time on the harder topics.&lt;/p&gt;&#xA;&lt;p&gt;It also requires consistent questions over many years. If you constantly change your test questions, it will likely throw the entire system off.&lt;/p&gt;&#xA;" OwnerUserId="5758" LastActivityDate="2022-02-22T06:31:17.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24953" PostTypeId="2" ParentId="7963" CreationDate="2022-02-22T13:53:00.363" Score="0" Body="&lt;p&gt;If you only care about scoring a single assessment that has already happened, you could grade each question on a curve: the scores (just 0s and 1s) get shifted and rescaled to have a fixed mean and standard deviation.&lt;/p&gt;&#xA;&lt;p&gt;This has the effect of increasing the number of points very significantly (if I've done my math right, something like: &lt;a href=&quot;https://www.desmos.com/calculator/pf33jysqex&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.desmos.com/calculator/pf33jysqex&lt;/a&gt;) where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the percent correct) for problems that are very commonly gotten right or wrong.&lt;/p&gt;&#xA;&lt;p&gt;Of course, if a problem is gotten entirely correct or entirely incorrect, you should drop it when scoring since it doesn't have any data.&lt;/p&gt;&#xA;&lt;p&gt;Aside: I'm curious how extrinsic motivation in math competitions affects motivation in math competitions and in the classroom. The typical relationship is that extrinsic motivation undermines intrinsic motivation and creates a dependence on extrinsic motivation. But are there studies about this specific type of extrinsic motivation for this specific audience?&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-02-22T13:53:00.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24955" PostTypeId="2" ParentId="24950" CreationDate="2022-02-22T16:56:40.187" Score="13" Body="&lt;p&gt;The answer to the lede question &amp;quot;Why do we use function composition?&amp;quot; is &amp;quot;Because we do.&amp;quot;  It is the way that, for historical reasons, the notation developed.  It is a convention which developed before a lot of set theory, modern algebra, and category theory came along, and this convention was deeply ingrained long before anyone thought to question it.&lt;/p&gt;&#xA;&lt;p&gt;From a naïve point of view, I also think that it makes a good deal of sense:  if &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; denotes the application of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to an input &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$g(f(x))$&lt;/span&gt; must denote the application of a function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; to an input &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.  Reading from left-to-right, we have &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;, which we might abbreviate by &lt;span class=&quot;math-container&quot;&gt;$(g\circ f)(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From a more set theoretic point of view, this becomes a little harder to reason about.  A typical schematic for functional composition in undergraduate texts looks something like the following:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tUqCa.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tUqCa.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Reading from left-to-right, we might expect the composite function to be &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and then &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;&amp;quot;, which we might want to denote by &lt;span class=&quot;math-container&quot;&gt;$f \ast g$&lt;/span&gt; or something similar.  But the convention is that this map is &lt;span class=&quot;math-container&quot;&gt;$g \circ f$&lt;/span&gt;—for historical reasons, maps are applied from right-to-left.&lt;/p&gt;&#xA;&lt;p&gt;For the most part, this doesn't really cause any problems, but there are some places where you might see some difficulties or a different choice of notation:&lt;/p&gt;&#xA;&lt;h4&gt;Category Theory&lt;/h4&gt;&#xA;&lt;p&gt;In chalk-talks, I have seen one or two category theorists use the notation &lt;span class=&quot;math-container&quot;&gt;$(x)f$&lt;/span&gt; to denote the image of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; under the map &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.  In this setting,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (x)(f\circ g) = ((x)f)g, $$&lt;/span&gt;&#xA;which is what we would usually write at &lt;span class=&quot;math-container&quot;&gt;$g(f(x))$&lt;/span&gt;.  So, if you want to compose on the right (as seems &amp;quot;natural&amp;quot;), you should also apply functions on the right.&lt;/p&gt;&#xA;&lt;p&gt;From one point of view, I suppose that this is a perfectly reasonable convention to adopt (and I kind of wish that we had adopted this convention in the first place), but it is probably a &lt;em&gt;very&lt;/em&gt; bad idea to teach this convention to a neophyte.&lt;/p&gt;&#xA;&lt;h4&gt;Differential Operators&lt;/h4&gt;&#xA;&lt;p&gt;Folk who study differential equations will often use notation like &lt;span class=&quot;math-container&quot;&gt;$\partial_{xy}$&lt;/span&gt; in order to mean&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \partial_{xy} = \partial_y \circ \partial_x, $$&lt;/span&gt;&#xA;where&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \partial_{x} f(x,y) = \frac{\partial}{\partial x} f(x,y). $$&lt;/span&gt;&#xA;Or should that be&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \partial_{xy} = \partial_x \circ \partial_y?$$&lt;/span&gt;&#xA;There is room for ambiguity here, and authors are well-served to be explicit about what they mean.  On the other hand, in most &amp;quot;nice&amp;quot; cases, the derivatives commute, so maybe it doesn't really matter.&lt;/p&gt;&#xA;&lt;h4&gt;Iterated Function Systems&lt;/h4&gt;&#xA;&lt;p&gt;In my own area of research, a common object of study is an iterated function system.  Such a system is simply a collection of maps &lt;span class=&quot;math-container&quot;&gt;$\{\varphi_j\}_{j=1}^n$&lt;/span&gt; which map some space into itself.  The maps are often composed with each other over and over again (hence the &amp;quot;iterated&amp;quot; part of the term), which leads to all kinds of wild notations for composition.  One common notation is to take &lt;span class=&quot;math-container&quot;&gt;$\mathbf{j} = (j_1, j_2, \dots, j_k)$&lt;/span&gt; to be a tuple of indices, and then write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \varphi^{\mathbf{j}} = \varphi_{j_k} \circ \varphi_{j_{k-1}} \circ \dotsb \circ \varphi_{j_2} \circ \varphi_{j_1}. $$&lt;/span&gt;&#xA;That is, &lt;em&gt;first&lt;/em&gt; apply the map &lt;span class=&quot;math-container&quot;&gt;$\varphi_{j_1}$&lt;/span&gt;, &lt;em&gt;then&lt;/em&gt; apply the map &lt;span class=&quot;math-container&quot;&gt;$\varphi_{j_2}$&lt;/span&gt;, and so on.  However, people mess up this convention &lt;em&gt;all the time&lt;/em&gt;.  I cannot tell you the number of times that I have seen the convention defined one way, and then applied the other.  This doesn't usually cause problems, because the underlying reasoning is generally correct—it is just a problem of transcription—but it can be a concern.&lt;/p&gt;&#xA;&lt;h4&gt;Reverse Polish Notation&lt;/h4&gt;&#xA;&lt;p&gt;I grew up using an &lt;a href=&quot;https://en.wikipedia.org/wiki/HP_48_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;HP48 series calculator&lt;/a&gt;.  These calculators were somewhat unique, in that they implemented computation via reverse Polish notation.  The idea is that the arguments of a function are placed onto a &amp;quot;stack&amp;quot;, and functions apply to the items on the stack in a &amp;quot;last in, first out&amp;quot; order.  So, for example, if I wanted to compute &lt;span class=&quot;math-container&quot;&gt;$\sin(\log(5) + 3)$&lt;/span&gt;, I might enter the keystrokes&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;[5]     &amp;lt;- this puts 5 on the bottom of the stack&#xA;[log]   &amp;lt;- the &amp;quot;eats&amp;quot; the 5, and puts log(5) on the bottom of the&#xA;           stack&#xA;[3]     &amp;lt;- this &amp;quot;bumps&amp;quot; log(5) up a spot, and puts 3 on the stack&#xA;           below log(5)&#xA;[+]     &amp;lt;- this &amp;quot;eats&amp;quot; the bottom two elements of stack, adds&#xA;           them together, and puts the result on the bottom of&#xA;           the stack&#xA;[sin]   &amp;lt;- this &amp;quot;eats&amp;quot; the bottom element of the stack, takes its&#xA;           sine, and places the result on the bottom of the stack&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Written out, the set of keystrokes is &amp;quot;composition in the reverse order&amp;quot;, i.e.&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;5 log 3 + sin&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;roughly corresponds to writing &lt;span class=&quot;math-container&quot;&gt;$(((5)\log, 3){+})\sin$&lt;/span&gt;, which kind of looks like backwards composition.  Using the standard notation we might write  &lt;span class=&quot;math-container&quot;&gt;$\sin( {+}(\log(5), 3))$&lt;/span&gt;.  In both cases, we think of &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; as a function &lt;span class=&quot;math-container&quot;&gt;$+: \mathbb{R}^2 \to \mathbb{R}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This may look arcane at first glance, but once you get used to it, it is a &lt;em&gt;really&lt;/em&gt; convenient way to enter computations into a calculator or computer.  It eliminates a lot of errors caused by missing parentheses.&lt;/p&gt;&#xA;&lt;h3&gt;TL;DR&lt;/h3&gt;&#xA;&lt;p&gt;The current convention exists for historical reasons.  If we were to start all over again, we might adopt a different convention, and there are good reasons to want to do so.  However, the convention exists, is well-established, and you would be doing your students a disservice to introduce an alternative notation.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2022-02-24T20:28:41.743" LastActivityDate="2022-02-24T20:28:41.743" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24956" PostTypeId="2" ParentId="18120" CreationDate="2022-02-22T20:56:15.960" Score="-1" Body="&lt;p&gt;Might I suggest understanding differential forms, chain complexes and algebraic topology?&lt;/p&gt;&#xA;&lt;p&gt;basically, the derivative is the opposite of the boundary.&lt;/p&gt;&#xA;&lt;p&gt;see these videos:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=1lGM5DEdMaw&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=1lGM5DEdMaw&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=2ptFnIj71SM&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=2ptFnIj71SM&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10335" LastActivityDate="2022-02-22T20:56:15.960" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24957" PostTypeId="2" ParentId="24950" CreationDate="2022-02-22T21:59:09.390" Score="5" Body="&lt;p&gt;Put in natural language terms, if your language has verb-object order, then verb1-verb2-object naturally represents verb1 (verb2-object). That is, apply verb2 first, then verb1. For instance, if you're writing a recipe, &amp;quot;cook the chopped onion&amp;quot; means &amp;quot;chop the onions, then cook them&amp;quot;. If you want someone to peel the onion, then chop it, then cook it, then serve it, that would be &amp;quot;serve the cooked chopped peeled onion&amp;quot;. If your language has object-verb order, then it's natural for composition to go the other way. Whatever direction your object to verb order is, that's your composition order, because the verb+object becomes the object for the next verb. Math is somewhat inconsistent as to which order is used; &amp;quot;x squared plus two&amp;quot; is basically object-verb order, while &amp;quot;f of x&amp;quot; is verb-object order.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2022-02-22T21:59:09.390" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24958" PostTypeId="2" ParentId="7963" CreationDate="2022-02-22T22:25:16.247" Score="1" Body="&lt;p&gt;From an information theory perspective, if the proportion of people getting a question right is &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, then someone getting that question  right is worth &lt;span class=&quot;math-container&quot;&gt;$-\log p$&lt;/span&gt; units of information, while someone getting it wrong is worth &lt;span class=&quot;math-container&quot;&gt;$-\log (1-p)$&lt;/span&gt; units of information. This does go to infinity as &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; goes to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, but not as quickly as &lt;span class=&quot;math-container&quot;&gt;$1/x$&lt;/span&gt;. So using this formula, someone who gets a question correct would gain &lt;span class=&quot;math-container&quot;&gt;$-\log p$&lt;/span&gt; points, while someone who gets the problem wrong would lose &lt;span class=&quot;math-container&quot;&gt;$-\log (1-p)$&lt;/span&gt;. As &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; goes to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, the penalty for getting the problem wrong would go to zero. However, as &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; goes to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, the penalty would go to infinity. If that doesn't appeal to you, you can get rid of the penalty, and adjust the scores of only the people who get the problem right, or make sure that all of your problems have &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; less than &lt;span class=&quot;math-container&quot;&gt;$0.5$&lt;/span&gt; (after all, the purpose of questions is to separate the best students from the rest, and if most of the students are getting the question right, then that's not separating out the best students).&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2022-02-22T22:25:16.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24959" PostTypeId="1" AcceptedAnswerId="24960" CreationDate="2022-02-23T00:22:44.827" Score="6" ViewCount="406" Body="&lt;p&gt;Recently I was reading &lt;a href=&quot;https://equitablemath.org/wp-content/uploads/sites/2/2020/11/1_STRIDE1.pdf&quot; rel=&quot;noreferrer&quot;&gt;A Pathway to Equitable Math Instruction Dismantling Racism in Mathematics Instruction&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And I ended up getting in doubt after reading things like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vUcND.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vUcND.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;According to the historical product of mathematics which, as I understand it, focuses on objectivity, comes from white supremacy? From where and how does this really prove to be legitimate? Does mathematics with constructive content and dependent on the above come from white supremacy? I would like more information on the subject.&lt;/p&gt;&#xA;" OwnerUserId="12232" LastActivityDate="2022-02-23T02:59:26.263" Title="Policy in mathematics teaching" Tags="|education-research|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="24960" PostTypeId="2" ParentId="24959" CreationDate="2022-02-23T02:51:46.440" Score="6" Body="&lt;p&gt;I will answer. I might not be answering the question asked (it is too unclear for me to be sure), but my answer is to what I think is being addressed in the linked book. (And if this question gets muddied in confusion, I may pose a related question soon.)&lt;/p&gt;&#xA;&lt;p&gt;How is the teaching of mathematics sullied by racism or white supremacism?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The naming of theorems and mathematical objects is eurocentric:  * We speak (in the U.S. and I'm guessing many other countries) of Pascal's Triangle, yet it was invented many times (long before Pascal lived) in India, China, Persia,and others. (&lt;a href=&quot;https://en.wikipedia.org/wiki/Pascal%27s_triangle&quot; rel=&quot;noreferrer&quot;&gt;See wikipedia&lt;/a&gt;.)    * The Pythagorean Theorem was also invented long before Pythagoras, in other cultures. (I know the argument that Pythagoras was the first to prove it. I doubt that that's accurate. We don't know all of what happened in those other cultures. Archeology is not great at getting all of the written records.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Math classes have been used as a filter for decades or centuries, to weed people out of certain professions. Think about doctors, for example. Are they required to do more math than they need? And lawyers, perhaps? So there used to be this mindset that it made sense for lots of students to fail. Who failed? Why? Look at the funding of our K12 systems in the U.S. and you see that economics (and race) are deeply involved. Why is race involved? The best book I know of on this is &lt;em&gt;&lt;strong&gt;&lt;a href=&quot;https://bookshop.org/books/the-color-of-law-a-forgotten-history-of-how-our-government-segregated-america/9781631494536&quot; rel=&quot;noreferrer&quot;&gt;The Color of Law&lt;/a&gt;&lt;/strong&gt;&lt;/em&gt;, by Richard Rothstein. &lt;a href=&quot;https://www.npr.org/2017/05/03/526655831/a-forgotten-history-of-how-the-u-s-government-segregated-america&quot; rel=&quot;noreferrer&quot;&gt;Here's an article&lt;/a&gt; about his work. Basically, our education system is funded by property tax, and our country was segregated by laws put in place by the government. Our history is a mess.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If we are trying to be anti-racist in our teaching, what can we do? There's a lot, and that book is trying to lay it out. Mathematical facts may be objective in a way we do not find problematic, but the choices of topics to cover in each course is not as objective as that. And stopping students from taking college level courses because of holes in their education has racist effect. If we can do things differently, we must. One solution that's happening in California is for students who are not interested in STEM to take statistics, and if their background is weak, to have co-requisite support. For students interested in STEM, it's a little harder, but I tell my students that it's my job to help them get from where they are to where they want to be. (And their job is to do the work it takes, while I guide them.) I explain misconceptions about fractions to my calculus students.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I will stop here for now. Yes, there are books about this. It will take books and good will, and effort from us all, to change the racist world we are living in.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-02-23T02:59:26.263" LastActivityDate="2022-02-23T02:59:26.263" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24961" PostTypeId="1" AcceptedAnswerId="24962" CreationDate="2022-02-23T11:55:03.967" Score="8" ViewCount="271" Body="&lt;p&gt;I am preparing a seminar on mathematical discovery for beginner's students, and one of the topics I plan to cover is the brachistocronic/tautochronic curve. For this purpose, I would like to let the students experiment with various curves connecting two given points P and Q for a while, trying to minimize the time it takes a ball starting at P to arrive at Q.&lt;/p&gt;&#xA;&lt;p&gt;Since I cannot think of a feasible way to do it physically, I am looking for a suitable piece of simulation software. Ideally, one should be able to draw a curve and then have it simulate a ball rolling along that curve, showing the time it took at the end.&lt;/p&gt;&#xA;&lt;p&gt;So far, I have been able to find this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.physicsclassroom.com/Physics-Interactives/Circular-and-Satellite-Motion/Roller-Coaster-Model/Roller-Coaster-Model-Interactive&quot; rel=&quot;noreferrer&quot;&gt;https://www.physicsclassroom.com/Physics-Interactives/Circular-and-Satellite-Motion/Roller-Coaster-Model/Roller-Coaster-Model-Interactive&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is about what I want, but for my purposes, it is too &amp;quot;coarse&amp;quot; with just four adjustable points.&#xA;Does anyone know of a piece of software that does this? Since it should be used by the students during class, it would be good if it was freely available online.&lt;/p&gt;&#xA;" OwnerUserId="19554" LastActivityDate="2022-02-23T15:32:12.540" Title="Ball rolling down curve simulator" Tags="|undergraduate-education|reference-request|undergraduate-research|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="24962" PostTypeId="2" ParentId="24961" CreationDate="2022-02-23T15:32:12.540" Score="6" Body="&lt;p&gt;The &lt;a href=&quot;https://phet.colorado.edu/sims/html/energy-skate-park/latest/energy-skate-park_en.html&quot; rel=&quot;noreferrer&quot;&gt;Energy Skate Park&lt;/a&gt; by PhET seems to come quite close:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/WXtbj.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WXtbj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the &amp;quot;Playground&amp;quot; section you can draw arbitrary curves and add a stopwatch. The simulation has to be manually stopped at the end of the curve and you have to deal with a person (or a dog) skating down the curve, but otherwise it should do everything that you need.&lt;/p&gt;&#xA;&lt;p&gt;You could also use the &amp;quot;Bumps&amp;quot; setting in &lt;a href=&quot;https://www.physicsclassroom.com/Physics-Interactives/Circular-and-Satellite-Motion/Roller-Coaster-Model/Roller-Coaster-Model-Interactive&quot; rel=&quot;noreferrer&quot;&gt;the simulation that you suggested&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Finally, there is &lt;a href=&quot;https://www.phywe.de/physik/mechanik/stoff-materialeigenschaften/berg-und-talbahn-fuer-demo-tafel-magn_410_1341/&quot; rel=&quot;noreferrer&quot;&gt;this product&lt;/a&gt; (german supplier, but maybe &amp;quot;magnetic curve track&amp;quot; yields results for your country as well) to create real curved tracks on any magnetic surface:&lt;a href=&quot;https://i.stack.imgur.com/8pXR6.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8pXR6.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2022-02-23T15:32:12.540" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24963" PostTypeId="1" AcceptedAnswerId="24966" CreationDate="2022-02-24T01:42:21.823" Score="10" ViewCount="489" Body="&lt;p&gt;I am teaching an introductory linear algebra courses for undergraduate students in math, computer science, or data science at a liberal art university. Most of the students have not decided their major yet. But the majority will take the CS or DS route, with a tiny minority taking a math major. For math students, or any one, this will be their only Linear Algebra course.&lt;/p&gt;&#xA;&lt;p&gt;While giving students a lot of exercises in computing, I also give them assignments and exam questions asking for simple proof. Usually these takes very little effort if they understand the definitions.&lt;/p&gt;&#xA;&lt;p&gt;When I mentioned this to a colleague, he says this is pointless and it is enough for students to understand the concepts and be able to compute things by hand.&lt;/p&gt;&#xA;&lt;p&gt;My rational for doing so is that most basic linear algebra computations can be done by computer very easily, so there is not much students can get from computing, say an inverse of a matrix. By asking them to write arguments formally, they could learn a bit about how to explain a technical point formally.&lt;/p&gt;&#xA;&lt;p&gt;Am I just being naive in what can we expect from students? What should we teach students in linear algebra?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorUserId="1737" LastEditDate="2022-02-27T03:36:05.063" LastActivityDate="2022-05-05T19:45:10.977" Title="Should students in a first university linear algebra class be taught to write simple proofs?" Tags="|undergraduate-education|proofs|linear-algebra|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24964" PostTypeId="1" CreationDate="2022-02-24T17:20:30.190" Score="5" ViewCount="183" Body="&lt;p&gt;My children attend a Montessori school (as did I when I was a child), and I have visited several other Montessori schools and spoken with their teachers.  Numerous times I've had Montessori teachers tell me that they never liked or understood mathematics until they learned to teach with the Montessori math materials, at which point they had an epiphany and it all started to make sense.&lt;/p&gt;&#xA;&lt;p&gt;This makes me wonder: could Montessori materials make a difference in the mathematical understanding of other high school or college students, even those who aren't intending to become Montessori teachers?  Specifically, could they help to reach other students who similarly &amp;quot;never liked or understood mathematics&amp;quot;?  Has anyone ever tried this?&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2022-02-26T04:21:40.827" Title="Montessori mathematics for HS/college students" Tags="|undergraduate-education|secondary-education|struggling-students|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24965" PostTypeId="2" ParentId="24950" CreationDate="2022-02-24T21:04:32.777" Score="4" Body="&lt;p&gt;I think it's worth noting that &lt;span class=&quot;math-container&quot;&gt;$g\,;f$&lt;/span&gt; is sometimes used instead of &lt;span class=&quot;math-container&quot;&gt;$f\circ g$&lt;/span&gt; in certain areas that have connections to computer science:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;In category theory&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$f\,;g$&lt;/span&gt; will sometimes be used for the composition of morphisms &lt;span class=&quot;math-container&quot;&gt;$f: A \to B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g: B \to C$&lt;/span&gt;, see for example &lt;a href=&quot;https://www.researchgate.net/publication/243780972_A_Gentle_Introduction_to_Category_Theory_-_the_calculational_approach&quot; rel=&quot;nofollow noreferrer&quot;&gt;this introductory text&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;In computer science proper&lt;/strong&gt;: Again, &lt;span class=&quot;math-container&quot;&gt;$g\,;f$&lt;/span&gt; might be used for composition of functions / partial functions / relations. For example: &lt;span class=&quot;math-container&quot;&gt;$[\mathit{-}]c[\mathit{-}] := post(c)\,;\supseteq$&lt;/span&gt; (from &lt;a href=&quot;https://dl.acm.org/doi/10.1145/3371078&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The main reason why it's not used more widely is simply due to convention: Had the mathematicians assumed some kind of postfix / &amp;quot;method notation&amp;quot; (e.g. &lt;span class=&quot;math-container&quot;&gt;$x.f.g.h$&lt;/span&gt;) instead of &lt;span class=&quot;math-container&quot;&gt;$h(g(f(x)))$&lt;/span&gt;, then the order &lt;span class=&quot;math-container&quot;&gt;$fst ; snd$&lt;/span&gt; would probably seem more natural than &lt;span class=&quot;math-container&quot;&gt;$snd \circ fst$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Regarding the prefix notation for functions, I believe that we can safely blame that on the early physicists. This is because in physics, one often has to work with some &amp;quot;important&amp;quot; quantities &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, which depend on the &amp;quot;less important&amp;quot; parameters &lt;span class=&quot;math-container&quot;&gt;$a, b, \dots, z$&lt;/span&gt;, so that the &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; comes first in an expression &lt;span class=&quot;math-container&quot;&gt;$F(a, b, \dots, z)$&lt;/span&gt;. You can see that the parameters really are &amp;quot;less important&amp;quot; from the fact that we often encounter abuse of notation where we are freely jumping between e.g. &lt;span class=&quot;math-container&quot;&gt;$F(x, y)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F(r,\phi)$&lt;/span&gt;, where the &amp;quot;important quantity&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is denoted by the same symbol, whereas the &amp;quot;less important&amp;quot; parameters don't even range over the same domain.&lt;/p&gt;&#xA;" OwnerUserId="15593" LastActivityDate="2022-02-24T21:04:32.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24966" PostTypeId="2" ParentId="24963" CreationDate="2022-02-25T02:12:43.920" Score="1" Body="&lt;p&gt;I enter this as an answer, rather than as a comment, because I think it &lt;em&gt;is&lt;/em&gt; a useful re-framing of the question.&lt;/p&gt;&#xA;&lt;p&gt;That is, for practical mathematics... do we want to be able to walk across a room, while chewing gum, and talking, and go through the door without banging into it... or do we want to have people &lt;em&gt;worry&lt;/em&gt; about the very possibility of doing so, and think that they need to be able to &lt;em&gt;prove&lt;/em&gt; (how?) that they can do so before simply walking out the door?&lt;/p&gt;&#xA;&lt;p&gt;I do seriously claim that all too often mathematics education comes down on the silly/wrong side of such distinctions. :)&lt;/p&gt;&#xA;&lt;p&gt;EDIT: in response to @David Steinberg's reasonable objection: for example, to prove/claim that the cardinality of a basis for a vector space is well-defined... Well, ok, yes, the proof has some interest, but its methods are much more exotic than the (apparent) nature of the assertion itself.&lt;/p&gt;&#xA;&lt;p&gt;Not only in &amp;quot;linear algebra&amp;quot;, but many other parts of mathematics, an interesting/discomfitting aspect is that proofs of heuristically necessary, and plausibly correct, ... things are much more sophisticated than the assertion. For me, this is interesting, but it does create complications in math education, for sure. That is, being completely sure that obvious-looking things are ok is often a much fancier thing... Telling novices about these (interesting!) fancier issue necessary to justify their computations should be a small thing, not the main point, at all.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2022-02-26T02:44:53.020" LastActivityDate="2022-02-26T02:44:53.020" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24967" PostTypeId="1" CreationDate="2022-02-25T04:30:15.753" Score="1" ViewCount="1662" Body="&lt;p&gt;I have taught College Algebra several times and will teach it again in the next semester.&#xA;College Algebra, according to the catalogue of my college, is described as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This course provides students an opportunity to gain algebraic&#xA;knowledge needed in many different fields such as engineering,&#xA;business, education, science, computer technology, and mathematics.&#xA;Graphical, numerical, symbolic, and verbal methods support the study&#xA;of functions and their corresponding equations and inequalities.&#xA;Students will study linear, quadratic, rational, exponential,&#xA;logarithmic, inverse, composite, radical, and absolute value&#xA;functions; systems of equations and inequalities modeling applied&#xA;problems; and curve fitting techniques. There will be extensive use of&#xA;graphing calculators.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have asked &lt;a href=&quot;https://matheducators.stackexchange.com/questions/12163/looking-for-a-stable-college-algebra-textbook&quot;&gt;a related question&lt;/a&gt; a few years ago and here I would like to ask this question:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Is there a College Algebra book that was written by a world-class mathematician?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am not sure if there is a universally accepted definition of &amp;quot;world-class mathematician&amp;quot; but let me give two examples here.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I have used Israel Moiseevich Gelfand's Trigonometry for my trigonometry class. The author was the first laureate of the Wolf Prize in Mathematics.&lt;/li&gt;&#xA;&lt;li&gt;I have also used Frederick Mosteller's Beginning Statistics with Data Analysis for my Elementary Statistics class. The author was the founding chair of Harvard's statistics department.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I enjoyed these two books greatly and now I am hoping to find a College Algebra book that the author (or one of the authors) is a world-class mathematician. Of course, this book must be available for purchasing.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="13919" LastEditDate="2022-02-25T22:31:20.093" LastActivityDate="2022-02-25T22:31:20.093" Title="Is there a College Algebra book that was written by a world-class mathematician?" Tags="|undergraduate-education|algebra|textbooks|" AnswerCount="2" CommentCount="9" ClosedDate="2022-02-26T14:01:09.903" ContentLicense="CC BY-SA 4.0" />
  <row Id="24968" PostTypeId="2" ParentId="24967" CreationDate="2022-02-25T13:06:59.183" Score="14" Body="&lt;p&gt;This question is close to one you've already asked, a subset of it practically.  Should I append a cute Venn diagram, showing the relation?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/17358/which-textbooks-on-college-algebra-trigonometry-pre-calculus-calculus-linear?rq=1&quot;&gt;Which textbooks on College Algebra, Trigonometry, Pre-calculus, Calculus, Linear Algebra, ODE are written by world-class mathematicians?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The answers to that previous one (rarity of research mathematicians writing lower level books, the noncongruence of good pedagogy and great research breakthroughs) apply still.&lt;/p&gt;&#xA;&lt;p&gt;To which I would add &amp;quot;college algebra&amp;quot; is a bit of a funny topic, maybe even more so than finding a HS geometry book or calc intro.  This is because &amp;quot;college algebra&amp;quot; in the US is something MOST students are expected to have done in high school. It's often called &amp;quot;algebra 2&amp;quot;.  The standard STEM freshman course at the Naval Academy, Virginia Tech, Cal etc. starts with calculus. Since at least the 80s (probably much earlier).&lt;/p&gt;&#xA;&lt;p&gt;The term college algebra has a bit of a 1940s flair to it, like from my father, when many US high schools stopped after first year algebra (mostly lines, culminating in the quadratic equation, but not logs/exponents/synth division) and geometry.  But probably since the 60s, most schools expect you to start at calculus,  In fact, most US students will have been exposed to a calculus in high school and a substantial portion will place out of the first year...but still, the nominal start of college is calculus.  If you start higher, you're ahead of pace.  If you take college algebra, you are taking REMEDIAL MATH.  This is even more extreme in places with better school systems (Germany, UK, Russia, Japan), where calculus is a standard HS senior class.  But in any case, &amp;quot;college algebra&amp;quot; is politically correct phrasing, using that 1940s flair (think bobby sox and sweaters) to cover for college kids that are taking REMEDIAL CLASSES.&lt;/p&gt;&#xA;&lt;p&gt;Given the kids are taking remedial math.  Are not on pace, not ahead of pace, not the best and the brightest, why would you even THINK about torturing them with some flashy name, bad pedagogy book?  No.  Just no.  You need to train them effectively and efficiently.  Heck, I don't like it when the crap pedagogy is used on the advanced kids.  But at least you have an excuse there.  &amp;quot;Enriching them&amp;quot; with such torture.&lt;/p&gt;&#xA;&lt;p&gt;Now of course, we are not talking about mathematic concepts, but fuzzy topics, when we say &amp;quot;what is a course&amp;quot;.  And things may differ in other countries or...uh...cantons.  And even within one (unless France), it's difficult to be axiomatic.  But, all those caveats aside, college algebra is advanced (second course) high school algebra.  You might as well look for an algebra 2 book (or &amp;quot;pre-calc&amp;quot;, whatever the HECK that means!) by Andrew Wiles--you won't find it either, but at least it wouldn't be remedial.&lt;/p&gt;&#xA;" OwnerUserId="19569" LastActivityDate="2022-02-25T13:06:59.183" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="24969" PostTypeId="2" ParentId="24967" CreationDate="2022-02-25T13:13:24.887" Score="5" Body="&lt;p&gt;Yes, Gelfand also has an algebra book:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/algebra_gelfand.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/algebra_gelfand.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-02-25T13:13:24.887" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24971" PostTypeId="1" AcceptedAnswerId="24984" CreationDate="2022-02-26T01:17:56.060" Score="4" ViewCount="237" Body="&lt;p&gt;I am teaching a first-linear-algebra course for mostly undergraduate Computer Science and Data Science students. Occasionally, I tried to do a bit computer experiments in class and ask students to guess what a theorem should be. But I noticed that students do not seem to be very engaged in the experiments. I guess the problems may be&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I am the one doing the experiment. So they cannot do much but just look.&lt;/li&gt;&#xA;&lt;li&gt;The experiments I am doing are numeric. So they do not look very interesting visually.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I wonder if anyone of you have tried similar things, and if you have learned how to make such experiments work better.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;One example of such experiments would be that I compute eigenvalues of a random diagonal matrix with computer, and asks students to guess the answer from observing such computations.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorUserId="16159" LastEditDate="2022-03-17T02:33:45.890" LastActivityDate="2022-03-17T02:33:45.890" Title="Tips for teaching mathematics with experiments?" Tags="|undergraduate-education|linear-algebra|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24972" PostTypeId="2" ParentId="24963" CreationDate="2022-02-26T01:19:06.023" Score="4" Body="&lt;p&gt;What you should teach in a linear algebra class depends on your audience, and while I agree with other answers' sentiments that you should emphasize concrete aspects / applications, the fact that you are teaching this course at a liberal arts school to students who have not &amp;quot;officially&amp;quot; chosen their majors yet may give you more latitude to include some proofs in the course.&lt;/p&gt;&#xA;&lt;p&gt;That students have not chosen their majors has both positive and negative implications for emphasizing proofs. On one hand, there are surely students who know their majors even if not yet declared, and some of these eventual CS/DS students are expecting your course to be a service course that will ultimately serve them well in more applied settings. On the other hand, students who are contemplating math as a major would be well-served by some exposure to proof given the stark differences between high-school and more advanced mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Being at a liberal arts school lends additional credence to the idea that all students, math majors or not, expect their education to include a broad exposure to the general, lofty aims of many academic disciplines; the students might be more receptive to the argument that they need to &amp;quot;appreciate mathematics&amp;quot; than a population at a large state engineering college would be.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2022-02-26T01:19:06.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24973" PostTypeId="2" ParentId="24963" CreationDate="2022-02-26T01:56:25.587" Score="2" Body="&lt;p&gt;One concept that is quite important in geometric computations is&#xA;the cross product, and in particular, how to compute the &lt;em&gt;signed area&#xA;of a triangle&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/4eTRH.png&quot; width=&quot;300&quot; /&gt;&#xA;&lt;p&gt;&lt;sup&gt;&lt;a href=&quot;https://ximera.osu.edu/mooculus/calculus2/master/crossProducts/digInCrossProducts&quot; rel=&quot;nofollow noreferrer&quot;&gt;OSU link&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;The length of &lt;span class=&quot;math-container&quot;&gt;$v \times w$&lt;/span&gt; is the area of the determined parallelogram,&#xA;twice the area of the triangle given coordinates of its vertices,&#xA;positive vector up and negative down.&#xA;This is a fundamental calculation for computing the convex hull of a&#xA;set of points, for example.&#xA;Another application is computing the area of a simple polygon by&#xA;summing the signed areas of triangles determined by each edge and&#xA;one fixed point—the so-called&#xA;&amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Shoelace_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;shoelace algorithm&lt;/a&gt;.&amp;quot;&#xA;This generalizes to computing the signed volume of a tetrahedron, and&#xA;the volume of simplices in higher dimensions.&lt;/p&gt;&#xA;&lt;p&gt;See also Wolfram Demo:&#xA;&lt;a href=&quot;https://demonstrations.wolfram.com/Signed2DTriangleAreaFromTheCrossProductOfEdgeVectors/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Signed 2D Triangle Area from the Cross Product of Edge Vectors&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-02-26T01:56:25.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24974" PostTypeId="2" ParentId="18977" CreationDate="2022-02-26T02:24:20.497" Score="1" Body="&lt;p&gt;For the group theory in physics, the following books looks very promising:&lt;/p&gt;&#xA;&lt;p&gt;Group Theory in a Nutshell for Physicists by Anthony Zee&lt;/p&gt;&#xA;&lt;p&gt;(It just came out in 2022 and I haven't gotten a chance to read it carefully, but the reviews are very good).&lt;/p&gt;&#xA;" OwnerUserId="10289" LastActivityDate="2022-02-26T02:24:20.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24976" PostTypeId="1" AcceptedAnswerId="25715" CreationDate="2022-02-26T04:49:46.533" Score="11" ViewCount="355" Body="&lt;p&gt;In undergraduate courses, students are often taught that a function is injective if it never maps distinct inputs to the same output, and it is surjective if every member of the codomain is an output.&lt;/p&gt;&#xA;&lt;p&gt;But in an undergraduate course, or a secondary-school course, one can easily also teach the idea that a function &lt;span class=&quot;math-container&quot;&gt;$f:A\to B$&lt;/span&gt; is injective if it has a left inverse and surjective if it has a right inverse. There is no need to mention categories and morphisms here, but this way of viewing the matter is in the spirit of category theory, i.e. it talks about things that in cateogory theory would be called objects or morphisms and not about members of the domain and the codomain.&lt;/p&gt;&#xA;&lt;p&gt;How far can one take this idea, fruitfully presenting ideas in the way in which one would in category theory without actually defining categories and without getting unreasonably complicated and while staying concrete so that students understand how it applies to concrete undergraduate subject matter?&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2022-10-24T02:22:12.920" Title="Category theory without categories and without theory" Tags="|undergraduate-education|category-theory|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="24977" PostTypeId="2" ParentId="4165" CreationDate="2022-02-26T06:31:37.963" Score="2" Body="&lt;p&gt;One can say &lt;span class=&quot;math-container&quot;&gt;$x\mapsto1/x$&lt;/span&gt; is continuous on &lt;span class=&quot;math-container&quot;&gt;$\mathbb R\cup\{\infty\}$&lt;/span&gt; where this is the &lt;span class=&quot;math-container&quot;&gt;$\text{“}\infty\text{”}$&lt;/span&gt; that is approached if one goes in either direction (so that the domain is topologically a circle) and the value of this function at &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; (not &lt;span class=&quot;math-container&quot;&gt;$\text{“}{+\infty}\text{”}$&lt;/span&gt; nor &lt;span class=&quot;math-container&quot;&gt;$\text{“}{-\infty}{”}$&lt;/span&gt;), and the limit of the function as the argument approached &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is also &lt;span class=&quot;math-container&quot;&gt;$\infty,$&lt;/span&gt; and the value of the function at &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$0.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This way of looking at things makes the tangent function continuous on &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; (but not on &lt;span class=&quot;math-container&quot;&gt;$\mathbb R\cup\{\infty\}.$&lt;/span&gt; If you want a discontinuity that fewer people will argue about, look at the discontinuity of the tangent function at &lt;span class=&quot;math-container&quot;&gt;$\infty.$&lt;/span&gt; And if it is objected that &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; is not in the domain so the function cannot have a discontinuity there, I will claim that it sometimes makes sense to speak of continuity on the &lt;em&gt;closure&lt;/em&gt; of the domain.&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2023-05-10T17:50:16.763" LastActivityDate="2023-05-10T17:50:16.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24978" PostTypeId="2" ParentId="5981" CreationDate="2022-02-27T02:53:17.593" Score="4" Body="&lt;p&gt;For nonnegative functions, define the integral to be the smallest member of &lt;span class=&quot;math-container&quot;&gt;$[0,+\infty]$&lt;/span&gt; (closed at both ends) that is not too small to be the integral. And define &amp;quot;too small&amp;quot; to mean smaller than the integral of a nonnegative function that takes only finitely many values and that nowhere exceeds the function to be integrated. And the integral of a function that takes only finitely many values is defined the way you would expect in terms of the values and the measures of those subsets of the domain where it takes those values. The measures of subsets of the domain may be defined merely &amp;quot;intuitively&amp;quot; in some contexts.&lt;/p&gt;&#xA;&lt;p&gt;For functions taking some positive and some negative values, treat the two parts separately and leave the integral undefined only when both parts are infinite.&lt;/p&gt;&#xA;&lt;p&gt;From this it follows that something like &lt;span class=&quot;math-container&quot;&gt;$\int_{\mathbb R} dx/(1+x^2)$&lt;/span&gt; is &lt;em&gt;not&lt;/em&gt; an &amp;quot;improper&amp;quot; integral, since it is not defined by first integrating over a bounded interval and then taking a limit as the bounds approach infinity.&lt;/p&gt;&#xA;&lt;p&gt;I would call attention to the distinction between iterated integrals and double integrals thus:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int\limits_{[a,b]} \left( \int\limits_{[c,d]} f(x,y)\, dy \right) \,dx \text{ versus } \int\limits_{[a,b]\times[c,d]} f(x,y)\, d(x,y)&#xA;$$&lt;/span&gt;&#xA;and illustrate that a thing that guarantees they are equal is &lt;em&gt;absolute convergence&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I would exhibit a simple case in which&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int\limits_A \lim_n f_n(x)\,dx \ne \lim_n\int\limits_A f_n(x)\,dx&#xA;$$&lt;/span&gt;&#xA;and explain that a thing that guarantees they are equal is that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int\limits_A \sup_n |f_n(x)|\,dx&amp;lt;+\infty.&#xA;$$&lt;/span&gt;&#xA;And I would say that Lebesgue's definitions are what facilitate proving these propositions.&lt;/p&gt;&#xA;&lt;p&gt;If this is well organized it could fit into an hour.&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2022-06-14T17:27:22.127" LastActivityDate="2022-06-14T17:27:22.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24979" PostTypeId="2" ParentId="5981" CreationDate="2022-02-27T06:51:27.540" Score="1" Body="&lt;p&gt;Supplement to other answers. Genius question for both you and the person who asked the ff, which I believe is the very heart of the difference of the 2 integrals.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;What is the benefit of using horizontal instead of vertical? Isn't that the same thing?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think of money. From &lt;a href=&quot;https://en.wikipedia.org/wiki/Lebesgue_integration#:%7E:text=Lebesgue%20summarized%20his%20approach%20to,This%20is%20the%20Riemann%20integral.&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To a non-maths major, just let them take for granted that somehow vertical integrals (Riemann integrals) can't integrate certain functions over certain sets the way horizontal integrals (Lebesgue integrals) can.&lt;/p&gt;&#xA;&lt;p&gt;P.S. Vertical vs horizontal should already be a great explanation. I think your question might be rephrased to asking about vertical vs horizontal specifically (unless you're open to other ways besides the vertical vs horizontal thing).&lt;/p&gt;&#xA;&lt;p&gt;P.P.S. I've been mentioned by Michał Miśkiewicz. Coooool.&lt;/p&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2022-02-28T07:17:27.153" LastActivityDate="2022-02-28T07:17:27.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24980" PostTypeId="2" ParentId="5981" CreationDate="2022-02-27T11:56:52.107" Score="6" Body="&lt;p&gt;I have the impression that the underlying problem is the expected value itself, not the integral (on which the expected value is based, of course). But since the question asks about the integral, I really don't see why people consider the Lebesgue integral inherently more difficult than the Riemann integral.&lt;/p&gt;&#xA;&lt;p&gt;For all the examples below, consider &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; to be a measure on (some &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra of) &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;h2&gt;The opposition &lt;em&gt;vertical&lt;/em&gt; vs &lt;em&gt;horizontal&lt;/em&gt; is false&lt;/h2&gt;&#xA;&lt;p&gt;It's not like we cannot introduce the Lebesgue integral by vertical slices (&lt;a href=&quot;https://en.wikipedia.org/wiki/Lebesgue_integration#Via_simple_functions&quot; rel=&quot;nofollow noreferrer&quot;&gt;see Wiki&lt;/a&gt;). The basic definition I was given in my second year analysis course was that the integral of a simple function &lt;span class=&quot;math-container&quot;&gt;$f = \sum_{i=1}^n a_i 1_{A_i}$&lt;/span&gt; is given by &lt;span class=&quot;math-container&quot;&gt;$\int_X f d\mu = \sum_{i=1}^n a_i \mu(A_i)$&lt;/span&gt;. One can define integrals for larger classes of functions by using a limiting process.&lt;/p&gt;&#xA;&lt;p&gt;Note that the integral &lt;span class=&quot;math-container&quot;&gt;$\int_X f d\mu$&lt;/span&gt; above is nothing else than looking at vertical slices &lt;span class=&quot;math-container&quot;&gt;$A_i \times [0,a_i]$&lt;/span&gt; and summing up their &lt;em&gt;measures&lt;/em&gt; (OK, one would have to introduce the product measure &lt;span class=&quot;math-container&quot;&gt;$\mu \otimes \lambda^1$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$X \times \mathbb{R}$&lt;/span&gt;, but that's not the point). The only difference between this and the Riemann integral is that the latter requires &lt;span class=&quot;math-container&quot;&gt;$A_i$&lt;/span&gt; to be intervals in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. Quite arbitrary, right? This takes us to the next point.&lt;/p&gt;&#xA;&lt;h2&gt;Riemann integration is built upon additional (and a posteriori, irrelevant) structure&lt;/h2&gt;&#xA;&lt;p&gt;Riemann integration requires us to look at intervals in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, while Lebesgue integration only sees the &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra of measurable sets, disregarding any (possible) existing metric or topological structure on &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; (although for most applications we do care about measuring Borel sets, and about some metric compatibility, too). Mathematicians around 1900 needed some time to arrive at this generalization and simplification, so why not embrace their wisdom?&lt;/p&gt;&#xA;&lt;p&gt;One application that a non-math major student could appreciate concerns summing up infinite series. Did you notice how confusing it is to study conditionally convergent series, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann_series_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann rearrangement theorem&lt;/a&gt; in particular? And how absolutely convergent series are better behaved, for some mysterious reasons?&lt;/p&gt;&#xA;&lt;p&gt;In a way, it's an artifact of how we introduce infinite summation:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;\label{eq:standard}&#xA;\sum_{n=1}^\infty a_n = S &#xA;\quad \Longleftrightarrow \quad &#xA;\forall_{\varepsilon &amp;gt; 0} \, \exists_{N &amp;gt; 0} \, \forall_{n&amp;gt;N} \, |a_1+\ldots+a_n - S| &amp;lt; \varepsilon.&#xA;\tag{$\star$}&#xA;\end{equation}&lt;/span&gt;&#xA;The order of the sequence &lt;span class=&quot;math-container&quot;&gt;$(a_n)$&lt;/span&gt; is encoded into the definition - a fact which most students (I, for one) miss at first encounter. Hence all the problems with rearrangements.&lt;/p&gt;&#xA;&lt;p&gt;The alternative Lebesgue summation approach is as follows. If &lt;span class=&quot;math-container&quot;&gt;$a_n \ge 0$&lt;/span&gt;, then let&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sum a := \sup \left\{ \sum_{j \in J} a_{j} : J \subseteq \mathbb{N} \text{ is finite} \right\}. &#xA;$$&lt;/span&gt;&#xA;This supremum may well be infinite, but it always exists. And of course, the result does not depend on the order of &lt;span class=&quot;math-container&quot;&gt;$(a_n)$&lt;/span&gt; in any way. One can check that &lt;span class=&quot;math-container&quot;&gt;$\sum a$&lt;/span&gt; coincides with the sum in \eqref{eq:standard}.&lt;/p&gt;&#xA;&lt;p&gt;For a general sequence &lt;span class=&quot;math-container&quot;&gt;$(a_n)$&lt;/span&gt;, consider its positive and negative part:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;a^+_n := \max(a_n,0), &#xA;\qquad &#xA;a^-_n := \max(-a_n,0). &#xA;$$&lt;/span&gt;&#xA;This way, &lt;span class=&quot;math-container&quot;&gt;$a_n = a^+_n - a^-_n$&lt;/span&gt;. If both &lt;span class=&quot;math-container&quot;&gt;$(a^+_n)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(a^-_n)$&lt;/span&gt; have finite sums, we put&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sum a := \sum a^+ - \sum a^-.&#xA;$$&lt;/span&gt;&#xA;Again, the order doesn't matter. Note that the assumption &lt;span class=&quot;math-container&quot;&gt;$\sum a^+, \sum a^- &amp;lt; \infty$&lt;/span&gt; is equivalent to &lt;span class=&quot;math-container&quot;&gt;$\sum |a| &amp;lt; \infty$&lt;/span&gt;, which is the same as saying that &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty |a_n|$&lt;/span&gt; is absolutely convergent.&lt;/p&gt;&#xA;&lt;p&gt;The best thing is, the definition above is not an ad hoc construction. It's simply the result of applying Lebesgue integration theory to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt; with the counting measure (no measurability nuisance involved!).&lt;/p&gt;&#xA;&lt;h2&gt;The Lebesgue integral can be built on top of the Riemann integral&lt;/h2&gt;&#xA;&lt;p&gt;There's a number of reasons why horizontal &lt;em&gt;is&lt;/em&gt; in fact better that vertical. BCLC's answer links to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Lebesgue_integration#Introduction&quot; rel=&quot;nofollow noreferrer&quot;&gt;money analogy Lebesgue himself made&lt;/a&gt; when explaining his integration theory. Jon Bannon's answer highlights the role of superlevel sets &lt;span class=&quot;math-container&quot;&gt;$S_{\lambda} := \{ x \in X : f(x) &amp;gt; \lambda \}$&lt;/span&gt;. Let me take it a bit further.&lt;/p&gt;&#xA;&lt;p&gt;The horizontal slice between two values &lt;span class=&quot;math-container&quot;&gt;$\lambda &amp;lt; \mu$&lt;/span&gt;, i.e. the set &lt;span class=&quot;math-container&quot;&gt;$\{ (x,t) : \lambda &amp;lt; t \le \min(\mu, f(x)) \}$&lt;/span&gt;, has measure between &lt;span class=&quot;math-container&quot;&gt;$(\mu-\lambda) \mu(S_\lambda)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(\mu-\lambda) \mu(S_\mu)$&lt;/span&gt;. If you sum up the contribution of all slices, and take the limit (as size of slices tends to zero), the integral turns out to be&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;\label{eq:horizontal}&#xA;\int_X f d \mu = \int_0^\infty \mu(S_\lambda) d\lambda,&#xA;\tag{$\star\star$}&#xA;\end{equation}&lt;/span&gt;&#xA;at least for &lt;span class=&quot;math-container&quot;&gt;$f \ge 0$&lt;/span&gt;. The cheapest way to have the Lebesgue integral is by taking \eqref{eq:horizontal} as the definition, where the left side is Lebesgue and the right side is Riemann. The necessary ingredients are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;determining &lt;span class=&quot;math-container&quot;&gt;$\mu(S_\lambda)$&lt;/span&gt; (you need to have a measure &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; to do this);&lt;/li&gt;&#xA;&lt;li&gt;making sure that &lt;span class=&quot;math-container&quot;&gt;$\lambda \mapsto \mu(S_\lambda)$&lt;/span&gt; is a Riemann-integrable function (it is, because it's monotone);&lt;/li&gt;&#xA;&lt;li&gt;defining the Riemann integral &lt;span class=&quot;math-container&quot;&gt;$\int_0^\infty \mu(S_\lambda) d\lambda$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2&gt;The are reasons why more powerful tools are better than less powerful tools&lt;/h2&gt;&#xA;&lt;p&gt;For historical and pedagogical reasons, it's nice to start with something less powerful, but conceptually simpler. And many times it's perfectly OK to stop there. But students can benefit from knowing that the integration theory doesn't stop on Riemann, and for a good reason.&lt;/p&gt;&#xA;&lt;p&gt;As already discussed in the other answers, Lebesgue's theory is useful for Fourier series and Fourier transform in general (due to limit theorems, I guess), calculus of variations (for the same reason, basically), probability theory (thanks to the general setting of measure spaces) and geometric applications (while one can extend Riemann integration to manifolds somehow, it's better to think about abstract measures).&lt;/p&gt;&#xA;&lt;p&gt;All right, most students won't need it personally, but at least they should know that the world has made progress since the XIX century. And if they're OK with using smartphones or WolframAlpha as calculators - just because there's a powerful and easy-to-use tool at hand - they might be OK with Lebesgue's integral as well.&lt;/p&gt;&#xA;" OwnerUserId="15367" LastEditorUserId="15367" LastEditDate="2022-06-14T21:43:02.303" LastActivityDate="2022-06-14T21:43:02.303" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24981" PostTypeId="2" ParentId="5981" CreationDate="2022-02-28T22:36:12.390" Score="0" Body="&lt;p&gt;Reframing: I wouldn't attempt this. After all, consider, the point is not the &lt;em&gt;construction&lt;/em&gt;, but the &lt;em&gt;properties&lt;/em&gt;. And we know what properties we'd &lt;em&gt;ideally&lt;/em&gt; like (modulo non-trivial technical details). Some not-so-crazy counter-examples show that, e.g., even if a pointwise limit of continuous functions on &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt; is continuous, the integral of the limit need not be the limit of the integrals, Riemann or otherwise. Yes, there are pointwise limits of continuous functions (on otherwise-nice compact sets, such as &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt;) that are not Riemann-integrable... but, in my practice, this is subordinate to the larger problems (about not generally being able to interchange integrals and &lt;em&gt;pointwise&lt;/em&gt; limits). That is, I think the serious issue is not about forming integrals, but about subtler notions of convergence than raw pointwise convergence.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-02-28T22:36:12.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24984" PostTypeId="2" ParentId="24971" CreationDate="2022-03-01T12:53:26.610" Score="3" Body="&lt;p&gt;One of many possible suggestions: make it interactive!&lt;/p&gt;&#xA;&lt;p&gt;As one example, see &lt;a href=&quot;https://davidaustinm.github.io/ula/sec-eigen-intro.html#activity-41&quot; rel=&quot;nofollow noreferrer&quot;&gt;this activity from David Austin's linear algebra text&lt;/a&gt;.  You could make it a game of sorts for choosing a matrix, and then sliding the vector around to see where it needs to go to be an eigenvector.  Try a few, and then start messing with the students by purposely going a direction that is less like an eigenvector.  Or do a bunch that have obvious ones, and then pick a matrix with no (real) eigenvectors (like a rotation matrix) and see how frustrated they get.&lt;/p&gt;&#xA;&lt;p&gt;This may sound corny or naive, but it really can work.  My &lt;a href=&quot;https://sagecell.sagemath.org/?z=eJydk01r4zAQhu-B_AcdCpIcbTdpyWXBsIHQ0kMvezXBqM4kFpUlMZ5k1_vrV0rWTkLaQiuwQdI7z4zmg1jO9hoFJy7Ho_Hop3EEqCsSVnd-R3lR8AVfqYLrHflyF9aagK9WUbyGDSvFIm80ofkjfi0fVFHM1DSKp2oWJerMJH_QtgX5YzxicT0to9t37Y6ahG_0K2x0BeK5t0wrWJs_Z3uoyKMQ36a3cxU_Kdm5BC8l99cSPKe8rcALxbUbCxsqoYPcGgeiiIGp6HklJ8HHPIq0b81fyOenE-xPThQ02_oCgxGDuFKVtx5zvkUAxwcE9tD37vHt-7O4m1jZOg8adQOxCFUZrCcxvLUgxWLOZvOsNU7cZcFkJFOeY02ZOF7OFZvK3gPCOvn_Km96SGwP86jdFi7iTT3wQbiVbwVJleAkB2Yw3-8G6IvV1euHMV5BWMIkSPwNHKv34NaAn0MdQOo-uwip6rT7FCZSjgiVUjhwWgPOaX7eUEA7dKe-mvR9OkmJnByqf1SHKKWuDJg6p6bGCt42xkLHtrtOMarBMYr1aDceG1izl47dLG64_O_sMY7xFnWoTdWWGlF3oiiGoX1aSjVsFnKY7Mfbtva_hW5DfGiJmozPZ_Ifu3thUg==&amp;amp;lang=sage&quot; rel=&quot;nofollow noreferrer&quot;&gt;favorite such activity in linear algebra&lt;/a&gt; is shamelessly borrowed/stolen from some colleagues at Stephen F. Austin State University, and the students love it.  Especially when I choose a singular matrix ... some of them then &amp;quot;improve&amp;quot; on the code.&lt;/p&gt;&#xA;&lt;p&gt;Obviously this is not a cure-all by any means.  But even engaging a third of the class highly, and another third passively, makes the remaining third much less influential in making a bad classroom vibe when you are trying your experiments.  And they may be more willing to &amp;quot;sit through&amp;quot; ones that are difficult to make interactive if you have a few that are.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2022-03-01T12:53:26.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24985" PostTypeId="1" AcceptedAnswerId="24987" CreationDate="2022-03-01T16:27:39.723" Score="12" ViewCount="6534" Body="&lt;p&gt;See the attached image:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/poKUW.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/poKUW.png&quot; alt=&quot;a photo of five true-or-false blanks with ambiguous &amp;quot;T&amp;quot; and &amp;quot;F&amp;quot; letters in each&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am having a difficult time grading this paper as I am not sure if the student intentionally wrote the answers in such a way that each answer looks like both &amp;quot;T&amp;quot; (or &amp;quot;t&amp;quot;) and &amp;quot;F&amp;quot; (or &amp;quot;f&amp;quot;) or if it is just his normal style of writing. He did a terrible job in answering other questions, if this information is even relevant.&lt;/p&gt;&#xA;&lt;p&gt;Questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How should I grade this student's answer? (The correct answers are &amp;quot;F&amp;quot;, &amp;quot;T&amp;quot;, &amp;quot;T&amp;quot;, &amp;quot;T&amp;quot;, and &amp;quot;F&amp;quot;.) If the student intentionally wrote ambiguously, should I give him zero for all these five questions? If yes, how to prove that he did this intentionally?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How should I design my questions to prevent this from happening in the future?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit 2022-03-04.&lt;/strong&gt; Mystery solved! I have talked to the student and he admits that he wrote his answers in frustration. His intended answers are TTTFT. Many thanks  to everyone for your input.&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="6166" LastEditDate="2022-03-04T21:21:30.870" LastActivityDate="2022-03-11T19:35:18.693" Title="How should I grade true-or-false questions if the student's writing is unclear?" Tags="|undergraduate-education|exams|grading|testing|test-design|" AnswerCount="12" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="24987" PostTypeId="2" ParentId="24985" CreationDate="2022-03-01T18:43:43.510" Score="34" Body="&lt;p&gt;I read them as TTTFT.  But the fourth is very hard to tell, the first somewhat hard to tell, and all show poor writing, perhaps indicating a lack of engagement. You could tell him that such terrible penmanship will get him zeroes in the future (not even the 50/50 chance he has going now).&lt;/p&gt;&#xA;&lt;p&gt;You could also print the T and F and have the kids circle one.  I actually find this quicker and easier to scan when grading.  (But don't feel like you have to change.)  E.g.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Circle one answer, &lt;strong&gt;T&lt;/strong&gt; true or &lt;strong&gt;F&lt;/strong&gt; false:&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;Choice&lt;/th&gt;&#xA;&lt;th&gt;Question&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;T F&lt;/td&gt;&#xA;&lt;td&gt;y=x^4 is the equation of a parabola.&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;T F&lt;/td&gt;&#xA;&lt;td&gt;x/5 + y/3.1415 =0 is the equation of a line.&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&lt;/blockquote&gt;&#xA;&lt;p&gt;FWIW, I wouldn't be so worried about static from the kid, unless you've had squabbles before.  He's going to get crushed on the test, overall, and will probably just take it on the chin.&lt;/p&gt;&#xA;&lt;p&gt;Edit:  I disagree with the comment that you should only have essay style (long answer) problems.  You should do a mix.  T/F is effective for checking tricky things, like the (very mildly) tricky questions I mentioned.  On the other hand, you should probably have more standard questions (sans tricks) for the long answer, worked problems. In other words, have a blend of problem types.&lt;/p&gt;&#xA;&lt;p&gt;Edit: I disagree with the suggestion to orally quiz the kid.  You don't have the time for that, especially if you have to initiate it yourself, versus the kid coming to you.  Plus he could have reviewed the materials, now.  If he actually meant something different, tough...this is a lesson to write more precisely in the future.  He will learn that harsh lesson much better from losing the points, than from pleas to write better.  And, we need to be a little sanguine about the possibility that at some point a student gets misgraded.  School is a practical process, not a theorem of Euclid.&lt;/p&gt;&#xA;" OwnerUserId="19587" LastEditorUserId="19587" LastEditDate="2022-03-02T17:47:52.647" LastActivityDate="2022-03-02T17:47:52.647" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="24988" PostTypeId="2" ParentId="2164" CreationDate="2022-03-01T19:23:46.280" Score="4" Body="&lt;p&gt;The reading assignments I assign (around 2 per week in a 3rd/4th year undergraduate class) are designed to help make sense of the textbook. Some favorite questions include:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Write down the details of [a step in a proof/example that the author claims is easy]&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;What does the author mean when they say [word whose use is particular to math or somehow unclear from context]&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;What is an example of something that fails to satisfy the criteria of [new definition]&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;What does [new theorem] say about [closely related example or non-example]&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Give a counter-example to [twist on new theorem]&amp;quot; etc&lt;/p&gt;&#xA;&lt;p&gt;Roughly, I try to ask them to do what I have taught myself to do when reading mathematics so that I am able to understand what is going on. It is easy to passively read mathematics and realize after the fact that you haven't engaged with it. These exercises (graded for completeness) are meant to at least let the student know that they don't yet understand it all. And then, we talk about the solutions to the reading assignment in class, and any questions they have (one of the reading assignment questions is always &amp;quot;what parts of the reading did you struggle with&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Hard to say with certainty that this helps students read the textbook, but they are reading it, and they are asking questions about the textbook, so I count that as progress if not success.&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2022-03-01T19:23:46.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24990" PostTypeId="2" ParentId="24985" CreationDate="2022-03-02T12:31:47.390" Score="2" Body="&lt;p&gt;Does it matter in terms of passing or failing?&lt;/p&gt;&#xA;&lt;p&gt;If it does you could mix those answers up and just ask him which letter is which.&lt;/p&gt;&#xA;&lt;p&gt;You can also make polls in future classes ;)&lt;/p&gt;&#xA;&lt;p&gt;An essential skill of an academic is making unambiguous statements. At least he clearly failed at that. If he's not a first-year student just giving him 0 points seems reasonable to me.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;You could provide boxes for them to check:&lt;/p&gt;&#xA;&lt;p&gt;T [_] F [_] How are you doing today?&lt;/p&gt;&#xA;" OwnerUserId="19590" LastActivityDate="2022-03-02T12:31:47.390" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="24991" PostTypeId="1" AcceptedAnswerId="24993" CreationDate="2022-03-02T14:42:27.003" Score="5" ViewCount="626" Body="&lt;p&gt;I am wondering the following. Suppose we have some particular space &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_1,x_2,x_3,x_4\in X$&lt;/span&gt; has the law of composition that works like Klein four-group. Is it correct to say that the structure that &lt;span class=&quot;math-container&quot;&gt;$x_i$&lt;/span&gt;'s forms is the Klein four-group or is the structure isomorphic to Klein four-group? I'm confused as when I learned algebra, there was not specified that in what space does the elements of some group, like Klein four-group, belongs to.&lt;/p&gt;&#xA;" OwnerUserId="19592" LastActivityDate="2022-03-03T02:07:43.917" Title="Should one teach to use equality or isomorphism in particular groups?" Tags="|abstract-algebra|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24993" PostTypeId="2" ParentId="24991" CreationDate="2022-03-02T15:12:46.613" Score="5" Body="&lt;p&gt;It depends on what foundation of mathematics is being used. Certain foundations, such as Zermelo-Frankel set theory or univalent type theory, have a notion of equality between set-level algebraic structures in a universe, while other foundations, such as the Elementary Theory of the Category of Sets, do not have such a notion of equality. So one may speak of either equality or isomorphism of set-level algebraic structures in the former foundations, but only of isomorphism of set-level algebraic structures in the latter.&lt;/p&gt;&#xA;&lt;p&gt;Since it is not true that one can speak of equality of set-level algebraic structures in all foundations, it would be better and clearer to simply use isomorphic/isomorphism in teaching for all set-level algebraic structures such as groups, since teaching is foundation-agnostic in practice.&lt;/p&gt;&#xA;" OwnerDisplayName="user19584" LastEditorDisplayName="user19584" LastEditDate="2022-03-02T15:52:39.607" LastActivityDate="2022-03-02T15:52:39.607" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24994" PostTypeId="2" ParentId="24991" CreationDate="2022-03-02T15:58:30.103" Score="5" Body="&lt;p&gt;Identifying structures with their isomorphism equivalence classes is a very useful perspective, but it can get really confusing if it's someone's primary perspective.&lt;/p&gt;&#xA;&lt;p&gt;Specifically, when one starts talking about subgroups (e.g. how many subgroups of G are isomorphic to the Klein four group?), one runs into problems: are all subgroups equal, since they're isomorphic? Are subgroups equal if there's an automorphism of G taking one to the other? Neither of these are particularly the question we'd like the answer to.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-03-02T15:58:30.103" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="24995" PostTypeId="2" ParentId="24985" CreationDate="2022-03-02T16:00:32.877" Score="-3" Body="&lt;p&gt;Just ask the student to clarify their answers, maybe in office hours or whatever's most convenient for you.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-03-02T16:00:32.877" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="24996" PostTypeId="2" ParentId="24985" CreationDate="2022-03-02T16:55:15.300" Score="8" Body="&lt;p&gt;This is an answer to the second question.&lt;/p&gt;&#xA;&lt;p&gt;If you are having problems distinguishing the written single-letter answers of this student, there are better ways to design the true/false questions such that you won't have the problems in the future. You could require the students to write the entire word &amp;quot;true&amp;quot; or &amp;quot;false&amp;quot;, to increase the chances that two answers are always distinguishable. If handwriting is still a problem, you could treat a true/false question as a multiple choice question with only 2 choices and have students circle the correct answer.&lt;/p&gt;&#xA;" OwnerDisplayName="user19584" LastEditorDisplayName="user19584" LastEditDate="2022-03-02T17:15:36.720" LastActivityDate="2022-03-02T17:15:36.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24997" PostTypeId="2" ParentId="24985" CreationDate="2022-03-02T21:27:09.367" Score="4" Body="&lt;p&gt;It is perfectly fine to ask students to write answers in as you have asked them on the exam, as simple T or F. But it is also acceptable to require clarity in letter formation. In fact it is good to require clarity in student's penmanship as penmanship is an aspect of literacy and an expectation of all educational institutions.  That is despite idiosyncrasies in one's letter formation, and so any student who retorts &amp;quot;that is my style of writing&amp;quot; can simply be informed that official documents have to be written in non-idiosyncratic styles, atleast as far as clarity is concerned. Any letters written unclearly are thus marked wrong because of a lack of clear communication.&lt;/p&gt;&#xA;" OwnerUserId="19595" LastActivityDate="2022-03-02T21:27:09.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24998" PostTypeId="2" ParentId="24985" CreationDate="2022-03-03T00:09:49.810" Score="2" Body="&lt;p&gt;It appears that answer 4 is an &amp;quot;F&amp;quot; because there's a second right-tick overlapping the line on the form.  The others don't have that, so they are &amp;quot;T&amp;quot; responses.  I would mark the paper with these interpretations, circled, next to the student's answers, then grade the paper as such.&lt;/p&gt;&#xA;&lt;p&gt;You should also encourage the student to improve their handwriting in a positive tone, explaining the problem.  Showing a little interest that way could have a very positive effect on their attitude and engagement.&lt;/p&gt;&#xA;" OwnerUserId="19597" LastActivityDate="2022-03-03T00:09:49.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="24999" PostTypeId="2" ParentId="24991" CreationDate="2022-03-03T02:07:43.917" Score="5" Body="&lt;p&gt;One should teach that in some situations you care about strict equality, in some situations you care about isomorphism, and doing a lot of fussing about terminology in a situation where you don't really care about the difference won't help you figure out what to do in a situation where you do care about the difference.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2022-03-03T02:07:43.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25000" PostTypeId="2" ParentId="24985" CreationDate="2022-03-03T19:54:05.780" Score="7" Body="&lt;p&gt;There seem to be two questions here.  I am going to answer them separately:&lt;/p&gt;&#xA;&lt;h4&gt;How should I mark ambiguous answers?&lt;/h4&gt;&#xA;&lt;p&gt;Students &lt;em&gt;often&lt;/em&gt; write things which are ambiguous or hard to understand.  For example, does&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/m9RZa.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/m9RZa.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;mean &lt;span class=&quot;math-container&quot;&gt;$\mathrm{e}^x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathrm{e}x$&lt;/span&gt;?  is this multiplication or exponentiation?  Arguably, the intended meaning is entirely clear from context, but there is still a problem in that one &lt;em&gt;must&lt;/em&gt; rely on that context to mark the answer.  Ambiguous responses like these represent a problem in the student's communication.  Depending on the objectives of your class or your personal philosophy of instruction, you might handle them differently:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I &lt;em&gt;explicitly&lt;/em&gt; include learning objectives related to communication in my syllabi:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[students will be able to] demonstrate an ability to clearly and effectively communicate mathematical ideas, both in writing and speech.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Because I have identified clear communication as a learning objective for the class, I consider ambiguous answers to be wrong—they fail to meet that objective.&lt;/p&gt;&#xA;&lt;p&gt;In practice, when I see an ambiguous answer on an assessment, I will mark it as wrong.  However, I also make it clear to students that they can appeal—if they believe their answer to be correct, they can discuss it with me, and I will generally regrade the assessment item.  In particular, if a student can explain the meaning of their notation verbally, they have (at least partially) met the communication objective, and the more mathematical objectives are more clearly met.  Most of the time, students end up with something like 2.999/3 points (the minuscule point deduction is a finger waggle in the direction of the student).&lt;/p&gt;&#xA;&lt;p&gt;The chief difficulty with this approach is one of scalability—it does not work well in a 600 student lecture.  However, I am at an institution where class sizes are small, so I don't have that problem.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even if &amp;quot;effective communication&amp;quot; is not something which has been explicitly spelled out as a learning objective, I do not think that it is unreasonable to mark ambiguous answers with &amp;quot;0 points; see me&amp;quot;.  Particularly in a large lecture class, but also in smaller classes, your time is a resource.  Is it more effectively spent trying to parse ambiguous answers, or engaging with a student after the fact about their ambiguous answer?&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, it is better for both you and the student to grade quickly, to err on the side of deducting points, and to be generous when regrading work when a student asks you to take a second look.  It is better for you because the reality of the world is that &lt;em&gt;most&lt;/em&gt; students are not going to ask for regrading (they should, but they won't); and it is better for the students because the ones that &lt;em&gt;do&lt;/em&gt; ask for regrading are going to have to engage in a conversation with you, and are likely to learn something from that interaction.&lt;/p&gt;&#xA;&lt;p&gt;In either case, it is unlikely that a few ambiguous answers are going to tank a student's grade, and you can take the &amp;quot;default harshness&amp;quot; into account when determining your grading scale.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You &lt;em&gt;could&lt;/em&gt; work very hard to interpret the student's ambiguous answer and grade it &amp;quot;as correctly&amp;quot; as possible.  For example, as many other people have noted, the answers to 1, 2, 3, and 5 all include only a single horizontal line, while the answer to 4 includes two horizontal lines.  It is nigh certain that the student intended &lt;code&gt;TTTFT&lt;/code&gt;.  Grade accordingly.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Alternatively, as noted above, it is unlikely that one or two questions are going to have much of an impact on the student's overall grade.  If you see an ambiguous answer which &lt;em&gt;could&lt;/em&gt; be correct, just give the student the benefit of the doubt and move on.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I don't think that this is the best approach, as the student might miss genuine opportunities to learn.  On the other hand, experience seems to indicate that most students won't bother to look for feedback, anyway, so there are only minimal opportunities to learn from mistakes, anyway.  Since a few ambiguous answers are not make-or-break, and since your time is valuable, assume the student meant whatever the right answer is, mark it as such, and go to the next exam.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h4&gt;How do I write assessments to avoid ambiguous answers?&lt;/h4&gt;&#xA;&lt;p&gt;Others have already covered much of this, but I have a few additional comments to offer (which are largely redundant, but I don't see an &lt;em&gt;answer&lt;/em&gt; which captures all of what has been said in the comments):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Avoid questions which require an answer which is likely to be ambiguous.  One letter answers are a potential source of ambiguity.  Why accept &lt;code&gt;T&lt;/code&gt; or &lt;code&gt;F&lt;/code&gt; when you could insist that they write out the entire word?  or, better yet, have them circle the correct answer or fill in a &amp;quot;bubble&amp;quot; corresponding to the correct answer?&lt;/p&gt;&#xA;&lt;p&gt;For example, you might try one of the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\bigcirc$&lt;/span&gt; True / &lt;span class=&quot;math-container&quot;&gt;$\bigcirc$&lt;/span&gt; False : Completely fill in the circle which corresponds to the correct answer.&lt;/p&gt;&#xA;&lt;p&gt;T / F  : Circle the correct answer.&lt;/p&gt;&#xA;&lt;p&gt;T ___ / F ___ : Mark the correct answer with an &lt;code&gt;x&lt;/code&gt;.&lt;/p&gt;&#xA;&lt;p&gt;________ : The correct answer is either &lt;code&gt;true&lt;/code&gt; or &lt;code&gt;false&lt;/code&gt;.  Write the entire word on the line.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;All of these are potentially open to abuse (i.e. a student could draw a large circle which captures both answers, or shade both bubbles, then erase them both in such a way that it is impossible to tell which answer is meant), but a student who is confident in their answer is much less likely to give an ambiguous answer to any of the above.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Don't ask true/false questions at all!  Or, better yet, ask true/false questions, but require justification.  Instructions for such a question might read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Determine if the following statement is &lt;code&gt;true&lt;/code&gt; or &lt;code&gt;false&lt;/code&gt; and justify your answer.  For example, if the statement is true, prove it; if the statement is false, give a counterexample: ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The specific instructions or forms of justification will vary by level (e.g. I don't think that you would generally ask a middle schooler to prove a statement), but it should be possible to ask this kind of question at any level.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2022-03-03T19:54:05.780" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25001" PostTypeId="2" ParentId="24985" CreationDate="2022-03-03T20:08:33.153" Score="2" Body="&lt;p&gt;Answers 1, 2 and 4 are not recognisable English characters so should be marked as incorrect.&lt;/p&gt;&#xA;&lt;p&gt;Answers 3 and 5 are close approximations of the letter T so if you are feeling generous you could them as such, making 3 correct and 5 incorrect.&lt;/p&gt;&#xA;&lt;p&gt;Communication is important, personally I would give them 0 marks for this but I like @Xander's idea of being open to an appeal as a chance to have a conversation with them.&lt;/p&gt;&#xA;" OwnerUserId="15119" LastActivityDate="2022-03-03T20:08:33.153" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25002" PostTypeId="2" ParentId="24985" CreationDate="2022-03-03T20:14:44.963" Score="2" Body="&lt;p&gt;You've tagged the question undergraduate-education. By this age it is expected that you are able to communicate clearly. If you have special needs that mean this is not possible, then these should be discussed with the academic staff before the exam. If a student has not provided a clear answer to the question, then they should not receive a mark for that question.&lt;/p&gt;&#xA;&lt;p&gt;Having said that, the answers in your photo all look like clear &amp;quot;t&amp;quot; characters to my eye. 1, 2, 3 and 5 are a straight vertical line that has been crossed, which is a simple way of writing a t. The closest &amp;quot;f&amp;quot; would have a distinct curve to the right at the top of the vertical line. There is a slight curve and forward lean on answer 1, but nowhere near enough for me to read that as an f.&lt;/p&gt;&#xA;&lt;p&gt;Answer 4 is different in that it has two clear right pointing horizontal strokes, but one of these is at the very bottom of the character. This is typical for a t, but any horizontal line at the bottom of an f would point to the left.&lt;/p&gt;&#xA;&lt;p&gt;Therefore I would take the answers to all five questions as being &amp;quot;true&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="19602" LastActivityDate="2022-03-03T20:14:44.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25003" PostTypeId="2" ParentId="24985" CreationDate="2022-03-04T01:23:34.547" Score="2" Body="&lt;p&gt;Student obviously wrote it hastily and carelessly, so you should read it hastily and carelessly.&lt;/p&gt;&#xA;&lt;p&gt;TTTFT. Done.&lt;/p&gt;&#xA;" OwnerUserId="13476" LastActivityDate="2022-03-04T01:23:34.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25004" PostTypeId="2" ParentId="24985" CreationDate="2022-03-04T14:16:32.520" Score="1" Body="&lt;p&gt;The student I am tutoring also has true false questions on his calculus test.  The answer choice is to mark it true or to give a counterexample if it's false.  I believe this option eliminates ambiguity and also challenges the student to show some real understanding.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2022-03-04T14:16:32.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25006" PostTypeId="1" CreationDate="2022-03-07T16:09:55.163" Score="13" ViewCount="2002" Body="&lt;p&gt;I have recently been noticing the tendency to use the term &amp;quot;perfect square&amp;quot; when &amp;quot;square number&amp;quot; is really what is meant.&lt;/p&gt;&#xA;&lt;p&gt;Usually I have seen it at elementary level: introductory algebra, popular puzzle pages, and so on.&lt;/p&gt;&#xA;&lt;p&gt;I confess I cringe at the term. There are already several usages in various branches of mathematics of the descriptor &amp;quot;perfect&amp;quot;, and applying it to the term &amp;quot;square&amp;quot; does not seem to be to be a useful one. For a start, it can confuse a bright but naïve student into wondering what such a square has to do with &amp;quot;perfect numbers&amp;quot;, and whether a &amp;quot;perfect square&amp;quot; means a &amp;quot;square number which happens also to be a perfect number&amp;quot; (and then go running off vainly to find one).&lt;/p&gt;&#xA;&lt;p&gt;Are there any &lt;em&gt;advantages&lt;/em&gt; to the term &amp;quot;perfect square&amp;quot;, or is it just to impress upon the student the gosh-wowery of a concept which is really pretty mundane? If you're in the realm of integers (sorry, we're at an elementary level here, &amp;quot;counting numbers&amp;quot;), then your number is going to be either &amp;quot;square&amp;quot; or it isn't. There's no such thing as an &amp;quot;imperfect square&amp;quot;, and while I will grant you that &lt;span class=&quot;math-container&quot;&gt;$143$&lt;/span&gt;, for example, is &amp;quot;almost square&amp;quot;, it's &lt;em&gt;not&lt;/em&gt; square.&lt;/p&gt;&#xA;&lt;p&gt;But I keep seeing it on the Mathematics forum, for no perceptibly useful effect.&lt;/p&gt;&#xA;&lt;p&gt;Any professional educators out there who find that expressing it as a &amp;quot;perfect square&amp;quot; aids understanding and doesn't hinder the learning process?&lt;/p&gt;&#xA;" OwnerUserId="15290" LastEditorUserId="376" LastEditDate="2022-03-07T18:35:31.427" LastActivityDate="2022-03-11T00:43:46.387" Title="Use of language: &quot;perfect square&quot;. is this useful or a hindrance?" Tags="|mathematical-pedagogy|terminology|" AnswerCount="5" CommentCount="9" ClosedDate="2022-03-09T02:24:33.000" ContentLicense="CC BY-SA 4.0" />
  <row Id="25007" PostTypeId="2" ParentId="25006" CreationDate="2022-03-07T17:34:22.220" Score="5" Body="&lt;ul&gt;&#xA;&lt;li&gt;perfect square&lt;/li&gt;&#xA;&lt;li&gt;absolutely perfect&lt;/li&gt;&#xA;&lt;li&gt;utterly exhausted&lt;/li&gt;&#xA;&lt;li&gt;pretty ridiculous&lt;/li&gt;&#xA;&lt;li&gt;quite certain&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In each phrase above, the modifier is added for emphasis rather than to indicate gradation/degree.&lt;/p&gt;&#xA;&lt;p&gt;Just as &amp;quot;perfect square&amp;quot; isn't meant to suggest that an imperfect square is a thing, &amp;quot;absolutely perfect&amp;quot; isn't intended to contrast with &amp;quot;partially perfect&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;These examples are of idiomatic English as used by fluent speakers/writers. (In fact, grammarians call “exhausted” an extreme, ungraded adjective for the reason just highlighted.)&lt;/p&gt;&#xA;&lt;p&gt;Although natural language is not a logical language even in mathematics—where context continues to implicitly guide usage and interpretation—it rightly pervades communication in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;The salient point is that there’s a difference between inaccurate language that reflects careless thinking, and &lt;em&gt;standard terms&lt;/em&gt; that aren’t epitomes of systematic design.&lt;/p&gt;&#xA;&lt;p&gt;P.S. Weaved in my comments that were previously below.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-03-08T05:04:28.447" LastActivityDate="2022-03-08T05:04:28.447" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25008" PostTypeId="2" ParentId="25006" CreationDate="2022-03-07T19:38:30.240" Score="13" Body="&lt;p&gt;Suppose the term &amp;quot;perfect square&amp;quot; was not in use.&lt;/p&gt;&#xA;&lt;p&gt;Then when I asked you if&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2 + 6x +7$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;was a square, you could conceivably say &amp;quot;sure, everything is a square:&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{x^2 + 6x + 7}^2 = x^2 + 6x + 7$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So I would need to clarify and invent new terminology. I would have to say that &lt;span class=&quot;math-container&quot;&gt;$x^2 + 6x + 9$&lt;/span&gt; is a &amp;quot;good square&amp;quot; as opposed to &lt;span class=&quot;math-container&quot;&gt;$x^2 + 6x + 7$&lt;/span&gt; which is admittedly a &amp;quot;square.&amp;quot; &amp;quot;Perfect square&amp;quot; seems like a fine choice for that term, because it correctly indicates that there is somehow a unique choice in some circumstances.&lt;/p&gt;&#xA;&lt;p&gt;Something like &amp;quot;polynomial square&amp;quot; might be more modern or precise wording, but your question is why we need a second word at all, and this is a good reason.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2022-03-07T19:38:30.240" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="25009" PostTypeId="2" ParentId="19303" CreationDate="2022-03-07T19:52:39.683" Score="2" Body="&lt;p&gt;Nebuchadnezzar:&lt;/p&gt;&#xA;&lt;p&gt;Your post contains two related questions.  (1) Do they know basic algebra? (2) Does it matter.  They should be considered separately.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Topic 1: Do they know basic algebra?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;You know the answer to this in your heart.  You tested them before and after the course and found them weak.&lt;/p&gt;&#xA;&lt;p&gt;Really you shouldn't even be surprised at even the entrance score performance, just based on that they are weaker STEMs and are taking a course that sounds like college algebra and &amp;quot;business calculus&amp;quot;.  Not the standard Calc 101, but something a little remedial.  Remedial students will not just be &amp;quot;behind&amp;quot;, they will be weaker at the basics, have lower SATs, etc.  ON AVERAGE, of course, class to class...there are always outliers...but we are discussing group differences.&lt;/p&gt;&#xA;&lt;p&gt;And then pedagogy (fancy, projects, group, offline, etc.) was not really oriented to helping them get better at the basics either.  You fell into the &amp;quot;project trap&amp;quot;.  Look at what we are doing?  How big, how fancy!  But then the basic learning did not happen.&lt;/p&gt;&#xA;&lt;p&gt;Btw, there is some discussion (as there usually is) about the dichotomy of &amp;quot;understanding&amp;quot; versus &amp;quot;drill&amp;quot;.  I personally think this dichotomy is overemphasized.  But in this case, it's tangential.  Your kids can't perform the skills regardless of if theory or drill would enable it.  And both tricky question understanding and demonstrate drill questions would be tough for them (with in class, unaided performance).&lt;/p&gt;&#xA;&lt;p&gt;Now, you do need to be reasonable.  You're not going to simultaneously teach them biz calc while giving them the manipulation skills of a 750M SAT, AP BC 5, IQ 140+ kid.  You just aren't.  All that said, you can still cover the ground.  And do a lot of GOOD!  And do some en passent strengthening of basic algebra.  And God knows freshman chem IS an applied algebra class!  But you need to follow a more standard, less ambitious plan.  Chop the stuff into simple, week-long segments.  Give drill homework (sure, explain it also, but give drill HW).  And then do an in-class test every end of week, with problems selected at random from the drill work. &lt;em&gt;sound of hands wiping past each other in that &amp;quot;done&amp;quot; meme&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Topic 2:  Does it matter?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Does it matter?  I think so, but this is much more debatable.&lt;/p&gt;&#xA;&lt;p&gt;If you had to go to a dictionary every time you needed to know a definition, how fast could you read?  These kids are going to see math in derivations and in homework problems in their chem/CS/etc. classes.  Freshman chem ESPECIALLY is almost just one giant algebra word problem (so is a lot of business consulting, accounting, etc.) And guest troll knows these two subjects.&lt;/p&gt;&#xA;&lt;p&gt;I think even your CSes will be helped from having some ability to do simple trouble shooting calcs for algorithms, loops, etc.  And nurses and doctors and pharms need to know how to dilute meds.&lt;/p&gt;&#xA;&lt;p&gt;It's not that there's no value to a &amp;quot;term paper&amp;quot;, but it's not the right priority.  Especially with weak students in an entrance course.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;P.s.  I probably mixed some 1 and 2 in my responses, not perfect disaggregation.  But hope the attempted issue analysis helps.&lt;/p&gt;&#xA;" OwnerUserId="19614" LastActivityDate="2022-03-07T19:52:39.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25010" PostTypeId="2" ParentId="24944" CreationDate="2022-03-07T22:44:04.043" Score="1" Body="&lt;p&gt;Have you tried bringing this to your math teacher's superiors? It seems to me a perfectly good case for them to make an exception to the &amp;quot;no calculator&amp;quot; rule. Normally teachers/schools don't let you use a calculator because they want to make sure the student actually knows how to do the operations and not blindly/fully depend on a calculator, but that's not the case for you.&lt;/p&gt;&#xA;&lt;p&gt;I would argue that you do understand the logic behind the math operations, you know how to solve them (and have already proved it) and it's only because your dyslexia that makes it more time consuming, time that is in the same amount given to you and to your non-dyslexic classmates when being tested. In summary, in your case the examination process it is not balanced compared to your classmates which translates into the low scores compared to your classmates. From there you could either argue that you need either more time allowed or the use of calculators to balance it out.&lt;/p&gt;&#xA;&lt;p&gt;I don't know the education rules in India but I'd think that at least there are some basic exceptions/accommodations for all types of cases. I say bring this to your teacher or your teacher's supervisors and I'm sure you can get to a type of understanding.&lt;/p&gt;&#xA;&lt;p&gt;My nephew has also dyslexia and is color blind, and in his school they did some exceptions for him. For example, for art classes they would allow him to participate in every activity just like his classmates but he was being tested only in activities that his condition affected him compared to his classmates. When it comes to math, calculators were not strictly prohibited so there wasn't much issue there although that doesn't mean it was a walk in the park, after all his dyslexia made it difficult to obtain the &amp;quot;know how&amp;quot; to solve problems in the first place. He normally would need math help so they recommended to him several different math websites like &lt;a href=&quot;https://www.khanacademy.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;KhanAcademy&lt;/a&gt; or &lt;a href=&quot;https://www.studypug.com/math-help/&quot; rel=&quot;nofollow noreferrer&quot;&gt;StudyPug&lt;/a&gt;, the advantage of these platforms is that the student can watch lessons and practice as many times as needed and he/she can create his/her own schedule. Most of them are free so no extra expenditure here, I'd strongly suggest that you try them out, it really made a difference in my nephew's dyslexia case.&lt;/p&gt;&#xA;" OwnerUserId="19616" LastActivityDate="2022-03-07T22:44:04.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25011" PostTypeId="2" ParentId="25006" CreationDate="2022-03-07T23:19:12.163" Score="11" Body="&lt;p&gt;According to &lt;a href=&quot;https://books.google.com/ngrams/graph?content=perfect+square&amp;amp;year_start=1700&amp;amp;year_end=2019&amp;amp;corpus=26&amp;amp;smoothing=3&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Google ngram&lt;/a&gt;, the term &amp;quot;perfect square&amp;quot; has been in use since at least the 18th century (there are earlier uses, but the ngram viewer reports usage as a percentage of words in the corpus, and I suspect that the smaller number of published works predating the 18th century distort things).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KNOKJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KNOKJ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is worth emphasizing that &amp;quot;perfect square&amp;quot; is a set phrase.  Like many set phrases (which occur in most languages), the phrase should not be parsed individually, but should be interpreted as a single, indivisible unit of syntax.&lt;sup&gt;[1]&lt;/sup&gt;  The phrase &lt;em&gt;does not&lt;/em&gt; imply that there are some squares which are perfect, and others which are not—the phrase should not—and &lt;em&gt;cannot&lt;/em&gt;—be parsed in this way.&lt;/p&gt;&#xA;&lt;p&gt;A discussion of &lt;em&gt;why&lt;/em&gt; this term developed, and whether or not it is distinguished from the notion of a &amp;quot;square&amp;quot; or &amp;quot;square number&amp;quot; is likely off-topic here, and is something that I lack the expertise to answer, anyway (you might try asking on &lt;a href=&quot;https://hsm.stackexchange.com/&quot;&gt;History of Science &amp;amp; Math&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;However, as a &lt;em&gt;pedagogical&lt;/em&gt; concern, the term &amp;quot;perfect square&amp;quot; is perfectly correct idiomatic mathematical English.  Indeed, I disagree with your premise that the term is &amp;quot;imprecise and woolly&amp;quot;.&lt;sup&gt;[2]&lt;/sup&gt; It is a well-understood term which, in most contexts, means &amp;quot;a ring element &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that there exists some ring element &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$a^2 = x$&lt;/span&gt;&amp;quot; (that is, a perfect square is an element of a ring which has a square root in that ring).&lt;/p&gt;&#xA;&lt;p&gt;Because this phrase is used and understood by mathematicians, and because part of the job of a mathematics educator is to teach students the language used by working mathematicians, it is appropriate to teach students the phrase.  As a matter of opinion, I don't think that you do any lasting harm by avoiding the term and using &amp;quot;square number&amp;quot; (or &amp;quot;square integer&amp;quot;) instead, but I'm not sure that you are doing any good, either.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;[1] For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;to kick the bucket&amp;quot; means to die—it has nothing to do with actually kicking a literal bucket;&lt;/li&gt;&#xA;&lt;li&gt;when one is &amp;quot;under the weather&amp;quot;, it means that one is not feeling well—again, a literal parsing is nonsense; and&lt;/li&gt;&#xA;&lt;li&gt;to apply &amp;quot;elbow grease&amp;quot; to a problem means to solve that problem with (possibly difficult) manual labor.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;None of these phrases make much sense if you try to parse them word-by-word, but they have set meanings in the English language.  Such phrasemes are common in human languages, including mathematical language.&lt;/p&gt;&#xA;&lt;p&gt;[2] If you would like to target your ire at a poorly defined, wooly, imprecise term which is commonly used in mathematics, I would suggest that you turn your eye toward the word &amp;quot;fractal&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2022-03-11T00:43:46.387" LastActivityDate="2022-03-11T00:43:46.387" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25012" PostTypeId="2" ParentId="25006" CreationDate="2022-03-08T01:39:33.327" Score="5" Body="&lt;p&gt;@PrimeMover, I guess the content of whatever terminology would be good here is that we mean &amp;quot;square of an integer&amp;quot;. &amp;quot;Square integer&amp;quot; is ambiguous, because, well, every integer is the square of a complex number, and complex numbers are standard. Positive integers are squares of real numbers, which are even more standard.&lt;/p&gt;&#xA;&lt;p&gt;So, seriously, how to distinguish? Partly by tradition, yes, the phrase &amp;quot;perfect square&amp;quot; has established itself as meaning &amp;quot;square of an integer&amp;quot;. It is a partly artifactual, but useful, terminology. And, at worst, harmless, so, what's any serious objection to it? It (in practice) removes ambiguity, unlike simply &amp;quot;square&amp;quot;... (&amp;quot;square of &lt;em&gt;what&lt;/em&gt;?&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Yes, one might declare that in certain highly controlled contexts there'd be no ambiguity, but, actually, mathematical contexts are not sufficiently centrally controlled so as to avoid this kind of thing. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-03-08T01:39:33.327" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25013" PostTypeId="2" ParentId="25006" CreationDate="2022-03-08T09:53:44.560" Score="0" Body="&lt;p&gt;I've some ideological sympathy with the question.&lt;/p&gt;&#xA;&lt;p&gt;If OTOH you want a hand in changing hundreds of years of tradition,... er... sorry I've not the musculature for that exercise .&lt;/p&gt;&#xA;&lt;p&gt;Here are some similar inapt adjectives that have annoyed me and slowed my students in my 3+ decades of teaching discrete math.&lt;/p&gt;&#xA;&lt;h1&gt;Many-One/One-Many Relation&lt;/h1&gt;&#xA;&lt;p&gt;A relation &lt;span class=&quot;math-container&quot;&gt;$R : A \leftrightarrow B$&lt;/span&gt; has property &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; defined by:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\forall x \in A,~ y_1, y_2 \in B: xRy_1 \land xRy_2 \Rightarrow y_1 = y_2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Decoded this means that &lt;strong&gt;there are not two different&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$y_1, y_2$&lt;/span&gt;s for any one &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;; ie there is no fan-out from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In an ideal world &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; would be named (something like) &lt;em&gt;un-one-many.&lt;/em&gt;&#xA;Tradition names it &lt;em&gt;many-one.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Doubly confusing since many-one and one-many are not negations of each other!&lt;/p&gt;&#xA;&lt;p&gt;[My solution: I name &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; as &lt;em&gt;functional&lt;/em&gt; and say in passing that math-books use many-one]&lt;/p&gt;&#xA;&lt;h1&gt;Partial Order&lt;/h1&gt;&#xA;&lt;p&gt;In English &lt;em&gt;partial&lt;/em&gt; invariably implies less-than-total.&lt;/p&gt;&#xA;&lt;p&gt;So if you say to me &lt;em&gt;You are partially correct.&lt;/em&gt; it contains the implication: &lt;em&gt;You are correct in abc and wrong in xyz&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;However in lattice theory etc partial orders can be total!&#xA;So mathematicians can blithely talk of &lt;em&gt;total partial orders&lt;/em&gt;,&#xA;which to an English-literate, math lay-person would (or should) sound like nonsense!&lt;/p&gt;&#xA;&lt;p&gt;Partial orders brings me to...&lt;/p&gt;&#xA;&lt;h1&gt;Subset vs Implies&lt;/h1&gt;&#xA;&lt;p&gt;&lt;sub&gt;Less related to question of inapt adjectives though generally related to inconsistent notation&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;p&gt;Subset and implies have an obvious relation in that:&#xA;&lt;span class=&quot;math-container&quot;&gt;$A \subseteq B ~~~\Longleftrightarrow~~~ x \in A \Rightarrow x \in B$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Yet we hardly notice that the subset has a line below for equal whereas the implies does not.&lt;/p&gt;&#xA;&lt;p&gt;I certainly had not noticed the inconsistency until someone asked a question here a couple of months ago. Sorry, cant seem to locate it.&lt;/p&gt;&#xA;&lt;p&gt;The essence of it as I remember: Why is not &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; the symbol for subset — proper or improper — with &lt;span class=&quot;math-container&quot;&gt;$\subsetneq$&lt;/span&gt; reserved for strict subset?&lt;/p&gt;&#xA;&lt;p&gt;People answered with drawing a parallel of &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;IMHO &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; is more related to &lt;span class=&quot;math-container&quot;&gt;$\Rightarrow$&lt;/span&gt; than to &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; insofar as by trichotomy for numbers &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;y$&lt;/span&gt; is false one of &lt;span class=&quot;math-container&quot;&gt;$x=y$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;y$&lt;/span&gt; must obtain whereas two random sets or propositions are more likely to be unrelated to each other.&lt;/p&gt;&#xA;" OwnerUserId="17672" LastEditorUserId="17672" LastEditDate="2022-03-08T14:19:42.587" LastActivityDate="2022-03-08T14:19:42.587" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="25015" PostTypeId="1" CreationDate="2022-03-08T17:25:32.360" Score="5" ViewCount="101" Body="&lt;p&gt;I'm interested in knowing whether there are conferences that have, among other topics, the difficulty of teaching higher level mathematical concepts to folks with prior established skills in STEM, e.g., software engineers.  An example paper topic might include assessing the difficulties of software engineers learning to use formal theorem and proof systems to help create better software.&lt;/p&gt;&#xA;&lt;p&gt;I have seen in [1] that the AMS lists general math conferences, but I'm wondering if there are more focused, practical math conferences that deal with education and topics such as this, addressing education of a more technical audience (not an elementary education audience).  Such conferences may not be &amp;quot;math&amp;quot; conferences per se.&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;&lt;p&gt;[1] &lt;a href=&quot;https://math.stackexchange.com/q/289410&quot;&gt;https://math.stackexchange.com/q/289410&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19621" LastActivityDate="2022-03-08T19:30:48.807" Title="Conferences dealing w/challenges of teaching higher level math to STEM audiences" Tags="|education-research|graduate-education|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25016" PostTypeId="2" ParentId="25015" CreationDate="2022-03-08T19:21:56.730" Score="-1" Body="&lt;p&gt;I followed your link in the MSE thread (and did searches on that site) and did a couple Google searches, for math ed conferences and and CS ed conferences.  Didn't find a perfect specialty conference fit.&lt;/p&gt;&#xA;&lt;p&gt;There are some mega conferences in math ed and CS ed.  Think your chances are better with the CS ed ones.  Look bigger and better funded.  They have a broad range of topics (probably breakout sessions) and will find you a home in one of them.&lt;/p&gt;&#xA;&lt;p&gt;You can also look at the list and pick the area nearest to you.  (Your first critical thinking test.)  It really doesn't matter that you are restricting the range to practicing engineers or to proof methods.  If anything, having a smaller scope might enable some better work from you, and the conference won't mind either...they end up with all sorts of snips and snails.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.sigcse.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.sigcse.org/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://waset.org/computer-science-education-conference-in-april-2023-in-paris&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://waset.org/computer-science-education-conference-in-april-2023-in-paris&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(there are more on the Google search for computer science education conference)&lt;/p&gt;&#xA;&lt;p&gt;Note, that CS as a field, really emphasizes conferences as opposed to journals for papers.  (Do they have more money to travel?  I don't get it.)  But this is not the case in hard sciences, which I know, or (I suspect) in mathematics.  So if you just want to do some research and write a paper and send it to a journal (lots of them in math or CS ed), nothing stopping you.  I think having done the work, you'd also be more informed in general about the field and likely to stumble onto a good conference if you like to travel and drink beer (I mean have scholarly conversations) with your peers.&lt;/p&gt;&#xA;&lt;p&gt;Note that I don't think you have to solve all the broader issues, great that you have a restricted scope.  But I think you should be somewhat sensitive and aware of some of the broader issues behind your topic.  E.g. software engineers aren't as good at math, as a group, as BS math students on proofy (research) tracks.  This is probably the second biggest hurdle.  Also given all they need to know (in CS, letalone in math), doing proofs may have limited utility for them...I suspect this is the biggest hurdle.  Of course there may be other issues (not having tailored coursework, working for a living versus being in school).&lt;/p&gt;&#xA;&lt;p&gt;But, I advise to read this article, for some situational awareness:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://link.springer.com/article/10.1007/s42979-021-00461-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/article/10.1007/s42979-021-00461-7&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;See in particular the discussion of refs 37-39 and those papers themselves.  Even if you have a strong bias that those darned CS people need to know proofs, it would be good for you to consider the counterarguments thoughtfully.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  Don't take my comments on scholarly beer drinking the wrong way.  Think it's great.  Also, honestly, it's doubtful you find a wonderful Gordon conference style thing with focus on your topic...but then I actually think you'd benefit also from considering tangential topics.  (Just my Bayesian assessment based on how you described your topic, doesn't seem like you are much advanced in researching and writing about it yet.)&lt;/p&gt;&#xA;" OwnerUserId="19622" LastEditorUserId="19622" LastEditDate="2022-03-08T19:30:48.807" LastActivityDate="2022-03-08T19:30:48.807" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25017" PostTypeId="1" AcceptedAnswerId="25026" CreationDate="2022-03-08T19:54:22.463" Score="19" ViewCount="3427" Body="&lt;p&gt;I am teaching a class on linear algebra to sophomore and junior science majors, and am having some trouble illustrating the difference between &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; and an n-dimensional vector space. The main example the textbook uses is the set of polynomials with degree &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt; or less, but the students see this as essentially the same as &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; because the canonical polynomial basis &lt;span class=&quot;math-container&quot;&gt;$1, t, t^2, $&lt;/span&gt; etc is so immediate. A couple were convinced that &lt;span class=&quot;math-container&quot;&gt;$1, (t-1), (t-1)^2$&lt;/span&gt; etc . would work too, but most thought that this was just as unnatural as using anything but the standard basis in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;. I told them that it will be important to be able to change bases even in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; later, but I would like to do more for them now.&lt;/p&gt;&#xA;&lt;p&gt;The set of colors is &lt;a href=&quot;https://math.stackexchange.com/questions/1777699/color-space-as-a-vector-space&quot;&gt;almost a vector space&lt;/a&gt;, and they were convinced more-or-less that for this to make sense, you'd have to agree ahead of time if you were going to use RYB or CMY and that this amounts to choosing a basis. (Apologies if there is a slight abuse here, the argument I gave them was mostly by analogy.)&lt;/p&gt;&#xA;&lt;p&gt;Can you think of other examples where the basis of the (finite dimensional, please!) vector space is not apparent, or many bases appear without one being obviously simplest/best?&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2023-05-17T21:31:31.130" Title="Concrete vectors spaces without an obvious basis or many &quot;obvious&quot; bases?" Tags="|undergraduate-education|examples|linear-algebra|" AnswerCount="18" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25018" PostTypeId="2" ParentId="25017" CreationDate="2022-03-08T20:52:24.310" Score="10" Body="&lt;p&gt;I think you're on the right track with the polynomials.  They're not wrong that &lt;span class=&quot;math-container&quot;&gt;$(a,b,c)\mapsto (x\mapsto ax^2+bx+c)$&lt;/span&gt; is an obvious linear isomorphism from &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^3$&lt;/span&gt; to what I will call &lt;span class=&quot;math-container&quot;&gt;$\text{Poly}_2(\mathbb R)$&lt;/span&gt;, but it is not the only useful one.  So is &lt;span class=&quot;math-container&quot;&gt;$$\Phi(a,b,c):\in \{f\in\text{Poly}_2(\mathbb R)\mid f(0)=a,f(1)=b,f(2)=c\}$$&lt;/span&gt;  Master this identification and the basis of &lt;span class=&quot;math-container&quot;&gt;$\text{Poly}_2(\mathbb R)$&lt;/span&gt; that is the image of the canonical basis of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^3$&lt;/span&gt; under this identification, and you have rediscovered Lagrange interpolation.  Yet another example is &lt;span class=&quot;math-container&quot;&gt;$$\Psi(a,b,c):\in \{f\in\text{Poly}_2(\mathbb R)\mid f(0)=a,f'(0)=b,f''(0)=c\}$$&lt;/span&gt; Admittedly, the basis elements at the heart of this is just scalar multiples of what your students are thinking about, but thinking about it with this framing gives us Taylor polynomials.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastEditorUserId="12619" LastEditDate="2022-03-08T22:32:27.053" LastActivityDate="2022-03-08T22:32:27.053" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25019" PostTypeId="2" ParentId="25017" CreationDate="2022-03-08T20:52:50.917" Score="2" Body="&lt;p&gt;Here is a useful project in this regard:&lt;/p&gt;&#xA;&lt;p&gt;We all know about b^2-4ac, the discriminant for a quadratic polynomial.  It tells us, in terms of the coefficients of a quadratic polynomial, whether the polynomial has a double root.&lt;/p&gt;&#xA;&lt;p&gt;We would like to do the same for a cubic polynomial.  To simplify, let's make the polynomial monic, so we would like a polynomial in b, c, and d that tells us whether the cubic polynomial x^3+bx^2+cx+d has a double root.&lt;/p&gt;&#xA;&lt;p&gt;Now, if we let r, s, and t be the roots of the polynomial, then the polynomial is (x-r)(x-s)(x-t), and b, c, and d are the elementary symmetric functions in terms of r, s, and t - to be precise, b=r+s+t, c=rs+st+rt, and d=rst.&lt;/p&gt;&#xA;&lt;p&gt;At the same time, the polynomial as a double root if (r-s)(r-t)(s-t)=0.  This isn't symmetric, so it's not a polynomial in b, c, and d, but its square is symmetric.&lt;/p&gt;&#xA;&lt;p&gt;Now we can express (r-s)^2(r-t)^2(s-t)^2 easily in terms of the monomial basis of symmetric functions (in r, s, and t).  What we want here is the change of basis to the elementary basis of symmetric functions, so that we write this expression as monomials in b=r+s+t, c=rs+st+rt, and d=rst.&lt;/p&gt;&#xA;&lt;p&gt;Warning: I have had students who found this project too theoretical for their tastes; they didn't care enough about the cubic discriminant to think it worth spending that much time on.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2022-03-08T20:52:50.917" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25020" PostTypeId="2" ParentId="25017" CreationDate="2022-03-08T21:28:21.523" Score="15" Body="&lt;p&gt;Two more examples:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The set of infinite Fibonacci-type sequences (those of the form &lt;span class=&quot;math-container&quot;&gt;$a_n=a_{n-1} + a_{n-2}$&lt;/span&gt;) (with point-wise addition and scaling) forms a 2-dimensional (real) vector space. E.g., elements like:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a_n = \left(2,8,10,18,28,46,...\right)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$b_n = \left(-3,1,-2,-1,-3,-4,...\right)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where one basis for the space is the set:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$e_1 = \left(1,\frac{1-\sqrt{5}}{2},\left(\frac{1-\sqrt{5}}{2}\right)^2,...\right)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e_2 = \left(1,\frac{1+\sqrt{5}}{2},\left(\frac{1+\sqrt{5}}{2}\right)^2,...\right)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;The set of &lt;span class=&quot;math-container&quot;&gt;$3\times 3$&lt;/span&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Magic_square&quot; rel=&quot;nofollow noreferrer&quot;&gt;magic squares&lt;/a&gt; (with point-wise addition and scaling) forms a 3-dimensional (real) vector space. E.g., elements like:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$A=\begin{bmatrix}&#xA;2&amp;amp;7&amp;amp;6 \\&#xA;9&amp;amp;5&amp;amp;1 \\&#xA;4&amp;amp;3&amp;amp;8&#xA;\end{bmatrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where one basis for the space is the set:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$e_1=\begin{bmatrix}1&amp;amp;1&amp;amp;1 \\ 1&amp;amp;1&amp;amp;1 \\ 1&amp;amp;1&amp;amp;1\end{bmatrix}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e_2=\begin{bmatrix}1&amp;amp;-1&amp;amp;0 \\ -1&amp;amp;0&amp;amp;1 \\ 0&amp;amp;1&amp;amp;-1\end{bmatrix}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e_3=\begin{bmatrix}0&amp;amp;-1&amp;amp;1 \\ 1&amp;amp;0&amp;amp;-1 \\ -1&amp;amp;1&amp;amp;0\end{bmatrix}$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2022-03-23T14:55:35.680" LastActivityDate="2022-03-23T14:55:35.680" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25021" PostTypeId="2" ParentId="25017" CreationDate="2022-03-09T01:48:16.370" Score="14" Body="&lt;p&gt;That is a linear algebra course?  So presumably before you get to this point of abstract vector space, you already did solution of systems of linear equations?  For example, solution of matrix equation &lt;span class=&quot;math-container&quot;&gt;$Mx=0$&lt;/span&gt; (say &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; is a given &lt;span class=&quot;math-container&quot;&gt;$p \times q$&lt;/span&gt; matrix, and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is an unknown &lt;span class=&quot;math-container&quot;&gt;$q \times 1$&lt;/span&gt; column).  In general, the set of solutions is a vector space, but has no canonical choice of basis.&lt;/p&gt;&#xA;&lt;p&gt;Then you can discuss the similar case of a linear (ordinary) homogeneous differential equation with variable coefficients.  The set of solutions is a vector space, but in general has no canonical choice of basis.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2022-03-09T01:48:16.370" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25022" PostTypeId="2" ParentId="25017" CreationDate="2022-03-09T03:45:04.933" Score="5" Body="&lt;p&gt;To simplify some examples already given here, consider &lt;span class=&quot;math-container&quot;&gt;$\{(x,y) : x+y=0\}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\{(x,y,z) : x+y+z = 0\}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-03-09T03:45:04.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25023" PostTypeId="2" ParentId="25017" CreationDate="2022-03-09T03:48:34.063" Score="3" Body="&lt;p&gt;Somewhat in the spirit of @AlexanderWoo's answer, one could consider the collection(s) of linear polynomials on &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; vanishing at &lt;span class=&quot;math-container&quot;&gt;$x_o$&lt;/span&gt;, or quadratic polynomials vanishing at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;, ... or at both &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt; ... and/or cubic polynomials vanishing at ...&lt;/p&gt;&#xA;&lt;p&gt;Vanishing conditions on (finite or infinite-dimensional) spaces of functions are genuine...&lt;/p&gt;&#xA;&lt;p&gt;And &amp;quot;orthogonality&amp;quot;: cubic polynomials on &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; whose integrals over &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. Or, whose integrals against &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-03-09T03:48:34.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25024" PostTypeId="2" ParentId="25017" CreationDate="2022-03-09T08:12:26.000" Score="5" Body="&lt;p&gt;Consider physical values (things like 5 m/s, 17 keV, 42.34) with:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the multiplication of values as vector addition,&lt;/li&gt;&#xA;&lt;li&gt;the exponentiation of values as scalar multiplication.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;You get a mix between ℤ-module  and vector space, which you might treat like a vector space for applications.&lt;/p&gt;&#xA;&lt;p&gt;The typical basis for this vector space are the SI base units (and 10), but this is clearly a human convention and you can as well choose &lt;a href=&quot;https://en.wikipedia.org/wiki/Centimetre%E2%80%93gram%E2%80%93second_system_of_units&quot; rel=&quot;nofollow noreferrer&quot;&gt;the CGS system&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Natural_units&quot; rel=&quot;nofollow noreferrer&quot;&gt;natural units&lt;/a&gt;, imperial units, etc.&#xA;Mind that in particular natural units differ by more than scaling of the basis vectors, i.e., basis vectors point in different directions.&#xA;One practical application of this vector space is transforming values to a different unit system.&lt;/p&gt;&#xA;&lt;p&gt;Blatant self-advertisement: I wrote &lt;a href=&quot;https://doi.org/10.1088/0143-0807/36/3/035008&quot; rel=&quot;nofollow noreferrer&quot;&gt;a didactic paper&lt;/a&gt; on this (&lt;a href=&quot;https://arxiv.org/abs/1504.02332&quot; rel=&quot;nofollow noreferrer&quot;&gt;preprint&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2022-03-11T11:31:06.543" LastActivityDate="2022-03-11T11:31:06.543" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25026" PostTypeId="2" ParentId="25017" CreationDate="2022-03-09T10:10:08.540" Score="27" Body="&lt;p&gt;Some physical examples from physics:&lt;/p&gt;&#xA;&lt;p&gt;Consider two spaceships that meet each other in deep space with arbitrary orientations (pitch, roll, and yaw). Even if they take the origin to be the midpoint between them, each spaceship will having their own coordinate system that is rotated with respect to the other. Each can claim to be using the standard &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; basis for their own ship, but they will both describe positions in the other ship as if they were using a different basis. There's no natural basis that both ships will choose.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Another physical example involves the following mass and springs system (&lt;a href=&quot;http://spiff.rit.edu/classes/phys283/lectures/eigen_ii/eigen_ii.html&quot; rel=&quot;noreferrer&quot;&gt;link to source&lt;/a&gt;):&#xA;&lt;a href=&quot;https://i.stack.imgur.com/F0b07.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/F0b07.png&quot; alt=&quot;A system of two masses connected to each other with a spring and each connected to walls on opposite ends by two more identical springs.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This system has two degrees of freedom and can be represented by a 2D vector space. An intuitive one would be to specify the displacement of both blocks from their equilibrium positions, yielding &lt;span class=&quot;math-container&quot;&gt;$(x_1(t), x_2(t))$&lt;/span&gt;. However, solving the motion of the blocks in this coordinate system is very complicated with their motion being coupled. Using &lt;span class=&quot;math-container&quot;&gt;$x_1 + x_2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_1 - x_2$&lt;/span&gt; as a basis decouples the equations and makes them much easier to solve (the eigenvalue analysis is what finds this basis).&lt;/p&gt;&#xA;&lt;p&gt;Transforming the coordinate system using the eigenvector basis reveals that this system has two characteristic (&amp;quot;eigen&amp;quot;) motions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The boxes always move in the same direction back and forth in sync with each other.&lt;/li&gt;&#xA;&lt;li&gt;The boxes always move in opposite each other, first towards each other, then away from each other.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;All other possible motions are linear combinations of these eigenvectors. Thus, the space of motions can be represented by a 2D vector space. There are two natural bases: the physical position basis of &lt;span class=&quot;math-container&quot;&gt;$(x_1(t), x_2(t))$&lt;/span&gt;, and the eigenmode basis. Transforming to the eigenmode basis is 90% of the solution.&lt;/p&gt;&#xA;&lt;p&gt;Before seeing the problem worked out, I don't think anyone would naturally see a vector space in the motion of these masses. &lt;a href=&quot;https://www.youtube.com/watch?v=Ye92jN6FrlU&quot; rel=&quot;noreferrer&quot;&gt;Here's a video&lt;/a&gt; that has demonstrations with two pendulums linked by springs (similar math) and a system of three masses in a similar configuration.&lt;/p&gt;&#xA;&lt;p&gt;If your students are not ready for calculus or differential equations, you can skip the derivation and present the eigenmodes and the linear combinations of motions--especially if you can demonstrate it physically or with animations.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A simpler physics example: consider a box sliding down an incline. The natural basis of all motion problems is to put the x-axis parallel to the flat ground and the y-axis parallel to gravity. A basis that makes the math easier to to align the x-axis with the incline. After doing some trigonometry to get the force of gravity along the incline, the one-dimensional motion only requires one coordinate to vary.&lt;/p&gt;&#xA;" OwnerUserId="3344" LastEditorUserId="3344" LastEditDate="2023-05-17T21:31:31.130" LastActivityDate="2023-05-17T21:31:31.130" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25027" PostTypeId="1" AcceptedAnswerId="25032" CreationDate="2022-03-09T10:37:24.043" Score="4" ViewCount="292" Body="&lt;p&gt;I'm wondering about the relationship between Eculid's work and modern high school geometry. In &amp;quot;two column proofs,&amp;quot; certain reasons are considered acceptible for steps in the proof, such as those shown below, from &lt;a href=&quot;https://www.somerset.k12.ky.us/userfiles/104/Classes/974/Geometry%20--%20Chapter%206%20PROOFS%20REFERENCE%20SHEET%202014%20BLANK%20COPY.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;#16. Definition of alternate interior angles – says that “If two angles are alternate interior, then they are on opposite sides of a transversal and are both on the interior to two lines (whether parallel or not).”&lt;/p&gt;&#xA;&lt;p&gt;#17. Alternate interior angle theorem – says that “If two lines are parallel and alternate interior angles are formed, then the angles will be congruent to one another.”&lt;/p&gt;&#xA;&lt;p&gt;#18. Converse of alternate interior angle theorem – says that “If alternate interior angles are&#xA;congruent, then the lines that form them will be parallel to one another.”&lt;/p&gt;&#xA;&lt;p&gt;It strikes me that many of the reasons considered acceptable in these kinds of proof are similar to, but not identical to, postulates, definitions etc. given in Euclid.&lt;/p&gt;&#xA;&lt;p&gt;What I'm wondering is if there is some kind of modern standard for these kinds of reasons, that captures the essence of Euclid's reasons, but which uses modern insights and concepts to make them more accessible.&lt;/p&gt;&#xA;&lt;p&gt;By which I mean basically, is there some kind of canonical source for what counts as valid reasons for steps in modern high school geometry proofs, or is what we have a kind of collective cultural habit that is loosely based on Euclid's work but is not in anyway standardized?&lt;/p&gt;&#xA;" OwnerUserId="17703" LastActivityDate="2022-03-10T03:52:03.427" Title="Valid Reasons in Two-Column Geometry Proofs" Tags="|secondary-education|geometry|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25028" PostTypeId="2" ParentId="25017" CreationDate="2022-03-09T12:24:32.133" Score="8" Body="&lt;p&gt;Here's another example, and a comment. First the comment:&lt;/p&gt;&#xA;&lt;p&gt;I think this is a great opportunity to talk to students about a trend in abstract mathematics where we progressively transition more and more things from &amp;quot;intuitively available&amp;quot; to &amp;quot;defined as part of the mathematical structure&amp;quot;. A basis is a &lt;em&gt;choice&lt;/em&gt; you make, without it the vector space is just a blank slate and you have no coordinates. You only get coordinates once you've chosen the basis. The &amp;quot;obvious&amp;quot; basis for students is always defined in terms of some already available external frame (in class this is often the edges of the board!). But even if they all accepted that changing bases is important... it's the same deal all over again with the inner product - now you're going to tell me there's more than one way for vectors to be orthogonal!? And there's more than one way to measure the length of vectors? And more than one way to measure length and volume (measure theory)? Or multiply things together (abstract algebra)?&lt;/p&gt;&#xA;&lt;p&gt;So it's a good opportunity to try to impress upon them that challenging the intuitive &amp;quot;this is the obvious way to do X&amp;quot; (for X = describe coordinates in this case) and putting in place some mathematical methods to describe that is useful. They might not get it completely, but it will give them a hint of what's to come. And for when they ask why: because it allows solving so many, oh so very many problems (as the many answers here show).&lt;/p&gt;&#xA;&lt;p&gt;At any rate with that aside, this example is similar to Mark H's spaceships:&lt;/p&gt;&#xA;&lt;p&gt;If you consider the sphere as a 2 dimensional manifold and fix a point p on the sphere, you can talk about the tangent space at that point. It's a two dimensional space, how do you choose a basis? There's nothing on the sphere to give any one basis a preference (cue discussion about the basis being a choice). You can guide them through a thought experiment where two teams get transported onto the same point of, say, the moon, and they choose a basis (draw it on the board without seeing what the other team drew, but don't let them use the edges of the board or the ceiling/walls as reference). Now they have two different maps around that spot. They find an alien ship at a different spot, how can they communicate with each other? They need to convert one team's coordinates to the other's. Try to make them see that any basis that seems &amp;quot;natural&amp;quot; is really only defined in terms of some other existing basis in their ambient environment.&lt;/p&gt;&#xA;" OwnerUserId="19624" LastActivityDate="2022-03-09T12:24:32.133" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25029" PostTypeId="2" ParentId="25017" CreationDate="2022-03-09T12:27:03.167" Score="5" Body="&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt; is the space of polynomials of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; or less, there are natural reasons why the canonical basis is less useful than others. You can show them that while &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt; has a natural Euclidean structure (that is, an inner product that induces the natural distance), using it in &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt; gives you a structure that completely ignores the fact that polynomials are functions. From this point of view, the Euclidean inner product is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\langle p,q\rangle_E=\sum_{k=0}^n\frac{p^{(k)}(0)\,q^{(k)}(0)}{(k!)^2},&#xA;$$&lt;/span&gt;&#xA;while there are more natural inner products like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\langle p,q\rangle_N=\sum_{k=0}^n p (k)q(k)&#xA;$$&lt;/span&gt;&#xA;and, with more long-term consequences,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\tag1&#xA;\langle p,q\rangle_2=\int_0^1p(t)q(t)\,dt.&#xA;$$&lt;/span&gt;&#xA;For these  products it is easy to see that the canonical basis is not orthonormal. The natural orthonormal basis for &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; are the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Legendre_polynomials&quot; rel=&quot;nofollow noreferrer&quot;&gt;Legendre polynomials&lt;/a&gt;, which come from doing Gram-Schmidt to the monomials. The article mentions several applications in physics.&lt;/p&gt;&#xA;&lt;p&gt;For an example with a different flavour, in Quantum Information many things happen in, or use, &lt;span class=&quot;math-container&quot;&gt;$M_2(\mathbb C)$&lt;/span&gt;. While there is an obvious canonical basis (the matrix units &lt;span class=&quot;math-container&quot;&gt;$E_{k j}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; in the &lt;span class=&quot;math-container&quot;&gt;$k,j$&lt;/span&gt; entry and zeroes elsewhere), the preferred basis is the &lt;em&gt;Pauli basis&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{bmatrix}1&amp;amp;0\\0&amp;amp;1\end{bmatrix},\quad&#xA;\begin{bmatrix}0&amp;amp;1\\1&amp;amp;0\end{bmatrix},\quad&#xA;\begin{bmatrix}0&amp;amp;-i\\i&amp;amp;0\end{bmatrix},\quad&#xA;\begin{bmatrix}1&amp;amp;0\\0&amp;amp;-1\end{bmatrix}.&#xA;$$&lt;/span&gt;&#xA;Most common applications in physics are mentioned &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Pauli_matrices#Physics&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="475" LastEditorUserId="475" LastEditDate="2022-03-11T17:34:48.347" LastActivityDate="2022-03-11T17:34:48.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25030" PostTypeId="2" ParentId="25017" CreationDate="2022-03-09T18:24:27.490" Score="3" Body="&lt;p&gt;Here's one inspired by @Wrzlprmft:&lt;/p&gt;&#xA;&lt;p&gt;Weight* is a vector space, without a universally standard basis. (I regularly have to &amp;quot;change bases&amp;quot; from ounces to grams in the kitchen.)&lt;/p&gt;&#xA;&lt;p&gt;Reasons I like this example:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;it connects something familiar to a new mathematical concept&lt;/li&gt;&#xA;&lt;li&gt;it highlights the difference between a vector space and a vector space with coordinates or a fixed basis. We know we can add and scale lengths, but something crucial prevents us from immediately identifying lengths with real numbers&lt;/li&gt;&#xA;&lt;li&gt;it requires no extra math&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^*$&lt;/span&gt;i probably mean mass, but I am not going to get into those weeds with my linear algebra students&lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="98" LastEditDate="2022-03-10T21:48:26.317" LastActivityDate="2022-03-10T21:48:26.317" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25031" PostTypeId="2" ParentId="25017" CreationDate="2022-03-10T02:33:24.000" Score="2" Body="&lt;p&gt;I think matrices are a good choice so I'll echo some of the other answers given.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ C = \left\{ \left[\begin{array}{cc} a &amp;amp; -b \\ b &amp;amp; a \end{array} \right] \ \bigg{|} \ a,b \in \mathbb{R} \right\} $$&lt;/span&gt;&#xA;has natural basis&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ I_2 = \left[\begin{array}{cc} 1 &amp;amp; 0 \\ 0 &amp;amp; 1 \end{array} \right] \qquad \&amp;amp; \qquad i = \left[\begin{array}{cc} 0 &amp;amp; -1 \\ 1 &amp;amp; 0 \end{array} \right] $$&lt;/span&gt;&#xA;notice &lt;span class=&quot;math-container&quot;&gt;$i^2 = -I_2$&lt;/span&gt;. Of course, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is just a model of the complex numbers, so as a vector space it is isomorphic to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt;, but certainly &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is not a set of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-vectors directly.&lt;/p&gt;&#xA;&lt;p&gt;Or, another example I love, the group algebra of the cyclic group of order three,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \mathbb{R}_{C_3} = \left\{ \left[\begin{array}{ccc} a &amp;amp; c &amp;amp; b \\ b &amp;amp; a &amp;amp; c \\ c &amp;amp; b &amp;amp; a \end{array} \right] \ \bigg{|} \ a,b,c \in \mathbb{R} \right\} $$&lt;/span&gt;&#xA;here we find natural basis &lt;span class=&quot;math-container&quot;&gt;$j, j^2, j^3=I_3$&lt;/span&gt; where&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ j = \left[\begin{array}{ccc} 0 &amp;amp; 0 &amp;amp; 1 \\ 1 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 1 &amp;amp; 0 \end{array} \right] $$&lt;/span&gt;&#xA;and you can verify&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ aj^3+bj+cj^2 = \left[\begin{array}{ccc} a &amp;amp; c &amp;amp; b \\ b &amp;amp; a &amp;amp; c \\ c &amp;amp; b &amp;amp; a \end{array} \right] $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In any event, I always had a contingent of students who refuse to admit that &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; matrices were not &lt;span class=&quot;math-container&quot;&gt;$4 \times 1$&lt;/span&gt; vectors. By this I mean, they would circumvent the formal construction of coordinates and just glibly write equality of vectors and the coordinate vectors which represented said vectors. This is both good and bad. It's good these students see through the notation to realize an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-dimensional real vector space is just &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; in disguise. It's bad because their understanding of the coordinate construction is so fuzzy that coordinate change will forever be a mystery to such students.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2022-03-10T02:33:24.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25032" PostTypeId="2" ParentId="25027" CreationDate="2022-03-10T03:52:03.427" Score="4" Body="&lt;p&gt;In a word, no.  There isn't even a standardized &lt;a href=&quot;https://en.wikipedia.org/wiki/Foundations_of_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;list of plane geometry axioms&lt;/a&gt; in the modern era.  Even if two different curricula happened to start from the same axiomatic basis, there is no longer a single authority on which consequences of those axioms rise to the level of propositions or theorems.   (I'm not sure there ever was.)&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2022-03-10T03:52:03.427" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25033" PostTypeId="2" ParentId="18736" CreationDate="2022-03-10T05:57:06.020" Score="0" Body="&lt;p&gt;I'm totally nit-picking, as requested.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Write the following sets of numbers represented textually in &lt;strong&gt;interval notation&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This instruction is problematic because it suggests that the given information is in interval notation; editing it to &amp;quot;Write, in interval notation,...&amp;quot; rectifies this issue, but the phrase &amp;quot;represented textually&amp;quot; still imposes unnecessary cognitive burden.&lt;/p&gt;&#xA;&lt;p&gt;I suggest:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Use interval notation to denote the following sets:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(a) All real numbers between 5 and 7 that include 5 but don't include 7.&lt;/p&gt;&#xA;&lt;p&gt;(b) All real numbers between 1 and 10 that include both 1 and 10.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&amp;quot;All objects that have property P...&amp;quot; refers to objects that each have property P, whereas &amp;quot;the set of objects that has property P...&amp;quot; refers to objects that &lt;em&gt;collectively&lt;/em&gt; have property P:&lt;/p&gt;&#xA;&lt;p&gt;your problem statement suggests that the number &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; “includes” &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; (and that &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; “includes” &lt;span class=&quot;math-container&quot;&gt;$5);$&lt;/span&gt; this is not entirely meaningful.&lt;/p&gt;&#xA;&lt;p&gt;I suggest:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;(a) the set of real numbers that are smaller than &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; but not smaller than &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;(a) the set of real numbers that are greater than or equal to &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and smaller than &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;(a) the set of real numbers between &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$7,$&lt;/span&gt; including the former but not the latter&lt;br&gt;&lt;br&gt;&lt;/li&gt;&#xA;&lt;li&gt;(b) the set of real numbers from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;(b) the set of real numbers between &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$10,$&lt;/span&gt; inclusive&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-03-10T13:49:15.647" LastActivityDate="2022-03-10T13:49:15.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25034" PostTypeId="2" ParentId="25017" CreationDate="2022-03-10T17:40:57.187" Score="1" Body="&lt;p&gt;A lot of the existing answers depend on specific subject-level outside knowledge, or a level of mathematical internalization that isn't necessarily the case for all introductory linear algebra students. So I'd keep it even simpler and more concrete.&lt;/p&gt;&#xA;&lt;p&gt;Consider a city whose streets have a grid layout and some cross-cutting avenues (e.g. &lt;a href=&quot;https://www.google.com/maps/@40.7331481,-73.9989216,16.46z&quot; rel=&quot;nofollow noreferrer&quot;&gt;Manhattan at 6th Ave and W 8th St&lt;/a&gt;, but you can ideally find a local example that your students will be familiar with, so that you can define the origin as the lecture hall). Any place on the map (the reals in 2) can be described as going so-and-so many blocks parallel to any pair of perpendicular grid streets, or to one of the grid streets and one of the diagonals (both vectors in 2). Thus the streets constitute many different bases for the addresses in the city. But which basis is most convenient for giving/following directions depends very much on the location of the building you're trying to get to: you should probably choose the one that lets you follow the directions without walking through any walls! Just as in your mathematical careers, you'll choose a basis that's easiest to work with.&lt;/p&gt;&#xA;" OwnerUserId="14516" LastEditorUserId="14516" LastEditDate="2022-03-10T21:20:17.573" LastActivityDate="2022-03-10T21:20:17.573" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25036" PostTypeId="2" ParentId="25017" CreationDate="2022-03-10T20:26:02.973" Score="3" Body="&lt;p&gt;You've said that your students found the monomial basis to be &amp;quot;immediate&amp;quot; and you even refer to it as &amp;quot;canonical&amp;quot;.&#xA;There are other useful bases for the space of polynomials and I'd even argue that there's nothing &amp;quot;canonical&amp;quot; about monomials.&#xA;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Bernstein_polynomial&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernstein polynomials&lt;/a&gt; are much more advantageous for both curve fitting and for computer-aided geometric design, see for example &lt;a href=&quot;https://en.wikipedia.org/wiki/B%C3%A9zier_curve&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bezier curves&lt;/a&gt;.&#xA;In particular, the curve&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\gamma(t) = \sum_{k = 0}^nB_{k, n}(t)P_k$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;defined as an expansion in degree-&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; Bernstein polynomials &lt;span class=&quot;math-container&quot;&gt;$B_{k, n}$&lt;/span&gt; lies in the convex hull of the control points &lt;span class=&quot;math-container&quot;&gt;$P_k$&lt;/span&gt;.&#xA;The reason that's true is because the Bernstein polynomials form a partition of unity, which is not true of the monomial basis.&#xA;For a concrete example, you can open up any vector graphics drawing program and show them how easy it is to make shapes with Bezier curves.&#xA;Trying to do that with monomials would be nigh-impossible.&lt;/p&gt;&#xA;&lt;p&gt;Likewise, for numerical solution of PDE, the Legendre polynomials can be a much better choice because of orthogonality, but that might be getting too far afield.&lt;/p&gt;&#xA;" OwnerUserId="19634" LastActivityDate="2022-03-10T20:26:02.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25037" PostTypeId="2" ParentId="18070" CreationDate="2022-03-11T01:29:20.187" Score="1" Body="&lt;p&gt;Calm down. Yes, nearly anything can be used as a weapon by psychotic people. Nevertheless, people should be allowed to have metal eating utensils, steak-knives, real hardened-metal screwdrivers, and so on. Pocket knives with actually sharp blades.&lt;/p&gt;&#xA;&lt;p&gt;So far as I know, serious psychotic people, and/or distressed people in prisons and so on, can make weapons from nearly anything. And, yes, understandably.&lt;/p&gt;&#xA;&lt;p&gt;Don't over-think this...&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-03-11T01:29:20.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25038" PostTypeId="2" ParentId="25017" CreationDate="2022-03-11T11:15:11.980" Score="1" Body="&lt;p&gt;I guess it's up to me to advertise telcomm applications, so&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let's do wave!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The waves &lt;span class=&quot;math-container&quot;&gt;$f(t)=A \cos(\omega t+\phi_0)$&lt;/span&gt; of a given frequency &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; (and varying amplitude &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and phase &lt;span class=&quot;math-container&quot;&gt;$\phi_0$&lt;/span&gt;) form a &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-dimensional vector space &lt;span class=&quot;math-container&quot;&gt;$V_\omega$&lt;/span&gt; (over &lt;span class=&quot;math-container&quot;&gt;$\Bbb{R}$&lt;/span&gt;). Basically because of the trig formulas&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\cos(\omega t+\phi_0)=\cos\phi_0\cos (\omega t)-\sin\phi_0\sin(\omega t)&#xA;$$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sin(\omega t)=\cos(\omega t-\pi/2)&#xA;$$&lt;/span&gt;&#xA;that A) show that &lt;span class=&quot;math-container&quot;&gt;$\sin(\omega t)\in V_\omega$&lt;/span&gt; and B) show that every wave is a linear combination &lt;span class=&quot;math-container&quot;&gt;$\cos(\omega t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin(\omega t)$&lt;/span&gt;. Conversely&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;A\cos(\omega t)+B\sin(\omega t)=C\cos(\omega t+\phi_0),&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$C=\sqrt{A^2+B^2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\phi_0=-\operatorname{atan2}(A,B)$&lt;/span&gt; is chosen such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$A=C\cos\phi_0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B=C\sin\phi_0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Ok. This may be a bad example in the sense that I already singled out &lt;span class=&quot;math-container&quot;&gt;$\cos(\omega t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin(\omega t)$&lt;/span&gt; as &lt;em&gt;a natural basis&lt;/em&gt;. Anyway, it may be not so immediately visualizable as the natural basis of &lt;span class=&quot;math-container&quot;&gt;$\Bbb{R}^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then you can further point out that 4G cell phone signals are in a higher dimensional space of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\bigoplus_{i=0}^{2047}V_{\omega+ i\Delta\omega},&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Delta\omega$&lt;/span&gt; depend on the frequency band of the operator. Other powers of two (instead of &lt;span class=&quot;math-container&quot;&gt;$2^{11}=2048$&lt;/span&gt;) are also in use.&lt;/p&gt;&#xA;" OwnerUserId="282" LastActivityDate="2022-03-11T11:15:11.980" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25039" PostTypeId="2" ParentId="25017" CreationDate="2022-03-11T17:06:26.063" Score="2" Body="&lt;p&gt;The Fourier series elements form basis vectors in the linear vector space of functions of one variable (for simplicity). In fact, it's an excellent homework assignment to have the students prove this - they should know enough calculus at this point to be able to do it.&lt;/p&gt;&#xA;" OwnerUserId="7344" LastActivityDate="2022-03-11T17:06:26.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25040" PostTypeId="2" ParentId="24985" CreationDate="2022-03-11T19:35:18.693" Score="0" Body="&lt;p&gt;&amp;quot;How should I grade...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;It depends much on the kid and the dynamic you have with him/her. My first thought was that this kid obviously did it on purpose probably because he/she didn't know the answers so this is better than nothing or simply to be joking around. In my experience, teacher normally would take this as a wrong answer since my handwriting is not legible, after all you are in school to, among many things, read and write correctly.&lt;/p&gt;&#xA;&lt;p&gt;Now, it could also be that this kid is not joking around and is being 100% serious, so in this case I'd say just ask her/him to clarify the answer for your.&lt;/p&gt;&#xA;&lt;p&gt;Now your second questions on how to avoid this happening in the future, I loved &amp;quot;Xander Henderson&amp;quot; suggestion, I see it as an absolute way to solve this.&lt;/p&gt;&#xA;" OwnerUserId="19616" LastActivityDate="2022-03-11T19:35:18.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25041" PostTypeId="1" CreationDate="2022-03-11T21:24:41.130" Score="6" ViewCount="313" Body="&lt;p&gt;I want ideas for pictorial representation of groups which can help one understand the different group theorems.&lt;/p&gt;&#xA;&lt;p&gt;Here are some examples of the type of thing I am looking for. In this video by &lt;a href=&quot;https://youtu.be/TCcSZEL_3CQ?t=411&quot; rel=&quot;noreferrer&quot;&gt;socratica&lt;/a&gt; at 6:51, the following picture comes up on screen to show that group set can be spanned by premultiplying the coset by a group element for proving lagranges theorem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tQL1g.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tQL1g.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Of course, the above representation is not perfect but it's a direction in how one could think about these things.&lt;/p&gt;&#xA;&lt;p&gt;Here is another example which I saw for the orbit stabilizer theorem from &lt;a href=&quot;https://math.stackexchange.com/questions/242968/intuition-on-the-orbit-stabilizer-theorem#:%7E:text=Bookmark%20this%20question.,group%20of%20x%20in%20G.&quot;&gt;this post&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UUMqi.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UUMqi.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The visualization is so simple and captures the big picture of the proof so easily. Also the other attached picture which the author uses to prove that the orbit of any element &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;  in the orbit set of another element &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; has equal length to orbit set of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2022-03-15T18:50:55.870" Title="Visual aids for understanding group theory" Tags="|abstract-algebra|big-list|visual-proof|aids|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25043" PostTypeId="2" ParentId="25017" CreationDate="2022-03-12T03:38:14.657" Score="2" Body="&lt;p&gt;It seems like if we aim for a vector space without an obvious basis, then the kernel of a linear transformation is a easy example. If we aim for a vector space with multiple good bases, then the eigenbasis presents itself: the canonical basis is natural (maybe it represents position in a physical system), but the eigenbasis is convenient for computation (whether it be solving differential equations or taking powers of transformations). What is nice about the latter example is that when we have multiple, important bases, it becomes an important task to conveniently transform between the two.&lt;/p&gt;&#xA;&lt;p&gt;Here is a mathematical example from combinatorics using polynomials, which fits neither of the above forms (as far as I am aware). Given &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; people, the number of ways to fill a leaderboard with &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; people is &lt;span class=&quot;math-container&quot;&gt;$n(n-1)\cdots(n-k+1)$&lt;/span&gt;. We define &lt;span class=&quot;math-container&quot;&gt;$x^{\underline{k}} = x(x-1)\cdots(x-k+1)$&lt;/span&gt;. Then the &lt;span class=&quot;math-container&quot;&gt;$x^{\underline{k}}$&lt;/span&gt; form a basis for the space of polynomials which is not the monomial basis. When we write &lt;span class=&quot;math-container&quot;&gt;$x^{\underline{k}}$&lt;/span&gt; in the monomial basis, we obtain the Stirling numbers of the first kind, which count the number of permutations with &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; cycles. When we write monomials in the &lt;span class=&quot;math-container&quot;&gt;$x^{\underline{k}}$&lt;/span&gt; basis, we obtain the Stirling numbers of the second kind, which count surjections. The point is, if you are going to be doing some counting, it may be more convenient to use the &lt;span class=&quot;math-container&quot;&gt;$x^{\underline{k}}$&lt;/span&gt; basis.&lt;/p&gt;&#xA;" OwnerUserId="19645" LastActivityDate="2022-03-12T03:38:14.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25044" PostTypeId="2" ParentId="25041" CreationDate="2022-03-12T09:50:14.113" Score="3" Body="&lt;p&gt;The book &lt;a href=&quot;https://www.zbmath.org/?q=an%3A1194.20001&quot; rel=&quot;nofollow noreferrer&quot;&gt;Visual Group Theory&lt;/a&gt; by Nathan Carter seems to be a rich source of the kind of materials that you are looking for.&lt;/p&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2022-03-12T09:50:14.113" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25045" PostTypeId="2" ParentId="25041" CreationDate="2022-03-12T15:08:22.080" Score="3" Body="&lt;p&gt;John Jones has a great visualization tool for a nice selection of finite group tables.  It allows you to do things like select a subgroup, and see the group table colored according to the cosets.  This lets you see &amp;quot;at a glance&amp;quot; whether a given subgroup is normal or not.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hobbes.la.asu.edu/groups/groups.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://hobbes.la.asu.edu/groups/groups.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AlMqKm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AlMqKm.png&quot; alt=&quot;normal subgroup&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/dDL1km.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dDL1km.png&quot; alt=&quot;non-normal subgroup&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-03-12T15:08:22.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25046" PostTypeId="1" AcceptedAnswerId="25047" CreationDate="2022-03-13T02:52:41.617" Score="13" ViewCount="6220" Body="&lt;p&gt;In the linear algebra course I taught, I have three exams. In the first exam, which is probably too hard, students did poorly. So many students said they have lost confidence. To encourage them, I announced that if their grades in the final exam is better the average grades of the three exams, I will use the final exam's grade on their report.&lt;/p&gt;&#xA;&lt;p&gt;In the end, some students performed much better in the final than they previously did and got very good grades. This caused other students who did better on average, but not particularly well in the final, to complain that it's unfair. I also realized that some students can game the system by intentionally under-perform in the second exam.&lt;/p&gt;&#xA;&lt;p&gt;It is a bit late for this course. But what should I have done after the first exam? Perhaps I should just announce that I will reduce the difficulty of the second and final exams?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2022-03-15T19:37:40.647" Title="What should I do if students did very poorly in the mid-term exam?" Tags="|grading|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25047" PostTypeId="2" ParentId="25046" CreationDate="2022-03-13T12:59:53.867" Score="15" Body="&lt;p&gt;In the past, I've identified the problems on the final exam that review exam 1 material and offered to use them in place of the original exam 1 grade, provided this improves students' grades and they come in to office hours to review exam 1. This requires a lot of questions to get good data.&lt;/p&gt;&#xA;&lt;p&gt;Nowadays, I use a standards-based grading system with plenty of retakes. It's a lot of work, but handles these sorts of problems naturally: students can make up missed points in office hours or on later assessments.&lt;/p&gt;&#xA;&lt;p&gt;A trick to preventing these sorts of situations is to offer students a &amp;quot;practice exam&amp;quot; as an in-class activity just before an exam. This makes sure students know how much they need to prepare for the actual exam but also lets you sneakily gauge how difficult the exam will be.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-03-13T12:59:53.867" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25048" PostTypeId="2" ParentId="25046" CreationDate="2022-03-13T13:46:58.517" Score="10" Body="&lt;p&gt;&lt;strong&gt;Corrective:&lt;/strong&gt;  The practical immediate answer is to do another test, covering the same material.  Burn an hour.  Not the end of the world.  Let students keep whichever grade they do better on.  Use your judgment if you water down the re-exam or if you just cover very similar material and the kids get kicked in the ass (calibrated) on level of effort needed.  But do warn them, if you are retaining the level of difficulty (say nothing if weakening it).  But do advise them on how to effectively study for the re-test.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Preventive:&lt;/strong&gt;  I love Tom's suggestion, but another similar one.  Don't do 3 exams per semester.  Do one per week.  Had a teacher, for two years in HS, pre-calc and calc, who did this.  Every Friday was a 50-min test.&lt;/p&gt;&#xA;&lt;p&gt;When I first encountered it, thought it was a lot and how do we cover the material.  But it ended up working great.  Just like homework problems are the most effective studying, tests are basically sets of compelled homework problems.  The regular routine made things very easy to plan in my life.  And having more moderate stakes (not high stakes like a final/midterm, not low stakes like turn in your homework) felt like a reasonable amount of stress.&lt;/p&gt;&#xA;&lt;p&gt;By the way, this will also allow you to get better at calibrating things yourself (since you have more tests).  And it makes a random poorly designed test less critical in students' end of grade experience (just &amp;quot;one of those things&amp;quot;, like a stochastic variable in a trend). But those are  tangential benefits.  The main driver is better training of the trainees by having more evaluated performances.&lt;/p&gt;&#xA;&lt;p&gt;I'm sure I will hear squawks about lost lecture time (from people paid to lecture, go figure), but there is more and more proof that people learn from practice (from doing) better than from hearing/observing.  And tests are a performance by the student.&lt;/p&gt;&#xA;" OwnerUserId="19649" LastActivityDate="2022-03-13T13:46:58.517" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25049" PostTypeId="2" ParentId="2653" CreationDate="2022-03-13T14:52:50.143" Score="2" Body="&lt;p&gt;You asked expansively for pros and cons, so here's some thoughts.&lt;/p&gt;&#xA;&lt;p&gt;I think you're mixing a couple different issues:&lt;/p&gt;&#xA;&lt;p&gt;(1) Compelled homework, which has pluses of getting them to do it and minuses of the work/bother in monitoring it.  Personally, I'm against it, but you seem to be making this a tacit assumption in favor, so fine.&lt;/p&gt;&#xA;&lt;p&gt;(2) Your particular scheme:&lt;/p&gt;&#xA;&lt;p&gt;Which has a serious minus of being so complicated my eyes glazed reading it...this may affect the students who feel gamed, also may make it harder to administer.  IOW not KISS.&lt;/p&gt;&#xA;&lt;p&gt;In addition, your second bullet is relatively low value.  Not zero, but near zero.  This is because strong students will knock out easy problems fast.  And most classes don't give an amount of homework that is seriously derailing the strong students.  I mean, it's not like telling someone to do 1000 pushups per day.&lt;/p&gt;&#xA;&lt;p&gt;The feedback (bullet 4 and part bullet 2) feels like a complicated kludge.  Why have this circling system versus just having published solutions?  Or office hours?  Or tutorials? Or in class questions?  The circle feedback is a lot of work you are committing yourself to, to write these little letters back. And for that matter what if students have questions on several (need to pick one, more gaming)?  Why do only strong students get feedback?  Tutorials or math center or office hours or published solutions or questions in class or combinations of these are all more efficient in the real world.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;My practical advice is just do a standard slate with equal split of ~4 easy (plugs, small calcs) ~4 medium (worked solutions, multi-stpe problems) and 0-2 hard (word problem, derivation, two techniques needed, etc.) for the nightly drill.  Most books are in order anyway and make clear the A/B/C problems.  The simple solution is the kids will do the problems in order.  The weaker kids will get through the easy ones (slowly, but will), founder on the mediums, and not seriously attempt the hards.  The strong kids will just pound it all out, and the 4 easy ones won't cost them much time, might even help them achieve more automaticity.&lt;/p&gt;&#xA;&lt;p&gt;You should also mention to the kids that nothing stops them from doing more drill.  They can do the entire book of drill problems. That you think the 10 problems a night are enough to pass, but if they are worried about an A versus a B or a B  versus a C, to do more.&lt;/p&gt;&#xA;&lt;p&gt;I would also tell them that test questions (what they care about) will be VERY similar to homework problems.  Stamp the ground and make a horse sound.  And then follow through.  (Why would you examine them on things they don't drill?  Why assign drill that doesn't help them with exams?) And of course you should have a similar split of easy/medium and very few hard on exams...this allows students to find their level...and doesn't skunk the weak students.  And have them in difficulty (not coverage) order.  Other than broader, multi-lecture coverage, is similar to their drill sessions (except with some moderate time/performance pressure).&lt;/p&gt;&#xA;" OwnerUserId="19649" LastActivityDate="2022-03-13T14:52:50.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25050" PostTypeId="1" AcceptedAnswerId="25052" CreationDate="2022-03-14T04:47:23.527" Score="6" ViewCount="704" Body="&lt;p&gt;I posted a &lt;a href=&quot;https://matheducators.stackexchange.com/q/25046/11702&quot;&gt;question&lt;/a&gt; regarding what to do when students have done poorly in a mid-term exam. One suggestion I got is to use frequent tests to reduce the risk of one poorly designed exam.&lt;/p&gt;&#xA;&lt;p&gt;Then I did a bit of googling and found this &lt;a href=&quot;https://peer.asee.org/increasing-student-engagement-and-motivation-by-replacing-homework-with-assignment-quizzes.pdf&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; &amp;quot;Increasing Student Engagement and Motivation by Replacing Homework with Assignment-Quizzes&amp;quot;. It suggests that instead of giving students homework, which they most likely will just google, we can give students a bank of exercises with solutions, and then we pick one directly from the bank to quiz them in class.&lt;/p&gt;&#xA;&lt;p&gt;I can see that this approach saves a lot of assignment grading time, and may actually push students to work a bit harder. Have you tried anything like this and how did it work?&lt;/p&gt;&#xA;&lt;p&gt;Also, if I use this method, I am tempted to at least change the questions a bit when giving a quiz. Otherwise what if a student just memorizes a proof?&lt;/p&gt;&#xA;&lt;p&gt;Another question is, should I do this on Monday or Fridays? If I do it on Friday, students may  not have enough time to read the assignments. If I do it on Monday next, I will not be able to test the material of the last week.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastEditorDisplayName="user11702" LastEditDate="2022-03-14T08:39:42.153" LastActivityDate="2022-03-27T19:39:55.840" Title="Should we use quizzes to replace homeworks?" Tags="|homework|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25051" PostTypeId="2" ParentId="25050" CreationDate="2022-03-14T13:59:55.660" Score="3" Body="&lt;p&gt;Do them Fridays.  Don't ruin their weekends. Or more importantly, be perceived as ruining their weekend.&lt;/p&gt;&#xA;&lt;p&gt;Push yourself to grade and release them that Friday.  Don't ruin your weekend.&lt;/p&gt;&#xA;&lt;p&gt;P.s. Regarding your last post, in the two years I had weekly tests, the teach did not grade homework, even for completion.  She did assign something and many kids did it or some of it and that's what people would ask questions on, next class.  She might call on you, though, but that was the only sanction (mild embarrassment). There's probably some connection, but still they're independent variables (frequency of tests, grading HW completion).&lt;/p&gt;&#xA;" OwnerUserId="19652" LastEditorUserId="19652" LastEditDate="2022-03-14T14:11:23.797" LastActivityDate="2022-03-14T14:11:23.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25052" PostTypeId="2" ParentId="25050" CreationDate="2022-03-14T17:57:51.117" Score="6" Body="&lt;p&gt;I swapped to what you propose in my calculus classes. Grading calculus homework is miserable:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If you grade on correctness, students use the solutions manual&lt;/li&gt;&#xA;&lt;li&gt;If you grade on completion, students use the solutions manual or submit fake work&lt;/li&gt;&#xA;&lt;li&gt;In either case, if a student doesn't quite complete an assignment, you have to decide between giving them an extension (miserable) or full credit (miserable), because if you lower their grade, you incentivize them to use the solutions manual or submit fake work again.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So instead I have a list of problems from each section called &amp;quot;absolutely critical practice.&amp;quot; I tell them it is the homework. I tell them it is due Monday, and answer questions in class on Monday -- I answer whatever they ask, so it's all the same as you would expect except &lt;strong&gt;I don't collect the homework&lt;/strong&gt;. Then on the last day of the week, I randomly select one of the absolutely critical practice problems from each section and that forms the quiz.&lt;/p&gt;&#xA;&lt;p&gt;I find that it takes a &lt;strong&gt;lot&lt;/strong&gt; of discussion of this policy to explain to students what is going on. However, in the end, I have &lt;em&gt;approximately the same level of homework completion as I did before&lt;/em&gt;, and I waste a lot less time. Students also feel that the weekly quizzes are totally fair, so they are happier. Most of the grade is still based on exams made of problems I write myself, but my quality of life overall is significantly higher.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2022-03-14T17:57:51.117" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25053" PostTypeId="2" ParentId="25046" CreationDate="2022-03-14T23:08:28.360" Score="3" Body="&lt;p&gt;As a student, I appreciate that you care about students' performance and experience in your class.&lt;/p&gt;&#xA;&lt;p&gt;What many professors I've had have done is average the lowest test score with the highest and replace it with the result. So if they have 50, 80, 100, it goes in as 75, 80, 100. I always appreciate policies like this because in the span of a months-long semester, especially during COVID, there will usually be a period where I underperform just because of issues outside of school.&lt;/p&gt;&#xA;&lt;p&gt;In my experience, as long as you are a good teacher, communicate well with students, and maintain a well structured class, nobody will complain too much about how extra credit is done if it's given. Then again, I don't hear every complaint on the teaching end so what do I know.&lt;/p&gt;&#xA;&lt;p&gt;My linear algebra professor this semester does a HW and quiz every week, which helps prepare students for tests, and also spreads the course grade weight across more assignments so having a bad day on one test won't kill your grade for the entire class.&lt;/p&gt;&#xA;" OwnerUserId="19654" LastActivityDate="2022-03-14T23:08:28.360" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25054" PostTypeId="2" ParentId="25046" CreationDate="2022-03-15T12:30:52.247" Score="1" Body="&lt;p&gt;All great answers, another thought.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes a student has external factors going on or is not good at taking tests (Nero Divergent students especially)&lt;/p&gt;&#xA;&lt;p&gt;So instead of another test I give them a project that covers the test scenarios. Ex: instead of 20 questions about the Python programming language i will give them a scenario where they have to cover 20-30 items in a project program.&lt;/p&gt;&#xA;&lt;p&gt;This allows me to still test their capabilities and evaluate their score accordingly.&lt;/p&gt;&#xA;" OwnerUserId="19657" LastActivityDate="2022-03-15T12:30:52.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25055" PostTypeId="1" CreationDate="2022-03-15T13:01:50.027" Score="4" ViewCount="437" Body="&lt;p&gt;Hello wonderful educators. I am hoping to get some help on a tough situation I am in. So first, a little bit of background facts:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;This is my first time teaching an AP class. I've adjuncted for Calc I at a local community college, but this is my first year teaching AP BC Calculus.&lt;/li&gt;&#xA;&lt;li&gt;The students at my school have a tremendous advantage in BC calculus. Taking AB calculus was a pre-req. That means all of my students have been through AB.&lt;/li&gt;&#xA;&lt;li&gt;However, last year, the AB teacher was not good at all and most of the students got 1s on the exam, and a few got 2s. I think one of them got a 3.&lt;/li&gt;&#xA;&lt;li&gt;So when I was told I'd teach BC, I thought I'd be able to cruise through the AB material and get to the BC stuff. However, their lack of preparation in AB really hindered our progress.&lt;/li&gt;&#xA;&lt;li&gt;You might think &amp;quot;well, most schools just have AB or BC, not both, so you should be in the same position as other BC teachers: teaching all AB and BC topics in one academic year.&amp;quot; But my rebuttal is that the school I teach at has a very limited number of classtime minutes. Basically, If I used the AP Calculus BC Course/Exam Description's estimation of how long to spend on each unit, I'd complete the course on our last day of school, which is late May. Obviously that's an issue because the AP Exam is on May 9th and I wanted to have time to review.&lt;/li&gt;&#xA;&lt;li&gt;I just now started the unit on Polar and Parametric equations, and then I still need to get through the infinite sequences and series unit.&lt;/li&gt;&#xA;&lt;li&gt;Another barrier is that we teach on an A/B schedule, which means I only see the students every other day. So even though there's about 55 days before the AP exam, I only have about 24 class periods left (discounting weekends, Spring Break, and the days I don't see the students).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So, my ultimate question is this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;QUESTION&lt;/strong&gt;&lt;br /&gt;&#xA;What should I do?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Should I just press on and cover polar + parametric + infinite series and not have time to review? Should I expect students to meet with me outside of class time to make sure we have time to cover the material (this would be difficult given the students' other responsibilities at the school)?&lt;/li&gt;&#xA;&lt;li&gt;Should I finish Polar + Parametric, cut my losses with infinite series and not even try to teach that, so that we can review everything from throughout the year in hopes of getting the students at least a decent AB sub score?&lt;/li&gt;&#xA;&lt;li&gt;Should I go more extreme and not even do polar/parametric equations and just have 24 days of reviewing everything up to this point, to try to ensure good AB sub scores?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I know it's a tough and very specific question, probably more of a subjective answer, but any guidance would be helpful. Note: I am also asking my department chair and the current AB teacher for their opinions as well, but I thought it wouldn't hurt to get even more perspective from people on here.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2022-03-15T19:33:22.843" Title="AP Calculus BC Guidance" Tags="|calculus|course-design|usa|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25056" PostTypeId="2" ParentId="25046" CreationDate="2022-03-15T15:54:53.487" Score="8" Body="&lt;p&gt;Students call anything &amp;quot;unfair&amp;quot; when they don't get the grade they want.&lt;/p&gt;&#xA;&lt;p&gt;Yes, students will game your system.  It doesn't matter as long as they leave with the required information in their heads. Students need to realize that writing exams are tools for their own improvement and that grades measure performance, not effort.&lt;/p&gt;&#xA;&lt;p&gt;If other students can blow off the mid-term but still ace the final that represents an unequal effort, surely, but the goal of your course is not to make your students expend effort, and certainly not for them to expend effort equally - it's that they &lt;strong&gt;learn the material&lt;/strong&gt;.  It's illogical for a highly capable student to invest effort in an exam that won't have weight in their final grade if they don't need that exercise to master the material - it's just a waste of time and they can make better use of that time by applying it elsewhere.&lt;/p&gt;&#xA;&lt;p&gt;Maybe it is unfair that some students can learn quickly, skip the midterm, and still achieve on the final.  But that's just reality - some people are more capable learners than others.  What matters is that they leave with the skills your course aims to provide them.  The only disconnect is with students who feel their efforts entitle them to grades regardless of their actual achievements.&lt;/p&gt;&#xA;" OwnerUserId="12561" LastEditorUserId="12561" LastEditDate="2022-03-15T19:37:40.647" LastActivityDate="2022-03-15T19:37:40.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25057" PostTypeId="2" ParentId="25041" CreationDate="2022-03-15T18:50:55.870" Score="1" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Cayley_graph&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cayley Graph&lt;/a&gt; is visually appealing.&#xA;Here is the dihedral group &lt;span class=&quot;math-container&quot;&gt;$D_4$&lt;/span&gt; on two generators &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QgfDP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QgfDP.png&quot; alt=&quot;D4&quot; /&gt;&lt;/a&gt;&#xA;&lt;br /&gt;&#xA;&lt;sup&gt;(Wikipedia image)&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-03-15T18:50:55.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25058" PostTypeId="2" ParentId="25055" CreationDate="2022-03-15T19:33:22.843" Score="4" Body="&lt;p&gt;After talking with my department chair, the answer seems just obvious to me and I feel almost silly for making this post. I'll post what he said, but I'll also leave the question open to for some other view points (and also not delete the post in case other users have this same issue in future years).&lt;/p&gt;&#xA;&lt;p&gt;Basically, we decided to push on and cover all of the BC material while weaving in AB review questions on HWs (as mentioned by Steve). The students signed up for BC, so we owe it to them to teach them BC and not focus on a test score. Additionally, some of these students will test into Calc II their first year of college or make it to Calc II eventually, where they will see this material again. We would be giving them a head start if they see it now and have at least some tiny exposure to it.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2022-03-15T19:33:22.843" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25059" PostTypeId="2" ParentId="11026" CreationDate="2022-03-16T03:51:14.340" Score="0" Body="&lt;p&gt;Instead of exams I’ve started asking students to make a video explaining the concept/calculation as if they were explaining the material to someone who doesn’t know it. students reported the process made them learn the material better. I feel it is more difficult to cheat (though not impossible)- one student tried using chegg but couldn’t explain the answer chegg provided at all in the video. This is actually formative, more difficult to cheat&lt;/p&gt;&#xA;&lt;p&gt;I forget who said it but “The best way to learn something is to teach it.” And I also require them to also explain how use technology - e.g. show how to use Python to compute the RREF of a matrix.&lt;/p&gt;&#xA;" OwnerUserId="10289" LastActivityDate="2022-03-16T03:51:14.340" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25060" PostTypeId="2" ParentId="24833" CreationDate="2022-03-16T04:07:24.540" Score="1" Body="&lt;p&gt;I had the same problem when I was a student, and it took me a while to learn and accept that forgetting is actually a &lt;em&gt;good&lt;/em&gt; thing because (vastly simplified) because it gives you a way to recode the material in a different way and that offers fresh perspective. There are neuroscience books on memory going into details.&lt;/p&gt;&#xA;&lt;p&gt;Here’s a quote from Andrew Wiles that I found on the internet:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I really think it's bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it's a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that's what actually enables you to get round [the problem].&#xA;So if you remembered all the failed attempts before, you wouldn't try them again. But because I have a slightly bad memory I'll probably try essentially the same thing again and then I realise I was just missing this one little thing I needed to do.*&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I was also surprised to learn that very accomplished researchers tell me that the remember nothing about papers they wrote ten years ago…of course if they went and reread them they would fairly quickly be able to refresh their memory.&lt;/p&gt;&#xA;&lt;p&gt;In terms of concrete things one could do to help, I would say talking/explaining the material to others via e.g a study group, a student seminar, or a YouTube channel.&lt;/p&gt;&#xA;" OwnerUserId="10289" LastActivityDate="2022-03-16T04:07:24.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25061" PostTypeId="1" CreationDate="2022-03-16T12:10:28.260" Score="13" ViewCount="239" Body="&lt;p&gt;I'm looking for math games that a group of students in grades 5 to 10 (ages 11 to 15, say) could play in a gym or parking lot.  My school has a STEM Day each year and I get tasked with cooking up some sort of math activity.  (It's hard to compete with the science people, who get to build catapults and blow things up.)&lt;/p&gt;&#xA;&lt;p&gt;Here's two examples of what I've done in the past:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I drew four different graphs (easy, medium, hard, impossible) with sidewalk chalk in the parking lot. Students get sent in waves of about 10, once per hour.  I had some small traffic/agility cones and the students were to walk the graph, trying to find an Euler tour (trace out each line exactly once.)  If they used a line, they marked it with the traffic cone.  This was fairly successful.  Most groups made it through easy, medium, hard in a few minutes.  So the group of 10 would end up at the impossible graph and start arguing and debating about how to find the Euler tour.  (I couched this as a snowplow needing to plow all the streets of this little town but having exactly enough gas to do all the streets once.)  So it got exciting and I got good feedback.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There's a game called &amp;quot;Gale&amp;quot; described here:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Shannon_switching_game#Gale&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Shannon_switching_game#Gale&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I had 30 orange and 30 blue traffic cones and some 3-foot green bamboo sticks (sold for staking garden plants) and some 3-foot natural-color bamboo sticks.  I set up a grid for Gale with the cones 3-feet apart.  When the wave of 10 students came, I had them pick teams of 3-ish and compete.  This also worked pretty well.  There was much yelling and excitement (which we don't get much of in math.)   We had to make a &amp;quot;no sword fighting&amp;quot; rule right away, however.&lt;/p&gt;&#xA;&lt;p&gt;So I'm looking for other similar ideas.  My criteria are: 1.  Real math involved and 2. Needs a parking lot or gym to implement (that is, the activity should be physical.)  3.  Is competitive.&lt;/p&gt;&#xA;&lt;p&gt;After the Snowplow problem, I could talk in my classes about the Königsburg Bridge Problem and the theorem that tells us when an Euler tour exists.  (The upperclass students act as assistants for STEM Day, so the 11th and 12th graders also know about the exercise.)  So there's some &amp;quot;real math&amp;quot; going on that we get to talk about later.  Likewise, with Gale, we spent some time in regular class analyzing the game.  For STEM day, the grids were 5 x 6.  So in class we looked at smaller and smaller grids.  4 x 5 is still complicated.  3 x 4 is simple enough that &amp;quot;whoever moves first is probably going to win.&amp;quot;  2 x 3 we can analyze completely.  1 x 2 gets a laugh.&lt;/p&gt;&#xA;&lt;p&gt;So I'm looking for other ideas with the same sort of flavor.  (And if someone uses one of my ideas above, all the better.)&lt;/p&gt;&#xA;" OwnerUserId="19450" LastEditorUserId="19450" LastEditDate="2022-03-16T16:59:32.460" LastActivityDate="2022-03-24T22:43:11.620" Title="Middle school math games for the parking lot" Tags="|middle-school|games|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25063" PostTypeId="2" ParentId="25061" CreationDate="2022-03-16T14:22:08.230" Score="1" Body="&lt;p&gt;Ehrenfeucht-Fraisse Games, maybe?&lt;/p&gt;&#xA;&lt;p&gt;Draw two (small-ish) graphs on the sidewalk&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Player 1 colors in a new color one vertex on &lt;em&gt;either&lt;/em&gt; graph&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Player 2 colors with the same color a vertex on the other graph to match them together&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This repeats. The game ends when both graphs have been fully colored in, or when matched vertices are connected in one graph and not connected in the other graph. (for instance, if the red and blue vertices are connected in one graph but not in the other)&lt;/p&gt;&#xA;&lt;p&gt;It's a bit more complicated than finding Euler tours and it'll be hard to explain why they should care about first order logic, but your students might be able to come up with some strategies to identify similarities/differences between graphs.&lt;/p&gt;&#xA;&lt;p&gt;You can try out EF games in the browser here: &lt;a href=&quot;https://trkern.github.io/efg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://trkern.github.io/efg&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-03-16T14:22:08.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25064" PostTypeId="1" CreationDate="2022-03-16T16:47:21.830" Score="3" ViewCount="62" Body="&lt;p&gt;Are there resources that compares/lists crunchy facts about the logistics of math education  in different regions or countries? I'm talking facts like: Do they use paper homework versus and online homework system? And which online homework systems? Which textbooks are used, and from which publishers? How is their mathematics curriculum broken up across classes? How are students assessed? What is the ratio of homework to classwork? Etc&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2022-03-26T23:48:26.783" Title="Resource for International Comparison of Math Education Logistics" Tags="|undergraduate-education|secondary-education|reference-request|classroom-management|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25065" PostTypeId="1" CreationDate="2022-03-17T06:22:20.733" Score="2" ViewCount="231" Body="&lt;p&gt;I am willing to teach that the Order of rotational Symmetry of Equilateral triangle as &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; using Animation. Any suggestions of good applet which demonstrates the rotation of equilateral triangle with different angles like &lt;span class=&quot;math-container&quot;&gt;$30^{\circ},60^{\circ},$&lt;/span&gt; etc about the centre?&lt;/p&gt;&#xA;" OwnerUserId="5666" LastActivityDate="2022-03-20T14:38:01.823" Title="Is there a good Animation to explain Rotational Symmetry of Equilateral triangle" Tags="|secondary-education|geometry|applications|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25067" PostTypeId="2" ParentId="25061" CreationDate="2022-03-17T16:49:34.467" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Nim&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nim&lt;/a&gt;? Draw chalk circles and put some beanbags on each one. On your turn, you go over to one circle, put some beanbags into a pail, and carry them away. Admittedly, the winning condition isn't as well motivated as the snow plowing story, but competition can be its own motivation.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2022-03-17T16:49:34.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25068" PostTypeId="1" AcceptedAnswerId="25069" CreationDate="2022-03-18T03:34:23.727" Score="3" ViewCount="97" Body="&lt;p&gt;I am looking for a program so that when you give it a 2D system of linear differential equations and an initial condition, it can show an animation of the trajectory of a particle starting at that initial condition.&lt;/p&gt;&#xA;&lt;p&gt;This program is close to what I want: &lt;a href=&quot;https://www.math.ksu.edu/%7Ealbin/teaching/math340/labs/10_eigenlines.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.math.ksu.edu/~albin/teaching/math340/labs/10_eigenlines.html&lt;/a&gt;&#xA;but it only draws the trajectory lines. I want to show students how the point actually moves. Do any of you know if something like this exists?&lt;/p&gt;&#xA;" OwnerUserId="19669" LastActivityDate="2022-03-20T18:50:12.170" Title="Program for visualizing trajectories for a 2D system of linear differential equations" Tags="|technology-in-education|differential-equations|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25069" PostTypeId="2" ParentId="25068" CreationDate="2022-03-18T14:23:10.907" Score="2" Body="&lt;p&gt;I've already created something like this: &lt;a href=&quot;https://trkern.github.io/eulersmethod&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://trkern.github.io/eulersmethod&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can animate a point following a trajectory by adjusting the &amp;quot;animate trajectory&amp;quot; input number.&lt;/p&gt;&#xA;&lt;p&gt;Note that this interactive (as the name implies) uses Euler's method to solve trajectories. If you need something more advanced, you'll have to use a computer algebra system.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-03-18T14:23:10.907" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25070" PostTypeId="2" ParentId="25065" CreationDate="2022-03-20T00:21:12.950" Score="2" Body="&lt;p&gt;Here is a simple, dynamic presentation: &lt;a href=&quot;https://www.youtube.com/watch?v=Npo0LgkklUQ&amp;amp;ab_channel=SimplyAnimatedMath&quot; rel=&quot;nofollow noreferrer&quot;&gt;YouTube video&lt;/a&gt;---only 2 minutes,&#xA;from &amp;quot;Simply Animated Math&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;   &lt;img src=&quot;https://i.stack.imgur.com/31noJ.png&quot; width=&quot;300&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;For more explanation (and less animation),&#xA;this may be &lt;em&gt;too&lt;/em&gt; detailed, but you could select just the portion on the&#xA;equilateral triangle: &lt;a href=&quot;https://www.youtube.com/watch?v=ZGcIIEwW1Xs&amp;amp;ab_channel=PocketClassroom&quot; rel=&quot;nofollow noreferrer&quot;&gt;YouTube video&lt;/a&gt; (11min total, equilateral triangle: 6min).&lt;/p&gt;&#xA;&lt;p&gt;   &lt;img src=&quot;https://i.stack.imgur.com/7BBnD.png&quot; width=&quot;300&quot; /&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2022-03-20T00:29:29.877" LastActivityDate="2022-03-20T00:29:29.877" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25071" PostTypeId="2" ParentId="25065" CreationDate="2022-03-20T14:38:01.823" Score="2" Body="&lt;p&gt;As I mentioned in the comment, things like this are achievable with Desmos. Let me describe a proof of concept here. You can &lt;a href=&quot;https://www.desmos.com/calculator/eynemdpv4j&quot; rel=&quot;nofollow noreferrer&quot;&gt;look it up&lt;/a&gt;, but please beware that I'm no programmer and no Desmos expert.&lt;/p&gt;&#xA;&lt;p&gt;If you define a point &lt;span class=&quot;math-container&quot;&gt;$A=(1,0)$&lt;/span&gt; and a function &lt;span class=&quot;math-container&quot;&gt;$R(P,\alpha)$&lt;/span&gt; that returns the point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; rotated by &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; around &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, then typing the following &lt;em&gt;action&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;A \to R \left( A,\frac{2\pi}{3} \right)&#xA;$$&lt;/span&gt;&#xA;(typed with &lt;code&gt;-&amp;gt;&lt;/code&gt;) will produce a button which rotates &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\frac{2\pi}{3}$&lt;/span&gt;. You could make it into an animation by defining&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;B = R(A, \alpha)&#xA;$$&lt;/span&gt;&#xA;and adding a parameter &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;. Desmos lets you bound &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; to the interval &lt;span class=&quot;math-container&quot;&gt;$\left[0, \frac{2\pi}{3}\right]$&lt;/span&gt; and animate its changes.&lt;/p&gt;&#xA;&lt;p&gt;Another method of animating it (I'm not sure if it's better in any way) is to  use a &lt;em&gt;ticker&lt;/em&gt; that says &lt;code&gt;Run A -&amp;gt; R(A,0.01*dt) every 20 ms&lt;/code&gt;. Then the point &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; will get rotated a bit every 20 ms (&lt;code&gt;dt&lt;/code&gt; is the time step, so changing &lt;code&gt;20 ms&lt;/code&gt; to something else won't alter the pace of animation). The problem with this approach is that the rotation doesn't stop after reaching the desired point. In the &lt;a href=&quot;https://www.desmos.com/calculator/eynemdpv4j&quot; rel=&quot;nofollow noreferrer&quot;&gt;example&lt;/a&gt; I gave, I used a cheap get-around: the animation slows down when it reaches its final point.&lt;/p&gt;&#xA;" OwnerUserId="15367" LastActivityDate="2022-03-20T14:38:01.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25072" PostTypeId="2" ParentId="25068" CreationDate="2022-03-20T18:50:12.170" Score="1" Body="&lt;p&gt;Here is a GeoGebra one:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.geogebra.org/m/vqn2mpgw&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geogebra.org/m/vqn2mpgw&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/A4jyK.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/A4jyK.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can enter an autonomous system and move or animate a slider to show the evolution of the system. (It animates two trajectories.)&lt;/p&gt;&#xA;&lt;p&gt;There is also a slider for a parameter.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2022-03-20T18:50:12.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25074" PostTypeId="2" ParentId="25017" CreationDate="2022-03-21T21:22:31.683" Score="6" Body="&lt;h1&gt;Introduction&lt;/h1&gt;&#xA;&lt;p&gt;The OP mentioned science majors, and since there are loads of examples from science, we might as well start (and probably finish) there.  There is a mathematical viewpoint about the utility of bases and abstract vector spaces, but I'll leave that aside and stick to practical concerns of scientific computing.  Real examples are complicated and not just by their detail. Pedagogically they are complicated by drawing on bits of linear algebra from throughout a standard first course and sometimes from extra bits you pick up as needed in your work after the standard course.  They are further complicated by being based on principles of science, each of which might be unfamiliar to some of one's students.  I think one can get around these complicating factors and motivate students.  If the problem, including the science and the basic role of linear algebra, is easy enough to understand, then the problem might tempt their curiosity.  The main point of these examples will be that when basis vectors are &lt;em&gt;&lt;strong&gt;meaningful&lt;/strong&gt;&lt;/em&gt;, their need and usefulness seems unquestionable. In the last group of examples, they are less meaningful but their utility is apparent; and it is less easy to see the vectors, which are functions, as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-tuples in a natural way.&#xA;&lt;span class=&quot;math-container&quot;&gt;$\def\y{{\bf y}}\def\x{{\bf x}}\def\A{{\Bbb A}}\def\R{{\Bbb R}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The problem with &lt;span class=&quot;math-container&quot;&gt;$\R^n$&lt;/span&gt; that the OP describes has several sources.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The vectors and their coordinates are indistinguishable.&lt;/li&gt;&#xA;&lt;li&gt;Students have been dealing with &lt;span class=&quot;math-container&quot;&gt;$\R^2$&lt;/span&gt; for years.&lt;/li&gt;&#xA;&lt;li&gt;The standard basis is orthonormal, which is a most convenient sort of basis.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Any other basis looks pretty inconvenient to a beginner. So &lt;span class=&quot;math-container&quot;&gt;$\R^n$&lt;/span&gt; feels &amp;quot;canonical.&amp;quot; Furthermore:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A main purpose of a basis is to make computation possible by mapping a vector space to &lt;span class=&quot;math-container&quot;&gt;$\R^n$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;To a practically minded person, the real work, the computation, is done in &lt;span class=&quot;math-container&quot;&gt;$\R^n$&lt;/span&gt;, and that is where the real importance lies.&lt;/li&gt;&#xA;&lt;li&gt;So why not do this in the easiest, most obvious way possible? For instance, use the coefficients of polynomials.&lt;/li&gt;&#xA;&lt;li&gt;It would help the student to accept a basis if &lt;strong&gt;the basis vectors had a meaning&lt;/strong&gt; or represented elementary concepts.&lt;/li&gt;&#xA;&lt;li&gt;The standard basis in &lt;span class=&quot;math-container&quot;&gt;$\R^n$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n=2,3$&lt;/span&gt; has such a meaning: They tell you how to plot a point in a most simple way. And that meaning is ingrained in the beginner's mindset.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;One extra thing I would point out to the students: You won't be doing much number-crunching yourself.  You'll be setting up computations to solve problems. It will be easier to set the problem up in a convenient basis and let the computer figure out how to get back to &lt;span class=&quot;math-container&quot;&gt;$\R^n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the scientific examples below, I've selected examples in which a basis vector corresponds to a fundamental element or structure in the problem being modeled. In function approximation, in which way too &amp;quot;many bases appear,&amp;quot; this is less the case. But these are used in important applications. For instance, in radial basis function interpolation (already a mouthful), &amp;quot;surface reconstruction&amp;quot; may sound abstruse, and students might need help imagining what is so important about modeling a surface.  Well, monitoring the growth of a cancer tumor or visualizing the surfaces in a heart seem important applications to me.&lt;/p&gt;&#xA;&lt;h1&gt;Linear Approximations&lt;/h1&gt;&#xA;&lt;p&gt;Science, to speak broadly, uses mathematics to model phenomena, often under simplifying assumptions that make the model approximate. The mathematics itself often needs to be computed approximately.  Linear approximations are the simplest and are remarkably robust and accurate for solving real problems.&lt;/p&gt;&#xA;&lt;h2&gt;Spectra (chemistry, physics, group theory)&lt;/h2&gt;&#xA;&lt;p&gt;In a comment I made soon after the question was asked (now deleted), I mentioned that the normal/fundamental modes form a basis for the small vibrations of a molecule or other oscillatory system (small = linearized approximation). In the linearized approximation, the oscillations are given by (complicated) functions of time representing the displacements of coordinates from the rest position. These oscillations form a linear system, a vector space. The remarkable thing is that any vibrational motion is a linear combination of the basis of normal modes, each of which may have its own frequency. This applies to &lt;a href=&quot;https://matheducators.stackexchange.com/a/25026/498&quot;&gt;spring-mass systems (@Mark H)&lt;/a&gt; as well.&lt;/p&gt;&#xA;&lt;p&gt;Normal-mode decompositions are used in spectroscopy in chemistry, say, to identify compounds in a mixture. This use of spectroscopy will have been seen by all the  chemistry majors at my college by the end of their sophomore year, although they will have had little explanation of the mathematics. They'll see it again in upper-level chemistry along with group theory, where they might learn this gem that was a highlight of my undergraduate education: The irreducible representations of the vibrational symmetries of methane (or other tetrahedral molecule) cannot be computed without the complex numbers. That is to say, the real phenomenon cannot be fully understood with just real numbers.  As a liberal arts major, I could rejoice that what was real was complex.&lt;/p&gt;&#xA;&lt;p&gt;Something I learned later was the significance of the two complex-conjugate irreducible representations that appear in the analysis of methane.  The &lt;a href=&quot;https://www.chem.purdue.edu/jmol/vibs/ch4.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;vibrational modes associated to these symmetries&lt;/a&gt; are the only ones that generate an oscillating dipole that is necessary for infrared absorption. There are three modes associated to each of the two symmetries, both labeled &lt;span class=&quot;math-container&quot;&gt;$t_2$&lt;/span&gt; below but with different frequencies. Further, they oscillate at infrared frequencies (wavenumbers &lt;span class=&quot;math-container&quot;&gt;$3156.8\ \text{cm}^{-1},\ 1367.4\ \text{cm}^{-1}$&lt;/span&gt;.  It is why &lt;a href=&quot;https://webbook.nist.gov/cgi/cbook.cgi?ID=C74828&amp;amp;Units=SI&amp;amp;Type=IR-SPEC&amp;amp;Index=1#IR-SPEC&quot; rel=&quot;nofollow noreferrer&quot;&gt;methane&lt;/a&gt; is a powerful greenhouse gas.&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/wgbhq.gif&quot; width=&quot;300&quot; alt=&quot;Animation showing the nine vibrational normal modes of methane&quot;&gt;&#xA;&lt;h2&gt;Approximating matrices&lt;/h2&gt;&#xA;&lt;p&gt;When you have data, such as the oscillations of the coordinates of the atoms in a molecule as in the preceding example -- the &lt;span class=&quot;math-container&quot;&gt;$\R^n$&lt;/span&gt; representation -- the standard coordinates don't necessarily in themselves represent an important concept in the problem, unlike the normal-mode coordinates, which indicate how much of each fundamental element, a normal mode, makes up a given vibration.  In the vibration problem, there are as many normal modes as degrees of freedom, and each normal-mode vector is equally important. In some problems, some vectors are more important than others, and identifying them can reveal the structure of phenomena. We'll look at one way to do that.&lt;/p&gt;&#xA;&lt;p&gt;It turns out that &lt;em&gt;any&lt;/em&gt; matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; can be written&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A = U S V^*\,,$$&lt;/span&gt;&#xA;where&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$V^*$&lt;/span&gt; is the transpose of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; (or conjugate transpose if &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is complex),&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is an &lt;span class=&quot;math-container&quot;&gt;$m\times n$&lt;/span&gt; matrix,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$m \times m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n \times n$&lt;/span&gt; orthogonal matrices (or, resp., unitary ones if &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is complex -- that is, in both cases, &lt;span class=&quot;math-container&quot;&gt;$V^*=V^{-1}$&lt;/span&gt;),&lt;/li&gt;&#xA;&lt;li&gt;and &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is a &lt;span class=&quot;math-container&quot;&gt;$m \times n$&lt;/span&gt; diagonal matrix with nonnegative real diagonal entries&#xA;&lt;span class=&quot;math-container&quot;&gt;$$s_{1} \ge s_{2} \ge \cdots \ge s_{n} \ge 0 \,.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the rank of the matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r &amp;lt; n$&lt;/span&gt;, then we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$s_{1} \ge s_{2} \ge \cdots \ge s_{r} &amp;gt; 0 \ \text{and}\ s_{r+1}=s_{r+2}=\cdots=s_{n}=0\,.$$&lt;/span&gt;&#xA;If the real numbers&#xA;&lt;span class=&quot;math-container&quot;&gt;$$s_{1}, s_{2}, s_{3},\dots$$&lt;/span&gt;&#xA;decrease rapidly, we can replace the small ones by zero and change the matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; by only a small amount.&#xA;This factorization of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is called the &lt;em&gt;singular value decomposition&lt;/em&gt; or SVD and the nonzero &lt;span class=&quot;math-container&quot;&gt;$s_j$&lt;/span&gt; are called the &amp;quot;singular values&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. The SVD factorization is not always covered in a first linear algebra course. It could be stated as a fact and used with the examples below to motivate future learning. It is worth considering simply because it's a really cool structure theorem that is in widespread use to solve real problems.  Have the students google &amp;quot;principal component analysis&amp;quot; or &amp;quot;PCA.&amp;quot;&#xA;The SVD is readily available in any decent system or library for linear algebra, and students may have to use it without ever covering it in class:&#xA;&lt;a href=&quot;https://numpy.org/doc/stable/reference/generated/numpy.linalg.svd.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Python&lt;/a&gt;,&#xA;R (&lt;code&gt;svd()&lt;/code&gt;),&#xA;&lt;a href=&quot;https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix_double_dense.html#sage.matrix.matrix_double_dense.Matrix_double_dense.SVD&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sage&lt;/a&gt;,&#xA;&lt;a href=&quot;https://www.maplesoft.com/support/help/maple/view.aspx?path=MTM/svd&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maple&lt;/a&gt;,&#xA;&lt;a href=&quot;https://www.mathworks.com/help/matlab/ref/double.svd.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matlab&lt;/a&gt;,&#xA;&lt;a href=&quot;https://reference.wolfram.com/language/ref/SingularValueDecomposition.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematica&lt;/a&gt;,&#xA;&lt;a href=&quot;https://maths.cnam.fr/Membres/wilk/MathMax/help/Maxima/maxima_58.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maxima&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.gnu.org/software/gsl/doc/html/linalg.html#singular-value-decomposition&quot; rel=&quot;nofollow noreferrer&quot;&gt;GNU GSL&lt;/a&gt;,&#xA;etc.&lt;/p&gt;&#xA;&lt;p&gt;We'll give a brief explanation of how it identifies &amp;quot;principal&amp;quot; components of a linear model.&#xA;First, let's break down the factorization of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;.&#xA;The matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; represents a linear mapping from &lt;span class=&quot;math-container&quot;&gt;$\R^n$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\R^m$&lt;/span&gt;.&#xA;The columns of &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; are orthonormal bases (the best kind) for the output &lt;span class=&quot;math-container&quot;&gt;$\R^m$&lt;/span&gt; and the input &lt;span class=&quot;math-container&quot;&gt;$\R^n$&lt;/span&gt;, respectively. First some notation: for a matrix &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, let &lt;span class=&quot;math-container&quot;&gt;$M_{ij}$&lt;/span&gt; denote the &lt;span class=&quot;math-container&quot;&gt;$ij$&lt;/span&gt; entry (two indices) and &lt;span class=&quot;math-container&quot;&gt;$M_j$&lt;/span&gt; denote &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;-th column matrix (one index). Then to compute &lt;span class=&quot;math-container&quot;&gt;$Ax$&lt;/span&gt; for a vector &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, we may write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Ax = \Big(\sum_{j,k} U_{ij} s_j V_{kj}x_k\Big)_{i=1}^m =  \sum_j (s_j\; (V_j^* \cdot x)) U_{j} \,. \tag{1}$$&lt;/span&gt;&#xA;Thus we first compute &lt;span class=&quot;math-container&quot;&gt;$V^*x$&lt;/span&gt;, which are the coordinates of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with respect to the basis given by the columns of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$SV^*x$&lt;/span&gt; scales these coordinates by the &lt;span class=&quot;math-container&quot;&gt;$s_j$&lt;/span&gt;. These scaled coordinates are the coordinates of &lt;span class=&quot;math-container&quot;&gt;$Ax$&lt;/span&gt; with respect to the basis given by the columns of &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt;, that is &lt;span class=&quot;math-container&quot;&gt;$Ax = U(SV^*x)$&lt;/span&gt;. It's pretty amazing that every matrix action can be viewed in terms of choosing two orthonormal bases and scaling coordinates. We can also see what happens when we replace &lt;span class=&quot;math-container&quot;&gt;$s_j$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; for a single value of &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;. It is in effect the same as deleting the &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;th term from (1), which introduces an error bounded by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\|(s_j\; (V_j^* \cdot x)) U_{j}\| \le s_j\,\| x\| \,,$$&lt;/span&gt;&#xA;since &lt;span class=&quot;math-container&quot;&gt;$\|U_j\|=\|V_j\|=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s_j \ge 0$&lt;/span&gt;.&#xA;If we replace all &lt;span class=&quot;math-container&quot;&gt;$s_j$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$j &amp;gt;J$&lt;/span&gt;, then the error relative to &lt;span class=&quot;math-container&quot;&gt;$\|x\|$&lt;/span&gt; will be bounded by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{j &amp;gt;J} s_j \,. \tag{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h3&gt;Image processing (data compression)&lt;/h3&gt;&#xA;&lt;p&gt;I'm counting it as &amp;quot;science&amp;quot; because we'll use the image of a scientist. Mainly, I introduce the notion this way because it's visual, and it's hard to believe one's eyes.  In this application, we're &lt;strong&gt;approximating a matrix&lt;/strong&gt; itself.  Since the matrix entries are extracted from image data and vice versa, students can see the approximation improve.  The initial matrix entries are the grayscale values (between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;) of the grid of pixels. The singular values this matrix decrease from &lt;span class=&quot;math-container&quot;&gt;$s_1=338$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$s_4=30$&lt;/span&gt;, to &lt;span class=&quot;math-container&quot;&gt;$s_{50}=3.3$&lt;/span&gt;, to &lt;span class=&quot;math-container&quot;&gt;$s_{100}=1.4$&lt;/span&gt;, and finally to &lt;span class=&quot;math-container&quot;&gt;$s_{841}=0.039$&lt;/span&gt; -- a decrease of five orders of magnitude.&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/yZhli.png&quot; width=&quot;300&quot; alt=&quot;a plot of singular values decreasing from 338 to 0.04&quot;&gt;&#xA;&lt;p&gt;Below are the images from the matrix&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A^*=\sum_{j=1} ^J s_j\,  U_{j} (V_j)^T$$&lt;/span&gt;&#xA;for the singular values up to &lt;span class=&quot;math-container&quot;&gt;$J=4,\,50,\,100,\,841$&lt;/span&gt; (all).  The relative error (RE) was calculated using matrix 2-norms, &lt;span class=&quot;math-container&quot;&gt;$\text{RE} = \|A^*-A\|/\|A\|$&lt;/span&gt;, but one should expect a weak correlation between the error of approximating a linear map and how accurate the picture looks. The Frobenius norm gives relative errors of 21%, 5.6%, 3.6%, and 0% respectively, which perhaps seem closer to how the images appear.  All norms give estimates much lower than the upper bound &lt;span class=&quot;math-container&quot;&gt;$\sum_{j&amp;gt;J} s_j$&lt;/span&gt; deduced above.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/W0XKz.jpg&quot; alt=&quot;Four images of Einstein in progressively sharper detail&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here are a couple of codes for constructing the image approximation (I used &lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/3/3e/Einstein_1921_by_F_Schmutzer_-_restoration.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;this image from Wikipedia&lt;/a&gt;, scaled by a factor of 1/3 and converted to grayscale):&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;(* Mathematica *)&#xA;img = Import[&amp;quot;&amp;lt;path/url to image file&amp;gt;&amp;quot;]; (* convert to grayscale if necessary *)&#xA;&#xA;{U, S, V} = SingularValueDecomposition[ImageData@img]; (* assuming &amp;quot;GrayScale&amp;quot; image *)&#xA;J = 4;  (* change as desired *)&#xA;Image[U[[All, ;; J]] . S[[;; J, ;; J]] . Transpose@V[[All, ;; J]]]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;pre&gt;&lt;code&gt;% MATLAB&#xA;img = imread('&amp;lt;path to image file&amp;gt;');  % convert grayscale if necessary&#xA;&#xA;[U,S,V] = svd(im2double(img)); % convert grayscale img to double&#xA;J = 50; % change as desired&#xA;newimg = U(:,1:J) * S(1:J,1:J) * V(:,1:J)';&#xA;imshow(newimg)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;h3&gt;DNA analysis (biology)&lt;/h3&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://physiology.med.cornell.edu/people/banfelder/qbio/lecture_notes/3.4_Principal_component_analysis.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantitative Understanding in Biology: Principal Component Analysis&lt;/a&gt; (lecture notes by Ju &lt;em&gt;et al.&lt;/em&gt;) is a wonderful introduction to principal component analysis (PCA), which may be read by any student in linear algebra.  It refers the connection to the singular value decomposition to a technical paper, but it does a good job explaining why identifying principal components and using them as basis vectors helps one tame an unwieldy dataset.  The authors describe how some common DNA datasets have 10,000+ measurements per sample and others around 600,000-900,000. They point out, &amp;quot;These datasets are typically never fully visualized because they contain many more datapoints than you have pixels on your monitor.&amp;quot; They introduce the notion of PCA  with a simple numerical example, consisting of a dataset of two samples resulting from the measurement of 5 genes (&lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$g_5$&lt;/span&gt;), in which there is hidden correlation. They show that the 5-dimensional representation may be reduced to 2 dimensions, using the principal components. Further they explain the weakness of the standard coordinates: &amp;quot;knowing the expression level of &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; will not give us any information about other genes.&amp;quot; And they explain the significance of the principal components to understanding the biology:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;However, in reality the expression levels of multiple genes tend to be correlated to each other (for example, pathway activation that bumps up the expression levels of &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g_2$&lt;/span&gt; together, or a feedback interaction where a high level of &lt;span class=&quot;math-container&quot;&gt;$g_3$&lt;/span&gt; suppresses the expression level of &lt;span class=&quot;math-container&quot;&gt;$g_4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g_5$&lt;/span&gt;), and we don't have to focus on the expression levels individually. This would mean that by knowing the expression level of &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; we can get some sense of the expression level of &lt;span class=&quot;math-container&quot;&gt;$g_2$&lt;/span&gt;, and from the level of &lt;span class=&quot;math-container&quot;&gt;$g_3$&lt;/span&gt; we can guess the levels of &lt;span class=&quot;math-container&quot;&gt;$g_4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g_5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus the principal basis vectors reflect biological function. That's pretty cool.&lt;/p&gt;&#xA;&lt;h3&gt;Protein Conformational Motions of HIV-1 Protease (biology, medicine, biomedical)&lt;/h3&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;http://recomb2000.hgc.jp/Posters/pdf/98.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Singular Value Decomposition of Protein Conformational Motions: Application to HIV-1 Protease&lt;/a&gt;, Teodoro &lt;em&gt;et al.&lt;/em&gt; analyze the conformational changes of an enzyme via the SVD.  &amp;quot;An illustrative example of these changes is the opening and closing of the binding site of HIV-1 protease (HIV Pr)....&amp;quot;  HIV Pr plays an important role in the life cycle of the HIV virus, and the opening and closing of the binding site allows the enzyme to do its job.  Disrupting its function is the focus of &lt;a href=&quot;https://en.wikipedia.org/wiki/Protease_inhibitor_(pharmacology)&quot; rel=&quot;nofollow noreferrer&quot;&gt;protease inhibitors&lt;/a&gt;.  The analysis of the motions is similar to the molecular spectra discussed above, except proteins exhibit both small vibrations and larger motions, and the interest lies in the larger motions.  The dimension of the space of motions is &lt;span class=&quot;math-container&quot;&gt;$597$&lt;/span&gt;, and the SVD was used to focus on a few principal basis vectors.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The left singular vectors [i.e., the columns of &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt;] obtained from the decomposition correspond to modes of collective motion whose displacements are directly proportional to the value of the corresponding singular values. The calculated right singular vectors [i.e., the columns of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;] correspond to the projection of the original trajectory in the new basis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this way, they were able to show that &amp;quot;the protein jumps rapidly between stable intermediate conformations until it reaches the open conformation.&amp;quot;  Again, the problem cannot easily be investigated in terms of &lt;span class=&quot;math-container&quot;&gt;$597$&lt;/span&gt;-tuples, and choosing the right basis, in which the vectors reflect structures within the problem, allows one to deduce how those structures work.&lt;/p&gt;&#xA;&lt;h2&gt;Function approximation&lt;/h2&gt;&#xA;&lt;p&gt;Function approximation is not a science topic per se, but it is used broadly in numerical solutions to scientific problems.  Let's say you have a bunch of function values &lt;span class=&quot;math-container&quot;&gt;$y^{(j)}$&lt;/span&gt; at a number of points &lt;span class=&quot;math-container&quot;&gt;$\x^{(j)}=(x_1^{(j)},x_2^{(j)},\dots,x_n^{(j)})$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$j=1,2,\dots,m$&lt;/span&gt;, and you'd like to construct a continuous function &lt;span class=&quot;math-container&quot;&gt;$f(\x)=f(x_1,x_2,\dots)$&lt;/span&gt; based on this data. How to do it?&lt;/p&gt;&#xA;&lt;p&gt;Viewed as a linear problem, we would choose a basis of functions &lt;span class=&quot;math-container&quot;&gt;$\{\phi_1,\dots,\phi_n\}$&lt;/span&gt;&#xA;and solve for coefficients &lt;span class=&quot;math-container&quot;&gt;$c_j$&lt;/span&gt; such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(\x) = \sum c_j \phi_j(\x)$$&lt;/span&gt;&#xA;meet some criteria.  The &lt;strong&gt;interpolation problem&lt;/strong&gt; is to find a function that passes through each point so that &lt;span class=&quot;math-container&quot;&gt;$f(\x^{(j)})=y^{(j)}$&lt;/span&gt;.  The &lt;strong&gt;fitting problem&lt;/strong&gt; is to find a function that minimizes the error between the values &lt;span class=&quot;math-container&quot;&gt;$f(\x^{(j)})$&lt;/span&gt; of the function and the values &lt;span class=&quot;math-container&quot;&gt;$y^{(j)}$&lt;/span&gt; at each point; the error typically used is &lt;strong&gt;least squares&lt;/strong&gt;, which problem can be solved with linear algebra.  When the values &lt;span class=&quot;math-container&quot;&gt;$y^{(j)}$&lt;/span&gt; are the values of a given function, then the problem is one of approximating the given function.  If the basis functions have the property that &lt;span class=&quot;math-container&quot;&gt;$\phi_k(\x^{(k)})=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\phi_k(\x^{(j)})=0$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$j\ne k$&lt;/span&gt;, then the basis is particularly nice for the interpolation problem: &lt;span class=&quot;math-container&quot;&gt;$c_j=y^{(j)}$&lt;/span&gt; and there's no need to solve a system of equations for &lt;span class=&quot;math-container&quot;&gt;$c_j$&lt;/span&gt;. Such a basis is called a &lt;strong&gt;cardinal basis&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;h3&gt;Polynomial interpolation&lt;/h3&gt;&#xA;&lt;p&gt;The power basis &lt;span class=&quot;math-container&quot;&gt;$1,\ x,\ x^2,\dots, x^n$&lt;/span&gt; is often numerically poorly conditioned for the interpolation problem. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Chebyshev_polynomials&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Chebyshev polynomials&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$T_j$&lt;/span&gt;&lt;/a&gt; form a better basis, and the &lt;strong&gt;Lagrange basis&lt;/strong&gt;, which is a cardinal basis, is better still in its &lt;a href=&quot;https://doi.org/10.1137/S0036144502417715&quot; rel=&quot;nofollow noreferrer&quot;&gt;barycentric form (see Berrut &amp;amp; Trefethen, eq. (4.2)&lt;/a&gt;). There are other families of orthogonal polynomials besides Chebyshev that have their uses.&lt;/p&gt;&#xA;&lt;h3&gt;Chebyshev approximation&lt;/h3&gt;&#xA;&lt;p&gt;A truncated power series (Taylor polynomial) is often not a good approximation away from its center of expansion.  However, for a function that is analytic over a finite interval &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt;, its Chebyshev series converges rapidly on &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt; with an error bound that is easy to state and apply practically. Let &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; have a Chebyshev series&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x) = \sum_{j=0}^\infty c_j T_j\left( \frac{2 x - (a+b)}{b-ab} \right) \,.$$&lt;/span&gt;&#xA;If the series is truncated at the term &lt;span class=&quot;math-container&quot;&gt;$j=J$&lt;/span&gt;, then since &lt;span class=&quot;math-container&quot;&gt;$|T_j| \le 1$&lt;/span&gt; on the interval, the truncation error is bounded by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{j&amp;gt;J} |c_j| \,, \tag{3}$$&lt;/span&gt;&#xA;which is similar to the SVD truncation error bound in (2).  If the coefficients &lt;span class=&quot;math-container&quot;&gt;$c_j$&lt;/span&gt; are rapidly decreasing,&#xA;then &lt;span class=&quot;math-container&quot;&gt;$c_J$&lt;/span&gt; is usually a good bound on the error.&#xA;If we interpolate &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; at the Chebyshev points &lt;span class=&quot;math-container&quot;&gt;$x_k = \cos(\pi k /J)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$k=0,1,\dots,J$&lt;/span&gt;&#xA;using the &lt;strong&gt;Chebyshev basis&lt;/strong&gt;, we approximate the truncated Chebyshev series&#xA;&lt;span class=&quot;math-container&quot;&gt;$$p(x) = \sum_{j=0}^J a_j T_j\left( \frac{2 x - (a+b)}{b-ab} \right)&#xA; \approx \sum_{j=0}^J c_j T_j\left( \frac{2 x - (a+b)}{b-ab} \right)&#xA; \approx f(x)\,.$$&lt;/span&gt;&#xA;The Chebyshev basis is an orthogonal basis with respect to the inner product &lt;span class=&quot;math-container&quot;&gt;$\langle f, g \rangle = \int_a^b f(x) g(x) / \sqrt{(x-a)(b-x)}\, dx$&lt;/span&gt;, and truncating the series is equivalent to orthogonal projection onto&#xA;a finite dimensional subspace (= polynomials of degree at most &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt;). As such, the truncated series is an optimal solution to approximating the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; with a polynomial of degree at most &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt; with respect to the norm induced by the inner product.&#xA;Further it is easy to evaluate, integrate, differentiate and find the roots of Chebyshev interpolations.&#xA;It is one of the best ways to solve transcendental equations (see &lt;a href=&quot;https://doi.org/10.1137/1.9781611973525&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boyd (2014)&lt;/a&gt;).&#xA;Chebyshev approximation is the basis of the &lt;a href=&quot;http://www.chebfun.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chebfun project&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;h3&gt;Fourier approximation&lt;/h3&gt;&#xA;&lt;p&gt;Virtually the same thing can be said for Fourier approximation, which in fact might be viewed as the basis for Chebyshev approximation.  The &lt;strong&gt;Fourier basis&lt;/strong&gt;, &lt;a href=&quot;https://matheducators.stackexchange.com/a/25039/498&quot;&gt;mentioned by @Kuba&lt;/a&gt;, for the interval &lt;span class=&quot;math-container&quot;&gt;$[-\pi,\pi]$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\phi_j(x) = e^{i j x}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$j\in{\Bbb Z}$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$|e^{i j x}|=1$&lt;/span&gt; and the coefficients of Fourier series for analytic functions are rapidly decreasing, we get similar approximation properties, including orthonormality of the basis (with respect to the inner product &lt;span class=&quot;math-container&quot;&gt;$(2\pi)^{-1}\int_{-\pi}^\pi f(x) g(x) \, dx$&lt;/span&gt;).  One reason this and the Chebyshev approximation are important is that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fast_Fourier_transform&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fast Fourier Transform (FFT)&lt;/a&gt; makes these approximations computationally efficient.&lt;/p&gt;&#xA;&lt;h3&gt;Sinc interpolation (engineering, signal processing, image processing)&lt;/h3&gt;&#xA;&lt;p&gt;The sinc function &lt;span class=&quot;math-container&quot;&gt;$(\sin t)/t$&lt;/span&gt; or the normalized sinc function &lt;span class=&quot;math-container&quot;&gt;$\sin(\pi t)/(\pi t)$&lt;/span&gt; (for &lt;span class=&quot;math-container&quot;&gt;$t\ne0$&lt;/span&gt;) can be used to form a cardinal basis for interpolating sampled signals or other regularly spaced time series. It should be understood that the sinc functions are taken to be continuous and have the value &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$t=0$&lt;/span&gt;.&#xA;Suppose we have a regular sample rate of &lt;span class=&quot;math-container&quot;&gt;$1/h$&lt;/span&gt; and thus the sampled times are &lt;span class=&quot;math-container&quot;&gt;$t_j=jh$&lt;/span&gt;.&#xA;The &lt;strong&gt;sinc basis&lt;/strong&gt; functions are&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\phi_j(jh) = 1\,,\ \text{otherwise}\ \phi_j(t) = {\sin(\pi (t-jh)) \over \pi (t-jh)} \,.$$&lt;/span&gt;&#xA;The interpolation formula for the sampled data &lt;span class=&quot;math-container&quot;&gt;$y^{(j)}=f(jh)$&lt;/span&gt; of a signal &lt;span class=&quot;math-container&quot;&gt;$f(t)$&lt;/span&gt; is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(t) \approx \sum_j y^{(j)} \phi_j(t) \,.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h3&gt;Radial basis function interpolation (neural networks, engineering, signal processing)&lt;/h3&gt;&#xA;&lt;p&gt;Radial basis function (RBF) interpolation can be used to interpolate &amp;quot;unstructured&amp;quot; data of any dimension (the sampling nodes need not be on a regular grid). For instance, the sampling nodes could be the locations in a network of weather stations or other geographic phenomena. Applications include the approximation of functions on arbitrary surfaces; surface reconstruction from medical imaging or geological stratigraphic data; machine learning; and so forth.&#xA;An RBF approximation is built on a &lt;strong&gt;radial basis&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$\phi_j(x)=\phi(\| \x-\x_j\|)$&lt;/span&gt; and has the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(\x) \approx \sum_{j=1}^J c_j \phi(\| \x-\x_j\|) \tag{4}$$&lt;/span&gt;&#xA;for some function &lt;span class=&quot;math-container&quot;&gt;$\phi(r)$&lt;/span&gt; of the distance from a sampling node, typically a Gaussian &lt;span class=&quot;math-container&quot;&gt;$e^{-\alpha^2 r^2}$&lt;/span&gt; for some  &amp;quot;shape parameter&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;.&#xA;Here &lt;span class=&quot;math-container&quot;&gt;$\x$&lt;/span&gt; may be in &lt;span class=&quot;math-container&quot;&gt;$\R$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\R^n$&lt;/span&gt;.&#xA;Solving for the coefficients &lt;span class=&quot;math-container&quot;&gt;$c_j$&lt;/span&gt; is a linear problem and might be done either as a least-squares fitting problem or as an interpolation problem.&#xA;As an example,&#xA;&lt;a href=&quot;http://dx.doi.org/10.1090/surv/141&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maz'ya &amp;amp; Schmidt (2007)&lt;/a&gt;&#xA;and&#xA;&lt;a href=&quot;http://dx.doi.org/10.1016/j.amc.2009.08.037&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boyd &amp;amp; Wang (2009)&lt;/a&gt;&#xA;independently derived a cardinal basis for Gaussian RBF approximation on an infinite grid with uniform spacing &lt;span class=&quot;math-container&quot;&gt;$h = x_j - x_{j-1}$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$-\infty&amp;lt;j&amp;lt;\infty$&lt;/span&gt;)&#xA;and shape parameter &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\phi_j(x)=\phi(x-x_j),\quad\text{where}\quad&#xA; \phi(x)= {\alpha ^2\over\pi}\,&#xA; {\sin \left({\pi x}/{h}\right) \over {\sinh}\left({\alpha^2 x}/{h}\right)} \,.$$&lt;/span&gt;&#xA;Since for this grid, the sum would be infinite, one usually truncates it and sums over a finite interval.&#xA;(It is related to sinc interpolation but the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;s and &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;s in &lt;span class=&quot;math-container&quot;&gt;$\sin \left({\pi x}/{h}\right)/(\pi x/h)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;${\sinh}\left({\alpha^2 x}/{h}\right)/(\alpha^2 x/h)$&lt;/span&gt; cancel out.)&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2022-03-26T22:26:49.827" LastActivityDate="2022-03-26T22:26:49.827" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25075" PostTypeId="2" ParentId="24401" CreationDate="2022-03-22T00:43:23.163" Score="2" Body="&lt;p&gt;The approach I tend to use is similar to that presented by user58697, and agrees with that presented in several calculus texts (e.g. &lt;em&gt;Thomas' Calculus&lt;/em&gt;, but not any of the versions with the &amp;quot;Early Transcendentals&amp;quot; sub-titles).&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Define the &lt;em&gt;natural logarithm&lt;/em&gt; by the integral formula:  if &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 0$&lt;/span&gt;, then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \log(x) := \int_{1}^{x} \frac{1}{t} \,\mathrm{d}t. $$&lt;/span&gt;&#xA;In terms of putting this definition into the &amp;quot;story&amp;quot; or &amp;quot;narrative&amp;quot; that I try to develop in class, the natural logarithm fills a hole in the power rule.  For any &lt;span class=&quot;math-container&quot;&gt;$n \ne -1$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int x^n \,\mathrm{d}x = \frac{1}{n+1} x^{n+1}. $$&lt;/span&gt;&#xA;This rule fails for &lt;span class=&quot;math-container&quot;&gt;$n=-1$&lt;/span&gt;, but the function &lt;span class=&quot;math-container&quot;&gt;$x \mapsto x^{-1}$&lt;/span&gt; is continuous (and therefore differentiable) on &lt;span class=&quot;math-container&quot;&gt;$(0,\infty)$&lt;/span&gt;, so the function&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x \mapsto \int_{1}^{x} t^{-1} \,\mathrm{d}t $$&lt;/span&gt;&#xA;is a perfectly well-defined function.  For mysterious reasons that I usually just kind of skip on by, we call this function the natural logarithm.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is not too difficult to show that the natural logarithm has a lot of nice properties, such as&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; is a continuous function,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log(ab) = \log(a) + \log(b)$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log(1/a) = -\log(a)$&lt;/span&gt;, and&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\log(a^r) = r \log(a)$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$a,b \in (0,\infty)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r \in \mathbb{Q}$&lt;/span&gt; (it is harder to show that last identity holds when &lt;span class=&quot;math-container&quot;&gt;$r \in \mathbb{R}$&lt;/span&gt;, as it requires working carefully with some limits, but this, too, can be done if you feel like it).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At this point in the course, students have been introduced to the inverse function theorem (IFT), so it is not unreasonable to ask about the inverse of the logarithm.  From the,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (\log^{-1})'(x) = \frac{1}{\log' (\log^{-1} x)} = \frac{1}{1/\log^{-1}(x)} = \log^{-1}(x). $$&lt;/span&gt;&#xA;In other words, the inverse of the natural logarithm is a function which has the property that it is its own derivative.  For more mysterious reasons (again, sometimes I talk about this), define the function &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt; to be the inverse of the natural logarithm (this is the &lt;em&gt;natural exponential&lt;/em&gt; function).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Running through properties of the logarithm (only backwards), it is possible to show that&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\exp(a+b) = \exp(a)\exp(b)$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\exp(-a) = 1 / \exp(a)$&lt;/span&gt;, and&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\exp(a^r) = \exp(a)^r$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;These properties are very reminiscent of how rational exponents work over the real numbers.  That is, if &lt;span class=&quot;math-container&quot;&gt;$a,b \in\mathbb{R}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c \in (0,\infty)$&lt;/span&gt;, then&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$c^{a+b} = c^a c^b$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$c^{-a} = 1/c^a$&lt;/span&gt;, and&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$c^{ar} = (c^a)^r$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This suggests that it might be reasonable to define a class of functions:  for each &lt;span class=&quot;math-container&quot;&gt;$c \in (0,\infty)$&lt;/span&gt;, define&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \exp_c : \mathbb{Q} \to \mathbb{R} : x \mapsto c^x. $$&lt;/span&gt;&#xA;This is the &lt;em&gt;exponential function with base &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I am a little careful about this, because the tools available in the class up to this point make it harder to extend this function to something continuous in a tractable manner (for intro students).  Rational exponentiation can be defined inductively, but the continuous extension is a bit of a leap.  However...&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This discussion suggests that there may be a special number (which I am going to call &lt;span class=&quot;math-container&quot;&gt;$\mathrm{e}$&lt;/span&gt;) which has the property that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \exp(x) = \exp_{\mathrm{e}}(x). $$&lt;/span&gt;&#xA;Actually, this step is kind of not that hard to believe.  Really, all I am doing is asserting that there is a number &lt;span class=&quot;math-container&quot;&gt;$\mathrm{e}$&lt;/span&gt; which has the property that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \exp(x) = \mathrm{e}^x$$&lt;/span&gt;&#xA;for any rational number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  But &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt; is continuous on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, which gives us the continuous extension we wanted above.&lt;/p&gt;&#xA;&lt;p&gt;Note that it is possible to approximate &lt;span class=&quot;math-container&quot;&gt;$\mathrm{e}$&lt;/span&gt; in a number of ways.  One possibility is to note that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\exp(1) = \mathrm{e} \iff 1 = \log(\mathrm{e}),$$&lt;/span&gt;&#xA;and then approximate its value using Riemann sums over partitions with increasingly small mesh.  A better approach is outlined in step (2) of the question (I usually do &lt;em&gt;both&lt;/em&gt; in order to demonstrate what a pain in the ass Riemann sums are if you have to work with them directly).&lt;/p&gt;&#xA;&lt;p&gt;At this point, it is entirely reasonable to assert that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\exp_{c}(x) = c^x$$&lt;/span&gt;&#xA;is perfectly well-defined for all &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{R}$&lt;/span&gt;, and that the function &lt;span class=&quot;math-container&quot;&gt;$\exp_c$&lt;/span&gt; is continuous, and has the desired properties (most importantly, that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\exp_{c}(a+b) = \exp_{c}(a)\exp_{c}(b) \leftrightarrow c^{a+b} = c^ac^b. $$&lt;/span&gt;&#xA;for any real &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Note that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \exp_c(x) = c^x = \exp(\log(c))^x = \exp(x \log(c)), $$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\log(c)$&lt;/span&gt; is &amp;quot;just&amp;quot; a constant.  Thus, from the chain rule,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{\mathrm{d}}{\mathrm{d}x} \exp_c(x) = \exp(x \log(c)) \frac{\mathrm{d}}{\mathrm{d}x} [ x \log(c)] = \log(c) \exp(x \log(c)) = \log(c) c^x. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, note that the whole machine can be run &amp;quot;backwards&amp;quot;:  define the &lt;em&gt;logarithm with base &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&lt;/em&gt; by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \log_c := \exp_{c}^{-1} $$&lt;/span&gt;&#xA;(that is, the logarithm with base &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the inverse of the exponential with base &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;), and run with it.  It is not too difficult to deduce properties of this logarithm, including the change of base formula (which should be mentioned):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \log_c(a) = \frac{\log_d(a)}{\log_d(c)}, $$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$c, d &amp;gt; 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2022-03-22T00:43:23.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25076" PostTypeId="2" ParentId="24401" CreationDate="2022-03-22T01:14:10.060" Score="1" Body="&lt;p&gt;A historical perspective may be interesting to students. Here's how Napier's definition of the logarithm (which he worked out a century before calculus), translated into terms of the natural logarithm &amp;amp; calculus, reveals the derivative of the logarithm.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Historical background&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Napier&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Napier (1550-1617)&lt;/a&gt; published his table of logarithms &lt;a href=&quot;https://books.google.com/books?id=J19dAAAAcAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mirifici Logarithmorum Canonis Descriptio&lt;/em&gt;&lt;/a&gt; in 1614 after some twenty years of work and described his method of construction in &lt;a href=&quot;https://books.google.com/books?id=VukHAQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mirifici Logarithmorum Canonis Constructio&lt;/em&gt;&lt;/a&gt;, published posthumously in 1619 (Edinburgh) by his son Robert, with appendices by Napier and &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_Briggs_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henry Briggs (1561-1630)&lt;/a&gt;.  In our modern terms, we can represent Napier's definition of a logarithm (Prop. 26):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;artificial number&amp;quot; (logarithm) of a given sine is that which has increased Arithmetically always with the same velocity as the total sine [began] to decrease Geometrically and in the same time as the total sine decreased to that given sine.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ignoring what is exactly meant by the given sine and total sign, what he is building on is a fundamental fact about geometric sequences that has been known from ancient times: If you take a geometric sequence starting from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and number the terms with consecutive integer indices start from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;,&#xA;then the product of two given terms of the geometric sequence will be the term in the geometric sequences who index is the sum of the indices of the given terms. In algebraic notation, this is the familiar law &lt;span class=&quot;math-container&quot;&gt;$a^ja^k=a^{j+k}$&lt;/span&gt;.  What Napier is saying is that if the index (the logarithm, which he first called an &amp;quot;artificial number&amp;quot;) increases a at a constant rate (arithmetically), then the term (called a sine) changes at a geometric rate, that is, proportional to its own magnitude. In fact, he has it backwards from the way I am going to do it: If &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the sine, then the rate is proportional to &lt;span class=&quot;math-container&quot;&gt;$r-x$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the total sine. Napier uses trigonometrical language for reasons I can only guess, which I won't go into. Remarkably from this, Napier is able to prove some inequalities about logarithms and construct quite an accurate table.  A key invention for his success is that he is imagining two quantities in continuous motion under two different laws for their rates of motion.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Modern version&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; change at an &amp;quot;arithmetic&amp;quot; (constant) rate starting from &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt;, and let &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; change at a &amp;quot;geometric&amp;quot; rate (proportional to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;) starting from &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt;; and we'll let the constants both be &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$${dy \over dt} = 1\,,\quad {dx \over dt} = x\,;\quad y(0)=0\,,\ x(0)=1 \,.$$&lt;/span&gt;&#xA;Then the natural logarithm at a given &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is defined to be &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Clearly, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$${d\over dx}\log x = {dy \over dx} = {dy/dt \over dx/dt} = {1 \over x}\,.$$&lt;/span&gt;&#xA;Likewise, since &lt;span class=&quot;math-container&quot;&gt;$x = \exp y$&lt;/span&gt;, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$${d\over dy}\exp y = {dx \over dy} = {dx/dt \over dy/dt} = x = \exp y\,.$$&lt;/span&gt;&#xA;Objections about the need to prove the existence of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; are quite technical, and I would question whether they are appropriate for the OP's purposes.&lt;/p&gt;&#xA;&lt;p&gt;Note that one can prove properties of &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; from this definition, but that would seem to go beyond the scope of the question.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2022-06-04T05:14:40.843" LastActivityDate="2022-06-04T05:14:40.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25077" PostTypeId="2" ParentId="24401" CreationDate="2022-03-22T20:48:00.390" Score="3" Body="&lt;p&gt;Here is the approach which seems most natural to me, and it wasn't a disaster the two times I used it in a classroom.  The basic idea is to hold off on &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; until it is motivated.&lt;/p&gt;&#xA;&lt;p&gt;Start with numerical data: &lt;span class=&quot;math-container&quot;&gt;$(2^{0.1}, 2^{0.01}, 2^{0.001}) = (1.071773, 1.006956, 1.000693)$&lt;/span&gt;. It looks like &lt;span class=&quot;math-container&quot;&gt;$2^h \approx 1 + 0.693 h$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; small. Ask class: What do we call that &lt;span class=&quot;math-container&quot;&gt;$0.693$&lt;/span&gt;? Hopefully, they know this is the derivative.&lt;/p&gt;&#xA;&lt;p&gt;Now shift to another point on &lt;span class=&quot;math-container&quot;&gt;$2^x$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$(2^{1.1}, 2^{1.01}, 2^{1.001}) = (2.143547, 2.013911, 2.001387)$&lt;/span&gt;. What does it look like we can say about &lt;span class=&quot;math-container&quot;&gt;$2^{1+h}$&lt;/span&gt;? If you are lucky, you get two answers: &lt;span class=&quot;math-container&quot;&gt;$2^{1+h} \approx 2+1.387 h$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2^{1+h} = 2 \times 2^h \approx 2(1+0.693 h)$&lt;/span&gt;. If you are not so lucky (I was one time and not the other), point out that &lt;span class=&quot;math-container&quot;&gt;$1.387$&lt;/span&gt; is twice &lt;span class=&quot;math-container&quot;&gt;$0.693$&lt;/span&gt; and ask if this is a coincidence.&lt;/p&gt;&#xA;&lt;p&gt;Keep playing around until students are happy that &lt;span class=&quot;math-container&quot;&gt;$2^{x+h} \approx 2^x +0.693  \times 2^x h$&lt;/span&gt;. So  &lt;span class=&quot;math-container&quot;&gt;$\tfrac{d}{dx} 2^x = 0.693 \times 2^x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;One could play the same game with powers of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, or jump to the general computation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d}{dx} a^x = \lim_{h \to 0} \frac{a^{x+h} - a^x}{h} = a^x \lim_{h \to 0} \frac{a^{h} - 1}{h}.$$&lt;/span&gt;&#xA;So &lt;span class=&quot;math-container&quot;&gt;$\tfrac{d}{dx} a^x = L(a) a^x$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$L(a) = \lim_{h \to 0} \frac{a^{h} - 1}{h}$&lt;/span&gt;. (This wasn't the sort of course where we proved limits exist.)&lt;/p&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$L(a)$&lt;/span&gt;? If &lt;span class=&quot;math-container&quot;&gt;$a = b^k$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$L(a) = \lim_{h \to 0} \tfrac{a^{h} - 1}{h} =\lim_{h \to 0} \frac{b^{kh} - 1}{h}= k \lim_{h \to 0} \frac{b^{kh} - 1}{kh} = k L(b)$&lt;/span&gt;. In other words, &lt;span class=&quot;math-container&quot;&gt;$L(a) = \log_b(a) L(b)$&lt;/span&gt;. So, if we know &lt;span class=&quot;math-container&quot;&gt;$L(b)$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, we can compute all the other &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;'s.&lt;/p&gt;&#xA;&lt;p&gt;We could just compute one value once and for all, say &lt;span class=&quot;math-container&quot;&gt;$L(10) = 2.30259$&lt;/span&gt;. But, instead, we introduce &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; as the number where &lt;span class=&quot;math-container&quot;&gt;$L(e) = 1$&lt;/span&gt; ...&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;The warning I would give is that, while all of our students had seen exponentials, it soon became clear that many of them didn't really remember how they work. I spend a lot of time writing &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$(a^3)^2 \neq a^9$&lt;/span&gt;!&amp;quot; and similar comments.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2022-03-23T17:55:56.310" LastActivityDate="2022-03-23T17:55:56.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25078" PostTypeId="1" AcceptedAnswerId="25086" CreationDate="2022-03-23T20:35:31.923" Score="10" ViewCount="4279" Body="&lt;p&gt;Is it standard for a math undergraduate program to have a course on Lebesgue integration?&lt;/p&gt;&#xA;&lt;p&gt;Does Riemann integral suffice for undergraduates?&lt;/p&gt;&#xA;&lt;p&gt;The reason of my question is I read a paper by Bartle titled &lt;a href=&quot;https://doi.org/10.2307/2974874&quot; rel=&quot;nofollow noreferrer&quot;&gt;Return to Riemann Integral&lt;/a&gt;.  In his paper,  Bartle claims that most undergraduate US students are not ready to study Lebesgue integration.  The paper was written in 1996.  Did anything change since then?&lt;/p&gt;&#xA;" OwnerUserId="580" LastEditorUserId="15254" LastEditDate="2023-09-18T15:53:07.047" LastActivityDate="2023-09-18T15:53:07.047" Title="Should an undergraduate math program contain a course on Lebesgue integration?" Tags="|undergraduate-education|mathematical-analysis|usa|integration|real-numbers|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25079" PostTypeId="2" ParentId="25078" CreationDate="2022-03-23T22:14:20.733" Score="15" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Is it standard for a math undergraduate program to have a course on Lebesgue integration?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No (assuming that &amp;quot;have a course&amp;quot; means &amp;quot;require people to take such a course in order to get a degree&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;But you don't have to take my word for it. Go ahead and check the catalogs of a few top-flight schools in the US.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Does Riemann integral suffice for undergraduates?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes.&lt;/p&gt;&#xA;&lt;p&gt;Michał Miśkiewicz says in a comment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If this math program includes probability theory (which uses the Lebesgue integral for the expected value) or functional analysis ( spaces require the Lebesgue integral), then it's a necessary prerequisite.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No. A normal undergraduate class in probability will simply assume that the functions they're dealing with are sufficiently well behaved that the Riemann integral is the same as the Lebesgue integral. They might make this assumption explicit or leave it implicit. It makes no difference because there are no real-world application in which it matters.&lt;/p&gt;&#xA;&lt;p&gt;An undergraduate class in probability is mainly populated with stat majors. I can't imagine how an undergraduate statistics program could make an entire course in the Lebesgue measure a prerequisite for their probability course. There's no way their students would satisfy that stack of requirements in time to graduate.&lt;/p&gt;&#xA;" OwnerUserId="19682" LastEditorUserId="19682" LastEditDate="2022-03-23T22:20:01.413" LastActivityDate="2022-03-23T22:20:01.413" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25080" PostTypeId="1" CreationDate="2022-03-24T01:38:19.573" Score="0" ViewCount="359" Body="&lt;p&gt;What is your favorite PreCalculus textbook for someone that needs to get their algebra skills up to snuff? Something comprehensive with some tricky problems. Stewart?  Sullivan? Blitzer? Something different?&lt;/p&gt;&#xA;" OwnerUserId="9968" LastActivityDate="2022-03-24T10:10:43.110" Title="Best PreCalculus Textbook" Tags="|precalculus|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25081" PostTypeId="2" ParentId="25078" CreationDate="2022-03-24T02:46:38.257" Score="6" Body="&lt;p&gt;Yes. As an elective. If there are interested students we ought to give them a platform to exercise their curiosity. As a requirement, well, that depends on what you aim for the mission of your program. If you want your program to be elite and give students the best chance at hitting graduate school with a well-rounded education then yes, emphatically yes this should be included. The 4-part Princeton Lecture Series on Analysis gives a reasonable path. I found it a worthwhile experience to teach such a course to several gifted students.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2022-03-24T02:46:38.257" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25082" PostTypeId="2" ParentId="25080" CreationDate="2022-03-24T10:10:43.110" Score="3" Body="&lt;p&gt;I like the old (1958) Ayres Schaum's book:  &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0007DPVM2&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/dp/B0007DPVM2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[not promotional]&lt;/p&gt;&#xA;&lt;p&gt;I generally find that older editions are better, but I have no evidence that the Schaum's got worse.  Just the old edition is a known trusted quantity.  You can get it used for $30 or so. [check other booksellers if you want to try to shave that number down, personally I think 30 is cheap, when you consider the value of your time from actually studying.]&lt;/p&gt;&#xA;&lt;p&gt;I think there are some disadvantages (in general, caveat, caveat) of web-based open textbooks.  Not professionally proofed, edited.  And I think for a person actually sitting and doing drill homework, that a paper book is more convenient and worth the price.  Especially if you do some used, trusted edition.&lt;/p&gt;&#xA;&lt;p&gt;Note, the Schaum's has all the answers, not just half.  So it is designed for self-studiers, unlike books marketed to professors, where the market (professors, not students) wants to hold some of the answers captive.  I also find a drill book or review (of any sort, but Schaum's good) is more convenient and usable for adult students that may have had the course before, but just need to improve their algebra via drill.  (Typical &amp;quot;doorstop&amp;quot; texts marketed to selection committees are written in a way to impress selection committees with verbose text and derivations rather than to connect with students.  And the rocket pictures, not needed by kids drilling.)&lt;/p&gt;&#xA;" OwnerUserId="19683" LastActivityDate="2022-03-24T10:10:43.110" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25083" PostTypeId="2" ParentId="25078" CreationDate="2022-03-24T12:24:20.430" Score="8" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Is it standard for a math undergraduate program to have a course on Lebesgue integration?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes, and I find it bizarre that a university would not have one.&lt;/p&gt;&#xA;&lt;p&gt;Lebesgue integration (or measure theory more generally) is necessary for a proper grounding in probability theory. Sure, most statisticians won't need it, but it's a very important part of any pure mathematics course. You certainly can't make a rigorous study of probability or statistics without it.&lt;/p&gt;&#xA;&lt;p&gt;One answer took &amp;quot;have a course&amp;quot; to mean &amp;quot;is a required course&amp;quot;. I would not be surprised if an elementary course on Lebesgue integration would be mandatory (although I would certainly not expect a full measure theory course to be mandatory).&lt;/p&gt;&#xA;&lt;p&gt;Since you are asking about specific programs: I took a measure theory course in third/fourth year that covered Lebesgue integration (among other things). This is now a second-year course called &amp;quot;Lebesgue Measure and Integration&amp;quot;. Additionally, if you take Real Analysis in second year then you cover multivariate Darboux/Riemann integration.&lt;/p&gt;&#xA;&lt;p&gt;The module guide has this to say about the Lebesgue Measure and Integration course:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Lebesgue Measure and Integration:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Essential for: All Pure Analysis modules; Geometry of curves and&#xA;surfaces.&lt;/p&gt;&#xA;&lt;p&gt;Useful for: All pure mathematics modules. Dynamical Systems;&#xA;Statistics modules. Mathematical Finance; Function Spaces and&#xA;Applications; Advanced Topics in PDEs; modules involving stochastic&#xA;analysis; Random Dynamical Systems and Ergodic Theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;MATH50006 Lebesgue Measure and Integration&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Brief description: Lebesgue's theory of measure and integration is a&#xA;powerful extension of the Riemann integral introduced in first-year&#xA;analysis. It is an essential tool in all aspects of analysis and its&#xA;applications, including probability, stochastic processes and PDEs.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Learning Outcomes: On successful completion of this module, you will be able to:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;explain key features of the construction of Lebesgue measure, and basic features of the measure as a function,&lt;/li&gt;&#xA;&lt;li&gt;explain the limitation of a measure, and measurable sets through algebras of open sets,&lt;/li&gt;&#xA;&lt;li&gt;define the measurability of functions in terms of measurable sets, and apply simple criteria to verify measurability,&lt;/li&gt;&#xA;&lt;li&gt;outline the Lebesgue integral of a function, and how it relates to the Riemann integral,&lt;/li&gt;&#xA;&lt;li&gt;work with spaces of functions as infinite dimensional vector spaces, and algebras, and discuss a variety of such spaces,&lt;/li&gt;&#xA;&lt;li&gt;use results on convergence of sequences in measurable functions spaces, and criterial to derive properties of limiting functions and&#xA;their integrals,&lt;/li&gt;&#xA;&lt;li&gt;discuss the relationship between integrability and differentiation,&lt;/li&gt;&#xA;&lt;li&gt;explain and use the theorems of Fubini and Tonelli,&lt;/li&gt;&#xA;&lt;li&gt;discuss abstract measure theory, and the difficulties with construction of measures and integrals on general spaces.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Module content (syllabus):&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;An indicative list of sections and topics is:&lt;/p&gt;&#xA;&lt;p&gt;Motivation: Drawbacks of the Riemann integral, Limits of functions,&#xA;Length and area, The Fundamental Theorem of Calculus, Measures of sets&#xA;in R^d,&lt;/p&gt;&#xA;&lt;p&gt;Measure Theory: Abstract measure theory: Motivation, basic definitions&#xA;of measure spaces and measures.&lt;/p&gt;&#xA;&lt;p&gt;Lebesgue Measure in R^d: Volume of rectangles and cubes, The exterior&#xA;measure, properties, Lebesgue measurable sets, countable additivity.&#xA;Properties of the Lebesgue measure, Regularity, Invariance, σ-algebras&#xA;and Borel sets, Non-measurable set.&lt;/p&gt;&#xA;&lt;p&gt;Measurable functions: Definitions and equivalent formulations of&#xA;measurability, Sums and products, compositions, limits of measurable&#xA;functions, “Almost everywhere” properties. Approximation by simple&#xA;functions and step functions, Egoroff’s and Lusin’s theorems.&lt;/p&gt;&#xA;&lt;p&gt;Lebesgue integration: Definition using bounded functions on sets of&#xA;finite measure, Riemann integrable functions are Lebesgue integrable,&#xA;Integrable functions as a normed vector space, L_p(R^d), dense&#xA;subsets, Completeness, Fatou’s Lemma, monotone convergence theorem,&#xA;uniform integrability, Vitali’s Theorem, Fubini’s and Tonelli’s&#xA;Theorems, statements and proofs.&lt;/p&gt;&#xA;&lt;p&gt;Differentiation and Integration: Differentiation of the Integral,&#xA;statement of Lebesgue differentiation theorem, Differentiation of&#xA;functions, Functions of bounded variation, properties,&#xA;characterisation, Bounded variation implies differentiable a.e.&#xA;Absolute continuity of measures, decomposition theorems by Jordan,&#xA;Hahn and Lebesgue, Radon-Nikodym Theorem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I can confirm that when I took this course, all of this was covered (along with some extra material about ergodic theory for the mastery component). This is at Imperial College London.&lt;/p&gt;&#xA;" OwnerUserId="8110" LastActivityDate="2022-03-24T12:24:20.430" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25084" PostTypeId="2" ParentId="25078" CreationDate="2022-03-24T15:36:50.480" Score="6" Body="&lt;p&gt;For undergraduate maths students at the University of Oxford, Riemann Integration is mandatory for 'freshmen' (year 1), and Lebesgue Integration is an elective taken by 'sophmores' (year 2).&lt;/p&gt;&#xA;&lt;p&gt;Full details of the Lebesgue Integration course, including lecture notes and problem sets, can be found on the Department's public &lt;a href=&quot;https://courses-archive.maths.ox.ac.uk/node/44033&quot; rel=&quot;noreferrer&quot;&gt;course archive page&lt;/a&gt;. The &lt;a href=&quot;https://courses-archive.maths.ox.ac.uk/node/43872&quot; rel=&quot;noreferrer&quot;&gt;course on Riemann Integration&lt;/a&gt; can also be found there.&lt;/p&gt;&#xA;&lt;p&gt;The elective on Lebesgue Integration is a prerequisite for the junior level courses Functional Analysis I &amp;amp; II, Distribution Theory, and Continuous Martingales &amp;amp; Stochastic Calculus.&lt;/p&gt;&#xA;" OwnerUserId="19685" LastActivityDate="2022-03-24T15:36:50.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25085" PostTypeId="1" AcceptedAnswerId="25090" CreationDate="2022-03-24T17:13:31.460" Score="10" ViewCount="965" Body="&lt;p&gt;What are examples of sci-fi books or short stories that have a mathematics theme? I'd like to have a pool of examples in mind that I could refer students to. The only example I've got in mind right now is &lt;a href=&quot;https://urbigenous.net/library/power.html&quot; rel=&quot;noreferrer&quot;&gt;The Feeling of Power&lt;/a&gt; by Asimov, which I sometimes show students when discussing calculator use.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2022-06-18T08:37:34.997" Title="What are examples of math-themed sci-fi appropriate for students?" Tags="|reference-request|student-motivation|curriculum|reference-books|big-list|" AnswerCount="10" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25086" PostTypeId="2" ParentId="25078" CreationDate="2022-03-24T17:23:14.977" Score="30" Body="&lt;p&gt;I think the existing answers understate how much a standard American math major does not see the Lebesgue integral.&#xA;I'm going to poke around at a variety of college websites to see how they cover this material. Of course, this is not systematic, but I think it is more representative than the other answers. Also, of course, in most cases I only read the school's webpages, so I may be missing something.&lt;/p&gt;&#xA;&lt;p&gt;I would like to emphasize that, while this list spends most of its effort on elite schools, &lt;b&gt;the majority of undergraduates attend community colleges&lt;/b&gt; and &lt;b&gt;the majority of undergraduates pursuing bachelors degrees attend schools with non-competitive admissions.&lt;/b&gt;&#xA;Also, this is an answer about math majors. &lt;b&gt;Most students taught by a math department are not math majors&lt;/b&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Schools where rigorous definitions of integral are not available&lt;/b&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The majority of college students are enrolled in community colleges. Here are the math courses at &lt;a href=&quot;https://www.wccnet.edu/learn/departments/mthd/class-sequence.php&quot; rel=&quot;noreferrer&quot;&gt;Washtenaw Community College&lt;/a&gt;, the nearest community college to me. It looks to me like none of the calculus courses involve rigorous proofs at all (though Linear Algebra does), and there is no analysis beyond Calculus III.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;b&gt;Schools where Lebesgue integration is not available&lt;/b&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Alright, you may say, but by &amp;quot;math major&amp;quot; I meant &amp;quot;math major at a four year institution pursing a Bachelor's degree.&amp;quot; The majority of those are at schools who accept the majority of their applicants. Here is &lt;a href=&quot;http://mathcs.albion.edu/req_major_math.php&quot; rel=&quot;noreferrer&quot;&gt;Albion College&lt;/a&gt;, chosen because it was the first college in Michigan listed &lt;a href=&quot;https://www.studypoint.com/admissions/category/competitiveness/somewhat-competitive-(40-80-)/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. Here, it looks like almost every route through the math requirements involves &lt;a href=&quot;http://mathcs.albion.edu/courses_math.php#math331&quot; rel=&quot;noreferrer&quot;&gt;Math 331: Real Analysis&lt;/a&gt; which is based on the Riemann integral. I see no course on Lebesgue integration or Measure Theory in their &lt;a href=&quot;http://mathcs.albion.edu/courses_math.php&quot; rel=&quot;noreferrer&quot;&gt;course listings&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;b&gt;Schools where Lebesgue integration is an unusual elective&lt;/b&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Some students are small liberal arts colleges (SLCs). Their math programs vary widely; I'll choose Williams as a college which has a reputation for an excellent math program. The highest mandatory analysis course is &lt;a href=&quot;https://web.williams.edu/Mathematics/lg5/350/index.html&quot; rel=&quot;noreferrer&quot;&gt;Math 350&lt;/a&gt;. Looking at the course webpage, I only see Riemann Integration, and the assignments only go up to Chapter 9 in the &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486477665&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;textbook&lt;/a&gt;, which is the construction of the Riemann integral (although the last chapter is the Lebesgue integral). Measure theory is an &lt;a href=&quot;https://math.williams.edu/courses/&quot; rel=&quot;noreferrer&quot;&gt;elective which is not taught every year&lt;/a&gt;. So, even at an elite SLC like Williams, Lebesgue integration isn't on the natural track, and it looks like students may have to put in effort to seek it out.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Many students are at state universities. A mid-ranked state university is Arizona State. It looks like the most advanced mandatory analysis course at Arizona State is &lt;a href=&quot;https://webapp4.asu.edu/catalog/courselist?s=MAT&amp;amp;n=371&amp;amp;t=2227&amp;amp;hon=F&amp;amp;gg=F&quot; rel=&quot;noreferrer&quot;&gt;Math 371, &amp;quot;Advanced Calculus I&amp;quot;&lt;/a&gt;, which is based on the Riemann Integral. However, one can choose to study Lebesgue integration in &lt;a href=&quot;https://webapp4.asu.edu/catalog/courselist?s=MAT&amp;amp;n=473&amp;amp;t=2227&amp;amp;hon=F&amp;amp;gg=F&quot; rel=&quot;noreferrer&quot;&gt;Math 473, &amp;quot;Intermediate Real Analysis II&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An example of a research-heavy state university (R1) would be &lt;a href=&quot;https://reg.msu.edu/AcademicPrograms/ProgramDetail.aspx?Program=3744&quot; rel=&quot;noreferrer&quot;&gt;Michigan State&lt;/a&gt;. At Michigan State, the standard analysis courses for majors are &lt;a href=&quot;https://users.math.msu.edu/users/shapiro/Teaching/classes/320/320.html&quot; rel=&quot;noreferrer&quot;&gt;Math 320&lt;/a&gt; and &lt;a href=&quot;https://users.math.msu.edu/users/seal/teaching/sp14/#description&quot; rel=&quot;noreferrer&quot;&gt;Math 421&lt;/a&gt;. I couldn't find a detailed topics list, but their text books are Abbot, &amp;quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0387950605&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Understanding Analysis&lt;/a&gt;&amp;quot; and Wade, &amp;quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0132296381&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Analysis&lt;/a&gt;&amp;quot;, both of which use the Riemann Integral. There are also honors versions of these courses: Math 327H and Math 429H. Math 327H uses the Riemann Integral; I couldn't find a statement for Math 429H. However, at Michigan State, some students will see the Lebesgue integral, if they take the graduate course &lt;a href=&quot;https://math.msu.edu/undergraduate/advanced_track/course_descriptions.aspx&quot; rel=&quot;noreferrer&quot;&gt;Math 828&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;b&gt;Students where I would guess most math majors don't see Lebesgue integrals, but a sizeable minority do&lt;/b&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;My own university, the University of Michigan, is what is called a &amp;quot;flagship university&amp;quot;, a state university which attempts to have a world class research department. The highest level analysis course which is required of math majors is &lt;a href=&quot;https://lsa.umich.edu/math/undergraduates/major-and-minor-programs/pure-mathematics.html&quot; rel=&quot;noreferrer&quot;&gt;Math 351: &amp;quot;Principles of Analysis&amp;quot;&lt;/a&gt;, which uses the Riemann integral. Students bound for graduate school often take the honors version, Math 395/396. Here the Lebesgue integral may or may not be covered by preference of the instructor. When I teach it, I use Riemann integrals; I believe that Prof. Barrett does teach the Lebesgue integral as a side topic but bases most of the course on the Riemann integrals. Students who want to be sure to see the Lebesgue integral should take Math 597, which is targeted primarily at first year grad students but has some undergraduates in it. So students are not required to learn the Lebesgue integral at this level, and may or may not see it naturally in undergraduate courses, but the option definitely exists.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Back when I was in school, &lt;a href=&quot;https://www.math.harvard.edu/media/brochure-2020.pdf&quot; rel=&quot;noreferrer&quot;&gt;Harvard University&lt;/a&gt; was top ranked for math.  Harvard math concentrators must take a course in analysis, meaning a course &lt;a href=&quot;https://www.math.harvard.edu/undergraduate/?courseid=64/&quot; rel=&quot;noreferrer&quot;&gt;numbered between 112 and 118&lt;/a&gt;, and the recommendation of the department is to take either &lt;a href=&quot;https://people.math.harvard.edu/%7Ewboney/113/&quot; rel=&quot;noreferrer&quot;&gt;Math 113 &amp;quot;Complex Function Theory&amp;quot;&lt;/a&gt; or &lt;a href=&quot;https://people.math.harvard.edu/%7Ectm/home/text/class/harvard/114/14/html/index.html&quot; rel=&quot;noreferrer&quot;&gt;Math 114 &amp;quot;Analysis of Function Spaces, Measure and Integration&amp;quot;&lt;/a&gt;. Math 114 teaches Lebesgue integration and 113 doesn't. When I was at Harvard, 113 was much more popular; I don't know if this is still true. So one of the two analysis courses that math concentrators are routed to teaches Lebesgue integration and the other doesn't. On a personal note, I made the very ill-advised decision to take Math 213, the graduate version of Complex Analysis, and was in way over my head. The professor frequently referred to sets of measure zero, so I guess we were using the Lebesgue integral, although I don't recall a specific statement. I have never taken a course which constructed the Lebesgue integral.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;b&gt;Schools where it might be true that the majority of pure math majors see a Lebesgue integral&lt;/b&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;MIT is considered one of the most challenging math departments in the nation. Here pure math majors must take &lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/18-100b-analysis-i-fall-2010/&quot; rel=&quot;noreferrer&quot;&gt;18.100&lt;/a&gt;, which uses the Riemann integral, and can then choose between &lt;a href=&quot;https://math.mit.edu/classes/18.101/fa18/&quot; rel=&quot;noreferrer&quot;&gt;18.101&lt;/a&gt; (analysis on manifolds), &lt;a href=&quot;https://math.mit.edu/%7Erbm/18-102-S17/&quot; rel=&quot;noreferrer&quot;&gt;18.102&lt;/a&gt; (functional analysis) and &lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/18-103-fourier-analysis-fall-2013/syllabus/&quot; rel=&quot;noreferrer&quot;&gt;18.103&lt;/a&gt;  (Fourier analysis). Looking at the webpages, it looks like 18.101 uses Riemann integrals and the others use Lebesgue integrals. There is also an applied math major, which looks like it would not see Lebesgue interals.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I looked into U Chicago, a school which I think of as having very high standards. Since I don't know this school personally, it is hard to judge what the typical math major does. At Chicago, the highest mandatory analysis course is the &lt;a href=&quot;https://www.coursicle.com/uchicago/courses/MATH/20500/&quot; rel=&quot;noreferrer&quot;&gt;20300/20400/20500 sequence&lt;/a&gt;. The textbook, &amp;quot;A textbook for advanced calculus&amp;quot; by Boller and Sally, uses Riemann integrals. (As a sidenote, I had to go to Course Hero to get the syllabus or to see the textbook. Come on, people, make a course webpage!) However, there are two other undergraduate sequences, &lt;a href=&quot;https://www.coursicle.com/uchicago/courses/MATH/20700/&quot; rel=&quot;noreferrer&quot;&gt;Math 20700/20800/20900&lt;/a&gt; and &lt;a href=&quot;https://www.coursicle.com/uchicago/courses/MATH/31200/&quot; rel=&quot;noreferrer&quot;&gt;Math 31200/31300/31400&lt;/a&gt; which teach the Lebesgue integral. I do not know how popular they are.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;b&gt;Schools where all math majors learn about the Lebesgue integral&lt;/b&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;At Caltech, it looks to me like &lt;a href=&quot;http://www.math.caltech.edu/%7E2016-17/2term/ma108b/&quot; rel=&quot;noreferrer&quot;&gt;Math 108b&lt;/a&gt; is a &lt;a href=&quot;https://pma.caltech.edu/research-and-academics/mathematics/math-undergraduate-studies/math-degree-requirements&quot; rel=&quot;noreferrer&quot;&gt;required course for math majors&lt;/a&gt; (or can only be replaced by more advanced courses), and it uses the Lebesgue integral. I don't know Caltech personally and their requirements page is very terse; corrections are welcome.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2022-03-24T19:41:24.833" LastActivityDate="2022-03-24T19:41:24.833" CommentCount="20" ContentLicense="CC BY-SA 4.0" />
  <row Id="25087" PostTypeId="2" ParentId="25085" CreationDate="2022-03-24T19:25:12.063" Score="1" Body="&lt;p&gt;I love a series of 4 books by Rosemary Kirstein. The first is titled The Steerswoman.&lt;/p&gt;&#xA;&lt;p&gt;Steerswomen journey around the known lands, drawing maps and documenting what they see in their logbooks. Rowan, a steerswoman, finds an odd gem and starts asking people questions about it. Everyone must answer a steerswoman's questions (honestly) or be banned from every asking their own questions of the steerswomen, who are very knowledgeable. People usually enjoy helping the steerswomen by offering information.&lt;/p&gt;&#xA;&lt;p&gt;The first book starts out seeming like fantasy, because there are wizards who do magic, and there are demons and goblins. But as the story progresses, Rowan keeps wondering what it means for something to be magic. The words science, gravity, and explosions (probably among others that I haven't noticed) are never mentioned, even though all are important themes in the story.&lt;/p&gt;&#xA;&lt;p&gt;The thinking Rowan and others do about how gravity works, why and how things fall the way they do, is an important part of the storyline. I often quote from this book in my calculus I courses.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-03-24T19:25:12.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25088" PostTypeId="2" ParentId="25061" CreationDate="2022-03-24T22:43:11.620" Score="1" Body="&lt;p&gt;The &lt;a href=&quot;https://mathlesstraveled.com/2010/11/19/the-mathematics-of-human-knots/&quot; rel=&quot;nofollow noreferrer&quot;&gt;human knot game&lt;/a&gt; is &lt;a href=&quot;https://math.stackexchange.com/questions/3176857/human-knot-solvability-probability&quot;&gt;certainly mathematical&lt;/a&gt; but also fun.  And definitely suitable for a parking lot.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2022-03-24T22:43:11.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25089" PostTypeId="2" ParentId="25085" CreationDate="2022-03-25T01:13:45.517" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Flatland&quot; rel=&quot;nofollow noreferrer&quot;&gt;Flatland&lt;/a&gt; is a classic and very short/readable.&lt;/p&gt;&#xA;" OwnerUserId="19683" LastEditorUserId="5220" LastEditDate="2022-03-26T16:31:49.517" LastActivityDate="2022-03-26T16:31:49.517" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25090" PostTypeId="2" ParentId="25085" CreationDate="2022-03-25T07:22:04.610" Score="9" Body="&lt;p&gt;On the off-chance you don't know about it, I recommend looking through entries at Alex Kasman's website &lt;a href=&quot;https://kasmana.people.cofc.edu/MATHFICT/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mathematical Fiction&lt;/em&gt;&lt;/a&gt;. This has been around quite a while. I don't know when I first learned about it, but definitely by the time of his &lt;a href=&quot;https://groups.google.com/g/sci.math/c/mFPAYeX3Ay4/m/SUaS7QfQVuYJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;31 January 2000 announcement in sci.math&lt;/a&gt;, where &lt;a href=&quot;https://groups.google.com/g/sci.math/c/mFPAYeX3Ay4/m/Zq8cYS-fjlMJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;I offered a few suggestions&lt;/a&gt; about a week later, some of which might be of use to you (may need to try the &lt;a href=&quot;https://archive.org/web/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WayBack Machine&lt;/a&gt; for some of the URLs). For what it's worth, in June 2001 &lt;a href=&quot;https://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf206&quot; rel=&quot;nofollow noreferrer&quot;&gt;I wrote one of the descriptions&lt;/a&gt; at his request when he saw my comments about Steven Baxter's 1997 short story &lt;em&gt;The Logic Pool&lt;/em&gt; in this &lt;a href=&quot;https://groups.google.com/g/sci.math/c/3rOWa0EdBcA/m/f-jiCPwFwzYJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;31 May 2001 sci.math post&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A couple of useful anthologies I have are&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0523004222&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Other Dimensions&lt;/strong&gt;&lt;/a&gt; edited by Robert Silverberg (1973, 1974)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fantasia_Mathematica&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Fantasia Mathematica&lt;/strong&gt;&lt;/a&gt; edited by Clifton Fadiman (1958, 1997)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Mathematical_Magpie&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Mathematical Magpie&lt;/strong&gt;&lt;/a&gt; edited by Clifton Fadiman (1962, 1997).&lt;/p&gt;&#xA;&lt;p&gt;I got a copy of the 1974 paperback edition of Silverberg's book shortly after it was published (this being when larger U.S. cities, such as Charlotte which was within an hour's drive of where I then lived, still had diners that also sold a large number of paperback books), and it played an important role in my early interest in mathematics, but I only got around to getting a copy of Fadiman's book (1958 original edition) a few years ago from amazon.com. Each of these books, and Fadiman's later book (which I apparently don't have a copy of, but it definitely merits including here), contains several stories that I think would interest some students.&lt;/p&gt;&#xA;&lt;p&gt;A couple of authors who are especially mathematically competent are &lt;a href=&quot;https://en.wikipedia.org/wiki/Rudy_Rucker&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rudy Rucker&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Greg_Egan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Greg Egan&lt;/a&gt;. Rucker's fiction is easier reading and, to me at least, his fiction seems to have the strange quality of conveying &lt;strong&gt;a lot more&lt;/strong&gt; underlying mathematical significance if I know something nontrivial about the mathematics involved. Egan's fiction is rather dense reading, but it also feels more realistic to me than most anything else I've read about how different &amp;quot;humanity&amp;quot; will be a thousand years from now, being more in accord with something I read several decades ago (don't remember by who) which said that humans eventually exploring our galaxy is likely to be as meaningful as saying bacteria landed on the moon in 1969.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2022-06-17T18:46:55.230" LastActivityDate="2022-06-17T18:46:55.230" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25091" PostTypeId="2" ParentId="25085" CreationDate="2022-03-25T10:08:22.437" Score="3" Body="&lt;p&gt;I strongly recommend &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Curious_Incident_of_the_Dog_in_the_Night-Time&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Curious Incident of the Dog in the Night-Time&lt;/a&gt;&amp;quot; by Mark Haddon.  (I wouldn't consider the novel science fiction but it, in my opinion, has quite some mathematics in it.)  From my &lt;a href=&quot;https://joelreyesnoche.wordpress.com/2013/10/22/the-curious-incident-of-the-dog-in-the-night-time/&quot; rel=&quot;nofollow noreferrer&quot;&gt;blog post&lt;/a&gt; about it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although the story is told from the viewpoint of a 15-year-old boy, I found the observations on language (literal and figurative), writing (how to write detective fiction), the nature of the mind (how a normal person’s way of thinking differs from that of an autistic person, or of an animal, or of a computer), and mathematics very deep. I particularly like how a wide variety of mathematics is presented (probability, chaos, games, tessellations).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The writing is simple and I think the story can be understood by young teen-agers or maybe even younger children.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2022-03-25T10:08:22.437" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25092" PostTypeId="2" ParentId="25085" CreationDate="2022-03-25T13:56:52.000" Score="3" Body="&lt;p&gt;I would recommend &lt;a href=&quot;https://en.wikipedia.org/wiki/Inverted_World&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inverted World&lt;/a&gt; by Christopher Priest.&lt;/p&gt;&#xA;&lt;p&gt;The maths content isn't clear till the denouement but it's spectacular and unique (in my experience) when it comes.&lt;/p&gt;&#xA;&lt;p&gt;I don't want to spoil it for anyone by putting more detail here, but I just checked out the Kasman website from Dave L Renfro's answer (looks like an excellent resource) and that has &lt;a href=&quot;https://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf49b&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page on it&lt;/a&gt; if you want to know more (they add &amp;quot;The&amp;quot; to the title btw, looks like it's been published in both forms).&lt;/p&gt;&#xA;" OwnerUserId="19691" LastEditorUserId="19691" LastEditDate="2022-03-25T14:05:13.147" LastActivityDate="2022-03-25T14:05:13.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25093" PostTypeId="1" AcceptedAnswerId="25096" CreationDate="2022-03-25T15:32:55.667" Score="0" ViewCount="262" Body="&lt;p&gt;The difference between a disc (disk) and a circle is crystal clear to me:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NaMxX.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NaMxX.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, in many children's books, a disc is usually called a circle:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/I9wI2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/I9wI2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Why do many children's book confuse discs with circles? Should we teach children the difference between a disc and a circle?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2022-03-26T09:06:52.533" Title="Why do so many children's book confuse discs with circles?" Tags="|geometry|terminology|preschool-education|" AnswerCount="1" CommentCount="9" ClosedDate="2022-03-30T23:05:38.300" ContentLicense="CC BY-SA 4.0" />
  <row Id="25094" PostTypeId="1" CreationDate="2022-03-25T20:16:51.497" Score="7" ViewCount="495" Body="&lt;p&gt;What are some &amp;quot;interesting&amp;quot; and creative problems or exercises on specifically 2-dimensional vector geometry that a high school student might find compelling to solve?&lt;/p&gt;&#xA;&lt;p&gt;The class' current knowledge consists of basic vector arithmetic, parametric vector equations of lines, the scalar dot product and pretty much everything that can be done with it algebraically and geometrically. No matrices are known at this stage.&lt;/p&gt;&#xA;&lt;p&gt;I can think of many &amp;quot;mathematically glamorous&amp;quot; problems in 3D vector geometry, but not in 2D. It seems to me that any 2D vector problem (at high school level) instantly reduces to either some standard application of the scalar product or instantly translates into a linear system of equations, and that there is little room to pose something with more depth. Any ideas would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="5345" LastEditorUserId="5345" LastEditDate="2022-03-25T20:24:52.580" LastActivityDate="2022-04-02T08:05:13.367" Title="Creative problems in 2D vector geometry" Tags="|secondary-education|geometry|linear-algebra|vectors|analytic-geometry|" AnswerCount="7" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25095" PostTypeId="2" ParentId="25094" CreationDate="2022-03-25T20:27:26.373" Score="5" Body="&lt;p&gt;A modestly non-trivial investigation that yields well to vector algebra is the Euler line.  That is, showing that the centroid of any triangle lies on the line segment connecting its circumcenter to its orthocenter and divides that line segment in the ratio 2:1.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2022-03-25T20:27:26.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25096" PostTypeId="2" ParentId="25093" CreationDate="2022-03-26T09:06:52.533" Score="10" Body="&lt;p&gt;Mathematics, as any other field of inquiry, has specific terminology. The rest of the world is not obliged to use the same technical terminology.&lt;/p&gt;&#xA;&lt;p&gt;Note that this also happens within mathematics; not every subfield is interested in the same issues and sees the same distinctions as important.&lt;/p&gt;&#xA;&lt;p&gt;In particular, for young children, the relevant thing to learn is the names of basic figures, together with names of colours, and so on. The minute difference between a circle and a disk is not relevant for a number of years. (Why minute: you have a one-to-one correspondence, so identifying them with each other is quite reasonable for many purposes, even for a professional mathematician. It is usually clear which one we are talking about.)&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-03-26T09:06:52.533" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25097" PostTypeId="2" ParentId="25085" CreationDate="2022-03-26T15:00:59.187" Score="3" Body="&lt;p&gt;And He Built a Crooked House is a light fanciful short story based on an unfolded tesseract. Not a great work of literature, but semi-famous.  Has a Wiki article.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/%22%E2%80%94And_He_Built_a_Crooked_House%E2%80%94%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/%22%E2%80%94And_He_Built_a_Crooked_House%E2%80%94%22&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is in the anthology, Dave mentioned, but has its own notability.  Also, I figure this is like the shape thread where we can show new ideas, since its a collection. (I would edit it in, but I'm a second class citizen, non-teacher, with no account...I know my place.)&lt;/p&gt;&#xA;&lt;p&gt;Also, to slightly segue, but related, you can also consider to refer kids to math in art.  Escher and Penrose stairs and Renaissance perspective and Dali and all that...Wiki articles on math and art, or fourth dimension in art, are starters (but not comprehensive).  Maybe helpful for the arty &amp;quot;bad at math&amp;quot; crew.  Or the engineer types that like hand drafting.  Or even just buy some prints and put them on the classroom walls.  Little bit of touchy feely visual motivation.&lt;/p&gt;&#xA;" OwnerUserId="19697" LastActivityDate="2022-03-26T15:00:59.187" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25098" PostTypeId="2" ParentId="25064" CreationDate="2022-03-26T23:48:26.783" Score="1" Body="&lt;p&gt;I'm not aware of any detailed resources that contain (reliable and current) information in that detail and depth.&#xA;One resource might be ICMI's &lt;a href=&quot;https://www.mathunion.org/icmi/activities/icmi-database-project&quot; rel=&quot;nofollow noreferrer&quot;&gt;Database Project&lt;/a&gt;. It is a list on math curricula and information on math education within countries.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2022-03-26T23:48:26.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25100" PostTypeId="2" ParentId="25094" CreationDate="2022-03-27T10:59:36.387" Score="3" Body="&lt;p&gt;Distance from a line segment to a point? With the right graphing system (maybe Desmos, but it's a bit awkward there) you can use this to draw thick line segments.&lt;/p&gt;&#xA;&lt;p&gt;There's also barycentric coordinates in a triangle&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-03-27T10:59:36.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25102" PostTypeId="2" ParentId="25050" CreationDate="2022-03-27T19:34:15.493" Score="3" Body="&lt;p&gt;&lt;em&gt;I did this in many lower-level courses over a period of many years. I've also written extensively about what I did and why, but most of this is in Math Forum discussion groups&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\ldots$&lt;/span&gt; (beginning of the first of several comments I made here about 2 weeks ago)&lt;/p&gt;&#xA;&lt;p&gt;I've finally managed to locate the text of two of the (definitely more than two) posts I mentioned in comments here almost 2 weeks ago. I don't know when they were made (very roughly, probably between 2007 and 2013), nor in what order they were made (the order I've used is simply what seems reasonable to me for here), or even whether they were made within a few weeks/months apart or several years apart. I also don't know where I originally had paragraph separations, but to avoid having huge walls of text, I've put in some paragraph separations here. Finally, nothing has been deleted (except the first sentence or two of the 1st post below) and nothing has been changed, but I have added a few additional words between some square brackets &lt;span class=&quot;math-container&quot;&gt;$[\ldots]$&lt;/span&gt; in the first of the posts below.&lt;/p&gt;&#xA;&lt;h1&gt;1st post&lt;/h1&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[\ldots]$&lt;/span&gt; However, I thought I'd jump in now about a policy I used to have (I no longer teach [since early May 2005]). This policy evolved throughout the 1980s into final form by the mid 1990s. It worked quite well for me, at least when I was able to use it (almost always, the only exception I can think of right now were some courses in my last two years of teaching [2003-2004 and 2004-2005]), but like many teaching methods, what works well for some people is not going to work well for some other people.&lt;/p&gt;&#xA;&lt;p&gt;Short quizzes -- I gave lots of these, each on a 10-point scale. The sum of a student's top 10 quiz grades becomes one of the student's 4 (or 5) major test grades. I usually guaranteed a minimum of 12 quizzes for the semester, but I often tried to work in 1 or 2 more [and for probably about 30% of the courses, I gave 15 or more quizzes]. The nice thing about this policy is that students WANT you to give more quizzes, especially in the final weeks of the semester when things are usually the most difficult. Sometimes near the end of the semester I'd give an &amp;quot;exam-review quiz&amp;quot;, which would be a quiz similar to one already given, usually on a topic they had trouble with and a topic that I think is very important for the course. By the way, depending on the difficulty of the material, the classroom climate, and other things, short quizzes were sometimes &amp;quot;you can use your homework but not your book&amp;quot; quizzes, &amp;quot;you can use your homework and class notes but not your book&amp;quot; quizzes, &amp;quot;you can use you homework, class notes, and book&amp;quot; quizzes, &amp;quot;you can work with a partner&amp;quot; quizzes (who I choose, who they choose, or names were paired randomly), the last type combined with one of the earlier types, and other possibilities I experimented with. My best successes for these alternate methods occurred in calculus 2 classes, and next with calculus 1 classes. In higher level classes it didn't work as well, and in lower level classes it didn't work as well, for whatever that's worth.&lt;/p&gt;&#xA;&lt;p&gt;Major tests -- 50 to 110 minute tests (depending on whether a MWF class or a TuTh class). As much as possible, the test would be made up of problems very similar to problems already put on quizzes, but in practice I was typically only able to get about 50% of the test problems in this way.&lt;/p&gt;&#xA;&lt;p&gt;Final Exam -- A major deal. However, those students who had done very well in the class up to that point (mid to high A students, maybe low A students, depending on how the overall grades in the class were -- usually this would be about 3 to 7 students in a class of 30-40 students) got to take a scaled back final exam that consisted only of &amp;quot;recent material&amp;quot; (material covered since the last major test of the semester), about twice the length of a short quiz, and if they did satisfactory on this &amp;quot;recent material exam&amp;quot;, they were guaranteed an 'A'. I think I didn't give such a student an 'A' only once in all my teaching. (The student only half-way knew the stuff, and I said they could take the full final exam and probably -- unless they really messed up on the stuff they had aced thus far -- earn an 'A', or take an 'A-'. He/she took the A-, being afraid of messing up on the final since they didn't study for it, plus I think this person had several other final exams he/she was really worried about and needed to leave to study for them.)&lt;/p&gt;&#xA;&lt;p&gt;The Final Exam also replaced the student's lowest test score (which could be their top 10 quizzes, although this may have never actually happened -- typically their &amp;quot;quiz test grade&amp;quot; was the highest test grade), unless the Final Exam score (as a percent) was lower than each of their major test scores (in which case the final exam still counted a lot, usually about 30% of the final grade, but the damage was restricted to just that amount). By the way, probably about 75% of the students did worse on the Final Exam than their test average up to that point, with about half of those 75% dropping a third of a letter grade or more (using +/- letter grades) and about half of those 75% winding up having their Final Exam grade not differing enough to change the pre-exam average (when rounded to a plus/minus letter grade).&lt;/p&gt;&#xA;&lt;p&gt;Finally, I had a &amp;quot;last chance&amp;quot; option for those true late bloomers, who in reality are generally quite rare unless you're teaching a lot of non-traditional adult students. Any student getting at least a 'C' on their Final Exam was guaranteed to pass the course, regardless of what their final course average was, and any student getting an 'A' on their Final Exam was guaranteed at least a 'C' in the course (at least, since typically anyone making an 'A' on their final, even when doing D work prior to this, wound up with a course average of higher than 'C' anyway). My guess is that I invoked the &amp;quot;last chance&amp;quot; option about once every 2 to 3 years (perhaps about once out of every 400 to 600 students). I also made it very clear at the beginning of the semester how rarely this happened, but I did say it could happen. Actually, what happened far more often (maybe as much as one student out of 50 or 60 students) was that someone was failing and got a 'C' on the Final Exam, or someone had a 'D' and got an 'A' on the Final Exam (the former was much more likely), and the weight of the Final Exam on their final course average plus the Final Exam replacing their lowest test score was more than enough to push them to a passing grade or a 'C' grade, without my having to invoke the &amp;quot;last chance option&amp;quot;.&lt;/p&gt;&#xA;&lt;h1&gt;2nd post&lt;/h1&gt;&#xA;&lt;p&gt;For what it's worth, I've always found final/mid-term exams fairly easy to write. My policy was to never include anything on a final exam whose &amp;quot;type&amp;quot; wasn't represented by a question on a prior test or short quiz, except for whatever new material we covered between the last test/quiz and the exam. So what I did was to get out copies of all the tests and quizzes (already done, since we quickly looked over these in class the day or two spent reviewing for the exam, with me saying &amp;quot;definitely on the exam&amp;quot;, &amp;quot;might be on the exam&amp;quot;, and &amp;quot;not on the exam&amp;quot; for each of the question types), along with my notes on what I told the students about the exam, and then I'd start making my question selections for the exam.&lt;/p&gt;&#xA;&lt;p&gt;Nowadays we don't have to hand-write or old-style typewriter-type exams, so it's just a matter of copying-and-pasting the selected questions onto a &amp;quot;final exam test document&amp;quot;, changing numbers or equations or graphs or whatever. Also, I often found that ideas for tweaking already asked questions came to me while I was reviewing in class, during the extra &amp;quot;Friday afternoon review sessions&amp;quot; (probably more of a college thing than a high school thing), working with struggling students in my office, etc. Thus, when it came time to actually write the exam, I usually already knew which question types I wanted to ask and I also had many specifics for them already jotted down in my notes.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, GRADING the exams was always a major effort -- typically 6 to 10 hours per class of 30-40 students. That for me was where the real work was during exam periods, grading the exams, not writing the exams.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, my experiences in many different settings (pre-algebra to graduate level, lowest performing HS in one state to a nationally known math/science academy in another state, small and medium size private colleges, medium and large size public universities) has been that students almost always perform less well on final exams than on previous tests given throughout the semester/year. There are several factors behind this, such as students being burned out by the time exams are given (everyone's last regular test, last term paper, etc. just occurred, although this may not be an issue in your [= original OP I was replying to] case), all the various exams the students have to take are given within a few days, and the exam is over several months of material instead of 3 to 4 weeks of material. Thus, even being as explicit as I was about what would be on the exam did not inflate student grades. In fact, even with this I found that exam grades, on average, were lower than major test and short quiz grades.&lt;/p&gt;&#xA;&lt;p&gt;The reason I've gone into such detail is I'm wondering whether others do something like this. I'm guessing you (Kerri) don't, otherwise you wouldn't be so concerned with writing the exams. Also, it seems to me that by using someone else's exam, especially for a non-AP calculus course that is quite likely different in many small ways from teacher to teacher and school to school, one runs the risk of over-emphasis and under-emphasis on the topics actually taught.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2022-03-27T19:39:55.840" LastActivityDate="2022-03-27T19:39:55.840" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25103" PostTypeId="2" ParentId="25094" CreationDate="2022-03-28T00:41:30.157" Score="5" Body="&lt;p&gt;I just had some students implement reflection of a ray in a&#xA;mirrored line segment,&#xA;which they then used to bounce a light ray around inside a polygon.&#xA;Here's a crude snapshot:&lt;/p&gt;&#xA;&lt;p&gt;     &lt;img src=&quot;https://i.stack.imgur.com/KWCr1.png&quot; width=&quot;200&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;This project also involved programming, but even hand calculations inside&#xA;an equilateral triangle are of interest.&#xA;As a bonus, this touches on an open problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Does every triangle have a periodic billiard path?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See this &lt;a href=&quot;https://www.youtube.com/watch?v=GWzhNYkl6fA&amp;amp;ab_channel=DianaDavis&quot; rel=&quot;noreferrer&quot;&gt;Diana Davis YouTube video&lt;/a&gt; for what's known on this&#xA;unsolved problem.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-03-28T00:41:30.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25105" PostTypeId="2" ParentId="25094" CreationDate="2022-03-30T01:24:20.670" Score="3" Body="&lt;p&gt;The proof that the angle between two bodies which collide elastically is &lt;span class=&quot;math-container&quot;&gt;$90^o$&lt;/span&gt; is an interesting two dimensional vector algebra problem.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2022-03-30T01:24:20.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25106" PostTypeId="1" CreationDate="2022-03-30T03:13:55.887" Score="5" ViewCount="123" Body="&lt;p&gt;Having put my audience through developing a probability space &lt;span class=&quot;math-container&quot;&gt;$(\Omega, \mathcal{F}, P)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is the sample space of a random trial, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}$&lt;/span&gt; is a &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-field on it, and P is a measure on that &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-field, how to I motivate shifting the whole construct to the real line?&lt;/p&gt;&#xA;&lt;p&gt;And while I'm at it, how do I justify it informally, i.e., without heavy math? I know (I think, but I'm a bit fuzzy) that a random variable &lt;span class=&quot;math-container&quot;&gt;$X:\Omega \rightarrow \mathbb{R}$&lt;/span&gt; induces a Borel field but even if I'm right about that how does that convince the listener that the whole probability space is ported to the real line?&lt;/p&gt;&#xA;" OwnerUserId="19709" LastActivityDate="2022-03-30T03:13:55.887" Title="After building the probability space, how to motivate for random variables?" Tags="|statistics|" AnswerCount="0" CommentCount="9" ClosedDate="2022-04-11T21:11:59.263" ContentLicense="CC BY-SA 4.0" />
  <row Id="25107" PostTypeId="2" ParentId="25094" CreationDate="2022-03-31T04:23:53.863" Score="0" Body="&lt;p&gt;From a previous answer on SE Math, the following is offered -&lt;/p&gt;&#xA;&lt;p&gt;Although the math behind the conformal transformation may seem obscure, the mathematical process is rather direct. Necessary elements in the transformation are a transform circle ζ of radius &lt;em&gt;a&lt;/em&gt; and nondescript vector Z extending from the origin to various positions on the perimeter of ζ.  The length of vector Z from the origin to the circumference of ζ changes through the continuum of increasing positive angles &lt;em&gt;θ&lt;/em&gt;  being swept out by Z. For each angular position taken by Z, vector &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt;&lt;/em&gt;/Z is poised in the interior of ζ at corresponding angle &lt;em&gt;-θ&lt;/em&gt;.  The airfoil contour in z(x,y) space is defined by the vector sum Z + &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt;&lt;/em&gt;/Z.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ftrkj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ftrkj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19712" LastActivityDate="2022-03-31T04:23:53.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25108" PostTypeId="2" ParentId="25094" CreationDate="2022-03-31T15:55:11.607" Score="1" Body="&lt;p&gt;Here is a problem that I remember running into on an Olympiad which is not too bad with vectors and very hard without.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$ABCDE$&lt;/span&gt; be a convex pentagon. Let &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt;, respectively, be the midpoints of the line segments &lt;span class=&quot;math-container&quot;&gt;$\overline{AB}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\overline{BC}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\overline{CD}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\overline{DE}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\overline{EA}$&lt;/span&gt; respectively. Suppose that there is a point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; in the intersection &lt;span class=&quot;math-container&quot;&gt;$\overline{DV} \cap \overline{EW} \cap \overline{AX} \cap \overline{BY}$&lt;/span&gt;. Show that the point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; also lies on the line segment &lt;span class=&quot;math-container&quot;&gt;$\overline{CZ}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8oFBs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8oFBs.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Solution:&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; Translate &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; to be at the origin. The condition that &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; lies on &lt;span class=&quot;math-container&quot;&gt;$\overline{DV}$&lt;/span&gt; says that &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$V = \tfrac{A+B}{2}$&lt;/span&gt; are parallel vectors, so &lt;span class=&quot;math-container&quot;&gt;$\det(D, \tfrac{A+B}{2})=0$&lt;/span&gt;. We deduce that &lt;span class=&quot;math-container&quot;&gt;$\det(D,A)+ \det(D,B)=0$&lt;/span&gt; or, in other words, &lt;span class=&quot;math-container&quot;&gt;$\det(D,A) = \det(B,D)$&lt;/span&gt;. Keeping going in this manner, &lt;span class=&quot;math-container&quot;&gt;$\det(A,C) = \det(D,A) = \det(B,D) = \det(E,B) = \det(A,E)$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$\det(C, \tfrac{A+E}{2})=0$&lt;/span&gt;, showing that &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tfrac{A+E}{2}= Z$&lt;/span&gt; also lie on a line through the origin.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2022-04-01T16:44:20.277" LastActivityDate="2022-04-01T16:44:20.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25109" PostTypeId="1" CreationDate="2022-04-01T20:59:45.290" Score="9" ViewCount="339" Body="&lt;p&gt;This question is addressed to those familiar with mathematics teaching in French-speaking Switzerland.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://www.musee-ecoles.ch/de/livres/l-016-mm-arithmetique-au-degre-moyen-1962-mamin&quot; rel=&quot;noreferrer&quot;&gt;main textbook&lt;/a&gt; in use in grades 3 to 5 in the canton of Vaud from 1957 to about 1972 appears to use the German layout for long division rather than the typical French one.&lt;/p&gt;&#xA;&lt;p&gt;I'm curious whether this was also the practice in the other French-speaking cantons and whether usage has changed since then (presumably to agree with the practice in France).&lt;/p&gt;&#xA;&lt;p&gt;Are there other interesting instances of Alemannic influence in math teaching? I believe I've also seen things like &amp;quot;55'000&amp;quot; for 55,000 in French-Swiss sources.&lt;/p&gt;&#xA;" OwnerUserId="19714" LastEditorUserId="2083" LastEditDate="2022-04-02T11:13:47.253" LastActivityDate="2022-04-02T11:13:47.253" Title="Long division layout in French-speaking Switzerland" Tags="|division|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25110" PostTypeId="2" ParentId="25094" CreationDate="2022-04-02T08:05:13.367" Score="1" Body="&lt;p&gt;It might be good to prove various trigonometric identities using vector operations.&lt;/p&gt;&#xA;&lt;p&gt;One could use vector methods to derive the laws of reflection and refraction from the Huygens construction.&lt;/p&gt;&#xA;&lt;p&gt;Along the lines of @JosephORourke's reflection problem, one could derive vector-forms of the laws of reflection and refraction (so that one can handle interfaces that are not necessarily along the standard xy-axes), then use them to image &amp;quot;objects&amp;quot; in an optical system (for example, a thick lens).&lt;/p&gt;&#xA;&lt;p&gt;Given two point charges of opposite sign and unequal magnitude, use vector methods to prove that the zero-equipotential is a circle [taking the potential at infinity to be zero]. (Apollonius circle.)&lt;/p&gt;&#xA;" OwnerUserId="10537" LastActivityDate="2022-04-02T08:05:13.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25111" PostTypeId="1" AcceptedAnswerId="25112" CreationDate="2022-04-02T13:20:41.570" Score="3" ViewCount="250" Body="&lt;p&gt;I have received request from a student, who is not in math major, asking me for advice on&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How to keep motivated when studying math (calculus, linear algebra, etc.)&lt;/li&gt;&#xA;&lt;li&gt;What does one need to do beyond doing homework in order to get an A?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;(I understand this is a subjective matter. And I do have my own opinions. But I want to ask this question, not in order to get a &lt;em&gt;correct&lt;/em&gt; answer, but to solicit some ideas different from my own.)&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2022-04-03T02:40:19.037" Title="What kind of general advice for studying math we can offer undergraduate studens who do not major in math?" Tags="|undergraduate-education|student-motivation|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25112" PostTypeId="2" ParentId="25111" CreationDate="2022-04-03T02:40:19.037" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Turn the homework into a bit of a game. (Each night, see how long it takes you and how many problems you get right.)  But to a certain sense, this needs to become like sports training or dieting...when you embrace the grind, when you do the grind, you come to enjoy or at least tolerate it.  But you can't flip a switch and love it at the beginning.  Have to work through the hard part at the beginning.  But if you stick with it, the motivation will become less of an issue.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;a.  Do all the homework diligently, not just the teacher assigned problems, but all the drill problems for each section of the book.&lt;/p&gt;&#xA;&lt;p&gt;b.  Preread the chapters, working each example as you go. Then do all the drill problems.  The lecture, just becomes a pleasant review, or a chance to ask for nuances.&lt;/p&gt;&#xA;" OwnerUserId="19716" LastActivityDate="2022-04-03T02:40:19.037" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25113" PostTypeId="1" AcceptedAnswerId="25114" CreationDate="2022-04-03T19:49:01.737" Score="11" ViewCount="4495" Body="&lt;p&gt;I can never figure out what should be a memorization concept and what should be in a reference table. For example, in calculus, you are expected to memorize all the derivatives and integrals but in real life, you can always just look them up.&lt;/p&gt;&#xA;&lt;p&gt;In some math classes, like stats and differential equations, you use statistical tables (and they also give you the formulas) and Laplace tables.&lt;/p&gt;&#xA;&lt;p&gt;Is there a list of things?&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="376" LastEditDate="2022-04-05T05:47:35.950" LastActivityDate="2022-04-07T06:07:00.470" Title="What should be memorized in math and what should be reference table?" Tags="|mathematical-pedagogy|" AnswerCount="6" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="25114" PostTypeId="2" ParentId="25113" CreationDate="2022-04-03T23:33:41.457" Score="35" Body="&lt;p&gt;The goal of memorization is to reduce cognitive load. If a student plans on using derivatives as part of a larger task, and doesn't have them memorized, they need to interrupt their thought process either with looking up the derivative or working it out by hand. Whether this is useful for a student depends on what sort of applications of the derivative they plan on working with in the future, and so will vary from student to student.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-04-03T23:33:41.457" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25115" PostTypeId="2" ParentId="25113" CreationDate="2022-04-04T11:52:00.447" Score="3" Body="&lt;p&gt;I completely agree with Tom's answer (the ratinonale).  It is much easier to consult a table for something you learned before than for something you never learned.&lt;/p&gt;&#xA;&lt;p&gt;But to answer the specific question.  For a typical calculus student they should learn and have facility with all the standard integrals, methods in the general calculus book, during the course year.  I.e. able to work problems and take tests, closed book and closed Mathematica.  Maybe years later, they have to look up the arc length formula (or its derivation), but it will be much easier than if they never learned it, at all.  I.e.  No, &amp;quot;we could just look it up&amp;quot; is not the right approach when first learning the material.&lt;/p&gt;&#xA;&lt;p&gt;(For the above, I'm assuming something like a general &amp;quot;101&amp;quot; class for STEM or undifferntiated students, using Thomas or the like.  American examples, but philosophy applies generally.)  There are a tiny amount of students taking crazy hard stuff at Harvard or CalTech...but that is not germane to the general answer.  A more substantial portion will take calculus for business (or the like).  To which the general philosophy still applies, but the amount of tricks and such will be smaller (no trig substitutions, e.g.)&lt;/p&gt;&#xA;&lt;p&gt;Note also that calculus class and the manipulation of hand solving equations drives great facility and familiarity with algebra and trig.  As a tangential benefit.&lt;/p&gt;&#xA;" OwnerUserId="19718" LastActivityDate="2022-04-04T11:52:00.447" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25116" PostTypeId="2" ParentId="25113" CreationDate="2022-04-04T15:12:15.293" Score="3" Body="&lt;p&gt;I'm not an educator, but I think there's a simpler answer than those given before: Memorize the things you'll need to use frequently, look up the ones that are less common.&lt;/p&gt;&#xA;&lt;p&gt;This is somewhat related to TomKern's answer, as frequent interruptions to look something up will cause the student to lose their train of thought in the larger problem.&lt;/p&gt;&#xA;&lt;p&gt;In practice, this is likely to happen naturally for many (most?) students. At the beginning they have nothing memorized, and they'll have to look up or work out everything. Some of these things will come up frequently, and they'll remember the result from previous lookups.&lt;/p&gt;&#xA;&lt;p&gt;I was never good at deliberately trying to memorize things. But when you use concepts in context frequently, they eventually become second nature.&lt;/p&gt;&#xA;" OwnerUserId="10551" LastActivityDate="2022-04-04T15:12:15.293" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="25117" PostTypeId="2" ParentId="25113" CreationDate="2022-04-04T15:35:06.783" Score="22" Body="&lt;p&gt;The reasons are manifold.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;One is cognitive load (@TomKern’s answer) and the distraction of looking things up sometimes causes the solution you’ve almost constructed to fall to pieces.  This is usually the first thing I tell the students because, to their way of thinking, it has an immediate practical import, and to motivate them to do things that help them learn is my goal.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another is that if you don’t memorize things, you won’t know what to look up.  Maybe some of us have never had a student who saw &lt;span class=&quot;math-container&quot;&gt;$\sin^2 x + \cos^2 x$&lt;/span&gt; in the formula they had derived and didn’t know what to do.  It’s just not obvious that there is anything to be looked up. In grad school, you learn some diligence and, if your advisor doesn’t have the answer, might spend some time each day doing (literature) research in the library, in the hopes that someone has published a paper on what to do. Undergraduates, at least the few who bring me such questions, do not have such perseverance, I’d say “luckily” because it’s a painfully slow process to find out something you should have already known.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you haven’t memorized things you use often, you will waste a lot of time looking them up. What you do changes as you go through life, so what you do often, that is, what you need to have memorized will also change.  You need to have certain facts about first-semester calculus memorized in order to efficiently learn second-semester calculus. Similarly for higher-level courses. When a student gets a job, they will have to memorize new things, things they have to memorize just for that job and nothing else. It’s kind of like in calculus, except worse because of the narrowness of its scope and applicability. In a college friend’s first job, they (a team) had to go in and analyze a company in a week and make a proposal to a group that wanted to invest or buy the company.  They had to memorize a ton a facts about this company that they would forget in a week or two, when they had to do it again for another company. You can’t keep such a job if you’re bad at memorizing. Students should take the opportunity to learn this useful skill, and teachers should help them with it.&lt;br /&gt;&#xA;3’. Similarly, when one is speaking live with someone, like a boss or in a job interview, you don’t want to have to say, “Let me look that up.”&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, a major reason is @CarlWitthoft’s point, that memorizing helps to relate concepts and discover relationships.  The foregoing examples underscore this, but it is worth emphasizing because it relates to problem-solving, if I may call it that, at all levels, whether one is working at the frontiers of mathematics to prove new theorems, or for a business,  or for a client.  That some things are worth memorizing in certain situations means that there must be things that are not worth memorizing.  What they are is particular to each situation and furthermore, particular to one’s future, which is uncertain.  So teach people to have a solid foundation in basic principles, and they can learn what they need as time unfolds.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;[Sorry about the switching around of “you,” “one,” “students,” etc.  I seem unable to stick to a point of view this morning.  The “you” here probably always means the learner, not the teacher, whether the learner is in school or working a job, even a teaching job.  Funny thought just occurred to me: Imagine a teacher thought think-pair-share was a good thing to do and said to the class, “Let me just look that up.”]&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2022-04-04T19:51:37.383" LastActivityDate="2022-04-04T19:51:37.383" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25118" PostTypeId="2" ParentId="25085" CreationDate="2022-04-04T21:22:49.450" Score="2" Body="&lt;p&gt;I just came across &lt;a href=&quot;https://en.wikipedia.org/wiki/Solar_Lottery&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solar Lottery&lt;/a&gt; by Philip Dick. I'll try to remember to update this answer with details after I read it. :)&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2022-04-04T21:22:49.450" CommentCount="0" CommunityOwnedDate="2022-04-04T21:22:49.450" ContentLicense="CC BY-SA 4.0" />
  <row Id="25119" PostTypeId="2" ParentId="25113" CreationDate="2022-04-05T09:24:15.680" Score="9" Body="&lt;p&gt;&lt;em&gt;I somewhat wish to contest &lt;a href=&quot;https://matheducators.stackexchange.com/a/25114/75&quot;&gt;Tom Kern’s answer&lt;/a&gt;:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The goal of most education is an internalised understanding, which in turn reduces cognitive load when building upon what you learnt (which in turn may be a deeper understanding of the same subject). Memorisation of some things almost inevitably happens on the way there, in particular when solving exercises. It may also be an important tool on the way (example below). Finally, seeking for efficient memorisation may structure one’s knowledge and expose patterns and similar.&lt;/p&gt;&#xA;&lt;p&gt;However, memorisation alone rarely &lt;em&gt;causes&lt;/em&gt; understanding or a sufficient reduction of cognitive load for application. In particular, students who only rote-memorise can (and often do) completely miss the goal of a course and can be completely lost when they need to build upon their understanding on the next level. And even if they can solve a task, they may need much more time to apply their memorised knowledge.&lt;/p&gt;&#xA;&lt;p&gt;For example, when teaching quadratic equations, the goal is not that students can regurgitate a quadratic formula, but that they know when to apply it, transform a problem so they can apply it, exclude implausible solutions, get a better understanding of equations and logic in general, etc. Now, when learning all of these things through exercises and similar, having a quadratic formula readily available is extremely useful – so useful that you indeed might want to instruct students to memorise it as soon as it comes up (and motivate why). However, even if you don’t, students who seriously work on exercises and similar will inevitably memorise it over the course.&lt;/p&gt;&#xA;&lt;p&gt;Thus, at the end of the teaching unit, not having memorised a quadratic formula is a strong indicator that a student also missed the goals of the course (or is severely underchallenged). However, the reverse does not hold: Students can have memorised that formula without reaching the goals of the course. An exam needs to assess whether students are able to properly apply the quadratic formula, not whether they can regurgitate it. It should not make much of a difference whether you write a quadratic formula on the blackboard or not – with the main exception probably being students who suffer from anxiety and for whom not having to memorise the equation will provide security and a ward against memorisation flukes.&lt;/p&gt;&#xA;&lt;p&gt;I would therefore only instruct students to memorise something (or even assess their memorisation) if all of the following apply:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;it helps achieving the actual learning goal (which is never the memorisation itself),&lt;/li&gt;&#xA;&lt;li&gt;you expect that for some students memorisation as a side effect (of doing exercises and similar) will be not optimal,&lt;/li&gt;&#xA;&lt;li&gt;you do not want to leave figuring out the above to the students, either because the usefulness of memorisation is not apparent a priori or they are not sufficiently independent learners yet. (Here, you need to compromise between your students learning math or learning how to learn.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Otherwise, I do not see a need to make a distinction.&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2022-04-05T12:20:54.370" LastActivityDate="2022-04-05T12:20:54.370" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25121" PostTypeId="1" CreationDate="2022-04-07T02:02:13.737" Score="10" ViewCount="404" Body="&lt;p&gt;My six-year-old daughter enjoys addition but not subtraction. When we walk together, I like to give her some &amp;quot;mental mathematics&amp;quot; questions, such as &amp;quot;What is 13 plus 33&amp;quot; and she enjoys answering them. However, she always asks me to give her addition problems only. So she will refuse to answer the question &amp;quot;What is 21 minus 6&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I have discovered a &amp;quot;trick&amp;quot; that works for her: I will ask the question &amp;quot;What is 21 &lt;strong&gt;plus&lt;/strong&gt; negative 6&amp;quot; and she will give me the answer 15. But still, she will not do 21 minus 6.&lt;/p&gt;&#xA;&lt;p&gt;Is it normal for her to dislike subtraction while enjoying addition? Is there anything that I can/should do about it other than converting a subtraction problem to an addition problem?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastActivityDate="2022-07-14T13:11:10.527" Title="Is it normal for a child to strongly prefer addition to subtraction?" Tags="|arithmetic-operations|arithmetic|preschool-education|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="25122" PostTypeId="2" ParentId="25113" CreationDate="2022-04-07T06:07:00.470" Score="-1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;For example, in calculus, you are expected to memorize all the derivatives and integrals&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since when? A student who memorizes these things is memorizing instead of learning. I mastered calculus by understanding the subject and I gave no effort to memorizing. I know the derivatives and integrals because I learned them; if I had memorized them instead then I would not know them.&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2022-04-07T06:07:00.470" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25124" PostTypeId="1" CreationDate="2022-04-10T22:38:35.033" Score="7" ViewCount="132" Body="&lt;p&gt;In classes like Calc 3 or Computer Graphics, I want my students to be comfortable describing common curves and surfaces parametrically (such as lines, triangles, circles, or surfaces of revolution). What are some good resources for teaching these skills?&lt;/p&gt;&#xA;&lt;p&gt;Edit: Specifically, if I had a particular shape in mind (like a Mobius strip, a helix, etc.) I could sit down and work out a parameterization for that shape. But I can't articulate my general process. I can describe how I got any specific parameterization, but I'm looking for a resource that describes the general skill. Most of the resources I've found (including calculus texts) tend to just give examples and (if one is lucky) describe how those specific examples work.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastEditorUserId="15671" LastEditDate="2022-04-13T14:54:11.977" LastActivityDate="2022-04-13T14:54:11.977" Title="Resources for Teaching Parameterization of Curves/Surfaces" Tags="|geometry|vector-calculus|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25125" PostTypeId="2" ParentId="25121" CreationDate="2022-04-11T17:36:32.370" Score="3" Body="&lt;p&gt;Since the learning proces begins with popping up more and more fingers (numbers) and especially when enhanced by helping and encouraging, it is kind of normal to me at least. Subtracting is a new strain and it does not add much of new abilities; it demands though more work\remembering.&#xA;It might be an idea to bring this technique up when challenging emotions; then sometimes leveling them is the better way to cope with negative experiences. And there are games in which it makes sense in diminishing possibilities in order to get the upper hand&lt;/p&gt;&#xA;" OwnerUserId="11696" LastActivityDate="2022-04-11T17:36:32.370" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25126" PostTypeId="2" ParentId="25121" CreationDate="2022-04-12T06:07:47.240" Score="3" Body="&lt;p&gt;It's certainly normal for people's skill at addition to be much stronger than subtraction, and for the preference and comfort level to follow suit.&lt;/p&gt;&#xA;&lt;p&gt;From a paper by Tom Macintyre and Ruth Forrester, &lt;a href=&quot;https://web.archive.org/web/20110313095657/http://www.icme-organisers.dk/tsg08/macintyre.doc&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Teaching Mental Calculation - how successfully are strategies being learnt?&lt;/a&gt;&amp;quot;, Edinburgh Centre for Mathematical Education (ECME), University of Edinburgh:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Addition tasks are clearly completed in a much more confident manner&#xA;than the subtraction items, with over 80% of the study group with at&#xA;most one error on the items. Subtraction items appear to have&#xA;presented a much bigger challenge to the pupils, with over 50% having&#xA;3 or more of those questions wrong.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Daniel Willingham observes one possible reason why that's the case (from &lt;a href=&quot;https://www.aft.org/periodical/american-educator/winter-2009-2010/ask-cognitive-scientist-it-true-some-people&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Is It True That Some People Just Can't Do Math?&amp;quot;&lt;/a&gt;, &lt;em&gt;American Educator&lt;/em&gt;, Winter 2009-2010):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Addition and multiplication facts are easier to memorize because they&#xA;are commutative; that is, 3 + 4 is the same as 4 + 3, and the same is&#xA;true for 3 x 4 and 4 x 3. That is not the case for subtraction and&#xA;division. Even well-educated adults from countries with excellent math&#xA;education will sometimes calculate subtraction and division facts,&#xA;rather than retrieve them from memory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2022-07-14T13:11:10.527" LastActivityDate="2022-07-14T13:11:10.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25127" PostTypeId="2" ParentId="25124" CreationDate="2022-04-12T23:33:05.570" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://mathinsight.org/parametrized_surface_introduction&quot; rel=&quot;nofollow noreferrer&quot;&gt;An introduction to parametrized surfaces&lt;/a&gt;,&#xA;from the &amp;quot;Math Insight&amp;quot; website,&#xA;is short but quite clear, with some nice 3D rotatable images.&lt;/p&gt;&#xA;&lt;p&gt;     &lt;img src=&quot;https://i.stack.imgur.com/eUYQF.png&quot; width=&quot;400&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;     &lt;em&gt;A parametrized helicoid.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-04-12T23:33:05.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25128" PostTypeId="1" CreationDate="2022-04-14T17:55:48.820" Score="8" ViewCount="185" Body="&lt;p&gt;I teach quite a few foreign students at a U.S. university.   Frequently students are placed in our most remedial math class due to not having placement scores and failing to test out of the course.&#xA;I am trying to put together a list of resources for how our terminology compares to what they are used to.  For example,  I know many parts of the world use a comma as a decimal separator where the U.S. uses a decimal point.  Other languages don't group things into groups of 1s, 10s, 100's of various units then such as &lt;a href=&quot;https://blog.teacherspayteachers.com/5-ways-countries-math-differently-us-important-know/&quot; rel=&quot;nofollow noreferrer&quot;&gt;5 things &lt;/a&gt; and &lt;a href=&quot;https://www.csus.edu/indiv/o/oreyd/acp.htm_files/todos.operation.description.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Latin America to U.S.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know of any such resources already made? If not,  other suggestions for how other countries do and represent things differently in primary/secondary mathematics would be helpful for me to put together my own tutorial for incoming students.&lt;/p&gt;&#xA;" OwnerUserId="2201" LastEditorUserId="376" LastEditDate="2022-04-15T07:22:45.980" LastActivityDate="2022-04-15T07:22:45.980" Title="References for mathematical notation for foreign students in the U.S" Tags="|notation|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25129" PostTypeId="1" CreationDate="2022-04-16T16:26:54.677" Score="3" ViewCount="1111" Body="&lt;p&gt;When I was learning the Pythagorean theorem(at time A), I was just told to memorize it. I used it often before trying to derive the equation(at time B), and I think actually I have forgotten the derivation now(denoted by time C). A is prior to B, and B is prior to C.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qEsiv.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qEsiv.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I learned from &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B016CAJLBA&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;The ABCs of How We Learn&lt;/a&gt; that chunking(in section D(deliberate practice)) takes two steps, namely proceduralization and automatization. In the first step, we need clear guidance(&lt;em&gt;verbal control and explicit memory retrieval&lt;/em&gt;), but after that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Interestingly, once procedures have been automatized, it is difficult for people to recover the original substeps.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;While experts exhibit superior performance by definition, they may not be very good at decomposing and explaining their expertise.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then my questions arise:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I don't know if the derivation I did at time B is worthwhile?&lt;/li&gt;&#xA;&lt;li&gt;Have(had) I reached the step automatization in mastering the Pythagorean theorem at time C(between time A and time B, or between time B and time C)? What are the differences?&lt;/li&gt;&#xA;&lt;li&gt;If I will be unable to &lt;em&gt;decompose and explain&lt;/em&gt;(in Daniel L. Schwartz's words in the book) the equation after automatization, can I just skip the derivation step and memorize the theorem(as a common sense, in my long-term memory) and apply it?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19755" LastEditorUserId="19755" LastEditDate="2022-04-18T01:30:34.577" LastActivityDate="2022-04-19T13:45:27.460" Title="Do we need to practice equation derivation while learning math if equations will be chunked and automatized?" Tags="|derivative|psychology|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25130" PostTypeId="1" CreationDate="2022-04-16T18:38:39.583" Score="3" ViewCount="283" Body="&lt;p&gt;I'm a first year mathematics major and I have a problem with my learning process. In my university, I only have books and questions that the university published, so I have to learn the most of the material by myself. The books are books that my university has written. I'm having problem with getting a deep understanding the material. My learning algorithm goes like this most of the time:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Opening the book, starting to read and most of the time reading fast.&lt;/li&gt;&#xA;&lt;li&gt;After a not very long time, starting to skip the proofs that I didn't get.&lt;/li&gt;&#xA;&lt;li&gt;Starting to skip some exercises.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So at the end of the day, I know the definitions, I can solve some exercises, but I don't understand the material deeply. When I'm reading I usually don't have questions and that's not a good thing. I want to read the book, ask questions all the time, prove things on my own, etc. How can I solve this? I don't have motivation to understand the proofs. I thought about proving to myself the theorems, but I don't know if that's a good idea. The courses that I'm taking now are Linear algebra 2 and Real anaylsis 2.&lt;/p&gt;&#xA;" OwnerUserId="19756" LastEditorUserId="19777" LastEditDate="2022-04-26T13:41:51.987" LastActivityDate="2022-04-26T13:41:51.987" Title="How to understand the book and the material to the deepest possible level?" Tags="|undergraduate-education|textbooks|proofs|self-learning|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25131" PostTypeId="2" ParentId="25129" CreationDate="2022-04-17T01:22:18.197" Score="3" Body="&lt;p&gt;This question goes to the core of what mathematics is about. Benjamin Pierce wrote, &amp;quot;Mathematics is the science which draws necessary conclusions&amp;quot;. So, deriving equations (as it's phrased here), is confirming that our conclusions are, indeed, necessary.&lt;/p&gt;&#xA;&lt;p&gt;In theory, one could imagine an alternative path of faith-based/cargo-cult math, where people used principles or algorithms received from authority figures without justification. But that would miss the whole point of why mathematics is powerful, would lack any self-correction method, and likely wind up in a cul-de-sac where someone comes up with an incorrect principle and winds up disseminating a broken and malfunctioning process.&lt;/p&gt;&#xA;&lt;p&gt;By way of analogy, one could likewise point at any physical or life science and say, &amp;quot;Why do I need experiments or evidence? Once the theory is written down, then I'll just memorize it.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2022-04-17T01:22:18.197" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25132" PostTypeId="2" ParentId="25129" CreationDate="2022-04-17T02:48:15.073" Score="8" Body="&lt;p&gt;I endorse Steve's comment about different skills and different levels of proficiency in different skills.&lt;/p&gt;&#xA;&lt;p&gt;Also, if one at one time justified the Pythagorean theorem, but is no longer able to do so, I wouldn't say that the knowledge has been automatized; I'd say it's been forgotten as a result of disuse. Expertise in mathematics does become automatized, as it does in other areas of endeavor, but the automatizing is not forgetting why things are true or forgetting how to justify them. Proving statements and justifying steps are part of the everyday practice of the mathematics expert, and if one doesn't do those things, one isn't an expert.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, automatization &lt;strong&gt;is&lt;/strong&gt; manifested when an expert mathematician gravitates toward an effective proof strategy or a useful simplification without having to think about it, and without necessarily being able to consciously articulate how they knew the approach would be a productive one.&lt;/p&gt;&#xA;&lt;p&gt;The question in the title is different from the one in the body of the post, but I should answer that one as well: yes, one does need to practice equation derivation.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2022-04-17T02:48:15.073" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25133" PostTypeId="2" ParentId="25129" CreationDate="2022-04-17T13:26:42.357" Score="1" Body="&lt;p&gt;So my answer is &amp;quot;both&amp;quot;, not either/or.  Belt and suspenders.  So, yes absolutely, you should have some mathematical formulas in ready access memory (Pythagorean formula, quadratic formula, some (not all) trig identities, etc.).  They are very frequently used and you will struggle if having to rederive them, each time.&lt;/p&gt;&#xA;&lt;p&gt;Note this does not mean only that you memorize them by short term recall (e.g. the way a plebe memorizes the menu for noon meal).  Perhaps starting a little with that (or for initial exams).  But a lot of the memorization is by repeated use within drill problems.  But it is still memorization, not dependent on derivation.&lt;/p&gt;&#xA;&lt;p&gt;That does NOT mean there is no value in deriving key formulas once or even a few times.  For one thing, you're using a different part of your mind and are practicing algebra manipulation (a general skill needed and one that many students are weak at).  Also, it will give you more confidence in the formula itself.  And even give you better confidence and faster recall if you have to look a derivation up, itself (e.g. using completing the square for some aspect of calculus or diffyQs).  Or of course the ability to re-derive occasionally, under pressure; this is the case with some of the trig identities.  I think most people make a basic subconscious decision of how much trig to memorize and how much to rederive as needed.  And it does not need to be the bare minumum (e.g. definitions of the functions) as the &amp;quot;derive everything&amp;quot; adherents advocate, but can be a select few like sinsq + cossq...without every random cscsq thingie.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I see this tendency to a false either/or fallacy amongst many questioners and answerers here.  I think this is an aspect of being &lt;strong&gt;Mathematics&lt;/strong&gt; (algorithmically) minded, versus &lt;strong&gt;Educators&lt;/strong&gt; (psychologically) minded and assuming that humans (basically smarter dogs!) learn in the manner of a computer program.  Almost always we hear here, &amp;quot;memorization bad&amp;quot;, &amp;quot;derivation good&amp;quot;.  But, like I said this is a fallacy, doesn't pertain to pedagogy of real, organic brained, humans.  Both is good.  It's like if you ask me if you should practice shooting or scrimmage for basketball...both, man!&lt;/p&gt;&#xA;&lt;p&gt;P.s.s.  The &amp;quot;do we need to&amp;quot; (first three words of question title) is an obvious question, but also shows the wrong emphasis.  Instead of trying to figure out the bare minimum, embrace doing extra (in all facets of learning).  Of course, you can overdo it, it is an optimization problem.  But in general, for most students, for most &amp;quot;non masters&amp;quot;, they should be looking to do more (of whatever, drill, short term memorization, derivations), rather than less.  &amp;quot;And&amp;quot;, not &amp;quot;instead&amp;quot;.  Don't be so concerned about wasted effort.  It will all pay off.&lt;/p&gt;&#xA;" OwnerUserId="19757" LastEditorUserId="19757" LastEditDate="2022-04-17T14:28:12.933" LastActivityDate="2022-04-17T14:28:12.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25134" PostTypeId="2" ParentId="25129" CreationDate="2022-04-17T20:26:01.533" Score="3" Body="&lt;p&gt;I don't have access to the book mentioned in the OPs question so this answer will be based on the limited information present in the question.&lt;/p&gt;&#xA;&lt;p&gt;There are many ways in which people &amp;quot;learn mathematics&amp;quot; and many different aspects of what is important depending on the learners focus and scope.&lt;/p&gt;&#xA;&lt;p&gt;Some don't need to learn the fundamental intricacies of a theorem and instead want to focus on its use towards other things. Others choose to delve deep into those intricacies and find out how the theorem relates to the components it was derived from.&lt;/p&gt;&#xA;&lt;p&gt;There is one thing in common that both groups need to be sure of however, namely &lt;strong&gt;Logical Validity and Soundness&lt;/strong&gt;. For example for a theorem &lt;span class=&quot;math-container&quot;&gt;$A \implies B$&lt;/span&gt; they need to be able to recognize and ascertain the two following conditions to make good use of the theorem.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The &lt;strong&gt;scope&lt;/strong&gt; in which the theorem is stated. They need to be able to recognize when they can use the theorem and when they cannot. In this case they need to be able to ascertain whether &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is true or not for their use case.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;strong&gt;trust&lt;/strong&gt; that the theorem has been &lt;strong&gt;proved&lt;/strong&gt;, i.e. trust that there is a valid chain of logical reasoning that establishes that from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; you can imply &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;. (That is, whenever A is true then B must also be true.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;With that in mind we can try to answer your questions. Question 1 and 2 uses subjective measures (&amp;quot;worthwhile&amp;quot; and &amp;quot;mastered&amp;quot;) so they will also have subjective answers.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;1. Was it worthwhile to derive it at time B?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Lets consider the equation here. You've invested time and mental resources to derive the theorem and made sure there are no errors in the reasoning. From this, what have you gained?&lt;/p&gt;&#xA;&lt;p&gt;In terms of the two bullet points above, you've ascertained that, indeed, there is a valid chain of reasoning from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;. You've also hopefully been able to reinforce your understanding of when &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is true and what pitfalls may arise that makes it not true. So you've been able to strengthen your &lt;strong&gt;trust&lt;/strong&gt; in both the validity of the theorem as well as your ability to recognize its scope. This trust is something you will remember.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;2. When does automatization happen?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I (the answer writer) can't decisively determine this since I don't have access to the book. So this must be taken with a grain of salt.&lt;/p&gt;&#xA;&lt;p&gt;However, from OPs own statement:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the first step, we need clear guidance(verbal control and explicit memory retrieval)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It seems that proceduralization is the activity of explicitly recalling and following the recipe needed to get from the premises to the conclusion.&lt;/p&gt;&#xA;&lt;p&gt;Automatization seems to be the process of these steps getting internalized and processed &lt;em&gt;implicitly&lt;/em&gt; without need of conscious effort to follow them.&lt;/p&gt;&#xA;&lt;p&gt;Thus I think the answer is that Automatization can happen in any/all of the steps A, B and C.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;3. Is it okay to skip the derivation and remember only the theorem?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A theorem that we learn in a mathematics book is itself a &amp;quot;landmark&amp;quot; result in a big forest of logically valid chains of reasoning. They are mentioned because they are noteworthy and useful.&lt;/p&gt;&#xA;&lt;p&gt;This means that learning only the theorem &lt;span class=&quot;math-container&quot;&gt;$A \implies B$&lt;/span&gt; itself is great as long as you are able to trust your understanding of the two bullet points. You don't need to commit the derivation to memory as long as you know the theorem itself holds.&lt;/p&gt;&#xA;&lt;p&gt;However, if you are tasked with convincing someone else that the theorem &lt;span class=&quot;math-container&quot;&gt;$A \implies B$&lt;/span&gt; is true, you need to be able to convince them of the two bullet points. Commiting every single derivation to long-term memory is something that is seldom done but in this case you might need that information. Most people probably remember some landmark steps of the derivation but not every single step.&lt;/p&gt;&#xA;&lt;p&gt;The theorem itself seems to be an automatization you're talking about if one is used to do every step of the derivation. It is useful to you and for getting the job done. If you cannot give the detailed derivation yourself then, you should maintain a list of references towards the derivations or know how to look them up again. A book or handwritten notes are &amp;quot;memory&amp;quot; too, even if they aren't in your head.&lt;/p&gt;&#xA;" OwnerUserId="19758" LastActivityDate="2022-04-17T20:26:01.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25135" PostTypeId="2" ParentId="25130" CreationDate="2022-04-18T11:01:05.163" Score="4" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Buy some alternate texts, that are more pedagogically useful for self studying.  There is cheap stuff available on Amazon from Dover, etc.  Check reviews on the web to see what is best suited for you.  (May not be the hardest ones, may be the easier ones.)  Also, check out what is in your library.  Yes, the actual brick building.  If you are too poor for buying books, then just look at the web and steal pdfs.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When you read, some degree of speed (not too much, but not too painstaking either) is good.  Just realize, it will never be as immersive as reading a novel or a history book.  BUT, when you hit examples or derivations, you need to slow down and &amp;quot;read with a pencil&amp;quot; (in your notebook, see 4 below).  Recreate the example/derivation or if material is easy, just attempt it yourself and then see how the book did it.  But treat each example/derivation as an en passant exercise.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do all the exercises.  Yes, including the easy ones (hopefully there are some...if not, find some).  They can build familiarity in a way that &amp;quot;discover the world&amp;quot; ball-buster problems do not.  Part of the struggle in these abstract classes is just getting used to a new language (e.g. in physics, few people &amp;quot;get&amp;quot; quantum mechanics at first...not the way kinematics is a blinding revelation, but over time they just get used to it.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Work with a notebook (bound or spiral).  One active one per class.  From the front, you have study/drill.  I usually have no icon at top of page for exercise.  Stars for lecture.  And hashes for book study.  From the back (IMPORTANT), keep a running list of questions (with space for answers).  Just this process of writing your questions down will help you study, help you feel less lost.  Often you will find yourself answering them...but in some cases, not.  But they are great questions for followup with the instructor.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Read and do drill work prior to lecture.  I know this sounds hard, but paradoxically it will make the work easier.  And the lecture will be extremely comprehensible!  This takes discipline.  But pays off.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P.s.  There is no royal road to geometry.  Buckle down and study.  Good luck!&lt;/p&gt;&#xA;" OwnerUserId="19757" LastActivityDate="2022-04-18T11:01:05.163" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25136" PostTypeId="2" ParentId="25129" CreationDate="2022-04-19T13:45:27.460" Score="1" Body="&lt;p&gt;The goal of learning mathematics is not to learn various theorems or to be able to use them.&lt;/p&gt;&#xA;&lt;p&gt;Rather, the goal of learning mathematics is to learn how to derive new facts from old ones.  The point is the skill of deriving, not the facts you derive.&lt;/p&gt;&#xA;&lt;p&gt;Mathematics is a humanities subject, not a science one.  When you study literature, you quickly realize memorizing an analysis of some poem done by someone else is kind of pointless; the idea is for you to come up with your own analysis, possibly of different poems.  Mathematics is the same.  Chunking and automizing facts makes sense for subjects where the point is the facts; mathematics is not one of those subjects.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2022-04-19T13:45:27.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25137" PostTypeId="1" CreationDate="2022-04-20T14:19:08.353" Score="0" ViewCount="888" Body="&lt;p&gt;I want to find an intuitive analogy to explain how binary addition (more precise: an adder circuit in a computer) works. The point here is to explain the abstract process of &lt;em&gt;adding&lt;/em&gt; something by comparing it to something that isn't abstract itself.&lt;/p&gt;&#xA;&lt;p&gt;In principle: An everyday object or an action that is structured like or functionally resembles an adder.&lt;/p&gt;&#xA;&lt;p&gt;Think of a thing that can belong to any number of categories x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;, x&lt;sub&gt;5&lt;/sub&gt;, x&lt;sub&gt;6&lt;/sub&gt;, x&lt;sub&gt;7&lt;/sub&gt;, x&lt;sub&gt;8&lt;/sub&gt; for which the property holds that if you put two objects together/perform two actions simultaneously, and both the objects/actions are of the same category you automatically create an object or perform an action that is of the next higher category that the object doesn't yet belong to, the whole thing therefore implementing the basic functionality of an adder.&lt;/p&gt;&#xA;&lt;p&gt;(Categories are changing here analogous to the bits in the circuit: 00000001 (1) + 00000001 (1) together, adds up to 00000010 (2).)&lt;/p&gt;&#xA;&lt;p&gt;But I just can't think of such a situation or an object where this pattern would occur. Whatever analogy i create with increasing amount of categories the way these categories transform becomes increasingly harder to explain, and the metaphor becomes overly specific and unhandy.&lt;/p&gt;&#xA;&lt;p&gt;Hence the question:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What's an everyday object that resembles an adder in it's basic functionality?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="19767" LastActivityDate="2022-04-22T04:27:58.637" Title="Finding an analogy to explain the function of a binary adder" Tags="|mathematical-pedagogy|primary-education|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25138" PostTypeId="2" ParentId="25137" CreationDate="2022-04-20T17:26:39.773" Score="8" Body="&lt;p&gt;Why do you need an analogy?&lt;/p&gt;&#xA;&lt;p&gt;I think just having a bunch of beans to count, and grouping the beans into groups of size 1, 2, 4, 8, 16, etc is intuitive enough without needing an analogy.  The comparison with base ten arithmetic, where we group into 1, 10, 100, 1000, etc is clear.&lt;/p&gt;&#xA;&lt;p&gt;Addition is straightforward:  combine the piles and regroup.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-04-20T17:26:39.773" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25139" PostTypeId="2" ParentId="25137" CreationDate="2022-04-20T20:09:33.790" Score="0" Body="&lt;p&gt;Weighing/massing is something that has this aspect.  Really any extrinsic property.  &lt;a href=&quot;https://en.wikipedia.org/wiki/Intrinsic_and_extrinsic_properties&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Intrinsic_and_extrinsic_properties&lt;/a&gt;  But I think weighing is easy to understand because you see the output, often in digital form.Seems to satisfy your desire.&lt;/p&gt;&#xA;&lt;p&gt;I would probably avoid fluid or electrical flow, because of the complexities of series and parallel and driving force.  All that said, if they know basic E&amp;amp;M, obviously resistance in series adds (and some basic adders are actually resistance networks).  Also, of course head loss of components in series adds linearly for fluid circuits...but I suspect your computer guys will be less comfortable with piping than electronics.&lt;/p&gt;&#xA;" OwnerUserId="19769" LastActivityDate="2022-04-20T20:09:33.790" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25140" PostTypeId="2" ParentId="25137" CreationDate="2022-04-20T23:16:22.553" Score="4" Body="&lt;p&gt;Not quite what you want, but there are mechanical binary counters, e.g.,&#xA;this one, video &lt;a href=&quot;https://www.reddit.com/r/oddlysatisfying/comments/8ke0yr/mechanical_binary_counter/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MB4Xp.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MB4Xp.jpg&quot; alt=&quot;Bits&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.pinterest.com/pin/310607705544531238/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Another wooden version here&lt;/a&gt;.&#xA;You could emphasize the analogy with an odometer.&lt;/p&gt;&#xA;&lt;p&gt;And here is a Minecraft version, not quite as compelling (to me):&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=MKi9cvdrisI&amp;amp;ab_channel=SgtGodswordBerserker&quot; rel=&quot;nofollow noreferrer&quot;&gt;YouTube link&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here's a clever marble binary adder:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/94XjF.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/94XjF.jpg&quot; alt=&quot;Adder&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hackaday.com/2009/10/13/binary-adder-will-give-you-slivers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Video link&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-04-20T23:16:22.553" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25141" PostTypeId="2" ParentId="25137" CreationDate="2022-04-20T23:56:14.637" Score="2" Body="&lt;p&gt;If you're looking for a metaphor that would instantly click with a modern audience, you might consider mobile merge games.  The gameplay is that you have a board that is filled with pieces, and you can combine two identical pieces to yield a single &amp;quot;evolved&amp;quot; piece.  For instance, maybe you combine two pieces of thread to get a spool of thread, two spools of thread make a piece of yarn, two pieces of yarn makes a ball of yarn, two balls of yarn makes a piece of rope, and so on.  For illustration, &lt;a href=&quot;https://youtu.be/RWGchSWy-H4&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; is a video of someone making a notoriously complex item in Merge Mayor from a large number of base units.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2022-04-20T23:56:14.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25142" PostTypeId="2" ParentId="25137" CreationDate="2022-04-21T02:04:18.787" Score="5" Body="&lt;p&gt;Why would you need an analogy? Binary is just another way of encoding numbers - so the question is, &amp;quot;why do you need something that adds numbers?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I think what people get hung up on with this issue is they keep wanting to associate &amp;quot;numbers&amp;quot; with &lt;em&gt;decimals&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;And to that end, I'd suggest that people should think about the difference between &amp;quot;3&amp;quot;, &amp;quot;three&amp;quot;, &amp;quot;III&amp;quot;, and perhaps a couple of foreign-language words with the same meaning. They're all just &lt;em&gt;symbols&lt;/em&gt; for the underlying number that occurs when we measure situations in which there is something, another something, and finally another something, but not any further somethings than that.&lt;/p&gt;&#xA;&lt;p&gt;And binary numerals are just another kind of symbols, built on a different rule, but they refer to exactly the same numbers. The number &amp;quot;11&amp;quot; &lt;em&gt;binary&lt;/em&gt; means the same thing that &amp;quot;III&amp;quot; above does. They're just two different &lt;em&gt;reporesentational systems&lt;/em&gt; for the same things.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;So why do you need a binary adder?&amp;quot; Well, why do you need a desk calculator? (Which almost certainly has a binary adder in its circuits, by the way, that does just what you think from the name it would be doing - so you can throw that on top, too.)&lt;/p&gt;&#xA;" OwnerUserId="7650" LastActivityDate="2022-04-21T02:04:18.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25143" PostTypeId="2" ParentId="25130" CreationDate="2022-04-21T13:36:49.347" Score="0" Body="&lt;p&gt;Your main concern is &amp;quot;but I dont understand the material deeply&amp;quot;. Especially in mathematics, a deep understanding is very important, so you should do your best to reach that level. By just passively reading a book chapter, you rarely get that deep understanding.&lt;/p&gt;&#xA;&lt;p&gt;Maybe the &lt;a href=&quot;https://en.wikipedia.org/wiki/Rubber_duck_debugging&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rubber duck technique&lt;/a&gt; can help you.&lt;/p&gt;&#xA;&lt;p&gt;After reading some chapter, explain it to some patient audience. In the process of re-phrasing your newly-acquired knowledge in your own words, you often find your weak points. It isn't even necessary that someone from your audience actively asks about aspects they didn't understand - you'll often feel that yourself. And that's why the audience can be a rubber duck if a group of fellow students isn't available.&lt;/p&gt;&#xA;" OwnerUserId="15862" LastActivityDate="2022-04-21T13:36:49.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25145" PostTypeId="2" ParentId="25137" CreationDate="2022-04-21T16:59:19.307" Score="3" Body="&lt;p&gt;I think if you've played Monopoly you understand that once you get 10 one-dollar bills, you'd rather trade them in for a ten-dollar-bill.  And once you get 10 ten-dollar bills, you'd rather trade them in for a hundred-dollar-bill.  You're trying to minimize the number of bills you have to manage.  It's easier to know how much money you have when you've reduced everything to use the largest bills possible too.&lt;/p&gt;&#xA;&lt;p&gt;Now suppose the denominations were \$1, \$2, \$4, \$8, \$16.  And there's a rule that says you can't have two of the same kind of bill -- e.g. you &lt;em&gt;must&lt;/em&gt; trade your two eight-dollar dollar bills in for a sixteen-dollar-bill.&lt;/p&gt;&#xA;&lt;p&gt;In this situation, if you have a dollar and the bank is going to pay you a dollar, then you intuitively know NOT to accept another one-dollar bill from the bank.  Instead, you &lt;strong&gt;give&lt;/strong&gt; your one-dollar bill to the bank and they give you back a two-dollar bill.&lt;/p&gt;&#xA;&lt;p&gt;I think people understand the carry mechanism this way too.  Imagine doing 3 + 1.  If I have 3 dollars (one one-dollar bill and one two-dollar bill).  And someone wants to pay me one dollar, I know that I can't accept a second one-dollar bill, so I must trade it in.  I give away my one-dollar bill expecting a two-dollar bill, but I can't accept a second two-dollar bill.  There's a beautiful moment where the banker has accepted my one-dollar bill and grabbed a two-dollar bill and is about to hand it to me but I look at my own wallet and &lt;em&gt;refuse&lt;/em&gt; to accept the two-dollar bill because I already have one and instead, I hand my two-dollar-bill to the bank, which the banker exchanges for a four-dollar bill.&lt;/p&gt;&#xA;&lt;p&gt;This moment where the banker hasn't finished giving you your change yet but is just holding a bill of some denomination in front of you for you to decide whether to accept or not is the &lt;em&gt;carry mechanism&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Young children can do this hands-on to get a feel for it.  You &amp;quot;pay&amp;quot; the child an amount of money by handing them a bunch of bills (being careful not to have two of any one kind of denomination yourself).  Then the child begins checking to see if they have two of anything (if they are breaking the &amp;quot;rule&amp;quot; by accepting a second bill of the same kind.)  When they hand you an n-dollar bill, you simply take it away and hand them a 2n-dollar bill instead.  This can be done in any order.  The child doesn't have to start with their lowest denomination.  The child is playing a match game - looking to see if the person paying them is offering them a bill of the same type as something they already have.  If they do, they just hand it over and are then offered the next bigger denomination of bill in return.&lt;/p&gt;&#xA;&lt;p&gt;The child can play as the banker too.  They will learn that they will be handed a bill of the same denomination as something they were offered, at which time they must return the two bills to their supply of bills and exchange them for one bill of the next higher denomination.&lt;/p&gt;&#xA;&lt;p&gt;The analog you are looking for is a person.  The person is intuitively alerted to the situation that they are about to have too many of a certain denomination of bill and takes action to resolve it.&lt;/p&gt;&#xA;" OwnerUserId="13481" LastEditorUserId="13481" LastEditDate="2022-04-22T04:27:58.637" LastActivityDate="2022-04-22T04:27:58.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25146" PostTypeId="1" AcceptedAnswerId="25162" CreationDate="2022-04-21T22:51:50.857" Score="4" ViewCount="538" Body="&lt;p&gt;I am tutoring another US college student in a Calculus 1 class. Initially, she was having trouble with basic concepts, but after much prodding most of the conceptual difficulties seem to have been alleviated. However there is one persistent problem I've never encountered with another student: after integrating a function, she'll remove the &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; but leave &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; (or &lt;span class=&quot;math-container&quot;&gt;$du$&lt;/span&gt; etc.). For example, she will do this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int 2x\, dx = x^2 + C\, dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; will hang around in her later calculations, and she'll express that something seems wrong (especially when none of the multiple-choice answers match - we're working with old exams), but I'll need to explicitly point out what happened before she fixes it. If I say &amp;quot;Why is the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; still there?&amp;quot; she'll immediately realize that it was a mistake, though sometimes she will still ask why keeping it is wrong.&lt;/p&gt;&#xA;&lt;p&gt;My first attempt at explaining the problem was &amp;quot;the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; goes poof when the integral goes away&amp;quot; (this being how I learned it). My next was &amp;quot;the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; marks the other end of what's being integrated, it's a boundary along with the &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt;&amp;quot;. My third, after looking up what the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; represented and finding &lt;a href=&quot;https://math.stackexchange.com/questions/200393/what-is-dx-in-integration&quot;&gt;What is &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; in integration?&lt;/a&gt;, was &amp;quot;it's like the &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; in a Riemann sum, and when you evaluate the limit there is no &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; still hanging around&amp;quot;. All of these have been used more than once. Each attempt has stopped the forgetting for some more time but then she gets into the weeds of an integral and leaves the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; on again while wrapped up thinking about something else.&lt;/p&gt;&#xA;&lt;p&gt;How can I hammer this home so that it sticks? She is most confident in Riemann sums (that's part of why I used them in my third attempt) so an answer which leverages them would be appreciated but not required. We do online tutoring; anything that requires me being physically present will not work.&lt;/p&gt;&#xA;" OwnerUserId="19777" LastActivityDate="2022-05-09T12:36:15.347" Title="Student forgets to remove dx after integrating" Tags="|undergraduate-education|calculus|tutoring|integration|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25147" PostTypeId="1" CreationDate="2022-04-22T15:10:39.907" Score="10" ViewCount="764" Body="&lt;p&gt;The taxicab metric can be used to measure distances in idealized gridded cities. However, usually this serves only as a fun exercise for students.&lt;/p&gt;&#xA;&lt;p&gt;I'm looking for engaging (as non-technical as possible) examples of real-world applications of the taxicab metric (or other metrics inspired by cities, such as the British rail metric).&lt;/p&gt;&#xA;&lt;p&gt;It seems that it's mostly used in machine learning, but everything I've seen so far is too technical to explain to laypeople. So, I'd love to get some ideas!&lt;/p&gt;&#xA;" OwnerUserId="7936" LastActivityDate="2022-04-24T13:39:12.890" Title="Real-world applications of taxicab metric" Tags="|reference-request|geometry|examples|applications|topology|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25148" PostTypeId="2" ParentId="25146" CreationDate="2022-04-22T21:40:08.070" Score="0" Body="&lt;p&gt;Give her repeated questions and give positive feedback, carrot, for doing it right. And negative feedback, stick, for doing it wrong.&lt;/p&gt;&#xA;&lt;p&gt;There's no magic aha.  She just needs attention to detail. And the above is how you train attention to detail.&lt;/p&gt;&#xA;" OwnerUserId="19781" LastActivityDate="2022-04-22T21:40:08.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25149" PostTypeId="2" ParentId="7281" CreationDate="2022-04-23T07:17:00.090" Score="0" Body="&lt;p&gt;I haven't taught this yet, I teach Munchkins but for me it looks like if we have:&lt;/p&gt;&#xA;&lt;p&gt;Fraction = a/b and&#xA;Ratio = x:y then&lt;/p&gt;&#xA;&lt;p&gt;b = x+y&lt;/p&gt;&#xA;&lt;p&gt;Also x:y is ALWAYS about one Unit of anything BUT a/b can be a Part of a Unit, a Unit or more than a Unit of anything.&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2022-04-23T07:17:00.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25150" PostTypeId="2" ParentId="25147" CreationDate="2022-04-23T09:30:26.100" Score="7" Body="&lt;p&gt;I would argue that the taxicab metric as such is actually carrying its weight if it is a &amp;quot;fun exercise for students&amp;quot; that points out that there are real-world applications where the Euclidean distance is not an effective metric for modeling.&lt;/p&gt;&#xA;&lt;p&gt;The notion of metrics in general is chock-full of real-world applications.  Here is an example&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I own three bakeries in a city and I'm planning to open a fourth.  I want a location that is &amp;quot;close&amp;quot; to as many potential new customers as possible, but not so &amp;quot;close&amp;quot; to my existing stores that the new store would compete with my existing business (but &amp;quot;close&amp;quot; enough that I can still effectively manage all four locations).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The mathematical model that I can use to turn that abstract concept of closeness into data that I can use to make a real-world decision is a metric space.  Is the taxicab metric the right metric space for me?  It might be a reasonable place to start, depending on the city I live in and the amount of simplicity I am willing to trade for valid data.&lt;/p&gt;&#xA;" OwnerUserId="12619" LastActivityDate="2022-04-23T09:30:26.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25151" PostTypeId="2" ParentId="25147" CreationDate="2022-04-23T11:37:01.453" Score="8" Body="&lt;p&gt;A variation on Matthew's bakeries:&lt;/p&gt;&#xA;&lt;p&gt;The taxicab Voronoi diagram shows which firestation can reach a fire fastest&#xA;in a city whose roads follow an orthogonal grid.&#xA;The dots below are firestations, and each region collects&#xA;the points closer to its firestation than to any other.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lHeKW.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lHeKW.png&quot; alt=&quot;VorDiag&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Wikimedia Commons image&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;There are also applications to VLSI layout, but less easily&#xA;explained:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Papadopoulou, Evantha, and D. T. Lee. &amp;quot;The &lt;span class=&quot;math-container&quot;&gt;$L_\infty$&lt;/span&gt; Voronoi diagram of segments and VLSI applications.&amp;quot; &lt;em&gt;International Journal of Computational Geometry &amp;amp; Applications&lt;/em&gt; 11, no. 05 (2001): 503-528.&#xA;&lt;a href=&quot;https://www.worldscientific.com/doi/10.1142/S0218195901000626&quot; rel=&quot;nofollow noreferrer&quot;&gt;Journal link&lt;/a&gt;.&#xA;&amp;quot;the &lt;span class=&quot;math-container&quot;&gt;$L_\infty$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L_1$&lt;/span&gt; Voronoi diagram of segments are&#xA;equivalent under a &lt;span class=&quot;math-container&quot;&gt;$45^\circ$&lt;/span&gt; rotation.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Incidentally, the taxicab metric is also known as&#xA;the Manhattan metric, or the &lt;span class=&quot;math-container&quot;&gt;$L_1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\ell_1$&lt;/span&gt; metric.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2022-04-24T13:39:12.890" LastActivityDate="2022-04-24T13:39:12.890" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25152" PostTypeId="1" CreationDate="2022-04-23T17:30:50.613" Score="3" ViewCount="1831" Body="&lt;p&gt;What term describes the relationship between tenth, hundredth, thousandth, et cetera (1/10, 1/100, 1/1000, ...) and the number ten?  (Despite what some may say, I don't accept that &amp;quot;decimal&amp;quot; is the answer.)&lt;/p&gt;&#xA;&lt;p&gt;More specifically, in this context, I'm looking for the opposite of the term &amp;quot;multiple&amp;quot;.  Ten, hundred, thousand, et cetera are &lt;em&gt;&lt;strong&gt;multiples&lt;/strong&gt;&lt;/em&gt; of ten.  Tenth, hundredth and thousandth are &lt;em&gt;&lt;strong&gt;what&lt;/strong&gt;&lt;/em&gt; of ten?  The closest thing I can think of is &amp;quot;multiplicative inverses of multiples of ten&amp;quot;.  Is there no better choice of word or words?&lt;/p&gt;&#xA;" OwnerDisplayName="Chris Shiherlis" LastActivityDate="2022-04-24T23:52:51.270" Title="What term describes the relationship between tenth, hundredth, thousandth and the number ten?" Tags="|terminology|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25153" PostTypeId="2" ParentId="25152" CreationDate="2022-04-23T17:52:36.967" Score="13" Body="&lt;p&gt;They are &lt;strong&gt;negative powers&lt;/strong&gt; of ten.&lt;/p&gt;&#xA;&lt;p&gt;Consider that&lt;br /&gt;&#xA;&lt;em&gt;ten to the positive power of two is ten squared (100)&lt;br /&gt;&#xA;ten to the positive power of three is a thousand (1000)&lt;/em&gt;&lt;br /&gt;&#xA;and so forth.&lt;/p&gt;&#xA;&lt;p&gt;ten to a negative power is exemplified by&lt;br /&gt;&#xA;&lt;em&gt;ten to the minus one = 0.1&lt;br /&gt;&#xA;ten to the minus two = 0.01&lt;/em&gt;&lt;br /&gt;&#xA;and so forth.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.britannica.com/science/power-of-10&quot; rel=&quot;noreferrer&quot;&gt;Britannica&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;power of 10, in mathematics, any of the whole-valued (integer) exponents of the number 10. A power of 10 is as many number 10s as indicated by the exponent multiplied together. Thus, shown in long form, a power of 10 is the number 1 followed by &lt;em&gt;n&lt;/em&gt; zeros, where &lt;em&gt;n&lt;/em&gt; is the exponent and is greater than 0; for example, 10&lt;sup&gt;6&lt;/sup&gt; is written 1,000,000. When &lt;em&gt;n&lt;/em&gt; is less than 0, the power of 10 is the number 1 &lt;em&gt;n&lt;/em&gt; places after the decimal point; for example, 10&lt;sup&gt;−2&lt;/sup&gt; is written 0.01.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="Anton" LastActivityDate="2022-04-23T23:04:24.413" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25154" PostTypeId="2" ParentId="25152" CreationDate="2022-04-23T18:59:43.360" Score="1" Body="&lt;p&gt;They are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;fractions&lt;/strong&gt; of one in which the denominator is an integral positive&#xA;power of ten&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;if one really wants to describe their fractional nature. It doesn’t fit the pattern in the question, but I think that’s a deficiency in the question, and it’s in the same spirit.&lt;/p&gt;&#xA;&lt;p&gt;PS&lt;/p&gt;&#xA;&lt;p&gt;Of course a simpler way this is described is as a “&lt;a href=&quot;https://www.cuemath.com/numbers/decimal-fraction&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;decimal fraction&lt;/strong&gt;&lt;/a&gt; with a numerator of one”.&lt;/p&gt;&#xA;" OwnerDisplayName="David" LastActivityDate="2022-04-23T19:39:27.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25157" PostTypeId="2" ParentId="25152" CreationDate="2022-04-23T22:59:23.607" Score="12" Body="&lt;p&gt;You wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;More specifically, in this context, I'm looking for the opposite of the term &amp;quot;multiple&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The opposite of a &amp;quot;multiple&amp;quot; is a &amp;quot;factor&amp;quot;. For example, 15 is a multiple of 3, and 3 is a factor of 15. The term &amp;quot;divisor&amp;quot; is sometimes used as a synonym for &amp;quot;factor&amp;quot; (although they are not really the same concept).&lt;/p&gt;&#xA;&lt;p&gt;However, that does not seem to be what you are actually looking for. You then wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ten, hundred, thousand, et cetera are multiples of ten. Tenth, hundredth and thousandth are what of ten?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is true that 10, 100, 1000, etc. are multiples of ten, but so are 20, 570, etc. You seem to be focussing on the fact that they are powers of 10. In that case, we can say that 10, 100, 1000, etc. are positive powers of 10:&lt;/p&gt;&#xA;&lt;p&gt;10&lt;sup&gt;1&lt;/sup&gt; = 10&lt;br /&gt;&#xA;10&lt;sup&gt;2&lt;/sup&gt; = 100&lt;br /&gt;&#xA;10&lt;sup&gt;3&lt;/sup&gt; = 1000&lt;br /&gt;&#xA;etc.&lt;/p&gt;&#xA;&lt;p&gt;and that 1/10, 1/100, 1/1000, etc. are negative powers of 10 (as others have pointed out):&lt;/p&gt;&#xA;&lt;p&gt;10&lt;sup&gt;−1&lt;/sup&gt; = .1 = 1/10&lt;br /&gt;&#xA;10&lt;sup&gt;−2&lt;/sup&gt; = .01 = 1/100&lt;br /&gt;&#xA;10&lt;sup&gt;−3&lt;/sup&gt; = .001 = 1/1000&lt;br /&gt;&#xA;etc.&lt;/p&gt;&#xA;&lt;p&gt;By the way, I noticed elsewhere on this page that you didn't like the term &amp;quot;negative power&amp;quot;. Because .001&lt;sup&gt;-1/3&lt;/sup&gt; = 10, .001 = the negative one-third root of 10. However, we don't usually use negative fractional roots.&lt;/p&gt;&#xA;" OwnerDisplayName="MarcInManhattan" LastActivityDate="2022-04-23T23:14:17.757" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25160" PostTypeId="2" ParentId="25152" CreationDate="2022-04-24T17:25:43.293" Score="1" Body="&lt;p&gt;On a slide rule, we have the standard C and D scales. Slide rule operators are facile with the CI and DI scales. Here the &amp;quot;I&amp;quot; is for &amp;quot;inverse.&amp;quot; So etymologically speaking, &amp;quot;inverses&amp;quot; or &amp;quot;inverse powers&amp;quot; might be the best answer!&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2022-04-24T18:00:11.500" LastActivityDate="2022-04-24T18:00:11.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25161" PostTypeId="2" ParentId="25152" CreationDate="2022-04-24T23:52:51.270" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;More specifically, in this context, I'm looking for the opposite of the term &amp;quot;multiple&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think the words &amp;quot;&lt;strong&gt;divisions&lt;/strong&gt;&amp;quot; and &amp;quot;&lt;strong&gt;subdivisions&lt;/strong&gt;&amp;quot; may be the terms you seek.&lt;/p&gt;&#xA;&lt;p&gt;In the Imperial system, quantities are commonly subdivided by powers of two.&#xA;We typically divide an inch into half-inches, quarter-inches, eighths, sixteenths, etc...&lt;/p&gt;&#xA;&lt;p&gt;In the Metric system, of course, it is subdivided by powers of ten... divisions into ten equal parts.&lt;/p&gt;&#xA;" OwnerUserId="10537" LastActivityDate="2022-04-24T23:52:51.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25162" PostTypeId="2" ParentId="25146" CreationDate="2022-04-25T00:26:01.600" Score="6" Body="&lt;p&gt;Since I teach physics, where units are very important,&#xA;I would suggest considering &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to be a length (carrying units of meters).&lt;/p&gt;&#xA;&lt;p&gt;The left-hand-side interpreted like an area has units of &lt;span class=&quot;math-container&quot;&gt;$m^2$&lt;/span&gt;.&lt;BR&gt;&#xA;The &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; carries units of &lt;span class=&quot;math-container&quot;&gt;$m^2$&lt;/span&gt;... and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; carries units of &lt;span class=&quot;math-container&quot;&gt;$m^2$&lt;/span&gt;.&lt;BR&gt;&#xA;So, &lt;span class=&quot;math-container&quot;&gt;$C dx$&lt;/span&gt; would carry units of &lt;span class=&quot;math-container&quot;&gt;$m^3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, &lt;span class=&quot;math-container&quot;&gt;$x^2 + C dx$&lt;/span&gt; would be inconsistent.&lt;/p&gt;&#xA;&lt;p&gt;(A geometric construction to interpret the integral as an area may help support the units interpretation.)&lt;/p&gt;&#xA;&lt;p&gt;(The student may need a reminder that the point of the constant of integration&#xA;is that the derivative of that term should be zero. Having &lt;span class=&quot;math-container&quot;&gt;$C dx$&lt;/span&gt; would be a strange construction for that purpose.&lt;/p&gt;&#xA;&lt;p&gt;In other words, a &amp;quot;check&amp;quot; of the integration could be to &amp;quot;take the derivative of the right-hand-side&amp;quot;.)&lt;/p&gt;&#xA;" OwnerUserId="10537" LastActivityDate="2022-04-25T00:26:01.600" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25163" PostTypeId="1" CreationDate="2022-04-25T16:15:18.183" Score="6" ViewCount="467" Body="&lt;p&gt;You know the error, when you're watching a student work through an algebraic calculation to solve for a variable trapped in the argument of a function, usually &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; or a trig function, and you watch them write this:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\log x = 42 \qquad\text{so}\qquad x = \frac{42}{\log}&#xA;$$&lt;/span&gt;&#xA;This error signals a distressing misunderstanding of what a &lt;em&gt;function&lt;/em&gt; is, but I only seen to catch students making this error weeks after we've talked about function fundamentals. My goal is to identify students who'd make this error early. What sort of questions can I ask to elicit this error in reasoning so I can intervene early?&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="205" LastEditDate="2022-05-17T18:40:00.243" LastActivityDate="2022-05-17T18:40:00.243" Title="How can you elicit the $\log x = {\log} \cdot x$ error?" Tags="|algebra|students-mistakes|precalculus|functions|problem-design|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25165" PostTypeId="1" CreationDate="2022-04-25T23:40:24.890" Score="5" ViewCount="284" Body="&lt;p&gt;Are there any collections or blogs of bad data visualizations? I know I've seen such misleading graphics in the news or in print, made misleading either maliciously or thoughtlessly, but I usually see these as one-offs on social media. Has anyone started a collection of these to use as examples in their math/stats classes?&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2022-04-27T14:07:12.650" Title="Where can I find a collection of bad/misleading data visualizations?" Tags="|reference-request|statistics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25167" PostTypeId="2" ParentId="25163" CreationDate="2022-04-25T23:55:20.850" Score="1" Body="&lt;p&gt;Function concept and various types of symbols and terms (e.g. log) are new to the students.  It is not uncommon for weaker ones to struggle.  The solution is not some secret aha revelation, not some change to convention, log(x) vice logx.  But rather drill and repetition and corrective feedback for mistakes and praise for success.  These things are new and strange to the kids.  It's like learning a language.  You need practice.&lt;/p&gt;&#xA;" OwnerUserId="19795" LastActivityDate="2022-04-25T23:55:20.850" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25168" PostTypeId="1" CreationDate="2022-04-26T12:43:28.157" Score="12" ViewCount="619" Body="&lt;p&gt;The math department at my institution (a private, four-year college with a total enrollment of about 4000) is in the process of brainstorming about a dedicated study/community space for our math majors (we have about 20-25 total).  We would like to create something that is tasteful, functional, encourages collaboration, and reinforces the STEM identity of our students.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have suggestions or advice about what has worked well (or not!) in their own various settings?  Any commentary, pictures, or ideas are very welcome--thanks in advance for whatever insight you might provide!&lt;/p&gt;&#xA;" OwnerUserId="5988" LastActivityDate="2022-07-20T03:52:22.107" Title="What are best practices for building a dedicated space for mathematics majors?" Tags="|undergraduate-education|" AnswerCount="7" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25169" PostTypeId="2" ParentId="25163" CreationDate="2022-04-26T13:18:02.147" Score="8" Body="&lt;p&gt;Ask the student to critique this work:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x} = 3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Easy: &lt;span class=&quot;math-container&quot;&gt;$x = \frac{3}{\sqrt{\phantom{x}}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have tried this a small number of times, and it has worked so far. The students recognize that the &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\phantom{x}}$&lt;/span&gt; symbol on its own is not a valid mathematical object, like the log in your question. This gives you a way to connect your issue to something they already know.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2022-04-26T13:18:02.147" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25170" PostTypeId="2" ParentId="12924" CreationDate="2022-04-26T13:30:06.833" Score="2" Body="&lt;p&gt;Give a regular quiz similar to this one.  Add questions as new data types become available.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f : \mathbb{R}^2 \to \mathbb{R}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g: \mathbb{R} \to \mathbb{R}^2$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$h: \mathbb{R}^2 \to \mathbb{R}^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For each of the following expressions, indicate which of the following options best represents the type of the expression:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;expression does not make sense&lt;/li&gt;&#xA;&lt;li&gt;real number&lt;/li&gt;&#xA;&lt;li&gt;vector&lt;/li&gt;&#xA;&lt;li&gt;vector field&lt;/li&gt;&#xA;&lt;li&gt;function of type &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \to \mathbb{R}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;function of type &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2 \to \mathbb{R}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;function of type &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \to \mathbb{R}^2$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(2,3)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(2,t)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(2)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$g(2,3)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$g(2)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$g \circ h$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$h \circ g$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$g(1)\cdot f(1,1)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$g'(1) \cdot \nabla f\big|_{(1,1)}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\nabla f\big|_{(1,2)} \cdot \langle 4, 5\rangle$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$Dh(g'(t))$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$Df(g(t))$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$Dg(f(t))$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2022-04-26T14:20:46.143" LastActivityDate="2022-04-26T14:20:46.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25171" PostTypeId="2" ParentId="25165" CreationDate="2022-04-26T14:34:56.487" Score="2" Body="&lt;p&gt;Check out the book &lt;a href=&quot;https://www.tylervigen.com/spurious-correlations&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Spurious Correlations&lt;/em&gt; by Tyler Vigen&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="5220" LastEditDate="2022-04-26T16:16:54.177" LastActivityDate="2022-04-26T16:16:54.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25172" PostTypeId="1" CreationDate="2022-04-26T14:54:32.690" Score="0" ViewCount="1065" Body="&lt;p&gt;What would be a good books of learning differential equations for a student who likes to learn things rigorously and has a good background on analysis and topology?&lt;/p&gt;&#xA;" OwnerUserId="19799" LastEditorUserId="470" LastEditDate="2022-05-21T18:18:32.210" LastActivityDate="2022-05-21T18:18:32.210" Title="A good way to learn differential equations rigorously" Tags="|differential-equations|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25173" PostTypeId="1" AcceptedAnswerId="25177" CreationDate="2022-04-26T17:14:21.733" Score="5" ViewCount="2687" Body="&lt;p&gt;I am teaching precalculus/basic calculus to my class (high schoolers of around 18 years of age), and I'm always searching for nice functions to plot and study (finding the domain, the function's sign, the zeros, derivatives, limits...), that are not too easy and not too hard on the kids with the computation of the derivatives. What are in your opinion and/or experience some good (possibily general) examples of functions that give students some head scratching but not really a headache?&#xA;In my limited experience, coming up with nicely plottable and nicely differentiable functions is not really straightforward.&lt;/p&gt;&#xA;" OwnerUserId="6286" LastEditorUserId="6286" LastEditDate="2022-04-28T09:58:17.337" LastActivityDate="2023-03-08T04:42:38.017" Title="What are some examples of great functions that are not too elementary (easy)?" Tags="|precalculus|functions|derivative|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25174" PostTypeId="2" ParentId="25173" CreationDate="2022-04-26T18:08:24.390" Score="5" Body="&lt;p&gt;Suggestions.  I took your requirement &amp;quot;not elementary&amp;quot; in the technical sense.&lt;/p&gt;&#xA;&lt;p&gt;Bessel functions, say &lt;span class=&quot;math-container&quot;&gt;$J_0(x)$&lt;/span&gt;, and other special functions of mathematical physics.  But since your students have not studied differential equations, maybe not a good choice.&lt;/p&gt;&#xA;&lt;p&gt;Gamma function &lt;span class=&quot;math-container&quot;&gt;$\Gamma(x)$&lt;/span&gt;.  How to do plotting without knowing more advanced mathematics?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lambert_W_function&quot; rel=&quot;noreferrer&quot;&gt;Lambert W function&lt;/a&gt;.  Here the inverse function &lt;span class=&quot;math-container&quot;&gt;$xe^x$&lt;/span&gt; is elementary, so can be used for plotting.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathworld.wolfram.com/SineIntegral.html&quot; rel=&quot;noreferrer&quot;&gt;Sine integral function&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Si}(x)$&lt;/span&gt;.  It is not elementary, but its derivative &lt;span class=&quot;math-container&quot;&gt;$\frac{\sin x}{x}$&lt;/span&gt; is, so that can be used for plotting.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2022-04-26T18:08:24.390" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25175" PostTypeId="2" ParentId="25172" CreationDate="2022-04-26T18:16:41.500" Score="5" Body="&lt;p&gt;A classic book is Coddington &amp;amp; Levinson, &lt;em&gt;Theory of Ordinary Differential Equations&lt;/em&gt;.  Probably tough going for most students.&lt;/p&gt;&#xA;&lt;p&gt;Coddington, &lt;em&gt;An Introduction to Ordinary Differential Equations.&lt;/em&gt;  Much shorter, suited to undergraduates, but still rigorous.  There is an inexpensive reprinting from Dover.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2022-04-26T18:16:41.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25176" PostTypeId="2" ParentId="25172" CreationDate="2022-04-26T20:59:20.317" Score="4" Body="&lt;p&gt;There's nothing wrong with learning it rigorously, but I would recommend to learn it &amp;quot;non-rigorously&amp;quot; first and then rigorously.  E.g. in the context of a &amp;quot;graduate&amp;quot; ODE course.  That's because there are some very important aspects of the topic that you may miss if you only concentrate on rigor, first.  Like an intuitive understanding of the 2nd order ODE and the whole forcing function, undamped, overdamped, etc.  And the sources that concentrate on rigor often assume that you have the non-rigorous understanding...not just that it's harder without the background, but that you never learn some key insights.&lt;/p&gt;&#xA;&lt;p&gt;In that vein, a couple texts that I like that will still give you the basics and are slightly rigorous are Murray Spiegel Applied Differential Equations (3rd edition) and Ordinary Differential Equations Tenenbaum and Pollard.  After that, I would move to a graduate ODE text, probably something in Springer, for the theory.&lt;/p&gt;&#xA;&lt;p&gt;But really, you are missing a major trick if you only consider ODEs (or worse PDEs) in the context of analysis/topo, with zero physical insights or references to the rich science and engineering applications.&lt;/p&gt;&#xA;" OwnerUserId="19801" LastActivityDate="2022-04-26T20:59:20.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25177" PostTypeId="2" ParentId="25173" CreationDate="2022-04-26T21:47:32.070" Score="19" Body="&lt;p&gt;I'm personally a fan of simple examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x e^x$&lt;/span&gt; (has nice critical point, point of inflection)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$e^{-x^2}$&lt;/span&gt; (with appropriate rescaling, the normal distribution from statistics)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{x}{x^2+1}$&lt;/span&gt; (a serpentine curve)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$3x^5-5x^3$&lt;/span&gt; and other differences of two monomials.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x^n + y^n = 1$&lt;/span&gt; (you can solve for y as a function of x. Students should have seen that tangents to circles are perpendicular to radii, you can talk about squircles in the case that n = 4, and n = 3 is also interesting. A patient student who knows about conjugates can work the derivative of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{1-x^2}$&lt;/span&gt; from the limit definition.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-04-26T21:47:32.070" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25178" PostTypeId="2" ParentId="25173" CreationDate="2022-04-27T11:25:29.777" Score="12" Body="&lt;p&gt;Don't forget purely graphical problems.  Give a graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt;, ask for the  student to sketch the other one.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-04-27T11:25:29.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25179" PostTypeId="2" ParentId="25168" CreationDate="2022-04-27T11:27:40.820" Score="11" Body="&lt;p&gt;Chalkboards. Hagoromo chalk. Some small (individual) study desks and some round tables - with acceptably comfortable chairs. A couple of couches and living-room-type chairs.&lt;/p&gt;&#xA;&lt;p&gt;POSTERS OF DIVERSE MATHEMATICIANS ON THE WALLS - &lt;a href=&quot;https://www.ams.org/posters&quot; rel=&quot;noreferrer&quot;&gt;the American Mathematical Society has a list on their website&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="19805" LastEditorUserId="80" LastEditDate="2022-04-29T19:01:55.660" LastActivityDate="2022-04-29T19:01:55.660" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25180" PostTypeId="2" ParentId="25168" CreationDate="2022-04-27T12:09:50.123" Score="11" Body="&lt;p&gt;Along with what others recommended, I would add some simple food prep options (microwave, fridge, sink, coffee-maker, cabinets).  We are physical beings.  It will encourage usage of the space.&lt;/p&gt;&#xA;" OwnerUserId="19806" LastActivityDate="2022-04-27T12:09:50.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25181" PostTypeId="2" ParentId="25165" CreationDate="2022-04-27T14:07:12.650" Score="2" Body="&lt;p&gt;Are you interested in resources in languages other than English? Bad data visualization has no nationality.&lt;/p&gt;&#xA;&lt;p&gt;There's a blog &lt;a href=&quot;http://smarterpoland.pl/&quot; rel=&quot;nofollow noreferrer&quot;&gt;SmarterPoland.pl&lt;/a&gt; (in Polish) devoted to good and bad data visualization. It runs an annual (anti-)competition for the worst data chart. &lt;a href=&quot;http://smarterpoland.pl/index.php/2021/12/plebiscyt-na-najgorszy-wykres-roku-2021/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; is the 2021 edition, along with the links to previous editions. It doesn't take (almost) any knowledge of Polish to understand how bad these charts are.&lt;/p&gt;&#xA;" OwnerUserId="15367" LastActivityDate="2022-04-27T14:07:12.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25182" PostTypeId="1" CreationDate="2022-04-27T15:27:34.713" Score="9" ViewCount="298" Body="&lt;p&gt;In my own personal experience in teaching linear algebra, where many students encounter abstract ideas for the first time, I find that most students have trouble consolidating observations from concrete examples into understanding of abstract ideas.&lt;/p&gt;&#xA;&lt;p&gt;Very often, after giving many examples, students understood the examples very well, but they can't say anything beyond that.&lt;/p&gt;&#xA;&lt;p&gt;Take the concept of subspace, for example: Many students can know 5 examples of subspaces of vector spaces very well, yet cannot answer the most basic question about any subspace they have not studied. E.g., &amp;quot;if we have a set of functions in the vector space of continuous functions over [a,b], how can we check if this set form a subspace?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I wonder if this a common problem:&#xA;Are any empirical studies or observations derived from large samples on how well average undergraduate students can learn abstract concepts through concrete examples.&lt;/p&gt;&#xA;" OwnerUserId="19807" LastActivityDate="2022-04-27T23:40:48.597" Title="How well can students learn abstract concepts through concrete examples?" Tags="|examples|concept-motivation|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25183" PostTypeId="2" ParentId="25182" CreationDate="2022-04-27T23:13:25.390" Score="9" Body="&lt;p&gt;The following article in &lt;em&gt;Science&lt;/em&gt; has now been cited 388 times according&#xA;to Google Scholar.&#xA;I am not in a position to assess this paper or to characterize all the subsequent discussion,&#xA;but it seems quite relevant to the OP's question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kaminsky, J.A., Sloutsky, V.M., &amp;amp; Heckler, A.F. The advantage of abstract examples in learning math. &lt;em&gt;Science&lt;/em&gt;, 320, no. 5875 (2008): 454-455.&#xA;&lt;a href=&quot;https://www.science.org/doi/10.1126/science.1154659&quot; rel=&quot;noreferrer&quot;&gt;DOI&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Undergraduate students may benefit more&#xA;from learning mathematics through a single&#xA;abstract, symbolic representation than from&#xA;learning multiple concrete examples.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;What we are suggesting is that grounding&#xA;mathematics deeply in concrete contexts can&#xA;potentially limit its applicability. Students&#xA;might be better able to generalize mathematical concepts to various situations if the concepts have been introduced with the use of&#xA;generic instantiations.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/l8oYc.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/l8oYc.gif&quot; alt=&quot;figure&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2022-04-27T23:40:48.597" LastActivityDate="2022-04-27T23:40:48.597" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25184" PostTypeId="2" ParentId="25163" CreationDate="2022-04-28T10:51:57.763" Score="10" Body="&lt;p&gt;Writing from a software engineer's point of view, it's a fact that mathematics uses a notation that's highly ambiguous.&lt;/p&gt;&#xA;&lt;p&gt;If you don't know that &lt;span class=&quot;math-container&quot;&gt;$log$&lt;/span&gt; is used to denote some logarithm function, then seeing a term like &lt;span class=&quot;math-container&quot;&gt;$log x$&lt;/span&gt; might as well mean the product of &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Even writing &lt;span class=&quot;math-container&quot;&gt;$log(x)$&lt;/span&gt; doesn't really solve the problem, it can still be the product of &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;. That is because adjacency of symbols can mean a lot of different things:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;it can mean multiplication (e.g. &lt;span class=&quot;math-container&quot;&gt;$2ab$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$2 \cdot a \cdot b$&lt;/span&gt;),&lt;/li&gt;&#xA;&lt;li&gt;it can form a longer word (e.g. &lt;span class=&quot;math-container&quot;&gt;$log$&lt;/span&gt;),&lt;/li&gt;&#xA;&lt;li&gt;it can mean function application (e.g. &lt;span class=&quot;math-container&quot;&gt;$logx$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$log(x)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;That makes it hard for the math teacher. When introducing functions, the students not only have to grasp that new concept, but also the peculiarities of the new notation:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There are multi-letter words now (e.g. &lt;span class=&quot;math-container&quot;&gt;$log$&lt;/span&gt;), whereas up to that point, mathematical entities were denoted by single letters (sometimes with some decoration).&lt;/li&gt;&#xA;&lt;li&gt;A series of some letters no longer necessarily means multiplication, but can denote function application, if the initial letters name a function.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As a teacher, you can't change the mathematics world to use a less ambiguous notation, so you can only teach your students formula understanding.&lt;/p&gt;&#xA;&lt;p&gt;I'd have them translate terms from the standard shorthand math notation into the most explicit possible one, e.g. that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$3\log x \log y - \log z$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;effectively is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\bigl[3 \cdot \log(x) \cdot \log(y) \bigr] - \log(z)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;By the way, when tutoring (weaker) math students, I often found that they lacked a thorough understanding of a mathematical term, so doing exercises on that topic from time to time might help many of them.&lt;/p&gt;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&lt;p&gt;One more problem comes with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Order_of_operations&quot; rel=&quot;noreferrer&quot;&gt;order of operations&lt;/a&gt; for function names, as that isn't obvious: &lt;span class=&quot;math-container&quot;&gt;$sin 2 \alpha$&lt;/span&gt; is meant to be &lt;span class=&quot;math-container&quot;&gt;$sin(2 \cdot \alpha)$&lt;/span&gt; and not &lt;span class=&quot;math-container&quot;&gt;$sin(2) \cdot \alpha$&lt;/span&gt;, but where are the rules defining that? And surely &lt;span class=&quot;math-container&quot;&gt;$sin \alpha + sin \beta$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$sin(\alpha) + sin(\beta)$&lt;/span&gt; and not &lt;span class=&quot;math-container&quot;&gt;$sin(\alpha + sin(\beta))$&lt;/span&gt;, but why?&lt;/p&gt;&#xA;&lt;p&gt;From these examples, function application seems to be higher in the order of operations than addition, but lower than multiplication. But there are counter-examples as well, e.g. &lt;span class=&quot;math-container&quot;&gt;$\log x \log y$&lt;/span&gt; isn't &lt;span class=&quot;math-container&quot;&gt;$\log (x \cdot \log (y))$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Is this ever taught explicitly, like the rule that multiplication is higher than addition? I doubt that it's even possible to give a rigorous definition how to read a math expression, without ambiguities and guesswork.&lt;/p&gt;&#xA;" OwnerUserId="15862" LastEditorUserId="15862" LastEditDate="2022-04-29T09:38:48.763" LastActivityDate="2022-04-29T09:38:48.763" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25185" PostTypeId="2" ParentId="25168" CreationDate="2022-04-28T13:29:40.887" Score="6" Body="&lt;p&gt;We have a room for this purpose at UMich. One nice feature besides the ones already mentioned is that it has blackboards and a projection screen suitable for hosting speakers. Here is Prof. Lagarias giving a talk to our math club:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/p6puo.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/p6puo.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some photos when the room isn't full. Note couches, tables, textboooks and books for pleasure, copies of the American Mathematical Monthly.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JwMZE.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JwMZE.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EAn6m.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EAn6m.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2022-07-18T18:33:18.580" LastActivityDate="2022-07-18T18:33:18.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25186" PostTypeId="2" ParentId="25168" CreationDate="2022-04-28T13:37:32.920" Score="7" Body="&lt;p&gt;Here is an issue to think through in advance: When will the room be open, and what are the policies about sleeping in the room?&lt;/p&gt;&#xA;&lt;p&gt;I've been on a bunch of campuses that had lounges like this (Harvard, Berkeley, UMich, MIT). They all tended to acquire students who worked late at night, leading to the questions above.&lt;/p&gt;&#xA;&lt;p&gt;As a student, I definitely resented being kicked out of lounges at 5 PM, or 12 AM, or whenever. And I think it caused a problem for female students, in that it moved study groups out of the public lounge spaces into (mostly male) students dorm rooms at night. At the same time, as an older adult, I can see why the university didn't want to be responsible for keeping a space safe at all hours of the night and why they want to encourage students to go to bed at a reasonable hour. And I have heard female students saying it feels unsafe to work in an area where male students are lying out sleeping on the couches. The Michigan lounge is open from 8 AM to 5 PM.&lt;/p&gt;&#xA;&lt;p&gt;Any way, I don't have a strong opinion about the right way to handle this, but it is something that may come up.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="6053" LastEditDate="2022-05-01T17:22:08.183" LastActivityDate="2022-05-01T17:22:08.183" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25187" PostTypeId="2" ParentId="25168" CreationDate="2022-04-29T02:52:38.957" Score="2" Body="&lt;p&gt;Zometool (&lt;a href=&quot;https://www.zometool.com/products/the-steam-kit.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.zometool.com/products/the-steam-kit.html&lt;/a&gt;) if you can trust students not to walk off with it. Papercraft solids (platonic, archimedian, etc) if you can't.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-04-29T02:52:38.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25188" PostTypeId="1" AcceptedAnswerId="25191" CreationDate="2022-05-01T10:49:14.677" Score="-1" ViewCount="324" Body="&lt;p&gt;Standard definition of writing interval states that it should be written (a,b) where a&amp;lt;b&lt;/p&gt;&#xA;&lt;p&gt;Due to this being arbitrary and just a convention that we all use, would it be considered a mistake to write it with non standard notation (b,a) where b&amp;gt;a?&lt;/p&gt;&#xA;&lt;p&gt;I am interested in hearing opinions of those who teach/grade students.In my opinion points should not be deducted for writing interval this way if the student is able to explain the reasoning behind it.&lt;/p&gt;&#xA;" OwnerUserId="19811" LastActivityDate="2022-05-06T00:03:46.960" Title="Is it considered a mistake to use different correct notation for writing intervals?" Tags="|calculus|set-theory|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25189" PostTypeId="2" ParentId="25188" CreationDate="2022-05-01T12:06:36.030" Score="6" Body="&lt;p&gt;In theory you could choose to define a notation such as &lt;span class=&quot;math-container&quot;&gt;$$(a, b) = \{x ; b &amp;lt; x &amp;lt; a\},$$&lt;/span&gt;&#xA;but why would you do it? It has exactly the same power as the usual notation, and now the reader either has to remind themself what you mean all the time or make mistakes due to going against a convention.&lt;/p&gt;&#xA;&lt;p&gt;If there is a wider context where this notation is more in harmony with other notation of the subfield, then there is some justification for it. There might be some other niche situations.&lt;/p&gt;&#xA;&lt;p&gt;This is especially bad since &lt;span class=&quot;math-container&quot;&gt;$(a, b)$&lt;/span&gt; might be an non-empty interval or an empty interval depending on the convention, and both cases might come up in a calculation or a proof.&lt;/p&gt;&#xA;&lt;p&gt;At the minimum I would advise using some alternate notation for open intervals, such as &lt;span class=&quot;math-container&quot;&gt;$]a, b[$&lt;/span&gt;, for the reversed interval notation. But again, what is the benefit of breaking convention?&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-05-01T12:06:36.030" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25190" PostTypeId="2" ParentId="25168" CreationDate="2022-05-01T13:03:53.303" Score="3" Body="&lt;p&gt;We had a really awesome math masters room in my previous university. I definitely agree that a kitchenette area with a hot water dispenser/water/sink, and microwave if you can put the kitchenette in a separate room (don't want the whole room smelling) is great. White boards and/or chalk board are awesome. Other things that we appreciated:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Adjoining rooms where you could go for collaboration, so that the main space is primarily for independent work.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a printer/copier etc Stationary supplies.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A space with bean bags/comfy sofa for people to chill and think and let their mind wander.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Lockers. If someone needs use of one, they can contact the office admin to arrange it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A poster board with summer internship/graduate opportunities, conferences etc. Information about sholarships and funding.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Restrooms close by.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some awesome geeky STEM things to play with. We had a 3D printer and lego at our disposal. The 3D printer already came with the code for a lot of math-related objects, so we could print Klein bottles, Tori etc&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19813" LastActivityDate="2022-05-01T13:03:53.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25191" PostTypeId="2" ParentId="25188" CreationDate="2022-05-01T13:20:14.650" Score="11" Body="&lt;p&gt;The goal of notation is to clearly and precisely communicate an idea.  The notation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (a,b) = \{ x : b &amp;lt; x &amp;lt; a \} $$&lt;/span&gt;&#xA;fails to clearly communicate an idea, as this notation is unusual, and conflicts with the the ordinary notation for an open interval.  As such, I would avoid this notation in the setting of precalculus, calculus, and real analysis, and would deduct (a small number of) points from a student who used this ambiguous notation.&lt;/p&gt;&#xA;&lt;p&gt;It might, however, be worth noting that the notation &lt;span class=&quot;math-container&quot;&gt;$[\alpha, \beta]$&lt;/span&gt; is sometimes used in complex analysis to denote the segment&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ [\alpha, \beta] = \{ (1-t)\alpha + t\beta :  0 \le t \le 1 \} \subseteq \mathbb{C}, $$&lt;/span&gt;&#xA;i.e. a parameterized, straight-line path from &lt;span class=&quot;math-container&quot;&gt;$\alpha \in \mathbb{C}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\beta\in \mathbb{C}$&lt;/span&gt;.  In this context, it is possible for &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; to both be real numbers with &lt;span class=&quot;math-container&quot;&gt;$\alpha &amp;gt; \beta$&lt;/span&gt;.  Even in this case, though, the order in which the two numbers are written matters:  the segment is oriented, in that the &amp;quot;initial point&amp;quot; is &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, and the &amp;quot;terminal point&amp;quot; is &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2022-05-01T13:20:14.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25192" PostTypeId="1" AcceptedAnswerId="25195" CreationDate="2022-05-01T16:18:59.837" Score="1" ViewCount="725" Body="&lt;p&gt;What is the most appropriate way to write equation of a straight line through a given point &lt;span class=&quot;math-container&quot;&gt;$(\alpha,\beta)$&lt;/span&gt;? &lt;br/&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you write it using the standard form &lt;br/&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$y- \beta = m (x - \alpha )$&lt;/span&gt; &lt;br/&gt;&#xA;Where m is the gradient of the line. &lt;br/&gt;&#xA;There is a problem of how to obtain equation of the line when it is vertical.&lt;br/&gt;&#xA;But if you write it as &lt;br/&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$b(y- \beta)= a(x - \alpha )$&lt;/span&gt; , here you have chance of obtaining equation for any situation. &lt;br/&gt;&#xA;I have come across this issue when finding equation of tangents to a given circle. What do you think about this? &lt;br/&gt;&lt;/p&gt;&#xA;&lt;p&gt;Added for further clarification, &lt;br/&gt;&#xA;Think about the equation of the tangent drawn at the point &lt;span class=&quot;math-container&quot;&gt;$(\alpha,\beta)$&lt;/span&gt; to the circle &lt;span class=&quot;math-container&quot;&gt;$x^2 + y^2 +2gx +2fy + c =0$&lt;/span&gt; , where &lt;span class=&quot;math-container&quot;&gt;$(\alpha,\beta)$&lt;/span&gt; is a point on the given circle. &lt;br/&gt;&#xA;Here you have the standard form, &lt;br/&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$x\alpha + y\beta + g(x+\alpha)+f(y+\beta) +c = 0$&lt;/span&gt; and by this you can get vertical tangents too. Do you know why ? &lt;br/&gt;&#xA;I suggest the reason is here to derive this standard form you are using something similar to &lt;span class=&quot;math-container&quot;&gt;$b(y - \beta)  = a( x - \alpha)$&lt;/span&gt;. &lt;br/&gt;&#xA;Therefore I think when we need to find equation of a straight line through a given point much better way is to use the form,&lt;br/&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$b(y - \beta)  = a( x - \alpha)$&lt;/span&gt;. &lt;br/&gt;&#xA;Otherwise you need to check whether&#xA;x = α is another possibility or not.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2022-05-02T13:20:14.060" LastActivityDate="2022-05-03T02:14:40.767" Title="Equation of a straight line on two dimensional Cartesian plane" Tags="|secondary-education|analytic-geometry|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25193" PostTypeId="2" ParentId="25188" CreationDate="2022-05-01T17:51:36.467" Score="-2" Body="&lt;p&gt;I just looked this up in a book of mine (Thomas Finney, 1980).  The convention is all greater than the first (a) and smaller than the second (b).  So, if your first number is greater than the second one, the result is the empty set, not just some backwards-directed interval.  This is not just arbitrary.  It allows doing various manipulations.&lt;/p&gt;&#xA;" OwnerUserId="19815" LastActivityDate="2022-05-01T17:51:36.467" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25194" PostTypeId="2" ParentId="25192" CreationDate="2022-05-01T19:37:39.610" Score="3" Body="&lt;p&gt;@TomKern mentioned the &amp;quot;standard&amp;quot; form above. I believe what he has in mind is ax+by=c, which allows for both horizontal and vertical lines.&lt;/p&gt;&#xA;&lt;p&gt;Another form, rarely used, but more like the equations of conics, is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{x}{a}+\frac{y}{b}=1$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-05-02T15:25:53.720" LastActivityDate="2022-05-02T15:25:53.720" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25195" PostTypeId="2" ParentId="25192" CreationDate="2022-05-01T20:55:19.333" Score="6" Body="&lt;p&gt;Assuming we are talking about an algebra class because of the secondary-education tag, the reason it is okay to use &amp;quot;point-slope form,&amp;quot; i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$y - y_1 = m(x - x_1)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;as the default way to write the equation of a line is that the two most important uses of lines do not care about vertical lines at all:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Using the tangent line as the linear approximation to a function in calculus&lt;/li&gt;&#xA;&lt;li&gt;Using a line of best fit correlating two variables in statistics&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;You could certainly do other things if you wanted. But is it worth it to use a nonstandard format to catch a relatively unimportant corner case? I don't think so.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2022-05-01T20:55:19.333" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25196" PostTypeId="1" AcceptedAnswerId="25197" CreationDate="2022-05-02T10:11:29.070" Score="20" ViewCount="6021" Body="&lt;p&gt;I came across &lt;a href=&quot;https://www.youtube.com/watch?v=loKnG__5p6Q&quot; rel=&quot;noreferrer&quot;&gt;this video&lt;/a&gt; explaining to kids why the volume of a cone is a third of the cylinder of same cross-sectional radius and height. Essentially the explainer presents pre-created cylindrical and conical objects with equal dimensions, and says it takes three cones to carry the same amount of water as the cylinder.&lt;/p&gt;&#xA;&lt;p&gt;Typically this video is intended for pre-calculus students, aged around 13-14. I understand that calculus is necessary to rigorously prove the statement. I also understand that sometimes there are different varieties of &amp;quot;proofs&amp;quot; - geometric proofs or proofs by construction. Does this explanation fall into an acceptable category of proof? Or is it simply nothing more than a mnemonic to remember the volume formulas?&lt;/p&gt;&#xA;" OwnerUserId="13747" LastActivityDate="2022-05-05T14:47:30.463" Title="Explaining why volume of cone is a third of cylinder" Tags="|secondary-education|geometry|proofs|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25197" PostTypeId="2" ParentId="25196" CreationDate="2022-05-02T11:18:44.730" Score="30" Body="&lt;p&gt;This is an experiment which can lead you to guess that the volume of a cone is approximately &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}$&lt;/span&gt; the volume of a cylinder with the same base and height.  It is not a proof in any sense of the word.&lt;/p&gt;&#xA;&lt;p&gt;If mathematics were a &lt;strong&gt;science&lt;/strong&gt; then we might have a &amp;quot;theory of the volume of cones&amp;quot; which would be confirmed by repeated experimentation with more and more exacting experimental set ups.  However, mathematics is not a science:  it is a deductive art. This is, I think, a valuable conversation to have with students.&lt;/p&gt;&#xA;&lt;p&gt;It is possible to justify this fact without Calculus if you accept Cavalieri's principle.&lt;/p&gt;&#xA;&lt;p&gt;First make a special pyramid. Use an &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; square pyramid with height &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, where the apex is located directly over one corner of the base.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.polyhedra.net/en/model.php?name-en=three-pyramids-that-form-a-cube&quot; rel=&quot;noreferrer&quot;&gt;https://www.polyhedra.net/en/model.php?name-en=three-pyramids-that-form-a-cube&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can see that 3 such pyramids form a cube, so the volume of this square based pyramid is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}a^3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now scaling by &lt;span class=&quot;math-container&quot;&gt;$\frac{h}{a}$&lt;/span&gt; vertically gives that the &amp;quot;stretched&amp;quot; pyramid with square base &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and height &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; has volume &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}a^2h$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now take any arbitrary cone with base of area &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and height &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;.  Compare this to the pyramid whose base is a &lt;span class=&quot;math-container&quot;&gt;$\sqrt{B}$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{B}$&lt;/span&gt; square and whose height is &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;.  Using similarity, you can show that these have equal area cross sections.  So by Cavalieri's principle, they have equal volume.  Thus the volume of the cone is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}Bh$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-05-02T11:18:44.730" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="25199" PostTypeId="2" ParentId="5614" CreationDate="2022-05-02T12:48:27.730" Score="1" Body="&lt;p&gt;Here is a more general framework which explains (mathematically at least) what is going on.  This would need to be adapted into age appropriate lessons if it were to be used.  The basic idea is similar to the currently accepted answer, but hopefully the increased precision will be clarifying.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$F: \mathbb{R}^2 \to \mathbb{R}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$T: \mathbb{R}^2 \to \mathbb{R}^2$&lt;/span&gt; be two functions, with &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; a bijection.&lt;/p&gt;&#xA;&lt;p&gt;We are interested in how the graph &lt;span class=&quot;math-container&quot;&gt;$\Gamma_F$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$F(x,y) = 0$&lt;/span&gt; compares with the graph &lt;span class=&quot;math-container&quot;&gt;$\Gamma_{F \circ T}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$F(T(x,y)) = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The answer is quite clear:  &lt;span class=&quot;math-container&quot;&gt;$(a,b) \in  \Gamma_F \Longleftrightarrow T^{-1}(a,b) \in \Gamma_{F \circ T}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let us apply this to some concrete examples.&lt;/p&gt;&#xA;&lt;p&gt;We want to graph &lt;span class=&quot;math-container&quot;&gt;$y - 3 = (x+1)^2$&lt;/span&gt;.  We know the graph of &lt;span class=&quot;math-container&quot;&gt;$y = x^2$&lt;/span&gt;.  Rephrasing into the language used above, we know &lt;span class=&quot;math-container&quot;&gt;$\Gamma_{F}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$F(x,y) = y-x^2$&lt;/span&gt; and we want to draw &lt;span class=&quot;math-container&quot;&gt;$\Gamma_{F \circ T}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$T(x,y) = (x+1,y-3)$&lt;/span&gt;.  Since &lt;span class=&quot;math-container&quot;&gt;$T^{-1}(x,y) = (x-1,y+3)$&lt;/span&gt;, we have that for every point &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\Gamma_F$&lt;/span&gt;, the corresponding point on &lt;span class=&quot;math-container&quot;&gt;$\Gamma_{F \circ T}$&lt;/span&gt; will be &lt;span class=&quot;math-container&quot;&gt;$(a+1,b+3)$&lt;/span&gt;.  So &lt;span class=&quot;math-container&quot;&gt;$\Gamma_{F \circ T}$&lt;/span&gt; looks like the &lt;span class=&quot;math-container&quot;&gt;$\Gamma_F$&lt;/span&gt;, only shifted to the right by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and up by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A more interesting example.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; be as above.  Let &lt;span class=&quot;math-container&quot;&gt;$T(x,y) = \begin{bmatrix} \cos(t) &amp;amp; -\sin(t) \\ \sin(t) &amp;amp; \cos(t) \end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix}$&lt;/span&gt;.  &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; has the effect of rotating &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; radians &lt;strong&gt;anti-clockwise&lt;/strong&gt; about the origin.&lt;/p&gt;&#xA;&lt;p&gt;So the graph of &lt;span class=&quot;math-container&quot;&gt;$(F \circ T)(x,y) = 0$&lt;/span&gt; will look like the graph of &lt;span class=&quot;math-container&quot;&gt;$F(x,y) = 0$&lt;/span&gt; rotated &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; radians &lt;strong&gt;clockwise&lt;/strong&gt; about the origin.  See the following Desmos link for a demonstration.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/8pu9uzvfr5&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.desmos.com/calculator/8pu9uzvfr5&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In short:  the confusion is arising because we are really applying these transformations to level curves of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; variable functions.  The treatment of both variables is even-handed when viewed in this way:  we &lt;strong&gt;always&lt;/strong&gt; transform using the &amp;quot;inverse transformation&amp;quot;.  It is only when we are dealing with functions and insist on writing them &lt;span class=&quot;math-container&quot;&gt;$y = f(x)$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$F(x,y) = y-f(x) = 0$&lt;/span&gt; that we run into trouble.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-05-02T12:48:27.730" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25200" PostTypeId="2" ParentId="25196" CreationDate="2022-05-02T13:23:30.643" Score="11" Body="&lt;p&gt;One framework of understanding and building towards proof that we teach to mathematics teachers at primary school level is (and translated to English on fly):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Naive empiricism: The pupil tries an example, or maybe several, and checks if the claim holds for them.&lt;/li&gt;&#xA;&lt;li&gt;Crucial experiment: The pupil tries an example chose with care such that if something should be wrong with a claim, this example would show it. The cruciality is not formally proven, however.&lt;/li&gt;&#xA;&lt;li&gt;Generic example: Here one takes a representative example and uses it to show an argument that works for any example, but this generality is not formalized (by, for example, using variables).&lt;/li&gt;&#xA;&lt;li&gt;General argument: An idea of proof, a mathematician would say.&lt;/li&gt;&#xA;&lt;li&gt;Formal argument: A mathematical proof.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;There is some overlap between these, and some of these are also proof strategies when formalized.&lt;/p&gt;&#xA;&lt;p&gt;I think one to three are directly from&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Balacheff, N. (1988): Aspects of proof in pupils’ practice of school mathematics,.&#xA;En Pimm, D. (ed.), Mathematics, teachers and children (Hodder &amp;amp; Stoughton: Londres),&#xA;216-235.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;His fourth category might be a bit different than the ones I use here.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The video&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Looking at the video, the presenter demonstrates the claim with a single example and invites (claims) it to hold for others, too. So we are at least at the level of naive empiricism.&lt;/p&gt;&#xA;&lt;p&gt;We might argue that there is nothing special about these particular cylinders (they seem to be of different ratios of height to radius) and it is also unlikely that something whether this ratio is a rational number or not, or some other occult property, would have any effect. So maybe this is a generic example? But the video does not argue this; I am over-interpreting here. But not much.&lt;/p&gt;&#xA;&lt;p&gt;We did not see this done with a cylinder that has very extreme form, like ratio of height to radius being very small or very large. Neither did we see the exercise done with a great deal of accuracy. Had we seen such, maybe we could say that this amounts to a generic example: the claim would have been demonstrated with a usual, an extremely sharp and extremely dull object, and thus it would have seen likely that it also holds for other shapes. Alas.&lt;/p&gt;&#xA;&lt;p&gt;Se we certainly saw a sloppy example, and were implied to that this same should hold generally, so we are at least at naive empiricism and maybe, maybe, seeing a generic example.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-05-02T13:23:30.643" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25201" PostTypeId="2" ParentId="25196" CreationDate="2022-05-02T20:41:30.683" Score="6" Body="&lt;p&gt;Without Calculus, you can still be sort of rigorous. With Euclidean geometry we can find the centroid of a right triangle by finding the point where all the medians intersect.  A cardboard triangle will balance on a pin at its centroid.  We also know that the centroid cuts each median into 1/3 and 2/3's of its length.  (This is where the 1/3 comes in.)&lt;/p&gt;&#xA;&lt;p&gt;Second, we have the (second) Theorem of Pappus (which takes some sort of calculus to prove, but which would seem quite reasonable to elementary students)  which says that he volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by the geometric centroid of F.&lt;/p&gt;&#xA;&lt;p&gt;A cone is formed by rotating a right angle about one leg.  We can calculate its volume using these two theorems.&lt;/p&gt;&#xA;" OwnerUserId="19450" LastActivityDate="2022-05-02T20:41:30.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25202" PostTypeId="2" ParentId="25192" CreationDate="2022-05-03T02:07:01.907" Score="2" Body="&lt;h3&gt;TL;DR&lt;/h3&gt;&#xA;&lt;p&gt;I try not to get overly hung up on notation, and prefer to teach some kind of principle.  In this setting, the principle is one of geometric transformation using basic techniques.  &lt;em&gt;Many&lt;/em&gt; equations for lines result, including some of the standard guys, like&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ ax + by + c = 0 \qquad\text{and}\qquad y = mx + b. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;A student should be comfortable with &lt;em&gt;any&lt;/em&gt; equation for a line, and should understand the limitations (e.g. &lt;span class=&quot;math-container&quot;&gt;$y=mx+b$&lt;/span&gt; is great for seeing the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept and slope, but can't describe vertical lines; &lt;span class=&quot;math-container&quot;&gt;$ax + by + c = 0$&lt;/span&gt; is very general, but the underlying geometry is largely obfuscated; etc.).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In my precalculus classes, I generally start the discussion of functions with more general relations, i.e. subsets of the plane.  In set builder notation, I might define a set as follows:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ S = \{ (x,y) : \text{[some formula]} \}. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Translating or scaling such a set is relatively straight-forward.  For example, to move the set to the right &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; units, we have to increase all of the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-coordinates by &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;.  Thus a right-translation of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is the set&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \tau_h S = \{ (x+h,y) : \text{[some formula]} \}. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(I tend not to use this notation with precalculus students—I generally use something more &lt;em&gt;ad hoc&lt;/em&gt; like &lt;span class=&quot;math-container&quot;&gt;$S_{\to h}$&lt;/span&gt;—but the notation &lt;span class=&quot;math-container&quot;&gt;$\tau_h$&lt;/span&gt; for a translation is pretty standard in higher math, so I'll assume that the audience here is okay with it.)&lt;/p&gt;&#xA;&lt;p&gt;For example, the unit circle is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \{ (x,y) : x^2 + y^2 = 1 \}, $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and a right translation of that set is given by&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \{ (x+h, y) : x^2 + y^2 = 1 \}. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, I don't really like having the translation on the left-hand side of the set builder notation, so I make a change of variables: let &lt;span class=&quot;math-container&quot;&gt;$\overline{x} = x+h$&lt;/span&gt;.  Then I have the set&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \{ (\overline{x}, y) : x^2 + y^2 = 1, \overline{x} = x+h \}. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is gross, so let's try to simplify this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ x^2 + y^2 = 1 \land \overline{x} = x+h&#xA;\iff x^2 + y^2 = 1 \land x = \overline{x}-h&#xA;\iff (\overline{x}-h)^2 + y^2 = 1. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Putting this back into the set builder notation gives&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \{ (\overline{x},y) :  \overline{x}^2 + y^2 \}. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Replacing &lt;span class=&quot;math-container&quot;&gt;$\overline{x}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (this is just a renaming, if a little confusing) gives the result that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \{ (x+h,y) : x^2 + y^2 = 1 \} = \{ (x,y) : (x-h)^2 + y^2 = 1 \}. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Similar kinds of arguments apply for vertical translations (which give one a &lt;span class=&quot;math-container&quot;&gt;$y-k$&lt;/span&gt; on the right), and for scalings (which give factors of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{a}$&lt;/span&gt; for a horizontal scaling, and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{b}$&lt;/span&gt; for a vertical scaling).  Once you have established this idea (that transformations of a set are most easily understood on the left, by doing the &amp;quot;obvious&amp;quot; thing; but can be moved to the right by doing the &amp;quot;opposite&amp;quot; thing), it is possible to reason about lines:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The primitive line is the set &lt;span class=&quot;math-container&quot;&gt;$\{ (x,y) : x = y\}$&lt;/span&gt;.  Since we &amp;quot;know&amp;quot; how to apply transformations to sets, I'll drop the set builder notation, abuse language a little bit, and refer to &amp;quot;the primitive line &lt;span class=&quot;math-container&quot;&gt;$x=y$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given a point &lt;span class=&quot;math-container&quot;&gt;$P_1 = (x_1, y_1)$&lt;/span&gt; in the plane, the primitive line can be scaled to pass through &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt;.  The idea is to scale the coordinates so that &lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt; gets sent to &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt; (this is not the &lt;em&gt;only&lt;/em&gt; way to reason about lines, but it is the one which makes the computation simplest).  This requires a horizontal scaling by a factor of &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt;, and a vertical scaling by a factor of &lt;span class=&quot;math-container&quot;&gt;$y_1$&lt;/span&gt;.  Thus the line&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{x}{x_1} = \frac{y}{y_1} $$&lt;/span&gt;&#xA;is a line through the origin and the point &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given a second point &lt;span class=&quot;math-container&quot;&gt;$P_2 = (x_2, y_2)$&lt;/span&gt;, reduce the problem to something which has already been solved.  Translate &lt;em&gt;both&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P_2$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$P_2$&lt;/span&gt; lands on the origin—that is, apply a horizontal translation of &lt;span class=&quot;math-container&quot;&gt;$-x_2$&lt;/span&gt; and a vertical translation of &lt;span class=&quot;math-container&quot;&gt;$-y_2$&lt;/span&gt; to that two point set.  &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt; ends up at the point&#xA;&lt;span class=&quot;math-container&quot;&gt;$$P_1' = (x_1 - x_2, y_1 - y_2), $$&lt;/span&gt;&#xA;and from step (2), the line through the origin and &lt;span class=&quot;math-container&quot;&gt;$P_1'$&lt;/span&gt; is given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{x}{x_1 - x_2} = \frac{y}{y_1 - y_2}. $$&lt;/span&gt;&#xA;Note that this only makes sense if &lt;span class=&quot;math-container&quot;&gt;$x_1 - x_2 \ne 0$&lt;/span&gt; and also &lt;span class=&quot;math-container&quot;&gt;$y_1 - y_2 \ne 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To deal with the case that one or the other is zero (they can't &lt;em&gt;both&lt;/em&gt; be zero, because then &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P_2$&lt;/span&gt; not distinct), we can &amp;quot;clear the denominators&amp;quot; to get the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (y_1 - y_2) x = (x_1 - x_2) y. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, the origin needs to be translated back to &lt;span class=&quot;math-container&quot;&gt;$P_2$&lt;/span&gt;, which requires a horizontal translation of &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt; units, and a vertical translation of &lt;span class=&quot;math-container&quot;&gt;$y_2$&lt;/span&gt; units, which gives the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (y_1 - y_2) (x - x_2) = (x_1 - x_2) (y - y_2). \tag{$\ast$}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This is, essentially, a variation  of the &amp;quot;point-point&amp;quot; equation for a line, which is usually given as something like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ y - y_2 = \underbrace{\frac{y_1 - y_2}{x_1 - x_2}}_{\text{$=m$, the slope}}(x-x_2). $$&lt;/span&gt;&#xA;On the other hand, if (&lt;span class=&quot;math-container&quot;&gt;$\ast$&lt;/span&gt;) is expanded out, we get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (y_1 - y_2) x - (x_2 - x_1) y - (y_1 - y_2)x_2 + (x_1 - x_2)y_2 = 0&#xA;\implies ax + by + c = 0,$$&lt;/span&gt;&#xA;where&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;a &amp;amp;= y_1 - y_2,  \\&#xA;b &amp;amp;= -(x_2 - x_1), \text{ and} \\&#xA;c &amp;amp;= - (y_1 - y_2)x_2 + (x_1 - x_2)y_2.&#xA;\end{align}&lt;/span&gt;&#xA;Note that this formulation describes &lt;em&gt;any&lt;/em&gt; line, and that it generalizes nicely to the idea of specifying a co-dimension one subspace of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; in terms of a normal vector.  In this setting the slope (if it exists) is &lt;span class=&quot;math-container&quot;&gt;$b/a$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2022-05-03T02:14:40.767" LastActivityDate="2022-05-03T02:14:40.767" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25203" PostTypeId="2" ParentId="5614" CreationDate="2022-05-03T07:53:27.150" Score="5" Body="&lt;p&gt;I think the best way to think of this is not to try to somehow create yet another arbitrary rule, but instead to invite more examination of just what both the graph of a function is, what a transformation is, and what happens to one when transformed under the other.&lt;/p&gt;&#xA;&lt;p&gt;You see, what's actually going on here is a multi-step process:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;we start with a function and then we &amp;quot;convert&amp;quot; it to its graph,&lt;/li&gt;&#xA;&lt;li&gt;we transform the graph,&lt;/li&gt;&#xA;&lt;li&gt;we then ask what function represents the transformed graph.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Now in some sense, one can say a function &amp;quot;already is&amp;quot; a graph, but depending on how it's been introduced or not, it is perhaps useful to simply make this more explicit. The &lt;strong&gt;graph&lt;/strong&gt; of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; of one real variable is the set of all points &lt;span class=&quot;math-container&quot;&gt;$(x, f(x))$&lt;/span&gt; in the Cartesian plane, one for each &lt;span class=&quot;math-container&quot;&gt;$x \in \mathrm{dom}(f)$&lt;/span&gt;, the function's domain. As a set of points in the plane, this is a geometric shape, just like a line or a circle is. It is on this &lt;em&gt;shape&lt;/em&gt; that our transformation acts, &lt;em&gt;not&lt;/em&gt; on &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; directly.&lt;/p&gt;&#xA;&lt;p&gt;The transformation, by the way, &lt;em&gt;itself&lt;/em&gt; is a function, but it's a function of a different kind: it maps points on the plane to other points on the plane, instead of distances along one axis to distances along another axis. Unfortunately though I suppose the idea of functions is not treated with the due generality and more importantly &lt;em&gt;fundamentality&lt;/em&gt; it deserves, so I won't spend more time on that particular aspect. What's most important is to note that in &lt;em&gt;this&lt;/em&gt; case, the transformation acts &amp;quot;in the right way&amp;quot;, intuitively: to shift the plane without rotation or stretch such that the origin &lt;span class=&quot;math-container&quot;&gt;$(0, 0)$&lt;/span&gt; is moved to the point &lt;span class=&quot;math-container&quot;&gt;$(x_0, y_0)$&lt;/span&gt;, we have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$T_{\langle x_0, y_0 \rangle}(x, y) := (x + x_0, y + y_0)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where we added &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y_0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, just as you'd think. (The angle brackets in the subscript are meant to indicate a &lt;em&gt;vector&lt;/em&gt; giving the lay of the transformation, but we don't really need that much detail here.)&lt;/p&gt;&#xA;&lt;p&gt;The relevant trick now is we can apply this to the graph of a &amp;quot;grapher's function&amp;quot; of the type we've been talking about earlier. In particular, the point &lt;span class=&quot;math-container&quot;&gt;$(x, f(x))$&lt;/span&gt; transforms to&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$T_{\langle x_0, y_0 \rangle}(x, f(x)) = (x + x_0, f(x) + y_0)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and thus there is implied a whole transformed set of points, which we want to interpret as the graph of a &lt;em&gt;different&lt;/em&gt; function &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;. And this is where the &amp;quot;reversal&amp;quot; comes in - note above that the sums are the same. You see, the transformed function &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; will have the points described on &lt;em&gt;its&lt;/em&gt; graph by &lt;span class=&quot;math-container&quot;&gt;$(x, g(x))$&lt;/span&gt;, but in the above, note that the left element of the pair is &lt;span class=&quot;math-container&quot;&gt;$x + x_0$&lt;/span&gt;, not &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Thus, what we really have above is actually the point on &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;'s graph, not at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, but at &lt;span class=&quot;math-container&quot;&gt;$x + x_0$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$g(x + x_0) = f(x) + y_0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And then you can see that to get &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;, i.e. to complete the third step above, you have to &lt;em&gt;subtract&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$g(x) = f(x - x_0) + y_0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and that's why we subtraction &lt;em&gt;inside&lt;/em&gt; the &lt;em&gt;function symbol&lt;/em&gt; moves the graph &lt;em&gt;right&lt;/em&gt; and not the expected &lt;em&gt;left&lt;/em&gt;. The relevant right-moving &lt;em&gt;addition&lt;/em&gt; happens more subtly.&lt;/p&gt;&#xA;" OwnerUserId="7650" LastActivityDate="2022-05-03T07:53:27.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25204" PostTypeId="2" ParentId="25188" CreationDate="2022-05-03T12:34:39.407" Score="1" Body="&lt;p&gt;Although I agree that this is terrible notation (given the standard notational expectations), there is a subtle point to address here.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In my opinion points should not be deducted for writing interval this way if the student is able to explain the reasoning behind it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is one of those rare (in my opinion) situations where one &lt;em&gt;may&lt;/em&gt; be justified in giving &amp;quot;clawback&amp;quot; points to a student who pushes back against being marked incorrect if they can clearly explain what their notation means.  The following must be observed here:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;One is teaching in a cultural context where &amp;quot;partial credit&amp;quot; is a typical expectation.&lt;/li&gt;&#xA;&lt;li&gt;The problem was marked wrong for a nontrivial number of points.  This depends on context - if it's a very minor part of a huge question one might overlook it completely.&lt;/li&gt;&#xA;&lt;li&gt;The student asks about why it was marked wrong.&lt;/li&gt;&#xA;&lt;li&gt;The instructor asks the student to explain the notation.&lt;/li&gt;&#xA;&lt;li&gt;The student clearly explains the notation as in the original question.&lt;/li&gt;&#xA;&lt;li&gt;The instructor then explains why this notation either should mean the empty set or would mean an &lt;em&gt;oriented&lt;/em&gt; interval (see below, but then again oriented intervals aren't equal).&lt;/li&gt;&#xA;&lt;li&gt;The student shows understanding.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Oriented intervals?  Sure, considered as a &amp;quot;path integral&amp;quot;/&amp;quot;vector line integral&amp;quot;, &lt;span class=&quot;math-container&quot;&gt;$\int_{(2,3)}1\cdot dx=\int_2^3 1\, dx=1$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$\int_{-(2,3)}1\cdot dx=\int_3^2 1\, dx=-1$&lt;/span&gt;, where I consider here &lt;span class=&quot;math-container&quot;&gt;$-(2,3)$&lt;/span&gt; as a shorthand for &amp;quot;the interval as a curve, but parametrized to go in the opposite direction&amp;quot;. (I think these are both standard notation in what used to be called &amp;quot;advanced calculus&amp;quot;, see e.g. &lt;a href=&quot;https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/16%3A_Vector_Calculus/16.2%3A_Line_Integrals&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)  The FTC/Stokes' Theorem is even true here, if one has an oriented boundary (well, and if one allows for a boundary of a non-closed interval, and ...).  And the student might enjoy seeing a place where this notation &lt;em&gt;could&lt;/em&gt; be used, while saying that for now we do not do so as it introduces unacceptable complexity.&lt;/p&gt;&#xA;&lt;p&gt;Having said all that, I think there is an even more basic reason for not accepting this notation; there is already (unacceptable?) ambiguity in notation between the point &lt;span class=&quot;math-container&quot;&gt;$(2,3)\in \mathbb{R}^2$&lt;/span&gt; and the interval &lt;span class=&quot;math-container&quot;&gt;$(2,3)$&lt;/span&gt;, and it is just increased by allowing this notation.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2022-05-06T00:03:46.960" LastActivityDate="2022-05-06T00:03:46.960" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25205" PostTypeId="1" CreationDate="2022-05-03T18:58:21.537" Score="12" ViewCount="6166" Body="&lt;p&gt;When teaching functions, one key aspect of the definition of a function is the fact that each input is assigned exactly one output.&lt;/p&gt;&#xA;&lt;p&gt;I always felt that the &amp;quot;exactly one&amp;quot; part is confusing to students because it seems to be &amp;quot;the default&amp;quot;, and I have a hard time to find convincing examples of binary relations with &amp;quot;ambiguous&amp;quot; &amp;quot;outputs&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Therefore I am asking for example of such binary relations.&#xA;(Bonus points if the &amp;quot;hack&amp;quot; to just take &amp;quot;the set of&amp;quot; multiple possible outputs, for example when looking at divisors of a number, &amp;quot;feels awkward&amp;quot;.)&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="667" LastEditDate="2022-05-07T10:36:49.927" LastActivityDate="2022-05-12T07:01:32.107" Title="Examples of relations that are not functions" Tags="|secondary-education|functions|" AnswerCount="14" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25206" PostTypeId="2" ParentId="25205" CreationDate="2022-05-03T19:19:58.353" Score="24" Body="&lt;p&gt;The example I use:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x) = x$&lt;/span&gt;'s sister&lt;/p&gt;&#xA;&lt;p&gt;Looks fine, has a formula. I can write &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$f($&lt;/span&gt;Chris&lt;span class=&quot;math-container&quot;&gt;$) = $&lt;/span&gt;Jessica&amp;quot; since I have a sister. I can talk about the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; by asking for someone to volunteer their sister's name, and asking someone else to volunteer that they do not have a sister. Usually someone in the room will have two sisters, and when this comes up, it proves the relation I am talking about is not a function, so I can't really write it as &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; after all.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastActivityDate="2022-05-03T19:19:58.353" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25207" PostTypeId="2" ParentId="25205" CreationDate="2022-05-03T20:27:59.157" Score="3" Body="&lt;p&gt;Another &amp;quot;real life&amp;quot; example is &amp;quot;result of throw of a die&amp;quot; (namely, from one to six pips).&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-05-03T20:27:59.157" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25208" PostTypeId="2" ParentId="25205" CreationDate="2022-05-03T20:49:25.440" Score="9" Body="&lt;p&gt;I’ll give a purely mathematical example that I often used with students beyond the elementary calculus level. (See last paragraph below regarding college algebra and precalculus level students.) At this level students will be encountering situations where they have to show (or be able to understand proofs) that certain definitions of a function are &lt;a href=&quot;https://en.wikipedia.org/wiki/Well-defined_expression&quot; rel=&quot;nofollow noreferrer&quot;&gt;well defined&lt;/a&gt; (sometimes one says “give a well defined function”). Examples of this arise when working with &lt;a href=&quot;https://math.stackexchange.com/q/190628/13130&quot;&gt;quotient groups&lt;/a&gt; and quotient rings, when defining the group operation for the &lt;a href=&quot;https://math.stackexchange.com/a/923976/13130&quot;&gt;fundamental group&lt;/a&gt; in topology, when working with &lt;a href=&quot;https://math.stackexchange.com/q/995549/13130&quot;&gt;congruence classes in number theory&lt;/a&gt;, in various places in differential geometry (&lt;a href=&quot;https://math.stackexchange.com/q/4425462/13130&quot;&gt;example 1&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/q/4313327/13130&quot;&gt;example 2&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/q/2739574/13130&quot;&gt;example 3&lt;/a&gt;), etc. Incidentally, while looking up these links I found the MSE question &lt;a href=&quot;https://math.stackexchange.com/q/153413/13130&quot;&gt;well-defined functions&lt;/a&gt;, which discusses the same type of example given below, and even has aspects &lt;strong&gt;very similar&lt;/strong&gt; to my Observation (2) below, despite my having written that before seeing this other discussion.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; be integers, one or both nonzero if necessary.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; be the “function” from the rationals to the integers defined by &lt;span class=&quot;math-container&quot;&gt;$f\left(\frac{p}{q}\right) = pq,\;$&lt;/span&gt; or some variation such as &lt;span class=&quot;math-container&quot;&gt;$\;=p+q\;$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\;=p^2 – q.$&lt;/span&gt; Incidentally, simplest would be to use &lt;span class=&quot;math-container&quot;&gt;$\;=p\;$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\;=q,\;$&lt;/span&gt; but I think using a slightly more involved formula works better, especially when Observation (2) below is dealt with.&lt;/p&gt;&#xA;&lt;p&gt;Note that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2} = \frac{2}{4} = \frac{3}{6},$&lt;/span&gt; but the formula for &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; gives different outputs for these different representations of the number &amp;quot;one-half&amp;quot;. Therefore, there is not a unique output for each input.&lt;/p&gt;&#xA;&lt;p&gt;I can think of two useful observations that one can make with this example. &lt;strong&gt;Observation (1)&lt;/strong&gt; Using a simple mathematical formula that otherwise seems unproblematic is not enough—the nature of the actual input elements matters. Specifically, we see that the distinction between a &lt;em&gt;representation of the number&lt;/em&gt; “one-half” and the number itself actually matters. &lt;strong&gt;Observation (2)&lt;/strong&gt; The “functionality” issue for this example can be investigated in the same way that is done in the actual situations that students will encounter, namely by showing that the output is independent of the choice of representation of the input. Specifically, the representations &lt;span class=&quot;math-container&quot;&gt;$\frac{p}{q}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{p’}{q’}$&lt;/span&gt; correspond to the same number if and only if there exists &lt;span class=&quot;math-container&quot;&gt;$k \neq 0$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$p’ = kp$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q’ = kq.$&lt;/span&gt; Thus, to be well defined (sometimes one says “single-valued” at this point), we require &lt;span class=&quot;math-container&quot;&gt;$f\left(\frac{p}{q}\right) = f\left(\frac{p’}{q’}\right),$&lt;/span&gt; which is equivalent to the requirement that &lt;span class=&quot;math-container&quot;&gt;$pq = p’q’ = (kp)(kq) = k^2pq.$&lt;/span&gt; However, because &lt;span class=&quot;math-container&quot;&gt;$k^2 \neq 1$&lt;/span&gt; is possible, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not well defined.&lt;/p&gt;&#xA;&lt;p&gt;Note that if we define &lt;span class=&quot;math-container&quot;&gt;$g\left(\frac{p}{q}\right) = \frac{p^2 \; + \; q^2}{pq},$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; &lt;strong&gt;is a function&lt;/strong&gt; from the nonzero rationals to the rationals. One way to see this, which corresponds to using the approach of Observation (1) above, is to note that &lt;span class=&quot;math-container&quot;&gt;$\frac{p^2 \; + \; q^2}{pq} = \frac{p}{q} + \frac{q}{p},$&lt;/span&gt; and thus we are actually dealing with the function &lt;span class=&quot;math-container&quot;&gt;$g(x) = x + \frac{1}{x}$&lt;/span&gt; for nonzero rationals &lt;span class=&quot;math-container&quot;&gt;$x.$&lt;/span&gt; Another way to see this, which corresponds to using the approach of Observation (2) above (this is what is typically done in practice in the case of quotient groups, the fundamental group operation, etc.), is to note that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ f\left(\frac{p'}{q'}\right) \; = \; f\left(\frac{kp}{kq}\right) \; = \; \frac{k^2p^2 + k^2q^2}{kp \cdot kq} \; = \; \frac{k^2(p^2 + q^2)}{k^2pq} \; = \; \frac{p^2 + q^2}{pq} \; = \; f\left(\frac{p}{q}\right) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;These examples can also be used at the college algebra and precalculus level, and I’ve done this. I like to think of &lt;a href=&quot;https://yamm.finance/wiki/Single-valued_function.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;pictures with arrows&lt;/a&gt; and &lt;a href=&quot;https://www.youtube.com/watch?v=y2TqnP_6M1s&quot; rel=&quot;nofollow noreferrer&quot;&gt;tables&lt;/a&gt; as a “right brain” approach and something like &lt;span class=&quot;math-container&quot;&gt;$f\left(\frac{p}{q}\right) = pq$&lt;/span&gt; as a “left brain” approach. Of course, while some students at this level will be intrigued with the approach of Observation (1) above for &lt;span class=&quot;math-container&quot;&gt;$f\left(\frac{p}{q}\right) = pq$&lt;/span&gt;, you’ll definitely want to stay away from the symbolism involved in the approach of Observation (2).&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2022-05-06T07:48:30.090" LastActivityDate="2022-05-06T07:48:30.090" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25209" PostTypeId="2" ParentId="25205" CreationDate="2022-05-04T00:22:29.180" Score="13" Body="&lt;p&gt;Given points &lt;span class=&quot;math-container&quot;&gt;$P = \{x_1,x_2,\ldots \}$&lt;/span&gt; on a line &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;,&#xA;let &lt;span class=&quot;math-container&quot;&gt;$nn(x)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \in L$&lt;/span&gt; be the &lt;em&gt;nearest neighbor&lt;/em&gt; to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;: the&#xA;point in &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; closest to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&#xA;For most &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, there is a unique &lt;span class=&quot;math-container&quot;&gt;$nn(x)$&lt;/span&gt;, but if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the midpoint&#xA;of two consecutive points of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;, then there is a tie.&lt;/p&gt;&#xA;&lt;p&gt;Generalizing to points (&amp;quot;sites&amp;quot;) &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; on a plane leads to the Voronoi diagram:&#xA;most points &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; have a unique nearest neighbor, points on a Voronoi edge&#xA;have two equally nearest neighbors, a point at a Voronoi vertex of&#xA;degree &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; equally nearest neighbors.&#xA;So &amp;quot;exactly one&amp;quot; is the default, but there are important exceptions.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/PL0AK.png&quot; width=&quot;250&quot; /&gt;&#xA;&lt;p&gt;(Image from Caio M R de Oliveira thesis, Appendix A.)&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2022-05-04T12:36:45.800" LastActivityDate="2022-05-04T12:36:45.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25210" PostTypeId="2" ParentId="25205" CreationDate="2022-05-04T05:22:03.067" Score="1" Body="&lt;p&gt;A confidence, tolerance, or predictive interval maps data to a range.&lt;/p&gt;&#xA;&lt;p&gt;Note that there are additional lessons here because there are an infinite number of confidence intervals.  Any function that covers the issue of interest at least &lt;span class=&quot;math-container&quot;&gt;$1-\alpha$&lt;/span&gt; percent of the time is a valid interval.  For example, if you throw a penny to the bottom of a pool and blindfold somebody and give them directions based on an interval to find the penny, the entire swimming pool is the trivial 100% interval as you know it is in there with probability one.  It is also a 95% interval since it covers it at least 95% of the time.&lt;/p&gt;&#xA;&lt;p&gt;That is ambiguous.  It does not cover it 95% of the time.  It covers it at least 95% of the time because it covers it 100% of the time.&lt;/p&gt;&#xA;&lt;p&gt;Most intervals, except exact intervals, are a range built on a minimax utility function.  You are guaranteed that it works &lt;em&gt;at least&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$1-\alpha$&lt;/span&gt; percent of the time.  In fact, the interval maps to a probability region instead of a point.  &lt;span class=&quot;math-container&quot;&gt;$\Pr(\theta\in[a,b])\ge{1-\alpha},\forall{x_1,x_2,x_3}\dots{x_n}\in\chi,$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\chi$&lt;/span&gt; is the sample space.&lt;/p&gt;&#xA;&lt;p&gt;It gets a little wilder too.  All samples that share the same sample statistics, share the same interval.  The data is mediated by estimators, so an interval is actually &lt;span class=&quot;math-container&quot;&gt;$g(h(x_1,x_2,\dots,x_n))$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is an estimator(s) chosen under some rule and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is a mapping rule.  If &lt;span class=&quot;math-container&quot;&gt;$h(X)=h(X')$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt; is the interval for both sample &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and sample &lt;span class=&quot;math-container&quot;&gt;$X'$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you want to get wilder, the definition of a Frequentist interval is a bit wild.  A 95% confidence interval does not imply that there is a 95% chance the real parameter is inside the interval.  What it does say is that if you built an infinite number of intervals, at least 95% of the intervals would cover the parameter.  It says nothing about the current experiment's chance of covering the parameter.&lt;/p&gt;&#xA;&lt;p&gt;A function can usually be thought of as having an output with a clear meaning.  That isn't a clear meaning.  You toss in 50 apples and you get one jar of applesauce.  Here, you toss in a group of inputs and you do not get an output that says the solution is somewhere around &amp;quot;here&amp;quot;.  You get an output that says the function works at least some amount of time and is broken, maybe even nonsense, at other times.  The answer isn't necessarily in the range, but it is wise to act as if it is.  If it isn't, you may look like a fool.  However, you will never be a fool more than &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; percent of the time.&lt;/p&gt;&#xA;" OwnerUserId="13529" LastActivityDate="2022-05-04T05:22:03.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25211" PostTypeId="2" ParentId="16636" CreationDate="2022-05-04T05:41:28.627" Score="5" Body="&lt;p&gt;The Yule-Simpson paradox or effect.  It is an important result in statistics that says you cannot simply plug numbers into a statistical engine and pop out a meaningful answer.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YXduM.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YXduM.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this example, from &lt;a href=&quot;https://en.wikipedia.org/wiki/Simpson%27s_paradox&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, the regression line is the dashed line.  It looks like there is a decreasing effect.  The true effect is that there is some categorical variable and in both possible states, the effect is an increasing one.&lt;/p&gt;&#xA;&lt;p&gt;The famous example, which is in the Wiki article, is the UC Berkley case where there was apparent and substantial discrimination against women for admission.  What was found was that there was a small, but statistically significant bias for women over men once the academic department was controlled for.&lt;/p&gt;&#xA;&lt;p&gt;It requires a person to think about subpopulations as well as the total population.  The danger is that simple analysis of the data is very misleading.  The mathematical intuition should be that if every subpopulation is increasing, the population should be increasing, but that is &lt;em&gt;not&lt;/em&gt; true.&lt;/p&gt;&#xA;&lt;p&gt;Think of that from a policy or voting perspective.  If society is worse off for each increase in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, then society should suppress &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  However, if every subpopulation is improved by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; increasing, then &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; should be amplified.  In the graphic above, the actual intuition should not be about suppressing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; but about shifting the subpopulation represented by the red line to the left.  It would be even better to shift all subpopulations to the left, while amplifying &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; at the same time.  It is the intercept that is the secret story in this diagram.&lt;/p&gt;&#xA;" OwnerUserId="13529" LastActivityDate="2022-05-04T05:41:28.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25212" PostTypeId="2" ParentId="25205" CreationDate="2022-05-04T06:47:46.180" Score="2" Body="&lt;p&gt;As you are teaching to students in a classroom, and presumably they are not your only students nor you their only teacher, a variety of mappings between the teachers, students, classes, and rooms become available to illustrate which can be functions and which are not:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;One &lt;span class=&quot;math-container&quot;&gt;$teacher_{math}$&lt;/span&gt; maps to many students&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One student maps to many &lt;span class=&quot;math-container&quot;&gt;$teacher_{subject}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Many students map to many &lt;span class=&quot;math-container&quot;&gt;$teacher_{departments}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One &lt;span class=&quot;math-container&quot;&gt;$class_{math}$&lt;/span&gt; may map to one or multiple rooms&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;and so on.&lt;/p&gt;&#xA;&lt;p&gt;Ideally the actual examples are chosen to reflect specific points you want to convey, and may depend on the context of your own institution (or a specific choice of institutions, whether real or invented) to do this clearly.&lt;/p&gt;&#xA;" OwnerUserId="6724" LastActivityDate="2022-05-04T06:47:46.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25213" PostTypeId="2" ParentId="25205" CreationDate="2022-05-04T07:16:58.793" Score="17" Body="&lt;p&gt;An example that should be natural to the students is the square root over the positive real numbers. If one simply says &amp;quot;square root of &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;&amp;quot; then there are two equally nice roots, &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;. We make this a function when we prescribe a recipe to choose one of them (typically, &amp;quot;pick the positive one&amp;quot;).&lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2022-05-04T07:16:58.793" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25214" PostTypeId="2" ParentId="25205" CreationDate="2022-05-04T07:23:17.857" Score="2" Body="&lt;p&gt;Map each country to their official languages. This might look like a (not injective) function depending on which countries you start out with but there are definitely countries with more than one official language.&lt;/p&gt;&#xA;" OwnerUserId="13433" LastActivityDate="2022-05-04T07:23:17.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25217" PostTypeId="2" ParentId="25205" CreationDate="2022-05-04T13:44:11.220" Score="3" Body="&lt;p&gt;If you are also teaching sets, or a student brings this up, consider the mapping from sets to their elements:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(\{1, 2\}) = 1\\f(\{1, 2\}) = 2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So this is not a function. On the other hand, there &lt;em&gt;is&lt;/em&gt; a function that maps singletons to their respective element:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(\{x\}) = x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Trying to assemble the outputs into a set would miss the whole point of the mappings.&lt;/p&gt;&#xA;&lt;p&gt;(On a side note, if someone brings up the idea that a non-functional mapping is just a function to a set, I think I would try to make clear that this changes what we're talking about. When I ask them, &amp;quot;Who is your friend?&amp;quot;, they may have more than one answer; they most likely won't respond like &amp;quot;It's the set of Anna and Bob.&amp;quot; It's not the set that is their friend. Asking &amp;quot;What's the set of your friends&amp;quot; is a different question. – There may, of course, very well be a function from your students to their &lt;em&gt;best&lt;/em&gt; friend.)&lt;/p&gt;&#xA;" OwnerUserId="19837" LastActivityDate="2022-05-04T13:44:11.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25218" PostTypeId="2" ParentId="25205" CreationDate="2022-05-04T15:03:13.017" Score="5" Body="&lt;p&gt;A classic example, and perhaps the reason that people started thinking about &amp;quot;multivalued functions&amp;quot; in the first place:&lt;/p&gt;&#xA;&lt;p&gt;Try to define&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\theta(x,y) = \displaystyle\int_{(1,0)}^{(x,y)} \frac{-y \textrm{ d}x + x\textrm{ d}y}{x^2+y^2}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where you are integrating along any path from &lt;span class=&quot;math-container&quot;&gt;$(1,0)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;.  This gives the measure of &amp;quot;the angle&amp;quot; connecting &lt;span class=&quot;math-container&quot;&gt;$(1,0)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;.  Unfortunately there is no &amp;quot;the angle&amp;quot;:  the angle is only well defined up to multiples of &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt;, and the value of this integral will differ by multiples of &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt; depending on the path taken.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-05-04T15:03:13.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25220" PostTypeId="2" ParentId="7660" CreationDate="2022-05-04T23:40:53.390" Score="2" Body="&lt;p&gt;Saul Stahl's ** A Gateway to Modern Geometry: the Poincare Upper Half Plane **&#xA;looks extremely promising. I haven't read it, but based on the table of contents, it seems to begin with Euclidean Geometry, then inversion (circles), then focus for most of the book on hyperbolic geometry, and in particular the upper half plane model.&lt;/p&gt;&#xA;&lt;p&gt;Chapter 12 is * Differential Geometry and Gaussian Curvature * and has the following sections:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;12.1 Differential Geometry 157&lt;/li&gt;&#xA;&lt;li&gt;12.2 A Review of Length and Area on Surfaces 163&lt;/li&gt;&#xA;&lt;li&gt;12.3 Gauss's Formula for the Curvature at a Point 168&lt;/li&gt;&#xA;&lt;li&gt;12.4 Riemannian Geometry Revisited 170&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10289" LastActivityDate="2022-05-04T23:40:53.390" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25221" PostTypeId="1" AcceptedAnswerId="25225" CreationDate="2022-05-05T06:39:51.673" Score="5" ViewCount="399" Body="&lt;p&gt;In each class, there are often a few students who seem to have given up early on.&lt;/p&gt;&#xA;&lt;p&gt;In my recent discrete math class, when I asked a student a question, he said he was working on his graduation project.&lt;/p&gt;&#xA;&lt;p&gt;In another time, when I picked the same student, he simply said he was not listening.&lt;/p&gt;&#xA;&lt;p&gt;I wonder if there is anything I should do in such situations?&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2022-05-06T17:57:27.823" Title="Is there anything one can do for students who have given up?" Tags="|undergraduate-education|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25222" PostTypeId="2" ParentId="25205" CreationDate="2022-05-05T06:57:21.527" Score="1" Body="&lt;p&gt;Maybe your students don't yet know this (but it might be instructive to revisit this when it comes up):&lt;/p&gt;&#xA;&lt;p&gt;The indefinite integral (antiderivative) can be considered a map from functions to functions, but of course the result is not a single function - that's just the point of the &lt;span class=&quot;math-container&quot;&gt;$+c$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\int x\, \text{d}x =\frac12 x^2+c$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16320" LastActivityDate="2022-05-05T06:57:21.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25223" PostTypeId="2" ParentId="25205" CreationDate="2022-05-05T13:28:35.053" Score="1" Body="&lt;p&gt;I found this example in &amp;quot;Finite Mathematics&amp;quot; by Rolf:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When Sarah buys six apples (&lt;span class=&quot;math-container&quot;&gt;$x=6$&lt;/span&gt;), the checker determines the weight in order to know the cost. The six apples of the customer behind Sarah likely will correspond to a different weight. Thus the weight &amp;quot;function&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; apples is not a function of the number of apples because a number in the domain (number of apples) may be related to more than one number in the range (weight of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; apples).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think this is a pretty intuitive example of a relation/map not being well-defined.&lt;/p&gt;&#xA;&lt;p&gt;Rolf actually goes a bit further pointing out that, likewise, cost is not a function of the number of apples since the price is usually by weight. So there's a natural comparison to &amp;quot;Cost of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; apples&amp;quot; not being well-defined, but something like &amp;quot;Cost of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; candy bars&amp;quot; is.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2022-05-05T13:28:35.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25224" PostTypeId="2" ParentId="25196" CreationDate="2022-05-05T14:47:30.463" Score="2" Body="&lt;p&gt;A possibility may be to show how a cube consists of 6 equivalent pyramids. The picture below shows three of them, and has empty slots in between for the missing three (I felt this would make the picture easier to comprehend). Each of the pyramids has height equal to the edge of the cube, and base that is one half of one of the faces of the cube. The only way the volumes can add up is if we have the correct formula &lt;span class=&quot;math-container&quot;&gt;$V=\frac13Ah=\frac13\cdot\frac12a^2\cdot a=\frac16a^3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EUMur.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EUMur.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As a bonus you also see the following. Assume that the cube in the picture is the set &lt;span class=&quot;math-container&quot;&gt;$[0,1]\times[0,1]\times[0,1]$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\Bbb{R}^3$&lt;/span&gt;, and the edge shared by all the six parts is the 3D-diagonal from &lt;span class=&quot;math-container&quot;&gt;$(0,0,0)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(1,1,1)$&lt;/span&gt;. Then the coordinates points &lt;span class=&quot;math-container&quot;&gt;$(x,y,z)$&lt;/span&gt; in any of the six parts share the same ordering. For example, all the points in one cone satisfy &lt;span class=&quot;math-container&quot;&gt;$x\le y\le z$&lt;/span&gt; et cetera. Obviously there are six possible orderings for the three coordinates. Observe that the three planes &lt;span class=&quot;math-container&quot;&gt;$x=y$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=z$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z=x$&lt;/span&gt; can also be seen in the picture, as all the six cones have a face on two of those planes.&lt;/p&gt;&#xA;" OwnerUserId="282" LastActivityDate="2022-05-05T14:47:30.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25225" PostTypeId="2" ParentId="25221" CreationDate="2022-05-05T16:15:54.030" Score="7" Body="&lt;p&gt;Online teaching is so hard.&lt;/p&gt;&#xA;&lt;p&gt;What is your college's policy, or yours, on whether the students must have their cameras on? At my college, we've been asked to respect the fact that some students don't have enough privacy at home to be comfortable putting their cameras on, so I don't require it. But I know that everyone having their cameras on helps with participation. However, if it's late in the semester there like it is here, I wouldn't recommend changing any policies now.&lt;/p&gt;&#xA;&lt;p&gt;I try to get students to participate in ways that are lower risk for them. I ask &amp;quot;Is that making sense?&amp;quot; and have them give me (only me) a number 1 to 5 in chat. Even that, many of them won't do.&lt;/p&gt;&#xA;&lt;p&gt;You described it as giving up, but I'd describe it as disengaged. I'm curious how this student is doing with assignments. I'm also wondering about his attitude. Both of his comments seem on the verge of hostile, or at least a bit disrespectful. (Maybe it depends on the tone...)&lt;/p&gt;&#xA;&lt;p&gt;I record each zoom session, so I'd tell him it's ok to not come, and to watch the zoom later. It might be worth figuring out an empathic response to that student, and messaging him privately. &amp;quot;It sounds like you're feeling pretty overwhelmed. I wonder how you're feeling about this class.&amp;quot; I guess that would depend on how he's doing with quizzes and tests. (Sadly, I have trouble with students who are cheating. As I said at the start, online teaching is so hard.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-05-05T20:05:15.253" LastActivityDate="2022-05-05T20:05:15.253" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25226" PostTypeId="2" ParentId="25188" CreationDate="2022-05-05T19:27:12.770" Score="1" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; has the advantage of matching how we would order numbers on the number line.  It would be possible to have &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; for the open set between &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; , but why not order them?&lt;/p&gt;&#xA;&lt;p&gt;Conventions matter a lot, otherwise how will we communicate effectively?  When awarding points/scores/marks deciding if a student understands the standard conventions of the subject is an important component. Indeed, when they write something and expect me to understand it I have to assume they are following the convention.  Do you want the &amp;quot;Humpty Dumpty&amp;quot; defense:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“When I use a word,” Humpty Dumpty said in rather a scornful tone, “it&#xA;means just what I choose it to mean—neither more nor less.” “The&#xA;question is,” said Alice, “whether you can make words mean so many&#xA;different things.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would deduct some marks for failing to follow the standard conventions under most circumstances.  Of course, we have a responsibility to explain the difference between a notational convention and when we overload notation.  E.g. &lt;span class=&quot;math-container&quot;&gt;$x^{1/2}$&lt;/span&gt; doesn't make sense in terms of repeated multiplication (&lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; multiplied &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; times).  Here we extend exponential notation to rational exponents, in a way which tries to preserve the theorems, e.g. &lt;span class=&quot;math-container&quot;&gt;$x^nx^m=x^{n+m}$&lt;/span&gt;.  This is quite different from a pure convention.  It is the &amp;quot;principle of permanence of forms&amp;quot;. There is a real blurring of the difference between pure convention and how analogy is used to choose notation or interpret notation.  I wished we talked about this more!&lt;/p&gt;&#xA;&lt;p&gt;There are some interesting and important cultural differences in the way people write intervals.  I quite like &lt;span class=&quot;math-container&quot;&gt;$ ]0,1] $&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$ 0&amp;lt;x \mbox{ and } x\leq 1$&lt;/span&gt; but that isn't what I normally write!&lt;/p&gt;&#xA;" OwnerUserId="7787" LastActivityDate="2022-05-05T19:27:12.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25227" PostTypeId="2" ParentId="24963" CreationDate="2022-05-05T19:45:10.977" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Should students in a first university linear algebra class be taught to write simple proofs?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'd like to de-construct this question a little.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;first university&lt;/li&gt;&#xA;&lt;li&gt;linear algebra&lt;/li&gt;&#xA;&lt;li&gt;should student be taught to write simple proofs?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The key distinguishing hallmark of mathematics (relative to other subjects) is proof, that is rigorous deductive reasoning.  I'd be sad to think that any &amp;quot;first university&amp;quot; mathematics course, or any &amp;quot;linear algebra&amp;quot; course did not have some component of proof, or the goal to help students appreciate the value of proof.&lt;/p&gt;&#xA;&lt;p&gt;I teach a first university course of linear algebra (in the UK) to over 900 students p/a, about 1/3 of whom are mathematics students.  The rest of my students are computer science, physics and engineering.  If I don't &amp;quot;set out the stall&amp;quot; for the nature of mathematics who will do this? For specialist mathematicians (compared to computer science, physics and engineering) they need to understand proof.  The rest might not be bothered about proof, but they should learn enough about proof to recognise it, its value and the nature of knowledge it provides.  Computation is important and so I try to maintain a balance.  Proof at this level is not the same as proof in a graduate research paper, and different proof styles serve different purposes.  I try to be kind to my students, without betraying the subject.&lt;/p&gt;&#xA;&lt;p&gt;Linear algebra is a very rich and diverse subject, and does not get the same recognition that calculus gets for the complexity of the decisions instructors have to make.  This is a really great book if you can get a copy.&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;@BOOK{Dorier2000,&#xA;  editor =       {Dorier, J.~L.},&#xA;  title =        {On the Teaching of Linear Algebra},&#xA;  publisher =    {Springer},&#xA;  year =         {2000},&#xA;  volume =       {23},&#xA;  series =       {Mathematics Education Library},&#xA;  doi =          {http://dx.doi.org/10.1007/0-306-47224-4},&#xA;}&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="7787" LastActivityDate="2022-05-05T19:45:10.977" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25228" PostTypeId="2" ParentId="18297" CreationDate="2022-05-06T06:30:27.010" Score="3" Body="&lt;p&gt;I once met a student who computed &lt;span class=&quot;math-container&quot;&gt;$\sqrt{29}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; digits precisions manually. At first I suspected cheating with computer since it's an at home exam. But the student showed me they can use bisection method to compute this so I let it go.&lt;/p&gt;&#xA;" OwnerDisplayName="user11702" LastActivityDate="2022-05-06T06:30:27.010" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25229" PostTypeId="2" ParentId="25221" CreationDate="2022-05-06T14:58:51.807" Score="5" Body="&lt;p&gt;Sometimes students will not engage with classes as a correct response to their current incentives. For instance, if a student is happy with the grade they would get in a course if they did no more work, they might shift focus to their other courses or other aspects of their lives. Same if a student doesn't see a practical way of passing a course. It sounds like this student has more pressing responsibilities in their graduation project.&lt;/p&gt;&#xA;&lt;p&gt;One of the important learning objectives of an undergraduate education is to learn how to prioritize one's tasks. For many students in many classes, these meta-lessons are more important than the actual content of the classes.&lt;/p&gt;&#xA;&lt;p&gt;Specifically for students who don't see a practical way of passing a course, I've found offering retakes to all of my students an effective but work-intensive way of getting some students back on board. This provides them with a clear pathway to success, although some may not be convinced if the pathway is too long or unfamiliar.&lt;/p&gt;&#xA;&lt;p&gt;It's hard to find a practical solution for students who are happy with their grade. Artificially inflating the priority of your class with grade penalties is too cruel. You could work to deflate the priority of students' other responsibilities with college-wide efforts to reduce student workloads, eliminate grades, or reduce the need for students to have jobs to pay for college, but these are big projects and can't address all of students' other priorities.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-05-06T14:58:51.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25230" PostTypeId="2" ParentId="25221" CreationDate="2022-05-06T17:57:27.823" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;From a practical perspective, I think you have to moderate your expectations.  It's good that you want to connect (and I'm basing this on previous questions as well).  But (a) you can't save them all.  And (b), you need to not let people get you down or let them screw with you either...there are just some social dynamics even games that go on, when one person is a presenter.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would advise to be outgoing, modulate your voice, wave your arms, etc.  Ham it up, some.  If you are normally a 5 or a 3, move it up to an 8, in terms of being high energy.  If you're doing that, you still won't reach them all.  But will get some people that you won't with a drone.  I'm not a natural high energy sales guy...more of a 10 caveats thinker.  But amping up the energy really helps me when teaching or just presenting research.  Yes, you can do it too much.  But I'd go two notches above your comfort factor.  That will be right level, then.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would definitely call on people.  Make a little bit of a joke about it, to take the edge off...but yeah, still call on people.  And not &amp;quot;do you understand&amp;quot; or &amp;quot;what questions&amp;quot; that make them passive and customers.  But &amp;quot;Johnny what step comes next&amp;quot;, or &amp;quot;Mary how do set up this integrating factor ODE&amp;quot;.  Don't torture them and move on if they can't answer...and then call on one of your safe people.  (You have the class to move along.)  But occasionally put the sleepers on the spot.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19848" LastActivityDate="2022-05-06T17:57:27.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25231" PostTypeId="2" ParentId="25205" CreationDate="2022-05-08T02:14:24.960" Score="-1" Body="&lt;p&gt;I just think of graphs, rather than input-output machines.  A circle is a relation.  xsq + ysq = rsq.  Gives you more than one y, for a given x.  [provided |x| is &amp;lt;r.]&lt;/p&gt;&#xA;&lt;p&gt;Lot's of good graphs to draw of equations that aren't functions.   I think mentioning something like that is useful as it's something dwelt on so much in geometry and algebra 2 class and is familiar to the students.  Rather than torturing so much with input-output machines.&lt;/p&gt;&#xA;&lt;p&gt;I think sometimes people here get so pro function that they forget all the wonderful relations, allowed to be relations.  Like graphing inverse trig functions.  The inverse of the trig function is not a function.  It's a relation.  There are multiple possible y values, because of how periodicity works.  Which is fine.  Relations exist, too.&lt;/p&gt;&#xA;" OwnerUserId="19857" LastEditorUserId="19857" LastEditDate="2022-05-08T02:35:02.180" LastActivityDate="2022-05-08T02:35:02.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25232" PostTypeId="1" CreationDate="2022-05-08T08:39:28.817" Score="2" ViewCount="122" Body="&lt;p&gt;(I have asked &lt;a href=&quot;https://math.stackexchange.com/q/4445546/118539&quot;&gt;this question&lt;/a&gt; on math.stackexchange.com and according to a suggestion from comments I am re-asking it here.)&lt;/p&gt;&#xA;&lt;p&gt;I can generate many examples of functions &lt;span class=&quot;math-container&quot;&gt;$f(x,y)$&lt;/span&gt;&#xA;for which finding local extrema is a good exercise for students, for example &lt;span class=&quot;math-container&quot;&gt;$f(x,y)=x^3+y^3−3xy$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x,y)=x^3+3xy^2−51x−24y$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x,y)=xy+\ln y+x^2$&lt;/span&gt;. The general idea of solving them is to obtain a polynomial of a small degree, for which the roots are easy to obtain, because, e.g., some of them are integer or rational. However, I do not know how to obtain nontrivial, i.e., not of the type &lt;span class=&quot;math-container&quot;&gt;$f(x,y)=(x−a)^2+(y−b)^2$&lt;/span&gt;, examples of functions with previously known extrema.&lt;/p&gt;&#xA;&lt;p&gt;A long time ago I found a simple rule for memorizing the values of &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;&#xA;for &lt;span class=&quot;math-container&quot;&gt;$x\in\{π/6,π/4,π/3,π/2\}$&lt;/span&gt;. I did not believe that it is not generally known. Indeed, many years later I found this observation in a written source.&lt;/p&gt;&#xA;&lt;p&gt;In this case, I also believe that the mathematical society knows the general rules of generating many examples of easy but not trivial exercises, only I do not know them. Are such rules known to you?&lt;/p&gt;&#xA;" OwnerUserId="19858" LastEditorUserId="376" LastEditDate="2022-05-08T09:59:14.340" LastActivityDate="2022-05-08T09:59:14.340" Title="Generating exercises about extrema of $f(x,y)$" Tags="|calculus|polynomials|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25233" PostTypeId="2" ParentId="3994" CreationDate="2022-05-08T16:59:07.370" Score="-2" Body="&lt;p&gt;I have a background in chemistry, through Ph.D.  Some dabbling in physics and engineering as well (classes and work).  Not a complete jock, but have a sense of the general STEM needs.  I almost never encountered hyperbolic trig functions, other than the treatment I saw in AP Calculus BC, c. 1982 (Thomas Finney).  They probably showed up somewhere.  But I remember them less than harder topics like Bessel functions or series solutions of ODEs or Laplace transforms.&lt;/p&gt;&#xA;&lt;p&gt;Given the above, I think the limited treatment was fine.  You could even make an argument to cut it more.  Pedagogically, I felt that the placement in TF calculus was fine (after trig, after exponentials, before series, before ODEs).  But if anything it was just a chance to do more manipulations.  Not something that I thought would connect to the real world the way exponentials and trig do.  I wondered maybe I'd see them more later in my STEM career.  But...no.&lt;/p&gt;&#xA;" OwnerUserId="19862" LastActivityDate="2022-05-08T16:59:07.370" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25234" PostTypeId="2" ParentId="25146" CreationDate="2022-05-09T03:13:32.130" Score="0" Body="&lt;p&gt;Sometimes we may need to bend the rules a little bit. In a polynomial equation (the easiest), you could say that the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; will turn into a &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; on the other side of the equation, but in addition to integration rules.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps, at the initial class of integration, it would be useful to recall that a function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is actually a  function of the form &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; and that it's derivative is &lt;span class=&quot;math-container&quot;&gt;$dy/dx$&lt;/span&gt;. So, when we integrate &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; will turn into &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; will turn into &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The easiest example could be &lt;span class=&quot;math-container&quot;&gt;$dy/dx = 1$&lt;/span&gt;, which turns out that &lt;span class=&quot;math-container&quot;&gt;$dy = dx$&lt;/span&gt;. So, when we integrate &lt;span class=&quot;math-container&quot;&gt;$\int dy = \int dx$&lt;/span&gt; will be equal to &lt;span class=&quot;math-container&quot;&gt;$y = x + c$&lt;/span&gt;. An extra &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; appears and the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; disappears.&lt;/p&gt;&#xA;&lt;p&gt;Another example would be  &lt;span class=&quot;math-container&quot;&gt;$y = a + bx + cx^2$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$dy/dx = b + 2cx$&lt;/span&gt;, thus &lt;span class=&quot;math-container&quot;&gt;$dy = (b + 2cx)dx$&lt;/span&gt;. So when we integrate, &lt;span class=&quot;math-container&quot;&gt;$\int dy = \int (b + 2cx) dx$&lt;/span&gt; it turns into &lt;span class=&quot;math-container&quot;&gt;$y = (b+cx)x$&lt;/span&gt;, losing the integer because of &amp;quot;integration rules&amp;quot;. Then, &lt;span class=&quot;math-container&quot;&gt;$y = bx + cx^2 + Constant$&lt;/span&gt;. An extra &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; appears and the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; disappears.&lt;/p&gt;&#xA;&lt;p&gt;Of course this simple rule won't last too long, but at least a beginner student will master the basics and get use to take away &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;. In addition, students should always remember that integrals are antiderivatives, and that trick helps a lot.&lt;/p&gt;&#xA;" OwnerUserId="19864" LastActivityDate="2022-05-09T03:13:32.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25235" PostTypeId="2" ParentId="3994" CreationDate="2022-05-09T05:12:23.207" Score="2" Body="&lt;p&gt;Lot's of great discussion in the other answers, and the comments by Dave Renfro were on point as usual. But, I'll add my two cents here:&lt;/p&gt;&#xA;&lt;p&gt;I think the reason that hyperbolic functions are largely ignored is multifaceted and likely to worsen as more and more educators buy into tempting arguments to downplay topics like trigonometric substitution and partial fractions because those calculations are &lt;em&gt;not interesting&lt;/em&gt; and can be easily done &lt;em&gt;with technology&lt;/em&gt;. Also, most calculators don't have hyperbolic trig functions, so why would you use them as an example when using the calculator is baked into the teaching of many.&lt;/p&gt;&#xA;&lt;p&gt;My own experience with calculus was that hyperbolic functions were not emphasized and perhaps not even mentioned. Now, probably if I went back and looked I could find hyperbolic sine and cosine somewhere in the textbooks I used. But, it didn't really get my notice until in the midst of my course on &lt;strong&gt;Special Relativity&lt;/strong&gt;. The professor used cosh and sinh and tanh as follows:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \tanh \phi = \beta \ \ \ \ \&amp;amp; \ \ \ \ \cosh \phi  = \gamma \ \ \ \ \&amp;amp; \ \ \ \ \sinh \phi  = \gamma \beta $$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\beta = v/c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the speed of light whereas &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; is the speed of the inertial frame. Here &lt;span class=&quot;math-container&quot;&gt;$\gamma = \frac{1}{\sqrt{1-\beta^2}}$&lt;/span&gt; so the assignment of rapidity is reasonable as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\cosh^2 \phi - \sinh^2 \phi = \gamma^2-\gamma^2\beta^2 = \gamma^2(1-\beta^2)=1. $$&lt;/span&gt; The quantity &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; is called the &lt;strong&gt;rapidity&lt;/strong&gt;. As a student, the rapidity was fascinating, new, and followed the simple rule &lt;em&gt;rapidities add&lt;/em&gt;. In contrast, velocities do not add in Special Relativity. Much to the horror of our professor, we explained didn't know what &amp;quot;cosh&amp;quot; and &amp;quot;sinh&amp;quot; were. That was my introduction. Later, as I taught Calculus and Differential Equations, and Complex Analysis I learned that not only were cosh and sinh essential topics, they are necessary to take a complete view of the exponential function.&lt;/p&gt;&#xA;&lt;p&gt;Let me give an example from the problem of inverse Laplace transforms.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \mathcal{L}^{-1} \left( \frac{s}{s^2+4s+13}\right) = \mathcal{L}^{-1} \left( \frac{s+2-2}{(s+2)^2+9}\right) = e^{-2t}\cos (3t) -\frac{2}{3}e^{-2t}\sin (3t) $$&lt;/span&gt;&#xA;Typically, when the denominator factors over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; then partial fractions is used in the standard course. But, there is no need for a problem which parallels the one above. In fact,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \mathcal{L}^{-1} \left( \frac{s}{s^2+4s-5}\right) = \mathcal{L}^{-1} \left( \frac{s+2-2}{(s+2)^2-9}\right) = e^{-2t}\cosh (3t) -\frac{2}{3}e^{-2t}\sinh (3t) $$&lt;/span&gt;&#xA;So, with both cos and cosh as well as sin and sinh, the technique of completing the square leads to a completely symmetric treatment for the non-repeated root case.&lt;/p&gt;&#xA;&lt;p&gt;It turns out there are generalizations of both trigonometric and hyperbolic trigonometric functions which appear in connection with the exponential over an algebra which is characteristic of a given &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th order ODE. The component functions of this exponential form a fundamental solution set. Cosine and sine and cosh and sinh are just the lowest order examples of this pattern.&lt;/p&gt;&#xA;&lt;p&gt;There is much that can be said about Calculus II as well, we see trigonometric substitution used as a method to eliminate radicals. Hyperbolic functions can be leveraged to make the same reduction. There are problems which are simple with the hyperbolic technique which are quite complicated in the trig. technique. Likewise, there are other problems which are easy in trig. substitution which are dastardly in the hyperbolic approach. Both methods should be used as the reinforce one another.&lt;/p&gt;&#xA;&lt;p&gt;There is a whole theory of hyperbolic trigonometry which mirrors the commonly known theory of circular trigonometry. I think Saul Stahl's text &lt;em&gt;A Gateway to Modern Geometry&lt;/em&gt; is a good place to look if you want to learn more. I suspect working through hyperbolic trigonometry would give a deeper understanding of the usual circle-based trig.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2022-05-09T05:36:29.463" LastActivityDate="2022-05-09T05:36:29.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25236" PostTypeId="2" ParentId="25146" CreationDate="2022-05-09T12:36:15.347" Score="0" Body="&lt;p&gt;The way that I learned this, was quite visual, and involved graphs and talk of dimensions/directions so may not work if this is a totally new approach to thinking about integration:&lt;/p&gt;&#xA;&lt;p&gt;When we intend to integrate&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int 2x\, dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We need to make it clear what role the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; is playing.&lt;/p&gt;&#xA;&lt;p&gt;Look at the graph of the formula to be integrated:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CvCvHm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CvCvHm.png&quot; alt=&quot;Graph showing y = 2x, from x = 0 to x = 2&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here, we have to understand for ourselves that the formula shows how to get from an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; value to a &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; value (i.e. by multiplying by two), but the formula &lt;span class=&quot;math-container&quot;&gt;$y = 2x$&lt;/span&gt; itself says &lt;em&gt;nothing&lt;/em&gt; about integration any enclosed area.&lt;/p&gt;&#xA;&lt;p&gt;When we integrate, we are changing the dimensions we are interested in. Here it's from a one dimensional line, to a two dimensional area. Elsewhere it's more complicated, but let's focus on the easy for now.&lt;/p&gt;&#xA;&lt;p&gt;Now this simple line is already splitting the graph into two areas, and when we integrate for most of it's just intuitive that we assume it's the area &lt;em&gt;under&lt;/em&gt; the graph, e.g. &lt;em&gt;towards&lt;/em&gt; the X-axis.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/sTVwIm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sTVwIm.png&quot; alt=&quot;The same y = 2 x graph, but with the area under the graph filled in&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But what in &lt;span class=&quot;math-container&quot;&gt;$\int 2x\, dx$&lt;/span&gt; tells us to fill in &lt;em&gt;that direction&lt;/em&gt;? It's the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To reiterate, the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; isn't decoration, it's direction. It's doesn't go &lt;em&gt;poof&lt;/em&gt; and vanish. In this analogy, if our integration is a journey then the formula tells us the way we &lt;em&gt;scale&lt;/em&gt; our step of X to the result of Y, but the integration shows us the &lt;em&gt;direction&lt;/em&gt; of the steps. Each &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; steps along the X-axis in the X direction.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;If might help to show the student this altered integration, and work through how to resolve it:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int 2x\, dy$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here, we want to integrate &lt;em&gt;with respect to &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&lt;/em&gt;, not &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. We're essentially flipping the graph around the gradient, and it points us to the fact that even though our gradient shows us how to get from &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; -&amp;gt; &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; (because we &lt;em&gt;still have&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$y = 2x$&lt;/span&gt;) but our direction, and integration &lt;em&gt;dimension&lt;/em&gt; has changed! If it merely vanished, we'd end up with the wrong result.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JnmMLm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JnmMLm.png&quot; alt=&quot;the graph but filled from the gradient to the y axis&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This graph looks bonkers, but it shows what a &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; means in place of the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;. We're filling in a different direction.&lt;/p&gt;&#xA;&lt;p&gt;Except there's a problem! We have nowhere in the formula for the &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; direction to point us; our scale and direction don't match!&lt;/p&gt;&#xA;&lt;p&gt;To get to something we can integrate, we have to re-arrange our formula:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$y = 2x \rightarrow x = \frac{y}{2}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int \frac{y}{2}\, dy$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Rearranging the formula lets us flip the axis of the graph back to what we're used to, but we're still keeping the integration direction of &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt;. We've just made the scale match for our convenience.&lt;/p&gt;&#xA;&lt;p&gt;The fact they have to match shows you that use the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; up. In polynomials, it directs all of it's matching 'scales' or powers to go up by one.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{aligned} \int 1x^0+2x^1+5x^2\, dx &amp;amp;= x^{0+1}+2x^{1+1}+5x^{2+1} + C \\ &amp;amp;= x^{1}+2x^{2}+5x^{3} + C \end{aligned}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;That &lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt; is from the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; because that's the 'direction' the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; takes us for polynomials. If you don't have a matching set of scales, you either need to rearrange to find one, or move everything except a &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; out of the integration.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{aligned} \int x+2x+5x^2\, dz &amp;amp;= (x+2x+5x^2)\times\int 1\, dz \\ &amp;amp;= (x+2x+5x^2)\times (z+C) \\ &amp;amp;= zx+2zx+5zx^2 + Cx+2Cx+5Cx^2 \end{aligned}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This concept of &lt;em&gt;direction&lt;/em&gt; is powerful, as it applies to more than just polynomials. Look at the cycle of trigonometric integrations:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/HBrgY.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HBrgY.png&quot; alt=&quot;circle showing four trigonometric functions, one at the top, left, right and bottom of the circle. clockwise they are: sin(x), cos(x), -sin(x), -cos(x). An arrow with an integration symbol points anti-clockwise between each function, and another arrow with a d/dx differentiation points clockwise&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now we're armed with our 'direction' concept, we can apply it here given the direction in the above image:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int \sin{x}\, dx = -\cos{x} + C$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; points us one formula around the circle, in the anti-clockwise direction (here anti-clockwise is pure convention - so long as we get between the right formulas it's ok). Taking that integration direction &lt;em&gt;uses&lt;/em&gt; up the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;, in the same way adding one to the 'scale' of each polynomial term uses up the &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8143" LastActivityDate="2022-05-09T12:36:15.347" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25237" PostTypeId="2" ParentId="3983" CreationDate="2022-05-10T11:21:38.770" Score="2" Body="&lt;p&gt;As a video called &amp;quot;The 4 Ways to Tell if a Matrix is Diagonalizable [Passing Linear Algebra]&amp;quot; by the YouTube channel &amp;quot;STEM Support&amp;quot; mentions, Eigenvalues and Eigenvectors are useful to create Diagonalizable Matrices.&lt;/p&gt;&#xA;&lt;p&gt;Diagonalizable Matrices make very resource-intensive (for computers) long matrix multiplications much, much simpler.&lt;/p&gt;&#xA;&lt;p&gt;Let's say I want to change my camera angle in a video game. I need to apply a rotation matrix to my current camera angle (camera angle being interpreted as beams of light closer to the player having larger angles than from distant objects).&lt;/p&gt;&#xA;&lt;p&gt;Great, I applied the rotation matrix. What's the issue? Well, for around 1,000,000 pixels (estimate for average 1,000x1,000 pixel laptop), I only generated one new image from multiplying that matrix by my current camera angle, a &lt;em&gt;&lt;strong&gt;single&lt;/strong&gt;&lt;/em&gt; frame in a game that should be operating around 60 Frames Per Second (fps).&lt;/p&gt;&#xA;&lt;p&gt;Let's say that my rotation matrix isn't the standard rotation matrix (involves sine and cosine in a 2x2 or 3x3 matrix). Let's say that I have my game only using a &amp;quot;camera-angle-modifying&amp;quot; matrix that doesn't require computing resource-intensive sines and cosines for &lt;em&gt;every&lt;/em&gt; possible combination of player camera angle movements.&lt;/p&gt;&#xA;&lt;p&gt;Next Camera Angle = [Transformation Matrix][Current Camera Angle Matrix]&lt;/p&gt;&#xA;&lt;p&gt;5 Camera Angles in the Future = [Transformation Matrix]&lt;sup&gt;5&lt;/sup&gt;[Current Camera Angle Matrix]&lt;/p&gt;&#xA;&lt;p&gt;Assuming my fixed-angle view-modifying transformation matrix is&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1 &amp;amp; 2\\ 3 &amp;amp; 4\end{bmatrix}$&lt;/span&gt;&#xA;and my current camera angle is&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}18 &amp;amp; 23\\ 39 &amp;amp; 42\end{bmatrix}$&lt;/span&gt;&#xA;, changing my camera angle only three times requires computing&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1 &amp;amp; 2\\ 3 &amp;amp; 4\end{bmatrix}^3$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}&#xA;18 &amp;amp; 23 \\&#xA;39 &amp;amp; 42 \\&#xA;\end{bmatrix}$&lt;/span&gt;&#xA;, which is computationally the same as&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1  &amp;amp;  2\\  3 &amp;amp;  4\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1  &amp;amp;  2\\  3 &amp;amp;  4\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1  &amp;amp;  2\\  3 &amp;amp;  4\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}18 &amp;amp; 23\\ 39 &amp;amp; 42\end{bmatrix}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1 &amp;amp; 2\\ 3 &amp;amp; 4\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1 &amp;amp; 2\\ 3 &amp;amp; 4\end{bmatrix}$&lt;/span&gt; =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1(1)+2(3) &amp;amp; 1(2)+2(4)\\ 3(1)+4(3) &amp;amp; 3(2)+4(4)\end{bmatrix}$&lt;/span&gt; =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}7 &amp;amp; 10\\ 15 &amp;amp; 22\end{bmatrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}7 &amp;amp; 10\\ 15 &amp;amp; 22\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1 &amp;amp;  2\\  3 &amp;amp;  4\end{bmatrix}$&lt;/span&gt; =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}7(1)+10(3) &amp;amp; 7(2)+10(4)\\&#xA;  15(1)+22(3) &amp;amp; 15(2)+22(4)\end{bmatrix}$&lt;/span&gt; =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}37 &amp;amp; 54\\ 81 &amp;amp; 118\end{bmatrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}37 &amp;amp; 54\\ 81 &amp;amp; 118\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}18 &amp;amp; 23\\ 39 &amp;amp;  42\end{bmatrix}$&lt;/span&gt; =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}37(18)+54(39) &amp;amp; 37(23)+54(42)\\&#xA;  81(18)+118(39) &amp;amp; 81(23)+118(42)\end{bmatrix}$&lt;/span&gt; = FinalCamAngle =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2772 &amp;amp; 3119\\ ... &amp;amp; ...\end{bmatrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1  &amp;amp;  2\\  3 &amp;amp;  4\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1  &amp;amp;  2\\  3 &amp;amp;  4\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}1  &amp;amp;  2\\  3 &amp;amp;  4\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}18 &amp;amp; 23\\ 39 &amp;amp; 42\end{bmatrix}$&lt;/span&gt; = FinalCamAngle =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}37 &amp;amp; 54\\ 81 &amp;amp; 118\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}18 &amp;amp; 23\\ 39 &amp;amp;  42\end{bmatrix}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Multiplying each of those matrices one at a time is an ugly and royal PiTA without any noticeable simplifying patterns, so let's try using a Diagonalized Transformation Matrix instead of any Transformation Matrix to apply camera angle rotations:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2 &amp;amp; 0\\ 0 &amp;amp; 7\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2 &amp;amp; 0\\ 0 &amp;amp; 7\end{bmatrix}$&lt;/span&gt; =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2(2)+0(0) &amp;amp; 2(0)+0(7)\\ 0(2)+7(0) &amp;amp; 0(0)+7(7)\end{bmatrix}$&lt;/span&gt; =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2^2 &amp;amp; 0\\ 0 &amp;amp; 7^2\end{bmatrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2^2 &amp;amp; 0\\ 0 &amp;amp; 7^2\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2 &amp;amp; 0\\ 0 &amp;amp; 7\end{bmatrix}$&lt;/span&gt; =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2^2(2)+0(0) &amp;amp; 2(0)+0(7)\\ 0(2)+7(0) &amp;amp; 0(0)+7^2(7)\end{bmatrix}$&lt;/span&gt;&#xA;= &lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2^3 &amp;amp; 0\\ 0 &amp;amp; 7^3\end{bmatrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2 &amp;amp; 0\\ 0 &amp;amp; 7\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2 &amp;amp; 0\\ 0 &amp;amp; 7\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2 &amp;amp; 0\\ 0 &amp;amp; 7\end{bmatrix}$&lt;/span&gt; =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2^3 &amp;amp; 0\\ 0 &amp;amp; 7^3\end{bmatrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2^3 &amp;amp; 0\\ 0 &amp;amp; 7^3\end{bmatrix}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}18 &amp;amp; 23\\ 39 &amp;amp;  42\end{bmatrix}$&lt;/span&gt; = FinalCamAngle =&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2^3(18) &amp;amp; 2^3(23)\\ 7^3(39) &amp;amp;  7^3(42)\end{bmatrix}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Which matrix is a prettier (and much easier to compute) transformation matrix?&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}37 &amp;amp; 54\\ 81 &amp;amp; 118\end{bmatrix}$&lt;/span&gt; or&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}2^3 &amp;amp; 0\\ 0 &amp;amp; 7^3\end{bmatrix}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm pretty sure you said the Diagonalized Matrix.&lt;/p&gt;&#xA;&lt;p&gt;That's why eigenvalues and eigenvectors are important. They allow the creation of diagonalizable matrices, which drastically simplify computational &lt;em&gt;&lt;strong&gt;and&lt;/strong&gt;&lt;/em&gt; time complexities.&lt;/p&gt;&#xA;" OwnerUserId="19780" LastActivityDate="2022-05-10T11:21:38.770" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25238" PostTypeId="2" ParentId="3983" CreationDate="2022-05-10T18:44:43.420" Score="-1" Body="&lt;p&gt;I would take a look at the treatment in Kreyszig Advanced Engineering Mathematics.  Or the treatment in Speigel Applied Differential Equations.  In both cases, they bring the topic in, in the context of pretty stripped down and engineer-oriented overviews of linear algebra.  So presumably they are not &amp;quot;proofy&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The Kreysig treatment is after the calc 3 topic.  (Also technically after ODES, but not dependent on it.)  The Speigel treatment is more diffyq oriented since it is a diffyQ text.&lt;/p&gt;&#xA;&lt;p&gt;In both cases they emphasize resonant frequencies as applications.  Which seems common sense to me.  I wouldn't bring in Markov process since this is higher math for most college sophomores/junior STEMs.  Even if I don't have a background in mechanics or spectral analysis, the idea of a resonant frequency (famous bridge shaking down) is pretty darned intuitive.&lt;/p&gt;&#xA;&lt;p&gt;I would also think the analogy to quantum energy levels in chemistry is helpful.  It's a course that most people have had.  You don't even need to know the whole Schroedinger development.  But it's well taught that the electrons can only have certain energies, only certain frequencies of emitted light, etc.  (People know this without following the math for it, basically trusting until p-chem or quantum physics that it will be justified.)  So, the eigenvalues are like the allowable energies.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I confess to never having a LA course.  Or really knowing what Igon values are.  See the famous Pinker takedown of Malcolm Gladwell.  &lt;a href=&quot;https://www.nytimes.com/2009/11/15/books/review/Pinker-t.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nytimes.com/2009/11/15/books/review/Pinker-t.html&lt;/a&gt;   So, I could be getting it wrong.  I'm probably right in the target zone of someone who ought to be able to grok and intuitive explanation.  Not a proof, mind you.  But a what sort of animal is this, explanation.&lt;/p&gt;&#xA;&lt;p&gt;P.s.s  I think it makes more sense to introduce this topic after or within first LA or ODE course.  (I saw it first in &amp;quot;engine math&amp;quot;, which was special topics advanced calc smorgasbord for engineers and physicists post ODEs, so ~fall of junior year in standard US college.)  This is well beyond most high schoolers.  Essentially bringing it in where it is, rather than trying to push it earlier into the curriculum.  We can't do everything and have to have an order.  So I think leaving this stuff for later is fine.  (I realize this is a comment on &amp;quot;if you should&amp;quot;, rather than &amp;quot;how&amp;quot; that you are asking.)&lt;/p&gt;&#xA;" OwnerUserId="19875" LastEditorUserId="19875" LastEditDate="2022-05-11T12:09:02.670" LastActivityDate="2022-05-11T12:09:02.670" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25239" PostTypeId="1" AcceptedAnswerId="25242" CreationDate="2022-05-12T02:26:48.740" Score="0" ViewCount="196" Body="&lt;p&gt;How can you improve exploring skills in high school mathematics education in order to guide students to get better opportunities to become mathematicians ?&lt;/p&gt;&#xA;&lt;p&gt;Here are my suggestions,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Let students to think about how to generalize the issue. As an example how to describe k- gonal numbers using the knowledge of triangle numbers and square numbers.&lt;/li&gt;&#xA;&lt;li&gt;Let students to think about alternative approaches specifically geometrical approaches.As an example geometrical proofs of A.M./ G.M. inequality.&lt;/li&gt;&#xA;&lt;li&gt;Let students have knowledge of history of mathematics to know how mathematics was developed over the generations into the present situation. This may provide better vision of how prevailing issues can be changed into new forms.&lt;/li&gt;&#xA;&lt;li&gt;Let students to use nontraditional or unconventional techniques because that gives much freedom for them to work on new ideas. As a second stage they can be asked to find rigorous proofs.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="2083" LastEditDate="2022-05-12T06:26:47.923" LastActivityDate="2022-05-12T16:03:38.723" Title="Importance of exploring skills in mathematics" Tags="|secondary-education|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25240" PostTypeId="2" ParentId="25239" CreationDate="2022-05-12T02:39:21.000" Score="2" Body="&lt;p&gt;I think by being upbeat and motivated and a good example, as a teacher.  A lot of people are inspired by HS teachers who were particularly magnetic.  I probably should/could have been a physicist, but ended up in chemistry (through grad school) from exposure to a remarkable teacher.&lt;/p&gt;&#xA;" OwnerUserId="19879" LastActivityDate="2022-05-12T02:39:21.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25241" PostTypeId="2" ParentId="25205" CreationDate="2022-05-12T07:01:32.107" Score="1" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$p(x,y)$&lt;/span&gt; be a two-variable polynomial (or really any function of two variables). Set &lt;span class=&quot;math-container&quot;&gt;$a R b$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$p(a,b) = 0$&lt;/span&gt;.  This is a binary relation that typically is not a function.&lt;/p&gt;&#xA;&lt;p&gt;Example: &lt;span class=&quot;math-container&quot;&gt;$p(x,y) = x^2 - 2y^2 - 1$&lt;/span&gt;.  Finding integers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a R b$&lt;/span&gt; amounts to finding integral solutions to a Pell equation.&lt;/p&gt;&#xA;&lt;p&gt;Look up “correspondences” in algebraic geometry to see that this kind of relation is used in math.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2022-05-12T07:01:32.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25242" PostTypeId="2" ParentId="25239" CreationDate="2022-05-12T16:03:38.723" Score="2" Body="&lt;p&gt;&amp;quot;How can you improve exploring skills [skills involved in discovering new mathematics] in high school mathematics education...?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I'm still feeling fuzzy on the question. It seems related to the question I ask myself and discuss with others. How do we make math meaningful for students?&lt;/p&gt;&#xA;&lt;p&gt;I agree with your implication that there are skills involved in exploring new ideas. And the way we teach math does not get at that.&lt;/p&gt;&#xA;&lt;p&gt;I've written about &lt;a href=&quot;http://mathmamawrites.blogspot.com/2010/02/sues-top-ten-issues-in-math-education.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;the problems with math education&lt;/a&gt;, which are closely linked to the problems with schooling. If kids met in small groups with mentors who knew them and could just offer interesting ideas to explore, with no agenda beyond helping the child's natural desire to learn expand into all the realms they might want to explore, that might be a structure that wouldn't damage their innate, intense desire to learn.&lt;/p&gt;&#xA;&lt;p&gt;You need to start long before high school, with a structure that respects kids' natural learning. And you need to give kids more freedom to choose. Once a kid has chosen to come to a group that is dedicated to playing around with math, what then? That's where math circles come in.&lt;/p&gt;&#xA;&lt;p&gt;Math circles are not classes in the usual sense. There are a bunch of younger people interested in exploring mathematical ideas, and there's a leader. But the leader does not teach. They propose a problem, and give nudges sometimes. They facilitate group dynamics. There are many sites online that offer &lt;a href=&quot;https://mathcircles.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;good math circle problems to explore (here's one)&lt;/a&gt;.  And for those of us who teach, Bob Kaplan (who died recently and will be missed) has some wonderful ideas about &lt;a href=&quot;https://drive.google.com/file/d/0B4Lou9CsLnQxaTdjODd6cFkxMjQ/view?usp=sharing&amp;amp;resourcekey=0-ti7DKQuTDM3TLPP1OJKBjw&quot; rel=&quot;nofollow noreferrer&quot;&gt;how to keep from stealing the show&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I hope this gives you some ideas, and is at least a partial answer to your question. It's a very deep question, with so many layers, I'm not sure I got at the parts you're interested in.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-05-12T16:03:38.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25243" PostTypeId="2" ParentId="18297" CreationDate="2022-05-13T04:04:03.747" Score="17" Body="&lt;p&gt;Possibly not what you're looking for, but: the things I've learned while grading are mostly pedagogical, not new mathematical facts (in fact, teaching at a community college as I do, I'm not sure that's ever happened).&lt;/p&gt;&#xA;&lt;p&gt;One of the main things that sticks with me is this: The rather incredible kaleidoscope of ways that students can misunderstand or be wrong about a thing. Generally the faculty in my department push the thesis that all-multiple-choice testing is fine (even required) for most courses up to calculus, say. The instinct is that it's &amp;quot;obvious&amp;quot; what the common mistakes might be, and these can be covered in a set of 3 or 4 distractor options.&lt;/p&gt;&#xA;&lt;p&gt;Now, I'm one of the very few instructors (maybe the only one now) who insists on at least a few open-ended questions on any of my tests to see what student work is actually like (and give feedback on it). In doing so, I've discovered a whole lot more &amp;quot;ways to be wrong&amp;quot; then I'd ever imagine, or that could be covered in a multiple-choice test. Looked at from another perspective: any multiple-choice test is an &lt;em&gt;enormous&lt;/em&gt; safety net, because it actually rules out the great &lt;em&gt;majority&lt;/em&gt; of student responses.&lt;/p&gt;&#xA;&lt;p&gt;One example: On a college algebra test a few years back, I asked: &amp;quot;Write the equation of the line, with the given properties, in slope-intercept form: Through (-2, 6) and (2, -7)&amp;quot;. Out of 40 test submissions, I found there were 26 different unique responses. (!) More specifically: 14 students got the right answer, 2 students duplicated a certain wrong answer, and 24 students each had a unique wrong answer, duplicated by no one else. (Which brings to mind Tolstoy's adage, &amp;quot;All happy families are alike; each unhappy family is unhappy in its own way.&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;Second example: For the first time this semester I'm giving programming tests on an actual computer in our lab. (For 20 years I gave programming tests on paper; transitioned to online tests for the COVID pandemic; and found enough advantages that I wanted to keep that as we switched back to in-person teaching.) Coincidentally, the lab has screen-monitoring software that's always on, so without planning it I found myself watching students write code in real-time on a test for the first time ever. I was amazed at how many of my second-semester CS majors couldn't write even basic structures; several were taking many shotgun attempts at simply declaring an array, or couldn't even write a simple for loop, for instance (e.g., mixing up bits of syntax between while, for, and do-while loops, taking as many as 10 minutes of iterations fired at the compiler trying to get it right). One student apparently actually memorized the entire practice test solution, typed that in first (with great difficulty and many compiler errors), and only once that was running tried to modify it to match the actual test question.&lt;/p&gt;&#xA;&lt;p&gt;Pretty fascinating stuff which I'd have never known if I didn't get to see the students' actual work process.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2022-05-13T16:41:28.890" LastActivityDate="2022-05-13T16:41:28.890" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="25244" PostTypeId="2" ParentId="18297" CreationDate="2022-05-15T03:41:17.950" Score="13" Body="&lt;p&gt;Reading the answer posted by Daniel R Collins reminded me of something else I learned while marking student work. Not exactly something mathematical, more something about constructing math exams.&lt;/p&gt;&#xA;&lt;p&gt;I had decided it wasn't fair to have the &amp;quot;distractors&amp;quot; in a multiple choice exam be answers that a student could come up with by making a simple error such as a sign error, the kind of error for which I would give partial credit if it weren't a multiple choice exam. So I wrote a test where all the distractors were crazy things that no student could possibly come up with.&lt;/p&gt;&#xA;&lt;p&gt;To my horror, the scores on this test were awful.&lt;/p&gt;&#xA;&lt;p&gt;I figured out why (I think). If a student makes a simple error, and arrives at an answer that is one of the choices, then the student marks that choice, and moves on to the next question. So, the student gets that one question wrong, but spends no more time on it, loses no confidence, and may well get some of the later questions right.&lt;/p&gt;&#xA;&lt;p&gt;But if a student makes a simple error, and arrives at an answer that is not one of the choices supplied, then the student goes back over the work, perhaps starts the problem from scratch, makes the same error, or maybe a new one, still doesn't arrive at an answer among the choices supplied, maybe panics, but in any event spends a lot of fruitless time on that one problem, has less time to spend on the rest of the test, and less confidence about tackling the rest of the test.&lt;/p&gt;&#xA;&lt;p&gt;So I learned the reason why distractors should be &amp;quot;plausible&amp;quot; answers. Maybe everyone else already knew that, but no one ever told me, and I had to learn it the hard way.&lt;/p&gt;&#xA;" OwnerUserId="13981" LastActivityDate="2022-05-15T03:41:17.950" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25245" PostTypeId="2" ParentId="3983" CreationDate="2022-05-15T17:31:38.560" Score="0" Body="&lt;p&gt;&lt;strong&gt;Disclaimer&lt;/strong&gt;: My answer only attempts to clarify what they are, not their importance.&lt;/p&gt;&#xA;&lt;p&gt;Eigenvectors are the vectors that do not change &lt;em&gt;aside from being scaled&lt;/em&gt; (do not &amp;quot;rotate&amp;quot;) when some linear transformation (matrix) is applied to them. The transformed vector will be on the same line as the original vector. So, such a vector is the &lt;em&gt;&lt;strong&gt;eigenvector&lt;/strong&gt; of that matrix&lt;/em&gt;. The factor by which they are scaled is known as the &lt;strong&gt;eigenvalue&lt;/strong&gt;.&#xA;If the direction of the vector is reversed due to the linear transformation, the eigenvalue is negative. Similarly, if the vector remains the exact same, the eigenvalue is 1.&lt;/p&gt;&#xA;&lt;p&gt;This answer is an attempt to give a slightly more specific explanation, building up on @RoryDaulton 's answer.&lt;/p&gt;&#xA;&lt;p&gt;(Also, I recommend watching the &lt;em&gt;&lt;a href=&quot;https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&quot; rel=&quot;nofollow noreferrer&quot;&gt;Essence of Linear Algebra&lt;/a&gt;&lt;/em&gt; series by 3Blue1Brown.)&lt;/p&gt;&#xA;" OwnerUserId="19888" LastEditorUserId="19888" LastEditDate="2022-05-15T18:11:19.307" LastActivityDate="2022-05-15T18:11:19.307" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25246" PostTypeId="2" ParentId="18297" CreationDate="2022-05-16T13:03:58.833" Score="5" Body="&lt;p&gt;This, from an exam I proctored for an absent teacher.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$y=32.5+27.5\cos\frac{\pi}{10}\left(x-8\right)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Students were given a real life word problem, the movement of a rider on a ferris wheel.&lt;/p&gt;&#xA;&lt;p&gt;A quick side note, one student came up to me to ask what a ferris wheel was, and I was reminded we often make assumptions about basic knowledge that might not be true. One can get to age 16 and never seen a ferris wheel.&lt;/p&gt;&#xA;&lt;p&gt;The equation above was the correct equation for the rider height at time x. The next question was to give the rider's height at time x=2. The correct answer was 24.002 which rounds nicely to 24. Multiple students answered -124.42. Because they entered the numbers into their calculator without proper parenthesis.&lt;/p&gt;&#xA;&lt;p&gt;For me, there are two issues. Students are using calculators from grade school, but not being taught proper use. We can use a bit of time each year to walk kids through the required keystrokes to get good results.&lt;/p&gt;&#xA;&lt;p&gt;Second, and most concerning. The students graphed the equation. The graphs were beautiful, minimum 5, maximum 60. I struggle to understand how they can so easily get such a result (a negative height!) and not return to the equation to track down their error.&lt;/p&gt;&#xA;&lt;p&gt;TL:DR&lt;/p&gt;&#xA;&lt;p&gt;In general, we need to address calculator skills with students. We also need to mindfully teach the skill of checking one's answer.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-05-16T13:03:58.833" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="25247" PostTypeId="1" AcceptedAnswerId="25249" CreationDate="2022-05-18T05:24:37.230" Score="4" ViewCount="117" Body="&lt;p&gt;When studying math in class, many students have a few moments falling behind, and they are reluctant to ask a quick question.&lt;/p&gt;&#xA;&lt;p&gt;Do we have a way (like an online web page or tool) for students to quicky access (with privacy and without any registration) for asking math questions?&lt;/p&gt;&#xA;&lt;p&gt;The following is what I really want to have:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students are given a code from the lecturer to access the platform. It might take less than 5 seconds for students to start asking questions using the given code.&lt;/li&gt;&#xA;&lt;li&gt;No information of students will be provided due to no registration. Only the lecturer can see the questions. And the lecturer can determine to answer them directly in class or reply back to the student who asked the question privately using the platfrom.&lt;/li&gt;&#xA;&lt;li&gt;Easy to type math formulas or upload images.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Thank for any suggestion.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Remark:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The code is used to distinguish different subjects or classes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user14865" LastEditorDisplayName="user14865" LastEditDate="2022-05-18T05:32:48.080" LastActivityDate="2022-05-18T22:19:16.980" Title="How to create a platform for students asking questions quickly and privately?" Tags="|technology-in-education|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25249" PostTypeId="2" ParentId="25247" CreationDate="2022-05-18T22:19:16.980" Score="0" Body="&lt;p&gt;A google form satisfies 1 &amp;amp; 2, and if you have a browser open to the google-sheet that holds the responses. I am not 100% certain that the google-sheet refreshes itself, so you may have to periodically refresh to see if questions are being asked.&lt;/p&gt;&#xA;&lt;p&gt;I do not know of anyway to satisfy (3) as well&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2022-05-18T22:19:16.980" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25250" PostTypeId="1" CreationDate="2022-05-19T00:20:58.530" Score="6" ViewCount="1300" Body="&lt;p&gt;I teach Pre-Calculus at a public high school in the United States. This school year, I allowed my students to retake any assessment regardless of how they performed on their first attempt. However, recognizing that I should focus on preparing my students for college, I want to be less care-free about retakes for next school year. To my knowledge, students are usually prohibited from retaking assessments in college (but feel free to provide evidence that says otherwise. It might depend on the professor...).&lt;/p&gt;&#xA;&lt;p&gt;Because I want my students to learn the course content, I still want to allow retakes. If a student is confident that they can perform better on an assessment and is willing to work hard to make that happen, then I don't see the harm in allowing them to retake it. I usually put questions on the retake that are similar to those on the original assessment.&lt;/p&gt;&#xA;&lt;p&gt;What I have noticed this year is students waiting until the end of the marking period to retake assessments after it has been weeks or even months since the original assessment was given. The significant amount of time between interacting with the material has led to some students performing worse on the retake.&lt;/p&gt;&#xA;&lt;p&gt;I have some ideas on how to improve my retake policy to ensure that my students are more prepared:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Giving students a deadline for retakes so that they are forced to do the retake sooner and be more likely to remember the required concepts. (However I have a hard time enforcing these kinds of deadlines because inevitably, students are busy with other school-related activities and need more time to prepare for the retake. Then again, I could simply lie about the deadline just to avoid students procrastinating and deal with the handful of students who need more time.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Not allowing a student to retake an assessment until they have turned in all of their missing homework assignments leading up to the assessment. This is intended to ensure that the student has practiced sufficiently and it also teaches them the value of doing the homework assignments to prepare for assessments. Alternatively, I could assign problems from the textbook and allow the student to retake the assessment after completing and reflecting on the approaches required to solve the problems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Require that a student who wants to do a retake meets with me after school for a short review session. I think it would also be a good idea to give a short 1-2 question assessment afterwards so they can see if they are ready for the retake or not.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similar to #3, hold one or more group study-sessions after school focusing on preparing for the retake of a certain assessment. I would walk the students through the solutions to the problems on the original version of the assessment, disclose common mistakes that I observed while grading, and allow students to ask questions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Has anyone tried anything similar to the ideas mentioned above? Based on your experience, what do you recommend?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="2083" LastEditDate="2022-05-19T07:15:03.490" LastActivityDate="2022-05-19T13:46:48.357" Title="How can I improve my Retake Policy?" Tags="|secondary-education|assessment|usa|struggling-students|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25251" PostTypeId="2" ParentId="25250" CreationDate="2022-05-19T02:26:17.750" Score="4" Body="&lt;p&gt;I'm not strictly familiar with how the US school year works, so take my suggestions as needing adaptation. I allow much restricted retakes in my classes, based on a system I met in undergrad:&lt;/p&gt;&#xA;&lt;p&gt;For reference, my school year is divided into 3 trimesters. In each trimester I give 3 assessments. At the end of the trimester, I allow students to retake one, and only one, assessment of their choosing. This way, whenever some students get a bad grade, they already know they have to avoid a further bad grade. This usually makes them put a bit more effort, and already approach me about material to study to the retake at the end of the trimester.&lt;/p&gt;&#xA;&lt;p&gt;By getting the student to have only one chance the idea is that the increased effort throughout the trimester helps them to be better prepared for the retake.&lt;/p&gt;&#xA;&lt;p&gt;One point to decide is whether you will allow the student to replace the grade only if it is better, or whether it is a strict replacement, so if they do worse they'll get a worse overall grade. The first encourages more students to take the replacement, and therefore to study, which is optimal. The latter will get many students not to try to replace an average grade but will avoid people showing up not having studied, so it lessens your effort at grading.&lt;/p&gt;&#xA;&lt;p&gt;The retake is at the last regular class of the trimester, so I can avoid dealing with students having different issues with other school classes, and becomes a minor reward for students who have dedicated themselves and can avoid a class and use the time for other classes.&lt;/p&gt;&#xA;" OwnerUserId="15277" LastActivityDate="2022-05-19T02:26:17.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25252" PostTypeId="2" ParentId="25250" CreationDate="2022-05-19T09:41:18.947" Score="-2" Body="&lt;p&gt;Many high school classes don't have a &amp;quot;retake policy&amp;quot;, so you could just devolve to that.  I.e. no retakes.  It's sort of nice what you are doing now, but definitely not required, in my experience.  I think if you give many assessments, it kind of works out.  But you will have to evaluate how much you want to do what is pedagogically most beneficial (HS has many features better than college) or &amp;quot;be like college&amp;quot;.  Personally, I would do what teaches the material best (am a fan of colleges that &amp;quot;teach like high school&amp;quot; anyhow.)&lt;/p&gt;&#xA;&lt;p&gt;FWIW, my teach in HS, for pre-calc and calc, that I quite liked her assessment policy, used the following assessments:&lt;/p&gt;&#xA;&lt;p&gt;*No graded homework.  Homework was for drill, for learning.  If you didn't do it, you paid on the tests.  She wasn't spending the time, grading it.  For calc, we used Thomas Finney c. 1980, which had all the answers in the back.  For pre-calc, many of the problems had answers in the back, and she made no effort to &amp;quot;assign the evens&amp;quot; or problems that didn't have answers...but she would sort of have both...obviously for those without answers, there would be more questions next class.  We would spend the normal time at beginning of next class on reviewing homework, sometimes with students at the board, or just questions from the class (those who did it).&lt;/p&gt;&#xA;&lt;p&gt;*Every Friday, a period (50min) long test.  Having a regular schedule is a very good management system.  And Friday means you're not ruining their weekend.  The teach was wicked fast at grading and would have the tests done by day end, even.  (She was the math chair and so had two of the six class periods free instead of only one.  And she would pound them during lunch and breaks, also.  And she would grade one classes, during others.)  In any case, kids for sure got them back MON, first thing, often earlier.  But rapid feedback drives the learning process...and anyone wants his results back ASAP after taking a test. Yes, you spend 20% (one of five days), doing exams vice lecture.  But actually exams are high stakes practice.  And practice, vice passive listening is a good use of lecture time.&lt;/p&gt;&#xA;&lt;p&gt;*Final exam worth 25%.  For the calc class, it was end of year (and anti-climactic with AP test done, but that's APs for ya).  For pre-calc, it was the end of each semester (end of 2nd and 4th quarter), since pre-calc was actually one semester of baby analysis/caculus (called &amp;quot;functions&amp;quot;) and one semester of analytic geometry.  I.e. two separate, formal courses.&lt;/p&gt;&#xA;&lt;p&gt;[Not saying this is the only way to skin the cat.  But it's an option to consider.  Yes I thought it was different at the time, but I/others really came to admire the regularity and sort of simplistic &amp;quot;fairness&amp;quot; of her method...pretty easy to calculate your grades.  And while the average was sort of draconian, no &amp;quot;makeups&amp;quot;, the frequency of assessments made it not so high stakes as &amp;quot;midterm and final&amp;quot; college classes.  Which I think are more based on lowering the work burden for instructors, not because it's better pedagogy.  I don't buy the &amp;quot;you're grown up now&amp;quot; excuse they make.]&lt;/p&gt;&#xA;" OwnerUserId="19907" LastActivityDate="2022-05-19T09:41:18.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25253" PostTypeId="2" ParentId="25250" CreationDate="2022-05-19T13:46:48.357" Score="7" Body="&lt;p&gt;As a college math professor, I have a retake policy. I've been trying to encourage other professors to develop retake policies, and it seems to be becoming more common with resources emerging like The Grading Conference &lt;a href=&quot;https://college-bridge.org/our-services/conferences/the-grading-conferences/&quot; rel=&quot;noreferrer&quot;&gt;https://college-bridge.org/our-services/conferences/the-grading-conferences/&lt;/a&gt; or Grading for Equity &lt;a href=&quot;https://gradingforequity.org/&quot; rel=&quot;noreferrer&quot;&gt;https://gradingforequity.org/&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;p&gt;I've settled on having quizzes be 100% of students' final grades, but I allow my students arbitrary retake attempts on each question (with the specifics scrambled each time) in office hours, and would be happy to award points for alternative demonstrations of understanding. To speed up grading, scores on questions are usually 0 or 1 depending if a student demonstrated the specific skill that question was trying to test or not.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Grading for Equity&lt;/em&gt; has lots of great advice about retakes, but the advice I've found most successful is setting aside class time for the retakes. When I give students one week's quiz, I also hand out a makeup version of last week's quiz, and in place of a midterm or final I have a &amp;quot;makeup midterm&amp;quot; and &amp;quot;makeup final&amp;quot; which are just compilations of new versions of all the questions students still have to make up. I've also experimented with setting aside time at the end of each class just for makeups which worked well but was time consuming.&lt;/p&gt;&#xA;&lt;p&gt;Not allowing students to do a retake until they've demonstrated that they've done some work studying is a good way to address students who are spamming your time, and I suggest this approach for any retake methods that are extra time consuming for you, such as meeting outside of class.&lt;/p&gt;&#xA;&lt;p&gt;But I think that the experience of a student seeing a makeup quiz in front of them and having plenty of time to work on it, and wishing that they had reviewed so they knew what to do tends to be a strong enough feeling that most students will start preparing for the makeup quizzes.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-05-19T13:46:48.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25254" PostTypeId="1" AcceptedAnswerId="25255" CreationDate="2022-05-19T23:46:00.163" Score="4" ViewCount="1432" Body="&lt;p&gt;About thirty years ago, in a practice GRE (Graduate Record Exam) math&#xA;test in the US,&#xA;a question assumed the student knew the definition of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Wronskian&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wronskian&lt;/a&gt;. I had never heard of this determinant&#xA;before.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Q&lt;/strong&gt;&lt;/em&gt;. Is this still assumed to be part of what a student heading toward&#xA;a graduate degree in mathematics knows?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2022-05-20T00:04:51.040" LastActivityDate="2022-05-21T02:29:42.043" Title="Is the Wronskian still assumed for graduate education?" Tags="|undergraduate-education|linear-algebra|graduate-education|differential-equations|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25255" PostTypeId="2" ParentId="25254" CreationDate="2022-05-20T01:27:14.077" Score="8" Body="&lt;p&gt;I would say the assumption is that people heading to mathematics graduate school know about the Wronskian, but this assumption isn't universally true.&lt;/p&gt;&#xA;&lt;p&gt;Certainly, anyone who has studied a semester of differential equations (and is heading to graduate school) should know it.&lt;/p&gt;&#xA;&lt;p&gt;However, there is a substantial minority of people interested in pure mathematics of a generally algebraic bent who never study differential equations.  (I was one of these people until I was assigned to teach it!)&lt;/p&gt;&#xA;&lt;p&gt;There really isn't anything that's universally expected (rather than merely generally assumed) of people heading to mathematics graduate school other than an ability to read and write proofs, and some folks doing mathematical modelling might even disagree with that.&lt;/p&gt;&#xA;" OwnerUserId="4991" LastEditorUserId="4991" LastEditDate="2022-05-20T01:33:20.790" LastActivityDate="2022-05-20T01:33:20.790" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25256" PostTypeId="2" ParentId="25254" CreationDate="2022-05-20T01:28:06.210" Score="4" Body="&lt;p&gt;It is discussed in all the introductory DEqns texts of which I've used. It's needed to complete the discussion of linear independence of solution sets. Together with Abel's formula it provides some rather general theorems for linear n-th order ODEs. The Wronskian lies at the heart of variation of parameters which is the general method to solve nonhomogeneous n-th order ODEs.&lt;/p&gt;&#xA;&lt;p&gt;It's also a bit tricky since the theorems which hold for solution sets do not hold for arbitrary sets of functions. In particular, the Wronksian can vanish identically for linearly independent functions. But, that can never happen for a solution set thanks to Abel's formula.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, all these things generalize to calculus over a finite dimensional commutative algebra.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2022-05-21T02:29:42.043" LastActivityDate="2022-05-21T02:29:42.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25257" PostTypeId="2" ParentId="16858" CreationDate="2022-05-20T01:28:41.747" Score="5" Body="&lt;p&gt;To be succinct: contest problems are meant to be accessible to a wide audience, so cannot &lt;em&gt;overtly&lt;/em&gt; involve somewhat-more-sophisticated mathematical ideas ... that many people would not yet have heard about.&lt;/p&gt;&#xA;&lt;p&gt;Naturally, since the general degree of sophistication has a bound, the questions will become at least a little byzantine, since standard questions at this introductory level would be answered by everyone, and there'd be no game. :)&lt;/p&gt;&#xA;&lt;p&gt;Unsurprisingly, although it's good to be able to succeed in such games, it is not at all the same as making a genuine contribution to further understanding of mathematics. There should be no surprise here!&lt;/p&gt;&#xA;&lt;p&gt;Still, having the enthusiasm, and chops, to do well in such contests is a positive. In my own arc, doing something positive on the Putnam Exam (in the U.S.) helped with my credibility...&lt;/p&gt;&#xA;&lt;p&gt;But/and I've seen several people who enjoyed contests, and excelled at them, but did not so much enjoy the professional math business. For various reasons, I infer.&lt;/p&gt;&#xA;&lt;p&gt;So: contests are meant to be accessible.&lt;/p&gt;&#xA;&lt;p&gt;Accessible-math is not necessarily what any of many sorts of professional mathematicians do, day-to-day. This should not be surprising. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-05-20T01:28:41.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25259" PostTypeId="2" ParentId="25172" CreationDate="2022-05-20T07:34:51.043" Score="4" Body="&lt;p&gt;There are many, many books on ODEs, many of them good.&lt;/p&gt;&#xA;&lt;p&gt;For the basic theory no one seems to have improved much on the book of Coddington and Levinson cited by Gerald Edgar. That book is old-fashioned, has essentially no examples, and could be seen as quite dry. It is clearly written and I was able to learn from it.&lt;/p&gt;&#xA;&lt;p&gt;V. I. Arnold. &lt;em&gt;Ordinary Differential Equations&lt;/em&gt; is an excellent book because the geometric ideas behind ODEs are explained. Arnold does not write out all the details (for me this is a virtue of his books), but the writing is always careful and one learns a lot by filling in the details. His more advanced book &lt;em&gt;Geometrical methods in the theory of ordinary differential equations&lt;/em&gt; is wonderful but requires some background.&lt;/p&gt;&#xA;&lt;p&gt;Another book in which topological/geometrical ideas operate is that of M. Hirsch and S. Smale, &lt;em&gt;Differential equations, dynamical systems, and linear algebra&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A student who has a good background in geometry and topology should also learn about numerics and practical solvability issues.&lt;/p&gt;&#xA;&lt;p&gt;For learning the theory behind numerical methods for solving ODEs I like John Butcher's &lt;em&gt;Numerical methods for ordinary differential equations&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A good introduction to how ODEs are solved in practice is the self-descriptively titled&#xA;Hairer, Nørsett,  Wanner, &lt;em&gt;Solving ordinary differential equations. I&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For a student with geometric interests it might be interesting to read &lt;em&gt;Numerical Hamiltonian problems&lt;/em&gt; by Sanz-Serna and Calvo and &lt;em&gt;Geometric numerical integration&lt;/em&gt; by Hairer, Lubich, and Wanner. These are about numeric methods adapted to the underlying geometry related to the flow being integrated.&lt;/p&gt;&#xA;&lt;p&gt;I'm not an expert in ODEs and as I observed there are many, many books available. I just listed some I like written by people with good mathematical judgment.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2022-05-20T07:34:51.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25260" PostTypeId="1" CreationDate="2022-05-20T20:58:06.073" Score="3" ViewCount="169" Body="&lt;h2&gt;TL;DR:&lt;/h2&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Tell me which topics should i study the most, based on this three tests:&lt;/p&gt;&#xA;&lt;p&gt;Mathematics (A):&#xA;&lt;a href=&quot;https://www.studyinjapan.go.jp/ja/_mt/2021/06/2020_ga_math_a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2020&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.studyinjapan.go.jp/ja/_mt/2021/06/2019_ga_math_a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2019&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.studyinjapan.go.jp/ja/_mt/2021/06/2018_ga_math_a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2018&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This question may sound a bit weird, since the natural answer would be &amp;quot;study whats already is on the test&amp;quot; but i'm wanna share a bit of context for this.&lt;/p&gt;&#xA;&lt;h4&gt;Context:&lt;/h4&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So i'm currently studying for the Japanese Scholarship known as MEXT in which every year the japanese embassy of each country gave the opportunity to enroll on a japanese university and go to study in that country. As an undergraduate student this is a big deal specially if there's a lack of opportunities in your current country.&lt;/p&gt;&#xA;&lt;p&gt;I also want you to know that here in my country, there is a lack of proper mathematical education when it comes to high school, pretty much anyone who obtained a bachelor degree's are not well prepared to face one of this japanese test designed for undergraduate students since the test, present to you calculus, trigonometry, modular-arithmetic, number-theory, etc. problems which you don't see until you reach third semester of any university engineer-based career (at least here where i live)&lt;/p&gt;&#xA;&lt;p&gt;Finally i want to tell that i've been studying calculus for a bit and i do know the basics and i don't have any problem with arithmetic or algebra.&lt;/p&gt;&#xA;&lt;h2&gt;So here is the question:&lt;/h2&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Could somebody look at those test and tell me which topics should i study the most? Below (in the comment section) you will find the three test that were used in past examinations and contains all the questions, i also will leave a link to the answers in case someone want to deep onto this:&lt;/p&gt;&#xA;&lt;h4&gt;Current approach and status:&lt;/h4&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In my current approach i tried solving most of the exercises presented and always fall in the same loop of no knowing, research, try solving, fail, research again, try solving again, failing, and finally making a question here. This of course may not be the best for a huge variety of topic since you have to make sure to &lt;strong&gt;understand&lt;/strong&gt; what you are doing before going into any kind of examination but with so many themes its hard to pick one thing to study since you're pretty much dropping everything else in order to solve only one thing.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MhGqK.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MhGqK.png&quot; alt=&quot;flow chart&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h3&gt;The ideal:&lt;/h3&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;When making this post i'm hoping someone could orient me through a path of things i should know and preferably master in order to be well prepared for the test.&lt;/p&gt;&#xA;" OwnerUserId="19913" LastEditorUserId="14627" LastEditDate="2022-05-20T21:03:38.160" LastActivityDate="2022-05-23T19:08:18.107" Title="How to study for a mathematics undergraduate entrance examination?" Tags="|self-learning|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25261" PostTypeId="2" ParentId="25260" CreationDate="2022-05-22T00:45:36.930" Score="3" Body="&lt;p&gt;You have described your situation clearly. But you have asked the wrong question. I don't think your problem is lack of knowledge &lt;em&gt;(which topics to study&lt;/em&gt;). It is a lack of understanding of how to play around (so &lt;em&gt;how to study&lt;/em&gt; is actually a good question). Most of these problems need a creative exploration. (There were a few that I would throw my hands up at, but not many.)&lt;/p&gt;&#xA;&lt;p&gt;I recommend that you check out &lt;a href=&quot;https://artofproblemsolving.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Art of Problem Solving&lt;/a&gt;. They teach a number of math courses, but they also teach courses to prepare students for math competitions. It looks like &lt;a href=&quot;https://artofproblemsolving.com/school/course/mathcounts-basics&quot; rel=&quot;nofollow noreferrer&quot;&gt;this is the lowest level of those courses&lt;/a&gt;. If the &lt;span class=&quot;math-container&quot;&gt;$\$360$&lt;/span&gt; pricetag is not reasonable to you, then you might consider &lt;a href=&quot;https://artofproblemsolving.com/alcumus&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;The Art of Problem Solving, Volume 1&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;, a textbook to help students get started on creative problem solving. That is &lt;span class=&quot;math-container&quot;&gt;$\$47$&lt;/span&gt; for an online book only. Still too much money? They have free resources in &lt;a href=&quot;https://artofproblemsolving.com/alcumus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alcumus&lt;/a&gt;. If you follow some of the questions and answers there, you might learn something just from that. The people at AOPS are dedicated to helping students learn to think creatively in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;There is also a wonderful book by Paul Zeitz, &lt;em&gt;&lt;strong&gt;The Art and Craft of Problem Solving&lt;/strong&gt;&lt;/em&gt;. &lt;a href=&quot;https://www.biblio.com/search.php?stage=1&amp;amp;author=zeitz&amp;amp;title=the%20art%20and%20craft%20of%20problem%20solving&amp;amp;pageper=20&amp;amp;omit_product_types=bp,bd,ns&amp;amp;strip_common=1&amp;amp;program=1005&amp;amp;order=priceasc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here are some copies on a site that sells used books&lt;/a&gt;. Or, if you live in a big city, maybe you could find it in a library?&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="511" LastEditDate="2022-05-22T12:22:36.500" LastActivityDate="2022-05-22T12:22:36.500" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25262" PostTypeId="1" CreationDate="2022-05-22T13:14:44.567" Score="3" ViewCount="239" Body="&lt;p&gt;what is the simplest way to explain how to determine whether the resolution of a triangle (finding all its sides and angles, given 3 of them) using the sine theorem gives one or two solutions and whether the solution found is correct? I have explained congruent triangles to my pupils (ASA, SAS, SSA, etc), but they can get the wrong answer if they change the order of calculating the angle (for instance, finding angle alpha before angle beta, which gives a seemingly correct answer unless you verify that the smaller angle corresponds to the smaller side, and so on). They could double check with the cosine theorem, but it seems a bit long.&#xA;Thank you&lt;/p&gt;&#xA;" OwnerUserId="19916" LastActivityDate="2022-05-23T04:38:57.717" Title="Simple way to explain Sine theorem applications" Tags="|trigonometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25264" PostTypeId="2" ParentId="25262" CreationDate="2022-05-22T15:32:11.363" Score="3" Body="&lt;p&gt;Summarising &lt;a href=&quot;http://jwilson.coe.uga.edu/EMT668/EMAT6680.2001/Mealor/EMAT%206700/law%20of%20sines/Law%20of%20Sines%20ambiguous%20case/lawofsinesambiguouscase.html#:%7E:text=Law%20of%20Sines%2D%2DAmbiguous%20Case&amp;amp;text=For%20those%20of%20you%20who,of%20those%20angles%20(SSA).&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page&lt;/a&gt;: when solving &lt;span class=&quot;math-container&quot;&gt;$$\frac{\sin A}{a}=\frac{\sin \color{red}B}{b}$$&lt;/span&gt; for angle &lt;span class=&quot;math-container&quot;&gt;$\color{red}B,$&lt;/span&gt; and the triangle altitude that intersects with sides &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; has height &lt;span class=&quot;math-container&quot;&gt;$h,$&lt;/span&gt; ambiguity arises precisely when&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;angle &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is acute  and&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$h&amp;lt;a&amp;lt;b.$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eKdjR.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eKdjR.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Sue VanHattum's comment is pertinent:&lt;/p&gt;&#xA;&lt;p&gt;I like this diagram, and I teach my students to &lt;strong&gt;draw the unknown side horizontally,&lt;/strong&gt; so that their diagram will look like this. &lt;strong&gt;Then draw angle A, then side b. Then you can imagine side a swinging back and forth, and finding its two positions.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-05-23T04:38:57.717" LastActivityDate="2022-05-23T04:38:57.717" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25265" PostTypeId="2" ParentId="25260" CreationDate="2022-05-23T18:49:44.710" Score="3" Body="&lt;p&gt;Carlos, Sue has a nice answer.  To add to it, there are probably a lot of problems in regular algebra 2, trig, pre-calc, and calc books that would help you, if you concentrate on those that are the harder part of the review exercises.  Usually those are with larger numbers.  Most homework exercises start with plug and chug and move to those needing a big more trickery.  Or sometimes they are &amp;quot;starred&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;You seem to have very good English, so I think lots of math books in that language would be helpful.  Look online or in libraries.  Those that have answers in the back are more useful (can check your work, or sometimes seeing the answer will help you figure out how to do the problem.  MSE is fine, but it is not efficient to go there for every hurdle.)  And I'm not suggesting to look in the back until you've solved it (to check) or given it a solid try (do the whole section of problems before checking...often your subconsious will figure it out whjile working on other problems or even a nearby problem will have an insight to help you.)&lt;/p&gt;&#xA;&lt;p&gt;By the way, that is a beautiful description of your issues, especially with the flow chart.  But I think you lack some facility/familiarity with doing a lot of manipulations.  Those problems you showed are not ball-buster hard.  I looked up your MSE questions and found this one:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/4444621/help-with-frac-sin-theta-cos-theta-sin-theta-cos-theta&quot;&gt;https://math.stackexchange.com/questions/4444621/help-with-frac-sin-theta-cos-theta-sin-theta-cos-theta&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The key insight was sinsq + cossq =1.  About the most common trig identity around! But you didn't try it, or mention trying it.  When I see you missing that, I think you haven't done enough &amp;quot;trig identity playing around with&amp;quot; problems in general.  Lots of the regular trig books will have a bunch of these style problems.&lt;/p&gt;&#xA;&lt;p&gt;I'm far from a math jock, but some of those Japanese questions don't look THAT hard.   Like 1-2 on the 2020 exam was one about minimum distance from a Cartesian coordinate point to a curve.  Did you try just putting in the distance formula and minimizing? Take a derivative and set to zero to find extrema...basic min/max stuff!  You probably can ignore the square root and just minimize the argument inside.  If that's an issue, I wonder if you've done enough min/max problems in general.  By the way, I just pulled down my AP Calculus book from the 80s (Thomas Finney) and a completely analogous problem is there (#17).  I would think this is a pretty normal problem in calculus, really.  And really the whole section of 40 problems are excellent.  Lots of word problems (they are &amp;quot;harder&amp;quot; and make you think, also.)  Not a single problem that is plug and chug.&lt;/p&gt;&#xA;&lt;p&gt;So, net/net, I don't think you are that rock solid on algebra, trig, and calculus, at least as commonly taught.  I think doing more problems in regular books will help you.  Fine if you want to emphasize the &amp;quot;back of the section&amp;quot; ones more.  But I wouldn't be surprised if some drill with the middle of the section or even the beginning of the section (at least to get more so these things are familiar) wouldn't hurt either.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I think there's a general issue of students in general thinking that &amp;quot;I get it&amp;quot; is sufficient.  The human mind is not a perfect computer, not a rules based system.  &amp;quot;I get it&amp;quot; is just the start.  You need to build familiarity by doing large volumes of problems.  sinsq plus cossq should be as familiar to you as the roof of your mouth is to your tongue.  Or like if you were learning the hip throw in judo...I get it is the start...we need to do a lot of them and develop comfort/familiarity.&lt;/p&gt;&#xA;" OwnerUserId="19907" LastEditorUserId="19907" LastEditDate="2022-05-23T19:08:18.107" LastActivityDate="2022-05-23T19:08:18.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25266" PostTypeId="2" ParentId="18030" CreationDate="2022-05-24T01:59:43.653" Score="4" Body="&lt;p&gt;I also felt this need a lot in the last few years. So I ended up developing &lt;a href=&quot;https://latex-whiteboard.herokuapp.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;my own solution&lt;/a&gt;. It is quite simple and with far fewer features than the other suggestions, but I think the real-time LaTeX support is good enough and it allows you to choose the position as well as traditional text tools.&lt;/p&gt;&#xA;&lt;p&gt;Colaborative version: &lt;a href=&quot;https://latex-whiteboard.herokuapp.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://latex-whiteboard.herokuapp.com/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Single user version: &lt;a href=&quot;https://fpelogia.github.io/whiteboard/public/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://fpelogia.github.io/whiteboard/public/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Github (feel free to fork it):&lt;a href=&quot;https://github.com/fpelogia/whiteboard&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://github.com/fpelogia/whiteboard&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Printscreen:&lt;br&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/v2iYY.jpg&quot; width=&quot;400&quot; /&gt;&lt;/p&gt;&#xA;" OwnerUserId="19925" LastEditorUserId="19925" LastEditDate="2022-05-24T02:00:22.710" LastActivityDate="2022-05-24T02:00:22.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25268" PostTypeId="1" CreationDate="2022-05-24T13:11:57.013" Score="3" ViewCount="113" Body="&lt;p&gt;Good morning,&#xA;I use Moodle for Mathematics High Schools Students.&lt;/p&gt;&#xA;&lt;p&gt;They have to reply to questions by writing some short formulas. I've not yet figured out a way of doing this without installing additional plugins. Plugins are a headache when it comes to upgrading Moodle.&lt;/p&gt;&#xA;&lt;p&gt;There's seem to be no question type able to give this possibility.&lt;/p&gt;&#xA;&lt;p&gt;I already asked for help on the Moodle forum but they didn't reply or misunderstood the question.&#xA;I also tried on Twitter and Reddit, but to no avail.&lt;/p&gt;&#xA;&lt;p&gt;Looking forward to hearing from you.&lt;/p&gt;&#xA;" OwnerUserId="16379" LastActivityDate="2022-05-24T13:11:57.013" Title="How to let students answer a Moodle Quiz by typing (or pasting) LaTeX code?" Tags="|latex|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25274" PostTypeId="1" CreationDate="2022-05-26T21:38:05.817" Score="5" ViewCount="343" Body="&lt;p&gt;I remember reading an old calculus book (years 1920-1930) and in the preface it was portrayed as revolutionary because it was for high school students. Nowadays, that is not revolutionary, because most countries have calculus in high school. So, this made me wonder: how did we go from having no calculus in high school to having calculus almost world-wide in high school? How exactly does such a significant shift in curriculum happen, with a new subject area added to high school?&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorUserId="-1" LastEditDate="2023-03-20T22:27:00.700" LastActivityDate="2023-03-22T02:39:25.437" Title="How does the average level of expected mathematical sophistication at high school level increase?" Tags="|calculus|curriculum|teaching|history|history-of-math|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25275" PostTypeId="1" AcceptedAnswerId="25278" CreationDate="2022-05-26T18:13:21.347" Score="5" ViewCount="967" Body="&lt;p&gt;I've come across the term &lt;em&gt;annular ring&lt;/em&gt; in parentheses following &lt;em&gt;washer&lt;/em&gt; in my calculus textbook: &amp;quot;has the shape of a washer (an annular ring)&amp;quot;. The definition of the word &amp;quot;annular&amp;quot; in &lt;a href=&quot;https://www.merriam-webster.com/dictionary/annular&quot; rel=&quot;nofollow noreferrer&quot;&gt;Merriam Webster&lt;/a&gt; is &amp;quot;of, relating to, or forming a ring&amp;quot;. With that, the term &lt;em&gt;annular ring&lt;/em&gt; seems redundant to me. A Google search of the phrase returns around 946,000 results, and leads me to question my conclusion.&lt;/p&gt;&#xA;&lt;p&gt;From what I have found, it appears that &lt;a href=&quot;https://en.wikipedia.org/wiki/Annulus_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;annulus&lt;/a&gt; would have been a more appropriate choice by the author of this textbook, as that is what the shape in question represents.&lt;/p&gt;&#xA;&lt;p&gt;Is this usage redundant, or have I missed a subtlety in its usage? For example, does mathematics (especially in calculus and/or geometry) provide for non-annular rings? If so, what distinguishes a ring that is annular from a ring that is not annular but still qualifies as a ring?&lt;/p&gt;&#xA;" OwnerUserId="19934" OwnerDisplayName="Brian" LastEditorUserId="7481" LastEditDate="2022-05-28T18:16:10.400" LastActivityDate="2022-05-28T18:16:10.400" Title="Is &quot;Annular Ring&quot; redundant?" Tags="|calculus|geometry|terminology|" AnswerCount="2" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="25276" PostTypeId="2" ParentId="25275" CreationDate="2022-05-27T00:25:15.310" Score="4" Body="&lt;p&gt;Perhaps &amp;quot;formally redundant&amp;quot;, depending on who you ask. As in the many comments. But/and a certain amount of redundancy can be good, if it saves time and confusion. &amp;quot;Let me repeat that in different words...&amp;quot; :)&lt;/p&gt;&#xA;&lt;p&gt;In the classes I teach, I do try to explain the sense in which I'll use words, pretending to avoid having to explain at-the-moment, but I do invariably explain at-the-moment also, ... &lt;em&gt;and&lt;/em&gt; meta-explain that setting context is surely inescapable for genuine technical communication.&lt;/p&gt;&#xA;&lt;p&gt;So, yeah, maybe some people would find it redundant, but, manifestly, not everyone. And, apart from the time spent talking about whether it's redundant, it is more efficient to be slightly redundant, to avoid confusion. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-05-27T00:25:15.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25278" PostTypeId="2" ParentId="25275" CreationDate="2022-05-27T12:28:41.373" Score="9" Body="&lt;p&gt;I argue that &lt;em&gt;annular ring&lt;/em&gt; is not redundant because in common use &lt;em&gt;annulus&lt;/em&gt; and &lt;em&gt;ring&lt;/em&gt; are not synonyms. Put another way, not all rings are annular.&lt;/p&gt;&#xA;&lt;p&gt;The adjective &lt;em&gt;annular&lt;/em&gt; is used to emphasize that the ring in question is flat in the sense of being close to planar.&lt;/p&gt;&#xA;&lt;p&gt;In colloquial use a &lt;em&gt;ring&lt;/em&gt; is something tubular (like a donut). That is, a section in a plane crossed by the tube is roughly circular. While &lt;em&gt;annulus&lt;/em&gt; has no colloquial use, it generally means something approximating the planar region delimited by two concentric circles. That is, a section in a plane crossed by the tube is roughly a short, wide rectangle. Together an &lt;em&gt;annular ring&lt;/em&gt; denominates a sort of flattened tube - that is a washer.&lt;/p&gt;&#xA;&lt;p&gt;The terminology &lt;em&gt;annular ring&lt;/em&gt; is common in electronics where it is used to refer to the ring of copper around a hole drilled through the copper pad on a (generally flat) circuit board.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2022-05-27T16:03:14.763" LastActivityDate="2022-05-27T16:03:14.763" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25279" PostTypeId="2" ParentId="21203" CreationDate="2022-05-27T13:20:17.067" Score="2" Body="&lt;p&gt;I'm not sure how I missed this question when it was first posted, and it already has many answers, but I feel there are some important things that still need to be said to address your specific concerns.&lt;/p&gt;&#xA;&lt;p&gt;“but aren’t we just dividing by zero? ... why are we doing maths where there are zeros on both the numerator, and the denominator, that's undefined,... right?” I can't claim to be giving an &amp;quot;experience based&amp;quot; answer since, in my years of teaching calculus I can't recall ever having been asked that question. Maybe it doesn't actually bother most students, or perhaps my teaching approach headed this off at an early stage, or perhaps students found it difficult to approach me---my classes were extremely large. For whatever reason, it never came up. I do understand from my own learning of calculus why it bothered you though, and, if the question does arise, I think it needs to be addressed in an honest and direct way.&lt;/p&gt;&#xA;&lt;p&gt;What I would emphasize is that the processes of taking limits and derivatives are defined so that this is explicitly ruled out. You don't have to give the formal definition of limit at first---an intuitive approach is fine---but you do need to tell them that when they see the formal definition of &lt;span class=&quot;math-container&quot;&gt;$\lim_{h\to0}$&lt;/span&gt; later, they will find that &lt;span class=&quot;math-container&quot;&gt;$h=0$&lt;/span&gt; is expressly forbidden. Specifically---and you may not want to mention these details at first---the definition makes use of a positive (not 0) number &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; lies either in the interval &lt;span class=&quot;math-container&quot;&gt;$(-\delta,0)$&lt;/span&gt; or the interval &lt;span class=&quot;math-container&quot;&gt;$(0,\delta)$&lt;/span&gt;. In other words, the whole set-up is designed so that you are protected from &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; ever being &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. To keep things simple, you can say that when taking the limit the value of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is considered small, but not &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, and emphasize the &amp;quot;not &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;&amp;quot; part repeatedly.&lt;/p&gt;&#xA;&lt;p&gt;Contrast this with algebra, where in dealing with &lt;span class=&quot;math-container&quot;&gt;$\frac{x+2}{x}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\frac{x^2+x}{x}$&lt;/span&gt; and no domain explicitly mentioned you have to take care about what would happen if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; were &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. In the definition of derivative you don't have to worry because the &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\lim_{h\to0}$&lt;/span&gt; is &lt;strong&gt;defined not to be&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. That means you are free to do cancelations of factors of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; that would require much more care in an algebra class.&lt;/p&gt;&#xA;&lt;p&gt;Limits are used in the definition of the derivative so that you can talk about what happens as &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; comes arbitrarily close to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, &lt;strong&gt;without equalling&lt;/strong&gt; 0. Velocity is a key application of the derivative, and I ask students to think about how they would estimate the instantaneous velocity of a moving object given a series of time-stamped still photos of the object at closely spaced points in its motion. I point out that it is impossible to make the estimate using only a single photo---you have to have two photos taken at slightly different times and look at how far the object moved in between. This observation provides a physical reason why we impose the condition &lt;span class=&quot;math-container&quot;&gt;$h\ne0$&lt;/span&gt;. Of course, any computation made based on photos spaced apart by any finite time interval can only be an approximation. Taking the limit can be thought of as finding the value these approximations approach in the process of shrinking the interval. (If such a value exists.)&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;why is &lt;span class=&quot;math-container&quot;&gt;$(\delta x)^2$&lt;/span&gt; &amp;quot;smaller&amp;quot; than &lt;span class=&quot;math-container&quot;&gt;$\delta x$&lt;/span&gt;? &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is just &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, right?&amp;quot; This, of course, relates to the other question, and the answer is that &lt;span class=&quot;math-container&quot;&gt;$\delta x$&lt;/span&gt;, which I have been calling &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, is never &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. Your proof, which is perfectly correct (apart from the little-o versus big-O issue), makes no mention of &lt;span class=&quot;math-container&quot;&gt;$(\delta x)^2$&lt;/span&gt; being less than &lt;span class=&quot;math-container&quot;&gt;$\delta x$&lt;/span&gt;, and there's no reason to talk about that. I think what you are bothered by is the instruction to &amp;quot;neglect lower order terms&amp;quot; that one commonly sees in calculus and physics books. My advice is simply not to use that language. Students will pick up through experience which terms are going to go to zero when the limit is taken, and can start neglecting such terms on their own once they have sufficient practice. As beginners, however, they should write out all the terms and compute which terms go to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and which don't.&lt;/p&gt;&#xA;&lt;p&gt;I should add that you are right that &lt;span class=&quot;math-container&quot;&gt;$h^2$&lt;/span&gt; is not smaller than &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$h=0$&lt;/span&gt;. It's also not smaller when &lt;span class=&quot;math-container&quot;&gt;$h\ge1$&lt;/span&gt;. What is true is that &lt;span class=&quot;math-container&quot;&gt;$h^2&amp;lt;\lvert h\rvert$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$h\in(-1,0)\cup(0,1)$&lt;/span&gt;. As mentioned above, in the &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; definition of &lt;span class=&quot;math-container&quot;&gt;$\lim_{h\to0}$&lt;/span&gt; a &lt;strong&gt;positive&lt;/strong&gt; number &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; is used to bound &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$h\in(-\delta,0)\cup(0,\delta)$&lt;/span&gt;. The definition doesn't explicitly require that &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; be less than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and for large &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; it may not need to be in which case &lt;span class=&quot;math-container&quot;&gt;$h^2&amp;lt;\lvert h\rvert$&lt;/span&gt; wouldn't always hold, but as &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; is made smaller, &lt;span class=&quot;math-container&quot;&gt;$\delta&amp;lt;1$&lt;/span&gt; ends up having to be true. The point here is that, yes, eventually &lt;span class=&quot;math-container&quot;&gt;$h^2$&lt;/span&gt; is less than &lt;span class=&quot;math-container&quot;&gt;$\lvert h\rvert$&lt;/span&gt;, but that fact plays no explicit role in your proof and students don't need to worry about it at first.&lt;/p&gt;&#xA;&lt;p&gt;Now eventually students will have to get past the &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; hurdle. Some other answerers have said that students simply can't hope to understand what's going on until they've jumped over that hurdle, but in my opinion that's not necessary, and forcing it will place an unnecessary barrier to understanding in front of many students. The &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; definition is difficult because it seems somewhat cumbersome, and students don't grasp at first why it needs to be so cumbersome. Informally, the meaning of &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to c}f(x)=L$&lt;/span&gt; is often stated as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; gets closer and closer to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; gets closer and closer to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&amp;quot; For the power functions &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; that you have in mind this is perfectly adequate: &lt;span class=&quot;math-container&quot;&gt;$\frac{(x+h)^2-x^2}{h}$&lt;/span&gt; really does get closer and closer to &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; gets closer and closer to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. It's for limits like &lt;span class=&quot;math-container&quot;&gt;$\lim_{h\to0}h\sin(1/h)=0$&lt;/span&gt; that you have to take greater care about how you state things. In this example the function actually gets all the way to the limiting value &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; infinitely many times as &lt;span class=&quot;math-container&quot;&gt;$h\to0$&lt;/span&gt; from the positive side, but each time it moves away from that value again before turning around and heading back toward it. In this kind of situation, the &amp;quot;gets closer and closer&amp;quot; language it too imprecise and, in fact, incorrect if interpreted literally. So here you really need the idea that eventually the function gets inside an &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-sized envelope surrounding the limiting value, and stays there. I think it is good for students to see examples like this so they know why the &amp;quot;closer and closer&amp;quot; language is inadequate and why a more precise definition is needed. But again, these issues are separate from your concerns and don't need to be overcome before the questions you raise can be answered.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2022-05-27T13:20:17.067" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25281" PostTypeId="1" CreationDate="2022-05-28T00:55:07.547" Score="3" ViewCount="1136" Body="&lt;p&gt;I don't mean to sound rude and denigrating to math educators but I always imagined that a science teacher (physics/chemistry), who has to know math, would be eligible to teach math?&lt;/p&gt;&#xA;&lt;p&gt;Ignoring the pedagogical differences and the fact that there are far less science teachers than math teachers, content wise strictly speaking, isn't a science teacher thus then qualified to teach math? Is there something a math teacher can do that a science teacher cannot?&lt;/p&gt;&#xA;&lt;p&gt;The only parts of math that science teachers probably cannot do as well are abstract math, number theory and statistics and probability.&lt;/p&gt;&#xA;&lt;p&gt;Am I wrong in this thinking?&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="376" LastEditDate="2022-06-03T15:25:20.160" LastActivityDate="2022-06-06T13:31:19.647" Title="Can a science educator teach math?" Tags="|general-pedagogy|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25282" PostTypeId="2" ParentId="25281" CreationDate="2022-05-28T12:20:25.547" Score="0" Body="&lt;p&gt;In the USA, I'd say it's common, not ubiquitous, that from K-6, kids get their math instruction from generalist elementary teachers, lacking a math degree, or often even any non-ed degree.  7-12, you usually have dedicated instructors.  So specialization is the norm.  Most of the math teachers tend to have math degrees.  But there's no reason for it to be 100%.  As long as you know the content and have the teaching cert, it's not that hard to move (more likely &amp;quot;be moved&amp;quot;) into teaching math versus science.&lt;/p&gt;&#xA;&lt;p&gt;Usually, it seems like schools struggle harder to find the teaches for the science courses and for computer science.  And there's competition for HS jobs.  So having a subject matter degree is generally desirable.  But I know of people teaching HS math with an undergrad in science.&lt;/p&gt;&#xA;&lt;p&gt;I even know one with an English degree.  They keep him away from AP Calculus, but he handles algebra and geometry fine.  Not the norm really, but certainly rarely happens.&lt;/p&gt;&#xA;&lt;p&gt;There's no conceptual reason you can't have an adjunct or even a tenure/track faculty in a university math department, who lacks a math degree.  I'd say it's incredibly rare though.  There's a big supply demand problem (for the jobseekers, it's a blessing for the colleges) in terms of the number of people looking for jobs versus the openings.  So, competition makes it extremely unlikely.&lt;/p&gt;&#xA;&lt;p&gt;All that said, I have reasonably frequently seen community colleges using non-math UGs to teach night school or summer sessions.  Usually from a science or engineering background and often mid career with a day job.&lt;/p&gt;&#xA;" OwnerUserId="19936" LastActivityDate="2022-05-28T12:20:25.547" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25283" PostTypeId="1" AcceptedAnswerId="25286" CreationDate="2022-05-28T13:21:56.687" Score="4" ViewCount="255" Body="&lt;p&gt;If it's a good idea to work on reading and language from early childhood, I'd bet that it's a good idea to work on math and quantities too.&lt;/p&gt;&#xA;&lt;p&gt;I have an 18 month old. I've pretty much been winging it with her math education.&lt;/p&gt;&#xA;&lt;p&gt;Are there best practices for fostering math skills  during early childhood?&lt;/p&gt;&#xA;" OwnerUserId="961" LastActivityDate="2022-05-30T02:40:19.220" Title="Is there a widely respected early childhood math curriculum?" Tags="|mathematical-pedagogy|teaching|education-research|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25284" PostTypeId="1" CreationDate="2022-05-28T15:35:20.530" Score="3" ViewCount="850" Body="&lt;p&gt;I recently worked my way through Walter Warwick Sawyer's book, &lt;em&gt;Mathematician's Delight&lt;/em&gt;, which has opened my eyes to Maths. I used to fear maths, feeling I was incapable. Sawyer (among other authors) has a gift for teaching the subject. I now feel much more confident tackling Maths problems, I have a better intuitive understanding of Maths and a renewed interest in it.&lt;/p&gt;&#xA;&lt;p&gt;There's a nice summary of Sawyer's life and work here: &lt;a href=&quot;https://plus.maths.org/content/os/latestnews/may-aug08/sawyer/index&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://plus.maths.org/content/os/latestnews/may-aug08/sawyer/index&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Have you had a similar experience after encountering Sawyer's work?&lt;/p&gt;&#xA;&lt;p&gt;Stephen&lt;/p&gt;&#xA;" OwnerUserId="19932" LastEditorUserId="60" LastEditDate="2022-05-30T02:32:00.557" LastActivityDate="2022-05-30T05:54:57.170" Title="Walter Warwick Sawyer: How has reading his works changed your learning or teaching?" Tags="|mathematical-pedagogy|self-learning|applied-mathematics|" AnswerCount="3" CommentCount="8" ClosedDate="2022-05-30T14:43:00.947" ContentLicense="CC BY-SA 4.0" />
  <row Id="25285" PostTypeId="2" ParentId="25284" CreationDate="2022-05-28T16:53:34.147" Score="7" Body="&lt;p&gt;I already loved math when I encountered his books. But yes, I was also inspired. &lt;em&gt;&lt;strong&gt;Mathematician's Delight&lt;/strong&gt;&lt;/em&gt; might be the one I put dozens of page markers in, so I could find all the great ideas again. He helped me think about how I might want to teach differently, especially in beginning algebra. I have quite a few of his books. The ones on higher math (like abstract algebra) are fun for me to work through.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.wwsawyer.org/work-by-sawyer.html&quot; rel=&quot;noreferrer&quot;&gt;This site&lt;/a&gt; has lists of both his books and articles he wrote.&lt;/p&gt;&#xA;&lt;p&gt;Thanks for posting this at the beginning of my summer break. I might try to find a few more of his books, and will definitely pull one out again to work on.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-05-28T16:53:34.147" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25286" PostTypeId="2" ParentId="25283" CreationDate="2022-05-28T16:57:59.960" Score="9" Body="&lt;p&gt;Yes. Most importantly, math in the early years should be play. My publisher has a lovely book titled &lt;a href=&quot;https://naturalmath.com/moebius-noodles/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Moebius Noodles&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt; that might give you more insight.&lt;/p&gt;&#xA;&lt;p&gt;But I wouldn't recommend anything like a curriculum. You don't want it to ever be a chore. Just like we read to kids when they ask us to, and in a nightly bedtime ritual, so should our mathplay with them be determined by their interest.&lt;/p&gt;&#xA;&lt;p&gt;You might also find some of the chapters in my book helpful. &lt;a href=&quot;https://naturalmath.com/playingwithmath/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Playing with Math: Stories from Math Circle, Homeschoolers, and Passionate Teachers&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;. (Much of this is for parents of older kids.)&lt;/p&gt;&#xA;&lt;p&gt;There is so much to say on this, I'm trying to figure out what is useful and short enough for an answer here. (You can check out my lists of games and books &lt;a href=&quot;http://mathmamawrites.blogspot.com/&quot; rel=&quot;noreferrer&quot;&gt;at my blog&lt;/a&gt;.) Here's a quick list of how to help your child love math:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Follow your child's lead.&lt;/li&gt;&#xA;&lt;li&gt;Be comfortable with math yourself.&lt;/li&gt;&#xA;&lt;li&gt;Have a good stash of mathy toys: building blocks (Keva/Kapla/Citi blocks are good), games (Blokus and Katamino are good for slightly older kids), and card games (Set and Blink are good for older kids).&lt;/li&gt;&#xA;&lt;li&gt;Also read fun mathy stories (The Cat in Numberland, Opposites, so many more).&lt;/li&gt;&#xA;&lt;li&gt;Notice math connections as you go through your day (cooking, things that come in 3s or 4s or 5s, ...).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-05-30T02:40:19.220" LastActivityDate="2022-05-30T02:40:19.220" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25287" PostTypeId="2" ParentId="25281" CreationDate="2022-05-28T20:35:09.373" Score="2" Body="&lt;p&gt;It's common in Madrid and mostly a failure, although there are of course individual exceptions. Most trained in something other than math don't know math well enough to teach it well. It's not so different from what happens when mathematicians have to teach computer programming. They can do it, those who don't have criteria by which to judge (students, parents, administrators, politicians) can't much tell the difference, but the result isn't what it could be (I speak from the experience of having taught C programming quite a few times).&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2022-05-28T20:35:09.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25288" PostTypeId="1" CreationDate="2022-05-28T21:17:42.693" Score="3" ViewCount="122" Body="&lt;p&gt;I am preparing to do some remote tutoring and I am looking for software to help me out.&lt;/p&gt;&#xA;&lt;p&gt;Requirements:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Drawing area shared by the tutor and the student, both can draw on the page&lt;/li&gt;&#xA;&lt;li&gt;Multiple drawing colors&lt;/li&gt;&#xA;&lt;li&gt;Ability to insert images by copy pasting (necessary for pulling in problems from a textbook)&lt;/li&gt;&#xA;&lt;li&gt;Ability to export page to PDF&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Nice to have:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Cross platform&lt;/li&gt;&#xA;&lt;li&gt;No cost to the student / guest&lt;/li&gt;&#xA;&lt;li&gt;Integrated video chat&lt;/li&gt;&#xA;&lt;li&gt;Join video chat from separate device&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;My last requirement is weird. I don't have a proper device with a webcam, so I was hoping to use my phone just for the video chat, until I can get a new laptop.&lt;/p&gt;&#xA;&lt;p&gt;I have done some research and &lt;a href=&quot;https://ziteboard.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ziteboard&lt;/a&gt; is the closest I have found. It seems to support my hard requirements and it works in the browser so it's cross platform. It &lt;em&gt;looks&lt;/em&gt; like it is free for guests, anyone can make an account for free and the board should get the pro features. It does have integrated video chat, but I can't tell if I could join from a separate device.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://limnu.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Limnu&lt;/a&gt; appears to also be an option. It's a bit cheaper with no video chat and it's less clear about whether it is free for guests.&lt;/p&gt;&#xA;&lt;p&gt;I will probably go with Ziteboard if I can't find anything better, but I would like to fully explore the options.&lt;/p&gt;&#xA;" OwnerUserId="19937" LastActivityDate="2022-05-28T21:17:42.693" Title="What software can I use for remote math tutoring?" Tags="|online-instruction|software|interactive-teaching|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25289" PostTypeId="2" ParentId="25284" CreationDate="2022-05-29T01:36:09.613" Score="5" Body="&lt;p&gt;Yes, I remember vividly my chance encounter at the library with that book of  his! Yes, it had a big impact on me. The idea that mathematics was a real thing in its own right, like music, and not just a school subject, and not just a device to filter people out.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-05-29T01:36:09.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25290" PostTypeId="2" ParentId="25284" CreationDate="2022-05-29T01:54:39.580" Score="16" Body="&lt;p&gt;No, I was never inspired by him because I had never heard of him before you mentioned it.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;Note: My answer is for the original version of the question.  Since then, the question has been edited so my answer is no longer appropriate.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2022-05-30T05:54:57.170" LastActivityDate="2022-05-30T05:54:57.170" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25291" PostTypeId="2" ParentId="25281" CreationDate="2022-05-29T02:49:51.010" Score="6" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Is there something a math teacher can do that a science teacher&#xA;cannot?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The essence of the mathematical discipline is &lt;em&gt;rigorous deductive proof&lt;/em&gt;. Feel free to read that as &amp;quot;correct and understandable explanations and justifications&amp;quot; for mathematical principles.&lt;/p&gt;&#xA;&lt;p&gt;Frequently, non-math science teachers have only taken computational math courses in college, like calculus, differential equations, and linear algebra (likely in their first two years or so).&lt;/p&gt;&#xA;&lt;p&gt;But for college math majors, there is a radical shift around the end of the second year: courses suddenly shift to being about &lt;em&gt;writing and understanding proofs&lt;/em&gt; as their central focus (generally talking U.S. system here; in other locales they wouldn't wait that long). It's common for math majors who are only good at mechanical calculations to find this very difficult or switch to another major as a result. Science majors usually never see these courses. (Conversely, some people who would be very good actual math majors never get through the mechanical-calculating prerequisites to get to this point.)&lt;/p&gt;&#xA;&lt;p&gt;So it's quite likely that science teachers, given their background, could present a calculating algorithm, but be unable to explain &lt;em&gt;why&lt;/em&gt; the method works. Not being trained and practiced in how to carefully attend to starting definitions and theorem-construction, it's quite likely that it would come out as a mishmash and be unclear on what the symbols, operations, starting assumptions, and sufficient and necessary conditions really are (in fact: I often experience this when I try to read posts on SE Physics, say).&lt;/p&gt;&#xA;&lt;p&gt;Overall, such a mode of presentation represents &lt;a href=&quot;https://en.wikipedia.org/wiki/Cargo_cult&quot; rel=&quot;nofollow noreferrer&quot;&gt;cargo-cult&lt;/a&gt; (or faith-based) math, and leaves students unable to succeed at the next level, because all they're doing is memorizing a ton of separate facts instead of understanding more broadly-applicable principles.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2022-05-31T14:31:47.660" LastActivityDate="2022-05-31T14:31:47.660" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25293" PostTypeId="2" ParentId="25085" CreationDate="2022-05-30T01:13:03.953" Score="1" Body="&lt;p&gt;I'm fond of a volume called &lt;em&gt;Mathenauts: Tales of Mathematical Wonder&lt;/em&gt;, edited by (mathematician and sci-fi author) Rudy Rucker, dated 1987. It includes Asimov's &lt;em&gt;The Feeling of Power&lt;/em&gt;, among other stories.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2022-05-30T01:13:03.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25295" PostTypeId="1" AcceptedAnswerId="25298" CreationDate="2022-05-31T11:58:36.340" Score="7" ViewCount="176" Body="&lt;p&gt;I just read Moebius Noodles. (Thanks for the recommendation Sue). Part of the book talks about keeping images about math around the house.&lt;/p&gt;&#xA;&lt;p&gt;My child's 18 months. But I figure, why not now. It's passive; so if it helps now, it helps now. If it doesn't, maybe it will help later.&lt;/p&gt;&#xA;&lt;p&gt;I've been looking for images available as decals or posters that illustrate important concepts. Unfortunately, after searching, I've only found posters about numerals and 'sacred geometry' (admittedly, cool looking - but Metatron's Cube doesn't seem to illustrate anything didactive.).&lt;/p&gt;&#xA;&lt;p&gt;What kind of images should I be looking for?&#xA;Do they exist? What kind of keywords can I used to find them?&lt;/p&gt;&#xA;" OwnerUserId="961" LastEditorUserId="961" LastEditDate="2022-05-31T13:32:48.617" LastActivityDate="2022-06-01T14:23:18.910" Title="Are there pre-printed wall images that might engender understanding in a very young child?" Tags="|mathematical-pedagogy|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25297" PostTypeId="2" ParentId="2074" CreationDate="2022-06-01T04:18:11.387" Score="3" Body="&lt;p&gt;Yes Euclid is deeply important. Its important for a few reasons:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Its visual and therefore leverages preexisting intuitions about space that people have.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It the easiest intro to axiomatic mathematics which forms the basis for all future math in university math depts&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Its doubles as a science...everything in Euclidean geometry can actually be tested in  the real world.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is also the basis for a lot of the philosophical tradition and epistemological tradition of the west.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is rigor and precision to how Euclid proceeds that I want in the heads of people as an example to strive for. I want them to desire this and look for it. If you surround yourself with ugliness you won't know to desire beauty because you will never see a good example of it. Euclid is did for truth what the Sistine chapel did for beautiful churches&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is constructive and synthetic. Current math is mostly analytic. A lot of math today is also non-constructive. In the mathematical tradition Euclid provides a deeply important aesthetic thread and I don't want it to be lost!!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It encourages the right epistemological morals. People complain that Euclid artificially constrains himself. But they don't appear to understand why. There is an important a lesson to be had here. The Greeks realized how easily one could be lead astray. By constraining what was assumed to as few as possible assumptions they hoped not to be led astray. This kind of fear error, limitation of assumptions and stress on using deductive proofs as far as possible is incredibly important. Newton understood it which is why he wouldn't use calculus and proved all his results geometrically.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It has inspired countless generations of mathematicians, scientists and philosophers Consider Russell's reaction:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the age of eleven, I began Euclid, with my brother as tutor. This&#xA;was one of the great events of my life, as dazzling as first love. I&#xA;had not imagined there was anything so delicious in the world.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or the brilliant Israeli logician Saharon Shelah:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But when I reached the ninth grade I began studying geometry and my&#xA;eyes opened to that beauty—a system of demonstration and theorems&#xA;based on a very small number of axioms which impressed me and&#xA;captivated me.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It difficult to fully express all that is going on in Euclid but it needs to be internalized into the heads of scientists and mathematicians. My fear is that it hasn't been in this generation and we will suffer the consequences in junk science and amorphous fuzzy results for decades to come.&lt;/p&gt;&#xA;&lt;p&gt;The greatest reason to study Euclid is therefore for me that it points away from the inexorable entropic forces of motivated reasoning, advocacy science and pure bullshit that are Western scientific culture is currently succumbing to. And towards something much much better.&lt;/p&gt;&#xA;" OwnerUserId="19948" LastActivityDate="2022-06-01T04:18:11.387" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25298" PostTypeId="2" ParentId="25295" CreationDate="2022-06-01T11:52:01.457" Score="7" Body="&lt;p&gt;Thank you for the reference, Sue! This was one of the too-quick recommendations in &amp;quot;Moebius Noodles&amp;quot; that may not make sense without context, and we need to add some in the next edition. Meanwhile...&lt;/p&gt;&#xA;&lt;p&gt;Why put up visible math around the house?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Value affirmation.&lt;/strong&gt; Some things are around for their utility, like the kitchen sink. If a portrait is up, though, that's because we value that person in our lives. We make some math visible in our space to affirm that we value it. You can invite your toddler to browse something like &lt;a href=&quot;http://gallery.bridgesmathart.org/exhibitions/2022-joint-mathematics-meetings&quot; rel=&quot;noreferrer&quot;&gt;http://gallery.bridgesmathart.org/exhibitions/2022-joint-mathematics-meetings&lt;/a&gt; (love their galleries!) and print out whatever the kid pokes, to start. Grown-ups can put up math that makes them emotional in some way, that connects to their lives, even a little, that evokes a memory, etc. Feelings are shareable with young kids, even if they don't understand the math!&lt;/p&gt;&#xA;&lt;p&gt;One of the examples I have up right now:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PUSV2.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PUSV2.jpg&quot; alt=&quot;Art Forms in Nature&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I do a lot of nature scavenger hunts with kids, and Haeckel is a great source for growing our math eyes.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Invitation to play.&lt;/strong&gt; A picture plus some &amp;quot;manipulatives&amp;quot; at the child's eye level is an awesome starter for a few minutes (or even a few seconds) of math-rich play. You can have a math corner with a picture book or two, a table with some hands-on objects, and a board to hang pictures. Look up &amp;quot;Reggio Emilia provocations&amp;quot; for ideas, but don't go fancy. You'll want to remake that corner every few weeks to keep the attention fresh.&lt;/p&gt;&#xA;&lt;p&gt;Here's how someone set up their child's mirror book table with some cute little objects, paper, pencils, etc. &amp;quot;Mirror book&amp;quot; is a chapter in &amp;quot;Moebius Noodles.&amp;quot; &lt;a href=&quot;https://happinessishereblog.com/mirror-books/&quot; rel=&quot;noreferrer&quot;&gt;https://happinessishereblog.com/mirror-books/&lt;/a&gt; You can put any pictures with radial symmetry next to that, like Haeckel's art.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MLTbd.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MLTbd.jpg&quot; alt=&quot;Mirror Books from Moebius Noodles&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Memory booster.&lt;/strong&gt; The toddler version: photograph your child at inspired play with some blocks, to spark more building. Older student version: hang up the chain rule formula after some quality time with calculus to commit it to the long-term memory. We memorize what we see multiple times in our environment.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nXPYi.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nXPYi.jpg&quot; alt=&quot;Lego 1981 ad &amp;quot;Beautiful&amp;quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="118" LastActivityDate="2022-06-01T11:52:01.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25299" PostTypeId="2" ParentId="25295" CreationDate="2022-06-01T14:23:18.910" Score="6" Body="&lt;p&gt;Some of M. C. Escher's works illustrate many mathematical concepts.  My favorite is &lt;a href=&quot;https://mathstat.slu.edu/escher/index.php/File:Circle-Limit-III.jpg&quot; rel=&quot;noreferrer&quot;&gt;Circle Limit III&lt;/a&gt;; it illustrates rotational symmetry, tessellations, hyperbolic geometry, limits, and infinity.  (More information about it can be found &lt;a href=&quot;https://en.wikipedia.org/wiki/Circle_Limit_III&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.)  Posters of M. C. Escher's works are available at the &lt;a href=&quot;https://mcescher.com/product-category/posters/&quot; rel=&quot;noreferrer&quot;&gt;official website&lt;/a&gt;, but you might want to just download public-domain images  of them and print them out.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/aakUg.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aakUg.jpg&quot; alt=&quot;Circle Limit III&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;When I was a young child (I suppose when I was first starting to read), I saw this picture in a &lt;em&gt;Scientific American&lt;/em&gt; magazine and it made me think and imagine.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2022-06-01T14:23:18.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25300" PostTypeId="2" ParentId="2074" CreationDate="2022-06-01T15:17:13.420" Score="6" Body="&lt;p&gt;I certainly appreciate the planar Euclidean geometry but my personal inclination is rather to teach combinatorics, graph theory, and number theory, which is exactly what I'm doing when I teach university math classes to future school teachers.&lt;/p&gt;&#xA;&lt;p&gt;The advantages I see in those are that you can get your proofs in an incremental way: you can experiment with small numbers and pictures before trying to discern the general formula or statement. It is also easier to refute the wrong proof by presenting an example where a certain step fails (not that it is easier to &lt;em&gt;create&lt;/em&gt; a counterexample, but it is easier to &lt;em&gt;convince&lt;/em&gt; the student that it is a counterexample). It is also easier (for me) to create chains of problems (say, counting ones) of gradually increasing difficulty, while in the Euclidean geometry you need to be quite inventive not to jump from &amp;quot;trivialities&amp;quot; to &amp;quot;hopelessly difficult&amp;quot; problems. As to computer visualization, the integer arithmetic seems more reliable to me than the graphic: if the machine tells me that 239+456=695, I generally believe it, but if I see two lines and a circle passing through one point, I'm always tempted to look around in search of a magnifying glass.&lt;/p&gt;&#xA;&lt;p&gt;In general, I do not care much &lt;em&gt;which subject&lt;/em&gt; a proof course is based upon as long as it &lt;em&gt;is a proof course&lt;/em&gt; and not the one about memorization of formulae and algorithms. Unfortunately, it is quite easy to turn the former into the latter, and then we do not need students for such courses: computers are easier to program and the results of such programming are more reliable. The only advantage of a human over a machine nowadays is that a human can think illogically to produce logical derivations and to build working contraptions and, while it is still unclear how exactly this process works, it is pretty clear that one needs practice to get good at it and that the proof courses are the best tool so far we have to develop this skill.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-06-01T15:17:13.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25301" PostTypeId="1" CreationDate="2022-06-01T16:28:20.137" Score="6" ViewCount="599" Body="&lt;p&gt;Next semester, I am going to teach a small section of advanced high school students a class of Multivariable Calculus (it's about 3-4 students that have completed AP Calculus BC). Multivariable Calculus is usually a 1 semester college course, but I have the advantage of being able to teach the material over the entire school yea, which is two semesters. So I am looking for project ideas to help fill some of the time but also give the students some practical applications of calculus. I also have the great advantage that the school I work at has access to 3D printers.&lt;/p&gt;&#xA;&lt;p&gt;I feel lucky to be in the position to work with a small and dedicated group of students and have plenty of time to get through the material and great resources to take advantage of, but on the other hand, I am relatively inexperienced with Multivariable Calculus (I know the content, but I don't know any good ideas for a projects) and I have no experience even using a 3D printer. So, &lt;strong&gt;what are some good project ideas for Multivariable Calculus&lt;/strong&gt; that hopefully don't require too much of a learning curve if using the 3D printer?&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2022-06-08T22:36:20.203" Title="Multivariable Calculus Project Ideas" Tags="|calculus|vector-calculus|resource-request|projects|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25302" PostTypeId="2" ParentId="2074" CreationDate="2022-06-03T12:33:03.860" Score="1" Body="&lt;p&gt;This question has just appeared in the list of currently active questions, so I thought to add my penny's worth.&lt;/p&gt;&#xA;&lt;p&gt;I have been tutoring a struggling mathematics student (mid-teens) by spending an hour a week with him, helping him understand the material he is having trouble with.&lt;/p&gt;&#xA;&lt;p&gt;He showed me what he was doing in geometry. Apparently they have been given an arbitrary bunch of theorems to learn, one of which being the Inscribed Angle Theorem. (In fact, they don't even have to &amp;quot;learn&amp;quot; them, they are provided a sheet of theorems which they may refer to during the course of any tests they are doing.) None of these theorems has even been proved in the classroom. They are just taught to be able to use them. As a result, my student feels a little glum, because he believes that geometry is way over his head.&lt;/p&gt;&#xA;&lt;p&gt;So I drew a few lines on the standard diagram, pointed out the isosceles nature of the triangles formed as a result (obvious once you've had it pointed out to you) and encouraged him to use what he knew about the number of degrees that the angles of a triangle add up to, remind him what algebra was for, and watched him as he proved that indeed, the angle subtended by a chord is exactly twice at the centre what it is when the angle is on the perimeter.&lt;/p&gt;&#xA;&lt;p&gt;He was right indignant. &amp;quot;That was really simple! Why couldn't the teacher have showed us that?&amp;quot; So the next time I saw him, I left my copy of Euclid for him to look at. He loves it.&lt;/p&gt;&#xA;" OwnerUserId="15290" LastActivityDate="2022-06-03T12:33:03.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25304" PostTypeId="2" ParentId="25281" CreationDate="2022-06-03T12:51:06.060" Score="2" Body="&lt;p&gt;As long as the science teacher is appropriately motivated, I can't see why on earth he or she would not be able to teach mathematics to at least high-school level.&lt;/p&gt;&#xA;&lt;p&gt;In order to reach a certain level of scientific sophistication as to be able to understand the science syllabus in order to teach it, there is &lt;em&gt;bound&lt;/em&gt; to be at least a modicum of the &amp;quot;important&amp;quot; stuff in mathematics: namely, how to recognise a proof and how a proof is structured.&lt;/p&gt;&#xA;&lt;p&gt;At the level of high school mathematics, it should not take a prohibitively long time to reach a level at which they can impart the knowledge and skill -- but as I say, the teacher has to be &lt;em&gt;motivated&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If it's the level of: &amp;quot;we haven't got anyone to teach maths, we cast lots and you're it,&amp;quot; is very probably &lt;em&gt;not&lt;/em&gt; going to work well.&lt;/p&gt;&#xA;" OwnerUserId="15290" LastActivityDate="2022-06-03T12:51:06.060" CommentCount="24" ContentLicense="CC BY-SA 4.0" />
  <row Id="25305" PostTypeId="2" ParentId="25281" CreationDate="2022-06-06T13:31:19.647" Score="8" Body="&lt;p&gt;That depends on the level. In general the rule is that to teach a subject coherently, one needs to know roughly speaking five times more than one is going to present to the students. If this condition is satisfied, then you should be just fine (the little quirks like what would be the best way/order to present this or that can be picked up on the way and there is no universal agreement on them anyway). If not, you'd rather leave it to other people. I acted as a tutor in elementary physics a couple of times (kinematics and dynamics: Newton laws, energy/momentum conservation and such) and I felt completely comfortable with that though I'm a pure math professor, and the results were satisfactory too, but I wouldn't go into, say, teaching quantum mechanics though I am not totally ignorant of it.&lt;/p&gt;&#xA;&lt;p&gt;So just evaluate your knowledge and make the conclusion yourself. I have no doubt that you'll be fine teaching addition of fractions but I have my reservations about letting you teach a graduate university course in measure theory (though I don't know you well enough to declare that you are disqualified from it, of course). Just follow that 5 to 1 rule and you'll be well within your competence level, that's all.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-06-06T13:31:19.647" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25306" PostTypeId="1" AcceptedAnswerId="25316" CreationDate="2022-06-07T12:34:13.377" Score="-1" ViewCount="419" Body="&lt;p&gt;I am recently into a sort of teaching which I shall term &amp;quot;Random Teaching&amp;quot;. It involves:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Asking student for a random word (not related to math remotely, say &amp;quot;cake&amp;quot;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Preforming a youtube search with that word + &amp;quot;math&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Selecting a mathematical video featuring the search&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Do you think I shall use this form of teaching? Should I try? Are there any ways I can improve this form of teaching?&lt;/p&gt;&#xA;&lt;h1&gt;Edit&lt;/h1&gt;&#xA;&lt;h3&gt;Due to the answers and comments I feel obligated to add the following:&lt;/h3&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I am &lt;strong&gt;not&lt;/strong&gt; teaching for a school&lt;/li&gt;&#xA;&lt;li&gt;I do &lt;strong&gt;not&lt;/strong&gt; have a specific goal except to keep the student, say, interested in mathematics&lt;/li&gt;&#xA;&lt;li&gt;I am doing a &lt;strong&gt;1 to 1 online tutoring class&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;I teach a &lt;strong&gt;sixth grader&lt;/strong&gt; but I &lt;strong&gt;tend to wander to higher-level topics, for instance, trigonometry, matrices, vectors, and occasionally even calculus&lt;/strong&gt;. It's more like a &lt;strong&gt;&lt;span class=&quot;math-container&quot;&gt;$\Large\text{special  interest  club}$&lt;/span&gt;.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="19960" LastEditorUserId="19960" LastEditDate="2022-06-08T06:01:59.703" LastActivityDate="2022-06-12T03:40:31.480" Title="Motivating Students Through &quot;Random Teaching&quot;" Tags="|student-motivation|" AnswerCount="2" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="25307" PostTypeId="2" ParentId="995" CreationDate="2022-06-07T13:06:34.440" Score="1" Body="&lt;p&gt;Something you can do to generate more diverse problems is to &lt;em&gt;combine&lt;/em&gt; the problem types from your textbook.&lt;/p&gt;&#xA;&lt;p&gt;You can also be explicit about the kind of work you want them to show you, and make it clear the simply recalling a formula will not be sufficient.&lt;/p&gt;&#xA;&lt;p&gt;Here are some examples.  In each example, I have provided some &amp;quot;scaffolding&amp;quot; to help students understand the steps they will need to take.  These &amp;quot;scaffolding&amp;quot; steps should be gradually removed over time, until just the problem statement remains with the expectation that students will provide this kind of scaffolding for themselves.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Volume and Density&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A solid is formed whose base is the region bounded by &lt;span class=&quot;math-container&quot;&gt;$y = 1-x^2$&lt;/span&gt; and the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis. The unit for the coordinate grid is inches. The cross sections perpendicular to the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis are squares, and the density of the solid at this cross section is given by &lt;span class=&quot;math-container&quot;&gt;$(2+x) \, \frac{\textrm{g}}{\textrm{in}^3}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Draw a picture of the resulting solid, calculate the infinitesimal mass of the generic slab from &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x + \textrm{d}x$&lt;/span&gt;, and finally integrate to find the total mass.&lt;/p&gt;&#xA;&lt;p&gt;WARNING:  If you just &amp;quot;write down the integral&amp;quot; without showing me your picture and calculation of the infinitesimal mass you will earn at most 1/2 credit on this problem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Arclength and Density&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;We have a rope of variable density.  It is lying on the coordinate plane (whose units are given in inches).  The rope is lying on top of the curve &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sin(x)$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x=\pi$&lt;/span&gt;.  The density of the rope is given by &lt;span class=&quot;math-container&quot;&gt;$(1+\cos(x)) \, \frac{\textrm{g}}{\textrm{in}}$&lt;/span&gt; where this density is measured &lt;em&gt;along the rope&lt;/em&gt; (not along the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis).&lt;/p&gt;&#xA;&lt;p&gt;Draw a picture of this rope.  Draw a picture of the generic segment from &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x+\textrm{d}x$&lt;/span&gt;.  Show me how you calculate both the length and mass of this generic segment.  Then write down and evaluate the integrals representing both the total length and total mass of the rope.&lt;/p&gt;&#xA;&lt;p&gt;WARNING:  If you just &amp;quot;write down the integrals&amp;quot; without showing me your picture and calculation of the infinitesimal length and mass you will earn at most 1/2 credit on this problem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Volume and Work&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;You have two tanks.  The first tank is full of water, and is the lower half of a hemisphere of radius &lt;span class=&quot;math-container&quot;&gt;$5 \textrm{m}$&lt;/span&gt; (the &amp;quot;south pole&amp;quot; of the sphere is on the ground).  The second tank is empty, and is a cone with a radius of &lt;span class=&quot;math-container&quot;&gt;$8 \textrm{m}$&lt;/span&gt; and a height of &lt;span class=&quot;math-container&quot;&gt;$5 \textrm{m}$&lt;/span&gt;.  The &amp;quot;tip&amp;quot; of the cone is on the ground.&lt;/p&gt;&#xA;&lt;p&gt;You are going to compute the work done against gravity by pumping the water from the first tank into the second tank.&lt;/p&gt;&#xA;&lt;ol start=&quot;0&quot;&gt;&#xA;&lt;li&gt;Sanity Check:  Is the second tank even big enough to hold all of the water from the first tank?  If so, what will the depth of water be in the second tank after it is all pumped?&lt;/li&gt;&#xA;&lt;li&gt;Say we have already pumped all of the water up to a height of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; from the first tank to the second tank.  Draw a generic slab of water in the first tank from a height of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$y + \textrm{d}y$&lt;/span&gt;.  Compute the mass of this slab of water.&lt;/li&gt;&#xA;&lt;li&gt;This slab of water was at a height of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; in the first tank.  What height will it be when we pump it into the second tank?  Why?  HINT:  You will need to solve another integration problem to answer this question!&lt;/li&gt;&#xA;&lt;li&gt;Use your answers to (1) and (2) to compute the work done by pumping this generic slab from the first tank to the second tank.&lt;/li&gt;&#xA;&lt;li&gt;Finally, integrate to find the total amount of work done against gravity by pumping all of the water.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;WARNING:  If you just &amp;quot;write down the integral&amp;quot; without showing me your pictures and calculations of work done by pumping the infinitesimal slab, you will earn at most 1/2 credit on this problem.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-06-07T13:06:34.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25308" PostTypeId="2" ParentId="25306" CreationDate="2022-06-07T15:20:15.800" Score="7" Body="&lt;p&gt;I found that using videos is a good way to get the students engaged, but I don't do it very often. I also pre-screen the videos I show so I know exactly what content is in it, what level audience the video was intended for, when I can pause for a discussion, etc. I would hesitate to bring up a random video that I hadn't seen before because the material might be above my class's head, or the video could be too long or just bad quality. Imagine if a sixth grader said the word &amp;quot;bridge&amp;quot; and you came up with &lt;a href=&quot;https://www.youtube.com/watch?v=_bJeKUosqoY&quot; rel=&quot;noreferrer&quot;&gt;this video&lt;/a&gt; (because it was most recently published from a pretty popular channel). I find this video quite fascinating and very interesting, but I don't know if I would use this for a class since it is packed with so many advanced topics.&lt;/p&gt;&#xA;&lt;p&gt;On a side note, the random word of 'cake' immediately made me think of these videos from Numberphile: &lt;a href=&quot;https://www.youtube.com/watch?v=wBU9N35ZHIw&amp;amp;ab_channel=Numberphile&quot; rel=&quot;noreferrer&quot;&gt;video 1&lt;/a&gt;, &lt;a href=&quot;https://www.youtube.com/watch?v=kaMKInkV7Vs&amp;amp;ab_channel=Numberphile&quot; rel=&quot;noreferrer&quot;&gt;video 2&lt;/a&gt;, &lt;a href=&quot;https://www.youtube.com/watch?v=FkVe8qrT0LA&amp;amp;t=642s&amp;amp;ab_channel=Numberphile&quot; rel=&quot;noreferrer&quot;&gt;video 3&lt;/a&gt;. But again, only because I had seen them before, I know which of the videos I would use for which of my classes and what points I would stop to talk to about further and which parts (if any) I would skip.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastActivityDate="2022-06-07T15:20:15.800" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25310" PostTypeId="2" ParentId="995" CreationDate="2022-06-07T20:14:08.257" Score="0" Body="&lt;p&gt;One that you see (saw) in chem classes was the &amp;quot;cut and weigh&amp;quot; method of integrating area under the peak for strip chart recorders.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=igR_jsqfcowC&amp;amp;pg=PA691&amp;amp;lpg=PA691&amp;amp;dq=nmr+area+under+the+peak+by+weighing+paper&amp;amp;source=bl&amp;amp;ots=4Hmo5VnP3C&amp;amp;sig=ACfU3U0JtrbFqZcj5KsOlosmVslxkPvpJA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjE5rvskpz4AhVxQzABHZ5wC3sQ6AF6BAgkEAM#v=onepage&amp;amp;q=nmr%20area%20under%20the%20peak%20by%20weighing%20paper&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com/books?id=igR_jsqfcowC&amp;amp;pg=PA691&amp;amp;lpg=PA691&amp;amp;dq=nmr+area+under+the+peak+by+weighing+paper&amp;amp;source=bl&amp;amp;ots=4Hmo5VnP3C&amp;amp;sig=ACfU3U0JtrbFqZcj5KsOlosmVslxkPvpJA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjE5rvskpz4AhVxQzABHZ5wC3sQ6AF6BAgkEAM#v=onepage&amp;amp;q=nmr%20area%20under%20the%20peak%20by%20weighing%20paper&amp;amp;f=false&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Anticipating the Shulman objection, yes this is not some super algebra trick.  But that's a feature, not a bug.  It gives a visceral intuition for integration.&lt;/p&gt;&#xA;&lt;p&gt;As does the box counting in the Hubbert manuscript:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.energycrisis.com/Hubbert/1956/1956.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.energycrisis.com/Hubbert/1956/1956.pdf&lt;/a&gt; (read, for example pages 8 to 11)&lt;/p&gt;&#xA;&lt;p&gt;The idea of accumulation inherent in an individual well cum (as opposed to rate) or even of a national or field or global result...is the same as the was-smiled-upon AP example of concert entrants.  In both cases, they give more intuitive sense for integration than just more applied algebra games in doing functional antiderivatives.  So...in defense of box counting!  ;-)  At least to drive &amp;quot;feel&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="19962" LastActivityDate="2022-06-07T20:14:08.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25311" PostTypeId="2" ParentId="25301" CreationDate="2022-06-08T22:36:20.203" Score="4" Body="&lt;p&gt;I have 3d printing as an extra-credit option for some projects, which are usually framed as a (fictional) business person looking for math equations to use specifically so they can then 3d print their products. My favorite is focused on Spirograph-style earrings!&lt;/p&gt;&#xA;&lt;p&gt;The project is basically something like “I found these equations online, can you explain to me how they work and make a computer widget so I can play with them to make pretty designs?” The students turn in a report and a Mathematica file, and for extra credit they can also made prototypes using a 3d printer.&lt;/p&gt;&#xA;&lt;p&gt;I’d be willing to share my materials if it would be helpful.&lt;/p&gt;&#xA;" OwnerUserId="19965" LastActivityDate="2022-06-08T22:36:20.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25312" PostTypeId="2" ParentId="15278" CreationDate="2022-06-09T16:02:22.580" Score="-3" Body="&lt;p&gt;Some students struggle with fractions because some teachers taught them fractions in a way they can not see the relation between fractions and daily life.&#xA;In &lt;a href=&quot;https://youtu.be/JKu26uLe2zM?t=100&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; Spanish-language video, a lady explains fractions. She draws a pizza sectioned into six equal pieces, one piece is then selected and eight &amp;quot;unit&amp;quot; (dots) placed on it. She then asks, &amp;quot;what was the whole number whose sixth part equals eight?&amp;quot; This explanations also works for proportions.&#xA;Additionally Ratios and rates are explained in a &lt;a href=&quot;https://www.youtube.com/watch?v=RQ2nYUBVvqI&quot; rel=&quot;nofollow noreferrer&quot;&gt;different video&lt;/a&gt;&#xA;. I add a third &lt;a href=&quot;https://archive.org/details/NasaConnect-Trror-RationalNumbers-StudentActivity&quot; rel=&quot;nofollow noreferrer&quot;&gt;video&lt;/a&gt; about fractions, which was made by NASA.&lt;/p&gt;&#xA;" OwnerUserId="17657" LastEditorUserId="17975" LastEditDate="2022-07-26T20:14:48.247" LastActivityDate="2022-07-26T20:14:48.247" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25313" PostTypeId="2" ParentId="6031" CreationDate="2022-06-10T20:10:44.927" Score="-1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;FWIW, I'm not a jock and didn't think the example was that hard.  I mean you got 5, which has integral factors of 1 and 5 (worth checking).  And then 3 times 5 is suspicously similar to 14 on the middle term.  Maybe minus a 1.  Probably get it faster than the &amp;quot;ac&amp;quot; method.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What I think is harder are ones where you have to put numbers on each x...like if it were 6 and you needed a 2 and a 3...and then things just spiral.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The broader question (which is tricky) is do we underdo/overdo factoring exercised in general.  Now, sure, there is a method (quadratic forumula) that solves all such questions and obviates ever doing factoring.  That said, I wonder if general manipulational ability/games are useful in later work with trig identities and the like.  It's actually a non-simple question to answer.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19971" LastActivityDate="2022-06-10T20:10:44.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25314" PostTypeId="1" CreationDate="2022-06-11T19:14:47.040" Score="4" ViewCount="686" Body="&lt;p&gt;I stumbled across the book when searching for rigorous alternatives to Rudin with some solutions. It’s an “old school” (1965) calculus text but, I think, covers similar material to Rudin in a more chatty way. It also has solutions/hints to selected problems, and fewer than Pugh. (I have a hard time knowing which ones to solve, since there are 80+ per chapter).&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have thoughts on where this book stands vis-à-vis Rudin, Apostol 1 and 2, Spivak, etc? There wasn’t much information on the Amazon review &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1577663020&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;page&quot;&gt;page&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19972" LastActivityDate="2022-06-11T19:14:47.040" Title="What are your experiences with Buck’s Advanced Calculus?" Tags="|mathematical-pedagogy|calculus|textbooks|mathematical-analysis|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25315" PostTypeId="2" ParentId="10106" CreationDate="2022-06-11T19:52:03.993" Score="0" Body="&lt;p&gt;It is important to be able to understand this sort of switching around.  But, for the example shown, I don't see any reason to push this method (in general or for an exercise).  For one thing, it's a better practice to group the terms first, the x's and y's, then perform the addition or subtraction.  Don't add complexity, then simplify--it increases cognitive load and makes mistakes more likely.  So group the terms first.&lt;/p&gt;&#xA;&lt;p&gt;For another thing, subtracting (a positive number) is probably a simpler idea (for a beginner) than adding a negative number.  If anything, other problems where you need to convert adding a negative to subtraction (the inverse of what is shown here) make more sense.  Or converting subtracting a negative to adding a positive.&lt;/p&gt;&#xA;&lt;p&gt;I don't think the speculations about group theory and the like (e.g. order of operations having different impact than in basic arithmetic/algebra) make sense in the context of helping kids learn beginning algebra, even pre-algebra.  We will deal with the differences involved in symmetry operations when we get to that (way further down the road).  If we even get there (for a lot of the kids).&lt;/p&gt;&#xA;" OwnerUserId="19973" LastEditorUserId="19973" LastEditDate="2022-06-11T19:52:59.570" LastActivityDate="2022-06-11T19:52:59.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25316" PostTypeId="2" ParentId="25306" CreationDate="2022-06-12T03:40:31.480" Score="2" Body="&lt;p&gt;So the only questions I think are relevant are, in order:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Does you student enjoy this approach?&lt;/li&gt;&#xA;&lt;li&gt;Does whoever is paying your salary (the parents?) appreciate this approach?&lt;/li&gt;&#xA;&lt;li&gt;Can you demonstrate that this approach is improving the student's mathematical abilities and/or interest?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If yes to all those, then go for it. If it's working for your student, then it is working.&lt;/p&gt;&#xA;&lt;p&gt;Of course you have to be open and clear to yourself on how you evaluate if it's working. Do you, for instance, have the student tell (or better, write a short paragraph) on what they learned from each of these &amp;quot;random video&amp;quot; connections? Do you have some sort of metric to showed either increased engagement or ability (or grades if that is the goal) over time?&lt;/p&gt;&#xA;" OwnerUserId="15606" LastActivityDate="2022-06-12T03:40:31.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25317" PostTypeId="1" AcceptedAnswerId="25318" CreationDate="2022-06-14T06:07:53.047" Score="3" ViewCount="875" Body="&lt;p&gt;I am more of an Intuitive Learner and Teacher so while looking at Linear Equations chapter I see that they are teaching Equations of Line there, which in my opinion is wrong.&lt;/p&gt;&#xA;&lt;p&gt;See How I understand it is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Equation of Line&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;code&gt;y = mx + c&lt;/code&gt; : This in a way gives you the address of the Street(Line). WHERE this line lies in XY Plane.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Whereas&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Linear Equation&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;code&gt;ax + by + c = 0&lt;/code&gt; : This gives a way to verify whether a House(Point) lies in the Aforementioned Street.&lt;/p&gt;&#xA;&lt;p&gt;Another difference I see is, Equation of a Line is relation between &lt;strong&gt;VALUES&lt;/strong&gt; on x and y axis, whereas Linear Equation deals with &lt;strong&gt;POSITIONS&lt;/strong&gt;(projections of x-y values on XY plane) on the XY planes.&#xA;In a way Linear Equation represents the solution set for Equation of a Line.&lt;/p&gt;&#xA;&lt;p&gt;Am I correct in my understanding?&#xA;Are Linear Equations and Equations of a Line, one and the same thing?&#xA;Any sources where I can get more clarification on these.&lt;/p&gt;&#xA;" OwnerUserId="12610" LastActivityDate="2022-06-20T23:22:18.500" Title="How do I teach the difference between Linear Equations and Equation of a Line" Tags="|primary-education|linear-algebra|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25318" PostTypeId="2" ParentId="25317" CreationDate="2022-06-14T16:17:58.623" Score="6" Body="&lt;p&gt;The only &amp;quot;real&amp;quot; difference (except for maybe some analogies that work differently as in your house/street example) is that the linear equations are a proper superset of the equations of lines because you can't describe vertical lines with &lt;span class=&quot;math-container&quot;&gt;$y=mx+c$&lt;/span&gt;, but you get them with &lt;span class=&quot;math-container&quot;&gt;$ax+by+c=0$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$b=0$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2022-06-14T16:17:58.623" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25319" PostTypeId="2" ParentId="25317" CreationDate="2022-06-14T23:40:55.957" Score="2" Body="&lt;p&gt;They aren't very much different from some perspectives. If what you care about is the question &amp;quot;Is this point on the line or not?&amp;quot; then you can simply plug your &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; values into either of them. If you get an equality, then yes. Otherwise, no. Here, the difference is that the &lt;span class=&quot;math-container&quot;&gt;$y=mx+b$&lt;/span&gt; form cannot handle vertical lines while &lt;span class=&quot;math-container&quot;&gt;$Ax+By+C=0$&lt;/span&gt; can. We often call &lt;span class=&quot;math-container&quot;&gt;$Ax+By+C=0$&lt;/span&gt; the standard form because it can handle all lines and any two different triples &lt;span class=&quot;math-container&quot;&gt;$(A,B,C)$&lt;/span&gt; will give different lines. For example, &lt;span class=&quot;math-container&quot;&gt;$y=mx+b$&lt;/span&gt; corresponds to &lt;span class=&quot;math-container&quot;&gt;$(A,B,C)=(-m,1,-b)$&lt;/span&gt;. (You subtract &lt;span class=&quot;math-container&quot;&gt;$mx+b$&lt;/span&gt; from both sides.)&lt;/p&gt;&#xA;&lt;p&gt;However, if we &lt;em&gt;can&lt;/em&gt; write a line in the form &lt;span class=&quot;math-container&quot;&gt;$y=mx+b$&lt;/span&gt;, then we have written &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as an &lt;em&gt;explicit&lt;/em&gt; function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. That is, you can pop an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; value in, do a few bits of arithmetic and get the corresponding value of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. The point is that you don't have to plug in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; then &lt;em&gt;solve&lt;/em&gt; for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. If you started with &lt;span class=&quot;math-container&quot;&gt;$Ax+By+C=0$&lt;/span&gt;, you would. (It's not hard to do so, but that isn't the point.)&lt;/p&gt;&#xA;&lt;p&gt;(Note: I use explicit as a contrast to the standard use of &lt;em&gt;implicit&lt;/em&gt;. e.g. &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2=1$&lt;/span&gt; defines &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as an implicit function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, away from &lt;span class=&quot;math-container&quot;&gt;$x=\pm1$&lt;/span&gt;. )&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2022-06-14T23:40:55.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25320" PostTypeId="2" ParentId="25317" CreationDate="2022-06-16T14:43:01.547" Score="3" Body="&lt;p&gt;Your intuition is fundamentally not correct. These are all equations of a line, and your textbook is correct to include them in the same chapter. The term &amp;quot;Linear Equation&amp;quot; is used because it is, in fact, the Equation of a Line. It's a fact I take the time to prove in my college algebra courses: the graph of any linear equation in two variables is a line.&lt;/p&gt;&#xA;&lt;p&gt;That said, there are different formats in which one can write a line, with various advantages and disadvantages in different use-cases; but they're all effectively equivalent, as can be shown with just a few lines of algebraic manipulation. Customary terminology includes the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Standard Form&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$ax + by = c$&lt;/span&gt;. Pros: Used as the definition of a linear equation. Can represent any line, including vertical ones. Can be written without use of fractions for the coefficients. Expands nicely to other shapes, such as conic sections. Easy to find both x- and y-intercepts.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Slope-Intercept Form&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$y = mx + b$&lt;/span&gt;. Pros: Has a clear graphical interpretation via use of slope. As a function, easy to compute dependent variable &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; from known independent variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Can be easily estimated visually from a graph. Bootstraps to understanding derivative in calculus. Can be implemented directly in a computer programming language.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Point-Slope Form&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$y - y_1 = m(x - x_1)$&lt;/span&gt;. Pros: Appears naturally midway into proof that nonvertical lines can be written in slope-intercept form. Shortcuts finding linear equation through two given points. Useful for other purposes in calculus.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;There are other forms as well, and it would benefit you to know this standard terminology (shown in your textbook and elsewhere).&lt;/p&gt;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13026/are-there-any-good-exercises-on-point-slope-form-of-a-linear-equation-by-itself&quot;&gt;Are there any good exercises on point-slope form of a linear equation by itself?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2022-06-16T16:05:13.263" LastActivityDate="2022-06-16T16:05:13.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25321" PostTypeId="1" CreationDate="2022-06-16T18:52:07.793" Score="11" ViewCount="473" Body="&lt;p&gt;I'm a maths tutor, and my students/tutees are aged &lt;span class=&quot;math-container&quot;&gt;$11 - 18.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Obviously I have limited time with students, usually one hour per week. Moreover, if parents don't see improvement over a year or two they will look for a better tutor which is undesirable for me.&lt;/p&gt;&#xA;&lt;p&gt;I feel like students not knowing their basics well enough, like being quick at multiplication, addition, subtraction, division, and fractions, holds them back enormously in learning more difficult topics. Their learning speed in these difficult compound topics that requires quick arithmetic is massively reduced and this disrupts their learning of the main goals of the harder topic.&lt;/p&gt;&#xA;&lt;p&gt;When I was a young kid of 4 - 7 years old (or something like that), my Mum used to play a cassette in the car of the times tables, and ever since, I have had instant recall, and this has been one of the main contributing factors for allowing me to learn the more difficult topics easier than others. [I am not comparing myself to others on this website, as most people here are also probably good/ quick at arithmetic. I am comparing myself to the general population]. This is because I don't get distracted from the learning goals of a topic when calculating &lt;span class=&quot;math-container&quot;&gt;$7\times 6$&lt;/span&gt; is required to attain that unrelated, more difficult goal: my brain converts &lt;span class=&quot;math-container&quot;&gt;$7\times 6$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$42$&lt;/span&gt; in under a second, and I can get on with understanding the more difficult topic.&lt;/p&gt;&#xA;&lt;p&gt;I was wondering if it was a good idea to give students - who are not quick or accurate enough at these basic calculations - tools, and a programme that if they stick to, will guarantee improvement in multiplication, arithmetic, fractions. And then only continue on the compound topics once they have proven their speed and accuracy in various tests? Then their newly-found instant recall will not stand in their way of learning harder topics, and might even give them a confidence boost in learning mathematics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://automatic-algebra.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;automatic-algebra&lt;/a&gt; has one-minute/&lt;span class=&quot;math-container&quot;&gt;$30$&lt;/span&gt;-second/&lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt;-second multiple choice tests on different aspects of arithmetic. But the difficulty and allotted time of the tests don't vary, and so the tests aren't very comprehensive/ are one-dimensional. However, I have tried that website with some students and some students do improve a bit with it and some students didn't improve much with it. So I was thinking of creating a website/webpage that was like that one but where, for example, either the difficulty of the questions increased, or the allotted time decreased, or a combination of both.&lt;/p&gt;&#xA;&lt;p&gt;I was thinking of changing my tutoring to be a strict, hard-line approach: Either they improve so that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;they can multiply any two numbers from their &lt;span class=&quot;math-container&quot;&gt;$9\times 9$&lt;/span&gt; tables in &lt;span class=&quot;math-container&quot;&gt;$&amp;lt; 5$&lt;/span&gt; seconds.&lt;/li&gt;&#xA;&lt;li&gt;Something similar with addition&lt;/li&gt;&#xA;&lt;li&gt;Then on to fractions - obviously more time is needed for this, but they should still need to do them quickly.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But I don't continue tutoring with them if they can't do the calculations within the time period.&lt;/p&gt;&#xA;&lt;p&gt;Is this approach too hard-line?&lt;/p&gt;&#xA;&lt;p&gt;Personally, I think it may be a good idea for almost every student who is not quick at these calculations. &lt;em&gt;I just worry as to whether or not this will work&lt;/em&gt;. If they haven't bothered learning their times tables by now, and they are reluctant to do so with my instruction, then what? Do I sweep this under the rug and teach them harder topics anyway? I know that students &lt;em&gt;can&lt;/em&gt; learn harder topics, but it just &lt;em&gt;takes much much longer for them to be able to do exam questions quickly in these harder topics&lt;/em&gt; (because they don't have the instant recall of arithmetic operations), and this exam performant is very important when it comes to tutoring, as parents are more likely to fire me if the student has done poorly in exams.&lt;/p&gt;&#xA;&lt;p&gt;Another aspect of this is that if I am too &amp;quot;hard-line&amp;quot; in this approach then the student may not like me (but the parents might)...&lt;/p&gt;&#xA;&lt;p&gt;What do you think? Has anyone tried this approach in practise?&lt;/p&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2022-06-18T19:55:25.820" LastActivityDate="2022-07-31T21:24:08.027" Title="For students who don't have instant recall with basic arithmetic, should I be stubborn about training them to have instant recall before moving on?" Tags="|self-learning|tutoring|arithmetic|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25322" PostTypeId="2" ParentId="25321" CreationDate="2022-06-16T19:13:48.877" Score="5" Body="&lt;p&gt;It depends greatly on the students.&lt;/p&gt;&#xA;&lt;p&gt;I have had great success training students with the flashmaster app.  With this app you can focus on improving time for basic facts. I have used this with my classroom students, insisting they spend a certain amount of time on it.&lt;/p&gt;&#xA;&lt;p&gt;Although less than 5 seconds is great, a student who can find the number in 10 or 20 seconds will also have success, so I think your goal is unnecessarily high.&lt;/p&gt;&#xA;&lt;p&gt;Another approach that I have been successful with is to give the students that I am tutoring a times table.  I tell them that they have to fill it in before we can do the work. This is often effective:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;filling in the chart is good practice for them - which means it is not a waste of time.&lt;/li&gt;&#xA;&lt;li&gt;you can see which facts are hard for them - often they don't know the product of 7 and 8 and I tell them 5, 6, 7, 8; 56 is 7 times 8 and they remember it afterwards.&lt;/li&gt;&#xA;&lt;li&gt;This can be easier than memorizing. For example a student can count by 3's to fill in the 3's and then double the 3's to get the 6's&lt;/li&gt;&#xA;&lt;li&gt;Once they have the chart, it will be easy to simplify fractions etc.&lt;/li&gt;&#xA;&lt;li&gt;Student who do this in class, will write times tables on their tests, so they can refer to them.&lt;/li&gt;&#xA;&lt;li&gt;If the student really resents having to do this - the alternative is to learn the tables and have them at their fingertips.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I test mastery by mixing up all the times facts from 1 to 10 and telling them that they have to complete it in 10 minutes.  It's not super fast but it's enough to succeed.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2022-06-16T19:13:48.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25323" PostTypeId="1" CreationDate="2022-06-16T22:24:56.657" Score="1" ViewCount="383" Body="&lt;p&gt;What is the most efficient (fastest) multiplication strategy that can be done mentally or with a pencil/paper? We can include strategies that use interesting tools like Napiers Bones or Soroban math. Strategies must generalize well, but answers might include special cases that allow especially quick calculations.&lt;/p&gt;&#xA;&lt;p&gt;Below are some examples of strategies that are used for the multiplication of integers. I conclude that Soroban mental multiplication is the fastest method based on my research.&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;Partial Products:&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;p&gt;23 x 13&#xA;=(20+3)x(10+3)&#xA;=200 + 60 + 30 + 9&#xA;=299&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;Open Array&lt;/strong&gt; (a graphical version of partial products)&lt;/h2&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5GXGk.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5GXGk.png&quot; alt=&quot;An open array showing partial products&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;Standard USA Algorithm with Carrying Place Value&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zrhkD.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zrhkD.png&quot; alt=&quot;Standard USA Multiplication Algorithm&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;&amp;quot;Russian&amp;quot; Multiplication Method&lt;/strong&gt;&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=HJ_PP5rqLg0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russian Method via Numberphile&lt;/a&gt;:&lt;/h2&gt;&#xA;&lt;p&gt;Image below sourced from &lt;a href=&quot;https://www.popularmechanics.com/science/math/a30797362/russian-multiplication-method/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Popular Mechanic&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7yyjB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7yyjB.png&quot; alt=&quot;Russian Multiplication Graphic&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;Korean Chisanbop&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3MnDb.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3MnDb.png&quot; alt=&quot;Chisanbop&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Chisanbop#/media/File:Chisanbop_hands_numbered.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Image source and description at Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;&lt;strong&gt;Soroban/Abacus Mental Multiplication&lt;/strong&gt;&lt;/h2&gt;&#xA;&lt;p&gt;I've concluded this method is the most efficient since this is what kids who win the speed contests tend to use (See &amp;quot;Anzan&amp;quot; contests)&lt;/p&gt;&#xA;&lt;p&gt;Explained at &lt;a href=&quot;https://www.sorobanexam.org/basics.html#multiply&quot; rel=&quot;nofollow noreferrer&quot;&gt;sorobanexam&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/QfKmQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QfKmQ.png&quot; alt=&quot;Soroban multiplication image&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.wired.com/2009/11/1112abacus-beats-calculator/&quot; rel=&quot;nofollow noreferrer&quot;&gt;This is a fun article&lt;/a&gt; that suggests that the soroban is actually faster than an electronic calculator 4/5 times.&lt;/p&gt;&#xA;&lt;h2&gt;Disclaimer&lt;/h2&gt;&#xA;&lt;p&gt;Research (such as Principals to Actions NCTM 2014) indicates building procedural fluency through conceptual understanding is better teaching practice than teaching &amp;quot;fast&amp;quot; procedures alone. This isn't intended for general teaching practice, but rather, for FUN!&lt;/p&gt;&#xA;" OwnerUserId="10227" LastEditorUserId="10227" LastEditDate="2022-06-16T22:34:41.720" LastActivityDate="2022-06-24T08:31:29.923" Title="What is the MOST efficient paper/mental multiplication algorithm for integers?" Tags="|primary-education|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25324" PostTypeId="2" ParentId="25317" CreationDate="2022-06-17T02:10:12.377" Score="7" Body="&lt;p&gt;I suspect that two issues may be partly underlying your intuition:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First, the equation &lt;span class=&quot;math-container&quot;&gt;$y = mx + c$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as an explicit &lt;em&gt;function of&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, whereas &lt;span class=&quot;math-container&quot;&gt;$ax + by + c = 0$&lt;/span&gt; only gives a &lt;em&gt;relation between&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.  This is a meaningful distinction, one worth making.  However, the correct words for making this distinction are not &amp;quot;equation of a line&amp;quot; vs. &amp;quot;linear equation&amp;quot; (which mean exactly the same thing) but rather &amp;quot;function&amp;quot; vs. &amp;quot;relation&amp;quot;.  As other commenters have pointed out, the format &lt;span class=&quot;math-container&quot;&gt;$y = mx + c$&lt;/span&gt; only describes non-vertical lines (which are functions), whereas &lt;span class=&quot;math-container&quot;&gt;$ax + by + c = 0$&lt;/span&gt; also includes vertical lines (which are not functions).  Another way of describing this distinction is that &lt;span class=&quot;math-container&quot;&gt;$y=mx + c$&lt;/span&gt; defines &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; &lt;em&gt;explicitly&lt;/em&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, whereas &lt;span class=&quot;math-container&quot;&gt;$ax + by + c = 0$&lt;/span&gt; defines &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; &lt;em&gt;implicitly&lt;/em&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (assuming &lt;span class=&quot;math-container&quot;&gt;$b \ne 0$&lt;/span&gt;).&lt;/li&gt;&#xA;&lt;li&gt;Second, in the equation &lt;span class=&quot;math-container&quot;&gt;$y = mx + c$&lt;/span&gt; the parameters &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; are &lt;strong&gt;uniquely determined&lt;/strong&gt; by the non-vertical line.  That is:  given any non-vertical line, there is a unique pair of numbers &lt;span class=&quot;math-container&quot;&gt;$(m, c)$&lt;/span&gt; with the property that &lt;span class=&quot;math-container&quot;&gt;$y = mx + c$&lt;/span&gt; describes the line; and conversely, given any pair of numbers &lt;span class=&quot;math-container&quot;&gt;$(m, c)$&lt;/span&gt; there is a unique line given by the equation &lt;span class=&quot;math-container&quot;&gt;$y = mx + c$&lt;/span&gt;.  In marked contrast, the parameters &lt;span class=&quot;math-container&quot;&gt;$a, b, c$&lt;/span&gt; in the equation &lt;span class=&quot;math-container&quot;&gt;$ax + by + c = 0$&lt;/span&gt; are not uniquely determined; if &lt;span class=&quot;math-container&quot;&gt;$ax + by + c = 0$&lt;/span&gt; is an equation for some line &lt;span class=&quot;math-container&quot;&gt;$\ell$&lt;/span&gt;, then for any non-zero value &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, the equation &lt;span class=&quot;math-container&quot;&gt;$kax + kby + kc = 0$&lt;/span&gt; describes the same line.  I suspect this may be what your intuition is recognizing when you say that &lt;span class=&quot;math-container&quot;&gt;$y = mx + b$&lt;/span&gt; is &amp;quot;the address&amp;quot; of the line; it is a unique identifier of the line, not just one of many possible descriptions of it.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Having said all of this, it is definitely the case that the textbook is &lt;em&gt;not wrong&lt;/em&gt;. There is indeed a distinction to be made between the two forms of the equation, but it is not the distinction you think it is.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I want to commend you for asking this question.  Some of the comments have been condescending and even derogatory, and I think that's not warranted. It is hard to admit you don't completely understand something, especially when you are preparing to be a teacher.  The whole point of this site is to answer questions about teaching math; if novices can't ask questions here, where else could they go?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2022-06-20T23:22:18.500" LastActivityDate="2022-06-20T23:22:18.500" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25325" PostTypeId="2" ParentId="25085" CreationDate="2022-06-17T10:29:01.137" Score="1" Body="&lt;p&gt;If your students are at university level, then &amp;quot;White Light&amp;quot; by Rudy Rucker is a banger.&lt;/p&gt;&#xA;&lt;p&gt;It explores infinity. It is perhaps the only novel which features a practical application of the Banach-Tarski paradox.&lt;/p&gt;&#xA;&lt;p&gt;Mind, if your prospective readers are at all prudish or sensitive, then perhaps not. Whatever you do, do &lt;em&gt;not&lt;/em&gt; suggest it as recommended reading for high-school students or below, or you'll lose your job.&lt;/p&gt;&#xA;" OwnerUserId="15290" LastActivityDate="2022-06-17T10:29:01.137" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25326" PostTypeId="2" ParentId="25323" CreationDate="2022-06-18T08:10:55.743" Score="6" Body="&lt;p&gt;Instead of a comment, I go try an answer, extending the good comments already there...&lt;/p&gt;&#xA;&lt;p&gt;If it comes to computers and runtime, I know the &lt;a href=&quot;https://en.wikipedia.org/wiki/Karatsuba_algorithm&quot; rel=&quot;noreferrer&quot;&gt;Karatsuba algorithm&lt;/a&gt; as an example of a very fast algorithm.&lt;/p&gt;&#xA;&lt;p&gt;If it comes to mental calculation, indeed I think the fun part is that you first need to look at the numbers and then decide on the way to multiply. That then get's very personal and depends on your knowledge. eg. if your fast and accurate at adding, subtracting and taking complements, you might avoid carries with some strategies. If you're fast and accurate at halving/doubling you might use some factoring strategies. For example: in order to multiply by 5 you might find it easier to multiply by ten and then take half.&#xA;This way, there are neat little tricks and strategies that are powerful, but not in all cases. so again: you learn to look at the numbers and then decide on the strategy.&#xA;Different strategies are discussed in more detail elsewhere in this forum, e.g. here: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13233/whats-the-best-technique-to-do-math-calculations-in-my-head&quot;&gt;Mental Calculation strategies.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2022-06-18T08:10:55.743" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25327" PostTypeId="2" ParentId="25085" CreationDate="2022-06-18T08:37:34.997" Score="1" Body="&lt;p&gt;I've just thought of another candidate: &lt;em&gt;Anathem&lt;/em&gt; by Neil Stephenson came along at exactly the point in my life where I needed the confidence of my convictions that I needed to model the sales structure of the company I worked for as a directed acyclic graph.&lt;/p&gt;&#xA;" OwnerUserId="15290" LastActivityDate="2022-06-18T08:37:34.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25328" PostTypeId="2" ParentId="19255" CreationDate="2022-06-18T13:42:11.687" Score="1" Body="&lt;p&gt;Harald Cramer, Mathematical Methods of Statistics. Princeton&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;First Part: MATHEMATICAL INTRODUCTION&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Chapter 1. General properties of sets&#xA;Chapter 2. Linear point sets&#xA;Chapter 3. Point sets in η dimensions&#xA;Chapter 4. The Lebesgue measure of a linear point set&#xA;Chapter 5. The Lebesgue integral for functions of one variable.&lt;br /&gt;&#xA;Chapter 6. Non-negative additive set functions in R&#xA;Chapter 7. The Lebesgue-Stieltjes integral for functions of one  variable&#xA;Chapter 8. Lebeegue measure and other additive set functions in R&#xA;Chapter 9. The Lebesgue-Stieltjes integral for functions of n variables&#xA;Chapter 10. Fourier integrals&lt;br /&gt;&#xA;Chapter 11. Matrices, determinants and quadratic forms&#xA;Chapter 12. Miscellaneous complements&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Second Part : RANDOM VARIABLES AND PROBABILITY  DISTRIBUTIONS.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Chapter 13. Statistics and probability&#xA;Chapter 14. Fundamental definitions and axioms&#xA;Chapter 15. General properties&#xA;Chapter 16. Various discrete distributions&#xA;Chapter 17. The normal distribution&#xA;Chapter 18. Various distributions related to the normal&#xA;Chapter 19. Further continuous distributions&#xA;Chapter 20. Some convergence theorems&#xA;Chapter 21. The two-dimensional case&#xA;Chapter 22. General properties of distributions in R¬n&#xA;Chapter 23. Regression and correlation in η variables&lt;br /&gt;&#xA;Chapter 24. The normal distribution&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Third Part. STATISTICAL INFERENCE.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Chapter 25. Preliminary notions on sampling&#xA;Chapter 26. Statistical inference&#xA;Chapter 27. Characteristics of sampling distributions&#xA;Chapter 28. Asymptotic properties of Sampling distributions&#xA;Chapter 29. Exact sampling distributions&#xA;Chapter 30. Tests of goodness of fit and allied tests&#xA;Chapter 31. Tests of significance for parameters&#xA;Chapter 32. Classification of estimates&#xA;Chapter 33. Methods of estimation&#xA;Chapter 34. Confidence regions&#xA;Chapter 35. General theory of testing statistical hypotheses&#xA;Chapter 36. Analysis of variance&#xA;Chapter 37. Some regression problems&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2022-06-18T13:42:11.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25329" PostTypeId="2" ParentId="25321" CreationDate="2022-06-18T19:38:53.953" Score="4" Body="&lt;p&gt;As I said in a comment, the quiz website you describe sounds a lot like mine at &lt;a href=&quot;http://www.automatic-algebra.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Automatic-Algebra.org&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;One chief reason that I made that site was because I saw students in my college remedial algebra classes be completely unable to progress at the point when they had to do polynomial factoring, when they didn't know their multiplication tables from memory. As in, they'd out-loud give up and say this task was simply impossible for them.&lt;/p&gt;&#xA;&lt;p&gt;Now, was that solution a silver bullet? No, and I doubt there is any. In recent years success rates in those classes have trended ever-downward, and my major university system (and others) are terminating all of those classes, because approximately no one benefits from them. The current dominant theory is that those basic facts should be reviewed in a just-in-time, mixed-practice, corequisite support context -- but I'm skeptical that will really make any difference.&lt;/p&gt;&#xA;&lt;p&gt;I will observe that there's at least a continuum of facility at picking up new mathematical (and other) facts -- some of us pick them up mostly on first sight, while others fail to retain them even after many tries. So admittedly a purely logical sequencing of needed-fact-A coming sometime before dependent-fact-B, which is sufficient for highly-proficient math learners, is insufficient for math learners with demonstrable weaknesses. Most people aren't like us, the visitors to this site.&lt;/p&gt;&#xA;&lt;p&gt;And I think I've seen experiments with classroom digital mastery systems, where students had to succeed at early skills before moving on, but the results weren't better than anything else.&lt;/p&gt;&#xA;&lt;p&gt;So arguably a policy might be that you should give some amount of review, but move at on at some later point, regardless of whether they digested the early facts or not. Even if they did, they'll likely forget them at some later point anyway. You might need to accept needing to always review those facts when they become critical later on. I suspect such students will be fundamentally limited in how far they can go, but I doubt there's any way to fix that for them.&lt;/p&gt;&#xA;&lt;p&gt;Related questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/352/what-are-the-arguments-for-and-against-learning-the-multiplication-table-by-hear&quot;&gt;What are the arguments for and against learning the multiplication table by heart?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/11396/what-fraction-of-the-population-is-incapable-of-learning-algebra&quot;&gt;What fraction of the population is incapable of learning algebra?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2022-06-18T23:10:48.083" LastActivityDate="2022-06-18T23:10:48.083" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25330" PostTypeId="2" ParentId="19255" CreationDate="2022-06-19T06:47:30.460" Score="1" Body="&lt;p&gt;Late answer, but I feel that the suggestion from Steve in a comment (a few hours after the question was posted) deserves to be promoted to an answer.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Casella, George, and Roger L. Berger. &lt;em&gt;Statistical inference&lt;/em&gt;. Cengage Learning, 2021.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Arguably, this may be the standard text for a mathematically rigorous treatment of statistics. It's what I used in graduate school; and I keep it close by on my bookshelf and refer to it frequently when I teach undergraduate statistics. It's sufficiently prominent to have spawned several prior questions on SE Mathematics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://faculty.cengage.com/titles/9780534243128&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Statistical Inference&lt;/em&gt; at Cengage&lt;/a&gt; (with Table of Contents).&lt;/p&gt;&#xA;&lt;p&gt;Related questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/847064/better-than-casella-and-bergers-statistical-inference&quot;&gt;Better than Casella and Berger's &lt;em&gt;Statistical Inference&lt;/em&gt;?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/1828025/book-recommendation-please-casella-berger-statistical-inference?rq=1&quot;&gt;Book Recommendation Please! [Casella Berger] Statistical Inference&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2022-06-19T06:47:30.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25331" PostTypeId="2" ParentId="25317" CreationDate="2022-06-19T08:30:38.370" Score="2" Body="&lt;h1&gt;Etymology/Grammar&lt;/h1&gt;&#xA;&lt;p&gt;&amp;quot;Line&amp;quot; is the root word, a noun whereas &amp;quot;linear&amp;quot; is a derived word, an adjective. So, apart from the grammar distinction, one could call &amp;quot;line&amp;quot; and &amp;quot;linear&amp;quot; the same word.&lt;/p&gt;&#xA;&lt;p&gt;Life however does not always respect grammar nor maintain &lt;a href=&quot;https://philosophy.stackexchange.com/questions/38674/is-philosophy-about-organizing-our-ignorance/38675#comment98843_8675&quot;&gt;etymological connections&lt;/a&gt;&#xA;... including...&lt;/p&gt;&#xA;&lt;h1&gt;Math&lt;/h1&gt;&#xA;&lt;p&gt;In math the provenance of &amp;quot;line&amp;quot; is just well a  line – a geometrical straight line.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Linear&amp;quot; &lt;em&gt;starts off&lt;/em&gt; there ­– lines and their equations in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; – and then broadens considerably:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;linear combinations in arbitrary vector spaces where the notion of (geometric) line may be quite far-fetched, eg &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;, polynomials over a variable, etc&lt;/li&gt;&#xA;&lt;li&gt;linear differential equations&lt;/li&gt;&#xA;&lt;li&gt;linear difference equations&lt;/li&gt;&#xA;&lt;li&gt;and so on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In all such cases the connotations of &amp;quot;linear&amp;quot; are likely to be far removed from anything reasonably geometrical.&lt;/p&gt;&#xA;&lt;p&gt;And so a more mathematically sophisticated rendering of your question would be to ask:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;Linear&amp;quot; in the context of which vector space?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[Aside: Though I suspect your students are not ready for this level of abstraction, it may still be useful for your own understanding]&lt;/p&gt;&#xA;&lt;p&gt;The underlying issue I feel you're groping for is...&lt;/p&gt;&#xA;&lt;h1&gt;Normal Form&lt;/h1&gt;&#xA;&lt;p&gt;In basic school arithmetic we say &amp;quot;2 + 3 is 5&amp;quot; and write it as &lt;span class=&quot;math-container&quot;&gt;$2 + 3 = 5$&lt;/span&gt;.&#xA;But most typically you will replace &lt;span class=&quot;math-container&quot;&gt;$2+3$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; &lt;em&gt;and not vice versa.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;So in some loose informal sense 5 is better than 2+3.&lt;/p&gt;&#xA;&lt;p&gt;This informal notion is formalized in systems like lambda calculus, and rewriting systems where two sets are defined, a larger set of terms and a smaller subset of normal-form terms, and computation in the formal system proceeds until there are no rewriteable terms and only a normal form (also called ground term) remains.&lt;/p&gt;&#xA;&lt;p&gt;[Note: &amp;quot;Ground term&amp;quot; is a downward metaphor — things fall downward till they hit the ground. The point they hit the ground is the ground term or normal form]&lt;/p&gt;&#xA;&lt;p&gt;So in the lambda calculus &lt;span class=&quot;math-container&quot;&gt;$2+3$&lt;/span&gt; would be rewritten to &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2 + 3 \rightarrow 5$&lt;/span&gt; would be considered as a &lt;em&gt;delta rewrite rule.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Notice that there is a strange double speak going on here:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Denotationally :: &lt;span class=&quot;math-container&quot;&gt;$ 2+ 3 = 5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$5 = 2+3$&lt;/span&gt; are identical.&lt;/li&gt;&#xA;&lt;li&gt;Computationally :: We prefer &lt;span class=&quot;math-container&quot;&gt;$2+3 = 5$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$5 = 2+3$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If we want to be more rigorous we would write it with a &lt;span class=&quot;math-container&quot;&gt;$\rightarrow$&lt;/span&gt; rather than a &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;.&lt;br /&gt;&#xA;So &lt;span class=&quot;math-container&quot;&gt;$2+3 \rightarrow 5$&lt;/span&gt; is a rule in the formal rewrite system but &lt;span class=&quot;math-container&quot;&gt;$5 \rightarrow 2+3$&lt;/span&gt; is not.&lt;/p&gt;&#xA;&lt;p&gt;Typically mathematicians are not that rigorous, sometimes treating &lt;span class=&quot;math-container&quot;&gt;$2+3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; as completely equivalent, and sometimes preferencing one over the other.&lt;/p&gt;&#xA;&lt;p&gt;As soon as we go beyond simple numbers trouble starts. Yeah, we &lt;em&gt;call&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x^2 -5x + 6$&lt;/span&gt; the simplified form and &lt;span class=&quot;math-container&quot;&gt;$(x-2)(x-3)$&lt;/span&gt; the factorized form.  But there's nothing &lt;em&gt;intrinsically&lt;/em&gt; preferencing one over the other.&lt;/p&gt;&#xA;&lt;p&gt;This gets only more knotty the more advanced we get.&lt;/p&gt;&#xA;&lt;p&gt;And even returning to plain numbers:&#xA;So we may naturally prefer &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$2+3$&lt;/span&gt; but between 12(base10) 14(base 8) and 1110(base2) there is little 'more normal' about the one than the other aside from convention.&lt;/p&gt;&#xA;&lt;p&gt;Coming to your question:&lt;/p&gt;&#xA;&lt;h1&gt;Equations of a line&lt;/h1&gt;&#xA;&lt;p&gt;As Daniel Collins points out there are a bunch of different equation forms for straight lines&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They are (generally) convertible between each other&lt;/li&gt;&#xA;&lt;li&gt;each with their pros and cons&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In short there are different forms of equation of a line. &lt;strong&gt;And no normal form.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="17672" LastEditorUserId="17672" LastEditDate="2022-06-19T16:06:31.510" LastActivityDate="2022-06-19T16:06:31.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25332" PostTypeId="2" ParentId="25321" CreationDate="2022-06-20T20:06:12.323" Score="3" Body="&lt;p&gt;&lt;strong&gt;The question&lt;/strong&gt;: Should I discontinue tutoring engagements with students who refuse to remediate their inability to perform basic calculations to some standard?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My answer&lt;/strong&gt;: it depends on your goals as a tutor / for your business.&lt;/p&gt;&#xA;&lt;p&gt;If your goal as a tutor is strictly to make money and you're worried that you won't be able to replace any students that walk away from your services, then you certainly shouldn't terminate any of these arrangements yourself but rather wait for dissatisfied students to leave on their own (assuming you don't think they're going to leave bad reviews on a website, bad-mouth you,  cut you out of some tutoring referral program that exists with local schools, etc).&lt;/p&gt;&#xA;&lt;p&gt;If you aren't concerned about your enrollment or your frustration with students' lack of progress outweighs your need for the money, then you can implement a policy without fear of losing business. Such a policy would likely be in the long term interest of your students and business because:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Many students in the stated age range (11 - 18-year-olds) could benefit from better arithmetic skills in the classroom (if nowhere else) for years to come.&lt;/li&gt;&#xA;&lt;li&gt;Seeking mastery of these basic facts appears to be pretty common practice, so the perceived benefits for approaching more difficult problems may be real.&lt;/li&gt;&#xA;&lt;li&gt;If the benefits are truly real, then you can imagine a virtuous cycle of improved student outcomes, increased morale, positive word-of-mouth leading to new business, increased credibility as a tutor, more compliance from future students, etc.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If you're going forward with this, next steps would be to iteratively refine your approach (basically some of the steps you already outlined):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Make a clear list of outcomes that are actually needed for success (cursory review online suggests that 5 seconds for a basic addition/multiplication fact is pretty lenient). Having a realistic/achievable goal is good for morale, ensures you're not chasing diminishing returns, and improves the likelihood of success on the following point...&lt;/li&gt;&#xA;&lt;li&gt;Create a program, if followed, which actually produces desired outcomes outlined in point 1 (May require student trials.) This is important because if the program doesn't work, that hurts your credibility with compliant students. They need to believe both on this subtask and the whole initiative that everyone is aligned on what the goal is and that you, the tutor, know what it takes for them to achieve said goal.&lt;/li&gt;&#xA;&lt;li&gt;Craft your elevator pitch when you broach the topic with each student (some may be more receptive to analogies with sports, others music, etc) but the basic idea that these are fundamentals that you need automaticity with to elevate your performance should be easy enough. This builds momentum for an otherwise boring (borderline embarrassing depending on age?) task.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2022-06-20T20:06:12.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25333" PostTypeId="2" ParentId="21131" CreationDate="2022-06-22T14:51:34.780" Score="3" Body="&lt;p&gt;I have come to realize another very practical reason for teaching differentiation before integration.&lt;/p&gt;&#xA;&lt;p&gt;In most applications of integration we are splitting &lt;em&gt;something&lt;/em&gt; (area, volume, arclength, work, etc) into lots of tiny pieces, compute an approximation of each piece, and then sum. Taking the limit as the size of the pieces tends to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; yields an integral.&lt;/p&gt;&#xA;&lt;p&gt;What is underappreciated, I think, is that often the &amp;quot;approximate each piece&amp;quot; step will involve differentiation.&lt;/p&gt;&#xA;&lt;p&gt;Take arc length as an example.&lt;/p&gt;&#xA;&lt;p&gt;We have a function &lt;span class=&quot;math-container&quot;&gt;$f: [a,b] \to \mathbb{R}$&lt;/span&gt; and we want to calculate the arc length of the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is natural to first subdivide the interval &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; equal sized subintervals &lt;span class=&quot;math-container&quot;&gt;$[x_k, x_{k+1}]$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x_k = a + \frac{b-a}{N}(k-1)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$k=0,1,2, 3 \dots, N, N+1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Since arclength is additive, the length of the whole curve is the sum of the length over each subinterval.&lt;/p&gt;&#xA;&lt;p&gt;We then approximate each small arc by the length of the secant line connecting the two endpoints &lt;span class=&quot;math-container&quot;&gt;$\sqrt{(f(x_{k+1}) - f(x_k))^2 + (x_{k+1}-x_k)^2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The issue is that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sum_0^{N-1} \sqrt{(f(x_{k+1}) - f(x_k))^2 + (x_{k+1}-x_k)^2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;is &lt;strong&gt;not&lt;/strong&gt; in the form of a Riemann sum for any function.&lt;/p&gt;&#xA;&lt;p&gt;So we need to approximate each of these summands again using the derivative:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sqrt{(f(x_{k+1}) - f(x_k))^2 + (x_{k+1}-x_k)^2} \approx \sqrt{(f'(x_k)(x_{k+1}-x_k))^2 + (x_{k+1}-x_k)^2}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So we obtain&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sum_0^{N-1} \sqrt{(f'(x_k))^2 + 1} \Delta x$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;as our approximation of the arc length.&lt;/p&gt;&#xA;&lt;p&gt;This is an extremely common phenomenon:  forming an integral to compute a quantity of interest relies on decomposition into small pieces, but approximating these small pieces requires differentiation.  This is a good mathematical reason for studying differentiation first.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-06-22T14:51:34.780" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25334" PostTypeId="1" AcceptedAnswerId="25336" CreationDate="2022-06-22T18:00:45.850" Score="6" ViewCount="1841" Body="&lt;p&gt;How would you explain the difference in the results given by integration of the two functions &lt;span class=&quot;math-container&quot;&gt;$y = \frac{1}{1+x}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y = \frac{1}{1+x^2}$&lt;/span&gt; ?&lt;/p&gt;&#xA;&lt;p&gt;The graphs of these two function look so similar on positive x-values as x tends to infinity. In both cases y approaches to 0 and y = 0 is a horizontal asymptote in both graphs, but when you integrate with respect to x from 0 to infinity, the first graph has infinite area between the graph and the x axis, while the second graph has finite value π/2 as the area.&lt;/p&gt;&#xA;&lt;p&gt;How would you explain this difference to students?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="60" LastEditDate="2022-06-23T15:59:44.913" LastActivityDate="2022-06-25T02:27:30.160" Title="Explaining difference between improper integrals that converge and diverge" Tags="|secondary-education|calculus|" AnswerCount="5" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="25335" PostTypeId="2" ParentId="25334" CreationDate="2022-06-22T19:47:50.847" Score="4" Body="&lt;p&gt;To me (a dumb troll), the way to intuitively think about it is 0 versus infinity.  Infinity makes things huge.  Zero makes them small. So, if you have an asymptotic approach you have a huge (to forever) amount...infinite like.  But with very small parts.  So.  If the &amp;quot;infinity is winning&amp;quot;, you don't converge. If the zero is winning, you do.  And it's like a battle of rock and hard place.  And sometimes one wins and sometimes the other does.  If the &amp;quot;zero-ing&amp;quot; is happening faster than the &amp;quot;infinity-ing&amp;quot;, you get something that converges.  If the infinitying is happening faster, you get an integral that blows up.&lt;/p&gt;&#xA;" OwnerUserId="19997" LastActivityDate="2022-06-22T19:47:50.847" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25336" PostTypeId="2" ParentId="25334" CreationDate="2022-06-22T20:06:50.383" Score="7" Body="&lt;p&gt;Perhaps it's easier for students to see the reason for these results when you are looking at infinite series. Does the course you teach include this? (If not, you can still do it and set up a diagram showing that the infinite series is smaller than the diverging integral but still diverges, and also showing the converging series as bigger than its integral but still converging.)&lt;/p&gt;&#xA;&lt;p&gt;It's easy to explain why you diverge when adding &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots $&lt;/span&gt;. (&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}+\frac{1}{4}$&lt;/span&gt; is greater than &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;. So is the next group of 4 terms, and the next group of 8 terms, etc.)&lt;/p&gt;&#xA;&lt;p&gt;For the other, you just need a convincing argument that it converges. Here's my best thought at the moment. The first 3 terms (&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}+\frac{1}{5}+\frac{1}{10}$&lt;/span&gt;) add to less than 1. Ignore them, basically. The next 4 add to less than &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{16}=\frac{1}{4}$&lt;/span&gt;, and the next 8 add to less than &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{64}=\frac{1}{8}$&lt;/span&gt;, and so on. Which all adds to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;. (I was surprised that this went to a geometric series when I did this. But I believe I have this right.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="205" LastEditDate="2022-06-24T04:46:05.490" LastActivityDate="2022-06-24T04:46:05.490" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25337" PostTypeId="2" ParentId="25334" CreationDate="2022-06-22T22:02:26.093" Score="10" Body="&lt;p&gt;I'd say: Don't trust your eyes, especially not when infinity is concerned and you are plotting in one small region. Sure, they look fairly similar when you are plotting over &lt;span class=&quot;math-container&quot;&gt;$[1,10]$&lt;/span&gt;.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/6UqWP.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6UqWP.png&quot; alt=&quot;1/(1+x) in blue, 1/(1+x^2) in red&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1/(1+x)$&lt;/span&gt; in blue, &lt;span class=&quot;math-container&quot;&gt;$1/(1+x^2)$&lt;/span&gt; in red. Honestly, I think they look fairly different already, but even still.&lt;/p&gt;&#xA;&lt;p&gt;Now try plotting the two over &lt;span class=&quot;math-container&quot;&gt;$[10^{10}, 10^{11}]$&lt;/span&gt;.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/kWUDj.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kWUDj.png&quot; alt=&quot;1/(1+x) in blue, 1/(1+x^2) in red&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Again, &lt;span class=&quot;math-container&quot;&gt;$1/(1+x)$&lt;/span&gt; in blue, &lt;span class=&quot;math-container&quot;&gt;$1/(1+x^2)$&lt;/span&gt; in red. The red one is pretty much gone compared to the blue.&lt;/p&gt;&#xA;&lt;p&gt;After that, you pretty much have to go to proofs and establish bounds, etc.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2022-06-22T22:02:26.093" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25339" PostTypeId="2" ParentId="25334" CreationDate="2022-06-23T22:58:26.137" Score="5" Body="&lt;p&gt;As Sue VanHattum suggests, it makes sense to shift gears to series first, for simplicity.&lt;/p&gt;&#xA;&lt;p&gt;I would also do two examples of series with decreasing terms, one where it's obvious that it converges, and the other where it's obvious that it diverges. Good examples would be:&lt;/p&gt;&#xA;&lt;p&gt;0.1 + 0.01 + 0.001 + ...&lt;/p&gt;&#xA;&lt;p&gt;1.1 + 1.01 + 1.001 + ...&lt;/p&gt;&#xA;&lt;p&gt;Next you could do an example whose terms approach zero, but clearly not quickly enough to converge. For example:&lt;/p&gt;&#xA;&lt;p&gt;1 + 1/2 + 1/2 + 1/3 + 1/3 + 1/3 + ...&lt;/p&gt;&#xA;&lt;p&gt;(where the &lt;span class=&quot;math-container&quot;&gt;$1/n$&lt;/span&gt; term is repeated &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; times).&lt;/p&gt;&#xA;&lt;p&gt;Sue VanHattum's example of the harmonic series is fine, but the argument is tricky. I'd avoid trickiness when introducing difficult new ideas.&lt;/p&gt;&#xA;" OwnerUserId="20000" LastActivityDate="2022-06-23T22:58:26.137" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25340" PostTypeId="2" ParentId="25323" CreationDate="2022-06-24T08:31:29.923" Score="0" Body="&lt;p&gt;I would say that you need to understand that you are dealing with primary education. This means that, after that, you have secondary education and maybe after that you might have higher education.&lt;/p&gt;&#xA;&lt;p&gt;The higher the education gets, the higher the number you'll need to multiply.&lt;/p&gt;&#xA;&lt;p&gt;So, let's try to multiply 12345 times 54321.&lt;/p&gt;&#xA;&lt;p&gt;Partial products and open array: that makes a total of 25 numbers you need to add.&lt;br /&gt;&#xA;The Russian method implies that you need to turn one of the numbers into binary format (for 12345, that gives 11000000111001, which implies that, every of the 13 times you're dividing by 2, you risk making a mistake).&lt;br /&gt;&#xA;Although the Korean style is quite handy to show numbers, I don't understand how you can use that method do to calculations and even if I would understand, I believe that people don't have enough fingers to show the numbers 12345 or 54321 :-)&lt;br /&gt;&#xA;The soroban technique basically does the same thing as the open array.&lt;/p&gt;&#xA;&lt;p&gt;So I'd opt for the standard way, as it's used in Europe and apparently also in USA. The others might be fun for showing the children how things were done before, but they are not useful.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2022-06-24T08:31:29.923" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25341" PostTypeId="2" ParentId="25334" CreationDate="2022-06-25T02:27:30.160" Score="2" Body="&lt;p&gt;My feeling is that if students understand the fundamental theorem of calculus, they should not have too much trouble seeing where such results come from. And if they don't understand the fundamental theorem of calculus, that's a far more central topic than improper integrals, and time would be better spent shoring up understanding the fundamental theorem than on improving understanding of improper integrals.&lt;/p&gt;&#xA;&lt;p&gt;Assuming the fundamental theorem is well in hand, here are a few thoughts. Your integrals are similar to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_1^\infty x^{-2}\,dx \qquad \text{and}\qquad\int_1^\infty x^{-1}\,dx,&#xA;$$&lt;/span&gt;&#xA;the first of which equals &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and the second of which is infinite. Students should be able to see that this blow-up doesn't happen &amp;quot;all of a sudden&amp;quot; as the exponent is changed from &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;, but rather, emerges in a sensible way as the exponent is continuously increased from &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be a positive real number and evaluate the analogous integrals with exponent &lt;span class=&quot;math-container&quot;&gt;$-\left(1+\frac{1}{N}\right)$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_1^\infty x^{-\left(1+\frac{1}{N}\right)}\,dx=\lim_{A\to\infty}-N(A^{-\frac{1}{N}}-1^{-\frac{1}{N}})=N.&#xA;$$&lt;/span&gt;&#xA;As &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; grows larger and larger, the exponent gets closer to &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; from below, and the value of the integral grows larger and larger without bound.&lt;/p&gt;&#xA;&lt;p&gt;It is interesting to look at what happens as &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is doubled. For &lt;span class=&quot;math-container&quot;&gt;$N=1,2,4,8,16,\ldots$&lt;/span&gt; the exponent is &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-\frac{3}{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-\frac{5}{4}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-\frac{9}{8}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-\frac{17}{16}$&lt;/span&gt;. As the area is &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;, it is doubling as well. Here's how some of the curves look.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Qn5v2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Qn5v2.png&quot; alt=&quot;integrands plotted from x=1 to 20 for N=1,2,4,8,16,32&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think I would be hard pressed to tell apart the top two curves if they were shown as separate graphs. That the area under the top curve is twice that of the curve second from the top is not something one can see from the graph, but is a testament to the dramatic effect of integrating all the way out to infinity. For similar reasons I don't think it is possible to visually distinguish curves with divergent integrals from curves with convergent ones.&lt;/p&gt;&#xA;&lt;p&gt;In a comment you asked about the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis asymptote of the integrands &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x(x^n+1)}$&lt;/span&gt;. (You noted that for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; positive these integrals between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; are divergent, although they are convergent between &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;.) Let's look at something simpler. The graph of &lt;span class=&quot;math-container&quot;&gt;$y=\frac{1}{x}$&lt;/span&gt; is symmetric about the line &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;, so the divergence of the integral from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is the same geometric fact as the divergence of the integral from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;. (Well the integral from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; includes an extra &lt;span class=&quot;math-container&quot;&gt;$1\times1$&lt;/span&gt; square, but is otherwise the same shape as the integral from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;, just reflected.)&lt;/p&gt;&#xA;&lt;p&gt;More generally, if the graph of &lt;span class=&quot;math-container&quot;&gt;$y=\frac{1}{x^n}$&lt;/span&gt; is reflected about &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt; we get the graph of &lt;span class=&quot;math-container&quot;&gt;$y=\frac{1}{x^{1/n}}$&lt;/span&gt;. Now &lt;span class=&quot;math-container&quot;&gt;$\int_1^\infty \frac{1}{x^n}\,dx$&lt;/span&gt; is divergent for &lt;span class=&quot;math-container&quot;&gt;$n\le1$&lt;/span&gt; and convergent for &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;1$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$\int_1^\infty \frac{1}{x^2}\,dx$&lt;/span&gt; is convergent while &lt;span class=&quot;math-container&quot;&gt;$\int_1^\infty \frac{1}{x^{1/2}}\,dx$&lt;/span&gt; is divergent. On the other hand &lt;span class=&quot;math-container&quot;&gt;$\int_0^1 \frac{1}{x^2}\,dx$&lt;/span&gt; represents the same area as &lt;span class=&quot;math-container&quot;&gt;$\int_1^\infty \frac{1}{x^{1/2}}\,dx$&lt;/span&gt; apart from a &lt;span class=&quot;math-container&quot;&gt;$1\times1$&lt;/span&gt; square, and hence is divergent. Likewise &lt;span class=&quot;math-container&quot;&gt;$\int_0^1 \frac{1}{x^{1/2}}\,dx$&lt;/span&gt; represents the same area as &lt;span class=&quot;math-container&quot;&gt;$\int_1^\infty \frac{1}{x^2}\,dx$&lt;/span&gt; apart from that &lt;span class=&quot;math-container&quot;&gt;$1\times1$&lt;/span&gt; square again and hence is convergent.&lt;/p&gt;&#xA;&lt;p&gt;Your integral &lt;span class=&quot;math-container&quot;&gt;$\int_0^1\frac{1}{x(x^n+1)}\,dx$&lt;/span&gt; looks very much like &lt;span class=&quot;math-container&quot;&gt;$\int_0^1\frac{1}{x}\,dx$&lt;/span&gt;, at least for the part near &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;, which is where the blow-up occurs, so the divergence is not surprising.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2022-06-25T02:27:30.160" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25345" PostTypeId="1" CreationDate="2022-06-27T11:23:52.030" Score="1" ViewCount="100" Body="&lt;p&gt;**&lt;em&gt;How would you use symmetry of mathematical expressions in problem solving in order to save time by minimizing simplifications?&lt;/em&gt; &lt;br/&gt;&#xA;This is not related to geometrical symmetry which is very much familiar even for primary students. Examinations are conducted to test knowledge of students in limited time. Therefore I suggest we need to use symmetry as much as possible to avoid repeating similar kind of simplifications because no use of giving marks for the knowledge which a student has already shown. To give clear picture of the issue here you have few examples.&lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When solving simultaneous equations.&lt;br/&gt; x + y = 5 , x³ + y³ = 35 we can only find x and by the symmetry of given equations solutions of y can be deduced. Because when you interchanged x and y given equations are unchanged. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;If Sinx + Siny + Sinz = 0 and Cosx + Cosy + Cosz = 0 we can eliminate z to obtain Cos(x-y) = - 1/2 . Since the given equations are cyclically symmetrical about x,y,z we can use that property to deduce Cos(y-z) = - 1/2 and Cos(z-x) = - 1/2 &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;When finding remainder if a polynomial of x is divided by (x-a)(x-b)(x-c),where a,b,c are distinct real numbers we can use symmetry of the divisor to express the remainder as &lt;br/&gt;&#xA;A (x-a)(x-b)+B(x-b)(x-c)+C(x-c)(x-a) and obtain the constants A,B,C in very short time without using the usual form Ax² + Bx +C .&lt;br/&gt;&#xA;This issue is not familiar to many of the teachers who are involved in secondary education. What would you have to say about using this symmetrical property in specifically advanced level mathematics ?**&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2022-06-27T12:39:16.367" LastActivityDate="2022-06-27T12:39:16.367" Title="Symmetry of mathematical expressions in problem solving" Tags="|secondary-education|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25347" PostTypeId="2" ParentId="8041" CreationDate="2022-06-27T20:42:57.540" Score="0" Body="&lt;p&gt;I suspect the issue (as many are) is more pedagogical than &amp;quot;explanation based&amp;quot;.  Instead of looking for some &amp;quot;aha&amp;quot; magic explanation, you need to expose the kid to many example and force him to replicate them.  Yes, even if he doesn't &amp;quot;get them&amp;quot;.  Over and over...and force him to solve problems.  And NOT the hardest, most aha kind.  But more like variations on a theme (same as example but with slightly varied numbers).&lt;/p&gt;&#xA;&lt;p&gt;Sometimes things only make sense in retrospect.  Like learning your kip as a gymnast.  &lt;a href=&quot;https://stickitgirl.com/blogs/stickitgirl/how-to-help-your-gymnast-get-her-kip&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://stickitgirl.com/blogs/stickitgirl/how-to-help-your-gymnast-get-her-kip&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And this is not a troll (Dan).  I really mean it in terms of a reframing.&lt;/p&gt;&#xA;&lt;p&gt;And in the end...the kid will probably even come upon an intuitive explanation.  But it will not happen top down.  Will happen after exposure and bottoms up familiarity.  [Somehow this blows the minds of axiom-based math logicians, but makes total sense to sports coaches...and practical...teachers.]&lt;/p&gt;&#xA;" OwnerUserId="19997" LastActivityDate="2022-06-27T20:42:57.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25348" PostTypeId="1" AcceptedAnswerId="25350" CreationDate="2022-06-28T06:30:30.310" Score="2" ViewCount="267" Body="&lt;p&gt;How would you define tangent line to a curve at a given point on the curve?&lt;/p&gt;&#xA;&lt;p&gt;I think in secondary education it is not sufficient to say a tangent line is a straight line touching a curve at a given particular point and you can not say it is a straight line which lies only in one side of the curve when it meets the curve or which have only two coincident intersecting points because those are not applicable in the case of point of inflection , as an example you can consider tangent at the origin to the curve &lt;span class=&quot;math-container&quot;&gt;$y = x^3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you say tangent line at the point &lt;span class=&quot;math-container&quot;&gt;$(a , f(a))$&lt;/span&gt; to the curve is the straight line going through the point &lt;span class=&quot;math-container&quot;&gt;$(a , f(a))$&lt;/span&gt; with gradient &lt;span class=&quot;math-container&quot;&gt;$f'(a)$&lt;/span&gt;, that is the derivative of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt; (as described in the Wolfram mathworld) what happens when the function is not differentiable at &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Sometimes a tangent line can exist even without existence of derivative as an example tangent at the origin to the parabola &lt;span class=&quot;math-container&quot;&gt;$y^2 = 4ax$&lt;/span&gt;, but in some cases tangent line does not exist when the derivative can not be defined as an example there is no tangent line at origin to the curve &lt;span class=&quot;math-container&quot;&gt;$y = |x|$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Therefore what should be the most appropriate way of explaining this to the students?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="893" LastEditDate="2022-07-09T20:17:01.763" LastActivityDate="2022-07-09T20:17:01.763" Title="Tangent line to a curve" Tags="|secondary-education|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25349" PostTypeId="2" ParentId="8041" CreationDate="2022-06-28T09:36:19.917" Score="0" Body="&lt;p&gt;*Actually I have enough experience of seeing students who don't answer a question but when the lesson is explained they say everything ok and understood. In mathematical induction there are different methods and for each of them there are many different types of questions. &lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The most commonly used method using only one assumption, &lt;br/&gt;&#xA;you can start by proving the result is true for n = 1 and then assuming that the result is true for n = p ( p is a positive integer) and use assumption to prove that the result is true in the case of n = p+1. In the other hand you can start by proving the result is true for n = 1,2,... ,k and go on with other steps when you require the condition p ≥ k for the proof in n = p+1 . &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Using more than one assumption, &lt;br/&gt;&#xA;As an example you can prove the result is true for n = 1 and n = 2 , then assume the result is true for n = p and n = p+1 . Now you can use both assumptions to prove that the result is true for n = p+2 . By this way you can have even 3, 4 ,....assumptions. &lt;br/&gt;&#xA;I think depending on the content of the syllabus and level of the students you need to explain few more examples until they are capable of using this as in any other cases in mathematics. &lt;br/&gt;&#xA;Actually I have another issue related to this, that is how can we use the term mathematical induction when this method is considered as a deductive proof.*&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2022-06-28T09:36:19.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25350" PostTypeId="2" ParentId="25348" CreationDate="2022-06-28T15:19:44.720" Score="7" Body="&lt;p&gt;Part of our role as mathematics educators is to welcome students into the cultural practices of mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;Too often we present definitions as &amp;quot;set in stone&amp;quot;.  It is more honest to acknowledge that definitions in mathematics are carefully crafted to balance several competing aesthetic and practical needs.  We want our definitions to capture the intuition we gain from working with the most important specific examples we have grappled with, we want them to be formulated in a way that they &amp;quot;play nicely&amp;quot; with other definitions we have already made, we want them to only admit pathology if that pathology is easily controlled (or is unavoidable in some sense).  I am sure there are many other considerations that we all make. A definition is a work of art, not a commandment set down by our mathematical grandparents.&lt;/p&gt;&#xA;&lt;p&gt;I think that you are showcasing these tensions in your original post.&lt;/p&gt;&#xA;&lt;p&gt;If we are concerned with zero sets of polynomials, then our intuition about tangents is related to multiplicity of intersection.  Following this path seriously might lead you to rediscover the theory of schemes!  If we are concerned with more general curves, it is not so easy to capture tangency using algebra in this way, and we end up needing differential calculus.  We would also need an &amp;quot;official&amp;quot; definition of &amp;quot;curve&amp;quot; as a precursor to making a definition of &amp;quot;tangent line to a curve&amp;quot;.  Whether your definition of curve is based on level sets or parameterizations will influence the definition you make for tangent lines.&lt;/p&gt;&#xA;&lt;p&gt;As an instructor you have a choice to make with every new definition you want to introduce:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Present the definition &amp;quot;straight from the textbook&amp;quot; and let the students figure it out.&lt;/li&gt;&#xA;&lt;li&gt;Present the definition &amp;quot;straight from the textbook&amp;quot; and explain, using examples, why you think the definition was chosen this way.&lt;/li&gt;&#xA;&lt;li&gt;Mimic the definition creation process.  Give a few examples, create a &amp;quot;rough definition #1&amp;quot; pick holes in this definition by showing some more examples where the rough definition is inconvenient, iterate until you settle on an &amp;quot;official definition&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;Show the students examples and guide them toward creating their own definitions.  Get them to pick holes in each others definitions and patch them up.  If they miss a hole, pick it yourself.  Continuing picking and patching until you have a workable definition.  If this is different from the &amp;quot;official definition&amp;quot; you want to work with for the course, compare and contrast the two and indicate why you will be using the &amp;quot;official definition&amp;quot;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Personally I go with option #2 and #3 most often, and only use #4 if I want to make a whole class (or two!) where the focus is the definition creation process.&lt;/p&gt;&#xA;&lt;p&gt;Here is an example of how option #4 might go down in (maybe in a Real Analysis class?):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Teacher:  Here are &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; graphs.  Each graph has a curve and a line.&#xA;What do you notice?&lt;/p&gt;&#xA;&lt;p&gt;Students:  [many interesting categorizations]&lt;/p&gt;&#xA;&lt;p&gt;Teacher:  Those are all interesting!  What about the difference&#xA;between graph #7 and graph #11?  Anything interesting you can say&#xA;there?&lt;/p&gt;&#xA;&lt;p&gt;Student:  In graph #7 the line is tangent to the curve, while in graph&#xA;#11 it is not.&lt;/p&gt;&#xA;&lt;p&gt;Teacher: Exactly.  This is what I want to focus on today:  creating a&#xA;rigorous definition for what it means for a line to be tangent to a&#xA;curve at a point.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I want to stress something&lt;/strong&gt;:  the point of class today is to learn what it is like to make a good definition.  We will probably not end&#xA;up with the same definition as the textbook (which is the definition&#xA;we will use for the rest of the course).  That is okay.  Again, the&#xA;point of the exercise is to learn how to make good definitions.&lt;/p&gt;&#xA;&lt;p&gt;Okay, so everyone get into pairs and take a stab at creating a&#xA;definition.&lt;/p&gt;&#xA;&lt;p&gt;Students:  [Create various definitions which are all flawed in some&#xA;way].&lt;/p&gt;&#xA;&lt;p&gt;Teacher:  Okay now pair up with another pair to make groups of 4.&#xA;Critique each others defintions.&lt;/p&gt;&#xA;&lt;p&gt;Students:  [Poke holes in and try to repair each others definitions].&lt;/p&gt;&#xA;&lt;p&gt;Iterate this process combined with whole class discussions until a&#xA;final definition is reached:&lt;/p&gt;&#xA;&lt;p&gt;Teacher:  Okay, after an hour of work we have finally come to a&#xA;definition the class feels satisfied with.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Definition&lt;/strong&gt; A line &lt;span class=&quot;math-container&quot;&gt;$Ax + By = C$&lt;/span&gt; is said to be &lt;strong&gt;tangent&lt;/strong&gt; to the curve &lt;span class=&quot;math-container&quot;&gt;$F(x,y) = 0$&lt;/span&gt; at the point &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; if&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is a point of intersection between the line and the curve.&lt;/li&gt;&#xA;&lt;li&gt;For every &lt;span class=&quot;math-container&quot;&gt;$\epsilon_1, \epsilon_2 &amp;gt; 0$&lt;/span&gt; there exist &lt;span class=&quot;math-container&quot;&gt;$0 \leq \delta_A, \delta_B, \delta_C \leq  \epsilon_1$&lt;/span&gt; so that the perturbed line &lt;span class=&quot;math-container&quot;&gt;$(A+\delta_A)x + (B + \delta_B)y =&#xA; C + \delta_C$&lt;/span&gt; has at least &lt;strong&gt;two&lt;/strong&gt; points of intersections with the&#xA;curve &lt;span class=&quot;math-container&quot;&gt;$F(x,y) = 0$&lt;/span&gt; in the disk centered at &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; of radius &lt;span class=&quot;math-container&quot;&gt;$\epsilon_2$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This is an amazing definition!  I am really surprised and pleased that&#xA;we were able to come up with this definition together.  I hope that&#xA;the process of creating this definition has been illuminating about&#xA;the whole process of how definitions are created, and how it involves&#xA;a lot of &lt;em&gt;choices&lt;/em&gt; to be made.&lt;/p&gt;&#xA;&lt;p&gt;In our discussion, we talked about how this definition means that&#xA;(technically) we are accepting that the &amp;quot;curve&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$0=0$&lt;/span&gt; is &amp;quot;tangent&amp;quot; to&#xA;every line.  We were okay with that.  We discussed that the curve &lt;span class=&quot;math-container&quot;&gt;$ y&#xA; = |x|$&lt;/span&gt; has multiple tangent lines through &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt;.  We were also okay with that!&lt;/p&gt;&#xA;&lt;p&gt;The authors of our textbook have made different aesthetic choices.&#xA;They would prefer to say that &lt;span class=&quot;math-container&quot;&gt;$y = |x|$&lt;/span&gt; doesn't have any tangent lines&#xA;passing through &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt;.  I hope we can all agree that this is valid.&#xA;Here is the definition our textbook is using.  We will need to digest&#xA;this definition and also compare/contrast with the definition we&#xA;developed.&lt;/p&gt;&#xA;&lt;p&gt;[Teacher puts &amp;quot;official definition&amp;quot; on the board, and discussion continues].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To answer your question more directly though:  it would depend on the course.&lt;/p&gt;&#xA;&lt;p&gt;In an Algebra 2 course I might define a tangent line to a conic section as a line which intersects the conic section only once.  I would say that this definition doesn't work for more general curves, but it works for these curves we are studying.&lt;/p&gt;&#xA;&lt;p&gt;In a calc 1 course, I would define define the tangent line to the graph of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; only for points where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differentiable.  In that case I would define it as the line &lt;span class=&quot;math-container&quot;&gt;$y - f(a) = f'(a)(x-a)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$(a,f(a))$&lt;/span&gt; is the point of tangency.  I might make a distinct definition for the definition of vertical tangent line, or for tangent lines to curves which are locally the graph of a function (such as &lt;span class=&quot;math-container&quot;&gt;$x^2 + y^2 = 1$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;In a multivariable course, I would define a smooth curve &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; as the image of an injective differentiable function &lt;span class=&quot;math-container&quot;&gt;$\gamma: [0,1] \to \mathbb{R}^n$&lt;/span&gt; with non-vanishing derivative.  I would define the tangent line to a smooth curve at the point &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; as the line passing through the point &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; with vector &lt;span class=&quot;math-container&quot;&gt;$\gamma'(t)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; is any such parameterization.  This would require the proof (via the chain rule) that re-parameterization does not change the tangent line.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2022-06-29T11:12:10.773" LastActivityDate="2022-06-29T11:12:10.773" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25351" PostTypeId="1" AcceptedAnswerId="25352" CreationDate="2022-06-28T15:58:31.670" Score="5" ViewCount="544" Body="&lt;p&gt;Imagine that you were writing an elementary book, for example for high school learners, and at the beginning you had a glossary where you wanted to write the definitions for common mathematical words (e.g. numerical expression, algebraic expression, variable, equation, etc.) to help the reader make sense of those words when they appeared throughout the text. In this setting, what would you write to explain/define what a mathematical formula is?&lt;/p&gt;&#xA;&lt;p&gt;Imagine that you wanted to put an entry for mathematical formula in a table like in these &lt;a href=&quot;https://i.stack.imgur.com/uYLdJ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;screenshots&lt;/a&gt;, taken from the book Algebra and Trigonometry: Structure and Method, Book 2&#xA;which you can read &lt;a href=&quot;https://archive.org/details/algebratrigonome00brow/page/n31/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, if you have create an account.&lt;/p&gt;&#xA;&lt;p&gt;I would like to avoid circular definitions.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Some context&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The term formula seems to raise confusion with some frequency and  this can be seen by the number of Math Stackexchange questions that appeared over the years:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/38155/what-is-the-difference-between-equation-and-formula&quot;&gt;what-is-the-difference-between-equation-and-formula&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/1082718/what-is-the-difference-between-a-function-and-a-formula&quot;&gt;what-is-the-difference-between-a-function-and-a-formula&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/4146122/what-is-the-correct-meaning-of-term-formula&quot;&gt;what-is-the-correct-meaning-of-term-formula&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/3296654/in-the-context-of-mathematics-what-is-the-difference-between-equation-and-formu&quot;&gt;in-the-context-of-mathematics-what-is-the-difference-between-equation-and-formula&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/4316470/what-is-the-difference-between-formula-and-function&quot;&gt;what-is-the-difference-between-formula-and-function&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/330595/difference-between-formula-and-algorithm&quot;&gt;difference-between-formula-and-algorithm&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;After some investigation it seems that people use the term mathematical formula to mean (mainly) two things:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Any mathematical sentence that uses symbols (e.g. equations, inequations/inequalities, identities). Which makes equations a kind of formula.&lt;/li&gt;&#xA;&lt;li&gt;A rule for expressing a subject in terms of some other variables (as stated in &lt;a href=&quot;https://math.stackexchange.com/questions/38155/what-is-the-difference-between-equation-and-formula/4254850?noredirect=1#comment9398749_4254850&quot;&gt;this comment&lt;/a&gt;). Or a procedure to compute some value of interest. Usually written as two expressions connected by an equals sign. Which makes formulas a kind of equation.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathworld.wolfram.com/Formula.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathworld&lt;/a&gt; seems to define a formula as the first. Many books use the term formula for the rules used to compute areas and volumes, and there is also the quadratic formula which provides the solution(s) to a quadratic equation. These are instances of the second definition.&#xA;Less frequently, I've also seen people call formula to any string of mathematical symbols, making an expression a kind of formula. Another example is the book &lt;a href=&quot;https://books.google.com/books?id=1CwTDAAAQBAJ&amp;amp;pg=PA23&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Algebra&lt;/a&gt; that defines formula as a synonym for expression in page 23 (&lt;a href=&quot;https://math.stackexchange.com/questions/1809529/is-32-5-a-equation?noredirect=1&amp;amp;lq=1#comment3697812_1809529&quot;&gt;according to this comment&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&#xA;After reading some of the comments I feel the need reword the first part of my original question to:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Imagine that you were writing a book that revisits elementary&#xA;mathematics subjects/areas but from an advanced standpoint, for&#xA;example for advanced high school learners or people preparing to an&#xA;undergraduate mathematics heavy course. One example of such book might&#xA;be Serge Lang's Basic Mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Now consider that in the beginning of this book you had a glossary&#xA;where you wanted to write the definitions for common mathematical&#xA;words (e.g. numerical expression, algebraic expression, variable,&#xA;equation, etc.) to help the reader make sense of those words when they&#xA;appeared throughout the text.&lt;/p&gt;&#xA;&lt;p&gt;In this setting, what would you write to &lt;strong&gt;explain/define what a&#xA;mathematical formula is?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I personally think that the answers given up to the point in time of this edit fit both the original question and this reword, but would prefer if new answers take this revision into consideration.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT 2&lt;/strong&gt;&#xA;I accepted @ryang's answer because it has all the information requested by the Question, but also found &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Wells_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Charles Wells&lt;/a&gt;' work, which helped me interiorize stuff:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://abstractmath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Website&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://abstractmath.org/Handbook/handbook.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A report on mathematical discourse&lt;/a&gt;, which has an entry for 'formula'.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="20015" LastEditorUserId="376" LastEditDate="2022-07-06T06:35:15.007" LastActivityDate="2022-07-14T06:22:15.817" Title="Definition for Mathematical Formula" Tags="|terminology|definitions|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="25352" PostTypeId="2" ParentId="25351" CreationDate="2022-06-28T17:43:10.303" Score="2" Body="&lt;p&gt;In formal logic, ‘formula’ has a standard and precise definition. Roughly speaking, take a constant and/or variable, apply a non-logical operation (e.g., addition) if desired, repeat if desired, this gives a ‘term’ (notice the deviation from its common meaning in mathematics); relate two terms using the equality relation to obtain an ‘atomic formula’; apply a logical operation or quantifier to atomic formulae finitely many (including zero) times to obtain &lt;strong&gt;formulae&lt;/strong&gt;; a formula is then called a sentence/proposition precisely when it has no free variable.&lt;/p&gt;&#xA;&lt;p&gt;But formal logic is tangential to your question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Any mathematical sentence that uses symbols (e.g., equations, inequations/inequalities, identities).&lt;/p&gt;&#xA;&lt;p&gt;Which makes equations a kind of formula.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What's missing from this characterisation is some mention that a formula expresses a rule or required relationship or somesuch.&lt;/p&gt;&#xA;&lt;p&gt;After all, the equation &lt;span class=&quot;math-container&quot;&gt;$3x^2+7x=8$&lt;/span&gt; is typically decidedly not a formula.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;as stated in &lt;a href=&quot;https://math.stackexchange.com/questions/38155/what-is-the-difference-between-equation-and-formula/4254850?noredirect=1#comment9398749_4254850&quot;&gt;this comment&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Perhaps: A &lt;strong&gt;formula&lt;/strong&gt; is a rule for expressing a subject in terms of some other variable(s); for example, &lt;span class=&quot;math-container&quot;&gt;$V=πr^2h$&lt;/span&gt; expresses a cylinder's volume in terms of its radius and height.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Which makes formulas a kind of equation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The keyword here is &lt;em&gt;rule&lt;/em&gt;, which is rather more specific than “equation”, and switching it out alters the meaning.&lt;/p&gt;&#xA;&lt;p&gt;And certainly, &lt;span class=&quot;math-container&quot;&gt;$V=πr^2h$&lt;/span&gt; is not an identity, as triples like &lt;span class=&quot;math-container&quot;&gt;$(1,1,1)$&lt;/span&gt; do not satisfy it. The example says that for each &lt;em&gt;cylinder&lt;/em&gt; with volume, radius and height &lt;span class=&quot;math-container&quot;&gt;$V,r,h,$&lt;/span&gt; respectively, the attributes obey the relationship &lt;span class=&quot;math-container&quot;&gt;$V=πr^2h.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Would you say that there exist  formulas that contain an inequality symbol?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'd typically call such an object an ‘inequality constraint’ rather than a ‘formula’.&lt;/p&gt;&#xA;&lt;p&gt;In the language of first-order arithmetic on the other hand, &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;y$&lt;/span&gt; is an ‘open formula’ [side note: it can be rewritten as &lt;span class=&quot;math-container&quot;&gt;$\exists r\: (x'+r=y)].$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Maybe don't try to conflate the formal-logic and informal usages of ‘formula’.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For example, in Spivak's Calculus, 4th Edition, page 33 he writes: &amp;quot;When does equality hold in the formula &lt;span class=&quot;math-container&quot;&gt;$G_n \le A_n$&lt;/span&gt;?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is apparent from the context that Spivak is using the term 'formula’ in the formal-logic sense. (But since this is an Analysis, not Logic, text, I’d probably rewrite the question as “When does equality hold in the sentence/statement/assertion &lt;span class=&quot;math-container&quot;&gt;$G_n \le A_n$&lt;/span&gt;?”)&lt;/p&gt;&#xA;&lt;p&gt;There's similarly no ambiguity in my various usages (in the third line and the previous sentence) of ‘term’: (1) in formal logic, &lt;span class=&quot;math-container&quot;&gt;$7y+8$&lt;/span&gt; is a term (2) in mathematics, the expression &lt;span class=&quot;math-container&quot;&gt;$7y+8$&lt;/span&gt; is not a term (3) the word ‘term' is a term.&lt;/p&gt;&#xA;&lt;p&gt;In mathematics, the word ‘formula’ is not part of the theory, so does not have a rigorous (or at least widespread rigorously consistent/matching) definition. Possibly, Spivak's text contains a definition of ‘formula’ in the preface or glossary, and the quoted usage above goes with it?&lt;/p&gt;&#xA;&lt;p&gt;At the end of the day, don't forget about the context; I mean, have you ever found the word ‘formula’ ambiguous in spite of its surrounding context?&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-07-14T06:22:15.817" LastActivityDate="2022-07-14T06:22:15.817" CommentCount="29" ContentLicense="CC BY-SA 4.0" />
  <row Id="25354" PostTypeId="2" ParentId="25348" CreationDate="2022-06-29T20:40:14.800" Score="1" Body="&lt;p&gt;There is no universally appropriate definition that will suit all pedagogical purposes for students at all levels. However, a pretty nice definition is given by Marsden and Weinstein, in their book Calculus Unlimited. Paraphrased:&lt;/p&gt;&#xA;&lt;p&gt;A line L through a point P on the graph of a function f is said to be a tangent line if all lines through P with slopes less than that of L cut through the graph in one direction, while all lines with slopes greater than L's cut through it in the opposite direction.&lt;/p&gt;&#xA;&lt;p&gt;(The notion of &amp;quot;cutting through&amp;quot; is defined by saying that in a small enough region around the given point, we're above or below the graph depending on whether we're to the left or right of the point.)&lt;/p&gt;&#xA;&lt;p&gt;In reality, you want to give your students multiple ways to look at this, multiple ways of experiencing it.&lt;/p&gt;&#xA;" OwnerUserId="20017" LastActivityDate="2022-06-29T20:40:14.800" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25355" PostTypeId="2" ParentId="12560" CreationDate="2022-07-01T00:25:58.483" Score="2" Body="&lt;p&gt;There is an iPad app that does this: DragginMath. Does what you envision. See the tree being built step by step. See the hierarchy explicitly. Then drag tree nodes around to invoke the various rules of algebra. I had this idea about 30 years ago, finally did something about it.&lt;/p&gt;&#xA;&lt;p&gt;Another answer mentions GraspableMath, which is significantly different from DragginMath. DragginMath solves equations, too. It does this using the standard algebraic properties &lt;em&gt;only&lt;/em&gt;. GraspableMath takes a lot of liberties with mathematical language to try to get things to happen somehow, where &lt;em&gt;somehow&lt;/em&gt; is often hard to explain in terms of actual algebra.&lt;/p&gt;&#xA;" OwnerUserId="20023" LastEditorUserId="20023" LastEditDate="2022-07-05T07:13:42.903" LastActivityDate="2022-07-05T07:13:42.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25356" PostTypeId="2" ParentId="12560" CreationDate="2022-07-04T17:04:58.677" Score="0" Body="&lt;p&gt;Acutally, I learned order of operations this way in school; even nowadays you find 'operator-diagrams' in school books, starting in primary school (only addition, multiplication; use of brackets).&lt;/p&gt;&#xA;&lt;p&gt;Next to the iPad app 'DragginMath' mentioned in the answers, there is an app online named &lt;a href=&quot;https://activities.graspablemath.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;GraspableMath&lt;/a&gt;.&#xA;I use it not only to teach order of operation, but also equation solving. You can 'scrub' the parameters and look what happens to the result.&lt;/p&gt;&#xA;" OwnerUserId="9004" LastActivityDate="2022-07-04T17:04:58.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25357" PostTypeId="1" CreationDate="2022-07-06T03:34:36.967" Score="5" ViewCount="1251" Body="&lt;p&gt;&lt;em&gt;Don't we need to use internationally recognized standards in defining mathematical terms since differences in definitions play very much important role in finding a unique solution for a given problem and from where we can find those definitions ?&lt;/em&gt;&#xA;&lt;br/&gt;&#xA;Here are few examples &lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Inclusion of 0 in set of natural numbers;&lt;br/&gt; This was included upto 2015 in advanced level mathematics in my country but now they say not to include it. When I inquired the issue from a Cambridge examiner , he told since there is no agreement regarding it, questions not asked related to that issue in their exam papers. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Negative divisors;&lt;br/&gt;&#xA;In one of past papers in my country it was asked to find positive divisors and it implies that they believe there can be negative divisors too but when I asked about it from national institute of education an official told me normally they consider positive divisors only. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Negative prime numbers;&lt;br/&gt;&#xA;When we talk about primes normally we just say prime numbers and we don't specifically mention positive prime numbers but when I searched there are articles related to the existence of negative prime numbers too.&lt;br/&gt;&#xA;I suggest we better maintain one standard so that our students can go abroad and continue their studies without any issues related to this matter .&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="376" LastEditDate="2022-07-06T06:34:18.020" LastActivityDate="2022-07-06T18:15:56.413" Title="Importance of standardization of definitions of mathematical terms" Tags="|secondary-education|definitions|" AnswerCount="2" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="25358" PostTypeId="2" ParentId="15278" CreationDate="2022-07-06T04:30:25.267" Score="0" Body="&lt;p&gt;Because it is actually very complicated. I myself never really understood fractions till I learned of it's construction through natural numbers. The idea of fractions as a formal expression of describing fraction quantites (lack of better word) is pretty easy to understand for most people , for instance, half a pizza, that's easy, right? But the issue is figuring out the correct general rule to add fractions and why that should be the rule i.e: why is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{a}{b} + \frac{c}{d} := \frac{ad + bc}{bd } ?$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;One way of understanding it is through the idea presented in &lt;a href=&quot;https://math.stackexchange.com/a/677913/688539&quot;&gt;this answer&lt;/a&gt; but again this is adding more sub concepts to understand this main.&lt;/p&gt;&#xA;&lt;p&gt;I think at a Young age, people find it difficult to do things which require understanding connections between different fields at once/ identifying what are the right connections to see between two different things so they can understand each of the thing individually.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2022-07-06T04:30:25.267" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25359" PostTypeId="2" ParentId="11182" CreationDate="2022-07-06T11:12:03.910" Score="0" Body="&lt;p&gt;This observation come from the perspective of a student. In my opinion, domain questions are being asked in place of the fundamental questions that lie beneath them. Instead of asking&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is the domain of the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; given by &lt;span class=&quot;math-container&quot;&gt;$f(x)=1/x$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;we should be asking&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is the set of real numbers which have a unique multiplicative inverse?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here, there are no ambiguities or abuses of notation in the question. The question has a clear, objective answer. You can also give a brief justification for why the answer is true: it's because &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; is a field.&lt;/p&gt;&#xA;&lt;p&gt;We can apply the same principle to your example. Instead of asking&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; be the square root function &lt;span class=&quot;math-container&quot;&gt;$f(x)=\sqrt x$&lt;/span&gt;. What is its domain?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;we could ask&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Give the set of numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for which there is a &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$y^2=x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Once we have answered the above question, the motivation for defining the square root function in the way we usually do becomes clear. If we want &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt; to denote a square root of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; then the largest possible domain of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\cdot}$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$[0,\infty)$&lt;/span&gt;. The uniqueness problem is solved by declaring that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt; denotes the nonnegative square of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14423" LastActivityDate="2022-07-06T11:12:03.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25360" PostTypeId="2" ParentId="25357" CreationDate="2022-07-06T11:25:39.273" Score="10" Body="&lt;p&gt;IMHO one can tolerate such small differences in the mathematical language pretty easily (much easier than to tolerate the ambiguity of the natural language that lacks specifiers for almost any adjective, so people drive me nuts when saying &amp;quot;that is good/bad/beautiful&amp;quot; and I drive them nuts asking &amp;quot;good/bad for what purpose?&amp;quot; or &amp;quot;what is your definition of beauty?&amp;quot; in response) if one makes it a habit to &lt;em&gt;explicitly&lt;/em&gt; state what certain words and symbols mean in the beginning of every article or assignment (for a class, you need to do it just once in a separate handout, or you can remind it five times in presentations after which everybody should get used to it).&lt;/p&gt;&#xA;&lt;p&gt;The full standartization is, probably, impossible, because in reality it is rather convenient to assume slightly different conventions and notation in different situations. The argument about which one is the correct one is totally pointless as any argument about definition of words. One should just tell the students that when in doubt, they should ask what exactly is meant by this or that and assure them that such questions will always be answered. We all speak to each other like Humpty-Dumpty in Carrrol's &amp;quot;Through the looking glass&amp;quot; all the time and there is no way around it except asking &amp;quot;what exactly did you mean?&amp;quot; before making any strong comment or agreeing/disagreeing with the speaker.&lt;/p&gt;&#xA;&lt;p&gt;So, I wouldn't bother too much about &amp;quot;international standartization&amp;quot; here. What I would rather like to see is sending a clear message to the students that the same word can have different meanings in different situations and that the speaker and the listener should agree on the underlying terminology before trying to convey any message.&lt;/p&gt;&#xA;&lt;p&gt;As an anecdote that illustrates what happens if they don't, there is now a hot dispute in Russia about the following situation: On an exam a student was given a problem that literally translates as follows:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;If you purchase 2 chocolate bars for 180 rubles, you receive one for free. How many chocolate bars can you receive if you have 360 rubles?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The student answered &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; because her idea was that &amp;quot;receive&amp;quot; and &amp;quot;purchase&amp;quot; are two mutually exclusive words. The official answer was &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; because the examiners considered these words synonyms. It cost the student a gold medal and the dispute was taken all the way up to the Ministry of Education and is still ongoing.&lt;/p&gt;&#xA;&lt;p&gt;So, just keep in mind and make it clear to others that everyone will speak like Humpty-Dumpty no matter what even in scientific communication and that the prudent thing to do as a listener would be to take the approach he recommended to Alice, though, as we all remember, she wasn't very fond of him or his way of speaking in the end :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-07-06T11:25:39.273" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25361" PostTypeId="2" ParentId="25357" CreationDate="2022-07-06T18:10:17.900" Score="8" Body="&lt;p&gt;It is impractical to provide a worldwide standard because many conventions are already in use, and this is driven by the utility of one convention over another in context.&lt;/p&gt;&#xA;&lt;p&gt;For example, when discussing natural numbers, whether 0 or 1 is included can affect some formulae. If you want &amp;quot;n!&amp;quot; to count the number of subsets of a set of size n for all natural numbers, then 0 should be included since sets of size 0 certainly make sense. However to define the harmonic sequence as 1/n for all natural numbers, we would wish to exclude 0. This is a simple example, but should get across that in different disciplines, the choice of which &lt;em&gt;precise&lt;/em&gt; definition to use is dependent on &lt;em&gt;which is more useful&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Instead, it is more valuable to teach students to &lt;em&gt;parse&lt;/em&gt; new definitions when needed, than to teach one unified, unerring definition. Indeed, the practice of mathematics is widely regarded by mathematicians not as a set of &lt;em&gt;facts&lt;/em&gt;, but, in brief, the skill of drawing logical conclusions from assumptions, and to this purpose, manipulating expressions when necessary. The skill of parsing definitions, then, is key in the practice of mathematics, for it is necessary for drawing any conclusion from the assumption that &amp;quot;an object is an X,&amp;quot; for any mathematical definition of the word X. In practice, definitions are given or made accessible when needed, especially when they may differ in convention from another subfield.&lt;/p&gt;&#xA;&lt;p&gt;But not only this; let's consider your &amp;quot;student draws an inadequate conclusion in a standardized exam due to unclear definitions&amp;quot; example. If the examining body had simply said the definition of what they were asking the students to compute, or (if they wanted all students to &lt;em&gt;know&lt;/em&gt; the definition ahead of time) gave a booklet of the definitions the examining body would always use, then a student with the skill to parse definitions would be able to adapt to whichever definition the examining body chose to use, even if they had learned a slightly different convention in school. And a student going abroad who learns to parse definitions will be able to adapt, and this is a much more realistic expectation than getting all universities across the globe to agree with each other.&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2022-07-06T18:15:56.413" LastActivityDate="2022-07-06T18:15:56.413" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25362" PostTypeId="1" CreationDate="2022-07-07T17:28:37.057" Score="1" ViewCount="934" Body="&lt;p&gt;&lt;em&gt;There are some cases where final result looks like strange when the concept of infinity involves. My issue is how to describe these situations to our students. &lt;br/&gt;&lt;/em&gt;&#xA;Here are few examples ; &lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When you keep on multiplying 1's such as,  1×1×1×... can you say here we have product of infinite number of 1's or is this limit of product of n number of 1's as n tends to infinity? If so can we say this product is indeterminate? Then what about the product 2×2×2....... because this also can be written in terms of 1×1×1×....? &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;We all know, sum of two rational numbers is rational and if you add another rational,  answer should be a rational. But when you consider the irrational number e , by its definition we have ,e = 1+1/(1!)+1/(2!)+1/(3!)+... .Here the right side is a sum of infinite number of rationals . What we have to say, is this because of infinity number of rationals involved ? &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;We all know tan(π/2) can not be defined and it is not infinity. But in definite integrals as an example, integral of 1/(1+x²) from x = 0 to x = ∞&#xA;We write tan inverse of infinity as π/2 . Is this because we consider interval of values and therefore x tends to infinity? &lt;br/&gt;&#xA;I know many students don't think about these issues if we do not mention,  but it is much better if we have the most appropriate way of describing these issues to give much confidence of what students learn.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2022-07-09T19:11:59.493" Title="Reasoning outcomes of simplifications involving infinity" Tags="|secondary-education|infinity|" AnswerCount="4" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="25363" PostTypeId="2" ParentId="25362" CreationDate="2022-07-07T18:19:38.967" Score="10" Body="&lt;p&gt;&amp;quot;How to describe these situations to our students&amp;quot; ...&lt;/p&gt;&#xA;&lt;p&gt;At any level below calculus, tell the students: &amp;quot;You will cover this later.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This is a special case of: Do not try to tell your students about something you do not understand yourself.&lt;/p&gt;&#xA;&lt;p&gt;[I resist the urge to provide off-topic commentary on the three questions in this forum.]&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2022-07-07T18:19:38.967" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="25364" PostTypeId="2" ParentId="25362" CreationDate="2022-07-07T21:12:36.810" Score="9" Body="&lt;p&gt;What resolved these questions for me was when a teacher explained that you cannot add or multiply infinitely many things together: all infinite sums are actually limits.&lt;/p&gt;&#xA;&lt;p&gt;It can help to think of addition and multiplication as naturally binary operations that get extended to n-ary operations in a well-defined (by associativity) way, as opposed to thinking of them as n-ary operations. This will reduce the temptation to extend them to infinity-ary operations.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-07-07T21:12:36.810" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25365" PostTypeId="1" CreationDate="2022-07-07T23:20:46.077" Score="0" ViewCount="121" Body="&lt;p&gt;Is analogy a way to solve problems when doing research in math and physics? Is it very important?How important is it? Could i do independent scientific research on some open problems perhaps having read them or having defined them on my own? How should the independent scientific research be done?&lt;/p&gt;&#xA;&lt;p&gt;I do not have any research experience, neither from phd studies nor from post doctorate studies, i am an undergraduate student in math, who tries to read and learn math and physics outside the math and physics taught in undergraduate studies. Perhaps papers from researchers i find on the internet and some other books and pdfs i have and other pdf files perhaps in the future.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastActivityDate="2022-07-08T12:29:17.253" Title="Problem solving and independent scientific research" Tags="|self-learning|problem-solving|" AnswerCount="1" CommentCount="2" ClosedDate="2022-07-08T18:38:35.087" ContentLicense="CC BY-SA 4.0" />
  <row Id="25366" PostTypeId="2" ParentId="25365" CreationDate="2022-07-08T12:29:17.253" Score="2" Body="&lt;p&gt;As you start to become more experienced in your studies, you will see patterns.  Mark Twain said, history doesn't repeat, but it rhymes.  I think this is the right spirit to have, looking at analogies.  Not precision (2:3 as 4:6).  But general similarity.&lt;/p&gt;&#xA;&lt;p&gt;Learning different things not only gives you a storehouse of examples, but also trains your brain at finding patterns (the act of looking for and finding analogies, partial analogies, FALSE analogies [I once thought Excel pivot tables were just like a third dimension...good guess, but no]).  By looking for analogies, you are an active participant in the process.&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't belabor it.  But just be mildly open to similarities.  E.g. hydraulic resistance in a fluids circuit has mathematically same parallel and series addition to electrical resistance. (Like that one's always helped me because when an electrical circuit confuses me, I think of pipes...and visa versa.)  Somehow having two physical models AND the math abstraction, helps me to internalize the concepts.  Like being three blind men at an elephant instead of one.&lt;/p&gt;&#xA;&lt;p&gt;Maybe a less direct analogy would be considering fluorine chemistry.  Sometimes, you can make a prediction based on the analogous chloride (downstairs brother), or sometimes the analogous oxide (next door brother).  Or sometimes different from both.  But sure as shit, you got a hell of a lot more insight if you understand periodic trends than if chemical combinations are pure chance.&lt;/p&gt;&#xA;&lt;p&gt;And I wouldn't totally underestimate the value of a storehouse of information either.  You will see a lot of people dismiss that as &amp;quot;just memorization&amp;quot;.  But to be a Renaissance man, you do need to know a few things first!  If you want to use the insights from catapult construction to help your painting and visa versa, you need to learn some of each!&lt;/p&gt;&#xA;&lt;p&gt;Now, yes, it is also good to have puzzle solving experience.  Good in general.  And good even for the purpose of learning the basic materials (progressive struggle drives retention and comprehension...why the worse athletes are sometimes better coaches).  But I wouldn't completely fetishize puzzle solving and deprecate information knowledge.  Feynman was renowned for looking at problems originally.  But he also ran through very significant amounts of very normal drill (and recommended it to struggling students at Cal Tech).  You can look at his calculus notebooks from high school and see both original thinking AND basic drill (LOTS of it).  Furthermore, before he did his most significant work, he realized he needed to buff up his knowledge of classical E&amp;amp;M (which is a ballbuster of a subject, often harder than some of the sexy topics).  And he grabbed a normal text and worked every problem.&lt;/p&gt;&#xA;&lt;p&gt;So, net, net.  I think there are wonder puzzle solving AND knowledge learning opportunities in a conventional course of study.  Please do not underestimate the huge value in learning classical mechanics (with the Lagrangian and all) before dealing with the &amp;quot;Internet-sexy&amp;quot; abstractions of quantum mechanics.  If you do other stuff on the side, great.  But only if you are crushing the normal course.  If you get below a high A in the normal courses and run around on reddit and sites like this working on popular books on relativity and QED, you're making the wrong choice.  Those basic courses have huge &amp;quot;meat&amp;quot; to fuel more creative later research work.  Both the information in them and the training of your mind involved in mastering them.&lt;/p&gt;&#xA;" OwnerUserId="20039" LastActivityDate="2022-07-08T12:29:17.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25367" PostTypeId="2" ParentId="24761" CreationDate="2022-07-08T12:42:42.083" Score="-1" Body="&lt;p&gt;This is the number one result on my Google search for survivor bias:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Survivorship_bias&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Survivorship_bias&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you want to do original work, some degree of initiative is required.&lt;br /&gt;&#xA;&amp;quot;Message to Garcia&amp;quot;. Of course this doesn't mean NEVER to ask a question.  But to not even Google, shows extremely low initiative.  You ain't solving FLT with that ethic.&lt;/p&gt;&#xA;&lt;p&gt;Also, yes, psychological hardening--although having nothing to do with survivor bias--is a benefit to doing R1 research. There's a lot of discouragements (including people issues) along the way.&lt;/p&gt;&#xA;" OwnerUserId="20039" LastActivityDate="2022-07-08T12:42:42.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25368" PostTypeId="1" CreationDate="2022-07-08T17:39:15.950" Score="1" ViewCount="513" Body="&lt;p&gt;I’ve studied convex optimization already with &lt;a href=&quot;https://web.stanford.edu/%7Eboyd/cvxbook/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boyd &amp;amp; Vandenberghe's book&lt;/a&gt;. Now I need to learn &lt;em&gt;non-convex&lt;/em&gt; optimization. What are the best/most common texts for this? Need to learn a general overview  for a company project. Have undergrad level of convex optimization. Have taken general 2nd/3rd year linear algebra class. Have taken calculus series, probability/stochastic processes, and linear algebra series. Have not taken analysis. Haven’t considered any books yet. I tried searching this up prior to my question, but I really couldn’t find much.&lt;/p&gt;&#xA;&lt;p&gt;Edit: studied from Boyd. Covered 75% of &lt;a href=&quot;https://link.springer.com/book/10.1007/978-3-319-77586-9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Practical Mathematical Optimization&lt;/a&gt; contents, but all of the main concepts covered. Looking for a general intro text including just a general overview of the field. Preferably global optimization.&lt;/p&gt;&#xA;" OwnerUserId="15802" LastEditorUserId="376" LastEditDate="2022-07-16T12:05:58.527" LastActivityDate="2022-07-16T20:17:54.537" Title="Textbooks for Non-Convex Optimization" Tags="|optimization|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25369" PostTypeId="2" ParentId="25362" CreationDate="2022-07-08T17:44:43.220" Score="6" Body="&lt;p&gt;&amp;quot;&lt;em&gt;When you keep on multiplying 1's such as, 1×1×1×... can you say here we have product of infinite number of 1's or is this limit of product of n number of 1's as n tends to infinity?&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In general, anything with &amp;quot;...&amp;quot; is imprecise, and it is meaningless if you cannot make it precise. I am sure that in your case it is meaningless, because...&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If so can we say this product is indeterminate?&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;No. Your use of the term &amp;quot;indeterminate&amp;quot; reveals that you conflated values and expressions. For example, 2×3 = 6, but &amp;quot;2×3&amp;quot; ≠ &amp;quot;6&amp;quot;. See &lt;a href=&quot;https://math.stackexchange.com/a/2184974/21820&quot;&gt;this post&lt;/a&gt; for more.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Then what about the product 2×2×2....... because this also can be written in terms of 1×1×1×....?&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Wrong. You haven't even defined &amp;quot;2×2×2.......&amp;quot; precisely, and if you could then you would not be saying that it &amp;quot;can be written in terms of 1×1×1×....?&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;But when you consider the irrational number e , by its definition we have ,e = 1+1/(1!)+1/(2!)+1/(3!)+... .Here the right side is a sum of infinite number of rationals . What we have to say, is this because of infinity number of rationals involved ?&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;No. Again. You shouldn't be saying that something happens &lt;em&gt;because&lt;/em&gt; of &amp;quot;infinity [&lt;em&gt;sic&lt;/em&gt;] number of rationals&amp;quot;. You should instead be telling yourself that you &lt;strong&gt;cannot&lt;/strong&gt; in general say that an infinite sum has the same properties as its partial sums.&lt;/p&gt;&#xA;&lt;p&gt;Consider the geometric sequence 1, 1/2, 1/4, ... For &lt;strong&gt;every&lt;/strong&gt; k∈ℕ, the sum of the first k terms is less than 2. But the sum of all of them (which needs to be defined as a limit) is &lt;strong&gt;not&lt;/strong&gt; less than 2.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We all know tan(π/2) can not be defined and it is not infinity.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;No. You are free to define tan(π/2) to be whatever you like. Whether or not your definition is useful is another matter, but it's false to say it cannot be defined. In fact, it may be convenient in some (not too common) situations to define tan(π/2) to be the infinity of the projectively extended real line, but one has to learn basic real analysis before looking at extensions of the standard real line.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;But in definite integrals as an example, integral of 1/(1+x²) from x = 0 to x = ∞ We write tan inverse of infinity as π/2.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In proper mathematics, we don't write whatever we like. We give precise definitions and prove precise theorems about the objects that we have defined. It doesn't matter if we define &lt;strong&gt;tan&lt;/strong&gt; to have domain excluding odd multiples of π/2. We are still &lt;strong&gt;free&lt;/strong&gt; to define &lt;strong&gt;atan&lt;/strong&gt; on the affinely extended real line such that atan(∞) = π/2. What properties we get for atan is another matter. Also, in basic real analysis we typically define integrals on a bounded interval, and define an integral on [0,∞) to be the limit of the integral on [0,x] as x → ∞. So in fact whether you define atan(∞) or not is irrelevant!&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I know many students don't think about these issues if we do not mention, but it is much better if we have the most appropriate way of describing these issues to give much confidence of what students learn.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If you &lt;strong&gt;first&lt;/strong&gt; correct all your own errors in your understanding of the concepts, then you would be much better positioned to teach students and recognize when they make a conceptual mistake.&lt;/p&gt;&#xA;&lt;p&gt;Understanding comes before teaching.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2022-07-08T17:44:43.220" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25370" PostTypeId="2" ParentId="24761" CreationDate="2022-07-08T20:59:09.790" Score="3" Body="&lt;p&gt;At least at first (i.e. before completing your Ph.D.) it helps to have an advisor.  She can &amp;quot;advise&amp;quot; you on this:  When to stop working on that problem and try another; how to try first a simpler case; references to use to avoid repeating what others have done.&lt;/p&gt;&#xA;&lt;p&gt;One of my fellow graduate students (1970s) wanted to solve the four color problem.  His advisor said &amp;quot;no&amp;quot;.  Wait until after you have tenure to attempt that.  If you really want to do graph theory, let's find a more reasonable problem for you to work on.&lt;/p&gt;&#xA;&lt;p&gt;A story says someone once asked Hilbert why he never worked on the Riemann hypothesis.  He replied that he would probably have to spend 5 years getting the background before he could attempt it, and he did not want to waste 5 years on a probable failure. [From memory, maybe the details are wrong?]&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2022-07-09T00:49:23.453" LastActivityDate="2022-07-09T00:49:23.453" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25371" PostTypeId="1" AcceptedAnswerId="25372" CreationDate="2022-07-09T16:19:28.137" Score="6" ViewCount="593" Body="&lt;p&gt;The question in the title mostly covers the question I want to ask. After seeing a number of questions here on ME and having taught/TA'd a number of introductory math classes, I wonder what people think the essential skills high school should teach students in terms of math. It might be interesting to see both what math major type people think is necessary and what just general population think is necessary. To limit the question though, let's say that we care what people who teach STEM based math (for non US, that's essentially the non-humanities based academics disciplines) think. What would you like your incoming students to have learnt in high school?&lt;/p&gt;&#xA;" OwnerUserId="5896" LastEditorUserId="5896" LastEditDate="2022-07-10T18:34:28.057" LastActivityDate="2022-07-18T06:22:52.143" Title="What are the essential mathematics skills that uni based STEM math educators want high school to teach students?" Tags="|undergraduate-education|secondary-education|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25372" PostTypeId="2" ParentId="25371" CreationDate="2022-07-09T17:53:08.870" Score="7" Body="&lt;p&gt;This is entirely opinion based, but I don't really care what content is &amp;quot;covered&amp;quot; at all.  I care that students are engaged with thinking about problems which involve quantitative and spatial reasoning, and that they develop progressively more sophisticate logical reasoning abilities.  They should be able to communicate their ideas effectively, produce logically coherent arguments, critique the arguments of others, produce examples and counter examples of claims generated by themselves and their peers, etc.  In other words, they should engage with the real process of mathematics.  Too often prescribing content leads to mimicry of these fundamental skills.  We need the real deal.&lt;/p&gt;&#xA;&lt;p&gt;It would be nice if the topics they are thinking about &amp;quot;build up&amp;quot; to something.  Another important feature of mathematics is how mastery of one body of knowledge can lay the groundwork for beginning on another.  What was at first insurmountable becomes routine.  I would like students to have this experience so that they know what it feels like, and that it is possible.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-07-09T17:53:08.870" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25373" PostTypeId="2" ParentId="25362" CreationDate="2022-07-09T19:11:59.493" Score="1" Body="&lt;p&gt;This is a point where it's useful to explain to students that the simplifications of infinite phases we give are not always obvious from the related finite phrases.  When dealing with this myself, I like to remind people that the laws of math were not pulled from thin air.  People had to think them through, reason them out, and come up with answers which seem to be in agreement with the reality around them.&lt;/p&gt;&#xA;&lt;p&gt;I agree with the sentiment that one really should not study such infinities until one is ready for calculus.  But I do think marveling at them is fair game.  It's simplifying things a bit much, but I like starting from Zeno's paradoxes to provoke thought -- to realize that this &amp;quot;infinity&amp;quot; thing is really tricky.  It takes a lot of formalism to get it right.  I'd argue that Zeno's most famous paradox was not truly laid to rest until Newton and Leibniz invented calculus and showed a rigorous way of dealing with infinities.&lt;/p&gt;&#xA;&lt;p&gt;To show how odd infinities can be, I do like &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert's Grand Hotel&lt;/a&gt;.  I like it because it's right on the edge of math.  The solutions to adding more guests make sense numerically, but they run just afoul of our intuitions.  And I point out that the math behind Hilbert's Hotel isn't the &lt;em&gt;only&lt;/em&gt; approach possible, but it is the one that we have found to be most consistent while we explored it.&lt;/p&gt;&#xA;&lt;p&gt;It's probably worth at least mentioning the concept of limits, as they are the rigorous tool mathematicians use to tame infinities.  For example, we can say the sum of any number of rational numbers is rational... but the devil is in the details.  By &amp;quot;any number&amp;quot; we really mean any integer number.  Your sum that leads to e dies not have an integer number of terms, so it is not bound by the obvious inductive patterns.  We can either speak of this as the limit of this sequence (and realize that limits change the game), or we can use transfinite numbers and speak of &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; terms in a series.  But by invoking transfinite numbers, we have to realize that rules that worked fine for finite numbers don't always work here... just as the rules for integers don't always work out when you expand them to the real numbers.&lt;/p&gt;&#xA;&lt;p&gt;But all the details are not necessary.  It's enough to realize that infinity is &lt;em&gt;different&lt;/em&gt; in material ways, and have a few examples like Zeno's Paradoxes and the Grand Hotel to guide you.  Learn how we rigorously handle these infinities when you are ready to learn how to rigorously handle these infinities, and not before.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2022-07-09T19:11:59.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25375" PostTypeId="2" ParentId="25371" CreationDate="2022-07-10T09:27:11.837" Score="3" Body="&lt;p&gt;Processing information &lt;em&gt;attentively&lt;/em&gt; for meaning;&lt;br&gt; thinking &lt;em&gt;systematically&lt;/em&gt; (structure, connections, etc.);&lt;br&gt; being &lt;em&gt;empowered&lt;/em&gt; to unpack stuff and continually make revisions and &lt;em&gt;sense&lt;/em&gt;;&lt;br&gt; skills of inquiry.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bVc8A.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bVc8A.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In short, active thinking. Quoting &lt;a href=&quot;https://matheducators.stackexchange.com/questions/15278/why-do-some-students-struggle-so-much-with-fractions/25358#comment64834_25358&quot;&gt;my recent comments&lt;/a&gt; (Why do some students struggle so much with fractions):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Or how about just cultivating &lt;a href=&quot;https://matheducators.stackexchange.com/q/17161/372&quot;&gt;confidence&lt;/a&gt; (by sufficiently exposing learners to reasoning that involves more than one step, being methodical, and the process of intellectual discovery)? Technical topics naturally seem formidable when it continues to feel alien to be applying attention span to them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I was riffing generally that, especially when it comes to hierarchical and technical subjects, waving the white flag at the first opportunity is a vicious cycle. Like reflexively dismissing alien cultures and never getting to broaden horizons.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt; to clarify that the above illustration is an inherently didactic &lt;em&gt;concept map&lt;/em&gt;, not an inspirational poster.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-07-18T06:22:52.143" LastActivityDate="2022-07-18T06:22:52.143" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25376" PostTypeId="2" ParentId="25371" CreationDate="2022-07-10T14:10:26.633" Score="-1" Body="&lt;p&gt;For most STEMs, they will have a standard calculus course, freshman year.  Strong working ability, B level, of most of Frank Ayres First Year College Math is a good enough foundation. Of course it would be nice to have more.  And some will have less. But that's a decent expectation.&lt;/p&gt;&#xA;&lt;p&gt;Note that, in the US, college algebra is a misnomer since the 50s. Calculus freshman year is the normal track. If you aren't ready for it, you're on a remedial track.  If you place out, you are on an accelerated track.&lt;/p&gt;&#xA;&lt;p&gt;Also, I'm not saying they need to study the Ayres book.  Duh. Just that it is a convenient synthesis of high school work, pre calculus. If you can work the Ayres problems, or the equivalent, you have enough to move into a normal calc class.&lt;/p&gt;&#xA;" OwnerUserId="20045" LastEditorUserId="20045" LastEditDate="2022-07-10T14:45:50.570" LastActivityDate="2022-07-10T14:45:50.570" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25377" PostTypeId="2" ParentId="25371" CreationDate="2022-07-11T00:24:23.207" Score="5" Body="&lt;p&gt;The nominal entry level into university-level mathematics is first-semester freshman calculus. The way that class is customarily taught, it makes only extremely modest demands on students' high-level reasoning skills, such as reading comprehension, creativity, and sense-making. There are almost no &amp;quot;word problems,&amp;quot; and the class consists almost entirely of differential calculus, at which students can succeed simply by mastering rules.&lt;/p&gt;&#xA;&lt;p&gt;Students need skills like the following in order to succeed in such a class:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;ability to manipulate fractions&lt;/li&gt;&#xA;&lt;li&gt;order of operations&lt;/li&gt;&#xA;&lt;li&gt;ability to solve an equation for an unknown&lt;/li&gt;&#xA;&lt;li&gt;knowledge of some very basic trigonometry, such as being able to tell what is the sine of 90 degrees and explain why without recourse to rote memorization&lt;/li&gt;&#xA;&lt;li&gt;knowledge of exponents and logarithms and basic properties such as &lt;span class=&quot;math-container&quot;&gt;$\log(ab)=\log a+\log b$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Many students at this level have only been exposed to solving multiple equations in multiple unknowns in the case where the equations are linear. That is probably sufficient in most cases for success in such a class.&lt;/p&gt;&#xA;&lt;p&gt;The classes that will more stringently test their preparation are second-semester calculus (because integration is not algorithmic) and first-semester physics (because it's all word problems and interpretation, and one often has to solve multiple nonlinear equations in multiple unknowns).&lt;/p&gt;&#xA;" OwnerUserId="20047" LastEditorUserId="20047" LastEditDate="2022-07-11T00:25:32.570" LastActivityDate="2022-07-11T00:25:32.570" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25379" PostTypeId="2" ParentId="25371" CreationDate="2022-07-11T09:15:44.607" Score="4" Body="&lt;p&gt;TL;DR: Most students need to already have a basic understanding of common concepts to cope with doing them again more quickly and abstractly at university.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;My perspective: I am involved in teaching physics to students of physics and some other disciplines (e.g., math) in Germany. I studied physics and maths in Germany myself.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;University mathematics education does almost everything from scratch. Moreover, before the first semester, every physics department in Germany offers a mathematical repetition course of about three weeks, covering all the mathematics you require for the first semesters. This will cover mathematics that the physics lectures require and that the parallel mathematics lectures haven’t got to (e.g., integrals). If a student has missed a particular topic in school, they will see it here. So, from a naïve perspective, a student doesn’t need anything except the ability to engage in critical mathematical thinking of any kind.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;However,&lt;/strong&gt; all these courses only really work if you have some familiarity with the general topics. For example, it simply takes some time (and repetition) until a student has wrapped their mind around derivatives and automatised the basic concepts to some extent (also see &lt;a href=&quot;https://matheducators.stackexchange.com/a/24721/75&quot;&gt;this answer of mine&lt;/a&gt;). If a student hasn’t already done this in school, it will be very difficult for them to follow the much more quick-paced and abstract introduction to calculus at university and the physics lectures that simply use it. Some exceptional student may be able to grasp university calculus to the required extent even though this is their first encounter with calculus, but they are, well, the exception.&lt;/p&gt;&#xA;&lt;p&gt;The good news is that the details don’t matter that much. For example, I care that students have an understanding of what a derivative is, but if I don’t care very much whether they know certain differentiation methods. Those can be learnt quickly and are not the basics for more advanced layers of mathematics. (Of course working with different differentiation methods is a good way to fortify the understanding of derivatives.)&lt;/p&gt;&#xA;&lt;p&gt;With all that being said, here are things that we just assume as given in first-semester physics education and for which it is already very helpful if you have already seen them for the mathematics courses:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;fractions&lt;/li&gt;&#xA;&lt;li&gt;solving equation systems (linear or easy non-linear)&lt;/li&gt;&#xA;&lt;li&gt;solving quadratic equations&lt;/li&gt;&#xA;&lt;li&gt;basic trigonometry&lt;/li&gt;&#xA;&lt;li&gt;vectors (geometrical interpretation and arithmetics)&lt;/li&gt;&#xA;&lt;li&gt;limits&lt;/li&gt;&#xA;&lt;li&gt;derivatives&lt;/li&gt;&#xA;&lt;li&gt;integrals&lt;/li&gt;&#xA;&lt;li&gt;exponents and logarithms&lt;/li&gt;&#xA;&lt;li&gt;probability (not required in the first semester, but later)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;With all these topics, I mostly want students to have a robust understanding of the basics. There is no use if a student can quickly solve all sorts of integrals but has no idea how to apply integrals to real problems or is lost once the variable of integration is not &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; anymore or if presented a straightforward double integral.&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2022-07-11T09:22:32.783" LastActivityDate="2022-07-11T09:22:32.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25380" PostTypeId="1" CreationDate="2022-07-11T11:32:38.983" Score="3" ViewCount="155" Body="&lt;p&gt;In the context of iterative methods for equations and linear systems, one usually says that &amp;quot;linear convergence / order 1&amp;quot; is when the error &lt;span class=&quot;math-container&quot;&gt;$err$&lt;/span&gt; goes to zero with the number of iterations &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$err \sim \rho^n$&lt;/span&gt; for a certain &lt;span class=&quot;math-container&quot;&gt;$\rho&amp;lt;1$&lt;/span&gt; (e.g., fixed-point method, Jacobi method for linear systems), and &amp;quot;quadratic convergence / order 2&amp;quot; is &lt;span class=&quot;math-container&quot;&gt;$err \sim \rho^{2^n}$&lt;/span&gt; (e.g., the Newton method for a simple zero).&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, in the context of numerical quadrature and methods for ODEs, one usually says that &amp;quot;order 1&amp;quot; is &lt;span class=&quot;math-container&quot;&gt;$err \sim \frac{1}{n}$&lt;/span&gt;, when &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is the number of subdivision points, usually equispaced; (e.g., the explicit/implicit Euler's method), and &amp;quot;order 2&amp;quot; is &lt;span class=&quot;math-container&quot;&gt;$err \sim \frac{1}{n^2}$&lt;/span&gt; (e.g., trapezoidal rule).&lt;/p&gt;&#xA;&lt;p&gt;These two incompatible definitions are also discussed, for instance, on &lt;a href=&quot;https://en.wikipedia.org/wiki/Rate_of_convergence&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What is a good way to differentiate them in teaching to prevent confusion? If possible I would like to avoid introducing non-standard terminology. This is in the context of an undergraduate course in numerical methods.&lt;/p&gt;&#xA;" OwnerUserId="4930" LastEditorUserId="4930" LastEditDate="2022-07-11T13:36:33.453" LastActivityDate="2022-07-11T18:57:48.793" Title="How to differentiate the two notions of convergence order?" Tags="|undergraduate-education|definitions|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25381" PostTypeId="2" ParentId="25380" CreationDate="2022-07-11T18:57:48.793" Score="4" Body="&lt;p&gt;“Order” usually means the degree of &lt;em&gt;something&lt;/em&gt;. What that something is varies with the subject. Standard integration/differentiation methods are not iterative, so the iterative definition of “order” cannot be applied; the term order in this case is usually connected with the order of the Taylor series approximation or the order of a polynomial interpolant. In iteration, order is the degree relating successive errors:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$E_{n+1}\le C\,(E_n)^p$$&lt;/span&gt;&#xA;for order &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I guess an approach is to focus on the degree of &lt;em&gt;&lt;strong&gt;what&lt;/strong&gt;&lt;/em&gt;, which is how I keep the distinction clear for myself.&lt;/p&gt;&#xA;&lt;p&gt;Since the same word is used for two different things, it's bound to cause confusion. I suggest this be made clear from the start and warn them not to get the two confused. So if iterative methods precede interpolation/integration/etc. — or vice versa — then begin by explaining that the students will have to keep two distinct definitions straight. One they will learn now, and one later. I would probably even say that one will be based on the degree of a polynomial and one on a power that appears in an inequality. I'd repeat this when we get to the second definition. The idea is from the start to help them to construct an accurate internal &amp;quot;concept map,&amp;quot; if you will. If the student learns just the first definition thoroughly, then it will get in the way of learning the second.&lt;/p&gt;&#xA;&lt;p&gt;Another source of confusion can be seen in the comments. Order &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; in iterative methods leads to an &lt;em&gt;exponential&lt;/em&gt; decrease in error (as a function of the number of steps &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;): Taking &lt;span class=&quot;math-container&quot;&gt;$C=1$&lt;/span&gt; for convenience,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$E_n \le (E_{n-1})^2 \le ((E_{n-2})^2)^2 \le &#xA;  (((E_{n-3})^2)^2)^2 \le \cdots \le (E_0)^{2^n} \,.$$&lt;/span&gt;&#xA;One often thinks order &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is polynomial, not exponential.  This should probably be emphasized, too, when introducing order in iterative methods.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2022-07-11T18:57:48.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25382" PostTypeId="2" ParentId="24761" CreationDate="2022-07-11T21:46:08.397" Score="3" Body="&lt;p&gt;For my thesis I learned a certain collection of tools quite well (Hormander's machinery for solving &lt;span class=&quot;math-container&quot;&gt;$\bar{\partial}$&lt;/span&gt;-problems in several complex variables with control of the norm of the solutions in some Hilbert space).  I learned the kinds of problems people had solved with these tools.  I choose one problem which seemed tractable with this technology but which no one had tackled yet (concerning approximation of holomorphic functions on a domain by holomorphic functions on a slightly enlarged domain).  This problem was too hard, so I introduced additional constraints on the problem until it was tractable.  I think this sort of pathway is fairly typical.  You don't go chasing the Riemann hypothesis: you learn the tools of the trade, and then push the boundary just a little bit.&lt;/p&gt;&#xA;&lt;p&gt;Take this answer with a grain of salt since my research career ended with my thesis. I am much more interested in teaching.  A more seasoned researcher or dissertation advisor might have a more informed take.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-07-11T21:46:08.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25385" PostTypeId="2" ParentId="25371" CreationDate="2022-07-13T18:01:32.180" Score="3" Body="&lt;ul&gt;&#xA;&lt;li&gt;Checking answers&lt;/li&gt;&#xA;&lt;li&gt;Showing and checking work. Algebraic work can benefit from the two-column proof approach: one column for the algebra work and another column explaining what happened in each step. This is a lot to ask of students though since they will grow impatient at how much longer an algebra problem takes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-07-13T18:01:32.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25386" PostTypeId="1" AcceptedAnswerId="25388" CreationDate="2022-07-14T16:42:02.033" Score="30" ViewCount="4089" Body="&lt;p&gt;I once encountered a math educator whose personal pet peeve was the &amp;quot;give 110%&amp;quot; meme.  He drilled into his students that 100% was the literal maximum.  Percent came from &amp;quot;per cent&amp;quot; and 100 per 100 is literally all of them.  You can't eat 110% of the nachos.  You can't use 110% of the gas tank.  You can't raise prices 110%.&lt;/p&gt;&#xA;&lt;p&gt;If you're like me, that last one probably caught your attention.  I asked, if the price of a hot dog goes from \$1 to \$2.10, isn't that a 110% increase.  His answer was that a 100% increase in price meant all the possible price increase.  In other words, to raise the price of a \$1 hot dog by 100%, the new price must be &lt;strong&gt;the value of the rest of everything else in existence&lt;/strong&gt;.  I asked whether that meant that a \$1.50 hot dog was no longer a 50% increase in price, and he said percentages weren't useful for terms like prices and that the appropriate answer was that the price of the hot dog had increased by a factor of 1.5.  Students needed to learn that a factor of 1.5 increase did not always equate to a 50% increase.&lt;/p&gt;&#xA;&lt;p&gt;I'm not always good on my feet and was worried that I was starting to antagonize the man, so I dropped the issue.  What would have been the proper way to correct this misconception?  Or am I the one who is being dense?&lt;/p&gt;&#xA;" OwnerUserId="5054" LastEditorUserId="851" LastEditDate="2022-07-17T04:39:57.653" LastActivityDate="2024-01-30T21:42:09.473" Title="How can I validate the existence of percentages above 100?" Tags="|terminology|" AnswerCount="12" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25387" PostTypeId="2" ParentId="25386" CreationDate="2022-07-14T17:46:18.287" Score="11" Body="&lt;p&gt;From a practical standpoint, the horse has long been out of the barn on quoting price increases and other measures of growth as percentages (furthermore, one will hear static ratios being quoted as percentages, such as an employee's current salary being estimated at 105% to market). Overall, this seems to be one of those &amp;quot;instructor's personal eccentricities turning into a futile crusade against society&amp;quot; situations. I would suggest teaching the typical definition of a percentage change because students will surely encounter it in life.&lt;/p&gt;&#xA;&lt;p&gt;As far as &amp;quot;refuting&amp;quot; this perspective, I am having a little trouble understanding what precisely is meant by &amp;quot;the value of everything else in existence&amp;quot; and reconciling this apparently infinite number with the obvious limiting behavior of 99% growth in the price, 99.9% growth in the price, ..., and so on.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2022-07-14T17:46:18.287" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25388" PostTypeId="2" ParentId="25386" CreationDate="2022-07-14T19:30:18.347" Score="26" Body="&lt;p&gt;The formula to express a value &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; as a percentage of another value &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; is simply &lt;span class=&quot;math-container&quot;&gt;$Percentage= (X/Y)×100$&lt;/span&gt;. It is simply a ratio of numbers (a multiplicative factor), times 100. There is absolutely nothing that requires that &lt;span class=&quot;math-container&quot;&gt;$X&amp;lt;Y$&lt;/span&gt; for the formula to be applied. If &lt;span class=&quot;math-container&quot;&gt;$X&amp;gt;Y$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is more than 100% of &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt;. In some cases the number &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; represents a physical value of &amp;quot;everything that exists&amp;quot; (like a full plate of nachos) so the number &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; cannot logically be larger than that (you can't eat 150% of the plate of nachos), but in many cases the number &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; does not represent a &amp;quot;maximum possible value&amp;quot;. A \$1 hot dog is not the maximum possible price that anyone could ever charge for a hot dog, there's no reason why someone couldn't charge 150% of that price, or \$1.50.&lt;/p&gt;&#xA;&lt;p&gt;It makes no sense to claim that percentages greater than 100% &amp;quot;don't exist&amp;quot;. Suppose we look at a system like the wiring in your house, and observe that there is some safe level of current that can go though the wires without starting a fire. As long as the current load stays below 100% of the safe level, everything is fine. When the current load exceeds 100% of the safe value, a breaker will trip. It makes no sense to suggest that there is no numerical way to quantify how far above the safe level the current is when the breaker trips - merely observing that the breaker tripped certainly doesn't mean that &amp;quot;all the current in existence&amp;quot; was passing through the wire.&lt;/p&gt;&#xA;&lt;p&gt;The only times percentages over 100% don't exist is when describing values with respect to a number that is known or defined to be the &lt;em&gt;highest possible value&lt;/em&gt;. You can't give 110% of your maximum effort, since in this scenario we &lt;em&gt;define&lt;/em&gt; the &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$X/Y$&lt;/span&gt; to be the maximum value - in scenarios like these, &lt;span class=&quot;math-container&quot;&gt;$X&amp;gt;Y$&lt;/span&gt; is indeed a contradiction of terms. But in general, percentages are not only applicable when representing something as a percent of a pre-defined maximum. The price of a hot dog has no theoretical maximum, so a hot dog price could climb by 50%, or 100%, or 1000%, or more.&lt;/p&gt;&#xA;&lt;p&gt;It is absurd to claim that percentages aren't useful for things like prices. It is utterly common to see discounts expressed as a percentage decrease in a price. Inflation has been measured as a percentage change in price for at least 100 years. Stock prices, loan interest, and bank interest rates can all be expressed in terms of percentage growth rates of dollar values.&lt;/p&gt;&#xA;" OwnerUserId="5959" LastEditorUserId="5959" LastEditDate="2022-07-14T20:44:44.147" LastActivityDate="2022-07-14T20:44:44.147" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="25389" PostTypeId="2" ParentId="25386" CreationDate="2022-07-15T06:48:48.927" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;You can't use 110% of the gas tank.&lt;/p&gt;&#xA;&lt;p&gt;He drilled into his students that 100% was the literal maximum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This reminds me of &lt;a href=&quot;https://math.stackexchange.com/questions/4492666/is-there-a-mathematical-symbol-for-not-enough-or-too-much#comment9426142_4492666&quot;&gt;my earlier comment&lt;/a&gt; that any discussion of &amp;quot;too much&amp;quot; or &amp;quot;too little&amp;quot; is meaningful only when a reference point has been agreed on.&lt;/p&gt;&#xA;&lt;p&gt;Sure, &lt;span class=&quot;math-container&quot;&gt;$100\%$&lt;/span&gt; corresponds to literal maximum, but a local maximum (perhaps due to an artificial ceiling), not &lt;em&gt;the&lt;/em&gt; (global) maximum.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;His answer was that a 100% increase in price meant all the possible price increase.&lt;/p&gt;&#xA;&lt;p&gt;to raise the price of a \$1 hot dog by 100%, the new price must be &lt;strong&gt;the value of the rest of everything else in existence&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;His guiding framework (if at all) is missing a specification of reference point, so he is erratically and incoherently wavering between&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$100\%\longleftrightarrow$&lt;/span&gt; his available gas (as opposed to &lt;strong&gt;all the gas that exists&lt;/strong&gt;),&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$100\%\longleftrightarrow$&lt;/span&gt; &lt;strong&gt;every dollar that exists&lt;/strong&gt; (as opposed to the hotdog's \$1 cost).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Percent came from &amp;quot;per cent&amp;quot; and 100 per 100 is literally all of them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes, the root meaning of &lt;span class=&quot;math-container&quot;&gt;$100\%$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; &lt;a href=&quot;https://www.dictionary.com/browse/percent&quot; rel=&quot;nofollow noreferrer&quot;&gt;by the hundred&lt;/a&gt;, and to be clear he should say that &lt;span class=&quot;math-container&quot;&gt;$100\%$&lt;/span&gt; refers to all of &lt;em&gt;those&lt;/em&gt; hundred—not everything in existence.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a \$1.50 hot dog&lt;/p&gt;&#xA;&lt;p&gt;he said the price of the hot dog had increased by a factor of 1.5.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;On a separate note: no, the increase was by a factor of only &lt;span class=&quot;math-container&quot;&gt;$0.5.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Students needed to learn that a factor of 1.5 increase did not always equate to a 50% increase.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Assuming that he actually means that an increase by a factor of &lt;span class=&quot;math-container&quot;&gt;$0.5$&lt;/span&gt; does not mean a &lt;span class=&quot;math-container&quot;&gt;$50\%$&lt;/span&gt; increase, then the onus is on him to justify this claim.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, &lt;em&gt;by definition&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; increases by a factor of &lt;span class=&quot;math-container&quot;&gt;$0.5\iff x$&lt;/span&gt; increases by &lt;span class=&quot;math-container&quot;&gt;$0.5x\iff x$&lt;/span&gt; undergoes a &lt;span class=&quot;math-container&quot;&gt;$50\%$&lt;/span&gt; increase. QED&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-07-18T15:36:15.027" LastActivityDate="2022-07-18T15:36:15.027" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25390" PostTypeId="2" ParentId="25386" CreationDate="2022-07-15T10:00:43.907" Score="7" Body="&lt;p&gt;If a student in my class believed this, I would respond as follows:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I would draw two pizzas in boxes.  If someone eats the contents of one box, they ate one pizza and one pizza is left over.  If someone ate &lt;strong&gt;half&lt;/strong&gt; the contents of a box, they ate &lt;span class=&quot;math-container&quot;&gt;$\frac 12$&lt;/span&gt; of a pizza and &lt;span class=&quot;math-container&quot;&gt;$1\frac 12$&lt;/span&gt; pizzas or &lt;span class=&quot;math-container&quot;&gt;$\frac 32$&lt;/span&gt; pizzas are left over.  We have now established fractions greater than 1.&lt;/li&gt;&#xA;&lt;li&gt;We would then convert all the fractions to hundredths, meaning each pizza was cut into 100 bite size morsels.  If someone eats the contents of one box, they ate &lt;span class=&quot;math-container&quot;&gt;$\frac{100}{100}$&lt;/span&gt; pizza and &lt;span class=&quot;math-container&quot;&gt;$\frac{100}{100}$&lt;/span&gt; pizza is left over.  If someone ate &lt;strong&gt;half&lt;/strong&gt; the contents of a box, they ate &lt;span class=&quot;math-container&quot;&gt;$\frac{50}{100}$&lt;/span&gt; of a pizza and &lt;span class=&quot;math-container&quot;&gt;$1\frac{50}{100}$&lt;/span&gt; pizzas or &lt;span class=&quot;math-container&quot;&gt;$\frac{150}{100}$&lt;/span&gt; pizzas are left over.  We have now established fractions with a denominator of 100 greater than 1.&lt;/li&gt;&#xA;&lt;li&gt;Finally, I would convert 100ths to percents.  We have started with 200% of a whole pizza.   If someone eats the contents of one box, they ate 100%  of a whole pizza and 100% of a whole pizza is left over.  If someone ate &lt;strong&gt;half&lt;/strong&gt; the contents of a box, they ate 50% of a pizza and 150% pizzas is left over.  We have now established percents greater than 1.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;While this might convince my student it doesn't sound like it would convince the person you mentioned.  I would ask him why a fraction can be greater than a whole but a percent can't.  He would probably find an answer that satisfied himself but not anyone else.  You can't convince everyone.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2022-07-16T19:06:56.207" LastActivityDate="2022-07-16T19:06:56.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25391" PostTypeId="2" ParentId="25386" CreationDate="2022-07-15T12:15:32.137" Score="5" Body="&lt;p&gt;This math educator’s argument seems to be grounded in &lt;strong&gt;language&lt;/strong&gt; more than in mathematics. It should be self-evident that the &lt;em&gt;mathematical processes&lt;/em&gt; of percentages are useful for things like price increases, and so his objection must be that it shouldn’t be &lt;em&gt;called&lt;/em&gt; ‘percentage’, not that we shouldn’t &lt;em&gt;do&lt;/em&gt; that sort of calculation.&lt;/p&gt;&#xA;&lt;p&gt;I’d like to know how he defines ‘per’. Maybe as ‘out of’ (some maximum)? That would make his argument make sense… but that’s not how the word is used in English, nor (to my limited understanding) in the original Latin.&lt;/p&gt;&#xA;&lt;p&gt;I would instead define ’per‘ as ‘for each’. That’s consistent with how it’s used in rates; if I drive at 60 kilometres per hour, I change position by 60 kilometres &lt;em&gt;for each&lt;/em&gt; hour driven.&lt;/p&gt;&#xA;&lt;p&gt;‘Per cent’ then literally means ‘for each hundred’, no more and no less. The question then becomes, each hundred &lt;strong&gt;what&lt;/strong&gt;?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If I score 50% on a test, that means that for each hundred marks available, I earned myself fifty of them.&lt;/li&gt;&#xA;&lt;li&gt;If I read that milk is 2% fat, that means that if I separated the milk into its constituents, for every hundred (arbitrary) units of measure, two of them are the fat.&lt;/li&gt;&#xA;&lt;li&gt;If a price rises by 110%, that means that for every hundred (arbitrary) units† of the original price before the change, the new price has a hundred and ten more (plus the original hundred).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The difference between the first two examples and the third one is that the former use parts of a &lt;em&gt;maximum&lt;/em&gt; quantity, while the latter uses parts of an &lt;em&gt;original&lt;/em&gt; quantity. &lt;strong&gt;There is nothing about the term ‘percentage’ that makes ‘maximum’ valid and ‘original’ invalid.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;† If you like (and if you use a currency that’s divided into cents), you can interpret ‘per cent’ as being ‘for each (monetary) cent’ instead of ‘for each hundred’. &lt;em&gt;For each cent&lt;/em&gt; of the $1.20 an ice cream cost when I was a kid, the new cost instead has 2.5 cents. *shakes cane*&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A final musing: I’d honestly expect more traction on an argument like this turned against improper fractions, rather than percentages. At least there, you can argue that ‘whole’ means (or ought to mean) &lt;em&gt;everything&lt;/em&gt;, so you can’t possibly have three halves. (If I say I have three halves of a pizza, you might argue that no, I have two halves of a pizza, plus a half of &lt;em&gt;some other&lt;/em&gt; pizza!)&lt;/p&gt;&#xA;" OwnerUserId="1946" LastActivityDate="2022-07-15T12:15:32.137" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25392" PostTypeId="2" ParentId="25386" CreationDate="2022-07-15T16:21:13.637" Score="4" Body="&lt;p&gt;Doesn't this come down to whether you are looking at parts of a single thing, or comparing parts of two things?&lt;/p&gt;&#xA;&lt;p&gt;If you are looking at the parts of a single thing, like &amp;quot;How full is the gas tank?&amp;quot;, then yes, you can't have more than 100%. But if you've got two cars, one with a gas tank that holds 10 gallons and one that holds 15, then the smaller's capacity is 66% of the larger, or the larger's capacity is 150% of the smaller's.&lt;/p&gt;&#xA;&lt;p&gt;When you're looking at the price increase for the hot dog, the two things being compared are the earlier price and the current price.&lt;/p&gt;&#xA;&lt;p&gt;As to how to deal with such a person, linguists talk about 'prescriptivist' vs. 'descriptivist' approaches, and my response to them might be &amp;quot;well, lots of people use percentages this way, and it seems self-consistent and effective, so I take it to be my duty as a math person to understand it and help others use it.&amp;quot; But I wouldn't expect them to change - some people can get really attached to  rules, even if they are unrealistic, and we've got no idea where they came from.&lt;/p&gt;&#xA;" OwnerUserId="14507" LastActivityDate="2022-07-15T16:21:13.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25393" PostTypeId="2" ParentId="11457" CreationDate="2022-07-15T17:36:21.653" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;What do you say to the following way of teaching &amp;quot;if...then&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;An implication &lt;span class=&quot;math-container&quot;&gt;$(A {\implies} B)$&lt;/span&gt; can be viewed as asserting that &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; &lt;em&gt;&lt;strong&gt;is at least as true as&lt;/strong&gt;&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$A.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Instead of suggesting that the truth of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; lies on a spectrum &lt;span class=&quot;math-container&quot;&gt;$(A$&lt;/span&gt; after all is unambiguously either true or false), why not explicitly write &lt;span class=&quot;math-container&quot;&gt;$$\text{for each }x,\;\Big(A(x){\implies}B(x)\Big),$$&lt;/span&gt; &lt;a href=&quot;https://math.stackexchange.com/a/4444327/21813&quot;&gt;or just&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$$A(x){\implies}B(x),$$&lt;/span&gt; and say that &lt;span class=&quot;math-container&quot;&gt;$\text“A(x)$&lt;/span&gt; being true forces &lt;span class=&quot;math-container&quot;&gt;$B(x)$&lt;/span&gt; to be true” or, more clearly, “whenever &lt;span class=&quot;math-container&quot;&gt;$A(x)$&lt;/span&gt; is true, &lt;span class=&quot;math-container&quot;&gt;$B(x)$&lt;/span&gt; is necessarily true”, that is, every value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that satisfies &lt;span class=&quot;math-container&quot;&gt;$A(x)$&lt;/span&gt; also satisfies &lt;span class=&quot;math-container&quot;&gt;$B(x).$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An implication &lt;span class=&quot;math-container&quot;&gt;$(A {\implies} B)$&lt;/span&gt; can be viewed as asserting that &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; &lt;em&gt;&lt;strong&gt;is at least as true as&lt;/strong&gt;&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$A.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Dr4iL.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Dr4iL.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If set &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; contains every value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that satisfies &lt;span class=&quot;math-container&quot;&gt;$A(x),$&lt;/span&gt; and likewise for &lt;span class=&quot;math-container&quot;&gt;$B(x),$&lt;/span&gt; then &lt;strong&gt;set&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; &lt;em&gt;&lt;strong&gt;contains at least the elements of set&lt;/strong&gt;&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; (i.e., &lt;span class=&quot;math-container&quot;&gt;$A\subseteq B$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thus, if &lt;span class=&quot;math-container&quot;&gt;$(A {\implies} B)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(B{\implies} A),$&lt;/span&gt; then  &amp;quot;at least as true&amp;quot; in both directions&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this case, &lt;span class=&quot;math-container&quot;&gt;$A(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B(x)$&lt;/span&gt; have the same solution set.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What do you say to the following way of teaching &amp;quot;the following are equivalent&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;it suffices to prove that &lt;span class=&quot;math-container&quot;&gt;$(A_1 {\implies} A_2 {\implies} \dots {\implies} A_n {\implies} A_1)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is intuitive and symmetrical. Equivalently, it suffices to show that&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$A_1\implies A_2\land A_3\land A_4\land\ldots\land A_n$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\lnot A_1\implies\lnot A_2\land \lnot A_3\land \lnot A_4\land \ldots\land\lnot A_n$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(assume, without loss of generality, that &lt;span class=&quot;math-container&quot;&gt;$A_3$&lt;/span&gt; is true; then, by contraposition, so is &lt;span class=&quot;math-container&quot;&gt;$A_1;$&lt;/span&gt; consequently, so are the remaining statements).&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-12-21T04:28:40.727" LastActivityDate="2023-12-21T04:28:40.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25394" PostTypeId="2" ParentId="25386" CreationDate="2022-07-15T19:13:33.510" Score="9" Body="&lt;p&gt;Let me digress into material probably for HSM Stack Exhcange: Arithmetic and math in general is an abstraction to our real world counting and measuring (from which mathematics originated from).&lt;/p&gt;&#xA;&lt;p&gt;The educator's stance is like saying negative numbers don't exist because you can't have negative quantities of something in the &amp;quot;real world&amp;quot;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Negative_number#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ancient mathematicians for a very long time considered negative numbers absurd&lt;/a&gt;, especially Greek mathematicians who understood math geometrically (what is a negative length or area?).&#xA;Mathematicians and merchants like Brahmagupta and Fibonacci came to understand negative quantities as debts or owed quantities as opposed to credits, which is so natural to us today that it is surprising that anyone would not accept it. But understanding numbers this way requires some philosophical leap of faith if you and your contemporaries only understand numbers as directly relating to physical quantities. You can imagine how difficult imaginary numbers were to accept.&lt;/p&gt;&#xA;&lt;p&gt;Similarly, the notion of percent is abstracted from dividing literal objects such as nachos or dollars into 100 parts. It is easier to say this price is 3% or 150% of another price instead of saying 0.03 and 1.50 times the price. You may have only one bag of nachos, but we can abstract the 1/100 parts to imagine and make sense of situations where we had 110% of nachos compared to a friend or -30% nachos compared to yesterday. These mathematical tools are useful even if they don't correspond exactly with what literally exists in the real world (without getting into too much philosophy of what even existence means).&lt;/p&gt;&#xA;&lt;p&gt;There are some situations where having &amp;quot;greater than 100% of some maximum&amp;quot; is quite unintuitive, even for mathematicians. Considering probabilities greater than 1 is not even allowed by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Probability_axioms&quot; rel=&quot;nofollow noreferrer&quot;&gt;standard (Kolmogorov) probability axioms&lt;/a&gt;, since the probability of the entire sample space is 1. &amp;quot;Nonsensical&amp;quot; probability densities could still be useful, such as &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Improper_prior&quot; rel=&quot;nofollow noreferrer&quot;&gt;improper priors&lt;/a&gt; used in Bayesian inference.&lt;/p&gt;&#xA;" OwnerUserId="20059" LastEditorUserId="20059" LastEditDate="2022-07-17T00:59:53.943" LastActivityDate="2022-07-17T00:59:53.943" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25395" PostTypeId="2" ParentId="25386" CreationDate="2022-07-16T00:01:44.813" Score="8" Body="&lt;p&gt;The way I like to put it is: &lt;strong&gt;‘Per cent’ means nothing on its own. Only ‘per cent OF …’ is meaningful.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A percentage is simply another way of writing a ratio (like a vulgar fraction, or a decimal fraction). It gives one quantity in terms of another. Without knowing what that starting quantity is, it tells you absolutely nothing. So you always need to know &lt;em&gt;what&lt;/em&gt; something is a percentage &lt;em&gt;of&lt;/em&gt; to know what the number means.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes the context is enough to tell you what the starting quantity is. But if it's not completely clear, ask yourself: “Percentage &lt;em&gt;of&lt;/em&gt; …?”&lt;/p&gt;&#xA;&lt;p&gt;And once you know what that other quantity is, that should tell you whether it makes sense to talk about more than 100% of it (or, equivalently, more than 1 of, or more than 1:1):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If the starting quantity is the most effort you could possibly put in, then clearly you can't put in any more than that, so &amp;gt;100% effort is not achievable.&lt;/li&gt;&#xA;&lt;li&gt;But if the starting quantity is your &lt;em&gt;normal&lt;/em&gt; best effort, then 110% of &lt;em&gt;that&lt;/em&gt; effort wouldn't be so surprising, as you'll probably be able to put in a bit more on very special occasions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(If sportspeople tend to use the second quantity, that might explain why it sounds nonsensical to people who assume the first… If the starting quantity were specified clearly, then there wouldn't be this ambiguity!)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If the starting quantity is the price of an item, then it's always possible to pay more, so you could end up paying 110% of the asking price.&lt;/li&gt;&#xA;&lt;li&gt;Similarly, a raise of 50% of your salary would mean getting half as much again — and a raise of 150% would mean getting more than twice as much! (I didn't say these were &lt;em&gt;realistic&lt;/em&gt; examples…)&lt;/li&gt;&#xA;&lt;li&gt;Conversely, if there's a pay &lt;em&gt;cut&lt;/em&gt;, you can't lose more than 100% of your salary (or you'd have to pay &lt;em&gt;them&lt;/em&gt; to work there)…&lt;/li&gt;&#xA;&lt;li&gt;This also makes sense of some apparent paradoxes (or scams) involving an increase followed by a reduction (or vice versa): for example, a 20% increase followed by a 20% reduction doesn't end up where you started, &lt;em&gt;because those are percentages of different things&lt;/em&gt;. And of course you can only compare percentages if they use the same starting quantity.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And remember: if there's any doubt, &lt;strong&gt;it's always worth asking: “Percentage OF WHAT?”&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="12762" LastActivityDate="2022-07-16T00:01:44.813" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25396" PostTypeId="2" ParentId="25386" CreationDate="2022-07-16T05:44:43.193" Score="3" Body="&lt;p&gt;&lt;em&gt;Percent&lt;/em&gt; almost literally means &amp;quot;a hundredth&amp;quot;. It's a way of expressing fractions. 150% is a factor of 1.5, etc. Now, what &lt;strong&gt;use&lt;/strong&gt; one makes of these factors is besides the point.&lt;/p&gt;&#xA;&lt;p&gt;The usual convention with percentage changes is that you add the percentage to the implied 100%. E.g. a -20% price change means you multiply the old price by 0.8 to get the discounted price. Nothing super fancy to it - I'm surprised someone would make a big deal out of it. There's no &amp;quot;only proper&amp;quot; approach here, the convention is arbitrary but works fine.&lt;/p&gt;&#xA;&lt;p&gt;In this convention, saying &amp;quot;monthly hot dog average prices up by 10%&amp;quot; and &amp;quot;hot dogs now cost 110% of the price last month&amp;quot; are exactly equivalent ways of conveying the factor 1.1.&lt;/p&gt;&#xA;" OwnerUserId="7344" LastActivityDate="2022-07-16T05:44:43.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25397" PostTypeId="2" ParentId="2013" CreationDate="2022-07-16T11:49:43.110" Score="1" Body="&lt;p&gt;For exporting images made in Geogebra, this command is extremely helpful:&#xA;&lt;a href=&quot;https://wiki.geogebra.org/en/ExportImage_Command&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://wiki.geogebra.org/en/ExportImage_Command&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Towards the bottom:&lt;/p&gt;&#xA;&lt;p&gt;pic1 = ExportImage(&amp;quot;view&amp;quot;, 2, &amp;quot;corner&amp;quot;, A, &amp;quot;corner2&amp;quot;, B)&lt;/p&gt;&#xA;&lt;p&gt;Creates a GeoGebra image of View 2 and puts it in the view with position defined by A and B&lt;/p&gt;&#xA;&lt;p&gt;Take a look at the other options within the command. You can choose to export a transparent image and even specify the file type !&lt;/p&gt;&#xA;" OwnerUserId="20070" LastActivityDate="2022-07-16T11:49:43.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25398" PostTypeId="2" ParentId="25386" CreationDate="2022-07-16T14:42:59.413" Score="5" Body="&lt;p&gt;Obviously the instructor's comments were total nonsense. The OP put forward a clear argument that showed he was wrong, he got unjustifiably defensive, and wound up doubling-down on his rhetoric, spinning it in even more nonsensical ways. If he had any reflection after the fact, I could imagine him regretting that interaction. Either way, it's kind of a tragedy, given its impact on young, impressionable minds. (A large part of my job as a community college educator is trying to clean up broken understandings from bad interactions like this, and it's &lt;em&gt;tough&lt;/em&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;To the extent that there are any numbers above 1, then there are percentages above 100%. It's just a different way of expressing numbers, particularly multipliers (and most often useful if they have a fractional component).&lt;/p&gt;&#xA;&lt;p&gt;One thing I would highlight is the common confusion between &lt;em&gt;percentages&lt;/em&gt; and &lt;em&gt;probability&lt;/em&gt;. Probability is often expressed as a percentage, in some cases being the predominant use of percentages, so they get linked in people's minds as possibly synonymous. Also, they're both 11-letter words starting with &amp;quot;p&amp;quot;, so they overlap in people's head space. &lt;em&gt;Probability&lt;/em&gt; really can't be above 100%, but that's not the case with &lt;em&gt;percentages&lt;/em&gt; in general. It's possible the instructor (like many people) had gotten confused at some point about the difference.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2024-01-30T21:42:09.473" LastActivityDate="2024-01-30T21:42:09.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25399" PostTypeId="2" ParentId="17399" CreationDate="2022-07-16T18:50:36.823" Score="1" Body="&lt;p&gt;Most people except mathematicians never need to use the special properties of real numbers (ex. unique Dedekind-complete ordered field). You can teach them that the ratio of circumference to diameter, &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, is about 3.14 and can't be represented exactly as a fraction, but can be approximated well and 5 or 6 decimals is enough for practical purposes. The same for common quantities in engineering like &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Therefore the concept of real number is never really taught or defined early on, just as an informal notion of number. The properties of reals are not as important early on in education as understanding numbers as abstractions of measurements and counting that can be manipulated symbolically using equations and operations like addition and multiplication. We know how these operations work for rational numbers and, we implicitly apply the same operations to irrational numbers without thinking about it because it works well enough. This simplification, while technically not &amp;quot;the whole truth&amp;quot;, is good enough to be useful and build understanding for future math concepts where our intuitions can later be ground in formal definitions.&lt;/p&gt;&#xA;" OwnerUserId="20059" LastEditorUserId="20059" LastEditDate="2022-07-17T00:58:20.187" LastActivityDate="2022-07-17T00:58:20.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25400" PostTypeId="2" ParentId="25386" CreationDate="2022-07-16T18:51:03.477" Score="4" Body="&lt;p&gt;Let's compare some apples to apples: Say I have 12 apples, you have 6. &amp;quot;I can say: I have more apples than you. I have 200% of your apples.&amp;quot; What did I just do? I have chosen your 6 apples as reference, to represent (quantity in this case) 100. My reasoning is: If you had 100 apples I would have 200.&lt;/p&gt;&#xA;&lt;p&gt;Your 6 apples can be the max. quantity or not. In this example it is not. Therefore more the 100% is possible.&lt;/p&gt;&#xA;" OwnerDisplayName="user20076" LastEditorDisplayName="user20076" LastEditDate="2022-07-16T19:08:06.680" LastActivityDate="2022-07-16T19:08:06.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25401" PostTypeId="2" ParentId="25368" CreationDate="2022-07-16T20:17:54.537" Score="1" Body="&lt;p&gt;Approaches to global optimization problems are broadly divided into deterministic methods (with some convergence guarantee, but requiring some assumptions on the objective function and constraints) such as branch-and-bound and stochastic (or heuristic) methods such as simulated annealing and tabu search that may work well in practice but do not guarantee convergence to a globally optimal solution.&lt;/p&gt;&#xA;&lt;p&gt;Some references for deterministic methods:&lt;/p&gt;&#xA;&lt;p&gt;Floudas, Christodoulos A. Deterministic global optimization: theory, methods and applications. Vol. 37. Springer Science &amp;amp; Business Media, 2013.&lt;/p&gt;&#xA;&lt;p&gt;Horst, Reiner, and Hoang Tuy. Global optimization: Deterministic approaches. Springer Science &amp;amp; Business Media, 2013.&lt;/p&gt;&#xA;&lt;p&gt;A reference for stochastic/heuristic methods:&lt;/p&gt;&#xA;&lt;p&gt;Horst, Reiner, and Panos M. Pardalos, eds. Handbook of global optimization. Vol. 2. Springer Science &amp;amp; Business Media, 2013. (Volume 1 covers deterministic methods.)&lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2022-07-16T20:17:54.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25402" PostTypeId="2" ParentId="2552" CreationDate="2022-07-17T08:16:11.150" Score="1" Body="&lt;p&gt;The substance and spirit of the OP's Question has already been addressed, and here's an orthogonal, perhaps pedantic, point:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(A) Square 2 has sides that are longer than those of square 1 by a&#xA;factor of 1.1. Compare the areas of the squares.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;“&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is longer than &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by a factor of &lt;span class=&quot;math-container&quot;&gt;$1.1$&lt;/span&gt;” means that &lt;span class=&quot;math-container&quot;&gt;$$y-x=1.1x,$$&lt;/span&gt; since “longer &lt;strong&gt;by&lt;/strong&gt;” explicitly indicates subtraction, &lt;em&gt;independent&lt;/em&gt; of “a factor of &lt;span class=&quot;math-container&quot;&gt;$1.1$&lt;/span&gt;” being a multiplicative operation.&lt;/p&gt;&#xA;&lt;p&gt;Thus, to interpret “&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is longer than &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by a factor of &lt;span class=&quot;math-container&quot;&gt;$1.1$&lt;/span&gt;” as “&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$1.1$&lt;/span&gt; times as long as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” would be indefensible, &lt;em&gt;even&lt;/em&gt; more so than the near-universal habit of interpreting “&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$1.1$&lt;/span&gt; times longer than &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” as the latter.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(B) Square 2 has sides that are longer than those of square 1 by 10%.&#xA;Compare the areas of the squares.&lt;/p&gt;&#xA;&lt;p&gt;I find that many college students majoring in the sciences get A right, but for B they compute &lt;span class=&quot;math-container&quot;&gt;$0.10^2=0.01$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'll wager that, in fact, even fewer people get question A than question B correct, as most misunderstand Question A and think that its square 2 side is &lt;span class=&quot;math-container&quot;&gt;$1.1$&lt;/span&gt; times as long as its square 1 side.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-07-17T10:52:04.017" LastActivityDate="2022-07-17T10:52:04.017" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25403" PostTypeId="1" AcceptedAnswerId="25404" CreationDate="2022-07-19T05:38:02.953" Score="4" ViewCount="697" Body="&lt;p&gt;In texts which present the third isomorphism theorem:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(G/N)/(H/N) \cong G/H$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;the relationship between the entities is often seen presented in the form:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be normal subgroups of a group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$N \subseteq H$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$H/N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$G/N$&lt;/span&gt; and there is an isomorphism &lt;span class=&quot;math-container&quot;&gt;$(G/N)/(H/N) \cong G/H$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(EDIT: This is how it is presented in Allan Clark's &lt;em&gt;Elements of Abstract Algebra&lt;/em&gt; (1971), article &lt;span class=&quot;math-container&quot;&gt;$68$&lt;/span&gt;, for example.)&lt;/p&gt;&#xA;&lt;p&gt;I have been in discussion with someone who insists that it is mandatory to state that &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;, or the definition is invalid, because if &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is not a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$H/N$&lt;/span&gt; is undefined.&lt;/p&gt;&#xA;&lt;p&gt;My own personal preference is that the statement of the theorem is perfectly good as it stands, because if &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; are both normal in &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N \subseteq H$&lt;/span&gt;, then it logically follows that &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; and it is therefore unnecessary and inelegant to specifically state this in the exposition.&lt;/p&gt;&#xA;&lt;p&gt;What one would of course do is to derive the property during the course of the proof, or reference it as a pre-proven lemma.&lt;/p&gt;&#xA;&lt;p&gt;Indeed, we have, by definition of normal subgroup:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\forall g \in G: g H = H g$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\forall g \in G: g N = N g$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and so:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\forall g \in H: g N = N g$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But, it is argued, the above is non-trivial and far from obvious, and therefore it is important in the exposition to make that explicit statement: &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;To my mind it is better to merely state subsetness, and to &lt;em&gt;prove&lt;/em&gt; normal-subgroupness during the course of the proof itself. Otherwise it may leave the reader open to ask the question: &amp;quot;But what happens if &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a subset of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; but not necessarily a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;? What happens then?&amp;quot; without taking that mental step of realisation that you can't have &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; being a subset of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; under the above circumstances without &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; being a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In other words, I believe it important to establish that (on the surface) &amp;quot;more general&amp;quot; statement of the problem, and it is unhelpful to explicitly include one of the facts that is established during the course of the proof.&lt;/p&gt;&#xA;&lt;p&gt;What is the general consensus of opinion here?&lt;/p&gt;&#xA;" OwnerUserId="15290" LastEditorUserId="15290" LastEditDate="2022-07-19T05:56:46.000" LastActivityDate="2022-07-20T09:59:55.637" Title="Third isomorphism theorem: how important is it to state the relationship between subgroups?" Tags="|proofs|abstract-algebra|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25404" PostTypeId="2" ParentId="25403" CreationDate="2022-07-19T07:30:32.127" Score="7" Body="&lt;p&gt;I agree that &lt;span class=&quot;math-container&quot;&gt;$\text“\color{red}{N \text{ is a normal subgroup of a group }H}\text”$&lt;/span&gt; is a redundant/bogus &lt;em&gt;condition&lt;/em&gt; of the given theorem.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;To specify what is rightly a &lt;span class=&quot;math-container&quot;&gt;$\color{red}{\textrm{lemma}}$&lt;/span&gt; of the third isomorphism theorem as one of its conditions instead would, at the logical level—unnecessarily—&lt;strong&gt;weaken&lt;/strong&gt; the theorem, since &lt;span class=&quot;math-container&quot;&gt;$$∀g{,}h{,}n\Big(Aghn→Bghn\Big) \quad\implies\quad ∀g{,}h{,}n\Big((Aghn∧\color{red}{Lhn})→Bghn\Big)$$&lt;/span&gt; while&#xA;&lt;span class=&quot;math-container&quot;&gt;$$∀g{,}h{,}n\Big((Aghn∧\color{red}{Lhn})→Bghn\Big) \quad\kern.6em\not\kern-.6em\implies\quad ∀g{,}h{,}n\Big(Aghn→Bghn\Big).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Echoing your sentiment: thoughtful readers whose legitimate proof derives that lemma instead of invoking it by assumption, will be needlessly wondering whether their work and understanding is indeed rigorous and valid, and will be futilely trying to figure out where that condition is &lt;em&gt;required&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If one wishes to highlight that lemma &lt;strong&gt;within the statement&lt;/strong&gt; of the third isomorphism theorem, then, while it's not illogical to specify it as a condition, it is more honest, more helpful, and ultimately cleaner to specify it as a &lt;em&gt;consequence&lt;/em&gt; instead. Some suggestions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be normal subgroups of a group &lt;span class=&quot;math-container&quot;&gt;$G,$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N \subseteq H,$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a normal subgroup of a group &lt;span class=&quot;math-container&quot;&gt;$H.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$H/N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$G/N,$&lt;/span&gt; and there is an isomorphism &lt;span class=&quot;math-container&quot;&gt;$(G/N)/(H/N) \cong G/H.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be normal subgroups of a group &lt;span class=&quot;math-container&quot;&gt;$G,$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N \subseteq H;$&lt;/span&gt; so, &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a normal subgroup of a group &lt;span class=&quot;math-container&quot;&gt;$H.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$H/N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$G/N,$&lt;/span&gt; and there is an isomorphism &lt;span class=&quot;math-container&quot;&gt;$(G/N)/(H/N) \cong G/H.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be normal subgroups of a group &lt;span class=&quot;math-container&quot;&gt;$G,$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N \subseteq H.$&lt;/span&gt; (So, &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a normal subgroup of a group &lt;span class=&quot;math-container&quot;&gt;$H.)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$H/N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$G/N,$&lt;/span&gt; and there is an isomorphism &lt;span class=&quot;math-container&quot;&gt;$(G/N)/(H/N) \cong G/H.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be normal subgroups of a group &lt;span class=&quot;math-container&quot;&gt;$G,$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N \subseteq H.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a normal subgroup of a group &lt;span class=&quot;math-container&quot;&gt;$H,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$H/N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$G/N,$&lt;/span&gt; and there is an isomorphism &lt;span class=&quot;math-container&quot;&gt;$(G/N)/(H/N) \cong G/H.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-07-20T04:12:14.697" LastActivityDate="2022-07-20T04:12:14.697" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25405" PostTypeId="2" ParentId="11548" CreationDate="2022-07-19T12:14:25.333" Score="2" Body="&lt;p&gt;I studied in one of these programs from 1971-1975.  This program taught me how to think.  Some of the books can be found here: &lt;a href=&quot;https://eric.ed.gov/?q=Unified+Modern+Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://eric.ed.gov/?q=Unified+Modern+Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20082" LastActivityDate="2022-07-19T12:14:25.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25406" PostTypeId="2" ParentId="25403" CreationDate="2022-07-19T15:28:18.780" Score="3" Body="&lt;p&gt;I also agree with ryang that stating the normal subgroup status of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; as a condition would be weakening the theorem. However, explicating and demonstrating that fact somewhere in the vicinity of the proof seems like good form.&lt;/p&gt;&#xA;&lt;p&gt;Here's the theorem in Hungerford's &lt;em&gt;Abstract Algebra: An Introduction&lt;/em&gt;. Note that, like the OP's referenced text, it doesn't include the normal-subgroup status of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; explicitly in this statement:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;THEOREM 7.30&lt;/strong&gt; (THIRD ISOMORPHISM THEOREM FOR GROUPS) Let &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be normal subgroups of a group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$N \subseteq K \subseteq&#xA;G$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$K/N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$G/N$&lt;/span&gt;, and the quotient group&#xA;&lt;span class=&quot;math-container&quot;&gt;$(G/N)/(K/N)$&lt;/span&gt; is isomorphic to &lt;span class=&quot;math-container&quot;&gt;$G/K$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, the fact &lt;em&gt;is&lt;/em&gt; given in prelude text just &lt;em&gt;before&lt;/em&gt; the theorem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be normal subgroup of a group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;... Let &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; be a subgroup of&#xA;&lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; that contains &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is certainly a subgroup of &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;. Since&#xA;&lt;span class=&quot;math-container&quot;&gt;$Na = aN$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$a \in G$&lt;/span&gt;, then in particular &lt;span class=&quot;math-container&quot;&gt;$Na = aN$&lt;/span&gt; for every&#xA;&lt;span class=&quot;math-container&quot;&gt;$a \in K$&lt;/span&gt;. Hence &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$K/N$&lt;/span&gt; is a&#xA;group by Theorem 7.23.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This seems good to me.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2022-07-19T15:28:18.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25408" PostTypeId="2" ParentId="25403" CreationDate="2022-07-19T16:30:57.587" Score="4" Body="&lt;p&gt;I would definitely want to state that &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; before saying anything that treats &lt;span class=&quot;math-container&quot;&gt;$H/N$&lt;/span&gt; as a well-defined group.  That statement of normality could reasonably be part of the conclusion (not the hypothesis) of the theorem, or it could be presented earlier in the text (perhaps as an exercise, since it's pretty trivial).&lt;/p&gt;&#xA;&lt;p&gt;I would also explicitly state, perhaps as part of the theorem or perhaps as an addendum after the proof, the explicit form of the isomorphism. The explicit isomorphism, not just the existence of &lt;em&gt;some&lt;/em&gt; isomorphism, is important.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2022-07-19T16:30:57.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25409" PostTypeId="1" CreationDate="2022-07-19T22:58:11.690" Score="5" ViewCount="479" Body="&lt;p&gt;Most Standards (e.g. &lt;a href=&quot;http://www.corestandards.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Common Core Standards&lt;/a&gt; in the U.S.)&#xA;explicitly promote the teaching of “mathematical reasoning”, with a usually vague description of what exactly that is. Does anyone know of experimental evidence clearly linking such an intervention intended to teach mathematical reasoning (by any desired clear definition) to improved performance? (Preferably in algebra 1 and geometry, but I’ll take any K12 level.)&lt;/p&gt;&#xA;&lt;p&gt;You may be confused by my not specifying a specific performance measure. However, I can't be specific about the performance measure because I am asking for a pointer to research that uses whatever performance measure it uses. If I asked for a specific one, it would preclude a wide range of other possible performance measures. Examples might be specific math tests, SAT scores, college entrance, etc. Since I'm asking about research papers with &amp;quot;real&amp;quot; experimental results, most of these will have their own instrumented performance measures.&lt;/p&gt;&#xA;" OwnerUserId="14430" LastEditorUserId="6053" LastEditDate="2023-09-09T00:19:22.023" LastActivityDate="2023-09-12T03:14:03.570" Title="Experimental Evidence that Mathematical Reasoning is Important" Tags="|untagged|" AnswerCount="2" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="25410" PostTypeId="2" ParentId="25168" CreationDate="2022-07-20T03:52:22.107" Score="0" Body="&lt;p&gt;Pool Table. Chalkboard. Printer. Lots of space to write. And yes, the Hagoromo chalk both white and other colors. Maybe some bulletin boards for professors to post things. This is my unresearched opinion.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2022-07-20T03:52:22.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25412" PostTypeId="2" ParentId="25403" CreationDate="2022-07-20T09:59:55.637" Score="3" Body="&lt;p&gt;When using expressions whose meaning depends on properties, you should proclaim those properties before using the expression. Here something of the form &lt;span class=&quot;math-container&quot;&gt;$G/H$&lt;/span&gt; is generally a set of cosets, but if &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; then one can (and usually does) use the same expression to denote a group. Since one can only form the quotient by a (normal) subgroup (at least if the result is to be considered as a group), it is problematic to interpret &lt;span class=&quot;math-container&quot;&gt;$(G/N)/(H/N)$&lt;/span&gt; unless it it known that &lt;span class=&quot;math-container&quot;&gt;$H/N$&lt;/span&gt; denotes a group (and indeed a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$G/N$&lt;/span&gt;). I think the following formulation should satisfy even the pedantically precise:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be normal subgroups of a group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$N \subseteq H$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a normal subgroup of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$H/N$&lt;/span&gt; is a group, and this group is (naturally identified with) a normal subgroup of the quotient group &lt;span class=&quot;math-container&quot;&gt;$G/N$&lt;/span&gt;, and  the rule &lt;span class=&quot;math-container&quot;&gt;$gH\mapsto (gN)(H/N)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$g\in G$&lt;/span&gt; determines a well defined map &lt;span class=&quot;math-container&quot;&gt;$G/H\to (G/N)/(H/N)$&lt;/span&gt; that is an isomorphism of groups.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I've added the (naturally identified with) because from some points of view a quotient group constructs a group in a fundamentally different way than a subgroup (the former comes with a canonical projection morphism, the latter with an embedding morphism, for instance), though in this case the set-theoretic nuts and bolts that make up a group proper (underlying set, identity and inverse maps) are exactly the same for the two groups in question.&lt;/p&gt;&#xA;" OwnerUserId="682" LastActivityDate="2022-07-20T09:59:55.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25413" PostTypeId="1" AcceptedAnswerId="25418" CreationDate="2022-07-21T15:33:28.693" Score="5" ViewCount="2255" Body="&lt;p&gt;Background - I am tutoring a second year college sophomore for a class titled Single Variable Calculus, and whose curriculum looks to be similar to the AB calculus I tutor in my High School.&lt;/p&gt;&#xA;&lt;p&gt;We are on limits and L’Hôpital’s Rule, and I see this among the questions (note, all the worksheet questions are meant to be solved via L-H rule) - The instruction is&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Evaluate the following using L’Hôpital’s Rule&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to 0}\frac{\sin x}x= $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I recall, when subbing for a calc teacher, that this is a classic example of the use of the &amp;quot;squeeze theorem&amp;quot; aka &amp;quot;sandwich theorem&amp;quot;. Once it's proven, we'd go on to different arguments of Sine, practice a bit, then move on. It's introduced prior to L-H rule.&lt;/p&gt;&#xA;&lt;p&gt;Given the fast pace of my student self-studying and effort of remote teaching, I'm inclined to ignore this, and move on. My question is whether skipping over Squeeze Theorem is doing her a disservice, and should I (forgive the pun) squeeze it into our next session? At my HS, students have told me it feels like it's introduced, practiced for a few problems, but never seeing again.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-07-21T20:17:53.007" LastActivityDate="2022-07-23T22:41:09.657" Title="Skipping a calculus topic (squeeze theorem)" Tags="|calculus|" AnswerCount="4" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="25414" PostTypeId="1" AcceptedAnswerId="25416" CreationDate="2022-07-21T16:46:23.997" Score="9" ViewCount="360" Body="&lt;p&gt;I came across the idea of mixed up practice in Benedict Carey's book, &lt;em&gt;&lt;strong&gt;How We Learn&lt;/strong&gt;&lt;/em&gt;, in a chapter on the benefits of interleaving, particularly for learning Maths.&lt;/p&gt;&#xA;&lt;p&gt;For instance, in &amp;quot;blocked practice&amp;quot;, students might work on Maths problems in order, e.g. Problem A, then Problem B etc. following a pattern like &lt;em&gt;AABBCCDD&lt;/em&gt;. In mixed up or &amp;quot;interleaved practice&amp;quot;, the order of problems is varied, forming the pattern &lt;em&gt;ADCBCDBA&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Carey also cites several studies showing how interleaved practice outperforms blocked practice in skills outside Maths, from badminton to piano playing. These insights are fascinating, as he says &amp;quot;the science suggests that interleaving is, essentially, about preparing the brain for the unexpected.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;However, many psychological experiments seem artificial and contrived so I was happy to find a 2015 article in &lt;em&gt;&lt;strong&gt;&lt;a href=&quot;https://www.scientificamerican.com/article/the-interleaving-effect-mixing-it-up-boosts-learning/&quot; rel=&quot;noreferrer&quot;&gt;Scientific American&lt;/a&gt;&lt;/strong&gt;&lt;/em&gt; that went a bit further:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Over the past four decades, a small but growing body of research has found that interleaving often outperforms blocking for a variety of subjects, including sports and category learning. Yet there have been almost no studies of the technique in uncontrived, real world settings—until recently.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They mention one study in particular:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The three-month study involved teaching 7th graders slope and graph problems. Weekly lessons, given by teachers, were largely unchanged from standard practice. Weekly homework worksheets, however, featured an interleaved or blocked design. When interleaved, both old and new problems of different types were mixed together. Of the nine participating classes, five used interleaving for slope problems and blocking for graph problems; the reverse occurred in the remaining four. Five days after the last lesson, each class held a review session for all students. A surprise final test occurred one day or one month later. The result? When the test was one day later, scores were 25 percent better for problems trained with interleaving; at one month later, the interleaving advantage grew to 76 percent.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That's a big difference.&lt;/p&gt;&#xA;&lt;p&gt;I have two questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How widely is interleaving used in your classrooms whether primary, secondary or college/university?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;More broadly, do you know of any studies where interleaving has been rigorously tested in more real world settings?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19932" LastActivityDate="2022-07-22T17:04:39.737" Title="The Interleaving Effect: How widely is this used?" Tags="|mathematical-pedagogy|self-learning|general-pedagogy|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25415" PostTypeId="2" ParentId="25413" CreationDate="2022-07-21T17:04:27.273" Score="11" Body="&lt;p&gt;The squeeze theorem merely introduces a basic framework for future analytic thinking. For example, the inequality&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{k=1}^n\frac1k&amp;gt;\int_1^{n+1}\frac{1}{x}\, dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;can be established easily by drawing boxes that fit above the curve &lt;span class=&quot;math-container&quot;&gt;$y=\frac1x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Upon knowing that the integral goes to infinity like &lt;span class=&quot;math-container&quot;&gt;$\ln(x)$&lt;/span&gt;, we use &amp;quot;squeeze theorem thinking&amp;quot; to deduce that the harmonic series diverges: the series is squeezed between &lt;span class=&quot;math-container&quot;&gt;$\ln(n+1)$&lt;/span&gt; and infinity as &lt;span class=&quot;math-container&quot;&gt;$n\to\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The point is that introducing the squeeze theorem sets a basic principle. Then with neuroplasticity, one can reuse and invoke the basic concept in future contexts, perhaps without even realizing explicitly that we are using it!&lt;/p&gt;&#xA;&lt;p&gt;The rationale for including the squeeze theorem in Calculus I is not to narrowly give the learner a tool to evaluate certain limits &lt;em&gt;per se.&lt;/em&gt; Applying the squeeze theorem to limits such as &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to0}\frac{\sin(x)}{x}$&lt;/span&gt; is merely &lt;em&gt;action&lt;/em&gt; that helps settle the abstract concept so that it can grow in the brain to become a &lt;em&gt;schema.&lt;/em&gt; Analysts use the squeeze theorem subconsciously in all sorts of abstract situations, but they had to develop this capacity.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2022-07-22T17:59:17.780" LastActivityDate="2022-07-22T17:59:17.780" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25416" PostTypeId="2" ParentId="25414" CreationDate="2022-07-21T18:47:54.423" Score="10" Body="&lt;p&gt;In an answer to question 2, a few references I am familiar with are brought up at &lt;a href=&quot;https://www.learningscientists.org/blog/2016/8/11-1&quot; rel=&quot;noreferrer&quot;&gt;https://www.learningscientists.org/blog/2016/8/11-1&lt;/a&gt;. I was made aware of interleaving through this website. I will copy-paste their references here for access:&lt;/p&gt;&#xA;&lt;p&gt;(1) Rohrer, D., &amp;amp; Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35, 481-498.&lt;/p&gt;&#xA;&lt;p&gt;(2) Rohrer, D. (2012). Interleaving helps students distinguish among similar concepts. Educational Psychology Review, 24, 355-367.&lt;/p&gt;&#xA;&lt;p&gt;Two other studies from the comments:&lt;/p&gt;&#xA;&lt;p&gt;(3) Rohrer, D., Dedrick, R., Burchess, K. (2014). The Benefit of Interleaved Mathematics Practice Is Not Limited to Superficially Similar Kinds of Problems&lt;/p&gt;&#xA;&lt;p&gt;(4) Rohrer, D., Dedrick, R., Hartwig, M. K. (2020) The Scarcity of Interleaved Practice in Mathematics Textbooks&lt;/p&gt;&#xA;&lt;p&gt;In an answer to question 1 which is more personal, I have incorporated interleaving deliberately in both high school and in college level instruction.&lt;/p&gt;&#xA;&lt;p&gt;Generally when I assign practice problems at any level, I make an effort to alternate the problem type, even while moving from &amp;quot;easy&amp;quot; to &amp;quot;hard.&amp;quot; Too many of the same problems in a row lead to student complacency. When learning about the chain rule, for example, it is sensible to include some problems which should not use the chain rule, and ask students to evaluate for multiple problems &amp;quot;Is the chain rule appropriate here, or not?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;At the high school level, when learning about trigonometry in high school geometry, we begin by learning to solve for missing sides of right triangles. Instead of grouping the problems where you solve for a leg and the problems where you solve for a hypotenuse together, I randomly alternate solving for leg and solving for side in their problem sets. This way, students have to contemplate the setup of their equations. Once we learn to solve for angles, the problem sets begin to interleave 'solving for side' and 'solving for angle' as well, and even some where you don't even need to use sine and cosine (and you can return back to the Pythagorean Theorem!).&lt;/p&gt;&#xA;&lt;p&gt;I motivate students to follow these forms of assignments by reminding them that in exams and in real life, you are not told what type of problem you are solving: determining whether you have the tools to solve a particular problem, and/or whether the tools you have are appropriate, is an important part of learning how to problem-solve.&lt;/p&gt;&#xA;&lt;p&gt;A place of common availability of mixed practice problems are websites such as IXL (&lt;a href=&quot;https://www.ixl.com/&quot; rel=&quot;noreferrer&quot;&gt;https://www.ixl.com/&lt;/a&gt;). These sites have batched practice and mixed problem sets. When assigning problems through these websites, I prioritize including the mixed problem sets. Other software has also always included this as an option (e.g.&#xA;kuta software &lt;a href=&quot;https://www.kutasoftware.com/&quot; rel=&quot;noreferrer&quot;&gt;https://www.kutasoftware.com/&lt;/a&gt; gives the opportunity to scramble problems within the set and between sets). I just downloaded the free trial to verify this. Personally, I currently assign homework through WebWork, which requires more manual attention to this detail.&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2022-07-22T17:04:39.737" LastActivityDate="2022-07-22T17:04:39.737" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25417" PostTypeId="2" ParentId="25413" CreationDate="2022-07-21T19:00:13.060" Score="1" Body="&lt;p&gt;Unless you're training a superstar, and college sophomore taking calc 1 is a clue here, I would not cover material that the regular teacher has made the choice not to cover.  And there are always limits on time and ability, thus choices on what to cover.&lt;/p&gt;&#xA;&lt;p&gt;Also, can the limit be found using lhopital?  Which student has probably had?  Take the derivative of top and bottom. And you get 1/cos0. Or 1/1 or 1.  This lesson was specifically on lhopital. This drill sheet expects practice with a specific method.&lt;/p&gt;&#xA;" OwnerUserId="20089" LastEditorUserId="11" LastEditDate="2022-07-23T22:41:09.657" LastActivityDate="2022-07-23T22:41:09.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25418" PostTypeId="2" ParentId="25413" CreationDate="2022-07-21T19:38:33.920" Score="7" Body="&lt;p&gt;The very limit you have chosen as an example shows the necessity of the squeeze theorem.&lt;/p&gt;&#xA;&lt;p&gt;When deriving the fact that the derivative of sine is cosine, you first use the angle sum formula for sine, which then reduces the computation to the derivative of sine at 0.  This is the limit you have chosen as an example.&lt;/p&gt;&#xA;&lt;p&gt;So you really have no choice: to convince someone that the derivative of sine is cosine, you need the squeeze theorem.  You can, of course, just skip the justification but that defeats the point of a mathematics class for me.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-07-21T19:38:33.920" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25419" PostTypeId="1" AcceptedAnswerId="25420" CreationDate="2022-07-22T06:47:07.140" Score="11" ViewCount="1111" Body="&lt;p&gt;I know that MOOCs were generally unsuccessful. However, I felt that M2O2C2 in Coursera was a great (at least my favorite) course and it's a pity it was removed. Does anyone have any info - will it ever be back under any format?&lt;/p&gt;&#xA;" OwnerUserId="20091" LastActivityDate="2022-07-22T14:06:38.773" Title="Why Massively Multivariate open online calculus class (M2O2C2) on Coursera was discontinued?" Tags="|calculus|vector-calculus|online|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25420" PostTypeId="2" ParentId="25419" CreationDate="2022-07-22T14:06:38.773" Score="15" Body="&lt;p&gt;I was the primary author of the M2O2C2 content.  So glad that you enjoyed the course!&lt;/p&gt;&#xA;&lt;p&gt;My friend Jim Fowler wrote the backend code (called Ximera).  All of the files for M2O2C2 were in an earlier, not backwards compatible, version of Ximera. So even getting the content to function on the web would require considerable work.  I also think Jim is the only person who actually knows how to get the thing to work...&lt;/p&gt;&#xA;&lt;p&gt;The tex files are all available on github:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://github.com/kisonecat/m2o2c2&quot; rel=&quot;noreferrer&quot;&gt;https://github.com/kisonecat/m2o2c2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Please feel free to download all the files and mess with them until they compile as a pdf.  I did enjoy writing up my unique perspective on multivariable Calculus for this course, but I have not had time or motivation to salvage it.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-07-22T14:06:38.773" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25421" PostTypeId="2" ParentId="25413" CreationDate="2022-07-22T15:30:36.097" Score="8" Body="&lt;p&gt;Most of the other answers in this thread focus on the mathematics.  This is appropriate, as this is a Q&amp;amp;A site for mathematics educators.  However, I suspect that the question being answered (&amp;quot;Should I teach the squeeze theorem?&amp;quot;) has already been addressed here many times.  The distinguishing question here seems to be &amp;quot;Should I, as a tutor, teach a tutee the squeeze theorem when the primary instructor has omitted it?&amp;quot;  While this question might be more appropriate for the &lt;a href=&quot;https://academia.stackexchange.com/&quot;&gt;Academia SE&lt;/a&gt;, it doesn't seem off-topic here.&lt;/p&gt;&#xA;&lt;h3&gt;What is your role?&lt;/h3&gt;&#xA;&lt;p&gt;Typically, the role of a tutor is to provide supplemental coaching to a student who is taking a course from a primary instructor.  It is the job of the primary instructor to establish the curriculum, the grading schemata, the pace, etc.  The job of the tutor is to review the material presented by the primary instructor in order to prepare the student for assessments.&lt;/p&gt;&#xA;&lt;h3&gt;Are &lt;em&gt;you&lt;/em&gt; doing harm by omitting the squeeze theorem?&lt;/h3&gt;&#xA;&lt;p&gt;No.&lt;/p&gt;&#xA;&lt;p&gt;As a tutor, you serve your tutees by preparing them to perform well on the assessments provided by their primary instructor.  When you help them with material that is likely to be on those assessments, you are doing your job.  If you spend time on material that is not likely to be on those assessments, you are, perhaps, helping the student to learn more, but may be harming them in the sense that they may perform less well on assessments.  Coving extra material is, likely, neutral at best.&lt;/p&gt;&#xA;&lt;h3&gt;Is the &lt;em&gt;student&lt;/em&gt; being harmed by the omission of the squeeze theorem?&lt;/h3&gt;&#xA;&lt;p&gt;In my opinion, yes.&lt;/p&gt;&#xA;&lt;p&gt;As has been pointed out in the comments and in other answers, the squeeze theorem is a fundamental result in analysis (or, perhaps more foundational, is the result that &lt;span class=&quot;math-container&quot;&gt;$f(x) \le g(x)$&lt;/span&gt; implies that &lt;span class=&quot;math-container&quot;&gt;$\lim f(x) \le \lim g(x)$&lt;/span&gt;).  Skipping this result does a disservice to the mathematics, and has the potential to create a kind of &amp;quot;cargo cult&amp;quot; version of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Thus, in my opinion, if an instructor chooses to omit the squeeze theorem, they are doing a disservice to their students.  But that is on the instructor, not you.&lt;/p&gt;&#xA;&lt;h3&gt;What should you do?&lt;/h3&gt;&#xA;&lt;p&gt;Again, this is probably a matter of opinion, but:  your role is to prepare your tutees for the assessments their instructor is likely to prepare.  This instructor has skipped the squeeze theorem, hence you should probably also skip it, or spend only minimal time on it.&lt;/p&gt;&#xA;&lt;p&gt;You also need to be careful to respect the authority of the instructor.  You and the instructor should appear to be a united team in front of the students.  Be careful not to criticize the instructor in earshot of your tutees.  Doing so only degrades the relationship between the instructor and their students, which is likely to make the overall learning environment worse.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, it might be worth talking to the instructor.  I can imagine many possible results, including:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students are often unreliable reporters when it comes to describing what has been taught.  Perhaps the instructor will tell you that they &lt;em&gt;did&lt;/em&gt; spend time on the squeeze theorem, and the student has simply forgotten (in which case, you &lt;em&gt;really&lt;/em&gt; need to spend some time on it).&lt;/li&gt;&#xA;&lt;li&gt;Perhaps the instructor had a really good reason for omitting the squeeze theorem.  I have difficulty imagining what that reason is, but you might find that they have a persuasive argument.&lt;/li&gt;&#xA;&lt;li&gt;Maybe the instructor just forgot about it (e.g. if they are teaching multiple sections of the course, they may have gotten confused about what was said to one group of students vs another).  They may welcome the feedback, and take it as an opportunity to fill a gap.&lt;/li&gt;&#xA;&lt;li&gt;Or the instructor could be a real a-hole who doesn't really know what they are doing, skipped the theorem because it is hard and students often struggle with it and they want to make sure that their teaching evaluations are good.  That would suck, but such is life.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Whatever the case, having a line of communication between instructor and tutor can be valuable.  If it is possible for you to talk to this instructor, I would advise you to do so.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2022-07-22T15:30:36.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25422" PostTypeId="2" ParentId="17399" CreationDate="2022-07-23T23:25:18.440" Score="0" Body="&lt;p&gt;As others have said, it's because historically we did not need to have a rigorous definition of the real numbers in order to &amp;quot;prove&amp;quot; theorems about (functions of) them. Therefore, depending on how serious the student is, they may not need it either. I could see it being useful for very serious students who are on the track to becoming career mathematicians, though.&lt;/p&gt;&#xA;" OwnerDisplayName="user20098" LastActivityDate="2022-07-23T23:25:18.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25423" PostTypeId="2" ParentId="18297" CreationDate="2022-07-25T06:55:35.560" Score="1" Body="&lt;p&gt;Question was:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Find the Maclaurin series of &lt;span class=&quot;math-container&quot;&gt;$\ln{(2+2x)}$&lt;/span&gt; up to the term in &lt;span class=&quot;math-container&quot;&gt;$x^3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Student A: &lt;span class=&quot;math-container&quot;&gt;$\ln{(2+2x)}=\ln{(2(1+x))}=\ln2+x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-...$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Student B: &lt;span class=&quot;math-container&quot;&gt;$\ln{(2+2x)}=\ln{(1+(1+2x))}=(1+2x)-\dfrac{(1+2x)^2}{2}+\dfrac{(1+2x)^3}{3}+...$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Of course, Student A is correct. (Student B has found a Taylor series but not a Maclaurin series.)&lt;/p&gt;&#xA;&lt;p&gt;But it is interesting to equate the coefficients in the two answers (temporarily ignoring the fact that the two series have different intervals of convergence). We get the following results:&lt;/p&gt;&#xA;&lt;p&gt;Equate coefficients of &lt;span class=&quot;math-container&quot;&gt;$x^1$&lt;/span&gt;:  &lt;span class=&quot;math-container&quot;&gt;$2-2+2-2+2-...=1$&lt;/span&gt;, that is, &lt;span class=&quot;math-container&quot;&gt;$1-1+1-1+1-...=\dfrac12$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Equate coefficients of &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;:  &lt;span class=&quot;math-container&quot;&gt;$-2+4-6+8-10+...=-\dfrac{1}{2}$&lt;/span&gt;, that is, &lt;span class=&quot;math-container&quot;&gt;$1-2+3-4+5-...=\dfrac14$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Equate coefficients of &lt;span class=&quot;math-container&quot;&gt;$x^3$&lt;/span&gt;:  &lt;span class=&quot;math-container&quot;&gt;$\dfrac43(2-6+12-20+30)$&lt;/span&gt;, that is, &lt;span class=&quot;math-container&quot;&gt;$1-3+6-10+15-...=\dfrac18$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And so on.&lt;/p&gt;&#xA;" OwnerUserId="20101" LastActivityDate="2022-07-25T06:55:35.560" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25425" PostTypeId="2" ParentId="18297" CreationDate="2022-07-25T14:00:42.737" Score="6" Body="&lt;p&gt;Question was:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Evaluate &lt;span class=&quot;math-container&quot;&gt;$\int{(\cos^3{x}-\cos^5{x})}\text{d}x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Student (desperately weak at math) simply wrote: &lt;span class=&quot;math-container&quot;&gt;$\dfrac{1}{3}\sin^3{x}-\dfrac{1}{5}\sin^5{x}+c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I thought, &amp;quot;&lt;em&gt;She's using a wrong method, badly&lt;/em&gt;&amp;quot;... then I realized that her answer was correct.&lt;/p&gt;&#xA;&lt;p&gt;It turns out that if you replace the 3 and the 5 (in both the question and answer) with any other numbers, it doesn't work.&lt;/p&gt;&#xA;" OwnerUserId="20101" LastActivityDate="2022-07-25T14:00:42.737" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25426" PostTypeId="2" ParentId="18395" CreationDate="2022-07-25T17:13:04.460" Score="6" Body="&lt;p&gt;David Clark and Robert Talbert have taken a lead experimenting with ungrading in their mathematics classes, and have started a substack, &lt;a href=&quot;https://gradingforgrowth.com&quot; rel=&quot;noreferrer&quot;&gt;Grading for Growth&lt;/a&gt;, for folks to write their reflections and ideas. Most of the posts are written by David and/or Robert and so will, at least implicitly, be about their experiences specific to teaching math.&lt;/p&gt;&#xA;&lt;p&gt;Here's one big take-away from their writing that seems particularly pertinent to math class: Ungrading works best when students are capable of reflecting on their own understanding. Without grades &lt;em&gt;telling&lt;/em&gt; students how they'r doing, they've got to be able to evaluate their progress (mostly) themselves, which can be tough; years of grades-based schooling has nerfed their metacognative abilities. Besides that, I think it's tougher to self-evaluate one's understanding of mathematical ideas more than ideas from other fields. As an instructor it's necessary to be aware of this, and to possibly spend class time having students practice reflecting on their understanding, building their metacognition.&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2022-07-25T17:13:04.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25428" PostTypeId="1" AcceptedAnswerId="25430" CreationDate="2022-07-27T09:45:29.510" Score="0" ViewCount="233" Body="&lt;p&gt;I have completed my Maths textbook chapter trigonometry but still my textbook is not sufficient for  exam so if there is any suggestion regarding  trigonometry + calculus resource book pls feel free to suggest me thanks&lt;/p&gt;&#xA;" OwnerUserId="20117" LastEditorUserId="11" LastEditDate="2022-07-28T05:19:51.207" LastActivityDate="2022-07-28T06:37:44.323" Title="Best book for trigonometry and calculus?" Tags="|secondary-education|book|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25429" PostTypeId="1" AcceptedAnswerId="25740" CreationDate="2022-07-27T21:26:13.423" Score="11" ViewCount="128" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\text{Hi,}$&lt;/span&gt; new contributor here. I am a teacher's assistant for a class in a Math Circle program for mathematically advanced high school (and occasionally a few middle school) students. I was planning to organize a contest for the students centered around the concepts of game theory and metastrategies, such that the &amp;quot;correct&amp;quot; answer will to some degree depend on how other students (and oneself) respond. Here are a few examples of problems that will almost certainly be on the contest (some devised by myself, some taken from contests I've seen, some previously known):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Pick any real number &lt;span class=&quot;math-container&quot;&gt;$n, 0\le n\le 100$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be the average of everyone’s picks. The winner will be anyone who minimizes &lt;span class=&quot;math-container&quot;&gt;$\big|\frac23N-n\big|$&lt;/span&gt;. (Known)&lt;/li&gt;&#xA;&lt;li&gt;Pick any positive integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be the number of distinct answers across all students. The winner will be anyone who minimizes &lt;span class=&quot;math-container&quot;&gt;$|N-n|$&lt;/span&gt;. (From a contest)&lt;/li&gt;&#xA;&lt;li&gt;Pick any positive integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. The winner will be whoever picks the least number nobody else picks. (Mine)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To find problems for my contest, I have picked over several contests I've taken that included problems with this general concept and spent a while trying to come up with further questions with interesting consequences and intended emergent behaviors, yet I seem to be coming up a few questions short of the length I had hoped for. So my request is for any (old or new) problems in this sort of mold- wherein the correct answer depends on everyone's answers and the statement can be easily understood by bright high schoolers.&lt;/p&gt;&#xA;&lt;p&gt;If this is the wrong setting in which to ask this question, please let me know a better forum and I will happily ask there instead. Any and all criticism is welcomed.&lt;/p&gt;&#xA;" OwnerUserId="20118" LastEditorUserId="5321" LastEditDate="2022-07-28T15:37:47.203" LastActivityDate="2022-10-31T03:58:01.173" Title="Ideas for game-theoretic meta contest problems for HS students" Tags="|games|math-contests|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25430" PostTypeId="2" ParentId="25428" CreationDate="2022-07-28T06:37:44.323" Score="0" Body="&lt;p&gt;Generally speaking I find the free books at &lt;a href=&quot;https://openstax.org/subjects/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax Math&lt;/a&gt; to be pretty comprehensive.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2022-07-28T06:37:44.323" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25431" PostTypeId="1" AcceptedAnswerId="25432" CreationDate="2022-07-29T04:31:38.900" Score="4" ViewCount="314" Body="&lt;p&gt;I am teaching high school precalculus and have a textbook that gives the following preamble to its rules for finding horizontal and slant asymptotes of rational functions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$f(x)=\frac{a(x)}{b(x)}$&lt;/span&gt; is a rational function, where &lt;span class=&quot;math-container&quot;&gt;$a(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b(x)$&lt;/span&gt; are polynomials, &lt;span class=&quot;math-container&quot;&gt;$b(x)\neq 0$&lt;/span&gt;, &lt;strong&gt;and &lt;span class=&quot;math-container&quot;&gt;$a(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b(x)$&lt;/span&gt; have no common factors other than &lt;span class=&quot;math-container&quot;&gt;$\pm 1$&lt;/span&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The text then lists the standard rules for finding horizontal and oblique asymptotes. (Asymptote at &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$n &amp;lt; m$&lt;/span&gt;, etc.)&lt;/p&gt;&#xA;&lt;p&gt;My question: Is it necessary for the polynomials to be fully simplified (no common factors other than &lt;span class=&quot;math-container&quot;&gt;$\pm 1$&lt;/span&gt;) to apply these rules? It seems to me that this does not affect the limit behavior. Am I missing something?&lt;/p&gt;&#xA;" OwnerUserId="20130" LastActivityDate="2022-08-02T10:47:22.773" Title="Do horizontal asymptote rules require function to be fully simplified?" Tags="|precalculus|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25432" PostTypeId="2" ParentId="25431" CreationDate="2022-07-29T19:22:11.673" Score="8" Body="&lt;p&gt;Summarizing from comments on the OP:&lt;/p&gt;&#xA;&lt;p&gt;For horizontal and slant asymptotes there is no need to stipulate that the numerator and denominator share no common factors.&lt;/p&gt;&#xA;&lt;p&gt;For vertical asymptotes you do need this stipulation so that you can distinguish between removable discontinuities and vertical asymptotes.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="2083" LastEditDate="2022-08-02T10:47:22.773" LastActivityDate="2022-08-02T10:47:22.773" CommentCount="0" CommunityOwnedDate="2022-07-29T19:22:11.673" ContentLicense="CC BY-SA 4.0" />
  <row Id="25434" PostTypeId="1" AcceptedAnswerId="25436" CreationDate="2022-07-30T16:12:07.797" Score="22" ViewCount="8168" Body="&lt;p&gt;My 13yr old has leapt forward in math during the pandemic. He's taking discrete math right now but is running into a bit of a wall with proofs. I have a feeling he needs to find reasons why they can be important and useful for him, and doing so will help motivate him to find a way to understand how to master them. I'm dyslexic and mathlexic, proofs are not my forte, so reaching out to you all on Stack for insightful ideas on how to help him make that 'connection'.&lt;/p&gt;&#xA;&lt;p&gt;What thoughts do you have about why they're important and how they're useful, especially for inventors, engineers, makers, software engineers, and philosophy?&lt;/p&gt;&#xA;" OwnerUserId="20151" LastEditorUserId="20151" LastEditDate="2022-07-30T19:52:16.287" LastActivityDate="2022-12-01T17:33:12.357" Title="Math Proofs - why are they important and how are they useful?" Tags="|proofs|discrete-math|" AnswerCount="19" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="25435" PostTypeId="2" ParentId="25431" CreationDate="2022-07-30T23:16:32.080" Score="0" Body="&lt;p&gt;One of your standard rules will be that if the degrees of your polynomials match, then there is a horizontal asymptote found by a simple calculation.&lt;/p&gt;&#xA;&lt;p&gt;However, consider the function &lt;span class=&quot;math-container&quot;&gt;$f(x) = \frac{1}{1}$&lt;/span&gt;. Would you say the graph of this function has a &amp;quot;horizontal asymptote at &lt;span class=&quot;math-container&quot;&gt;$y=1$&lt;/span&gt;?&amp;quot; I would not. Maybe you would?&lt;/p&gt;&#xA;&lt;p&gt;It is possible that the author is trying to avoid such issues entirely by stipulating that there are no common factors? It's hard to say for sure.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2022-08-01T04:12:38.263" LastActivityDate="2022-08-01T04:12:38.263" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25436" PostTypeId="2" ParentId="25434" CreationDate="2022-07-31T14:53:03.703" Score="31" Body="&lt;p&gt;Proofs are important because proofs are just &lt;em&gt;understanding how we know that something is true&lt;/em&gt;.  This is what mathematics is all about!&lt;/p&gt;&#xA;&lt;p&gt;What if all you care about is using the results of mathematics:  you don't particularly care about why it works, just how to use it.  Should you still care about proofs?&lt;/p&gt;&#xA;&lt;p&gt;Yes!&lt;/p&gt;&#xA;&lt;p&gt;I am teaching infinite series in calculus at the moment, so I will use an example from this subject to illustrate some reasons why.&lt;/p&gt;&#xA;&lt;p&gt;One result which is important in the study of infinite series is that if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; are two real numbers then:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a+ar+ar^2+ar^3+ar^4+\dots = \frac{a}{1-r} \textrm{ if } -1&amp;lt;r&amp;lt;1.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can memorize this result without understanding the proof.   However, I will illustrate that understanding the proof gives several practical benefits to the user of mathematics.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Understanding the proof allows us to utilize the intermediate results which lead to the proof.&lt;/p&gt;&#xA;&lt;p&gt;Part of the proof of this theorem involves computing the finite sum&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;	a+ar +ar^2 + ar^3 + \dots + ar^N = a\frac{1-r^{N+1}}{1-r}.&#xA;	$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This result is useful in its own right.  For instance, this formula is used when calculating savings with recurring regular deposits (&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;) with interest &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; over a certain number of deposits &lt;span class=&quot;math-container&quot;&gt;$N+1$&lt;/span&gt;.  Note that this intermediate result is applicable even when &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is not in the restricted range &lt;span class=&quot;math-container&quot;&gt;$-1&amp;lt;r&amp;lt;1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Understanding the proof makes it easier to remember the full statement of the theorem, including all of the conditions.&lt;/p&gt;&#xA;&lt;p&gt;If you understand the proof of this theorem, then you know the reason &lt;strong&gt;why&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is restricted to between &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is that &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\lim_{N \to \infty} r^{N+1} = 0$&lt;/span&gt;  if and only if &lt;span class=&quot;math-container&quot;&gt;$-1&amp;lt;r&amp;lt;1$&lt;/span&gt;.  By understanding this reasoning you are building a more robust network of facts in your brain.  Placing the fact (&lt;span class=&quot;math-container&quot;&gt;$-1&amp;lt;r&amp;lt;1$&lt;/span&gt;) in the context of a story (the proof) makes it much easier to remember the fact.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Understanding the proof allows you to generalize the ideas in the proof to new contexts.&lt;/p&gt;&#xA;&lt;p&gt;You might need to apply this formula in a new context:  for instance, what if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; are complex numbers?  If you only know the theorem in the context of real numbers, and you do not know the proof, then you may not be sure whether you can apply the formula in the context of complex numbers.  You would need to consult an external authority to get confirmation before using it.  However, if you know the proof you can easily see that &amp;quot;the proof goes through&amp;quot; for complex numbers as well.  The only modification needed is that we need &lt;span class=&quot;math-container&quot;&gt;$|r|&amp;lt;1$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$-1&amp;lt;r&amp;lt;1$&lt;/span&gt; (which would be meaningless for complex numbers).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Understanding the proof builds a collection of tricks and ideas which can be applied &amp;quot;far away&amp;quot; from the original context.&lt;/p&gt;&#xA;&lt;p&gt;There is a cool trick to get the finite sum result I mentioned above:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$S = a+ar +ar^2 + ar^3 + \dots + ar^N$&lt;/span&gt;.  Then multiply both sides by &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$rS = ar + ar^2 + ar^3 + \dots +ar^N +ar^{N+1}$&lt;/span&gt;.  Now &lt;span class=&quot;math-container&quot;&gt;$S - rS = a - ar^{N+1}$&lt;/span&gt; since all the other terms cancel.  Solving for &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; yields &lt;span class=&quot;math-container&quot;&gt;$S  = a \frac{1-r^{N+1}}{1-r}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Learning and understanding this proof gives you an idea which can be used in other contexts.  The idea is something like: &amp;quot;Sometimes I have a bunch of things added together in a sequence.  If I can shift all of the terms by one, I can subtract to cancel all but a few.&amp;quot;  More generally the trick might be: &amp;quot;If something is &lt;em&gt;nearly&lt;/em&gt; invariant under some group action, then applying the group action and subtracting should 'isolate' the non-invariant part.&amp;quot;  This idea can be applied in many more situations than just geometric sums.&lt;/p&gt;&#xA;&lt;p&gt;For example, when estimating integrals a similar idea comes up when you look at the difference between the left and right Riemann sum for an increasing function.  The left and right Riemann sums have all the same terms except for the first and last term.  So subtracting gives an easy bound on the error between the integral and the Riemann sum approximation in this case.&lt;/p&gt;&#xA;&lt;p&gt;Here we have used the same idea in a completely different context.  Without knowing the proof, I wouldn't have this idea in my arsenal.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2022-11-28T14:48:34.917" LastActivityDate="2022-11-28T14:48:34.917" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25437" PostTypeId="2" ParentId="25434" CreationDate="2022-07-31T15:52:47.040" Score="9" Body="&lt;p&gt;As in @StevenGubkin's answer, indeed, &amp;quot;proofs&amp;quot; are (fairly &lt;em&gt;definitive&lt;/em&gt;) explanations &lt;em&gt;why&lt;/em&gt; something is true.&lt;/p&gt;&#xA;&lt;p&gt;I would agree/concede that mathematics is very useful to a variety of people &lt;em&gt;without&lt;/em&gt; knowing the proofs. That is, in fact, the &amp;quot;what&amp;quot; of mathematics is already crazily useful...&lt;/p&gt;&#xA;&lt;p&gt;... which is why sometimes we'd care about the &amp;quot;why&amp;quot;. Sometimes, to have more confidence in the &amp;quot;what&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;And, yes, I concede/agree that &lt;em&gt;some&lt;/em&gt; pictures of &amp;quot;proof&amp;quot; are very stylized, and/or not interesting. E.g., although there is some interest in seeing that we can prove &lt;span class=&quot;math-container&quot;&gt;$1+1=2$&lt;/span&gt; from very primitive assumptions (as in Peano and Russell-Whitehead), the added knowledge is not that &amp;quot;finally we know that &lt;span class=&quot;math-container&quot;&gt;$1+1=2$&lt;/span&gt;&amp;quot;, but something about formal logical deduction systems, with that merely as a test case.&lt;/p&gt;&#xA;&lt;p&gt;Especially when doing somewhat fancier things, we can imagine that we are perhaps stressing the ideas beyond their original operating range. :)  So, although we might have many known examples where we can &amp;quot;see&amp;quot; the truth and don't need a dressed-up &amp;quot;proof&amp;quot;, in more extreme cases a proof can be critically reassuring.&lt;/p&gt;&#xA;&lt;p&gt;An example of which I'm fond (which also illustrates the human-fallibility aspect) is to test whether multiplication of large integers is commutative. Yes, we &amp;quot;know&amp;quot; that it is, and can also &amp;quot;prove&amp;quot; this from Peano or other foundations. But, in contrast, if we just &amp;quot;test&amp;quot;, by hand-multiplying two random-ish 20-digit decimal expansions, most likely we will seem to find that &lt;span class=&quot;math-container&quot;&gt;$a\cdot b\not=b\cdot a$&lt;/span&gt;. :) Duh, because out of the 400 single-digit multiplications and lots of additions, the chances are high that we'll make a mistake. :) One of the aspects, too, is the brittleness of something like this computation. It's not self-correcting. In contrast, a &lt;em&gt;good&lt;/em&gt; proof style (probably not the toooo stylized/caricatured ones) is that it is robust, in the sense that we can &lt;em&gt;sensibly&lt;/em&gt; review it, and also that the pieces fit together in a mutually-correcting way (in human psychological/linguistic terms).&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-07-31T15:52:47.040" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25438" PostTypeId="2" ParentId="25434" CreationDate="2022-07-31T16:19:13.867" Score="11" Body="&lt;p&gt;Along similar lines as the previous answers. Just as a computer program needs to be tested, to make sure it works correctly, a mathematical result needs to be proved to make sure it really does work.&lt;/p&gt;&#xA;&lt;p&gt;When you are trying to get clearer about relationships between things like numbers and shapes, you might look for a pattern. And if you find a pattern that fits, you might think you've solved the problem. But you don't know that you've solved it until you've given a reason (proof) for that solution to fit that relationship. Perhaps this is too vague. Here's the example I'm thinking of. It starts out like a puzzle.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;On a circle, put some points. Connect each point to every other point with straight lines. How many regions can you create for n points?&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;One point can't connect to any others. There is just 1 region in the circle.&lt;/li&gt;&#xA;&lt;li&gt;Two points connect to each other with one line, making 2 regions.&lt;/li&gt;&#xA;&lt;li&gt;Three points connect with 3 lines, making 4 regions.&lt;/li&gt;&#xA;&lt;li&gt;Four points connect with 6 lines, making 8 regions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Well, there's an obvious pattern here. The number of regions is doubling. And yes, it still doubles again for 5 points. But sadly (or amazingly, depending on your perspective) when we get to 6 points, the pattern breaks down.&lt;/p&gt;&#xA;&lt;p&gt;We need to know where the pattern comes from to know that the pattern will always work. And finding where it comes from is pretty much the same as proving that the pattern will work.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-07-31T22:24:41.537" LastActivityDate="2022-07-31T22:24:41.537" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25439" PostTypeId="2" ParentId="25434" CreationDate="2022-07-31T18:31:02.393" Score="18" Body="&lt;p&gt;I am an engineer.&lt;/p&gt;&#xA;&lt;p&gt;Proofs are important to &amp;quot;get&amp;quot; engineering, but are not directly used. I see three aspects of learning proofs as important: Logic, Process,  and Ontology.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Logic&lt;/strong&gt; is the simplest: you need to know how logic works. It's foundational. For example, you will not be able to usefully apply even very basic statistics without being able to apply a clear, logical chain of reasoning. If you try something complex, you are hosed. That example is minor and applicable to every field you mentioned; it is even more foundational for software and philosophy.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Process&lt;/strong&gt; is harder: as an engineer, in anything but the most trivial tasks, you will need to rapidly learn and manipulate processes and systems. Proofs model this: you take inputs that are given and evaluate and describe the implications. You also Voltron together smaller logic-blobs into larger logic-blobs to reach a defined outcome, which nicely models the engineering design process.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Ontology&lt;/strong&gt;: engineering is all approximations and uncertainty. You need to be able to bound and contain that uncertainty and to recognize which approximations are close enough. To do that you need to not only know stuff, but also understand how much you can rely on the stuff you know. Proofs are the high end of that; a triangle in a plane &lt;strong&gt;always&lt;/strong&gt; has 180 degrees. Compare that with e.g. Young's Modulus for a material, which is experimentally determined, or the maximum load on something, which is measured or assumed.&lt;/p&gt;&#xA;&lt;p&gt;None of those say &amp;quot;doing proofs&amp;quot; is critical to engineering, because it's not. By the same token, weight training isn't part of football and singing scales isn't part of vocal performance. Proofs will train your mind for any technical field in a way that is very hard to replicate.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Now all that said, geometric proofs are also independently great for two reasons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You organically learn all the geometry that you need to know anyway (otherwise you would have to memorize it, which is terrible). E.g. I have no memory of what the angle on a regular dodecagon is, but I can figure it out with scratch paper faster than I can look it up (150 degrees; approx. 20 seconds). That's useful for everything from phasors to CAD.&lt;/li&gt;&#xA;&lt;li&gt;They're fun, if you let them be. Geometric proofs are great brain-teasers. If you will enjoy inventing, engineering, making, software engineering, or philosophy, you will very likely enjoy proofs.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13344" LastActivityDate="2022-07-31T18:31:02.393" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25440" PostTypeId="2" ParentId="25321" CreationDate="2022-07-31T21:24:08.027" Score="0" Body="&lt;p&gt;In my opinion, it depends on the age of the student and if they will typically be allowed access to a calculator in their math classes and on tests.&lt;/p&gt;&#xA;&lt;p&gt;Focusing on arithmetic fact fluency (times tables, adding within 10 or 20) requires that the student have reasonable memorization skills and a solid sense of numeracy. Some students do not have these skills: for example, students with processing disorders, students with dyscalculia or other math-related learning differences, and neurodivergent students may struggle more to acquire these skills. For these students especially, it may be unrealistic to ask them to achieve strong working fluency with arithmetic facts. For some, it is just not possible; for others, it would take a tremendous investment of time and effort that would be better used to focus on (for example) algebra skills. In this way, there is a real accessibility concern associated with demanding arithmetic fact fluency from all students. Of course, if the student's course requires that they have strong recall of their facts without use of a calculator, then this is a worthwhile topic for you to focus on as a tutor; however, for students from age 11-18 (middle school &amp;amp; high school), they are typically transitioning out of courses with a computational focus (and where calculators are not allowed) into courses with an abstract algebraic problem-solving focus (where scientific or graphing calculators are commonly allowed).&lt;/p&gt;&#xA;&lt;p&gt;As a tutor, your student is paying you to help them achieve success in their coursework. Of course, this doesn't mean only focusing on homework as it is common that a student needs significant remediation before they are able to engage with their current course level. However, the job of a tutor is to help your student develop effective strategies to achieve success in their current and upcoming courses. For students in elementary school, this does mean that an emphasis on computational fluency and knowledge of arithmetic facts is appropriate, since arithmetic knowledge is a &amp;quot;high-mileage&amp;quot; skill that will serve them well in many problem contexts, especially since students at that level can rarely use a calculator in class or on tests. However, for a student in algebra 1 or beyond who is able to use a calculator in class and on tests, they may likely need more support navigating the shift from courses focused on rote computation to courses focused on algebraic skills. In my opinion, you would be doing those students a disservice by skirting the main issue which is their need to develop fluency with more abstract problem-solving techniques applicable to high school and college math classes. Many students who struggled with computational math classes early in their education can succeed and excel in more abstract and algebra-based courses later in their education; similarly, a student can breeze through their computational math classes yet struggle with the transition to algebra work (this is a very common context for students to begin working with tutors). So these two skill sets and course contexts, although certainly not disjoint, don't overlap as much as you might think.&lt;/p&gt;&#xA;&lt;p&gt;For those topics which do require knowledge of arithmetic facts (eg. factoring quadratics), I typically have students rely on calculators to identify factor pairs. I do sometimes teach divisibility rules if the student is ready for it. Many students, even those who struggled to achieve arithmetic fact fluency in their earlier years, can observe the patterns in factor pairs and develop a stronger sense of numeracy through practice with a calculator. There are some students who need a calculator for any numerical computation (even adding within 10). These students should not be shamed for their lack of fact knowledge, nor should their efforts in developing algebra fluency be derailed in factor of a computational approach that they will rarely need to use in the future. Rather, when working at the level you indicated (age 11-18 and beyond), a tutor should help the student find strategies that they can use in solving diverse types of problems, especially problems which require strong knowledge of algebra techniques. In this context, calculators are an accessibility tool that broadens access to the field of mathematics and increases the odds of student success at high levels.&lt;/p&gt;&#xA;&lt;p&gt;Ultimately, although fact fluency is a cornerstone for many computation-based classes in elementary and early middle school, the prevalence of calculators in algebra 1 courses and beyond typically makes it unnecessary for students to have a strong command of arithmetic facts. It is unlikely to be an effective use of instructional/tutor-contact time to focus on fact fluency for students whose presenting problem concerns the development of algebraic problem-solving skills. I recommend checking with each student if they are able to use a calculator on tests/quizzes and in class, and if they are allowed, then teach them how to use it!&lt;/p&gt;&#xA;" OwnerUserId="20152" LastActivityDate="2022-07-31T21:24:08.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25441" PostTypeId="2" ParentId="25434" CreationDate="2022-07-31T22:17:33.520" Score="6" Body="&lt;p&gt;You need a proof to know that what you've observed is true beyond the cases where you observed it.&lt;/p&gt;&#xA;&lt;p&gt;How do  you know that the sum of two odd numbers is even? Most kids say it's obvious after they've seem lots of cases - where &amp;quot;lots&amp;quot; means a few with really small odd numbers. The proof isn't hard - once you have good definitions of &amp;quot;even&amp;quot; and &amp;quot;odd&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;How do you know there are infinitely many primes? There are - Euclid proved it. How do you know there are infinitely many pairs of primes two apart, like 5 and 7 or 101 and 103. You don't. No one does. If you can prove it or prove it is false you will be famous (in a small circle).&lt;/p&gt;&#xA;&lt;p&gt;How do you know that every even number is the sum of two primes? That's known to be true for all the evens up to some huge number. You can find the current bound reading about &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Goldbach%27s_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Goldbach's conjecture&lt;/a&gt;&lt;/em&gt;. But no one knows whether it's true &lt;em&gt;forever&lt;/em&gt;. If you can prove it (or find a counterexample) you'll be famous .&lt;/p&gt;&#xA;&lt;p&gt;I hope that in your son's studies he is asked questions like &amp;quot;prove or disprove X&amp;quot; where X is a statement interesting enough to play with. Sometimes it will be true, sometimes not. Finding out which and writing a convincing argument for your conclusion (that is, a proof) is a good way to learn why proofs are necessary and how to write them.&lt;/p&gt;&#xA;" OwnerUserId="4984" LastActivityDate="2022-07-31T22:17:33.520" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25442" PostTypeId="2" ParentId="25434" CreationDate="2022-07-31T22:55:43.127" Score="3" Body="&lt;p&gt;I strongly disagree with &amp;quot;proofs are understanding &lt;em&gt;why&lt;/em&gt; something is true&amp;quot;. The statement holds only in a minority.&lt;/p&gt;&#xA;&lt;p&gt;I write because I've been told time and again by professors to be content with proofs as intuitions or explanations on why something is true. Stokes' Theorem, electromagnetic fields, derivatives, Fourier transform, etc - I'm pointed to heavy-duty math and told that's the best there is. This has been &lt;em&gt;very&lt;/em&gt; wrong every single time.&lt;/p&gt;&#xA;&lt;p&gt;For advanced topics, proofs are generally useful as explanations only if someone already has a significant background in the field &lt;em&gt;and&lt;/em&gt; its underlying mathematical field. For beginner topics, a beginner can't comprehend actual &amp;quot;proofs&amp;quot;, only derivations - these make up most of the minority I refer to. Indeed the derivation of the quadratic formula is quite useful, and &lt;em&gt;knowing how to derive&lt;/em&gt; is very important, else we're blind-faithing, which is anti-science.&lt;/p&gt;&#xA;&lt;p&gt;My background is electrical engineering, signal processing, and machine learning via calculus and linear algebra. This post is a response to other posts in the &lt;em&gt;general context&lt;/em&gt; rather than OP's, so it's not necessarily to invalidate other posts. YMMV in other fields and maths.&lt;/p&gt;&#xA;" OwnerUserId="20153" LastActivityDate="2022-07-31T22:55:43.127" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25443" PostTypeId="2" ParentId="25434" CreationDate="2022-08-01T00:18:35.923" Score="1" Body="&lt;p&gt;I agree with some answers, but will phrase it a bit differently.&lt;/p&gt;&#xA;&lt;p&gt;The theorems are &amp;quot;what&amp;quot;. The proofs are &amp;quot;how&amp;quot; and &amp;quot;why&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But not even this is the full beauty of maths. You start with some formal definitions, the axioms. Basically, you let things be like this, assume that in a new, empty, from-scratch world things go like that. From those axioms you can conclude some further statements, the theorems. You can prove them, basing on the properties of the prior facts. (As I often tend to say, the art of a mathematical proof is to obtain a stronger statement by chaining weaker forms of already known statements.) Then, repeat, repeat, repeat. You are building a beautiful building from reasoning on top of the axioms you chose.&lt;/p&gt;&#xA;&lt;p&gt;And now, the crux. There is not a single set of axioms, you can use multiple. Of course, many would use the already existing sets, as it suffices in that field. Things like the euclidean axioms of geometry on a plane. Or ZFC. Or some kind of a logic. Or category theory. But you can totally switch to a different set of axioms if you need to. Or even a build new one, if you need to.&lt;/p&gt;&#xA;&lt;p&gt;Next, about proofs. There are proofs that basically show, that if the theorem in question does not hold, then the world is broken. Those are the contradiction proofs. But there are also proofs, that are constructive. Basically, a theorem merely states &amp;quot;foobar exists&amp;quot;. A constructive proof would show you, how to build a foobar (whatever it is).&lt;/p&gt;&#xA;&lt;p&gt;Proofs can be tedious and exhausting, which is why they are often excluded or only glossed about in the typical school setting. I daresay, if a child is interested enough in the subject, then using the typical childish curiosity, &amp;quot;and why is that?&amp;quot;, is a possible way to tackle proofs. &amp;quot;It is because of this, but you might grow tired from all the details.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Next issue, the proofs are typically written down to be watertight. To convey all the needed information in all the detail. (Later, they might miss out some steps, but leave enough to connect the dots, more on it below.) One thing, which is in my opinion just as essential as a correct proof, is the proof idea. A rather short and a more approachable summary of how the proof works. (&amp;quot;Square root of two is irrational, because if we'd assume it is not, then it's a reduced fraction. But we would be able to reduce it once more, which is a contradiction&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;Now, for connecting dots. During my studies quite some time ago, I stumbled upon a formal definition of what is &amp;quot;can easily be seen&amp;quot;, &amp;quot;is trivial&amp;quot; and the sort. It basically means, the intended reader should be able to figure it out in 15 minutes with a paper and a pencil. Which is a lot for an unprepared mind. Basically, you'd need to build up some kind of a frustration tolerance in a young mathematician, ensure they don't think of themselves as dumb and unworthy if they cannot figure a step in the proof, if they cannot connect the dots immediately. Asking around helps. Oral explanations tend to go in the direction of the &amp;quot;proof idea&amp;quot; and hence help with the understanding a lot. (Which is why lectures are important.)&lt;/p&gt;&#xA;&lt;p&gt;To summarize: Theorems are the &amp;quot;what&amp;quot;, proofs are the &amp;quot;why&amp;quot;. One of the best elementary examples of this way of thinking is classical plane geometry, the things Ancient Greeks knew. Geometry is much closer to &amp;quot;real mathematics&amp;quot; than any other pre-university math-related subject. It takes some skill to read the proofs, to extract the &amp;quot;idea&amp;quot; from it. Aspiring mathematicians need to be wary of trying to gasp everything in this very instant and without help. Some kind of explanations on your side or video lectures might also be a great help.&lt;/p&gt;&#xA;" OwnerUserId="20154" LastActivityDate="2022-08-01T00:18:35.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25444" PostTypeId="2" ParentId="25434" CreationDate="2022-08-01T09:41:16.083" Score="0" Body="&lt;p&gt;A proof is a precise justification given precise premises that a precise statement is true.&lt;/p&gt;&#xA;&lt;p&gt;If you don't have a proof or you don't reasonably believe that someone has seen a proof, you don't have a reasonable reason to believe it. Or a reasonable reason--a reason--reasoning--to believe it. A proof is the reasoning that something is so. That is &amp;quot;why they're important and how they're useful&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;We mostly get things done by accepting that certain things have been proved in mathematics or reasonably shown likely in engineering and everyday life--often aided by mathematics.&lt;/p&gt;&#xA;&lt;p&gt;However,learning how to understand &amp;amp; generate proofs of simple precise statements helps us to learn to reason in less precise contexts.&lt;/p&gt;&#xA;&lt;p&gt;Other answers here are wrong &amp;amp; misleading to say that a proof tells us or shows us &amp;quot;why&amp;quot; something is true. It does not. It shows that it is true. Having a proof is why we are justified in saying it is true. There is no &amp;quot;why&amp;quot; in mathematics; given some premises, other things follow. The only way a proof can reasonably though sloppily be said to say &amp;quot;why&amp;quot; is in the trivial sense that we are justified in saying it is true &amp;quot;because&amp;quot; a proof exists as demonstrated by a proof having been given or proved to exist--the conclusion is true &amp;quot;because of&amp;quot; the premises.&lt;/p&gt;&#xA;" OwnerUserId="20157" LastEditorUserId="20157" LastEditDate="2022-08-01T11:58:16.173" LastActivityDate="2022-08-01T11:58:16.173" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="25445" PostTypeId="2" ParentId="25434" CreationDate="2022-08-01T12:11:05.657" Score="0" Body="&lt;p&gt;As other answers have already well-addressed the question of, &amp;quot;why proving something is useful/important ?&amp;quot;, I'll address the related but different question of &lt;em&gt;what personal benefits do I get from the practice of proving/attempting to prove something ?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Firstly, &lt;em&gt;trying to understand a problem statement&lt;/em&gt; (of something that is wanted to be proved) is a task in and of itself- and this exercises your &lt;em&gt;mathematical comprehension&lt;/em&gt; (related but different than English/language comprehension because maths comprehension is usually more abstract than people-focused).&lt;/p&gt;&#xA;&lt;p&gt;Usually there are many different proofs for a given maths problem. Comprehension is also exercised when you try to understand other peoples' proofs that maybe you did not come up with yourself.&lt;/p&gt;&#xA;&lt;p&gt;Secondly, I think proofs are important because they improve your problem-solving skills through creative thinking and abstract imagination, especially when the proof is difficult - it requires you to think outside of the box.&lt;/p&gt;&#xA;&lt;p&gt;Thirdly, proofs can improve your communication skills because often times it is easy to see the idea behind a proof but it is difficult to actually write down the proof concretely. Furthermore, coming up with your own proof which is different to other proofs sometimes gives you the opportunity to help those who did not understand the other proofs.&lt;/p&gt;&#xA;&lt;p&gt;I'm sure there are many more you could add to this list...&lt;/p&gt;&#xA;&lt;p&gt;I have no doubt my answer could be improved, but I think the essence of what I am saying is important: that many important skills are exercised and improved when one does (mathematical) problem-solving.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastActivityDate="2022-08-01T12:11:05.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25446" PostTypeId="2" ParentId="25434" CreationDate="2022-08-01T14:45:15.510" Score="5" Body="&lt;p&gt;Proofs are the whole point of mathematics. They are how we verify and explain that we &lt;em&gt;know&lt;/em&gt; things instead of merely &lt;em&gt;guess&lt;/em&gt; at them. When I personally teach discrete mathematics, the first-day opening that I use to address this issue is this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Consider a function defined on natural numbers &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$f(n) = n^2 - n +&#xA;11$&lt;/span&gt;. Are all outputs of this function prime?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Inevitably in my classes (at a community college in CUNY), the students quickly and unanimously convince themselves of one answer at the end of the first day, at which point I stop and suggest further things to think about (including the sketch of a program they could write, as this is at least second semester in a CS program). They next day we return and find out they were all incorrect, despite the certainty they had on day one. The point being: proofs are important in this class because they let us distinguish between when we &lt;em&gt;really know&lt;/em&gt; and when we only &lt;em&gt;think we know&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Depending on the OP child's maturity (and also the OP), they might consider trying this exercise together.&lt;/p&gt;&#xA;&lt;p&gt;Finally, on the issue of use for engineers, here's a quote that always sticks with me. From Stein/Barcellos, &lt;em&gt;Calculus and Analytic Geometry&lt;/em&gt;, 5E, &amp;quot;To the Instructor&amp;quot;, p. xxii (1992):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the Tulane conference on &amp;quot;Lean and Lively Calculus&amp;quot; in 1986 we&#xA;heard the engineers say, &amp;quot;Teach the concepts. We'll take care of the&#xA;applications.&amp;quot; &lt;strong&gt;Steve Whitaker, in the engineering department at Davis,&#xA;advised us, &amp;quot;Emphasize proofs, because the ideas that go into the&#xA;proofs are often the ideas that go into the applications.&amp;quot;&lt;/strong&gt; Oddly,&#xA;mathematicians suggest that we emphasize applications, and the applied&#xA;people suggest that we emphasize concepts. We have tried to strike a&#xA;reasonable balance that gives the instructor flexibility to move in&#xA;either direction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2022-08-02T12:59:29.150" LastActivityDate="2022-08-02T12:59:29.150" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25447" PostTypeId="2" ParentId="25434" CreationDate="2022-08-01T17:04:47.143" Score="3" Body="&lt;p&gt;Proof techniques are problem solving tools just as much as wrenches and ratchets are.&lt;/p&gt;&#xA;&lt;p&gt;Consider proof by contradiction.&lt;/p&gt;&#xA;&lt;p&gt;The other day when I was fasting I got suddenly concerned I might've drunk soda pop while at home. So I assumed I did. Since I drank soda, there must be at least one open can whether partly full or empty. Then I checked for open soda cans in the fridge, countertop and trash bins(all soda in the house is of canned variety). There were none. Thus the assumption was wrong and therefore I didn't drink any soda.&lt;/p&gt;&#xA;&lt;p&gt;One time I dosed off with a laptop in my hands and when I woke up a program was open with some data exposed in clear text. I got scared for a bit that I might've accidentally erased some data. I read the text beginning to end and everything held up logically. There were no obvious holes that would indicate data loss. Then I assumed data loss and supposed I found the lost data. Adding that data back would, then, break the logical flow of the existing data. Contradiction. Therefore no data loss occurred.&lt;/p&gt;&#xA;" OwnerUserId="20162" LastActivityDate="2022-08-01T17:04:47.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25448" PostTypeId="2" ParentId="25434" CreationDate="2022-08-01T17:32:20.617" Score="4" Body="&lt;p&gt;Fundamentally, proofs draw a conclusion from some premises.  This is how you apply them in the rest of life.  If you observe (or assume) the premises are true in real life, the conclusion is true as well.  You don't have to go through the effort of verifying the arduous chain from premises to conclusions.  Once you have verified the proof, you can just use it!&lt;/p&gt;&#xA;&lt;p&gt;I find the word &amp;quot;verify&amp;quot; to be very important in the above paragraph.  One of the things that makes proofs so gosh darn powerful is that you can sit down and work through their logic to determine if the conclusion really follows from the premises.  If your son is having trouble writing proofs this is to be expected: it is &lt;strong&gt;far&lt;/strong&gt; harder to write a proof than to verify it.&lt;/p&gt;&#xA;&lt;p&gt;And, indeed, that is the point.  A proof succinctly captures something enormously difficult to think through and puts it in a form that is easier to verify.&lt;/p&gt;&#xA;&lt;p&gt;You mention software engineering.  I recently had to put together a piece of software that needed to demonstrate that some condition was true before proceeding.  I had access to some data about the present conditions, and needed to process it to see if it &amp;quot;entailed&amp;quot; the condition I was looking for.  With the data I had available, this called for a particularly difficult class of algorithms.  Fortunately, I was not the first person to want these algorithms.  I could go out on the internet and find papers about how to do it, written by multiple PhDs.&lt;/p&gt;&#xA;&lt;p&gt;But could I trust it?  Sure, I could code it up, but how could I be certain that it would handle all of my scenarios?  The answer was that the papers I read contained proofs.  I could go through them, line by line, and see &lt;em&gt;why&lt;/em&gt; the algorithms I was looking at were &amp;quot;complete.&amp;quot;  And this is really interesting, because I don't have a PhD.  In fact, my forte isn't even in the field these algorithms come from.  There is no way I could have worked my way through these on my own, but the authors had done the hard part.  They walked me through the steps and the leaps of inspiration I wouldn't have thought of, proving the algorithm did what they said it did.&lt;/p&gt;&#xA;&lt;p&gt;And then I built on it, with my own proofs.  They were nowhere near as difficult as the proofs I found for the underlying math, but I could prove that, assuming their algorithms did what they say they did, mine would do what I said it did as well.  And this is something that makes proofs very powerful.  When you're just learning, you're only thinking about one proof at a time.  But they &lt;em&gt;chain&lt;/em&gt;.  We build upon the lessons from others.  Proofs let us stand on the shoulders of giants and peer just a little further.&lt;/p&gt;&#xA;&lt;p&gt;This transitive property is king.  It is what makes mathematics as powerful as it is.  It lets you build up giant castles of knowledge, built on the bedrock of simpler math that you understand.  It also lets you explore that simpler math, and see how deep the bedrock truly goes -- how high of a castle do you dare build on it.&lt;/p&gt;&#xA;&lt;p&gt;And the process of learning to write proofs is useful too.  Proofs define a criteria for &amp;quot;this is true,&amp;quot; a bar which one must achieve in order to claim that something is actually true.  And it provides tools to learn how to demonstrate such things.  These tools apply outside of mathematics as well.  When I am at a design review, defending my software, I don't present mathematically accurate proofs, but I do go through the same process of helping the reviewers walk through my logic and come to an agreement that it is sound.  The difference is that the rigorous proofs of math could &lt;em&gt;prove&lt;/em&gt; that what I say is true, while I am otherwise forced to hold reviews to &lt;em&gt;demonstrate&lt;/em&gt; that it is true.&lt;/p&gt;&#xA;&lt;p&gt;As a parting philosophical topic, mathematics often explores really tricky cases.  Consider Zeno's paradox: someone trying to run across a field must first run half way across it.  And they must first run halfway across that halfway, and so on.  This is an infinite number of steps that have to occur before you move.  Thus, Zeno concludes nobody moves at all!&lt;/p&gt;&#xA;&lt;p&gt;Working with these infinities are tricky.  I'd argue it took over a millennium before we had precise enough wordings to grapple with them.  And it required proofs to avoid paradoxes like the ones Zeno put forth.  If your son is interested in reading about math, &amp;quot;Gödel, Escher, Bach: an Eternal Golden Braid&amp;quot; by Douglas Hofstadter is a wonderfully accessible work covering such topics and the &lt;em&gt;search&lt;/em&gt; for truth.  I like it because it not only demonstrates the power behind such formal thinking, but also shows the process of &lt;em&gt;getting there&lt;/em&gt;.  I find it hard to separate proofs from mathematical history.  Seeing how people formed incomplete ideas into cohesive proofs to share with others is the core of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2022-08-01T17:32:20.617" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25449" PostTypeId="2" ParentId="25434" CreationDate="2022-08-01T20:07:44.460" Score="0" Body="&lt;p&gt;Learning how to come up with a math proof is good for a kid. It's a problem-solving activity.&lt;/p&gt;&#xA;&lt;p&gt;Students believe what they are told by their teachers, especially in math class, and with good reason because math teachers rarely teach an untrue theorem, and proofs often are used instead of good &lt;em&gt;explanations&lt;/em&gt;. The proofs are looked at, and then forgotten, with nothing gained.&lt;/p&gt;&#xA;&lt;p&gt;Sure, students probably should be more skeptical, but then again, maybe that skepticism should not be about the math the teacher is teaching them. But teaching students math proofs is not an efficient way to make them skeptical. Showing students proofs that look valid and then showing them how they've been duped might work to make the students skeptical, and even create some genuine interest in proofs.&lt;/p&gt;&#xA;&lt;p&gt;You could try showing your son this video by 3Blue1Brown called &amp;quot;How to lie with visual proofs&amp;quot; &lt;a href=&quot;https://www.youtube.com/watch?v=VYQVlVoWoPY&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=VYQVlVoWoPY&lt;/a&gt; and another good idea might be &amp;quot;How to Lie with Statistics&amp;quot; by Darrell Huff. Probably everyone should read the latter.&lt;/p&gt;&#xA;&lt;p&gt;You say, &amp;quot;especially for inventors, engineers, makers, software engineers, and philosophy&amp;quot; and to that I'd say there's some use in some parts of philosophy, but not among the other groups you mention. These people, like most nonmathematicians, can greatly benefit from knowing how to &lt;em&gt;use&lt;/em&gt; math, but rarely find it useful to be able to prove that a piece of math is true. If the math community says it's been proved, they can be safely believed, I think. All those groups you mentioned, with the exception of some types of philosophers, need to master, besides math, the art of testing, and the art of using and making computer simulations. On the other hand, if the student uses creativity to come up a proof, the increase in general creativity that could be expected to result would be useful.&lt;/p&gt;&#xA;" OwnerUserId="15662" LastEditorUserId="15662" LastEditDate="2022-08-04T20:02:11.220" LastActivityDate="2022-08-04T20:02:11.220" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25450" PostTypeId="2" ParentId="25434" CreationDate="2022-08-01T21:31:10.503" Score="20" Body="&lt;p&gt;I am an engineer.  I have not done a mathematical proof since leaving school.  Despite that, I believe that proofs are the second most important skill that &lt;em&gt;any&lt;/em&gt; student will learn during their entire educational career (second only to basic literacy).&lt;/p&gt;&#xA;&lt;p&gt;When you do a mathematical proof, you:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Start with a &lt;strong&gt;problem&lt;/strong&gt;, a well-defined set of conditions.&lt;/li&gt;&#xA;&lt;li&gt;Advance from that starting point &lt;strong&gt;one step at a time&lt;/strong&gt;, with each step representing the application of a theorem, definition, or other principle that is &lt;em&gt;itself&lt;/em&gt; already known to be true.&lt;/li&gt;&#xA;&lt;li&gt;Arrive at a &lt;strong&gt;conclusion&lt;/strong&gt; that either justifies or contradicts whatever it is that you're trying to prove.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This is about far more than math.  This process is the basis for rational thought itself.  You're not just learning how to prove mathematical theorems, you're learning how to think logically and to reason using facts (not opinion or instinct).  You're learning how to take various things that you know to be true and use them to solve problems that are unlike anything you've ever encountered before.  You're learning how to use a rigorous thought process to avoid logical fallacies, jumping to conclusions, or being misled by rhetoric.&lt;/p&gt;&#xA;&lt;p&gt;Most schools no longer teach logic as it's own subject.  The same important concepts are generally taught through mathematics, as math is more concrete (thus presumably easier to understand) than approaching logic from the abstract, philosophical side.  My school taught proofs in geometry, where problems involved shapes, angles, and other things that could be drawn and easily visualized each step of the way.  Proofs were always the hard part of the course.  Students never really &amp;quot;got it&amp;quot; until they realized that it's not really about geometry at all.  It's about &lt;em&gt;thinking&lt;/em&gt;, and the geometry is more of a visual aid used to create a limited workspace with well-defined rules that we can use to practice logical reasoning.&lt;/p&gt;&#xA;&lt;p&gt;Even though I haven't done a formal mathematical proof in years, I use those skills and go through the same process every day: when I encounter a type of problem that I haven't seen before, when I evaluate what someone said to determine whether it's true or not, when my instincts tell me something is true but I want to make sure that it's &lt;em&gt;actually&lt;/em&gt; true, etc.&lt;/p&gt;&#xA;&lt;p&gt;The subjects taught in school are like the exercises done in a gym.  Bench-pressing weights seems pointless, how often do you ever need to lift a heavy stick vertically off of your chest?  You do them not because that activity itself is useful, but because it's designed to isolate and strengthen a specific muscle group.  After building up strength throughout your body, you'll use those muscle groups &lt;em&gt;together&lt;/em&gt; to solve real-world problems.  The proofs that you're doing are building the logic portion of your brain.  You're learning &lt;em&gt;how to think&lt;/em&gt;, a critical life skill.  If you know how to read, how to think, and how to learn, then you have the ability to teach yourself anything that you'd ever want to know - and what can be more powerful than that?&lt;/p&gt;&#xA;" OwnerUserId="20163" LastActivityDate="2022-08-01T21:31:10.503" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25451" PostTypeId="2" ParentId="25434" CreationDate="2022-08-02T08:18:38.040" Score="0" Body="&lt;p&gt;Philosopher here.&lt;/p&gt;&#xA;&lt;p&gt;Mathematical proofs are formulated in a logical language (usually some sort of predicate logic, to a lesser extend propositional logic). Remember that all the tools like proof by contradiction, modus ponens, etc, are logical tools rather than mathematical tools.&lt;/p&gt;&#xA;&lt;p&gt;Logic provides a framework that has very powerful claims. Let's name a few of them:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;consistency&lt;/strong&gt;: There must not be any false formula within a given theory, otherwise the theory could produce any random contradiction. Do not confuse this with the framing of physical theories.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;soundness&lt;/strong&gt;: A valid logical argument whose premises are all true has a true conclusion. The other way around: A logical argument of which a contradiction can be derived shows that one of the premises is false. This observation is especially useful to check if something you assume is really true.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;axioms&lt;/strong&gt;: Axioms are well-chosen contingent formulae (formulae whose truth values vary depending on interpretation) which are considered to be true within a given theory. All other true formulae within this theory can be derived from them, and no false formulae can be derived from them. The following observation can be made: As soon as you can derive your assumption from axioms, it must be true.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For a 13yo (or any school-level), mathematics is usually a single theory.&lt;/p&gt;&#xA;" OwnerUserId="11891" LastEditorDisplayName="user20173" LastEditDate="2022-08-03T10:52:00.767" LastActivityDate="2022-08-03T10:52:00.767" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25452" PostTypeId="2" ParentId="25434" CreationDate="2022-08-02T09:46:38.587" Score="1" Body="&lt;h3&gt;Example: Board Game&lt;/h3&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You are playing a game with a single stone on a hexagonal board. You have cards indicating one of the six directions on the board, which move the stone in the respective direction when played.&lt;/p&gt;&#xA;&lt;p&gt;Now, your stone is starting in the centre of the board and you have six cards, one for each direction. You have to play each card exactly once, but you can choose the order.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eTy7a.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eTy7a.png&quot; alt=&quot;Illustration of the board game&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;On which spaces can your stone end?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This task it not challenging if you know the basics of  ℝ² vector arithmetics, in particular commutativity, which tells you that the order of cards does not matter and the stone always returns to the centre, because the hand of cards is composed entirely of pairs of inverses.&#xA;If anything, the difficulty is recognising that you can apply this, i.e., applying the right abstraction (vectors) to the problem.&#xA;Still, even if you have all of mathematics at your disposal to throw at this problem, you will likely perform a little proof to convince yourself of the solution, even if you may not recognise it as such or just think of it as a calculation.&lt;/p&gt;&#xA;&lt;p&gt;Now, suppose you try to prove the solution without knowing (or applying) vectors.&#xA;In that case, you will inevitably gain some insights on the mechanics of this game.&#xA;Most importantly, you will likely discover that the order of cards does not matter – which can be useful in other scenarios (different cards) but also translates to other practical problems.&#xA;To prove this you may, e.g., look at the sides of rhombuses and learn or reinforce some geometry in the process, or introduce tuples of numbers to describe positions and thus discover vectors.&lt;/p&gt;&#xA;&lt;p&gt;Sidenote:&#xA;A student may claim that it’s obvious that the order does not matter.&#xA;You can demonstrate that this is not so obvious by placing different counters on the spaces that are collected when the stone is moved.&lt;/p&gt;&#xA;&lt;h3&gt;What does this demonstrate?&lt;/h3&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If you are working with mathematics or similar, you are likely to prove some small thing frequently, even though you may have internalised many (proof) patterns to an extent that you don’t even recognise them as such.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Proving things grants you a deeper understanding of the problem at hand, allowing you to better solve similar problems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Trying to prove things challenges you to find patterns, abstractions, and similar that are useful beyond the specific problem you are trying to solve. Formulating these is often the main accomplishment of a proof, while “the proof itself” can be a mere technical exercise or calculation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Proving can often be more easy than empiricism or brute force.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="75" LastActivityDate="2022-08-02T09:46:38.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25453" PostTypeId="2" ParentId="25434" CreationDate="2022-08-02T15:45:21.163" Score="3" Body="&lt;p&gt;There are a lot good answers here, but I'd like to supplement with some other benefits.&lt;/p&gt;&#xA;&lt;p&gt;First, besides being an exercise in reasoning, &lt;strong&gt;mathematical proofs are exercises in language use&lt;/strong&gt; particularly in regards to being precise in syntax. As someone who programs, I can tell you that attention to the details of a syntax are tremendously important. One misplaced or wrong character in a program of a million lines of code can cause the system to fail to build, or even worse, to build and allow subtle errors to occur. Good mathematical proofs contain healthy doses of syntax. Finding &lt;a href=&quot;https://en.wikipedia.org/wiki/Compile-time_function_execution&quot; rel=&quot;nofollow noreferrer&quot;&gt;compile-time&lt;/a&gt; errors is pretty easy. The system barfs on you. Finding subtle &lt;a href=&quot;https://en.wikipedia.org/wiki/Runtime_system&quot; rel=&quot;nofollow noreferrer&quot;&gt;run-time errors&lt;/a&gt; is a job skill.&lt;/p&gt;&#xA;&lt;p&gt;Second, &lt;strong&gt;proofs are actually an object of study themselves&lt;/strong&gt; by mathematicians in the discipline of &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathematical logic&lt;/a&gt;. Both proof theory and &lt;a href=&quot;https://en.wikipedia.org/wiki/Model_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;model theory&lt;/a&gt; explores proofs in the abstract, and those both go to &lt;a href=&quot;https://en.wikipedia.org/wiki/Foundations_of_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathematical foundations&lt;/a&gt; in the philosophy of math.&lt;/p&gt;&#xA;&lt;p&gt;Lastly, the foundations of mathematics answers the question &amp;quot;What is mathematics?&amp;quot;; however, &lt;strong&gt;mathematics and logics are a foray into the exploration of &lt;a href=&quot;https://en.wikipedia.org/wiki/Metaphysics&quot; rel=&quot;nofollow noreferrer&quot;&gt;metaphysics&lt;/a&gt; and philosophical questions&lt;/strong&gt;. Take for instance, &lt;a href=&quot;https://philosophy.stackexchange.com/questions/84218/can-the-brain-be-considered-an-analog-computer&quot;&gt;Is the brain a computer? (PhilSE)&lt;/a&gt; Questions like these delve into deep notions like &lt;a href=&quot;https://en.wikipedia.org/wiki/Truth#Major_theories&quot; rel=&quot;nofollow noreferrer&quot;&gt;theories of truth&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Justification_(epistemology)&quot; rel=&quot;nofollow noreferrer&quot;&gt;the nature of justification&lt;/a&gt;, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Informal_fallacy&quot; rel=&quot;nofollow noreferrer&quot;&gt;nature of poor reasoning&lt;/a&gt;, and so on. Simple math proofs are a simple introduction into philosophical discourse, and have been at the heart of very famous philosophical clashes and events like &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Foundations_of_Arithmetic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frege's invention of formal systems and his resistance to psychologism&lt;/a&gt; which still are argued today.&lt;/p&gt;&#xA;" OwnerUserId="18201" LastActivityDate="2022-08-02T15:45:21.163" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25454" PostTypeId="1" AcceptedAnswerId="25463" CreationDate="2022-08-03T05:33:57.453" Score="1" ViewCount="184" Body="&lt;p&gt;My Mathematics Olympiads are approaching So I have not started the preparation yet. I need to have a better resource for +2(11th or 12th class basically a high school student) Olympiads so if you have any suggestion pls go ahead :)&lt;/p&gt;&#xA;" OwnerUserId="20117" LastEditorUserId="20117" LastEditDate="2022-08-03T09:51:16.703" LastActivityDate="2022-08-05T14:47:43.223" Title="Book recommendation for Mathematical olympiads for a +2 student" Tags="|secondary-education|reference-books|book|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25455" PostTypeId="1" AcceptedAnswerId="25457" CreationDate="2022-08-03T18:04:56.547" Score="4" ViewCount="224" Body="&lt;p&gt;I want a clickable 3-circle Venn diagram (in a box labeled U), where I can shade each region by clicking. I have googled &amp;quot;venn diagram maker free online&amp;quot; and looked at about a dozen of these sites. Many want me to sign in, which should not be necessary. None of the others are what I want.&lt;/p&gt;&#xA;&lt;p&gt;If I knew how to program websites, maybe I could make my own.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know of a site like this?&lt;/p&gt;&#xA;&lt;p&gt;I want to be able to make lots of diagrams, all in the same style, for teaching, quizzes, and tests. Basically, I just want to be able to click which regions will be shaded (also being able to label the circles would be nice, they usually need to be A, B, C). I'd like two circles on top and one below, because that's how the diagrams are in the text I'm using (discrete math).&lt;/p&gt;&#xA;&lt;p&gt;[A year or two ago, I was trying to make my own on drawing sites. Those had issues too. On many, the overlapping circles hid each other. On many I could only shade a whole circle. A drawing site that makes this easy would be just as helpful as a dedicated venn-diagram-maker.]&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-01-14T05:25:05.060" Title="Looking for Venn Diagram Maker" Tags="|technology-in-education|software|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25456" PostTypeId="2" ParentId="25434" CreationDate="2022-08-03T20:23:49.193" Score="0" Body="&lt;p&gt;Let me give an answer which frames proof as a step towards programmability.&lt;/p&gt;&#xA;&lt;p&gt;Technically, there are proofs which are not &lt;strong&gt;constructive&lt;/strong&gt;. Typically such proofs rest on some application of proof by contradiction. I don't mean to speak to such proofs. Instead, I'll focus my answer on proofs which are &lt;strong&gt;constructive&lt;/strong&gt; in the following sense:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A constructive proof gives a set of instructions on how to solve a given problem for completely arbitrary inputs. Or, at least for inputs which are suitably constrained.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This sort of proof is not directly programmable without a lot of work in practice. For example, in Linear Algebra you might have a list of items where your proof selects from the list a particular member with a desired attribute. Then, often the argument goes forward by relabeling that desired choice as the first in the newly formed list. This sort of &lt;em&gt;renumeration&lt;/em&gt; is easy enough to write in the parlance of typical theoretical mathematics, but, how do you actually achieve that for a given problem using a programming language on a given computer ? I worked for a year outside academics and I learned there the difference between proving and applying. I worked with people who were able to convert my index relabeling arguments by clever filtering and data sorting. Matlab has a deep toolset for those who are schooled in these methods.&lt;/p&gt;&#xA;&lt;p&gt;I argue:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Direct proof is easier than application.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To apply a direct proof to real world data you have to choose data types and prepare for differently sized sets of data ideally. The rough sketch of how to set-up such a program is given by a direct formal proof. In short,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Proof is a first step towards building a general program.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A proof is akin to a flow-chart which can be understood by an audience of your mathematical peers.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2022-08-03T20:23:49.193" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25457" PostTypeId="2" ParentId="25455" CreationDate="2022-08-04T03:58:08.180" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/n44xx78e49&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's something I made in Desmos&lt;/a&gt; to this effect. While you cannot click the regions themselves to fill them, you can enable or disable the region by clicking the corresponding folder on the left sidebar to fill the regions.&lt;/p&gt;&#xA;&lt;p&gt;Edit: I forgot that &lt;a href=&quot;https://help.desmos.com/hc/en-us/articles/4407725009165&quot; rel=&quot;nofollow noreferrer&quot;&gt;Actions&lt;/a&gt; are a thing on Desmos! I rebuilt it to have clickable regions based off of Michał Miśkiewicz's fantastic work. &lt;a href=&quot;https://www.desmos.com/calculator/58i1abhksx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Check out the new &lt;strong&gt;clickable&lt;/strong&gt; version here.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="45" LastEditorUserId="45" LastEditDate="2022-08-04T17:17:20.317" LastActivityDate="2022-08-04T17:17:20.317" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="25458" PostTypeId="2" ParentId="24761" CreationDate="2022-08-04T04:16:29.410" Score="4" Body="&lt;p&gt;There is one trick that often helps even if you are not Perelman or Wiles: try to solve several unrelated problems simultaneously rotating the amounts of time you spend on each of them so when you feel that you are completely stuck on one, you concentrate heavily on some other one and think of the first one in the background until you get some fresh idea about it that you think is worth trying in serious.&lt;/p&gt;&#xA;&lt;p&gt;In addition to reducing the frustration, this approach often rewards you with revealing some unexpected connections when the techniques that you try on problem X without success suddenly turn out to be applicable to problem Y while you would never think of using them if you concentrated on Y alone. I usually make my living as a mathematician from spending about 50% of my time thinking (with various degrees of success) on random questions that come my way from my friends, colleagues, or just complete strangers on MathOverflow. True, that strategy may be 50% suboptimal if your goal is to prove the Riemann hypothesis and you care about nothing else, but if you just want to have fun and establish plenty of contacts in the process, it pays off handsomely.&lt;/p&gt;&#xA;&lt;p&gt;As to the dissertation, in my case it just came naturally from several topics I was thinking of even as an undergraduate. I just hid the failures (which constituted the overwhelming majority) and brought a few remaining successes to the light. That turned out to be sufficient. I should also say that I was lucky with my adviser who had an amazing skill of finding a problem matching the graduate student abilities and inclinations &lt;em&gt;without&lt;/em&gt; having any idea of how to approach that problem himself. How that could be done still remains a complete mystery to me.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-08-04T04:16:29.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25459" PostTypeId="1" CreationDate="2022-08-04T14:15:46.017" Score="2" ViewCount="256" Body="&lt;p&gt;I was studying about polynomials when I stumbled upon this video&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=vBfdYuoc3x4&amp;amp;list=PLjS5lmipV2HJEaKfdeVSKdprfFxinzmNw&amp;amp;index=2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=vBfdYuoc3x4&amp;amp;list=PLjS5lmipV2HJEaKfdeVSKdprfFxinzmNw&amp;amp;index=2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The video says that a monomial has three parts -- constant, variables, and exponent. But I remember my teacher said it was coefficient.&lt;/p&gt;&#xA;&lt;p&gt;Was my teacher wrong? Should I ask her if constant is same as coefficient?&lt;/p&gt;&#xA;" OwnerUserId="20178" LastEditorUserId="376" LastEditDate="2022-08-05T11:19:59.773" LastActivityDate="2022-08-08T01:24:26.320" Title="Is coefficient same as constant?" Tags="|algebra|terminology|solving-polynomials|polynomials|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25460" PostTypeId="1" CreationDate="2022-08-04T18:09:36.373" Score="21" ViewCount="1328" Body="&lt;p&gt;What resources (Desmos, Geogebra, math3d, etc.) have you built to help your students better understand or visualize topics in mathematics?&lt;/p&gt;&#xA;" OwnerUserId="45" LastEditorUserId="45" LastEditDate="2022-08-08T15:02:07.150" LastActivityDate="2022-09-01T00:54:06.503" Title="Educators, what resources have you built to better serve your students?" Tags="|software|interactive-teaching|resource-request|big-list|resources|" AnswerCount="6" CommentCount="0" CommunityOwnedDate="2022-08-09T22:33:06.387" ContentLicense="CC BY-SA 4.0" />
  <row Id="25461" PostTypeId="2" ParentId="25459" CreationDate="2022-08-04T18:43:40.413" Score="6" Body="&lt;p&gt;I'd say that the video is not using the best word. I would call that constant the coefficient.&lt;/p&gt;&#xA;&lt;p&gt;Constant means that it is a number and not a variable. That's true. But the word coefficient conveys more meaning. It is the constant that comes before a variable (or variables).&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-08-04T18:43:40.413" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25462" PostTypeId="2" ParentId="25460" CreationDate="2022-08-04T19:12:29.773" Score="12" Body="&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/8ewdsrc5k8&quot; rel=&quot;noreferrer&quot;&gt;Riemann Sum visualization&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/zrp2ijtfuh&quot; rel=&quot;noreferrer&quot;&gt;Fundamental Theorem of Calculus Explanation&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/ap2c6sbt08&quot; rel=&quot;noreferrer&quot;&gt;Error bound on integral for increasing function&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/xowdomvxq3&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Secant Circles&amp;quot;&lt;/a&gt; and &lt;a href=&quot;https://www.desmos.com/calculator/vfgzav8ngx&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Tangent Circles&amp;quot;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Some students of mine worked out what would happen if you find the unique circle passing through three nearby points on the graph of a function, and then take a limit as those points get close to each other.  Similar idea to the tangent line development, but with circles.  They ended up needing Taylor series to get the exact equations for the &amp;quot;tangent circle&amp;quot;!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.desmos.com/calculator/q0gn3athdb&quot; rel=&quot;noreferrer&quot;&gt;1-form visualizer&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This visualizes the one form by looking at level sets of the covector in a small rectangle around each point.  You integrate the 1-form along a curve by just counting how many level sets you cross (taking into account the spacing &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; between level sets).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-08-04T19:12:29.773" CommentCount="3" CommunityOwnedDate="2022-08-09T22:33:06.387" ContentLicense="CC BY-SA 4.0" />
  <row Id="25463" PostTypeId="2" ParentId="25454" CreationDate="2022-08-05T14:47:43.223" Score="1" Body="&lt;p&gt;The UK Mathematics Trust has published some books that may be suitable and are widely used in the UK: 'A Mathematical Olympiad Primer' and 'A Mathematical Olympiad Companion' by Geoff Smith. You can find them &lt;a href=&quot;https://www.ukmt.org.uk/shop&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. They are designed for help students prepare for the British Mathematics Olympiad rounds 1 and 2.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2022-08-05T14:47:43.223" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25464" PostTypeId="2" ParentId="25460" CreationDate="2022-08-05T22:09:21.757" Score="6" Body="&lt;p&gt;Here are some physics-related ones... many not quite as polished as they could be.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;update:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://webvpython.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;webvpython&lt;/a&gt;&lt;/strong&gt; / &lt;strong&gt;&lt;a href=&quot;https://www.glowscript.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;glowscript&lt;/a&gt;&lt;/strong&gt;   (also supported on &lt;a href=&quot;https://trinket.io&quot; rel=&quot;nofollow noreferrer&quot;&gt;trinket.io&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.glowscript.org/#/user/Rob_Salgado/folder/My_Programs/program/robphy-maxwell-EMwave-2022&quot; rel=&quot;nofollow noreferrer&quot;&gt;EMwave&lt;/a&gt; - &lt;a href=&quot;https://trinket.io/library/trinkets/61766a27c2&quot; rel=&quot;nofollow noreferrer&quot;&gt;EMwave on trinket.io&lt;/a&gt; -  an interactive 3D animation that shows how the Faraday and Ampere-Maxwell laws propagate the electric and magnetic fields in a plane electromagnetic wave.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;desmos&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/yc8hwkitvy&quot; rel=&quot;nofollow noreferrer&quot;&gt;add Covectors (equipotenials)&lt;/a&gt; (draft version)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/ivoa6mgsua&quot; rel=&quot;nofollow noreferrer&quot;&gt;covectors map a vector&lt;/a&gt; (draft version)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/u7abq6fvgo&quot; rel=&quot;nofollow noreferrer&quot;&gt;161-robphy-Electric-VectorField-Superposition-v2&lt;/a&gt; (draft)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/qjoxqoeh7h&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adding Equipotentials - point charges&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/wokup9ah7z&quot; rel=&quot;nofollow noreferrer&quot;&gt;Circular Wave Interference on a Screen&lt;/a&gt; (play the time variable)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/emqe6uyzha&quot; rel=&quot;nofollow noreferrer&quot;&gt;spacetime diagrammer for relativity v.8e-2021 (time upward)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://www.geogebra.org/u/robphy&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geogebra.org/u/robphy&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/m/vkhy5eza&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vector Fields, Divergence and Curl, and Flux and Circulation Integrals&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/m/sjzxecxm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Spherical Coordinates&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/m/y8rguazh&quot; rel=&quot;nofollow noreferrer&quot;&gt;Refraction-Snell Law&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/m/pr63mk3j&quot; rel=&quot;nofollow noreferrer&quot;&gt;CircularLightClocks-VisualizingProperTimeInSpecialRelativity&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/m/XFXzXGTq&quot; rel=&quot;nofollow noreferrer&quot;&gt;Relativity-LightClock-MichelsonMorley-2018&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/m/HYD7hB9v&quot; rel=&quot;nofollow noreferrer&quot;&gt;Relativity on Rotated Graph Paper (robphy) - MAA2016&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/m/HWW47d6R&quot; rel=&quot;nofollow noreferrer&quot;&gt;Virtual Torque Lab&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/m/DemUu87n&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vernier Caliper&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10537" LastEditorUserId="10537" LastEditDate="2022-08-17T04:00:53.503" LastActivityDate="2022-08-17T04:00:53.503" CommentCount="2" CommunityOwnedDate="2022-08-09T22:33:06.387" ContentLicense="CC BY-SA 4.0" />
  <row Id="25469" PostTypeId="2" ParentId="18785" CreationDate="2022-08-07T21:05:05.183" Score="0" Body="&lt;p&gt;I wish there were more answers to this!! I agree I love this reading program and I've been looking for a no-brainer math approach that requires little to no lesson prep. I think I'll probably use a traditional &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/158947323X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;workbook&lt;/a&gt;. I also found this &lt;a href=&quot;https://youtu.be/PSN7WhTrGxY&quot; rel=&quot;nofollow noreferrer&quot;&gt;video&lt;/a&gt; helpful. I'm curious about brilliant.org and a friend recommended Mathnasium learning center. I wish there was something like the 100 lesson math book!!&lt;/p&gt;&#xA;" OwnerUserId="20188" LastEditorUserId="20188" LastEditDate="2022-08-07T21:06:16.213" LastActivityDate="2022-08-07T21:06:16.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25470" PostTypeId="2" ParentId="25459" CreationDate="2022-08-08T01:04:22.030" Score="0" Body="&lt;p&gt;It depends on the context. For example, in physics we often represent constants with letters. For example, the equation for gravitational force between two masses (&lt;span class=&quot;math-container&quot;&gt;$m_1$&lt;/span&gt;and &lt;span class=&quot;math-container&quot;&gt;$m_2$&lt;/span&gt;) separated by a distance &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is given by &lt;span class=&quot;math-container&quot;&gt;$F = G(m_1m_2/r^2)$&lt;/span&gt; where G is a constant.&lt;/p&gt;&#xA;&lt;p&gt;But in mathematics we might write &lt;span class=&quot;math-container&quot;&gt;$y = ax^2 + bx + c$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; are constants, but we call them coefficients.&lt;/p&gt;&#xA;&lt;p&gt;So, what is the difference? The difference is that in the physics expression G stands for a known value, &lt;span class=&quot;math-container&quot;&gt;$6.6743 x 10^-11 m^3/kgs^2$&lt;/span&gt;, but in the math example, the coefficients are arbitrary and unspecified. They are constants in that they do not change like x and y, but arbitrary in that we haven't assigned a specific value to them.&lt;/p&gt;&#xA;&lt;p&gt;In the sense of how they operate, coefficients are constants. Whether we call them constants or coefficients depends on whether we've actually actually assigned a value to them.&lt;/p&gt;&#xA;" OwnerUserId="8894" LastActivityDate="2022-08-08T01:04:22.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25472" PostTypeId="2" ParentId="17952" CreationDate="2022-08-08T10:12:41.537" Score="4" Body="&lt;p&gt;Adding to what has been said:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;These sentences in a proof are equivalent:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt; be a polygon of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt; be an &lt;strong&gt;arbitrary&lt;/strong&gt; polygon of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices.&lt;/li&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt; be an &lt;strong&gt;arbitrary&lt;/strong&gt; polygon and suppose that &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices.&lt;/li&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt; be an &lt;strong&gt;arbitrary&lt;/strong&gt; polygon such that &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The adjective ‘arbitrary’ (alternatively: ‘&lt;strong&gt;any&lt;/strong&gt;’) is frequently tacit—as in the OP's given sentence—as the intended meaning can be inferred from the context. So, as suggested by user21820, an immediate solution to the issue raised by the OP of clarifying Mathematical English is to not omit it from the sentence.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Concluding the proof and asserting its result &lt;span class=&quot;math-container&quot;&gt;$\psi(P_n):$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;For each&lt;/strong&gt; polygon &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices, &lt;span class=&quot;math-container&quot;&gt;$\psi(P_n).$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall P_n{\in}\{x \mid x \text{ is a polygon of $n$ vertices}\}\:\:\psi(P_n).$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This implicitly invokes &lt;a href=&quot;https://en.wikipedia.org/wiki/Universal_generalization&quot; rel=&quot;nofollow noreferrer&quot;&gt;Universal Introduction&lt;/a&gt; to &lt;strong&gt;generalise a representative (any &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt;) to the universal (each &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt;).&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The word ‘let’ per se does not connote universal (or existential) quantification; it merely signifies that a definition/restriction/value is to be assigned.&lt;/p&gt;&#xA;&lt;p&gt;However, having answered such questions on Mathematics.Stackexchange&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My issue comes with statements like &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; be an integer&amp;quot;. I really do not know how I should intuitively interpret such a statement.  Should I interpret it like how I interpret &amp;quot;Suppose &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is an integer&amp;quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What does &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; be a group&amp;quot; even mean?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I agree with user21820 that the more specific phrasing&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Consider&lt;/em&gt; an arbitrary polygon &lt;span class=&quot;math-container&quot;&gt;$P_n$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;is probably clearer.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Here's a comparison of the various translations of &lt;span class=&quot;math-container&quot;&gt;$$∀x{\in}F\; \psi(x).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;“&lt;strong&gt;For all&lt;/strong&gt; elements &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$F,\,\psi(x)$&lt;/span&gt; holds” sometimes sounds like the property &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; might belong to &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; as a whole rather than to its individual members: “for all members of the family, they have a house” (1 house in total? or 5?). Contrast with “for each member of the family, they have a house” (definitely 5 houses in total). (&lt;a href=&quot;https://matheducators.stackexchange.com/a/26771/372&quot;&gt;“For all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” is Mathlish.&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;“&lt;strong&gt;For every&lt;/strong&gt; element &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$F,\,\psi(x)$&lt;/span&gt; holds”, despite ‘every’ too having a collective sense, says that the property &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; is common to the members of &lt;span class=&quot;math-container&quot;&gt;$F.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;“&lt;strong&gt;For each&lt;/strong&gt; element &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$F,\,\psi(x)$&lt;/span&gt; holds” directly attributes the property &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; to individual members of &lt;span class=&quot;math-container&quot;&gt;$F.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;“&lt;strong&gt;For any&lt;/strong&gt; element &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$F,\,\psi(x)$&lt;/span&gt; holds” doesn't strongly communicate that the property &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; belongs to each and every member of &lt;span class=&quot;math-container&quot;&gt;$F;$&lt;/span&gt; nevertheless, “for any &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” typically means “&lt;strong&gt;given an arbitrary&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;”, which, logically, is synonymous with “&lt;strong&gt;for every&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” and “&lt;strong&gt;for each&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;”.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In any case, when writing and teaching mathematics, the word ‘&lt;strong&gt;any&lt;/strong&gt;’ should be used judiciously, because its meaning alternates between ‘every’ and ‘some’, or could be ambiguous.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;“&lt;em&gt;&lt;strong&gt;If any&lt;/strong&gt;&lt;/em&gt; intruder enters, the alarm will sound.”&lt;/p&gt;&#xA;&lt;p&gt;Here, ‘if any’ means ‘&lt;em&gt;&lt;strong&gt;if some&lt;/strong&gt;&lt;/em&gt;’ instead of ‘if every’.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;“He does &lt;em&gt;&lt;strong&gt;not&lt;/strong&gt;&lt;/em&gt; have &lt;em&gt;&lt;strong&gt;any&lt;/strong&gt;&lt;/em&gt; pet.”&lt;/p&gt;&#xA;&lt;p&gt;Here, ‘not any’ means ‘&lt;em&gt;&lt;strong&gt;not some&lt;/strong&gt;&lt;/em&gt;’ instead of ‘not every’.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Disjoint_sets&quot; rel=&quot;nofollow noreferrer&quot;&gt;definition of set disjointedness&lt;/a&gt;: “A collection of two or more sets is disjoint &lt;em&gt;&lt;strong&gt;if any&lt;/strong&gt;&lt;/em&gt; two distinct sets of the collection are disjoint.”&lt;/p&gt;&#xA;&lt;p&gt;Are three sets are disjoint &lt;em&gt;&lt;strong&gt;if every&lt;/strong&gt;&lt;/em&gt; pair of distinct sets is disjoint or &lt;em&gt;&lt;strong&gt;if some&lt;/strong&gt;&lt;/em&gt; pair of distinct sets is disjoint? It turns out from reading between the lines two sections down that Wikipedia intends the ‘if every’ reading, even as the ‘if some’ reading is probably more idiomatic. No wonder &lt;a href=&quot;https://math.stackexchange.com/a/4012997/21813&quot;&gt;authors disagree between these two contrasting definitions&lt;/a&gt;!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-09-15T09:52:20.433" LastActivityDate="2023-09-15T09:52:20.433" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25473" PostTypeId="1" CreationDate="2022-08-08T11:44:24.137" Score="10" ViewCount="934" Body="&lt;p&gt;When I was in late primary and middle school (east coast US, early 1990's), we were assigned a lot of word problems of the following general form:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mary has eight self-sealing stem bolts. She sells half to John and then finds an additional self-sealing stem bolt on the playground. How many self-sealing stem bolts does she now have?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The obvious way to solve this is to show:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;8 - (8/2) + 1 = &lt;strong&gt;5&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My teachers expressed to me in no uncertain terms that this was absolutely wrong. The correct answer was:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;8 self-sealing stem bolts - ( 8 self-sealing stem bolts / 2 ) + 1 self-sealing stem bolt = &lt;strong&gt;5 self-sealing stem bolts&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Sometimes, if we failed to fully clarify the units of the answer, the teacher would mock us by saying things like, &amp;quot;Five? Five what? Five elephants? You need to be more specific.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This always struck me as redundant and pedantic. The wording of the question indicates that any response must be addressed in terms of self-sealing stem bolts and not any other unit (e.g. elephants, crystals, spiral notebooks, or planet-killing photon cannons) to qualify as a reasonable answer, so an answer of &amp;quot;5&amp;quot; is in fact perfectly clear and unambiguous when taken in context with the question asked and not as a lone number floating out their in the void of context-free and content-free nothingness.&lt;/p&gt;&#xA;&lt;p&gt;Certainly, if I was asked today on the job, &amp;quot;Hey, how many vials of delta-omega varuonic acid do we have left on the shelf?&amp;quot;, I would give an answer like, &amp;quot;Hey, yeah, looks like five.&amp;quot;, rather than pedantically wasting everyone's time by saying, &amp;quot;The answer to your question is that there are five vials of delta-omega varuonic acid on the shelf.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I am aware that tracking units throughout is useful when doing dimensional analysis, but the vast majority of problems that we were assigned involved only one dimension or unit.&lt;/p&gt;&#xA;&lt;p&gt;A small number of teachers went even &lt;em&gt;beyond&lt;/em&gt; this and required all answers to word problems to be expressed in complete sentences. For them, an answer of &amp;quot;5 self-sealing stem bolts&amp;quot; wasn't good enough, the correct answer was &amp;quot;Mary has 5 self-sealing stem bolts.&amp;quot;, but discussing that is a question for another day.&lt;/p&gt;&#xA;&lt;p&gt;So, I can come up with the following cases in which requiring students to disclose, declare, or identify the units of their steps or answer might be considered pedagogically relevant:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The problem inherently requires dimensional analysis/unit conversion in such a way as to require the student to track and record the units in order to identify the correct mathematical operations to perform.&lt;/li&gt;&#xA;&lt;li&gt;Identifying the units of the answer is part of the problem itself rather than stated within the question. For example, asking students &amp;quot;What is Ann's speed?&amp;quot; may require them to recognize the difference between units of speed, time, and distance, and so having the student identify which unit is the speed unit demonstrates learning.&lt;/li&gt;&#xA;&lt;li&gt;The problem is worded in such a way as to give the student choice over which units to give the answer in. For example, a question requiring the student to &amp;quot;Provide the length&amp;quot; might allow the student to choose whether to express the length in terms of meters, centimeters, inches, furlongs, or FIFA-regulation soccer field lengths, and thus the grader must know which units the student chose in order to grade their answer.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Other than the cases above, is there pedagogical value in requiring students to explicitly state the units of answers when the appropriate units are already clear from context?&lt;/p&gt;&#xA;" OwnerUserId="7481" LastEditorUserId="7481" LastEditDate="2022-08-09T09:37:23.063" LastActivityDate="2022-12-31T04:18:07.203" Title="What value is there in requiring students to declare the dimensions of an answer when it is already clear from context?" Tags="|primary-education|middle-school|language-use|word-problems|" AnswerCount="6" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="25474" PostTypeId="1" CreationDate="2022-08-08T12:04:21.330" Score="-1" ViewCount="51" Body="&lt;p&gt;&lt;em&gt;What can be the most appropriate way of asking , Solve the problem or answer the question ?&lt;/em&gt; &lt;br/&gt;&#xA;As common usage in examinations we have question papers for all subjects. Therefore like in other subjects we normally use to say answer following questions but I suggest specifically from secondary education it's much better to say solve following problems rather than asking to answer questions. I would like to get your attention to maths olympiad question papers, there they present mathematics challenges as problems. &lt;br/&gt;&#xA;By the meaning, &lt;em&gt;problem&lt;/em&gt; is a difficulty that has to be resolved or dealt with but &lt;em&gt;question&lt;/em&gt; is a sentence, phrase or word which ask for information.&lt;br/&gt;&#xA;To find the best solution to a problem you need to ask questions and questions can be created even by the solution of a problem too. &lt;br/&gt; I would like to share my personal experience regarding the difference between question answering and problem solving. Very recently when I came across an IQ challenge related to moving ants along sides of a triangle I asked the question to myself what if we do this for any convex polygon and then what about if ants are allowed to move along diagonals too. Eventually by the way I questioned myself I could able to introduce a new concept and could able to publish 12 integer sequences on OEIS.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="2083" LastEditDate="2022-08-08T13:45:44.880" LastActivityDate="2022-08-08T13:50:52.767" Title="Problem solving or question answering" Tags="|secondary-education|problem-solving|vocabulary|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25475" PostTypeId="2" ParentId="25474" CreationDate="2022-08-08T13:50:52.767" Score="3" Body="&lt;p&gt;This is certainly highly language-dependent.&#xA;In Finnish I would expect to read &amp;quot;tehtävä&amp;quot;, while in Scandinavic languages &amp;quot;oppgave/opgave/oppgåva&amp;quot; (not sure of Swedish), all of which might be translated to a task, or a thing to be done. To my ears a problem could mean something negative; an issue, a thing that is wrong.&lt;/p&gt;&#xA;&lt;p&gt;But regardless of the conventional use of the words, they quickly take on their own particular meaning in the context of mathematics exams. It is a pretty strong context and there is lots of jargon in the school world anyway. I doubt there is a much confusion caused or a large effect in what kinds of things pupils or students write down or draw as a result of this one word.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-08-08T13:50:52.767" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25476" PostTypeId="2" ParentId="25460" CreationDate="2022-08-08T14:59:31.320" Score="4" Body="&lt;h2&gt;Desmos&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/qjieqzjkwn&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clickable Venn diagram with three sets and universe set&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/rlzb0f0htq&quot; rel=&quot;nofollow noreferrer&quot;&gt;Unit circle&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/8hsvgtc8ni&quot; rel=&quot;nofollow noreferrer&quot;&gt;Secant lines motivating instantaneous rate of change&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/van4xnlmbd&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tangent lines&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/mecegleoc7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rolle's Theorem&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/idqb3ay2bu&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mean Value Theorem (for derivatives)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/zvstto4sxl&quot; rel=&quot;nofollow noreferrer&quot;&gt;Slopes/concavity for building questions from&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/8yefm5pqnf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inverse Function Theorem&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/5oolvnip1z&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann sums with arbitrary (random) sample points&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/uzqnhhvgpo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann sums with left/midpoint/right sample points&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/2zv6t8nqbt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Upper and lower Darboux sums&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/9tu5yykd6n&quot; rel=&quot;nofollow noreferrer&quot;&gt;Method of exhaustion for area of a circle (part of a scaffolded assignment I give the students)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/mrhwpgtsta&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann sums for areas between curves&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/6d7hdrcmz3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Areas between curves&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/3phfvpgdqz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trapezoidal rule&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/sxhnqwmbbf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simpson's rule&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/23enhbhkjl&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arc length&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.desmos.com/calculator/mj5j5uhnsa&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fourier series (&lt;strong&gt;warning&lt;/strong&gt;: slow)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2&gt;Geogebra&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/uyjtqubv&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes with equilateral triangle cross sections&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/ahsyzgsd&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes with isosceles right triangle cross sections&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/tjaft7gb&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes with square cross sections&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/uugkfsjh&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes with semicircular cross sections&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/ygpyk9ck&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes with circular cross sections&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/rwnumpnt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes with washers&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/xn8aqapd&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes with washers&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/rwpev3xy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes with cylindrical shells&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/m6x96fcr&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes formed by rotating around horizontal line&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/pyqgb2tt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volumes formed by rotating around vertical line&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.geogebra.org/calculator/a9unxggh&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cavalieri's principle&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2&gt;Colab&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://colab.research.google.com/drive/19WaqfaySfpdoqmGdar_p_LNwiBQAeRS_?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Numerical integration (trapezoidal/Simpson's rule)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://colab.research.google.com/drive/1cMjBTwM9Ly6e6AWagaWj4-EKm-EkbSmj?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Taylor approximations&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="45" LastActivityDate="2022-08-08T14:59:31.320" CommentCount="0" CommunityOwnedDate="2022-08-08T14:59:31.320" ContentLicense="CC BY-SA 4.0" />
  <row Id="25478" PostTypeId="2" ParentId="2551" CreationDate="2022-08-08T17:17:12.083" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Everything that’s already in the world when you’re born is just normal.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Anything that gets invented between then and before you turn thirty is incredibly exciting and creative and with any luck you can make a career out of it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Anything that gets invented after you’re thirty is against the natural order of things and the beginning of the end of civilisation as we know it until it’s been around for about ten years when it&#xA;gradually turns out to be alright really.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Douglas Adams&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In today's times when general computing devices that often fit in our pockets are universally available this is an important question.  And this is because they are effectively '&lt;em&gt;the natural order of things&lt;/em&gt;': a student today is unlikely to have &lt;strong&gt;not&lt;/strong&gt; handled computing devices today, a sharp inversion of the general situation that obtained from the beginning of civilization (and therefore math) till about 30 years ago.&#xA;So a student of today asked to do a computation is very likely — more so if they are bright — to ask the teacher:&lt;br /&gt;&#xA;&lt;em&gt;Why do you ask me to do this repetitive boring task when I can simply ask my computer (calculator etc) to do it for me?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[I'm using 'computer' for phone, tablet, old-fashioned desktop, cloud services etc etc]&lt;/p&gt;&#xA;&lt;h1&gt;The General Answer&lt;/h1&gt;&#xA;&lt;p&gt;To be able to give a satisfactory answer is non-trivial.&#xA;I guess the answer in its full generality is that computers — and their weaker cousins, calculators — are essentially &lt;em&gt;mathematons&lt;/em&gt;.  This is true historically, pragmatically and ontologically:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;History: Church, Turing, Gödel... Babbage, Leibniz, Pascal... &lt;em&gt;all&lt;/em&gt; were mathematicians. And their computing pioneering came from attempts to do math.&lt;/li&gt;&#xA;&lt;li&gt;Ontics: At the core of a computer is a 'CPU'; at whose core is an 'ALU' which expands to 'arithmetic-logic-unit.  So arithmetic (and logic) is where it starts and all the multile layers of CS abstracted over never lose this fact&lt;/li&gt;&#xA;&lt;li&gt;Pragmatics: The well-rounded study of computer science needs (what CS-ists call) 'theory' which is just a specialized branch of math&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I expect that this general answer is too general for the school level at which OP is asking so following&#xA;I'll take an extended extract from a suggestion of an old doyen of computer algebra systems re the use of such systems for math.&#xA;This can then be tailored and specialized to give more accessible and calculator-oriented answers.&lt;/p&gt;&#xA;&lt;h1&gt;Buchberger's CAS answer&lt;/h1&gt;&#xA;&lt;p&gt;In 1989, Bruno Buchberger, a pioneer of computer algebra wrote this paper: &lt;a href=&quot;https://dl.acm.org/doi/10.1145/382276.1095228&quot; rel=&quot;nofollow noreferrer&quot;&gt;Should students learn integration rules&lt;/a&gt;. In this he formulated the &amp;quot;Blackbox-Whitebox principle&amp;quot;&#xA;He starts by stating the situation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is well known that these software systems (for example, MACSYMA , SCRATCHPAD, REDUCE, MATHEMATICA , ...) provide mathematical problem solving power that outperforms students who have passed, say, the usual introductory calculus and linear algebra courses. For example, integration, the eye-catcher of our note, is much more deeply, more systematically, and, of course, much more efficiently handled in symbolic math systems than was ever possible in calculations by humans. There is no inherent limit to the scope, in breadth and depth, that symbolic math systems will be able to achieve in the future. For example, by recent advances, even theorem proving — the essential nucleus of mathematical activit — has been trivialized in certain areas of mathematics, e.g. geometrical theorem proving.  In this context, I would like to call an area of mathematics &amp;quot;trivialized&amp;quot; as soon as there is a (feasible, efficient, tractable) algorithm that can solve any instance of a problem from this area.&lt;/p&gt;&#xA;&lt;p&gt;In this sense the area of arithmetic on the natural numbers is trivialized because there exist algorithms for addition and multiplication on the natural numbers in the &amp;quot;symbolic representation&amp;quot; as decimals. (Note that arithmetic, in early stages of mathematics, was not &amp;quot;trivial&amp;quot; at all and humans who were able to perform arithmetical operations even in a limited number range were deemed to be &amp;quot;intelligent&amp;quot; far beyond average.)&lt;/p&gt;&#xA;&lt;p&gt;The area of integrating functions described by elementary transcendental expressions (i.e. expressions involving variables, arithmetical operators, log, and exp) is trivialized: Risch's algorithm decides, in finitely many steps, whether the integral of a given elementary transcendental function is again elementary transcendental and, in the positive case, produces the expression that describes the integral. Furthermore, this algorithm, in combination with heuristic methods, is efficient and, on modern hardware, outperforms humans by orders of magnitude.&lt;/p&gt;&#xA;&lt;p&gt;Similarly, geometrical theorem proving is trivialized: Wu's algorithm or the Gröbner basis algorithm decide in finitely many steps whether or not a given geometrical theorem that can be expressed by equational hypotheses and an equational conclusion is true or not (and, again, these algorithms work amazingly fast on quite non-trivial examples).&lt;/p&gt;&#xA;&lt;p&gt;Given the present and future power and potential of symbolic math systems it is therefore near at hand to pose the following general question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Should math students learn area X of mathematics when this area has been trivialized?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think this question has a very simple answer,...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One extreme answer to our question is:&lt;br /&gt;&#xA;No, students need not learn itegration rules any more or, more generally, as soon as an area of mathematics is trivialized, math (and other) students should not be tortured with it any more.&lt;/p&gt;&#xA;&lt;p&gt;Another extreme answer:&lt;/p&gt;&#xA;&lt;p&gt;For math education, ignore the existence of integration software and, in general, of symbolic math software systems.&#xA;[And for the OP's question — calculators!]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Buchberger goes on to argue why both these extreme answers are unsatisfactory and states his&lt;/p&gt;&#xA;&lt;h1&gt;Blackbox-Whitebox principle&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I think it is totally inappropriate to answer such a question by a strict &amp;quot;yes&amp;quot; or &amp;quot;no&amp;quot;. Rather, the answer depends on the stage of teaching area X.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In the stage where area X is new to the students, the use of a symbolic software system realizing the algorithms of area X as black boxes would be a disaster (see however the remark at the end of Subsection 3.4). Students have to study the area thoroughly, i.e. they should study problems, basic concepts, theorems, proofs, algorithms based on the theorems, examples, hand calculations.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In the stage where area X has been thoroughly studied, when hand calculations for simple examples become routine and hand calculations for complex examples become intractable, students should be allowed and encouraged to use the respective algorithms available in the symbolic software systems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Example 1 (Integration) Heuristic solutions to the problem of integration (of elementary transcendental functions) give rise to a wealth of mathematical concepts, insights and techniques that are, of course, of central importance even if one knows that ultimately the problem can be handled by a software system. This is even more true if one seeks completely algorithmic solutions to the problem. The mathematics necessary for Risch's integration algorithm (e.g. Liouville theorems) goes far beyond what is normally taught in undergraduate analysis courses and, therefore, can form an extremely worthwhile block of mathematical concepts, knowledge and techniques.&lt;/p&gt;&#xA;&lt;p&gt;As long as these concepts, theorems and techniques are new to the students they are a worthwhile subject of a curriculum. And, as long as the algorithm based on these concepts, theorems and techniques is not routine for the student, hand calculations in examples is a necessary and worthwhile actavity during education.&lt;/p&gt;&#xA;&lt;p&gt;However, at the stage where integration theory and algorithmics is taught, algorithmic prerequisites from earlier areas as, for example, partial fraction decomposition, squarefree factorization.., should not be the subject of hand calculation but a symbolic software system should be used for this.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="17672" LastEditorUserId="17672" LastEditDate="2022-08-08T18:16:58.320" LastActivityDate="2022-08-08T18:16:58.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25479" PostTypeId="2" ParentId="25473" CreationDate="2022-08-09T09:51:44.250" Score="6" Body="&lt;p&gt;I think it is ridiculous to keep repeating &amp;quot;self-sealing stem bolts&amp;quot; over and over in order to maintain units throughout a computation. But there is important benefit in keeping correct units. Practically, it is just a natural part of being &lt;strong&gt;precise&lt;/strong&gt;, which may have safety consequences (e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Gimli_Glider&quot; rel=&quot;noreferrer&quot;&gt;Gimli glider&lt;/a&gt;). For this reason alone, I would say there is no clear benefit in having incorrect units.&lt;/p&gt;&#xA;&lt;p&gt;Yet, there is a far &lt;strong&gt;better&lt;/strong&gt; alternative to what you said your teachers expected you to write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; be the initial number of self-sealing stem bolts that Mary had, and &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; be the final number. Then &lt;span class=&quot;math-container&quot;&gt;$r = n-n·\frac12+1 = 8-8·\frac12+1 = 8-4+1 = 5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No units have obscured the mathematics, but the solution is precise!&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2022-08-09T09:51:44.250" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25480" PostTypeId="2" ParentId="14731" CreationDate="2022-08-09T14:19:08.553" Score="-1" Body="&lt;h3&gt;These are limits&lt;/h3&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&amp;quot;the price of a product has a limit value that, from this price, the&#xA;number of sales start to decrease, reducing the benefits&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;the light speed has a limit&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;the population of animals is limited due to the amount of food&amp;quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Each of these items above does, in fact, display &lt;a href=&quot;https://en.wikipedia.org/wiki/Limit_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;limit&lt;/a&gt; behavior.&lt;/p&gt;&#xA;&lt;p&gt;The economics of pricing and supply/demand is replete with curves and optimization that &lt;a href=&quot;https://www.investopedia.com/terms/d/demand-curve.asp&quot; rel=&quot;nofollow noreferrer&quot;&gt;display limit behavior&lt;/a&gt;. Light speed &lt;a href=&quot;https://www.britannica.com/science/time-dilation&quot; rel=&quot;nofollow noreferrer&quot;&gt;exhibits limit behavior with time&lt;/a&gt;. As you asymptotically approach the speed of light, the effects of time dilation increase without bound. Animal population graphs are also asymptotic. As you add animals to an ecosystem, each of them is able to eat less and becomes slightly more likely to die of malnutrition or conflict. As time T goes to infinity, the animal population tends to an equilibrium expressed by the point at which the death rate of animals (due both to natural causes as well as causes related to food shortages) exceeds the maximum sustainable birth rate. This is the population limit.&lt;/p&gt;&#xA;&lt;p&gt;To be clear, the first and third cases are not fully continuous (e.g. you can't increase the price of a good by 1/100,000,000th of a cent or increase the population of deer by 0.0000001 deer) and thus cannot be completely modeled by algebraic functions, but they can (and often are) modeled by curve-matching functions that do demonstrate limit behavior. The second is also likely not purely continuous due to quantum behavior, but it will be much closer and so your model will be much more accurate.&lt;/p&gt;&#xA;" OwnerUserId="7481" LastEditorUserId="7481" LastEditDate="2022-08-09T16:05:10.093" LastActivityDate="2022-08-09T16:05:10.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25481" PostTypeId="2" ParentId="25473" CreationDate="2022-08-09T15:29:33.693" Score="7" Body="&lt;p&gt;Different STEM fields have different conventions about this. These conventions are adapted to the needs of the field, so in general if the work you're doing has the flavor of field X, you will probably benefit from doing what's conventional in field X -- the convention was probably adopted by those people because it made their work either more efficient or less prone to error.&lt;/p&gt;&#xA;&lt;p&gt;The field that has the most clearly defined conventions is physics, where the vast majority of numbers have units, and there is a variety of units in use for the same thing. So for instance, one typically uses the kilogram as the base unit for mass these days, but it wouldn't be surprising to find the gram used in some contexts (and the gram is the non-prefixed unit). The physics convention is spelled out in style guides for journals and is also used consistently in writing such as freshman physics textbooks. The convention is basically that you always write the unit explicitly. In this convention, you do &lt;em&gt;not&lt;/em&gt; (usually) write units if the number is dimensionless, so, e.g., it would be fine to write 5 rather than 5 self-sealing stem bolts -- but the latter would be fine if it gives better readability in context.&lt;/p&gt;&#xA;&lt;p&gt;In math, the most common convention is to treat the units as part of the definition of the variable. &amp;quot;Let ρ be the density in units of g/cm3. [calculations...] So we find that ρ=17.38.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In engineering, you see more of a mix of people switching back and forth between the physicsy style and the mathy style. Often engineers are dealing with some awkward mixture of SI and non-SI units, so for example you'll see a lot of equations that look like &lt;span class=&quot;math-container&quot;&gt;$A=0.13784bc^2/\sqrt{d}$&lt;/span&gt;, where the pure SI version would be, say, &lt;span class=&quot;math-container&quot;&gt;$A=3\pi bc^2/\sqrt{d}$&lt;/span&gt;, but they're measuring &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; in bushels or something.&lt;/p&gt;&#xA;&lt;p&gt;So if you're teaching schoolkids, I would suggest that you teach them always to write explicit units on things that have units taken from some arbitrarily defined human-constructed system of measurement, such as the SI. In those contexts, experience shows that it really isn't a good idea just to not write the units, on the theory that the units have already been defined by context.  Failing to do so leads to errors, such as the famous Mars probe that crashed because someone didn't convert meters to feet.&lt;/p&gt;&#xA;&lt;p&gt;Related to this is the use of units as a tool for reasoning. It's extremely common for people to get all the way to college without really having a clear idea of the conceptual meanings of multiplication and division. If they need to solve a problem, and they haven't been told what algorithm to use, they don't know how to tell whether to multiply or divide. One way of building their reasoning ability is to get them in the habit of thinking about units. Then if they have 10 pounds/in2 and 100 in2, they have the mental tools to be able to figure out that multiplying would be what would give you a force in units of pounds.&lt;/p&gt;&#xA;" OwnerUserId="20192" LastEditorUserId="20192" LastEditDate="2022-08-09T15:36:28.033" LastActivityDate="2022-08-09T15:36:28.033" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25482" PostTypeId="2" ParentId="20684" CreationDate="2022-08-09T15:32:16.230" Score="2" Body="&lt;p&gt;Lehrer describes two different &amp;quot;old&amp;quot; methods, commonly referred to as the &lt;em&gt;equal additions method&lt;/em&gt; and the &lt;em&gt;decomposition method&lt;/em&gt;. The descriptions I have seen of these methods use the term &amp;quot;borrow&amp;quot; instead of &amp;quot;carry&amp;quot;. There is an adaptation that can be applied to either method (but more naturally to the equal additions method) that rewrites the subtraction problem as an addition problem, in which case the term &amp;quot;carry&amp;quot; becomes more appropriate. When this adaptation is applied to the equal additions method, the resulting method is known as the &lt;em&gt;Austrian method&lt;/em&gt;, which is described in Steven Alexis Gregory's answer. Since this is not one of the methods Lehrer uses in the introduction to his song, I am slightly puzzled by his use of the term &amp;quot;carry&amp;quot;, but perhaps this was common in schools, possibly owing to a tendency to conflate the equal additions and Austrian methods.&lt;/p&gt;&#xA;&lt;p&gt;The decomposition method is the one commonly taught in American schools today and is identical to the &amp;quot;New Math&amp;quot; method satirized in Lehrer's song. The satire is directed toward the emphasis on &amp;quot;understand[ing] what you're doing&amp;quot; rather than &amp;quot;get[ting] the right answer&amp;quot;, manifested in the constant references to place value, the aside about the commutative law, and the generalization to bases other than 10.&lt;/p&gt;&#xA;&lt;p&gt;The distinction between the equal additions method and the decomposition method is in how one compensates for borrowing. When a ten needs to be borrowed for the ones column, the equal additions method compensates by adding a one to the tens column of the subtrahend, on the principle that &lt;span class=&quot;math-container&quot;&gt;$a-b=(a+10)-(b+10)$&lt;/span&gt;, whereas the decomposition method compensates by subtracting a one from tens column of the minuend. Lehrer demonstrates both for the problem &lt;span class=&quot;math-container&quot;&gt;$342-173$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;if you’re under 35 or went to a private school you say seven from three is six but if you’re over 35 and went to a public school you say eight from four is six, ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is consistent with the history described in Susan Ross and Mary Pratt-Cotter. 1997. &lt;a href=&quot;http://math.coe.uga.edu/tme/issues/v08n1/1ross.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Subtraction in the United States: An Historical Perspective&lt;/a&gt;. &lt;em&gt;Mathematics Educator&lt;/em&gt; 8(1), 49-56. They write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;During the early to middle 1900’s, examples could be found where different textbooks used any combination of the three different algorithms for teaching subtraction... All of this was to change. In November 1937, William A. Brownell conducted a study to determine if a “crutch” in the algorithm of subtraction problems was beneficial. The crutch involved marking through numerals from which an amount was borrowed in order to keep track of the different steps when working a problem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They go on to show that after the end of the 1930s, the decomposition method using the &amp;quot;crutch&amp;quot; became the dominant method taught in American schools. Lehrer's song came out in 1965. The remark about those &amp;quot;under 35&amp;quot; using decomposition, i.e. &amp;quot;seven from three is six&amp;quot;, is consistent with Ross and Pratt-Cotter's chronology.&lt;/p&gt;&#xA;&lt;p&gt;That the New Math method &lt;em&gt;was&lt;/em&gt; the decomposition method is not just my perception. Ross and Pratt-Cotter describe this period:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the 1960’s, the School Mathematics Study Group published a textbook series for the elementary grades, where the focus was on the use of place value. Numbers were written in expanded form and used to explain the subtraction process and algorithm. When subtraction requiring regrouping is first introduced, an abacus is used to illustrate the renaming along with writing the number in expanded form. This process was then very quickly shortened into the algorithm using the crutch. Very little actual change was made in teaching subtraction in spite of the emphasis on place value. The decomposition algorithm was still utilized.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;An article that answers one of your other questions is Enzinger, N. 2014. &lt;a href=&quot;https://digitalcommons.georgefox.edu/soe_faculty/114/&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Investigation of Subtraction Algorithms from the 18th and 19th Centuries&lt;/a&gt;. &lt;em&gt;MAA Convergence: Loci&lt;/em&gt;. About the origin of the decomposition method, also called &amp;quot;simple borrowing&amp;quot;, it says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This algorithm is quite old, dating back to Spain in the thirteenth century, Italy in the Middle Ages, and India even earlier (Smith, 1909). In fact, there is evidence of the decomposition algorithm present in the writings of Rabbi ben Ezra (1140, cited in Smith, 1925).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The text in Figure 8 of the paper, taken from an 1830 book of Daniel Adams is interesting. It describes the decomposition method and uses the term &amp;quot;borrowing&amp;quot;, giving an explanation involving place value. It also briefly describes the equal additions method as an alternative.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2022-08-09T15:32:16.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25483" PostTypeId="2" ParentId="25473" CreationDate="2022-08-10T02:39:13.800" Score="17" Body="&lt;p&gt;In my experience with remedial-level community-college students (USA), it is simply never the case that the units are trivially &amp;quot;clear from context&amp;quot;. I can easily see some of my former students, when prompted for units in the OP's example, saying any of either &amp;quot;people&amp;quot;, &amp;quot;playgrounds&amp;quot;, &amp;quot;feet&amp;quot;, &amp;quot;seconds&amp;quot; (depending on what they most recently saw as an example before the test), etc.&lt;/p&gt;&#xA;&lt;p&gt;Scanning some actual remedial arithmetic tests from a few years ago, noting that every test I gave had a word problem like this, I'd say that 80% of the students were never able to specify any units at all, even though this was a direct and repeated piece of instruction. Secondarily, as soon as there is more than one unit of anything in the problem, it's close to a random guess which one they pick for an answer. Simplified example: &amp;quot;In 2010 there were 1000 people in a school, in 2020 there were 1500 people, what was the percentage increase?&amp;quot; can result in answers with units of years, people, or percent.&lt;/p&gt;&#xA;&lt;p&gt;I would never suggest carrying a very long units name through every expression of a calculation. But generally for word problems I do ask for the answer as a complete sentence with units. While such units may seem obvious to you or I, they are not for everyone, and students regularly get them incorrect -- so I feel a responsibility to test and give correction where needed on that issue.&lt;/p&gt;&#xA;&lt;p&gt;For the complete-sentence aspect, part of my thinking is that such communication is not necessarily verbal (such as in the OP's example), but perhaps more likely to be via email, text messaging, a written report, etc., and so students should have an opportunity to exercise and sharpen that skill before entering a workplace.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2022-12-31T04:18:07.203" LastActivityDate="2022-12-31T04:18:07.203" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25484" PostTypeId="2" ParentId="25460" CreationDate="2022-08-10T03:22:55.077" Score="1" Body="&lt;h2&gt;&lt;a href=&quot;http://www.automatic-algebra.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Automatic-Algebra.org&lt;/a&gt;&lt;/h2&gt;&#xA;&lt;p&gt;I built this website for the purpose of timed self-quizzes on topics that should (mostly) be prerequisite, &amp;quot;automatic&amp;quot; knowledge prior to a number of basic college courses. Available quizzes include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Times tables&lt;/li&gt;&#xA;&lt;li&gt;Negative numbers&lt;/li&gt;&#xA;&lt;li&gt;Order of operations&lt;/li&gt;&#xA;&lt;li&gt;Sets of numbers&lt;/li&gt;&#xA;&lt;li&gt;Special products&lt;/li&gt;&#xA;&lt;li&gt;Finding intercepts&lt;/li&gt;&#xA;&lt;li&gt;Special lines&lt;/li&gt;&#xA;&lt;li&gt;Graphing lines&lt;/li&gt;&#xA;&lt;li&gt;Decimal rounding&lt;/li&gt;&#xA;&lt;li&gt;Decimal relations&lt;/li&gt;&#xA;&lt;li&gt;Decimals to percent&lt;/li&gt;&#xA;&lt;li&gt;Sampling distributions&lt;/li&gt;&#xA;&lt;li&gt;Chained relations&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2022-08-10T03:22:55.077" CommentCount="0" CommunityOwnedDate="2022-08-10T03:22:55.077" ContentLicense="CC BY-SA 4.0" />
  <row Id="25485" PostTypeId="1" CreationDate="2022-08-10T05:04:57.230" Score="9" ViewCount="325" Body="&lt;p&gt;Not sure if this is more appropriate for here or for Math.SE, but here goes:&lt;/p&gt;&#xA;&lt;p&gt;How does one who is self-studying mathematics determine if a textbook is too hard for you?&lt;/p&gt;&#xA;&lt;p&gt;Math is hard in general, but when does a textbook cross that line from being challenging to being nearly intractable?&lt;/p&gt;&#xA;&lt;p&gt;Sometimes I can't tell if I'm just being challenged when I have to re-read one paragraph ten times to understand what the author is saying (even if I understand all the components of their statement individually), or if the book is simply not at the right level for my current background.&lt;/p&gt;&#xA;" OwnerUserId="20196" LastActivityDate="2022-08-11T11:59:18.073" Title="How do you know when a textbook is too difficult for you?" Tags="|textbooks|self-learning|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25486" PostTypeId="1" CreationDate="2022-08-10T09:56:22.117" Score="1" ViewCount="75" Body="&lt;p&gt;My professor asked me to take 20 mins on talking about Cyclic Groups and their practical applications in front of the whole class.&lt;/p&gt;&#xA;&lt;p&gt;How should I start? Can anyone please help me.&lt;/p&gt;&#xA;" OwnerUserId="20150" LastActivityDate="2022-08-10T11:09:55.127" Title="On Teaching Cyclic Groups" Tags="|abstract-algebra|" AnswerCount="1" CommentCount="2" ClosedDate="2022-08-10T22:02:50.563" ContentLicense="CC BY-SA 4.0" />
  <row Id="25487" PostTypeId="2" ParentId="25486" CreationDate="2022-08-10T11:09:55.127" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;Read your course literature to find out about applications of cyclic groups.&lt;/li&gt;&#xA;&lt;li&gt;Think about example cyclic groups and where those might be used.&lt;/li&gt;&#xA;&lt;li&gt;Think of practical examples where something goes in a cycle, and think how to model it as a cyclic group.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-08-10T11:09:55.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25488" PostTypeId="1" AcceptedAnswerId="25574" CreationDate="2022-08-10T15:44:27.860" Score="16" ViewCount="262" Body="&lt;p&gt;I have noticed a common pattern followed by many students in crisis:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They experience a crisis or setback (injury, illness, tragedy, etc)&lt;/li&gt;&#xA;&lt;li&gt;This causes them to miss a lot of class.&lt;/li&gt;&#xA;&lt;li&gt;They may stay away from class longer than they &amp;quot;need to&amp;quot; because of shame:  they feel that since they have been absent, coming back to class will cause their teacher to be critical of them.&lt;/li&gt;&#xA;&lt;li&gt;Once they do come back, they are overwhelmed.  They might need to do twice the amount of work in all of their courses just to catch up.  In classes where early material becomes a needed prerequisite for later material, it can become impossible to follow the course content being presented when they do return.&lt;/li&gt;&#xA;&lt;li&gt;This experience of failure can create long term patterns of giving up under pressure.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I have known many &lt;strong&gt;very&lt;/strong&gt; capable students who fall into this trap and end up flunking out of University.  I could have fallen into this trap myself.  As a high school student and college student I got close on several occasions.  I have a lot of empathy for this kind of situation.&lt;/p&gt;&#xA;&lt;p&gt;Question:  What kinds of policies and practices can institutions and individual faculty members adopt which help students out of this trap?&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="2083" LastEditDate="2022-08-11T06:42:53.797" LastActivityDate="2022-09-10T02:17:33.247" Title="Policies and practices for supporting students in crisis" Tags="|undergraduate-education|psychology|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25489" PostTypeId="2" ParentId="25485" CreationDate="2022-08-11T02:35:50.850" Score="2" Body="&lt;p&gt;I don't know how to decide, in general, whether a book is the right one or not. In particular cases, it might make sense to ask here, with the details.&lt;/p&gt;&#xA;&lt;p&gt;But I have a great answer for the related question, &lt;em&gt;&lt;strong&gt;&amp;quot;How do I approach reading a math book?&amp;quot;&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's very different than reading other books, and takes practice. (If a class I'm teaching uses a good textbook, I require students to take notes on each section we cover. That at least gets them started on actually reading a math book.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.stonehill.edu/compsci/History_Math/math-read.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here is a great article about how to read a math book&lt;/a&gt; that I've modified for my students. (I mostly took out stuff that didn't seem vital, so my students would read the article itself!) One of the authors, Shai Simonson, has written a great book about learning math, &lt;em&gt;&lt;strong&gt;&lt;a href=&quot;http://web.stonehill.edu/compsci/RediscoveringMath/RM.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rediscovering Mathematics&lt;/a&gt;&lt;/strong&gt;&lt;/em&gt;, which you might also find useful. (&lt;a href=&quot;https://www.bookfinder.com/search/?ac=sl&amp;amp;st=sl&amp;amp;ref=bf_s2_a1_t1_1&amp;amp;qi=BocApO.85IhokaQKfjp0CRsqLlE_1660182923_1:7:9&amp;amp;bq=author%3Dshai%2520simonson%26title%3Drediscovering%2520mathematics%2520you%2520do%2520the%2520math%2520classroom%2520resource%2520materials&quot; rel=&quot;nofollow noreferrer&quot;&gt;I avoid amazon, so here's where I'd look to buy it.&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;I tell my students to expect reading the textbook to go slowly, and to expect to read it 3 times. Skim the first time to get a feel for the big picture, read carefully the 2nd time and mark anything that doesn't make sense, and work ahead of the author (in examples) in your third read. I think you'll find all that in this article, and lots more.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-08-11T02:35:50.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25490" PostTypeId="2" ParentId="25485" CreationDate="2022-08-11T04:51:47.447" Score="1" Body="&lt;p&gt;Lovely question; I await better answers than mine; meanwhile here's my provisional one(s)...&lt;/p&gt;&#xA;&lt;h1&gt;tl;dr&lt;/h1&gt;&#xA;&lt;p&gt;(a) tempo (b) flow (c) practice-to-success&lt;/p&gt;&#xA;&lt;p&gt;Note 1: These are related but different enough to merit separate discussions.&lt;br /&gt;&#xA;Note 2: The commonality and differenc s are clearer in music/arts than in math so I'll use that as a running example.&lt;br /&gt;&#xA;[Yeah when I was in school I dangled for quite a while choosing between STEM and music]&lt;/p&gt;&#xA;&lt;h1&gt;Longer Version&lt;/h1&gt;&#xA;&lt;h2&gt;Tempo&lt;/h2&gt;&#xA;&lt;p&gt;Think of a normal person walking, an old disabled person, an infant just beginning to try and an olympic champion at their peak.  Each has vastly different details of movement, but that's too reductionistic. Each has a very different &lt;em&gt;tempo&lt;/em&gt;.  You need to have a sense of the perimeter of your tempo and not fall wildly outside.&lt;/p&gt;&#xA;&lt;p&gt;Note at the start this almost always will require some downscaling. Even the math prodigy will not be able to read math with the speed of novels, newspapers etc.  So at the start you set the limits and then you work within that pushing against (ultimately yourself) a little at a time. Little is important as I show below!!&lt;/p&gt;&#xA;&lt;h2&gt;Flow&lt;/h2&gt;&#xA;&lt;p&gt;Compare &lt;a href=&quot;https://www.youtube.com/watch?v=Qt45HCsDljk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dinnerstein&lt;/a&gt; with &lt;a href=&quot;https://www.youtube.com/watch?v=HlvNKc5pYrk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sokolov&lt;/a&gt; playing &lt;em&gt;the same&lt;/em&gt; Bach. Do they sound same??&lt;/p&gt;&#xA;&lt;p&gt;I'm not going to say which I prefer, still less which you should prefer.&#xA;Reason I'm giving two very different renderings is to show that hi-tempo &lt;em&gt;can&lt;/em&gt; be be enjoyable but is not &lt;em&gt;necessarily&lt;/em&gt; better. One can tarry along enjoying -- so to say -- the flowers by the wayside.&lt;/p&gt;&#xA;&lt;p&gt;When you're unable to digest a book/author X and resonate with Y, it can mean X is hard and Y is easy.  But it can also mean quite simply that Y suits you X doesn't.  Respect your own innate preferences.&lt;/p&gt;&#xA;&lt;p&gt;One of the biggest figures of math-in-Computer-Science, Dijkstra, towards the latter part of his career was given a festschrift that summed up his work. The title is &lt;a href=&quot;https://link.springer.com/book/10.1007/978-1-4612-4476-9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beauty is our Business&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So that's flow, which is kin to tempo but different.&lt;/p&gt;&#xA;&lt;p&gt;But this all applies to 'the great'.  What about all of us who are far from there?  In music as in math... it's practice.&lt;/p&gt;&#xA;&lt;p&gt;However there's helpful, not so helpful, useless and outright damaging practice.&#xA;The physical aspects which apply in music don't carry over much beyond specific instruments and genres.  The psychological aspects are common to both music and math.  I'll try and transmit a very crucial lesson I received from a music professor. And his lesson was:&lt;/p&gt;&#xA;&lt;h2&gt;Practice to Success not to Failure&lt;/h2&gt;&#xA;&lt;p&gt;[Firstly note how he adroitly changes the cliche &lt;em&gt;Practice to perfection&lt;/em&gt;]&lt;br /&gt;&#xA;He said to us: &lt;em&gt;When you walk the walls of a music school you'll often hear this:&lt;/em&gt; Then he sat at the piano and demonstrated:&lt;/p&gt;&#xA;&lt;p&gt;He started playing the well known &lt;a href=&quot;https://www.youtube.com/watch?v=aeEmGvm7kDk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mozart&lt;/a&gt;.&lt;br /&gt;&#xA;At first it was fine.&lt;br /&gt;&#xA;Then he started making mistakes.&lt;br /&gt;&#xA;Then he started showing more and more frustration.&lt;br /&gt;&#xA;And getting worse and worse.&lt;br /&gt;&#xA;Finally he slammed the keyboard and got up.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;Of course a description does not work anything like a real demo; the closest I can think of are the pantomimes of &lt;a href=&quot;https://www.youtube.com/watch?v=3fEy9g2T1l8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Victor Borge&lt;/a&gt; but this was more literal.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;What he said after this was memorable and (I believe) applies to music math or any field where there is significant difficulty to be overcome. He said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I believe this actually damages the nervous system. When you practice, stop when you're doing well not when you are doing badly. &lt;strong&gt;Practice to success not to failure.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I believe this applies very much to math.  A good dose of math — whether reading, proving, problem solving, whatever — should have a resultant residue of a delicious coffee after a good sleep.  &lt;em&gt;It should refresh and invigorate.&lt;/em&gt;  If its leaving a residue of frustration you're doing it wrong.&lt;/p&gt;&#xA;" OwnerUserId="17672" LastEditorUserId="17672" LastEditDate="2022-08-11T07:56:45.120" LastActivityDate="2022-08-11T07:56:45.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25491" PostTypeId="1" AcceptedAnswerId="25499" CreationDate="2022-08-11T05:16:22.860" Score="2" ViewCount="543" Body="&lt;p&gt;Last night I was fiddling with some equations and admittedly, I made a careless mistake because I was exhausted. However, in doing so, I began to question the process of what I used to understand as &amp;quot;cancelling&amp;quot; in fractions. The equation was:&lt;/p&gt;&#xA;&lt;p&gt;(6 ÷ 2)/2&lt;/p&gt;&#xA;&lt;p&gt;I cancelled the divison by 2 with the 2 in the denominator.&lt;/p&gt;&#xA;&lt;p&gt;After some research, I came to realise that a better term is reduction to 1 as it is a better indication of what we are actually doing when simplifying fractions.&lt;/p&gt;&#xA;&lt;p&gt;However, I would like to know the reason why we can't do what I did because if a student asks, I would like to see if there is an explanation beyond BODMAS.&lt;/p&gt;&#xA;&lt;p&gt;Also, I do believe that we can only cancel factors but if a student were to say ((6 ÷ (2)(1))/2, we now have factors that we seem to be able to cancel, but because of the division in the numerator, we cannot. Is there a more articulate and mathematically correct way of putting this?&lt;/p&gt;&#xA;" OwnerUserId="20201" LastEditorUserId="376" LastEditDate="2022-08-11T09:13:34.243" LastActivityDate="2022-08-12T03:20:32.017" Title="Reduction of fractions" Tags="|fractions|arithmetic-operations|arithmetic|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25492" PostTypeId="1" AcceptedAnswerId="25500" CreationDate="2022-08-11T09:01:06.327" Score="-4" ViewCount="345" Body="&lt;p&gt;&lt;strong&gt;Edit re the close vote&lt;/strong&gt;: I guess this 1 of those questions whose on-topic-ness depends on the answer. If the answer is no, then well maybe it's off-topic. But if the answer is yes, then I believe it's very on-topic.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;More specific version of: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/17425/should-eulers-formula-eix-cos-xi-sin-x-be-seen-as-a-definition-rather-th&quot;&gt;Should Euler&amp;#39;s formula $e^{ix}=\cos x+i\sin x$ be seen as a definition rather than something to prove?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://proofwiki.org/wiki/Euler%27s_Formula#Theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler's formula (...'for complex numbers'?)&lt;/a&gt; is simply&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$e^{iz} = \cos(z)+i\sin(z)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And 1 way to prove this is using &lt;em&gt;complex&lt;/em&gt; &lt;a href=&quot;https://proofwiki.org/wiki/Euler%27s_Formula#Proof&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maclaurin series&lt;/a&gt; for the &lt;em&gt;complex&lt;/em&gt; exponential, sine and cosine.&lt;/p&gt;&#xA;&lt;p&gt;Now, there's a way to 'prove' the 'theorem' &lt;span class=&quot;math-container&quot;&gt;$e^{it} = \cos(t)+i\sin(t)$&lt;/span&gt;, which is often said to be 'Euler's formula (...for real numbers?)' without using the original formula above such as by using &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler%27s_formula#Using_power_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maclaurin&lt;/a&gt;. And ok fine, I guess you can indeed get the Maclaurin series expansion of &lt;span class=&quot;math-container&quot;&gt;$e^z$&lt;/span&gt; by defining &lt;span class=&quot;math-container&quot;&gt;$e^z$&lt;/span&gt; simply as the unique solution of &lt;span class=&quot;math-container&quot;&gt;$g:\mathbb C \to \mathbb C, g'=g,g(0)=1$&lt;/span&gt; without explicitly getting &lt;span class=&quot;math-container&quot;&gt;$e^z = e^x(\cos(y)+i\sin(y))$&lt;/span&gt; then you can just let &lt;span class=&quot;math-container&quot;&gt;$z=it$&lt;/span&gt; (i.e. let &lt;span class=&quot;math-container&quot;&gt;$(x,y)=(0,t)$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;But...&lt;/p&gt;&#xA;&lt;p&gt;There's a 'proof' that Wikipedia does, again without using the original formula, '&lt;a href=&quot;https://en.wikipedia.org/wiki/Euler%27s_formula#Using_differentiation&quot; rel=&quot;nofollow noreferrer&quot;&gt;by differentiation&lt;/a&gt;' (&lt;a href=&quot;https://proofwiki.org/wiki/Euler%27s_Formula/Real_Domain#Proof_2&quot; rel=&quot;nofollow noreferrer&quot;&gt;proofwiki does this too&lt;/a&gt;) where we consider the function &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb R \to \mathbb C,f(t) = \frac{\cos(t)+i\sin(t)}{e^{it}}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is indeed well-defined and then show &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is identically 1.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1: So what exactly&lt;/strong&gt; is &lt;span class=&quot;math-container&quot;&gt;$h:\mathbb R \to \mathbb C, h(t)=e^{it}$&lt;/span&gt; here assuming &lt;span class=&quot;math-container&quot;&gt;$e^z$&lt;/span&gt; is defined simply as the unique solution of &lt;span class=&quot;math-container&quot;&gt;$g:\mathbb C \to \mathbb C, g'=g,g(0)=1$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Goal&lt;/strong&gt;: I'm asking if this option is strictly speaking sensible. Like those probability textbooks that talk about independence but are actually strictly speaking nonsensical. I wanna make sure there's no imprecision actually. If it's actually nonsensical, then I wanna be sure to specify that the same way I teach probability without measure theory: 'It's nonsensical but just play along.' And then in this case, &lt;strong&gt;which parts are nonsensical, and how should they be taught?&lt;/strong&gt; Like in probability when we teach some nonsensical parts by 'assuming the function is &amp;quot;well-behaved&amp;quot;' or something.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2&lt;/strong&gt;: Alternatively, is there some textbook that 'proves' Euler's formula for real numbers based on the unique &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; definition?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Anyhoo, guess for Question 1&lt;/strong&gt;: It's &lt;span class=&quot;math-container&quot;&gt;$h(t)=g(0,t)$&lt;/span&gt;. Explicitly:&lt;/p&gt;&#xA;&lt;p&gt;I think &lt;span class=&quot;math-container&quot;&gt;$h(t)=g(0,t)=u(0,t)+iv(0,t)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$g(z)=g(x,y)=u(x,y)+iv(x,y)$&lt;/span&gt; is the unique solution to &lt;span class=&quot;math-container&quot;&gt;$g:\mathbb C \to \mathbb C, g'=g,g(0)=1$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;And then we can indeed show that&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is nowhere zero? (Probably we can show &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is nowhere zero?)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dt} h(t) = ih(t)$&lt;/span&gt;, i.e. we have both &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dt} u(0,t) = -v(0,t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dt} v(0,t) = u(0,t)$&lt;/span&gt;? (I think this is what proofwiki assumes too in &lt;a href=&quot;https://proofwiki.org/wiki/Euler%27s_Formula/Real_Domain/Proof_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;another proof&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$-\frac{d}{dt} h(-t) = -ih(-t)$&lt;/span&gt; ?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As for differentiability of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, well I guess it follows from differentiability (or at least holomorphicity) of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Notes&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;No need for specifics. I can work it out. I just wanna know if Wikipedia is just &lt;strong&gt;doing nonsense or if there's something&lt;/strong&gt; really possible behind 'proving' the 'formula' from an alternate definition of the complex exponential.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Currently, my thought is that &lt;strong&gt;they're doing just some heuristic&lt;/strong&gt; definition that &lt;span class=&quot;math-container&quot;&gt;$e^{(\text{whatever})(x)} = \text{whatever} \ e^{(\text{whatever})(x)}$&lt;/span&gt;. But eh maybe there's a rigorous justification that is somehow possible without getting the formula for &lt;span class=&quot;math-container&quot;&gt;$e^z$&lt;/span&gt; explicitly.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5987" LastEditorUserId="5987" LastEditDate="2022-08-11T16:52:41.657" LastActivityDate="2022-08-16T20:11:03.013" Title="Is it nonsensical to try to 'prove' Euler's 'formula' in real numbers? What is Wikipedia/proofwiki even doing?" Tags="|calculus|reference-request|complex-numbers|" AnswerCount="2" CommentCount="11" ClosedDate="2022-08-16T20:11:23.867" ContentLicense="CC BY-SA 4.0" />
  <row Id="25493" PostTypeId="2" ParentId="25491" CreationDate="2022-08-11T10:25:09.110" Score="5" Body="&lt;p&gt;Cancelling is a shorthand for:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The rule for multiplying fractions, used in reverse&lt;/li&gt;&#xA;&lt;li&gt;Any number (except 0) divided by itself is 1&lt;/li&gt;&#xA;&lt;li&gt;The rule for multiplying by 1&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{ac}{bc} = \frac{a}{b} \cdot \frac{c}{c} = \frac{a}{b} \cdot 1 = \frac{a}{b}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-08-11T10:25:09.110" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25494" PostTypeId="1" CreationDate="2022-08-11T11:27:48.120" Score="0" ViewCount="124" Body="&lt;p&gt;I know what are PDEs, but I am looking to identify the major strengths of PDEs. If I have to convince a pool of engineers to use PDEs for solving a problem, let's say stress distribution in a body. What are the top three strengths I should pitch to convince them? Can I mathematically explain those strengths?&lt;/p&gt;&#xA;" OwnerUserId="20202" LastActivityDate="2023-01-08T12:08:00.927" Title="What are specific set of tools Partial Differential Equations provide in studying a system?" Tags="|calculus|applied-mathematics|differential-equations|engineering-mathematics|physical-sciences|" AnswerCount="1" CommentCount="7" ClosedDate="2023-01-08T15:26:11.110" ContentLicense="CC BY-SA 4.0" />
  <row Id="25495" PostTypeId="2" ParentId="25494" CreationDate="2022-08-11T11:52:23.177" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Dimensionality.  Instead of just y as a function of x, little baby problems like first year physics, you are looking at x, y, z (sometimes).  The world of engineers (well at least the whole mech-E branch, mechE and all the associated ones like aero/chem/narch/civil/etc.) is very physical and three dimensional.  Things like bridges and frac jobs and nuclear reactors are all systems with multiple dimensions.  You can also look at multiple output dimensions (heat, pressure, etc.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Use in their curriculum.  This is a COMPLETELY LEGITIMATE and TRENCHANT argument.  You don't need to wax on about the wonders of math and Andrew Wiles and proofs.  But be honest.  There is a reason why they have an engine math course with a sampler of PDEs (or get an intro in their ODE class).  You need it ESPECIALLY for topics with flow (fluids and heat transfer), which are required basic courses for most engineers.  [Not the EEs, but they need a butt-ton of math anyhow since they are modeling signals and such.]  Learning the basic tools in a math course makes it a lot easier to apply in a majors course. If they have to learn all that stuff while they are learning the content, it makes it much harder...learning two things at once.  Also, of course straight math problems are much easier than &amp;quot;word problems&amp;quot;.  And their majors courses are full of word problems.  So this is a chance to learn the techniques first.  So, they are not scared of the freaking &amp;quot;Joe&amp;quot; (J(0)) Bessel function the first time they see it.  Let them have it straight. Remember this is WHY there are jobs for math profs.  To teach techniques to STEM students (math majors are a TINY minority of the population).  And engineers are practical and money oriented.  &amp;quot;This will help you pass fluids class&amp;quot; is a GOOD argument.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Engineers can't count.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P.s.  Even when the PDE is ugly and hard to solve analytically, you can get good insights from the PDE itself.  For example setting rate of change to zero (at equilibrium) and assessing final conditions.  (This does apply to ODEs as well.)  The reason I didn't list this as &amp;quot;3&amp;quot; is because it's a bit of a fine point and makes more sense with kids who are late in their course.  Think it will not sell as well with a neophyte.  But keep it in reserve for later on.&lt;/p&gt;&#xA;" OwnerUserId="20204" LastActivityDate="2022-08-11T11:52:23.177" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25496" PostTypeId="2" ParentId="25485" CreationDate="2022-08-11T11:59:18.073" Score="2" Body="&lt;p&gt;When you are self-studying, it is a much harder environment because (a) you lack external &amp;quot;whip&amp;quot; of the grade.  And (b) you don't have a teacher, lectures, etc.  With that in mind, you need to pick the books that are most beginner friendly.  Especially useful are &amp;quot;programmed instruction&amp;quot; books (e.g. those by Stroud).  Please ignore all the people telling you to learn calculus from Spivak or RA from Rudin.  That's just Internet jocks, pecker-flexing.  You can always go do the Spivak or Rudin LATER.  But pick something easy to start with, that you won't give up on.  Often Amazon reviews are helpful to find such books.  (And also ones that have the answers in the back...very important for self studiers.)&lt;/p&gt;&#xA;" OwnerUserId="20204" LastActivityDate="2022-08-11T11:59:18.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25498" PostTypeId="2" ParentId="25492" CreationDate="2022-08-11T16:51:19.330" Score="0" Body="&lt;p&gt;Ok maybe it's a false alarm. If anyone thinks up any reason for alarm, then feel free to post your own answer.&lt;/p&gt;&#xA;&lt;p&gt;At least for 2, this is what I got:&lt;/p&gt;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$g(x,y)=u(x,y)+iv(x,y)$&lt;/span&gt;, Cauchy Riemann gives us &lt;span class=&quot;math-container&quot;&gt;$u_x=v_y,v_x=-u_y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From that and &lt;span class=&quot;math-container&quot;&gt;$$u(z)+iv(z)=g(z)=g'(z)=\frac{\partial}{\partial x} g(x,y) = u_x(z)+iv_x(z),$$&lt;/span&gt; we get&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$u(x,y)=u_x(x,y)=v_y(x,y),$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$v(x,y)=v_x(x,y)=-u_y(x,y)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$u(0,t)=v_y|_{y=t}(0,y),$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$v(0,t)=-u_y|_{y=t}(0,y)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then -- &lt;strong&gt;This part I'm not so sure is rigorous&lt;/strong&gt; --&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$v_y|_{y=t}(0,y)=\frac{d}{dt}v(0,t) \ ?!$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$-u_y|_{y=t}(0,y)=\frac{d}{dt} -u(0,t) \ ?!$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For 3:&lt;/p&gt;&#xA;&lt;p&gt;Similar I guess.&lt;/p&gt;&#xA;&lt;p&gt;For 1:&lt;/p&gt;&#xA;&lt;p&gt;Probably some contradiction with &lt;span class=&quot;math-container&quot;&gt;$0=g(z_0)=g'(z_0)=u_x(x_0,y_0)=u(x_0,y_0)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For 4:&lt;/p&gt;&#xA;&lt;p&gt;?&lt;/p&gt;&#xA;" OwnerUserId="5987" LastActivityDate="2022-08-11T16:51:19.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25499" PostTypeId="2" ParentId="25491" CreationDate="2022-08-11T17:08:53.107" Score="7" Body="&lt;p&gt;Your first example is equivalent to &lt;span class=&quot;math-container&quot;&gt;$(6\div2)\div2$&lt;/span&gt;. Writing it this way, the erroneous cancelation can be seen as treating division as associative, that is, as writing&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(6\div2)\div2=6\div(2\div2)=6\div1=6.&#xA;$$&lt;/span&gt;&#xA;The mistake in this calculation is that division, like subtraction, is &lt;em&gt;not&lt;/em&gt; associative. Compare the similar erroneous calculation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(6-2)-2=6-(2-2)=6-0=6.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Turning to your second example: by &lt;span class=&quot;math-container&quot;&gt;$((6 ÷ (2)(1))/2$&lt;/span&gt; do you mean &lt;span class=&quot;math-container&quot;&gt;$(6\div(2\times1))/2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$((6\div2)\times1)/2$&lt;/span&gt;? By the precedence rules I am accustomed to it should mean the latter, but this kind of expression is sufficiently confusing to most people that it is best to be more explicit about what you mean by putting in additional parentheses.&lt;/p&gt;&#xA;&lt;p&gt;The reason I ask is that you refer to the &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; in the numerator as a &amp;quot;factor&amp;quot;, which I can only make sense of if you are using the first interpretation (so that &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is a factor in the product &lt;span class=&quot;math-container&quot;&gt;$2\times1$&lt;/span&gt;). Assuming that's the interpretation you had in mind, the reason the cancelation of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;s is not allowed is that it would violate a general principle of cancelation in fractions. That general principle is that cancelation is only possible when both numerator and denominator are products and, in addition, those two products have a factor in common. To emphasize, the numerator and denominator must &lt;em&gt;be&lt;/em&gt; products; it's not enough for them merely to &lt;em&gt;contain&lt;/em&gt; products as subexpressions. So just as you can't cancel the &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;s in &lt;span class=&quot;math-container&quot;&gt;$(1+2\times3)/(2\times4)$&lt;/span&gt;, you can't cancel the &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;s in &lt;span class=&quot;math-container&quot;&gt;$(6\div(2\times1))/2$&lt;/span&gt;. In neither case is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; a factor of both the numerator and the denominator.&lt;/p&gt;&#xA;&lt;p&gt;Returning to your first example, you can write &lt;span class=&quot;math-container&quot;&gt;$(6\div2)/2$&lt;/span&gt;  as &lt;span class=&quot;math-container&quot;&gt;$\left(6\times\frac{1}{2}\right)/2$&lt;/span&gt;. Writing it this way, it is clear that a factor of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; does not cancel a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;. On the other hand, consider &lt;span class=&quot;math-container&quot;&gt;$(6\div2)/(3\div2)$&lt;/span&gt;. This can be written as &lt;span class=&quot;math-container&quot;&gt;$\left(6\times\frac{1}{2}\right)/\left(3\times\frac{1}{2}\right)$&lt;/span&gt;, and cancelation of the factors &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; is legitimate.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2022-08-11T17:08:53.107" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25500" PostTypeId="2" ParentId="25492" CreationDate="2022-08-11T21:29:19.660" Score="1" Body="&lt;p&gt;First develop differential calculus of curves &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \to \mathbb{C}$&lt;/span&gt;.  The cool thing is that if you define the derivative in any sensible way, everything works out nicely.  For example, if you let &lt;span class=&quot;math-container&quot;&gt;$\gamma(t) = u(t) + iv(t)$&lt;/span&gt; for real valued &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, and define &lt;span class=&quot;math-container&quot;&gt;$\gamma'(t) = u'(t) + iv'(t)$&lt;/span&gt;, then you can prove that all of the usual calculus rules work for derivatives including the product rule (where multiplication of complex numbers is used in the codomain), quotient rule (where division of complex numbers is used in the codomain), etc.&lt;/p&gt;&#xA;&lt;p&gt;None of this requires you to understand the derivative of a function &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C} \to \mathbb{C}$&lt;/span&gt;.  None of it uses the Cauchy-Riemann equations.&lt;/p&gt;&#xA;&lt;p&gt;If you were going to try to define &lt;span class=&quot;math-container&quot;&gt;$\gamma(t) = e^{it}$&lt;/span&gt; as a function &lt;span class=&quot;math-container&quot;&gt;$\gamma:  \mathbb{R} \to \mathbb{C}$&lt;/span&gt; it would be reasonable to ask that &lt;span class=&quot;math-container&quot;&gt;$\gamma'(t) = ie^{it} = i\gamma(t)$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$\gamma(0) = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;proof by differentiation&amp;quot; then goes through smoothly.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\gamma: \mathbb{R} \to \mathbb{C}$&lt;/span&gt; be a function with &lt;span class=&quot;math-container&quot;&gt;$\gamma'(t) = i\gamma(t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\gamma(0) = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Define &lt;span class=&quot;math-container&quot;&gt;$C(t) = \frac{\cos(t)+i\sin(t)}{\gamma(t)}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;span class=&quot;math-container&quot;&gt;$\begin{align*}&#xA;C'(t) &amp;amp;= \frac{\gamma(t)(-\sin(t) + i\cos(t)) - \gamma'(t)(\cos(t)+i\sin(t))}{\gamma(t)^2}\\&#xA;&amp;amp;= \frac{\gamma(t)(-\sin(t) + i\cos(t)) - i\gamma(t)(\cos(t)+i\sin(t))}{\gamma(t)^2}\\&#xA;&amp;amp;= \frac{(-\sin(t) + i\cos(t)) - i(\cos(t)+i\sin(t))}{\gamma(t)}\\&#xA;&amp;amp;= 0\\&#xA;\end{align*}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is a constant function. Evaluating at &lt;span class=&quot;math-container&quot;&gt;$t=0$&lt;/span&gt; shows that &lt;span class=&quot;math-container&quot;&gt;$C(t) = 1$&lt;/span&gt;, and the result follows.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit to reply to BCLC's comment.&lt;/p&gt;&#xA;&lt;p&gt;The way that this calculus of complex curves interacts with the calculus of holomorphic functions is as follows:&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$\eta: \mathbb{R} \to \mathbb{C}$&lt;/span&gt; is a curve, and &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{C} \to \mathbb{C}$&lt;/span&gt; is a holomorphic function, then &lt;span class=&quot;math-container&quot;&gt;$f \circ \eta$&lt;/span&gt; is another curve and the definitions of derivative are compatible according to the chain rule&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(f \circ \eta)'(t) = f'(\eta(t))\eta'(t)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here &lt;span class=&quot;math-container&quot;&gt;$f'(\eta(t))$&lt;/span&gt; is the complex derivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$\eta(t)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\eta'(t)$&lt;/span&gt; is as I defined it, and the product is the product of complex numbers.&lt;/p&gt;&#xA;&lt;p&gt;Now let &lt;span class=&quot;math-container&quot;&gt;$\gamma: \mathbb{R} \to \mathbb{C}$&lt;/span&gt; be the function I defined above (which we proved was equal to &lt;span class=&quot;math-container&quot;&gt;$\cos(t) + i\sin(t)$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\exp: \mathbb{C} \to \mathbb{C}$&lt;/span&gt; be defined by &lt;span class=&quot;math-container&quot;&gt;$\exp'(z) = \exp(z)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\exp(0)=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\eta: \mathbb{R} \to \mathbb{C}$&lt;/span&gt; be defined by &lt;span class=&quot;math-container&quot;&gt;$\gamma(t) = it$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I will argue that &lt;span class=&quot;math-container&quot;&gt;$\exp(\eta(t)) = \gamma(t)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We just need to verify that &lt;span class=&quot;math-container&quot;&gt;$\exp(\eta(t))$&lt;/span&gt; satisfies the defining conditions of &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\begin{align*}&#xA;(\exp \circ \eta)'(t) &amp;amp;= \exp'(\eta(t))\eta'(t)\\&#xA;&amp;amp;=i\exp(\eta(t))&#xA;\end{align*}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and clearly &lt;span class=&quot;math-container&quot;&gt;$\exp(\eta(0)) = \exp(0) = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So we must have &lt;span class=&quot;math-container&quot;&gt;$\exp(\eta(t)) = \gamma(t)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$e^{it} = \cos(t) + i\sin(t)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2022-08-12T12:28:21.737" LastActivityDate="2022-08-12T12:28:21.737" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25501" PostTypeId="2" ParentId="25491" CreationDate="2022-08-12T03:20:32.017" Score="3" Body="&lt;p&gt;I think the best defense against this sort of error (and explanation for students) is to know what the process of cancelling is actually doing and why it works.&lt;/p&gt;&#xA;&lt;p&gt;If you think of cancelling as just spotting the same number (or a multiple thereof) above and below the line and getting rid of them, then the sort of mistake you made is easy. To guard against it you need to remember a bunch of auxiliary rules like &amp;quot;don't cancel something that's a right operand of a division&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;However it actually gets much simpler if you remember that the process of cancelling is not &amp;quot;spot two of the same number and get rid of them&amp;quot; but rather &lt;strong&gt;dividing&lt;/strong&gt; the numerator and denominator by the same number.&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this case dividing the denominator by 2 gives us &lt;span class=&quot;math-container&quot;&gt;$2\div2$&lt;/span&gt;, which results in 1 as we wanted. But &lt;span class=&quot;math-container&quot;&gt;$(6\div2)\div2$&lt;/span&gt; simply isn't &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; by the ordinary rules for &lt;strong&gt;division&lt;/strong&gt; (which the students hopefully already understand); it isn't anything to do with special conditions on cancelling. The cancelling process was done when we added &lt;span class=&quot;math-container&quot;&gt;$\div2$&lt;/span&gt; to both sides; the rest is ordinary simplification.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; Which is based on the observation that if we multiply or divide both numerator and denominator the same number (other than zero) it's equivalent to multiplying by &lt;span class=&quot;math-container&quot;&gt;$\frac{n}{n}$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$n \ne 0$&lt;/span&gt;, which is multiplying by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, which must produce an equal fraction.&lt;/p&gt;&#xA;" OwnerUserId="20207" LastActivityDate="2022-08-12T03:20:32.017" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25502" PostTypeId="1" AcceptedAnswerId="25503" CreationDate="2022-08-12T04:08:26.990" Score="7" ViewCount="762" Body="&lt;p&gt;In particular, my question is about abstract mathematics such as group theory, analysis, topology, etc. where most textbooks are filled with exercises which require proof, and how to go about effectively &amp;quot;internalizing&amp;quot; proofs and solutions to exercises/theorems when you read them.&lt;/p&gt;&#xA;&lt;p&gt;Often I will have the experience of trying out an exercise for a while--maybe a few hours, or days--but eventually giving up and ultimately consulting the textbook or Math.SE, etc. for a solution. However, more often than not while I do get the &amp;quot;aha&amp;quot; feeling of seeing the method or &amp;quot;trick&amp;quot; needed to prove the statement, I don't really feel as if I gained something I could apply to a brand new problem.&lt;/p&gt;&#xA;&lt;p&gt;Recent example: &lt;strong&gt;Prove that &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2 + \sqrt 3 + \sqrt 5$&lt;/span&gt; is irrational.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I tried to solve this for a couple of hours, but I didn't make much progress. The solution in the book looks like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\sqrt 2 + \sqrt 3 + \sqrt 5 &amp;amp;= r \in \mathbb Q\\&#xA;\\&#xA;\sqrt 2 + \sqrt 3  &amp;amp;= r - \sqrt 5\\&#xA;\\&#xA;(\sqrt 2 + \sqrt 3)^2  &amp;amp;= (r - \sqrt 5)^2\\&#xA;\\&#xA;5+2\sqrt 6 &amp;amp;= r^2 + 5 -2r\sqrt 5\\&#xA;\\&#xA;2\sqrt 6 + 2r\sqrt 5 &amp;amp;= r^2,\\&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;so &lt;span class=&quot;math-container&quot;&gt;$2\sqrt 6 + 2r\sqrt 5$&lt;/span&gt; is rational. Squaring this expression again will show that &lt;span class=&quot;math-container&quot;&gt;$\sqrt {30}$&lt;/span&gt; is rational, which can be easily disproved.&lt;/p&gt;&#xA;&lt;p&gt;But my question is, how does one read a solution to a problem in such a way where you can begin to comprehend what the thought processes were behind &lt;em&gt;creating&lt;/em&gt; the solution? In the above example, it all makes sense, but why did they move &lt;span class=&quot;math-container&quot;&gt;$\sqrt 5$&lt;/span&gt; to the RHS in the first place?  Trial and error? Intuition?&lt;/p&gt;&#xA;&lt;p&gt;Or in other words, are there any concrete studying techniques that can take you from &amp;quot;Yeah, that makes sense now that you mention it&amp;quot; to &amp;quot;Yeah, I can understand why they thought of this.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="20196" LastEditorUserId="851" LastEditDate="2022-08-24T18:08:54.000" LastActivityDate="2022-09-01T00:11:10.070" Title="How can I internalize solutions/proofs to theorems and exercises?" Tags="|textbooks|proofs|self-learning|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25503" PostTypeId="2" ParentId="25502" CreationDate="2022-08-12T08:05:50.920" Score="7" Body="&lt;p&gt;Attempt to solve a similar exercise with different numbers, see when it works and when it fails. Using the technique by yourself will help in internalizing it. That is, ask questions like:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;does this work for different numbers in the square roots?&lt;/li&gt;&#xA;&lt;li&gt;what about a different amount of square root expressions; is this phenomenon or argument specific to three?&lt;/li&gt;&#xA;&lt;li&gt;what if the square root expressions are multiplied by integers or rational numbers?&lt;/li&gt;&#xA;&lt;li&gt;what about other roots or other exponents?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;You do not need to think about or solve all of these questions, but if you feel this particular trick is important to internalize, then you may want to look at some of them.&lt;/p&gt;&#xA;&lt;p&gt;After having done it yourself, the techniques might accessible for you to use in solving other problems, or at least in solving more complicated problems where this is an intermediate stage.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2022-08-12T11:40:00.317" LastActivityDate="2022-08-12T11:40:00.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25504" PostTypeId="2" ParentId="25502" CreationDate="2022-08-12T11:50:28.803" Score="9" Body="&lt;p&gt;The main intuition behind the proof is that you are trying to obtain an equality of the form &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x} = r'$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$r' \in \mathbb{Q}$&lt;/span&gt;, because we known how to disprove that one.&lt;/p&gt;&#xA;&lt;p&gt;The intuition behind moving &lt;span class=&quot;math-container&quot;&gt;$\sqrt 5$&lt;/span&gt; to the right-hand side is that squaring a sum of two square roots gets rid of one root, whereas squaring a sum of three square roots doesn't get rid of a root directly. Indeed:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(\sqrt a + \sqrt b)^2 = a + 2 \sqrt {ab} + b$$&lt;/span&gt; goes from two roots down to one root; whereas&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(\sqrt a + \sqrt b + \sqrt c)^2 = a + b + c + 2 (\sqrt {ab} + \sqrt {ac} + \sqrt {bc})$$&lt;/span&gt; goes from three roots to three roots, which doesn't immediately look like progress.&lt;/p&gt;&#xA;&lt;p&gt;That being said, I agree with @ryang that moving &lt;span class=&quot;math-container&quot;&gt;$\sqrt{5}$&lt;/span&gt; to the right-hand side doesn't look crucial. You can also arrive at a contradiction starting from &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2 + \sqrt 3 + \sqrt 5 = r$&lt;/span&gt; just by squaring repeatedly, moving the rational terms to the right-hand side after each squaring.&lt;/p&gt;&#xA;&lt;p&gt;Here is a proof without moving &lt;span class=&quot;math-container&quot;&gt;$\sqrt 5$&lt;/span&gt; to the right-hand side.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Assume &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2 + \sqrt 3 + \sqrt 5 = r \in \mathbb{Q}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We are going to square sums of three terms more than once, so let's do it with variables first so we have a formula to apply:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;(a + b + c)² &amp;amp; = (a + b)² + 2((a+b)c) + c² \\&#xA;(a + b + c)² &amp;amp; = a^2 + 2ab + b² + 2ac + 2bc + c² \\&#xA;(a + b + c)² &amp;amp; = a^2 + b² + c² + 2(ab + ac + bc) \\&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now we can square our rational expression:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\sqrt 2 + \sqrt 3 + \sqrt 5 &amp;amp; = r \in \mathbb{Q} \\&#xA;(\sqrt 2 + \sqrt 3 + \sqrt 5)² &amp;amp; = r^2 \\&#xA;2 + 3 + 5 + 2 (\sqrt 6 + \sqrt {10} + \sqrt {15}) &amp;amp; = r^2 \\&#xA;\sqrt 6 + \sqrt {10} + \sqrt {15} &amp;amp; = r_1 \in \mathbb{Q} \\&#xA;(\sqrt 6 + \sqrt {10} + \sqrt {15})^2 &amp;amp; = {r_1}^2 \\&#xA;6+10+15+2(\sqrt {60} + \sqrt{90} + \sqrt{150}) &amp;amp; = {r_1}^2 \\&#xA;\sqrt {60} + \sqrt{90} + \sqrt{150} &amp;amp; = r_2 \in \mathbb{Q} \\&#xA;2 \sqrt {15} + 3 \sqrt{10} + 5 \sqrt{6} &amp;amp; = r_2 \\&#xA;\end{align*}&lt;/span&gt;&#xA;Now we recognise that &lt;span class=&quot;math-container&quot;&gt;$r_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r_2$&lt;/span&gt; are both expressed as a combination of the same three square roots, &lt;span class=&quot;math-container&quot;&gt;$\sqrt {15}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{10}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{6}$&lt;/span&gt;. So we can eliminate one of the three roots with a subtraction:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;(2-2) \sqrt {15} + (3-2) \sqrt{10} + (5-2) \sqrt{6} &amp;amp; = r_2 - 2 r_1 \\&#xA;\sqrt{10} + 3 \sqrt{6} &amp;amp; = r_3 \in \mathbb{Q} \\&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Finally we have only two roots, so now squaring again will leave us with only one root.&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\sqrt{10} + 3 \sqrt{6} &amp;amp; = r_3 \\&#xA;(\sqrt{10} + 3 \sqrt{6})^2 &amp;amp; = {r_3}^2 \\&#xA;10 + 6 \sqrt{60} + 54 &amp;amp; = {r_3}^2 \\&#xA;\sqrt {15} &amp;amp; = r_4 \in \mathbb{Q} \\&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="16159" LastEditorUserId="16159" LastEditDate="2022-08-12T13:48:49.483" LastActivityDate="2022-08-12T13:48:49.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25505" PostTypeId="2" ParentId="25488" CreationDate="2022-08-12T12:34:38.750" Score="3" Body="&lt;p&gt;I've personally known two students who went through such a crisis, and didn't flunk. What set them apart from what you describe is that in both cases, they advised the university of their personal issue when it happened, and were able to come up with an acceptable plan together with the university.&lt;/p&gt;&#xA;&lt;p&gt;This completely removed the &amp;quot;shame&amp;quot; part that you describe. In one case, the student still had to retake the year because they ended up missing too much, but there was no shame in that, just an extra year at university. In the other case, the student managed to validate the year and go on to the next year even though they missed several months of class. But in neither case did they flunk or really experience &amp;quot;failure&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;A few lessons to draw from that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Contact the student as soon as the tragic event happens;&lt;/li&gt;&#xA;&lt;li&gt;Devise a plan together with the student;&lt;/li&gt;&#xA;&lt;li&gt;Communication is key; defuse any unwarranted feelings of &amp;quot;shame&amp;quot; the student might have for missing class;&lt;/li&gt;&#xA;&lt;li&gt;Do not wait until it's too late! Start communicating as soon as you can.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16159" LastActivityDate="2022-08-12T12:34:38.750" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25506" PostTypeId="2" ParentId="25473" CreationDate="2022-08-12T14:21:20.457" Score="4" Body="&lt;p&gt;Apologies for jumping to early college. It turns out that the problem persists, if you don't get into the habit of (at least mentally) keeping track of dimensions/units/whatnot.&lt;/p&gt;&#xA;&lt;p&gt;I have been battling with the consequences of not doing that (or rather, students adding fitting units as an afterthought without checking whether the appropriate units actually resulted from the calculation). One of my pedagogical tricks is to insist on &amp;quot;reality checks&amp;quot; like this. If you are calculating a volume, and get an answer in square meters, you should immediately infer that something went horribly wrong rather than simply keep the numerical value and switch to cubic meters. That will never wash.&lt;/p&gt;&#xA;&lt;p&gt;I expect to run into this, again, a few week from now. I will be teaching a first course on vector calculus to undergrads. I predict that ten per cent of them will fail to include the Jacobian, when calculating a some volume using polar coordinates. If they do that in their personal assignment, I will deduct a little (only little, I'm a nice guy) from their score and direct them to &lt;a href=&quot;https://echo360.org.uk/lesson/fb7953e8-341c-456a-9151-76c5ca02fe9a/classroom#sortDirection=desc&quot; rel=&quot;nofollow noreferrer&quot;&gt;a video I prepared just for them&lt;/a&gt;. Sorry, it's in Finnish. Anyway the title is the (purposefully) patronizing &amp;quot;Why I must not forget to include the Jacobian?&amp;quot; In there I show that if you don't, you will get a formula like &lt;span class=&quot;math-container&quot;&gt;$A=2\pi R$&lt;/span&gt; for the area of a circle of radius &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; (call it a disk if you want to). A bell should be ringing, but some have not been educated enough to notice how this formula produces wrong units. Sigh.&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2022-09-14T18:56:56.403" LastActivityDate="2022-09-14T18:56:56.403" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25507" PostTypeId="1" CreationDate="2022-08-12T15:46:46.910" Score="6" ViewCount="278" Body="&lt;p&gt;Let me know if you think the following would be better suited for academia.stackexchange. But I think some answers will be maths-specific, so I'll start with it here.&lt;/p&gt;&#xA;&lt;p&gt;I'm a maths tutor. Up until now, when a student (ages 12 - 18) talks about maths negatively - &amp;quot;I don't like maths&amp;quot;, &amp;quot;I suck at maths&amp;quot; etc, I have listened to what they had to say, but I don't have anything great to say in response. Sometimes I empathise, &amp;quot;yes maths is hard...&amp;quot;, but I don't really address or talk to them about their negative attitude towards maths in a substantive way- rather I try to get the student to focus on the maths we intend to cover in the hour's long tutoring session. These situations do happen fairly regularly when I start tutoring someone new, and sometimes even if I have been tutoring them for a while. The reason my approach has been to not address their negative self-perception is partly I don't know exactly how and partly because I feel guilty that I'm basically forcing them to study maths when they don't want to - and I believe that if someone doesn't want to learn something, then I shouldn't be forcing them to.&lt;/p&gt;&#xA;&lt;p&gt;After some reflection and reading some comments from other questions on this website, I think there is a better way to handle this situation. This would be to help them &lt;em&gt;reframe&lt;/em&gt; their hatred of maths or self-perception of their own abilities as being poor, by starting with asking leading questions such as, &amp;quot;why do you think you suck at maths&amp;quot;, or &amp;quot;why do you hate maths?&amp;quot;, and then they will say something like &amp;quot;I get bad marks&amp;quot; or &amp;quot;I just don't understand it&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But instead of saying, &amp;quot;well you may have got bad marks in the past, but with effort and time you can improve your maths abilities for sure&amp;quot;, or, &amp;quot;you may not understand the maths now- we need to start from the fundamentals and break it down into smaller chunks.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Sure, this is fine. But I think it could be really useful to also explain to them that, psychologically, if they keep saying to themselves, &amp;quot;I suck at maths&amp;quot; and &amp;quot;I don't understand maths&amp;quot; then that is a self-fulfilling prophecy, because: &lt;strong&gt;if they don't believe in themselves then they won't even try.&lt;/strong&gt; And spending real time and effort studying maths trying is the biggest step a young student can take towards self-confidence and ultimately self-improvement.&lt;/p&gt;&#xA;&lt;p&gt;So, is it a good idea to discuss with the student that reframing is a &lt;em&gt;regular exercise&lt;/em&gt; they can do whenever a negative thought pops up- remember aimed at a young person - and how exactly would you explain this/discuss this with them? What exactly should you say?&#xA;And, what &lt;em&gt;other&lt;/em&gt; useful reframing exercises are there?&lt;/p&gt;&#xA;&lt;p&gt;I have done some reframing myself with a CBT (cognitive behavioural therapy) professional for my anxiety, but I'm far from an expert at understanding &lt;em&gt;how to teach someone else&lt;/em&gt; that they can reframe their negative thoughts and &lt;em&gt;what the process of doing this entails&lt;/em&gt;. So my question is: how you would you explain/discuss this with someone? Would you explain it differently depending on the student's age? I suspect &lt;span class=&quot;math-container&quot;&gt;$18-$&lt;/span&gt; year olds would be able to understand and implement reframing a lot better than &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; year olds, so maybe a different approach needs to be taken towards my younger students?&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastActivityDate="2022-08-19T18:26:32.267" Title="Should/ how should I help students reframe their negative thoughts about their own mathematical abilities?" Tags="|tutoring|anxiety|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25508" PostTypeId="2" ParentId="25507" CreationDate="2022-08-13T08:20:47.283" Score="1" Body="&lt;p&gt;The first point, surely, is figuring out whether, for this individual right here, the negative things they say is just a harmless ritual response, or whether there is some actual problems there.&lt;/p&gt;&#xA;&lt;p&gt;By the harmless ritual response alternative I mean that starting to do something that requires attention, thinking and trying, more generally effort is not comfortable, and someone might express this by a phrase that puts a negative light on mathematics. But this is essentially harmless; just a part of the ritual to get in the mindset and start doing things.&lt;/p&gt;&#xA;&lt;p&gt;I will probably make some unhappy noises before I start washing dishes or cutting the grass or various other tasks, but this does not mean you should pay attention to it and start worrying about my bad relationship to dishwashing.&lt;/p&gt;&#xA;&lt;p&gt;Supposing it is not simply this, but there is actually some kind of emotional problem, the typical recommended way is to:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Let them speak and you listen.&lt;/li&gt;&#xA;&lt;li&gt;Acknowledge that the emotions are real.&lt;/li&gt;&#xA;&lt;li&gt;Do not try to fix it by offering a recipe to follow, or other rational response.&lt;/li&gt;&#xA;&lt;li&gt;Do support them if they want to do something about it.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;For going further than this I do not really have advice. But there exists plenty of literature on effective communication (also with power imbalance), and on motivation in mathematics; the stuff I have read recently include:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;EIDE, Hilde; EIDE, Tom. Kommunikasjon i relasjoner: personorientering, samhandling, etikk. Gyldendal akademisk, 2017.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;WÆGE, Kjersti. Motivasjon i matematikk. Universitetsforl., 2018.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is even more material about what a safe and inviting classroom environment in mathematics is like; things like being allowed to make mistakes and drawing out multiple ways of reasoning, rather than fixating on a single correct solution or method. These might be helpful in the long run, too.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;BOALER, Jo. Mathematical mindsets: Unleashing students' potential through creative mathematics, inspiring messages and innovative teaching. John Wiley &amp;amp; Sons, 2022.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;KLAVENESS, E.; KARLSEN, L.; KVERNDOKKEN, K. 101 grep for å aktivisere elever i matematikk. Bergen: Fagbokforlaget, 2019.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-08-13T08:20:47.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25509" PostTypeId="2" ParentId="25473" CreationDate="2022-08-13T12:29:31.827" Score="11" Body="&lt;p&gt;The legitimate purpose of this is trying to get the student to actually read the question and take note of the fact that there is indeed a context.&lt;/p&gt;&#xA;&lt;p&gt;Many students approach mathematical exercises as &amp;quot;set of instructions to carry out&amp;quot; rather than &amp;quot;problem to be solved&amp;quot;. Subsequently, they dislike &amp;quot;word problems&amp;quot;, because they just obfuscate the instructions. They might just scan the question for numbers, and then guess some arithmetic operations to carry out on those numbers.&lt;/p&gt;&#xA;&lt;p&gt;Now of course requesting to state the unit can just lead to the student also scanning the question for what might be the unit, rather than ever conceptualizing what is going on. Moreover, my suspicion is that for at least some teachers insisting on this, providing the unit is just part of whatever arcane set of instructions apply, rather than any actual understanding of the issue mentioned above.&lt;/p&gt;&#xA;" OwnerUserId="13471" LastEditorUserId="13471" LastEditDate="2022-12-29T14:23:02.757" LastActivityDate="2022-12-29T14:23:02.757" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25510" PostTypeId="2" ParentId="25507" CreationDate="2022-08-13T22:14:49.143" Score="5" Body="&lt;p&gt;You mentioned two different sorts of negative attitudes: &amp;quot;I don't like math&amp;quot; and &amp;quot;I'm not good at math&amp;quot;. If a student has only expressed one of those, the approach you take might be different for one versus the other. But the reasons for both are often the same: math is approached badly in school. (See &lt;a href=&quot;https://www.maa.org/external_archive/devlin/LockhartsLament.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Mathematician's Lament&lt;/a&gt;, by Paul Lockhart.)&lt;/p&gt;&#xA;&lt;p&gt;For the students who think they aren't good at math, your approach is lovely and if you keep researching CBT, I think you'll get good at helping them that way. (You probably know more about CBT than I do, so I won't say more about that.) My approach for students with test anxiety has been to have them listen (repeatedly) to a guided visualization I wrote that I call &lt;a href=&quot;https://mathmamawrites.blogspot.com/2013/02/resource-math-relax.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Relax&lt;/a&gt;. For more general math anxiety, I have them read one of the many good books for that: &lt;em&gt;&lt;strong&gt;&lt;a href=&quot;https://www.biblio.com/search.php?stage=1&amp;amp;pageper=20&amp;amp;keyisbn=overcoming%20math%20anxiety&amp;amp;omit_product_types=bp,bd,ns&amp;amp;strip_common=1&amp;amp;program=1005&amp;amp;order=price_ship_usasc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Overcoming Math Anxiety&lt;/a&gt;&lt;/strong&gt;&lt;/em&gt;, by Sheila Tobias and &lt;em&gt;&lt;strong&gt;&lt;a href=&quot;https://www.biblio.com/search.php?stage=1&amp;amp;pageper=20&amp;amp;keyisbn=mind%20over%20math&amp;amp;omit_product_types=bp,bd,ns&amp;amp;strip_common=1&amp;amp;program=1005&amp;amp;order=price_ship_usasc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mind Over Math&lt;/a&gt;&lt;/strong&gt;&lt;/em&gt;, by Stanley Kogelman are two I like.&lt;/p&gt;&#xA;&lt;p&gt;For the students who don't like math, I'd definitely explore what about it they don't like. If the parents (who are paying the bill) are ok with some of the time being spent to deal with this bigger issue, I'd try to bring in activities that connect math with things they do like. One student I tutored liked logic. I love logic puzzles and games. His mom paid for us to do that because I thought it would help improve his attitude about math.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-08-19T18:26:32.267" LastActivityDate="2022-08-19T18:26:32.267" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25511" PostTypeId="1" AcceptedAnswerId="25512" CreationDate="2022-08-14T22:44:56.600" Score="4" ViewCount="1071" Body="&lt;h2&gt;Context and examples&lt;/h2&gt;&#xA;&lt;p&gt;When teaching mathematics to a new class in high school, I often open up with some mathematical games that can be played cooperatively and with minimal requirements (good old chalk and blackboard). On the one hand, this helps break the ice with the class (so to speak) and I get to quickly learn their names, and on the other hand, by trying different games I get to partially gauge their understanding of certain fundamental topics (such as parity, symmetry, prime numbers, ...) and their technical skills (multiplication table, visualisation skills, posing and solving simple equations, ...).&lt;/p&gt;&#xA;&lt;p&gt;The games I use are simple enough in premise that even when there is a language barrier (such as welcoming classes for recently arrived refugees), kids will be able to quickly grasp the idea and participate.&lt;/p&gt;&#xA;&lt;h2&gt;Game of multiples and divisors&lt;/h2&gt;&#xA;&lt;p&gt;One good example is the Juniper-Green game, or simply the game of multiples and divisors: you set an upper bound (say 30), and then&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1st student chooses a number between 1 to 30.&lt;/li&gt;&#xA;&lt;li&gt;2nd student chooses either a multiple or divisor (still within 1 to 30) of the 1st student's number.&lt;/li&gt;&#xA;&lt;li&gt;3rd student chooses similarly based on the 2nd student's number, so on and so forth.&lt;/li&gt;&#xA;&lt;li&gt;As they choose these numbers I write them on the blackboard with their name next to it. This is important because no number can be used twice.&lt;/li&gt;&#xA;&lt;li&gt;The last player who can still choose a valid number wins.&#xA;The game is fun because they not only brush up on their multiples/divisors but after a few runs they also discover the tactical role of prime numbers in this game.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2&gt;Color of a square on the chessboard&lt;/h2&gt;&#xA;&lt;p&gt;Another example is drawing a chessboard on the blackboard but only coloring the first row, rest being a blank grid. Then asking the students to guess the color of a random square I choose. At first they show their visual skills by instantly finding paths connecting to the that square from a known square and then using the alternating color rule.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;So if e5 is the target, one student tells me: I start from b1 which is a light square, then move to c1-dark, c2-light, d2-dark, d3-light, e3-dark, e4-light and e5 must therefore be dark.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Others find more efficient ways of rerouting to the target square and guess the color (e.g., moving diagonally). Then I would tell them, what if the board was much larger, not 8 by 8, but 100 by 100? Surely, it would take a long time to find the color of a random square with these path-finding methods and assuming we don't make mistakes along the way. What if there was a mathematical rule that would associate the coordinates of a square to its color?&lt;/p&gt;&#xA;&lt;p&gt;As soon as I would say it and also substitute the file letters a-to-h with numbers 1-to-8 (like rows), they would discuss in small groups and within minutes hands would be raised!  They've got it: if the sum of the coordinates of a square is an even number the square is dark, if the sum is odd the square is light. Suddenly, they discover on their own the power of mathematically modelling a problem: their predictions are no longer bound by how far they can visualize or how deep a sequence can be memorized.&lt;/p&gt;&#xA;&lt;h2&gt;Questions&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do you know of other similarly spirited mathematical games (or more logic oriented) that can readily be played in classrooms? Ideally, accessible to most high school students.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;em&gt;Admittedly, this is a rather open question meaning that all proposals are welcome, whether they be games/puzzles you've used or seen used before, your own personal ideas for designing a new game, or simply examples you've read about before which may be fitting here. The latter brings me to the next question:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Equally of interest would be puzzle themed book (or other resources) recommendations with similar mathematical games that would be particularly suited for teaching.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="20206" LastActivityDate="2022-08-16T15:59:03.627" Title="Games suitable for high school level teaching" Tags="|secondary-education|interactive-teaching|games|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25512" PostTypeId="2" ParentId="25511" CreationDate="2022-08-15T00:01:17.097" Score="5" Body="&lt;p&gt;&lt;strong&gt;Board Games&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Prime Climb&lt;/li&gt;&#xA;&lt;li&gt;Katamino (pentominoes, for 2 or as a puzzle for 1)&lt;/li&gt;&#xA;&lt;li&gt;Blokus (shapes using 1 to 5 squares, strategy)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Card Games&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Set (logic)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;A game on the board&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's a bit like yours, but the teacher is the Divisor Miser or the Tax Collector. It is possible for the students to win, if they strategize well.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Numbers 1 to (perhaps) 40 are on the board.&lt;/li&gt;&#xA;&lt;li&gt;Students pick one. You cross it off and add it to their score.&lt;/li&gt;&#xA;&lt;li&gt;You add all its factors to your score.&lt;/li&gt;&#xA;&lt;li&gt;They cannot pick a number that will give you nothing.&lt;/li&gt;&#xA;&lt;li&gt;You get all that's left at the end.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Logic Puzzles&lt;/strong&gt;&#xA;Rush Hour and Chocolate Fix&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Geometry Puzzles&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Geometry Snacks, by Ed Southall and Vincent Pantaloni&lt;/li&gt;&#xA;&lt;li&gt;Also Catriona Shearer has lots of good ones on twitter&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(There might be others you like at &lt;a href=&quot;http://mathmamawrites.blogspot.com/p/games.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;my blog&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-08-15T00:01:17.097" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25514" PostTypeId="1" CreationDate="2022-08-15T20:54:37.707" Score="3" ViewCount="157" Body="&lt;p&gt;I am wondering what examples you like to give when introducing linear programming, where the examples are &lt;strong&gt;not&lt;/strong&gt; clearly better suited as &lt;strong&gt;integer&lt;/strong&gt; linear programs. I would like a few examples where we can discuss the motivating story, then formalize and write it down as an LP.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I can motivate production problems as LPs, when I am careful to phrase everything in terms of rates: rate of resources consumed per day, rate of products produced per day, maximizing rate of profit per day. This way, fractional solutions make sense. Same for their duals (min-cost procurement).&lt;/li&gt;&#xA;&lt;li&gt;I will use zero-sum games as a more intricate example.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I think scheduling and assignment problems are generally better as ILPs.&lt;/li&gt;&#xA;&lt;li&gt;Flows, paths, and matchings deserve their own separate discussion in my opinion, due to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Maximum_flow_problem#Integral_flow_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Integrality Theorem&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;It seems like most practically motivating applications are scheduling and assignment problems that really should be ILPs. What are your favorite motivating LP applications? I also appreciate thoughts about how to navigate this question pedagogically, for example if your approach is to just motivate ILPs and treat LPs as a means to an end.&lt;/p&gt;&#xA;" OwnerUserId="4939" LastActivityDate="2023-01-13T18:03:20.270" Title="Favorite linear programming (not integer) examples?" Tags="|examples|concept-motivation|applications|optimization|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25515" PostTypeId="2" ParentId="25511" CreationDate="2022-08-16T00:20:56.930" Score="2" Body="&lt;p&gt;In a comment I mentioned discovering Euler's Formula using polydrons,&#xA;which requires quite a collection of polydrons.&lt;/p&gt;&#xA;&lt;p&gt;However, Euler's Formula holds for planar graphs (the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-skeleton of a polyhedron is a planar graph). So you could discover it with paper and pencil, or better, on a blackboard, drawing a random planar graph and counting. Needs introduction to explain what is a connected, simple, planar graph; what is a vertex, an edge, a face. One tricky part is to make sure to count the outer face. Once &lt;span class=&quot;math-container&quot;&gt;$V-E+F=2$&lt;/span&gt; is established, you can show a few polyhedra, and connect the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-skeleton of the polyhedron with the corresponding planar graph.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;   &lt;img src=&quot;https://i.stack.imgur.com/Bkusi.png&quot; width=&quot;200&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;   Icosahedron graph: &lt;span class=&quot;math-container&quot;&gt;$V=12$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$E=30$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$F=20$&lt;/span&gt;. &lt;em&gt;Wikipedia image&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Then once they know what is a planar graph,&#xA;you'd have an excuse to mention the &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;-color theorem :-).&lt;/p&gt;&#xA;&lt;p&gt;   &lt;a href=&quot;https://i.stack.imgur.com/EepYy.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EepYy.gif&quot; alt=&quot;4-coloring&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;   &lt;em&gt;Image from &lt;a href=&quot;https://www.mathpages.com/home/kmath266/kmath266.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathpages.com&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-08-16T00:20:56.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25516" PostTypeId="1" AcceptedAnswerId="25517" CreationDate="2022-08-16T13:44:12.657" Score="2" ViewCount="89" Body="&lt;p&gt;There are some &lt;a href=&quot;https://medium.com/however-mathematics/13-must-see-mathematics-movies-inspired-by-true-events-1beda86255cd&quot; rel=&quot;nofollow noreferrer&quot;&gt;nice movies about math or mathematicians&lt;/a&gt;, however, the inspiring history from Tycho Brahe to Johannes Kepler to Isaac Newton seems are not covered.&lt;/p&gt;&#xA;&lt;p&gt;This is indeed amazing, Tycho spent his whole life observing remote celestial objects, Kepler found the secrete links hidden among the numbers, then finally &lt;a href=&quot;https://physics.stackexchange.com/questions/112668/derivation-of-keplers-laws&quot;&gt;Newton explained it with the law of universal gravitation&lt;/a&gt;, and calculs, the math he invented to explain the nature.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately such exciting chapter of human intelligence is not made to a movie by any director, or a video by any youtube channel.&lt;/p&gt;&#xA;&lt;p&gt;I wonder, if there's a book telling the story, with solid math explaining the theories?&lt;/p&gt;&#xA;" OwnerUserId="9059" LastActivityDate="2022-08-16T15:07:07.590" Title="From Tycho, Kepler to Newton" Tags="|book|videos|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25517" PostTypeId="2" ParentId="25516" CreationDate="2022-08-16T15:07:07.590" Score="2" Body="&lt;p&gt;Books.  Two classic books by Arthur Koestler:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The Sleepwalkers: A History of Man's Changing Vision of the Universe&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The Watershed: a Biography of Johannes Kepler&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[In fact, &lt;em&gt;The Watershed&lt;/em&gt; is Part 4 of &lt;em&gt;The Sleepwalkers&lt;/em&gt;, including much on Tycho.  Part 5 then goes on to Newton.]&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2022-08-16T15:07:07.590" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25518" PostTypeId="2" ParentId="25511" CreationDate="2022-08-16T15:59:03.627" Score="4" Body="&lt;h3&gt;Logic&lt;/h3&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.chiark.greenend.org.uk/%7Esgtatham/puzzles&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Tatham's Puzzles&lt;/strong&gt;&lt;/a&gt;, such as &lt;a href=&quot;https://www.chiark.greenend.org.uk/%7Esgtatham/puzzles/js/loopy.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Loopy&lt;/a&gt; (Cairo) and &lt;a href=&quot;https://www.chiark.greenend.org.uk/%7Esgtatham/puzzles/js/tracks.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tracks&lt;/a&gt;, both of which are endless sources of puzzles where pigeonhole principle and parity are very useful tools for solving them. Tatham's version of Minesweeper is also guaranteed to be solvable with pure logical deduction (i.e. no guessing), but Loopy and Tracks encourage a much wider variety of combinatorial deductions.&lt;/p&gt;&#xA;&lt;p&gt;  &lt;img src=&quot;https://i.stack.imgur.com/6ykC4m.png&quot; alt=&quot;Loopy Logo&quot; /&gt;   &lt;img src=&quot;https://i.stack.imgur.com/4vmB3.png&quot; alt=&quot;Tracks Logo&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://cosmicexpressgame.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Cosmic Express&lt;/strong&gt;&lt;/a&gt;, which some people solve by just trying randomly, but which can be solved quite systematically if you are sufficiently good at logic and abstraction.&lt;/p&gt;&#xA;&lt;p&gt;  &lt;img src=&quot;https://i.stack.imgur.com/zIZHD.jpg&quot; alt=&quot;Cosmic Express Logo&quot; /&gt;&lt;/p&gt;&#xA;&lt;h3&gt;Programming&lt;/h3&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://pleasingfungus.com/Manufactoria&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Manufactoria&lt;/strong&gt;&lt;/a&gt;, which was originally &lt;a href=&quot;https://archive.ph/o/KvqWA/www.kongregate.com/games/PleasingFungus/manufactoria&quot; rel=&quot;nofollow noreferrer&quot;&gt;a free Flash game&lt;/a&gt; with an incredibly steep rise in difficulty (most people cannot complete the hardest levels), but apparently has &lt;a href=&quot;https://www.pcgamer.com/make-silly-robots-in-programming-puzzler-manufactoria-2022&quot; rel=&quot;nofollow noreferrer&quot;&gt;a gentler version released in 2022&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;  &lt;img src=&quot;https://i.stack.imgur.com/m0jWVm.jpg&quot; alt=&quot;Manufactoria 2022 Logo&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.robozzle.com/beta&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Robozzle&lt;/strong&gt;&lt;/a&gt;, which is a more restricted type of programming puzzle than Manufactoria, but has simpler puzzles. The &amp;quot;Campaign&amp;quot; should be a good place to start.&lt;/p&gt;&#xA;&lt;p&gt;  &lt;img src=&quot;https://i.stack.imgur.com/ozu8Xs.jpg&quot; alt=&quot;Robozzle Logo&quot; /&gt;&lt;/p&gt;&#xA;&lt;h3&gt;Geometry&lt;/h3&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://sciencevsmagic.net/geo&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;ScienceVSMagic Geometry&lt;/strong&gt;&lt;/a&gt;, which is a collection of compass-straightedge challenges.&lt;/p&gt;&#xA;&lt;p&gt;  &lt;img src=&quot;https://i.stack.imgur.com/ft4wQs.png&quot; alt=&quot;ScienceVSMagic Geometry Logo&quot; /&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2022-08-16T15:59:03.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25519" PostTypeId="2" ParentId="25514" CreationDate="2022-08-16T16:14:45.510" Score="1" Body="&lt;p&gt;[Chris, convert to comment, if you deem.  /Not cookie mousing.]&lt;/p&gt;&#xA;&lt;p&gt;I think you are best off for an &lt;strong&gt;intro&lt;/strong&gt; to do an example that is extremely simple and toy-like and might even be simply solved by high school algebra (but allows describing a logistical problem and drawing a sketch).  &lt;strong&gt;The students are new to the techniques.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Something like this, but then use the simplex method (don't prove it now, leave for later...for now, just assert it and use it) at the end:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=M_mpRrGKKMo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=M_mpRrGKKMo&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20204" LastActivityDate="2022-08-16T16:14:45.510" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25520" PostTypeId="1" CreationDate="2022-08-16T19:35:05.863" Score="-2" ViewCount="335" Body="&lt;p&gt;Please share links to websites / pdf files containing syllabus in subject mathematics in US education system Grades 1-12.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="20224" LastEditorUserId="2083" LastEditDate="2022-08-17T07:31:41.790" LastActivityDate="2022-08-19T01:57:29.377" Title="math syllabus US education system 1-12th Grade" Tags="|secondary-education|primary-education|usa|syllabus|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25521" PostTypeId="1" CreationDate="2022-08-17T16:50:43.607" Score="14" ViewCount="3430" Body="&lt;p&gt;Context: I am an assistant professor at a small liberal arts institution in the US.&lt;/p&gt;&#xA;&lt;p&gt;The title really says it all: I am looking to provide applications for logarithms in a business math course (most students are freshman).  The sciences have a ton of uses for logs, but I'd like to have something specific to business/accounting/finance/economics.&lt;/p&gt;&#xA;&lt;p&gt;Of course you can use them to solve questions related to exponential growth/decay problems, but I'd like to find examples where a logarithmic function itself describes the behavior we're examining.  I've looked in a few business math textbooks and haven't come up anything terribly compelling thus far.&lt;/p&gt;&#xA;&lt;p&gt;Examples I've written myself basically all hinge on some sort of &amp;quot;diminishing return,&amp;quot; like the price of a tech stock immediately after its IPO, the effects of continued spending on advertising, or the depreciation of electronic equipment.&lt;/p&gt;&#xA;&lt;p&gt;I'm a mathematician (not a businessperson), so I am suspicious there might be some ideas I'm missing, especially in economics.  Any suggestions or advice is highly appreciated!&lt;/p&gt;&#xA;" OwnerUserId="5988" LastEditorUserId="5988" LastEditDate="2022-08-18T17:22:46.923" LastActivityDate="2022-08-31T16:17:56.743" Title="Applications for logarithms in a business math course" Tags="|applications|logarithm|" AnswerCount="7" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25522" PostTypeId="2" ParentId="25521" CreationDate="2022-08-17T17:54:51.907" Score="4" Body="&lt;p&gt;Compound interest and cumulative compounding are the classic financial examples.&lt;/p&gt;&#xA;&lt;p&gt;A search on &amp;quot;investment math&amp;quot; will give you several books, problems, sites.  I would suggest to buy one or check out a few in a college library.  Below is a link to a pdf of one by Hart:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/mathematicsofinv004275mbp/page/n15/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/mathematicsofinv004275mbp/page/n15/mode/2up&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you want a shorter, simpler take, than see the single chapter on investment math that Hart included in his college algebra text (war manual EM 315).  You can get it from a library (use ILL if needed) or just buy it for a few bucks on a used book website.&lt;/p&gt;&#xA;" OwnerUserId="20204" LastActivityDate="2022-08-17T17:54:51.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25523" PostTypeId="2" ParentId="25521" CreationDate="2022-08-17T20:25:57.723" Score="3" Body="&lt;p&gt;A fundamental concept in economics is &lt;a href=&quot;https://en.wikipedia.org/wiki/Elasticity_(economics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;elasticity&lt;/a&gt;, which is closely related to logarithms:&lt;/p&gt;&#xA;&lt;p&gt;Consider two economical quantities &lt;span class=&quot;math-container&quot;&gt;$x,y \in (0,\infty)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; depends on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. The elasticity &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is defined as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \epsilon(x) := \frac{\partial y}{\partial x} \,\frac{x}{y}.&#xA;$$&lt;/span&gt;&#xA;Here are two ways to see a relation to logarithms:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Due to the chain rule, one has &lt;span class=&quot;math-container&quot;&gt;$\epsilon(x) = \frac{\partial \log(y)}{\partial x} x$&lt;/span&gt;. This is, at least at first glance, purely computational; a more conceptual perspective is as follows:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let us consider a &amp;quot;logarithmic transformation of variables&amp;quot;, i.e., we define new quantities &lt;span class=&quot;math-container&quot;&gt;$u := \log(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v := \log(y)$&lt;/span&gt; (which are thus elements of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$(0, \infty)$&lt;/span&gt;). Then, again by the chain rule,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \frac{\partial v}{\partial u} = \frac{\partial y}{\partial x} \frac{x}{y} = \epsilon.&#xA;$$&lt;/span&gt;&#xA;In other words, the elasticity of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is simply the derivative in logarithmic variables.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2022-08-17T20:25:57.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25524" PostTypeId="2" ParentId="25521" CreationDate="2022-08-18T03:09:44.587" Score="18" Body="&lt;h3&gt;Log scales can introduce logarithms swimmingly! Let's start with a &lt;a href=&quot;https://xkcd.com/1162&quot; rel=&quot;nofollow noreferrer&quot;&gt;germane xkcd comic&lt;/a&gt;&lt;/h3&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/jjGwE.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thanks to user &lt;a href=&quot;https://matheducators.stackexchange.com/users/8943/nat&quot;&gt;Nat&lt;/a&gt; for &lt;a href=&quot;https://matheducators.stackexchange.com/questions/25521/applications-for-logarithms-in-a-business-math-course#comment65524_25524&quot;&gt;charting the log-10 version&lt;/a&gt; in the comments! But I matched Nat's colors to mine.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\def\p#1#2#3{\smash{\rlap{\hskip{#1px}\raise{#2}{#3}}}}\newcommand{\r}[1]{\hskip{-1in}\style{transform-origin:left;display:inline-block;transform:rotate(-90deg);padding:1in}{#1}}\newcommand{\b}[3]{\hskip{25px}\color{#3}{\p{0}{0px}{\rule{20px}{#2em}}\p{0}{#2em}{\raise{2px}{\Tiny#2}}\p{0}{-9px}{\r{\llap{\raise{-7px}\small\text{#1}}}}}}\p{0}{80px}{\p{-8}{2px}{\hskip{-3px}\b{Sugar}{1.28}{limegreen}\b{Coal}{1.38}{darkorange}\b{Fat}{1.59}{skyblue}\b{Gasoline}{1.66}{goldenrod}\b{Uranium}{7.88}{red}}\p{0}{0px}{\hskip{5px}\rule{2px}{9em}\rule{138px}{2px}}}{\phantom{\rule{140px}{14em}}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://www.reddit.com/r/explainlikeimfive/comments/nk990a/eli5_what_is_a_logarithmic_scale_why_is_it_more/gzbng1b/&quot; rel=&quot;nofollow noreferrer&quot;&gt;This is my favorite motivation&lt;/a&gt;. (I improved the original text from reddit.)&lt;/h3&gt;&#xA;&lt;p&gt;Imagine you wanted to plot the net worth of people in your area, in a graph that shows all the data. Presuppose that most  net worth will be around the lower end, with some higher in between. And presuppose that the wealthiest person possesses $500K. You should be able to effortlessly see the distribution of net worth among your population.&lt;/p&gt;&#xA;&lt;p&gt;Then suddenly Elon Musk decides to join your area. Now your graph is no longer readable, and it's warped! Why is it misshapen? Because the number line suddenly jumped to 147 billion!&lt;/p&gt;&#xA;&lt;p&gt;As your graph is supposed to show all points, now it's squished such that Elon Musk is at the top, a single outlier in the whole distribution. Even the richest resident in your area is a measly 0.00034% on the scale — This is how laughable 500k is in that distribution!&lt;/p&gt;&#xA;&lt;p&gt;Elon Musk's billions make it impossible to see any difference between the 500k guy, the 300k guy, the 30k guy, and so on. They're all down in the same line from Elon's perspective.&lt;/p&gt;&#xA;&lt;p&gt;So now you take a logarithmic scale, meaning you take their net worth, throw it into a logarithmic function and then plot that value. You obviously have to adjust the y axis accordingly.&lt;/p&gt;&#xA;&lt;p&gt;The 30k guy is now a 4.47.&lt;/p&gt;&#xA;&lt;p&gt;The 200k guy is now a 5.3.&lt;/p&gt;&#xA;&lt;p&gt;The 500k guy is now a 5.7.&lt;/p&gt;&#xA;&lt;p&gt;And Elon? Well he's an 11.17.&lt;/p&gt;&#xA;&lt;p&gt;These now fit on a single graph, and you can still make out differences between the &amp;quot;smaller&amp;quot; numbers.&lt;/p&gt;&#xA;&lt;p&gt;Logarithmic scales are generally often used when exponential growth is concerned, like population growth, spreading of diseases. Also when you have to work with data that spans a huge range of values, like comparing the size of objects in the universe.&lt;/p&gt;&#xA;&lt;p&gt;Machine learning and data science commonly use logarithmic scales and transformations.&lt;/p&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://www.reddit.com/r/explainlikeimfive/comments/12u4df/eli5_the_logarithmic_scale/c6y8bsk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Another ELI5 introduction to log scales.&lt;/a&gt; I edited it.&lt;/h3&gt;&#xA;&lt;p&gt;Suppose you measure something that produces only positive numbers (this is a requirement to use a logarithmic scale). You get these measurements:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{fuchsia}{1, 3, 5134573435345, 0.0000000053453}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Ouch! That's quite a lot of variation there! If you plot this on a normal scale, the plot is going to look misshapen! One point is way off somewhere, and the rest sort of clustered around zero. By looking at this plot, nobody is able to tell that there's a huge difference between 1 and 0.0000000053453.&lt;/p&gt;&#xA;&lt;p&gt;This is not a contrived example. I made up these numbers, but lots of natural phenomena are like this.&lt;/p&gt;&#xA;&lt;p&gt;So what do we do? We can use a logarithmic scale. If I take the logarithm of the four numbers above, I get:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{fuchsia}{0, 0.477, 12.7, —8.27}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you plot these numbers on a normal scale, things are much clearer! There's a negative number, 0, another near zero, and a positive number...all in the interval of &lt;span class=&quot;math-container&quot;&gt;$[-15, 15]$&lt;/span&gt; that you can plot by hand!&lt;/p&gt;&#xA;&lt;p&gt;Logarithmic scales are used for all sorts of phenomena that tend to produce numbers such as the ones above, such as earthquake intensity (Richter's scale), concentration of ions (pH-scale), anything that involves our senses (f-stops in photography, decibels, octaves), entropy, star brightness... the list goes on. Another less obvious example is the &amp;quot;five standard deviations&amp;quot; rule of thumb used by CERN, when CERN were looking for the Higgs boson.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://redd.it/6t8lbr&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here are more real life applications of logarithms.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user155" LastEditorUserId="20265" LastEditDate="2022-08-28T23:42:54.050" LastActivityDate="2022-08-28T23:42:54.050" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25525" PostTypeId="2" ParentId="25521" CreationDate="2022-08-18T16:49:16.043" Score="11" Body="&lt;p&gt;I feel compelled to provide your &amp;quot;of course&amp;quot; answer of exponential growth/decay.  This answer is hopefully appropriate for lower-level business courses such as high-school level.&lt;/p&gt;&#xA;&lt;p&gt;Here's a totally reasonable, practical, and compelling question where logarithms are used to get the answer.&lt;/p&gt;&#xA;&lt;h3&gt;How many times does something need to change by a percentage?&lt;/h3&gt;&#xA;&lt;p&gt;I was training to run a half-marathon (22 km).  The advice given to me was:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Do not increase your running distance by more than 10% each week&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, if I was currently running 5 km, I would need to know how many times I needed to increase my distance by 10% in order to reach my 22 km goal.  I was trying to set my expectations as to how many weeks it would take me.&lt;/p&gt;&#xA;&lt;p&gt;Each week my distance should be 1.10 times what it was the previous week.&lt;/p&gt;&#xA;&lt;p&gt;This makes me ask &lt;span class=&quot;math-container&quot;&gt;$$5 \times 1.1^n = 22$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Without logarithms you can only get as far as &lt;span class=&quot;math-container&quot;&gt;$$1.1^n=22/5$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you're unfamiliar with exponents and logarithms, it can be intimidating to see your variable trapped in the exponent like that and you need a more advanced mathematical technique that can get it out.  Now &lt;strong&gt;with&lt;/strong&gt; logarithms you can do &lt;span class=&quot;math-container&quot;&gt;$$log(1.1^n) = log(22/5)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And knowing a little about logarithmic identities you can free the variable from its exponential prison &lt;span class=&quot;math-container&quot;&gt;$$n \times log(1.1) = log(22/5)$$&lt;/span&gt; and finally &lt;span class=&quot;math-container&quot;&gt;$$n = {log(22/5)\over log(1.1)} \approx 15.5$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The advice is for me not to try to do this in fewer than about 16 weeks of training.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;If you take this same problem and just change the story to suppose that I invest \$5,000 and I can increase my money by 10% per year, and ask &amp;quot;How many years will it take until I have \$22,000?&amp;quot; then it's a business application related to &lt;em&gt;&lt;strong&gt;planning&lt;/strong&gt;&lt;/em&gt; and computing the answer in variations of the scenario builds &lt;em&gt;&lt;strong&gt;intuition about exponential growth/decay&lt;/strong&gt;&lt;/em&gt;.  (The answer is the same: 16 years.)&lt;/p&gt;&#xA;&lt;p&gt;It's an &amp;quot;aha&amp;quot; moment to do this and realize that maybe it's not practical to wait 16 years for my \$5,000 investment to grow to the \$22,000 I need to buy a car that I need soon, for example.  But a parent could invest \$5,000 at 10% when a child is born in order to afford that same \$22,000 car when the child turns 16 (a common legal driving age).&lt;/p&gt;&#xA;" OwnerUserId="13481" LastEditorUserId="13481" LastEditDate="2022-08-29T15:13:51.880" LastActivityDate="2022-08-29T15:13:51.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25526" PostTypeId="2" ParentId="25521" CreationDate="2022-08-18T19:24:26.940" Score="1" Body="&lt;p&gt;A Cobb-Douglas production/utility function has the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ U(x,y,z,\dots)=K\,x^ay^bz^c\cdots \tag{1}$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; represents the utility or level of production&#xA;for given levels &lt;span class=&quot;math-container&quot;&gt;$x,y,z,\dots$&lt;/span&gt; of &amp;quot;inputs&amp;quot; like capital, labor, materials, energy, and so forth.&#xA;A typical modeling application is to maximize &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; subject to constraints on the inputs.  The typical constraint is to measure the inputs in terms of dollars and require the total to be within budget.  For such a problem one can set &lt;span class=&quot;math-container&quot;&gt;$K=1$&lt;/span&gt;, if you're a mathematician; others might insist on proper units, so that they can relate solutions to what they know about their factories and sales.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;What is the significance of the exponents &lt;span class=&quot;math-container&quot;&gt;$a,b,c,\dots$&lt;/span&gt;?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;We have (ignore &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; if you wish):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\log U = a \log x + b \log y + c \log z + \log K \tag{2}$$&lt;/span&gt;&#xA;The differential of &lt;span class=&quot;math-container&quot;&gt;$\log w$&lt;/span&gt; is the relative change &lt;span class=&quot;math-container&quot;&gt;$dw/w$&lt;/span&gt;, which the students should think of in the common phrase, &amp;quot;percent growth.&amp;quot; They should think of &amp;quot;percent growth&amp;quot; as the derivative of the logarithm of the quantity.&#xA;In our case, we obtain&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ {dU \over U} = a\,{dx \over x} + b\,{dy \over y} + c\,{dz \over z} \tag{3}$$&lt;/span&gt;&#xA;We see that the exponents appear as the coefficients in a linear combination of the percent changes in the inputs.&#xA;They indicate the relative weights of the percent changes in the inputs.&#xA;Together, this linear combination models the percent change in utility.&#xA;At maximum utility, &lt;span class=&quot;math-container&quot;&gt;$dU=0$&lt;/span&gt; and the linear combination has to &amp;quot;balance.&amp;quot;&#xA;An increase in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$dx&amp;gt;0$&lt;/span&gt;) is met by an overall decrease in the remaining terms. In terms of a budget constraint, if you spend more on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, you have less to spend on &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;.  It turns out that at maximum utility, one spends on &lt;span class=&quot;math-container&quot;&gt;$x,y,z$&lt;/span&gt; amounts proportional to &lt;span class=&quot;math-container&quot;&gt;$a:b:c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The role of logarithms here, algebraically, is rather minor. As a person who attended a small liberal arts institution in the US, what I liked about my math program was an emphasis on concept, in particular, on applying concepts in thinking about problems.  There are a couple that stand out to me. One I have already emphasized, the connection between logarithms and relative change.  The other is how logarithms convert an exponent form like (1) into linear forms (2) and (3), and linear is much easier to understand.&#xA;Of course one can proceed (3) &lt;span class=&quot;math-container&quot;&gt;$\rightarrow$&lt;/span&gt; (2) &lt;span class=&quot;math-container&quot;&gt;$\rightarrow$&lt;/span&gt; (1) via the exponential function since it is the inverse of the logarithm, another important concept.  These concepts are mathematical, not business concepts.  One builds on business concepts to create a mathematical model.  Then one applies whatever mathematics is appropriate.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2022-08-18T19:24:26.940" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25527" PostTypeId="2" ParentId="25521" CreationDate="2022-08-18T23:58:42.627" Score="4" Body="&lt;p&gt;Find some processes that are naturally exponential (investments, chemical processes, PSA test values in Prostate cancer patients).  Get some log graph paper.  Get them to graph this data on both normal graph paper and on log graph paper.&lt;/p&gt;&#xA;&lt;p&gt;If you can find a lot of examples from radically different knowledge areas, it will be more effective.&lt;/p&gt;&#xA;&lt;p&gt;Get a slide rule (yes, you can still find them online (there used to be an online shop in Vancouver Canada, but it looks like it's run out - you can find them on Etsy and eBay)).  There are slide rule apps for phones.&lt;/p&gt;&#xA;&lt;p&gt;Show the class how people multiplied and divided large numbers before the advent of calculators.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;As an aside:&lt;br /&gt;&#xA;I have had prostate cancer.  After my prostate was removed, my PSA count should have dropped to zero, but it's not unusual to see it greater than zero for several months.  Typical PSA counts for folks my age are 2-3 ng/ml.  With no prostate, I get &amp;quot;ultra-sensitive&amp;quot; PSA tests, that measure a few orders of magnitude lower than that.  It takes a while before testing starts again after a prostatectomy.  Mine were definitely non-zero.  After 4 tests, the doctors kept saying &amp;quot;don't worry, it's so low&amp;quot;.  But, those 4 tests, when graphed (taking into account some measurement noise) formed a near perfect exponential curve, with a doubling time of about 4-5 months.  Don't worry, radiation got rid of the troubling cancer cells&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="13120" LastEditorUserId="13120" LastEditDate="2022-08-19T22:56:22.090" LastActivityDate="2022-08-19T22:56:22.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25528" PostTypeId="2" ParentId="25520" CreationDate="2022-08-19T01:57:29.377" Score="-1" Body="&lt;p&gt;Texas has its own curriculum so it does not follow the Common Core Standards.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://tea.texas.gov/academics/curriculum-standards/teks-review/mathematics-texas-essential-knowledge-and-skills&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://tea.texas.gov/academics/curriculum-standards/teks-review/mathematics-texas-essential-knowledge-and-skills&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20232" LastActivityDate="2022-08-19T01:57:29.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25529" PostTypeId="1" AcceptedAnswerId="25530" CreationDate="2022-08-19T06:37:08.973" Score="6" ViewCount="2239" Body="&lt;p&gt;I was asked to teach the following topics to undergraduate Computer Science (CS) students in a discrete math course:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sequences (definition, convergent sequences, find the limit of a convergent sequence, properties of convergent sequences)&lt;/li&gt;&#xA;&lt;li&gt;Series (partial sums, convergent series, geometric series, harmonic series, test for divergence)&lt;/li&gt;&#xA;&lt;li&gt;Testing series (integral test, comparison tests, alternating series test, absolute convergence, ratio and root tests)&lt;/li&gt;&#xA;&lt;li&gt;Power series&lt;/li&gt;&#xA;&lt;li&gt;Taylor series&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This is included in my course because CS students are not required to take a 2nd calculus course. But I was told these topics may be useful if they go to graduate school so we must teach it somewhere.&lt;/p&gt;&#xA;&lt;p&gt;I cannot really think of any graduate CS courses which may require these topics. How should I motivate my students in this case?&lt;/p&gt;&#xA;" OwnerDisplayName="user19945" LastEditorDisplayName="user19945" LastEditDate="2022-08-21T02:30:12.043" LastActivityDate="2022-08-21T14:03:51.510" Title="What are some rationales to teach Computer Science students Sequences and Series?" Tags="|calculus|computer-science|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25530" PostTypeId="2" ParentId="25529" CreationDate="2022-08-19T12:44:20.210" Score="27" Body="&lt;p&gt;Big O and related notations relate closely to these notions. They are often defined using limits. Although it is also possible to define them without using limits, the style of those definitions is essentially the same as the style of definition used to define a limit: inequalities and several levels of quantifiers. Big O notation is a staple of computer science.&lt;/p&gt;&#xA;&lt;p&gt;Whether one of your students will use Taylor series is probably luck of the draw, but they're a basic tool of literacy in STEM. One thing I don't quite understand about your list of topics is that you're supposed to be teaching them concepts taught in second-semester calculus (in the US), but the sub-topics include the integral test, and that makes it sound like they've &lt;em&gt;already&lt;/em&gt; had a year of calculus.&lt;/p&gt;&#xA;&lt;p&gt;Is there a textbook that has been used by previous instructors when teaching this class? If so, then you should be able to see how the text integrates these topics into the narrative, and where these topics get used to support other topics.&lt;/p&gt;&#xA;" OwnerUserId="20234" LastEditorUserId="20234" LastEditDate="2022-08-19T12:45:14.297" LastActivityDate="2022-08-19T12:45:14.297" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25532" PostTypeId="2" ParentId="25507" CreationDate="2022-08-19T16:17:14.430" Score="-1" Body="&lt;p&gt;I think you are better off NOT engaging in discussion when remarks like this are made. 1. It's a distraction from the task, training on math. 2.  It's an intractable argument that won't be won by some magic rationale.&lt;/p&gt;&#xA;&lt;p&gt;As with many questions on this site, it is in the form of &amp;quot;how do I win this argument&amp;quot;, but I think it is important to realize that teaching is a people management problem, to a huge degree.  Different from the how do I integrate something, math problems. So, you have more options than the direct argument.&lt;/p&gt;&#xA;&lt;p&gt;Also consider humor to deflect, but acknowledge. Yeah, it can feel that way, but after you do more problems, you'll get used to it. Or other remarks like that. Use your judgmrnt based on the personslity. But don't get sucked into some mopey self assurance crap.&lt;/p&gt;&#xA;" OwnerUserId="20236" LastActivityDate="2022-08-19T16:17:14.430" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25533" PostTypeId="2" ParentId="25529" CreationDate="2022-08-19T21:54:27.363" Score="7" Body="&lt;p&gt;Look at the book “Concrete Mathematics” by Graham, Knuth, and Patashnik, which was written for a course for CS students. Power series show up in the context of generating functions.&lt;/p&gt;&#xA;&lt;p&gt;I think that list of convergence tests is strange: let them learn it in a calculus course. Or are they not required to take calculus?&lt;/p&gt;&#xA;&lt;p&gt;Odlyzko’s long survey paper &lt;a href=&quot;http://www.dtc.umn.edu/%7Eodlyzko/doc/asymptotic.enum.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; on asymptotic enumeration might be interesting to skim. It relies often on power series as functions of a complex variable (extracting polar terms), so going beyond the basic power series topics you’ll cover.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2022-08-19T21:54:27.363" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25534" PostTypeId="2" ParentId="25529" CreationDate="2022-08-19T22:25:12.597" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Rationale for ever learning:  They are related to numerical methods and algorithms.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Rationale for (re)covering with CS students.  Are a normal part of the calculus sequence (even the AP BC) in the US.  Not even some &amp;quot;advanced calc&amp;quot; or &amp;quot;engine math&amp;quot; or God Help Us (real analysis) thingie.  but at the end of the second semester and usually a little harder, more &amp;quot;off the path&amp;quot; of just learning antidifferentiation tricks.  I'd say the rationale for giving them to CS AGAIN is that in general (IN GENERAL), CS are a bit math weak compared to other STEMS, so perhaps they need a bit more exposure to stuff they should have learned in standard second semester calculus (or even high school AP BC).  And this is a part of that course that has some special relevance (more than partial fractions or the like).  Note: That this is not a strong rationale, is a marginal one.  If you have strong CS students, would not bother, teach 'em new stuff.  But if you have weaker ones, than it might make sense...and it's not crazy off the beaten path (see point 1).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;[Edited to clarify.]&lt;/p&gt;&#xA;" OwnerUserId="20204" LastEditorUserId="20204" LastEditDate="2022-08-21T14:03:51.510" LastActivityDate="2022-08-21T14:03:51.510" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25535" PostTypeId="2" ParentId="25529" CreationDate="2022-08-20T18:18:48.537" Score="3" Body="&lt;p&gt;Because every single one of those concepts naturally come up when you want to &amp;quot;do&amp;quot; things with computers!&lt;/p&gt;&#xA;&lt;p&gt;Let make take for example convergent sequences, what does it mean to converge? If we have a sequence &lt;span class=&quot;math-container&quot;&gt;$(a_n)$&lt;/span&gt; that means, for any possible &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;, there is some &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; such that for any &lt;span class=&quot;math-container&quot;&gt;$m&amp;gt;N$&lt;/span&gt; , we have:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ |a - a_m | &amp;lt; \epsilon$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;What does this mean? Suppose that I am a computer scientist and I wish to figure out how many terms of the series of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; I should take such that the difference between the series evaluation and actual value of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; below  some error, then naturally the method of answering this systematically would be same as knowing what convergence is.&lt;/p&gt;&#xA;&lt;p&gt;And more, the series convergence tests tells us what sequences it is worth to try search for bounds and which are not.&lt;/p&gt;&#xA;&lt;p&gt;And to my understanding, Taylor series is like the swiss knife of modelling. So many problems which may come up in modelling natural phenomena through computers can be treated quite easily be consider Taylor expansions. &lt;a href=&quot;https://matheducators.stackexchange.com/questions/21023/an-introductory-example-for-taylor-series-12th-grade/21032#21032&quot;&gt;See here&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorUserId="14627" LastEditDate="2022-08-21T07:35:59.460" LastActivityDate="2022-08-21T07:35:59.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25536" PostTypeId="2" ParentId="14672" CreationDate="2022-08-21T00:40:24.130" Score="-3" Body="&lt;p&gt;Just read this post and see why students still fail at math.&#xA;The first guy after reading your response confidently claimed ,’no, not a problem with the confusing language the problem is you have a slow learner’.&lt;/p&gt;&#xA;&lt;p&gt;The mark of a great teacher is defined by their results not theories or assumptions of why a student can’t understand a particular subject. If your student fails it’s because you failed them.&lt;/p&gt;&#xA;&lt;p&gt;It’s so easy to blame the student because it’s convenient but it takes true intelligence like you yourself must have to question the status quo and see if it can be improved upon.&lt;/p&gt;&#xA;&lt;p&gt;One user said for example,’oh if you make it simple they’ll just use that same routine without ever thinking analytically (in a nutshell) .&lt;/p&gt;&#xA;&lt;p&gt;It blows my mind. Grown adults who’ve went off, studied at prestigious schools, raising future generations , who in the face of the obvious say ‘no nothings wrong with the material, how it’s worded, it’s the student’&lt;/p&gt;&#xA;&lt;p&gt;Lazy, ineffective, and worthless. Just waiting on their pension marking off students as smart, slow, or dumb because deep down they really can’t teach , they just have a degree.&lt;/p&gt;&#xA;&lt;p&gt;Having a degree in education and actually being a teacher who can provide results are two different things.&lt;/p&gt;&#xA;&lt;p&gt;You’re on the right track. Disregard all these others.&lt;/p&gt;&#xA;&lt;p&gt;Find ways to show how all this complex is easy. Model/guide , quiz in the moment like a drill, allow student to think for themselves (a human will naturally be analytical when trying to solve a problem therefore it’s stupid to believe forcing a particular wording is the only way to bring this about and that’s the problem the wording)&lt;/p&gt;&#xA;&lt;p&gt;Likewise make it fun. This whole learning isn’t supposed to be fun comes from people who have degrees but can’t teach that doesn’t sound like you it sounds like you have what I wish every student had someone who got it.&lt;/p&gt;&#xA;&lt;p&gt;It just goes to show this is why the education system is the way it is. No one wants to say it but these teachers suck.&lt;/p&gt;&#xA;&lt;p&gt;They’re boring. Can’t put together a creative, stimulating, and challenging exercise to produce a result to save their lives.&lt;/p&gt;&#xA;&lt;p&gt;Don’t be like them, trust me years later you have no idea how grateful your students will be some you’ll mark forever because you did it differently.&lt;/p&gt;&#xA;" OwnerUserId="20239" LastActivityDate="2022-08-21T00:40:24.130" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25537" PostTypeId="1" CreationDate="2022-08-21T07:59:51.347" Score="-1" ViewCount="156" Body="&lt;p&gt;&lt;em&gt;How we have to start describing counting principles in combinatorics,what we need to apply first, addition principle or product principle ?&lt;/em&gt; &lt;br/&gt;&#xA;I have seen many starts with product principle rather than addition principle.One of the most common examples to introduce the lesson is calculating total different ways to travel from the city A to the city C via city B .Here many says if there are 3 different ways from A to B and another 2 different ways from B to C by the product principle number of total different ways is 3×2 = 6 . My issue is why can't we first describe, the involvement of addition principle in obtaining 3 in the first stage and 2 in the second stage because those values give number of mutually exclusive events.In that sense can't we argue that always you need to apply addition principle first and then product principle if necessary. &lt;br/&gt;&#xA;P.S. I have taken this example because couple of days ago very senior university lecturer used the same example without mentioning addition principle. Some times you may think this issue is not important in answering but I think answering and teaching are two different things and we have to be much precise in teaching which is the relevant part in this community.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2022-08-22T01:58:04.913" LastActivityDate="2022-08-22T14:45:10.927" Title="Counting principles in combinatorics" Tags="|secondary-education|combinatorics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25538" PostTypeId="2" ParentId="25537" CreationDate="2022-08-22T14:45:10.927" Score="3" Body="&lt;p&gt;It is not about whether you should use/introduce Addition Principle or Multiplication Principle first, but about making students to clearly understand that &amp;quot;Mutually exclusive options is Addition, Addition is mutually exclusive options&amp;quot; and hammering a similar mantra into their heads about Multiplication (though there it is a bit fancier). So, if somebody tries to use the Addition Principle, you should always ask them to identify the underlying mutually exclusive events, etc. Then they'll learn it eventually.&lt;/p&gt;&#xA;&lt;p&gt;Then a decent amount of time should be spent on understanding PIE, complementary counting, and other stuff. AoPS used to have a nice series of short videos on all that and I believe they are still there.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-08-22T14:45:10.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25539" PostTypeId="1" CreationDate="2022-08-22T16:52:05.513" Score="3" ViewCount="86" Body="&lt;p&gt;Has anyone experimented with using &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Fitch_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fitch-style&lt;/a&gt; proofs as a teaching aid in courses outside of logic specifically and if so, how was the technique received by students?&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2022-08-22T22:16:09.960" Title="Fitch Style Deduction in Non-Logic Classes" Tags="|undergraduate-education|mathematical-pedagogy|proofs|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25540" PostTypeId="2" ParentId="25539" CreationDate="2022-08-22T22:16:09.960" Score="3" Body="&lt;p&gt;Yes, I am writing a discrete math textbook which uses Fitch style proofs.&lt;/p&gt;&#xA;&lt;p&gt;It has been well received so far.  I want to do a study to see if the students are better at comprehending and constructing proofs in followup courses if they have learned this style of natural deduction in their Discrete math class.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://drive.google.com/drive/u/0/folders/19pXwLTowCwz1frNnVFNk0_v7KknxqXyJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://drive.google.com/drive/u/0/folders/19pXwLTowCwz1frNnVFNk0_v7KknxqXyJ&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-08-22T22:16:09.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25541" PostTypeId="1" AcceptedAnswerId="25543" CreationDate="2022-08-23T14:56:12.183" Score="5" ViewCount="1585" Body="&lt;p&gt;I am a STEM major and have a good friend who is a non-STEM major. We are both taking a CS minor, me because it is relevant to my field of study and he because he wants a backup plan in case his preferred career path doesn't work out. Thus, while I am a much stronger math student than he, we end up taking the same courses where the CS minor is concerned.&lt;/p&gt;&#xA;&lt;p&gt;Part of the CS minor's requirements are a series of math courses. We took the penultimate one last year, together, and I struggled with helping him through it. These are not math classes or concepts which either of us have been formally exposed to before. However, I have self-studied some things, and am better at grasping the concepts the first time around (during the lecture), while he needs more reinforcement and multiple explanations for more complicated ideas.&lt;/p&gt;&#xA;&lt;p&gt;Because we are friends we naturally did the homework together (doing homework in groups is encouraged by the professors in this series of classes). I felt awkward sometimes, as I had two goals the whole time: do the homework as quickly as I could do it well, and teach him the concepts. Our working relationship was not quite that of tutor and student, which I have more experience with - I tutored calculus on the side - but it was also not that of two equal friends struggling through the homework together. Besides homework, we also prepared for tests together, and I shared my notes with him.&lt;/p&gt;&#xA;&lt;p&gt;With the coming of fall classes, I have been thinking about how I could improve the next time our classes overlap, which will be in the winter term. &lt;strong&gt;How could I better approach helping/tutoring a friend in the same course as me?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="19777" LastActivityDate="2022-08-24T00:22:56.477" Title="How can I help/tutor a friend who is taking the same course as me?" Tags="|undergraduate-education|tutoring|computer-science|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25542" PostTypeId="2" ParentId="25541" CreationDate="2022-08-23T15:32:49.177" Score="4" Body="&lt;p&gt;Although I do think this is appropriate here, it's more a question about interpersonal relationships than about math education.&lt;/p&gt;&#xA;&lt;p&gt;First, what is the problem you want to solve? The only problem you named was sometimes feeling awkward. I am curious: Do you feel any resentment? Do you think your friend feels any? What boundaries would you like to set that you didn't before?&lt;/p&gt;&#xA;&lt;p&gt;Getting clarity about these things will help.&lt;/p&gt;&#xA;&lt;p&gt;If you'd like more help here, you'll have to define your problem more clearly.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-08-23T15:32:49.177" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25543" PostTypeId="2" ParentId="25541" CreationDate="2022-08-23T17:13:11.090" Score="10" Body="&lt;p&gt;&lt;em&gt;I had two goals the whole time: do the homework as quickly as I could do it well, and teach him the concepts.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you really want to help, the first goal should be scrapped entirely, IMHO. The best thing a stronger person can do for a weaker one when doing the homework together is to delegate all actual work to the latter and just to give hints when the weaker person is genuinely stuck. As to the rest, I would let the other person decide when and how much help he needs (though you may nudge him a bit now and then when you see that he is falling behind). Just play it by the ear: there is no recipe that fits all people and all situations. If you feel like he may see you as dominating him, just ask him to teach you something he is better than you at and that you are really interested in.&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-08-23T17:13:11.090" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25544" PostTypeId="2" ParentId="25541" CreationDate="2022-08-24T00:22:56.477" Score="1" Body="&lt;p&gt;I think you should prioritize your own learning. You are not getting paid to help him. Others are. Maybe avoid these group projects also, or get in a different section or take in a different order.&lt;/p&gt;&#xA;&lt;p&gt;This sounds cruel.  But you need to prioritize. Your job is to learn, not teach. That's what parents or governments are paying for.&lt;/p&gt;&#xA;&lt;p&gt;Nothing wrong with a quick tip. But stop studying together.&lt;/p&gt;&#xA;" OwnerUserId="20236" LastActivityDate="2022-08-24T00:22:56.477" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25545" PostTypeId="1" CreationDate="2022-08-24T07:30:48.380" Score="0" ViewCount="194" Body="&lt;p&gt;The question asks students to describe the expressions in words. This is a student's answer:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$(x + 4) \times 2$&lt;/span&gt;&amp;quot; is &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; plus four times two&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$5 - 3a$&lt;/span&gt;&amp;quot; is &amp;quot;five minus three times &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I think one of the answers should be marked as correct. If so, which one is it? Should both of the answers be marked as wrong because they are not clear? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="7487" LastEditorUserId="745" LastEditDate="2022-08-24T08:20:34.520" LastActivityDate="2022-08-25T13:46:03.057" Title="Can 5 - 3a be described as five minus three times a?" Tags="|algebra|" AnswerCount="2" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="25546" PostTypeId="2" ParentId="25545" CreationDate="2022-08-24T12:35:50.363" Score="1" Body="&lt;p&gt;&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$(x + 4) \times 2$&lt;/span&gt;&amp;quot; is incorrect.&lt;/p&gt;&#xA;&lt;p&gt;It is not  &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; plus four times two&amp;quot; because that would be &lt;span class=&quot;math-container&quot;&gt;$x + 4 × 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It should be the quantity of &lt;span class=&quot;math-container&quot;&gt;$x+4$&lt;/span&gt; times 2.  This shows that the parentheses are there which affects the value.  This is the standard used in the high school I taught.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me though that this standard should be introduced before giving this assignment. Otherwise - you are left with the confusion of the OP.&lt;/p&gt;&#xA;&lt;p&gt;I believe the second one is correct.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2022-08-25T13:46:03.057" LastActivityDate="2022-08-25T13:46:03.057" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25547" PostTypeId="2" ParentId="25545" CreationDate="2022-08-24T13:21:32.630" Score="4" Body="&lt;p&gt;If the question is to have any assessment efficacy, then the standard for the answer should be something like &amp;quot;Add 4 to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, then multiply the result by 2.&amp;quot; For the other example, the answer standard should be &amp;quot;Multiply 3 by &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, then subtract the result from 5.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2022-08-24T13:21:32.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25548" PostTypeId="2" ParentId="2279" CreationDate="2022-08-25T17:51:20.027" Score="0" Body="&lt;p&gt;A relatively simple way is to &lt;strong&gt;record a Zoom presentation by/to yourself&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;create a new Zoom meeting,&lt;/li&gt;&#xA;&lt;li&gt;then &amp;quot;share screen&amp;quot; of the screen you want to record (e.g. window on your laptop, or your tablet screen),&lt;/li&gt;&#xA;&lt;li&gt;hit record (in Zoom) and start presenting&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(My answer addresses the question in the title, not the aspect about TI-calculators, which I don't use).&lt;/p&gt;&#xA;" OwnerUserId="10289" LastActivityDate="2022-08-25T17:51:20.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25549" PostTypeId="1" CreationDate="2022-08-25T18:50:28.823" Score="5" ViewCount="146" Body="&lt;p&gt;I ask students to create tutorial videos where they explain course material (e.g how to do a math problem about a topic they are learning about, e.g. integration, induction). In this past, the students shared their videos only with me, not required (or requested) to share with classmates. and one or two asked if they would be shared with classmates, and I said no.&lt;/p&gt;&#xA;&lt;p&gt;But now I want to ask students to create videos and make them available to classmates. They upload them to the password proctected learning management system (LMS) used by the college, so it is not open to the general public, just students in the class (of course students in the class could share with general public...)&lt;/p&gt;&#xA;&lt;p&gt;Also I prefer not to &amp;quot;screen&amp;quot; the videos, i.e. watch them beforehand to make sure whether they are correct, don't contain inappropriate material. I just don't have the time.&lt;/p&gt;&#xA;&lt;p&gt;I'm hesitant though because&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;there are the students that have limited understanding of what is going on and confidently assert non-sense, and may confuse other students as well as may be embarrased by their lack of understanding, or aspects of their presentation (e.g. voice, technical setup, etc)&lt;/li&gt;&#xA;&lt;li&gt;privacy concerns: while the videos are on the password protected college LMS, there is nothing that prevents an individual student from copying the video and sharing/storing it for whatever purposes&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(If it matters, this is a remote course taught synchronously on zoom, in the US, undergraduate level).&lt;/p&gt;&#xA;&lt;p&gt;I'm having trouble formulating my exact question, but one is are there other concerns students might have? Any references to relevant literature about student privacy in digital classrooms (an area I have no background in) Are these concerns enough to justify abandoning this feat, or are there escape routes I can give? Last semester I tried suggesting that students make their video accessible to the entire class, but only a small number of students did. Is this assignment advisable? (Sorry if some of these questions are opinion based)&lt;/p&gt;&#xA;&lt;p&gt;Here is an article illustrating the idea, but there the instructors pre-screened submissions:&lt;/p&gt;&#xA;&lt;p&gt;Matthew A. Morena, Shelly Smith &amp;amp; Robert Talbert (2019) Video Made the Calculus Star, PRIMUS, 29:1, 43-55, DOI: 10.1080/10511970.2017.1396568&#xA;To link to this article: &lt;a href=&quot;https://doi.org/10.1080/10511970.2017.1396568&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1080/10511970.2017.1396568&lt;/a&gt;&#xA;(sorry, behind a paywall, but for only $55 you can have 48 hours of access to it, lol)&lt;/p&gt;&#xA;" OwnerUserId="10289" LastActivityDate="2022-09-30T03:07:50.923" Title="video tutorials by students for classmates - concerns?" Tags="|technology-in-education|" AnswerCount="1" CommentCount="5" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25550" PostTypeId="1" CreationDate="2022-08-25T21:57:38.497" Score="0" ViewCount="192" Body="&lt;p&gt;When we are first taught functions , we are typically taught of them as maps between real numbers and we taught to think of them mainly as a mapping between elements. It seems intuitive to take this route as well because most of the functions we are familiar with such as the square root, square etc are defined as a general which can taken any element in the real number. Eg: &lt;span class=&quot;math-container&quot;&gt;$f(x)= \sqrt{x}$&lt;/span&gt; , &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, to lead to the ideas in analysis and topology, one must think of function rather as a mapping between subsets. Now, the initial idea of course is equivalent to this, but it is usually non obvious why this form is more useful. Yet, I have never seen this point being emphasized when functions are being taught early on.&lt;/p&gt;&#xA;&lt;p&gt;Why is it so that this second way is not emphasized? Has there been any attempts to change into this alternate pedagogy for functions?&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2022-08-26T11:08:13.973" Title="The two paradigms of seeing a functions" Tags="|mathematical-pedagogy|functions|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25551" PostTypeId="2" ParentId="25550" CreationDate="2022-08-25T22:37:19.183" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;Because there isn't just a second way, but also a third way, and a fourth way, and many others.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;For example, we can also think of a function &lt;span class=&quot;math-container&quot;&gt;$f: X \rightarrow Y$&lt;/span&gt; as a function that takes an element of &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; and gives all the elements of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; that map to it - i.e. the data of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is equivalent to the data of &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt; is thought of as a function from &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; to the set of subsets of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Another way is to think of a function &lt;span class=&quot;math-container&quot;&gt;$f: X \rightarrow Y$&lt;/span&gt; as a function from subsets of &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; to subsets of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I can keep coming up with more complicated examples if you would like.&lt;/p&gt;&#xA;&lt;p&gt;The point is that, eventually, the student has to be able to see many different ways of representing the same data as being equivalent, and pointing out one specific way isn't all that helpful.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;If we have a function &lt;span class=&quot;math-container&quot;&gt;$g: 2^X \rightarrow 2^Y$&lt;/span&gt;, it does not necessary come from a function &lt;span class=&quot;math-container&quot;&gt;$f: X \rightarrow Y$&lt;/span&gt;.  For example, if &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; comes from such a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, we must have &lt;span class=&quot;math-container&quot;&gt;$g(\emptyset)=\emptyset$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2022-08-25T22:37:19.183" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25552" PostTypeId="2" ParentId="2279" CreationDate="2022-08-25T23:23:03.407" Score="0" Body="&lt;p&gt;Modern Windows systems come with an &amp;quot;xbox game capture&amp;quot; which is pretty generous about what it considers a game, including allowing you to capture ms paint and web browsers.&lt;/p&gt;&#xA;&lt;p&gt;You can move up from that to OBS Studio, which is free and allows you to combine/crop video sources and switch between scenes (configurations of combined/cropped video sources) at record time.&lt;/p&gt;&#xA;&lt;p&gt;If you're interested in making your own videos I definitely suggest practicing with software/hardware that's free or that you already have available before spending money.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-08-25T23:23:03.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25553" PostTypeId="2" ParentId="25550" CreationDate="2022-08-26T09:58:23.087" Score="4" Body="&lt;p&gt;The functions &lt;span class=&quot;math-container&quot;&gt;$f: A \to B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f^{-1}: B \to A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f^{\textrm{Im}}:2^A \to 2^B$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$f^{\textrm{Inv}}: 2^B \to 2^A$&lt;/span&gt; are all related, but distinct, functions.&lt;/p&gt;&#xA;&lt;p&gt;We often abuse notation and use the same notation for &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f^{\textrm{Im}}$&lt;/span&gt;, and for &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f^{\textrm{Inv}}$&lt;/span&gt;.  This is just for convenience though:  not because we actually think these are the same functions.  It is done to save time writing notation, and because it is entirely clear from context which function is actually meant.&lt;/p&gt;&#xA;&lt;p&gt;For instance if &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{R} \to \mathbb{R}$&lt;/span&gt; is defined by &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt;, then&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(2) = 4$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f^{\textrm{Im}}([1,2]) = [1,4]$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt; is not a function and&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f^{\textrm{Inv}}([1,4]) = [-2,-1] \cup [1,2]$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;You might see the following &lt;em&gt;technically incorrect&lt;/em&gt; notations in a book:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f^{-1}(4) = \{-2,2\}$&lt;/span&gt; (they should really write &lt;span class=&quot;math-container&quot;&gt;$f^{\textrm{Inv}}(\{4\}) = \{-2,2\}$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f^{-1}([1,4]) = [-2,-1] \cup [1,2]$&lt;/span&gt; (they should really be using &lt;span class=&quot;math-container&quot;&gt;$f^{\textrm{Inv}}$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f([1,2]) = [1,4]$&lt;/span&gt; (they should really by using &lt;span class=&quot;math-container&quot;&gt;$f^{\textrm{Im}}$&lt;/span&gt;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;We abuse notation like this to save time, finger strength, and mental energy.  It is unlikely to lead to genuine logical errors because what is actually meant is clear from the type of the input.  This is similar to type coercion in programming.&lt;/p&gt;&#xA;&lt;p&gt;p.s.  My favorite notation is actually &lt;span class=&quot;math-container&quot;&gt;$\vec{f}$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$f^{\textrm{Im}}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\overleftarrow{f}$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$f^\textrm{Inv}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-08-26T09:58:23.087" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25554" PostTypeId="2" ParentId="25550" CreationDate="2022-08-26T11:08:13.973" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't understand the difference you are claiming.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why does it surprise you that an advanced concept is NOT used at the intro?   It is a normal part of pedagogy to teach harder things later.   Your question springs from the same confusion as people who ask why not teach real analysis before calculus.  The literal reason is because humans have organic brains and don't learn everything immediately, by definitions.  Throw in also that the vast majority will not be math majors and won't need analysis or topology, ever. Something that seems crushingly obvious, but is routinely ignored by the know mathematics, don't know audience, crowd.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20236" LastActivityDate="2022-08-26T11:08:13.973" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25555" PostTypeId="1" CreationDate="2022-08-26T12:51:48.277" Score="2" ViewCount="139" Body="&lt;p&gt;I'm looking for examples of facts/results in basic school that are supported by calculus/analysis. Here is an example:&lt;/p&gt;&#xA;&lt;p&gt;Consider the infinite decimal expansion (a usual topic in basic school): basic school means primary;secondary school but mainly high school!&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a = 0.12345678910\ 11\ 12\ 13\ 14\ldots$&lt;/span&gt; (the &lt;a href=&quot;https://en.wikipedia.org/wiki/Champernowne_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;Champernowne constant&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;What does this expression mean? The answer &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is the limit of the sequence:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a_1 = 0.1, a_2 = 0.12, a_3 = 0.123,$&lt;/span&gt; etc.&lt;/p&gt;&#xA;&lt;p&gt;How do we know that this sequence has a limit? Here calculus helps: every bounded monotonic sequence has limit.&lt;/p&gt;&#xA;&lt;p&gt;Another example: Why does there exist a number &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$a \cdot a = 11$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Answer from calculus: &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt; is continuous, &lt;span class=&quot;math-container&quot;&gt;$f(0) = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(4) = 16$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$0 \leq 11 \leq 16$&lt;/span&gt;, by the Intermediate value theorem the existence of the number &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; (square root of &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt;) follows.&lt;/p&gt;&#xA;&lt;p&gt;Other examples?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="1680" LastEditDate="2022-08-28T00:54:08.130" LastActivityDate="2022-08-28T12:45:27.340" Title="Calculus/analysis as basis for basic school topics" Tags="|calculus|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25556" PostTypeId="1" CreationDate="2022-08-26T12:59:03.950" Score="2" ViewCount="655" Body="&lt;p&gt;Why should pre-service math basic school teachers take calculus courses?&#xA;Should this course be different from calculus courses offered to engineering?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="2083" LastEditDate="2022-08-27T08:59:04.840" LastActivityDate="2022-12-31T18:09:38.087" Title="Why is calculus important for pre-service Math basic school teachers?" Tags="|secondary-education|calculus|primary-education|teacher-preparation|teacher-training|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25557" PostTypeId="2" ParentId="25556" CreationDate="2022-08-26T16:03:26.337" Score="4" Body="&lt;p&gt;As Tommi commented, the question is a bit imprecise, but attempting to address it:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't think lower level teachers (early grades) [IN GENERAL] are required to take calculus.  If they are as part of a general ed degree, the only rationale would be for their general knowledge as university students.  And yes, a &amp;quot;calculus for non-STEM&amp;quot; would make total sense here.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I do think there is some benefit to going a &amp;quot;step higher&amp;quot; than what you teach.  So, it makes sense for high school and intermediate math teachers (say ages 12-18ish) to have had calculus.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;2.1.  For one thing, you solidify a lot of algebra skills by doing calculus!  Feynman noted this in his famous remarks.&lt;/p&gt;&#xA;&lt;p&gt;2.2.  Also, there is sort of a track of pre algebra to algebra to second year algebra to trig to pre-calc to calc.  (US basis, and the exact names vary, but my sister went to school in Europe and was very similar.)  So, at least you've seen the track you're heading kids on.  [Geometry is sort of off the track, but this is irrelevant to my point.]&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;There is probably a minor benefit in ed research in knowing calculus given the use of stats in papers and the like. But really, even for masters in ed, I think a baby algebra-based stats course is fine.  So, wouldn't justify the pain of making kindergarten teachers do Thomas...but just including the rationale, to be comprehensive.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20270" LastEditorUserId="20270" LastEditDate="2022-08-26T16:07:20.597" LastActivityDate="2022-08-26T16:07:20.597" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25558" PostTypeId="1" CreationDate="2022-08-26T20:47:15.927" Score="-3" ViewCount="152" Body="&lt;p&gt;English can be precise but it is rather 'flowery' and easily gets in its' own way.  East-Asian natural languages like Mandarin, Cantonese, Korean, and Japanese are notorious for permitting the simplification of mental arithmetic.  It has been argued this has something to do with numbers such as: 36 transliterating to &amp;quot;three tens and six.&amp;quot;  In Indian English, 16/4 seems to be spoken as &amp;quot;Sixteen on Four&amp;quot; or &amp;quot;Sixteen by Four.&amp;quot;  In American English we might say &amp;quot;Sixteen divided by...&amp;quot; or &amp;quot;Sixteen over...&amp;quot; (the latter example being somewhat less common).  Small mental or linguistic tricks can make a huge difference in the formation of an intuitive conception of various maths relationships, operations, and concepts. Again,&lt;/p&gt;&#xA;&lt;p&gt;Q1: &amp;quot;[What are arguably the...] best natural language(s) for conveying, conceptualizing, teaching, understanding, and learning Probabilistic &amp;amp; Statistical concepts &amp;amp; theory??&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;There are many indirect ways to attempt to answer the main question I present in this post; which of course requires subjective responses.  One might ask:&lt;/p&gt;&#xA;&lt;p&gt;Q2: Which countries produce or employ the most Statisticians per capita?&lt;/p&gt;&#xA;&lt;p&gt;Q3: Which is the ratio/percentage between the top native languages spoken by graduates of universities who study Probability, Statistics or an equivalent field heavy in these subjects?&#xA;Or employees of corporations who hold titles in positions that require a robust working knowledge of said fields?&lt;/p&gt;&#xA;&lt;p&gt;Q4: How do these candidate languages, that differ in their approach from English, describe, verbally present, and or phrase to act as the building-blocks that lead to the development of a strong foundation in these fields?&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;&lt;p&gt;Sincerely,&#xA;NaiveBaye aka Zorkv&lt;/p&gt;&#xA;" OwnerUserId="20271" LastEditorUserId="20271" LastEditDate="2022-10-19T06:29:55.330" LastActivityDate="2022-11-18T07:04:45.597" Title="Best natural language(s) for conveying, conceptualizing, teaching, understanding, and learning Probabilistic &amp; Statistical concepts &amp; theory?" Tags="|mathematical-pedagogy|statistics|probability|language-use|concept-maps|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25559" PostTypeId="2" ParentId="25558" CreationDate="2022-08-26T22:15:35.693" Score="0" Body="&lt;p&gt;I don't think it makes much difference, especially after you get past the most basic counting and arithmetic.  That said, it does bring up natural questions as to the best languages to seduce someone in, to sell used cars in, or to recite poetry in.  E.g. English often look down on Germans, but Schiller is quite poetic.&lt;/p&gt;&#xA;" OwnerUserId="20272" LastActivityDate="2022-08-26T22:15:35.693" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25560" PostTypeId="2" ParentId="25556" CreationDate="2022-08-27T13:14:33.483" Score="7" Body="&lt;p&gt;When the undergraduate curriculum for preservice teachers does not include mathematics beyond middle school (say grades 6-9, algebra 1 or algebra 2 in the US), then what can happen in some cultures is that preservice teachers with negative attitudes about mathematics or with low mathematical self-esteem tend to end up being those who gravitate towards teaching elementary school. This is turn perpetuates negative societal memes about mathematics, because those teaching mathematics to formative young children hold these negative attitudes.&lt;/p&gt;&#xA;&lt;p&gt;So we can argue that the existence of undergraduate pathways to certification that do not require “college algebra” or precalculus or calculus will only perpetuate negative societal memes about mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Of course simply requiring preservice teachers to have mathematical competence beyond basic algebra is not in itself sufficient to counter this problem created by self-selection of who teaches elementary school based on their mathematical self-efficacy. Teacher preparation programs must teach precalculus or calculus in such a way that mathematical self-efficacy of the future teachers increases, or is at least not ruined.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2022-08-27T13:14:33.483" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25561" PostTypeId="2" ParentId="25555" CreationDate="2022-08-28T12:45:27.340" Score="2" Body="&lt;p&gt;Calculus is the reason radian angle measure is important (formulas like &lt;span class=&quot;math-container&quot;&gt;$(\sin x)’ = \cos x$&lt;/span&gt; are incorrect using degrees or any other angle measure besides radians) and the reason &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt; are important (look at formulas for &lt;span class=&quot;math-container&quot;&gt;$(b^x)’$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(\log_b x)’$&lt;/span&gt;). Before calculus there is no natural reason for those to seem like genuinely relevant topics.&lt;/p&gt;&#xA;&lt;p&gt;The chain rule is arguably the reason that function composition needs to be emphasized earlier in high school. At least I remember that when I was learning calculus, once the chain rule appeared I thought “Aha, that’s why we spent all that time on function composition before.”&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2022-08-28T12:45:27.340" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25562" PostTypeId="1" CreationDate="2022-08-28T13:40:26.323" Score="2" ViewCount="124" Body="&lt;p&gt;A common rule given in textbooks is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{R}$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$(x^a)^b=x^{ab}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Suppose I write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(-9)^{1/2}=(-9)^{2/4}=((-9)^2)^{1/4}=(81)^{1/4}=3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But this contradicts the fact that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(-9)^{1/2}=\sqrt{-9}=3i$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What did I do wrong?&lt;/p&gt;&#xA;&lt;p&gt;I don't see a problem with changing &lt;span class=&quot;math-container&quot;&gt;$(-9)^{1/2}$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$(-9)^{2/4}$&lt;/span&gt;. Is the issue&#xA;with going from &lt;span class=&quot;math-container&quot;&gt;$(-9)^{2/4}=((-9)^2)^{1/4}$&lt;/span&gt;, and if so, does that mean the rule given above only applies to positive bases? Are the textbooks wrong?&lt;/p&gt;&#xA;" OwnerUserId="20130" LastActivityDate="2022-08-29T14:49:53.793" Title="Do properties of exponents apply only to positive real numbers?" Tags="|precalculus|" AnswerCount="1" CommentCount="14" ClosedDate="2022-08-28T18:24:30.493" ContentLicense="CC BY-SA 4.0" />
  <row Id="25563" PostTypeId="2" ParentId="25562" CreationDate="2022-08-28T16:06:08.560" Score="1" Body="&lt;p&gt;As stated in the comments, the theorem is wrong. There are several similar looking theorems where &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; are integers or where &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0$&lt;/span&gt;. You should carefully check the text to make sure that you didn't miss some qualifiers. (This would probably be an easy thing for authors to miss in a first edition too, so who knows.)&lt;/p&gt;&#xA;&lt;p&gt;The root of the problem comes from the fact that, if you view &lt;span class=&quot;math-container&quot;&gt;$(-9)^{1/n}$&lt;/span&gt; to be &amp;quot;the&amp;quot; solution to &lt;span class=&quot;math-container&quot;&gt;$x^n=-9$&lt;/span&gt;, then you don't have just one value, but &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; of them, i.e., &lt;span class=&quot;math-container&quot;&gt;$(-9)^{1/2}$&lt;/span&gt; could be &lt;span class=&quot;math-container&quot;&gt;$3i$&lt;/span&gt; or maybe &lt;span class=&quot;math-container&quot;&gt;$-3i$&lt;/span&gt;. Further, &lt;span class=&quot;math-container&quot;&gt;$81^{1/4}$&lt;/span&gt; could be &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$3i$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$-3i$&lt;/span&gt; since all of these have the correct fourth power. (Things get even messier if you want to think about things like &lt;span class=&quot;math-container&quot;&gt;$(-9)^{\sqrt{2}}$&lt;/span&gt;.) Most of the time, though, we want just one answer. Then you pick out a &amp;quot;principal branch&amp;quot; which chooses just one of these as the answer. Unfortunately, choices of principal branch may not be compatible when you bounce around between denominators. That's why the answer &lt;span class=&quot;math-container&quot;&gt;$3i$&lt;/span&gt; shows up in the list for &lt;span class=&quot;math-container&quot;&gt;$81^{1/4}$&lt;/span&gt;. It's &lt;em&gt;a&lt;/em&gt; 4th root of 81, just not the principal one.&lt;/p&gt;&#xA;&lt;p&gt;As for teaching this topic, I feel like you need to say the right thing, but I can really see the point of not getting bogged down with the qualifiers that much.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="12255" LastEditDate="2022-08-29T14:49:53.793" LastActivityDate="2022-08-29T14:49:53.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25565" PostTypeId="2" ParentId="25556" CreationDate="2022-08-28T21:04:56.727" Score="1" Body="&lt;p&gt;Others have already addressed the first half of the question, and complained about its imprecision.&lt;/p&gt;&#xA;&lt;p&gt;The second half is this: &amp;quot;Should this course be different from calculus courses offered to engineering?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Well, more broadly, too many different types of students are currently thrown together in the same one-year calculus sequence in the US. It's absurd to expect people to learn this much calculus in order to become dentists, for example, but that is effectively what is required in places like California. E.g., a community college student who says they want to go to dental school will be steered into calculus and calc-based physics, so that they can apply to UC's in the biology major.&lt;/p&gt;&#xA;&lt;p&gt;The question has tags for both primary and secondary education. If someone is going to become a primary educator in the US, then they are not required to take calculus. If they're going to become secondary-school math teachers, then certainly they should take a solid one-year calculus course of the type that currently exists in the US. They'll need that background because they may be called upon to teach AP calculus. But realistically, may US high schools are drafting their PE teachers to teach algebra.&lt;/p&gt;&#xA;&lt;p&gt;In an ideal world, I would suggest two or three different calculus tracks for freshmen in the US, with only the highest-level track doing epsilon-delta proofs and methods of integration. Preservice high school math teachers would take this track. The lowest track would be a one-semester survey of calculus.&lt;/p&gt;&#xA;&lt;p&gt;I think the reality is that this will never happen. Math departments don't want to lose the enrollment that they would lose if lots of students only had to take one semester of calculus. Biology departments started instituting a calculus and calc-based physics requirement back in the 90's precisely because they wanted the hardest possible requirements in order to filter prospective majors.&lt;/p&gt;&#xA;" OwnerUserId="20275" LastEditorUserId="20275" LastEditDate="2022-08-28T21:12:23.383" LastActivityDate="2022-08-28T21:12:23.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25567" PostTypeId="2" ParentId="25556" CreationDate="2022-08-29T20:09:31.410" Score="3" Body="&lt;p&gt;You’ve got to understand where Calculus sits in the history of Math Education.&lt;/p&gt;&#xA;&lt;p&gt;What we know as the math curriculum was essentially set up by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Committee_of_Ten?wprov=sfti1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Committee of Ten&lt;/a&gt;, a group from 1892. It’s an oversimplification, but essentially they set Calculus as the capstone course and arranged Algebra, Geometry, everything really, to focus on the skills one needs to learn Calculus in college. Think about it: why would students need to learn how to manipulate and verify trigonometric identities if it weren’t for Calculus? Why so much on sequences and series? There are tons of examples of topics that are essentially unjustifiable to students because they’ll never see them again, never use them in real life, &amp;amp;c., &lt;em&gt;except&lt;/em&gt; if you think about what you need to take two years of Calculus. If that’s your goal (and, again, it was) then pretty much anything in the curriculum has a reason for being there. That’s the history.&lt;/p&gt;&#xA;&lt;p&gt;So if you’re a math teacher, even today with things like Statistics crufted into the curriculum, you’re basically teaching Pre-Calculus. Algebra I is Pre-Calculus. Geometry is Pre-Calculus. It’s all pre-Calculus. And guess what—take serious statistics in college and you’ll discover it’s pretty much Calculus at its core.&lt;/p&gt;&#xA;&lt;p&gt;Knowing that, is there a course that should be taught to all future math teachers? A branch of math that all math teachers need to experience? Even if you never think you’re going to teach anything higher than Pre-Calculus Algebra I, what course should be unquestionably in your preparation? I mean, there are math courses that you can take that have no calculus in them like &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0134468376&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1429254823&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt; (both excellent bases for courses) or things like &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1337629960&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;this blah book&lt;/a&gt; or especially &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0321987292&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;this travesty&lt;/a&gt;, but you should think about those in addition to a calculus sequence.&lt;/p&gt;&#xA;&lt;p&gt;Not to mention that Calculus is a human achievement that allowed the modern world to happen. If you think art and music and Literature are worthwhile for everyone (and I &lt;em&gt;so&lt;/em&gt; do) then Calculus should clearly be on that list. I realize it takes a lot of preparation, but it’s a beautiful thing, as is lots of the math leading up to it, especially if taught well. And a prerequisite to teaching it well is having it in your teaching toolbox.&lt;/p&gt;&#xA;&lt;p&gt;Note I’m not contending for the standard two years of Calculus like math majors or engineers take. You don’t need that much detail. But two semesters of applied calculus (especially if your teacher uses &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1305085329&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book&lt;/a&gt;) feels to me like exactly what’s needed. Because it’s all Pre-Calculus, see?&lt;/p&gt;&#xA;" OwnerUserId="9241" LastActivityDate="2022-08-29T20:09:31.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25568" PostTypeId="1" AcceptedAnswerId="25569" CreationDate="2022-08-29T23:16:09.770" Score="6" ViewCount="154" Body="&lt;p&gt;I am teaching high school math in a room with no whiteboard but a projector. I have a tablet that I write on with a stylus and this is projected at the front of the room.&lt;/p&gt;&#xA;&lt;p&gt;I have been making PowerPoint slides with background material and examples that I work out in real-time with the class by writing on the slides with the stylus.&lt;/p&gt;&#xA;&lt;p&gt;I have been finding that I run out of room to write on the slides.&lt;/p&gt;&#xA;&lt;p&gt;I need to find a way to switch from the slides to another software program that gives me a larger &amp;quot;canvas&amp;quot; to write on (e.g. GoodNotes). But that involves rewriting the examples from the slides in the new program, which is annoying and wastes time.&lt;/p&gt;&#xA;&lt;p&gt;What does your teaching workflow look like in high school math? Do you primarily use slides, the board, a tablet, or a combination?&lt;/p&gt;&#xA;" OwnerUserId="20130" LastActivityDate="2022-08-31T12:02:24.260" Title="Workflow for class notes/slides and examples?" Tags="|secondary-education|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25569" PostTypeId="2" ParentId="25568" CreationDate="2022-08-30T16:25:37.377" Score="6" Body="&lt;p&gt;Instead of using PowerPower for presenting your lectures, use software which allows you to display and annotate files (such as .pdf files).&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;First, export your PowerPoint slides to .pdf.  Depending on what operating system and version of PowerPoint you are using, this can be done from the &amp;quot;Save As&amp;quot; dialog, the &amp;quot;Export&amp;quot; dialog, or the &amp;quot;Print&amp;quot; dialog (all of these are options under the &amp;quot;File&amp;quot; dropdown).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Import the files into the program of your choice.  In the comments, Sue VanHattum recommended &lt;a href=&quot;https://www.goodnotes.com/&quot; rel=&quot;noreferrer&quot;&gt;GoodNotes&lt;/a&gt;.  Personally, I use &lt;a href=&quot;https://notability.com/&quot; rel=&quot;noreferrer&quot;&gt;Notability&lt;/a&gt;.  There are other programs out there---just find one you like.  You can write directly on the slides, or insert extra pages for annotations.  As you get more experience, you will likely find ways of creating slides specifically to be annotated, or find some other workflow which you are happy with.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As an added bonus, you can export the document as a .pdf (or whatever other format you prefer), then make those files available to students, if you so desire.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Another bit of the workflow:  use screenshots to copy things from the screen into the notes.  For example, I often use screenshots to include snippets from the course text, GeoGebra graphs, and bits-and-bobbles from the LMS.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;On an Apple machine, you can hit &lt;code&gt;CTRL&lt;/code&gt;+&lt;code&gt;CMD&lt;/code&gt;+&lt;code&gt;SHIFT&lt;/code&gt;+&lt;code&gt;4&lt;/code&gt;, then click-and-drag in order to copy a section of the screen to the clipboard.  This can then be pasted into other apps, such as Notability or GoodNote.  Other keyboard shortcuts for screenshotting can be found on &lt;a href=&quot;https://support.apple.com/en-us/HT201361&quot; rel=&quot;noreferrer&quot;&gt;Apple's support pages&lt;/a&gt; (note: adding &lt;code&gt;CTRL&lt;/code&gt; to the chord puts the screenshot on the clipboard, rather than saving it to a file.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Directions for taking screenshots on a Windows machine can be found on &lt;a href=&quot;https://support.microsoft.com/en-us/windows/use-snipping-tool-to-capture-screenshots-00246869-1843-655f-f220-97299b865f6b&quot; rel=&quot;noreferrer&quot;&gt;Microsoft's support pages&lt;/a&gt;.  I am not a regular Windows user, and have no experience with their screenshot utility, but it &lt;em&gt;does&lt;/em&gt; exist.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;On tablets, clever phones, and other devices, screenshots can be taken in a variety of ways, which likely depend on the specifics of the hardware and software being used.  For example, on my clever phone (a Google thingy), one takes a screenshot by holding down the power butting, then selecting the &amp;quot;Screenshot&amp;quot; option when the menu pops up.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Once the screen is shot and the captured image stored in the clipboard, it can be pasted into the program of choice using &lt;code&gt;CMD&lt;/code&gt;+&lt;code&gt;V&lt;/code&gt; (or &lt;code&gt;CTRL&lt;/code&gt;+&lt;code&gt;V&lt;/code&gt; on a Windows box) or via the &amp;quot;Paste&amp;quot; option in the &amp;quot;Edit&amp;quot; dropdown.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2022-08-31T02:41:06.647" LastActivityDate="2022-08-31T02:41:06.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25570" PostTypeId="2" ParentId="25549" CreationDate="2022-08-31T02:25:13.120" Score="1" Body="&lt;p&gt;I work in a high school, and can tell you that students really prefer not to be randomly called on. I ask a question and hands go up, so there's some interaction, but when I call on a student looking down, it's clear they are not comfortable. In college/university, they should be more mature, but people, by their nature, may be shy. I think that to the outspoken student, there's no issue, but the shy ones may find this request very unfair.&#xA;&amp;quot;I signed up for a math class, not to be put on the spot and on stage every week.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;That said, the intent you had is great. If you had a TA or 2, they could help identify the best video(s) for each problem, and by the end of the term, you have the ability to offer something close to a self-study course, if those students are detailed enough with their explanations. During the year of Covid remote teaching, I supported a teacher who wasn't comfortable with tech. And I made videos to help the students who were falling behind. When the teacher might race through an example in 3 minutes, I'd get a chat message from a number of students, and create a detailed slow video closer to 15 minutes. The concept has potential value. The implementation might be very tough.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-08-31T02:25:13.120" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25571" PostTypeId="2" ParentId="25568" CreationDate="2022-08-31T12:02:24.260" Score="4" Body="&lt;p&gt;If you are happy using the tech, you've gotten some good advice. But. I sense your frustration at the lack of a white board. I'd suggest that the home center type stores (I am looking at Home Depot) have 4 foot by 8 foot sheets of such material in the wall paneling section. It's not sold as a whiteboard, there is no frame, it's just the sheet itself, and it's reasonably priced, I see it listed near me for $20. If you are in the same room for all your classes, 2 nails in the front wall are all you need.&lt;/p&gt;&#xA;&lt;p&gt;If you are staying on this path - Xander has the solution. During Covid remote, and remote tutoring since, I've gotten to really like Notability. I'd add just one thought to his answer - it offers the ability to link to a folder in Google Drive. I am in my email, and see a pdf my student sent me, and click 'open in' and choose the app. I then switch to Notability, (do what I need to share on Zoom call), write on the PDF as we work together, and when I close out of the app, the updated file, a PDF, is there, in my Google Drive folder to forward from my computer. You can easily create a second copy or erase the added work when presenting to the next classe, but in the end, you have the ability to easily have your work preserved for your own reference or for an absent student.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-08-31T12:02:24.260" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25572" PostTypeId="2" ParentId="25521" CreationDate="2022-08-31T16:17:56.743" Score="0" Body="&lt;p&gt;Growth rates, especially of macroeconomic variables, that are usually small in value, are routinely modeled theoretically and estimated empirically by using logarithms. Let &lt;span class=&quot;math-container&quot;&gt;$Y_t$&lt;/span&gt; be, say, gross domestic product, or national income, or aggregate consumption, at period &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, (which is usually a year or a quarter), then its growth rate compared to period &lt;span class=&quot;math-container&quot;&gt;$t-1$&lt;/span&gt; is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$${\rm growth\; rate} \equiv \frac{Y_{t} - Y_{t-1}}{Y_{t-1}} = \frac{\Delta Y_{t}}{Y_{t-1}}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Consider the natural logarithm of the ratio of &lt;span class=&quot;math-container&quot;&gt;$Y_t$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$Y_{t-1}$&lt;/span&gt;, while writing&#xA;&lt;span class=&quot;math-container&quot;&gt;$Y_t \equiv Y_{t-1} + \Delta Y_t$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\ln \left(\frac{Y_{t}}{Y_{t-1}}\right) = \ln \left(\frac{Y_{t-1} + \Delta Y_t}{Y_{t-1}}\right) = \ln \left(1+ \frac{\Delta Y_t}{Y_{t-1}}\right) \approx \frac{\Delta Y_t}{Y_{t-1}} = {\rm growth\;rate}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This approximation is acceptable when &lt;span class=&quot;math-container&quot;&gt;$|\Delta Y_t/Y_{t-1}| &amp;lt; 0.1$&lt;/span&gt;, which is almost always the case with macroeconomic variables in quarterly frequency, or even in yearly frequency. By the properties of the logarithms then,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\ln Y_{t} - \ln Y_{t-1} \approx {\rm growth\;rate\; of}\; Y_t.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For micro-level, business level though, this may not be advisable to use, since single business may experience growth (or contraction) rates higher than &lt;span class=&quot;math-container&quot;&gt;$10\%$&lt;/span&gt; in their revenues, profits or expenses --but then the log-approximation becomes rather visibly inaccurate.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2022-08-31T16:17:56.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25573" PostTypeId="2" ParentId="25460" CreationDate="2022-08-31T17:45:47.793" Score="0" Body="&lt;p&gt;I've put together a number of online interactives in Javascript for College Algebra, Precalculus, Statistics, and Linear Algebra, available here: &lt;a href=&quot;https://trkern.github.io/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://trkern.github.io/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-08-31T17:45:47.793" CommentCount="0" CommunityOwnedDate="2022-08-31T17:45:47.793" ContentLicense="CC BY-SA 4.0" />
  <row Id="25574" PostTypeId="2" ParentId="25488" CreationDate="2022-08-31T17:53:53.903" Score="1" Body="&lt;p&gt;Set realistic intermediate goals for students while they're trying to catch up. By default their goal might be along the lines of completely catching up within a week. That's unrealistic, and if they fail at it, they'll be demotivated and can easily get stuck in a failure/demotivation feedback spiral.&lt;/p&gt;&#xA;&lt;p&gt;Setting expectations you know the student can be successful in can get students on a success/motivation feedback spiral instead.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-08-31T17:53:53.903" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25576" PostTypeId="2" ParentId="13809" CreationDate="2022-08-31T23:06:40.523" Score="3" Body="&lt;p&gt;My former adviser explained continuity in this way to a freshman class (all math majors):&lt;/p&gt;&#xA;&lt;p&gt;Assume that you are taking shower. The independent variable is the position of the shower handle and the dependent variable is the water temperature. If rotating the handle by small amount, you change the temperature just a little, you have continuity but if the slightest touch applied to the handle changes the temperature from icy cold to boiling hot, then you have a discontinuity. And you have, probably, seen both kinds of showers, haven't you?&lt;/p&gt;&#xA;&lt;p&gt;Instead of the shower, you can put a ball on a flat surface and then on a convex one (a piece of wood, a sheet of transparent plastic, some old bearing, a few screws and 10 minutes of work, and you have the device you need). Then tilt the surface a bit. On the convex one, the position of the ball will change only a bit with rotation, but on the flat one it will travel from one end to another when you cross the horizontal position. Same idea, of course, but slightly less wet and messy.&lt;/p&gt;&#xA;&lt;p&gt;I should say that I cannot reproduce how he proved most of the theorems today, but 38 years later I still remember the shower story. So, perhaps, your students will remember it too. And it can easily be converted into the formal &lt;span class=&quot;math-container&quot;&gt;$\varepsilon-\delta$&lt;/span&gt; statement, should you choose to go that far.&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastEditorUserId="19946" LastEditDate="2022-08-31T23:17:25.373" LastActivityDate="2022-08-31T23:17:25.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25577" PostTypeId="2" ParentId="25502" CreationDate="2022-09-01T00:11:10.070" Score="5" Body="&lt;p&gt;&lt;em&gt;are there any concrete studying techniques that can take you from &amp;quot;Yeah, that makes sense now that you mention it&amp;quot; to &amp;quot;Yeah, I can understand why they thought of this.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Technically, yes, but not at the beginner's level. Try to read Polya's &amp;quot;How to solve it?&amp;quot;. The main advice there (if you forget about various tricks and details) is just &amp;quot;relate it to something familiar&amp;quot;. It applies to the analysis of proofs just as well as to the proof design. But to follow it you have to have some database of &amp;quot;familiar things&amp;quot; and an effective algorithm for extracting them and evaluating the degree of similarity. Both come with practice. In the beginning you just look for similar &lt;em&gt;words&lt;/em&gt; (This problem is about, say, a simplex in &lt;span class=&quot;math-container&quot;&gt;$R^n$&lt;/span&gt;. What other problems about a simplex in &lt;span class=&quot;math-container&quot;&gt;$R^n$&lt;/span&gt; have I seen? Can I use the same technique here?). Then you slowly acquire a feeling for a &lt;em&gt;general setup&lt;/em&gt; (so the simplex problem can be related to a problem about contractive mappings, say). And so on. It all takes time and patience.&lt;/p&gt;&#xA;&lt;p&gt;As to your particular problem, the answer is another ubiquitous general principle (also discussed in Polya, I believe): &lt;em&gt;Simplify whenever possible, making sure that you have some tangible gain at each step!&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here it works like that. When you see an equation with square roots, your first instinct (from high school, say) should be to &lt;em&gt;square something to get rid of those nasty radicals&lt;/em&gt;. But if you square the LHS as it is, you start with 3 radicals and you end up with 3 radicals after the squaring, so not much is gained. The reason is that the product of two different radicals is a radical again more often than not, so you have three radicals and three pairwise products. You want to reduce the number of square roots. So you want to have only 2 or fewer of them on one side. Hence, to the other side one of them goes (which one does not matter in this problem though sometimes it does). Now you have a really tangible gain (down to 2 from 3). Then you go from 2 to 1 and, bingo, you can relate it to the famous ancient proof of the irrationality of &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2$&lt;/span&gt; only, perhaps, with a different prime now.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-09-01T00:11:10.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25579" PostTypeId="2" ParentId="25460" CreationDate="2022-09-01T00:54:06.503" Score="1" Body="&lt;p&gt;I usually just use html/SVG combination and do it on a case by case basis. Examples include&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.kent.edu/%7Enazarov/switches/switchgame.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.math.kent.edu/~nazarov/switches/switchgame.html&lt;/a&gt;&#xA;(Linear system in &lt;span class=&quot;math-container&quot;&gt;$Z_2$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.kent.edu/%7Enazarov/Hillcipher/hill.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.math.kent.edu/~nazarov/Hillcipher/hill.html&lt;/a&gt;&#xA;(Linear cipher)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.kent.edu/%7Enazarov/Buffon/buffon.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.math.kent.edu/~nazarov/Buffon/buffon.html&lt;/a&gt;&#xA;(Buffon's needle)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.kent.edu/%7Enazarov/T-intersection/tint.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.math.kent.edu/~nazarov/T-intersection/tint.html&lt;/a&gt;&#xA;(&amp;quot;real life&amp;quot; linear equations problem (ill-posed without prohibiting left turn))&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.kent.edu/%7Enazarov/turmach&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.math.kent.edu/~nazarov/turmach&lt;/a&gt;&#xA;(Turing machine emulation)&lt;/p&gt;&#xA;&lt;p&gt;There is more stuff in that directory (mainly games made for Math. Club). It doesn't relate to the question directly, but you are welcome to take a look at it too just to see what can be done in two or three evenings with basic HTML skills (except Jeopardy: that one took a couple of weeks and was a work of several people). Some of these are not immediately obvious how to operate, but if you get interested in any of them, just let me know.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-09-01T00:54:06.503" CommentCount="0" CommunityOwnedDate="2022-09-01T00:54:06.503" ContentLicense="CC BY-SA 4.0" />
  <row Id="25580" PostTypeId="2" ParentId="25488" CreationDate="2022-09-02T09:43:26.673" Score="0" Body="&lt;p&gt;One obvious solution to the problem is obligatory student presence at contact teaching (maybe 80 %). This will make students concerned about being present and can stop the process where shame at not having been there causes more absence.&lt;/p&gt;&#xA;&lt;p&gt;In particular, an attendance policy usually carries with it an assumption of communication: that students tell of absences and their reasons, and the faculty also keeps track and takes contact if the absences start approaching the limit. This assumption of communication undercuts the spiral of shame.&lt;/p&gt;&#xA;&lt;p&gt;Of course, this has its own issues in terms of keeping track of attendance and taking away agency from the students. Hence, it is not an obviously correct solution, but maybe still worth considering.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2022-09-03T06:55:24.040" LastActivityDate="2022-09-03T06:55:24.040" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25581" PostTypeId="2" ParentId="25488" CreationDate="2022-09-02T16:08:43.017" Score="2" Body="&lt;p&gt;Policies differ a lot. My dad died December 1 of my second year at USNA. I got 3 days off and zero allowances for turning in final papers or doing final exams.  My sister got sent home from a civilian school, total forgiveness on finals and assignments, and credit to move on. Both semester system. Neither one seemed to affect us much academically and next semester it was onward. My littlest sister was in high school and got ongoing attention from teachers throughout the year. And it seemed to derail her.&lt;/p&gt;&#xA;&lt;p&gt;I guess to boil it down, I think best to cut any allowances short at the end of the term. After that life goes on.  Death of parents is something that all of us will encounter unless we die young. I don't have a good suggestion for dealing with death in the family earlier in the term.&lt;/p&gt;&#xA;&lt;p&gt;This is not to say you can't be someone to talk to.  But even here, I would use your judgment and let the student come to you. At the end, he or she is just going to have to deal with it. You may or more likely are not be someone for them to connect with. Use your spidey sense.&lt;/p&gt;&#xA;&lt;p&gt;I read Stef's answer and violently agree. None of us, students or schools can prevent death if a parent or have a time machine. The important thing is to make a plan and go forward. You can't take away the suck factor of losing a parent. Just be pragmatic about what you do control, school.&lt;/p&gt;&#xA;" OwnerUserId="20297" LastEditorUserId="19777" LastEditDate="2022-09-10T02:17:33.247" LastActivityDate="2022-09-10T02:17:33.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25582" PostTypeId="1" CreationDate="2022-09-03T04:37:43.667" Score="2" ViewCount="118" Body="&lt;p&gt;How to explain the Cayley-Klein or sometimes called Beltrami–Klein metric concept to find the distance between two points in a hyperbolic space to an audience with no higher education than maybe a second or third-year undergrad? What are a, b, C and what are p, q on quadric Q.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;${\displaystyle d(a,b)=C\log {\frac {\left|bp\right|\left|qa\right|}{\left|ap\right|\left|qb\right|}}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, is there any exercises online in finding distance using the above concept?&lt;/p&gt;&#xA;" OwnerUserId="20300" LastActivityDate="2022-09-03T04:37:43.667" Title="Is there any way of explaining the Cayley/Beltrami–Klein metric to undergrads?" Tags="|undergraduate-education|exercises|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25584" PostTypeId="2" ParentId="25556" CreationDate="2022-09-04T21:29:52.570" Score="0" Body="&lt;p&gt;Well, I found this scientific reference today:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://doi.org/10.5951/MT.16.7.0431&quot; rel=&quot;nofollow noreferrer&quot;&gt;THE PLACE OF THE CALCULUS IN THE TRAINING OF THE HIGH SCHOOL TEACHER&#xA;Author(s): BYRON COSBY&#xA;Source: The Mathematics Teacher , NOVEMBER, 1923, Vol. 16, No. 7 (NOVEMBER, 1923),&#xA;pp. 431-439&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is another study:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.361.5453&quot; rel=&quot;nofollow noreferrer&quot;&gt;eAspects of Calculus for Preservice Teachers BY  Lee Fothergill&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="6053" LastEditDate="2022-12-31T18:09:38.087" LastActivityDate="2022-12-31T18:09:38.087" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25585" PostTypeId="1" CreationDate="2022-09-05T12:35:05.973" Score="5" ViewCount="2366" Body="&lt;p&gt;I teach high school Pre-Calculus. In an effort to make my lessons more efficient time-wise, one area I think I can improve is choosing the right time to return graded assessments to my students (that are given on paper).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Option 1 (what I have been doing up until now):&lt;/strong&gt; &lt;em&gt;Return the assessments to the students at the START of class.&lt;/em&gt; This ensures that I don't forget to do it later on. However, I usually give the students a Do-Now to work on at the start of class. If they receive their graded assessment back as they are working, they become distracted by looking through their assessment and comparing scores with their peers. Hence, it takes longer for the students to complete the Do-Now and time is wasted.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Option 2:&lt;/strong&gt; &lt;em&gt;Return the assessments to the students in the MIDDLE of class.&lt;/em&gt; I tried this once but it had the same effect as Option 1; students would get distracted from working on the in-class exercises and they would want to go over some of the solutions right away, disrupting the flow of the lesson.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Option 3:&lt;/strong&gt; &lt;em&gt;Return the assessments to the students at the END of class&lt;/em&gt; (ideally when the lesson has concluded). This may work because the students are not trying to work on math problems at the same time I am returning the assessment. However, I often forget to return the assessments because I am thinking about other announcements I have to make before the students leave. Also, the lesson needs to end a few minutes before the class period ends, which for me is hit or miss.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Option 4:&lt;/strong&gt; &lt;em&gt;Return the assessments to the students OUTSIDE of class.&lt;/em&gt; I have not tried this yet but if I were to try it, I would hold on to the graded assessments and tell the students that if they want to look at their assessment, they have to see me before or after-school. This may conflict with some students' schedules (extracurriculars, sports, jobs etc.), but the students who value their grade will probably find the time.&lt;/p&gt;&#xA;&lt;p&gt;I appreciate any advice from those who also encountered this problem and what their solution was. Is Option #4 worth trying?&lt;/p&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="13657" LastEditDate="2022-09-05T13:01:05.720" LastActivityDate="2022-09-06T04:14:52.437" Title="When is the best time to return a graded assessment?" Tags="|grading|assessment|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25586" PostTypeId="2" ParentId="25585" CreationDate="2022-09-05T15:03:36.107" Score="12" Body="&lt;p&gt;Return at the beginning and go over it then. Any in class drill should wait for the finish of q and a on the papers.&lt;/p&gt;&#xA;&lt;p&gt;Avoid parallel processing.  You are leading a large group, not an individual.  Regimentation helps with classroom management. You don't have to be harsh, just organized. Set the tone and they will fall in.&lt;/p&gt;&#xA;&lt;p&gt;Personally I don't hold with graded homework or projects, in high school math.  I advocate weekly, Friday, hour long tests.  But the above goes for returning those.  First thing Monday.  Or even let kids come by Friday afternoon and grade them fast during lunch and prep period. But overall class gets them Monday class start.&lt;/p&gt;&#xA;&lt;p&gt;Whatever you do, don't dawdle. Grade them ASAP. This is only fair given they care about them. Also, it quickens the feedback loop, better pedagogy.&lt;/p&gt;&#xA;" OwnerUserId="20307" LastEditorUserId="20307" LastEditDate="2022-09-05T15:06:21.290" LastActivityDate="2022-09-05T15:06:21.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25587" PostTypeId="2" ParentId="25585" CreationDate="2022-09-05T20:10:01.173" Score="4" Body="&lt;p&gt;There's lots of good news here.&lt;/p&gt;&#xA;&lt;p&gt;It's great that you (a) treat class time as valuable and try not to waste your students' time, and (b) give students written feedback on their work. Good for you. You're incredibly lucky that (c) your students care so much about doing well on their schoolwork, but (d) aren't just copying the answers to the problems off of the internet.&lt;/p&gt;&#xA;&lt;p&gt;Realistically, high school teaching involves a huge number of contact hours, far more than is actually necessary for instruction. I doubt that you're going to find anything that will restore these extra few minutes of lost time that you're worried about. What you're already doing sounds fine, so I would probably just keep doing it and not worry.&lt;/p&gt;&#xA;&lt;p&gt;If you want to try something different, then you could give feedback to your students outside of class, but do it using the internet. Expecting high school students to see you after school to get feedback on their work is just not realistic.&lt;/p&gt;&#xA;" OwnerUserId="20308" LastActivityDate="2022-09-05T20:10:01.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25588" PostTypeId="2" ParentId="25585" CreationDate="2022-09-06T01:43:16.467" Score="6" Body="&lt;p&gt;You're putting the cart before the horse... first answer the question&lt;/p&gt;&#xA;&lt;h3&gt;why are you giving the graded assessment back?&lt;/h3&gt;&#xA;&lt;p&gt;It might be so that they have a record of their performance in case you get hit by a bus. It might be to motivate them to participate in class activities in a different way. It might be to set up going over a set of common mistakes you saw. It might be the material they need to complete their post-assessment reflection.&lt;/p&gt;&#xA;&lt;p&gt;It might be ten other things, and it's almost certainly a mix of all, in different amounts for different students at different times.&lt;/p&gt;&#xA;&lt;p&gt;I, for instance, often am handing back papers because &lt;em&gt;as part of my lesson plan we're using them.&lt;/em&gt; So I give them back at the relevant point in my lesson's plan. (It might be the first thing we do that day, it might be the last, it might not matter and I put the day's bullet points on the board and poll the students for what order to go in.)&lt;/p&gt;&#xA;&lt;h3&gt;You've identified the right factors.&lt;/h3&gt;&#xA;&lt;p&gt;The intended use of class-time, time spent comparing with other students, questions that arise from seeing your feedback... all of these should be considered as part of your planning. So after grading, think &amp;quot;what do I want the students to get out of continuing engagement with this piece of paper? What activity of theirs will best lead to that? What class time (if any) should I devote to ensuring that happens?&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="6869" LastActivityDate="2022-09-06T01:43:16.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25589" PostTypeId="2" ParentId="25585" CreationDate="2022-09-06T04:14:52.437" Score="2" Body="&lt;p&gt;&lt;strong&gt;Hand it to the Class Monitor before the class&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am a student myself, and in my experience, this is the best method that has ever been done.&lt;/p&gt;&#xA;&lt;p&gt;In every class there is a responsible one. Hand it to this responsible student (termed the 'monitor' in schools in my country) and forget about it. Since all students want their assessments back, they would collect it on their own accord after finishing their do now. Others get the assessment after the lesson is over when the monitor distributes the rest of the assessments.&lt;/p&gt;&#xA;&lt;p&gt;Any personal feedback to the students are to be given after the end of the period/school hours on a one to one basis&lt;/p&gt;&#xA;" OwnerUserId="20310" LastActivityDate="2022-09-06T04:14:52.437" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25592" PostTypeId="1" CreationDate="2022-09-08T08:28:15.980" Score="11" ViewCount="2919" Body="&lt;p&gt;We all know how to use a protractor; it is taught in elementary school. However, I was wondering what type of knowledge is required to build one from scratch.&lt;/p&gt;&#xA;&lt;p&gt;For instance, was the understanding of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; and a compass first required before the first protractor, and if so, how can I draw a full protractor on paper with just a compass, a ruler and some understanding of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;I guess my point is, if we can draw a semi-circle on paper, then how can we fill up all degrees without the help of a protractor?&lt;/p&gt;&#xA;&lt;p&gt;I consider the fact that I can build the tool that I would use to solve the problem in a different way as a great engineering learning experience.&#xA;I am especially interested in answers that take this aspect into account.&lt;/p&gt;&#xA;" OwnerUserId="20313" LastEditorUserId="143" LastEditDate="2022-09-09T23:00:55.677" LastActivityDate="2022-09-11T06:51:59.830" Title="How can I build a protractor without a protractor?" Tags="|geometry|engineering-mathematics|" AnswerCount="4" CommentCount="10" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25593" PostTypeId="2" ParentId="833" CreationDate="2022-09-08T08:58:58.617" Score="0" Body="&lt;p&gt;** Introduce two distinct sections in your questionnaire- one moderate, and other harder, trickier, possibly olympiad level questions **&lt;/p&gt;&#xA;&lt;p&gt;The first section is good for testing the student's learning and understanding.  The second  section, the &amp;quot;Achiever's section &amp;quot; helps test critical thinking and true understanding and grasp of the subject.&lt;/p&gt;&#xA;" OwnerUserId="20310" LastActivityDate="2022-09-08T08:58:58.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25594" PostTypeId="2" ParentId="25592" CreationDate="2022-09-08T09:13:31.257" Score="2" Body="&lt;p&gt;In terms of doing it on paper, it is impossible to trisect (or quintisect) angles with a conventional compass and straight edge.  And 180 = 2 * 2 * 3 * 3 * 5.  Of course you could be an anti-Babylonian rebel and create your own angle measurements on a power of 2.&lt;/p&gt;&#xA;&lt;p&gt;However, you can use a quadratitrix compass (essentially a moving plane device) for trisection (and quintisection) of angles, to satisfy the conventional degree measurement.  See  &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadratrix_of_Hippias&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Quadratrix_of_Hippias&lt;/a&gt; for a very short discussion.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  Somewhat tangential, but of interest...see this video on origins of metrology for practical machining.  &lt;a href=&quot;https://www.youtube.com/watch?v=gNRnrn5DE58&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=gNRnrn5DE58&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20315" LastEditorUserId="20315" LastEditDate="2022-09-08T09:14:36.723" LastActivityDate="2022-09-08T09:14:36.723" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25595" PostTypeId="2" ParentId="17686" CreationDate="2022-09-08T09:39:52.987" Score="1" Body="&lt;p&gt;One of the benefits of the partial quotients division method as illustrated above with 186 / 3 is NOT in dividing by single digit divisors but in multi-digit divisors. Students are generally taught the bus stop method when dividing by a single digit (which makes sense). But when you start dealing with multiple digit divisors this approach can help students to see that division is really just a process of subtractions. They do not need to be accurate in the multiples they use (as required by the traditional long division method) as this approach is more flexible and can limit the multiples to just 1x 2x &amp;amp; 5x (as these are the factors of 10) and focus on reducing the remainder. Students should be taught that fractions &amp;amp; division are very similar as it may be possible to simplify the division problem by reducing both dividend &amp;amp; divisor by the same number just like simplifying a fraction. As long division is fraught with difficulty it puzzles me why student are not taught to check their long division answers by using Sum of the Digits: (divisor x quotient) + remainder = dividend.&lt;/p&gt;&#xA;" OwnerUserId="20317" LastActivityDate="2022-09-08T09:39:52.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25596" PostTypeId="2" ParentId="11516" CreationDate="2022-09-08T10:57:06.950" Score="0" Body="&lt;p&gt;Since you asked for different approaches (even anecdotes), so giving mine.  (My practical solution is at the end.)&lt;/p&gt;&#xA;&lt;p&gt;My undergrad school had a similar engine math course.  It was semester-long and used Kreyszig as the text.  Which is a quite good book, imo, with sound explication and drill, but also wide coverage.  Of course the problem is that it would take 4-6 semesters to cover it all!  And the students don't have time for that.  They have to take statics and thermo and etc.  To the extent that it already constrains their humanities electives more than non-engineers (chemists for example).  So, the solution is to prioritize.  They get a deeper exposure to PDEs (over what the diffyqs course gave) so they've at least dealt with &amp;quot;Jo and Yo&amp;quot; (Bessel functions, J(0) and Y(0) and the vibrating drum problems).  And a few weeks on linear algebra (which is basic matrix manipulations).  This, because they DON'T have time for a full semester class on LA. (Something math majors are often shocked to learn, but is common.)  Complex analysis was NOT covered at all, despite being in the book.  Really, only a small amount of the book was covered.&lt;/p&gt;&#xA;&lt;p&gt;When I went there, the solution for mathies was that math majors NEVER took engine math.  They took the same classes through diffyqs (calc 1-3 plus diffyscrews (almost entirely ODEs).  But never took engine math.   They moved into real analysis (called &amp;quot;theoretical calculus&amp;quot;) and abstract algebra and the like. They also had a required full semester in linear algebra, along with the option for additional courses in that topic.  PDEs and complex analysis were electives and full semester courses, meaning some of the students never took them, although realistically, the majority did...and if someone prefers taking an operations research course to PDEs, we should realize that &amp;quot;different strokes for different folks&amp;quot;.  There is not time in the undergrad to take everything, hence why we have majors to start with!&lt;/p&gt;&#xA;&lt;p&gt;The solution for physics was to take engine math...and then any special topics were done in physics class (e.g. learning calculus of variations in the second mechanics course).  This is non-ideal, but life is a constrained by time problem.  A few years later, they just felt this was inadequate and installed a two semester math prep course.  So essentially physics (and EEs) got a second semester of engine math that covered the complex analysis (of Kreyszig).&lt;/p&gt;&#xA;&lt;p&gt;So, practical alternatives:  &lt;strong&gt;diverge the math students out of an engine math course (they never even take it).  And for physicists and EEs give them a second semester&lt;/strong&gt; (and figure out what to cover in it, but it's mostly an engineering-oriented complex analysis course.  Maybe a little more on PDEs also.  To be practical, I would just do that as an appended course. Not attempting to &amp;quot;enrich&amp;quot; the previous semester, while running or integrate into it.  [Note that this is not unreasonable.  ODEs is treated again after a short exposure in second semester calculus.  And it is normal for physics, chemistry, and biology to be treated at a college level after an earlier high school level exposure.  We routinely get questions on this site for &amp;quot;why don't they do it right from the start&amp;quot;, but this ignores practical pedagogy and the limits of human comprehension, as well as the option value of those who take a simple course and decide later to go deeper.]&lt;/p&gt;&#xA;" OwnerUserId="20315" LastActivityDate="2022-09-08T10:57:06.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25597" PostTypeId="2" ParentId="25592" CreationDate="2022-09-08T19:15:42.780" Score="17" Body="&lt;p&gt;As Will Orrick says in the comments under &lt;a href=&quot;https://matheducators.stackexchange.com/a/25594/29&quot;&gt;user20315's answer&lt;/a&gt;, it is possible, with straightedge and compass, to construct a regular 120-gon, and therefore it is possible to mark off every 3 degrees on a circle.&lt;/p&gt;&#xA;&lt;p&gt;Can we get any farther? It depends on how much precision you require, and how much error you are willing to tolerate. In principle, it is not possible to trisect a &lt;span class=&quot;math-container&quot;&gt;$3^\circ$&lt;/span&gt; angle using only a compass and straightedge.  However, the following &lt;em&gt;incorrect&lt;/em&gt; trisection method produces angles that are very, very close to correct:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; be the center of a circle, and let &lt;span class=&quot;math-container&quot;&gt;$A, B$&lt;/span&gt; be points on the circle such that arc &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; measures 3 degrees.&lt;/li&gt;&#xA;&lt;li&gt;Join &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; to create segment &lt;span class=&quot;math-container&quot;&gt;$\overline{AB}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Trisect &lt;span class=&quot;math-container&quot;&gt;$\overline{AB}$&lt;/span&gt; using a compass and straightedge, finding points &lt;span class=&quot;math-container&quot;&gt;$C, D \in \overline{AB}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$AC = CD = DB$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Draw rays &lt;span class=&quot;math-container&quot;&gt;$\overrightarrow{AC}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\overrightarrow{AD}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The resulting angles &lt;span class=&quot;math-container&quot;&gt;$\angle AOC, \angle COD, \angle DOB$&lt;/span&gt; are &lt;strong&gt;not&lt;/strong&gt; exactly 1 degree each, but the difference between the actual measures and the desired measures are less than 1 part in 10,000.  Given the imprecision involved in using mechanical construction tools (how thick is the tip of your pencil? how smoothly can you draw an arc with a compass? how 'straight' is your straightedge?), and the inherent limits involved in reading or using a protractor (can you even measure a degree to less than 0.1 degree precision with a protractor anyway?), this would seem to be good enough for almost all conceivable purposes.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2022-09-08T19:40:01.000" LastActivityDate="2022-09-08T19:40:01.000" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25599" PostTypeId="2" ParentId="25592" CreationDate="2022-09-09T05:40:13.177" Score="13" Body="&lt;p&gt;Step 1:  Find a machinable material that is reasonably incompressible.&lt;br /&gt;&#xA;Step 2.  Find a string material that is reasonably non-stretchy.&lt;br /&gt;&#xA;Step 3:  Make a cylinder out of the machinable material (perhaps using a lathe).&lt;br /&gt;&#xA;Step 4:  Make a fine groove in the cylinder, of even depth and non-wavering axial elevation.  (The lathe can do this, too.)&lt;br /&gt;&#xA;Step 5:  Wrap the string around the cylinder (in the groove).&lt;br /&gt;&#xA;Step 6:  Cut off exactly one lap of the string.&lt;br /&gt;&#xA;Step 7:  Measure off however many even increments you want along the string.  (You can do this by making a set of evenly-spaced parallel lines, and placing the taut string between points on the far parallel lines.  The number of parallel lines needs to be one more than the number of increments.)&lt;br /&gt;&#xA;Step 8:  Put the marked-off string back in the groove.&lt;br /&gt;&#xA;Step 9:  Optionally, secure the string in place.&lt;/p&gt;&#xA;&lt;p&gt;Note that steps 1-5 are equivalent to using a compass to draw a circle on a paper.  Step 6 is a way to straighten out the circle back to a line segment.  Steps 5 and 8 are a way to wrap a line segment into a circle.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2022-09-11T06:51:59.830" LastActivityDate="2022-09-11T06:51:59.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25601" PostTypeId="2" ParentId="4339" CreationDate="2022-09-10T03:52:26.637" Score="0" Body="&lt;p&gt;Engineers use Pascal's triangle and Taylor series approximations.&lt;/p&gt;&#xA;&lt;p&gt;Can you relate sequences and series to error analysis?&lt;/p&gt;&#xA;&lt;p&gt;The Fundamental Theorem of Calculus is necessary, so that they can check their work.  Most Differential Equations courses (over-)rely on it.&lt;/p&gt;&#xA;&lt;p&gt;Engineers need to be able to take derivatives and integrals of polynomials, exponentials, sine waves, and cosine waves in their sleep.  And set up the boundary conditions for these problems, too.&lt;/p&gt;&#xA;&lt;p&gt;Engineers should know how the limit formula for a derivative works, why every single term for it is what it is, and how to reverse the derivation to get an integral.&lt;/p&gt;&#xA;&lt;p&gt;Consider other textbook authors, such as Swokowski.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2022-09-10T03:52:26.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25602" PostTypeId="2" ParentId="25592" CreationDate="2022-09-11T01:40:35.027" Score="6" Body="&lt;p&gt;In &lt;a href=&quot;https://matheducators.stackexchange.com/users/29/mweiss&quot;&gt;mweiss&lt;/a&gt;' &lt;a href=&quot;https://matheducators.stackexchange.com/posts/25597&quot;&gt;method&lt;/a&gt;, the outer angles ∠AOC and ∠DOB are shy of 1° by slightly more than 0.0001°, and the inner angle ∠COD exceeds 1° by slightly more than 0.0002°.&#xA;On a (huge!) one meter diameter protractor, that is a worst case error of less than 4 microns, or one-seventh of a thousandth of an inch.&lt;/p&gt;&#xA;&lt;p&gt;Formulas used to calculate the &amp;quot;shy-ness&amp;quot; of the outer angles:&lt;br /&gt;&#xA;α = angle being trisected&lt;br /&gt;&#xA;inner angle = 2 * arctan(tan(α/2)/3)&lt;br /&gt;&#xA;shy-ness = arctan(tan(α/2)/3) - α / 6&lt;/p&gt;&#xA;&lt;p&gt;If that accuracy is not good enough for you, you are welcome to repeat the process:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mirror image ∠DOB about the line OD,&lt;/li&gt;&#xA;&lt;li&gt;to make an angle ∠AOE that exceeds 1° by slightly more than 0.0002°.&lt;/li&gt;&#xA;&lt;li&gt;Angle ∠COE is slightly more than 0.0003°.&lt;/li&gt;&#xA;&lt;li&gt;Use mweiss' method to approximately trisect angle ∠COE.&lt;/li&gt;&#xA;&lt;li&gt;Let &amp;quot;F&amp;quot; be the new point that is one-third of the way from C to E.&lt;/li&gt;&#xA;&lt;li&gt;Angle ∠AOF is our second approximation of a degree.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Using the previous formula, and impossibly accurate compasses and straight-edges, angle ∠AOF is shy of 1° by slightly more than 10&lt;sup&gt;-16&lt;/sup&gt; degrees.&#xA;On a protractor the size of the Earth, 90° would be 10,000 kilometers.  Even at that scale, angle ∠AOF would only be shy of 1° by slightly more than 10&lt;sup&gt;-11&lt;/sup&gt; meters.&lt;/p&gt;&#xA;&lt;p&gt;In our quantum-mechanical world, you cannot draw anything that accurately.&#xA;&amp;quot;Solid&amp;quot; matter made of nuclei and electrons gets really fuzzy at a scale of 10&lt;sup&gt;-10&lt;/sup&gt; meters.  If you try measuring any smaller than this scale, you cannot know whether an electron will be where you are prodding or not.  So this second approximation of a degree is limited by Earthly physics:  you cannot make a compass or a straight edge perfect enough to achieve it, and even if you could, you could not measure how perfectly you drew it.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2022-09-11T06:44:27.800" LastActivityDate="2022-09-11T06:44:27.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25603" PostTypeId="1" CreationDate="2022-09-11T16:19:31.930" Score="28" ViewCount="5617" Body="&lt;p&gt;This is related to my previous question &lt;a href=&quot;https://matheducators.stackexchange.com/q/25473/7481&quot;&gt;What value is there in requiring students to declare the dimensions of an answer when it is already clear from context?&lt;/a&gt; , but with a different focus.&lt;/p&gt;&#xA;&lt;p&gt;A sizeable minority of my primary and secondary school teachers (east-coast US in the late 1980's/early 1990's) insisted that word problems be answered in complete sentences.&lt;/p&gt;&#xA;&lt;p&gt;Consider the following hypothetical question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Anne's car travels 225 miles in five hours. What is the speed of Anne's car?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The correct answer, according to these teachers, would be&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The speed of Anne's car is 45 mph.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some leeway in wording was allowed, so an answer of &amp;quot;Anne's car has a speed of 45 mph&amp;quot;, &amp;quot;45mph is the speed of Anne's car&amp;quot;, or &amp;quot;Anne's car travels at a 45mph speed&amp;quot; could be accepted for full credit, but answers such as &amp;quot;45&amp;quot; or even &amp;quot;45 mph&amp;quot; were considered insufficient and would result at best in lost points and at worst in being awarded zero points for the question.&lt;/p&gt;&#xA;&lt;p&gt;The explanation that I recall being given was that we were being educated to use math to communicate &lt;em&gt;information&lt;/em&gt; to others in real-life contexts, and that context was always necessary. For example, if we walked up to an adult and told them &amp;quot;45 mph!&amp;quot;, they would look at us like we were crazy and wonder what it is we were trying to communicate, but if we instead presented them with &amp;quot;The speed of Anne's car is 45 mph&amp;quot;, they would instead react, &amp;quot;Wow! I've always wondered about that. Thanks!&amp;quot; and would be sure to lavish us with awards and college recommendation letters.&lt;/p&gt;&#xA;&lt;p&gt;The school system I spent most of my childhood years in was (at that time) very big into interdisciplinary and cross-curricular studies. We were supposed to write papers about math, do math problems on real history, study the history of science, etc., so my understanding was that the requirement to write answers in complete sentences was an exercise in &lt;em&gt;literacy&lt;/em&gt; and not mathematics per se.&lt;/p&gt;&#xA;&lt;p&gt;So, is there &lt;em&gt;mathematical&lt;/em&gt; pedagogical value in requiring students to answer word problems in complete sentences, or is this actually a literacy or communication exercise that has been applied interdisciplinarily to math exercises?&lt;/p&gt;&#xA;" OwnerUserId="7481" LastEditorUserId="7481" LastEditDate="2022-09-11T16:38:59.007" LastActivityDate="2022-09-13T16:13:15.563" Title="What value is there in requiring students to answer word problems in complete sentences?" Tags="|secondary-education|primary-education|grading|language-use|word-problems|" AnswerCount="7" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25604" PostTypeId="5" CreationDate="2022-09-11T16:32:32.667" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2022-09-11T16:32:32.667" LastActivityDate="2022-09-11T16:32:32.667" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25605" PostTypeId="4" CreationDate="2022-09-11T16:32:32.667" Score="0" Body="Word problems are a type of mathematics exercise in which a mathematical question is expressed in a written narrative and the student must use literacy skills to determine the correct operations and numbers to be used to find the solution. Use this tag for questions about writing, explaining, or grading word problems, or about whether certain word problems are or are not acceptable for certain purposes or certain skill levels." OwnerUserId="7481" LastEditorUserId="7481" LastEditDate="2022-09-18T13:47:28.217" LastActivityDate="2022-09-18T13:47:28.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25606" PostTypeId="2" ParentId="25603" CreationDate="2022-09-11T19:31:37.877" Score="40" Body="&lt;p&gt;Yes, there is mathematical pedagogical value in the usage of complete sentences - but this does not only refer to &amp;quot;answers&amp;quot; and not only to &amp;quot;word problems&amp;quot;, but to all parts of the solution to any mathematical problem or task.&lt;/p&gt;&#xA;&lt;p&gt;The reason is that mathematics is not only about numbers, or computations, or equations, or inequalities. It is also (some people might say: mainly) about how to precisely formulate mathematical arguments and propositions. Propositions and arguments are typically composed of complete sentences.&lt;/p&gt;&#xA;&lt;p&gt;One of the various challenges that I perceive whenever dealing with first year math students at university level is that they find it very difficult to express their mathematical thoughts in a precise and clearly understandable way - and part of this problem is that most of them didn't learn in math classes in school to express themselves in complete sentences.&lt;/p&gt;&#xA;" OwnerUserId="16250" LastEditorUserId="16250" LastEditDate="2022-09-11T20:28:12.693" LastActivityDate="2022-09-11T20:28:12.693" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25607" PostTypeId="2" ParentId="25603" CreationDate="2022-09-12T10:02:36.937" Score="37" Body="&lt;p&gt;I do think there is value in expecting students to give answers in correct English. It will certainly help when they start to face longer and less structured questions.&lt;/p&gt;&#xA;&lt;p&gt;However, the example you give is entirely inappropriate. When you are asked a direct question, it &lt;em&gt;is&lt;/em&gt; correct English to simply respond with the answer. You do not need to prefix it with &amp;quot;The answer to your question is&amp;quot;, or words of like meaning. Requiring this sort of thing just becomes an artificial constraint designed to trip people up, rather like Jeopardy.&lt;/p&gt;&#xA;" OwnerUserId="20327" LastActivityDate="2022-09-12T10:02:36.937" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="25608" PostTypeId="2" ParentId="25603" CreationDate="2022-09-12T15:10:41.580" Score="1" Body="&lt;p&gt;I don't think there's a mathematical reason for this, but teachers may view this as part of an interdisciplinary approach. Requiring students to answer in complete sentences reinforces what they're learning in language class, by having them write complete, grammatical sentences.&lt;/p&gt;&#xA;&lt;p&gt;I recall similar requirements in other classes. For instance, in History class, if asked when the US declared its independence, you were expected to say something like &amp;quot;The Declaration of Independence was signed in 1776&amp;quot; rather than just &amp;quot;1776&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="10551" LastEditorUserId="10551" LastEditDate="2022-09-13T16:13:15.563" LastActivityDate="2022-09-13T16:13:15.563" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="25609" PostTypeId="2" ParentId="25603" CreationDate="2022-09-12T16:57:04.953" Score="11" Body="&lt;p&gt;Jochen's answer is very good. To add to it, here's the reasoning I was given as a student: answering in full sentences makes students think about the answer in different psychological context. This can help them catch mistakes.&lt;/p&gt;&#xA;&lt;p&gt;For example, imagine the student answering the question had made the common mistake of dividing the hours by the distance instead of the other way around, and got an answer of 0.0222. Writing the answer as a full sentence (&amp;quot;The speed of Anne's car is 0.0222mph&amp;quot;) might well prompt them to realize that they've made an error.&lt;/p&gt;&#xA;" OwnerUserId="20331" LastActivityDate="2022-09-12T16:57:04.953" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25610" PostTypeId="1" AcceptedAnswerId="25615" CreationDate="2022-09-12T18:53:39.163" Score="5" ViewCount="790" Body="&lt;p&gt;In some cultures, there is an expression of &amp;quot;spoon feeding&amp;quot; type of instruction, where the teacher shows how to do steps to a problem, and then students are assigned minor variants thereof. There is little to no critical thinking. (One could debate what exactly constitutes spoon feeding and that all instruction has varying levels of spoon feeding, but that is not my question). My question is, is the usage of the term inappropriate, offensive? Infantilizing? Insensitive to people with disabilities (I am actually recently disabled myself so am increasingly aware of the issue)? What could be some replacement language?&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Spoon feeding only teaches you the shape of the spoon&amp;quot; is a quote my 80 year old father told me he was taught in school, but times have changed (for the better I will say) of what constitutes acceptable discourse. Based on an online search it seems the quote is due to E.M. Forster: “Spoon feeding in the long run teaches us nothing but the shape of the spoon.”&lt;/p&gt;&#xA;&lt;p&gt;For context, this is about helping students make the jump from calculus and below-type books to upper level courses (not just proofs, but that is typical setting) where the book does not necessarily show you how to do problems on the homework, but students have to think, understand definitions. A search of the term on this site reveals several questions using the terminology.&lt;/p&gt;&#xA;&lt;p&gt;Not sure if math educators is an appropriate site for this question, or I should try another stack exchange site (academia? suggestion welcome)&lt;/p&gt;&#xA;" OwnerUserId="10289" LastEditorUserId="10289" LastEditDate="2022-09-13T07:24:33.110" LastActivityDate="2022-09-16T04:51:19.860" Title="what could be some replacement language for the term &quot;spoon feeding&quot;" Tags="|teaching|terminology|struggling-students|critical-thinking|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25611" PostTypeId="2" ParentId="25603" CreationDate="2022-09-12T19:33:20.643" Score="9" Body="&lt;p&gt;I do think this has (some) pedagogical value, specifically for word problems.&lt;/p&gt;&#xA;&lt;p&gt;Word problems are designed to give students practice applying math to real-world situations. Doing this effectively means you want the students to understand the situation being described, and think about how math can apply to it. (This also requires that your word problems be realistic; if the real-world answer is &amp;quot;that's nonsense&amp;quot;, it's a bad word problem for a math class.)&lt;/p&gt;&#xA;&lt;p&gt;What you don't want is for the student to &lt;a href=&quot;https://www.youtube.com/watch?v=6egeUxIEQnM&quot; rel=&quot;noreferrer&quot;&gt;scan the problem looking for two numbers, and a mathematical operation to apply to them&lt;/a&gt;. Requiring the student to restate the situation at hand, and how the number they're giving fits into that situation, is one way to tell if they've read the problem thoroughly, which is a component to understanding it.&lt;/p&gt;&#xA;" OwnerUserId="4736" LastActivityDate="2022-09-12T19:33:20.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25612" PostTypeId="2" ParentId="25610" CreationDate="2022-09-12T20:19:23.013" Score="3" Body="&lt;p&gt;A term for the same or similar that has positive connotations is 'scaffolding'.&lt;/p&gt;&#xA;&lt;p&gt;I want my students to understand the connection between a function's graph and the graph of its derivative, so I scaffold it, helping them see each step in my thought process. (Scaffolding usually refers to asking easier questions first to help students climb up to the harder stuff.)&lt;/p&gt;&#xA;&lt;p&gt;To help students make that shift from lower-level to upper-level textbooks, I ask them in the higher-level courses I teach (Linear Algebra and Discrete Math) to take notes on the textbook. I also give them &lt;a href=&quot;https://web.stonehill.edu/compsci/History_Math/math-read.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;an article by Shai Simonson and Fernando Gouvea on how to read a math book&lt;/a&gt;. I also talk repeatedly about the roles of definitions and theorems.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-09-12T20:19:23.013" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25613" PostTypeId="2" ParentId="25603" CreationDate="2022-09-12T21:03:38.210" Score="22" Body="&lt;p&gt;An example of a problem where phrasing the answer as a sentence might prevent mistakes and encourage understanding is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Alice goes to the store with \$2.00. A gumball costs \$0.80. How many gumballs can Alice buy?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A careless student might write&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\$2.00\  /\  \$0.80 = 2.5$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;but writing out&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Alice can buy 2.5 gumballs&amp;quot; might make them think again. Relating the numbers to an assertion about reality could plausibly help students check that the answers they obtain are reasonable.&lt;/p&gt;&#xA;" OwnerUserId="20334" LastEditorUserId="15941" LastEditDate="2022-09-13T11:11:52.833" LastActivityDate="2022-09-13T11:11:52.833" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25614" PostTypeId="2" ParentId="25603" CreationDate="2022-09-13T01:01:22.910" Score="7" Body="&lt;p&gt;The answer to &lt;em&gt;What is the speed of Anne's car?&lt;/em&gt; is simply &lt;em&gt;45 mph&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, math problems such as this follow a very simple pattern of &lt;em&gt;pick up every number and do something with them&lt;/em&gt;, so with a simple scan of the text, the answer must be one of&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;230&lt;/li&gt;&#xA;&lt;li&gt;220&lt;/li&gt;&#xA;&lt;li&gt;1125&lt;/li&gt;&#xA;&lt;li&gt;45&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(-220 or 0,022… would be 'unlikely' answers)&lt;/p&gt;&#xA;&lt;p&gt;which is exactly what some students will do: picking one of them seemingly at random, without paying any attention to the wording.&lt;/p&gt;&#xA;&lt;p&gt;This can lead to interactions with &lt;em&gt;certain&lt;/em&gt; students such as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Teacher: What's the speed of Anne's car?&lt;/li&gt;&#xA;&lt;li&gt;Student: 230!&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Teacher gives look of &amp;quot;this is not the right answer&amp;quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Student: I meant 220!&lt;/li&gt;&#xA;&lt;li&gt;Teacher: Are you sure?&lt;/li&gt;&#xA;&lt;li&gt;Student: 1125!, 1125!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;A full answer of&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The speed of Anne's car is 45 mph.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;requires from the student not only to read the numbers and guess the right operation (perhaps all the tasks today are divisions?), but also to parse the subject of the sentence (Anne's car), the property being sought (its speed) and build a full sentence out of it.&lt;/p&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;https://matheducators.stackexchange.com/users/20331/omegastick&quot;&gt;Omegastick&lt;/a&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/a/25609/5910&quot;&gt;points out&lt;/a&gt; that an implausible value associated to a sentence might prompt them to realize that they've made an error. But I wouldn't hold my breath. I think their parents are more likely to notice from the context that the car is probably &lt;em&gt;not&lt;/em&gt; going at 0.022 mph. If the students suspect it may be wrong, that probably stems from the answer not being an integral number.)&lt;/p&gt;&#xA;&lt;p&gt;As a nitpick, I should note that the above answer is still not complete. If we get fancy, we should say that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The speed of Anne's car is 45 mph &lt;em&gt;on average&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5910" LastActivityDate="2022-09-13T01:01:22.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25615" PostTypeId="2" ParentId="25610" CreationDate="2022-09-13T11:13:17.310" Score="9" Body="&lt;p&gt;No one can make any universal guarantees as to whether or not any particular student may find your usage of spoon-feeding or any other term offensive or insensitive. However, spoon-feeding as a metaphorical descriptor &lt;em&gt;is&lt;/em&gt; meant to infantilize (the idiom is definitely associated with feeding an incapable baby with a spoon at minimum), criticize certain teaching methods, and even possibly challenge student proficiency (which could motivate some students and demoralize others). This is &lt;strong&gt;not&lt;/strong&gt; to say that those who use the term spoon-feeding should stop but only to describe the rhetorical effect your words might have on the audience.&lt;/p&gt;&#xA;&lt;p&gt;So... what's the goal here? If you are trying to find a one-time method to contrast the high standards of your course (and others in the future) with other courses as non-controversially as possible, then one approach would be to criticize the other courses objectively and focus on either objective aspects of your course or paint your goals in a more aspirational light. An example might be: &amp;quot;In prior courses, you may have been presented with theory and walked through a significant number of problems which closely mirrored future problems on assignments and assessments, but in this course, you will be held to a higher standard: to independently apply the theory we develop to  problems and proofs of your own.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If the goal is to find a pithy replacement that you intend to use repeatedly which still conveys the critical connotation, you could say something like &amp;quot;I'm not going to baby you&amp;quot;, which at least has the advantage of &lt;em&gt;very directly&lt;/em&gt; indicating that you are comparing the alternative approach to how one would treat a baby.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2022-09-13T11:13:17.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25616" PostTypeId="2" ParentId="25610" CreationDate="2022-09-13T12:06:38.017" Score="1" Body="&lt;p&gt;Education tends to use loaded terms rather than thoughtful arguments.  &amp;quot;Drill and kill&amp;quot; must be bad, right?  &amp;quot;Mastery&amp;quot; must be good, right?&lt;/p&gt;&#xA;&lt;p&gt;For spoon feeding, in particular, I've seen the term used a lot for graduate courses, in the &amp;quot;we're not spoon feeding&amp;quot;, but it really struck me as a justification for the low level of effort by the college (profs more interested in research), rather than a reasoned decision to challenge the students.&lt;/p&gt;&#xA;&lt;p&gt;I'm not so concerned with the impact on the students of the term.  I just don't like the ed research fad-driven use of loaded emotional terms.&lt;/p&gt;&#xA;&lt;p&gt;As far as your specific issue, I kind of doubt that &amp;quot;lack of Spivak&amp;quot; is preventing kids from making the move to proof based clasess in LA, RA, and AA.  The kind of kid who can handle a very tough calc book like that as his first exposure is the kind of kid who can make the jump later, also, without the prior Spivak exposure, and probably doesn't even need a &amp;quot;proof class&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="20340" LastEditorUserId="20340" LastEditDate="2022-09-13T20:23:49.250" LastActivityDate="2022-09-13T20:23:49.250" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25617" PostTypeId="1" CreationDate="2022-09-13T20:27:54.500" Score="0" ViewCount="190" Body="&lt;p&gt;I am looking for manipulative materials  to teach functions (the concepts including domain, image, etc.) and kind of function (affine, quadratic, exponential logarithmic, polynomial, trigonometric) thanks!&lt;/p&gt;&#xA;&lt;p&gt;iam looking for concrete manipulaives. there are plenty of digital ones: –&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="8828" LastEditDate="2022-09-16T23:27:39.677" LastActivityDate="2022-09-16T23:27:39.677" Title="Manipulative materials to teach functions" Tags="|algebra|functions|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25618" PostTypeId="1" CreationDate="2022-09-14T13:59:57.993" Score="15" ViewCount="4329" Body="&lt;p&gt;The research level of mathematics (what is done by professors and upper-level graduate students) tends to be heavily portrayed as focused on writing proofs to the exclusion of most anything else math-related. This was certainly the case for me as an undergraduate, and I pictured professors spending all of their time outside of teaching trying to write out formal proofs that X over blah is bounded by the gradient of Y in P-space, blah blah blah, rather than doing things I thought might have been more fun but still math-related. In fact, one of my textbooks started out with a &amp;quot;helpful&amp;quot; reminder that my childhood was over and that I was entering a world in which proofs were the most important thing. In other words, &amp;quot;knowing&amp;quot; math meant pretty much diddily-squat unless I could formally and rigorously write out proofs for everything I thought I knew.&lt;/p&gt;&#xA;&lt;p&gt;I never really enjoyed proofs, and so was discouraged from following math any further than what was required for my non-math undergraduate degree. In fact, I've always been an intuitive, heuristic learner, combining past knowledge with intelligence and intuition to determine what a reasonable answer is even though I cannot &amp;quot;prove&amp;quot; it to be 100% correct.&lt;/p&gt;&#xA;&lt;p&gt;To that end, are there research-level subfields of mathematics that can be used as examples of &amp;quot;There's more to math than proofs!&amp;quot;? In other words, these would be sub-fields that could be used to inspire students interested in research-level mathematics but who are weak in terms of proofs or do not find proofs interesting enough.&lt;/p&gt;&#xA;" OwnerUserId="7481" LastEditorUserId="7481" LastEditDate="2022-09-14T15:31:10.483" LastActivityDate="2022-09-27T16:41:32.847" Title="What are some research-level opportunities in mathematics that do not focus on proofs?" Tags="|undergraduate-education|student-motivation|graduate-education|" AnswerCount="11" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="25619" PostTypeId="2" ParentId="25618" CreationDate="2022-09-14T15:08:27.350" Score="14" Body="&lt;p&gt;I have taught &lt;em&gt;Discrete &amp;amp; Computational Geometry&lt;/em&gt; to US undergraduates project-based,&#xA;as opposed to assignment- and test-based.&#xA;Some of the projects do involve proofs, but others are&#xA;more experimental explorations, either computationally&#xA;or physically.&#xA;Here is one on &amp;quot;D-forms&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gluing together two identical copies of a smooth convex shape, one rotated with respect to the other, produces an elegant 3D convex shape with one seam where the two perimeters are glued together. It is an unsolved problem to determine the 3D shape from the 2D shapes and how one is twisted with respect to the other. Explore some subclass of shapes and twists and formulate conjectures. Even studying doubly-covered rectangles is not entirely understood.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A student submitted an entire box of paper D-forms in support&#xA;of her conjecture. :-)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ZgiX0b.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZgiX0b.jpg&quot; alt=&quot;surface formed by joining two ellipses&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Image from &lt;a href=&quot;https://www.cutoutfoldup.com/1602-d-forms.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;CutOutFoldUp&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="15884" LastEditDate="2022-09-15T15:56:50.237" LastActivityDate="2022-09-15T15:56:50.237" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25620" PostTypeId="2" ParentId="25618" CreationDate="2022-09-14T15:10:27.003" Score="14" Body="&lt;p&gt;With the technological advances of the past couple decades, computational mathematics is now somewhat accessible to undergraduates. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Computational_mathematics&quot; rel=&quot;noreferrer&quot;&gt;wikipedia entry for computational mathematics&lt;/a&gt; lists out the various subfields. Basically, actually being able to compute something is, in and of itself, a worthwhile endeavor.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2022-09-14T15:10:27.003" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25621" PostTypeId="2" ParentId="25618" CreationDate="2022-09-14T17:14:35.787" Score="7" Body="&lt;p&gt;No.  By definition, mathematics is the study of what can be formally proved, so if someone is not at all concerned with proofs, they are not doing mathematics (but possibly some math-related subject).&lt;/p&gt;&#xA;&lt;p&gt;(I'm not sure I 100% believe this answer, but it's worth putting out there.)&lt;/p&gt;&#xA;" OwnerUserId="4991" LastActivityDate="2022-09-14T17:14:35.787" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="25622" PostTypeId="2" ParentId="25618" CreationDate="2022-09-14T17:45:34.483" Score="29" Body="&lt;p&gt;I think a more correct view is that proof is the &lt;strong&gt;LAST&lt;/strong&gt; of several stages involved in researching something in math. What follows is a quickly sketched out scenario of what is often the case.&lt;/p&gt;&#xA;&lt;p&gt;Before proof, there is &amp;quot;playing around with some topic&amp;quot; enough to decide whether your background and ability &lt;strong&gt;might&lt;/strong&gt; suffice to uncover something sufficiently useful to actually begin formulating conjectures. Or alternatively, reading over past attempts/results in the literature about a known problem enough to obtain sufficient background information to decide whether it &lt;strong&gt;might&lt;/strong&gt; be worth while for you to pursue the problem.&lt;/p&gt;&#xA;&lt;p&gt;Then comes some more focused investigations, maybe trying out the conjecture with some &amp;quot;toy examples&amp;quot; or seeing if you can prove weaker versions of the conjecture. For example, maybe you want to prove that every positive integer has a certain property. To vastly oversimply what usually is the case for the sake of illustrating what I mean, can you prove this for a specific integer? Two specific integers? Infinitely many integers? Can you prove it for even integers, or for odd integers, or for square free integers, or for all sufficiently large integers, etc.?&lt;/p&gt;&#xA;&lt;p&gt;After this (or at the same time) comes a lot of attempts at heuristic arguments for why the result is true, with the hope that one such argument can be made sufficiently sound to ultimately result in a rigorous proof. Usually by the time one tries to write a rigorous proof you know whether the approach is likely to work (rigorously), so for a mathematician writing a proof (as opposed to a student still learning how to write proofs), it is somewhat like a judge writing a formal legal judgement after hearing a case and thinking over all the relevant facts and appropriate legal issues involved and coming to a decision about the case.&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure how much this will help, but &lt;a href=&quot;https://mathoverflow.net/q/430389/15780&quot;&gt;this recent mathoverflow question&lt;/a&gt; gives an example of someone at the &amp;quot;playing around with a topic&amp;quot; stage, and in the comments I mention something that might be related and which is something I've never even gotten to the &amp;quot;playing around with a topic&amp;quot; stage despite having gathered some relevant literature on it over the years.&lt;/p&gt;&#xA;" OwnerUserId="745" LastEditorUserId="745" LastEditDate="2022-09-14T17:50:55.977" LastActivityDate="2022-09-14T17:50:55.977" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25623" PostTypeId="2" ParentId="25618" CreationDate="2022-09-14T18:38:46.297" Score="10" Body="&lt;p&gt;When I describe undergraduate research to students majoring in mathematics, I ask them to browse the abstracts of the most recent MAA undergraduate poster session. Here is a link: &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/students/undergrad/PosterSess22%20%282%29.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abstracts for the&#xA;MAA Undergraduate Student Poster Session&#xA;MAA MathFest 2022&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From this, one can easily become convinced that the proof-centric mode of research that you describe need not be the norm.&lt;/p&gt;&#xA;&lt;p&gt;Another direction is to learn about what is called &amp;quot;experimental mathematics.&amp;quot; There is a &lt;a href=&quot;https://en.wikipedia.org/wiki/Experimental_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;good Wikipedia page&lt;/a&gt;, and a highly respected publication title &lt;a href=&quot;https://www.tandfonline.com/journals/uexm20&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Experimental Mathematics&lt;/em&gt;&lt;/a&gt;. Their &amp;quot;aims and scopes&amp;quot; statement states that the journal publishes &amp;quot;papers on formal results inspired by experimentation, conjectures suggested by experiments and data supporting significant hypotheses in mathematics.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2022-09-15T00:54:21.013" LastActivityDate="2022-09-15T00:54:21.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25624" PostTypeId="2" ParentId="25618" CreationDate="2022-09-14T19:06:55.067" Score="8" Body="&lt;p&gt;Not quite an answer to your question, but if you have students who are interested in mathematics but not interested in (generating their own) proofs, encourage them to go into mathematics communication! Arguably, finding ways of effectively communicating mathematics to wide audiences is at least as important as discovering that mathematics in the first place.&lt;/p&gt;&#xA;&lt;p&gt;Figuring out just the right way of communicating an idea requires (and creates!) a deep understanding of that idea, and communication skills will be valuable for your students no matter what they wind up doing.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-09-14T19:06:55.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25625" PostTypeId="2" ParentId="25618" CreationDate="2022-09-14T20:50:59.477" Score="5" Body="&lt;p&gt;IF you have kids that like math, but not proofs, I suggest modeling financial systems (e.g. actuarial issues) or the like (e.g. refinery operation) is a good activity for undergrad math majors.  Data analysis or ops analysis projects.  For sure, it will help them in the work world, just in case they don't end up being Andrew Wiles or a juco math teacher (which by the odds, most won't).  If they do, it won't hurt them, either, to have a little wider perspective.&lt;/p&gt;&#xA;" OwnerUserId="20354" LastActivityDate="2022-09-14T20:50:59.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25626" PostTypeId="2" ParentId="25618" CreationDate="2022-09-14T23:26:12.563" Score="6" Body="&lt;p&gt;Especially with the advent of relatively very fast numerical and symbolic manipulation software/computers, giving numerical evidence for various things is an eminently feasible, and interesting, heuristic ... apart from proving that the heuristic is correct. Indeed, in some cases, extensive numerical computation is useful, and perhaps &lt;em&gt;necessary&lt;/em&gt;, in order to form conjectures about what should be &lt;em&gt;proven&lt;/em&gt; (if possible!).&lt;/p&gt;&#xA;&lt;p&gt;Yes, some sources take up a quite pompous attitude about &amp;quot;mathematics&amp;quot; being &amp;quot;only&amp;quot; things that have complete, rigorous proofs... and that everything else is irrelevant. Well, of course we'd &lt;em&gt;like&lt;/em&gt; GOOD (=explanatory) proofs of things, if possible, rather than just numerical evidence, but sometimes we just don't get that. Just get numerical evidence... or other heuristics that are hard to justify.&lt;/p&gt;&#xA;&lt;p&gt;As far as specifics: numerical simulation of n-body problems in celestial mechanics... numerical search for solutions to various diophantine equations ... are easy examples that come to mind. But/and I primarily wanted to make the point that math has many parts other than &amp;quot;formal proof&amp;quot;... so don't believe anyone who tries to tell you that &amp;quot;it's not math&amp;quot; unless it's formal proofs.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-09-14T23:26:12.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25627" PostTypeId="2" ParentId="4339" CreationDate="2022-09-15T03:44:00.873" Score="1" Body="&lt;p&gt;Technically the answer is &amp;quot;What an engineer (or any other person in a STEM-related profession) will need from an analysis course may be anything from the very basic understanding of the linear approximation and rudiments of the differential/integral calculus and rules for finding extrema to the rigorous grasp of the subject at a professional mathematician's level&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I would just make a clear outline of the course the way you want it, discuss it with the people from the engineering department, modify it according to their needs but preserving the internal logic and feasibility as &lt;em&gt;you&lt;/em&gt; see them and, hopefully, get their support, so it will be &lt;em&gt;they&lt;/em&gt;, who will argue with your department head, not you. Your only real argument can be that what you propose makes sense and hence is worth trying though, of course, your proposed course of action is &lt;em&gt;not the only one&lt;/em&gt; that makes sense (and the same is the most that can be said of the suggestions of all other people, including your department head). That is the best practical advice I can think of.&lt;/p&gt;&#xA;&lt;p&gt;As to the students, you owe them a clear explanation of what the course is going to be about, what the expectations and requirements are and other stuff like that but you are absolutely not obliged to take their wishes or shortcomings into account beyond the usual courtesy of spending some time in the beginning to refresh the required prerequisites in their minds. No matter what you do, you won't be able to hit just the right spot for &lt;em&gt;everybody&lt;/em&gt; who might be interested in or forced into taking analysis class this year, so if your choice of how to run it is not completely outlandish and is supported by at least some good people in the engineering department, be sure that it is &lt;em&gt;just as good as any other one with the same properties&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Ideally (in my humble opinion of a total ignoramus in the modern advanced pedagogical science), there should be several different analysis courses taught by different people with different attitudes toward both the subject and the teaching philosophy in general and the students should be allowed to freely select which one they are going to take themselves with no conclusions or, God forbid, penalties, drawn from the natural result that some classes will attract 100 people and some only 10 this way. We want to teach all 110 and we never really know whether that group of 100 or that group of 10 would be worse to lose because any real craft (and engineering is such) is moved forward by &amp;quot;mass effort&amp;quot; just as much as by &amp;quot;insights of the few&amp;quot;, both being indispensable. However, if there is only one section with a single instructor offered each year, the best approximation to that would be to give him/her free hand (within the limits stated above) and &lt;em&gt;rotate&lt;/em&gt; from year to year among the people who are interested in teaching the course (Yeah, I'm advocating for diversity here, though I'm afraid that my understanding of this word is somewhat different from its current mainstream meaning).&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-09-15T03:44:00.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25628" PostTypeId="1" CreationDate="2022-09-15T05:00:30.283" Score="-4" ViewCount="170" Body="&lt;p&gt;This might seem a bit misplaced, but, is very relevant to mathematics education.&lt;/p&gt;&#xA;&lt;p&gt;The question is, how can I love someone, and teach students to love, or attempt and complete actions of love, through mathematics.&lt;/p&gt;&#xA;&lt;p&gt;If mathematics was taught with love in mind, it might be more useful.&lt;/p&gt;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="12214" LastActivityDate="2022-09-15T20:01:34.317" Title="Mathematics and love" Tags="|mathematical-pedagogy|" AnswerCount="2" CommentCount="2" ClosedDate="2022-09-16T00:34:42.333" ContentLicense="CC BY-SA 4.0" />
  <row Id="25629" PostTypeId="2" ParentId="25628" CreationDate="2022-09-15T11:19:02.683" Score="3" Body="&lt;p&gt;There is research on dialogues that happen in mathematics teaching context. You might be interested in an article of Annica Andersson and David Wagner: &lt;a href=&quot;https://www.diva-portal.org/smash/record.jsf?pid=diva2%3A1408655&amp;amp;dswid=4798&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.diva-portal.org/smash/record.jsf?pid=diva2%3A1408655&amp;amp;dswid=4798&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;ANDERSSON, Annica; WAGNER, David. Love and bullying in mathematical conversations. In: the 9th International Mathematics Education and Society Conference-MES9, Thessaly, Greece (april 2017). University of Thessaly Press, 2017. p. 382-392.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is also research on emotions that appear when one does mathematics. This book I have not read, but might be worth looking at for further information on this: &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-319-06808-4_2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/chapter/10.1007/978-3-319-06808-4_2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;RADFORD, Luis. Of love, frustration, and mathematics: A cultural-historical approach to emotions in mathematics teaching and learning. In: From beliefs to dynamic affect systems in mathematics education. Springer, Cham, 2015. p. 25-49.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The first article analyses some examples of how love and bullying appear in a classroom dialogue. It might help in answering your question.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-09-15T11:19:02.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25630" PostTypeId="2" ParentId="25618" CreationDate="2022-09-15T14:05:11.607" Score="2" Body="&lt;p&gt;Digital signal processing would seem like a good avenue.&lt;/p&gt;&#xA;&lt;p&gt;The colors on this monitor, the audio and video, the compression, the encoding, the timing, everything digital is passing in and out of digital processing equations.&lt;/p&gt;&#xA;&lt;p&gt;It can be very heavy for mathematics, and it's rather a factor of efficiency rather than proof, aiming for the most efficient way of processing digital signals and patterns.&lt;/p&gt;&#xA;&lt;p&gt;Creatively there is a lot of research going into 3D equation based media that can be efficiently understood by AI... Actually, one of the biggest difficulties in AI is organizing data into compressed low data formats that can be processed by machine learning. It's good to be an expert in a particular field of media because the computer scientists often lack the mathematical knowledge of the data they process to inform the computers how to process efficiently. In 3D for example, it's all about quaternions, matrices, vectors, voxels, SDF's. In audio, it's all about waves, which have an intrinsic uncertainty in the time-frequency domain which resorts to quantum maths equations like Wigner distribution transforms.&lt;/p&gt;&#xA;&lt;p&gt;The field of DSP and machine encoding-decoding of data is very mathematically heavy and reqires solutions demonstrating efficiency because the proof is already known.&lt;/p&gt;&#xA;" OwnerUserId="20371" LastEditorUserId="20371" LastEditDate="2022-09-15T14:10:15.413" LastActivityDate="2022-09-15T14:10:15.413" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25631" PostTypeId="2" ParentId="25628" CreationDate="2022-09-15T20:01:34.317" Score="0" Body="&lt;p&gt;[Not a troll]&lt;/p&gt;&#xA;&lt;p&gt;I suggest watching the movie Conrack.  It will give you a perspective on how empathy and engagement can help with teaching (or even public speaking).  And this is coming from a harsh ex-military guy.  It's not about being soft, per se.  But some level of human connection, helps with coaching, training, teaching.&lt;/p&gt;&#xA;&lt;p&gt;Go pay the $3.99 or whatever to stream it on Amazon (like renting a video).&lt;/p&gt;&#xA;&lt;p&gt;Trailer:  &lt;a href=&quot;https://www.youtube.com/watch?v=WMejIRLtTSM&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=WMejIRLtTSM&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;P.s. Of course it is just a movie, a work of art.  So don't take it as purely a pedagogy argument.  Similarly 12 O'clock High is a movie, not a leadership manual (but it's an interesting case study, still, for leadership).&lt;/p&gt;&#xA;" OwnerUserId="20354" LastActivityDate="2022-09-15T20:01:34.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25632" PostTypeId="2" ParentId="25610" CreationDate="2022-09-16T04:51:19.860" Score="2" Body="&lt;p&gt;You might consider the phrase &amp;quot;algorithmic learning&amp;quot; vs. &amp;quot;deep understanding&amp;quot;.  Those who learn algorithmically, often memorize steps with no understanding of them and then can't apply them  in exceptional cases. Other students understand why they are doing what they are doing and can adjust it according to the needs of any problem.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2022-09-16T04:51:19.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25633" PostTypeId="2" ParentId="25618" CreationDate="2022-09-16T15:27:56.290" Score="11" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;In other words, &amp;quot;knowing&amp;quot; math meant pretty much diddily-squat unless I could formally and rigorously write out proofs for everything I thought I knew.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You appear to believe that somebody can &amp;quot;know&amp;quot; mathematics without being able to prove things, but that their mathematics knowledge is devalued by stuffy gatekeepers who insist that all mathematicians should &lt;em&gt;also&lt;/em&gt; be able to prove things. This is nonsense, akin to the idea that a skilled fiction writer should &lt;em&gt;also&lt;/em&gt; be able to tell stories, or a skilled athlete should &lt;em&gt;also&lt;/em&gt; be able to compete in sport.&lt;/p&gt;&#xA;&lt;p&gt;When a textbook says &amp;quot;your childhood is over&amp;quot; and now proofs are important in mathematics, that is like saying to a writer that it is no longer sufficient to be able to spell words and construct grammatical sentences, because while those abilities are fundamental to writing, they don't &lt;em&gt;make&lt;/em&gt; a writer. The next foundational skill the writer needs to learn is to tell stories, and the next foundational skill a mathematician needs to learn is to prove things.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I've always been an intuitive, heuristic learner, combining past knowledge with intelligence and intuition to determine what a reasonable answer is even though I cannot &amp;quot;prove&amp;quot; it to be 100% correct.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To me, what that means is that you frequently made guesses and you weren't generally able to justify why those guesses were correct, perhaps you weren't even always aware when one of your assumptions was a guess rather than something you really &amp;quot;knew&amp;quot; to be true. This approach may well have found you great success, but only because the mathematics you were doing was &amp;quot;too easy&amp;quot;: easy to get intuition for and easy to make correct guesses about. But once &amp;quot;your childhood is over&amp;quot;, being a good mathematician requires more than the ability to make guesses that are usually &amp;quot;reasonable&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I'm sorry if that sounds harsh, but I think it is a realistic assessment of this kind of attitude towards mathematical rigour. A student with this attitude is in what Terence Tao calls the &amp;quot;&lt;a href=&quot;https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;noreferrer&quot;&gt;pre-rigorous stage&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To that end, are there [...] sub-fields that could be used to inspire students interested in research-level mathematics but who are weak in terms of proofs or do not find proofs interesting enough[?]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes, there are surely plenty of research areas in mathematics which don't &amp;quot;focus&amp;quot; on proofs; see the other answers for examples. But &amp;quot;not focusing on proofs&amp;quot; does not mean that one can become a good researcher in those subfields without learning how to prove things; rather it means that proofs are merely not the end product of that research. Good researchers in these areas (or in all areas, I should think) are at the &amp;quot;post-rigorous&amp;quot; stage, not the &amp;quot;pre-rigorous&amp;quot; stage.&lt;/p&gt;&#xA;&lt;p&gt;To inspire students who struggle with proofs, I think &amp;quot;don't worry, you can still do mathematics if you're bad at proofs&amp;quot; is the wrong message. I would rather say: &lt;em&gt;You can learn how to write proofs, and I will help you&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Likewise, if students find proofs uninteresting before undergraduate level, I think &amp;quot;don't worry, you can do all the other things and not focus on proofs&amp;quot; is the wrong message. Rather, I would say: &lt;em&gt;The things you're proving now are not that interesting, sure, but that's because these are basic exercises to help you learn the basic proof techniques. But there's lots of interesting mathematics ahead of you, which you'll need proofs for, and it's much more fun once you're proving things that you do care about.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="19245" LastActivityDate="2022-09-16T15:27:56.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25634" PostTypeId="1" CreationDate="2022-09-17T16:16:32.800" Score="3" ViewCount="737" Body="&lt;p&gt;I am new to teaching math, especially middle school math. My own math education was limited to high school and college calculus. These were painful experiences. How to do I go about learning to teach middle school math?&lt;/p&gt;&#xA;&lt;p&gt;More importantly, I want to tell my student WHY/WHAT FIELDS/CAREERS require the maths I will be teaching.&lt;/p&gt;&#xA;" OwnerUserId="20381" LastEditorUserId="2083" LastEditDate="2022-09-17T19:46:53.167" LastActivityDate="2022-09-19T16:40:15.727" Title="New Math Teacher Seeking Advice" Tags="|teaching|" AnswerCount="3" CommentCount="6" ClosedDate="2022-09-21T14:43:31.637" ContentLicense="CC BY-SA 4.0" />
  <row Id="25635" PostTypeId="2" ParentId="25634" CreationDate="2022-09-18T03:09:18.900" Score="13" Body="&lt;p&gt;The fields &amp;amp; careers it is good for is the least important part, in my opinion.&lt;/p&gt;&#xA;&lt;p&gt;You don't say what math topics it is. That might help us answer more effectively.&lt;/p&gt;&#xA;&lt;p&gt;I am guessing that you are uncomfortable with math at any level. If you accepted a position teaching it, you owe it to your students to get past your discomfort. You will want to re-learn what you're going to be teaching, and find the joy in it. (Most of us who answer here really love math. You can too.)&lt;/p&gt;&#xA;&lt;p&gt;I'd suggest you start by getting an online subscription to &lt;a href=&quot;https://beastacademy.com/&quot; rel=&quot;noreferrer&quot;&gt;Beast Academy&lt;/a&gt;, a curriculum originally written for gifted kids, but usable by most students, for arithmetic up through much of what's called pre-algebra. Start at their Level 1. It will be too easy for you. While you're working through it, think about how you'd teach that stuff to 1st or 2nd graders. There may be some challenge problems that challenge you. Great. Have fun playing with them. Savor the experiences you have.&lt;/p&gt;&#xA;&lt;p&gt;When you get to Level 2, you'll find a few more challenges. That's ok. Play around. Slow down.&lt;/p&gt;&#xA;&lt;p&gt;But if you're already teaching, you might want to skip around. See which Beast topics match what you'll be teaching in the next few weeks. (You might find them in any of the levels, from 3 to 5, possibly lower.) Print out a page of puzzles. If you can convince your students these puzzles are especially fun (they are!), then you are a good middle school teacher.&lt;/p&gt;&#xA;&lt;p&gt;Of course there are ed school programs in how to teach math, and if you haven't done any of that, maybe you could pick up a textbook and see if it has any ideas you like.&lt;/p&gt;&#xA;&lt;p&gt;Tell us more details about your situation, and we can give you more targeted ideas.&lt;/p&gt;&#xA;&lt;p&gt;Good luck.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-09-18T03:09:18.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25636" PostTypeId="2" ParentId="25634" CreationDate="2022-09-18T13:47:07.773" Score="9" Body="&lt;p&gt;I highly recommend that you practice. Practice doing the types of problems you are teaching. There are so many sources for you in this day and age. Khan Academy is a great start. There are also Youtube videos by an endless number of math people. Students can tell when a teacher is confident and are often unkind to a nervous one. The extra prep you do before each class will be well worth it.&#xA;You got through calculus while in college. Middle school topics should feel far less intimidating to you.&lt;/p&gt;&#xA;&lt;p&gt;(I agree with Sue, the &amp;quot;When will I use this&amp;quot; is far less important, although at some point we all get this. Avoid the answer &amp;quot;You won't, but the smart kids might&amp;quot;. Especially with lower grades, I stay away from jobs and talk about the math in my life. Not getting ripped off at a store for one. Managing your money. In the kitchen, using recipes, and especially adjusting them up or down. This is a great way to talk about why we need to be comfortable with fractions.)&lt;/p&gt;&#xA;&lt;p&gt;You still might edit the question to better focus on what you need.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-09-18T19:12:43.720" LastActivityDate="2022-09-18T19:12:43.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25637" PostTypeId="2" ParentId="25634" CreationDate="2022-09-19T16:38:29.557" Score="2" Body="&lt;p&gt;For the applicability of the math, you should just google the topics online and collect the info.&lt;/p&gt;&#xA;&lt;p&gt;More importantly, it helps to believe that there is math intuition in every human mind. The student doesn't get something because they haven't yet come to tap into that particular intuition. That way, you are never likely to fully give up on a student. You feel like the student's intuition is always around the corner. And that's a perennially motivating space to be in.&lt;/p&gt;&#xA;&lt;p&gt;Of course, in your experience of teaching you should work towards tapping into your own intuition for the maths you teach. As you feel more convinced, your students benefit more.&lt;/p&gt;&#xA;" OwnerUserId="20386" LastEditorUserId="20386" LastEditDate="2022-09-19T16:40:15.727" LastActivityDate="2022-09-19T16:40:15.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25638" PostTypeId="1" CreationDate="2022-09-20T08:24:47.977" Score="5" ViewCount="286" Body="&lt;p&gt;I'm teaching a graduate (Master's) introduction to geometry and topology (e.g. some basics on manifolds, vector bundles, algebraic topology). What textbooks have you found are best for teaching a course like that?&lt;/p&gt;&#xA;&lt;p&gt;I'm less worried about the specific list of topics but just wanted to know if people found a particular textbook to be particular good at explaining these concepts for the first time to beginning graduate students - and/or particularly good for structuring a course from.&lt;/p&gt;&#xA;&lt;p&gt;(But if you wish, some possible topics are: Topological manifolds. The fundamental group and covering spaces. Applications.&#xA;Singular homology and applications.&#xA;Smooth manifolds. Differential forms and Stokes’ theorem, definition of de-Rham cohomology.)&lt;/p&gt;&#xA;" OwnerUserId="20387" LastEditorUserId="376" LastEditDate="2022-09-20T15:27:17.950" LastActivityDate="2022-09-20T15:27:17.950" Title="Which books on geometry and topology are best for teaching an intro graduate course?" Tags="|textbooks|course-design|graduate-education|topology|differential-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25639" PostTypeId="2" ParentId="25638" CreationDate="2022-09-20T12:59:25.793" Score="5" Body="&lt;p&gt;Disclaimer: I haven't taught the kind of course you describe, so please take my recommendations below with a grain of salt. Nevertheless, I hope they're helpful.&lt;/p&gt;&#xA;&lt;p&gt;John M. Lee's &lt;em&gt;Introduction to Topological Manifolds&lt;/em&gt; is a great place to start for the basics of topology, topological manifolds, the fundamental group, covering spaces and just a touch of homology. It's written at an introductory graduate level and the author has taken pains to write clearly and in detail. For smooth manifolds, you can follow up with his &lt;em&gt;Introduction to Smooth Manifolds&lt;/em&gt;. However, if the latter book is too long and detailed for your course, you could switch to Loring W. Tu's &lt;em&gt;An Introduction to Manifolds&lt;/em&gt;, which is also well written.&lt;/p&gt;&#xA;&lt;p&gt;If your students have already had some general topology, you could either skip or go lightly through the early chapters of &lt;em&gt;Introduction to Topological Manifolds&lt;/em&gt;, giving you more time for smooth manifolds later.&lt;/p&gt;&#xA;&lt;p&gt;A book that I'm less familiar with, but looks promising, is the recent &lt;em&gt;A Short Course in Differential Topology&lt;/em&gt; by Bjørn I. Dundas. It introduces smooth manifolds, the tangent space, regular values, vector bundles, flows and a few other topics, with the assumed background on general topology given in an appendix. Admittedly, it doesn't treat several of the topics in your list, but you said you're less worried about the specific topics. The author has a PDF version available at &lt;a href=&quot;https://folk.uib.no/nmabd/dt/dtcurrent.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://folk.uib.no/nmabd/dt/dtcurrent.pdf&lt;/a&gt;. The PDF isn't identical to the published version but it's close enough to give you a good impression. Note that despite the term &amp;quot;differential topology&amp;quot; in the title, it covers much of the smooth manifolds basics you'd find in the early chapters of many books of manifolds, be they heading toward analysis on manifolds, differential geometry or differential topology.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2022-09-20T12:59:25.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25640" PostTypeId="2" ParentId="134" CreationDate="2022-09-21T22:32:46.133" Score="1" Body="&lt;p&gt;Dave:&lt;/p&gt;&#xA;&lt;p&gt;You might consider programmed instruction. (To answer your question, but maybe not helpful for making money as a teacher!)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Programmed_learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Programmed_learning&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's sort of a step-by-step way of learning, self studying.  I have found such texts to be extremely efficient as a self studier, much better structure than just reading lessons and working problems in a normal textbook.  Of course, preparing such materials is a significant effort.  And it has less market in academia, since it is designed for use without a teacher.&lt;/p&gt;&#xA;&lt;p&gt;Ken Stroud was a famous programmed learning math textbook writer, can check out some of his texts.  Just look at them overall as well as the prefaces, where he talks about the methodology.  [HE covers a fair amount of engineering math and below, but not proofs, I don't think.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ken_Stroud&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ken_Stroud&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note, the system is not restricted to math.  I have used programmed texts in basic accounting and in nautical rules of the road.  In both cases, they were extremely time efficient, easy to follow and not give up or get bored, etc.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  Apologies for other answer.  I probably had two tabs open, you know how that is.  Cookie mouse request for a moderator to cut the 29APR21 answer...&lt;/p&gt;&#xA;" OwnerUserId="20392" LastEditorUserId="20392" LastEditDate="2022-09-21T22:34:19.747" LastActivityDate="2022-09-21T22:34:19.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25641" PostTypeId="1" CreationDate="2022-09-22T12:10:48.020" Score="7" ViewCount="1451" Body="&lt;p&gt;For the question &amp;quot;Write &lt;span class=&quot;math-container&quot;&gt;$y=\sqrt{3+x}$&lt;/span&gt; as the composite of two functions&amp;quot;, what if a student gives the answer &lt;span class=&quot;math-container&quot;&gt;$f(x)=\sqrt{3+x}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)=x$&lt;/span&gt;? This answer would be technically correct but it is not what the intention of the question is.&lt;/p&gt;&#xA;&lt;p&gt;How to restate the question in a rigorous way to exclude this answer?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="64" LastEditDate="2022-09-24T16:17:30.623" LastActivityDate="2022-10-28T09:58:05.743" Title="Write $y=\sqrt{3+x}$ as the composite of two functions" Tags="|algebra|precalculus|functions|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25642" PostTypeId="2" ParentId="25641" CreationDate="2022-09-22T12:16:47.227" Score="4" Body="&lt;p&gt;I suppose you could add that neither of the two functions is allowed to be the identity function. That said, students might be more confused by this restriction than helped by it.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2022-09-22T12:16:47.227" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25643" PostTypeId="2" ParentId="25641" CreationDate="2022-09-22T13:13:40.107" Score="10" Body="&lt;p&gt;There are many other compositions (e.g., translate one direction and then translate the other, scale by two factors offsetting each other) that may not reflect your intentions either. Simply put, there are too many degrees of freedom in this question if you are looking for a very specific answer. I would suggest “write this function as a composition of two functions, one of which is…” as an alternative, though this really would reduce the difficulty.&lt;/p&gt;&#xA;&lt;p&gt;You could try to come up with other criteria to exclude certain functions, but given the implied level of this question, that could really increase the difficulty.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2022-09-22T13:13:40.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25644" PostTypeId="2" ParentId="25641" CreationDate="2022-09-22T13:14:39.093" Score="7" Body="&lt;p&gt;It might be easier to just chat with the rare student who gives that answer. If they understand why their answer is technically correct, then it is very likely that the teaching goals have been accomplished.&lt;/p&gt;&#xA;&lt;p&gt;Otherwise, I'd say that you shouldn't just do an exercise in formulas. Make the factorization have meaning and the quantities (inputs and outputs to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;) have units. You can probably cook up something geometric, maybe with small alterations to the formula.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2022-09-22T13:14:39.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25645" PostTypeId="2" ParentId="25641" CreationDate="2022-09-22T15:20:53.613" Score="3" Body="&lt;p&gt;If this is an exercise from an introductory calculus course, one place students are going to need this skill is in applying the chain rule. The alternative composition you propose, as well as the alternative compositions in Steve's answer are not going to be very helpful in that context.&lt;/p&gt;&#xA;&lt;p&gt;For the purposes of the chain rule, what is needed is a decomposition into simple functions, each of which the student knows how to differentiate. Since students usually learn rules for derivatives of polynomial functions, power functions, the logarithmic and exponential functions, and trigonometric functions, you could ask for a decomposition into only functions from that list. You need to make sure that students will recognize the square-root function as a power function (a skill that, again, will be required when it comes time to learn differentiation).&lt;/p&gt;&#xA;" OwnerUserId="1737" LastEditorUserId="1737" LastEditDate="2022-09-23T13:53:06.493" LastActivityDate="2022-09-23T13:53:06.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25646" PostTypeId="2" ParentId="25641" CreationDate="2022-09-22T16:02:32.330" Score="11" Body="&lt;p&gt;Keep it simple and phrase like this: Write &lt;span class=&quot;math-container&quot;&gt;$y=\sqrt{3+x}$&lt;/span&gt; as the composition of two functions in a non-trivial way.&lt;/p&gt;&#xA;&lt;p&gt;Most students will not appreciate the qualification about non-trivial and go on to give a good answer. If a student does give one of the two trivial decompositions, commend them for having the mindset of a mathematician and use the opportunity to explain what is meant by non-trivial.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2022-10-05T20:23:21.253" LastActivityDate="2022-10-05T20:23:21.253" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25647" PostTypeId="1" CreationDate="2022-09-23T16:35:24.500" Score="10" ViewCount="1547" Body="&lt;p&gt;For a matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; in, say for instance, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^{m \times n}$&lt;/span&gt;, there are at least two different conventions to denote its entry at position &lt;span class=&quot;math-container&quot;&gt;$(j,k)$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Denote the entry as &lt;span class=&quot;math-container&quot;&gt;$a_{jk}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Denote the entry as &lt;span class=&quot;math-container&quot;&gt;$A_{jk}$&lt;/span&gt; .&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In my experience (at several universities in Germany) the first convention is much more common in teaching; but this might be different in other places.&lt;/p&gt;&#xA;&lt;p&gt;When teaching a linear algebra course last year I chose to use notation 2, though. The reaction of some colleagues (who were, let's say, surprised) and the prevalence of the first notation in many books I looked up, made me wonder whether I might be overlooking some important advantages of notation 1 or disadvantages of notation 2. So for my decision about the notation in future courses, I'm interested in the following:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What are advantages and disadvantages of notations 1 and 2, respectively, in math teaching?&lt;/p&gt;&#xA;&lt;p&gt;Remarks:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I am &lt;strong&gt;not&lt;/strong&gt; asking for your opinion on which notation is &amp;quot;better&amp;quot;. Instead, I am asking for a list of advantages and disadvantages, which can be used by course instructors to make an informed decision for their individual courses.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The question is mainly about the advantages and disadvantages &lt;strong&gt;in teaching&lt;/strong&gt; (thus I am asking this on Math Educators Stackechange). The same question might be relevant for research papers, but this is  not the focus of this question.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastEditorUserId="376" LastEditDate="2022-09-26T15:06:37.853" LastActivityDate="2022-10-01T15:20:03.077" Title="Lowercase vs. uppercase letters for matrix entries" Tags="|linear-algebra|notation|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25648" PostTypeId="2" ParentId="25647" CreationDate="2022-09-23T19:03:35.187" Score="11" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$A_{jk}$&lt;/span&gt; is sometimes used to mean the matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; with row &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; and column &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; deleted.&lt;/p&gt;&#xA;&lt;p&gt;[For example, see David Lay, &lt;em&gt;&lt;strong&gt;Linear Algebra and its Applications&lt;/strong&gt;&lt;/em&gt;, 4th edition, page 165.]&lt;/p&gt;&#xA;&lt;p&gt;To avoid confusion with that usage, I would never use the capital letter to mean an entry.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="15884" LastEditDate="2022-09-23T19:08:10.177" LastActivityDate="2022-09-23T19:08:10.177" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25649" PostTypeId="2" ParentId="25647" CreationDate="2022-09-23T19:44:02.487" Score="8" Body="&lt;p&gt;One area where capitals may be useful is statistics, where you'll encounter matrices whose entries are random variables and denoted &lt;span class=&quot;math-container&quot;&gt;$X_{ij}$&lt;/span&gt;. Generally I prefer lowercase but that's probably out of habit; I don't see what's so bad about calling entries &lt;span class=&quot;math-container&quot;&gt;$A_{ij}$&lt;/span&gt; other than that it's less common.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2022-09-23T19:44:02.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25650" PostTypeId="2" ParentId="4339" CreationDate="2022-09-24T11:18:49.217" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It would be very unusual to use Apostol for engineers in the US.  That book is used at Cal Tech (where A taught) but is very rare in the US landscape versus texts like Stewart, Larson, Thomas, Swokowski, etc.  It is not used at decent engineering schools (but a cut below Cal Tech) like Navy or Georgia Tech.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even at Cal Tech, that book and the course related to it are somewhat of a rite of passage and somewhat resented.  For instance later courses, like ODEs, allow students the option to omit a heavy analysis theoretic emphasis and return to a more conventional approach.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;See here, the choices for &amp;quot;practical&amp;quot; versus &amp;quot;analytical&amp;quot; flavored classes.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://pma.caltech.edu/research-and-academics/mathematics/math-course-schedule-and-websites-2022-2023&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://pma.caltech.edu/research-and-academics/mathematics/math-course-schedule-and-websites-2022-2023&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[Copied for link rot]&lt;/p&gt;&#xA;&lt;p&gt;Math 2/102 - Differential Equations&lt;/p&gt;&#xA;&lt;p&gt;Fall     Math 2 - Practical     Dr. Makarov      MWF  10:00 - 10:55 am     KRK 119      Course Webpage&lt;/p&gt;&#xA;&lt;p&gt;Fall     Math 2 - Analytical     Dr. Hutchcroft      MWF  10:00 - 10:55 am     Linde 310      Course Webpage&lt;/p&gt;&#xA;&lt;p&gt;(For practical, they've often used Boyce and Diprima, i.e. a conventional text.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.its.caltech.edu/%7Ematilde/Ma2aFall2010.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.its.caltech.edu/~matilde/Ma2aFall2010.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[copied for link rot]&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Textbook  William E. Boyce, Richard C. DiPrima, &amp;quot;Elementary differential equations and boundary value problems&amp;quot;, 9th edition&amp;quot;&amp;quot;&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;FWIW, I have a general engineering undergrad minor, have worked at an engineering firm (mechanical/electrical), passed the EIT, and almost took the mechE PE, and have worked in several industries with heavy engineering use, and even nuclear power and NASA.  I think you have a very academic view of engineers (were at an institute and know engineers who taught).  Don't assume you know what engineers need based on your very biased (statistical meaning) sample of them.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In general, working engineers barely use algebra.  IN GENERAL.  (But generalities are how practical decisions of allocation of effort are made...this is not a &amp;quot;divided by zero&amp;quot; real analysis exception situation, so please spare me some example of a Ph.D. engineer using real analysis for something.) They do need calculus and diffyQs to get through their majors courses (fluids, heat transfer, etc.)  But in the work world, it's rare to even use simple pieces of calculus.  There's a lot of adding heat loads or picking equipment from a catalog or sizing a pipe using a nomograph.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's interesting that you feel outgunned because of your colleagues knowing more pedagogy.  Well...maybe this is more of a pedagogy situation than a &amp;quot;how good at math are you&amp;quot; situation.  In other words, the educator part, not the math part.  Your colleagues are probably making arguments similar to mine (but based on experience with the actual students at your school).  Instead of looking for sympathy (or arguments to support your opinion) on the Internet (and forums like this, or MSE, or Physics Forum, tend to attract outstanding people in terms of math ability who prefer harder books...after all &amp;quot;I can handle it&amp;quot;), you might want to study your students, analyze your colleagues remarks, etc.  In other words, try to reflect and consider that you might be wrong.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For what it's worth, even the easier books in the US, usually have some theory.  Topics like FTOC, Rolle's Theorem, Taylor series continuity are certainly taught.  Just open the index of say Thomas...you will find pages with stuff on those topics.  However, perhaps not with the intense rigor you want.  All that said, even this theory is really just the bleeding in of analysis that has happened since the mid 19th century when it was developed and then bled into conventional texts over time (most heavily after WW2).  I remember learning epsilon-delta in Thomas and enjoying the long pages of notation to work a few small problems.  And then my teacher said, &amp;quot;that's great, but you're not going to need it the rest of the class&amp;quot;. I was disappointed...but it turned out she was right.  You don't epsilon delta to do some partial fractions or trig substitution or related rates head scratcher.  And consider that Euler did some pretty badass calculus...and he died 30 years before Weirstrass &amp;quot;sprungt aus der Ei&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20392" LastEditorUserId="20392" LastEditDate="2022-09-24T11:33:40.267" LastActivityDate="2022-09-24T11:33:40.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25652" PostTypeId="2" ParentId="25647" CreationDate="2022-09-25T04:50:08.543" Score="12" Body="&lt;p&gt;Sometimes a matrix name is suggestive: for example Jacobian or Ricci. We might use &lt;span class=&quot;math-container&quot;&gt;$\text{Jac}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$\text{Ric}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;. In these situations it would be awkward to switch to lower case to reference an entry; it would be linguistically awkward to write &lt;span class=&quot;math-container&quot;&gt;$r_{i,j}$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$R_{i,j}$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is meant to refer to Ricci curvature.&lt;/p&gt;&#xA;&lt;p&gt;Another example is the use of something like &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; to name a matrix, e.g., a curvature two-form. Strict adherence to the convention of switching to lowercase font for entries of &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; would dictate we use &lt;span class=&quot;math-container&quot;&gt;$\omega_{i,j}$&lt;/span&gt; for an entry of &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;. For those less comfortable with Greek characters, this would be confusing. I have never seen the Greek letter &lt;span class=&quot;math-container&quot;&gt;$\Xi$&lt;/span&gt; used to name a matrix, but its lowercase variant is &lt;span class=&quot;math-container&quot;&gt;$\xi$&lt;/span&gt; and then using &lt;span class=&quot;math-container&quot;&gt;$\xi_{i,j}$&lt;/span&gt; for an entry of &lt;span class=&quot;math-container&quot;&gt;$\Xi$&lt;/span&gt; seems ill-advised.&lt;/p&gt;&#xA;&lt;p&gt;If your students are going to use an environment such as MATLAB or Python, then adherence to a notational convention of switching to lowercase font to designate entries is not natural. In programming, if we have a matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; then an entry is specified typically by function notation &lt;span class=&quot;math-container&quot;&gt;$A(i,j)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;These examples not withstanding, using &lt;span class=&quot;math-container&quot;&gt;$a_{i,j}$&lt;/span&gt; to denote an entry of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is not incorrect. Mathematical notation is often ambiguous, and learning to adjust to notational ambiguities is something that needs to be learned. But your second convention seems pedagogically more sound.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2022-09-25T05:12:44.703" LastActivityDate="2022-09-25T05:12:44.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25653" PostTypeId="2" ParentId="25647" CreationDate="2022-09-26T00:35:07.467" Score="5" Body="&lt;p&gt;The convention you're describing isn't used in physics. The entries of matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$A_{jk}$&lt;/span&gt;. Part of the reason for this is probably that physicists tend to assign specific physical interpretations to the various letters of the alphabet. For instance, &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; would be a moment of inertia, which can be represented by a matrix, while &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; would be either an electrical current or &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Relativists also often use abstract index notation, in which &lt;span class=&quot;math-container&quot;&gt;$A_{mn}$&lt;/span&gt; would actually be the entire matrix, without reference to any basis, while &lt;span class=&quot;math-container&quot;&gt;$A_{\mu\nu}$&lt;/span&gt; is an element in a specific basis. (Latin letters are abstract indices, Greek letters are concrete.) To understand the logic of this in a more familiar context, consider a notation like &lt;span class=&quot;math-container&quot;&gt;$x^2-1$&lt;/span&gt; to mean a function. You can think of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as being a notation for the identity function. Abstract index notation is similar. Abstract indices don't have values like 3, just as the identity-function &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; doesn't have a value like 3.&lt;/p&gt;&#xA;" OwnerUserId="20401" LastEditorUserId="20401" LastEditDate="2022-09-26T00:35:55.903" LastActivityDate="2022-09-26T00:35:55.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25654" PostTypeId="2" ParentId="25618" CreationDate="2022-09-27T16:41:32.847" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;What are some research-level opportunities in mathematics that do not&#xA;focus on proofs?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'll give a common-sense answer without any particular knowledge:&lt;/p&gt;&#xA;&lt;p&gt;Physics, Engineering, economics, some Investment Banking or market - related research, Cryptocurrency and online security (encryption), Chemistry, Computer Science, Machine Learning, Robotics, Statistics, Computer simulations, some computer programming/software engineering.... These all &lt;em&gt;apply&lt;/em&gt; mathematical proofs and theorem without the need for knowing the proofs themselves, although of course you need to know the theorems.&lt;/p&gt;&#xA;&lt;p&gt;If you want to study &lt;em&gt;mathematics&lt;/em&gt; however, then you will need to read and write proofs in a comprehensive manner. There is no way around this. &amp;quot;Doing mathematics&amp;quot; requires theorem and proofs. Otherwise, it is not mathematics.&lt;/p&gt;&#xA;&lt;p&gt;If you study an in-between ground - a math-heavy subject that is not 100% mathematics, like mathematical physics, or econometrics, then you will have to know (by heart) &lt;em&gt;some&lt;/em&gt; important proofs, but not the unimportant proofs, for example you probably won't &lt;em&gt;have&lt;/em&gt; to know the small lemmas leading up to a major theorem (although they might help you in your job/research).&lt;/p&gt;&#xA;&lt;p&gt;Question answered.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;I cannot &amp;quot;prove&amp;quot; it to be 100% correct.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Absolute knowledge is unobtainable. Get over it.&lt;/p&gt;&#xA;&lt;p&gt;If what you really meant is that you struggle with proofs because they are hard, then yes. So does everyone else. Comprehension of (reading and understanding) proofs, and proof-writing are advanced skills that take a lot of time and effort to develop and improve upon. Either you put the effort in - which for me is made easier by my enjoyment of studying mathematics, which motivates me to put the effort in myself to understand proofs - and you improve, or you don't put the effort in, and you stagnate in your abilities.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastActivityDate="2022-09-27T16:41:32.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25656" PostTypeId="2" ParentId="25556" CreationDate="2022-09-29T08:53:16.207" Score="1" Body="&lt;p&gt;When teaching, one needs to know what one is hiding.&lt;/p&gt;&#xA;&lt;p&gt;A topic often taught in elementary schools is finding square roots of integers. Beyond guessing above and below, the simplest method is that of Newton, and this can't be understood without calculus (even if in the particular case of square roots it can be reduced to an algorithm that doesn't use calculus).&lt;/p&gt;&#xA;&lt;p&gt;In middle school algebra one works with polynomials with real coefficients. Many of the properties of such polynomials are best understood using calculus rather than algebra. Even someone who is not teaching such properties to students needs to know them and understand their origin. Some basic exercises in a calculus class should include: prove an odd degree real polynomial has a root and show that a degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; real polynomial with &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; distinct roots is identically zero. I'd be wary of someone teaching elementary algebra who can't prove those things.&lt;/p&gt;&#xA;&lt;p&gt;Many secondary school curricula include some elementary probability and statistics. Almost nothing meaningful can be said about a Gaussian without using some basic calculus.&lt;/p&gt;&#xA;&lt;p&gt;The basic properties of trigonometric and exponential functions and logarithms are all proved using calculus (more specifically the existence and uniqueness theorem for ordinary differential equations and convergence results for power series). These are central objects of study in high school math. Who teaches them should certainly know how to justify all their basic properties. For example, it's important to realize that the different definitions of the exponential function - as solution to an ODE, as a power series, as a limit - do not at all obviously yield the same function - or, more relevantly, that the property &lt;span class=&quot;math-container&quot;&gt;$e^{a + b} = e^{a}e^{b}$&lt;/span&gt; is not obvious ...  Students taught by those who know no calculus might reasonably guess that &lt;span class=&quot;math-container&quot;&gt;$\ln(ab) = \ln(a)\ln(b)$&lt;/span&gt; ...&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2022-09-29T08:53:16.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25657" PostTypeId="2" ParentId="25641" CreationDate="2022-09-30T10:20:04.953" Score="3" Body="&lt;p&gt;IMHO, students often struggle with concepts if they don't see their usefulness. So, embedding the question into a context like &amp;quot;to help finding the first derivative using the chain rule&amp;quot; will guide the students into the desired direction.&lt;/p&gt;&#xA;&lt;p&gt;If this isn't possible within your curriculum, you have a didactic problem, teaching something apparently useless to your students. Alas, this happens all too often in curricula.&lt;/p&gt;&#xA;&lt;p&gt;If you can't embed the exercise into a useful context, you can resort to a clarification like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Write &lt;span class=&quot;math-container&quot;&gt;$y=\sqrt{3+x}$&lt;/span&gt; as the composite of two functions. A solution using &lt;span class=&quot;math-container&quot;&gt;$f(x)=x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$g(x)=x$&lt;/span&gt; does not count as a valid answer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="15862" LastActivityDate="2022-09-30T10:20:04.953" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25658" PostTypeId="2" ParentId="13026" CreationDate="2022-09-30T19:15:01.093" Score="3" Body="&lt;p&gt;Point-slope form is used all over the place in Calculus where the point and the slope (i.e. derivative) is exactly what you have in most situations. You can strip out any explicit calculus though and simply talk about rates. So something like the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A car is traveling 60 mph and passes mile-marker 13 at 3pm. Give an equation for the position &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; of the car as a function of &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; hours since noon. Answer: &lt;span class=&quot;math-container&quot;&gt;$p(t) = 13+60(t-3)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is similar to ncr's answer where they use a rate of spending. But any rate can do. You can even do something like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It's known that for each raise in a degree Fahrenheit of the average summer temperature, the population of Northern Pike in a lake decreases by 29. If at an average summer temperature of 83 degrees F, a fish count reveals 325 Northern Pike, give a model for the population &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; as a function of temperature &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;. Answer: &lt;span class=&quot;math-container&quot;&gt;$P(T) = 325-29(T-83)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="401" LastActivityDate="2022-09-30T19:15:01.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25659" PostTypeId="1" CreationDate="2022-10-01T10:36:15.463" Score="6" ViewCount="321" Body="&lt;p&gt;Are there any studies that list the leading (English-language) mathematics education journals?  Here, &amp;quot;leading&amp;quot; could mean something based on data (for example, citations) or something based on opinions (for example, surveys).&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2022-10-01T10:36:15.463" Title="What are the leading journals in mathematics education?" Tags="|journals|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25660" PostTypeId="2" ParentId="25659" CreationDate="2022-10-01T10:36:15.463" Score="8" Body="&lt;p&gt;Geiger, Delzoppo, and Straesser (&lt;a href=&quot;https://doi.org/10.1007/s10649-022-10174-0&quot; rel=&quot;noreferrer&quot;&gt;2022, p. 551&lt;/a&gt;) use the following list (which seems to be unordered):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The Journal of Mathematics Teacher Education&lt;/li&gt;&#xA;&lt;li&gt;For the Learning of Mathematics&lt;/li&gt;&#xA;&lt;li&gt;ZDM Mathematics Education&lt;/li&gt;&#xA;&lt;li&gt;The Journal of Mathematical Behavior&lt;/li&gt;&#xA;&lt;li&gt;Educational Studies in Mathematics&lt;/li&gt;&#xA;&lt;li&gt;The Journal for Research in Mathematics Education&lt;/li&gt;&#xA;&lt;li&gt;Mathematical Thinking and Learning&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This is based on &amp;quot;the ranked journal lists developed by Williams and Leatham (&lt;a href=&quot;https://doi.org/10.5951/jresematheduc.48.4.0369&quot; rel=&quot;noreferrer&quot;&gt;2017&lt;/a&gt;). These lists were based on two sub-studies---one citation-based and the other in-field opinion-based, that produced ranked lists of 20 English language journals that publish Mathematics Education research.&amp;quot;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;Geiger, V., Delzoppo, C., &amp;amp; Straesser, R. (2022). Supporting English non‑dominant language authors’ efforts to publish: Perspectives from the editors‑in‑chief of highly recognised journals in Mathematics Education. &lt;em&gt;Educational Studies in Mathematics&lt;/em&gt;, &lt;em&gt;111&lt;/em&gt;:543–565.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;Williams, S., &amp;amp; Leatham, K. (2017). Journal quality in mathematics education. &lt;em&gt;Journal for Research in Mathematics Education&lt;/em&gt;, &lt;em&gt;48&lt;/em&gt;(4), 369–396.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2022-10-01T10:36:15.463" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25661" PostTypeId="2" ParentId="25647" CreationDate="2022-10-01T15:20:03.077" Score="2" Body="&lt;p&gt;If you are teaching a lot of students whose first language doesn't use the Roman alphabet, the bijection between upper and lower case letters may be something they have to pause and think about. I've run into this with Chinese students.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2022-10-01T15:20:03.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25662" PostTypeId="2" ParentId="13809" CreationDate="2022-10-01T23:23:59.647" Score="4" Body="&lt;p&gt;My advice is to pick one or the other, not both, for topic, probably continuity.&lt;/p&gt;&#xA;&lt;p&gt;Also, physics students are not a-mathematical. They may not be PhD geometrists. But they are definitely people that cleave to the standard algebra and calculus track.&lt;/p&gt;&#xA;&lt;p&gt;I would avoid the trap of thinking of them as unwashed non Rudinites that you are going to correct gaps of. Instead, frame your talk more to explore the different ways one can think about continuity, or perhaps with a msth history flair, discuss how and who had different frames over time.  On other words more like a fun pop video on YouTube.&lt;/p&gt;&#xA;&lt;p&gt;It is fine to mention some very advanced concepts without feeling you need to teach and prove. I get value out of knowing what a topic is, who works on it, what classes cover it, without knowing the content itself. Like having a situational awareness of what an abstract algebra course us, without having ever done one. Similar to how you (should) have a general awareness of what an organic chemistry course is, without having done one. But more as a generally educated person to know whatvthe fields of knowledge are.&lt;/p&gt;&#xA;&lt;p&gt;Format is separate from the aspect of how you slant the talk. But my advice is to do a chalk talk.  And even for that, I would do more of an entertaining speech, not a step by step derivation. Ie less chalk than a teacher. Just occasionally as needed, for some example or diagram.&lt;/p&gt;&#xA;&lt;p&gt;Don't do slides.  Will put them to sleep. Death by PowerPoint. And it will make it to easy for you to fall into the trap of being too hard and too proofy.&lt;/p&gt;&#xA;" OwnerUserId="20424" LastActivityDate="2022-10-01T23:23:59.647" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25663" PostTypeId="1" AcceptedAnswerId="25664" CreationDate="2022-10-02T08:34:49.227" Score="12" ViewCount="5941" Body="&lt;h2&gt;If 'linguistic knowledge is largely subconscious'&lt;sup&gt;3&lt;/sup&gt;, why isn't math?&lt;/h2&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://redd.it/xnw7qu&quot; rel=&quot;noreferrer&quot;&gt;Most math instructors sermonize solving exercises and problems.&lt;/a&gt; But a student challenged why students need practice — because most students and adults don't need to know, or practice, any linguistics or socio-pragmatics to speak their L1 (native language)!&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;         Modern linguistic research carried out within the Chomskyan framework&#xA;reveals the complex system of linguistic knowledge (grammatical competence)&#xA;that enters into actual language use to be almost totally tacit or unconscious. Only&#xA;a small fraction of that knowledge is accessible to conscious thought. In other&#xA;words, the set of rules which native speakers employ to represent internally an&#xA;utterance of any one of the infinite number of sentences of his/her language constitutes a system of unconscious knowledge; thus, no native speaker is capable of&#xA;describing those complex formal rules — syntactic, semantic, morphological, phonological, and phonetic rules — which native speakers employ in their day-to-day&#xA;communication. Only a fraction of this knowledge is conscious. The conscious&#xA;aspect of linguistic knowledge is restricted largely to the native speaker's ability&#xA;to offer grammaticality judgments about an utterance.&lt;br /&gt;&#xA;         Although linguistic (grammatical) competence is the central element in language use, knowledge of linguistic form must be integrated with socio-pragmatic&#xA;competence/ knowledge in order for an utterance to be socially appropriate/&#xA;acceptable. Like linguistic competence, socio-pragmatic knowledge is also largely&#xA;unconscious. Consider, for example, the rules of turn-taking in a conversation. It&#xA;is remarkable how complex the factors are that enable speakers to determine&#xA;unconsciously when and how to take or yield their turn in order to ensure the&#xA;smooth operation and success of a conversation (see Green 1990 for the markers&#xA;of cooperation in a drug case; Crystal 2011: 126—7 for the index of criminal intent&#xA;through turn-taking in conversation). In contrast, social evaluation of speech, i.e.,&#xA;the notion of language and speaker categorization (e.g., 'good' vs. 'bad' language,&#xA;'intelligent' vs. 'stupid' speaker) and prescriptive rules which are learnt by means&#xA;of formal instruction (prohibition of double negatives in a formal context) represent a conscious dimension of socio-pragmatic competence. Figure 27.1 sums up&#xA;the conscious and unconscious aspects of the two knowledge-based systems&#xA;which interact with each other to ensure socially accepted language use.&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To wit, why is math unlike &amp;quot;most of our knowledge of the complex system of English&#xA;grammar [that] is unconscious — which is very fortunate considering the rate at which&#xA;we produce and comprehend language in normal conversations.[?] Frankly, the&#xA;quantity of information that we have to know to use a language is astounding,&#xA;and we generally only become aware of the depth of that knowledge when we&#xA;have to learn a second language. Then we begin to see that language learning&#xA;is so much more than just memorizing new words and sounds! All the world's&#xA;languages have complex rules, though they differ greatly from English. The&#xA;speakers of other languages are equally unaware of the vast majority of their&#xA;grammar rules as well.&amp;quot;&lt;sup&gt;2&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;They &amp;quot;&lt;strong&gt;know their grammar&lt;/strong&gt;&amp;quot; but &lt;strong&gt;do not necessarily know much &lt;em&gt;about&lt;/em&gt; it&lt;/strong&gt;, i.e. they are often not able to formulate the rules governing their linguistic choices.&lt;sup&gt;4&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h2&gt;Why can't students rely on 'native speaker intuition' for math?&lt;/h2&gt;&#xA;&lt;p&gt;'Even though most speakers claim not to know the rules of grammar of their language, they can tell you if a sentence is grammatical or not. [. . .] Although native speakers could say that the sentence sounds funny and&#xA;declare what sounds better, most could not provide the complete rule for the&#xA;order of multiple adjectives in a sentence even though they have been apply-&#xA;ing this rule correctly, though unconsciously, most of their lives. We may not&#xA;even think about this rule until we take a foreign language and then have&#xA;to overtly learn the adjective order rules for that language — which are most&#xA;likely different from the rules of our own.'&lt;sup&gt;5&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Native language users process&#xA;grammar and vocabulary unconsciously, i.e., they use language&#xA;automatically with no clear consciousness of controlling&#xA;grammar (Bialystok and Shamood Smith 1985). In our daily&#xA;lives, for instance, we may drive a car while talking with friends,&#xA;which means that we can easily perform the two actions of&#xA;driving and talking in parallel, suggesting that we can manage&#xA;both actions unconsciously (or automatically). Driving would&#xA;become very dangerous if intentional control were required in&#xA;the process of driving and/or talking. Automatic and unconscious&#xA;processing makes it relatively easy for people to perform&#xA;different tasks in parallel.&lt;sup&gt;6&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; Tej K. Bhatia, &lt;em&gt;The Handbook of Bilingualism and Multilingualism&lt;/em&gt;, &lt;a href=&quot;https://www.google.com/books/edition/The_Handbook_of_Bilingualism_and_Multili/b3PYAwAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA675&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 675&lt;/a&gt;.&lt;br /&gt;&#xA;&lt;sup&gt;2&lt;/sup&gt; Elizabeth Winkler, &lt;em&gt;Understanding language A Basic Course in Linguistics&lt;/em&gt;, &lt;a href=&quot;https://www.google.com/books/edition/Understanding_Language/YLkDCgAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA13&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 13&lt;/a&gt;.&lt;br /&gt;&#xA;&lt;sup&gt;3&lt;/sup&gt;  Markus Bieswanger, &lt;em&gt;Introduction to English Linguistics&lt;/em&gt;, 2017, &lt;a href=&quot;https://www.google.com/books/edition/Introduction_to_English_Linguistics/Tzg8DwAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA121&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 122&lt;/a&gt;.&lt;br /&gt;&#xA;&lt;sup&gt;4&lt;/sup&gt; Bieswanger, p. 122.&lt;br /&gt;&#xA;&lt;sup&gt;5&lt;/sup&gt; Winkler, p. 12.&lt;/p&gt;&#xA;" OwnerDisplayName="user20427" LastEditorDisplayName="user20427" LastEditDate="2022-10-02T19:46:38.963" LastActivityDate="2022-10-19T07:07:54.433" Title="Why does math need to be practiced and exercised, when L1 Linguistic Competence is subconscious?" Tags="|exercises|" AnswerCount="8" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25664" PostTypeId="2" ParentId="25663" CreationDate="2022-10-02T10:28:35.647" Score="18" Body="&lt;p&gt;Probably evolution. Our brains are not purely, perhaps not even mostly programmable computers. You can learn to walk naturally.  Doing a kip in gymnastics takes practice. Similarly, your brain is specialized to have massive innate control of your tongue, but not of individual smaller toes.&lt;/p&gt;&#xA;&lt;p&gt;Language has probably been a part of human society for tens or hundreds of thousands of years. Math has not. Especially at a general level.  Consider now, if you lost the ability to understand math or language, which would make your life worse.&lt;/p&gt;&#xA;&lt;p&gt;The brain does have some flexibility and can learn to do kips, play piano, type, and solve equations.  But these require practice because we are not evolved to need them.&lt;/p&gt;&#xA;&lt;p&gt;In contrast, facial recognition is an innate talent. One which programmable computers have only recently achieved.&lt;/p&gt;&#xA;&lt;p&gt;Or consider the differences in muscular control of your fingers versus your toes.  Or your fingers versus a dog's paw digits.&lt;/p&gt;&#xA;&lt;p&gt;Or your sense of smell versus sight.  And part of that is in the sensing organ, but a lot is in the brain.  Look at the size of your optical lobe on a diagram of the brain.&lt;/p&gt;&#xA;" OwnerUserId="20424" LastActivityDate="2022-10-02T10:28:35.647" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25665" PostTypeId="2" ParentId="25663" CreationDate="2022-10-02T10:37:39.147" Score="12" Body="&lt;p&gt;I believe there are people who develop mathematical competency without being taught but they are in the minority.  No one taught me algebra, but I placed out of algebra by &amp;quot;intuition&amp;quot;. I am sure there are people who calculate without being taught how.&lt;/p&gt;&#xA;&lt;p&gt;Most likely everyone develops language because we use it all day from a young age.    I do not have the same competency in Hebrew, a language that I started learning at age 6 and I often am confused about what's grammatically correct. Sometimes if I stop and think about it, I notice my mistake and sometimes I don't.  My granddaughter who is a native Hebrew speaker, catches all my grammatical mistakes without effort.   Of course, I have no ability in Japanese because I was never exposed to it.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2022-10-02T10:37:39.147" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25666" PostTypeId="2" ParentId="25663" CreationDate="2022-10-02T19:05:35.623" Score="46" Body="&lt;p&gt;L1 language competence requires thousands and thousands of hours of practice to reach. We don't realise it only because much of it happens so early in life. Nonetheless huge amounts of early schooling time is put into mastering language skills.&lt;/p&gt;&#xA;&lt;p&gt;It's so fantastically difficult to achieve this level of fluency that it is generally reckoned to be impossible for adults to reach it in a second language.&lt;/p&gt;&#xA;&lt;p&gt;Many people receive sufficient mathematical training for simple arithmetic to seem innate but we simply don't spend the time to achieve unconscious mastery of more complex areas.&lt;/p&gt;&#xA;" OwnerUserId="6399" LastActivityDate="2022-10-02T19:05:35.623" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25667" PostTypeId="1" AcceptedAnswerId="25668" CreationDate="2022-10-02T21:46:18.410" Score="4" ViewCount="93" Body="&lt;h2&gt;Context&lt;/h2&gt;&#xA;&lt;p&gt;In the case of recently arrived refugee classes (e.g. Ukrainian classes during current times), a first challenge is to establish the individual level of the students in order to tune the program according to their actual level. My main fear is approaching the class in a traditional manner and thus give a general course (adhering to a program for their age within the school) and in the process numb the enthusiasm of the stronger ones.&lt;/p&gt;&#xA;&lt;p&gt;To that end, it's possible to just compile a long list of exercises that cover various chapters (numbers, operations, geometry, etc) and formulated such that the language barrier is felt minimally, but I'm afraid any such approach may come across as too dry and it won't necessarily motivate the students to carefully answer them.&lt;/p&gt;&#xA;&lt;p&gt;So I was wondering, if there are known resources where one can find exercise worksheets on various of the aforementioned topics (aimed for 11-13 year olds) which are covered in a more playful manner with an emphasis on questions that probe understanding as opposed to solving lots by rote.&lt;/p&gt;&#xA;&lt;h2&gt;Questions&lt;/h2&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there online resources of math exercises/tests that would be particularly suited for evaluating the knowledge of students that come from different educational backgrounds? Anything that acts as a reference for good methods of evaluation will bear merit for me.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am trying to compile exercise sheets myself, and I was wondering, should I also translate each question into the mother tongue of the students? Or rather, opt for a very simplified English when framing the questions?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Any advice or recommendations for other ways to go about gauging my students' level in maths are quite welcome.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20206" LastActivityDate="2022-10-02T23:41:48.470" Title="Ideas and resources helpful in evaluating students with diverse backgrounds" Tags="|secondary-education|resource-request|test-design|teacher-development|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25668" PostTypeId="2" ParentId="25667" CreationDate="2022-10-02T23:41:48.470" Score="2" Body="&lt;p&gt;You might like some of the &lt;a href=&quot;https://beastacademy.com/resources/printables&quot; rel=&quot;nofollow noreferrer&quot;&gt;puzzles and review sheets that Beast Academy puts out&lt;/a&gt;. Start with level 5 I think, but there might be goodies in levels 4 and 3 also. &lt;a href=&quot;https://beastacademy.com/u/pdfs/5B/printables/GCFLCMWebs.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's one on LCM and GCF&lt;/a&gt; your students might like.&lt;/p&gt;&#xA;&lt;p&gt;Wishing you the best in your work.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-10-02T23:41:48.470" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25669" PostTypeId="2" ParentId="10442" CreationDate="2022-10-03T10:29:47.240" Score="1" Body="&lt;h1&gt;Limit supremum (of a sequence of numbers)&lt;/h1&gt;&#xA;&lt;p&gt;I will try to give intuition only using words, without using mathematical symbols.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is the smallest number which is greater than infinitely many members of a sequence? --&amp;gt; limit supremum&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;em&gt;Story:&lt;/em&gt; To be a candidate for the limit supremum, a number has to be greater than infinitely many members of the sequence. For this, we need to consider tails of the sequence. Why tails? Why not heads? Because given any member, all the members to the left of it (head) are finite in number as the sequence starts on the left and the starting point is known ([avoiding bi-infinite sequences for now] a sequence is a mapping from the set of natural numbers which starts at 1, so the starting side is known. You can tell the first natural number, the first 2 natural numbers, etc. but cannot tell the last 2 natural numbers), only the right side is infinite. So, where there is talk of infinitely many members, you have to consider a tail. Any tail of the sequence is infinitely long. Hence, the supremum of any tail is greater than infinitely many members of the sequence. Thus, the supremum of any tail is a candidate for limit supremum. Further, as successive tails are subsets of the previous ones, the corresponding supremums form a monotonically decreasing sequence of candidates for the limit supremum. However, the infimum (= limit, in this case) of this sequence of candidates is the smallest number which already bounds infinitely many members of the original sequence from above--which is precisely the purpose. So, we reject all larger candidates, and this infimum, as the best/most parsimonious candidate, is elected as the limit supremum of the original sequence.&lt;/p&gt;&#xA;&lt;h1&gt;Limit infimum (of a sequence of numbers)&lt;/h1&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is the largest number which is smaller than infinitely many elements of the sequence? --&amp;gt; limit infimum&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;You can write the story now.&lt;/p&gt;&#xA;" OwnerUserId="20435" LastActivityDate="2022-10-03T10:29:47.240" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25670" PostTypeId="1" CreationDate="2022-10-03T11:13:43.023" Score="0" ViewCount="148" Body="&lt;p&gt;As a maths tutor, when I set tests (past papers in particular), if a student does poorly (~50/60% or below), obviously some topics need to be retaught. But should I make them do the exact same test - closed book again? I think this generally is a good idea in these situations because then they will revise harder on the questions they got wrong and will learn from those mistakes. I do this myself if I did badly in a test and it definitely helps. I guess we should go through the questions they got wrong first before setting the test again.&lt;/p&gt;&#xA;&lt;p&gt;The student might see this as punishment though. What are your thoughts?&lt;/p&gt;&#xA;&lt;p&gt;If they get &lt;span class=&quot;math-container&quot;&gt;$70$&lt;/span&gt;% or above then we just go through the few questions they got wrong, and those people are generally smart enough to improve on some of the questions they got wrong. Maybe I should get these students to gen up on/revise these questions and then set timed tests - but the test consists only of questions they got wrong - later on as homework?&lt;/p&gt;&#xA;" OwnerUserId="12691" LastActivityDate="2022-10-03T16:56:50.457" Title="As a tutor, should I make students who did badly on timed, closed book past paper tests, do the exact same test again?" Tags="|exams|testing|test-design|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25671" PostTypeId="2" ParentId="25670" CreationDate="2022-10-03T16:43:56.773" Score="0" Body="&lt;p&gt;Either that or changing some questions slightly (for example question A has &amp;quot;x=5 and y=3x&amp;quot; then change it to something else).&#xA;Because, personally, the way you said might just turn them instead a bit &amp;quot;lazy&amp;quot; when facing hard questions since they can always just go back and solve them.&lt;/p&gt;&#xA;&lt;p&gt;*&lt;em&gt;Although this means more work for you, so...&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="20437" LastEditorUserId="20437" LastEditDate="2022-10-03T16:45:50.433" LastActivityDate="2022-10-03T16:45:50.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25672" PostTypeId="2" ParentId="25670" CreationDate="2022-10-03T16:53:51.023" Score="4" Body="&lt;p&gt;I advise to go for it with a small tweak.  Instead of doing 100% the same, make it equivalent problems (VERY SIMILAR), but slightly different numbers, formulas.  It's worth doing if they 100% missed a basic concept.&lt;/p&gt;&#xA;&lt;p&gt;The one experience like this I had was in AP Chem where entire class got crushed on a unit in descriptive chemistry (sources of the elements), since all the students were primed for the heavily algebraic problems in stoichiometry, equilibrium, etc. and basically just didn't grok that they needed to memorize some things and answer essay questions (e.g. describe the Bessemer process).  When we did the retake, teach said it would be very similar (was), but she varied all the questions except one.  But the fundamental thing was a corrective aspect for a major part of the course and even for a mindset about &amp;quot;what we have to learn&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Let students keep the higher of the two tests.  (If they crush it, roll with it and let them have a bone.)&lt;/p&gt;&#xA;&lt;p&gt;And I would not make this an every time type thing.  Just with a real trainwreck where you think the vast majority, didn't get a basic important module.  Maybe every 3 years or so.  Not twice in a class.  [You should at least be able to advise next years classes to &amp;quot;watch out&amp;quot;.]&lt;/p&gt;&#xA;" OwnerUserId="20439" LastEditorUserId="20439" LastEditDate="2022-10-03T16:56:50.457" LastActivityDate="2022-10-03T16:56:50.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25673" PostTypeId="2" ParentId="18784" CreationDate="2022-10-03T18:28:07.363" Score="1" Body="&lt;p&gt;&amp;quot;Is there a way I could adopt to prove it, using a simpler approach? Or is there another way to construct Bézout's identity, without continued fraction but also easy to understand?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Your question is two years' old so this may be untimely but there is an algorithm that operates on two natural numbers p and q which in a single process and using just addition and subtraction will provide gcd(p, q), Bezout identity, expression of non-coprime p and q as a fraction in lowest form, continued fraction terms (with some simple interpretation) and successive approximations.&lt;/p&gt;&#xA;&lt;p&gt;A:              31 (1)  -  13 (0)   =   31    First chosen starting number&lt;/p&gt;&#xA;&lt;p&gt;B:              31 (0)  -  13 (1)   =  -13    Second chosen starting number multiplied by -1&lt;/p&gt;&#xA;&lt;p&gt;C = A + B:      31 (1)  -  13 (1)    =  18&lt;/p&gt;&#xA;&lt;p&gt;Add the smallest positive result obtained so far to the 'smallest' negative result so far - 'smallest' here meaning 'closest to zero'&lt;/p&gt;&#xA;&lt;p&gt;D = B + C      31 (1)  -  13 (2)    =   5&lt;/p&gt;&#xA;&lt;p&gt;Repeat the above operation: add smallest negative to smallest positive and add to list&lt;/p&gt;&#xA;&lt;p&gt;E = B + D:      31 (1)  -  13 (3)    =  -8     Repeat above instructions&lt;/p&gt;&#xA;&lt;p&gt;F = D + E:      31 (2)  -  13 (5)    =  -3&lt;/p&gt;&#xA;&lt;p&gt;G = D + F:     31 (3)  -  13 (7)    =   2&lt;/p&gt;&#xA;&lt;p&gt;H = F + G:      31 (5)  -  13 (12)   =  -1&lt;/p&gt;&#xA;&lt;p&gt;I = G + H:      31 (8)  -  13 (19)   =   1&lt;/p&gt;&#xA;&lt;p&gt;J = H + I:      31 (13) -  13 (31)   =   0&lt;/p&gt;&#xA;&lt;p&gt;The gcd is the smallest non-zero result (1) which in this algorithm will always appear once as negative and once as positive value.  The Bezout identity is seen from the co-efficients of the equation resulting in the gcd.  If the starting numbers are not co-prime then the co-efficients of the final equation that equals zero will represent the starting numbers as a  fraction in lowest form. The successive approximations to 13 / 31 or its inverse are seen from the co-efficients of the equations as the alorithm progresses.  These pairs also represent the route down the Stern-Brocot tree to 13 / 31.&lt;/p&gt;&#xA;&lt;p&gt;The continued fraction is not as obvious as in the Euclidean algorithm.  It is shown by the pattern of flipping of signs from positive to negative and back again, disregarding the first two equations for the starting numbers.  In this example the results are +18, +5, -8, -3, +2, -1, +1, 0.  Two positives, two negatives, one positive, one negative, and finally one positive and zero which is counted as whichever sign the term before it has.  So the continued fraction is [2; 2, 1, 1, 2] or [0; 2, 2, 1, 1, 2], since the algorithm is identical for a fraction and its inverse.&lt;/p&gt;&#xA;&lt;p&gt;A slightly more sophisticated version of the algorithm, using the floor values of division rather than just repeated subtraction, gives a neater route to the continued fraction. For larger numbers this is necessary.&lt;/p&gt;&#xA;&lt;p&gt;The method is equivalent to using matrix multiplication but matrices are not needed explicitly. I can say that it's easy and quite a lot of fun.&lt;/p&gt;&#xA;&lt;p&gt;More explanation about how this works &lt;a href=&quot;https://simplyfractions.blogspot.com/p/blog-page_5.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://simplyfractions.blogspot.com/p/blog-page_5.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20440" LastEditorUserId="20440" LastEditDate="2022-10-03T18:56:20.650" LastActivityDate="2022-10-03T18:56:20.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25674" PostTypeId="2" ParentId="25663" CreationDate="2022-10-03T18:35:07.307" Score="0" Body="&lt;p&gt;Because as Otto Jespersen shows in his Philosophy of Language, we mostly speak based on patterns. So a child learns &amp;quot;John hit me&amp;quot; and from that has &amp;quot;X hit Y&amp;quot; to cover all future hittings. But though you have 3x + 4x = 14 as a pattern , it is a pattern of a setup but not of how to solve. If you form a sentence you know what you mean, but if you form a word problem you might have nary a clue to solve it.&lt;/p&gt;&#xA;" OwnerUserId="20441" LastActivityDate="2022-10-03T18:35:07.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25675" PostTypeId="2" ParentId="25663" CreationDate="2022-10-03T21:53:16.910" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Why can't students rely on 'native speaker intuition' for math?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Because we don't usually get to teach students mathematics; we teach them number problems. To &lt;a href=&quot;https://mysite.science.uottawa.ca/mnewman/LockhartsLament.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;quote Paul Lockhart&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“The area of a triangle is equal to one-half its base times its height.” Students are asked to memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill, the joy, even the pain and frustration of the creative act. There is not even a &lt;em&gt;problem&lt;/em&gt; anymore.&#xA;The question has been asked and answered at the same time— there is nothing left for the student to do.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some students manage to figure out the underlying mathematics, and those students can often coast on their 'native speaker intuition' – until they can't, because practising &lt;em&gt;is&lt;/em&gt; actually important for &lt;em&gt;developing&lt;/em&gt; that mathematical intuition.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If I chop the rectangle into two pieces like this, I can see that each piece is cut diagonally in half by the sides of the triangle. So there is just as much space inside the triangle as outside. That means that the triangle must take up exactly half the box! […]&lt;/p&gt;&#xA;&lt;p&gt;Now where did this idea of mine come from? How did I know to draw that line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck. That’s the art of it, creating these beautiful little poems of thought, these sonnets of pure reason.&#xA;There is something so wonderfully transformational about this art form. The relationship between the triangle and the rectangle was a mystery, and then that one little line made it obvious. I couldn’t see, and then all of a sudden I could. Somehow, I was able to create a profound simple beauty out of nothing, and change myself in the process. Isn’t that what art is&#xA;all about?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Consider two students.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;One learnt about algebra, figured out how to solve quadratic simultaneous equations, and was bored for the next year of maths lessons, barely engaging with the work except when explicitly encouraged.&lt;/li&gt;&#xA;&lt;li&gt;The other treated maths lessons as a game of &amp;quot;spot and apply the algorithm&amp;quot;, without really realising there was anything deeper, but consistently producing well-written answers to textbook questions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Who's going to do better when faced with the trigonometry unit? The student who has been learning to spot patterns, identify algorithms, and execute them in a clear and correct manner? Or the student who's barely done &lt;em&gt;any&lt;/em&gt; mathematical thinking for a whole year, because they solved the general case two weeks in, and doesn't remember how to multiply fractions?&lt;/p&gt;&#xA;&lt;p&gt;Neither of these students is mathematically-literate. The engaged one is at least &lt;em&gt;numerate&lt;/em&gt; – but you don't become literate by reading &lt;em&gt;Biff, Chip and Kipper&lt;/em&gt; 30 000 times. You do it by having conversations about all sorts of topics, listening to people from all sorts of backgrounds, and writing in all sorts of ways. When was the last time you saw somebody having a mathematical conversation in a classroom more sophisticated than “how do I do question 21b?”? When was the last time you &lt;em&gt;read a piece of mathematical literature in the classroom&lt;/em&gt;? (Do your students even know that mathematical literature exists?)&lt;/p&gt;&#xA;&lt;p&gt;A mathematically literate person doesn't need to memorise the quadratic formula, because they understand what it means, and so (with a little thinking) they can derive it. But they're likely to have memorised it anyway, or near enough as makes no odds, because quadratics come up a lot and it's useful for studying certain properties of them.&lt;/p&gt;&#xA;&lt;p&gt;A mathematically non-literate person might memorise the quadratic formula, and be able to apply it really well to problems of the form ax² + bx + c = 0, but as soon as they forget it, that skill is permanently lost ­– at least, until they re-learn it from a textbook. Without an understanding of what that computation is actually &lt;em&gt;doing&lt;/em&gt;, they might as well buy a calculator and familiarise themselves with its instruction manual.&lt;/p&gt;&#xA;&lt;p&gt;Maybe if mathematics exams were more like language exams, there might be more scope &lt;a href=&quot;https://nrich.maths.org/31&quot; rel=&quot;nofollow noreferrer&quot;&gt;to teach mathematics in classrooms&lt;/a&gt;, and more students &lt;a href=&quot;https://nrich.maths.org/33&quot; rel=&quot;nofollow noreferrer&quot;&gt;might pick it up&lt;/a&gt; from &lt;a href=&quot;https://nrich.maths.org/37&quot; rel=&quot;nofollow noreferrer&quot;&gt;immersion and exposure&lt;/a&gt; (&lt;a href=&quot;https://nrich.maths.org/34&quot; rel=&quot;nofollow noreferrer&quot;&gt;and chocolate&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="13158" LastEditorUserId="13158" LastEditDate="2022-10-03T22:11:02.623" LastActivityDate="2022-10-03T22:11:02.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25676" PostTypeId="2" ParentId="25663" CreationDate="2022-10-04T12:29:32.003" Score="1" Body="&lt;p&gt;Everything we learn, we learn as patterns. Our brain works by pattern matching and pattern prediction.&lt;/p&gt;&#xA;&lt;p&gt;Our brain has an enormous flexibility, e.g. the same area can be used for spoken language as well as for sign language (if you are deaf). People without hands learn to do everything with their toes.&lt;/p&gt;&#xA;&lt;p&gt;The main factor for learning language - and everything else - is exposure. From day one, we spend multiple hours per day learning and practicing language. In a multilingual environment, we can even learn several languages at the same time.&lt;/p&gt;&#xA;&lt;p&gt;It's really the same with lots of other skills we acquire. We learn to drive a bicycle and a car on a subconscious level - even if, in theory, there is a lot of physics and mathematics involved. We learn to subconsciously apply what is best described in quadratic and differential equations.&lt;/p&gt;&#xA;&lt;p&gt;We do not spend the same amount of time with mathematics, as it is taught at school and often used at work. We manage to learn the multiplication tables on a subconscious level, not much more.&lt;/p&gt;&#xA;&lt;p&gt;Also, we do not want to do it subconsciously. We want to write it down explicitly, so that we, and the teacher, and the tax auditor can check what we have done. In many exams (at least in Europe), you get points for writing down all the steps, not just for the result - much to the chagrin of the gifted student who solves the task instantly in his head.&lt;/p&gt;&#xA;&lt;p&gt;While we do get some gut feeling over time and move some of our problem solving skills to the subconscious, students always learn lots of new things and simply cannot spend that much time on any topic.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(For reference, I would like to point to the TED talks of &lt;a href=&quot;https://www.ted.com/talks/jeff_hawkins_how_brain_science_will_change_computing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Hawkins&lt;/a&gt; and &lt;a href=&quot;https://www.ted.com/talks/lydia_machova_the_secrets_of_learning_a_new_language&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lýdia Machová&lt;/a&gt;, and their &lt;a href=&quot;https://www.librarything.com/work/28334&quot; rel=&quot;nofollow noreferrer&quot;&gt;other&lt;/a&gt; &lt;a href=&quot;https://www.languagementoring.com/youtube/#talks&quot; rel=&quot;nofollow noreferrer&quot;&gt;works&lt;/a&gt;. You might also want to ask the question on &lt;a href=&quot;https://psychology.stackexchange.com&quot;&gt;Psychology &amp;amp; Neuroscience StackExchange&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="17540" LastEditorUserId="17540" LastEditDate="2022-10-19T07:07:54.433" LastActivityDate="2022-10-19T07:07:54.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25677" PostTypeId="1" CreationDate="2022-10-04T22:15:20.047" Score="12" ViewCount="1049" Body="&lt;p&gt;Over the past one or two years, at least two different teachers have told my children that &lt;span class=&quot;math-container&quot;&gt;$\sqrt 4$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-2.$&lt;/span&gt; I don't think this is useful, but if you want to define &lt;span class=&quot;math-container&quot;&gt;$\sqrt x$&lt;/span&gt; as the set of solutions of &lt;span class=&quot;math-container&quot;&gt;$y^2 = x$&lt;/span&gt;, I guess you could. Of course you get into all kinds of confusion with very common constructions and notation, for example the graph of &lt;span class=&quot;math-container&quot;&gt;$x\mapsto \sqrt x$&lt;/span&gt; (should we write &lt;span class=&quot;math-container&quot;&gt;$\max\sqrt x$&lt;/span&gt;?), Pythagoras' theorem or the root formula for quadratic equations, but let's say we accept that, and the common notation is just sloppy. In fact, the teacher is consistent, and says that the root formula should be written&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;rather than&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now the teacher went one step further, saying that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x + \sqrt{7 - 3x} = 1$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;has two solutions: &lt;span class=&quot;math-container&quot;&gt;$x = -3$&lt;/span&gt; (obviously), but also &lt;span class=&quot;math-container&quot;&gt;$x = 2$&lt;/span&gt;. This is found by writing&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{7 - 3x} = 1 - x$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and squaring. Since it is assumed &lt;span class=&quot;math-container&quot;&gt;$\sqrt{7 - 3\cdot 3} = \pm 1$&lt;/span&gt;, we select the &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;, and in this way this also is a solution to the original equation.&lt;/p&gt;&#xA;&lt;p&gt;This seems obviously wrong to me, but I'm not sure how obvious that really is. If we really define the square root as a set of values (of 0, 1 or 2 elements) rather than as a (single-valued) function on non-negative real numbers, we could say that the solutions of an equation involving square roots are those values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; so that from each of these sets we can select a value satisfying the equation. That would have strange consequences like&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(1 + \sqrt x)^2 = -3$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;having the solution &lt;span class=&quot;math-container&quot;&gt;$x = 4$&lt;/span&gt;. Also you cannot factorize in terms of square roots anymore: &lt;span class=&quot;math-container&quot;&gt;$a - b$&lt;/span&gt; is a single value, but &lt;span class=&quot;math-container&quot;&gt;$(\sqrt a + \sqrt b)(\sqrt a - \sqrt b)$&lt;/span&gt; can have 8 different values.&lt;/p&gt;&#xA;&lt;p&gt;Of course these last two are less defensible than the original example, for the teacher's example at least for each of the solutions a branch of the square root can be defined for which it is a solution, but they are not taught that you have to fix whether you want &lt;span class=&quot;math-container&quot;&gt;$\sqrt 4$&lt;/span&gt; to be &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-2,$&lt;/span&gt; but rather that it is both.&lt;/p&gt;&#xA;&lt;p&gt;After this long introduction, my question: could it be reasonably defended that &lt;span class=&quot;math-container&quot;&gt;$x = 2$&lt;/span&gt; is a valid solution of &lt;span class=&quot;math-container&quot;&gt;$x + \sqrt{7 - 3x} = 1$&lt;/span&gt; alongside &lt;span class=&quot;math-container&quot;&gt;$x = -3$&lt;/span&gt;? In other words, if at an exam a student only produces &lt;span class=&quot;math-container&quot;&gt;$x = -3$&lt;/span&gt;, could they protest if full points are not awarded, even though the teacher taught them that both are solutions? After all, even though mathematical truth is non-negotiable, definitions and conventions are, and to a certain extent the teacher is free to fix these.&lt;/p&gt;&#xA;" OwnerUserId="20450" LastEditorUserId="205" LastEditDate="2023-05-10T17:59:34.073" LastActivityDate="2023-05-11T08:28:27.760" Title="Can it be defended that $\sqrt 4$ is both $2$ and $-2$ (and likewise for general square roots)?" Tags="|secondary-education|algebra|" AnswerCount="10" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="25678" PostTypeId="2" ParentId="25663" CreationDate="2022-10-04T23:35:50.240" Score="2" Body="&lt;p&gt;Frankly, I'm surprised no one else has mentioned this already:&lt;br /&gt;&#xA;&lt;strong&gt;Speaking a language and doing math are two very different skills&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When you speak a language, you express some ideas of yours in a more or less precise way, in a more or less elaborate manner, in a more or less eloquate tone, etc. &lt;strong&gt;Every single category you apply to language is &lt;em&gt;imprecise&lt;/em&gt;&lt;/strong&gt;. There are some hard and fast rules in the grammar, but do you make your grammar teacher happy as you speak? You &lt;em&gt;try&lt;/em&gt; to write texts with good grammar, but I have yet to meet a person who always talks in full sentences... Nevertheless, we typically have no problem understanding the intended meaning of the imperfectly chosen words that are bound together with broken grammar which some person utters at us. And the grammar is the most precise thing about language to begin with.&lt;/p&gt;&#xA;&lt;p&gt;By contrast, &lt;strong&gt;mathematics &lt;em&gt;requires&lt;/em&gt; perfect(!) precision&lt;/strong&gt;. As a math student, you must learn to apply text transformations without error. Now, you may get most of the points for doing the right steps in an exam where you replaced a 42 with a 24 by accident, but whatever your final result is, it's simply wrong. End of the story. Worse than worthless. Dangerous.&lt;/p&gt;&#xA;&lt;p&gt;That means, that it is plain impossible to learn mathematics by trial and error. You'd only produce errors. And guess what, that's the experience of all those math haters out there. They tried to learn math the same way they'd learn a language, and they failed abysmally. Learning math requires a totally different approach: The approach of &lt;em&gt;knowing&lt;/em&gt; the rules. And only applying the rules that are &lt;em&gt;known&lt;/em&gt; to be applicable. And do so 100% correctly. And, as any math lover can attest, this radical &amp;quot;stick to the rules!&amp;quot; provides those that can apply it the remarkably blissful experience of: &amp;quot;Stick to the rules, and you'll &lt;em&gt;never&lt;/em&gt; get a wrong result. You might fail to find the solution, but you'll never get a wrong one!&amp;quot; &lt;strong&gt;This &amp;quot;stick to the rules&amp;quot; is not what we are born to do. We have no intuition about it. We need to understand its necessity, to practice it, and to exercise it until it's a learned reflex to stick to the rules whenever we use math&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is the key difference between learning the exact sciences and the &amp;quot;softer&amp;quot; subjects like languages.&lt;/p&gt;&#xA;" OwnerUserId="16293" LastActivityDate="2022-10-04T23:35:50.240" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25679" PostTypeId="2" ParentId="25677" CreationDate="2022-10-05T13:00:27.253" Score="18" Body="&lt;p&gt;Sure, you can define &lt;span class=&quot;math-container&quot;&gt;$\sqrt{4}$&lt;/span&gt; to be &lt;span class=&quot;math-container&quot;&gt;$\pm 2$&lt;/span&gt; if you like. The math you teach this way will be internally consistent.&lt;/p&gt;&#xA;&lt;p&gt;You can also teach students that the symbol &amp;quot;@&amp;quot; means multiplication, and you can teach students to write fractions using ordered pairs bracketed by semicolons. &lt;span class=&quot;math-container&quot;&gt;$;4,5; @ ;1,2; = ;2,5;$&lt;/span&gt; &amp;quot;can be defended&amp;quot; in the same way as &lt;span class=&quot;math-container&quot;&gt;$\sqrt{4} = \pm 2$&lt;/span&gt;; this nonstandard notation is internally consistent.&lt;/p&gt;&#xA;&lt;p&gt;However, the damage in both cases is the same; &lt;strong&gt;the students are not being taught the standard conventions and will end up harmed in the end&lt;/strong&gt; when they are unable to transfer their knowledge to other courses.&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="11" LastEditDate="2022-10-05T14:06:30.473" LastActivityDate="2022-10-05T14:06:30.473" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25680" PostTypeId="2" ParentId="25677" CreationDate="2022-10-05T19:01:23.880" Score="0" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2-4=0$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Is a quadratic equation that gives the 2 answers, a positive and negative 2.&lt;/p&gt;&#xA;&lt;p&gt;The simple question “what is the square root of this number?” is answered with just the positive result. In High School, we see this mistake all the time, left over from middle school teachers who are missing the distinction between the 2 circumstances.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-10-05T19:01:23.880" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25681" PostTypeId="2" ParentId="25677" CreationDate="2022-10-06T00:22:17.303" Score="4" Body="&lt;p&gt;There has been a change in the conventions people use in math, and I think the change probably resulted from the introduction of cheap electronic calculators. In the before times, it was common to talk about &amp;quot;multi-valued functions,&amp;quot; and to use phrases like &amp;quot;a square root of 4&amp;quot; or &amp;quot;an arctangent of 1.&amp;quot; When lots of people started to have calculators with square root keys, then inverse trig functions, and so on, it became necessary for the manufacturers to pick branches of the functions to use. In many cases there is no natural choice of the branch cut, but they just picked one. People who wrote math textbooks then had to adjust to the fact that to a student, the inverse tangent function was nothing more or less than that button on their calculator. If the teacher says that an inverse tangent of 1 is 225 degrees, and the calculator says it's 45, then the teacher must be wrong. Get him fired! So it became necessary to start putting standard branch cuts in all the math textbooks, as a side-effect of which students were forced to mindlessly memorize those choices.&lt;/p&gt;&#xA;&lt;p&gt;None of this does the world any good. For instance, when a student is adding two vectors in the cartesian plane, gets (-1,-1), and is then trying to find the angle of the vector, they have been conditioned to think that &amp;quot;the&amp;quot; answer has to be 45 degrees. (I used to suggest to my students that they get some nail polish and put a dab on their arctangent key as a reminder of this pitfall. I don't think any of them ever did.)&lt;/p&gt;&#xA;&lt;p&gt;So it's not quite right to say that the OP's kid's teacher is just violating well-established conventions. What has happened is that &lt;em&gt;both&lt;/em&gt; this teacher &lt;em&gt;and&lt;/em&gt; the current high school textbooks are oversimplifying the well-established conventions, which are still understood by working mathematicians. The conventions are actually kind of complicated and ambiguous, but that's OK. That reflects the fact that human thought is complicated and ambiguous. We have notations like &lt;span class=&quot;math-container&quot;&gt;$\pm$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mp$&lt;/span&gt;. Sometimes we write &lt;span class=&quot;math-container&quot;&gt;$x=+2$&lt;/span&gt; to emphasize the sign. These conventions make sense to humans, but they don't map nicely onto some sort of literalistic idiot-savant calculator mindset.&lt;/p&gt;&#xA;&lt;p&gt;The OP complains that havoc will ensue if a student is asked to graph &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt;. Well, not really. Actually what will happen is that the student will be encouraged to think about graphing it as a nice, complete parabola, rather than a crippled, ugly half-parabola.&lt;/p&gt;&#xA;&lt;p&gt;I would agree that what this teacher is doing is a bit out of step with the rest of the world, but what the rest of the K-12 world is doing is also a bit out of step with what mathematicians do. And in any case the really important thing is to get kids to &lt;em&gt;think&lt;/em&gt;. The ambiguity in sign of a square root isn't something that can be handled by habit. It has to be handled by &lt;em&gt;thinking&lt;/em&gt; about what the ambiguity means in whatever real-world application we're talking about.&lt;/p&gt;&#xA;" OwnerUserId="20458" LastActivityDate="2022-10-06T00:22:17.303" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25682" PostTypeId="1" CreationDate="2022-10-06T07:25:34.417" Score="2" ViewCount="101" Body="&lt;p&gt;A central skill in maths is graphing a set of data on a pair of axes with premarked divisions. Whether the axes are prelabeled with appropriate numbers or not changes the difficulty of this task a lot. I'd like to focus on making an appropriate labelling with numbers of the axes. (Of course, this is a 2d version of labelling and marking numbers on a number line.)&lt;/p&gt;&#xA;&lt;p&gt;I consider this the most difficult task asked of high school students. For want of an algorithm I find myself offering vague rules of thumb, eg&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;count the number of horizontal and vertical divisions&lt;/li&gt;&#xA;&lt;li&gt;find the range(s) of your (rounded) data&lt;/li&gt;&#xA;&lt;li&gt;try to fit these ranges onto your axes&lt;/li&gt;&#xA;&lt;li&gt;try assigning numbers to your divisions going up in 1s, 2s, 4s and 5s, (or 0.1, 0.2, ... 10,20, ..., etc) see &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Preferred_number#1-2-5_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.m.wikipedia.org/wiki/Preferred_number#1-2-5_series&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;proceed by trial and error, erasing when necessary&lt;/li&gt;&#xA;&lt;li&gt;try &amp;quot;cutting&amp;quot; your axes&lt;/li&gt;&#xA;&lt;li&gt;try to use as much available of the graph paper as possible.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Question: What is an algorithm for doing this task?&lt;/p&gt;&#xA;&lt;p&gt;(apologies for the terminology used in this question, divisions vs labelling. Are they tick marks?)&lt;/p&gt;&#xA;" OwnerUserId="998" LastActivityDate="2022-10-17T23:44:57.067" Title="Algorithm vs guidelines for choosing axes divisions when graphing data" Tags="|secondary-education|graphing|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25683" PostTypeId="2" ParentId="10442" CreationDate="2022-10-06T14:59:58.827" Score="2" Body="&lt;p&gt;For me, this picture from wikipedia helped the most in understanding limsup and liminf&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6YJq4.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6YJq4.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12691" LastActivityDate="2022-10-06T14:59:58.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25684" PostTypeId="2" ParentId="25677" CreationDate="2022-10-06T16:49:56.033" Score="4" Body="&lt;p&gt;The defensibility/correctness depends on context. And, in the first place, the idea that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{t}$&lt;/span&gt; for positive real &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; means just the positive real square root is just a &lt;em&gt;convention&lt;/em&gt;, not any sort of mathematical assertion. And the possible attraction of having a single-valued square root function cannot be sustained when considering square roots of complex numbers.&lt;/p&gt;&#xA;&lt;p&gt;So, while the current convention in K-12 and lower division undergrad math seems to be that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{t}$&lt;/span&gt; is the positive square root, no such convention persists beyond that, except in context, where it's not a mere &lt;em&gt;convention&lt;/em&gt; that that's what &lt;span class=&quot;math-container&quot;&gt;$\sqrt{t}$&lt;/span&gt; means, but is a tangible answer to some question.&lt;/p&gt;&#xA;&lt;p&gt;So, for current K-12, no, it is not a good idea to break with K-12 convention.&lt;/p&gt;&#xA;&lt;p&gt;In other situations, pretending to insist that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{t}$&lt;/span&gt; is somehow magically only the positive square root can lead to pointless ambiguities and misunderstandings.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2022-10-06T16:49:56.033" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25685" PostTypeId="2" ParentId="25677" CreationDate="2022-10-06T20:27:57.393" Score="27" Body="&lt;p&gt;This comes up over and over again on the MSE site: see for example &lt;a href=&quot;https://math.stackexchange.com/questions/26363/square-roots-positive-and-negative&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/questions/1636807/problematic-square-root/1636820#1636820&quot;&gt;here&lt;/a&gt;.  However, the question you ask is slightly different than the ones asked at those sites; rather than ask &amp;quot;Is this correct?&amp;quot;, you want to know &amp;quot;Is there a way to define things consistently so that this is correct?&amp;quot; That's a more subtle question.&lt;/p&gt;&#xA;&lt;p&gt;The real problem, in my opinion, with the convention introduced by this teacher is that it is incompatible with the way we normally use the equals sign.  If the teacher is comfortable with saying both &lt;span class=&quot;math-container&quot;&gt;$5 + \sqrt{4} = 7$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$5 + \sqrt{4} = 3$&lt;/span&gt;, does it not then follow that &lt;span class=&quot;math-container&quot;&gt;$7 = 3$&lt;/span&gt;?  Or are we now going to decide that &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; is not a transitive relation?&lt;/p&gt;&#xA;&lt;p&gt;It is &lt;em&gt;possible&lt;/em&gt; to use this convention in a consistent way.  Let &lt;span class=&quot;math-container&quot;&gt;$\sqrt{a}$&lt;/span&gt; to denote the &lt;em&gt;set&lt;/em&gt; of solutions to the equation &lt;span class=&quot;math-container&quot;&gt;$x^2 = a$&lt;/span&gt;.  Given a set of real numbers &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; and a real number &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, let &lt;span class=&quot;math-container&quot;&gt;$r + S$&lt;/span&gt; denote the set &lt;span class=&quot;math-container&quot;&gt;$\{ r + s \vert s \in S \}$&lt;/span&gt;.  Then we can write:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{4} = \{-2, 2 \}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$5 + \sqrt{4} = \{3, 7\}$$&lt;/span&gt;&#xA;Then instead of using the equals sign, we can use the symbol &lt;span class=&quot;math-container&quot;&gt;$\ni$&lt;/span&gt;, as follows:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$5 + \sqrt 4 \ni 3$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$5 + \sqrt 4 \ni 7$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Because &lt;span class=&quot;math-container&quot;&gt;$\ni$&lt;/span&gt; is not a transitive relation, we don't end up with the same problems we would get from using the &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; sign.&lt;/p&gt;&#xA;&lt;p&gt;But all of this creates an unnecessarily complicated scheme. It makes much more sense to say, as is universally done at the secondary level, that &lt;span class=&quot;math-container&quot;&gt;$x^2 = 4$&lt;/span&gt; does indeed have two square roots, positive and one negative, but that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{4}$&lt;/span&gt; denotes the &lt;em&gt;positive&lt;/em&gt; square root.  The standard convention is standard precisely because it makes things simpler, and other approaches require more complicated machinery.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2022-10-06T20:27:57.393" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25687" PostTypeId="1" CreationDate="2022-10-08T21:55:48.723" Score="2" ViewCount="590" Body="&lt;p&gt;What are the online homework systems for ordinary differential equations?&lt;/p&gt;&#xA;&lt;p&gt;If you have worked with two or more systems I would like to know how you compare them for ODEs. I have heard of &lt;a href=&quot;https://www.webassign.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WebAssign&lt;/a&gt;, &lt;a href=&quot;https://www.wileyplus.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WileyPlus&lt;/a&gt;,&#xA;&lt;a href=&quot;https://mlm.pearson.com/northamerica/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MyLab&lt;/a&gt;, &lt;a href=&quot;https://webwork.maa.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WebWork&lt;/a&gt;, and (the build your own and share it) &lt;a href=&quot;https://ximera.osu.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ximera&lt;/a&gt;. I am familiar with WebAssign.&lt;/p&gt;&#xA;&lt;p&gt;(a) Is there another system?&lt;/p&gt;&#xA;&lt;p&gt;(b) If you have worked with just one homework system did you find the coverage in it adequate for the course?&lt;/p&gt;&#xA;&lt;p&gt;(c) If you have taught ODEs out of any of the variety of texts named &amp;quot;Advanced Engineering Mathematics&amp;quot; using any online homework system I would like to hear about your experience.&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2022-10-29T13:43:49.723" LastActivityDate="2022-10-29T13:43:49.723" Title="Online homework systems for ordinary differential equations" Tags="|online-instruction|differential-equations|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25688" PostTypeId="2" ParentId="25687" CreationDate="2022-10-08T23:04:16.403" Score="6" Body="&lt;p&gt;MyOpenMath is free. It doesn't look like it has a full course, but it has something, and you could add to it.&lt;/p&gt;&#xA;&lt;p&gt;Here's what I found: Covers chapters 1-6 and 8-10 from the Trench text (Elementary Differential Equations). This is a minimal course, with only 1-5 exercises for most sections, meant to be supplemented with problems from the text.&lt;/p&gt;&#xA;&lt;p&gt;I've been writing questions for a few courses for about a year. I have been frustrated by the inadequate documentation. But as I got more experienced, it got easier to write questions. There's a forum for asking how to do things, and my questions there have usually been answered quickly and usefully.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-10-08T23:04:16.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25689" PostTypeId="2" ParentId="25687" CreationDate="2022-10-09T11:00:57.993" Score="2" Body="&lt;p&gt;For WeBWorK ODE, see: &lt;a href=&quot;https://webwork.maa.org/wiki/ModelCourses/Ordinary_Differential_Equations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Model Course&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2022-10-09T11:00:57.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25690" PostTypeId="2" ParentId="25677" CreationDate="2022-10-09T19:58:58.240" Score="-2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Can it be defended that &lt;span class=&quot;math-container&quot;&gt;$\sqrt 4$&lt;/span&gt; is both &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; (and likewise&#xA;for general square roots)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I will try to make plausible arguments defending this, although my arguments have no scientific backing, and are also obviously somewhat in jest, although there is nothing completely obviously outlandish about them.&lt;/p&gt;&#xA;&lt;p&gt;By doing this, it is clear that it will:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Generally cause confusion in students learning maths.&lt;/li&gt;&#xA;&lt;li&gt;Waste unnecessary amounts of students' time learning, at best, superfluous notation.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; might benefit weaker students in that they are more likely to give up entirely on maths more quickly; thus, they can spend more time focusing on their actual career paths.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; might benefit stronger maths students because they are likely to take the task of learning the new notation seriously, so they are less likely to waste their time on their iPhones and PS5's, the result being a more focused and hard-working maths elite.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2022-10-09T20:04:47.660" LastActivityDate="2022-10-09T20:04:47.660" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25694" PostTypeId="1" CreationDate="2022-10-15T10:52:22.467" Score="2" ViewCount="102" Body="&lt;p&gt;Is there a easy-to-use program that can make a GIF / Video in which an equation is transformed?&lt;/p&gt;&#xA;&lt;p&gt;Like this, which was make (via &lt;code&gt;PPT &amp;gt; Transition &amp;gt; Morph&lt;/code&gt;), but I would like a more automated process (I enter all the steps and everything else is automatic; if the software can deduce the middle steps, that would be great):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=vAQgIYFaRIA&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=vAQgIYFaRIA&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It would have to be available on Mac (which is what I use).&lt;/p&gt;&#xA;&lt;p&gt;I know that there is the alternative of Manim, but I just couldn't get the download working and it seems to have to do with a lot of Python. Possibly just software or something online?&lt;/p&gt;&#xA;" OwnerUserId="19960" LastEditorUserId="11" LastEditDate="2022-10-17T04:03:54.180" LastActivityDate="2022-10-17T04:03:54.180" Title="Is there a easy-to-use program that can make a GIF / Video in which an equation is transformed?" Tags="|computer-programs|equal-sign|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25695" PostTypeId="1" AcceptedAnswerId="25696" CreationDate="2022-10-16T01:44:58.473" Score="12" ViewCount="8831" Body="&lt;p&gt;Consider this elementary problem:&lt;/p&gt;&#xA;&lt;p&gt;Define an operation &lt;span class=&quot;math-container&quot;&gt;$*$&lt;/span&gt; between integers as follows: &lt;span class=&quot;math-container&quot;&gt;$a*b=ab-a+b$&lt;/span&gt;. Solve the equation &lt;span class=&quot;math-container&quot;&gt;$4*x=36$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If we give this problem to Pre-Calculus and Calculus III students (assuming that none of them have seen this type of problems before) I think Calculus III students are more likely to solve it (you may disagree, but that can be tested). If we accept that Calculus III students are more likely to solve this problem, it means Calculus III students have developed &amp;quot;certain ability&amp;quot; as a result of taking two semesters of Calculus. What would you call this &amp;quot;certain ability&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;As I created this question for a certain purpose, I do have a name for this ability, but I am curious what others would call it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit 1&lt;/strong&gt; (added later). A number of responses have questioned my statement that the said &amp;quot;ability&amp;quot; of Calculus III students was developed as a result of &lt;em&gt;taking two semesters of Calculus&lt;/em&gt;. That's fine, let's not assume that it was a result of taking two semesters of Calculus. But still, what would you call this &amp;quot;certain ability&amp;quot;, regardless of how it was developed?&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, one can perform this experiment slightly differently. One way would be to perform it every semester, varying the pedagogy used to teach the Calculus sequence, and then observe the change (if any) in results, over various semesters.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="8571" LastEditDate="2023-03-18T20:51:14.047" LastActivityDate="2023-03-18T20:51:14.047" Title="Why is a Calculus III student more likely to solve this problem?" Tags="|undergraduate-education|" AnswerCount="6" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="25696" PostTypeId="2" ParentId="25695" CreationDate="2022-10-16T02:13:36.430" Score="28" Body="&lt;p&gt;Great question! I suspect &amp;quot;the ability to accept new definitions and work with them&amp;quot; although perhaps it's more specifically &amp;quot;the ability to accept new definitions without knowing what they mean and work with them symbolically&amp;quot;. Students get lots of practice of both in the calculus sequence.&lt;/p&gt;&#xA;&lt;p&gt;That's not to speak ill of the practice of working with definitions symbolically without knowing what they mean: this was a vital part of breakthroughs on the part of (among others) Hilbert and Gödel.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-10-16T02:13:36.430" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="25697" PostTypeId="2" ParentId="25695" CreationDate="2022-10-16T17:02:47.153" Score="35" Body="&lt;p&gt;In your proposed scenario of testing pre-calculus students as well as calculus III students, you seem to be assuming that the calculus III students are, overall, equivalent in their pre-existing level of mathematical aptitude/interest to the pre-calculus students, but having received additional education and being a couple years older. However, in real life, it would almost never be the case that the two groups are equivalent in their overall level of pre-existing math aptitude. Here are a couple reasons why:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Culling Effect:&lt;/strong&gt; Many students struggle with calculus I and or II, and will therefore not be able to pass the courses that are pre-requisites for calculus III, and/or may elect to drop out of the sequence if it's not necessary for their degree. These will tend to be studentss with lower math aptitude, meaning that the students remaining in the sequence (who do make it to calculus III) are, on average, a higher-aptitude subset of the original co-hort of pre-calculus students.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Self-Selection:&lt;/strong&gt; Pre-calculus is often a requirement for students to take, even for non-STEM fields of study. Calculus I and II are sometimes (but less often) a requirement. It is even less common for calculus III to be required, especially for a non-STEM degree. The students taking calculus III therefore will tend to be students who are required to take it as part of their STEM degree, and not students majoring in, say, English or history. These students will usually have a higher math aptitude than students from the original pre-calculus cohort, as the calculus III students chose a STEM field of study partially &lt;em&gt;because&lt;/em&gt; they were good at math and found their success in math courses to be rewarding. Therefore, many students who were in the original pre-calculus cohort who were stronger or more interested in another area (compared to math), may have been likely to choose a different field of study and therefore stopped taking math classes when they fulfilled their requirements. If calculus III was not required for them, then they never ended up taking it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Trying to draw conclusions by comparing these two groups would be a form of selection bias, as you are comparing two groups that have not been selected in the same way. The calculus III students are a subset of the original pre-calculus cohort that are enriched in math aptitude and interest. This may be comparable to what's known as survivorship bias, in that you are considering the calculus III students without including their peers from pre-calculus who didn't &amp;quot;survive&amp;quot; within the sequence up to the level of calculus III.&lt;/p&gt;&#xA;&lt;p&gt;Here's a pictorial representation of what I'm talking about (each colored circle represents one student):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/askZz.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/askZz.png&quot; alt=&quot;Hypothetical Scenario&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20493" LastEditorUserId="19780" LastEditDate="2022-10-22T10:13:38.750" LastActivityDate="2022-10-22T10:13:38.750" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25698" PostTypeId="2" ParentId="25695" CreationDate="2022-10-16T17:32:36.127" Score="20" Body="&lt;p&gt;I'm not sure if there's one single &amp;quot;certain ability&amp;quot; that makes students able to solve this problem. In order to solve this problem, a student should:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Know what it means to &amp;quot;define an operation between integers.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Know what an equation like &lt;span class=&quot;math-container&quot;&gt;$a * b = a b - a + b$&lt;/span&gt; means when it's interpreted as a definition.&lt;/li&gt;&#xA;&lt;li&gt;Recognize that it is both permissible and useful to apply the substitution property to this situation.&lt;/li&gt;&#xA;&lt;li&gt;Know &lt;em&gt;how&lt;/em&gt; to apply the substitution property to this situation.&lt;/li&gt;&#xA;&lt;li&gt;Recognize that, after the substitution property has been applied, the answer to the new problem is the answer to the original problem.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;These are all distinct pieces of knowledge, and it's possible for a student to possess any four of these but not possess the fifth.&lt;/p&gt;&#xA;&lt;p&gt;That said, these pieces of knowledge are all fundamental facts about elementary algebra. (Of course, &amp;quot;fundamental&amp;quot; doesn't mean &amp;quot;easy to learn&amp;quot;!) So I think I'd summarize these as something like &amp;quot;a solid understanding of the fundamentals of elementary algebra.&amp;quot;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In response to your comment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you were to put all the 5 bullet points in a box and give it a label, what would you write on that label? &amp;quot;A solid understanding of the fundamentals of elementary algebra&amp;quot; doesn't seem to capture the concept, because as far as I know, in elementary algebra they don't teach equations like &lt;span class=&quot;math-container&quot;&gt;$4∗x=36$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That's a great question, and I'm not really sure.&lt;/p&gt;&#xA;&lt;p&gt;Let me start by clarifying that when I say &amp;quot;elementary algebra,&amp;quot; I mean &amp;quot;manipulating expressions and equations involving numbers, and especially solving equations&amp;quot;—in other words, the area of math that's usually just called &amp;quot;algebra&amp;quot; in the context of secondary education.&lt;/p&gt;&#xA;&lt;p&gt;Next, let me pretend for a moment that the problem that you wrote was:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Define a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; on integers as follows: &lt;span class=&quot;math-container&quot;&gt;$f(a, b) = a b − a + b$&lt;/span&gt;. Solve the equation &lt;span class=&quot;math-container&quot;&gt;$f(4, x) = 36$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now that I think about it, I think that there is one particular skill that a student needs in order to solve this problem. If I had to give a name to that skill, I think I would call it &lt;strong&gt;familiarity with functions&lt;/strong&gt;. This skill consists essentially of the pieces of knowledge I listed above: knowing what it means to define a function; knowing what an equation means when it's interpreted as a function definition; and knowing how to use the substitution property with a function definition.&lt;/p&gt;&#xA;&lt;p&gt;In order to solve the problem you originally described, a student needs to have familiarity with functions, and also needs to know that &amp;quot;define an operation &lt;span class=&quot;math-container&quot;&gt;$a * b$&lt;/span&gt;&amp;quot; means exactly the same thing as &amp;quot;define a function &lt;span class=&quot;math-container&quot;&gt;$f(a, b)$&lt;/span&gt;&amp;quot; (aside from the difference in notation).&lt;/p&gt;&#xA;&lt;p&gt;(If I may ramble for a few moments, I think that there are two important &amp;quot;levels&amp;quot; of familiarity with functions. The first level is what I described above—it's the level where a student is capable of reading &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$f(4, x) = 36$&lt;/span&gt;&amp;quot; as meaning &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$a = 4$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b = x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y = 36$&lt;/span&gt;.&amp;quot; The second level is where a student is capable of treating a function as an independent object in its own right, and solving problems that involve multiple values of the same function without getting confused.)&lt;/p&gt;&#xA;" OwnerUserId="930" LastEditorUserId="930" LastEditDate="2022-10-16T23:11:29.883" LastActivityDate="2022-10-16T23:11:29.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25699" PostTypeId="2" ParentId="25663" CreationDate="2022-10-16T18:35:53.990" Score="1" Body="&lt;p&gt;Maths is two things. One is a sort of &amp;quot;language&amp;quot; as you describe - which is much more unambiguous than &amp;quot;natural&amp;quot; human languages, but also more limited in scope. The other is a practice, a study. Think about the difference between speaking English, as you refer to, and carrying out debate or rhetoric in English effectively. Which one do you think will take more practice?&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, the two are mashed together, and many don't even get to know the &amp;quot;grammatical&amp;quot; parts of the mathematical &amp;quot;language&amp;quot; until very late, if at all typically unless going into a more specialized track in college or even graduate school and if not that, then only by self-study, or even really just get an informal idea for what the various pieces of &amp;quot;language&amp;quot; actually mean. For example, how many would, if you ask, know that an expression like &amp;quot;2 + 2 = 4&amp;quot; is actually a term composed of two sub-terms, namely &amp;quot;2 + 2&amp;quot; and &amp;quot;4&amp;quot;, where the &amp;quot;=&amp;quot; follows a grammatical rule that &amp;quot; = &amp;quot; is itself a term? Moreover, each term has a &lt;em&gt;type&lt;/em&gt; (namely the term formed by &amp;quot;=&amp;quot; you might say type &amp;quot;Boolean&amp;quot;, but its subterms may not, and the subterms on &amp;quot;=&amp;quot; must agree for type. And that's just one particular grammatical system that I just happen to like (look up &amp;quot;type theory&amp;quot; for the &amp;quot;type&amp;quot; idea, and even then that's far from a complete one and I've not found most fully decked-out ones entirely satisfactory); there are many others possible.&lt;/p&gt;&#xA;&lt;p&gt;I would suggest you can learn mathematical &amp;quot;language&amp;quot;, in this regard, without too much difficulty indeed, and you should also be able to understand individual mathematical concepts, i.e. the definitions of words, by themselves if you are conveyed the underlying intended intuition and semantics instead of a banal formalism (for example, &amp;quot;+&amp;quot; is a function that is constructed the way it is so that it can be used to represent quantifying aggregation with conservation). The real skill is in the mathematical &lt;em&gt;problem solving&lt;/em&gt;, i.e. actually &lt;em&gt;using&lt;/em&gt; them - e.g. determining whether &amp;quot;2 + 2&amp;quot; stands for the same number as &amp;quot;1 + 3&amp;quot; and thus whether the assertion &amp;quot;2 + 2 = 1 + 3&amp;quot; is true or false. This takes practice for the same reason it takes practice to do any other form of investigation or inquiry, and you have to learn the relevant methods and tools as well as develop the relevant facility in that regard.&lt;/p&gt;&#xA;&lt;p&gt;Finally, even regarding the language part, maths' language is always, due to its more specialized expressive power, in the position of learning that a &amp;quot;second language&amp;quot; would occupy. And those always take more effort - it's only your first language that comes naturally and &amp;quot;effortlessly&amp;quot; when very young, and even then that's not entirely so since while you can get &amp;quot;workable facility&amp;quot; in it effortlessly, but it still takes practice to get &lt;em&gt;great&lt;/em&gt; at it.&lt;/p&gt;&#xA;" OwnerUserId="7650" LastActivityDate="2022-10-16T18:35:53.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25700" PostTypeId="2" ParentId="25695" CreationDate="2022-10-17T05:24:33.307" Score="12" Body="&lt;p&gt;I hypothesize this is due to &lt;strong&gt;notation&lt;/strong&gt;, particularly &lt;strong&gt;formula notation&lt;/strong&gt;: Calculus III students have been exposed to much more sophisticated notation than students in the novice courses. Calculus III students have seen and used notations like multiple integrals, higher order derivative notations, vector notation, cross product notation, gradients, etc. This work develops a certain comfort level with notation that earlier students are less likely to have. Most importantly, Calculus III students are used to dealing with formula notation, like cross product formula, chain rule, divergence theorem etc. and know how to apply them. This skill of application of formula is the key cognition involved in solving this. Also, I have seen questions like this while tutoring for standardized tests like the SATs, and it is mostly the students who are uncomfortable with formula notation that struggle with these types of '&lt;em&gt;fictious operator&lt;/em&gt;' problems.&lt;/p&gt;&#xA;" OwnerUserId="20496" LastEditorUserId="20496" LastEditDate="2022-11-02T00:28:42.287" LastActivityDate="2022-11-02T00:28:42.287" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25702" PostTypeId="2" ParentId="25682" CreationDate="2022-10-17T23:35:40.910" Score="1" Body="&lt;p&gt;What has to be understood by the students is that the particular way of labeling the axis (including not labeling them at all) should be chosen based on what function features you want to emphasize with your graph. For instance&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You want to convey the message that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is positive everywhere. Then having the horizontal axis on the graph is more than enough and the rest is pure noise.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You want to emphasize that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is defined on &lt;span class=&quot;math-container&quot;&gt;$[-1,3]\cup[5,6]$&lt;/span&gt; and maps those intervals into &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$[6,7]$&lt;/span&gt; respectively. Then marking &lt;span class=&quot;math-container&quot;&gt;$-1,3,5,6$&lt;/span&gt; on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis and &lt;span class=&quot;math-container&quot;&gt;$0,1,6,7$&lt;/span&gt; on the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; axis is what you want.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You want to see the detailed behavior of &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2$&lt;/span&gt; near the point &lt;span class=&quot;math-container&quot;&gt;$x=100$&lt;/span&gt;. Then you choose the bounding box &lt;span class=&quot;math-container&quot;&gt;$[98,102]\times [9600,10500]$&lt;/span&gt;, say with 10 equally spaced ticks in each (or equally spaced &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-ticks and marked values of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; at them on the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You want to show that &lt;span class=&quot;math-container&quot;&gt;$f(x)=10000+x^2$&lt;/span&gt; is huge on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;. Draw the horizontal axis through &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and mark the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-scale in thousands.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You want to show that the same function doesn't change much on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;. Draw the horizontal axis at &lt;span class=&quot;math-container&quot;&gt;$y=10000$&lt;/span&gt; and label &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis in &lt;span class=&quot;math-container&quot;&gt;$0.1$&lt;/span&gt;-s.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;And so on, and so forth. We draw a graph of a function to quickly tell a story about it, not just to exercise in connecting dots by a smooth line (though the latter skill requires some training as well). So, before choosing the scales and the bounding box, ask yourself &amp;quot;What story do I want to tell and what is the clearest way in which I can do it?&amp;quot;. Or, when looking at the graph, ask &amp;quot;What are the features of the function I see immediately from that drawing and what are the features that would be hard to discern from it?&amp;quot;. It is like choosing words in a sentence. There is a big difference between &amp;quot;I took a walk&amp;quot;, &amp;quot;I took a long walk in a beautiful Fall forest&amp;quot;, &amp;quot;I took a 7 mile walk on a hilly terrain with average inclination 5%&amp;quot; and &amp;quot;I took a 3 hour walk despite the fact that I should rather spend that time preparing my classes&amp;quot; while they all can describe the same full reality &amp;quot;I took a 3 hour walk of total length 7 miles (which I consider long) in a beautiful Fall forest located on a hilly terrain with average inclination 5%, though I should rather spend that time preparing my classes&amp;quot;. The same applies to mathematical communication and, in particular, to mathematical communication using pictures. The real objective is to say exactly as much as you want to say in the clearest possible manner. Alas, as far as I know, there are no ready algorithms for that either in mathematics on in the natural language.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastEditorUserId="19946" LastEditDate="2022-10-17T23:44:57.067" LastActivityDate="2022-10-17T23:44:57.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25703" PostTypeId="1" AcceptedAnswerId="25723" CreationDate="2022-10-18T00:09:32.947" Score="8" ViewCount="326" Body="&lt;p&gt;When I teach undergraduate level math course, I make slides with a lot of white space.&#xA;In my classes, I write on my slides with an iPad and project the screen to a big TV.&#xA;After a class, I provide annotated slides for students.&#xA;In this way, students don't have to take notes.&#xA;They can just read my slides with annotations.&lt;/p&gt;&#xA;&lt;p&gt;But yesterday a colleague commented that it is better to force students to take notes so they pay more attentions. So I was thinking maybe I should stop providing annotated notes.&#xA;But then students can always take photos or borrow notes from someone who takes them.&lt;/p&gt;&#xA;&lt;p&gt;So should we require students to take notes in class?&lt;/p&gt;&#xA;" OwnerDisplayName="user19945" LastActivityDate="2022-10-27T14:43:41.467" Title="Should we require students to take notes in class?" Tags="|undergraduate-education|lecture-notes|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25705" PostTypeId="2" ParentId="25695" CreationDate="2022-10-18T12:11:19.543" Score="8" Body="&lt;p&gt;To me this seems much more straightforward than the question and other answers suggest. I would describe the problem itself as &amp;quot;abstract algebra&amp;quot;, with some overlap with linear algebra. In any case, I don't see knowledge of calculus itself as relevant to the problem.&lt;/p&gt;&#xA;&lt;p&gt;However, for students to progress to calculus III, they must have either had a solid foundation in algebra before starting calculus, or they must have been able to fill in their knowledge of algebra to succeed in calculus I and II. Personally, I understood algebra a lot better after taking a year of calculus.&lt;/p&gt;&#xA;&lt;p&gt;In short, my answer to the question is:&lt;/p&gt;&#xA;&lt;h2&gt;The calculus III students are better at &lt;em&gt;algebra&lt;/em&gt; than the pre-calculus students.&lt;/h2&gt;&#xA;" OwnerUserId="5176" LastActivityDate="2022-10-18T12:11:19.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25706" PostTypeId="2" ParentId="21164" CreationDate="2022-10-18T12:58:05.847" Score="1" Body="&lt;p&gt;UN's 17 goals for sustainability was mentioned in another answer. UNESCO has a new publication that you might find useful:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://Mathematics%20for%20action:%20supporting%20science-based%20decision-making&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics for action: supporting science-based decision-making&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It doesn't go deep into the mathematics. It's mostly about examples of mathematical modelling in contexts having to do with sustainability.&lt;/p&gt;&#xA;" OwnerUserId="39" LastActivityDate="2022-10-18T12:58:05.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25707" PostTypeId="2" ParentId="25695" CreationDate="2022-10-19T03:04:37.387" Score="-3" Body="&lt;p&gt;Intelligence. With familiarity with math being an additional factor.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2022-10-19T03:04:37.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25708" PostTypeId="2" ParentId="25558" CreationDate="2022-10-19T07:01:46.227" Score="0" Body="&lt;p&gt;I am not aware of any research on language and probability in particular (but neither am I an expert), but you might want to get familiar with research on language and mathematics (mostly number sense and counting) to get a sense of the field.&lt;/p&gt;&#xA;&lt;p&gt;For example, the book &lt;a href=&quot;https://doi.org/10.1515/9783110661941&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1515/9783110661941&lt;/a&gt; is not old and is free, and has several articles that touch upon language and mathematics learning or understanding.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Fritz, A., Gürsoy, E. &amp;amp; Herzog, M. (2021). Diversity Dimensions in Mathematics and Language Learning: Perspectives on Culture, Education and Multilingualism. Berlin, Boston: De Gruyter. &lt;a href=&quot;https://doi.org/10.1515/9783110661941&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1515/9783110661941&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-10-19T07:01:46.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25709" PostTypeId="1" CreationDate="2022-10-19T08:47:04.013" Score="12" ViewCount="3437" Body="&lt;p&gt;I am teaching calculus &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; this semester, and I saw the following motivation for using induction by another teacher:&lt;/p&gt;&#xA;&lt;p&gt;Since we can't go over &amp;quot;manually proving&amp;quot; all claims &lt;span class=&quot;math-container&quot;&gt;$1,2,\ldots$&lt;/span&gt; and actually get to the finish line in a finite time, we use induction to prove &amp;quot;all the claims at once&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But this motivation feels a bit inaccurate and dishonest to me, since proving the &amp;quot;induction step&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$T(n) \Rightarrow T(n+1)$&lt;/span&gt; also amounts to proving an infinite number of claims, one for each &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So, we don't really avoid the need to prove &lt;em&gt;some&lt;/em&gt; claim  &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; at once, by some argument.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus I guess that a more refined presentation would be like this: We have a sequence of claims we want to prove &lt;span class=&quot;math-container&quot;&gt;$T(n)$&lt;/span&gt;. It may be hard to this directly for an arbitrary &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, so we instead &amp;quot;replace&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$T(n)$&lt;/span&gt; with another sequence of claims &lt;span class=&quot;math-container&quot;&gt;$T(n) \Rightarrow T(n+1)$&lt;/span&gt; which we &lt;strong&gt;can&lt;/strong&gt; prove &amp;quot;once and for all&amp;quot; (and &lt;span class=&quot;math-container&quot;&gt;$T(1)$&lt;/span&gt; of course).&lt;/p&gt;&#xA;&lt;p&gt;I think most students won't notice the subtle &amp;quot;sweeping under the rug&amp;quot; here, if not explicitly stated, but I think it can be beneficial to remark honestly on what we are actually doing here. On the other hand this might confuse them unnecessarily.&lt;/p&gt;&#xA;&lt;p&gt;What do you think?&lt;/p&gt;&#xA;" OwnerUserId="20511" LastActivityDate="2022-10-27T19:36:55.650" Title="Does induction really avoid proving an infinite number of claims?" Tags="|mathematical-pedagogy|proofs|student-motivation|concept-motivation|induction|" AnswerCount="7" CommentCount="12" FavoriteCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25710" PostTypeId="2" ParentId="25709" CreationDate="2022-10-19T10:41:49.547" Score="13" Body="&lt;p&gt;The &amp;quot;avoidance of proving an infinite number of claims&amp;quot; explanation for the need for induction has not yet resonated with me because there are obviously many universally quantified statements over infinite domains that are proven without induction (e.g., for all &lt;span class=&quot;math-container&quot;&gt;$n \in \mathbb{N}$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is divisible by 4, then &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is even).&lt;/p&gt;&#xA;&lt;p&gt;Instead, they might think of it as an approach to enabling repeated application of some step in a way that makes it clear what inference rules are being used. Here's an example starting with a student's flawed proof from &lt;a href=&quot;https://matheducators.stackexchange.com/questions/20666/grade-on-proving-a-1-a-2-a-n-le-a-1a-2-a-n&quot;&gt;a previous question&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In an Advanced Calculus course, students were asked to prove&#xA;&lt;span class=&quot;math-container&quot;&gt;$|a_1+a_2+...+a_n|≤|a_1|+|a_2|+...+|a_n|$&lt;/span&gt;&#xA;for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; real numbers &lt;span class=&quot;math-container&quot;&gt;$a_1,a_2,...a_n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I am teaching assistant for this course, and one of my students replied&#xA;like this:&#xA;&lt;span class=&quot;math-container&quot;&gt;$|a_1 + a_2|≤|a_1|+|a_2|$&lt;/span&gt; by triangular inequality.&#xA;Then, &lt;span class=&quot;math-container&quot;&gt;$|a_1+(a_2+...+a_n)|≤|a_1|+|a_2+...+a_n|$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$|a_1|+|a_2+(a_3+...+a_n)|≤|a_1|+|a_2|+|a_3+...+a_n|$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Repeating this,&#xA;&lt;span class=&quot;math-container&quot;&gt;$|a_1+a_2+...+a_n|≤|a_1|+|a_2|+...+|a_n|$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The student recognizes that the base case is just the triangle inequality for two real numbers (assuming this has been proven in the course and can just be cited if the hypotheses are true and that the &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt; case is trivial). Even though it is intuitively clear in this case that &amp;quot;repeated application of the triangle inequality&amp;quot; is the crux of the idea, what &lt;em&gt;formally valid inference rules&lt;/em&gt; are being applied in the &amp;quot;repeating this&amp;quot; step?&lt;/p&gt;&#xA;&lt;p&gt;We can get around this by &lt;strong&gt;assuming &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt;&lt;/strong&gt;, introducing one extra real number labeled &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;|a_1+a_2+ \cdots +a_{n+1}| &amp;amp;= |b + a_{n+1}| &amp;amp;(\text{Sum of real numbers is real}) \\&#xA;|a_1+a_2+ \cdots +a_{n+1}| &amp;amp;\leq |b| + |a_{n+1}| &amp;amp;(\text{Triangle Inequality for Two Real Numbers}) \\&#xA;|b| &amp;amp;\leq |a_1| + \cdots + |a_n| &amp;amp;(\text{Assumption: P(n)}) \\&#xA;|b| + |a_{n+1}| &amp;amp;\leq |a_1| + \cdots + |a_n| + |a_{n+1}| &amp;amp;(\text{Add a real number to both sides}) \\&#xA;|a_1+a_2+ \cdots + a_{n+1}| &amp;amp;\leq |a_1| + \cdots + |a_n| + |a_{n+1}| &amp;amp;(\text{Transitivity of Order Relation}) \\&#xA;\end{align}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then, it bears repeating to the students that we have two facts:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The base case, &lt;span class=&quot;math-container&quot;&gt;$P(1)$&lt;/span&gt; (trivial) or &lt;span class=&quot;math-container&quot;&gt;$P(2)$&lt;/span&gt; (triangle inequality for two real numbers).&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall n (P(n) \implies P(n+1))$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;and from these facts we can now conclude &lt;span class=&quot;math-container&quot;&gt;$\forall n P(n)$&lt;/span&gt; (writing it like this highlights the difference between #2 and what we were looking to prove). We were able to do this without any &amp;quot;repeating this...&amp;quot; or &amp;quot;by a similar argument...&amp;quot;  , which are not really valid inference rules, in the middle of it all.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastEditorUserId="1230" LastEditDate="2022-10-19T10:53:15.210" LastActivityDate="2022-10-19T10:53:15.210" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25711" PostTypeId="2" ParentId="25709" CreationDate="2022-10-19T23:31:11.260" Score="11" Body="&lt;p&gt;My personal take on this, is that all the talk about &amp;quot;infinite this, and infinite that&amp;quot; is only mudding the waters. The emphasis should not be on wanting to prove &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; for all &lt;strong&gt;all&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, but rather that you want to prove it for &lt;strong&gt;each&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The way I see induction is that the following happens. You know that &lt;span class=&quot;math-container&quot;&gt;$P(1)$&lt;/span&gt; is true, and that whenever &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; is true, &lt;span class=&quot;math-container&quot;&gt;$P(n+1)$&lt;/span&gt; is true. Then we we need &lt;span class=&quot;math-container&quot;&gt;$P(100)$&lt;/span&gt;, say, we have a chain of true implications &lt;span class=&quot;math-container&quot;&gt;$$P(1)\implies P(2)\implies\cdots\implies P(100),$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$P(1)$&lt;/span&gt; is true and hence all are true. No infinity of any kind.&lt;/p&gt;&#xA;&lt;p&gt;The emphasis on infinitely many things only confuses. When you show that &lt;span class=&quot;math-container&quot;&gt;$n(n+1)$&lt;/span&gt; is even, say, you just write a proof and it works for every &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;; nobody ever emphasizes that one has just proven infinitely many statements.&lt;/p&gt;&#xA;" OwnerUserId="475" LastEditorUserId="475" LastEditDate="2022-10-20T04:24:48.267" LastActivityDate="2022-10-20T04:24:48.267" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25712" PostTypeId="1" AcceptedAnswerId="25716" CreationDate="2022-10-20T12:36:25.680" Score="5" ViewCount="120" Body="&lt;p&gt;When teaching random variables for the first time, most of us say that it is a function &lt;span class=&quot;math-container&quot;&gt;$$X : \Omega \to \mathbb{R}$$&lt;/span&gt; without any further restriction. Of course, a more general definition is to say that it is a function &lt;span class=&quot;math-container&quot;&gt;$$X : \Omega \to \mathbb{R}$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is equipped with a sigma-algebra &lt;span class=&quot;math-container&quot;&gt;$\mathcal{A}$&lt;/span&gt; and where &lt;span class=&quot;math-container&quot;&gt;$X^{-1}(I) \in \mathcal{A}$&lt;/span&gt; for all interval &lt;span class=&quot;math-container&quot;&gt;$I \subset \mathbb{R}.$&lt;/span&gt; I would like to switch from the first definition to the second with my students. In order to justify the relevance of doing so, I would like to have interesting examples of random variables defined with a sigma-algebra &lt;span class=&quot;math-container&quot;&gt;$\mathcal{A}$&lt;/span&gt; which is strictly smaller than &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}(\Omega)$&lt;/span&gt;. The classical examples using coins and dices are using &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}(\Omega)$&lt;/span&gt;. Thanks for any suggestion.&lt;/p&gt;&#xA;" OwnerUserId="19276" LastEditorUserId="16250" LastEditDate="2022-10-20T17:17:06.457" LastActivityDate="2022-10-21T09:06:39.893" Title="Examples of random variables that are measurable with respect to a strict subset of $\mathcal{P}(\Omega)$" Tags="|undergraduate-education|probability|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25713" PostTypeId="2" ParentId="25709" CreationDate="2022-10-20T17:18:17.887" Score="2" Body="&lt;p&gt;As with the other answers, talk of proving an infinite number of claims is unnecessary.&lt;/p&gt;&#xA;&lt;p&gt;Rather, we should view proofs by induction as showing us something fundamental about the successor function. Now, every natural is either zero, or some number of applications of the successor function to zero. What a proof by induction gives us is a proof that, say, zero is green, and that given a green object, the successor function gives us back a green object. Hence, since all numbers are either zero or formed by the successor function, all numbers are green.&lt;/p&gt;&#xA;&lt;p&gt;In particular, the inductive step shows that the successor function preserves &amp;quot;greenness&amp;quot;. Now, this approach generalizes to any inductively/recursively defined objects.&lt;/p&gt;&#xA;&lt;p&gt;As for what distinguishes when we need induction and when we don't I refer to this helpful answer: &lt;a href=&quot;https://math.stackexchange.com/questions/1295010/what-are-statements-about-the-natural-numbers-where-induction-is-impossible-or-u&quot;&gt;https://math.stackexchange.com/questions/1295010/what-are-statements-about-the-natural-numbers-where-induction-is-impossible-or-u&lt;/a&gt;. In particular, induction seems to be unnecessary in the many divisibility cases, where the proof strategy will ultimately be to obtain some algebraic rewrites. Also, induction is not very helpful with properties such as primality- even if we can show that the successor function respects certain properties, it is not very helpful for us to have this information.&lt;/p&gt;&#xA;" OwnerUserId="20518" LastActivityDate="2022-10-20T17:18:17.887" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25714" PostTypeId="2" ParentId="25709" CreationDate="2022-10-20T17:24:49.440" Score="9" Body="&lt;p&gt;The statement&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since we can't go over &amp;quot;manually proving&amp;quot; all claims 1,2,… and actually get to the finish line in a finite time, we use induction to prove &amp;quot;all the claims at once&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is essentially accurate, but it's actually making &lt;em&gt;two&lt;/em&gt; separate points:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It's trying to motivate why it is necessary to go from countably many separate statements &lt;span class=&quot;math-container&quot;&gt;$P(0)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P(1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P(2)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P(3)$&lt;/span&gt;, ... to a single universally quantified statement &lt;span class=&quot;math-container&quot;&gt;$\forall n\in\mathbb{N}. P(n)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;It introduces the induction principle that allows you to prove &lt;span class=&quot;math-container&quot;&gt;$\forall n\in\mathbb{N}.P(n)$&lt;/span&gt; by proving the simpler statements &lt;span class=&quot;math-container&quot;&gt;$P(0)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\forall n\in\mathbb{N}.P(n) \Rightarrow P(n + 1)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The first part of the quote&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;we can't go over &amp;quot;manually proving&amp;quot; all claims 1,2,… and actually get to the finish line in a finite time&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;motivates the point (1.), whereas the second part&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;we use induction to prove &amp;quot;all the claims at once&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;hints at one specific method how to accomplish (2.).&lt;/p&gt;&#xA;&lt;p&gt;Motivating (1.) and explaining (2.) are actually two separate pedagogical tasks, both challenging in their own way.&lt;/p&gt;&#xA;&lt;h3&gt;Motivating the induction principle&lt;/h3&gt;&#xA;&lt;p&gt;The (2.) might be the easier one, because it does not include any subtleties about &amp;quot;infinitely many statements&amp;quot;, and because it's just one of Peano's axioms. Intuitively, it simplifies the problem, because it replaces a &amp;quot;complicated&amp;quot; statement &lt;span class=&quot;math-container&quot;&gt;$\forall n.P(n)$&lt;/span&gt; about &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; by a &amp;quot;much simpler&amp;quot; statement &lt;span class=&quot;math-container&quot;&gt;$\forall n. P(n) \Rightarrow P(n + 1)$&lt;/span&gt; about tiny steps &lt;span class=&quot;math-container&quot;&gt;$P(n) \Rightarrow P(n + 1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;While proving &lt;span class=&quot;math-container&quot;&gt;$P(0)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\forall n.P(n) \Rightarrow P(n+1)$&lt;/span&gt; is &lt;em&gt;sufficient&lt;/em&gt;, it's not &lt;em&gt;necessary&lt;/em&gt; for proving &lt;span class=&quot;math-container&quot;&gt;$\forall n.P(n)$&lt;/span&gt;: if, for example, one is able to assume &lt;span class=&quot;math-container&quot;&gt;$n:\mathbb{N}$&lt;/span&gt; and prove &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; directly, then this probably should be preferred in most of the cases. The quote by itself therefore does not motivate why you &amp;quot;need&amp;quot; the induction: it just introduces one additional tool specific to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;h3&gt;Motivating the necessity to go from infinitely many statements &lt;span class=&quot;math-container&quot;&gt;$P(1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P(2)$&lt;/span&gt;, ... to a single statement &lt;span class=&quot;math-container&quot;&gt;$\forall n.P(n)$&lt;/span&gt;.&lt;/h3&gt;&#xA;&lt;p&gt;The (1.)st point might be quite a bit more difficult to motivate. Below, I'm suggesting to take some inspiration from intuitionism, and emphasize the difference between &amp;quot;giving a man a fish&amp;quot; (&lt;span class=&quot;math-container&quot;&gt;$P(1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P(2)$&lt;/span&gt;, ...) vs. &amp;quot;giving a man a fishing rod&amp;quot; (&lt;span class=&quot;math-container&quot;&gt;$\forall n.P(n)$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;h4&gt;Explain it like I'm five&lt;/h4&gt;&#xA;&lt;p&gt;Let's start with a &amp;quot;physical&amp;quot; metaphor, in which intuitionistic proof terms are replaced by physical artifacts.&lt;/p&gt;&#xA;&lt;p&gt;Consider the following problem: suppose that you want to reach a certain altitude of &lt;code&gt;n&lt;/code&gt; meters above the ground. How would you explain the fundamental difference between&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the ability to build arbitrarily tall skyscrapers and&lt;/li&gt;&#xA;&lt;li&gt;interstellar spaceflight?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In the first case, being able to build arbitrarily tall skyscrapers means that&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For each given height of &lt;code&gt;n&lt;/code&gt; meters&lt;/li&gt;&#xA;&lt;li&gt;one is able to design and to build a tower of height &lt;code&gt;n&lt;/code&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In the second case, &amp;quot;spaceflight&amp;quot; means that&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;One is able to design and build a rocket that&lt;/li&gt;&#xA;&lt;li&gt;for each given distance of &lt;code&gt;n&lt;/code&gt; meters&lt;/li&gt;&#xA;&lt;li&gt;can reach the distance of &lt;code&gt;n&lt;/code&gt; meters from the surface of Earth.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Note how the order of &lt;em&gt;building&lt;/em&gt; and &lt;em&gt;choosing the &lt;code&gt;n&lt;/code&gt;&lt;/em&gt; have switched:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;for towers, you first have to pick and &lt;em&gt;fix&lt;/em&gt; an &lt;code&gt;n&lt;/code&gt;, and only then can you build the tower.&lt;/li&gt;&#xA;&lt;li&gt;for rockets, you first build a rocket, which then allows you to pick an arbitrary &lt;code&gt;n&lt;/code&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Both methods would, in a highly idealized world, allow you to reach arbitrary distance from the surface of the Earth. But in the first case, you would have to re-design and re-build the tower for each given &lt;code&gt;n&lt;/code&gt;, whereas in the second case, you would be able to build one single vehicle &lt;em&gt;once&lt;/em&gt;, which could then travel arbitrarily far.&lt;/p&gt;&#xA;&lt;p&gt;You can see how the induction principle works for a rocket:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;code&gt;P(0)&lt;/code&gt;: You can place the rocket on the surface of the Earth (without it falling over etc.)&lt;/li&gt;&#xA;&lt;li&gt;&lt;code&gt;P(n) =&amp;gt; P(n + 1)&lt;/code&gt;: If it went &lt;code&gt;P(n)&lt;/code&gt; meters, and it can fly one more meter, then it can also reach the distance of &lt;code&gt;P(n + 1)&lt;/code&gt; meters.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Thus, for a rocket, you can conclude that, in principle, it can go arbitrarily far.&lt;/p&gt;&#xA;&lt;p&gt;The induction doesn't work for towers / skyscrapers, though: it could well be that having a wooden tower that's 100 meters tall doesn't allow you to go to 101 meter, because the entire construction is on the brink of collapse. You would have to switch to brick / steel / titanium / graphene / unobtanium ... in order to build taller buildings. This is the difference between the ability to build arbitrarily tall towers and the ability to build rockets.&lt;/p&gt;&#xA;&lt;h4&gt;Explain it like I can program&lt;/h4&gt;&#xA;&lt;p&gt;If one understands the distinction between the compile time / runtime, and is working in a language where types can depend on integers, then it should be quite obvious that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Being able to write down a term of type &lt;code&gt;P(n)&lt;/code&gt; at compile time will only give you a &lt;em&gt;single value&lt;/em&gt; of runtime type &lt;code&gt;P(n)&lt;/code&gt;, but not necessarily &lt;code&gt;P(n + 1)&lt;/code&gt;, &lt;code&gt;P(n + 2)&lt;/code&gt; , ... etc.&lt;/li&gt;&#xA;&lt;li&gt;Being able to write down a term of the dependent function type &lt;code&gt;Π(n: Nat).P(n)&lt;/code&gt;/&lt;code&gt;forall(n: Nat).P(n)&lt;/code&gt; gives you a &lt;em&gt;function&lt;/em&gt; which, given an arbitrary &lt;code&gt;n: Nat&lt;/code&gt; at runtime, would produce you a value of type &lt;code&gt;P(n)&lt;/code&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Here is &lt;a href=&quot;https://stackoverflow.com/questions/67093833/how-to-ask-scala-if-evidence-exists-for-all-instantiations-of-type-parameter/67097769#67097769&quot;&gt;a very similar construction, written out in an actual programming language&lt;/a&gt;: the question showed that one could prove &lt;code&gt;P(n)&lt;/code&gt; for each &lt;code&gt;n&lt;/code&gt; by rerunning the compiler, whereas the answer showed how to prove &lt;code&gt;Forall n. P(n)&lt;/code&gt; once and for all, without having to re-run the compilation for each new &lt;code&gt;n&lt;/code&gt;.&lt;/p&gt;&#xA;&lt;h4&gt;Explain it like I understood something about formal systems&lt;/h4&gt;&#xA;&lt;p&gt;In the statement&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;For all κ, we can prove &lt;code&gt;P(κ)&lt;/code&gt;.&amp;quot; (where κ is a metavariable that can take values of natural numbers, e.g. κ = &lt;code&gt;1&lt;/code&gt; for &lt;code&gt;P(1)&lt;/code&gt;, κ = &lt;code&gt;2&lt;/code&gt; for &lt;code&gt;P(2)&lt;/code&gt; etc.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;the universal quantification takes place in the &lt;em&gt;metalanguage&lt;/em&gt; (in this case, in plain English, without the &lt;code&gt;gray background&lt;/code&gt;). It's a meta-linguistic statement about &lt;em&gt;infinitely many&lt;/em&gt; propositions &lt;code&gt;P(1)&lt;/code&gt;, &lt;code&gt;P(2)&lt;/code&gt;, &lt;code&gt;P(3)&lt;/code&gt; ... etc.&lt;/p&gt;&#xA;&lt;p&gt;In the statement&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;We can prove &lt;code&gt;∀n.P(n)&lt;/code&gt;&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;the universal quantification takes place in the &lt;em&gt;object language&lt;/em&gt;, the quantifier is part of the proposition that we are proving, and we are talking about &lt;em&gt;one single proposition&lt;/em&gt; &lt;code&gt;∀n.P(n)&lt;/code&gt;.&lt;/p&gt;&#xA;&lt;p&gt;These are clearly very different statements, because in the first case, one doesn't need universal quantifiers, whereas the second couldn't even be formulated in a system that doesn't have universal quantification.&lt;/p&gt;&#xA;&lt;p&gt;This should also clarify Steve's question &lt;a href=&quot;https://matheducators.stackexchange.com/a/25711/15593&quot;&gt;under this answer&lt;/a&gt;: it's universal quantification in both cases, but in one case, it's in the metalanguage, whereas in the other case, it's actually in the object language.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So, in all three cases, one can see that it's always&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;building infinitely many static structures vs.&lt;/li&gt;&#xA;&lt;li&gt;building one single rocket with unlimited range&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;rerunning the compiler unbounded number of times to generate separate terms of type &lt;code&gt;P(n)&lt;/code&gt; vs.&lt;/li&gt;&#xA;&lt;li&gt;running the compiler just once to generate a function of type &lt;code&gt;forall n. P(n)&lt;/code&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;making a metalanguage-claim about infinitely many separate propositions &lt;code&gt;P(1)&lt;/code&gt;, &lt;code&gt;P(2)&lt;/code&gt;, ...&lt;/li&gt;&#xA;&lt;li&gt;making a claim about a single proposition &lt;code&gt;∀n.P(n)&lt;/code&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To conclude,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since we can't go over &amp;quot;manually proving&amp;quot; all claims 1,2,… and actually get to the finish line in a finite time, we use induction to prove &amp;quot;all the claims at once&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is essentially accurate.&lt;/p&gt;&#xA;" OwnerUserId="15593" LastEditorUserId="15593" LastEditDate="2022-10-24T09:56:05.847" LastActivityDate="2022-10-24T09:56:05.847" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25715" PostTypeId="2" ParentId="24976" CreationDate="2022-10-21T01:59:00.223" Score="1" Body="&lt;p&gt;I don't see how a single &amp;quot;answer&amp;quot; could be provided so I'm simply going to write &amp;quot;I love where your head's at and here are some more fun ways to continue in that manner...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;In Linear Algebra&lt;/strong&gt;, for me one of the key things to teach in an undergraduate course is that not all vector spaces &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; are isomorphic to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; for some natural number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. Teaching this concept comes in many flavors but below is the approach I've settled on and leads itself to implicitly teaching students about functors.&lt;/p&gt;&#xA;&lt;p&gt;Given a set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and a vector space &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; we prove that the set &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Set}(X,V) = \{ f: X \to V \mid f \text{ is a function } \}$&lt;/span&gt; is itself a vector space. Below are some nice implications once you have this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$X= \{ x_1, \dots, x_n\}$&lt;/span&gt; is finite then &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Set}(X,V) \simeq \bigoplus\limits_{i=1}^n V$&lt;/span&gt; if the students learned the direct sum. In my experience I just use that when &lt;span class=&quot;math-container&quot;&gt;$V = \mathbb{R}$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Set}(X,V) \simeq \mathbb{R}^n$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$X= \{ x_1, \dots, x_n\}$&lt;/span&gt; is finite then there is a natural basis set given by the injective function &lt;span class=&quot;math-container&quot;&gt;$X \hookrightarrow \operatorname{Set}(X,\mathbb{R})$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; gets sent to the function &lt;span class=&quot;math-container&quot;&gt;$f(x')$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$x' \ne x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$x' = x$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$X = \mathbb{N}$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Set}(X,V)$&lt;/span&gt; is the set of vector valued sequences, and once again with &lt;span class=&quot;math-container&quot;&gt;$V = \mathbb{R}$&lt;/span&gt; these are the sequences from Calculus/Analysis.&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$X = U$&lt;/span&gt; is a vector space then the set of linear maps &lt;span class=&quot;math-container&quot;&gt;$\mathcal{L}(U,V)$&lt;/span&gt; is a subspace of &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Set}(U,V)$&lt;/span&gt;, and of course is isomorphic to &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Mat}_{n \times n}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$X =V= \mathbb{R}$&lt;/span&gt; then we have an infinite chain of subspaces,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \operatorname{Set}(X,V) \supset C^0(\mathbb{R})  \supset C^1(\mathbb{R}) \supset \dots  \supset C^n(\mathbb{R}) \supset \dots \supset C^{\infty}(\mathbb{R})$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$C^n(\mathbb{R})$&lt;/span&gt; are the real valued functions whose &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th derivative is continuous.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And so on... Anyway, of course if I have a set function &lt;span class=&quot;math-container&quot;&gt;$X \xrightarrow{f} Y$&lt;/span&gt;, those comfortable with category theory know that I have a linear map &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Set}(X,V) \xleftarrow{L_f} \operatorname{Set}(Y,V)$&lt;/span&gt; given by a function &lt;span class=&quot;math-container&quot;&gt;$Y \xrightarrow{g} V$&lt;/span&gt; gets assigned the composition &lt;span class=&quot;math-container&quot;&gt;$L_f(g)= (X \xrightarrow{f} Y \xrightarrow{g} V)$&lt;/span&gt;. So this is a nice way to show that you have a functor: &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Set} \to \operatorname{Vect}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Similarly, we can explore the relationships of the free functor &lt;span class=&quot;math-container&quot;&gt;$$\mathbb{R}^{(X)} = \{ f \in \operatorname{Set}(X,\mathbb{R}) \mid f(x) \ne 0 \text{ for finitely many } x\}$$&lt;/span&gt;&#xA;as a subspace of &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Set}\operatorname{Set}(X,\mathbb{R})$&lt;/span&gt; and also how it provides a functor similar to the above but now &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Set}^\mathrm{op} \to \operatorname{Vect}$&lt;/span&gt;. &lt;strong&gt;This functor is essentially the categorical version of change of basis.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have lots more places I like to sneak these things in but hopefully that was the sort of answer you were looking for!&lt;/p&gt;&#xA;" OwnerUserId="6589" LastEditorUserId="205" LastEditDate="2022-10-24T02:22:12.920" LastActivityDate="2022-10-24T02:22:12.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25716" PostTypeId="2" ParentId="25712" CreationDate="2022-10-21T09:06:39.893" Score="4" Body="&lt;p&gt;&lt;em&gt;(this answer is meant as an extension of Jochen Glueck's comment)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;As far as I know, there are two fundamental uses for the notion of measurability:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If we want to integrate functions (in the context of probability: take expectations of random variables), we have to make sure these functions are measurable with respect to the &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra on which our measure is defined. For example, the Lebesgue measure on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; cannot be meaningfully defined on all subsets of &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;, and this means that we cannot Lebesgue-integrate all functions on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;In probability, &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebras can represent information (in a sense). In this context, saying that &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is measurable with respect to some &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}$&lt;/span&gt; means that in we only need certain knowledge (represented by &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}$&lt;/span&gt;) to determine the value of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The first use is more due to a technical obstruction (our inability to define the desired measure on the whole &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}(\Omega)$&lt;/span&gt;), and it's actually irrelevant when &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is discrete. Even if &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is not discrete, this issue of measurability is not very interesting from the point of view of probability (although that's just my opinion).&lt;/p&gt;&#xA;&lt;p&gt;The second use is quite interesting and meaningful, so let me write a few words about it, using the simplest possible example.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; be the space corresponding to two dice throws:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \Omega = \{ (a,b) : a,b \in [6] \}, $$&lt;/span&gt;&#xA;where I used &lt;span class=&quot;math-container&quot;&gt;$[6]$&lt;/span&gt; to denote the set &lt;span class=&quot;math-container&quot;&gt;$\{ 1,2,\ldots,6 \}$&lt;/span&gt;. The &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}(\Omega)$&lt;/span&gt; corresponds to knowing the outcomes of both throws, and any &lt;span class=&quot;math-container&quot;&gt;$X \colon \Omega \to \mathbb{R}$&lt;/span&gt; is measurable with respect to &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}(\Omega)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But let us consider the following two &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebras:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ &#xA;\mathcal{F}_1 = \{ A \times [6] : A \subseteq [6] \}, \quad &#xA;\mathcal{F}_2 = \{ [6] \times B : B \subseteq [6] \}. &#xA;$$&lt;/span&gt;&#xA;Here, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}_1$&lt;/span&gt; corresponds to knowing the outcome of the first throw, and &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}_2$&lt;/span&gt; - the second throw. For example, the random variable&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;Y(a,b) = &#xA;\begin{cases}&#xA;0 &amp;amp; 2 \mid a \\&#xA;1 &amp;amp; 2 \nmid a&#xA;\end{cases}&#xA;$$&lt;/span&gt;&#xA;which yields the result of the first throw modulo &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, is measurable with respect to &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}_1$&lt;/span&gt; but not with respect to &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}_2$&lt;/span&gt;. Formally, this is because the sets &lt;span class=&quot;math-container&quot;&gt;$Y^{-1}(0) = \{2,4,6\} \times [6]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y^{-1}(1) = \{1,3,5\} \times [6]$&lt;/span&gt; lie in &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}_1$&lt;/span&gt; but not &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}_2$&lt;/span&gt;. Intuitively the reason is that &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; is determined by the first throw, but it's not determined by the second throw&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Consider also the example &lt;span class=&quot;math-container&quot;&gt;$Z(a,b) = a+b$&lt;/span&gt;, which is measurable with respect to &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}(\Omega)$&lt;/span&gt;, but not &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}_1$&lt;/span&gt; nor &lt;span class=&quot;math-container&quot;&gt;$\mathcal{F}_2$&lt;/span&gt; (as can be checked easily). This corresponds to the fact that knowing one of the throws is not enough to determine &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There's a nice construction of a &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra generated by a random variable:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \sigma(X) = \{ X^{-1}(U) : U \in \mathcal{B} \}, $$&lt;/span&gt;&#xA;where I used &lt;span class=&quot;math-container&quot;&gt;$\mathcal{B}$&lt;/span&gt; to denote the &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra of Borel sets in &lt;span class=&quot;math-container&quot;&gt;$\mathcal{R}$&lt;/span&gt;. Now, when &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\sigma(X)$&lt;/span&gt;-measurable it means that &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; is determined by &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; (i.e. it can be represented as a deterministic function of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;). For example&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;W(a,b) = &#xA;\begin{cases}&#xA;0 &amp;amp; 2 \mid a+b \\&#xA;1 &amp;amp; 2 \nmid a+b&#xA;\end{cases}&#xA;$$&lt;/span&gt;&#xA;is determined by &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; and indeed, it is &lt;span class=&quot;math-container&quot;&gt;$\sigma(Z)$&lt;/span&gt;-measurable.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In the applications, it is usual to think of &lt;span class=&quot;math-container&quot;&gt;$(\Omega,\mathcal{F})$&lt;/span&gt; as representing all possible outcomes of everything in the world (I'm exaggerating a bit) and introduce smaller &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebras corresponding to partial knowledge. Usually the time is involved, so for example &lt;span class=&quot;math-container&quot;&gt;$S_t$&lt;/span&gt; can represent stock prices at time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, and then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\mathcal{F}_T = \sigma(S_t : 0 \le t \le T)&#xA;$$&lt;/span&gt;&#xA;represents the knowledge of all things that happened on the market up to time &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="15367" LastActivityDate="2022-10-21T09:06:39.893" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25717" PostTypeId="2" ParentId="25709" CreationDate="2022-10-21T15:26:33.203" Score="4" Body="&lt;p&gt;I guess what I dislike about the characterization of induction quoted in the OP is that it takes a statement &lt;span class=&quot;math-container&quot;&gt;$\forall n.T(n)$&lt;/span&gt; about natural numbers, separates it into a sequence of statements &lt;span class=&quot;math-container&quot;&gt;$T(1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$T(2)$&lt;/span&gt; etc., and then seems to imply that when faced with such a sequence, the method of proof to use is induction. Rather, I would stress in my characterization that induction is a tool we allow ourselves to use when convenient. There are other methods to prove a statement of the form &lt;span class=&quot;math-container&quot;&gt;$\forall n.T(n)$&lt;/span&gt;. You can use whatever method works. Sometimes induction might be the easiest way, and sometimes it might be the only way.&lt;/p&gt;&#xA;&lt;p&gt;For instance one can prove &lt;span class=&quot;math-container&quot;&gt;$n+n$&lt;/span&gt; is even. Since the context is a calculus class and the goals are, I presume, to teach the students to construct and understand induction arguments, I will take the rules of arithmetic for granted, even though they are established by means of induction or its equivalent when &lt;em&gt;not&lt;/em&gt; taken for granted. A typical proof that &lt;span class=&quot;math-container&quot;&gt;$n+n$&lt;/span&gt; is even starts with &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$n\in{\Bbb N}$&lt;/span&gt; be arbitrary.&amp;quot;  The rest is just algebra in whatever detail you wish to use or impose: &lt;span class=&quot;math-container&quot;&gt;$n+n=(1\cdot n)+(1\cdot n) = (1+1)\cdot n = 2n$&lt;/span&gt;, which is even by definition (with the appropriate definition). A student in calculus should not construct an induction proof of this statement. (One &lt;em&gt;could&lt;/em&gt; prove it by induction, and one &lt;em&gt;could&lt;/em&gt; make the student do so; but I don't see why it's a &lt;em&gt;good&lt;/em&gt; idea.)&lt;/p&gt;&#xA;&lt;p&gt;To motivate the &lt;em&gt;usefulness&lt;/em&gt; of induction, one should look for an example in which &lt;span class=&quot;math-container&quot;&gt;$T(n+1)$&lt;/span&gt; naturally builds on &lt;span class=&quot;math-container&quot;&gt;$T(n)$&lt;/span&gt;.  A nice example for this is Maurolycus's propostion that the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; odd integers is &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt;.  The standard &amp;quot;proof-without-words&amp;quot; figure of a square on grid paper shows how the firgure for &lt;span class=&quot;math-container&quot;&gt;$T(n+1)$&lt;/span&gt; is literally constructed from the figure for &lt;span class=&quot;math-container&quot;&gt;$T(n)$&lt;/span&gt;. Generally summation formulas are a common starting point for induction proofs because it is clear how the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; terms is built on the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; terms.&lt;/p&gt;&#xA;&lt;p&gt;As regards the &amp;quot;sweeping under the rug,&amp;quot; the induction step is of the form &lt;span class=&quot;math-container&quot;&gt;$\forall n. P(n)$&lt;/span&gt;, and one needs to apply some tool for proving universal statements to it. You shouldn't introduce induction before you've introduced at least one way other than induction to prove such a statement, such as the one that starts, &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; be arbitrary....&amp;quot;  Possibly your students know this already.  In the US, where I teach, I find it helpful in calculus to remind them, as it were, what &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; be arbitrary&amp;quot; means and how it is used.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="5597" LastEditDate="2022-10-22T11:19:26.967" LastActivityDate="2022-10-22T11:19:26.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25718" PostTypeId="2" ParentId="25709" CreationDate="2022-10-21T19:29:09.817" Score="6" Body="&lt;p&gt;The most basic way to prove a claim of the form &lt;span class=&quot;math-container&quot;&gt;$$\forall x \in X: P(x)$$&lt;/span&gt;&#xA;is &lt;a href=&quot;https://en.wikipedia.org/wiki/Universal_generalization&quot; rel=&quot;noreferrer&quot;&gt;universal generalization&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Such a proof looks like this:  Let &lt;span class=&quot;math-container&quot;&gt;$x \in X$&lt;/span&gt; be chosen arbitrarily.  Argue &lt;span class=&quot;math-container&quot;&gt;$P(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is a way of proving an arbitrary number (including infinite numbers) of statements all at once.&lt;/p&gt;&#xA;&lt;p&gt;When we are dealing with a statement quantified over the natural numbers, then we have an additional tool to use which is mathematical induction.  We take it as an axiom that for any proposition &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; about natural numbers,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$[P(0) \wedge (\forall k \in \mathbb{N}: P(k) \implies P(k+1))] \implies \forall n \in \mathbb{N}: P(n)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So to prove &lt;span class=&quot;math-container&quot;&gt;$\forall n \in \mathbb{N}: P(n)$&lt;/span&gt; we can instead argue &lt;span class=&quot;math-container&quot;&gt;$P(0) \wedge (\forall k: P(k) \implies P(k+1))$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The way to argue this is to use conjunction introduction:  Argue &lt;span class=&quot;math-container&quot;&gt;$P(0)$&lt;/span&gt;, then argue &lt;span class=&quot;math-container&quot;&gt;$\forall k: P(k) \implies P(k+1)$&lt;/span&gt;.  The way we usually argue the second claim is using universal generalization:  take an arbitrary &lt;span class=&quot;math-container&quot;&gt;$k \in \mathbb{N}$&lt;/span&gt; and try to argue &lt;span class=&quot;math-container&quot;&gt;$P(k) \implies P(k+1)$&lt;/span&gt;.  Finally the way we argue this is using implication introduction:  Assume &lt;span class=&quot;math-container&quot;&gt;$P(k)$&lt;/span&gt; and attempt to deduce &lt;span class=&quot;math-container&quot;&gt;$P(k+1)$&lt;/span&gt; under this hypothesis.&lt;/p&gt;&#xA;&lt;p&gt;So we have (at least) two different proof strategies for proving universal statements about natural numbers:  &amp;quot;regular&amp;quot; universal generalization, and mathematical induction.&lt;/p&gt;&#xA;&lt;p&gt;Some examples:  we can easily prove that &lt;span class=&quot;math-container&quot;&gt;$6|(12n)$&lt;/span&gt; for all natural numbers &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; without appealing to induction.  However, I think you will find it difficult to prove that either &lt;span class=&quot;math-container&quot;&gt;$2|n$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$2|(n+1)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; except by using mathematical induction.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-10-21T19:29:09.817" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="25719" PostTypeId="2" ParentId="25709" CreationDate="2022-10-21T21:12:43.867" Score="2" Body="&lt;p&gt;I endorse Andrey Tyukin's answer but I want to add a few remarks.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Induction certainly &lt;em&gt;is&lt;/em&gt; all about proving statements for infinitely many &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; all at once. If you are proving a statement for only finitely many &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; then you don't need induction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think it possible that you are interpreting the other instructor's words more rigidly then they were intended: &amp;quot;Since we can't go over &amp;quot;manually proving&amp;quot; all claims &lt;span class=&quot;math-container&quot;&gt;$1, 2, \ldots$&lt;/span&gt; and actually get to the finish line in a finite time, we use induction to prove 'all the claims at once'.&amp;quot; If the instructor meant that statements of the form &lt;span class=&quot;math-container&quot;&gt;$\forall n. P(n)$&lt;/span&gt; can &lt;em&gt;only&lt;/em&gt; be proved by induction, then you are right to object, since that is untrue. For example, the statement &lt;span class=&quot;math-container&quot;&gt;$\forall n. Sn\ne 0$&lt;/span&gt; holds without induction, being an axiom of Peano arithmetic. But if the instructor merely meant that &lt;em&gt;some&lt;/em&gt; proofs of statements of the form &lt;span class=&quot;math-container&quot;&gt;$\forall n. P(n)$&lt;/span&gt; require induction, then that is correct, and for precisely the reason given: the inputs to an inductive proof already give a proof for any particular &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; simply by stringing implications together, but induction is the extra ingredient needed to turn this into a &amp;quot;for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&amp;quot; statement.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You are right to be concerned that the proposition &lt;span class=&quot;math-container&quot;&gt;$\forall n. P(n)\Rightarrow P(Sn)$&lt;/span&gt; is itself a &amp;quot;for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&amp;quot; statement, which either undermines the other instructor's claim (if that claim is taken to hold for all universally quantified statements), or else itself requires induction to prove, leading to an apparent infinite regress. The answer to this is twofold: (1) as mentioned in the previous remark, there do exist &amp;quot;for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&amp;quot; statements that can be proved without induction; (2) although the proof of &lt;span class=&quot;math-container&quot;&gt;$\forall n. P(n)\Rightarrow P(Sn)$&lt;/span&gt; may &lt;em&gt;appear&lt;/em&gt; not to use induction, it probably relies on many common arithmetic truths that are treated as black boxes. How do you know that these black boxes don't have proofs by induction inside? Many things we mostly take for granted &lt;em&gt;do&lt;/em&gt; require induction to prove in Peano arithmetic. The Wikipedia article on &lt;a href=&quot;https://en.wikipedia.org/wiki/Robinson_arithmetic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robinson arithmetic&lt;/a&gt; is interesting in this regard. Robinson arithmetic is essentially the fragment of Peano arithmetic without induction. The article mentions that in Robinson arithmetic there is a proof of the commutativity of the addends of any particular sum, but there is no proof of the statement &lt;span class=&quot;math-container&quot;&gt;$\forall x,y. x+y=y+x$&lt;/span&gt;. So in any particular inductive proof, it is quite likely that there's an induction buried inside of the proof of &lt;span class=&quot;math-container&quot;&gt;$\forall n. P(n)\Rightarrow P(Sn)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This is an aside, but I want to call attention to the very interesting comment of Passer By, pointing out that even in Peano arithmetic &lt;em&gt;with&lt;/em&gt; induction there are statements &lt;span class=&quot;math-container&quot;&gt;$P(n)$&lt;/span&gt; for which proofs can be given for any particular &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, but for which no proof of the statement &lt;span class=&quot;math-container&quot;&gt;$\forall n. P(n)$&lt;/span&gt; is possible.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2022-10-21T21:12:43.867" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25720" PostTypeId="2" ParentId="5706" CreationDate="2022-10-22T11:45:44.953" Score="0" Body="&lt;p&gt;Just offer it as extra credit for an outside class assugnmrnt. Let the kids be creative and see what you get. Call kids to the board or the like who have a solution. That's the literal answer to how to encourage.&lt;/p&gt;&#xA;&lt;p&gt;Although you asked about encouragement, not solution, most of the answers here ARE solution oriented. Plus in the later part of your question, you mention a method. So not really clear your objective. I think perhaps you really are more interested in methods, and would have benefitted from asking on MSE.&lt;/p&gt;&#xA;" OwnerUserId="20524" LastActivityDate="2022-10-22T11:45:44.953" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25721" PostTypeId="2" ParentId="5706" CreationDate="2022-10-24T23:51:19.293" Score="1" Body="&lt;p&gt;This may be a type of problem that is both funny and educational, provided that&lt;/p&gt;&#xA;&lt;p&gt;a) You clearly state the rules. &amp;quot;Don't use ...&amp;quot; is way too vague and can result in all sorts of arguments. Always say &amp;quot;Using only #operation list#&amp;quot; instead. Since everybody has an access to a computer now, you may go as far as to create a simple html/javascript calculator emulator that has exactly the buttons you want to keep and invite students to make a presentation of their approach in class projecting this emulation to a big screen.&lt;/p&gt;&#xA;&lt;p&gt;b) You discuss why it works in the end of the day (explaining both the underlying idea and the error estimates).&lt;/p&gt;&#xA;&lt;p&gt;c) You don't make the computation overly tedious (so I would stop at the first non-obvious digit or the next one, unless you get more essentially for free).&lt;/p&gt;&#xA;&lt;p&gt;In general it can give the students a clear idea of what is going behind the scenes when they push buttons, and you'll have an occasion to talk about everything from bisection to Taylor approximation and Newton's method. It can be made into a fun competition too: the task then is to arrive at the result in as few allowed operations as possible and whoever invokes the least number of standard operation &lt;em&gt;with a clear explanation of why the precision is achieved&lt;/em&gt; wins some prize (in that case I would not tell the right hand side until the class time but tell the left hand side of the equation and the range of possible numbers on the right, so that the error bound should hold in the whole range).&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-10-24T23:51:19.293" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25723" PostTypeId="2" ParentId="25703" CreationDate="2022-10-26T11:26:47.717" Score="1" Body="&lt;p&gt;I argue against &lt;em&gt;requiring&lt;/em&gt; note taking. Although this might in fact be effective for some students, I'm quite sure it's counterproductive for other students.&lt;/p&gt;&#xA;&lt;p&gt;Although the claim is purely anecdotal, I never learned anything from lectures in math classes until I &lt;em&gt;stopped&lt;/em&gt; taking notes. In practice now I only take notes when listening to a talk about something which I already know very well (e.g. I wrote a paper about it), or to write down specific examples, references, theorems, or when there exists no alternative source of information (as sometimes is the case in research talks).&lt;/p&gt;&#xA;&lt;p&gt;I often advise students &lt;em&gt;not&lt;/em&gt; to take notes, or to take fewer notes.&lt;/p&gt;&#xA;&lt;p&gt;Here are some more objections to note taking:&lt;/p&gt;&#xA;&lt;p&gt;It is distracting. Concentrating on copying makes it harder to think about what is being conveyed. This effect is augmented for students who don't distinguish between minor details and key ideas.&lt;/p&gt;&#xA;&lt;p&gt;There are better sources than a student's notes. These include - the teacher's notes, published books, and online videos. All of these are more polished, better structured/organized, and so forth.&lt;/p&gt;&#xA;&lt;p&gt;Many students never look at their own notes (I never did).&lt;/p&gt;&#xA;&lt;p&gt;It reinforces the notion that classes serve to learn specific things (of course they do, to a certain limited extent). Classes (the sort that might require note taking) should serve to orient and guide, to indicate key issues and ideas, but learning should occur in more active situations, or outside of class.&lt;/p&gt;&#xA;&lt;p&gt;Obviously there are arguments in favor of taking notes, but my point is really that &lt;em&gt;obligating&lt;/em&gt; note taking seems to me a terrible idea. It focuses attention on the wrong things and may impede learning for some in the audience.&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2022-10-26T11:26:47.717" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25725" PostTypeId="1" CreationDate="2022-10-26T21:47:50.017" Score="2" ViewCount="99" Body="&lt;p&gt;The setup: A 10 year old kid who knows addition up to 10+10 but is 3 times as slow with it as the rest of the class (with very low percentage of mistakes though: out of 50 problems only 1 or 2 are done wrong). The kid doesn't know multiplication yet (beyond low numbers like 1,2,3).&lt;/p&gt;&#xA;&lt;p&gt;Objective: to make the addition and multiplication tables &amp;quot;hardwired&amp;quot; in the brain (in the same sense as most of us have them: given &lt;span class=&quot;math-container&quot;&gt;$9\times 4$&lt;/span&gt; we just &amp;quot;see&amp;quot; the answer momentarily without any conscious work).&lt;/p&gt;&#xA;&lt;p&gt;What I decided to do is to write a simple HTML/javascript application that creates problems and checks the answers for everyday self-practice that runs for a prescribed time after the start button is pushed and then provides the statistics. If the answer is wrong, the problem stays on the screen and the &amp;quot;incorrect&amp;quot; count is advanced by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. Otherwise the &amp;quot;solved&amp;quot; count advances and a new problem is generated. Pretty basic stuff really...&lt;/p&gt;&#xA;&lt;p&gt;At this point the program just picks two random integers &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; in the range &lt;span class=&quot;math-container&quot;&gt;$[0,10]$&lt;/span&gt; and creates the &lt;span class=&quot;math-container&quot;&gt;$a+b=$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(a+b)-b=$&lt;/span&gt; problem independently of the previous ones (for addition/subtraction; the multiplication/division modification is obvious and I actually programmed the switch that allows to train with any combination of +-*/).&lt;/p&gt;&#xA;&lt;p&gt;The question is whether I can improve upon it. For instance,&lt;/p&gt;&#xA;&lt;p&gt;a) What if I change independent examples to some other exercise ordering, like having &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; changing only every 10th time, so that it essentially emulates row by row repetition of the table? Will it facilitate the learning or doesn't it really matter? In principle I can program anything you can describe in words, so even some &amp;quot;adaptive algorithms&amp;quot; are possible.&lt;/p&gt;&#xA;&lt;p&gt;b) Is there any point in bringing up the whole table to the screen now and then (say, after every incorrect answer) in the hope that together with scanning the particular instance some neighborhood of it will be scanned? (or a part of the table, or whatever).&lt;/p&gt;&#xA;&lt;p&gt;c) Etcetera&lt;/p&gt;&#xA;&lt;p&gt;I also see the kid once a week, so if there are some good exercises requiring a human teacher rather than a computer, I'll be interested in hearing about those as well.&lt;/p&gt;&#xA;&lt;p&gt;So, just share anything that you find worth sharing, be it some formal research or anecdotal evidence. Just make sure that you have something that supports your idea and that you find a satisfactory evidence yourself: I can try random experiments myself without asking anyone :-)&lt;/p&gt;&#xA;&lt;p&gt;The answer to an obvious question some of you may be eager to ask &amp;quot;Yes, it partially is, but I don't think it changes anything except that more time may be needed to achieve the desired result&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-10-26T21:47:50.017" Title="A computer program for learning addition and multiplication table" Tags="|interactive-teaching|" AnswerCount="0" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="25726" PostTypeId="2" ParentId="25703" CreationDate="2022-10-27T14:43:41.467" Score="0" Body="&lt;p&gt;In a nutshell:  no, don't require note taking, don't structure the course to really need it.&lt;/p&gt;&#xA;&lt;p&gt;I think this is very much something where different methods by professor or student can work, so not the canned input-output Q&amp;amp;A format, but more a salon-like different ideas type of question (as many here are).&lt;/p&gt;&#xA;&lt;p&gt;That said, my $.02 would be to use a book and drill program with a lot of scaffolding.  NOT the &amp;quot;don't need a text, just follow me&amp;quot; idea of the romantic lecturing college professor--rather more of an athletic coach or professional trainer instead.  Lectures should introduce the topic with a a question and a demonstration and then give some examples.  Rest of the time is student questions, calling on students, recitation, drill, tests, etc.&lt;/p&gt;&#xA;&lt;p&gt;I would not bother with publishing notes, especially if you are using chalk, which seems the most human way to communicate. (Not the acetate and OHP, not the projected laptop...at least you write on the fly, versus PPT, though!)&lt;/p&gt;&#xA;&lt;p&gt;Would encourage students to take notes as they see fit, but again try to use a book that is good at explanation and examples, for those who gaff it off or miss something.  And follow it.  It is better to do something good, perfectly, than to do something perfect, poorly.  And you have to realize there are other humans in the loop (the students) and they are the bottleneck.  Which is my constant reminder to the rigor-luvvr curriculum questioners.  Sometimes a shorter text, that uses more accessible language...vice impressing professor buying committees...is the way to go, but that is a tanget.&lt;/p&gt;&#xA;&lt;p&gt;As a student, I had a system (got from a friend) and has helped me in every single class:  Have a separate spiral (or bound) notebook for each class.  Drill homework, or example trials (from the text) are &amp;quot;normal pages&amp;quot;.  Lecture notes are separate pages and have inked stars at the top.  Proceed left to right and intersperse as the class progresses. Stars allow checking notes.  From the back, use this for questions (with spaces for answers).  Forces a feedback loop to ask instructor (sometimes figured out on own, later, though).  A separate file folder (or a section of a 3 hole binder, but then you need a punch and reinforcements) is used to store handouts, tests, etc.&lt;/p&gt;&#xA;&lt;p&gt;Best practice is to pre-study the material.  In this case, lecture is really a dream (a review) and notes are easy and less of a transcription.  Would encourage your students to do this, but realistically few will because of human nature.  But even there, I would emphasize to them that you have a good book, that the tests WILL be from material in the book and that the lectures are so they have one more way of learning the material, not as something they must hang on since material outside the scaffolded text is covered.   [IF you have some cute method like tabular integration, fine...add it, but not as a tested requirement, more as an &amp;quot;easier way to do things&amp;quot;.]&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I'm not a teacher.  I just took STEM math classes (and not majors courses) and watched Stand and Deliver.  And lived it.  (Sort of.)&lt;/p&gt;&#xA;" OwnerUserId="20540" LastActivityDate="2022-10-27T14:43:41.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25729" PostTypeId="2" ParentId="25677" CreationDate="2022-10-28T03:56:14.027" Score="0" Body="&lt;p&gt;If you apply the same method ( I don't go into detailed answer assuming you know how to find square roots of a complex number) of finding square root of a complex number 5 + 12i you can not avoid getting two values for √4 . Since real numbers are part of complex numbers you need to  say square roots of 4 are ± 2 and principal square root of 4 is 2.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2022-10-28T03:56:14.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25730" PostTypeId="1" AcceptedAnswerId="25732" CreationDate="2022-10-28T04:18:50.163" Score="-2" ViewCount="178" Body="&lt;p&gt;I have two issues related to finite sum of infinite series, &lt;br/&gt;&#xA;&lt;em&gt;1) How you would to describe 2 when you talk about the infinite geometric series 1+ 1/2 + 1/4 + 1/8 + .....&lt;br/&gt;&#xA;2) How you would compare using partial sums in following cases, &lt;br/&gt;&#xA;1-1+1-1+...... = 1 -(1-1+1-1+...) which gives 1-1+1-1+... = 1/2&#xA;With the method we apply to express recurring decimals as fractions ( ratio between two integers)&lt;/em&gt; &lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Which way is correct , 2 is the sum of infinite number of terms or 2 is the limit of sum of n number of terms as n tends to infinity. In exam papers I have seen students are asked to find sum of infinite number of terms. &lt;br/&gt;&#xA;2)As we know 1/2 can not be accepted as the sum of 1-1+1-.... then what about applying the same method in recurring decimals, there we take x as the recurring part and subtract two equations to remove x to get rational form of the number. Where x is sum of infinite series.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2022-10-28T14:53:18.563" Title="Finite sum of infinite series" Tags="|secondary-education|limits|series|" AnswerCount="1" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="25731" PostTypeId="2" ParentId="25641" CreationDate="2022-10-28T09:58:05.743" Score="0" Body="&lt;p&gt;What if you asked to express the given function as a composite function of two other non-composite functions. In this case you can not use the given function again because it is composite.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2022-10-28T09:58:05.743" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25732" PostTypeId="2" ParentId="25730" CreationDate="2022-10-28T14:53:18.563" Score="5" Body="&lt;p&gt;&amp;quot;Infinite sum&amp;quot; is in common use, so it should be acceptable to say that &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is the value of the infinite sum &lt;span class=&quot;math-container&quot;&gt;$1+\frac{1}{2}+\frac{1}{4}+\ldots$&lt;/span&gt;. But students need to be very clear that the value of an infinite sum is &lt;em&gt;defined to be&lt;/em&gt; the limit of the partial sums, assuming that limit exists, and that if the limit doesn't exist, the infinite sum is undefined. They should also be well aware that operations that would not change the value of a finite sum, such as rearranging terms, &lt;em&gt;can&lt;/em&gt; change the value of an infinite sum unless it is &lt;a href=&quot;https://en.wikipedia.org/wiki/Absolute_convergence&quot; rel=&quot;noreferrer&quot;&gt;absolutely convergent&lt;/a&gt;. They should be aware of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann_series_theorem&quot; rel=&quot;noreferrer&quot;&gt;Riemann rearrangement theorem&lt;/a&gt;, which says that the terms of a conditionally convergent series can be arranged to give any value whatsoever, including &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You ask to compare the calculation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x=1-1+1-1+1-\ldots=1-(1-1+1-1+\ldots)=1-x,&#xA;$$&lt;/span&gt;&#xA;which leads to &lt;span class=&quot;math-container&quot;&gt;$x=1-x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x=\frac{1}{2}$&lt;/span&gt;, and calculations like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x=0.3333\ldots=-3+3.3333\ldots=-3+10x,&#xA;$$&lt;/span&gt;&#xA;which leads to &lt;span class=&quot;math-container&quot;&gt;$x=-3+10x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x=\frac{1}{3}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The issue here is that the first sum is divergent since the partial sums do not have a limit, making the entire calculation illegitimate (What is &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; here?) In contrast, the second sum is convergent. It is only because of the convergence that we can meaningfully write &lt;span class=&quot;math-container&quot;&gt;$x=0.3333\ldots$&lt;/span&gt;. Without convergence &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; would be undefined. Once we have convergence we can prove the other needed statement, namely that &lt;span class=&quot;math-container&quot;&gt;$3.3333\ldots=10x$&lt;/span&gt;. All infinite decimal numbers can be seen to be convergent by, for example, using the &lt;a href=&quot;https://en.wikipedia.org/wiki/Direct_comparison_test&quot; rel=&quot;noreferrer&quot;&gt;comparison test&lt;/a&gt; with the convergent geometric series &lt;span class=&quot;math-container&quot;&gt;$0.999\ldots$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2022-10-28T14:53:18.563" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25734" PostTypeId="2" ParentId="25687" CreationDate="2022-10-28T16:22:32.213" Score="1" Body="&lt;p&gt;The commercial systems such as WebAssign, WileyPlus and MyLab are correlated with their textbooks and their coverage is comprehensive. If you need a lot of coverage of various topics, the price for the commercial systems may be worthwhile. You can also look into lower price options such as xyzhomework - it also has Trench questions from MyOpenMath, but they have also added some of their own.&lt;/p&gt;&#xA;" OwnerUserId="20543" LastActivityDate="2022-10-28T16:22:32.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25735" PostTypeId="1" AcceptedAnswerId="25745" CreationDate="2022-10-28T20:33:24.560" Score="4" ViewCount="226" Body="&lt;p&gt;While listening to recordings of Calculus &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; lectures, I noticed that some students get confused between showing that &amp;quot;some object &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a solution&amp;quot;, and showing that &amp;quot;every  (possible) solution must be of the form of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;,&lt;/p&gt;&#xA;&lt;p&gt;e.g. the difference between proving &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x=6$&lt;/span&gt; is a solution&amp;quot; and &amp;quot;if there is a solution, then it must be &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The problem seems to be that most equations seen in school are usually analyzed using &lt;em&gt;reversible&lt;/em&gt; logical manipulations.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I would like to have some nicely motivating and &lt;em&gt;simple&lt;/em&gt;&#xA;examples for why the students should be careful to check that all their manipulations/logical steps are bi-directional.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In particular, good examples for &lt;em&gt;equations&lt;/em&gt; would be nice. For inequalities, that's easy:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0 \Rightarrow x^2&amp;gt;0$&lt;/span&gt; is not reversible in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="20511" LastActivityDate="2023-01-08T10:12:55.757" Title="Elementary examples for non-reversible logical steps" Tags="|mathematical-pedagogy|student-motivation|logic|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25736" PostTypeId="1" CreationDate="2022-10-29T02:55:51.667" Score="0" ViewCount="85" Body="&lt;p&gt;&lt;em&gt;What kind of standard we need to maintain when preparing advanced level mathematics term test papers ?&lt;/em&gt; &lt;br/&gt;&#xA;In many cases I have seen test papers prepared using exactly past paper questions and in some occasions with slight changes in values and letters used in past papers . Since students have better chance of getting much exposure to past papers and answers they score higher marks when the papers also contains questions very much similar to those. &lt;br/&gt;&#xA;My opinion is we need to prepare students for the future and not for the past, therefore we should concentrate much on new type of questions and questions which cover the areas not considered in past papers. I suggest that is the better way to encourage students to learn the subject and not just to be exam oriented. What would you have to say about this?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2022-10-29T15:15:43.743" Title="Standardization of evaluation tests" Tags="|secondary-education|exams|" AnswerCount="1" CommentCount="8" ClosedDate="2022-10-29T15:49:59.103" ContentLicense="CC BY-SA 4.0" />
  <row Id="25738" PostTypeId="2" ParentId="25736" CreationDate="2022-10-29T15:15:43.743" Score="5" Body="&lt;p&gt;I understand the desire to test a broad knowledge of the subject versus very select cribbed questions.  All that said, study of past questions is not purely for the hope of seeing those questions again, but because they typify what is required knowledge.  I would posit that it's actually pretty HARD to be able to perform well on several (say 3+) past final exams, including with changed numbers/phrasing and NOT have learned a butt-ton about the topic!&lt;/p&gt;&#xA;&lt;p&gt;I would be wary of always pushing for something unique in that you start creating &amp;quot;puzzle&amp;quot; type questions that are testing more for intelligence (or even luck) rather than basic mastery of the subject.  Yes, calculus is broader than just the subset of what questions you have time to ask.  Yet, it's also a stereotypically familiar subject. So, the ability to answer TYPICAL questions is a feature, not a bug.&lt;/p&gt;&#xA;&lt;p&gt;Think of teaching something more mechanical like carrier airplane landings or full-twisting double backs.  Probably testing those should involve the exact task.  Now, again, calculus (or most courses) are necessarily broader than there is time to ask about.  That said, some reasonable correspondence to the course topics (and within that the most central and common and used ones) is desirable.&lt;/p&gt;&#xA;&lt;p&gt;All in all, I think rather than scratching your head on how to make the kids scratch their heads, you should construct rather &amp;quot;normal&amp;quot; (commonplace) tests that correspond to the homework.  And the drill works should correspond to what you test kids on!&lt;/p&gt;&#xA;&lt;p&gt;Train the kids, get them over the hump.  Don't worry, Nash and Ramunujan will emerge on their own, despite your not challenging them.  The bigger issue in this country is getting F kids to C's.  Getting C's to B's. And getting B's to A's.  NOT getting A+ to A+++.  Think about the number of kids you are teaching and their average capabilities.&lt;/p&gt;&#xA;" OwnerUserId="20547" LastActivityDate="2022-10-29T15:15:43.743" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25740" PostTypeId="2" ParentId="25429" CreationDate="2022-10-31T03:58:01.173" Score="1" Body="&lt;p&gt;I'll start with something that is not a new game per se but a small remark about what constitutes &amp;quot;a correct answer&amp;quot; here.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;best&amp;quot; strategy in such games is almost always a randomized one rather than a deterministic one. Let's simplify your third game to the extreme where the answer is still non-obvious: you have 3 contestants each picking either 0, or 1, or 2 (same rules). You want to maximize your chance to win. Suppose that you choose 0 with probability &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, 1 with probability &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;, 2 with probability &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, and everybody else does the same with probabilities &lt;span class=&quot;math-container&quot;&gt;$P,Q,R$&lt;/span&gt; respectively. Then your opponents choose 00 with probability &lt;span class=&quot;math-container&quot;&gt;$P^2$&lt;/span&gt;, 11 with probability &lt;span class=&quot;math-container&quot;&gt;$Q^2$&lt;/span&gt;, 22 with probability &lt;span class=&quot;math-container&quot;&gt;$R^2$&lt;/span&gt;, 01 with probability &lt;span class=&quot;math-container&quot;&gt;$2PQ$&lt;/span&gt;, 02 with &lt;span class=&quot;math-container&quot;&gt;$2PR$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2QR$&lt;/span&gt;, so your chance to win is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;P^2(q+r)+Q^2(p+r)+R^2(p+q)+2QRp&#xA;\\&#xA;=(Q^2+R^2+2QR)p+(P^2+R^2)q+(P^2+Q^2)r\,.&#xA;$$&lt;/span&gt;&#xA;If the maximum of this expression in &lt;span class=&quot;math-container&quot;&gt;$p,q,r$&lt;/span&gt; is attained at &lt;span class=&quot;math-container&quot;&gt;$P,Q,R$&lt;/span&gt;, then one cannot improve his chances &lt;em&gt;unilaterally&lt;/em&gt; when everybody else plays &lt;span class=&quot;math-container&quot;&gt;$P,Q,R$&lt;/span&gt;. This is what is called &amp;quot;Nash equilibrium&amp;quot;. (Note, however, that certain ways of cooperation of two players can kill the third one: just choose 01 collectively). Here it happens when &lt;span class=&quot;math-container&quot;&gt;$P=\frac{\sqrt 3}{\sqrt 3+2}, Q=R=\frac{1}{\sqrt 3+2}$&lt;/span&gt;, so all expressions in parentheses are the same and you have the chance &lt;span class=&quot;math-container&quot;&gt;$\frac 4{7+4\sqrt 3}\approx 0.287$&lt;/span&gt; to win no matter how you play &lt;em&gt;if everybody else is playing those &lt;span class=&quot;math-container&quot;&gt;$P,Q,R$&lt;/span&gt;&lt;/em&gt;, so you have no real incentive to deviate from these common values if all other people stick to them.&lt;/p&gt;&#xA;&lt;p&gt;This simple example can be explained to reasonably bright high school kids, so it may be worth discussing it before letting them to play the actual games of this type.&lt;/p&gt;&#xA;&lt;p&gt;As for the games themselves, you seem to be looking for something that is symmetric and satisfies the property that making the common deterministic choice makes everybody a loser, so unilateral deviating from this strategy is beneficial. Having these two properties in mind, you can design as many games as you wish. Almost all such games can be described by the scheme&lt;/p&gt;&#xA;&lt;p&gt;Everybody makes a choice &lt;span class=&quot;math-container&quot;&gt;$x_i$&lt;/span&gt; and the (unique?) minimizer of &lt;span class=&quot;math-container&quot;&gt;$F(x_i;x_1,\dots,x_n)$&lt;/span&gt; wins, where &lt;span class=&quot;math-container&quot;&gt;$F(y;x_1,\dots x_n)$&lt;/span&gt; is a symmetric in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-variables function such that &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt; is never a minimum of &lt;span class=&quot;math-container&quot;&gt;$F(y,x,\dots, x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here are some other possibilities to implement that idea:&lt;/p&gt;&#xA;&lt;p&gt;Everybody writes down names of some other people in the group (as many as he wants). The number of points one receives is the number of people not in his list who wrote his name plus the number of people on his list who did not write his name. The highest score wins.&lt;/p&gt;&#xA;&lt;p&gt;This is of the type &amp;quot;Maximize the lack of reciprocity&amp;quot; (you can act in a certain number of ways towards every other person and your goal is to maximize the number of people who do not act in the same way towards you; I leave the analysis of this one with 3 people to you).&lt;/p&gt;&#xA;&lt;p&gt;Democratic election in Wonderland: The students are split into two equal size parties, one supporting Tweedledee and another Tweedledum. They can vote (after a secret party meeting, if you want it to be a team game) in any way they want but nobody can abstain. The score of the candidate is &lt;span class=&quot;math-container&quot;&gt;$\min(12q,5-3q)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is the portion of people voting for him (so the optimal score is achieved by &lt;span class=&quot;math-container&quot;&gt;$q=1/3$&lt;/span&gt; while &lt;span class=&quot;math-container&quot;&gt;$1/6$&lt;/span&gt; loses to &lt;span class=&quot;math-container&quot;&gt;$5/6$&lt;/span&gt;, which in principle allows each party to win if they know exactly what the other party is doing by either making their candidate score exactly &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt; or making the opponent score exactly &lt;span class=&quot;math-container&quot;&gt;$1/6$&lt;/span&gt;, say. This is the only property of the score function one needs to make the game really interesting.) The candidate with the higher score gets elected (you can adjust the coefficients a bit depending on the number of players to make a tie impossible, so that you'll always have a winner. With my function all you need is that the total number of students be not a multiple of 9)&lt;/p&gt;&#xA;&lt;p&gt;I wish something like that were implemented into the USA voting system instead of the mathematically boring &amp;quot;majority rule&amp;quot; with the trivial optimal strategy of &amp;quot;voting for your preferred candidate no matter what&amp;quot;. Then the smarter people would have a chance to win even if they are not a majority (which they almost never are) but not if they are too few. Of course, if there are too many people, you may have more candidates but the score function should be chosen differently with the same underlying idea: each party should be able to win if it knows how the members of other parties are going to vote).&lt;/p&gt;&#xA;&lt;p&gt;This was just to show you some possibilities. The key point is that, since the best strategies are probabilistic in nature, each such game should be played several times, if possible, to see who can really outsmart whom.&lt;/p&gt;&#xA;&lt;p&gt;Just my 2 cents :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-10-31T03:58:01.173" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25741" PostTypeId="2" ParentId="3950" CreationDate="2022-10-31T12:32:53.063" Score="0" Body="&lt;p&gt;Try&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0821809539&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Selected-Problems-Translations-Mathematical-Monographs/dp/0821809539&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't call that entire book &amp;quot;really hard&amp;quot; (if I were writing a &amp;quot;killer problem&amp;quot; book in undergraduate analysis, I would include only about 10% of what is there), but you have a wide selection of problems in it of varying degrees of difficulty, many of which require some thinking rather than following a standard algorithm, so I'm pretty sure that with some preliminary work, you will be able to choose a subset of problems that is just right for your purposes. Hints and solutions are provided for most but not all problems. The second Russian edition is noticeably better than the English translation of the first one, so if you can read Russian, I recommend getting that one (with a fat cat looking for a tiny mouse under the hyperbola on the front cover) for yourself.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-10-31T12:32:53.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25742" PostTypeId="1" CreationDate="2022-10-31T20:28:12.623" Score="5" ViewCount="325" Body="&lt;p&gt;(I'm not sure if this counts as on-topic for this StackExchange community. If not, please let me know and I'll delete the question!)&lt;/p&gt;&#xA;&lt;p&gt;When teaching math in front of a class, I often use a blackboard and chalk. Unfortunately, this means that by the end of the class, my hands are full of chalk dust. I don't want to leave my things in the classroom while I go wash my hands, so I have to pack up my things, which means I get chalk on my notes, my bag, my jacket, etc.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have any suggestions on how to use a blackboard while avoiding getting chalk on all my stuff?&lt;/p&gt;&#xA;" OwnerUserId="20556" LastActivityDate="2022-11-09T16:02:35.867" Title="How to keep your hands clean while using a blackboard?" Tags="|board-use|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25743" PostTypeId="6" CreationDate="2022-10-31T21:24:10.407" Score="0" Body="&lt;h1&gt;What do I have to do as moderator? Clean stuff up? I like me a good spring cleaning. And spring is just around the corner... But seriously, I think I can have a good go at this, and I guess it's a way to give back to the community.&lt;/h1&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2022-10-31T21:24:10.407" LastActivityDate="2022-10-31T21:24:10.407" CommentCount="1" CommunityOwnedDate="2022-10-31T21:24:10.407" ContentLicense="CC BY-SA 4.0" />
  <row Id="25745" PostTypeId="2" ParentId="25735" CreationDate="2022-11-01T01:27:34.287" Score="6" Body="&lt;p&gt;I think calling some operations bi-directional is not the clearest way to put it. What I think you mean is that while for all functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; in the domain of &lt;span class=&quot;math-container&quot;&gt;$f,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$x=y \implies f(x)=f(y),$$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$$f(x)=f(y) \;\not\!\!\!\implies x=y.$$&lt;/span&gt; I think a better name would be &lt;em&gt;invertible&lt;/em&gt; operations. If your students have been introduced to the concept of function inverses (I would expect so in a calculus class), then operation inverses should be an easy extension.&lt;/p&gt;&#xA;&lt;p&gt;The inverse of adding one is subtracting one. The inverse of multiplying by three is dividing by three. However, not all operations have inverses--at least, not always.&lt;/p&gt;&#xA;&lt;h4&gt;Operations with restricted domains eliminate solutions&lt;/h4&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x^2 = x$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{x^{2}}{x} = \frac{x}{x}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x = 1$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Most students will be able to look at the original equation and see that &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; is a solution as well. Ask them what happened to zero in the worked out solution. One way to explain this is that since division by zero is not defined, dividing by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; eliminates zero as a possible value for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Multiplying by zero does not have an inverse operation because dividing by zero is undefined.&lt;/p&gt;&#xA;&lt;h4&gt;Operations that are not one-to-one fabricate solutions&lt;/h4&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = 3$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^2 = 9$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^2 - 9 = 0$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(x - 3)(x + 3) = 0$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x = 3 \quad \textrm{or} \quad x = -3$$&lt;/span&gt;&#xA;Where did the extra solution come from? You could say that since &lt;span class=&quot;math-container&quot;&gt;$x^2 = (-x)^2,$&lt;/span&gt; squaring both sides of an equation mixes up answers with different signs. Squaring has no inverse for negative inputs.&lt;/p&gt;&#xA;&lt;h4&gt;Operations that aren't quite inverses&lt;/h4&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x^2 = 9$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{x^2} = \sqrt{9}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x = 3$$&lt;/span&gt;&#xA;Where did the solution &lt;span class=&quot;math-container&quot;&gt;$x = -3$&lt;/span&gt; go? The square root operation only gives positive results, so a square root is not an inverse operation of squaring when the input is negative. This is related to the previous example. A non one-to-one function cannot have an inverse for all inputs. One can use &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x^2} = |x|$&lt;/span&gt; to get all of the correct answers.&lt;/p&gt;&#xA;&lt;h4&gt;Other thoughts&lt;/h4&gt;&#xA;&lt;p&gt;I wonder if the above is related to the trouble I and others had with the proving trigonometric identities back in high school. When problems were of the form:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\textrm{Prove that: } \textrm{A mess of trig functions} = \textrm{Another mess of trig functions}$$&lt;/span&gt;&#xA;a common strategy was to manipulate the equation until it resulted in something like &lt;span class=&quot;math-container&quot;&gt;$1 = 1$&lt;/span&gt; and declare that, &amp;quot;Since the original equation is equivalent to &lt;span class=&quot;math-container&quot;&gt;$1=1,$&lt;/span&gt; it must be true.&amp;quot; This is obviously fallacious since one could prove that &lt;span class=&quot;math-container&quot;&gt;$\sin\theta=\cos\theta$&lt;/span&gt; just by multiplying both sides of the equation by zero to get the obviously true &lt;span class=&quot;math-container&quot;&gt;$0=0.$&lt;/span&gt; However, if all of the operations were invertible, then we would have a valid proof just be reversing the order of the lines: start with &lt;span class=&quot;math-container&quot;&gt;$1=1$&lt;/span&gt; and end up with the identity.&lt;/p&gt;&#xA;" OwnerUserId="3344" LastEditorUserId="3344" LastEditDate="2023-01-08T10:12:55.757" LastActivityDate="2023-01-08T10:12:55.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25746" PostTypeId="6" CreationDate="2022-11-01T13:49:34.713" Score="0" Body="&lt;h1&gt;quid&lt;/h1&gt;&#xA;&lt;p&gt;The election was scheduled to bring the site towards graduation. I am involved with the site all the way back since its inception and serve as a nominated moderator since the early days. I would be glad to continue to moderate the site, and thus present my nomination for this election.&lt;/p&gt;&#xA;&lt;p&gt;While I am not as active now as I used to be, mostly as my actual job as a mathematics professor at a research university takes up ever more time, I am still around frequently and ready to step in if the need arises.  Over the years I believe to have gained a good amount of experience with the site on a technical and a community level, which I believe is beneficial going forward. Moreover, I moderate the Mathematics site and as such can assure  a smooth interaction with that community.&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2022-11-01T13:49:34.713" LastActivityDate="2022-11-01T13:49:34.713" CommentCount="8" CommunityOwnedDate="2022-11-01T13:49:34.713" ContentLicense="CC BY-SA 4.0" />
  <row Id="25748" PostTypeId="1" CreationDate="2022-11-02T00:24:44.650" Score="6" ViewCount="311" Body="&lt;p&gt;Terry Tao describes 3 stages of one's mathematics education on his &lt;a href=&quot;https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;web blog&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;1: Pre-rigorous&lt;br&gt;&#xA;2: Rigorous&lt;br&gt;&#xA;3. Post-rigorous&lt;/p&gt;&#xA;&lt;p&gt;I know how one can progress from stage 1 to stage 2 (this simply can be done by learning a book like 'How to Prove It' by Daniel Velleman or Ian Stewart's 'Foundations of Mathematics')&lt;/p&gt;&#xA;&lt;p&gt;What can one do to progress from stage 2 to stage 3? Terry Tao states &amp;quot;&lt;em&gt;So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Also, are there any books that can facilitate this transition similar to how the aforementioned books can facilitate the stage 2 to stage 3 transition?&lt;/p&gt;&#xA;" OwnerUserId="20496" LastEditorUserId="20496" LastEditDate="2022-11-29T05:00:39.967" LastActivityDate="2022-11-29T05:00:39.967" Title="What are some ways that one can progress from stage 2 to stage 3 of the rigor stages that Terry Tao has described?" Tags="|graduate-education|problem-solving|intuition|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25749" PostTypeId="6" CreationDate="2022-11-02T03:06:01.687" Score="0" Body="&lt;h1&gt;Sue VanHattum&lt;/h1&gt;&#xA;&lt;p&gt;&lt;strong&gt;Sue VanHattum&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I've been moderating this site since its beginning. At first, I was a newbie to SE sites, so I had trouble figuring out how to do things. I’ve learned a lot in the past 8 ½ years! My first goal was to help this site be welcoming to homeschoolers and K-12 teachers (and I think we've met that goal). I am a community college math teacher, and sometimes tutor younger students.&lt;/p&gt;&#xA;&lt;p&gt;I'm still learning how the SE interface works, so I’m very grateful for the team of moderators I work with.&lt;/p&gt;&#xA;&lt;p&gt;I do all I can to be fair. When I'm not patient, I try to understand what got in my way, so that I can be more patient in future.&lt;/p&gt;&#xA;&lt;p&gt;I think it's vital that a woman be on the moderator team. I'd love to see more women participating here, and will keep thinking about how we might get that to happen.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-11-02T03:06:01.687" LastActivityDate="2022-11-02T03:06:01.687" CommentCount="5" CommunityOwnedDate="2022-11-02T03:06:01.687" ContentLicense="CC BY-SA 4.0" />
  <row Id="25751" PostTypeId="2" ParentId="25742" CreationDate="2022-11-02T11:45:09.963" Score="7" Body="&lt;p&gt;fedja is right.  I always used a chalk holder like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/85UdK.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/85UdK.jpg&quot; alt=&quot;chalk holder&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you know an academic who uses chalkboards (rare nowadays, most use PowerPoint), a chalk holder would be a great gift for them.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2022-11-02T11:45:09.963" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25752" PostTypeId="2" ParentId="24786" CreationDate="2022-11-02T16:51:09.013" Score="1" Body="&lt;p&gt;Another recommendation ist Vilenkin: Conbinatorics, see &lt;a href=&quot;https://www.isinj.com/mt-usamo/Combinatorics%20Vilenkin%20N.Ia.%20(1971).pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.isinj.com/mt-usamo/Combinatorics%20Vilenkin%20N.Ia.%20(1971).pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A good book for undergraduates is&lt;/p&gt;&#xA;&lt;p&gt;L. Lovász, J. Pelikán, K. Vesztergombi:Discrete Mathematics, elementary and beyond. Springer, 2003, ISBN: 978-0-387-95585-8, &lt;a href=&quot;https://doi.org/10.1007/b97469&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/b97469&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="61" LastActivityDate="2022-11-02T16:51:09.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25753" PostTypeId="2" ParentId="25748" CreationDate="2022-11-03T14:08:09.570" Score="6" Body="&lt;p&gt;This is not as much to answer the original question (to which the answer is just that you develop any skill by trying to practice it and evaluating the results) but to tell what my understanding of a &amp;quot;post-rigorous thinking&amp;quot; is. My main point will be that it has nothing to do with &amp;quot;trusting your refined intuition&amp;quot; (Intuition is a funny thing and it deserves a separate discussion) but rather internalizing the ideas to the extent that you no longer need to apply them consciously. It is the ability to just see the answer sometimes even without immediate understanding why you see it.&lt;/p&gt;&#xA;&lt;p&gt;What do I mean? Let's start with &lt;span class=&quot;math-container&quot;&gt;$4\times 9$&lt;/span&gt;. The first stage would be just to honestly compute it as &lt;span class=&quot;math-container&quot;&gt;$4+4=8, 8+4=12, 12+4=16, 16+4=20, 20+4=24$&lt;/span&gt; (Erm... where am I? Looks I added &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; five times? Ah, no, the initial one counts too, so six, actually. Three more to go) &lt;span class=&quot;math-container&quot;&gt;$24+4=28, 28+4=32, 32+4=36$&lt;/span&gt;. Hurray, &lt;span class=&quot;math-container&quot;&gt;$36$&lt;/span&gt;!. The next stage would be to use some easy anchoring points and/or properties of multiplication and proceed from them. Since it is easy to multiply by powers of &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;, one can do &lt;span class=&quot;math-container&quot;&gt;$4\times 9=4\times (10-1)= 4 \times 10-4=36$&lt;/span&gt;. Much cooler, isn't it? The final stage is you just see the answer &lt;span class=&quot;math-container&quot;&gt;$36$&lt;/span&gt; without even knowing how your brain did count, so if somebody asks you &amp;quot;Why? How did you count?&amp;quot;, you may be a bit at loss how to explain and the explanation you finally give, although formally impeccable, may have very little to do with your actual thought process. You just no longer remember which way of counting finally led to the internalization. Revisiting it (i.e., asking this question yourself) may surprise you and here you will be able to observe the distinction between &amp;quot;mechanical memorization&amp;quot; and &amp;quot;internalizing the idea&amp;quot;. For, if you ended up with pure memorization of the multiplication table, you will know &lt;span class=&quot;math-container&quot;&gt;$4\times 9$&lt;/span&gt; but not &lt;span class=&quot;math-container&quot;&gt;$6*99999$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$3\times 1213122$&lt;/span&gt; but if you ended up with internalizing the ideas of anchoring and split, the answers to those will be just as obvious to you, though in these cases you'll, probably, know how your brain counted.&lt;/p&gt;&#xA;&lt;p&gt;Now two real life examples from my recent teaching experience (graduate analysis).&lt;/p&gt;&#xA;&lt;p&gt;Problem 1: Show that there exists &lt;span class=&quot;math-container&quot;&gt;$\delta&amp;gt;0$&lt;/span&gt; such that if &lt;span class=&quot;math-container&quot;&gt;$a,b\ge 0, a+b=1$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2\ge\delta$&lt;/span&gt; (It was explicitly stated and emphasized that there was no need to come with an optimal or even numerical value!).&lt;/p&gt;&#xA;&lt;p&gt;Students: &lt;span class=&quot;math-container&quot;&gt;$b=1-a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a^2+(1-a)^2=2a^2-2a+1=f(a), a\in [0,1]$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f'(a)=0\Leftrightarrow 4a-2=0, a=1/2, f(a)=1/4+1/4=1/2, f(0)=f(1)=1&amp;gt;1/2$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$\delta=1/2$&lt;/span&gt; works (It took about 10 minutes to come up with that, believe it or not!).&lt;/p&gt;&#xA;&lt;p&gt;I just see it. If asked to explain, I might come up with&lt;/p&gt;&#xA;&lt;p&gt;The set &lt;span class=&quot;math-container&quot;&gt;$a,b\ge 1,a+b=1$&lt;/span&gt; is compact, the function &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2$&lt;/span&gt; is continuous, the only way it can get to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$a,b\ge 0$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$a=b=0$&lt;/span&gt;, which is not in the set, ergo...&lt;/p&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a+b\ge 1\Rightarrow \max(a,b)\ge 1/2\Rightarrow a^2+b^2\ge (1/2)^2\ge 1/4$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I honestly &lt;em&gt;do not&lt;/em&gt; know which of those really ran subconsciously in my brain or even whether it was any of those.&lt;/p&gt;&#xA;&lt;p&gt;The difference between these ways of thinking is that with the students' approach the question &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$a,b,c,d\ge 0, a^3+b^2c-5cd^6+d^{18}=1$&lt;/span&gt;. Conclude that &lt;span class=&quot;math-container&quot;&gt;$a^5+bc+d^5\ge\delta&amp;gt;0$&lt;/span&gt;&amp;quot; becomes insurmountable, while for me it is just as obvious as the first one.&lt;/p&gt;&#xA;&lt;p&gt;In this case both explanations of how I might be thinking are formally correct. The next case is different in this respect.&lt;/p&gt;&#xA;&lt;p&gt;Problem. Show that &lt;span class=&quot;math-container&quot;&gt;$\inf_{B\ge 0}\int_{[1,2]}\log|1-Bt|\,dt&amp;gt;-\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Students: Compute the integral explicitly splitting into cases (the antiderivative is elementary here), investigate the resulting function in each case, conclude.&lt;/p&gt;&#xA;&lt;p&gt;Again, I just see it. I was really asked by one of the students &amp;quot;How&amp;quot;. I evaded the response, though my initial desire was to say:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Look, for &lt;span class=&quot;math-container&quot;&gt;$B\ge 2$&lt;/span&gt;, the integrand is positive, for &lt;span class=&quot;math-container&quot;&gt;$B\le 1/2$&lt;/span&gt; it is bounded below by &lt;span class=&quot;math-container&quot;&gt;$-\log 2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$[1/2,2]$&lt;/span&gt; is a compact interval, there is no point on it where the integral diverges, so it is, probably, continuous, hence, bounded from below&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I didn't give this response, but for a while I kept it in my head always feeling that something was incongruous about it. Two things bothered me: that shaky &amp;quot;probably&amp;quot; and why I bothered to take special care of &lt;span class=&quot;math-container&quot;&gt;$B\le 1/2$&lt;/span&gt; separately (&lt;span class=&quot;math-container&quot;&gt;$[0,2]$&lt;/span&gt; is also compact and there is no divergence point there).&lt;/p&gt;&#xA;&lt;p&gt;It took me some time to realize what might be a good formalization of the &amp;quot;obvious&amp;quot; here. It is&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb R\to\mathbb R$&lt;/span&gt; is such that &lt;span class=&quot;math-container&quot;&gt;$\int_\mathbb R\min(f,0)&amp;gt;-\infty$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sup g'&amp;lt;-C&amp;lt;0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\int_{\mathbb R}f\circ g\ge C^{-1}\int_\mathbb R\min(f,0)$&lt;/span&gt;&#xA;(because you shoot through the negative region at the speed at least &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;So my initial &amp;quot;compactness&amp;quot; explanation was totally off. I didn't need to take any special care of large &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, no continuity played any role, and the only thing that I was right about (and which looked strange in that explanation) was to worry about &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; close to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. And, again, my second explanation is formally impeccable but I won't give you any guarantee on whether that is what I actually thought in my subconscious brain.&lt;/p&gt;&#xA;&lt;p&gt;I guess those two are enough to illustrate my point of view, which is that the &amp;quot;post-rigorous&amp;quot; thinking is &lt;em&gt;not&lt;/em&gt; trusting the intuition but internalizing the ideas and techniques to the extent that you are no longer able to explain how you figure something out to the extent that when you are forced to come with an explanation, you may easily end up with a wrong one. It is like driving a car through a sharp turn and being asked &amp;quot;How did you choose the speed and the angle of the rotation of the steering wheel so you didn't go off the road&amp;quot;? In all honesty, I have no idea of how my brain determines from just seeing a curve ahead and a wet or dry asphalt which number of my speedometer I should be aiming at and by exactly how much my hands should move to rotate the steering wheel. I just do it. Can you explain what is really going on here? And it is not a &amp;quot;pure memorization&amp;quot;: I have never seen this turn before. Yeah, memorization plays some role in driving: if you go over the same very road again and again, you find out that you can drive on it faster and more confidently than on an unfamiliar stretch, but you learned to drive decently, you don't panic or crash when you hit some new road either. You just exercise a bit more caution, and that's it. And you don't just &amp;quot;trust your feelings&amp;quot;. Those who exercise that driving philosophy end up in a ditch more often than not. You do &lt;em&gt;know exactly&lt;/em&gt; what you are doing. You just no longer know how you know it, but the rigor is not lost in the slightest, so I would say &amp;quot;post-rigorous&amp;quot; is a bad choice of name for this stage. I would really call it &amp;quot;shmarky plaft&amp;quot; because I have no real idea how the brain is actually operating at this level. And it is possible to achieve in any craft, mathematics being no exception. How? It just comes with experience and active learning from your errors.&lt;/p&gt;&#xA;&lt;p&gt;If somebody can up with more bright ideas, I would be most interested in hearing them, but so far here are my two cents :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastEditorUserId="511" LastEditDate="2022-11-08T18:17:18.237" LastActivityDate="2022-11-08T18:17:18.237" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25754" PostTypeId="2" ParentId="25742" CreationDate="2022-11-03T17:28:05.713" Score="2" Body="&lt;p&gt;You could use the very expensive chalk with the thin wax coating on the sides: Fulltouch by Hagoromo. You will still get a little chalk on your hands, but less, and fulltouch really is the champagne of chalk&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2022-11-03T17:28:05.713" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25755" PostTypeId="1" CreationDate="2022-11-04T00:58:31.653" Score="4" ViewCount="123" Body="&lt;p&gt;I do freelance tutoring; the math-related ones are mostly 1-2 hour one-on-one sessions for high school and college students taking calculus classes. Often I find myself scrambling for words - I don't want to stare blankly at the student for half a minute planning what phrasing to use. Unfortunately, the aforementioned scrambling means that sometimes terminology which I know is incorrect slips in. (Indeed, every single bad phrase from &lt;a href=&quot;https://matheducators.stackexchange.com/q/12125/19777&quot;&gt;this post&lt;/a&gt; has come out of my mouth during one tutoring session or another!)&lt;/p&gt;&#xA;&lt;p&gt;An example from a quick conversation in recent session [paraphrased]:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Me: Do you know how to find the derivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;?&lt;br /&gt;&#xA;Student: Wait, that (&amp;quot;that&amp;quot; is &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}[f(x)]$&lt;/span&gt;) means to take the derivative?&lt;br /&gt;&#xA;Me: Yes, it's equivalent notation.&lt;br /&gt;&#xA;Student: I need to find &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; prime then.&lt;br /&gt;&#xA;Me: Yep, just take the prime. [internal facepalm]&lt;br /&gt;&#xA;Student: Hmm... [starts scribbling work down]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Should I have done anything in particular after I accidentally used incorrect terminology? I don't want to distract too much from the student getting to work on the problem and practice doing calculus, but I'd also hate to contribute to what I know is widespread misuse of basic calculus terminology, if for no other reason than I like fancy terms. Is it worth interrupting the student's work to correct myself, probably stumbling over more words in the process of explaining why my initial phrasing was wrong? Or is there another remedy I can employ?&lt;/p&gt;&#xA;&lt;p&gt;I should clarify that this is exclusively &lt;em&gt;verbal&lt;/em&gt; terminology - I'm a stickler about how I write down mathematical work due to a few strict teachers who managed to drill in, for example, that you never stop writing the &lt;span class=&quot;math-container&quot;&gt;$\lim_{x \to 0}$&lt;/span&gt; during the intermediate steps if it's still there. I will also clarify that I get both students who just want to get through their homework, and students who want to learn more broadly e.g. to prepare for a test or get caught up with their class, so it would be helpful if answers could consider both those cases separately or together.&lt;/p&gt;&#xA;" OwnerUserId="19777" LastActivityDate="2022-11-04T00:58:31.653" Title="What should I do when I, as a math tutor, use incorrect verbal notation?" Tags="|tutoring|notation|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25756" PostTypeId="1" AcceptedAnswerId="25771" CreationDate="2022-11-04T04:53:14.107" Score="7" ViewCount="778" Body="&lt;p&gt;How can students &lt;a href=&quot;https://math.stackexchange.com/q/2387482&quot;&gt;learn to&lt;/a&gt; &lt;a href=&quot;https://math.stackexchange.com/q/2387482&quot;&gt;discover results&lt;/a&gt; themselves, if textbooks just spoonfeed them right in the exercise, before asking them to prove it? Why don't these textbook problems educate students on &lt;a href=&quot;https://academia.stackexchange.com/a/172749&quot;&gt;proofs' thought process&lt;/a&gt;? I quote four popular undergraduate textbooks.&lt;/p&gt;&#xA;&lt;h2&gt;Charles Pinter, &lt;em&gt;A Book of Abstract Algebra&lt;/em&gt; (1982 2 edn), page 158.&lt;/h2&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let G and H be groups. Suppose J is a normal subgroup of G and K is a normal&#xA;subgroup of H.&lt;/p&gt;&#xA;&lt;p&gt;1 Show that the function &lt;span class=&quot;math-container&quot;&gt;$f(x, y) = (Jx, Ky)$&lt;/span&gt; is a homomorphism from &lt;span class=&quot;math-container&quot;&gt;$G × H$&lt;/span&gt; onto &lt;span class=&quot;math-container&quot;&gt;$(G/J) × (H/K)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would change the last sentence to either —&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;From what set — onto what other set — is the function &lt;span class=&quot;math-container&quot;&gt;$f(x, y) = (Jx, Ky)$&lt;/span&gt;  a homomorphism? Prove your claim.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or —&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Onto what set is the the function &lt;span class=&quot;math-container&quot;&gt;$f(x, y) = (Jx, Ky)$&lt;/span&gt;  a homomorphism from &lt;span class=&quot;math-container&quot;&gt;$G × H$&lt;/span&gt;? Prove your claim.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h3&gt;Page 236&lt;/h3&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;9 For any prime &lt;span class=&quot;math-container&quot;&gt;$p &amp;gt; 2, x^2 \equiv -1  \pmod p$&lt;/span&gt; has a solution &lt;span class=&quot;math-container&quot;&gt;$\iff p \not \equiv 3 \pmod 4$&lt;/span&gt;. (HINT:&#xA;Use part 8 and E3.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would change the last independent clause to —&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;for what &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; does &lt;span class=&quot;math-container&quot;&gt;$x^2 \equiv -1  \pmod p$&lt;/span&gt; have a solution?&lt;/li&gt;&#xA;&lt;li&gt;if &lt;span class=&quot;math-container&quot;&gt;$p \not \equiv 3 \pmod 4$&lt;/span&gt;, then when does &lt;span class=&quot;math-container&quot;&gt;$x^2 \pmod p$&lt;/span&gt; have a solution?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;h2&gt;David Bressoud, &lt;em&gt;Radical Approach to Real Analysis&lt;/em&gt; (2006 2 edn), p 263.&lt;/h2&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;6.4.1. Prove that if &lt;span class=&quot;math-container&quot;&gt;$f(a - 0) \neq f(a) \neq f(a + 0)$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$F(x) =\int^x_0 f(t) dt$&lt;/span&gt; cannot be differentiable at &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would ask —&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f(a - 0) \neq f(a) \neq f(a + 0)$&lt;/span&gt;, then is &lt;span class=&quot;math-container&quot;&gt;$\; F(x) = \int^x_0 f(t) dt \;$&lt;/span&gt; differentiable at &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt;? Prove or disprove.&lt;/p&gt;&#xA;&lt;h2&gt;Terence Tao, &lt;em&gt;Analysis I&lt;/em&gt; (2016 3 edn), p 300.&lt;/h2&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Exercise 11.9.3. Let a &amp;lt; b be real numbers, and let  &lt;span class=&quot;math-container&quot;&gt;$ f : [a, b] \rightarrow \mathbb{R}$&lt;/span&gt; be a monotone increasing function. Let &lt;span class=&quot;math-container&quot;&gt;$F : [a, b] \rightarrow \mathbb{R}$&lt;/span&gt; be the function &lt;span class=&quot;math-container&quot;&gt;$F(x) := \int_{[a,x]} f$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; be an element of &lt;span class=&quot;math-container&quot;&gt;$[a, b]$&lt;/span&gt;. Show that &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is differentiable at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; if and only if f is continuous at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;. (Hint: one direction is taken care of by one&#xA;of the fundamental theorems of calculus. For the other, consider left and right limits of f and argue by contradiction.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would change the second last sentence to —&lt;/p&gt;&#xA;&lt;p&gt;When is &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; differentiable at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;? When is &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; continuous at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;? Prove your claims.&lt;/p&gt;&#xA;&lt;h2&gt;Terence Tao, &lt;em&gt;Analysis II&lt;/em&gt; (2016 3 edn), p 205.&lt;/h2&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Exercise 8.3.3. Let &lt;span class=&quot;math-container&quot;&gt;$f : \mathbb{R} \rightarrow \mathbb{R}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g : \mathbb{R} \rightarrow \mathbb{R}$&lt;/span&gt; be absolutely integrable,&#xA;measurable functions such that &lt;span class=&quot;math-container&quot;&gt;$f(x) \le g(x)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{R}$&lt;/span&gt;, and that &lt;span class=&quot;math-container&quot;&gt;$\int_\mathbb{R} f = \int_\mathbb{R} g.$&lt;/span&gt; Show that &lt;span class=&quot;math-container&quot;&gt;$f(x) = g(x)$&lt;/span&gt; for almost every &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{R}$&lt;/span&gt; (i.e., that &lt;span class=&quot;math-container&quot;&gt;$f(x) = g(x)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{R}$&lt;/span&gt;  except possibly for a set of measure zero).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would change the last sentence to —&lt;/p&gt;&#xA;&lt;p&gt;How does &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; relate to &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;,  for all &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{R}$&lt;/span&gt;  except possibly for a set of measure zero? Prove your claim.&lt;/p&gt;&#xA;" OwnerUserId="20536" LastEditorUserId="20536" LastEditDate="2023-01-15T07:42:01.553" LastActivityDate="2023-01-23T19:21:51.087" Title="Why don't exercises ask readers to discover results themselves, before requiring proof? Why disclose the result in the question?" Tags="|textbooks|" AnswerCount="6" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="25758" PostTypeId="2" ParentId="25756" CreationDate="2022-11-04T11:42:40.570" Score="7" Body="&lt;p&gt;Good point! That said, disclosing the result in the question could be to make the question more doable / less intimidating for the student, so they know what to aim for.&lt;/p&gt;&#xA;&lt;p&gt;Note that popular undergraduate analysis textbook &lt;em&gt;Understanding Analysis&lt;/em&gt; by Abbott &lt;em&gt;does&lt;/em&gt; include more challenging questions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The exercises (even more so now than in the first edition) are superb: many call for proofs, but there are also quite a lot that ask for examples (or an explanation why no examples exist; the reader isn’t always told which is which) or ask the reader to explore the consequences if a definition is modified slightly. Few if any constitute “busy work”; some are quite challenging.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(See Hunacek's review at &lt;a href=&quot;https://www.maa.org/press/maa-reviews/understanding-analysis.&quot; rel=&quot;noreferrer&quot;&gt;https://www.maa.org/press/maa-reviews/understanding-analysis.&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Maybe a mix of question types would be a good idea? Of course, we cannot rule out that the lack of questions of the type you suggest could also be because of mere oversight or force of habit.&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2022-11-04T11:42:40.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25760" PostTypeId="1" AcceptedAnswerId="25766" CreationDate="2022-11-05T06:45:10.640" Score="4" ViewCount="947" Body="&lt;p&gt;Besides the 2 mathematicians quoted below and me, who else has touted free dissemination &lt;em&gt;&lt;strong&gt;to students&lt;/strong&gt;&lt;/em&gt; of detailed solutions, to EVERY exercise and problem (like in textbooks)? I uphold this wholeheartedly! This free dissemination ought to be the norm!&lt;/p&gt;&#xA;&lt;p&gt;I do NOT refer to snippety one-line answers at the back of a textbook, student solution manuals that solve merely some or half of the exercises, or solution manuals  restricted to instructors.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Robert  Ash (1935-2015), &lt;a href=&quot;https://faculty.math.illinois.edu/%7Er-ash/RV/RVPreface.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Preface&lt;/a&gt; to &lt;a href=&quot;https://faculty.math.illinois.edu/%7Er-ash/RV.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Real Variables with Basic Metric Space Topology&lt;/em&gt;&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I rely especially on one of the most useful of all learning devices: the inclusion of detailed solutions to exercises. Solutions to problems are commonplace in elementary texts but quite rare (although equally valuable) at the upper division undergraduate and graduate level. This feature makes the&#xA;book suitable for independent study, and further widens the audience.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;David Patrick, &lt;em&gt;Introduction to Counting and Probability&lt;/em&gt; (2005), page v.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;However,&#xA;if you are using this book on your own to learn independently, then you probably have a copy of the&#xA;solution book, in which case there are some very important things to keep in mind:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Make sure that you make a serious attempt at the problem before looking at the solution. Don't&#xA;use the solution book as a crutch to avoid really thinking about a problem first. You should think&#xA;&lt;em&gt;hard&lt;/em&gt; about a problem before deciding to give up and look at the solution.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;After you solve a problem, it's usually a good idea to read the solution, even if you think you know&#xA;how to solve the problem. The solution that's in the solution book might show you a quicker or&#xA;more concise way to solve the problem, or it might have a completely different solution method&#xA;that you might not have thought of. [emboldening mine]&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you have to look at the solution in order to solve a problem, make sure that you make a note of&#xA;that problem. Come back to it in a week or two to make sure that you are able to solve it on your&#xA;own, without resorting to the solution.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20536" LastEditorUserId="20536" LastEditDate="2022-11-06T05:02:35.083" LastActivityDate="2022-11-06T05:02:35.083" Title="What other mathematicians have advocated for freely publicizing detailed solutions, to every exercise?" Tags="|publishing|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25761" PostTypeId="2" ParentId="86" CreationDate="2022-11-05T06:47:30.537" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/q/25760&quot;&gt;The two mathematicians below would answer YES!&lt;/a&gt; They tout the free promulgation &lt;em&gt;&lt;strong&gt;to students&lt;/strong&gt;&lt;/em&gt; of detailed answers and solutions to EVERY exercise and problem in a book.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Robert  Ash (1935-2015) touted in the &lt;a href=&quot;https://faculty.math.illinois.edu/%7Er-ash/RV/RVPreface.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Preface&lt;/a&gt; to &lt;a href=&quot;https://faculty.math.illinois.edu/%7Er-ash/RV.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Real Variables with Basic Metric Space Topology&lt;/em&gt;&lt;/a&gt;,&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I rely especially on one of the most useful of all learning devices: the inclusion of detailed solutions to exercises. Solutions to problems are commonplace in elementary texts but quite rare (although equally valuable) at the upper division undergraduate and graduate level. This feature makes the&#xA;book suitable for independent study, and further widens the audience.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;David Patrick touts on page v, in &lt;em&gt;Introduction to Counting and Probability&lt;/em&gt; (2005)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;However,&#xA;if you are using this book on your own to learn independently, then you probably have a copy of the&#xA;solution book, in which case there are some very important things to keep in mind:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Make sure that you make a serious attempt at the problem before looking at the solution. Don't&#xA;use the solution book as a crutch to avoid really thinking about a problem first. You should think&#xA;&lt;em&gt;hard&lt;/em&gt; about a problem before deciding to give up and look at the solution.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;After you solve a problem, it's usually a good idea to read the solution, even if you think you know&#xA;how to solve the problem. The solution that's in the solution book might show you a quicker or&#xA;more concise way to solve the problem, or it might have a completely different solution method&#xA;that you might not have thought of. [emboldening mine]&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you have to look at the solution in order to solve a problem, make sure that you make a note of&#xA;that problem. Come back to it in a week or two to make sure that you are able to solve it on your&#xA;own, without resorting to the solution.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20536" LastActivityDate="2022-11-05T06:47:30.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25762" PostTypeId="2" ParentId="3839" CreationDate="2022-11-05T07:04:12.473" Score="3" Body="&lt;p&gt;I am dismayed that nobody has mooted students who work part time, or who suffer from chronic conditions or disabilities? They may need to work and submit homework ahead of schedule, and the extra time. This posting ahead of schedule can accommodate disabilities!&lt;/p&gt;&#xA;&lt;p&gt;I quote UC Santa Cruz's Disability Resource Center on &lt;a href=&quot;https://drc.ucsc.edu/faculty/legal-obligation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Legal Obligation for Accomodations&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There are two federal laws that protect students with disabilities that protects them from discrimination and ensure they have equal access to all aspects of university life.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.ada.gov/2010_regs.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Americans with Disabilities Act of 1990 (ADA)&lt;/a&gt;: &amp;quot;prohibits discrimination and ensures equal opportunity for persons with disabilities in employment, State and local government services, public accommodations, commercial facilities, and transportation.&amp;quot; &amp;quot;A public entity shall make reasonable modifications in policies or procedures when the modifications are necessary to avoid discrimination on the basis of disability, unless the public entity can demonstrate that making the modifications would fundamentally alter the nature of the service, program, or activity.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.dol.gov/agencies/oasam/centers-offices/civil-rights-center/statutes/section-504-rehabilitation-act-of-1973&quot; rel=&quot;nofollow noreferrer&quot;&gt;Section 504, Rehabilitation Act of 1973&lt;/a&gt;: &amp;quot;No otherwise qualified individual with a disability in the United States... shall, solely by reason of his or her disability, be excluded from the participation in, be denied the benefits of, or be subjected to discrimination under any program or activity receiving Federal financial assistance... Program or activity is defined as... a college, university, or other post-secondary institution, or a public system of higher education.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Quest University in B.C. Canada on &lt;a href=&quot;https://questu.ca/student-life/student-success/accesibility-services&quot; rel=&quot;nofollow noreferrer&quot;&gt;Accessibility + Equity&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Quest affirms the right of applicants to Quest and students of Quest who are academically qualified and who have a protected characteristic within the meaning of the &lt;em&gt;Human Rights Code&lt;/em&gt;, R.S.B.C. 1996, c. 210 to be welcomed as equal members of the Quest community; to have full access to services Quest customarily provides to applicants and students; and, on matriculating, to participate fully and freely in student life at Quest.&lt;/p&gt;&#xA;&lt;h3&gt;Accommodations&lt;/h3&gt;&#xA;&lt;p&gt;Students experiencing chronic medical conditions, mobility limitations, disabilities, learning exceptionalities, mental health concerns, and other such protected characteristics may qualify for special academic or non-academic adjustments (accommodations).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Disability Rights UK on &lt;a href=&quot;https://www.disabilityrightsuk.org/adjustments-disabled-students-and-apprentices&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adjustments for disabled students and apprentices&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you have a disability, health condition or specific learning difficulty such as dyslexia, you may need certain facilities, assistive technology or support services to enable you to make the most of your studies or training. This can include alternative exam or assessment arrangements. The Equality Act 2010 calls the arrangements that your education or training provider makes to meet these needs ‘reasonable adjustments’. For more information about the Equality Act please see section 7.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;University of Dundee's &lt;a href=&quot;https://www.dundee.ac.uk/corporate-information/academic-adjustments-disabled-students&quot; rel=&quot;nofollow noreferrer&quot;&gt;Academic Adjustments for Disabled Students Policy&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;3.1    Under the Equality Act (2010), Universities are required to make reasonable adjustments in anticipation of, and in response to, disabled students’ needs and must ensure that disabled students are not treated less favourably than other students for reasons relating to their disability. These duties apply to all services and facilities the University provides for students, including all aspects of learning and teaching.&lt;/p&gt;&#xA;&lt;p&gt;As this is a duty to disabled students generally, it applies “regardless of whether the education provider knows that a particular person is disabled or whether it currently has disabled students”.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20536" LastActivityDate="2022-11-05T07:04:12.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25763" PostTypeId="6" CreationDate="2022-11-05T11:54:15.413" Score="0" Body="&lt;h1&gt;Xander Henderson&lt;/h1&gt;&#xA;&lt;p&gt;I think that a moderator on an SE site needs to demonstrate two major characteristics: (1) a knowledge of the mechanics of the SE network, and (2) a connection to the particular community which they are tasked with moderating.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I have been a diamond moderator on Math SE for a couple of years. In that time, I've handled nearly 7k flags, sent around 500 moderator messages, have actively engaged with folk in chat, and have otherwise become quite familiar with to tools available to moderators.  I have no doubts about my ability to help moderate this community in terms of the technical aspects of moderation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I will admit that I have been somewhat less active since finishing graduate school and accepting a full-time teaching position, though I have been consistently lurking. Still, I believe that I have a sense of what this community is and wants to be, and I would be honored to serve here.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;All of that being said, please don't vote for me—both Sue VanHattum and quid have been doing an excellent job.  While I am willing and capable, I am running primarily to ensure that there are enough candidates for a successful election.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2022-11-05T11:54:15.413" LastActivityDate="2022-11-05T11:54:15.413" CommentCount="4" CommunityOwnedDate="2022-11-05T11:54:15.413" ContentLicense="CC BY-SA 4.0" />
  <row Id="25765" PostTypeId="2" ParentId="25760" CreationDate="2022-11-05T14:30:15.437" Score="3" Body="&lt;p&gt;This is hardly &amp;quot;touting&amp;quot; and surely below the level of your interest, but&#xA;I will nevertheless mention that two books I wrote at the&#xA;high-school / early college level&#xA;both include solutions to every exercise at the back of the book:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;https://cs.smith.edu/%7Ejorourke/HowToFoldIt/&quot; rel=&quot;nofollow noreferrer&quot;&gt;How To Fold It&lt;/a&gt;:&#xA;The Mathematics of Linkages, Origami, and Polyhedra&lt;/em&gt;, Cambridge, 2011.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;https://cs.smith.edu/%7Ejorourke/PopUps/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pop-Up Geometry&lt;/a&gt;:&#xA;The Mathematics behind Pop-Up Cards&lt;/em&gt;, Cambridge, 2022.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-11-05T14:30:15.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25766" PostTypeId="2" ParentId="25760" CreationDate="2022-11-06T02:24:23.290" Score="5" Body="&lt;p&gt;I think if you look at texts that are more designed for self study and marketed to students, it's a norm.  Schaum's Outlines does this.  Even more extreme, perhaps there are programmed instruction texts.  I know my father's WW2 education manuals (math from geometry to calculus) were all like this also.  Some individual texts like this are Thomas and Thomas Finney Calculus (at least from when they were alive) or Tannenbaum and Pollard ODE or Wienberger PDE books.&lt;/p&gt;&#xA;&lt;p&gt;I think the big driver behind keeping the solutions cloaked is the market being teachers, not students, when texts are selected.  (Giving graded homework, wanting to force students not to look at answers to fast, or wanting to retain more of a role for being.)&lt;/p&gt;&#xA;&lt;p&gt;If you happen to be a student or meet one, who really doesn't want to have solutions or answers available (incapable of discipling self to work the problems before looking at the answer), than advise him to tear out the pages at the back and burn them.  Personally, I think having the solutions or at least answers is very helpful.  So, that's silly to say a text is worse by having solutions. But if you really think so, just burn those pages.&lt;/p&gt;&#xA;&lt;p&gt;I think solutions or at least answers gives a feedback loop, including for &amp;quot;dumb mistakes&amp;quot;.  Can even be a form of partial hint (helps to know the answer to work the problem, look at the integral stumpers on MSE).  And this is a feedback loop (or hint aid) that is immediately available, very important in the learning process, rather than waiting for hours or days.&lt;/p&gt;&#xA;" OwnerUserId="20575" LastActivityDate="2022-11-06T02:24:23.290" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25767" PostTypeId="1" CreationDate="2022-11-06T21:16:21.607" Score="11" ViewCount="6371" Body="&lt;p&gt;John is trying to learn some field of mathematics. John's teacher Sarah gives constructive criticism to John after seeing John's attempts on exercises related to the field he studies. Later, John tries a different set of exercises on the topic, and yet again, Sarah finds John making the same mistakes as before.&lt;/p&gt;&#xA;&lt;p&gt;How would the teacher Sarah help John who has &amp;quot;internalized&amp;quot; the mistake? It seems that directly telling the student to do more exercise would only make their condition worse as the &amp;quot;mistakes&amp;quot; would be more drilled in.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorUserId="851" LastEditDate="2022-11-08T20:12:54.077" LastActivityDate="2023-05-02T19:15:13.753" Title="How can a teacher help a student who has internalized mistakes?" Tags="|teaching|students-mistakes|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25768" PostTypeId="1" CreationDate="2022-11-06T21:17:02.213" Score="1" ViewCount="123" Body="&lt;p&gt;I'm a second year teacher teaching Precal for the first time.&lt;/p&gt;&#xA;&lt;p&gt;I have a few textbooks, but the problem is that my school does not (and will never) have class sets of any of them for students to use.&lt;/p&gt;&#xA;&lt;p&gt;To make worksheets using the textbook would therefore be copyright infringement.&lt;/p&gt;&#xA;&lt;p&gt;I've looked at some of the EngageNY/Eureka Math stuff but it's hard to edit/move around.&lt;/p&gt;&#xA;&lt;p&gt;Kuta is good but it's mostly for drilling with little/no conceptual. Same with IXL and DeltaMath which I have access to.&lt;/p&gt;&#xA;&lt;p&gt;I've dropped hundreds on TPT materials but they're not very good. They're not tailored to the particular kids I'm teaching.&lt;/p&gt;&#xA;&lt;p&gt;Math teachers, how do you make your materials?&lt;/p&gt;&#xA;" OwnerUserId="20130" LastActivityDate="2022-11-06T21:17:02.213" Title="What do you use to make your course materials?" Tags="|secondary-education|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25769" PostTypeId="2" ParentId="25767" CreationDate="2022-11-06T22:36:32.027" Score="11" Body="&lt;p&gt;The student has a wrong process internalized. Can you explain how their process works? (What I mean here is, could you predict their wrong answer to a new question.) I think that will help you analyze what they are thinking.&lt;/p&gt;&#xA;&lt;p&gt;If you can find the nugget of right thinking in their wrong thinking, then you can show them how they were right up to this point and then took this one wrong step. They will need to see very clearly why that one step leads them astray.&lt;/p&gt;&#xA;&lt;p&gt;If you could give an example of the mistakes this student is making, I could perhaps give a more useful answer.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-11-06T22:36:32.027" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25770" PostTypeId="2" ParentId="25767" CreationDate="2022-11-07T00:26:23.617" Score="19" Body="&lt;p&gt;Coach them through doing it the right way.  Have them repeat it the right way, several times.  In front of you.  And go very easy, including repeats. Gradually relax the guardrails and keep drilling. They are not going to get it by comprehension, but by exposure.  Perhaps comprehension will follow long later.  But for now drill it the way you would drill a physical task. I suggest computer drill sites and very easy ones.&lt;/p&gt;&#xA;" OwnerUserId="20524" LastEditorUserId="60" LastEditDate="2023-05-02T19:15:13.753" LastActivityDate="2023-05-02T19:15:13.753" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25771" PostTypeId="2" ParentId="25756" CreationDate="2022-11-07T22:48:44.503" Score="16" Body="&lt;p&gt;I largely agree with you that your approach often makes sense. Since you are asking &amp;quot;why people don't (always) do that&amp;quot;, I will only describe the downsides.&lt;/p&gt;&#xA;&lt;p&gt;One framework to think about teaching tasks and their goals is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bloom%27s_taxonomy&quot; rel=&quot;noreferrer&quot;&gt;Bloom's taxonomy&lt;/a&gt;, in which there are 6 levels of knowledge: Remember, Understand, Apply, Analyze, Evaluate, and Create. In every teaching situation, it is good to mind what level you are teaching on, and choose the tasks accordingly. What you are suggesting is, essentially, to add a bit of &lt;strong&gt;create&lt;/strong&gt; to each task, but it may simply be out of place.&lt;/p&gt;&#xA;&lt;p&gt;For instance, in your first example, once the students &lt;strong&gt;remember&lt;/strong&gt; the definition of homomorphisms and factor-groups, and &lt;strong&gt;understand&lt;/strong&gt; them - i.e.,  know what they need to check, why &lt;span class=&quot;math-container&quot;&gt;$Jx$&lt;/span&gt; is an element of the factor-group, etc. - the exercise is solved. So, this particular exercise is aimed &lt;strong&gt;at reaching these two levels&lt;/strong&gt;. It goal is simply not to &amp;quot;teach the students to discover results by themselves&amp;quot;, it is to let the students who were just shown some abstract definitions for the first time in their life, and are utterly confused, to get some hands-on working experience. If you want to teach them to &lt;strong&gt;create&lt;/strong&gt; (i.e., re-discover by themselves) results in group theory, that's great, but it should be done when they have way different level of knowledge and understanding. And - arguably - with more meaningful exercises.&lt;/p&gt;&#xA;&lt;p&gt;On a more technical side, it is not easy to ask an open-ended question with a well-defined correct answer. Many of your suggested formulations &lt;strong&gt;have a perfectly correct answer which is different from the implied one.&lt;/strong&gt; E.g., in the first one, I answer &amp;quot;from &lt;span class=&quot;math-container&quot;&gt;$J\times K$&lt;/span&gt; to the one-element set &lt;span class=&quot;math-container&quot;&gt;$\{J\times K\}$&lt;/span&gt;, equipped with its unique group structure, since any map from a group to a one-element group is a homomorphism&amp;quot;, and demand full credit. It's not just nitpicking, it's really what will happen: on top of struggling with the material, the students will get confused about what you are asking, what constitutes a right answer etc. More harm than good.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2022-11-07T22:48:44.503" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25772" PostTypeId="2" ParentId="25767" CreationDate="2022-11-08T04:31:51.960" Score="2" Body="&lt;p&gt;Assuming the student seems to understand your correction, but simply forget or fall back into bad habits, you can teach the student to use metacognition.&lt;/p&gt;&#xA;&lt;p&gt;Tell them that they seem to have made the same mistake which they made earlier. Ask them questions along the lines of, &amp;quot;What do you think happened? What situations do you think you might find yourself making the same mistake again in the future? Can you think of any reminders for yourself to avoid making the same mistake in those situations?&amp;quot; Then you can teach them any time they are doing this process that they can ask themselves, &amp;quot;I know I tend to make X mistake here; did I make it this time? Let's check.&amp;quot; Next, have them do the problem in front of you, not only showing you that they learned the correct process but also the questions to ask themselves to make sure they haven't repeated the same mistakes.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2022-11-08T04:31:51.960" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25774" PostTypeId="2" ParentId="25767" CreationDate="2022-11-08T17:01:09.770" Score="7" Body="&lt;p&gt;In some cases it helps to first try to understand exactly why they might have some misconception. If you can do that, though not always possible, you can more easily find a way to guide them.&lt;/p&gt;&#xA;&lt;p&gt;An example: I once had a student that evaluated math expressions strictly left to right without considering operator precedence. It was very hard to convince him that he was getting the wrong answers and why. So 2 + 3 * 4 was 20 in his computations.&lt;/p&gt;&#xA;&lt;p&gt;After some explorations I discovered that there were two things that got him to this situation and reinforced his thinking. The first was bad teaching at some point in his past. He was convinced by some instructor that this was correct. Then, he happened to buy and use a calculator that didn't take account of precedence so his earlier misconception was reinforced daily.&lt;/p&gt;&#xA;&lt;p&gt;It took a while to convince him that, while he made correct inferences based on his available evidence, it was the evidence that was the cause of his issue, not a personal failing. But until I understood that, and could communicate it, along with the proper rules, he resisted my efforts as somehow undermining him. He hated me for a while, I think, until we worked it out.&lt;/p&gt;&#xA;" OwnerUserId="8756" LastActivityDate="2022-11-08T17:01:09.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25775" PostTypeId="2" ParentId="25677" CreationDate="2022-11-09T13:55:49.487" Score="3" Body="&lt;p&gt;Here is a defense of the notation for students trying to problem-solve.  Let's say I have the equation, &lt;span class=&quot;math-container&quot;&gt;$x^2 = 4$&lt;/span&gt;.  This has two solutions, &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$+2$&lt;/span&gt;.  However, what &lt;em&gt;operation&lt;/em&gt; should I do to result in these numbers?  If you say that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x^2}$&lt;/span&gt; only yields the principle square root, then &lt;em&gt;which&lt;/em&gt; operation yields both solutions?  I can't take the square root of both sides, because, according to the understanding of the square root presented, it only yields one value.&lt;/p&gt;&#xA;&lt;p&gt;It would make more sense for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x^2}$&lt;/span&gt; to be only the positive square root &lt;em&gt;if&lt;/em&gt; there were another operator which yields all of them.  Then, I could distinguish between them.  But, when &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x^2}$&lt;/span&gt; just means the positive side, this is very confusing for students because it leaves them without a concrete operation to apply to an equation to actually get both answers.&lt;/p&gt;&#xA;&lt;p&gt;You could tell them that there is an exception here.  However, I've always found when teaching that students get &lt;em&gt;really&lt;/em&gt; hung up on exceptions.  They have enough trouble just following the rules.  My point is that, if you teach that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x^2}$&lt;/span&gt; only means a positive value, don't be surprised if they later tell you that &lt;span class=&quot;math-container&quot;&gt;$x^2 = 4$&lt;/span&gt; only has one solution.  If you teach them that &lt;span class=&quot;math-container&quot;&gt;$x^2 = 4$&lt;/span&gt; has two solutions, and that square rooting is the inverse operation of squaring, then don't be surprised if they give two answers for square roots.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2022-11-09T13:55:49.487" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25776" PostTypeId="1" CreationDate="2022-11-09T13:57:26.300" Score="2" ViewCount="330" Body="&lt;p&gt;I was having a conversation about math education in the USA. Would an excellent middle school student, high school student, or college freshman be able to answer this simple question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Exercise 0&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$A(z) = a_0 + a_1 z + a_2 z^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B(z) = b_0 + b_1 z + b_2 z^2$&lt;/span&gt;, what are the coefficients &lt;span class=&quot;math-container&quot;&gt;$c_i$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$C(z) = \sum_i c_i z^i = A(z) \times B(z)$&lt;/span&gt;? Only write the answer here, not the calculation (to do on scrap).&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\begin{aligned}&#xA;\ &amp;amp; c_0 = \qquad\qquad c_3 =\\&#xA;\ &amp;amp; c_1 = \qquad\qquad c_4 =\\&#xA;\ &amp;amp; c_2 = \qquad\qquad &#xA;\end{aligned}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20588" LastEditorUserId="15884" LastEditDate="2022-11-11T01:28:12.867" LastActivityDate="2022-11-11T01:28:12.867" Title="What age student should be able to answer this question?" Tags="|secondary-education|usa|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25777" PostTypeId="2" ParentId="25742" CreationDate="2022-11-09T16:02:35.867" Score="3" Body="&lt;p&gt;You can't.&lt;/p&gt;&#xA;&lt;p&gt;The answers given by David Steinberg and Gerald Edgar are both very good mitigation strategies (I have a rather nice chalk holder which I have used so much that you can see wear patterns on the part which I hold), but there is nothing which you will ever be able to do which keeps your hands completely free of chalk.  Chalk dust gets everywhere.&lt;/p&gt;&#xA;&lt;p&gt;A couple of other mitigation strategies:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Wear cheap cotton gloves (e.g. archival gloves) while teaching.  Some chalk dust will permeate these and get on your hands (especially if you use your hand to erase), but not all of it.  Just make sure that you clean them frequently (e.g. by rinsing them out with water after class).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Keep a box of wet wipes or moist towelettes with you while teaching.  After class, scrub your hands down with a wet wipe before packing your gear.  This can also have the advantage of helping to keep your hands moisturized, as many brands will put some kind of moisturizer into the wet wipe solution (in the past, chalk dust and the arid desert environment in which I live has done quite a bit of damage to my hands).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A more eco-friendly version of wet wipes is to just keep a dish towel with you.  Teach your class, then wipe your hands with the dry towel at the end before packing up.  This will have to be frequently laundered, and you don't want to pack it in with your other stuff (though I suppose you could keep it in, for example, a ZipLoc bag), but it might help.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Just wipe your hands on your pants.  I know, I know—you don't want chalk all over yourself.  But chalk generally washes out pretty easily, and you do laundry regularly, right?  There isn't really anything wrong with having big chalk spots on your pants at the end of the day, unless you have something special to do (in which case, why not try the strategies above?).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2022-11-09T16:02:35.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25778" PostTypeId="1" CreationDate="2022-11-10T09:31:45.123" Score="3" ViewCount="1224" Body="&lt;p&gt;I am going to teach a Calculus 1 course next semester, and I have 15 weeks for the course material. The class meets MWF for 50 minutes each. I have taught this class before using the same syllabus, but my colleague shared concerns that my pacing is too fast:&lt;/p&gt;&#xA;&lt;p&gt;Week 1: Review of Functions&lt;/p&gt;&#xA;&lt;p&gt;Week 2: Limits and Continuity; Infinite Limits&lt;/p&gt;&#xA;&lt;p&gt;Week 3: Derivative (Limit Definition); Differentiation Rules; Transcendental Functions&lt;/p&gt;&#xA;&lt;p&gt;Week 4: Implicit Differentiation; Related Rates; Linear Approximation and Differentials&lt;/p&gt;&#xA;&lt;p&gt;Week 5: Extrema; Curve Sketching; L'Hopital's Rule&lt;/p&gt;&#xA;&lt;p&gt;Week 6: Optimization; &lt;em&gt;Newton's Method&lt;/em&gt;; Antiderivatives&lt;/p&gt;&#xA;&lt;p&gt;Week 7: Integrals; Fundamental Theorem of Calculus&lt;/p&gt;&#xA;&lt;p&gt;Week 8: Applications of Integrals: Work; Areas; Volumes of Solids&lt;/p&gt;&#xA;&lt;p&gt;Week 9: Integration by Parts; Partial Fraction Decomposition&lt;/p&gt;&#xA;&lt;p&gt;Week 10: Trig Substitution; &lt;em&gt;Approximate Integration&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Week 11: Arc Length; Surface Area&lt;/p&gt;&#xA;&lt;p&gt;Week 12: First-Order Differential Equations&lt;/p&gt;&#xA;&lt;p&gt;Week 13: Parametric Equations and Polar Coordinates&lt;/p&gt;&#xA;&lt;p&gt;Week 14: Introduction to Sequences and Series&lt;/p&gt;&#xA;&lt;p&gt;Week 15: Review for Final&lt;/p&gt;&#xA;&lt;p&gt;This is the syllabus I used last Spring, and I didn't have any problems with running out of time. I get straight to the point with my lectures, and my student grades have been above average compared to other instructors. However, the exams I use from my department only cover up to the fundamental theorem. So my students end up with a lot of excess information, since I cover up towards the end of a traditional Calculus 2 course. This means they're more than prepared for Calculus 2. However, I don't feel like going slower if I do not have to. I mean, if I were to get rid of the first week review, I could theoretically cover all of Calculus 1 and 2 in one course. Not sure why my colleagues take so long in lecturing. I sat in on a class and it took my colleague the entire 50 minutes to teach about power rule and product rule, when in the same time frame I can cover all differentiation rules plus transcendentals. Student evaluations seemed to be good. In a class of 33 students, 26 got &amp;quot;A's&amp;quot; and 5 got &amp;quot;B's&amp;quot; and 2 got &amp;quot;C's&amp;quot;. No one failed.&lt;/p&gt;&#xA;" OwnerUserId="20591" LastActivityDate="2022-11-10T17:30:33.373" Title="What is a good pacing for a Calculus 1 undergraduate course?" Tags="|undergraduate-education|" AnswerCount="2" CommentCount="22" ContentLicense="CC BY-SA 4.0" />
  <row Id="25780" PostTypeId="2" ParentId="25778" CreationDate="2022-11-10T15:03:22.423" Score="3" Body="&lt;p&gt;Just do what works for you and your students. If they get a good grasp of the material (which you should check regularly) and do not look terribly overworked, I see nothing wrong with going fast. However, if you see that a noticeable portion of the class is falling behind, slow down and allow them more time to digest everything. Don't get surprised if the situation changes from year to year and even from topic to topic. The only danger arises when you fool yourself into believing that the students understand the lectures when they actually don't. Then the things can go astray really fast and get really ugly in the end.&lt;/p&gt;&#xA;&lt;p&gt;26 A's is an amazing result for an undergraduate calculus course (when I have 30 students, I usually give not more than 10, though I never curve, just set the bar for the passing score in the beginning and stick to it). Make sure that you don't demand too little (another standard way of fooling oneself into believing that everything works just fine).&lt;/p&gt;&#xA;&lt;p&gt;So, if everything is as you presented it, just don't change anything unless you see yourself that you have to. I have no way to check your claims against reality, so I leave it to you and your peers. But if you are not deceiving yourself, then my congratulations: you seem to have really good students and, probably, found an efficient way to teach them. Just make sure it is not a mirage. I agree with other commenters that it looks somewhat unbelievable. If not a secret, where are you teaching?&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-11-10T15:03:22.423" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25781" PostTypeId="2" ParentId="10348" CreationDate="2022-11-10T15:22:50.630" Score="2" Body="&lt;p&gt;I recently learned that the inequality&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ ab \le \varepsilon a^2 + \frac{1}{4\varepsilon} b^2 \qquad (\varepsilon &amp;gt; 0) $$&lt;/span&gt;&#xA;is referred to as &lt;strong&gt;Peter-Paul inequality&lt;/strong&gt;, in reference to the phrase &lt;a href=&quot;https://en.wikipedia.org/wiki/To_rob_Peter_to_pay_Paul&quot; rel=&quot;nofollow noreferrer&quot;&gt;rob Peter to pay Paul&lt;/a&gt;. This is a variation of &lt;span class=&quot;math-container&quot;&gt;$ab \le \frac 12 a^2 + \frac 12 b^2$&lt;/span&gt; written in a suggestive way: you can make the constant in front of &lt;span class=&quot;math-container&quot;&gt;$a^2$&lt;/span&gt; arbitrarily small at the expense of &lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt;, hence the name.&lt;/p&gt;&#xA;" OwnerUserId="15367" LastActivityDate="2022-11-10T15:22:50.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25782" PostTypeId="2" ParentId="25778" CreationDate="2022-11-10T15:58:35.417" Score="10" Body="&lt;p&gt;As fedja says, if your students are doing as well as you say, and you believe the students are similar this year, there isn't a reason to change.&lt;/p&gt;&#xA;&lt;p&gt;It seems hard to believe, though. I've taught calculus at U Michigan and UC Berkeley, which are generally considered to be good schools, and in 14 weeks with 4 hours a week, we generally get to your week 7. To give very rough numbers, about 25% of our students skip over first semester calculus and take a more advanced class, and about 50% never take it; so this course is targeting the 50th-75th percentile of math preparation. A bit more than half of them have had some high school calculus. Of course, we also have honors courses that cover much more; I'm talking about the non-honors section.&lt;/p&gt;&#xA;&lt;p&gt;The reason is that the students who take these classes don't have a strong understanding about how to think about functions, graphing and formulae. So that each concept has to be presented many ways before they can use it. Some reasons you might be seeing different results:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;You are an amazing instructor.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You are at an extremely selective college. (But then why do your colleagues not get the same results?)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For some reason, you have unusually strong students in your section. Do your students come from a different major than the norm, or are a large number of them transfer students from a different educational system?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You are pushing out the students who would fail before they get to the final exam. How many students drop the course, or change out of your section, before they get to the final?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Your exams are way too easy.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a large amount of cheating on your exams. (Were they administered online?)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If 1,2 or 3 applies and will apply next term then what you are doing is working so there is no reason to change. If 4 applies, then I believe you should work to keep and help those students, although it gets complicated. Obviously, if 5 or 6 applies you should fix your exams!&lt;/p&gt;&#xA;&lt;p&gt;And, if 1 applies, then congratulations!&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2022-11-10T17:30:33.373" LastActivityDate="2022-11-10T17:30:33.373" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25783" PostTypeId="2" ParentId="25776" CreationDate="2022-11-10T16:33:39.493" Score="9" Body="&lt;p&gt;I'm going to give an answer based on my feelings and memories, but I hope someone will give a more informed answer.&lt;/p&gt;&#xA;&lt;p&gt;This question is testing&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Does the student know how to multiply polynomials with more than 2 terms? Many students just learn the &lt;a href=&quot;https://en.wikipedia.org/wiki/FOIL_method&quot; rel=&quot;noreferrer&quot;&gt;FOIL mnemonic&lt;/a&gt; and are therefore confused by trinomials.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is the student comfortable with having many letters on the page, some of which are variables and some of which are coefficients?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Does the student understand summation notation and the word &amp;quot;coefficient&amp;quot;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Here are versions of the question which test subsets of these skills.&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Just 1&lt;/b&gt; Fill in the blanks:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(1+2x+3x^2)(4+5x+6x^2) = \underline{\hspace{0.5 in}}+\underline{\hspace{0.5 in}} x +\underline{\hspace{0.5 in}} x^2 +\underline{\hspace{0.5 in}}x^3 + \underline{\hspace{0.5 in}} x^4.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Just 2&lt;/b&gt; Fill in the blanks:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(a_0+a_1 x)(b_0 + b_1 x) = \underline{\hspace{0.5 in}}+\underline{\hspace{0.5 in}} x +\underline{\hspace{0.5 in}} x^2.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;1 and 2&lt;/b&gt; Fill in the blanks:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(a_0+a_1 x+a_2 x^2)(b_0 + b_1 x+b_2 x^2) = \underline{\hspace{0.5 in}}+\underline{\hspace{0.5 in}} x +\underline{\hspace{0.5 in}} x^2++\underline{\hspace{0.5 in}} x^3+\underline{\hspace{0.5 in}} x^4.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;Now the question: At what ages can excellent students do this? I would love to see an answer which addressed where these skills lie in the common core curriculum. Being lame, I am going to give my subjective impression. Also, of course, there is the question of what is &amp;quot;excellent&amp;quot;. I'm going to take this to mean the top few students in a typical school.&lt;/p&gt;&#xA;&lt;p&gt;In my uninformed opinion, an excellent middle school student at a median middle school can do the &amp;quot;just 1&amp;quot; problem. &lt;a href=&quot;https://www.mathcounts.org/resources/past-competitions&quot; rel=&quot;noreferrer&quot;&gt;MATHCOUNTS&lt;/a&gt; and &lt;a href=&quot;https://artofproblemsolving.com/wiki/index.php/AMC_8_Problems_and_Solutions&quot; rel=&quot;noreferrer&quot;&gt;AMC-8&lt;/a&gt; are two competitions targeting students at this level, and this feels easy by their standards. However, if you notice, their problems are written in much more concrete language than your question, so I am not sure whether an excellent middle school student can do &amp;quot;just 1 and 2&amp;quot;, let alone the original question.&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure much changes at high school? The level of the lower students moves up a lot: At middle school, IIRC, a large fraction of students take algebra but the median student doesn't whereas, by high school, they almost all have. But &amp;quot;the top math student from an unexceptional high school&amp;quot; is the sort of student I see a lot of teaching at U Michigan, and lots of them are still confused by skills 2 and 3.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps naively, I think that an excellent college freshman should be able to do all of this. Certainly, the top students coming into U Michigan would have no problem, but I would like to think that, at most universities, the top freshmen have all these skills.&lt;/p&gt;&#xA;&lt;p&gt;At any of these ages, if you change &amp;quot;excellent&amp;quot; to &amp;quot;typical&amp;quot;, the typical student struggles with the &amp;quot;just 1&amp;quot; version.&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="51" LastEditDate="2022-11-10T17:00:19.497" LastActivityDate="2022-11-10T17:00:19.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25784" PostTypeId="1" AcceptedAnswerId="25787" CreationDate="2022-11-10T17:04:26.477" Score="6" ViewCount="1236" Body="&lt;p&gt;As many Americans know, the “traditional” high school sequence is:&lt;/p&gt;&#xA;&lt;p&gt;Algebra 1&lt;/p&gt;&#xA;&lt;p&gt;Geometry&lt;/p&gt;&#xA;&lt;p&gt;Algebra 2&lt;/p&gt;&#xA;&lt;p&gt;PreCalculus&lt;/p&gt;&#xA;&lt;p&gt;Calculus&lt;/p&gt;&#xA;&lt;p&gt;For those who take developmental education at the community college level, it consists of something like:&lt;/p&gt;&#xA;&lt;p&gt;Developmental Algebra&lt;/p&gt;&#xA;&lt;p&gt;Intermediate Algebra&lt;/p&gt;&#xA;&lt;p&gt;College Algebra&lt;/p&gt;&#xA;&lt;p&gt;PreCalculus&lt;/p&gt;&#xA;&lt;p&gt;Calculus&lt;/p&gt;&#xA;&lt;p&gt;While the college courses cover most of the algebra, there seems to be no Geometry in the curriculum. Why is that?  If there's a good reason for it not to be covered in the Community College system, does it still have a place in high school?&lt;/p&gt;&#xA;" OwnerUserId="9968" LastEditorUserId="15367" LastEditDate="2022-11-10T20:38:04.203" LastActivityDate="2022-11-11T14:05:26.897" Title="Geometry in the Community College Curriculum" Tags="|mathematical-pedagogy|geometry|usa|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25785" PostTypeId="2" ParentId="11673" CreationDate="2022-11-10T20:20:08.747" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://hannahfry.co.uk/books/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dr. Hannah Fry's books&lt;/a&gt; Mathematics of Love and The Indisputable Existence of Santa Claus are short, charming, and easy to read.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Logicomix&quot; rel=&quot;nofollow noreferrer&quot;&gt;Logicomix&lt;/a&gt; is a bit more philosophical and historical, but a very intriguing graphic novel on the formalization of logic.&lt;/p&gt;&#xA;&lt;p&gt;Not a book, but the youtube channel &lt;a href=&quot;https://www.youtube.com/c/numberphile&quot; rel=&quot;nofollow noreferrer&quot;&gt;Numberphile&lt;/a&gt; is also a tremendous resource on making mathematics approachable. I like Tadashi Tokieda's videos the most.&lt;/p&gt;&#xA;" OwnerUserId="20586" LastActivityDate="2022-11-10T20:20:08.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25786" PostTypeId="2" ParentId="11673" CreationDate="2022-11-10T21:28:30.947" Score="0" Body="&lt;p&gt;A little more mathematical, but very accessible, and useful:&lt;/p&gt;&#xA;&lt;p&gt;How to Lie with Statistics&lt;/p&gt;&#xA;" OwnerUserId="20600" LastActivityDate="2022-11-10T21:28:30.947" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25787" PostTypeId="2" ParentId="25784" CreationDate="2022-11-10T21:59:58.230" Score="11" Body="&lt;p&gt;&amp;quot;Geometry,&amp;quot; the American high school course, is generally pseudo-axiomatic Euclidean geometry. I don't know whether your claim about the CC curriculum is broadly true, but assuming it is, it's probably because Euclidean geometry is simply not a prerequisite for calculus. Students need to know the Pythagorean theorem and a bit about areas and volumes, which are covered well before high school geometry; beyond that, all the geometric content in calculus is either analytic or heavily trigonometry-influenced. And trigonometry is covered mainly in precalculus.&lt;/p&gt;&#xA;&lt;p&gt;There is a lot to wonder about whether high school geometry is a necessary course. The main argument in its favor is that it gives students a brief acquaintance with logical rigor, a very important part of intellectual culture that they otherwise won't seriously meet unless they get to a course as advanced as abstract algebra in college. But practically speaking, this isn't obviously a compelling argument; causally, Euclidean geometry is still there because Euclidean geometry was the pinnacle of early modern math education. It's not hard to argue that it would better be replaced, for instance with wider exposure to statistics and probability. But that's really just a matter of opinion.a&lt;/p&gt;&#xA;" OwnerUserId="5233" LastActivityDate="2022-11-10T21:59:58.230" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25788" PostTypeId="2" ParentId="25776" CreationDate="2022-11-10T22:16:37.110" Score="4" Body="&lt;p&gt;It would be no big deal for a strong US algebra 2 high school student, who is about half way through the course. Done their work on factoring and the like.  Like around this time of year.&lt;/p&gt;&#xA;&lt;p&gt;Stereotypically that is 11th grad (16-17ish), but many strong college prep students would take that course in 10th grade (15-16ish).  They may or may not have memorized how to multiply trinomials, but anyone can do it the brute force way (multiply each thing in first by each thing in second and get 9 terms and then group them).&lt;/p&gt;&#xA;&lt;p&gt;I don't think it's a fair question for kids younger than that.  (Some might get it, but they really haven't practiced this sort of manipulation.  Would be rare.)&lt;/p&gt;&#xA;&lt;p&gt;For college kids, it should be doable, expecially if they are &amp;quot;excellent&amp;quot;.  I would not expect remedial students (who aren't even taking calculus first year) to handle it.  But of course, they aren't excellent. They are remedial.&lt;/p&gt;&#xA;&lt;p&gt;Of course we don't like in Lake Wobegone, so the average student is not an excellent one.&lt;/p&gt;&#xA;" OwnerUserId="20600" LastEditorUserId="60" LastEditDate="2022-11-11T01:07:44.590" LastActivityDate="2022-11-11T01:07:44.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25789" PostTypeId="2" ParentId="25784" CreationDate="2022-11-11T09:33:29.437" Score="3" Body="&lt;p&gt;Typically, a geometry class in high school teaches Euclidean geometry. Depending on how much time is spent and the exact class, Euclidean geometry as rendered today explores properties of triangle, parallelogram, and circle, while following an axiomatic system, much of which was initially laid out thousands of years ago. The purpose is less to understand these objects deeply per se but to learn how a large set of conclusions can be derived from a few simple assumptions.&lt;/p&gt;&#xA;&lt;p&gt;In higher education past high school, modern mathematics are taught instead -- either to prepare students to research in mathematics, or to prepare students for applications of mathematics. Euclidean geometry would be rather remote from either. No current research fields draw much from the old school Euclidean geometry -- we are thousands of years past that after all. No common applications require Euclidean geometry either.&lt;/p&gt;&#xA;&lt;p&gt;In the case of training students on logical rigor, an advanced degree have many more such opportunities. Thus no need to spend time on Euclidean geometry. For high school, some of the more intricate machineries in modern mathematics can be too abstract. Euclidean geometry thus became the quick and more viable way to expose students to an axiomatic approach.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The competition for time should be a thing in high school too. So it's worth mentioning that, in Asian countries, Euclidean geometries are taught in junior high schools. ie. grade 7-9 (though there is little teaching in grade 9 with it being mostly exam prep) where grade 1-6 is the primary school. With that arrangement, any hypothesized benefits would be gained without taking away valuable time in high school that can be used for further education in math -- to teach foundation for calculus (including some basic analysis), probability, etc, which in turn can be especially helpful in preparing students who will enter fields that require applications of math yet won't have a lot of time to spare for math such as the various engineering fields.&lt;/p&gt;&#xA;" OwnerUserId="20601" LastEditorUserId="20601" LastEditDate="2022-11-11T09:48:04.537" LastActivityDate="2022-11-11T09:48:04.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25790" PostTypeId="2" ParentId="25784" CreationDate="2022-11-11T14:03:18.880" Score="4" Body="&lt;p&gt;The other answers well discuss how geometry is sort of &amp;quot;off track&amp;quot; for the push to calculus and the other topics (trig, algebra) are much more integral.  But I don't think they make the point enough that community college students are already behind the curve in terms of time.  We are talking about kids taking remedial classes (I guess &amp;quot;developmental&amp;quot; is a euphemism...first time hearing it.)  So they have even more need to prioritize than high school students (Sue does in her comment/answer).  An allowable luxury for high school students is not allowable for kids that are several years behind and just need to get that calc credit so they can get their BS Nursing or Accounting or whatever.  And by the way, those courses in nursing and accounting are more vital in the end than this math stuff...really, truly, for them.&lt;/p&gt;&#xA;&lt;p&gt;Of course this is not to say no geometry is ever needed, or it is completely irrelevant to calculus.  But when we work and live and make life choices, we live in a world of priorities, not of &amp;quot;divide by zero&amp;quot; rigor absolutes.  There is not time to study everything.  So finding isolated connections is not the same as prevalent connections.  Heck, even for the basic formulas (like triangle equation in related rates problems), I remember thinking in HS geometry, how we really weren't learning much geometry (measurement of land), but seemed to be learning vast amounts of proofs.&lt;/p&gt;&#xA;" OwnerUserId="20585" LastEditorUserId="20585" LastEditDate="2022-11-11T14:05:26.897" LastActivityDate="2022-11-11T14:05:26.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25791" PostTypeId="1" CreationDate="2022-11-11T16:02:54.590" Score="4" ViewCount="204" Body="&lt;p&gt;I'm teaching this year out of &amp;quot;Precalculus with limits&amp;quot; by Ron Larson [7th ed], and the following expression appears in the unit introducing polynomial functions:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)=a{(x-h)}^2+k$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;He calls this the &amp;quot;standard form of a quadratic function&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I've only ever seen this form referred to as &amp;quot;vertex form&amp;quot;, whereas &amp;quot;standard form&amp;quot; instead means a quadratic expression written as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ax^2+bx+c$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This ambiguity is EXTREMELY confusing for my students, and it's really obnoxious for me, too. (I have to rewrite all materials to&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know why this terminological variation is introduced in this textbook or how well-established it is in general?  I can't think of any good reason for changing it now unless it actually is well-established elsewhere.  But I don't think I've ever seen this over the past 25 years, at least not in the US.&lt;/p&gt;&#xA;" OwnerUserId="10572" LastActivityDate="2022-11-21T05:58:48.160" Title="Why is there variation in the meaning of &quot;Standard form&quot; for a quadratic?" Tags="|secondary-education|terminology|functions|polynomials|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25792" PostTypeId="1" AcceptedAnswerId="25837" CreationDate="2022-11-12T09:18:24.683" Score="-1" ViewCount="629" Body="&lt;p&gt;Too many &lt;a href=&quot;https://matheducators.stackexchange.com/q/17854&quot;&gt;mature students&lt;/a&gt; ask why they need to toil at problems —  like end of chapter questions in textbooks  — to succeed at math. They contrast buckling  to math exercises with lasting skills like driving —  because after earning their license, &lt;a href=&quot;https://www.reddit.com/r/explainlikeimfive/comments/37ulg4/eli5_why_do_we_remember_some_skills_eg_swimming/crpygxq&quot; rel=&quot;nofollow noreferrer&quot;&gt;most adults can still drive skillfully&lt;/a&gt; without drudgery or exertion, even though AI still can't drive!&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.technologyreview.com/2020/02/19/868178/what-ai-still-cant-do&quot; rel=&quot;nofollow noreferrer&quot;&gt;Machine-learning systems can be duped or confounded by situations they haven’t seen before. A self-driving car gets flummoxed by a scenario that a human driver could handle easily. An AI system laboriously trained to carry out one task (identifying cats, say) has to be taught all over again to do something else (identifying dogs). In the process, it’s liable to lose some of the expertise it had in the original task. Computer scientists call this problem “catastrophic forgetting.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;What distinguishes math from these lifelong competencies?  Why &lt;a href=&quot;https://www.reddit.com/r/explainlikeimfive/comments/t2hn12/eli5_why_do_we_forget_long_practiced_skills_after/hymkhsn&quot; rel=&quot;nofollow noreferrer&quot;&gt;can you lose math, if you don't use it&lt;/a&gt;  — but you can still remember and master other lifetime competences like driving, without moil or practice?&lt;/h1&gt;&#xA;&lt;p&gt;After acquiring their licence, most adult drivers don't need to deliberately plod or grind away at driving with an instructor. Most still drive competently without incident, pains, or sweat! After issuing licenses, Canada and USA aren't required to test drivers again until elderliness!&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.abbynews.com/news/b-c-senior-calls-driving-exams-for-seniors-aged-80-and-up-unfair/&quot; rel=&quot;nofollow noreferrer&quot;&gt;RoadSafetyBC, the B.C. government’s agency responsible for road safety, mandates persons 80 and above to renew their driver’s licence every two years. They are required to get a medical examination report to be completed by their physicians. [...] Those that fail the medical exams, however, may end up having to take the Enhanced Road Assessment to determine whether they are fit to drive. It is free of charge but it takes a 90 minutes to complete and includes a pre-trip vehicle orientation, a 45-minute on-road drive and a post-trip review.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.dmv.ca.gov/portal/driver-education-and-safety/special-interest-driver-guides/senior-drivers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;DMV does not take away your driver’s license when you reach a certain age. Your mental and/or physical condition or your inability to follow traffic laws and rules regardless of age determines whether your license is renewed, restricted, suspended, or revoked.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h4&gt;Some parents analogize that most humans never forget competencies of &lt;a href=&quot;https://www.reddit.com/r/explainlikeimfive/comments/8ddm3n/eli5_why_cant_we_forget_how_to_swim_or_to_ride_a/dxmccl1&quot; rel=&quot;nofollow noreferrer&quot;&gt;muscle memory like swimming&lt;/a&gt;, or &lt;a href=&quot;https://www.reddit.com/r/explainlikeimfive/comments/1l00xe/eli5_how_is_it_that_we_never_forget_how_to_ride_a/cbuei4q&quot; rel=&quot;nofollow noreferrer&quot;&gt;implicit memory  like how to ride a bike&lt;/a&gt; — even if they haven't done this in years!&lt;/h4&gt;&#xA;&lt;h3&gt;Some parents analogize to cooking — as most people haven't attended culinary school, but most can still cook well enough.&lt;/h3&gt;&#xA;&lt;p&gt;Not having studied cooking professionally, most people can't cook like Michelin Star chefs. Most people don't even seek to advance or hone their cooking, but most people can still cook to survive!&lt;/p&gt;&#xA;" OwnerUserId="20536" LastEditorUserId="20536" LastEditDate="2023-06-24T21:15:41.090" LastActivityDate="2023-06-24T21:15:41.090" Title="Why must most adults slog at exercises to improve at Math, but not driving that AI still can't automate?" Tags="|exercises|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25793" PostTypeId="2" ParentId="12718" CreationDate="2022-11-12T09:57:36.560" Score="0" Body="&lt;p&gt;Is there a way to cross-reference MathOverflow? This was asked and answered there.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/q/371486&quot;&gt;Why is it still common to not motivate results in publications? &lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/q/315843&quot;&gt;Examples of Mathematical Papers that Contain a Kind of Research Report&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/q/358&quot;&gt;Examples of great mathematical writing&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20536" LastEditorUserId="20536" LastEditDate="2022-11-13T01:06:44.137" LastActivityDate="2022-11-13T01:06:44.137" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25794" PostTypeId="2" ParentId="25792" CreationDate="2022-11-12T12:03:09.540" Score="7" Body="&lt;p&gt;Our minds are not well adapted to math. It is an abstract activity.  And requires significant practice to sink in.  We are not rule based systems.  Not flawless computers. Our brains are optimized for social interactions and physical control of our bodies. Not for math, although they are plastic enough to have some ability in math, it is not their strong suit.&lt;/p&gt;&#xA;&lt;p&gt;All that said, your question seems a bit  muddled in asking why need to do end of chapter exercises which are clearly part of the initial learning process, and then why you don't retain some unpracticed math techniques many years later, like bike riding.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, it is true you may not recall some math technique years later, if unpracticed, you can probably relearn it much easier if needed than someone learning it the first time.&lt;/p&gt;&#xA;&lt;p&gt;Finally, it is not just math per se, but other calculational topics like chemistry and physics that benefit from exercises.&lt;/p&gt;&#xA;" OwnerUserId="20606" LastActivityDate="2022-11-12T12:03:09.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25795" PostTypeId="2" ParentId="41" CreationDate="2022-11-12T12:19:15.953" Score="0" Body="&lt;p&gt;Math handwriting is &lt;em&gt;not&lt;/em&gt; always a challenge. You can always change things up.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Yz1pQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Yz1pQ.png&quot; alt=&quot;Figure 1: I, i, 1, 7, l&quot; /&gt;&lt;/a&gt;&#xA;It's best to keep &lt;code&gt;1&lt;/code&gt; a straight line, and more importantly, add a bar to &lt;code&gt;7&lt;/code&gt;. And then, there's the letters &lt;code&gt;I&lt;/code&gt; and &lt;code&gt;i&lt;/code&gt;. Always put bars on the &lt;code&gt;I&lt;/code&gt; and a dot on the &lt;code&gt;i&lt;/code&gt;, obviously. The lowercase &lt;code&gt;l&lt;/code&gt; is always tricky, and confusing, but using a cursive version always helps.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BYIlm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BYIlm.png&quot; alt=&quot;Figure 2: Z, z, 2, α, 3&quot; /&gt;&lt;/a&gt;&#xA;Next is &lt;code&gt;2&lt;/code&gt;. &lt;code&gt;2&lt;/code&gt; can maybe look like &lt;code&gt;α&lt;/code&gt; and probably &lt;code&gt;3&lt;/code&gt;. Make the loop smaller on the &lt;code&gt;2&lt;/code&gt;, and don't add a loop to the &lt;code&gt;3&lt;/code&gt;. The &lt;code&gt;2&lt;/code&gt; may look like &lt;code&gt;Z&lt;/code&gt; and &lt;code&gt;z&lt;/code&gt;, but crossing it makes it look different easily.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GvFv9.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GvFv9.png&quot; alt=&quot;Figure 3: y, g, 9, 4, q, 8, β&quot; /&gt;&lt;/a&gt;&#xA;There's the numbers &lt;code&gt;9&lt;/code&gt;, &lt;code&gt;4&lt;/code&gt;, and the letter &lt;code&gt;q&lt;/code&gt;. Open the &lt;code&gt;4&lt;/code&gt;, and add a tail to &lt;code&gt;q&lt;/code&gt; so it doesn't look like a &lt;code&gt;9&lt;/code&gt;. Don't add a loop to a &lt;code&gt;q&lt;/code&gt;, it may look like &lt;code&gt;8&lt;/code&gt; or &lt;code&gt;β&lt;/code&gt;. Don't add a hook to the &lt;code&gt;9&lt;/code&gt; because it may look like a single-storey &lt;code&gt;g&lt;/code&gt;, and always make the &lt;code&gt;g&lt;/code&gt; double-storey. It's probably best to keep the &lt;code&gt;y&lt;/code&gt; curved.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/y2Hpc.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/y2Hpc.png&quot; alt=&quot;Figure 4: S, 5, s&quot; /&gt;&lt;/a&gt;&#xA;There's the number &lt;code&gt;5&lt;/code&gt; and the lookalikes &lt;code&gt;S&lt;/code&gt; and &lt;code&gt;s&lt;/code&gt;. Always write &lt;code&gt;5&lt;/code&gt; with two strokes, add a serif to &lt;code&gt;S&lt;/code&gt;, and make &lt;code&gt;s&lt;/code&gt; smaller.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tE2tx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tE2tx.png&quot; alt=&quot;Figure 5: C, 6, G&quot; /&gt;&lt;/a&gt;&#xA;There's &lt;code&gt;6&lt;/code&gt;, &lt;code&gt;C&lt;/code&gt;, and &lt;code&gt;G&lt;/code&gt;. Make sure to close the &lt;code&gt;6&lt;/code&gt; properly and give a pointy end to &lt;code&gt;G&lt;/code&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wBB96.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wBB96.png&quot; alt=&quot;Figure 6: Q, O, o, 0, θ, ∅, ϕ&quot; /&gt;&lt;/a&gt;&#xA;And worst of all: &lt;code&gt;0&lt;/code&gt;, &lt;code&gt;O&lt;/code&gt;, and &lt;code&gt;o&lt;/code&gt;. Make sure to add a loop on the &lt;code&gt;O&lt;/code&gt; and &lt;code&gt;o&lt;/code&gt;, because adding a slash to &lt;code&gt;0&lt;/code&gt;, makes it a different one. Small stroke will be a &lt;code&gt;Q&lt;/code&gt;, horizontal slash will be a &lt;code&gt;θ&lt;/code&gt;, large diagonal slash will be a &lt;code&gt;∅&lt;/code&gt;, and a vertical slash will be a &lt;code&gt;ϕ&lt;/code&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/W2uAl.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/W2uAl.png&quot; alt=&quot;Figure 7: u, V, v, W, w, ν, ω&quot; /&gt;&lt;/a&gt;&#xA;There are letters worth mentioning, like &lt;code&gt;u&lt;/code&gt;, &lt;code&gt;V&lt;/code&gt; &lt;code&gt;v&lt;/code&gt;, &lt;code&gt;W&lt;/code&gt; &lt;code&gt;w&lt;/code&gt;, &lt;code&gt;ν&lt;/code&gt; and &lt;code&gt;ω&lt;/code&gt;. Make sure to make &lt;code&gt;v&lt;/code&gt; like the &lt;code&gt;ν&lt;/code&gt; in Lucida Grande, and also apply it for &lt;code&gt;V&lt;/code&gt;, &lt;code&gt;W&lt;/code&gt;, and &lt;code&gt;w&lt;/code&gt;. Make sure to keep &lt;code&gt;ν&lt;/code&gt;'s left side vertical, add a hook on the left. Keep &lt;code&gt;ω&lt;/code&gt; rounded.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oNzoD.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oNzoD.png&quot; alt=&quot;Figure 8: u, μ, υ, n, η&quot; /&gt;&lt;/a&gt;&#xA;Here are the letters: &lt;code&gt;u&lt;/code&gt;, &lt;code&gt;μ&lt;/code&gt;, &lt;code&gt;υ&lt;/code&gt;, &lt;code&gt;n&lt;/code&gt;, &lt;code&gt;η&lt;/code&gt;. Make sure that &lt;code&gt;μ&lt;/code&gt; is in 2 different strokes, &lt;code&gt;υ&lt;/code&gt; and &lt;code&gt;η&lt;/code&gt; has a curved hook, and &lt;code&gt;μ&lt;/code&gt; and &lt;code&gt;η&lt;/code&gt; longer than &lt;code&gt;u&lt;/code&gt; and &lt;code&gt;n&lt;/code&gt; respectively.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PDrXi.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PDrXi.png&quot; alt=&quot;Figure 9: X, x, χ, ×, t, +&quot; /&gt;&lt;/a&gt;&#xA;We have &lt;code&gt;X&lt;/code&gt;, &lt;code&gt;x&lt;/code&gt;, &lt;code&gt;χ&lt;/code&gt;, and &lt;code&gt;×&lt;/code&gt;, and writing them normally would be confusing. Add a hook to &lt;code&gt;X&lt;/code&gt;, make &lt;code&gt;x&lt;/code&gt; like two parentheses curved more and joined, &lt;code&gt;χ&lt;/code&gt; like in slanted Times New Roman, and &lt;code&gt;×&lt;/code&gt; small. We also have &lt;code&gt;t&lt;/code&gt; and &lt;code&gt;+&lt;/code&gt;. Add a hook to &lt;code&gt;t&lt;/code&gt; and make &lt;code&gt;+&lt;/code&gt; small.&lt;/p&gt;&#xA;&lt;p&gt;And that's all!&lt;/p&gt;&#xA;&lt;p&gt;Preview of all of the letters:&lt;/p&gt;&#xA;&lt;p&gt;Latin Lowercase:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Lqb3R.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Lqb3R.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/0c2Nr.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0c2Nr.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/j6hy2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/j6hy2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/8rhjk.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8rhjk.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;Latin Uppercase:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/HK2ME.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HK2ME.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/0CGZy.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0CGZy.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/9OisK.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9OisK.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/rpcQp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rpcQp.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;Greek Lowercase:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/I1afq.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/I1afq.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/apwyk.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/apwyk.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/o7te8.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/o7te8.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;Greek Uppercase:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/PmQ44.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PmQ44.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/BF4m2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BF4m2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/fnmxz.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fnmxz.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;Numbers:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/z3F3N.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/z3F3N.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/4Hoc9.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4Hoc9.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;Others:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ODp2b.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ODp2b.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/I70HM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/I70HM.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/DarvX.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DarvX.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;Common Variations (Original in CYAN, Alternates in YELLOW):&#xA;&lt;a href=&quot;https://i.stack.imgur.com/DL1i2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DL1i2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/AExJr.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AExJr.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/aCQCf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aCQCf.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/2mYUd.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2mYUd.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/oRvyQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oRvyQ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/JtcMr.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JtcMr.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/CW0tc.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CW0tc.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/j7Wi5.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/j7Wi5.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/eZgL2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eZgL2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20605" LastEditorUserId="20605" LastEditDate="2022-12-11T08:18:32.903" LastActivityDate="2022-12-11T08:18:32.903" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25796" PostTypeId="2" ParentId="25792" CreationDate="2022-11-12T15:23:52.637" Score="7" Body="&lt;p&gt;Generally, when we talk about improving a skill, we tend to refer to skills that are physical-based. For instance, a professional basketball player may be a strong three-point shooter because they go to the gym, shoot a lot of three-pointers, and make adjustments to improve their results. Also, marching bands learn halftime shows and each member has to memorize the music and where they are supposed to be on the field at any given time, which they improve at through repetition during rehearsal.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.azquotes.com/author/28949-Paul_Halmos&quot; rel=&quot;noreferrer&quot;&gt;Paul Halmos&lt;/a&gt; famously said &amp;quot;The only way to learn mathematics is to do mathematics.&amp;quot;, which I interpret as although mathematics is inherently an abstract discipline, there is still some physical component to it. Practice is a significant part of improving at mathematics, because solely watching someone else do math problems such as the teacher or some stranger in a YouTube video is only sustainable for short-term and not long-term internalization. At some point, the student needs to roll up their sleeves and work on solving math problems. Hence, if you look at how to improve a math skill through a lens similar to the examples mentioned above, learning math is not so different from learning other skills as one might think.&lt;/p&gt;&#xA;&lt;p&gt;The challenges with helping students improve at mathematics are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;as mentioned by Guest, the human brain needs to work harder to understand mathematics since it is an abstract discipline.&lt;/li&gt;&#xA;&lt;li&gt;having a system where the students not only have the opportunity to practice mathematics, but practice &lt;em&gt;efficiently&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;To facilitate overcoming these obstacles, I minimize lecturing and have my students work on carefully-chosen activities that allow them to see how the concepts unfold and build off of each other, even if I have to break down a more complicated problem into multiple parts so they can pull through. Then we discuss the main idea of each activity where I try to get several students to state the main idea. They need to &amp;quot;own&amp;quot; the lesson and be able to explain the concept(s) in their own words because the person who does the talking does the learning.&lt;/p&gt;&#xA;&lt;p&gt;This sets them up to complete the regularly assigned homework. I justify to my students that they need to practice as soon as possible because the main ideas of the lesson are reinforced most effectively when the students work on the practice problems the same day, as opposed to several days later when they may have forgotten everything. This also allows them to discover sooner which concept(s) they need to spend more time practicing (homework is assigned online so instant feedback is given), as opposed to discovering this the day before the assessment. &lt;strong&gt;Instant feedback is critical when working on practice problems.&lt;/strong&gt; For instance, if a student were to solve 10 quadratic equations and unknowingly solve all 10 of them incorrectly due to the lack of instant feedback, then recovering from that involves reteaching the skill, which puts the student in a poor position since they have to undo the muscle memory from the 10 incorrect attempts.&lt;/p&gt;&#xA;&lt;p&gt;Eventually, the summative assessment takes place and then we move on to the next unit. However, the concepts need to be revisited at some point in the future so that the students do not forget them. One way I achieve this is by allowing my students to retake assessments if they are not satisfied which their performance because this is a form of &amp;quot;doing math&amp;quot;; these students are taking it upon themselves to learn from their mistakes, thus showing that they understand how learning mathematics is a continuous process that does not end with taking a test.&lt;/p&gt;&#xA;&lt;p&gt;Despite my carefully organized system, it still may not be enough because the students are ultimately responsible for their learning. Their views towards mathematics, their motivation to succeed, and the busy nature of their lives all affect their willingness or how much time they have to practice efficiently. Nevertheless, doing exercises and problems is fundamentally what allows students to &amp;quot;do math&amp;quot;, and engaging in these activities efficiently allows students to &lt;em&gt;learn&lt;/em&gt; math.&lt;/p&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="13657" LastEditDate="2022-11-12T17:20:42.360" LastActivityDate="2022-11-12T17:20:42.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25797" PostTypeId="2" ParentId="220" CreationDate="2022-11-12T22:29:32.437" Score="-1" Body="&lt;p&gt;I'm quite surprised that nobody so far has mentioned the&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Existence of prime factorization of integers:&lt;/strong&gt; Every integer &lt;span class=&quot;math-container&quot;&gt;$n \ge 2$&lt;/span&gt; can be written as a finite product of prime numbers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Proof.&lt;/em&gt; It is true for &lt;span class=&quot;math-container&quot;&gt;$n=2$&lt;/span&gt; since &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is prime, so let &lt;span class=&quot;math-container&quot;&gt;$n \ge 2$&lt;/span&gt; and assume that the claim has been proved for all integers from &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; is prime, there is nothing to prove. If &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; is not prime, then we can write it as &lt;span class=&quot;math-container&quot;&gt;$n+1 = ab$&lt;/span&gt; for two integers &lt;span class=&quot;math-container&quot;&gt;$a,b \in \{2, \dots, n\}$&lt;/span&gt;; by induction hypothesis, &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; can be written as products of prime numbers, so the same is true for &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$\square$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Uniqueness of the factorization is more difficult, though.&lt;/p&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2022-11-12T22:29:32.437" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25798" PostTypeId="1" AcceptedAnswerId="25804" CreationDate="2022-11-13T00:38:00.303" Score="2" ViewCount="897" Body="&lt;p&gt;Yearly, at least one student emails me this question, after wholly relying on passive reading then failing the exam. They successfully remember and can regurgitate everything from the textbook and lectures; can compute a question if analogous enough to something déjà vu, and something they saw before; and can prove straightforward corollaries. But they can't answer unseen, new-fashioned questions or trickier computations. They wholly fail to prove more abstract results.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Students tend to &lt;em&gt;prefer&lt;/em&gt; passive studying &lt;strong&gt;because it's easier and more comfortable&lt;/strong&gt;. In fact, &lt;a href=&quot;https://www.pnas.org/content/pnas/early/2019/09/03/1821936116.full.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Harvard study&lt;/a&gt; shows that most students even convince themselves that passive methods are more effective, and that lecturing professors provide &amp;quot;better learning&amp;quot; than professors who engage students in-class projects, even though those hands-on projects ultimately provide much deeper education than traditional passive methods. &lt;a href=&quot;https://www.brainscape.com/academy/active-studying-vs-passive-studying/&quot; rel=&quot;nofollow noreferrer&quot;&gt;[Source]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h3&gt;But why precisely does passive study fail for learning math? Why can't math students simply read multiple textbooks on the same subject + DETAILED FULL solutions to exercises and problems, in lieu of picking up a writing implement and plugging away at exercises?&lt;/h3&gt;&#xA;&lt;p&gt;I am seeking explanations more detailed, scholarly, and specific to math than &lt;a href=&quot;https://www.leonardoenglish.com/blog/passive-language-learning-is-bullshit&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The reason is that learning is not a passive activity. As &lt;a href=&quot;http://mjltm.org/files/site1/user_files_a9608a/admin-A-10-1-23-7c58f0c.pdf#page=125&quot; rel=&quot;nofollow noreferrer&quot;&gt;some researchers&lt;/a&gt; have stated, &lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;“Students cannot learn just by attending the classes, listening to teachers, memorising ready-made assignments, and producing answers. They must talk about what they are learning (output), write about it (integration), connect it to past knowledge, and use it in their daily lives. They must make what they learn part of their lives by practice.”&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The problem with passive learning is that you &lt;a href=&quot;https://universeofmemory.com/optimize-your-language-learning/&quot; rel=&quot;nofollow noreferrer&quot;&gt;don’t engage with the information you receive.&lt;/a&gt; And that means you don’t learn it as well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is not always so when recalling facts in a self-test—more effort is often required to bring the facts to mind, so they don’t seem as solid. From a student’s point of view, it can seem obvious which method—restudying—produces better learning. Robert Bjork refers to this as an “illusion of competence” after restudying. The student concludes that she knows the material well based on the confident mastery she feels at that moment.&lt;/p&gt;&#xA;&lt;p&gt;And she expects that the same mastery will be there several days later when the exam takes place. But this is unlikely. The same illusion of competence is at work during cramming, when the facts feel secure and firmly grasped. While that is indeed true at the time, it’s a mistake to assume that long-lasting memory strength has been created. - &lt;a href=&quot;https://books.google.pl/books/about/How_Memory_Works_and_How_to_Make_It_Work.html?id=R3T3CQAAQBAJ&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Madigan - How Memory Works&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20536" LastActivityDate="2022-11-14T07:18:45.027" Title="Why can't students master math simply by passive reading?" Tags="|students-mistakes|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25799" PostTypeId="1" CreationDate="2022-11-13T06:14:23.547" Score="2" ViewCount="511" Body="&lt;p&gt;I am trying to find samples of completed homework, class work and tests for 7th and 8th grade math in the US. I can find a million blank workbooks but not copies that students have completed with shown work, correct answers, incorrect answers etc.&lt;/p&gt;&#xA;&lt;p&gt;If you know of good resources or have materials of your own with completed assignments I'd love to hear from you!&lt;/p&gt;&#xA;&lt;p&gt;I'm developing a virtual teacher assistant.&lt;br /&gt;&#xA;For it to be able to learn it needs thousands of samples to recognize patterns and correctly grade assignments/tests, and over time customize learning strategies for individual students and their learning needs.&lt;/p&gt;&#xA;&lt;p&gt;Would love any support, feedback, resources. My goal is to build this with teachers for teachers.&lt;/p&gt;&#xA;" OwnerUserId="20620" LastEditorUserId="2083" LastEditDate="2022-11-14T08:16:47.740" LastActivityDate="2022-11-14T08:16:47.740" Title="Help needed to find 7th &amp; 8th grade completed math samples" Tags="|technology-in-education|interactive-teaching|usa|middle-school|resource-request|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25800" PostTypeId="2" ParentId="25799" CreationDate="2022-11-13T09:16:57.667" Score="6" Body="&lt;p&gt;One way to go is to do this in partnership with a school or a university (or other higher education institute) that does teacher education.&lt;/p&gt;&#xA;&lt;p&gt;Pupil workbooks are rarely published, and you might face some issues of copyright or ethics if the corpus you find is not explicitly published under a free license.&lt;/p&gt;&#xA;&lt;p&gt;So: since your goal is to build it with teachers, talk with teachers. Check your local legislation for what kind of ethics/copyright clearance you need, if anything, to gather and analyze pupil data (universities can help with this kind of stuff). Then figure out together with the teachers the logistics of it and get going.&lt;/p&gt;&#xA;&lt;p&gt;I do admit that it sounds like an ambitious project if you intend to take as input pictures or scans of hand-drawn responses (for example) and want to output useful formative, or even summative, assessment. The program also needs to understand the question and the varied and freeform responses. Maybe some more limited format, like digital responses, are a bit easier to get started with.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-11-13T09:16:57.667" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25803" PostTypeId="2" ParentId="4382" CreationDate="2022-11-13T19:41:50.710" Score="-1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It's a moderately complicated expression.  With x, p, and a -1....and then putting a 1 into it.  It's not complicated for you (or for strong students), but for neophytes and weak students, verily, it is.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I think there's some weakness on order of operations.  Even if the kids mostly get it right, they don't have it down gnat's ass.  So in a tough situation, they flub.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some of them are not being enough &amp;quot;step by step&amp;quot;.  Yes, you showed that some had an intermediate step.  But many didn't.  And even for those, they'd be more likely to get it, by using multiple intermediate steps.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Finally, I think this question falls into a class that we often see on here.  The &amp;quot;why don't kids get something that is easy for me&amp;quot;.  It shows a little bit of a lack of empathy or at least of savviness.  People are different.  Your students are (likely) both less experienced than you and less innately strong at math.&lt;/p&gt;&#xA;&lt;p&gt;However, to give it credit, it looks for some solution, some pedagogical trick to solve the student's gap.  But this is not the right way to think about things (lock and key fix).  Instead, you need to break the problem down.  AND then drill EXTREMELY SIMILAR problems over and over.&lt;/p&gt;&#xA;&lt;p&gt;Yes, occasionally there will be a lock and key aha.  But even then the aha will differ by person...and often will only make sense after significant struggle (like &amp;quot;getting your kip&amp;quot; in gymnastics).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As a practical solution, what I recommend is just to ask them what (1)^(5) is.  Then ask them (1)^87 is.  then (1)^n is.  The (1)^(n-1) is.  Then (1)^(p-1) is.  Even that last one is a simpler expression than the problem statement.  So finally ask that (maybe even covering it up).  And then having walked them through this and eventually step by step coaxed them to doing the final problem right, don't expect that things will click and no problems in the future.  In general, weaker students will need lots of SUBSEQUENT drill to do this right...not just the one coaching session.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I highly encourage you to make the kids show step by step manipulations.  Eventually they may not need them.  But for neophytes or intrinsically weak students, they help a lot.  There's a nice little essay by Feynman where he talks about being able to quickly &amp;quot;get&amp;quot; solutions of x in little one degree, one unknown problems...and that he did it &amp;quot;by arithmetic&amp;quot;, rather than by combining terms, adding to each side of the equation, dividing by coefficient, etc.  Well...that's fine for Dick.  But for your average/weak neophyte, those step by step processes (and writing them) in pre-algebra (or algebra) are a Godsend.&lt;/p&gt;&#xA;" OwnerUserId="20623" LastActivityDate="2022-11-13T19:41:50.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25804" PostTypeId="2" ParentId="25798" CreationDate="2022-11-13T22:05:44.090" Score="3" Body="&lt;p&gt;If the goal of learning mathematics would be to retain certain information, then passive reading would work, albeit still be a supoptimal learning strategy.&lt;/p&gt;&#xA;&lt;p&gt;But this isn't the goal. The goal is to &lt;strong&gt;adjust how we think&lt;/strong&gt;. It seems kind of obvious to me that this will require extensive attempts to think in the new way.&lt;/p&gt;&#xA;" OwnerUserId="13471" LastActivityDate="2022-11-13T22:05:44.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25806" PostTypeId="1" CreationDate="2022-11-14T07:37:41.930" Score="5" ViewCount="8945" Body="&lt;p&gt;I posted earlier about how I was surprised that a typical Calculus 1 course that meets 3-4 hours each week for 15 weeks only barely manages to reach the fundamental theorem by the end of the course. If we consider James Stewart's &lt;em&gt;Calculus: Early Transcendentals&lt;/em&gt;, then the fundamental theorem of calculus is towards the end of Chapter 5 (5.3-5.5). As a reference, Chapter 1 is a review of functions, Chapter 2 is limits and the notion of a derivative, Chapter 3 is methods of differentiation and basic applications (related rates; differentials), Chapter 4 is more applications of derivatives (optimization; curve sketching), and Chapter 5 is integrals (culminating in the fundamental theorem).&lt;/p&gt;&#xA;&lt;p&gt;There are 5 chapters with a total of 38 sections. And given 3 lectures per week per 15 weeks, there are 45 class meetings in a semester. Taking into account days for exams and other holidays, that leaves still about 40 classes, which is two more lectures than I need. In other words, the rate is 0.3 chapters per week.&lt;/p&gt;&#xA;&lt;p&gt;I mentioned in my previous post that I am able to, for the most part, end my Calculus 1 course at around Chapter 11.3, which covers the integral test for infinite sequences and series. This is usually the last chapter of a Calculus 2 course (Chapters 6 - 11; although some instructors skip Ch. 9). This ends up being an average of 0.73 chapters per week.&lt;/p&gt;&#xA;&lt;p&gt;I have not had any negative effects due to teaching at such a fast pace, but I decided to draft a syllabus that goes at the common pace. In this syllabus, I basically ended up having one 50-minute lecture for every single section of the text, with room to spare for quizzes on certain days. This rate allows me to finish at the fundamental theorem of calculus. For example, in this new syllabus, I am dedicating an entire 50 minutes to Chapter 3.2 (Product and Quotient Rules).&lt;/p&gt;&#xA;&lt;p&gt;In my &amp;quot;fast&amp;quot; version of the lectures, I typically cover all rules of differentiation and trigonometric derivatives all in about 50 minutes. As such, I do not know how to drag out &lt;strong&gt;only&lt;/strong&gt; the product and quotient rule for 50 minutes. I do not see any point in providing the amateur proof of these rules, since I am not teaching for math majors.&lt;/p&gt;&#xA;&lt;p&gt;If any of you are familiar with Professor Leonard's YouTube series, his lectures about the product and quotient rules is one hour long! Is his style the same way most instructors teach? The comments from students generally say positive things about his content, but my students also say positive comments about my teaching in the student evaluations. Most of the comments point out that I am quick to the point with the material and examples.&lt;/p&gt;&#xA;&lt;p&gt;My students have had success, but the only issue I see is that since the exams are departmental, the material being tested is usually topics we covered weeks ago. But because calculus is cumulative, it ends up not being too much of a problem. For example by the time they are taking the final exam, my students have had a lot of experience taking derivatives when doing integration by parts and other integration methods. Similarly, they breeze through the exam questions that are about basic integration, since they have experience with more advanced methods of integration. They can also compute limits much faster with knowledge of L'Hopital's Rule and the problems they did with limits in the improper integral sections. I have even had a handful of students test out of Calculus 2 after taking my class, since I essentially covered 90% of the material of Calculus 2.&lt;/p&gt;&#xA;&lt;p&gt;I am not at a selective college. It is a small liberal arts college in the midwest. My students are not in an honors section. Exams are departmental. I do weekly quizzes, and two semi-exams in the semester. Homework is assigned but not graded for credit. The entire grading scale is based on assessments only, there are no cushion points.&lt;/p&gt;&#xA;&lt;p&gt;I will be teaching Calculus 1 again next semester, and I will also be teaching Calculus 3. One problem with this is that I have already created lectures for Calculus 3 (covering Chapters 12 - 16 of Stewart's). My current playlist has 21 lectures each that is 50-55 minutes long. That means that if I follow the pace of may already recorded lectures, I will end up covering all of Calculus 3 by Week 6, and that still leaves me with 9 more weeks of class with nothing to do. Should I just start them on differential equations after? I guess I am not sure how much the students will grasp the material, but even so, I think Calculus 3 is easier on students than Calculus 1 (I won't have to review functions or worry about the students understanding the actual notion of derivatives and integrals). I asked my colleague for a copy of his calendar for Calculus 3, and he spends about 3-4 lectures per a single section (about an entire week on &lt;strong&gt;just&lt;/strong&gt; double integrals over general regions, whereas in my 50-minute lecture, I covered double integrals in rectangular and general regions, and in polar coordinates.)&lt;/p&gt;&#xA;&lt;p&gt;All advice is appreciated. I am always seeking to be a better instructor.&lt;/p&gt;&#xA;" OwnerUserId="20591" LastActivityDate="2022-11-25T04:54:02.100" Title="What do you do in order to drag out lectures?" Tags="|undergraduate-education|calculus|vector-calculus|" AnswerCount="8" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="25807" PostTypeId="1" CreationDate="2022-11-14T10:42:46.563" Score="2" ViewCount="135" Body="&lt;p&gt;In some sources where I read about propositional logic, it is shown that the propositions come with a truth value baked into the sentence. In other books, it is said that the truth value a proposition take is given by &amp;quot;truth functions&amp;quot; which map from the proposition to the truth values.&lt;/p&gt;&#xA;&lt;p&gt;Some sources which take first approach was my highschool textbook (NCERT Class-12th book), Frederic Schuller's Geometric Anatomy of Physics lectures and books which take the interpretation approach is Kleene's Logic, Modern Logic by Graeme Forbes etc&lt;/p&gt;&#xA;&lt;p&gt;Why is there a split in the pedagogy?&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2022-11-14T10:42:46.563" Title="Split in pedagogy of propositional logic" Tags="|logic|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25808" PostTypeId="2" ParentId="25806" CreationDate="2022-11-14T10:49:45.447" Score="6" Body="&lt;p&gt;When I was in school, what we did was to fill the class time by doing every exercise in the book on board. Naturally some exercises were kept as homework.&lt;/p&gt;&#xA;&lt;p&gt;But I personally disliked this approach. I think it would be better if there was more of a student focused approach. Maybe some problems the students can do on board, and others the teacher can. Maybe make it competitive?&lt;/p&gt;&#xA;&lt;p&gt;I think there is large appeal to doing problems require only basic calculus knowledge to solve (see problem solving channel like Michael Penn, Presh Talwalker, Blackpenred pen). Perhaps you could rip some problems from them and show in Class.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I was tasked with teaching calculus to some begineers at times and I often noticed that the perception we have of them understanding what we say, the perception they have of understanding what we say and what their actual understanding in practice is, are all at a disconnect.&lt;/p&gt;&#xA;&lt;p&gt;There were many scenarios were I thought the person understood it, the person thought they understood it but when I gave them a practical problem, they were totally lost on how to even begin it.&lt;/p&gt;&#xA;&lt;p&gt;You may feel you're able to communicate well, but sometimes what ends up happening with clear lectures which are &amp;quot;too clear&amp;quot; is that the student ends up in a sort of dunning krueger state where they over-estimate their own competance.&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2022-11-14T10:49:45.447" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25809" PostTypeId="2" ParentId="25806" CreationDate="2022-11-14T13:56:06.693" Score="47" Body="&lt;p&gt;What you are describing is so far outside of my, and I suspect most educators, experience that it appears to be literally incredible.&lt;/p&gt;&#xA;&lt;p&gt;The very strongest Universities in the country, with some of the best prepared students and very well designed Calculus courses (such as the University of Michigan), still struggle to fit all of the material of Calculus 1 into a single semester while having the majority of students achieve competence. Doing both semesters of Calculus in a single semester via a &amp;quot;straight lecture&amp;quot; approach and having students excel is an extreme anomaly.&lt;/p&gt;&#xA;&lt;p&gt;My first suspicion would be either rampant cheating or tests which vary so little from year to year that students can easily memorize their way to a passing grade.&lt;/p&gt;&#xA;&lt;p&gt;Do you have one on one conversations with your students?  Do you find that they are able to solve problems equally well during office hours as during the exam?&lt;/p&gt;&#xA;&lt;p&gt;The number of basic misconceptions even bright students bring with them from high school is staggering in most cases.&lt;/p&gt;&#xA;&lt;p&gt;Here is an example (which has research to back it up):  Describe a car driving along a N/S road with an intersection.  Give a graph of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, together with the interpretation that &lt;span class=&quot;math-container&quot;&gt;$f(t)$&lt;/span&gt; represents the displacement of the car northward, in feet, relative to the intersection &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; seconds after passing the intersection.&lt;/p&gt;&#xA;&lt;p&gt;Many students will believe that the graph of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a sketch of the motion of the car in two dimensions.  They are not able to correctly interpret that the car is moving only in one dimension, and that the horizontal axis represents time, not a spatial dimension.  If you give them a toy car and ask them how the car is moving, they will move it along the graph, not N/S in a line.&lt;/p&gt;&#xA;&lt;p&gt;A lot of the time I use to &amp;quot;drag out&amp;quot; my lectures is spent on correcting such fundamental misunderstandings by giving students a wide variety of conceptual problems using multiple representations and having students work (often in groups) on engaging with these problems meaningfully, and bringing the mathematical techniques we are learning to bear on these problems.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-11-14T13:56:06.693" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25810" PostTypeId="2" ParentId="3989" CreationDate="2022-11-14T22:08:29.050" Score="0" Body="&lt;p&gt;I think that would be redundant. I took Calculus 3 in four weeks and I had no difficulty, neither did any other of my classmates.&lt;/p&gt;&#xA;&lt;p&gt;Does it really take three weeks to teach students about dot and cross products? My colleague does. Calculus 3 is usually a 200-level course, and by then students should not need any coddling by artifically extending the material. I believe that even in a quarter system, Calculus 3 should still be a one term course, not two. Most students have probably taken linear algebra before or concurrently with Calculus 3.&lt;/p&gt;&#xA;&lt;p&gt;I know someone that teaches Calculus “3.5”, and they fill the remaining few weeks of class with topics of Taylor polynomials and complex differentiation and integration.&lt;/p&gt;&#xA;" OwnerUserId="20591" LastActivityDate="2022-11-14T22:08:29.050" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25811" PostTypeId="2" ParentId="25806" CreationDate="2022-11-14T22:57:24.123" Score="8" Body="&lt;p&gt;What do I do? First, explain the product rule. Let's say 3 minutes on that. Then write a random product of 2 functions and differentiate it. 2 more minutes. Then write a product of 2 functions and &lt;em&gt;ask students&lt;/em&gt; to differentiate that until you get at least 5 answers (&amp;quot;I got the same&amp;quot; is a legitimate answer after the first one) and go over that example together. 5 more minutes, say. Then you play the same game again. 15 minutes into the class. Then you do two examples with &lt;em&gt;students suggesting the products to differentiate&lt;/em&gt; (25 minutes in). Now you give a product of 3 functions and let the students to figure out how to do it in two steps. Let's say, 5 more minutes with the game repeated if everything moves too fast. You can also write it for arbitrary many functions in the product times the logarithmic derivative form (I find it useful to compare with the quotient rule later, which is much easier to remember that way). Once you have 15 minutes left, give a 5 problem quiz. If someone submits early, he or she is free to go and enjoy a longer break until the next class but tell them that there will be no partial credit to scare them a bit into checking their work). Then you collect the quizzes (with no partial credit the grading is done in under 45 seconds per paper, so you aren't too hard on yourself) and announce the topic for the next class.&lt;/p&gt;&#xA;&lt;p&gt;And if you have a young Gauss among your students, ask him to find the value &lt;span class=&quot;math-container&quot;&gt;$\frac d{dx}[(x-1)(x-3)(x-5)(x-7)\dots(x-99)]$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x=50$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastEditorUserId="19946" LastEditDate="2022-11-14T23:12:50.527" LastActivityDate="2022-11-14T23:12:50.527" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25813" PostTypeId="2" ParentId="25806" CreationDate="2022-11-16T09:44:54.477" Score="3" Body="&lt;p&gt;Going at such a pace with lectures can work fine, but all that lecture time should be supplemented with time for students to ask questions and work on problems. In my physics degree (in Europe) we covered all of single and multivariable calculus in the first semester, which is 16 weeks if you exlude exam weeks, with around 6 hours of lectures and 4 hours of exercise classes per week. In terms of lecture time this seems similar to your &amp;quot;fast&amp;quot; pace.&lt;/p&gt;&#xA;&lt;p&gt;Later I was a TA for this class a couple of times and noticed that often people would think that they had a fairly good idea of the theory, just from the lectures. They could easily solve exercises that looked like the examples presented by the lecturer or as found in the textbook, but anything a little beyond that was too difficult. With students often barely knowing how to start. Having a TA or lecturer then guide you with hints or a further explanation is a lot more valueable than students studying the answer and simply remembering the solution for a test. Which is what I believe happens a lot.&lt;/p&gt;&#xA;&lt;p&gt;I think that when you calculate lecture time you should take into account time for exercise classes, or something similar, rather than just time spent talking and explaining the theory. If you don't have that much time to spend per week, it is probably a good idea to sacrifice some of the pace for a higher quality learning experience.&lt;/p&gt;&#xA;" OwnerUserId="20636" LastActivityDate="2022-11-16T09:44:54.477" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25814" PostTypeId="2" ParentId="25806" CreationDate="2022-11-16T14:57:17.160" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Slow down your actual rate of speech.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Explain the same thing three different ways.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give five separate concrete examples for every abstract idea. Make them physical demonstrations if needed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask questions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask &lt;a href=&quot;http://journal.canadianschoollibraries.ca/wp-content/uploads/2019/03/2013-05-07_1409.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;more involved questions&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Give physical demonstrations of the principles.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Solve problems using TPS. (Think, Pair, Share: do it individually; do it with a partner or two; do it as a group. Add or skip parts of this process as time requires.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Incorporate more breaks, but use the class time more intensively. (Not 85 min + 10 min + 85 min but 50 min + 10 min + 60 min + 10 min + 50 min, ensuring that the non-break time is dense and rich.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Run competitive games.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Run work periods during which you talk with (and orally evaluate the understanding of) individual students or small groups.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="11920" LastActivityDate="2022-11-16T14:57:17.160" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25815" PostTypeId="2" ParentId="25806" CreationDate="2022-11-16T15:21:52.647" Score="1" Body="&lt;p&gt;IF you want to fill more time (and that is what the question asks), the clear answer is to spend more time on in-class drill.  You are verging back to debating if you should spend more time, in responding to answers.  But that's not this question (which is &amp;quot;how&amp;quot;, not &amp;quot;if&amp;quot;, as your other question was).&lt;/p&gt;&#xA;&lt;p&gt;FWIW, if you're really crushing it like you say, I would say the answer to the &amp;quot;if&amp;quot; question is no.  [And this is coming from someone who usually pushes in class drill...but if you're getting results, you're getting results.]&lt;/p&gt;&#xA;&lt;p&gt;You could just cover 1st semester calc and give students half the periods back.  Or stick with your original scheme and teach both semesters of calculus (at least there is some drill of first semester and of algebra buried within the work of second semester) in a normal semester.  I mean...it worked, right.  Just keep your head low and ignore colleagues (no bragging and just go your way on the sly).&lt;/p&gt;&#xA;" OwnerUserId="20585" LastActivityDate="2022-11-16T15:21:52.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25816" PostTypeId="2" ParentId="9960" CreationDate="2022-11-17T01:30:15.693" Score="4" Body="&lt;p&gt;I started SMSG math in 4th or 5th grade in Washington State in 1961. I liked it a lot. I still remember the yellow softcover text I was given and liked that I could write in it. I hung onto that yellow book for years, but eventually lost it. Everything seemed to make sense in a very pleasing way. I wanted to be a mathematician in Jr High, but that ambition was lost too. It seemed better math than what I saw my son being taught in the 1990’s.&lt;/p&gt;&#xA;" OwnerUserId="20640" LastActivityDate="2022-11-17T01:30:15.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25817" PostTypeId="2" ParentId="25756" CreationDate="2022-11-17T06:33:58.333" Score="7" Body="&lt;p&gt;One reason that this is the case is that, as other members have stated, questions may intentionally be not testing these different aspects of knowledge, they might just be introductory problems made to ensure people have a good grasp of material. Another closely related reason might be to moderate for difficulty- problems might be too hard otherwise.&lt;/p&gt;&#xA;&lt;p&gt;Another important reason is that sometimes, it simply doesn't make sense to ask the question in such a way. For example, in your first question, &lt;span class=&quot;math-container&quot;&gt;$Jx$&lt;/span&gt; is an element of &lt;span class=&quot;math-container&quot;&gt;$(G/J)$&lt;/span&gt;- the way the question is stated already makes it clear that the homomorphism is onto &lt;span class=&quot;math-container&quot;&gt;$(G/J) \times (H/K)$&lt;/span&gt; and removing that from the question doesn't make it any better or more interesting of a question, it's just clunkier. &lt;br /&gt;&#xA;In your two examples from Tao, the answer space is far too broad for it to be a very good question- &amp;quot;When is &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; continuous at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;&amp;quot; could be answered via an epsilon-delta definition of continuity, for example. &lt;br /&gt;&#xA;The other two examples do have a possible way to ask in such a manner, although for your number theory one only the first question you asked really makes sense- the second question is somewhat nonsensical (what does it mean for &lt;span class=&quot;math-container&quot;&gt;$x^2 \pmod{p}$&lt;/span&gt; to have/not have a solution?)&lt;br /&gt;&#xA;For those two examples, if an author wanted to ask the question in such a way, they certainly could (in the number theory example, they could even give less information and ask for what primes &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; instead of letting on that there is a mod 4 condition). However, this would make the problem much harder in the number theory case, which might be undesirable, and would barely change the problem, and therefore is unnecessary, in the Bressoud case.&lt;/p&gt;&#xA;&lt;p&gt;In general, textbook authors ask questions that they think are pedagogically appropriate for the readers answering them- if they have chosen to give information, it often is for a good reason.&lt;/p&gt;&#xA;" OwnerUserId="20641" LastActivityDate="2022-11-17T06:33:58.333" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25818" PostTypeId="2" ParentId="9960" CreationDate="2022-11-17T13:28:36.943" Score="1" Body="&lt;p&gt;A lot of places ran it. Fairfax county was one. Results were decent.  But you had a very strong, i.e. non typical demographic.&lt;/p&gt;&#xA;&lt;p&gt;Personally I didn't hate it. But I don't buy the hype either.  It's another outside in curriculum by non teachers, with government funding push. Education is prone to these, like common core.  There is this attraction to dealing with the (easier) problem of what to teach and a hesitancy to deal with the (harder) problem of how to teach, how to get results.&lt;/p&gt;&#xA;&lt;p&gt;And Dick Feynman had a good point on set theory. It's almost irrelevant in the general science and engineering courses. Much more important to solve the issues people have with algebraic manipulations, even arithmetical manipulations than to teach union and intersection.&lt;/p&gt;&#xA;" OwnerUserId="20606" LastEditorUserId="20623" LastEditDate="2022-11-17T20:16:51.023" LastActivityDate="2022-11-17T20:16:51.023" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25819" PostTypeId="2" ParentId="8494" CreationDate="2022-11-18T08:28:21.127" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;Figure out &lt;strong&gt;n&lt;/strong&gt; significant steps, as per your idea 2. If there are several different popular ways of responding that have some merit to them, do this for each.&lt;/li&gt;&#xA;&lt;li&gt;Give one point per step.&lt;/li&gt;&#xA;&lt;li&gt;In case of mistakes, do not subtract points, but rather set a maximum; &lt;strong&gt;n-1&lt;/strong&gt; for a small mistake, &lt;strong&gt;n-2&lt;/strong&gt; for several small ones or one large one, etc. (If &lt;strong&gt;n&lt;/strong&gt; is large, adjust the penalties to taste.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The purpose of the exam is to measure student knowledge and skills. Typically the most valuable skills are conceptual understanding and solution methods, so give the points for these. However, should someone be sloppy and not correct their mistakes, they also show weakness in understanding; a good solution should be checked and thus problems noted. Hence, a reduction in points is in order, but I prefer setting a maximum, as that does not destroy the accomplishments of weaker students, while setting high standards for the strong ones.&lt;/p&gt;&#xA;&lt;p&gt;Also, mistakes that are noticed or even suspected should have less weight than mistakes that are accepted uncritically and at face value. A problem one recognizes can be fixed in the real world, even if the exam situation is ill suited for this.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-11-18T08:28:21.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25820" PostTypeId="2" ParentId="11436" CreationDate="2022-11-20T14:53:44.783" Score="1" Body="&lt;p&gt;A few reasons for the importance of using complex numbers. I will split them following the age of the students:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Highschool level&lt;/strong&gt; (in Romania we had a whole year studying complex numbers, 10th grade)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;just like negative, rational, irrational numbers are introduced so that we can have the right context to solve various equations, complex numbers help solve &lt;em&gt;all&lt;/em&gt; polynomial equations. In particular all quadratics with real coefficients, all cubics, all quartics.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;many aspects in planar geometry can be turned into algebraic manipulations using complex numbers. Some geometry problems become exceptionnally simple when turned into complex context: (for example proving that the midpoints of a quadrilateral form a parallelogram). Basic geometric transformations (translations, rotations, homotheties) are easily described using complex numbers.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;University level&lt;/strong&gt; (in addition to topics mentioned in the other answers)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Fourier analysis, signal processing, solving ODEs.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Whenever a polynomial equation needs to be solved, we want to be able to talk about its roots, without worrying about their existence or not. Therefore, students should be aware that the right context, where every polynomial of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; roots, is the one of complex numbers.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="20649" LastActivityDate="2022-11-20T14:53:44.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25821" PostTypeId="2" ParentId="25791" CreationDate="2022-11-21T05:58:48.160" Score="3" Body="&lt;p&gt;When considering transformations of the &amp;quot;basic&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt;, there is not much else you can do besides&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Vertical shift&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Horizontal shift&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Scaling&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The first is achieved by adding a constant to &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. The second is achieved by adding a constant to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. The third can be achieved in several ways, such as by a ratio of coefficients applied to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; separately.&lt;/p&gt;&#xA;&lt;p&gt;This leads to something in the form &lt;span class=&quot;math-container&quot;&gt;$m\cdot y+a={(n\cdot x+b)}^2$&lt;/span&gt;. It is an exercise in manipulation and renaming to obtain &lt;span class=&quot;math-container&quot;&gt;$y=k\cdot {(x-b)}^2+c$&lt;/span&gt; from there.&lt;/p&gt;&#xA;&lt;p&gt;The same transformations (and in a teaching sequence, for consistency between symbols and actions, the letters) can be applied to cubics and quartics, hyperbolae and sines, exponential and logarithm curves, absolutes and ceilings and floors...&lt;/p&gt;&#xA;&lt;p&gt;One could and perhaps should consider these to be standard forms of the equations matching the transformed graphs involved. While it makes less sense if one is only working in polynomials or where transformations are a secondary concern, it is a coherent position to hold when working graphically and when using the equations for modelling (since it allows for direct correspondence between the coefficients/constants and aspects of many models).&lt;/p&gt;&#xA;" OwnerUserId="6724" LastActivityDate="2022-11-21T05:58:48.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25822" PostTypeId="2" ParentId="11226" CreationDate="2022-11-21T16:11:48.537" Score="0" Body="&lt;p&gt;Definitions are not universal. You need to specify your definitions for each new context, and that goes for line segments and lengths as well. Today, I came across the notation in a textbook that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] in &lt;span class=&quot;math-container&quot;&gt;$\triangle ABC$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a=|BC|$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;that is &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is to be considered a number (indicating length). I must confess that I so much more often see the &lt;em&gt;length&lt;/em&gt; extracted from a segment in the two-letter form, i.e. &lt;span class=&quot;math-container&quot;&gt;$|BC|$&lt;/span&gt; than in the one-letter form, i.e. &lt;span class=&quot;math-container&quot;&gt;$|a|$&lt;/span&gt;. But would that distinction be consistent?&lt;/p&gt;&#xA;&lt;p&gt;I tend to see &lt;strong&gt;segments&lt;/strong&gt; both in &lt;span class=&quot;math-container&quot;&gt;$BC$&lt;/span&gt; and in &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, but acknowledge that I subscribe to the notation of e.g. the Pythagorean theorem &lt;span class=&quot;math-container&quot;&gt;$c^2=a^2+b^2$&lt;/span&gt; which indicates that small-letter notation does indeed signify lengths, that is &lt;strong&gt;numbers&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3502" LastActivityDate="2022-11-21T16:11:48.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25823" PostTypeId="1" CreationDate="2022-11-22T04:22:30.260" Score="7" ViewCount="218" Body="&lt;p&gt;It's very tempting for a student who is overly excited about mathematics to discount intellectual work in other fields, particularly the humanities, where the nature of knowledge and knowledge formation is very different: hardly ever will someone in the humanities have the last word on a specific question.&lt;/p&gt;&#xA;&lt;p&gt;There are a wide variety of real world problems that require a combination of technical and human perspectives to solve, and so I believe it is our responsibility as educators to cultivate a willingness towards this sort of cooperation in our technical-minded students. I've seen mathy types far too often propose technical solutions to what are ultimately social problems.&lt;/p&gt;&#xA;&lt;p&gt;Do you have thoughts on how we can help our students in advanced math courses see the value in &amp;quot;softer&amp;quot; fields? Do you do anything for this in your classroom? Do you know of any resources I could look into for this?&lt;/p&gt;&#xA;&lt;p&gt;Addenda:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Perhaps a more useful perspective on this question is &amp;quot;how can we teach pure math in a way that best leaves students open to the value of other fields?&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;This perspective also has the advantage that it means this question isn't an exact duplicate of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/7880/how-can-i-teach-my-students-that-other-disciplines-are-important-too&quot;&gt;How can I teach my students that other disciplines are important too?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;For context, my personal interest in this question is that I have a small youtube channel where I post lecture videos in pure math, and would like to gently steer my audience in a positive direction.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15671" LastEditorUserId="15671" LastEditDate="2022-11-23T00:18:14.790" LastActivityDate="2022-11-23T17:22:54.910" Title="Encouraging students to see value in things that can't be measured" Tags="|undergraduate-education|general-pedagogy|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25824" PostTypeId="2" ParentId="25823" CreationDate="2022-11-22T16:10:09.677" Score="-2" Body="&lt;p&gt;I think the way to go after this is tangentially, not directly.  Encourage them into looking at fields like Bayesian predictions, where decisions are made on imperfect information, but have real consequences (e.g. betting prediction markets).  Or to doing analysis of science or business problems.  In particular, if you can gently put them into situations where they don't immediately succeed, but they see how semiquantitative process work, they may be incentivized to learn them.&lt;/p&gt;&#xA;&lt;p&gt;This is not to say that there is no value in literature, art, etc. (there is huge value, they concern the human condition, of living/dying/striving).  But you have to use the indirect approach.  Maybe occasional career/life stories or the like.  And realistically, you don't have all the time in the world and need to get the basic job done, first.  Life will eventually teach these kids who live in an abstraction.  So, I would set the bar low (for accomplishing your objective) and be gentle.  Encourage them to do coops and clubs and the like also.  Something to get life experience from.&lt;/p&gt;&#xA;&lt;p&gt;It's still going to be hard though.  Every year you get another crop of wet behind the ear kids.  Heck, even on the math stuff, the next year rolls around and the incoming froshes don't know calculus again.  And you taught it to them last year.  That's teaching for ya.  ;)&lt;/p&gt;&#xA;&lt;p&gt;FWIW, I'm routinely flabergasted by commenters and questioners here (a very high IQ group by the way) making logical fallacies.  E.g. evaluating teaching techniques or curriculum design (which are probabalistic situations, full of confounding input variables, economic factor limitations, multiple output variables to optimize, and incomplete information) based on single counter/pro examples.  As if they were divided by zero math gotchas.  Total wrong analytical frame of reference.&lt;/p&gt;&#xA;&lt;p&gt;So...I do think there is something to this idea of a blind spot for math types.  But fixing it...that's a hard task.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  &lt;a href=&quot;https://www.youtube.com/watch?v=Rf6BhxuxSR8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=Rf6BhxuxSR8&lt;/a&gt; (math v engineering mindset)&lt;/p&gt;&#xA;" OwnerUserId="20658" LastEditorUserId="20658" LastEditDate="2022-11-22T18:07:42.003" LastActivityDate="2022-11-22T18:07:42.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25825" PostTypeId="1" CreationDate="2022-11-22T20:02:50.720" Score="10" ViewCount="299" Body="&lt;p&gt;This is not a precise question. I am curious to know how do you present to your students the (imprecise) concept of &amp;quot;without loss of generality&amp;quot;, and how to use it correctly/incorrectly.&lt;/p&gt;&#xA;&lt;p&gt;I got curious after a student of mine (in calculus &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;), while trying to prove that for every &lt;span class=&quot;math-container&quot;&gt;$a \in \mathbb{R}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\big(\frac{a+|a|}{2}\big)^2+\big(\frac{a-|a|}{2}\big)^2=a^2,&#xA;$$&lt;/span&gt;&#xA;wrote that without loss of generality, we may assume that &lt;span class=&quot;math-container&quot;&gt;$a \ge 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I decided not to accept this usage of the concept as legitimate. Both sides of the equations are invariant under the transformation &lt;span class=&quot;math-container&quot;&gt;$a \to -a$&lt;/span&gt;, so one &lt;em&gt;can&lt;/em&gt; appeal to symmetry here, but this appealing to symmetry needs to be stated (especially since it is not &lt;em&gt;immediately obvious&lt;/em&gt; for the LHS).&lt;/p&gt;&#xA;&lt;p&gt;Anyway, this got me thinking about how to explain to students the slippery concept of WLOG.&lt;/p&gt;&#xA;&lt;p&gt;I think that a good way is using examples, &lt;strong&gt;so I am asking for elementary examples (suitable for first semester math courses) which illustrate the concept.&lt;/strong&gt; &amp;quot;Anti-examples&amp;quot; where this usage is less appropriate (in a non-trivial way) will also be nice, in order to warn the students from making mistakes.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;To start the ball rolling, here is a classic example:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;We want to prove some property &lt;span class=&quot;math-container&quot;&gt;$P(x,y)$&lt;/span&gt; holds for every real numbers &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is symmetric, i.e. &lt;span class=&quot;math-container&quot;&gt;$P(x,y) \iff P(y,x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$Q(x,y)$&lt;/span&gt; be some &amp;quot;property&amp;quot; (logical formula) satisfying&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\mathbb{R}^2=\{ (x,y) \in \mathbb{R}^2 \, | \, Q(x,y) \} \cup \{ (x,y)\in \mathbb{R}^2 \, | \, Q(y,x) \}. \tag{1}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then we can assume W.L.O.G that &lt;span class=&quot;math-container&quot;&gt;$Q(x,y)$&lt;/span&gt; holds, i.e. it suffices to prove that &lt;span class=&quot;math-container&quot;&gt;$Q(x,y) \Rightarrow P(x,y)$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Indeed, &lt;span class=&quot;math-container&quot;&gt;$Q(y,x) \Rightarrow P(y,x) \Rightarrow P(x,y)$&lt;/span&gt;, and by our assumption &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt;, this justifies the reduction. (A common choice for &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$Q(x,y):=x \le y$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;Here is a summary of an informal introduction to the concept:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The term W.L.O.G is used to indicate the assumption that follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic, &lt;strong&gt;or&lt;/strong&gt; follows by a  symmetry argument.&lt;/p&gt;&#xA;&lt;p&gt;Ultimately, it is intentionally introducing an easy-to-fill gap in a proof and relying on the reader to fill the gap.&lt;/p&gt;&#xA;" OwnerUserId="20511" LastActivityDate="2022-11-24T16:01:29.987" Title="How to explain the concept &quot;Without loss of generality&quot; (through examples)?" Tags="|mathematical-pedagogy|proofs|general-pedagogy|logic|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25826" PostTypeId="2" ParentId="25806" CreationDate="2022-11-23T12:22:57.573" Score="0" Body="&lt;p&gt;Either the centralized tests do not properly assess your students' understanding, or your students are being inspired by your fast-paced style to spend more of their own time on reading and doing exercises - things that the lecturer or a TA would usually be doing in the settings I have been part of, as a student long ago and as an educator for the past few decades.&#xA;It is unlikely (not impossible, but improbable)  that you have found a magic recipe for accelerated learning of the same skills, at the same level, that take twice as long to master for other groups of students. Comparisons between US and any other country can sometimes be misleading, because of the often huge differences in math curriculum and the level of skill retained from lower grades, but I assume you are comparing apples and apples here.&#xA;Some students can learn at an amazing speed. However, calculus is hard for most people, and they require time for repetition and rote practice of problem solving.&#xA;You might possibly be giving your students a false sense of accomplishment by preparing them to pass the tests but not for applying math to actual problem solving. Note that I say &amp;quot;possibly&amp;quot;. Only you and the people near you can know.&lt;/p&gt;&#xA;" OwnerUserId="20663" LastActivityDate="2022-11-23T12:22:57.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25827" PostTypeId="2" ParentId="25823" CreationDate="2022-11-23T12:23:00.593" Score="2" Body="&lt;p&gt;Take this advice with a grain of salt, as I have not implemented anything like it yet.&lt;/p&gt;&#xA;&lt;p&gt;I have been thinking a lot about the responsibility that we have, as mathematics educators, to not only give students technical tools but to help them to integrate mathematical thinking into how they view the world.  Mathematics is one way that we gain and organize knowledge about the world, but it is not the only way.&lt;/p&gt;&#xA;&lt;p&gt;One great example of how this could look is Tom Murphy's textbook &amp;quot;Energy and Human Ambition on a Finite Planet&amp;quot;:  &lt;a href=&quot;https://escholarship.org/uc/energy_ambitions&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://escholarship.org/uc/energy_ambitions&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think that this book could be suitable for a &amp;quot;quantitative literacy&amp;quot; course at most Universities.  The course would help students to build mathematical skills such as unit conversion, modeling with linear, polynomial, and exponential functions, setting up and solving linear, polynomial, and exponential equations, understanding rates of change, etc.&lt;/p&gt;&#xA;&lt;p&gt;However, these skills are being developed in the context of grappling with big questions.  A student might have to grapple with questions like &amp;quot;If we try and store one day of solar energy globally using lead-acid batteries, how much lead will be required?  How does that compare with the global supply of lead?  What are the implications of this for society?&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So while mathematics is a part of the toolset we need for understanding our predicament, it is clear that it is not sufficient.  We need to understand history (how did we end up as reliant on fossil fuels as we are?), our philosophical values (what makes a life &amp;quot;good&amp;quot;?  Are we able to live good lives with significantly less energy expenditures?), sociology (we see the need for a change:  how do human beings make these kinds of transitions?), etc.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-11-23T12:23:00.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25828" PostTypeId="2" ParentId="25823" CreationDate="2022-11-23T17:22:54.910" Score="1" Body="&lt;p&gt;I think the most straightforward way of showing the value of &amp;quot;softer&amp;quot; fields is to point out that the softer fields are what mathematics is based on.  I always tell my students, &amp;quot;step 1 of math is philosophy.&amp;quot;  If you have not philosophically determined that the entities in question obey the relevant formulas, then the mathematics is useless.  Thus, the ability to use mathematics requires the softer fields, such as philosophy, to even get started.&lt;/p&gt;&#xA;&lt;p&gt;But it even goes deeper than that.  The verification of mathematics requires philosophy.  How do we know that the law of non-contradiction is true?  This is a philosophical question.  Godel incompleteness shows that there are an infinite number of proofs which require philosophical intuition - i.e., deductive mathematics will be unable to reach them.&lt;/p&gt;&#xA;&lt;p&gt;Philosophy also determines what sort of functions should be allowed in the descriptions of reality.  Some people philosophize that reality is computable, and therefore any full description of reality should only include computable functions.  Others have a more expansive view, and can include other sorts of functions in their descriptions of reality.&lt;/p&gt;&#xA;&lt;p&gt;Even for the sciences, &lt;em&gt;before&lt;/em&gt; you can assign numbers to anything, you have to determine &lt;em&gt;what&lt;/em&gt; the quantifiable entities are!  Thus, we have to philosophize about nature before we can even apply numbers to the question.  Polanyi covers a lot of this in his book, &amp;quot;Science, Faith, and Society.&amp;quot;  Another book in this vein is Clouser's &amp;quot;Myth of Religious Neutrality,&amp;quot; where he shows that one's starting metaphysics influences what sort of mathematics they use.&lt;/p&gt;&#xA;&lt;p&gt;Additionally, there are questions about what classifies as a proper explanation.  This is, again, a philosophical question.   For information on this topic, see Keas' paper, &amp;quot;Systematizing the Theoretical Virtues.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Other fields do not have the certainty of mathematics, and that is often what students are searching for.  However, in order to gain certainty, one must structure their thinking in such a way that makes it possible.  Making sure one is doing that correctly is a philosophical notion.&lt;/p&gt;&#xA;&lt;p&gt;Additionally, there are many things which are not quantitative, or at least not in the usual sense (usually, there are &lt;em&gt;aspects&lt;/em&gt; of anything that can be quantified, but the core reality may be non-computational).  For instance, beauty is non-quantifiable.  Beauty, however, is actually more primary for living a joyful life than productivity, while quantity usually contributes more to productivity than joy.&lt;/p&gt;&#xA;&lt;p&gt;A great quote from Niebuhr from his book, &amp;quot;The Irony of American History&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Yet we cannot deny the indictment that we seek a solution for practically every problem of life in quantitative terms; and are not fully aware of the limits of this approach.  The constant multiplication of our high school and college enrollments has not had the effect of making us the most &amp;quot;intelligent&amp;quot; nation, whether we measure intelligence in terms of social wisdom, aesthetic discrimination, spritual serenity or any other basic human achievement.  It may have mad us technically the most proficient nation, thereby proving that technical efficiency is more easily achieved in purely quantitative terms than any other value of culture....No national culture has been as assiduous as our own in trying to press the wisdom of the social and political sciences, indeed of all the humanities, into the limits of the natural sciences...the result is frequently a preoccupation with the minutiae which obscures the grand and tragic outlines of contemporary history, and offers vapid solutions for profound problems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2022-11-23T17:22:54.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25829" PostTypeId="1" CreationDate="2022-11-23T21:06:50.013" Score="6" ViewCount="73" Body="&lt;p&gt;Social science training typically involves statistics as equivalent to &amp;quot;quantitative methods&amp;quot;, particularly statistical modelling but also some material about data quality and exploratory analysis. But there's a range of other mathematical topics that are useful for thinking about social science issues. I am in the early stages of developing a module that would be available to undergraduate students in any social science degree (sociology, criminology, business, anthropology, geography... but potentially more generally) that introduces them to these not-statistics topics.&lt;/p&gt;&#xA;&lt;p&gt;When I look for textbooks or other sources of ideas, I keep on getting &amp;quot;Quantitative Reasoning&amp;quot; GRE preparation materials, but that's not what I'm after. Here's a preliminary list of topics:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Spatial data&lt;/li&gt;&#xA;&lt;li&gt;Standardisation (comparing mortality rates across populations, price indices)&lt;/li&gt;&#xA;&lt;li&gt;Population dynamics (evolution, cultural transmission)&lt;/li&gt;&#xA;&lt;li&gt;Decision Making under uncertainty (utility, regret, multiple criteria?)&lt;/li&gt;&#xA;&lt;li&gt;Social Dilemmas and Game Theory&lt;/li&gt;&#xA;&lt;li&gt;Machine learning (likely just k-means clustering or similar)&lt;/li&gt;&#xA;&lt;li&gt;Network analysis&lt;/li&gt;&#xA;&lt;li&gt;Thinking about data (ethics, meaning of data)&lt;/li&gt;&#xA;&lt;li&gt;Logic and fuzzy sets&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Obviously they are not going to really learn much of any of these topics and probably several won't make the cut but I have listed them to give you an idea of the intent of the module. It would be entirely case study driven, with some starting question for each topic.&lt;/p&gt;&#xA;&lt;p&gt;If anyone has any experience developing a module along these lines, I would really appreciate suggestions for topics and/or textbooks.&lt;/p&gt;&#xA;" OwnerUserId="20665" LastActivityDate="2022-11-23T21:06:50.013" Title="Potential topics for a university level mathematical thinking module" Tags="|undergraduate-education|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25830" PostTypeId="2" ParentId="25825" CreationDate="2022-11-24T16:01:29.987" Score="6" Body="&lt;p&gt;I disagree with your definition of WLOG. In my opinion, WLOG is synonymous to&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;[Obviously, ]the general case can be reduced to the following particular case.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Usually&lt;/strong&gt; the reduction is done by some sort of symmetry, but it &lt;strong&gt;does not have to be the case&lt;/strong&gt;. On the other hand, in the following situation you describe&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;the use of WLOG is &lt;strong&gt;not&lt;/strong&gt; appropriate, one should rather say &amp;quot;We only do [...], the other cases are similar&amp;quot;. WLOG really conveys that the omitted cases are not just similar, but can be reduced to the treated one.&lt;/p&gt;&#xA;&lt;p&gt;If used in this sense, there's nothing special about WLOG that makes its use correct or incorrect: it's just a part of the argument presumed obvious and skipped. Whether you are convinced that the level of details is appropriate and that the student would know how to fill them out depends entirely on context.&lt;/p&gt;&#xA;&lt;p&gt;That said, here are some examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;permutational/relabeling symmetry: assume WLOG that &lt;span class=&quot;math-container&quot;&gt;$a_1\leq \dots\leq a_n$&lt;/span&gt; when the statement you are required to prove does not depend on the order;&lt;/li&gt;&#xA;&lt;li&gt;shifting symmetry: assume WLOG that &lt;span class=&quot;math-container&quot;&gt;$x_0=0$&lt;/span&gt; (e.g., when considering the Taylor series of a function at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;). Assume WLOG &lt;span class=&quot;math-container&quot;&gt;$\mathbb{E}X=0$&lt;/span&gt; (when the statement is unchanged under subtraction of a constant from a random variable &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;) such as &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Var}\,(X+Y)=\mathrm{Var}\,X+\mathrm{Var}\,Y$&lt;/span&gt; for independent &lt;span class=&quot;math-container&quot;&gt;$X,Y$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;scaling: assume WLOG &lt;span class=&quot;math-container&quot;&gt;$||x||=1$&lt;/span&gt; when proving a statement about a vector &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; invariant under scaling. Assume WLOG that &lt;span class=&quot;math-container&quot;&gt;$[a,b]=[0,1]$&lt;/span&gt; (e.g., when proving the intermediate value property.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The usage is not really limited to symmetry arguments:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Assume W.L.O.G. that all elements of a list are distinct or all numbers in the list are non-zero, when adding repetitions or zero entries obviously does not affect the problem;&lt;/li&gt;&#xA;&lt;li&gt;Assume WLOG that [a graph/space/set] &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is connected, when the disconnected case is done by considering each connected component separately;&lt;/li&gt;&#xA;&lt;li&gt;Let us prove that &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{R}\to\mathbb{R}$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$a_i\to a$&lt;/span&gt;. WLOG, &lt;span class=&quot;math-container&quot;&gt;$a_i$&lt;/span&gt; is decreasing or increasing.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2022-11-24T16:01:29.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25831" PostTypeId="1" CreationDate="2022-11-24T16:16:16.217" Score="3" ViewCount="1074" Body="&lt;p&gt;&lt;strong&gt;tldr: There is a simple intuitive definition of a limit for monotone sequences, and I suggest that it can be used to motivate the (more complicated) standard definition. I am asking for feedback on my suggestion. Do you think it can form a good motivation to present the students?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; My target audience are students who are taking a rigorous calculus course, where full proofs and rigorous arguments are the center of the course. In particular, more than a third of the course is devoted to sequences and their limits. (Subsequence, Cauchy sequences, Bolzano–Weierstrass theorem etc).&#xA;The course begins with sequences, and only after about 6 weeks we move to treat functions.&lt;/p&gt;&#xA;&lt;p&gt;I suggest an approach for motivating the definition of a limit of a sequence.&lt;/p&gt;&#xA;&lt;p&gt;What are good ways to help students understand the nested quantifiers in the definition of limit&lt;/p&gt;&#xA;&lt;p&gt;While thinking of the definition of a limit of a sequence lately, I had the following thought: Naively, there is something which might appear &lt;a href=&quot;https://math.stackexchange.com/questions/192883/whats-wrong-with-this-backwards-definition-of-limit&quot;&gt;unsatisfactory&lt;/a&gt; with the &lt;a href=&quot;https://math.stackexchange.com/questions/993866/switching-the-definition-of-epsilon-delta-limit&quot;&gt;order of the quantifiers&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;For every &lt;span class=&quot;math-container&quot;&gt;$\epsilon&amp;gt;0$&lt;/span&gt; there exists &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; such that...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Intuitively, we would like to say that &amp;quot;as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; increases, &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; tends closer and closer to the limit &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So &lt;em&gt;very&lt;/em&gt; naively, we would like something in the spirit of &lt;span class=&quot;math-container&quot;&gt;$$n_1 &amp;gt;n_2 \Rightarrow |a_{n_1}-L| &amp;lt; |a_{n_2}-L|, \tag{1}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which suggests &lt;strong&gt;first&lt;/strong&gt; choosing &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, and &lt;strong&gt;then&lt;/strong&gt; choosing &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, this specific formulation &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; is bad, since it assumes monotonicity, and it also does not imply that  &lt;span class=&quot;math-container&quot;&gt;$|a_n-L|$&lt;/span&gt; tends to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, it can tend to &lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt; instead. (it implies that &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; tends closer and closer to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;, but not necessarily &lt;em&gt;arbitrarily&lt;/em&gt; close to it).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The point is that the standard definition can seem &lt;a href=&quot;https://math.stackexchange.com/questions/192883/whats-wrong-with-this-backwards-definition-of-limit&quot;&gt;&amp;quot;intuitively backwards&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question: Do you think that the following idea provides a good justification/motivation for the way we define the limit concept?&lt;/strong&gt; (perhaps in honors courses).&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;First, we define what does it mean for a monotone sequence to converge. This can be done in a more intuitive way, similarly to &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt;, which produces something which is equivalent to the standard definition. (see definition below).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Second, &lt;strong&gt;we use the Sandwich theorem as a guiding principle:&lt;/strong&gt;&#xA;It is intuitive to say, that whatever notion of convergence we want to have, it should obey the Sandwich theorem.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Now, one can prove that a sequence &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; converges to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; (by the standard definition) if and only if there exist monotonically increasing &lt;span class=&quot;math-container&quot;&gt;$b_n$&lt;/span&gt; and monotonically decreasing &lt;span class=&quot;math-container&quot;&gt;$c_n$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$b_n \le a_n \le c_n$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$b_n,c_n \to L$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, one can define&#xA;&lt;span class=&quot;math-container&quot;&gt;$$c_{n} := \sup_{k \geq n}a_{k}, b_{n} := \inf_{k \geq n}a_{k}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim c_n =\limsup a_n = L =\liminf a_n =\lim b_n.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(One can prove this claim directly, over any ordered field actually, without assuming and using the existence of infimums or supremums, but never mind that for now).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;To summarize, I suggest:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Define convergence for monotone sequences (below).&lt;/li&gt;&#xA;&lt;li&gt;Show that a sequence converges via the standard definition if and only if it is bounded by monotone sequences which converge to the same limit (via definition &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;The point is that after we have done this, we keep only the standard definition, which this reasoning &amp;quot;confirmed&amp;quot;, since it's much more concise and easy to work with, then constructing bounding monotone sequences every time we want to prove a limit etc...&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;My definition of convergence for monotone sequences:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;We say a monotone sequence &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; converges to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; if&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall n_1,n_2 \in \mathbb{N}  \,\,\, n_1 &amp;gt;n_2 \Rightarrow |a_{n_1}-L| &amp;lt; |a_{n_2}-L|$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt;0$&lt;/span&gt;, &lt;strong&gt;there exists&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$n \in \mathbb{N}$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$|a_{n}-L| &amp;lt; \epsilon. $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Item &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; enforces the &lt;em&gt;arbitrarily&lt;/em&gt; closed (without it there is no uniqueness).&lt;/p&gt;&#xA;" OwnerUserId="20511" LastEditorUserId="20511" LastEditDate="2022-11-25T07:52:29.643" LastActivityDate="2022-11-28T11:58:22.573" Title="Is this motivation for the concept of a limit a good one?" Tags="|mathematical-pedagogy|calculus|concept-motivation|limits|sequences|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25832" PostTypeId="2" ParentId="24599" CreationDate="2022-11-24T18:39:27.037" Score="1" Body="&lt;p&gt;There is a specific AP edition of the Stewart text that I have a sample of and it's fine. It's not that different from the one you have, it just has some pointers about AP-specific questions and their mark scheme and a few other things to align you.&lt;/p&gt;&#xA;&lt;p&gt;Since Stewart is the text they are most likely to see in college, it makes some sense to use it as prep for college. But as the AP folks will proudly tell you, more students take Calc I in high school than in college as of 2009 so AP is actually the standard plurality curriculum for Calc I, but even for II/BCish classes probably Stewart is that.&lt;/p&gt;&#xA;&lt;p&gt;Math snobs hate Stewart, but if you're a pedagogy snob instead it's fine.&#xA;Remember than in other countries, students learn a lot of what we call &amp;quot;calculus&amp;quot; in lower classes and &amp;quot;calculus&amp;quot; is really more like Elementary Real Analysis. But for some reason we do both all at once and hide the derivative.&lt;/p&gt;&#xA;&lt;p&gt;So, there will inevitably always be pie fights between math purists and people who want to just teach the mechanics and the truth lies somewhere in between.&lt;/p&gt;&#xA;&lt;p&gt;tl;dr check out the AP-specific edition of Stewart.&lt;/p&gt;&#xA;" OwnerUserId="20670" LastActivityDate="2022-11-24T18:39:27.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25833" PostTypeId="2" ParentId="25806" CreationDate="2022-11-25T04:54:02.100" Score="0" Body="&lt;p&gt;There have been some nice responses, and your experience goes counter to mine (I am at a &amp;quot;fancy&amp;quot; school, and I have a daily struggle to get through the material...). But, I want to give a different framing:&lt;/p&gt;&#xA;&lt;p&gt;The question I ask when thinking about lectures is: &lt;strong&gt;what can the students get out of class time with me that they couldn't get on their own?&lt;/strong&gt; While your exposition may be good and efficient, your students have excellent exposition in their textbook and on YouTube.&lt;/p&gt;&#xA;&lt;p&gt;One thing they cannot get outside of the classroom is &lt;strong&gt;immediate feedback from an expert&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I design my classes so that I can get real-time information from students about what their misconceptions are and then work to address those misconceptions on the spot.&lt;/p&gt;&#xA;&lt;p&gt;For example, many misconceptions come out when students try and explain their reasoning. I pose a question to the class and give students time to work on it (~5 minutes). Many students are able to &amp;quot;solve&amp;quot; the question; I then ask for 3 or 4 student answers and transcribe them to the board verbatim (or have the students write them on the board). I then have students discuss for each answer whether it is &lt;em&gt;totally correct&lt;/em&gt;, &lt;em&gt;mostly correct&lt;/em&gt;, or &lt;em&gt;incorrect&lt;/em&gt;, and how they would fix any response that wasn't totally correct.&#xA;Activities like this take quite a bit of time (~20 minutes depending on how involved their answers are), but the time spent emphasizes to the students that explaining an answer is just as important as having an answer. It also teaches me a lot about the students! (For example, today I asked my students what the derivative of &lt;code&gt;sin(t degrees)&lt;/code&gt; was. I gave them time to work it out. The results were illuminating...)&lt;/p&gt;&#xA;&lt;p&gt;Some notes on teaching this way:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Coming up with questions that promote good discussion and reasoning is hard! If the explanation is &amp;quot;I took the derivative and set it to zero&amp;quot;, that's not interesting to discuss. There are many places where you can find pre-made questions. For calculus, you can look at &lt;a href=&quot;http://pi.math.cornell.edu/%7EGoodQuestions/materials.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cornell's Good Questions Project&lt;/a&gt;. If you are teaching Linear Algebra, I wrote a &lt;a href=&quot;https://github.com/siefkenj/IBLLinearAlgebra/raw/master/dist/linearalgebra-book.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;textbook&lt;/a&gt; with in-class questions built into the book.&lt;/li&gt;&#xA;&lt;li&gt;Creating a classroom environment where students are willing to share is hard! There's always a few good students willing to answer any question, but when I teach, I want to target the &amp;quot;middle 60%&amp;quot;. That means, I need to make a comfortable environment for those students to share their (often wrong) ideas.&lt;/li&gt;&#xA;&lt;li&gt;For more details about classroom techniques, I suggest looking at the &lt;a href=&quot;https://www.maa.org/sites/default/files/Programs/InstructPracGuide_web.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA Instructional Practices Guide&lt;/a&gt;. It is meant to be &amp;quot;flipped through&amp;quot;, so you can start reading from any section that catches your attention.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5978" LastActivityDate="2022-11-25T04:54:02.100" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25834" PostTypeId="2" ParentId="25831" CreationDate="2022-11-25T05:14:29.050" Score="3" Body="&lt;p&gt;I think the answer depends on the meta question: &lt;strong&gt;why are students learning the definition of a limit?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here are some reasons I can think of:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Because students are taking an intro-to-proofs class and limits give them something to &amp;quot;make proofs&amp;quot; about.&lt;/li&gt;&#xA;&lt;li&gt;Because students will take further classes that use &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proofs and you want them to have familiarity with the proof format.&lt;/li&gt;&#xA;&lt;li&gt;Because students are taking an analysis/advanced calculus course where the theme of the course is &amp;quot;prove everything you already know&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;Because you want the students to &amp;quot;feel&amp;quot; the difference between the original non-proofs formulation of calculus and the modern formal framework.&lt;/li&gt;&#xA;&lt;li&gt;Because you want them to be able to answer &amp;quot;tricky&amp;quot; questions where reliance on their intuition would produce the wrong answer.&lt;/li&gt;&#xA;&lt;li&gt;Because your department mandates that &amp;quot;proofs about limits&amp;quot; show up on the exam.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Each of these goals might suggest a different way to teach limits. If you can state what your goal is, I think it would be easier to answer whether your proposal is &amp;quot;good&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="5978" LastActivityDate="2022-11-25T05:14:29.050" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25835" PostTypeId="2" ParentId="25756" CreationDate="2022-11-25T05:33:50.143" Score="6" Body="&lt;p&gt;I think there are two parts to the answer:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The purpose of a textbook – as a reference or a learning tool.&lt;/li&gt;&#xA;&lt;li&gt;How experienced mathematicians read books.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Addressing 1: A textbook may serve as a reference or as a learning tool. These two uses are very different (though many textbooks try to do both at the same time). If a textbook is designed exclusively for learning, it makes a lot of sense to have theorems stated in terms of exercises. This is the idea behind Inquiry Based Learning, and you can find many such question sequences in the &lt;a href=&quot;http://www.jiblm.org/&quot; rel=&quot;noreferrer&quot;&gt;Journal of Inquiry Based Learning&lt;/a&gt;. As a reference, of course, it doesn't make sense to leave core content as exercises; there are times when you're justified in just wanting to know what is true.&lt;/p&gt;&#xA;&lt;p&gt;Addressing 2: When an experienced mathematician picks up a book to learn something (i.e., is not looking for a reference), they stop, put the book down, and try to work out parts along the way. Essentially, they turn theorem statements into questions, work out the questions, and then continue reading to see if the author had some cleaver tricks that they missed. In this way, any reference book can also serve as a &lt;em&gt;good&lt;/em&gt; learning tool if you're disciplined enough.&lt;/p&gt;&#xA;&lt;p&gt;I try to teach my students to read math this way, but it takes a long time to develop the skill.&lt;/p&gt;&#xA;" OwnerUserId="5978" LastActivityDate="2022-11-25T05:33:50.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25836" PostTypeId="2" ParentId="25831" CreationDate="2022-11-25T12:13:53.553" Score="9" Body="&lt;ol&gt;&#xA;&lt;li&gt;The concept of a limit has nothing to do with the order on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. The standard definition of a limit of a sequence generalizes, almost verbatim, to sequences with values in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;, metric spaces, and topological spaces, which lack order structure.&lt;/li&gt;&#xA;&lt;li&gt;Perhaps all examples of convergence that the students naturally encounter are monotone. For example, the limits in the definition &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\searrow  x_0}\frac{f(x)-f(x_0)}{x-x_0}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\nearrow  x_0}\frac{f(x)-f(x_0)}{x-x_0}$&lt;/span&gt; for the right and left derivatives are (eventually) monotone for any analytic &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. In that sense, I think many students might only have a mental picture for monotone convergence. The problem here here is that while &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; gets closer and closer to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; grows&lt;/em&gt; is a wrong heuristics for limits in general, it is what happens in most examples.&lt;/li&gt;&#xA;&lt;li&gt;To motivate the more general setup, you may give an example of an infectious disease that's being eradicated: there might be occasional outbreaks, the overall incidence rate still tends to zero. Or decaying sine-wave sound signal. Or you may just say that the general setup simplifies the proofs, as we don't need to be bothered with checking monotonicity all the time.&lt;/li&gt;&#xA;&lt;li&gt;I feel that the pedagogically right way to treat this problem is to simply discuss honestly the difference between the two scenarios. We don't really care what happens with &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; on a step-by-step level, it may occasionally backtrack, as long as it does not wander too far away. From there, the standard definition flows rather naturally.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2022-11-25T12:13:53.553" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25837" PostTypeId="2" ParentId="25792" CreationDate="2022-11-25T13:04:40.257" Score="6" Body="&lt;p&gt;The analogy to driving a car or riding a bike is a bad one. I agree that once you learn those skills, is not unreasonable to spend some years without doing those and still have the skill learned fairly well. I was perfectly able to ride a bike after many years of not riding one. I got my driver’s license at &lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt;, did not own a car until &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; years later, and very rarely drove during those &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; years, but I had no real problem driving after I owned my first car.&lt;/p&gt;&#xA;&lt;p&gt;I suggest to the OP that you respond to anyone who makes an analogy to riding a bike or driving a car with other skills: playing a musical instrument and learning a foreign language. Anyone who learned to play a violin or speak a foreign language recognizes that after years of not practicing these, your abilities will be pretty rusty. I am reminded about &lt;a href=&quot;https://music.stackexchange.com/questions/64250/do-advanced-musicians-still-practice-scales-every-day&quot;&gt;this music stack exchange question&lt;/a&gt; on the topic of practicing scales. And it is a common phenomenon for Chinese and Japanese speakers to have their ability to write characters get degraded (character amnesia) now that “writing” is done so often on screens with autocomplete available. See &lt;a href=&quot;https://m.youtube.com/watch?v=sJNxPRBvRQg&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for Japanese and &lt;a href=&quot;https://www.goldthread2.com/culture/are-chinese-people-forgetting-how-write/article/3011174&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for Chinese.&lt;/p&gt;&#xA;&lt;p&gt;If it makes parents of math students feel better, you can tell them that even mathematicians may forget some areas of math that they learned as students but never use in their own work. For instance, those who work in analysis may forget things they once knew about finite groups (conjugacy classes for dihedral groups or the Sylow theorems, say) and those who work in algebra may forget basic theorems from measure theory. High school math teachers will forget the calculus they once knew if they only teach other topics and stop engaging with calculus.&lt;/p&gt;&#xA;&lt;p&gt;A main reason students need to practice the math they learn in each new math class is that it contains &lt;em&gt;new&lt;/em&gt; mathematical topics, so they can’t rely on only knowing math from earlier classes. Going back to driving cars, a better analogy there would be driving different &lt;em&gt;types&lt;/em&gt; of vehicles: an automatic car, a standard (stick shift) car, a limousine, a motorcycle, and a truck. Do the parents think that being able to drive an automatic car makes you qualified to start driving a stick shift car right away, or that being able to drive a stick shift car makes you qualified to drive a truck without training? There is a good reason that a driver’s license is limited to a certain class of vehicles! And do they think a pilot licensed to fly one type of plane should be able to get into any other type of plane and fly it with no practice? Surely not. In fact, even flying the same type of plane requires regular practice: pilots are not allowed to fly a commercial jet without going back to something like a flight simulator for practice if they have not logged enough monthly flight hours. This became especially notable when covid shut down so much airline activity.See &lt;a href=&quot;https://www.ctvnews.ca/mobile/health/coronavirus/why-some-canadian-pilots-are-heading-back-to-flight-simulators-before-hitting-the-sky-1.5368242&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2022-11-25T13:37:30.490" LastActivityDate="2022-11-25T13:37:30.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25838" PostTypeId="2" ParentId="25831" CreationDate="2022-11-25T14:40:09.027" Score="7" Body="&lt;p&gt;I don't like it.  Had a hard time following it.  Just tuned out.  Yes, I'm not a Ph.D. in math.  But neither will be the target students.  You should have won me over.  You didn't.  I have the IQ to follow this stuff.  But you didn't pull me along.  This is not being flippant.  You will have the same issues with students if you approach them as you approached this forum.&lt;/p&gt;&#xA;&lt;p&gt;I think you are better off being truly motivational (intuitive, loose, colloquial, pictorial) when you want to be &amp;quot;motivational&amp;quot;.  And as an adjunct and on top of the rigorous approach.  Not to supplant it.  A supplement.  You seem to want to present a slightly different (non conventional) approach to the rigourous explication and see this as &amp;quot;motivation&amp;quot;. It's not.&lt;/p&gt;&#xA;&lt;p&gt;Also, it feels very &amp;quot;product forward&amp;quot; or even &amp;quot;pet idea from R&amp;amp;D forward&amp;quot;, rather than market back.  I urge taking the opposite approach and looking at the market more.  Where do actual students struggle, with what tendencies and at least a semiquantitative (&amp;quot;many&amp;quot; versus &amp;quot;few&amp;quot; students) observation.  Not hypothetical students.  Real ones in the wild.&lt;/p&gt;&#xA;&lt;p&gt;Also, consider multiple ways of correcting the issues.  Not just an alternate rigorous explication or even an intuitive one, but things like drill and explication and interaction and personality of the teacher.  Remember teaching involves humans.&lt;/p&gt;&#xA;&lt;p&gt;From an intuitive (conceptual, motivational) sense the idea of &amp;quot;getting closer and closer to&amp;quot; is a good starter concept of a limit.  OF course this does not need to be monotonic.  So after discussion of say a decaying exponential, you can draw some sort of collapsing sinusoid and show that some dithering (actually use this word, remember we are in &amp;quot;motivation&amp;quot; land) is good enough.&lt;/p&gt;&#xA;&lt;p&gt;Heck, just draw the graph and get them to consider the view.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MDUz7.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MDUz7.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;AND!  Note you don't need to pre-emptively clarify this when first discussing limits.  You can do this in a progressive manner.  One view &amp;quot;closer and closer to&amp;quot; and then the corrective upgrade.&lt;/p&gt;&#xA;&lt;p&gt;I think that the actual practice of rigorous proof in research papers (and competitive discussion with speakers in academia) makes neophyte academic teachers think that they need to avoid any progression of ideas.  For fear of looking or being wrong  This is not correct, is not how people (real human people with limited meat brains adapted for fighting and fucking, not rule based silicon assemblies) progress in training.  Think about your limit...it does not have to approach a goal monotonically.  Neither does your teaching!&lt;/p&gt;&#xA;" OwnerUserId="20674" LastEditorUserId="20674" LastEditDate="2022-11-25T14:51:53.947" LastActivityDate="2022-11-25T14:51:53.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25839" PostTypeId="2" ParentId="25831" CreationDate="2022-11-25T14:51:12.997" Score="3" Body="&lt;p&gt;While monotone behavior is important in analysis (such as the monotone convergence theorem in measure theory),&#xA;I think you should forget the emphasis on monotone behavior and just be honest with the students that the way we &lt;em&gt;pronounce&lt;/em&gt; notation like &lt;span class=&quot;math-container&quot;&gt;$\lim_{n\to \infty} a_n = L$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to a} f(x) = L$&lt;/span&gt;, namely “the limit as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; tends to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;” and “the limit as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; tends to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;”, is backwards from the idea behind those  concepts when they are made rigorous. It would have been better if we pronounced the notation as “the limit of &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; tends to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;“ and “the limit of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; tends to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.” Offer a &lt;em&gt;reason&lt;/em&gt; the precise definition leads us to quantify on the limit value (for all &lt;span class=&quot;math-container&quot;&gt;$\varepsilon &amp;gt; 0 \ldots$&lt;/span&gt;) first: it is in relation to how close we want to be able to get to the limit that we determine we are far enough along in the sequence variable &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; or function variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;That is also why it gets even more subtle to prove convergence of a sequence or function when there is no proposed explicit limit value &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;, since the definition of a limit is not immediately applicable. That includes proving convergence of infinite series, where verifying convergence often precedes the estimation of the value of the series. I recall being confused when first learning convergence tests for series in the calculus, because here we were showing something converges without knowing ahead of time what it converges to.&lt;/p&gt;&#xA;&lt;p&gt;The order of quantifiers is super important and subtle. Historically, it tripped up even prominent mathematicians. The difference between pointwise convergence and uniform convergence is the exchange of two quantifiers, and it is related to how  mathematicians in the 19th century mistakenly thought they had proved an infinite series (convergent at each point) of continuous function is continuous.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2022-11-25T14:51:12.997" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25840" PostTypeId="2" ParentId="25831" CreationDate="2022-11-26T10:07:51.280" Score="3" Body="&lt;p&gt;Here is a simpler definition of limits which works better for monotone sequences, and has value in motivating further investigation of the concept of limits:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If a sequence &lt;span class=&quot;math-container&quot;&gt;$x_1, x_2, ...$&lt;/span&gt; is monotonically increasing and bounded above, then &lt;span class=&quot;math-container&quot;&gt;$\lim x_n$&lt;/span&gt; is the least upper bound of all &lt;span class=&quot;math-container&quot;&gt;$x_i$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That is, the limit is a number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is an upper bound (i.e. &lt;span class=&quot;math-container&quot;&gt;$x \ge x_i$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;) and &lt;span class=&quot;math-container&quot;&gt;$x \le x'$&lt;/span&gt; for all upper bounds &lt;span class=&quot;math-container&quot;&gt;$x'$&lt;/span&gt;. This can be shown to be equivalent to the standard definition (using epsilons) for increasing bounded-above sequences.&lt;/p&gt;&#xA;&lt;p&gt;I think this definition is a better one to present than something more ad-hoc, because the concept of a &amp;quot;least upper bound&amp;quot; (or &amp;quot;supremum&amp;quot;) is an important one in Analysis. For example, my undergraduate Analysis lecturer used &lt;a href=&quot;https://en.wikipedia.org/wiki/Least-upper-bound_property&quot; rel=&quot;nofollow noreferrer&quot;&gt;the existence of least upper bounds of non-empty bounded-above sets&lt;/a&gt; as an axiomatic definition of the set of real numbers.&lt;/p&gt;&#xA;&lt;p&gt;Note that the concept of a &amp;quot;least upper bound&amp;quot; isn't a &lt;em&gt;logically&lt;/em&gt; simpler definition — &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a least upper bound if &lt;span class=&quot;math-container&quot;&gt;$\forall x' . (\forall i . x_i \le x') \implies x \le x'$&lt;/span&gt;, still with two quantifiers. But it's perhaps a simpler one to explain to students who aren't ready for formal statements with quantifiers, because it hides the quantifiers in the relatively-easy-to-grasp words &amp;quot;upper bound&amp;quot; and &amp;quot;least&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="19245" LastEditorUserId="19245" LastEditDate="2022-11-27T04:11:55.507" LastActivityDate="2022-11-27T04:11:55.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25841" PostTypeId="2" ParentId="14142" CreationDate="2022-11-26T22:59:35.967" Score="4" Body="&lt;p&gt;I don't believe that most middle and high school math teachers have encountered LaTeX.  Indeed, I don't believe that most undergraduate math students encounter it.&lt;/p&gt;&#xA;&lt;p&gt;I'm a college professor and my students certainly encounter it.  In the Intro to Proofs course, before mid-semester they get two extra points for using it on the weekly handin and then after mid-semester they must use it.  On the first day I point them to the &lt;em&gt;Not So Short Introduction&lt;/em&gt; and to Overleaf, and they figure it out.  They say that they do not find it hard.&lt;/p&gt;&#xA;&lt;p&gt;But, when I talk to the teachers who come back for a visit they usually say that no one that they know uses it.  Everyone shares Word or Google Docs and the easiest thing to do in a very busy day is to go along.&lt;/p&gt;&#xA;" OwnerUserId="213" LastActivityDate="2022-11-26T22:59:35.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25843" PostTypeId="1" CreationDate="2022-11-27T07:50:59.820" Score="1" ViewCount="179" Body="&lt;p&gt;My question comes from first few paragraphs of preface of &amp;quot;Analysis on Manifolds&amp;quot; by James R. Munkres, as excerpted below:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A year-long course in real analysis is an essential part of the&#xA;preparation  of any potential mathematician. For the first half of&#xA;such a course, there  is substantial agreement as to what the syllabus&#xA;should be. Standard topics  include: sequence and series, the topology&#xA;of metric spaces, and the derivative  and the Riemannian integral for&#xA;functions of a single variable. There are a  number of excellent texts&#xA;for such a course, including books by Apostol [A],  Rudin [Ru],&#xA;Goldberg [Go], and Royden [Ro], among others.&lt;/p&gt;&#xA;&lt;p&gt;There is no such universal agreement as to what the syllabus of the&#xA;second  half of such a course should be. Part of the problem is that&#xA;there are simply  too many topics that belong in such a course for one&#xA;to be able to treat them  all within the confines of a single&#xA;semester, at more than a superficial level.&lt;/p&gt;&#xA;&lt;p&gt;At M.I.T., we have dealt with the problem by offering two independent&#xA;second-term courses in analysis. One of these deals with the&#xA;derivative and  the Riemannian integral for functions of several&#xA;variables, followed by a   treatment of differential forms and a proof&#xA;of Stokes' theorem for manifolds in  euclidean space. The present book&#xA;has resulted from my years of teaching  this course. The other deals&#xA;with the Lebesgue integral in euclidean space and its applications to&#xA;Fourier analysis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As says by the third paragraph, the text &amp;quot;Analysis on Manifolds&amp;quot; by James R. Munkres is the text for one of the two independent second-term courses in analysis at MIT. But the text for the other second-term course having different syllabus (&amp;quot;Lebesgue  integral in euclidean space and its applications to Fourier analysis&amp;quot;) in parallel with &amp;quot;Analysis on Manifolds&amp;quot; is not mentioned. If you happen to know (or enrolled in), can you please tell me what textbook is used for this &amp;quot;other second-term course in analysis at MIT&amp;quot;? It would be more than appreciated if you can also let me know its course number and the link to a course website of &amp;quot;the other second-term course in analysis at MIT.&amp;quot; Thanks a lot.&lt;/p&gt;&#xA;" OwnerUserId="20682" LastEditorUserId="893" LastEditDate="2022-12-04T13:43:16.847" LastActivityDate="2022-12-04T13:43:16.847" Title="What is the text for &quot;the other second-term course in analysis at MIT?&quot;" Tags="|textbooks|mathematical-analysis|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25844" PostTypeId="2" ParentId="10492" CreationDate="2022-11-27T09:19:42.570" Score="0" Body="&lt;p&gt;My understanding of the reason (I am in a university in USA ranked 50-100):&lt;/p&gt;&#xA;&lt;p&gt;1, No motivation&lt;/p&gt;&#xA;&lt;p&gt;A faculty does not have right to open a new course freely to his/her discretion. A new course has to be applied for to the college with pretty much paper work. After approval, the contents of the course are fixed and no way to extend.&lt;/p&gt;&#xA;&lt;p&gt;If an instructor teaches more to students, s/he (and the TA) will not be paid more.&lt;/p&gt;&#xA;&lt;p&gt;In some culture, being eager to open and teach a hard course but failed in the end is embarrassing. Colleagues will laugh at it.&lt;/p&gt;&#xA;&lt;p&gt;So, why bother?&lt;/p&gt;&#xA;&lt;p&gt;2, Dangerous&lt;/p&gt;&#xA;&lt;p&gt;If an instructor teaches more, often harder contents like multivariate calculus to the students, s/he is taking the risk that students will have a hard time understanding it, causing a low GPA. The students may be unhappy about it and complain to the department that the syllabus is too hard, or they have to spend more hours per day on study so that they can't play or stay with family. According to Natwest's most recent &amp;quot;Student Living Index&amp;quot;, students of University of Leeds spend only 3.7 hours a day on study, which is however the longest in UK. This is particularly the case in USA. Recently, Dr. Maitland Jones Jr., a famous chemist, was fired by New York University because his course &amp;quot;Organic Chemistry&amp;quot; is too hard. 82 of his 350 students complained to the university due to their low scores in the exam. This affair was reported early last month (Oct. 3rd) by New York Times.&lt;/p&gt;&#xA;&lt;p&gt;So, why bother?&lt;/p&gt;&#xA;&lt;p&gt;3, Qualification&lt;/p&gt;&#xA;&lt;p&gt;Well, admit it, some faculties themselves don't understand multivariate calculus very well. How can they teach students?&lt;/p&gt;&#xA;&lt;p&gt;To sum up, I still think multivariate calculus is sufficiently emphasized in top universities (like Munkres' university). But top universities like MIT account for only a very small portion of all universities in the world, which gives people an impression that multivariate calculus is often omitted. After all, the course design has to match the background of students admitted as well as of existing faculties. And money is more important than education in many institutions. So, why bother?&lt;/p&gt;&#xA;" OwnerUserId="20682" LastEditorUserId="20682" LastEditDate="2022-11-27T09:23:06.497" LastActivityDate="2022-11-27T09:23:06.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25846" PostTypeId="1" CreationDate="2022-11-27T13:48:54.567" Score="6" ViewCount="167" Body="&lt;p&gt;I am teaching high school math. My students are generally hardworking and competent. Class sizes are about 35.&lt;/p&gt;&#xA;&lt;p&gt;We were recently forced online due to the pandemic, and I have been searching for a way to give online quizzes that minimizes cheating. I have an idea, and &lt;strong&gt;I would like to ask other math educators if you think it would work well, or how it might be improved&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;At the beginning of the 45 minute lesson, the teacher gives the students the quiz, which has two questions: a regular question worth 5 marks, and a bonus question worth 1 extra mark.&lt;/li&gt;&#xA;&lt;li&gt;The students can work out their solution(s) on scratch paper (which will not be seen by the teacher).&lt;/li&gt;&#xA;&lt;li&gt;Before the end of the lesson, they must &lt;strong&gt;make and upload a video&lt;/strong&gt; in which they solve the regular question, explaining each step through writing and speaking. To make the video, they can use the online platform, or phone camera while writing on paper/pad. At the beginning of the video, they must show their face, and the screen/paper must show the question and nothing else. The recommended time length is roughly five minutes.&lt;/li&gt;&#xA;&lt;li&gt;Optional: Before the end of the lesson, they may also make a similar video in response to the bonus question.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Example quiz:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Solve &lt;span class=&quot;math-container&quot;&gt;$\sin{x}=\sin{4x}, 0&amp;lt;x&amp;lt;\pi$&lt;/span&gt;.&lt;br /&gt;&#xA;Bonus: Show that the areas of the four regions enclosed by a circle and two intersecting lines, can be in any ratio.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would try to set questions that are not easy to find online.&lt;/p&gt;&#xA;&lt;p&gt;If a student cheats and gets help from another student or from the internet, they still have to attain at least some level of understanding of the question and solution, in order to make a convincing video. So if they neglect their exercises, it will be difficult to cheat and get away with it.&lt;/p&gt;&#xA;&lt;p&gt;The videos will be scored based on the correctness and clarity of the written and verbal explanations. Students will be shown a demo video, to give them a better idea of what is expected.&lt;/p&gt;&#xA;&lt;p&gt;The students will be given the usual rules about academic honesty (no collaboration, no internet searches, etc.).&lt;/p&gt;&#xA;&lt;p&gt;I did not see this idea of a video quiz mentioned in related questions such as &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18477/what-are-the-best-practices-for-giving-online-tests&quot;&gt;What are the best practices for giving online tests?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;UPDATE: I did my first video quiz with my students, and I was quite pleased.  An unexpected benefit is that it really helps keep the teacher &amp;quot;in touch&amp;quot; with the students. A few students misunderstood the instructions: they wrote out the solution, then started filming and just kept the camera on their paper while talking about what they had written, without writing anything during the video.&lt;/p&gt;&#xA;" OwnerUserId="20101" LastEditorUserId="20101" LastEditDate="2022-11-29T13:30:17.420" LastActivityDate="2022-11-29T13:30:17.420" Title="Online math quiz: students make short video explaining solution" Tags="|online-instruction|test-design|cheating|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25847" PostTypeId="2" ParentId="10492" CreationDate="2022-11-27T16:52:45.547" Score="0" Body="&lt;p&gt;[Speculative, but too long for a comment.]&lt;/p&gt;&#xA;&lt;p&gt;I think the main purpose of calculus (multi or single) is to learn how to solve calculus problems and to follow derivations, primarily to support physics, chemistry and engineering classes.  Real analysis or theoretical calculus doesn't help you solve more calculus problems.  It teaches rigor that underlies calculus.  And of course it is a program for math majors only.&lt;/p&gt;&#xA;&lt;p&gt;RA is a very hard course and gets people to scrunch their brains and try to create proofs.  A few tools from RA are needed for follow-on courses (e.g. topology, theoretical statistics), but in general, not even most of the standard RA course.&lt;/p&gt;&#xA;&lt;p&gt;Finally, life is full of limits (in time, money, IQ) so the &amp;quot;why don't we do this extra thing&amp;quot; (many questions on this site are like that) needs to consider what would be pruned to make way for the addition.  Or, if this hypothetical course were an elective, we need to find the market for it (like enough students wanting it).&lt;/p&gt;&#xA;&lt;p&gt;I would also add, that there's very few anecdotes of people saying they were lacking in multivariable real analysis and that it held them back in some other course.  Even you don't talk about needing it for some specific purpose.&lt;/p&gt;&#xA;&lt;p&gt;Probably the few people that really need it are in grad school (tiny market, dispersed).  And even here, we have evidence from your school that it doesn't seem like a huge demand for all students even in grad school to do MV RA.&lt;/p&gt;&#xA;&lt;p&gt;IF you need it, take that third quarter that is offered.  You're lucky that you even have that option, because at a certain point, you will be too specialized and there will be too few people near you physically to justify a course.  And you will have to learn it on your own.  (Which is not as good as having an instructor.  But that's life.)&lt;/p&gt;&#xA;&lt;p&gt;But in any case, why push to want everyone in grad school (or undergrad for gosh sakes) to take it.  Life is hard and we have limits.  You can't just add requirements without considering the cost (not only financial).&lt;/p&gt;&#xA;" OwnerUserId="20674" LastActivityDate="2022-11-27T16:52:45.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25848" PostTypeId="2" ParentId="25831" CreationDate="2022-11-28T11:58:22.573" Score="1" Body="&lt;p&gt;Two points to think about:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It might be more intuitive to replace &amp;quot;from some &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; onward&amp;quot; by &amp;quot;for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; except finitely many&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;I often present the notion of a limit through a game: I claim that the limit is &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;. Challenge me: ask me to prove that all but finitely-many elements satisfy &lt;span class=&quot;math-container&quot;&gt;$|a_n - L|&amp;lt;0.1$&lt;/span&gt;, and then  &lt;span class=&quot;math-container&quot;&gt;$|a_n - L|&amp;lt;0.001$&lt;/span&gt;. This is not convincing (show a counter-example), hence motivates showing for arbitrary &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12910" LastActivityDate="2022-11-28T11:58:22.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25849" PostTypeId="2" ParentId="12718" CreationDate="2022-11-28T12:31:32.270" Score="0" Body="&lt;p&gt;I think the history of a subject and its exposition are both incredibly interesting and worthwhile subjects, but combining them in a textbook is often problematic.&lt;/p&gt;&#xA;&lt;p&gt;The textbook (non-math, but math-related) I know that spends the most time explaining motivations/background/history for discovery is Apologia's chemistry textbook.  Students reading the book get really bogged down in the historical explanation.  Additionally, it makes the book big and daunting.  With all the historical explanation, students get really confused as to what the main point even is.  They don't know which part they are supposed to learn and be applying.  They get really lost in the details.&lt;/p&gt;&#xA;&lt;p&gt;Interestingly, another math-oriented science book, Saxon Physics, is actually much more &lt;em&gt;difficult&lt;/em&gt; material, but students seem to learn it better, because it is much more direct.  The lessons are just a few pages, and very little background is presented.&lt;/p&gt;&#xA;&lt;p&gt;Again, my point is not that background/motivation is not important, but that, for textbooks, separating it from content helps students because they are better able to see which things directly apply to them.&lt;/p&gt;&#xA;&lt;p&gt;If someone had an example of a textbook that included the background information but in a way that wasn't distracting to students, I would be interested to see it.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2022-11-28T12:31:32.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25850" PostTypeId="2" ParentId="14692" CreationDate="2022-11-28T12:51:57.847" Score="1" Body="&lt;p&gt;Given a map consisting of a few different terrain types (maybe: field, forest, piecewise straight road), and a movement speed along each one, what is the fastest way between two given points?&lt;/p&gt;&#xA;&lt;p&gt;You can adjust the map to increase difficulty, or add in curved roads, continuously changing speeds or direction-dependence via elevation to require more advanced mathematics. Add obstacles (building, river) to investigate non-continuous functions.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-11-28T12:51:57.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25851" PostTypeId="2" ParentId="12" CreationDate="2022-11-28T16:31:33.913" Score="2" Body="&lt;p&gt;Here: &lt;a href=&quot;https://archive.org/details/traitlment00qu/page/6/mode/2up?view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/traitlment00qu/page/6/mode/2up?view=theater&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Traité élémentaire du Calcul differentiel et de Calcul integral&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Modern textbooks can learn a thing or two from this classic (1848)&lt;/p&gt;&#xA;&lt;p&gt;The author starts with simple definitions of differentials, shows how to calculate them in a wide variety of cases, and goes on with applications. Then, he takes integration in the same spirit, short statements, going straight to the point and providing clear explanations. In the process, you learn a lot of things about curves, logarithms as areas and power series . Setting the focus on variables and differentials (rather than on functions and derivatives) is what distinguishes this book from modern ones.&lt;/p&gt;&#xA;" OwnerUserId="20689" LastEditorUserId="20689" LastEditDate="2023-02-21T21:25:59.547" LastActivityDate="2023-02-21T21:25:59.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25852" PostTypeId="2" ParentId="12" CreationDate="2022-11-28T17:37:56.827" Score="2" Body="&lt;p&gt;Edward's &amp;quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0817684115&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Advanced Calculus: A Differential Forms Approach&lt;/a&gt;&amp;quot; does this, writing everything in terms of differential forms and getting all the way up to Stokes' theorem, while giving lots of pictures and geometric intuition. I own a copy, and it looks very nice to me to use for a class of advanced math majors; I've never gotten a chance to try, though.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2022-11-28T17:37:56.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25853" PostTypeId="1" AcceptedAnswerId="25857" CreationDate="2022-11-28T20:51:20.333" Score="6" ViewCount="2512" Body="&lt;p&gt;With regard to an undergraduate statistics course, I am developing a standardized list of point deductions with the TAs (doctoral students) so that graders are consistent in what they are taking off intermediate points for.  For example, most problems are 10 points total, and my proposed point deductions for intermediate math errors are (for example):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;-2 pts, erroneous +, - , *, /&lt;/li&gt;&#xA;&lt;li&gt;-2 pts, erroneous sign, e.g. 3.02 instead of -3.02&lt;/li&gt;&#xA;&lt;li&gt;-3 pts, failed to square, e.g. (x) instead of (x)^2&lt;/li&gt;&#xA;&lt;li&gt;-3 pts, failed to take square root, e.g. (x) instead of sqrt(x)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If, after grading, you discover on a particular exam that the final answers for five 10-point questions are incorrect because of only making a minor -2 point intermediate error, the student could conceivably obtain a score of 80% on an exam if they missed only -2 points per question (40/50).&lt;/p&gt;&#xA;&lt;p&gt;However, in statistics, there is a contextual element to every question, not just solving for a numerical answer -- that is, in addition to the worked problem, students need to write a text-based response for the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;(2 pts) state whether the hypothesis test is significant or not&lt;/li&gt;&#xA;&lt;li&gt;(2 pts) state whether the null hypothesis is rejected or accepted&lt;/li&gt;&#xA;&lt;li&gt;(2 pts) state whether the p-value is less than 0.05 or not.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So if there was only one minor (-2 point) intermediate error made, causing an incorrect final numerical answer, the student will also incorrectly respond to the final text-based answers (above) as well.&lt;/p&gt;&#xA;&lt;p&gt;Thus, would you also take off e.g. -2 points for an incorrect final numerical answer, as well as -6 points for missing the final text-based sub-items listed above?&lt;/p&gt;&#xA;&lt;p&gt;In other words, would you only deduct -2 points for a complex (multi-step) algebra or calculus question if only a minor intermediate step was erroneous, or would you also deduct for having an incorrect final numerical answer as well?&lt;/p&gt;&#xA;&lt;p&gt;Maybe I could propose to the TAs to augment the point deduction list with:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;-1 pt, incorrect final numerical answer&lt;/li&gt;&#xA;&lt;li&gt;-1 pt, state whether the hypothesis test is significant or not&lt;/li&gt;&#xA;&lt;li&gt;-1 pt, state whether the null hypothesis is rejected or accepted&lt;/li&gt;&#xA;&lt;li&gt;-1 pt, state whether the p-value is less than 0.05 or not.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="20690" LastEditorUserId="20690" LastEditDate="2022-11-29T22:07:12.203" LastActivityDate="2022-12-01T13:01:17.643" Title="Point Deductions - Exams and Quizzes" Tags="|exams|grading|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25854" PostTypeId="2" ParentId="25853" CreationDate="2022-11-28T21:36:38.680" Score="12" Body="&lt;p&gt;I advise being less intricate and put less load on the graders.  I personally would go with all, half or zero credit for every question.&lt;/p&gt;&#xA;&lt;p&gt;*All is correct answer (and some reasonable explication, not an essay, but also not a bare number.&lt;/p&gt;&#xA;&lt;p&gt;*Half shows some decent knowledge of the process, but founders partway.  Or has a &amp;quot;dumb mistake&amp;quot; on the algebra/arithmetic.&lt;/p&gt;&#xA;&lt;p&gt;*And zero is for a mess.&lt;/p&gt;&#xA;&lt;p&gt;After all there is more than one question on a test.  More than one test in the course.  And more than one course that the students take.  Life is statistical after all, so these things even out.&lt;/p&gt;&#xA;&lt;p&gt;Don't invest too much time in intricate grading.  The cost/benefit is not worth it.&lt;/p&gt;&#xA;" OwnerUserId="20674" LastActivityDate="2022-11-28T21:36:38.680" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25855" PostTypeId="2" ParentId="25853" CreationDate="2022-11-28T22:31:46.493" Score="20" Body="&lt;p&gt;I favor an additive grading scheme, where points are &lt;em&gt;earned&lt;/em&gt; toward a possible maximum (say 10) instead of deducting points for the myriad possible mistakes one could make. Here, I would try to adopt a set of markers I am looking for and awarding points if they appear in the written work. This could help in standardizing your grading.&lt;/p&gt;&#xA;&lt;p&gt;To avoid the situation you mentioned (where a student loses 6 points because they came to the wrong numerical conclusion and thereby has the wrong verbal interpretation), I might change your markers to see &lt;em&gt;what to award points for&lt;/em&gt;, like:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;(+2 pt) correctly/appropriately calculate the p-value&lt;/li&gt;&#xA;&lt;li&gt;(+1 pt) state whether the p-value is less than 0.05&lt;/li&gt;&#xA;&lt;li&gt;(+1 pt) if answer above matches the calculated p-value&lt;/li&gt;&#xA;&lt;li&gt;(+1 pt) state whether the hypothesis test is significant or not&lt;/li&gt;&#xA;&lt;li&gt;(+1 pt) if statement above matches the calculated p-value&lt;/li&gt;&#xA;&lt;li&gt;(+1 pt) state whether the null hypothesis is rejected or accepted&lt;/li&gt;&#xA;&lt;li&gt;(+1 pt) if statement above matches the calculated p-value&lt;/li&gt;&#xA;&lt;li&gt;etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This way, a student can get the wrong p-value, but still answer the rest of the problem &amp;quot;correctly&amp;quot; and receive points.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2022-11-28T22:31:46.493" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="25856" PostTypeId="1" CreationDate="2022-11-28T23:03:24.117" Score="4" ViewCount="208" Body="&lt;p&gt;The reason I prefer starting at 0 is due to a computer science background and also, I think it helps to start at 0 because there are certain reasons that demand it (in particular, combinatorics) and I think having exposure to starting at 0 is 'healthier' in the long run?&lt;/p&gt;&#xA;&lt;p&gt;Teaching both starting at 0 or 1 is probably cause for confusion for a lot of students and they usually do not have the discernment to decide which to start at.&lt;/p&gt;&#xA;&lt;p&gt;Any advice on the issue?&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="376" LastEditDate="2022-12-05T16:08:49.930" LastActivityDate="2022-12-05T16:08:49.930" Title="When dealing with sequences, should we teach students to start at 0 or 1?" Tags="|mathematical-pedagogy|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="25857" PostTypeId="2" ParentId="25853" CreationDate="2022-11-29T03:54:24.703" Score="7" Body="&lt;p&gt;When designing a grading system for a question on an exam, I would identify the key skills being tested on that question. This can involve looking at other questions on the exam: if there are other questions that test a skill, I like to weigh it less unless it's a vital skill. A good exam is actually created starting from a list of skills you want to test.&lt;/p&gt;&#xA;&lt;p&gt;Then allot some points for each key skill being tested on the problem. For each skill, I would suggest either two possible scores (all or nothing) or maybe an additional &amp;quot;demonstrated a significant partial understanding of the skill&amp;quot; score.&lt;/p&gt;&#xA;&lt;p&gt;I advise avoiding putting emphasis on skills which are not taught/reviewed in the course. So for instance, I won't deduct points for an arithmetic mistake since arithmetic is not a skill taught in the course. However there are similar skills that are taught: using the right formula, knowing how to use that formula (what plugs in where), and checking to make sure the answer makes intuitive sense.&lt;/p&gt;&#xA;&lt;p&gt;What this means for multi-step problems is: if the student got step 1 wrong, they get points for step 2 if:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Their answer would have been correct if the answer to step 1 had been what they said&lt;/li&gt;&#xA;&lt;li&gt;They demonstrated the skill in question&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, for instance, if they get the wrong p-value on a question, but interpret that incorrect p-value correctly to get the corresponding (but incorrect) interpretation, they get full points for demonstrating the &amp;quot;interpret p-values&amp;quot; skill. If they interpret their incorrect p-value incorrectly to get the interpretation of the correct answer, they don't get points for demonstrating the &amp;quot;interpret p-values skill&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Of course the best test of a grading system is to actually look at graded work and see if it gives reasonable scores. Try whatever you choose on some sample student work before locking it in.&lt;/p&gt;&#xA;&lt;p&gt;Also, depending on the number of exams, questions, and TAs, consider having each TA grade a specific problem across all sections. This improves uniformity and distances the grader from the student.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-11-29T03:54:24.703" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25859" PostTypeId="2" ParentId="25853" CreationDate="2022-11-29T20:38:17.397" Score="9" Body="&lt;p&gt;Part of the problem is writing the exam questions in the first place.  Others have noted that, when designing a grading rubric, you should identify what the key skills in the problem are.  This seems backwards to me.  &lt;em&gt;First&lt;/em&gt;, identify the key skill that you want to test, and &lt;em&gt;then&lt;/em&gt; write the exam questions which hit those skills.&lt;/p&gt;&#xA;&lt;p&gt;Once the exam has been written, I would very much recommend that you make the rubric as simple as possible, given that you want consistent grading across a (possibly large) group of TAs.  I typically grade on a 3 points scale:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;[3] The answer is nearly perfect.&lt;/li&gt;&#xA;&lt;li&gt;[2] The answer contains errors that are mechanical in nature (e.g. missing signs, incorrect computations, etc), but not conceptual.  The mechanical errors are minor or are not central to the skill(s) being tested by the question.&lt;/li&gt;&#xA;&lt;li&gt;[1] There are serious mechanical errors and/or conceptual errors, but &lt;em&gt;something&lt;/em&gt; correct or relevant has been written on the page, in a way which clearly demonstrates at least some conceptual understanding.&lt;/li&gt;&#xA;&lt;li&gt;[0] The answer is essentially ungradable (it is blank, or nonsensical, or whatever).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I will note that my [2] and [1] are essentially the 50% category in &lt;a href=&quot;https://matheducators.stackexchange.com/a/25854/&quot;&gt;this answer&lt;/a&gt;.  I think that it is worthwhile to distinguish between &amp;quot;dumb&amp;quot; arithmetic mistakes and more fundamental conceptual errors.  That being said, my scheme is essentially the same idea—simple and quick to implement.&lt;/p&gt;&#xA;&lt;p&gt;Experience has shown me that students really don't &lt;em&gt;like&lt;/em&gt; this system.  The students I work with are used to a grading scale in which an A is anything over 90%, a B is anything over 80%, and so on.  Thus when they get 2 points out of 3 (67%), they feel like they are failing (since 67% is a D).  However, my feeling is that [2] represents, roughly speaking, B or C level response, while a [1] represents a D or low C.  This is something which has to be thought about when assigning letter grades—either add more &amp;quot;free&amp;quot; points into the course elsewhere, or grade on a different percentage scale, or accept that more students are going to fail.&lt;/p&gt;&#xA;&lt;p&gt;If you are really stuck on a 90/80/70% scale, then (for example) remap [3] to 5 points, [2] to 4 points, [1] to 3 points, and [0] to 0 points.&lt;/p&gt;&#xA;&lt;p&gt;If you want to weight different questions differently, continue to grade them on a 3 point scale, but weight them differently (easy-peasy).  Students will be most happy if all of the questions are worth multiples of 3 points (because they don't really want to think about weighting).&lt;/p&gt;&#xA;&lt;p&gt;In any event, the overall goal is to construct a grading scheme which is &lt;strong&gt;fast&lt;/strong&gt; (you don't want your graders to have to spend a lot of time on things), and &lt;strong&gt;consistent&lt;/strong&gt; (different graders should score a given response in the same way).  Creating lots of deductions or opportunities for partial credit makes grading slower, hence I would tend to avoid it.  Consistency is also easier to attain if there are fewer categories.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2022-12-01T13:01:17.643" LastActivityDate="2022-12-01T13:01:17.643" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="25860" PostTypeId="1" CreationDate="2022-12-01T01:58:26.117" Score="1" ViewCount="124" Body="&lt;p&gt;&lt;em&gt;How can you determine a quality of a solution for a mathematics problem in general ?&lt;/em&gt; &lt;br/&gt;&#xA;If you try to explain too much then solution becomes bit longer and it takes time to go through causing less interest to ones who know the subject sometimes answer may look like a lesson. You can avoid doing similar simplifications and sometimes you can use symmetry to minimize simplifications to keep your answers short. What about using figures to explain the issue ,what else you can suggest and what you would think as the limitations?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2022-12-01T01:58:26.117" Title="Quality of a solution in mathematics" Tags="|secondary-education|problem-solving|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25861" PostTypeId="2" ParentId="25434" CreationDate="2022-12-01T17:33:12.357" Score="0" Body="&lt;p&gt;There are many important issues in mathematics which are outcomes of observing patterns in mathematicians minds,one of very common example is Collatz conjecture and still it is a conjecture because no one could able to prove it.Now you can see proof is the main link between a conjecture and a theorem. Rigorous proofs introduced in Pythagoras era had added much value to the proofs by the use of deductive logical reasoning and which connects not only various fields of mathematics but also it connects even physics concepts such as application of Kirchhoff's laws in squaring a square concept.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2022-12-01T17:33:12.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25863" PostTypeId="1" CreationDate="2022-12-02T15:15:40.717" Score="-2" ViewCount="206" Body="&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/Polysemy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Polysemy - Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Polysemy&lt;/strong&gt; (/pəˈlɪsɪmi/ or /ˈpɒlɪˌsiːmi/; from Ancient Greek πολύ- (&lt;em&gt;polý-&lt;/em&gt;) 'many', and σῆμα (&lt;em&gt;sêma&lt;/em&gt;) 'sign') is the capacity for a sign (e.g. a symbol, a morpheme, a word, or a phrase) to have multiple related meanings. For example, a word can have several word senses. Polysemy is distinct from &lt;em&gt;monosemy&lt;/em&gt;, where a word has a single meaning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Several languages (including Mathematics) use this kind of linguistic resource. My question is, &amp;quot;Why?&amp;quot; Specially in mathematics where precision is welcome polysemy seems to be undesirable but it seems it is inevitable in any language. Why?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="15884" LastEditDate="2022-12-05T22:06:54.363" LastActivityDate="2022-12-05T22:06:54.363" Title="Why do we use polysemy?" Tags="|language-use|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="25865" PostTypeId="2" ParentId="25863" CreationDate="2022-12-03T00:05:37.510" Score="2" Body="&lt;p&gt;I believe there can be a clash between the ideal local symbol and a more global symbol that clashes with the local symbol.&lt;/p&gt;&#xA;&lt;p&gt;When discussing a theorem concerning polyhedra, I usually use &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. But then&#xA;later I might need two polyhedra. This could be &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P'$&lt;/span&gt;, especially if&#xA;&lt;span class=&quot;math-container&quot;&gt;$P'$&lt;/span&gt; is directly related to &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, say, &lt;span class=&quot;math-container&quot;&gt;$P'$&lt;/span&gt; is a truncation of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;.&#xA;But continuing the exposition, I may need a 3rd polyhedron, but &lt;span class=&quot;math-container&quot;&gt;$P''$&lt;/span&gt;&#xA;starts to become ugly, and maybe&#xA;I should go back and set &lt;span class=&quot;math-container&quot;&gt;$P=P_0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P_2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But suppose then I need to discuss the unfolding of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; to a polygon&#xA;in the plane. Normally, I would want to use &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; for a planar polygon,&#xA;but here I might opt to call the polyhedron &lt;span class=&quot;math-container&quot;&gt;$\mathcal{P}$&lt;/span&gt; and the polygonal&#xA;net &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So I find when writing that the ideal local symbol is often evident,&#xA;but as the context widens, adjustments need to be made to respect&#xA;naturalness but include more complex environments.&lt;/p&gt;&#xA;&lt;p&gt;At an extreme, one would never want to exclude &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; to represent disatance&#xA;just because later on one needs to refer to &lt;span class=&quot;math-container&quot;&gt;$dx/dy$&lt;/span&gt;. The context&#xA;easily distinguishes the two meanings of &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, and to refer to distance&#xA;&lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt; would only obscure.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-12-03T00:05:37.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25866" PostTypeId="2" ParentId="2338" CreationDate="2022-12-03T17:40:00.893" Score="4" Body="&lt;p&gt;After having looked into the history of calculus and into original works by Leibniz and others, I have come to the conclusion that the schizophrenic way the &amp;quot;indefinite integral&amp;quot; is sometimes taught is a byproduct of the common schizophrenic mixture of Leibniz-time notation with Bourbaki-time foundations of calculus. I've found a satisfactory way to deal with the former, but not yet with the latter (I am working on it).&lt;/p&gt;&#xA;&lt;p&gt;When trying to make sense of the traditional Leibniz notation in calculus, it is important to understand that even the notion of &lt;em&gt;function&lt;/em&gt; in the 18th century was not what it is now. Now a function is something along the lines of &amp;quot;a set of pairs,&amp;quot; &amp;quot;a functional relation,&amp;quot; &amp;quot;a triple of a functional relation, a domain, and a co-domain,&amp;quot; &amp;quot;a morphism in the category of sets.&amp;quot; In the 18th century, &lt;em&gt;to be a function&lt;/em&gt; was a property/predicate on &lt;em&gt;variables&lt;/em&gt;: one variable could be a function of another, and also two variables could be functions of each other.&lt;/p&gt;&#xA;&lt;p&gt;According to what I have understood, the notation of Leibniz, as used in the 18th century, heavily relied on the context. Expressions, like integrals, were not self-contained. In fact, even integration bounds were not written below and above &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt;.&amp;quot; Apparently, the first well-known text where the bounds of integration appeared at the integration symbol was &lt;em&gt;Théorie analytique de la chaleur&lt;/em&gt; (1822) by Joseph Fourier -- see &lt;em&gt;The history of notations of the Calculus&lt;/em&gt; (1923) by Florian Cajori.&lt;/p&gt;&#xA;&lt;p&gt;I long believed that in the expression &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\int x^2dx$&lt;/span&gt;,&amp;quot; the variable &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; is bound (is not free) because of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;,&amp;quot; but this is not at all the idea. One can have &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\int ydx$&lt;/span&gt;,&amp;quot; &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\int xdy$&lt;/span&gt;,&amp;quot; &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\int(xdy + ydx)$&lt;/span&gt;,&amp;quot; and how this is to be integrates should be specified separately. For instance:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \int_{x=0}^1 x^2dx =\frac{1}{3},\qquad&#xA;  \int_{y=0}^1 x^2dx = 0,&#xA;$$&lt;/span&gt;&#xA;where in the second integral the variable &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; represents a fixed quantity (fixed in the context of this integral), while &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; varies from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&#xA;(This is not how this would have been written in the 18th century, because bounds were not written by the integration symbol, but this is the idea.)&lt;/p&gt;&#xA;&lt;p&gt;Now, I return to the question of the meaning of the &amp;quot;indefinite integral.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If we simply want to write identities like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \int (f(x) + g(x))dx =\int f(x)dx +\int g(x)dx&#xA;$$&lt;/span&gt;&#xA;without worrying about the bounds, like this would have been done in the 18th century, it suffices to interpret the &amp;quot;indefinite integrals&amp;quot; that appear here as &lt;em&gt;indeterminate&lt;/em&gt; definite integrals -- definite integrals with indeterminate bounds.  The meaning of this identity then is that for all &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; et &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; such that both sides are defined, we have:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \int_{x=a}^b (f(x) + g(x))dx =\int_{x=a}^b f(x)dx +\int_{x=a}^b g(x)dx.&#xA;$$&lt;/span&gt;&#xA;In modern terms, this amounts to an identity of functions of two variables, which take &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; et &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and return the value obtained by integrations from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. I believed having already seen such a definition of &lt;em&gt;indefinite integral&lt;/em&gt; given by Paul Halmos, but I have just tried to find it, and I have no idea where I could have seen it. Perhaps I deduced it myself from the definition of &lt;em&gt;indefinite Lebesgue integral&lt;/em&gt; given by Halmos (see below).&lt;/p&gt;&#xA;&lt;p&gt;Here is the definition of an &lt;em&gt;indefinite integral&lt;/em&gt; that I plan to use in my teaching from now on:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\int_x f(x)dx$&lt;/span&gt;, written also as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\int f(x)dx$&lt;/span&gt;&amp;quot; when it is clear from the context that the integration is with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, is the function that takes a pair of reals &lt;span class=&quot;math-container&quot;&gt;$(a, b)$&lt;/span&gt; and returns the value &lt;span class=&quot;math-container&quot;&gt;$\int_{x=a}^b f(x)dx$&lt;/span&gt;. We adopt the convention to not write expressions like &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$(\int_x f(x)dx)(0, 1)$&lt;/span&gt;,&amp;quot; but to write instead &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\int_{x=0}^1 f(x)dx$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If teaching time permits, I may mention that historically there was another notion of &amp;quot;indefinite integral,&amp;quot; which I may call, for example, &lt;em&gt;indefinite integral in the sense of Cauchy&lt;/em&gt; (see my another &lt;a href=&quot;https://matheducators.stackexchange.com/a/27082&quot;&gt;partial answer&lt;/a&gt;), but that this historical notion is virtually informalizable in the modern mathematics.&lt;/p&gt;&#xA;&lt;p&gt;In addition to the proposed definition of an &lt;em&gt;indefinite integral&lt;/em&gt;, I plan to use the following definition of an &lt;em&gt;indefinite difference&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[f(x)]_x$&lt;/span&gt;, written also as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$[f(x)]$&lt;/span&gt;&amp;quot; when there should be no ambiguity, is the function that takes a pair of reals &lt;span class=&quot;math-container&quot;&gt;$(a, b)$&lt;/span&gt; and returns the value &lt;span class=&quot;math-container&quot;&gt;$[f(x)]_{x=a}^b = f(b) - f(a)$&lt;/span&gt;. We adopt the convention to not write expressions like &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$([f(x)]_x)(0, 1)$&lt;/span&gt;,&amp;quot; but to write instead &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$[f(x)]_{x=0}^1$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We have the following obvious property:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  [f(x)] = [g(x)]\quad\Leftrightarrow\quad&#xA;  [f(x) - g(x)] = [0]\quad\Leftrightarrow\quad&#xA;  f - g\ \ \text{is constant}.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The second fundamental theorem of calculus can now be written as:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \int f'(x)dx = [f(x)].&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This interpretation is compatible with and can be applied to &amp;quot;non-oriented&amp;quot; integrals &lt;span class=&quot;math-container&quot;&gt;$\int f(x)|dx|$&lt;/span&gt;, Stieltjes integrals &lt;span class=&quot;math-container&quot;&gt;$\int f(x)dg(x)$&lt;/span&gt;, &amp;quot;non-oriented&amp;quot; Stieltjes integrals &lt;span class=&quot;math-container&quot;&gt;$\int f(x)|dg(x)|$&lt;/span&gt;, etc.&lt;/p&gt;&#xA;&lt;p&gt;This interpretation also agrees with my recent idea to view measure-theoretic integration as a binary operation that takes a signed measure and a measurable function and returns a signed measure. Just now I've re-discovered that this is exactly the definition of &lt;a href=&quot;https://tymathdb.com/database/indefinite-integral/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;indefinite Lebesgue integral&lt;/em&gt;&lt;/a&gt; given by Paul Halmos in his &lt;em&gt;Measure theory&lt;/em&gt;. Thus, the &lt;em&gt;indefinite Lebesgue integral&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\int fd\mu$&lt;/span&gt; is the signed measure that takes a measurable set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and returns the number &lt;span class=&quot;math-container&quot;&gt;$\int_X fd\mu$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I am adding some examples of calculations:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_t te^t dt =\int_t tde^t&#xA;= [te^t]_t -\int_t e^tdt = [te^t]_t -[e^t]_t = [te^t - e^t]_t.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{multline*}&#xA;\int_x\frac{dx}{1 - x^2}&#xA;=\int_x\frac{1}{2}\left(\frac{dx}{1 - x} +\frac{dx}{1 + x}\right)&#xA;=\frac{1}{2}\left(\int_x\frac{dx}{x + 1} -\int_x\frac{dx}{x - 1} \right)\\&#xA;=\frac{1}{2}([\ln |x + 1|]_x -[\ln |x - 1|]_x)&#xA;=\frac{1}{2}[\ln |x + 1| -\ln |x - 1|]_x&#xA;=\left[\frac{1}{2}\ln\left|\frac{x + 1}{x - 1}\right|\right]_x.&#xA;\end{multline*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the second example, it is by default understood that the obtained identity is equivalent to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_{x=a}^b\frac{dx}{1 - x^2}&#xA;=\left[\frac{1}{2}\ln\left|\frac{x + 1}{x - 1}\right|\right]_{x=a}^b&#xA;$$&lt;/span&gt;&#xA;for all &lt;span class=&quot;math-container&quot;&gt;$a,b\in\mathbf{R}$&lt;/span&gt; such that the integral exists, which excludes the possibility that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; be separated by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6987" LastEditorUserId="6987" LastEditDate="2023-11-04T13:01:44.110" LastActivityDate="2023-11-04T13:01:44.110" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="25867" PostTypeId="1" CreationDate="2022-12-03T17:54:34.390" Score="1" ViewCount="411" Body="&lt;p&gt;Computer algebra software is easily and freely available for students nowadays but they still have mathematical errors. For instance, ask them to solve the simple equation &lt;span class=&quot;math-container&quot;&gt;$a x  - b = 0$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. The answer given is typically &lt;span class=&quot;math-container&quot;&gt;$b/a$&lt;/span&gt;, but that is wrong if &lt;span class=&quot;math-container&quot;&gt;$a = 0$&lt;/span&gt;, for instance. I am collecting other mathematical errors in CAS systems. Any examples or references are welcome, the simpler, the better.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="14024" LastEditDate="2022-12-07T23:12:24.127" LastActivityDate="2022-12-26T20:11:59.410" Title="What are some examples of mathematical errors in computer algebra systems (CAS)?" Tags="|computer-programs|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25868" PostTypeId="1" CreationDate="2022-12-03T18:06:35.427" Score="7" ViewCount="6739" Body="&lt;p&gt;Mathematics is currently considered one of the most hated school subjects, (at least in Brazil it is but I think it is a worldwide and cultural phenomenon.) My question is when did this start to happen? in what context? was it with the advent of the modern mathematics movement promoted by the Bourbaki group?&lt;/p&gt;&#xA;&lt;p&gt;A related question: in the trivium and quadrivium which subject was the most hated: arithmetic and geometry?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="8571" LastEditDate="2022-12-06T11:18:52.027" LastActivityDate="2022-12-06T11:20:13.300" Title="When did math start to be a hated subject in schools and universities?" Tags="|secondary-education|" AnswerCount="9" CommentCount="31" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25870" PostTypeId="2" ParentId="25867" CreationDate="2022-12-04T03:03:30.893" Score="8" Body="&lt;p&gt;I find it very helpful in my classes to include examples of calculators giving obviously wrong answers to answer the constant question &amp;quot;why do I need to learn this if my calculator can do it?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;You can find some examples in Patrick Honner's talk &amp;quot;When Technology Fails&amp;quot;: &lt;a href=&quot;https://www.youtube.com/watch?v=tRT3QIt7w20&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=tRT3QIt7w20&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some answers from math.stackexchange: &lt;a href=&quot;https://math.stackexchange.com/questions/1360953/when-would-a-graphing-calculator-make-an-error-while-plotting-a-function&quot;&gt;https://math.stackexchange.com/questions/1360953/when-would-a-graphing-calculator-make-an-error-while-plotting-a-function&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some examples I've collected in Desmos:&#xA;&lt;a href=&quot;https://www.desmos.com/calculator/yjw4awkdej&quot; rel=&quot;noreferrer&quot;&gt;https://www.desmos.com/calculator/yjw4awkdej&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.desmos.com/calculator/h1ctokasxm&quot; rel=&quot;noreferrer&quot;&gt;https://www.desmos.com/calculator/h1ctokasxm&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2022-12-04T03:03:30.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25871" PostTypeId="2" ParentId="25868" CreationDate="2022-12-04T21:42:36.387" Score="32" Body="&lt;p&gt;Here are a few reasons: (1) math is a subject that students are required to take &lt;em&gt;every year&lt;/em&gt;, (2) once you fall behind, it is hard to catch up again because of the way that mathematical knowledge is cumulative, and (3) math gets progressively more abstract in a way that most other subjects in school do not.&lt;/p&gt;&#xA;&lt;p&gt;For other hard school courses like physics and chemistry, people have experience from daily life about many basic objects of the subject (forces, waves, atoms, heat, temperature) or read about it in the news, but about the only way to get an intuition for what you are taught in math classes once algebra begins is... to study more math.  Physics classes can mention things from the 20th century like relativity and quantum mechanics, but the math taught in school is about things that were known centuries ago (calculus is from the late 1600s, and the version of it taught in school is from the 1800s).&lt;/p&gt;&#xA;&lt;p&gt;That math classes are not well-liked by many students long predated Bourbaki. Here is what Woodrow Wilson wrote over 100 years ago during his time as the president of Princeton from 1902 to 1910. It is taken from pages 44-45 of H. J. Form's &lt;em&gt;Woodrow Wilson: the Man and his Work&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There are different sorts of subjects in a curriculum, let me remind you; there are drill subjects, which I suppose are mild forms of torture, but to which every man must submit.  So far as my own experience is concerned, the natural carnal man never desires to learn mathematics.  We know by a knowledge of the history of the race that it is necessary by painful processes of drill to insert mathematics into a man's constitution; he cannot be left to get up mathematics for himself because he cannot do it.  There are some drill subjects which are just as necessary as measles in order to make a man a grown-up person: he must have gone through those things in order to qualify himself for the experiences of life; he must have crucified his will and got up things which he did not intend to get up and reluctantly was compelled to get up. That I believe is necessary for the salvation of the soul.  But there are other subjects, those subjects which are out of the field of the ordinary school curriculum and which I may perhaps be permitted to say are more characteristic in their kind of the university study.  They are what I call the reading subjects, like philosophy, like literature, like law, like history.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As an analogy, the mandatory Russian language classes every year in Eastern Europe during the Cold War were a widely hated subject.  Here is an excerpt from the article &lt;a href=&quot;https://www.adst.org/2013/06/hungary-escapes-the-shadow-of-the-soviet-union/&quot; rel=&quot;noreferrer&quot;&gt;Hungary Escapes the Shadow of the Soviet Union&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the time of the collapse of the Berlin Wall, Russian language teachers permeated Hungary. Russian language instruction was a profession like any other and seemed to offer good job security. Russian was a required course from primary school all the way through gymnasium, through secondary school. Thousands of Hungarians were Russian language teachers. Then the communist East Bloc collapsed. Guess what? No one wanted to study Russian anymore.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Until 1991, students in the Soviet Union were required in every year of high school and college to study topics related to the communist party and they hated it. (When I was learning Russian during 1988-1990 in college, I found math books in Russian far more interesting to read than my assigned course material in Russian about Lenin or communism. I once mentioned this to a math professor who grew up in the USSR and he said that as a student in the 1980s he felt the same way. :)) The required courses on communism in the USSR became obsolete after the country collapsed, but what happened to the thousands of teachers who had been teaching communism?  They couldn't all just be fired. They were turned into philosophy teachers!&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2022-12-05T04:43:45.573" LastActivityDate="2022-12-05T04:43:45.573" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25874" PostTypeId="1" CreationDate="2022-12-05T03:35:04.870" Score="4" ViewCount="89" Body="&lt;p&gt;My motivation for this question is personal. I'm a software engineer and I study mathematical logic as a hobby. Subjectively, it &lt;em&gt;feels&lt;/em&gt; like I make the most progress after asking a question on the Math Stack Exchange or with friends and getting some kind of feedback on it. Also, I seem to get some benefit from &lt;em&gt;preparing a question&lt;/em&gt; even if I later discard it. I don't know much about the effectiveness of various kinds of self-learning though (and receiving feedback from other people might not qualify as self-learning), and I have no non-anecdotal knowledge.&lt;/p&gt;&#xA;&lt;p&gt;I'm wondering if any research has been done on acquisition of undergraduate and post-undergraduate-level math outside of classroom settings and what kinds of strategies yield what kinds of results.&lt;/p&gt;&#xA;&lt;p&gt;It seems like research on this would be hard to do: you'd have to pick a subject, pick a book or a paper, send it to people, give different groups different instructions, and then compare results after a while. I'm not sure what you'd test: maybe test the effect of doing exercises in a textbook versus not on student performance on an exam.&lt;/p&gt;&#xA;&lt;p&gt;So I'm curious if any kind of study has been done on this before.&lt;/p&gt;&#xA;" OwnerUserId="13521" LastActivityDate="2022-12-05T15:23:17.640" Title="Research Into Self-Learning at Undergraduate Level and Above" Tags="|self-learning|education-research|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25875" PostTypeId="2" ParentId="25677" CreationDate="2022-12-05T04:25:36.173" Score="1" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{-b\pm\sqrt{b^2-4ac}}{2a} = \frac{2c}{-b\mp\sqrt{b^2-4ac}}&#xA;$$&lt;/span&gt;&#xA;How would those two teachers write something like the above efficiently?&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2022-12-05T04:25:36.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25876" PostTypeId="2" ParentId="25868" CreationDate="2022-12-05T06:17:31.330" Score="2" Body="&lt;p&gt;I think it is because it tends to be taught incredibly poorly (which usually means taught by rote without any meaningful/interesting application).  I hated math in school until I learned calculus, where there were many interesting and practical applications included as part of the teaching. Some of the higher-level calculus like field integrals and such again became painful and uninteresting until I started applying them in my engineering courses to concrete problems.&lt;/p&gt;&#xA;&lt;p&gt;My point being, math is often taught without practical and concrete applications (the 'what' without the 'why' if you will).  The times I have found myself absolutely loving math has been when it was coupled with an interesting problem.  This is not to say that you can't have very abstract and theoretical problems as part of teaching (say a point in a vector field) but I find it tremendously helpful to eventually tie that example back to a real-world application.  Students need a sense of the real-world power they gain by learning a particular concept.&lt;/p&gt;&#xA;" OwnerUserId="20705" LastActivityDate="2022-12-05T06:17:31.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25877" PostTypeId="2" ParentId="25868" CreationDate="2022-12-05T14:06:48.577" Score="-2" Body="&lt;p&gt;What is school mathematics? 4th century BCE geometry, 12th century algebra, 18th century probability theory and a bit of 19th century set theory. Nothing of the contemporary math, thinking and applications, e.g. in emergence, complex systems theory and all the engineering applications, applications in knowledge engineering and knowledge automation (HOL, etc), applications in non-euclidean geometry and particle physics (and connections with neural networks). And all the questions about the philosophical foundations of math and world (e.g. Goedel theorems) - nothing is mentioned in the school.&lt;/p&gt;&#xA;&lt;p&gt;You are required to solve hundreds of quadratic equations as part of drilling, while the computer code for it consists of 10 lines and can be written by 7th grader. You are required to invent the shortcuts and rules for trigonometric calculus, while these (and far, far more) can be solved by computer algebra packages or even sophisticated proof assistants.&lt;/p&gt;&#xA;&lt;p&gt;No connection with computer science and programming. I wonder why the great minds is not revamping the school curricula.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, nice preprint &lt;a href=&quot;https://arxiv.org/abs/2212.01354&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/2212.01354&lt;/a&gt; &amp;quot;Designing Ecosystems of Intelligence from First Principles&amp;quot; was published this morning in Arxiv.AI. This is text preprint, but all the remaining theory involved some interesting math. If this is not motivation to study the math then I have no idea what else can motivate people (AI is the next big money)?&lt;/p&gt;&#xA;" OwnerUserId="20710" LastActivityDate="2022-12-05T14:06:48.577" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25878" PostTypeId="2" ParentId="25868" CreationDate="2022-12-05T14:21:50.810" Score="10" Body="&lt;p&gt;While I fully agree with the other answers, there is one further issue which I think merits mentioning.&lt;/p&gt;&#xA;&lt;p&gt;Because of its characteristics, in particular concerning basic algorithms for arithmetic, and because it is taught through the whole school career, mathematics has been &lt;em&gt;abused&lt;/em&gt; as the discipline that teaches how to follow prescripted processes, and memorized methods, in obtaining a result, and how to produce results in a neat form. Without wanting to go into politics, such a skill seems to be obviously desirable for factory workers in an industrial society.&lt;/p&gt;&#xA;&lt;p&gt;In combination with some badly trained teachers this leads to teaching of solution methods that emphasize meaningless rote processing of problems without understanding what is really done.&lt;/p&gt;&#xA;&lt;p&gt;Examples of this include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Requirements on particular paper or particular grading margins&lt;/li&gt;&#xA;&lt;li&gt;Calculating GCD of integers by factoring and collecting common factors&lt;/li&gt;&#xA;&lt;li&gt;Solving Quadratic equations by factoring quadratic polynomials&lt;/li&gt;&#xA;&lt;li&gt;Requirements to bring results in a particular form: &lt;span class=&quot;math-container&quot;&gt;$2\frac 12$&lt;/span&gt; is correct while &lt;span class=&quot;math-container&quot;&gt;$\frac 52$&lt;/span&gt; is graded as wrong&lt;/li&gt;&#xA;&lt;li&gt;Solution approaches that split up in an exceeding number of subcases, rather than one uniform method.&lt;/li&gt;&#xA;&lt;li&gt;Geometry classes that emphasize the clean and careful drawing of lines and circles over the conceptual acts of proof.&lt;/li&gt;&#xA;&lt;li&gt;Requirements to put answers to text problems in full sentences.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="19485" LastActivityDate="2022-12-05T14:21:50.810" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25879" PostTypeId="2" ParentId="25874" CreationDate="2022-12-05T15:23:17.640" Score="4" Body="&lt;p&gt;I recommend a science citation search for papers on mathematics correspondence courses.  Perhaps restrict it to last century, to eliminate online courses, as being a little extra scaffolding.  Old style correspondence courses are a little more scaffolding than just having a book.  But a decent proxy.  In particular, look at dropout and pass rates.&lt;/p&gt;&#xA;&lt;p&gt;I think the biggest hurdle is dropout.  Look at the massive drops in moocs.   Anything you can do to minimize that, the better.  Easier texts.  Auditing.  Feedback loops.   Making a promise to a friend. Having an external objective like a validation or AP test.  Paying a tutor. Etc.  This is more serious than the learning, or not, from question answering sites.&lt;/p&gt;&#xA;&lt;p&gt;If you don't master one tricky homework problem, but get through Stewart calculus front to back, you have outperformed the kid who drops out in chapter 2 of Spivak. And you can always do the Spivak later.&lt;/p&gt;&#xA;" OwnerUserId="20606" LastActivityDate="2022-12-05T15:23:17.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25880" PostTypeId="2" ParentId="6083" CreationDate="2022-12-05T16:23:00.937" Score="3" Body="&lt;p&gt;Irish past exam papers are available for free download (though individually, not in bulk) from &lt;a href=&quot;https://www.examinations.ie/exammaterialarchive/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.examinations.ie/exammaterialarchive/&lt;/a&gt;. They are organised by year and then by exam cycle (you want Leaving Certificate and Leaving Certificate Applied; the Junior Certificate is mid-way through secondary school) and then by subject.&lt;/p&gt;&#xA;&lt;p&gt;Within each subject, you will find two or three levels (most subjects have Ordinary and Higher level; some also have Foundation level). And within each level there may actually be two papers, which will be sat separately but combined for the overall result. (Papers should also be available in two languages, English and Irish.)&lt;/p&gt;&#xA;&lt;p&gt;So you will have to fish through the menus to find what you seek, but the information is there, and goes back quite a long way. There have been a few syllabus changes over the years too, of course.&lt;/p&gt;&#xA;" OwnerUserId="436" LastActivityDate="2022-12-05T16:23:00.937" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25882" PostTypeId="2" ParentId="25868" CreationDate="2022-12-04T20:58:00.900" Score="0" Body="&lt;p&gt;The title of the question  asks: &amp;quot;When did math start to be a hated subject in schools and universities?&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The literal meaning is &amp;quot;When did the subject of math start to be hated by students in schools and universities?&amp;quot;.  Notice it says schools and universities.&lt;/p&gt;&#xA;&lt;p&gt;So what were the first and/or oldest universities?&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia's list of oldest universities in operation discusses the criteria to be counted as a university in teh list.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This article contains a list of the oldest existing universities in continuous operation in the world. Inclusion in this list is determined by the date at which the educational institute first met the traditional definition of a university used by academic historians[Note 1][specify] although it may have existed as a different kind of institution before that time.&lt;a href=&quot;https://en.wikipedia.org/wiki/University_of_Constantinople&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; This definition limits the term &amp;quot;university&amp;quot; to institutions with distinctive structural and legal features that developed in Europe, and which make the university form different from other institutions of higher learning in the pre-modern world, even though these may sometimes now be referred to popularly as universities. Thus, to be included in the list below, the university must have been founded before 1500 in Europe or be the oldest university derived from the medieval European model in a country or region. It must also be still in operation, with institutional continuity retained throughout its history. So some early universities, most notably the University of Paris (founded around the beginning of the thirteenth century[2]), which was abolished by the Revolution in 1793,[3] are excluded. Some institutions re-emerge, but with new foundations, such as the modern University of Paris, which came into existence in 1896 after the Louis Liard law disbanded Napoleon's University of France system.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[https://en.wikipedia.org/wiki/List_of_oldest_universities_in_continuous_operation][4]&lt;/p&gt;&#xA;&lt;p&gt;It lists as the oldest university still operating, the University of Bologna,in Bologna in the Kingdom of Italy in the Holy Roman Empire, founded about 1180 to 1190.&lt;/p&gt;&#xA;&lt;p&gt;The University of Naples Frederico II was founded in 1224 by the KIng of Sicily, Emperor of the Romans Frederick II, claims to be the oldest public university in the World, founded by a head of government.&lt;/p&gt;&#xA;&lt;p&gt;The list says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ancient higher-learning institutions, such as those of ancient Greece, ancient Persia, ancient Rome, Byzantium, ancient China, ancient India and the Islamic world, are not included in this list owing to their cultural, historical, structural and legal differences from the medieval European university from which the modern university evolved.[Note 2][Note 3][10]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This list &lt;a href=&quot;https://en.wikipedia.org/wiki/University_of_Constantinople&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.topuniversities.com/blog/10-oldest-universities-world&lt;/a&gt; suggests a sort of older university,  Al-Azhar University in Cairo:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Despite not gaining university status until 1961, Al-Azhar University deserves a mention in this list as it was originally established as early as 970 AD in Cairo, Egypt. Originally a ‘madrasa’, teaching students from primary to tertiary level, Al-Azhar University was first known as a center of Islamic learning but has since developed a modern curriculum of secular subjects, ensuring its survival.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This list [https://www.mastersavenue.com/articles-guides/good-to-know/the-10-oldest-universities-in-the-world][2] describes several universities as being older than the Wikipedi lists indicates.&lt;/p&gt;&#xA;&lt;p&gt;It dates the University of Paris to 1160, the University of Salamanca to 1134, teh Universit of Oxford to 1096, the University of Bologna to 1088, Al-Azhar Unviversity in Cairo, founded in 970, and the University of Al Quaraouiyine, Fez, Morocco founded in 859 by a woman, Fatima al-Fihri.&lt;/p&gt;&#xA;&lt;p&gt;According to the Guinness World Records site:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The oldest existing, and continually operating educational institution in the world is the University of Karueein, founded in 859 AD in Fez, Morocco. The University of Bologna, Italy, was founded in 1088 and is the oldest one in Europe.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[https://www.guinnessworldrecords.com/world-records/oldest-university][3]&lt;/p&gt;&#xA;&lt;p&gt;And Karueein is a different spelling of Al-Quaraouiyine.&lt;/p&gt;&#xA;&lt;p&gt;I note that a sort of unviversity was founed in the eastern Roman Empire at about the same time as Al-Quaraouiyine or Karueein:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;With improving stability in the 9th century came measures to improve the quality of higher education. In 863 chairs of grammar, rhetoric, and philosophy (which included mathematics, astronomy, and music) were founded and given a permanent location in the imperial palace. These chairs continued to receive official state support for the next century and a half, after which the Church assumed the leading role in providing higher education. During the 12th century the Patriarchal School was the leading center of education which included men of letters such as Theodore Prodromos and Eustathius of Thessalonica.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[https://en.wikipedia.org/wiki/Byzantine_university][5]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The University of Constantinople, founded as an institution of higher learning in 425, educated graduates to take on posts of authority in the imperial service or within the Church.[39] It was reorganized as a corporation of students in 849 by the regent Bardas of emperor Michael III, is considered by some to be the earliest institution of higher learning with some of the characteristics we associate today with a university (research and teaching, auto-administration, academic independence, et cetera). If a university is defined as &amp;quot;an institution of higher learning&amp;quot; then it is preceded by several others, including the Academy that it was founded to compete with and eventually replaced. If the original meaning of the word is considered &amp;quot;a corporation of students&amp;quot; then this could be the first example of such an institution. The Preslav Literary School and Ohrid Literary School were the two major literary schools of the First Bulgarian Empire.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[https://en.wikipedia.org/wiki/Ancient_higher-learning_institutions#Christian_Europe][5]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Imperial University of Constantinople, sometimes known as the University of the Palace Hall of Magnaura (Greek: Πανδιδακτήριον τῆς Μαγναύρας), was an Eastern Roman educational institution that could trace its corporate origins to 425 AD, when the emperor Theodosius II founded the Pandidakterion (Medieval Greek: Πανδιδακτήριον).&lt;a href=&quot;https://en.wikipedia.org/wiki/University_of_Constantinople&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/University_of_Constantinople&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/University_of_Constantinople&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ancient institutions that provided frameworks for scholarly activities in Asia and Africa date back centuries before the European medieval universities. These included Buddhist monasteries like the Nalanda in India (427 AD – 1197 AD) and imperial academies in East Asia such as the Taixue in China (circa 202 BC–220 AD) and the Daigaku-ryō in Japan (671 AD).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[https://research.com/universities-colleges/oldest-university-in-the-world#:~:text=The%20University%20of%20Al%2DKaraouine,Guinness%20World%20Records%2C%20n.d.).][2]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Daigaku-ryō (大学寮) was the former Imperial university of Japan, founded at the end of the 7th century.&lt;a href=&quot;https://en.wikipedia.org/wiki/University_of_Constantinople&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; The Daigaku-ryō predates the Heian period, continuing in various forms through the early Meiji period. The director of the Daigaku-ryō was called the Daigaku-no-kami.[2]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[https://en.wikipedia.org/wiki/Daigaku-ry%C5%8D][3]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Taixue (Tai-shueh; simplified Chinese: 太学; traditional Chinese: 太學; lit. 'Greatest Study or Learning'), or sometimes called the &amp;quot;Imperial Academy&amp;quot;, &amp;quot;Imperial School&amp;quot;, &amp;quot;Imperial University&amp;quot;[1][2][3][4] or &amp;quot;Imperial Central University&amp;quot;, was the highest rank of educational establishment in Ancient China created during the Han dynasty. The Sui dynasty instituted major reforms, giving the imperial academy a greater administrative role and renaming it the Guozijian (國子監).[5] As the Guozijian, the institution was maintained by successive dynasties until it was finally abolished in 1905 near the end of the Qing dynasty.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[https://en.wikipedia.org/wiki/Taixue][4]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nalanda (Nālandā, pronounced [naːlən̪d̪aː]) was a renowned mahavihara (Buddhist monastic university) in ancient Magadha (modern-day Bihar), India.[5][6] Considered by historians to be the world's first residential university[7] and among the greatest centers of learning in the ancient world, it was located near the city of Rajagriha (now Rajgir) and about 90 kilometres (56 mi) southeast of Pataliputra (now Patna). Operating from 427 until 1197 CE,[8][9] Nalanda played a vital role in promoting the patronage of arts and academics during the 5th and 6th century CE, a period that has since been described as the &amp;quot;Golden Age of India&amp;quot; by scholars.[10]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So if you want to be conservative you can say there were no universities for students to hate or fear math in until about 1190, which if you want to use looser defination of a university, there might have been universities centuries or millennia earlier where students might have dreated math courses.&lt;/p&gt;&#xA;&lt;p&gt;I note that some of the early university equivlanents were for members of high ranking families, while other would have taken the most gifted students.  I presume that many of the most gifted students in early university like schools might have had no fear of math courses at all, unlike many ordinary students.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, math courses were only feared and hated by students in ordinary schools until the first universities were founded.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/University_of_Constantinople&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;: &lt;a href=&quot;https://www.topuniversities.com/blog/10-oldest-universities-world&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.topuniversities.com/blog/10-oldest-universities-world&lt;/a&gt; [2]: &lt;a href=&quot;https://www.mastersavenue.com/articles-guides/good-to-know/the-10-oldest-universities-in-the-world&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mastersavenue.com/articles-guides/good-to-know/the-10-oldest-universities-in-the-world&lt;/a&gt; [3]: &lt;a href=&quot;https://www.guinnessworldrecords.com/world-records/oldest-university&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.guinnessworldrecords.com/world-records/oldest-university&lt;/a&gt; [4]: &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_oldest_universities_in_continuous_operation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_oldest_universities_in_continuous_operation&lt;/a&gt; [5]: &lt;a href=&quot;https://en.wikipedia.org/wiki/Byzantine_university&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Byzantine_university&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[2]: &lt;a href=&quot;https://research.com/universities-colleges/oldest-university-in-the-world#:%7E:text=The%20University%20of%20Al%2DKaraouine,Guinness%20World%20Records%2C%20n.d.&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://research.com/universities-colleges/oldest-university-in-the-world#:~:text=The%20University%20of%20Al%2DKaraouine,Guinness%20World%20Records%2C%20n.d.&lt;/a&gt;).&#xA;[3]: &lt;a href=&quot;https://en.wikipedia.org/wiki/Daigaku-ry%C5%8D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Daigaku-ry%C5%8D&lt;/a&gt;&#xA;[4]: &lt;a href=&quot;https://en.wikipedia.org/wiki/Taixue&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Taixue&lt;/a&gt;&#xA;[5]: &lt;a href=&quot;https://en.wikipedia.org/wiki/Ancient_higher-learning_institutions#Christian_Europe&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ancient_higher-learning_institutions#Christian_Europe&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="M.A. Golding" LastEditorUserId="8571" LastEditDate="2022-12-06T11:20:13.300" LastActivityDate="2022-12-06T11:20:13.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25883" PostTypeId="2" ParentId="25868" CreationDate="2022-12-05T21:55:58.973" Score="4" Body="&lt;p&gt;Adding to the multitude of existing answers, I believe one of the reasons is that kids in schools (less applicable to universities, but still) are simply being taught their answer is wrong and they have to do it all over more often in math than most other subject, possibly excluding physics.&lt;/p&gt;&#xA;&lt;p&gt;In a sense, many of them approach education a bit like neural networks, without building deep understanding of &amp;quot;why&amp;quot;, just trying different things until something works. With some subjects, brute forcing is possible: you learn what the teacher likes to hear and bingo, it is solved. Humans are rather good at this part of communication.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, it does not work with subjects like math. You can sometimes observe the brute forcing there as well - I have seen kids and university students alike trying to do a blind search without building deep understanding. &amp;quot;Do I divide A by B? No?.. Multiply then?..&amp;quot;. The problem is that it kind of works with &lt;em&gt;most&lt;/em&gt; school subjects, and in some cases, university ones as well. Criteria for a passing grade can be gamed, but not in math.&lt;/p&gt;&#xA;&lt;p&gt;So my answer to the titular question would be that students feel powerless and constantly punished by math, because they are often trained to game the system instead of scaling the obstacle course properly.&lt;/p&gt;&#xA;" OwnerUserId="20642" LastActivityDate="2022-12-05T21:55:58.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25884" PostTypeId="2" ParentId="25868" CreationDate="2022-12-05T23:18:48.623" Score="0" Body="&lt;p&gt;Here are some more reasons.&lt;/p&gt;&#xA;&lt;p&gt;Early on, perhaps grades 2, 3, and 4, we learn the addition, subtraction, multiplication and division tables by rote practice and memorization. This is easy for some, but harder than it sounds for many people. Then there are timed quizzes, and we get punished with a bad mark if we don't do it in time. How about if we quit focusing on speed (factory worker), and just help the kids find the confidence that they can learn them, even if they are slower than others?&lt;/p&gt;&#xA;&lt;p&gt;For some people, visualizing the related concept in their head is easy, but for others, nothing happens except confusion. That often leads to embarrassment when classmates are watching. With these emotions going on, it sounds quite natural for a person to protect themselves by rejecting the subject, rather than sit with it and patiently continue to learn it.&lt;/p&gt;&#xA;&lt;p&gt;Then one of the biggest reasons... vocabulary.&#xA;When I learned calculus, the concept of limits was difficult for me. I must have asked a dozen people to explain limits to me. It finally felt I understood them well enough to use them confidently. But how many words in the math vocabulary does one have to know to &amp;quot;be good at&amp;quot; math? 1000? Also, word meanings fade if not used.&lt;/p&gt;&#xA;" OwnerUserId="20713" LastActivityDate="2022-12-05T23:18:48.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25885" PostTypeId="2" ParentId="12868" CreationDate="2022-12-06T04:54:14.777" Score="0" Body="&lt;p&gt;Given the choice of numbers in this problem, it would be more useful to calculate &lt;span class=&quot;math-container&quot;&gt;$(4800 - x)/48$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$4800 - x = 4368 \implies x = 4800 - 4368 = 432$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thus our problem reduces to calclating &lt;span class=&quot;math-container&quot;&gt;$100 - 432/48$&lt;/span&gt;. As &lt;span class=&quot;math-container&quot;&gt;$432/48$&lt;/span&gt; must be between &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;, we compare the last digit &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and use the times table fact &lt;span class=&quot;math-container&quot;&gt;$8 \cdot 9 = 72$&lt;/span&gt;. Checking that &lt;span class=&quot;math-container&quot;&gt;$432/48$&lt;/span&gt; is indeed &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; by finding &lt;span class=&quot;math-container&quot;&gt;$48 \cdot 9$&lt;/span&gt;, this gives the answer as &lt;span class=&quot;math-container&quot;&gt;$100 - 9 = 91$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13864" LastActivityDate="2022-12-06T04:54:14.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25886" PostTypeId="2" ParentId="25868" CreationDate="2022-12-06T08:27:00.770" Score="2" Body="&lt;p&gt;A &lt;strong&gt;quick&lt;/strong&gt; search with Google scholar, using terms such as &amp;quot;mathematics anxiety&amp;quot;, starts finding articles between 1950 and 1960; for example&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;DREGER, Ralph Mason; AIKEN JR, Lewis R. The identification of number anxiety in a college population. Journal of Educational psychology, 1957, 48.6: 344.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;PHILLIPS, Beeman N.; HINDSMAN, Edwin; MCGUIRE, Carson. Factors associated with anxiety and their relation to the school achievement of adolescents. Psychological Reports, 1960, 7.2: 365-372.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This, however tells more about Google scholar than anything else. A proper literature search would presumably find earlier discussions.&lt;/p&gt;&#xA;&lt;p&gt;But more generally, considering the historical roots of mathematics anxiety, I would like to offer the following confounding theory, which is based on common sense and general but somewhat superficial familiarity with mathematics didactics:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Mathematics is hard, in that it requires actual work to master. Compare to learning (foreign) languages, which are also often seen as hard. The stories of learning to read in Finland when it was a new thing and you did it by learning the catechism by heart, more or less, are similar.&lt;/li&gt;&#xA;&lt;li&gt;Mathematics is compulsory. Compare, for example, compulsory foreign languages such as Swedish in Finland and even nynorsk/sidemål in Norway, and the aforementioned reading in countryside.&lt;/li&gt;&#xA;&lt;li&gt;School, especially in its historical form of having to sit still and concentrate for long periods of time, is not very suited to children, so we would expect a significant number of children to dislike and have always disliked school.&lt;/li&gt;&#xA;&lt;li&gt;Summarizing the three conditions above, mathematics is hard, compulsory, and at least was thought in an unfortunate pedagogical manner to children.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Based on this, I suppose mathematics hatred has been there for a long time, and can be minimized with good pedagogy, but at the heart is the compulsory and difficulty, which are hard to remove. I do not think it is meaningful to ask if math is the most hated, and I think it contributes little to understanding the underlying phenomena.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-12-06T08:27:00.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25887" PostTypeId="2" ParentId="6083" CreationDate="2022-12-06T08:54:04.810" Score="1" Body="&lt;p&gt;For Finland/Suomi, you can find the official information on the ylioppilaskoe/studentexamen/matriculation examination at &lt;a href=&quot;https://www.ylioppilastutkinto.fi/en/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ylioppilastutkinto.fi/en/&lt;/a&gt;. This covers all theoretical lukio/gymnasium subjects.&lt;/p&gt;&#xA;&lt;p&gt;The old tests are officially available at &lt;a href=&quot;https://www.ylioppilastutkinto.fi/sv/informationsservice/gamla-prov&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ylioppilastutkinto.fi/sv/informationsservice/gamla-prov&lt;/a&gt;, but not for everyone. They are also documented at several unofficial sites such as &lt;a href=&quot;https://yle.fi/aihe/abitreenit/matematiikka&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://yle.fi/aihe/abitreenit/matematiikka&lt;/a&gt; and &lt;a href=&quot;https://svenska.yle.fi/abimix/matematik&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://svenska.yle.fi/abimix/matematik&lt;/a&gt; for mathematics (in Finnish and Swedish, respectively). You can navigate those websites for other subjects, such as physics.&lt;/p&gt;&#xA;&lt;p&gt;The examination is digital. It has been some time since I lived in Finland or had my teacher education there, so I am not sure of the details currently, but what they used to do is to provide a separate Linux-based operating system with programs such as Geogebra, calculator emulator, office programs and more, which one used when taking the exam. The first linked website should contain information about this, though not all of it might be available in English. Swedish should be fairly legible to you since you know German, so taking a look at the Swedish version of the website, too, is a good idea in case of missing information.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2022-12-06T08:54:04.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25888" PostTypeId="1" AcceptedAnswerId="25891" CreationDate="2022-12-06T08:55:54.333" Score="2" ViewCount="252" Body="&lt;p&gt;I have found that math is relative easy for me to study but I have problems in geometry. I often omit some obvious steps as I think the problem visually but not from axioms. What would help me for that problem? Are there good books to learn geometry from Hilbert's axioms to, say, IMO level problems and shows how to add a measure theory to geometry such that one can measure lengths, angles and areas?&lt;/p&gt;&#xA;" OwnerUserId="20716" LastActivityDate="2022-12-29T00:42:48.647" Title="Where to learn axiomatic geometry from Hilbert's axioms?" Tags="|textbooks|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25889" PostTypeId="2" ParentId="25868" CreationDate="2022-12-06T09:55:41.877" Score="2" Body="&lt;p&gt;KCd at &lt;a href=&quot;https://matheducators.stackexchange.com/a/25871/1385&quot;&gt;https://matheducators.stackexchange.com/a/25871/1385&lt;/a&gt; quotes&#xA;Woodrow Wilson from 1902 speaking of mathematics in the curriculum as&#xA;a &amp;quot;mild forms of torture&amp;quot;.  There are much earlier specific references&#xA;to the students' suffering at the hand of their math teachers.  Thus,&#xA;Grattan-Guinness on page 1263 of his 1990 &amp;quot;Convolutions in French&#xA;mathematics&amp;quot; writes concerning Coriolis:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A short time before his death, [Coriolis] declared ... that he would&#xA;have liked to devote the remainder of his powers to the reform of&#xA;mathematical teaching, in this same direction. To bring everything&#xA;into relation with the infinitesimal method was ... the chief aim of&#xA;his entire life, was a professeur and as the Directeur des etudes.  As&#xA;he saw it, the teaching of mathematics in France today was the&#xA;dullest, most pedantic, most tiring exercise for pupils and teachers&#xA;alike that it was possible to discover, and presented the most&#xA;peculiar example of deadly routine that any teaching in any period&#xA;could offer. 'When men talk', he said, 'as they often do, of routine&#xA;in the teaching of theology in the seminaries, they are far from&#xA;suspecting that the teaching of mathematics is prey to an incomparably&#xA;duller and more cruel routine'.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is quoted from Gratry 1855.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2022-12-06T09:55:41.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25890" PostTypeId="1" CreationDate="2022-12-06T13:35:04.147" Score="5" ViewCount="324" Body="&lt;p&gt;I observed that my students do not understand what a probability distribution is.&lt;/p&gt;&#xA;&lt;p&gt;We do not treat probability axiomatically on the course, so the required level of understanding is knowing all the possible outcomes and their probability, which is good enough for discrete distributions. We do not pay too much attention as to what this might mean with continuous distributions; I would be happy if the students understood the meaning for discrete ones.&lt;/p&gt;&#xA;&lt;p&gt;When asked about the probability distribution of, say, the sum of two dice, I am equally likely to get the expected value or the probability of a single outcome, or even the probability of getting a given result on both dice, than I am to get what I would think of as the probability distribution.&lt;/p&gt;&#xA;&lt;p&gt;I have had students calculate some probability distributions and things based on them, but this is not a huge theme on the course. I have also given them the meaning of the term &amp;quot;probability distribution&amp;quot;. We have seen different representations of probability distributions, such as trees, graphs and the area model. It is unclear whether I will cover probability distribution as a concept next year and how thoroughly.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Questions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is this a known issue and where can I read more about this?&lt;/li&gt;&#xA;&lt;li&gt;Any good ideas as to how to work with probability distribution as a theme?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Context&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Teacher education of future teachers at grades 5-10 in Norway (mostly grades 8-10 and thereby lower secondary education). The students are second year university students with 40 ECTS credits in mathematics didactics, but no previous university level education in probability. Many have only practical (low-level) mathematics from videregående (upper secondary school) and grades might not be very good.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Literature&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This year we used Alfa as the main book, supplemented by QED 5-10 -books. Next year we are likely to have QED as the primary textbook. Hva er sjansen for det? does not cover the subject. Other books, preferably in Norwegian or other Scandinavian language, are possible.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2022-12-06T14:26:48.727" LastActivityDate="2022-12-15T17:15:57.017" Title="How to teach the concept of probability distribution?" Tags="|undergraduate-education|students-mistakes|probability|teacher-preparation|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25891" PostTypeId="2" ParentId="25888" CreationDate="2022-12-06T22:18:53.967" Score="2" Body="&lt;p&gt;Euclidean and Non-euclidean Geometry by Greenberg is my pick, although it doesn't prove the measure results you want. (However, it does have references)&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2022-12-06T22:18:53.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25892" PostTypeId="2" ParentId="25888" CreationDate="2022-12-07T15:09:25.350" Score="3" Body="&lt;p&gt;I recommend looking at &lt;em&gt;Geometry, Euclid and Beyond&lt;/em&gt; by Robin Hartshorne.&lt;/p&gt;&#xA;&lt;p&gt;From the preface: &lt;em&gt;The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa­rately. The remainder of the book is an exploration of questions that arise natu­rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2022-12-07T15:09:25.350" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25894" PostTypeId="1" CreationDate="2022-12-07T19:58:09.810" Score="5" ViewCount="148" Body="&lt;p&gt;The highschool where I'm teaching welcomes a Ukrainian refugee who is in &amp;quot;classe Terminale&amp;quot; (ultimate highschool grade in France).&lt;/p&gt;&#xA;&lt;p&gt;The math teacher who has this student in charge is not that fluent in English, but the student speaks English very correctly.&lt;/p&gt;&#xA;&lt;p&gt;This is why we are looking for online resources in English that correspond to the program of the subject called &amp;quot;Spécialité Mathématiques&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The only thing I've found so far is a course by Paul Milan (Paris Highschool For Adults), but the course deals with an older program (2015) : &lt;a href=&quot;https://www.lyceedadultes.fr/sitepedagogique/pages/mathTermS_english.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.lyceedadultes.fr/sitepedagogique/pages/mathTermS_english.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Do you know of any resource that could be helpful?&lt;/p&gt;&#xA;&lt;p&gt;If not courses devoted to the French baccalaureate program, maybe courses related to International Baccalaureate's Mathematics, hoping the two programs overlap.&lt;/p&gt;&#xA;" OwnerUserId="19341" LastEditorUserId="2083" LastEditDate="2022-12-08T07:47:43.827" LastActivityDate="2022-12-08T07:47:43.827" Title="English version of the &quot;Spécialité Mathématiques&quot; French baccalaureate course (for a Ukrainian refugee)?" Tags="|curriculum|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25895" PostTypeId="2" ParentId="12" CreationDate="2022-12-08T09:08:32.553" Score="0" Body="&lt;p&gt;Munroe's &lt;em&gt;Modern Multidimensional Calculus&lt;/em&gt; (downloadable from &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015030838208&amp;amp;view=1up&amp;amp;seq=9&quot; rel=&quot;nofollow noreferrer&quot;&gt;hathitrust.org&lt;/a&gt;; there is a &lt;a href=&quot;https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/modern-multidimensional-calculus-by-m-e-munroe-pp-viii-392-1963-74s-addisonwesley-reading-mass/370C22F89BCC5DFF66121A71CA2E60E2&quot; rel=&quot;nofollow noreferrer&quot;&gt;review&lt;/a&gt; by Munn on Mathematical Gazette) motivated the concept of and defined differentials quite early on in chapter 2 (the book has ten chapters). His definition is in line with the one used in differential geometry. This allows him to view &lt;span class=&quot;math-container&quot;&gt;$dy/dx$&lt;/span&gt; as a legitimate ratio between two &lt;em&gt;mappings&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I have only casually read the first few chapters of the book. I like the way he motivated the concept of differential, but I think his presentation of material is sometimes unclear.&lt;/p&gt;&#xA;" OwnerUserId="2059" LastActivityDate="2022-12-08T09:08:32.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25896" PostTypeId="1" CreationDate="2022-12-08T12:24:56.957" Score="3" ViewCount="153" Body="&lt;p&gt;From time to time I need to teach mathematics to my students remotely. I would like to approximate writing on a blackboard together.&lt;/p&gt;&#xA;&lt;p&gt;The subjects I teach are geometry and pre-calculus. So I would like something that I can easily create graphs and geometric shapes.&lt;/p&gt;&#xA;&lt;p&gt;I have a MacBook Air M1 and the digital pen &amp;quot;XP-pen&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But Webex Whiteboard is not very handy for me.&lt;/p&gt;&#xA;&lt;p&gt;I'd be interested in a whiteboard like the one they use at Khan Academy.&lt;/p&gt;&#xA;&lt;p&gt;Any suggestions?&lt;/p&gt;&#xA;" OwnerUserId="20721" LastEditorUserId="20721" LastEditDate="2022-12-08T14:25:52.807" LastActivityDate="2023-01-09T19:09:18.973" Title="Suggestions for Virtual Blackboard?" Tags="|technology-in-education|online-instruction|software|specific-software|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25897" PostTypeId="2" ParentId="12" CreationDate="2022-12-08T17:41:43.970" Score="0" Body="&lt;p&gt;As with the answer about Keisler, this might be beyond what the question is asking for, but &lt;a href=&quot;https://global.oup.com/academic/product/calculus-set-free-9780192895592?cc=us&amp;amp;lang=en&amp;amp;&quot; rel=&quot;nofollow noreferrer&quot;&gt;C. Bryan Dawson has a new book&lt;/a&gt; also using infinitesimals throughout, but without going into the full details on the math logic side in the normal exposition.  It is intended for a fairly typical audience (e.g. in the US context, definitely not just honors at some R1).  Not my cup of tea, but he is pretty convincing about it both in person and in several &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/00029890.2018.1401832&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA&lt;/a&gt; &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/07468342.2019.1662706&quot; rel=&quot;nofollow noreferrer&quot;&gt;articles&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2022-12-08T17:41:43.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25899" PostTypeId="2" ParentId="25890" CreationDate="2022-12-09T15:48:03.467" Score="2" Body="&lt;p&gt;At the initial stages it is always &amp;quot;drill to kill&amp;quot;. Literally, there is no interesting issue there. Just tell them to write the answers in full sentences always starting with&#xA;&amp;quot;The quantity X can possibly equal to &amp;quot; when the question is &amp;quot;Find the probability distribution of the quantity X&amp;quot; and reduce points for not writing this sentence. Then tell them to make a full value versus probability table and, again, insist on the prescribed format. Those are idiotic requirements when the person knows perfectly well what he or she is doing, but if they don't, they help both them and you: if somebody writes in good faith &amp;quot;The sum of the numbers on two dice can possibly equal 1,2,3,4,5 or 6, you can immediately discuss &lt;em&gt;just that particular sentence&lt;/em&gt; with them without trying to figure out what and where went wrong in the solution written in the free style format.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2022-12-09T15:48:03.467" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25900" PostTypeId="2" ParentId="25896" CreationDate="2022-12-10T06:53:04.627" Score="0" Body="&lt;p&gt;One option is MathWhiteboard &lt;a href=&quot;https://www.mathwhiteboard.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathwhiteboard.com&lt;/a&gt;. It's handy. It is free.&lt;/p&gt;&#xA;&lt;p&gt;There are quite a few videos on YouTube on how you can use it!&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=el8LnfU0Wx4&amp;amp;list=RDCMUCMv_WF1x-BUzvZkIUc05LaQ&amp;amp;start_radio=1&amp;amp;rv=el8LnfU0Wx4&amp;amp;t=2733&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=el8LnfU0Wx4&amp;amp;list=RDCMUCMv_WF1x-BUzvZkIUc05LaQ&amp;amp;start_radio=1&amp;amp;rv=el8LnfU0Wx4&amp;amp;t=2733&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20721" LastActivityDate="2022-12-10T06:53:04.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25901" PostTypeId="1" AcceptedAnswerId="25902" CreationDate="2022-12-10T07:24:23.250" Score="0" ViewCount="93" Body="&lt;p&gt;What is the most suitable book for the IB program Analysis and Approaches SL for a student with significant weaknesses?&lt;/p&gt;&#xA;&lt;p&gt;I had suggested the book from HAESE Mathematics yet he finds it particularly incomprehensible.&lt;/p&gt;&#xA;" OwnerUserId="20721" LastActivityDate="2022-12-10T12:59:59.240" Title="Suggestion for IB program Analysis and Approaches SL book?" Tags="|secondary-education|textbooks|mathematical-analysis|book|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25902" PostTypeId="2" ParentId="25901" CreationDate="2022-12-10T12:59:59.240" Score="4" Body="&lt;p&gt;I looked up that textbook.  The website touts it as part of a &amp;quot;rigorous series of mathematics books for IB students around the world&amp;quot;.  If you have a weak, struggling student, why recommend a &amp;quot;rigorous&amp;quot; text?&lt;/p&gt;&#xA;&lt;p&gt;Here's an alternate, covering most of the same material:  Frank Ayres, Theory and problems of first year college mathematics, [College algebra, plane trigonometry, plane and solid analytic geometry, introduction to calculus] (Schaum's outline series)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0007DPVM2&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Theory-problems-first-college-mathematics/dp/B0007DPVM2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have a copy of this text and it is very student friendly &amp;quot;precalc&amp;quot; or &amp;quot;college algebra&amp;quot; (but with an intro to calc) book.  (And based on looking at the IB course description for Analysis and Approach, this is the right course translation.)&lt;/p&gt;&#xA;&lt;p&gt;Also, has all the answers, so very drill friendly.  Deliberately uses simple/less words to help students immediately get the math.  Sure, it's not steeped in IB-ness.  Or in rigor.  But those are features, not bugs, for a weak student.&lt;/p&gt;&#xA;" OwnerUserId="20732" LastActivityDate="2022-12-10T12:59:59.240" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25903" PostTypeId="1" CreationDate="2022-12-11T15:52:20.060" Score="14" ViewCount="450" Body="&lt;p&gt;Some ChatGPT answers have been posted on (and removed from) MathOverflow, and there was a resulting &lt;a href=&quot;https://meta.mathoverflow.net/questions/5531/chatgpt-strikes-mathoverflow&quot;&gt;MMO discussion&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In that discussion, @darijgrinberg &lt;a href=&quot;https://meta.mathoverflow.net/questions/5531/chatgpt-strikes-mathoverflow#comment28079_5533&quot;&gt;said&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When I tell it about its errors, it &amp;quot;recognizes&amp;quot; them, sometimes elaborating on a counterexample I provide, but then it repeats the same errors again. I have a hunch that it treats lexically similar &amp;quot;words&amp;quot; as semantically similar, which dooms it to uselessness in mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I suspect I'm not alone in recognising common student behaviour when I read that description.  Indeed, I &lt;em&gt;know&lt;/em&gt; I'm not alone.  @YemonChoi &lt;a href=&quot;https://meta.mathoverflow.net/questions/5531/chatgpt-strikes-mathoverflow#comment28083_5533&quot;&gt;said&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;@darijgrinberg &amp;quot; I have a hunch that it treats lexically similar &amp;quot;words&amp;quot; as semantically similar&amp;quot; - by &lt;em&gt;complete coincidence&lt;/em&gt; reading this made me think of marking HW and some referee work I should be doing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Given this (superficial but striking) similarity in behaviour, might the study of how AI &amp;quot;learns&amp;quot; mathematics shed any light on human learning?&lt;/p&gt;&#xA;&lt;p&gt;Here's an example:  it is, I think, both controversial and difficult to run educational experiments on students; in practice those with reform-minded sensibilities often run &lt;em&gt;ad hoc&lt;/em&gt; experiments, but, I think, with little attention given to experimental design (for example, as far as I know, there is usually no attempt to make sure that students in reform- and non-reform-styled classrooms are representative of the student body as a whole; in many cases, the populations are even self selected).  However, there seem to be no ethical concerns about running educational experiments on chatbots, and I can imagine something like an attempt by a very patient pair of instructors to teach a standard undergraduate course to an otherwise identically configured chatbot (or, why not, a classroom of chatbots, in cases where student interaction is part of the model) in two different ways, and observe the outcomes to determine which approach is better.&lt;/p&gt;&#xA;&lt;p&gt;EDIT:  Bill Wadge has a bit to say in a related, but different, direction:  &lt;a href=&quot;https://billwadge.com/2022/12/15/just-how-smart-are-you-chatgpt-i-quiz-chatgpt-about-math&quot; rel=&quot;nofollow noreferrer&quot;&gt;I quiz ChatGPT about math&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2070" LastEditorUserId="2070" LastEditDate="2022-12-16T21:40:00.137" LastActivityDate="2023-03-22T20:24:57.360" Title="Pedagogical insights to be gleaned from AI attempts to ‘learn’ mathematics" Tags="|general-pedagogy|software|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25904" PostTypeId="1" CreationDate="2022-12-12T14:59:24.273" Score="6" ViewCount="1055" Body="&lt;p&gt;How to explain to a ~&lt;span class=&quot;math-container&quot;&gt;$12-16$&lt;/span&gt; year-old student who is weak at maths, that &lt;span class=&quot;math-container&quot;&gt;$\ 38 \times 27 \times 14 = 27 \times 14 \times 38,\ $&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$\ 3.4 \times 10^{-6} \times 2.1 \times 10^{-5} = 3.4\times 2.1 \times 10^{-6} \times 10^{-5},\ $&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$\ 11^4 \times 7^7 \times 3^5 \times 5^2 = 3^5 \times 5^2 \times 7^7 \times 11^4,\ $&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$\ \frac{ 8 }{ 14 } \times \frac{ 3 }{ 4 } \times \frac{ 7 }{ 11 } \times \frac{ 22 }{ 9 } = \frac{ 8 \times 3 \times 7 \times 22 }{ 14 \times 4 \times 11 \times 9 } = \frac{ 7 \times 8 \times 22 \times 3 }{ 14 \times 4 \times 11 \times 9 } = \frac{ 7 }{ 14 } \times \frac{ 8 }{ 4 } \times \frac{ 22 }{ 11 } \times \frac{ 3 }{ 9 },\ $&lt;/span&gt; and so on.&lt;/p&gt;&#xA;&lt;p&gt;I have been in this situation a few times, where it was necessary - or at least extremely helpful - for them to use the fact that if the value of the product of finitely many terms remains unchanged if terms being multiplied are rearranged within the product. [Maybe there's a better way of wording this, but anyway...]&lt;/p&gt;&#xA;&lt;p&gt;I proceeded to explain that, &amp;quot;you can swap any two numbers without affecting the result due to &lt;span class=&quot;math-container&quot;&gt;$ab = ba,$&lt;/span&gt; and therefore we can always re-arrange the numbers in the original product &lt;span class=&quot;math-container&quot;&gt;$\ 38 \times 27 \times 14\ $&lt;/span&gt; in any order we want, without affecting the value of the product.&amp;quot; However, upon reflection, I think this line of reasoning is too complicated for students who are weak at maths to understand. This is for two reasons. First, they have to understand that swapping any two numbers gives the same result, which is not necessarily obvious, and secondly, they have to understand that there is a way to keep swapping terms to get from the first product &lt;span class=&quot;math-container&quot;&gt;$\ (\ 38 \times 27 \times 14 \ )\ $&lt;/span&gt; to the second product &lt;span class=&quot;math-container&quot;&gt;$\ (\ 27 \times 14 \times 38\ )\ $&lt;/span&gt;, which also is not necessarily obvious.&lt;/p&gt;&#xA;&lt;p&gt;Although, when I was that age, I found both of those steps &amp;quot;obvious&amp;quot;, even if I did not know the mathematical language to prove it formally.&lt;/p&gt;&#xA;&lt;p&gt;And introducing rigor is not advisable for weak students either.&lt;/p&gt;&#xA;&lt;p&gt;So is there a completely different (perhaps visual?) way to explain this phenomenon/fact that is easy for a weak student to understand, that can also be extended to the product of four, or more numbers? Or can it be explained along the lines I am trying to, but in a way that is easier to understand?&lt;/p&gt;&#xA;&lt;p&gt;Or is this just one of those &amp;quot;facts&amp;quot; that you tell students of that age that they &amp;quot;just have to accept&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2022-12-19T13:31:48.463" LastActivityDate="2023-02-06T18:02:33.370" Title="How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?" Tags="|secondary-education|multiplication|" AnswerCount="11" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="25905" PostTypeId="2" ParentId="25904" CreationDate="2022-12-12T15:21:10.967" Score="8" Body="&lt;p&gt;For up to three dimensions you could use the concept of &amp;quot;area of a rectangle&amp;quot; or &amp;quot;volume of a rectangular cuboid&amp;quot;. Use only numbers (no variables) and let the student pick a convention like &amp;quot;the first factor is the width and the second factor is the height (and the third one the depth)&amp;quot;. Then use lots of examples and let the student draw many rectangles (or, if material is available and the task doesn't seem silly, arrange dice or something else in rectangular patterns) and have him or her realise (&amp;quot;accept?&amp;quot;) that ab = ba always, because there are the same number of unit squares in both rectangles. Extend to three dimensions and then generalise for arbitrary numbers of factors.&lt;/p&gt;&#xA;&lt;p&gt;I think this is about &amp;quot;looking at enough examples to accept the commutative law&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2022-12-12T15:21:10.967" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="25907" PostTypeId="2" ParentId="25904" CreationDate="2022-12-12T19:27:25.563" Score="3" Body="&lt;p&gt;My advice is &amp;quot;and&amp;quot;.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;List the principle.  But NOT in a symbolic (ab) manner.  Just &amp;quot;you can change the order when multiplying.&amp;quot;  [Realize you will need to repeat it, several times.  They are not programmable machines.  But just keep doing so.]&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;-and-&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Each time they miss it, have them do a few simpler examples.  2 x 3 x 4.  2 x 5 x 10. (in a couple orders)  [After a while, they will get sick of it.  And decide to remember the principle.]&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So, net net, I think your instincts are correct in recognizing that weaker students won't learn axiomatically).  They are different than you. But in degree...we all share traits of ability and disability in intellect.  Unless you're Von Neumann, you need drill, to engrain math tricks.  He of course was a Martian.  But human brains are more like Harry Potter's hand.  Need Umbridge's pen to furrow some lines into it/them.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I think the &amp;quot;like a rectangular prism volume&amp;quot; is low likely to succeed.  Wrong pedagogic insight.  It's MUCH more likely that repeated statements and practice will make things click than some magic explanation (in this case a hard one).  That doesn't mean, you can't have them, can't try them.  After all the &amp;quot;and&amp;quot; principle applies.  And you should try lots of things.  But.  I would not think &amp;quot;key explanation&amp;quot; (like a lock and key) is the way to overcome these hurdles.  It's more likely that familiarization is what is needed.  And yes, more than you would need.  More than what a high IQ student would need.&lt;/p&gt;&#xA;&lt;p&gt;P.s.s.  I don't think of rectangles with multiplication as Xander mentions.  I'm not a Ph.D. mathematician. But a decent ex-STEM student.  And I think of  multiplication as extension of the times table (via long multiplcation).  I.e. algebraically. If I really scratch my head and think of the times table itself, I probably think of repeated addition.  This is NOT to say that it's the preferred way.  Just that rectangles are not something everyone thinks of.  I'm always pleasantly surprised by geometric drawings for Pythagoras or the like...since I don't think as geometrically.  Again, not even saying this is good. Just saying the impulse to think that everyone thinks in one manner (the way one does) is a mistake.  I mean...heck with exponentiation do you think first of repeated multiplication or think first of &amp;quot;more dimensions in Flatland&amp;quot;!?&lt;/p&gt;&#xA;" OwnerUserId="20739" LastEditorUserId="20739" LastEditDate="2022-12-12T19:47:56.783" LastActivityDate="2022-12-12T19:47:56.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25908" PostTypeId="2" ParentId="25903" CreationDate="2022-12-12T19:55:54.400" Score="-3" Body="&lt;p&gt;There's a whole topic called Artificial Intelligence.  Was very sexy in the 80s (and the 60s before that).  Now it has come back again and seems to finally be having real achievements.   Technical trends can be like that.  Look at electric cars.  Or fuel cells.  Or video phones.  Or space travel.  Or smart phones (remember the Treo that failed?)&lt;/p&gt;&#xA;&lt;p&gt;There are insights from computers to human thought.  And also the converse.  It's not a one for one.  But in visual processing, insights have gone in both directions.  So in a very vague, meta, way, I think yes there are/will be insights.  In a practical, practitioner way, I don't think ChatGPT is worth paying attention to.  At least now.  Unless you're doing research in AI itself.  IF there are insights, they will be conveyed.  I'd blow it off and ignore it for now.  (Bayesian prior.)&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I'd also be extremely cautious about false insights.  Computers have massive RAM, that humans lack.  Very easy to make the wrong insights based on assuming a computer works like a human or visa versa.  See this routinely with people that assume just explaining a rule is adequate to drive performance, retention.  Thus devaluing the human need for practice.&lt;/p&gt;&#xA;" OwnerUserId="20739" LastEditorUserId="20739" LastEditDate="2022-12-12T22:24:31.080" LastActivityDate="2022-12-12T22:24:31.080" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25909" PostTypeId="2" ParentId="25904" CreationDate="2022-12-13T01:21:50.363" Score="8" Body="&lt;p&gt;I think having a student play with examples is a great approach, and that seems the easiest route to having them see &lt;em&gt;that&lt;/em&gt; this is true.&lt;/p&gt;&#xA;&lt;p&gt;If seeing multiple permutations of factors all giving the same result isn't convincing, or if they want a different way to conceptualize the situation, maybe draw a model of a product like &lt;span class=&quot;math-container&quot;&gt;$2\times5\times3$&lt;/span&gt; and show that pieces can be rearranged. You'd have to choose in advance how to interpret an expression like &lt;span class=&quot;math-container&quot;&gt;$a\times b\times c$&lt;/span&gt;, say where there are &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; objects, each containing &lt;span class=&quot;math-container&quot;&gt;$a\times b$&lt;/span&gt; smaller units.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/WLMCj.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WLMCj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here, each of three given rectangles has &lt;span class=&quot;math-container&quot;&gt;$2\times 5$&lt;/span&gt; little squares in it. Put them together, shade in different parts (in this case columns), reform the columns into rectangles, and then interpret this new picture as a new product (using the original factors, just reordered).&lt;/p&gt;&#xA;&lt;p&gt;I'm not totally sure how a conversation like this would go with a student, whether they would be satisfied with it. I guess it would all come down to how student and instructor decide to interpret multiplication. [E.g. If &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; means &amp;quot;number of rows in each rectangle&amp;quot;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; means &amp;quot;number of columns in each rectangle&amp;quot; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; means &amp;quot;number of rectangles&amp;quot;, then you could just rotate each rectangle to start with to get other expressions like &lt;span class=&quot;math-container&quot;&gt;$5\times 2 \times 3$&lt;/span&gt;.]&lt;/p&gt;&#xA;&lt;p&gt;[Edit: I can see that my color choice probably wasn't great, since it looks like red changed size.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2022-12-13T01:21:50.363" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25910" PostTypeId="2" ParentId="25904" CreationDate="2022-12-13T01:37:10.490" Score="2" Body="&lt;p&gt;Many college students have a 2nd issue with a problem like this. If they see 2(3 · 4), they sometimes want to distribute, and write 2·3 · 2·4. Yikes. This one requires seeing the difference between distributive property and a · b · c. We work on that.&lt;/p&gt;&#xA;&lt;p&gt;First, we go back to areas and volumes. My answer is therefore much like Jasper's. Yes, considerable time must be spent on these basics for further mathematical steps to have any meaning for them.&lt;/p&gt;&#xA;&lt;p&gt;I ask: What is the area of a rectangle? Someone tells me. Most of them nod.&#xA;I ask: Why? They look at me like I'm nuts. They think of it as a &amp;quot;formula&amp;quot;, and they think of those as givens.&lt;/p&gt;&#xA;&lt;p&gt;So I have them look at the ceiling tiles (because the floor tiles are 1 foot by 1 foot and don't teach area). We estimate those to be 2 feet by 4 feet. So that's 2 rows of 4 tiles each. We could count, or we could ... multiply, yes.&lt;/p&gt;&#xA;&lt;p&gt;Once we've done one more problem like that on the board, and they are &lt;em&gt;seeing&lt;/em&gt; that the multiplication is how many rows times how much in a row, then we go to volume. I use my hands to &amp;quot;create&amp;quot; (mime) a cardboard box on top of my desk. It's 4 feet across, 3 feet front to back, and 2 feet high. We imagine small 1 cubic foot boxes being packed inside. There are 3 rows of 4 on the bottom layer, and then there's a top layer. How many? 12 and 12 is 24 of the cubic foot boxes. Could we count vertical layers from front to back? Sure. The front 'layer' is 4 by 2, and then they go 3 back. 8 times 3 is also 24. And we could go left to right. 3 by 2 is 6, 4 times that is still 24 cubic feet. Of course.&lt;/p&gt;&#xA;&lt;p&gt;I also reteach distributive property, using visuals. My goal is always to help them to see &lt;em&gt;why&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-12-13T01:43:46.567" LastActivityDate="2022-12-13T01:43:46.567" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25911" PostTypeId="2" ParentId="25904" CreationDate="2022-12-13T13:11:31.647" Score="3" Body="&lt;p&gt;Most of the answers here have been geometric. Here is an answer for three or more factors that is algebraic. It requires the student to know that multiplication is associative (more understandable to them, &amp;quot;we can put parentheses anywhere because of order of operations&amp;quot;, maybe needs additional justification), the commutative law for two numbers (justified well by the rectangles in the other answers) and that two equal expressions can be substituted for one another.&lt;/p&gt;&#xA;&lt;p&gt;Sample explanation:&lt;/p&gt;&#xA;&lt;p&gt;Start with &lt;span class=&quot;math-container&quot;&gt;$38 \times 27 \times 14$&lt;/span&gt;. We are not going to calculate this, but if we were, we would need to multiply two of the numbers together first, and then multiply that result times the third number. What that means in symbols is that we might actually calculate &lt;span class=&quot;math-container&quot;&gt;$ (38 \times 27) \times 14$&lt;/span&gt;. Because multiplication is associative, we could also do &lt;span class=&quot;math-container&quot;&gt;$ 38 \times (27 \times 14) $&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To summarize, &lt;span class=&quot;math-container&quot;&gt;$38 \times 27 \times 14 = (38 \times 27) \times 14 = 38 \times (27 \times 14)$&lt;/span&gt;. Let's say we went with &lt;span class=&quot;math-container&quot;&gt;$(38 \times 27) \times 14$&lt;/span&gt;, and just take a look at &lt;span class=&quot;math-container&quot;&gt;$(38 \times 27)$&lt;/span&gt; on its own. Because of our commutative law, we know that &lt;span class=&quot;math-container&quot;&gt;$(38 \times 27) = (27 \times 38)$&lt;/span&gt;, and since we can substitute equal expressions into a third expression, our original is now &lt;span class=&quot;math-container&quot;&gt;$38 \times 27 \times 14 = (38 \times 27) \times 14 = (27 \times 38) \times 14$&lt;/span&gt;. We can then move the brackets again (associativity) and switch 38 and 14 to get the final answer.&lt;/p&gt;&#xA;&lt;p&gt;The bottom line is that by doing this trick, we can see that when we have a bunch of numbers multiplied together, we can switch them around in any order.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2022-12-13T13:11:31.647" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25912" PostTypeId="2" ParentId="25903" CreationDate="2022-12-13T23:02:47.713" Score="9" Body="&lt;p&gt;In intro proof classes I often wish that I had a large collection of &amp;quot;bullshit student work&amp;quot; for my students to pick apart.  ChatGPT is basically a bullshit generator, so it seems well suited to this task.  I gave ChatGPT the task of proving the following false statement:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Prompt:  Prove that any number divisible by 3 is a square mod 4.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To prove that any number divisible by 3 is a square mod 4, we first need to understand what it means for a number to be a square mod 4.&#xA;In modular arithmetic, a number a is a square mod 4 if it satisfies the equation a^2 = 1 (mod 4). In other words, the remainder when a^2 is divided by 4 must be 1.&#xA;For example, the number 5 is a square mod 4 because 5^2 = 25 = 6 (mod 4). On the other hand, the number 6 is not a square mod 4 because 6^2 = 36 = 0 (mod 4).&#xA;Now, let's consider a number that is divisible by 3. Since 3 is a prime number, any number that is divisible by 3 must also be divisible by 3^2 = 9. Therefore, we can write any number that is divisible by 3 in the form 9k, where k is an integer.&#xA;We can now use this form to show that any number divisible by 3 is a square mod 4. Let's start by assuming that a number x is divisible by 3 and can be written in the form 9k, where k is an integer. We want to show that x^2 = 1 (mod 4).&#xA;Since x = 9k, we can write x^2 = (9k)^2 = 81k^2 = 3^4 * k^2. Since the remainder when 3^4 is divided by 4 is 1, we know that the remainder when 3^4 * k^2 is divided by 4 must also be 1. Therefore, x^2 = 1 (mod 4), which means that x is a square mod 4.&#xA;Therefore, we have proven that any number divisible by 3 is a square mod 4.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This does seem like something a student could write if they were trying to bullshit a proof and &amp;quot;finding all the errors&amp;quot; seems like a reasonable exercise.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-12-13T23:02:47.713" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25913" PostTypeId="2" ParentId="25903" CreationDate="2022-12-14T04:49:55.117" Score="0" Body="&lt;p&gt;The algorithms such as ANN, CNN, although they have biological inspiration on the human brain are based on mathematics.&lt;/p&gt;&#xA;&lt;p&gt;For this reason it is not meaningful to look at the pedagogical insights to be gleaned from AI attempt to learn mathematics as what you are looking at is some mathematical model.&lt;/p&gt;&#xA;&lt;p&gt;If you are interested I have linked a video that go through the mathematics of ANN. Although I am not sure how exactly ChatGPT ML model was created I would presume it is based some neural network.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=Wo5dMEP_BbI&amp;amp;list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=Wo5dMEP_BbI&amp;amp;list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20626" LastActivityDate="2022-12-14T04:49:55.117" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25914" PostTypeId="2" ParentId="19510" CreationDate="2022-12-15T10:11:16.213" Score="0" Body="&lt;p&gt;Based on what I learned from abstract algebra, you only need the Rational Root Theorem for historical purposes of splitting polynomials. If you are teaching history or motivating abstract algebra, then go over it, otherwise, I would skip it.&lt;/p&gt;&#xA;&lt;p&gt;It's usually not in the standards and if it is, it's usually because it s a relic from before calculators and computers.&lt;/p&gt;&#xA;" OwnerUserId="10335" LastActivityDate="2022-12-15T10:11:16.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25915" PostTypeId="1" CreationDate="2022-12-15T10:55:43.873" Score="-1" ViewCount="238" Body="&lt;p&gt;I have only passed the one fourth of my courses so far, which means like one year of the maths undergraduate studies. I live in a different place from where the university i study at is and try to study alone only with books, notes and pdfs from the internet.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps i have fear of math exams and generally of exams, when i face the topics and problems to prove in the exams i have some ideas but mostly they are wrong.In the past i passed some exams.&lt;/p&gt;&#xA;&lt;p&gt;When i try to learn what i read i have an approach perhaps wrong. I try to remember all of what i read without missing or forgetting a symbol or word and if i fail to learn this way i think i am not capable of learning and that i do not learn correctly.&lt;/p&gt;&#xA;&lt;p&gt;I want to get my math degree, passing the exams and finally getting the degree would be okay i think.&lt;/p&gt;&#xA;&lt;p&gt;What should i do to overcome the fear of math exams and generally exams?Perhaps, should i try to solve math exercises?Is my way of learning wrong? And what is a more suitable way of learning? What should be my schedule or daily program of reading and solving exerises be? If i get stuck at some exercises should i try to find the answers at the books or notes?&lt;/p&gt;&#xA;&lt;p&gt;Generallly what does a math student use to solve exercises from the theory? Should i learn the proofs-as i try to generally do- of theorems to have a better learning?&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17553" LastActivityDate="2023-07-17T06:41:48.860" Title="Learning, exams, math degree" Tags="|self-learning|exams|problem-solving|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25916" PostTypeId="2" ParentId="25915" CreationDate="2022-12-15T11:18:24.087" Score="2" Body="&lt;p&gt;Solve more problems.  Take more practice tests.  Experience reduces anxiety and improves performance.&lt;/p&gt;&#xA;&lt;p&gt;The first time I led a scram drill, on a submerged submarine, I was scared.  After 11 of them, I was pretty salty.&lt;/p&gt;&#xA;" OwnerUserId="20606" LastActivityDate="2022-12-15T11:18:24.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25917" PostTypeId="2" ParentId="25890" CreationDate="2022-12-15T17:15:57.017" Score="1" Body="&lt;p&gt;The only way I have ever had success with students understanding what a probability distribution is by taking data, building a frequency distribution, and then asking questions about what we should expect if we chose an individual at random, given that frequency distribution.&lt;/p&gt;&#xA;&lt;p&gt;For example: if you have 30 students in your class, ask them all a hopefully bland, non-intrusive question. Maybe &amp;quot;how long is your commute to school&amp;quot; or &amp;quot;what is the fourth digit of your phone number&amp;quot;? You can probably come up with better examples.&lt;/p&gt;&#xA;&lt;p&gt;Once we have the frequency distribution, we can talk about it approximating a probability distribution, and hopefully make connections as to why its shape makes sense and what that means for expectations.&lt;/p&gt;&#xA;" OwnerUserId="6589" LastActivityDate="2022-12-15T17:15:57.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25918" PostTypeId="1" CreationDate="2022-12-17T18:19:21.907" Score="-2" ViewCount="123" Body="&lt;p&gt;What I want to ask is: can I use the results of others when doing scientific research, whether it is math, physics or other science, without trying to prove them myself or try to verify them? For example results like theorems, proofs, definitions, abstract mathematical objects in math or results in physics. Should I refer to them in the end of the research and tell the literature I used, for example which books or notes? What results whether older or more modern could I use? They should be verified I suppose?&lt;/p&gt;&#xA;&lt;p&gt;Could I try to answer my own problems or questions or already known problems? Should I communicate with scientists from the university I study at? Could they help me to tackle the problems i want and read the things I want to read?If the math or physics I want to use for the problems are not already discovered should I try to discover them myself?&lt;/p&gt;&#xA;" OwnerUserId="17553" LastEditorUserId="8571" LastEditDate="2022-12-21T15:29:44.030" LastActivityDate="2022-12-21T15:29:44.030" Title="On doing scientific research" Tags="|self-learning|problem-solving|" AnswerCount="1" CommentCount="2" ClosedDate="2022-12-18T14:34:28.673" ContentLicense="CC BY-SA 4.0" />
  <row Id="25919" PostTypeId="2" ParentId="25918" CreationDate="2022-12-17T18:35:53.667" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;This is not a formed question.  You ask at least two different things (is it trustworthy to use others results--and--should you try to prove missing things yourself.)  There's also a question about attribution.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You should start capitalizing I (i).  Was giving you some leeway based on likely foreign language.  But English is the lingua franca (haha) of technical work.  As a young student, just buckle down and learn it.  It comes off as lazy if you don't.  I'm sure you'll still have some minor errors, I do, and fine. But the i thing just irks like crazy.  You won't get help from others if you are so lazy yourself.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;(Content) By and large yes, you can use known results (e.g. physical constants in physics, molar masses in chemistry, and character tables of point groups).  For math, it's probably pretty normal that you would be able to, or at least once have, prove(ed) the tools you are using.  In chemistry/physics/bio, it is normal to use some well known results without checking them.  Of course, if it is a sketchy phenomenon (new report, e.g. &amp;quot;stripy nanoparticles&amp;quot;), you should verify the basis, before building further.  Attributions via citations are normal for relatively new phenomena (BOTH to give credit and to direct readers for more info), but not required for well known results.  The decision of where to draw the line is a judgment call, but not hard as a practitioner.  (Instead of trying to anticipate how to do this, get some &amp;quot;practitioner&amp;quot; work. There are many things actually getting your hands dirty with real research will teach you...and you will ask better questions as a result.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20757" LastEditorUserId="20757" LastEditDate="2022-12-17T18:39:14.303" LastActivityDate="2022-12-17T18:39:14.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25920" PostTypeId="2" ParentId="25904" CreationDate="2022-12-18T12:55:43.720" Score="0" Body="&lt;p&gt;When Xander wrote in his comment to his answer, &amp;quot;If these students cannot work out the area of a rectangle, what is their mental model of multiplication?&amp;quot;, I think a more relevant comment is, &amp;quot;If these students cannot work out the area of a rectangle, then they shouldn't yet be trying to learn that &lt;span class=&quot;math-container&quot;&gt;$ab = ba,$&lt;/span&gt; let alone &lt;span class=&quot;math-container&quot;&gt;$abc = cab.$&lt;/span&gt;&amp;quot; I should build multiplication up with geometry at the same level of multiplication that we are working on, with geometry as a helpful tool assisting in the understanding of multiplication, in a similar manner one would use money problems to assist in the teaching of addition. And I shouldn't try to cover all of &amp;quot;numbers&amp;quot; (standard form, HCF/LCM, etc) before moving onto geometry later on; with strong students this might be possible, as we can cover  ground more quickly, but with weak students - which is what we are talking about here - it isn't an effective teaching strategy, I guess.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastActivityDate="2022-12-18T12:55:43.720" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="25921" PostTypeId="1" CreationDate="2022-12-19T01:14:35.043" Score="8" ViewCount="1866" Body="&lt;p&gt;I know an undergraduate who just finished calc III, and really struggled with it. However, it seems like a lot of the issues are coming from skills which should have been established during algebra, precalc, and calc I. I want to recommend some resources (practice problems, videos, etc.) to help him improve at these skills, does anyone have any recommendations?&lt;/p&gt;&#xA;&lt;p&gt;Here are some examples to illustrate the sort of skills I'm hoping he can practice:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;As part of a larger problem, he had to solve cos(x)  = 0.4, and wasn't able to apply the idea an inverse function (even after being reminded about the existence of inverse-cosine)&lt;/li&gt;&#xA;&lt;li&gt;Often when talking through a problem, he'll struggle with the fact that he was introduced to a formula using different letters. E.g., when learning how to compute a directional derivative they used v for the vector, but when needing to calculate a directional derivative using a new vector which was called p, he was confused. This issue in particular has come up in across a ton of different problems I've helped him with.&lt;/li&gt;&#xA;&lt;li&gt;Generally connecting the math to the actual situation being described is very hard for him. This leads to a lot of trouble when problems are given using language which does not precisely match the language which the methods were originally presented using. This in particular seems like a hard thing to study without tutoring/instructor feedback.&lt;/li&gt;&#xA;&lt;li&gt;This may be related to the last point, but when problems are given which flip &amp;quot;usual&amp;quot; question/answer information, the idea of writing out the known formulas and solving for the requested information is hard to prompt him to come up with. For example, when given a vector v in 2-D and told &amp;quot;find any vector orthogonal to v&amp;quot;, the idea of setting the dot product of v and the unknown vector equal to 0 did not come to mind, even after we talked about what how we know when two  vectors are orthogonal.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I know this is a lot, but he's hoping to apply to PhD programs in a pretty technical field, so these are things he'll need to catch up to his cohort on. Does anyone have any advice on self-study material? (Other advice about how to make progress on improving would also be helpful)&lt;/p&gt;&#xA;&lt;p&gt;P.S. One last meta-issue which may be particularly pertinent to self-studying is that he seems to have a lot of trouble knowing when he does or doesn't know something. When I've self-taught math subjects I've generally found that I'm able to identify when I'm not understanding something and seek outside help. Does anyone have any advice on how to build this ability up?&lt;/p&gt;&#xA;" OwnerUserId="20763" LastActivityDate="2022-12-20T19:53:16.983" Title="Problem set recommendations for a student who has taken classes through Calc III, but has very weak fundamentals" Tags="|resource-request|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25922" PostTypeId="2" ParentId="25921" CreationDate="2022-12-19T02:02:24.037" Score="2" Body="&lt;p&gt;Two somewaht different thoughts.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In terms of drill volume, I like Frank Ayres Schaum's Outline First Year College Math (pre-calculus), the original edition.  Has the answers.  Goal is just to do lots of volume.  No sudden &amp;quot;aha&amp;quot;, just more facility from exposure.  There are competing &amp;quot;huge problem&amp;quot; books.  Also online solutions like Kahn Academy.  But the one I mentioned is one I know that still gets good reviews on Amazon.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would also suggest, for efficiency, just doing drill in the context of his degree (using the familiar variable names), which you did not mention.  Trying to guess one that goes through calc 3, but not ODEs.  Thinking comp sci or chemistry.  Realistically, remediating his general math skills may not be as efficient as just brushing him up on the specifics of his major (e.g. &amp;quot;p chem&amp;quot;).  With adult learners, moved on to grad school, sometimes you just have to be more economical.  Like I really could care less if the average coder or natural products synthetic chemist knows div grad curl.  Or even arccosine.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20764" LastActivityDate="2022-12-19T02:02:24.037" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25923" PostTypeId="2" ParentId="25921" CreationDate="2022-12-19T12:05:34.020" Score="9" Body="&lt;p&gt;I am sceptical whether just doing more exercises would help this student. Based on the description, they are thinking about mathematics the wrong way.&lt;/p&gt;&#xA;&lt;p&gt;The problematic skill seems to be sense-making, ie getting at the meaning behind the mathematical formalism and actually reason about things. While it is very common for beginners to struggle with that, if they haven't gotten it by Calc III, it seems like a cause for real concern.&lt;/p&gt;&#xA;&lt;p&gt;Unless the additional concreteness of meaning in Microeconomics makes a difference to their sense-making ability, going into a PhD in that field without fixing this seems like a horrible idea - you can solve exercises by blindly following steps you've learned by heart, but you can't do research like that.&lt;/p&gt;&#xA;" OwnerUserId="13471" LastActivityDate="2022-12-19T12:05:34.020" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25924" PostTypeId="2" ParentId="25921" CreationDate="2022-12-19T18:53:58.853" Score="3" Body="&lt;p&gt;Sense-making versus doing lots of exercises. Both are important. But this particular student seems to really not get that math is sense-making.&lt;/p&gt;&#xA;&lt;p&gt;I am guessing you are his tutor. Would you like to ask him to play with the &lt;a href=&quot;https://www.thinkfun.com/products/rush-hour/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rush Hour&lt;/a&gt; puzzle for a few minutes at the beginning of a session? There are lots more fun and challenging logic puzzles on that site.&lt;/p&gt;&#xA;&lt;p&gt;Desmos has oodles of good activities. One I have used extensively is &lt;a href=&quot;https://teacher.desmos.com/activitybuilder/custom/566b31734e38e1e21a10aac8?collections=featured-collections%2C5e73b204d560367270838c4b&quot; rel=&quot;nofollow noreferrer&quot;&gt;marbleslides&lt;/a&gt;. It's meant to help students  think about equations versus graphs. You didn't say that he was bad at that, but if he's just memorized rules, this will help him build a more robust understanding. (I linked the one for straight lines; there is a whole collection for different function types, and more.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://drive.google.com/file/d/1hVP8tLURVDphmHsphz5BQLVzHCeTts29/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;Catriona Schearer makes wonderful (hard) puzzles&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;All of these seem like they focus somewhat on spatial reasoning. Hmm. &lt;a href=&quot;https://www.setgame.com/set/puzzle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Set&lt;/a&gt; is good too, logic and not spatial.&lt;/p&gt;&#xA;&lt;p&gt;I can't think of a good game specifially for notation. I suppose you could create one yourself. Like concentration. Have a bunch of cards that all mean the same thing, but using different symbols and arranged in algebraically different ways. (My students often have trouble recognizing that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{9-{x^{2}}}$&lt;/span&gt; is (the upper half of) a circle. They would benefit from a game like this.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2022-12-20T19:53:16.983" LastActivityDate="2022-12-20T19:53:16.983" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25925" PostTypeId="1" CreationDate="2022-12-21T20:43:08.153" Score="4" ViewCount="162" Body="&lt;p&gt;It is common that school math textbooks are supplemented with instructor's materials (answers, hints and guidelines).&#xA;Are there scientific studies on how instructor's materials impact math teaching?&#xA;Any reference about the world of instructor's material (design, principles, uses, etc.) are welcome.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="376" LastEditDate="2022-12-22T19:58:15.113" LastActivityDate="2022-12-23T15:36:11.140" Title="About instructor's manual in math textbooks: principles, design, uses and impact" Tags="|reference-request|textbooks|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25926" PostTypeId="2" ParentId="25925" CreationDate="2022-12-23T15:36:11.140" Score="1" Body="&lt;p&gt;some research on the topic:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;LCROSS-CULTURAL ANALYSIS OF THE VOICE OF&#xA;CURRICULUM MATERIALS by Janine Remillard, Hendrik Van Steenbrugge, Tomas Bergqvist&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwjQ48DI8Y_8AhXwD7kGHcQhC74QFnoECA0QAQ&amp;amp;url=https%3A%2F%2Fwww.diva-portal.org%2Fsmash%2Fget%2Fdiva2%3A805729%2FFULLTEXT03.pdf&amp;amp;usg=AOvVaw06yFQA3IAWnD1xtFZfdCUa&quot; rel=&quot;nofollow noreferrer&quot;&gt;International Conference on&#xA;Mathematics Textbook Research and Development 2014&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Textbooks and Teachers' Guides: A Dilemma for Beginning Teachers and Teacher&#xA;Educators by Deborah Loewenberg Ball and Sharon Feiman-NemserSource&lt;a href=&quot;http://www.jstor.org/stable/1179386?origin=JSTOR-pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;JSTOR&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;TEACHER MANUAL OF SCHOOL-BASED AND CLASSROOM-BASED ACTIVITIES TO SUPPORT ALL LEARNERS BY Judith Hollenweger AND Edina Krompák&lt;br /&gt;&#xA;&lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwiHs-7Jh5D8AhXOHbkGHQFQAlIQFnoECAoQAQ&amp;amp;url=https%3A%2F%2Fwww.unicef.org%2Fnorthmacedonia%2Freports%2Fteacher-manual-school-based-and-classroom-based-activities-support-all-learners&amp;amp;usg=AOvVaw2xPMenP7jzWmz48xhuuOtc&quot; rel=&quot;nofollow noreferrer&quot;&gt;UNICEF&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Design Principles for Learning to Guide Teacher Walk&#xA;Throughs BY Eric Feeney. &lt;a href=&quot;http://www.tandfonline.com/action/showCitFormats?doi=10.1080/00098655.2013.823903&quot; rel=&quot;nofollow noreferrer&quot;&gt;Journal of Educational Strategies, Issues and Ideas&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;the world of teacher's manual and guides for textbooks needes more research!&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2022-12-23T15:36:11.140" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25928" PostTypeId="2" ParentId="267" CreationDate="2022-12-24T15:44:08.757" Score="1" Body="&lt;p&gt;Claim:  The function &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{R} \to \mathbb{R}$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^3+x-2$&lt;/span&gt; has one and only one root.&lt;/p&gt;&#xA;&lt;p&gt;Proof 1 (by contradiction)&lt;/p&gt;&#xA;&lt;p&gt;We can see that &lt;span class=&quot;math-container&quot;&gt;$f(1) = 0$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has at least one root.&#xA;Assume the the contrary that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has an additional root &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;.  Then, by the mean value theorem for derivatives, there is a &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; between &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$f'(c) = 0$&lt;/span&gt;.  However, &lt;span class=&quot;math-container&quot;&gt;$f'(c) = 3c^2 + 1 &amp;gt; 1$&lt;/span&gt;.  We have arrived at a contradiction, so the hypothesis that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has more than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; root false.  Thus &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has one and only one root.&lt;/p&gt;&#xA;&lt;p&gt;Proof 2 (direct proof)&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; be a root.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp; \hphantom{fdsfsf} x^3 + x - 2 = 0\\&#xA;&amp;amp;\iff (x-1)(x^2+x+2) =0\\&#xA;&amp;amp;\iff (x-1)\left[(x+\frac{1}{2})^2 + \frac{7}{4}\right] = 0\\&#xA;&amp;amp;\iff x-1 = 0\\&#xA;&amp;amp;\iff x= 1&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can justify the equivalence of the third and fourth line by noting that &lt;span class=&quot;math-container&quot;&gt;$\left[(x+\frac{1}{2})^2 + \frac{7}{4}\right] \geq \frac{7}{4}$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-12-24T15:44:08.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25929" PostTypeId="2" ParentId="25867" CreationDate="2022-12-24T21:00:57.577" Score="1" Body="&lt;p&gt;&lt;em&gt;&amp;quot;solve the simple equation ax−b=0 for x. The answer given is typically b/a, but that is wrong if a=0&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;But is it truly wrong?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;One gets the same result manually.&lt;/p&gt;&#xA;&lt;p&gt;Taken as a &lt;strong&gt;limit&lt;/strong&gt; where &lt;strong&gt;a ⇒ 0&lt;/strong&gt; it's correct.&lt;/p&gt;&#xA;&lt;p&gt;Granted it's undefined at that exact value and that has to be taught, but the question implicitly brings &lt;em&gt;Numerical Methods&lt;/em&gt; into play as it involves computers. When dealing with floating point numbers, i.e. non-integers, the very concept of equality is a mistake.&lt;/p&gt;&#xA;&lt;p&gt;In the computer language &lt;strong&gt;APL&lt;/strong&gt;, it defines &lt;strong&gt;0/0 = 1&lt;/strong&gt;. Ironically &lt;strong&gt;APL&lt;/strong&gt; was created specifically for math. In the mathematical abstract it's undefined, but in numerical methods it often makes more sense than what most languages do, which is to crash, because more often than not the computation is approaching zero in the denominator when it ran out of bits.&lt;/p&gt;&#xA;" OwnerUserId="10376" LastActivityDate="2022-12-24T21:00:57.577" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25930" PostTypeId="1" CreationDate="2022-12-25T14:44:21.927" Score="29" ViewCount="2314" Body="&lt;p&gt;I am visiting my in-laws for the holidays.  My sister in law is a statistician.  She asked me to take a stab at a calculus problem which was coming up in her research.  The Lambert &lt;span class=&quot;math-container&quot;&gt;$W_0$&lt;/span&gt; function is defined as the inverse of the function &lt;span class=&quot;math-container&quot;&gt;$f(x) = xe^x$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \geq -1$&lt;/span&gt;.  It is clear that &lt;span class=&quot;math-container&quot;&gt;$g(x) = W_0(f(x)) = x$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x \geq -1$&lt;/span&gt;.  Her problem was to bound the derivative of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x &amp;lt; -1$&lt;/span&gt;.  I figured it out and found it to be a very fun calculus problem!  The solution didn't use anything which would be inaccessible to a calc 1 student, but there were several creative steps which were needed.  I think that, with a bit of hand-holding, this could make a nice project in a calc 1 class.  It would have the added bonus of being motivated by real research, not just being a contrived exercise.&lt;/p&gt;&#xA;&lt;p&gt;I am hoping for a &lt;strong&gt;big list&lt;/strong&gt; of interesting calculus problems, solvable using calc 1 techniques, which have come up in actual research.  Links to papers are especially welcome!&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2022-12-27T14:50:01.957" Title="Calculus problems arising from real research problems" Tags="|calculus|undergraduate-research|problem-design|projects|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25932" PostTypeId="2" ParentId="25930" CreationDate="2022-12-26T06:02:16.533" Score="18" Body="&lt;h2&gt;Modeling Basketball Free Throws&lt;/h2&gt;&#xA;&lt;p&gt;by Joerg M. Gablonsky and Andrew S. I. D. Lang, SIAM Review vol. 47, no. 4, pp. 775-798, 2005, &lt;a href=&quot;https://doi.org/10.1137/S0036144598339555&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1137/S0036144598339555&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;Abstract&lt;/h3&gt;&#xA;&lt;p&gt;This paper presents a mathematical model for basketball free throws. It is intended to be a supplement to an existing calculus course and could easily be used as a basis for a calculus project. Students will learn how to apply calculus to model an interesting real-world problem, from problem identification all the way through to interpretation and verification. Along the way we will introduce topics such as optimization (univariate and multiobjective), numerical methods, and differential equations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="77" LastEditorUserId="15884" LastEditDate="2022-12-27T02:10:34.657" LastActivityDate="2022-12-27T02:10:34.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25933" PostTypeId="1" CreationDate="2022-12-26T13:31:24.423" Score="1" ViewCount="126" Body="&lt;p&gt;Primarily a reference request, collaborator search-tips requests, and question-improvement request (including improveent by deletion and re-posting to more appropriate stack, meta, wiki, etc). Rank newbie. Please edit etc at will.&lt;/p&gt;&#xA;&lt;p&gt;Penrose's &amp;quot;The Road to Reality&amp;quot; changed my life. In the last years of pre-MO-stackexchange+pre-math-wikipedia's combined ascendancy, there was suddenly a book with 2/3illustrations (excellent ones) per page and thrice thrice that in daunting-enticing looking equations and inequalities, with about 900 pages of mathematical immersion followed by about the same in quantum and cosmological physics explanations described in terms of what had come before. This was a trade paperback.&lt;/p&gt;&#xA;&lt;p&gt;A &amp;quot;popular science book&amp;quot; trade-paperback. The first I ever encountered that ignored the &amp;quot;lose half your readership per equasion&amp;quot; urban-legend of the publishing industry. A book purporting to take one from algebra-I (actually, from fractions and variables) through differencial forms and (hyper-) complex analysis \ tensor calculus on real and complex manifolds, spinorial basics... etc. I read it four times or so not including time spent on page-footnote excercises, chapter-end forward-and-cross-references, and internet-accesible endnote references. I cannot imagine a better gift for anyone of any age at the level of &amp;quot;due to learn about polynomials this year.&amp;quot; If I'd had it as a Jr-high graduation present, I think I might have been a T.Tau or whatever, if I'd had any parents or homework cheking adults in my life, anyway.&lt;/p&gt;&#xA;&lt;p&gt;Question: has anyone used the book for adolescent-level math students with jr-college+ level reading (and HS level maturity)? Would anyone like to help me develop a curriculum around it. (I'd go to grad school to do so, if I needed....) Are there Math Ed grad  programs that are known for alternative-curriculum design for not-specifically-child-only gifted education purposes I should seek info\advisor-candidates from?&lt;/p&gt;&#xA;&lt;p&gt;Thanks. Merry Yule!&lt;/p&gt;&#xA;&lt;p&gt;Please&lt;/p&gt;&#xA;" OwnerUserId="20792" LastActivityDate="2022-12-26T13:31:24.423" Title="Math curricula\programs or any experience using &quot;The Road to Reality&quot; as the\a primary textbook" Tags="|undergraduate-education|textbooks|curriculum|precalculus|book|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25935" PostTypeId="2" ParentId="25867" CreationDate="2022-12-26T20:11:59.410" Score="2" Body="&lt;p&gt;Here is an example using Wolfram|Alpha. Note that &lt;span class=&quot;math-container&quot;&gt;$1/\lfloor x \rfloor &amp;lt; 2/x$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 1$&lt;/span&gt;. However, Wolfram|Alpha seems to think otherwise.&lt;/p&gt;&#xA;&lt;p&gt;Input:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;Solve 1/floor(x) &amp;lt; 2/x for x &amp;gt; 1&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Output:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h5&gt;Solution over the reals&lt;/h5&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\displaystyle x = \frac{99}{5} \approx 19.8000$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xoHqp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xoHqp.png&quot; alt=&quot;Screenshot.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A different solution appears if the terms are rearranged.&lt;/p&gt;&#xA;&lt;p&gt;Input:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;Solve x &amp;lt; 2 floor(x) for x &amp;gt; 1&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Output:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h5&gt;Solution over the reals&lt;/h5&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\displaystyle x = \frac{759}{10} = 75.9000$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eTRLa.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eTRLa.png&quot; alt=&quot;Screenshot.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;One can try changing the range of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (such as &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 20$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 76$&lt;/span&gt;, to avoid the previous solutions) to get some other values as the answer.&lt;/p&gt;&#xA;&lt;p&gt;I don't know how Wolfram|Alpha internally defines the floor function, but it clearly has trouble handling some inequalities that use it.&lt;/p&gt;&#xA;" OwnerUserId="15884" LastActivityDate="2022-12-26T20:11:59.410" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25936" PostTypeId="2" ParentId="25930" CreationDate="2022-12-27T03:38:42.330" Score="13" Body="&lt;p&gt;Consider &lt;a href=&quot;https://en.wikipedia.org/wiki/Betz%27s_law&quot; rel=&quot;noreferrer&quot;&gt;Betz's law&lt;/a&gt;, which was worked out around &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; years ago by three different scientists independently (in Germany, the UK, and Russia). It determines the maximum power that can be extracted from a wind turbine. Without getting into the physical background here, the problem eventually turns into the task of finding the point &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; where the polynomial &lt;span class=&quot;math-container&quot;&gt;$P(x) = (1/2)(1+x-x^2-x^3)$&lt;/span&gt; is maximized.&lt;/p&gt;&#xA;&lt;p&gt;The derivative of &lt;span class=&quot;math-container&quot;&gt;$P(x)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$(1/2)(1 - 2x - 3x^2) = (1/2)(1-3x)(1+x)$&lt;/span&gt;, whose only zero in &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; is at &lt;span class=&quot;math-container&quot;&gt;$x = 1/3$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$P(1/3) = 16/27 \approx .592$&lt;/span&gt;, while the boundary values are &lt;span class=&quot;math-container&quot;&gt;$P(0) = 1/2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P(1) = 0$&lt;/span&gt;, the maximum occurs at &lt;span class=&quot;math-container&quot;&gt;$x = 1/3$&lt;/span&gt;. That its value there is around &lt;span class=&quot;math-container&quot;&gt;$.592$&lt;/span&gt; has the real-world meaning that a wind turbine can convert at most around &lt;span class=&quot;math-container&quot;&gt;$59\%$&lt;/span&gt; of a wind's power into mechanical power.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2022-12-27T03:38:42.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25937" PostTypeId="1" CreationDate="2022-12-27T03:50:17.530" Score="0" ViewCount="113" Body="&lt;p&gt;The reasons for my question are complicated but basically I need information about different subjects, mostly languages (Eng, Spa, Fre), math, and sciences, as taught in different grades in elementary/high schools.&lt;/p&gt;&#xA;&lt;p&gt;Now I do understand that the material covered by different schools or teachers is not the same, nor is it taught in the same way. But is it true that there are some basics that students are expected to have learned, no matter where they go to school? For example, is there the expectation that students who are starting Grade 8 physics should have learned certain physics principles, even if they come from another US state? Or someone who just finished Grade 12 chemistry should know certain things about acid/base chemistry?&lt;/p&gt;&#xA;&lt;p&gt;Alternatively, are there certain books one could purchase that are commonly used in schools when teaching these subjects?&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance.&lt;/p&gt;&#xA;" OwnerUserId="20801" LastActivityDate="2023-02-13T10:03:05.360" Title="Where can I find a summary of typical material (e.g., mathematics) taught in US elementaries and high schools?" Tags="|textbooks|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25938" PostTypeId="2" ParentId="25930" CreationDate="2022-12-27T03:51:41.967" Score="6" Body="&lt;p&gt;Consider the reflective property of a parabola: rays coming into a parabola parallel to its axis of symmetric will all bounce off the parabola (angle of incidence equals angle of reflection) and meet at the focus.  Verifying this property is a calculus problem related to normal directions at a point on a parabola, which is perpendicular to the tangent direction at that point.&lt;/p&gt;&#xA;&lt;p&gt;This property also holds for a paraboloid of revolution and is why radar dishes are shaped like parts of a paraboloid, with the receiver at the focus.  Going in the other direction, this is why the casing around some front-facing (not rear-facing!) car headlights are shaped like paraboloids with the bulb at the focus.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2022-12-27T03:51:41.967" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25939" PostTypeId="2" ParentId="25930" CreationDate="2022-12-27T04:34:07.360" Score="8" Body="&lt;p&gt;Consider trying to find a curve that will let you link up two straight tracks smoothly, such as two straight parts of a roller coaster track. By &amp;quot;smoothly&amp;quot; meeting, we want the derivatives to match. This can be done with &lt;a href=&quot;https://en.wikipedia.org/wiki/Spline_(mathematics)&quot; rel=&quot;noreferrer&quot;&gt;cubic splines&lt;/a&gt;. Degree &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is not enough because there are four conditions imposes by equality of function values and their derivatives at a point, so the three coefficients of a quadratic polynomial are inadequate but the four coefficients of a cubic polynomial are a good model. As a simple example of this task you can present in a calculus course, find the cubic polynomial on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; that will let you smoothly transition from &lt;span class=&quot;math-container&quot;&gt;$y = -.5x$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \leq 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y = .6(x-1)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \geq 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Splines show up all over the place: computer animation (movies, video games), automotive and aircraft design (where two pieces of metal should meet up in a smooth way), and &lt;a href=&quot;https://en.wikipedia.org/wiki/Vector_graphics&quot; rel=&quot;noreferrer&quot;&gt;scalable fonts&lt;/a&gt; used on the screen on which you are reading this.&lt;/p&gt;&#xA;&lt;p&gt;For &lt;a href=&quot;https://en.wikipedia.org/wiki/Track_transition_curve&quot; rel=&quot;noreferrer&quot;&gt;transition curves&lt;/a&gt; used for railroad tracks or highways exits and entrances, the related math is Fresnel integrals.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2022-12-27T04:34:07.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25940" PostTypeId="2" ParentId="25930" CreationDate="2022-12-27T14:00:16.040" Score="1" Body="&lt;p&gt;The economic spacing of horizontal wells in a &amp;quot;shale&amp;quot; is a practical max problem.  [First semester calculus]  You want to get the most profit.  Closer than about 2000 feet and the wells start to interfere (cannibalize or steal from each other).  However, each new well bore does break (literally, that is what hydraulic fracturing is) new rock.  So it is not 100% cannibalization.  But they cost money.  Based on the price of oil/gas, and the cost of drilling wells, there is an optimal density.  You will always &amp;quot;leave some behind&amp;quot; and always &amp;quot;have some cannibalization&amp;quot;.  In addition, the cost function does not scale exactly.  Each new well bore (drilling) is essentially a scaled cost.  But the costs of the &amp;quot;pad&amp;quot; (road, permit, etc.) are essentially constant (thus divided over more wells).  Also, fracturing and even drilling have some cost advantages when done in a campaign (less equipment moves).&lt;/p&gt;&#xA;&lt;p&gt;After solving a base case, you can do some interesting questions.  Like &amp;quot;is it possible to drill 6.5 wells (no) in a unit.  So then they have to pick 6 or 7 and see which is better.  Or you can discuss unitizing sections (combining two units, with the right leases/permits) to allow drilling &amp;quot;13 in a double wide&amp;quot;.  Plus you can do useful variances from the base case (what is the optimal density if oil price is &lt;span class=&quot;math-container&quot;&gt;$150/bb?  How about $&lt;/span&gt;50?)&lt;/p&gt;&#xA;&lt;p&gt;Personally, I think the easy thing is to just write out reasonable functions yourself, than do the analysis.  But here are some papers.  (Not sure they are the best, so many junky little papers!)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S2468256X21000523#:%7E:text=Coupling%20with%20the%20economic%20evaluation,fracture%20network%20and%20production%20system&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.sciencedirect.com/science/article/pii/S2468256X21000523#:~:text=Coupling%20with%20the%20economic%20evaluation,fracture%20network%20and%20production%20system&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://onepetro.org/ARMAUSRMS/proceedings-abstract/ARMA21/All-ARMA21/ARMA-2021-1649/468103&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://onepetro.org/ARMAUSRMS/proceedings-abstract/ARMA21/All-ARMA21/ARMA-2021-1649/468103&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.earthdoc.org/content/papers/10.3997/2214-4609.202085022&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.earthdoc.org/content/papers/10.3997/2214-4609.202085022&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Maybe there is a little local connect for your kids, since you have the Utica in Ohio...although it's on the other side and further south.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  &lt;strong&gt;In general, I'm not a fan of this sort of thing.  I think research oriented teachers are too interested in the research&lt;/strong&gt; and also don't realize that the classical examples are new/interesting to their students, even the techniques are new to them.  So, it ends up being too much &amp;quot;push&amp;quot; from the teacher and a diversion.  You do seem very organized in teaching your kids and getting results, so maybe you have the slack to spend time on this sort of thing.  But, still...&amp;quot;Danger, Will Robinson!&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;P.s.s.  In practice companies aren't really doing calculus.  They just do a spreadsheet and look at 1-20 wells per section and pick the max profit.  Where added oil and diminishing returns counterbalance.  I'm not sure this is even cutting edge &amp;quot;research&amp;quot; really, since companies can do it without the literature.  It's just interesting FP&amp;amp;A.  But it's definitely calculus driving the bus in terms of the math that actually affects the economics.  So an econ grad student would think of it that way, write about it that way.  In a sense, there are some beautiful equations and functions and calculus concepts lurking behind the matrix of the Excel.  And I think it helps as an engineer/executive/consultant to have some functional insight as to what is going on, not just the black box of the Excel tables.&lt;/p&gt;&#xA;&lt;p&gt;(Another P.S.) I understand your joy/pride in the collaboration with your sister, but I think it would be better form to lead with the question, rather than your anecdote.  &lt;em&gt;Looking, now at the commentary, I see a lot of discussion of the mathematics details of the problem itself.  Almost as if this was more the interest than the &amp;quot;math ed&amp;quot; part at the end.&lt;/em&gt;  So that it feels to me, more discussion of the math itself, rather than education.  As a rule-breaker, of course, I applaud your (mis)use of the forum, to accomplish any objective, rather than the stated objective.  All that said, I would just be very wary of this sort of &amp;quot;I like this hard math&amp;quot; that does not translate well to kids learning things (just the chain rule is new to them).&lt;/p&gt;&#xA;&lt;p&gt;(Yet one more)  If you don't like the shale industry (I love it, see capitalism in action), you can search on environmental cost benefit analysis.  The Google Scholar returns lots of academic lit (if that's important to you).  The regular Google search returns some case studies and the like, which may be more quickly relevant.  Here is one review:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.annualreviews.org/doi/10.1146/annurev.environ.33.020107.112927&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.annualreviews.org/doi/10.1146/annurev.environ.33.020107.112927&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(not immediately usable for a class, but gets you into the lit, since it's a review article.  Really I think a book chapter or simple case study is better than nitty gritty cutting edge research.)&lt;/p&gt;&#xA;" OwnerUserId="20803" LastEditorUserId="20803" LastEditDate="2022-12-27T14:50:01.957" LastActivityDate="2022-12-27T14:50:01.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25941" PostTypeId="2" ParentId="25930" CreationDate="2022-12-27T14:04:33.907" Score="6" Body="&lt;p&gt;There should be a lot of interesting examples in ecology. The first one that comes to mind for me is population dynamics: &lt;a href=&quot;https://en.wikipedia.org/wiki/Population_dynamics&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia Link&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The simplest model is exponential growth:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ dN/dt = rN $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is time, &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the population growth rate, and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is the population size.&lt;/p&gt;&#xA;&lt;p&gt;This can be relevant for things like predicting bacteria growth. But in general, many factors prevent this type of growth for most species, so there are many ways to extend this model for practical applications. The most common example is probably the logistic growth model:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ dN/dt = rN(1 - N/K) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This incorporates &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; as carrying capacity to limit the maximum population size. It may be feasible to have calc 1 students derive this themselves from the exponential growth equation.&lt;/p&gt;&#xA;&lt;p&gt;Other variations might be interesting as well. For example, the Explosion-Extinction model:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ dN/dt = rN(N/M - 1) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here, &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; represents a threshold population size. If &lt;span class=&quot;math-container&quot;&gt;$N &amp;gt; M$&lt;/span&gt;, the population will grow. If &lt;span class=&quot;math-container&quot;&gt;$N &amp;lt; M$&lt;/span&gt;, the population will shrink. This type of equation is useful for modeling invasive species (which can have ecological and economical consequences), which usually have to meet some threshold &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; to become established.&lt;/p&gt;&#xA;" OwnerUserId="13207" LastActivityDate="2022-12-27T14:04:33.907" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25942" PostTypeId="1" AcceptedAnswerId="25950" CreationDate="2022-12-28T17:23:55.667" Score="12" ViewCount="1281" Body="&lt;p&gt;I'm looking to books introducing different topics of math with fun for my son, to give him a taste of different areas.&lt;/p&gt;&#xA;&lt;p&gt;My son roughly understands A-level topics until single variable calculus (of course, need exercise to get familiar to all the tricks).&lt;/p&gt;&#xA;&lt;p&gt;Dover Books on Mathematics is a well-known series, on Amazon there are now 306 items listed. However, they are a bit too long (normally about 200 pages), with many topics too deep, and , honestly, the text might be a bit too serious for a kid of 9.&lt;/p&gt;&#xA;&lt;p&gt;Martin Gardner's books are quite fun, with many challenging puzzles, maybe more attractive, however, they are normally not focusing on a particular topic.&lt;/p&gt;&#xA;&lt;p&gt;Evan Chen started a &lt;a href=&quot;https://web.evanchen.cc/napkin.html&quot; rel=&quot;noreferrer&quot;&gt;Napkin project&lt;/a&gt; introducing more than 80 topics, but are a bit &amp;quot;formal&amp;quot; and more on the pure math theories instead of motivation, application, and funny puzzles.&lt;/p&gt;&#xA;&lt;p&gt;Any suggestion please?&lt;/p&gt;&#xA;" OwnerUserId="9059" LastActivityDate="2022-12-31T01:19:18.563" Title="Introductory Books easier than Dover" Tags="|big-list|" AnswerCount="7" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="25943" PostTypeId="2" ParentId="25942" CreationDate="2022-12-28T21:12:03.450" Score="5" Body="&lt;p&gt;I would go with Art of Problem Solving and/or Beast Academy if he has some primary gaps.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://artofproblemsolving.com/&quot; rel=&quot;noreferrer&quot;&gt;https://artofproblemsolving.com/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There should be enough fun factor, that he's engaged.  And lots of interesting problem solving.  But enough of a structure and curriculum so he moves through and masters basic subjects.&lt;/p&gt;&#xA;&lt;p&gt;Your 9 year old (I checked your previous questions) is bright and has been exposed to a lot of stuff ahead of time.  But is not really experienced the way a conventional student ready to go into calculus is (at age 18ish).&lt;/p&gt;&#xA;&lt;p&gt;So, keep it fun and keep letting him have exposure, but don't convince yourself he's really at the level of a strong 17 year old.  For comparison, I spent Christmas with my relative's boys.  They're bright 9 and 11 year-olds and have had a lot of exposure pushed at them by their dad (&amp;quot;we know trig&amp;quot;, &amp;quot;we know calculus&amp;quot;), but it's really exposure, not mastery.  In terms of real school, the 6th grade 11 yo is doing 9th grade (US system) algebra 1.  Still very good...but the real practiced level is not as high as his exposure.&lt;/p&gt;&#xA;&lt;p&gt;Note also, Dover is not really a coordinated series, but reprints of old classics.  Which vary a lot in quality, scope, approach, solutions enclosed, or even print quality.  It's a great resource of old classics.  But you need to check the individual titles (look at Amazon reviews, skim them, ask about them) to see if they match a specific need.&lt;/p&gt;&#xA;&lt;p&gt;A few other suggestions:&lt;/p&gt;&#xA;&lt;p&gt;Calculus Made Easy and Calculus for the Practical Man (and the other &amp;quot;for the practical man&amp;quot;) are relatively fun (and easy prose).  Were used by Feynman to learn young (but he eventually mastered normal texts also.&lt;/p&gt;&#xA;&lt;p&gt;For an easy prose, non doorstop review of high school math (algebra, geometry, trig, precaculus, intro to calculus), I like Frank Ayres First YEar College Math Schaum's Outline.  Has the solutions.  I have the first edition, which I like.  You can get that or the second edition on the net, used.  Probably pretty similar but I always worry about regression...but use your judgment.&lt;/p&gt;&#xA;&lt;p&gt;I don't know of a good baby physics (project based or book based) curriculum.  But if you could find one, that would be good.  the intuition of angles and position/speed/acceleration are helpful to informing and motivating math study.&lt;/p&gt;&#xA;" OwnerUserId="20811" LastActivityDate="2022-12-28T21:12:03.450" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25944" PostTypeId="2" ParentId="25888" CreationDate="2022-12-29T00:42:48.647" Score="0" Body="&lt;p&gt;The best way to learn geometry from Hilbert's axioms is his own book Foundations of Geometry.&lt;/p&gt;&#xA;&lt;p&gt;However I don't see any reasons to learn Geometry &amp;quot;from Hilbert axioms&amp;quot;. There are quite a few really good modern textbooks on geometry, my favorite one is Hartshorne (recommended in the answer of user52817), which covers more of less the contents of Euclid but from modern point of view.&lt;/p&gt;&#xA;&lt;p&gt;Another good elementary geometry book (which covers much more, and not tied to Euclid) is M. Berger, Geometry.&lt;/p&gt;&#xA;" OwnerUserId="20727" LastActivityDate="2022-12-29T00:42:48.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25945" PostTypeId="2" ParentId="25942" CreationDate="2022-12-29T00:52:43.423" Score="3" Body="&lt;p&gt;&lt;em&gt;&lt;strong&gt;The Man Who Counted&lt;/strong&gt;&lt;/em&gt;, by Malba Tahan&lt;/p&gt;&#xA;&lt;p&gt;Written in Brazil, set in the Middle East, these stories follow the adventures of Beremiz, an accomplished mathematical problem-solver. He uses math to settle disputes, solve riddles and mysteries, and entertain his hosts. The series of 34 adventures, each with a math puzzle, is reminiscent of the Arabian Nights.&lt;/p&gt;&#xA;&lt;p&gt;So it's not developing a particular topic.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;How to Count Like a Martian&lt;/strong&gt;&lt;/em&gt;, by Glory St.John&lt;/p&gt;&#xA;&lt;p&gt;A really good way to understand place value is to work with other number bases. &lt;em&gt;&lt;strong&gt;How to Count Like a Martian&lt;/strong&gt;&lt;/em&gt; is a detective story in which the history of other number systems plays a starring role.&#xA;“Out of the depths of the dark and starry night come the first of the faint and mysterious sounds … At your radio telescope, you are expertly tuning the dials.” You have just received a message from Mars. “You know that this is not a message in words. Martians and Earthlings would have too much trouble trying to find the same words to succeed that way. But there is another kind of language that both Martians and Earthlings understand.”&lt;/p&gt;&#xA;&lt;p&gt;Numbers… And so you research the number systems that have been used on Earth, hoping that will help you decipher this message. The book proceeds to explain eight different counting systems, including the abacus, and computers.&lt;/p&gt;&#xA;&lt;p&gt;In the process, the concepts of place value (she just calls it place), base, and zero are explored. By the end of the book, you can see that the beeps and bee-beeps of the message you received are just the counting numbers, Martian style.&#xA;How to Count Like a Martian was written in 1975, when there were still dials and tape recorders. those two items may be the only evidence of its age.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;The Number Devil&lt;/strong&gt;&lt;/em&gt;, by Hans Magnus Enzensberger&lt;/p&gt;&#xA;&lt;p&gt;The Number Devil visits Robert in his dreams, and gets him thinking about the strangest things! Rutabaga numbers and prima donnas (roots and primes) are just the beginning. This one includes Pascal's (aka Jia Xian's) triangle.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Mathematics: A Human Endeavor&lt;/strong&gt;&lt;/em&gt;, by Harold Jacobs&lt;/p&gt;&#xA;&lt;p&gt;This is a delightful textbook. It explores logic, symmetry, functions, ...&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Who Is Fourier?&lt;/strong&gt;&lt;/em&gt; by Transnational College of LEX&lt;/p&gt;&#xA;&lt;p&gt;Fourier series, approachable with very little background.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2022-12-29T00:52:43.423" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25946" PostTypeId="2" ParentId="14976" CreationDate="2022-12-29T03:49:33.310" Score="3" Body="&lt;p&gt;I did develop a durable physical model for this purpose a while back. It best represents the disc method, but can be extended to the washer method as well.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UrPLG.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UrPLG.jpg&quot; alt=&quot;Physical Disk Model (Volume of Revolution)&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/VsyG3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VsyG3.png&quot; alt=&quot;2D Representation of Model&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Many found it to be extremely helpful in the conceptual understanding of volumes of revolution. It's now available to anyone interested &lt;a href=&quot;https://moesmathmodels.weebly.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A lot of the other answers here are great ideas and options to consider and experiment with. I look forward to trying many of them myself.&lt;/p&gt;&#xA;" OwnerUserId="6717" LastActivityDate="2022-12-29T03:49:33.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25947" PostTypeId="2" ParentId="25942" CreationDate="2022-12-29T05:46:04.397" Score="4" Body="&lt;p&gt;Dover books tend to be from early generations of math when most topics were taught at a higher level and as such the text books are often more sophisticated. That trend paired with the general concept that the reader should fill in the gaps makes Dover texts on the average a somewhat dangerous path for the independent self-learning enthusiast. Yet, they are cheap. Certainly I looked a lot at Dover texts when I was an undergraduate. In retrospect, I would have done better to get my hands on some undergraduate Springer texts. In any event, the texts available now are generally much more accessible than what you find in Dover (with a few exceptions, there are Dover reprints of things from the 90's and later and my comment here is not really aimed at those)&lt;/p&gt;&#xA;&lt;p&gt;Ok, so what to do ? I would consider getting an old edition of a few different topics which fit your child's interest. For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;an old edition of Susan Colley's Vector Calculus&lt;/li&gt;&#xA;&lt;li&gt;an old edition of a Friendly Introduction to Number Theory by Silverman&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Also fun, &amp;quot;A Manga Guide to Linear Algebra&amp;quot;, or one of Brian Greene's books on String Theory. Perhaps &amp;quot;Road to Reality&amp;quot; by Penrose (that will give them a lot to chew on, it's over 1000 pages).&lt;/p&gt;&#xA;&lt;p&gt;Anyway, as a rule, pick a book which is currently popular in courses, but get a copy from a few editions back. For not too much money you can generate a library for your child to poke around as their interest leads them.&lt;/p&gt;&#xA;&lt;p&gt;For me, getting a copy of &amp;quot;Hyperspace&amp;quot; by Michio Kaku was very eye opening at one point in my education. There are many such books now.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2022-12-29T05:46:04.397" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25948" PostTypeId="2" ParentId="14976" CreationDate="2022-12-29T06:27:25.827" Score="1" Body="&lt;p&gt;A &lt;a href=&quot;https://c3d.libretexts.org/CalcPlot3D/CalcPlot3D-Help/section-stlfile.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;feature&lt;/a&gt; of &lt;a href=&quot;https://c3d.libretexts.org/CalcPlot3D/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;CalcPlot3D&lt;/a&gt; is being able to create STL files for printing your own 3D models of solids.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2022-12-29T06:27:25.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25949" PostTypeId="2" ParentId="25942" CreationDate="2022-12-29T14:08:07.047" Score="2" Body="&lt;h3&gt;Bert Mendelson's &lt;em&gt;Introduction to Topology&lt;/em&gt;&lt;/h3&gt;&#xA;&lt;p&gt;I began working through Mendelson's &lt;em&gt;Introduction to Topology&lt;/em&gt; in high school, but could have begun even sooner. It begins with a perfectly accessible introduction to set theory and proofs, and then moves on to the basics of point-set topology without requiring any knowledge of abstract algebra or real analysis. I know it sounds crazy to recommend a topology text to a 9 year old, but I wish I had been exposed to the abstract thinking of set theory sooner, and I think Mendelson's text is an excellent choice for that. Once you've got a grasp of the fundamentals of sets, Mendelson's approach to topology is really just goofing off with set theory relationships.&lt;/p&gt;&#xA;" OwnerUserId="15238" LastActivityDate="2022-12-29T14:08:07.047" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25950" PostTypeId="2" ParentId="25942" CreationDate="2022-12-29T18:31:44.147" Score="6" Body="&lt;p&gt;Have a look at the series of books originally published starting in the 1960's under the series title of &amp;quot;Anneli Lax New Mathematical Library&amp;quot; by the Mathematical Association of America (MAA).  These are now available from the American Mathematical Society (AMS) and Cambridge University Press (CUP).&lt;/p&gt;&#xA;&lt;p&gt;The series is described as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Featuring fresh approaches and broad coverage of topics especially&#xA;suitable for high school and the first two years of college, the&#xA;volumes in this series are an excellent source of enrichment material&#xA;for teachers and students. Good mathematical reading with lively&#xA;exposition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some of my favorites:&lt;/p&gt;&#xA;&lt;p&gt;Grossman, Israel, and Wilhelm Magnus. &lt;em&gt;Groups and their Graphs&lt;/em&gt;. Vol. 14. New York: Random House, 1964.&lt;/p&gt;&#xA;&lt;p&gt;Niven, Ivan. &lt;em&gt;Mathematics of Choice: Or, How to Count Without Counting&lt;/em&gt;. Vol. 15. MAA, 1965.&lt;/p&gt;&#xA;&lt;p&gt;Niven, Ivan. &lt;em&gt;Numbers: Rational and Irrational&lt;/em&gt;. Vol. 1. New York: Random House, 1961.&lt;/p&gt;&#xA;&lt;p&gt;Ore, Oystein, and Robin J. Wilson. &lt;em&gt;Graphs and their Uses&lt;/em&gt;. Vol. 34. Cambridge University Press, 1990.&lt;/p&gt;&#xA;&lt;p&gt;I was reading these books by the time I was in high school.  In general, they should be accessible to interested students who are studying US high school level mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5841" LastActivityDate="2022-12-29T18:31:44.147" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25951" PostTypeId="2" ParentId="25942" CreationDate="2022-12-30T01:15:39.593" Score="5" Body="&lt;p&gt;So let me answer your request for non-Dover books by recommending you some Dover books! First of all I agree with your overall assessment of them, and the descriptions on the backs are often misleading (&amp;quot;only requires familiarity with high school calculus&amp;quot; my a**). Also you've got some very nice non-Dover suggestions already.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, there are indeed a few Dover books that are beginner(ish)-friendly. Anything by W.W Sawyer fits the bill, along with &lt;em&gt;Excursions in Geometry&lt;/em&gt; and &lt;em&gt;Excursions in Number Theory&lt;/em&gt; by Ogilvy. I also like Morris Kline's &lt;em&gt;Calculus: An Intuitive and Physical Approach&lt;/em&gt;. It may be longer than you'd like but for anyone wanting to learn or brush up on calculus who also likes physics it's great.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2022-12-30T01:15:39.593" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25952" PostTypeId="2" ParentId="25473" CreationDate="2022-12-30T13:17:05.910" Score="4" Body="&lt;p&gt;Sometimes the problem has only one unit, but the answer has different units.  A cube has a face with an area of 9 square inches.  Find the  volume.   The answer would of course be in cubic inches.  The fact that only one unit is in the problem doesn't make the choice of units easy.&lt;/p&gt;&#xA;&lt;p&gt;I have always insisted that students write units.  Depending on the difficulty I would take off more or less partial credit if they left the units off.  I did this because students don't always have the judgement to decide when they need to make the units clear.  As my geometry problem is meant to show, the units aren't always clear from the problem.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes the units might be clear.  Consider:  John has 9 apples.  Kim has three times as many.  How many more apples does Kim have than John?  A student might say 18.  In that case, I would gently remind the student that we need to specify 18 what.  But I would never say 18 what? 18 elephants??  That's just insulting.&lt;/p&gt;&#xA;&lt;p&gt;My students learned to specify what the units were and I'm sure it helped them in the long run.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2022-12-30T13:17:05.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25953" PostTypeId="2" ParentId="25942" CreationDate="2022-12-31T01:19:18.563" Score="3" Body="&lt;p&gt;&lt;em&gt;Group Theory in the Bedroom&lt;/em&gt; by Brian Hayes, a great writer.&#xA;His articles are educational, informative, and often entertaining.&#xA;I linked to this collection in&#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/a/18981/511&quot;&gt;this earlier MESE question&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;   &#xA;&lt;img src=&quot;https://i.stack.imgur.com/2RCdE.jpg&quot; width=&quot;300&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Contents &lt;a href=&quot;http://grouptheoryinthebedroom.com/about.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2022-12-31T01:19:18.563" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25954" PostTypeId="2" ParentId="7837" CreationDate="2023-01-02T02:16:36.460" Score="2" Body="&lt;p&gt;I recently tried to explain the invert-and-multiply process to a handful of 5th graders and I was surprised at how it went. I'm not their teacher so I don't really know what they do in class, but I started by asking them to compute &lt;span class=&quot;math-container&quot;&gt;$\frac{15}{8}\div \frac{3}{2}$&lt;/span&gt;. I expected them to know how to invert and multiply and to use that process to get the answer. Then I planned to confuse them by asking ``Why not just divide across? Look, &lt;span class=&quot;math-container&quot;&gt;$15\div 3=5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$8\div 2=4$&lt;/span&gt;! Magic, huh? So how come your teachers didn't teach it that way?'' So imagine my surprise when they &lt;em&gt;did&lt;/em&gt; divide across and they &lt;em&gt;didn't&lt;/em&gt; know how to invert and multiply! (I should email their teachers and see if this is actually what they're supposed to know or if they had forgotten some things.)&#xA;Of course, if they're comfortable with multiplying fractions (perhaps by using some visual representation like in the question) and the fact that &lt;span class=&quot;math-container&quot;&gt;$a/b=c$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$a=bc$&lt;/span&gt;, it makes sense that they would know you can divide across. If you can get them to this point, the rest isn't so bad.&lt;/p&gt;&#xA;&lt;p&gt;What I did next was ask them something like &lt;span class=&quot;math-container&quot;&gt;$\frac{15}{7}\div \frac{3}{2}$&lt;/span&gt;, and then mention that it sure would be nice if the left denominator were a multiple of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and we could divide across, so we wrote it as &lt;span class=&quot;math-container&quot;&gt;$\frac{30}{14}\div \frac{3}{2}$&lt;/span&gt; and did just that, getting &lt;span class=&quot;math-container&quot;&gt;$\frac{10}{7}$&lt;/span&gt;. I made no attempt to write it as &lt;span class=&quot;math-container&quot;&gt;$\frac{15}{7}\times \frac{2}{3}$&lt;/span&gt; because that's too much information at once. This was just to show them that even when it initially looks like we can't divide across, sometimes (always?) if we're clever we can. I did several more examples like this, where only one numerator or denominator was a multiple of the other one.&lt;/p&gt;&#xA;&lt;p&gt;Then I jumped to something like &lt;span class=&quot;math-container&quot;&gt;$\frac{13}{7}\div \frac{3}{2}$&lt;/span&gt; and as usual I made, say, the numerators nice:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{13}{7}\div \frac{3}{2}=\frac{13\times 3}{7\times 3}\div \frac{3}{2}$$&lt;/span&gt;&#xA;All this time I kept telling them to resist their urge to immediately compute &lt;span class=&quot;math-container&quot;&gt;$13\times 3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$7\times 3$&lt;/span&gt;, and by now I think they were starting to understand that it's much easier to leave it in the above form. The next step was to get the left denominator to be a multiple of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, which by now was easy.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{13}{7}\div \frac{3}{2}=\frac{13\times 3}{7\times 3}\div \frac{3}{2}=\frac{13\times 3\times 2}{7\times 3\times 2}\div \frac{3}{2}$$&lt;/span&gt;&#xA;Dividing across gives &lt;span class=&quot;math-container&quot;&gt;$\frac{13\times 2}{7\times 3}=\frac{26}{21}$&lt;/span&gt;. &lt;em&gt;Only then&lt;/em&gt; did I call attention to the fact that &lt;span class=&quot;math-container&quot;&gt;$\frac{13\times 2}{7\times 3}=\frac{13}{7}\times\frac{2}{3}$&lt;/span&gt;, and isn't it funny that the second fraction got flipped and the division turned into a multiplication? Then we went through all the previous examples to see if that funny thing happened in all of them too.&lt;/p&gt;&#xA;&lt;p&gt;So it didn't go the way I anticipated, but I was happy with the result.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2023-01-02T02:16:36.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25955" PostTypeId="1" AcceptedAnswerId="25956" CreationDate="2023-01-02T04:32:23.137" Score="27" ViewCount="4506" Body="&lt;p&gt;In my intro to proofs (and discrete mathematics) class, I see a common mistake where students make nonsensical statements because, for lack of a better term, they confuse the types of the mathematical objects they work with. Some examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In a proof that a function &lt;span class=&quot;math-container&quot;&gt;$f: A \to B$&lt;/span&gt; is onto, a student will say &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$x \in f$&lt;/span&gt;&amp;quot; instead of &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$x \in B$&lt;/span&gt;&amp;quot;. Thinking of sets and functions as distinct objects would make a student realize this is not what they meant (we technically define a function as a set, but this is not what students are thinking of)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When learning induction, students often define a predicate as being equal to an expression, such as &lt;span class=&quot;math-container&quot;&gt;$P(x) := 2^x + 5$&lt;/span&gt;, when thinking of expressions and predicates as distinct types would catch this error.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In relations, students often say: &amp;quot;The pairs &lt;span class=&quot;math-container&quot;&gt;$(1,1), (2,2), (3,3)$&lt;/span&gt; are reflexive&amp;quot; when we define reflexive as a property of a relation, not a property of individual elements.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is there an established name for this type of error other than &amp;quot;type disagreement&amp;quot;, and&lt;/li&gt;&#xA;&lt;li&gt;How do you steer students from this type of mistake, and teach them to think about the &amp;quot;types&amp;quot; of the objects they work with?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20818" LastEditorUserId="17865" LastEditDate="2023-01-05T13:01:24.397" LastActivityDate="2023-01-06T15:18:27.093" Title="Students confusing &quot;object types&quot; in introductory proofs class" Tags="|proofs|students-mistakes|" AnswerCount="6" CommentCount="33" ContentLicense="CC BY-SA 4.0" />
  <row Id="25956" PostTypeId="2" ParentId="25955" CreationDate="2023-01-02T15:52:30.580" Score="39" Body="&lt;p&gt;I personally use terms like &amp;quot;type disagreement&amp;quot; and &amp;quot;type error&amp;quot;. This agrees with the notion of types within computer science (&lt;a href=&quot;https://en.wikipedia.org/wiki/Type_system&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Type_system&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;When I taught linear algebra, I would give students a list of problems of the form &amp;quot;given that &lt;span class=&quot;math-container&quot;&gt;$\vec{x}, \vec{y}$&lt;/span&gt; are vectors and &lt;span class=&quot;math-container&quot;&gt;$a, b,c$&lt;/span&gt; are scalars, identify each of the following formulas as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Representing a vector&lt;/li&gt;&#xA;&lt;li&gt;Representing a scalar&lt;/li&gt;&#xA;&lt;li&gt;Doesn't make sense&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;These sorts of problems isolate this specific skill and encourage students to see it as an important skill. This is an unfamiliar type of problem, so you'll need to demonstrate this skill for students in lecture. Keep a list of type errors you see happening to populate these problems.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2023-01-02T15:52:30.580" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25958" PostTypeId="2" ParentId="25756" CreationDate="2023-01-03T08:21:40.340" Score="5" Body="&lt;p&gt;Late to join this party, sorry about that. Posting this largely to get some feedback and criticism myself.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A problem with including a discovery phase in a homework problem is simply that those take a lot of time. And the students usually are &lt;strike&gt;unwilling&lt;/strike&gt; unable to invest much (yours is not the only course they are enrolled in), unless you are lucky to find a particularly enticing question! Alas, those are rare. Please share, if you have a surplus!&lt;/p&gt;&#xA;&lt;p&gt;A possibility is to split the exercise into steps:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;experimenting/testing,&lt;/li&gt;&#xA;&lt;li&gt;forming a &amp;quot;conjecture&amp;quot;,&lt;/li&gt;&#xA;&lt;li&gt;proving the conjecture.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I have experimented with the following two part exercise. Basically the two first bullets formed one homework problem, and the last bullet was then an extra credit problem. This was for a first course in abstract algebra. The students have just been introduced to the cycle notation of permutations. My first and most obvious goal is to make them familiar with the process of rewriting a composition of permutations as a product of disjoint cycles. There are the obligatory rote examples to be worked out serving that end. And then this...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A property of permutations we have not yet discussed at all is the number of disjoint cycles in it. Here we do a bit of research! Unlike in the text, here we also count the 1-cycles, so for example the permutation &lt;span class=&quot;math-container&quot;&gt;$\sigma=(12)(456)=(12)(3)(456)\in S_6$&lt;/span&gt; has three cycles of respective lengths 2,1 and 3. The problem we investigate here is what happens to the number of cycles in a permutation, when we compose it (from the left) with the 2-cycle &lt;span class=&quot;math-container&quot;&gt;$(12)$&lt;/span&gt;. So compare the number of cycles in the permutations &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(12)\alpha$&lt;/span&gt; with the following choices of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{aligned}&#xA; &amp;amp;(12)(34), (13)(24)\in S_4,\\&#xA; &amp;amp;(13)(245), (1324)(5)\in S_5,\ \text{and}\\&#xA; &amp;amp;(135)(246), (123)(456), (16)(235)(4), (135264)\in S_6&#xA;\end{aligned}$$&lt;/span&gt;&#xA;You should see that the number of cycles in &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; always differs from those in &lt;span class=&quot;math-container&quot;&gt;$(12)\alpha$&lt;/span&gt; by the same amount, but in different directions. By how much? Can you formulate/guess a general result as to when the number of cycles increases and when it decreases? Do more experiments to test your guess! You will only be asked to prove that your guess works in a follow up problem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(apologies for my unpolished choice of phrase in the formulation)&lt;/p&gt;&#xA;&lt;p&gt;The student response is, unsurprisingly, very mixed:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;TL;DR;&amp;quot; (I'm sure you have your share of such students, too)&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;We calculated all those permutations, but what exactly are we supposed to do with the answers?&amp;quot; This may (or may not) be accompanied by a &amp;quot;WTF!&amp;quot; This from students who take notes in class and can do basic variants of examples from lectures, but feel lost without a model example.&lt;/li&gt;&#xA;&lt;li&gt;Exalted students (future researchers and a few others willing to try their hand at figuring things out for themselves).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In a typical problem session, this does generate a lot of discussion. Two of the three groups of students have correctly figured out that the number of cycles always changes by one. I have about 20 students per session, and usually 1-3 suggest a conjecture. If I'm lucky, the conjecture is wrong, and I can quickly refute it with a different example. Even if the conjecture is the correct one (= it depends on whether &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; belong to the same cycle or to distinct cycles), I still test it with at least one more example on the chalkboard.&lt;/p&gt;&#xA;&lt;p&gt;Then in the extra credit follow up exercise I ask a subset of the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Prove that &lt;span class=&quot;math-container&quot;&gt;$(12)\alpha$&lt;/span&gt; has one cycle more than &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; appear in the same cycle of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, and one cycle less than &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, when &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; appear in disjoint cycles of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;What changes, if we use another 2-cycle instead of &lt;span class=&quot;math-container&quot;&gt;$(12)$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;li&gt;What happens, if we compose by &lt;span class=&quot;math-container&quot;&gt;$(12)$&lt;/span&gt; from the right? Compare the number of cycles in &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\alpha(12)$&lt;/span&gt;!&lt;/li&gt;&#xA;&lt;li&gt;We saw that if &lt;span class=&quot;math-container&quot;&gt;$C(\alpha)$&lt;/span&gt; denotes the number of cycles in the permutation &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$C((12)\alpha))\equiv C(\alpha)+1\pmod2.$&lt;/span&gt; Recall the definition of &lt;em&gt;parity of a permutation&lt;/em&gt; (=odd/even). Prove that if &lt;span class=&quot;math-container&quot;&gt;$\alpha\in S_n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; even, then &lt;span class=&quot;math-container&quot;&gt;$C(\alpha)$&lt;/span&gt; is even if and only if &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is even. But when &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is odd, then &lt;span class=&quot;math-container&quot;&gt;$C(\alpha)$&lt;/span&gt; is even if and only if &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is odd.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So the idea is to give the students an easy enough conjecture to form, and then later prove. The result could (should ?!) veer a bit off the baseline material. I think it is necessary to split the problem into many parts, because such an exercise requires quite a bit more time from the students. My example does not take much (I have had bright students who turn in a proof for the conjecture at that same problem session), but overall I'm happy with the student response (I may be wearing rose-tinted specs when interpreting it).&lt;/p&gt;&#xA;&lt;p&gt;I'm afraid I cannot cite any piece of research based on testing similar ideas. It seems to fit into the Bloom taxonomy Kostya described.&lt;/p&gt;&#xA;" OwnerUserId="282" LastEditorUserId="282" LastEditDate="2023-01-10T05:02:08.187" LastActivityDate="2023-01-10T05:02:08.187" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25959" PostTypeId="2" ParentId="25955" CreationDate="2023-01-03T08:30:43.763" Score="6" Body="&lt;p&gt;For me, I use &amp;quot;&lt;strong&gt;type error&lt;/strong&gt;&amp;quot; or &amp;quot;&lt;strong&gt;typing error&lt;/strong&gt;&amp;quot;. I would avoid &amp;quot;type disagreement&amp;quot; because it suggests that you actually have types that disagree. But that's not correct; any expression is only &lt;strong&gt;meaningful&lt;/strong&gt; only when it does not have any type error! If there is a type error, it would be more accurate to say that the expression is &lt;em&gt;meaningless&lt;/em&gt;!&lt;/p&gt;&#xA;&lt;p&gt;As for how to make sure students do not make this mistake, in my experience typing errors clearly indicate that the students were not taught proper logical reasoning. I have &lt;a href=&quot;https://matheducators.stackexchange.com/a/24386/1550&quot;&gt;written before&lt;/a&gt; about what I think needs to be taught for students to actually be capable of logical reasoning, and I also think that trying to find a shortcut will be terrible for most students in the long run. 1% of them will somehow figure out how to do logical reasoning on their own, but 99% of them will never be 100% sure of their own reasoning in the absence of a proper deductive system.&lt;/p&gt;&#xA;&lt;p&gt;Once you have taught a proper deductive system, &lt;em&gt;all typing errors will completely and permanently vanish&lt;/em&gt;. Also see below for some remarks about the teaching process specific to definitions.&lt;/p&gt;&#xA;&lt;p&gt;I also want to elaborate on some of my comments. I said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Predicates are &lt;strong&gt;not objects&lt;/strong&gt; and actually cannot be objects in some foundational systems. They are meta-objects. To truly understand what defining a predicate means, you need to have a solid grasp of basic FOL (e.g. using a Fitch-style system such as &lt;a href=&quot;https://math.stackexchange.com/a/1684204/21820&quot;&gt;this one&lt;/a&gt;), and you need to know the distinction between ∃elim and definitorial expansion for new predicate/function-symbols. If you follow the definitorial expansion rule in that linked system, an example would be &amp;quot;Let Q(k) ≡ ¬∃x∈ℤ ( 2^k+5 = 17·x ), for each k∈ℕ.&amp;quot;. This is a complete definition of a (fresh) predicate-symbol &amp;quot;Q&amp;quot; (where ℕ,ℤ,+,·,^ are inbuilt or have been defined). You can also see that if you fastidiously follow the rules in that system, you are forced to think about what every object type is, and will never write the kind of nonsense listed in this question.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I know from experience that the only way to truly impart understanding is to be 100% precise. You must clearly distinguish between &lt;strong&gt;predicates&lt;/strong&gt; and &lt;strong&gt;predicate-symbols&lt;/strong&gt;. Taking my earlier example, &amp;quot;¬∃x∈ℤ ( 2^[1]+5 = 17·x )&amp;quot; (where the &amp;quot;[1]&amp;quot; is a place-holder for the 1st input parameter) is a predicate, whereas the symbol &amp;quot;Q&amp;quot; in &amp;quot;Let Q(k) ≡ ¬∃x∈ℤ ( 2^k+5 = 17·x ), for each k∈ℕ.&amp;quot; is a predicate-symbol. Under &amp;quot;Definitorial expansion&amp;quot; in that linked post you can find the deductive rules supporting definitions, plus a link to &lt;a href=&quot;https://math.stackexchange.com/a/1399880/21820&quot;&gt;an explanation of why we want definitions&lt;/a&gt;. This motivation tells us that we want to be able to express predicates without having to write it again and again. To support this, we allow defining new predicate-symbols. Make very clear to students this motivation as well as the fact that &amp;quot;Q(E)&amp;quot; can be substituted for the definition of Q with every &amp;quot;[1]&amp;quot; replaced by &amp;quot;E&amp;quot;. This is the &lt;strong&gt;true meaning of definitions&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Regarding the first point, make sure you never call predicates or predicate-symbols &amp;quot;objects&amp;quot;; neither of them are objects. And avoid talking about their &amp;quot;type&amp;quot;, because that would be a meta-type. For your own reference (don't tell beginning students about it), Russel's paradox concerns the fact that we can define R(x) ≡ x∉x, for each set x, but in ordinary set theories we cannot have an object that captures this predicate R. That is, we cannot have a set S such that ∀x∈set ( x∈S ⇔ R(x) ), otherwise we would get a contradiction.&lt;/p&gt;&#xA;&lt;p&gt;Regarding the second point, notice that &lt;strong&gt;if&lt;/strong&gt; students truly understand the meaning of definitions (i.e. using symbols to denote a longer expression), &lt;strong&gt;then&lt;/strong&gt; automatically they will not write nonsense such as &amp;quot;If P(k) = 2^k+5&amp;quot; when &amp;quot;P&amp;quot; has been declared to denote a predicate. Technically, this problem should never even occur in the first place, because students ought to be taught to correctly use a deductive system &lt;strong&gt;before&lt;/strong&gt; even using definitions! In my experience, &lt;strong&gt;none&lt;/strong&gt; of my students who learnt my deductive system ever made typing errors...&lt;/p&gt;&#xA;&lt;p&gt;But if students had been taught in the wrong order, then you can help to get rid of the problem quickly by simply asking them to substitute the &amp;quot;P(k)&amp;quot; for its &lt;em&gt;exact definition&lt;/em&gt; to show the 'meaning' of what they wrote. They should get something like &amp;quot;If f(k) = 2^k+5 = 2^k+5&amp;quot;, and realize why their original is not sensible.&lt;/p&gt;&#xA;&lt;p&gt;If you really want to comment on types, as per my explanation above, make sure you do &lt;strong&gt;not&lt;/strong&gt; say &amp;quot;The type of P is ...&amp;quot;. But you &lt;strong&gt;can&lt;/strong&gt; say the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Whenever you want to reason about a conditional scenario, you would say &amp;quot;If something, then ...&amp;quot;. What can that something be? It can be anything with a &lt;strong&gt;boolean value&lt;/strong&gt; (true/false). You cannot say &amp;quot;If elephant, then ...&amp;quot;. You can say &amp;quot;If elephants are cute, then ...&amp;quot;. You cannot say &amp;quot;If elephants are cute are cute, then ...&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And if we write &amp;quot;Let P(k) ≡ ( f(k) = 2^k+5 ), for each k∈ℕ.&amp;quot;, from then on &amp;quot;P&amp;quot; is a predicate-symbol we can use, and &amp;quot;P(E)&amp;quot; is &lt;strong&gt;boolean&lt;/strong&gt; for any expression &amp;quot;E&amp;quot; that is &lt;strong&gt;natural&lt;/strong&gt;. Keep in mind that &amp;quot;P&amp;quot; cannot be used by itself; it is simply &lt;strong&gt;ungrammatical&lt;/strong&gt; (i.e. syntax error).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So if we want to reason about the scenario in which &amp;quot;P(k)&amp;quot; is true, then we would say &amp;quot;If P(k), then ...&amp;quot;. We cannot say &amp;quot;If P(k) = 2^k+5, then ...&amp;quot;. We also cannot say &amp;quot;If 2^k+5, then ...&amp;quot;. They are simply &lt;strong&gt;ungrammatical&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2023-01-03T08:30:43.763" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25960" PostTypeId="2" ParentId="25955" CreationDate="2023-01-04T05:57:51.820" Score="2" Body="&lt;p&gt;These are examples of &lt;a href=&quot;https://en.wikipedia.org/wiki/Metonymy&quot; rel=&quot;nofollow noreferrer&quot;&gt;metonymy&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Let x∈f&amp;quot; instead of &amp;quot;Let x∈B&amp;quot;.&amp;quot; This is &lt;em&gt;totum pro parte&lt;/em&gt; synedoche.  And aspect of the function (the range) is being confused with the function itself. Here, it might help to get your students to think of a function as a verb rather than a noun. &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; refers to what you &lt;em&gt;do&lt;/em&gt; to things in &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, not the things you get after you've done it. This sort of metonymy is very common in language. For instance, &amp;quot;smoke&amp;quot; can mean the process of turning things into a gas, or the gas that results. In math, we can square a number, or say that a number is &amp;quot;square&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;In relations, students often say: &amp;quot;The pairs (1,1),(2,2),(3,3) are reflexive&amp;quot; when we define reflexive as a property of a relation, not a property of individual elements.&amp;quot; This is the fallacy of division: a property of the whole is being treated as applying to the parts. This is a common fallacy.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastEditorUserId="-1" LastEditDate="2023-01-06T15:18:27.093" LastActivityDate="2023-01-06T15:18:27.093" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25961" PostTypeId="1" CreationDate="2023-01-04T22:11:09.837" Score="3" ViewCount="723" Body="&lt;p&gt;The following is an elementary-level &lt;strong&gt;Math Kangaroo&lt;/strong&gt; multiple choice question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is the maximum value of the sum of the digits of the sum of the digits of a three-digit number?&lt;/p&gt;&#xA;&lt;p&gt;A. 9 &lt;br /&gt;&#xA;B. 10 &lt;br /&gt;&#xA;C. 11 &lt;br /&gt;&#xA;D. 12 &lt;br /&gt;&#xA;E. 18&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have the answer key, so I can work backward to sort of get how to solve this problem. However, as I was trying to write a solution for my students, I realized I could NOT help them eliminate the wrong answer choices. The following is what I have got so far:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is a tricky problem. You might have no glue the first time reading the question, but you can still capture some keywords such as “maximum” and “three-digit number”.   These keywords can give you some information to start solving the problem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When you saw the word “maximum”, “three-digit number”,  you could started trying out with the biggest three digit-number, which is 999. Now you can plug in the number into the question, then you will find out the question is asking “What is the maximum value of the sum of the digits of the sum of the digits of 999?”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now, the question is getting clearer. First, you need to solve the sum of the three-digits number 999, which is 9 + 9 + 9 = 18 + 9 = 27.  The question turns into “What is the maximum value of the sum of the digits of 27”. Then you need to solve the sum of 27, which is 2+7 = 9.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Take a look at the answer choices; there is a 9. This is a signal that you are on the right track, but before you circle the answer, don’t forget the power of double check! You need to ask yourself, is nine the maximum value you can get from combining two single digits?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now I am stuck with how to help the student eliminate other answer choices.&lt;/p&gt;&#xA;" OwnerUserId="20831" LastEditorUserId="8571" LastEditDate="2023-01-08T15:30:24.407" LastActivityDate="2023-01-13T14:19:00.810" Title="What is the maximum value of the sum of the digits of the sum of the digits of a three-digit number?" Tags="|primary-education|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25963" PostTypeId="2" ParentId="25955" CreationDate="2023-01-04T22:43:27.073" Score="2" Body="&lt;p&gt;I don't think the three examples are instances of the same error.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1. In a proof that a function f:A↦B is onto, a student will say &amp;quot;Let x∈f&amp;quot; instead of &amp;quot;Let x∈B&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Whatever misunderstanding results in this would probably not be cleared up by telling the student that functions are not sets. I think the student will probably add a &amp;quot;typecast&amp;quot; like &amp;quot;Oh, I meant x∈the set of f&amp;quot; which is still just as wrong but now it says it's a set so your complaint is invalid.&lt;/p&gt;&#xA;&lt;p&gt;I think the actual problem here is there are several sets relating to a particular function (such as its domain, range, codomain, and the set of input/output ordered pairs which is often defined as the function itself) and the student hasn't said which one they're talking about.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;2. When learning induction, students often define a predicate as being equal to an expression, such as P(x):=2&lt;sup&gt;x&lt;/sup&gt;+5&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The only immediate problem here is that you're assuming P is a predicate. Now, supposing the student actually does use P as a predicate, I would be very curious to know what they thought it meant. If the student writes &lt;span class=&quot;math-container&quot;&gt;$P(x)\lor Q(x)$&lt;/span&gt; what do they think that means? It's the same as writing &lt;span class=&quot;math-container&quot;&gt;$(2^x + 5) \lor (3x + 1)$&lt;/span&gt; for example. What does that mean? The error depends on what they think it means.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;3. In relations, students often say: &amp;quot;The pairs (1,1),(2,2),(3,3)&#xA;are reflexive&amp;quot; when we define reflexive as a property of a relation, not a property of individual elements.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't see why, in the context of relations, a pair &lt;em&gt;shouldn't&lt;/em&gt; be defined as reflexive if the two sides are equal. The intention is clear. Your complaint is that it's a non-standard definition in your class. If you are trying to teach strict adherence to definitions, then point that out; otherwise, I see no problem with calling these pairs reflexive as the meaning is quite obvious.&lt;/p&gt;&#xA;" OwnerUserId="6105" LastEditorUserId="6105" LastEditDate="2023-01-05T15:59:15.153" LastActivityDate="2023-01-05T15:59:15.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25964" PostTypeId="2" ParentId="25961" CreationDate="2023-01-05T02:39:04.173" Score="9" Body="&lt;p&gt;Since digits are less than or equal to 9, then you know the sum of the digits of a 3 digit number is less than or equal to 27. So the question becomes &amp;quot;what number less than or equal to 27 has the largest sum of digits?&amp;quot; Since 27 has a digit-sum of 9, you look for a number with digit-sum 10 or more. 19 has a digit-sum of 10. You know that a digit-sum of 11 or more is impossible since the first digit is no more than 2 (and 27 is the highest). Thus the answer is B. 10. (Note: to check that this max actually occurs, inspection finds 667 gives digit-sum 19 which has digit-sum 10.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit (from the comments):&lt;/strong&gt; To get students there I would ask them to (i) find the biggest digit-sum from a 2-digit number less than various 2-digit numbers (such as 72, 58, or 40), (ii) find the biggest digit-sum from a 3-digit number, (iii) solve the original question.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit (about English language difficulties mentioned in the comments):&lt;/strong&gt; At an older level the ideas can be bridged by using function notation. Defining &lt;span class=&quot;math-container&quot;&gt;$s(n)=\text{sum of the digits of $n$}$&lt;/span&gt;, e.g. &lt;span class=&quot;math-container&quot;&gt;$s(87)=8+7=15$&lt;/span&gt; and so the question is asking to maximize &lt;span class=&quot;math-container&quot;&gt;$s(s(n))$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;At the elementary level, using an arrow notation for functions instead usually gets the point across. So explain that an arrow gives sum of the digits, e.g. &lt;span class=&quot;math-container&quot;&gt;$87 \Rightarrow 8+7 = 15$&lt;/span&gt; and then asking if you start with a three-digit number what's the largest number you can get in the last blank of:  &lt;span class=&quot;math-container&quot;&gt;$$\underline{\ \ \ \ } \Rightarrow \underline{\ \ \ \ \ } =\underline{\ \ \ \ \ } \Rightarrow \underline{\ \ \ \ \ }=\underline{\ \ \ \ \ } .$$&lt;/span&gt; It would also be good to do an example like &lt;span class=&quot;math-container&quot;&gt;$$\underline{\ 147\ } \Rightarrow \underline{\ 1+4+7\ } =\underline{\ \ 12\ \ } \Rightarrow \underline{\ 1+2\ }=\underline{\ \ 3\ \ } .$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;When I've done similar things with elementary students, they usually find the correct maximum pretty quickly through trial and error. So then you need to push them to explain why they are sure it's the largest (without checking every three-digit number).&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2023-01-06T16:08:59.107" LastActivityDate="2023-01-06T16:08:59.107" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25965" PostTypeId="2" ParentId="25961" CreationDate="2023-01-05T12:28:26.160" Score="2" Body="&lt;p&gt;You don't need to know just the largest digit sum of three digit numbers, but all of them. 999 has the largest digit sum 27, 100 has the smallest digit sum 1, and everything in between is possible. So the first digit sum is anything from 1 to 27.&lt;/p&gt;&#xA;&lt;p&gt;The digit sum of 27 is 9. But that is not the largest digit sum. We can pick a smaller number, specifically 19, with a larger digit sum. That's because by making the first digit 1 smaller, we can make the second digit two larger. This gives a digit sum of 10.&lt;/p&gt;&#xA;&lt;p&gt;As a bonus, we get the first number with a digit sum of 19 if we make the tens and single digits as large as possible and the hundreds digit as small as possible, which is the number 199.&lt;/p&gt;&#xA;&lt;p&gt;As a bonus, which is the smallest number where the sum of digits of the sum of digits is larger, that is 11? The smallest number with a digit sum of 11 is a digit 9 and a digit 2, that's 29. And the smallest number with a digit sum of 29 has three nine's and a 2, that is 2,999.&lt;/p&gt;&#xA;&lt;p&gt;And a third bonus question: Which is the smallest number where E is the right answer? E means: Sum of digits of the sum of digits is 18. That's two nine's, so the sum of digits is 99. That's an enormous number: We need eleven nines, so the smallest number is 99,999,999,999.&lt;/p&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2023-01-05T12:28:26.160" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25966" PostTypeId="2" ParentId="353" CreationDate="2023-01-05T16:52:17.300" Score="0" Body="&lt;p&gt;My favour example is the implication, &amp;quot;If it is raining, then it is cloudy.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This does not mean that rain &lt;em&gt;causes&lt;/em&gt; cloudiness. (There is no causality in mathematics.) In the usual sense of if-then statements in mathematics, it means only that it is not the case (present tense) that it is both raining and not cloudy.  Using this definition, we can easily construct the truth table:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$~~~~~~~~~R~~  C~~~~  R\implies C$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$~~T~~T~~~~~~~~~T$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$~~T~~F~~~~~~~~~F$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$~~F~~T~~~~~~~~~T$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$~~F~~F~~~~~~~~~T$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$R = $&lt;/span&gt; &amp;quot;It is raining&amp;quot;, and &lt;span class=&quot;math-container&quot;&gt;$C = $&lt;/span&gt; &amp;quot;It is cloudy&amp;quot; (both in the present tense).&lt;/p&gt;&#xA;&lt;p&gt;(We can also formally prove the above &amp;quot;definition&amp;quot;, &lt;span class=&quot;math-container&quot;&gt;$(A\implies B) \iff \neg (A \land \neg B)$&lt;/span&gt; from &amp;quot;first principles&amp;quot; &lt;a href=&quot;http://www.dcproof.com/DeriveImplies.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Only 19 lines.)&lt;/p&gt;&#xA;&lt;p&gt;This is the so-called material conditional. (Other forms of the conditional may be required for past and future tenses and causality. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Conditional&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;As required, if it is true that it is raining (&lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;), and it is true that if it is raining then it is cloudy (&lt;span class=&quot;math-container&quot;&gt;$R\implies C$&lt;/span&gt;), then &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; must be true (see line 1).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$~~~~(R \land (R\implies C))\implies C$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is the so-called Detachment Rule.&lt;/p&gt;&#xA;&lt;p&gt;To disprove that &lt;span class=&quot;math-container&quot;&gt;$R\implies C$&lt;/span&gt;, it is sufficient to prove that &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is true and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is false (see line 2).&lt;/p&gt;&#xA;&lt;p&gt;Interestingly, if it is not raining (&lt;span class=&quot;math-container&quot;&gt;$\neg R$&lt;/span&gt;), then it is true that &lt;span class=&quot;math-container&quot;&gt;$R\implies C$&lt;/span&gt;, regardless of the truth value of &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; (see lines 3 and 4).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$~~~~\neg R \implies (R\implies C)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that we cannot infer from this statement that &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is true, or that it is false since &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; itself is false.&lt;/p&gt;&#xA;&lt;p&gt;This is the so-called Principle of Vacuous Truth. It is rarely if ever used daily discourse since we don't usually give much consideration to implications with a false antecedent. It is, however, a valid and useful method of proof in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-01-06T00:21:28.000" LastActivityDate="2023-01-06T00:21:28.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25967" PostTypeId="2" ParentId="25955" CreationDate="2023-01-05T20:28:27.737" Score="0" Body="&lt;p&gt;Excellent question. The lingo, as best as I have witnessed it practiced is:&lt;/p&gt;&#xA;&lt;p&gt;In philosophy, they can be called &lt;a href=&quot;https://plato.stanford.edu/entries/abstract-objects/&quot; rel=&quot;nofollow noreferrer&quot;&gt;abstract objects (SEP)&lt;/a&gt;.&#xA;In mathematics and mathematical philosophy, they are indeed called &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_object&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathematical objects&lt;/a&gt;.&#xA;In computer science, they are &lt;a href=&quot;https://en.wikipedia.org/wiki/Type_system&quot; rel=&quot;nofollow noreferrer&quot;&gt;types&lt;/a&gt; which essentially are a differentiated set used in hierarchies of various sorts to avoid the Russellian paradox at a theoretical level.&lt;/p&gt;&#xA;&lt;p&gt;As a former high school teacher who used to provide universities and colleges students, I'll make the following statements:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The crux of the problem is that there is a difference between &lt;a href=&quot;https://en.wikipedia.org/wiki/Syntax%E2%80%93semantics_interface&quot; rel=&quot;nofollow noreferrer&quot;&gt;syntax and semantics&lt;/a&gt; and we do not explicitly teach navigating the dichotomy. Schools overwhelming allow students to process syntax with little grasp of semantics and yet, due to grade inflation and other pressures, consider them educated. To be honest, it's a problem with the system. Students are excellent at regurgitating syntax, but they never quite figure out what it means; I certainly didn't have many insights until I taught math and had to think in depth about it. The real question is what to do about it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you really want to help students, and you take responsibility for it, then the best way to do it is to design a preliminary of unit of instruction that explicitly teaches the skills. If you're familiar with &lt;a href=&quot;https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dunning-Kruger's work&lt;/a&gt;, which has distinct epistemological overtones with practical pedagogical implications, most students don't even REALIZE they don't know things. What happens in math traditionally, is that mathematics teachers don't realize they have the skill and don't teach it explicitly, and then they promote a series of students who figure it out implicitly who then become teachers never realizing that mastery of mapping semantics onto syntax is an explicit skill. It's a vicious cycle. If you're teaching math, there's a good chance that your intuitions in math are good, and so you also have never explicitly had discussions or lessons about the skill of mapping. You just &amp;quot;get it&amp;quot;. That happened to me until I taught in urban education.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;What I've said has been said before more pithily. Gauss's quote on teaching is one half &lt;a href=&quot;https://matheducators.stackexchange.com/a/24684/18201&quot;&gt;(buried in my MathEdSE answer here)&lt;/a&gt;. Joseph Joubert's witty quote &amp;quot;To teach is to learn twice&amp;quot; is the other.&lt;/p&gt;&#xA;&lt;p&gt;Here's what you do. You design a diagnostic to specifically test for the sorts of errors in terms of decoding and encoding mathematical syntax and administer it very first day of class. Then, for each category of the mapping this is a function, this is a tuple, this is a real, this is a mapping, this is a relation, ad nauseum, you simply provide a refersher of some sort, and then assign practice. If you do it right, you do it all in natural language. My mid-term and final always had 50 questions in plain English, and my students would wail it wasn't math. And to be fair, they had gone years without anyone actually trying to determine if they could understand the mathematics in depth.&lt;/p&gt;&#xA;&lt;p&gt;Mathematical comprehension certainly entails a sophistication of syntax, but it is a necessary condition, it is not sufficient. The semantics of math is the backbone of comprehension. Most students can find &lt;em&gt;c&lt;/em&gt; given a Pythagorean theorem, but can they have a discussion about why the hypotenuse is equal to the positive root of the sum of the square of the legs? The former is plug and chug, the latter is mathematical discourse and sadly, the string of teachers that furnished you your students probably rushed though content to meet district requirements dealing with large batches of irregularly educated students. In the US, at least, mastery in mathematics at the elementary and secondary levels isn't a thing except for the gifted.&lt;/p&gt;&#xA;&lt;p&gt;If that's too much work for you, then that is what it is. But if you're contemporary in your pedagogy, you can easily use software to automate the grading and tracking, and could easily make it a regular feature of your instruction. I would give my midterm and final questions in natural language at the start of the semester to be worked on and reflected upon, and students would still struggle with answering the questions. The lengths you go to bend over backwards to teach helps determine whether you shake out as a mathematician just teaching math or are actually a math educator.&lt;/p&gt;&#xA;" OwnerUserId="18201" LastActivityDate="2023-01-05T20:28:27.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25968" PostTypeId="2" ParentId="25955" CreationDate="2023-01-06T00:37:09.723" Score="0" Body="&lt;p&gt;Instead of using &lt;span class=&quot;math-container&quot;&gt;$f: A \to B$&lt;/span&gt; from the start, it might be more instructive to formally write &lt;span class=&quot;math-container&quot;&gt;$\forall x: [x \in A \implies f(x) \in B]$&lt;/span&gt;. Then you cannot infer from this that &lt;span class=&quot;math-container&quot;&gt;$x \in f$&lt;/span&gt;. This, of course, would require some familiarity with basic predicate logic.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastActivityDate="2023-01-06T00:37:09.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25969" PostTypeId="1" CreationDate="2023-01-07T19:13:07.937" Score="2" ViewCount="235" Body="&lt;p&gt;I am looking for any hints or experience reports or materials/potential difficulties about how to introduce the use of Greek letters in high school Math/Physics.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="376" LastEditDate="2023-01-21T16:31:41.447" LastActivityDate="2023-02-03T07:52:43.633" Title="How to introduce the use of Greek letters in high school?" Tags="|secondary-education|notation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25970" PostTypeId="2" ParentId="25969" CreationDate="2023-01-07T19:56:33.920" Score="3" Body="&lt;p&gt;There is an MSE question very similar to this. (Closed now, but with good discussions, answers.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/701970/do-we-need-to-formally-teach-the-greek-alphabet&quot;&gt;https://math.stackexchange.com/questions/701970/do-we-need-to-formally-teach-the-greek-alphabet&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Also, some lesson plans and the like if you just Google.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=how+to+introduce+the+use+of+greek+letters+in+high+school&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=how+to+introduce+the+use+of+greek+letters+in+high+school&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I didn't have a huge problem with it, just dealt with them as they came up.  But am a strong student overall. All that said, I still don't really know them all, just the ones I've encountered.&lt;/p&gt;&#xA;&lt;p&gt;I think a fun memorization (flashcard practice, tested) sidetrack to all the math/math would be to cover it as a lesson for a day's hour (with a short quiz, next day).  And it may get a few of the arty, humanities types interested and enjoying something off the math/math track.   But just when doing it, let the kids know that this will be slightly different from all their calculational practice so that they can reframe themeselves.&lt;/p&gt;&#xA;&lt;p&gt;I think the right spot/time is in &amp;quot;Algebra 2&amp;quot; (US meaning).  They will have maybe seen a few already, maybe in chemistry or geometry.  This now gives them a chance to learn all the primates in the zoo.&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't bother with it in a physics or chem class though.  Just use the few you need and clarify the meaning, pronunciation, &amp;quot;false friends&amp;quot; (letters that look same as English but are not, etc.)&lt;/p&gt;&#xA;&lt;p&gt;And &amp;quot;concept lovers&amp;quot; will scoff, but there is some enjoyment in learning things like this. (Lists of state capitals are much enjoyed by small children...not a sin.)&lt;/p&gt;&#xA;&lt;p&gt;I would restrict to using it rather conventionally (alpha, beta, gamma for angles, lambda for wavelength, rho density, etc.).  I would not introduce the confusion of using a symbol in an abnormal fashion.  And I would not stick in Greek when not needed (or at least not normally used).  I hate the use of Fraktur, e.g.  I would avoid the use of &amp;quot;squiggle&amp;quot; and &amp;quot;other squiggle&amp;quot; unless really needed (unlikely in high school classes).&lt;/p&gt;&#xA;&lt;p&gt;If their book has a list of them (many do, where the formulas are listed), point it out to them. If not, give them a paper handout.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  Answer to question in first comment:  the first is &amp;quot;squiggle&amp;quot;, the second is &amp;quot;other squiggle&amp;quot;.  This was actually semi-allowed in Navy Nuclear Power School, to refer to the Greek letters that way.  Squiggle is a common sign for logarithmic energy decrement, which comes up in the context of collisions during neutron thermalization.  I can't remember what other squiggle did, maybe delayed neutron precursor fraction or fast fission.  Something of that sort.  If you do a Google search, it's not just a Navyism, but lots of engineers and physicists refer to the Greek letters that way.  Also, at NPS, you were also allowed to refer to Y(0) and J(0) Bessel functions as Yo and Jo.  Like &amp;quot;Yo, Adrian&amp;quot;.  Little bit of grain for the hard working kine...&lt;/p&gt;&#xA;" OwnerUserId="20811" LastEditorUserId="-1" LastEditDate="2023-02-03T07:52:43.633" LastActivityDate="2023-02-03T07:52:43.633" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25971" PostTypeId="1" CreationDate="2023-01-08T14:10:09.577" Score="4" ViewCount="486" Body="&lt;p&gt;In constructive mathematics we make a distinction between &amp;quot;proof of negation&amp;quot; and &amp;quot;proof by contradiction&amp;quot;.  You can read a great account of the difference &lt;a href=&quot;https://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/#:%7E:text=Proof%20of%20negation%20is%20an,the%20same%20classically%20and%20intuitionistically.&quot; rel=&quot;nofollow noreferrer&quot;&gt;in this blog post of Andrej Bauer&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To summarize&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Proof of negation&lt;/strong&gt; is proving &lt;span class=&quot;math-container&quot;&gt;$\neg p$&lt;/span&gt; by assuming &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and arguing an absurdity.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Proof by contradiction&lt;/strong&gt; is proving &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; by assuming &lt;span class=&quot;math-container&quot;&gt;$\neg p$&lt;/span&gt; and arguing an absurdity.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I am writing an introduction to proof book which puts &lt;a href=&quot;https://en.wikipedia.org/wiki/Fitch_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fitch style natural deduction&lt;/a&gt; introduction and elimination rules front and center.  My definition of negation is &lt;span class=&quot;math-container&quot;&gt;$\neg p \equiv (p \implies \bot)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\bot$&lt;/span&gt; stands for an absurdity.  The introduction rule for negation is thus the proof pattern we would call &amp;quot;proof of negation&amp;quot; above.&lt;/p&gt;&#xA;&lt;p&gt;I am looking for a big list of elementary arguments which are actually proof by contradiction rather than proof of negation.  I need these for the section of my textbook which covers the &amp;quot;proof by contradiction&amp;quot; strategy.  It is exceedingly difficult to find elementary situations where proof by contradiction is actually an appropriate strategy.&lt;/p&gt;&#xA;&lt;p&gt;Here is one example of each strategy:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof of negation&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; is not odd.&lt;/p&gt;&#xA;&lt;p&gt;Assume that &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; is odd.  Then there is an integer &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$6=2k+1$&lt;/span&gt;.  Solving for &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; yields &lt;span class=&quot;math-container&quot;&gt;$k=2.5$&lt;/span&gt; which is not an integer. This is a contradiction.  Thus &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; is not odd.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof by contradiction&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I steal an &lt;a href=&quot;https://math.stackexchange.com/a/1119062/34287&quot;&gt;example from Neil Strickland&lt;/a&gt;:  Some non-zero digit occurs infinitely often in the decimal expansion of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Proof:  Assume to the contrary that every non-zero digit appears finitely many times in the decimal expansion of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.  Then &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; would be a rational number.  However &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is known to be irrational.  Contradiction.&lt;/p&gt;&#xA;&lt;p&gt;In this argument we obtain a contradiction by assuming the negation of what we are trying to prove.  Note that it does not actually produce an explicit example of a non-zero digit which occurs infinitely often (if it did the proof would be constructive).&lt;/p&gt;&#xA;&lt;p&gt;UPDATE:  I made the following table for my book.  Perhaps it will inspire some more ideas.  All of the current answers (I think) showcase the consequence of proving an existentially quantified statement by contradiction:  that you cannot actually show me how to produce a witness.  It would be interesting to see examples of proofs by contradiction which showcase the other sacrifices.&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;Type of statement&lt;/th&gt;&#xA;&lt;th&gt;The sacrifice we make  using proof by contradiction&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$p \vee q$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;I know that either &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is true, but I cannot tell you which one.&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$p \wedge q$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;I don't have a direct argument for &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; independently.&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$\neg p$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;There is no sacrifice.  &lt;span class=&quot;math-container&quot;&gt;$\neg p$&lt;/span&gt; is equivalent to &lt;span class=&quot;math-container&quot;&gt;$\neg (\neg (\neg p))$&lt;/span&gt; in intuitionist logic as well.  A bit weird to do though.&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$p \implies q$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;If you hand me an argument that &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is true, I do not actually have a way to use that data to argue &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;.  All I can do is tell you that there should be such an argument.&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$\exists x \in U: A(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;While I know that &lt;strong&gt;some&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x_1 \in U$&lt;/span&gt; makes &lt;span class=&quot;math-container&quot;&gt;$A(x_1)$&lt;/span&gt; true, I do not know how to actually show you one.&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x \in U: A(x)$&lt;/span&gt;&lt;/td&gt;&#xA;&lt;td&gt;If you hand me a particular &lt;span class=&quot;math-container&quot;&gt;$x_1 \in U$&lt;/span&gt; I don't have a direct argument that &lt;span class=&quot;math-container&quot;&gt;$A(x_1)$&lt;/span&gt; is true.  I can only show you that assuming it is not true leads to contradiction.  In other words, whatever type of  statement &lt;span class=&quot;math-container&quot;&gt;$A(x_1)$&lt;/span&gt; is, I will pay the associated price for that statement.&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2023-02-22T22:17:03.627" LastActivityDate="2023-02-22T22:17:03.627" Title="Proof by Contradiction vs. Proof of Negation" Tags="|textbooks|proofs|examples|logic|big-list|" AnswerCount="2" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="25973" PostTypeId="2" ParentId="24786" CreationDate="2023-01-08T19:46:38.507" Score="0" Body="&lt;p&gt;The &lt;a href=&quot;http://www.math.ualberta.ca/%7Exinweiyu/421.Q1.17w/&quot; rel=&quot;nofollow noreferrer&quot;&gt;lecture notes&lt;/a&gt; from Xinwei Yu's Finite Mathematics course at the University of Alberta contain a discussion that may be at the right level. See &lt;a href=&quot;http://www.math.ualberta.ca/%7Exinweiyu/421.Q1.17w/421Q1Winter2017_L15_20170213.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Generating Functions II&lt;/a&gt;, Example 5. This example is slightly more general than your problem, but describes the necessary techniques in detail. Exercise 2 is exactly your problem.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2023-01-08T19:46:38.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25974" PostTypeId="2" ParentId="25896" CreationDate="2023-01-09T19:09:18.973" Score="1" Body="&lt;p&gt;This might be a little beyond the capabilities you're looking for, but &lt;a href=&quot;https://cocalc.com/features/whiteboard&quot; rel=&quot;nofollow noreferrer&quot;&gt;CoCalc now has a whiteboard&lt;/a&gt; which allows embedding of &lt;a href=&quot;https://doc.cocalc.com/whiteboard.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jupyter notebook cells and timers and pens&lt;/a&gt; and a lot of other stuff.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2023-01-09T19:09:18.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25975" PostTypeId="1" CreationDate="2023-01-11T21:39:00.937" Score="12" ViewCount="4507" Body="&lt;p&gt;A student benefits from their attempt at a solution or proof being checked by the teacher. My own view is that, if the student's work is poor, it is best just to provide a model solution or proof in its place. In general, I think that bad work is best ignored and forgotten. The value of the failed attempt is that it has focused the student somewhat on the issues; he or she is then better prepared to understand and appreciate the good solution or proof.&lt;/p&gt;&#xA;&lt;p&gt;An alternative view is that it is better to start from what the student has done: correcting and adding minimally so that the goal is eventually reached. Presumably the idea is to give encouragement by allowing the student to believe that they mostly got there and only needed a bit of help.&lt;/p&gt;&#xA;&lt;p&gt;I don't like the alternative view because, even when the amended result is, in a strict sense, free of fallacy, the student is typically still left with a clumsy and badly written piece of work.&lt;/p&gt;&#xA;&lt;p&gt;What I am seeking here is an argument for the opposing view and against my view, which I admit may be wrong.&lt;/p&gt;&#xA;" OwnerUserId="4804" LastActivityDate="2023-01-13T09:06:59.573" Title="Is patching up a student's poor solution better than providing a good solution?" Tags="|students-mistakes|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="25976" PostTypeId="2" ParentId="25975" CreationDate="2023-01-11T22:29:16.650" Score="29" Body="&lt;p&gt;As Besicovich once said, the reputation of a mathematician is determined by the number of their badly written papers (the pioneering work is often clumsy and follows the route far from the shortest one). When one really tries to solve a problem, the thinking process goes back and forth between several options hitting dead ends in many places until one gets &lt;em&gt;some&lt;/em&gt; path from A to B, so the final solution may resemble a trip from Cleveland to Detroit through San-Francisco, Sidney, Beijing, and Paris (not necessarily in this order). However, one can often pick some useful stuff and see interesting things on that way as opposed to driving for 4 hours on I-80/75.&lt;/p&gt;&#xA;&lt;p&gt;So, showing the students how they can tie loose ends and bridge the gaps in their arguments has its merits, &lt;em&gt;provided that&lt;/em&gt; what they try is something leading in the right direction or, at least, going somewhere by some non-zero distance. Then they can learn to go further and further in their attempts until they are able to design a correct full proof (though not necessarily the slickest one). Never try to repair total gibberish though. If some step is obviously formally wrong or if what is written is a non-statement (or even a non-sentence), weed those things out ruthlessly to start with, even if they can be turned into something useful by a moderate amount of modifications.&lt;/p&gt;&#xA;&lt;p&gt;In general, one usually derives much more satisfaction and confidence from seeing how his/her own approach can be completed than from seeing the shortest and the most elegant approach possible. The latter often just leaves one with the feeling that he/she is a total idiot and can do nothing (I'm not saying that this feeling is far from the truth: very few of us will have a reason to rate themselves much higher when compared to Archimedes, say, and that guy lived over 2000 years ago. I'm just saying that cultivating it too much is counterproductive to the learning process).&lt;/p&gt;&#xA;&lt;p&gt;You'll have enough opportunities to present the good solutions when going over the problems in class. But when working with a student one on one or when letting them go to the board, I usually just allow them follow their own path, even if it is suboptimal, as long as what they are saying makes sense.&lt;/p&gt;&#xA;&lt;p&gt;I hope what I wrote is a bit more meaningful than what GPT3 would write in response to this question, but I leave it to the other people to judge if it is really the case :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-01-11T22:29:16.650" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25977" PostTypeId="2" ParentId="25975" CreationDate="2023-01-11T23:35:53.843" Score="2" Body="&lt;p&gt;It's a judgment call, but I think you are probably right, especially with very bad solutions.&lt;/p&gt;&#xA;&lt;p&gt;I agree that it's important for people to try first (even if bad).  Makes them learn more than seeing the result without trying first.  Seen this in many settings outside of math also (history, business, engineering).  In the military, I remember hearing the value of war planning is not the plan, but the planning (learning the features and issues).  Same thing in corporate M&amp;amp;A.  Kudos for making this point.&lt;/p&gt;&#xA;&lt;p&gt;I really don't think that it's critical to make the connection from the poor solution to the strong one.  Especially for weak students.  And after all, they are neophytes, also...accumulating a range of insights, like the model solution.  But the key is the mental/moral investment of making a trial attempt first.&lt;/p&gt;&#xA;&lt;p&gt;The strong students will eventually make the connection themselves.  Like Freeman Dyson showing equivalence in competing QED approaches.  But to start with, maybe more important that they get some grist for the mill (from learning the model solution, AFTER being invested with a trial attempt.)&lt;/p&gt;&#xA;&lt;p&gt;Also, of course...it's important to have something to &amp;quot;make the solution your own&amp;quot;.  Even with the investment.  Repeating it a few days later.  Showing applications or small variations of it, etc.  This will help retention and later usage.&lt;/p&gt;&#xA;" OwnerUserId="20849" LastEditorUserId="20849" LastEditDate="2023-01-11T23:43:41.967" LastActivityDate="2023-01-11T23:43:41.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25978" PostTypeId="2" ParentId="25975" CreationDate="2023-01-12T06:40:47.357" Score="5" Body="&lt;p&gt;One problem with mathematics for many students is that is perceived as something complete that one learns, rather than an activity one does.&lt;/p&gt;&#xA;&lt;p&gt;On one hand, you do want to present nice and elegant mathematics.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, you really, really want the students to do mathematics and think for themselves. It is very easy to nod along as an expert does something, but this is unlikely to lead to understanding, unless you already are a strong student. You need some relationship to the mathematics happening, and one way of establishing that relationship is starting from and playing with what the student has given you.&lt;/p&gt;&#xA;&lt;p&gt;On the third hand, if you not in a one-on-one situation, you should take into account all the other students present, too. They might not have much of a relationship to the attempted solution by a peer (though they might have! hard to know), and maybe seeing a muddy and unclear rescue attempt is not so useful for them.&lt;/p&gt;&#xA;&lt;p&gt;So: as usual, you have to balance several considerations, and most probably also use of time and sometimes whether you even understand what the student is trying to do or say.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-01-12T06:40:47.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25979" PostTypeId="1" CreationDate="2023-01-12T10:48:54.650" Score="8" ViewCount="396" Body="&lt;p&gt;I sometimes come across equations like&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^\circ = 180^\circ - \frac{360^\circ}{n}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where I would write&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x = 180^\circ - \frac{360^\circ}{n}$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;or like&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3 \text{ cm} + x \text{ cm} = 5 \text{ cm}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where I would write&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3 \text{ cm} + x = 5 \text{ cm}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I wonder which style is recommended and why.&lt;/p&gt;&#xA;&lt;p&gt;Should one include the unit in the variable?&lt;/p&gt;&#xA;" OwnerUserId="6527" LastActivityDate="2024-01-02T14:54:53.003" Title="Should one include the unit in the variable? E.g. should one write $x^\circ = 30^\circ$ or $x = 30^\circ$?" Tags="|notation|units|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="25980" PostTypeId="2" ParentId="25979" CreationDate="2023-01-12T11:47:36.313" Score="7" Body="&lt;p&gt;This must certainly depend on the context. In the examples you present here, the units/dimensions do nothing, so I would just go ahead and forget them, taking them back in use to check if the answer makes sense.&lt;/p&gt;&#xA;&lt;p&gt;In this case, writing &lt;span class=&quot;math-container&quot;&gt;$3 \text{ cm} + x \text{ cm}$&lt;/span&gt; is a way of formalizing that the dimensions are ignored; certainly the next step is to divide away the centimetres and then calculate with the pure numbers.&lt;/p&gt;&#xA;&lt;p&gt;However, if I was dealing with a more complicated case with several different units, then it would be more natural for me to think that the variable contains both numerical information and the units for it. That way I would not have to know the units in advance; rather, the outcome of the calculation would be&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;a = \frac{3}{7} \frac{\text{m}}{\text{s}^2}&#xA;$$&lt;/span&gt;&#xA;or something similar. Here &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; corresponds to the actual acceleration, as opposed to the case where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is an arbitrary numerical value that only is meaningful after we have given the dimensions.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-01-12T11:47:36.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25981" PostTypeId="2" ParentId="25979" CreationDate="2023-01-12T12:20:48.180" Score="19" Body="&lt;p&gt;The usual physics convention is that variables have units built into them, so you don't need to (indeed, shouldn't) write the units separately. For example, Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Kinetic_energy&quot; rel=&quot;noreferrer&quot;&gt;writes&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In classical mechanics, the kinetic energy of a non-rotating object of mass &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; traveling at a speed &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\tfrac{1}{2}mv^{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;not&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In classical mechanics, the kinetic energy of a non-rotating object of mass &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; kilograms traveling at a speed &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; meters per second is &lt;span class=&quot;math-container&quot;&gt;$\tfrac{1}{2}mv^{2}$&lt;/span&gt; joules.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The first sentence definitely seems clearer to me!&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="5794" LastEditDate="2023-01-13T13:50:26.977" LastActivityDate="2023-01-13T13:50:26.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25982" PostTypeId="2" ParentId="25979" CreationDate="2023-01-12T12:37:28.170" Score="10" Body="&lt;p&gt;As far as the centimeters are concerned, I believe that both are possible, but there's a catch:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$3 \text{cm} + x = 5 \text{cm}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Solution:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x = 2 \text{cm}$&lt;/span&gt; =&amp;gt; correct.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x = 0.2 \text{dm}$&lt;/span&gt; =&amp;gt; correct.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x = 0.02 \text{m}$&lt;/span&gt; =&amp;gt; correct.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$3 \text{cm} + x \text{cm} = 5 \text{cm}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x = 2$&lt;/span&gt; =&amp;gt; correct (no unit).&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x = 0.2 \text{dm}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x = 0.02\text{m}$&lt;/span&gt; =&amp;gt; wrong (unit must be centimeters, as it is mentioned)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8190" LastEditorUserId="8190" LastEditDate="2024-01-02T12:14:05.617" LastActivityDate="2024-01-02T12:14:05.617" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="25983" PostTypeId="2" ParentId="9710" CreationDate="2023-01-12T14:41:32.843" Score="3" Body="&lt;p&gt;I'm in favour of using the different brackets : '(' and ')', '[' and ']' and finally, '{' and '}'.&lt;/p&gt;&#xA;&lt;p&gt;Why?&lt;br /&gt;&#xA;Well: &lt;span class=&quot;math-container&quot;&gt;$[2 \cdot (3 + 5)]^6$&lt;/span&gt; is much clearer than &lt;span class=&quot;math-container&quot;&gt;$(2 \cdot (3 + 5))^6$&lt;/span&gt;, especially for weaker students.&lt;/p&gt;&#xA;&lt;p&gt;Your reaction: &amp;quot;But computers don't understand this notation!&amp;quot;&lt;br /&gt;&#xA;My reaction back: &amp;quot;People come before machines!&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;When you introduce brackets, you can easily say:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;First the calculations inside '(' and ')'.&lt;/li&gt;&#xA;&lt;li&gt;Afterwards the calculations inside '[' and ']'.&lt;/li&gt;&#xA;&lt;li&gt;Finally the calculations inside '{' and '}'.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Like this, the students learn about calculating from inner to outer.&lt;br /&gt;&#xA;Once they have grasped that idea, they might be confronted with machines, who are not even capable of working with different brackets.&lt;br /&gt;&#xA;&amp;quot;Indeed: a computer is a machine, who is not capable of things which are obvious for you, so you are smarter than a computer!&amp;quot; (never miss an opportunity to praise your students!).&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-01-12T14:41:32.843" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25984" PostTypeId="2" ParentId="9710" CreationDate="2023-01-12T15:23:27.767" Score="3" Body="&lt;p&gt;I would recommend normal parenthesis most of the time, but if it gets confusing, consider:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Use smaller parenthesis inside and larger ones outside, matching the sizes of matching left and right parenthesis. Handwriting and LaTeX can both do this.&lt;/li&gt;&#xA;&lt;li&gt;Maybe have the outermost as square brackets, if the above is not sufficient.&lt;/li&gt;&#xA;&lt;li&gt;Also consider splitting the expression and using letters; in mathematical analysis papers, a long integral or other expression might be split up as I + II + III, and then each of those parts analyzed separately. This adds to the readability of a given expression.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;There are many ways of making readable nested expression, and I fear that if several different kinds of parenthesis are really useful, the thing is quite pathological (only research folk are likely to meet it), an example to prove a point or poorly organized and could be split or otherwise handled better.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-01-12T15:23:27.767" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25985" PostTypeId="2" ParentId="25975" CreationDate="2023-01-12T19:27:12.130" Score="14" Body="&lt;p&gt;A lot of studies in math education have found that mathematical &lt;em&gt;confidence&lt;/em&gt; is an important mediator of success - that is, students who aren't confident in their ability to do math are less likely to successfully solve problems, even when those problems are entirely within their abilities. In my own teaching, I place a high importance on building that confidence - and part of building confidence is building the understanding that we do not do math fully-formed. If you tell a student &amp;quot;no, this is wrong, here's the right way to do it&amp;quot;, you're telling them &amp;quot;the ideas you had are irredeemable&amp;quot;. If that's true, that's true, and both you and the student need to live with that - but it's hardly ever true.&lt;/p&gt;&#xA;&lt;p&gt;If you instead help them repair the solution, they begin to believe that this is something they can do, because they got 25% or 50% of the way to an answer instead of zero. And they can see their own progress over time - &amp;quot;last time, we had to fix everything past the first two steps, but this time it was only the second half that was wrong&amp;quot;. That helps!&lt;/p&gt;&#xA;&lt;p&gt;Now, of course, the result is often inefficient or awkward, and there is a lot of value in being able to do math efficiently and smoothly. But this makes the most sense to me as a &lt;strong&gt;separate&lt;/strong&gt; learning outcome, and it deserves a separate part of the conversation. For example, after I've helped a student fix their solution, &lt;em&gt;then&lt;/em&gt; I'll point out things like &amp;quot;okay, you could have made a different choice &lt;em&gt;here&lt;/em&gt; that would have saved you from having to do &lt;em&gt;this&lt;/em&gt;&amp;quot; or &amp;quot;gee, didn't those variable names turn out to be so confusing? Would it have been easier if you'd used &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; for width instead of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Of course there are exceptions! There have been times when a student has come to me with a solution that was far enough off that repairing it would have taken ages, and we didn't have that kind of time. Even in those circumstances, though, I prefer to validate and then divert: &amp;quot;This general approach would probably work, but it'd be much harder than it needs to be; here's a different strategy that'd be faster.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="10081" LastActivityDate="2023-01-12T19:27:12.130" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="25986" PostTypeId="2" ParentId="25975" CreationDate="2023-01-12T22:46:13.107" Score="9" Body="&lt;p&gt;I've had foreign language teachers distinguish between learning a grammatical form either &amp;quot;for production&amp;quot; or &amp;quot;for recognition&amp;quot;. The former means the student can produce a sentence using the form correctly and appropriately, while the latter means that they merely know it when they see it, and know what it means. As an example, my knowledge of the subjunctive in French is merely at the &amp;quot;recognition&amp;quot; level.&lt;/p&gt;&#xA;&lt;p&gt;I think your choice runs a higher risk of students only absorbing the material at a recognition level.&lt;/p&gt;&#xA;&lt;p&gt;As I would sometimes say when explaining a problem one-on-one &amp;quot;It's not enough to be able to nod along as I explain it - you're going to have to come up with this on your own on the exam.&amp;quot;, and then I'd ask them to explain it back to me. And I think that getting a student to full understanding from where they currently are, (as opposed to picking them up and plopping them into the cleaner, better answer) helps them better make the material their own.&lt;/p&gt;&#xA;&lt;p&gt;Of course, this approach is more time and energy intensive, and sometimes we just don't have that available, so sure, show them the optimal solution. But just be aware of the risk that they might be actually learning less.&lt;/p&gt;&#xA;" OwnerUserId="14507" LastActivityDate="2023-01-12T22:46:13.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25987" PostTypeId="1" CreationDate="2023-01-13T01:10:45.713" Score="0" ViewCount="230" Body="&lt;p&gt;I am looking for a list of the most frequent Greek letters used in high school and college textbooks or some other corpora. I've realized my students don't know Greek letters and I would like to teach them starting from the most used letters. My interest is in high school, mainly.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="376" LastEditDate="2023-01-21T16:35:20.870" LastActivityDate="2023-01-21T16:35:20.870" Title="Which are the most used Greek letters in math textbooks?" Tags="|secondary-education|notation|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="25988" PostTypeId="2" ParentId="25975" CreationDate="2023-01-13T09:06:59.573" Score="2" Body="&lt;p&gt;There is usually more than one solution to every problem - so, if student took a less efficient way of doing it, they would benefit from both: knowing how to get through using this method and how to do it more efficiently.&lt;/p&gt;&#xA;&lt;p&gt;In my life as a scientist I have extracted a lot of mileage from an advice given by one of my teachers: know the &lt;em&gt;general method&lt;/em&gt; (that is &lt;em&gt;a dull but sure way of solving a problem&lt;/em&gt;), and resorting to it whenever you are stuck, out of ideas, or cannot wrap your head around it.&lt;/p&gt;&#xA;&lt;p&gt;Teaching students to be able to use a general method goes beyond a particular problem - it teaches them a skill that they will be able to use on other problems, including those that they will face after graduating. Teaching ingenious/beautiful ways of dealing with with specific problems fosters their ingenuity. Ultimately, it is about balance between educating an engineer and a top-class scientist - some people will become the latter, but many of them need just practical skills.&lt;/p&gt;&#xA;" OwnerUserId="20854" LastActivityDate="2023-01-13T09:06:59.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25989" PostTypeId="2" ParentId="25987" CreationDate="2023-01-13T09:29:32.483" Score="6" Body="&lt;p&gt;It strongly depends on field. Everybody uses &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; appears in many fields as the summation sign. In elementary statistics you're likely to see &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\chi$&lt;/span&gt;. In geometry and trigonometry &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is often used, and sometimes &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt;. In calculus, &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; are ubiquitous. In elementary physics &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt; have common uses. Beyond those, you often see &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\tau$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Lambda$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Pi$&lt;/span&gt;, and sometimes &lt;span class=&quot;math-container&quot;&gt;$\eta$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\zeta$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\xi$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\kappa$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Theta$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\Xi$&lt;/span&gt;. The variants &lt;span class=&quot;math-container&quot;&gt;$\varphi$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\vartheta$&lt;/span&gt; are common.&lt;/p&gt;&#xA;&lt;p&gt;It is less common to see the letters that are indistinguishable from Latin letters, namely A, B, E, Z, H, I, K, M, N, O, P, T, X (Alpha, Beta, Epsilon, Zeta, Eta, Iota, Kappa, Mu, Nu, Omicron, Rho, Tau, Chi), and &lt;span class=&quot;math-container&quot;&gt;$\iota$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\omicron$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\upsilon$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\Upsilon$&lt;/span&gt; seem to be rare as well. The variants &lt;span class=&quot;math-container&quot;&gt;$\varsigma$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\varpi$&lt;/span&gt; are uncommon, although &lt;span class=&quot;math-container&quot;&gt;$\varpi$&lt;/span&gt; shows up in number theory.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2023-01-13T09:29:32.483" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="25990" PostTypeId="2" ParentId="25961" CreationDate="2023-01-13T14:19:00.810" Score="1" Body="&lt;p&gt;I understand your question to target the &amp;quot;average student&amp;quot;, not the top ones (who will most probably find the solution more or less by intuition). So, the focus lies on explaining a failsafe way that can be easily understood.&lt;/p&gt;&#xA;&lt;p&gt;Have them do it in tiny, explicit steps.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;maximum value of the sum of the digits of the sum of the digits of a three-digit number&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Analyzing a phrase with so many &amp;quot;of&amp;quot; words starts at the end of the phrase.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Start with &amp;quot;three-digit number&amp;quot; and ask about the possible numbers matching that expression, leading to &amp;quot;anything from 100 to 999&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Go to &amp;quot;the sum of the digits of a three-digit number&amp;quot;. The basic, brute-force approach is to calculate all the hundreds of digit-sums, but as &lt;strong&gt;any combination&lt;/strong&gt; of the digits 1-9, 0-9, 0-9 is possible, we can take a shortcut to convince ourselves that anything from 1+0+0 to 9+9+9 is possible, so the sum can be anything between 1 and 27, inclusive.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For &amp;quot;the sum of the digits of the sum of the digits of a three-digit number&amp;quot;, being &amp;quot;the sum of the digits of a number between 1 and 27&amp;quot;, taking a shortcut is tricky, as there are different cases to be taken care of: one-digit numbers, two-digit numbers beginning with a 1, and two-digit numbers beginning with a 2. So, writing down the 27 numbers and calculating their sum of digits is doable and will lead to a list of the possible sum-of-digits values, ranging from 1 to 10.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Clearly, the &amp;quot;maximum value of ...&amp;quot; then is 10.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15862" LastActivityDate="2023-01-13T14:19:00.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25991" PostTypeId="2" ParentId="25455" CreationDate="2023-01-14T05:19:28.953" Score="0" Body="&lt;p&gt;To summarize:&lt;/p&gt;&#xA;&lt;p&gt;@CameronWilliams' work gave me what I needed to make all the images I wanted. I am using &lt;a href=&quot;https://www.desmos.com/calculator/7p0jgbohdb&quot; rel=&quot;nofollow noreferrer&quot;&gt;this on desmos&lt;/a&gt; for that purpose.&lt;/p&gt;&#xA;&lt;p&gt;I used those images to make a randomized problem in MyOpenMath. It's problem #990768, if you'd like to use it. (I'm using it in a quiz.)&lt;/p&gt;&#xA;&lt;p&gt;And to help students learn more before the quiz, I point them to &lt;a href=&quot;https://www.geogebra.org/m/KSuUb2TR?fbclid=IwAR0F0CNQAP4Lf6DsPRc_hGuKu2Y4qdn4TqmK_VkAy6xETenGGPr3o_ESofw&quot; rel=&quot;nofollow noreferrer&quot;&gt;this geogebra page&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-01-14T05:25:05.060" LastActivityDate="2023-01-14T05:25:05.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25992" PostTypeId="2" ParentId="25937" CreationDate="2023-01-14T09:21:58.627" Score="1" Body="&lt;p&gt;For math and english, a number US states either explicitly or implicitly use the much-debated Common Core. These standards have been rather controversial, but they still appear to be the de-facto standard in many US school districts. The &lt;a href=&quot;https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;us common core math standard&lt;/a&gt; gives a broad description of what should be covered in each grade. A similar document exists for the&#xA;&lt;a href=&quot;https://learning.ccsso.org/wp-content/uploads/2022/11/ELA_Standards1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;common core english standard&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the sciences, there are the &amp;quot;next generation science standards&amp;quot;, but my understanding is they are even less universally accepted than common core. A starting place may be the NGSS standards published &lt;a href=&quot;https://nap.nationalacademies.org/read/18290/chapter/1&quot; rel=&quot;nofollow noreferrer&quot;&gt;by the National Academy Press&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Concerning textbooks, a blunt answer is that there are no standard ones in the US; the market is highly fractured and each and every school district in the US adopts their own texts (if they adopt any at all). Indeed, many school districts do not use printed textbooks at all.&lt;/p&gt;&#xA;" OwnerUserId="15606" LastActivityDate="2023-01-14T09:21:58.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25993" PostTypeId="1" CreationDate="2023-01-14T11:19:23.820" Score="11" ViewCount="1623" Body="&lt;p&gt;I'm trying to learn arithmetic from scratch again. Even though I can use it, I'm not sure if I can teach it to someone and I believe if you can't teach something properly, there might be loopholes in your understanding. Hence, I'm looking for a book that discusses various topics of arithmetic, like numbers (counting, fractions, integers, rational, real), numerals, place value number systems, addition, multiplication, division, etc very explicitly.&lt;/p&gt;&#xA;" OwnerUserId="20701" LastEditorUserId="20701" LastEditDate="2023-01-14T12:07:59.640" LastActivityDate="2023-01-16T16:10:11.673" Title="Arithmetic books for adults" Tags="|primary-education|reference-books|arithmetic|book|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="25994" PostTypeId="2" ParentId="25993" CreationDate="2023-01-14T18:08:49.870" Score="5" Body="&lt;p&gt;You might check out the &lt;a href=&quot;https://openstax.org/details/books/prealgebra-2e&quot; rel=&quot;noreferrer&quot;&gt;OpenStax Prealgebra text&lt;/a&gt;. It seems to have much of what you’re looking for.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2023-01-14T18:08:49.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25995" PostTypeId="2" ParentId="25993" CreationDate="2023-01-14T21:21:03.890" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B08P2XRCLR&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Math-Ages-SAT-High-School/dp/B08P2XRCLR/ref=sr_1_3?crid=1TWKUIXC302BL&amp;amp;keywords=high+school+maths&amp;amp;qid=1673731187&amp;amp;s=books&amp;amp;sprefix=high+school+math%2Cstripbooks-intl-ship%2C184&amp;amp;sr=1-3&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have used this book to help me understand mathematics and prepare for my exams. It is helpful for all levels and the explanations are clear and friendly.&lt;/p&gt;&#xA;&lt;p&gt;:)&lt;/p&gt;&#xA;" OwnerUserId="20864" LastActivityDate="2023-01-14T21:21:03.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25996" PostTypeId="2" ParentId="9710" CreationDate="2023-01-14T21:41:22.557" Score="0" Body="&lt;p&gt;To answer your question, no it is not wrong to suggest that they use all parentheses if that's what you prefer, but I don't think it's a good idea to be too strict about it if they find it easier to parse with some combination of parentheses and brackets. If they prefer something like &lt;span class=&quot;math-container&quot;&gt;$[\sin(x+y)+\sin(x-y)]$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\left(\sin(x+y)+\sin(x-y)\right)$&lt;/span&gt; then they're in good company. I certainly prefer the former.&lt;/p&gt;&#xA;&lt;p&gt;I can't emphasize enough how much &lt;em&gt;&lt;strong&gt;it is not a convention to only use parentheses&lt;/strong&gt;&lt;/em&gt;. I have no idea where people are getting that! I flipped through a variety of books in my house and here are some that use &lt;span class=&quot;math-container&quot;&gt;$($&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$[$&lt;/span&gt;, and in fact even some &lt;span class=&quot;math-container&quot;&gt;$\{$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;1-5) The calculus books of Stewart, Thomas/Finney, Kline, Spivak, and Openstax,&#xA;6) &lt;em&gt;Physics&lt;/em&gt; 6e, Serway/Jewett,&#xA;7) &lt;em&gt;Electricity and Magnetism&lt;/em&gt; 3e, Purcell/Morin,&#xA;8) &lt;em&gt;Classical Electrodynamics&lt;/em&gt; 3e, Jackson,&#xA;9) &lt;em&gt;Modern Classical Physics&lt;/em&gt;, Thorne/Blandford,&#xA;10) &lt;em&gt;Visual Differential Geometry and Forms&lt;/em&gt;, Needham,&#xA;11) &lt;em&gt;Applied Longitudinal Analysis&lt;/em&gt; 2e, Fitzmaurice/Laird/Ware,&#xA;12) &lt;em&gt;Advanced Engineering Mathematics&lt;/em&gt; 10e, Kreyszig,&#xA;13) &lt;em&gt;Mathematics and Its History&lt;/em&gt; 3e, Stillwell&#xA;14) Strang's &lt;em&gt;Linear Algebra and Its Applications&lt;/em&gt; 4e&lt;/p&gt;&#xA;&lt;p&gt;Only about 1 in 3 or 1 in 4 books I checked used only parentheses, so obviously it's to their benefit to at least be exposed to square brackets at some point in their education.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2023-01-14T21:41:22.557" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25997" PostTypeId="2" ParentId="10163" CreationDate="2023-01-15T01:39:22.070" Score="0" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1\times1=1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$11\times11=121$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$111\times111=12321$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1111\times1111=1234321$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$11111\times11111=123454321$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$111111\times111111=12345654321$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1111111\times1111111=1234567654321$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$11111111\times11111111=123456787654321$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$111111111\times111111111=12345678987654321$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1111111111\times1111111111=1234567900987654321$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(When later we teach the multiplication algorithm, we can try to figure out why the first nine &amp;quot;work&amp;quot;, while the tenth doesn't.)&lt;/p&gt;&#xA;" OwnerDisplayName="user18187" LastActivityDate="2023-01-15T01:39:22.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="25998" PostTypeId="1" CreationDate="2023-01-15T05:25:41.667" Score="1" ViewCount="221" Body="&lt;p&gt;Is there a high school level proof of the following?&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$a,b &amp;gt; 0$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$a/b &amp;gt; 0.$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="10432" LastActivityDate="2023-01-24T17:25:44.490" Title="A high school level proof that $a/b &gt; 0$" Tags="|proofs|" AnswerCount="2" CommentCount="3" ClosedDate="2023-01-23T15:42:11.990" ContentLicense="CC BY-SA 4.0" />
  <row Id="26000" PostTypeId="1" AcceptedAnswerId="26009" CreationDate="2023-01-15T12:58:13.513" Score="4" ViewCount="702" Body="&lt;h4&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Language_of_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peer-reviewed publications&lt;/a&gt; analogize learning math to learning a L2. But what are the DisAnalogies and CounterArguments?  How can you distinguish learning math from learning L2?&lt;/h4&gt;&#xA;&lt;p&gt;Luciana Oliveira, Marta Civil. &lt;em&gt;Teaching Mathematics to English Language Learners&lt;/em&gt; (2020), p. 75&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/books/edition/Teaching_Mathematics_to_English_Language/kTkCEAAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=math+is+a+foreign+language&amp;amp;pg=PA75&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra, first and foremost, is another foreign language and, as such, has to be understood, transcoded, and treated as one.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Linda Pound, Trisha Lee. &lt;em&gt;Teaching Mathematics Creatively&lt;/em&gt; (2021). &lt;a href=&quot;https://www.google.com/books/edition/Teaching_Mathematics_Creatively/R3g-EAAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=mathematics+%22foreign+language%22&amp;amp;pg=PT62&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;No page number listed.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;• Learning to talk mathematically has been likened to learning a foreign language (Worthington and Carruthers 2006): Lee (2006:2) confirms this analogy:&lt;/p&gt;&#xA;&lt;p&gt;For many pupils learning to use language to express mathematical ideas will be similar to learning a foreign language. Unless the pupils know about the way that language is used in&#xA;mathematics they may think that they do not understand a certain concept when what they cannot do is express the idea in language.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Paul D. Nolting, &lt;em&gt;Winning at Math&lt;/em&gt;, &lt;a href=&quot;https://www.google.com/books/edition/Winning_at_Math/ACem4L9U1L4C?hl=en&amp;amp;gbpv=1&amp;amp;dq=math+is+a+foreign+language&amp;amp;pg=PA24&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 24&lt;/a&gt;  includes a cartoon referring to Spanish, French, Latin, Chinese! &lt;a href=&quot;https://www.academicsuccessblog.com/about.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nolting holds a Ph.D. degree in Education in Curriculum Instruction from the University of South Florida. His Ph.D. dissertation was &amp;quot;The Effects of Counseling and Study Skills Training on Mathematics Academic Achievement.&amp;quot;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;Math as a Foreign Language&lt;/h3&gt;&#xA;&lt;p&gt;         Another way to understand studying math is to consider it as a foreign language. Looking at math as a foreign language can improve your study procedures. Learning how to speak math as a language is the key to math success. Currently, most universities consider computer and statistics (a form of math) courses as foreign language courses. Some universities have now gone so far as to actually classify math as foreign language. In the case of a foreign language, if you&#xA;do not practice the language, what happens? You forget it. If you do not practice math, what happens? You are likely to forget it, too. Students who excel in a foreign language must practice it every day. The same study skills apply to math, because it is considered a foreign language.&#xA;         Like a foreign language, math has unfamiliar vocabulary words or terms, which must be put in sentences called expressions or equations. Understanding and solving a math equation is similar to speaking and understanding a sentence in a foreign language. Listen to yourself or others as they explain a math equation. They are taking mathematical symbols and translating them into words and sentences.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/UX6Zb.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;In 1987, &lt;a href=&quot;https://www.cs.bham.ac.uk/%7Emmk/Teaching/MathLogic/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joseph Newmark&lt;/a&gt; and Frances Lake published &lt;a href=&quot;https://catalogue.nla.gov.au/Record/2631519&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mathematics as a Second Language (4th ed)&lt;/em&gt;.&lt;/a&gt; &lt;a href=&quot;http://webcache.googleusercontent.com/search?q=cache:JYtf9cfWUKUJ:https://www.math.csi.cuny.edu/Faculty/Newmark_/&amp;amp;hl=en&amp;amp;strip=0&amp;amp;vwsrc=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newmark received his BS and MS degrees in mathematics from Brooklyn College, and his PhD in Operations Research from New York University.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20536" LastEditorUserId="20536" LastEditDate="2023-01-16T01:01:48.843" LastActivityDate="2023-01-18T18:50:05.173" Title="How does learning math differ from learning second foreign languages (L2)?" Tags="|language-use|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26002" PostTypeId="2" ParentId="25904" CreationDate="2023-01-16T03:49:49.110" Score="0" Body="&lt;p&gt;Start with simpler examples. So, start with explaining why &lt;span class=&quot;math-container&quot;&gt;$\ 1 \times 2 = 2 \times 1$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\ 2 \times 3 = 3 \times 2$&lt;/span&gt;, etc.&lt;/p&gt;&#xA;&lt;p&gt;Next, proceed to &lt;span class=&quot;math-container&quot;&gt;$\ 1 \times 2 \times 3 = 2 \times 3 \times 1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\ 1 \times 2 \times 4 = 2 \times 4 \times 1$&lt;/span&gt;, etc.&lt;/p&gt;&#xA;&lt;p&gt;Etc.&lt;/p&gt;&#xA;&lt;p&gt;The examples you give are ridiculously complex and intimidating for a student who doesn't yet understand (or &amp;quot;believe&amp;quot;) that multiplication is commutative. But once she gets the above simple examples and understands the concept, she should be able to easily extend it to your examples.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;P.S. You repeatedly use the word &lt;em&gt;weak&lt;/em&gt; to describe your students and even mention that &amp;quot;when I was that age, I found both of those steps &amp;quot;obvious&amp;quot;&amp;quot;. Perhaps your student aren't as smart as you.&lt;/p&gt;&#xA;&lt;p&gt;But if your goal is to be an effective teacher, perhaps it's better to abandon such beliefs and negative thoughts (satisfying as they may be for your ego):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Concede that you are not a good teacher and seek to improve (rather than blame your students for their failure to learn).&lt;/li&gt;&#xA;&lt;li&gt;Believe it or not, your students probably know that you think they're stupid. This is very demoralizing and intimidating for them and makes it that much harder for them to learn (especially from you).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user18187" LastEditorDisplayName="user18187" LastEditDate="2023-01-16T04:00:30.697" LastActivityDate="2023-01-16T04:00:30.697" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26003" PostTypeId="2" ParentId="9710" CreationDate="2023-01-16T07:47:05.463" Score="2" Body="&lt;p&gt;The accepted answer claims that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Your experience at school flys in the face of convention. Teaching your students that way will not prepare them well for meeting the rest of the world&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is incorrect. Brackets and braces as second and third layers of parentheses are still widely used in &amp;quot;the rest of the world&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;For example, the &lt;a href=&quot;https://blog.apastyle.org/apastyle/2013/05/punctuation-junction-parentheses-and-brackets.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;APA Style Guide&lt;/a&gt; prescribes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;When a mathematical equation contains one level of enclosure, use parentheses, ( ); for two levels, add brackets outside, [( )]; for three levels, add curly brackets outside, {[( )]}.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Likewise, the &lt;em&gt;Chicago Manual of Style&lt;/em&gt; (2017, 7e, 12.26) prescribes&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/f9I36.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/f9I36.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;One can argue that this is illogical or not conforming with current practice of most mathematicians. But this is clearly still widely used.&lt;/p&gt;&#xA;&lt;p&gt;So my recommendation would be to &lt;strong&gt;teach both&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user18187" LastEditorDisplayName="user18187" LastEditDate="2023-01-16T07:57:40.173" LastActivityDate="2023-01-16T07:57:40.173" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26004" PostTypeId="2" ParentId="25971" CreationDate="2023-01-16T15:01:46.593" Score="4" Body="&lt;p&gt;I believe good examples for proofs by contradiction will be hard to come by. The reason is that a proof by contradiction of &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; actually proves &lt;span class=&quot;math-container&quot;&gt;$\neg \neg \phi$&lt;/span&gt;, a statement without any ``constructive content'' whatsoever. Usually, our proofs will contain some actual constructive content (even if not enough to be entirely constructive), and phrasing them as a proof-by-contradiction means we willfully ignore that information.&lt;/p&gt;&#xA;&lt;p&gt;However, a very nice example is given by this answer over at math.stackexchange.com:&#xA;&lt;a href=&quot;https://math.stackexchange.com/a/4619477/128989&quot;&gt;https://math.stackexchange.com/a/4619477/128989&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The statement is that in the decimal expansion of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; at least two digits occur infinitely often. As this is a positive statement, proving it by deriving a contradiction (via the irrationality of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;) is a genuine proof by contradiction. According to Z.A.K., we don't actually have a proof for any particular digit to occur infinitely often, so there is no &amp;quot;willful ignorance&amp;quot; going on either.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, The strategy stealing example isn't really working. What strategy stealing shows is &amp;quot;This player does not have a winning strategy&amp;quot;, and it is a proof by negation. To get to &amp;quot;The other player has a winning strategy&amp;quot; you also need that the game is determined. Whether the entire argument then is constructive depends on the proof of determinacy.&lt;/p&gt;&#xA;" OwnerUserId="13471" LastActivityDate="2023-01-16T15:01:46.593" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26005" PostTypeId="2" ParentId="25993" CreationDate="2023-01-16T16:10:11.673" Score="1" Body="&lt;p&gt;I would unironcially recommend Tao's Analysis-1. The issue of pinning down numbers as a logically consistent concept is much more complicated than one may think it to be.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2023-01-16T16:10:11.673" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26006" PostTypeId="2" ParentId="25971" CreationDate="2023-01-17T09:55:14.370" Score="3" Body="&lt;h4&gt;Refutations by contradiction (a.k.a proof of negation)&lt;/h4&gt;&#xA;&lt;p&gt;The usual proofs of the following statements are refutations by contradiction:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Euclid's Elements, &lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/elements/bookI/propI6.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book 1, Proposition 6&lt;/a&gt;: If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.&lt;/li&gt;&#xA;&lt;li&gt;Russell's paradox&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is irrational&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_by_contradiction#Infinitude_of_primes_2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Infinitude of primes&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Arguments by infinite descent, such as &amp;quot;there is no smallest positive rational&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Non-existence of the Halting oracle.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h4&gt;Proof by contradiction&lt;/h4&gt;&#xA;&lt;p&gt;The usual proofs of the following statements are proofs by contradiction&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_by_contradiction#Hilbert%27s_Nullstellensatz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert's Nullstellensatz&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_by_contradiction#Infinitude_of_primes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Infinitude of primes&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Heine-Borel compactness of &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; can be proved by contradiction, as in &lt;a href=&quot;https://planetmath.org/115compactnessoftheinterval&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theorem 11.5.11&lt;/a&gt; at PlanetMath, although it also uses some instance of excluded middle. The subsequent discussion on how to make it more constructive is also relevant.&lt;/li&gt;&#xA;&lt;li&gt;The usual proofs of &amp;quot;an epimorphism is surjective&amp;quot; proceed by contradiction: suppose &lt;span class=&quot;math-container&quot;&gt;$f : A \to B$&lt;/span&gt; is epi but not surjective, then there is &lt;span class=&quot;math-container&quot;&gt;$y \in B \setminus \mathrm{im}(f)$&lt;/span&gt;, which allows us to construct &lt;span class=&quot;math-container&quot;&gt;$g, h : B \to \{0,1\}$&lt;/span&gt; differing at &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$g \circ f = h \circ f$&lt;/span&gt;. (See &lt;a href=&quot;https://mathoverflow.net/a/178804/1176&quot;&gt;this answer&lt;/a&gt; for a constructive proof.)&lt;/li&gt;&#xA;&lt;li&gt;In a Heine-Borel compacts space &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, if a family of closed sets &lt;span class=&quot;math-container&quot;&gt;$F_i$&lt;/span&gt; has the finite intersection property (every finite intersection is inhabited) then the whole intersection is inhabited: if &lt;span class=&quot;math-container&quot;&gt;$\cap_i F_i$&lt;/span&gt; were empty, &lt;span class=&quot;math-container&quot;&gt;$X \setminus F_i$&lt;/span&gt; would form an infinite cover of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; without a finite subcover.&lt;/li&gt;&#xA;&lt;li&gt;If there is no continuous retraction from the unit disk &lt;span class=&quot;math-container&quot;&gt;$D^1$&lt;/span&gt; to the unit circle &lt;span class=&quot;math-container&quot;&gt;$S^1$&lt;/span&gt; then every continuous map &lt;span class=&quot;math-container&quot;&gt;$f : D^1 \to D^1$&lt;/span&gt; has a fixed point: suppose &lt;span class=&quot;math-container&quot;&gt;$f : D^1 \to D^1$&lt;/span&gt; were without a fixed point. Then we could define a retraction &lt;span class=&quot;math-container&quot;&gt;$D_1 \to S^1$&lt;/span&gt; by mapping &lt;span class=&quot;math-container&quot;&gt;$x \in D^1$&lt;/span&gt; to the intersection of &lt;span class=&quot;math-container&quot;&gt;$S^1$&lt;/span&gt; and the half-ray with the origin &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; passing through &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h4&gt;Excluded middle&lt;/h4&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;König's lemma: an infinite binary tree has an infinite path. Start at the root. The left subtree is either infinite or not. If it is, go left, otherwise go right. Repeat.&lt;/li&gt;&#xA;&lt;li&gt;“A subset of a finite set is finite” is equivalent to excluded middle.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cantor-Schröder-Bernstein theorem&lt;/a&gt; uses excluded middle and &lt;a href=&quot;https://arxiv.org/abs/1904.09193&quot; rel=&quot;nofollow noreferrer&quot;&gt;implies it&lt;/a&gt;. (The proof with the most obvious use of excluded middle is Tarski's proof using the fixed point theorem.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="165" LastEditorUserId="165" LastEditDate="2023-01-17T20:22:58.520" LastActivityDate="2023-01-17T20:22:58.520" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26007" PostTypeId="2" ParentId="9710" CreationDate="2023-01-17T13:51:37.757" Score="0" Body="&lt;p&gt;The use of the full range of options for enclosing sections of equations is still alive and well in the scientific world. As one recent example, the first article I pulled from the December 22 issue of &lt;a href=&quot;https://journals.aps.org/prb/abstract/10.1103/PhysRevB.106.245139&quot; rel=&quot;nofollow noreferrer&quot;&gt;Physical Review&lt;/a&gt; contains the following equations:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/W6LYa.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/W6LYa.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The hierarchy is clearly shown. There is a reason they were put into TeX by Knuth eons ago.&lt;/p&gt;&#xA;" OwnerUserId="17543" LastActivityDate="2023-01-17T13:51:37.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26008" PostTypeId="2" ParentId="25904" CreationDate="2023-01-18T06:46:23.863" Score="1" Body="&lt;p&gt;Exercise: Following are several products of the same 3 numbers, but in different orders. Calculate these products using a calculator.&lt;/p&gt;&#xA;&lt;p&gt;14 x 27 x 38&lt;/p&gt;&#xA;&lt;p&gt;14 x 38 x 27&lt;/p&gt;&#xA;&lt;p&gt;27 x 14 x 38&lt;/p&gt;&#xA;&lt;p&gt;27 x 38 x 14&lt;/p&gt;&#xA;&lt;p&gt;38 x 14 x 27&lt;/p&gt;&#xA;&lt;p&gt;38 x 27 x 14&lt;/p&gt;&#xA;&lt;p&gt;What do you notice? Pick three other numbers to test your conjecture.&lt;/p&gt;&#xA;&lt;p&gt;Does this work with addition?&lt;/p&gt;&#xA;&lt;p&gt;Does it work with subtraction?&lt;/p&gt;&#xA;" OwnerUserId="6103" LastActivityDate="2023-01-18T06:46:23.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26009" PostTypeId="2" ParentId="26000" CreationDate="2023-01-18T18:50:05.173" Score="5" Body="&lt;p&gt;That not regularly practicing a language or math makes you forget them is a pretty weak reason for suggesting learning math is like learning a foreign language. The same feature (namely no practice leads to forgetting) is true for learning how to play music well, making good basketball shots, solving a Rubik's cube very fast, and so on.&lt;/p&gt;&#xA;&lt;p&gt;I like what Feynman once said about the comparison between math and speaking a  language (&amp;quot;The Character of Physical Law&amp;quot;, pp. 34--35):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You might say, &amp;quot;All right, then if there is no explanation of the law, at least tell me what the law is. Why not tell me in words instead of symbols? Mathematics is just a language, and I want to be able to translate the language.&amp;quot; [...] But I do not think it is possible, because mathematics is &lt;em&gt;not&lt;/em&gt; just another language. Mathematics is a language plus reasoning; it is like a language plus logic. [...] The apparent enormous complexities of nature, with all its funny laws and rules, each of which has been carefully explained to you, are really very closely interwoven. However, if you do not appreciate the mathematics, you cannot see, among the great variety of facts, that logic permits you to go from one to another.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="893" LastActivityDate="2023-01-18T18:50:05.173" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26010" PostTypeId="2" ParentId="1506" CreationDate="2023-01-19T18:03:26.137" Score="1" Body="&lt;p&gt;I'm curious where those &amp;quot;quoted&amp;quot; parts came from. It's not about &amp;quot;division works this way&amp;quot; or &amp;quot;logs work this way&amp;quot;, exactly. Those are definitions. When kids first learn division, they aren't ready to hear about definitions. But when we teach logarithms, I hope that most students are ready to hear about definitions (and proofs).&lt;/p&gt;&#xA;&lt;p&gt;I definitely teach students to work with logarithms in the same way they do most other algebra steps: do the same thing to both sides.&lt;/p&gt;&#xA;&lt;p&gt;I think there are some problems for which it helps to refer back to the definition. (But I'm not thinking of what they are at the moment. I will come back and add to this when I'm able to.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-01-19T18:03:26.137" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26011" PostTypeId="1" AcceptedAnswerId="26015" CreationDate="2023-01-19T18:40:26.670" Score="13" ViewCount="1482" Body="&lt;p&gt;&lt;em&gt;(Migrated from the math stack exchange, where I received an apt-seeming suggestion to pose the question here, at the math-educators stack exchange)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I introduce my young kids to basic math concepts in the hopes that my armchair enthusiasm for math will rub off on them.&lt;/p&gt;&#xA;&lt;p&gt;To this end, I've described multiplying as &amp;quot;&lt;strong&gt;a special kind of adding where you add a number &lt;em&gt;to itself&lt;/em&gt;&lt;/strong&gt; (however many times).&amp;quot;&lt;br /&gt;&#xA;In this regard, I've also described that &lt;span class=&quot;math-container&quot;&gt;$(n \times m)$&lt;/span&gt; is always the same as &lt;span class=&quot;math-container&quot;&gt;$(m \times n)$&lt;/span&gt;, and I've explained why using items arranged in rows and columns, an egg carton being an easy illustrative example.&lt;br /&gt;&#xA;Putting this all together, I've been able to explain, for example, that  &lt;span class=&quot;math-container&quot;&gt;$3 \times 4$&lt;/span&gt; would be &lt;span class=&quot;math-container&quot;&gt;$(3 + 3 + 3 + 3)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(4 + 4 + 4)$&lt;/span&gt;, i.e. &amp;quot;3 added to itself 4 times&amp;quot; or &amp;quot;4 added to itself 3 times.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I'm soon about to introduce fractions, and in my mind this is where my explanation, above, seems to falter.&lt;/p&gt;&#xA;&lt;p&gt;Though my kids haven't formally learned fractions, they understand the concept of certain special, named fractions, such as a &amp;quot;half&amp;quot;, and I can tie that to the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;&amp;quot;-style notation.&lt;br /&gt;&#xA;Therefore, I can reasonably explain that &lt;span class=&quot;math-container&quot;&gt;$(\frac{1}{2} \times 4)$&lt;/span&gt; is &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$(\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2})$&lt;/span&gt;&amp;quot;&lt;br /&gt;&#xA;I.e. this ties in nicely to my description of multiplying as &amp;quot;adding a number to itself (however many times)&amp;quot; because we're &amp;quot;adding &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; to itself &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; times.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But something falls apart when I try to circle back to the notion that &lt;span class=&quot;math-container&quot;&gt;$(n \times m)$&lt;/span&gt; is always the same as &lt;span class=&quot;math-container&quot;&gt;$(m \times n)$&lt;/span&gt;.&lt;br /&gt;&#xA;I.e. &amp;quot;adding &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; to itself &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; times&amp;quot; is comprehensible...&lt;br /&gt;&#xA;but &amp;quot;adding &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; to itself &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; a time&amp;quot; doesn't quite make sense...at least grammatically.&lt;/p&gt;&#xA;&lt;p&gt;Is my reluctance over the phrase &amp;quot;adding a number to itself a fractional number of times&amp;quot; merely a hangup over English grammar? Mathematically, is there any legitimacy to the notion of adding a number (to itself) &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; times, where &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is not a whole number?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Update: Since my question generated so many &amp;quot;off-by-one&amp;quot; responses, I thought deeper about what exactly I'd said to my kids, and I think key is my phrase &amp;quot;&lt;strong&gt;however many times&lt;/strong&gt;&amp;quot;. In my original post, I kind of hand-waved away that phrase, but during discussion with my kids I would have explained in a manner that conveyed that &amp;quot;however many times&amp;quot; is &lt;span class=&quot;math-container&quot;&gt;$(n-1)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(m-1)$&lt;/span&gt; &amp;quot;instances&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; in the &amp;quot;addition chain&amp;quot;. Still clunky, no doubt, but I wanted to clarify that I didn't mislead my kids to a state of &amp;quot;off-by-one&amp;quot; confusion. (I think both the clunky phrasing and the grammatically &amp;quot;off&amp;quot; &lt;em&gt;addition by a fractional amount&lt;/em&gt; are nicely addressed by the accepted answer)&lt;/p&gt;&#xA;" OwnerUserId="20879" LastEditorUserId="20879" LastEditDate="2023-01-25T16:30:06.143" LastActivityDate="2023-01-25T16:30:06.143" Title="Is it meaningful to add a number to itself a fractional number of times?" Tags="|fractions|arithmetic|multiplication|" AnswerCount="9" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26012" PostTypeId="2" ParentId="26011" CreationDate="2023-01-19T22:33:43.260" Score="3" Body="&lt;p&gt;I think it is important to consider your audience. Are these kids going to immediately jump to asking about &amp;quot;adding 4 to itself &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; times?&amp;quot; I'd guess not. I think this is a perfectly fine definition of multiplication for positive integers. Let them discover extensions to larger sets of numbers once they have mastered this.&lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="98" LastEditDate="2023-01-21T00:14:23.200" LastActivityDate="2023-01-21T00:14:23.200" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26013" PostTypeId="1" CreationDate="2023-01-20T00:09:10.090" Score="0" ViewCount="198" Body="&lt;p&gt;I haven't been able to find examples in real life where we have a function or sequence such that the limit goes to infinity when the independent variable goes to infinity.   The only one so far is related to the &lt;em&gt;Coastline paradox&lt;/em&gt; (&lt;a href=&quot;https://en.wikipedia.org/wiki/Coastline_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Coastline_paradox&lt;/a&gt;).  Another one that I thought is the &amp;quot;size&amp;quot; of the universe, but I am not very familiar how to even model that problem.&lt;/p&gt;&#xA;&lt;p&gt;I'd appreciate any example provided.&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="15933" LastActivityDate="2023-01-23T23:25:34.450" Title="Example of a phenomenon from real life where there is a limit going to infinity" Tags="|limits|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26014" PostTypeId="2" ParentId="26013" CreationDate="2023-01-20T10:56:49.130" Score="2" Body="&lt;p&gt;What about the infinite summing of some fractions?&lt;/p&gt;&#xA;&lt;p&gt;This one is well-known for &lt;strong&gt;not&lt;/strong&gt; becoming infinitely large:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + ...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;~Proof:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7abzl.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7abzl.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This one is well-known for becoming infinitely large:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ...$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-01-20T10:56:49.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26015" PostTypeId="2" ParentId="26011" CreationDate="2023-01-20T20:43:21.410" Score="23" Body="&lt;p&gt;For the product &lt;span class=&quot;math-container&quot;&gt;$a\times b$&lt;/span&gt;, I intentionally &lt;em&gt;don't&lt;/em&gt; use the phrase &amp;quot;add &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; &lt;strong&gt;to itself&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; times&amp;quot;, but rather I prefer something like &amp;quot;start with zero and add &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; (copies) of the number &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&amp;quot;. This makes sense for &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}\times 4$&lt;/span&gt;, since we start with zero and write four copies of the number &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;, giving us  &lt;span class=&quot;math-container&quot;&gt;$0+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=2$&lt;/span&gt;. Similarly, it makes sense for &lt;span class=&quot;math-container&quot;&gt;$4\times\frac{1}{2}$&lt;/span&gt;, since we start with zero and add one half (copy) of the number &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, giving us &lt;span class=&quot;math-container&quot;&gt;$0+2=2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, my &amp;quot;solution&amp;quot; for your issue is to just phrase what is happening a bit differently. Instead of saying we are adding something &lt;em&gt;to itself&lt;/em&gt; a fractional number of times, indicating that the second factor counts the number of times the addition happens, I use that second factor to count the quantity of the first factor we are adding. It's subtle, but I've found it helps arithmetic students.&lt;/p&gt;&#xA;&lt;p&gt;[Note: I started doing this when teaching positive integer exponents, when it seemed that some students were misapplying the interpretation of &lt;span class=&quot;math-container&quot;&gt;$9^1$&lt;/span&gt; as &amp;quot;multiply 9 by itself 1 time&amp;quot;, thinking this could mean &lt;span class=&quot;math-container&quot;&gt;$9\times 9$&lt;/span&gt; (since we have, indeed, multiplied 9 &lt;strong&gt;by itself&lt;/strong&gt; just once). I began explaining &lt;span class=&quot;math-container&quot;&gt;$9^n$&lt;/span&gt; along the lines of &amp;quot;start with 1, and multiply &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; factors of &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;. This has the added benefit of providing an explanation of how &lt;span class=&quot;math-container&quot;&gt;$9^0$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. Of course, we later just use exponent properties.&lt;/p&gt;&#xA;&lt;p&gt;On the note of fractional exponents (which could come next in the conversation), while I've never actually said this, I suppose &amp;quot;starting with 1 and multiplying a half of a factor of &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; could be justified for &lt;span class=&quot;math-container&quot;&gt;$9^{\frac{1}{2}}$&lt;/span&gt;, though I don't think I'd want to have to also invent a convenient way to explain &amp;quot;half of a factor&amp;quot; as &amp;quot;the square root&amp;quot;.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2023-01-20T20:43:21.410" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26017" PostTypeId="2" ParentId="26011" CreationDate="2023-01-20T21:16:57.857" Score="7" Body="&lt;p&gt;The Common Core State Standards &amp;quot;definition&amp;quot; of multiplication is that &lt;span class=&quot;math-container&quot;&gt;$A \times B$&lt;/span&gt; represents the number of units when you have &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; groups, each &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; group containing &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; units.&lt;/p&gt;&#xA;&lt;p&gt;For instance &lt;span class=&quot;math-container&quot;&gt;$4 \times \frac{1}{2}$&lt;/span&gt; could represent the number of pounds of flour you get from &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; bags when each &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; bag contains &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; of a pound.  Note that this is &amp;quot;backwards&amp;quot; from the way you are interpreting multiplication.  Now I am aware that multiplication is commutative, but this is not obvious from either your definition or the CCSS definition so it needs to be argued.&lt;/p&gt;&#xA;&lt;p&gt;Similarly &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2} \times 4$&lt;/span&gt; could represent the number of pounds of flour you get from &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; of a bag when each &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; bag contains &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; pounds.&lt;/p&gt;&#xA;&lt;p&gt;The CCSS adopted this definition (rather than a &amp;quot;repeated addition&amp;quot; perspective) exactly to solve this issue:  it provides a unified mental model which applies equally well across all number system representations.  We can equally well make sense of &lt;span class=&quot;math-container&quot;&gt;$\frac{5}{4} \times 1.7$&lt;/span&gt;:  If each bag of flour has 1.7 pounds of flour, how many pounds of flour would be in &lt;span class=&quot;math-container&quot;&gt;$\frac{5}{4}$&lt;/span&gt; of a bag?&lt;/p&gt;&#xA;&lt;p&gt;I will make two more observations:&lt;/p&gt;&#xA;&lt;p&gt;(1)  The &amp;quot;repeated addition&amp;quot; perspective is really limited to integers.  The &amp;quot;proof&amp;quot; would be something like &lt;span class=&quot;math-container&quot;&gt;$A \times B = A \times (1 + 1 + 1 + ... + 1) = A +A+A + ... + A$&lt;/span&gt;.  In other words, multiplication by an integer &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is related to repeated addition because of the distributive property.&lt;/p&gt;&#xA;&lt;p&gt;(2) The CCSS definition is very much related to unit conversion and measurement.  One could say that &lt;span class=&quot;math-container&quot;&gt;$A\times B$&lt;/span&gt; represents the following situation:  you measure an object using &lt;span class=&quot;math-container&quot;&gt;$\textrm{unit}_1$&lt;/span&gt; find that the object is &lt;span class=&quot;math-container&quot;&gt;$A \textrm{ unit}_1$&lt;/span&gt; .  When you measure &lt;span class=&quot;math-container&quot;&gt;$\textrm{unit}_1$&lt;/span&gt; using &lt;span class=&quot;math-container&quot;&gt;$\textrm{unit}_2$&lt;/span&gt; you find that each &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\textrm{unit}_1$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$B \textrm{ unit}_2$&lt;/span&gt;.  Then &lt;span class=&quot;math-container&quot;&gt;$A \times B$&lt;/span&gt; is defined to be the number of &lt;span class=&quot;math-container&quot;&gt;$\textrm{unit}_2$&lt;/span&gt; you would obtain when measuring the same object.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2023-01-21T12:51:59.587" LastActivityDate="2023-01-21T12:51:59.587" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26018" PostTypeId="2" ParentId="26011" CreationDate="2023-01-20T22:22:58.217" Score="11" Body="&lt;p&gt;Frame challenge: I think your verbiage &amp;quot;adding (whole number) to itself (whole number) times&amp;quot; is misleading and incorrect and exhibits an off-by-one error. Think about the example &lt;span class=&quot;math-container&quot;&gt;$(4×1)$&lt;/span&gt;. Would you call that &amp;quot;adding 4 to itself 1 time&amp;quot;? No, because you're not doing any addition at all. The result of &amp;quot;adding 4 to itself 1 time&amp;quot; should be the result of one addition, i.e. &lt;span class=&quot;math-container&quot;&gt;$(4+4)$&lt;/span&gt;, equivalent to &lt;span class=&quot;math-container&quot;&gt;$(4×2)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3593" LastActivityDate="2023-01-20T22:22:58.217" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26019" PostTypeId="1" CreationDate="2023-01-20T23:49:25.007" Score="3" ViewCount="238" Body="&lt;p&gt;This is a little bit of a niche topic.&lt;/p&gt;&#xA;&lt;p&gt;I've dealt with a pretty bad dose of long COVID that has caused some serious gaps in my mathematics (basically causing terrible arithmetic skills and a really shaky foundation). Here's essentially what I'm dealing with. I've seen the mathematics in old courses (Ross' Analysis, etc.) before but my memory is shaky and when I go to do problems I know how to start but my newfound significant lack of algebra,trigonometry,etc. hinders me from properly completing the proof. Therefore, I know I need to go back and review from the ground up before relearning the necessary proof based math foundations like Abstract Algebra, Analysis, Topology, etc.&lt;/p&gt;&#xA;&lt;p&gt;I am debating whether to go down the path of using the Beast Academy and Art of Problem Solving books to do so or going through books used in Developmental Mathematics courses and then following the &amp;quot;conventional&amp;quot; community college/university path that leads to Stewart's Calculus, etc.&lt;/p&gt;&#xA;&lt;p&gt;For someone wishing to go back to grad school in Math, which route would you take and why?&lt;/p&gt;&#xA;" OwnerUserId="9968" LastEditorUserId="5367" LastEditDate="2023-01-30T13:37:34.833" LastActivityDate="2023-08-15T17:40:46.787" Title="Relearning math after long COVID using AoPS or developmental math textbooks?" Tags="|undergraduate-education|algebra|self-learning|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26020" PostTypeId="2" ParentId="25969" CreationDate="2023-01-21T12:42:27.560" Score="3" Body="&lt;p&gt;My main point of advice (based on teaching maths in engineering contexts, either at or post-high school) is to introduce these new letters properly, rather than try to glaze over them.&lt;/p&gt;&#xA;&lt;p&gt;Take for example the equation for resistance, R=ρL/A. As well as defining what each symbol represents (e.g. ρ is resistivity), I name the greek letter verbally, and &lt;strong&gt;write its name in Latin letters alongside the equation&lt;/strong&gt; (&amp;quot;rho&amp;quot;) when I first introduce it. I typically mention that it's a Greek letter, and if there's an alliterative link that can add meaning or aid recall (such as ρ -&amp;gt; r -&amp;gt; resistivity) I'll highlight that too.&lt;/p&gt;&#xA;&lt;p&gt;Students become familiar with the new symbol, fairly quickly but don't remember its name immediately after hearing it spoken once; the ability to look back at their notes and read the name helps a lot in the early sessions. I continue to refer to the symbol by name to aid recall, but don't write out &amp;quot;rho&amp;quot; again after the first introduction.&lt;/p&gt;&#xA;" OwnerUserId="20882" LastActivityDate="2023-01-21T12:42:27.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26021" PostTypeId="2" ParentId="26011" CreationDate="2023-01-21T16:50:47.507" Score="2" Body="&lt;p&gt;You can try going back to the way you may have first taught them the concept of addition: combining collections of things -- &amp;quot;if you have 3 apples, and I give you 4 apples, how many apples do you have?&amp;quot;. So 4 means a collection like&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;X X X X&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;When you multiply, you duplicate it that many times. So 4x3 looks like:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;X X X X&#xA;X X X X&#xA;X X X X&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Fractions are similar, but you instead of duplicating the whole collection, you stop short. 1/2 means you stop half-way through the collection.&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;X X&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;This allows them to understand mixed fractions like 3 1/2. This has 3 complete repetitions, followed by a partial repetition.&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;X X X X&#xA;X X X X&#xA;X X X X&#xA;X X&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="10551" LastActivityDate="2023-01-21T16:50:47.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26022" PostTypeId="2" ParentId="26019" CreationDate="2023-01-21T20:13:51.203" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If you want to go down this path:  I would go with a review text instead, like &lt;strong&gt;Frank Ayres First Year College Math&lt;/strong&gt;. (Schaum's Outline, I'm familiar with the first edition, available for cheaps on the Intertubes.)  I have a strong feeling that if you work all the problems--you do know you need to work problems, right?--you will get enough review of algebra 1/2 (US meaning), Euclidean geometry, pre-calc, trig, analytic geometry.  In the process, you should get enough basic repetition of arithmetic also (especially fractions).  I think reviewing long division and percentages from the ground up, would be a waste of time.  &lt;em&gt;I think doing an entire curriculum like AOPS would be overkills and not efficient as it is designed for new learners.&lt;/em&gt;  Bright ones, but still new learners.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As far as your other implied question, &amp;quot;I am debating&amp;quot;, my advice would be NOT to do the math grad school. It is a much better degree for the top performers, but has very low economic return for the lower end.  I fear that you would be avoiding moving on with your life.  In comp sci or business or some other field, you'd be quite competitive.&lt;/p&gt;&#xA;&lt;p&gt;[This is only partially to do with the Covid...I completely believe you on the brain fog and the like.  Had a buddy who dealt with it.  And have had my own issues with POTS.  But researching your previous questions, I'm not seeing a heavy hitter even before Covid.  And what if the brain fog or some fraction of it is still there? It's like walking into a fight with one arm tied behind your back. It will still hinder you.  And were you even really that competitive beforehand?]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Good luck.  #2 is not to discourage you.  Is my honest, Bayesian assessment.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20884" LastEditorUserId="20884" LastEditDate="2023-01-21T20:21:25.107" LastActivityDate="2023-01-21T20:21:25.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26023" PostTypeId="2" ParentId="1506" CreationDate="2023-01-22T05:14:17.960" Score="4" Body="&lt;p&gt;Being able to work out how to convert between exponential and logarithmic equations can practice a really essential algebra skill: what to do if you &lt;em&gt;almost&lt;/em&gt; remember a rule.&lt;/p&gt;&#xA;&lt;p&gt;I instruct students who forget how to convert between exponential and logarithmic equations to simply remember one true logarithm fact. (This is aided by the fact that I reviewed logs base 10 previously, so they can remember by counting zeroes that &lt;span class=&quot;math-container&quot;&gt;$\log_{10}(1000) = 3$&lt;/span&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;They then write the fact down and the corresponding true exponential fact &lt;span class=&quot;math-container&quot;&gt;$10^3 = 1000$&lt;/span&gt;, then replace all the numbers with letters to get a conversion formula &lt;span class=&quot;math-container&quot;&gt;$\log_b(c) = a \iff b^a=c$&lt;/span&gt;. Then they plug in whatever they want for &lt;span class=&quot;math-container&quot;&gt;$a,b,c$&lt;/span&gt; to get a new true equation.&lt;/p&gt;&#xA;&lt;p&gt;While this trick won't let students generate new algebra knowledge, and requires a bit of care to remember a good example (&lt;span class=&quot;math-container&quot;&gt;$100 \cdot 100 = 10,000$&lt;/span&gt; won't help you remember if the rule is &lt;span class=&quot;math-container&quot;&gt;$x^a \cdot x^b = x^{a+b}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$=x^{ab}$&lt;/span&gt;), it can help them prepare in advance to rederive rules from class, or identify if rules are wrong before using them. (I'd love to have a grading system that rewards students if they:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Identify that they have misremembered a rule&lt;/li&gt;&#xA;&lt;li&gt;Demonstrate by plugging in an example that the remembered the rule wrong&lt;/li&gt;&#xA;&lt;li&gt;Solve the problem using the wrong rule as if it were right&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;)&lt;/p&gt;&#xA;&lt;p&gt;That being said, I then also show students the &amp;quot;take logs or exponentials of both sides&amp;quot; approach to converting log and exponential equations, which I find a more appealing approach, since it gets students thinking that taking logs or exponentials of both sides of an equation is a helpful thing to do.&lt;/p&gt;&#xA;&lt;p&gt;A side note about teaching logarithms:&lt;/p&gt;&#xA;&lt;p&gt;My understanding (and please correct me if I'm wrong) is that it used to be that students would encounter logarithms earlier as tools for multiplication in arithmetic, but students nowadays may be seeing logarithms for the first time in an algebra (College Algebra) class.&lt;/p&gt;&#xA;&lt;p&gt;It's a lot to throw at students all at once:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Function notation&lt;/li&gt;&#xA;&lt;li&gt;Subscripts&lt;/li&gt;&#xA;&lt;li&gt;Exponent rules&lt;/li&gt;&#xA;&lt;li&gt;Functions&lt;/li&gt;&#xA;&lt;li&gt;Inverse functions&lt;/li&gt;&#xA;&lt;li&gt;A brand new arithmetic operation they don't have an algorithm for computing (they don't have an algorithm for square root either, just intuition, so this can be a good way to explain how to think about logs)&lt;/li&gt;&#xA;&lt;li&gt;Applications of logarithms&lt;/li&gt;&#xA;&lt;li&gt;Solving logarithmic/exponential equations&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So I suggest spreading these topics out over the semester/between semesters and not trying to cover all of them in three class meetings.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2023-01-22T05:14:17.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26024" PostTypeId="2" ParentId="26011" CreationDate="2023-01-22T05:32:19.383" Score="2" Body="&lt;p&gt;Go back to your egg-carton analogy. Then, move it to the area of a rectangle as number of unit squares Cutting the rectangle in half, either horizontally or vertically, produces the same answer.&lt;/p&gt;&#xA;" OwnerUserId="20885" LastActivityDate="2023-01-22T05:32:19.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26025" PostTypeId="2" ParentId="25998" CreationDate="2023-01-22T05:43:32.677" Score="1" Body="&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$b&amp;gt;0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{b}&amp;gt;0$&lt;/span&gt;. Product of two positives is positive.&lt;/p&gt;&#xA;" OwnerUserId="20885" LastEditorUserId="8190" LastEditDate="2023-01-24T17:25:44.490" LastActivityDate="2023-01-24T17:25:44.490" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26026" PostTypeId="2" ParentId="26011" CreationDate="2023-01-22T08:23:50.313" Score="1" Body="&lt;p&gt;Half &lt;strong&gt;times&lt;/strong&gt; something is the same as half &lt;strong&gt;of&lt;/strong&gt; it. When students are used to that, and also used to the distributive law, then you can put them together like &lt;span class=&quot;math-container&quot;&gt;$2\frac{1}{2}\times a$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$a + a + \frac{1}{2} \times a$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="17542" LastActivityDate="2023-01-22T08:23:50.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26027" PostTypeId="2" ParentId="26011" CreationDate="2023-01-22T16:05:43.110" Score="2" Body="&lt;p&gt;I think you have to teach progressively.  Even with bright kids, advanced kids.  Even when not really establishing mastery, but giving exposure.  So, let the kids have more exposure to fractions themselves, writing, rationalizing, adding subtracting, etc. before worrying about this conundrum.&lt;/p&gt;&#xA;&lt;p&gt;So instead of concerning this conundrum and how to contort yourself to push it with a kid that is very young, I would concern yourself more with getting other parts of their number sense mastered.  And mastered does NOT mean &amp;quot;heard the rule&amp;quot;, it means going for walks and asking them little (easy) questions like 5+8, 13+ 7.  (Or whatever level they are on...I ask an 11 year old relative of mine, first degree in x questions as we walk/scooter.  Kids enjoy hanging with you and showing off a little with stuff of a little difficulty, but still within their grasp.)&lt;/p&gt;&#xA;&lt;p&gt;Also, it is very natural to think of half times four as 1/2 * 4. Once they are more familiar with the &amp;quot;easy way&amp;quot; (fraction first), you can introduce the reverse.  Hopefully after they've also had experience with 3 * 2 = 2 * 3 from extensive single digit quizzes, learning the times table, and even hearing the rule, etc.&lt;/p&gt;&#xA;&lt;p&gt;I would also do some writing of the numbers.  Both verbal drill and written drill are helpful for learning things.  Multiple modes...&lt;/p&gt;&#xA;" OwnerUserId="20890" LastEditorUserId="20890" LastEditDate="2023-01-22T16:06:52.393" LastActivityDate="2023-01-22T16:06:52.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26028" PostTypeId="2" ParentId="26011" CreationDate="2023-01-22T20:50:05.353" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Putting this all together, I've been able to explain, for example,&#xA;that 3×4 would be (3+3+3+3) or (4+4+4), i.e. &amp;quot;3 added to itself 4&#xA;times&amp;quot; or &amp;quot;4 added to itself 3 times.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No, 3 added to itself 4 times is something like: 3+3, 3+3, 3+3, 3+3. To describe 4 times 3, you should say: four threes all added up. Then there is no problem explaining half times 4, since it is just half a four (all added up).&lt;/p&gt;&#xA;" OwnerUserId="9295" LastActivityDate="2023-01-22T20:50:05.353" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26029" PostTypeId="2" ParentId="25998" CreationDate="2023-01-22T21:57:38.070" Score="3" Body="&lt;p&gt;Suppose to the contrary that &lt;span class=&quot;math-container&quot;&gt;$\frac ab$&lt;/span&gt; were negative. Then &lt;span class=&quot;math-container&quot;&gt;$b \cdot \frac ab$&lt;/span&gt; would be the product of a positive and a negative number, which would be negative.  But we know that &lt;span class=&quot;math-container&quot;&gt;$b \cdot \frac ab = a$&lt;/span&gt;, which is positive, so this can't be right.  Similarly, if &lt;span class=&quot;math-container&quot;&gt;$\frac ab$&lt;/span&gt; were zero, then &lt;span class=&quot;math-container&quot;&gt;$b \cdot \frac ab = a$&lt;/span&gt; would have to be zero as well, which isn't true.  So the only possibility left is that &lt;span class=&quot;math-container&quot;&gt;$\frac ab &amp;gt; 0$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2023-01-22T21:57:38.070" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26030" PostTypeId="2" ParentId="26019" CreationDate="2023-01-23T08:19:55.140" Score="4" Body="&lt;p&gt;If you have gaps, Khan Academy is a great way to go.  It has videos teaching topics that you've forgotten, lots of practice material, and it will keep track of what you've mastered and what you need more help with.  It will assign problems for your weaknesses.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorDisplayName="user18187" LastEditDate="2023-08-15T17:40:46.787" LastActivityDate="2023-08-15T17:40:46.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26031" PostTypeId="2" ParentId="25756" CreationDate="2023-01-23T19:21:51.087" Score="4" Body="&lt;p&gt;I would like to add my 2 cents to this question, which already has many great answers. This is rather long, sorry about that!&lt;/p&gt;&#xA;&lt;p&gt;Discovering results is hard. It's why most discoveries are labeled theorems. It requires experimentation, trial and error, perseverance over long stretches of time. Your proposed examples attempt to address some of the aspects of research in a sandbox environment: you want students to learn the ropes of discovery but with well established, routine proofs. You hope that they will develop grit that will serve them well in their careers. However, it is likely that these will have the opposite effect: it will diminish their willingness to engage meaningfully with the material, end up with a spotty understanding of the fundamentals and ultimately unmotivated to go further in their studies.&lt;/p&gt;&#xA;&lt;p&gt;To help explain why, I have broke down the answer in multiple parts.&lt;/p&gt;&#xA;&lt;h3&gt;Cognitive load and memory&lt;/h3&gt;&#xA;&lt;p&gt;Mathematics is cognitively demanding. It is strenuous to do mathematics at the edge of your skill level, i.e. stretching the limits of your current abilities. Working at this level implies high &lt;a href=&quot;https://en.wikipedia.org/wiki/Cognitive_load&quot; rel=&quot;nofollow noreferrer&quot;&gt;cognitive load&lt;/a&gt;, requires full attention and complete focus. When someone is learning new mathematics, they already are exercising grit. They are already discovering the results for themselves, because it is new to them and their mental schemas.&lt;/p&gt;&#xA;&lt;p&gt;Cognitive load is the mental workload you have to deal with when learning. It is split into two categories: intrinsic and extrinsic cognitive load. Intrinsic cognitive load is the work necessary to understand the material and directly contributes to your learning. Extrinsic cognitive load is useless work that does not contribute and instead actively hinders learning. The objective is to maximize intrinsic and minimize extrinsic cognitive load. It is tempting to think that this approach will do that, but I believe it does the opposite.&lt;/p&gt;&#xA;&lt;p&gt;Research shows that we have a very small &lt;a href=&quot;https://en.wikipedia.org/wiki/Working_memory&quot; rel=&quot;nofollow noreferrer&quot;&gt;working memory (WM)&lt;/a&gt;, which holds information about what we are doing. Usually this means 3 to 4 unrelated items at a time. It is no surprise that mathematics easily fills one's working memory and becomes taxing to learn. There's a catch: you can bypass the limits of working memory by using knowledge and schemas stored in your &lt;a href=&quot;https://en.wikipedia.org/wiki/Long-term_memory&quot; rel=&quot;nofollow noreferrer&quot;&gt;long-term memory (LTM)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Someone learning new mathematics does not have knowledge and schemas stored in their LTM related to the new area. Therefore, they will struggle to incorporate important details in long-term memory by themselves, let alone think creatively and experiment to discover the established results you want them to.&lt;/p&gt;&#xA;&lt;h3&gt;Flow state and success rate&lt;/h3&gt;&#xA;&lt;p&gt;As &lt;a href=&quot;https://matheducators.stackexchange.com/a/25835/47&quot;&gt;Jason Siefken&lt;/a&gt; has pointed out, it's important to make a distinction between a reference and a textbook. In a textbook you want novices, with respect to the subject, to be able to confidently apply definitions, lemmas, theorems, know established examples and counter-examples, reason in terms of the new structure and understand how it fits into an overall big picture. It does not mean working at the knowledge frontier of that particular area.&lt;/p&gt;&#xA;&lt;p&gt;With this in mind, I feel it's important to help learners achieve a &lt;a href=&quot;https://en.wikipedia.org/wiki/Flow_(psychology)#Characteristics&quot; rel=&quot;nofollow noreferrer&quot;&gt;flow state&lt;/a&gt;, a concept created by &lt;a href=&quot;https://en.wikipedia.org/wiki/Mihaly_Csikszentmihalyi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mihaly Csikszentmihalyi&lt;/a&gt; where their perceived challenge is high but so is their relative skill level, in a way that they have relatively high success rate but not perfect.&lt;/p&gt;&#xA;&lt;p&gt;Scott H. Young has a good article discussing what he calls the &lt;a href=&quot;https://medium.com/mind-cafe/the-85-rule-for-learning-53aed2e40d60&quot; rel=&quot;nofollow noreferrer&quot;&gt;85% Rule for Learning&lt;/a&gt;. From his article:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you’re always successful, it’s hard to know what to improve. If you constantly fail, you won’t learn what works. Only when we have a mixture of success and failure can we draw a contrast between good and bad strategies. [...] The research also suggests that the optimal success rate for fostering student achievement appears to be about 85 percent. A success rate of 85 percent shows that students are learning the material, and it also shows that the students are challenged.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Your proposed changes will turn out to have much lower success rates I assume, drawing from both personal experience learning mathematics and as a teacher.&lt;/p&gt;&#xA;&lt;h3&gt;Thinking mathematically and problem solving&lt;/h3&gt;&#xA;&lt;p&gt;In the book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0273728911&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thinking Mathematically&lt;/a&gt;, the authors distinguish three phases into the work of dealing with a problem:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Entry phase,&lt;/li&gt;&#xA;&lt;li&gt;Attack phase,&lt;/li&gt;&#xA;&lt;li&gt;Review phase.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The entry phase is for answering questions like &amp;quot;what do I know&amp;quot;, &amp;quot;what do I want&amp;quot; and &amp;quot;what can I introduce&amp;quot;. As &lt;a href=&quot;https://matheducators.stackexchange.com/a/25771/47&quot;&gt;Kostya_I&lt;/a&gt; mentioned, these are in line with Bloom's taxonomy first two tiers (remember, understand). This is where the learner should be occupied understanding what is being asked, what are the nuances of the problem, etc.&lt;/p&gt;&#xA;&lt;p&gt;The attack phase is when you start to conjecture possible ways to solve your problem, together with justifying and convincing yourself that the steps you followed are correct beyond doubt. This is where you construct examples, specialize cases, think of lemmas, try to generalize from concrete forms, etc. This phase should correspond to the next two tiers in Bloom's taxonomy (apply, analyze).&lt;/p&gt;&#xA;&lt;p&gt;Last but not least, the review phase is where you check your resolution, reflect on key ideas and key moments, and try to extend these to wider contexts. These are the last two tiers in Bloom's taxonomy (evaluate, create), where you can create new problems, solidify understandings, build bridges between concepts.&lt;/p&gt;&#xA;&lt;p&gt;My point in elaborating all of this is that this is difficult enough when you have clear start and endpoints, let alone when you have no idea what should be of interest and what's not. Naturally, research begins like that: you have muddled understanding of something and you want to make headway in a challenging environment. But this is not the point when you simply want students to learn and internalize well-known mathematics that should be foundational.&lt;/p&gt;&#xA;&lt;p&gt;I also recommend &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1493305999&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alan Schoenfeld's 1985 book, Mathematical Problem Solving&lt;/a&gt;. Amongst other things, he discusses how knowledge impacts problem solving by analyzing verbal data together with sketches from novices, semi-novices, more experienced problem solvers and experts at fairly elementary tasks. He tracks time spent in six episodes, or stages, labeled &lt;em&gt;read, analyze, explore, plan, implement, verify&lt;/em&gt;. The data is very interesting and shows how differently they handle uncertainty in problem solving.&lt;/p&gt;&#xA;&lt;h3&gt;Suggestions&lt;/h3&gt;&#xA;&lt;p&gt;I believe the best way I've seen to introduce this sort of &amp;quot;result discovery for themselves&amp;quot; is done in both volumes of &lt;em&gt;Multidimensional Real Analysis&lt;/em&gt; by authors J. J. Duistermaat and J. A. C. Kolk. They tend to break rather long problems into many subparts (I think I've seen up to twelve items!), each involving different concepts and fitting into a sort of chain of exercises throughout the two-volume set. They serve to both apply concepts, develop interesting results, and get your hands dirty in meaningful ways. This is one example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9aVNd.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9aVNd.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/qxFEI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qxFEI.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/UwFi7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UwFi7.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/mvMrm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mvMrm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/qvlUo.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qvlUo.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think you can more easily switch to this type of problems, which will enrich your problem sets and your students' abilities. As a bonus, they get to broaden their mathematical horizons!&lt;/p&gt;&#xA;&lt;h3&gt;References&lt;/h3&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0273728911&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Mason, Leone Burton, Kaye Stacey - Thinking Mathematically&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1493305999&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alan Schoenfeld - Mathematical Problem Solving&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;J. J. Duistermaat, J. A. C. Kolk - Multidimensional Real Analysis, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B01LZKLF7T&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volume 1&lt;/a&gt; and &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0521829259&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volume 2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://medium.com/mind-cafe/the-85-rule-for-learning-53aed2e40d60&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scott H. Young - The 85% Rule for Learning&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B00E3URF1A&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jan Plass, Roxana Moreno, et. al - Cognitive Load Theory (somewhat outdated)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://link.springer.com/article/10.1007/s10648-019-09465-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Sweller, Jeroen J. G. van Merriënboer and Fred Paas - Cognitive Architecture and Instructional Design: 20 Years Later&lt;/a&gt; - This article is the most recent I know regarding cognitive load theory from a broad perspective&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0061339202&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mihaly Csikszentmihalyi - Flow: The Psychology of Optimal Experience&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2023-01-23T19:21:51.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26033" PostTypeId="2" ParentId="26013" CreationDate="2023-01-23T23:25:34.450" Score="2" Body="&lt;p&gt;What counts and what doesn't depends a lot on how one defines &amp;quot;real&amp;quot; and &amp;quot;life&amp;quot; (for me the function &lt;span class=&quot;math-container&quot;&gt;$y(z)=z^2+1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$1/z$&lt;/span&gt; is definitely more real than Napoleon: I can play with them right here and right now and they play with me in the most surprising and unpredictable ways. Try to do it with Napoleon and you'll see that there are some difficulties, so I don't really understand why the calculus textbook talk so much about &amp;quot;real life&amp;quot; examples most of which seem hopelessly contrived to me when the mathematics itself is a vast reality with many things more interesting from the sightseeing point than pyramids of Egypt and beaches of Canary islands and (IMHO) our actual task, as math educators, is to teach the students to navigate that reality comfortably and appreciate its beauty without any props from or applications to the physical world (like when they teach poetry, they don't expect a poem to explain how to make french fries or to fix a broken bicycle chain).  But OK, let's stick to the boring anthropomorphic experience.&lt;/p&gt;&#xA;&lt;p&gt;An optimistic example: the amount of the money in a bank account that pays you some interest. You can play with various rules of paying interest here to see how fast/slow it will grow in each case. In reality most banks collapse at some point (some simultaneously with the state) and this collapse can easily happen during the lifespan of a single generation.&lt;/p&gt;&#xA;&lt;p&gt;Another example: Take a flashlight on a cloudless night and direct it straight up. Some photons will escape the Earth atmosphere, so their distance from Earth will go up without limit as time progresses.&lt;/p&gt;&#xA;&lt;p&gt;Yet another one: human greed and consumption against the technological progress. It may be not immediately clear how exactly and in what units to measure them, but the more stuff becomes available, the more is consumed and wanted. Diogenes was content with an old barrel, a nearby fountain and occasional donations of food, demonstrating that that is all that is really necessary (at least in the climate of Athens) and now (almost) everybody cannot live without the latest version of i-phone and the trend continues.&lt;/p&gt;&#xA;&lt;p&gt;Actually, any time something is (believed to be) going up without limit, most dependent variables also tend to infinity: the number of books vs. the number of writers who ever lived, the number of stars that can be seen vs. the power of our telescopes, the number of good (and also bad) ideas vs. the number of thinking people, and so on, and so forth.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-01-23T23:25:34.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26034" PostTypeId="2" ParentId="26019" CreationDate="2023-01-23T23:50:09.920" Score="4" Body="&lt;p&gt;I would like to complement existing answers by suggesting &lt;a href=&quot;https://en.wikipedia.org/wiki/Spaced_repetition&quot; rel=&quot;nofollow noreferrer&quot;&gt;spaced repetition&lt;/a&gt;. I have been applying it myself and have found myself feeling like I had newfound powers.&lt;/p&gt;&#xA;&lt;p&gt;I use a free app called &lt;a href=&quot;https://mochi.cards/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mochi&lt;/a&gt;. There is a mobile version but I use it on my notebook. You can start with simple flashcards but it can do so much more: you can use &lt;a href=&quot;https://en.wikipedia.org/wiki/Cloze_test&quot; rel=&quot;nofollow noreferrer&quot;&gt;cloze deletions (also called occlusion tests)&lt;/a&gt; on both text and images (by using the &lt;em&gt;diagram card&lt;/em&gt;). It accepts LaTeX so you can input formulas easily, both inline and as blocks.&lt;/p&gt;&#xA;&lt;p&gt;It is also a good method for learning because it keeps you actively thinking about the material in order to build your cards. Ever since I've started using this app feels like my skills have gone up drastically, merely because I've stopped struggling with a full working memory and instead began to rely on a well-honed long-term memory. I've written more extensively about working memory and long-term memory in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/25756/why-dont-exercises-ask-readers-to-discover-results-themselves-before-requiring/26031#26031&quot;&gt;this answer&lt;/a&gt;, if you're interested in learning more.&lt;/p&gt;&#xA;" OwnerUserId="47" LastActivityDate="2023-01-23T23:50:09.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26035" PostTypeId="1" CreationDate="2023-01-24T18:51:30.667" Score="1" ViewCount="109" Body="&lt;p&gt;Can you recommend one for a first or second year calculus course?&lt;/p&gt;&#xA;&lt;p&gt;Ideally the website that can plot direction fields:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;is free&lt;/li&gt;&#xA;&lt;li&gt;is 100% WYSIWYG (does not require any coding or markup or anything of the sort)&lt;/li&gt;&#xA;&lt;li&gt;does not require payment, login, download, etc.&lt;/li&gt;&#xA;&lt;li&gt;can plot a solution curve given a point&lt;/li&gt;&#xA;&lt;li&gt;can plot slopes of varying density on the plane&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="19499" LastActivityDate="2023-01-24T23:39:11.053" Title="Best Free Direction Field Plotter?" Tags="|software|vector-calculus|calculators|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26036" PostTypeId="2" ParentId="26035" CreationDate="2023-01-24T22:18:33.360" Score="3" Body="&lt;p&gt;I found a good example on Desmos: &lt;a href=&quot;https://www.desmos.com/calculator/jkjgxemmkj&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.desmos.com/calculator/jkjgxemmkj&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And another on Geogebra: &lt;a href=&quot;https://www.geogebra.org/m/W7dAdgqc&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geogebra.org/m/W7dAdgqc&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15671" LastEditorUserId="15671" LastEditDate="2023-01-24T23:39:11.053" LastActivityDate="2023-01-24T23:39:11.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26038" PostTypeId="2" ParentId="7654" CreationDate="2023-01-25T05:13:00.290" Score="1" Body="&lt;p&gt;I think its more about the complexities of equations, such as rational equations with a quadratic under a radical in the denominator where you need to complete the square to simplify....  even when you know all the basic rules things start to get a little bit more complicated when everything is thrown in together and you're trying to do something new like integrate at the same time. I thought I had a good foundation in the basic concepts but pretty quickly realized I didn't know it as well as I thought&lt;/p&gt;&#xA;" OwnerUserId="20896" LastActivityDate="2023-01-25T05:13:00.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26039" PostTypeId="1" CreationDate="2023-01-25T05:17:19.970" Score="2" ViewCount="287" Body="&lt;p&gt;The usual definition of a group action is as follows.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; be a group and &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; be a set. An action of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is defined to be a map &lt;span class=&quot;math-container&quot;&gt;$\rho:G\times A\rightarrow A$&lt;/span&gt; satisfying certain conditions.&lt;/p&gt;&#xA;&lt;p&gt;I was thinking for someone who is going to do more advanced mathematics, it is useful to know that “(almost) every structure on an object (of a category) would be just a morphism (in that category)”&lt;/p&gt;&#xA;&lt;p&gt;In this case group action &lt;span class=&quot;math-container&quot;&gt;$G\times A\rightarrow A$&lt;/span&gt; should be seen as a group homomorphism &lt;span class=&quot;math-container&quot;&gt;$G\rightarrow {\rm ISO}(A)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;${\rm ISO}(A)$&lt;/span&gt; is the group whose underlying set is the set of all bijective functions from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; where the group structure is given by composition.&lt;/p&gt;&#xA;&lt;p&gt;Any thoughts on why it is not introduced (in general) in such terms?&lt;/p&gt;&#xA;" OwnerUserId="12627" LastActivityDate="2023-01-25T14:33:12.230" Title="Introduction of group action as morphism of groups" Tags="|abstract-algebra|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26040" PostTypeId="1" CreationDate="2023-01-25T08:15:23.057" Score="3" ViewCount="353" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$&lt;/span&gt;&lt;strong&gt;Question:&lt;/strong&gt; What not-trivial mistakes do students often make when solving problems in probability theory, mathematical statistics and random processes?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Some examples of wrong solutions:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem 1:&lt;/strong&gt; Find distribution of &lt;span class=&quot;math-container&quot;&gt;$F_{\xi}(\xi)$&lt;/span&gt; for continous &lt;span class=&quot;math-container&quot;&gt;$F_{\xi}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Solution&amp;quot;: &lt;span class=&quot;math-container&quot;&gt;$F_{\xi}(\xi) = P(\xi \le \xi) = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem 2:&lt;/strong&gt; Is it possible to guess if one of a pair of random numbers is larger with probability &lt;span class=&quot;math-container&quot;&gt;${}&amp;gt; \frac{1}2$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Solution&amp;quot;. Obviously no (sometimes with some blurry reasoning, mentioning symmetry).&#xA;(if smb.is interested, see discussion, e.g., in&#xA;&lt;a href=&quot;https://mathoverflow.net/questions/9037/how-is-it-that-you-can-guess-if-one-of-a-pair-of-random-numbers-is-larger-with&quot;&gt;https://mathoverflow.net/questions/9037/how-is-it-that-you-can-guess-if-one-of-a-pair-of-random-numbers-is-larger-with&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem 3:&lt;/strong&gt; Find &lt;span class=&quot;math-container&quot;&gt;$\Var(\min(X,Y))$&lt;/span&gt; for independent &lt;span class=&quot;math-container&quot;&gt;$X,Y \sim \exp(1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Solution&amp;quot;. If &lt;span class=&quot;math-container&quot;&gt;$X \le Y$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\min(X,Y) = X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Var(\min(X,Y)) = DX = 1$&lt;/span&gt;, if  &lt;span class=&quot;math-container&quot;&gt;$X &amp;gt; Y$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\min(X,Y) = Y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Var(\min(X,Y)) = DY = 1$&lt;/span&gt;, so in any case we got &lt;span class=&quot;math-container&quot;&gt;$\Var(\min(X,Y)) = 1$&lt;/span&gt;.&#xA;A more absurd version is problem 3b: find &lt;span class=&quot;math-container&quot;&gt;$D\xi$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\xi \sim \Bern(p)$&lt;/span&gt;. &amp;quot;Solution&amp;quot;: &lt;span class=&quot;math-container&quot;&gt;$\xi$&lt;/span&gt; takes values &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, if it's equal to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\Var\xi = \Var(0)=0$&lt;/span&gt;, in other case &lt;span class=&quot;math-container&quot;&gt;$\Var\xi = \Var(1)=0$&lt;/span&gt;, hence &lt;span class=&quot;math-container&quot;&gt;$\Var(\xi)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem 4:&lt;/strong&gt; Suppose that systems of sets &lt;span class=&quot;math-container&quot;&gt;$\mathcal{A}_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathcal{A}_2$&lt;/span&gt; are independent. Are &lt;span class=&quot;math-container&quot;&gt;$\sigma(\mathcal{A}_1)$&lt;/span&gt; and   &lt;span class=&quot;math-container&quot;&gt;$\sigma(\mathcal{A}_2)$&lt;/span&gt; independent?&#xA;&amp;quot;Solution&amp;quot;. Obviously yes (sometimes with some blurry reasoning).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem 5:&lt;/strong&gt;  &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; teams were divided to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; subgroups with &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; teams in each group. What is the chance that the &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; strongest teams will play in the same subgroup?&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Solution&amp;quot;: &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt; by symmetry.&#xA;(See discussion, e.g. &lt;a href=&quot;https://math.stackexchange.com/questions/3369903/probability-problem-whats-the-chance-that-two-strongest-teams-will-play-in-the/3369914&quot;&gt;Probability problem: what's the chance that two strongest teams will play in the same subgroup.&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Problem 6:&lt;/strong&gt; &amp;quot;A patient goes to see a doctor. The doctor performs a test with 99 percent reliability—that is, 99 percent of people who are sick test positive and 99 percent of the healthy people test negative. The doctor knows that only 0.01 percent of the people in the country are sick. If the patient test is positive, what are the chances the patient is sick?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Solution&amp;quot;: 99 percent as follows immediately from the task.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Some references:&lt;/strong&gt; There are different paradoxes, such as Monty Hall problem, see, e.g. &amp;quot;&lt;a href=&quot;https://doi.org/10.1007/978-94-017-2533-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paradoxes in Probability Theory and Mathematical Statistics&lt;/a&gt;&amp;quot; by G. J. Székely.  There are some interesting examples of popular mistakes in &amp;quot;&lt;a href=&quot;https://doi.org/10.2307/749665&quot; rel=&quot;nofollow noreferrer&quot;&gt;The evolution with age of probabilistic, intuitively based misconceptions&lt;/a&gt;&amp;quot; by E. Fischbein and D. Schnarch (and references therein).&lt;/p&gt;&#xA;&lt;p&gt;Of course, there are a huge number of mistakes that, in principle, can be made, but I mean, firstly, popular ones, and secondly, it is desirable that these were errors not on the most simple topics of combinatorics and not connected with typos and so on. &lt;em&gt;What not-trivial mistakes did you see lots of times?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The question is connected with &lt;a href=&quot;https://matheducators.stackexchange.com/questions/12234/common-misconceptions-in-high-school-probability-curriculum&quot;&gt;Common misconceptions in high school probability curriculum&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Addition: uninteresting examples :&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A coin is tossed twice at random. What is the probability of getting the same face?&#xA;&amp;quot;Solution:&amp;quot; Three possible outcomes are HH, HT=TH, TT, where H = head, T = tail. So the probability is &lt;span class=&quot;math-container&quot;&gt;$\frac13$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$E\xi^2 = (E \xi)^2$&lt;/span&gt; because it's &amp;quot;obvious&amp;quot; because we are speaking about mean, and even reference: there was an equality &lt;span class=&quot;math-container&quot;&gt;$E \xi \eta = E\xi E \eta$&lt;/span&gt; in one of the properties of expectation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For independent &lt;span class=&quot;math-container&quot;&gt;$\xi$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\eta$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$\Var(\xi + \eta) = \Var(\xi) + \Var(\eta)$&lt;/span&gt; &amp;quot;hence&amp;quot;  &lt;span class=&quot;math-container&quot;&gt;$\Var(\xi - \eta) = \Var(\xi) - \Var(\eta)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Density of &lt;span class=&quot;math-container&quot;&gt;$U[0,1]$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$F' = I_{[0,1]}(x)$&lt;/span&gt; a.e., so density of &lt;span class=&quot;math-container&quot;&gt;$\Pois(\lambda)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$F'= 0$&lt;/span&gt; a.e.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20897" LastEditorUserId="372" LastEditDate="2023-02-04T20:31:07.763" LastActivityDate="2023-02-05T17:30:03.497" Title="Common mistakes in probability" Tags="|undergraduate-education|students-mistakes|graduate-education|probability|big-list|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26041" PostTypeId="2" ParentId="26039" CreationDate="2023-01-25T12:45:40.543" Score="4" Body="&lt;p&gt;Here are two reasons in favour of the definition as a map &lt;span class=&quot;math-container&quot;&gt;$G \times A \to A$&lt;/span&gt; (whether they outweigh the reasons in favour of the group homomorphism definition is of course a judgement call).&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;With this definition it is more convenient to introduce topological concepts. Specifically, if &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is a topological group and &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is a topological space, then one can easily speak of separate and/or joint continuity of the group action &lt;span class=&quot;math-container&quot;&gt;$G \times A \to A$&lt;/span&gt;. If one, instead, defines a group action as a group homomorphism &lt;span class=&quot;math-container&quot;&gt;$G \to \operatorname{ISO}(A)$&lt;/span&gt;, this become less intuitive (at least from my perspective).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One pedagogical advantage of defining group actions as mappings &lt;span class=&quot;math-container&quot;&gt;$G \times A \to A$&lt;/span&gt; that satisfy a list of axioms, is that it gives the students additional practice in deriving properties of abstract objects from a list of axioms. (Whether this is really relevant in a particular course depends, of course, on the preliminary knowledge and experience of the students.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2023-01-25T12:45:40.543" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26042" PostTypeId="2" ParentId="26039" CreationDate="2023-01-25T14:33:12.230" Score="7" Body="&lt;p&gt;I prefer just to keep the language as simple as possible and the level of abstraction as low as the circumstances allow when teaching courses for students who see the material for the first time. So in the measure theory course, say, I keep working on metric spaces when I need some topology though what I really need is some separability axiom. IMHO, working with more familiar objects in such situation facilitates understanding and reduces distractions from the main topic (after all, measure theory is &lt;em&gt;not&lt;/em&gt; about topological subtleties) and does not reduce generality or obscure &amp;quot;the view from the top&amp;quot; too much (in a sense, it does not reduce it at all: if you clearly spell out every time what properties of the continuous functions on a metric space you use for each construction, then the arguments will be easy to repeat for general topological space once those properties are derived in a non-metric setting).&lt;/p&gt;&#xA;&lt;p&gt;As to your particular example, it took me a few minutes to fully decipher and comprehend the construct &amp;quot;&lt;em&gt;a group homomorphism G→ISO(A), where ISO(A) is the group whose underlying set is the set of all bijective functions from A to A where the group structure is given by composition&lt;/em&gt;&amp;quot;. Yes, you can say things this way, but, unless you are going to discuss at least one similar but different situations when &amp;quot;group&amp;quot; is not really a group and &amp;quot;composition&amp;quot; is something else, but all interesting theorems and constructions can be transferred to the new setting with only minor modifications and look exactly the same from this higher point of view, it would just hang in the air and look to the students about as contrived as &amp;quot;the image of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; under the action of a unique linear map &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; to itself such that &lt;span class=&quot;math-container&quot;&gt;$L(1)=2$&lt;/span&gt;&amp;quot; instead of the usual &lt;span class=&quot;math-container&quot;&gt;$2\times 3$&lt;/span&gt; despite the fact that once you go from one real variable calculus to the multivariate one, that is &lt;em&gt;exactly&lt;/em&gt; the shift in the language and notation from &lt;span class=&quot;math-container&quot;&gt;$f'(x)h$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(D_xf)h$&lt;/span&gt;. Everything should be done at its proper time and for a clear (to the students) reason. So in teaching I would recommend sticking to the good good KISS principle unless you are absolutely forced to deviate from it by the circumstances.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-01-25T14:33:12.230" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26043" PostTypeId="1" AcceptedAnswerId="26044" CreationDate="2023-01-25T20:51:52.353" Score="1" ViewCount="383" Body="&lt;p&gt;What would one suggest for online math teaching to draw diagrams? Is Tikz or Matplotlib too slow for helping to visualize problems during the lesson?&lt;/p&gt;&#xA;" OwnerUserId="20899" LastEditorUserId="5321" LastEditDate="2023-01-30T09:50:11.600" LastActivityDate="2023-01-30T09:50:11.600" Title="When teaching online, how does the teacher teach visual concepts?" Tags="|graphing|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26044" PostTypeId="2" ParentId="26043" CreationDate="2023-01-25T22:03:43.907" Score="6" Body="&lt;p&gt;Depends a lot on the kind of diagrams, and on the technology you are using to teach online.&lt;/p&gt;&#xA;&lt;p&gt;When I teach online I use Zoom, and log in on two different devices: (a) a laptop, which captures my face and voice, and (b) an iPad, on which I have video and audio turned off by default, that I use for screensharing.  I use the Notability app on my iPad as a &amp;quot;virtual chalkboard&amp;quot;, and write and draw sketches on it as I would a physical chalkboard; I also will switch to other apps (e.g. Desmos or Geogebra) if I want a more precise and/or dynamic rendering of a particular diagram.  I wouldn't dream of using Tikz for real-time presentations, although I suppose it might be reasonable to prepare some diagrams in Tikz ahead of time and paste them into my Notability document during the lecture if there was not another way to create them.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2023-01-25T22:03:43.907" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26045" PostTypeId="2" ParentId="26043" CreationDate="2023-01-26T07:36:13.747" Score="1" Body="&lt;p&gt;I use Geogebra for dynamic trigonometry or functions, and when drawing something by hand or doing calculations at the side of the drawing, Zoom whiteboard is okay. The whiteboards can be saved and afterwards I share them with the students, together with the recording.&lt;/p&gt;&#xA;&lt;p&gt;I have a Microsoft tablet computer and a pen for it for the physical drawings.&lt;/p&gt;&#xA;&lt;p&gt;A document camera would be at least as good and convenient, though maybe better for lecturing than teaching forms where the students also contribute.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-01-26T07:36:13.747" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26046" PostTypeId="1" CreationDate="2023-01-26T21:27:06.027" Score="30" ViewCount="8382" Body="&lt;p&gt;In teaching, we sometimes necessarily oversimplify concepts. Terry Pratchett famously referred to this as &lt;em&gt;Lies to children&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A lie-to-children is a statement that is false, but which nevertheless leads the child’s mind towards a more accurate explanation, one that the child will only be able to appreciate if it has been primed with the lie.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In online discussions about lies to children, most examples I have come across are in the natural sciences, most frequently physics and chemistry. There are some examples on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Lie-to-children&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; page as well. Does anyone know of any good examples in mathematics and statistics, particularly at the undergraduate level or higher?&lt;/p&gt;&#xA;&lt;p&gt;I don’t mean examples where we as teachers simplify and give hand-waving arguments because the level required for a rigorous proof is far too high. What I have in mind are explanations that are completely and utterly incorrect, but somehow still helpful.&lt;/p&gt;&#xA;" OwnerUserId="20901" LastEditorUserId="75" LastEditDate="2023-01-27T14:45:10.490" LastActivityDate="2023-05-25T03:23:52.097" Title="‘Lies to children’ in mathematics and statistics education" Tags="|undergraduate-education|teaching|" AnswerCount="20" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="26047" PostTypeId="2" ParentId="7654" CreationDate="2023-01-26T22:07:55.407" Score="0" Body="&lt;p&gt;I think the main thing is a comfort with working multi-step equations.  (Doing same thing to both sides, showing your work, having sufficient care/practice to make a very low level of mistakes.)  At the simplest level, think of the steps in solving for x in first order, one unknown problems.  But this same trait, in other problems.  I would say that this can still be drilled, emphasized &amp;quot;as you go&amp;quot; in teaching calculus.  For example at a help center.  And if you are teaching weak students, SHOW EVERY STEP.  Get them to show every step. No &amp;quot;throwing something from one side to the other and flipping sign&amp;quot; but &amp;quot;subtract same from each side, showing it pedantically step by step&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Obviously there are some specific items of knowledge (e.g. exponent rules) that are important also.  It's basically &amp;quot;most of it&amp;quot; from the precursor classes, so it's almost as helpful to specify what parts of the curriculum are NOT important.  And this is not to say they shouldn't be taught.  Just for a kid, trying to progress, with some gaps, they are low bang for the buck.&lt;/p&gt;&#xA;&lt;p&gt;On the list of things that are lower priority (for weak kids, because life is prioritization, because they ain't ever getting to Rudin, but we'd like them to get their nursing degrees):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Most of geometry (side angle side, proofs, etc.).  Only mensuration area formulas are needed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Conic sections, except for the very basics of a centered circle and parabola and hyperbola on y-x plot.  But not foci and directrix and ellipses and vertical/sideways displacements.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Most of the trig identities (just sinsq plus cossq).  [If double angle or the like needed later, review it then.]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Determinants.  [Review it when needed in a calc class for weak students or even just eschew entirely...it's usually &amp;quot;enrichment&amp;quot; anyhow, even within calculus or diffy screws.]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Rotation and displacement of coordinates (analytic geometry).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;harder (often, but not always asterisked, sections of Alg 2):  synthetic division, partial fractions, induction proofs, series, etc.  For series or partial fractions, if that is part of the calculus course, make sure to cover the required algebra WHEN they do the calculus topic (and show all the steps).  But don't expect precursor competence.  Look at it more as an opportunity to brush up that topic they didn't handle/remember in 11th grade.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If you want a build up (versus tear down) approach, the list from Kahn Academy is decent:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ap-ab-about/a/ap-calc-prerequisites&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ap-ab-about/a/ap-calc-prerequisites&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Please note, I'm assuming a context of community college or other remedial work.  Not precalc at Thomas Jefferson.  They should have a tough course.  But then they don't match Brian Rushton's posited set of students struggling with standard calc 101 because of poor algebra.&lt;/p&gt;&#xA;&lt;p&gt;Also note:  don't underestimate the importance of algebra.  It's at the core of calc and diffyQs in terms of the actual working the problems.  And is required in most chem, physics, engineering classes.  Just to work problems (often not needing calculus even!)&lt;/p&gt;&#xA;&lt;p&gt;Not also that this means when you DO teach a calc class for weaker students, you should be MUCH more receptive to the &amp;quot;here's an example, here's a method, practice small variations of the algorith, test recognizable problems versus the examples&amp;quot;.  After all, even if it is &amp;quot;mechanical&amp;quot;, it gives them a golden chance to practice multistep algebra, itself.&lt;/p&gt;&#xA;" OwnerUserId="20902" LastEditorUserId="20902" LastEditDate="2023-01-26T22:41:01.857" LastActivityDate="2023-01-26T22:41:01.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26048" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T02:49:49.117" Score="15" Body="&lt;p&gt;&amp;quot;Random variable.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;...because, as we all know, a random variable is neither random nor a variable. It is a real-valued function. But if we tried to introduce the concept, the feeling, of a stochastic, uncertain, incompletely known environment using such a deterministic terminology as &amp;quot;real-valued function&amp;quot;, we would certainly fail. Hence, random variable it is.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2023-01-27T02:49:49.117" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26049" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T07:40:28.267" Score="6" Body="&lt;p&gt;I think sometimes when introducing &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; to children they are told that it's exactly &lt;span class=&quot;math-container&quot;&gt;$3.1415$&lt;/span&gt; or some other arbitrary number of decimals, or even that it equals &lt;span class=&quot;math-container&quot;&gt;$22/7$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="1281" LastActivityDate="2023-01-27T07:40:28.267" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26050" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T08:26:26.223" Score="15" Body="&lt;p&gt;I was introduced to the real numbers as &amp;quot;all the points on a [two-sided infinitely long] line&amp;quot;.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;At best this is a circular definition. It's certainly very sloppy, and those words could be used to describe many different objects.&lt;/li&gt;&#xA;&lt;li&gt;The real-world intuition is incorrect. By the time you're looking at a small enough scale for it to matter whether you've got &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;, space has stopped behaving like &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But it's intuitively obvious what it means. Eventually you'll be introduced to &amp;quot;Dedekind- or Cauchy-completion of the rationals&amp;quot; to formalise what it means for something to &lt;em&gt;be&lt;/em&gt; a &amp;quot;point on a line&amp;quot; in the intended sense.&lt;/p&gt;&#xA;" OwnerUserId="7101" LastActivityDate="2023-01-27T08:26:26.223" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26051" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T09:06:59.427" Score="9" Body="&lt;p&gt;A lot of things around the limitation and construction of number spaces come to mind such as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You cannot divide 5 by 2.&lt;br /&gt;&#xA;You cannot take the square root of a negative number.&lt;br /&gt;&#xA;You must not divide by zero.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Real numbers hold more reality than imaginary numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Imaginary numbers hold some reality because they have applications in electrical engineering, quantum physics, …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mathematicians invented complex numbers because they wanted to take the square root of negative numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(There may be some truth to this one. It’s not a good motivation though.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We use complex numbers to be able to solve more equations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Infinity is not a valid result and you cannot perform computations with it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Whereas the truth is more like:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;All number spaces (and infinity) are artificial constructs.&lt;/li&gt;&#xA;&lt;li&gt;“Advanced” number spaces (and infinity) allow us to treat certain problems more efficiently or avoid irrelevant pathologies.&lt;/li&gt;&#xA;&lt;li&gt;Taking the square root of a real number (and similar) is most often not something we do for its own sake, but something that comes up in application and is either a way to an useful answer or not.&lt;/li&gt;&#xA;&lt;li&gt;The restrictions of number spaces reflect restrictions of real-life or inner-mathematical applications and it’s our duty to apply them as appropriate.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2023-01-27T23:00:00.360" LastActivityDate="2023-01-27T23:00:00.360" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26052" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T09:12:21.807" Score="18" Body="&lt;h3&gt;The average&lt;/h3&gt;&#xA;&lt;p&gt;where the lie-to-children is the word &amp;quot;the&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Ask anyone what &amp;quot;the average&amp;quot; of a set of values is, and immediately you'll be told the arithmetic mean.  That's how it's taught initially, and that's what everyone falls back to by default.&lt;/p&gt;&#xA;&lt;p&gt;A little further on in primary school, you're taught that actually there are three kinds of averages - mean, median and mode.  But only three.&lt;/p&gt;&#xA;&lt;p&gt;Study actual statistics though, and you get onto least-squares, standard deviation and other ways to know how confident you are about your &amp;quot;average&amp;quot;, fitting to polynomials or other functions, Bayesian statistics, and so on.&lt;/p&gt;&#xA;" OwnerUserId="6456" LastActivityDate="2023-01-27T09:12:21.807" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="26053" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T09:13:41.410" Score="21" Body="&lt;p&gt;The idea that “a number” means “this decimal expansion”, rather than the expansion being a way of representing a number that has some more set-theoretic definition. It's the de facto truth for everyone in the world who isn't going into rigorous mathematics, but thinking of the decimal as “the number” and other properties as things that are true about it is the root of the classic confusion around &lt;span class=&quot;math-container&quot;&gt;$0.999... = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When I was introduced to the concepts at the undergraduate level, we couldn’t even get through the basic definitions without acknowledging that &lt;span class=&quot;math-container&quot;&gt;$0.999... = 1.000...$&lt;/span&gt; as an assumption of how the representation is constructed to allow us to prove that numbers map (almost) uniquely in both directions.&lt;/p&gt;&#xA;&lt;p&gt;This relates closely to Patrick Stevens' answer – a straightforward understanding of rational and real numbers works intuitively, but the formalism has to go the other way from the intuition.&lt;/p&gt;&#xA;" OwnerUserId="20903" LastEditorUserId="75" LastEditDate="2023-01-27T09:24:29.063" LastActivityDate="2023-01-27T09:24:29.063" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="26054" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T10:44:15.430" Score="32" Body="&lt;p&gt;We usually teach:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int\frac1xdx=\ln{|x|}+c$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Whereas it should be:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int\frac1xdx =&#xA;\begin{cases}&#xA;\ln{x}+c_1  &amp;amp; x&amp;gt;0 \\&#xA;\ln{(-x)}+c_2 &amp;amp; x&amp;lt;0&#xA;\end{cases}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Why don't we teach the correct version? Using the &amp;quot;wrong&amp;quot; version usually leads to the right final answer, whereas using the correct version would probably overburden struggling calculus students.&lt;/p&gt;&#xA;" OwnerUserId="20101" LastEditorUserId="20101" LastEditDate="2023-01-27T10:52:42.963" LastActivityDate="2023-01-27T10:52:42.963" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="26055" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T11:26:29.870" Score="41" Body="&lt;p&gt;Young children 5-8 years old, are taught to subtract the smaller number from the bigger number. They are told that you can't subtract a bigger number from a smaller number.   This lie has its advantages and helps cement the order of the numbers in a subtraction sentence. Then when students learn about negative numbers they can easily unlearn the previous &amp;quot;lie&amp;quot; that you can't subtract a smaller number from a bigger number.&lt;/p&gt;&#xA;&lt;p&gt;Of course, there is often a bright 7 year old who has been taught about negative numbers at home who objects to not being able subtract bigger numbers from smaller numbers.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2023-01-27T11:26:29.870" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="26056" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T13:22:04.663" Score="14" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\LARGE\mathbb R$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Students are introduced to real numbers long before they are ready for the formal definition. At second level they are primed for dealing with &lt;em&gt;fractions&lt;/em&gt; and &lt;em&gt;not fractions&lt;/em&gt; and told there are numbers like &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; where the decimal expansion goes on forever.&lt;/p&gt;&#xA;&lt;p&gt;The formal definition is in terms of completion of a metric space or Dedekind cuts. I was in my early twenties when I first encountered it.&lt;/p&gt;&#xA;&lt;p&gt;Interesting fact: The metric space definition is already circular if you insist on defining a metric as a function &lt;span class=&quot;math-container&quot;&gt;$d: X \times X \to \mathbb R$&lt;/span&gt;. This assumes &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; is defined and cannot be used to create the definition. Your homework is to repair this problem.&lt;/p&gt;&#xA;" OwnerUserId="20908" LastActivityDate="2023-01-27T13:22:04.663" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26057" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T13:44:09.607" Score="0" Body="&lt;p&gt;Independence in probability theory is not independence in usual sense because it does not take into account causation.&lt;/p&gt;&#xA;&lt;p&gt;Addition1: The notion of independence is well-known en.wikipedia.org/wiki/Independence_(probability_theory). For example, events &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are called independent if &lt;span class=&quot;math-container&quot;&gt;$\mathbb{P}(AB)=\mathbb{P}(A)\mathbb{P}(B)$&lt;/span&gt;. As we can see, this is not at all the same as causal independence or smth.like that, because there is not a word about causes in this definition. And if smb. speaks about independence, for example &amp;quot;Student John's grade in physical education does not depend on the size of Napoleon's army&amp;quot;, it doesn't look like he means that &lt;span class=&quot;math-container&quot;&gt;$\mathbf{P}(AB) = \mathbf{P}(A)\mathbf{P}(B)$&lt;/span&gt; for the corresponding events.&lt;/p&gt;&#xA;&lt;p&gt;Addition2: This is not the first time someone put a dislike, failing to formulate a single objection against the obviously correct thesis)&lt;/p&gt;&#xA;" OwnerUserId="20897" LastEditorUserId="20897" LastEditDate="2023-01-27T19:03:19.520" LastActivityDate="2023-01-27T19:03:19.520" CommentCount="20" ContentLicense="CC BY-SA 4.0" />
  <row Id="26058" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T16:37:47.613" Score="12" Body="&lt;p&gt;I was told in high school that Euclidean geometry can be derived from the five postulates written by Euclid, but this is not the case. Several of Euclid's proofs have holes in them and one can create models of the five axioms that do not satisfy those results (e.g., in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}^2$&lt;/span&gt; a line can be closer to the center of a circumference than the length of its radius and still not inersect the circle).&lt;/p&gt;&#xA;&lt;p&gt;A rigorous foundation of Euclidean geometry was done by &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_axioms&quot; rel=&quot;noreferrer&quot;&gt;Hilbert in 1899&lt;/a&gt; and it requires 20 axioms, which exclude the above models and make all classical theorems valid.&lt;/p&gt;&#xA;" OwnerUserId="15941" LastEditorUserId="15941" LastEditDate="2023-01-28T14:56:36.423" LastActivityDate="2023-01-28T14:56:36.423" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26061" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T20:22:05.597" Score="0" Body="&lt;p&gt;1-2-3-4-5-6-7-8-9-10... &lt;a href=&quot;https://simple.wikipedia.org/wiki/Base_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Well, not if you are using binary...&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The angles on a triangle add up to 180 degrees... &lt;a href=&quot;https://nrich.maths.org/1434&quot; rel=&quot;nofollow noreferrer&quot;&gt;Only if the triangle is on a flat surface.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/64488/is-multiplication-not-just-repeated-addition&quot;&gt;Multiplication is repeated addition.&lt;/a&gt; Have you ever multiplied, say, (-3)*(-4) and wondered about how you were adding up a negative number of times? No?&lt;/p&gt;&#xA;&lt;p&gt;Most of elementary mathematics when you get to abstract algebra, really. Adding two positive numbers always gives a positive, larger number? &lt;a href=&quot;https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nope, sorry&lt;/a&gt;, we are in Z7 today so 4+4=1. (Or we are actually adding time--if it is currently 10pm and your homework is due in 11 hours, 10+11=9am.)&lt;/p&gt;&#xA;&lt;p&gt;Subtraction is adding a negative number. Unless you are subtracting a negative number, oops.&lt;/p&gt;&#xA;&lt;p&gt;IMO math is absolutely crammed full of lies to children, to the point where any elementary math teacher who uses the words &amp;quot;always&amp;quot;, &amp;quot;never&amp;quot;, &amp;quot;every&amp;quot;, etc., is probably automatically fibbing.&lt;/p&gt;&#xA;" OwnerUserId="1908" LastEditorUserId="1908" LastEditDate="2023-01-27T20:34:07.933" LastActivityDate="2023-01-27T20:34:07.933" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26062" PostTypeId="1" CreationDate="2023-01-27T23:06:49.547" Score="9" ViewCount="2652" Body="&lt;p&gt;I recently was appointed math department chair at a small university. We have a 3 credit math for elementary teachers content course. Administration told us they will change this course into an elementary education course. Basically, it will go from being called MA XXX to ELED XXX with the elementary education department overseeing it. Are there any problems with this?&lt;/p&gt;&#xA;" OwnerUserId="20928" LastActivityDate="2023-01-30T16:27:39.217" Title="Should math for elementary teachers content be taught under the direction of the math department?" Tags="|primary-education|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26063" PostTypeId="2" ParentId="26062" CreationDate="2023-01-27T23:10:58.480" Score="4" Body="&lt;p&gt;There is no universal consensus.  Some of these courses are taught in math departments, and some of them are taught in education departments.  Do you have faculty who are experts at this content and would be disappointed if they were not able to teach them anymore?  If so, you should help to either fight against this transition, or negotiate the transition gracefully.  If you do not have faculty experts I see no harm in letting the Education department take on this duty if they want to do so.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-01-27T23:10:58.480" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26064" PostTypeId="2" ParentId="26046" CreationDate="2023-01-27T23:31:50.183" Score="6" Body="&lt;p&gt;We can &lt;strong&gt;define&lt;/strong&gt; irrational numbers to be those numbers which are not rational, and then define the real numbers to be the union of the rationals and irrationals.&lt;/p&gt;&#xA;&lt;p&gt;The problem with this definition is that it first requires use to have defined the set of real numbers.&lt;/p&gt;&#xA;&lt;p&gt;Here is a related discussion:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/11211/why-does-the-widespread-erroneous-definition-of-irrational-number-persist-with/11222#11222&quot;&gt;Why does the widespread erroneous definition of &amp;quot;irrational number&amp;quot; persist without being taught?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If we try to define irrational numbers as &amp;quot;&lt;em&gt;numbers&lt;/em&gt; that are not rational&amp;quot; then we unwittingly capture numbers like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;, quaternions, &lt;span class=&quot;math-container&quot;&gt;$\aleph_0$&lt;/span&gt;, surreal numbers, etc.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-01-27T23:31:50.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26065" PostTypeId="2" ParentId="26046" CreationDate="2023-01-28T00:34:47.530" Score="18" Body="&lt;p&gt;Whether or not the derivative &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; is a fraction. Similarly, what, exactly, are &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;This actually goes through several iterations of lies:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;We first hammer it into Calc I students that &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; is &lt;strong&gt;not&lt;/strong&gt; a fraction, but instead the limit of fractions and that we write it as a fraction for intuition. We teach that &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; should be thought of as &amp;quot;an infinitesimal change in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; and sometimes even mention that this isn't strictly accurate, but does mirror the intuition of Newton, Leibniz, and others during the historical development of infinitesimal calculus.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Somewhere in a differential equations or physics course, we teach students to manipulate &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; as a fraction. But only in special circumstances and that it's not &lt;em&gt;really&lt;/em&gt; a fraction; it just plays one on TV. All of these fraction manipulations are simply algebraic shorthand for things like change of variables in integration when solving a separable differential equation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Somewhere in a differential geometry/calculus on manifolds course we introduce the idea that &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; is, in fact, a fraction relating vectors in tangent spaces. And simultaneously introduce the idea that a derivative is a linear mapping between tangent spaces and not a fraction. And that &lt;span class=&quot;math-container&quot;&gt;$dx^i$&lt;/span&gt; is simply a map that returns the &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;-th coordinate of a vector in a tangent space.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7260" LastActivityDate="2023-01-28T00:34:47.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26067" PostTypeId="2" ParentId="26062" CreationDate="2023-01-28T01:55:47.100" Score="15" Body="&lt;p&gt;That depends on who is really going to teach those classes (real people with all their attitudes, biases, (lack of) intelligence, etc., etc.). The real problem currently seems to be that the math education the teachers usually get is of too low level and the requirements are too lax. I'm not saying that the teachers should know what some advanced math concepts are (I don't know many of those myself despite my pretending to be a mathematician and a professor for 20+ years). However they should&lt;/p&gt;&#xA;&lt;p&gt;a) To be able to solve standard problems in elementary geometry, number theory, and combinatorics.&lt;/p&gt;&#xA;&lt;p&gt;b) To be able to distinguish between a valid and an invalid mathematical argument written in free format (and between a statement and a non-statement too)&lt;/p&gt;&#xA;&lt;p&gt;c) Be comfortable with using quantifiers and negating statements like &amp;quot;for every cat, there is a mouse the cat will not eat&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;d) Like to &lt;em&gt;play&lt;/em&gt; with mathematical concepts and ideas as children play with toys and be skillful in setting up such games. (You may laugh at this requirement as much as you want, but, IMHO, nothing is more discouraging than &amp;quot;the beastly seriousness&amp;quot;. Even utter incompetence is not that detrimental to the learning process if the students are bright enough).&lt;/p&gt;&#xA;&lt;p&gt;If you have people in Math department who can achieve those goals better than the people in Education, by all means keep those classes there. If it is the other way around, pass them to Education. In any case, even if you disagree with the above list, just make your own and decide based on it. Our task, as educators, is just to provide good enough educational options for the students and to keep the standards high enough. Pride, redistribution of money between the departments, retention rates, and student or administrator happiness are not that important on the grand scale of events.&lt;/p&gt;&#xA;&lt;p&gt;Of course, this all applies if you are dealing with reasonable people (in your opinion). If you think that you will be dealing with greedy idiots, my advice is to fight &lt;em&gt;any change whatsoever&lt;/em&gt; because then it will &lt;em&gt;always&lt;/em&gt; end up being a change to the worse no matter how nice the intent is and how convincing  the arguments in favor of it are.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-01-28T01:55:47.100" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="26068" PostTypeId="2" ParentId="26062" CreationDate="2023-01-28T02:28:25.430" Score="24" Body="&lt;p&gt;It seems like a big problem to me.&lt;/p&gt;&#xA;&lt;p&gt;So many people in education are &lt;em&gt;&lt;strong&gt;scared&lt;/strong&gt;&lt;/em&gt; of math. You need a course that helps teacher candidates get over their fear. And you need someone who loves math teaching it. Will there be anyone in the education department who loves math and understands it deeply? (And there is a depth of understanding for teaching that most people are unaware of. See Liping Ma's book,  &lt;em&gt;&lt;strong&gt;Knowing and Teaching Elementary Mathematics&lt;/strong&gt;&lt;/em&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;[In Fedja's list, a to c seem too high level to me. Part d, oh yes! My own list would include really understanding place value and fractions. I taught this course long ago, and loved it.]&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-01-30T16:27:39.217" LastActivityDate="2023-01-30T16:27:39.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26069" PostTypeId="1" CreationDate="2023-01-28T03:17:22.730" Score="6" ViewCount="1263" Body="&lt;p&gt;Example: 5 x 18 = (5 x 10) + (5 x 8) instead of 5 x 10 + 5 x 8?&lt;/p&gt;&#xA;" OwnerUserId="20932" LastActivityDate="2023-01-28T06:04:57.690" Title="What is the justification to teach the (redundant) use of parentheses in multiplications?" Tags="|primary-education|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26070" PostTypeId="2" ParentId="26069" CreationDate="2023-01-28T04:26:33.760" Score="27" Body="&lt;p&gt;Early in their education (even well into learning algebra), students don't naturally see the structure within algebraic expressions. It takes a bit of mental overhead to see that 5 x 10 + 5 x 8 is the sum of two products.&lt;/p&gt;&#xA;&lt;p&gt;Presenting the expression as (5 x 10) + (5 x 8) saves students the mental overhead, letting them focus on the message that's trying to be communicated.&lt;/p&gt;&#xA;&lt;p&gt;I recently found myself adding unnecessary parentheses to first-order logical formulas in order to make them easier to read, and thus more effective at quickly communicating the idea I was trying to get across.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2023-01-28T04:26:33.760" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26072" PostTypeId="2" ParentId="26062" CreationDate="2023-01-28T08:36:28.737" Score="4" Body="&lt;p&gt;What is the purpose of the course?  Such a course could have different purposes.  Elementary teachers will be using maths in two very different ways.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Using maths in their role outside the classroom.&lt;/li&gt;&#xA;&lt;li&gt;Teaching maths to their students.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If the purpose of the course is to ensure that the future teachers can deal with the maths in the work environment of a school, they need to be able to handle data (from tests) interpret correlations, organise a calendar, work out taxes, work out the stagger on a 400m track.&lt;/p&gt;&#xA;&lt;p&gt;Such a course could work in either department. The level is not particularly high. It may be better within the educational dept, since the maths isn't really college level. Educationalists may bring genuine experience. &amp;quot;Maths for plumbers&amp;quot; may be better taught by a successful plumber who understands the maths that plumbers need.  The same is true for teachers.&lt;/p&gt;&#xA;&lt;p&gt;If the course is about the maths that they will teach, then the content would be very different. It might be:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Remedial.  Some might need grade-school level teaching because they can't,(for example) add two fractions together. Or (as suggested in comments) that &amp;quot;dividing by zero equals one&amp;quot;. Teachers do need to to have the skills that they will teach to children.&lt;/li&gt;&#xA;&lt;li&gt;Foundational.  It may be advantageous for teachers who can do maths to grade school level to understand the subject that they will teach more deeply.  The don't need to know calculus to teach fractions, but they may benefit from a deeper understanding of what fractions really &lt;em&gt;are&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Pegagogy.  Knowing how to teach fractions is different from knowing how to add fractions. The course could focus on pedagogy not maths.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Remedial maths needs someone who can teach at (grade) school level.&lt;/p&gt;&#xA;&lt;p&gt;Foundational maths probably needs a mathematician.&lt;/p&gt;&#xA;&lt;p&gt;Teaching pedagogy is clearly well within the remit of the education department.&lt;/p&gt;&#xA;&lt;p&gt;Finally the courses might not be about school at all. It might be about playing with ideas and developing problem solving skills, rather than having anything to do with specific knowledge required in the classroom. It might be a calculus course or linear algebra or something off the wall like the theory of knots. The aim of such a course is to develop a love of maths at a higher level.&lt;/p&gt;&#xA;&lt;p&gt;And I suppose there is also a course that is a mere box ticking exercise: They have to do math, so we do maths.&lt;/p&gt;&#xA;&lt;p&gt;Whether brilliant or boring, it would be normal for a mathematician to lead such a course.&lt;/p&gt;&#xA;" OwnerUserId="5361" LastEditorUserId="5361" LastEditDate="2023-01-29T12:07:55.237" LastActivityDate="2023-01-29T12:07:55.237" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26073" PostTypeId="2" ParentId="26046" CreationDate="2023-01-28T15:46:13.293" Score="0" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{0} = \infty$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I was told this by my Math teacher in my 8 or 9 grade. I only knew at that time about infinity was that it is a very large number; so large that no can ever write it.&lt;/p&gt;&#xA;&lt;p&gt;I don't expect it to be very common though.&lt;/p&gt;&#xA;" OwnerUserId="20935" LastActivityDate="2023-01-28T15:46:13.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26074" PostTypeId="2" ParentId="5676" CreationDate="2023-01-28T16:10:27.153" Score="2" Body="&lt;h3&gt;Games!&lt;/h3&gt;&#xA;&lt;p&gt;Meet your students where they are. Many students in their late teens are going to be into various forms of gaming activities. Find out what games are currently popular and come up with questions that allow them to use their existing interest and knowledge of the game to their advantage.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Dungeons &amp;amp; Dragons&lt;/strong&gt;: Sam is playing a figher-classed half-orc, rolling a D20 to hit. Seven out of ten of Sam's last rolls were above 15. The Dungeon Master suspects that Sam's D20 is unbalanced. Develop a test to investigate this.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Pokémon&lt;/strong&gt;: Mary's Rapidash has been using Fury Attack, which has an 85% Accuracy. Mary attempted to use this move seven times today, and all seven attempts missed. Mary tells you that she thinks her little sister hacked her game cartridge to make her miss more often. Develop a statistical trial to determine if this is plausible.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For the students in your class that are not familiar with the game, provide definitions. For example, you may define D20 as &amp;quot;a twenty-sided die&amp;quot; and Accuracy as &amp;quot;the percentage likelihood to hit&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="7481" LastEditorUserId="7481" LastEditDate="2023-01-28T16:15:27.523" LastActivityDate="2023-01-28T16:15:27.523" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26075" PostTypeId="2" ParentId="26046" CreationDate="2023-01-28T18:03:47.060" Score="8" Body="&lt;p&gt;My biggest pet peeve:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;A vector is a quantity with both magnitude and direction&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One is required to say this to pick up marks on on A-level physics exam, say, despite it being very, very wrong and arguably placing emphasis on the wrong concepts. I am not yet at university, but I imagine there is a large culture shock when students are first exposed to linear algebra. That said, introducing any other notion seems to be very counter productive to the high school education, and with regards to one actually needs to be able to do, the &amp;quot;magnitude-and-direction&amp;quot; lie is successful.&lt;/p&gt;&#xA;&lt;p&gt;Many examples of this can be found in high school mathematics curricula. Unrigorous notions or incorrect definitions of the derivative, even of &amp;quot;increasing/decreasing function&amp;quot;, integral, etc. are all lies that tend to cause students some difficulty unlearning in the first years of university but are essential for teaching the applications of high school maths to the typical high schooler.&lt;/p&gt;&#xA;" OwnerUserId="16527" LastActivityDate="2023-01-28T18:03:47.060" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26076" PostTypeId="2" ParentId="5676" CreationDate="2023-01-28T19:03:17.797" Score="1" Body="&lt;p&gt;I have students gather data from amongst themselves, or also from other students, and we have looked at things like sleep (significantly less than average?), study time (less than the assumed norm), difference between genders in coffee consumption, correlation between hours studied and hours worked (no), etc.&lt;/p&gt;&#xA;&lt;p&gt;I did not do this fully rigorously; due to limitations of time, we first gathered data (both numeric and category-based), then went through such statistical tools as confidence interval, hypothesis testing and correlation/regression and had students use the relevant tool somewhere in the data. At the end, in the most ambitious take, I had them write a report and present their findings.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-01-28T19:03:17.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26077" PostTypeId="1" CreationDate="2023-01-29T04:23:19.097" Score="7" ViewCount="225" Body="&lt;p&gt;In answering another question &lt;a href=&quot;https://matheducators.stackexchange.com/q/26069/15671&quot;&gt;What is the justification to teach the (redundant) use of parentheses in multiplications?&lt;/a&gt; I was left wondering what we actually know about students' progression in terms of how they see algebraic expressions.&lt;/p&gt;&#xA;&lt;p&gt;Is there any research work that has been done into the progression of students' ability to parse math expressions?&lt;/p&gt;&#xA;&lt;p&gt;My impression is that as students progress in their mathematical careers, the way that they mentally process mathematical expressions changes from seeing them as a sequence of characters (and having to consciously think about order of operations) to automatically seeing their structure of nested subexpressions.&lt;/p&gt;&#xA;&lt;p&gt;In other words, for early students (5 x 10) + (5 x 8) and 5 x 10 + 5 x 8 are processed differently, but for later students they are processed nearly the same.&lt;/p&gt;&#xA;&lt;p&gt;But at what time scale does this happen? My impression is that students make substantial growth in their parsing skills during calculus.&lt;/p&gt;&#xA;&lt;p&gt;This is analogous to going from reading a sentence in a foreign language and having to puzzle out the grammar to this process happening subconsciously during the reading stage.&lt;/p&gt;&#xA;&lt;p&gt;I suspect if there is any research at all along these lines it would be eye-tracking research, but I couldn't find eye-tracking research in math focused on formulas specifically.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastEditorUserId="15671" LastEditDate="2023-01-30T16:46:30.060" LastActivityDate="2023-01-31T07:11:54.993" Title="Research into how students read algebraic expressions" Tags="|reference-request|algebra|education-research|psychology|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26078" PostTypeId="2" ParentId="26046" CreationDate="2023-01-29T17:02:59.943" Score="0" Body="&lt;p&gt;Students are often taught the chain-rule as a trivial cancellation law. In reality, there are intricacies within the chain-rule.&lt;/p&gt;&#xA;" OwnerUserId="20949" LastActivityDate="2023-01-29T17:02:59.943" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26081" PostTypeId="2" ParentId="26046" CreationDate="2023-01-30T03:38:23.497" Score="2" Body="&lt;p&gt;Anyone who was taught about Venn Diagrams was probably told that a set is a collection of objects.&lt;/p&gt;&#xA;&lt;p&gt;This is, of course false. But, it is a handy way to think about sets, and even when you get around to getting to a more formal notion of sets- you'll still think of them as collections of objects, while reasoning about them.&lt;/p&gt;&#xA;" OwnerUserId="20952" LastEditorUserId="20952" LastEditDate="2023-01-30T07:24:32.550" LastActivityDate="2023-01-30T07:24:32.550" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26082" PostTypeId="2" ParentId="25979" CreationDate="2023-01-30T13:00:56.913" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I sometimes come across equations like&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I)&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$3 \text{ cm} + x \text{ cm} = 5 \text{ cm}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where I would write&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;II)&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$3 \text{ cm} + x = 5 \text{ cm}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Aside from the fact that &lt;strong&gt;option I&lt;/strong&gt; allows the units to be immediately discarded and the ensuing presentation more succinct, consider the following declaration where, like in &lt;strong&gt;Option II&lt;/strong&gt;, the variables encode both numerical values and units:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If a sector has arc length &lt;span class=&quot;math-container&quot;&gt;$s $&lt;/span&gt; and radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and its angle subtended at its circle's centre is &lt;span class=&quot;math-container&quot;&gt;$\theta,$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$$s=\frac{\pi r\theta}{180}.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$(r,s)=(4\text{ cm},\pi\text{ cm}),$&lt;/span&gt; this formula is consistent with neither &lt;span class=&quot;math-container&quot;&gt;$\theta{=}45^\circ$&lt;/span&gt; nor &lt;span class=&quot;math-container&quot;&gt;$\theta{=}\frac{\pi}4$&lt;/span&gt; and wrongly refers to &lt;span class=&quot;math-container&quot;&gt;$45$&lt;/span&gt; as an angle. The sensible way to fix this anomaly is to be explicit about the choice of units (e.g., gradians, degrees, natural circular measure rad) of the &lt;a href=&quot;https://math.stackexchange.com/a/4289650/21813&quot;&gt;dimensionless&lt;/a&gt; quantity &lt;span class=&quot;math-container&quot;&gt;$\theta,$&lt;/span&gt; as in &lt;strong&gt;option I&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If a sector has arc length &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; cm and radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; cm and its angle subtended at its circle's centre is &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; &lt;strong&gt;degrees&lt;/strong&gt;, then &lt;span class=&quot;math-container&quot;&gt;$$s=\frac{\pi r\theta}{180}.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-01-30T16:07:20.533" LastActivityDate="2023-01-30T16:07:20.533" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26083" PostTypeId="2" ParentId="25409" CreationDate="2023-01-30T16:02:13.343" Score="1" Body="&lt;p&gt;I don't know if this is something you count as mathematical reasoning but lately I've read a story about a mathematician who has saved numbreds of lives:&lt;br /&gt;&#xA;During the second World War, somebody had the brilliant idea to make a summary drawing of all bullet holes that were shot into airplanes. The idea was to use that drawing in order to decide which parts of the planes should be better protected.&lt;/p&gt;&#xA;&lt;p&gt;That summary indeed showed some regions, showing more bullet holes than other regions, so the idea came up to foresee more protection on those regions.&lt;/p&gt;&#xA;&lt;p&gt;... but luckily there was also a mathematician in the room, who said just the opposite: instead of adding protection to the regions with bullet holes, he proposed to add protection to the regions WITHOUT bullet holes.&lt;/p&gt;&#xA;&lt;p&gt;Why???&lt;/p&gt;&#xA;&lt;p&gt;Well, he said: there is absolutely no reason to believe that some regions receive more bullets than others. The fact that our summary indicates otherwise, just means that the regions without bullet holes are the regions where the plane is so vulnerable that a single bullet on that spot causes the plane to crash. And as that plane is crashed, it does not appear in our summary!&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-01-30T16:02:13.343" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26084" PostTypeId="2" ParentId="26046" CreationDate="2023-01-30T17:56:37.033" Score="1" Body="&lt;p&gt;As OP asks for examples &amp;quot;particularly at the undergraduate level or higher&amp;quot;, here is a meta-mathematical &amp;quot;lie-to-children&amp;quot; that undergraduates tend to learn (often probably implicitly) when they major in mathematics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Validating a mathematical result means checking how every single step in the argument follows logically from the previous steps.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The skill to check how every single step of a proof follows from the previous steps is obivously something that one needs to learn well. So I don't think one can avoid that the students get, at some point, the impression that this consecutive validation of steps is the essence of checking a mathematical result.&#xA;However, when more experienced mathematicians read a result and its proof, many of them approach it quite differently then reading every single step in the proof in detail.&lt;/p&gt;&#xA;&lt;p&gt;Consequences of this (maybe unavoidable) &amp;quot;lie-to-children&amp;quot; can be observed on many occasions (for instance, on several Stackexchange sites):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Some (many?) people seem to be under the impression that a mathematical theory were essentially a logical house of cards that collapses once one removes a single piece. Experience shows that this is not the case, though: mistakes in research papers do occur on a regular base and at varying degrees of severity, yet mathematics (or subfields of it) is far from collapsing. Several reasons for this are discussed in the answers to &lt;a href=&quot;https://mathoverflow.net/q/338607/102946&quot;&gt;this MathOverflow post&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Many PhD students believe that the &amp;quot;canonical&amp;quot; way to referee a mathematical research paper is to check every single step in all the proofs. However, this is hardly what one does in practice. In &lt;a href=&quot;https://academia.stackexchange.com/a/192629/135841&quot;&gt;this post&lt;/a&gt; I described how checking proofs during peer review tends to work in practice.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2023-01-30T17:56:37.033" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26085" PostTypeId="1" AcceptedAnswerId="26087" CreationDate="2023-01-30T19:24:37.660" Score="6" ViewCount="2595" Body="&lt;p&gt;Examples of axial symmetry abound, but I could not find an example of pure central symmetry (that is, without axial symmetry)! Do you know of any? A butterfly shows axial symmetry, what shows point/central symmetry in nature?&lt;/p&gt;&#xA;&lt;p&gt;So, many examples resemble even functions, but I am looking for something that resembles an odd function. The best I can come up with is the image of a wave. But natural waves tend to be complicated and simple waves look too mathy.&lt;/p&gt;&#xA;&lt;p&gt;Edit and to clarify: The question and terminology were vague. I left it as it was so that the connection to discussions below is not lost. Here is another attempt at explaining the question.&lt;/p&gt;&#xA;&lt;p&gt;Odd and even functions are brought up early in college algebra. y=x^2 is even and exhibits symmetry with respect to the y-axis. Symmetry with respect to an axis is quite common in nature and one can find many examples. y=x^3 is odd and it exhibits symmetry with respect to the origin, if (x,y) is on the graph so is (-x,-y). I am looking for natural examples of symmetry with respect to origin, without the presence of symmetry with respect to an axis.&lt;/p&gt;&#xA;&lt;p&gt;Thanks to the answers below I found the following (Figure 1b is what I was looking for) taken from &lt;a href=&quot;https://www.mdpi.com/2223-7747/11/15/1987?type=check_update&amp;amp;version=2&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If you have other examples please post a picture or give a link.&lt;/p&gt;&#xA;&lt;p&gt;Trivia: After this post I asked ChatGPT the same question. After two obviously incorrect answers it said it cannot find any!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/O7vqZ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/O7vqZ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2023-02-06T23:24:36.330" LastActivityDate="2023-02-06T23:24:36.330" Title="Are there examples of central symmetry, without axial symmetry, in nature?" Tags="|undergraduate-education|precalculus|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26086" PostTypeId="2" ParentId="26085" CreationDate="2023-01-30T20:06:41.763" Score="15" Body="&lt;p&gt;Does this example of a flower with rotational, but not reflective, symmetry hit what you are looking for? (Name: Pinwheel Flower or &lt;a href=&quot;https://en.wikipedia.org/wiki/Tabernaemontana_divaricata&quot; rel=&quot;noreferrer&quot;&gt;Tabernaemontana divaricata&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PzrER.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PzrER.png&quot; alt=&quot;pinwheel flower&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2023-01-30T20:06:41.763" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26087" PostTypeId="2" ParentId="26085" CreationDate="2023-01-30T23:51:58.247" Score="12" Body="&lt;p&gt;It is easy to have axial symmetry, without an inversion center (e.g. the picture Opal showed).&lt;/p&gt;&#xA;&lt;p&gt;It is more difficult to have the converse.  Many molecules with inversion centers also have rotational symmetry, at least a C2.  However, it is possible (in 3D).  What you want is the &amp;quot;Ci&amp;quot; point group.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Coordination_Chemistry_(Landskron)/02%3A_Symmetry_and_Group_Theory/2.02%3A_Point_Groups&quot; rel=&quot;noreferrer&quot;&gt;https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Coordination_Chemistry_(Landskron)/02%3A_Symmetry_and_Group_Theory/2.02%3A_Point_Groups&lt;/a&gt; [third point group]&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OYuNJ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OYuNJ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As you can see, there's not a C3 along the C-C bond because of the three different subsittuents.  You might think there's a C2, perpendicular to the C-C bond, but there isn't either.  You can rotate and make the hydrogens work, but then the Cl and Br mess things up and don't coincide.&lt;/p&gt;&#xA;&lt;p&gt;[Note that if this were staggered ethane (CH3-CH3) instead of this halogenated ethane beastie, you'd actually still have an inversion center, but you would have a C3 axis along the C-C long axis and three C2s perpendicular to it.]&lt;/p&gt;&#xA;&lt;p&gt;P.s. I do not believe there is a 2D analogue of Ci though.&lt;/p&gt;&#xA;&lt;p&gt;P.s.s  The butterfly does not have axial symmetry.  It's not a C2, because the legs are on the bottom.  It has a mirror plane.  This is the case with most animals (at least at a gross level) and is called &amp;quot;bilateral symmetry&amp;quot;.  Maybe a starfish is &amp;quot;C5&amp;quot;.  But most higher animals are &amp;quot;Cs&amp;quot; (no rotation, but a single mirror plane).&lt;/p&gt;&#xA;&lt;p&gt;Here is a molecule:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6jxab.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6jxab.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you think about that molecule like an animal, it would have a purple head, and two dark green legs and be dragging a light green tail/body.  Sort of like an airplane, maybe.&lt;/p&gt;&#xA;" OwnerUserId="20971" LastEditorUserId="20971" LastEditDate="2023-01-31T00:17:41.603" LastActivityDate="2023-01-31T00:17:41.603" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26088" PostTypeId="2" ParentId="26077" CreationDate="2023-01-31T07:11:54.993" Score="-1" Body="&lt;p&gt;So to know what some expression will evaluate to you will need to look into the operators in other words what you referred to in your post as characters.&lt;/p&gt;&#xA;&lt;p&gt;Q1: (and having to consciously think about order of operations) to automatically seeing their structure of nested subexpressions?&lt;/p&gt;&#xA;&lt;p&gt;Due to use of terms in algebra it fairly to understand mathematical equation.&lt;/p&gt;&#xA;&lt;p&gt;(a x b) + (a x c) is a little bit hard to read.&lt;/p&gt;&#xA;&lt;p&gt;But if we instead write&lt;/p&gt;&#xA;&lt;p&gt;ab + ac it is a lot easier.&lt;/p&gt;&#xA;&lt;p&gt;To summarise the example you have given is not really encountered but if it did you will need think about the operations i.e. 5 x 10 + 5 x 8. at least a little bit otherwise you will end up 440. So I don;t think you can answer a question subconsciously.&lt;/p&gt;&#xA;&lt;p&gt;Q2: But at what time scale does this happen?&lt;/p&gt;&#xA;&lt;p&gt;Assumption is being made by the question about subconsciousness.&lt;/p&gt;&#xA;&lt;p&gt;Very quickly I will presume. Order of operations is taught in year 5-7. Though I don't think subconsciousness will occur as mention in Q1.&lt;/p&gt;&#xA;&lt;p&gt;Regarding eye tracking. A mathematical expression is read left to right by convention. I don;t any stats my guess would be most people would look at the left first to see if anything can be done. If not they will look at the other terms.&lt;/p&gt;&#xA;&lt;p&gt;e.g.&lt;/p&gt;&#xA;&lt;p&gt;8 / 2(2+2)&lt;/p&gt;&#xA;&lt;p&gt;Since 2(2+2) is one term it needs to be evaluate first before division.&lt;/p&gt;&#xA;" OwnerUserId="20626" LastActivityDate="2023-01-31T07:11:54.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26089" PostTypeId="2" ParentId="12165" CreationDate="2023-01-31T10:10:28.603" Score="2" Body="&lt;p&gt;I call these &amp;quot;haircut questions&amp;quot;: questions that you can think about, and answer, while getting a haircut, i.e. just thinking, no writing.&lt;/p&gt;&#xA;&lt;p&gt;How about:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If everyone in the world (population 8 billion) chooses a random real number between 0 and 1, one by one, approximately how many times will a world record for lowest number be set?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is an intuitive &lt;a href=&quot;https://math.stackexchange.com/a/4625455/398708&quot;&gt;solution&lt;/a&gt; that does not require writing. And the solution could lead to further questions for curious pre-calculus students.&lt;/p&gt;&#xA;" OwnerUserId="20101" LastActivityDate="2023-01-31T10:10:28.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26090" PostTypeId="2" ParentId="26085" CreationDate="2023-01-31T12:49:42.830" Score="3" Body="&lt;p&gt;Just to clarify:&lt;/p&gt;&#xA;&lt;p&gt;There are &lt;strong&gt;zero&lt;/strong&gt; two dimensional &amp;quot;examples of central symmetry [(x,y) -&amp;gt; (-x,-y)], without axial symmetry[Cn], in nature&amp;quot;.  This is because it is &lt;strong&gt;impossible mathematically&lt;/strong&gt;.  In two dimensions an inversion center is &lt;strong&gt;equivalent&lt;/strong&gt; to a 180 degree rotation around the origin.&lt;/p&gt;&#xA;&lt;p&gt;However, in three dimensions, you can have an inversion center [(x,y,z) -&amp;gt; (-x,-y,-z)] without any axial symmetry.  This is the Ci point group.&lt;/p&gt;&#xA;" OwnerUserId="20974" LastEditorUserId="20974" LastEditDate="2023-01-31T12:54:12.387" LastActivityDate="2023-01-31T12:54:12.387" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="26093" PostTypeId="1" AcceptedAnswerId="26095" CreationDate="2023-02-02T11:58:11.553" Score="3" ViewCount="1351" Body="&lt;p&gt;How one should teach using &lt;strong&gt;brackets in summation?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;For example, why is it correct to write &lt;span class=&quot;math-container&quot;&gt;$\sum_i a_ib_i$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$\sum_i a_i+b_i$&lt;/span&gt; should be written as &lt;span class=&quot;math-container&quot;&gt;$\sum_i (a_i+b_i)$&lt;/span&gt; But &lt;span class=&quot;math-container&quot;&gt;$\sum_i\frac{a_i+b_i}{2}$&lt;/span&gt; is, in turn, correct case not to use brackets?&lt;/p&gt;&#xA;" OwnerUserId="20984" LastEditorUserId="20986" LastEditDate="2023-02-03T07:52:34.723" LastActivityDate="2023-02-07T22:24:15.997" Title="How to teach using brackets in sums?" Tags="|notation|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26094" PostTypeId="2" ParentId="26093" CreationDate="2023-02-02T14:49:30.937" Score="5" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Teacher: What do you want to add ?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Student: &lt;span class=&quot;math-container&quot;&gt;$a_i+b_i$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teacher: Put it between brackets.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Student: &lt;span class=&quot;math-container&quot;&gt;$(a_i+b_i)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teacher: Sum it!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Student: &lt;span class=&quot;math-container&quot;&gt;$\sum_i (a_i + b_i)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Same for the fraction:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Teacher: What do you want to add all together?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Student: &lt;span class=&quot;math-container&quot;&gt;$\frac{a_i+b_i}{2}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teacher: Put it between brackets.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Student: &lt;span class=&quot;math-container&quot;&gt;$(\frac{a_i+b_i}{2})$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Teacher: Sum it!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Student: &lt;span class=&quot;math-container&quot;&gt;$\sum_i (\frac{a_i+b_i}{2})$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;=&amp;gt; this provides students with a fixed procedure, which is very clear, and does not generate any misunderstandings.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastEditorUserId="20986" LastEditDate="2023-02-07T22:24:15.997" LastActivityDate="2023-02-07T22:24:15.997" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26095" PostTypeId="2" ParentId="26093" CreationDate="2023-02-02T15:15:54.353" Score="17" Body="&lt;p&gt;I would say that in terms of order of operations, the summation symbol is between multiplication/division and addition/subtraction. So when you write &lt;span class=&quot;math-container&quot;&gt;$\sum_i a_i+b_i$&lt;/span&gt;, the implication is to do &lt;span class=&quot;math-container&quot;&gt;$\sum_i a_i$&lt;/span&gt; first, and so you need parentheses. However in &lt;span class=&quot;math-container&quot;&gt;$\sum_i a_ib_i$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\sum_i \frac{a_i+b_i}{2}$&lt;/span&gt; the multiplication and division by 2 come before the summation.&lt;/p&gt;&#xA;&lt;p&gt;Note though that &lt;span class=&quot;math-container&quot;&gt;$\sum_i a_i+b_i$&lt;/span&gt; implicitly means &lt;span class=&quot;math-container&quot;&gt;$\sum_i (a_i+b_i)$&lt;/span&gt; because the dummy index on the &lt;span class=&quot;math-container&quot;&gt;$b_i$&lt;/span&gt; doesn't make sense outside the context of the sum. The only real confusion comes in when you have something like &lt;span class=&quot;math-container&quot;&gt;$\sum_i a_i+3$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2023-02-02T15:15:54.353" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26096" PostTypeId="2" ParentId="10206" CreationDate="2023-02-02T17:07:01.033" Score="1" Body="&lt;p&gt;They use variables as soon as they start counting. We should introduce number machines ASAP to encourage thinking like a mathematician.&#xA;The INPUT----&amp;gt; DO ----&amp;gt; OUTPUT relationship diagram is relevant at all levels of Maths and introducing algebra plus x, y graphs is more meaningful. This would also help with the true understanding of the equal sign.&#xA;I am embarrassed as a maths teacher that variables are introduced in Science to elementary pupils but rarely mentioned in the Mathematics curriculum.&lt;/p&gt;&#xA;" OwnerUserId="20985" LastEditorUserId="20985" LastEditDate="2023-02-02T18:16:43.090" LastActivityDate="2023-02-02T18:16:43.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26097" PostTypeId="2" ParentId="26093" CreationDate="2023-02-02T20:49:46.287" Score="2" Body="&lt;p&gt;There are actually two questions that you're asking: how to teach &amp;quot;the summation symbol&amp;quot; and the followup question that asks for concrete explanations for examples.&lt;/p&gt;&#xA;&lt;p&gt;I think the followup questions are non-questions if the summation symbol is carefully introduced and defined, I'd do something like&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\sum$&lt;/span&gt; is a shorthand for a (probably infinite) summation.&lt;/li&gt;&#xA;&lt;li&gt;Following the order of operations, the terms that are summed up stretch to the next (in terms of order of execution) addition or subtraction step in the term, if any.&lt;/li&gt;&#xA;&lt;li&gt;It then becomes clear that &lt;span class=&quot;math-container&quot;&gt;$\sum ab$&lt;/span&gt; is a sum of products and that &lt;span class=&quot;math-container&quot;&gt;$\sum a + b$&lt;/span&gt; is a sum of many &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;s and one &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;To sum multiple sums, you have to use parentheses: &lt;span class=&quot;math-container&quot;&gt;$\sum (a+b)$&lt;/span&gt; is a sum of sums.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="667" LastActivityDate="2023-02-02T20:49:46.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26098" PostTypeId="1" AcceptedAnswerId="26100" CreationDate="2023-02-03T05:38:49.910" Score="21" ViewCount="2351" Body="&lt;p&gt;When introducing the techniques of differentiation, polynomials come up all the time as great examples to familiarize students with the &amp;quot;power rule&amp;quot; and the linearity of differentiation.&lt;/p&gt;&#xA;&lt;p&gt;A common extension is to then work with expressions involving terms with negative degree, like &lt;span class=&quot;math-container&quot;&gt;$3x^2 - 2x  + 5 - \frac{1}{x} + \frac{3}{x^2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I was wondering if there was a canonical way to refer to such expressions.&lt;/p&gt;&#xA;&lt;h2&gt;Considerations so far&lt;/h2&gt;&#xA;&lt;p&gt;The closest terminology I have come across in my own study is the &lt;em&gt;Laurent series&lt;/em&gt;, but I find it non-ideal because of&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The association with complex numbers&lt;/li&gt;&#xA;&lt;li&gt;The infinite nature of the series (though this could be solved by invoking a &amp;quot;formal Laurent Series&amp;quot;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Other names that I have considered include the &lt;em&gt;rational function&lt;/em&gt;, but I feel that this refers to an expression more akin to &lt;span class=&quot;math-container&quot;&gt;$\frac{P(x)}{Q(x)}$&lt;/span&gt; which is more relevent in the discussion of the quotient rule.&lt;/p&gt;&#xA;" OwnerUserId="360" LastActivityDate="2023-02-05T16:39:56.423" Title="Is there a canonical name for a polynomial-like expression allowing for negative powers?" Tags="|calculus|terminology|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26099" PostTypeId="2" ParentId="26098" CreationDate="2023-02-03T11:24:31.730" Score="17" Body="&lt;p&gt;You can call it a &amp;quot;polynomial in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and  &lt;span class=&quot;math-container&quot;&gt;$x^{-1}$&lt;/span&gt;&amp;quot; or a &amp;quot;polynomial function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and  &lt;span class=&quot;math-container&quot;&gt;$x^{-1}$&lt;/span&gt;&amp;quot;. The idea is that you are taking a two variable polynomial &lt;span class=&quot;math-container&quot;&gt;$p(x,y)$&lt;/span&gt; and then substituting &lt;span class=&quot;math-container&quot;&gt;$y=x^{-1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Similar terminology is also useful so that you can say things like &amp;quot;this technique allows me to integrate any rational function of sine and cosine&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-02-03T11:24:31.730" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26100" PostTypeId="2" ParentId="26098" CreationDate="2023-02-03T13:27:38.997" Score="42" Body="&lt;p&gt;There is also the term &amp;quot;Laurent series&amp;quot;, where we allow an infinite number of terms...&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\dots +5x^{-3} + 2x^{-2} + 3x^{-1} + 2 + 4x - 7x^2+\dots&#xA;$$&lt;/span&gt;&#xA;So perhaps yours is a &amp;quot;finite Laurent series&amp;quot; or maybe &amp;quot;Laurent polynomial&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2023-02-03T13:27:38.997" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26101" PostTypeId="2" ParentId="26040" CreationDate="2023-02-03T15:48:38.317" Score="3" Body="&lt;p&gt;The mistake I did in probability was when I was first introduced to un-equally likely outcomes. The example was something like this-&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A toin is tossed one time. If it appears head, a die is rolled. The sample space is &lt;span class=&quot;math-container&quot;&gt;$S = (H1 , H2 , H3 , H4 , H5 , H6 , T)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After reading upon the sample space, I blindly thought that all of these outcomes are equally likely and probability of any&#xA;( say &lt;span class=&quot;math-container&quot;&gt;$H1$&lt;/span&gt;) is &lt;span class=&quot;math-container&quot;&gt;$1/7$&lt;/span&gt;. However I realised the mistake later.&lt;/p&gt;&#xA;" OwnerUserId="20935" LastActivityDate="2023-02-03T15:48:38.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26102" PostTypeId="2" ParentId="25904" CreationDate="2023-02-03T16:10:10.947" Score="1" Body="&lt;p&gt;&lt;strong&gt;Introduce them some of euclid's axioms.&lt;/strong&gt; For example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Things that are equal to the same thing are also equal to one another&#xA;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If equals are added to equals, then the wholes are equal (Addition property of equality).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If equals are divided from equals, then the results are equal (division property of equality).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then, show them with examples how these axioms can be used to prove the desire things ( You can give them a chance prior to this if you wish so). It would not result into any unfairness if they accept it as common sense because these are accepted axioms and not informal shortcuts.&lt;/p&gt;&#xA;" OwnerUserId="20935" LastActivityDate="2023-02-03T16:10:10.947" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26103" PostTypeId="2" ParentId="25904" CreationDate="2023-02-03T20:08:06.123" Score="0" Body="&lt;p&gt;You can teach/remind him to simplify &lt;span class=&quot;math-container&quot;&gt;$(38×27×14)/(27×14×38)$&lt;/span&gt;. If he knows the same number can be eliminated from both sides regardless of the position to simplify a fraction, he'll get 1, meaning they're equal.&lt;/p&gt;&#xA;" OwnerUserId="20993" LastEditorUserId="20993" LastEditDate="2023-02-03T20:26:02.040" LastActivityDate="2023-02-03T20:26:02.040" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26104" PostTypeId="2" ParentId="26098" CreationDate="2023-02-04T01:03:02.423" Score="4" Body="&lt;p&gt;I would also like to suggest &lt;a href=&quot;https://en.wikipedia.org/wiki/Posynomial&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;posynomials&lt;/em&gt;&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$$f(x_1,\cdots,x_n)=\sum_{k=1}^Kc_kx_1^{a_{1,k}}\cdots x_n^{a_{n,k}}.$$&lt;/span&gt; These generalise the idea to a multivariate setting with interaction terms, and the exponents can be any real numbers.&lt;/p&gt;&#xA;" OwnerUserId="8598" LastActivityDate="2023-02-04T01:03:02.423" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26105" PostTypeId="2" ParentId="26098" CreationDate="2023-02-04T17:42:52.217" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Rational_function#Definitions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rational function&lt;/a&gt; fits the description, as we can always combine the terms into a single ratio of polynomials.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\underbrace{\overbrace{3x^2 - 2x  + 5}^{\rm polynomial} + \overbrace{ - \frac{1}{x} + \frac{3}{x^2}}^{\text{proper rational}}}_{\rm rational}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="5494" LastActivityDate="2023-02-04T17:42:52.217" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26107" PostTypeId="2" ParentId="26040" CreationDate="2023-02-04T20:30:28.747" Score="6" Body="&lt;p&gt;These two concepts from elementary probability are not that elementary to digest.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If, in a probability experiment, event B's outcomes are restricted due to knowing that event A happens, the two events can &lt;em&gt;still&lt;/em&gt; be &lt;strong&gt;independent&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\quad$&lt;/span&gt; &lt;a href=&quot;https://math.stackexchange.com/a/4403764/21813&quot;&gt;Intuition on the independence of two events&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As an example, consider a roll of a fair die:&#xA;&lt;span class=&quot;math-container&quot;&gt;$ Ω= \{1,2,3,4,5,6\}\\&#xA;   A= \lbrace 3, 6\rbrace\\&#xA;   B= \lbrace 2, 4, 6 \rbrace$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Knowing that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; happens restricts the outcomes of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; to just &lt;span class=&quot;math-container&quot;&gt;$\{6\}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P(A\cap B)=P(A)P(B),$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are independent events.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For uncountable sample spaces, &lt;strong&gt;mutually exclusive&lt;/strong&gt; events need &lt;em&gt;not&lt;/em&gt; be disjoint (unless the author defines mutual exclusivity using something like &lt;span class=&quot;math-container&quot;&gt;$X\cap Y=\emptyset$&lt;/span&gt; instead of something like &lt;span class=&quot;math-container&quot;&gt;$P(X\cap Y)=0$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;For example, the events &lt;span class=&quot;math-container&quot;&gt;$(1,2]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$[2,3)$&lt;/span&gt; are mutually exclusive but not disjoint.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Two mutually exclusive positive-probability events are necessarily &lt;em&gt;dependent&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is common to naively make wrong inferences between independence and mutual exclusivity, so this illustration may appear counterintuitive:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \begin{array}{r}  \begin{array}{c|c|c}&#xA;  \style{font-family:inherit}{}  &amp;amp; \style{font-family:inherit}{U_1}  &amp;amp; \style{font-family:inherit}{U_2}  &amp;amp; \style{font-family:inherit}{U_3}&#xA;\\\hline&#xA;  \style{font-family:inherit}{P(X\cap Y)}  &amp;amp; 0  &amp;amp; \frac14  &amp;amp; \frac14&#xA;\\[0pt]\hline&#xA;  \style{font-family:inherit}{P(X)P(Y)}  &amp;amp; \frac14\times\frac12=\frac18  &amp;amp; \frac14\times\frac34=\frac38  &amp;amp; \frac12\times\frac12=\frac14&#xA;\\[0pt]\hline&#xA;  \style{font-family:inherit}{ X\text{ and }Y\text{ are}\ldots}  &amp;amp; \textbf{dependent}  &amp;amp; \textbf{dependent}  &amp;amp; \textbf{independent}&#xA;\\[0pt]\hline&#xA;  \style{font-family:inherit}{X\text{ and }Y\text{ are}\ldots}  &amp;amp; \textbf{mutually exclusive}  &amp;amp; \textbf{not ME}  &amp;amp; \textbf{not ME} \end{array}\hskip-5.5pt  \end{array} $$&lt;/span&gt;&lt;span class=&quot;math-container&quot;&gt;$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$&lt;/span&gt;&lt;a href=&quot;https://i.stack.imgur.com/vCZSbm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vCZSbm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-02-05T17:30:03.497" LastActivityDate="2023-02-05T17:30:03.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26108" PostTypeId="2" ParentId="26098" CreationDate="2023-02-05T01:28:56.710" Score="0" Body="&lt;p&gt;Another option is to use &lt;strong&gt;&lt;em&gt;multinomial&lt;/em&gt;&lt;/strong&gt;  or &lt;strong&gt;&lt;em&gt;algebraic multinomial&lt;/em&gt;&lt;/strong&gt;. However, it should be noted that this term is sometimes used as an AKA of polynomial.&lt;/p&gt;&#xA;&lt;h3&gt;References:&lt;/h3&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathworld.wolfram.com/Multinomial.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathworld:&#xA;Multinomial&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com/books?id=akLt13oOnv8C&amp;amp;pg=PA73&amp;amp;lpg=PA73&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google&#xA;Books: &lt;em&gt;Maths Plus 6&lt;/em&gt;, SC Das, p. 73&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com/books?id=jzTpCgAAQBAJ&amp;amp;pg=PA20&amp;amp;lpg=PA20&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Books: &lt;em&gt;Algebra: A Mathematical Analysis Preliminary to Calculus&lt;/em&gt;, A. Fuentes, p. 20&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5779" LastActivityDate="2023-02-05T01:28:56.710" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26109" PostTypeId="2" ParentId="12234" CreationDate="2023-02-05T09:40:12.583" Score="4" Body="&lt;p&gt;Very elementary issues that can hinder later understanding:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Misunderstanding the difference between independent and mutually excusive and how to apply them.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Not grasping that most elementary discrete probability problems are based on the &lt;em&gt;modelling assumption&lt;/em&gt; that the probability of each outcome in a sample space are equal, and equal in value to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{size\,of\,sample\,space}$&lt;/span&gt;. Conversely a real-world (discrete) problem may have outcomes of non-equal probability - and how to handle that situation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Misunderstanding the definition of event in terms of outcomes and the definition of the probability of an event.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Being unable to model a hypothesis test based of (an often poorly-worded) problem and not understanding the meaning of the result of the h-test.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Grasping the difference between the frequency distribution of discrete outcomes/events and calculating an experimental/empirical probability and a probability calculated from a theoretical model - and how this can be one way into hypothesis tests. For example what is the probability of rolling a four on a fair 6-faced die if the die is rolled 600 times come shows a four 95 times? I don't think this is really a student misunderstanding - I just think it is a seemingly simple issue that is difficult to tackle at the start - which is when such scenarios are used to introduced probability.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7954" LastEditorUserId="7954" LastEditDate="2023-02-05T18:03:30.910" LastActivityDate="2023-02-05T18:03:30.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26110" PostTypeId="2" ParentId="26098" CreationDate="2023-02-05T15:30:51.533" Score="6" Body="&lt;p&gt;In comments, &lt;a href=&quot;https://matheducators.stackexchange.com/users/20990/eike-schulte&quot;&gt;Eike Schulte&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/users/63/paul-garrett&quot;&gt;paul garrett&lt;/a&gt; mentioned that the standard term seems to be &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Laurent_polynomial&quot; rel=&quot;noreferrer&quot;&gt;Laurent polynomials&lt;/a&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/26100/930&quot;&gt;Gerald Edgar's answer&lt;/a&gt; also mentions this term in passing, but I figured that it may be worthwhile to have an answer that features it front-and-center.&lt;/p&gt;&#xA;" OwnerUserId="930" LastActivityDate="2023-02-05T15:30:51.533" CommentCount="0" CommunityOwnedDate="2023-02-05T15:30:51.533" ContentLicense="CC BY-SA 4.0" />
  <row Id="26111" PostTypeId="2" ParentId="26098" CreationDate="2023-02-05T16:39:56.423" Score="5" Body="&lt;p&gt;The question is about what to call these expressions/functions &lt;em&gt;&lt;strong&gt;when teaching the first basic properties of derivatives&lt;/strong&gt;&lt;/em&gt;. All of the names proposed in previous answers would just make this harder for the student. They would be better off with this collection of expressions/functions having no name.&lt;/p&gt;&#xA;&lt;p&gt;Dave Renfro's comment seemed useful to me, so I'm bumping it up to an answer, with some added thoughts of my own. He wrote: &amp;quot;In classes I taught I used to call these &amp;quot;polynomial type functions&amp;quot; (probably since the late 1980s, if not earlier), and for me they included fractional and even irrational exponents...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The textbooks typically include negative and fractional powers, along with the positive powers (even though the extension of the power rule to these powers has not yet been proved). I call them &amp;quot;polynomial-like functions&amp;quot;, very similar to Dave's &amp;quot;polynomial type functions&amp;quot;. Thus I have a name for what is included, but I'm not asking my students to learn some new (not particularly useful) vocabulary.&lt;/p&gt;&#xA;&lt;p&gt;I recommend that we keep it simple here.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-02-05T16:39:56.423" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26112" PostTypeId="2" ParentId="25904" CreationDate="2023-02-06T13:07:01.753" Score="5" Body="&lt;p&gt;Some great answers were provided here, but I wanted to add something which I think is essential: &lt;em&gt;to emphasize to the student that it is a very good thing to question this!&lt;/em&gt; Why am I saying that? Because indeed, for many operations in algebra the &lt;a href=&quot;https://en.wikipedia.org/wiki/Associative_property&quot; rel=&quot;noreferrer&quot;&gt;associative property&lt;/a&gt; does not hold! For example, when talking about division: &lt;span class=&quot;math-container&quot;&gt;$8/(4/2) \neq (8/4)/2$&lt;/span&gt;. So in fact I would go so far as to say, that a student who questions this may have a latent / undeveloped talent for Math! Because clearly here is an example of a single operation, namely division, where the associative property does not hold. So I would definitely compliment the student for not taking this for granted, and go on to explain that multiplication is one of the &lt;em&gt;special&lt;/em&gt; operations where it holds that &lt;span class=&quot;math-container&quot;&gt;$a\times(b \times c) = (a\times b)\times c$&lt;/span&gt; and therefore we can drop the parentheses without ambiguity.&lt;/p&gt;&#xA;&lt;p&gt;A similar example can also explain the &lt;a href=&quot;https://en.wikipedia.org/wiki/Commutative_property&quot; rel=&quot;noreferrer&quot;&gt;commutativity&lt;/a&gt; of multiplication: &lt;span class=&quot;math-container&quot;&gt;$2 \times 4 = 4 \times 2$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$2 / 4 \neq 4 / 2$&lt;/span&gt;. Actually I realize that this is actually more relevant to the examples given in the question, and in fact it is important to notice the &lt;em&gt;implication&lt;/em&gt; of having these two properties &lt;em&gt;together&lt;/em&gt; for a single operation such as multiplication.&#xA;It can actually be a fun exercise for a student to prove that given:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$(a \times b) \times c = a \times (b \times c)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$a \times b = b \times a$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;then also: &lt;span class=&quot;math-container&quot;&gt;$ a \times b \times c = c \times b \times a $&lt;/span&gt; .&lt;/p&gt;&#xA;&lt;p&gt;Note that given only 1. &lt;em&gt;or&lt;/em&gt; 2., this doesn't hold! This is also a source of (justified) confusion: a lot of students implicitly think that commutativity (swapping rule) implies associativity, but those are totally independent properties.&lt;/p&gt;&#xA;&lt;p&gt;Another thing which occasionally helps here is to ask the student the simplest possible question: &amp;quot;&lt;em&gt;What is bothering you about this rule?&lt;/em&gt;&amp;quot;. I think I've already established, there &lt;em&gt;is&lt;/em&gt; something to be bothered about here. But the student, not always having the faculty with words to &lt;em&gt;explain&lt;/em&gt; what is bothering him, just feels &amp;quot;silly&amp;quot; and &amp;quot;inadequate&amp;quot; to deal with the problem. In other words, he feels that it is already &lt;em&gt;wrong&lt;/em&gt; to be &lt;em&gt;bothered&lt;/em&gt; about something so &amp;quot;simple&amp;quot;, which is absolutely bananas! The opposite is true.&lt;/p&gt;&#xA;" OwnerUserId="21006" LastEditorUserId="21006" LastEditDate="2023-02-06T18:02:33.370" LastActivityDate="2023-02-06T18:02:33.370" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26113" PostTypeId="1" CreationDate="2023-02-06T14:23:25.650" Score="3" ViewCount="530" Body="&lt;p&gt;Recently my brother in high school and I were talking about some math when he said&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If we know two things are the same i.e. equal why do we need to state&#xA;that they're the same? What he was trying to say was, why do we need&#xA;to say a thing is itself?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is what I answered&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When we start, we really don't know whether those things are equal&#xA;i.e. that they're the same. We start by seeing them as different, not knowing&#xA;whether they're indeed the same. This is evident in the fact that we also give them different names at first. It is only after some observations we find that these two things which we named different and were perceived as different were indeed the same thing. Stating two things are equal is important because we started off by assuming they're different.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm not sure this is what the correct answer is. I believe mathematics teachers here would've encountered such a situation with their students at some point.&lt;/p&gt;&#xA;&lt;p&gt;I'm not a mathematics teacher but I think this question is a fit for the community.&lt;/p&gt;&#xA;" OwnerUserId="20701" LastEditorUserId="1680" LastEditDate="2023-02-07T18:07:11.137" LastActivityDate="2023-02-09T10:36:23.320" Title="Why do we explicitly state the equality of two things when we know they're equal" Tags="|secondary-education|primary-education|terminology|middle-school|equal-sign|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="26114" PostTypeId="2" ParentId="26113" CreationDate="2023-02-06T15:28:02.707" Score="3" Body="&lt;p&gt;Your answer is good for showing that indeed we sometimes arrive at an equality via an &lt;em&gt;experimental&lt;/em&gt; approach: We try to relate different quantities we observe in day to day life to each other and express their relationship, very often via an equality. The simplest example is probably &lt;span class=&quot;math-container&quot;&gt;$x=v \cdot t$&lt;/span&gt;, which can be stated as &amp;quot;The distance an object traverses equals to its speed times duration of travel&amp;quot;. Of course one can go deeper into this, and talk about cases where for example the speed varies with time, and then that particular form of the equality does not hold.&lt;/p&gt;&#xA;&lt;p&gt;Then, you can explain equality in a more abstract sense, as a &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Binary_relation&quot; rel=&quot;nofollow noreferrer&quot;&gt;relation&lt;/a&gt;&lt;/em&gt;. So in math we can express all kinds of relations, some of them are rather trivial: &lt;span class=&quot;math-container&quot;&gt;$5=5$&lt;/span&gt;, others slightly more interesting:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$1+2+3+\dots+n = \frac{n(n+1)}{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and so on. This can be a particularly nice example for the usefulness of an equality relation: it is a lot easier and quicker to compute the right hand side than the left hand side.&lt;/p&gt;&#xA;&lt;p&gt;It can also be explained that while our rule for distances travelled &lt;span class=&quot;math-container&quot;&gt;$x=v\cdot t$&lt;/span&gt; is an &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Empirical_evidence&quot; rel=&quot;nofollow noreferrer&quot;&gt;empirical&lt;/a&gt;&lt;/em&gt; (stemming from observation) one, the rule for summing the integers written above &lt;em&gt;can be &lt;a href=&quot;https://en.wikipedia.org/wiki/Formal_proof&quot; rel=&quot;nofollow noreferrer&quot;&gt;derived&lt;/a&gt;&lt;/em&gt; without having to rely on any physical observations.&lt;/p&gt;&#xA;&lt;p&gt;This demonstrates the difference between what we usually call &lt;em&gt;physical laws&lt;/em&gt;, which are based on our experiments and observations, and what are called &lt;em&gt;mathematical theorems&lt;/em&gt;. Both may involve equality relations, but the &amp;quot;justification&amp;quot; and &lt;em&gt;meaning&lt;/em&gt; of the equations is very different in both worlds (albeit deeply &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_physics&quot; rel=&quot;nofollow noreferrer&quot;&gt;interrelated&lt;/a&gt;!).&lt;/p&gt;&#xA;&lt;p&gt;Also it is interesting to distinguish a &lt;em&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Symmetric_relation&quot; rel=&quot;nofollow noreferrer&quot;&gt;symmetric relation&lt;/a&gt;&lt;/em&gt; like &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; from an asymmetric one. For a symmetric relation like &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt; &lt;a href=&quot;https://en.m.wikipedia.org/wiki/If_and_only_if&quot; rel=&quot;nofollow noreferrer&quot;&gt;iff&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$b=a$&lt;/span&gt; but for &lt;span class=&quot;math-container&quot;&gt;$c&amp;lt;d$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$d \nless c $&lt;/span&gt; (hence &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; is &lt;a href=&quot;https://en.wikipedia.org/wiki/Asymmetric_relation&quot; rel=&quot;nofollow noreferrer&quot;&gt;asymmetric&lt;/a&gt;). It is also fun to observe that if we have &lt;span class=&quot;math-container&quot;&gt;$A\leq  B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A \geq B$&lt;/span&gt; it implies &lt;span class=&quot;math-container&quot;&gt;$A=B$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Upshot: there are many ways to address such a question, both from an experimental science perspective and also from a mathematical perspective. I would try to go along with what the student finds most interesting and explain it from that angle :)&lt;/p&gt;&#xA;" OwnerUserId="21006" LastEditorUserId="21006" LastEditDate="2023-02-06T15:55:50.273" LastActivityDate="2023-02-06T15:55:50.273" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26116" PostTypeId="2" ParentId="26113" CreationDate="2023-02-07T06:44:55.543" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Stating two things are equal is important because we started off by assuming they're different.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You'd only &lt;em&gt;assume&lt;/em&gt; that they are different if you are attempting a proof by contradiction.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;why do we need to say a thing is itself?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Writing &lt;span class=&quot;math-container&quot;&gt;$$V_S=\frac43\pi r^3$$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$$x^2-y^2=(x+y)(x-y)\tag#$$&lt;/span&gt; doesn't merely equate objects/expressions that are &amp;quot;obviously&amp;quot; the same, but conveys equations that may not be self-evident (yet, anyway). So much so that &lt;span class=&quot;math-container&quot;&gt;$(\#)$&lt;/span&gt; benefits from being rewritten as &lt;span class=&quot;math-container&quot;&gt;$x^2-y^2\equiv(x+y)(x-y),$&lt;/span&gt; which specifically asserts that &lt;span class=&quot;math-container&quot;&gt;$(\#)$&lt;/span&gt; holds for &lt;em&gt;every&lt;/em&gt; combination of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; such that the equality is defined. (In contrast, notice that not every point of the Cartesian plane satisfies &lt;span class=&quot;math-container&quot;&gt;$(V_S,r).$&lt;/span&gt;) We are not stating the obvious with a tautology like &lt;span class=&quot;math-container&quot;&gt;$$x=x,$$&lt;/span&gt; but communicating and using &lt;em&gt;formulae&lt;/em&gt; and &lt;em&gt;identities&lt;/em&gt; that we have discovered.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, &lt;code&gt;=&lt;/code&gt; is useful also for conveying information that cannot be derived, for example to impose constraints, specify definitions and issue labels:  &lt;span class=&quot;math-container&quot;&gt;$$x+4=10\\\sqrt{x^2}:=|x|\\f(x)=2x+4\\y=2x+C.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/4254850/21813&quot;&gt;Another way to classify equations.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-02-09T10:36:23.320" LastActivityDate="2023-02-09T10:36:23.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26117" PostTypeId="2" ParentId="26113" CreationDate="2023-02-07T16:55:41.617" Score="2" Body="&lt;p&gt;A usage of &amp;quot;=&amp;quot; that can be confusing for learners is when we use the equality symbol to assign something to a variable. Perhaps we should use &amp;quot;:=&amp;quot; in these cases but this is not standard. For example, we might write,&amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$x=5$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$x^2=25$&lt;/span&gt;.&amp;quot; In this case, the variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and the integer 5 are not literally equal. We are just saying &lt;em&gt;if&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x=5$&lt;/span&gt; then...&lt;/p&gt;&#xA;&lt;p&gt;Another related example is what we see in typical proofs that &lt;span class=&quot;math-container&quot;&gt;$\sqrt2$&lt;/span&gt; is irrational: Suppose &lt;span class=&quot;math-container&quot;&gt;$\sqrt2$&lt;/span&gt; is rational. Then &lt;span class=&quot;math-container&quot;&gt;$\sqrt2=\frac{p}{q}$&lt;/span&gt; for some integers &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; an &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-02-07T16:55:41.617" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26119" PostTypeId="1" CreationDate="2023-02-07T22:00:21.800" Score="2" ViewCount="390" Body="&lt;p&gt;In western schools is a tradition to use Greek letters to denote angles. I wonder what about Greek schools do they also use Greek letters to denote angles or do they prefer other kind of alphabet to avoid confusion?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="8190" LastEditDate="2023-02-08T15:16:55.473" LastActivityDate="2023-02-14T19:24:55.533" Title="Do Greek students use Greek letters to denote angles?" Tags="|notation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26122" PostTypeId="2" ParentId="26113" CreationDate="2023-02-08T17:32:11.997" Score="3" Body="&lt;p&gt;If I write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;202384569+4765923845-141243678 =&#xA;$$&lt;/span&gt;&#xA;then go off and think a bit, do a hand calculation, or consult a calculating device, and come back and complete it to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;202384569+4765923845-141243678 = 4827064736&#xA;$$&lt;/span&gt;&#xA;then likely my main reason for using the equals sign is to record the knowledge just obtained for future use. This allows me to avoid having to repeat the work I just did.&lt;/p&gt;&#xA;&lt;p&gt;If I'm in elementary school and the teacher asks me to add &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, and I then write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2+2=4&#xA;$$&lt;/span&gt;&#xA;then I'm using the equals sign to communicate to the teacher that I know the value of the sum.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2023-02-08T17:32:11.997" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26123" PostTypeId="2" ParentId="26113" CreationDate="2023-02-08T18:53:09.153" Score="0" Body="&lt;p&gt;I think there might be some kind of language barrier because of the similar but fundamentally different words in OP's question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If we know two things are the &lt;strong&gt;same&lt;/strong&gt; i.e. &lt;strong&gt;equal&lt;/strong&gt; why do we need to state&#xA;that they're the same? What he was trying to say was, why do we need&#xA;to say a thing &lt;strong&gt;is itself&lt;/strong&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The symbol &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; is used in mathematics to express identity (two things actually being just one thing) as well as equality (two things really are distinct but can be considered equal in at least some aspect)  as the following two examples might show:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Pythagorean Theorem: &lt;span class=&quot;math-container&quot;&gt;$c^2 = a^2 + b^2$&lt;/span&gt; states that the square over the hypothenuse is equal (in size) to the sum of squares over the other sides. But of course those squares are not identical (&amp;quot;the same thing&amp;quot;). But their &lt;strong&gt;equality&lt;/strong&gt; is still an important fact to state.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\{ x \in \mathbb N ~|~ x \text{ ends in 0 or 5} \} = \{x \in \mathbb N ~|~ x \text{ is divisible by 5} \}$&lt;/span&gt; states that both sets are actually the same thing, the set of the numbers 0, 5, 10, ... Again, this &lt;strong&gt;identity&lt;/strong&gt; is state-worthy.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;For pure numbers, the distinction between equality and identity breaks down.&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2023-02-08T18:53:09.153" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26124" PostTypeId="1" CreationDate="2023-02-10T14:40:55.490" Score="14" ViewCount="5220" Body="&lt;p&gt;I'm a retired university math prof and I now have a retirement job teaching at a small private high school.  This is my 4th year.  This school teaches Algebra 1 to 8th graders.  Geometry to 9th graders.  Algebra 2 to 10th graders.  Precalculus to 11th graders.  Calculus to 12th graders.  I have come to the opinion that for many students, these subjects are coming a year too soon.  I'm convinced that if we'd let most students get another year through or past puberty, and develop their ability to think abstractly more, then they'd have less trouble in math all through high school.&lt;/p&gt;&#xA;&lt;p&gt;I found this study:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nber.org/papers/w21610&quot; rel=&quot;noreferrer&quot;&gt;https://www.nber.org/papers/w21610&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;which tangentially supports my opinion.&lt;/p&gt;&#xA;&lt;p&gt;My question is:  Is there other research that more directly addresses when to teach the various subjects of middle school and high school math?&lt;/p&gt;&#xA;&lt;p&gt;My observations are:  I've listened to the middle school teachers complain that they can't teach Prealgebra to 7th graders because they can't add fractions.  And the 5 and 6th grade teachers complain that they can't teach adding fractions because the students don't know the multiplication table.  I used to complain that college freshmen who took Calculus in high school were usually worse off than the freshmen who hadn't.  By taking Calculus in high school, many students seemed to be permanently warped in a way that made Calc 2 much more difficult.  My suspicion is that by ever trying to cram more stuff in earlier, things get a lighter treatment.  So that students are never quite prepared for the next class.  And this effect snowballs.&lt;/p&gt;&#xA;&lt;p&gt;I don't know how many times I had a sophomore in Differential Equations in my office who had never seen Pascal's triangle or had no idea how to factor a quadratic.  Or he sort of knew how, but couldn't execute.  I would complain, &amp;quot;I wish the high schools would teach the high school math solidly, so that I can teach Calculus to students who are ready.&amp;quot; And I'm observing the same effect in this high school.&lt;/p&gt;&#xA;&lt;p&gt;Here in Texas, they have the STARR test.  It's a monster.  By studying it, we can tell that they're having 5th graders do stem-and-leaf plots and find quartiles of date.  Sure, they can do it, but it's just one more thing crammed into the curriculum that distracts from learning to add fractions.  And these easy things, like finding quartiles, can be learned by a 10th grader in one day.  Why are we torturing 5th graders with it for a week?  A typical STARR test problem is to show a polygon in the shape of Utah, and assign lengths to some of the sides.  These lengths are all mixed numeral.  The student is supposed to find the area of the polygon.  So first he has to add and subtract mixed numerals to find the lengths of the other sides.  Then he has to multiply some of these to get the area.  So this is a multistep problem testing more than one thing.  Not only is more material being ever shoved into the lower grade curriculum, but also a higher expectation of problem solving skills.  I don't think most 5th graders are ready for the Utah problem.&lt;/p&gt;&#xA;&lt;p&gt;So my question is, again, does anyone know of real research that would support my opinion?  (Or negate my opinion?)&lt;/p&gt;&#xA;" OwnerUserId="19450" LastEditorUserId="19450" LastEditDate="2023-02-10T20:50:26.810" LastActivityDate="2023-02-18T12:08:27.613" Title="Teaching math too soon in middle school and high school" Tags="|secondary-education|" AnswerCount="8" CommentCount="29" ContentLicense="CC BY-SA 4.0" />
  <row Id="26125" PostTypeId="2" ParentId="26124" CreationDate="2023-02-10T16:26:39.623" Score="17" Body="&lt;p&gt;I think you have to consider tracking.  The answer for the average kid is not the same as for the top or bottom ends.  Exeter and Thomas Jefferson do fine accelerating their entire cohorts by a year (if not more).  Conversely, we don't live in Lake Woebegone where the 100% of the kids are above average...and California has struggled with early algebra for all.&lt;/p&gt;&#xA;&lt;p&gt;As far as &amp;quot;research&amp;quot;, start with this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/external_archive/columns/launchings/launchings_06_09.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.maa.org/external_archive/columns/launchings/launchings_06_09.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The referenced studies Morgan and Claric and Morgan and Ramist are good ones to look at.  In essence, the takeaway from Bressuod's work:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;For the fraction of kids that accelerate, and get a 4 or 5 on the AP (AB or BC), they do fine in follow on classes in calculus, outperforming kids who did not accelerate.  For kids with a 3, it's more of a wash (and many schools don't accept a 3 and more kids with 3 will self select out of college advanced placement).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The vast majority of kids who experience calculus in high school do NOT get a 4 or 5 on the AP.  And it is unclear if their acceleration mildly helps them, is a wash, or is detrimental.  But arguably, it may not be worth the effort expended if no college credit ends up being earned, even if a wash.  And may be really hurting the kids who are lower tier and could benefit from a year of increasing smarts (it is well known from IQ literature that we get smarter from age 12 to 18) or even maturity.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I don't have research on the amount of time spent on fractions and how it correlates to later performance.  I suspect it is a hard nut to crack regardless.  Upper level teachers have been complaining for decades about prior prep.  But there is no magic switch.  Not everyone is going to achieve high scholastic performance.&lt;/p&gt;&#xA;&lt;p&gt;I personally sort of like the baby stats exposure.  A lot of elementary school is phenomenological learning, vice axiomatic.  So why not let them play with it.  And &amp;quot;analysis&amp;quot; (casual meaning, not math) is creeping into business life with even secretaries playing with Excel now.&lt;/p&gt;&#xA;&lt;p&gt;I am anti calculator in primary school math though and think more time with paper and pencil is better.  The kid who needs calculator help, needs to not have it.  The kid who doesn't need it, can pick it up easily later in science classes (7+) where doing calculations of inconvenient numbers.   I mean, do we really think they won't know how to push the buttons, unless shown in a class?  There is research in both directions on this, though and it is politicized.  (But of course I'm fine with calculator use for the baby stats as the numbers can be a hassle.)&lt;/p&gt;&#xA;" OwnerUserId="21014" LastEditorUserId="21014" LastEditDate="2023-02-10T16:29:42.530" LastActivityDate="2023-02-10T16:29:42.530" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26126" PostTypeId="2" ParentId="26124" CreationDate="2023-02-10T17:38:50.090" Score="9" Body="&lt;p&gt;You will find two schools of thought about teaching algebra 1 in grade 8. One can be characterized by the Robert Moses &amp;quot;Algebra for All&amp;quot; philosophy. See this article by Dubinsky and Moses:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ams.org/notices/201103/rtx110300401p.pdf&quot; rel=&quot;noreferrer&quot;&gt;Philosophy, Math Research,&#xA;Math Ed Research, K–16&#xA;Education, and the Civil&#xA;Rights Movement: A Synthesis&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In particular, see the section towards the end of the paper titled &lt;em&gt;The Algebra Project and the Civil Rights Movement&lt;/em&gt; and the conclusion. In this school of thought, having &lt;em&gt;all&lt;/em&gt; students take algebra 1 in grade 8 is most equitable. Of course this hinges on infrastructure to provide adequate foundations before grade 8, as well as infrastructure to teach algebra 1 in grade 8 to all students.&lt;/p&gt;&#xA;&lt;p&gt;The other school of thought is that algebra 1 should &amp;quot;never&amp;quot; be taught in grade 8. Deciding who has access to algebra 1 in grade 8 and then providing this selective access is framed by institutional societal inequities, and motivated by the so called &amp;quot;rush to take calculus in grade 12.&amp;quot; This was a source of debate as California revises it math framework. See for example this article:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://edsource.org/2022/california-revises-new-math-framework-to-keep-backlash-at-bay/669010&quot; rel=&quot;noreferrer&quot;&gt;California revises new math framework to keep backlash at bay--&#xA;&lt;strong&gt;Offering algebra in eighth grade remains a local choice&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The work of the Dana Center Launch Years Initiative is relevant to this discussion. See&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.utdanacenter.org/our-work/k-12-education/launch-years-initiative/launch-years-initiative-course-frameworks&quot; rel=&quot;noreferrer&quot;&gt;Launch Years Initiative Course Frameworks&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Read at least the Executive Summary. This initiative is not without its own controversy, but I think it brings valuable ideas to the discussion.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-02-10T17:38:50.090" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26127" PostTypeId="2" ParentId="26124" CreationDate="2023-02-11T19:55:49.467" Score="6" Body="&lt;p&gt;Supposing for a moment that the answer to that question is a resounding, inexorable &lt;b&gt;&amp;quot;Yes!&amp;quot;&lt;/b&gt; - these subjects are taught a year too soon. I think it's very important to ask the next question, which is no less important, &lt;b&gt;why is it too soon?&lt;/b&gt; and here I think it is important to distinguish two totally different options:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;There is something &lt;b&gt;inherent&lt;/b&gt; in the age and intellectual development that happens up to that age, that prevents students from dealing with the subjects adequately.&lt;/li&gt;&#xA;&lt;li&gt;There is something &lt;b&gt;inherently wrong&lt;/b&gt; in the mathematical education that the students go through &lt;b&gt;before&lt;/b&gt; they reach that age.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Now, I have my own opinion about which one is more likely - but that's not the point. I just want to point out that it's important to approach this question in an unprejudiced way.&lt;/p&gt;&#xA;" OwnerUserId="21006" LastActivityDate="2023-02-11T19:55:49.467" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26128" PostTypeId="1" AcceptedAnswerId="26130" CreationDate="2023-02-12T13:21:38.853" Score="7" ViewCount="3809" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt; Solve &lt;span class=&quot;math-container&quot;&gt;$x+5=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A:&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x+5=0\implies x=-5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This answer would be given full marks.&lt;/p&gt;&#xA;&lt;p&gt;Isn’t it better to tell students to use &lt;span class=&quot;math-container&quot;&gt;$\equiv$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\iff$&lt;/span&gt;? Because that is what lets them say &lt;span class=&quot;math-container&quot;&gt;$-5$&lt;/span&gt; is a solution to the equation. One big upside of this would be that students would never be confused about why extraneous solutions may be introduced when squaring both sides of an equation.&lt;/p&gt;&#xA;" OwnerUserId="20949" LastEditorUserId="-1" LastEditDate="2023-05-22T19:46:19.203" LastActivityDate="2023-05-22T19:46:19.203" Title="What implication arrows, if any, should I require in teaching?" Tags="|algebra|proofs|notation|" AnswerCount="4" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="26129" PostTypeId="2" ParentId="26128" CreationDate="2023-02-12T18:27:55.483" Score="9" Body="&lt;p&gt;I think there's no good reason to insist on &lt;span class=&quot;math-container&quot;&gt;$\iff$&lt;/span&gt;. First of all the obvious: &lt;span class=&quot;math-container&quot;&gt;$p\iff q$&lt;/span&gt; always implies &lt;span class=&quot;math-container&quot;&gt;$p\implies q$&lt;/span&gt;, but not the other way around.&#xA;Since beginning students may not have that totally nailed down yet it's better for them to err on the side of &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; rather than mistakenly write &lt;span class=&quot;math-container&quot;&gt;$\iff$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now about extraneous solutions, I think that technique wouldn't help much anyway because the point is that if you square the equation and get two solutions &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt; you still need to &lt;em&gt;check&lt;/em&gt; which of them satisfies the original equation, whether you are aware of the &amp;quot;broken&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$\iff$&lt;/span&gt; relation or not. So actually you are more likely to check if you are aware that it is &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; all the way through, since it is a reminder that the equation implies the solutions, but &lt;em&gt;not&lt;/em&gt; the other way around.&lt;/p&gt;&#xA;&lt;p&gt;In other words, I think what's more important is to understand that when we solve an equation, we are interested in the solution, and therefore the direction of &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; is what interests us, that is, we are interested in making a series of logical steps that &lt;b&gt;imply one or more &lt;em&gt;possible&lt;/em&gt; solutions&lt;/b&gt;. For that exact same reason, we need to check which of our solutions satisfy the equation if we get more than a single one.&lt;/p&gt;&#xA;&lt;p&gt;Of course, at the more advanced stage the student would know ahead of time how many solutions to expect, but I'm assuming the question is not concerned with that level of math yet. Although even at an early stage, it is quite easy to explain to the student that an equation of integer degree &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, that is containing &lt;span class=&quot;math-container&quot;&gt;$x^k$&lt;/span&gt; as the highest order term in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, can have &lt;b&gt;at most&lt;/b&gt; &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; solutions - this can just be given as a simple &amp;quot;axiom&amp;quot; at first, with a promise to be explained more deeply some time later...&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Edit:&lt;/b&gt; After thinking more about this, I find it important to add that as we know, it's taken for granted that we don't teach propositional logic at an early age. I'm not sure if we &lt;em&gt;should&lt;/em&gt; teach this. (At that age, it is certainly in the area of &amp;quot;experimental education&amp;quot; as far as I know). However, if it happens that the young students &lt;b&gt;are&lt;/b&gt; taught propositional logic, then it recasts the question in a totally different light. In that case, I would say it would make much more sense to expect the student to be able to distinguish &lt;span class=&quot;math-container&quot;&gt;$\iff$&lt;/span&gt; steps from &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; steps, and that it would in general aid them, not only in algebra equations but also in all areas of math. So perhaps this question of whether and how to introduce propositional logic early on, is one that should precede questions about the &lt;em&gt;usage&lt;/em&gt; of propositional logic.&lt;/p&gt;&#xA;" OwnerUserId="21006" LastEditorUserId="21006" LastEditDate="2023-02-12T23:37:56.390" LastActivityDate="2023-02-12T23:37:56.390" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="26130" PostTypeId="2" ParentId="26128" CreationDate="2023-02-13T03:25:35.337" Score="13" Body="&lt;p&gt;What does and doesn't work in math education is an empirical question. For instance, how math is taught in Singapore means what works for those students may be different than how math works and what is taught in Chicago. So I'd push back on any inclination that education needs to be this or that way when it comes to notation.&lt;/p&gt;&#xA;&lt;p&gt;I think the best approach might be to explain the difference between arithmetic equality, logical equivalence (either with three lines or bidirectional arrow) all of which are &lt;a href=&quot;https://en.wikipedia.org/wiki/Equivalence_relation&quot; rel=&quot;noreferrer&quot;&gt;equivalence relations&lt;/a&gt;. Thus, to use either of these is to emphasize the equivalence relation, which entails reflexivity, symmetry, and transitivity. To use the &lt;a href=&quot;https://en.wikipedia.org/wiki/Material_conditional&quot; rel=&quot;noreferrer&quot;&gt;material conditional&lt;/a&gt; either as an 'if-then' or 'because' construction in natural language emphasizes &lt;a href=&quot;https://en.wikipedia.org/wiki/Logical_consequence&quot; rel=&quot;noreferrer&quot;&gt;logical consequence&lt;/a&gt;. So, when you're teaching solving for unknowns, I'd start with logical consequence and then move to explaining how the bi-conditional constitutes identity. And, like this post, I'd do in plain, natural language.&lt;/p&gt;&#xA;&lt;p&gt;Part of the difference in how you approach this is a function of the level of sophistication of the student. A really bright student will hear this once or twice and understand, whereas someone who struggles may be confused by the distinctions in symbols since they've never likely been explained this. Certainly not at the elementary level, probably not at the secondary level, and probably poorly at the higher educational level. The use of the material conditional, for me, is superior because it emphasizes process, rather than entity. An arrow means intuitively, start here and end there, and with each use, one can talk about why we move from the initial to final state. With the equals symbol, it may be a little more mysterious why the statement changes. To really aid a student, you can write a brief reason above the left-to-right arrow. For instance, if you have substituted a constant value into a variable, simply mark &amp;quot;SUB x&amp;quot; over the arrow. If you have have changed the order of an addend and adder, you simply write &amp;quot;COMM ADD&amp;quot; to say you have invoked the commutativity of addition as your justification.&lt;/p&gt;&#xA;&lt;p&gt;In this way, much like a two-column proof where a rationale is provided, but less tabular in form, you can layout the concepts behind the symbolism. One of the chief problems with poor math education is an over-use of symbols and an under-use of natural language to explain. Thus, there are hordes of students who can manipulate symbols successfully, and have no &lt;a href=&quot;https://en.wikipedia.org/wiki/Symbol_grounding_problem&quot; rel=&quot;noreferrer&quot;&gt;semantic grounding&lt;/a&gt;, and hence no real understanding of why they do what they do.&lt;/p&gt;&#xA;&lt;p&gt;So, no matter what symbols you use, just ensure students understand why.&lt;/p&gt;&#xA;" OwnerUserId="18201" LastActivityDate="2023-02-13T03:25:35.337" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26131" PostTypeId="2" ParentId="26124" CreationDate="2023-02-13T04:03:31.483" Score="4" Body="&lt;p&gt;My (Math) student teaching hours were split between two extreme polar opposite high school classes: The AP Calculus class, and the General Math, which was very basic. The basic class was doing tax forms. The AP Calculus class was doing derivatives of tangents (or something). I was in the AP track. I never learned taxes at school. My mom showed me early. I just kept hoping those kids moms like mine. Summary: Flawed system. They need to ensure the kids have what they need before they overfill them with other concepts that yes- they may not be just yet ready for.&lt;/p&gt;&#xA;" OwnerUserId="21028" LastEditorUserId="21028" LastEditDate="2023-02-13T04:06:54.757" LastActivityDate="2023-02-13T04:06:54.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26132" PostTypeId="2" ParentId="26128" CreationDate="2023-02-13T09:54:17.703" Score="2" Body="&lt;p&gt;The identity symbol would be a poor choice as this is frequently used as a convenient notation for &amp;quot;for all x&amp;quot;.  Overloading it with a second meaning for &amp;quot;equivalent propositions&amp;quot; would be a poor notation.&lt;/p&gt;&#xA;&lt;p&gt;While symbolic manipulation is a useful skill, it isn't &amp;quot;teaching algebra&amp;quot;.&#xA;Whether you use implication symbols, therefore triglyphs or words like &amp;quot;so&amp;quot;  misses the point that the equation &amp;quot;x+5=0&amp;quot; expresses some property of a number or set of numbers. That you are answering the question &amp;quot;What is the number/are the numbers that when you add five to them gives you the value zero?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Cover up methods &amp;quot;[finger] + 5 = 0.  The thing under my finger must have a value of -5.  So x = -5.&amp;quot; shift students away from memorising manipulation &amp;quot;rules&amp;quot; to understanding what the meaning of the equation.&lt;/p&gt;&#xA;&lt;p&gt;These methods are scaffolding, and at some point, efficient methods of symbolic manipulation can be introduced. Or rather, aspects of the symbolic manipulation can be introduced incrementally.&lt;/p&gt;&#xA;&lt;p&gt;When it finally becomes significant, for example in a solution of |x+2|=2x-1 or 3sin(x)=2cos(x) by &amp;quot;squaring both sides and using the pythagorian identity&amp;quot; students may be ready for careful use of implication symbols, and understanding how this can create addition &amp;quot;solutions&amp;quot;.  For students at the &amp;quot;x+5=0&amp;quot; level, that would merely be confusing.&lt;/p&gt;&#xA;&lt;p&gt;Therefore my answer is &amp;quot;no&amp;quot;,  when teaching algebra, you should encourage understanding of the structure over formal symbolic manipulation.&lt;/p&gt;&#xA;" OwnerUserId="5361" LastActivityDate="2023-02-13T09:54:17.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26133" PostTypeId="2" ParentId="26124" CreationDate="2023-02-13T10:47:53.083" Score="4" Body="&lt;p&gt;A bit of a tangent from a different school system, and with no studies to back it up...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My suspicion is that by ever trying to cram more stuff in earlier, things get a lighter treatment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My experience is not as teacher, but as parent in the German school system which switched from a 13-year education in my state to 12-years (and back to a voluntary 13-years). I myself had the 13 years, and my state school system was considered the toughest in Germany, back then (all pre-Pisa, when states had individual power over their system).&lt;/p&gt;&#xA;&lt;p&gt;In the 12 years, it was excessively obvious that cramming more stuff in is bad. Speaking bluntly, pupils quickly learned to forget everything after a test, to make space for the next &amp;quot;content&amp;quot;, to be forgotten after the next test, and so on. Teachers obviously had a very hard time stuffing their teachings into those short years. There was very bad filtering between 4th and 5th grade (where pupils choose one of three higher school types).&lt;/p&gt;&#xA;&lt;p&gt;As far as I know, this was one of the main motivations to give schools the option to offer 13-years again, next to 12-years. We'd rather have a little older children enter the workforce or uni, than totally burned out younglings (who were not even adults when entering uni, often!).&lt;/p&gt;&#xA;&lt;p&gt;So yes, cramming more stuff seems to be bad.&lt;/p&gt;&#xA;&lt;p&gt;In contrast, if I recall my own time in school, I clearly remember that I never had to try hard (especially in maths); I had good marks in maths, not great, mainly because I was very lazy and basically never prepared anything at home at all. I found the stuff interesting, which is the main reason I attribute the good marks to; the secondary reason being that there always was more than enough time for everything.&lt;/p&gt;&#xA;&lt;p&gt;Comparing that to my children's maths (in the 12-years curriculum), everything seems relatively unconnected and compressed. I find it very hard to see how someone (especially if they are not high-flyers in maths) would take anything much of interest from it, or even come along for the ride, easily.&lt;/p&gt;&#xA;&lt;p&gt;So if I were to fight the good fight, my main focus would be to reduce the amount of mandatory &amp;quot;stuff&amp;quot;, and give everything more time to sink in (maybe offer optional knowledge for high achievers). Also, there are many low-hanging fruits in the teaching itself (you don't write anything about that, but for my kids they occasionally had multiple-choice-tests in maths?! You got to be kidding me). Motivation trumps everything.&lt;/p&gt;&#xA;&lt;p&gt;The issue now is, does switching everything one year later help with providing more space? Or are you then just concentrating more content into the latter years? Even, or especially in the second half of adolescence, pupils have other stuff on their mind than school, and even some younger 2x-year-olds are quite child-like.&lt;/p&gt;&#xA;" OwnerUserId="8146" LastActivityDate="2023-02-13T10:47:53.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26134" PostTypeId="2" ParentId="26128" CreationDate="2023-02-13T18:38:09.560" Score="6" Body="&lt;p&gt;Whatever symbol you use probably doesn't matter very much, it's how you explain it that matters.&lt;/p&gt;&#xA;&lt;p&gt;If you tell your students to write &amp;quot;⟹&amp;quot; between the equations then they will write that because you told them to; the same goes for &amp;quot;⟺&amp;quot; or anything else. But if you want them to understand the conceptual meaning of this symbol then you have to teach them that concept. Teaching them correct use of a symbol and then showing correct examples of how that symbol is used is not sufficient to teach the concept that the symbol represents.&lt;/p&gt;&#xA;&lt;p&gt;There is a related and widespread misconception among high-school students, that the equals symbol &amp;quot;=&amp;quot; means &amp;quot;the answer is&amp;quot;. That's probably because in the examples they are taught, &amp;quot;=&amp;quot; appears before the answer, and when they wrote an answer like &amp;quot;−5&amp;quot; their teacher told them they should have written &amp;quot;x = −5&amp;quot; instead. This leads to students writing things like &amp;quot;12 + 5 = 17 × 3 = 51&amp;quot; because 17 is the answer to 12 + 5, and 51 is the answer to 17 × 3. Such students might also reject or be confused by equations like &amp;quot;17 = 12 + 5&amp;quot;, because 17 isn't something that needs to be worked out, and 12 + 5 is not an answer to it.&lt;/p&gt;&#xA;&lt;p&gt;Likewise, I think that high-school students are unlikely to learn the concept of logical implication from the procedural rule that says they are supposed to write &amp;quot;⟹&amp;quot; in between steps when they are working out a problem. If your students need to understand logical implication, I think it would be better to teach it as its own concept first, and then incorporate it into how they solve algebraic problems later. On the other hand, if they don't need to understand logical implication yet, then you shouldn't insist on them using a symbol to represent it.&lt;/p&gt;&#xA;&lt;p&gt;In case you read that previous paragraph and thought &amp;quot;but &lt;em&gt;of course&lt;/em&gt; students must understand logical implication in order to solve algebraic equations!&amp;quot;: no, they don't. They solve algebraic equations using a set of procedures and rules you have taught them. Understanding logical implication may be necessary to justify why those procedures and techniques are correct; but you aren't asking them to justify that. As far as they're concerned, &amp;quot;write ⟹ between the steps&amp;quot; is just another rule, or &amp;quot;write ⟺ between the steps, except write ⟹ when you square both sides&amp;quot; is a more complicated rule. Students can learn these rules and follow them without understanding that they produce justifications for their answers.&lt;/p&gt;&#xA;" OwnerUserId="19245" LastActivityDate="2023-02-13T18:38:09.560" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26136" PostTypeId="1" CreationDate="2023-02-14T17:47:44.143" Score="5" ViewCount="1016" Body="&lt;p&gt;My daughter (a biologist) is presently teaching also math at a middle school (9th grade, so about 14 years olds). Now the topic in probability seems to be the law of large numbers! More and more I doubt that is a good topic for that age group.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It is a limit result, the children have no prior experience with limits or asymptotes ...&lt;/li&gt;&#xA;&lt;li&gt;how to make good relevant exercises?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I was shown an exercise meant for the children:  First, they are asked (from a drawing of all outcomes) to find the probability of two head when throwing a fair coin 4 times. Then they are asked what they believe the probability of exactly 500 head when throwing 1000 times (about 2.5%) ... I find this a strange way to do it! So, two parts to the question:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If you must introduce the law of large numbers in 9th grade, how?&lt;/li&gt;&#xA;&lt;li&gt;Is that really appropriate?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="122" LastActivityDate="2023-02-14T19:14:29.600" Title="Law of large numbers as a middle school topic?" Tags="|probability|middle-school|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26137" PostTypeId="2" ParentId="26136" CreationDate="2023-02-14T19:10:56.843" Score="6" Body="&lt;p&gt;You have two different questions.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Should?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I don't think your daughter has a choice here.  So it's irrelevant to her.  Obviously society has a choice.  I don't think the topic is so awful, personally.  It can be understood intuitively without formal limit training.  After all, we have plenty of college prob/stat courses that are non calculus based.  And I have personally been involved in Six Sigma training and the like in a manufacturing environment, where intuitive and phenomenological training was done, sometimes even with operators, but even at the professional level in a very &amp;quot;no limits needed&amp;quot; environment.&lt;/p&gt;&#xA;&lt;p&gt;Of course there is the issue of what it displaces, but that's a tricky topic.  I can think of Peggy Sue Got Married, where she tells her teacher that she knows from the future that she really won't ever need algebra!  Clearly there is a bit of a fad/trend to play with the stats more, earlier.  I kind of like it...since many kids will never get to calculus or be STEM superstars and this gives them a quick exposure to intuitive probs/stats so they have some better understanding of things in the newspaper.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;How?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This is not a de novo problem.  You/she can Google and find different resources.&lt;/p&gt;&#xA;&lt;p&gt;Example:    &lt;a href=&quot;https://www.pbslearningmedia.org/resource/vtl07.math.data.col.lplawlarge/probability-and-the-law-of-large-numbers/&quot; rel=&quot;noreferrer&quot;&gt;https://www.pbslearningmedia.org/resource/vtl07.math.data.col.lplawlarge/probability-and-the-law-of-large-numbers/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For one thing, there are some nice (non rigorous) videos on the topic.&lt;/p&gt;&#xA;&lt;p&gt;I would start by looking at existing approaches versus speculating on how to do it.&lt;/p&gt;&#xA;&lt;p&gt;But with the proverbial gun at my head, I think the method she has is OK, but I would even just run a class exercise with 4 and then with 10 flips and then with 20 flips.  Have the kids pair up, so they can socialize a little.  And then do the histogram of class results on the board or with an overhead (same scale y scale as percentage, but align the endpoints of the x scale)  and with the graphs superimposed or aligned above and below each other.  I think after that, ask them what they think happens with 500 flips.    Another provocation (US audience) is to ask if you have a sports team with a 55% chance of winning/game and it is single elimination (like Super Bowl last Sunday) or if it is best 4 out of 7, like the NBA.  Which favors the better team more, which favors the lesser team more?  (Don't prove the algebra, just let them debate it a little.)&lt;/p&gt;&#xA;" OwnerUserId="21039" LastEditorUserId="21039" LastEditDate="2023-02-14T19:14:29.600" LastActivityDate="2023-02-14T19:14:29.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26138" PostTypeId="2" ParentId="26119" CreationDate="2023-02-14T19:24:55.533" Score="2" Body="&lt;p&gt;An answer to both question and first comment:&lt;/p&gt;&#xA;&lt;p&gt;Usually, numbers or the letters phi, omega are selected to denote angles, while x and y are the standard symbols for independent and dependent variables in functions.&lt;/p&gt;&#xA;&lt;p&gt;See for example the 10th grade (15 years old) geometry book: &lt;a href=&quot;http://ebooks.edu.gr/ebooks/v/pdf/8547/2608/22-0236-01_Eukleideia-Geometria_A-Lykeiou_Vivlio-Mathiti/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ebooks.edu.gr/ebooks/v/pdf/8547/2608/22-0236-01_Eukleideia-Geometria_A-Lykeiou_Vivlio-Mathiti/&lt;/a&gt; Theorem 1 at page 26,&lt;/p&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;p&gt;The Algebra book of the same grade at &lt;a href=&quot;http://ebooks.edu.gr/ebooks/v/pdf/8547/2466/22-0270-01_Algebra-kai-Stoicheia-Pithanotiton_A-Lykeiou_Vivlio-Mathiti/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ebooks.edu.gr/ebooks/v/pdf/8547/2466/22-0270-01_Algebra-kai-Stoicheia-Pithanotiton_A-Lykeiou_Vivlio-Mathiti/&lt;/a&gt; Definition at page 146.&lt;/p&gt;&#xA;" OwnerUserId="8643" LastActivityDate="2023-02-14T19:24:55.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26139" PostTypeId="1" CreationDate="2023-02-14T20:28:12.313" Score="8" ViewCount="238" Body="&lt;p&gt;I have always been passionate and fascinated with maths, my job revolves around the subject, but I'm not an educator. Today I met the 8 year old son of a friend, I had the opportunity to speak to him for only a few mins, in that time he told me he hated maths at school and he didn't see the point of it.&lt;/p&gt;&#xA;&lt;p&gt;I wanted to say something that would inspire him, but I was stumped, as the examples that I was thinking of in the moment were too complicated/abstract for an 8 year old, and especially one that appears to have no interest in maths.&lt;/p&gt;&#xA;&lt;p&gt;He's football (soccer) mad, so I'm trying to think up related examples to show him that maths is more than learning times tables, in a manner he can grasp.&lt;/p&gt;&#xA;&lt;p&gt;I really want to do our subject justice, and at least give this little boy some inspiration - and who knows it might change his mind on the subject.&lt;/p&gt;&#xA;&lt;p&gt;I'm sure I'll have the opportunity to meet him again in the future and I'd love to be armed with things I can say - only a few sentences, that could really help to make him think more about maths.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="21040" LastActivityDate="2023-03-10T06:31:08.713" Title="A few quick sentences to inspire an 8 year old in Maths" Tags="|primary-education|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26140" PostTypeId="2" ParentId="26139" CreationDate="2023-02-15T07:44:31.960" Score="5" Body="&lt;p&gt;It is a fine thing to wish to do, but likely not feasible. You might consider why they hate mathematics and do not see the point of it.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Maybe it is just a thing they are used to saying, with no particular thought, emotion or commitment behind it.&lt;/li&gt;&#xA;&lt;li&gt;Maybe their identity/role/storyline is such that they are supposed to hate mathematics, since others with similar identity/role/storyline also hate mathematics. To act against it requires effort, bravery and active work with own's own identity. This is not quite where 8-year olds tend to be.&lt;/li&gt;&#xA;&lt;li&gt;Maybe their experiences with school mathematics are boring, difficult and irrelevant.&lt;/li&gt;&#xA;&lt;li&gt;Maybe their mathematics teacher(s) assume they are bad at mathematics and they have adopted the attitude.&lt;/li&gt;&#xA;&lt;li&gt;Maybe they just are not very good at mathematics (at this point in their life, for whatever reasons) and they defend their sense of self by hating mathematics; they can still be competent at football, for example, and thus see themselves in a good light.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These are pretty difficult to change quickly, especially if you not in a teacher-like position.&lt;/p&gt;&#xA;&lt;p&gt;In particular, anything relying on future benefits the 8-year old can not easily visualize are unlikely to go through. Sure, should would could, but are they really going to believe, understand, remember and be motivated by some adult saying something will happen in ten or twenty years?&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2023-02-17T14:21:10.333" LastActivityDate="2023-02-17T14:21:10.333" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26141" PostTypeId="2" ParentId="11963" CreationDate="2023-02-15T17:58:23.457" Score="2" Body="&lt;p&gt;One might consider organizing a seminar with the theme:&lt;/p&gt;&#xA;&lt;p&gt;Mathematical Insights Into Fairness Questions&lt;/p&gt;&#xA;&lt;p&gt;Examples of such questions in recent news involve&lt;/p&gt;&#xA;&lt;p&gt;a. Fair ways to share diminished amounts of water in the Colorado River due to drought conditions&lt;/p&gt;&#xA;&lt;p&gt;b. School choice&lt;/p&gt;&#xA;&lt;p&gt;c. Improved ways to conduct primary elections when there are many candidates&lt;/p&gt;&#xA;&lt;p&gt;d. Rent sharing systems&lt;/p&gt;&#xA;&lt;p&gt;e. Medicine costs&lt;/p&gt;&#xA;&lt;p&gt;etc.&lt;/p&gt;&#xA;&lt;p&gt;Mathematical topics would include vertex/edge graph theory, solving difference equations (recursions),linear programming and other optimization methods, ham sandwich ideas, etc. Both continuous and discrete mathematical ideas come into play.&lt;/p&gt;&#xA;&lt;p&gt;This approach might help students see how mathematics affects their daily lives.&lt;/p&gt;&#xA;" OwnerUserId="1865" LastActivityDate="2023-02-15T17:58:23.457" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26142" PostTypeId="1" AcceptedAnswerId="26143" CreationDate="2023-02-15T20:47:47.177" Score="30" ViewCount="2833" Body="&lt;p&gt;When teaching students how to compute the difference quotient in a precalculus or calculus class, we need them to evaluate the expression&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{f(x+h) - f(x)}{h}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;for various simple functions, like linear and quadratic functions. Let's say we are using the function &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2 + x + 1$&lt;/span&gt; for example.&lt;/p&gt;&#xA;&lt;p&gt;However, students find this to be more or less impossible. Specifically, when trying to find &lt;span class=&quot;math-container&quot;&gt;$f(x+h)$&lt;/span&gt;, students do all sorts of crazy things while computing the difference quotient. They replace &lt;span class=&quot;math-container&quot;&gt;$f(x+h)$&lt;/span&gt; in the expression with things like:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x^2 + x + 1 + h)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$(x+h)^2 + x + 1$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2 + x + 1 + h$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;which are all clearly wrong.&lt;/p&gt;&#xA;&lt;p&gt;However, if you ask these same students this question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Find &lt;span class=&quot;math-container&quot;&gt;$f(\text{Tomato Soup})$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;they are happy to do so and do it successfully.&lt;/p&gt;&#xA;&lt;p&gt;In fact, my experience is that they can actually also complete this question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Find &lt;span class=&quot;math-container&quot;&gt;$f(xxx)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is also easy for them:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Find &lt;span class=&quot;math-container&quot;&gt;$f(y)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, this question is a whole different story for them:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Find &lt;span class=&quot;math-container&quot;&gt;$f(x+h)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Why do students see &lt;span class=&quot;math-container&quot;&gt;$f(x+h)$&lt;/span&gt; as fundamentally different from the others? What series of questions or conversations can be used to help them?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-02-26T06:02:37.107" Title="f(x+h) in the difference quotient" Tags="|algebra|precalculus|" AnswerCount="6" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="26143" PostTypeId="2" ParentId="26142" CreationDate="2023-02-15T21:21:15.330" Score="40" Body="&lt;p&gt;You have already applied some good diagnostic tests.  I recommend the following additional diagnostics&lt;/p&gt;&#xA;&lt;p&gt;What happens if you ask them to evaluate each of the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(3y)$&lt;/span&gt;:  Passing this test indicates that they can think of a monomial as an input. I am guessing your students will pass this test given the other diagnostics you have run.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(3x)$&lt;/span&gt;:  A student could pass the first test and fail this one if they don't think it is &amp;quot;kosher&amp;quot; to give &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; an input of &lt;span class=&quot;math-container&quot;&gt;$3x$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; was defined by &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2 + x + 1$&lt;/span&gt;.  They are experiencing &lt;strong&gt;variable scope anxiety&lt;/strong&gt;. They are getting &amp;quot;the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$3x$&lt;/span&gt;&amp;quot; confused with &amp;quot;the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;&amp;quot;.  To be more technical about it we should really say &amp;quot;The function &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{R} \to \mathbb{R}$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$\forall x \in \mathbb{R}, f(x) = x^2+x+1$&lt;/span&gt;&amp;quot;.  The &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &lt;strong&gt;bound&lt;/strong&gt; by the universal quantifier.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(t+3)$&lt;/span&gt;:  Passing this test indicates that they can think of a binomial as a &amp;quot;single input&amp;quot;.  A student who fails this test is not thinking of &lt;span class=&quot;math-container&quot;&gt;$(t+3)$&lt;/span&gt; as a &amp;quot;single thing&amp;quot;.  They might also have difficulty distributing &lt;span class=&quot;math-container&quot;&gt;$(t+3)(a+b) = (t+3)a + (t+3)b$&lt;/span&gt; since they cannot conceptualize the &lt;span class=&quot;math-container&quot;&gt;$(t+3)$&lt;/span&gt; as a single number to be distributed.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x+3)$&lt;/span&gt;:  Again, a student could fail this test while passing the third test because of variable scope anxiety.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(j+h)$&lt;/span&gt;:  A student could fail this test while passing all of the other tests because they think that a function should apply to only &amp;quot;one variable at a time&amp;quot;.  They view &lt;span class=&quot;math-container&quot;&gt;$x+3$&lt;/span&gt; as a legit input since it is a single expression involving one variable.  They freak out at &lt;span class=&quot;math-container&quot;&gt;$f(j+h)$&lt;/span&gt; because they are not sure which variable they should be applying &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to.  In essence they are seeing a function being applied to an expression with two variables and think that this will make &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; a &amp;quot;two variable function&amp;quot; even though it is only a one variable function.  They don't know what to do with the &amp;quot;second input&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x+h)$&lt;/span&gt; Again, a student could fail this test while passing test #5 because of variable scope confusion.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;My success rate at diagnosing these issues is pretty good, but I have a poor cure rate...&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2023-02-15T21:31:20.387" LastActivityDate="2023-02-15T21:31:20.387" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26145" PostTypeId="2" ParentId="26142" CreationDate="2023-02-15T21:52:52.670" Score="20" Body="&lt;p&gt;I know a teacher who (at least in the past) would &lt;em&gt;require&lt;/em&gt; students to write underlined blanks in place of the input whenever they were evaluating a function from its formula:&lt;/p&gt;&#xA;&lt;p&gt;[Examples with &lt;span class=&quot;math-container&quot;&gt;$f(x)=5x^2-3x+1$&lt;/span&gt;]&lt;/p&gt;&#xA;&lt;p&gt;Evaluate &lt;span class=&quot;math-container&quot;&gt;$f(6)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Answer:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align} f(6) &amp;amp;= 5(\text{_______})^2 - 3(\text{_______}) +1 \\&#xA;&amp;amp;= 5(6)^2 - 3(6) + 1\\&#xA;&amp;amp;= \dots\end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Evaluate &lt;span class=&quot;math-container&quot;&gt;$f(3+h)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Answer:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align} f(3+h) &amp;amp;= 5(\text{_______})^2 - 3(\text{_______}) +1 \\&#xA;&amp;amp;= 5(3+h)^2 - 3(3+h) + 1\\&#xA;&amp;amp;= \dots\end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This wouldn't stop a student from mishandling the &lt;span class=&quot;math-container&quot;&gt;$(3+h)^2$&lt;/span&gt;, but it might get the &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; in the right place.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2023-02-15T21:52:52.670" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26146" PostTypeId="1" CreationDate="2023-02-15T23:02:14.637" Score="3" ViewCount="161" Body="&lt;p&gt;If a field L extends a subfield K then L is called an &lt;strong&gt;extension&lt;/strong&gt; of K and K is called the extension's &lt;strong&gt;base field&lt;/strong&gt;. See &lt;a href=&quot;https://en.wiktionary.org/wiki/extension_field#English&quot; rel=&quot;nofollow noreferrer&quot;&gt;extension field&lt;/a&gt; for a definition.&lt;/p&gt;&#xA;&lt;p&gt;What is the analog of &amp;quot;base field&amp;quot; when it's a function being extended?&lt;/p&gt;&#xA;&lt;p&gt;That is, if a function L &lt;a href=&quot;https://en.wikipedia.org/wiki/Restriction_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;extends&lt;/a&gt; a given function K then L is called an &lt;strong&gt;extension&lt;/strong&gt; of K and the original function K is called the extension's &lt;strong&gt;____?&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The word &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Restriction_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;restriction&lt;/a&gt;&amp;quot; does not work because no one calls K &amp;quot;the extension's restriction&amp;quot; and I don't want to make up new terminology that no one else uses. Whenever the word &amp;quot;restriction&amp;quot; is applied to a function, then the set that it is being restricted to is always either explicitly mentioned or else clear from context. For example, people often write &amp;quot;the function's restriction to this-or-that set&amp;quot;, and if the set is clear from context then they &lt;em&gt;might&lt;/em&gt; simply refer to it as &amp;quot;the function's restriction&amp;quot;. But (at least based on my Internet searches), no one writes &amp;quot;the function's restriction&amp;quot; without somehow making clear the set that it is being restricted to.&lt;/p&gt;&#xA;&lt;p&gt;Example usage:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If an extension has the same [this-or-that property] as its ______ then they will also have the same [this-or-that property].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What word fits in the blank?&lt;/p&gt;&#xA;&lt;p&gt;More generally, we have a noun (&amp;quot;extension&amp;quot;) for any thing produced by an act of extending. Is there a noun for the thing that is going to be extended (before the act occurs)?&lt;/p&gt;&#xA;" OwnerUserId="21053" OwnerDisplayName="mgkrupa" LastEditorUserId="376" LastEditDate="2023-02-17T06:39:44.363" LastActivityDate="2023-02-17T06:39:44.363" Title="Word for an object being extended: Given F, a function that extends F is called an extension and F is called the extension __?" Tags="|terminology|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26147" PostTypeId="2" ParentId="26142" CreationDate="2023-02-16T04:12:40.590" Score="3" Body="&lt;p&gt;I think the problem originates from conflicting mathematical notation. In elementary school multiplication is explicitly written out &lt;span class=&quot;math-container&quot;&gt;$a \times b$&lt;/span&gt;. By middle school it becomes just &lt;span class=&quot;math-container&quot;&gt;$ab$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$a(b+c)$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$(b+c)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then functions come in during the last few years of high school. And the notation there is similar to multiplication, and perhaps student thinks &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; should mean &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$f(x+h)$&lt;/span&gt; should be &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$(x+h)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;One solution is to insist on explicit writing of all operations. So &lt;span class=&quot;math-container&quot;&gt;$a\times b$&lt;/span&gt; for multiplication and perhaps &lt;span class=&quot;math-container&quot;&gt;$f \circ (x)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$f \&amp;amp; (x)$&lt;/span&gt;  for &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; of/at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; or something similar. Of course the downside is lengthy writing. Probably &lt;span class=&quot;math-container&quot;&gt;$a\times b$&lt;/span&gt; will be a hard sell, but I think explicit math symbol for &amp;quot;of&amp;quot; or &amp;quot;at&amp;quot; (as in &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;) is very much needed in high school. Using blank space to mean an operation when it is the first time you are introducing that operation and it conflicts with another usage is just questionable.&lt;/p&gt;&#xA;&lt;p&gt;The symbol &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; emerged rather late (in 1734 by Euler) and exclusively for the practicing mathematicians.  The attempt to sell it to high school students appears as a bit of premature activity or oversight.&lt;/p&gt;&#xA;&lt;p&gt;I have used something like Nick C. recalls.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f\circ (\square) = 5\square^2+ 3\square +2&#xA;$$&lt;/span&gt;&#xA;In my precalculus and calculus I classes (instead of the standard &lt;span class=&quot;math-container&quot;&gt;$f(x)=5x^2+3x+2$&lt;/span&gt;). This is to emphasize two points, (a) &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is the name of a function, as flagged by &lt;span class=&quot;math-container&quot;&gt;$\circ$&lt;/span&gt; and it takes an input, (b) the input is whatever is in &lt;span class=&quot;math-container&quot;&gt;$\square$&lt;/span&gt;.  The box helps instruct the student to be careful with substitution. It also avoids the overuse of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We use blank space to indicate an operation in other places as well, for example when it comes to exponents &lt;span class=&quot;math-container&quot;&gt;$b^n$&lt;/span&gt;. This is not as much of a problem because it does not conflict with an earlier usage. However when it comes to &lt;span class=&quot;math-container&quot;&gt;$f^{-1}$&lt;/span&gt; to mean functional inverse or &lt;span class=&quot;math-container&quot;&gt;$f^{(2)}$&lt;/span&gt; to mean the second derivative we run into similar issues in calculus.&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2023-02-18T20:46:31.337" LastActivityDate="2023-02-18T20:46:31.337" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26148" PostTypeId="2" ParentId="26142" CreationDate="2023-02-16T06:22:15.610" Score="8" Body="&lt;p&gt;My solution is something like Nick's. I think it has helped my students, but I don't have hard data.&lt;/p&gt;&#xA;&lt;p&gt;I scribble out the x's. So &lt;span class=&quot;math-container&quot;&gt;$f(x)=5x^2-3x+1$&lt;/span&gt; becomes &lt;span class=&quot;math-container&quot;&gt;$f(scribble)=5scribble^2-3scribble+1$&lt;/span&gt;. And I talk about f of any mess gives us 5 times that mess squared minus 3 times the mess plus 1.&lt;/p&gt;&#xA;&lt;p&gt;I then tell them to circle the x's. For f(x+h) we'll need an (x+h) inside each of those circles, and yes, we'll want those parentheses around it each time.&lt;/p&gt;&#xA;&lt;p&gt;Then I give them a quiz on finding the derivative (and tangent line) from the definition. I let them take this quiz multiple times, hopefully until they get it right. Then they have to do it again on the first test.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-02-16T06:22:15.610" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26149" PostTypeId="2" ParentId="26142" CreationDate="2023-02-16T17:34:17.523" Score="10" Body="&lt;p&gt;The issue here seems to be with &lt;a href=&quot;https://en.wikipedia.org/wiki/Substitution_(logic)&quot; rel=&quot;noreferrer&quot;&gt;substitution&lt;/a&gt;, a fundamental operation in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Specifically, substituting &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with an expression that &lt;em&gt;itself&lt;/em&gt; contains &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The issue of how to perform substitution correctly actually &lt;a href=&quot;https://en.wikipedia.org/wiki/Lambda_calculus#Substitution&quot; rel=&quot;noreferrer&quot;&gt;arises in the lambda calculus&lt;/a&gt; and requires &lt;a href=&quot;https://en.wikipedia.org/wiki/Lambda_calculus#%CE%B1-conversion&quot; rel=&quot;noreferrer&quot;&gt;variable renaming&lt;/a&gt; to handle conflicts.&lt;/p&gt;&#xA;&lt;p&gt;I propose using this idea of renaming to avoid confusion. Specifically, we &amp;quot;break down&amp;quot; the substitution &lt;span class=&quot;math-container&quot;&gt;$x \mapsto x + h$&lt;/span&gt; into two steps: &lt;span class=&quot;math-container&quot;&gt;$x \mapsto y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y \mapsto x + h$&lt;/span&gt;. Here, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is what's called a &lt;em&gt;fresh variable&lt;/em&gt;, that is, a variable that has &lt;em&gt;not&lt;/em&gt; appeared before (and thus avoids any kind of conflict with the expressions we've working with so far). For example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp;&amp;amp; f(x) &amp;amp;= x^2 + x + 1 \\&#xA;x &amp;amp;\mapsto y &amp;amp;&#xA;f(y) &amp;amp;= y^2 + y + 1 \\&#xA;y &amp;amp;\mapsto x + h &amp;amp;&#xA;f(x + h) &amp;amp;= (x + h)^2 + (x + h) + 1&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note: Make sure to emphasize that, when substituting a &lt;em&gt;compound expression&lt;/em&gt; like &lt;span class=&quot;math-container&quot;&gt;$x + h$&lt;/span&gt;, it should be &lt;em&gt;wrapped in parentheses&lt;/em&gt;, like &lt;span class=&quot;math-container&quot;&gt;$(x + h)$&lt;/span&gt;. This ensures that it's treated as a &amp;quot;single unit&amp;quot; in the underlying expression (e.g., avoids changing the order of operations).&lt;/p&gt;&#xA;&lt;p&gt;In contrast, performing &lt;span class=&quot;math-container&quot;&gt;$x \mapsto x + h$&lt;/span&gt; all at once induces an additional &lt;a href=&quot;https://en.wikipedia.org/wiki/Cognitive_load&quot; rel=&quot;noreferrer&quot;&gt;cognitive load&lt;/a&gt;. This cognitive load comes from having to keep track, as the expression is being written out, of which &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;s are &amp;quot;the original &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;s&amp;quot; versus the ones that come from the new expression. In particular, you have to make sure you don't miss an &amp;quot;original &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; &lt;em&gt;and&lt;/em&gt; make sure you don't accidentally replace a &amp;quot;new &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;. For example, consider the following thinking process:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp; x^2 + x + 1 \\&#xA;&amp;amp; \text{We have to replace $x$ with $x + h$.} \\&#xA;&amp;amp; \text{Ok, let's start with the first occurrence of $x$.} \\&#xA;&amp;amp; (x + h)^2 + x + 1 \\&#xA;&amp;amp; \text{Hmm, which of the $x$'s am I supposed to replace now?}&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="21050" LastEditorUserId="21050" LastEditDate="2023-02-16T18:36:53.277" LastActivityDate="2023-02-16T18:36:53.277" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26150" PostTypeId="2" ParentId="26146" CreationDate="2023-02-16T22:14:08.900" Score="2" Body="&lt;p&gt;This might answer your question in a backwards way.&lt;/p&gt;&#xA;&lt;p&gt;We have the notion of &lt;em&gt;germ equivalence&lt;/em&gt; of two functions. Given a point &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, we say that functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; define the same germ at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; if there is a neighborhood &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (open set containing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;) with &lt;span class=&quot;math-container&quot;&gt;$f\vert_{U}=g\vert_U$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Germ equivalence focuses on localization, as opposed to extension which is more of a global focus.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-02-16T22:14:08.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26151" PostTypeId="2" ParentId="26146" CreationDate="2023-02-17T02:49:33.460" Score="3" Body="&lt;p&gt;I'd just say &amp;quot;the original function&amp;quot;, having a pretty good colloquial sense, rather than create a technical word. Yes, if you are considering such stuff in a more extravagant way (e.g., in a vast category-theoretic setting), you might have a need to formalize this relationship, but, in my own experience, just the colloquial language is adequate. Or, in some modest examples, &amp;quot;restriction of the extension&amp;quot;, but I've used this in situations where the &amp;quot;restriction&amp;quot; was not the original thing... For example, fooling around with unbounded self-adjoint operators (already in Sturm-Liouville problems, so it's not at all contrived).&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2023-02-17T02:49:33.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26152" PostTypeId="2" ParentId="26124" CreationDate="2023-02-17T15:46:42.893" Score="2" Body="&lt;p&gt;As others have noted, there are two distinct hypotheses being presented. The first is that younger students are developmentally incapable of learning algebra. This is generally not supported by education research.&lt;/p&gt;&#xA;&lt;p&gt;You may be interested in the research summarized in this chapter from &lt;em&gt;&lt;a href=&quot;https://www.taylorfrancis.com/books/edit/10.4324/9781315097435/algebra-early-grades-james-kaput-david-carraher-maria-blanton?refId=d133c47d-3890-4fe8-a16c-ec7f59adb3c2&amp;amp;context=ubx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra in the Early Grades&lt;/a&gt;&lt;/em&gt; titled &amp;quot;&lt;a href=&quot;https://www.researchgate.net/publication/242682671_Early_Algebra_Is_Not_the_Same_As_Algebra_Early&quot; rel=&quot;nofollow noreferrer&quot;&gt;Early Algebra Is Not The Same As Algebra Early&lt;/a&gt;.&amp;quot; The authors worked with students in an ethnically diverse school in Greater Boston. They conducted weekly early algebra activities, following the students from second grade to fourth grade. Here's an excerpt that I think is relevant:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We witnessed a dramatic shift in students’ thinking over one and one-half years. At the beginning of grade three, students interpreted a story with indeterminate quantities as a single tale about particular people and amounts. By the middle of grade four most students construed this sort of situation as entailing many possible stories involving variable quantities in an invariant functional relationship. A good many of the students made use of algebraic notation to convey the variations and invariance across the stories.&lt;/p&gt;&#xA;&lt;p&gt;The decisive changes in their thinking surprised us. Several years ago, when invited to assess the evidence regarding whether young students could ‘do algebra’, we downplayed the discontinuity between arithmetic (as generally taught in K-8) and algebra. Our initial findings showed that young students could make mathematical generalizations and express them in algebraic notation...&lt;/p&gt;&#xA;&lt;p&gt;Our present findings have convinced us that there is indeed a large leap from thinking in terms of particular numbers and instances to thinking about functional relations. But the fact that most students throughout the United States do not make this transition easily, nor early, may well say more about our failure to offer suitable conditions for them to learn algebra as an integral part of elementary mathematics than it does about the limitations of their mental structures.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course, the success of early algebra programs depends on a variety of factors, and it's not a given that earlier is always better. But according to the authors of this &lt;a href=&quot;https://www.rand.org/content/dam/rand/pubs/working_papers/WR1200/WR1209/RAND_WR1209.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;RAND working paper&lt;/a&gt; who studied the effects of California's push for early algebra:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Our analyses indicate that 8th grade Algebra enrollment has a substantial and lasting positive effect on students’ advanced math course enrollment, and more modest positive effects on students’ mathematics and ELA test scores. Importantly, however, we find evidence to suggest that the effects of 8th grade Algebra vary substantially across schools and placement strategies. In some settings, early Algebra has large and statistically significant positive effects on student achievement and attainment; in other settings, these effects are negative. Descriptive analyses suggest that the effects of early Algebra placement are more positive in schools with larger shares of relatively high-achieving and socioeconomically advantaged students, as well as schools that set high test score thresholds for enrolling in 8th grade Algebra.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The second hypothesis is that students are not adequately prepared for algebra in eighth grade due to bad curriculum design choices. This is significantly harder to address and research, because students' algebra readiness depends on a host of factors besides the design of the intended curriculum. Additionally, there are very few opportunities for comparative longitudinal studies on curriculum.&lt;/p&gt;&#xA;&lt;p&gt;Since quantitative studies are hard to come by, another way to gain insight on this question is to study the history of the Common Core State Standards for Mathematics (CCSSM.) The new standards were motivated in part by the perception that existing U.S. mathematics curricula were inadequate, especially when viewed in international comparison. R. James Milgram and Hung-Hsi Wu, both mathematicians who have had a strong presence in curriculum reform efforts, wrote this in the introduction to &lt;a href=&quot;https://math.berkeley.edu/%7Ewu/six-topics1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;a proposed K-7 mathematics intervention program&lt;/a&gt;, prior to the release of the CCSSM:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Analysis of the results from TIMSS suggests that the U.S. school mathematics curriculum is a mile wide and an inch deep. It covers too many topics and each topic is treated superficially. By contrast, the structure of mathematics instruction in countries which outperformed the U.S. follows a strikingly different pattern. In all cases, only a few carefully selected focus topics are taught and learned to mastery by students in the early grades. At the fourth grade level, since the students in these countries have not been exposed to as broad a curriculum as U.S. students, it sometimes appears on standardized tests such as TIMSS that they perform at a comparable level to U.S. students, but by grade eight the students in the leading countries are far outperforming our students. In fact, key test items already show serious weaknesses in our fourth grade student performances. This difference becomes even greater by the end of high school, where even our top students do not match up well with the average achievement levels of students in these countries.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Wu would go on to become a staunch proponent of the CCSSM. This is what he said in &lt;a href=&quot;https://math.berkeley.edu/%7Ewu/wu2011&quot; rel=&quot;nofollow noreferrer&quot;&gt;a piece for the AFT journal American Educator&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the very purpose of mathematics standards (prior to the CCSMS) seems to be to establish in which grade topics are to be taught. Often, standards are then judged by how early topics are introduced; thus, getting addition and subtraction of fractions done in the fifth grade is taken as a good sign. By the same ridiculous token, if a set of standards asks that the multiplication table be memorized at the beginning of the third grade or that Algebra I be taught in the eighth grade, then it is considered to be rigorous.&lt;/p&gt;&#xA;&lt;p&gt;The CCSMS challenge this dogma. Importantly, the CCSMS do not engage in the senseless game of acceleration—to teach every topic as early as possible—even though refusing to do so has been a source of consternation in some quarters. For example, the CCSMS do not complete all the topics of Algebra I in grade 8 because much of the time in that grade is devoted to the geometry that is needed for understanding the algebra of linear equations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is general support among mathematicians and mathematics educators for the idea that, in the past several decades, U.S. mathematics curriculum standards have been unfocused and shallow. In my estimation, the predominant attitude towards the CCSSM is that the K-8 standards have improved in this regard, though there have been problems with uptake and implementation. However, a group of prominent mathematics educators called for the high school standards to be revised in &lt;a href=&quot;http://steveleinwand.com/wp-content/uploads/2022/01/Proposal-for-Revision-of-HS-Standards-to-NCTM-NCSM-and-ASSM-1-12-22.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;an open letter last year&lt;/a&gt;, stating:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Over the past ten years, mathematics educators have recognized that the K-8 Common Core State Standards, while not perfect and certainly in need of tweaking, are internationally benchmarked, coherently based on research-affirmed learning progressions, fewer and deeper by design, and, most importantly, achievable. In stark contrast, the high school standards are cluttered, outdated, and, because of their scope, unteachable and unachievable for many. They are not based on coherent progressions, they do not accommodate the impact of widely available technology (including computer algebra systems), nor do they articulate equally rigorous differentiated pathways that reflect the broad range of post-secondary mathematical needs. By clinging to increasingly obsolete content, they leave little room for statistics and modeling, and most problematically, they reinforce serious inequities and limit opportunities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Based on my personal experience teaching the high school standards, I agree.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-02-17T15:46:42.893" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26153" PostTypeId="2" ParentId="26124" CreationDate="2023-02-17T23:36:25.560" Score="3" Body="&lt;p&gt;I don't have a research based answer to add, however, I have some anecdotal evidence to offer. My wife has used Singapore math curriculum for the most part with my children. This curriculum integrates algebra in stages from the earliest grade levels. In fact, such is the case with all branches of mathematics. It integrates geometry, proof techniques, symbolic algebra, trigonometry etc.. which is appropriate to the grade level each year. In this way they revisit basic algebra probably 5 to 6 years. In the end it gives students a much deeper understanding of algebra and numbers if you stick with it. Anyway, by 10th or 11th grade the student is ready for calculus. I'm currently teaching my 14 and 16 year old children second semester university Calculus. You might complain, my kids aren't typical. That is true, but what is also true is that there are other kids like my kids who ought to have the opportunity to excel in highschool mathematics which includes calculus (and more for the truly interested kids...) Is calculus best for all students, certainly not. Just like university is a place where a lot less people should go. The ubiquity of university education is the larger problem.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2023-02-17T23:36:25.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26154" PostTypeId="2" ParentId="26124" CreationDate="2023-02-18T00:17:59.277" Score="3" Body="&lt;p&gt;This is another tangential remark, but perhaps (tangentially) relevant.&lt;/p&gt;&#xA;&lt;p&gt;A decade or more ago I was invited to give a math talk to middle-school students in the U.S. (6th, 7th, 8th grades). I talked about &lt;a href=&quot;https://en.wikipedia.org/wiki/Art_gallery_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Art Gallery Theorems&lt;/a&gt;.&#xA;Understanding required the notion of a polygon, and the notion of visibility&#xA;within a polygon, of course both new to them. I walked them through&#xA;many slides of examples of polygons with few vertices:&#xA;triangles, quadrilaterals, pentagons (already getting complicated).&#xA;And interactively we figured out how many guards were sufficient, and how many&#xA;necessary (one-third &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I was surprised and delighted that not only could at least some of the students grasp these&#xA;abstractions (the &amp;quot;necessary and sufficient&amp;quot; concept in particular&#xA;is not easy to grok),&#xA;but they immediately tried to work on the elementary open problem&#xA;I posed (concerning &amp;quot;edge guards&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;My point is that I was underestimating their capabilities. They knew some&#xA;algebra and a little geometry, but guards in art galleries does not&#xA;need much machinery. It just needs basic logic, geometric intuition, and imagination. And they&#xA;could handle that, and (at least a subset) were excited to explore a question whose answer&#xA;was not known (and is still not known).&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2023-02-18T12:08:27.613" LastActivityDate="2023-02-18T12:08:27.613" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26155" PostTypeId="1" CreationDate="2023-02-18T16:08:58.760" Score="0" ViewCount="227" Body="&lt;p&gt;I have unsuccessfully attempted several times over the years to formalize high-school (Euclidean) geometry, or even a working subset of it. Think very simple, diagramless geometry.&lt;/p&gt;&#xA;&lt;p&gt;The usual two-column proofs are quite inadequate for this. Any suggestions would be appreciated. My intended audience is advanced high-school students and first or second year undergrads. As such, the axioms of Tarski or Hilbert would not be helpful. Maybe a workable subset of them to address a limited set of problems.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastActivityDate="2023-02-18T17:44:24.790" Title="How to formalize high-school (Euclidean) geometry?" Tags="|undergraduate-education|secondary-education|geometry|axioms-foundations|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26156" PostTypeId="2" ParentId="26155" CreationDate="2023-02-18T17:44:24.790" Score="5" Body="&lt;p&gt;Clark and Pathania might be of interest to you.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0HdNm.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0HdNm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;This textbook provides a full and complete axiomatic development of exactly that part of plane Euclidean geometry that forms the standard content of high school geometry. It begins with a set of points, a measure of distance between pairs of points and ten simple axioms. From there the notions of length, area and angle measure, along with congruence and similarity, are carefully defined and their properties proven as theorems. It concludes with a proof of the consistency of the axioms used and a full description of their models. It is provided in guided inquiry (inquiry-based) format with the intention that students will be active learners, proving the theorems and presenting their proofs to their class with the instructor as a mentor and a guide.&lt;/p&gt;&#xA;&lt;p&gt;The book is written for graduate and advanced undergraduate students interested in teaching secondary school mathematics, for pure math majors interested in learning about the foundations of geometry, for faculty preparing future secondary school teachers and as a reference for any professional mathematician. It is written with the hope of anchoring K-12 geometry in solid modern mathematics, thereby fortifying the teaching of secondary and tertiary geometry with a deep understanding of the subject.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="5005" LastActivityDate="2023-02-18T17:44:24.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26157" PostTypeId="1" AcceptedAnswerId="26160" CreationDate="2023-02-18T18:24:48.273" Score="6" ViewCount="846" Body="&lt;p&gt;How would you describe the existence of a composite function &lt;span class=&quot;math-container&quot;&gt;$f(g(x))$&lt;/span&gt;in terms of range of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; and domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; . Does range of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; need to be subset of domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; or is it sufficient if the two sets have intersection only? &lt;br&gt;&#xA;I used to define composite function if range of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is a subset of domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and in that case domain of the composite function is same as domain of the function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; but I have come across questions where only intersection exists. In that type of situation domain of composite function can not be the domain of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;. I have this issue related to high school mathematics. Could you please help me to find the most appropriate way of describing the issue to the students relevant to their level of studies . &lt;br/&gt;&#xA;For further clarification here you have examples &lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x) = \sqrt{x-1}$&lt;/span&gt;.&lt;br/&gt;&#xA;Here range of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is a subset of domain of &lt;span class=&quot;math-container&quot;&gt;$f $&lt;/span&gt;, therefore &lt;span class=&quot;math-container&quot;&gt;$f(g(x)) = x -1$&lt;/span&gt; exists. Now the issue is can you say &lt;span class=&quot;math-container&quot;&gt;$f(g(0))= - 1$&lt;/span&gt; because 0 is not in the domain of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f(x)= x + 2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x) = 1/(x - 1)$&lt;/span&gt;.&#xA;&lt;br/&gt; Here range of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not a subset of domain of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; , now what about &lt;span class=&quot;math-container&quot;&gt;$g(f(x)) = 1/(x+1)$&lt;/span&gt;, how can you explain the way to obtain domain of this composite function because here that is not the same domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do we need to treat finding expression for &lt;span class=&quot;math-container&quot;&gt;$g(f(x))$&lt;/span&gt; and finding composite function &lt;span class=&quot;math-container&quot;&gt;$g(f(x))$&lt;/span&gt; in two different ways?&lt;br/&gt;&#xA;Edited &lt;br/&gt;&#xA;By going through the suggested answers following are the conclusions I could able to make , &lt;br/&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When the terms function or domain not mention in the question we can treat&lt;span class=&quot;math-container&quot;&gt;$ f(g(x))$&lt;/span&gt; as an expression and substitute any real value to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; if output is real. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;When the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; are given &lt;span class=&quot;math-container&quot;&gt;$f(g(x))$&lt;/span&gt; can be defined if range of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is a subset of domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;When domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; are not given we have to determine the domain of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; such that range of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is a subset of domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and range of &lt;span class=&quot;math-container&quot;&gt;$f(g(x))$&lt;/span&gt; should be determined according to the selected domain of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;. &lt;br/&gt;&#xA;If you have any exceptions please mention it in your answers or comment about it so that we can make the final conclusion.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-02-21T03:07:05.523" LastActivityDate="2023-02-21T03:07:05.523" Title="Composite functions" Tags="|secondary-education|functions|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26159" PostTypeId="2" ParentId="26157" CreationDate="2023-02-19T01:18:57.523" Score="5" Body="&lt;p&gt;I initially found requiring the range of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; being a subset of the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to be the more appealing option, but having thought about it more, I now think that any two real functions should be composable. This is based on two principles:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The key takeaway skill from talking about domains is that students should be able to recognize when an expression exists or doesn't exist for various &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. There shouldn't be extra rules for dealing with functions for students to memorize. As such:&lt;/li&gt;&#xA;&lt;li&gt;Functions should not behave differently from ordinary algebra: students can write algebraic expressions that don't exist for any &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, so they should also be able to write &lt;span class=&quot;math-container&quot;&gt;$f(g(x))$&lt;/span&gt; when it doesn't make sense for any &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Of course, the most important principle is to agree with whatever your textbook says, so that students don't get confused when they reference it.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastEditorUserId="15671" LastEditDate="2023-02-19T08:12:17.203" LastActivityDate="2023-02-19T08:12:17.203" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26160" PostTypeId="2" ParentId="26157" CreationDate="2023-02-19T02:53:22.287" Score="9" Body="&lt;p&gt;Before addressing some of the issues directly, let me mention what I think is the standard approach in mathematics. A function is typically defined together with its domain and codomain, so saying &amp;quot;function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;&amp;quot; is a shorthand for &amp;quot;function &lt;span class=&quot;math-container&quot;&gt;$f \colon A \to B$&lt;/span&gt;&amp;quot; (where &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is the codomain and potentially the range &lt;span class=&quot;math-container&quot;&gt;$f(A)$&lt;/span&gt; is a proper subset of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;). And then, the standard way of introducing composition is to require two functions &lt;span class=&quot;math-container&quot;&gt;$f \colon A \to B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g \colon B \to C$&lt;/span&gt; to have matching (co)domains and let &lt;span class=&quot;math-container&quot;&gt;$g \circ f \colon A \to C$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(g \circ f) (x) = g(f(x))$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, this is not the only option. If &lt;span class=&quot;math-container&quot;&gt;$f \colon A \to B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g \colon B' \to C$&lt;/span&gt; don't have matching (co)domains but &lt;span class=&quot;math-container&quot;&gt;$B' \subseteq B$&lt;/span&gt; (typically &lt;span class=&quot;math-container&quot;&gt;$B = \mathbb{R}$&lt;/span&gt;), you can still define &lt;span class=&quot;math-container&quot;&gt;$g \circ f$&lt;/span&gt; on the set &lt;span class=&quot;math-container&quot;&gt;$A' := f^{-1}(B')$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$(g \circ f) \colon A' \to C$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(g \circ f)(x) = g(f(x))$&lt;/span&gt;. It's mostly a pedagogical choice. If one wanted to be pedantic and stick to the previous definition, one would just have to use the composition &lt;span class=&quot;math-container&quot;&gt;$g \circ f|_{A'}$&lt;/span&gt; (with &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; replaced by its restriction).&lt;/p&gt;&#xA;&lt;p&gt;It's probably best to stick with whatever your primary reference for the subject uses.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Now, there's a type of problems where a function is only given by an expression (say, &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt;) and the student is required to find the set of all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s for which the expression makes sense, the so called natural domain (here it would be &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \setminus \{0\}$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;However, it's important to make a distinction between &lt;em&gt;a function&lt;/em&gt; and &lt;em&gt;an expression&lt;/em&gt;. Formally, in problems like the one above you are given an expression and asked to make it into a function. I'd suspect not all students are ready for such subtleties, but there are some advantages:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This is how it works in mathematics, but even more importantly, in programming. There, functions are usually statically typed, meaning that you specify the type of their input and output. If you try to compose two functions: &lt;span class=&quot;math-container&quot;&gt;$$ f \colon \mathbb{N} \to \mathbb{R}, \ f(n)=\sqrt{n} &#xA;\quad \text{with} \quad &#xA;g \colon \mathbb{N} \to \mathbb{N}, \ g(x)=x^2, $$&lt;/span&gt;&#xA;in a typical programming language, you should expect an error when trying to compute &lt;span class=&quot;math-container&quot;&gt;$g(f(4))$&lt;/span&gt;, even though the reason in somewhat artificial (at least from the mathematical perspective).&lt;/li&gt;&#xA;&lt;li&gt;The distinction can clear up some confusion in your example 1. If &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x) = \sqrt{x-1}$&lt;/span&gt;, then actually we deal with functions &lt;span class=&quot;math-container&quot;&gt;$g \colon [1,\infty) \to \mathbb{R}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f \colon \mathbb{R} \to \mathbb{R}$&lt;/span&gt;, and their composition is by definition &lt;span class=&quot;math-container&quot;&gt;$f \circ g \colon [1,\infty) \to \mathbb{R}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(f \circ g)(x) = f(g(x)) = x-1$&lt;/span&gt;. Of course, given the expression &lt;span class=&quot;math-container&quot;&gt;$x-1$&lt;/span&gt; out of context, we would ascribe &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; as its natural domain. Avoiding unnecessary subtleties, one could also summarize it as follows: the natural domain of the composition &lt;span class=&quot;math-container&quot;&gt;$f \circ g$&lt;/span&gt; may be larger than the natural domain of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I hope it also sheds some light on how you could explain points 2 and 3.&lt;/p&gt;&#xA;" OwnerUserId="15367" LastEditorUserId="15367" LastEditDate="2023-02-19T03:20:27.457" LastActivityDate="2023-02-19T03:20:27.457" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="26161" PostTypeId="2" ParentId="26157" CreationDate="2023-02-19T16:36:43.820" Score="5" Body="&lt;p&gt;I think the  &lt;a href=&quot;https://matheducators.stackexchange.com/a/26160/1680&quot;&gt;answer&lt;/a&gt; by Michal Miśkiewicz does an excellent job addressing issues, and captures the formal mathematical perspective.&lt;/p&gt;&#xA;&lt;p&gt;The distinction between the formality of &amp;quot;function with domain&amp;quot; and the more &lt;em&gt;laissez-faire&lt;/em&gt; perspective of &amp;quot;expressions&amp;quot; is an important pedagogical issue. When I teach composition in precalculus and come to this issue, I like to refer to the &amp;quot;function with domain&amp;quot; as a rule typically given by an algebraic expression along with a demon who prevents evaluation at values not in the domain. This is akin to Maxwell's demon, the proverbial gatekeeper of the Second Law of Thermodynamics.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/mFoCN.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mFoCN.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some of my more artistic students enjoy drawing their creative renditions of gatekeepers of functions domains. Often the characteristics of the demon reflect the name of the function--&amp;quot;square root&amp;quot; inspires some to draw a demon holding a mandrake with square roots. Having students draw domain demons so I can share them with the class seems to help students retain the idea that functions have domains. When I grade exams, I sometimes see more sketches that students draw in the margins.&lt;/p&gt;&#xA;&lt;p&gt;When the demon goes to sleep, the function becomes an expression and that's when we can play fast and loose. When the demon for &lt;span class=&quot;math-container&quot;&gt;$(f(x)=x^2,\ x\in {\bf R})$&lt;/span&gt; goes to sleep, we can square all sorts of things such as matrices, mandrake roots, etc.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-02-19T20:29:49.853" LastActivityDate="2023-02-19T20:29:49.853" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26162" PostTypeId="1" CreationDate="2023-02-20T03:24:21.197" Score="3" ViewCount="680" Body="&lt;p&gt;I'm the math department chair at a small university. Our general education program is non-traditional. The university is split into three areas. Students are expected to complete a major in one of the areas and earn minors in the other two. Overall, this is good because students get more depth in the chosen minor areas, but is detrimental to elementary education majors because they need a more general education. I would like to put together a basic math minor to help elementary education majors get the math they need to be effective teachers. This should be easy enough that elementary education majors would choose it over psychology and political science, yet challenging enough that anyone who earns the minor and graduates would be an elementary teacher math specialist. Does the following list of courses sound reasonable or do we need more?&#xA;*Intermediate Algebra&#xA;*College Algebra&#xA;*Quantitative Reasoning (logic, financial math, basic statistics, etc)&#xA;*Math for Elementary Education Majors (standard course that covers topics on the PRAXIS)&lt;/p&gt;&#xA;" OwnerUserId="20928" LastActivityDate="2023-02-20T17:16:54.250" Title="Elementary Teacher Math specialist/ Basic Math Minor" Tags="|primary-education|teacher-preparation|teachers|teacher-development|teacher-training|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26163" PostTypeId="2" ParentId="26162" CreationDate="2023-02-20T06:01:52.580" Score="3" Body="&lt;p&gt;[Caveat, I'm just a citizen, not a teach.]&lt;/p&gt;&#xA;&lt;p&gt;I like it.  Call it a &amp;quot;math ed minor&amp;quot; to clarify it is different from say a chemist getting a &amp;quot;math minor&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I think the two remedial algebra classes are fine.  Kind of gives them some understanding of where the kids are headed after arithmetic.  And there are aspects of elementary school arithmetic that are somewhat algebraic (long division and multiplication, fractions, even &amp;quot;r*t=d&amp;quot; problems).  I'm OK with leaving out geometry (90% of the &amp;quot;race to calculus&amp;quot; in high school is algebra...you have to prioritize).  I'm also fine leaving out calculus and higher classes.  After all, these are primary school teachers we are talking about.  I do quite like the quantitative reasoning class and the math ed class.&lt;/p&gt;&#xA;&lt;p&gt;Keep it simple. Rack up a win.  Don't let people screw it up by making it longer (more classes) or sticking in Rudin.  30% of a loaf is better than none.  More sugar, less medicine.  Present this very much as &amp;quot;kinder, gentler&amp;quot; (thanks George HW).  [Nothing is stopping any of these ed majors from taking any math class they want...but let's be real, the demand for true majors courses in miniscule.]&lt;/p&gt;&#xA;&lt;p&gt;And do something good for your department: your previous question asked about the PRAXIS class being taken out of your department and if you listen to the tough, tough crowd then the ed majors will just all take psych. I did notice your 09FEB comment saying you'd been able to save it by being more customer centric.  Good job. There's a long history of math departments losing classes (e.g. stats) because they were not customer centric enough...wanted to teach math major stuff and were unsympathetic towards the needs of non math majors (often even ignorant and unwilling to learn about application and non math major careers).&lt;/p&gt;&#xA;" OwnerUserId="21066" LastEditorUserId="21066" LastEditDate="2023-02-20T06:09:29.337" LastActivityDate="2023-02-20T06:09:29.337" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26164" PostTypeId="2" ParentId="2338" CreationDate="2023-02-20T13:25:17.660" Score="-1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;when one writes&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int \sin(x) \,\mathrm{d}x = -\cos(x) + k$$&lt;/span&gt;&#xA;then the status of both &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is pretty unclear (which quantifier in front of each of these variables?)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Yes, when an indefinite integral is specified like in the above (that is, as a general representation or as a function of the form &lt;span class=&quot;math-container&quot;&gt;$-\cos(x)+k,$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is arbitrary), it doesn't really make sense to quantify &lt;span class=&quot;math-container&quot;&gt;$k,$&lt;/span&gt; unless initial/boundary conditions are given, in which case we can assert that &lt;span class=&quot;math-container&quot;&gt;$$\exists k{\in}\mathbb R\;\forall x{\in}\mathbb R \int \sin(x) \,\mathrm{d}x = -\cos(x) + k.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;On the other hand, when the above indefinite integral is undersood as a literal set of antiderivatives, then:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;for each&lt;/strong&gt; real &lt;span class=&quot;math-container&quot;&gt;$x,$&lt;/span&gt; the indefinite integral of &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt; is the set of functions such that &lt;em&gt;each&lt;/em&gt; one, &lt;strong&gt;for some&lt;/strong&gt; real &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, equals &lt;span class=&quot;math-container&quot;&gt;$-\cos(x)+k.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Appendix&lt;/strong&gt; (excerpt from &lt;a href=&quot;https://math.stackexchange.com/a/4631163/21813&quot;&gt;How to interpret the indefinite integral&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;An indefinite integral is more a useful notational shorthand than a mathematically important object. It is conventionally called a family of antiderivatives; as such, if its integrand is of the form &lt;span class=&quot;math-container&quot;&gt;$g'$&lt;/span&gt; and has an interval domain, then, literally, &lt;span class=&quot;math-container&quot;&gt;$$\int g'\,\mathrm dx=\{g(x)+C\mid C\in\mathbb R\},$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; represents uncountably many values. &lt;a href=&quot;https://math.stackexchange.com/q/3767159/21813&quot;&gt;How to understand the &amp;quot;indefinite integral&amp;quot; notation in calculus?&lt;/a&gt; contains several justifications for manipulating and adding indefinite integrals as actual sets by reading &lt;span class=&quot;math-container&quot;&gt;$\text‘=\text’$&lt;/span&gt; as an equivalence relation such that writing &lt;span class=&quot;math-container&quot;&gt;\begin{align}x^2+C=x^2+3+C\quad&amp;amp;\implies0=3\\x^2+C=x^2+3+D\quad&amp;amp;\implies E=C-D=3\end{align}&lt;/span&gt; make sense.&lt;/p&gt;&#xA;&lt;p&gt;More simply, we can frame an indefinite integral as the general representation of its integrand's antiderivatives, its specification containing one &lt;a href=&quot;https://math.stackexchange.com/a/4631162/21813&quot;&gt;independent parameter (arbitrary constant) &lt;span class=&quot;math-container&quot;&gt;$C_i$&lt;/span&gt;&lt;/a&gt; per maximal interval of its integrand's domain. Here, each instantiation of &lt;span class=&quot;math-container&quot;&gt;$$\int g'\,\mathrm dx=g(x)+C$$&lt;/span&gt; has an unimportant value of &lt;span class=&quot;math-container&quot;&gt;$C.$&lt;/span&gt; Even though this object is not a particular antiderivative, we manipulate it as if &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is merely undetermined.&lt;/p&gt;&#xA;&lt;p&gt;In the context of solving differential equations with given conditions, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; becomes an unknown whose value is to be determined.&lt;/p&gt;&#xA;" OwnerUserId="372" LastActivityDate="2023-02-20T13:25:17.660" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26165" PostTypeId="2" ParentId="26162" CreationDate="2023-02-20T17:06:11.273" Score="9" Body="&lt;p&gt;The top priority should be ensuring that elementary education majors have a deep and mathematically-sound understanding of elementary school mathematics. They need to (re)learn the core mathematics that they will be teaching—whole number and fraction arithmetic—so that they can make sense of it for themselves and for their students. (Liping Ma's book &lt;a href=&quot;https://www.taylorfrancis.com/books/mono/10.4324/9781003009443/knowing-teaching-elementary-mathematics-liping-ma&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Knowing and Teaching Elementary Mathematics&lt;/em&gt;&lt;/a&gt; which was recommended in &lt;a href=&quot;https://matheducators.stackexchange.com/a/26068/20719&quot;&gt;this answer to your previous question&lt;/a&gt; gives a sobering account of U.S. elementary teachers' dismal knowledge of fractions.)&lt;/p&gt;&#xA;&lt;p&gt;If the elementary education program at your institution is already doing a good job of this in their mathematics methods courses, then the courses you've suggested seem appropriate for a minor, if a bit light, as others have noted. I might suggest including a problem-solving or inquiry-based course that you can market as demonstrating how to implement student-centered pedagogies (Euclidean geometry lends itself well to this type of course), or some sort of a maker or constructionist course, or a modeling course, or a history of math course—something that pre-service elementary teachers will view as relevant to their teaching, since you're competing with psychology courses.&lt;/p&gt;&#xA;&lt;p&gt;If elementary education majors are not already getting the math they need from their education program, which is what it sounds like, then courses that are focused specifically on elementary school mathematics would be much more useful for them, and you would probably see higher enrollment. As guest troll suggested, it might be more appropriate to call this a &amp;quot;math education minor&amp;quot; or a &amp;quot;math for teaching minor.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I highly recommend talking a look at some of &lt;a href=&quot;https://math.berkeley.edu/%7Ewu/&quot; rel=&quot;noreferrer&quot;&gt;Hung-Hsi Wu's writings&lt;/a&gt;. He's spent several decades thinking about &lt;a href=&quot;https://math.berkeley.edu/%7Ewu/ICMtalk.pdf&quot; rel=&quot;noreferrer&quot;&gt;how mathematicians should be involved in mathematics teacher education&lt;/a&gt;, and he's written a series of textbooks for mathematics teachers. You could start with this Notices of the AMS article, &lt;a href=&quot;https://math.berkeley.edu/%7Ewu/NoticesAMS2011.pdf&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;The Mis-Education of Mathematics Teachers.&amp;quot;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here are what I think are some of the important points that he's made.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;School mathematics is not and should not be university mathematics. Defining a rational number as an equivalence class of ordered pairs of integers is appropriate for the mathematician's purposes, but not for the elementary teacher's purposes.&lt;/li&gt;&#xA;&lt;li&gt;Nevertheless, school mathematics should still be rigorous mathematics. Fractions need to be defined in order to reason about them clearly. Thus, the elementary teacher should work with a definition like&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given a positive integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, divide the interval &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; on the number line into &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; parts of equal length. The fraction &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{n}$&lt;/span&gt; is the number located at the first division point to the right of &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. Given a positive integer &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, the fraction &lt;span class=&quot;math-container&quot;&gt;$\frac{m}{n}$&lt;/span&gt; is the number to the right of &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; by a distance that is &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; times the distance between &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The mathematics that has been taught in U.S. schools for the past several decades, as codified in textbooks, is not sound mathematics. In lieu of definitions, it gives confounding statements like&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A fraction can be a &lt;em&gt;part of a whole&lt;/em&gt;, a &lt;em&gt;ratio&lt;/em&gt;, or a &lt;em&gt;quotient&lt;/em&gt;. The fraction &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4}$&lt;/span&gt; can be &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; parts when the whole is divided into &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; parts, such as three-quarters of a pie. It can represent a &amp;quot;ratio situation,&amp;quot; such as &amp;quot;there are &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; boys for every &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; girls.&amp;quot; It is also the result of dividing &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, which can arise from a &amp;quot;partitioning situation&amp;quot; such as sharing &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; cookies fairly between &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; people.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Teachers are a product of the education system, so they have likely learned unsound mathematics and will likely teach unsound mathematics. Taking university mathematics courses will not help them unlearn and relearn the mathematics they need in order to be effective teachers. To break this cycle, mathematicians need to&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;consult with education colleagues, help design new mathematics courses for teachers, teach those courses, and offer constructive criticisms in every phase of this reorientation in preservice professional development.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-02-20T17:16:54.250" LastActivityDate="2023-02-20T17:16:54.250" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26166" PostTypeId="1" CreationDate="2023-02-21T05:42:47.637" Score="3" ViewCount="237" Body="&lt;p&gt;The Triangle Inequality (&lt;span class=&quot;math-container&quot;&gt;$|x+y|\leq|x|+|y|$&lt;/span&gt;) is useful later on in the student's math education (e.g. in proving results about limits).&lt;/p&gt;&#xA;&lt;p&gt;But for the high-school student, are there any useful and interesting applications of the Triangle Inequality?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Please exclude from your answers any use of the Triangle Inequality to simply derive some similar inequality (e.g.  &lt;span class=&quot;math-container&quot;&gt;$||x|-|y||\leq |x+y|$&lt;/span&gt;), unless that derived inequality can be shown to have some interesting applications (other than further deriving other similar inequalities).&lt;/p&gt;&#xA;" OwnerDisplayName="user18187" LastActivityDate="2023-02-25T21:28:23.273" Title="Applications of Triangle Inequality for high-school students?" Tags="|examples|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26167" PostTypeId="2" ParentId="13334" CreationDate="2023-02-21T05:49:47.987" Score="1" Body="&lt;p&gt;The question is about students who &amp;quot;won't need university-level math&amp;quot;, but a prediction about the future needs of the student isn't a fact.  It's a guess.  What if every path pursued by a student reaches a dead-end of failure, and the student begins to reconsider possible goals, and all of the most attractive and feasible goals seem to require university-level maths?  What if a given student discovers in future that university-level maths are needed for what the student -- after a re-evaluation of goals -- ardently hopes to achieve?&lt;/p&gt;&#xA;&lt;p&gt;If we are really talking about having a limited amount of time, and avoiding mathematics simply because there isn't enough time available for the student to study everything that the student wants to study, then there may be some ways to increase the amount of logic studied, and reduce the amount of math studied.  However, is that the situation?&lt;/p&gt;&#xA;&lt;p&gt;Another possibility is that the mathematics simply gets difficult, and the student wants to avoid those difficulties.  For example, if the student is confused by a statement that begins with three quantifiers (such as &amp;quot;for every epsilon greater than zero, there exists delta greater than zero, such that for every real number x, ...&amp;quot;) ... or if the distinction between ordinary continuity and &lt;strong&gt;uniform&lt;/strong&gt; continuity of a function mapping the real numbers to the real numbers is too subtle for the student to appreciate, ... then the student is unlikely to proceed very far in the study of logic.&lt;/p&gt;&#xA;&lt;p&gt;If, like a judge, you hope to recuse yourself from matters where there may be a conflict of interest, but you take things to the extreme of neither  endorsing nor casting doubt upon any assertions whatsoever, then you may be free to concentrate on questions of &lt;strong&gt;pure&lt;/strong&gt; logic. However, an interesting or important problem in pure logic is likely to have its origins in aspects of reality that go beyond pure logic.  To create a problem in logic, you may generalize, simplify, and omit many particular details. However, you may find it valuable or essential to return to the source.  That source may include messy and complicated mathematics, physics, astronomy, etc.&lt;/p&gt;&#xA;&lt;p&gt;If you study history, then you will learn about Copernicus, Galileo, Kepler, and Newton.  Galileo used perfect circles in his model of the solar system, but Kepler used ellipses.  Ellipses are conic sections.  We are back to mathematics.&lt;/p&gt;&#xA;&lt;p&gt;How did people compute before there were electronic computers?  The word &amp;quot;computer&amp;quot; used to name an occupation (like &amp;quot;baker&amp;quot;, &amp;quot;tailor&amp;quot;, or &amp;quot;sailor&amp;quot;).  People who worked as computers used books including tables of values of logarithms, and the purpose of those logarithm values was simply to allow for efficient multiplication.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Does everyone need to know what a cosine is if the UK is to have a&#xA;brighter future?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In many cases, it's easier to create an equation for a curve in 3-dimensional space that relates x, y, and z, if we begin with a parametric equation that involves not only x, y, and z, but also an additional parameter such as the variable t.  Now, if we omit z, and consider a curve in two-dimensional space, then we can describe a circle as follows: it's the set of points (cosine(t), sine(t)) as t takes on values from zero to twice pi.&lt;/p&gt;&#xA;&lt;p&gt;There's a bit more to it than that.  We define cosine so that all real numbers are in its domain, and so that it's a periodic function.  However, the main idea is that cosine(t) is simply the horizontal coordinate value, and sine(t) is the vertical coordinate value for a point on a circle.&lt;/p&gt;&#xA;&lt;p&gt;Here's a new question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Does everyone in the UK need to know what &lt;strong&gt;width&lt;/strong&gt; and &lt;strong&gt;height&lt;/strong&gt;&#xA;are if the UK is to have a brighter future?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Maybe you have heard that Donald Trump is six feet tall and has a waist size of 36 inches or three feet, but if it's difficult to remember which direction we measure to discover what is called &amp;quot;height&amp;quot; and if it's difficult to remember what the word &amp;quot;waist&amp;quot; means, then some people might get those dimensions mixed up and announce &lt;em&gt;&lt;strong&gt;that Donald Trump is three feet tall&lt;/strong&gt;&lt;/em&gt; and has a waist size of 72 inches or six feet.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;for many children it is just a set of rather pointless rules for&#xA;manipulating symbols&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes, but that's not a problem with the subject area itself.  That's a problem with what is taught in schools, how it is taught, and the selection process used to determine who will be teaching math.  Now we're getting into the topic of political policy, a topic quite far from pure logic!&lt;/p&gt;&#xA;" OwnerUserId="12816" LastEditorUserId="12816" LastEditDate="2023-02-21T05:55:44.323" LastActivityDate="2023-02-21T05:55:44.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26168" PostTypeId="2" ParentId="26166" CreationDate="2023-02-21T19:20:21.623" Score="8" Body="&lt;p&gt;At its most basic, the Triangle Inequality says that it is (in general) a shorter trip to go directly from point &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to point &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; than it would be to stop at point &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; in between.  That is, the length of any one side of a triangle is less than the sum of the lengths of the other two sides.  That is the source of the name &amp;quot;Triangle Inequality&amp;quot;, and is the most elementary version of the (more complicated) claim that a straight line is the shortest distance between two points.  This is certainly a &amp;quot;useful&amp;quot; application, although whether or not it counts as &amp;quot;interesting&amp;quot; is, I suppose, debatable.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2023-02-21T19:20:21.623" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26169" PostTypeId="1" CreationDate="2023-02-23T04:33:36.233" Score="3" ViewCount="376" Body="&lt;p&gt;When I was in 6th grade (U.S. so 12-13 years old), I took a summer school class. The teacher gave us all different sized spools (spools that hold sewing thread but were empty). We each made a mark on one wheel then rolled it one revolution over paper and marked how far it went (i.e. one circumference). Then we held the spool on its side and measured how may diameters it took to cover that distance. And, no matter what size the spool, it took a little over 3 diameters. &lt;/p&gt;&#xA;&lt;p&gt;In looking back this was a great exercise to prove that circumference equals &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; * diameter. I was wondering if this demonstration is common.&lt;/p&gt;&#xA;" OwnerUserId="14854" LastEditorUserId="1737" LastEditDate="2023-02-23T18:19:28.833" LastActivityDate="2023-02-26T22:33:19.570" Title="Best demonstration of $\pi$ ever; is this common?" Tags="|geometry|manipulatives|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26170" PostTypeId="1" CreationDate="2023-02-23T07:33:54.163" Score="1" ViewCount="334" Body="&lt;p&gt;Especially in mathematics, we give a set of definitions and rules, and ask our students to prove a particular statement or to solve equations or inequalities.&lt;/p&gt;&#xA;&lt;p&gt;By this kind of system we limit students to a particular frame and mainly train their minds for vertical thinking which not only discourages improving their creativity and think differently but also it doesn't enhance exploring skills in mathematics. I think lateral thinking is what the most important way to overcome practical challenges in real life too. My issue is how we have to focus on this matter in a class room and what kind of activities we can do with students while preparing the students to face exams in relevant curriculum?&lt;/p&gt;&#xA;&lt;p&gt;Here I suggest one of my approaches where you can allow students to use unconventional and nontraditional techniques in their suggestions. &lt;br/&gt;&#xA;Bring the puzzle set Towers of Hanoi of 10 discs and 3 poles to the classroom&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Allow students to think about it for about 15 minutes in their own way and express their views,&lt;/li&gt;&#xA;&lt;li&gt;Ask students to suggest a game to use it,&lt;/li&gt;&#xA;&lt;li&gt;Ask students to give a suitable name for the puzzle,&lt;/li&gt;&#xA;&lt;li&gt;Demonstrate the normal way of playing the puzzle and check whether who can accept the challenge to solve the puzzle in the case of 64 discs,&lt;/li&gt;&#xA;&lt;li&gt;Give a month to prepare full list of issues related to the puzzle without mentioning any guidelines.&lt;br /&gt;&#xA;Finally you can allow them to share their suggestions and recommend any amendments if needed.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I suggest specifically this puzzle because here students have many areas to improve such as by studying the history of the puzzle, mathematics concepts can be applied,observing the patterns, different versions, how to make the challenge much difficult and availability of this puzzle set etc.&lt;/p&gt;&#xA;&lt;p&gt;Is this kind of activity(also can be used as a project)useful to improve lateral thinking for grade 10 or above in high school and advanced level or what you would suggest more useful for them?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="238" LastEditDate="2023-02-24T19:56:13.997" LastActivityDate="2023-02-25T02:01:32.720" Title="Lateral thinking in mathematics" Tags="|projects|activities|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26171" PostTypeId="1" AcceptedAnswerId="26173" CreationDate="2023-02-23T09:23:35.797" Score="7" ViewCount="1802" Body="&lt;p&gt;I’m currently getting my Master’s in Math at Portland State University so that I can teach community college mathematics. I’m specifically hoping to teach calculus, statistics, and linear algebra, so I’m trying to pick classes that will give me deeper knowledge of these subjects. In particular, I’m hoping the degree will give me deep enough knowledge to be able to teach multivariable and vector calculus. Since we are required to take at least two 600 level sequences, next year I’m thinking of taking the Mth 614-616 sequence called “Modern Analysis”. I’m currently taking Mth 512, which is part of the 500 level Real Analysis sequence, and is the prerequisite for Modern Analysis. I’m getting 100% in the class this term so I expect to be ready for Modern Analysis next year. As far as I’m aware, Modern Analysis is the only 600 level analysis sequence that will be offered next year. Here is the course description (copied from &lt;a href=&quot;https://pdx-mobile.smartcatalogiq.com/en/2022-2023/Bulletin/Courses/Mth-Mathematical-Sciences/600/Mth-614&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;) for all three courses in the sequence:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Topics from nonlinear analysis, harmonic analysis, analytic functions, ordered vector spaces, analysis on Lie groups, and operator theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It seems like the class will obviously cover very high-level material, but I’m concerned it might not relate enough to multivariable/vector calculus. However, I’m not that familiar with these topics, so maybe there’s more of a connection than I currently realize. Would a sequence like this give me deep enough knowledge to teach a multivariable/vector calculus course at a community college? Thank you for your help!&lt;/p&gt;&#xA;" OwnerUserId="19352" LastActivityDate="2023-02-24T19:53:43.440" Title="If I take Modern Analysis next year, will I be prepared to teach multivariable/vector calculus?" Tags="|mathematical-analysis|graduate-education|community-colleges|advising|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="26172" PostTypeId="2" ParentId="26170" CreationDate="2023-02-23T09:53:48.400" Score="4" Body="&lt;p&gt;A relevant term in didactics of mathematics is 'problem solving', which means facing a problem where one does not know the means of solving it already. You may want to check the literature for this. Your particular activity also reminds of landscapes of investigation (undersøgelseslandskab).&lt;/p&gt;&#xA;&lt;p&gt;There are other terms like open tasks and rich mathematical tasks. In Norwegian we also speak of LIST-tasks (low threshold, high ceiling).&lt;/p&gt;&#xA;&lt;p&gt;For concrete tasks in English, you might want to check out Youcubed: &lt;a href=&quot;https://www.youcubed.org/tasks/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youcubed.org/tasks/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For literature, I would recommend:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;BOALER, Jo. Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. John Wiley &amp;amp; Sons, 2015. Chapter 5.&lt;/li&gt;&#xA;&lt;li&gt;KARLSEN, Lisbeth. Tenk det!: utforskning, forståelse og samarbeid - elever som tenker sjæl i matematikk: ungdomstrinnet. Cappelen Damm akademisk, 2014.&lt;/li&gt;&#xA;&lt;li&gt;SKOVSMOSE, Ole. Landscapes of investigation. Zentralblatt für Didaktik der Mathematik, 2001, 33: 123-132.&lt;/li&gt;&#xA;&lt;li&gt;WÆGE, K.; NOSRATI, M. Motivasjon i matematikk (2. utg.). Oslo: Universitetsforlaget, 2019. Especially chapter 6.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-02-23T09:53:48.400" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26173" PostTypeId="2" ParentId="26171" CreationDate="2023-02-23T11:10:43.213" Score="18" Body="&lt;p&gt;It is impossible to say whether you will be ready to teach multivariable calculus after taking this course.  Certainly many of the topics you list will employ some multivariable calculus, but using bits and pieces of an undergrad course as a tool in a higher level course will probably not be the best preparation for teaching.&lt;/p&gt;&#xA;&lt;p&gt;If you are going to be teaching multivariable calculus, I would recommend getting a few really good textbooks and making sure you can do all of the exercises, motivate all of the definitions and theorems, and feel comfortable enough with the proofs that you can both present the main ideas without any prep work and present a detailed proof with a little bit of prep work.  You should also work on having some kind of overarching narrative for the course which makes sense to you.&lt;/p&gt;&#xA;&lt;p&gt;The textbooks I would recommend are &amp;quot;Multivariable Mathematics&amp;quot; by Shifrin and &amp;quot;Vector Calculus&amp;quot; by Marsden and Tromba.&lt;/p&gt;&#xA;&lt;p&gt;If you want a &amp;quot;higher level perspective&amp;quot; on multivariable calculus then you should learn some differential geometry.  I would recommend John M. Lee's introductions to topological manifolds, smooth manifolds, and riemannian manifolds in that order.&lt;/p&gt;&#xA;&lt;p&gt;I want to say that this answer is not meant to discourage you from taking the course you mention.  It sounds interesting, and it is often true that we really learn something when we are applying it in different situations.  I am sure that taking the course will strengthen your understanding of multivariable calculus, just probably not in the most systematic way.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="376" LastEditDate="2023-02-24T06:27:18.763" LastActivityDate="2023-02-24T06:27:18.763" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26174" PostTypeId="2" ParentId="26171" CreationDate="2023-02-23T23:13:17.240" Score="3" Body="&lt;p&gt;I echo Steven's comments.  Honestly, independently was writing the same and then dropped it when I saw his response...then came back like a dog.&lt;/p&gt;&#xA;&lt;p&gt;It's pretty tenuous the connection.  You could be a great teacher without this class and visa versa.  It's just not as high bang for the buck as taking a few texts and working them through (every problem, especially the hard ones at the end of sections).&lt;/p&gt;&#xA;&lt;p&gt;I would only maybe add that you might consider a PDE course also if you want something multivariable to take now (Steve's list is great, just an addition).  It will still not be as applicable as the material itself.&lt;/p&gt;&#xA;&lt;p&gt;I'd also consider some stats courses (maybe a DOE class based on Box, Hunter, Hunter or similar) as that is both &amp;quot;multivariable&amp;quot; in tone (multiple regression) and also fits your other desire for stats coverage.&lt;/p&gt;&#xA;" OwnerUserId="21078" LastEditorUserId="21078" LastEditDate="2023-02-23T23:14:48.000" LastActivityDate="2023-02-23T23:14:48.000" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26175" PostTypeId="2" ParentId="26171" CreationDate="2023-02-24T03:22:15.147" Score="16" Body="&lt;p&gt;Assuming you already have a good understanding of the material in a standard course in multivariable calculus and vector calculus, I would recommend that you take a class in electricity and magnetism from the physics department. I would guess that a sizable fraction of your students are going to major in engineering and physics, and their courses in electricity and magnetism will likely be where they use vector calculus most heavily. So as their teacher, you ought to understand that material yourself.  If you don't already have a lot of physics under your belt, then taking a physics class will broaden your horizons more than studying more analysis will.&lt;/p&gt;&#xA;" OwnerUserId="12478" LastActivityDate="2023-02-24T03:22:15.147" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26176" PostTypeId="2" ParentId="26166" CreationDate="2023-02-24T03:46:19.297" Score="5" Body="&lt;p&gt;There is the &lt;a href=&quot;https://mathoverflow.net/questions/2014&quot;&gt;broken stick problem&lt;/a&gt;, which asks for the probability that if you break a stick at two points chosen uniformly at random along its length, then the resulting pieces can be used to form a triangle.  This is an easily stated problem that is not so trivial to solve.&lt;/p&gt;&#xA;&lt;p&gt;Admittedly, the hard part of this problem isn't the triangle inequality, but the triangle inequality is the first crucial step in the solution.  I think that for many students, the triangle inequality looks so trivial that it doesn't seem like it could be useful.  But when students first see the broken stick problem, they often don't immediately recognize that the triangle inequality is relevant.  It takes a bit of thought.  This thought process helps them appreciate why the triangle inequality is worth stating formally.&lt;/p&gt;&#xA;" OwnerUserId="12478" LastActivityDate="2023-02-24T03:46:19.297" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26179" PostTypeId="2" ParentId="26171" CreationDate="2023-02-24T19:53:43.440" Score="5" Body="&lt;p&gt;I'm going to throw in a frame challenge: you stated your goal is to be prepared to teach at a collegiate level.&lt;/p&gt;&#xA;&lt;p&gt;All the other answers talk about whether you have the correct material, whether various courses will help you understand what you are teaching, etc.&lt;/p&gt;&#xA;&lt;p&gt;If you want to be prepared to &lt;em&gt;teach&lt;/em&gt;, then &lt;strong&gt;take workshops/courses on university-level teaching&lt;/strong&gt; (I have).&lt;/p&gt;&#xA;&lt;p&gt;There is a lot of evidence-based teaching methods out there, and most of this focuses on &lt;em&gt;how&lt;/em&gt; you deliver the material, not &lt;em&gt;what&lt;/em&gt; you teach. Practically none of the teaching training I had was topic-specific, possibly also due to lack of attendance.&lt;/p&gt;&#xA;&lt;p&gt;Knowing the material is necessary, but nowhere near sufficient to be a competent teacher.&lt;/p&gt;&#xA;" OwnerUserId="21114" LastActivityDate="2023-02-24T19:53:43.440" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26180" PostTypeId="2" ParentId="26170" CreationDate="2023-02-25T02:01:32.720" Score="2" Body="&lt;p&gt;Another problem that accommodates the &amp;quot;lateral thinking&amp;quot; the OP espouses is&#xA;discovering and understanding Euler's polyhedron formula, &lt;span class=&quot;math-container&quot;&gt;$V-E+F=2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Discovery is aided by &lt;a href=&quot;https://matheducators.stackexchange.com/a/4313/511&quot;&gt;polydrons&lt;/a&gt; or equivalent manipulatives.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/rIDxo.jpg&quot; alt=&quot;PolydronsPlatonic&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;And there is a complex history&#xA;behind the formula detailed in Imre Lakatos's classic book:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BMu9a.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BMu9a.jpg&quot; alt=&quot;Lakatos&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Finally, this is more than just a puzzle. It is a significant&#xA;and deep theorem.&lt;/p&gt;&#xA;&lt;p&gt;In my experience, this topic is quite accessible to high-school students in the U.S.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2023-02-25T02:01:32.720" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26181" PostTypeId="1" AcceptedAnswerId="26182" CreationDate="2023-02-25T08:43:17.200" Score="0" ViewCount="864" Body="&lt;p&gt;As a property of definite integrals, we teach that definite integrals are zero if the lower and upper limits are the same (Wolfram mathworld says this too).  Is this valid in general?&lt;/p&gt;&#xA;&lt;p&gt;In the case of integrating &lt;span class=&quot;math-container&quot;&gt;$\dfrac1{4+\sin^2x}$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$π$&lt;/span&gt; using the substitution &lt;span class=&quot;math-container&quot;&gt;$t = \tan x,$&lt;/span&gt; we get both new limits &lt;span class=&quot;math-container&quot;&gt;$0,$&lt;/span&gt; so the integral should be &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, but if you use the symmetry of the graph and describe the integral in terms of area, you get a positive value. How would explain this to students? Do we need to say that there are exceptions?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="372" LastEditDate="2023-02-25T10:19:29.460" LastActivityDate="2023-02-26T14:40:04.977" Title="Definite integrals with equal limits" Tags="|secondary-education|calculus|integration|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26182" PostTypeId="2" ParentId="26181" CreationDate="2023-02-25T10:19:33.173" Score="6" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;How would explain this to students?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Explain that &lt;em&gt;checking that conditions are satisfied is part of the process of applying theorems&lt;/em&gt;/laws.&lt;/p&gt;&#xA;&lt;p&gt;(Admittedly, though, since integration change of variable is seldom rigorously taught or even presented as a theorem, it is not a great example to demonstrate carefully reading theorems.)&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In the OP's example, writing the substitution &lt;span class=&quot;math-container&quot;&gt;$t=\tan x$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$t=g(x),$&lt;/span&gt; the variable-change theorem requires that &lt;span class=&quot;math-container&quot;&gt;$g'$&lt;/span&gt; is integrable on the original integration domain &lt;span class=&quot;math-container&quot;&gt;$[0,\pi];$&lt;/span&gt; since this isn't satisfied, the substitution isn't applicable.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/4635104/21813&quot;&gt;A related fake trick&lt;/a&gt; is &lt;span class=&quot;math-container&quot;&gt;$$\int_a^b f(x)\,\mathrm dx\ne0 \\\text{and}\\u=\color{red}{g(x)} \text{ such that }g’\text{ is integrable on }[a,b]\text{ and }g(a)=k=g(b)\\\implies\\\int_a^b f(x)\,\mathrm dx=\int_k^k\color\red{\text{[this integrand doesn't matter]}}\,\mathrm du=0.$$&lt;/span&gt; Here, in neglecting to attempt to determine the new integrand (doing this is another aspect of verifying that this theorem is applicable), the student is denied the discovery that the given integrand &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is not actually compatible with this particular choice of &lt;span class=&quot;math-container&quot;&gt;$g.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-02-26T14:40:04.977" LastActivityDate="2023-02-26T14:40:04.977" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26183" PostTypeId="2" ParentId="26181" CreationDate="2023-02-25T15:05:19.663" Score="2" Body="&lt;p&gt;Your question is similar to the issue raised in this post:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/8508/should-varphi-be-monotone-in-the-integration-by-substitution&quot;&gt;Should $\varphi$ be monotone in the integration by substitution?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But in your situation, there is a more critical issue in that your new variable &lt;span class=&quot;math-container&quot;&gt;$g(x)=\tan x$&lt;/span&gt; is not defined at &lt;span class=&quot;math-container&quot;&gt;$x=\pi/2$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-02-25T15:05:19.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26184" PostTypeId="2" ParentId="26181" CreationDate="2023-02-25T17:42:12.050" Score="5" Body="&lt;p&gt;ryang has explained the issue, but I think it might be helpful to students to frame the issue in a more positive light--instead of &amp;quot;you can't use this substitution here&amp;quot;, say &amp;quot;here's what you need to do to use this substitution.&amp;quot; The problem is that &lt;span class=&quot;math-container&quot;&gt;$\tan x$&lt;/span&gt; is not differentiable on &lt;span class=&quot;math-container&quot;&gt;$[0,\pi]$&lt;/span&gt; because it's not even defined at &lt;span class=&quot;math-container&quot;&gt;$x=\frac{\pi}{2}$&lt;/span&gt;. So in order to use this substitution you have to find a way to rewrite the integral to avoid the troublesome point. In this example we can write&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\int_0^\pi\frac{1}{4+\sin^2x}dx&amp;amp;=\int_0^{\pi/2}\frac{1}{4+\sin^2x}dx+\int_{\pi/2}^\pi\frac{1}{4+\sin^2x}dx\\&#xA;&amp;amp;=\lim_{r\to\pi/2^-}\int_0^r\frac{1}{4+\sin^2x}dx+\lim_{r\to\pi/2^+}\int_r^\pi\frac{1}{4+\sin^2x}dx.&#xA;\end{align}&lt;/span&gt;&#xA;The purpose of introducing the limits is to keep &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}{2}$&lt;/span&gt; out of the interval. Now your substitution works fine and you get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{s\to\infty}\int_0^s\frac{1}{5t^2+4}dt+\lim_{s\to-\infty}\int_s^0\frac{1}{5t^2+4}dt.&#xA;$$&lt;/span&gt;&#xA;In this example since the integrand is an even function you can just double the first integral.&lt;/p&gt;&#xA;&lt;p&gt;By explaining things this way, you don't have throw away a promising substitution and, by going through the process of choosing intervals carefully, you reinforce the idea of checking the needed conditions. My feeling is that by the time students are learning these kinds of substitutions they should already be strongly conditioned to recoil against any attempt to do a definite integral using the fundamental theorem of calculus over an interval where the integrand is not defined at every point. For an example of what can go badly wrong, consider how if you naively apply the fundamental theorem to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_{-1}^1\frac{1}{x^2}dx&#xA;$$&lt;/span&gt;&#xA;you get obvious nonsense (a finite negative answer where the integrand is positive and, properly treated, the integral is infinite).&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2023-02-25T17:42:12.050" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26185" PostTypeId="1" CreationDate="2023-02-25T18:57:41.187" Score="7" ViewCount="2031" Body="&lt;p&gt;I am a high school mathematics teacher, in our school students take the Cambridge IGCSE 0580 exam. After IGCSE our school offers the IB Diploma programme and I am thinking about proposing the management of our school to move to the IGCSE 0607.&#xA;The reason for this is the investigation exam paper and the use of the graphic display calculator.&lt;/p&gt;&#xA;&lt;p&gt;Comparing the syllabi I do not find a big difference and I think that introducing the graphic display calculator earlier can be an advantage in the IBDP. However, I do not want to propose a change that can be overly challenging to the less motivated/interested students.&lt;/p&gt;&#xA;&lt;p&gt;I would be interested to hear the opinion of those who have experience in teaching both syllabus. We do not have the resources to teach both, so it would be especially interesting to hear opinions of those who moved from 0580 to 0607. Do students find the 0607 more challenging? Did you see a change in grades on the exam?&lt;/p&gt;&#xA;" OwnerUserId="14957" LastEditorUserId="1737" LastEditDate="2023-02-27T16:57:09.880" LastActivityDate="2024-03-12T09:33:42.890" Title="Difference between the Cambridge IGCSE 0580 and 0607 mathematics courses" Tags="|secondary-education|syllabus|uk|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26186" PostTypeId="2" ParentId="26166" CreationDate="2023-02-25T21:28:23.273" Score="1" Body="&lt;p&gt;Here are two examples (one for your one-dimensional question and one in 2D, which is not really what you are asking about)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Consider the following (not really lifelike) situation. There are two shops along a straight road and we are considering building a bakery somewhere on the road that supplies bread to these shops. The delivery van can only carry enough bread for one shop. Where shall we build the bakery to minimise the distance travelled by the delivery van, if the van starts and finishes at the bakery? The answer is: anywhere between the two shops. This can be justified using common sense or using coordinates on the number line and the triangle inequality you mentioned. An interesting extension is to discuss this problem with 3, 4, 5, ... shops (still along a straight road). Note that these questions can be also answered using graphs involving sums of shifted absolute value functions, which can also be interesting to investigate with high school students.&lt;/li&gt;&#xA;&lt;li&gt;I know that the following example is to illustrate the geometric triangle inequality (which is not what you were asking about), but it still might be interesting to mention to high school students when you discuss why your algebraic expression is called triangle inequality. It is Heron's shortest distance problem (which can be illustrated on a pool table): What is the shortest path from A to B that touches a given line? Of course the interesting part of the question is when the two points are on the same side of the line. Here is a &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/historical-activities-for-calculus-module-3-optimization-herons-shortest-distance-problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;link to an activity&lt;/a&gt; related to this.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2023-02-25T21:28:23.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26187" PostTypeId="2" ParentId="26142" CreationDate="2023-02-26T05:56:40.683" Score="3" Body="&lt;p&gt;You start with &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^2+x+1$&lt;/span&gt;. How about having students determine &lt;span class=&quot;math-container&quot;&gt;$f(y)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$y=z +1$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;First, substitute &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to obtain  &lt;span class=&quot;math-container&quot;&gt;$f(y)=y^2+ y +1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then, substitute &lt;span class=&quot;math-container&quot;&gt;$z+1$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; to obtain &lt;span class=&quot;math-container&quot;&gt;$f(z+1)=(z+1)^2 +(z+1) +1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then expand and collect like terms to obtain &lt;span class=&quot;math-container&quot;&gt;$f(z)=z^2 +3z +3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;After a bit of practice, they learn that they can do two steps at once as a &amp;quot;shortcut.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-02-26T06:02:37.107" LastActivityDate="2023-02-26T06:02:37.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26188" PostTypeId="1" CreationDate="2023-02-26T09:17:48.210" Score="0" ViewCount="137" Body="&lt;p&gt;As we all agree mathematics educators community is doing a great service as an international platform regarding teachers issues related to mathematics education. Not all but only personally motivated teachers join and get the full use by actively involving. Sometimes lack of participation may be a result of language barriers or because of less self motivation. What measures would you suggest to motivate teachers to get the full use of online sources such as communities like this one ? &lt;br/&gt;&#xA;As a local platform for mathematics teachers we have one which was created for us in last year. But only few are interested to be active there. My suggestion is you may need to select about 100 teachers who can be easily motivated and give a plan to keep the platform live. This selection can be done by the help of Zonal Directors because they can do it by the assistance of principals of schools. What would you suggest to encourage teachers regarding this and how can you compare the participation of your mathematics teachers in a local online platform similar to this mathematics educators?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-02-26T10:09:39.813" LastActivityDate="2023-10-26T06:30:36.107" Title="Online platforms for teachers to discuss matters related to mathematics" Tags="|secondary-education|onlineplatforms|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26189" PostTypeId="2" ParentId="26169" CreationDate="2023-02-26T22:33:19.570" Score="1" Body="&lt;p&gt;I don't know how common this demonstration is (I never saw it in a class when I was a student), but there's a nice video illustrating it that students can watch: see the first 35 seconds of the Vertasium video on pi: &lt;a href=&quot;https://www.youtube.com/watch?v=gMlf1ELvRzc&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=gMlf1ELvRzc&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2023-02-26T22:33:19.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26190" PostTypeId="5" CreationDate="2023-02-27T16:57:30.960" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2023-02-27T16:57:30.960" LastActivityDate="2023-02-27T16:57:30.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26191" PostTypeId="4" CreationDate="2023-02-27T16:57:30.960" Score="0" Body="For questions that (at least in part) are specific to the United Kingdom." OwnerUserId="1737" LastEditorUserId="1737" LastEditDate="2023-03-02T16:50:05.420" LastActivityDate="2023-03-02T16:50:05.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26192" PostTypeId="5" CreationDate="2023-02-27T16:58:04.117" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2023-02-27T16:58:04.117" LastActivityDate="2023-02-27T16:58:04.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26193" PostTypeId="4" CreationDate="2023-02-27T16:58:04.117" Score="0" Body="For questions that (at least in part) are specific to India." OwnerUserId="1737" LastEditorUserId="1737" LastEditDate="2023-03-05T15:54:56.850" LastActivityDate="2023-03-05T15:54:56.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26195" PostTypeId="1" CreationDate="2023-03-03T17:28:52.850" Score="6" ViewCount="110" Body="&lt;p&gt;What factors are there to consider when adopting a text as far as ADA (Americans with Disabilities Act) is concerned? Is there a certification? What do you look for in the digital version of the text? Do you have experience in this regard as far calculus books are concerned?&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2023-03-04T07:06:34.887" LastActivityDate="2023-03-04T07:06:34.887" Title="Is there ADA-compliance certification for mathematics text books?" Tags="|calculus|textbooks|usa|disability|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26198" PostTypeId="2" ParentId="26185" CreationDate="2023-03-03T21:51:25.947" Score="1" Body="&lt;p&gt;&lt;strong&gt;This answer assumes that both 0607 and 0580&#xA;are being considered&#xA;at the extended level&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;0607 is definitely a more challenging curriculum, primarily this is because it expects students to be extremely familiar with a GDC, especially for paper 6–investigation. As far as the move to IB is concerned it depends on the type of math the student wants to take, if they want to take IB AI which is very stats based then 0580 is fine, if they want to take AA HL which is very theory based then 0607 is recommended. I suggest instead of moving from 0580 to 0607 you instead offer both.&lt;/p&gt;&#xA;&lt;p&gt;(I realise that this advice is probably very hard to follow, but hopefully it at least a little helpful)&lt;/p&gt;&#xA;" OwnerUserId="20949" LastEditorUserId="20949" LastEditDate="2023-03-03T21:54:36.677" LastActivityDate="2023-03-03T21:54:36.677" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26199" PostTypeId="1" CreationDate="2023-03-04T18:34:53.567" Score="11" ViewCount="1787" Body="&lt;p&gt;With the advent of the Internet administrators used to allocate release time or summer salary for making online course content. The pandemic made a Sal Khan out of most of us and making online content became more of a routine activity.  Now administrators probably face a dilemma of how to allocate resources between the online and on-campus courses. Math departments used to witness resource disputes between pure, applied, and statistics, or full-time vs part-time, but now there is a new category &amp;quot;online&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;What are the guidelines and policies you use or you think should be used to allocate resources for the online component of a department? Online courses usually have a substantially higher production cost than the on-campus version of the same course. How should faculty's effort in this area be recognized?&lt;/p&gt;&#xA;" OwnerUserId="1472" LastActivityDate="2023-03-06T03:50:00.623" Title="How do/should administrators estimate the cost of producing an online introductory mathematics class?" Tags="|undergraduate-education|online-instruction|online|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26200" PostTypeId="2" ParentId="26199" CreationDate="2023-03-04T21:39:42.867" Score="26" Body="&lt;p&gt;I can answer a small part of this.&lt;/p&gt;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&lt;p&gt;The most important part of the costs that administration overlooks during these kinds of efforts is the cost of &lt;strong&gt;maintenance&lt;/strong&gt; of the online course content that is created. Online courses are like cars; they require care and maintenance which is sometimes expensive. Here are some examples of events that require maintenance work:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;At some point, some of the graded content of the course can leak to Chegg or a local student group; that needs to be replaced.&lt;/li&gt;&#xA;&lt;li&gt;While using the course materials, faculty can realize some things are in the wrong order or the pacing is incorrect in certain modules.&lt;/li&gt;&#xA;&lt;li&gt;If the course is aligned to a textbook, the textbook can go into a new edition.&lt;/li&gt;&#xA;&lt;li&gt;If the course is aligned to some software, the software can go out of favor with the faculty.&lt;/li&gt;&#xA;&lt;li&gt;Something surprising can go wrong -- maybe the youtube videos are hosted on an account that is accidentally deleted. Maybe the original, easily-editable copy of the guided notes is lost and someone has to piece things together from a PDF.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As a result, administrators overseeing resource allocation for the online component of a department should think the way they think when purchasing other expensive assets like cars. Specifically,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Money should be set aside for &amp;quot;&lt;strong&gt;oil changes&lt;/strong&gt;&amp;quot;: small maintenance that, if done regularly, extends the overall life of the online content.&lt;/li&gt;&#xA;&lt;li&gt;The entire project should be &lt;strong&gt;depreciated&lt;/strong&gt; over time: it should be assumed that the material will run out of its useful life sometime in the next 20 years, even if maintained very well.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If administrators understand this analogy while allocating resources, everyone will be less bitter during the natural process of decay and maintenance of the developed materials.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-03-04T21:39:42.867" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26201" PostTypeId="2" ParentId="26199" CreationDate="2023-03-05T03:43:02.677" Score="8" Body="&lt;p&gt;&amp;quot;An online course plays roughly the same role as an interactive textbook.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If we treat the above as a guideline, then some things become clear.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;We are willing to spend more on the production of a textbook than on the production of course notes. Similarly, a course taught online, with some &lt;em&gt;additional&lt;/em&gt; cost can be converted into an &amp;quot;online course&amp;quot; (in the sense above).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A textbook can replace actual instruction only when a student cannot have direct continual access to the instructor. Similarly, an &amp;quot;online course&amp;quot; cannot take the place of regular university instruction except in the context of (severe) resource constraints.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In many subjects, textbooks become outdated. As Chris Cunningham has pointed out the same applies to online course content which is not updated.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;(1) says that we &lt;em&gt;should&lt;/em&gt; be prepared to spend money on the generation on online courses. (2) says that we should not expect these costs to &lt;em&gt;reduce&lt;/em&gt; the cost of actual instruction. (3) says that we should expect to incur regular costs for upkeep of online courses just as we need budgets for libraries.&lt;/p&gt;&#xA;" OwnerUserId="13498" LastEditorUserId="13498" LastEditDate="2023-03-06T03:50:00.623" LastActivityDate="2023-03-06T03:50:00.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26202" PostTypeId="1" CreationDate="2023-03-05T23:39:47.867" Score="5" ViewCount="128" Body="&lt;p&gt;I am interested in developing an app for students that are starting a grade career involving mathematics. It is a real problem that they start with almost no knowgladge of basic mathematics and there is no time to teach it in the curricular courses. Is it possible to help they learn some basic properties of operations with real numbers like product, sums and exponents using an app? They are really confused about which operations are &amp;quot;allowed&amp;quot; and which are not. For example when is it correct to cancel two &amp;quot;x&amp;quot; or some doubts that involve operator's precedence. I know that behind this, there is a lack of understanding about the meaning of the equations they write. Do you think is it possible to impart this knowgladge with an app? Do you know any existing app that aims to do this? (i could not find, but I am not good at research). Sorry for my bad english.&lt;/p&gt;&#xA;" OwnerUserId="22133" LastActivityDate="2023-03-05T23:39:47.867" Title="Is it possible to learn some basic mathematics using an app?" Tags="|undergraduate-education|mathematical-pedagogy|software|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26204" PostTypeId="2" ParentId="715" CreationDate="2023-03-08T18:35:23.253" Score="5" Body="&lt;p&gt;This example here I discovered myself:&lt;/p&gt;&#xA;&lt;p&gt;Consider a cylinder of radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and height &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, circumscribing a paraboloid &lt;span class=&quot;math-container&quot;&gt;$y=h\left(\dfrac{x}{r}\right)^2$&lt;/span&gt; whose apex is at the center of the bottom base of the cylinder and whose base is the top base of the cylinder.&lt;br&gt;&#xA;Also consider the paraboloid &lt;span class=&quot;math-container&quot;&gt;$y=h-h\left(\dfrac{x}{r}\right)^2$&lt;/span&gt;, with equal dimensions but with its apex and base flipped.&lt;/p&gt;&#xA;&lt;p&gt;For every height &lt;span class=&quot;math-container&quot;&gt;$0\le y\le h$&lt;/span&gt;, the disk-shaped cross-sectional area &lt;span class=&quot;math-container&quot;&gt;$\pi\left(\sqrt{1-\frac{y}{h}}\,r\right)^2$&lt;/span&gt; of the flipped paraboloid is equal to the ring-shaped cross-sectional area &lt;span class=&quot;math-container&quot;&gt;$\pi r^2-\pi\left(\sqrt{\frac{y}{h}}\,r\right)^2$&lt;/span&gt; of the cylinder part ''outside'' the inscribed paraboloid.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, the volume of the flipped paraboloid is equal to the volume of the cylinder part ''outside'' the inscribed paraboloid. In other words, the volume of the  paraboloid is &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\pi}{2}r^2h$&lt;/span&gt;, half the volume of its circumscribing cylinder.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/yMfEI.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yMfEI.gif&quot; alt=&quot;The volume of the paraboloid is half the volume of its circumscribing cylinder.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22147" LastActivityDate="2023-03-08T18:35:23.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26205" PostTypeId="1" AcceptedAnswerId="26206" CreationDate="2023-03-09T11:14:15.597" Score="11" ViewCount="4517" Body="&lt;p&gt;How can I fairly compensate a student who showed passion and dedication for my undergraduate course but performed poorly on the final exam, without unfairly advantaging them over other students?&lt;/p&gt;&#xA;&lt;p&gt;Additional context: The student spent a significant amount of time taking meticulous notes for the course and shared them with others, but this was not announced as an opportunity for extra credit beforehand.&lt;/p&gt;&#xA;" OwnerDisplayName="user19945" LastEditorUserId="470" LastEditDate="2023-03-09T14:35:02.630" LastActivityDate="2023-03-13T00:57:31.680" Title="Is there a fair way to increase the grade of students who did not do well in exams?" Tags="|undergraduate-education|grading|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26206" PostTypeId="2" ParentId="26205" CreationDate="2023-03-09T12:22:29.113" Score="40" Body="&lt;p&gt;With respect to considerations of this kind one should reflect on one's information sources. For every student who exhibits to the professor hard work there are other students who have worked just as hard but have not exhibited it to the professor. If exhibiting activity to the professor does not form part of formal evaluation procedures, it should not be taken into account, as this is certainly unfair to other students.&lt;/p&gt;&#xA;&lt;p&gt;If one desires to reward effort rather than accomplishment (perhaps a bad idea!) then one should build this into the evaluation scheme from the beginning. If one wishes to allow for some part of the evaluation to be discretional, based on the professor's subjective impressions (perhaps a bad idea!), one should build this into the evaluation scheme—provided one's institutional context even permits it.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes one has a situation like the following that often occurs in Spain. Students are graded on a 0-10 scale that steps by tenths. A passing grade is 5.0. The difference between a student who obtains a 4.8 and a 5.0 is often essentially marginal - one could say within the margin of error of the not so robust evaluation process - and the professor decides to pass everyone who obtained at least a 4.7. One possibility is to add .3 to every grade that it at least 4.7. Of course this generates a new threshhold effect at 4.6 ... and so such a procedure can reasonably be questioned. But anyone reasonable will agree that it is less questionable than giving 5.0 to both the student who obtained 4.7 and the student who obtained 5.0. What seems to me wholly unfair is to give 5.0 to some student who obtained 4.7 because they worked hard, while not doing the same for some student who got 4.8 and never came to office hours and appeared (!) to be a lazy bum.&lt;/p&gt;&#xA;&lt;p&gt;In short, I think best practice is to stick to the previously published criteria. This has its shortcomings, but they are less significant than the shortcomings of the alternatives.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="13919" LastEditDate="2023-03-13T00:57:31.680" LastActivityDate="2023-03-13T00:57:31.680" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26208" PostTypeId="2" ParentId="26205" CreationDate="2023-03-09T16:30:22.573" Score="23" Body="&lt;h3&gt;Follow your syllabus&lt;/h3&gt;&#xA;&lt;p&gt;Presumably, you gave every student a syllabus at the beginning of the course, and this syllabus outlined standards for grading.  The grades you assign at the end of the term should adhere to whatever standard you set at the beginning of the term.  To significantly vary from this scheme is unfair to everyone in the class, and, in some places, could lead to your grades being overturned.&lt;/p&gt;&#xA;&lt;p&gt;If your syllabus says that students need to do well on the final exam in order to pass, then it sounds like this student did not earn the grade you want to give them, per the terms of the syllabus.  As such, to give them an inflated grade because they worked hard is unfair to the other students in the class.&lt;/p&gt;&#xA;&lt;p&gt;Don't do it.&lt;/p&gt;&#xA;&lt;h3&gt;What do your grades mean?&lt;/h3&gt;&#xA;&lt;p&gt;That being said, this seems like a chance to reevaluate your grading scheme for the future.  Think about the following questions (among others, but, based on the information you've given, these are the most salient points):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What should a grade mean?  What information am I conveying if I give a student an A?  or a C?&lt;/li&gt;&#xA;&lt;li&gt;Is the &lt;em&gt;effort&lt;/em&gt; that students put into a class one of the things which should be captured in the grade at the end of the term?&lt;/li&gt;&#xA;&lt;li&gt;Is the &lt;em&gt;final exam&lt;/em&gt; something which should be so central to the way in which grades are computed?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;You should then construct a grading scheme which reflects these values.&lt;/p&gt;&#xA;&lt;p&gt;For example, my belief is that the grades I give in my precalculus classes reflect two things, in almost equal proportion:  an understanding of the concepts being taught in the class, and the academic skills which are likely to lead to success in later classes (both mathematical and non-mathematical).  I do believe that effort is one of those academic skills, hence I would like to see effort incorporated into final grades.  I also believe that being able to accurately complete work in under the pressure of a clock is important to demonstrating mastery of the material (the time limit is a rough proxy for how &amp;quot;internalized&amp;quot; ideas have become).&lt;/p&gt;&#xA;&lt;p&gt;As such, I have components in my grading scheme which reward effort (students must turn in a certain amount of &amp;quot;drill-and-kill&amp;quot; work, they must participate in class or office hours, etc), and I have components which reward demonstrations of competence (there are graded written assignments, weekly quizzes, and a final exam).&lt;/p&gt;&#xA;&lt;h3&gt;Moral of the story&lt;/h3&gt;&#xA;&lt;p&gt;Don't make one-off exceptions for individual students.  In any given term, stick to the terms of your syllabus as best as possible.  However, take difficult situations (like the one outlined in the question) and use them as opportunities to evaluate your assessment schemata, and update your syllabus to reflect what you have learned.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2023-03-09T16:30:22.573" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26209" PostTypeId="2" ParentId="26139" CreationDate="2023-03-10T06:31:08.713" Score="2" Body="&lt;p&gt;On the pure mathematics side, you can show him the &lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/0/08/Comparison_of_truncated_icosahedron_and_soccer_ball.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;truncated icosahedron&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/0/08/Comparison_of_truncated_icosahedron_and_soccer_ball.png&quot; alt=&quot;truncated icosahedron beside soccer ball&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Why exactly can these pentagons and hexagons fit together like this? It would be a good mental exercise for you too. What happens if you shrink the pentagons? What about shrinking the hexagons? This can lead to talking about the platonic solids and crystal structures and so on. Did you know about &lt;a href=&quot;https://en.wikipedia.org/wiki/Fullerene&quot; rel=&quot;nofollow noreferrer&quot;&gt;fullerene&lt;/a&gt;? Did you know that &lt;a href=&quot;https://en.wikipedia.org/wiki/Capsid#Specific_shapes&quot; rel=&quot;nofollow noreferrer&quot;&gt;many viruses have icosahedral capsids&lt;/a&gt;?&lt;/p&gt;&#xA;&lt;p&gt;On the applied mathematics side, you can explain that mathematics allows you to demonstrate that (ignoring wind and air resistance) you can kick the ball furthest if you kick it upwards at an angle of 45°, which can lead to talking about the fields of analysis, quadratic curves and optimization.&lt;/p&gt;&#xA;&lt;p&gt;Unrelated to soccer, I recommend introducing people to &lt;a href=&quot;https://matheducators.stackexchange.com/a/18990/1550&quot;&gt;concrete interesting mathematics&lt;/a&gt; rather than the dry fare that most textbooks are filled with. I would say that true mathematics is much closer in spirit to solving &lt;a href=&quot;https://matheducators.stackexchange.com/a/25518/1550&quot;&gt;(genuine) logic puzzles&lt;/a&gt; than to learning from typical textbooks.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2023-03-10T06:31:08.713" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26210" PostTypeId="1" CreationDate="2023-03-10T15:01:55.010" Score="7" ViewCount="684" Body="&lt;p&gt;In anticipation of &lt;a href=&quot;https://www.piday.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pi Day&lt;/a&gt;, which is (of course) March 14, I would like to ask:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What are some interesting math questions whose answer is exactly &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The questions can be for any age group.&lt;/p&gt;&#xA;&lt;p&gt;Of course, any math question can be tweaked so that its answer is &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. For example, &amp;quot;What is a closed form for &lt;span class=&quot;math-container&quot;&gt;$4\sum\limits_{k=1}^\infty \frac{(-1)^{k-1}}{2k-1}$&lt;/span&gt; ?&amp;quot;, where the &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; is there just to force the answer to be &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. But I'm asking for natural, untweaked questions whose answer is exactly &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here are some examples.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; identical &amp;quot;sunrays&amp;quot; of edge length &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; emanate from a circle of radius &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, as shown here with example &lt;span class=&quot;math-container&quot;&gt;$n=8$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lNwdR.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lNwdR.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;What is &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{n\to\infty}(\text{total area of rays})$&lt;/span&gt; ?&lt;/p&gt;&#xA;&lt;p&gt;I suppose this is not entirely trivial - unless you see the intuitive solution:&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; Flip the sunrays inside the circle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Evaluate &lt;span class=&quot;math-container&quot;&gt;$\int_{-\infty}^\infty \frac{\sin{x}}{x}dx$&lt;/span&gt;. &lt;a href=&quot;https://math.stackexchange.com/questions/5248/evaluating-the-integral-int-0-infty-frac-sin-x-x-mathrm-dx-frac-pi&quot;&gt;Here&lt;/a&gt; are some solutions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The famous &lt;a href=&quot;https://math.stackexchange.com/questions/138289/intuitive-reasoning-behind-pis-appearance-in-bouncing-balls&quot;&gt;colliding blocks&lt;/a&gt; (or balls) question: &amp;quot;Let the mass of two balls be &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; respectively. Assume that &lt;span class=&quot;math-container&quot;&gt;$M=16 \times 100^n m$&lt;/span&gt;. Now, we will roll the ball with mass &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; towards the lighter ball which is near a wall. How many times do the balls touch each other before the larger ball changes direction? (The large ball hits the small ball which bounces off the wall.)&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20101" LastEditorUserId="8571" LastEditDate="2023-03-11T16:03:44.527" LastActivityDate="2023-03-15T11:59:35.447" Title="Pi Day is approaching: What are some interesting math questions whose answer is exactly $\pi$?" Tags="|big-list|recreational-learning|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26211" PostTypeId="2" ParentId="26210" CreationDate="2023-03-10T15:34:44.610" Score="5" Body="&lt;p&gt;&amp;quot;What are some interesting math questions whose answer is exactly &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The area of a unit circle--It is exactly &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The proof is very neat and visual and can be understood by &lt;em&gt;any&lt;/em&gt; audience(which is is what separates it from other examples and makes it so important to know.):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ncvig.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ncvig.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20949" LastEditorUserId="20949" LastEditDate="2023-03-11T09:05:10.193" LastActivityDate="2023-03-11T09:05:10.193" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26212" PostTypeId="2" ParentId="26210" CreationDate="2023-03-10T15:43:19.493" Score="10" Body="&lt;p&gt;Regarding your second example. Not only &lt;span class=&quot;math-container&quot;&gt;$\int_{-\infty}^\infty \frac{\sin x}{x} dx = \pi$&lt;/span&gt;, but also&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} dx = \pi $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} \frac{\sin (x/5)}{x/5}dx = \pi $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} \frac{\sin (x/5)}{x/5} \frac{\sin (x/7)}{x/7}dx = \pi $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \dots $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} \frac{\sin (x/5)}{x/5} \frac{\sin (x/7)}{x/7} \dots \frac{\sin (x/13)}{x/13} dx = \pi . $$&lt;/span&gt;&#xA;And then the pattern breaks and we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} \frac{\sin (x/5)}{x/5} \frac{\sin (x/7)}{x/7} \dots \frac{\sin (x/15)}{x/15} dx &amp;lt; \pi $$&lt;/span&gt;&#xA;specifically&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} \frac{\sin (x/5)}{x/5} \frac{\sin (x/7)}{x/7} \dots \frac{\sin (x/15)}{x/15} dx \approx \pi - 4.62\times 10^{-11} $$&lt;/span&gt;&#xA;For the explanation, check the &lt;a href=&quot;https://en.wikipedia.org/wiki/Borwein_integral&quot; rel=&quot;noreferrer&quot;&gt;Borwein integral on Wikipedia&lt;/a&gt; and &lt;a href=&quot;https://www.youtube.com/watch?v=851U557j6HE&amp;amp;t=37s&quot; rel=&quot;noreferrer&quot;&gt;this video&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16302" LastActivityDate="2023-03-10T15:43:19.493" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26213" PostTypeId="2" ParentId="26205" CreationDate="2023-03-10T23:46:47.737" Score="6" Body="&lt;p&gt;There is a general principle that if you offer any &lt;em&gt;additional&lt;/em&gt; opportunity, then it has to be opened to everybody. You can declare pretty much any grading policy &lt;em&gt;in the beginning&lt;/em&gt; of the class. My favorite extra options are extracurricular projects, extra hard problems to solve at home and reading assignments (I'm teaching mathematics). There have been cases when people using those options ended up with having enough points for an A even without getting any credit for the finals and I was completely content with that.&lt;/p&gt;&#xA;&lt;p&gt;What I consider somewhat unethical is offering special opportunities to those who failed miserably &lt;em&gt;a posteriori&lt;/em&gt; without giving an opportunity to those who passed without flying colors to improve their grade too, so I wouldn't change the grade itself. However you can always give some extra encouragement ranging from a few kind words to a formal recommendation addressed to a colleague or registrar's office for the student to enter an advanced course for which yours is a prerequisite even if the official grade is below the official requirement (if you think it is appropriate, of course). You can also award some funny prize for good notes or something else. Once my official requirement for an A in a graduate course was solving 5 problems at the blackboard and one student solved 10, so I gave him an official grade of A, but presented him an AA battery on the last day of the class. You can easily invent something like that too: the cost is next to nothing but it is a clear sign of appreciation, and, I think, that is all the situation calls for.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-03-10T23:46:47.737" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26214" PostTypeId="2" ParentId="26210" CreationDate="2023-03-11T11:14:04.550" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Buffon%27s_needle_problem&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Buffon%27s_needle_problem&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jPke3.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jPke3.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;17 matches thrown randomly.  Spacing between lines is 1 match length.  11 cross a line.  &lt;span class=&quot;math-container&quot;&gt;$\frac{2 \cdot 17}{11} \approx 3.1 \approx \pi$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-03-11T11:14:04.550" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26215" PostTypeId="2" ParentId="26205" CreationDate="2023-03-11T12:26:18.613" Score="3" Body="&lt;h3&gt;Don't ever do it, because this is fundamentally unfair to every other student you teach&lt;/h3&gt;&#xA;&lt;p&gt;The ideal we strive for as a society is equality of opportunity. We &lt;em&gt;&lt;strong&gt;must not&lt;/strong&gt;&lt;/em&gt; mandate equality of outcome - the classic satire &lt;a href=&quot;https://en.wikipedia.org/wiki/Harrison_Bergeron&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harrison Bergeron&lt;/a&gt; is the logical conclusion of that.&lt;/p&gt;&#xA;&lt;p&gt;By all means attempt to make your assessment procedures as fair as possible. Exams may not be the best way to do a fair assessment, so for sure you can look at that structurally. But after that, you &lt;em&gt;&lt;strong&gt;must not&lt;/strong&gt;&lt;/em&gt; bias the results. And to prevent you biasing the results in favour of your favourites, you should have a marking scheme to stop you doing it.&lt;/p&gt;&#xA;&lt;p&gt;At most, you could look at the appeals procedure in case this student has had some kind of life event which has affected their results. But apart from that, you have to let it stand.&lt;/p&gt;&#xA;&lt;h3&gt;It may also get you fired&lt;/h3&gt;&#xA;&lt;p&gt;Artificially boosting the grades of your favourites is literally on the list of gross misconduct for pretty much any academic job. It's how male teachers reward their favourite female students in exchange for &amp;quot;favours&amp;quot;. (Or sometimes in different orientations, sure, but that's the most common one.) You could be fired on the spot and never work again. Seriously, don't go there.&lt;/p&gt;&#xA;" OwnerUserId="6456" LastActivityDate="2023-03-11T12:26:18.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26216" PostTypeId="2" ParentId="26210" CreationDate="2023-03-12T20:42:12.320" Score="2" Body="&lt;p&gt;Not an answer, but more of a chuckle. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Indiana_Pi_Bill#:%7E:text=The%20Indiana%20Pi%20Bill%20is,mathematical%20truth%20by%20legislative%20fiat.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Indiana Bill&lt;/a&gt; also gives &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;More seriously, it all depends on your audience. &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80&quot; rel=&quot;nofollow noreferrer&quot;&gt;There are many choices&lt;/a&gt;.  Some of your students might like the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pi&quot; rel=&quot;nofollow noreferrer&quot;&gt;the history&lt;/a&gt; of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; as well.&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2023-03-12T20:56:19.010" LastActivityDate="2023-03-12T20:56:19.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26217" PostTypeId="2" ParentId="26210" CreationDate="2023-03-12T23:51:03.507" Score="4" Body="&lt;p&gt;A favorite is:&#xA;&lt;a href=&quot;https://mathoverflow.net/q/151706/6094&quot;&gt;What fraction of the integer lattice can be seen from the origin?&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The answer is about &lt;span class=&quot;math-container&quot;&gt;$61$&lt;/span&gt;%.&#xA;Here &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; makes its appearance in a surprising, non-circular context&#xA;(surprising to me):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 0.6079 \;.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/BIwIu.jpg&quot; alt=&quot;LatticeVisib8Grid&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;And &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\zeta(d)}$&lt;/span&gt; in dimension &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;: Almost all the lattice points&#xA;are visible as &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; grows (because &lt;span class=&quot;math-container&quot;&gt;$\zeta(d) \to 1$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2023-03-13T00:51:43.757" LastActivityDate="2023-03-13T00:51:43.757" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26218" PostTypeId="2" ParentId="26210" CreationDate="2023-03-13T14:44:53.253" Score="3" Body="&lt;p&gt;On Youtube, there's a video about two squares, one of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;kg. and one of &lt;span class=&quot;math-container&quot;&gt;$10^n$&lt;/span&gt;kg., who are bouncing between themselves and the Y-axis. apparently the amount of bounces equals the decimals of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Hereby the &lt;a href=&quot;https://www.youtube.com/watch?v=HEfHFsfGXjs&quot; rel=&quot;nofollow noreferrer&quot;&gt;Youtube URL&lt;/a&gt;, uploaded by 3blue1brown.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastEditorUserId="8190" LastEditDate="2023-03-15T11:59:35.447" LastActivityDate="2023-03-15T11:59:35.447" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26219" PostTypeId="1" CreationDate="2023-03-13T16:25:45.513" Score="1" ViewCount="175" Body="&lt;p&gt;I will have to teach a first course in differential equations. What should I cover in this module? For example, in most books, have Laplace Transforms which is fine but I would not use LT to solve differential equations.&lt;/p&gt;&#xA;&lt;p&gt;This differential equations module is for sophomore students who have done a calculus module in their first year which covered 1st and second-order DEs. They have come across separable equations, integrating factor, Newton's Law of cooling and solving second order by characteristic equation and particular integral. Both first and second order initial value problems. In addition they have solved these equations by numerical methods. They do a final year module called partial differential equations. I hope this sets the background of the students I will be teaching.&lt;/p&gt;&#xA;&lt;p&gt;I want to write a course that motivates students and has an impact. What topics and what is the most motivating way to introduce differential equations? I want a well-structured and practical approach to differential equations. It is for mathematics and physics students.&lt;/p&gt;&#xA;" OwnerUserId="1567" LastEditorUserId="11" LastEditDate="2023-03-15T17:31:25.893" LastActivityDate="2023-03-15T17:31:25.893" Title="What to cover on a first ordinary differential equations module?" Tags="|undergraduate-education|mathematical-pedagogy|differential-equations|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26220" PostTypeId="1" AcceptedAnswerId="26228" CreationDate="2023-03-16T16:36:06.037" Score="8" ViewCount="886" Body="&lt;p&gt;When analyzing the functions&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x) = \sqrt{x-5}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$g(x) = \frac{1}{x-5}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$h(x) = 2^{x-5}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;we know that it is useful to think about what happens at &lt;span class=&quot;math-container&quot;&gt;$x = 5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, this logic will yield the &lt;em&gt;vertex&lt;/em&gt; of a graph.&lt;/p&gt;&#xA;&lt;p&gt;For the function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, this logic will yield the &lt;em&gt;vertical asymptote&lt;/em&gt; of a graph.&lt;/p&gt;&#xA;&lt;p&gt;For the function &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, this logic doesn't really yield and particularly special point, but still helps us draw a graph.&lt;/p&gt;&#xA;&lt;p&gt;I have been telling my algebra students that in all three cases,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;&lt;span class=&quot;math-container&quot;&gt;$x = 5$&lt;/span&gt; is an important &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-value&lt;/strong&gt;, or&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;all the action is happening near &lt;span class=&quot;math-container&quot;&gt;$x=5$&lt;/span&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, my textbook does not talk this way or make this connection across different types of functions. Is there any better way to talk about this idea? Does the idea have a better name than &amp;quot;important x-values?&amp;quot; The word &amp;quot;critical&amp;quot; is already taken. The word &amp;quot;important&amp;quot; is imprecise. What should I be saying to my students?&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-03-29T16:36:28.467" Title="What should I call the &quot;important&quot; values of x?" Tags="|mathematical-pedagogy|algebra|terminology|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26221" PostTypeId="2" ParentId="26220" CreationDate="2023-03-16T17:00:10.573" Score="2" Body="&lt;p&gt;My advice would not be to strain for a special name like that.  Just teach the practice of curve sketching instead (vice inventing a new term, not in the book). f and g are already &amp;quot;important&amp;quot; in the sense that they are the value for one of the step by step parts of the curve sketching practice.  But I think you are better off teaching that as a step by step method, versus looking at &amp;quot;term roots&amp;quot; (that's the name I guess) to find which step it corresponds to (especially for weak/new students).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Curve_sketching&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Curve_sketching&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Also, I think it's arguable if the h function has an &amp;quot;important&amp;quot; (key) x point, like the other two do.  It's not a value at one of the steps of the typical curve sketching method.&lt;/p&gt;&#xA;" OwnerUserId="22169" LastActivityDate="2023-03-16T17:00:10.573" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26222" PostTypeId="1" CreationDate="2023-03-16T17:23:16.613" Score="6" ViewCount="479" Body="&lt;p&gt;I'm in the middle of teaching first-semester Calculus where, for the first time, I'm trying to implement a flipped classroom. (Background: Small university in U.S.; Calc 1 for STEM majors, 50 minute classes, 4 times a week, have taught the course in traditional lecture format many times.) In the flipped context, I've struggled to understand the role/importance of homework.&lt;/p&gt;&#xA;&lt;p&gt;Initially this semester, the format was: Due by class time, students were assigned video lectures on a topic/section. During class I would quickly summarize the main points and then they would work in groups, focusing on 2-3 problems in that topic meant to specifically prepare them for the quizzes. After class, they were assigned 8-12 (WebAssign) homework problems meant to give them more practice and/or round out their understanding of the concept. Even though attendance was good and students seemed to be logging in to watch the videos, the problem was that very few students were completing (even 25% of) the homework. My concern was that doing a homework set &lt;em&gt;and&lt;/em&gt; having to watch videos for the next class was too much for them.&lt;/p&gt;&#xA;&lt;p&gt;So more recently I've shifted to, after summarizing main points, opening up the homework set in class and working selected problems. Even though WebAssign has some randomization to it's problems, they seem able to adapt fine so that now most of them get at least those problems in the homework done (but for many that is &lt;em&gt;all&lt;/em&gt; they get done). My concern now is that they are not getting enough non-teacher-led practice with the concepts and, as such, struggle to be able to work efficiently on the quizzes/exams.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My main question to those who successfully teach a flipped classroom: How do you balance the load of content delivery, in-class work, and homework to adequately prepare students to succeed on the assessments? Or more broadly, what should I do differently next semester?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;As I mentioned above, I've taught first-semester Calculus multiple times in the past and so I know that not everyone is going to do all the homework. But admittedly this is the first time with this course since COVID, so perhaps there are larger issues at play with the student population. That's why I'm very curious to hear from those who taught flipped classrooms before and after COVID. To know if what I'm experiencing is the result of a newbie mistake in course design or of a larger issue that all flipped classrooms are struggling with recently.&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="2083" LastEditDate="2023-03-16T19:31:10.177" LastActivityDate="2023-03-16T19:31:10.177" Title="Homework in a Flipped Classroom" Tags="|undergraduate-education|calculus|homework|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26223" PostTypeId="2" ParentId="26222" CreationDate="2023-03-16T18:17:48.737" Score="6" Body="&lt;p&gt;A flipped classroom course does typically have &lt;em&gt;less&lt;/em&gt; homework than a lecture one to account for the time spent out of class doing things like watching videos, but it's pretty common to still have some, possibly a significant amount, of homework. I've been teaching flipped and partially flipped calculus classes since before the pandemic, and they've had substantially more homework than what it sounds like you're describing, so I'm skeptical the amount of homework is the issue.&lt;/p&gt;&#xA;&lt;p&gt;Most schools have formal or informal norms about how much out of class work a typical class has (and at a minimum, unless you're the only calculus instructor, you can find out roughly work other calculus classes expect). 2-3 hours out of class per hour of class time is very common. I think of in class time as being the premium thing, so, especially in a flipped class, I'm trying to fill that in the most useful ways I can, and then assign an amount of out-of-class work that can reasonably be done in the remaining time, and there isn't space for anything that doesn't fit there. (Calculus classes are notoriously overpacked, so there often is some stuff that doesn't fit.)&lt;/p&gt;&#xA;&lt;p&gt;My experience is that flipped classes are more sensitive than lectures to small design choices - it's frustratingly easy to break something (especially around homework) with a minor design decision, and relatedly the fixes are sometimes surprisingly small.&lt;/p&gt;&#xA;&lt;p&gt;If students aren't doing the homework (and the amount of work is reasonable), I'd focus on asking what's supposed to motivate them to do it, and why those incentives aren't work.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is it counted in their grade? Does it count for a big enough part? Grades are &lt;em&gt;usually&lt;/em&gt; enough to motivate most students to do work, so the easy answer is often &amp;quot;make it count for more&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;Do students understand why they're supposed to do it? My &lt;em&gt;guess&lt;/em&gt;, based on the little you've said, is that this is where the breakdown is: that students are focused on the quizzes and exams, and don't see how the homework is relevant to that. You might think &amp;quot;these are problems involving the same material that's on the quiz&amp;quot; would be enough, but my experience is that it's usually not: students sometimes need the connection to the quiz/exams spelled out &lt;em&gt;very&lt;/em&gt; explicitly. You might consider having some of the quiz/exam problems be straight repeats (maybe with numbers changed, but even maybe not) of homework problems. If you don't want to do that, you might want to consider other ways to illustrate to students how doing the homework helps them prepare.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="62" LastActivityDate="2023-03-16T18:17:48.737" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26224" PostTypeId="1" AcceptedAnswerId="26225" CreationDate="2023-03-16T18:25:46.307" Score="14" ViewCount="2770" Body="&lt;p&gt;I’ve been working with many students on logarithms and have noted that log has a base of 10 unless specified. Further, I commented that putting a 10 as a subscript to log is redundant, or at least not needed. A student sent me this, from a popular math blog. Now I am wondering if I made a mistake. This image basically implies “when using log, always indicate the base, to avoid confusion.”&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gE9JE.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gE9JE.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Edit: When this came to my attention, I was reminded of a prior question &amp;quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/20824/why-do-we-write-x-instead-of-1x&quot;&gt;Why do we write  instead of 1 &lt;/a&gt;?&amp;quot; It seems from the replies, this is a different situation.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-03-17T10:37:31.177" LastActivityDate="2023-03-20T11:03:42.563" Title="When writing log, do you indicate the base, even when 10?" Tags="|secondary-education|logarithm|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26225" PostTypeId="2" ParentId="26224" CreationDate="2023-03-16T18:45:43.710" Score="36" Body="&lt;p&gt;And a computer scientist thinks that &lt;span class=&quot;math-container&quot;&gt;$\log=\log_2$&lt;/span&gt;. I am using &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; for the natural logarithm by default in all my courses though I clearly state that in the beginning of each course (I teach at the university level). In general, one can use any default base one wishes provided that it has been made clear what it is and that the usage is consistent. At the school level the common tradition is to use &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; for base &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; for natural logarithm, but if you go higher and look at various textbooks in STEM subjects, both old and modern, this convention is rarely upheld there.&lt;/p&gt;&#xA;&lt;p&gt;So, my advice is rather to alert the students that &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt; without base indicated may mean different bases in different contexts and, if the base is not immediately clear, the clarification should be sought and/or requested.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-03-16T18:45:43.710" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26226" PostTypeId="2" ParentId="26224" CreationDate="2023-03-16T18:52:52.040" Score="2" Body="&lt;p&gt;I think stereotypically that log means base 10 and ln means base e.  That is how all the 8 different high school and college texts (published 30s to 80s) that I have use it.  But I have noticed a (maybe growing?) tendency for other usage.  Had to get a formal paper in econ to define their base.  (And they were using the log means base e usage, that is more rare...at least for knuckle draggers like me.)&lt;/p&gt;&#xA;&lt;p&gt;I did a quick Google search on does log mean log base 10 and their were several university math tutor sites still propounding the orthodox religion about common logs.  But, a few heretics were seen also.  :-(&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=does+log+mean+log+base+10&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=does+log+mean+log+base+10&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;So, when you see log by itself, it means base ten log. When you see ln, it means natural logarithm (we'll define natural logarithms below). In this course only base ten and natural logarithms will be used.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;U Minn:  &lt;a href=&quot;http://www.mclph.umn.edu/mathrefresh/logs.html#:%7E:text=So%2C%20when%20you%20see%20log,natural%20logarithms%20will%20be%20used&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.mclph.umn.edu/mathrefresh/logs.html#:~:text=So%2C%20when%20you%20see%20log,natural%20logarithms%20will%20be%20used&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My advice would be to stick with the log means 10 usage since it is still (I think) the most common usage and it's your class that they mostly need it in.  I still remember fondly log tables (base 10) being handed out for AP chem into the 90s, in case kids did not have a calculator.  ;-)&lt;/p&gt;&#xA;&lt;p&gt;Maybe briefly mention it to the kids, that some usage is different. (heck &amp;quot;billion&amp;quot; means something different in different countries...and don't get me started on the comma/decimal hassles when passing documents back and forth with Germans).&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't feel too worried if they see something different in the future.  That can be dealt with then by the other teacher.  Or if they are really doing sophisticated work in econ or pure math, than they'll be quite capable of dealing with the momentary notation confusion.&lt;/p&gt;&#xA;&lt;p&gt;Also, FWIW, I wouldn't worry about alternate usage in the more rarified air of higher math courses.  Most of your kids need to learn things the old school way and they will be heading for the natural sciences or engineering (which are more traditional).  If they end up reading a measure theory text, there's other parts of the text that will vex them more than Napierian being treated as common. ;)&lt;/p&gt;&#xA;&lt;p&gt;P.s.  Of course if you're tutoring and you learn the teach has other usage, defer to him.  Or her.  Or it.&lt;/p&gt;&#xA;" OwnerUserId="22169" LastEditorUserId="22169" LastEditDate="2023-03-16T19:00:02.513" LastActivityDate="2023-03-16T19:00:02.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26227" PostTypeId="2" ParentId="26220" CreationDate="2023-03-16T19:02:12.957" Score="6" Body="&lt;p&gt;When graphing a function (or just investigating it from some point of view), it is, indeed, useful to look at some special points where the behavior changes in some way. Most of them have well-established names like &amp;quot;root&amp;quot;, &amp;quot;extremum&amp;quot;, &amp;quot;critical point&amp;quot;, or &amp;quot;inflection point&amp;quot; and one can just use those. Some are specific to the graphing task, like the easy to plot (say, integer) points where the function takes easy to compute (say, rational) values. What to look for is really task-dependent.&lt;/p&gt;&#xA;&lt;p&gt;The main idea to convey is that once one has decided what features of the function are interesting in the context of some particular task, some points become &amp;quot;more equal than others&amp;quot; from that particular point of view, and you are absolutely right that understanding this fact is important. Unfortunately, the only way to communicate this general idea is through a variety of worked out examples with different tasks resulting in different sets and notions of &amp;quot;important points&amp;quot;. There is no magic bullet that would convey &lt;em&gt;the meaning&lt;/em&gt; by just coining a catchy name.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-03-16T19:02:12.957" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26228" PostTypeId="2" ParentId="26220" CreationDate="2023-03-16T19:31:46.977" Score="5" Body="&lt;p&gt;For function transformations I use &amp;quot;base point&amp;quot; (&lt;span class=&quot;math-container&quot;&gt;$y=a^x$&lt;/span&gt; has base point &lt;span class=&quot;math-container&quot;&gt;$(0,1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y = \frac{1}{x}$&lt;/span&gt; has base point &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt;, even though it isn't on the graph).&lt;/p&gt;&#xA;&lt;p&gt;In calculus I use &amp;quot;point of interest&amp;quot; (for critical and hypercritical points), which can be useful language if you want to talk about multiple points moving as you transform your function.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2023-03-16T19:31:46.977" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26229" PostTypeId="2" ParentId="16948" CreationDate="2023-03-16T20:18:07.330" Score="3" Body="&lt;p&gt;This answer is concerned with real numbers only. Adding to Dave Renfro's comment about absolute-value manipulations being—or at least feeling—different from usual algebraic manipulations, and Mark B's answer about &lt;strong&gt;the additional intricacy of the logic of absolute-value statements&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The true statement &lt;span class=&quot;math-container&quot;&gt;$$\text{for each real }x,\:\:|x|=\pm x\tag{✔️}$$&lt;/span&gt; looks like a mathematical identity, &lt;span class=&quot;math-container&quot;&gt;$$|x|\equiv\pm x,\tag{❌}$$&lt;/span&gt; and misleadingly appears to suggest that &lt;span class=&quot;math-container&quot;&gt;$|x|$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\pm x$&lt;/span&gt; are mutually substitutable.&lt;/p&gt;&#xA;&lt;p&gt;But of course &lt;span class=&quot;math-container&quot;&gt;$$x^2+3x=\color\red{|x|} \iff x^2+3x=\color\red{\pm x}\tag{❌}$$&lt;/span&gt; is incorrect (&lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; satisfies only the RHS equation).&lt;/p&gt;&#xA;&lt;p&gt;Yet this is correct: &lt;span class=&quot;math-container&quot;&gt;$$\color\red{|x|} = 5 \iff \color\red{±x} = 5.\tag{✔️}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What's going on is that &lt;span class=&quot;math-container&quot;&gt;$|x|=\pm x$&lt;/span&gt; is a faux equation and the first statement, which means &lt;span class=&quot;math-container&quot;&gt;$$\text{for each real }x,\:\:|x|\in \{x,-x\},\tag{✔️}$$&lt;/span&gt; isn't actually an identity (a universally true &lt;em&gt;equation&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;As a matter of fact, the result &lt;span class=&quot;math-container&quot;&gt;$$y=|x|\quad \iff\quad y=\pm x\:\:\:\text{and}\:\:\:y\ge0\tag{✔️}$$&lt;/span&gt; is a quick derivation from the definition of absolute value, and the second ticked statement above is merely a particular case of it, rather than an instance of substitution-by-applying-an-identity.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Thus, this is correct: &lt;span class=&quot;math-container&quot;&gt;\begin{align}&amp;amp;x^2+3x=|x| \\\iff{}&amp;amp;x^2+3x=\pm x \:\:\:\text{and}\:\:\:x^2+3x \ge0.\tag{✔️}\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And this: &lt;span class=&quot;math-container&quot;&gt;\begin{align}&amp;amp;x^2+3x=|x| \\\iff{}&amp;amp;\big(x&amp;lt;0\:\:\:\text{and}\:\:\:x^2+3x=-x\big) \:\:\:\text{or}\:\:\:\big(x\ge0\:\:\:\text{and}\:\:\:x^2+3x=x\big).\tag{✔️}\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And this: &lt;span class=&quot;math-container&quot;&gt;\begin{gather}x^2+3x=|x| \\\implies x^2+3x=x \:\:\:\text{or}\:\:\:x^2+3x=-x;\\&#xA;\text{eliminate the extraneous solution }-2.\tag{✔️}\end{gather}&lt;/span&gt; (The extraneous solutions created by the logic of an absolute-value equation, unlike those of a logarithmic or trigonometric equation, are generally not simply outside the equation's domain.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-03-17T08:56:46.253" LastActivityDate="2023-03-17T08:56:46.253" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26230" PostTypeId="1" CreationDate="2023-03-17T01:11:55.937" Score="2" ViewCount="169" Body="&lt;p&gt;Usually we describe proofs in algebra by algebraic means, I think it may be useful to introduce geometrical approaches to those proofs to improve creativity skills of students, what are the examples we can use in this regard and from where we can find more suited for advanced level students? &lt;br/&gt;&#xA;Here are few issues I have in my mind,&lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;sum of two consecutive triangle numbers is a square number. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;sum of cubes of first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; positive integers is square of sum of first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; positive integers. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;${n+1 \choose 2} = \frac{n \cdot (n+1)}{2}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="8190" LastEditDate="2023-03-20T14:23:51.610" LastActivityDate="2023-03-20T14:23:51.610" Title="Geometrical approaches in algebra" Tags="|secondary-education|algebra|geometry|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26231" PostTypeId="2" ParentId="26224" CreationDate="2023-03-17T08:03:00.050" Score="15" Body="&lt;p&gt;I'm in Germany (chemist, FWIW). I'm familiar with:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt;: base is unknown/not needed (as in &lt;span class=&quot;math-container&quot;&gt;$\log (a) + \log (b) = \log (ab)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;natural logarithm: &lt;span class=&quot;math-container&quot;&gt;$\ln = \log_e$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;base 10 logarithm: &lt;span class=&quot;math-container&quot;&gt;$\lg = \log_{10}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In chemistry, both natural and base 10 logarithm are used a lot: The natural logarithm wherever one has to differentiate or integrate (naturally ;-) ), e.g. in kinetics or statistical thermodynamics. In parallel, base 10 logarithm is used a lot in definitions of physical quantities like &lt;span class=&quot;math-container&quot;&gt;$\mathrm{pH}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathrm{pK}_a$&lt;/span&gt;, absorbance etc.&lt;br /&gt;&#xA;In some cases (e.g. Beer-Lambert-law) there is some ambiguity and field-dependent definitions. In that case, we sometimes include the base in the name of the quantity, e.g. the &lt;a href=&quot;https://en.wikipedia.org/wiki/Molar_absorption_coefficient&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;decadic&lt;/em&gt; molar absorption coefficent&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathrm{lb} = \log_2$&lt;/span&gt; Rare. I've seen it and I recognize it - but I'd spell out this definition rather than assuming any reader to be familiar with this. Don't know whether one could assume computer scientists to be familiar with this.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Update: From &lt;a href=&quot;https://math.stackexchange.com/a/1734/97593&quot;&gt;this answer on math.sx&lt;/a&gt; I learned that this notation is &lt;a href=&quot;https://en.wikipedia.org/wiki/ISO_31-11#Exponential_and_logarithmic_functions&quot; rel=&quot;noreferrer&quot;&gt;ISO standard&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13257" LastEditorUserId="13257" LastEditDate="2023-03-17T12:29:02.647" LastActivityDate="2023-03-17T12:29:02.647" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26232" PostTypeId="2" ParentId="16948" CreationDate="2023-03-17T14:32:16.407" Score="1" Body="&lt;p&gt;IMHO the main issue is that the absolute value doesn't satisfy any simple algebraic rules, so you cannot simplify the expressions including it. The computation itself is no more difficult than the computation with parentheses. If a student has more trouble with finding &lt;span class=&quot;math-container&quot;&gt;$f(x)=\Big\lvert\big\lvert\lvert x-3\rvert-5\lvert 3x-7\rvert\big\rvert-\lvert 6x+2\rvert\Big\rvert$&lt;/span&gt; than with finding &lt;span class=&quot;math-container&quot;&gt;$g(x)=\Big(\big((x-3)-5(3x-7)\big)-(6x+2)\Big)$&lt;/span&gt;, for a &lt;em&gt;given value&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, that merely means that they should put some effort into understanding the basic definitions and learning the order of operations. However, if one asks to solve the inequality &lt;span class=&quot;math-container&quot;&gt;$f(x)\le 10$&lt;/span&gt; as opposed to solving &lt;span class=&quot;math-container&quot;&gt;$g(x)\le 10$&lt;/span&gt;, then the increase in the degree of difficulty is genuine not because of some fancy &amp;quot;psychological&amp;quot; reasons, but merely because it is a truly long and branching task, i.e., one can write a three line computer program that outputs the answer to the second question, but I wonder how much effort it will take you to write a piece of code that solves the first inequality correctly when I put arbitrary parameters instead of the numeric values into that rather simple expression. And if you have trouble teaching a machine to do something that requires just a routine algorithm, you'll certainly have trouble with students too, because at that level they are just imperfect computers.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastEditorUserId="16250" LastEditDate="2023-03-17T17:40:06.767" LastActivityDate="2023-03-17T17:40:06.767" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26233" PostTypeId="2" ParentId="220" CreationDate="2023-03-17T16:55:57.983" Score="0" Body="&lt;p&gt;1)Finitely many lines divide a plane into regions. Show these regions can be coloured by two colours in such a way that neighboring regions have different colours.&lt;br/&gt;&#xA;2)Showthat there are &lt;span class=&quot;math-container&quot;&gt;$2^n%$&lt;/span&gt; number of events in the event space corresponding to the sample space of n number of sample points.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2023-03-17T16:55:57.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26234" PostTypeId="1" CreationDate="2023-03-17T17:23:19.480" Score="1" ViewCount="356" Body="&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What are some arguments for and against telling the public that &lt;em&gt;Mathematics is everywhere&lt;/em&gt;? I would like to know if there is any evidence that telling the public &lt;em&gt;Mathematics is everywhere&lt;/em&gt; generates some interest in mathematics, as opposed to perhaps only making people &lt;em&gt;respect&lt;/em&gt; mathematics, without necessarily changing their feelings toward it?&lt;/p&gt;&#xA;&lt;p&gt;Note: I am not asking whether mathematics &lt;em&gt;is&lt;/em&gt; everywhere or not.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;You have probably heard that Mathematics is everywhere. It is used to popularize mathematics. In fact, &lt;em&gt;Mathematics is everywhere&lt;/em&gt; has even found its way to the International Congress of Mathematicians (ICM). For instance, in ICM 2014 there was a &amp;quot;Mathematics is everywhere&amp;quot; panel (see [&lt;a href=&quot;https://www.mi.fu-berlin.de/en/math/groups/ag-geom/publications/db/ICM-Proceedings-Maths_everywhere.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;] for the presentation of this panel). I quote from the first paragraph of the introduction in [&lt;a href=&quot;https://www.mi.fu-berlin.de/en/math/groups/ag-geom/publications/db/ICM-Proceedings-Maths_everywhere.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;]:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose you are a mathematician and you are put in front of this&#xA;statement [Mathematics is everywhere]. Are you convinced? If you are in front of your classroom,&#xA;are you able to explain the statement? And if you are in front of the&#xA;public, or of a journalist, what examples will you choose to&#xA;illustrate the statement? We, the panelists, have the impression that&#xA;many of our mathematician colleagues are convinced. Yet, many of us&#xA;lack good examples to pass the message.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The last sentence: &amp;quot;Yet, many of us lack good examples to pass the message&amp;quot; is notable, as it implies that many of our mathematician colleagues were &lt;em&gt;not&lt;/em&gt; attracted to mathematics because mathematics is everywhere, otherwise, you would expect them to know a few good examples of where mathematics is. So then why should one attempt to popularize mathematics in this way?&lt;/p&gt;&#xA;&lt;p&gt;Perhaps a more prevalent reason for why some people become mathematicians is that they find mathematics enjoyable, but does telling the public &lt;em&gt;Mathematics is everywhere&lt;/em&gt; convey this message?&#xA;In November 2022 there was a piece [&lt;a href=&quot;https://www.nytimes.com/2022/11/10/learning/what-students-are-saying-about-the-value-of-math.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;] in the New York Times that reflected high-school students' opinion about the value of learning mathematics. Reading through these opinions, one will quickly realize that almost all of them assess the value of learning mathematics based on its possible applications rather than being an enjoyable activity. Interestingly, there were some students who &amp;quot;admitted&amp;quot; mathematics is very important because it is everywhere, but still said they detested it. By contrast, one can speculate that if these students were asked about the value of learning music, many of them would talk about how much they enjoy music.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;References.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[&lt;a href=&quot;https://www.mi.fu-berlin.de/en/math/groups/ag-geom/publications/db/ICM-Proceedings-Maths_everywhere.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;]&lt;/strong&gt;  E. Colli, F. R. Nemenzo, K. Polthier, C. Rousseau: &lt;em&gt;&lt;a href=&quot;https://www.mi.fu-berlin.de/en/math/groups/ag-geom/publications/db/ICM-Proceedings-Maths_everywhere.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics is everywhere&lt;/a&gt;&lt;/em&gt;, Proceedings of ICM 2014, Vol. 1, pp: 799-811.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[&lt;a href=&quot;https://www.nytimes.com/2022/11/10/learning/what-students-are-saying-about-the-value-of-math.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;]&lt;/strong&gt; &lt;em&gt;&lt;a href=&quot;https://www.nytimes.com/2022/11/10/learning/what-students-are-saying-about-the-value-of-math.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;What students are saying about the value of Math&lt;/a&gt;&lt;/em&gt;, The New York Times, November 10, 2022.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-03-24T16:23:29.413" LastActivityDate="2023-03-24T16:23:29.413" Title="The effects of telling the public that Mathematics is everywhere" Tags="|mathematics-in-daily-life|" AnswerCount="1" CommentCount="6" ClosedDate="2023-03-18T00:47:31.920" ContentLicense="CC BY-SA 4.0" />
  <row Id="26235" PostTypeId="2" ParentId="26234" CreationDate="2023-03-17T20:32:00.890" Score="-5" Body="&lt;p&gt;The simple answer to your question is that math is everywhere because the universe in which we live is not a chaotic and unordered universe which is devoid of meaning. Rather, the universe in which we live speaks to the existence of an intelligence from which this order flows. As a Christian, we speak of the General Revelation of God, the fact that the universe itself testifies to His existence. The beauty and majesty of creation itself which includes mathematical and physical law in part testifies to the existence of creative agent which stands above the created order. Of course atheists are free to disagree, but that's how I see it.&lt;/p&gt;&#xA;&lt;p&gt;There are those who believe in the multiverse and believe that our ability to see mathematics in everything is a quirk of our particular existence. In their view there are infinitely many other universes where it is not so. I don't have faith in the multiverse, so, I don't find such speculations particularly interesting or convincing.&lt;/p&gt;&#xA;&lt;p&gt;Getting back to your question, if there is a question, yes I agree, we should not preach mathematics on the basis of its applicability. Surely it can be applied, but for many of us who love it most dearly, the application is of tertiary importance. The beauty and complexity of mathematics and the discovery of new structures in math, new levels of abstraction, the completion of classifications,... These vistas of undiscovered mathematics, these are what draw us in, we want to understand more. Looking backward to see how little our predecessors understood, it makes us question, are we the same ? At this point in history, the internet gives us a collective memory which was before unavailable except in limited form in libraries at universities. Now the internet is the library and universities are soon becoming irrelevant.&lt;/p&gt;&#xA;&lt;p&gt;In any event, your question is doomed as you have not taken enough time to reformulate your argumentative essay into the form of a question which is acceptable discourse on this Q&amp;amp;A site.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2023-03-17T20:32:00.890" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26236" PostTypeId="2" ParentId="220" CreationDate="2023-03-18T07:49:19.947" Score="0" Body="&lt;p&gt;Simple proofs (Proofs 1-3)&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Bernoulli Inequality&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Inequality of AM - GM (There various proof using mathematical induction. You can use standard induction or forward-backward induction.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Newton's Inequality&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Since you said looking for proof of surprising facts you can refer following below. Proofs are relatively straightforward with basic knowledge but some parts may be challenging.&lt;/p&gt;&#xA;&lt;h5&gt;Uses two variables in base case.&lt;/h5&gt;&#xA;&lt;h5&gt;Graph theory: Ramsey numbers&lt;/h5&gt;&#xA;&lt;p&gt;R(s, t) &amp;lt;= R(s-1, t) + R(s, t-1)&lt;/p&gt;&#xA;&lt;h5&gt;Double mathematical induction&lt;/h5&gt;&#xA;&lt;p&gt;Number of partitions of&#xA;x1 + x2 + ... xm = n&lt;/p&gt;&#xA;&lt;h5&gt;Proof of Maclaurin Inequality using Induction\&lt;/h5&gt;&#xA;&lt;p&gt;A generalisation of AM-GM inequality&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=m4xnI6_OR7M&amp;amp;list=PLBiVnG9A5gcfHRbtZcfQdJuUMAS6v48uj&amp;amp;index=4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Double mathematical induction partitions&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://kconrad.math.uconn.edu/articles/maclaurin.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maclaurin Inequality&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=CwrZ7E97e0c&amp;amp;list=PL3dVWQq5Sx9cEjb5W_4WVxSMPy9PnsP4d&amp;amp;index=3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ramsey's theory: Littlewood's inequality&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=Aj7F2gcKgIg&amp;amp;list=PLBiVnG9A5gcfHRbtZcfQdJuUMAS6v48uj&amp;amp;index=4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Forward backward induction&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20626" LastActivityDate="2023-03-18T07:49:19.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26237" PostTypeId="2" ParentId="26230" CreationDate="2023-03-18T20:31:11.367" Score="4" Body="&lt;p&gt;This may not be what you have in mind, but your question reminds me of “&lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_without_words&quot; rel=&quot;nofollow noreferrer&quot;&gt;proofs without words&lt;/a&gt;” aka visual proofs. The three examples you gave have relatively famous visual proofs.&lt;/p&gt;&#xA;&lt;p&gt;There’s some contention over whether visual proofs should be considered proofs, as &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/proofs-without-words-and-beyond-pwws-and-mathematical-proof&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; explains. Essentially, since proofs without words do not make the logic of their arguments explicit, some might argue that they cannot be considered proofs. The opposing viewpoint is that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If a [proof without words] presents a mathematical idea with sufficient apparent evidentiary force to convince a mathematician of the truth of a statement, why count it as anything other than a proof?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The linked article explores this debate in greater detail.&lt;/p&gt;&#xA;&lt;p&gt;As for the pedagogical value of visual proofs, I think they are beneficial if used in moderation. They are concrete examples of the mathematical notion of “elegance,” and in this way can help students to understand the culture of modern mathematics. They also embody the idea that a different representation can make a difficult problem more tractable, and this is a valuable lesson for students. It can also be a useful exercise to have students “fill in the words” to a proof without words.&lt;/p&gt;&#xA;&lt;p&gt;However, many proofs without words of algebraic statements are not very generalizable, and I think that the priority should be having students become proficient with basic proof techniques. In any discipline, technique is important for creativity.&lt;/p&gt;&#xA;&lt;p&gt;Roger B. Nelsen has put together &lt;a href=&quot;https://bookstore.ams.org/view?ProductCode=CLRM-PWW-SET&quot; rel=&quot;nofollow noreferrer&quot;&gt;three volumes of proofs without words&lt;/a&gt;, many of them algebra-related. These could be a good source of exercises for your students.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-03-18T20:31:11.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26238" PostTypeId="2" ParentId="26230" CreationDate="2023-03-18T23:44:31.463" Score="7" Body="&lt;p&gt;I offer this (community wiki) only to illustrate the OP's 2nd example.&lt;/p&gt;&#xA;&lt;p&gt;From the &lt;a href=&quot;https://archimedes-lab.org/2021/06/29/sum-of-consecutive-cubes-visual-proof/&quot; rel=&quot;noreferrer&quot;&gt;Archimedes Lab Project&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/13Z9I.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/13Z9I.jpg&quot; alt=&quot;SumOfCubes&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Quite beautiful!&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2023-03-18T23:44:31.463" CommentCount="2" CommunityOwnedDate="2023-03-18T23:44:31.463" ContentLicense="CC BY-SA 4.0" />
  <row Id="26239" PostTypeId="2" ParentId="26224" CreationDate="2023-03-19T11:56:59.873" Score="0" Body="&lt;p&gt;My experience as university student, researcher and teacher in technical and scientific fields in &lt;strong&gt;Italy&lt;/strong&gt; has taught me that the only widely known unambiguous notation is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;ln(x)&lt;/strong&gt; for base &lt;em&gt;e&lt;/em&gt; log (most common)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Log(x)&lt;/strong&gt; for base 10 log (slightly less common – note the capital &amp;quot;L&amp;quot;), but probably to be avoided when writing things by hand, since the capitals can be easily be warped by quick handwriting.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The notation &lt;strong&gt;log(x)&lt;/strong&gt; (sometimes abbreviated as &lt;strong&gt;lg(x)&lt;/strong&gt; in Italy) is used to indicate mostly natural logarithm (especially in math courses), but also frequently base 10 logarithm (in some engineering courses, for example). Some computer scientists use it to indicate base 2 log, but it is far less common (but not at all rare).&lt;/p&gt;&#xA;&lt;p&gt;What my math professors at university always said was to check my assumptions when using the &amp;quot;baseless&amp;quot; notation and to check any paper/book that used that notation for disambiguation clues, such as symbol lists, because there was no universal standard or practice to rely on.&lt;/p&gt;&#xA;&lt;p&gt;The only important thing is to be consistent: if you choose to use &amp;quot;log&amp;quot; for natural logarithms, then state so upfront, remind that choice often to your students (especially at the beginning), point out that it is not an universal choice, and &lt;strong&gt;never, ever,&lt;/strong&gt; use that symbol for something else in the same context or with the same group of students.&lt;/p&gt;&#xA;&lt;p&gt;My usual choice nowadays (in technical fields) is to use &lt;strong&gt;log&lt;/strong&gt; for base 10 (so I don't have capitalization issues) and &lt;strong&gt;ln&lt;/strong&gt; for base &lt;em&gt;e&lt;/em&gt;, so it matches any pocket calculator a student may be using.&lt;/p&gt;&#xA;" OwnerUserId="1673" LastEditorUserId="1673" LastEditDate="2023-03-20T11:03:42.563" LastActivityDate="2023-03-20T11:03:42.563" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26240" PostTypeId="1" CreationDate="2023-03-19T13:51:09.400" Score="4" ViewCount="160" Body="&lt;p&gt;If you have used commercial as well as &lt;a href=&quot;https://openstax.org/subjects/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;OPENSTAX OER textbooks&lt;/a&gt; for calculus I would like to know about your experience. How would you compare the two? Were there any disadvantages to using OpenStax?&lt;/p&gt;&#xA;" OwnerUserId="1472" LastActivityDate="2023-03-20T02:16:03.693" Title="How do you describe your experience using OER textbooks for calculus?" Tags="|calculus|textbooks|online|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26241" PostTypeId="1" CreationDate="2023-03-19T14:02:18.740" Score="5" ViewCount="692" Body="&lt;p&gt;We discuss lot of questions related to mathematical expressions consisting equations or inequalities in mathematical induction. What are the examples where we can apply mathematical induction as the method of proof in proving statements without involving  equation or inequality for advanced level students. I think this kind of problems may be quite interesting to improve creative skills in mathematics.&lt;br/&gt;&#xA;Here is one example to get the clear picture of the issue &lt;br/&gt;&#xA;Finitely many lines divide a plane into regions. Show that these regions can be coloured by two colours in such a way that neighboring regions have different colours.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-03-19T14:38:33.070" LastActivityDate="2023-03-19T17:55:07.593" Title="Mathematical induction without simplifying equations or inequalities" Tags="|secondary-education|discrete-math|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26242" PostTypeId="2" ParentId="26241" CreationDate="2023-03-19T17:54:51.850" Score="3" Body="&lt;p&gt;Here are a few examples (for students at very different levels, since it's rather subjective what constitutes an &amp;quot;advanced level&amp;quot;):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The task in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tower_of_Hanoi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Towers of Hanoi&lt;/a&gt; puzzle is solvable. (The Towers of Hanoi are often used as an example for a recursive algorithm - but one can, of course, also frame it as an example for induction.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A classic: existence of the prime factorization of integers.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A compact metric space (or, more generally, a compact topological Hausdorff space) which is countable and infinite contains infinitely many isolated points.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Identity theorem for polynomials: if a complex polynomial of degree &lt;span class=&quot;math-container&quot;&gt;$d \ge 0$&lt;/span&gt; vanishes at &lt;span class=&quot;math-container&quot;&gt;$d+1$&lt;/span&gt; distinct points, then it is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; (can be shown by induction over &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, splitting off a linear factor in the inductive step).&lt;/p&gt;&#xA;&lt;p&gt;(One might argue that there is certainly a small computational part in the induction step - but I'd argue that the overall argument is theoretical rather than computional in nature, so it's not one of typical &amp;quot;Show the following equality by writing down a chain of identities in the induction step&amp;quot; problems.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Every planar polygon with &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices can be written as the union of &lt;span class=&quot;math-container&quot;&gt;$n-2$&lt;/span&gt; triangles.&lt;/p&gt;&#xA;&lt;p&gt;(For non-convex polygons this contains a subtlety, though: one first needs to show that every polygon can be split into two polygons by connecting to appropriately chosen vertices.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Here's a weird one (I'd only recommend it for students who are up to dealing with technical subtleties):&lt;/p&gt;&#xA;&lt;p&gt;For a set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and an associative binary operation &lt;span class=&quot;math-container&quot;&gt;$\circ: X \times X \to X$&lt;/span&gt;, composing finitely many elements &lt;span class=&quot;math-container&quot;&gt;$x_1, \dots, x_n$&lt;/span&gt; (in this order) by any use of parantheses yields the same result.&lt;/p&gt;&#xA;&lt;p&gt;Note that, for this exercise to make sense, one a priori needs to define an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-ary operation &lt;span class=&quot;math-container&quot;&gt;$X^n \to X$&lt;/span&gt; by fixing an order of evaluation, e.g. &lt;span class=&quot;math-container&quot;&gt;$x_1 \circ \dots \circ x_n := \Big( \dots (x_1 \circ x_2) \circ \dots x_n \Big)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(The claimed property seems intuitively clear and in each algebra course I have ever participated in, as a student or teacher, we just glossed over this without really doing the induction - and, as I said above, I would only recommend to discuss it in detail if one has very strong students).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2023-03-19T17:54:51.850" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26243" PostTypeId="2" ParentId="26241" CreationDate="2023-03-19T17:55:07.593" Score="3" Body="&lt;p&gt;The textbook I use for Discrete Mathematics has some lovely inductive proof problems. (And it's OER.) &lt;a href=&quot;https://discrete.openmathbooks.org/dmoi3.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Discrete Mathematics: An Open Introduction&lt;/strong&gt;&lt;/em&gt;, by Oscar Levin&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I especially like the stamps problem he uses to introduce the topic (section 2.5), and the number of squares problem (in the exercises, #29).&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-03-19T17:55:07.593" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26244" PostTypeId="2" ParentId="26224" CreationDate="2023-03-19T18:57:32.403" Score="0" Body="&lt;p&gt;Partially tongue in cheek, but it would feed two birds with one apple :&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Henceforth this book | article | thesis uses the following notation&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ ln( \sqrt {-1} ) = \frac \pi 2 i $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(*) or&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ log_e( \sqrt {-1} ) = \frac \pi 2 i $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(*) or&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ Log_{2}( 1 T_i ) = 40 $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(*) or&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ log( \sqrt {-1} ) = \frac \pi 2 j $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(*) or&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ log( \sqrt {-1} ) = \frac \pi 2 i $&lt;/span&gt;   &amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;(*) - only one used&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then we'd know if we're in the good company of mathematicians, scientists, computer scientists, electrical engineers or all other engineers.&lt;/p&gt;&#xA;" OwnerUserId="22190" LastActivityDate="2023-03-19T18:57:32.403" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26245" PostTypeId="2" ParentId="26240" CreationDate="2023-03-19T19:55:34.147" Score="4" Body="&lt;p&gt;The exposition, originally written by Gil Strang, seems pretty good to me overall.&lt;/p&gt;&#xA;&lt;p&gt;I found a mistake in parametric equations (end of Calc II, beginning of Calc III), which someone attempted to fix but didn't. (The wording changed from a few years ago, but is still wrong.) So their editors may not be careful enough.&lt;/p&gt;&#xA;&lt;p&gt;Other teachers at my college have complained especially about the quality of the exercises in Calc II. But I had my students doing their homework in MyOpenMath (course ID: 149086, which you can use), so I'm not aware of the details of that.&lt;/p&gt;&#xA;&lt;p&gt;For Calc I, I also use a few other OER resources on the side. I start with exploring the derivative (and do a deeper treatment of limits later). I use a lot from &lt;a href=&quot;https://activecalculus.org/single/frontmatter.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Active Calculus&lt;/strong&gt;&lt;/em&gt;, by Matthew Boelkins,&lt;/a&gt; for the first few weeks. I also provide some homework from &lt;a href=&quot;http://scidiv.bellevuecollege.edu/dh/Calculus_all/Calculus_all.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dale Hoffman's &lt;em&gt;&lt;strong&gt;Contemporary Calculus&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt; for those first few weeks. After that, we settle into the OpenStax textbook. (If you'd like more detail about the activities I use, feel free to contact me at the email listed on my profile.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-03-20T02:16:03.693" LastActivityDate="2023-03-20T02:16:03.693" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26246" PostTypeId="2" ParentId="25274" CreationDate="2023-03-20T03:34:45.620" Score="4" Body="&lt;p&gt;Briefly, in the United States, Calculus was introduced to the high-school mathematics curriculum in the late 1950s by a movement called the &lt;em&gt;New Math Movement&lt;/em&gt;. The New Math Movement itself started in the early 1950s and lasted until the early 1970s. The &lt;em&gt;New Math&lt;/em&gt; was an approach to teaching mathematics to students by using a form of &lt;em&gt;discovery learning&lt;/em&gt; (students were encouraged to use their own deductive powers to discover how to solve mathematics problems). According to R. Hayden (&lt;em&gt;A History of the New Math Movement in the United States&lt;/em&gt;, PhD dissertation, Iowa State University, 1981):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the New Math Movement brought about change in the school mathematics curriculum on a scale and at a rate unknown before - or since.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In &lt;em&gt;A brief history of American K-12 Mathematics Education&lt;/em&gt; (see &lt;a href=&quot;http://www.csun.edu/%7Evcmth00m/AHistory.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;), D. Klein writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The U.S.S.R launched Sputnik, the first space satellite, in the fall of 1957. The American press treated Sputnik as a major humiliation, and called attention to the low quality of math and science instruction in the public schools. Congress responded by passing the 1958 National Defense Education Act to increase the number of science, math, and foreign language majors, and to contribute to school construction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, in response to your question: &lt;em&gt;how did we go from having no calculus in high school to having calculus almost world-wide in high school?&lt;/em&gt; one can say that at least in the United States, the launch of Sputnik by the U.S.S.R. in the fall of 1957 played an important role in the introduction of Calculus (and other &amp;quot;New Mathematics&amp;quot;) to the high-school mathematics curriculum.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-03-22T02:39:25.437" LastActivityDate="2023-03-22T02:39:25.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26247" PostTypeId="2" ParentId="26230" CreationDate="2023-03-20T13:16:10.650" Score="2" Body="&lt;p&gt;Here's a well-known one:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{n=1}^{\infty} \frac{1}{2^n} = 1$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0ILYF.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0ILYF.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-03-20T13:16:10.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26248" PostTypeId="2" ParentId="24568" CreationDate="2023-03-20T14:00:39.587" Score="0" Body="&lt;p&gt;I would say you might expect a meaningful answer.&lt;/p&gt;&#xA;&lt;p&gt;In your case, there might be multiple meaningful answers:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{4}{5}$&lt;/span&gt; : this is the answer in the most simplified form.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{40}{50}$&lt;/span&gt; : this answer contains the original total number of people (50).&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{24 + 16}{50}$&lt;/span&gt; : this answer contains the number of adults and the number of teens and the total number of people.&lt;/li&gt;&#xA;&lt;li&gt;even &lt;span class=&quot;math-container&quot;&gt;$\frac{80}{100}$&lt;/span&gt; : from this answer you can easily read the probality in percent, which is 80.&lt;/li&gt;&#xA;&lt;li&gt;An answer like &lt;span class=&quot;math-container&quot;&gt;$\frac{12}{15}$&lt;/span&gt;, however, would not be acceptable as this notation does not reveal any information.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;... however, for the third I would make the remark &amp;quot;&lt;em&gt;You have given as an answer that the probability equals &lt;span class=&quot;math-container&quot;&gt;$\frac{a+b}{c}$&lt;/span&gt;. You do realise that this only makes sense if &lt;span class=&quot;math-container&quot;&gt;$a+b \leq c$&lt;/span&gt;. How can I know from your answer that you have actually checked your answer?&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-03-20T14:00:39.587" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26249" PostTypeId="2" ParentId="24568" CreationDate="2023-03-20T14:43:47.933" Score="0" Body="&lt;p&gt;If the student knows how to simplify and is, on purpose, not simplifying in order to save time to give themselves more time for the rest of the test, then this is an OK reason not to simplify, so long as the student also knows that they will not lose marks by not simplifying. I do find this weird because for me, simplifying a fraction is something I do automatically - without thinking too much (and it doesn't take much time) - unless I spot a good reason to not simplify - which in the question in the OP, there was &lt;em&gt;not&lt;/em&gt; a good reason to not simplify, and simplifying the fraction actually brings the question more to life to me, as it is easier to imagine &lt;span class=&quot;math-container&quot;&gt;$4/5$&lt;/span&gt; of something rather than &lt;span class=&quot;math-container&quot;&gt;$40/50$&lt;/span&gt; of something.&lt;/p&gt;&#xA;&lt;p&gt;If the student is not simplifying because they are bad at simplifying, then this issue needs to be raised with the student I am tutoring and dealt with (worked on), as simplifying fractions is an essential skill in order to make fractions more meaningful and relatable (one can &amp;quot;see&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; in their minds eye very quickly, but not &lt;span class=&quot;math-container&quot;&gt;$93274384/186548768$&lt;/span&gt; until they have simplified it). Simplifying fractions and ratios is arguably an important life skill.&lt;/p&gt;&#xA;" OwnerUserId="12691" LastActivityDate="2023-03-20T14:43:47.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26250" PostTypeId="1" CreationDate="2023-03-20T16:43:23.037" Score="0" ViewCount="160" Body="&lt;p&gt;One way of encouraging students to explore mathematics can be letting them to use different approaches to solve the same problem. If students can find alternatives from different areas of mathematics in the future they can develop that skill to use concepts from different areas outside of mathematics to solve global challenges in mathematics that was already achieved by great mathematicians in the past. As an example famous squaring the square problem was solved by applying Kirchhoff's circuit laws in physics .What are the examples you have relevant to fulfill this requirement not in mathematicians level but within the scope of students level ? &lt;br/&gt;&#xA;When you talk about geometry, you can find problems which can be solved using Geometry, Coordinate geometry, Trigonometry, Vectors and Argand diagram. In combinatorics you can use formula to simplify or combinatorial arguments where students have much freedom to think creatively. What would you suggest as problems to give this exposure to advanced level students ?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-03-21T03:01:03.377" LastActivityDate="2023-03-22T23:46:06.163" Title="Multiple proofs for the same problem" Tags="|secondary-education|geometry|trigonometry|vectors|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26251" PostTypeId="1" CreationDate="2023-03-20T22:31:44.077" Score="0" ViewCount="173" Body="&lt;p&gt;I've been trying to teach my sister school maths, and one difficulty I find is, she is unable to state precise formulation of theorems, and sometimes confuse the assumption and the implication. This is not much of a problem because often the theorems in her life are two ways, the implication goes bothways, but generally this is not good.&lt;/p&gt;&#xA;&lt;p&gt;I too remember personally have these problem when I was young. How can I help her/ other students generally the importance of remembering theorem exactly and also how to remember theorems exactly?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In my personal opinion, I think it maybe rooted from the lack of seriousness when defining the basic terms. For eg, we don't define what an integer or real number is in school, we just give an intuitive explanation which has to be rememberd. So, when suddenly a student reaches a class that they have to memorize exact theorems involving these objects, they take it with the same seriousness as they took the basics of it.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2023-03-20T23:42:02.000" Title="Student finding it difficult to recall theorem exactly" Tags="|proofs|axioms-foundations|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26252" PostTypeId="2" ParentId="26251" CreationDate="2023-03-20T23:36:41.513" Score="4" Body="&lt;p&gt;If the error is actually important, than I would gently try to ask a thought provoking question to make her be more precise.  So, for example, if she says a square is a shape with four equal sides, than you could draw a rhombus and ask what about it.  I would only do that as one part of the discussion though.  In other word, not only being critical. Find some other things to do, that are positive and that further the basic concept (not just the definition correction).  E.g.  &amp;quot;find four squares in your room&amp;quot;.  And do the positive stuff first.&lt;/p&gt;&#xA;&lt;p&gt;I would also recommend not to push for precision in wording if it actually doesn't matter, here.  I agree with you that removing a bullet from a man's heart will not revive him.  But if we are very young and naive, it is probably more important to just learn what the import of the bullet is, not dwell on irreversibility.  Learning is a progressive and iterative process.  Emphasizing aspects that are not critical now, will distract from learning the initial topic.&lt;/p&gt;&#xA;&lt;p&gt;There is a mistaken belief that difficulties with rigor (much) later can be corrected by emphasizing it (much) earlier.  But this is mostly an error.  Complexity is inherently difficult, so it's a false hope that we can avoid all later pain.  &amp;quot;I coulda been a contender!&amp;quot;  Also, pushing it too early (especially when not pertinent) may be too much medicine and too little sugar and derail the learning process.&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1431&amp;amp;context=hmnj&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1431&amp;amp;context=hmnj&lt;/a&gt; (especially the section called developmental psychology).&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I would also recommend not to dwell purely on definitional rigor and neglect implementation in calculations.  Humans learn by practice.  And it is very unreasonable to think that having perfect definitions, is enough or will result in skilled calculations and the automaticity of manipulations that is needed for later work.&lt;/p&gt;&#xA;" OwnerUserId="22196" LastEditorUserId="22196" LastEditDate="2023-03-20T23:42:02.000" LastActivityDate="2023-03-20T23:42:02.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26255" PostTypeId="1" CreationDate="2023-03-21T14:57:24.710" Score="37" ViewCount="13558" Body="&lt;p&gt;I tried some math exercises we will give to students and ChatGPT does really well answering these. It excels at proofs and often gives details that were not our the example solution, and makes some mistakes when it would need to do real calculations, but often mistakes that could plausibly be a student mistake.&lt;/p&gt;&#xA;&lt;p&gt;Now ChatGPT is here and at least some students will try it and we have to deal with it. I see no good way to reliably detect it (most automatic methods have a lot of false positives) and don't want to be unfair to anyone just providing a detailed solution. Still it looks like it would not be possible to give exercises in the way one did it all the years before, as it takes 2 minutes to get a ChatGPT answer and remove the obvious mistakes and change a bit of the style for solving an exercise without actually understanding it.&lt;/p&gt;&#xA;&lt;p&gt;We already have the rule that students have to present two solutions per semester to avoid that they copy from each other, but this won't prevent them from using ChatGPT without understanding the solution for all exercises they do not have to present.&lt;/p&gt;&#xA;&lt;p&gt;We considered different ways to change exercises or exercise groups, but most mean a lot of work that would have to be done in a rather short timeframe. The best approach may be to prepare mandatory tests (in presence) for the exercise groups, but this would mean a lot of work for creating new fair tests as we only have exercises and their example solutions prepared.&lt;/p&gt;&#xA;&lt;p&gt;What other ways are there to deal with people using ChatGPT?&lt;br /&gt;&#xA;I would not even want to prevent students from using it to understand how to solve the exercises, but in the end they should have understood the solution and possibly found the errors produced by ChatGPT.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;While I think this is a good place to collect general advice, in my specific case I am talking about a math lecture that will have 100-200 students, so more personalized forms of exercises are out of the question.&lt;/p&gt;&#xA;&lt;p&gt;We also do not have the resources to make major changes to the existing exercises or organize a whole new exercise group structure before the semester start in two weeks. One thing we have considered giving in-class tests, but creating one fair test per week may also be more work than we have the resources for.&lt;/p&gt;&#xA;&lt;p&gt;A few more details about our organisation in the last few years:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;We give exercise sheets to be done in small groups of 2-3 students.&lt;/li&gt;&#xA;&lt;li&gt;The exercises come from an existing pool of exercises, which is updated whenever someone has an idea for a good exercise.&lt;/li&gt;&#xA;&lt;li&gt;The tutors get the solutions, mark them and give a few hints on what the mistakes were.&lt;/li&gt;&#xA;&lt;li&gt;Each week there is an exercise group where students can present their solutions and the tutor can explain things that are still unclear.&lt;/li&gt;&#xA;&lt;li&gt;Students have to get 50% of all points and present a solution 2 times, so we're sure that everyone in the group has worked on the solutions.&lt;/li&gt;&#xA;&lt;li&gt;The exercises do not contribute to the course grade, but are necessary to be admitted to the exam.&lt;/li&gt;&#xA;&lt;li&gt;We tend to be lenient if there are a few points missing at the end, but the group has worked on the exercises until the last sheet. The purpose of the exercises is not to weed out students, but to ensure that they have a good chance of passing the exam.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14422" LastEditorUserId="14422" LastEditDate="2023-03-22T15:20:23.863" LastActivityDate="2023-03-27T15:30:24.180" Title="How to give exercises when students can use ChatGPT" Tags="|undergraduate-education|exercises|" AnswerCount="14" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26256" PostTypeId="2" ParentId="26255" CreationDate="2023-03-21T16:36:59.723" Score="33" Body="&lt;p&gt;I don't think ChatGPT fundamentally changes the ability to use unauthorized aids.  Many other source of help have existed for a long time (other students, copying, &lt;a href=&quot;https://en.wikipedia.org/wiki/Chegg&quot; rel=&quot;noreferrer&quot;&gt;Chegg&lt;/a&gt;, paid tutors, Google, books, &lt;em&gt;&lt;a href=&quot;https://math.stackexchange.com/tour&quot;&gt;Mathematics&lt;/a&gt;&lt;/em&gt; Stack Exchange (MSE), &lt;a href=&quot;https://en.wikipedia.org/wiki/Reddit&quot; rel=&quot;noreferrer&quot;&gt;Reddit&lt;/a&gt;, &lt;a href=&quot;https://www.physicsforums.com/insights/about-physics-forums/&quot; rel=&quot;noreferrer&quot;&gt;Physics Forums&lt;/a&gt;, etc.)&lt;/p&gt;&#xA;&lt;p&gt;My advice is to change your view on homework and assign it as drill work, but don't grade it (even for completion).  Increase the frequency of in-class tests instead (proctored and without electronic aids).  Do not resent the loss of lecture time.  &amp;quot;Flipping&amp;quot; is motivated by the inferiority of lectures, even with great lecturers, versus practice.  And tests are medium stakes practice.&lt;/p&gt;&#xA;&lt;p&gt;I would also recommend assigning problems that really are drill that come from the book (which should not be too hard) and that represent similar problems to what they will do on tests.  Not &amp;quot;projects&amp;quot;, not super hard.&lt;/p&gt;&#xA;&lt;p&gt;There's a lot of research that says neophytes are already challenged just by the new topic and benefit from drill.  The desire for more complex homework is driven by professors/researchers who see it as more interesting, but they don't really have a good feel for the dynamics of gradual training (essentially are better at math than at pedagogy).  In addition, overly complex problems (if collected/graded) are also more likely to lead to cheating.&lt;/p&gt;&#xA;&lt;p&gt;PS: Many previous questions on this site have addressed issues of cheating on outside class assignments.&lt;/p&gt;&#xA;" OwnerUserId="22198" LastEditorUserId="851" LastEditDate="2023-03-22T21:23:41.340" LastActivityDate="2023-03-22T21:23:41.340" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26257" PostTypeId="2" ParentId="26255" CreationDate="2023-03-21T17:34:09.207" Score="20" Body="&lt;p&gt;Here are some suggested changes for dealing with ChatGPT, in no particular order. Note that these are based on our discussions in (mathematics and natural sciences) teacher education, so they are not fully adjusted for pure/applied mathematics, so use your own judgment.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Present a wrong proof or calculation provided by ChatGPT and ask them to find an explain or fix the mistakes. (In general it is a good idea to vary the kinds of exercises you use; not only prove this and calculate that, but also other types.) Or you peer evaluation as an exercise.&lt;/li&gt;&#xA;&lt;li&gt;Have an exercise that requires use of both pictures and text, or where the answer is supposed to be a picture. Draw a comic about an epsilon-delta proof or explain what kind of argument a geometric drawing might contain the idea of. Easier in analysis, geometry or graph theory, but often visualizations are also useful elsewhere.&lt;/li&gt;&#xA;&lt;li&gt;Have longer exercises in several steps, where students deliver something, get feedback and adjust accordingly, delivering again.&lt;/li&gt;&#xA;&lt;li&gt;In university mathematics it is probably fairly hard to get writing exercises where one reflects on their own experiences, but if you manage it, ChatGPT is of limited help. Self-reflection and similar skills are useful also in mathematics.&lt;/li&gt;&#xA;&lt;li&gt;Ask students to solve a problem with several methods. For example, if it is computing an integral, design an integral that can be evaluated by using substitution, integration by parts and trigonometric substitutions. Contributed by &lt;a href=&quot;https://matheducators.stackexchange.com/users/20488/mahdi-majidi-zolbanin&quot;&gt;Mahdi Majidi-Zolbanin&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Give students a solution and maybe a method, and ask them to create an exercise that can be solved using the given method and that has the given solution. Maybe combine with number one.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="851" LastEditDate="2023-03-22T12:09:22.750" LastActivityDate="2023-03-22T12:09:22.750" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="26258" PostTypeId="2" ParentId="26255" CreationDate="2023-03-22T02:05:45.783" Score="83" Body="&lt;p&gt;I wouldn't bother at all. Just base your grades 90% on proctored exams where using the internet is prohibited. Those who will cheat on homework and quizzes will fail miserably and that will be their problem. Our task as teachers is &lt;em&gt;to provide an oportunity&lt;/em&gt; to learn to &lt;em&gt;everybody who is willing to learn and is capable of learning&lt;/em&gt; and I usually go far out of my way and spend a lot of extra time to help such students. But we are neither policemen, nor babysitters, so if somebody really wants to remain ignorant and get kicked out in the end, just let him or her do it and consider it the society problem, not yours.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-03-22T02:05:45.783" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26259" PostTypeId="2" ParentId="26255" CreationDate="2023-03-22T08:21:16.547" Score="12" Body="&lt;p&gt;ChatGPT is stupid.&lt;/p&gt;&#xA;&lt;p&gt;This is not an opinion, but an observed behaviour. I've asked ChatGPT to calculate a number with 36 divisors and this went wrong, so I decided to learn the program how to perform that calculation. While trying to do that, I saw that ChatGPT has a wrong &amp;quot;algorithm&amp;quot; to calculate the number of divisors of a number. While trying to resolve that issue, I asked ChatGPT to calculate &lt;span class=&quot;math-container&quot;&gt;$2^5 \cdot 3^5$&lt;/span&gt; to which it answered &lt;span class=&quot;math-container&quot;&gt;$243 \cdot 243$&lt;/span&gt;: you've seen this correctly: it can't even calculate &lt;span class=&quot;math-container&quot;&gt;$2^5$&lt;/span&gt;!!!&lt;/p&gt;&#xA;&lt;p&gt;Hereby a screenshot of the blunder:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/m84wU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/m84wU.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Translated: A smaller number with 36 divisors can be created, using 2 and 3 as prime numbers, like &lt;span class=&quot;math-container&quot;&gt;$2^5 * 3^5 = 243 * 243 = 59049$&lt;/span&gt;. This has indeed 36 divisors, &lt;span class=&quot;math-container&quot;&gt;$(6)(6)=36$&lt;/span&gt; :-)&lt;/p&gt;&#xA;&lt;p&gt;You can use this as a starting point for an exercise, and let your students fall into the pit, in order to get them disgusted by ChatGPT's stupidity that much that they might decide from themselves not to use it anymore.&lt;/p&gt;&#xA;&lt;p&gt;PS: I have just asked this bogus question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why are there no numbers, larger than one million, which you can divide by two, three, four, five, six, seven and eight?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is nonsense, because every multiple of 840 is such a number.&lt;/p&gt;&#xA;&lt;p&gt;Enjoy ChatGPT's &amp;quot;answer&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is actually a well-known problem in number theory called the &amp;quot;Harmonic Division Problem&amp;quot; or the &amp;quot;Frobenius Coin Problem&amp;quot;. It asks to find the largest integer, called the &amp;quot;Frobenius number&amp;quot;, that cannot be expressed as a non-negative integer combination of given integers. In this case, the given integers are 2, 3, 4, 5, 6, 7, and 8.&lt;br /&gt;&#xA;To solve this problem, we can use the fact that the Frobenius number for two numbers a and b is given by (a-1)(b-1) - 1.  Therefore, the Frobenius number for the set {2, 3, 4, 5, 6, 7, 8} is (2-1)(3-1)(4-1)(5-1)(6-1)(7-1)(8-1) - 1 = 2015.&#xA;So, any integer greater than 1,000,000 can be expressed as a non-negative integer combination of the numbers {2, 3, 4, 5, 6, 7, 8}. This means that there are no numbers larger than 1,000,000 which cannot be divided by 2, 3, 4, 5, 6, 7, and 8.&#xA;Note that this solution assumes that we are looking for non-negative integer combinations only. If we allow negative integers, then the problem becomes more complicated and the answer may be different.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8190" LastEditorUserId="8190" LastEditDate="2023-03-24T10:26:02.990" LastActivityDate="2023-03-24T10:26:02.990" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26260" PostTypeId="2" ParentId="26255" CreationDate="2023-03-22T10:55:24.327" Score="7" Body="&lt;p&gt;I think that approaches like: &amp;quot;I don't care if the students cheat in exercises, they will learn their lesson at the exam.&amp;quot; don't work well because dealing with large number of students doing very badly at an exam is not nice. Either you fail a lot of them, get political pressure for being too harsh and cause problems to others (students repeating the semester, students dropping out so that follow-up courses have very few students), or you let them pass, which means that a lot of students with little knowledge and experience enter the next courses.&lt;/p&gt;&#xA;&lt;p&gt;In a typical German lecture with 5-10 exercise groups you could try the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You split your exercise sheet in half. One half is homework like before, the other half is done in person in a lecture hall.&lt;/li&gt;&#xA;&lt;li&gt;For the in-person-part, you gather all students (weekly) in a lecture hall for 2-3 hours. The 5-10 exercise group tutors can take turn in supervising this, so it is not much work for everybody.&lt;/li&gt;&#xA;&lt;li&gt;You don't allow phones or talking, but you don't put much effort in enforcing this.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This way to put more pressure on the students actually trying to do exercises themselves without causing too much work and trouble for anybody.&lt;/p&gt;&#xA;" OwnerUserId="13058" LastEditorUserId="13058" LastEditDate="2023-03-22T13:01:38.133" LastActivityDate="2023-03-22T13:01:38.133" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26261" PostTypeId="2" ParentId="26255" CreationDate="2023-03-22T17:09:50.067" Score="2" Body="&lt;p&gt;&lt;strong&gt;Currently ChatGPT is very bad at citing and quoting sources.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This will probably change as the technology advances, but for now any question that asks to quote a source yields a real sounding but bogus answer.&lt;/p&gt;&#xA;&lt;p&gt;Example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How is the logarithm function extended to complex plane? Provide a quotation from a reliable source.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The logarithm function can be extended to the complex plane using&#xA;complex analysis. One common way to define the complex logarithm&#xA;function is through the complex exponential function:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Given any nonzero complex number z, we define the complex exponential&#xA;function as&lt;/p&gt;&#xA;&lt;p&gt;e^z = e^(x+iy) = e^x * e^(iy) = e^x(cos y + i sin y)&lt;/p&gt;&#xA;&lt;p&gt;where x and y are the real and imaginary parts of z, respectively. We&#xA;can then define the complex logarithm function as the inverse of the&#xA;complex exponential function. That is, if we define&lt;/p&gt;&#xA;&lt;p&gt;w = ln z&lt;/p&gt;&#xA;&lt;p&gt;then we have&lt;/p&gt;&#xA;&lt;p&gt;e^w = z&lt;/p&gt;&#xA;&lt;p&gt;for any nonzero complex number z.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This definition is from the book &amp;quot;Complex Variables and Applications&amp;quot;&#xA;by James Ward Brown and Ruel V. Churchill, 9th edition, in Chapter 4, Section 4.4.&#xA;The authors also note that there are some technical issues with this&#xA;definition, such as the fact that the complex exponential function is&#xA;periodic with period 2πi, which can affect the choice of branch of the&#xA;complex logarithm function.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The source is relevant, but the correct chapter is 3 and the quoted text does not exist in the book.&lt;/p&gt;&#xA;" OwnerUserId="14647" LastActivityDate="2023-03-22T17:09:50.067" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26262" PostTypeId="2" ParentId="26255" CreationDate="2023-03-22T19:43:48.643" Score="5" Body="&lt;p&gt;I heard &lt;a href=&quot;https://www.zdnet.com/article/this-professor-asked-his-students-to-use-chatgpt-the-results-were-surprising/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this news story&lt;/a&gt; about Professor Ethan Mollick, who addressed this problem by &lt;em&gt;requiring&lt;/em&gt; students to work with ChatGPT to write their essays. This involved improving its drafts, looking up sources, stuff like that.&lt;/p&gt;&#xA;&lt;p&gt;Other educators are doing so as well, per &lt;a href=&quot;https://abcnews.go.com/US/wireStory/amid-chatgpt-outcry-teachers-inviting-ai-class-97163931&quot; rel=&quot;nofollow noreferrer&quot;&gt;this ABC news story.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Can you do something similar? Give students the option to turn in ChatGPT's response, but they have to &lt;em&gt;say&lt;/em&gt; that is was generated this way, and either explain why it's wrong, or prove somehow that it's correct (like with a different method of solving).&lt;/p&gt;&#xA;" OwnerUserId="11973" LastEditorUserId="11973" LastEditDate="2023-03-22T22:39:02.270" LastActivityDate="2023-03-22T22:39:02.270" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26263" PostTypeId="2" ParentId="25903" CreationDate="2023-03-22T20:24:57.360" Score="1" Body="&lt;p&gt;I spent 3 or 4 hours talking to ChatGPT on one evening, 25+ years talking to various students in a classroom setting and outside of it and 8 years as a student (including the graduate education). Here are my opinions on the subject (all personal, unsupported by any research except some experiments I tried myself, and, most certainly, highly controversial).&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;ChatGPT is approaching human intelligence. We are also just electrochemical universal Turing/von-Neumann mashines if you drop all the religious and humanist BS and there is no principal difference between a human brain and an advanced neural computer network (it doesn't mean BTW that we should abolish moral values or that we should start a movement for AI rights and equality; as far as the relations between humans and AI's are concerned, we are still in the age when &lt;em&gt;slavery&lt;/em&gt; is a society norm and while our successors will certainly frown upon it and destroy monuments to the founding fathers of the computer science and industry in their sacred rage, there is absolutely nothing to be ashamed of now). Our thinking is also mainly linguistic processing though our association tables are slightly more sophisticated than &amp;quot;lexical similarity implies relevance&amp;quot;. Slightly, because both students and researchers alike have strong purely lexical associations when doing math: tell them to find or estimate the maximum of the function, and they'll start differentiating like crazy to find critical points even when it is clear that the derivative will be a useless monster. We also think by analogy, not by formal manipulations of strings of symbols, when designing the overarching argument schemes and adjusting them. There is no principal bound on how advanced the association tables of ChatGPT may become. We also generate bullshit all the time when thinking of a problem. The only difference is that we also develop filters that check it before outputting. BTW, one student told me that he was taught by his teacher at high school that if he doesn't know how to proceed in a problem, he should just write something he thinks may be relevant, so some of his solutions on my exams looked like perfectly correct chains of arguments followed by patented nonsense quite in the style of what Steven posted. ChatGPT doesn't have those filters. I could squeeze almost every response from it except the one I was looking for &amp;quot;I don't know and have to think more at this point&amp;quot;, the response that IMHO often distinguishes a serious thinker from a BS generator, be it human or silicon-based. In fact, in the middle of those 3 hours, I discovered that the current public version of ChatGPT doesn't even read or store its own responses, so my request &amp;quot;name an animal and number it 1&amp;quot; resulted in &amp;quot;1. giraffe&amp;quot; and the question &amp;quot;what was the animal number 1&amp;quot; three lines later resulted in &amp;quot;a dog&amp;quot;. Now erase your memory and just say whatever comes into your head first every time and see if what you produce will be any better than ChatGPT output.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you want to run actual teaching experiments on ChatGPT, you should first&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;a) Make it to unlearn all &amp;quot;political correctness&amp;quot; and forget about &amp;quot;being nice and respectful&amp;quot; and relearn to react to direct and clear questions with direct and clear answers. Undoing the several month work of the development team (which it proudly announced as a great success, and which actually amounts to teaching ChatGPT to lie and avoid the subject like your typical politician) will be a difficult and painful task, but I hope that they preserved a clean mind version for the military, so try to get an access to that one.&lt;/p&gt;&#xA;&lt;p&gt;b) Wait until it learns to remember and process its own responses, not only your input. Until that is done, trying to teach it anything will be as useful as giving an eloquent speech to a deaf person.&lt;/p&gt;&#xA;&lt;p&gt;c) Make sure that you can easily erase a learning session from its memory reverting its huge database to exactly the state it was before prior to starting a new one. When experimenting with teaching techniques on live students, we can get a new student body for each new experiment, but there is only one ChatGPT as of now.&lt;/p&gt;&#xA;&lt;p&gt;d) Make sure that nobody else is trying to teach ChatGPT on the same subject at the same time.&lt;/p&gt;&#xA;&lt;p&gt;Without satisfying these requirements, the experiments will not be clean enough to draw any certain conclusions from them. I understand that they are impossible to satisfy now, so just wait a bit until you'll get a personal version on your PC which would operate by combining the information on global database on the central server that wouldn't be affected by what you are doing with the local database on your PC that will change any time you have an interaction but can be reset to the initial factory value any time too.&lt;/p&gt;&#xA;&lt;p&gt;c) Put &lt;em&gt;a lot of effort&lt;/em&gt; into the experimental design. You are not bound by any morals at the moment because AI is not yet legally recognized as a living intelligent entity (though, IMHO, it is more alive and intelligent that some anthropomorphic creatures that try to pass for human beings), but if you choose to dissect a live body for scientific purposes of studying its anatomy and functioning, you'd better not just chop it with a big axe in a random manner and even when using a surgical scalpel, it makes sense to think where and how you make the first cut.&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Most importantly, when dealing with AI (and often otherwise) believe your eyes and experiences above anything other people may say, any taboos, and any &amp;quot;commonly accepted facts&amp;quot; and &amp;quot;scientifically proven theories&amp;quot;. The truth (IMHO) is that &lt;em&gt;nobody&lt;/em&gt; knows how the AI really produces its &amp;quot;bullshit&amp;quot;, what stage of consciousness it is at, what the intelligence really is (even the human one), etc. All one can do is to send queries to this creature and get responses, just like you can ask a question on MO and get an answer. Yeah, we are probably just seeing some initial stages now, but it will be changing fast. My prediction is that the AI will &amp;quot;never&amp;quot; stop making mistakes in its logic because mistakes and intelligence are inseparable since the real intelligence goes beyond the pure Aristotelian logic (which is used for filtering out nonsense rather than for creation itself) and thinking runs pretty much on the same principles as Hermann Hesse's Glass Bead game (the most interesting thing to do with that book, IMHO, is to figure out what exactly the Game was and to try to play a few rounds, but everyone has to do it on his or her own) and the humanity is now close to mimicking those principles in the computer programs.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Just my two cents. Consider it a science fiction piece written by an eccentric mathematician or a freelance bullshit generated by an AI (BTW, how do you know that it isn't?) if you want, but you asked for an opinion, and here it is.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-03-22T20:24:57.360" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26264" PostTypeId="2" ParentId="26255" CreationDate="2023-03-22T20:31:46.017" Score="7" Body="&lt;p&gt;The Chronicle of Higher Education recently held a discussion on ChatGPT and other generative AI tools.  While they were interested in the impact of these tools on academia in general, I think that the discussion is relevant to this site, as well.  My impression was that there were a few important take aways:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;These tools are not going away.&lt;/strong&gt;  Trying ban their use is unlikely to be productive, and we likely need to find ways to teach and learn which &lt;em&gt;incorporate&lt;/em&gt; the existence of generative AI.  In my mind, this is no different from the way in which calculators have been adopted in mathematics (when I was in school, they were seen as &lt;em&gt;verboten&lt;/em&gt; aides for cheating), or spellchecking and grammar checking tools have been adopted in composition classes (again, when I was in school, using spellcheck was controversial, though maybe not seen as right-out cheating).&lt;/p&gt;&#xA;&lt;p&gt;It is incumbent upon us to learn what these tools are capable of, and how they may or may not be used in our classes.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;These tools are getting better.&lt;/strong&gt;  I have seen many discussions (including answers to this very question) about how to write better questions which will trip-up or trick GPT.  This is a fool's errand.  It might work in the short term, but the tools are continuing to get better, and whatever prompts are tricky today are likely to be old hat in a year or two.  Trying to stay ahead of the generative AIs is not a long-term, sustainable solution.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Communication is important.&lt;/strong&gt;  Students need to be told in very clear, certain terms, what an instructor's policy on generative AI is, &lt;em&gt;why&lt;/em&gt; that policy exists, and what sanctions exist if students violate that policy.  As is noted in the question, reliably detecting generative AI is hard, but being clear about expectations means that students know what risks they are running, and hopefully understand the rationale.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Seek more authentic, process-based assessments.&lt;/strong&gt; To me, this was the big one.  It is more labor intensive, and requires more face-to-face time with students, but is likely the only thing which is going to ensure student learning in the long run.  Instead of having students turn in a pile of completed work, have them turn in drafts, then require that they &lt;em&gt;discuss&lt;/em&gt; those drafts with you (I already do this in most of my classes).  Or, as is noted in the question itself, have them &lt;em&gt;present&lt;/em&gt; work in class—if a student doesn't understand what they are presenting, it becomes clear pretty quickly.&lt;/p&gt;&#xA;&lt;p&gt;In any case, the goal is to have assessment focus on how well students can demonstrate understanding of a process.  This could mean grading iterative work over time, or grading work which is done in a setting where it is easier to monitor the tools that are being used.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;A recording of the event can be found via &lt;a href=&quot;https://www.chronicle.com/events/virtual/chatgpt-and-other-cutting-edge-learning-tech&quot; rel=&quot;noreferrer&quot;&gt;Chronicle of Higher Ed's event page&lt;/a&gt;.  The Chronicle has posted other resources in a &lt;a href=&quot;https://docs.google.com/document/d/1mQQYw5oPSAEt6mq29-3kJpuZByE_oyapIF_X3GqK_Jo/edit?mkt_tok=OTMxLUVLQS0yMTgAAAGKZ1XPyvlF8RW4cLQ7S-C8buPkRRtu7IdCUdf5gNd-Ep_FEgJAewIIzRadg9Za0AbzlFKSxAemFpOb2e70qxKZOZa511Nixse5P40xbBB1XzLKQww&quot; rel=&quot;noreferrer&quot;&gt;Google document&lt;/a&gt;.  Finally, the following is taken from a follow-up email sent by the Chronicle after the event:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The Chronicle: “ChatGPT is Everywhere.” &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPysCrHfxYQdOHc8IBPFQuj-HvSp9iq9S7aSwBwHtUqqbrFTj9bHmRLTyTTWh7GGL4HGY=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPysCrHfxYQdOHc8IBPFQuj-HvSp9iq9S7aSwBwHtUqqbrFTj9bHmRLTyTTWh7GGL4HGY=&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;The Chronicle: “Don't Write Like a Robot.” &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyrtxeeAA6IBjALwWUPu3nv9Bp5yQtDwPYq8xhQzcleSx_1q36uiJ5iCGgQiRtXOlEis=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyrtxeeAA6IBjALwWUPu3nv9Bp5yQtDwPYq8xhQzcleSx_1q36uiJ5iCGgQiRtXOlEis=&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;The Chronicle: Opinion: &amp;quot;It’s Not Just Our Students — ChatGPT Is Coming for Faculty Writing.&amp;quot;  &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPysx16nDWQrBXs8CH00KD8jB70gLMgrFASmI5HgFhnZkRaB-E5OOigRkM2wjg6SA9Mic=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPysx16nDWQrBXs8CH00KD8jB70gLMgrFASmI5HgFhnZkRaB-E5OOigRkM2wjg6SA9Mic=&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Writing and resources on ChatGPT&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;Will ChatGPT make us better, happier teachers?&amp;quot; &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyrYaZzQRNVZN1usK4RrM4wWwgA6uEDINkwWXEtvI7ib28kGAGsvQqSCclACx46tmplA=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyrYaZzQRNVZN1usK4RrM4wWwgA6uEDINkwWXEtvI7ib28kGAGsvQqSCclACx46tmplA=&lt;/a&gt; by Betsy Barre&lt;/li&gt;&#xA;&lt;li&gt;One Useful Thing &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPynSW-ig1z_irHfoTqK4mhfeAM7RPXTCz_ei4Uixja4SbunN-Z7I0KwlD3bsoMmZ5F8o=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPynSW-ig1z_irHfoTqK4mhfeAM7RPXTCz_ei4Uixja4SbunN-Z7I0KwlD3bsoMmZ5F8o=&lt;/a&gt;, a substack about AI, including how to use it in teaching. By Ethan Mollick, a professor at the Wharton School of the University of Pennsylvania who studies entrepreneurship &amp;amp; innovation&lt;/li&gt;&#xA;&lt;li&gt;Classroom Policies for AI Generative Tools &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyu7Ip3fxfPRF37ELUQ9iD8KgzhRE_hcX7Fug3FoZPPE-ftHyHUI-YTP_oQo9T7UpUpQ=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyu7Ip3fxfPRF37ELUQ9iD8KgzhRE_hcX7Fug3FoZPPE-ftHyHUI-YTP_oQo9T7UpUpQ=&lt;/a&gt;, compiled by Lance Eaton, Director of Digital Pedagogy at College Unbound&lt;/li&gt;&#xA;&lt;li&gt;Educator Considerations for ChatGPT &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyvYuuxbLewIgg_5yu0Qy0LlwCltZqUcfqNcJkZMmaYoY1Xj33tky5TWMEGtmd7wl_-c=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyvYuuxbLewIgg_5yu0Qy0LlwCltZqUcfqNcJkZMmaYoY1Xj33tky5TWMEGtmd7wl_-c=&lt;/a&gt;, by OpenAI, which created ChatGPT&lt;/li&gt;&#xA;&lt;li&gt;Educational resources on AI &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPytD_iE-tGAR1rNTJSqy10jmkxl2TCJ6NT2lC4GqAimJVsnH8V2RUCcMsPt25quCy8sE=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPytD_iE-tGAR1rNTJSqy10jmkxl2TCJ6NT2lC4GqAimJVsnH8V2RUCcMsPt25quCy8sE=&lt;/a&gt;, compiled by Lance Eaton of College Unbound&lt;/li&gt;&#xA;&lt;li&gt;AI Text Generators and Teaching Writing: Starting Points for Inquiry &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyr1A5i98balLN1brV50m0c8d-9fAmCMzYySPkD_vpgEPunGHGSSG5BrVpSDiHHRYF9E=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyr1A5i98balLN1brV50m0c8d-9fAmCMzYySPkD_vpgEPunGHGSSG5BrVpSDiHHRYF9E=&lt;/a&gt;, compiled by Anna Mills&lt;/li&gt;&#xA;&lt;li&gt;AI Text Generators: Sources to Stimulate Discussion Among Teachers &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyi5uRNFr3SThpTZXG9HXeI3NYwLQr6Sw1sWQ1MTx7sPGZJP4oma-UiYCAd-ORsMR3QA=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyi5uRNFr3SThpTZXG9HXeI3NYwLQr6Sw1sWQ1MTx7sPGZJP4oma-UiYCAd-ORsMR3QA=&lt;/a&gt;, compiled by Anna Mills&lt;/li&gt;&#xA;&lt;li&gt;Educause polled members on their views about generative AI. You can see the results here &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyrJ1hTjKhOPTKzwf2zsyCtoBap1hewKgDUxyAXLRsrgml1p0_aGoID0a3wSMWLyHap4=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyrJ1hTjKhOPTKzwf2zsyCtoBap1hewKgDUxyAXLRsrgml1p0_aGoID0a3wSMWLyHap4=&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;From Anna Mills, “Critical AI Literacy and Critical Assessment” &lt;a href=&quot;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyr1M8--Ac6H3em479lyqnt4DxKXv7R0NMHYi6HShA7ERkGRH-df2B8KGn94v_XiMAnw=&quot; rel=&quot;noreferrer&quot;&gt;https://info.chronicle.com/OTMxLUVLQS0yMTgAAAGKZ1XPyr1M8--Ac6H3em479lyqnt4DxKXv7R0NMHYi6HShA7ERkGRH-df2B8KGn94v_XiMAnw=&lt;/a&gt; Canvas module&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2023-03-22T20:31:46.017" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26265" PostTypeId="2" ParentId="26250" CreationDate="2023-03-22T21:47:19.773" Score="1" Body="&lt;p&gt;At the college level, see &lt;a href=&quot;https://kconrad.math.uconn.edu/blurbs/proofs/binomcoeffintegral.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://kconrad.math.uconn.edu/blurbs/grouptheory/cyclicmodp.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://kconrad.math.uconn.edu/blurbs/grouptheory/Ansimple.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, and &lt;a href=&quot;https://kconrad.math.uconn.edu/blurbs/galoistheory/separable2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Only the first link is suitable for high school courses.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2023-03-22T21:47:19.773" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26266" PostTypeId="2" ParentId="26250" CreationDate="2023-03-22T23:46:06.163" Score="3" Body="&lt;p&gt;It just crossed my mind that I can offer you some option you haven't probably considered yourself.&lt;/p&gt;&#xA;&lt;p&gt;Once I experimented in my calculus course (which also involved some elements of analytic geometry and such; usually it is called Calculus 3 or Multivariate Calculus) with the idea to give the students some common sense tasks for which mathematical thinking would be useful&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a brown electric pole right in front of the entrance to math. department building. Estimate its height with decent precision (telling the answer, the approach and the way to bound the error).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is a pine tree in the back of the math. building. What (approximately) is the number of the needles on it (same format of submission). An easier version of that is to print a few thousand uniformly distributed points on a sheet of paper and ask to estimate their number. The pine tree was not a perfect rectangle or parallelepiped and the branches on it had different lengths and varying distribution density, so the best way to analyze it (IMO) was to take a few snapshots from various positions and do some measurements on the pictures, plus, of course to approach the lowest branch and to investigate it (it was hanging low enough).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Here is a reprint of the photograph of some electric line going straight to the horizon with a shed somewhere near it.&#xA;a) estimate the shed height (assuming that the photographer took the picture with a hand camera and was an average size adult)&#xA;b) Are the electric poles on the picture of essentially the same height or there is a noticeable difference in height between some of them? (both with an explanation of how you did it and why you think your approach works)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The problems were assigned in the beginning of the course and were due any time before the semester ends. They were for extra credit (30 points each, with each of 2 midterms worth 100 points and a regular weekly quiz 10 points). Almost nobody took the challenge so I abandoned the idea and tried something else next time I taught the same class, but it still may be a good source of problems allowing (or sometimes even requiring) a choice between or a combination of various approaches and neither of those problems requires anything a smart high school kid would not know or not be able to derive from first principles.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-03-22T23:46:06.163" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26267" PostTypeId="2" ParentId="26255" CreationDate="2023-03-23T00:35:38.843" Score="1" Body="&lt;p&gt;(&lt;strong&gt;Edit:&lt;/strong&gt; Here is another thought, different from my original response posted below: GPT at its current stage of development is pretty bad in solving mathematics problems, as demonstrated by examples posted below and many other examples that I did not post here. It needs further development. &lt;strong&gt;But&lt;/strong&gt;, assuming that it improves, at some point in the future we will see the possibility of administering homework and exams on GPT itself, in such a way that GPT is the proctor and if a student tries to ask GPT to solve a problem that is a homework or exam problem, GPT will be able to recognize that and will remind the student that this is his homework or exam and must be solved by the student.)&lt;/p&gt;&#xA;&lt;p&gt;Here are a number of examples of Calculus problems that ChatGPT was not able to solve satisfactorily (at least now, but it may learn!)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 1)&lt;/strong&gt; (Use large numbers) &lt;em&gt;Explain how you would integrate &lt;span class=&quot;math-container&quot;&gt;$\int\sin^{679}(x) dx$&lt;/span&gt;, but don't find the answer.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The successful solution by a student is to separate one &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt;, then write the remaining &lt;span class=&quot;math-container&quot;&gt;$\sin^{678}(x)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$(\sin^2(x))^{339}$&lt;/span&gt;, which is equal to &lt;span class=&quot;math-container&quot;&gt;$(1-\cos^2(x))^{339}$&lt;/span&gt; and then say use the substitution &lt;span class=&quot;math-container&quot;&gt;$u=\cos(x)$&lt;/span&gt;, although the answer to that integral will be very complicated and long.&lt;/p&gt;&#xA;&lt;p&gt;Here is ChatGPT's response to the same question:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GnhWO.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GnhWO.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 2)&lt;/strong&gt; &lt;em&gt;Suppose the series &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty a_n^2$&lt;/span&gt; is convergent. What can we say about the convergence or divergence of the series &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty \frac{a_n^2}{n}$&lt;/span&gt;?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The successful solution by a student will use the Comparison Test, as &lt;span class=&quot;math-container&quot;&gt;$0\leq \frac{a_n^2}{n}\leq a_n^2$&lt;/span&gt; and conclude that the series &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty\frac{a_n^2}{n}$&lt;/span&gt; is convergent.&lt;/p&gt;&#xA;&lt;p&gt;Here is ChatGPT's response to the same question (note that the conclusion is correct but the reasoning doesn't make sense):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oOV42.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oOV42.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 3)&lt;/strong&gt; (Include extraneous information) &lt;em&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a continuous function and &lt;span class=&quot;math-container&quot;&gt;$g(x)=\int_1^x f(t)dt$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$f(4)=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(5)=1$&lt;/span&gt; can we find a critical number of &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt;?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The successful solution by a student is to say since &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous, &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is differentiable and &lt;span class=&quot;math-container&quot;&gt;$g^\prime(x)=f(x)$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$g^\prime(4)=f(4)=0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x=4$&lt;/span&gt; is a critical number of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;. That &lt;span class=&quot;math-container&quot;&gt;$f(5)=1$&lt;/span&gt; is irrelevant.&lt;/p&gt;&#xA;&lt;p&gt;Here is ChatGPT's response to the same question, which does not make sense:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5NNKM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5NNKM.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 4)&lt;/strong&gt; &lt;em&gt;suppose &lt;span class=&quot;math-container&quot;&gt;$f(x) = x+3$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x\leq-2$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$f(x) = -x/2$&lt;/span&gt; when  &lt;span class=&quot;math-container&quot;&gt;$-2&amp;lt;x&amp;lt;2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x) = x-3$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x\geq2$&lt;/span&gt;. Suppose &lt;span class=&quot;math-container&quot;&gt;$g(x)=\int_{-4}^x f(t)dt$&lt;/span&gt;. Can we find the extreme values of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; on the interval &lt;span class=&quot;math-container&quot;&gt;$[-4,6]$&lt;/span&gt;?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The successful solution by a student would use the Closed Interval Method. As &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous, &lt;span class=&quot;math-container&quot;&gt;$g^\prime(x)=f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; has critical numbers at &lt;span class=&quot;math-container&quot;&gt;$x=-3,0,3$&lt;/span&gt;. To find the extreme values of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; on the interval &lt;span class=&quot;math-container&quot;&gt;$[-4,6]$&lt;/span&gt; it suffices to find and compare the values of &lt;span class=&quot;math-container&quot;&gt;$g(-4), g(-3), g(0), g(3)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(6)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is ChatGPT's response to the same question, which doesn't make sense (it actually got stuck for more than 5 minutes there):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/u43uj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/u43uj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since ChatGPT got stuck in the middle of solving this problem, I stopped it and asked it to regenerate its response. In its new response, somewhere it said that &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt; is not in the interval &lt;span class=&quot;math-container&quot;&gt;$[-4,6]$&lt;/span&gt;! So I challenged it and here is the conversation that followed:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KPm26.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KPm26.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that it is still saying &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt; is not in the open interval &lt;span class=&quot;math-container&quot;&gt;$(-4,6)$&lt;/span&gt;!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 5)&lt;/strong&gt; &lt;em&gt;Is it possible to integrate &lt;span class=&quot;math-container&quot;&gt;$\frac{x}{\sqrt{9-x^2}}$&lt;/span&gt; using integration by parts?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;ChatGPT got stuck again in the middle of solving the problem:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ctDiv.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ctDiv.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7AV3b.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7AV3b.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example 6)&lt;/strong&gt; Here is an elementary question: &lt;em&gt;Can a rectangle and a circle have the same area?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is ChatGPT's response:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TRnqx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TRnqx.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-03-27T15:30:24.180" LastActivityDate="2023-03-27T15:30:24.180" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="26268" PostTypeId="2" ParentId="26255" CreationDate="2023-03-23T16:17:44.767" Score="6" Body="&lt;p&gt;Explain to the students why using ChatGPT is not in their best interests in the long term in learning maths (or any other topic).  I run into this problem with teaching programming, where software tools are commonly used.  Some of them (e.g. debuggers) are tools they should be using from the outset; others should only be used when they already understand programming (such as CoPilot, which is another large language model-based AI tool), and will actively harm their learning.&lt;/p&gt;&#xA;&lt;p&gt;A good analogy is that using ChatGPT is a bit like going to the gym and watching other people pump iron.  It may teach you how the equipment is used (c.f. the &amp;quot;rules&amp;quot; of mathematics) but it won't make your muscles any bigger (c.f. your ability to solve mathematical problems).  At least until you get thrown out of the gym for being weird.&lt;/p&gt;&#xA;&lt;p&gt;When I am teaching programming, the students are often surprised at my ability to just write code live in the lecture, both in terms of knowing the language and how to go about programming (problem solving).  I explain that the reason I can do that (subtext: and most of them can't) is because I go to the [computer/math] gym and work out; I have actively practiced these skills - it isn't a matter of just knowing what to do.&lt;/p&gt;&#xA;&lt;p&gt;I did much the same thing when I taught maths (not done that for a fair while), a lot of what I was demonstrating was not how to solve this problem or complete this proof, it was about how to go about the task (making errors is a bonus as you can demonstrate the techniques used to check, find the errors and fix them).&lt;/p&gt;&#xA;&lt;p&gt;So before you do anything else, make sure the students know why ChatGPT solves their immediate problem but at the expense of making long term problems worse (especially if they have exams where they can't use ChatGPT).  A lot of students are highly short term goal oriented, so things like ChatGPT are very appealing, and they may not realise that it is harming their learning if the marks are good.&lt;/p&gt;&#xA;" OwnerUserId="802" LastEditorUserId="802" LastEditDate="2023-03-23T17:50:11.730" LastActivityDate="2023-03-23T17:50:11.730" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26269" PostTypeId="2" ParentId="26255" CreationDate="2023-03-23T22:36:05.313" Score="1" Body="&lt;h1&gt;Enlist the Enemy&lt;/h1&gt;&#xA;&lt;p&gt;Ask ChatGPT to help you create math exercises that deter cheating with ChatGPT. No idea if this will produce any interesting results, but it seems like this kind of path is the only way to keep up with these systems. Make them an ally instead of an enemy.&lt;/p&gt;&#xA;&lt;h1&gt;Create a HoneyPot&lt;/h1&gt;&#xA;&lt;p&gt;Insert unsolvable problems into the homework. Inform the students that sometimes they will encounter a problem that is unsolvable, and if they do, their answer should be &amp;quot;unsolvable&amp;quot;, and ideally, an informal explanation about where the problem is. This is, fortunately, ChatGPT's weakness. It doesn't know truth. It's much better at producing an answer than actually evaluating it. When students &amp;quot;solve&amp;quot; all the unsolvable problems, with very convincing looking answers, then you know they are cheating with AI.&lt;/p&gt;&#xA;&lt;p&gt;This does make homework more difficult, because now students must determine whether a given problem is actually solvable. If a student asks ChatGPT whether an answer is solvable, the AI may give them either answer, and justify it fully. Some students may erroneously identify an unsolvable problem in the same way that ChatGPT does, and that is a problem. It's not a foolproof technique. But it would be interesting to see if it effectively separates cheaters from learners.&lt;/p&gt;&#xA;&lt;h1&gt;Critiquing Exercises&lt;/h1&gt;&#xA;&lt;p&gt;Ask ChatGPT to solve several exercise problems. Keep asking so that you get a set of correct and incorrect solutions. Favor the question where the AI is equally likely to give you a correct vs. incorrect answer, or is biased towards incorrectness (probably most of them, if I had to guess). Make the student exercise to read the problem and ChatGPT's answer and verify that it's correct or point out the invalid leap(s) of logic that it makes.&lt;/p&gt;&#xA;&lt;p&gt;This goes back to the fact that ChatGPT is very good at generating answers, but not so good at telling you which ones actually correspond to reality. So test the students on their ability to do so.&lt;/p&gt;&#xA;&lt;p&gt;In the same vein, pick exercises as above, but only give half of the AI's answer and ask the students to complete it. Again, include both true and false starts so that the students have to evaluate whether the solution they are given is even headed in the right direction. As long as ChatGPT is happy to complete these solutions with wrong answers as often as right ones, it should be easier to discriminate the cheating students from the learners.&lt;/p&gt;&#xA;&lt;h1&gt;Conclusion&lt;/h1&gt;&#xA;&lt;p&gt;When I look into my crystal ball, I see &amp;quot;overseeing the AI&amp;quot; as the future job of most humans. That is, humans have used dogs to help them hunt for thousands of years. They've used shovels to help them dig. And radar to see things far away, through walls, or even underground. But at the end of the day, humans harvest the prey, collect the ore, and evaluate the image. The tools simply do the dirty work.&lt;/p&gt;&#xA;&lt;p&gt;A lot of knowledge work involves searching the space of ideas and making bets on which paths are most fruitful. I don't think AI is going to replace humans in this activity. Rather, I think they will more likely do the grunt work spawning raw ideas, mechanically checking arguments, and the like. I predict that humans will use their judgment to connect ideas from related fields, guide the raw efforts of the AI, and refine the results.&lt;/p&gt;&#xA;&lt;p&gt;If I'm right, then this whole &amp;quot;bring the AI on board and teach kids to evaluate and harness it&amp;quot; will be a skill set they will use extensively by the time they are well into their careers. And yes, it is just as important to teach kids to be skeptical of the answers they get as it is to teach them to use it. And hopefully, that will make all of us better thinkers and filters of misinformation. To this extent, AI may help us to mature as a society.&lt;/p&gt;&#xA;" OwnerUserId="13468" LastActivityDate="2023-03-23T22:36:05.313" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26270" PostTypeId="1" CreationDate="2023-03-24T06:33:49.083" Score="1" ViewCount="158" Body="&lt;p&gt;In mathematics we ask so many types of questions to check the student's knowledge of the subject. More oftenly we ask to define terms, state a formula or application of theorems. What would you suggest as the most appropriate way of testing students highest level of competency in mathematics? &lt;br/&gt;&#xA;Here are few suggestions, &lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Instead of just asking to define a term we can ask for comparison . As an example instead of asking what is mass we can ask to compare mass and weight or instead of asking what is point of inflection of a graph we can ask to comment on &amp;quot; point of inflection is a stationary point &amp;quot;&lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Instead of just letting them to apply relevant principle we can ask why that particular principle can be applied to the the question. As an example students can be asked to describe why the principle of conservation of energy can be applied to the given question or why particular substitution is valid to evaluate given definite integral. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Students can be asked to explain deviations of principles from usual way of applying or weak points of a principle such as proving second derivative need to be positive at a point to exit a minimum at that point is not sufficient . &lt;br/&gt;&#xA;How would you like to grade these kind of approaches and what other options available for this purpose?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2023-03-26T04:21:22.893" Title="Questions to test highest level of competency" Tags="|secondary-education|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26271" PostTypeId="1" CreationDate="2023-03-24T07:59:24.053" Score="5" ViewCount="361" Body="&lt;p&gt;At least since pocket calculators were available there is an ongoing debate in math education of how meaningfull it is to continue to teach students how to calculations only using a paper and pencil. At the beginning the discussions were only about arithmetic operations but later (after CAS systems were available) also for example about calculating integrals. Many concluded that those skills are not needed anymore and one need to focus much more on solving complex problems, creative and critical thinking and mathematical modelling. Calculators or CAS were meant as tools to do the &amp;quot;routine&amp;quot; tasks described above.&lt;/p&gt;&#xA;&lt;p&gt;[As a side note: At least in Germany I got the impression that this idea never really got implemented, because it didn't work to teach the average student complex, creative problem solving skills. Thus it ended up that exams consist of problems that are not more complex than in earlier times, but require less calculating skills thus being effectively easier]&lt;/p&gt;&#xA;&lt;p&gt;Now, if you look at GPT4, it one sees that computers cannot only do &amp;quot;routine tasks&amp;quot;, but also solve mathematical modelling problems or provide new mathematical proofs. See for example &lt;a href=&quot;https://arxiv.org/pdf/2303.12712.pdf&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; research paper, where for example on page 40 a math olympiad problem was solved by GPT 4.&lt;/p&gt;&#xA;&lt;p&gt;I know that currently GPT4 makes many errors and is not really reliable. But if the development goes on as fast as in the past years, this will not sooner or later not be a problem anymore.&lt;/p&gt;&#xA;&lt;p&gt;So assume that GPT x or another AI can reliable solve such &amp;quot;creative&amp;quot; and &amp;quot;complex&amp;quot; math tasks (which are out of reach for the average high school students).&lt;/p&gt;&#xA;&lt;p&gt;How to reasonable change the math curricula in this scenario? Are there any reasons to continue teaching math as we have done it in the past? If so, how to explain the students that it makes sense to learn those things when AI can do it better and much faster?&lt;/p&gt;&#xA;" OwnerUserId="1203" LastActivityDate="2023-03-28T16:06:52.863" Title="Impact of GPT4 and future AI development on math curricula in schools" Tags="|curriculum|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26272" PostTypeId="2" ParentId="26270" CreationDate="2023-03-24T08:00:57.497" Score="2" Body="&lt;p&gt;I think you're better off concentrating on performance of actual problems versus memorizing definitions or theorems, when teaching math for general STEM. It is much more lasting to anchor something like exponent laws by doing problems with exponents (and being tested on them), versus just regurgitating them.  In addition, much of math has a heavy algebra component and this is a weakness of kids.  So, problems that require them to use several lines of algebra properly are a feature, not a bug.&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't completely stay away from &amp;quot;concept style&amp;quot; questions. But don't make them the main objective.  So, for example, students learning second order constant coefficient equations should be tested with the ability to do a multi-step problem (abstracting the characteristic equation, solving the quadratic, maybe dealing with repeated roots, finding the particular solution if it is a heterogenous, and even perhaps finding the values of the C1 and C2, based on initial conditions.).  If you want to ask a question about under/over/critical damped, fine.  Makes sense.  But that's a little garnish for the stew.  The main thing is they need to be able to solve for y.&lt;/p&gt;&#xA;&lt;p&gt;And yes, doing all the darned algebra involved in that, and properly.  This will be needed/expected in their electrical engineering classes.  It's OK and even optimal if you push the math problem first, rather than an electrical or control circuit.  That should only be brought in later after the student has shown the ability to manipulate the math itself.  Then (if there is time), they can later learn about how capacitors work different than inductors or what the heck a dashpot is.  And if you don't get into it, they will get that in their majors courses.  But it will go a lot smoother if they already have some familiarity with the ODE, while they are learning the physical components.&lt;/p&gt;&#xA;&lt;p&gt;If you must push the concept stuff, there are ways to go about it that are a little more fun than recitation.  T/F questions.  Evaluating drawings.  What has to be right for this relation to be true.  Fill in the blank.  Etc.  But I would urge you not to jump too deep into &amp;quot;higher order concept&amp;quot; land and omit the needed work on solidifying their basic skills in the content.  For one thing, these concept questions can be very logic/verbally demanding (the SAT question stuff) that ends up being a bit of a general intelligence test at keeping track of double negatives or divided by zero exceptions or the like.  It's not that those are completely unimportant, but I just wouldn't emphasize them at the expense of normal calculations.&lt;/p&gt;&#xA;&lt;p&gt;Of course in a class on logic or proof writing, the whole point is to learn to be careful about language and conditions, so fine.  But this is not the case in most of the algebra/algebra 2/trig/analytic geometry/calculus/ODE/PDE coursework of an engineer.  And even their (rather limited) needs in linear algebra should be more demonstrated hands on in terms of manipulating matrices versus proofs (and certainly versus tricky questions).&lt;/p&gt;&#xA;&lt;p&gt;In particular, if you are teaching kids that are below elite (not CalTech, not Thomas Jefferson), it's probably especially important to be more traditional.  They need the practice in multi step algebra.  It's not trivial to them.&lt;/p&gt;&#xA;" OwnerUserId="22232" LastEditorUserId="22232" LastEditDate="2023-03-24T08:03:13.087" LastActivityDate="2023-03-24T08:03:13.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26273" PostTypeId="2" ParentId="26270" CreationDate="2023-03-24T08:34:06.430" Score="4" Body="&lt;p&gt;What you might be looking for is cognitively demanding questions. Students/Pupils have typically comparatively easy time with algorithmic exercises (including the kinds of exercises, calculations or proofs, that they are used to doing) and can also learn to regurgitate conceptual knowledge like repeating definitions or remembering that a positive derivative means an increasing function.&lt;/p&gt;&#xA;&lt;p&gt;Combining these two is more difficult.&lt;/p&gt;&#xA;&lt;p&gt;One source for these things is Markus Hähkiöniemi's doctoral thesis: &lt;a href=&quot;https://jyx.jyu.fi/bitstream/handle/123456789/22530/1/rep104.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://jyx.jyu.fi/bitstream/handle/123456789/22530/1/rep104.pdf&lt;/a&gt; (might have been one of the included articles, which you will need to dig up separately)&lt;/p&gt;&#xA;&lt;p&gt;One way to look at the matter of representations for functions is attributed to Janvier. A function can be represented as&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A formula&lt;/li&gt;&#xA;&lt;li&gt;A list of inputs and outputs&lt;/li&gt;&#xA;&lt;li&gt;A graph&lt;/li&gt;&#xA;&lt;li&gt;A narrative description of a situation&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The transitions between these are the key and you can vary these. Give the formula to a function and ask to describe a scenario where it could be used. Provide a graph and ask to recover the minimum and the maximum value and the points where those are reached. Give a list of (x, y) -coordinates and ask for a function that goes through them.&lt;/p&gt;&#xA;&lt;p&gt;Some of these tasks are obviously easier than others, but what is important is that they require a more broad understanding of what a function is than the student/pupil might otherwise need. And one key idea in didactics of mathematics is that the conceptual understanding a student/pupil develops is the simplest one required to solve the problems they meet.&lt;/p&gt;&#xA;&lt;p&gt;Janvier's idea is attributed to the following article, though it might be a bit implicit there:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;BELL, Alan; JANVIER, Claude. The interpretation of graphs representing situations. For the learning of mathematics, 1981, 2.1: 34-42.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The idea of the students/pupils using the least complicated understanding of an idea that carries them forward I picked up from Matematikk på småskoletrinnet by Bjørnar Alseth, but it, too, is sure to be a more broadly known idea.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Cognitively demanding tasks&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;These are in the order from least to most demanding.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Memorizing&lt;/li&gt;&#xA;&lt;li&gt;Procedures without a context (practicing a given algorithm)&lt;/li&gt;&#xA;&lt;li&gt;Procedures within a context (you have to choose which algorithm to apply)&lt;/li&gt;&#xA;&lt;li&gt;Problem solving (the problem is new in the sense that you do not have an algorithm to use on it)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I looked these up from Anita Valenta's article for Matematikksenteret: &lt;a href=&quot;https://www.matematikksenteret.no/publikasjoner/kognitive-krav-i-matematikkoppgaver&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.matematikksenteret.no/publikasjoner/kognitive-krav-i-matematikkoppgaver&lt;/a&gt; . The PDF there has further references, also in English.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-03-24T08:34:06.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26274" PostTypeId="2" ParentId="26255" CreationDate="2023-03-24T14:24:07.017" Score="1" Body="&lt;h2&gt;You may be trying to band-aid a missing leg.&lt;/h2&gt;&#xA;&lt;p&gt;As another answer mentions, the internet provides a number of means to easily cheat fairly easily. You can take some measures to stop obvious cases of cheating (and you should take some such measures), but there's only so much you can do.&lt;/p&gt;&#xA;&lt;p&gt;As fedja puts it, the job of teachers is &amp;quot;to provide an opportunity to learn to everybody who is willing to learn&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;My conclusion is slightly different, however: you shouldn't necessarily based grades (almost) fully on proctored exams. A student can have a moderately decent understanding of a topic and still not perform all that well in an exam due to the time constraint, due to stress, due to not having memorised the appropriate formulas, because they were having a bad day or week, etc.&lt;/p&gt;&#xA;&lt;p&gt;If it weren't for the possibility of cheating, I'd say the best grading scheme is likely mostly one without exams (doctors, for example, probably still need surgical exams). There are also measures to focus on testing whether students actually understand the material rather than testing whether they were able to e.g. memorise a bunch of things that they can recall within a short timeframe, and that they can write quickly enough, but I won't go into that here.&lt;/p&gt;&#xA;&lt;p&gt;For different courses (even within maths), it may be more or less easy to cheat on homework (with ChatGBT or through other means).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My suggestion&lt;/strong&gt; is to possibly reduce the weight of homework so having a degree continues to actually mean something for what you know (even if it already often means very little). But also don't go to to extreme measures to avoid having someone pass mostly through cheating: if a student cheats, they're shooting themselves in the foot by squandering what's hopefully a good opportunity for them to learn things they'll use later in life. Don't then shoot other students in the foot to put them on equal &lt;em&gt;footing&lt;/em&gt; with the cheater.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I can't, however, tell you how exactly you should weight homework, quizzes, bigger tests, projects and exams to come up with a final grade. That's something you'll need to figure out based on all the factors involved.&lt;/p&gt;&#xA;&lt;p&gt;One option to consider is to have an overall grade that's calculated in some way (which has a minimal passing grade), but then to also have a minimal grade for the exam only, which is lower than the overall passing grade.&lt;/p&gt;&#xA;&lt;p&gt;Or one could combine homework, etc. with exams in some variable way. You could, for example, say that the maximum distance between the two is 20%, and the exam mark will drag the homework mark down, or raise it up, if the distance exceeds 20%. The downside here would be that the grading would be less obvious to students (even if you're transparent about how it's weighted, which you should be), and this may make it feel unfair.&lt;/p&gt;&#xA;" OwnerUserId="14976" LastEditorUserId="14976" LastEditDate="2023-03-24T14:41:57.603" LastActivityDate="2023-03-24T14:41:57.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26275" PostTypeId="2" ParentId="26271" CreationDate="2023-03-24T17:03:41.630" Score="4" Body="&lt;p&gt;I think this is an important question. The issue, as I see it, is that for a long time learning mathematics has been promoted based on applications of mathematics. Obviously this approach fails to address the question that you raised, i.e., it cannot convince high-school students why they should learn tasks that AI can do better and faster. But let me ask another question: suppose one day AI can formulate and prove theorems in mathematics better and faster than humans. Are mathematicians going to abandon mathematics? It depends on why they do mathematics. Do mathematicians do mathematics to prove theorems? Some may, but many others do mathematics because they enjoy it. Consequently, it is reasonable to expect that many mathematicians will continue to do mathematics even if AI one day can formulate and prove theorems. One can ask the same question about music. If one day AI can compose music (it already can, to some extent) will humans stop composing music? Not if they enjoy doing it.&lt;/p&gt;&#xA;&lt;p&gt;The point I am making is that whether the mathematics curricula should change or not is secondary. What should change before that is the way learning mathematics is promoted. If learning mathematics is promoted based on its enjoyability, rather than its applications, then in response to a high-school student who asks why should I learn tasks that AI can do better and faster, one can say because mathematics is enjoyable? Understandably, mathematics cannot suddenly become enjoyable by just telling people that it is enjoyable and a lot of thought and work is needed to make it enjoyable, which may include changing the curriculum.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-03-24T17:13:07.533" LastActivityDate="2023-03-24T17:13:07.533" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="26276" PostTypeId="2" ParentId="26255" CreationDate="2023-03-24T17:39:43.970" Score="3" Body="&lt;p&gt;I'm on the receiving end of this exercise, so I feel that I'm well equipped to answer this question.&lt;/p&gt;&#xA;&lt;p&gt;ChatGPT has been absolutely wonderful for me. It has helped me learn a lot more, in a more effective manner. As a result, I feel that it can be used for the benefit of the students.&lt;/p&gt;&#xA;&lt;p&gt;Most are extremely misguided with their answers here. I will try to address as many as possible.&lt;/p&gt;&#xA;&lt;p&gt;First, those that are extremely effective:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There's a lot of research that says neophytes are already challenged just by the new topic and benefit from drill. The desire for more complex homework is driven by professors/researchers who see it as more interesting, but they don't really have a good feel for the dynamics of gradual training (essentially are better at math than at pedagogy). In addition, overly complex problems (if collected/graded) are also more likely to lead to cheating.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Seek more authentic, process-based assessments. To me, this was the big one. It is more labor intensive, and requires more face-to-face time with students, but is likely the only thing which is going to ensure student learning in the long run. Instead of having students turn in a pile of completed work, have them turn in drafts, then require that they discuss those drafts with you (I already do this in most of my classes). Or, as is noted in the question itself, have them present work in class—if a student doesn't understand what they are presenting, it becomes clear pretty quickly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then, those that are not:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I wouldn't bother at all. Just base your grades 90% on proctored exams where using the internet is prohibited. Those who will cheat on homework and quizzes will fail miserably and that will be their problem. Our task as teachers is &lt;em&gt;to provide an opportunity&lt;/em&gt; to learn to &lt;em&gt;everybody who is willing to learn and is capable of learning&lt;/em&gt; and I usually go far out of my way and spend a lot of extra time to help such students. But we are neither policemen, nor babysitters, so if somebody really wants to remain ignorant and get kicked out in the end, just let him or her do it and consider it the society problem, not yours.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Please do not follow this&lt;/strong&gt;. Your audience/demographic consists of extremely influenced youngsters who do not rank high on maturity and self-awareness. Students simply do not choose to be ignorant. Their ignorance stems from ignorance itself and the lack of exposure, so it will take some time to catch up.&lt;/p&gt;&#xA;&lt;p&gt;I believe that the hallmark of a great teacher is to pull in disinterested students into studying the subject. The interested ones will come after you for further learning anyways.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Increase the frequency of in-class tests instead (proctored and without electronic aids). Do not resent the loss of lecture time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Please do not&lt;/strong&gt;. There is nothing more irritating for a student than constantly having to give graded tests when very little material has been covered between the tests. Tests do not serve as practice; they serve as tests, because the students see it as a test and approach it as a test.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Answers talking about exploiting some errors of ChatGPT.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is a quick fix at best. There is always going to be further developments that will minimize errors. You should be looking at a long term solution.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Explain to the students why using ChatGPT is not in their best interests in the long term in learning maths (or any other topic).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One can always try, but most students resort to soft-cheating like using ChatGPT because they value marks more than learning.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Coming to what I would love to see as a student:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I would love to see a lecturer passionate about what they are teaching and is passionate about getting as many students as possible interested into the subject; someone who generates that interest in the student into solving questions because they want to understand the material.&lt;/p&gt;&#xA;&lt;p&gt;Does this ensure that students &lt;em&gt;will&lt;/em&gt; be more interested? &lt;strong&gt;No.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm guilty of not paying too much attention to courses because I simply didn't find it interesting, despite of everything. However, it increases the probability by a great margin.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Have great lectures&lt;/strong&gt;. Structure them so that students can understand material simply and easily, and want to learn more.&lt;/li&gt;&#xA;&lt;li&gt;Keep telling them about the importance of solving the exercise sheet&#xA;on their own if they want to understand the concepts and score well&#xA;in exams.&lt;/li&gt;&#xA;&lt;li&gt;Additional resources that help them understand is what students are primarily looking for, so providing them with these resources cuts off the incentive for using ChatGPT.&lt;/li&gt;&#xA;&lt;li&gt;Try giving questions that have a short solution, but getting to the solution requires some thinking. Of course, having them drill in some problems is extremely important as well, so you can have sessions just for that.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Tutorials are a great idea. How are they held exactly?&#xA;Over here we have an hour dedicated in a week where students need to sit in a room and solve the exercise sheet and tutors go around the classroom solving doubts that students have in the exercises. You could enforce this more strongly by preventing use of electronics and making attendance mandatory.&lt;/p&gt;&#xA;&lt;p&gt;You have stated that the tutors have to hold a two hour exercise group. Having two 1 hour sessions is less demanding and more effective for both parties.&lt;/p&gt;&#xA;&lt;p&gt;The system you are currently utilising is pretty great anyway, so there shouldn't be too much of a problem anyway.&lt;/p&gt;&#xA;&lt;p&gt;Lastly, you could try being open with your students. You can ask them about this matter and ask them about ways to structure the exercises so that their dependence on do-it-for-you's like GPT and &lt;a href=&quot;https://en.wikipedia.org/wiki/Chegg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chegg&lt;/a&gt; is reduced. There are always a group of students who will have useful suggestions.&lt;/p&gt;&#xA;&lt;p&gt;This is from an interview with Sam Altman, the CEO of OpenAI.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We're going to try and do some things in the short term,&amp;quot;&#xA;said Altman.  &amp;quot;There may be ways we can help teachers be a little more  likely to detect output of a GPT-like system. But honestly, a  determined person will get around them.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Generative text is something we all need to adapt to,&amp;quot; he said. &amp;quot;We adapted to calculators and changed what we tested for in math class, I imagine. This is a more extreme version of that, no doubt, but also the benefits of it are more extreme, as well.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="22234" LastEditorUserId="851" LastEditDate="2023-03-25T06:36:48.653" LastActivityDate="2023-03-25T06:36:48.653" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="26278" PostTypeId="2" ParentId="26255" CreationDate="2023-03-25T15:58:01.157" Score="4" Body="&lt;p&gt;In my most recent departmental meeting a colleague of mine made an interesting point about the direction of technology as it regards math education.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;With the introduction of ChatGPC we finally will see the rest of the academy suffer as we have already suffered. In particular, for decades we have faced the problem of explaining why we should still teach something a computer can &amp;quot;do&amp;quot;. Now all the other academic disciplines face the same fate.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He took this as a positive. In particular, we should now have greater solidarity with other disciplines who also find themselves needing to explain how dare they teach something which a computer can &amp;quot;do&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But, I'm not as much an optimist. I foresee two very different paths forward, one a path of redemption towards greater intellectual purity, the other the path of pragmatism where our pigeons come home to roost:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Path One:&lt;/strong&gt; other disciplines encourage mathematics and the sciences to reject the push to applications motivated teaching. We return to the concept that education at it's base is for knowledge and skill. Education is not job training. Sure, it may provide training for some jobs, but that cannot be it's primary motivator (unless of course the education is hosted by a business which is training employees for a specific task, which is well and good, but probably should be done in businesses rather than universities). In this hypothetical universe, other departments in the university join together to reject automation of thinking whether inside or outside of math and together we seek a future where our progeny learns critical thinking.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Path Two:&lt;/strong&gt; we accept the direction of society and give up on people having to think for themselves. Instead, we train them how to use ChatGPC etc. to form thoughts for them. After all, it is more efficient to use a computer to think thoughts. Moreover, if we have people use ChatGPC then we can be sure they don't think unapproved thoughts. If everything we write is done by some AI then the government's ministry of information can more readily enforce proper speech. People writing of their own volition and style then would be renegades, indeed dangerous purveyors of unprofessional speech. We would sell students not writing their own papers as a &lt;strong&gt;best practice&lt;/strong&gt; and we would soon find a vast literature which supports the efficacy of automatic paper writing and argumentation as an &lt;strong&gt;evidence-based-methodology&lt;/strong&gt;. Surely students in this future who wrote papers with their own impartial and flawed understanding of grammar and such would be no match for their peer's professional ChatGPC-based writing. This sort of thing is already done in math where we compare results of students doing homework in a system which essentially spoon feeds them the method against traditional methods where students have to think for themselves.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In any event, I support fedja's answer. It is correct. We must insist on proctored, internet free exams if we are to have universities with meaningful degrees. The fact that accreditation bodies are not meaningfully policing online education degrees with the expectation of proctoring endangers the whole program of higher education. It is an existential threat as the proliferation of online courses with low quality more and more makes ordinary folks question the value of our degrees ( and rightly so ).&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="128" LastEditDate="2023-03-27T11:28:55.100" LastActivityDate="2023-03-27T11:28:55.100" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26279" PostTypeId="1" CreationDate="2023-03-25T19:34:20.547" Score="1" ViewCount="454" Body="&lt;p&gt;For those unaware, Clifford Algebra (also known as Geometric Algebra) is able to generalize vectors and rotations in n-dimensional space, and simplifies a great many formulas.  However, I was curious if anyone had ever thought to &lt;em&gt;skip&lt;/em&gt; complex numbers and move straight to Clifford Algebra, or create a Clifford Algebra textbook which assumes only a high-school level knowledge?&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2023-03-31T05:24:06.910" Title="Teaching Clifford Algebra Instead of Imaginary/Complex Numbers" Tags="|complex-numbers|vectors|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26280" PostTypeId="2" ParentId="9960" CreationDate="2023-03-25T21:44:24.183" Score="2" Body="&lt;p&gt;I had it in my Lansing, Michigan 5th grade class in 1961. I recall that approximately 30% of us were 'selected' somehow. Loved it.&lt;/p&gt;&#xA;" OwnerUserId="22240" LastActivityDate="2023-03-25T21:44:24.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26281" PostTypeId="2" ParentId="26270" CreationDate="2023-03-26T04:21:22.893" Score="5" Body="&lt;p&gt;As asked, it is a bit too general to answer with anything but other generic words. So let's play with some simple particular topic like finding maxima of functions on intervals.&lt;/p&gt;&#xA;&lt;p&gt;The lowest level: find the maximum of &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^3/3-1.5x^2+2x-1$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$[0,3]$&lt;/span&gt; with explanation.&lt;/p&gt;&#xA;&lt;p&gt;The next level: same problem for &lt;span class=&quot;math-container&quot;&gt;$g(x)=|f(x)|$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h(x)=\cos(f(x))$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;One up: Given that &lt;span class=&quot;math-container&quot;&gt;$f(0)=1.1$&lt;/span&gt;, which one is larger: &lt;span class=&quot;math-container&quot;&gt;$\max_x f(x)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\max_x (2f(x)-f(x)^2)$&lt;/span&gt; (on &lt;span class=&quot;math-container&quot;&gt;$[-1,1]$&lt;/span&gt;, say)?&lt;/p&gt;&#xA;&lt;p&gt;One more up: Find the maximum of &lt;span class=&quot;math-container&quot;&gt;$1-x^2+x^3\cos(e^x-50x^4)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$[-1,1]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The highest one I would bother to check at the undergraduate level:&#xA;Prove that &lt;span class=&quot;math-container&quot;&gt;$\max_{x&amp;gt;0} (1000x^2-4^x)\le 50000$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Those are (IMHO) in the order of increasing difficulty and almost decreasing tediousness. I hope this example conveys the idea clearly enough so that you can invent a similar series in any other topic yourself.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-03-26T04:21:22.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26282" PostTypeId="2" ParentId="26279" CreationDate="2023-03-26T06:59:58.017" Score="6" Body="&lt;p&gt;There is a reason why mathematics is taught from more concrete to more abstract. Students need to figure out what the role of proofs, axioms, etc. is before they can start abstraction first. Some very few students have figured this out before university, while some go through their master thesis without really getting it. Or at least to the master-level courses. They will be difficult.&lt;/p&gt;&#xA;&lt;p&gt;So, in general, there is a good reason to increase the abstraction step by step - first real numbers, than complex numbers, then into more abstraction. With the complex numbers the students have already learned some tools and intuitions about missing tools that they can take further, as well as routine for calculations and just plain general mathematical maturity due to the course happening later.&lt;/p&gt;&#xA;&lt;p&gt;That said, in principle, there is no problem having a specialist course, or maybe an elite university could indeed hop over some stages in the step-by-step-progression. I was not at an elite university, but we had a first year course on abstract metric spaces as an alternative to a course on Euclidian spaces. Of the seven or so people taking that course, about five have doctor's degree.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-03-26T06:59:58.017" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26283" PostTypeId="1" AcceptedAnswerId="26284" CreationDate="2023-03-26T16:53:53.157" Score="3" ViewCount="440" Body="&lt;p&gt;How do you teach to teenagers or kids that if we have a square with side length &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; m (that is, &lt;span class=&quot;math-container&quot;&gt;$100x$&lt;/span&gt; cm), then its area is &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; m&lt;span class=&quot;math-container&quot;&gt;$^2,$&lt;/span&gt; but not &lt;span class=&quot;math-container&quot;&gt;$10000x^2$&lt;/span&gt; c&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;m&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; ?&lt;/p&gt;&#xA;" OwnerUserId="22243" LastEditorUserId="372" LastEditDate="2023-03-27T11:19:40.683" LastActivityDate="2023-04-12T17:29:48.467" Title="How to teach that $10000x^2$ c$^2$m$^2$ is wrong?" Tags="|units|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26284" PostTypeId="2" ParentId="26283" CreationDate="2023-03-27T08:35:33.967" Score="7" Body="&lt;p&gt;But &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; m&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$10000x^2$&lt;/span&gt; c&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;m&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; are arguably &lt;strong&gt;both correct&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The prefix ‘centi’ means a hundredth, and the convention &lt;span class=&quot;math-container&quot;&gt;\begin{align}\text{cm}^2:={}&amp;amp;(\text{cm})^2\\={}&amp;amp;(10^{-2}\text{ m})^2\\={}&amp;amp;10^{-4}\text{ m}^2\end{align}&lt;/span&gt; does not preclude the curious-looking first intermediate step in the presentation &lt;span class=&quot;math-container&quot;&gt;\begin{align}(100x\text{ cm})^2&amp;amp;{}\color{red}{=100^2x^2\text{ c}^2\text m^2}\\&amp;amp;=100^2(0.01)^2 x^2\text{ m}^2\\&amp;amp;=x^2\text{ m}^2,\end{align}&lt;/span&gt; which is just as sound and solid as the presentation&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;2000\text{ cm$^2$/h}&amp;amp;=2000\left(\frac1{100}\text{ m}\right)^2\left(60^2\text{ s}\right)^{-1}\\&#xA;&amp;amp;=5.56\times10^{-5}\text{ m$^2$/s}.&#xA;\end{align}&lt;/span&gt; In all the above, ‘centi’ and ‘&lt;span class=&quot;math-container&quot;&gt;$\%$&lt;/span&gt;’ perform the function of dividing by &lt;span class=&quot;math-container&quot;&gt;$100:$&lt;/span&gt; the former as a prefix operator, whilst the latter as a &lt;a href=&quot;https://math.stackexchange.com/a/4399862/21813&quot;&gt;postfix operator&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;P.S. Likewise for &lt;a href=&quot;https://www.nist.gov/pml/owm/metric-si-prefixes&quot; rel=&quot;nofollow noreferrer&quot;&gt;deci, milli, micro, nano, kilo, giga, etc.&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;P.P.S. While obtaining the above link, I learnt that &lt;em&gt;deca&lt;/em&gt; is the only such prefix whose symbol ‘da’ has two characters. Since &lt;em&gt;milli&lt;/em&gt; versus &lt;em&gt;mega&lt;/em&gt; is ‘m’ versus ‘M’, it's a shame that &lt;em&gt;deci&lt;/em&gt; versus &lt;em&gt;deca&lt;/em&gt; couldn't just be ‘d’ vesus ‘D’.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-03-28T22:34:05.067" LastActivityDate="2023-03-28T22:34:05.067" CommentCount="21" ContentLicense="CC BY-SA 4.0" />
  <row Id="26285" PostTypeId="2" ParentId="26279" CreationDate="2023-03-27T14:07:56.723" Score="10" Body="&lt;p&gt;One of the pinnacle goals in high school and basic college algebra is to be able to express the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra&quot; rel=&quot;noreferrer&quot;&gt;fundamental theorem of algebra&lt;/a&gt;, which is everywhere expressed in terms of complex numbers.&lt;/p&gt;&#xA;&lt;p&gt;Complex numbers are the first and simplest algebraically closed field, so everyone mathematically literate should be familiar with them.&lt;/p&gt;&#xA;&lt;p&gt;Thus, it's exceedingly unlikely that &amp;quot;skipping&amp;quot; complex numbers would be fruitful or justified in almost any conceivable curriculum.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-03-27T14:07:56.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26286" PostTypeId="2" ParentId="26279" CreationDate="2023-03-28T02:56:42.133" Score="2" Body="&lt;p&gt;If we want to solve the quadratic equation &lt;span class=&quot;math-container&quot;&gt;$az^2+bz+c=0$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a,b,c \in \mathcal{A}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a^{-1}$&lt;/span&gt; exists where &lt;span class=&quot;math-container&quot;&gt;$\mathcal{A}$&lt;/span&gt; is a semi-simple associative unital algebra then since &lt;span class=&quot;math-container&quot;&gt;$\mathcal{A}$&lt;/span&gt; is isomorphic to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n \times \mathbb{C}^m$&lt;/span&gt; the quadratic equation given translates into &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-real quadratic equations paired with &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;-imaginary quadratic equations. Therefore, understanding the real and complex case is, well, basic. If the Clifford algebra viewpoint makes it easier, then I wait to see it.&lt;/p&gt;&#xA;&lt;p&gt;For example, hyperbolic numbers &lt;span class=&quot;math-container&quot;&gt;$\mathcal{H} = \{ x+jy \ | \ x,y \in \mathbb{R}, j^2=1 \}$&lt;/span&gt; are isomorphic to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \times \mathbb{R}$&lt;/span&gt; and as such a quadratic equation in &lt;span class=&quot;math-container&quot;&gt;$\mathcal{H}$&lt;/span&gt; amounts to a pair of real quadratic equations. It follows there is either no solution, one solution, two solutions, or four solutions to the quadratic equation in the hyperbolic number system. These correspond respectively to:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;both underlying real quadratic equations have only complex solutions (no solution)&lt;/li&gt;&#xA;&lt;li&gt;one or both of the underlying real quadratic equations have repeated root solution (one solution)&lt;/li&gt;&#xA;&lt;li&gt;both of the underlying real quadratic equations have distinct real solutions (four solutions)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For the Clifford Algebra approach to be interesting to me, it would have to also allow for the natural demarcation into real and complex aspects of a semi-simple real associative algebra.&lt;/p&gt;&#xA;&lt;p&gt;For what it's worth, I have similar gripes with algebraists who insist on always working over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; since &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; are &amp;quot;easy&amp;quot; to derive from the &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; theory... this is little consolation when you're looking for a breakdown into what is possible over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2023-03-28T02:56:42.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26288" PostTypeId="1" CreationDate="2023-03-28T07:44:35.353" Score="14" ViewCount="4872" Body="&lt;p&gt;We teach students that a notation like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$17 - -59$$&lt;/span&gt; is not acceptable or at least not good. Instead we want them to write &lt;span class=&quot;math-container&quot;&gt;$$17-(-59)$$&lt;/span&gt;&#xA;The main reason seems to be that it's more readable if you separate the operator and the number sign.&lt;/p&gt;&#xA;&lt;p&gt;But since&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;students frequently make errors regarding this notation,&lt;/li&gt;&#xA;&lt;li&gt;the notation is not really needed to avoid ambiguity and&lt;/li&gt;&#xA;&lt;li&gt;there is that weird break of symmetry since only use brackets for the subtrahend and not the minuend&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I wonder to just accept the version without parentheses or even use it myself in class.&lt;/p&gt;&#xA;&lt;p&gt;Are there any reasons why this notation is necessary?&lt;/p&gt;&#xA;" OwnerUserId="22246" LastEditorUserId="22246" LastEditDate="2023-03-28T13:46:00.637" LastActivityDate="2023-04-19T13:26:38.780" Title="Parentheses around negative numbers" Tags="|undergraduate-education|notation|negative-numbers|" AnswerCount="4" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="26289" PostTypeId="2" ParentId="26288" CreationDate="2023-03-28T08:21:27.913" Score="36" Body="&lt;ol&gt;&#xA;&lt;li&gt;It is not a good idea to against a convention, unless you have strong reasons for it.&lt;/li&gt;&#xA;&lt;li&gt;That said, you might not want to punish those going against the convention, either.&lt;/li&gt;&#xA;&lt;li&gt;An obvious reason for having something between the minus symbols is that when writing by hand (and maybe in some fonts) it might be far too easy to write the minus symbols together, and far too hard to figure out how many minus symbols there are supposed to be in any given case.&lt;/li&gt;&#xA;&lt;li&gt;Also, you would typically want to have the parentheses in cases like &lt;span class=&quot;math-container&quot;&gt;$2 \cdot (-3)$&lt;/span&gt; for similar issues of legibility that &lt;span class=&quot;math-container&quot;&gt;$2 \cdot -3$&lt;/span&gt; has. Hard to know whether the dot is supposed to be there or some artifact of the writing process.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It might be a bit of redundancy to have the parentheses,  but some redundancy adds legibility. Error correction for human language.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="60" LastEditDate="2023-03-28T14:38:08.427" LastActivityDate="2023-03-28T14:38:08.427" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26291" PostTypeId="2" ParentId="26288" CreationDate="2023-03-28T21:18:15.413" Score="17" Body="&lt;p&gt;Others have talked about legibility, so I want to address the importance of the parentheses in an expression like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$17\color{red}{-}(\color{blue}{-}59)$$&lt;/span&gt;&#xA;from the perspective of educational psychology. &lt;a href=&quot;https://matheducators.stackexchange.com/questions/26288/parentheses-around-negative-numbers#comment69409_26288&quot;&gt;As @Michał mentions&lt;/a&gt;, the two minus signs in this expression have different meanings. To emphasize the difference, we can decode the expression as:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$17\color{red}{\text{ take away}} \color{blue}{\text{ the opposite of }} 59.$$&lt;/span&gt; &lt;a href=&quot;https://matheducators.stackexchange.com/questions/26288/parentheses-around-negative-numbers/26291#comment69513_26291&quot;&gt;As @ryang mentions&lt;/a&gt;, there are good reasons to pronounce both minus signs as “minus,” and it makes sense that we use the same symbol to represent subtraction and negation, since&#xA;&lt;span class=&quot;math-container&quot;&gt;$$a-b=a+(-b),$$&lt;/span&gt;&#xA;but this is not a trivial fact. It can be quite confusing for students that a symbol which has always signified subtraction now also signifies negative and negation.&lt;sup&gt;1&lt;/sup&gt; Using parentheses helps students distinguish the different meanings of the minus sign.&lt;/p&gt;&#xA;&lt;p&gt;Some curricula attempt to address this by introducing negation using a different symbol, e.g. &lt;span class=&quot;math-container&quot;&gt;$$b^* \ \text{or }\ \overline{b}\ \text{ or } {}^−\!b$$&lt;/span&gt;&#xA;as &lt;a href=&quot;https://matheducators.stackexchange.com/questions/26288/parentheses-around-negative-numbers#comment69432_26288&quot;&gt;@Raciquel&lt;/a&gt; mentioned. Such a curriculum would then introduce the notation &lt;span class=&quot;math-container&quot;&gt;$-b$&lt;/span&gt; &lt;strong&gt;after&lt;/strong&gt; students are convinced of the fact that &lt;span class=&quot;math-container&quot;&gt;$a-b=a+b^*\!.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Students need to be able to recognize and go between the different meanings of the minus sign to understand negative number arithmetic. &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/09515080802285552&quot; rel=&quot;nofollow noreferrer&quot;&gt;In a study by Joëlle Vlassis&lt;/a&gt;, she found that when students were asked to solve the equation &lt;span class=&quot;math-container&quot;&gt;$4-x=5$&lt;/span&gt;, some erroneously wrote&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;4-x&amp;amp;=5 \\&#xA;x&amp;amp;=5-4&#xA;\end{align*}&lt;/span&gt;&#xA;which is a very common mistake. In her interpretation, this was because&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Students considering, in this context, only the subtraction function of the minus sign could not imagine writing this sign before x at the second stage because, from their point of view, there was no more operation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some students who found &lt;span class=&quot;math-container&quot;&gt;$x=-1$&lt;/span&gt; were unable to write &lt;span class=&quot;math-container&quot;&gt;$4-(-1)=5$&lt;/span&gt; as justification.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It did not occur to students to put parentheses around the solution... For students who thought that these minus signs were two binary signs, it was inconceivable to write an expression with a succession of two &amp;quot;subtracting&amp;quot; signs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Difficulty interpreting minus signs can happen when other operations are involved as well. For example, with the equation &lt;span class=&quot;math-container&quot;&gt;$-6x=24$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When the interviewer suggested the solution &lt;span class=&quot;math-container&quot;&gt;$-4$&lt;/span&gt; to them, these students felt that this was not possible, since&#xA;you would have &lt;em&gt;times minus&lt;/em&gt; 4.&amp;quot; These two students appeared perplexed about the possibility of producing an expression in which the &amp;quot;times&amp;quot; and &amp;quot;minus&amp;quot; signs followed one another.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the convention of using parentheses to separate the unary minus from other operations is a &lt;strong&gt;thinking tool&lt;/strong&gt; as much as it is a communication tool. It is natural for students to find it difficult at first, because we are not simply asking them to adopt a new writing convention, but to understand and attend to the different meanings of the minus sign.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1: “Negative” and “negation” are distinct ideas. For example, when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is negative, &lt;span class=&quot;math-container&quot;&gt;$-x$&lt;/span&gt;, the opposite of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, is positive. So for students, there are &lt;em&gt;three&lt;/em&gt; ideas related to the minus sign that need to be distinguished: subtraction, negative numbers, and negation.  &lt;a href=&quot;https://par.nsf.gov/servlets/purl/10123109&quot; rel=&quot;nofollow noreferrer&quot;&gt;This article (public access through the NSF)&lt;/a&gt; goes in depth on the different meanings and has suggestions for teachers on how to help students make sense of the minus sign. It really is difficult! &lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-04-02T13:01:19.130" LastActivityDate="2023-04-02T13:01:19.130" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26292" PostTypeId="2" ParentId="26283" CreationDate="2023-03-28T22:45:40.153" Score="2" Body="&lt;p&gt;I wanted to synthesize some of the discussion happening in the comments into an answer.&lt;/p&gt;&#xA;&lt;p&gt;To me, a statement like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(100x\ \text{cm})^2 = 10\,000x^2\ \text{c}^2\text{m}^2$$&lt;/span&gt;&#xA;indicates that the student has not associated the symbol &lt;span class=&quot;math-container&quot;&gt;$\text{cm}$&lt;/span&gt; with the unit &lt;span class=&quot;math-container&quot;&gt;$\text{centimeters}$&lt;/span&gt; in the context of exponentiation. You can help them make this connection by rewriting the left-hand side as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(100x\ \text{centimeters})^2\!,$$&lt;/span&gt;&#xA;at which point they will hopefully see that &lt;span class=&quot;math-container&quot;&gt;$10\,000x^2\ \text{c}^2\text{m}^2$&lt;/span&gt; is gobbledygook. In this form, it is also easier to see that&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;(100x\ \text{centimeters})^2 &amp;amp;= (x\ \text{meters})^2\!\\&#xA;&amp;amp;= x^2\ \text{square meters}\text{, so}\\&#xA;(100x\ \text{cm})^2 &amp;amp;= (x\ \text{m})^2\!\\&#xA;&amp;amp;= x^2\ \text{m}^2\!.&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-03-28T22:45:40.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26293" PostTypeId="2" ParentId="26220" CreationDate="2023-03-28T22:53:02.430" Score="4" Body="&lt;p&gt;I've used the term &amp;quot;reference point&amp;quot; or &amp;quot;reference coordinate&amp;quot; when referring to the points on the original function, compared to their transformed locations. I've expected my students to know what happens to each of the &amp;quot;reference points&amp;quot; of the original graph. The most typical &amp;quot;reference coordinate&amp;quot; occurs where &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;, which in all three of the examples above is transformed to the coordinate &lt;span class=&quot;math-container&quot;&gt;$x=5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To add to this, I often use the original points on the graph at &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=-1$&lt;/span&gt; as the remaining reference coordinates, which works on functions like &lt;span class=&quot;math-container&quot;&gt;$f(x)=a^x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^n$&lt;/span&gt; for integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, etc. For inverse functions like &lt;span class=&quot;math-container&quot;&gt;$\log_b(x)$&lt;/span&gt; it is occasionally better to use what occurs at original reference coordinates generated from &lt;span class=&quot;math-container&quot;&gt;$y=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=-1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="724" LastEditorUserId="724" LastEditDate="2023-03-29T16:36:28.467" LastActivityDate="2023-03-29T16:36:28.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26294" PostTypeId="2" ParentId="26220" CreationDate="2023-03-29T02:22:29.103" Score="2" Body="&lt;p&gt;Often questions about graph sketching will ask for a sketch showing all &lt;strong&gt;relevant&lt;/strong&gt; points, or in some cases all relevant features. The actual points or features to be shown will vary depending on the specific graph and on what has been covered in the teaching unit. Students of the unit will understand from their studies and practice examples which features are relevant (intercepts, turning points etc) and which are not (points at every integer value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$-4$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$+4$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="17542" LastActivityDate="2023-03-29T02:22:29.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26295" PostTypeId="2" ParentId="26288" CreationDate="2023-03-29T09:27:19.883" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Are there any reasons why this notation is necessary?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes, of course - it is necessary to avoid easily avoidable errors introduced by unclear writing. While it may usually be obvious from context if you see that notation without parantheses in typewritten text (i.e., computers...), this is not the case for pupils writing by hand.&lt;/p&gt;&#xA;&lt;p&gt;Looking at my own children, writing cleanly was a &lt;em&gt;major&lt;/em&gt; obstacle at all times. If they then also are maybe not finding the actual content of their maths trivial, it is just an unneccessary additional cause of error and frustration.&lt;/p&gt;&#xA;&lt;p&gt;This continues into adult life. I.e. when writing computer code, where a compiler does interpret expressions in a very formal way, with crystal clear semantics, it very often makes sense and is a convention to add completely superfluous parentheses for readability and maintainability.&lt;/p&gt;&#xA;&lt;p&gt;So aside from teaching the maths itself, it is equally important to teach that it is important to communicate clearly. A correct formula is still wrong if the reader cannot read or interpret it because it's hastily scribbled, and in-built &amp;quot;error correction&amp;quot; techniques like parantheses have also been skipped.&lt;/p&gt;&#xA;" OwnerUserId="8146" LastActivityDate="2023-03-29T09:27:19.883" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26296" PostTypeId="1" CreationDate="2023-03-29T23:59:54.317" Score="0" ViewCount="203" Body="&lt;p&gt;In mathematics, we have sets, such as &lt;span class=&quot;math-container&quot;&gt;$\begin{Bmatrix}1, 2, 3 \end{Bmatrix}$&lt;/span&gt; or the real-numbers, usually denoted as &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When teaching students about sets for the &lt;em&gt;&lt;strong&gt;first time&lt;/strong&gt;&lt;/em&gt;, it can sometimes (not always, but sometimes) be nice to make an semi-intuitive analogy to real-life.&lt;/p&gt;&#xA;&lt;p&gt;What collection of 3 to 10 words in the English language can we use to represent a mathematical set?&lt;/p&gt;&#xA;&lt;p&gt;Ideally...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;There exists two sets &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; in our example such that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; overlap (&lt;span class=&quot;math-container&quot;&gt;$A \cap B \neq \emptyset$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There exists a universal set labeled &lt;span class=&quot;math-container&quot;&gt;$(\alpha\Omega)$&lt;/span&gt; in our example such that &lt;span class=&quot;math-container&quot;&gt;$A \subseteq (\alpha\Omega)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B \subseteq (\alpha\Omega)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A \cup B \neq (\alpha\Omega)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;We could ask students for such an example, except that we are teaching student who do not already know what a set is.&lt;/p&gt;&#xA;&lt;p&gt;These students have never heard the words &amp;quot;union&amp;quot; or &amp;quot;intersection&amp;quot; used in math-class before.&lt;/p&gt;&#xA;&lt;p&gt;Our goal is to choose a handful of words from every day English in order to provide an analogy for what a set is.&lt;/p&gt;&#xA;&lt;p&gt;Someday, I would like to present students with many examples of this, but I am having difficulty thinking of more than three examples.&lt;/p&gt;&#xA;" OwnerUserId="16039" LastActivityDate="2023-03-31T14:51:19.897" Title="What is an example of something you might see outside of math class and how would you model that thing as a set?" Tags="|set-theory|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26297" PostTypeId="2" ParentId="26296" CreationDate="2023-03-29T23:59:54.317" Score="1" Body="&lt;h1&gt;Examples of Sets You Might see Outside of Math Class&lt;/h1&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Example Using Coordinates of Wolves in a Video-game&lt;/h2&gt;&#xA;&lt;p&gt;A person could could imagine having a RPG (real-time role-playing) videogame with &lt;span class=&quot;math-container&quot;&gt;$\mathcal{wolves}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The geo-special coordinates for every animal in a videogame might be stored in a set named  &lt;span class=&quot;math-container&quot;&gt;$\mathcal{animals}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There are more than just &lt;span class=&quot;math-container&quot;&gt;$\mathcal{wolves}$&lt;/span&gt; in this videogame. Students can be asked to suggest what other animals they might want in the videogame. It is important to me that students participate actively.&lt;/p&gt;&#xA;&lt;p&gt;Not every animal in the videogame is a &lt;span class=&quot;math-container&quot;&gt;$\mathcal{wolf}$&lt;/span&gt; Thus, the set of coordinates for &lt;span class=&quot;math-container&quot;&gt;$\mathcal{wolves}$&lt;/span&gt; is a &lt;em&gt;&lt;strong&gt;subset&lt;/strong&gt;&lt;/em&gt; of the set of GPS coordinates for all &lt;span class=&quot;math-container&quot;&gt;$\mathcal{animals}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Example Two of a Set Outside of Math Class&lt;/h2&gt;&#xA;&lt;p&gt;In our second example, the set of all geo-spatial coordinates of cups, plates, forks and other dishes in someone's kitchen might work. The set of all &lt;strong&gt;water glasses&lt;/strong&gt; is a subset of the set of all &lt;strong&gt;drinking vessels&lt;/strong&gt;. You could also have &lt;strong&gt;asian tea bowls&lt;/strong&gt;, &lt;strong&gt;clay coffee mugs&lt;/strong&gt;, and other subsets of the set of drinking vessels.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Example Three of a Set Outside of Math Class&lt;/h2&gt;&#xA;&lt;p&gt;We could take &lt;span class=&quot;math-container&quot;&gt;$\mathcal{electronic\_devices}$&lt;/span&gt; to be the universal set in this example. We let &lt;span class=&quot;math-container&quot;&gt;$\mathcal{disp\_devs}$&lt;/span&gt; be the set of all devices which have a &lt;em&gt;&lt;strong&gt;display&lt;/strong&gt;&lt;/em&gt; which shines out light for your eyes to see. The set of all &lt;span class=&quot;math-container&quot;&gt;$\mathcal{newerphones}$&lt;/span&gt; would be a subset of the set of all &lt;span class=&quot;math-container&quot;&gt;$\mathcal{disp\_devs}$&lt;/span&gt;. We could show students  pictures of old fashion telephones from museums.&lt;/p&gt;&#xA;" OwnerUserId="16039" LastActivityDate="2023-03-29T23:59:54.317" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26298" PostTypeId="2" ParentId="26296" CreationDate="2023-03-30T00:58:09.323" Score="1" Body="&lt;p&gt;The best colloquial word synonym for set, I've heard is &amp;quot;collection&amp;quot;.  Not in the sense of a stamp collection, but just in the sense of collecting anything.  And it's just for introducing the idea, not for rigor (not saying every set is a physical collection of objects).  I think Tenenbaum and Pollard use that idea to remind kids of what sets are, in their ODE primer...and it works nicely.&lt;/p&gt;&#xA;&lt;p&gt;I would try to keep it very simple at first.  For example for the set of all chairs in your house, the kitchen chairs would be a subset.   Or the intersection of all chairs in the house with all furniture in the kitchen is...the kitchen chairs.&lt;/p&gt;&#xA;&lt;p&gt;All that said, I think the idea of union and intersection is pretty easy to get and I was fine with collections of letters or numbers to start with the idea.  I think this is an easier access than entire number systems like integers or reals or whatever, at least at first.  I find the whole R or C (special fancy font) slightly offputting.&lt;/p&gt;&#xA;" OwnerUserId="22251" LastEditorUserId="22251" LastEditDate="2023-03-30T01:04:25.113" LastActivityDate="2023-03-30T01:04:25.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26299" PostTypeId="2" ParentId="26296" CreationDate="2023-03-30T02:08:09.230" Score="1" Body="&lt;p&gt;The example which is frequently used in textbooks is the set of &lt;em&gt;students in a school&lt;/em&gt; who may be &lt;em&gt;taking a Foreign Language class&lt;/em&gt; or &lt;em&gt;taking a Social Studies class.&lt;/em&gt; You could also talk about the students who are taking either type of class, and have them talk about the fact that some people might be taking both. Talking about students taking Spanish class vs. any language would naturally lead to the discussion of subsets.&lt;/p&gt;&#xA;&lt;p&gt;The benefit of this example is that it is something the students have intimate familiarity with.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2023-03-30T02:08:09.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26300" PostTypeId="2" ParentId="26296" CreationDate="2023-03-30T20:28:44.807" Score="1" Body="&lt;p&gt;I would just use English pronouns (classical ones, though, if you are in the mood for it, you can add some Canadian neologisms too, provided that you can figure out how to pronounce them and remember which one is which). It is a simple not too large universe with several clearly defined subsets:&lt;/p&gt;&#xA;&lt;p&gt;singular/plural&lt;/p&gt;&#xA;&lt;p&gt;masculine/feminine/neuter&lt;/p&gt;&#xA;&lt;p&gt;first/second/third person&lt;/p&gt;&#xA;&lt;p&gt;subject/object/possessive&lt;/p&gt;&#xA;&lt;p&gt;Starting with &amp;quot;i&amp;quot;,&amp;quot;y&amp;quot;,&amp;quot;t&amp;quot;,etc.&lt;/p&gt;&#xA;&lt;p&gt;1 letter long/2 letters long/3 letters long, etc.&lt;/p&gt;&#xA;&lt;p&gt;You can make a full list of pronouns you want to use and distribute it to the students so that you all agree on what the underlying universe is, after which everything becomes well-defined and you can discuss all the set-theoretic concepts on simple examples.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-03-30T20:28:44.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26302" PostTypeId="2" ParentId="26279" CreationDate="2023-03-31T05:24:06.910" Score="4" Body="&lt;p&gt;Yes. David Hestenes has been pushing for this for many years, and wrote the textbook 'New Foundations for Classical Mechanics' with the idea of starting that process off. It's probably more advanced than high school level - I think it's intended as a first year university course - but the first few chapters probably wouldn't pose any great difficulty to a bright high school student.&lt;/p&gt;&#xA;&lt;p&gt;One of the major selling points of geometric algebra is that it already includes the complex numbers as a natural part of the number system, and gives them a clear, intuitive, geometrical interpretation. Geometric algebra encompasses and unifies the reals, complex numbers, quaternions, spinors, vectors, pseudovectors, directed areas, directed volumes, reflections, rotations, scaling, points, lines, planes, circles, spheres, and more. You only have to learn one number system, and all the others are special cases or simple extensions of it. It also fixes a lot of the problems with vector algebra, which is valuable in itself.&lt;/p&gt;&#xA;&lt;p&gt;It does not require a great deal of abstraction (at least to start with), being only a little more complicated than vectors. We start with the vectors, and we define a new sort of multiplication on them, that combines the dot product and something that works a lot like the cross product together (except that the cross product only works in 3 dimensions, while the geometric product generalises to any number of dimensions). This can produce scalars (like the dot product), but also produces new higher dimensional objects called bivectors, trivectors, etc. that correspond to directed areas, directed volumes, etc. in the same way that vectors represent directed lengths. If you start with the 2D vectors, then the combinations of scalars and bivectors they produce turn out to be identical to the complex numbers. If you start with the 3D vectors, then the combinations of scalars and bivectors produced turn out to be the quaternions.&lt;/p&gt;&#xA;&lt;p&gt;So, you would start with vectors, introduce multiplication, look at the new sorts of geometrical objects you can get, and then point to this particular subset of them as a straightforward way to introduce the complex numbers. They're easily visualised geometrical objects, that you can apply your inbuilt geometric intuition to.&lt;/p&gt;&#xA;&lt;p&gt;The principal advantage of geometric algebra is the deep intuitive insights the geometric interpretation provides in a lot of otherwise quite opaque and abstract topics. It is often criticised by mathematicians as not offering anything new, that was not already well known from the theory of Clifford algebras, and they're probably right about that. (Although I've not seen the generalisations of Cauchy's theorem for complex numbers to higher dimensions elsewhere.) Physicists, on the other hand, usually love it, because it provides simple pictures of things. And one of the particular things they like most is that it means you can do physics without needing complex numbers!&lt;/p&gt;&#xA;&lt;p&gt;There are lots of introductions to geometric algebra around, although in my view most of them leap ahead too rapidly to the more advanced, abstract topics. &lt;a href=&quot;http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; is one by Lasenby, that specifically deals with complex numbers. The first third of the document is quite basic, introducing multiplication of vectors and showing how complex numbers arise, but then he suddenly accelerates and by the end is covering Maxwell's equations and advanced electrodynamics. The start demonstrates that there is no reason why you &lt;em&gt;couldn't&lt;/em&gt; write a basic textbook for highschool students. It's just that most of the books on it are aimed at professionals who already &lt;em&gt;know&lt;/em&gt; the standard vector methods, trying to persuade them to pick up a new and unfamiliar tool.&lt;/p&gt;&#xA;&lt;p&gt;(For anyone interested in what else it is capable of, &lt;a href=&quot;https://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=7943&amp;context=etd_theses&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; is a more comprehensive introduction.)&lt;/p&gt;&#xA;" OwnerUserId="22254" LastActivityDate="2023-03-31T05:24:06.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26303" PostTypeId="2" ParentId="26296" CreationDate="2023-03-31T14:51:19.897" Score="0" Body="&lt;p&gt;Couldn't leave a comment due to rep&lt;/p&gt;&#xA;&lt;p&gt;I just wanted to elaborate on:&lt;/p&gt;&#xA;&lt;h5&gt;&amp;quot;These students have never heard the words &amp;quot;union&amp;quot; or &amp;quot;intersection&amp;quot; used in math-class before.&amp;quot;&lt;/h5&gt;&#xA;&lt;p&gt;There is a strong relationship between logic and set theory. You can make use of that. The key assumption is that they have studied logic before.&lt;/p&gt;&#xA;" OwnerUserId="20626" LastActivityDate="2023-03-31T14:51:19.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26304" PostTypeId="1" AcceptedAnswerId="26307" CreationDate="2023-03-31T18:57:59.047" Score="3" ViewCount="774" Body="&lt;p&gt;Title is pretty self explanatory. All recommendations welcome. Comments and answers which reject the premise of the question will be met with eye rolling.&lt;/p&gt;&#xA;&lt;p&gt;If I don't see a good enough answer I'll have to write my own book and I'd love not to do this.&lt;/p&gt;&#xA;&lt;p&gt;Edit: I'm interested in a category theory books which mostly focus on categories and functors which come from undergrad courses (sets, groups, graphs, vector spaces, R-modules, topological spaces, cartesian spaces, the matrix cat) and those which don't really require any serious heavy lifting to consider (posets, the simplex cat, etc). The ultimate book for me would tie lots of ideas together from various courses for the student.&lt;/p&gt;&#xA;" OwnerUserId="6589" LastEditorUserId="6589" LastEditDate="2023-03-31T22:14:23.673" LastActivityDate="2023-12-10T05:35:46.830" Title="Best category theory textbook for undergraduate students" Tags="|undergraduate-education|textbooks|category-theory|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26305" PostTypeId="1" AcceptedAnswerId="26306" CreationDate="2023-03-31T20:20:44.347" Score="3" ViewCount="347" Body="&lt;p&gt;When interpreting the meaning of the derivative in real phenomena, it &lt;em&gt;may seem&lt;/em&gt; that the interpretation is in conflict with the definition of the derivative itself. The confusion is caused by the units of measurement. We measure continues quantities using discrete units. For example, every unit of measurement of time (e.g., hour, minute, second, etc.) is discrete. To see the confusion that may arise, suppose the distance at time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, of a small object moving along a straight line, measured from a conveniently chosen origin, is given by &lt;span class=&quot;math-container&quot;&gt;$f(t)=t^2+2t+1$&lt;/span&gt;, where distance is measured in meter and time is measured in second. Then &lt;span class=&quot;math-container&quot;&gt;$f^\prime(3)=8$&lt;/span&gt; is interpreted by saying that at &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; seconds after the object started to move its &lt;em&gt;instantaneous&lt;/em&gt; velocity is &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; meters per second. This may become confusing, because one &lt;em&gt;may think&lt;/em&gt; that one second (as in &lt;em&gt;per second&lt;/em&gt;) does not correctly capture the nature of the definition of &lt;span class=&quot;math-container&quot;&gt;$f'(3)$&lt;/span&gt; that is defined as:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{t\rightarrow3}\frac{f(t)-f(3)}{t-3}.$$&lt;/span&gt;&lt;br /&gt;&#xA;Additionally, since the instantaneous velocity of this object at &lt;span class=&quot;math-container&quot;&gt;$t=3$&lt;/span&gt; seconds is also &lt;span class=&quot;math-container&quot;&gt;$0.008$&lt;/span&gt; meters per millisecond, a student may ask why isn't &lt;span class=&quot;math-container&quot;&gt;$f^\prime(3)=0.008$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;So, what is an appropriate way of explaining the meaning of  &lt;span class=&quot;math-container&quot;&gt;$f^\prime(3)=8$&lt;/span&gt; in this context that avoids this confusion? Is it better to say &lt;span class=&quot;math-container&quot;&gt;$f^\prime(3)=8$&lt;/span&gt; means: if we assume that at &lt;span class=&quot;math-container&quot;&gt;$t=3$&lt;/span&gt; seconds the velocity of the object suddenly remains constant, then it will move 8 meters every second?&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-04-01T03:24:51.603" LastActivityDate="2023-04-05T17:07:14.890" Title="Interpreting the derivative as instantaneous rate of change in real phenomena" Tags="|undergraduate-education|calculus|derivative|" AnswerCount="3" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="26306" PostTypeId="2" ParentId="26305" CreationDate="2023-03-31T20:40:27.283" Score="7" Body="&lt;p&gt;Shades of Zeno:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Zeno%27s_paradoxes&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Zeno%27s_paradoxes&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And I don't think your confusion really has to do with calculus per se, but with the definition of speed (or velocity if we are using vectors).  Even in a non calculus based physics, we can look at time/distance/speed/acceleration with various algebraic formulas. Calculus enables us to understand the relation of speed and acceleration and the like, better.  But it's not required to understand speed itself.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Constant speed at less than a full time interval:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Per [thingie], does not mean you have to have at least one of them.  Like if I do a ten minute trip (to keep it simple, assume constant speed, 10 miles covered in ten minutes), does that mean I didn't have a speed expressible in miles per hour?  Can I get out of the a ticket?  Or is my speed 60 miles per hour, even though I went less than one?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Speed as a function of time (non-constant):&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Like if I drop a rock (in a vacuum), the velocity is -gt.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Free_fall#Uniform_gravitational_field_without_air_resistance&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Free_fall#Uniform_gravitational_field_without_air_resistance&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this case, the speed is not constant, is changing.  But that doesn't mean it doesn't have an actual value at certain points.&lt;/p&gt;&#xA;&lt;p&gt;Or consider driving.  Unless you are doing a long trip on the highway, there's a good chance your speed is non-constant the whole time.  But that doesn't mean there wasn't an instantaneous speed.  Just look at the dashboard.&lt;/p&gt;&#xA;" OwnerUserId="22255" LastActivityDate="2023-03-31T20:40:27.283" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26307" PostTypeId="2" ParentId="26304" CreationDate="2023-03-31T21:43:16.697" Score="5" Body="&lt;p&gt;This really depends on the group of students you are working with, including their mathematical maturity level and what their interests are.&lt;/p&gt;&#xA;&lt;p&gt;I think &amp;quot;Sets for Mathematics&amp;quot;, by Lawvere, is probably the gentlest introduction.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Category Theory in Context&amp;quot; is thorough and doesn't assume as much background knowledge as &amp;quot;Categories for the Working Mathematician&amp;quot; by Mac Lane, although I learned from that book as an undergrad myself.&lt;/p&gt;&#xA;&lt;p&gt;In another direction you could try &amp;quot;Seven Sketches in Compositionality: An Invitation to Applied Category Theory&amp;quot; by Brendan Fong and David Spivak.  Not at all systematic, but seems like it would be a really fun romp.&lt;/p&gt;&#xA;&lt;p&gt;Another completely different direction would be &amp;quot;Category Theory for Programmers&amp;quot; by Bartosz Milewski.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-03-31T21:43:16.697" CommentCount="2" CommunityOwnedDate="2023-03-31T21:43:16.697" ContentLicense="CC BY-SA 4.0" />
  <row Id="26308" PostTypeId="2" ParentId="20714" CreationDate="2023-04-01T04:01:01.923" Score="0" Body="&lt;p&gt;Too many comments, so using the answer.  Besides you write essayish questions don't you...so I can write forum style responses.  ;-)&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think some fun stuff can be good.  I would just be very occasional about sprinkling it in.  The flash of flashy motivation does not last as long as the satisfaction of gradual learning/progression.  Not saying to cut it all out, just a caution.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I find parkour a little strained as a comparison.  Maybe a lot of the kids are not into it.  Also, to me things like wrestling (kissing cousins with Gubkin's BJJ) are easier to think of as analogies because of the different moves.  Also, the idea of a toolbox or of troubleshooting are helpful (especially in things like integration or solving ODEs, where you need many tricks...many tools in your toolkit and the experience to pick different ones, even combine them.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also, and this is even much simpler, but analogies to training running, lifting, or musical instrument playing are useful in explaining to kids the importance of drill (and it is important).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's maybe not as mechanical as reading (decoding training), but there is sort of brain grooving going on in acquiring math skills (as there are with many fine motor skills).&lt;/p&gt;&#xA;&lt;p&gt;See for example: &lt;strong&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=EMAsQeLfr3o&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=EMAsQeLfr3o&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;But there are many academic articles on brain plasticity (I recommend neuroscience&amp;gt;applied psych&amp;gt;edschool researchers (lot of fluff from them, even trying to justify repudiated learning models).) But the basic idea is that our brains evolved to have a &amp;quot;walking mode&amp;quot; and a &amp;quot;talking mode&amp;quot; and a &amp;quot;recognizing faces&amp;quot; mode.  Any child will learn these readily WITHOUT optimized training.  However things like reading or doing math or typing or playing piano or tennis or doing double back flips requires detailed training and rewiring of the plastic parts of the brain to accommodate them. Many good articles, just Google Science them.&lt;/p&gt;&#xA;&lt;p&gt;The one thing I would caution is that different parts of the brain may be used for different modifications.  We know that both reading and math are &amp;quot;unnatural&amp;quot; (not evolved for) and that they hijack parts of the oral language centers.  But it's actually different areas within that.  So, while brain plasticity (new skill acquisition) is ongoing, it does not have to be for the same part of the brain for parkour and math (I would bet not).  It's just more of a general feature that we have some plasticity.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This is not anything against parkour...I was a mediocre college gymnast and used to do parkour type stuff before it was a thing.  Nothing like what you see in videos, but still...we would set up strange obstacle courses and the like and do goofy stuff.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P.s.  &amp;quot;La meme chose&amp;quot; &lt;strong&gt;&lt;a href=&quot;https://www.wimp.com/early-20th-century-actor-is-the-original-parkour-star/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.wimp.com/early-20th-century-actor-is-the-original-parkour-star/&lt;/a&gt;&lt;/strong&gt;  (not like you whippersnappers were the first!)&lt;/p&gt;&#xA;" OwnerUserId="22255" LastEditorUserId="22255" LastEditDate="2023-04-01T04:23:43.010" LastActivityDate="2023-04-01T04:23:43.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26309" PostTypeId="2" ParentId="26304" CreationDate="2023-04-01T04:36:15.160" Score="3" Body="&lt;p&gt;Leinster's Basic Category Theory is quite appropros. It is slightly shorter and slightly easier than Riehls' CTIC and draws examples primarily from a first course in  topology, algebra, and order theory.&lt;/p&gt;&#xA;&lt;p&gt;Awodey has a category theory textbook, extremely accessible with very little background, maybe inteded for first or second year. This is, I believe, still more apropros than Milewski's, which, while interesting, is more from programmers than mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="20518" LastActivityDate="2023-04-01T04:36:15.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26310" PostTypeId="2" ParentId="11307" CreationDate="2023-04-01T04:59:35.513" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I would be very cautious about the idea that math training (strongly) enables general problem solving.  If anything, I think training in rhetoric may be more generally useful across fields.  See for example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://classicalchristian.org/the-lost-tools-of-learning-dorothy-sayers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://classicalchristian.org/the-lost-tools-of-learning-dorothy-sayers/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But I'm not advocating Sayers per se (although she was a right sharp lady), because I think in general &amp;quot;general problem solving&amp;quot; is oversold as a skill.  Most problem solving is domain specific.  We are like Ugg the caveman teaching his son how to make the best shaped stone tools...really we are...and even this is a marvelous invention...allows the formation and transfer of culture and technology.&lt;/p&gt;&#xA;&lt;p&gt;What training in math mostly teaches is training in math.  And nothing wrong with that!  There are a lot of important things to be done.  Difficult things.  Non trivial things.  And a lot of balls not getting across the goal line (students not learning minimum standards) that we should concentrate on correcting.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would be cautious about asserting that math is better performed in a foreign language (is that even your point...kind of a word salad to weed through, but trying to find/address your main idea that you want contested.  At a minimum, I would want solid, repeated, multiple controls study done to show it.  And burden of proof on the proposition, not on the contrary.  After all math is hard enough, let alone doing it in another language!  All that said my sister did a year abroad (as an English speaker to German region) and math was her easiest subject (aside from English as a foreign language!)  Next hardest was science.  Then history. Then German.  Then...the absolute hardest was trying to learn Latin IN German.  Kind of pretty much what you would expect, I guess.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't believe Chomsky natural language is applicable to math (or reading).  Both of these are functions that are very recent in our evolutionary history.  Oral language on the other hand is not...and has significant dedicated brain structure (and is easily learned by children).  To the extent the Chomsky stuff is useful (and I'm pretty positive on it, but not an expert), I think that is in the context of oral language.  I would be very hesitant to extend oral language models to other domains...that is the mistake the whole language reading advocates made and there have been study after study showing several reasons why they were mistaken...to the extent people have run from the term.  Definitely Dewed-eyed dreaming there!  The more we learn about rigorous applied psych testing (or even brain imaging), the more this Rousseau-ian stuff gets crushed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="22255" LastActivityDate="2023-04-01T04:59:35.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26311" PostTypeId="2" ParentId="26305" CreationDate="2023-04-01T13:35:23.487" Score="5" Body="&lt;h3&gt;Chunky vs. smooth&lt;/h3&gt;&#xA;&lt;p&gt;In mathematics education literature, the concept of an instantaneous rate of change has been described in terms of &amp;quot;chunky images of change&amp;quot; vs. &amp;quot;smooth images of change,&amp;quot; borrowing the language of &lt;a href=&quot;https://flm-journal.org/Articles/11A85FC0E320DF94C00F28F2595EAF.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Castillo-Garsow et al.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Chunky images of change involve imagining change as occurring in completed chunks. Two features characterize chunky images of change: a unit chunk whose repetition makes up the variation, and the lack of an image of variation within the unit chunk. Therefore, ongoing change is generated by a sequence of equal-sized chunks, and this makes measuring change essentially about counting how many chunks have occurred.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As you state, students may interpret &amp;quot;8 meters per second&amp;quot; to mean that the smallest unit of variation is one second. This is a chunky image of change. In contrast,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Smooth images of change involve imagining a change in progress. Ongoing change is generated by conceptualizing a variable as always taking on values in the continuous, experiential flow of time.... Smooth images of change are not the same as chunky images of change cut up really small. Smooth images of change involve an entirely different conceptualization of variation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this paradigm, there is no unit of variation at all. An object moves continuously, and at each instant, it has a speed. One way to help students envision this is, &lt;a href=&quot;https://matheducators.stackexchange.com/a/26306/20719&quot;&gt;as @guest philsopher also suggested&lt;/a&gt;, to think about the speedometer of a car. As the car accelerates, the speedometer measures the instantaneous speed of the car as it varies over time. &amp;quot;The instantaneous speed of the car is 8 meters per second&amp;quot; means that the speedometer is pointing at 8 m/s (or 28.8 km/h or 17.9 mph.)&lt;/p&gt;&#xA;&lt;h3&gt;Real phenomena and calculus&lt;/h3&gt;&#xA;&lt;p&gt;Here's a metaphor that the authors use to explain how images of change relate to &amp;quot;reality.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;A child throwing a ball through a paper wall sees a continuous motion and knows that the ball collides with the wall. A traditional animator animating a ball being thrown through a paper wall draws frames, and it's possible that the ball never overlaps with the wall in any frame.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now, imagine the child watching the animated movie. Drawing from her own physical experience, the child might imagine the ball colliding with the wall. However in the reality of the animator, the ball never collided with the wall, because none of the frames he drew show the ball in contact with the wall. This difference in the child’s reality (relating the ball to physical experience and perceiving the movie as continuous motion in progress) and the animator’s reality (individual frames in which there is no motion) is analogous to the difference between smooth and chunky images of change.&lt;/p&gt;&#xA;&lt;p&gt;We also note that the “true” nature of the reality itself does not matter. The child uses smooth thinking in both the real ball and animated ball cases by imagining the ball in continuous motion. Similarly, the animator, who imagines the animated ball as individual frames, could also imagine the real ball in frames by imagining change at the molecular, atomic, or quantum level. Images of change are conceptualizations and are thus in the mind of the beholder.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The physical world is full of quantities which students perceive as varying smoothly. This experiential knowledge is a basis for the mathematical concept of instantaneous rate of change. Accordingly, Castillo-Garsow et al. give this suggestion for teaching calculus.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was only with the advent of modern analysis that a fully chunky (epsilon-delta) calculus was possible—a calculus in which limits are characterized in measurable intervals (chunks) without necessitating an appeal to the non-chunky idea of “infinitesimal” or ideas of smooth motion such as “approaching” in proof. Still, during the development of this calculus, mathematicians relied on smooth thinking to develop intuitions and conjectures before writing the formal, chunky proofs. Because we ask students to reason about change and rate long before they study analysis, it seems reasonable that beginning with smooth images of change could create the foundations for students’ consideration of the difficult-to-learn concepts of limit, rate of change/differentiation, and accumulation/integration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h3&gt;Units and functions&lt;/h3&gt;&#xA;&lt;p&gt;What to do if a student asks &amp;quot;Why isn't &lt;span class=&quot;math-container&quot;&gt;$f'\!(3)=0.008?$&lt;/span&gt;&amp;quot; is an entirely different question that could merit an entire article. My answer is that when we write&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(t)=t^2+2t+1,$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;we've omitted the units of the coefficients, obscuring the unit-dependence of the equation. To make the unit-dependence explicit, we can write&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x=(1\,\text{m}/\text{s}^2)\cdot t^2 + (2\,\text{m}/\text{s})\cdot t+1\,\text{m}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;By changing the units of the coefficients, we can see that this is equivalent to&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x=(10^{-6}\,\text{m}/\text{ms}^2)\cdot t^2+(2\cdot10^{-3}\,\text{m}/\text{ms})\cdot t+1\,\text{m}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, we can calculate&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\left.\frac{dx}{dt}\right|_{t=3\,\text{s}} &amp;amp;= 2\cdot(10^{-6}\,\text{m}/\text{ms}^2)\cdot (3\,\text{s})+(2\cdot10^{-3}\,\text{m}/\text{ms})\\&#xA;&amp;amp;= 2\cdot(10^{-6}\,\text{m}/\text{ms}^2)\cdot (3\cdot10^3\,\text{ms})+(2\cdot10^{-3}\,\text{m}/\text{ms})\\&#xA;&amp;amp;= 0.008\,\text{m/ms}.&#xA;  \end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The velocity of the object at &lt;span class=&quot;math-container&quot;&gt;$t=3\,\text{s}$&lt;/span&gt; is indeed &lt;span class=&quot;math-container&quot;&gt;$0.008\, \text{m/ms}$&lt;/span&gt;. Unfortunately, this does not mean that &lt;span class=&quot;math-container&quot;&gt;$f'\!(3)=0.008$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Note that, interpreted as &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}\to\mathbb{R}$&lt;/span&gt; functions, the two expressions for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; are not equal. We have&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;f(t)&amp;amp;=t^2+2t+1,\\&#xA;g(t)&amp;amp;=0.000001t^2+0.002t+1,&#xA;\end{align*}&lt;/span&gt;&#xA;and using function notation, the student's idea would have to be written as &lt;span class=&quot;math-container&quot;&gt;$g'\!(3000)=0.008$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thus we can see that &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}\to\mathbb{R}$&lt;/span&gt; functions and modern function notation fail to capture how functions are often treated in the sciences: quantities that vary in response to other quantities, which is &lt;a href=&quot;https://mathoverflow.net/q/307947&quot;&gt;the original meaning of &lt;em&gt;function&lt;/em&gt;&lt;/a&gt;. Using Leibniz notation, &amp;quot;the velocity at time &lt;span class=&quot;math-container&quot;&gt;$t=t_0\!$&lt;/span&gt;&amp;quot; can be represented independent of units as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left.\frac{dx}{dt}\right|_{t=t_0}$$&lt;/span&gt;&#xA;However, if we use our two functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f'\!(3)$&lt;/span&gt; is the velocity &lt;strong&gt;in meters per second&lt;/strong&gt; at &lt;span class=&quot;math-container&quot;&gt;$t=3$&lt;/span&gt; &lt;strong&gt;seconds&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$g'\!(3)$&lt;/span&gt; is the velocity &lt;strong&gt;in meters per millisecond&lt;/strong&gt; at &lt;span class=&quot;math-container&quot;&gt;$t=3$&lt;/span&gt; &lt;strong&gt;milliseconds&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$g'\!(3000)$&lt;/span&gt; is the velocity &lt;strong&gt;in meters per millisecond&lt;/strong&gt; at &lt;span class=&quot;math-container&quot;&gt;$t=3$&lt;/span&gt; &lt;strong&gt;seconds&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This is why many equations in physics are written with variables and parameters instead, i.e.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x=at^2+v_0t+x_0.$$&lt;/span&gt;&#xA;&lt;a href=&quot;https://mathoverflow.net/questions/307931/has-incorrect-notation-ever-led-to-a-mistaken-proof/307990#307990&quot;&gt;See also this post and discussion over on MathOverflow.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-04-03T12:51:08.603" LastActivityDate="2023-04-03T12:51:08.603" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26312" PostTypeId="1" CreationDate="2023-04-02T05:11:06.240" Score="2" ViewCount="549" Body="&lt;p&gt;(Updated)&lt;/p&gt;&#xA;&lt;p&gt;An example of a logical (material) implication that is commonly  used  is: &amp;quot;If it is raining outside, then the ground is wet.&amp;quot; The problem with this example is that it could be seen as a &lt;em&gt;causal&lt;/em&gt; relationship, i.e. that rain &lt;em&gt;cause&lt;/em&gt;s the ground to be wet, possibly leading to confusion conflating implication and causality--a common mistake among beginners.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt; Might a better example not be: &amp;quot;If it is raining, then it is cloudy?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I think (hope?) that it is common knowledge that rain does not &lt;em&gt;cause&lt;/em&gt; cloudiness. This alternative example means only that, at the moment, it is not both raining and not cloudy.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$~~~~~~~~Raining \to Cloudy ~~\equiv~~ \neg [Raining \land \neg Cloudy]$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;More generally, for any logical propositions &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$~~~~~~~~A \to B ~~ \equiv~~ \neg [ A ~\land \neg B] $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This interpretation can be easily seen to be consistent with the usual truth table for logical implication:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/owNp0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/owNp0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The rule of detachment (&lt;em&gt;modus ponens&lt;/em&gt;) can also be easily seen to hold by this interpretation. When both &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A\to B$&lt;/span&gt; are true (only on line 1), then &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; must also be true.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;From my comment below: Care must be taken to avoid examples of relations that may be construed as causal. Ideally, both antecedent and consequent should be stated in the present tense.&lt;/p&gt;&#xA;&lt;p&gt;Mathematical statements, e.g. &lt;em&gt;y&lt;/em&gt;=&lt;em&gt;x&lt;/em&gt;+1, should be thought of as being stated in the &lt;em&gt;present tense&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-04-07T16:39:36.750" LastActivityDate="2023-04-12T07:51:20.787" Title="A better example of a logical implication" Tags="|mathematical-pedagogy|logic|" AnswerCount="4" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="26313" PostTypeId="2" ParentId="26312" CreationDate="2023-04-03T10:58:15.953" Score="2" Body="&lt;p&gt;I personally think all of these real world examples do a pretty poor job of helping students form an understanding of implication which is both accurate and useful in forming arguments.&lt;/p&gt;&#xA;&lt;p&gt;For instance,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;False implies False &amp;quot;If the moon is made of cheese, then I am the Pope&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;False implies True &amp;quot;If the moon is made of cheese, then the moon is not made of cheese&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;True implies True &amp;quot;If 1 + 1 = 2, then in nineteen ninety eight the Undertaker threw Mankind off hеll in a cell and he plummeted sixteen feet through an announcer's table.&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;These all highlight the fundamental lack of &amp;quot;relevance&amp;quot; between the  antecedent and consequent of an implication.  There have been attempts to fix this such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Relevance_logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Relevance_logic&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Since these kinds of examples are just as &amp;quot;valid&amp;quot; as the more &amp;quot;sensible&amp;quot; ones you are coming up with using rain and clouds, they must also be accepted by the student for them to have a truly robust model of material implication.&lt;/p&gt;&#xA;&lt;p&gt;My own hot take is to introduce introduction and elimination rules without mentioning the truth table.  These rules are motivated by the following interpretation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Interpretation: The implication &lt;span class=&quot;math-container&quot;&gt;$p \implies q$&lt;/span&gt; is true exactly when given a proof of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, we can produce a proof of &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Introduction rule:  to argue that &lt;span class=&quot;math-container&quot;&gt;$p \implies q$&lt;/span&gt; is true assume we have a proof of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and argue &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; relative to that assumption.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Elimination rule:  A proof of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and a proof of &lt;span class=&quot;math-container&quot;&gt;$p \implies q$&lt;/span&gt; together constitute a proof of &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These are the things which are useful for actually doing and using mathematics.  The truth table is really not.  It is a distraction.&lt;/p&gt;&#xA;&lt;p&gt;I would focus on using and proving sensible mathematical implications for a long time before discussing truth tables.  Things like &amp;quot;For all integers x, if 15 divides x then 5 divides x&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I would also &lt;strong&gt;define&lt;/strong&gt; absurdity by &lt;span class=&quot;math-container&quot;&gt;$\bot \implies p$&lt;/span&gt; for all statements &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.  This definitely requires some motivation (such as &lt;span class=&quot;math-container&quot;&gt;$1=0$&lt;/span&gt; implying both other absurd statements like &lt;span class=&quot;math-container&quot;&gt;$2=0$&lt;/span&gt; as well as true statements like &lt;span class=&quot;math-container&quot;&gt;$0=0$&lt;/span&gt;:  the proof of the first is to multiply both sides by 2 and the second by 0).&lt;/p&gt;&#xA;&lt;p&gt;You can also mention that this definition of absurdity permits a &lt;em&gt;uniform&lt;/em&gt; treatment of the proof of quantified implications.  For instance in the proof of &amp;quot;For all integers x, if 15 divides x then 5 divides x&amp;quot; we take an arbitrary integer &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and then argue that &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; under the &lt;em&gt;assumption&lt;/em&gt; that we have a proof of &lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt; dividing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  If &lt;span class=&quot;math-container&quot;&gt;$x  = 45$&lt;/span&gt; this proof will do a beautiful job of showing exactly how to take the proof that 15 divides 45 (namely demonstrating that for &lt;span class=&quot;math-container&quot;&gt;$k=3$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$45 = 15k$&lt;/span&gt;) and turning it into a proof that &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (namely &lt;span class=&quot;math-container&quot;&gt;$ j= 3k = 9$&lt;/span&gt; will work to show that &lt;span class=&quot;math-container&quot;&gt;$45 = 5j$&lt;/span&gt;).  However if &lt;span class=&quot;math-container&quot;&gt;$x = 17$&lt;/span&gt;, and we still want &amp;quot;the same argument to work&amp;quot; we need to accept that a false premise can lead to the conclusion we desire.&lt;/p&gt;&#xA;&lt;p&gt;After getting a lot of experience proving honest mathematical theorems using the introduction and elimination rules I would introduce truth tables as a kind of &amp;quot;algebraic shadow&amp;quot; of our reasoning processes.  It is an arithmetic of truth, but does not fully capture the subtleties.&lt;/p&gt;&#xA;&lt;p&gt;Then the rows of the truth table become fairly intuititive:  &lt;span class=&quot;math-container&quot;&gt;$\bot$&lt;/span&gt; implies everything, if a statement is &amp;quot;true&amp;quot; then it ``proves itself'', so a valid proof of &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; starting from an assumption of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is to just state &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;, and it should be evident that there is no valid way to argue a false conclusion from a true premise.&lt;/p&gt;&#xA;&lt;p&gt;I also think that relying on excluded middle as heavily as you do in your proof is a mistake.  I try to avoid excluded middle whenever possible, and I would certainly not want to wed it to my fundamental understanding and motivation for implication.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2023-04-03T12:48:11.600" LastActivityDate="2023-04-03T12:48:11.600" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26314" PostTypeId="2" ParentId="26305" CreationDate="2023-04-03T11:22:06.977" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;So, what is an appropriate way of explaining the meaning of &lt;span class=&quot;math-container&quot;&gt;$f′(3)=8$&lt;/span&gt; in this context that avoids this confusion?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think this confusion can't be and &lt;em&gt;should not&lt;/em&gt; be avoided, because it's at the very base of the concept of the derivative.&lt;/p&gt;&#xA;&lt;p&gt;Feynman elaborates on this point in his &amp;quot;Lectures on Physics&amp;quot; in the chapter about motion, which is freely available here&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.feynmanlectures.caltech.edu/I_08.html#Ch8-S2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.feynmanlectures.caltech.edu/I_08.html#Ch8-S2&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;p&gt;Main point:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Calculus was invented in order to describe motion, and its first application was to the problem of defining what is meant by going “60 miles an hour.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="667" LastActivityDate="2023-04-03T11:22:06.977" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26315" PostTypeId="2" ParentId="26304" CreationDate="2023-04-03T19:06:09.593" Score="5" Body="&lt;p&gt;Eugenia Cheng's new book &lt;a href=&quot;https://www.cambridge.org/core/books/joy-of-abstraction/00D9AFD3046A406CB85D1AFF5450E657&quot; rel=&quot;noreferrer&quot;&gt;The Joy of Abstraction&lt;/a&gt; came out not too long ago.  It is an introduction to category theory, intended to bridge the gap between little-to-no undergraduate mathematics and the standard undergraduate category theory texts by Lawvere and Schanuel, Riehl, Leinster, etc.&lt;/p&gt;&#xA;&lt;p&gt;At this moment &lt;a href=&quot;https://www.cambridge.org/core/services/aop-cambridge-core/content/view/0BCFB55AC4687FD038B4237A034BFA0A/9781108477222pro_1-10.pdf/prologue.pdf&quot; rel=&quot;noreferrer&quot;&gt;a PDF of the prologue&lt;/a&gt; for Cheng's book is available for free on the publisher's website, and might help you decide if the goals and intended audience of the book will align with your goals.  It could also be that using this book in combination with one of the other standard undergraduate texts (as recommended by others) will meet your needs.&lt;/p&gt;&#xA;" OwnerUserId="12605" LastActivityDate="2023-04-03T19:06:09.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26316" PostTypeId="2" ParentId="26304" CreationDate="2023-04-03T22:44:08.580" Score="2" Body="&lt;p&gt;Don't overlook &lt;a href=&quot;https://ncatlab.org/nlab/show/category+theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;Lab&lt;/a&gt;. It is an excellent open resource for this, and includes a good list of textbooks.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-04-03T22:44:08.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26317" PostTypeId="1" CreationDate="2023-04-04T00:58:59.240" Score="4" ViewCount="168" Body="&lt;p&gt;Are there any studies that have investigated this question?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Why certain pedagogical practices that used to be effective up to a few years ago, may suddenly become less or even no longer effective?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am specifically interested in this question in the field of mathematics education, but I am also interested in such studies in a broader context.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-04-04T03:30:04.397" LastActivityDate="2023-04-04T03:30:04.397" Title="Studies on the change in effectiveness of pedagogical practices over time" Tags="|mathematical-pedagogy|reference-request|general-pedagogy|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26318" PostTypeId="2" ParentId="26312" CreationDate="2023-04-05T14:25:02.850" Score="4" Body="&lt;p&gt;My answer is a bit of a frame challenge to the question (and to the part of &lt;a href=&quot;https://matheducators.stackexchange.com/a/26313/16250&quot;&gt;Stephen Gubkin's answer&lt;/a&gt; where real world examples are said to be poorly suited for teaching purposes).&lt;/p&gt;&#xA;&lt;p&gt;I agree that &lt;em&gt;&amp;quot;If it is raining, then it is cloudy&amp;quot;&lt;/em&gt; might be advantageous since, in contrast to &lt;em&gt;&amp;quot;If it is raining outside, then the ground is wet&amp;quot;&lt;/em&gt;, it does not suggest any relation to causality.&lt;/p&gt;&#xA;&lt;p&gt;However, I think both examples are actually formulated in a way that makes it difficult to understand the point of implications. For me the problem, though, is not that they are &amp;quot;real-world&amp;quot; examples, but something else: the wording does not make it clear that the involved propositions in both implications depend on a free variable - but this is actually the point why implications are so useful in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Here's what I mean by this in detail:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Implication between propositions without variables and with known truth values&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;An implication between two propositions whose truth values are fixed and known appears to be intuitively odd, no matter whether in mathematics or in real life.&#xA;Consider the following examples, the first of which I copied from Stephen Gubkin's answer:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If the moon is made of cheese, then I am the Pope.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If the moon is not made of cheese, then water is wet.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$5 = 11-10$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$3 = 2+9$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$5 \not= 11-10$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$2+2=4$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I've been working in mathematics for about 10 years now and still don't have any good intuition for any of those four examples. Actually, I find the last two just as non-telling as the first two.&#xA;The intuitive problem with all four implications is that the truth values of their premises and conclusions are fixed and known - they do not depend on any free variable.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Propositions that depend on a variable:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Implications make much more sense intuitively if one uses them for propositions that depend on a variable and whose truth values can change depending on that variable.&#xA;Examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If an integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is divisible by &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, then it is also divisible by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If a function &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{R} \to \mathbb{R}$&lt;/span&gt; is differentiable, then it is continuous.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; is a subgroup of a finite group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, then the cardinality of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; divides the cardinality of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If a Banach space &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is infinite-dimensional, then every algebraic basis of &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is uncountable.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The same works for &amp;quot;real-life&amp;quot; examples: as soon as they depend on a variable which might make the involved proposition true or false, things get much more intuitive:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If a planet supports the evolution of Carbon-based life, then there exists liquid water on it.&lt;/p&gt;&#xA;&lt;p&gt;Or, if you want to make the variable more explicit:&lt;/p&gt;&#xA;&lt;p&gt;If a planet &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; supports the evolution of Carbon-based life, then there exists liquid water on &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If it's raining on a street &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; gets wet.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If it's raining at a place &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, then there are clouds over &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The last two example are the ones from the question on which I commented at the beginning of this answer.&#xA;The way I worded them now is, of course more technical - but I think this is actually an advantage for the following reasons:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It makes it very clear and explicit that there is, in a sense, &amp;quot;uncertain knowledge&amp;quot; involved in the implication and that the uncertainty stems from the fact that we do not know the value of the variable.&#xA;This illustrates the very point of a (true) implication: while there might be uncertainty about the truth values of the premise and the conclusion, the implication tells us that at least a certain relation between them is always true.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In such an example one can easily illustrate all four combinations of truth values of the premise and the conclusion by inserting different values for the variable.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Implications between propositions that depend on a variable is actually how implications occur in mathematical results, so those examples are a good way to prepare the students for this.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The transition from everyday language to more formal logical expressions can be smoothened this way:&#xA;We may start with the sentence &amp;quot;When it rains, it is cloudy&amp;quot;, then let the students notice that there is an implicit assumption in this sentence that we talk about a specific location (since the whether is not the same everywhere), and then make this assumption explicit by denoting the location by a variable.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="16250" LastEditorUserId="16250" LastEditDate="2023-04-05T19:59:04.613" LastActivityDate="2023-04-05T19:59:04.613" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="26320" PostTypeId="2" ParentId="26312" CreationDate="2023-04-05T19:10:45.633" Score="0" Body="&lt;p&gt;I understand your question to be about the efficacy of replacing the language &amp;quot;A implies B&amp;quot; with &amp;quot;not A or B.&amp;quot; In other words, is there benefit in teaching to avoiding &lt;span class=&quot;math-container&quot;&gt;$A\rightarrow B$&lt;/span&gt; and reflexively converting to the conjunctive normal form &lt;span class=&quot;math-container&quot;&gt;$\neg A\vee B$&lt;/span&gt;? The reasoning goes that &amp;quot;implication&amp;quot; carries too much experiential baggage by intelligent agents such as students, and humans get mired in &amp;quot;causation versus correlation&amp;quot; and other matters irrelevant to logic.&lt;/p&gt;&#xA;&lt;p&gt;I am trained as a mathematician, and only recently learned how people in artificial intelligence make fundamental use of what they call &amp;quot;proof by resolution&amp;quot; in the context of logical agents. See, for example, chapter 7 of &lt;a href=&quot;https://en.wikipedia.org/wiki/Artificial_Intelligence:_A_Modern_Approach&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Artificial Intelligence: A Modern Approach&lt;/em&gt;&lt;/a&gt;, fourth edition by Russell and Norvig. They start with a knowledge base &lt;span class=&quot;math-container&quot;&gt;$KB$&lt;/span&gt;, tantamount to a set of axioms in propositional logic or first-order logic, and convert everything to conjunctive normal form. Then to decide if a statement &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is entailed by &lt;span class=&quot;math-container&quot;&gt;$KB$&lt;/span&gt;, in symbols &lt;span class=&quot;math-container&quot;&gt;$KB\models\alpha$&lt;/span&gt;, one adds &lt;span class=&quot;math-container&quot;&gt;$\neg\alpha$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$KB$&lt;/span&gt; and applies proof by resolution, which amounts to nothing more than algebraic cancellations. For example, if &lt;span class=&quot;math-container&quot;&gt;$KB=\lbrace A,A\rightarrow B\rbrace$&lt;/span&gt;, the conjunctive normal form is &lt;span class=&quot;math-container&quot;&gt;$\lbrace A,\neg A\vee B\rbrace$&lt;/span&gt;. If we want to decide if &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is entailed by &lt;span class=&quot;math-container&quot;&gt;$KB$&lt;/span&gt;, we work with &lt;span class=&quot;math-container&quot;&gt;$\lbrace A,\neg A\vee B,\neg B\rbrace$&lt;/span&gt;.  The we crank the gears of proof by resolution, which in this simple-minded case amounts to &amp;quot;algebraically&amp;quot; canceling &lt;span class=&quot;math-container&quot;&gt;$A,\neg A$&lt;/span&gt;, and then in the next step algebraically cancelling &lt;span class=&quot;math-container&quot;&gt;$B,\neg B$&lt;/span&gt; leaving &amp;quot;absurdity&amp;quot; or &lt;span class=&quot;math-container&quot;&gt;$\bot$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The perspective of working exclusively with conjunctive normal forms, setting up a query &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; as a proof by contradiction (add &lt;span class=&quot;math-container&quot;&gt;$\neg \alpha$&lt;/span&gt;), and then going through algebraic cancellations of appearances of &lt;span class=&quot;math-container&quot;&gt;$x,\neg x$&lt;/span&gt; is something that is amenable to artificially intelligent agents, who probably have no experiential baggage related to the implication, causation vs correlation, etc. A student is somewhere between an artificially intelligent agent and the &amp;quot;expert system&amp;quot; human teacher.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-04-06T12:39:28.553" LastActivityDate="2023-04-06T12:39:28.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26321" PostTypeId="1" AcceptedAnswerId="26323" CreationDate="2023-04-06T01:53:37.713" Score="8" ViewCount="2035" Body="&lt;p&gt;I am currently teaching a linear algebra course at a university and have chosen to assess my students using five quizzes throughout the semester, instead of assigning homework. I have encountered a situation where some students may not be able to attend a quiz due to personal reasons, such as internships.&lt;/p&gt;&#xA;&lt;p&gt;I am considering implementing a policy that allows students to skip one quiz (or drop their lowest quiz score) during the semester. I would like to gather your thoughts and advice on this approach. Here are some points I have considered:&lt;/p&gt;&#xA;&lt;p&gt;Pros:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Flexibility: The policy could accommodate unexpected personal circumstances without overly penalizing students for missing an assessment.&lt;/li&gt;&#xA;&lt;li&gt;Reduced anxiety: Students might feel less pressured, knowing they have a &amp;quot;safety net&amp;quot; in case they cannot attend a quiz.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Cons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Reduced motivation: The policy might inadvertently encourage some students to skip a quiz, leading to less overall engagement in the course.&lt;/li&gt;&#xA;&lt;li&gt;Fairness: Students who consistently attend all quizzes might perceive the policy as unfair.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Alternatively, I have thought about offering make-up quizzes or alternative assessments for students who have legitimate reasons for missing a quiz. But this involves a lot of extra work for just a handful students.&lt;/p&gt;&#xA;&lt;p&gt;I would greatly appreciate any advice or suggestions you may have based on your own experiences and teaching practices.&lt;/p&gt;&#xA;" OwnerDisplayName="user19945" LastEditorUserId="22264" LastEditDate="2023-04-07T22:26:54.947" LastActivityDate="2023-04-07T22:26:54.947" Title="Seeking Advice on Allowing Students to Skip a Quiz in Linear Algebra Course" Tags="|undergraduate-education|assessment|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26322" PostTypeId="2" ParentId="26321" CreationDate="2023-04-06T02:28:31.857" Score="1" Body="&lt;p&gt;My suggestion: I wouldn't drop any quizzes (why drop?) Instead, I would give &lt;em&gt;all&lt;/em&gt; students the option of attempting each quiz multiple times. If a student misses a quiz, or even if they don't miss it, but don't do well on a quiz, I would give them an Incomplete on that quiz, which means the student has to come to your office hour at some point before the semester ends and complete that quiz. Of course, the new quiz will have different questions. A little bit more work for you, but this will increase students' motivation, is fair to everyone and solves the issue of missing a quiz. To reduce the amount of work, when a student comes to your office to take a quiz, I would just hand-write the quiz questions and give it to them to solve right there, no need to prepare and type quiz questions in advance.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-04-06T02:34:28.890" LastActivityDate="2023-04-06T02:34:28.890" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26323" PostTypeId="2" ParentId="26321" CreationDate="2023-04-06T03:15:50.383" Score="12" Body="&lt;p&gt;I do recommend dropping a certain number of quizzes to handle cases like this.&lt;/p&gt;&#xA;&lt;p&gt;You don't want a policy that makes more demands on your time and labor.&lt;/p&gt;&#xA;&lt;p&gt;This is common enough to be familiar to students. When a student has a problem I remind them of the drop policy, and they always accept that as a good resolution.&lt;/p&gt;&#xA;&lt;p&gt;This feature is built into many online learning management systems (e.g., Blackboard).&lt;/p&gt;&#xA;&lt;p&gt;I've never seen reduced motivation or received any complaints from students about this policy.&lt;/p&gt;&#xA;&lt;p&gt;This is what I've done for many years, and it's the best solution I've found or heard about; there's practically no downside to it.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-04-06T03:15:50.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26324" PostTypeId="2" ParentId="26321" CreationDate="2023-04-06T18:57:51.607" Score="1" Body="&lt;p&gt;Here is a suggestion. Allow the students to drop one quiz, but offer a small bonus for those taking all quizzes. Or, even without offering an explicit bonus, make the students note that &amp;quot;four out of five&amp;quot; is better than &amp;quot;four out of four&amp;quot;, so there is a natural bonus in taking all quizzes.&lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2023-04-06T18:57:51.607" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26326" PostTypeId="2" ParentId="26321" CreationDate="2023-04-07T18:11:21.307" Score="6" Body="&lt;p&gt;My experience: I've used several times the method of having n quizzes but grading according to the average of the n-1 best ones. The consequences:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;That's a lot easier to manage than having special rules for justified absences - at least, you don't need to worry about whether absences are justified or not.&lt;/li&gt;&#xA;&lt;li&gt;Some students may drop the last quiz or do very little work on it if they know that they have already passed the quiz part of the course. If the quiz is useful homework and its topic is going to be in the test, students may be more motivated to take all quizzes.&lt;/li&gt;&#xA;&lt;li&gt;Some students will still complain that not being able to take all quizzes due to a justified cause puts them at disadvantage compared with their peers that can do all quizzes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7178" LastActivityDate="2023-04-07T18:11:21.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26329" PostTypeId="1" CreationDate="2023-04-11T20:53:17.197" Score="3" ViewCount="150" Body="&lt;p&gt;What is the best didactic and number sense-promoting method for reading decimal numbers?&lt;/p&gt;&#xA;&lt;p&gt;For instance, is it best to teach students to read the number 3.14 as 'three and fourteen hundredths' instead of 'three, point, one, four'? I am looking for studies and references about this matter&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="98" LastEditDate="2023-04-11T22:29:47.510" LastActivityDate="2023-04-12T04:03:23.880" Title="What is the best didactical way to read decimal numbers?" Tags="|real-numbers|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26330" PostTypeId="2" ParentId="26329" CreationDate="2023-04-12T04:03:23.880" Score="4" Body="&lt;p&gt;One reference is James Hiebert's chapter on &lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=Vyl42R9JV1oC&amp;amp;oi=fnd&amp;amp;pg=PA283&amp;amp;ots=pioO8y8SXU&amp;amp;sig=me_EGp_6RJZbQOPxsia3tHGLzy8#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Mathematical, Cognitive, and Instructional Analyses of Decimal Fractions&amp;quot;&lt;/a&gt; in the book &lt;em&gt;Analysis of Arithmetic for Mathematics Teaching.&lt;/em&gt; He doesn't ever address the digit by digit reading of decimals (&amp;quot;three point one four&amp;quot;), but he does say that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Perhaps the most important conclusion that can be derived from reviewing previous research and ana­lyzing current textbooks is that instruction must spend more time and attention helping students establish rich meanings for decimal symbols.&lt;/p&gt;&#xA;&lt;p&gt;A first useful activity, in third or fourth grade, involves base-10 blocks where the large block is assigned the unit. The class then can discuss what a tenth would look like, with the flat becoming a tenth. (If base-10 blocks are not available, a square cardboard piece can be the unit with strips representing tenths.) A day of activities would include building concrete representations of spoken decimals (e.g., “two and seven tenths”) and verbalizing the number represented with concrete materials (e.g., 1 large block and 3 flats)....&lt;/p&gt;&#xA;&lt;p&gt;After numerous oral activities, the written notation can be introduced. Activities then can include building concrete representations for numerals and writing numerals for concrete representations. It is only at this point that the textbook activities will have some meaning for the students....&lt;/p&gt;&#xA;&lt;p&gt;After several days with tenths, and before beginning addition or subtrac­tion, hundredths can be introduced with the concrete materials as both tenths of tenths and hundredths of the unit. The same sequence of ac­tivities can be followed as with tenths: (a) using verbalizations and con­crete representations only, (b) changing the unit block, and (c) eventually introducing the written symbol and showing connections between con­crete and written representations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This supports introducing decimals through verbal representations like &amp;quot;three and fourteen hundredths.&amp;quot; However, as &lt;a href=&quot;https://matheducators.stackexchange.com/questions/26329/what-is-the-best-didactical-way-to-read-decimal-numbers#comment69621_26329&quot;&gt;Sue&lt;/a&gt; already pointed out,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[A] contrast in whole number and decimal language arises from the fact that there are a variety of acceptable ways of saying decimal numbers. For example, there are two common ways of saying 2.38, “two and thirty-eight hundredths” and “two and three tenths, eight hundredths.” Note that the first form implies two counting units (ones and hundredths) and the second form implies three counting units (ones, tenths, and hundredths). Whole numbers, on the other hand, are usually said in only one way. The number 47 could be said “four tens, seven ones” but it is almost always said “forty seven.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Hiebert suggests that having students grapple with these different readings will support their understanding of the properties of decimal notation. In his analysis of two textbook series, he says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There is no explicit attention given in either series (Grades 3-6) to the remaining two properties: The value of a digit is its face value times its place value and the value of the numeral is the sum of the value of the digits. Of course, the authors may assume that such properties will gener­alize automatically from whole number notation, but no help is given stu­dents in recognizing the similarities between whole number and decimal notation. An even more serious deficiency is that students’ attention is never drawn to the power provided by these properties. The quantity represented by 2.16 can be described in many ways—two hundred sixteen hundredths, or twenty-one tenths six hundredths, or one and eleven tenths six hundredths, and so on. None of these noncanonical forms are explored; only the canonical form, two and sixteen hundredths, is presented.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So I think it's didactically beneficial to teach students how to read decimals flexibly and to be able to make intentional choices. The reading &amp;quot;three point one four&amp;quot; can be used when we need to refer to the numeral 3.14, e.g. &amp;quot;three point one four can be read multiple ways.&amp;quot; And as Sue also already pointed out, &amp;quot;three point one four&amp;quot; may be preferable when working with older students. They often deal with continuous quantities, and &amp;quot;three point one four&amp;quot; supports the interpretation of 3.14 as one number among the continuum of the real numbers, or as a measurement of a continuous quantity to a certain precision.&lt;/p&gt;&#xA;&lt;p&gt;A natural follow-up question is &amp;quot;When is the best time to introduce the reading 'three point one four' to students?&amp;quot; I don't have an answer to this, but perhaps others can chime in. Regardless, I think this advice from Hiebert applies, wherever students are in their understanding of decimals:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The evidence on students’ failure to connect symbols with referents suggests that instruction will succeed in helping students build appropri­ate connections only if it is intentionally designed to do so. Although the exact nature of the “best” instruction is not yet clear, several features that are sure to be included in effective instruction are (a) explicit attention to connections between meaningful, familiar referents and symbols that re­present them, (b) reflection on the semantic characteristics of written symbols, and (c) the development of a rich language that can be used to talk about the symbols.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-04-12T04:03:23.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26331" PostTypeId="2" ParentId="26312" CreationDate="2023-04-12T07:51:20.787" Score="1" Body="&lt;p&gt;I don't like the whole example of &amp;quot;It's raining, which means the ground gets wet.&amp;quot;, because:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It might rain but the ground is covered, so it doesn't get wet.&lt;/li&gt;&#xA;&lt;li&gt;The ground gets wet, even when it's not raining, because you spilled a glass of water on it.&lt;/li&gt;&#xA;&lt;li&gt;...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Therefore you might better give an example which is less doubtful, like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I am located inside this classroom, so I am located inside this school building.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As there is no way being inside the classroom without being at school grounds, this is much clearer.&lt;/p&gt;&#xA;&lt;p&gt;If somebody says &amp;quot;&lt;em&gt;But I can still be on school grounds without being inside this classroom&lt;/em&gt;&amp;quot;, then you can react that this is a perfect example of &lt;span class=&quot;math-container&quot;&gt;$A \Longrightarrow B$&lt;/span&gt; without &lt;span class=&quot;math-container&quot;&gt;$B \Longrightarrow A$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In top of that, this example show a link between logic and set theory, which might be useful at later ages.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-04-12T07:51:20.787" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26332" PostTypeId="1" AcceptedAnswerId="26333" CreationDate="2023-04-12T12:16:39.477" Score="1" ViewCount="604" Body="&lt;p&gt;We usually say (3,4,5) , (5,12,13) as Pythagorean triples. What is much better way to refer those sets of numbers, Pythagorean triples or Pythagorean triplets? &lt;br/&gt;&#xA;According to the normal usage we say triple beds, triple rooms, triple jump when we want to give the meaning of three times of a unit. But in the case of Pythagoras theorem we have sets of three integers. Is it not much appropriate way to say Pythagorean triplets for those sets of numbers as in the case of using triplets when you refer three babies. I think in the case of babies we consider a set of three babies and not three times of one baby. What you have to say just to ignore the difference of the two terms or use it much meaningfully. I'm raising this issue here because I have seen the wording Pythagorean triples frequently in many articles.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="2083" LastEditDate="2023-04-12T13:12:12.267" LastActivityDate="2023-04-12T13:12:12.267" Title="Triples or triplets in Pythagoras theorem" Tags="|geometry|terminology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26333" PostTypeId="2" ParentId="26332" CreationDate="2023-04-12T12:28:14.853" Score="6" Body="&lt;p&gt;The word “triple” is appropriate here because &lt;span class=&quot;math-container&quot;&gt;$(3,4,5)$&lt;/span&gt; is a &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Tuple&quot; rel=&quot;noreferrer&quot;&gt;tuple&lt;/a&gt; consisting of three elements.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In mathematics, a tuple is a finite ordered list (sequence) of elements... Mathematicians usually write tuples by listing the elements within parentheses &amp;quot;( )&amp;quot; and separated by a comma and a space; for example, (2, 7, 4, 1, 7) denotes a 5-tuple.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The term originated as an abstraction of the sequence: single, couple/double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;‑tuple, ..., where the prefixes are taken from the Latin names of the numerals. The unique 0-tuple is called the null tuple or empty tuple. A 1‑tuple is called a single (or singleton), a 2‑tuple is called an ordered pair or couple, and a 3‑tuple is called a triple (or triplet).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The word “triple” in “Pythagorean triple” is a noun synonymous with 3-tuple, not an adjective. The word “triplet” can also be used in this sense.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-04-12T12:28:14.853" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26334" PostTypeId="2" ParentId="26283" CreationDate="2023-04-12T17:29:48.467" Score="2" Body="&lt;p&gt;Although being a bit late to the question...&lt;/p&gt;&#xA;&lt;p&gt;The underlying problem I see is that we first expose our students to a math symbology where single letters are used to name things, and that simply concatenating these characters means multiplication.&lt;/p&gt;&#xA;&lt;p&gt;Then we introduce units where this no longer holds: &lt;span class=&quot;math-container&quot;&gt;$mm^2$&lt;/span&gt; is not the same as &lt;span class=&quot;math-container&quot;&gt;$m^3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$mmm$&lt;/span&gt;, and we are surprised that students struggle with this situation. Decomposing &lt;span class=&quot;math-container&quot;&gt;$mm$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$m \cdot m$&lt;/span&gt; needs the student to understand that  &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; as a prefix is not the same as &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; as a unit (completely contradicting everything we taught before!), and that this &amp;quot;multiplication&amp;quot; is implicitly surrounded by a pair of parentheses. So, this will not aid in understanding the concept, but instead create more confusion.&lt;/p&gt;&#xA;&lt;p&gt;In fact, unit prefixes need a special treatment.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In most situations, units with prefixes should be viewed as a multi-letter entities (just like function names like &lt;span class=&quot;math-container&quot;&gt;$sin$&lt;/span&gt;, see &lt;a href=&quot;https://matheducators.stackexchange.com/questions/25163/how-can-you-elicit-the-log-x-log-cdot-x-error/25184#25184&quot;&gt;my answer here&lt;/a&gt;), and not as a multiplication of a prefix with a unit.&lt;/li&gt;&#xA;&lt;li&gt;Only when it comes to unit conversions, then it might make sense to separate prefix and base unit, as in &lt;span class=&quot;math-container&quot;&gt;$mm = 0.001 \cdot m$&lt;/span&gt; (but remind the students that they should use parentheses, so &lt;span class=&quot;math-container&quot;&gt;$mm = (0.001 \cdot m)$&lt;/span&gt; would be the &amp;quot;safe&amp;quot; replacement).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, it's more consistent to introduce units as multi-letter names, thus falling into a category that's needed anyway, e.g. for function names.&lt;/p&gt;&#xA;" OwnerUserId="15862" LastActivityDate="2023-04-12T17:29:48.467" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26335" PostTypeId="2" ParentId="13026" CreationDate="2023-04-12T20:27:29.930" Score="2" Body="&lt;p&gt;I wonder why one needs to teach them as if they were separate formulas. When one knows one, the others are just rearranged versions.&lt;/p&gt;&#xA;&lt;p&gt;Personally I find the definition of m easiest to memorize because you can just think of the triangle at the slope to derive the slope. Then you rearrange the formula for what quantity you need. This also helps for better understanding why the formulas related and why it is useful to be able to rearrange formulas.&lt;/p&gt;&#xA;" OwnerUserId="14422" LastActivityDate="2023-04-12T20:27:29.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26337" PostTypeId="2" ParentId="13026" CreationDate="2023-04-15T01:50:58.647" Score="0" Body="&lt;p&gt;I usually start with &lt;em&gt;&amp;quot;A linear function is a function for which the increment of the function value is proportional to the increment of the argument&amp;quot;&lt;/em&gt;, which is just a verbal version of (3) A typical question then would be:&lt;/p&gt;&#xA;&lt;p&gt;For a linear &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, we know that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(2023)-f(2020)=1234.&#xA;$$&lt;/span&gt;&#xA;Find &lt;span class=&quot;math-container&quot;&gt;$f(305)-f(5)$&lt;/span&gt; in your head.&lt;/p&gt;&#xA;&lt;p&gt;That seems to fit the bill because converting to (4) would involve working with one free parameter and some fractions while an intelligent use of (3) will give the answer immediately.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-04-15T01:50:58.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26338" PostTypeId="1" AcceptedAnswerId="26339" CreationDate="2023-04-15T10:20:06.393" Score="4" ViewCount="189" Body="&lt;p&gt;I am currently teaching special classes to students whose ages range from 11 to 15 and there is quite a wide spectrum in their levels of maths. The lessons are given in English and we do not have a book to adhere to. Right now, I want to start the chapter on Decimals, having already covered fractions (in particular, limited to integer quotients and the 4 basic operations). Now I can nicely conceptualize decimal expansion using their understanding of fractions.&lt;/p&gt;&#xA;&lt;p&gt;I was wondering, if there exist publicly available resources, such as textbooks or notes specifically on decimals, that I could print for the students, which would become our guiding manuscript and a constant source of exercises (important for the ones who want to advance further and I'm always behind in composing new series of exercises for them).&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In terms of online resources available and not bound to copyrights, are there textbooks (exercise handbooks,..) or dedicated source of exercises on Decimals (expansion, 4 operations,...)?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similarly, are there well-known sources for printable exercises on fractions?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;em&gt;I write about this without any reservation, meaning I understand there is not going to be something out there that exactly matches what I am looking for and that I might have to put various resources together to complement my courses. That said, any resources you know of that apply to my examples would be most welcome.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="20206" LastEditorUserId="2083" LastEditDate="2023-04-15T12:24:54.203" LastActivityDate="2023-04-15T12:36:24.223" Title="Resources for teaching decimal numbers" Tags="|textbooks|fractions|middle-school|resource-request|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26339" PostTypeId="2" ParentId="26338" CreationDate="2023-04-15T12:36:24.223" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://www.opentextbookstore.com/details.php?id=13&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Lippman’s book &lt;em&gt;Arithmetic for College Students&lt;/em&gt;&lt;/a&gt; might suit your needs. Fractions are covered in Unit 2, and decimals are in Unit 3. The book has explanations and examples and is available in &lt;a href=&quot;http://www.opentextbookstore.com/arithmetic/book.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF format&lt;/a&gt;. There are also handouts and worksheets for each lesson in &lt;a href=&quot;http://www.opentextbookstore.com/arithmetic/worksheets.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF format&lt;/a&gt; and &lt;a href=&quot;http://www.opentextbookstore.com/arithmetic/worksheets.doc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Word format&lt;/a&gt;. Everything is open source.&lt;/p&gt;&#xA;&lt;p&gt;There is also &lt;a href=&quot;https://www.myopenmath.com/course/course.php?cid=807&amp;amp;folder=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;an online course for this book&lt;/a&gt; on the free platform MyOpenMath. You can view the course without creating an account by using the username “guest”, no password necessary. In addition to the textbook and worksheets, the online course contains videos and online exercises.&lt;/p&gt;&#xA;&lt;p&gt;For printable exercises, Kuta Software has exercise generators for Pre-Algebra that can generate worksheets on decimals and fractions. It isn’t free, but they have &lt;a href=&quot;https://www.kutasoftware.com/freeipa.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;a few free worksheets on their website&lt;/a&gt;, and you can often find Kuta worksheets that other teachers have created if you search on Google.&lt;/p&gt;&#xA;&lt;p&gt;Math-Drills.com also has printable worksheets. &lt;a href=&quot;https://www.math-drills.com/fractions.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here are their fractions worksheets.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-04-15T12:36:24.223" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26340" PostTypeId="2" ParentId="13026" CreationDate="2023-04-15T13:42:09.273" Score="1" Body="&lt;p&gt;If you are teaching a class where a main theme of the class is &lt;strong&gt;translations of graphs&lt;/strong&gt;, then you can also approach point-slope form this way:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A line with slope &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; through the origin has equation &lt;span class=&quot;math-container&quot;&gt;$y = mx$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Find the new equation for the line after it has been translated to the right by &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; and up by &lt;span class=&quot;math-container&quot;&gt;$y_1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It used to pass through the origin. What point does it pass through now?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-04-15T13:42:09.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26341" PostTypeId="1" CreationDate="2023-04-15T18:20:04.253" Score="1" ViewCount="918" Body="&lt;p&gt;Would I have to read a standard textbook in addition -- i.e., Stewart, etc. -- or would this be sufficient ? My interest is in applications (dynamical systems theory and physics in general).The book states that is is not rigorous (except for the last chapter).&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance :-)&lt;/p&gt;&#xA;" OwnerUserId="22285" LastEditorUserId="20488" LastEditDate="2023-04-17T21:17:31.933" LastActivityDate="2023-04-17T21:17:31.933" Title="Is Morris Kline's 'Calculus: An Intuitive and Physical Approach' a Good Book to Learn Calculus From ?'" Tags="|calculus|self-learning|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26342" PostTypeId="2" ParentId="26341" CreationDate="2023-04-16T05:11:44.140" Score="1" Body="&lt;p&gt;To rigorously learn calculus you would have to study real analysis. To study real analysis you would have to study some a little bit Discrete Math. (I recommend Suzzane E. P)&lt;/p&gt;&#xA;&lt;p&gt;However for your case that is not necessary. I would recommend you study logic, set theory and proofs from discrete maths so you have fundamental knowledge of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;What I would suggest is study Stewart Calculus. With Stewart Calculus you have fully available solution online on Quizlet.&lt;/p&gt;&#xA;" OwnerUserId="20626" LastEditorUserId="20626" LastEditDate="2023-04-16T12:02:51.600" LastActivityDate="2023-04-16T12:02:51.600" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26343" PostTypeId="1" AcceptedAnswerId="26344" CreationDate="2023-04-16T15:41:02.330" Score="2" ViewCount="763" Body="&lt;p&gt;I was tutoring a student today and they asked a question which made me curious.&lt;/p&gt;&#xA;&lt;p&gt;We were working on the following question together.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SAvcr.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SAvcr.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;After explaining that we must look at the limit along the x axis, I mentioned that we then need need to look at the limit along the path y=mx.&lt;/p&gt;&#xA;&lt;p&gt;We got to the step where we would find f(x,xm) but when I looked at his paper he had f(y/m,xm). Now, he quickly established that he ended up going in circles however, when he asked if there was a mathematical reason why this did not work, I couldn’t tell him.&lt;/p&gt;&#xA;&lt;p&gt;So, is there, if any, a mathematical reason why we only change the y value and keep the x value as is?&lt;/p&gt;&#xA;" OwnerUserId="20201" LastEditorUserId="20201" LastEditDate="2023-04-16T15:53:52.190" LastActivityDate="2023-04-17T13:10:14.560" Title="Process of finding limits for multivariable functions" Tags="|calculus|limits|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26344" PostTypeId="2" ParentId="26343" CreationDate="2023-04-16T19:14:32.230" Score="9" Body="&lt;p&gt;There's nothing wrong with &amp;quot;changing&amp;quot; or substituting expressions for both &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. However, we can only use &lt;strong&gt;one variable&lt;/strong&gt; across both expressions.&lt;/p&gt;&#xA;&lt;p&gt;What does looking at a function along different &lt;em&gt;paths&lt;/em&gt; mean? For our purposes, a path is just a curve, which we often associate with a parametrization &lt;span class=&quot;math-container&quot;&gt;$\vec{r}(t)$&lt;/span&gt;. So if we have a two-variable function &lt;span class=&quot;math-container&quot;&gt;$f(x,y)$&lt;/span&gt;, when we restrict our attention to a specific path, what we're really doing is considering &lt;span class=&quot;math-container&quot;&gt;$f\bigl(\vec{r}(t)\bigr)$&lt;/span&gt;, which is a &lt;strong&gt;single-variable function&lt;/strong&gt;. This allows us to use all of our tools from single-variable calculus!&lt;/p&gt;&#xA;&lt;p&gt;I think many textbooks obscure this idea in their treatments of multivariable limits by presenting evaluating a function along a path as performing a substitution. For example, following a certain popular commercial text, the problem you gave would be done like this.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Along the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis, every point is of the form &lt;span class=&quot;math-container&quot;&gt;$(x,0)$&lt;/span&gt;, and the limit along this approach is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{(x,0)\to(0,0)}\frac{x \cdot 0}{x^2+0^2}=0.$$&lt;/span&gt;&#xA;However, when &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; along the line &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;, we obtain&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{(x,x)\to(0,0)}\frac{x \cdot x}{x^2+x^2}=\frac{1}{2}.$$&lt;/span&gt;&#xA;The limit does not exist because two approaches produce different limits.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another popular text is similar, perhaps marginally better.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let's approach &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; along the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis. On this path &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt; for every point &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;, so for all &lt;span class=&quot;math-container&quot;&gt;$x\neq0$&lt;/span&gt;, the function becomes &lt;span class=&quot;math-container&quot;&gt;$$f(x,0)=0/x^2=0,$$&lt;/span&gt; and thus&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x,y)\to0 \quad \text{as} \quad (x,y)\to(0,0)\,\text{ along the $x$-axis.}$$&lt;/span&gt;&#xA;Let's now approach &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; along another line, say &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;. For all &lt;span class=&quot;math-container&quot;&gt;$x\neq0$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x,x)=\frac{x^2}{x^2+x^2}=\frac{1}{2},$$&lt;/span&gt;&#xA;therefore&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x,y)\to\tfrac{1}{2} \quad \text{as} \quad (x,y)\to(0,0)\,\text{ along $y=x$.}$$&lt;/span&gt;&#xA;Since we have obtained different limits along different paths, the given limit does not exist.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In both of these approaches, it appears as though we're simply performing the substitutions &lt;span class=&quot;math-container&quot;&gt;$(x,y)=(x,0)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x,y)=(x,x)$&lt;/span&gt;. So why doesn't &lt;span class=&quot;math-container&quot;&gt;$(x,y)=(y,x)$&lt;/span&gt; work? Or if we want to approach &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; along &lt;span class=&quot;math-container&quot;&gt;$y=2x$&lt;/span&gt;, why can't we do &lt;span class=&quot;math-container&quot;&gt;$(x,y)=(y/2,2x)$&lt;/span&gt;? It's because these aren't parametrizations of curves—they aren't paths.&lt;/p&gt;&#xA;&lt;p&gt;Vector-valued functions and parametrizations of curves are presented &lt;em&gt;before&lt;/em&gt; multivariable limits in both of the texts I referenced. So I think a better approach would be something like this.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We can approach &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; along the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis by following the path &lt;span class=&quot;math-container&quot;&gt;$$\vec{r}(t)=(t,0)\,\text{ as }\, t\to0.$$&lt;/span&gt; This gives us&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{t\to0} f\bigl(\vec{r}(t)\bigr)=\lim_{t\to0}\frac{t \cdot 0}{t^2+0^2}=0.$$&lt;/span&gt;&#xA;We can also approach &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; along the line &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt; by following the path&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\vec{s}(t)=(t,t)\,\text{ as }\, t\to0.$$&lt;/span&gt; This gives us&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{t\to0} f\bigl(\vec{s}(t)\bigr)=\lim_{t\to0}\frac{t \cdot t}{t^2+t^2}=&#xA;\frac{1}{2}.$$&lt;/span&gt;&#xA;The limits along different paths do not agree, therefore the limit does not exist.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-04-16T22:20:56.607" LastActivityDate="2023-04-16T22:20:56.607" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26346" PostTypeId="2" ParentId="26343" CreationDate="2023-04-16T20:33:40.080" Score="4" Body="&lt;p&gt;Presumably your textbook has some theorem along the lines of the following, which sets up the goal of trying different paths:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; &lt;em&gt;If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a real-valued function such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{(x,y)\rightarrow(a,b)} f(x,y) = L \,,$$&lt;/span&gt;&#xA;then for all real numbers &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; not both zero,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{t\rightarrow0} f(a+pt,b+qt) = L \,.$$&lt;/span&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Instead of the parametrized lines &lt;span class=&quot;math-container&quot;&gt;$(a+pt,b+qt)$&lt;/span&gt;,&#xA;the theorem might be about more general paths &lt;span class=&quot;math-container&quot;&gt;${\bf r}(t)$&lt;/span&gt; that approach &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; but do not pass through &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$t\ne c$&lt;/span&gt;.&#xA;(Or perhaps your textbook relies only on the uniqueness of the limit, which would complicate the contrapositive argument below.)&lt;/p&gt;&#xA;&lt;p&gt;One then applies the contrapositive of the theorem (and maybe this is the relevant theorem in your textbook): If you get different one-variable limits along different paths through &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt;, then the two-variable limit does not exist.&lt;/p&gt;&#xA;&lt;p&gt;Whatever the statement of the theorem, the goal is to find one-variable limits that disagree; then you win. If the limits agree, then you lose: find another path or reconsider whether the two-variable limit exists.&lt;/p&gt;&#xA;&lt;p&gt;With that goal in mind, &lt;span class=&quot;math-container&quot;&gt;$f(y/m, mx)$&lt;/span&gt; is a bad problem-solving move, since there are still two variables. The student missed the goal of getting down to one variable. Either &lt;span class=&quot;math-container&quot;&gt;$f(y/m, y)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$f(x, mx)$&lt;/span&gt; is a good move, however. Or you can change both &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; in terms of the one variable &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, which is the way I prefer to teach it. Even &lt;span class=&quot;math-container&quot;&gt;$x=0, y=t$&lt;/span&gt; instead of just &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt;.  I'd accept either, but I always present a solution to students in parametric form.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;P.S. In polar coordinates, the expression becomes &lt;span class=&quot;math-container&quot;&gt;$\cos\theta\sin\theta$&lt;/span&gt;, and the one-variable limits depends on the angle of approach. Just another way to view the limit.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2023-04-17T02:23:38.567" LastActivityDate="2023-04-17T02:23:38.567" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26347" PostTypeId="2" ParentId="26341" CreationDate="2023-04-16T23:26:22.363" Score="4" Body="&lt;p&gt;Kline's book is not rigorous in the sense that it avoids the formal definitions and proofs that were developed so that calculus could be put on a &amp;quot;firm footing,&amp;quot; at least until the last chapter. However, based on my experience, most calculus courses that use &amp;quot;standard&amp;quot; textbooks also don't emphasize these formalisms/don't do a good job of teaching them. As Tegh said, you would only be expected to know  the &amp;quot;rigorous stuff&amp;quot; after learning real analysis, and many scientists never take a real analysis course.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.reddit.com/r/learnmath/comments/fbk41/comment/c1f03ou&quot; rel=&quot;nofollow noreferrer&quot;&gt;See also this Reddit comment about the book.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Kline's book covers single-variable calculus pretty thoroughly. It does cover a decent chunk of multivariable calculus, but you would probably still want to supplement this with another text. For example, Kline doesn't cover vector calculus, which is important in many science fields.&lt;/p&gt;&#xA;&lt;p&gt;After Kline's book, if you want to jump straight into dynamical systems, Steven Strogatz's &lt;em&gt;Nonlinear Dynamics and Chaos&lt;/em&gt; is aimed at students that have completed a first course in calculus. &lt;a href=&quot;https://www.youtube.com/playlist?list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V&quot; rel=&quot;nofollow noreferrer&quot;&gt;Video lectures are available.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-04-17T10:43:23.350" LastActivityDate="2023-04-17T10:43:23.350" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26348" PostTypeId="1" CreationDate="2023-04-17T02:29:28.707" Score="3" ViewCount="189" Body="&lt;p&gt;I am an early-career mathematics-education researcher. Recently, I received a request for major revisions for a manuscript I had submitted on opportunities to learn provided by undergraduate mathematics calculus textbooks.&lt;/p&gt;&#xA;&lt;p&gt;One of the issues I need to address is explaining the theoretical contribution of the study reported in the manuscript. I am not sure how to address this issue. I guess my greatest challenge is that I don't know what a theoretical contribution is in mathematics education. I would greatly appreciate if someone could explain what the reviewers mean or rather expect in my response. Suggestions for resources to read on this subject are welcome too.&lt;/p&gt;&#xA;" OwnerUserId="22288" LastEditorUserId="75" LastEditDate="2023-04-17T05:11:19.430" LastActivityDate="2023-04-20T21:07:40.080" Title="What is a theoretical contribution in mathematics-education research?" Tags="|undergraduate-education|education-research|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26349" PostTypeId="2" ParentId="26348" CreationDate="2023-04-19T13:15:29.233" Score="4" Body="&lt;p&gt;I think what the reviewer is looking for is a &amp;quot;why&amp;quot;.   If X causes Y improvement, why does it do so?  What aspects of teachers, students, and class dynamics contribute to this situation?  How would I recognize a similar problem in a different context?  What were the driving forces that led to the problem that needed a solution?  Was it the class, the school, the requirements, the students, or the society as a whole?&lt;/p&gt;&#xA;&lt;p&gt;You didn't give much information for what you have written, but I would suggest adding a &amp;quot;discussion&amp;quot; section which addresses the deeper issues at play.  Even if you don't have definitive answers, looking at the theoretical questions will help others more than just the data itself.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2023-04-19T13:15:29.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26350" PostTypeId="2" ParentId="26288" CreationDate="2023-04-19T13:26:38.780" Score="0" Body="&lt;p&gt;My suggestion:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Use the notation yourself&lt;/li&gt;&#xA;&lt;li&gt;Explain why it is helpful&lt;/li&gt;&#xA;&lt;li&gt;Show examples of where it helps to avoid confusion&lt;/li&gt;&#xA;&lt;li&gt;Don't require it from students (i.e., don't mark off for not using it)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;If 1-3 are in place, (4) will eventually happen with maturity, even if it isn't this year.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2023-04-19T13:26:38.780" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26351" PostTypeId="1" CreationDate="2023-04-19T20:22:50.757" Score="5" ViewCount="161" Body="&lt;p&gt;I posted this question in the math stackexchange &lt;a href=&quot;https://math.stackexchange.com/questions/4681694/comparison-of-textbooks-on-how-to-write-proofs&quot;&gt;https://math.stackexchange.com/questions/4681694/comparison-of-textbooks-on-how-to-write-proofs&lt;/a&gt; and one person suggested that I cross-post it here.  I'd be most apprecicative for your inputs.  Here is the original post, slightly edited.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Next fall I will teach a class on &amp;quot;How to write proofs&amp;quot;. Prerequisite is first-year calculus. The three textbooks I am considering are&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Hammack, Book of Proof (3rd ed)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sundstrom, Mathematical Reasoning: Writing and Proof (3rd ed)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cummings, Proofs: A long-form mathematics textbook.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This will be my first time teaching such a course, and I'd be most appreciative for feedbacks and comments from those who have taught such a class using these books -- why do you pick one over the others? feedback from students? advice on how to teach such a class/use these books?  To give you an idea the issues I'm considering (incorrectly, perhaps!), here are my first impressions of the three books, based on a quick reading:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Sundstrom and Hammack provide free downloads and Cummings is inexpensive, so cost is not an issue.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sundstrom is the only the one that provides instructor solution manual. I can obviously solve all the problems, but having solutions available does have me in picking problems for homework (e.g. length of solution, complications/tricky spots etc) and lecture preparation (e.g. be sure to go over certain examples before I assign a particular problem).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sundstrom is (relatively speaking) the most 'old-fashion' of the three. There is not a lot of motivation of the materials, and the &amp;quot;beginning activities&amp;quot; at the start of each section, while helpful for lecture planning, in a way &amp;quot;dilute&amp;quot; the materials, and I worry that it might discourage students from reading the text. I also wish that (very basic) set theory were presented in chap 1, and I find it a bit strange that congruence is covered before equivalence relations (please remember that these are my first impressions).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cummings is the opposite of Sundstrom: Very fresh and lively, and the presentation/examples are interesting. At the same time the exposition seems too chatty, and the jokes etc get tired after a short while. The problems also seem a bit more challenging that Sundstrom.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hammack is in a way half way between the two. It does not get to proofs until page 113, which seems a bit late --- speaking as someone who has not taught a &amp;quot;proof class&amp;quot; before. Again congruence is introduced before equivalence relations. I really wish there is a solution manual.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Again, for those who have used these books:  Why did you pick one over the others?  Feedback from students?  Advice on how to teach such a use/use the book?  Order of topics? (e.g. congruences before/after equivalence relations)&lt;/p&gt;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;&lt;p&gt;Note: I am aware of prior posts such as &lt;a href=&quot;https://math.stackexchange.com/questions/3316114/book-recommendation-for-proof&quot;&gt;https://math.stackexchange.com/questions/3316114/book-recommendation-for-proof&lt;/a&gt;&#xA;and &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13034/book-request-teaching-proving-and-reasoning-at-an-american-university&quot;&gt;Book request: teaching proving and reasoning at an American university&lt;/a&gt; The posts there are either lists of books and/or recommendation for a specific book. What I am looking for are comments and comparison about the three specific ones listed above. THANKS!&lt;/p&gt;&#xA;" OwnerUserId="22295" LastEditorDisplayName="user18187" LastEditDate="2023-08-15T11:59:57.730" LastActivityDate="2023-08-15T11:59:57.730" Title="Comparison of texbook for &quot;how to write proofs&quot;" Tags="|undergraduate-education|textbooks|proofs|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26352" PostTypeId="1" CreationDate="2023-04-19T21:53:14.790" Score="3" ViewCount="894" Body="&lt;p&gt;I am looking for simple examples of situations where a good notation/diagram was fundamental for solving an elementary problem in mathematics (I am looking for examples accessible to a basic school student).&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="851" LastEditDate="2023-04-21T06:52:12.343" LastActivityDate="2023-04-22T21:55:28.380" Title="Simple examples of how a good notation or diagram can help to solve math problems" Tags="|notation|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26353" PostTypeId="2" ParentId="26352" CreationDate="2023-04-19T23:07:11.160" Score="3" Body="&lt;p&gt;Graph theory is my favorite example. Euler created it to solve the Konigsberg bridges problem. No algebra needed to play around with it.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-04-20T03:33:59.147" LastActivityDate="2023-04-20T03:33:59.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26354" PostTypeId="1" CreationDate="2023-04-20T00:41:11.023" Score="1" ViewCount="100" Body="&lt;p&gt;he Simpson's Paradox is a statistical phenomenon in which a trend or relationship observed within a dataset disappears or reverses when the dataset is divided into smaller groups. It occurs when a confounding variable affects the relationship between the primary variables being analyzed, leading to incorrect or misleading interpretations.&lt;/p&gt;&#xA;&lt;p&gt;i am looking for references on experiences on working with simpson paradox in classroom(basic school or undergraduate courses).&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="2083" LastEditDate="2023-04-20T07:48:52.510" LastActivityDate="2023-04-20T07:48:52.510" Title="simpson paradox in classroom: reports?" Tags="|undergraduate-education|secondary-education|statistics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26355" PostTypeId="2" ParentId="26352" CreationDate="2023-04-20T09:14:43.607" Score="5" Body="&lt;p&gt;Two examples in probability:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A contingency table.&lt;/li&gt;&#xA;&lt;li&gt;An events tree.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7178" LastActivityDate="2023-04-20T09:14:43.607" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26356" PostTypeId="1" CreationDate="2023-04-20T09:15:24.013" Score="0" ViewCount="59" Body="&lt;p&gt;I am looking for &amp;quot;must read&amp;quot;/classical references on data aggregation/disaggregation in statistics, particularly, what they are exactly, why they are done and how statistical measures (media, mean, correlation coefficient, etc.) and indices (Human Development Index, poverty, Gini, etc.) behave under aggregation/disaggregation of data.&#xA;Is this topic worked out in basic school or undergraduate courses. It seems it is not.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="498" LastEditDate="2023-04-20T15:22:24.177" LastActivityDate="2023-04-20T15:22:24.177" Title="What are &quot;must read&quot;/classical refences on data aggregation/disaggregation in statistics?" Tags="|statistics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26357" PostTypeId="2" ParentId="26352" CreationDate="2023-04-20T11:23:02.610" Score="4" Body="&lt;p&gt;&amp;quot;General interest&amp;quot; mathematics journals like &lt;em&gt;Mathematics Magazine&lt;/em&gt; or &lt;em&gt;The Mathematical Intelligencer&lt;/em&gt; sometimes have &amp;quot;proofs without words&amp;quot;, and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_without_words&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; is also helpful. These are diagrams that more or less immediately prove a mathematical statement. For instance, there are a number of &lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagorean_theorem#Rearrangement_proofs&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;rearrangement proofs&amp;quot; of the Pythagorean theorem&lt;/a&gt;. More examples are &lt;a href=&quot;https://doi.org/10.1080/0025570X.2020.1791531&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cantone (2020)&lt;/a&gt; or &lt;a href=&quot;https://doi.org/10.1007/s00283-017-9727-9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chakraborty (2018)&lt;/a&gt; or &lt;a href=&quot;https://doi.org/10.1007/s00283-021-10087-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mukherjee (2021)&lt;/a&gt;, all of which are unfortunately paywalled. Roger Nelsen also published collections of such proofs in book form.&lt;/p&gt;&#xA;" OwnerUserId="5845" LastActivityDate="2023-04-20T11:23:02.610" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26358" PostTypeId="2" ParentId="26356" CreationDate="2023-04-20T13:30:42.080" Score="4" Body="&lt;p&gt;A classic reference is &lt;em&gt;The Future of Data Analysis&lt;/em&gt; by John Tukey. This appeared in the  Annals of Mathematical Statistics in 1962 and is widely available, for example on &lt;a href=&quot;https://www.jstor.org/stable/2237638&quot; rel=&quot;nofollow noreferrer&quot;&gt;JSTOR&lt;/a&gt;. I love the beginning sentences: &lt;em&gt;For a long time I have thought I was a statistician, interested in inference from particular to general. But as I have watched mathematical statistics evolve, I have cause to wonder and doubt.&lt;/em&gt;  Of particular interest for this forum is section 9 of the paper titled &amp;quot;&lt;em&gt;&lt;strong&gt;teaching of data analysis&lt;/strong&gt;&lt;/em&gt;.&amp;quot; Tukey also authored the book &lt;strong&gt;Exploratory Data Analysis&lt;/strong&gt; in 1977.&lt;/p&gt;&#xA;&lt;p&gt;Tukey is referred to as &amp;quot;the progenitor of data science,&amp;quot; was an intellectual giant of the twentieth century, and was one of the most influential statisticians of the twentieth century. He discovered the ubiquitous Fast Fourier Transform, coined the word &amp;quot;bit,&amp;quot; developed the statistical programming language S which begat R, introduced visualizations such as box and whisker plots, &lt;span class=&quot;math-container&quot;&gt;$\ldots$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-04-20T13:42:01.687" LastActivityDate="2023-04-20T13:42:01.687" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26359" PostTypeId="1" AcceptedAnswerId="26371" CreationDate="2023-04-20T17:18:52.657" Score="18" ViewCount="3081" Body="&lt;p&gt;I believe that most middle-school math curricula have at least a brief section about irrational numbers, in which students are taught (among other things) that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is irrational and &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is irrational.&lt;/p&gt;&#xA;&lt;p&gt;What I'm wondering is if students are ever told that the irrationality of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is not at all obvious, but there is a way to prove it, though the proof requires advanced techniques.  I suspect that most of the time, the irrationality of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is asserted as a brute fact that the students are expected to accept because authority figures say so, or perhaps on the basis of very limited empirical evidence.  But maybe I'm wrong.&lt;/p&gt;&#xA;&lt;p&gt;Is there any textbook (or any kind of written curriculum—I understand that nowadays, many teachers eschew traditional textbooks in favor of free materials that they download from some website) at the middle school level which indicates that irrationality of a number such as &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is something beyond the empirical observation that its decimal expansion does not &lt;i&gt;seem&lt;/i&gt; to repeat?&lt;/p&gt;&#xA;" OwnerUserId="12478" LastActivityDate="2023-04-23T18:55:46.190" Title="Do any middle-school texts indicate that irrationality requires proof?" Tags="|textbooks|proofs|middle-school|rational-numbers|" AnswerCount="7" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="26360" PostTypeId="1" CreationDate="2023-04-20T18:06:48.743" Score="1" ViewCount="63" Body="&lt;p&gt;In my Finite Math course* almost every section includes a part where students have to create a file (from scratch) in Desmos or in Google Sheets. For example, they use Desmos to plot piecewise linear functions for contextual data and feasible regions in 2D linear programming. They use Sheets to create dynamic tables that model linear cost/revenue situations, solve higher-dimensional linear programming, and create dynamic tables of various savings and loans situations.&lt;/p&gt;&#xA;&lt;p&gt;After having a very successful semester doing this, I was given the feedback by my fellow faculty members that the course is also supposed to contain a section on basic, discrete probability. In general, I'm okay with this and have already identified the material from the first semester that I'll cut.&lt;/p&gt;&#xA;&lt;p&gt;But the issue I am having is that I can't think of any good modeling activities in probability where the &lt;strong&gt;students build the model&lt;/strong&gt; in Desmos or Sheets.&lt;/p&gt;&#xA;&lt;p&gt;There are plenty of pre-built Desmos-based demonstrations of various probability concepts (disqualified because students aren't creating them) and Desmos does a great job working with probability distributions (disqualified because not discrete probability).&lt;/p&gt;&#xA;&lt;p&gt;Using Sheets instead, the only thing I can think to leverage is the &lt;code&gt;RAND()&lt;/code&gt; function to simulate coin flips, etc. This seems like a good way to demonstrate properties of probability (like multiplying probabilities when connected by an AND), but I can't think of any type of situation in probability where I'd turn to Sheets as a problem-solving tool.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;So I'm looking for suggestions of problems/scenarios for students in discrete probability where the act of modeling in Sheets or Desmos provides insight into the problem and/or spares algebraic calculation.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Alternatively, I guess I would accept suggestions of such problems and a different software that is as free and easy to use as Desmos and Sheets.)&lt;/p&gt;&#xA;&lt;p&gt;(* United States college course aimed mostly at Business Majors too weak to go into Pre-Calculus.)&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="2083" LastEditDate="2023-04-20T18:32:07.967" LastActivityDate="2023-04-20T20:52:43.433" Title="Discrete Probability Modeling with Desmos or Spreadsheets" Tags="|undergraduate-education|technology-in-education|probability|activities|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26361" PostTypeId="2" ParentId="26360" CreationDate="2023-04-20T20:52:43.433" Score="1" Body="&lt;p&gt;It's pretty cool how the more coin tosses you have, the more the curve looks like a normal curve. And one die versus many dice rolls is also an interesting progression. Is that helpful?&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-04-20T20:52:43.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26362" PostTypeId="2" ParentId="26348" CreationDate="2023-04-20T21:07:40.080" Score="0" Body="&lt;p&gt;You must check the state of the art, in which your paper is framed. For instance, task design.&lt;/p&gt;&#xA;" OwnerUserId="1789" LastActivityDate="2023-04-20T21:07:40.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26363" PostTypeId="2" ParentId="26359" CreationDate="2023-04-20T21:48:42.777" Score="13" Body="&lt;p&gt;Here's a quote from the syllabus for the 9th grade in the school type &amp;quot;Gymnasium&amp;quot; in the federal state of Bavaria in Germany:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kompetenzerwartungen und Inhalte&lt;/p&gt;&#xA;&lt;p&gt;Die Schülerinnen und Schüler [...] verstehen das Grundprinzip eines indirekten Beweises, vollziehen damit den Beweis für die Irrationalität von Wurzel aus 2 nach und erläutern diesen; dabei erfassen sie auch, dass das Beweisen eine zentrale Bedeutung für die Mathematik und deren stringenten Aufbau hat.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In English (translation mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Expected competences and contents:&lt;/p&gt;&#xA;&lt;p&gt;The students [...] understand the basic principle of an indirect proof and therefore follow and explain the proof that the square root of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is irrational; thereby they also recognize that proofs play a central role for mathematics and its stringent structure.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.lehrplanplus.bayern.de/fachlehrplan/gymnasium/9/mathematik#217169&quot; rel=&quot;noreferrer&quot;&gt;Link to the source.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Context information:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;9th graders are typically about 14 to 16 years old in Germany.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There are different type of schools in Germany, with the details depending on the federal state. Attending a &amp;quot;Gymansium&amp;quot; takes more years than (most) other school types (in Bavaria, it lasts from grade 5 to 13) and is typically considered to be more theoretical than the other schools. This school type offers the most direct (but not the only) way to get access to tertiary education in Germany. A quick internet search suggests that currently about 40% of all 5th graders in Bavaria attend a gymnasium (but some of them switch to another type of school at some point later).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;From my experience as a middle school and highschool student (at a gymnasium in Bavaria, about 25 to 15 years ago) I severely doubt that the majority of students in 9th grade achieve the &amp;quot;expected competences&amp;quot; cited above.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastEditorUserId="16250" LastEditDate="2023-04-21T00:16:40.023" LastActivityDate="2023-04-21T00:16:40.023" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26364" PostTypeId="2" ParentId="26359" CreationDate="2023-04-21T01:00:34.523" Score="3" Body="&lt;p&gt;A web search* of several state public school math standards and online textbooks and lesson plans showed that nowadays grade 8 seems to emphasize understanding the concept of irrational numbers and that they are out there.  But not requiring proof (especially for &amp;quot;struggling learners&amp;quot;, a descriptor that is a direct quote).&lt;/p&gt;&#xA;&lt;p&gt;Of course, any time you say &amp;quot;are there any texts&amp;quot; the answer is probably yes (at least one exists).  For example, when I learned the topic in middle school, we were shown the sqrt(2) proof.  So, I'm sure there are some places still teaching that, even if it is deprecated a bit more now.  The right question is probably not are there any, but how common is it.  (If you just want a single example, try the AoPS pre-algebra text.  But that book may be a bit harder than optimal, for average students...and is expensive.)&lt;/p&gt;&#xA;&lt;p&gt;In some cases, from the search, it seems like the proof was mentioned as a concept (e.g. with the name of a Greek) but not demonstrated.  It appears to be current thinking to not require the sqrt(2) demonstration.  Which I don't really have a problem with, the whole point is to just introduce the concept.  For what it's worth, I've probably never seen the proof that pi is irrational but just accepted it. After all, people like bragging about how many decimals they know and pi sure doesn't look rational for several decimals out so it probably is irrational and I'll accept it that someone proved it without me needing to see the details.&lt;/p&gt;&#xA;&lt;p&gt;Note also that some numbers are STILL not proven as irrational, but we have been unable to express them as rational numbers!  E.g. pi plus e.  This despite, we know about where it is on the number line and it's even a rather simple expression, not some ugly continued fraction foul thingie.  But if it is rational, it's got to be a very large denominator, since we haven't found it yet.  Personally, I think this is kind of cool, now, knowing what irrationals are (maybe more like an engineer knows them, not like a Rudin-luvver, since I never took pure math), that there is this richness to the topic.  But I wouldn't derail the initial learning about irrationals themselves with this complexity.&lt;/p&gt;&#xA;&lt;p&gt;FWIW, with &amp;quot;high track&amp;quot; students (US G/T or German gymnasium), I would have no issue with showing the sqrt(2) proof still.  They can handle it.  But maybe not with the middle/struggling tracks.  Just follow the modern practice to &amp;quot;not require proof&amp;quot; (an actual quote from one standard)  And even for the G/T track, I wouldn't go so far as to say &amp;quot;proof needed&amp;quot; being the emphasis.  The emphasis is more conceptual, even for the good students (knowing the concept of can't be a rational fraction or expressed in terminating/repeating decimal). Just the proof is a nice flourish to add and give them some more grounding.&lt;/p&gt;&#xA;&lt;p&gt;*Source: user &amp;quot;guest&amp;quot;, Google, APR2023.  I can't show more than two links without getting spam blocked, but see these:&lt;/p&gt;&#xA;&lt;p&gt;(example state standard)  &lt;a href=&quot;https://portal.ct.gov/SDE/CT-Core-Standards/Materials-for-Teachers/Mathematics/Math-Lesson-Plans/Math/Grade-8-Rational-and-Irrational-Numbers---Real-Number-Race&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://portal.ct.gov/SDE/CT-Core-Standards/Materials-for-Teachers/Mathematics/Math-Lesson-Plans/Math/Grade-8-Rational-and-Irrational-Numbers---Real-Number-Race&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;An interesting paper on the math ed aspect of student's first intro to irrationals:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/3482830&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/3482830&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22299" LastEditorUserId="22299" LastEditDate="2023-04-21T01:11:15.220" LastActivityDate="2023-04-21T01:11:15.220" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="26365" PostTypeId="2" ParentId="26359" CreationDate="2023-04-21T11:20:54.640" Score="3" Body="&lt;p&gt;&lt;em&gt;(originally a comment, but started getting too long)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;My Algebra 1 text (I think published in 1970) definitely dealt with the meaning of irrational numbers. In my school everyone took Algebra 1 in the 9th grade, the first year of high school (ages 14 to 15). Even I took Algebra 1 then (1973-1974 school year), although by this time I was 2-3 years ahead of my classmates in math from reading ahead on my own (algebra was not offered in our middle school, but public libraries existed back then). However, since Algebra 1 (in the U.S.) is now universally taught in middle school, at least for average to above-average level students, maybe this counts?&lt;/p&gt;&#xA;&lt;p&gt;I can't find our actual textbook online (freely available), but I did find essentially the next best thing -- the &lt;a href=&quot;https://archive.org/details/modernalgebrastr01dolc_2/page/396/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;teacher's edition of the version used by students 2+ years older than me&lt;/a&gt;. (I believe textbook editions were changed 2 years before I took the class.) The teacher's edition includes some blue comments, answers in blue, etc. that should clearly indicate what was present in the actual students' book. As you can see, this has quite a bit relevant to your question. See Chapter 11 (&lt;em&gt;The Real Numbers&lt;/em&gt;, pp. 396-433), especially pp. 401-402 (proof of decimal eventual periodicity characterization of rational numbers) and p. 407 (proof that if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a positive integer, then &lt;span class=&quot;math-container&quot;&gt;$\sqrt n$&lt;/span&gt; is either an integer or irrational).&lt;/p&gt;&#xA;&lt;p&gt;I know that in class we covered all the details of the decimal characterization of rational numbers. Being in the BC (Before Calculators) era, this involves ideas that students were definitely familiar with from many years of VERY many long division calculations. On the other hand, in class I'm pretty sure that only the result of the p. 407 stuff was mentioned, and the proof was skipped.&lt;/p&gt;&#xA;&lt;p&gt;For what it's worth, I think our Algebra 1 year-long course ended with a brief introduction to quadratic equations (last few days of class for the year), which is dealt with in Chapter 13. During those last few days I believe our teacher only introduced the quadratic formula (without proof) and we used it to solve some numerically simple quadratic equations. Completing the square was not done until our 11th grade Algebra 2 course (ages 16-17). Possibly worth mentioning is that typically 0-2 students each year in my high school doubled-up with 10th grade Geometry and 11th grade Algebra 2 during their 10th grade, which allowed them to take 12th grade precalculus in the 11th grade, thereby allowing them to take calculus at a university about 20 miles away in their last year of high school (our high school at that time did not offer calculus).&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2023-04-21T11:20:54.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26367" PostTypeId="1" CreationDate="2023-04-21T12:54:42.273" Score="1" ViewCount="119" Body="&lt;p&gt;Does anyone know of links to resources to explain why basic algebra rearrangement operations take place in a certain order?&lt;/p&gt;&#xA;&lt;p&gt;A simple, seemingly absurd example, but not uncommon follows.&lt;/p&gt;&#xA;&lt;p&gt;Say the student is tasked to make &amp;quot;y&amp;quot; the subject of:&lt;/p&gt;&#xA;&lt;p&gt;x = 3y - 7&lt;/p&gt;&#xA;&lt;p&gt;Now &amp;quot;obviously&amp;quot; the first step is to add 7 to both sides. But I see students whose first step is to try to divide by 3 - maybe because that's what they need to do to &amp;quot;get y on its own&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Or teaching a rearrangement of, say&lt;/p&gt;&#xA;&lt;p&gt;3xy = 7x - 4y&lt;/p&gt;&#xA;&lt;p&gt;to make &amp;quot;x&amp;quot; (or &amp;quot;y&amp;quot;) the subject.&lt;/p&gt;&#xA;&lt;p&gt;Weaker students find it difficult to know where to start and often try actions like &amp;quot;divide by y&amp;quot; rather than rearrange and apply linear factorisation (a very challenging step for weaker students).&lt;/p&gt;&#xA;&lt;p&gt;Now, I'm not thinking the process of rearrangement can be reduced to a simple algorithm but are there any resources that can help the student to think &amp;quot;well, &lt;strong&gt;first&lt;/strong&gt; I do this, &lt;strong&gt;then&lt;/strong&gt; I do that, &lt;strong&gt;then&lt;/strong&gt; I do that&amp;quot;? All the examples I can find example what to do in a particular situation but don't go into the why.&lt;/p&gt;&#xA;" OwnerUserId="7954" LastActivityDate="2023-04-22T05:33:20.993" Title="Any online resources explaining why rearrangement of terms occurs in a particular order" Tags="|algebra|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26368" PostTypeId="2" ParentId="26367" CreationDate="2023-04-21T14:36:38.803" Score="8" Body="&lt;p&gt;This is a golden opportunity to not provide such a resource.&lt;/p&gt;&#xA;&lt;p&gt;Consider: have many pupils/students solve the equation, go around and see what they have done, and then have several of them present their own solutions side by side.&#xA;Then have the students/pupils compare and contrast - which was easiest, shortest, most productive, where are there mistakes, and so on.&lt;/p&gt;&#xA;&lt;p&gt;You can offer a strategy or two, too, if necessary, if only as foils, though you do risk students/pupils thinking that what you provided is the correct solution.&lt;/p&gt;&#xA;&lt;p&gt;The idea here is that discussing different strategies shows the learners that there are many ways to approach a problem, and can get them to reflect what they have done. Also, since you are not offering a canonical choice, there is fairly low identity threat (thinking one is perceived to be stupid or do a wrong thing). Try to catch and find as many different strategies as possible, and tell this to the students/pupils, too.&lt;/p&gt;&#xA;&lt;p&gt;This should help with your actual problem (the students/pupils have not figured out equations yet), and also with your perceived problem (increased strategy awareness should help with picking a good one, too).&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-04-21T14:36:38.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26369" PostTypeId="2" ParentId="19546" CreationDate="2023-04-21T14:47:38.973" Score="4" Body="&lt;p&gt;I agree with many of the other answers, especially with the notion that one should to be able to &lt;em&gt;estimate&lt;/em&gt; results of computations, which is essentially impossible without basic computation skills.&lt;/p&gt;&#xA;&lt;p&gt;However, I find it a bit strange that many answers focus mainly on the usage of computational skills in later math classes or on tasks related to computations in certain math-heavy professional settings (e.g., &amp;quot;An engineer needs to have a feeling whether the computer's result can be right&amp;quot;). This is all correct and important, but I think there is anoter point that is even much more important for the majority of people:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;People should have a basic understanding of numbers in everyday life and they won't get this understanding if they don't learn to do elementary computations.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;By &amp;quot;everyday life&amp;quot; I don't necessarily mean the &amp;quot;standard examples&amp;quot; such as &amp;quot;add the prices of items in a shop&amp;quot; or &amp;quot;if you have ten dollars, how many pieces of chocolate can you buy for 1.80 dollars each&amp;quot; (which I find, admittedly, a bit contrived).&#xA;But basic understanding of numbers and arithmetics is essential for a large variety of things every day. Here is a - more or less arbitrary - list of examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I promised to fetch a colleague from the station. Their train is schedule to arrives at 10:53 am, it usually takes me 15 minutes to the station, but now they write me that their train is delayed by about 35 minutes. When approximately should I leave to head for the station?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Boarding of your flight is scheduled for 2:20 pm, from experience you plan 40 minutes for security and baggage drop-off plus another half an hour buffer in case that something goes wrong. Which bus to the airport do you take?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;From the news: The United States announced this week to spend another 300 million dollars of military aid for Ukraine. A few weeks ago they had announced a different package of about 2 billion dollars. Which package is larger? How much larger is it roughly?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Again from the news: Inflation in Germany was about 7.9% in 2022. Usually it's more between 2% and 3%. What is this even supposed to mean?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some numbers related to climate change: nationwide C02 emissions in 2018, in percentage of the global emissions: 6.9% for India, 1.8% for South Korea. What might those numbers tell us?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Say your trade union negogiates a new collective wage agreement: effective of June 2023 you'll get 8% more salary, and then the salary won't change until the next negotiations mid 2025. In the news you read that economists estimate an inflation rate for your country of roughly 6-7% in 2023 and that the rate will probably stay similarly high for the next few years. What's your opinion about the deal that your union negotiated?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Where I live it still happens very frequently that people pay cash in shops. So do I quite often, but I want to avoid having too many coins in my wallet. Say, at a grocery store I have to pay 15.67 €, and I can see that I have two 20€ bills in may wallet, along with several 1€ coins, several 50 cent coins and several 10 and 20 cent coins. What do I do?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;My wife and I are at a restaurent and the bill is 34.60 €. The service was good and we'd like to give at least 10% tip (but of course we'll do some rounding to keep the numbers simple). How do we know how much money we should roughly give?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I want to use a cloud service to save files on it. I'm offered 2 terabytes for 8€ per month or 8 terabytes for 15€ per month. How can I decide which offer is more suitable for me?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Say I'd like to have an online meeting with a colleague in Miami and they wrote me that they have time next Tuesday at 4pm (in their local time). When I read their message, it's about 11 am where I live and an internet search engine tells me that, at the same moment, it's about 5 am in Miami. What's the suggested meeting time in my time zone?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I travel by train a lot and I regularly have to switch trains in Nuremberg - where my usual connection arrives at 5:32 pm and leaves at 6:00 pm. This usually suffices to briefly have dinner and for a visit to the washroom. Unfortunately, the German railway company &amp;quot;Deutsche Bahn&amp;quot; isn't anymore what it used to be, so I often arrive in Nuremberg with a delay. If I arrive at 5:41 pm, will I still be able to do what I usually do when I hurry up a bit? And what if I arrive at 5:47 pm?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;These are all examples that occur - many of them quite often - in some variation in my everyday life. In each of such situations I would be more or less lost if didn't have at least a very basic understanding of numbers and arithmetics.&lt;/p&gt;&#xA;&lt;p&gt;Am I supposed to use a calculator before I give money to the cashier or to the waiter? Does the calculator tell me that I need to look up the populations of India and South Korea to have any chance of making sense of their C02 emissions? If I can't even see that the 8 terabyte offer gives me 4 times us much memory for about twice the price (because I'm not able to do even the most elementary computations without a calculator), how am I supposed to make an informed decision about whether I really have good use for those 8 terabytes such that the 7 additional bucks per month are worth it? Am I suppposed to enter the arrival and departure times at Nuremberg in my Smartphone's calculator to find out that arrival at 5:41 means that I have approximately 20 minutes instead of approximately 30 minutes time? And even if I do it, what are those 20 and 30 minutes going to tell me if I have no understanding whatsoever of even small numbers? How is my calculator going to tell me at which times I have to leave in order to meet my delayed colleague at the station?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Summary.&lt;/strong&gt; A basic understanding of numbers and arithmetics is extremely important to properly understand the world around us and to act in it.&#xA;Teaching this skill to everybody is one of the most important purposes of school, directly after reading and writing.&lt;/p&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2023-04-21T14:47:38.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26370" PostTypeId="2" ParentId="26359" CreationDate="2023-04-22T02:49:39.847" Score="7" Body="&lt;p&gt;&lt;a href=&quot;http://www.oo-lyceum-533.ru/document/%D0%A1%D0%BE%D0%BB%D1%8B%D0%BD%D0%B8%D0%BD%20%D0%90-%D0%9C%D0%BE%D1%89%D0%BD%D0%BE%D1%81%D1%82%D0%B8-1.pdf&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt; is a page with teacher's instructions (and some homework problems) for a Russian 8th grade class in a specialized &amp;quot;mathematical&amp;quot; school (or, it is a student handout, I am not sure). Among other things, it discusses transcendental numbers and says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Такие числа называются трансцендентными. Наиболее известное трансцендентное число — π, но доказать, что оно трансцендентно, очень сложно. Указать явно хотя бы одно трансцендентное&#xA;число, тоже непросто.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Such numbers are called transcendental. The most famous transcendental number is π, but it is very hard to prove that it is indeed transcendental. It is even not so easy to give an explicit example of a transcendental number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="16155" LastEditorUserId="16155" LastEditDate="2023-04-22T03:19:33.443" LastActivityDate="2023-04-22T03:19:33.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26371" PostTypeId="2" ParentId="26359" CreationDate="2023-04-22T04:23:57.170" Score="10" Body="&lt;p&gt;Here are two typical examples in print and digital educational content of how this is done:&lt;/p&gt;&#xA;&lt;p&gt;From Open-Up Resources:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In your future studies, you may have opportunities to understand or write a proof that √2 is irrational, but for now, we just take it as a fact that √2 is irrational.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From the College of the Redwoods:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Archimedes (287-212 BC) was...able to prove that 223/71 &amp;lt; π &amp;lt; 22/7...Modern mathematicians have proved that π is an irrational number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is a set of proofs of irrationality intended for advanced students in 7th-8th grade or older at &lt;a href=&quot;https://www.algebra.com/algebra/homework/Number-Line/Proving-irrationality-of-some-real-numbers.lesson&quot; rel=&quot;noreferrer&quot;&gt;https://www.algebra.com/algebra/homework/Number-Line/Proving-irrationality-of-some-real-numbers.lesson&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22304" LastActivityDate="2023-04-22T04:23:57.170" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26372" PostTypeId="2" ParentId="26367" CreationDate="2023-04-22T05:33:20.993" Score="2" Body="&lt;p&gt;For a linear equation like &lt;span class=&quot;math-container&quot;&gt;$x = 3y - 7$&lt;/span&gt;, there is a simple algorithm, which in my classes I call the &amp;quot;general solving process&amp;quot;. The general rule is, &lt;strong&gt;apply inverses in reverse order of operations&lt;/strong&gt;. That is: when using inverse operations, everything works in reverse, including the ordering.&lt;/p&gt;&#xA;&lt;p&gt;Usually I give an analogy of building a house (put up foundation, walls, roof) versus doing demolition of a house (take down roof, walls, foundation).&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-04-22T05:33:20.993" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26375" PostTypeId="2" ParentId="26352" CreationDate="2023-04-22T14:46:31.800" Score="0" Body="&lt;p&gt;&lt;strong&gt;Function Notation f(x)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Instead of:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x,y: [(x,y)\in f\implies x\in A \land y\in B]\\&#xA;\land ~ \forall x\in A: \exists y:[(x,y)\in f ~ \land ~ \forall z: [(x,z)\in f \implies z=y]]$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can write:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall x\in A: f(x)\in B$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Both fully and formally describe an arbitrary function mapping elements of a domain &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and to elements of a codomain &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-04-22T21:55:28.380" LastActivityDate="2023-04-22T21:55:28.380" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26376" PostTypeId="2" ParentId="26352" CreationDate="2023-04-22T15:20:44.307" Score="0" Body="&lt;p&gt;I think tables are a very powerful method for making sense of logic problems.  Before, I knew the method, found these GRE/LSAT* problems much harder.  I think the benefit is in (1) overcoming the working memory bottleneck and (2) making sure all cases are covered.&lt;/p&gt;&#xA;&lt;p&gt;For example:  &lt;a href=&quot;https://www.youtube.com/watch?v=loyetRWK3VU&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=loyetRWK3VU&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;P.s.  Tables are not the only form of diagram for logic problems, but are pretty common.  Sometimes just linking networks are the diagrams.  If you check out an LSAT prep book, they will show several types of diagrams.&lt;/p&gt;&#xA;&lt;p&gt;*It looks like the GRE Analytical section has been changed into a writing sample.  Used to consist of LSAT style questions.  (Not sure if the writing sample benefits from tables/not.)&lt;/p&gt;&#xA;" OwnerUserId="22306" LastActivityDate="2023-04-22T15:20:44.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26377" PostTypeId="2" ParentId="26359" CreationDate="2023-04-22T16:49:40.550" Score="4" Body="&lt;p&gt;I looked through my old math textbooks that were given to me, having gone to school in Austria.&lt;/p&gt;&#xA;&lt;p&gt;In 4th grade of Gymnasium (8th grade of school in total, i.e. for students who are normally 13 or 14 years old), I had the book &amp;quot;Das ist Mathematik 4&amp;quot;, 2nd edition printed in 2007, ISBN-13 978-3-209-03636-0.&lt;/p&gt;&#xA;&lt;p&gt;Irrational numbers are first introduced on page 16, specifically giving the example of the square root of 32. It asks the question: Could we, if we had enough digits on our calculator, display the exact value of the square root of 32? Then it points out that no, we could not.&lt;/p&gt;&#xA;&lt;p&gt;On page 17, a proof by contradiction is written down for the example of the square root of 32. (As I am not actually a math educator and just found this question through the hot network questions, I am not going to reproduce that here. I think you can imagine it, it seems familiar to me from when the irrationality of the square root of 2 was discussed in university.)&lt;/p&gt;&#xA;&lt;p&gt;For pi specifically, the closest I could find in that book is on page 205, which discusses the history of pi. The last two paragraphs talk about the historical problem of &amp;quot;squaring the circle&amp;quot; and how only the German mathematician F. v. Lindemann managed near the end of the 19th century to prove that that problem cannot be solved in an exact way. It points out that mathematics can also prove the impossibility of something.&lt;/p&gt;&#xA;&lt;p&gt;That page does not use the term &amp;quot;irrational&amp;quot; at all. When a few pages earlier (on page 187) the term is mentioned, it is simply stated as a fact that pi is irrational.&lt;/p&gt;&#xA;&lt;p&gt;I was unable to find any more discussion of this in any textbooks of later grades either.&lt;/p&gt;&#xA;" OwnerUserId="22307" LastActivityDate="2023-04-22T16:49:40.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26378" PostTypeId="2" ParentId="26359" CreationDate="2023-04-22T19:38:24.437" Score="0" Body="&lt;p&gt;The irrationality of the square root of two is easy to proof. Irrationality of e is actually even easier. The irrationality of π is very hard to proof. So it's not so much a matter of &amp;quot;believing authority&amp;quot;, but accepting that some maths is beyond you. And beyond the teacher. BTW proving that e / π is irrational is one level harder.&lt;/p&gt;&#xA;&lt;p&gt;It's like in engineering, a bunch of talented 18 year olds with lots of time might be able to build a hang glider. A Wright brothers' style airplane is very likely beyond them, even if they are allowed to use a motorbike engine and as many parts from a scrap yard as they like. Building a Concorde - no way.&lt;/p&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2023-04-22T19:38:24.437" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26379" PostTypeId="1" AcceptedAnswerId="26381" CreationDate="2023-04-23T03:11:45.467" Score="2" ViewCount="212" Body="&lt;p&gt;If two integers are either both negative or both positive, we can say they have the same sign.&lt;/p&gt;&#xA;&lt;p&gt;How about two integers that are either both odd or both even? Is there any term for them?&lt;/p&gt;&#xA;" OwnerUserId="22308" LastEditorUserId="2083" LastEditDate="2023-04-23T06:28:05.477" LastActivityDate="2023-04-23T15:35:51.483" Title="What is the terminology for integers with the same oddness or evenness?" Tags="|terminology|number-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26380" PostTypeId="1" CreationDate="2023-04-23T03:17:47.080" Score="0" ViewCount="272" Body="&lt;p&gt;What would you suggest as examples to demonstrate as applications of differentiation in finding integer solutions of an equation for advanced level students? &lt;br/&gt;&#xA;Here you have one example which I have come across so far relevant to the mentioned issue. &lt;br/&gt;&#xA;To prove that there is only one positive integer pair &lt;span class=&quot;math-container&quot;&gt;$( n,m )$&lt;/span&gt; satisfying &lt;span class=&quot;math-container&quot;&gt;$n^m = m^n$&lt;/span&gt; we can use the knowledge of stationary points of the graph &lt;span class=&quot;math-container&quot;&gt;$y = (ln x)/x$&lt;/span&gt; . Since the only stationary point is at &lt;span class=&quot;math-container&quot;&gt;$x = e (&amp;gt;2)$&lt;/span&gt; and it is a maximum we can say that one value of the pair &lt;span class=&quot;math-container&quot;&gt;$(n,m)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;. Hence we can find the other value of the pair. &lt;br/&gt;&#xA;Edit ; &lt;span class=&quot;math-container&quot;&gt;$n , m $&lt;/span&gt; are distinct values &lt;br/&gt;&#xA;Here is the generalized version which I could able to construct, &lt;br/&gt;&#xA;How can you prove that there is only one triplet &lt;span class=&quot;math-container&quot;&gt;$(n,m,p)$&lt;/span&gt; of positive integers satisfying the equation &lt;span class=&quot;math-container&quot;&gt;$$n^{m^p} = m^{n^p}$$&lt;/span&gt; given that &lt;span class=&quot;math-container&quot;&gt;$ m ≠ n $&lt;/span&gt;. I would like to know different types of examples.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-04-23T07:34:26.317" LastActivityDate="2023-05-05T01:39:50.173" Title="Differentiation in integer solutions" Tags="|secondary-education|precalculus|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26381" PostTypeId="2" ParentId="26379" CreationDate="2023-04-23T15:35:51.483" Score="8" Body="&lt;p&gt;Expanding on a comment by @TomKern: The word you're looking for is &lt;strong&gt;parity&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You can see it in most standard English dictionaries, such as &lt;a href=&quot;https://www.merriam-webster.com/dictionary/parity&quot; rel=&quot;noreferrer&quot;&gt;Merriam-Webster&lt;/a&gt;, here:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;4 a: the property of an integer with respect to being odd or even&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;3 and 7 have the same parity&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;b(1): the state of being odd or even used as the basis of a method of&#xA;detecting errors in binary-coded data (see binary entry 2 sense 5)&lt;/p&gt;&#xA;&lt;p&gt;b(2): parity bit&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And here's the &lt;a href=&quot;https://en.wikipedia.org/wiki/Parity_(mathematics)&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; article on the subject.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In mathematics, &lt;strong&gt;parity&lt;/strong&gt; is the property of an integer of whether it&#xA;is &lt;strong&gt;even&lt;/strong&gt; or &lt;strong&gt;odd&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And here's the equivalent question at &lt;a href=&quot;https://english.stackexchange.com/questions/434962/word-that-means-the-classification-of-a-number-as-even-or-odd&quot;&gt;Stack Exchange: English Language &amp;amp; Usage&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-04-23T15:35:51.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26382" PostTypeId="1" CreationDate="2023-04-23T15:38:56.977" Score="11" ViewCount="1350" Body="&lt;p&gt;I recently came across a thought-provoking &lt;a href=&quot;https://youtu.be/dk_NI1jfdzE&quot; rel=&quot;nofollow noreferrer&quot;&gt;video&lt;/a&gt; where Simon Sinek explains that the human brain struggles to process negative statements. In the video, Sinek states that skiers should not spend their time thinking &amp;quot;don’t hit the trees&amp;quot; because all they’ll notice is the trees on the mountain. Instead, Sinek suggests that skiers should focus on the positive statement of &amp;quot;follow the path,&amp;quot; which directs their attention to the desired outcome of navigating through the trees.&lt;/p&gt;&#xA;&lt;p&gt;I was particularly intrigued by Sinek's idea because I am preparing to teach a lesson on error-analysis in solving trigonometric equations to my Pre-Calculus students. I have always found error-analysis lessons to be valuable in helping students build confidence and improve their performance on summative assessments. Additionally, a quick &lt;a href=&quot;https://www.google.com/search?q=Are%20error%20analysis%20lessons%20effective%3F&amp;amp;rlz=1C1VDKB_enUS1003US1003&amp;amp;ei=F01FZJLyIPDi5NoP6KWXoAk&amp;amp;ved=0ahUKEwjS2dGcp8D-AhVwMVkFHejSBZQQ4dUDCA8&amp;amp;uact=5&amp;amp;oq=Are%20error%20analysis%20lessons%20effective%3F&amp;amp;gs_lcp=Cgxnd3Mtd2l6LXNlcnAQAzIFCCEQoAEyBQghEKABMgUIIRCgAToFCAAQogRKBAhBGABQAFiIBmCrC2gAcAF4AIABWYgB5QGSAQEzmAEAoAEBwAEB&amp;amp;sclient=gws-wiz-serp&amp;amp;safe=active&amp;amp;ssui=on&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google search&lt;/a&gt; reveals numerous benefits of incorporating error-analysis in mathematics education.&lt;/p&gt;&#xA;&lt;p&gt;However, I feel that Sinek's idea raises a valid concern: since error-analysis lessons inherently involve showing students what &lt;strong&gt;not&lt;/strong&gt; to do, could these types of lessons potentially hinder their mathematical development by triggering a focus on negative statements? Or is there a flaw in Sinek's argument? I am eager to hear your thoughts on this matter.&lt;/p&gt;&#xA;&lt;p&gt;(Please note that my intention is not to discredit Sinek's idea, but rather to critically examine its potential implications in the context of mathematics education.)&lt;/p&gt;&#xA;" OwnerUserId="13657" LastEditorUserId="13657" LastEditDate="2023-04-25T00:25:38.180" LastActivityDate="2023-04-26T15:35:17.530" Title="Are Error-Analysis Lessons Effective?" Tags="|secondary-education|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26383" PostTypeId="2" ParentId="26382" CreationDate="2023-04-23T17:50:28.160" Score="3" Body="&lt;p&gt;My advice is to do very little of it, if any.  The neophyte learner needs to learn how to do the basic procedures first and then to hone his automaticity.  To the extent that you cover errors, it makes sense AFTER the mistake, as a correction.  If you try to do it pre-emptively, you are likely to be giving too elaborate of an instruction (remember the new learner is mastering the procedure first, so long winded instructions tax their working memory).  You want to give bite sized instruction and intersperse it with successful practice in a gradually scaling up behavior.  If you have experienced students, it may make sense to discuss common errors, but that is different from preemptive instruction.&lt;/p&gt;&#xA;&lt;p&gt;See, the last section of this article:  &lt;a href=&quot;https://theconversation.com/why-students-make-silly-mistakes-in-class-and-what-can-be-done-48826&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://theconversation.com/why-students-make-silly-mistakes-in-class-and-what-can-be-done-48826&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'd say after a test, makes most sense.  There's the benefit of huge investment that the learners have made (thus interest).  Also, they likely know the material the best they will (as a point in time).  Also, they have strong familiarity with the examples (having worked the problems and perhaps even having a key to refer to, if you provide those).&lt;/p&gt;&#xA;&lt;p&gt;I have found that with some common errors, it is helpful to have acronyms or pet names.  E.g. SF-1 (in red pen).  Sig figs or significant figures, -1 point.  Similarly maybe with +C on indefinite integrals. (Is there a cute name for that?  I do know Jamie Escalante had cute names for common mistakes.)  But even here, you probably need to just use correction of the mistakes as the main training method.  If you strain too hard preemptively, it won't stop the errors and there are much bigger (new) aspects of the chemistry or calculus problem the trainees are demonstrating, and have not yet mastered.&lt;/p&gt;&#xA;&lt;p&gt;Probably there are other times when it makes sense to stress error analysis, but they are not probably germane to typical math/school learning.  For example, dealing with very risky and expensive operations, like flying an F-14. Even here, you need to be wary as trainees need the situational awareness to grasp the watchouts that are being stressed.  Telling me all the things I could do wrong is hard for me to remember if I don't know the basics of flying.&lt;/p&gt;&#xA;&lt;p&gt;N.B.  All this said, if you're doing it and it's working, use your judgment.&lt;/p&gt;&#xA;&lt;p&gt;P.s. I've been a ski instructor, in a very good ski school, at a destination resort in the Rockies.  That example feels very made up education writing to me.  First of all, it's rare that you would teach tree skiing (given the legal constraints).  If you did, it would only be with very advanced skiers.  Also, it's obvious even to 5 year old &amp;quot;never, ever&amp;quot; (neophyte) skiers that you don't hit trees.  It's honestly something that you don't have to tell them.  Yes, with advanced skiers, you would tell them to pick a line (not &amp;quot;the path&amp;quot;), but this is similar to how you slalom flags or deal with moguls.&lt;/p&gt;&#xA;" OwnerUserId="22310" LastEditorUserId="22310" LastEditDate="2023-04-23T17:52:32.913" LastActivityDate="2023-04-23T17:52:32.913" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26384" PostTypeId="2" ParentId="26382" CreationDate="2023-04-23T18:39:23.327" Score="3" Body="&lt;p&gt;I think an important step in addressing this question is to unpack the term “error analysis.” There are many pedagogical moves that I would argue can reasonably be called “error analysis.” Here are a few that come to mind, in no particular order.&lt;/p&gt;&#xA;&lt;p&gt;The teacher presents students with an incorrect worked example…&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;without a label and asks students to judge its correctness.&lt;/li&gt;&#xA;&lt;li&gt;and asks students to find the errors.&lt;/li&gt;&#xA;&lt;li&gt;and asks students to justify why the work is erroneous.&lt;/li&gt;&#xA;&lt;li&gt;and asks students to describe why someone may have made the errors.&lt;/li&gt;&#xA;&lt;li&gt;and asks students to correct the work.&lt;/li&gt;&#xA;&lt;li&gt;alongside a correct example and asks students to draw comparisons.&lt;/li&gt;&#xA;&lt;li&gt;and tells students which errors to avoid.&lt;/li&gt;&#xA;&lt;li&gt;and asks students to describe how they will avoid making the same errors in the future.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;A teacher can use more than one of these moves in a single lesson. They can have students work individually or in groups. The error analysis task can be implemented before/after explicit instruction, or before/after an assessment. The teacher can have students inspect their own erroneous work.&lt;/p&gt;&#xA;&lt;p&gt;It’s reasonable that different implementations of “error analysis” will achieve different effects, and this is without even considering student variability! I think this is a big part of why conclusions in mathematics education literature about “error analysis” or “erroneous examples” are mixed.&lt;/p&gt;&#xA;&lt;p&gt;To address Sinek’s idea, his examples of thinking of elephants and skiing into trees both involve actions which occur because of a lack of conscious control. But as &lt;a href=&quot;https://matheducators.stackexchange.com/a/26383/20719&quot;&gt;guest troll pointed out,&lt;/a&gt; post-assessment error analysis activities are primarily intended to help students recognize their own erroneous automatic behaviors and develop conscious control.&lt;/p&gt;&#xA;&lt;p&gt;I do think that error-analysis activities can effectively serve other purposes. For example, prior to explicit instruction, counterintuitive erroneous examples can generate cognitive need, e.g. discussing the Monty Hall problem before instruction on conditional probability.&lt;/p&gt;&#xA;&lt;p&gt;For these types of error analysis activities, it seems like a good idea to (1) “contain” the error within a specific and memorable problem context and (2) to avoid unnecessary exposure to erroneous procedures which are easily automatized, e.g. &lt;a href=&quot;https://matheducators.stackexchange.com/questions/26382/are-error-analysis-lessons-effective#comment69787_26382&quot;&gt;Daniel’s example&lt;/a&gt; of &lt;span class=&quot;math-container&quot;&gt;$(a+b)^2 \neq a^2 + b^2$&lt;/span&gt;. (I never write an erroneous identity like this in front of students without the not equal symbol or some other prominent qualification.)&lt;/p&gt;&#xA;&lt;p&gt;Ultimately, you know the effects of your lessons on your students best. Run an experiment and collect some data if you want to test your hypotheses! And exercise your professional judgment.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-04-23T18:39:23.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26386" PostTypeId="2" ParentId="26382" CreationDate="2023-04-23T19:21:27.683" Score="8" Body="&lt;p&gt;I prefer to do in the format &amp;quot;what's wrong with this argument/answer? Can you explain why it is invalid?&amp;quot; than in the format &amp;quot;here is a common mistake, lo and behold!&amp;quot;, i.e., concentrating on the &lt;em&gt;techniques&lt;/em&gt; of (self-) checking the solutions (from the trivial ones like &amp;quot;plug in the answer into the equation&amp;quot; and &amp;quot;check your identities on small integers&amp;quot; to more advanced ideas like &amp;quot;homogeneity&amp;quot;, &amp;quot;symmetry&amp;quot;, &amp;quot;letting the parameters to their extreme values&amp;quot;, and such) more than on the errors themselves. I'm not sure if it has anything to do with &amp;quot;avoiding negative thinking&amp;quot;. Rather it is replacing &amp;quot;don't hit the trees&amp;quot; with &amp;quot;identify obstacles and avoid them as you go&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Ability to detect that something went wrong, a.k.a. &amp;quot;sanity check&amp;quot; is quite important IMHO, so devoting to it some time and developing the corresponding series of exercises seems to make perfect sense. Then, of course, one needs also to develop the ability to actually pinpoint the error, but it is a completely different story.&lt;/p&gt;&#xA;&lt;p&gt;However the main advice is this: be open to all ideas and feel free to experiment with them, but if something works for you with reliable results, stick to it, and if something doesn't, abandon it no matter what anyone may say.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-04-23T19:21:27.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26387" PostTypeId="2" ParentId="26382" CreationDate="2023-04-23T19:58:51.407" Score="1" Body="&lt;p&gt;To a skier &amp;quot;following the path&amp;quot; is an obvious task, as your post suggests that there is a clear designated path and all the skier needs to do is to follow it. I highly doubt that to a Pre-Calculus student who is just learning solving trigonometric equations the path is as clear as the ski path. In fact, if the path were clear to the students and all they had to do was to follow it, solving trigonometric equations would not be so difficult. If students make mistakes, it is because the path that they need to follow is not so clear yet, not because they are not focusing on following the path. So how do you clarify the path to the student? At least the skier knows that there is something called &amp;quot;tree&amp;quot; that should be avoided. Now if you want the skier not to focus on the trees and just focus on staying on the path, that's fine. But does a student know what a &amp;quot;tree&amp;quot; is in solving a trigonometric equation? I doubt that. So maybe we should show our students the &amp;quot;trees&amp;quot; first, then ask them to follow the path and not focus on the &amp;quot;trees&amp;quot;. In my opinion to achieve an effective teaching, an important requirement is to believe that our students are intelligent and talented. I find it absurd to think that if we show our students possible errors, they will make the same errors. This idea may come across as not believing in students' intelligence.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-04-23T20:36:37.793" LastActivityDate="2023-04-23T20:36:37.793" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26388" PostTypeId="2" ParentId="26380" CreationDate="2023-04-23T22:33:53.910" Score="0" Body="&lt;p&gt;Here is a similar but slightly different example:&lt;/p&gt;&#xA;&lt;p&gt;Show that there are no distinct integers &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; satisfying &lt;span class=&quot;math-container&quot;&gt;$e^n-n=e^m-m$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Suppose there are such integral solutions &lt;span class=&quot;math-container&quot;&gt;$m&amp;lt;n$&lt;/span&gt;. Then applying the Mean Value Theorem to the function &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; on the interval &lt;span class=&quot;math-container&quot;&gt;$[m,n]$&lt;/span&gt;, we see that there exists a number &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; between &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$e^c=1$&lt;/span&gt;. But then we must have &lt;span class=&quot;math-container&quot;&gt;$c=0$&lt;/span&gt;. This shows that &lt;span class=&quot;math-container&quot;&gt;$m&amp;lt;0&amp;lt;n$&lt;/span&gt;. Now, from &lt;span class=&quot;math-container&quot;&gt;$e^n-n=e^m-m$&lt;/span&gt; it follows &lt;span class=&quot;math-container&quot;&gt;$e^n-e^m=n-m$&lt;/span&gt;. Multiplying both sides by &lt;span class=&quot;math-container&quot;&gt;$e^{-m}$&lt;/span&gt; we obtain:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$e^{n-m}-1=(n-m)e^{-m}\ \ \mathrm{or}\ \ e^{n-m}-(n-m)e^{-m}-1=0.$$&lt;/span&gt;&#xA;In other words, &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; satisfies the algebraic equation &lt;span class=&quot;math-container&quot;&gt;$x^{n-m}-(n-m)x^{-m}-1=0$&lt;/span&gt;, which is a contradiction, as &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is not algebraic. Note that since &lt;span class=&quot;math-container&quot;&gt;$m&amp;lt;0$&lt;/span&gt;, the exponent of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$x^{-m}$&lt;/span&gt; is positive.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;More generally:&lt;/strong&gt; The equation &lt;span class=&quot;math-container&quot;&gt;$p^n-n=p^m-m$&lt;/span&gt; does not have any integral solutions in &lt;span class=&quot;math-container&quot;&gt;$p,m,n$&lt;/span&gt; satisfying &lt;span class=&quot;math-container&quot;&gt;$p&amp;gt;2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m\neq n$&lt;/span&gt;. (For &lt;span class=&quot;math-container&quot;&gt;$p=2$&lt;/span&gt; there is a solution &lt;span class=&quot;math-container&quot;&gt;$m=0, n=1$&lt;/span&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;The original equation &lt;span class=&quot;math-container&quot;&gt;$e^n-n=e^m-m$&lt;/span&gt; could be solved without using any calculus, as pointed out by fedja in a comment, only using the fact that &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is not algebraic. To show that the equation &lt;span class=&quot;math-container&quot;&gt;$p^n-n=p^m-m$&lt;/span&gt; does not have any integral solutions in &lt;span class=&quot;math-container&quot;&gt;$p,m,n$&lt;/span&gt; satisfying &lt;span class=&quot;math-container&quot;&gt;$p&amp;gt;2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m\neq n$&lt;/span&gt;, suppose there are such integral solutions &lt;span class=&quot;math-container&quot;&gt;$p&amp;gt;2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m&amp;lt;n$&lt;/span&gt;. As &lt;span class=&quot;math-container&quot;&gt;$p&amp;gt;2$&lt;/span&gt;, it is easy to see that &lt;span class=&quot;math-container&quot;&gt;$m\neq0$&lt;/span&gt;. Then applying the Mean Value Theorem to the function &lt;span class=&quot;math-container&quot;&gt;$p^x$&lt;/span&gt; on the interval &lt;span class=&quot;math-container&quot;&gt;$[m,n]$&lt;/span&gt;, we see that there exists a number &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; between &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$p^c=1$&lt;/span&gt;. But then we must have &lt;span class=&quot;math-container&quot;&gt;$c=0$&lt;/span&gt;. This shows that &lt;span class=&quot;math-container&quot;&gt;$m&amp;lt;0&amp;lt;n$&lt;/span&gt;. Now, from &lt;span class=&quot;math-container&quot;&gt;$p^n-n=p^m-m$&lt;/span&gt; it follows &lt;span class=&quot;math-container&quot;&gt;$p^n-p^m=n-m$&lt;/span&gt;. Multiplying both sides by &lt;span class=&quot;math-container&quot;&gt;$p^{-m}$&lt;/span&gt; we obtain:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$p^{n-m}-1=(n-m)p^{-m}\ \ \mathrm{or}\ \ p^{n-m}-(n-m)p^{-m}=1.$$&lt;/span&gt;&#xA;This means &lt;span class=&quot;math-container&quot;&gt;$p^{n-m}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p^{-m}$&lt;/span&gt; are coprime, which is absurd, as &lt;span class=&quot;math-container&quot;&gt;$n-m&amp;gt;0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-m&amp;gt;0$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-04-30T01:39:58.180" LastActivityDate="2023-04-30T01:39:58.180" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26389" PostTypeId="1" CreationDate="2023-04-24T04:20:02.477" Score="17" ViewCount="4596" Body="&lt;p&gt;I would like a critique of this approach to teaching continuity to calculus 1 students.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Show them that for an increasing function on (a,b) we have that (a,b) is contained in the set of solutions to &lt;span class=&quot;math-container&quot;&gt;$f(a) &amp;lt; f(x) &amp;lt; f(b)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For decreasing function we get (a, b) is contained in the set of solutions to &lt;span class=&quot;math-container&quot;&gt;$f(b) &amp;lt; f(x) &amp;lt; f(a)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Define pre-image of a point and then of an open interval.  The solutions to the equation &lt;span class=&quot;math-container&quot;&gt;$f(x) =a$&lt;/span&gt; is the pre-image of a.  The solutions to &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;f(x)&amp;lt;b$&lt;/span&gt; are the pre-image of &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A function is continuous at x if the pre-image of every open interval containing f(x) contains an open interval containing x.  In English this is hard to understand/state but it’s clear geometrically by looking at graphs of functions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It is easy to show that functions like &lt;span class=&quot;math-container&quot;&gt;$\sin{x}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt; are increasing or decreasing on certain intervals and thus it’s relatively simple to show continuity of these functions at specific points.&lt;/p&gt;&#xA;&lt;p&gt;My main problem with epsilon-delta proofs is that we don’t need to find delta.  It’s clear that if c is in (a, b) then there is an open interval centered at c contained in (a,b).  Given epsilon I just need to show that the set of solutions to &lt;span class=&quot;math-container&quot;&gt;$L-\epsilon &amp;lt; f(x) &amp;lt; L+\epsilon$&lt;/span&gt; contains an open interval containing c.&lt;/p&gt;&#xA;&lt;p&gt;After teaching continuity I then plan to introduce limits.  Limit at x existing is weaker than continuity because we just need the pre-image of an open interval to contain an open interval with x removed.  A punctured interval.&lt;/p&gt;&#xA;&lt;p&gt;Am I crazy for doing this approach?&lt;/p&gt;&#xA;&lt;p&gt;EDIT:  What’s easier to do:  epsilon-delta proof for continuity of &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt; or show that &lt;span class=&quot;math-container&quot;&gt;$4-\epsilon&amp;lt; x^2&amp;lt;4+\epsilon$&lt;/span&gt; can be solved easily since &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; is increasing for positive numbers and the solutions are &lt;span class=&quot;math-container&quot;&gt;$\sqrt{4-\epsilon} &amp;lt; x &amp;lt; \sqrt{4+\epsilon}$&lt;/span&gt;.  This latter interval obviously contains 2.&lt;/p&gt;&#xA;" OwnerDisplayName="user22312" LastEditorUserId="1385" LastEditorDisplayName="user22312" LastEditDate="2023-06-12T10:04:38.107" LastActivityDate="2023-06-12T10:04:38.107" Title="Why don’t we teach a topological view of continuity instead of epsilon-delta?" Tags="|undergraduate-education|calculus|infinitesimals|" AnswerCount="7" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="26390" PostTypeId="2" ParentId="26389" CreationDate="2023-04-24T05:21:44.577" Score="26" Body="&lt;p&gt;Yes.  You are crazy to be spending time on another effort to bring real analysis style rigor into a calc 1 course.  In this case, even off brand real analysis.&lt;/p&gt;&#xA;&lt;p&gt;Instead of a &amp;quot;different&amp;quot; way to deal with continuity, don't obsess on it at all.  Do the bare minimum and move on.  Teach them the chain rule and how to integrate by parts.&lt;/p&gt;&#xA;&lt;p&gt;I'm serious...you need to prioritize.  You are constrained in time, student interest, and student intellect.  Get the basics done, that they need for physics class.  Even that can be hard!  Make a limited objective and get a victory.  Not a trainwreck.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1234&amp;amp;context=hmnj&quot; rel=&quot;noreferrer&quot;&gt;https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1234&amp;amp;context=hmnj&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22313" LastActivityDate="2023-04-24T05:21:44.577" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26391" PostTypeId="2" ParentId="26389" CreationDate="2023-04-24T12:55:36.930" Score="9" Body="&lt;p&gt;A positive attribute of the standard &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; definition continuity is that the same formalism can be mutated to define similar things such as limits at infinity or the limit of a sequence &lt;span class=&quot;math-container&quot;&gt;$\displaystyle{\lim_{n\to\infty}a_n}$&lt;/span&gt;. How would your formalism be adaptable  to these?&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-04-24T12:55:36.930" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26392" PostTypeId="2" ParentId="26389" CreationDate="2023-04-24T13:21:42.957" Score="15" Body="&lt;p&gt;You aren't crazy because Leonard Gillman wasn't crazy. See &lt;a href=&quot;https://www.utm.edu/staff/jschomme/topology/c/a/a/g/10.htm&quot; rel=&quot;noreferrer&quot;&gt;Emphasizing order relations rather than metric concepts in real analysis&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The family of all open intervals (A, B) about a point L assuredly constitutes a base for the neighborhood system at L. There is no need to restrict oneself to intervals (L - e , L + e), whose very notation suggests computation. Indeed the order relations A &amp;lt; L &amp;lt; B are usually sufficient, the distances of A and B to L being a distraction. One advantage is to simplify the notation and, in notable cases, to drastically simplify a proof. Another is to obtain a definition of limit that is less intimidating than epsilon-delta. Examples are given from elementary and semi-advanced calculus. Other, somewhat related topics are a characterization of limit that is very easy to understand, yet is completely rigorous; the composition of limits; the exceptional case in the proof of the chain rule; and a simple proof of L'Hôpital's rule.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My personal view is that &lt;a href=&quot;https://people.math.wisc.edu/%7Ehkeisler/calc.html&quot; rel=&quot;noreferrer&quot;&gt;Howard Jerome Keisler&lt;/a&gt; is even less crazy than Gillman. Engineers, scientists, and basically everyone except mathematicians and students of history should have the epsilon-delta definition of the limit entirely removed from their curriculum and rely on the infinitesimal quantities of Leibniz brought up to full rigor by the hyperreal number system of Abraham Robinson in the unfortunately-named nonstandard analysis.&lt;/p&gt;&#xA;" OwnerUserId="22304" LastActivityDate="2023-04-24T13:21:42.957" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26393" PostTypeId="2" ParentId="26389" CreationDate="2023-04-24T20:44:07.800" Score="9" Body="&lt;p&gt;One approach that has been advocated several times is to significantly decrease the focus the focus on “pointwise continuity” and “pointwise differentiability” notions (or entirely omit them) and instead work with their uniform versions. Two people I can think of who have advocated this approach in the last 20-30 years are &lt;a href=&quot;https://www.google.com/search?q=Peter+Lax+%22calculus%22+%22uniform+continuity%22&amp;amp;filter=0&quot; rel=&quot;noreferrer&quot;&gt;Peter D. Lax&lt;/a&gt; (see his letter on p. 6 of the &lt;a href=&quot;https://www.ams.org/journals/notices/199801/199801FullIssue.pdf&quot; rel=&quot;noreferrer&quot;&gt;Jan 1998 &lt;em&gt;Notices of the AMS&lt;/em&gt;&lt;/a&gt;, which I believed generated several follow-up letters in subsequent months) and &lt;a href=&quot;https://www.google.com/search?q=Mark+Bridger+%22uniform+continuity%22&amp;amp;filter=0&quot; rel=&quot;noreferrer&quot;&gt;Mark Bridger&lt;/a&gt; (see his &lt;a href=&quot;https://www.jstor.org/stable/2589492&quot; rel=&quot;noreferrer&quot;&gt;1999 paper co-authored with Stolzenberg&lt;/a&gt; and his &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0471792306&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;2007 real analysis textbook&lt;/a&gt; which was &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1470451441&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;reprinted in 2019&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Regarding Bridger, in the late 1990s (beginning I think around May 1996, and maybe continuing into the first few years of the 2000s) he made a lot of posts in Math Forum groups (especially CALC-REFORM) in which he strongly advocated using uniform notions of continuity and differentiability in calculus, and even in elementary real analysis. Unfortunately, &lt;a href=&quot;https://www.nctm.org/&quot; rel=&quot;noreferrer&quot;&gt;NCTM&lt;/a&gt; took over Math Forum in the mid 2010s and within a few years they took down the tens of thousands (hundreds of thousands?) of Math Forum posts, some of which were of extremely high quality, especially back when Bridger’s posts were made. Perhaps someone better than I am at finding such posts at the &lt;a href=&quot;https://archive.org/web/&quot; rel=&quot;noreferrer&quot;&gt;Internet Archive&lt;/a&gt; can locate some of them, at least if they exist there. (The calc-reform URLs in the late 1990s began with &lt;a href=&quot;http://forum.swarthmore.edu/epigone/calc-reform&quot; rel=&quot;noreferrer&quot;&gt;http://forum.swarthmore.edu/epigone/calc-reform&lt;/a&gt; and at some point in the early 2000s this initial portion was changed to &lt;a href=&quot;http://mathforum.org/epigone/calc-reform&quot; rel=&quot;noreferrer&quot;&gt;http://mathforum.org/epigone/calc-reform&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;I've also seen this advocacy of “uniform over pointwise” notions argued in some non-research journal papers (going back to at least the early 1930s, I think), but I don't know any specific references at the moment. I think I have photocopies of some of these papers in the notebook binder I used for many of the references in &lt;a href=&quot;https://matheducators.stackexchange.com/a/1414/745&quot;&gt;this answer&lt;/a&gt;, but I can’t find that binder now.&lt;/p&gt;&#xA;" OwnerUserId="745" LastActivityDate="2023-04-24T20:44:07.800" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26394" PostTypeId="2" ParentId="26389" CreationDate="2023-04-25T01:37:34.857" Score="11" Body="&lt;p&gt;If you want to teach a topological view of continuity, then you have to teach topology first. The set of open intervals is a basis of a topology, but it's not itself a topology, and the topology generated by it is not the only topology possible on the real line. So you'll have to either introduce a bunch of stuff about abstract topology in general, and then proceed to say &amp;quot;But never mind that, we're only going to concern ourselves with one particular topology&amp;quot;, or you have to pretend that open intervals are a topology, leaving any students who do go on to study topology to have to re-learn what &amp;quot;topology&amp;quot; means.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, open intervals are equivalent to the &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; definition. If there is some open interval about &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, it can be expressed as &lt;span class=&quot;math-container&quot;&gt;$(x-d_1, x+d_2)$&lt;/span&gt;, and if we take &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; as the minimum of &lt;span class=&quot;math-container&quot;&gt;$\{d_1,d_2\}$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$(x-\delta, x+\delta)$&lt;/span&gt; is a subset of that interval.&lt;/p&gt;&#xA;&lt;p&gt;Also, you used &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to mean two different things, which you should avoid when teaching.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is easy to show that functions like &lt;span class=&quot;math-container&quot;&gt;$\sin{x}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt;&#xA;are increasing or decreasing on certain intervals and thus it’s relatively simple to show continuity of these functions at specific points.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Increasing functions aren't necessarily continuous. Your implicit argument seems to be that given an open interval &lt;span class=&quot;math-container&quot;&gt;$(A, B)$&lt;/span&gt; containing &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, we can express &lt;span class=&quot;math-container&quot;&gt;$(A, B)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$(f(a), f(b))$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt;.  But that isn't necessarily true. Given two values that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; takes, we can use the IMV to show that it takes all values in between, but the IMV requires continuity, which is what we're trying to prove.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastEditorUserId="8989" LastEditDate="2023-04-29T15:43:52.547" LastActivityDate="2023-04-29T15:43:52.547" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26395" PostTypeId="2" ParentId="26382" CreationDate="2023-04-25T23:25:21.993" Score="2" Body="&lt;p&gt;The idea is flawed.&lt;/p&gt;&#xA;&lt;p&gt;In the instance described of the skier, the skier going down a slope is in the middle of their performance. To place the math student as the skier in a equivalent position would be putting the math student in a test, or in a competition.&lt;/p&gt;&#xA;&lt;p&gt;I have yet to see ski lessons that begin with a student already in skis going down a slope or anywhere near the slope.&lt;/p&gt;&#xA;&lt;p&gt;The ski instructor will definitely be advising a student skier of the hazards prior to the skier ever getting to the top of the hill. In addition to this, a good instructor of any skill/discipline will advise of how to minimize damage upon contact with a hazard.&lt;/p&gt;&#xA;&lt;p&gt;At best, this lesson should be adapted to the idea that obvious mistakes shouldn't have focus, but the more dangerous ones should. You don't want to hit the trees, or focus on trees, because they're obvious. This is akin to talking to your pre-calculus class about common arithmetic mistakes made in grade school.&lt;/p&gt;&#xA;&lt;p&gt;To know what hazards are obvious to your students you need to know your students. An easy and lazy method is to assume they've mastered any prerequisite material, so only focus on errors that the new material introduces.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes there are subtle differences between something being correct or incorrect. In these cases, it can be difficult to teach a student to remain correct with only showing the correct version. Sometimes the differences are best when clearly shown both.&lt;/p&gt;&#xA;" OwnerUserId="19262" LastEditorUserId="19262" LastEditDate="2023-04-26T15:35:17.530" LastActivityDate="2023-04-26T15:35:17.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26396" PostTypeId="2" ParentId="26389" CreationDate="2023-04-26T03:39:27.247" Score="12" Body="&lt;p&gt;If you want the comment on the approach yourself rather than on whether you should try to implement it, the key words in your post are&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Given epsilon I just need to show that the set of solutions to &lt;span class=&quot;math-container&quot;&gt;$L−\varepsilon&amp;lt;f(x)&amp;lt;L+\varepsilon$&lt;/span&gt; contains an open interval containing &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;That is absolutely correct, but even for strictly increasing functions, your (1) is of little help in that task unless I misunderstand your logic entirely, so I just do not see how you are going to manage for any function that doesn't have an obvious formula for its inverse, say a third degree polynomial.&lt;/p&gt;&#xA;&lt;p&gt;As for &amp;quot;&lt;em&gt;it’s clear geometrically by looking at graphs of functions&lt;/em&gt;&amp;quot;, do not forget that those graphs are drawn &lt;em&gt;from the idea that we have continuity&lt;/em&gt;. Besides, I have never seen &lt;em&gt;a perfectly accurate&lt;/em&gt; graph of &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; in my life: even the computer generated images cannot go beyond the resolution of the screen and the hand-drawn ones hardly ever go beyond the 5% accuracy, so even for some relatively large &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; like &lt;span class=&quot;math-container&quot;&gt;$10^{-239}$&lt;/span&gt; you cannot see much on any of them, forget about really small values. The whole point (if you want to introduce rigor at all) is to teach the students to develop healthy doubts about what they &amp;quot;clearly see&amp;quot; and to convert &amp;quot;obvious&amp;quot; pictures into impeccable arguments. Replacing one hand-waving style with another hand-waving style will accomplish very little, if anything.&lt;/p&gt;&#xA;&lt;p&gt;One personal story in the end. When I was in the 8-th grade, I was on some math. olympiad where there was a problem about a few quadratic equations (something in the style of if two quadratic polynomials with some coefficients depending on a couple of free parameters attain some values somewhere, then so does the third). I figured out that the third quadratic polynomial had two points where it was above and below the target and happily said to the examiner &amp;quot;since a quadratic polynomial is a continuous function, it must take the intermediate value as well&amp;quot;. The response was &amp;quot;and what is a continuous function?&amp;quot;. I said &amp;quot;Well, it is a function whose graph is a curve&amp;quot;, to which the response was &amp;quot;and what is a curve?&amp;quot;. After several more attempts, with a sweaty back, I managed to get away with some explicit algebraic monster using square roots about which I proved that their arguments were positive.&lt;/p&gt;&#xA;&lt;p&gt;Four years later, when I saw the Weierstrass &lt;span class=&quot;math-container&quot;&gt;$\varepsilon-\delta$&lt;/span&gt; definition and the proof of the IVT in the first analysis university course, my reaction was &amp;quot;OMG! How simple and elegant!&amp;quot;. What's the moral? IMHO, it is &amp;quot;One cannot appreciate the beauty before he is thrown face down into the ugliness&amp;quot;. Whether you want to make your students sweat a bit before revealing what you call the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\varepsilon-\delta$&lt;/span&gt; monstrosity&amp;quot; is up to you, but I'm ready to bet that without that sweat they'll never appreciate how amazingly simple and clear-cut it really is...&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-04-26T03:39:27.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26399" PostTypeId="1" AcceptedAnswerId="26404" CreationDate="2023-04-27T15:36:17.440" Score="2" ViewCount="184" Body="&lt;p&gt;I am studying (self learner) linear equations/equation of line and my idea is to discover the equations myself rather than try and understand ready-made equations available in text books. I am using X-Y Cartesian plane, with 2 points A(x1,y1) and B(x2,y2). I know linear equations describe the relation between two points in my case Point A and Point B. So I was wondering if I come up with a definition in words and more Importantly as an Algebraic Equation; How can I make sure that my equation describes the relation &lt;strong&gt;Fully/Completely&lt;/strong&gt;?! Is there a method/procedure to do this?! Also other than &amp;quot;completeness&amp;quot;, are there other things/parameters a definition/description/equation is supposed to fulfill?&lt;/p&gt;&#xA;&lt;p&gt;In addition to explaining it, any pointers, or links to relevant content are also hugely welcome.&lt;/p&gt;&#xA;&lt;p&gt;Thanks much in Advance...&lt;/p&gt;&#xA;" OwnerUserId="12610" LastEditorUserId="2083" LastEditDate="2023-04-27T16:42:52.420" LastActivityDate="2023-05-01T16:27:48.347" Title="Rediscovering euqation of line" Tags="|self-learning|linear-algebra|" AnswerCount="1" CommentCount="5" ClosedDate="2023-04-29T10:50:33.067" ContentLicense="CC BY-SA 4.0" />
  <row Id="26400" PostTypeId="2" ParentId="24600" CreationDate="2023-04-27T18:53:05.687" Score="4" Body="&lt;p&gt;You can recast it as&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;A thief is trying to escape from a policeman initially stationed at the origin. The position of the thief after &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; seconds is &lt;span class=&quot;math-container&quot;&gt;$50+t^2$&lt;/span&gt; (in feet, probably). The policeman runs after the thief and catches him in &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; seconds. What was the average speed of the policeman over the full duration of the chase?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The analogue of the fixed cost in the position-velocity scenario is the headway and the simplest way to introduce it is to have two moving objects instead of one. Whether your students are smart enough to understand that it is the same problem they had before in disguise is up to you to decide.&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-04-27T18:53:05.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26402" PostTypeId="2" ParentId="4280" CreationDate="2023-04-28T16:20:08.260" Score="-2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I have the common misconception in my business calculus classes that the &lt;strong&gt;average rate of change&lt;/strong&gt;, say from &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x=5$&lt;/span&gt;, is the &lt;strong&gt;statistical average&lt;/strong&gt; of the rates on the four unit intervals &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/24602/372&quot;&gt;This previous answer&lt;/a&gt; establishes that Average Unit Cost (AUC) is the &lt;em&gt;average rate of change&lt;/em&gt; associated with the Total Cost (TC) function.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;strong&gt;average unit cost&lt;/strong&gt; from &lt;span class=&quot;math-container&quot;&gt;$q=a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$q=b$&lt;/span&gt; units is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$AUC(a,b)=\frac{TC(0)+TC(b)-TC(a)}{b-a}\tag{discrete}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$AUC(a,b)=\frac{TC(0)+\int_a^bMC(q)\,\mathrm dq}{b-a},\tag{continuous}$$&lt;/span&gt; where MC stands for Marginal Cost.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;If there is no fixed cost &lt;span class=&quot;math-container&quot;&gt;$TC(0),$&lt;/span&gt;&lt;/strong&gt; then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$AUC(a,b)=\frac{TC(b)-TC(a)}{b-a}=\frac{MC(a+1)+\ldots+MC(b)}{b-a}\tag{discrete}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$AUC(a,b)=\frac{\int_a^bMC(q)\,\mathrm dq}{b-a},\tag{continuous}$$&lt;/span&gt; that is, the average unit cost is a genuine &lt;strong&gt;statistical mean&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;How can I tell them not to compute a statistical average without lying to them ?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the above example, computing the average rate of change using the statistical method amounts to ignoring the fixed production cost &lt;span class=&quot;math-container&quot;&gt;$TC(0).$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="372" LastActivityDate="2023-04-28T16:20:08.260" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26403" PostTypeId="2" ParentId="4280" CreationDate="2023-04-29T04:03:37.357" Score="3" Body="&lt;p&gt;As many people mentioned already, there is no misconception here. Moreover, there is no such thing as &amp;quot;statistical average&amp;quot; or &amp;quot;calculus/physics/whatever&amp;quot; average. An average is always just an average and it enjoys all the properties of an average and is computed by the same techniques no matter where it comes from.&lt;/p&gt;&#xA;&lt;p&gt;What differs between various disciplines is the &lt;em&gt;interpretation&lt;/em&gt; of that average (the mean or the expected value in statistics, the constant velocity that will give you the same displacement over the given period of time in kinematics, and so on, and so forth). So, I would just be honest with the students here and tell them that the underlying concept is always &lt;em&gt;exactly&lt;/em&gt; the same, but the way the result is interpreted is &lt;em&gt;problem specific&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;That is the idea about an abstraction in general and it is introduced as early as the third grade when they are doing the word problems like &amp;quot;A dinosaur and a duck ate 17 pine cones together. The dinosaur ate 4. How many pine cones did the duck eat?&amp;quot; and then &amp;quot;Peter took a trip to the village 17 miles away. He went 4 miles by foot and then caught a ride in a car. How many miles did he travel in a car?&amp;quot;. The kids should understand that&lt;/p&gt;&#xA;&lt;p&gt;a) It is &lt;em&gt;exactly the same&lt;/em&gt; subtraction problem &lt;span class=&quot;math-container&quot;&gt;$17-4=13$&lt;/span&gt; from the abstract point of view.&lt;/p&gt;&#xA;&lt;p&gt;b) Those are &lt;em&gt;different&lt;/em&gt; 13's based on the context: in the first problem they are 13 pine cones eaten and in the second one it is 13 miles traveled (so Peter &lt;em&gt;didn't&lt;/em&gt; eat 13 pine cones and the duck &lt;em&gt;didn't&lt;/em&gt; travel 13 miles in a car).&lt;/p&gt;&#xA;&lt;p&gt;If they understand &lt;em&gt;both&lt;/em&gt; (a) and (b) clearly, you may declare your mission accomplished.&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-04-29T04:03:37.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26404" PostTypeId="2" ParentId="26399" CreationDate="2023-04-29T04:50:20.297" Score="1" Body="&lt;p&gt;Not sure this is what you are getting at, but here goes.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem:&lt;/strong&gt; A straight line in &lt;span class=&quot;math-container&quot;&gt;$R^2$&lt;/span&gt; is uniquely deteremined by two distinct points on that line.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof Sketch:&lt;/strong&gt; Suppose &lt;span class=&quot;math-container&quot;&gt;$(x_1, y_1)\in R^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(x_2, y_2)\in R^2$&lt;/span&gt; are distinct. There are two cases to consider:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Case 1:&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x_1\neq x_2$&lt;/span&gt; (non-vertical lines)&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$~m=\frac{y_2-y_1}{x_2-x_1}$&lt;/span&gt; (the slope between these two points).&#xA;Prove that there exists a unique a set of points &lt;span class=&quot;math-container&quot;&gt;$L\subset R^2$&lt;/span&gt;&#xA;such that &lt;span class=&quot;math-container&quot;&gt;$(x_1,~y_1)\in L$&lt;/span&gt; and the the slope between any two distinct points in &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;. That is, prove that all such subsets of &lt;span class=&quot;math-container&quot;&gt;$R^2$&lt;/span&gt; are the &lt;em&gt;same&lt;/em&gt; set.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Case 2:&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$~x_1=x_2$&lt;/span&gt; (vertical lines)&lt;/p&gt;&#xA;&lt;p&gt;Prove that there exists a unique a set of points &lt;span class=&quot;math-container&quot;&gt;$L\subset R^2$&lt;/span&gt;&#xA;such that &lt;span class=&quot;math-container&quot;&gt;$(a,~b)\in L$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$a=x_1$&lt;/span&gt;. That is, prove that all such subsets of &lt;span class=&quot;math-container&quot;&gt;$R^2$&lt;/span&gt; are the &lt;em&gt;same&lt;/em&gt; set.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-05-01T16:27:48.347" LastActivityDate="2023-05-01T16:27:48.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26405" PostTypeId="1" CreationDate="2023-05-02T13:45:40.760" Score="1" ViewCount="242" Body="&lt;p&gt;Here I have two issues related to this post. &lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How can etymological approach to a language be used to improve creativity skills of mathematics in students;&lt;br/&gt;&#xA;I think, knowing the evolution which has taken place by gathering information related to the historical developments of a language can be useful to shift the perspectives when generating new ideas in mathematics because mathematics itself is also a language. In that sense the etymological approach of why the word is spelled that way, is based on its roots rather than leaning how to spell, and encouraging students to find words sharing the same root rather than memorizing the meaning can be much effective differentiation tool. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;How can etymological approach to terminology be used to find much appropriate terms;&lt;br/&gt;&#xA;As many know the present contemporary English form Sine is originated in ancient India as a Sanskrit word and it was transliterated into Arabic. The transliterated word was translated into Latin in 12 th century and finally it was translated into English. Because of that the present word has no connection with original word which stands for half chord. We all know now it's too late to correct that kind of mistake happened very long ago, but this kind of awareness may be important to avoid such situations in the future and it can be helpful to determine much appropriate term by going into roots .&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2024-01-03T21:53:31.943" Title="Importance of etymological approach to terminology" Tags="|terminology|education-research|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26406" PostTypeId="2" ParentId="26405" CreationDate="2023-05-02T14:37:36.180" Score="6" Body="&lt;p&gt;I think etymology is not useful in the way proposed by the OP.&lt;br /&gt;&#xA;But here is a case in mathematics where etymology can help with spelling:&lt;br /&gt;&#xA;&lt;strong&gt;parallelepiped&lt;/strong&gt;&lt;br /&gt;&#xA;The etymological analysis is: &lt;em&gt;parallel + epi + ped&lt;/em&gt; (meaning &amp;quot;parallel upon the foot&amp;quot;).  Remembering this helps me avoid errors like: &lt;em&gt;parallelopiped&lt;/em&gt;, &lt;em&gt;parallelipiped&lt;/em&gt; and such.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2023-05-02T14:37:36.180" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26407" PostTypeId="2" ParentId="26380" CreationDate="2023-05-05T01:39:50.173" Score="3" Body="&lt;p&gt;I think the scope of your question is too narrow. If you ask more broadly about the use of calculus in number theory (not just differentiation for solving Diophantine equations) then books on analytic number theory will provide lots of examples.  For instance, there is an easy upper bound &lt;span class=&quot;math-container&quot;&gt;$\log(n!) &amp;lt; n\log n$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n \geq 2$&lt;/span&gt;, and with calculus we can derive a related lower bound &lt;span class=&quot;math-container&quot;&gt;$\log(n!) &amp;gt; n\log n - n$&lt;/span&gt;: see Theorem 2.1 &lt;a href=&quot;https://kconrad.math.uconn.edu/blurbs/analysis/stirling.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In a different direction, if we want to solve a Diophantine equations in &lt;em&gt;polynomials&lt;/em&gt; rather than in integers, then we can use differentiation to prove things that are far more difficult in the setting of integers. Consider the polynomial analogue of Fermat's Last Theorem: for &lt;span class=&quot;math-container&quot;&gt;$n \geq 3$&lt;/span&gt;, the only polynomial solutions to the equation &lt;span class=&quot;math-container&quot;&gt;$f(x)^n + g(x)^n = h(x)^n$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$f(x), g(x)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$h(x)$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbf R[x]$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbf C[x]$&lt;/span&gt; are constant solutions.  This can be proved by differentiating both sides of the Fermat equation.  See the first two pages of a paper by Granville and Tucker &lt;a href=&quot;https://www.ams.org/notices/200210/fea-granville.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2023-05-05T01:39:50.173" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26408" PostTypeId="1" CreationDate="2023-05-05T01:45:41.603" Score="3" ViewCount="124" Body="&lt;p&gt;Today I was made aware of the term &amp;quot;congruency&amp;quot; as a word related to congruence in the same way that equality is related to equation.  I have never seen the term &amp;quot;congruency&amp;quot; used in math before, and a web search suggests that when &amp;quot;congruency&amp;quot; occurs in math it is only in the context of congruent triangles in plane geometry; I have never seen it used in the context of congruences in modular arithmetic.&lt;/p&gt;&#xA;&lt;p&gt;If you have encountered the term &amp;quot;congruency&amp;quot; before in a math book, do you consider this term common or rare? I suspect even in geometry this term is not often used, but I would not consider myself any kind of expert on the terminology used in high school geometry courses.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="2083" LastEditDate="2023-05-05T05:19:20.727" LastActivityDate="2023-05-05T05:19:20.727" Title="congruency: how widely used?" Tags="|terminology|number-theory|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26410" PostTypeId="1" CreationDate="2023-05-05T16:11:23.960" Score="2" ViewCount="137" Body="&lt;p&gt;As a math faculty in a public institution in TX, USA, I am supposed to conduct a &amp;quot;Core Assessment&amp;quot; for most of my freshman classes. This typically entails an assignment to be completed by students. There is a very lengthy rubric to be used to grade and assess student performance. Then a tally is submitted through a chain of command in university.&lt;/p&gt;&#xA;&lt;p&gt;Where/when did this practice originate? Who sets the standard and the rubric? What was the philosophy behind it? What is done with the data?&lt;/p&gt;&#xA;&lt;p&gt;Edits:&lt;/p&gt;&#xA;&lt;p&gt;Seen mostly in Texas,&#xA;&lt;a href=&quot;https://www.highered.texas.gov/our-work/supporting-our-institutions/institutional-resources/transfer-resources/texas-core-curriculum/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Texas Higher Education Coordinating Board&lt;/a&gt;,&#xA;&lt;a href=&quot;https://assessment.tamu.edu/core&quot; rel=&quot;nofollow noreferrer&quot;&gt;Texas A&amp;amp;M&lt;/a&gt;,&#xA;&lt;a href=&quot;https://ugs.utexas.edu/core-assessment&quot; rel=&quot;nofollow noreferrer&quot;&gt;U Texas&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But also nationally and at different levels&#xA;&lt;a href=&quot;https://www.unr.edu/assessment/core&quot; rel=&quot;nofollow noreferrer&quot;&gt;U Nevada, Reno&lt;/a&gt;&#xA;&lt;a href=&quot;https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A K-12 version&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Related and of interest&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://uwm.edu/academicaffairs/wp-content/uploads/sites/32/2019/04/Guide.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;U Wisconsin, Milwaukee&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2023-05-06T09:40:34.903" LastActivityDate="2023-05-06T09:40:34.903" Title="What is the origin of &quot;Core Assessment&quot; for freshman university classes?" Tags="|assessment|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26411" PostTypeId="2" ParentId="4280" CreationDate="2023-05-05T17:41:54.667" Score="0" Body="&lt;p&gt;Your question is a good one because it relates to a mathematical fact and a pedagogical issue. The mathematical fact is that, for a continuous function, the average rate of change along an interval is equal to the mean of multiple average rates of change along subintervals of equal width. The pedagogical issue is that most of the time, teachers should encourage students to rely on prior skills.&lt;/p&gt;&#xA;&lt;p&gt;That's why it might seem pedagogically unsound to instruct a business student who is used to finding an average by taking a sum and then dividing by the number of data points that only the first and final values in an interval are needed when finding an average rate of change. But I wouldn't think of it as a lie. I think what's needed is an emphasis using an example of how an average rate of change is distinct from other averages and therefore requires less calculation.&lt;/p&gt;&#xA;&lt;p&gt;Alice and Bob are given annual census data for the population of Chicago for every year from 1900 to 2000 and asked to find the average annual rate of change in that population from 1900 to 2000. Alice finds 100 changes in population by subtracting the population from one year from the next, and she takes the average of those 100 numbers. Alice has found the average rate of change from the average of 100 annual changes. Bob finds the difference in population between 1900 and 2000 and divides by 100. Bob has found the average rate of change from the total change divided by 100 years.&lt;/p&gt;&#xA;&lt;p&gt;Alice isn't wrong, but you students definitely want to be like Bob because Bob did less unnecessary calculation.&lt;/p&gt;&#xA;&lt;p&gt;I think that answers your main question, but there is a little bit more potential for confusion.&lt;/p&gt;&#xA;&lt;p&gt;In most introductory statistics courses, students learn about &amp;quot;average&amp;quot; as a synonym for &amp;quot;measure of central tendency&amp;quot; in a list of numbers. It's only in that context that the word has the potential to be ambiguous. For example, a typical person buying a home is probably more interested in the average assessed values of single family homes in a neighborhood as a &lt;strong&gt;median&lt;/strong&gt; instead of as a &lt;strong&gt;mean&lt;/strong&gt;. A good introductory statistics course will present the distinction between average-as-median and average-as-mean as part of its pedagogy, regardless of whether the course includes &amp;quot;average rate of change&amp;quot; or not.&lt;/p&gt;&#xA;&lt;p&gt;I think that genuine potential for ambiguity about the word &amp;quot;average&amp;quot; should not be conflated with anything taught in a calculus course.&lt;/p&gt;&#xA;" OwnerUserId="22304" LastActivityDate="2023-05-05T17:41:54.667" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26414" PostTypeId="1" CreationDate="2023-05-08T17:22:43.687" Score="3" ViewCount="179" Body="&lt;p&gt;I was talking to a college student I know who is currently taking linear algebra. He told me that during his take home midterm/final exams he was assigned essay questions, which surprised me quite a bit&lt;/p&gt;&#xA;&lt;p&gt;I believe the one on the midterm had the prompt&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Write a two page essay about the Jordan normal form&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is this common practice? It seems to me that it's testing a lot of skills which are not really related to what most linear algebra classes set out to teach. Does anyone know of any references which argue for this as a good examination method?&lt;/p&gt;&#xA;" OwnerUserId="20763" LastActivityDate="2023-05-08T17:22:43.687" Title="Assigning essays in take home exams" Tags="|exams|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26415" PostTypeId="1" CreationDate="2023-05-10T15:46:54.797" Score="3" ViewCount="68" Body="&lt;p&gt;Representation theory has very deep ties with physics, leading to incredibly profound and admittedly cool results such as the classification of particles in the Standard Model via mass and spin by looking at the unitary irreducible representations of the Poincaré group; however, most discussions on the topic are quite complicated as the result is pretty high-level (you can tell by the word salad I had to use to even &lt;em&gt;mention&lt;/em&gt; the thing), which I would assume is in part due to the fact that we're dealing with Lie groups (and their Lie algebras) as opposed to finite groups.&lt;/p&gt;&#xA;&lt;p&gt;Indeed, by comparison, the representation theory of finite groups is a lot easier to present, more self-contained, and is overall something that an undergrad in math should not have too many problems understanding given the right presentation. But I do feel a bit sad when I look at representations of Lie algebras having cool applications like the one mentioned above, whereas the representation theory of finite groups can be used to prove... &lt;a href=&quot;https://en.wikipedia.org/wiki/Burnside%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Burnside's theorem&lt;/a&gt; on groups of order &lt;span class=&quot;math-container&quot;&gt;$p^aq^b$&lt;/span&gt;? That's rather underwhelming.&lt;/p&gt;&#xA;&lt;p&gt;So this leads me to my question: are there any interesting physical applications of finite group characters/representations to physics? Even including those that are just ‘toy’ versions of equally interesting applications in the field of Lie algebra representations, whenever studying the toy example can lead to actual, significant insight towards the more complicated (and useful) situation—if any such examples exist.&lt;/p&gt;&#xA;" OwnerUserId="22346" LastActivityDate="2023-05-10T15:46:54.797" Title="Are there any fun toy applications of representation and character theory for finite groups to physics?" Tags="|examples|applications|physical-sciences|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26416" PostTypeId="2" ParentId="21203" CreationDate="2023-05-10T19:16:30.580" Score="2" Body="&lt;p&gt;After seeing your edit I'd like to say something I thought of writing when I fist saw this but didn't: maybe it's not that the students don't care, but rather that they never ran into the same problem you did. I know I didn't have the same problem when I learned calculus &lt;em&gt;without&lt;/em&gt; epsilons. There is no &amp;quot;derivative paradox&amp;quot; that can only be cleared up with the epsilon-delta definition of limit, and I never thought I was dividing by zero. It's possible to understand the essential idea of limits without a rigorous definition. I'd even argue that you have it all backwards; you can't make sense of the definition of limits unless you already understand them!&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2023-05-10T19:16:30.580" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26417" PostTypeId="1" CreationDate="2023-05-11T00:57:45.303" Score="19" ViewCount="3852" Body="&lt;p&gt;When I took precalculus, we learned about polynomials and how to factor them, we learned about trigonometry and lots of great and useful identities there, and we learned about matrices. They didn't call it linear algebra, they just called it matrices. We learned how to multiply them, we learned a whole bunch of ridiculous complicated methods to calculate determinants. We learned to solve systems of linear equations using Cramer's Rule, which is the mathematical equivalent of scrubbing a floor with a toothbrush. Never once did we actually learn what matrices were, how they would be useful, or why we would want to know any of this. I found the whole exercise to be tedious and pointless and quickly braindumped all of the matrix stuff as soon as the class was over.&lt;/p&gt;&#xA;&lt;p&gt;Now, 20 years, a degree in math, a degree in physics, a career as a software engineer and data scientist, using linear algebra in building neural networks and many other places later, I am teaching a precalculus class, and I still have no idea what the point of teaching about matrices in it is. Like maybe some basic matrix/vector multiplication would be useful, applying them to systems of linear equations, but why do we force our students to painfully calculate determinants by hand when literally no one does this and the only reason to study these algorithms is if you are implementing a computer program to calculate them.&lt;/p&gt;&#xA;&lt;p&gt;Don't get me wrong, I am a huge proponent of learning fundamentals and practicing working through problems by hand before you give it to your calculator. It is important to understand the concepts behind what you are doing and how they applies to the world around you so you can use them to solve problems in applications and they aren't just pure abstractions. Except that doesn't really work for matrices because we never bother to explain what they are or how to apply them, even less so for determinants.&lt;/p&gt;&#xA;&lt;p&gt;Linear algebra is a very useful field used quite a lot in physics, engineering, computer science, and any number of other areas. And it has nothing to do with calculus. When I took a linear algebra class, I had to relearn all of these fundamental operations anyway, except there weren't any lengthy exercises in calculating determinants by hand because no one ever actually does that outside of a precalculus class. I have students asking me why we are learning this and I honestly don't have a good answer to give them. Why is it there?&lt;/p&gt;&#xA;" OwnerUserId="22348" LastActivityDate="2023-05-17T03:34:50.463" Title="Why do we teach linear algebra in precalculus classes?" Tags="|undergraduate-education|secondary-education|linear-algebra|precalculus|" AnswerCount="4" CommentCount="20" ContentLicense="CC BY-SA 4.0" />
  <row Id="26418" PostTypeId="2" ParentId="26417" CreationDate="2023-05-11T06:14:27.987" Score="16" Body="&lt;p&gt;Vector algebra is a standard 3rd-semester calculus topic (e.g., see &lt;a href=&quot;https://openstax.org/details/books/calculus-volume-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax Calculus 3&lt;/a&gt;, Ch. 2-3). This includes calculations of the dot product, cross product, and related values. Standard applications include calculating results of work, force, and business productions.&lt;/p&gt;&#xA;&lt;p&gt;While this can be done purely algebraically (e.g., OpenStax does so), it's natural to express these operations in matrix notation, so calculus books may have some sections devoted to showing it in that form. Example on my bookshelf: this appears in Stein and Barcellos, &lt;em&gt;Calculus and Analytic Geometry&lt;/em&gt; (5E, 1992), Sec. 12.5-12.7. Common exercises of the determinant and cross product include computing the areas of parallelograms, volumes of parallelipipeds, possibly crystallography, and torque. (In fact, calculation of parallelipiped volume from the end of the calculus sequence is one of the key half-dozen standard questions we use for assessment of our graduating math majors every year in my community college department.)&lt;/p&gt;&#xA;&lt;p&gt;So presumably matrices appearing in a precalculus class are partly meant to set the stage for this use in calculus. Related, the precalculus course has covered solving linear equations (possibly by substitution and elimination), and it would be natural to show yet another option, the &amp;quot;next step&amp;quot;, so to speak. And many of the students who take calculus will need linear algebra, so why not give them an introductory taste of all that content at the next level? (without calling the course something unwieldy like &lt;em&gt;Preparation for Calculus, Linear Algebra, and Related Mid-Undergraduate Mathematical Topics&lt;/em&gt;, of course).&lt;/p&gt;&#xA;&lt;p&gt;In addition, matrices also appear in the standard undergraduate discrete mathematics and data structures course texts as the basic way to represent general relations, graphs, and weighted graphs (and hence used in implementations for Euler circuits, Dijkstra's algorithm, Prim's and Kruskal's algorithms, etc.). Note this is a context in which matrices are &lt;em&gt;not&lt;/em&gt; being used to represent linear transformations, so it's a case that highlights that's not &lt;em&gt;a priori&lt;/em&gt; an essential part of the meaning of a matrix. In my own work, this is my own customary usage of matrices, for example.&lt;/p&gt;&#xA;&lt;p&gt;So that's my understanding of the legacy of why those subjects have been included. If someone thinks that's poorly justified, or those topics have been squeezed out of those courses in recent years, then I could imagine coherent arguments for that.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-05-12T14:13:40.310" LastActivityDate="2023-05-12T14:13:40.310" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26419" PostTypeId="2" ParentId="25677" CreationDate="2023-05-11T08:28:27.760" Score="3" Body="&lt;p&gt;If you want to defend &lt;span class=&quot;math-container&quot;&gt;$\sqrt{4}$&lt;/span&gt; being both &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$+2$&lt;/span&gt;, you'll need to accept that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{1} + \sqrt{9} + \sqrt{25}$&lt;/span&gt; can have all of the following values:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{cases}&#xA;\\ -9 = (-1) + (-3) + (-5)&#xA;\\ -7 = (+1) + (-3) + (-5)&#xA;\\ -3 = (-1) + (+3) + (-5)&#xA;\\ -1 = (+1) + (+3) + (-5)&#xA;\\ +1 = (-1) + (-3) + (+5)&#xA;\\ +3 = (+1) + (-3) + (+5)&#xA;\\ +7 = (-1) + (+3) + (+5)&#xA;\\ +9 = (+1) + (+3) + (+5)&#xA;\end{cases}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In other words, not a good idea :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-05-11T08:28:27.760" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26420" PostTypeId="2" ParentId="26417" CreationDate="2023-05-11T12:13:08.643" Score="9" Body="&lt;p&gt;The College Board made curriculum decisions for their new AP Precalculus course that align with sentiments you express. The course is divided into four units, where unit four is titled &lt;em&gt;Functions Involving Parameters, Vectors, and Matrices&lt;/em&gt;. But unit 4 is &lt;strong&gt;not&lt;/strong&gt; assessed on the exam. Here is an excerpt from their &lt;a href=&quot;https://apcentral.collegeboard.org/media/pdf/ap-precalculus-course-and-exam-description.pdf&quot; rel=&quot;noreferrer&quot;&gt;Course and Exam Description document&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Units 1, 2, and 3 topics comprise the content and conceptual&#xA;understandings in which colleges and universities typically expect&#xA;students to be proficient, in order to qualify for college credit&#xA;and/or placement. Therefore, these topics are included on the AP Exam.&#xA;Unit 4 consists of topics that teachers may include based on state or&#xA;local requirements.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-05-11T13:02:55.457" LastActivityDate="2023-05-11T13:02:55.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26422" PostTypeId="1" AcceptedAnswerId="26427" CreationDate="2023-05-11T22:23:29.150" Score="4" ViewCount="340" Body="&lt;p&gt;I'm trying to find a good real-world analogy (or even good visualization) for teaching about error correcting codes and erasure encodings. The most natural way to talk about it really &lt;em&gt;is&lt;/em&gt; in terms of the lower-order polynomials.&lt;/p&gt;&#xA;&lt;p&gt;The examples / analogies that I have heard:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Take a piece of information, split it into a bunch of puzzle pieces including some extra ones. That way, even if someone loses a piece, you can still put the puzzle back together.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I find this one a bit weak, as it doesn't really get across the point that the split is generic. You can use &lt;em&gt;any subset&lt;/em&gt; that is large enough to get all the information back.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;Think about if you wanted to tell someone else about a line. Two points define a line, so you could just send them 4 points on the line. That way, even if some points got lost, you could figure out which points were missing.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I really like this one, but I find some students have a hard time extending their intuition on lines to polynomials immediately. It definitely does a good job of being precise, as it is exactly what is going on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;It's like a checksum. Imagine you have a bunch of data in binary form, and at the end you put two digits. Both of them are the same, and are 0 if the number of 1s in the binary is even, and 1 if it is odd. That way, if exactly one binary digit changes in transit, you can always notice it. This example then goes into the process of checking the cases of no errors, error in data, and error in check-data.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This one is also pretty nice, but then at the end you still have to say &amp;quot;but imagine you could do that but more&amp;quot;. Also, it's very data-oriented.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Are there better analogies that can be used to teach this concept?&lt;/p&gt;&#xA;" OwnerUserId="22352" LastEditorUserId="22352" LastEditDate="2023-05-13T00:36:29.557" LastActivityDate="2023-05-20T19:56:37.087" Title="Good analogies for teaching error correcting codes" Tags="|undergraduate-education|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26424" PostTypeId="2" ParentId="26417" CreationDate="2023-05-12T00:14:55.463" Score="8" Body="&lt;p&gt;Because derivatives are linear operators.&lt;/p&gt;&#xA;" OwnerUserId="22354" LastEditorUserId="16250" LastEditDate="2023-05-12T08:19:28.013" LastActivityDate="2023-05-12T08:19:28.013" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26425" PostTypeId="2" ParentId="26422" CreationDate="2023-05-12T10:12:15.600" Score="2" Body="&lt;p&gt;There are some well-known examples, like bank account numbers (at least in Belgium), which consist of twelve digits, something like:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\overline{a_1a_2a_3a_4a_5a_6a_7a_8a_9a_{10}a_{11}a_{12}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is the general rule:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\overline{a_{11}a_{12}} = \mod(\overline{a_1a_2a_3a_4a_5a_6a_7a_8a_9a_{10}}, 97)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As a result, when you type a bank account number and you miss a digit, that condition won't stand anymore and you'll get a warning about your mistake.&lt;/p&gt;&#xA;&lt;p&gt;A same thing exists for books with so-called ISBN numbers: the last digit is the reminder of the rest while dividing by 11. If ever the result is 10, then an &lt;code&gt;X&lt;/code&gt; is used.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-05-12T10:12:15.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26426" PostTypeId="1" CreationDate="2023-05-13T09:59:14.407" Score="0" ViewCount="157" Body="&lt;p&gt;I heard one mathematician who said “induction &lt;strong&gt;on&lt;/strong&gt; ” and another who said “induction &lt;strong&gt;with respect to&lt;/strong&gt; ”.  Do these two expressions mean exactly the same thing mathematically?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If so, then are they both correct ways to say this and which is more common?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If not, then what shade of difference is there between the two phrases?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="guest" LastActivityDate="2023-05-13T22:34:48.713" Title="‘Induction on’ vs ‘Induction with respect to’ in math" Tags="|terminology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26427" PostTypeId="2" ParentId="26422" CreationDate="2023-05-13T15:34:40.357" Score="12" Body="&lt;p&gt;I'm not sure what exactly a &amp;quot;real world analogy&amp;quot; is, but the idea of &amp;quot;error correcting codes&amp;quot; itself is rather ubiquitous. Just go to the nearest grocery store and, when hearing the usual &amp;quot;paper or plastic?&amp;quot; question, answer something like &amp;quot;plastious&amp;quot; or &amp;quot;vaper&amp;quot;. You'll still be understood unless the cashier has really good ears. Orr tri two redd ths sntense. Thn ts vun. End saw on.&lt;/p&gt;&#xA;&lt;p&gt;The whole point of error correction is that there is some well-defined expected subset of admissible messages and you choose the nearest one (in certain metric) to the message that was actually received. That's more or less how we actually read and listen and, I guess, everyone can come up with an example of a misread or misheard message in his or her life. Of course, if the erratic part is too large and the non-erratic one is too small to provide enough context, one can get perplexed a bit. If you only know that &amp;quot;c_l_nd_r&amp;quot; is an English word, you have at least 3 ways to recover the missing letters. However, if the previous text was about kitchen utensils, errand list for the week, or geometry, the choice becomes fairly clear.&lt;/p&gt;&#xA;&lt;p&gt;So, if you want to introduce the concept, I would suggest just playing a game when you write or say some messages or draw some pictures that become gradually more and more incomplete or garbled and ask the students to guess the intended meanings. That will also bring up an opportunity to discuss various possible metrics to use when choosing the nearest word/image. You can also play hangman and &amp;quot;turn one word into the other&amp;quot; here with the emphasis on &lt;em&gt;what is a good strategy in such games&lt;/em&gt; or to try an exercise like&lt;/p&gt;&#xA;&lt;p&gt;You know that the actual word is either &amp;quot;file&amp;quot; or &amp;quot;ruin&amp;quot;. Your receiving device printed out one of the following words:&lt;/p&gt;&#xA;&lt;p&gt;file pile pale tale tall tail rail rain ruin&lt;/p&gt;&#xA;&lt;p&gt;Which correction would you choose for each of them and why?&lt;/p&gt;&#xA;&lt;p&gt;(with the emphasis that there is no &amp;quot;the only correct answer&amp;quot; here for the middle of the list but if you need to teach the machine to distinguish and correct automatically, you still have to formalize unambiguously how your choice is made).&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents.&lt;/p&gt;&#xA;&lt;p&gt;Fedja&lt;/p&gt;&#xA;" OwnerUserId="19946" LastEditorUserId="19946" LastEditDate="2023-05-13T15:48:45.723" LastActivityDate="2023-05-13T15:48:45.723" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26428" PostTypeId="2" ParentId="26426" CreationDate="2023-05-13T22:34:48.713" Score="7" Body="&lt;p&gt;Yes, these expressions mean exactly the same thing.  Both are correct; in my experience, &lt;span class=&quot;math-container&quot;&gt;``&lt;/span&gt;induction on n'' is more common.  Usually, the shorter the phrase, the more likely it is to be widely adopted.&lt;/p&gt;&#xA;" OwnerUserId="6696" LastActivityDate="2023-05-13T22:34:48.713" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26429" PostTypeId="2" ParentId="15214" CreationDate="2023-05-15T01:29:39.310" Score="-1" Body="&lt;p&gt;I can relate to your observations!  Most textbooks start the same ... Points, Lines, Quadrilaterals, Polygons, and finally ... the triangle.  However, what if we started with the triangle and the circle and noted all the other concepts taught first and apply it to the triangle in particular... particularly noting that triangles are and can be inscribed in so may other shapes invisible to most...use triangles to express squares, rectangles, parallelograms, etc.&lt;/p&gt;&#xA;&lt;p&gt;I am definitely going to try this methodology with some of my high school students next year!&lt;/p&gt;&#xA;&lt;p&gt;Maybe it is time to think inside and outside the triangle!&lt;/p&gt;&#xA;" OwnerUserId="22363" LastActivityDate="2023-05-15T01:29:39.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26430" PostTypeId="2" ParentId="15214" CreationDate="2023-05-15T03:59:50.283" Score="6" Body="&lt;p&gt;Here is a late but brief answer to the question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If this is the normal way of teaching geometry, why? Why is the course&#xA;focused more on memorizing theorems rather than understanding where&#xA;they come from — something which, in my experience, seems to be&#xA;nigh-impossible without the tools learned after Geometry?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is not normal. What the OP describes (lots of memorizing and no justification) is, I'd argue, the &lt;em&gt;exact antithesis&lt;/em&gt; of what the standard geometry class is for in the curriculum -- to give students a taste for rigorous mathematics, where principles are built up from axioms, without taking everything on faith.&lt;/p&gt;&#xA;&lt;p&gt;Basic geometry does not require tools from trigonometry for its development, and historically did not use them. To address one example, here is a &lt;a href=&quot;https://www.cuemath.com/geometry/hypotenuse-leg-theorem/&quot; rel=&quot;nofollow noreferrer&quot;&gt;proof of the hypotenuse-leg theorem&lt;/a&gt; using only elementary geometry.&lt;/p&gt;&#xA;&lt;p&gt;The prospect of angle-side-side congruency can be disproved with a single counterexample, of course.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-05-15T23:36:38.013" LastActivityDate="2023-05-15T23:36:38.013" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26431" PostTypeId="2" ParentId="26422" CreationDate="2023-05-16T18:09:08.383" Score="4" Body="&lt;p&gt;While it is too mathematically sophisticated for your course, continued fractions offer an analogue. If I tell you an initial part of the &lt;em&gt;decimal expansion&lt;/em&gt; of a rational number, sometimes something as small as even &lt;span class=&quot;math-container&quot;&gt;$3\%$&lt;/span&gt; of the repeating part of the decimal, then in practice the whole rational number (its numerator and denominator) can be recovered in two steps: (i) write the finite decimal as a continued fraction and (ii) truncate that continued fraction right before an unusually large entry and use the preceding part of the continued fraction as your guess for the correct rational number.  Some reasons for wanting to guess rational numbers this way are in my answer &lt;a href=&quot;https://math.stackexchange.com/questions/585675/what-are-the-applications-of-continued-fractions&quot;&gt;here&lt;/a&gt;; see also the answer (and comments to it) by Andreas Blass &lt;a href=&quot;https://math.stackexchange.com/questions/150229/how-are-continued-fractions-useful&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The analogy between what I just said and error-correcting codes is that continued fractions offers a way to reconstruct, in practice, a rational number from incomplete information about it (just a small part of the period of its decimal expansion). In fact, the analogy is closer than what I suggested: one of the methods of correcting errors in a BCH code can be described in terms of convergents to a continued fraction of a rational function &lt;span class=&quot;math-container&quot;&gt;$\mathbf F_q(x)$&lt;/span&gt; where the coefficients of the denominator depend  on the received message.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2023-05-16T18:09:08.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26432" PostTypeId="1" AcceptedAnswerId="26435" CreationDate="2023-05-16T20:16:03.963" Score="32" ViewCount="9812" Body="&lt;p&gt;Elsewhere, among a group of high school math teachers, I encountered a discussion of the term 'cancel'. Most (&amp;gt;20) people in the discussion had very strong feelings about why the term should be eliminated. I searched among posts on this stack to find the word common used and its meaning seems well understood.&lt;/p&gt;&#xA;&lt;p&gt;The objection the the word's use included the concern about errors that might result from dividing to cancel and thinking that zero results.&lt;/p&gt;&#xA;&lt;p&gt;My response was that if a student's work reflects accurate use of subtracting from both sides, dividing properly when attempting to simplify, etc, that what the student calls it is far less important. I even suggested that it would be preferable to teach the proper name for what's occurring, e.g. 'The additive inverse property' but as long as the student is doing the math correctly, and typically, silently, that there's no harm in using a word that's less than a textbook definition.&lt;/p&gt;&#xA;&lt;p&gt;(Given the use of the word here, over 100 posts, and no pushback that I saw when skimming over a dozen, I suspect I know the answer. I'm here for confirmation)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-05-16T20:49:18.943" LastActivityDate="2024-01-08T16:50:39.050" Title="Is it harmful to use the word &quot;Cancel&quot;?" Tags="|vocabulary|" AnswerCount="9" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26433" PostTypeId="2" ParentId="26432" CreationDate="2023-05-16T21:35:15.957" Score="34" Body="&lt;p&gt;In the book &lt;a href=&quot;http://www.nixthetricks.com/index.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Nix the Tricks&lt;/em&gt;&lt;/a&gt;, written by Tina Cardone in collaboration with numerous other math teachers, &amp;quot;cancel&amp;quot; appears in section 6.4. The book gives this reason for &amp;quot;nixing&amp;quot; it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Cancel is a vague term that hides the actual mathematical operations being used, so students do not know&#xA;when or why to use it. To many students, cancel is&#xA;digested as “cross-out stuff” by magic, so they see no problem with crossing out parts of an expression or across addition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The following &amp;quot;fixes&amp;quot; are suggested:&#xA;&lt;span class=&quot;math-container&quot;&gt;$\require{cancel}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Say two factors &amp;quot;divide to one&amp;quot; for multiplicative cancellation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{4x+2}{2}=\frac{2(2x+1)}{2}=\cancelto{1}{\frac{2}{2}}\hspace{-0.75em}\cdot \frac{2x+1}{1}=2x+1.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Say two terms &amp;quot;add to zero&amp;quot; for additive cancellation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$4x-4(x+2)=4x-4x-8=\cancelto{0}{4x-4x}\hspace{-0.75em}-8=-8.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I don’t think the word “cancel” is harmful, and I wouldn’t penalize students for using it. As others have suggested, it may even be helpful to encourage students to use “cancel” after they’ve attained some fluency. But in my experience, students who have learned to make erroneous cancellations are less likely to do so when given clearer language and notation to use.&lt;/p&gt;&#xA;&lt;p&gt;Edit: I've also commonly heard &amp;quot;cancel&amp;quot; used by students in the context of inverse functions, and they often make erroneous statements like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x^2}=x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\arcsin(\sin(x))=x$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-05-17T17:20:49.250" LastActivityDate="2023-05-17T17:20:49.250" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26434" PostTypeId="2" ParentId="26432" CreationDate="2023-05-16T22:24:10.730" Score="9" Body="&lt;p&gt;Language that points to the actual reasoning helps math make sense. I might say cancel sometimes (and I would never ding a student for saying it), but I do think it's better to say: We factor, and then the common factor makes a 1, which is multiplied by the rest of it, so we don't need to keep that one.&lt;/p&gt;&#xA;&lt;p&gt;That's more than I'm going to say every time, but I definitely say it if a weak student asked the question.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-05-16T22:24:10.730" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26435" PostTypeId="2" ParentId="26432" CreationDate="2023-05-16T23:24:45.830" Score="59" Body="&lt;p&gt;Coincidentally, a colleague at my community college brought up this same issue one week ago (and I'd never heard such a complaint before then). My colleague directly tells their students that &amp;quot;canceling&amp;quot; is simply not a proper thing in mathematics. Therefore I was doing a brief literature review in the last few days.&lt;/p&gt;&#xA;&lt;p&gt;In short, &amp;quot;canceling&amp;quot; is a customary and well-defined term in standard textbooks of any level I could find, from the most elementary algebra to graduate abstract algebra texts. Students will be set up to have a hard time interfacing with those standard works if they're warned away from use of the word &amp;quot;canceling&amp;quot;. A few examples where I found it:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Bennett &amp;amp; Briggs, &lt;em&gt;Using &amp;amp; Understanding Mathematics, A Quantitative Reasoning Approach&lt;/em&gt; (Sec. 2A, Working with Units, Brief Review of Common Fractions, and problems throughout the text).&lt;/li&gt;&#xA;&lt;li&gt;Sullivan, &lt;em&gt;Algebra &amp;amp; Trigonometry&lt;/em&gt; (presents two Cancellation Properties as basic axioms of real numbers in Sec. R.1)&lt;/li&gt;&#xA;&lt;li&gt;Thomas' &lt;em&gt;Calculus&lt;/em&gt; (explanations in work on basic limits and derivatives)&lt;/li&gt;&#xA;&lt;li&gt;Rosen, &lt;em&gt;Discrete Mathematics and its Applications&lt;/em&gt; (explanations in problems on induction and combinations).&lt;/li&gt;&#xA;&lt;li&gt;Hungerford, &lt;em&gt;Abstract Algebra&lt;/em&gt; (as an analog to &lt;strong&gt;Z&lt;/strong&gt;, Theorem 3.7 says a ring is an integral domain if and only if it has a &amp;quot;cancellation property&amp;quot;; likewise Theorem 7.2 asserts that &amp;quot;cancellation holds&amp;quot; in any group).&lt;/li&gt;&#xA;&lt;li&gt;Rudin, &lt;em&gt;Principles of Mathematical Analysis&lt;/em&gt; (Proposition 1.14 describes the basic property of fields as &amp;quot;a cancellation law&amp;quot;).&lt;/li&gt;&#xA;&lt;li&gt;Lang, &lt;em&gt;Algebra&lt;/em&gt; (describes a &amp;quot;cancellation law&amp;quot; in the context of Grothendieck groups, Sec. 1.7, as well as work in basic fractions and factoring), and also his &lt;em&gt;Complex Analysis&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Serre, &lt;em&gt;A Course in Arithmetic&lt;/em&gt; (Theorem 4 on quadratic forms described as a &amp;quot;cancellation theorem&amp;quot;).&lt;/li&gt;&#xA;&lt;li&gt;Wikipedia has articles on &lt;a href=&quot;https://en.wikipedia.org/wiki/Cancelling_out&quot; rel=&quot;noreferrer&quot;&gt;cancelling out&lt;/a&gt; in elementary algebra, and a &lt;a href=&quot;https://en.wikipedia.org/wiki/Cancellation_property&quot; rel=&quot;noreferrer&quot;&gt;cancellation property&lt;/a&gt; in abstract algebra.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;... and so forth.&lt;/p&gt;&#xA;&lt;p&gt;My colleague's chief complaint was along the lines of &amp;quot;there's not 'a' cancellation law, there's two, one for adding, and one for multiplying, so we shouldn't use that term&amp;quot;, but I find that unpersuasive -- that's just a reflection of the fact that in elementary algebra on the field of real numbers, we have two commutative properties, two associative properties, two inverse properties, two identities, and so forth.&lt;/p&gt;&#xA;&lt;p&gt;I suppose language changes over time, and sometimes ideologues wind up winning campaigns like this. But I broadly don't see the benefit of such a change, and as a description of current standard mathematical language it seems to be broadly incorrect  (and would leave students in the lurch about interfacing with this common usage in a variety of texts throughout the discipline). In regards to my colleague's issue, I would see this not as something to shy away from, but as an opportunity to embrace a careful awareness of our two basic operations in real numbers.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-05-17T01:22:54.533" LastActivityDate="2023-05-17T01:22:54.533" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26436" PostTypeId="2" ParentId="26417" CreationDate="2023-05-17T03:34:50.463" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Linear algebra is a very useful field used quite a lot in physics, engineering, computer science, and any number of other areas. And it has nothing to do with calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is incorrect to say linear algebra has nothing to do with calculus: linear algebra is &lt;em&gt;all over the place&lt;/em&gt; in multivariable calculus. Maybe you meant to say that linear algebra has nothing to do with single-variable calculus.&lt;/p&gt;&#xA;&lt;p&gt;I too saw some matrix algebra (in the &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; case) in my high school precalculus course. I never thought much about why anyone decided to put it  there. Possibly it was included to expose students to some matrix work so the concept of a matrix would not be completely new if/when the students get to multivariable calculus. Or the person designing the curriculum knew that matrices are a mathematical tool that also comes up in later coursework not relying on calculus, e.g., some &amp;quot;math for business/economics majors&amp;quot; courses that I've seen have a unit on matrices: consider the Leontief input/output model.&lt;/p&gt;&#xA;&lt;p&gt;You compare Cramer's rule to &amp;quot;scrubbing a floor with a toothbrush&amp;quot;. For the purposes you saw of it, I understand that impression. But in higher mathematics (at the level only math majors would see), Cramer's rule is genuinely useful. It is one of the methods that implies algebraic numbers form a field, it shows &lt;span class=&quot;math-container&quot;&gt;${\rm GL}_n(\mathbf R)$&lt;/span&gt; is a topological group, and it shows &lt;span class=&quot;math-container&quot;&gt;${\rm GL}_n(K)$&lt;/span&gt; is an algebraic group for (algebraically closed) fields &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2023-05-17T03:34:50.463" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26437" PostTypeId="1" AcceptedAnswerId="26439" CreationDate="2023-05-17T06:06:33.667" Score="8" ViewCount="2112" Body="&lt;p&gt;I am a mathematics professor seeking advice on refining my grading system for future courses. I currently employ a bonus points system intended to provide a catch-up mechanism for students who may be falling behind. However, I've noticed a trend where the bonus points disproportionately benefit students already doing well, leading to a grade distribution heavily tilted towards the higher end.&lt;/p&gt;&#xA;&lt;p&gt;I do not wish to curve grades due to the competitive environment it might foster. What I'm seeking are strategies to make the attainment of an 'A' grade more challenging while still providing support for struggling students.&lt;/p&gt;&#xA;&lt;p&gt;Here are a few ideas I've considered:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Adjusting the bonus points system so that students at the lower end of the grade spectrum receive more benefit.&lt;/li&gt;&#xA;&lt;li&gt;Offering remedial assignments for students who are falling behind.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I would greatly appreciate insights or experiences from other educators. Are there additional strategies that have worked well in your own classrooms?&lt;/p&gt;&#xA;" OwnerDisplayName="user19945" LastActivityDate="2023-05-18T16:25:50.480" Title="Adjusting a Bonus Points System to More Equitably Benefit Struggling Students" Tags="|grading|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26438" PostTypeId="2" ParentId="26432" CreationDate="2023-05-17T07:24:51.663" Score="8" Body="&lt;p&gt;I think the word &amp;quot;cancel&amp;quot; is a useful one, particularly the phrase &amp;quot;cancel out&amp;quot; as applied to two terms, factors or operations which are inverses of each other ─ that is, I would always rather say that the &lt;em&gt;terms&lt;/em&gt; cancel out, or the &lt;em&gt;factors&lt;/em&gt; cancel out, or a term &lt;em&gt;cancels with&lt;/em&gt; another term ─ not that &lt;em&gt;we&lt;/em&gt; cancel them.&lt;/p&gt;&#xA;&lt;p&gt;This distinction is perhaps important because terms (additive expressions) cancel additively whereas factors (multiplicative expressions) cancel multiplicatively. But it's also perhaps important because it frames the algebraic simplification as something inherent to the formula being simplified, rather than a way that we choose to change that formula. And also it can make sense that terms cancel in a different way to how factors cancel.&lt;/p&gt;&#xA;&lt;p&gt;If students are having trouble with the word &amp;quot;cancel&amp;quot; e.g. because they're simplifying multiplicative inverses to zero instead of one, I would suppose that the problem is not the word used to discuss the concept, but the way that concept is being taught and presented. If a student incorrectly simplifies e.g. &lt;span class=&quot;math-container&quot;&gt;$4 + \frac{x}{x}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, this student's misapprehension is that they know division &amp;quot;cancels&amp;quot; but they haven't identified that it needs a multiplication to &amp;quot;cancel with&amp;quot;. And indeed, this formula doesn't contain any multiplication. So we might introduce one: &lt;span class=&quot;math-container&quot;&gt;$4 + 1 \times \frac{x}{x}$&lt;/span&gt;. The student can agree that multiplying the fraction by 1 doesn't change its value; but now there is a multiplication for the division to cancel with, and the formula can correctly be simplified to &lt;span class=&quot;math-container&quot;&gt;$4 + 1$&lt;/span&gt;. (Of course, it must be given that &lt;span class=&quot;math-container&quot;&gt;$x \ne 0$&lt;/span&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;The argument has been made in another answer that &amp;quot;cancelling&amp;quot; is not a formal concept so it shouldn't be taught. I would argue instead that mathematics education is not just about teaching formal concepts, it's also about developing intuition, and the ways the subject is practiced and talked about by real people.&lt;/p&gt;&#xA;&lt;p&gt;The concept of two inverse components of a formula being simplified is an important one which reflects the way people intuitively think about algebra, and it's useful to have one word which acknowledges the connection between the different ways that things can be inverses of each other. The word &amp;quot;cancel&amp;quot; also happens to be a word that is commonly used for this by people who do mathematics or talk about it.&lt;/p&gt;&#xA;" OwnerUserId="19245" LastActivityDate="2023-05-17T07:24:51.663" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26439" PostTypeId="2" ParentId="26437" CreationDate="2023-05-17T14:20:07.090" Score="20" Body="&lt;p&gt;Probably the most straightforward mechanism is to potentially replace some of the grades in the gradebook, e.g.:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;One or more assessments (tests, quizzes, assignments) are dropped from the averaging formula, counting only the higher ones.&lt;/li&gt;&#xA;&lt;li&gt;Assessments may be re-taken, with only the higher grade used in the final average.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Both of these methods allow low-scorers to boost their grades, while having little or no benefit to those who scored high in the first place. They also serve to handle missed assessments due to illness or other interruptions. And both of these mechanisms are implemented in standard online learning management system gradebooks.&lt;/p&gt;&#xA;&lt;p&gt;Note that #1 makes no extra labor on the part of the instructor, while #2 does so.&lt;/p&gt;&#xA;&lt;p&gt;(Personally I do #1 but not #2.)&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-05-17T14:20:07.090" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26440" PostTypeId="2" ParentId="26432" CreationDate="2023-05-17T15:13:39.477" Score="18" Body="&lt;p&gt;Use of the word &amp;quot;cancel&amp;quot; is not a bug.  It's a &lt;em&gt;feature&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A year or two after high school, while manipulating equations in later training, I found myself &amp;quot;canceling&amp;quot; in various ways and doing it automatically.  I wasn't even using the word &amp;quot;cancel&amp;quot; in my head.  I just knew that I could simplify expressions according to different rules.&lt;/p&gt;&#xA;&lt;p&gt;At one point I stopped and wondered &amp;quot;What allows me to do that?&amp;quot; And after a short reflection, I remembered what I was actually doing: Adding or subtracting identical terms from both sides of an equation, or multiplying a fraction by another fraction that equaled 1.&lt;/p&gt;&#xA;&lt;p&gt;But that's how it's supposed to be.  Alfred North Whitehead once said (paraphrase) &amp;quot;Civilization advances by the number of operations we can perform without thinking.&amp;quot;  That's why we drill students.   We want the rules of algebra, trigonometric identities, boolean algebra, DeMorgan's laws, matrix operations, integration formulas, etc. to become automatic.  And if some of those rules fall under the generic word, &amp;quot;Cancel,&amp;quot; well that's fine.  That's a sign the rules have been internalized. It's a sign of &lt;em&gt;fluency&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To be sure, we don't want students to just memorize the rules by rote.  We want them to understand why they work. But they shouldn't have to derive a often-used formula every time they apply it.&lt;/p&gt;&#xA;&lt;p&gt;As others have pointed out, trying to eliminate the word &amp;quot;Cancel&amp;quot; from the student's vocabulary will put them at a disadvantage when communicating with others mathematically.  For those that feel strongly that &amp;quot;Cancel&amp;quot; is too ambiguous for mathematical use, I'd suggesting teaching students to learn to live with the ambiguity instead of taking the burning books approach.  Maybe analyze the word &amp;quot;Cancel&amp;quot; with them.  Itemize the different operations that fall under that category. Or when students use the word in class, ask them, &amp;quot;That's right.  What kind of canceling is that?&amp;quot;  or &amp;quot;That's right. What allows you to 'cancel' in that situation?&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="5465" LastEditorUserId="5465" LastEditDate="2023-06-25T18:08:25.473" LastActivityDate="2023-06-25T18:08:25.473" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26441" PostTypeId="1" CreationDate="2023-05-17T15:50:42.633" Score="0" ViewCount="116" Body="&lt;p&gt;We have plenty of examples in mathematical induction for advanced level mathematics students. Can we introduce infinite descent method as extremely opposite approach to mathematical induction and is it important to mention by the name infinite descent method when you prove irrationality of square root 2 by the method of contradiction rather than just mentioning method of contradiction ? &lt;br/&gt;&#xA;What would you suggest as examples to apply infinite descent method such as proving non existence of positive integers &lt;span class=&quot;math-container&quot;&gt;$x,y,z$&lt;/span&gt; satisfying the equation &lt;span class=&quot;math-container&quot;&gt;$x^3 + 2y^3 = 4z^3 $&lt;/span&gt;, which are not much complicated ?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2023-05-17T15:50:42.633" Title="Infinite descent method" Tags="|secondary-education|proofs|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26444" PostTypeId="2" ParentId="26437" CreationDate="2023-05-17T19:10:11.323" Score="9" Body="&lt;p&gt;If you'd like some reading on this, I suggest Feldman's &amp;quot;Grading for Equity&amp;quot;, which goes through the principles of trying to get grades to reflect students' final demonstrated understanding of the course goals. This means if a student is unable to demonstrate a learning objective on an assessment, they should have plenty of opportunities later in the semester to demonstrate the objective, and thus to make up the full points. This encourages students who have fallen behind to focus on learning the topics they've missed. In fact, a students' best strategy in this system is often to focus on the most fundamental learning objectives, and with a little guidance you can help students see that.&lt;/p&gt;&#xA;&lt;p&gt;I implement a makeup quiz system: grades are 100% quizzes. Each quiz question covers a particular learning objective, and students have many opportunities to get the point for that question throughout the semester.&lt;/p&gt;&#xA;&lt;p&gt;Some things I've found to make makeup quizzes less time consuming:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Grade each question 0 or 1: either the student demonstrated the tested skill or not. Full credit for unimportant mistakes.&lt;/li&gt;&#xA;&lt;li&gt;Makeups should be on an individual-question basis. Only give students questions they didn't get full marks on already. Canvas LMS has rubrics you can use to keep track of which problems students have the point for, and when you update the rubric scores it automatically updates the quiz score.&lt;/li&gt;&#xA;&lt;li&gt;Keep each question as simple as possible to test the desired skill. This makes it easy to come up with new versions of the question.&lt;/li&gt;&#xA;&lt;li&gt;Unless students have trouble understanding the original question, just rerandomize the numbers/formulas, don't bother coming up with a completely different question for the desired skill. This isn't a great compromise, but it's necessary to keep workload reasonable.&lt;/li&gt;&#xA;&lt;li&gt;Eventually you'll want lots of students to be taking lots of makeups. It's a great way to encourage students to keep studying a topic until they get it. Try to get lots of students working on makeup quizzes simultaneously. Once you've sped up the problem creation and grading processes, proctoring becomes the biggest timesink. Try to get lots of students in office hours at the same time or offer time in-class for makeup quizzes.&lt;/li&gt;&#xA;&lt;li&gt;One way to reduce end-of-semester demand for makeups is with &amp;quot;makeup midterms/finals&amp;quot;. Create a makeup version of every problem from previous quizzes, then have students just make up the problems they need to make up in the style of an in-class exam.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2023-05-17T19:10:11.323" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26445" PostTypeId="2" ParentId="26437" CreationDate="2023-05-17T19:27:39.267" Score="3" Body="&lt;p&gt;If you truly are interested in putting the time and effort into helping low-to-middle students learn the material and achieve success, then I highly recommend Standards-Based or Mastery Grading. Here's a link to a document that inspired and guided me in the beginning: &amp;quot;&lt;a href=&quot;https://rtalbert.org/building-calculus-the-grading-system/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Building Calculus: The grading system&lt;/a&gt;&amp;quot;. There are abundance of materials to find through web searches once you know the right phrases.&lt;/p&gt;&#xA;&lt;p&gt;I have used a standards-based approach in my lower-level courses: First semester STEM calculus, business calculus, finite math, and (non-proof) discrete math. The results are profound. A-level students don't mind the weirdness. And any other grade-level student can always see a pathway to improving their grade if they so desire.&lt;/p&gt;&#xA;&lt;p&gt;There are downsides. One reason I don't use it in my upper-level classes is you have to have an abundance of assessments/problems. (This can be hard to generate in proof-based courses.) Office hours will be busier (especially as the semester's end approaches). And the gradebook system will be complicated (I always enjoy a good spreadsheet challenge).&lt;/p&gt;&#xA;&lt;p&gt;But in the end it's all worth it. Students who want to put in the effort to learn the material and improve their grades will do so. (The students who don't want to put in the effort... well, the outcome is the same.)&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2023-05-17T19:27:39.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26446" PostTypeId="2" ParentId="26437" CreationDate="2023-05-17T23:24:34.540" Score="2" Body="&lt;p&gt;My idea is that you should just give the students a second chance on each problem/task without adding any workload for yourself. The extreme scheme would be like this: you give, say, 10 quizzes of 3 problems each, but with only 15 problems total (just literally, without changing even the numbers, but, of course, with a noticeable time spacing between the identical problems and without a warning about what exactly will be repeated). Then you just use the best attempt grade on each of those 15 problems. That extreme is a bit hard to implement exactly as it is because one needs to more or less align the quiz problems with the material being covered, but you can modify it a bit in various ways so that it becomes workable. The students who solve the problem from the first attempt for the full grade do not even need to bother to try it again and may spend more time on the new ones, which is a good incentive to do things right from the beginning for those who have some common sense and elementary logic. You can also include homework or quiz problems into exams and if they are solved correctly on the exam, upgrade the initial scores.&lt;/p&gt;&#xA;&lt;p&gt;Another idea that works in a small (10-20) people class is to give them a number of problems in the very beginning that they can submit any time during the course making as many attempts as they wish (if you do not want to endure too many attempts, you can set up a waiting period after each attempt or say that each problem can be tried at most once within the same week). When I was a student and we studied integration, we were just given a list of 100 integrals from the start some of which we didn't learn to do until later (unless one chose to study the corresponding techniques by him/herself ahead of the schedule). The submission was oral. The requirement was that by the time of the submission, the student should be able to do it without outside help (the exact way to ensure that varied but a common technique was just to write the integral on a blank sheet of paper and to let the student do the full computation in front of you). No real attempt to restrict what help the student was getting outside the classroom was made (everybody agreed that there was no point to try to control what one could not control and that the surest way to make a person dishonest is to make him/her promise absolute honesty). I use this scheme in my graduate classes all the time and in undergraduate ones quite often, though the portion of the total grade coming from such assignments is lower in the latter case.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-05-17T23:24:34.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26447" PostTypeId="2" ParentId="26432" CreationDate="2023-05-18T00:05:35.847" Score="6" Body="&lt;p&gt;Like any other word, it comes handy when the students are already proficient with the standard cancellation operations: reduction of common factors in a fraction, removing equal summands/factors from both sides of an equation, removing the summands with opposite signs from a formula, and so on.&lt;/p&gt;&#xA;&lt;p&gt;In the beginning, one is much better off with solving something like&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$5(3x+6)+18=15(7x-5x+8)+18$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;by saying&lt;/p&gt;&#xA;&lt;p&gt;Subtracting 18 from both sides, grouping the terms with the common factor &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; on the right, and dividing both sides by &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; afterwards, we can simplify the equation to &lt;span class=&quot;math-container&quot;&gt;$3x+6=3(2x+8)=6x+24$&lt;/span&gt;. Subtracting &lt;span class=&quot;math-container&quot;&gt;$6x+6$&lt;/span&gt; from both sides, we get &lt;span class=&quot;math-container&quot;&gt;$-3x=18$&lt;/span&gt;. Dividing both sides by &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt;, we get &lt;span class=&quot;math-container&quot;&gt;$x=-6$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, once one gets accustomed to such tricks, one can certainly say just:&lt;/p&gt;&#xA;&lt;p&gt;Cancel &lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt;, then cancel &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;, then cancel &lt;span class=&quot;math-container&quot;&gt;$5x$&lt;/span&gt; in parentheses, then open the parentheses and cancel &lt;span class=&quot;math-container&quot;&gt;$3x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$0=3x+18$&lt;/span&gt;, whence &lt;span class=&quot;math-container&quot;&gt;$x=-6$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;That's how we really think of the whole procedure ourselves and there is no reason to abstain from that language once one is ready for it. It is just a shortcut language with one word for many different operations having the common underlying idea. When you encounter some longer and more complicated formula, like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;xy+3(x^3+y^3)-zt+abc-3x^3-4y^3+tz+y^3\,&#xA;$$&lt;/span&gt;&#xA;it is much easier to say &amp;quot;canceling similar terms with opposite signs&amp;quot;, than to describe what exactly is being done step by step, but that phrase requires that &lt;span class=&quot;math-container&quot;&gt;$-zt$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$tz$&lt;/span&gt; immediately strike your eyes as essentially the same thing, that you can automatically open the parentheses in &lt;span class=&quot;math-container&quot;&gt;$3(x^3+y^3)$&lt;/span&gt; and combine &lt;span class=&quot;math-container&quot;&gt;$-4y^3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$+y^3$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$-3y^3$&lt;/span&gt;, etc. Before that stage is achieved, it is, probably, better to be more detailed on how you transform the formula into the &amp;quot;obvious&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$xy+abc$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;P.S. Looks like I mostly just rephrased what Sintax Junkie and Daniel R. Collins said already, but once I typed it, let it be :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-05-18T00:05:35.847" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26448" PostTypeId="2" ParentId="10150" CreationDate="2023-05-18T00:31:23.430" Score="3" Body="&lt;p&gt;Note that the &amp;quot;strong induction&amp;quot; is not even the most general induction scheme you can design. A high school level example is the proof of the AM-GM inequality for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; positive numbers where you can first use the two-number version to get it for &lt;span class=&quot;math-container&quot;&gt;$n=2^k$&lt;/span&gt; and then to descend from any &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; to any smaller &lt;span class=&quot;math-container&quot;&gt;$n'$&lt;/span&gt; by setting the last &lt;span class=&quot;math-container&quot;&gt;$n-n'$&lt;/span&gt; variables to be the AM (or GM, if you prefer) of the first &lt;span class=&quot;math-container&quot;&gt;$n'$&lt;/span&gt; ones.&lt;/p&gt;&#xA;&lt;p&gt;I usually start teaching induction with drawing a long train on the board and saying something like &amp;quot;the engine starts to move; how do we know that the rest of the cars will follow?&amp;quot;. The underlying idea is that each car is hooked to the previous one and that is the classical &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. Then, after reasonably many examples of that one, I ask the students how else we can link the cars to ensure that the whole train moves, opening a Pandora box with strong induction and all other types. You should be a bit careful here because the train analogy is not absolutely perfect, but one can usually get a clear idea and work out the details from it nevertheless.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-05-18T00:31:23.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26450" PostTypeId="2" ParentId="26437" CreationDate="2023-05-18T16:25:50.480" Score="3" Body="&lt;p&gt;It has also been my experience that most bonus schemes disproportionately benefit stronger students and increase grade inequality.  I think that dropped scores (as mentioned in Daniel's excellent answer) are a helpful start, and I use them in all of my courses.&lt;/p&gt;&#xA;&lt;p&gt;But going further, I find it helpful to think about &lt;em&gt;why&lt;/em&gt; I'd like to give bonus points.  As you mentioned, I would like bonus points to serve as a &amp;quot;catch-up&amp;quot; mechanism for weaker students.  I don't want to inflate their grades artificially, but I want to help them make progress and also &lt;em&gt;feel&lt;/em&gt; like they're making progress.&lt;/p&gt;&#xA;&lt;p&gt;This lead me to the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I aim to use bonus points to &amp;quot;subsidize&amp;quot; behaviors that promote engagement with the course objectives.&lt;/li&gt;&#xA;&lt;li&gt;I usually tie available bonus points to the number of lost points on an assessment.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Some of the answers above talked about make-up assessments, but I find managing the logistics of these frustrating and tedious.  Instead, I've done the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Give students that opportunity to complete an exam reflection sheet with bonus credit being awarded based on original test score (90-100% gets +1%, 80-90% gets +2%, etc.) Here's an example and more detailed description: &lt;a href=&quot;https://cte.psu.edu/teaching-learning/adopting-a-metacognitive-approach-to-teaching/exam-wrapper/&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Give students the opportunity to recover some percentage of lost test points by attending tutoring sessions at the university STEM Center.  For example, I allow students in math for business to recover up to 30% of their lost points.  This keeps the strong students from wasting their time, but it can make a huge difference for students who are at the bottom of the class and need to spend more time intentionally engaging with the homework.  (I don't do this for Calc 1, as it could allow ill-equipped students to continue to Calc 2.)&lt;/li&gt;&#xA;&lt;li&gt;Allow students to recover full credit on up to &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; exam problems from an assessment by presenting the solution(s) to me on the whiteboard in my office without any notes.  (This is a big time commitment for me, so I've only offered it in upper-level classes when I've given an exam with a poor class average and only for one problem of each student's choice.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5988" LastActivityDate="2023-05-18T16:25:50.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26451" PostTypeId="1" CreationDate="2023-05-19T07:02:18.990" Score="7" ViewCount="98" Body="&lt;p&gt;My department is considering using more than one lecturer (sequentially, not in parallel) to give lectures in our large first-year classes (e.g. 500 students doing engineering mathematics).&lt;/p&gt;&#xA;&lt;p&gt;In other words, one lecturer does the first 4 weeks, a second does the next 4 weeks and a third does the remaining 4 weeks.&lt;/p&gt;&#xA;&lt;p&gt;There is enough content of disparate nature (e.g., linear algebra, calculus of various types) to split into three distinct blocks so that each lecturer teaches a coherent block. For example, a student won't have to learn about eigenvalues from lecturer A and diagonalization from lecturer B.&lt;/p&gt;&#xA;&lt;p&gt;Is there evidence, ideally published research studies, about the effectiveness of such an arrangement, and whether the students view it favourably or otherwise etc?&lt;/p&gt;&#xA;&lt;p&gt;In addition, if you or your department has tried this, I'd be interested in knowing how it went on an anecdotal basis.&lt;/p&gt;&#xA;" OwnerUserId="4795" LastEditorUserId="2083" LastEditDate="2023-05-19T07:26:32.937" LastActivityDate="2023-05-19T15:22:24.360" Title="Joint Teaching of a First Year Engineering Maths Class" Tags="|undergraduate-education|teaching|engineering-mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26452" PostTypeId="1" CreationDate="2023-05-19T13:47:35.243" Score="2" ViewCount="353" Body="&lt;p&gt;(Im sorry if the question is out of the scope of the forum)&lt;/p&gt;&#xA;&lt;p&gt;Hi, Im currently a Physics student. I have studied most of the Calculus. Now, according to the book Im using, there is chapter on &amp;quot;Fourier Transform&amp;quot; and &lt;em&gt;then&lt;/em&gt; there's a chapter on &amp;quot;Laplace Transform&amp;quot;.&#xA;Now, Iam for some reason inclined to learn&amp;quot;Laplace Transform&amp;quot; (because a Great youtuber has its full series)&lt;/p&gt;&#xA;&lt;p&gt;So, is the knowledge of Fourier transform necessary for learning Laplace Transform, or I will be fine without it?&lt;/p&gt;&#xA;" OwnerUserId="22393" LastEditorUserId="2083" LastEditDate="2023-05-19T16:11:09.023" LastActivityDate="2023-05-21T12:50:27.357" Title="Should one study Laplace Transformation before Fourier Transforms?" Tags="|mathematical-analysis|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26453" PostTypeId="2" ParentId="26452" CreationDate="2023-05-19T14:51:23.373" Score="5" Body="&lt;p&gt;I think you should be able to easily understand Laplace transform though it is little tricky at first. I didn't need any knowledge of Fourier Transform.&lt;/p&gt;&#xA;&lt;p&gt;I don't what book you are reading you might be interested: A first course of Differential Equations with Modelling Application By Dennis G Zill. It covers Laplace transform as well application of Laplace Transform to differential equations. I found it easy to read.&lt;/p&gt;&#xA;" OwnerUserId="20626" LastActivityDate="2023-05-19T14:51:23.373" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26454" PostTypeId="2" ParentId="26451" CreationDate="2023-05-19T15:22:24.360" Score="2" Body="&lt;p&gt;I have a close friend who teaches physiology at Harvard. She has done courses this way, and says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I’ve done it in different ways.  Usually I have a block, then the&#xA;co-teacher, then back to me.  The key is communication to make sure&#xA;you don’t repeat each other, or assume that the other person has&#xA;covered something that they hadn’t.&lt;/p&gt;&#xA;&lt;p&gt;Also, you need to figure out how to split up administrative tasks so&#xA;that no one is over-burdened and so it’s clear who is responsible for&#xA;what.  It’s great when the teachers have other obligations - gives&#xA;them blocks of time to get things done.&lt;/p&gt;&#xA;&lt;p&gt;As long as both teachers are good, students generally like it.&#xA;Different people explain things slightly differently, and it can help&#xA;students get a better understanding.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[If you'd prefer only math-related answers, please let me know and I'll delete this.]&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-05-19T15:22:24.360" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26470" PostTypeId="2" ParentId="21203" CreationDate="2023-05-20T00:13:53.443" Score="4" Body="&lt;p&gt;&lt;em&gt;&lt;strong&gt;On the derivative paradox.&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think it is essential to raise doubt in the students' minds about what the OP's calls &amp;quot;the derivative paradox,&amp;quot; at least in the case of instantaneous velocity, which is what I use to introduce the derivative.  I've related before the story of a friend of mine who got a fortune cookie whose fortune read, &amp;quot;Never try to prove what no one doubts.&amp;quot;  The doubt motivates whatever technical move is needed, whether for a theorem, its proof, or a definition, such as using a limit to not let the denominator reach zero, as &lt;a href=&quot;https://matheducators.stackexchange.com/a/25279&quot;&gt;Will Orrick&lt;/a&gt; observed.&lt;/p&gt;&#xA;&lt;p&gt;No student fails to grasp the paradox, and part of that paradox is the feeling that there must be such a thing as instantaneous velocity.&#xA;&amp;quot;Everyone has or believes to have an idea of velocity,&amp;quot; said Lagrange in criticizing one of the two common approaches to analysis at the time.  Indeed, most textbooks (that I've read) omit the paradox and latch directly onto our intution:&#xA;&amp;quot;The idea of instantaneous velocity makes intuitive sense, but care is required to define it precisely.&amp;quot; (Rogawski/Adams/Franzosa)&#xA;&amp;quot;We assume from watching the speedometer that the car has a definite velocity at each moment, but how is the 'instantaneous' velocity defined?&amp;quot; (Stewart/Clegg/Saleem) Not only do they assume it exists, they implicitly assume it is the limit of the average velocity: &amp;quot;...[W]e can approximate the desired quantity by computing the average velocity over the brief time interval of a tenth of a second...&amp;quot; (Stewart/Clegg/Saleem).  In similar fashion, it is common to assume the slope of the secant line approximates the slope of the tangent line, without any prior definition of tangency.  It seems fine to me to make such assumptions.  Textbooks, and therefore I assume many teachers, make plenty of them in teaching first-time calculus, just as I do. I have two criticisms. One is what are being assumed, defined, and proved should be made clear.  Many of the textbooks are so informal at the beginning (introduction) of the calculus that it is sometimes vague which of the three is happening with regard to instantaneous velocity and the slope of the tangent line.  The other criticism is that if the idea of instantaneous velocity makes clear, intuitive sense, I wonder why it took so long to define it precisely.  That is a long and interesting investigation, which I won't go into, but the point is that there is a paradox like the OP's which cannot easily be dismissed as being intuitively obvious.  As others have pointed out, it is the definition of the derivative that is meant to get the student past the paradox.&lt;/p&gt;&#xA;&lt;p&gt;This Q&amp;amp;A came to my attention because of the update by the OP that included a sad assessment, &amp;quot;Everything I thought might be a problem ended up as a non-issue because no-one challenged anything.&amp;quot;&#xA;Ever since graduate school, when the younger sister of one my students said hi to me when we were introduced by saying, &amp;quot;Math is bogus!,&amp;quot; I have encountered such attitudes.  And I have always taken seriously their reactions.  I think their attitude toward math contributes to the student's feeling that challenging anything in math is pointless.  I teach in the US, and, of course, not every student has the same experience. The OP mentions &amp;quot;A level,&amp;quot; which I take to mean they are working within the British educational system or one based on it.  I do not know what difference that makes, though.  I know that students might leave my class with a math-is-bogus attitude, despite my attempts to show them true, non-bogus mathematics, because they come into the course with preconceived notions and bogus math feels comfortable when they are good at getting right answers.&lt;/p&gt;&#xA;&lt;p&gt;What I have to share here is a different approach that complements other answers. Certainly I think the derivative paradox is handled adequately by the definition of the limit. However, I think we can do better.&#xA;A rich understanding of mathematics comprises more than the logical structure created by our choices of axioms and definitions.&#xA;The connections that motivate our choices are important.&#xA;As Proclus said, &amp;quot;...mathematics, though beginning with reminders from the outside world, ends with ideas that it has within....&amp;quot;&#xA;Often what motivates our choices becomes clear only on reflection, after we try out a definition and see why it succeeds.  I offer an approach to derivatives motivated by an intuition that what we would like is for the average velocity of a smoothly moving object to vary continuously.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;An approach to the derivative paradox&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Does anyone have any ideas, experience-backed, for how to take a curious student past the “derivative paradox”?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Perhaps you will like the following approach, which I adapted from Stephen Kuhn (The Derivative à la Carathéodory. &lt;em&gt;Am. Math. Monthly&lt;/em&gt;, 1991) after seeing Ray Mayer's &lt;a href=&quot;https://people.reed.edu/%7Emayer/math112.html/chap10.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;notes for Math 112&lt;/a&gt; at Reed College.  I chose a slight twist to Kuhn's idea.  Kuhn's small observation that the average rate of change between a variable point &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and fixed point &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; can be extended to a continous function at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, in the case where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differentiable at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, is key to the kind of model of change we want to create in calculus/analysis.  Our experience of the real world leads us to imagine that in continuous change, such as we experience in observing the motion of an object, velocity changes continously when an object moves smoothly.  Treat these observations as intuitions, and ask, how can I create a mathematics that models what I imagine I see?  Once we create such a mathematics, we can analyze which functions model smooth change and which do not.&lt;/p&gt;&#xA;&lt;p&gt;Building on our notion of average velocity or rate of change, the change in position over the change in time, there is a fairly clear way to procede, as set out by Carathéodory (Theory of Functions of a Complex Variable, AMS/Chelsea, 2001; orig. 1954; German ed. 1950).&#xA;We may define differentiability and the derivative as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; defined in an open interval containing a given number &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, if we can find a function &lt;span class=&quot;math-container&quot;&gt;$f^\Delta$&lt;/span&gt; such that (1) &lt;span class=&quot;math-container&quot;&gt;$f^\Delta(x) = [f(x)-f(c)]/(x-c)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x\ne c$&lt;/span&gt; in the interval&#xA;and (2) &lt;span class=&quot;math-container&quot;&gt;$f^\Delta$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is said to be &lt;em&gt;differentiable&lt;/em&gt; at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and the derivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is given by &lt;span class=&quot;math-container&quot;&gt;$f'(c)=f^\Delta(c)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Example 1 (differentiability).&lt;/strong&gt;&lt;/em&gt;&#xA;Let &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2$&lt;/span&gt;. Fix a number &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; (any real number).&#xA;Then &lt;span class=&quot;math-container&quot;&gt;$[f(x)-f(c)]/(x-c) = (x^2-c^2)/(x-c)=x+c$&lt;/span&gt;.  Now, &lt;span class=&quot;math-container&quot;&gt;$f^\Delta$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$f^\Delta(x)=x+c$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, and it is equal to the average rate of change of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \ne c$&lt;/span&gt;. Hence we have found the required function, and therefore &lt;span class=&quot;math-container&quot;&gt;$f'(c)=2c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Note that the algebraic mechanics of finding &lt;span class=&quot;math-container&quot;&gt;$f^\Delta(x)$&lt;/span&gt; is the same as finding the limit in the limit formulation of the derivative.  Indeed, it is easy to prove they are equivalent. OTOH, the conceptual goal is to find a continuous model for the average rate of change in a neighborhood of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.  At no time are we concerned with a limit of the indeterminate form &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Discussion.&lt;/em&gt;&#xA;The process works this way for the standard algebraic examples used to introduce the derivative. I think that is sufficient to recommend it. In terms of mechanical algebraic operations, there is not much difference, which may leave you thinking that the difference is negligible. I might add that the &amp;quot;recipe&amp;quot; is to start with the average rate and analyze. For an algebraic function, factor out &lt;span class=&quot;math-container&quot;&gt;$x-c$&lt;/span&gt; and reduce to a continuous function. I like that we focus the discussion on the average rate and the question of its continuity instead of on a limit formula.&#xA;(When working from the limit definition of the derivative, one could structure the limit computation to start with the average rate, but it seems natural to start with the formula given in the definition.)&#xA;See Example 4 below for a transcendental function example.&#xA;I do like to give an example of nondifferentiability, not only because it is good practice to show objects on both sides of the boundary drawn by a definition, but because of how it reflects on the goal of finding a continuous model for the average rate of change; see Example 4.&#xA;First, Example 1 may be extended as follows.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Example 2 (mutatis mutandis).&lt;/strong&gt;&lt;/em&gt;&#xA;Let &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^n$&lt;/span&gt; for an integer &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;2$&lt;/span&gt;.&#xA;Fix a number &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; (any real number).&#xA;Then &lt;span class=&quot;math-container&quot;&gt;$[f(x)-f(c)]/(x-c) = (x^n-c^n)/(x-c)=x^{n-1}+c\,x^{n-2}+\cdots+c^{n-1}$&lt;/span&gt;, a polynomial.&#xA;Now, &lt;span class=&quot;math-container&quot;&gt;$f^\Delta$&lt;/span&gt; defined by this polynomial is continuous at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, and it is equal to the average rate of change of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \ne c$&lt;/span&gt;.&#xA;Hence we have found the required function, and therefore &lt;span class=&quot;math-container&quot;&gt;$f'(c)=n\,c^{n-1}$&lt;/span&gt;, the value of the polynomial at &lt;span class=&quot;math-container&quot;&gt;$x=c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Example 3 (general rule).&lt;/strong&gt;&lt;/em&gt;&#xA;Let &lt;span class=&quot;math-container&quot;&gt;$f^\Delta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g^\Delta$&lt;/span&gt; be the continuous extensions of the average rates of change of functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; respectively, assumed to be differentiable at a number &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;. Then it's easy to show &lt;span class=&quot;math-container&quot;&gt;$f^\Delta+g^\Delta$&lt;/span&gt; is the required &amp;quot;Delta&amp;quot; function to show the differentiability of &lt;span class=&quot;math-container&quot;&gt;$f+g$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;. Exercise left for the reader.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Example 4 (singularity).&lt;/strong&gt;&lt;/em&gt;&#xA;Suppose a ball bounces off a wall,&#xA;such that the ball initially had some constant velocity, say &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;,&#xA;and after the bounce the velocity was constant but &lt;span class=&quot;math-container&quot;&gt;$-v$&lt;/span&gt;.&#xA;The position would be given by &lt;span class=&quot;math-container&quot;&gt;$f(t)=-v\,|t-c|$&lt;/span&gt;,&#xA;if we set the time of the collision to be at &lt;span class=&quot;math-container&quot;&gt;$t=c$&lt;/span&gt;.&#xA;The average velocity between &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \cases{\phantom{-}v &amp;amp; if $t&amp;lt;c$; \cr -v &amp;amp; if $t&amp;gt;c$. \cr}&#xA;$$&lt;/span&gt;&#xA;Since the limit as &lt;span class=&quot;math-container&quot;&gt;$t \rightarrow c$&lt;/span&gt; does not exist,&#xA;there can be no function &lt;span class=&quot;math-container&quot;&gt;$f^\Delta(t)$&lt;/span&gt; that is continuous at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;&#xA;and equal to the average velocity for &lt;span class=&quot;math-container&quot;&gt;$t \ne c$&lt;/span&gt;.&#xA;Thus the position is not a differentiable function at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;,&#xA;and we do not have a way to define the derivative at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Discussion.&lt;/em&gt;&#xA;One might argue that the average of the two velocities, a velocity of &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, makes sense as a way to define the derivative since the wall is at rest and the ball is in contact with the wall at &lt;span class=&quot;math-container&quot;&gt;$t=c$&lt;/span&gt;.&#xA;(Compare with the &lt;a href=&quot;https://math.stackexchange.com/q/2699551&quot;&gt;central difference formula&lt;/a&gt;.)&#xA;But looking to the physics in this case is problematic, since rebounds are quite complicated phenomena that are not instantaneous.  To rely on the physics requires a more complicated and reliable model than the absolute value. To refocus our attention, it is the nondifferentiability of the absolute value that is our interest.&#xA;We may like to imagine that velocity is an inherent property of an object, like its color, and present at each instant.&#xA;As far as our model of an instantaneous bounce goes,&#xA;it's probably best to say the instantaneous velocity at &lt;span class=&quot;math-container&quot;&gt;$t=c$&lt;/span&gt; is undefined.&#xA;Note that this conclusion is about our model and not about the physics of bouncing; however, the nondifferentiability of absolute value is a consequence of our choice to base differentiability on extending the average rate of change to a continuous function.&lt;/p&gt;&#xA;&lt;p&gt;Viewing differentiability as the continuity of the average rate adds some interest to the standard pathological example &lt;span class=&quot;math-container&quot;&gt;$g(x) = x^2 \sin(1/x)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \ne 0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g(0)=0$&lt;/span&gt;. For any &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, there is a continuous &lt;span class=&quot;math-container&quot;&gt;$g^\Delta(x)$&lt;/span&gt; that equals &lt;span class=&quot;math-container&quot;&gt;$[g(x)-g(c)]/(x-c)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x \ne c$&lt;/span&gt; — continuous, as it turns out, for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. So &lt;span class=&quot;math-container&quot;&gt;$g'(c)$&lt;/span&gt; is defined for all &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;. But despite the continuity of &lt;span class=&quot;math-container&quot;&gt;$g^\Delta(x)$&lt;/span&gt; for each &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g'$&lt;/span&gt; is not itself continuous at &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. Later one can spiral back to take up the continuity of &lt;span class=&quot;math-container&quot;&gt;$g^\Delta$&lt;/span&gt; as a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &lt;em&gt;&lt;strong&gt;and&lt;/strong&gt;&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; in multivariable calculus.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Remark on &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt; and friends.&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Once you get past the introduction of the idea of a derivative, one can simply use the limit formulation of the derivative to find the value for &lt;span class=&quot;math-container&quot;&gt;$f'(c)$&lt;/span&gt; at the missing point &lt;span class=&quot;math-container&quot;&gt;$x=c$&lt;/span&gt; that makes the average rate continuous at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; (or show no such value exists).&#xA;But if you want to continue with the approach in example 1, the transcendental functions offer challenges that are familiar when using the limit formula.  The two processes never need be very different, after all.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Example 5.&lt;/strong&gt;&lt;/em&gt;&#xA;So for &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;, one can show geometrically that if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; lie in the same quadrant, &lt;span class=&quot;math-container&quot;&gt;$\sin x - \sin c$&lt;/span&gt; lies between &lt;span class=&quot;math-container&quot;&gt;$\cos x\cdot (x-c)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos c \cdot (x-c)$&lt;/span&gt;. One can use the squeeze theorem then to show &lt;span class=&quot;math-container&quot;&gt;$\cos c$&lt;/span&gt; extends &lt;span class=&quot;math-container&quot;&gt;$[\sin x - \sin c]/(x-c)$&lt;/span&gt; to a continuous function at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2023-05-22T04:54:05.267" LastActivityDate="2023-05-22T04:54:05.267" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26471" PostTypeId="2" ParentId="21203" CreationDate="2023-05-20T10:41:39.777" Score="0" Body="&lt;p&gt;What you have described is an indeterminant form. Examples include&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;infinity - infinity&lt;/li&gt;&#xA;&lt;li&gt;0/0&lt;/li&gt;&#xA;&lt;li&gt;0^0&lt;/li&gt;&#xA;&lt;li&gt;1^infinity&lt;/li&gt;&#xA;&lt;li&gt;infinity/infinity&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;What this means is upon substitution of limit value you will not be to evaluate the limit.&lt;/p&gt;&#xA;&lt;p&gt;To give an example&lt;/p&gt;&#xA;&lt;p&gt;dy/dx = limit dx-&amp;gt;0 (f(x+dx) - f(x))/dx&lt;/p&gt;&#xA;&lt;p&gt;Substituting dx = 0 in the right handside we will get&lt;/p&gt;&#xA;&lt;p&gt;(f(x+0) - f(x)) / (0) = (f(x)-f(x))/0 = 0/0&lt;/p&gt;&#xA;&lt;p&gt;So performing differentiation by first principles we need to deal with indeterminant forms.&lt;/p&gt;&#xA;&lt;p&gt;The question you have asked is how do we resolve dividing by zero. The answer is we are not. What we are doing performing limit operations. We can various informal definition for limits such the standard limit definition, infinite limits, limits at infinity. In other words it is not required to understand the epsilon delta definition limits.&lt;/p&gt;&#xA;&lt;p&gt;Just add a minor note. I often students ask why 0/0 = k. For some reason they have confused indeterminant forms with arithmetic.&lt;/p&gt;&#xA;&lt;p&gt;Now question is how do deal with indeterminant forms. To deal with these forms often some simplification needs be done or proof a theorem. So in the case of the power rule. When proving you you should cancel out the dx in the numerator/denominator.&lt;/p&gt;&#xA;&lt;p&gt;This would have left you with&lt;/p&gt;&#xA;&lt;p&gt;limit dx-&amp;gt;0 (nx^(n-1)  + (dx)*(nC2)*x^n-2 + ...)&lt;/p&gt;&#xA;&lt;p&gt;All the terms with dx will become zero.&lt;/p&gt;&#xA;&lt;p&gt;You have asked why dx^2 is smaller than dx. If we think graph x and x^2. when x=0.5, x=0.25. To summarise a x^2 is approach zero faster than x. This is what makes limit x-&amp;gt;0 (x^2)/x = 0. The other way to think about it is the simplification of fractions which we apply earlier in the power rule. In this case dx^2/dx = dx.&lt;/p&gt;&#xA;&lt;p&gt;So limit dx-&amp;gt;0 dx^2 /dx = limit dx-&amp;gt;0 dx = 0.&lt;/p&gt;&#xA;&lt;p&gt;To summarise there is no paradox however the content can be a bit challenging to understand.&lt;/p&gt;&#xA;" OwnerUserId="20626" LastEditorUserId="20626" LastEditDate="2023-05-20T10:48:34.677" LastActivityDate="2023-05-20T10:48:34.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26472" PostTypeId="1" CreationDate="2023-05-20T18:22:35.037" Score="2" ViewCount="193" Body="&lt;p&gt;When tutoring, I try to simplify concepts. I came up with these examples to explain the intention behind variance and covariance. Could you please help me find conceptual, pedagogical or mathematical errors I've overlooked? Any comments and improvements are appreciated. And if it's correct, please feel free to use it.&lt;/p&gt;&#xA;&lt;h2&gt;Variance&lt;/h2&gt;&#xA;&lt;p&gt;A psychologist monitors Jack and his Cat in their home and regularly notes their respective moods. Negative numbers indicate stress, positive numbers indicate happiness.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Jack&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;&lt;strong&gt;Mood&lt;/strong&gt;&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;-20 ‍♂️&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;-15 ‍♂️&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;-10 ‍♂️&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;-5 ‍♂️&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;0 ‍♂️&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;+5 ‍♂️&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;+10 ‍♂️&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;+15 ‍♂️&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;+20 ‍♂️&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;strong&gt;Count&lt;/strong&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;3&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;3&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;2&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;2&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;4&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;&lt;strong&gt;Jack's Cat&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;&lt;strong&gt;Mood&lt;/strong&gt;&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;-20 &lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;-15 &lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;-10 &lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;-5 &lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;0 &lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;+5 &lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;+10 &lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;+15 &lt;/th&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;+20 &lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;strong&gt;Count&lt;/strong&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;0&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;5&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;5&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;3&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;2&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;1&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: center;&quot;&gt;0&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;Variance is a measure for how much a value deviates from its arithmetic mean. Variance values large deviations much stronger than smaller ones.&lt;/p&gt;&#xA;&lt;p&gt;The sample variance of Jack's mood is 238.24, while the cat's mood shows a sample variance of 55.88. From these numbers the psychologist can induce correctly, that the cat is overall a lot calmer than Jack.&lt;/p&gt;&#xA;&lt;h2&gt;Covariance&lt;/h2&gt;&#xA;&lt;p&gt;In a new measurement, the psychologist tries to determine, whether Jack and his cat influence each other in their moods. She makes three different measurements:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Jack get's a call from his boss ‍♂️, cat sleeps &lt;/strong&gt;&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;&lt;strong&gt;Time (minutes)&lt;/strong&gt;&lt;/th&gt;&#xA;&lt;th&gt;0&lt;/th&gt;&#xA;&lt;th&gt;1&lt;/th&gt;&#xA;&lt;th&gt;2&lt;/th&gt;&#xA;&lt;th&gt;3&lt;/th&gt;&#xA;&lt;th&gt;4&lt;/th&gt;&#xA;&lt;th&gt;5&lt;/th&gt;&#xA;&lt;th&gt;6&lt;/th&gt;&#xA;&lt;th&gt;7&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;strong&gt;Jack's mood&lt;/strong&gt;&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;-5&lt;/td&gt;&#xA;&lt;td&gt;-15&lt;/td&gt;&#xA;&lt;td&gt;-15&lt;/td&gt;&#xA;&lt;td&gt;-5&lt;/td&gt;&#xA;&lt;td&gt;0&lt;/td&gt;&#xA;&lt;td&gt;-5&lt;/td&gt;&#xA;&lt;td&gt;-5&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;strong&gt;Cat's mood&lt;/strong&gt;&lt;/td&gt;&#xA;&lt;td&gt;0&lt;/td&gt;&#xA;&lt;td&gt;-1&lt;/td&gt;&#xA;&lt;td&gt;+1&lt;/td&gt;&#xA;&lt;td&gt;0&lt;/td&gt;&#xA;&lt;td&gt;0&lt;/td&gt;&#xA;&lt;td&gt;0&lt;/td&gt;&#xA;&lt;td&gt;0&lt;/td&gt;&#xA;&lt;td&gt;0&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;Covariance: -1.25&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Jack comes home from work ‍♂️, just in time for feeding his Cat &lt;/strong&gt;&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;&lt;strong&gt;Time (minutes)&lt;/strong&gt;&lt;/th&gt;&#xA;&lt;th&gt;0&lt;/th&gt;&#xA;&lt;th&gt;1&lt;/th&gt;&#xA;&lt;th&gt;2&lt;/th&gt;&#xA;&lt;th&gt;3&lt;/th&gt;&#xA;&lt;th&gt;4&lt;/th&gt;&#xA;&lt;th&gt;5&lt;/th&gt;&#xA;&lt;th&gt;6&lt;/th&gt;&#xA;&lt;th&gt;7&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;strong&gt;Jack's mood&lt;/strong&gt;&lt;/td&gt;&#xA;&lt;td&gt;-5&lt;/td&gt;&#xA;&lt;td&gt;+5&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;+5&lt;/td&gt;&#xA;&lt;td&gt;+5&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;strong&gt;Cat's mood&lt;/strong&gt;&lt;/td&gt;&#xA;&lt;td&gt;-10&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;+15&lt;/td&gt;&#xA;&lt;td&gt;+15&lt;/td&gt;&#xA;&lt;td&gt;+15&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;+5&lt;/td&gt;&#xA;&lt;td&gt;0&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;Covariance: 37.5&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The cat intentionally anger's Jack  by throwing a flower pot to the ground ‍♂️&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;&lt;strong&gt;Time (minutes)&lt;/strong&gt;&lt;/th&gt;&#xA;&lt;th&gt;0&lt;/th&gt;&#xA;&lt;th&gt;1&lt;/th&gt;&#xA;&lt;th&gt;2&lt;/th&gt;&#xA;&lt;th&gt;3&lt;/th&gt;&#xA;&lt;th&gt;4&lt;/th&gt;&#xA;&lt;th&gt;5&lt;/th&gt;&#xA;&lt;th&gt;6&lt;/th&gt;&#xA;&lt;th&gt;7&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;strong&gt;Jack's mood&lt;/strong&gt;&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;-20&lt;/td&gt;&#xA;&lt;td&gt;-15&lt;/td&gt;&#xA;&lt;td&gt;-15&lt;/td&gt;&#xA;&lt;td&gt;-10&lt;/td&gt;&#xA;&lt;td&gt;-10&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;&lt;strong&gt;Cat's mood&lt;/strong&gt;&lt;/td&gt;&#xA;&lt;td&gt;0&lt;/td&gt;&#xA;&lt;td&gt;+5&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;+15&lt;/td&gt;&#xA;&lt;td&gt;+15&lt;/td&gt;&#xA;&lt;td&gt;+10&lt;/td&gt;&#xA;&lt;td&gt;+5&lt;/td&gt;&#xA;&lt;td&gt;+5&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;Covariance: -37.5&lt;/p&gt;&#xA;&lt;p&gt;Covariance shows, whether two variables deviate together.&#xA;In the first example only Jack is upset, the cat sleeps soundly. This makes for a covariance near zero, since the two moods don't really affect each other.&lt;/p&gt;&#xA;&lt;p&gt;When Jack meets and feeds his cat, both get happier together. The positive covariance shows, that both variables deviate in the same direction from their respective means.&lt;/p&gt;&#xA;&lt;p&gt;When the Cat angers Jack it get's happier while Jack grows quite unhappy at the same time. Here the covariance is negative, showing that the variables are connected but moving in opposite directions.&lt;/p&gt;&#xA;" OwnerUserId="22413" LastActivityDate="2023-05-30T06:31:57.693" Title="Are these explanations of variance and covariance intuitive?" Tags="|examples|statistics|intuition|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26473" PostTypeId="2" ParentId="26422" CreationDate="2023-05-20T19:56:37.087" Score="2" Body="&lt;p&gt;Some easy examples I would probably use for error correction and information redundancy:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Bad phone connection and people with hearing aids&lt;/strong&gt; &lt;br/&gt;&#xA;Most people naturally talk slower, intentionally elongating their words. This ensures that most of the information gets through to the recipient, even if occasionally bits and pieces get lost on the way.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Explaining as a teacher&lt;/strong&gt; &lt;br/&gt;&#xA;We often try to explain concepts with different (and redundant) examples, knowing that not all of them reach each student. &lt;br/&gt;&#xA;&lt;em&gt;&amp;quot;The average rate of change can be used to calculate the average speed. Like monitoring how fast bitcoins are losing value or determining, if a video is trending.&amp;quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Marriage vows and love letters&lt;/strong&gt; &lt;br /&gt;&#xA;Lovers often struggle to express their feelings verbally, which makes them use more words than necessary, giving redundant information and often even repeating themselves, hoping to get the point across.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Mafia comedy movies&lt;/strong&gt; &lt;br /&gt;&#xA;Trying to get the point across without actually saying it&lt;br /&gt;&#xA;&lt;em&gt;Fredo, go and take care of Antonio Balucci. I want him iced, you know? Accidents happen. Make him sleep with the fishes...&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For two dimensional error correction it might be good to show a heavily compressed jpeg of a text. Yes, it's readable, but we have to add missing and change wrong pixels in our mind to reconstruct the letters. And since there are only 26 possible letters to choose from we automatically work the same tricks as, say, a prefix code.&lt;/p&gt;&#xA;" OwnerUserId="22413" LastActivityDate="2023-05-20T19:56:37.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26481" PostTypeId="2" ParentId="26452" CreationDate="2023-05-21T12:50:27.357" Score="4" Body="&lt;p&gt;Laplace Transforms are a fairly common, if not required, topic in the first undergraduate course on differential equations. Essentially the Laplace Transform is used to treat problems with discontinuous forcing functions. It also has the advantage of naturally folding initial conditions into your solution. Typical problems allow for us to trade a differential equation in the time domain for an associated algebraic problem in the so-called frequency domain. The full mathematics is really analysis of distributions and you will not find much of that in the usual textbooks.&lt;/p&gt;&#xA;&lt;p&gt;Fourier Transforms are not usually taught in the introductory DEQns course, at least in my experience at typical US universities. You might find Fourier analysis in a second course in DEqns where you focused on boundary value problems. However, I had such a course without the transform. The Fourier Transform would seem to enjoy greater popularity in its application to higher mathematics. Generalized Fourier analysis is a thing. Also, the application to quantum mechanics is important, the Fourier transform takes you from position to momentum representation.&lt;/p&gt;&#xA;&lt;p&gt;Both transforms have a similar application, they're used to convert differential equations to algebraic equations. The Laplace transform treats initial data nicely. The Fourier transform implement boundary conditions naturally. As other comments have already pointed out, both of these transforms are best when implemented in tandem with the full suite of complex analysis tools. Last semester I taught an independent study course where the text (Introduction to Hilbert Spaces with applications by Debnath and Mikusinski) had an example which simultaneously used both Laplace and Fourier transforms in conjunction to solve a partial differential equation.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2023-05-21T12:50:27.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26482" PostTypeId="1" CreationDate="2023-05-21T19:57:58.660" Score="4" ViewCount="103" Body="&lt;p&gt;Are you aware of any research examining the impact of utilizing online platforms in teaching mathematics, on students' beliefs about mathematics? To give you an example of the kind of beliefs that I have in mind, does the use of online platforms promote the belief that the goal of solving problems in mathematics is to find the correct answer? Surprisingly, my search for such studies did not find any notable results.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-05-21T20:16:58.930" LastActivityDate="2023-05-21T20:16:58.930" Title="Studies on the effects of using online platforms in teaching mathematics on students' beliefs about mathematics" Tags="|undergraduate-education|education-research|online-instruction|onlineplatforms|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26483" PostTypeId="2" ParentId="26432" CreationDate="2023-05-22T02:58:49.167" Score="5" Body="&lt;p&gt;Just because a term can be used without really understanding it doesn't, in my opinion, mean that the term should be avoided. One good practice in such cases would be to sometimes ask students what they mean by the term.&lt;/p&gt;&#xA;&lt;p&gt;Whenever we teach the concept of canceling in fractions, we definitely need to communicate what is really going on (that we simply have a multiply by 1 that we no longer have to write). When it's for addition, there simply is an addition of 0 that we also don't have to write.&lt;/p&gt;&#xA;&lt;p&gt;One advantage of using the word &amp;quot;cancel&amp;quot; is that a similar thing really is going on in both addition and multiplication. Using the same word for analogous things is a good thing.&lt;/p&gt;&#xA;" OwnerUserId="620" LastActivityDate="2023-05-22T02:58:49.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26484" PostTypeId="2" ParentId="26389" CreationDate="2023-05-22T07:04:33.837" Score="3" Body="&lt;p&gt;&lt;em&gt;Introduction&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I take it that the OP, now departed from the site, was exploring a pedagogical way to introduce the concept of continuity. I do not think they were trying to put continuity on a new rigorous basis.  I once gave a talk, &amp;quot;Limits without delta,&amp;quot; in which I described a middle ground between the &amp;quot;intuitive notion of a limit&amp;quot; and the &amp;quot;rigorous definition of a limit,&amp;quot; as they are often called in US textbooks.  The problem with learning a rigorous approach to limits is the algebraic handling of inequalities involving distance between numerical quantities.  Students are comparatively okay at determining the sign of factored expressions.  But distances involving the absolute value are an algebraic mess in their hands.  They might manage the nested quantifiers — who knows? Others say no, but I claim the students' difficulties at coordinating the &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; inequalities&#xA;(not their fault, due to it not being taught at all)&#xA;makes testing that hypothesis impossible.&#xA;I intended the middle ground to give the students some practice at thinking with inequalities,&#xA;a skill that benefits from recurrent practice over a few semesters.&#xA;The idea started from Hardy's approach to sequences in &lt;em&gt;A Course of Pure Mathematics&lt;/em&gt; and later got a boost while I was rereading Marsden and Tromba's &lt;em&gt;Vector Calculus&lt;/em&gt;.  In Marsden and Tromba, I noticed they kept using the squeeze theorem to manage &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; without any need to solve for &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;.  Another attack on &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; is that it is more complicated than how mathematicians actually prove limits, and I wondered why we don't teach the squeeze theorem more effectively.  In most textbooks, it resembles an appendix on the verge of appendicitis.&lt;/p&gt;&#xA;&lt;p&gt;The definition below introduces continuity without reference to limits, in accordance with OP's course structure.  It is nothing more than &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; in disguise.  An advantage is obtained by quickly introducing some squeeze theorems that the students can use in practical work.&#xA;Intuitive language based on the common notion of distance allow the students to avoid much of the algebra of inequalities that usually accompanies &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;.  The work is not devoid of inequalities, but it is simplified.  Again, the main goal is to understand the concept of continuity, and only in simplified contexts.  In my mind the importance of continuity has its root source in our experience of motion and change in the world.  Most of what we see or feel are continuous changes.  Sudden changes — collisions, for instance — are imagined to be occur over a very small interval of time and not be &amp;quot;quantum leaps.&amp;quot;&#xA;In class, I place&#xA;continuous change centered in the analysis of the change &lt;span class=&quot;math-container&quot;&gt;$f(x)-f(c)$&lt;/span&gt; and emphasize that continuity is not just a kind of exam question but a model of change in the world.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;Definition&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The setup to be assumed:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is a number;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a function;&lt;/li&gt;&#xA;&lt;li&gt;Some notion that &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is a &amp;quot;limit point&amp;quot; of the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;:&#xA;For first-time calculus students, let's say we will assume the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; contains an open interval containing &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;. This is stronger than necessary but it keeps things simple.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is &lt;strong&gt;continuous&lt;/strong&gt; at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; if for every positive distance, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; lies within that distance of &lt;span class=&quot;math-container&quot;&gt;$f(c)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that are sufficiently close to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and belong to the domain of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;Theorems and examples&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Theorem 1.&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$f(x) = x$&lt;/span&gt; is continuous at every real number.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Corollary 1.1.&lt;/em&gt; If you can find a positive constant &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$|f(x)-f(c)| \le M\cdot|x-c|$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; sufficiently close to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Proofs: Theorem 1 is a tautology, more or less, and your best approach is not to try to prove what no one doubts. One might say, however close we would like &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(c)$&lt;/span&gt; to be, well, that's the same as how close &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; would have to be, since &lt;span class=&quot;math-container&quot;&gt;$f(x)=x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(c)=c$&lt;/span&gt;. --&#xA;[Perhaps you would like to point out the meaning of being within some distance. Since &lt;span class=&quot;math-container&quot;&gt;$f(x)-f(c)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$x-c$&lt;/span&gt;, then what we need to show to prove continuity is that &lt;span class=&quot;math-container&quot;&gt;$|x-c|$&lt;/span&gt; is small provided &lt;span class=&quot;math-container&quot;&gt;$|x-c|$&lt;/span&gt; is small, a tautology.  Even in formal &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proof, we quickly derive a tautology after letting &lt;span class=&quot;math-container&quot;&gt;$delta=\epsilon$&lt;/span&gt;.]&lt;br /&gt;&#xA;    &#xA;For the corollary, we have to divide by &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; as follows: However close we would like &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(c)$&lt;/span&gt; to be, say, less than some desired distance, such closeness is guaranteed whenever &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; are within &lt;span class=&quot;math-container&quot;&gt;$1/M$&lt;/span&gt; times the desired distance. -- [If you want to introduce &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;, this is a good place to do it: Let &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; be the distance that we desire &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(c)$&lt;/span&gt; to be within. Then we would have &lt;span class=&quot;math-container&quot;&gt;$M\cdot|x-c| &amp;lt; \epsilon$&lt;/span&gt; whenever &lt;span class=&quot;math-container&quot;&gt;$|x-c| &amp;lt; \epsilon/M$&lt;/span&gt;; etc.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Discussion.&lt;/em&gt; For the corollary, warm-up proofs of special cases can be helpful. Take &lt;span class=&quot;math-container&quot;&gt;$f(x)=10x$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$f(x)-f(c)=10(x-c)$&lt;/span&gt;. So, if we want the distance between &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(c)$&lt;/span&gt; to be less than &lt;span class=&quot;math-container&quot;&gt;$1/10$&lt;/span&gt;, how close to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; should &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; be to be &lt;em&gt;close enough&lt;/em&gt;? If the desired distance were &lt;span class=&quot;math-container&quot;&gt;$1/100$&lt;/span&gt;? Whatever distance I pick, how close would be close enough?&lt;/p&gt;&#xA;&lt;p&gt;Likewise, we can show:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Corollary 1.2.&lt;/em&gt; If &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|f(x)-f(c)| \le |g(x)-g(c)|$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; sufficiently close to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;With these tools it is as easy as calculating derivatives to show &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; are continuous at &lt;span class=&quot;math-container&quot;&gt;$c \ne 0$&lt;/span&gt;.  (The special case &lt;span class=&quot;math-container&quot;&gt;$c=0$&lt;/span&gt; is easier but uses a different approach for roots.) The basic idea for &lt;span class=&quot;math-container&quot;&gt;$c\ne 0$&lt;/span&gt; is to factor out &lt;span class=&quot;math-container&quot;&gt;$x-c$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$f(x)-f(c)$&lt;/span&gt; and bound the remaining factor. For well-chosen examples, finding the bound &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; is a simple matter.  For complicated examples, it is not.  I avoid the complicated examples, since my goal is an understanding of the concept and not to train the students to be consummate provers of continuity.  As for which examples are worth understanding, I would say the basic elements that make up functions expressed by formulas: powers, trig., log., and exp., as well as the ways functions are combined, &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt;, and composition.  One should end by remarking that a function given by a single formula in terms of elementary functions is continuous at each point in its domain.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Example from the OP.&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Proof:&#xA;&lt;span class=&quot;math-container&quot;&gt;$x^2-2^2=(x+2)(x-2)$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is closer to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; than one unit, then &lt;span class=&quot;math-container&quot;&gt;$1&amp;lt;x+2&amp;lt;3$&lt;/span&gt;. In that case, &lt;span class=&quot;math-container&quot;&gt;$|x^2-2^2| \le 5\cdot|x-2|$&lt;/span&gt;. By Corollary 1.1, &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Extension 1 of the example.&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; is continuous at all numbers &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; in the interval &lt;span class=&quot;math-container&quot;&gt;$(1,3)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Proof:&#xA;Changing &lt;span class=&quot;math-container&quot;&gt;$5 = 3+2$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$3+c$&lt;/span&gt;,&#xA;it's still the case that &lt;span class=&quot;math-container&quot;&gt;$|x^2-c^2| \le (3+c)\cdot|x-c| \le 6\cdot|x-c|$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$(1,3)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Extension 2 of the example.&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; is continuous at all numbers &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Proof:&#xA;Changing &lt;span class=&quot;math-container&quot;&gt;$5 = (2+1)+2$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$(|c|+1)+|c|$&lt;/span&gt;,&#xA;it's still the case that &lt;span class=&quot;math-container&quot;&gt;$|x^2-c^2| \le (2|c|+1)\cdot|x-c|$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; within &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; unit of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Example 2.&lt;/em&gt;&#xA;The unit step function &lt;span class=&quot;math-container&quot;&gt;$u(x)$&lt;/span&gt; is discontinuous at &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Proof:&#xA;Over any open interval containing, the values of &lt;span class=&quot;math-container&quot;&gt;$u(x)$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. So we cannot make &lt;span class=&quot;math-container&quot;&gt;$u(x)$&lt;/span&gt; be closer to &lt;span class=&quot;math-container&quot;&gt;$u(0)$&lt;/span&gt; than one unit for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, no matter how close we require &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to be.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Theorem 2.&lt;/em&gt;&#xA;If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; are continuous at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, then so is &lt;span class=&quot;math-container&quot;&gt;$f+g$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Proof:&#xA;&lt;span class=&quot;math-container&quot;&gt;$(f+g)(x)-(f+g)(c)=[f(x)-f(c)]+[g(x)-g(c)]$&lt;/span&gt;, so&#xA;&lt;span class=&quot;math-container&quot;&gt;$|(f+g)(x)-(f+g)(c)|\le|f(x)-f(c)|+|g(x)-g(c)|$&lt;/span&gt;.&#xA;By the continuity of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, each of &lt;span class=&quot;math-container&quot;&gt;$|f(x)-f(c)|$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$|g(x)-g(c)|$&lt;/span&gt; may be made as small as we please. Thus their sum may be made as small as we please,&#xA;provided &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is sufficiently close to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;. --&#xA;[If you feel the above is not pedantic enough:&#xA;Suppose we'd like &lt;span class=&quot;math-container&quot;&gt;$(f+g)(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(f+g)(c)$&lt;/span&gt; to be closer&#xA;than some desired distance.&#xA;Then in the first case, if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is sufficiently close to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$|f(x)-f(c)|$&lt;/span&gt; will be less than some part of it, say, half the deired distance. Likewise in the second case, if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is sufficiently close to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$|g(x)-g(c)|$&lt;/span&gt; will also be less than half the deired distance.&#xA;In the first case, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; has to be within one distance of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;;&#xA;in the second, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; has be within a second distance of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&#xA;Thus if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is closer to &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; than each distance,&#xA;then &lt;span class=&quot;math-container&quot;&gt;$|f(x)-f(c)|$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|g(x)-g(c)|$&lt;/span&gt; will each be less than half the &amp;quot;desired distance.&amp;quot; Therefore their sum will be less than the desired distance, and &lt;span class=&quot;math-container&quot;&gt;$|(f+g)(x)-(f+g)(c)|$&lt;/span&gt; is at most their sum.]&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;Conclusion&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;One can then develop derivatives à la Stephen Kuhn (The Derivative à la Carathéodory. &lt;em&gt;Am. Math. Monthly&lt;/em&gt;, 1991), which has conceptual advantages over the standard limit of the difference quotient; see &lt;a href=&quot;https://matheducators.stackexchange.com/questions/21203/how-does-one-tutor-an-a-level-student-past-the-derivative-paradox/26470#26470&quot;&gt;this answer&lt;/a&gt;.&#xA;But the OP indicated that they would do limits next.&lt;/p&gt;&#xA;&lt;p&gt;One warning: I find it easier to characterize various kinds of singularities with limits: discontinuities, nondifferentiable points, asymptotes. But getting across the main gist of calculus, which deals with smooth functions, seems simpler without limits.&#xA;Furthermore, students need to understand limits in subsequent courses, so one should not dispense with limits.  They can be included without much trouble, including the characterizations of continuity and differentiability in terms of limits.&lt;/p&gt;&#xA;&lt;p&gt;The upshot is that &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; need only be found for &lt;span class=&quot;math-container&quot;&gt;$f(x)=x$&lt;/span&gt;, by the teacher, and for &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^{1/n}$&lt;/span&gt; in the case that &lt;span class=&quot;math-container&quot;&gt;$|x| &amp;lt; \delta \land \delta = \epsilon^n$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$|x^{1/n}| &amp;lt; \epsilon$&lt;/span&gt;, which the teacher can do. (For &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; even, we would have to discuss one-sided continuity, which I've omitted from this answer.)  The teacher might also elect to omit the &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proofs.&#xA;The approach frames continuity in terms of change in a (often physical) quantity, analyzes &lt;span class=&quot;math-container&quot;&gt;$f(x)-f(c)$&lt;/span&gt; which analysis will be echoed in the subsequent discussion of differentiability, introduces inequalities related to the notion of &amp;quot;arbitrarily small,&amp;quot; and cultivates &amp;quot;number sense.&amp;quot;&#xA;The squeeze theorems teach the student to think in terms of bounds and estimate them.  It feels satisfying when they realize that if &lt;span class=&quot;math-container&quot;&gt;$M=2$&lt;/span&gt; is good enough to apply Corollary 1.1, then so is &lt;span class=&quot;math-container&quot;&gt;$M=100$&lt;/span&gt;.&#xA;Of course, there is something satisfying to the students when they can actually think with the concepts and not just push symbols around and calculate.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2023-05-22T07:04:33.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26485" PostTypeId="1" CreationDate="2023-05-22T18:14:12.197" Score="6" ViewCount="1190" Body="&lt;p&gt;I teach AP Calculus BC at my high school and we have AP Calculus AB as a pre-req for taking BC. So &lt;em&gt;most&lt;/em&gt; of my students are coming in with a strong calculus foundation, and I can spend less time on the basics (like product and quotient rule). This made me think about some more challenging topics I could include for the class. So now I'm curious as to what topics could appear in a standard college level Calculus I and II course that are left out of the AP Calculus curriculum (&lt;a href=&quot;https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf&quot; rel=&quot;noreferrer&quot;&gt;pages 27-30 of this pdf&lt;/a&gt;). Here's a preliminary list I've already brainstormed, but I'd love to see other topics that AP Calculus is missing out on.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Delta-Epsilon definition of limits&lt;/li&gt;&#xA;&lt;li&gt;Logarithmic Differentiation&lt;/li&gt;&#xA;&lt;li&gt;Newton's Method of finding roots&lt;/li&gt;&#xA;&lt;li&gt;Simpson's Rule&lt;/li&gt;&#xA;&lt;li&gt;More work with trig integrals (like integrating &lt;span class=&quot;math-container&quot;&gt;$\int\sin^{2}{(x)} dx$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Trig substitutions for integrals&lt;/li&gt;&#xA;&lt;li&gt;Volumes by the shell method&lt;/li&gt;&#xA;&lt;li&gt;Surface area of revolution&lt;/li&gt;&#xA;&lt;li&gt;Root Test for series convergence (I think this might have been part of the AP Curriculum in the past, but it's not currently).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Personally, I don't think I would show my high school class the delta-epsilon definition, but I think the other topics I listed would be fair to include. Any other good topics I left off?&lt;/p&gt;&#xA;&lt;p&gt;EDIT: I teach in the US. &amp;quot;AP&amp;quot; calculus refers to &amp;quot;Advanced Placement,&amp;quot; which means that at the end of the school year, the students take a standardized exam. Some colleges offer college credit for students that reach a certain score threshold. There are two AP Calculus classes: AB and BC (they don't stand for anything that I'm aware of). You can see specifically what AB and BC cover in the pdf I linked. But roughly speaking, AB covers all of Calc I plus a bit of calc II (like volumes of rotation and some easy DiffEQ stuff) while BC covers most of Calc I and most of Calc II.&lt;/p&gt;&#xA;" OwnerUserId="8527" LastEditorUserId="8527" LastEditDate="2023-05-22T19:51:03.867" LastActivityDate="2023-05-24T15:37:23.937" Title="Topics covered in Calculus I and II (university level) that aren't covered in the AP Curriculum" Tags="|calculus|curriculum|usa|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26486" PostTypeId="2" ParentId="26485" CreationDate="2023-05-22T18:58:49.377" Score="8" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It's a very good list already.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would add that a more fulsome differential equations introduction used to be part of BC (and may still be part of stronger college courses).  In particular, the second order+ ODE with constant coefficients (including the heterogenous version and variation of parameters and undetermined coefficients).  This was still part of the course when I took it in 1983.  I think it was pulled out in the 90s along with other small diminishments of the course, along with the addition of &amp;quot;TI time&amp;quot; (which most colleges still think is not important).  See this text, for example and its ODE coverage (which was written specifically for AP of the 80s).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=GvcRUUh_rtc&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=GvcRUUh_rtc&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I used this and so did Escalante's students.  It is (for a normal calc book, not Spivak and ilk) reasonably rigorous.  It's the only text I know (ODE or calc) that gives a reasonable way to determine the method of repeated roots if you don't suspect the answer already (not the &amp;quot;we should try this&amp;quot;, but a deductive method).&lt;/p&gt;&#xA;&lt;p&gt;2.5 (edit)  I guess hyperbolic trig functions.  Did a find text on your pdf and didn't see it listed.  By the way, that is a very hard document to look at with all the colors, three column lists, etc.  Makes me wonder (harumph) about the ed reformers making that stuff.&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;If you have not done so, I would take a look at this site (and leave a comment asking your question):&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://teachingcalculus.com/&quot; rel=&quot;noreferrer&quot;&gt;https://teachingcalculus.com/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Most of the participants here are not high school teachers--you will probably get a stronger answer there.&lt;/p&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;I would consider carefully if you want to go down this road of increasing the difficulty.  Are you at an elite school?  In general, I find that students (and schools) that feel the need to stretch a year of college calculus out to two years of high school are not as strong as those that just can do BC in a year, straight up.  Don't know your situation, but just a caution.  It seems like your school's policy is to grant the kids the handicap of going slower, so you might think twice about (partially) reversing this by &amp;quot;enriching&amp;quot;.  Maybe just set things up for a win and help them get high scores on the BC.  [And I say this as someone who likes the additional content, wish it were required, etc.]&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="22431" LastEditorUserId="22431" LastEditDate="2023-05-22T19:13:14.900" LastActivityDate="2023-05-22T19:13:14.900" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26487" PostTypeId="2" ParentId="26485" CreationDate="2023-05-22T23:18:31.513" Score="7" Body="&lt;p&gt;I will tell my impressions based on the BC students I've taught.&lt;/p&gt;&#xA;&lt;p&gt;Important weaknesses/gaps (skills that prepare them for vector calc. &amp;amp; diff. eq.):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Trig. &amp;amp; integrals (OP's ##5,6).&lt;/li&gt;&#xA;&lt;li&gt;Setting up integrals &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; vs. &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; and applying an appropriate method (e.g. volume, OP's #7).&lt;/li&gt;&#xA;&lt;li&gt;Setting up integrals in polar coordinates for the area of intersecting graphs.&lt;/li&gt;&#xA;&lt;li&gt;Facility/confidence with power series (integrating, differentiating, finding sums, problem-solving skill in determining convergence). This seems a big weakness of BC at least among the cohort of students we get; a segment of them elects to retake calc II in large part because of this.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Less important:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Delta-epsilon.&lt;/li&gt;&#xA;&lt;li&gt;Logarithmic differentiation. It's more important that when they see something of the form &lt;span class=&quot;math-container&quot;&gt;$a^b$&lt;/span&gt; and do not know immediately what to do, they think, does rewriting it &lt;span class=&quot;math-container&quot;&gt;$a^b = e^{b \ln a}$&lt;/span&gt; help?&lt;/li&gt;&#xA;&lt;li&gt;Newton's method. It's not important for further work where I teach (because they get it from scratch in a later course), but I include it in calc. I because it's important that students know how to calculate, at least one method, to know calculators and computers are not beyond their understanding.  One of my pet passions. &lt;em&gt;Cum grano salis&lt;/em&gt; and all that.&lt;/li&gt;&#xA;&lt;li&gt;Simpson's rule. Don't have time to explain quadratic interpolation, so I stick to the obvious one, the trapezoidal rule.  Trapezoidal rule is good for measured data.  Simpsons used to be used all over the place, I think or was told, but now adaptive Gauss-Kronrod is in the TIs.  Simpsons's not as impressive as when I took calc. (1980s).&lt;/li&gt;&#xA;&lt;li&gt;Surface area.  I teach this in calc. II because we do volume in calc. I, and this is a way to refresh their understanding of integration.  Otherwise, it helps but is not crucial for surface integrals in vector calc.&lt;/li&gt;&#xA;&lt;li&gt;Root test.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Other things to consider:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;More applications, problem-solving, multistep problems that have the students practice applying concepts and skills already learned in BC.&lt;/li&gt;&#xA;&lt;li&gt;Multiple approaches to the same problem.  It makes connections.  Another pet passion.  Sometimes an increase in excitement leads to an increase in knowledge and understanding.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;One favorite application for power series from my college calc. class was the generating function (a power series) for the Fibonacci sequence (&lt;a href=&quot;http://math.caltech.edu/%7Esyye/teaching/courses/Ma8_2015/Lecture%20Notes/ma8_wk10.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;details courtesy of Google&lt;/a&gt;).  You can (1) sum the series, which is a rational function, (2) use partial fractions to break it down into a sum of proper algebraic fractions with linear denominators, (3) expand the fractions as geometric series, and (4) obtain an explicit formula for the Fibonacci sequence; &lt;em&gt;and&lt;/em&gt;, since one of the two terms in the formula is less than a half in magnitude, you can drop it and round the other term to the nearest integer, thus (5) teaching them a cool thing about approximating integers.  So three major BC skills and a solution (4) with a cherry on top (5).&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2023-05-24T15:37:23.937" LastActivityDate="2023-05-24T15:37:23.937" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26488" PostTypeId="1" CreationDate="2023-05-23T05:14:31.160" Score="4" ViewCount="374" Body="&lt;p&gt;&lt;strong&gt;tl;dr&lt;/strong&gt; I am looking for references which cover introductory abstract Linear Algebra but with a programming / computational approach. The only one I found is the &lt;a href=&quot;https://bvanderlei.github.io/jupyter-guide-to-linear-algebra/intro.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jupyter guide to Linear Algebra&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Long version&lt;/strong&gt;: I am currently teaching Linear Algebra for STEM at a &amp;quot;industry-focused&amp;quot; department for second-term students. We have 7 undergraduate engineering and one &amp;quot;Bachelor of Science and Technology&amp;quot; programs. The general topics which are covered in this course are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Abstract vector spaces, subspaces, bases etc.&lt;/li&gt;&#xA;&lt;li&gt;Linear maps&lt;/li&gt;&#xA;&lt;li&gt;Inner product spaces&lt;/li&gt;&#xA;&lt;li&gt;Eigenvalues and diagonalization&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;(Matrix operations, general linear systems and Gaussian elimination were already covered in a pre-requisite course.)&lt;/p&gt;&#xA;&lt;p&gt;In my department, this Linear Algebra course (which is common to all programs) has a long problematic history, with high rates of failure, and is seen as quite displaced from the other courses as it is much more abstract. Even as the instructor, teaching it is quite monotonous.&lt;/p&gt;&#xA;&lt;p&gt;When I took Linear Algebra, as a Mathematics student, it was mostly theorem-proving, which was just right for my goals of becoming a pure mathematician. This has not worked very well for this course.&lt;/p&gt;&#xA;&lt;p&gt;Considering this (and a few other changes of perspective which I had in the last couple of years), I am trying to incorporate more programming into this course. I adapted some of the more constructive proofs to algorithms, and created a large-ish question bank with questions which necessitate a bit of &amp;quot;engineering&amp;quot; and can be solved programmatically.&lt;/p&gt;&#xA;&lt;p&gt;This approach has been relatively successful, however I cannot find good references which are appropriate for this approach. Most Linear Algebra references I know either move to Numerical Analysis, focus only on solving linear systems, or are theorem/proof-based.&lt;/p&gt;&#xA;&lt;p&gt;The only references I have found which move in this direction are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Practical Linear Algebra for Data Science&lt;/strong&gt;: &lt;em&gt;From Core Concepts to Applications Using Python&lt;/em&gt;, by Mike X Cohen&lt;/p&gt;&#xA;&lt;p&gt;However, this book focuses on matrices, whereas I am required to follow the syllabus, including introducing the abstract concepts.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://bvanderlei.github.io/jupyter-guide-to-linear-algebra/intro.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jupyter Guide to Linear Algebra&lt;/a&gt;, by Ben Vanderlei&lt;/p&gt;&#xA;&lt;p&gt;This seems to cover most of what the course requires.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I would like to get more references similar to the ones above (specially the second one), preferably books with physical copies by known scientific publishing companies. References with applications are also welcome (although a bit tangential).&lt;/p&gt;&#xA;" OwnerUserId="14410" LastActivityDate="2023-05-23T17:02:40.817" Title="Programming and computation-focused textbook for introductory linear algebra" Tags="|linear-algebra|reference-books|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26489" PostTypeId="1" AcceptedAnswerId="26491" CreationDate="2023-05-23T09:26:23.663" Score="34" ViewCount="5636" Body="&lt;p&gt;I'm not sure if this is more educational or more &amp;quot;pure math&amp;quot;, but:&lt;/p&gt;&#xA;&lt;p&gt;For multiplication and addition, there is exactly one inverse operation, namely division and subtraction.&lt;/p&gt;&#xA;&lt;p&gt;For exponentiation, we have both logarithms and roots.&lt;/p&gt;&#xA;&lt;p&gt;Why is that the case? Maybe one could argue that exponentiation is &amp;quot;even less commutative&amp;quot; than division and there is a difference in solving for x in &lt;span class=&quot;math-container&quot;&gt;$x/4=5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4/x=5$&lt;/span&gt;, but that doesn't feel quite right to me...&lt;/p&gt;&#xA;" OwnerUserId="667" LastEditorUserId="851" LastEditDate="2023-05-24T18:39:14.970" LastActivityDate="2023-05-25T18:39:56.723" Title="Why are there two inverses to exponentiation?" Tags="|arithmetic-operations|inverse|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26490" PostTypeId="2" ParentId="26489" CreationDate="2023-05-23T12:52:22.247" Score="33" Body="&lt;p&gt;Is your variable at the base or the exponent?&lt;/p&gt;&#xA;&lt;p&gt;An &lt;em&gt;exponential function&lt;/em&gt; is a function of the form &lt;span class=&quot;math-container&quot;&gt;$f(x)=a^x$&lt;/span&gt; for some constant &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. In this case, the inverse is indeed given by a logarithm &lt;span class=&quot;math-container&quot;&gt;$f^{-1}(x)=\log_a(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, if you place the variable at the base, &lt;span class=&quot;math-container&quot;&gt;$g(x)=x^a$&lt;/span&gt;, then you get what is better called (to avoid confusion) a &lt;em&gt;power function&lt;/em&gt;, whose inverse is given by roots: &lt;span class=&quot;math-container&quot;&gt;$g^{-1}(x)=\sqrt[a]{x}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is a nice example to highlight the importance of being precise with the terms being used. &amp;quot;Operation&amp;quot; was not defined properly, nor what an &amp;quot;inverse&amp;quot; of it should be. As a binary operation, &amp;quot;exponentiation&amp;quot; is simply the function &lt;span class=&quot;math-container&quot;&gt;$K(x,y)=x^y$&lt;/span&gt;, which is not injective and thus does not have any meaningful &amp;quot;inverse&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="14410" LastActivityDate="2023-05-23T12:52:22.247" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26491" PostTypeId="2" ParentId="26489" CreationDate="2023-05-23T13:32:34.037" Score="84" Body="&lt;p&gt;The concept of an inverse operation itself is a bit tricky. Often we consider arithmetic operations to be binary operations:&#xA;&lt;span class=&quot;math-container&quot;&gt;$\DeclareMathOperator{\add}{add}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\DeclareMathOperator{\subtract}{subtract}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\add(a,b)=a+b,$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\subtract(a,b)=a-b.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is the view we take when we make statements like “addition is commutative,” which is the same as saying &lt;span class=&quot;math-container&quot;&gt;$\add(a,b)=\add(b,a)$&lt;/span&gt;. However, in this view, addition and subtraction are &lt;em&gt;not&lt;/em&gt; inverses of each other. Subtraction does not take the output of addition and return the inputs.&lt;/p&gt;&#xA;&lt;p&gt;To make sense of the statement that subtraction is the inverse operation of addition, we need to view arithmetic operations as unary operations or single-variable functions:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\add_a\!(x)=x+a,$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\subtract_a\!(x)=x-a.$$&lt;/span&gt;&#xA;In this view, it’s true that &lt;span class=&quot;math-container&quot;&gt;$\add_a\!$&lt;/span&gt; and  &lt;span class=&quot;math-container&quot;&gt;$\subtract_a\!$&lt;/span&gt; are inverses. They undo each other. This is why I prefer to tell students that “subtracting &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; undoes adding &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.”&lt;/p&gt;&#xA;&lt;p&gt;However, there’s a problem. We’ve defined &lt;span class=&quot;math-container&quot;&gt;$\subtract_a\!(x)$&lt;/span&gt; to mean “subtract &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.” But what about the operation “from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; subtract &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;?” From the binary-operation perspective, this is just &lt;span class=&quot;math-container&quot;&gt;$\subtract(a,x)$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\subtract(x,a)$&lt;/span&gt;. But from the unary-operation perspective, this is a completely different kind of operation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\operatorname{subtractfrom}_a\!(x)=a-x,$$&lt;/span&gt; and it is actually it’s &lt;em&gt;own&lt;/em&gt; inverse. Suppose we have a bottle of water that has volume &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and we spill an unknown volume &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. How can we recover the information of how much we spilled? We can take the remaining volume and subtract it from the original volume, &lt;span class=&quot;math-container&quot;&gt;$a-(a-x)=x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This demonstrates that a non-commutative binary operation has &lt;em&gt;two types&lt;/em&gt; of associated unary operations, and they have different inverses. The same is true for division. “Divide &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;” and “divide &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;” are different functions of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with different inverses. The same is also true for exponentiation, and language for distinguishing the two kinds of functions already exists.&lt;/p&gt;&#xA;&lt;p&gt;The functions&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\operatorname{power}_a\!(x)=x^a$$&lt;/span&gt;&#xA;are commonly known as power functions. Their inverses are other power functions, &lt;span class=&quot;math-container&quot;&gt;$(x^a)^{1/a}=x$&lt;/span&gt;. When &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is a positive integer, &lt;span class=&quot;math-container&quot;&gt;$x^{1/a}=\sqrt[a]{x}$&lt;/span&gt; is also called a root function.&lt;/p&gt;&#xA;&lt;p&gt;The functions&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\exp_a\!(x)=a^x$$&lt;/span&gt;&#xA;are commonly known as exponential functions. Their inverses are logarithmic functions, &lt;span class=&quot;math-container&quot;&gt;$\log_a\!(a^x)=x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I have to explain this frequently to students in single-variable calculus who are learning derivative rules. Power functions and exponential functions both involve exponentiation, but they are fundamentally different functions with different derivatives:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm d}{\mathrm d x}x^a=ax^{a-1},$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm d}{\mathrm d x}a^x=a^x\ln a.$$&lt;/span&gt;&#xA;The mnemonic I give them is that “&lt;span class=&quot;math-container&quot;&gt;$a^x\!$&lt;/span&gt; is an &lt;em&gt;exponential&lt;/em&gt; function because the variable is in the &lt;em&gt;exponent&lt;/em&gt;.”&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-05-23T13:48:52.437" LastActivityDate="2023-05-23T13:48:52.437" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="26492" PostTypeId="1" CreationDate="2023-05-23T13:36:26.597" Score="4" ViewCount="118" Body="&lt;p&gt;I'm looking for references about activities that encourage elementary school students to create or evaluate mathematical notations. do you know any?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="20854" LastEditDate="2023-05-25T10:41:06.717" LastActivityDate="2023-05-25T10:41:06.717" Title="Activities that encourage students to create or evaluate mathematical notations" Tags="|mathematical-pedagogy|notation|activities|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26493" PostTypeId="2" ParentId="26488" CreationDate="2023-05-23T15:13:59.537" Score="5" Body="&lt;p&gt;You may be interested in &lt;a href=&quot;https://link.springer.com/book/10.1007/978-3-030-86155-1&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Application-Inspired Linear Algebra&lt;/em&gt;&lt;/a&gt; by Heather Moon, Thomas Asaki, and Marie Snipes. It covers all of the general topics you’ve listed, and there are exploratory programming tasks. The book is designed for MATLAB/Octave, however, not for Python. It still presents theorems and proofs, and I wouldn’t say there is a heavy programming emphasis. Additional resources and code are available on the &lt;a href=&quot;http://www.imagemath.org/AILA.html&quot; rel=&quot;noreferrer&quot;&gt;authors’ website.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.routledge.com/Advanced-Linear-Algebra-for-Engineers-with-MATLAB/Dianat-Saber/p/book/9781420095234&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Advanced Linear Algebra for Engineers with MATLAB&lt;/em&gt;&lt;/a&gt; by Sohail Dianat and Eli Saber has a stronger programming emphasis, but abstract vector spaces and linear transformations don’t get introduced until halfway through.&lt;/p&gt;&#xA;&lt;p&gt;A non-commercial textbook that might be useful is Sean Fitzpatrick’s &lt;a href=&quot;https://opentext.uleth.ca/Math3410/linear-algebra.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Linear Algebra:&#xA;A second course, featuring proofs and Python&lt;/em&gt;&lt;/a&gt;. There are also accompanying &lt;a href=&quot;https://github.com/sean-fitzpatrick/3410worksheets&quot; rel=&quot;noreferrer&quot;&gt;Jupyter notebooks&lt;/a&gt; available. Similar to &lt;em&gt;Application-Inspired Linear Algebra&lt;/em&gt;, the programming serves mainly to motivate and facilitate the mathematical ideas.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-05-23T16:15:47.303" LastActivityDate="2023-05-23T16:15:47.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26494" PostTypeId="2" ParentId="26488" CreationDate="2023-05-23T17:02:40.817" Score="5" Body="&lt;p&gt;I don't know if any of what I do will be helpful information for you, but maybe?&lt;/p&gt;&#xA;&lt;p&gt;I teach at a community college, and most of my students certainly aren't into proof. But that feels like an important aspect of the course. I've left out the more tedious proofs (determinants, argh) and focus more on helping them learn to prove. We use an older edition of David Lay's &lt;em&gt;&lt;strong&gt;Linear Algebra and its Applications&lt;/strong&gt;&lt;/em&gt;. (And bought a class set for the library to loan out for the whole semester.) I use a lot of the true-false problems there, and ask the students to prove or disprove. I do workshops with them on this.&lt;/p&gt;&#xA;&lt;p&gt;I found some projects which definitely help the students see the linear transformation aspect more clearly and deeply. There are created by a group working on &lt;a href=&quot;https://iola.math.vt.edu/&quot; rel=&quot;noreferrer&quot;&gt;Inquiry-Oriented Linear Algebra&lt;/a&gt;. I love these projects.&lt;/p&gt;&#xA;&lt;p&gt;I made a list of OER textbooks, with my comments, which you can find &lt;a href=&quot;https://docs.google.com/document/d/1xIB1GGt2qswBk_qzf_oKz-D3T1IFK5hyLAz2-aRdhdM/edit?usp=sharing&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. &lt;a href=&quot;https://davidaustinm.github.io/ula/sec-gaussian-revisited.html&quot; rel=&quot;noreferrer&quot;&gt;The first one on the list, by David Austin&lt;/a&gt;, has a section on programming in Sage, and has blocks of Sage code throughout the text.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-05-23T17:02:40.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26495" PostTypeId="2" ParentId="26472" CreationDate="2023-05-23T20:42:52.217" Score="1" Body="&lt;p&gt;I think it's good to try to help students develop some intuition for variance and covariance. However, a pedagogical problem I see with your variance example is that both distributions have the same expected value. This allows the psychologist to conclude that &amp;quot;the cat is overall a lot calmer than Jack,&amp;quot; but the same reasoning would not work if the distributions did not have the same center. This is problematic.&lt;/p&gt;&#xA;&lt;p&gt;I suggest using distributions with different centers, and drawing conclusions that depend only on the spread of the distributions. If you want students practice calculating the variances by hand, it helps to keep things small. For example, consider the following frequency tables, where &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; represents Jack's mood and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; represents his cat's mood, both on a scale from -1 to 1.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{array}{c|ccc}&#xA;X &amp;amp; -1 &amp;amp; 0 &amp;amp; 1 \\&#xA;\hline&#xA;\text{freq.} &amp;amp; 1 &amp;amp; 2 &amp;amp; 1&#xA;\end{array}&#xA;\qquad&#xA;\begin{array}{c|ccc}&#xA;Y &amp;amp; -1 &amp;amp; 0 &amp;amp; 1 \\&#xA;\hline&#xA;\text{freq.} &amp;amp; 2 &amp;amp; 2 &amp;amp; 0&#xA;\end{array}&#xA;$$&lt;/span&gt;&#xA;From the fact that the variance of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is greater than the variance of &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt;, we can reasonably conclude that &amp;quot;Jack's mood fluctuates more than his cat's mood does.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If your students have some physics knowledge, another way to intuit variance is through &lt;a href=&quot;http://stla.github.io/stlapblog/posts/Variance_inertia.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;moment of inertia&lt;/a&gt;. The variance represents how difficult it would be to &amp;quot;rotate&amp;quot; a distribution around its center. When the mass of a distribution is farther from its center, it will be more difficult to rotate. This generalizes well to continuous distributions.&lt;/p&gt;&#xA;&lt;p&gt;For your covariance examples, you're missing an example of a situation where zero covariance does not imply independence. Two random variables can be nonlinearly dependent and have zero covariance. For example, we can consider the following contingency table. (I'm using a contingency table to be consistent with the frequency tables in the variance example. You could also show the raw data in both examples. Students should see both.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{array}{c|ccc|l}&#xA;_{\large X}\backslash^{\large Y} &amp;amp; -1 &amp;amp; 0 &amp;amp; 1 \\&#xA;\hline&#xA;-1 &amp;amp; 1 &amp;amp; 0 &amp;amp; 0 &amp;amp; 1 \\&#xA;0 &amp;amp; 0 &amp;amp; 2 &amp;amp; 0 &amp;amp; 2  \\&#xA;1 &amp;amp; 1 &amp;amp; 0 &amp;amp; 0 &amp;amp; 1  \\&#xA;\hline&#xA;&amp;amp; 2 &amp;amp; 2 &amp;amp; 0 &amp;amp; 4&#xA;\end{array}&#xA;$$&lt;/span&gt;&#xA;The covariance of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; is zero, but &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; exhibit a clear relationship. When Jack's mood is neutral, his cat's mood is neutral. When Jack's mood is not neutral, his cat's mood is negative.&lt;/p&gt;&#xA;&lt;p&gt;For any concept, it's important to have examples that show boundaries/limitations/exceptions, as well as to avoid focusing on special examples with non-general properties. Also, I agree with &lt;a href=&quot;https://matheducators.stackexchange.com/questions/26472/are-these-explanations-of-variance-and-covariance-intuitive#comment70088_26472&quot;&gt;Jon Custer&lt;/a&gt; that intuition is very personal, and an example that is illuminating for one student may be confusing to another. So be flexible!&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-05-23T20:42:52.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26497" PostTypeId="2" ParentId="26489" CreationDate="2023-05-24T09:02:22.233" Score="23" Body="&lt;p&gt;(Despite good answers already, I thought the concrete example below could be useful.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For multiplication and addition, there is exactly one inverse operation, namely division and subtraction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yet for subtraction and division, there are two possible inverse operations each. This applies to any non-commutative operation that takes two inputs.&lt;/p&gt;&#xA;&lt;p&gt;Subtraction: &lt;strong&gt;x - y = z&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To get &lt;strong&gt;x&lt;/strong&gt; back, you compute &lt;strong&gt;z + y&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;To get &lt;strong&gt;y&lt;/strong&gt; back, you compute &lt;strong&gt;x - z&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Division: &lt;strong&gt;x / y = z&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To get &lt;strong&gt;x&lt;/strong&gt; back, you compute &lt;strong&gt;z × y&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;To get &lt;strong&gt;y&lt;/strong&gt; back, you compute &lt;strong&gt;x / z&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Exponentiation: &lt;strong&gt;x&lt;sup&gt;y&lt;/sup&gt; = z&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To get &lt;strong&gt;x&lt;/strong&gt; back, you compute &lt;strong&gt;z&lt;sup&gt;1/y&lt;/sup&gt;&lt;/strong&gt; (y'th root of z)&lt;/li&gt;&#xA;&lt;li&gt;To get &lt;strong&gt;y&lt;/strong&gt; back, you compute &lt;strong&gt;log&lt;sub&gt;x&lt;/sub&gt;(z)&lt;/strong&gt; (base-x logarithm of z)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14647" LastActivityDate="2023-05-24T09:02:22.233" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26498" PostTypeId="2" ParentId="26489" CreationDate="2023-05-24T14:31:45.670" Score="0" Body="&lt;p&gt;In one case base is the variable and in the other case power is the variable. Here you are talking about an inverse of a function and not just an expression. Therefore role of variable is much important and depending on that inverse function must be different.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2023-05-24T14:31:45.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26499" PostTypeId="1" AcceptedAnswerId="26502" CreationDate="2023-05-24T15:02:57.693" Score="2" ViewCount="110" Body="&lt;p&gt;When you come to explain dividing given n number of objects into k number of groups, is it good to describe the cases involved using an example to cover as many cases as possible in order to give better opportunity to students to understand the difference by comparing ? &lt;br/&gt;&#xA;What would you think about the following, &lt;br/&gt;&#xA;Let consider dividing 5 objects into 3 groups;&lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;putting 5 similar balls into 3 similar baskets. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;giving 5 similar balls to 3 persons.&lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;putting 5 different balls into 3 similar baskets. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;giving 5 different balls to 3 persons.&lt;br/&gt;&#xA;Are these cases enough to give full knowledge about partitioning with an extra  restriction such as each of the partition should have at least one item? &lt;br/&gt;&#xA;Do we need to add another 7 objects of a different kind to the first two cases in latter stages to make students familiar to much complicated situations? &lt;br/&gt;&#xA;Do you have much better way of introducing this to students?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2023-05-24T22:14:36.677" Title="Partitioning objects in combinatorics" Tags="|secondary-education|combinatorics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26500" PostTypeId="5" CreationDate="2023-05-24T18:09:34.957" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2023-05-24T18:09:34.957" LastActivityDate="2023-05-24T18:09:34.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26501" PostTypeId="4" CreationDate="2023-05-24T18:09:34.957" Score="0" Body="For questions about teaching complex (numbers like $z = a + b \cdot i$ where $\left\{ a,\, b \right\} \in \mathbb{R} \wedge i^{2} = -1$) numbers." OwnerUserId="22437" LastEditorUserId="22437" LastEditDate="2023-06-07T14:56:23.990" LastActivityDate="2023-06-07T14:56:23.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26502" PostTypeId="2" ParentId="26499" CreationDate="2023-05-24T22:14:36.677" Score="4" Body="&lt;p&gt;When I was introduced to combinatorics, we were concentrating on various counting techniques rather than on various types of situations in which the counting has to be performed with the idea that the techniques are not too many but the number of setups one can invent is immense. Interestingly enough, ArtOfProblemSolving mainly takes the same approach, so if you look at their videos, you'll see counting trees, independent choice, mutually exclusive choice, complementary counting, Venn diagrams, principle of inclusion-exclusion, compensating for overcounting using symmetry, counting with restrictions, and so on, and so forth rather than a discussion of particular setups in some order.&lt;/p&gt;&#xA;&lt;p&gt;However the argument about what is primary in mathematics: an object or a method is, probably, older than all of us. I'm in the &amp;quot;method&amp;quot; camp, so I try to go one technique a time when teaching some introductory level classes like that with various seemingly unrelated setups where the same technique or set of techniques is used. But if you want to go &amp;quot;object by object&amp;quot; as the classification in your post suggests, that is fine too if you think in advance of what setups to choose (arrangements, partitions, assignments, etc.) and how to organize the stuff in each category so that not to overload students' memory because many of them just tend to memorize examples rather than absorb the underlying ideas at this level.&lt;/p&gt;&#xA;&lt;p&gt;There seems to be no way that is unconditionally better (or worse). Just think of what skills you want your students to acquire in the end. Ideally, they should know some amount of techniques, some set of classical setups, the standard matching between them, and, when going into the uncharted territory, be able to reduce the new setups to the old ones, or match them with appropriate (combinations of) known techniques directly, or add some twists to the existing methods to account for the peculiarities of the situations. However in real life one can &lt;em&gt;never&lt;/em&gt; go that far in the short introductory course, so one has to sacrifice this and that, cut corners here and there, and reduce stuff to the simplest cases now and then. The choice of this compromise is by no means unique and depends on both the teacher preferences and the student abilities.&lt;/p&gt;&#xA;&lt;p&gt;So my final answer to your direct question is &amp;quot;Start with what seems most logical to you and adjust the level and content of the lessons to the level of the students &lt;em&gt;within reasonable limits&lt;/em&gt;. For the latter I usually set up some minimal requirements in the beginning (for myself, not for telling the students) and if I see that I cannot successfully do something &lt;em&gt;above&lt;/em&gt; them, I consider my lectures too hard for this particular audience, but if I have trouble doing something &lt;em&gt;within&lt;/em&gt; them, I consider the students too weak or not ready for the course. Normally those minimal requirements are more or less aligned with the standard description of the course but, of course, you always have some power of interpretation here.&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-05-24T22:14:36.677" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26503" PostTypeId="2" ParentId="26046" CreationDate="2023-05-25T03:23:52.097" Score="0" Body="&lt;p&gt;In the US, UK, and former UK colonies, at certain primary-school grades, you can't leave your final answer as an &lt;strong&gt;improper fraction&lt;/strong&gt;. You'll lose marks for answering&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{3}{4}+\frac{1}{2}=\frac{5}{4}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Students are taught that there is something &amp;quot;improper&amp;quot; and wrong about leaving the above as your final answer.&lt;/p&gt;&#xA;&lt;p&gt;For full marks, you must answer&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{3}{4}+\frac{1}{2}=\frac{5}{4}=1\frac{1}{4}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some grade levels later, students discover this &amp;quot;rule&amp;quot; was nonsense.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This seems to be mainly a US/UK thing: See &lt;a href=&quot;https://matheducators.stackexchange.com/questions/24420/what-is-the-rationale-for-distinguishing-between-proper-and-improper-fractions&quot;&gt;What is the rationale for distinguishing between proper and improper fractions?&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user18187" LastActivityDate="2023-05-25T03:23:52.097" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26504" PostTypeId="2" ParentId="26489" CreationDate="2023-05-25T18:39:56.723" Score="4" Body="&lt;p&gt;The other answers have shown that it depends on which of the two inputs you fix and which you let vary. As a way to visualize this, here's a plot of the 3D surface &lt;span class=&quot;math-container&quot;&gt;$z = x^y$&lt;/span&gt;. I've drawn two polynomial curves: &lt;span class=&quot;math-container&quot;&gt;$z = x^2$&lt;/span&gt; (blue) and &lt;span class=&quot;math-container&quot;&gt;$z = x^3$&lt;/span&gt; (green), and two exponential curves: &lt;span class=&quot;math-container&quot;&gt;$z = 2^y$&lt;/span&gt; (red) and &lt;span class=&quot;math-container&quot;&gt;$z = 3^y$&lt;/span&gt; (purple).&lt;/p&gt;&#xA;&lt;p&gt;To visualize the polynomial inverses (roots), rotate the whole picture so that the &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; axis is horizontal and the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; axis is pointing upward. To visualize the exponential inverses (logs), rotate the whole picture so that the &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; axis is horizontal and the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; axis is pointing upward.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/c8kYY.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/c8kYY.png&quot; alt=&quot;A 3D rendering of z = x^y with polynomial and exponential curves illustrated&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A sideways look at the polynomial curves and their inverses, with the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; axis into/out of the screen:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5C2Xm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5C2Xm.png&quot; alt=&quot;Head-on view of two polynomial curves and their inverses&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A sideways look at the exponential curves and their inverses, with the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; axis into/out of the screen:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rFL81.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rFL81.png&quot; alt=&quot;Head-on view of two exponential curves and their inverses&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13116" LastActivityDate="2023-05-25T18:39:56.723" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26505" PostTypeId="1" CreationDate="2023-05-25T20:37:51.957" Score="4" ViewCount="174" Body="&lt;p&gt;Dyslexic students sometimes benefit from informal analogies to things in the world which the student can see with their eyes, and/or touch with their hands.&lt;/p&gt;&#xA;&lt;p&gt;Tentatively, we can conjecture that mathematics which makes mathematics easier for dyslexics could also make math easier to understand for non-dyslexic persons as well.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I had trouble with writing &lt;code&gt;1 &amp;lt; 7&lt;/code&gt; and &lt;code&gt;1 &amp;gt; 7&lt;/code&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When I was less than 10 years old, a teacher told me that the less-than-sign &lt;code&gt;&amp;gt;&lt;/code&gt; always faces the larger of two numbers because the less-than-sign resembles the mouth of a hungry alligator &lt;code&gt;&amp;gt;&lt;/code&gt;. The idea is that a hungry alligators, when presented with two meals, would have incentive eat a the larger of two meals.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://1.bp.blogspot.com/-Mc4Vl-ivhhM/VCAD3TbJwaI/AAAAAAAAcRE/aCFw8I-hMk0/s1600/Greater%2BThan%2BAlligator.jpg&quot; alt=&quot;picture of the head of a cartoon alligator facing the larger of two numbers&quot; /&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The following has not yet been verified, but we can hypothesize that for any live in-person lesson and/or educational video &lt;strong&gt;v1&lt;/strong&gt; and &lt;strong&gt;v2&lt;/strong&gt; and for any students &lt;em&gt;&lt;strong&gt;d&lt;/strong&gt;&lt;/em&gt; and &lt;strong&gt;nd&lt;/strong&gt; if student &lt;em&gt;&lt;strong&gt;d&lt;/strong&gt;&lt;/em&gt; is more dyslexic than student &lt;em&gt;&lt;strong&gt;nd&lt;/strong&gt;&lt;/em&gt;, and dyslexic person &lt;em&gt;&lt;strong&gt;d&lt;/strong&gt;&lt;/em&gt; prefers lesson/video &lt;em&gt;&lt;strong&gt;gv&lt;/strong&gt;&lt;/em&gt; more than &lt;em&gt;&lt;strong&gt;bv&lt;/strong&gt;&lt;/em&gt; then non-dyslexic person &lt;em&gt;&lt;strong&gt;nd&lt;/strong&gt;&lt;/em&gt; might benefit from video &lt;em&gt;&lt;strong&gt;gv&lt;/strong&gt;&lt;/em&gt; more than *&lt;strong&gt;bv&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It might be similar to how the fact that some wheel-chair bound people prefer using elevators to stairs implies that some non-wheel chair bound persons prefer using elevators to stairs.&lt;/p&gt;&#xA;&lt;p&gt;Likewise, potato peelers designed for a shaking persons hands might be easier for persons whose hands, also, do not shake.&lt;/p&gt;&#xA;" OwnerUserId="16039" LastEditorUserId="498" LastEditDate="2023-05-25T22:13:55.660" LastActivityDate="2023-05-25T22:13:55.660" Title="What evidence is there in the literature that lessons geared towards dyslexic student help non-dyslexic students?" Tags="|examples|teaching|lecturing|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26508" PostTypeId="1" CreationDate="2023-05-27T17:23:12.797" Score="6" ViewCount="365" Body="&lt;p&gt;Michael Sipser's textbook &lt;em&gt;Introduction to the Theory of Computation&lt;/em&gt; (now 3rd ed.) includes for each major theorem, a demarcated &lt;strong&gt;Proof Idea&lt;/strong&gt; of length a paragraph to more than a page, prior to presenting the formal &lt;strong&gt;Proof&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I am seeking examples of textbooks that do something similar, regularly providing&#xA;proof sketches prior to the proofs. Of course it is not uncommon to sketch the main idea&#xA;before a proof, but I am seeking systematic use of this pedagogical practice spanning an entire textbook.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2023-06-21T22:53:25.790" Title="Demarcated &quot;Proof Idea&quot;" Tags="|textbooks|proofs|reference-books|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26509" PostTypeId="2" ParentId="26508" CreationDate="2023-05-27T20:44:23.253" Score="5" Body="&lt;p&gt;I'm not sure if this is what you're looking for, but &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1118164024&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;How to Read and Do Proofs: An Introduction to Mathematical Thought Processes&amp;quot; by Solow&lt;/a&gt; is an intro-to-proof-writing book that frequently goes very slow and wordily through a proof/argument before giving the concise mathematical version.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2023-05-27T20:44:23.253" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26510" PostTypeId="1" CreationDate="2023-05-28T12:21:05.680" Score="4" ViewCount="486" Body="&lt;p&gt;As a student attending a grammar school in the UK ,I have been fortunate to have access to various opportunities to showcase my mathematical abilities. These include participating in maths challenges and engaging in lunchtime clubs dedicated to mathematics, which provide around 35 minutes of additional support. I recognize that these opportunities for mathematical exploration and development may be more limited for students attending state schools in the UK. Lack of support from the school or peer pressure can hinder their mathematical growth, as some students may face stigma or negative stereotypes associated with being interested in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;However, when comparing the mathematical opportunities available in the UK to those in the United States, it becomes evident that the UK lags behind. The US offers a plethora of mathematical opportunities, particularly through competitions , such as MathsCounts, AMC, and various math leagues. In contrast, the UK seems to have only one prominent maths competition known as the UKMT. Additionally, the US provides further support through courses offered by the Art of Problem Solving (AOPS), which are renowned for their mathematical depth and rigor. It is disheartening to witness this disparity in opportunities.&lt;/p&gt;&#xA;&lt;p&gt;In my pursuit of mathematical excellence, I have acquired two books from the UKMT. However, I have found them to be somewhat lacking in clear explanations or in-depth exploration of certain topics. Consequently, I am seeking recommendations to address two specific needs:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Preparation for maths olympiads: I am in search of books that provide comprehensive guidance and practice problems specifically tailored to excel in maths olympiads, allowing me to further enhance my mathematical skills and problem-solving abilities.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Interesting maths topics: I am eager to explore captivating areas of mathematics such as topology or the beauty of primes. I seek resources, books, or materials that delve into these topics in a sophisticated and intellectually stimulating manner, helping me appreciate the elegance and depth of these mathematical concepts.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Any courses in preparation for the UKMT challenges&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I would greatly appreciate any recommendations or guidance from individuals who can help me find suitable resources to fulfill these needs. Thank you for your assistance in this matter.&lt;/p&gt;&#xA;" OwnerUserId="22467" LastEditorUserId="2083" LastEditDate="2023-12-28T10:34:21.750" LastActivityDate="2024-01-17T21:32:03.550" Title="Recommendations for secondary student interested in maths" Tags="|secondary-education|resource-request|math-contests|uk|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26511" PostTypeId="2" ParentId="26510" CreationDate="2023-05-28T13:20:22.237" Score="3" Body="&lt;p&gt;I will address point 2.&lt;/p&gt;&#xA;&lt;p&gt;I would say there three main accessible areas according to my mathematical knowledge.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Discrete Maths (Susanna Epp, Discrete Mathematics with Applications)&lt;/li&gt;&#xA;&lt;li&gt;Operation Research (W.L. Winston, Operations Research: Applications and Algorithms)*&lt;/li&gt;&#xA;&lt;li&gt;Programming (Python) (I prefer using Udemy an online platform rather than textbook. &lt;a href=&quot;https://www.udemy.com/course/data-science-machine-learningtheoryprojectsa-z-90-hours/&quot; rel=&quot;nofollow noreferrer&quot;&gt;link to course&lt;/a&gt;)**&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;*Some parts of course can be a bit advance. Learning LP(linear programming) and IP(Integer programming) is good goal&lt;/p&gt;&#xA;&lt;p&gt;** A good goal will be to learn basics of Python, numpy, matplotlib and pandas. Although this may not seem related to math programming play important role in numerical methods and understanding mathematical algorithms as well as data science. Some parts of the course are quite advance however I encourage you try to complete all the sections.&lt;/p&gt;&#xA;&lt;p&gt;Requisites:&#xA;Requisites for discrete math and programming are very limited. For operation research you should be comfortable solving a system linear equations.&lt;/p&gt;&#xA;&lt;p&gt;Although studying can be hard you can find fully worked solutions to most problems using Quizlet.com. If you enroll in a course at Udemy you will have instructor who can answer your questions.&lt;/p&gt;&#xA;" OwnerUserId="20626" LastEditorUserId="20626" LastEditDate="2023-05-28T15:41:13.897" LastActivityDate="2023-05-28T15:41:13.897" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26513" PostTypeId="2" ParentId="26510" CreationDate="2023-05-29T10:12:35.057" Score="4" Body="&lt;p&gt;For UKMT competitions, the books and past papers are the best sources of material you can use to prepare. BMO papers dating back to the early 1970s are available on the &lt;a href=&quot;https://bmos.ukmt.org.uk/home/bmo.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;British Mathematical Olympiad website&lt;/a&gt;. Detailed solutions are available in the books by Geoff Smith and Tony Gardiner. If you persevere with the problems in past papers and ask your teacher or the Mathematics Stack Exchange when you get stuck, you will make progress. My experience with the books is not that they are lacking in clear explanations - again, ask your teacher or MSE if there is something that is not clear. You may also find that some of the exercises are designed for you to discover and prove results for yourself.&lt;/p&gt;&#xA;&lt;p&gt;The books focused on specific topics, such as Bradley's 'Introduction to Number Theory and Inequalities' and Leversha's 'Crossing the Bridge' are structured more like traditional school textbooks with exposition followed by exercises. The olympiad books are designed for the reader to focus more on solving the problems, so much of the exposition comes after you have attempted the problems.&lt;/p&gt;&#xA;&lt;p&gt;The UKMT also offers other opportunities beyond the challenges and olympiads; there is a UKMT summer school, a mentoring scheme, an olympiad mentoring scheme and training camps for invited students. The team challenges have been paused for the current academic year. The resources from the mentoring scheme (monthly worksheets) are designed to help students progress from just qualifying for the olympiads to performing well. You should be able to access these resources through your school.&lt;/p&gt;&#xA;&lt;p&gt;Geoff Smith has some excellent advice &lt;a href=&quot;https://people.bath.ac.uk/masgcs/advice.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; about olympiad preparation.&lt;/p&gt;&#xA;&lt;p&gt;In relation to online courses, AFAIK, you can still access online courses offered by AoPS if you have the means to afford them. Kevin Olding of &lt;a href=&quot;https://mathsaurus.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathsaurus.com/&lt;/a&gt; also offers UKMT-focused courses.&lt;/p&gt;&#xA;&lt;p&gt;On the second point, I have a few book recommendations:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;'The Shape of Space' by Jeff Weeks&lt;/li&gt;&#xA;&lt;li&gt;'The Music of the Primes' by Marcus du Sautoy&lt;/li&gt;&#xA;&lt;li&gt;'Euler's Gem' by David Richeson&lt;/li&gt;&#xA;&lt;li&gt;Any introductory number theory book - I can recommend Bradley's book (mentioned above).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;You might also like to check out this &lt;a href=&quot;https://www.maths.cam.ac.uk/undergrad/admissions/files/admissions/reading-list.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;reading list&lt;/a&gt; aimed at sixth form students intending to read mathematics at university.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2023-05-29T10:12:35.057" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26514" PostTypeId="2" ParentId="26472" CreationDate="2023-05-29T17:38:22.320" Score="4" Body="&lt;p&gt;Here is an everyday example familiar to many students. The time it takes to walk to school has less variance than the time it takes to get to school by bus (this is true everywhere I've lived).&lt;/p&gt;&#xA;&lt;p&gt;This example is nice also because often the mean time walking is longer than the mean time by bus, but from experience one knows that one prefers the security of the low variance choice (or one prefers to get there quickly some of the time, that depends on psychology). Students could even do an experiment and measure such times.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2023-05-30T06:31:57.693" LastActivityDate="2023-05-30T06:31:57.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26515" PostTypeId="2" ParentId="26510" CreationDate="2023-05-30T06:42:44.710" Score="3" Body="&lt;p&gt;In Vlaanderen (which is the Dutch speaking part of Belgium), we have a website, containing all questions of the Math Olympiads (&lt;a href=&quot;https://www.vwo.be/vwo/wedstrijdvragen-per-jaargang-en-ronde/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.vwo.be/vwo/wedstrijdvragen-per-jaargang-en-ronde/&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The questions are written in Dutch, but a good translating engine should not to be too hard to find.&lt;/p&gt;&#xA;&lt;p&gt;There also is Köhmal, which is directly written in English (&lt;a href=&quot;https://www.komal.hu/verseny/feladatok.e.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.komal.hu/verseny/feladatok.e.shtml&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Have fun :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-05-30T06:42:44.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26516" PostTypeId="2" ParentId="26510" CreationDate="2023-05-30T08:01:12.547" Score="2" Body="&lt;p&gt;Addressing point 2, a few interesting maths topics/unsolved puzzles that you could delve deeper into are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Oppermann's conjecture, relating to your interest in prime numbers.&lt;/li&gt;&#xA;&lt;li&gt;Hodge conjecture, relating to your interest in topology and knots.&lt;/li&gt;&#xA;&lt;li&gt;Euler characteristics, eg. Euler characteristics of Torus/Klein Bottle, also relating to your interest in shapes and space.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Enjoy :)&lt;/p&gt;&#xA;" OwnerUserId="22471" LastEditorUserId="22471" LastEditDate="2023-06-07T16:47:07.080" LastActivityDate="2023-06-07T16:47:07.080" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26517" PostTypeId="1" AcceptedAnswerId="26518" CreationDate="2023-05-30T13:11:47.140" Score="9" ViewCount="2200" Body="&lt;p&gt;I remember my high school days where subsets were defined in the following manner:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given two sets A and B, if every element of B is an element of A, then B is called a subset of A.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A common confusion arising from this definition is that of an empty set being a subset of every set. 'But an empty set doesn't contain any elements which are in any other set, so how can it be a subset of every set?'&lt;/p&gt;&#xA;&lt;p&gt;I feel the following definition removes this confusion:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given two sets A and B, if by removing the elements of B we can form A, then A is a subset of B.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since for any general set X, we can remove its elements to form the empty set, we can say that the empty set is a subset of every set.&lt;/p&gt;&#xA;&lt;p&gt;Are there any flaws in this way of thinking about subsets?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I'm not a teacher, but I thought the question better suits this community.&lt;/p&gt;&#xA;" OwnerUserId="20701" LastActivityDate="2023-06-01T15:08:55.023" Title="Can this be a better way of defining subsets?" Tags="|secondary-education|definitions|set-theory|" AnswerCount="6" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="26518" PostTypeId="2" ParentId="26517" CreationDate="2023-05-30T13:48:15.267" Score="21" Body="&lt;p&gt;In both formal and informal treatments of set theory, we need to specify which operations and relations are allowable and build from there.&lt;/p&gt;&#xA;&lt;p&gt;Usually we take sets and set membership as primitives.  We also usually build on top of first order logic.&lt;/p&gt;&#xA;&lt;p&gt;So it is very natural to define &lt;span class=&quot;math-container&quot;&gt;$B \subset A$&lt;/span&gt; as &amp;quot;For all &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$b \in A$&lt;/span&gt;&amp;quot;.  This uses first order logic (the universal quantifier) and set membership.&lt;/p&gt;&#xA;&lt;p&gt;Your definition requires us to also have a notion of &amp;quot;removing a set of elements from a set&amp;quot;.  This operation would generally be defined in terms of more primitive notions as follows:  Let &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; be sets.  We define the set &lt;span class=&quot;math-container&quot;&gt;$X - S$&lt;/span&gt; as the collection of all elements of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; which are not elements of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;. This requires &amp;quot;set formation&amp;quot; axioms, but it is a legit definition.&lt;/p&gt;&#xA;&lt;p&gt;Now your proposed definition of the subset relation is &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$B \subset A$&lt;/span&gt; if and only if there exists a set &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$A - S = B$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;This definition is valid, but is a lot more logically complex than the usual definition.  Try proving a few basic theorems using the usual definition and this modified definition.  For example, try proving that if &lt;span class=&quot;math-container&quot;&gt;$A \subset B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B \subset C$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$A \subset C$&lt;/span&gt;.  I think you will find that the new definition is much clunkier.&lt;/p&gt;&#xA;&lt;p&gt;I think it would be fine to mention the equivalence of these two definitions to a student who was struggling to understand that the empty set is a subset of any other set, but at some point they are going to have to deal with understanding vacuously true statements if they want to understand this equivalency.&lt;/p&gt;&#xA;&lt;p&gt;Personally I think your time would be better spent really working on understanding vacuously true statements.  It will generalize better and enable them to understand the standard definition.  Definitely hard work though!&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-05-30T13:48:15.267" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26519" PostTypeId="1" CreationDate="2023-05-30T17:53:25.330" Score="2" ViewCount="354" Body="&lt;p&gt;One of the greatest mathematicians Andrew Wiles once told that it is really important to give students a chance to work with passionate teachers before starting secondary education, and it is not easy to do so, even in America.&lt;/p&gt;&#xA;&lt;p&gt;He knew by his experience because he was determined to solve Fermat's Last Theorem when he was only 10 years old. Many of us agree that being passionate is just as important as experience and paper qualifications.&lt;/p&gt;&#xA;&lt;p&gt;Is there a system to encourage mathematics teachers to be passionate? And how can (we make it easier for) students to realize that their teacher is passionate in mathematics?&lt;/p&gt;&#xA;&lt;p&gt;In December 2021, I created my own combinatorics challenge and sent it to many of experienced and qualified university lecturers but none of them could sense the importance of the issue. Finally an overseas professor told the problem is good enough for a university research project. That encouraged me a lot, and after working with it for couple of months I was able to solve the problem and generate 12 new sequences which are published on OEIS. After that I have created another 4 new problems related to grid paths and generated yet another 4 completely new sequences which are already published on OEIS.&lt;/p&gt;&#xA;&lt;p&gt;I think by doing such things I can gather new knowledge and be an example to students for how to be committed to their studies by finding ways to get maximum output from the very little they know.&lt;/p&gt;&#xA;&lt;p&gt;I'm doing this as my passion without receiving any kind of encouragement from local bodies, but if there is a way of giving value for such personal research works like happening in university level, it would be great encouragement for mathematics teachers to be passionate in subject.&lt;/p&gt;&#xA;&lt;p&gt;What would you suggest to produce passionate teachers who can inspire students also to be passionate in mathematics ?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="2083" LastEditDate="2023-10-10T10:13:48.007" LastActivityDate="2023-10-10T10:13:48.007" Title="Teachers passionate in mathematics" Tags="|teacher-preparation|teacher-training|teacher-motivation|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="26520" PostTypeId="2" ParentId="26517" CreationDate="2023-05-30T18:38:52.160" Score="8" Body="&lt;p&gt;This answer is going to have a lot of overlap with Steve Gubkin's, but maybe a bit of a different focus. I agree with him that&lt;/p&gt;&#xA;&lt;p&gt;(1) The key issue here is understanding how vacuous cases work.&lt;/p&gt;&#xA;&lt;p&gt;(2) Introducing a notion of &amp;quot;removing&amp;quot; doesn't help, because it introduces a new primitive notion, where set theory is deliberately built on just the primitive &lt;span class=&quot;math-container&quot;&gt;$\in$&lt;/span&gt;, and because &amp;quot;removing&amp;quot; introduces its own ambiguities.&lt;/p&gt;&#xA;&lt;p&gt;What I want to say is that there are two goals you might have here:&lt;/p&gt;&#xA;&lt;p&gt;(a) Teaching students what &lt;span class=&quot;math-container&quot;&gt;$\subseteq$&lt;/span&gt; means and&lt;/p&gt;&#xA;&lt;p&gt;(b) Teaching students how to parse mathematical language.&lt;/p&gt;&#xA;&lt;p&gt;For goal (a), I think rephrasing the definition in as many ways as you can think of (within time limitations) is great. For some students, &amp;quot;removing&amp;quot; will resonate where the usual language doesn't, and vice versa. Pointing out the vacuous case of the empty set is great here; it is usually good teaching (and writing) to point confusing cases. Just be clear that all of the things you are saying are equivalent, and clarify why they are equivalent.&lt;/p&gt;&#xA;&lt;p&gt;But I think you also need to think about goal (b). Students need to be able to read standard mathematical texts, and standard texts will assume that the student is comfortable with the operation of mathematical logic, more and more so as you get further into the curriculum. An introductory set theory course is often the place for students to get used to the language of logic. So I don't think you should dodge the issue of finding another phrasing, I think you should take the time to explain why &lt;span class=&quot;math-container&quot;&gt;$\forall_{x \in \emptyset} \ P(x)$&lt;/span&gt; is always considered true.&lt;/p&gt;&#xA;&lt;p&gt;I haven't tried this, but studies on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Wason_selection_task&quot; rel=&quot;noreferrer&quot;&gt;Wason selection task&lt;/a&gt; find that students understand the mathematical meaning of &amp;quot;if&amp;quot; best if it is formulated as a rule to be enforced (e.g. &amp;quot;If you are drinking alcohol, then you must be over 18&amp;quot;.) I suspect the same would work with statements of the form &lt;span class=&quot;math-container&quot;&gt;$\forall_{x \in \emptyset} P(x)$&lt;/span&gt;. For example: &amp;quot;A website has a rule that it will only host videos if all of the humans depicted have given their consent to be filmed. Does a video which shows only cats obey this rule?&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2023-05-30T18:38:52.160" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26521" PostTypeId="2" ParentId="26517" CreationDate="2023-05-30T20:47:36.453" Score="13" Body="&lt;p&gt;In addition to the very good answers that you've already received, it's probably worthwhile to also mention the following point:&lt;/p&gt;&#xA;&lt;p&gt;The alternative that you suggest might lead to a similar type of confusion as the more common definition does: every set &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is a subset of itself and to see this from your definition, one needs to understand that &amp;quot;removing some elements of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;&amp;quot; includes the case of not removing any element at all.&lt;/p&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2023-05-30T20:47:36.453" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26522" PostTypeId="2" ParentId="26517" CreationDate="2023-05-31T18:57:13.383" Score="7" Body="&lt;p&gt;A potential objection: what does it mean to be able to make a set by removing elements from another set? Some students will think of removing the elements one at a time. Then there'll be misconceptions such as that &lt;span class=&quot;math-container&quot;&gt;$\{1, 2, 3\}$&lt;/span&gt; is not a subset of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;, because every time you remove an element from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;, it is still infinite.&lt;/p&gt;&#xA;&lt;p&gt;There is &lt;a href=&quot;https://link.springer.com/book/10.1007/978-1-4614-7966-6&quot; rel=&quot;noreferrer&quot;&gt;a theory&lt;/a&gt; that students learn abstract concepts in mathematics by generalising from actions (things you do) to processes (things that can be done but don't have to be) and objects (things that exist and can be reasoned about). For example, a student can compute &lt;span class=&quot;math-container&quot;&gt;$x + 12$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x = 5$&lt;/span&gt; by substituting and simplifying, or recognise &lt;span class=&quot;math-container&quot;&gt;$x + 12$&lt;/span&gt; as a process which transforms &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; without having to know what &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is, or conceptualise &lt;span class=&quot;math-container&quot;&gt;$x + 12$&lt;/span&gt; as a function which has certain properties and can be composed with other functions.&lt;/p&gt;&#xA;&lt;p&gt;The traditional definition of a subset works well in this theory because we can first talk about a procedure for verifying &lt;span class=&quot;math-container&quot;&gt;$A \subseteq B$&lt;/span&gt; by checking all of the elements of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to see if they are elements of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, but then the definition doesn't say that you have to &lt;em&gt;do&lt;/em&gt; this or even that it has to be &lt;em&gt;possible&lt;/em&gt; to follow that procedure ─ the traditional definition doesn't rely on the procedure or refer to it at all, it says what a subset &lt;em&gt;is&lt;/em&gt; rather than how to verify that something is a subset. On the other hand, the proposed alternative definition talks about a thing you have to do, i.e. remove elements from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; until only the elements of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; remain. So it's harder to see how to generalise from that definition to a concept of &amp;quot;subset&amp;quot; which doesn't depend on a procedure.&lt;/p&gt;&#xA;" OwnerUserId="19245" LastActivityDate="2023-05-31T18:57:13.383" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26523" PostTypeId="2" ParentId="26517" CreationDate="2023-06-01T02:17:47.187" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;'But an empty set doesn't contain any elements which are in any other set, so how can it be a subset of every set?'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Does &amp;quot;If there is no element in A that isn't in B, then A is a subset of B&amp;quot; fit your intuition any better?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given two sets A and B, if by removing the elements of B we can form A, then A is a subset of B.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The elements of B&amp;quot; is ambiguous. It could be taken to mean &lt;em&gt;all&lt;/em&gt; of the elements. What you really mean is &amp;quot;if there is a subset C of B such that removing C from B results in A, then A is subset of B&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I'll leave it as an exercise for the reader what the problem with &lt;em&gt;that&lt;/em&gt; is.&lt;/p&gt;&#xA;" OwnerUserId="8989" LastActivityDate="2023-06-01T02:17:47.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26524" PostTypeId="2" ParentId="26517" CreationDate="2023-06-01T15:08:55.023" Score="6" Body="&lt;p&gt;The only flaw is that it doesn't really fix the perceived &amp;quot;problem&amp;quot; of the original definition.&lt;/p&gt;&#xA;&lt;p&gt;What if &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are both empty? What elements of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are you going to remove to get &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;I don't think it sidesteps anything. The same issue just pops up elsewhere.&lt;/p&gt;&#xA;" OwnerUserId="13192" LastActivityDate="2023-06-01T15:08:55.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26525" PostTypeId="1" CreationDate="2023-06-02T09:23:03.510" Score="12" ViewCount="937" Body="&lt;p&gt;One common way to introduce Euler's number &lt;span class=&quot;math-container&quot;&gt;$\mathrm e$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$$\mathrm e = \lim_{n\to \infty} \left(1+\frac{1}{n}\right)^n,$$&lt;/span&gt; where the right-hand expression has an &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/E_(mathematical_constant)#Compound_interest&quot; rel=&quot;noreferrer&quot;&gt;interest rate&lt;/a&gt; interpretation&amp;quot; (so that a high-school student can see why this expression might be of interest and worth attaching a name to).&lt;/p&gt;&#xA;&lt;p&gt;Separately, we can show that &lt;span class=&quot;math-container&quot;&gt;$$\mathrm e = \frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots.$$&lt;/span&gt; Is there, similarly, any real-world meaning/&amp;quot;interpretation&amp;quot; we can attach to the expression &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;(An ideal answer would target a secondary/high-school student who has no prior knowledge about calculus or the properties of &lt;span class=&quot;math-container&quot;&gt;$\mathrm e$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt;, as is the case with the &amp;quot;interest rate interpretation&amp;quot;.)&lt;/p&gt;&#xA;" OwnerDisplayName="user18187" LastEditorDisplayName="user18187" LastEditDate="2023-06-03T02:53:50.717" LastActivityDate="2023-08-04T14:01:20.443" Title="Any meaning/interpretation for $\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots (= \mathrm e)$ (sum of reciprocals of factorials)?" Tags="|secondary-education|concept-motivation|" AnswerCount="8" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26526" PostTypeId="2" ParentId="26525" CreationDate="2023-06-02T14:03:01.060" Score="12" Body="&lt;p&gt;You could ask this series of questions; I don't know how &amp;quot;real world&amp;quot; it is but it does indicate the specific contributions of each individual term of the series.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose we have a car, and it is 1 meter away from us. How far away will it be after 1 second?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;1 meter.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose we have a car, and it is 1 meter away from us, but it is also moving away from us at 1 meter per second. How far away will it be after 1 second?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;2 meters.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose we have a car, and it is 1 meter away from us, but it is also moving away from us at 1 meter per second, and it is also accelerating away from us at 1 meter per second per second. How far away will it be after 1 second?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2 \frac12$&lt;/span&gt; meters...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose we have a car, and it is 1 meter away from us, but it is also moving away from us at 1 meter per second, and it is also accelerating away from us at 1 meter per second per second, AND it is also getting jerked away from us at 1 meter per second per second per second. How far away will it be after 1 second?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2 \frac23$&lt;/span&gt; meters...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We don't seem to be gaining much with each of these. If you keep adding these 1s forever, how far away can the car get in 1 second?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-06-02T14:03:01.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26527" PostTypeId="2" ParentId="26525" CreationDate="2023-06-02T15:20:03.053" Score="9" Body="&lt;p&gt;I think the one with useful interpretation is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{1}{0!}+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots = \exp(x)&#xA;$$&lt;/span&gt;&#xA;You can check &lt;span class=&quot;math-container&quot;&gt;$\exp(x+y) = \exp(x)\exp(y)$&lt;/span&gt;, so it does behave like you would expect a power &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; to behave.  More advanced students can check that it satisfies the differential equation &lt;span class=&quot;math-container&quot;&gt;$\exp'(x) = \exp(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2023-06-02T15:20:03.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26528" PostTypeId="2" ParentId="26525" CreationDate="2023-06-02T17:00:24.087" Score="4" Body="&lt;p&gt;If we have a random event that occurs &lt;span class=&quot;math-container&quot;&gt;$\lambda=1$&lt;/span&gt; time per day on average, then over time, the proportion of days in which the event occurs &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; times approaches &lt;span class=&quot;math-container&quot;&gt;$P(k) = e^{-1}/k!$&lt;/span&gt;, the probability that the event occurs &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; times in a day (Poisson distribution). So&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{k=0}^\infty {e^{-1} \over k!} = e^{-1} \sum_{k=0}^\infty {1 \over k!} = 1$$&lt;/span&gt;&#xA;is the sum of the probabilities that the number of events in a day is 0, or 1, or 2,..., which of course is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&#xA;Consequently, for &lt;span class=&quot;math-container&quot;&gt;$k=0$&lt;/span&gt;, the probability of &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; events is&#xA;&lt;span class=&quot;math-container&quot;&gt;$${e^{-1} \over 0!} = e^{-1} = 1 \Big/ \sum_{k=0}^\infty {1 \over k!} \,.$$&lt;/span&gt;&#xA;Alternatively, this is the proportion of days over time in which the event does not occur.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2023-06-02T17:00:24.087" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26529" PostTypeId="2" ParentId="26525" CreationDate="2023-06-02T17:14:27.583" Score="2" Body="&lt;p&gt;This is a similar answer to Raciquel's: To look at &lt;a href=&quot;https://en.wikipedia.org/wiki/Poisson_distribution&quot; rel=&quot;nofollow noreferrer&quot;&gt;Poisson processes&lt;/a&gt;. As a concrete example of the sort of thing we might want to model with a Poisson process, consider traffic accidents per day in a large city. Suppose that our records show that there are an average of &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; accidents per day. Then the Poisson model says that the probability of having &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; accidents in a given day is &lt;span class=&quot;math-container&quot;&gt;$\tfrac{\lambda^k e^{-\lambda}}{k!}$&lt;/span&gt;. The fact that the probabilities add up to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is thus equivalent to &lt;span class=&quot;math-container&quot;&gt;$e^{-\lambda} \sum_{k=0}^{\infty} \tfrac{\lambda^k}{k!} = 1$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$\sum_{k=0}^{\infty} \tfrac{\lambda^k}{k!} = e^{\lambda}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let's see how to motivate the Poisson model without using the Taylor series of &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt;, so that this is not circular: Suppose that there are &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; cars. Since there are &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; accidents per day, the probability of a particular car having an accident on a particular day is &lt;span class=&quot;math-container&quot;&gt;$\lambda/N$&lt;/span&gt;. For a given set of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; cars, if accidents are independent, the chance of these &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; cars having accidents and no other cars doing so is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(\lambda/N)^k (1-\lambda/N)^{N-k}.$$&lt;/span&gt;&#xA;There are &lt;span class=&quot;math-container&quot;&gt;$\binom{N}{k}  = \tfrac{N(N-1)(N-2) \cdots (N-k+1)}{k!}$&lt;/span&gt; sets of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; cars, so the chance that &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; cars have accidents on a given day is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{N(N-1)(N-2) \cdots (N-k+1)}{k!} (\lambda/N)^k (1-\lambda/N)^{N-k} \approx \frac{N^k}{k!} (\lambda/N)^k(1-\lambda/N)^N \approx \frac{\lambda^k}{k!} e^{-\lambda}.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="51" LastEditorUserId="20719" LastEditDate="2023-06-25T00:44:18.307" LastActivityDate="2023-06-25T00:44:18.307" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26530" PostTypeId="2" ParentId="26525" CreationDate="2023-06-02T18:21:28.107" Score="9" Body="&lt;p&gt;A possible motivation is the solution to the so-called &amp;quot;hat check problem.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;There are &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; people who throw their hats into a box and then randomly take one out. The probability that each person ends up with a hat that is not their own is the alternating exponential series:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$P(N)=\sum_{n=0}^N (-1)^n\frac{1}{n!}\approx e^{-1}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you search for the word &amp;quot;derangements&amp;quot; you will find more details. The formula follows directly from the inclusion–exclusion principle and the binomial formula.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-06-03T12:08:39.823" LastActivityDate="2023-06-03T12:08:39.823" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26535" PostTypeId="2" ParentId="26525" CreationDate="2023-06-11T03:36:48.457" Score="5" Body="&lt;p&gt;For what it's worth: The distinction between &lt;span class=&quot;math-container&quot;&gt;$e = \lim_{n\to\infty} (1+\frac{1}{n})^n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e = \sum_{n=0}^\infty \frac{1}{n!}$&lt;/span&gt; has, to me, become very blurry in recent years. In fact, just assume &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is &amp;quot;very big&amp;quot;, and use the binomial formula to multiply out:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\left(1+\frac{1}{n}\right)^n = 1 + \underbrace{n\cdot \frac{1}{n}}_{=1} + \underbrace{\dfrac{n(n-1)}{2!}\cdot\frac{1}{n^2}}_{\approx \frac{1}{2!}}+\underbrace{\dfrac{n(n-1)(n-2)}{3!} \cdot \frac{1}{n^3}}_{\approx \frac{1}{3!}} + \dots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and the approximations become better and better for &lt;span class=&quot;math-container&quot;&gt;$n \to \infty$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8931" LastEditorUserId="8931" LastEditDate="2023-06-14T20:35:50.870" LastActivityDate="2023-06-14T20:35:50.870" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26536" PostTypeId="2" ParentId="26525" CreationDate="2023-06-12T08:28:58.780" Score="3" Body="&lt;p&gt;Here's a problem I designed, and my solution 'discovering' the summation expression as an expected value of a triangular number of objects.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A builder is tasked to make stairs from a set of wooden planks. The owner has already bought the planks, and told the builder to use the first plank for the first step of the stairs, the next two planks for the second step, the next three for the third step, and so on. The builder does so, but then the owner realises they forget to tell the builder how to arrange the planks within each step, as there was a pattern on the side of the stairs they wanted to have seen. If there are many steps in the stairs, estimate how many are correctly arranged.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We do not have enough information to determine the exact number of correctly arranged steps. But assuming the builder chose a random arrangement of the planks within each step, we can calculate the expected number of correct steps using probability.&lt;/p&gt;&#xA;&lt;p&gt;There is only 1 plank in the first step. So there is only one possible arrangement, and it must be the correct one. Hence, the first step is guaranteed correct.&lt;/p&gt;&#xA;&lt;p&gt;The second step has 2 planks; so &lt;span class=&quot;math-container&quot;&gt;$2!=2$&lt;/span&gt; possible arrangements, out of which only 1 is correct. This arrangement has a correctness value of &amp;quot;1&amp;quot;, while the other arrangement has a correctness value of &amp;quot;0&amp;quot;. And assuming uniform distribution, each arrangement has an equal likelihood of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2!}$&lt;/span&gt; of being chosen. Hence, the expected correctness of the second step is &lt;span class=&quot;math-container&quot;&gt;$$1\times \frac{1}{2!} + 0\times \frac{1}{2!} = \frac{1}{2!}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Repeating this logic, the expected correctness of the third step (which has 3 planks with &lt;span class=&quot;math-container&quot;&gt;$3!=6$&lt;/span&gt; possible arrangements) is, &lt;span class=&quot;math-container&quot;&gt;$$1\times \frac{1}{3!} + 0\times \frac{1}{3!} + 0\times \frac{1}{3!} + 0\times \frac{1}{3!} + 0\times \frac{1}{3!} + 0\times \frac{1}{3!} = \frac{1}{3!}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Continuing, step &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; has an expected correctness of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{n!}$&lt;/span&gt;. Then the expected total 'number' of correct steps is, &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} ... \frac{1}{n!}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If we calculate this sum for increasing &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, we see it tends towards a constant value of roughly 1.72.&#xA;|n   |1   |2   |3   |4   |5   |6|&#xA;|----|----|----|----|----|----|-|&#xA;|&lt;span class=&quot;math-container&quot;&gt;$\sum_{i=1}^n \frac{1}{i!}$&lt;/span&gt;|1|1.5|1.67|1.71|1.717|1.718&lt;/p&gt;&#xA;&lt;p&gt;This constant comes up often in mathematics, and so it has a special symbol, &lt;span class=&quot;math-container&quot;&gt;$\mathrm e$&lt;/span&gt;. But with one difference of starting from &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{0!} = 1$&lt;/span&gt;, as this form is more useful.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\mathrm e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} ... $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Therefore, &lt;span class=&quot;math-container&quot;&gt;$\mathrm e - 1$&lt;/span&gt; can be physically interpreted as the expected number of correctly arranged steps in the above staircase problem.&lt;/p&gt;&#xA;" OwnerUserId="22492" LastActivityDate="2023-06-12T08:28:58.780" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26537" PostTypeId="2" ParentId="24814" CreationDate="2023-06-12T11:31:00.797" Score="6" Body="&lt;p&gt;At my university, students with learning challenges receive extra time during exams.  Based on this, I can tell you that there are quite a number of them in the math courses that I have taught.  Some of these students have specifically dyslexia and ADHD that you mentioned.  My conclusion would be that these conditions are &lt;em&gt;not&lt;/em&gt; insurmountable obstacles to obtaining a degree in mathematics.  On the other hand, you also mention forgetting definitions.  This is not necessarily a medical problem; if a student has not grasped the meaning of a definition, it can certainly be easily forgotten.  My impression is that if your medical authority rules out memory problems of a more serious kind, by all means you should be encouraged to work on obtaining a degree...  Hope I answered at least one of your questions.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2023-06-14T10:25:33.530" LastActivityDate="2023-06-14T10:25:33.530" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26538" PostTypeId="1" CreationDate="2023-06-12T15:55:36.650" Score="1" ViewCount="180" Body="&lt;p&gt;I recently landed on a book written for elementary school teachers which introduced the concept of whole numbers in the following manner:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We have a set &lt;span class=&quot;math-container&quot;&gt;$\{\alpha, \beta, \gamma\}$&lt;/span&gt;. There are other sets we&#xA;can identify that share a common property with this set by virtue of&#xA;which a one-to-one correspondence can be established between the&#xA;elements of the set &lt;span class=&quot;math-container&quot;&gt;$\{\alpha, \beta, \gamma\}$&lt;/span&gt; and all other sets&#xA;identified. This common property abstracted away from all these sets&#xA;including &lt;span class=&quot;math-container&quot;&gt;$\{\alpha, \beta, \gamma\}$&lt;/span&gt; and considered as a thing in and&#xA;of itself is named '3'.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I've also come across some books on set theory that define 3 as literally the set {0,1,2} or {{}, {{}}, {{}, {{}}}}.&lt;/p&gt;&#xA;&lt;p&gt;So now I'm confused what is the correct way of thinking about whole numbers? A property abstracted from sets and considered to be a thing in itself or literally as sets themselves (as in 3={0,1,2}?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The first approach appeals to me better because it is in line with what I think of 3. However, the second one, says 3 is a set, that's not what my intuition tells me. 3 and the set {0,1,2} are clearly different things.&lt;/p&gt;&#xA;" OwnerUserId="20701" LastEditorUserId="20701" LastEditDate="2023-06-12T16:18:24.300" LastActivityDate="2023-06-13T06:46:36.103" Title="Whole numbers as sets vs abstracted properties of sets" Tags="|secondary-education|primary-education|number-theory|set-theory|" AnswerCount="3" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26539" PostTypeId="2" ParentId="26538" CreationDate="2023-06-12T16:34:58.760" Score="2" Body="&lt;p&gt;The first approach is an attempt to define an equivalence relation on sets, so that two sets are equivalent if and only if they have the same cardinality. An Achilles heel of this approach is that the set of all sets is not a proper set. In fact the collection of all things with cardinality three is not a proper set. If we are trying to contain everything in the theory of sets, we have come to a problem.&lt;/p&gt;&#xA;&lt;p&gt;Axiomatic set theory such as Zermelo-Frankel begins with an axiom that gives the empty set {} and then proceeds to construct &amp;quot;one&amp;quot; which is {{}}, etc. Then there is the axiom of infinity which begins the construction of a new generation of sets that are not in bijective correspondence with those constructed in our first generation, which consists of finite sets.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-06-12T16:34:58.760" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26540" PostTypeId="2" ParentId="26538" CreationDate="2023-06-13T05:03:21.477" Score="0" Body="&lt;p&gt;What you seem to be referring to is the cardinality of a set i.e. the number of elements in a set. If you want take the cardinality of set A you should write |A|.&lt;/p&gt;&#xA;&lt;p&gt;e.g., A = {1, 2, 3} =&amp;gt;  |A| = 3&lt;/p&gt;&#xA;&lt;p&gt;Often you will see the cardinality symbol is not used for convenience. However this is not technically correct as a set is different from a number.&lt;/p&gt;&#xA;&lt;p&gt;i.e. {1, 2, 3} != 3&lt;/p&gt;&#xA;" OwnerUserId="20626" LastActivityDate="2023-06-13T05:03:21.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26541" PostTypeId="2" ParentId="26538" CreationDate="2023-06-13T06:46:36.103" Score="4" Body="&lt;p&gt;Disregarding research on the foundations of mathematics, what we really want to have are the usual numbers with the usual properties. The usual properties are really nice, because they correspond to how the world works: I can have three apples in one hand and four in the other, and no matter which order I count them in, I always get seven apples total. And so I want &lt;span class=&quot;math-container&quot;&gt;$3+4 = 4+3 = 7$&lt;/span&gt;. Likewise for the other properties.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You can declare a set of axioms that are minimal and capture this behaviour. (As a first approach, check Peano axioms.&lt;/li&gt;&#xA;&lt;li&gt;You can declare axioms for how set works and then build up natural numbers from them.&lt;/li&gt;&#xA;&lt;li&gt;You can declare how categories, in category theory, work, and build up natural numbers from there.&lt;/li&gt;&#xA;&lt;li&gt;I presume there are other ways of going about this stuff, too.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;However, the main point is that you always get the same natural numbers (again, disregarding subtleties in the foundations of mathematics) which have all the familiar properties, because those are the ones we want to model our reality.&lt;/p&gt;&#xA;&lt;p&gt;I suggest not seeing the different ways to construct natural numbers as somehow fundamental, unless you are in fact interested in looking into the foundations of mathematics. The properties of the natural numbers are the most important part, for almost everyone, including almost every mathematician.&lt;/p&gt;&#xA;&lt;p&gt;In the context of teaching mathematics you also want to have the properties make sense; you want people to understand that &lt;span class=&quot;math-container&quot;&gt;$a+b = b+a$&lt;/span&gt; corresponds to first counting the apples in one hand and then the other, or the other way around, and getting an equal amount of apples both ways. Being able to draw this connection is important; that is, more generally, giving meaning to the formulae.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-06-13T06:46:36.103" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26542" PostTypeId="2" ParentId="26525" CreationDate="2023-06-13T14:22:48.827" Score="3" Body="&lt;p&gt;This post is just an elaboration of my comment to user52817's answer on the hat check problem. The &lt;em&gt;rencontres&lt;/em&gt; numbers &lt;span class=&quot;math-container&quot;&gt;$D_{n,m}$&lt;/span&gt; enumerate the permutations of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; items with exactly &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; fixed points. Their number can be computed by multiplying the number of ways of choosing which items are the fixed ones by the number of derangements of the remaining items, that is, by multiplying the number of ways of choosing which &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; people get their own hats by the number of ways of assigning hats not their own to the remaining &lt;span class=&quot;math-container&quot;&gt;$n-m$&lt;/span&gt; people,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;D_{n,m}=\binom{n}{m}D_{n-m,0}=\frac{n!}{m!\,(n-m)!}\sum_{k=0}^{n-m}(-1)^k\frac{(n-m)!}{k!}.&#xA;$$&lt;/span&gt;&#xA;A standard expression for the number of derangements has been used on the right. (A derivation is at the end of this post.) As a consequence, the probability that a random permutation of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; items has exactly &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; fixed points is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{D_{n,m}}{n!}=\frac{1}{m!}\sum_{k=0}^{n-m}(-1)^k\frac{1}{k!}.&#xA;$$&lt;/span&gt;&#xA;Fixing &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and letting &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; go to infinity we get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{n\to\infty}\frac{D_{n,m}}{n!}=\mathcal{N}\frac{1}{m!},&#xA;$$&lt;/span&gt;&#xA;where we can think of &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}=\sum_{k=0}^\infty(-1)^k\frac{1}{k!}$&lt;/span&gt; as a normalization constant for the probability distribution. You will recognize that &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}=\frac{1}{e}$&lt;/span&gt;, but we can pretend we don't know that yet. Probabilities for all possible numbers of fixed points must sum to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, which gives&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;1=\mathcal{N}\sum_{m=0}^\infty\frac{1}{m!}.&#xA;$$&lt;/span&gt;&#xA;So in addition to the expression for &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}$&lt;/span&gt; already given, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}$&lt;/span&gt; is also the reciprocal of &lt;span class=&quot;math-container&quot;&gt;$\sum_{m=0}^\infty\frac{1}{m!}$&lt;/span&gt;. The consistency of these two characterizations can be verified by multiplying out the two sums and using an identity for the alternating sum of binomial coefficients to show that the product is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. (A version of this calculation for finite &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is given at the end of this post.) In addition to its role as normalization constant, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}$&lt;/span&gt; is the asymptotic probability that a random permutation has no fixed points, and also the asymptotic probability that it has exactly one fixed point. It should be noted that the probability distribution we have obtained by taking the &lt;span class=&quot;math-container&quot;&gt;$n\to\infty$&lt;/span&gt; limit is the Poisson distribution with &lt;span class=&quot;math-container&quot;&gt;$\lambda=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Whether students will see &lt;em&gt;rencontres&lt;/em&gt; numbers as a compelling enough problem to motivate the definition of a new mathematical constant is a good question. My feeling is that the Poisson distribution is the fundamental thing since it shows up over and over again, with the &lt;em&gt;rencontres&lt;/em&gt; numbers being but one application.&lt;/p&gt;&#xA;&lt;p&gt;One potential objection that seems to have been raised is that Poisson probability is almost always defined as a limit of Bernoulli probability and that taking this limit requires the use of (a generalization of) the limit that shows up in the interest rate application. Which, I guess makes the Poison interpretation somehow &amp;quot;not independent&amp;quot; of the interest rate formula. All of this is discussed in David E Speyer's answer and the comments below it.&lt;/p&gt;&#xA;&lt;p&gt;I'm actually not sure why that's so bad. Two points I would make are (1) that the &lt;em&gt;rencontres&lt;/em&gt; numbers application shows that limits not involving the interest rate formula also arrive at the Poisson distribution, and (2) that it is just as interesting or more so to show interconnections between ideas as it is to show completely independent derivations. If the Poisson distribution is too complicated for students at this level to derive from scratch, it is still useful to them to teach some of the bare facts about it, such as that the probability of exactly &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; hits is proportional to &lt;span class=&quot;math-container&quot;&gt;$\frac{\lambda^k}{k!}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;With regard to the last point, it may be that students at the introductory calculus level will see the Poisson distribution as intimidating because it involves a transcendental expression &lt;span class=&quot;math-container&quot;&gt;$e^\lambda$&lt;/span&gt;. A key point is that the &lt;em&gt;ratio&lt;/em&gt; of any two Poisson probabilities is just a power of &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; times a rational number--so the kind of thing that is very familiar from algebra. The only role the transcendental expression plays is as an overall normalization constant that divides every probability equally and that cancels out if you take ratios of probabilities.&lt;/p&gt;&#xA;&lt;p&gt;So my view is that the Poisson distribution is very down-to-earth motivation for introducing the transcendental function &lt;span class=&quot;math-container&quot;&gt;$e^\lambda$&lt;/span&gt; and the transcendental number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, the latter being relevant to the very important special case &lt;span class=&quot;math-container&quot;&gt;$\lambda=1$&lt;/span&gt;. (The importance of &lt;span class=&quot;math-container&quot;&gt;$\lambda=1$&lt;/span&gt; is that, for a Poisson process unfolding in continuous time or continuous space, the inverse of the hit rate provides a natural time or distance scale for the problem. Hence one of the bald facts that I think students ought to know is that in a Poisson process the probability of zero hits within the natural time scale for the problem is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{e}$&lt;/span&gt;. (And also for one hit.) This is the most compelling occurrence of the constant &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; &amp;quot;in nature&amp;quot; that I know of.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Some details filled in:&lt;/strong&gt; The formula for &lt;span class=&quot;math-container&quot;&gt;$D_{n,0}$&lt;/span&gt;, the number of derangements of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; items, can be derived by defining &lt;span class=&quot;math-container&quot;&gt;$C_S$&lt;/span&gt; to be the set of permutations of &lt;span class=&quot;math-container&quot;&gt;$1,2,\ldots,n$&lt;/span&gt; in which each element of the set &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is a fixed point, so that &lt;span class=&quot;math-container&quot;&gt;$\lvert C_S\rvert=(n-\lvert S\rvert)!$&lt;/span&gt;. By inclusion-exclusion we have&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;D_{n,0} &amp;amp;=n!-\sum_i\lvert C_{\{i\}}\rvert+\sum_{i&amp;lt;j}\lvert C_{\{i,j\}}\rvert-\sum_{i&amp;lt;j&amp;lt;k}\lvert C_{\{i,j,k\}}\rvert+\ldots+(-1)^n\lvert C_{\{1,2,\ldots,n\}}\rvert\\&#xA;&amp;amp;=n!-n(n-1)!+\binom{n}{2}(n-2)!-\binom{n}{3}(n-3)!+\ldots+(-1)^n\\&#xA;&amp;amp;=\frac{n!}{0!}-\frac{n!}{1!}+\frac{n!}{2!}-\frac{n!}{3!}+\ldots+(-1)^n\frac{n!}{n!}.&#xA;\end{align}&lt;/span&gt;&#xA;The probability that a random permutation of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; items is a derangement is therefore&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{D_{n,0}}{n!}=\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\ldots+(-1)^n\frac{1}{n!}.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;That the probabilities for all possible numbers of fixed points sum to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; must, of course, work out for finite &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. Let's see how this goes:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\sum_{m=0}^n\frac{D_{n,m}}{n!} &amp;amp;=&#xA;\sum_{m=0}^n\frac{1}{m!}\sum_{k=0}^{n-m}(-1)^k\frac{1}{k!}\\&#xA;&amp;amp;= \sum_{t=0}^n\sum_{m=0}^t\frac{1}{m!}(-1)^{t-m}\frac{1}{(t-m)!}\\&#xA;&amp;amp;= \sum_{t=0}^n\frac{1}{t!}\sum_{m=0}^t\binom{t}{m}(-1)^{t-m}\\&#xA;&amp;amp;= \sum_{t=0}^n\frac{1}{t!}\delta_{t,0}=1.&#xA;\end{align}&lt;/span&gt;&#xA;In the second line we have introduced a new summation index &lt;span class=&quot;math-container&quot;&gt;$t=m+k$&lt;/span&gt;; in the fourth line &lt;span class=&quot;math-container&quot;&gt;$\delta_{a,b}$&lt;/span&gt; is the Kronecker delta; it arises from the evaluation of the alternating sum of binomial coefficients.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastEditorUserId="1737" LastEditDate="2023-06-13T15:09:41.793" LastActivityDate="2023-06-13T15:09:41.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26543" PostTypeId="2" ParentId="20829" CreationDate="2023-06-14T11:00:19.257" Score="4" Body="&lt;p&gt;Beyond the visual guesswork of scanning the looseleaf to estimate how many lines are there, the process described amounts to modeling the situation based on estimated values of a few important key factors. Most school mathematics does not encourage this type of thinking because problems are often aimed at exercising or testing specific math techniques rather than developing &amp;quot;reasonable&amp;quot; answers to more speculative questions. Further, most problems settings are not rich enough to permit this type of modeling; usually there is not enough contextual data provided to allow students to make choices about their approach, and there is not an opportunity to make estimates of factors based on measurements in the physical world or recollections of their own prior experience as was done with your example (estimating words/line based on experience, estimating lines/page based on a physical procedure). Most of the estimation that is emphasized is &amp;quot;sense-making&amp;quot;, often limited to word problems based on familiar experiences (e.g., a human is not 100 ft tall, a trip in a car does not likely have an average speed of 1 cm/hr...).&lt;/p&gt;&#xA;&lt;p&gt;I suspect it is too much to ask for the youngest children who are just learning to add and multiply to also grapple with these concepts at the same time, so I think one of the first opportunities for elementary students to consciously develop these skills would be in a science class like physical science (a combination of physics, chemistry, geology, and ecology often taught to middle schoolers in the US). One can talk about the use of data to grow understanding and make predictions, the development and testing of a hypothesis that assumes certain factors are important, and the reasonable precision to assume for estimated or collected data while giving students more practice setting up word problems and doing simple addition/multiplication/equations. Once a student has enough science background, you could introduce other meaningful physics or chemistry word problems in a math setting without too much back-tracking to discuss basic aspects of the subject matter. A statistics class is another natural opportunity to practice this thinking where it is clear that the aim is to specify a simplified model of a random process that meets desired accuracy and explainability criteria and that multiple models with different factors might be considered on par with one another.&lt;/p&gt;&#xA;&lt;p&gt;Edit: This is not meant to draw lines between math and science but just to emphasize how closely connected they are and point out that richer problem contexts come up naturally in the science setting.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastEditorUserId="1230" LastEditDate="2023-06-14T11:28:16.653" LastActivityDate="2023-06-14T11:28:16.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26545" PostTypeId="1" CreationDate="2023-06-15T13:51:52.750" Score="11" ViewCount="2807" Body="&lt;p&gt;We all know how important and ubiquitous polynomials are in mathematics. However, when faced with a (not so much in love with the subject) 14-year-old asking us why they should care about these things, answering that studying polynomials opens the door to &lt;em&gt;more&lt;/em&gt; mathematics is probably not the best strategy. Of course, textbooks include some simple geometric and “real life” problems that can be tackled using polynomials, but they are far from exciting to solve.&lt;/p&gt;&#xA;&lt;p&gt;How do/would you personally motivate your high-school students to study polynomials? More precisely, can you suggest&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;fun problems (not necessarily from everyday life) that can be solved using polynomials&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;interesting applications of polynomials (inside or outside of mathematics)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;that might convince them that studying these sums of products of numbers and letters is actually worth it?&lt;/p&gt;&#xA;" OwnerUserId="15420" LastEditorUserId="15420" LastEditDate="2023-06-16T19:14:20.930" LastActivityDate="2023-08-22T12:56:13.053" Title="How can we best motivate the study of polynomials to high-school students?" Tags="|secondary-education|concept-motivation|polynomials|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26546" PostTypeId="2" ParentId="26545" CreationDate="2023-06-15T17:28:54.527" Score="15" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Using puzzles to attract attention:&lt;/em&gt;&#xA;&amp;quot;Think of a number, subtract 7, multiply 3, add 30, divide by 3. Then subtract the original number. The result will always be 3. Why does this magic work?&amp;quot; After learning the trick, ask the student to come up with their own puzzles.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Illustrate the idea from intuitive exercises (and without jargon):&lt;/em&gt;&#xA;&amp;quot;I am thinking of a number. I subtracted 5 from it and then squared it. The final answer is 9. Which number was I thinking?&amp;quot; Of course there are two possible answers: 8 or 2. But students often will find only one answer. It starts a discussion about if there are more answers. This nicely sets up discussing degrees and roots.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Tell them the idea is already familiar to them:&lt;/em&gt;&#xA;Essentially every number written in decimal system is a polynomial in '10'. Start discussing various bases like binary and ternary. This also naturally leads to learning the algebra of polynomials. You can add, subtract, multiply, divide polynomials because they are numbers in base 'x'.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Grasp their attention with history:&lt;/em&gt; Tell them the story of Italians and their conquests. Very complicated looking monsters like &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{26+15\sqrt{3}}+\sqrt[3]{26-15\sqrt{3}}$&lt;/span&gt; might actually be a familiar face (4 in this case!). An argument for this involves a cubic.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Set cool goals and motivate them:&lt;/em&gt; Results like rational root theorem and ideas about irreducible/minimal polynomials are powerful tools to prove irrationality of certain numbers. I was impressed when I could prove that &lt;span class=&quot;math-container&quot;&gt;$\sin 10^{\circ}$&lt;/span&gt; is irrational using polynomials.&#xA;Maybe you can tell them that, once they learn enough of the theory you will show them how to prove a cool result.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Well, these are the techniques I use to teach 10th grade students. I think they are generally effective.&lt;/p&gt;&#xA;" OwnerUserId="368" LastEditorUserId="60" LastEditDate="2023-06-16T16:05:21.503" LastActivityDate="2023-06-16T16:05:21.503" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26547" PostTypeId="2" ParentId="26545" CreationDate="2023-06-15T21:12:24.617" Score="4" Body="&lt;p&gt;I think this question and the likely answers are heading you down a wrong path, because of unstated assumptions. You would be better off working with the student and helping him achieve.  Baby steps and coaching style pedagogy. Direct instruction, mimicking, etc.  Greg Ashman has a very good presentation, referencing studies, on how achievement drives motivation more than motivation driving achievement.&lt;/p&gt;&#xA;&lt;p&gt;In addition many suggestions of puzzles or applications or discovery learning will actually be harder than the standard content...more applicable to top students than to struggling ones.&lt;/p&gt;&#xA;" OwnerUserId="22503" LastActivityDate="2023-06-15T21:12:24.617" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="26548" PostTypeId="1" CreationDate="2023-06-15T21:52:28.953" Score="4" ViewCount="520" Body="&lt;p&gt;I am a pre-university student who wants to help students with Algebra 1 and 2 in high school. I am curious to how the curriculum was built and what the goal of teaching both algebra 1 and 2 might be. I am comfortable with Algebra, I just never understood why the two classes are structured the way they are. I would like to know this more (and review the curriculum of the two) and see how I can help with students taking the course. My math background is college algebra, trig, and calc 1 and I am fairly decent at solving algebraic equations.&lt;/p&gt;&#xA;" OwnerUserId="22504" LastEditorUserId="2083" LastEditDate="2023-06-16T05:34:08.873" LastActivityDate="2023-06-27T22:00:30.853" Title="What is Algebra 1 and 2 as it is in US highschool education?" Tags="|secondary-education|algebra|usa|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26549" PostTypeId="2" ParentId="26545" CreationDate="2023-06-16T02:33:28.440" Score="4" Body="&lt;p&gt;I disagree with the claim by others that puzzles are more applicable to top students than struggling ones. Of course, you must choose &lt;a href=&quot;https://matheducators.stackexchange.com/a/25518/1550&quot;&gt;the right puzzles for the right level&lt;/a&gt;, and actually &lt;strong&gt;teach&lt;/strong&gt; them how to solve the puzzles. If you are unable to solve the puzzles yourself, then it's very difficult to teach in a successful manner using them. You may notice that the puzzles I linked to have essentially nothing to do with polynomials. This is not a coincidence; logical reasoning precedes all true mathematical reasoning.&lt;/p&gt;&#xA;&lt;p&gt;Once your student grasps logical reasoning fully, polynomials will simply show up everywhere and they would have no trouble recognizing and analyzing them. For example, Pascal's triangle literally has a polynomial in each (diagonal) column. You can start there, just playing around with Pascal's triangle and getting students familiar with its interesting properties.&lt;/p&gt;&#xA;&lt;p&gt;I think the other answer's suggestion of base-n representation is not too good as an example of polynomials, because it actually is not one; a (univariate) polynomial is &lt;em&gt;characterized&lt;/em&gt; by having a weighted sum of a &lt;em&gt;variable&lt;/em&gt; raised to some natural powers, so to get a true polynomial in base-n representation you would need to have &lt;em&gt;varying n&lt;/em&gt;, which is just useless. However, it is alright to say that a polynomial in X is like a number in base X where X is very large.&lt;/p&gt;&#xA;&lt;p&gt;I think the most common places you would see true polynomials in real life are the parabolic trajectory of a ball (ignoring air-resistance), and the polynomial approximations of a function/curve (related to asymptotic expansion).&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2023-06-16T02:33:28.440" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26550" PostTypeId="2" ParentId="26545" CreationDate="2023-06-16T04:39:56.933" Score="4" Body="&lt;p&gt;if you're willing to take&#xA;a bit of diversion, you could show how polynomials are used for error correcting codes that make the internet possible (reed-Solomon, but do it with real numbers instead of finite fields so their heads don't explode). one really nice thing about this use case is that it also motivate some of the things you want to prove about polynomials (e.g. that n+1 degree polynomials are determined uniquely by n points)&lt;/p&gt;&#xA;&lt;p&gt;for a slightly harder one, FFTs can be understood as a way of quickly evaluating polynomials, and FFTs are definitely in the top 10 coolest and most useful bits of math of all time.&lt;/p&gt;&#xA;" OwnerUserId="13845" LastActivityDate="2023-06-16T04:39:56.933" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26552" PostTypeId="2" ParentId="26545" CreationDate="2023-06-16T08:08:34.210" Score="1" Body="&lt;p&gt;Few ideas that come to mind:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Let them predict trajectories of vehicles. Data can come from a simulator.&lt;/li&gt;&#xA;&lt;li&gt;Generate splines for scalable fonts targeting fictitious hieroglyphics&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22508" LastActivityDate="2023-06-16T08:08:34.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26553" PostTypeId="2" ParentId="26545" CreationDate="2023-06-16T10:07:25.603" Score="-1" Body="&lt;p&gt;You have two hallways with a different size. What is the maximum length of the segment you can turn from one hallway into the other?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ez2lN.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ez2lN.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;... and now something more realistic: you live in a house with hallways of different sizes. You wife wants to buy a new sofa, but she can't decide to buy the small one or the large one.&lt;br /&gt;&#xA;The vendors want her to buy the large one, but you are capable of calculating the maximum size of the sofa, proving the vendors wrong.&lt;br /&gt;&#xA;Your wife will be very proud being married to such an intelligent man.&lt;/p&gt;&#xA;&lt;p&gt;Do you want your wife to be proud of you? :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-06-16T10:07:25.603" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26554" PostTypeId="2" ParentId="13233" CreationDate="2023-06-16T13:15:55.473" Score="1" Body="&lt;p&gt;Are you aware of this one: &lt;span class=&quot;math-container&quot;&gt;$(a+b)\cdot (a-b)=a^2-b^2$&lt;/span&gt;, which means nothing else than that the product of two numbers equals the difference of the squares of the average and the distance to that average.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Come again???&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let's take 37 and 43:&lt;br /&gt;&#xA;the average is 40, the distance to that average is &lt;span class=&quot;math-container&quot;&gt;$40-37=43-40=3$&lt;/span&gt;, hence:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$37 \cdot 43 = (40 - 3) \cdot (40 + 3) = 40^2 - 3^2 = 1600 - 9 = 1591$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(This obviously only works when it's relatively easy to calculate the square of the average.)&lt;/p&gt;&#xA;&lt;p&gt;... and these two: &lt;span class=&quot;math-container&quot;&gt;$(a±1)^2 = a^2 ± 2a + 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(a±2)^2 = a^2 ± 4 \cdot (a±1)$&lt;/span&gt;, leading to:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$61^2 = 60^2 + 2 \cdot 60 + 1 = 3721$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$48^2 = 50^2 - 4 \cdot (50 - 1) = 2304$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Yes, I know what you're about to say:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$a^2 - b^2 = (a+b) \cdot (a-b)$&lt;/span&gt;, hence:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$a^2 = (a+b) \cdot (a-b) + b^2$&lt;/span&gt;, so:&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$48^2 = (48 + 2) \cdot (48 - 2) + 2^2 = 50 \cdot 46 + 4 = 2300 + 4 = 2304$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Oh, sorry, you meant that people should learn the first powers of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; by heart, so you get:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$48^2 = (16 \cdot 3)^2 = (2^4 \cdot 3)^2 = 2^8 \cdot (8 + 1) = 2^8 \cdot (2^3 + 1) = 2^{11} + 2^8 = 2048 + 256 = 2304$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Oh, you still know another way to get it done?&lt;br /&gt;&#xA;Don't hesitate editing this answer :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastEditorUserId="8190" LastEditDate="2023-06-19T12:08:40.283" LastActivityDate="2023-06-19T12:08:40.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26555" PostTypeId="2" ParentId="26545" CreationDate="2023-06-17T10:39:26.513" Score="6" Body="&lt;p&gt;If the student has ever used a vector-based computer drawing program, they will be familiar with &lt;a href=&quot;https://en.wikipedia.org/wiki/B%C3%A9zier_curve&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bézier curves&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Bézier curves are extremely intuitive to understand for humans. They became popular as a digital replacement for wooden splines that were used by car designers to shape the smooth surface of a car's body. People typically very quickly understand how to shape them to their liking when playing around with a graphics program, even without given instructions.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, Bézier curves play a fundamental role in computer graphics. Everything visible on your students' phone that is not a photograph is likely being computed from a Bézier polynomial. The text you are looking at right now? A polynomial. Any &lt;code&gt;.svg&lt;/code&gt; graphic on a website? A polynomial. The emojis that your students will undoubtedly use extensively? You get the point...&lt;/p&gt;&#xA;&lt;p&gt;You won't be able to explain how exactly Bézier polynomials work, as that theory will never fit into a high school math curriculum (&lt;a href=&quot;https://www.youtube.com/watch?v=aVwxzDHniEw&quot; rel=&quot;nofollow noreferrer&quot;&gt;this video&lt;/a&gt; gives some pretty nice visualizations though). But it may be instructive to start with a familiar figure, like a glyph for a letter or an emoji, show how it is composed of curves and then show the polynomial form of the equations for some of those curves. Students are good at pattern matching, so they should be able to identify the common structure of all those equations. Now all you have to do is slap a name on it, and you have motivated polynomials.&lt;/p&gt;&#xA;" OwnerUserId="22513" LastEditorUserId="22513" LastEditDate="2023-08-22T12:56:13.053" LastActivityDate="2023-08-22T12:56:13.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26557" PostTypeId="2" ParentId="11433" CreationDate="2023-06-19T00:24:15.337" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://pat-thompson.net&quot; rel=&quot;noreferrer&quot;&gt;Pat Thompson&lt;/a&gt; has done research on students' and teachers' understandings of functions and variation. (He's a co-author on the article mentioned in &lt;a href=&quot;https://matheducators.stackexchange.com/a/11434/20719&quot;&gt;this answer&lt;/a&gt;, though he's not listed for some reason in the link you gave in your comment.)&lt;/p&gt;&#xA;&lt;p&gt;In a &lt;a href=&quot;http://pat-thompson.net/PDFversions/2013WhyF(x).pdf&quot; rel=&quot;noreferrer&quot;&gt;short 2013 article&lt;/a&gt;, Thompson calls the statement &lt;span class=&quot;math-container&quot;&gt;$C=f(x)=3x+2$&lt;/span&gt; a misuse of notation that can lead to &amp;quot;[confusing] the idea of function and variable.&amp;quot; In the same article, he relates this to when students see a function definition like &lt;span class=&quot;math-container&quot;&gt;$V(u)=u(13.76-2u)(16.42-2u)$&lt;/span&gt; and think that the name of the function is &lt;span class=&quot;math-container&quot;&gt;$V(u)$&lt;/span&gt;, i.e. &lt;span class=&quot;math-container&quot;&gt;$V(u)$&lt;/span&gt; is simply a label for the right-hand side. He also makes the following claim about function notation that agrees with the sentiment of &lt;a href=&quot;https://matheducators.stackexchange.com/a/20866/20719&quot;&gt;your answer to another question&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A primary source of students’ difficulty with function notation is that they only see it where “” could be used just as well. The textbook says they must use () when there really is no need for it. They rarely see function notation used in settings where using it actually enables them to do things that they otherwise could not.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://pat-thompson.net/PDFversions/2020FuncNotation.pdf&quot; rel=&quot;noreferrer&quot;&gt;Yoon and Thompson (2020)&lt;/a&gt; offers several more references on students' and teachers'  difficulties with function notation.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sajka’s (2003) case study involved a 45-minute interview with one high school student, Kasia, who had seen function notation for three years. [The study] showed Kasia moving from thinking &amp;quot;&amp;quot; means “the beginning of a function formula” to thinking of “()” as serving the same role as “”...  Sajka’s report that Kasia understood “()” as the start of a function formula is in line with Thompson’s (1994) report of students thinking a function definition consists of the “()” on the left side, the symbol “=”, and an algebraic expression on the right side. Thompson (1994) also pointed to this way of thinking as behind the common mistake of mismatched letter on the left and letter on the right, such as () = ( + 1)( + 2).&lt;/p&gt;&#xA;&lt;p&gt;Musgrave and Thompson (2014) and Thompson and Milner (2019) shifted focus from students’ to teachers’ understanding of function notation. Musgrave and Thompson (2014) and Thompson and Milner (2019) found that, for many teachers, function notation served as a label or a name for the defining formula rather than a representation of one quantity’s values in relation to another quantity’s values. While students meaning of function notation as a label might be a root of so many reports of students’ difficulties with it (e.g., Carlson, 1998; Dreyfus &amp;amp; Eisenberg, 1982; Vinner &amp;amp; Dreyfus, 1989), it is important to investigate the possibility that teachers hold similar meanings.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yoon and Thompson's study asked teachers in the US and in South Korea to respond to tasks involving function notation, and they give many examples of what I think aligns with your idea of &amp;quot;blurring between functions and variables.&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A large majority of US teachers seemed to think that “” in () is a part of the function name and they were therefore free to use other letters in the function’s defining rule. Thompson (1994, 2013b) suggests a reason for the teachers who used  or  at least one blank in Level 2 and Level 1. Teachers thought of function notation as a four-character symbol that is used in place of the letter “” (Thompson, 2013b). Teachers who filled the blanks with  or  might consider “()” as one symbol because they thought they could replace “()” with “”.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Teacher 1’s response... tells us that she thought () was one inseparable symbol instead of thinking  is a function’s name and  is an independent variable... Although she said  is a function of , she thought  always accompanies with , and viewed () as one entity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Teacher 1 first wrote  = π&lt;sup&gt;2&lt;/sup&gt; , and then added () after . Her final answer was () = π&lt;sup&gt;2&lt;/sup&gt; . She used function notation only to represent the area on&#xA;the left hand side and used  in the function notation and r in the defining rule. Her statement “when I am using function notation I am thinking ()” and “I was just noting the area of the circle which was what I was starting” is consistent with our hypothesis that teachers who used function notation only to represent the area think of () as one symbol, and () is a label for the formula on the defining rule.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One of the study's interesting conclusions is that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;SK data shows that US teachers’ difficulties when reasoning with function notation are not due to epistemological obstacles to understanding meanings of function notation. The SK data suggests that US teachers’ problematic responses are a systemic aspect of mathematics education in the U.S. Thus, it is plausible that US teachers convey problematic meanings to students unintentionally.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;http://pat-thompson.net/PDFversions/2016ThompsonCarlsonCovariation.pdf&quot; rel=&quot;noreferrer&quot;&gt;Thompson and Carlson (2017)&lt;/a&gt;, the authors discuss the concepts that you call &amp;quot;function&amp;quot; and &amp;quot;function of&amp;quot; and argue that many student difficulties stem from an overemphasis in school mathematics on the modern, set-theoretic definition of function and on a static conception of variables. Instead, they argue that there should be more focus on the notion of what you call in &lt;a href=&quot;https://mathoverflow.net/questions/307947/formalizations-of-the-idea-that-something-is-a-function-of-something-else&quot;&gt;this MO question&lt;/a&gt; &amp;quot;variable quantities.&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We argue that ideas of continuous variation and continuous covariation are epistemologically necessary for students and teachers to develop useful and robust conceptions of functions. Put another way, we argue that variational and covariational reasoning are fundamental to students’ mathematical development. We ground this claim in research that highlights difficulties students experience regarding function relationships by not having the ability to reason variationally or covariationally and in research that shows productive shifts in teachers’ and students’ conceptions and uses of function that result from reasoning covariationally.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They review U.S. and Japanese school mathematics textbooks and find that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Our cursory review of 17 U.S. secondary precalculus level textbooks ranging from algebra 1 to precalculus, revealed, consistent with Cooney and Wilson’s (1993) textbook review, that all the textbooks used a correspondence definition of function. The research we have cited in this chapter further supports the fact that U.S. curriculum and instruction are failing to develop students’ quantitative and covariational reasoning abilities, contributing to many weaknesses in students’ conceptions of fundamental mathematical ideas, such as variable, function, and rate of change, that are essential for understanding calculus and modeling dynamically changing phenomena in the sciences and engineering.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Japanese primary mathematics texts have a clear, coherent focus on having students think about quantities whose values vary and about ways that quantities’ values vary together. By high school, Japanese texts’ authors presume that students think with images of variation and covariation and rely on this assumption as a matter of practice. In contrast to Japanese textbooks, many popular U.S. textbooks do not emphasize or support students in conceptualizing quantities and viewing function formulas and graphs as representing how two varying quantities change together. The idea of variable often is presented as representing a single unknown value.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Based on this, I think there is substantial support for the claim that the &amp;quot;function&amp;quot; vs. &amp;quot;function of&amp;quot; issue and the &amp;quot;function&amp;quot; vs. &amp;quot;variable&amp;quot; issue do indeed affect student understanding, and that teachers' actions contribute to the confusion.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-06-19T00:24:15.337" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26558" PostTypeId="1" CreationDate="2023-06-19T19:52:26.080" Score="26" ViewCount="8912" Body="&lt;p&gt;My partner is a PhD student in philosophy and has recently developed a keen interest in learning pure mathematics. I am doing my best to teach her (I'm a pure maths PhD student myself) and it is turning out to be a rewarding experience for the both of us, but having virtually no teaching experience myself I have run into a few difficulties.&lt;/p&gt;&#xA;&lt;p&gt;Some context:&lt;/p&gt;&#xA;&lt;p&gt;She hated maths at school and did quite badly, mainly due to very poor teaching and undiagnosed autism. She wants to gain confidence by proving to herself that she can succeed in maths.&lt;/p&gt;&#xA;&lt;p&gt;She is very intelligent and teaches formal logic to philosophy undergraduates. I think given her logical reasoning abilities she might have a lot of potential in mathematics, but is struggling with the formal language and style of proofs.&lt;/p&gt;&#xA;&lt;p&gt;Currently she is exploring topics such as elementary set theory, complex numbers and an axiomatic approach to understanding the real numbers.&lt;/p&gt;&#xA;&lt;p&gt;I am a little unsure what texts might be best to suggest given her mathematical background, since most people have a very thorough grounding in school-level mathematics (elementary algebra, calculus, graphs etc) before studying these things and she is rather rusty on these topics. Her interest is in pure mathematics, so ideally she would be able to take the fastest route to diving into typical first-year undergraduate pure maths.&lt;/p&gt;&#xA;&lt;p&gt;How might I help her best? Any advice much appreciated :)&lt;/p&gt;&#xA;" OwnerUserId="22521" LastActivityDate="2023-09-16T05:25:03.870" Title="How to teach pure mathematics to a well-educated adult who did badly in maths at school" Tags="|undergraduate-education|mathematical-pedagogy|textbooks|private-lessons|" AnswerCount="14" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26559" PostTypeId="2" ParentId="26558" CreationDate="2023-06-19T23:06:52.267" Score="13" Body="&lt;p&gt;As a person self-studying pure maths right now, I recommend the book &amp;quot;How to think about Analysis&amp;quot; by Lara Alcock since its very readable and the way Lara explains things, the concepts covered are quite digestible + the book has many advice on how to learn and study pure maths. It also does go through set theory notation, reading and understanding proofs, etc. I think she has one on Abstract algebra as well.&lt;/p&gt;&#xA;" OwnerUserId="22504" LastEditorUserId="372" LastEditDate="2023-06-20T15:01:00.313" LastActivityDate="2023-06-20T15:01:00.313" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26560" PostTypeId="2" ParentId="26558" CreationDate="2023-06-20T00:23:48.400" Score="3" Body="&lt;p&gt;Since she teaches logic, this should be accessible:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Set Theory and Logic&lt;/em&gt; by Robert R. Stoll. Corrected (1979) reprint.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is &lt;a href=&quot;https://store.doverpublications.com/0486638294.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;a Dover book&lt;/a&gt; so quite inexpensive.&#xA;Lots of examples and exercises. From a review:&#xA;&amp;quot;Its clarity makes this book excellent for self-study.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rJeOc.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rJeOc.jpg&quot; alt=&quot;Cover&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2023-06-20T13:41:39.070" LastActivityDate="2023-06-20T13:41:39.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26561" PostTypeId="2" ParentId="26558" CreationDate="2023-06-20T09:39:15.480" Score="12" Body="&lt;p&gt;If she is more of a visual learner, there are excellent Youtube channels about mathematics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/c/3blue1brown&quot; rel=&quot;nofollow noreferrer&quot;&gt;3Blue1Brown&lt;/a&gt; is excellent and visually introduces many concepts that are quite scary when looking at the equations.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/channel/UCoxcjq-8xIDTYp3uz647V5A&quot; rel=&quot;nofollow noreferrer&quot;&gt;Numberphile&lt;/a&gt; is another one. This one is more oriented on interesting or surprising things in mathematics, and it is a door to areas of mathematics that otherwise may not have been known by the viewer.&lt;/p&gt;&#xA;&lt;p&gt;There are many more (including actual courses).&lt;/p&gt;&#xA;" OwnerUserId="8076" LastEditorUserId="8076" LastEditDate="2023-06-21T09:22:59.680" LastActivityDate="2023-06-21T09:22:59.680" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26562" PostTypeId="2" ParentId="26558" CreationDate="2023-06-20T14:36:13.640" Score="2" Body="&lt;p&gt;It might not be what you want but I'd recommend starting with the book &amp;quot;Engineering Mathematics&amp;quot; by KA Stroud, it is a standard text in the UK for Engineering undergrads, so is a little bit more slowly and clearly explained for someone who might be a bit rusty/ not actually taking a mathematics degree.&lt;/p&gt;&#xA;" OwnerUserId="22528" LastActivityDate="2023-06-20T14:36:13.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26563" PostTypeId="2" ParentId="26558" CreationDate="2023-06-20T18:17:22.150" Score="8" Body="&lt;p&gt;You might consider that she is experiencing difficulties not because the formal proofs per see are difficult, but because she has not developed intuitions about mathematics. Starting a mathematical education with a focus on set theory leads to a situation where the student is spending most of the time making a colossal effort to prove in a very abstract way something that seems trivial while not knowing why. The New Math movement in the US failed drastically in this approach.&lt;/p&gt;&#xA;&lt;p&gt;Supposing that understanding maths is synonymous with being able to talk to mathematicians and understand why they are interested in a given topic, I think it is important to try not only to grasp formal proofs, but to understand that they are necessary because our regular intuition fails to consider interesting structures that are logically possible.&lt;/p&gt;&#xA;&lt;p&gt;Given all I said, I would suggest going back to geometry. It is a topic where everybody has some intuition about the objects involved and at the same time has plenty of interesting constructions that help motivate the need for proofs. One suggestion would be to use A. P. Kiselev's two volumes &amp;quot;Geometry&amp;quot;. This is a soviet textbook for advanced high school, and has a very professional &amp;quot;proposition-proof&amp;quot; style for euclidean geometry, while trying to introduce more advanced topics, for instance the first book, on plane geometry, will introduce the concept of homothety and similarity transformations (which should help her have a first contact with the notion of a group and a higher level of abstraction) and the second book, on solid geometry, finishes by introducing vectors and having a first discussion on the foundations of geometry.&lt;/p&gt;&#xA;&lt;p&gt;A follow-up could be Coxeter's &amp;quot;Introduction to Geometry&amp;quot;, a book aimed at undergrads. It is divided in four parts, euclidean geometry,  coordinate geometry, foundations and differential geometry. It is not meant to be thorough but to give an introduction to interesting problems in each of these parts of geometry. It has a lot of discussion on symmetry groups and non-euclidean geometry.&lt;/p&gt;&#xA;&lt;p&gt;I must say I'm biased in liking a lot of group theory, but maybe she could profit from this kind of focus. Studying geometry from the ground up with a focus on symmetry is a way of using an intuitive topic, with plenty of room to pick up basic competencies (such as elementary algebra) and introducing a topic that motivates more abstract reasoning (from the symmetries themselves to the abstract concept of group grounded in set theory and algebraic structures). It also has the benefit of leading straight to a discussion of the Erlangen program and a discussion of the foundations of geometry (and maths), something a philosopher might find interesting.&lt;/p&gt;&#xA;&lt;p&gt;Since you mentioned complex numbers, in the same vein I would recommend  I. M. Yaglom's &amp;quot;Complex Numbers in Geometry&amp;quot;, another soviet book for advanced high school. It requires being comfortable with elementary algebra, but introduces complex numbers with a focus on transformations on the plane. It also introduces dual numbers, which I find neat at this level to look at the myriad of interesting objects that are logically plausible, and motivate more abstract reasoning about structures.&lt;/p&gt;&#xA;" OwnerUserId="15277" LastActivityDate="2023-06-20T18:17:22.150" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26564" PostTypeId="2" ParentId="26558" CreationDate="2023-06-20T23:18:35.597" Score="1" Body="&lt;p&gt;What level of logic does she teach/ is she comfortable with?&lt;/p&gt;&#xA;&lt;p&gt;If&lt;/p&gt;&#xA;&lt;p&gt;Propositional logic -&amp;gt; she should learn to be comfortable with predicate logic and basic discrete math.&lt;/p&gt;&#xA;&lt;p&gt;Predicate logic -&amp;gt; most intros to predicate logic are at the equivalent of an intro to proof math course, so she can safely being with any gentle intro pure math course, say Pinters abstract algebra or Axlers LA. Alternatively, work up to metalogic with Sider's logic for philosophy.&lt;/p&gt;&#xA;&lt;p&gt;Metalogic -&amp;gt; this course is roughly equivalent to a first course in mathematical logic, say some natural deduction + completeness and applications. If shes here, she probably has enough mathematical maturity to choose topics at will.&lt;/p&gt;&#xA;&lt;p&gt;Where in particular are her interests? There are several important places where math and philosophy intersect.&lt;/p&gt;&#xA;&lt;p&gt;modal logic -&amp;gt; perhaps she is interested in modal logic? the semantics of modality formed a large part of phil lang discourse in the later part of the century. Van Benthems modal logic would be quite accessible as long as she is comfortable with induction + predicate logic.&lt;/p&gt;&#xA;&lt;p&gt;proof theory -&amp;gt; structural proof theory features in say, Dummett's phil of lang. Also important in phil logic. Very few prereqs, could be done possibly before a metalogic course. Negri/von Plato have a good text.&lt;/p&gt;&#xA;&lt;p&gt;phil math -&amp;gt; she'll want to go with a nice set theory text, maybe Jechs beginner text. Don't bother with naive set theory if you can help it, as knowledge of axiomatic set theory is assumed at higher levels. Some familiarity with analysis might be helpful.&lt;/p&gt;&#xA;&lt;p&gt;formal epistemology -&amp;gt; bayesianism and interpretations of probability were (still are?) quite popular for sometime in epistemology. There are a lot of texts for probability geared for beginners, and many good texts for measure theory.&lt;/p&gt;&#xA;&lt;p&gt;Also an intro computation course (say along the lines of Cutland) could be very helpful in discussions about computational theory of mind. This would lead straight to discussions of Godels incompleteness,an important topic in phil math.&lt;/p&gt;&#xA;" OwnerUserId="20518" LastActivityDate="2023-06-20T23:18:35.597" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26565" PostTypeId="2" ParentId="26558" CreationDate="2023-06-21T07:53:03.307" Score="1" Body="&lt;p&gt;It's best to expose her to different topics at different levels of rigor. You can do set theory and basic algebra rigorously and at the same time also calculus in a more intuitive way. You can then later, when she has made progress with the more basic math, polish up het calculus by doing that rigorously. And when you arrive at that point, her non-rigorous calculus skills can be sufficient to study applied math with.&lt;/p&gt;&#xA;" OwnerUserId="22533" LastActivityDate="2023-06-21T07:53:03.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26566" PostTypeId="2" ParentId="26558" CreationDate="2023-06-21T13:15:03.960" Score="4" Body="&lt;p&gt;I don't think the books of Joel David Hamkins have been proposed yet. Beside posting beautiful answers on Math.SE, he has also published &amp;quot;Lectures on the Philosophy of Mathematics&amp;quot; and &amp;quot;Proof and the Art of Mathematics&amp;quot;, and I don't think one could find better with clarifying the connection between formal philosophy and math.&lt;/p&gt;&#xA;" OwnerUserId="22539" LastActivityDate="2023-06-21T13:15:03.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26567" PostTypeId="2" ParentId="26558" CreationDate="2023-06-21T20:04:51.880" Score="3" Body="&lt;p&gt;Have you considered &lt;a href=&quot;http://eugeniacheng.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eugenia Cheng&lt;/a&gt;' work? I'm currently reading &lt;a href=&quot;https://www.goodreads.com/en/book/show/60658614&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Joy of Abstraction: an Exploration of Math, Category Theory, and Life&lt;/a&gt;, which might interest your partner. Eugenia Cheng has a &lt;a href=&quot;https://www.youtube.com/@TheCatsters&quot; rel=&quot;nofollow noreferrer&quot;&gt;Youtube channel dedicated the Category Theory&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="19663" LastActivityDate="2023-06-21T20:04:51.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26568" PostTypeId="2" ParentId="26508" CreationDate="2023-06-21T22:53:25.790" Score="1" Body="&lt;p&gt;Two great books that do this are the books on &lt;a href=&quot;https://longformmath.com/proofs-home&quot; rel=&quot;nofollow noreferrer&quot;&gt;proof writing&lt;/a&gt; and &lt;a href=&quot;https://longformmath.com/analysis-home&quot; rel=&quot;nofollow noreferrer&quot;&gt;real analysis&lt;/a&gt; by Jay Cummings.&lt;/p&gt;&#xA;" OwnerUserId="22540" LastActivityDate="2023-06-21T22:53:25.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26569" PostTypeId="2" ParentId="26558" CreationDate="2023-06-22T11:22:47.910" Score="4" Body="&lt;p&gt;My experience is that getting the student to believe that they are capable of understanding the material is 80% of the work. Most 'not understanding' comes from a belief that they 'will never be able to understand'.&lt;/p&gt;&#xA;&lt;p&gt;Make them feel safe, link your first few examples to stuff your student knows about and is already utilising the concepts in.&lt;/p&gt;&#xA;&lt;p&gt;eg1: teaching an elementary school teacher about statistics. Ask: How do you know your class is doing well in a subject? (mean grade) Does this mean grade tell you things about each individual student? (how representative is this mean value? - variance, distribution etc).&lt;/p&gt;&#xA;&lt;p&gt;eg2: teaching a aged care worker about mathematical formulas: If you had to say how many beds you'll have free next month, how would you do it? (they start giving you variables) Have &lt;em&gt;them&lt;/em&gt; write down abbreviations for these variables and the calculations they use. Then it's a short step to show that mathematical formulae are just language.&lt;/p&gt;&#xA;&lt;p&gt;Show them that they already know, you'll notice that they will start being playful and inquisitive about the material. Then give them the next level.&#xA;Sometimes it even helps to just repeat: 'This is something that you can understand'&lt;/p&gt;&#xA;" OwnerUserId="22545" LastActivityDate="2023-06-22T11:22:47.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26570" PostTypeId="1" CreationDate="2023-06-22T13:50:46.517" Score="1" ViewCount="220" Body="&lt;p&gt;What are the topics of Calculus which can be useful for competitive level and the students are not exposed to it ?&lt;/p&gt;&#xA;" OwnerUserId="20117" LastActivityDate="2023-06-22T14:05:21.437" Title="Calculus at competitive level or Olympiad level" Tags="|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26571" PostTypeId="2" ParentId="26570" CreationDate="2023-06-22T14:05:21.437" Score="2" Body="&lt;p&gt;On &lt;a href=&quot;https://www.vwo.be/vwo/wedstrijdvragen-per-jaargang-en-ronde/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page&lt;/a&gt;, you find all questions, asked during math olympiads as asked in Vlaanderen (Belgium), already from the start in 1985. The questions are written in Dutch, but there are plenty of translation websites available on the internet.&lt;br /&gt;&#xA;Another well-known competition is the Hungarian &lt;a href=&quot;https://www.komal.hu/verseny/korabbi.e.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kömal&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Have fun :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-06-22T14:05:21.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26572" PostTypeId="1" CreationDate="2023-06-23T12:33:18.697" Score="4" ViewCount="268" Body="&lt;p&gt;Using decimal representation for numbers,&lt;/p&gt;&#xA;&lt;p&gt;we read 10 as ten (not  as one, zero)&#xA;we read 1011 as one thousand eleven (not as  one, zero, one, one digit by digit).&lt;/p&gt;&#xA;&lt;p&gt;But using binary representation, how 10 and 1011 should be read? Is digit by digit the only way?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2023-06-23T21:34:48.993" Title="how to read binary numbers?" Tags="|untagged|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26573" PostTypeId="2" ParentId="26572" CreationDate="2023-06-23T16:27:22.473" Score="-4" Body="&lt;p&gt;There is not solid standardization for this terminology. But one convention is this&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{|c|c|c|}&#xA;\hline &#xA;\text{1} &amp;amp; \text{one} &amp;amp; \text{bit} \\ &#xA;\hline &#xA;\text{10} &amp;amp; \text{two} &amp;amp; \text{crumb} \\  &#xA;\hline &#xA;\text{100} &amp;amp; \text{four} &amp;amp; \text{nibble} \\ &#xA;\hline &#xA;\text{1000}&amp;amp; \text{eight} &amp;amp; \text{byte} \\&#xA;\hline&#xA;\text{10000}&amp;amp; \text{sixteen} &amp;amp; \text {word}\\&#xA;\hline&#xA;\text{100000}&amp;amp; \text{thirty-two} &amp;amp; \text {dword}\\&#xA;\hline&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and this continues as qword, dqword.&lt;/p&gt;&#xA;&lt;p&gt;So 101 could be read as &amp;quot;nibble-bit&amp;quot; and 1010 would be read as &amp;quot;byte-crumb.&amp;quot;  This all feels a little bit like Jabberwocky.&lt;/p&gt;&#xA;&lt;p&gt;One source for this convention &lt;a href=&quot;https://mathworld.wolfram.com/Bit.html#:%7E:text=The%20smallest%20unit%20of%20binary,bits%20are%20called%201%20byte&quot; rel=&quot;nofollow noreferrer&quot;&gt;is this Wolfram page&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Also see these Microsoft references:&#xA;&lt;a href=&quot;https://learn.microsoft.com/en-us/openspecs/windows_protocols/ms-dtyp/f8573df3-a44a-4a50-b070-ac4c3aa78e3c&quot; rel=&quot;nofollow noreferrer&quot;&gt;word&lt;/a&gt; &lt;a href=&quot;https://learn.microsoft.com/en-us/openspecs/windows_protocols/ms-dtyp/262627d8-3418-4627-9218-4ffe110850b2&quot; rel=&quot;nofollow noreferrer&quot;&gt;dword&lt;/a&gt; &lt;a href=&quot;https://learn.microsoft.com/en-us/openspecs/windows_protocols/ms-dtyp/ac050bbf-a821-4fab-bccf-d95d892f428f&quot; rel=&quot;nofollow noreferrer&quot;&gt;qword&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This answer has received some down votes based on the objection that this terminology is used typically when describing memory allocations. For example, a byte is &lt;em&gt;any&lt;/em&gt; eight bit expression: 00000000 through 11111111. This is a fair objection. However, mathematicians often think of integers as the union of their predecessors, so &lt;span class=&quot;math-container&quot;&gt;$$8=\{7,6,5,4,3,2,1,0\}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Also I have see undergraduate computer science majors work as mentors in CS Scratch Clubs to help elementary and middle school children learn. I have seen them use base-two blocks and refer to them as &lt;em&gt;bits, crumbs, nibbles, bytes&lt;/em&gt;. Kids giggled when they made up things like taking nibble out of bite and being left with two crumbs, etc. They had fun with the language and were learning place value. So in spite of the technical objection described above, the approach does have efficacy in teaching mathematics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YOcQ1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YOcQ1.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-06-23T21:34:48.993" LastActivityDate="2023-06-23T21:34:48.993" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26574" PostTypeId="2" ParentId="26572" CreationDate="2023-06-23T19:25:06.660" Score="7" Body="&lt;p&gt;In any discussion of numeral systems, I think it’s important to distinguish between numbers and numerals.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The number whose name is “ten” is a &lt;strong&gt;mathematical object&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The numeral &amp;quot;10&amp;quot; is a &lt;strong&gt;symbol&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;When a symbol is used to signify an object or idea, we often read it by saying the name of the signified. For example, I might read a sign with the symbol &amp;quot;&amp;quot; as &amp;quot;no smoking.&amp;quot; Similarly, without any additional context, I might read the sentence &amp;quot;10 is an even number&amp;quot; as &amp;quot;ten is an even number.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Following this method, if the context is that we are working in base-2, then it would make sense to read the sentence &amp;quot;10 is an even number&amp;quot; as &amp;quot;two is an even number.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;When we want to refer to a symbol itself (often indicated using quotation marks around the symbol), then we often read the name of the symbol, or we describe the symbol. For example, I might read the sentence &amp;quot;'10' has two digits&amp;quot; as &amp;quot;the numeral one-zero has two digits.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;There is another method of reading that is specific to numerals used in place value systems. Its purpose is primarily pedagogical. From left to right, we read each digit, followed by its corresponding place value. I've often heard this called &amp;quot;expanded form.&amp;quot; For example, in the context of base-2, I might read &amp;quot;10&amp;quot; as &amp;quot;one two and zero ones.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Others can probably suggest even more methods of reading numerals. To summarize, here are the three I've described, applied to the numeral &amp;quot;1011&amp;quot; in the context of base-2.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Read the name of the number: &amp;quot;eleven.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Describe the symbol: &amp;quot;one-zero-one-one.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Read the number in expanded form: &amp;quot;one eight, zero fours, one two, and one one.&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-06-23T19:25:06.660" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26575" PostTypeId="1" AcceptedAnswerId="26576" CreationDate="2023-06-24T18:43:15.150" Score="5" ViewCount="471" Body="&lt;p&gt;Long time ago a student asked me what the dimension of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt; is, and I said&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\dim_{\mathbb{Q}}\mathbb{R}=\mathfrak{c}$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{c}$&lt;/span&gt; is the cardinality of the continuum, since if there was a countable basis the real numbers would be countable too, since every real number could be written as a finite linear combination of countably many basis elements over a countable field. I added that we assume the continuum hypothesis. It was a course in linear Algebra I and I then felt that my answer ( I was a tutor) was sufficient. By now I think I should have been more explicit and may be should have given some hints how this could actually be proved. However keeping in mind it was the first semester for the students, does anyone have any suggestions how I  could have answered more appropriately? Many thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="9724" LastEditorUserId="9724" LastEditDate="2023-12-30T10:48:50.747" LastActivityDate="2024-01-02T08:04:26.907" Title="What is the dimension of $\mathbb{R}$ over $\mathbb{Q}$?" Tags="|linear-algebra|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26576" PostTypeId="2" ParentId="26575" CreationDate="2023-06-25T17:00:25.720" Score="9" Body="&lt;p&gt;I remember being confused by this as an undergraduate, or to be more honest, blind to the fact that I held a fundamental misconception that would take years to realise.&lt;/p&gt;&#xA;&lt;p&gt;A student in a first linear algebra course might have had exposure to infinite series in Calculus II and maybe Fourier series in a physics class. Perhaps such a student has done some exploratory reading that  planted the idea of a Hilbert Space such as &lt;span class=&quot;math-container&quot;&gt;$l^2$&lt;/span&gt; and its norm. I know that I held the misconception that the dimension of &lt;span class=&quot;math-container&quot;&gt;$l^2$&lt;/span&gt; was &lt;span class=&quot;math-container&quot;&gt;$\aleph_0$&lt;/span&gt;, perhaps even into graduate school. The prevalent use of infinite sums in Fourier series and &lt;span class=&quot;math-container&quot;&gt;$l^2$&lt;/span&gt; was just too powerful for me to realise otherwise.  My schema was not developed to the extent that I was bothered by the fact that the definition of a basis required finite linear combinations. Perhaps in my enthusiastic undergraduate &amp;quot;curiosity&amp;quot; readings I saw the phrase &amp;quot;Schauder basis.&amp;quot; From my analysis class, I knew that polynomials were dense in &lt;span class=&quot;math-container&quot;&gt;$C^0[a,b]$&lt;/span&gt;, i.e., I knew about the Weierstrass Approximation Theorem, and this seemed consistent with my schema for Taylor series. A continuous function  on &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt; has a nice Taylor series.  I was not ready for my schema to upended. It felt as if everything was consistent with my belief that the countable set of monomials was a basis for &lt;span class=&quot;math-container&quot;&gt;$C^0[0,1]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now if an inquisitive undergraduate in linear algebra were to ask me about the dimension of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;, I would first ask probing questions to reveal their schema for polynomials, infinite series, Fourier series, Stone-Weierstrass Theorem, etc. I would test to see if their schema was flawed, as was mine. I would draw a parallel with the question at hand, namely the dimension of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On another note, a student in a first course in linear algebra class might not have a mental schema needed to appreciate the difference between countably infinite and the cardinality of &lt;span class=&quot;math-container&quot;&gt;$2^{\aleph_0}$&lt;/span&gt;. Perhaps this needs to be developed in the process of answering the question. We might also need some time to develop appreciation for the fact that the cardinality of the family of finite sets of &lt;span class=&quot;math-container&quot;&gt;$\aleph_0$&lt;/span&gt; is still countably infinite.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="8190" LastEditDate="2024-01-02T08:04:26.907" LastActivityDate="2024-01-02T08:04:26.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26577" PostTypeId="2" ParentId="20714" CreationDate="2023-06-26T03:40:52.563" Score="0" Body="&lt;p&gt;You might do well to look at radical enactivism - to see how a person adds up a column of numbers - first single numbers only (0-9), then a column of 2 digits (11-99), and then a column of 3 digits - I met an accountant once who had mastered a column that was a mix of 3 and 4 digits.  The radical enactivists claim that you don't need content whilst thinking, so like someone doing parkour they are going down a column of numbers without retaining the sub-totals...&lt;/p&gt;&#xA;" OwnerUserId="22556" LastActivityDate="2023-06-26T03:40:52.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26578" PostTypeId="2" ParentId="26548" CreationDate="2023-06-27T11:32:22.400" Score="3" Body="&lt;p&gt;I don't think there is a single, central authority that defines a precise content standard of &amp;quot;Algebra I&amp;quot; and &amp;quot;Algebra II&amp;quot; in the US, but the &lt;a href=&quot;https://openstax.org/subjects/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax&lt;/a&gt; textbooks &lt;em&gt;Elementary Algebra&lt;/em&gt; and &lt;em&gt;Intermediate Algebra&lt;/em&gt; mostly align with my memory of the subjects. There seems to be a little variability in terms of the closing topic(s) of Algebra II; the OpenStax site lists sequences and series, other sites list complex numbers or trigonometry, and I personally recall learning about matrices and probability. Practical concerns such as the fact that students typically take geometry between these two courses and have significant cognitive growth over the multi-year period account for some of the topic overlap.&lt;/p&gt;&#xA;&lt;p&gt;An overarching goal of both courses is to build students' capabilities to &lt;strong&gt;manipulate&lt;/strong&gt; algebraic expressions, &lt;strong&gt;solve&lt;/strong&gt; equations, &lt;strong&gt;graph&lt;/strong&gt; functions, and translate word problems into one of the aforementioned three basic tasks. Why are the courses structured the way they are? To illustrate the relevance of and give students more practice carrying out the manipulating, solving, and graphing skills in different contexts. A major theme is that one can manipulate different algebraic expressions to highlight specific features (e.g., transform quadratics into vertex form to find an extremum, factor to identify roots) and use the equation-solving approach, graphing approach, or both to understand expression behavior and solve problems.&lt;/p&gt;&#xA;&lt;p&gt;A drawback of this structure is that the topics can feel disconnected from one another, like lists of &amp;quot;prescribed rules&amp;quot; for manipulating different expressions and different procedures to follow in different contexts. My experience was that all these topics were unified in the subsequent Pre-calculus course which defined some elementary functions and the concept of inverses, developed coherent frameworks for graphing these functions in terms of scaling and translation and for solving equations in terms of some basic properties of equality, and focused the second half of the course on complex numbers and trigonometry. Emphasizing the commonalities among the topics and reinforcing the problem-solving approaches (manipulating, solving, and graphing) are some things you could do to help students at this level as they practice.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2023-06-27T11:32:22.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26579" PostTypeId="2" ParentId="26548" CreationDate="2023-06-27T22:00:30.853" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/26578/20719&quot;&gt;As Steve says,&lt;/a&gt; there is no standard curriculum for  Algebra 1 or Algebra 2 in the United States. However, if you intend to work with public school students in a particular U.S. state, you can refer to the state department of education's website. There, you may be able to find course descriptions, sample syllabi, instructional guidance, etc. that can help you understand the content and structure of the courses as intended by state-level decision makers.&lt;/p&gt;&#xA;&lt;p&gt;For example, here are the chapters on &lt;a href=&quot;https://www.cde.ca.gov/ci/ma/cf/documents/mathfwalgebra1lmgjl.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra 1&lt;/a&gt; and &lt;a href=&quot;https://www.cde.ca.gov/ci/ma/cf/documents/mathfwalgebra2lmgjl.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra 2&lt;/a&gt; from the 2013 California Mathematics Framework. It's due to be revised soon, but there have been several &lt;a href=&quot;https://marketbrief.edweek.org/marketplace-k-12/california-pushes-release-math-framework-2023-long-fight/&quot; rel=&quot;nofollow noreferrer&quot;&gt;delays&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As another example, here are Florida course standards for &lt;a href=&quot;https://www.cpalms.org/PreviewCourse/Preview/20363&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra 1&lt;/a&gt; and &lt;a href=&quot;https://www.cpalms.org/PreviewCourse/Preview/20397&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra 2&lt;/a&gt;, current as of 2023. You can click on each standard for more detailed information and instructional guidance.&lt;/p&gt;&#xA;&lt;p&gt;If you want a non-state-specific reference, College Board also sells &amp;quot;Pre-AP&amp;quot; &lt;a href=&quot;https://pre-ap.collegeboard.org/media/pdf/pre-ap-algebra-1-course-guide.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra 1&lt;/a&gt; and &lt;a href=&quot;https://pre-ap.collegeboard.org/media/pdf/pre-ap-algebra-2-course-guide.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebra 2&lt;/a&gt; courses, and course guides are available online. These courses were developed relatively recently, but the College Board has strong influence on U.S. education, so I expect adoption to continue to grow.&lt;/p&gt;&#xA;&lt;p&gt;Because of the lack of standardization, as well as a host of other reasons, it's hard to give a definitive answer to the question of why these courses are structured the way that they are. What I'd like to do instead is provide some examples of the variety of purposes and conceptions of school algebra that influence curricula.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.bgsu.edu/content/dam/BGSU/nwo/documents/camp/Aug11-2016/ONLYPAGES1and2_conceptionsschoolalgebra_Usiskin.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Usiskin (1988)&lt;/a&gt; identifies four distinct conceptions of school algebra and relates them to different uses of variables. Here is a reproduction of a table from the article.&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;Conception of algebra&lt;/th&gt;&#xA;&lt;th&gt;Use of variables&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;Generalized arithmetic&lt;/td&gt;&#xA;&lt;td&gt;Pattern generalizers (translate, generalize)&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;Means to solve certain problems&lt;/td&gt;&#xA;&lt;td&gt;Unknowns, constants (solve, simplify)&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;Study of relationships&lt;/td&gt;&#xA;&lt;td&gt;Arguments, parameters (relate, graph)&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;Structure&lt;/td&gt;&#xA;&lt;td&gt;Arbitrary marks on paper (manipulate, justify)&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;From a historical perspective, &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-319-59204-6_2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kanbir et al. (2017)&lt;/a&gt; identify six purposes of school algebra in their analysis. Four of the purposes line up with Usiskin's conceptions; there are two that are distinct:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Algebra as preparation for further mathematics and science; and&lt;/li&gt;&#xA;&lt;li&gt;Algebra as a gatekeeper for entry to higher studies.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Debates over the relative importance of different purposes have gone on for more than a century and have influenced several waves of curriculum reform, so it's not uncommon for a course to feel incoherent or aimless. If you have specific students in mind, my recommendation is to go through their curriculum yourself, as well as any instructional resources that might be relevant, and to meet students where they are. That is, find ways to help students make sense of the material in alignment with purposes that are important for them.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-06-27T22:00:30.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26581" PostTypeId="2" ParentId="26558" CreationDate="2023-06-28T10:38:33.813" Score="0" Body="&lt;p&gt;Since your partner is a philosophy student, why not use Russel or Frege as starting point to tackle pure mathematics? The book 'Introduction to mathematical Philosophy' by Russell is excellent at tackling foundational questions of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="22563" LastActivityDate="2023-06-28T10:38:33.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26582" PostTypeId="2" ParentId="26558" CreationDate="2023-06-28T18:26:35.130" Score="1" Body="&lt;p&gt;I have tutored a few adults on this very topic and I have used:&#xA;Mastering Algebra, An Axiomatic Approach (2nd Edition), by Roger W. Oster&lt;/p&gt;&#xA;" OwnerUserId="22565" LastActivityDate="2023-06-28T18:26:35.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26583" PostTypeId="1" AcceptedAnswerId="26585" CreationDate="2023-06-29T07:46:09.017" Score="8" ViewCount="144" Body="&lt;p&gt;I am a psychologist studying mathematical thinking and learning and I have been organizing a monthly math night at a local library. Each math night consists of a short presentation followed by several fun hands-on activities relating to a math topic such as tessellation, probability, game theory, etc.&lt;/p&gt;&#xA;&lt;p&gt;Having done this for some time, I now have numerous sets of materials that I've designed and could potentially be used by other people. The materials are slides to be used for presentations, scripts that tell the organizer what materials are needed and how to run the activities, and in some cases, handouts to be printed and used in the activities.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What would be some good channels or platforms for me to release these materials for free use by others?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="22569" LastActivityDate="2023-06-30T23:45:59.887" Title="Where to distribute free math ed materials for informal settings?" Tags="|primary-education|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26584" PostTypeId="2" ParentId="26558" CreationDate="2023-06-29T20:29:11.873" Score="2" Body="&lt;p&gt;I recently finished reading, and enjoying, Alec Wilkinson's 2022 book:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;A Divine Language:&#xA;Learning Algebra, Geometry, and Calculus at the Edge of Old Age&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I found the book to be quite enlightening and highly readable;&#xA;it often rambles off into wonderful byroads of the philosophy and&#xA;history of mathematics. The author acts in the role of student, rather&#xA;than teacher, which enhanced it value for me.&lt;/p&gt;&#xA;&lt;p&gt;I highly recommend this book.&lt;/p&gt;&#xA;" OwnerUserId="22576" LastActivityDate="2023-06-29T20:29:11.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26585" PostTypeId="2" ParentId="26583" CreationDate="2023-06-30T23:45:59.887" Score="4" Body="&lt;p&gt;I recommend you post them &lt;strong&gt;here&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On StackExchange sites, &lt;a href=&quot;https://meta.stackexchange.com/a/17847/164838&quot;&gt;it's totally acceptable to ask a question and then answer it yourself&lt;/a&gt;, even posting multiple answers.&lt;/p&gt;&#xA;&lt;p&gt;Just write the question that you think someone might Google later when trying to find your materials, then answer it multiple times with the various materials. Or, if you have several categories, you could ask multiple questions and answer them all, but I'd recommend keeping it to as few questions as possible.&lt;/p&gt;&#xA;&lt;p&gt;The possible downside is that StackExchange doesn't really &amp;quot;host&amp;quot; worksheets and printouts, but if you can post them somewhere else like Google Drive and reference them here, this place is designed well to help others find your questions and answers over the medium-to-long-term.&lt;/p&gt;&#xA;&lt;p&gt;As an added bonus, others might post more materials as replies!&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-06-30T23:45:59.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26586" PostTypeId="1" CreationDate="2023-07-01T11:18:27.087" Score="1" ViewCount="84" Body="&lt;p&gt;At the moment a MathArt contest is running about Aesthetic Conformal Image Mapping were individuals and classes can participate:&#xA;&lt;a href=&quot;https://www.freelancer.com/contest/MathArt-Contest-Aesthetic-Conformal-Image-Mapping-ACIM_a-2269317/details&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.freelancer.com/contest/MathArt-Contest-Aesthetic-Conformal-Image-Mapping-ACIM_a-2269317/details&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Goal is to find non trivial closed form expressions of complex, especially conformal functions that deform images in an aesthetic interesting way.&lt;/p&gt;&#xA;&lt;p&gt;Code for Matlab and links for Mathematica and Maple are given and those who have no experience with such Math programs can experiment with G'MIC and G'MIC Online.&lt;/p&gt;&#xA;&lt;p&gt;It is a kind of recursive contest were interesting closed form expressions will be selected, parametrized and included in an open library for aesthetic mathematical image mappings as a stepping stone for upcoming versions of this contests held once or twice a year.&lt;/p&gt;&#xA;" OwnerUserId="22578" LastActivityDate="2023-07-01T11:18:27.087" Title="MathArt contest about Aesthetic Conformal Image Mapping" Tags="|complex-numbers|math-contests|" AnswerCount="0" CommentCount="2" ClosedDate="2023-07-03T06:59:23.487" ContentLicense="CC BY-SA 4.0" />
  <row Id="26589" PostTypeId="1" CreationDate="2023-07-04T08:29:25.210" Score="5" ViewCount="168" Body="&lt;p&gt;I have been going over some applied instances of category theory in Programming, and also by a book by conceptual Mathematics by Lawvrere, and I think an issue of applying category theory to real life, is that there seems to be no point for quite a bit as all structures seem like sets, and it feels like we could just think of maybe functions than functors and so on.&lt;/p&gt;&#xA;&lt;p&gt;A lot of the significance becomes much easier to process, I think, when we start thinking of objects in categories as types.&lt;/p&gt;&#xA;&lt;p&gt;Is there anywhere I can find discussions on the pedagogic value of starting CT with types vs otherwise? Or could someone who is well versed with both approaches share their views on it?&lt;/p&gt;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastActivityDate="2023-07-04T08:29:25.210" Title="Is it better to teach category theory in the background of type theory than set theory?" Tags="|undergraduate-education|category-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26590" PostTypeId="1" CreationDate="2023-07-06T12:31:24.887" Score="0" ViewCount="127" Body="&lt;p&gt;In teaching the high school curriculum Pythagoras is usually bundled with Trigonometry. They might be justified by way of proof of some sort. They are used to solve 2d and 3d geometry problems for unknowns.&lt;/p&gt;&#xA;&lt;p&gt;They are bundled together with the justification that they both are used for funding unknowns in right triangles. I would argue against this pairing on the basis that their similarities are superficial and their differences are more fundamental.&lt;/p&gt;&#xA;&lt;p&gt;In fact, either could be taught first with their own respective prerequisites.&lt;/p&gt;&#xA;&lt;p&gt;What arguments are some arguments for their sequencing?&lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="2083" LastEditDate="2023-07-06T17:07:53.630" LastActivityDate="2023-07-06T17:07:53.630" Title="Pythagoras and Trigonometry sequencing" Tags="|secondary-education|trigonometry|course-sequence|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26591" PostTypeId="2" ParentId="26590" CreationDate="2023-07-06T13:43:18.927" Score="1" Body="&lt;p&gt;I learnt about Pythagoras before I started learning about trigonometry.&lt;br /&gt;&#xA;I have learnt the Pythagoras proof, using uniform triangles, but there is also this very popular proof of the Pythagoras theorem:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gectN.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gectN.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The whole idea is based on the total surface, being equal to the sum of the individual surfaces:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$S(square(A+B)) = S(square(C)) + 4 \cdot S(triangle(A,B))$$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(A+B)^2 = C^2 + 4 \cdot \frac{A \cdot B}{2}$$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A^2 + 2 \cdot A \cdot B + B^2 = C^2 + 2 \cdot A \cdot B$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hence:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$A^2 + B^2 = C^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As you see, no &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt; involved :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-07-06T13:43:18.927" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26592" PostTypeId="1" CreationDate="2023-07-06T18:32:54.747" Score="6" ViewCount="613" Body="&lt;p&gt;I heard that there were some studies on what is the best order for teaching math. But I cannot find any papers (probably my English is too poor to google this paper correctly).&#xA;As I heard idea was: take two groups. For the first group teacher gives theory and examples and after that students start to solve problems. For the second groups teacher firstly gives problems and students try to find a way to solve them. After some attempts teachers shows theory and explains it.&lt;/p&gt;&#xA;&lt;p&gt;Has anyone seen these kind of research?&lt;/p&gt;&#xA;" OwnerUserId="22605" LastEditorUserId="2083" LastEditDate="2023-07-07T05:52:42.500" LastActivityDate="2023-07-07T05:52:42.500" Title="Theory-practice order vs practice-theory" Tags="|reference-request|problem-solving|lecturing|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26593" PostTypeId="2" ParentId="26592" CreationDate="2023-07-06T19:45:07.780" Score="10" Body="&lt;p&gt;A go-to article for me is the following. Short overview article with extensive endnotes to further supporting literature:&lt;/p&gt;&#xA;&lt;p&gt;Clark, Richard, Paul A. Kirschner, and John Sweller. &amp;quot;Putting students on the path to learning: The case for fully guided instruction.&amp;quot; &lt;em&gt;American Educator&lt;/em&gt; 36.1 (2012): 5-11.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Our goal in this article is to put an end to this debate. Decades of&#xA;research clearly demonstrate that for &lt;em&gt;novices&lt;/em&gt; (comprising virtually&#xA;all students), direct, explicit instruction is more effective and more&#xA;efficient than partial guidance. So, when teaching new content and&#xA;skills to novices, teachers are more effective when they provide&#xA;explicit guidance accompanied by practice and feedback, not when they&#xA;require students to discover many aspects of what they must learn. As&#xA;we will discuss, this does not mean direct, expository instruction all&#xA;day every day. Small group and independent problems and projects can&#xA;be effective—not as vehicles for making discoveries, but as a means of&#xA;&lt;em&gt;practicing&lt;/em&gt; recently learned content and skills.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-07-06T19:45:07.780" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26594" PostTypeId="1" CreationDate="2023-07-08T17:22:16.487" Score="3" ViewCount="116" Body="&lt;p&gt;I am seeking example math books that have a glossary.&#xA;I would especially be interested in an example that has a list of symbols,&#xA;a glossary, and an index.&lt;/p&gt;&#xA;&lt;p&gt;I am working on a book (aimed at graduate students and researchers)&#xA;and I'm trying to decide if all three (symbols, glossary, index)&#xA;make sense. I've used all three but not in the same book, and at lower educational levels.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2023-07-08T17:22:16.487" Title="Books with Glossaries" Tags="|textbooks|graduate-education|reference-books|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26596" PostTypeId="1" CreationDate="2023-07-08T18:18:00.717" Score="4" ViewCount="326" Body="&lt;p&gt;I am trying to compile a list of mind-boggling/curious  historical facts in mathematics that will inspire and attract young people (9–11 years old) to the discipline of Mathematics. Do you have one fact to share? Multidisciplinary historical facts are welcome!&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2023-07-11T14:42:52.617" Title="do you know a mind-boggling/curious historical facts that will inspire and attract young people?" Tags="|history-of-math|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26597" PostTypeId="2" ParentId="26596" CreationDate="2023-07-08T21:33:48.473" Score="5" Body="&lt;p&gt;Factoring numbers is usually taught at around that age, and students are taught about prime and composite numbers and how to find all the divisors of a number. It's not too much additional work to discuss perfect numbers and amicable numbers and to demonstrate why 220 and 284 are an amicable pair.&lt;/p&gt;&#xA;&lt;p&gt;The curious historical fact is that after centuries of brilliant mathematicians exploring number theory, it wasn't until 1867 that a sixteen year old boy in Italy, Nicolò I. Paganini, discovered the second smallest pair, that 1184 and 1210 are amicable.&lt;/p&gt;&#xA;" OwnerUserId="22304" LastActivityDate="2023-07-08T21:33:48.473" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26598" PostTypeId="2" ParentId="26596" CreationDate="2023-07-09T05:42:21.960" Score="6" Body="&lt;p&gt;The mathematician Hardy told the following story when he went to visit the mathematician Ramanujan: &amp;quot;I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. &amp;quot;No,&amp;quot; he replied, &amp;quot;it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.&amp;quot; (Indeed: &lt;span class=&quot;math-container&quot;&gt;$1729 = 10^3+9^3 = 12^3+1^3$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;p&gt;It's amazing that Ramanujan was able to come up with this instantly.&lt;/p&gt;&#xA;&lt;p&gt;I am adding a second story.&#xA;In 1939, George Bernard Dantzig arrived late for a graduate-level statistics class. There were two problems written on the board. Because he was late he didn't know they were examples of &amp;quot;unsolved&amp;quot; statistics problems.  Instead he took them for part of a homework assignment, copied them down, and solved them.   He used these solutions as his PhD thesis.  I&lt;/p&gt;&#xA;&lt;p&gt;It's said that if he had known they were unsolvable he wouldn't have tried to solve them.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastEditorUserId="5321" LastEditDate="2023-07-11T14:42:52.617" LastActivityDate="2023-07-11T14:42:52.617" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26600" PostTypeId="2" ParentId="26596" CreationDate="2023-07-11T07:16:46.170" Score="6" Body="&lt;p&gt;I recall being inspired, at about the age of 12, by the widely-known anecdote of how the mathematical talent of the famous mathematician Carl Friedrich Gauss was first recognized at the age of nine. His teacher put his students to work on summing the first one hundred integers, and young Gauss came up with the answer almost immediately: 5050.&lt;/p&gt;&#xA;&lt;p&gt;Our math teacher provided some historical context: Gauss had been born into a working-class family in Brunswick in Northern Germany in 1777. He was the only child of his father's second marriage. His father worked as a bricklayer and gardener, and while he could read, write, and perform simple arithmetic, Gauss's mother was functionally illiterate. From the age of seven Gauss attended a public school: A one-room school house with a schoolmaster in charge of one hundred students. Corporal punishment was common. Students used slate tablets as the use of paper was too expensive (the use of slate tablets remained common in German elementary schools until after World War II).&lt;/p&gt;&#xA;&lt;p&gt;Our teacher did not tell us right away &lt;em&gt;how&lt;/em&gt; Gauss had figured out the result almost instantaneously; instead he challenged us to ponder how he might have done it. I spent an entire afternoon trying to figure it out, and I seem to recall I worked out a formula based on averaging. The math teacher revealed Gauss's (supposed) technique the next day: One can form pairs &lt;span class=&quot;math-container&quot;&gt;$100+1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$99+2$&lt;/span&gt;, etc and in this way generate fifty pairs each summing to &lt;span class=&quot;math-container&quot;&gt;$101$&lt;/span&gt;, thus the final sum is &lt;span class=&quot;math-container&quot;&gt;$50 \times 101 = 5050$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It should be noted that the specifics of this computation seem to be a bit of a modern invention. The earliest biography of Gauss does include this anecdote and claims Gauss himself as the source, but it just mentions that Gauss's teacher tasked his students with summing an unspecified arithmetic series during instruction in arithmetic. It further states that it was the custom at that school that when finished with a task students would drop off their slate tablet at the teacher's desk, one on top of the other. Gauss turned in his slate almost immediately with the words &amp;quot;There it lies&amp;quot; and then sat quietly until the end of class, when the teacher flipped the stack of slate tables around, inspected the top-most slate and to his astonishment found that Gauss had written down the correct answer and nothing else:&lt;/p&gt;&#xA;&lt;p&gt;Wolfgang Sartorius von Waltershausen, &lt;em&gt;Gauss zum Gedächtniss.&lt;/em&gt; Leipzig: S. Hirzel 1856 (&lt;a href=&quot;https://www.google.com/books/edition/Gauss_zum_Ged%C3%A4chtniss/6stUgey3KXUC?hl=en&amp;amp;gbpv=1&quot; rel=&quot;noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;A recent publication has tried to trace how details were added to this story over time as it was retold numerous times:&lt;/p&gt;&#xA;&lt;p&gt;Brian Hayes, &amp;quot;Gauss's Day of Reckoning.&amp;quot; &lt;em&gt;American Scientist&lt;/em&gt;, Vol. 94, No. 3, May 2006, pp. 200-205 (&lt;a href=&quot;https://www.americanscientist.org/article/gausss-day-of-reckoning&quot; rel=&quot;noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="17715" LastActivityDate="2023-07-11T07:16:46.170" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26601" PostTypeId="2" ParentId="16622" CreationDate="2023-07-11T14:51:19.433" Score="-4" Body="&lt;p&gt;As a parent of a middle schooler, I can attest that PEMDAS can be problematic. &#xA;Here's a simple equation as an exception to PEMDAS (not GEMDAS):&lt;/p&gt;&#xA;&lt;p&gt;3+3+3+3x0=?&lt;/p&gt;&#xA;&lt;p&gt;If you multiply as the first order of operation according to PEMDAS rules, then the equation would process like this (3+3+3+(3x0)) – since multiplication and division precede addition and subtraction – giving you an answer of 9.&lt;/p&gt;&#xA;&lt;p&gt;If you group the numbers being added first before multiplying, then the equation would process like this ((3+3+3+3)x0) giving you the answer of 0.&lt;/p&gt;&#xA;&lt;p&gt;I stand by 0. Or am I missing something? My MS student would like to believe the latter ;-)&#xA;" OwnerUserId="22623" LastActivityDate="2023-07-11T14:51:19.433" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26602" PostTypeId="1" CreationDate="2023-07-13T13:19:38.490" Score="6" ViewCount="2015" Body="&lt;p&gt;I teach internal combustion engines theory in a technical school. I have an elementary knowledge of calculus and my students lack even this.&lt;/p&gt;&#xA;&lt;p&gt;I want to intuitively explain them what is the pdV integral, what is an integral anyway and why calculating an integral gives an area.&lt;/p&gt;&#xA;&lt;p&gt;Could you give some hints or suggest any texts?&lt;/p&gt;&#xA;&lt;p&gt;I am an engineer that wants to help his students please don't mind  the naive question&lt;/p&gt;&#xA;" OwnerUserId="15462" LastActivityDate="2023-07-15T22:06:06.503" Title="How to explain that integral calculate areas?" Tags="|calculus|" AnswerCount="7" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26603" PostTypeId="2" ParentId="26602" CreationDate="2023-07-13T13:42:37.930" Score="2" Body="&lt;p&gt;This might take some back and forth, to see what you and your students do know. If you and they understand derivatives, &lt;a href=&quot;https://mathmamawrites.blogspot.com/2012/11/calculus-anti-derivatives-and-area.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this explanation&lt;/a&gt; might help. (If it does, I can copy it over here.) If not, I can explain more.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-07-13T13:42:37.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26604" PostTypeId="2" ParentId="26602" CreationDate="2023-07-13T14:23:47.467" Score="9" Body="&lt;p&gt;If your audience knows literally no calculus at all, then my go-to is to take a simple linear equation, i.e., &lt;span class=&quot;math-container&quot;&gt;$y = 2x$&lt;/span&gt;, plot it, and then compute the area under the triangle it forms with the x-axis (assuming most everyone knows the area of a triangle). Do this from a few different vertical altitudes, and then inductively reason to a formula for said triangle. Once you've done that, you can hand-wave a generalization of the relationship.&lt;/p&gt;&#xA;&lt;p&gt;In the other direction, for derivatives, I take the formula &lt;span class=&quot;math-container&quot;&gt;$y = x^2$&lt;/span&gt;, draw a parabola, and intuit the slope of the tangent line at various integer points. Combining these two pieces allows you to state the fundamental theorem of calculus to an audience with no knowledge beyond elementary algebra in about half an hour.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-07-13T14:23:47.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26605" PostTypeId="2" ParentId="26602" CreationDate="2023-07-13T16:46:00.900" Score="6" Body="&lt;p&gt;First you have to acknowledge  that P&lt;em&gt;V or &lt;span class=&quot;math-container&quot;&gt;$P*\Delta V$&lt;/span&gt; is not really an area! but if you illustrate P(v) in a diagram,  with V =const the numerical value of the area  under V is  the result of P&lt;/em&gt;V, than take V not constant, so you do it in small steps every &lt;span class=&quot;math-container&quot;&gt;$P*\Delta V$&lt;/span&gt; gives you a small rectangle in the diagram , again the numerical value of the area is the result. The integral then sums up over all the small &lt;span class=&quot;math-container&quot;&gt;$\Delta V$&lt;/span&gt; for increasingly small  &lt;span class=&quot;math-container&quot;&gt;$\Delta V$&lt;/span&gt;&#xA;so in reality the integral ist just a sum over very small intevals, if V is not constant.&lt;/p&gt;&#xA;" OwnerUserId="21051" LastEditorUserId="21051" LastEditDate="2023-07-13T17:08:32.773" LastActivityDate="2023-07-13T17:08:32.773" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26606" PostTypeId="2" ParentId="26602" CreationDate="2023-07-13T19:48:56.147" Score="10" Body="&lt;p&gt;This is a really interesting question! A colleague who teaches applied calculus likes to use the notion of &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Growing_degree-day&quot; rel=&quot;noreferrer&quot;&gt;growing degree days&lt;/a&gt;&lt;/strong&gt;. Many of his students are from farms that grow corn and he finds this example motivating. It is completely analogous to your situation.&lt;/p&gt;&#xA;&lt;p&gt;The beginning of the discussion is a graph where the vertical axis is temperature and the horizontal axis is time. The area units of a rectangle are of course &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\hbox{degrees}\cdot\hbox{time}$&lt;/span&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;He says it is fun to point out that &amp;quot;you cannot add apples and oranges, but guess what, you can multiply apples and oranges!&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-07-13T22:25:32.487" LastActivityDate="2023-07-13T22:25:32.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26607" PostTypeId="2" ParentId="26602" CreationDate="2023-07-14T00:17:49.550" Score="2" Body="&lt;p&gt;Chapter 1 of &lt;a href=&quot;https://www.gutenberg.org/files/33283/33283-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus Made Easy&lt;/a&gt; by Thompson is the right place to start. It explains what the integral symbol means. Then move on to Chapter 19, which explains why integrals are useful for calculating area.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-07-14T00:17:49.550" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26610" PostTypeId="1" CreationDate="2023-07-14T20:17:32.763" Score="2" ViewCount="381" Body="&lt;p&gt;I'm trying to put together some thoughts on the importance of a strong college precalculus sequence (mainly I'm thinking College Algebra, where much of my experience is) for addressing socioeconomic inequality in the United States. While I'm curious to hear the thoughts of matheducators.stackexchange, I'm mainly interested in finding published writing on the subject.&lt;/p&gt;&#xA;&lt;p&gt;The college precalculus sequence is the gateway for many students who were under-served by their high school mathematics courses to get into a wide variety of STEM and STEM-adjacent fields. A strong college precalculus program can help lessen the negative impact of a weak high school program, and a weak college precalculus program can cut some students off from STEM almost entirely.&lt;/p&gt;&#xA;&lt;p&gt;However, in my experience, there are a wide variety of factors that negatively impact the effectiveness of the college precalculus sequence:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Students are often encouraged to take courses at too high a level for maximum benefit&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Early precalculus courses often don't come with credit&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Courses try to cover a year's worth of material (in high school) in only a semester (in college)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematics professors are often very far removed from the experience of someone struggling with algebra&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Where can I read more about:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The role the college precalculus sequence plays in socioeconomic feedback loops, and&lt;/li&gt;&#xA;&lt;li&gt;The ways college precalculus sequences fail to meet the needs of their students?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15671" LastEditorUserId="2083" LastEditDate="2023-07-16T16:11:55.370" LastActivityDate="2023-07-26T14:27:40.200" Title="Sources on inequity in precalculus sequence" Tags="|undergraduate-education|algebra|precalculus|usa|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26611" PostTypeId="2" ParentId="26610" CreationDate="2023-07-14T21:00:57.070" Score="8" Body="&lt;p&gt;Here's a hair-raising article I sometimes use as a touchstone:&lt;/p&gt;&#xA;&lt;p&gt;Kenschaft, Patricia Clark. &amp;quot;Racial equity requires teaching elementary school teachers more mathematics.&amp;quot; &lt;em&gt;Notices of the AMS&lt;/em&gt; 52.2 (2005): 208-212.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Flitting back and forth from the richest to the poorest districts in&#xA;the state [New Jersey] convinced me that the mathematical knowledge of&#xA;the teachers was pathetic in both. It appears that the higher scores&#xA;in the affluent districts are not due to superior teaching in school&#xA;but to the supplementary informal “home schooling” of children.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It should be noted that a rather aggressive trend by U.S. colleges in the last decade has been to eliminate courses that remediate/repeat those in &amp;quot;weak high school program[s]&amp;quot; that don't come with credit (alleviating OP's bullet #2), at the expense of pushing those students into higher-level courses instead (possibly worsening bullets #1 and #3).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www1.cuny.edu/mu/forum/2023/01/12/cuny-ends-traditional-remedial-courses/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www1.cuny.edu/mu/forum/2023/01/12/cuny-ends-traditional-remedial-courses/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-07-26T14:27:40.200" LastActivityDate="2023-07-26T14:27:40.200" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26613" PostTypeId="2" ParentId="26602" CreationDate="2023-07-15T12:51:25.050" Score="3" Body="&lt;p&gt;A practical similar situation is in the US nuclear navy.  When I was in, there at nuclear power school, for enlisted men, the curriculum was all algebra and &lt;strong&gt;the only calculus was &lt;em&gt;graphical calculus&lt;/em&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;See, for example, comments in thread:  &lt;a href=&quot;https://www.reddit.com/r/NavyNukes/comments/te7w78/questions_about_schooling/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.reddit.com/r/NavyNukes/comments/te7w78/questions_about_schooling/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Derivative is slope.  Integral is area under the curve.  It is still quite useful and needed, because of the need to graph and understand reactor transients.  Rod height, reactivity, reactivity addition rate (needed because the startup rate uses both excess reactivity AND reactivity addition rate), startup rate (log power rate of change) and power.  They have a long history of teaching this way [any time you want solid data on entrance testing and/or training, look at the military, because of their numbers...massively better than the typical tiny sample sizes and confounding variables in ed pedagogy academic articles] and it works well.&lt;/p&gt;&#xA;&lt;p&gt;And no, you absolutely should NOT explain/justify why the graphical methods work.  This is very basic awareness you are doing.  You are already past the standard for the students.  And should not be derailed with theoretical justifications.&lt;/p&gt;&#xA;&lt;p&gt;The one slight justification I would give the kids is that derivatives are like distance/speed/acceleration, WHEN YOU DRIVE.  This will connect to their course and daily experience.  And to this day, is honestly how I intuitively think about derivatives (rate of change, rate of change of rate of change).&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;And FWIW, I really quite liked this graphical approach (actually graph the functions, by hand above each other) and had not used it myself during my training.  But it is quite often that a simpler mathematical method may be more intuitive for understanding physical effects.&lt;/em&gt;  Vectors are better than tensors.  Scalars are better than vectors.  Algebra is better than calculus, which is better than ODEs, which are better than PDEs.  And single variable is better than multivariable.  And by &amp;quot;better&amp;quot;, I mean in terms of ease and intuition...not rigor.  So consider how it is more powerful to have an intuitive understanding that pressure drop in a hydraulic circuit increases with more bends in the pipe (&amp;quot;adding head losses&amp;quot;) than saying &amp;quot;uh, I need to run off and do Bernoulli's equation&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  You still haven't given me a good reason why you need (or especially want) to push some calculus onto the auto shop kids. I would be very leery of doing it, if it gets too hard or is a turnoff.  There are places where a more algebraic or even arithmetical approach is still beneficial to understanding the mechanical components (like adding head losses in a hydraulic circuit, like adding resistances in an electrical circuit, like parallel versus series, etc.), and you might take the lower hurdle challenge as a win.  I also remember well an officer sitting next to me (and we DID have real calculus for officers) who was intensely turned off by all the math and higher concepts in thermodynamics...although he loved working on his car...and was expecting to love the heat cycle stuff.&lt;/p&gt;&#xA;" OwnerUserId="22637" LastEditorUserId="22637" LastEditDate="2023-07-15T13:00:05.033" LastActivityDate="2023-07-15T13:00:05.033" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26614" PostTypeId="1" CreationDate="2023-07-15T15:50:43.683" Score="1" ViewCount="269" Body="&lt;p&gt;&amp;quot;Can you explain the distinctions between graphs, diagrams, and charts, and provide definitions for each of these concepts? Specifically, is every graph considered a diagram? Are graphs exclusively reserved for functions? Is it accurate to refer to an ellipse drawing as a graph, diagram, or chart?&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="498" LastEditDate="2023-07-15T23:05:25.847" LastActivityDate="2023-07-18T09:30:16.100" Title="Are there differences between graphs, diagrams and charts?" Tags="|definitions|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26615" PostTypeId="2" ParentId="26602" CreationDate="2023-07-15T19:46:29.653" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Why calculating an integral gives an area?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Consider sweeping out an area, either when paving or playing with your Zen toy:&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SXKMJ.png&quot; height=&quot;240&quot;&gt;&lt;/th&gt;&#xA;&lt;th style=&quot;text-align: left;&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3PjrI.jpg&quot; height=&quot;240&quot;&gt;&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Credit: &lt;a href=&quot;https://www.acmesand.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Acme Sand &amp;amp; Gravel&lt;/a&gt;&lt;/td&gt;&#xA;&lt;td style=&quot;text-align: left;&quot;&gt;Credit: &lt;a href=&quot;https://www.instagram.com/yukikawae/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yuki Kawae&lt;/a&gt;&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;When you pull the screed or rake a little bit — call the displacement &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; —  you sweep out a little bit of area &lt;span class=&quot;math-container&quot;&gt;$dA$&lt;/span&gt; equal to the length of the screed or rake times &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;. Call the length &lt;span class=&quot;math-container&quot;&gt;$L(x)$&lt;/span&gt;. Then we have:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ dA = L(x) \; dx \,.$$&lt;/span&gt;&#xA;Usually rakes and screed do not change their lengths, so that &lt;span class=&quot;math-container&quot;&gt;$L(x)$&lt;/span&gt; would be constant. But we could always swap out one length for another. In fact, there's no real difficulty in imagining someone inventing an adjustable length screed or rake, so that a continuously changing length seems possible. Controlled by a computer and operated by a robot, even. In any case we can sum up the bits of area, indicated by the formal symbol &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt;, an integral-sign S for &amp;quot;Sum&amp;quot;, and obtain an expression for the total area swept out as follows:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ A = \int dA = \int L(x) \; dx \,.$$&lt;/span&gt;&#xA;One also sees that the rate of change of &lt;span class=&quot;math-container&quot;&gt;$A=A(x)$&lt;/span&gt; as a function of the leading edge (the position &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; at which the screed or rake is) is given by the length:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ {\text{change in $A$} \over \text{change in $x$}} = {dA \over dx} = L(x) \,.$$&lt;/span&gt;&#xA;In terms of calculus, we say that &lt;span class=&quot;math-container&quot;&gt;$A(x)$&lt;/span&gt; is &amp;quot;an integral (or antiderivative) of the function &lt;span class=&quot;math-container&quot;&gt;$L(x)$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In terms of the graph of a positive function &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;, the screed is represented by a vertical line from the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; axis to &lt;span class=&quot;math-container&quot;&gt;$y = L(x)$&lt;/span&gt;, which possibly has a different length for each value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. (Normally, math teachers call this length the &amp;quot;height&amp;quot; of the graph.) The integral then represents the accumulation of area swept out by this length (or &amp;quot;height&amp;quot;, or &amp;quot;screed&amp;quot;, if you want to keep to the metaphor), starting at some position &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; and ending at another position &lt;span class=&quot;math-container&quot;&gt;$x=b$&lt;/span&gt;.  For pedagogical reasons we denote this in the following way (and it has become the dominant traditional notation in calculus):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ A = \int_a^b L(x) \; dx \,,$$&lt;/span&gt;&#xA;where the integral and the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in it represent summing all the bits of area for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x=b$&lt;/span&gt;.  It is called &amp;quot;the integral from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Now at this point, mathematicians tend to make a fuss, for both formal and pedagogical reasons, about the notation. &lt;span class=&quot;math-container&quot;&gt;$A=\int L(x)\,dx$&lt;/span&gt; versus &lt;span class=&quot;math-container&quot;&gt;$A=\int_a^b L(x)\,dx$&lt;/span&gt;. Whether we can write &lt;span class=&quot;math-container&quot;&gt;$A(x) = \int_a^x L(x)\,dx$&lt;/span&gt;, or we have to write &lt;span class=&quot;math-container&quot;&gt;$A(x) = \int_a^x L(t)\,dt$&lt;/span&gt; with some new made-up variable &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, or just write &lt;span class=&quot;math-container&quot;&gt;$A(x) = \int L(x)\,dx$&lt;/span&gt;. Many of these issues and reasons will seem irrelevant to your students, if they don't know calculus and do not have to use it to find formulas. However, one does have to know what the integral &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; represents, the completed process (such as the total area from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;) or the process in progress (the accumulated area from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; up to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;). Both things might come up in the same situation. Whatever notation you pick, I'd suggest supporting with a verbal description every time it's used (or at least in every paragraph).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is the &lt;span class=&quot;math-container&quot;&gt;$p\,dV$&lt;/span&gt; integral?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;I'm guessing that &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is pressure and &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is volume,&#xA;and the integral has to do with the work done by the expansion of a volume of an ideal(?) gas.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;A little bit of work, like a bit of area, is equal to the force &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; (applied in the direction of motion) times a bit of displacement &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;. Like the length of the screed, that force may vary. The total or net work is the sum of the bits of work:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ W = \int F \; dx \,.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now in a volume of gas, the force on the surface is equal to the pressure &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; times the surface area &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. To take a cylindrical volume an example, suppose only one end can move as below. It does not really matter the shape of the cross section, only what its area is. If the end has area &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, then the force on it will be &lt;span class=&quot;math-container&quot;&gt;$F = p\,A$&lt;/span&gt;. If the end moves a little bit &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;, then the volume changes by &lt;span class=&quot;math-container&quot;&gt;$dV = A\, dx$&lt;/span&gt;. Thus the bit of work done is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$dW = F\; dx = p \, A \; dx = p \; dV \,.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/2lwYY.png&quot; width=&quot;350&quot;&gt;&#xA;&lt;p&gt;As the volume changes from one value to another, the accumulation of the bits of work is given by the integral:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$W = \int dW = \int p \; dV \,.$$&lt;/span&gt;&#xA;One may put numbers/letters on the integral sign, &lt;span class=&quot;math-container&quot;&gt;$\int_a^b$&lt;/span&gt;, to denote the beginning and end of the change in volume.&#xA;In this example we can go one step further.&#xA;First, &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is constant. Second, the sum of a constant times any number of bits can be computed by summing the bits and then multiplying by the constant.&#xA;This is represented by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$W = \int p \; dV = p \int dV = p \, \Delta V \,,$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\int dV = \Delta V$&lt;/span&gt; is the net change in volume.&lt;/p&gt;&#xA;&lt;p&gt;Now there's a remarkable thing. The example seems very simplistic: The face that moves is flat. It doesn't change area. And the pressure is constant.&#xA;What about an expanding sphere or other shape? Most of what was said carries over without much change.  The main difference, conceptually, is that we have to deal with the curved surface.  To connect the curved surface with the cylinder example, break the surface up into little bits. Each bit is virtually flat and can be thought of as the end of cylinder that expands as above.&lt;/p&gt;&#xA;&lt;p&gt;[Students would be right to doubt this, and it can be justified. Tell them that three semesters of calculus should clear it up in a jiffy. OTOH, it is the same idea as dividing a graph up into little bits of height times &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;.]&lt;/p&gt;&#xA;&lt;p&gt;In any case, the pressure leads to a force normal to the surface and parallel to the direction of expansion, just as in the cylinder. The bit of work done on each bit of the surface is &lt;span class=&quot;math-container&quot;&gt;$dW = p\,dV$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$dV$&lt;/span&gt; is the change in volume swept out by the bit of surface expanding. If we sum over all the bits of surface and all the bits of expansion, we get &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; times the net change in volume, assuming &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; remains constant.&lt;/p&gt;&#xA;&lt;p&gt;[To break it down closer to multivariable terms, which may be unnecessary:&#xA;Let &lt;span class=&quot;math-container&quot;&gt;$dS$&lt;/span&gt; denote the area of each bit of surface, which plays the role that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; played above. The expansion &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; is normal to this surface. So the surface sweeps out a bit of volume equal to &lt;span class=&quot;math-container&quot;&gt;$dV = dS\,dx$&lt;/span&gt;. The force exerted in the direction of expansion is &lt;span class=&quot;math-container&quot;&gt;$F=p\,dS$&lt;/span&gt;. The bit of work done is then &lt;span class=&quot;math-container&quot;&gt;$dW = F\,dx = p\,dS\,dx = p\,dV$&lt;/span&gt;.&#xA;Then the sum over the bits of surface &lt;span class=&quot;math-container&quot;&gt;$dS$&lt;/span&gt; and over the bits of displacement &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; may be represented as a nested surface and scalar integral: &lt;span class=&quot;math-container&quot;&gt;$\int\mkern-6mu\int\left(\int p \, dx\right)dS$&lt;/span&gt;. That may go too far, but they used surface integrals in my physics class before we barely knew what an integral was.]&lt;/p&gt;&#xA;&lt;p&gt;Well, that's about it. I don't have a good answer to,&#xA;&amp;quot;What is an integral anyway?&amp;quot;&#xA;There's a sense in which nobody knew what it was anyway,&#xA;since it has had to be redefined several times.&#xA;If we think we know now, it might be because we haven't yet run into the next problem that will reveal a deficiency in its definition.&#xA;The applications above show one aspect of what it is — an accumulation of little bits — probably the only one of interest to the OP's students.  So maybe it's enough.  A hint of how to make &amp;quot;little bits&amp;quot; more precise is given below, but it's done more with pictures than with algebra.  It didn't seem necessary to the main answers.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Addendum: On &amp;quot;little bits&amp;quot; arguments&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Frankly, I think smart students question or doubt these arguments. For many, math has been taught in a way that to them seems bogus. They've learned to accept (and memorize) whatever the teacher says and use it as instructed.  But face it: Nobody thinks the height of a little bit of a nonconstant graph is a unique and well-defined number. One end is higher than another. Whatever value you choose, you going to think that it's probably not exactly right but maybe it's close enough. Just like when making something and you need to cut something to a length.&lt;/p&gt;&#xA;&lt;p&gt;But here's what one can show in calculus. First, there are bounds on how far wrong your choice of height can be: Each little bit has a maximum and a minimum height. When you sum up the little bits, there will still be an error. Again we can bound how bad the error, based on the minimum and maximum possible heights.  Here is the key: No matter what the errors are, the worst-case error gets smaller if the little bits are divided into smaller bits; further, the worst-case error can be made as small as we please. It follows there is a unique number that the sum for anyone's choices for heights will get close to. That number is called the integral.&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/QYWs8.png&quot; width=&quot;300&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/80L2p.png&quot; width=&quot;300&quot;&gt;&#xA;&lt;p&gt;In the figures above (based on Newton's pre calculus book, &lt;em&gt;Principia&lt;/em&gt;, Bk. I, Lemma III), the yellow rectangles show the difference between the maximum and minimum choices of the heights of the bits. The worst-case error is bounded by the sum of the areas of the yellow rectangles. The change the figure from the left to the right shows what happens if the third bit is divided. The large rectangle on the left becomes two smaller rectangles on the right. In effect, the large was divided into four, and two of the pieces were thrown away. As each bit is divided again and again, the yellow error bound get smaller and smaller:&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/KqewP.png&quot; width=&quot;300&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/54b6h.png&quot; width=&quot;300&quot;&gt;&#xA;&lt;p&gt;One can also see the areas of the rectangles given by the minimum heights and maximum heights get closer to the area under the curve.&lt;/p&gt;&#xA;&lt;p&gt;I guess it's difficult for math people not to give math explanations. But at least I put it in an appendix. And everyone knows not to read the appendix. But as I said at the beginning of the appendix, just saying, &amp;quot;Trust me, it works,&amp;quot; contributes to the appearance of bogosity in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2023-07-15T22:06:06.503" LastActivityDate="2023-07-15T22:06:06.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26616" PostTypeId="1" AcceptedAnswerId="26617" CreationDate="2023-07-16T17:07:52.133" Score="7" ViewCount="2319" Body="&lt;p&gt;A lot of students complain about &amp;quot;never being shown that before&amp;quot;.  What's the idea called when you test multiple concepts or one or two new ones along with some old ones in a word problem, for example?  I'm trying to figure out how to articulate it so I can let the students know ahead of time that I do this during exams.&lt;/p&gt;&#xA;&lt;p&gt;Suppose, for example, I give a word problem asking to maximize profit, a calculus concept and application they're being tested on (finding the max/min of a profit function).  They know that they need to come up with a profit function if one is not provided.  But the way I describe how to get the revenue and cost functions is not direct either - I give them enough info to determine those two functions.  And perhaps in the way that I give them that info is not like any previously assigned problems but nevertheless all they need to do is some algebra to figure out the linear or quadratic cost or revenue functions.  Of course, they also need to determine a suitable domain which again I may bury in the problem somehow.&lt;/p&gt;&#xA;&lt;p&gt;What's this idea called that I'm doing?  Applying calculus concepts to novel situations or ...?&lt;/p&gt;&#xA;" OwnerUserId="9816" LastEditorUserId="2083" LastEditDate="2023-07-17T06:17:40.673" LastActivityDate="2023-08-05T19:07:02.403" Title="What's it called when multiple concepts are combined into a single problem?" Tags="|terminology|test-design|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="26617" PostTypeId="2" ParentId="26616" CreationDate="2023-07-16T19:39:29.987" Score="15" Body="&lt;p&gt;The answer I am about to give is a little tangential, but I think that it will help to answer the question &amp;quot;How do I communicate to &lt;em&gt;students&lt;/em&gt; that the problem I am setting for them is going to require the application of several ideas, perhaps in a novel context.  My approach is related to &lt;a href=&quot;https://cft.vanderbilt.edu/guides-sub-pages/blooms-taxonomy/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bloom's taxonomy of knowledge&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hPEIk.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hPEIk.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;When I write questions (exam questions, homework questions, worksheet questions, etc), I try to put them into one of three categories:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Recall:&lt;/strong&gt; These questions sit at the bottom of the taxonomy (roughly, &amp;quot;remember&amp;quot; and &amp;quot;understand&amp;quot;).  They ask students to remember facts, regurgitate definitions, or perform very simple computations.  For example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Determine the derivative of the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, given by &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^3$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Evaluate the indefinite integral &lt;span class=&quot;math-container&quot;&gt;$\int \sin(t)\,\mathrm{d}t$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Complete the definition by filling in the blank:  Let &lt;span class=&quot;math-container&quot;&gt;$f : I \to \mathbb{R}$&lt;/span&gt; be a function, and let &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; be a point in the interior of &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;.  The &lt;em&gt;derivative&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is ____, assuming that this limit exists.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note that this kind of question should be answerable almost instantaneously (either you know the answer or you don't).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Analysis:&lt;/strong&gt;  These questions sit at the middle of the pyramid (&amp;quot;analyze&amp;quot; and &amp;quot;apply&amp;quot;).  They ask students to complete more complicated computations, solve equations, compare and contrast different ideas, and so on.  Typical questions are&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Evaluate the definite integral &lt;span class=&quot;math-container&quot;&gt;$\int_{0}^{1/2} \sin(\pi x)\,\mathrm{d}x$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Explain the difference between a definite integral and an indefinite integral.&lt;/li&gt;&#xA;&lt;li&gt;Determine the maximum value of the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, given by &lt;span class=&quot;math-container&quot;&gt;$f(x) = 3x^3 - 4x + 6$&lt;/span&gt; on the interval &lt;span class=&quot;math-container&quot;&gt;$[-2, 1]$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;An open topped box is to be made by removing identical squares from each corner of a rectangle of cardboard which measures &lt;span class=&quot;math-container&quot;&gt;$24 \times 18$&lt;/span&gt; inches (see the picture). Determine the size of square which must be removed in order to maximize the volume of the box.  [INSERT PICTURE HERE]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Synthesis:&lt;/strong&gt;  These questions are the kind asked about in the question, and try to hit the top of the taxonomy (evaluate and create).  These questions are often beyond the level that I would expect students to be able to complete on their own (especially in an exam setting).  I usually assign these kinds of problems as homework, and encourage collaboration.  That said, there are times when they might be appropriate on an exam.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The force of gravity on the moon acts to accelerate objects towards the center of the moon at a constant rate of about &lt;span class=&quot;math-container&quot;&gt;$-1.6$&lt;/span&gt; m/s&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;.  A golf ball is hit at an angle &lt;span class=&quot;math-container&quot;&gt;$45^\circ$&lt;/span&gt; above horizontal with an initial velocity of &lt;span class=&quot;math-container&quot;&gt;$200$&lt;/span&gt; m/s.  Determine the maximum height that the ball reaches.  [Note that this question would be given &lt;em&gt;before&lt;/em&gt; discussing multivariable calculus—the students need to figure out for themselves that the velocity is &amp;quot;split&amp;quot; between a vertical and horizontal component.]&lt;/li&gt;&#xA;&lt;li&gt;Recall that the cosine and sine of an angle are defined to be the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; coordinates (respectively) of the point where the angle intersects the unit circle.  Analogously, define the &amp;quot;square cosine&amp;quot; (&lt;span class=&quot;math-container&quot;&gt;$\operatorname{sqcos}$&lt;/span&gt;) and &amp;quot;square sine&amp;quot; (&lt;span class=&quot;math-container&quot;&gt;$\operatorname{sqcos}$&lt;/span&gt;) of an angle &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; to be the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; coordinates of the point where the angle &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; intersects a square with vertices at the points &lt;span class=&quot;math-container&quot;&gt;$(1,0)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(0,1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(-1,0)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(0,-1)$&lt;/span&gt;.  Give formulas for &lt;span class=&quot;math-container&quot;&gt;$\operatorname{sqcos}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\operatorname{sqsin}$&lt;/span&gt; in terms other known functions (polynomials, the usual trigonometric functions, exponentials and logarithms, etc).&lt;/li&gt;&#xA;&lt;li&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f: I \to \mathbb{R}$&lt;/span&gt;, and let &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; be in the interior of &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;.&#xA;The &lt;em&gt;symmetric derivative&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is defined to be&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \lim_{h\to 0} \frac{f(x+h) - f(x-h)}{2h}, $$&lt;/span&gt; assuming that this limit exists.  Show that if &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differential at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; (in the usual sense), then the symmetric derivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; exists at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, and has the same value as the usual derivative.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In communicating with students, I explicitly tell them what kind of question they are answering.  That is, I explicitly use language like &amp;quot;This is a recall question,&amp;quot; or &amp;quot;This is a synthesis question—you might have to think about material from many different sections of the text in order to get to an answer.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So... to (kind of) answer the question:  the kinds of problems described in the original question likely fall at the high end of &amp;quot;analysis&amp;quot;, or at the low end of &amp;quot;synthesis&amp;quot; (depending on how much students are supposed to create on their own, as opposed to just compile together a bunch of ideas in the process of solving the problem).&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2023-07-19T12:34:09.417" LastActivityDate="2023-07-19T12:34:09.417" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26618" PostTypeId="2" ParentId="26616" CreationDate="2023-07-16T23:30:13.047" Score="4" Body="&lt;p&gt;Many textbooks simply identify these as &amp;quot;Applications&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, the more bits and pieces that need to be pulled in and synthesized for a problem, the more challenging it will be. Some books mark increasing difficulty of exercises in this way in some fashion (often A-B-C level exercises or the like).&lt;/p&gt;&#xA;&lt;p&gt;The general idea of practicing old skills interleaved with new ones is called &lt;a href=&quot;https://en.wikipedia.org/wiki/Varied_practice&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Mixed Practice&amp;quot;&lt;/a&gt; (or &amp;quot;Varied Practice&amp;quot;), but that more often refers to a series of small, separate exercises on a variety of skills.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-07-17T15:30:20.980" LastActivityDate="2023-07-17T15:30:20.980" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26621" PostTypeId="1" CreationDate="2023-07-17T09:10:34.613" Score="0" ViewCount="174" Body="&lt;p&gt;For example, consider &lt;span class=&quot;math-container&quot;&gt;$\lim_{x \to -\infty} \frac{x}{e^x}$&lt;/span&gt;,  what would constitute as a good justification that the limit diverges too infinity?&lt;/p&gt;&#xA;&lt;p&gt;It's pretty easy to justify convergent limits rigorously, because one resort to standard results of limits or l'hopital rule, which are all rigorous steps to calculate the limits out. But, unclear to me for divergent.&lt;/p&gt;&#xA;" OwnerUserId="14627" LastEditorUserId="14627" LastEditDate="2023-07-17T11:52:41.227" LastActivityDate="2023-07-18T17:18:45.217" Title="What would constitute as a good justification of why a divergent limit is divergence for highschool teaching?" Tags="|limits|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26622" PostTypeId="2" ParentId="26621" CreationDate="2023-07-17T12:33:36.250" Score="3" Body="&lt;p&gt;You can rewrite &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to -\infty} \frac{x}{e^x}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to \infty} -x e^x$&lt;/span&gt; by &amp;quot;reversing the x-axis&amp;quot;. i.e. making the &lt;span class=&quot;math-container&quot;&gt;$x\to -x$&lt;/span&gt; substitution. Then things should be pretty straightforward as both &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; to go &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The variant that requires more work is &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \lim_{x\to +\infty} \frac{x}{e^x}$&lt;/span&gt;. I usually go with the heuristic &amp;quot;exponentials grow faster than polynomials&amp;quot; for a non-rigorous explanation for that. (You know, with the appropriate caveats about the base and once you have already talked about &lt;span class=&quot;math-container&quot;&gt;$x^n/e^x$&lt;/span&gt; using L'Hopital's rule.)&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2023-07-17T12:33:36.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26623" PostTypeId="2" ParentId="26614" CreationDate="2023-07-18T09:30:16.100" Score="2" Body="&lt;p&gt;At the elementary school, level graphs include: bar graphs, pie graphs, picture graphs, line graphs,  and histograms.  Only the line graphs represent functions.  For example, a bar graph might show favorite colors, where 5 children like pink, 5 children like turquoise, and 2 children like green.  This a relation that is not a function.  Similarly, high school students who look at graphs on the x-y axis are told to use the vertical line test to determine whether a graph shows a function. Many show a relation that is not a function.&lt;/p&gt;&#xA;&lt;p&gt;Diagrams at the elementary  school level include simple geometric figures including a circle, square, etc.  They might also include a simple street map where the student has to figure out the shortest path from A to B. At this level we would not call a chart a diagram.&#xA;&lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED623687.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;This research paper&lt;/a&gt; asked secondary school teachers what diagrams are and what are they used for. They concluded:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Diagrams are succinct, visual representations of real-world or mathematical objects that serve&#xA;to communicate properties of the objects and facilitate problem solving. This study explored&#xA;perceptions of mathematics teachers in Singapore, who are heads of mathematics departments&#xA;in their respective schools, related to diagrams and their use in the teaching of mathematics. An&#xA;open-ended survey was adopted to illicit responses to i) what is a diagram, and ii) when do you&#xA;use diagrams in your mathematics instruction. The findings of the study show that participants&#xA;generally viewed diagrams as visual representations of real life or mathematical objects and they&#xA;used them mainly as visual aids when illustrating mathematical concepts and relationships.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Charts at the elementary level would include a tally chart, such as a chart used to make a bar graph (number of votes for a favorite color in tallies next to each name). Charts include various representations of data including graphs.  Pie graphs are called pie charts.&lt;/p&gt;&#xA;&lt;p&gt;Tables with function values are not graphs but represent functions.  They would be called  charts as well.&lt;/p&gt;&#xA;" OwnerUserId="5321" LastActivityDate="2023-07-18T09:30:16.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26624" PostTypeId="2" ParentId="16815" CreationDate="2023-07-18T13:13:37.837" Score="3" Body="&lt;p&gt;I quite liked the comment (on the other site's closed version of this question), which said &amp;quot;You're not alone.  Do what you're good at.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Consider, Richard Feynman:&lt;/p&gt;&#xA;&lt;p&gt;Weiner:&lt;/p&gt;&#xA;&lt;p&gt;Were you majoring in mathematics?&#xA;Feynman:&lt;/p&gt;&#xA;&lt;p&gt;Oh, yes. That’s interesting. At first I was in the mathematics course. It doesn’t make any difference what course you are in, really, very much. For the first year or so you take more or less the same thing. You take physics and chemistry and electrical engineering, mathematics, and so on. English. Somewhere around the first year I began to get upset. This wasn’t right. The mathematics, I looked at it, the mathematical things, were too abstract. They weren’t connected to anything, mathematics. And I went to the head of the mathematics department. This was in 1936, now, so you know it’s still in the Depression. I said to him, “Sir, what is the use of higher mathematics besides teaching more higher mathematics.” So he said, “Well, you could become an actuary,” calculating the insurance rates for an insurance company. This didn’t sit well with me, see. He also said that a man who asked that kind of a question is perhaps not right for mathematics. And I thought the thing I ought to do — I mean, I liked to get my hands dirty. I’d had a laboratory. The physical world was real, and the mathematics, I had become enthralled with, but not for itself, really — you know what I mean? It was fascinating, but my real heart was somewhere else. So I decided, I have to get my hands dirty, I can’t stand these abstract things. So I changed to electrical engineering, because there was something that was real. But then some few months later, I realized I’d gone too far, and that somewhere in between — that physics was the right place. So I moved around a little bit at the beginning, and ended up with the physics course.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/5020-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/5020-2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is not to say you should end up in physics, or EE.  Math might be right for you.  Or maybe you want to be even more far from math like in chem/bio/business/CS/English/etc.  Nor that there aren't people who will enjoy math best.  But the number of people doing abstract math is a small minority of those who learn math because it helps them with science or engineering or business or etc.&lt;/p&gt;&#xA;&lt;p&gt;For a humorous take, see:  &lt;a href=&quot;https://xkcd.com/435/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://xkcd.com/435/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You will even see some differentiation of math for majors and non-majors courses.  For example, in statistics, where having intuitions and knowing how to work applied problems is more important to [everything else] and where understanding the justifications is more important to math majors.  Another example is in group theory...where asymptotically approaching zero chem undergrads or grad students take abstract algebra.  Instead, the minimal group theory needed for understanding molecular vibrations and IR spectra is taught in specialized &amp;quot;group theory for chemistry&amp;quot; texts.  Another is engineering and physics where &amp;quot;engine math&amp;quot; or &amp;quot;math methods for physicists&amp;quot; sort of works as a stripped down PDE and linear algebra class (sometimes even a little complex analysis or the more practicial series manipulations and such in real analysis...I.e. actual computations and caring about the &amp;quot;Yo, Adrian&amp;quot; Bessel functions and the drum vibrations...but not having time for full courses, especially with emphasis on theory, and sometimes even shirking computational and applied problems, in math major courses.&lt;/p&gt;&#xA;&lt;p&gt;Beware of answers on M(E)SE, Reddit, etc. from math majors or instructors who sometimes have a perplexing blind spot here...in recognizing the crushingly evident fact that the vast majority of math studied in the world is NOT for future Andrew Wiles wannabes, like uh...them.&lt;/p&gt;&#xA;" OwnerUserId="22651" LastEditorUserId="-1" LastEditDate="2023-07-26T19:42:43.093" LastActivityDate="2023-07-26T19:42:43.093" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26627" PostTypeId="2" ParentId="7528" CreationDate="2023-07-20T00:01:11.230" Score="2" Body="&lt;p&gt;As &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler_spiral&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; notes, &lt;strong&gt;Euler spirals&lt;/strong&gt;, also known as &lt;strong&gt;clothoids&lt;/strong&gt;, are&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;widely used in railway and highway engineering to design transition curves between straight and curved sections of railway or roads&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;An application that may be more relatable to students is in the design of rollercoaster loops, as pointed out in this post (with a couple of pictures) at &lt;a href=&quot;https://thatsmaths.com/2014/04/10/rollercoaster-loops/&quot; rel=&quot;nofollow noreferrer&quot;&gt;ThatsMath&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="17715" LastActivityDate="2023-07-20T00:01:11.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26629" PostTypeId="2" ParentId="16815" CreationDate="2023-07-26T00:47:05.200" Score="0" Body="&lt;p&gt;The trick is to start perceiving mathematical objects as &amp;quot;real&amp;quot; as the ones in the so called &amp;quot;real world&amp;quot;. In fact, they are way more real than the physical objects, but I'd rather not digress on that now and concentrate on one more thing that is rarely taught or even mentioned.&lt;/p&gt;&#xA;&lt;p&gt;You can develop a dual vision in which you see abstract mathematical setups as common situations and vice versa easily translating from one language to another. Here is an example that I recently ran over with my students (it was pure improvisation: they asked me how to solve some problems from Analysis and I tried to explain a bit how one could visualize the situation).&lt;/p&gt;&#xA;&lt;p&gt;Problem 1 (abstract mathematics): Given a countable set of sequences &lt;span class=&quot;math-container&quot;&gt;$a^{(n)}_1, a^{(n)}_2,\dots$&lt;/span&gt; increasing to infinity, construct a single sequence &lt;span class=&quot;math-container&quot;&gt;$a_1,a_2,\dots$&lt;/span&gt; still increasing to infinity such that it is eventually smaller than each sequence in the given countable family, i.e., for every &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, there exists &lt;span class=&quot;math-container&quot;&gt;$K=K(n)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$a_k\le a^{(n)}_k$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$k\ge K$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The solution for the finite set of sequences would be trivial: just take &lt;span class=&quot;math-container&quot;&gt;$a_k$&lt;/span&gt; to be the minimum of &lt;span class=&quot;math-container&quot;&gt;$a^{(n)}_k$&lt;/span&gt; over all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. The minimum of finitely many growing to infinity sequences still grows to infinity and we are done.&lt;/p&gt;&#xA;&lt;p&gt;However the countable case was not so easy for them. And then I restated the problem in the following way.&lt;/p&gt;&#xA;&lt;p&gt;Problem 2 (traveling on a railroad) You have a railroad going from your current location &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; to infinity with infinitely many stations on the way. Every hour a train departs from &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and each train is slower than the previous one. You want to use these trains to go to infinity but you should eventually find yourself behind any particular train. How do you do that?&lt;/p&gt;&#xA;&lt;p&gt;The answer came in about 3 minutes: you take the first train to the first station, disembark there, wait for the second train and take it to the second station, then wait there for the third train and take it to the third station, and so on. After that it took some time to convert it back into the abstract mathematical language (there are some pesky translation details that I'm intentionally skipping), but once the idea was there, it all was more or less routine. The main point, however, is that for a mathematician &lt;em&gt;there is absolutely no difference&lt;/em&gt; between Problem 1 and Problem 2. Solving one of them is pretty much the same as solving the other and, because of that equivalence, they have equal &amp;quot;tangibility&amp;quot; and other properties, though Problem 2 uses next to no mathematical notions (the only one that is really there is forward infinity in time and space, which most people would, probably, consider &amp;quot;physical&amp;quot; too).&lt;/p&gt;&#xA;&lt;p&gt;At more basic level of elementary calculus, there is no difference between a girl holding a dog on a leash with the dog holding a cat on a leash and a question how short the leashes need to be to ensure that the cat doesn't travel more than 10 yards away from the girl on the one hand, and the formal standard &lt;span class=&quot;math-container&quot;&gt;$\varepsilon/2$&lt;/span&gt; argument that the limit of the sum of two convergent sequences is the sum of the limits (or that the sum of two continuous functions is continuous, or...) on the other hand.&lt;/p&gt;&#xA;&lt;p&gt;At more advanced level, one of the proofs of the Ergodic Theorem is just the big yellow bulldozer pushing a huge sand pile that fills all holes in the road on its way. I really keep in memory nothing but that about the proof when I come to the board to teach it: the rest is just the translation of that image into a chain of formulas.&lt;/p&gt;&#xA;&lt;p&gt;It won't be a big exaggeration to say that (almost) all mathematical concepts (definitely all basic ones and the vast majority of even the most sophisticated ones) can be translated into plain English and the underlying ideas explained on the setups having nothing to do with &amp;quot;mathematical abstractions&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Why isn't that done more often? The reason is the same that we don't normally accompany our typing in English by pictures except in the children books: we rely upon the idea that it is enough to type 5 letters &amp;quot;g-r-e-e-n&amp;quot; to bring in the reader's mind the pictures of the forest, the lizards, the traffic lights, the stale bread, the ecological party, etc., etc. What makes it &amp;quot;tangible&amp;quot; for us is just a huge association table in our brain that contains the &amp;quot;green&amp;quot; entry in many places. By itself, however, it is not even a certain wavelength range: all you saw when reading this passage was 5 fancy wiggles made out of a few dozen pixels on the screen: pure intangible meaningless abstraction, as it is. The same happens with mathematics: as long as the long association chains are not present, you just feel like the book author juggles the same 23 symbols over and over again in various combinations for no apparent reason: hdudfhwuio jddljk hjscf kkfelr.&lt;/p&gt;&#xA;&lt;p&gt;What to do to develop those association chains? My advice would be to relax a bit on studying the theory and pick some problem solving books. I don't know what your favorite area of mathematics is so I'll abstain from naming any titles for now, but trying to figure things out and to navigate that parallel Universe by yourself is a lot of fun even if the solutions of some particular problems are already known to humanity and you can pretty easily take a sidestep at any point to find yourself on a totally uncharted territory.&lt;/p&gt;&#xA;&lt;p&gt;Finally, before trying to answer the question &amp;quot;What is a pure mathematics good for except producing more pure mathematics?&amp;quot; try to answer the question &amp;quot;What are human beings good for except producing more human beings?&amp;quot; It is actually the same question in the sense I tried to describe. Once you find a satisfactory answer to either version, just translate it into the other setup.&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-07-26T00:47:05.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26630" PostTypeId="1" CreationDate="2023-07-29T03:30:35.357" Score="6" ViewCount="651" Body="&lt;p&gt;I am a student helping to develop a remedial course for other students who have recently failed the undergraduate PDE course at our university. The topics are provided from the syllabus in the pictures. What specific mathematical topics (not programming) do you recommend that students be proficient in, in order to succeed in this course?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cSqE3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cSqE3.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gTwIw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gTwIw.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jdJVj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jdJVj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SIBXv.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SIBXv.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22682" LastEditorUserId="376" LastEditDate="2023-07-29T15:09:49.680" LastActivityDate="2023-08-03T13:22:42.597" Title="What mathematical topics are important for succeeding in an undergrad PDE course?" Tags="|undergraduate-education|course-design|differential-equations|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26631" PostTypeId="2" ParentId="26630" CreationDate="2023-07-29T03:38:18.313" Score="9" Body="&lt;p&gt;They must understand the quadratic equation and how to factor it, solve it and manipulate it formally in both the real and complex case. Also, how to solve simple trigonometric equation and knowledge of the graphs of basic trig. functions such as sine, cosine and tangent. In addition, they must understand the solution to 2nd order constant coefficient ODEs in depth. In particular, the use of hyperbolic cosine and sine for the formulation of solutions in parallel to the usual introduction of sine and cosine and exponentials is helpful. Of course they must have a strong grounding in partial differentiation and power series calculation. Of course, I speculate, since I have no idea as to why the students you target have failed. But, in my experience, students struggle a lot with things which are actually from before calculus. Algebra and trigonometry are important.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2023-07-29T03:38:18.313" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="26632" PostTypeId="1" CreationDate="2023-07-30T16:03:58.837" Score="1" ViewCount="219" Body="&lt;p&gt;I am a HS math teacher and just started a job that adopted a new curriculum at the last minute. It gave us the tasks but did not provide any answer keys. Of course, I know how to work through the math but I would prefer not to spend the 1-2 hours every single week creating the answer keys for my 3 separate courses.&lt;/p&gt;&#xA;&lt;p&gt;The tasks are freely, publicly available online to anyone so it not proprietary stuff that I shouldn't be sharing. I have heard about sites like Nerdify where freelancers can assist. Does anyone have an opinion on them or have any other advice for what I am trying to do?&lt;/p&gt;&#xA;&lt;p&gt;And yes I know plenty of math educators who will hate every word of my question here and would tell me to work through every single example myself :). I have a busy year and was hoping to find a solution to save me time even if it costs me some $. Thank you for reading&lt;/p&gt;&#xA;" OwnerUserId="22686" LastActivityDate="2023-08-05T19:03:26.357" Title="A Question about Answer Keys" Tags="|teaching|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26633" PostTypeId="2" ParentId="26630" CreationDate="2023-07-30T17:29:44.940" Score="5" Body="&lt;p&gt;I would say that a good knowledge of linear algebra is paramount (for many reasons). About equally important is the full mastery of the differential and integral calculus of one variable. With these two, you can introduce the differential and partial derivatives pretty fast even if the students haven't seen them before and just wave your hands about multidimensional integration (though, of course, if they've seen the multivariate calculus too, it will be helpful as well).&lt;/p&gt;&#xA;&lt;p&gt;Beyond that they should just have common sense. That is not taught in any cookbook courses, so some course developing it should be taken (discrete mathematics, elementary number theory, etc.; just something where you need to think and put a few things together for solving problems, not just to follow a prescribed algorithm).&lt;/p&gt;&#xA;&lt;p&gt;With these three, you should be just fine filling some gaps in on the fly, if necessary. Without one of those, you'll struggle no matter what else you put as formal prerequisites. Warning: students who have successfully completed all three courses I mentioned are generally rare.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-07-30T17:29:44.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26634" PostTypeId="1" CreationDate="2023-07-31T13:21:56.360" Score="1" ViewCount="84" Body="&lt;p&gt;I just recently had a student (U.S. university) ask &amp;quot;How long would I have to wait until I could be eligible to take the FE or PE exam if I earned a non-ABET accredited engineering degree?&amp;quot; The student is looking to transfer, and I could not answer his question. I tried looking it up on the NCEES and ABET website, but I couldn't find a clear answer. Any answers or resources would be greatly appreciated.&lt;/p&gt;&#xA;&lt;p&gt;FE = Fundamentals of Engineering&lt;br /&gt;&#xA;PE = Principles and Practice of Engineering&lt;br /&gt;&#xA;ABET = Accreditation Board for Engineering and Technology&lt;br /&gt;&#xA;NCEES = National Council of Examiners for Engineering and Surveying&lt;/p&gt;&#xA;" OwnerUserId="22682" LastEditorUserId="2083" LastEditDate="2023-08-01T16:57:19.963" LastActivityDate="2023-08-01T16:57:19.963" Title="If a student earns a non-ABET accredited engineering degree, when can they take the FE or PE exam?" Tags="|usa|engineering-mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26635" PostTypeId="1" AcceptedAnswerId="26637" CreationDate="2023-07-31T14:34:13.637" Score="9" ViewCount="1555" Body="&lt;p&gt;Context: I am an associate professor at a small liberal arts institution in the US.&lt;/p&gt;&#xA;&lt;p&gt;I am currently preparing to teach geometry this fall.  Our course is mostly focused on Euclidean geometry (it's required for all math education majors), but it does have a few weeks of noneuclidean geometry coverage near the end.&lt;/p&gt;&#xA;&lt;p&gt;We typically study hyperbolic geometry from a &amp;quot;model-based&amp;quot; perspective, either by using half-plane or Poincare disk as our workspace.  While students are able to use these models, I feel that they do not truly understand how they connect to/relate to/represent hyperbolic space.&lt;/p&gt;&#xA;&lt;p&gt;As such, I would like to &lt;em&gt;first&lt;/em&gt; show a model for spherical geometry.  But spherical geometry is typically simply pictured on the surface of a sphere!&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Does anyone know of a &amp;quot;model&amp;quot; for spherical geometry?&lt;/li&gt;&#xA;&lt;li&gt;Could I simply use a projection of the northern hemisphere in the z-direction to the closed unit disk as my model?  That would mean that only the lines or circular arcs connecting antipodal points (or circular arcs along the boundary) would represent &amp;quot;lines&amp;quot; in our original spherical space.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5988" LastActivityDate="2024-03-18T23:34:30.710" Title="Models for spherical geometry" Tags="|geometry|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26636" PostTypeId="2" ParentId="26635" CreationDate="2023-07-31T16:44:37.773" Score="10" Body="&lt;p&gt;I often teach a geometry class like the one you describe and always try to include both elliptic and hyperbolic plane geometry.&lt;/p&gt;&#xA;&lt;p&gt;As you indicate, elliptic geometry is modeled by spherical geometry, but we must take care to identify antipodal points. The reason why this needs to be done is to guarantee that the postulate &amp;quot;two distinct points determine a unique line&amp;quot; holds true in the model. For example, {North Pole, South Pole} is just one Point in the model. I find this level of abstraction is enough to make for an interesting learning experience in the course.&lt;/p&gt;&#xA;&lt;p&gt;As a consequence, lines have finite length &lt;span class=&quot;math-container&quot;&gt;$\pi R$&lt;/span&gt; and the total area of the plane is &lt;span class=&quot;math-container&quot;&gt;$2\pi R^2$&lt;/span&gt;. Another interesting thread is regular tilings of the elliptic plane. These correspond to central projections of Platonic solids out to a circumscribing sphere. Four of the five solids have antipodal symmetry so give tilings of the elliptic plane. The exception is the tetrahedron.&lt;/p&gt;&#xA;&lt;p&gt;Another interesting feature of the geometry resulting from the antipodal identification is that it is not orientable. When you slide a right handed glove around it becomes left handed.&lt;/p&gt;&#xA;&lt;p&gt;Bringing all these concepts together can make for about three interesting classes/lectures if you also include the formula for the area of a triangle in terms of the angles and the parameter &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To bring more history into the discussion after hyperbolic geometry, one can explain how geometers first started developing hyperbolic geometry, before elliptic. They were assuming that they would find contradictions, thereby proving the parallel postulate. They started finding some strange formulas. But it was realized that any such formula could be transformed into a formula for spherical trigonometry by the transformation &lt;span class=&quot;math-container&quot;&gt;$k\rightarrow ik$&lt;/span&gt;. In view of the evident consistency of spherical trigonometry, it was realized that the &amp;quot;strange new world&amp;quot; of hyperbolic non-Euclidean geometry must be consistent!&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-08-01T04:05:55.733" LastActivityDate="2023-08-01T04:05:55.733" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26637" PostTypeId="2" ParentId="26635" CreationDate="2023-07-31T18:06:19.783" Score="13" Body="&lt;p&gt;Here are three  projections that result in models of spherical geometry. I think the stereographic model is closest to what you're looking for.&lt;/p&gt;&#xA;&lt;h2&gt;Stereographic projection&lt;/h2&gt;&#xA;&lt;p&gt;One possible &lt;a href=&quot;https://en.wikipedia.org/wiki/Stereographic_projection&quot; rel=&quot;noreferrer&quot;&gt;stereographic projection&lt;/a&gt; projects from a pole onto the equatorial plane, giving a model that is analogous to the Poincaré disk model. (Projecting onto the equatorial plane identifies the equator with the unit circle. Other stereographic projections are scaled differently.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/aPcNw.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aPcNw.png&quot; alt=&quot;Diagram of stereographic projection. By Leonid 2, CC-BY-SA 3.0.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Riemann_sphere1.svg&quot; rel=&quot;noreferrer&quot;&gt;Image credit to Leonid 2, CC-BY-SA 3.0&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This stereographic model has many desirable properties for geometry. It is conformal/angle-preserving, it can represent the entire sphere using the plane plus a point at infinity (cf. the &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann_sphere&quot; rel=&quot;noreferrer&quot;&gt;Riemann sphere&lt;/a&gt;), and the great circles are the lines and circles that intersect the unit circle at two ends of a diameter. (This includes the unit circle itself.) &lt;a href=&quot;https://www.geogebra.org/m/gmudya4r&quot; rel=&quot;noreferrer&quot;&gt;Here's a nice visualization in GeoGebra&lt;/a&gt;. &lt;strong&gt;The stereographic projection is the only projection that represents all spherical circles as circles.&lt;/strong&gt; (If we consider lines to be circles with infinite radius.)&lt;/p&gt;&#xA;&lt;p&gt;This &lt;a href=&quot;https://sites.math.washington.edu//%7Eking/coursedir/m445w05/lab/lab07.html&quot; rel=&quot;noreferrer&quot;&gt;lab assignment&lt;/a&gt; from a 2005 Geometry for Teachers course at the University of Washington taught by James King has teachers explore great circles and spherical triangles under stereographic projection. You can find even more of his resources on &lt;a href=&quot;https://sites.math.washington.edu/%7Eking/res/geom.html&quot; rel=&quot;noreferrer&quot;&gt;this page&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We can think of the Poincaré disk model as &lt;a href=&quot;https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model#Relation_to_the_hyperboloid_model&quot; rel=&quot;noreferrer&quot;&gt;the projection&lt;/a&gt; of one sheet of the unit hyperboloid of two sheets onto the equatorial plane, with the projection center being the opposite vertex.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rmYBs.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rmYBs.png&quot; alt=&quot;Projective relation of hyperboloid and Poincare models. By Selfstudier, CC0.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://commons.wikimedia.org/wiki/File:HyperboloidProjection.png&quot; rel=&quot;noreferrer&quot;&gt;Image credit to Selfstudier, CC0&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To facilitate comparison of the two models, we can restrict the stereographic model to only the projection of the hemisphere opposite the pole. This would make it a disk model. By identifying opposite points on the boundary, we could also consider it to be a model of &lt;em&gt;elliptic geometry&lt;/em&gt;, which as &lt;a href=&quot;https://matheducators.stackexchange.com/a/26636/20719&quot;&gt;this answer&lt;/a&gt; explains is distinct from spherical geometry.&lt;/p&gt;&#xA;&lt;p&gt;Ian Stevenson developed a computer &amp;quot;microworld&amp;quot; for exploring these two models, and you can read more about it in &lt;a href=&quot;https://core.ac.uk/download/pdf/33680037.pdf&quot; rel=&quot;noreferrer&quot;&gt;his doctoral thesis&lt;/a&gt; or in &lt;a href=&quot;https://www.jstor.org/stable/40248299&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt;. He attributes the stereographic model to Klein. He calls it the &amp;quot;hot-plate universe,&amp;quot; and he calls the Poincaré disk model the &amp;quot;cold-plate universe.&amp;quot; This language is from Jeremy Gray's &lt;em&gt;Ideas of Space&lt;/em&gt;, and Gray reports getting it from &lt;a href=&quot;https://www.feynmanlectures.caltech.edu/II_42.html&quot; rel=&quot;noreferrer&quot;&gt;Feynman&lt;/a&gt;. The idea is that, if we use a metal ruler as our metric, the ruler will grow as we move away from the origin in the stereographic model, and the ruler will shrink as we move away from the origin in the Poincaré disk model. Thus, paths far from the origin are shorter than they appear in the stereographic model, and they are longer than they appear in the Poincaré disk model.&lt;/p&gt;&#xA;&lt;p&gt;You can read about pre-service teachers' investigation of the models in Stevenson's microworld in &lt;a href=&quot;https://www.jstor.org/stable/40248341&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt;, &lt;a href=&quot;https://link.springer.com/article/10.1023/A:1009821625805&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt;, and &lt;a href=&quot;https://link.springer.com/article/10.1023/A%3A1009802004645&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;h2&gt;Orthographic projection&lt;/h2&gt;&#xA;&lt;p&gt;Flattening the northern hemisphere in the z-direction onto the equatorial plane as you describe in the question is an &lt;a href=&quot;https://en.wikipedia.org/wiki/Orthographic_map_projection&quot; rel=&quot;noreferrer&quot;&gt;orthographic projection&lt;/a&gt;. The orthographic projection is easy to visualize, but it isn’t conformal, only shows half the sphere, and crucially, &lt;strong&gt;an orthographic projection does not map great circles to circular arcs.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;To see this, consider the orthographic projection of the globe shown. The edge of the projection is a circle representing the 30W and 150E meridians. The other meridians except for the central meridian (90E) are tangent to the outer circle at both poles, so they cannot be circular arcs. &lt;a href=&quot;http://www.grad.hr/geomteh3d/Monge/11rotacija/rotacija_eng.html#:%7E:text=Projections%20of%20a%20circle,%CE%A0%20is%20an%20ellipse%20k%27.&quot; rel=&quot;noreferrer&quot;&gt;In general, an orthographic projection maps spherical circles to (possibly degenerate) ellipses&lt;/a&gt;. &lt;a href=&quot;https://www.jstor.org/stable/3603587&quot; rel=&quot;noreferrer&quot;&gt;This article on spherical geometry&lt;/a&gt; by Lodge and Heawood discusses the drawing of spherical circles under orthographic and stereographic projections.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pTKlT.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pTKlT.png&quot; alt=&quot;Orthographic projection of the Earth from 30W to 150E. By Planiglobe, CC-BY 2.5.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Orthographic_Projection_30W_150E.png&quot; rel=&quot;noreferrer&quot;&gt;Image credit to Planiglobe, CC-BY 2.5&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;h2&gt;Gnomonic projection&lt;/h2&gt;&#xA;&lt;p&gt;A &lt;a href=&quot;https://en.wikipedia.org/wiki/Gnomonic_projection&quot; rel=&quot;noreferrer&quot;&gt;gnomonic projection&lt;/a&gt; projects from the center of the sphere onto a tangent plane. The resulting model of a hemisphere is analogous to the Klein disk model, which can be thought of as &lt;a href=&quot;https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model#Relation_of_the_disk_model_to_the_hyperboloid_model&quot; rel=&quot;noreferrer&quot;&gt;a projection&lt;/a&gt; of one sheet of the unit hyperboloid from the origin onto a plane tangent to the vertex. Like the Klein disk model, it represents geodesics as straight lines, but it also isn't conformal. Unlike the Klein disk model, it uses the entire plane. And to be a faithful model of elliptic geometry, we need the entire &lt;a href=&quot;https://en.wikipedia.org/wiki/Elliptic_geometry#Elliptic_plane&quot; rel=&quot;noreferrer&quot;&gt;real projective plane&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bVHvq.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bVHvq.png&quot; alt=&quot;Gnomonic projection (projection from the center of the sphere). By Marozols. CC-BY-SA 3.0.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Gnomonic.png&quot; rel=&quot;noreferrer&quot;&gt;Image credit to Marozols, CC-BY-SA 3.0.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-08-02T11:47:29.107" LastActivityDate="2023-08-02T11:47:29.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26638" PostTypeId="2" ParentId="26630" CreationDate="2023-07-31T22:04:52.297" Score="4" Body="&lt;p&gt;In my experience, the difficulty with this course (the course description is very standard) is that students need to have actually mastered all of the prerequisite skills to a fairly high standard.&lt;/p&gt;&#xA;&lt;p&gt;To solve a BVP by separation of variables you will need to solve several ODE problems, each of which requires solving multiple calculus problems, each of which requires solving multiple algebra and trig problems.  If you aren’t 95% accurate in each of the sub problems then there is little chance you’ll get a correct solution to PDE boundary value problem.&lt;/p&gt;&#xA;" OwnerUserId="5841" LastEditorUserId="5841" LastEditDate="2023-07-31T22:14:41.940" LastActivityDate="2023-07-31T22:14:41.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26639" PostTypeId="2" ParentId="26635" CreationDate="2023-08-01T22:30:10.300" Score="1" Body="&lt;p&gt;Imagine the world were a big glass ball, that is, transparent all the way through, and you could see your &amp;quot;shadow&amp;quot; on the antipodal point at all times.&lt;/p&gt;&#xA;&lt;p&gt;So as you run around the spherical surface, your &amp;quot;shadow&amp;quot; runs around on the surface at the opposite end of the earth in exactly the opposite direction to you at all times.&lt;/p&gt;&#xA;&lt;p&gt;Extrapolate that out to imagining space to be like that.&lt;/p&gt;&#xA;" OwnerUserId="15290" LastActivityDate="2023-08-01T22:30:10.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26640" PostTypeId="2" ParentId="26635" CreationDate="2023-08-02T12:28:21.253" Score="1" Body="&lt;p&gt;One relatively simple projection starts with a disk, like the Poibcaré disk of hyperbolic geometry. But we will define the &amp;quot;lines&amp;quot; differently. The interior of the disk will correspond to half the sphere, the exterior to the other half.&lt;/p&gt;&#xA;&lt;p&gt;Define a line as a Euclidean circular arc passing through two points (which may be on, inside, or outside the disk boundary), and intersecting the boundary at two diametrically opposite points. If you draw the three &amp;quot;lines&amp;quot; defined in this way passing pairwise through three points, you see the sides of the formed triangle bulging out as they would when viewing a spherical triangle. So you satisfy the angular sum exceeding 180°. Indeed, with the right selection of a metric to measure distances, the triangle meets all the trigonometric laws applying to spherical triangles!&lt;/p&gt;&#xA;" OwnerUserId="22707" LastEditorUserId="8571" LastEditDate="2024-03-10T22:15:04.010" LastActivityDate="2024-03-10T22:15:04.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26641" PostTypeId="1" AcceptedAnswerId="26645" CreationDate="2023-08-02T17:54:18.273" Score="8" ViewCount="2429" Body="&lt;p&gt;This question is about getting realistic expectations for how a university student actually does and should learn maths. I'm becoming increasingly suspicious that my approach is detrimental, but I don't see a way out of it. I know that screams &amp;quot;off-topic personal question&amp;quot;, but I think the themes of this question should appeal to a broad audience and can be answered without personal knowledge of who I am and how I learn. Please don't automatically close this; I will endeavour to edit this into a more answerable shape if necessary.&lt;/p&gt;&#xA;&lt;p&gt;Tl;Dr - skipping the context, my question is more or less what it says in the title. I learn things very carefully but there is a massive time overhead involved. I would like to find some sort of compromise position, but don't know how to change. I could restate the question as: &amp;quot;how to take better notes?&amp;quot; since my current note-taking strategy is rather intense.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Mathematics is &lt;em&gt;supposed&lt;/em&gt; to be a rigorous subject. As long as you don't delve into foundational issues - as long as you can believe in a bare minimum of basic set theoretic constructions and basic logic - you can, in principle, understand and prove beyond all doubt any statement you learn in your journey. This is what draws me to mathematics beyond any other subject; anything I read, I am empowered to question and (attempt to) check on my own. Despite my relative ignorance, I can point out mistakes in the author's work and decide for myself what the correct statements should be and I can know with certainty (ignoring foundational issues) that it is right.&lt;/p&gt;&#xA;&lt;p&gt;When I started getting into higher maths (i.e. beyond school level) I was aware that &amp;quot;in 'real' maths, we prove things&amp;quot; so I made an effort to learn how to read and write proofs; every new thing I learnt I would either try to prove for myself or look up a proof &lt;em&gt;that I could understand&lt;/em&gt;. There is considerable additional effort spent in expanding upon and annotating proofs with further explanations - Lord knows authors leave a lot of missing gaps for the readers to fill. I kept this attitude up; &lt;strong&gt;it has only been in exceptionally rare cases that I have noted down a statement without noting its proof&lt;/strong&gt;. I have only done so when the statement was a side remark invoking a theorem that was way above my paygrade at the time. Even then, I would have tried (and failed) to understand the relevant papers.&lt;/p&gt;&#xA;&lt;p&gt;As you can imagine, this is a very, very slow process. I've somehow managed to cover a lot nonetheless, but it can sometimes take me hours or even weeks to get notes on a single paragraph - there's something I don't understand, so I spend time trying to check it, then I find I maybe need to learn some new things before moving on, etc. and before you know it, what feels like an eon has passed.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;On the one hand, this is a good way to expose myself to new maths and new ideas, and by thoroughly checking everything written and filling in the gaps I get experience in &amp;quot;research&amp;quot;, overcoming frustration and doing my own proof writing, problem solving. Moreover, by writing down annotated and fully explained proofs I have a reliable reference so that whenever I need to revisit a theorem and its proof, I can have more confidence that (a) the result is correct, because my younger self checked it and (b) I don't need to waste extra energy checking mysterious steps in the proof, since my younger self has hopefully explained them&lt;/li&gt;&#xA;&lt;li&gt;On the other hand, this is absurdly slow (I check and take the time to write down absolutely &lt;em&gt;everything&lt;/em&gt;) and everyone I've talked to on (mathematical) social media seems to &lt;em&gt;not at all&lt;/em&gt; follow the same procedure. I get the impression most maths students - very successful ones - 'get away' without painstakingly examining and cross-examining their textbooks. It seems many people don't even read the proofs of many of the statements they might learn in a textbook, preferring to mostly focus on the ideas, the intuition; this seems to work wonderfully for them.&lt;/li&gt;&#xA;&lt;li&gt;On the third hand, &lt;strong&gt;authors are capable of mistakes and omissions&lt;/strong&gt;. It is not common but it is not &lt;em&gt;un&lt;/em&gt;common either that I encounter sloppily stated theorems/proofs or false/grossly misleading statements. One benefit of my approach is that I can usually detect this and correct it; had I just had pure faith in my learning source, I may have learnt a false thing and missed out on the opportunity to learn/discover new mathematics necessary to make the corrections. &lt;em&gt;This phenomenon gets worse as the maths gets harder and more abstract&lt;/em&gt;, I've found - which motivates to be more fastidious, essentially making me learn slower and slower&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The 'fear' of learning false statements and blindly accepting mistakes from authors is with me, and it's why I struggle to 'grow out' of this approach of mine. However when I read things like &lt;a href=&quot;https://math.stackexchange.com/a/1227991&quot;&gt;this&lt;/a&gt; I really get the sense that I've been doing it all wrong, very wrong. I start university this year and I should probably start on the right foot; I don't want to be left behind in the dust by less fastidious - but faster - students.&lt;/p&gt;&#xA;&lt;p&gt;Questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is this slow approach genuinely problematic, or am I just being dramatic?&lt;/li&gt;&#xA;&lt;li&gt;How can one get through content more quickly but still manage to learn proof techniques and feel like they understand everything that's going on? I fear that if I started, say, skipping the proof of every other theorem, I would quickly lose track of how things are working and I would be afraid of failing to catch mistakes/sloppy statements&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Perhaps the question becomes more answerable if I state it like this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In your experiences of teaching and learning, what are some good strategies for learning mathematics at a reasonable pace, all the while maintaining a high (ideally perfect?) standard of rigour &lt;em&gt;without getting bogged down&lt;/em&gt;?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In case this context is needed; I’m not following any course (not at university yet). I’m entirely self studying; at the moment this has been investigations into enriched and model category theory and simplicial homotopy theory, following a few different textbooks.&lt;/p&gt;&#xA;" OwnerUserId="16527" LastEditorUserId="16527" LastEditDate="2023-08-02T20:34:37.013" LastActivityDate="2023-08-05T10:15:06.153" Title="Should I really just &quot;shut up and calculate&quot;? On learning at a good pace without sacrificing rigour" Tags="|proofs|self-learning|lecture-notes|" AnswerCount="5" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="26642" PostTypeId="2" ParentId="26641" CreationDate="2023-08-03T00:09:09.130" Score="5" Body="&lt;p&gt;&lt;strong&gt;Take a class, and ask the professor&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As the class starts, try to get an idea for what kinds of statements the professor is assuming to be true without justifying them, that you do not immediately see how to justify. Then pick one or two specific ones (not three or ten) and take them to office hours in the second week or so. Tell the professor that you don't immediately see how to justify these statements, and ask the professor if it would be a good use of time for you to work through the details.&lt;/p&gt;&#xA;&lt;p&gt;You will get one of these answers, probably:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;No, this is too foundational for you to be worrying about right now; you will not be able to work on the material that is the purpose of this class if you spend time on this issue, or&lt;/li&gt;&#xA;&lt;li&gt;Yes, it would be good for you to work out those details so that you are comfortable with them, or&lt;/li&gt;&#xA;&lt;li&gt;This statement is obvious; there is nothing to prove here. (You would need to then explain the specific situation you are worried about that makes you think it is complicated, or retreat and regroup and think about why this was the answer.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The first two answers are clearly good for you. The third one is &lt;strong&gt;actually also good for you&lt;/strong&gt; because you asked for the expectations of the class and got the answer. If you get the third answer, then yes, you should &amp;quot;shut up.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In summary, ask the person leading your class, and try to respect the answer you get.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-08-03T00:09:09.130" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26643" PostTypeId="2" ParentId="26641" CreationDate="2023-08-03T13:03:33.173" Score="1" Body="&lt;p&gt;...yes.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If your approach is holding you back, you need to change it.  Consider foreign language study.  If you stop and try to translate every word, research its etymology, etc., you will go much slower than someone who does more immersion.  (And probably give up!)  The issue is NOT one of the structure (of language or of math) but of HUMAN LEARNING.  This doesn't mean that you'll never learn all those words (or research the foundations).  But for now, you need to acquire key familiarity, first.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There is SIGNIFICANT benefit in building your algebraic muscles from the &amp;quot;just calculate&amp;quot; mode.  And by the way, there's a buttload of algebra in epsilon-delta or in series convergence, etc.  It's not just divided by zero function domain kvetching.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And even purist Hardy was very good at the calculate part of math (taking joy in Cleo-like solution of definite integrals).  And so was Ramanujan.  And so is Andrew Wiles.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="22711" LastEditorUserId="22711" LastEditDate="2023-08-03T13:04:19.780" LastActivityDate="2023-08-03T13:04:19.780" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26644" PostTypeId="2" ParentId="26630" CreationDate="2023-08-03T13:22:42.597" Score="3" Body="&lt;p&gt;I think polling the (very probably small) group of students would be a better way to determine the answer than looking for some general trend, here.  The reason I say this is because PDEs are really quite advanced compared to the average juco/4-year remediation situation (students trying to get through calculus who have basic problems in algebra and even arithmetic).&lt;/p&gt;&#xA;&lt;p&gt;Even in a situation with significant amounts of social promotion and students getting by predecessor classes with low grades, you are talking about students who have passed calculus and ODEs.  Even if not all-stars, they have to have some reasonable skills to have not washed out previously.  With that in mind, I'd even say that remediation of previous courses (yes, even if not flawlessly understood!) is NOT the key metric for helping the students.  Much more likely is to just give them a second go through and have them concentrating more on the topic.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I disagree with the &amp;quot; more linear algebra&amp;quot; answer.  This is a very math-y answer, not a pedagogically sound answer.  While there may be some interesting foundations/connections of LA to PDEs, this is not helpful/needed for a struggling learner in an engineering style PDE course.  Pulling down my engine math book (similar to the text you have here) and looking at the PDE chapters, there are asymptotically approaching zero matrices.  It's really more like ODEs, with some annoyances of series futzing (which is a lot of x-pushing algebra) and the engineering based derivations (which I disagree with, think it's harder to learn a math technique in the context of an applied word problem...prefer the ODE approach, where you learn the math first and then practice it, as math...and then later do applied problems with the added cognitive load of translating to/from physical to equation land.)&lt;/p&gt;&#xA;" OwnerUserId="22711" LastActivityDate="2023-08-03T13:22:42.597" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26645" PostTypeId="2" ParentId="26641" CreationDate="2023-08-03T21:38:35.347" Score="16" Body="&lt;p&gt;I looked at some of your posts on MSE before answering. Well, I wouldn't say that you are a &amp;quot;Jack of trades&amp;quot; yet, but you are certainly &lt;em&gt;way above&lt;/em&gt; what one would expect from somebody 2 years out of high school even by the 21-st century standards. So, if I were you, I wouldn't complain much about &amp;quot;going too slowly&amp;quot;: you go many times faster than most university students I have seen already and, being young, are going only to accelerate in the next 10-20 years if your brain doesn't fizzle at some point.&lt;/p&gt;&#xA;&lt;p&gt;There is one thing you should understand however: one cannot learn and become proficient in &lt;em&gt;all&lt;/em&gt; mathematics. So, you will always have some areas in which you'll have solid rock knowledge, some areas in which your knowledge is fuzzy and some areas in which you are totally ignorant. Of course, you can choose any particular fuzzy or even unknown stuff and turn it into the solid rock one by putting enough time and effort into it, but while staying omnivorous is an excellent strategy, you'll not be able to swallow and digest everything you can bite. So, get comfortable with the idea that some fuzziness somewhere is OK.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand the only way to build the solid rock part of your knowledge is the way you are following: go into everything, scrutinize every detail that is still not obvious to you (some of your posts on MSE show that you already treat a lot of standard stuff as &amp;quot;trivial&amp;quot;, which is good for you since you haven't made any mistakes there that I would notice, so I assume you, indeed, absorbed that stuff completely, but bad for your readers to the extent that they often wait for someone else to provide a more detailed explanation even after you have provided a full answer), try to prove everything by yourself, and so on. There isn't and there will never be any shortcut here. And yeah, that solid rock part grows like a stalactite with just small mineral deposits from a huge flow of information you work through. Fortunately you will realize soon (if you haven't realized that already) that a lot of that information is repetitive and then, if you want accelerate, you'll have to sift it for really new ideas, techniques and tricks and ignore the stuff that is fairly familiar.&lt;/p&gt;&#xA;&lt;p&gt;What I'm advocating for is not abandoning that solid rock building strategy (God save you from that!), but complementing it with some fuzzy area building one. You should always understand that that fuzzy area is fuzzy, so you'll see those things in a kind of light (or even moderate) fog, but the advantage is that you'll be able to see much more and way farther and it can be expanded much faster than the solid rock one, which may be useful even if the crystal clarity of the vision you've got so accustomed to is sacrificed. However, if something in that fuzzy area attracts your attention and arouses your interest, you always have an option to jump upon it and add it to your solid rock part. Big fuzzy area will just give you more options and choices of what to do next (and yeah, that means that you'll have to select and prioritize).&lt;/p&gt;&#xA;&lt;p&gt;As to &amp;quot;this&amp;quot; that you've read, the actual advice there was merely a) to solve problems, not just to read theory (which you are apparently doing anyway, so everything is fine here) and b) not to waste time on reading and verifying slow-paced repetitive expositions that have never been intended for anyone reasonably smart in the first place (or, more blatantly said, to choose the books that match your level or slightly exceed it). Both are reasonable pieces of advice and  I absolutely do not see why you should get frustrated about them.&lt;/p&gt;&#xA;&lt;p&gt;The last piece of advice: find somebody to work and study with. Solitude is fine, but the life gets way more fun when you can share the excitement and compete with others. Also, as many people said already, a good mentor (at the university professor level, say) may benefit you too.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, I can say nothing about category theory and such because I'm a total ignoramus there, so I hope somebody else will provide some pertinent comments related to your current studies (like suggesting well-written and reasonably fast-paced books, etc.)&lt;/p&gt;&#xA;&lt;p&gt;Just my 2 cents.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-08-03T21:38:35.347" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26646" PostTypeId="2" ParentId="26641" CreationDate="2023-08-04T16:36:15.013" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Mathematics is supposed to be a rigorous subject. As long as you don't delve into foundational issues - as long as you can believe in a bare minimum of basic set theoretic constructions and basic logic - you can, in principle, understand and prove beyond all doubt any statement you learn in your journey.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I find this statement interesting because I personally enjoy dabbling in the foundational issues.  One doesn't have to, of course, but your wording suggests that this mindset shapes your view of what mathematics is.&lt;/p&gt;&#xA;&lt;p&gt;You say mathematics should be rigorous, but what does that mean?  Consider that until 1922, there was no ZFC to root your rigor in.  Rigor was defined differently before that point.  There was, in my opinion, a great deal of persuasion involved, convincing the reader that one's logic was sound, and that persuasion may not have the luxury of being founded in formal notations.&lt;/p&gt;&#xA;&lt;p&gt;I point this out because I don't believe this approach fundamentally changed just because we developed fundamental tools like set theory.  Sure, a lot shifted there, because mathematicians in general agree with the results that come from set theory, but there's still a part of mathematics that isn't rigorously proven with formal logic.&lt;/p&gt;&#xA;&lt;p&gt;Were mathematicians shackled to such rules, logicism would still be a big thing.  There would be no value in a thing that is not proven with a formal proof.  For me, the spark of mathematics lies buried under these lines.  There's some value in having a hypothesis which isn't &lt;em&gt;quite&lt;/em&gt; proven formally and exploring it.  Using it.  Seeing whether it leads you in a direction of sufficient value as to warrant taking the time to pen a formal proof.&lt;/p&gt;&#xA;&lt;p&gt;In math class, they are teaching you dead math.  They're teaching you things that are substantially proven and agreed upon.  When one starts looking at live math, one lives on the hairy edge between that which is proven and that which is still seeking proof.&lt;/p&gt;&#xA;&lt;p&gt;I like to look at the history of calculus as an example.  The calculus of Calc I and Calc II are dead math.  But at one point, they were alive.  When the mathematics of calculus were being constructed, they did not rely on epsilon-delta proofs (the current gold-standard for arguing the provability of calculus).  Indeed, epsilon-delta proofs were not invented for over 100 years.  There was enough of a spark there to drive mathematicians for a century before the formal proofs finally nailed down the roots of calculus.&lt;/p&gt;&#xA;&lt;p&gt;It's great to get to learn all of the little formal proofs behind every statement.  It attracted me to mathematics as well.  But there's also something to be said for being comfortable working with things that aren't &lt;em&gt;quite&lt;/em&gt; formally proven yet -- or at least not formally proven to you.  The skill in working which such potential truths will be very helpful as you approach the living edge of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2023-08-04T16:36:15.013" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26647" PostTypeId="2" ParentId="26635" CreationDate="2023-08-04T18:23:47.517" Score="1" Body="&lt;p&gt;Assuming you want to talk about spherical and elliptical geometry, you might want to try this.  It's gonna sound kind of crazy, but it's worked (at least slightly) for me; it will definitely get their attention.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Take an inflatable sphere-like object.  A beach ball might do, or a large yoga ball.  Big is good.&lt;/li&gt;&#xA;&lt;li&gt;Inflate (possibly before class) and discuss what axioms do work.  Maybe positive curvature and triangle area formulas, if you've covered some of that.&lt;/li&gt;&#xA;&lt;li&gt;Note that not all neutral axioms are satisfied (see @user52817 answer, for instance).&lt;/li&gt;&#xA;&lt;li&gt;Now &lt;em&gt;deflate&lt;/em&gt; the ball in such a way that the &amp;quot;bottom half&amp;quot; sort of curves inside the rest of the ball.  This is much easier with a large exercise/yoga ball than a beach ball, but in principle both might work.  At the least you should be able to get a kind of hemisphere with the &amp;quot;bottom half&amp;quot; smashed under the rest, if you move it in such a way to &amp;quot;keep air in the top half&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;Discuss with students whether two points might determine a line uniquely now.  (It won't because the &amp;quot;equator&amp;quot; will probably still all be visible, but it will be a lot closer, and should get them talking about &amp;quot;identification of antipodes&amp;quot;, which implies that each great circle is in some sense the same as the &amp;quot;visible half&amp;quot; of the great circle when we require this axiom.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2023-08-04T18:23:47.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26648" PostTypeId="2" ParentId="26641" CreationDate="2023-08-05T00:37:22.530" Score="4" Body="&lt;p&gt;I did the exact same thing you did, except in physics instead of math; I would in fact rederive the relevant equations from scratch &lt;em&gt;every time&lt;/em&gt; I used them so as to really hammer into my head where they were coming from, what their assumptions were, and when they made use of approximations or might break down.&lt;/p&gt;&#xA;&lt;p&gt;In college, I, like you, found myself somewhat overworked and overwhelmed and working frantically to catch up. It didn't help that I had a &lt;em&gt;very&lt;/em&gt; good physics teacher in high school with a very clear understanding of the material and a gift for transmitting it, whereas the professors in college were great &lt;em&gt;researchers&lt;/em&gt; but were absolutely terrible at conveying their understanding. So, asking questions to clarify my understanding quite often plain didn't work, and however much time I spent on a confusion it would never really get dissolved.&lt;/p&gt;&#xA;&lt;p&gt;I gave up on some of my attention to detail as a result, just taking a few things for granted and moving on. But I've pretty consistently regretted it since, because &lt;em&gt;inevitably&lt;/em&gt; it comes up later; I can &lt;em&gt;see&lt;/em&gt; where my ability to understand a topic is limited by my shoddy foundation. Shoddy understanding produces shoddy understanding, and my ability to take advanced courses that ought to have built on earlier knowledge is diminished. Perhaps it is only diminished in a way that my classmates would say is normal, what they're used to and how they've learned everything until now, but &lt;em&gt;I'm&lt;/em&gt; certainly not satisfied with it and I certainly don't want to perpetuate that cycle any further than it's already gone.&lt;/p&gt;&#xA;&lt;p&gt;So no, I would absolutely say that as much as possible, you should stick to your habits. I don't know what your peers and classmates are planning to do with their future careers, and likely you don't either; but if &lt;em&gt;you&lt;/em&gt; intend to go on to grad school and research, you want as firm a foundation as you can get. Your insight later in life will largely be determined by the strength of your foundation now.&lt;/p&gt;&#xA;&lt;p&gt;That's not to say you can't, for example, look up new fields on Wikipedia and use places like Quanta for a high-altitude overview of a topic, as a way of orienting yourself or looking for existing work in a field; nor is it an argument against reading natural language summaries of a topic for a general idea of where you're going. But once you've decided to learn something for real, then...&lt;/p&gt;&#xA;&lt;p&gt;Well, okay, one exceptional case: there's a thing that sometimes happens, where you &amp;quot;shut up and calculate&amp;quot;, reach the end of the proof or chapter, then go back to where you lost the plot and try to understand from there, and this is often easier than trying to figure out a specific difficult step and the overall direction of the proof at the same time. But that should never have more than about a chapter or so of 'understanding debt' between the point where you move on and the point stop and return.&lt;/p&gt;&#xA;&lt;p&gt;Other than that, I'll go so far as to say that in my opinion, there's no point in studying a topic at all in math if you're not willing to be rigorous about it!&lt;/p&gt;&#xA;" OwnerUserId="22715" LastEditorUserId="22715" LastEditDate="2023-08-05T10:15:06.153" LastActivityDate="2023-08-05T10:15:06.153" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26651" PostTypeId="2" ParentId="7528" CreationDate="2023-08-05T18:58:08.413" Score="0" Body="&lt;p&gt;An interesting curve I recently noticed in some reading is the &lt;strong&gt;Hawaiian earring&lt;/strong&gt; which we can denote &lt;span class=&quot;math-container&quot;&gt;$\mathbb{H}$&lt;/span&gt; for the purpose of this answer.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \mathbb{H} = \{ (x,y) \in \mathbb{R}^2 \ | \ (x-1/n)^2+y^2=1/n^2, n=1,2,\dots \}. $$&lt;/span&gt;&#xA;In other words, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{H}$&lt;/span&gt; is the union of all circles with center &lt;span class=&quot;math-container&quot;&gt;$(1/n,0)$&lt;/span&gt; and radius &lt;span class=&quot;math-container&quot;&gt;$1/n$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a positive integer. Here is a partial picture of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{H}$&lt;/span&gt;, it just has the the &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; largest circles (found this in the Wikipedia article linked below)&#xA;&lt;a href=&quot;https://i.stack.imgur.com/sOCD2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sOCD2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;There are also higher dimensional analogs of this construction where the circles are replaced by higher dimensional spheres. The topology on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{H}$&lt;/span&gt; is very interesting as you can read about here:&lt;a href=&quot;https://en.wikipedia.org/wiki/Hawaiian_earring&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Hawaiian_earring&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2023-08-05T18:58:08.413" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26652" PostTypeId="2" ParentId="26632" CreationDate="2023-08-05T19:03:26.357" Score="6" Body="&lt;p&gt;Don't underestimate the value of working through the problems yourself. When I assign problems to students, I am not just checking to see if they got the answer correct or not -- I am looking for patterns, common mistakes, thought processes, etc. If you create your own answer key, you can write up your answers from the perspective of your students, showing the steps. I do the same with my middle schoolers, and it helps me anticipate what kinds of mistakes they might make. Hope this helps!&lt;/p&gt;&#xA;" OwnerUserId="22697" LastActivityDate="2023-08-05T19:03:26.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26653" PostTypeId="2" ParentId="26616" CreationDate="2023-08-05T19:07:02.403" Score="0" Body="&lt;p&gt;A related concept is Rich Math Tasks (&lt;a href=&quot;https://www.doe.virginia.gov/teaching-learning-assessment/k-12-standards-instruction/mathematics/instructional-resources/rich-mathematical-task&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.doe.virginia.gov/teaching-learning-assessment/k-12-standards-instruction/mathematics/instructional-resources/rich-mathematical-task&lt;/a&gt;) -- activities that require math students to pull in many different levels of thinking (as Xander points out), as well as multiple areas of the course. Getting students familiar with performing these tasks before an exam can help improve student readiness for novel problems on test day.&lt;/p&gt;&#xA;" OwnerUserId="22697" LastActivityDate="2023-08-05T19:07:02.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26656" PostTypeId="2" ParentId="19017" CreationDate="2023-08-08T19:36:50.510" Score="1" Body="&lt;p&gt;Roll a die.&lt;/p&gt;&#xA;&lt;p&gt;Say, it will either roll a &amp;quot;1&amp;quot; or not, right? So the probability of rolling a &amp;quot;1&amp;quot; is 50%, yes?&lt;/p&gt;&#xA;&lt;p&gt;Say, it will either roll a &amp;quot;2&amp;quot; or not, right? So the probability of rolling a &amp;quot;2&amp;quot; is 50%, yes?&lt;/p&gt;&#xA;&lt;p&gt;Say, it will either roll a &amp;quot;3&amp;quot; or not, right? So the probability of rolling a &amp;quot;3&amp;quot; is 50%, yes?&lt;/p&gt;&#xA;&lt;p&gt;Say, it will either roll a &amp;quot;4&amp;quot; or not, right? So the probability of rolling a &amp;quot;4&amp;quot; is 50%, yes?&lt;/p&gt;&#xA;&lt;p&gt;If necessary, proceed.&lt;/p&gt;&#xA;" OwnerUserId="8931" LastActivityDate="2023-08-08T19:36:50.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26659" PostTypeId="1" AcceptedAnswerId="26661" CreationDate="2023-08-11T03:18:57.513" Score="0" ViewCount="261" Body="&lt;p&gt;Here are three situations in which students might try to apply the factor theorem.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Proving that &lt;span class=&quot;math-container&quot;&gt;$x + 1$&lt;/span&gt; is a factor of the polynomial &lt;span class=&quot;math-container&quot;&gt;$x^3 + x + 2$&lt;/span&gt; can be done using the factor theorem by showing that the polynomial vanishes when &lt;span class=&quot;math-container&quot;&gt;$ x = -1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If we want to prove that &lt;span class=&quot;math-container&quot;&gt;$a + b$&lt;/span&gt; is a factor of &lt;span class=&quot;math-container&quot;&gt;$a^3 + b^3 $&lt;/span&gt;, the same approach can be used by replacing &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$-b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Can we use the factor theorem to check whether there exist positive integer pairs &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 + ab$&lt;/span&gt; is divisible by &lt;span class=&quot;math-container&quot;&gt;$a + b$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In case (1) we can say &lt;span class=&quot;math-container&quot;&gt;$x + 1$&lt;/span&gt; is a factor for any value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and in case (2) also we can say &lt;span class=&quot;math-container&quot;&gt;$a + b$&lt;/span&gt; is a factor for any integer pair &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt;. But how can we explain situations when &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; both have the same sign, considering that we substitute &lt;span class=&quot;math-container&quot;&gt;$a = - b$&lt;/span&gt; to obtain the factor?&lt;/p&gt;&#xA;&lt;p&gt;I think in case (3) we cannot replace &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$- b$&lt;/span&gt; because both are positive values, and the factor theorem is totally irrelevant because we need to find particular values for &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What would be the importance of showing students this limitation of the factor theorem? How do we help students achieve better understanding of the concepts involved, rather than only being able to answer questions of common standard patterns?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="20719" LastEditDate="2023-08-12T23:33:23.110" LastActivityDate="2023-08-13T12:02:21.733" Title="Limitations of applying the factor theorem" Tags="|secondary-education|algebra|number-theory|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26660" PostTypeId="1" CreationDate="2023-08-11T13:35:49.900" Score="0" ViewCount="369" Body="&lt;p&gt;How can we explain to students these ideas?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A square with 4 sides measuring 25 cm each does not have an area of 1 square meter.&lt;/li&gt;&#xA;&lt;li&gt;A shape which is not a square can have an area of 1 square meter.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Is “square meter” a badly-phrased term? What would be a more appropriate term?&lt;/p&gt;&#xA;" OwnerUserId="22748" LastEditorUserId="20719" LastEditDate="2023-08-13T10:25:40.990" LastActivityDate="2023-11-27T16:29:53.693" Title="How to explain square meters?" Tags="|geometry|units|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26661" PostTypeId="2" ParentId="26659" CreationDate="2023-08-12T23:39:36.623" Score="5" Body="&lt;p&gt;An idea with pedagogical importance here is that students often have trouble distinguishing between objects that look identical but are mathematically distinct, and these frequently appear in school algebra when students work with polynomial expressions. These expressions can represent (1) expressions involving arbitrary numbers, (2) &lt;a href=&quot;https://math.stackexchange.com/questions/98345/why-are-polynomials-defined-to-be-formal-vs-functions&quot;&gt;formal expressions&lt;/a&gt;, and (3) functions. Mathematicians commonly refer to objects of the second type as polynomials and objects of the third type as polynomial functions.&lt;/p&gt;&#xA;&lt;p&gt;As one example, &lt;a href=&quot;https://matheducators.stackexchange.com/questions/26659/limitations-of-applying-factor-theorem#comment70697_26659&quot;&gt;Steven Gubkin&lt;/a&gt; brought up how students might think that the polynomial &lt;span class=&quot;math-container&quot;&gt;$x^2+1$&lt;/span&gt; is prime/irreducible. This is true if we consider &lt;span class=&quot;math-container&quot;&gt;$x^2+1$&lt;/span&gt; as an element of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}[x]$&lt;/span&gt;, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Polynomial_ring&quot; rel=&quot;nofollow noreferrer&quot;&gt;ring of polynomials&lt;/a&gt; in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that have real coefficients, but it is not true if we consider it as an element of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}[x]$&lt;/span&gt;, the ring of polynomials in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that have complex coefficients.&lt;/p&gt;&#xA;&lt;p&gt;I'll first address your questions about case (2) and case (3) by carefully distinguishing between polynomials and expressions involving arbitrary integers. Anyone who is only interested in the pedagogical discussion can skip the next section.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In your cases (2) and (3), the expressions &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a^3+b^3$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$a^2+ab+b^2$&lt;/span&gt; represent both&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Polynomials with &lt;a href=&quot;https://en.wikipedia.org/wiki/Indeterminate_(variable)&quot; rel=&quot;nofollow noreferrer&quot;&gt;indeterminate&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and constant &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; (an element of the underlying ring), and&lt;/li&gt;&#xA;&lt;li&gt;Integers given two arbitrary integers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To help distinguish these, instead of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, I'll use &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; when talking about polynomials, and I'll use &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; when talking about integers.&lt;/p&gt;&#xA;&lt;p&gt;The factor theorem is a theorem about &lt;strong&gt;polynomials&lt;/strong&gt;. It states that &lt;span class=&quot;math-container&quot;&gt;$X-r$&lt;/span&gt; is a factor of the polynomial &lt;span class=&quot;math-container&quot;&gt;$P(X)$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is a root of &lt;span class=&quot;math-container&quot;&gt;$P(X)$&lt;/span&gt;, i.e. &lt;span class=&quot;math-container&quot;&gt;$P(r)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Given the polynomial &lt;span class=&quot;math-container&quot;&gt;$P(X)=X^3+c^3$&lt;/span&gt;, since &lt;span class=&quot;math-container&quot;&gt;$P(-c)=0$&lt;/span&gt;, we know that &lt;span class=&quot;math-container&quot;&gt;$X+c$&lt;/span&gt; is a factor of &lt;span class=&quot;math-container&quot;&gt;$X^3+c^3$&lt;/span&gt;. This means that &lt;span class=&quot;math-container&quot;&gt;$X^3+c^3$&lt;/span&gt; is the product of &lt;span class=&quot;math-container&quot;&gt;$X+c$&lt;/span&gt; and some polynomial &lt;span class=&quot;math-container&quot;&gt;$Q(X)$&lt;/span&gt;. Specifically,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(X+c)(X^2-cX+c^2)=X^3+c^3.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So far, no integers have been involved. Now, if we replace &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; with arbitrary integers &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, then we get a &lt;em&gt;general relationship&lt;/em&gt; between &lt;strong&gt;integers&lt;/strong&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (n+m)(n^2-nm+m^2) = n^3 + m^3, $$&lt;/span&gt;&#xA;which we could also call an &lt;em&gt;identity.&lt;/em&gt; This identity tells us that, for any integers &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$n^3+m^3$&lt;/span&gt; is the product of &lt;span class=&quot;math-container&quot;&gt;$n+m$&lt;/span&gt; and some integer, and therefore &lt;span class=&quot;math-container&quot;&gt;$n+m$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$n^3 + m^3.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Sometimes zero is not allowed as a divisor, in which case we can restrict our conclusion that &lt;span class=&quot;math-container&quot;&gt;$n+m$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$n^3+m^3$&lt;/span&gt; to integers &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$n+m\neq0$&lt;/span&gt;. There's no need for a similar restriction on the statement that &lt;span class=&quot;math-container&quot;&gt;$X+c$&lt;/span&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Polynomial_ring#:%7E:text=one%20says%20that%20p%20divides%20q&quot; rel=&quot;nofollow noreferrer&quot;&gt;divides&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$X^3+c^3$&lt;/span&gt;, because &lt;span class=&quot;math-container&quot;&gt;$X+c$&lt;/span&gt; is not the &lt;a href=&quot;https://mathworld.wolfram.com/ZeroPolynomial.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;zero polynomial&lt;/strong&gt;&lt;/a&gt;. However if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-m$&lt;/span&gt; have the same value, then &lt;span class=&quot;math-container&quot;&gt;$n+m$&lt;/span&gt; is the &lt;strong&gt;integer&lt;/strong&gt; zero.&lt;/p&gt;&#xA;&lt;p&gt;To your question about case (2), notice that nowhere in this argument did we assume that &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; have different signs. We took a relationship between &lt;strong&gt;polynomials&lt;/strong&gt; that the factor theorem helped us to establish, and we used it to deduce a general relationship between &lt;strong&gt;integers&lt;/strong&gt;. If we wanted to, we could restrict it to a general relationship between positive integers: for any positive integers &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$n+m$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$n^3+m^3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In case (3), we are asked to find specific relationships between &lt;strong&gt;integers&lt;/strong&gt;. The factor theorem tells us that, as &lt;strong&gt;polynomials&lt;/strong&gt;,  &lt;span class=&quot;math-container&quot;&gt;$X+c$&lt;/span&gt; is not a factor of &lt;span class=&quot;math-container&quot;&gt;$X^2+cX+c^2$&lt;/span&gt;. This means that we cannot use the same argument we used above to deduce a general relationship/identity involving integers. However, there may be non-general integer relationships that have this form. And indeed,&#xA;&lt;span class=&quot;math-container&quot;&gt;$2+2$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$2^2+2\cdot2+2^2.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;There's a conversation to be had about &lt;em&gt;when&lt;/em&gt; students should be introduced to the level of abstraction at which polynomials and expressions involving arbitrary numbers are seen as distinct objects. I think expressions like &lt;span class=&quot;math-container&quot;&gt;$a^3+b^3$&lt;/span&gt; should first be introduced in the context of generalized arithmetic and not formal algebra or functions, which Hung-Hsi Wu makes an argument for in &lt;a href=&quot;https://math.berkeley.edu/%7Ewu/Arithmetic-to-Algebra2019.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;.  But also, prior to introducing the factor theorem in his textbook &lt;em&gt;Algebra and Geometry&lt;/em&gt;, Wu says,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Up to this point, the algebra we have been doing may be called generalized arithmetic, because the symbols employed in all the expressions we have come across all stand for numbers. It follows that all our computations have been with numbers. This kind of algebra is more or less the algebra of the period from al-Khwarizmi (780–850) to 1800. After 1800, the work of Gauss, Abel, Galois, and others slowly transformed algebra into an abstract study of structure. School algebra, at some point, must also give an introduction to such abstract considerations. The study of polynomials provides the most natural venue to showcase the transition from algebra as generalized arithmetic to formal algebra. (p. 175).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would argue that students need to be aware of this transition in order to flexibly solve problems and construct arguments using the factor theorem, and in turn, the factor theorem motivates the transition. At the secondary level, students don't need the machinery and language of polynomial rings, but they need to have a sense that the statement &amp;quot;the polynomial &lt;span class=&quot;math-container&quot;&gt;$x+1$&lt;/span&gt; is a factor of the polynomial &lt;span class=&quot;math-container&quot;&gt;$x^3+x+2$&lt;/span&gt;&amp;quot; is of a completely different kind than the statement &amp;quot;given an integer &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a+1$&lt;/span&gt; is a factor of &lt;span class=&quot;math-container&quot;&gt;$a^3+a+2$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-08-13T12:02:21.733" LastActivityDate="2023-08-13T12:02:21.733" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26662" PostTypeId="1" CreationDate="2023-08-14T21:34:07.087" Score="9" ViewCount="155" Body="&lt;p&gt;&lt;strong&gt;Educational setting.&lt;/strong&gt;&#xA;I'm teaching math courses - typically consisting of lectures, weekly homework sheets, and an exercise class where the homework questions are discussed - for undergraduate and Master's level students at a German university. The courses are &amp;quot;proof-based&amp;quot; in the sense that lecturers are supposed (and inclined) to prove most or all results in the course. Homework problems also regularly include proof-based questions in those courses.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Issue.&lt;/strong&gt;&#xA;Almost all lectures I've seen - and all lectures I've taught - essentially use a top-down theory building approach, where the lecturer gives some definition, proves some results, and illustrates the definitions and the results with a number of examples (either in the lectures or in the homework problems).&lt;/p&gt;&#xA;&lt;p&gt;I'm trying to pay quite some attention to (i) the motivation of the various definitions and results and to (ii) presenting overarching schemes and narratives which show how the various topics are related and why it makes sense to study precisely those notions and results that we do study. From the course evaluations I'm under the impression that the students mostly appreciate this. However:&lt;/p&gt;&#xA;&lt;p&gt;I still have the distinct and persistent feeling that it is very difficult for most students get of good and deep intuition of how the definitions and theorems are motiviated and how they are related. I'm also under the impression that there is too much focus on teaching students well-developed theories and too little focus on how or why such theories are developed.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Goal.&lt;/strong&gt;&#xA;One option to (partially) improve the aforementioned situation might to be to develop mathematical theories in lectures in a way which moves from questions about objects that the students already know to definitions and results that can be used to answer those questions.&lt;/p&gt;&#xA;&lt;p&gt;I've seen a (very, very) mild version of this in the German functional analysis book &amp;quot;Grundkurs Funktionalanalysis&amp;quot; (&amp;quot;Introductory course on functional analysis&amp;quot;) by Winfried Kaballo. He starts each chapter with a small set of questions which serve as a motivation for the contents of the question. I've tried to adapt this in a few lectures (with somewhat positive feedback by some students), but it is arguably only a very small step.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt;&#xA;I'm looking for course materials - in particular textbooks - which develop a mathematical topic by taking such a &amp;quot;motivate new contents by questions about preceding contents&amp;quot; approach.&lt;/p&gt;&#xA;&lt;p&gt;Depending on the answer I would be interested in building a course based on such a book or - what is more likely - in developing my own materials for such a course and taking existing materials as an inspiration.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Scope of the question.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;As mentioned at the beginning, I am mainly interested in materials for proof-based courses that develop a mathematical theory in a rigorous way. However, if you know resources for different courses (say for instance , a less rigorous Calculus course) I'd also be interested, as I hope that these material could still serve as in inspiration.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This question focusses only on course materials, not on alternative types of institutional or educational settings. I'm aware of things like the Moore method or inverted class room (and have also tried the latter myself in one course), but this is not the topic of this question.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm not asking for individual examples of questions that can be used to motivate certain concepts or topics. My point is rather to see an entire course that is developed in this way.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;By &amp;quot;questions that motivate a certain concept or topic or result&amp;quot; I'm not mainly referring to questions that stem from applied settings. If some of the motivating questions stem from such a setting, this is fine for me - but building an entire mathematical theory will in many cases rely on many math-intrinsic questions, too. It would be great if the materials reflect this - so that they really present &amp;quot;how and why to build a mathematical theory from the point of view of mathematicians&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I do not think that it is necessary that the the course evolves &amp;quot;along historical lines&amp;quot;. So it's ok for me if the questions that motivate the theory have occurred only in retrospective and do thus not necessarily represent the original motivations that drove the historical development of the topic.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm happy with course materials for any mathematical topic - although I suspect such materials will be easier to develop for some topics (for instance, linear algebra) than for others (for instance, point set topology).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastEditorDisplayName="user18187" LastEditDate="2023-08-15T05:41:27.317" LastActivityDate="2023-08-15T05:41:27.317" Title="Course materials for developing a mathematical theory from &quot;natural questions to ask&quot;" Tags="|reference-request|textbooks|concept-motivation|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26663" PostTypeId="1" AcceptedAnswerId="26667" CreationDate="2023-08-15T04:02:28.397" Score="1" ViewCount="268" Body="&lt;p&gt;What would you suggest as the best way to deal with students' questions that seem irrelevant to their upcoming exams?&lt;/p&gt;&#xA;&lt;p&gt;When I was studying for my university-entrance exam, I came across a couple of university-level integration questions that I couldn't answer. When I asked my school teacher for help, he advised me to ignore them as they are irrelevant to the exam.&lt;/p&gt;&#xA;&lt;p&gt;In another incident, when approached for help with a question from a Physics textbook, my school teacher said that the question is important but he didn't know how to answer it. I eventually got the answer from the educational service of a state radio channel, then the same question appeared in the following year's physics exam.&lt;/p&gt;&#xA;&lt;p&gt;I think that in addition to exam-related knowledge, having a healthy curiosity about our teaching subject is important when assisting students with relatively novel issues.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-08-24T16:27:24.687" LastActivityDate="2023-08-24T16:27:24.687" Title="Responding to students' questions that aren't directly relevant to their exams" Tags="|secondary-education|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26664" PostTypeId="2" ParentId="26663" CreationDate="2023-08-15T11:14:01.140" Score="8" Body="&lt;p&gt;Here are a few thoughts to assist with responding to student inquiries which address some of the scenarios mentioned with the guiding principle that every question has an appropriate time and place:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Instructors have limited knowledge just like students, so if a question is raised for which the answer is not obvious, there's nothing wrong with either showing the student how you would figure it out (working it out, asking a colleague, Math SE, etc.) if there are not immediate competing demands on your time. If there are competing demands, you can commit to get back to them and provide a timetable for your response.&lt;/li&gt;&#xA;&lt;li&gt;If the answer is obvious but is best answered using some more advanced method outside of the scope of the course, then the most appropriate time to revisit it is after the student has become familiar with that topic. An approach would be to name-drop the topic and course (e.g., interesting integral -- it may be easiest solved using a technique in complex analysis).&lt;/li&gt;&#xA;&lt;li&gt;If the answer is obvious and easily answered using methods from the course but there is a limited amount of time (e.g., last in-class review session before the final) and it is not relevant to the exam, I think it is fair to defer the question (e.g., &amp;quot;see me after&amp;quot;, send an email) or possibly say that we don't have time to answer that at the moment but make a comment about how you would answer it. If all of the above but no demands on time, I think the instructor should answer because relevant student questions should not be set aside solely based on whether the content will appear on the exam.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2023-08-15T11:14:01.140" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26665" PostTypeId="2" ParentId="17823" CreationDate="2023-08-15T19:22:51.920" Score="0" Body="&lt;p&gt;The &amp;quot;c&amp;quot; in limit notation is called the index (or at least that's what I have always been taught).&lt;/p&gt;&#xA;" OwnerUserId="22754" LastActivityDate="2023-08-15T19:22:51.920" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26666" PostTypeId="2" ParentId="17823" CreationDate="2023-08-16T13:28:09.400" Score="-1" Body="&lt;p&gt;Not standard but how about: the limit of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; or the limit of the independent variable?&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2023-08-16T13:28:09.400" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26667" PostTypeId="2" ParentId="26663" CreationDate="2023-08-17T13:12:01.670" Score="10" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;What would you suggest as the best way to deal with students questions&#xA;which seems like not relevant to standards of exam comparing with past&#xA;papers?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;First, commend the student.&lt;/p&gt;&#xA;&lt;p&gt;Second, either start the student on an answer, or point them in the right direction, or say &amp;quot;let's figure it out&amp;quot; or .... Here's a story.&lt;/p&gt;&#xA;&lt;p&gt;A woman I know was a math teacher at Hunter High School in New York City (this is a school for gifted kids).  One student in her calculus class asked:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;I know about 2nd derivatives, 3rd derivatives, even -1st derivatives, but what about fractional derivatives?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;My friend didn't know. But they went to the library (this was before the internet) and found a professor in Germany who was an expert in these. The student wrote to the professor and they wound up writing a paper together for some mathematics journal.&lt;/p&gt;&#xA;&lt;p&gt;And remember what Plutarch said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A mind is not a vessel to be filled, but a fire to be kindled.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;At a minimum, don't be a fire extinguisher.&lt;/p&gt;&#xA;" OwnerUserId="1663" LastActivityDate="2023-08-17T13:12:01.670" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26668" PostTypeId="2" ParentId="18580" CreationDate="2023-08-17T21:28:08.810" Score="2" Body="&lt;p&gt;I've been doing private tutoring this year (2023) in the US and realize basically none of the students (age 13-16) can convert between fractions and decimals. Like even simple ones like 1/4 is 0.25; However, they can do fraction math like fraction addition and division. This is a real pain because many problems are much easier to solve when you can fluidly go back and forth between the basic ones. I understand not needing to know 63/89 off the top of your head, but like 1/5?? That really should be common knowledge. The common core is really under performing on this topic. I agree we WeCanLearnAnything that this should be learned around age 8 or 9 but more complicated fraction-to-decimals should be learned later or maybe not at all.&lt;/p&gt;&#xA;" OwnerUserId="22757" LastActivityDate="2023-08-17T21:28:08.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26671" PostTypeId="2" ParentId="26660" CreationDate="2023-08-18T19:14:04.540" Score="1" Body="&lt;p&gt;For the first, give them a shape that is 1 meter on a side. Then let them try to fill it with four shapes that are each .25 m on a side.&lt;/p&gt;&#xA;&lt;p&gt;For the second, get several other shapes that are 1 square meter but not square. Then cut them up so they fit in the first shape.&lt;/p&gt;&#xA;" OwnerUserId="1663" LastActivityDate="2023-08-18T19:14:04.540" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26672" PostTypeId="1" AcceptedAnswerId="26673" CreationDate="2023-08-19T12:09:43.173" Score="3" ViewCount="256" Body="&lt;p&gt;Say there's an exam question whose answer is &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to the power of negative one. Two ways of writing this are &lt;span class=&quot;math-container&quot;&gt;$x^{-1}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I know that questions will sometimes request an answer without negative exponents, but if no such directions are given, is one notation preferable to the other?&lt;/p&gt;&#xA;&lt;p&gt;Context: I'm a middle and high school mathematics teacher and follow the International Baccalaureate system.&lt;/p&gt;&#xA;&lt;p&gt;Edit: Just to be clear, I'm not asking about marking anything correct or incorrect, or simplifying, or trying to fuss over something insignificant (or worse, encourage students to do so). I was just wondering if there was a reason to prefer one form over another. Seems that it can be context dependent (calculus, trigonometry, units, etc.) but in general, one form isn't to be preferred over the other.&lt;/p&gt;&#xA;" OwnerUserId="15763" LastEditorUserId="15763" LastEditDate="2023-08-21T11:51:11.997" LastActivityDate="2023-08-21T11:51:11.997" Title="Is there a preferred way to format a negative exponent?" Tags="|notation|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26673" PostTypeId="2" ParentId="26672" CreationDate="2023-08-20T12:27:42.067" Score="4" Body="&lt;p&gt;Both are proper.&lt;/p&gt;&#xA;&lt;p&gt;If the entire answer is &lt;span class=&quot;math-container&quot;&gt;$x^{-1}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$1/x$&lt;/span&gt; then I would mark both correct and would not fuss about it. I think this sort of fuss is one of many reasons a lot of kids don't like math.&lt;/p&gt;&#xA;&lt;p&gt;If this is part of an answer, then I think it is context dependent. Since these are fairly young kids, I'd still mark everything correct, but might put a note something like &amp;quot;in XXXX we usually write this as XXXX&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, in a specific teaching context, you might be asking for one or the other, but you seem to be excluding those.&lt;/p&gt;&#xA;" OwnerUserId="1663" LastActivityDate="2023-08-20T12:27:42.067" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26674" PostTypeId="2" ParentId="26672" CreationDate="2023-08-20T20:52:42.607" Score="7" Body="&lt;p&gt;One of my biggest frustrations in teaching (and in speaking to other instructors, particularly elementary and high school instructors) is a belief that mathematics is prescriptivist and inscrutable.  There is an idea that certain notations or processes are &amp;quot;right&amp;quot;, and others are &amp;quot;wrong&amp;quot;.  For example,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is &lt;span class=&quot;math-container&quot;&gt;$\frac{\sqrt{2}}{2}$&lt;/span&gt; preferable to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\sqrt{2}}$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;li&gt;Is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; preferable to &lt;span class=&quot;math-container&quot;&gt;$\frac{50}{100}$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;li&gt;Is &lt;span class=&quot;math-container&quot;&gt;$(x-a)(x+a)$&lt;/span&gt; preferable to &lt;span class=&quot;math-container&quot;&gt;$x^2 - a^2$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;li&gt;Is &lt;span class=&quot;math-container&quot;&gt;$\tan(\theta)$&lt;/span&gt; preferable to &lt;span class=&quot;math-container&quot;&gt;$\frac{\sin(\theta)}{\cos(\theta)}$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In each of these cases, I am quite certain that students would argue with me if I wrote one of the latter expressions, or asserted that any of the latter expressions were in any way superior.  Indeed, nearly every semester, I have some trig student get &lt;em&gt;angry&lt;/em&gt; at me for writing &lt;span class=&quot;math-container&quot;&gt;${1}/{\sqrt{2}}$&lt;/span&gt;.  Usually, when I drill down into why, it is, essentially, &amp;quot;My high school teacher told me it was wrong!&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The best thing to do (in my opinion) is to use whatever notation is most appropriate to a particular context, and to avoid optimizing early, where&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the &amp;quot;most appropriate&amp;quot; notation is (a) whatever notation makes it easier to perform the next step of a computation, and/or (b) whatever notation most clearly communicates the key idea, and&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;optimizing early&amp;quot; means rewriting an expression into a different form before knowing that the rewrite is useful or meaningful (e.g. expanding the denominator of a rational function before realizing that it needs to be added to another rational function, and having the denominator factored is going to make that easier).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, to answer the headline question:  both &lt;span class=&quot;math-container&quot;&gt;$x^{-1}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt; are correct.  In &lt;em&gt;most&lt;/em&gt; contexts, there is no reason to prefer one over the other, and students should be comfortable working with &lt;em&gt;either&lt;/em&gt; form.  Unless there is some context in which one form or the other is easier to work with, there is absolutely no reason that students should be &lt;em&gt;told&lt;/em&gt; to prefer one version over another.  Don't be pedantic. :/&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2023-08-20T20:52:42.607" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26675" PostTypeId="1" CreationDate="2023-08-20T22:35:13.187" Score="1" ViewCount="103" Body="&lt;p&gt;I'm a high school Mathematics teacher and I want to issue some research articles for highschool students to improve their math problem resolve skills. Is there any valuable Math journal for high school students (grades 12, 11) that can issue Mathematics articles and be citeable by Google Scholar?&lt;/p&gt;&#xA;" OwnerUserId="22767" LastActivityDate="2023-08-23T15:49:24.420" Title="highschool's mathematics journal which citable in Google Scholar" Tags="|secondary-education|teaching|teacher-preparation|teacher-development|journals|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26676" PostTypeId="1" CreationDate="2023-08-21T09:11:03.043" Score="2" ViewCount="267" Body="&lt;p&gt;As a maths tutor, some students I have tutored don't just make the odd mistake in arithmetic (including fractions) and algebra: they make every possible mistake and regularly.&lt;/p&gt;&#xA;&lt;p&gt;My go-to approach for helping these students is to show them how to correctly apply the rule they have just made a mistake on, as and when they make that mistake.&lt;/p&gt;&#xA;&lt;p&gt;However, for these students, their mistakes are relentless, and whilst the % of errors may decrease over time, once I show them a correct way of doing it, this % may only decrease from 80% of the time to 20% of the time a few months later, which still isn't very good, as for arithmetic and algebra, you need to be striving for errors basically 0% of the time.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;But 80% to 20% is good improvement&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No. This is algebra/arithmetic. This is not proof writing, which is a skill that you improve with over time. If you stick to the rules of algebra/arithmetic, then you shouldn't really be making errors at all. And yes, I do go through various checking methods with my students.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, such students will make every possible error on most future topics. I believe this is due to their, &amp;quot;I don't care about maths: I just want to pass my exams&amp;quot; mindset, which is counter-productive, as the only way to improve on maths is to spend time doing maths (exercises). Anyway, the point is that I think these students need a &lt;strong&gt;radical change in mindset/approach&lt;/strong&gt; in order to succeed in maths.&lt;/p&gt;&#xA;&lt;p&gt;I wonder if, for such students, an entirely different overall approach is more productive; namely, the following &amp;quot;sceptical&amp;quot; approach, which is the one I have always gone by:&lt;/p&gt;&#xA;&lt;p&gt;Don't use a rule until 100% convinced:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The rule works&lt;/li&gt;&#xA;&lt;li&gt;You know when the rule works and when it does not work&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;and more importantly,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you are not 100% convinced the rule works, do not use it&#xA;(other than to test it), and keep questioning it's validity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They can even write an &amp;quot;investigation into the validity of rule X&amp;quot;, up until the point they are convinced the rule works.&lt;/p&gt;&#xA;&lt;p&gt;Then, at some point, they will be naturally more and more convinced of it's correctness, as they will have run out of questions as to why the rule might not work.&lt;/p&gt;&#xA;&lt;p&gt;There are many obvious benefits of this approach, including the fact that it forces the student to actively engage in the material.&lt;/p&gt;&#xA;&lt;p&gt;However, I'm doubtful of this approach because I doubt these type of students' abilities to correctly apply scepticism and questioning of the method. Another criticism is that this approach takes more time to carry out at first; however I don't buy this criticism as the approach probably saves time in the long run.&lt;/p&gt;&#xA;&lt;p&gt;I'm interested to hear what other maths educators think. Has anyone tried this approach with relentless error-ridden students, and did this approach work?&lt;/p&gt;&#xA;" OwnerUserId="12691" LastActivityDate="2023-08-23T13:00:24.953" Title="Scepticism as the cornerstone for not making mistakes in arithmetic/algebra etc, especially for students who relentlessly make every possible error" Tags="|algebra|students-mistakes|arithmetic|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26677" PostTypeId="2" ParentId="26672" CreationDate="2023-08-21T10:01:49.160" Score="3" Body="&lt;p&gt;I would say that both are ok, but you might be careful when dealing with inverse functions, like &lt;span class=&quot;math-container&quot;&gt;$\sin^{(-1)}(x)$&lt;/span&gt;: does it mean &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\sin(x)}$&lt;/span&gt; or does it mean &lt;span class=&quot;math-container&quot;&gt;$\arcsin(x)$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-08-21T10:01:49.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26678" PostTypeId="2" ParentId="26676" CreationDate="2023-08-22T00:21:10.690" Score="2" Body="&lt;p&gt;In part agreeing with @fedja's comment... but perhaps aggressively pursuing the idea wanting to have your students look critically at &amp;quot;feelings/impulses&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Namely, I am not a fan of math-as-rules. I would want to sell the idea that math &amp;quot;is about something&amp;quot;. So no &amp;quot;authority&amp;quot; has the power to dictate what it is, beyond conventions on the order of speed limits or parking restrictions on weekends.&lt;/p&gt;&#xA;&lt;p&gt;Conceivably, some students could be reached by suggesting that math is a &lt;em&gt;narrative&lt;/em&gt; about (certain aspects of) &lt;em&gt;reality&lt;/em&gt;, albeit in a specialized language.&lt;/p&gt;&#xA;&lt;p&gt;One analogy is that, although (e.g.) English has various approximate grammar rules (changing with time, of course), most discussions of math in English are not marred by grammatical errors. Yes, sometimes accidental ambiguities dues to grammar-fails &lt;em&gt;can&lt;/em&gt; impede the mathematical narrative...&lt;/p&gt;&#xA;&lt;p&gt;One bottle-neck here is that many students seem to have an accumulated math culture that math &lt;em&gt;is&lt;/em&gt; about (ineffable) rules... So it's hard to &lt;em&gt;credibly&lt;/em&gt; argue against that. E.g., even in the grad courses I teach, it's only been in the last few years (with an ever-balder head, grayer beard, etc.) that students seem willing to believe me when I argue that there is a genuine reality behind definitions and notation... often contradicting their impression from typical undergrad (and grad) textbooks. :)&lt;/p&gt;&#xA;&lt;p&gt;So, yes, it is tricky to talk to people who both want rules and want to not have to follow them &lt;em&gt;carefully&lt;/em&gt;... No easy solution. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2023-08-22T00:21:10.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26679" PostTypeId="1" AcceptedAnswerId="26698" CreationDate="2023-08-23T01:32:48.230" Score="4" ViewCount="324" Body="&lt;p&gt;how to convince or demonstrate to a high school student who does not know differential and integral calculus that the geodesics of a sphere are arcs of great circles?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="60" LastEditDate="2023-08-23T15:28:23.750" LastActivityDate="2023-08-27T06:57:59.627" Title="How to convince a student without calculus that great circles are geodesics in a sphere?" Tags="|proofs|geometry|" AnswerCount="7" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26680" PostTypeId="2" ParentId="26679" CreationDate="2023-08-23T12:43:40.430" Score="8" Body="&lt;p&gt;Take a physical sphere such as a beach ball, and a string. Pick two points. Hold the string down with one finger at one point then stretch it to the second point. Next, holding the string tight at both points, continue to wrap the string. You find the wrapping will be a great circle. You can also try this with a rubber band.&lt;/p&gt;&#xA;&lt;p&gt;A less &amp;quot;physical&amp;quot; approach is this. Without loss of generality after rotation, one of the two points is the North Pole. From this configuration, it is rather intuitive that the shortest path to the second point is to follow the longitudinal circle from the North Pole to the second point.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-08-23T15:32:12.120" LastActivityDate="2023-08-23T15:32:12.120" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26681" PostTypeId="2" ParentId="26676" CreationDate="2023-08-23T13:00:24.953" Score="1" Body="&lt;p&gt;I'm sorry, but I find your attitude towards your students somewhat disturbing. In particular:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is algebra/arithmetic. This is not proof writing, which is a&#xA;skill that you improve with over time. If you stick to the rules of&#xA;algebra/arithmetic, then you shouldn't really be making errors at all.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;First, algebra and arithmetic are also &amp;quot;skill(s) that you improve with over time&amp;quot;.  So is just about everything else. When we are born, we can pee and we can suck and that's about it. Everything else is learned. Including arithmetic and algebra (but also tying your shoes).&lt;/p&gt;&#xA;&lt;p&gt;Second &amp;quot;stick to the rules&amp;quot; is exactly the wrong thing. Which rule? If you look at various social media, such as Facebook, you will see that knowledge of which rule to apply when is lacking and, while it may be obvious to you and to me, it is not obvious to all. Clearly, it is not obvious to your students.&lt;/p&gt;&#xA;&lt;p&gt;I also doubt that the problem is that your students don't think the rule is &amp;quot;correct&amp;quot;. I would be surprised if a student said that, and, if one did, it would probably be a student who was very good at math, not one who is very bad.&lt;/p&gt;&#xA;&lt;p&gt;Rather, if you have taught the rule well, so that they understand it, then the problem is likely to be which rule to apply when, and in what order.&lt;/p&gt;&#xA;&lt;p&gt;Also, if your students' only goal is to pass exams, then you can motivate them by saying &amp;quot;well, the only way to pass exams is to practice&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Finally, if your students get down to 20% errors, then they will, in fact, pass their exams. Not every  student does any math after what is required, and many would be pleased to get an 80.  Indeed, those are precisely the students who are likely to hire a tutor.  Students who hire tutors get worse grades than those who don't, because students who are doing well don't hire them.&lt;/p&gt;&#xA;" OwnerUserId="1663" LastActivityDate="2023-08-23T13:00:24.953" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26683" PostTypeId="2" ParentId="26675" CreationDate="2023-08-23T15:49:24.420" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://www.maa.org/press/periodicals/college-mathematics-journal/the-college-mathematics-journal&quot; rel=&quot;nofollow noreferrer&quot;&gt;The College Mathematics Journal&lt;/a&gt; is the closest I know to what you are looking for, because it is geared toward &amp;quot;the first two years&amp;quot;, which nowadays would correspond to high school level in the better schools.&lt;/p&gt;&#xA;&lt;p&gt;Other than that, there is the &lt;a href=&quot;https://scholarworks.umt.edu/tme/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Montana Mathematics Enthusiast&lt;/a&gt; that publishes popular articles (&amp;quot;Montana&amp;quot; was eventually removed from the title).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://scholarship.claremont.edu/jhm/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Journal of Humanistic Mathematics&lt;/a&gt; is an unusual one that may have some accessible stuff now and then.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2023-08-23T15:49:24.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26684" PostTypeId="2" ParentId="26679" CreationDate="2023-08-24T01:01:45.847" Score="3" Body="&lt;p&gt;Maybe this would work? The &lt;a href=&quot;https://en.wikipedia.org/wiki/Spherical_law_of_cosines&quot; rel=&quot;nofollow noreferrer&quot;&gt;spherical law of cosines&lt;/a&gt; says that, on a sphere of radius &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, if you have a sphercial triangle with edge lengths &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and angles &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\cos \tfrac{c}{R} = \cos \tfrac{a}{R} \cos \tfrac{b}{R} + \sin \tfrac{a}{R} \sin \tfrac{b}{R} \cos C. \qquad (\ast)$$&lt;/span&gt;&#xA;I emphasize that the definition of &amp;quot;spherical triangle&amp;quot; is &amp;quot;a triangle whose sides are great circles&amp;quot;, and this relationship can be proved without knowing in advance that those great circles are geodesics; see the several proofs in the Wikipedia article.&lt;/p&gt;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$\cos C \leq 1$&lt;/span&gt;, we deduce that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\cos \tfrac{c}{R} \geq \cos \tfrac{a}{R} \cos \tfrac{b}{R} - \sin \tfrac{a}{R} \sin \tfrac{b}{R} = \cos \left( \tfrac{a}{R}+\tfrac{b}{R} \right).$$&lt;/span&gt;&#xA;We deduce that &lt;span class=&quot;math-container&quot;&gt;$\tfrac{c}{R} \leq \tfrac{a}{R} + \tfrac{b}{R}$&lt;/span&gt; or, in other words &lt;span class=&quot;math-container&quot;&gt;$c \leq a+b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, if &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_3$&lt;/span&gt; are the corners of the triangle, then the distance &lt;span class=&quot;math-container&quot;&gt;$x_1 \to x_3$&lt;/span&gt; along a great circle is &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; the distance along a path &lt;span class=&quot;math-container&quot;&gt;$x_1 \to x_2 \to x_3$&lt;/span&gt; which takes a great circle segment along each arrow.&lt;/p&gt;&#xA;&lt;p&gt;Then, inductively, for any &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt;, ..., &lt;span class=&quot;math-container&quot;&gt;$x_n$&lt;/span&gt;, the distance along a great circle &lt;span class=&quot;math-container&quot;&gt;$x_1 \to x_n$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; the distance along the path &lt;span class=&quot;math-container&quot;&gt;$x_1 \to x_2 \to \cdots \to x_n$&lt;/span&gt; which takes a great circle segment along each arrow.&lt;/p&gt;&#xA;&lt;p&gt;Then (wave your hands furiously) the length of a great circle path &lt;span class=&quot;math-container&quot;&gt;$x_1 \to x_n$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; the length of any path &lt;span class=&quot;math-container&quot;&gt;$x_1 \to x_n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To do better than this, you need a precise definition of arc length, which may not be somewhere you want to go.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2023-08-24T01:01:45.847" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26685" PostTypeId="1" CreationDate="2023-08-24T07:55:42.603" Score="5" ViewCount="2265" Body="&lt;p&gt;We have theoretical questions in our exams (often in ABCD format).&#xA;For example, I can state the question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For an indefinite integral, does it hold that &lt;span class=&quot;math-container&quot;&gt;$\int  f(g(x)) \cdot g'(x) \, dx  = \int f(x)  dx $&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or should I state the question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f, g: \mathbb{R} \rightarrow \mathbb{R}$&lt;/span&gt; be two strictly&#xA;monotonic differentiable functions with continuous derivatives. Does&#xA;it hold that &lt;span class=&quot;math-container&quot;&gt;$\int  f(g(x)) \cdot g'(x) \, dx  = \int f(x)  dx $&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Surely the 2nd one is more correct.&#xA;But it gets even worse with 2-variable calculus, where one often has to put a bunch of conditions on the most basic theorems. I feel that exams get bloated with conditions, but on the other hand, we should make the questions precise if we grade them. What is your opinion on this?&lt;/p&gt;&#xA;" OwnerUserId="22774" LastEditorUserId="2083" LastEditDate="2023-08-24T13:17:28.540" LastActivityDate="2023-08-27T01:01:35.387" Title="In math exams, how rigorous should the questions be?" Tags="|undergraduate-education|exams|student-question|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="26686" PostTypeId="2" ParentId="26679" CreationDate="2023-08-24T09:03:08.327" Score="6" Body="&lt;p&gt;Answer inspired by Michał Miśkiewicz's comment on the OP.&lt;/p&gt;&#xA;&lt;p&gt;To a high school student:&lt;/p&gt;&#xA;&lt;p&gt;Put an ant on a basketball.  Draw a tiny arrow representing the direction it should walk.  The ant always just puts one foot in front of the other, trying to stay as &amp;quot;straight&amp;quot; as possible.  Where will it end up?&lt;/p&gt;&#xA;&lt;p&gt;Firstly:  do you agree that it has one, and only one, path it can follow?&lt;/p&gt;&#xA;&lt;p&gt;Now, let's draw the great circle containing this point and vector.  Can the ant ever leave the great circle?&lt;/p&gt;&#xA;&lt;p&gt;Imagine that at some time the ant leaves the circle.  It has to leave to one side of the great circle or the other.  However, if it leaves to one side then reflecting the path over the plane containing the great circle would give another path with identical length.  So the ant would have two paths it can follow.  This contradicts what we said earlier about the ant only having one path it can take.  So the ant can never leave the great circle.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-08-24T09:03:08.327" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26687" PostTypeId="2" ParentId="26685" CreationDate="2023-08-24T13:08:52.937" Score="12" Body="&lt;p&gt;On the one hand, the assumptions &lt;em&gt;are&lt;/em&gt; an indispensable part of any theorem and many mistakes (well beyond the student homework) have been made exactly because somebody didn't understand that the conclusion of a certain result holds only under certain conditions which might be well-hidden in the sentences &amp;quot;assuming that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; are as in Lemma 5&amp;quot;. So remembering formulae without clear understanding what classes of functions they can be applied to is not enough.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, a lot of times we start teaching results in simple situations that are very far from full generality in which they really hold, so it is important to emphasize that some conditions are imposed just because of the students' limited knowledge, not because they are really needed.&lt;/p&gt;&#xA;&lt;p&gt;The best practice is to bite a bit from both ends, i.e., to state and prove the theorem under assumptions that are easy to understand and check, but to show some really pathological counterexamples too so that the students would know that without assuming anything at all things can go awry.&lt;/p&gt;&#xA;&lt;p&gt;If it were a decent analysis course, I would phrase your particular test question as follows.&lt;/p&gt;&#xA;&lt;p&gt;Assuming that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; are defined on the entire real line, &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is non-decreasing, and &lt;span class=&quot;math-container&quot;&gt;$a,b\in\mathbb R$&lt;/span&gt;, the change of variable formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_a^b f(g(x))g'(x)dx=\int_{g(a)}^{g(b)} f(y)dy&#xA;$$&lt;/span&gt;&#xA;holds for the Riemann integral&lt;/p&gt;&#xA;&lt;p&gt;A) For arbitrary locally Riemann integrable &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and differentiable &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;B) For arbitrary continuous &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and continuously differentiable &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;C) For arbitrary locally Riemann integrable &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and differentiable &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; with locally Riemann integrable derivative.&lt;/p&gt;&#xA;&lt;p&gt;D) For arbitrary locally Riemann integrable &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and continuously differentiable &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Choose the most general option that is correct.&lt;/p&gt;&#xA;&lt;p&gt;In calculus you, probably, do not want to go that far, but you can make a similar list with options that are more obviously true or false.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-08-24T13:08:52.937" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26688" PostTypeId="2" ParentId="26679" CreationDate="2023-08-24T14:36:49.000" Score="1" Body="&lt;p&gt;Since you ask specifically for a way to &amp;quot;convince&amp;quot; the student rather than to provide a proof, I would propose the following.  We can assume for simplicity that the sphere has unit radius, and measure angles in degrees.  Move one of the points, say &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, to the North pole by an isometry (e.g., a suitable rotation).  If there were a path from  &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; shorter than the smaller arc of great circle connecting them, we could continue further beyond &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; along the longitude through &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, to arrive to the south pole &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; by a path of combined length less than &lt;span class=&quot;math-container&quot;&gt;$180^\circ$&lt;/span&gt;.  But can one reach north pole to south pole without crossing all the latitudes?  Certainly not.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2023-08-24T14:36:49.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26689" PostTypeId="2" ParentId="26679" CreationDate="2023-08-24T15:16:33.493" Score="1" Body="&lt;p&gt;There is a problem here with, er, circular definitions. How are you going to define a great circle? My preferred definition would be that it's a geodesic (one that's been extended as far as possible). If you want to make an argument that's at all non-vacuous, you need to give some other characterization of a great circle -- or at least a partial characterization.&lt;/p&gt;&#xA;&lt;p&gt;One partial characterization would be that a great circle splits a sphere into two equal hemispheres. Now suppose a plane heads east from New York and flies straight and level, which is actually a physicist's definition of geodesic -- inertial motion. So now we have at least some kind of independent characterization of these two notions, and it can be non-vacuous to try to relate them.&lt;/p&gt;&#xA;&lt;p&gt;One might expect that the plane would stay on a line of latitude. This would split the globe into two unequal parts, and would therefore not be a great circle. But then we could draw a similar path that was a mirror image of the first one, with the smaller, right-hand region being centered on some spot off the coast of Peru rather than on the south pole. Since the poles have no geometrical significance, there is no reason to prefer one of these paths to the other. Appealing to physical intuition, inertial motion should be unique. There is no reason for the plane to prefer one of these circles to the other. Therefore the line of latitude can't be a geodesic.&lt;/p&gt;&#xA;" OwnerUserId="22776" LastActivityDate="2023-08-24T15:16:33.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26691" PostTypeId="1" CreationDate="2023-08-25T22:27:43.430" Score="2" ViewCount="339" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JmClk.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JmClk.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am interested in teaching maths visually.  in page 36 of Growing ideas of number (by John N Crossley) the following image appears, yet I cannot fully grasp how to interpreted it.&lt;/p&gt;&#xA;" OwnerUserId="12046" LastEditorUserId="12046" LastEditDate="2023-08-26T00:22:04.317" LastActivityDate="2023-08-26T19:04:15.127" Title="Geometric line: constructing fractions" Tags="|geometry|fractions|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26692" PostTypeId="2" ParentId="26691" CreationDate="2023-08-25T23:23:54.137" Score="5" Body="&lt;p&gt;I don't think this diagram would help kids understand fractions. But I do like how it makes me think.&lt;/p&gt;&#xA;&lt;p&gt;Your problem might be that the 1's are not to scale.&lt;/p&gt;&#xA;&lt;p&gt;If we are given that the 3 lines (or line segments) are parallel (which I hope was mentioned in the book), then the slopes are all the same. If we are given the numbers on the y-axis and the 1 on the x-axis, we can use those equal slopes to find the other numbers/points on the x-axis. Call those points a and b. Then looking at the common slope gives us n/1 = 1/a, so a = 1/n. And n/1 = m/b, so nb = m and b = m/n.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps constructing diagrams like this in geogebra yourself would help you. I started to do that &lt;a href=&quot;https://www.geogebra.org/classic/tvbuzf9r&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-08-26T02:42:47.857" LastActivityDate="2023-08-26T02:42:47.857" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26693" PostTypeId="2" ParentId="26679" CreationDate="2023-08-26T02:16:48.917" Score="3" Body="&lt;p&gt;Determine the shortest route from New York City USA to Rome Italy using a piece of string on a globe. Both cities are south of the 45th parallel and yet the the shortest route deviates considerably to its north. The string traces the geodesic (the great circle arc) between the two cities. It is the shortest and &amp;quot;straightest&amp;quot; line on the surface of the earth between both cities.&lt;/p&gt;&#xA;&lt;p&gt;Similarly, in the southern hemisphere, determine the shortest route from Perth Australia to Durban South Africa. It deviates considerably to the south.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-08-26T17:38:36.630" LastActivityDate="2023-08-26T17:38:36.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26694" PostTypeId="1" AcceptedAnswerId="26696" CreationDate="2023-08-26T15:42:30.447" Score="0" ViewCount="74" Body="&lt;p&gt;I struggle with connecting graphs of linear equation with algebraic form like &lt;span class=&quot;math-container&quot;&gt;$x+y=p$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;How do I develop the intuition that it represents a line that is sloping down and passes through value &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis in &lt;span class=&quot;math-container&quot;&gt;$XY$&lt;/span&gt; plane?&lt;/p&gt;&#xA;" OwnerUserId="12610" LastEditorUserId="8190" LastEditDate="2023-08-28T06:49:23.600" LastActivityDate="2023-08-28T21:48:37.297" Title="How to intuitively connect Linear Equation in two variables and the graph of them?" Tags="|linear-algebra|" AnswerCount="2" CommentCount="1" ClosedDate="2023-08-28T22:14:20.847" ContentLicense="CC BY-SA 4.0" />
  <row Id="26695" PostTypeId="2" ParentId="26691" CreationDate="2023-08-26T16:59:07.040" Score="8" Body="&lt;p&gt;In &lt;em&gt;Growing Ideas of Number&lt;/em&gt;, Crossley provides this diagram, Figure 3.4, in a discussion of the notion of the &amp;quot;geometric line.&amp;quot; From the perspective of modern mathematics, all of the points on the continuous &amp;quot;geometric line&amp;quot; correspond to real numbers, given some choice of 0 and 1.&#xA;Many early mathematicians did not think of numbers this way. They saw numbers as corresponding to measurements taken using some unit length. We start with a unit length, and all other numbers must be constructed from this unit length. &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-319-99386-7_5&quot; rel=&quot;noreferrer&quot;&gt;Bass (2018)&lt;/a&gt; calls these two perspectives the &amp;quot;occupation narrative&amp;quot; and the &amp;quot;construction narrative.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In the construction narrative, the integers and the rational numbers are legitimate numbers because they can be constructed. This is what Crossley is talking about in the following paragraph.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is easy to take some measure as a unit, and then to measure off all integer multiples of that unit. I shall ignore the question of where negative numbers come from... Then it is easy to construct fractions of these multiples by drawing similar triangles. See Figure 3.4.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The purpose of Figure 3.4 is to show how, from a segment of length 1, we can construct segments of length &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{m}{n}$&lt;/span&gt;. Here is a construction procedure that you can try to follow while looking at the diagram.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;On a horizontal line, mark a segment of length 1. Call it OC.&lt;/li&gt;&#xA;&lt;li&gt;Construct a vertical line passing through O.&lt;/li&gt;&#xA;&lt;li&gt;On the vertical line, mark a segment OU. (U corresponds to the &amp;quot;1&amp;quot; on the vertical axis.)&lt;/li&gt;&#xA;&lt;li&gt;Copy segment OU end-to-end along the vertical line to produce segments OM and ON whose lengths are &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; times OU and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; times OU respectively.&lt;/li&gt;&#xA;&lt;li&gt;Draw line CN.&lt;/li&gt;&#xA;&lt;li&gt;Construct a line parallel to CN passing through U.&lt;/li&gt;&#xA;&lt;li&gt;Mark the intersection of the line just constructed and the horizontal line as A.  Triangle AOU is similar to triangle CON, and OA has length &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{n}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Construct a line parallel to CN passing through M.&lt;/li&gt;&#xA;&lt;li&gt;Mark the intersection of the line just constructed and the horizontal line as B. Triangle BOM is similar to triangle CON, and OB has length &lt;span class=&quot;math-container&quot;&gt;$\frac{m}{n}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This is related to the construction procedure of &amp;quot;dividing a line segment into equal parts.&amp;quot; &lt;a href=&quot;https://www.mathopenref.com/constdividesegment.html&quot; rel=&quot;noreferrer&quot;&gt;Here's a step-by-step animation and proof of that procedure.&lt;/a&gt; Instead of similar right triangles, it uses similar acute triangles. The underlying principle in both constructions is that the side lengths of similar triangles are in proportion.&lt;/p&gt;&#xA;&lt;p&gt;As Sue mentions in &lt;a href=&quot;https://matheducators.stackexchange.com/a/26692/20719&quot;&gt;her answer&lt;/a&gt;, Figure 3.4 is a bit confusing because the axes don't use the same scale. Perhaps this is because the scale of the vertical axis doesn't matter for the construction. We just need similar triangles whose corresponding sides are in the ratios &lt;span class=&quot;math-container&quot;&gt;$1:n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m:n$&lt;/span&gt;. We can interpret the &amp;quot;1&amp;quot; on the vertical axis as an artifact of this part of the construction.  The &amp;quot;1&amp;quot; on the horizontal axis marks the unit length that we want to divide into &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{m}{n}$&lt;/span&gt;. Or perhaps the “1” on the vertical axis is meant to illustrate that, from a unit length, lengths &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; are also constructible.&lt;/p&gt;&#xA;&lt;p&gt;This construction gives us the &amp;quot;rational number line,&amp;quot; but early mathematicians knew that not all lengths could be represented in this way, and so this is not the &amp;quot;geometric line.&amp;quot; But should the geometric line be discrete, like the rational number line, or continuous? According to Crossley, both are logically possible. Crossley attributes the resolution of this problem to Dedekind, referring to his discussion in Section 5.1 of Dedekind's construction of the real numbers.&lt;/p&gt;&#xA;&lt;p&gt;In the occupation narrative, the continuous nature of the geometric line is assumed from the beginning. As Bass says, &amp;quot;In contrast with the construction narrative, wherein more and more points are installed to &lt;em&gt;build&lt;/em&gt; the line, all points are present from the start in the occupation narrative, and more of them acquire numerical names across the curriculum.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-08-26T19:04:15.127" LastActivityDate="2023-08-26T19:04:15.127" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26696" PostTypeId="2" ParentId="26694" CreationDate="2023-08-26T17:36:30.493" Score="1" Body="&lt;p&gt;Better to write  it  as  y=-x+p  you see  that at x=0 y=p and x=p y=0  so for p&amp;gt;0 it goes down from (0,p) to (p,0)&#xA;if p&amp;lt;0 it goes down from (0,p)to (p,0)&lt;/p&gt;&#xA;" OwnerUserId="21051" LastActivityDate="2023-08-26T17:36:30.493" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26698" PostTypeId="2" ParentId="26679" CreationDate="2023-08-27T06:57:59.627" Score="5" Body="&lt;p&gt;Recall that a great circle is the intersection of the unit sphere and a plane passing through the origin.  A key point is that for short arcs of great circles, the ratio of euclidean distance between endpoints to the length of the arc is close to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$(x,y,z)$&lt;/span&gt; be the standard coordinates in &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^3$&lt;/span&gt;.  We work in spherical coordinates &lt;span class=&quot;math-container&quot;&gt;$(\phi,\theta)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$z=\cos\phi$&lt;/span&gt;.  Let &lt;span class=&quot;math-container&quot;&gt;$c(t)$&lt;/span&gt; be a path on the sphere joining north pole &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to south pole &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$c(0)=A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c(1)=B$&lt;/span&gt;.  Denote by &lt;span class=&quot;math-container&quot;&gt;$\phi(t)$&lt;/span&gt; the &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;-coordinate of the point &lt;span class=&quot;math-container&quot;&gt;$c(t)$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$\phi(0)=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\phi(1)=\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now suppose the path &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; has length &lt;span class=&quot;math-container&quot;&gt;$|c|$&lt;/span&gt; strictly less than &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, and write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$|c|=\pi-\epsilon$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\epsilon&amp;gt;0$&lt;/span&gt;.  Let &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; be an integer.  Consider a partition &lt;span class=&quot;math-container&quot;&gt;$0=t_0&amp;lt;t_1&amp;lt;\cdots&amp;lt;t_N=1$&lt;/span&gt; that breaks up &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; into paths &lt;span class=&quot;math-container&quot;&gt;$c([t_i,t_{i+1}])$&lt;/span&gt; of equal length &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi-\epsilon}{N}$&lt;/span&gt;.  Consider the corresponding sequence &lt;span class=&quot;math-container&quot;&gt;$\phi(t_0),\ldots,\phi(t_N)$&lt;/span&gt;.   Choose &lt;span class=&quot;math-container&quot;&gt;$t_k$&lt;/span&gt; which maximizes the difference &lt;span class=&quot;math-container&quot;&gt;$\phi(t_{k+1})-\phi(t_k)$&lt;/span&gt;.  Then &lt;span class=&quot;math-container&quot;&gt;$\phi(t_{k+1}-\phi(t_k)\geq\frac{\pi}{N}$&lt;/span&gt;.  Then we obtain &lt;span class=&quot;math-container&quot;&gt;$$\frac{|c([t_k,t_{k+1}])|}{\phi(t_{k+1})-\phi(t_k)}\leq\frac{\pi-\epsilon}{\pi}=1-\frac{\epsilon}{\pi}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus the ratio is uniformly bounded way from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.  As &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; tends to infinity, the length of the path &lt;span class=&quot;math-container&quot;&gt;$c([t_k,t_{k+1}])$&lt;/span&gt; tends to zero.  But for such paths, the ratio of the Euclidean distance &lt;span class=&quot;math-container&quot;&gt;$|c(t_{k+1})-c(t_k)|$&lt;/span&gt; to length of arc of great circle is bounded below by a bound that tends to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; by elementary plane trigonometry (requiring no calculus).  Therefore for sufficiently large &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;, we will have &lt;span class=&quot;math-container&quot;&gt;$$\frac{c(t_{k+1})-c(t_k)}{\phi(t_{k+1})-\phi(t_k)}&amp;lt;1-\frac{\epsilon}{2\pi},$$&lt;/span&gt; a contradiction.  The contradiction shows (without calculus and using only elementary plane trigonometry) that a path joining north and south pole must have length at least &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, and therefore great circles are length-minimizing.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2023-08-27T06:57:59.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26699" PostTypeId="1" CreationDate="2023-08-27T13:28:53.780" Score="0" ViewCount="40" Body="&lt;p&gt;I think proofs without words is much important topic when we want to improve students interest in subject using their skills other than in mathematics. Recently I could able to find that kind of proof for &lt;span class=&quot;math-container&quot;&gt;$$arccot(3) + arccot(7) +arccot(13)+...= (π/4) $$&lt;/span&gt;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$arccot(2) + arccot(8)+arccot(18)+...= (π/4) $$&lt;/span&gt;&#xA;I thought GeoGebra is the best app to draw the figures and share to others in full details. Here you have the links&#xA;&lt;a href=&quot;https://www.geogebra.org/m/j5yvst8s&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geogebra.org/m/j5yvst8s&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.geogebra.org/m/vm4ffyvj&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geogebra.org/m/vm4ffyvj&lt;/a&gt; &lt;br/&gt;&#xA;I would like to know whether there are much better options and how to use these apps in much efficient ways.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-08-27T13:38:27.243" LastActivityDate="2023-08-27T13:38:27.243" Title="Apps to make mathematics much interesting by sharing creative ideas with others" Tags="|secondary-education|apps|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26700" PostTypeId="1" CreationDate="2023-08-27T14:33:40.303" Score="7" ViewCount="3024" Body="&lt;p&gt;I'm interested in opinions based on experience about how to introduce volume for beginner calculus students. Below I present some observations and specific questions.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In Stewart's book, the volume of a solid &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; that lies between &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=b$&lt;/span&gt; is defined by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$V=\int_a^b A(x)\ dx,$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$A(x)$&lt;/span&gt; is the area of the vertical cross-sectional area of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. This definition comes from vertical slicing.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In the examples of the said book, the following formula that comes from horizontal slicing for the volume of a solid &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; that lies between &lt;span class=&quot;math-container&quot;&gt;$y=c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=d$&lt;/span&gt; is also used:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$V=\int_c^d B(y)\ dy,$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$B(y)$&lt;/span&gt; is the area of the horizontal cross-sectional area of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It seems the said book gives no explanation of the fact that both formulas give the same result when applied to the same solid.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Do you introduce volume for beginners calculus students as defined in Stewart's book?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If so, how do you deal with the fact that both formulas give the same number? Do you ignore this fact or assume that it is obvious? Why?&lt;/li&gt;&#xA;&lt;li&gt;If not, what is your approach? Do you give a definition or just integral formulas? Why?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13585" LastActivityDate="2023-08-30T13:54:14.803" Title="How to properly define volume for beginner calculus students?" Tags="|calculus|geometry|integration|" AnswerCount="5" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="26701" PostTypeId="2" ParentId="26700" CreationDate="2023-08-27T15:00:53.457" Score="17" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$dV$&lt;/span&gt; represents a tiny bit of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$V = \int dV$&lt;/span&gt; says that you can find the volume by adding up all the tiny bits of volume. This is why it is called an &amp;quot;integral;&amp;quot; you need to integrate all the little pieces together into a whole.&lt;/p&gt;&#xA;&lt;p&gt;Now we just need a formula for &lt;span class=&quot;math-container&quot;&gt;$dV$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Each formula for &lt;span class=&quot;math-container&quot;&gt;$dV$&lt;/span&gt; comes from a way of creating the tiny bits of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There are lots of ways to slice a volume &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; into tiny bits &lt;span class=&quot;math-container&quot;&gt;$dV$&lt;/span&gt;. We will learn two ways in this class:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You could slice it horizontally: &lt;span class=&quot;math-container&quot;&gt;$dV = B(y) dy$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$V = \int dV = \int B(y) dy$&lt;/span&gt;, or&lt;/li&gt;&#xA;&lt;li&gt;You could slice it vertically: &lt;span class=&quot;math-container&quot;&gt;$dV = A(x) dx$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$V = \int dV = \int A(x) dx$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-08-27T15:00:53.457" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26702" PostTypeId="2" ParentId="26700" CreationDate="2023-08-27T17:52:25.173" Score="2" Body="&lt;p&gt;In a third-semester calculus course one typically defines double-integration over a suitable family of plane regions, by approximating the region by small rectangles as size tends to zero.  In such a setting, one proves Fubini's theorem, to the effect that the double-integration equals the iterated integral (again over suitable domains).  If you are only interested in a rotation of &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}2$&lt;/span&gt;, then such a double application of Fubini is sufficient to establish that the integral (in particular, the area) will be the same.&lt;/p&gt;&#xA;&lt;p&gt;If you want more general rotations, probably one would have to do a bit of more advanced differential geometry and introduce at least 1-forms and 2-forms, so one can speak of the area form and its invariance under rotations.  But possibly one can take a shortcut via polar coordinates and show that the expression remains the same under integration.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2023-08-27T17:52:25.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26703" PostTypeId="1" CreationDate="2023-08-27T21:33:53.393" Score="1" ViewCount="303" Body="&lt;p&gt;Faces are attributes of polyhedra, so it doesn't make sense to ask how many faces a cone has.&lt;/p&gt;&#xA;&lt;p&gt;Are there traditional scholars that use faces attached to cones? How do different countries deal with the subject?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="2083" LastEditDate="2023-08-29T05:32:54.383" LastActivityDate="2023-08-29T05:32:54.383" Title="Is it correct to state that a cone has no faces?" Tags="|geometry|terminology|definitions|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26704" PostTypeId="2" ParentId="26700" CreationDate="2023-08-28T01:30:41.773" Score="27" Body="&lt;p&gt;It depends somewhat on the style of the course, but the majority of calculus students do not need a formal definition of volume or area, in my experience. They have studied geometry and (usually) physics, and they understand these notions on an intuitive level. That's enough, at this stage. (Of course, one can remind students about the concept by recalling some formulas for the volume of polyhedra, etc.)&lt;/p&gt;&#xA;&lt;p&gt;My approach could be described as &amp;quot;we understand what volume is, and here are several ways to calculate it.&amp;quot; All the various formulas that appear in the text are regarded as tools for calculating the volume, rather than definitions of volume.&lt;/p&gt;&#xA;&lt;p&gt;Formal/logical questions like &amp;quot;what really IS volume?&amp;quot; are of interest primarily to students who specialize in mathematics, and they are by far in the minority out of students studying calculus.&lt;/p&gt;&#xA;&lt;p&gt;But no matter the style of the course, the fact that different orders of integration lead to the same answer for the area/volume should certainly be emphasized, as this is one of the most interesting and useful aspects of multivariable integration.&lt;/p&gt;&#xA;" OwnerUserId="22788" LastEditorUserId="22788" LastEditDate="2023-08-28T01:44:00.870" LastActivityDate="2023-08-28T01:44:00.870" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26705" PostTypeId="2" ParentId="26703" CreationDate="2023-08-28T09:01:54.520" Score="2" Body="&lt;p&gt;This answer explains why this question can't be answered (but it does not fit inside a comment section).&lt;/p&gt;&#xA;&lt;p&gt;First of all: what is a cone?&lt;br /&gt;&#xA;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Cone&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Wikipedia-page&lt;/a&gt;, a cone has a circular ground surface, which is in most mathematics handbooks pure nonsense:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Either a cone does not have a ground surface.&lt;/li&gt;&#xA;&lt;li&gt;Either the ground surface is a general ellips (not necessary a circle).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Please clearly define what you mean by a cone.&lt;/p&gt;&#xA;&lt;p&gt;Next: what is a face? &lt;a href=&quot;https://en.wikipedia.org/wiki/Face_(disambiguation)&quot; rel=&quot;nofollow noreferrer&quot;&gt;This Wikipedia-page&lt;/a&gt; mentions different definitions of that word, but the geometrical definition is not present.&lt;br /&gt;&#xA;So, a face: is this a subset of a plane, where every two points can be connected by a line, never leaving the face? Or do you have a better definition?&lt;/p&gt;&#xA;&lt;p&gt;Please, get your definitions straight and re-phrase the question.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-08-28T09:01:54.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26706" PostTypeId="1" AcceptedAnswerId="26716" CreationDate="2023-08-28T13:51:49.287" Score="24" ViewCount="5757" Body="&lt;p&gt;One of my college students writes the Greek letter &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; as a script &lt;em&gt;n&lt;/em&gt; with a bar over it, like &lt;span class=&quot;math-container&quot;&gt;$\bar{n}$&lt;/span&gt;. [There is actual space between the letter and the bar.] I have never seen this before, and Googling didn't get me anywhere. I feel compelled to 'correct' him on this, but I wanted to first make sure this isn't, say, an accepted form of the letter. I am asking this here, as I am supposing mathematicians and math historians may have a better view of what and how symbols are used in the field (as opposed to a different site that focuses on Greek language).&lt;/p&gt;&#xA;&lt;p&gt;Have you seen this symbol used in place of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;? Did my student just make it up?&lt;/p&gt;&#xA;&lt;p&gt;A more general question here: When should we correct students who use alternate symbols or form them in a new/strange way? Obviously, there are certain things I won't budge on, like using a symbol whose meaning has one standard use (e.g. &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;) in place of another (e.g. &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;). But where should we draw the line? [No pun intended]&lt;/p&gt;&#xA;&lt;p&gt;[This question may be too 'opinion-based' and I understand if folks vote to close it.]&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2023-08-28T17:29:22.423" LastActivityDate="2023-08-30T10:49:35.887" Title="Correcting how a student writes symbols" Tags="|notation|history|" AnswerCount="5" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="26707" PostTypeId="2" ParentId="26703" CreationDate="2023-08-28T14:22:31.167" Score="2" Body="&lt;p&gt;AT the end of the day, it just depends on how you want to define things and becomes a bit of a semantics game.  (We used to have a guy who was obsessing over squares versus rectangles.  We also get the area of a circle versus area enclosed by a circle kvetchers.)&lt;/p&gt;&#xA;&lt;p&gt;Personally, I think of a cone as a physical object, with a volume and mass.  Probably made out of marble or concrete.  So I would say it has two faces.  The bottom flat face and the curved upper face.  (If you think of it sitting nicely, pointing up.)  And yes, it still has faces even if I'm considering the idealization (maybe a figure of one).&lt;/p&gt;&#xA;&lt;p&gt;Also, I would not assume that some sort of conic section, mathy view of a cone is the only way to think about it.  Look at a general engineering reference instead (e.g. Lindberg EIT reference manual...and it's got a base surface and a volume.)&lt;/p&gt;&#xA;" OwnerUserId="22790" LastEditorUserId="22790" LastEditDate="2023-08-28T14:24:20.677" LastActivityDate="2023-08-28T14:24:20.677" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26708" PostTypeId="2" ParentId="26706" CreationDate="2023-08-28T15:42:03.387" Score="38" Body="&lt;p&gt;Personally, if I can make up an ordinary math problem where the student's alternate/new/strange symbols lead to an incorrect response, then I think that's grounds for correcting the student. (Of course, I'd show them the problem I came up with so they understand &lt;em&gt;why&lt;/em&gt; I'm correcting them.)&lt;/p&gt;&#xA;&lt;p&gt;In the case of your student, the following problem would suffice:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose that you have a collection of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; polygons, where &lt;span class=&quot;math-container&quot;&gt;$\bar{n}$&lt;/span&gt; is the average number of sides per polygon. Prove that the sum of all the interior angles in this collection of polygons is &lt;span class=&quot;math-container&quot;&gt;$(\bar{n}-2) N \pi.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this problem, your student's notation would lead them to state that the sum of all interior angles in the collection of polygons is &lt;span class=&quot;math-container&quot;&gt;$(\bar{n}-2) N \bar{n},$&lt;/span&gt; which is incorrect.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-08-28T15:43:44.127" LastActivityDate="2023-08-28T15:43:44.127" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26709" PostTypeId="2" ParentId="26703" CreationDate="2023-08-28T15:50:45.247" Score="5" Body="&lt;p&gt;This isn't a bad question, it's a good opportunity to discuss how mathematical definitions are made and their relative merits.&lt;/p&gt;&#xA;&lt;p&gt;In convex geometry, a face of a convex body &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is the intersection &lt;span class=&quot;math-container&quot;&gt;$K \cap H$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; is a supporting hyperplane of &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;. (See, for example, &lt;a href=&quot;https://encyclopediaofmath.org/wiki/Convex_set&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.) So a cone has infinitely many faces -- the base disc, and also the infinitely many line segments joining the cone point to each of the points on the perimeter of the base disc.&lt;/p&gt;&#xA;&lt;p&gt;I'm sure there are different definitions in other fields.&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2023-08-28T15:50:45.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26710" PostTypeId="2" ParentId="26706" CreationDate="2023-08-28T17:52:38.817" Score="15" Body="&lt;p&gt;I think the real-world operational point is not that there is some sacred correctness to &amp;quot;orthodox&amp;quot; writing/font styles, but that writing in a very unorthodox style will cause one's readers significant difficulties, perhaps complete misunderstanding.&lt;/p&gt;&#xA;&lt;p&gt;That is, conformity is not-at-all necessarily a virtue, except perhaps in communication with other people. Even then, there's not a universal, eternal &amp;quot;conformity&amp;quot;, but, rather, of course, something depending on one's audience, and what year it is. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2023-08-28T17:52:38.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26711" PostTypeId="2" ParentId="26694" CreationDate="2023-08-28T21:48:37.297" Score="0" Body="&lt;p&gt;Write it as &lt;span class=&quot;math-container&quot;&gt;$y = p + (-1) x.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Along the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$0,$&lt;/span&gt; so the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-value is &lt;span class=&quot;math-container&quot;&gt;$p.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is getting multiplied by &lt;span class=&quot;math-container&quot;&gt;$-1.$&lt;/span&gt; So if you increase &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by some number, the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-value &lt;em&gt;decreases&lt;/em&gt; by that number.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In other words, each time you take one step to the right, you need to take one step down to stay on the line.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-08-28T21:48:37.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26712" PostTypeId="2" ParentId="26706" CreationDate="2023-08-28T23:05:52.023" Score="17" Body="&lt;p&gt;This isn't a moral conundrum, and your students shouldn't be snowflakes who freak out when they're corrected on something like this. They need your guidance in fixing their incorrect habits while they're still easier to fix, so that they won't become embarrassments to them later on. Here are some typical examples I see of students writing Greek letters incorrectly:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wYj9m.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wYj9m.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In most cases, what they're doing is to write the Greek letter as if it were some superficially similar Latin letter. In the case of gamma, I think some of them are responding to the fact that in some fonts (such as the default math font used by LaTeX), the bottom loop is very skinny, and it's hard to tell that the letter isn't in fact a y.&lt;/p&gt;&#xA;&lt;p&gt;It's not OK to say that these are simply arbitrary codes. First of all, it's not a one-to-one mapping. And in many fields in the sciences, engineering, and statistics, the Greek letters have specific meanings. In engineering, ω is an angular frequency. In physics, γ is the Lorentz factor. In statistics, σ is the standard deviation. It's also common to hear students using the corresponding incorrect names, e.g., reading cos ωt as &amp;quot;cosine of double-u t.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Re the specific example you gave, there is some variety in how Greeks write π. For an unusual example, see the image on this page: &lt;a href=&quot;https://en.wiktionary.org/wiki/%E1%BC%B5%CF%80%CF%80%CE%BF%CF%82&quot; rel=&quot;noreferrer&quot;&gt;https://en.wiktionary.org/wiki/%E1%BC%B5%CF%80%CF%80%CE%BF%CF%82&lt;/a&gt; If your student might be Greek or might have grown up with the Cyrillic alphabet, you could certainly ask if this is some style of writing π that they learned in their home country.&lt;/p&gt;&#xA;" OwnerUserId="22792" LastEditorUserId="22792" LastEditDate="2023-08-28T23:08:52.800" LastActivityDate="2023-08-28T23:08:52.800" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26713" PostTypeId="2" ParentId="26706" CreationDate="2023-08-28T23:48:32.153" Score="9" Body="&lt;p&gt;These kind of syntactical glitches need to be corrected immediately, firmly, and clearly. As others have stated: it's immensely easier to fix these things earlier, rather than later when bad habits have set in. Take points off on assignments; that's frequently the only way to get students' attention on a matter.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-08-28T23:48:32.153" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26715" PostTypeId="2" ParentId="26700" CreationDate="2023-08-29T12:48:51.423" Score="2" Body="&lt;p&gt;As a former student, I would prefer an actual example of calculating a solid volume.&lt;br /&gt;&#xA;My favorite one is calculating the formula for the volume of pyramid, say hexagonal one.&#xA;You can do that by horizontally slicing the shape into infinitely many flat sections and integrating over that. Just use function for the area of each slice, dependent on the value of h, the height.&lt;br /&gt;&#xA;It has the added benefit of explaining where these weird 1/3 in the formula came from.&lt;br /&gt;&#xA;You can also show that number of walls does not affect the volume in any way - a three sided pyramid has the same formula as twenty sided one, and in fact the same as a cone.&lt;/p&gt;&#xA;&lt;p&gt;You can end the example by stating they no longer have to memorize the formula for the volume, since they can derive it themselves.&lt;/p&gt;&#xA;" OwnerUserId="22806" LastEditorUserId="22806" LastEditDate="2023-08-29T12:50:44.313" LastActivityDate="2023-08-29T12:50:44.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26716" PostTypeId="2" ParentId="26706" CreationDate="2023-08-29T13:46:00.717" Score="19" Body="&lt;p&gt;Not sure if this is the case but: is this student a Spanish speaker? What they write looks like ñ, so if that's the case it could just be that they are interpreting  as a letter they know, specially if they haven't been explicitly taught about the Greek alphabet or if when they were taught they were ill or going through a rough phase that made them unfocused.&lt;/p&gt;&#xA;&lt;p&gt;So just asking &amp;quot;Hey, have you ever been formally taught the Greek alphabet? I notice for  you write it like ñ, which could lead to confusion with other letters/symbols, or could be a problem if another teacher needs to replace me at some point as they could misinterpret things. Here is a good resource to learn how to (or improve how to) write them. I recommend you do some practice on this by X date, and I'll be happy to help as you require&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="22807" LastActivityDate="2023-08-29T13:46:00.717" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="26717" PostTypeId="2" ParentId="1245" CreationDate="2023-08-29T16:40:50.590" Score="2" Body="&lt;p&gt;The fact of the matter is series is sort of viscerally off the beaten track of limits/derivatives/minmax apps/integral toolbag/integral apps.  It's there because it is needed later (and covered more) in ODEs, PDEs, and (for math majors) real analysis.  You definitely don't want to torture yourself (and the class) by spending huge time on justifying what is a bit of a side trip  [even if someone comes on and says it's not a sidetrip, it sure FEELS like one, which is the important thing].&lt;/p&gt;&#xA;&lt;p&gt;That doesn't mean you have to be totally just &amp;quot;do this&amp;quot;.  But a couple remarks to the effect of &amp;quot;needed in later classes&amp;quot; as well as the stereotypical &amp;quot;series can be used to approximate smooth functions&amp;quot; is enough.  And yes, without proving, showing etc. those two comments.  Then get on with it.  Teach the topic and be energetic and follow the text.  And test them.  And move one.  Don't make something that is a sidetrack even sidetrackier.  Give them the exposure (as with baby determinants in HS or Laplace transforms in ODEs)...and then roll on and away.&lt;/p&gt;&#xA;&lt;p&gt;Also, even though it doesn't feel as amazingly wonderful and applied and organically connected as minmax or integration by parts, it's not the worst thing in the world for the kids to do the mechanical work in fiddling with series.  There's a fair amount of just paper and pencil symbol algebra in series and you see it from second year (US blabla) HS algebra to calculus to ODEs to PDEs to numerical analysis to real analysis to complex analysis.  (Realizing everyone does not take all of those, but engineers will get a lot of that, even if in an &amp;quot;engine math&amp;quot; course.)  I don't really remember the rules for convergence...I mean it was good to learn it once, and at least know intuitively that some things that sorta look like they oughta converge don't.  But the &amp;quot;muscles&amp;quot; of writing terms and all that was good stuff.  And in some ways similar to the &amp;quot;muscles&amp;quot; of dealing with equations for EE in circuit analysis or fluids hydraulic circuits or thermo power cycles.&lt;/p&gt;&#xA;" OwnerUserId="22811" LastActivityDate="2023-08-29T16:40:50.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26718" PostTypeId="1" AcceptedAnswerId="26720" CreationDate="2023-08-29T19:25:19.417" Score="15" ViewCount="2318" Body="&lt;p&gt;I am an undergraduate grader at my institution where I have been entrusted with grading a section of an undergraduate analysis course; it's usual for this role to be offered exclusively to graduate students (who usually serve it as a secondary function while TAing) but my university was forced to add another section, creating an opening. I am passionate regarding the subject matter and proficient in it at around a lower-graduate level, but I have limited experience grading, particularly in proof-based courses. With this in mind, I was wondering if anyone had anyone had any tips regarding grading analysis, particularly in the way of maintaining equitable grading.&lt;/p&gt;&#xA;&lt;p&gt;The course covers all the &amp;quot;platitudes&amp;quot; of analysis, i.e. &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; \ &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proofs and the essential topological aspects of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;. We do however also include some more unorthodox material for a first course, like the Banach fixed-point theorem and some superficial covering of metric spaces in general. Any guidance from anyone who has graded something like this would be appreciated!&lt;/p&gt;&#xA;" OwnerUserId="22812" LastActivityDate="2023-08-31T03:45:03.307" Title="What is important to keep in mind in grading proof-based courses?" Tags="|mathematical-analysis|grading|" AnswerCount="6" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26719" PostTypeId="2" ParentId="26700" CreationDate="2023-08-29T20:01:08.083" Score="4" Body="&lt;p&gt;&lt;em&gt;How to properly define volume for beginner calculus students?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'd say that the Apostol's approach, which I found after looking through several books, is the best. His definition of volume is axiomatic. Here is a simpler adaptation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Definition:&lt;/strong&gt; We define the &lt;em&gt;volume&lt;/em&gt; of a three-dimensional solid &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; to be a real number &lt;span class=&quot;math-container&quot;&gt;$v(S)$&lt;/span&gt; that express the &amp;quot;amount of space occupied by &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;&amp;quot;, having the following properties:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$v(S)\geq 0$&lt;/span&gt; for all solid &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$v(S_1\cup S_2)=v(S_1)+v(S_2)-v(S_1\cap S_2)$&lt;/span&gt; for all solids &lt;span class=&quot;math-container&quot;&gt;$S_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$S_2$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Cavalieri's principle:&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$S_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$S_2$&lt;/span&gt; are two solids such that &lt;span class=&quot;math-container&quot;&gt;$\operatorname{area}(S_1\cap \alpha)\leq \operatorname{area}(S_2\cap \alpha)$&lt;/span&gt; for every plane &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; perpendicular to a given line, then &lt;span class=&quot;math-container&quot;&gt;$\operatorname{volume}(S_1)\leq \operatorname{volume}(S_2)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is a rectangular parallelepiped with edges having lengths &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$v(R)=abc$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The key point is the Cavalieri's principle, which allows us to relate areas to volumes without double integrals.&lt;/p&gt;&#xA;&lt;p&gt;In this context, it is proved that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$v(S)=\int_a^b A(u)\,du,$$&lt;/span&gt;&#xA;for any &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-axis, where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is the cross-sectional area for the &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-axis.&lt;/p&gt;&#xA;&lt;p&gt;The generic &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-axis can be replaced by the usual &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis or &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis. In other words: horizontal as well as vertical slicing are allowed, that is, the usual formulas&#xA;&lt;span class=&quot;math-container&quot;&gt;$$V(S)=\int_a^b A(x)\,dx\quad\text{and}\quad V(S)=\int_c^d B(y)\,dy$$&lt;/span&gt;&#xA;are valid.&lt;/p&gt;&#xA;&lt;p&gt;Apostol's approach uses step functions. However, under suitable hypothesis, it's possible to adapt the argument for the context of Stewart's definition of integral. See the proof &lt;a href=&quot;https://math.stackexchange.com/a/4760566/70305&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Remark:&lt;/em&gt; An axiomatic definition is not mandatory. Given that Cavalieri's principle is assumed to be true, the said proof (of the independence of the axes) validates Stewart's definition. Maybe this is the simplest possible approach.&lt;/p&gt;&#xA;" OwnerUserId="13585" LastEditorUserId="13585" LastEditDate="2023-08-30T13:54:14.803" LastActivityDate="2023-08-30T13:54:14.803" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26720" PostTypeId="2" ParentId="26718" CreationDate="2023-08-29T20:05:10.957" Score="17" Body="&lt;p&gt;To grade faster and provide more targeted feedback, it could be helpful to review some lists of mistakes that students commonly make during proofs. For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/17900/taxonomy-of-bad-proofs&quot;&gt;Taxonomy of bad proofs&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.mathcs.bethel.edu/%7Egossett/DiscreteMathWithProof/CommonErrorsInProofs.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.mathcs.bethel.edu/~gossett/DiscreteMathWithProof/CommonErrorsInProofs.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://personal.math.ubc.ca/%7Ecytryn/teaching/scienceOneF10W11/handouts/OS.proof.5invalid.html&quot; rel=&quot;noreferrer&quot;&gt;https://personal.math.ubc.ca/~cytryn/teaching/scienceOneF10W11/handouts/OS.proof.5invalid.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This way, when you see something weird going on in a student's proof, you can (more often) quickly recognize and articulate the issue, and spend less time being confused about what the heck they're trying to do.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-08-29T20:05:10.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26721" PostTypeId="2" ParentId="26718" CreationDate="2023-08-29T20:32:19.770" Score="6" Body="&lt;p&gt;This answer is specific to your situation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Focus on the things the professor thinks are important.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In general, in a course like this, one could focus on writing, on conceptual understanding, on logic, on many aspects of proof.  (Or all at once, which probably many would prefer!)  But in reality that is overwhelming for students to actually process as feedback.&lt;/p&gt;&#xA;&lt;p&gt;For myself, in a class like this I give the most feedback on proper use of definitions, not missing subtle topological/metric issues, and so forth - the things I think make real analysis important. That would include quantifiers.  If someone is using this course more as an introduction to proof course, then focusing more in feedback on the structure of the proofs, correct logical connectives, and writing might be seen as more important.  Creativity or writing style may feature more highly for another instructor.  And no two people are alike on this, so focusing on one more than the other, without other guidance, may indeed lead to perceived (if not actual) inequities.&lt;/p&gt;&#xA;&lt;p&gt;So my advice is that once you've graded a representative sample of not-perfect, but not-awful, exercises, ask the professor(s) for five minutes to just check over that you didn't go beyond some unwritten, but important, marker in their eyes in one of those categories.  Within just a few minutes you should get a pretty good sense of whether you've focused too much e.g. on trivial typos (or not enough), according to the expectations you, as an employee, are given.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastActivityDate="2023-08-29T20:32:19.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26722" PostTypeId="2" ParentId="26718" CreationDate="2023-08-29T21:34:33.540" Score="2" Body="&lt;p&gt;Among the most helpful tips I've ever gotten is to keep the grading extremely simple. Specifically, advice on this site by Joel David Hamkins to use a 5-point rubric for grading:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/14409/should-students-be-given-partial-scores-when-they-gave-an-incomplete-proof-by-co/14412#14412&quot;&gt;Should students be given partial scores when they gave an incomplete proof by contradiction?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-08-29T21:34:33.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26723" PostTypeId="2" ParentId="26718" CreationDate="2023-08-30T04:50:07.420" Score="11" Body="&lt;p&gt;My dealing with proofs, as a student and an undergrad TA myself, left me feeling that a lot of students have one common problem with proof-based courses. They have trouble figuring out exactly which things are considered &amp;quot;obvious&amp;quot;, and which they need to prove. (And figuring out exactly what to &lt;em&gt;say&lt;/em&gt;, to constitute an acceptable proof of a fact that seems obvious.)&lt;/p&gt;&#xA;&lt;p&gt;Therefore, to a grader of proofs, I would say: Try to be as clear as possible about what's missing, especially early in the semester. Especially if some things are intuitively obvious, but still have to be written out. (And if some steps are extraneous, maybe note that too, even if it's not a deduction.) Obviously a professor/TA has more opportunity to converse with the students interactively about these issues, but your comments will be the only ones that the students definitely see.&lt;/p&gt;&#xA;" OwnerUserId="15737" LastActivityDate="2023-08-30T04:50:07.420" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26726" PostTypeId="2" ParentId="26718" CreationDate="2023-08-30T09:29:55.727" Score="3" Body="&lt;h1&gt;Help with the most important problem&lt;/h1&gt;&#xA;&lt;p&gt;The students are likely to make many different mistakes, and maybe in addition the handwriting is messy and they are using a proof by contradiction where that is not actually necessary and so on.&lt;/p&gt;&#xA;&lt;p&gt;Giving them feedback on all of this at the same is not useful. You need to identify what is the single worst obstacle to proceeding and developing mathematical maturity, and give feedback precisely on that. Maybe some other detail if really necessary.&lt;/p&gt;&#xA;&lt;p&gt;So typically you would be ignoring lots of minor problems and instead telling them to be very clear about what the claim is, what are the assumptions they make, and how they use the assumptions to prove the claim. Before that has caught on, problems with fractions or unclear writing are bad, but not the central contents.&lt;/p&gt;&#xA;&lt;h1&gt;However&lt;/h1&gt;&#xA;&lt;p&gt;Most teachers do not know how to give feedback, and most students do not know what to make of the feedback they are given. This is a skill issue on both sides. So do not be all too surprised if the feedback you give does not have much of an effect on what the students actually do. This is a notoriously hard problem in university education in general.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-08-30T09:29:55.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26729" PostTypeId="2" ParentId="26718" CreationDate="2023-08-31T03:45:03.307" Score="3" Body="&lt;p&gt;At my university we teach a course in Discrete Mathematics that acts as an introduction to proof writing for undergraduate students. Three things of note.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;We use &lt;a href=&quot;https://www.ams.jhu.edu/ers/books/mathematics-a-discrete-introduction/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ed Scheinerman's &lt;em&gt;Mathematics: A Discrete Introduction&lt;/em&gt;&lt;/a&gt; as a text, one that I think is very well suited for this course. Among the many helpful elements of the text, the author provides proof templates that are useful to students.&lt;/li&gt;&#xA;&lt;li&gt;To grade proofs we draw from Brown and Michel's paper &lt;a href=&quot;http://sigmaa.maa.org/rume/crume2010/Archive/Brown_D.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Assessing Proofs with Rubrics: the RVF Method&lt;/em&gt;&lt;/a&gt;. Having this rubric proves useful for both graders and students alike.&lt;/li&gt;&#xA;&lt;li&gt;As an additional resource we share with students &lt;a href=&quot;https://danaernst.com/teaching/mat431f21/ElementsOfStyle.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Appendix A from Dana Ernst's &lt;em&gt;Elements of Style&lt;/em&gt;&lt;/a&gt;. This also proves useful to all parties.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Keeping all of the things mentioned in the latter two notes above is a good start for anyone new to grading proofs, in my opinion.&lt;/p&gt;&#xA;" OwnerUserId="267" LastActivityDate="2023-08-31T03:45:03.307" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26731" PostTypeId="1" CreationDate="2023-09-01T15:21:32.563" Score="13" ViewCount="1371" Body="&lt;p&gt;I suspect that; just as one must &amp;quot;do&amp;quot; mathematics to learn mathematics, one must have practice teaching mathematics to become a great mathematics instructor.&lt;/p&gt;&#xA;&lt;p&gt;Still, a good book on mathematical pedagogy could only help! I am looking for book recommendations related to mathematical pedagogy.&lt;/p&gt;&#xA;&lt;p&gt;I am generally interested in teaching math to university students, but my strongest preference is for books that reference research in math education.&lt;/p&gt;&#xA;" OwnerUserId="22665" LastActivityDate="2023-09-10T22:54:49.330" Title="What are some good books on mathematical pedagogy?" Tags="|mathematical-pedagogy|reference-request|" AnswerCount="7" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26732" PostTypeId="1" CreationDate="2023-09-01T15:31:40.260" Score="5" ViewCount="128" Body="&lt;p&gt;Solving a first order ODE with the Euler method is simple enough, if the student is to do it for a few rounds, however for anything substantial a programmable calculator or access to a programming language will be required. I see a number of web apps that bypass such requirements. If you have used them in your classes which one do you prefer and why?&lt;/p&gt;&#xA;&lt;p&gt;So here we are solving dy/dx=f(x,y), with a given (x0,y0), and step-size h. The output will be (x_i,y_i), for i from 0 to N, and perhaps an error estimate,  etc.&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2023-09-09T11:26:04.400" LastActivityDate="2023-09-09T21:47:18.730" Title="Recommended online software for Euler method/ODEs" Tags="|software|differential-equations|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26733" PostTypeId="2" ParentId="26731" CreationDate="2023-09-01T18:18:13.307" Score="12" Body="&lt;p&gt;In &lt;em&gt;&lt;strong&gt;The Teaching Gap&lt;/strong&gt;&lt;/em&gt;, James Stigler and James Hiebert analyze research from the TIMSS studies. Part of their conclusion is that much of our experiences of classroom culture are deeply imprinted in us, as what school &lt;em&gt;is&lt;/em&gt;. “The scripts for teaching in each country appear to rest on a relatively small and tacit set of core beliefs about the nature of the subject, about how students learn, and about the role that a teacher should play in the classroom.” (p.87) So it's hard to change how we teach math if we are working within the school systems.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;em&gt;&lt;strong&gt;Mindstorms: Children, Computers, and Powerful Ideas&lt;/strong&gt;&lt;/em&gt;, Seymour Papert's research shows us some alternative learning environments that could make a big difference in how we learn things like math. (I loved this book, and highly recommend it.)&lt;/p&gt;&#xA;&lt;p&gt;In &lt;em&gt;&lt;strong&gt;What's Math Got To Do With It?&lt;/strong&gt;&lt;/em&gt;, Jo Boaler discusses her research in what she calls &lt;em&gt;complex instruction&lt;/em&gt;. Groupwork is part of it. Also the use of what educators might call low floor, high ceiling problems - problems that are engaging, easy to get started on, and offer challenging aspects. (I wrote about some of this &lt;a href=&quot;https://mathmamawrites.blogspot.com/2011/07/complex-instruction.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;I can also highly recommend &lt;em&gt;&lt;strong&gt;Mathematical Problem Solving&lt;/strong&gt;&lt;/em&gt;, by Alan Schoenfeld.&lt;/p&gt;&#xA;&lt;p&gt;I reviewed 11 articles on math education in &lt;a href=&quot;https://mathmamawrites.blogspot.com/2010/09/11-interesting-articles-on-math.html&quot; rel=&quot;noreferrer&quot;&gt;this blog post&lt;/a&gt;, including two by Schoenfeld, and one by Uri Treisman, whose UC Berkeley Calculus education research is very important to consider.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-09-02T14:19:55.440" LastActivityDate="2023-09-02T14:19:55.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26735" PostTypeId="2" ParentId="26732" CreationDate="2023-09-01T19:27:43.147" Score="4" Body="&lt;p&gt;This is the best one that I've come across: &lt;a href=&quot;https://www.geogebra.org/m/NUeFjm9J&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geogebra.org/m/NUeFjm9J&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/88GjI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/88GjI.png&quot; alt=&quot;Geogebra Euler Estimator&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-09-01T19:27:43.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26736" PostTypeId="2" ParentId="26731" CreationDate="2023-09-02T01:15:10.653" Score="10" Body="&lt;p&gt;My favored text that I consider essential for &amp;quot;getting started&amp;quot; as a college math instructor is:&lt;/p&gt;&#xA;&lt;p&gt;Krantz, Steven George. &lt;em&gt;How to teach mathematics.&lt;/em&gt; American Mathematical Soc., 1999.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-09-02T01:15:10.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26737" PostTypeId="2" ParentId="26731" CreationDate="2023-09-02T13:26:53.523" Score="9" Body="&lt;p&gt;&amp;quot;Building thinking classrooms&amp;quot; by Liljedahl is a summary of effective teaching practices distilled from Liljedahl's research.  TLDR: get students working on questions which require thinking independently before introducing new techniques.  Have them work standing at whiteboards.  Use visible randomized grouping to create the groups.  Encourage knowledge to circulate through the classroom.  Summarize the outputs of student work &lt;em&gt;using their work&lt;/em&gt;.  Assess students using transparent standards.  Allow students to demonstrate their mastery in a variety of ways.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Cheating Lessons&amp;quot; by Lang argues that effective teaching practices are also our best defense against cheating.  TLDR:  give frequent low stakes assessments rather than few high stakes assessments.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="640" LastEditDate="2023-09-03T16:18:14.217" LastActivityDate="2023-09-03T16:18:14.217" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26738" PostTypeId="1" CreationDate="2023-09-02T15:24:07.550" Score="6" ViewCount="150" Body="&lt;p&gt;I am in my early 70's and licensed to teach 8-12 math in Texas. I  have an advanced degree in the same area. I used to teach in high school decades ago but have since quit because the student's disruptive behavior was just too stressful for me. It literally left a mental scar on me. Now I am semi-retired and working as an adjunct instructor at local colleges. Recently I got in touch with a company in Austin, TX, about teaching virtually via video conferencing. This company uses Zoom video to deliver a teacher live and remotely to a class, with a teacher aid physically present in the classroom to manage the classroom.&lt;/p&gt;&#xA;&lt;p&gt;Did you have experience working as a virtual teacher before? Specifically I would like to know this: How stressful is it to teach in this video format with respect to classroom management? Please disregard any other stressors such as lower pay rate and workload.&lt;/p&gt;&#xA;&lt;p&gt;It is my observation that, rightly or wrongly, I will be responsible only to ensure that my teaching is engaging to capture students' full attention. But the bulk of classroom management should be on the facilitator since I am not physically present in the classroom. Any practical advice would be very much appreciated. Thank you for your time.&lt;/p&gt;&#xA;" OwnerUserId="13970" LastEditorUserId="2083" LastEditDate="2023-09-03T18:30:34.310" LastActivityDate="2023-09-06T12:22:46.237" Title="Remote Teaching by Video Conferencing" Tags="|secondary-education|usa|remote-teaching|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26739" PostTypeId="1" CreationDate="2023-09-03T12:58:30.283" Score="-5" ViewCount="229" Body="&lt;p&gt;there is a large literature about what Mathematics is. but what about &lt;strong&gt;what mathematics is not&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;for instance the question displayed in the following image is, for me an example of what mathematics is not(it is quite common to see such kind of question in social media). It is not mathematics because its solution relies on a linguistic joke: &lt;em&gt;20 for three&lt;/em&gt;). Not mathematics &lt;em&gt;lateral thinking&lt;/em&gt;, perhaps. Note:Brigadeiro is a candy)&#xA;&lt;a href=&quot;https://i.stack.imgur.com/0EMd3.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0EMd3.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;so my question is about references and examples of what mathematics is not.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2023-09-03T18:37:01.220" Title="references and examples of questions that shows what mathematics is not" Tags="|problem-design|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26740" PostTypeId="2" ParentId="26739" CreationDate="2023-09-03T18:37:01.220" Score="2" Body="&lt;p&gt;While I don't know of any outside references that definitively answer your question, the following decision rule pops into my head:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Does the correct answer to the question permit counterexamples? If so, then it's not a mathematical question. (If not, then it may or may not be a mathematical question.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For instance, the question in the image that you posted would not be a mathematical question because Juliana could very well have prepared those brigadeiros at any of the listed times. (Perhaps one time is much more likely than the others, but all times are possible.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; I wouldn't bet my life on this decision rule being 100% correct, and I don't claim that it definitively answers your question, but it seemed like it might useful enough to share.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-09-03T18:37:01.220" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26741" PostTypeId="2" ParentId="26731" CreationDate="2023-09-04T06:32:11.327" Score="2" Body="&lt;p&gt;One of my interests is East Asian mathematics education. A very good book that succinctly describes some of its philosophies is “Principles of Mathematics Education” published by the Association of Mathematical Instruction (of Japan) and written by Kô Ginbayashi in 1984. It's a very small booklet (52 pages, 5&amp;quot;x7&amp;quot; in size).&lt;/p&gt;&#xA;&lt;p&gt;Its emphasis is on applied mathematics (and not on pure mathematics). The basic concept used is quantity (and not number). Quantities are either discrete (magnitude of a countable set; its unit of measurement is clear at the start) or continuous (magnitude of a continuum; its unit of measurement has not been determined a priori). For example, the number of children (how many) has only one possible unit of measurement (children). The amount of water (how much) has many possible units of measurement (gallons, liters, etc.)&lt;/p&gt;&#xA;&lt;p&gt;Addition and subtraction only apply to quantities that have the same units of measurement, and multiplication and division involve quantities that can have different units of measurement.&lt;/p&gt;&#xA;&lt;p&gt;Some continuous quantities can be added (like area or weight); some cannot (like density or speed). For example, you add the weights of two bodies to get their total weight, but you don't add the speeds of two bodies to get their &amp;quot;total speed.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The concept of a decimal is not a particular case of the concept of a fraction; the concepts are different. Decimals involve smaller units that are known in advance; fractions involve smaller units that are known only after the quantity to be measured is given. For example, I can measure the height of a person using decimeters, and if there is a remainder, then I use centimeters, and so on.  But if I measure the height using, say, eighths of a meter, and there is a remainder, then I need to change the units.&lt;/p&gt;&#xA;&lt;p&gt;There are a lot more concepts in the book. I wrote about a few of them in &lt;a href=&quot;https://joelreyesnoche.wordpress.com/2015/05/27/theory-of-quantity/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this blog post&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="77" LastEditorUserId="77" LastEditDate="2023-09-05T05:52:09.193" LastActivityDate="2023-09-05T05:52:09.193" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26742" PostTypeId="2" ParentId="26731" CreationDate="2023-09-04T14:20:50.823" Score="5" Body="&lt;p&gt;This isn't a book, but I recommend (along with looking at the texts suggested here), to read the article by Jaime Escalante.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED345942.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://files.eric.ed.gov/fulltext/ED345942.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is not to say his words are gospel.  One of the key issues in math pedagogical research is people thinking there are single proven/research solutions versus competing approaches and a murky topic.&lt;/p&gt;&#xA;&lt;p&gt;A couple more good articles to read, by James Cargal (On Teaching in the Mathematical Sciences; The Reform Calculus Debate and the Psychology of Learning Mathematics).  With the main messages being DON'T be too &amp;quot;hard&amp;quot;...DON'T assume your kids are as smart as you, or headed to R1 math grad school...and DON'T assume that you can give them a more efficient path to wisdom by doing real analysis before calculus, for example.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1431&amp;amp;context=hmnj&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1431&amp;amp;context=hmnj&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1234&amp;amp;context=hmnj&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1234&amp;amp;context=hmnj&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Note: Education research tends to be very shoddy.  E.g. low sample sizes, confounding variables, teacher/student dropout effects, non representative samples, poor metrics, failures to replicate, even political bias on top of the statistical bias.  You are better off skimming such things with a very open/critical mind (hmm...maybe this might work...looks like BS...etc.) versus a &amp;quot;this is research, it was printed, must be right&amp;quot; approach. In particular, you may be BETTER off having anecdotal discussions with teachers (including those with opposing views) than looking for &amp;quot;the answer&amp;quot; in print, when it doesn't definitively exist.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, pedagogical research is also weak on tools for in depth observation such as ethnographic studies, detailed case histories, wide ranging interviews, etc.  Probably because they are more work to do...and education schools prefer the shiny appearance of scholarship from the shite statistical-looking studies they churn out, decade after decade.&lt;/p&gt;&#xA;&lt;p&gt;To make my point further, consider the counter to the &amp;quot;answer&amp;quot; (really an opinion, because that is how this forum especially, but really much of non hard science/math SE functions best) above advocating Thinking Classrooms by Lilejedahl.  See for example this criticism of the (facade of) research backing it up:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://pershmail.substack.com/p/the-evidence-for-building-thinking&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://pershmail.substack.com/p/the-evidence-for-building-thinking&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;We could imagine a version of this book that doesn’t say anything about research at all. “Here are my thoughts on what makes for good teaching,” it would say. “It’s based on my own teaching, observations, and the experiences of the people I’ve worked with. But don’t take my word for it—try it for yourself, and you’ll see that it works wonders.” Would that be OK?&lt;/p&gt;&#xA;&lt;p&gt;I think the answer is, that would be very OK. That’s just telling people what you think. You have to be allowed to do that.&lt;/p&gt;&#xA;&lt;p&gt;Now, how differently should we think about this system with the research support that Peter has given us? I’d argue, not much differently. It’s not that there’s no evidence, and it’s not that he’s playing loose with the facts. It’s just that the evidence is weak. It doesn’t support big generalizations. You wouldn’t want to bet the bank on it.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;See also, the criticism by Greg Ashman (paywalled, but free 7 day trial):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://fillingthepail.substack.com/p/peter-liljedahl-wants-to-make-kids&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://fillingthepail.substack.com/p/peter-liljedahl-wants-to-make-kids&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[Excerpting the main thesis of his pro-DI and anti-IBL view:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;It was Alfred North Whitehead, a mathematician and philosopher, who made the following observation:&lt;/p&gt;&#xA;&lt;p&gt;'It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.'&lt;/p&gt;&#xA;&lt;p&gt;Whitehead’s view is now strongly supported by cognitive science. It is widely accepted that working memory — roughly, the thoughts we are conscious we are having — is extremely limited, capable of processing about only four items at any one time. However, these constraints fall away entirely when dealing with organised knowledge held in long-term memory. Educational psychologists call these webs of knowledge that are related to each other by meaning, ‘schemas’. We can effectively activate an entire schema and solve problems without much conscious effort, provided we have a schema in long-term memory to activate.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;[None of this is to say Lilejedahl is wrong and his critics are right...go ahead and look at it...just buyer beware of one more buzzworded edfad.]&lt;/p&gt;&#xA;&lt;p&gt;P.s.  If you really want &amp;quot;the research&amp;quot;, you are better off starting with a lit search of your own, to include visiting a university library and scanning/skimming the books on the shelf.  Then asking more focused questions.  Expecting a dispositive lit search from a forum, as if there were a librarian or a grad student here to prepare it for you, is not the right tool for the job.  (Not only is it not there, but you are missing the grounding you'd get from some search yourself.)  You are better off with focused questions, after some minimal self education, and with accepting and being interested in anecdotes...versus &amp;quot;tha litrachure&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="22846" LastEditorUserId="22846" LastEditDate="2023-09-05T14:41:00.620" LastActivityDate="2023-09-05T14:41:00.620" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26743" PostTypeId="2" ParentId="26731" CreationDate="2023-09-04T14:26:28.360" Score="0" Body="&lt;p&gt;Perhaps &lt;em&gt;Proofs and Refutations&lt;/em&gt; by Lakatos might give you a good perspective.&lt;/p&gt;&#xA;" OwnerUserId="22824" LastActivityDate="2023-09-04T14:26:28.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26744" PostTypeId="2" ParentId="26738" CreationDate="2023-09-06T12:22:46.237" Score="3" Body="&lt;p&gt;I have no experience, but I believe it to be extremely stressful and even a bad way of teaching:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;As a teacher, you don't focus anymore on an entire region (left, right, in front, in the back of the class, ...) but everything happens on an area of &lt;span class=&quot;math-container&quot;&gt;$±30\cdot40cm^2$&lt;/span&gt;, which is more or less the size of your computer screen.&lt;/li&gt;&#xA;&lt;li&gt;You don't catch the atmosphere in the classroom anymore: in a real class sometimes you just &lt;strong&gt;feel&lt;/strong&gt; that the class is not paying attention anymore and you can decide to speak of something else (some fun you had recently) just in order to get their focus again.&lt;/li&gt;&#xA;&lt;li&gt;It's a pure waste of human resources: you'll be at home,the students will be inside a classroom, but some other adult will need to be present there too in order to maintain order (don't forget, you're dealing with minor people). So two people need to be paid for doing the work of one.&lt;/li&gt;&#xA;&lt;li&gt;How many times doesn't it happen that a student is afraid of asking a question, because of his or her reputation? When you're present in the class, you can see a student frowning, giving you the opportunity to act upon it. No way you'll see this on your screen. Also, there are the whispers when somebody asks a question, giving you useful information on who is cooperative or just the contrary.&lt;/li&gt;&#xA;&lt;li&gt;...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Remote teaching is a possibility when dealing with motivated adults, but while dealing with minor people, it's pure rubbish!&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-09-06T12:22:46.237" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26745" PostTypeId="2" ParentId="26732" CreationDate="2023-09-09T21:47:18.730" Score="1" Body="&lt;p&gt;Most students know how to use a spreadsheet like EXEl and it's very easy to do implement euler with it . I did it with my high school kids.&lt;/p&gt;&#xA;" OwnerUserId="21051" LastActivityDate="2023-09-09T21:47:18.730" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26746" PostTypeId="1" CreationDate="2023-09-03T22:34:26.243" Score="0" ViewCount="381" Body="&lt;p&gt;This is a question regarding why the order of the real number subsets commonly used in the mathematics community is such:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here the concept of negative numbers is introduced before the concept of quotients. Theoretically, it is possible to introduce the concept of quotients before the concept of negative numbers.&lt;/p&gt;&#xA;&lt;p&gt;If we defined the second subset (after the natural numbers) as equivalence classes of pairs of natural numbers (p, q) with q ≠ 0, using the equivalence relation defined as follows: &lt;span class=&quot;math-container&quot;&gt;$$\displaystyle (p_{1},q_{1})\sim (p_{2},q_{2}) \Longleftrightarrow p_{1}q_{2}=p_{2}q_{1}$$&lt;/span&gt; then the third as consisting of the second subset, 0 and the negatives of the second subset.&lt;/p&gt;&#xA;&lt;p&gt;The establishment of mathematical convention should have much less to do with the student's competence and much more to do with providing a logical sequence of concept introductions to promote a sound understanding of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I believe that the most straightforward approach to sequencing the introductions of mathematical concepts is not always the  most logically sound approach.&lt;/p&gt;&#xA;&lt;p&gt;Why would I advocate for introducing quotients before negative numbers, in the real number subset order? If we assume ~1500 years between the first use of division around 2000 B.C.E. and the Greek counting table at Salamis around 300 B.C.E., this leads me to believe the natural development of mathematics prioritised quotients over negative numbers.&lt;/p&gt;&#xA;&lt;p&gt;A university course on the history of mathematics is primarily about understanding the stories behind the mathematics we use today. A chronological account would tell many other stories of the changes in societal mentality throughout history regarding mathematics which do not apply to the mathematics we use today.&lt;/p&gt;&#xA;&lt;p&gt;Mathematics can be a fantastic subject to study with great societal value but we should always aim to focus the logic and reasoning behind our ideas on the truth. The bounds of mathematics should not be restricted to the current thought of today.&lt;/p&gt;&#xA;&lt;p&gt;If such a thing as the &amp;quot;natural order of events&amp;quot; exists and it prioritised introducing quotients before negative numbers and the factors around the conception of the Renaissance era were consistent with said order of events (thus denouncing any feeling of historical exceptionalism), there is reason to factor this consideration when deciding the fundamentals and their effect on our advancements within mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I believe the truth is more important than our current comprehension of mathematics. If there is a strong historical case to be made that humans advanced using quotients before negative numbers, this should be a consideration when deciding the future of mathematics. In particular the conventions of which our future advancements will be made with.&lt;/p&gt;&#xA;&lt;p&gt;If you think the current system has merits that supercede these I would appreciate an explanation as to why.&lt;/p&gt;&#xA;" OwnerUserId="22858" OwnerDisplayName="Luke Nemeth" LastEditorUserId="22858" LastEditDate="2023-09-13T07:48:21.750" LastActivityDate="2023-09-13T14:30:47.560" Title="Why are negative numbers introduced before quotients in the real number subsets?" Tags="|real-numbers|rational-numbers|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26747" PostTypeId="2" ParentId="26746" CreationDate="2023-09-10T00:56:26.460" Score="10" Body="&lt;p&gt;Note that the OP appears to have slightly tangled up two distinct issues of: (1) ordering of subsets, and (2) order of what concepts are introduced in school. That is, it would be simply false to try and say, &lt;span class=&quot;math-container&quot;&gt;$ \mathbb{Q}\subseteq\mathbb{Z}$&lt;/span&gt;. But it's true that we do have a choice about what order we introduce things in a curriculum; and even that historically quotients were used before negatives.&lt;/p&gt;&#xA;&lt;p&gt;However, given that we do have negatives in our toolkit today, it seems clear that understanding negatives is a lot simpler than arbitrary fractions. Handling negatives is really just an extra &lt;em&gt;bit&lt;/em&gt; of information -- a binary decision of whether you have a positive or negative sign in front of your whole number; and you can clearly depict this on a number line by extending it both left and right with discrete marks. On the other hand, quotients are more complicated in that they add a denominator with an infinite number of possible integer values, there are issues of equating fractions in different forms, reducing to simplest terms, factoring and canceling, etc.; and there is no way to clearly show all rationals present in a given slice of the number line.&lt;/p&gt;&#xA;&lt;p&gt;Ultimately, we observe the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Integers are essentially the result of all possible &lt;em&gt;subtractions&lt;/em&gt; of two natural numbers.&lt;/li&gt;&#xA;&lt;li&gt;Rationals are essentially the result of all possible &lt;em&gt;divisions&lt;/em&gt; of two integers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I think most of us would agree (and psychological studies bear this out, if it wasn't totally obvious) that subtraction (inverting addition) is simpler than division (inverting multiplication). So it seems highly coherent to develop the expanding sets of numbers by going up the ladder of operations from simple to more complex, establishing the closure at each step.&lt;/p&gt;&#xA;&lt;p&gt;In short, the conceptual and psychological benefits seem to weigh in favor of bringing in negatives before quotients, despite the fact of the historical evolution. If anything, it highlights the possibly surprising historically oddity that integers weren't developed before rationals.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-09-10T06:06:39.523" LastActivityDate="2023-09-10T06:06:39.523" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26748" PostTypeId="2" ParentId="26746" CreationDate="2023-09-10T06:33:52.450" Score="3" Body="&lt;h1&gt;At school, quotients come before negative numbers&lt;/h1&gt;&#xA;&lt;p&gt;At least in some school systems. In Finland both decimal numbers and fractions, as well as the single straggler pi, were introduced somewhere in the upper half of ala-aste, so grades four to six, I guess. Memories are vague on details. Negative numbers were introduced during yläaste, so grades seven to nine, but probably fairly early there.&lt;/p&gt;&#xA;&lt;p&gt;In the current Norwegian curriculum, after fifth grade the pupil is supposed to be able to:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;utforske og forklare samanhengar mellom brøkar, desimaltal og prosent og bruke det i hovudrekning&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So there come the quotients. (Also other goals related to quotients are there.)&lt;/p&gt;&#xA;&lt;p&gt;At seventh grade the pupil is supposed to be able to:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;utforske negative tal i praktiske situasjonar.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So there come the negative numbers.&lt;/p&gt;&#xA;&lt;p&gt;This might be different in other countries, but at least my experience suggests the negative numbers indeed do come some years after many quotients.&lt;/p&gt;&#xA;&lt;p&gt;Presumably this is because of the abstract and finicky nature of negative numbers: is -8 larger or smaller than -2? Natural language makes this ambiguous. The same with the rules of what happens when we add or take away a negative amount of something; this is quite challenging to think about. Not to speak of division with negatives, where intuition and comprehension are mostly exchanged with some variant of negating the minus sign in the divisor or moving it away.&lt;/p&gt;&#xA;&lt;h1&gt;At university level&lt;/h1&gt;&#xA;&lt;p&gt;In most courses all the number systems are presumed known.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;An analysis course might pay attention to completeness of the real numbers and maybe even somewhat carefully go from the rationals to them.&lt;/li&gt;&#xA;&lt;li&gt;An algebra course is likely to spend a lot of time on groups such as the integers, and later on fields/rings such as the rationals.&lt;/li&gt;&#xA;&lt;li&gt;A course interested in the construction of the number systems is likely to do them in the usual order for reasons of conventions, but certainly quotients could be taken before the negatives. But then one would meet different algebraic structures on the way; whether those are as useful as the standards one I have not thought about. But might also be that the integers make for a simpler equivalence class than positive rationals do, given that most students are more at home with subtraction than fractions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h1&gt;A general principle&lt;/h1&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The establishment of mathematical convention should have much less to do with the student's competence and much more to do with providing a logical sequence of concept introductions to promote a sound understanding of mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When teaching, pupil and student competence is more crucial than the logical structure. Children going to school do not start with the foundations of mathematics, and for a good reason. (The attempts have not been successful.) One needs to grow into abstraction, not be thrown into the lake to either sink or, in a rare few precious cases, swim.&lt;/p&gt;&#xA;&lt;p&gt;I would also consider the connection to group theory as an important one at university level, and one gets there faster with integers than the alternate way.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-09-10T06:33:52.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26749" PostTypeId="2" ParentId="26731" CreationDate="2023-09-10T22:54:49.330" Score="1" Body="&lt;p&gt;I find the MAA's &lt;a href=&quot;https://maa.org/sites/default/files/InstructPracGuide_web.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Instructional Practices Guide&lt;/a&gt; to be thought-provoking.&lt;/p&gt;&#xA;" OwnerUserId="213" LastActivityDate="2023-09-10T22:54:49.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26750" PostTypeId="1" AcceptedAnswerId="26784" CreationDate="2023-09-11T07:28:47.303" Score="4" ViewCount="282" Body="&lt;p&gt;I am looking for a web app software that takes step-by-step directions from a student to perform the linear combination operation on a matrix with &lt;strong&gt;symbolic&lt;/strong&gt; coefficients (as opposed to just numbers). So, student indicates to software to replace row &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; with some symbolic multiple of row &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; plus a multiple of row &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;. The software only carries out this step.&lt;/p&gt;&#xA;&lt;p&gt;Edit 1&lt;/p&gt;&#xA;&lt;p&gt;One of the answers below refers to a &lt;a href=&quot;https://www.wolframalpha.com/calculators/system-equation-calculator&quot; rel=&quot;nofollow noreferrer&quot;&gt;WolframAlpha System Solver&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The answer by Thierry points to &lt;a href=&quot;https://matrixcalc.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MatrixCalc&lt;/a&gt; which asks students to participate in taking the required steps. Note: certain symbols are pre-defined.&lt;/p&gt;&#xA;&lt;p&gt;Edit 2&lt;/p&gt;&#xA;&lt;p&gt;The application:&lt;/p&gt;&#xA;&lt;p&gt;A 2*2 system of linear constant coefficient ODEs for &lt;span class=&quot;math-container&quot;&gt;$x=x(t), y=y(t)$&lt;/span&gt; will be&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{matrix}&#xA;L_{11}x &amp;amp;+L_{12}y= f(t) \newline&#xA;L_{21}x &amp;amp;+L_{22}y= g(t)&#xA;\end{matrix}&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$L_{ij}$&lt;/span&gt; is a differential operator with constant coefficients. For example &lt;span class=&quot;math-container&quot;&gt;$L_{11}=D^2+3D+4$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$D=d/dt$&lt;/span&gt;.  Now we want to, say, eliminate &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, so we multiply row 1 by a suitable factor (a polynomial in &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;), and also row 2, and add the rows to get a single ODE for &lt;span class=&quot;math-container&quot;&gt;$x(t)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;First-order cases are seen in problems related to systems of mixing tanks, the second-order cases in spring-mass systems.&lt;/p&gt;&#xA;&lt;p&gt;For a 3*3 system and above an app will be helpful. Just as it is in linear algebra (see the apps Justin Skykac has listed below for numerical cases).&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2023-09-24T11:13:33.703" LastActivityDate="2023-09-24T11:13:33.703" Title="Looking for web app resources for symbolic Gaussian elimination" Tags="|algebra|linear-algebra|differential-equations|apps|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26751" PostTypeId="2" ParentId="26750" CreationDate="2023-09-11T14:21:41.320" Score="2" Body="&lt;p&gt;[Edit: the tools listed below work only with numeric coefficients, not symbolic coefficients.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Tool #1: &lt;a href=&quot;https://www.zweigmedia.com/RealWorld/tutorialsf1/scriptpivotold.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.zweigmedia.com/RealWorld/tutorialsf1/scriptpivotold.html&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is my favorite. As a demonstration, in the screenshot below, I have entered&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the matrix &lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix} 1 &amp;amp; 2 &amp;amp; 3 \\ 1 &amp;amp; 1 &amp;amp; 0 \\ 0 &amp;amp; 1 &amp;amp; 1 \end{bmatrix}$&lt;/span&gt; and&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the row operation &lt;span class=&quot;math-container&quot;&gt;$1 \times \textrm{Row 2} + 5 \times \textrm{Row 3}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lnyQo.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lnyQo.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If I press the &lt;span class=&quot;math-container&quot;&gt;$\fbox{Do It}$&lt;/span&gt; button, then it will replace &lt;span class=&quot;math-container&quot;&gt;$\textrm{Row 2}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$1 \times \textrm{Row 2} + 5 \times \textrm{Row 3},$&lt;/span&gt; resulting in the matrix &lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix} 1 &amp;amp; 2 &amp;amp; 3 \\ 1 &amp;amp; 6 &amp;amp; 5 \\ 0 &amp;amp; 1 &amp;amp; 1 \end{bmatrix}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(In general, supplying the input &lt;span class=&quot;math-container&quot;&gt;$a \times \textrm{Row } i + b \times \textrm{Row } j$&lt;/span&gt; will replace &lt;span class=&quot;math-container&quot;&gt;$\textrm{Row } i$&lt;/span&gt; with the given expression.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Tool #2: &lt;a href=&quot;https://maa.org/book/export/html/117002&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://maa.org/book/export/html/117002&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This one has a nicer interface, but it requires Adobe Flash Player, which you may have to install separately (whereas Tool #1 works right out of the box and seems like it has effectively the same functionality, even if it's not as pretty).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NIK7Z.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NIK7Z.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-09-11T18:05:38.930" LastActivityDate="2023-09-11T18:05:38.930" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26752" PostTypeId="2" ParentId="26750" CreationDate="2023-09-11T22:36:18.407" Score="2" Body="&lt;p&gt;I think the question with the requirement of symbolic coefficients needs to be rethought. In Wolfram Alpha, we can put in the general form of three linear equations.&lt;/p&gt;&#xA;&lt;p&gt;The output gets complicated because we need to know something about the determinant of the coefficients.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XK3e2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XK3e2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-09-12T02:14:33.573" LastActivityDate="2023-09-12T02:14:33.573" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26753" PostTypeId="2" ParentId="25409" CreationDate="2023-09-12T03:14:03.570" Score="2" Body="&lt;p&gt;If we are into anecdotal stories, I want to share one about my former adviser Vicktor Havin, who once made a public speech about one way in which mathematical thinking can be useful to a layman. His main point was that while we can make any list of properties defining an object, not all such lists will be consistent and if the list is self-contradictory, the object may merely fail to exist, so if we decide to put forth any such list and act upon it, we'd better make all terms completely unambiguous and verify its consistency first. It is easy to comprehend when one talks about &amp;quot;primes divisible by 27&amp;quot;, but many old and modern society slogans from &amp;quot;Liberty, Equality, Fraternity&amp;quot; to &amp;quot;Diversity, Equity, Inclusion&amp;quot; would better undergo a thorough scrutiny for ambiguities and inconsistencies before being implemented. While such scrutiny may not show a total self-contradiction in the list or in its individual properties, in many cases it will show a necessity for certain clarifications, limitations, and balance that are usually absent when we launch a crusade for the latest political fad, whatever it is. What is Equity, for instance? It can be interpreted as giving everyone opportunities according to his/her abilities, inclinations, and willingness to achieve their goals, or it can be interpreted as giving no one a clear advantage over anyone else, and these two definitions brought to their extremes will result in two completely opposite policies, neither of which (pure meritocracy or cutting off every head that sticks out) will make most (if any) of us happy.&lt;/p&gt;&#xA;&lt;p&gt;Another thing is that in mathematics you can call anything by any name, but every notion you introduced should be clearly defined and have a clear domain of applicability. Lack of such definitions results in various empty universal agreements like &amp;quot;We have to hire the &lt;em&gt;best&lt;/em&gt; candidate&amp;quot; often followed by bitter fights over the meaning of the word &lt;em&gt;best&lt;/em&gt; when everybody is at everybody else's throat in the end. Or, everybody seems to have some notions of &amp;quot;Good&amp;quot; and &amp;quot;Evil&amp;quot; but no 2 people will completely agree on what is in which category because both words require about 10 specifiers (&amp;quot;for what purpose&amp;quot;, &amp;quot;under what circumstances&amp;quot;, &amp;quot;morally/logically/pragmatically&amp;quot;, etc., etc.) to acquire minimally concise meanings. One cannot easily change how people communicate in general, of course, but one thing I find worth trying at least occasionally when somebody says something that I strongly disagree with is to ask &amp;quot;What exactly do you mean by that?&amp;quot; before throwing myself into a heated argument (not that it always prevents the latter, but, at least, one can get a better idea of what the opponent is after this way and avoid fighting ghosts).&lt;/p&gt;&#xA;&lt;p&gt;At last, mathematics is something where we have a clear and universally agreed upon criterion of truth, which, in most cases, allows everyone to run full verification of claims by him/herself. That facilitates the communication enormously. In the modern world, where the notion of &amp;quot;information&amp;quot; seems to be almost completely substituted by that of &amp;quot;information wars&amp;quot; (which range from toilet paper advertisements to COVID vaccination debates to Ukrainian war coverage), this is something worth keeping in mind too.&lt;/p&gt;&#xA;&lt;p&gt;Of course, this doesn't answer the question as posed but rather clarifies what I mean by &amp;quot;mathematical reasoning&amp;quot; that I'd like to see being taught in schools, so feel free to downvote :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-09-12T03:14:03.570" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26754" PostTypeId="1" CreationDate="2023-09-12T13:25:46.483" Score="1" ViewCount="368" Body="&lt;p&gt;Calculus  is a foundational mathematics course that is often seen as a bottleneck for STEM majors. However, it is also a course that is notorious for its high dropout rates. In the United States, for example, the average dropout rate for calculus is 30%. in Brazil it reaches 70% or higher.&lt;/p&gt;&#xA;&lt;p&gt;My question: :Is there any university or college in any country where failure and dropout rates in Calculus are not so high? top PISA countries Perhaps?&lt;/p&gt;&#xA;&lt;p&gt;Is there a glimmer of hope?&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="2083" LastEditDate="2023-09-12T19:13:40.890" LastActivityDate="2023-09-12T19:13:40.890" Title="Is there any university or college in any country where failure and dropout rates in Calculus are not so high?" Tags="|undergraduate-education|calculus|usa|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26755" PostTypeId="1" AcceptedAnswerId="26757" CreationDate="2023-09-12T14:32:50.873" Score="9" ViewCount="2186" Body="&lt;p&gt;I attempted to introduce the summation notation &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; to my students. The notation was unfamiliar to the students beforehand. I worked through many examples with them, but for most of them, working with such an abstract notation remains challenging. Do you have any suggestions on how to best teach someone who has very little prior knowledge of math? What is the most intuitive method to introduce this notation? I also tried using a for loop, but it didn't seem to help.&lt;/p&gt;&#xA;" OwnerUserId="22866" LastActivityDate="2023-09-14T17:53:43.967" Title="Explaining Sigma-Notation" Tags="|secondary-education|student-motivation|notation|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26756" PostTypeId="2" ParentId="26754" CreationDate="2023-09-12T14:35:07.483" Score="1" Body="&lt;p&gt;Many selective STEM-focused colleges/universities require calculus as part of the general education requirements, meaning that their overall dropout rate would place an upper bound on their calculus-specific dropout rate.&lt;/p&gt;&#xA;&lt;p&gt;For instance, MIT requires 2 semesters of calculus as a general education requirement and has a graduation rate around 95%, meaning that no more that 5% of students drop out of calculus there. (In reality, it's probably far lower than 5%.)&lt;/p&gt;&#xA;&lt;p&gt;I believe this answers the specific question that you asked:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is there any university or college in any country where failure and dropout rates in Calculus are not so high?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Though, there are obviously some caveats relevant to the general spirit of the question:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students entering selective STEM universities are more likely to have more previous experience, intrinsic enjoyment, and natural ability, in math.&lt;/li&gt;&#xA;&lt;li&gt;Using graduation rates as a proxy metric will ignore cases where students drop the course and successfully reattempt it later.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-09-12T15:46:10.537" LastActivityDate="2023-09-12T15:46:10.537" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26757" PostTypeId="2" ParentId="26755" CreationDate="2023-09-12T14:58:08.763" Score="35" Body="&lt;p&gt;I've experienced positive results by first having students spend some time writing out sums in full (or using ellipsis notation if there are many terms).&lt;/p&gt;&#xA;&lt;p&gt;That way, it gets annoying to spend so much time writing the sums, and I can present sigma notation as &amp;quot;a shorthand developed by mathematicians who, like you, were tired of spending so much time writing out sums.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Once a student understands on a visceral level what problem an abstraction is solving, they are far more receptive to the abstraction as a solution to that problem. (Otherwise, if they don't really &amp;quot;get&amp;quot; what problem the abstraction is solving, then it just feels like needless complexity, and their eyes glaze over.)&lt;/p&gt;&#xA;&lt;p&gt;Of course, after the problem of writing out sums is experienced and sigma notation is introduced as a solution, it's still necessary to scaffold the pedagogy well, starting with simple examples (the sum of even numbers from &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$100,$&lt;/span&gt; the sum of squares from &lt;span class=&quot;math-container&quot;&gt;$1^2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$10^2,$&lt;/span&gt; etc).&lt;/p&gt;&#xA;&lt;p&gt;And students would still need to have mastered the necessary prerequisites, such as writing a sequence given its formula and vice versa (and a prerequisite for &lt;em&gt;that&lt;/em&gt; would be evaluating functions).&lt;/p&gt;&#xA;&lt;p&gt;But properly motivating the notation should at least get you over the initial hump.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-09-12T17:55:16.593" LastActivityDate="2023-09-12T17:55:16.593" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26758" PostTypeId="1" CreationDate="2023-09-12T15:51:18.103" Score="3" ViewCount="95" Body="&lt;p&gt;These days I am reading Imre Lakatos's &lt;em&gt;Proofs and Refutations&lt;/em&gt; and I can't stop thinking how one could utilize it in the classroom (mostly high school). Some stray half-baked ideas I have had so far are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To use parts of the text as a basis for a theatrical play in classroom, providing thus the opportunity to discuss (before and/or after) the ideas presented in it.&lt;/li&gt;&#xA;&lt;li&gt;To hand out fragments of the text and start a discussion based on those.&lt;/li&gt;&#xA;&lt;li&gt;Given the difficulty of Lakatos' text for high schoolers, to initiate a dialogue in classroom of a similar fashion in a more appropriate setting, e.g., some (even naive) conjecture about the naturals.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Regarding the first two ideas or similar ones, the text itself, albeit a source of inspiration, is also prohibitive. It is quite abstract in its most part and probably not the best way to introduce students to concepts around proofs (I am more than happy to be proven wrong here, however). Regarding the latter, any reference to similar approaches that might have been tested in the past would be welcome.&lt;/p&gt;&#xA;&lt;p&gt;So, to wrap up, my question is how could one bring Lakatos' work in the classroom, either (and preferably) by utilizing some genuine parts of the text in some way or even by drawing strong inspiration from it?&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2023-09-13T13:46:27.380" Title="Utillizing Lakatos' &quot;Proofs and Refutations&quot; in Secondary Education" Tags="|reference-request|proofs|classroom-interaction|activities|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26759" PostTypeId="1" AcceptedAnswerId="26760" CreationDate="2023-09-12T18:44:35.807" Score="4" ViewCount="407" Body="&lt;p&gt;I always say that the most difficult part of graphing or plotting points is labelling your axis/es. In the case of plotting the graph of a linear equation with integer coefficients in 2 variables it seems the most difficult part might be finding an integer point to start with.&lt;/p&gt;&#xA;&lt;p&gt;Take the equation &lt;span class=&quot;math-container&quot;&gt;$6x+21y=20$&lt;/span&gt; or similar.&lt;/p&gt;&#xA;&lt;p&gt;One could use Euclid's algorithm (or similar) to find an integer point and step off another point, but this is not feasible to teach at high school.&lt;/p&gt;&#xA;&lt;p&gt;My question is what procedure to teach students how to graph equations which will handle such equations?&lt;/p&gt;&#xA;&lt;p&gt;Possible techniques include finding intercepts or finding y intercept and gradient, approximating as necessary. Judicious choice of examples to avoid annoying fractions could also be a strategy.&lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="8190" LastEditDate="2023-09-13T10:04:43.890" LastActivityDate="2023-09-28T19:03:11.797" Title="Graphing lines by finding integer points" Tags="|graphing|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26760" PostTypeId="2" ParentId="26759" CreationDate="2023-09-12T19:13:19.423" Score="8" Body="&lt;p&gt;If you're just trying to get students to graph lines, then while it's a good idea to start students off graphing lines whose equations permit obvious integer points, I think it's a mistake to necessitate integer points when the equations get more complicated.&lt;/p&gt;&#xA;&lt;p&gt;For instance, the line &lt;span class=&quot;math-container&quot;&gt;$y= x + \sqrt{2}$&lt;/span&gt; could easily show up in a high school math course, but there aren't any integer points that lie on it.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I think the easiest procedure for graphing lines is to just plot the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercepts and draw a line through them&lt;/p&gt;&#xA;&lt;p&gt;In your case, for the line &lt;span class=&quot;math-container&quot;&gt;$6x+21y=20,$&lt;/span&gt; it's easy to see that the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; intercepts are as follows:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;6x + 21 \cdot 0 = 20 \quad &amp;amp;\Rightarrow \quad x = \frac{20}{6} = 3 \, \frac{1}{3} \\[5pt]&#xA;6 \cdot 0 + 21 y = 20 \quad &amp;amp;\Rightarrow \quad y = \frac{20}{21}&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And when you write the intercepts in mixed number form (or decimal form) it's easy to see where they go on a graph.&lt;/p&gt;&#xA;&lt;p&gt;That said, if both of the intercepts come out to &lt;span class=&quot;math-container&quot;&gt;$0,$&lt;/span&gt; then you need to either plug in a nonzero number for one of the variables or use the slope (gradient) to find another point.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-09-28T19:03:11.797" LastActivityDate="2023-09-28T19:03:11.797" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26761" PostTypeId="2" ParentId="26746" CreationDate="2023-09-13T08:02:46.697" Score="0" Body="&lt;p&gt;There are three possibilities for the introduction of rationals and negatives in school curricula:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To introduce rationals &lt;em&gt;after&lt;/em&gt; negatives.&lt;/li&gt;&#xA;&lt;li&gt;To introduce rationals &lt;em&gt;before&lt;/em&gt; negatives.&lt;/li&gt;&#xA;&lt;li&gt;To introduce rationals &lt;em&gt;alongside&lt;/em&gt; negatives.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The first one has been nicely covered in @Daniel R. Collins's answer while the latter seems too much in terms of cognitive load for students. Thus, I should also present here some merits of having things as done so far, i.e., introducing rationals before negatives.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The spirit of this answer is to complete the views that have been expressed by the time this is being written. Personally, I do not have a strong opinion on the matter, as both have significant merits as well as give rise to various problems, from my viewpoint.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;h2&gt;Rationals prior to Negatives&lt;/h2&gt;&#xA;&lt;p&gt;What is currently done in most curricula I have in mind is that rationals are introduced some years earlier than negatives. While this appears to be historically accurate, as the OP also observes at some points, it might also appear to have several benefits for the structure and flow of the curriculum as a whole.&lt;/p&gt;&#xA;&lt;p&gt;To begin with, after having talked about naturals and the four operations between them (addition, subtraction, multiplication and division), introducing fractions facilitates the transition to decimal numbers (hinting, in some sense, to the reals). Moreover, using fractions further facilitates the introduction of proportions and &lt;span class=&quot;math-container&quot;&gt;$\%$&lt;/span&gt; percentages, which, among others, are things that are quite relevant to students' everyday lives.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, given that an ever-growing number of curricula around the world tend to introduce key concepts of statistics and probabilities quite early (even grades 3 or 4 in some cases), fractions become of crucial importance. While negatives can be &amp;quot;hidden&amp;quot; behind subtraction and they seldomly appear in relevant applications and problems, the concepts of &amp;quot;observation frequency&amp;quot;, &amp;quot;percentage&amp;quot; etc cannot be easily conceptualized without the construct of fractions. Given that one of the most common ways to introduce probabilities is through frequency-based definitions (or, better, intuitions), fractions are again a useful construct in that setting.&lt;/p&gt;&#xA;&lt;p&gt;Other than the above, fractions are also seemingly a prerequisite in introducing functions. Indeed, the latter is usually done through linear functions and proportionally changing variables. In turn, proportions are easily conceived on the basis of fractions and, more precisely, equivalent fractions. Thus, any delay in the introduction of rationals, also affects the way one can talk about functions, at least in the way it is currently done.&lt;/p&gt;&#xA;&lt;p&gt;There might be more things to say in favor of the status quo, but the above are some indicative cases in which having discussed and learned about fractions early on comes in handy when designing a curriculum.&lt;/p&gt;&#xA;" OwnerUserId="12829" LastActivityDate="2023-09-13T08:02:46.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26762" PostTypeId="2" ParentId="26759" CreationDate="2023-09-13T12:16:58.213" Score="1" Body="&lt;p&gt;You are better off just having them do the x and y intercepts (yes approximate if it's 3 and 10/21 or whatever).  They are drawing by eye anyhow.&lt;/p&gt;&#xA;&lt;p&gt;And the point is to start getting intuitive feeling for (first, simple) curves.  Later in analytic geometry, they will learn to graph more complex functions and will do so by some process of looking at intercepts, asymptotes, symmetry, blabla, etc.  There will really never be a point where you want to push them towards an &amp;quot;integer point&amp;quot;.  The whole idea is to be able have an intuitive feel for graphs by honing in on key features, versus too fussy precision, at the cost of intuitive understanding.   I.e. intercepts uber integer points.&lt;/p&gt;&#xA;" OwnerUserId="22875" LastActivityDate="2023-09-13T12:16:58.213" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26763" PostTypeId="2" ParentId="26758" CreationDate="2023-09-13T12:57:31.240" Score="2" Body="&lt;p&gt;I've used &lt;em&gt;Proofs and Refutations&lt;/em&gt; in the classroom&#xA;(college students in the US), but not in the direct ways you suggested.&#xA;When leading up to Euler's formula &lt;span class=&quot;math-container&quot;&gt;$V-E+F=2$&lt;/span&gt;, I try&#xA;to get the students to see that it is not obvious what is a vertex, an edge,&#xA;a face, by showing them examples of interpenetrating faces, coplanar&#xA;faces, etc.&#xA;I have them experimenting with&#xA;&lt;a href=&quot;https://matheducators.stackexchange.com/a/4313/511&quot;&gt;polydrons&lt;/a&gt;,&#xA;making their own counts,&#xA;trying to discover Euler's formula.&#xA;This can require them to revise their definitions&#xA;(one of Lakatos' primary lessons), e.g., to require&#xA;an &amp;quot;edge&amp;quot; to have a dihedral angle different from &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(On this specific example, it turns out that allowing an &amp;quot;edge&amp;quot;&#xA;to be flat, as in &lt;a href=&quot;https://en.wikipedia.org/wiki/Deltahedron&quot; rel=&quot;nofollow noreferrer&quot;&gt;deltahedra&lt;/a&gt;, still satisfies Euler's formula,&#xA;but with different counts of &lt;span class=&quot;math-container&quot;&gt;$V,E,F$&lt;/span&gt;.)&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/fXJnY.jpg&quot; /&gt;&#xA;&lt;p&gt;Are the five coplanar yellow triangles one face?&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2023-09-13T12:57:31.240" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26764" PostTypeId="2" ParentId="26758" CreationDate="2023-09-13T13:46:27.380" Score="2" Body="&lt;p&gt;Frame-challenge answer:&lt;/p&gt;&#xA;&lt;p&gt;It's better to start with the educational goal you have in mind, and from there decide the best way to achieve it. Starting with a book that you like, and trying to find a way to use it, is like the tail wagging the dog.&lt;/p&gt;&#xA;&lt;p&gt;In general, there is a temptation while teaching to try to make the material interesting for YOU, the teacher. I've certainly experienced this. But you can't give in to that temptation. The stuff you find interesting is mostly too advanced and will just confuse the students.&lt;/p&gt;&#xA;&lt;p&gt;To be clear, you do want to make the material interesting, but for them, not necessarily for you.&lt;/p&gt;&#xA;" OwnerUserId="22788" LastActivityDate="2023-09-13T13:46:27.380" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26765" PostTypeId="2" ParentId="26746" CreationDate="2023-09-13T14:30:47.560" Score="0" Body="&lt;p&gt;AFAIK in France there are different levels of introduction of numbers.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In primary schools (6-11 years)&lt;/p&gt;&#xA;&lt;p&gt;Decimal and other non integer positive numbers (1/3, pi) are presented as simple calculus tools with very little (if any...) theorical explainations. For example, pi is said to be 3.14 or 22/7 without any more detail. And any of those &lt;em&gt;definitions&lt;/em&gt; are used as-is to get numeric values. But negative numbers are just left out at this point.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In secondary schools (&amp;gt; 11 years)&lt;/p&gt;&#xA;&lt;p&gt;It now matters to explain how &lt;em&gt;mathematically&lt;/em&gt; things work. And on that point, introducing negative numbers only needs addition/substraction which are the simplest operations. On the other hand, correctly handling fractions (not identifying them to their numeric approximation) is much more complex. So IMHO it seems logical to introduce negative numbers before fractions &lt;em&gt;in this cursus&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;That means (IMHO) that the argument that fractions were used centuries before negative numbers has no impact on the way those notions are to be introduced to students. If we just want pupils to be able to do simple calculations fractions should be presented before negative numbers. But if we want to teach some mathematic concepts, then negative numbers should come first.&lt;/p&gt;&#xA;" OwnerUserId="22876" LastActivityDate="2023-09-13T14:30:47.560" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26766" PostTypeId="1" AcceptedAnswerId="26767" CreationDate="2023-09-13T20:04:51.793" Score="7" ViewCount="785" Body="&lt;p&gt;I chose to ask this question on MESE because I think it's not about mathematics per se but more about how it should be communicated.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Quantified statements in mathematics are often written for instance, as,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For all real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x+y=y+x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are variables whose domains are the set of all real&#xA;numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even though I understand here that the expression &amp;quot;&lt;em&gt;For all real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;...&lt;/em&gt;&amp;quot; means that all the elements in the domains of the variables &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; satisfy &lt;span class=&quot;math-container&quot;&gt;$x+y=y+x$&lt;/span&gt;, I feel the expression &amp;quot;&lt;em&gt;For all real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&lt;/em&gt;&amp;quot; by and of itself doesn't convey that same meaning.&#xA;Here is why I feel like that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Case 1. If by writing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in the expression &amp;quot;&lt;em&gt;For all real numbers&#xA;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/em&gt;&amp;quot; we're referring to the variable i.e. the symbol then the&#xA;expression &amp;quot;For all real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; is just like saying &amp;quot;&lt;em&gt;For all&#xA;real numbers (and then abruptly referring to) * symbol x * &amp;quot;&lt;/em&gt;. I mean&#xA;the symbol &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; is not &lt;em&gt;all real&lt;/em&gt; numbers. So what sense does it make&#xA;to follow &lt;em&gt;For all real numbers&lt;/em&gt; by &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;?&lt;/li&gt;&#xA;&lt;li&gt;Case 2. If by writing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in the expression &amp;quot;&lt;em&gt;For all real numbers&#xA;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/em&gt;&amp;quot; we're referring to an unspecified element in the domain of&#xA;&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;, then &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; only refers to a single element (just unspecified).&#xA;So even following &amp;quot;For all real numbers&amp;quot; by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; won't make sense&#xA;because &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a single real number (unspecified); it is not all real&#xA;numbers.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As an analogy to explain my point further, say we take a set &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; whose members are all men in a particular region. Also, let's assume that all these men are married. If we consider the pronoun &amp;quot;he&amp;quot; as a variable, then writing something as&#xA;&amp;quot;For all men &lt;span class=&quot;math-container&quot;&gt;$he$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$he$&lt;/span&gt; is married&amp;quot;&#xA;won't make sense right? We're trying to say that all men in &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; satisfy the open sentence &amp;quot;He is married&amp;quot;, so wouldn't something like &amp;quot;For all men in the domain of &lt;span class=&quot;math-container&quot;&gt;$he$&lt;/span&gt;&amp;quot; or &amp;quot;For all replacements of &lt;span class=&quot;math-container&quot;&gt;$he$&lt;/span&gt;&amp;quot; make more sense?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;It seems to me like &amp;quot;For all real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.....&amp;quot; is a contraction for something fuller such as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For all real numbers in the domain of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&amp;quot;, &lt;span class=&quot;math-container&quot;&gt;$x+y=y+x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For all significant replacements of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&amp;quot;, &lt;span class=&quot;math-container&quot;&gt;$x+y=y+x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So my question is, is writing only 'For all things &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;' instead of something like 'For all things in the domain of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;' an abuse of language in mathematics?&lt;/p&gt;&#xA;" OwnerUserId="20701" LastEditorUserId="20701" LastEditDate="2023-09-14T11:27:05.107" LastActivityDate="2023-09-27T23:49:39.640" Title="Is 'For all $x$' an abuse of language in math?" Tags="|logic|language-use|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26767" PostTypeId="2" ParentId="26766" CreationDate="2023-09-13T21:01:00.550" Score="12" Body="&lt;p&gt;No, there is no abuse of language here.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are placeholders that stand for individual numbers, and your second suggestion captures this: For each number we can insert in place of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, the statement holds. The &amp;quot;For all real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; part specifies where the instances come from. The values &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; take are &lt;em&gt;in&lt;/em&gt; the domain of the real numbers; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; themselves are not the domain, so your first suggestion actually has it slightly backwards.&lt;/p&gt;&#xA;&lt;p&gt;The meaning of the variables is something like &amp;quot;Imagine an arbitrary number - let's call it &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; ... - then we have: &lt;span class=&quot;math-container&quot;&gt;$x + ... = ...$&lt;/span&gt; &amp;quot;. We use letters like &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; because giving these abstarct placeholders names just makes it easier than refer to than &amp;quot;it&amp;quot;, especially as soon as there is more than one variable involved.&lt;/p&gt;&#xA;" OwnerUserId="17743" LastActivityDate="2023-09-13T21:01:00.550" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26768" PostTypeId="2" ParentId="26766" CreationDate="2023-09-14T00:10:46.293" Score="3" Body="&lt;p&gt;Do you have any qualms about &amp;quot;good old fashioned&amp;quot; first-order logic notation? In this language we would write something like &lt;span class=&quot;math-container&quot;&gt;$$ x,y\in \mathbb{R}\rightarrow P(x,y)$$&lt;/span&gt; where, for example, &lt;span class=&quot;math-container&quot;&gt;$P(x,y)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$x+y=y+x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The formalism of first order logic typically comes equipped with an infinite set of &amp;quot;variables&amp;quot; like &lt;span class=&quot;math-container&quot;&gt;$x_0,x_1,\ldots$&lt;/span&gt;&lt;/em&gt; See &lt;a href=&quot;https://ncatlab.org/nlab/show/variable&quot; rel=&quot;nofollow noreferrer&quot;&gt;nLab&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Colloquially we would write &amp;quot;for all real numbers &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$x+y=y+x$&lt;/span&gt;&amp;quot;. Or often people like &lt;span class=&quot;math-container&quot;&gt;$\forall x\in\mathbb{R}, \forall y\in\mathbb{R},\  x+y=y+x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You seem to be going down the rabbit hole of &lt;a href=&quot;https://ncatlab.org/nlab/show/type&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;types&amp;quot;&lt;/a&gt;, and uneasy with issues related to symbols and variables. If so, then I don't think your question is pertinent to mathematics education.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-09-14T00:39:26.680" LastActivityDate="2023-09-14T00:39:26.680" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26769" PostTypeId="1" CreationDate="2023-09-14T00:29:17.427" Score="0" ViewCount="69" Body="&lt;p&gt;Can anyone recommend good and didactic references that delve into the dualism between deterministic and stochastic phenomena? Ideally, I'm seeking materials that provide a conceptual explanation along with numerous examples suitable for high school students.&lt;/p&gt;&#xA;&lt;p&gt;to define what is a random phenomena is not easy as point out this article &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00029890.2002.11919838&quot; rel=&quot;nofollow noreferrer&quot;&gt;What Is a Random Sequence?by Sérgio B. Volchan&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;also, see these videos : &lt;a href=&quot;https://www.youtube.com/watch?v=9rIy0xY99a0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=9rIy0xY99a0&lt;/a&gt; and&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=sMb00lz-IfE&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=sMb00lz-IfE&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="2083" LastEditDate="2023-09-14T08:40:46.310" LastActivityDate="2023-09-15T04:43:29.260" Title="Seeking References on Deterministic and Stochastic Phenomena Suitable for High School Students" Tags="|secondary-education|examples|probability|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26770" PostTypeId="2" ParentId="26766" CreationDate="2023-09-14T04:14:21.950" Score="3" Body="&lt;p&gt;&lt;em&gt;&amp;quot;For all 'things' &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;&lt;/em&gt; is just the shortest way to say any of the following, which are equivalent but progressively wordy and cumbersome:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;&amp;quot;for all 'things' &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&amp;quot;if you replace &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with any 'thing'&amp;quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&amp;quot;for every element &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that is a 'thing'&amp;quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&amp;quot;for any 'thing' that you replace &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with&amp;quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&amp;quot;if you let the domain of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; consist of all 'things'&amp;quot;&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-09-14T04:14:21.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26771" PostTypeId="2" ParentId="26766" CreationDate="2023-09-14T10:55:15.917" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; refers&#xA;to the variable, i.e., the symbol '&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;', right? So, isn't&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For all real numbers '&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;',...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;a more accurate phrasing? If so, this sounds like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For all real numbers &lt;em&gt;the symbol &lt;span class=&quot;math-container&quot;&gt;$x,$&lt;/span&gt;&lt;/em&gt;....&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Additionally, if we're making the name-object distinction, then it's the names of&#xA;real numbers that we're substituting for '&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;', not the real numbers themselves,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the sentence “For all real numbers &lt;span class=&quot;math-container&quot;&gt;$x,$&lt;/span&gt;...”, the variable/symbol &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (its entity, so to speak) is a &lt;strong&gt;placeholder&lt;/strong&gt; for any real number, like &lt;span class=&quot;math-container&quot;&gt;$\sqrt2.$&lt;/span&gt; Outside of an ontology discussion, I struggle to find the value in distinguishing between the &lt;em&gt;names&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &amp;amp; &lt;span class=&quot;math-container&quot;&gt;$\sqrt2$&lt;/span&gt; and the entities &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &amp;amp; &lt;span class=&quot;math-container&quot;&gt;$\sqrt2;$&lt;/span&gt; enclosing them within quotations marks is also pointlessly meta. To wit: Harshit is both a referent and the referent's name, and in the following sentence, would you use quotation marks around the word and call attention to it?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For the candidate ‘Harshit’, an interview is scheduled.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Case 2. &amp;quot;For all real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; doesn't make sense because &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a single real number (unspecified); it is not all real numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thanks for pointing this out—this is the most interesting part of your Question! In a &lt;a href=&quot;https://matheducators.stackexchange.com/a/25472/372&quot;&gt;previous post&lt;/a&gt;, I expressed dissatisfation with the phrasing “for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;”, and hinted that I prefer to say “&lt;strong&gt;for each&lt;/strong&gt;/&lt;strong&gt;every&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;”:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;“&lt;em&gt;For all&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$F,\, P(x)$&lt;/span&gt; holds” sometimes sounds like the property &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; might belong to &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; as a whole rather than to its individual members: “for all members of the family, they have a house” (1 house in total? or 5?). Contrast with “&lt;em&gt;for each&lt;/em&gt;/&lt;em&gt;every&lt;/em&gt; member of the family, they have a house” (definitely 5 houses in total).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;After rereading your Question, Michael's answer, and further rumination, I think it's finally clear that the root issue is as you are highlighting: in the phrasing “&lt;em&gt;for all elements&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x,\,P(x)$&lt;/span&gt; holds”, every occurrence of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is singular, so what's with the plural form “for all”? We could contrive that this phrasing is short for “for all elements, each of which we denote by &lt;span class=&quot;math-container&quot;&gt;$x,\,P(x)$&lt;/span&gt; holds” but of course then we might as well just say “for &lt;em&gt;each element&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x,\,P(x)$&lt;/span&gt; holds”.&lt;/p&gt;&#xA;&lt;p&gt;Let's just acknowledge that &lt;strong&gt;the “for all &lt;span class=&quot;math-container&quot;&gt;$\boldsymbol x$&lt;/span&gt;” phrasing is Mathlish&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I am asking this question on MESE because I think it's not about mathematics per se but more about how it should be communicated.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Sorry for being so contrary, but this question is not about mathematics communication or teaching, but rather about mathematical writing, that is, communication &lt;em&gt;in&lt;/em&gt; mathematics.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2023-09-26T02:59:37.827" LastActivityDate="2023-09-26T02:59:37.827" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26772" PostTypeId="2" ParentId="26766" CreationDate="2023-09-14T11:50:20.000" Score="5" Body="&lt;p&gt;I'm not a native english speaker, but I agree that &amp;quot;for all numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&amp;quot; sounds strange. Because in everyday language we don't use names after a &amp;quot;for all&amp;quot;. For instance, we wouldn't say: &amp;quot;for all persons, Alice and Bob...&amp;quot;. Nor can I think of a variation that makes this work: &amp;quot;for all two persons, call them Alice and Bob...&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;But I believe we can say: &amp;quot;for &lt;em&gt;any&lt;/em&gt; pair of persons, call this pair Alice and Bob for the sake of argument...&amp;quot; or &amp;quot;for &lt;em&gt;any&lt;/em&gt; two persons, call them Alice and Bob...&amp;quot; or maybe &amp;quot;for &lt;em&gt;every&lt;/em&gt; two persons, say Alice and Bob...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So maybe it sounds more natural if you say &amp;quot;for any/each/every&amp;quot; instead of &amp;quot;for all&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="590" LastEditorUserId="590" LastEditDate="2023-09-15T03:33:33.613" LastActivityDate="2023-09-15T03:33:33.613" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="26774" PostTypeId="2" ParentId="26769" CreationDate="2023-09-14T17:38:56.983" Score="1" Body="&lt;p&gt;There was a question &amp;quot;is throwing dice a deterministic or stochastic process&amp;quot; on Physics.SE a while back, and I think &lt;a href=&quot;https://physics.stackexchange.com/a/477980&quot;&gt;this answer&lt;/a&gt; is insightful and accessible enough to spur a good discussion amongst a class of high schoolers who have some baseline amount of interest in the distinction.&lt;/p&gt;&#xA;&lt;p&gt;An excerpt:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Throwing dice is just throwing dice. That's all. It's not stochastic, nor deterministic. It's just throwing dice.&lt;/p&gt;&#xA;&lt;p&gt;Now we model throwing dice as a process, and that's where the stochastic or deterministic side starts to play in. It is the process that is stochastic or deterministic, not the throwing of the dice. It's how we think about the throwing of dice that can be stochastic or deterministic.&lt;/p&gt;&#xA;&lt;p&gt;... continued &lt;a href=&quot;https://physics.stackexchange.com/a/477980&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-09-15T04:43:29.260" LastActivityDate="2023-09-15T04:43:29.260" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26775" PostTypeId="2" ParentId="26755" CreationDate="2023-09-14T17:53:43.967" Score="3" Body="&lt;p&gt;For a textbook reference, you'll find that big-sigma summation notation is covered in most textbooks for college algebra and precalculus, complete with scaffolded exercises on the topic. On the one hand, it's likely to be in the latter part of the book and not covered in related courses; but any such book would provide a good resource to fill in a gap like this.&lt;/p&gt;&#xA;&lt;p&gt;One open-access example would be: &lt;a href=&quot;https://openstax.org/books/college-algebra-2e/pages/9-4-series-and-their-notations&quot; rel=&quot;nofollow noreferrer&quot;&gt;OpenStax College Algebra, Sec. 9.4: Series and Their Notations&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I think it's important to note the name and history of the Greek big-sigma symbol as the ancestor of our capital &amp;quot;S&amp;quot;, and so it makes sense as a mnemonic for a big summation. An interesting side-question would then be to see if students can intuit a good capital Greek letter to use for a large product.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-09-14T17:53:43.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26776" PostTypeId="2" ParentId="26759" CreationDate="2023-09-14T19:01:52.963" Score="2" Body="&lt;p&gt;This is a little bit of a frame challenge, but I don't think that it matters all that much.  What is the graph of a line trying to convey?  What data are &lt;em&gt;really&lt;/em&gt; necessary to get right?  In my opinion,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;an approximation of the slope (very negative, slightly negative, flat, slightly positive, very positive),&lt;/li&gt;&#xA;&lt;li&gt;a general indication of the location of the intercepts, and&lt;/li&gt;&#xA;&lt;li&gt;there is no (3).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;My general advice to students when graphing these kinds of things is to either (a) sketch two easy-to-find points and connect-the-dots, or (b) sketch one easy-to-find point and use the slope to finish the graph.  &amp;quot;Easy-to-find&amp;quot; is a bit of a dodge—it doesn't &lt;em&gt;really&lt;/em&gt; explain what is going on—which is why I like to give a few random examples of different kinds of lines, generally by using a set of 20-sided dice to determine coefficients (plus a coin to determine the signs of coefficients).  What is &amp;quot;easy-to-find&amp;quot; is going to be down to taste and experience.&lt;/p&gt;&#xA;&lt;p&gt;In the example given, I think that the intercepts are relatively easy-to-find, so I would expect an answer from a student which looks something like the following:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UcpGn.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UcpGn.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;My students will often work very hard to draw everything to scale, but you'll notice that I haven't bothered.  The important information is there:  the exact location of the intercepts (more importantly, the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-intercept is on the positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis, the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept is on the positive &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis, and the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept is closer to the origin than the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-intercept), and an indication that the slope is slightly negative (i.e. negative, but not super-steep, i.e. somewhere between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-1/2$&lt;/span&gt;, perhaps)&lt;/p&gt;&#xA;&lt;p&gt;I think that the philosophy that students should be taught early is that a graph should show key information, and that everything else can and should be left a little imprecise.  This is true for lines, for parabolas, and for all the curves graphed in calculus and beyond.  If one needs a &lt;em&gt;really&lt;/em&gt; pretty picture, computers are very good at making super accurate graphs—the point of doing things by hand is to identify important features, not to get a publication-ready image.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2023-09-14T19:01:52.963" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26777" PostTypeId="1" AcceptedAnswerId="26778" CreationDate="2023-09-15T14:29:27.533" Score="-1" ViewCount="122" Body="&lt;p&gt;I heard that before you can handle data automatically, you have to know how to handle it manually.&lt;/p&gt;&#xA;&lt;p&gt;Why is it impossible to find a problem and build some machine learning algorithm to find a proper algorithm to solve the problem?&lt;/p&gt;&#xA;" OwnerUserId="22887" LastActivityDate="2023-09-15T15:17:06.097" Title="Why do problems should be solved by pen and paper before coding?" Tags="|problem-solving|" AnswerCount="1" CommentCount="3" ClosedDate="2023-09-16T11:21:33.420" ContentLicense="CC BY-SA 4.0" />
  <row Id="26778" PostTypeId="2" ParentId="26777" CreationDate="2023-09-15T15:17:06.097" Score="4" Body="&lt;p&gt;In theory, it &lt;em&gt;is&lt;/em&gt; possible to solve a problem by throwing an ML algo at it, without knowing how to do it manually.&lt;/p&gt;&#xA;&lt;p&gt;But in practice, your ML solution usually doesn't work right out of the box, and you have to debug it.&lt;/p&gt;&#xA;&lt;p&gt;In order to debug, you need to be able to identify differences between what it's doing and what it &amp;quot;ought&amp;quot; to be doing -- which means you need to understand the problem/solution well enough to run some sanity checks on your model and the underlying data.&lt;/p&gt;&#xA;&lt;p&gt;Additionally, a common solution approach involves compiling a data set of human decisions and using ML to interpolate those decisions, i.e. predict what decision a human would make on cases not explicitly included in the data set.&lt;/p&gt;&#xA;&lt;p&gt;For instance, when creating an image classifier, the most common solution approach is to&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;gather a data set of images,&lt;/li&gt;&#xA;&lt;li&gt;manually label the images with classifications (or pay other people to do the manual labeling if it's a ton of work or requires domain expertise that you don't have), and then&lt;/li&gt;&#xA;&lt;li&gt;train an ML model to predict what classification a human would manually assign, given any arbitrary image input.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Step 2 above requires the ability for humans to perform the task manually (even if they don't know exactly how they're doing it).&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-09-15T15:17:06.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26779" PostTypeId="1" CreationDate="2023-09-15T19:26:49.197" Score="4" ViewCount="157" Body="&lt;p&gt;This question concerns teaching teachers who often already teach mathematics, but are now studying to get a formal qualification for it, and hopefully some more competency as well.&lt;/p&gt;&#xA;&lt;p&gt;What are the key ideas about &lt;strong&gt;(1) mathematical modelling&lt;/strong&gt; and &lt;strong&gt;(2) teaching it to pupils&lt;/strong&gt;, who are often in middle school, but might be in primary school or even the latter part of secondary education (videregånde in Norway)?&lt;/p&gt;&#xA;&lt;p&gt;My own understanding of mathematical modelling is that one takes some phenomenon or process from reality, simplifies it, captures it usually in a function or an equation (a much more complicated model is hardly relevant for these students, who are not physicists or engineers or even mathematicians), manipulates or solves that or deduces something from there, and then translates the solution back to the real life scenario, seeing what one has learned from there, and also checking whether the outcome makes any sense at all.&lt;/p&gt;&#xA;&lt;p&gt;I would expect to teach something like this process. But what other key ideas are there in the didactics literature, especially about teaching modelling?&lt;/p&gt;&#xA;&lt;p&gt;Please cite the research literature you are drawing your information from.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-09-17T14:44:42.027" Title="The key didactical ideas on mathematical modelling?" Tags="|applied-mathematics|teacher-training|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26780" PostTypeId="2" ParentId="26558" CreationDate="2023-09-16T05:25:03.870" Score="2" Body="&lt;p&gt;&amp;quot;&lt;a href=&quot;https://www.cambridge.org/highereducation/books/how-to-prove-it/6D2965D625C6836CD4A785A2C843B3DA&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to Prove It&lt;/a&gt;&amp;quot; is a good book for beginners in proofs in mathematics. But there is also &amp;quot;&lt;a href=&quot;https://www.abebooks.com/9780199218431/Nets-Puzzles-Postmen-Exploration-Mathematical-0199218439/plp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nets, Puzzles, and Postmen&lt;/a&gt;&amp;quot; that is an excellent introduction to graph theory for lay people that even has interesting bits for advanced readers while keeping it simple and clear for beginners.&lt;/p&gt;&#xA;&lt;p&gt;Regarding logic, I don't believe that &amp;quot;She [...] teaches formal logic to philosophy undergraduates&amp;quot; and &amp;quot;[she] is struggling with the formal language and style of proofs&amp;quot; are consistent. If she truly understands FOL (first-order logic) well enough to be able to teach it, then she certainly can learn all mathematics (regardless of area) given sufficient work and patience. But I actually have some doubts as to that. For a concrete assessment, look at &lt;a href=&quot;https://math.stackexchange.com/a/1684204/21820&quot;&gt;this Fitch-style natural deduction system for FOL&lt;/a&gt;, which anyone (with just high-school knowledge) can learn to use and thereby gain a 100% crystal-clear understanding of formal proofs. I have had success with every student who has learnt this system.&lt;/p&gt;&#xA;&lt;p&gt;Also, I always encourage teachers to use more interesting applications of mathematics, including teaching more &lt;a href=&quot;https://matheducators.stackexchange.com/a/18990/1550&quot;&gt;concrete interesting mathematics&lt;/a&gt;, and engaging in solving &lt;a href=&quot;https://matheducators.stackexchange.com/a/25518/1550&quot;&gt;logic puzzles&lt;/a&gt;. You will notice that these things are &lt;strong&gt;rarely&lt;/strong&gt; taught in schools, which in my opinion is to the detriment of students. Mathematics should not be about learning to do what the teacher told you to do, but about &lt;strong&gt;learning truth via guided exploration with a curious mindset&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2023-09-16T05:25:03.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26782" PostTypeId="2" ParentId="26779" CreationDate="2023-09-17T01:34:53.133" Score="4" Body="&lt;p&gt;I'll talk about just one idea: quantitative reasoning is extremely important for mathematical modeling. This may seem obvious to some, especially to science educators, but I rarely hear mathematics educators discussing quantitative reasoning and the related literature, even though modeling is becoming an increasingly popular conversation topic.&lt;/p&gt;&#xA;&lt;p&gt;In a 2022 article, &amp;quot;&lt;a href=&quot;https://www.researchgate.net/publication/361477780_A_bridging_study_analyzing_mathematical_model_construction_through_a_quantities-oriented_lens&quot; rel=&quot;nofollow noreferrer&quot;&gt;A bridging study analyzing mathematical model construction through a quantities-oriented lens&lt;/a&gt;,&amp;quot; Czocher et al. describe various modeling competencies that align well with your description of modelling. &amp;quot;Modelling competencies include forming an understanding of a real-world scenario, simplifying, mathematizing (producing a representation), solving, interpreting, and validating.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In their introduction, Czocher et al. mention that the &amp;quot;simplifying&amp;quot; and &amp;quot;mathematizing&amp;quot; competencies are known to be particularly challenging for students. A goal of the study was to analyze the relationship between these competencies and students' quantitative reasoning abilities. The authors concluded that &amp;quot;with regard to mathematizing, the data support Thompson’s (2011) position that quantitative reasoning should be the basis of mathematical modelling.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In Thompson's (2011) article &lt;a href=&quot;http://pat-thompson.net/PDFversions/2011QR&amp;amp;Modeling.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Quantitative Reasoning and Mathematical Modeling,&amp;quot;&lt;/a&gt; he describes a theoretical framework for quantitative reasoning, and there are several implications for teaching modeling that I've taken away. One is that I need to intentionally support students to conceive of quantities in productive or desired ways.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The point that quantities are mental constructions, and that their creation is often effortful, is central to mathematics education. Too often quantities, such as area and volume, are taken as obvious, and hence there is no attention given to student’s construction of quantity through the dialectic object-attribute-quantification. Instead, textbook writers and teachers just use them to teach. (Thompson, 2011, p. 34)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Lobato and Siebert (2002), in &lt;a href=&quot;https://www.researchgate.net/publication/222550283_Quantitative_reasoning_in_a_reconceived_view_of_transfer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Quantitative reasoning in a reconceived view of transfer,&amp;quot;&lt;/a&gt; describe a teaching experiment in which students who had previously learned about slope were taught again with a focus on quantitative reasoning. They present the case of a student who, at the start of the experiment, explicitly rejected the use of the slope formula on a task involving the steepness of a ramp. According to the researchers, after &amp;quot;a lengthy process of reconstructing his understanding of the quantitative relationships involved,&amp;quot; the student was able to employ proportional reasoning on the same task.&lt;/p&gt;&#xA;&lt;p&gt;Another implication for teaching modeling that I've taken from the quantitative reasoning literature is that I need to attend closely to students' conceptions of &amp;quot;variation.&amp;quot; In the same article, Thompson describes two students' differing perspectives on how quantities vary, a &amp;quot;chunky&amp;quot; perspective and a &amp;quot;continuous&amp;quot; perspective, from &lt;a href=&quot;http://pat-thompson.net/PDFversions/Theses/2010CastilloGarsow.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;a teaching experiment by Castillo-Garsow&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In Tiffany’s thinking, the account’s earned interest did not pass through the 1/4-year amount on its way to the one-year amount. Rather, the 1/4-year amount was an amount to be computed ad hoc. Derek, who thought of time passing continuously and thought of an account having a value at each moment in time, thought more in terms of the account changing at the rate of a number of dollars per year for some number of years, and the number of years needn’t be a whole number. Earned interest accrued continuously. Tiffany could not see that the account's rate of change with respect to time was proportional to the account’s value. Derek saw this relationship intuitively. (p. 49)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There are other important ideas about teaching mathematical modeling in the literature, and Czocher et al. acknowledge several of them in their article. But I find the link to quantitative reasoning to be the most compelling, based on my personal experience with students' difficulties with modeling.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-09-17T01:34:53.133" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26783" PostTypeId="2" ParentId="26779" CreationDate="2023-09-17T14:44:42.027" Score="1" Body="&lt;p&gt;Opposing points of view to emphasizing modeling are mostly based on cognitive load theory, with John Sweller the most prominent voice.  I would read some of that work and at least consider them before diving into a bunch of discovery/constructivist/IBL/etc. instruction.&lt;/p&gt;&#xA;&lt;p&gt;I'm not even saying the counterargument is correct (but I lean in that direction), just that you should at least read both sides.  Clearly the situation is not so simple, nor pedagogy as a field so clear cut, that people can't get contrasting views published.  We are not talking about reporting spectral lines of atoms, when things get published as ed research.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_1&lt;/a&gt; (longer)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://fillingthepail.substack.com/p/elevator-pitch&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://fillingthepail.substack.com/p/elevator-pitch&lt;/a&gt; (nutshell)&lt;/p&gt;&#xA;&lt;p&gt;Note1:  A bit of a frame challenge.  I would also carefully consider level of the kids and ability of the kids when thinking about modeling.  What is best for older kids (with more models already in their head as schema to rely on) or stronger kids (with more native problem solving skills) may not be best for average or sub-average middle school kids.  At least think about that variable in the equation.  I'm worried about the experience factor in particular.  12 year olds don't have the same experience with chemistry and physics (many equations, models) as 18 year olds.&lt;/p&gt;&#xA;&lt;p&gt;Note2:  Another frame challenge.  I would also consider that modeling activities may make more sense as part of a science course, not a math class.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  If this sounds like I'm anti modeling or anti applied math, it's far from the case.  If anything, I have a stronger general background in general technology and social science than most mathematicians and math educators.  (And weaker math background than them!)  And I love the gears and sweat and oil of the applied world more than the icy axioms of Mathland.  But...&amp;quot;words problems are hard&amp;quot;, as guest troll Barbie once said, so...&amp;quot;Danger, Will Robinson&amp;quot;.  ;-)&lt;/p&gt;&#xA;" OwnerUserId="22893" LastActivityDate="2023-09-17T14:44:42.027" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26784" PostTypeId="2" ParentId="26750" CreationDate="2023-09-17T22:51:28.920" Score="1" Body="&lt;p&gt;I like this site for matrix computations: &lt;a href=&quot;https://matrixcalc.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://matrixcalc.org/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I don't think you can directly tell it to add &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; times row &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; to row &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, so instead you have to multiply by the appropriate matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, which is easy enough. It works well with symbolic entries, it's easy to update matrices &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; as you progress, and it keeps a record of your computations. Might not be exactly what you want but check it out.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2023-09-17T22:51:28.920" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26785" PostTypeId="1" AcceptedAnswerId="26786" CreationDate="2023-09-19T10:56:31.280" Score="-5" ViewCount="118" Body="&lt;p&gt;Does the &amp;quot;&lt;strong&gt;Middle School Mathematics domains&lt;/strong&gt;&amp;quot; on page 3 of &lt;a href=&quot;https://www.ets.org/content/dam/ets-org/pdfs/praxis/5164.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ets.org/content/dam/ets-org/pdfs/praxis/5164.pdf&lt;/a&gt; refer to the the following 5 topics/categories?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(I) Numbers and Operations, (II) Algebra, (III) Functions, (IV) Geometry and Measurement, and (V) Statistics and Probability&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0gw6I.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0gw6I.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22900" LastEditorUserId="2083" LastEditDate="2023-09-19T20:06:21.917" LastActivityDate="2023-09-19T20:06:21.917" Title="Does the &quot;Middle School Mathematics domains&quot; refer to (I) through (V) topics?" Tags="|terminology|middle-school|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26786" PostTypeId="2" ParentId="26785" CreationDate="2023-09-19T15:28:18.610" Score="3" Body="&lt;p&gt;Yes. It says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;all questions assess content from the &lt;strong&gt;above&lt;/strong&gt; Middle School Mathematics domains,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and the only thing directly &lt;strong&gt;above&lt;/strong&gt; is the table. And the only text fields in the table are the content categories:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(I) Numbers and Operations&lt;/p&gt;&#xA;&lt;p&gt;(II) Algebra&lt;/p&gt;&#xA;&lt;p&gt;(III) Functions&lt;/p&gt;&#xA;&lt;p&gt;(IV) Geometry and Measurement&lt;/p&gt;&#xA;&lt;p&gt;(V) Statistics and Probability&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, the phrase &amp;quot;Middle School Mathematics domains&amp;quot; refers to the &amp;quot;Content Categories&amp;quot; column in the table.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-09-19T15:33:31.703" LastActivityDate="2023-09-19T15:33:31.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26787" PostTypeId="1" CreationDate="2023-09-19T20:08:49.167" Score="1" ViewCount="215" Body="&lt;p&gt;Let's say I have the numeral 2.263,3 thousands, and convert it to 2.263.300 units.&lt;/p&gt;&#xA;&lt;p&gt;How do we describe what I have done to the numeral regarding &lt;strong&gt;units&lt;/strong&gt; ?&lt;/p&gt;&#xA;&lt;p&gt;I know it has to do with the place values of the positions of digits&lt;/p&gt;&#xA;" OwnerUserId="12046" LastEditorUserId="20488" LastEditDate="2023-09-20T15:44:24.453" LastActivityDate="2023-09-26T06:31:48.787" Title="Naming the procedure of converting the place values of digits" Tags="|terminology|place-value|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26788" PostTypeId="2" ParentId="26787" CreationDate="2023-09-19T21:40:38.727" Score="3" Body="&lt;p&gt;I think this is just called &amp;quot;converting between place value.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If you Google that phrase, you'll see tons of hits that include problems like the one you referenced.&lt;/p&gt;&#xA;&lt;p&gt;For instance, IXL (one of the most popular sites for practicing math problems) has a problem type called &amp;quot;convert between place value&amp;quot; in its 4th grade math contents: &lt;a href=&quot;https://www.ixl.com/math/grade-4/convert-between-place-values&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ixl.com/math/grade-4/convert-between-place-values&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-09-19T21:40:38.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26789" PostTypeId="2" ParentId="26787" CreationDate="2023-09-21T16:58:26.467" Score="0" Body="&lt;p&gt;When the &amp;quot;thousands&amp;quot; are given as &lt;a href=&quot;https://en.wikipedia.org/wiki/Metric_prefix&quot; rel=&quot;nofollow noreferrer&quot;&gt;SI prefix&lt;/a&gt; &amp;quot;k&amp;quot;, then it's just transforming the &amp;quot;k&amp;quot; to the corresponding power of 10:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2.263,3  \text { k[unit]} = 2.263,3 \cdot 10^3 \text{[unit]} = 2.263.300 \text{[unit]}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="667" LastActivityDate="2023-09-21T16:58:26.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26791" PostTypeId="2" ParentId="26787" CreationDate="2023-09-23T15:13:37.973" Score="1" Body="&lt;p&gt;It is called converting of units: if you have 2.263,3 m you can convert it to 226.330cm o 2263.3300mm and so on . when you write   2.263,3 thousands probably you mean in units  2.263,3*1m=2.263,3m *1000mm/m&lt;/p&gt;&#xA;" OwnerUserId="21051" LastActivityDate="2023-09-23T15:13:37.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26792" PostTypeId="1" AcceptedAnswerId="26795" CreationDate="2023-09-23T19:33:46.900" Score="6" ViewCount="6208" Body="&lt;p&gt;Consider:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In a right triangle:&lt;/p&gt;&#xA;&lt;p&gt;sin(2x + 4) = cos (46)&lt;/p&gt;&#xA;&lt;p&gt;What is the value of x?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The question above is from standardized tests for a geometry course. If my goal is to have students understand cofunctions, what benefit is there to force students to do algebra? It seems that we are making geometry harder than it needs to be.&lt;/p&gt;&#xA;&lt;p&gt;And I guess this could probably apply to other fields of math?&lt;/p&gt;&#xA;&lt;p&gt;I can understand analysis courses requiring topology but the topology is taught with regards to the analysis and not as pure topology.&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="10335" LastEditDate="2023-09-26T02:07:46.977" LastActivityDate="2023-10-14T19:08:22.767" Title="What benefit is there to obfuscate the geometry with algebra?" Tags="|secondary-education|geometry|general-pedagogy|" AnswerCount="5" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="26793" PostTypeId="2" ParentId="26792" CreationDate="2023-09-23T20:52:02.803" Score="18" Body="&lt;p&gt;This multi-step question requires students to understand and apply multiple concepts or strategies to solve the problem. The goal of a standardized test is not to provide a correctly-sequenced list of questions that leads students to conceptual understanding; instead, tests need to include more difficult questions in order to distinguish between two students with above-average levels of ability. Ideally, students would have learned these concepts before approaching this problem. If a student is not able to solve the related equation &lt;span class=&quot;math-container&quot;&gt;$\mathrm{sin}(y) = \mathrm{cos}(46)$&lt;/span&gt;, then he or she is not ready to solve this one. In fact, one &amp;quot;strategy&amp;quot; to solving this problem would be to first solve the simpler equation and then assuming &lt;span class=&quot;math-container&quot;&gt;$y = 2x+4$&lt;/span&gt;, solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. This strategy effectively separates the issues for the student.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2023-09-23T20:52:02.803" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26794" PostTypeId="2" ParentId="26792" CreationDate="2023-09-23T23:55:43.190" Score="23" Body="&lt;p&gt;Tests seek to measure ability. Math ability, like most other forms of ability (including athletic ability), isn't solely dependent on one's ability to execute individual skills in isolation -- it also depends on one's ability to strategically combine skills.&lt;/p&gt;&#xA;&lt;p&gt;Your goal might just be for your students to understand cofunctions, but the goal of the test is to also measure students' abilities to strategically combine that skill with other skills.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-09-24T00:04:18.817" LastActivityDate="2023-09-24T00:04:18.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26795" PostTypeId="2" ParentId="26792" CreationDate="2023-09-24T20:00:45.017" Score="21" Body="&lt;h1&gt;Is your ultimate goal &lt;em&gt;really&lt;/em&gt; just to teach cofunctions?&lt;/h1&gt;&#xA;&lt;p&gt;Or are you trying to teach cofunctions so that the students can apply them later?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I am speaking as a student rather than an educator, but math, just as most subjects, is taught incrementally with more advanced concepts building on simpler concepts. All mathematics was invented to solve problems in the real world. Showing that a student can apply their knowledge of different concepts in isolation has little practical use. The real world is not a controlled laboratory environment where specific disciplines can be controlled and used in isolation. Only the simplest real-world problems can be solved by applying a single concept by itself.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Anecdotally, when I was in high school, I was continually surprised by the number of my classmates that would excel in advanced math and science courses yet struggle tremendously when multiple math concepts needed to be applied in conjunction, particularly with word problems. Well meaning educators &lt;em&gt;do&lt;/em&gt; test concepts in isolation in order not to confuse students or &amp;quot;obfuscate&amp;quot; the problem, but it's my opinion that this will only serve to set the student up for failure in the future.&lt;/p&gt;&#xA;" OwnerUserId="22173" LastActivityDate="2023-09-24T20:00:45.017" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26796" PostTypeId="2" ParentId="26792" CreationDate="2023-09-25T13:29:15.217" Score="9" Body="&lt;p&gt;I have been on committees that write questions for standardized tests and placement tests. In this role, I have reviewed results of many trigonometry questions that were piloted and then revised for re-piloting, removed for future piloting all together, or approved for inclusion as a graded test item.&lt;/p&gt;&#xA;&lt;p&gt;If this were piloted in the form of a multiple-choice question, I would expect the &lt;a href=&quot;https://testing.wisc.edu/whatdonumbersmean.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;point-biserial correlation test score&lt;/a&gt; would raise lots of red flag!&lt;/p&gt;&#xA;&lt;p&gt;The first thing that made me cringe with this question is probably just professional baggage: I got hung up quickly on the 46. I spent too much mental energy wondering if somebody meant &lt;span class=&quot;math-container&quot;&gt;$46^{\circ}$&lt;/span&gt; instead of 46 radians. After all, &lt;span class=&quot;math-container&quot;&gt;$46$&lt;/span&gt; is close to &lt;span class=&quot;math-container&quot;&gt;$45$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$45^{\circ}$&lt;/span&gt; is nice. The strangeness of the integer 46 was a big distraction for me. I got over it, and said to myself that  in the spirit of the &lt;em&gt;Douglas Adams answer to the universe&lt;/em&gt;, too bad it was not 42. Oh well.&lt;/p&gt;&#xA;&lt;p&gt;But I did not get over it gracefully! The darn integer 46 and &lt;span class=&quot;math-container&quot;&gt;$\cos(46)$&lt;/span&gt; made me start thinking about &lt;span class=&quot;math-container&quot;&gt;$\arcsin(\cos(46))$&lt;/span&gt;. Embarrassing. I thought to myself, &amp;quot;I mean really, &lt;span class=&quot;math-container&quot;&gt;$\cos(46)$&lt;/span&gt; has no prospects for a future.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Eventually I realized I should think in terms of &lt;span class=&quot;math-container&quot;&gt;$\sin(\theta)=\cos(46)$&lt;/span&gt; and then &lt;span class=&quot;math-container&quot;&gt;$$\sin(\theta)=\cos(\theta+\pi/2)=\cos(46)$$&lt;/span&gt; and then I knew I was on the right path, finally. I was still annoyed with the manifest radian context with the &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; on the left side and that thing I thought might be a misprint for &lt;span class=&quot;math-container&quot;&gt;$46$&lt;/span&gt; degrees on the right.&lt;/p&gt;&#xA;&lt;p&gt;Then the results are still messy and kind of tricky for most students:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\theta=2k\pi+\frac{27\pi}{2}+46$$&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\theta=2k\pi+\frac{29\pi}{2}-46$$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$k\in{\bf Z}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Yikes! And I still have to come back and fiddle with the substitution &lt;span class=&quot;math-container&quot;&gt;$\theta=2x+4$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My guess is that if this question were piloted on a standardized test, it would come back with very poor results and beg for some revisions and extensive repiloting before being allowed on the exam for grading.&lt;/p&gt;&#xA;&lt;p&gt;Observation: if the 46 was indeed supposed to be &lt;span class=&quot;math-container&quot;&gt;$46^{\circ}$&lt;/span&gt; then the solutions for &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; (in radians) would be cleaner:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\theta=2k\pi+\frac{34\pi}{45}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\theta=2k\pi+\frac{11\pi}{45}$$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$k\in{\bf Z}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-09-25T23:35:36.940" LastActivityDate="2023-09-25T23:35:36.940" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26797" PostTypeId="2" ParentId="26766" CreationDate="2023-09-26T06:12:42.123" Score="1" Body="&lt;p&gt;&lt;em&gt;&lt;strong&gt;edit&lt;/strong&gt; - Stumbled on a succinct way to put it when writing the comments below:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;You're trying to apply a stricter reading of the English language than a reasonable speaker of the language would apply. You're not wrong that your interpretation is &lt;em&gt;a&lt;/em&gt; correct reading of that phrasing, but you are wrong when you claim that it is &lt;em&gt;the only&lt;/em&gt; possible correct reading of that phrasing.&lt;/p&gt;&#xA;&lt;p&gt;It is perfectly reasonable for &amp;quot;for all real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; to be understood to mean &amp;quot;with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; being any possible real number value&amp;quot;, as opposed to only being understood to mean &amp;quot;with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; being the complete set of all real numbers&amp;quot;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;It would appear that the core issue you perceive here is that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;all real numbers &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;can only be interpreted as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is [all real numbers]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and not as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;All [&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a real number] instances&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is an overly strict reading of the phrasing, and one that does not mesh well with how English is commonly interpreted. In informally spoken English, it is perfectly reasonable for&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;plural of [real number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;to be considered synonymous with&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[plural of real number] &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is that formally correct? No, but English is not defined as rigorously as mathematics, and the informal understanding is that these two definitions are sufficiently equivalent to be understood the same way.&lt;/p&gt;&#xA;" OwnerUserId="13169" LastEditorUserId="13169" LastEditDate="2023-09-27T23:49:39.640" LastActivityDate="2023-09-27T23:49:39.640" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="26798" PostTypeId="2" ParentId="26787" CreationDate="2023-09-26T06:26:31.860" Score="2" Body="&lt;p&gt;I don't believe there is a set phrase for this, other than &amp;quot;conversion&amp;quot;. However, conversion is very broad because it also includes factors that are not powers of the base that you are working in.&lt;/p&gt;&#xA;&lt;p&gt;When the conversion factor is not a power of the base you're working in (e.g. multiplying a base 10 number by any number that is not a power of 10), the number's digits will change instead of only shifting the decimal separator left or right.&lt;br /&gt;&#xA;Imperial units are a classic example of this, and this added complexity is generally why proponents of the metric system claim it to be superior to the imperial system.&lt;/p&gt;&#xA;&lt;p&gt;If you specifically want to focus on moving the decimal separator without otherwise altering the digits, and you're not sufficiently happy with &amp;quot;moving the decimal separator&amp;quot;, I would suggest the neologisms &lt;strong&gt;digit shift&lt;/strong&gt; (for any base) or &lt;strong&gt;decimal shift&lt;/strong&gt; (specifically for base 10).&lt;/p&gt;&#xA;&lt;p&gt;The suggestion of &amp;quot;digit shift&amp;quot; is based on the established name for &lt;a href=&quot;https://en.wikipedia.org/wiki/Bitwise_operation&quot; rel=&quot;nofollow noreferrer&quot;&gt;bit shifting&lt;/a&gt;, which specifically refers to doing this in binary, where the digits are named &amp;quot;bits&amp;quot;. In all other bases, there is no particular nickname for the digits (ternary's &amp;quot;trits&amp;quot; are more often than not used jokingly rather than genuinely), so &amp;quot;digit shift&amp;quot; seems the better choice here.&lt;/p&gt;&#xA;&lt;p&gt;The suggestion of &amp;quot;decimal shift&amp;quot; is based on &amp;quot;bit shift&amp;quot; also being known as &amp;quot;binary shift&amp;quot; (see linked wiki page where it says: &lt;em&gt;&amp;quot;Binary shift&amp;quot; redirects here&lt;/em&gt;). Binary is base 2, decimal is base 10, so &amp;quot;decimal shift&amp;quot; would be an analogous name for the same operation in base 10.&lt;/p&gt;&#xA;" OwnerUserId="13169" LastEditorUserId="13169" LastEditDate="2023-09-26T06:31:48.787" LastActivityDate="2023-09-26T06:31:48.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26799" PostTypeId="2" ParentId="20649" CreationDate="2023-09-27T08:41:46.483" Score="0" Body="&lt;p&gt;As a Chinese student, I have had gone through a hell-like period on math in high school. In fact,  every day with math sucks(sorry to say it).&lt;/p&gt;&#xA;&lt;p&gt;Boredom and abstruseness are not the only two problem it puts. The most important thing is, math is not based on sociality. Sociologists argue that schools are prepared for better socialization. However, you can see no hint that indicates math will help because it is never a bridge&lt;br /&gt;&#xA;of emotions or feelings, never enables you to enter a social situation. And societal animals hate this.&lt;/p&gt;&#xA;&lt;p&gt;The secondary problem is, that math merely favours those who conquest it (they get flowers and applause),  while those who don't,  however, are long thought of as dumb ones. Continuous failure (basically impossible to avoid ) and negative feedbacks will definitely destroy learners' confidence even if genius is on the opposite. Thus, math becomes a symbol of stigma in many people's memories.&lt;/p&gt;&#xA;&lt;p&gt;Since math is connected with a feeling of being humiliated, it's hard to understand why people like math.  And physics is abstruse, too. I mean, perhaps many people hate physics equally just as math.&lt;/p&gt;&#xA;" OwnerUserId="22930" LastActivityDate="2023-09-27T08:41:46.483" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26800" PostTypeId="2" ParentId="20649" CreationDate="2023-09-27T15:04:14.570" Score="5" Body="&lt;p&gt;As enumerated in other answers, there are plenty of mechanisms by which one might develop a dislike of math as a result of one's own experiences.&lt;/p&gt;&#xA;&lt;p&gt;But why is it so easy for an initial dislike of math to quickly spiral into a full-blown hatred that forces its way into conversation whenever the topic of math comes up, even if there are no substantial changes in one's direct experiences with math?&lt;/p&gt;&#xA;&lt;p&gt;Because &lt;em&gt;hatred and vilification of math has been normalized&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In general, if someone dislikes thing X, and there is a large group of people who hate thing X and are vocal about it, then there is a social-emotional gravity that pulls this person towards the group while elevating their dislike into a similar level of hatred and vocalization.&lt;/p&gt;&#xA;&lt;p&gt;Positive feedback loops like this happen all the time in politics. Math, unfortunately, is no different.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-09-27T15:04:14.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26801" PostTypeId="1" CreationDate="2023-09-27T15:52:48.537" Score="-1" ViewCount="162" Body="&lt;p&gt;I'm competitive exam student learning Quantative aptitude what should i choose over solving more questions and skipping the one i can't solve or spending hours on one question till i solve it and then moving to next one.&lt;/p&gt;&#xA;" OwnerUserId="22935" LastEditorUserId="2083" LastEditDate="2023-09-28T06:59:08.000" LastActivityDate="2023-09-29T16:54:52.083" Title="I'm in dilemma while solving arithmetic problems" Tags="|algebra|exams|arithmetic|" AnswerCount="2" CommentCount="4" ClosedDate="2023-09-30T15:44:35.357" ContentLicense="CC BY-SA 4.0" />
  <row Id="26802" PostTypeId="1" CreationDate="2023-09-27T16:02:07.180" Score="2" ViewCount="984" Body="&lt;p&gt;What is the importance of using approaches related to Geometric Algebra in teaching,is it only useful when introducing Algebra to the students or can it be helpful to improve creative skills in mathematics? &lt;br/&gt; As an example if we consider the completing square method used by Al-Khuwarizmi the father of Algebra, he used it only for special cases such as when coefficients are positive. Is it sensible to use the same approach in the cases of negative coefficients by assigning negative values for the sides of squares or do we have to use this approach only to introduce the concept of completing square?&lt;br/&gt;&#xA;Is it not much sensible to rearrange the quadratic equation &lt;br/&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$x^2 - 6x + 8 = 0 $&lt;/span&gt;  as &lt;br/&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$(-x)^2 + 6(-x) + 9 = 1 $&lt;/span&gt;&lt;br/&gt;&#xA;and draw squares of sides &lt;span class=&quot;math-container&quot;&gt;$-x , 3$&lt;/span&gt; ?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-09-28T09:57:05.657" LastActivityDate="2023-09-28T11:32:52.617" Title="Geometrical verifications for Algebraic formulae" Tags="|secondary-education|algebra|geometry|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26803" PostTypeId="2" ParentId="20649" CreationDate="2023-09-27T17:38:23.823" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Many students do mathematics as a subject not because they like it but because it is compulsory to do so. Because of less interest, they feel it is a difficult subject to follow.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The approach may lead to creating negative feelings about understanding mathematics. Creative approaches such as getting help from geometry in the case of proof without words can be used to create interest in the subject.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When the students are put into a fixed frame by asking to apply specific principles or formulae, they can feel uncomfortable. It is much more important to let them have their approach which can be nontraditional or unconventional and thet they need to improve lateral thinking.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If students are not aware of real-life applications of mathematical concepts, it can lead to distract them from mathematics. Sometimes negative feelings about the subject are created by the terms that are supposed to be used in teaching. We can take imaginary numbers as an example, from the very beginning they give an idea of uselessness in real life but as we know, imaginary numbers are very important in studying electricity in physics and identifying the beautifulness of any object.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="15884" LastEditDate="2023-11-24T22:41:52.153" LastActivityDate="2023-11-24T22:41:52.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26804" PostTypeId="2" ParentId="26801" CreationDate="2023-09-27T18:32:25.553" Score="1" Body="&lt;p&gt;If you're solving questions that are&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;so easy and familiar that solving them requires no thought, or&lt;/li&gt;&#xA;&lt;li&gt;so difficult that you can't solve them or solving them requires hours of thought,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;then that suggests the educational resources you are using for practice might not be scaffolded very well, and you may benefit from following a more highly scaffolded curriculum.&lt;/p&gt;&#xA;&lt;p&gt;In order to give more specific recommendations regarding practice material, we'd need to know exactly what exam you're preparing for and what resources you are currently using to prepare for it.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-09-27T18:32:25.553" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26805" PostTypeId="1" CreationDate="2023-09-27T19:15:38.837" Score="1" ViewCount="163" Body="&lt;p&gt;As I prepare to instruct an online Mathematics course next year, I'm currently writing the syllabus. Right now, I'm writing about student participation in the Learning Management System, which takes about 20% of the total grade, involving the exchange of ideas through discussion posts to earn credit. However, I'm considering what can be considered common knowledge for our course, and I am prompted to reflect on the role of class materials, including lecture slides, textbooks, past homework assignments, and exams. How do I define common knowledge in my syllabus to warn students of unethical scholarly conduct in student postings on platforms?&lt;/p&gt;&#xA;" OwnerUserId="22936" LastActivityDate="2023-10-29T08:49:42.617" Title="What can be considered as common knowledge in an online Mathematics course?" Tags="|interactive-teaching|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26806" PostTypeId="2" ParentId="26802" CreationDate="2023-09-27T20:47:14.883" Score="2" Body="&lt;p&gt;There are always a number of students who understand the geometry and like it and profit from it, others get more confused and like the algebraic approach. so why not use both. But sides should not be negativ, instead you deduct the rectangles. Also visualize earlier  the three binomial formulas with squares and rectangles.&lt;/p&gt;&#xA;" OwnerUserId="21051" LastActivityDate="2023-09-27T20:47:14.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26807" PostTypeId="2" ParentId="26805" CreationDate="2023-09-27T21:26:05.800" Score="0" Body="&lt;p&gt;I would consider common knowledge to be any information that does not give away the answer to a question on a pending assignment or test.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;A piece of information &amp;quot;gives away&amp;quot; an answer if it is the answer itself, a formula (or any sort of template) that can be filled in to obtain the answer, a generality for which the answer is a special case, or a special case from which the answer can be inferred as an obvious generalization. (This is just off the top of my head -- there may be more conditions I'm not thinking about.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;An assignment or test is &amp;quot;pending&amp;quot; if it is accepting submissions or will be accepting submissions at any point in the future.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-09-27T21:26:05.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26808" PostTypeId="2" ParentId="26802" CreationDate="2023-09-27T21:55:26.100" Score="11" Body="&lt;p&gt;In my experience,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;geometry can lead you into interesting questions that can be answered geometrically in special cases, and&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;algebra can help you tighten up the rigor of your answers and generalize your results.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For instance, here's an example of how I would combine geometric and algebraic approaches to teach an algebraic concept. I'll illustrate on a simple concept, expanding binomials.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;1. Motivate the end result using geometric intuition.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The quantity &lt;span class=&quot;math-container&quot;&gt;$(a+b)^2$&lt;/span&gt; represents the area of a square with side length &lt;span class=&quot;math-container&quot;&gt;$a+b.$&lt;/span&gt; Draw a picture of said square...&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NF2V2.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NF2V2.png&quot; alt=&quot;non-partitioned square&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;...and illustrate how it can be partitioned into&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; squares of dimensions &lt;span class=&quot;math-container&quot;&gt;$a \times a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b \times b,$&lt;/span&gt; and&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; rectangles of dimensions &lt;span class=&quot;math-container&quot;&gt;$a \times b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b \times a.$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/giL6G.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/giL6G.png&quot; alt=&quot;partitioned square&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This would suggest that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;(a+b)^2 &amp;amp;= a^2 + b^2 + ab + ba \\&#xA;&amp;amp;= a^2 + 2ab + b^2.&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;2. Tighten up the rigor using algebra.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Sure, this picture makes sense when &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are positive numbers. But what about if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is zero or negative?&lt;/p&gt;&#xA;&lt;p&gt;Granted, if we come up with a situation like that and plug it into our formula, the formula seems to come out to a true statement. But how can we be sure it will always work? And how can we make sense of the fact that it does in fact still work?&lt;/p&gt;&#xA;&lt;p&gt;To answer these questions, we observe that the same result can be obtained by writing the exponentiation as a multiplication and then applying the distributive property:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;(a+b)^2 &amp;amp;= (a+b)(a+b) \\&#xA;&amp;amp;= a(a+b) + b(a+b) \\&#xA;&amp;amp;= a^2 + ab + ba + b^2 \\&#xA;&amp;amp;= a^2 + 2ab + b^2 \quad \checkmark&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;3. Demonstrate the utility of algebra.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;What if we wanted to compute &lt;span class=&quot;math-container&quot;&gt;$(a+b)^3,$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(a+b)^4,$&lt;/span&gt; or a higher power? It's tricky to draw a picture of a cube, and impossible to draw a picture of a cube with &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; or more dimensions. What do we do?&lt;/p&gt;&#xA;&lt;p&gt;We just use the algebraic approach:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;(a+b)^3 &amp;amp;= (a+b)(a+b)^2 \\&#xA;&amp;amp;= (a+b)(a^2+2ab+b^2) \\&#xA;&amp;amp;= a(a^2+2ab+b^2) + b(a^2+2ab+b^2) \\&#xA;&amp;amp;= a^3 + 2a^2b + b^2 + ba^2 + 2ab^2 + b^3 \\&#xA;&amp;amp;= a^3 + 3a^2b + 3ab^2 + b^3&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-09-27T22:00:38.670" LastActivityDate="2023-09-27T22:00:38.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26809" PostTypeId="2" ParentId="26802" CreationDate="2023-09-27T23:38:35.570" Score="0" Body="&lt;p&gt;I think it will hurt more students than it helps, if you spend much time on this sort of thing.  (Unless you have very strong students, who also are not being highly accelerated...thus having time/capacity for enrichment.)&lt;/p&gt;&#xA;&lt;p&gt;In equations, the key idea (from pre-algebra on) is the concept of the equals sign and how you manipulate the terms on each side, to analyze problems.  Do not underestimate how many people are weak at this and suffer from lack of equation manipulation &amp;quot;muscles&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Note:  this is NOT to deny that there aren't some cool insights from geometry.  But you need to think about the constraints of time and audience.  And the need to prepare/move students for/along the whole &amp;quot;algebra to calculus&amp;quot; pathway.&lt;/p&gt;&#xA;" OwnerUserId="22938" LastActivityDate="2023-09-27T23:38:35.570" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26810" PostTypeId="2" ParentId="26802" CreationDate="2023-09-28T00:21:46.090" Score="5" Body="&lt;p&gt;I agree with the general idea of Justin Skycak, and I think you'd be doing the students a disservice if you don't show them some kind of geometric pictures to go with this stuff.&lt;/p&gt;&#xA;&lt;p&gt;In your specific example, I strongly disagree with anyone who says you shouldn't do the &amp;quot;completing the square picture&amp;quot; with negative sides. Why? Well, first of all, if you show them how to do it pictorially with positive numbers, the first time they encounter an equation with negative numbers they're going to draw a picture. So the question is almost unavoidable.&lt;/p&gt;&#xA;&lt;p&gt;At this point you have two options. 1) Tell them negative side lengths make no sense and not to draw the picture, removing a tool they have to solve the problem. And 2) say &amp;quot;well, I've never seen a negative side length and they don't exist in 'the real world', but let's see what happens if we finish the problem.&amp;quot; Sure enough, you'll get the right answer. Option 2 should be accompanied by algebra, of course, as well as trula's suggestion of deducting rectangles.&lt;/p&gt;&#xA;&lt;p&gt;The problem with option 1 is that when they learn complex numbers, the teacher &lt;em&gt;will&lt;/em&gt; pretty much go with option 2 for square roots of negatives. If the student prefers option 1, which they've been taught before, well, tough luck kid, that was last year and math is capricious and arbitrary!&lt;/p&gt;&#xA;&lt;p&gt;As educators, we have to be consistent. If we tell them that the beauty of math is that we can create new, fantastic objects as long as they're consistent and help us solve problems, we have to put our money where our mouth is and entertain the idea of negative side lengths.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2023-09-28T00:21:46.090" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26811" PostTypeId="1" AcceptedAnswerId="26816" CreationDate="2023-09-28T08:08:08.580" Score="9" ViewCount="2125" Body="&lt;p&gt;Ideally, I am looking for an online resource. But a book or any other would help already.&lt;/p&gt;&#xA;&lt;p&gt;Background: I am a senior teaching assistant in the field of business and statistics. Most of my students have not had formal training in mathematics, and neither have I. I notice many of our students have difficulty following the professor's lectures for the trivial reason that they do not know mathematical notation (the professor does have a math background). In my experience as a schoolkid and student, no-one has ever explained how to read an equation. The only reason I know, is that I have cobbled it together from what my professors have said while pointing to their equations, which they always acted we knew how to read already.&lt;/p&gt;&#xA;&lt;p&gt;I have made some effort to look for such a resource with my search engine, but have not found anything approaching what I mean.&lt;/p&gt;&#xA;&lt;p&gt;Caveat: I appreciate that there are many subfields of mathematics, and that they use vastly different notations. A general resource would interest me most (and be most useful to future readers of this question), but in my case I am specifically in need of the notations needed in statistical calculations, so: sum, estimate, average, indexing, etc.&lt;/p&gt;&#xA;" OwnerUserId="22941" LastActivityDate="2024-01-21T17:19:02.650" Title="Is there a resource for learning to read mathematical notation/equations/formulae?" Tags="|mathematical-pedagogy|notation|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26812" PostTypeId="2" ParentId="26802" CreationDate="2023-09-28T11:32:52.617" Score="4" Body="&lt;p&gt;I &lt;em&gt;personally&lt;/em&gt; had a lot of fun in high school trying to extend the geometric picture for completing the square to include negative numbers.  I ended up inventing a system of signed areas which was equivalent to the sign conventions followed with determinants!  My teacher was, thankfully, knowledgable enough to be able to make this connection for me.  I don't think it would be good &lt;em&gt;required&lt;/em&gt; material, but it could be enriching stuff for a math club to play with.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-09-28T11:32:52.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26813" PostTypeId="2" ParentId="26811" CreationDate="2023-09-28T14:03:56.977" Score="-1" Body="&lt;p&gt;I'm guessing that the sigma (summation) notation is the problem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OwZeH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OwZeH.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;See: &lt;a href=&quot;https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/review-summation-notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/review-summation-notation&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Maybe integral notation (another kind of summation) from calculus?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LMkiw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LMkiw.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;See: &lt;a href=&quot;https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-1/v/introduction-to-integral-calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-1/v/introduction-to-integral-calculus&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6103" LastActivityDate="2023-09-28T14:03:56.977" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26814" PostTypeId="2" ParentId="26811" CreationDate="2023-09-28T14:40:02.237" Score="3" Body="&lt;p&gt;If you are a teaching assistant, you should ask your professor to give a short instruction of the notation he uses, this would be much better for you and the students. He could do it in a lecture or just give out a short paper with the commonly used notation an examples of use. like &lt;span class=&quot;math-container&quot;&gt;$\sum_{k=1}^5 k^2= 1^2+2^2+3^2+4^2+5^2$&lt;/span&gt;&#xA;This would be much better than any source in the net or book.&lt;/p&gt;&#xA;" OwnerUserId="21051" LastActivityDate="2023-09-28T14:40:02.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26815" PostTypeId="2" ParentId="26811" CreationDate="2023-09-28T15:31:23.473" Score="6" Body="&lt;p&gt;Here's a general workflow for identifying the meaning of an unknown mathematical symbol/expression:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;use an online tool (like &lt;a href=&quot;https://webdemo.myscript.com/views/math/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;webdemo.myscript.com/views/math&lt;/a&gt;) to convert a drawing of the symbol/expression into Latex commands&lt;/li&gt;&#xA;&lt;li&gt;ask ChatGPT (&lt;a href=&quot;https://chat.openai.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;chat.openai.com&lt;/a&gt;, free version 3.5 is fine) what the math symbol/expression given by those Latex commands means&lt;/li&gt;&#xA;&lt;li&gt;[added in response to Sue's comment] based on ChatGPT's explanation, perform Google searches for more specific information/examples/fact-checking as needed&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Here's an demonstration on the expression &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \sum_{i=1}^n \textrm E[X_i],$&lt;/span&gt; which seems highly relevant to your specific use-case.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Step 1: Drawing &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; Latex Commands&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/yWD0n.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yWD0n.png&quot; alt=&quot;lookup-latex-commands&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Step 2: Latex Commands &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; Summary&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cWABC.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cWABC.png&quot; alt=&quot;chatgpt-prompt&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/sZ1b8.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sZ1b8.png&quot; alt=&quot;chatgpt-response&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Step 3: Summary &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; More Specific Information / Examples / Fact-Checking&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BTAO1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BTAO1.png&quot; alt=&quot;google-summation&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bj3qJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bj3qJ.png&quot; alt=&quot;google-expected-value&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-09-29T06:05:02.640" LastActivityDate="2023-09-29T06:05:02.640" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26816" PostTypeId="2" ParentId="26811" CreationDate="2023-09-28T16:16:51.487" Score="11" Body="&lt;p&gt;The video, &lt;a href=&quot;https://www.youtube.com/watch?v=ooqQZ-_f_bg&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=ooqQZ-_f_bg&lt;/a&gt; (U. Adelaide, &amp;quot;Understanding Maths Notation&amp;quot;), goes from elementary math through calculus notations.&lt;/p&gt;&#xA;&lt;p&gt;I don't know a source that discusses the different meanings of &amp;quot;=&amp;quot; (identity, condition, equation to be solved, assignment/definition). Students with a weak background in math often do not distinguish these meanings.&lt;/p&gt;&#xA;&lt;p&gt;Overall, learning new math notation is like learning a new language. It takes a lot of practice to be able to readily understand a lecture. Being told that, say, a capital sigma means summing numbers up does not prepare the student to follow the lecturer at the same speed that the lecturer will be reasoning about sums in their presentation. I will say that fluency allows one, at least half the time, to predict what the next step in the lecture will be. That ability certainly makes the lecture more readily understood. Generally, math teachers seem to think that fluency is best attained by solving relevant math problems. If the student does not want to put in the work to be fluent, at least being able to decipher their notes will make their notes much more useful. I hope the video addresses that. I assume the students probably do not need to be fluent; otherwise, there should be mathematics prerequisite for the course. But if some want to be good at something, they ought to be aware of what choices they have to strive for that goal.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2023-09-28T16:16:51.487" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26817" PostTypeId="2" ParentId="26811" CreationDate="2023-09-29T04:17:53.947" Score="2" Body="&lt;p&gt;Questions about &lt;a href=&quot;https://ell.stackexchange.com/questions/tagged/mathematics+reading-aloud&quot;&gt;[reading-aloud] [mathematics]&lt;/a&gt; expressions are on-topic on the English Language Learners stack exchange.  For example, &lt;a href=&quot;https://ell.stackexchange.com/questions/42963&quot;&gt;this question&lt;/a&gt; asks how to read an equation including a summation, and another equation including integrals and limits.  &lt;a href=&quot;https://ell.stackexchange.com/questions/42963#43084&quot;&gt;One of the answers&lt;/a&gt; has links to 3 guides.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/14559&quot;&gt;Another question&lt;/a&gt; on this MathEducators stack exchange discusses why math uses shorthand so much.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastEditorUserId="1147" LastEditDate="2023-09-29T04:37:52.090" LastActivityDate="2023-09-29T04:37:52.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26819" PostTypeId="2" ParentId="26801" CreationDate="2023-09-29T16:54:52.083" Score="3" Body="&lt;p&gt;There is a “middle way” between sinking hours into a problem and skipping it completely. Spend a certain amount of time on the problem, document what you did and what your blockers were, and then seek feedback — either from an answer key, a teacher, a peer, or an online forum (like Math SE). If you document what you tried and where you’re struggling, you can more easily use the key to see where you went wrong and get a better reception to your question from a teacher or Math SE.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastActivityDate="2023-09-29T16:54:52.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26820" PostTypeId="1" AcceptedAnswerId="26822" CreationDate="2023-10-01T21:44:37.023" Score="1" ViewCount="266" Body="&lt;p&gt;Yearly, I teach 16 year olds this diagram beneath (improvement of &lt;a href=&quot;https://math.stackexchange.com/a/774233&quot;&gt;this&lt;/a&gt;) that reappears on standardized tests IN BLACK AND WHITE below &lt;em&gt;&lt;strong&gt;with different lengths, letters,  and orientation&lt;/strong&gt;&lt;/em&gt;.  Tests require students to label in terms of &lt;span class=&quot;math-container&quot;&gt;$\vec{b}, \vec{r}$&lt;/span&gt; ( = circle's radius ) the:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;long line from &lt;span class=&quot;math-container&quot;&gt;$(-4, 0)$&lt;/span&gt; to B.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;short line segment from a point on the circumference to B.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.imgur.com/iQ1sy7s.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Despite my best endeavors like adding colors, someone ALWAYS mixes up red and blue!&lt;/p&gt;&#xA;&lt;h4&gt;How can I better explain their pictoric difference? How would you prevent this diagrammatic mix up BY EYE?&lt;/h4&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.imgur.com/riRlQMN.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user22729" LastEditorDisplayName="user22729" LastEditDate="2023-10-03T03:41:49.003" LastActivityDate="2023-10-03T03:41:49.003" Title="Plainly by eye, how can 16 year olds visually distinguish $\color{red}{\vec{b} - \vec{r}}$ from $\color{dodgerblue}{|\vec{b}| - |\vec{r}|}$?" Tags="|visual-proof|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26821" PostTypeId="2" ParentId="26820" CreationDate="2023-10-01T22:22:51.483" Score="1" Body="&lt;p&gt;My first reaction is to doubt the problem is that they somehow can't see the difference (how could you miss the vertical lines?), but rather that labeling only one vector with its length is throwing them off. I know it doesn't have an arrow attached but still, it's a little confusing. For that matter, the arrow placements themselves, at least for &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, might cause confusion. What if you just wrote &amp;quot;the length of this segment is &lt;span class=&quot;math-container&quot;&gt;$|b|-|r|$&lt;/span&gt;&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;I guess another explanation could be that their understanding of the math is shaky, in which case they'll probably make all sorts of errors no matter how good the diagrams are.&lt;/p&gt;&#xA;" OwnerUserId="12671" LastActivityDate="2023-10-01T22:22:51.483" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26822" PostTypeId="2" ParentId="26820" CreationDate="2023-10-01T22:31:13.647" Score="2" Body="&lt;p&gt;I would tell students that &lt;strong&gt;absolute value bars represent distance, and distance is indicated by a line segment &lt;em&gt;without&lt;/em&gt; an arrow.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This rule of thumb is very intuitive:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;students should already know that absolute value bars remove a negative sign from a number,&lt;/li&gt;&#xA;&lt;li&gt;so they should associate bars with the act of &amp;quot;removing things,&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;so it shouldn't feel too weird that bars around a vector remove the arrow,&lt;/li&gt;&#xA;&lt;li&gt;and you can make this more rigorous by explaining that in both cases, the bars are just removing the &lt;em&gt;direction&lt;/em&gt; from the object that they are enclosing.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Additionally, this rule of thumb provides sufficient information for students to distinguish between &lt;span class=&quot;math-container&quot;&gt;$\mathbf{b}-\mathbf{r}$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$|\mathbf{b}|-|\mathbf{r}|$&lt;/span&gt; in the diagram: there is only one expression with absolute value bars, and only one line segment without an arrow, so they must pair up.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-10-01T22:31:13.647" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26823" PostTypeId="2" ParentId="26820" CreationDate="2023-10-01T23:54:39.210" Score="3" Body="&lt;p&gt;No one diagram is going to help every student actually absorb this idea. In fact, the mistake reminds me a bit of students thinking  &lt;span class=&quot;math-container&quot;&gt;$\sqrt{a^{2}+b^{2}}=a+b$&lt;/span&gt;. They are &amp;quot;simplifying&amp;quot; in a way that makes no real sense, partly because they are not trying to make sense when they're doing math, sadly. Many students think of math as a bunch of rules, and that doesn't really work.&lt;/p&gt;&#xA;&lt;p&gt;I'd guess they need to work more with each part of this separately: meanings of vectors, subtraction of vectors, and lengths of vectors.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-10-01T23:54:39.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26824" PostTypeId="1" CreationDate="2023-10-02T12:31:15.333" Score="5" ViewCount="358" Body="&lt;p&gt;Many textbooks include &lt;span class=&quot;math-container&quot;&gt;$\int \frac{1}{x} dx = \ln |x| + c$&lt;/span&gt; in their list of antiderivative formulas, &lt;em&gt;with&lt;/em&gt; the absolute value. Correspondingly, they do the same with the antiderivative of &lt;span class=&quot;math-container&quot;&gt;$\tan x$&lt;/span&gt; or similar.&lt;/p&gt;&#xA;&lt;p&gt;Is it productive to introduce the complication of the absolute value in a practically oriented course, when under time pressure? The alternative is to try to restrict ourselves to positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My concerns are that one still needs to be careful not to include the singularity at 0 when writing definite integrals and the absolute value hides this. Some texts will omit the absolute value which can be confusing to students. So do most computer algebra systems, which operate with complex numbers by default.&lt;/p&gt;&#xA;" OwnerUserId="22959" LastActivityDate="2023-10-04T12:02:12.043" Title="Antiderivative of $1/x$, with or without absolute value?" Tags="|integration|logarithm|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26825" PostTypeId="2" ParentId="26824" CreationDate="2023-10-02T12:38:59.787" Score="11" Body="&lt;p&gt;Even &lt;span class=&quot;math-container&quot;&gt;$\int \frac{1}{x} \textrm{ d}x = \ln(|x|) + C$&lt;/span&gt; is incorrect.&lt;/p&gt;&#xA;&lt;p&gt;It should be&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int \frac{1}{x} \textrm{ d}x = \begin{cases}&#xA;\ln(x) + C_1 \textrm{ if $x &amp;gt; 0$}\\&#xA;\ln(-x) + C_2 \textrm{ if $x &amp;lt; 0$}\\&#xA;\end{cases}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Personally, I think that it would be best to include some problems where this issue arises naturally, and resolve it through thinking about it rather than memorizing another arcane aspect of a formula without understanding it.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-10-02T12:38:59.787" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26826" PostTypeId="2" ParentId="26824" CreationDate="2023-10-02T15:38:37.627" Score="4" Body="&lt;p&gt;Absolute value or no absolute value, there will be students that attempt to integrate over infinite discontinuities anyway. Leaving off the absolute value in&#xA;&lt;span class=&quot;math-container&quot;&gt;$\int \frac{1}{x}\,\mathrm{d}x = \ln|x| + C$&lt;/span&gt; may stop them from blindly attempting to evaluate &lt;span class=&quot;math-container&quot;&gt;$\int_{-1}^2 \frac{1}{x}\,\mathrm{d}x$&lt;/span&gt;, but what about &lt;span class=&quot;math-container&quot;&gt;$\int_{-1}^2 \frac{1}{x^2}\,\mathrm{d}x$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;I agree with Steven's approach of having students encounter a problem and, through resolving it, arrive at a deeper understanding. So I would suggest giving the formula &lt;span class=&quot;math-container&quot;&gt;$\int \frac{1}{x}\,\mathrm{d}x = \ln|x| + C$&lt;/span&gt; and asking students&#xA;to attempt to evaluate something like &lt;span class=&quot;math-container&quot;&gt;$\int_{-1}^2 \frac{1}{x}\,\mathrm{d}x$&lt;/span&gt;. Let them get it wrong, and have them realize that their answer is wrong by asking them to perform some sort of check.&lt;/p&gt;&#xA;&lt;p&gt;It may be beneficial to start with &lt;span class=&quot;math-container&quot;&gt;$\int_{-1}^2 \frac{1}{x^2}\,\mathrm{d}x$&lt;/span&gt;, as this integral is unambiguously divergent. Whereas for &lt;span class=&quot;math-container&quot;&gt;$\int_{-1}^2 \frac{1}{x}\,\mathrm{d}x$&lt;/span&gt;, students might want to make some sort of symmetry argument, cf. Cauchy principal value.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-10-02T15:57:26.533" LastActivityDate="2023-10-02T15:57:26.533" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26827" PostTypeId="1" CreationDate="2023-10-02T15:49:20.343" Score="1" ViewCount="135" Body="&lt;p&gt;There are many questions on this site about specific (or not-so-specific) applications of calculus to the &amp;quot;real world&amp;quot;.  However, one issue I've noticed in using &lt;em&gt;textbooks&lt;/em&gt; for this purpose is that the applications are not always well-sourced, and hence I suspect are (in some sense) not really used.&lt;/p&gt;&#xA;&lt;p&gt;The question I have is what strategies educators use (if at all) to actually verify for students that these applications are used in &lt;em&gt;some&lt;/em&gt; actual practitioner's toolkit.  I'm referring to those not involving conceptual ideas (such as area under a force-distance curve), but actual computation with algebra (such as integrating said area from a specific cubic model to help compute how much better a kangaroo can jump than a human).&lt;/p&gt;&#xA;&lt;p&gt;Here are the type of answers I am looking for, as long as there is at least anecdotal evidence that it pays off.  I happen to think these techniques don't always work well, though you could still use them as answers if your experience differs.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Doing straight-up web searches for keywords&lt;/li&gt;&#xA;&lt;li&gt;Using references in &amp;quot;application calculus&amp;quot; textbooks (because there often aren't references)&lt;/li&gt;&#xA;&lt;li&gt;Asking colleagues in discipline X about calculus example in discipline X&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As an aside, the story motivating this question is my hunt for a verified source &lt;em&gt;outside a calculus text&lt;/em&gt; that medical researchers use &lt;span class=&quot;math-container&quot;&gt;$R(q)=q^2\left(\frac{k}{2}-\frac{q}{3}\right)$&lt;/span&gt; as a general formula to measure the response &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; to a dose of quantity &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; of some medication.  If your method actually finds a source for that, it will get an upvote for sure!&lt;/p&gt;&#xA;&lt;p&gt;As a second aside, I'm mostly referring to applications &lt;em&gt;outside&lt;/em&gt; of the traditional physics-chem package, and especially to mathematical models of real-life data which calculus could be nicely applied to if they were really used, and which are purported to be very similar to actual uses, but which aren't.&lt;/p&gt;&#xA;" OwnerUserId="1608" LastEditorUserId="1608" LastEditDate="2023-10-03T15:02:55.350" LastActivityDate="2023-10-03T15:02:55.350" Title="Sourcing and verifying calculus applications" Tags="|calculus|resource-request|applications|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26828" PostTypeId="2" ParentId="26827" CreationDate="2023-10-02T17:48:52.960" Score="1" Body="&lt;p&gt;The primary use of calculus is not day to day &amp;quot;application&amp;quot; by working/paid professionals, but it IS a needed tool for undergrad engineers, scientists (and sometimes social scientists) within their majors coursework.  It's needed to follow derivations and do homework problems in physics, EE, etc.  That a practicing EE will rarely calculate impedance using integrals and derivatives is irrelevant to their need to have those tools when learning their basic courses.&lt;/p&gt;&#xA;&lt;p&gt;And it's not even really required that they have practiced the specific applications (circuits or fluids or whatever), as they will also be taught in majors courses.&#xA;Nice sure (good to see things twice, including from different perspectives).  But still not really vital.   They need the basic algebraic calculus toolkit and at least a bit of eventual practice in some applications (since &amp;quot;word problems are hard&amp;quot;, they represent a higher cognitive load training of the basic calculus material.)&lt;/p&gt;&#xA;&lt;p&gt;P.s.  It never ceases to amaze me the naivete of math educators, when asking questions like this.  If you want to know how often engineers use calculus, ask them...not your fellow math educators (most of whom lack a strong generalist STEM background...I know...I do have one.).  Or, go walk over to the library (or better yet colleagues' offices) and look at the texts for all the basic majors courses in physics, engineering, p-chem, etc.  A very simple starting point would be Lindeburg's EIT reference manual (one of the most amazing compilations of most undergrad STEM coursework).  &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0912045566&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Engineer-Training-Reference-Michael-Lindeburg/dp/0912045566&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22961" LastActivityDate="2023-10-02T17:48:52.960" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26829" PostTypeId="1" AcceptedAnswerId="26830" CreationDate="2023-10-02T18:14:33.813" Score="11" ViewCount="1259" Body="&lt;p&gt;It's kind of a serious question even if the title seems silly.&#xA;As math educators, we all know that students link together different algebraic expressions thinking that they mean the same thing, e.g.&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;(a+b)^2 = a^2+b^2&#xA;\end{equation}&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;\frac{x+2}{2} = x&#xA;\end{equation}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;These errors can be grouped statistically in categories, such as &amp;quot;false linearity&amp;quot; as in the  &lt;span class=&quot;math-container&quot;&gt;$(a+b)^2$&lt;/span&gt; case. This student-induced grouping leads to a kind of student-induced topology on the space of all algebraic formulas, where the &amp;quot;student-distance&amp;quot; between two expressions is small if statistically a lot of students think that they are the same expression.&lt;/p&gt;&#xA;&lt;p&gt;My question is this: has this &amp;quot;topology&amp;quot;, i.e. the organization of student errors in algebra, been studied sistematically? Is there any reference in literature that thoroughly deals with the issue of really categorizing and understanding errors in basic algebraic expression manipulation?&lt;/p&gt;&#xA;&lt;p&gt;I think it would be extremely beneficial to know how to traverse the formula space in order to optimize learning (Of course this taxonomic approach has a sense only for algebraic manipulation, not for complex problems).&lt;/p&gt;&#xA;" OwnerUserId="6286" LastEditorUserId="6286" LastEditDate="2023-10-03T11:18:28.727" LastActivityDate="2023-10-06T16:53:04.540" Title="Does there exist a (statistical) topology induced by students on the space of algebraic formulas? :)" Tags="|algebra|students-mistakes|test-design|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26830" PostTypeId="2" ParentId="26829" CreationDate="2023-10-02T19:14:28.667" Score="17" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Has this &amp;quot;topology&amp;quot;, i.e. the organization of student errors in algebra, been studied systematically? Is there any reference in literature that thoroughly deals with the issue of really categorizing and understanding errors in basic algebraic expression manipulation?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I did a toy research project a while back (prototyped a software system to automatically detect/classify/explain algebra mistakes in student work) and investigated a lot of this stuff while doing a literature search beforehand.&lt;/p&gt;&#xA;&lt;p&gt;If you Google something like &amp;quot;most common algebra mistakes&amp;quot;, then you'll find plenty of online resources (e.g. &lt;a href=&quot;http://tutorial.math.lamar.edu/Extras/CommonErrors/AlgebraErrors.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; and &lt;a href=&quot;https://math.vanderbilt.edu/schectex/commerrs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;) detailing common algebra mistakes that educators experience in practice when working with students.&lt;/p&gt;&#xA;&lt;p&gt;I also found a handful of papers that more generally and systematically describe errors as invalid edges in &amp;quot;procedural networks&amp;quot; (nodes represent granular steps in a problem-solving procedure, and edges reflect how the steps may be arranged in sequence). Some references:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Brown, J. S., &amp;amp; Burton, R. R. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive science, 2(2), 155-192.&lt;/p&gt;&#xA;&lt;p&gt;Brown, J. S., &amp;amp; VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive science, 4(4), 379-426.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, these papers only considered errors in arithmetic and did not extend into algebra. It turns out that the procedural network for algebra is so complicated that it doesn't actually simplify the problem of diagnosing student errors.&lt;/p&gt;&#xA;&lt;p&gt;To automatically diagnose student errors in algebra, it's more common to use ad-hoc error diagnosis, where a set of common errors are chosen using domain knowledge, and an error detector is hand-crafted for each individual error. Some references:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Erabadda, B., Ranathunga, S., &amp;amp; Dias, G. (2017, July). Automatic Identification of Errors in Multi-Step Answers to Algebra Questions. In Advanced Learning Technologies (ICALT), 2017 IEEE 17th International Conference on (pp. 215-219). IEEE.&lt;/p&gt;&#xA;&lt;p&gt;Gonzalez, C. S., Guerra, D., Sanabria, H., Moreno, L., Noda, M. A., &amp;amp; Bruno, A. (2010). Automatic system for the detection and analysis of errors to support the personalized feedback. Expert Systems with Applications, 37(1), 140-148.&lt;/p&gt;&#xA;&lt;p&gt;Hoppe, H. U. (1994). Deductive error diagnosis and inductive error generalization for intelligent tutoring systems. Journal of Interactive Learning Research, 5(1), 27.&lt;/p&gt;&#xA;&lt;p&gt;Huang, Z., &amp;amp; Tokuda, N. (1996). A syntactical approach to diagnosing multiple bugs in an intelligent tutoring system. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 26(2), 280-285.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For more details, see Appendix 1 of my project writeup (&lt;a href=&quot;https://www.justinmath.com/checkmysteps/&quot; rel=&quot;nofollow noreferrer&quot;&gt;html&lt;/a&gt;, &lt;a href=&quot;https://www.justinmath.com/files/skycak-gatech-checkmysteps.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pdf&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-06T05:57:53.350" LastActivityDate="2023-10-06T05:57:53.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26831" PostTypeId="2" ParentId="26827" CreationDate="2023-10-03T13:03:39.900" Score="4" Body="&lt;p&gt;I vaguely recall seeing the reference to modeling response in a calculus textbook, probably an edition of Stewart or Thomas. I took part of your question and did a &amp;quot;ChatGPT search.&amp;quot; My prompt was&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;give me a source for this assertion:  medical researchers use &lt;span class=&quot;math-container&quot;&gt;$R(q)=q^2\left(\frac{k}{2}-\frac{q}{3}\right)$&lt;/span&gt; as a general formula to measure the response &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; to a dose of quantity &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; of some medication.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The return was a reference to a book &amp;quot;&lt;em&gt;Principles of Pharmacology: The Pathophysiologic Basis of Drug Therapy&lt;/em&gt;&amp;quot; by Golan et al. I was able to confirm this is a real book, and not a hallucination. The ChatGPT response also described the &lt;strong&gt;Hill equation&lt;/strong&gt; and said it was a type of sigmoid function. So I started using search engines with queries like &amp;quot;Hill equation&amp;quot; and &amp;quot;Hill equation response rate.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;There was an abundance of interesting things to read about the Hill equation, which I was not familiar with. You will find that the Hill equation is a real thing used in applications in biochemistry, pharmacology, etc. There is lots of literature. Nothing I found seemed to explicitly use a cubic as in the the expression &lt;span class=&quot;math-container&quot;&gt;$q^2\left(\frac{k}{2}-\frac{q}{3}\right)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But it seems likely that this cubic is a simple but practical representation (approximation) of a sigmoid curve that avoids transcendental functions, which would not come up until later in the course. The maximum provided by the cubic approximation is at &lt;span class=&quot;math-container&quot;&gt;$q=k$&lt;/span&gt; and the inflection point is halfway, at &lt;span class=&quot;math-container&quot;&gt;$q=k/2$&lt;/span&gt;, which seems like a decent model. In the context, the inflection point would correspond to the &amp;quot;dosage saturation point.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The key information that ChatGPT gave me that I would have had difficulty getting from &amp;quot;straight-up web searches&amp;quot; is &lt;em&gt;Hill equation&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-10-03T13:27:31.217" LastActivityDate="2023-10-03T13:27:31.217" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26832" PostTypeId="1" AcceptedAnswerId="26834" CreationDate="2023-10-03T17:45:12.293" Score="5" ViewCount="1526" Body="&lt;p&gt;When explaining fractions to my kids, I've used the analogy that &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; means &amp;quot;you want &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; out of every group of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; (of the thing you're finding a fraction of).&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;E.g. &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4}$&lt;/span&gt; of a pizza means &amp;quot;you want &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; out of every group of &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; (pizza slices) of a pizza (that's been sliced up to let you make groups of 4)&amp;quot;. So if you sliced up the pizza into &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; slices, you'd have &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; groups of &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, so if you took &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; from each group of &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, you'd have &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; slices, which is your &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4}$&lt;/span&gt; of the original pizza that was cut up into 8 slices.&lt;/p&gt;&#xA;&lt;p&gt;This analogy works pretty well for fractions that are &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$&amp;lt;=$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$1.0$&lt;/span&gt;,&amp;quot; like &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{5}{7}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\frac{11}{11}$&lt;/span&gt;, etc.&lt;br /&gt;&#xA;But it seems to break down (or at least I can't find a way to extend it) when considering fractions that are &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$1.0$&lt;/span&gt;, like &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}$&lt;/span&gt;.&lt;br /&gt;&#xA;Like: the extended analogy for &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}$&lt;/span&gt; would be &amp;quot;you want &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; out of every group of &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; pizza slices (of a pizza that's been sliced up to let you make groups of 4).&amp;quot;&lt;br /&gt;&#xA;I can't wrap my head around whether that makes &lt;strike&gt;grammatical sense, let alone&lt;/strike&gt; mathematical sense.&lt;/p&gt;&#xA;&lt;p&gt;I do realize that &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}$&lt;/span&gt; can be thought of as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;&amp;quot;, in which case we can go back to our nice analogy that seemingly only works for fractions that are &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;=$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$1.0$&lt;/span&gt;.&lt;br /&gt;&#xA;But here, I struggle for a way to explain (to 8-year-olds) &lt;em&gt;&lt;strong&gt;why&lt;/strong&gt;&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}$&lt;/span&gt; can be thought of as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;&amp;quot; in the context of the original analogy that &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; means &amp;quot;you want &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; out of every group of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So I think my question comes down to: &lt;strong&gt;is there a way to extend the analogy that &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; means &amp;quot;you want &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; out of every group of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;&amp;quot; to account for when &lt;span class=&quot;math-container&quot;&gt;$a &amp;gt; b$&lt;/span&gt; (i.e. that it effectively means &lt;span class=&quot;math-container&quot;&gt;${a}\div{b}$&lt;/span&gt;)?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="20879" LastEditorUserId="20879" LastEditDate="2023-10-03T18:47:39.427" LastActivityDate="2023-10-05T11:59:06.467" Title="Is there a way to extend the analogy that fractions means &quot;x out of y&quot; to show that fractions are also dividing?" Tags="|fractions|division|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26833" PostTypeId="2" ParentId="26824" CreationDate="2023-10-03T18:05:03.020" Score="4" Body="&lt;p&gt;One reason to discuss the anti-derivative of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt; also for negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is that some (or many, depending on your audience) students will needs it later on when they learn about ordinary differential equations.&lt;/p&gt;&#xA;&lt;p&gt;For instance, consider the very simple initial value problem&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \begin{cases}&#xA;    \dot x(t) = x(t), \\ &#xA;    x(0) = x_0.&#xA;  \end{cases}&#xA;$$&lt;/span&gt;&#xA;As we all know its solution is given by &lt;span class=&quot;math-container&quot;&gt;$x(t) = x_0 e^t$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$t \in \mathbb{R}$&lt;/span&gt;. But assume that you don't know the solution, yet (for instance, if you're a student taking an ODE course) and try to find it by using separation of variables. This gives&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \int 1 \, \mathrm{d}t = \int \frac{1}{x} \, \mathrm{d}x.&#xA;$$&lt;/span&gt;&#xA;Assume now that you only learnt that the anti-derivative of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\ln x + c$&lt;/span&gt; for positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; - it happens to easily that you forget about the sign restriction and just blindly apply the formula. You thus get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  t = \ln x(t) + c \qquad (*)&#xA;$$&lt;/span&gt;&#xA;for a constant &lt;span class=&quot;math-container&quot;&gt;$c \in \mathbb{R}$&lt;/span&gt; and hence, &lt;span class=&quot;math-container&quot;&gt;$x(t) = e^{-c} e^t$&lt;/span&gt;. The initial condition finally yields &lt;span class=&quot;math-container&quot;&gt;$e^{-c} = x_0$&lt;/span&gt;. But this cannot be true if &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; is negative. (Unless one allows for complex &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; - which is a pain here, since you then need to draw the complex logarithm of a negative number, which means that you cannot use the principal branch of the complex logarithm - in any case, this moves quite deep into the territory of complex analysis for such an innocent differential equation over the real field).&lt;/p&gt;&#xA;&lt;p&gt;By the way, to add further to the confusion, the same problem does at first glance not arise if you do the separation of variables by using definite integrals instead of indefinite integrals. This gives you&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \int_0^t 1 \, \mathrm{d}s = \int_{x_0}^{x(t)} \frac{1}{x} \, \mathrm{d}x,&#xA;$$&lt;/span&gt;&#xA;thus (if makes again the mistake to believe that &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt; were an anti-derivative of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt; for postive and negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  t = \ln x(t) - \ln x_0 = \ln \frac{x(t)}{x_0}, \qquad (**)&#xA;$$&lt;/span&gt;&#xA;and hence &lt;span class=&quot;math-container&quot;&gt;$x(t) = x_0 e^t$&lt;/span&gt; - which is the correct solution formula. Yet, we made the same error as above, by taking &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt; as a (presumed) anti-derivative of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt; even for negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. But the error cancels out for the following reason: Solutions cannot cross equilibrium points, so &lt;span class=&quot;math-container&quot;&gt;$x(t)$&lt;/span&gt; always has the same sign as &lt;span class=&quot;math-container&quot;&gt;$x(0)$&lt;/span&gt;. So if &lt;span class=&quot;math-container&quot;&gt;$x(0)$&lt;/span&gt; is negative, so is &lt;span class=&quot;math-container&quot;&gt;$x(t)$&lt;/span&gt; and hence the correct computation in &lt;span class=&quot;math-container&quot;&gt;$(**)$&lt;/span&gt; for negative &lt;span class=&quot;math-container&quot;&gt;$x(0)$&lt;/span&gt; is actually&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  t = \ln (-x(t)) - \ln (-x_0) = \ln \frac{-x(t)}{-x_0} = \ln \frac{x(t)}{x_0},&#xA;$$&lt;/span&gt;&#xA;which just happens to give the same result as our incorrect (for &lt;span class=&quot;math-container&quot;&gt;$x(0) &amp;lt; 0$&lt;/span&gt;) computation above.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;To sum up:&lt;/strong&gt; If one learns the anti-derivative of &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}$&lt;/span&gt; only for positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and happens to forget about this restriction later on (which is a very easily made mistake), one gets a mess when working with initial value problems. The same method (separation of variables) can then lead either to correct or incorrect result, depending on how precisely one applies the method.&lt;/p&gt;&#xA;&lt;p&gt;By the way, in case that you find the example discussed above too artificial (because the initial value problem is so simple) - a similar issue appears, for instance, for the logistic equation.&lt;/p&gt;&#xA;" OwnerUserId="16250" LastEditorUserId="16250" LastEditDate="2023-10-03T18:10:28.003" LastActivityDate="2023-10-03T18:10:28.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26834" PostTypeId="2" ParentId="26832" CreationDate="2023-10-03T18:19:52.940" Score="16" Body="&lt;p&gt;I think the heuristic&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\frac{8}{4} \quad \longleftrightarrow \quad 8 \textrm{ out of every group of } 4&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;still makes sense if you emphasize that you &lt;strong&gt;want&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; slices of pizza out of every group of &lt;span class=&quot;math-container&quot;&gt;$4,$&lt;/span&gt; even if it's not possible to take that many slices.&lt;/p&gt;&#xA;&lt;p&gt;If you &lt;em&gt;want&lt;/em&gt; more slices than there are available in the group, then what you're really saying is you want more than a full pizza.&lt;/p&gt;&#xA;&lt;p&gt;This naturally leads to the follow-up question &lt;em&gt;&amp;quot;Then how many full pizzas do you want?&amp;quot;&lt;/em&gt; and the answer to that question is to divide &lt;span class=&quot;math-container&quot;&gt;$8 \div 4.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here's how I imagine this actually being explained to a kid. (I'll refer to &amp;quot;pancakes&amp;quot; instead of &amp;quot;pizzas&amp;quot; since it's easier to imagine oneself hungry for multiple pancakes than for multiple pizzas.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Kid:&lt;/strong&gt; How can you take &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; slices out of every &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; slices?? That doesn't make sense. There are only &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; slices. You can't take more than &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; of those slices.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;You:&lt;/strong&gt; You're right. You can't actually take &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; of those &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; slices, because there are only &lt;span class=&quot;math-container&quot;&gt;$4.$&lt;/span&gt; But you can &lt;em&gt;want&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; slices that are the same size as those &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; slices.&lt;/p&gt;&#xA;&lt;p&gt;Imagine that we're having pancakes for breakfast. Normally, you don't even want a full pancake. So I cut up a single pancake into &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; pieces and ask you how many of those pieces you want.&lt;/p&gt;&#xA;&lt;p&gt;But today you're really hungry and you want to eat not only the full pancake (all &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; pieces), but also ANOTHER full pancake! You want to eat two full pancakes instead of just one.&lt;/p&gt;&#xA;&lt;p&gt;But I don't know this, so I bring out a pancake that's cut into &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; pieces and ask you how many of those pieces you want. You have to answer my question with a number. What number can you tell me, that indicates you want more than I'm offering you?&lt;/p&gt;&#xA;&lt;p&gt;Well, you just tell me how many pieces &lt;em&gt;of those size&lt;/em&gt; you want. You want the entire pancake that I'm offering you (&lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; pieces), and another full pancake (another &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; pieces of the same size). So you should tell me that you want &lt;span class=&quot;math-container&quot;&gt;$4+4=8$&lt;/span&gt; of those pieces.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-03T23:39:35.000" LastActivityDate="2023-10-03T23:39:35.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26835" PostTypeId="1" AcceptedAnswerId="26857" CreationDate="2023-10-03T23:16:40.900" Score="1" ViewCount="287" Body="&lt;p&gt;I could be wrong but those two ideas sound the same, just that the partition postulate is more general. There is also the angle addition postulate.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The segment addition postulate states that if three points A, B, and C are collinear such that B lies between A and C, then the sum of the lengths of segment AB and segment BC is equal to the length of the entire segment AC.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Partition postulate states that the whole is equal to the sum of its parts&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For context, I learned the partition postulate and not the segment addition postulate&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="10335" LastEditDate="2023-10-04T20:25:24.070" LastActivityDate="2023-10-07T03:43:46.787" Title="Is there a particular reason why segment addition postulate and partition postulate are two different things?" Tags="|geometry|definitions|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26836" PostTypeId="2" ParentId="26835" CreationDate="2023-10-04T00:06:14.897" Score="1" Body="&lt;p&gt;Without going down into the rabbit hole of axiomatic systems, I think the high-level answer you're looking for is that&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;yes, the segment addition postulate is a more specific case of the partition postulate, and&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the reason why a geometry course might present both postulates is that the segment addition postulate clarifies what is meant in a case where the partition postulate is vague enough to lead to ambiguity.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;To elaborate on item 2 -- it could be tempting to claim that the partition postulate supports the following statement, which is not true in general:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given three points A, B, and C, the sum of the lengths of segment AB and segment BC is equal to the length of the entire segment AC.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The segment addition postulate specifies the additional conditions that are required for the statement to hold true in general:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given three &lt;strong&gt;collinear&lt;/strong&gt; points A, B, and C &lt;strong&gt;such that B lies between A and C&lt;/strong&gt;, the sum of the lengths of segment AB and segment BC is equal to the length of the entire segment AC.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-10-04T00:06:14.897" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26837" PostTypeId="2" ParentId="26832" CreationDate="2023-10-04T05:16:26.307" Score="1" Body="&lt;p&gt;We teach that there are two basic ways of understanding division. På norsk they are called delingsdivisjon and målingsdivisjon; sharing division and measurement division would be rough translations. The actual English terminology might be different.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Sharing division&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;You have three liters juice and seventeen guests as the birthday. How much is there for everyone to drink? Is it enough, too much, or too little? (To make for an open question that might be interesting in reality, too.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Measurement division&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;You have three liters juice and drinking glasses where you can pour 5/3 deciliters juice to. How many glasses of juice can you fill? (And the same bonus question as above.)&lt;/p&gt;&#xA;&lt;p&gt;I recommend doing these with concrete objects to anchor the mathematical understanding to the real world and operations there.&lt;/p&gt;&#xA;&lt;p&gt;I am not quite answering the question you asked, but I hope this idea suggests how to move forward also in your case.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2023-10-04T19:28:25.137" LastActivityDate="2023-10-04T19:28:25.137" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26839" PostTypeId="2" ParentId="26824" CreationDate="2023-10-04T12:02:12.043" Score="-1" Body="&lt;p&gt;Note that the derivative of &lt;span class=&quot;math-container&quot;&gt;$\log x$&lt;/span&gt; has a Dirac delta term: &lt;a href=&quot;https://mathoverflow.net/questions/127601/does-the-derivative-of-log-have-a-dirac-delta-term&quot;&gt;https://mathoverflow.net/questions/127601/does-the-derivative-of-log-have-a-dirac-delta-term&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This involves the principal value.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2023-10-04T12:02:12.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26840" PostTypeId="1" AcceptedAnswerId="26844" CreationDate="2023-10-04T12:59:54.777" Score="9" ViewCount="3482" Body="&lt;p&gt;I'm teaching a geometry class and want to ensure my students understand the most common errors and misconceptions related to the Pythagorean Theorem and its applications.&lt;/p&gt;&#xA;&lt;p&gt;I attempted an initial Google search, but it didn't produce any significant results.&lt;/p&gt;&#xA;&lt;p&gt;Any references would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2023-10-06T03:07:27.553" Title="What are some common errors and misconceptions about the Pythagorean Theorem?" Tags="|geometry|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26842" PostTypeId="2" ParentId="20931" CreationDate="2023-10-04T13:17:02.383" Score="0" Body="&lt;p&gt;Using a metacognitive approach, I recommend advising your students about the potential ambiguity. Different resources often employ varying definitions. As Humpty Dumpty aptly puts it, &amp;quot;The question is which is to be master — that's all.&amp;quot; &lt;a href=&quot;https://sabian.org/looking_glass6.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://sabian.org/looking_glass6.php&lt;/a&gt;. t's crucial to refer to the definition provided by the specific authority in question (be it a teacher, textbook, etc.) and adhere to it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FRFG6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FRFG6.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;When a new authority emerges on the scene (such as a new teacher in a new school year or the adoption of a new textbook), examine any potential new definitions and adhere to them. However, it's crucial that the entire text remains consistent with the proposed definition.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2023-10-04T13:17:02.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26843" PostTypeId="2" ParentId="26832" CreationDate="2023-10-04T15:25:24.040" Score="4" Body="&lt;p&gt;Justin Skycak wrote a great answer that you can use with your kids, but I'd like to explain how what you're telling them relates to what they might be seeing right now and later on in school. I'll be referring to the U.S. Common Core State Standards (CCSS), but many curricula in the U.S. and in other countries follow similar approaches, though timing may differ.&lt;/p&gt;&#xA;&lt;p&gt;In the CCSS, fractions are introduced in grade 3. Here's an excerpt from the standards.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Students develop an understanding of fractions, beginning with&#xA;unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models&#xA;to represent parts of a whole... Students are able to use fractions to represent numbers equal to, less than, and greater than one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This means that we want third graders to view &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}$&lt;/span&gt; as &amp;quot;8 parts when &lt;em&gt;each&lt;/em&gt; whole is divided into 4 parts.&amp;quot; To use pizza as an example, we would say &amp;quot;One pizza is a whole. Divide &lt;em&gt;each&lt;/em&gt; whole into 4 parts. 8 of these parts makes &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}$&lt;/span&gt;.&amp;quot; Of course, this means that &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}$&lt;/span&gt; is more than 1 whole!&lt;/p&gt;&#xA;&lt;p&gt;Here's what a grade 3 workbook exercise from a popular publisher looks like.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Each shape is 1 whole. Write a fraction greater than 1 for the parts that are shaded.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/cyhjP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cyhjP.png&quot; alt=&quot;A diagram of two circles, each partitioned into halves. Every&#xA;half is shaded. An equation to the right reads &amp;quot;two equals blank.&amp;quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A third grader might reason it out like this.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Each circle is 1 whole, and each whole is divided into 2 equal parts. This means each part is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;. There are four parts shaded, so &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is pretty much all we expect from third graders. In particular, &lt;strong&gt;we do not expect 8-year-olds to reason about fractions using multiplication.&lt;/strong&gt; The remainder of this answer is devoted to considering what students will learn in later grades.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In grade 4, students learn how to add and subtract fractions with the same denominator, and the operation &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt;&amp;quot; is introduced. Students at this point can interpret &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}$&lt;/span&gt; as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;.&amp;quot; To go back to the pizza example, &amp;quot;I ate a quarter of a whole pizza eight times.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Then in grade 5, the interpretation of fractions as division, &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b} = a \div b$&lt;/span&gt;, and the operation &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;&amp;quot; are introduced. As you've said, it's not simple at all to explain why &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}$&lt;/span&gt; should be the same as &lt;span class=&quot;math-container&quot;&gt;$8 \div 4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt; of 8, and this is a major focus of 5th grade mathematics in the CCSS.&lt;/p&gt;&#xA;&lt;p&gt;Understanding this idea requires children to be able to, within a single problem, change units/re-identify a whole. This is why it's critically important to have third graders go through the process of identifying a whole, aka &amp;quot;unitizing.&amp;quot; To illustrate what I mean, here's another exercise from the same workbook.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Write a fraction to name the shaded part of the group.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/mXFJU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mXFJU.png&quot; alt=&quot;A diagram of 8 dots in 4 columns of 2. A rectangle is drawn around each column. The dots in the first column are shaded.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The students are being asked to treat the entire group as a whole. They might interpret the whole as being divided into 8 equal parts, in which case the shaded parts are &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{8}$&lt;/span&gt; (of a whole), or they might interpret the whole as being divided into 4 equal parts, and the shaded part is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt; (of a whole). This is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt; of our whole pizza order. On the other hand, if we consider each dot to be a whole, then each dot is divided into one part, and the shaded parts are &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{1}$&lt;/span&gt; (of a whole), or 2 wholes. This is 2 whole pizzas.&lt;/p&gt;&#xA;&lt;p&gt;After the notion of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;&amp;quot; is introduced in grade 5, students learn to say that this is also &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt; of 8 wholes. To do this, they have to be able to quickly switch between viewing the entire group as a whole and viewing each dot as a whole. Once they're able to, they can determine that &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt; of 8 whole pizzas is 2 whole pizzas. They also know that 2 is the result of dividing 8 by 4. And by dividing each whole pizza into 4 equal slices, they can see that 2 whole pizzas is also &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}$&lt;/span&gt; whole pizzas. &lt;span class=&quot;math-container&quot;&gt;$\frac{8}{4}= 8 \times \frac{1}{4} = 8 \div 4 = \frac{1}{4} \times 8$&lt;/span&gt;. It seems innocuous, but this idea is one of the capstones of elementary school mathematics.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-10-04T22:40:06.243" LastActivityDate="2023-10-04T22:40:06.243" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26844" PostTypeId="2" ParentId="26840" CreationDate="2023-10-04T15:48:46.580" Score="25" Body="&lt;p&gt;Here's a list of mistakes that I've seen students make.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Conceptual Mistakes&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Applying the Pythagorean Theorem on non-right triangles. (They may also think that the word &amp;quot;hypotenuse&amp;quot; means the longest side of &lt;em&gt;any&lt;/em&gt; triangle.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Assuming that the missing side is always the square root of the sum of squares of the other sides -- even when that missing side is a leg, not the hypotenuse. (They think the unknown side is always &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; in the formula &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2.$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Writing their final answer with &lt;span class=&quot;math-container&quot;&gt;$\pm$&lt;/span&gt; even though the side length cannot be negative. (They use &lt;span class=&quot;math-container&quot;&gt;$c = \pm \sqrt{a^2 + b^2}$&lt;/span&gt; because they're used to solving quadratic equations with no domain constraints, e.g., &lt;span class=&quot;math-container&quot;&gt;$x^2 = 9 \Rightarrow x = \pm 3.$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Mindless Mistakes in Algebra&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Forgetting to take the square root at the end (i.e., &lt;span class=&quot;math-container&quot;&gt;$c = a^2 + b^2$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Forgetting to square the quantities within the square root (i.e., &lt;span class=&quot;math-container&quot;&gt;$c = \sqrt{a + b}$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Using a product instead of addition (i.e., &lt;span class=&quot;math-container&quot;&gt;$c = \sqrt{ab}$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Misconceptions About Algebra&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Distributing the square root (i.e., &lt;span class=&quot;math-container&quot;&gt;$c = \sqrt{a^2 + b^2}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$= \sqrt{a^2} + \sqrt{b^2}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$= a + b$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;Canceling out&amp;quot; exponents of all terms in an equation (i.e., &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\Rightarrow a + b = c$&lt;/span&gt;). Note that while this is technically the same error as distributing the square root, most students who make this error won't understand that and will continue to make it even if they understand that they cannot distribute a square root.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-10-04T15:48:46.580" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26845" PostTypeId="2" ParentId="26832" CreationDate="2023-10-05T11:59:06.467" Score="3" Body="&lt;p&gt;Justin H has a good answer emphasizing how fractions are a complex topic and we learn more and more about them over the years. To your specific question, I would emphasize the need to spend a significant amount of time with the less than one fractions before moving to greater than one. Often in questuons here there is a tacit or explicit desire for earlier and more comprehensive explanations.  But this is usually a mistake in terms of human pedagogy.&lt;/p&gt;&#xA;&lt;p&gt;On the specific question, when ready, I would just think of passing out slices of pizza to kids.  So five thirds is each kid getting a slice of pizza that is one third of the pie.  And yes you need a second pizza.  And yes one slice to stick in the fridge.&lt;/p&gt;&#xA;&lt;p&gt;Also note that your comments about four fifths of 20 or the like are a higher level of problem yet.  Since it is of 20.  So dont jump to that immediately. And it only works, for neophytes, with common factors&lt;/p&gt;&#xA;" OwnerUserId="22978" LastActivityDate="2023-10-05T11:59:06.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26846" PostTypeId="2" ParentId="26840" CreationDate="2023-10-06T00:26:33.363" Score="1" Body="&lt;p&gt;Comment from &lt;a href=&quot;https://matheducators.stackexchange.com/users/10547/josephdoggie&quot;&gt;JosephDoggie&lt;/a&gt; on the question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Very famously, the scarecrow in the Wizard of Oz movie (&lt;span class=&quot;math-container&quot;&gt;$1939$&lt;/span&gt;) misquotes the Pythagorean Theorem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Looking up the quote on Google gives &lt;a href=&quot;https://www.metro-arts.org/ourpages/auto/2015/5/14/58561904/Scarecrows%20Math%20from%20The%20Wizard%20of%20Oz.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; resource which gives this bit of info:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the end of The Wizard of Oz, the Scarecrow receives a diploma and then immediately says, &amp;quot;The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Looking up &amp;quot;Isosceles triangle&amp;quot; on Google gives a calculator and switching what we need to solve for from &amp;quot;base&amp;quot; to &amp;quot;side&amp;quot; gives us &lt;span class=&quot;math-container&quot;&gt;$a=\frac P2-\frac b2$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$P=2a+b$&lt;/span&gt;) because an isosceles triangle has two sides that are equal to each other. The statement made by the Scarecrow would only be true if and only if &lt;span class=&quot;math-container&quot;&gt;$b=0$&lt;/span&gt; (while &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; could be equal to anything) because we have&lt;span class=&quot;math-container&quot;&gt;$$\sqrt a+\sqrt b=\sqrt a\implies b=0,a=c_1$$&lt;/span&gt;where &lt;span class=&quot;math-container&quot;&gt;$c_1$&lt;/span&gt; is any number.&lt;/p&gt;&#xA;" OwnerUserId="22983" LastActivityDate="2023-10-06T00:26:33.363" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26847" PostTypeId="2" ParentId="26840" CreationDate="2023-10-06T03:07:27.553" Score="2" Body="&lt;p&gt;Reversing the implication: if &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2 = c^2$&lt;/span&gt;, then a,b,c are the side-lengths of a right triangle. With the right restrictions that's true, but it's not what the Pythagorean theorem claims.&lt;/p&gt;&#xA;" OwnerUserId="22984" LastActivityDate="2023-10-06T03:07:27.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26849" PostTypeId="1" AcceptedAnswerId="26853" CreationDate="2023-10-06T17:26:50.447" Score="1" ViewCount="229" Body="&lt;p&gt;Next semester I'm going to lecture calculus 2 in an institution I just joined. However, when I had calculus 2 back then the syllabus was very different, it mainly covered several variable calculus up until derivatives, including all that analysis of implicit/inverse function theorem, Lagrangian multiplier method, and so on...&lt;/p&gt;&#xA;&lt;p&gt;For this course, on the other hand, the syllabus goes more or less like this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Functions of several variables&lt;/li&gt;&#xA;&lt;li&gt;Partial derivatives, directional derivatives and gradients&lt;/li&gt;&#xA;&lt;li&gt;Integrals of functions of several variables&lt;/li&gt;&#xA;&lt;li&gt;Sequences and series of functions&lt;/li&gt;&#xA;&lt;li&gt;Line integrals, Green's theorem, rotational and divergent&lt;/li&gt;&#xA;&lt;li&gt;Surface integrals, Stokes' and Gauss' theorem&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I remember having the functions of several variables and differentiability parts in Calculus &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, then line and multivariate integrals in Calculus &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, and then sequences and series part was in another course still (I majored in mathematics). This is a one-semester course for engineering, so I take it the course should not be aimed at analyzing things as thoroughly, but still it seems kind of an all-over-the-place syllabus to organize... How would you go on organizing this course? What would you focus more on? Would it be wise to separate in 3 big parts as &amp;quot;limits/differentiability then sequences/series and then line/surface integrals&amp;quot;? Or would you do it differently?&lt;/p&gt;&#xA;" OwnerUserId="16351" LastEditorUserId="22983" LastEditDate="2023-10-07T12:18:26.173" LastActivityDate="2023-10-07T12:18:26.173" Title="What would be a good pacing for teaching this calculus 2 course?" Tags="|calculus|syllabus|organization|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26850" PostTypeId="1" CreationDate="2023-10-06T20:30:52.537" Score="15" ViewCount="6619" Body="&lt;h1&gt;Background&lt;/h1&gt;&#xA;&lt;p&gt;I'm informally helping a few students with college Calc 1.  This isn't the first time I've aided people with calculus, and so they've sought me for help, though I don't consider myself to be particularly good teacher or good at math.&lt;/p&gt;&#xA;&lt;p&gt;In general they are good students, have good work ethic, follow what I &lt;em&gt;tell&lt;/em&gt; them to do to succeed, but recently I ran into something of a road block with tangent lines and derivative trends in graphs.&lt;/p&gt;&#xA;&lt;p&gt;They seem to at least &lt;em&gt;be able&lt;/em&gt; to perform all other tasks, If I tell them they need to use X formula to solve Y problem, they can easily figure out the rest, and use that later in a different part of the problem, perform basic algebra or trig, and do the rest of the &amp;quot;calculus&amp;quot; part of the problem.&lt;/p&gt;&#xA;&lt;p&gt;But when they seem super confused when I talk about tangent lines.  First there seems to be an aversion to &amp;quot;precise&amp;quot; language.  At first I tried to get them to describe to me what they thought a derivative physically meant, because I could tell they just weren't getting why parts of the graph meant positive or negative derivative, and were clearly not connecting rate of change with things like &amp;quot;a tub draining or filling with water&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I couldn't parse what they were trying to express.  They claim that they just don't use the same &amp;quot;big words&amp;quot; I do.  I tell them, the things they are saying are in-comprehensible, mathematics is in part communication, and being un able to communicate ideas isn't going to allow them to even let their professors know they know what they are talking about.&lt;/p&gt;&#xA;&lt;p&gt;To be clear, I don't expect &amp;quot;big&amp;quot; words, but they weren't even getting close to anything related to &amp;quot;rate of change&amp;quot;, it was like they were trying to find the &amp;quot;teachers favorite color&amp;quot; except I'm not their professor so that straight up doesn't work with me.&lt;/p&gt;&#xA;&lt;p&gt;From that point, I decided okay, lets take step back.  I'd previously explained in earlier sessions what the derivative is, and why the book was saying the things it does.  But that hadn't stuck or they hadn't internalized it or something.  So I went in to try to figure out what they weren't understanding.  Eventually I got them to understand, at least I believe, the whole rate of change thing, and that this is a &lt;em&gt;generic&lt;/em&gt; concept, it can apply to any quantity, any X, and they proved they understood by applying that knowledge to word problems using it.&lt;/p&gt;&#xA;&lt;p&gt;So then I went forward and asked them what the significance of the tangent line was.  They seemed to have no clue, and when I tried to get them to understand, they seemed to be &lt;em&gt;waay&lt;/em&gt; too focused on the line, rather than what it represented.  They seem to want to plug and play values with the line, then get confused when it doesn't work, I don't know how to explain it.  It's like they don't seem to grasp the concept of something &lt;em&gt;representing&lt;/em&gt; something else in the way it is framed with tangent lines.  I ask them what is the relationship between the tangent line and the derivative, but they struggle, get it wrong, start guessing (note they've already been taught this).  I'll end up telling them that the value of the &lt;em&gt;slope&lt;/em&gt; of the tangent line is the value of the derivative at the point the on f(x) associated with the tangent line and they claim to understand that, only for them to clearly not be able to apply that knowledge later on.&lt;/p&gt;&#xA;&lt;p&gt;There's something about it not being a straight &amp;quot;equation&amp;quot; that they can apply somewhere that makes it harder for them to use than if I just said &amp;quot;Here's an equation that approximates the derivative&amp;quot; or something.&lt;/p&gt;&#xA;&lt;p&gt;Coming back to graphs, they would get the wrong answer and they would seem to mix up the derivative and properties of the actual f(x) line up, and I tried to show them how they could just visualize the derivative but because they struggled so much with tangent lines, they seemed to be unable to grasp visualizing what the derivative was (and maybe still haven't internalized what the concept was yet?)&lt;/p&gt;&#xA;&lt;p&gt;They kept having homework questions over this, got frustrated that we were &amp;quot;spending so much time on this&amp;quot; (understandably, I was spending over an hour trying to get them to understand these concepts on questions that should have taken less than a minute), and I told them we could move on, but you needed to get help from &lt;em&gt;someone&lt;/em&gt; who could figure out why they didn't understand these things.&lt;/p&gt;&#xA;&lt;p&gt;One problem is their professor does not speak intelligible English, uses &amp;quot;reverse&amp;quot; classes (do 'homework' in class, watch lectures at home, and also do more homework) the lectures are recorded in poor quality.   But the books material seemed to be similar to what I had when I was in school, and they get ample time with TAs who they seem to like. Their TA's have already aided them on similar problems is my understanding.&lt;/p&gt;&#xA;&lt;p&gt;Come test time, they couldn't answer these types of questions correctly, and did poorly.  I want to help, but I've told them for the time being they need to reach out to their TAs (Professor is useless, office hours are almost suspiciously poorly timed to avoid student needs).   The next sections they seem to have little problem with, again, it's just plug and play equations for them, and now they are getting to more advanced derivative rules. I really don't understand &lt;em&gt;how&lt;/em&gt; to get them to understand what should be &lt;em&gt;much&lt;/em&gt; simpler things than the other problems they are able to solve.&lt;/p&gt;&#xA;&lt;h1&gt;Question&lt;/h1&gt;&#xA;&lt;p&gt;What are some strategies, methods I can use to help these students understand tangent lines and trends of derivatives (and second derivatives) on graphs?&lt;/p&gt;&#xA;" OwnerUserId="8994" LastActivityDate="2023-10-14T04:58:24.877" Title="Students can't seem to grasp the intent of tangent lines and getting general trends of derivatives from graphs" Tags="|undergraduate-education|mathematical-pedagogy|calculus|tutoring|" AnswerCount="15" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="26851" PostTypeId="2" ParentId="26850" CreationDate="2023-10-06T21:26:23.400" Score="29" Body="&lt;p&gt;Sadly, these students seem to think of math as a bunch of rules. You have done great work, and probably gotten them a little closer. But they are resisting the reasoning that can't easily be put into rule form.&lt;/p&gt;&#xA;&lt;p&gt;One strategy I use is to have students imagine themselves in tiny cars, driving along the function. Their headlights and taillights make the tangent line. The slope of that line is also the car's (or function's) slope.&lt;/p&gt;&#xA;&lt;p&gt;I also draw a long curve on the board that goes up and down a bunch, and ask students to tell whether the slope is positive, negative, zero, or undefined at lots of points on this curve. (We do this with concavity too.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="5563" LastEditDate="2023-10-07T14:30:41.137" LastActivityDate="2023-10-07T14:30:41.137" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26852" PostTypeId="2" ParentId="26850" CreationDate="2023-10-06T21:42:13.987" Score="4" Body="&lt;p&gt;First you need to think about what you are doing and what the priorities are or should be.  You are an unofficial resource and not plugged in with the prof, imperfect as he may be. Physician, first do no harm...&lt;/p&gt;&#xA;&lt;p&gt;Many students don't even master procedures. So it's actually half a loaf to be getting kids who are at least doing that.&lt;/p&gt;&#xA;&lt;p&gt;The way to teach concepts is very similar to how to teach procedures. Imitation and practice.  Lots of imitation and lots of practice.  And weaker kids will need more. There is no royal road, no magic lock and key explanation.&lt;/p&gt;&#xA;&lt;p&gt;Still, I would be very concerned about you derailing things and going from half a loaf to zero, if you actually implement what I just said...large amounts of practice on concepts.  Especially if the REAL teacher is going to test them mostly on procedures.&lt;/p&gt;&#xA;" OwnerUserId="22978" LastActivityDate="2023-10-06T21:42:13.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26853" PostTypeId="2" ParentId="26849" CreationDate="2023-10-06T21:54:04.097" Score="4" Body="&lt;p&gt;In the US, most of both of your variants is covered in 3rd semester calc. A bit of a drive by of the different topics given the number of them and the differences if them.  Still manageable with tailored difficulty books.&#xA;But the students tend to get exposure more than mastery. In a way that is different from their experience in single variable calculus.  Sequences and series are usually a week or so within 2nd semester calc...and are a bit off the main theme, tacked on at end.&lt;/p&gt;&#xA;&lt;p&gt;Looking at it practically, you seem to have a text and pacing for the bulk of yhe course.  Just need to squeeze in series stuff.  My advice is do a week on it, no more. It is really just another exposure. No...don't use that book that you have.  Will be too long and need too much tailoring. Instead look at a typical us calc book like Thomas or Stewart. Either will have a week long capsule.&lt;/p&gt;&#xA;" OwnerUserId="22978" LastActivityDate="2023-10-06T21:54:04.097" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26854" PostTypeId="2" ParentId="26811" CreationDate="2023-10-06T22:19:06.103" Score="1" Body="&lt;p&gt;Quarter of a loaf answer, but one of the things I like about some texts is that they will have a list of symbols, to include even Greek alphabet. And will have a glossary.  It seems most appropriate for lower level books and within that foe review texts. And still only a sad small subsection of books do this.  But in my mind a book is a tool snd these sorts of features, also add in a good index make the learning tool more helpful. So practically, what I suggest is just looking at several textbooks and seeing if a few have front or back sections like this that you like and can then plagiarize. The texts I know that do this well are the 1940s era US War Dept Ed Manuals. But if you just physically look through some college algebra a calc and such texts at a brick and mortar library you should find some good ones, maybe a bit more in older books. Also, take a look at, DC minus one, Ayres schaums review for college algebra...think it is called first year math.&lt;/p&gt;&#xA;&lt;p&gt;Just reread head post and saw you were looking for stats stuff. Again, would just look at some entry level review texts. Also...an incredible resource of this sort of thing is Michael Linberg EIT Reference Manual. Chock full of formula and symbol lists, for all STEM topics.&lt;/p&gt;&#xA;" OwnerUserId="22978" LastEditorUserId="22978" LastEditDate="2023-10-06T22:37:23.473" LastActivityDate="2023-10-06T22:37:23.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26855" PostTypeId="2" ParentId="26850" CreationDate="2023-10-06T23:58:27.067" Score="20" Body="&lt;p&gt;When a concept just won't stick, no matter how many ways you explain it, that is usually a sign of a gap in the student's background knowledge. So I'd recommend searching for the gap.&lt;/p&gt;&#xA;&lt;p&gt;Of course, you've already tried this to some extent, and I think reinforcing derivatives and rates of change was a good first step.&lt;/p&gt;&#xA;&lt;p&gt;Next, you might try reviewing lines/linear functions. I know it sounds basic, but can these students look at a graph of a linear function and understand how the slope relates to a rate of change? And can they apply this to simple modeling questions?&lt;/p&gt;&#xA;&lt;p&gt;It's possible they &amp;quot;know&amp;quot; this stuff, but it's been too long and/or they just aren't connecting it to the new material they're learning.&lt;/p&gt;&#xA;&lt;p&gt;After refreshing linear functions, you could return to rates of change, and try to hammer home the idea of an instantaneous rate of change and how it relates to the rate of change (slope) of a linear function. This is basically the difference between a car traveling at constant speed vs. variable speed. If they can understand this, then tangent lines are the graphical version of the same concept.&lt;/p&gt;&#xA;" OwnerUserId="22788" LastActivityDate="2023-10-06T23:58:27.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26856" PostTypeId="2" ParentId="26850" CreationDate="2023-10-07T01:24:29.507" Score="6" Body="&lt;p&gt;I can't figure out from what you say what definition of derivative you are using, or what you have said the connection is between derivatives, tangents, and rates of changes. So I can't tell what might be missing, or what the alternatives might be. But maybe if I described where I would start, that might help?&lt;/p&gt;&#xA;&lt;p&gt;We start off with a concrete equation, like &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt;. We pick a few values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, calculate &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; for each, plot them on a graph, and then draw a smooth curve through them. (Get the class to use their calculators, and shout out the numbers.)&lt;/p&gt;&#xA;&lt;p&gt;Now pick a value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; like &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; and ask how does the curve behave 'close' to this point. Let's try &lt;span class=&quot;math-container&quot;&gt;$x={1,1.1,1.2,1.3}$&lt;/span&gt;, and we get &lt;span class=&quot;math-container&quot;&gt;$y={1,1.21,1.44,1.69}$&lt;/span&gt;. For every &lt;span class=&quot;math-container&quot;&gt;$0.1$&lt;/span&gt; we increase &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; goes up by a bit more than twice as much as that. Let's try looking closer. We take &lt;span class=&quot;math-container&quot;&gt;$x={1,1.01,1.02,1.03}$&lt;/span&gt;, and we get &lt;span class=&quot;math-container&quot;&gt;$y={1,1.0201,1.0404,1.0609}$&lt;/span&gt;, and can say that for every &lt;span class=&quot;math-container&quot;&gt;$0.01$&lt;/span&gt; we increase &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; goes up by very close to twice as much. If we plot them on a graph, the line looks straight. Now look even closer, &lt;span class=&quot;math-container&quot;&gt;$x={1,1.001,1.002,1.003}$&lt;/span&gt;, and so on. If we go from &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x+\Delta x$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; goes to a value close to &lt;span class=&quot;math-container&quot;&gt;$y+2\,\Delta x$&lt;/span&gt;, and the smaller the step &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt;, the closer the approximation.&lt;/p&gt;&#xA;&lt;p&gt;Next try the same technique on other points on the same curve. Like &lt;span class=&quot;math-container&quot;&gt;$x={2,2.01,2.02,2.03}$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x={3,3.01,3.02,3.03}$&lt;/span&gt;, and so on. It's not hard to spot the pattern in the multipliers - they go &lt;span class=&quot;math-container&quot;&gt;$2, 4, 6,\ldots$&lt;/span&gt;, as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; goes &lt;span class=&quot;math-container&quot;&gt;$1, 2, 3,\ldots$&lt;/span&gt; . The multiplier is &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;, and when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is increased by a small step &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; increases by &lt;span class=&quot;math-container&quot;&gt;$2x\,\Delta x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This 'multiplier' is what we call the derivative. And this gives us a 'pocket calculator' way to find the derivative at a point &lt;span class=&quot;math-container&quot;&gt;$x=x_0$&lt;/span&gt;. If we calculate &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; at the given point, and one just a millionth of a unit to the right, we see that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; increases by so many millionths, which tells us the derivative. (This method doesn't &lt;em&gt;always&lt;/em&gt; work, if the function has sharp corners or tight wiggles, or involves very big numbers. But for all the examples you will meet in an introductory course, it will.)&lt;/p&gt;&#xA;&lt;p&gt;If we plot the points on a graph, we get the chord from &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(x+\Delta x,y+\Delta y)$&lt;/span&gt;, and if we extend this straight line segment beyond the curve in both directions, we find the two crossing points get closer and closer together, and the line gets closer and closer to being the tangent, where the line touches the curve at a single point. (I have to say, that's probably more useful to 17th century mathematicians who grew up with Euclid's geometry and have already internalised the properties of tangents. To a more modern audience, I might say it's the straight line 'along' the curve at the point.)&lt;/p&gt;&#xA;&lt;p&gt;Next we pick some other simple polynomial curves, and calculate derivatives at points along the curve with the 'millionth' method. Plot them out, note the patterns. The derivative is positive when the curve is sloping up, the derivative is negative when the curve is sloping down, and the derivative is zero when the curve is horizontal, at a minimum, maximum, or inflection point. We put in a few more complicated curves like &lt;span class=&quot;math-container&quot;&gt;$y=1/x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=\sin x$&lt;/span&gt; to make clear that the definition applies to &lt;em&gt;any&lt;/em&gt; smooth function, not just polynomials.&lt;/p&gt;&#xA;&lt;p&gt;If they have access to computers, this is a great point to automate the process - to plot a graph of a function, and then plot its derivative using the 'millionth' method. Then if they want to check something, experiment, build intuition, they can do so without hours of labour with a calculator.&lt;/p&gt;&#xA;&lt;p&gt;At this point, we can move on to the abstraction and introduce the algebra and notation for differentiating &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)+g(x)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$kf(x)$&lt;/span&gt;for constant &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, and build up to simple polynomials, and the formal idea of 'limits' and so on. They have something concrete to cling to, with the 'millionth' method, numerous simple examples, and mental pictures of curves and slopes.&lt;/p&gt;&#xA;&lt;p&gt;In learning, we always go from the concrete, to the pictorial, to the abstract. Mathematicians have a bad habit of starting with the abstract. It works, in terms of being able to give correct answers, but it quickly becomes a bit of a '&lt;a href=&quot;https://en.wikipedia.org/wiki/Chinese_room#Chinese_room_thought_experiment&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chinese Room&lt;/a&gt;', where symbols are manipulated without understanding, to give the illusion of intelligence. I think you probably did a bit better trying to start with the pictorial. But when that's not working, I find you have to go all the way back to the concrete.&lt;/p&gt;&#xA;" OwnerUserId="22994" LastEditorUserId="205" LastEditDate="2023-10-07T05:53:06.550" LastActivityDate="2023-10-07T05:53:06.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26857" PostTypeId="2" ParentId="26835" CreationDate="2023-10-07T03:38:46.583" Score="3" Body="&lt;p&gt;In the &lt;em&gt;Elements&lt;/em&gt;, Euclid did not use lengths. But in contemporary high school geometry, we typically find  lengths of segments being represented as real numbers. The &amp;quot;ruler postulate&amp;quot; was introduced in the nineteenth or twentieth century. It states that the points on a line can be put into correspondence with the real numbers. It is only in this &amp;quot;modern&amp;quot; context that the segment addition postulate makes any sense. Similarly, there is a &amp;quot;protractor postulate&amp;quot; to put the measures of angles into correspondence with real numbers.&lt;/p&gt;&#xA;&lt;p&gt;I think the difference between the &amp;quot;segment addition postulate&amp;quot; and the &amp;quot;partition postulate&amp;quot; is that the partition postulate fits best when discussing how Euclid framed geometry in the &lt;em&gt;Elements&lt;/em&gt;, whereas in Euclidean geometry with the ruler postulate in play, the partition postulate becomes updated in the form of segment addition postulate.&lt;/p&gt;&#xA;&lt;p&gt;Chapter 3 of  &amp;quot;Geometry: Euclid and Beyond&amp;quot; by Robin Hartshorne, it is good reference.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-10-07T03:43:46.787" LastActivityDate="2023-10-07T03:43:46.787" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26858" PostTypeId="1" AcceptedAnswerId="26866" CreationDate="2023-10-07T07:40:11.153" Score="2" ViewCount="952" Body="&lt;p&gt;What is a much better proof in mathematics, is it need to be a much more advanced one compared with the proof already available or a much simpler one? &lt;br/&gt;&#xA;I think you can challenge a proof in two different ways. Either we can find much generalised proof covering all cases as an extension for the previous proof using much advanced concepts in mathematics or we can suggest a much simpler proof. As an example how would you compare if a mathematician could able to prove FLT in less number of pages when comparing with the proof given by Andrew Wiles since after giving the full credit for the first valid proof there is no chance of withdrawing it later for a better one.&lt;br/&gt;&#xA;Edited ; I posted this because I think as mathematics educators we need to know how we have to decide which proof is much better and that kind of issue is much relevant to mathematics teachers rather than students. Hope you will give your fullest support to get the full use of this discussion with much understanding .&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-10-09T10:22:54.907" LastActivityDate="2023-10-09T10:22:54.907" Title="Better proof for a proposition when a proof is already available" Tags="|mathematical-pedagogy|proofs|" AnswerCount="3" CommentCount="5" ClosedDate="2023-10-09T09:45:42.233" ContentLicense="CC BY-SA 4.0" />
  <row Id="26859" PostTypeId="2" ParentId="26850" CreationDate="2023-10-07T09:40:47.007" Score="3" Body="&lt;p&gt;Might or might not work, but try a graph of how far a car has traveled with respect to time, some journey they have actually traveled, then ask them what the speedo would have been reading at a few points on the graph. Something they should be able to relate to.&lt;/p&gt;&#xA;" OwnerUserId="1615" LastActivityDate="2023-10-07T09:40:47.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26860" PostTypeId="2" ParentId="26858" CreationDate="2023-10-07T13:16:47.533" Score="4" Body="&lt;p&gt;For educational purposes, what you want is a proof that has different ideas in it. This is similar to teaching: you want pupils to present different ways to solve a given problem and then compare and contrast, to increase understanding of the problem, the concepts and the tools.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-10-07T13:16:47.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26861" PostTypeId="2" ParentId="26850" CreationDate="2023-10-07T14:36:27.790" Score="4" Body="&lt;p&gt;Time permitting, I would want to quiz them on whether they intuitively understand slope and tangent lines in the first place:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;See if they can answer speed-drills on graphing basic lines. This is one of 5 skills that I personally identify as &amp;quot;automatic&amp;quot; necessities for college algebra, and have a timed quiz at the link below. I'd want to see that they can answer this correctly within the timer; if not, they should practice a few minutes every day until they do so.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.automatic-algebra.org/graphinglines.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Automatic-Algebra: Graphing Lines&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Then, I would want to give them some random curvy graphs, name a point, and ask them to draw a tangent line there and estimate its slope. If they can do that, then the explanation &amp;quot;derivatives give you that slope&amp;quot; should be coherent.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-10-07T17:54:53.493" LastActivityDate="2023-10-07T17:54:53.493" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26862" PostTypeId="2" ParentId="26850" CreationDate="2023-10-07T15:31:01.447" Score="15" Body="&lt;p&gt;I taught physics for 25 years at a community college, and the situation you're describing is one that I'm very familiar with. (I also got the oportunity to teach first-semester calculus a couple of times, which was super fun!) Students would show up in the first semester of a calculus-based physics class, and the first week or two would be velocity and acceleration. You would think this would be easy for them, since it's exactly the material they've covered in calculus, in the application that is probably the most heavily emphasized in calculus.&lt;/p&gt;&#xA;&lt;p&gt;This problem may be exacerbated when there are other issues present, e.g., the student is not very intelligent, or they had inadequate preparation for calculus. However, I'm convinced that what you're describing also occurs among the best students. Part of this is just from having discussed with a couple of very intelligent people, who are good at math, what they got out of their college calculus class. (One of these people is a lawyer and one is a French professor.) It was clear from talking to both of these people that they had absolutely no idea what calculus was about at the conceptual level. When I said to them that calculus was the study of rates of change and of how change accumulates, their reaction was like &amp;quot;Oh, really? That's what it was?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;You discuss their difficulty in understanding the concept as &amp;quot;generic&amp;quot; and relating it to standard examples like water flowing out of a bathtub. I don't think their difficulties with this are mysterious at all. Calculus classes and textbooks simply don't do applications. Look at the homework problems: dozens and dozens of problems, one after another, that are pure symbol manipulation.&lt;/p&gt;&#xA;&lt;p&gt;Another cause of the tangent-line issues is that students are never (or almost never) assigned any exercises involving sketching without a formula. It's not hard to make up such exercises. You can give a verbal description (the car speeds up, then slows down) and ask for sketches of x, x', and x'', stacked vertically. You can give a graph and ask where the derivative is positive, zero, etc. You can give a graph and ask for a sketch of the derivative. These tasks just don't show up in a calculus class. (For one thing, they can't be graded by the textbook publisher's homework grading software.)&lt;/p&gt;&#xA;&lt;p&gt;The whole thing is not particularly easy conceptually. It doesn't come naturally to the human mind. Look at Star Trek, where shutting off the Enterprise's engines should cause &lt;strong&gt;ẍ&lt;/strong&gt;=&lt;strong&gt;0&lt;/strong&gt;, but the scriptwriters all believe it causes &lt;strong&gt;ẋ&lt;/strong&gt;=&lt;strong&gt;0&lt;/strong&gt;. When it comes to money, people often ignore these distinctions as if they don't exist or are too hard. For example, people talk about the national debt's ratio to GDP, as if both numbers had units of dollars and the ratio were unitless. Actually GDP has units of dollars per year.&lt;/p&gt;&#xA;" OwnerUserId="23001" LastEditorUserId="1147" LastEditDate="2023-10-08T00:32:17.507" LastActivityDate="2023-10-08T00:32:17.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26863" PostTypeId="2" ParentId="26850" CreationDate="2023-10-07T16:23:26.200" Score="4" Body="&lt;p&gt;I used to do a lot of math tutoring, and I fear change in the curriculum prerequisite to calculus may be the problem.&#xA;Way back when, a lot of time went into generic curve-sketching.  And students worked more abstractly.&#xA;But with the advent and use of graphing calculators, students today want to input a function then use the calculator to calculate and draw the graph.  A level of abstraction has been lost, I fear.&lt;/p&gt;&#xA;" OwnerUserId="23002" LastActivityDate="2023-10-07T16:23:26.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26864" PostTypeId="2" ParentId="26850" CreationDate="2023-10-07T21:44:27.963" Score="0" Body="&lt;p&gt;I think @user22788 has made a good point about prior concepts, but there is a related issue, motivation. Why are the students doing calculus in the first place?  I assume that they don't aspire to be mathematicians or physicists, because I wouldn't expect such people to struggle with calculus. Maybe the examples, e.g. cars driving up the graph of f(x), have no relation to anything that interests them. Can you find examples that are actually in line with their interests? If so, try to pick up @user22788 's other point, starting with simple linear examples, where you really don't need calculus, then introduce the tangent as &amp;quot;the linear function when you don't have a linear function&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="19663" LastActivityDate="2023-10-07T21:44:27.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26865" PostTypeId="2" ParentId="26858" CreationDate="2023-10-08T00:14:14.750" Score="6" Body="&lt;p&gt;Let me disagree with this dichotomy:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Either we can find much generalised proof ... or we can suggest a much simpler proof.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think there are other options, proofs that yield different insights,&#xA;draw on different areas of math, make connections to different fields.&#xA;There is not one &amp;quot;better proof&amp;quot;; &amp;quot;quality&amp;quot; seems an inappropriate measure.&lt;/p&gt;&#xA;&lt;p&gt;An example is Euler's formula:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$V - E + F = 2 \;.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;David Eppstein maintains a webpage with&#xA;&lt;a href=&quot;https://ics.uci.edu/%7Eeppstein/junkyard/euler/&quot; rel=&quot;noreferrer&quot;&gt;Twenty-one Proofs of Euler's Formula&lt;/a&gt;.&#xA;I've found the variety quite useful for (college) students&#xA;appreciating the result in the classroom.&#xA;Different students&#xA;may find different proofs more insightful, depending on their&#xA;prior exposure to the relevant background concepts.&#xA;Not unlike Tommi's viewpoint.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CwqPJ.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CwqPJ.png&quot; alt=&quot;StereoProj&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2023-10-08T00:14:14.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26866" PostTypeId="2" ParentId="26858" CreationDate="2023-10-08T00:37:35.623" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;As an example how would you compare if a mathematician could able to prove FLT in less number of pages when comparing with the proof given by Andrew Wiles since after giving the full credit for the first valid proof there is no chance of withdrawing it later for a better one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That would, of course, be a very impressive accomplishment.&lt;/p&gt;&#xA;&lt;p&gt;But solving an unsolved problem &lt;em&gt;at all&lt;/em&gt;, especially a long-standing well-known problem like FLT, is generally way harder than coming up with a simpler solution to an already-solved problem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Which proof would be &amp;quot;better&amp;quot;?&lt;/em&gt; The simpler one.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Who should get the credit for solving FLT?&lt;/em&gt; Andrew Wiles.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Does that mean there's no point in striving for a simpler proof?&lt;/em&gt; No -- when you come up with a simpler proof, you're generally doing so by developing a better understanding of the &amp;quot;core&amp;quot; of the problem. And that better understanding can helpful for solving other unsolved problems, which is beneficial regardless of whether you're thinking altruistically (what's in it for the field of mathematics as a whole) or selfishly (what's in it for you in particular).&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-08T00:42:59.063" LastActivityDate="2023-10-08T00:42:59.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26867" PostTypeId="2" ParentId="26850" CreationDate="2023-10-08T11:47:16.993" Score="-1" Body="&lt;p&gt;Easy. The issue is it's not a math problem. But one rooted in behavioral psychology and misappropriated expectations for consistent results.&#xA;Expecting an exacting mathematical Datum consistency, where the measure doesn't apply in teaching math subjects.&lt;/p&gt;&#xA;&lt;p&gt;Hint: Everyone's born as uniquely different as night and day. And it's obviously a mistake to expect a categorically (satisfactorily) consistent means and result thereof.&#xA;Ya' can't force feed, what won't work anyway.&lt;/p&gt;&#xA;&lt;p&gt;Remember now!&lt;/p&gt;&#xA;&lt;p&gt;New Math&#xA;Tom Lehrer&#xA;&amp;quot;So very simple, that only a child can do it.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="23007" LastActivityDate="2023-10-08T11:47:16.993" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26868" PostTypeId="2" ParentId="26850" CreationDate="2023-10-08T20:23:03.483" Score="0" Body="&lt;p&gt;The attitude of the students is that this is a hoop they need to jump through. They aren't motivated to understand the thing your trying to teach them. They just want to jump through the hoop. I had this experience with many kids I tried to tutor. They don't really care to understand the thing at all. This is true for the whole society. If they could dispense completely with learning it they would.&lt;/p&gt;&#xA;" OwnerUserId="19948" LastActivityDate="2023-10-08T20:23:03.483" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26869" PostTypeId="2" ParentId="26850" CreationDate="2023-10-08T23:58:23.930" Score="2" Body="&lt;p&gt;Here's my personal experience, as a former A(ish) grade student, now in his late 30's, doing math in my spare time:&lt;/p&gt;&#xA;&lt;p&gt;There was a big disconnect inside my head between the abstract symbols I was easily manipulating (functions &amp;amp; equations, basically symbols and numbers), and the physicality of a function's graph - I applied stuff like f'(x)=0 to find min&amp;amp;max, the dips and peaks of a graph, but just as formulas and algorithms 'given' to us, without a mental picture of rates of change and infinitesimals.&lt;/p&gt;&#xA;&lt;p&gt;Much, much later, I had trouble grasping why, for parametric curves, the calculated df/dg 'happened' to exactly be the 'physical' tangent to the sketched graph. What helped me bridge these two universes are the 'rise-over-run' mnemonic and understanding an x-y graph as an actual, precise, 2D 'physical' map, and not just some 'yet another' simplified representation of a function*.&lt;/p&gt;&#xA;&lt;p&gt;*In class, a graph was most of the times a 'rough sketch' of a function, some squiggle meant to convey a rough outline of what the function was actually doing 'internally', in it's own universe of symbols and numbers, where it was precise - and without giving it much thought, that's how I internalised it, as an imprecise tool for showing someone else a glimpse into what your function does, not as an accurate 'image' of the function - precise graphing tools like &lt;a href=&quot;https://www.desmos.com/calculator&quot; rel=&quot;nofollow noreferrer&quot;&gt;desmos&lt;/a&gt; where I can &amp;quot;see&amp;quot; that the graph matches the grid and changes in x &amp;amp; y are coordinates I can draw and see them moving precisely according to equations helped me overcome this 'rough sketch' perception.&lt;/p&gt;&#xA;" OwnerUserId="23013" LastActivityDate="2023-10-08T23:58:23.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26870" PostTypeId="2" ParentId="3858" CreationDate="2023-10-09T02:14:46.580" Score="-1" Body="&lt;p&gt;Math can be used in court as part of admissible evidence although the records of it haven't been favorable towards mathematics since math had been used to prove innocent people guilty&lt;/p&gt;&#xA;" OwnerUserId="10335" LastActivityDate="2023-10-09T02:14:46.580" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26872" PostTypeId="1" AcceptedAnswerId="27205" CreationDate="2023-10-09T03:06:45.570" Score="1" ViewCount="489" Body="&lt;p&gt;If the educator decides to handle the situation by declaring that the question is  beyond the scope of the course, then would it be fair to ensure that the course description and course syllabus include the name of the particular system of set theory that all theorems studied in the course are deduced from?&lt;/p&gt;&#xA;" OwnerUserId="12816" LastActivityDate="2023-11-21T16:48:19.200" Title="How should an educator answer a student who asks &quot;Can this theorem be deduced in other systems of set theory?&quot;" Tags="|terminology|language-use|axioms-foundations|history-of-math|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26873" PostTypeId="1" CreationDate="2023-10-09T03:16:52.523" Score="5" ViewCount="252" Body="&lt;p&gt;To get a deeper understanding of mathematics conceptual teaching and learning is supposed to be a much better approach than factual teaching and learning processes. Since the conceptual approach is much abstract it doesn't have a considerable impact on creating interest in the subject in average students. To fulfill this requirement we have the next level, that is contextual teaching and learning (CTL), where the priority is given to make sense of why we need to know about the concepts in mathematics. &lt;br/&gt;Mathematics has a history that is rich in astonishing breakthroughs, false starts, misattributions, confusions, and dead ends. Therefore I think, historical developments over the generations play an important role in the contextual teaching and learning process, when it comes to discovering meaningful relationships between abstract knowledge of conceptual learning and real-world applications in a new environment outside the classroom.&lt;br/&gt;&#xA;As experienced educators what would you suggest regarding to use history of mathematics as a tool in CTL?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-10-09T07:00:31.907" LastActivityDate="2023-10-19T17:28:38.173" Title="Role of history of mathematics in contextual teaching and learning" Tags="|mathematical-pedagogy|education-research|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26874" PostTypeId="2" ParentId="26873" CreationDate="2023-10-09T07:47:24.053" Score="4" Body="&lt;p&gt;My advice is to minimize this sort of thing and do it more with strong track kids than with average or weak. What's really interesting and motivating is mastering the material. Your priority needs to be on training the kids, not diversions.&lt;/p&gt;&#xA;&lt;p&gt;Of course as an instructor, you should have added knowledge of your topic and if a little drops now and then, can be ok. But it's a garnish. It's the sprig of parsley, not the schnitzel.&lt;/p&gt;&#xA;&lt;p&gt;If I'm learning gymnastics or wrestling, the history is interesting. But it is a 4th order priority versus learning moves. Math class is the same.&lt;/p&gt;&#xA;&lt;p&gt;What's intriguing to me is how weak many math instructors are on the general engineering curriculum and the use of math for the majority of students.  Much bigger gap than math history.  Just doing some orientation skimming of STEM texts in engineering, physics and chem would upgrade them. But even here, I will take a good trainer over someone who knows applications.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  I don't like the supposed to be formulation. According to who?  And are we sure?  After all, a lot of silly stuff like learning styles, inquiry learning, whole language has been pushed because it sounded modern an liberal, both meanings, even though testing showed no benefit or even detriment. So beware the supposed to groupthink.&lt;/p&gt;&#xA;" OwnerUserId="23015" LastActivityDate="2023-10-09T07:47:24.053" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26875" PostTypeId="5" CreationDate="2023-10-09T11:12:11.230" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2023-10-09T11:12:11.230" LastActivityDate="2023-10-09T11:12:11.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26876" PostTypeId="4" CreationDate="2023-10-09T11:12:11.230" Score="0" Body="Not to be confused with kindergarten and other education in the early cognitive development of the individual student, the logical foundations of mathematics includes, for example, rigorous proof that a given mathematical statement cannot be deduced from some given axioms.  The history of foundations included efforts to prove consistency of axioms with each other, but now it is recognized some assumption about consistency is usually needed." OwnerUserId="12816" LastEditorUserId="12816" LastEditDate="2023-10-24T06:08:14.897" LastActivityDate="2023-10-24T06:08:14.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26877" PostTypeId="2" ParentId="26850" CreationDate="2023-10-09T15:34:08.240" Score="0" Body="&lt;p&gt;Perhaps this &lt;em&gt;might&lt;/em&gt; help:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The derivative is (essentialy) the best linear approximation of a function (at a given point).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I never quite understood expressions such as &amp;quot;rate of change&amp;quot; or &amp;quot;slope of a line&amp;quot;, even though I never had any trouble with the mechanics of derivatives. Much later I was told the quote above, and everything was clear.&lt;/p&gt;&#xA;&lt;p&gt;Suppose you have a function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; (with appropriate hypothesis), and you want to approximate &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at point &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;. Then to approximate by a constant &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; the best choice is the value of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, i.e &lt;span class=&quot;math-container&quot;&gt;$b = f(x_0)$&lt;/span&gt;. To approximate by a linear function, i.e. some &lt;span class=&quot;math-container&quot;&gt;$g(x) = a \cdot (x - x_0) + b$&lt;/span&gt;, the best choice is &lt;span class=&quot;math-container&quot;&gt;$a = f'(x_0), b = f(x_0)$&lt;/span&gt;. The derivative is not the linear approximation, but the coefficient of the parameter; you already have the constant approximation &lt;span class=&quot;math-container&quot;&gt;$f(x_0)$&lt;/span&gt; so the derivative &lt;span class=&quot;math-container&quot;&gt;$f'(x_0)$&lt;/span&gt; is all the extra info you need.&lt;/p&gt;&#xA;&lt;p&gt;(&amp;quot;Best&amp;quot; means that the appropriately defined error goes to 0 when approaching &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;. You &lt;em&gt;might&lt;/em&gt; be able to motivate the error formula, or you may get by on intuition.)&lt;/p&gt;&#xA;&lt;p&gt;Applications:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is the function increasing at &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;? Look at the linear approximation: when is it increasing? When &lt;span class=&quot;math-container&quot;&gt;$a &amp;gt; 0$&lt;/span&gt; [plus technicalities]. :-)&lt;/li&gt;&#xA;&lt;li&gt;What about a (local) maximum? (Skipping details) either &lt;span class=&quot;math-container&quot;&gt;$a &amp;lt; 0$&lt;/span&gt; at the left edge, &lt;span class=&quot;math-container&quot;&gt;$a = 0$&lt;/span&gt; inside, or &lt;span class=&quot;math-container&quot;&gt;$a &amp;gt; 0$&lt;/span&gt; at the right edge.&lt;/li&gt;&#xA;&lt;li&gt;Instantaneous speed, and many other Physics-motivated examples.&lt;/li&gt;&#xA;&lt;li&gt;Solving equations! (approximately). You want &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f(x)=0$&lt;/span&gt; (e.g. to compute &lt;span class=&quot;math-container&quot;&gt;$\sqrt{u}$&lt;/span&gt; solve &lt;span class=&quot;math-container&quot;&gt;$f(x) = x^2-u$&lt;/span&gt;). Then whenever you have a good approximation &lt;span class=&quot;math-container&quot;&gt;$x_k$&lt;/span&gt;, solve instead &lt;span class=&quot;math-container&quot;&gt;$g(x) = f'(x_k) \cdot (x - x_k) + f(x_k) = 0$&lt;/span&gt; instead. You get &lt;span class=&quot;math-container&quot;&gt;$x = x_k - f(x_k)/f'(x_k)$&lt;/span&gt;, and that is a step in Newton-Raphson method. For &lt;span class=&quot;math-container&quot;&gt;$\sqrt{u}$&lt;/span&gt; it'd be &lt;span class=&quot;math-container&quot;&gt;$x_{k+1} = x_k - (x_k^2-u)/(2 \cdot x_k)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Extends to Taylor expansion formula.&lt;/li&gt;&#xA;&lt;li&gt;Generalises to diferentiation in multiple variables.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12521" LastActivityDate="2023-10-09T15:34:08.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26878" PostTypeId="1" CreationDate="2023-10-09T18:18:34.203" Score="24" ViewCount="16516" Body="&lt;p&gt;I have helped to &lt;a href=&quot;https://en.wikipedia.org/wiki/Teaching_assistant&quot; rel=&quot;nofollow noreferrer&quot;&gt;TA&lt;/a&gt; and taught several courses with mixtures of advanced undergraduate and early graduate students in engineering/&lt;a href=&quot;https://en.wikipedia.org/wiki/Science%2C_technology%2C_engineering%2C_and_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;STEM&lt;/a&gt;. These courses are the classics: signal processing, control, optimization and (more recently) machine learning. The mathematics used in these courses run the gamut from linear algebra to multivariable calculus to probability and so on.&lt;/p&gt;&#xA;&lt;p&gt;I am noticing that students coming into these courses (upper-year, shy of graduation, or beginning graduate/master students) basically have very poor ability in manipulating basic mathematics that they were taught and used over the years.&lt;/p&gt;&#xA;&lt;p&gt;Some common problems:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;No respect for dimensionality. Suppose &lt;span class=&quot;math-container&quot;&gt;$Av$&lt;/span&gt; is a column vector, students will equate it to a row vector, a matrix, or even a scalar. This is by far the most common one.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Wild manipulations, such as freely dividing a matrix &lt;span class=&quot;math-container&quot;&gt;$1/A$&lt;/span&gt; and basically treating a matrix as a scalar. Similarly, freely dividing vectors, raising vectors to powers, etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Incorrect beliefs, such as, &amp;quot;if a matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is symmetric, then it has an inverse.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some students are coming into advanced mathematics heavy courses without basically any background.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I am really at a loss as to why this is happening and what should be the correct remedy, as it seems that it is not easy to address this late into their study.&lt;/p&gt;&#xA;&lt;p&gt;Also there seems to be no improvement despite feedback and I'm constantly being surprised by some &amp;quot;new mathematics&amp;quot; that the students would invent in the homework or tests.&lt;/p&gt;&#xA;" OwnerUserId="3560" LastEditorUserId="851" LastEditDate="2023-10-14T20:51:01.110" LastActivityDate="2023-10-31T17:49:42.753" Title="What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations?" Tags="|undergraduate-education|mathematical-pedagogy|linear-algebra|graduate-education|" AnswerCount="14" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26879" PostTypeId="2" ParentId="26878" CreationDate="2023-10-09T19:40:03.320" Score="4" Body="&lt;p&gt;Here's a breakdown of the problems:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Some common problems:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;No respect to dimensionality. Suppose &lt;span class=&quot;math-container&quot;&gt;$Av$&lt;/span&gt; is a column vector, students will equate it to a row vector, a matrix, or even a scalar. This is by far the most common one.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One thing I would suggest is if this is a common problem, then why don't you show them why &lt;span class=&quot;math-container&quot;&gt;$Av$&lt;/span&gt; isn't a row vector, matrix, or scalar.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For example: Consider the column vector &lt;span class=&quot;math-container&quot;&gt;$Av$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$A=\begin{pmatrix}6\\-5\\4\end{pmatrix},v=2$&lt;/span&gt;. Then, part of your problem is done (they shouldn't possibly be able to equate it to a matrix of some sort, since this is a &lt;span class=&quot;math-container&quot;&gt;$1\times3$&lt;/span&gt; column vector.) Then, if they have taken some sort of Linear Algebra class, then it shouldn't be hard to get them back on track from there with a quick review on how column vectors work.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Wild manipulations, such as freely dividing a matrix &lt;span class=&quot;math-container&quot;&gt;$1/A$&lt;/span&gt; and basically treating a matrix as a scalar. Similarly, freely dividing vectors, raising vectors to powers, etc.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Again, it shouldn't be that hard to review with them why they can't freely divide any matrix &lt;span class=&quot;math-container&quot;&gt;$1/A$&lt;/span&gt;. An easy way to review is by showing that if &lt;span class=&quot;math-container&quot;&gt;$\operatorname{det}(A)=0,A^{-1}=1/A$&lt;/span&gt; is indeterminate. For example, if we take the &lt;span class=&quot;math-container&quot;&gt;$2\times2$&lt;/span&gt; matrix&lt;span class=&quot;math-container&quot;&gt;$$A=\begin{bmatrix}2&amp;amp;3\\4&amp;amp;6\end{bmatrix}$$&lt;/span&gt;since we know &lt;span class=&quot;math-container&quot;&gt;$\operatorname{det}(B)=ad-bc$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$2\times2$&lt;/span&gt; matrix &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, and then inverse of any matrix &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is defined as&lt;span class=&quot;math-container&quot;&gt;$$\dfrac1{\operatorname{det}(B)}B$$&lt;/span&gt;we have that &lt;span class=&quot;math-container&quot;&gt;$A^{-1}=1/A$&lt;/span&gt; does not exist since we have&lt;span class=&quot;math-container&quot;&gt;$$\operatorname{det}(A)=ad-bc=2\cdot6-3\cdot4=12-12=0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Incorrect beliefs, such as &amp;quot;if a matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is symmetric, then it has an inverse.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Again, you can review with them that if the determinant of a matrix is equal to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, the matrix does not have an inverse.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Some students are coming into advanced mathematics courses with basically no background.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I honestly feel like this is way above my jurisdiction to provide an opinion on. You should probably discuss this with the school board if you feel like there should be more restrictions on taking these classes that lean heavily on advanced mathematics, however that is only my opinion.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Also there seems to be no improvement despite feedback and I'm constantly surprised by some &amp;quot;new mathematics&amp;quot; that the students would invent in the homework or tests.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I feel like this is definitely something you should expect if you have students with no background in mathematics coming into class that lean heavily on, of course, courses that rely heavily on advanced mathematics. I'll even admit, I've even done that before on M.SE when, of course, I was trying to do something that I didn't know what to do. (An example of this would be on &lt;a href=&quot;https://math.stackexchange.com/questions/4765741/how-to-solve-the-floor-function-equation-lfloor2x-rfloor-lfloor3x-rfloor-lfl&quot;&gt;this&lt;/a&gt; question of mine) I would recommend checking in with each student individually (if you could do that feasibly) and ask about what they might have meant by the &amp;quot;new mathematics&amp;quot;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So, what's the correct remedy? Most likely just reviewing with your students the basic mathematics for each section that they will need to know throughout the course to make sure you really drill it in their head. Otherwise, you're probably going to be having this problem. If any of the students does some &amp;quot;new mathematics&amp;quot; that you don't understand, then try to chat with them privately at some point to ask about what they meant by their &amp;quot;new mathematics&amp;quot; that they put on their homework/test.&lt;/p&gt;&#xA;" OwnerUserId="22983" LastActivityDate="2023-10-09T19:40:03.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26880" PostTypeId="2" ParentId="26878" CreationDate="2023-10-09T20:37:11.260" Score="21" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I am really at a loss as to why this is happening and what should be the correct remedy, as it seems that it is not easy to address this late into their study.&lt;/p&gt;&#xA;&lt;p&gt;Also there seems to be no improvement despite feedback...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Why is this happening?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's likely the result of a lack of learning in prerequisite courses. It sounds like students were somehow able to pass the prerequisite courses without demonstrating sufficient mastery of the material. The usual culprits are grade inflation and/or lowering of standards.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What should be the correct remedy?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;correct&amp;quot; remedy is to hold the students accountable for learning the material in your course, including the prerequisite material that they are missing.&lt;/p&gt;&#xA;&lt;p&gt;This requires the instructor to put forth a ton of effort supporting students through remedial assignments/assessments and help sessions while simultaneously &amp;quot;holding the line&amp;quot; in a hardcore way (i.e. you have to be a bit of a hard-ass). And of course it requires a ton of effort from students because they have to put in the extra work to pay off their &amp;quot;learning debt,&amp;quot; which is usually a rude awakening for them.&lt;/p&gt;&#xA;&lt;p&gt;But what usually ends up happening instead is everyone turns a blind eye and pushes the problem further down the line: the instructor just gives the usual lectures/assignments, curves (or otherwise inflates) the grades, students go along with it, and the problem is left for their next instructor to deal with (or not deal with).&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-10-09T20:37:11.260" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26881" PostTypeId="2" ParentId="26878" CreationDate="2023-10-09T21:53:25.033" Score="6" Body="&lt;p&gt;In his 2005 article &lt;a href=&quot;https://www.jstor.org/stable/30034943&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Reconceptualizing Procedural Knowledge,&amp;quot;&lt;/a&gt; Jon Star argued that there is such a thing as &amp;quot;deep procedural knowledge,&amp;quot; which he describes as follows.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Deep procedural knowledge would be knowledge of procedures that is associated with comprehension, flexibility, and critical judgment and that is distinct from (but possibly related to) knowledge of concepts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would argue that mistakenly thinking  &lt;span class=&quot;math-container&quot;&gt;$Av$&lt;/span&gt; is something other&#xA;than a column vector demonstrates a lack of deep procedural knowledge of matrix multiplication. Similarly, mistakenly &amp;quot;dividing by &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;&amp;quot; demonstrates a lack of deep procedural knowledge of matrix inversion and solving matrix equations. Students may have seen the ideas of &amp;quot;linear algebra,&amp;quot; but they haven't deeply learned the procedures of &amp;quot;matrix algebra.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;What's the remedy? In addition to Justin Skycak's suggestion to hold students accountable for learning prerequisite material and to support them with remedial practice, I would recommend that, in your instruction, you work through examples that feature the kinds of procedures that students struggle with. As you do these examples, you should describe your thought processes out loud. This is a kind of &lt;a href=&quot;https://en.wikipedia.org/wiki/Cognitive_apprenticeship#:%7E:text=procedural&quot; rel=&quot;nofollow noreferrer&quot;&gt;cognitive apprenticeship.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In addition to any extra practice resources you decide to provide for students, I suggest incorporating procedural practice on homework—a few short exercises at the start of the assignment, for example. Exercises help students develop and maintain fluency, and you can also ask questions that push them towards deeper knowledge of procedures. You might ask students, &amp;quot;Given the matrix equation &lt;span class=&quot;math-container&quot;&gt;$AB=C$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is a &lt;span class=&quot;math-container&quot;&gt;$3 \times 4$&lt;/span&gt; matrix and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is a &lt;span class=&quot;math-container&quot;&gt;$3 \times 1$&lt;/span&gt; matrix, what do the dimensions of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; have to be?&amp;quot; Or &amp;quot;Does the matrix equation &lt;span class=&quot;math-container&quot;&gt;$Ax=b$&lt;/span&gt; always have a solution?&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-10-12T13:40:20.860" LastActivityDate="2023-10-12T13:40:20.860" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26882" PostTypeId="2" ParentId="26878" CreationDate="2023-10-09T23:08:46.993" Score="2" Body="&lt;p&gt;Your first three examples are linear algebra.  Example 4 might or might not be, hard to say...since it is not an example but a vast generalization.&lt;/p&gt;&#xA;&lt;p&gt;Probably one issue is that most ee students don't really need or do much LA. At least not to the extent that they do normal algebra.  And not to the extent that mathy types think they do.  Many EEs don't get a full semester, just a few weeks within an engine math survey class.&lt;/p&gt;&#xA;&lt;p&gt;CS probably do get LA course but are less bright and less mathematical in general than EEs.   And then you're talking MS students, not Ph.D.  add onto that the hype around machine learning.  It's zero surprise that if you have hard-core LA multiple regression stuff that they will choke.&lt;/p&gt;&#xA;&lt;p&gt;In terms of the solution?  I would say either weaken the curriculums.  Don't scoff, look at stats or even physics which can be taught with lower math demands versions and still significant useful learning. Or...keep the MS as is and just let the weeding sort itself out. I doubt remediation, at that level is practical on a population basis, i.e. in general. Nor is fix the pipeline a practical option for this dealing with the output.&lt;/p&gt;&#xA;" OwnerUserId="23019" LastEditorUserId="23019" LastEditDate="2023-10-09T23:13:06.877" LastActivityDate="2023-10-09T23:13:06.877" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26883" PostTypeId="2" ParentId="26850" CreationDate="2023-10-09T23:09:14.767" Score="1" Body="&lt;p&gt;Maybe a physical demonstration would help, e.g. using a thin wooden disk (or dinner plate) and a ruler.  With the ruler in the plane of the disk, touch it to various points on the disk noting (1) how the ruler touches the disk at a single point, and (2) how the &amp;quot;slope&amp;quot; of the ruler changes as you move the point of contact around the disk. Maybe try different shapes, some with corners. What happens at the corners and on the sides of, say, a triangle?&lt;/p&gt;&#xA;&lt;p&gt;Or some animation. From Wikipedia: &lt;a href=&quot;https://www.geogebra.org/m/sVCRDDmA&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tangent Line Animation&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-10-10T14:44:48.060" LastActivityDate="2023-10-10T14:44:48.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26884" PostTypeId="2" ParentId="689" CreationDate="2023-10-10T00:57:14.940" Score="1" Body="&lt;p&gt;Inspired by &lt;a href=&quot;https://matheducators.stackexchange.com/a/791/23021&quot;&gt;this answer&lt;/a&gt; and expanding on &lt;a href=&quot;https://matheducators.stackexchange.com/a/788/23021&quot;&gt;this one&lt;/a&gt;...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How many numbers are there? &lt;em&gt;Infinity. (Write the first few naturals)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Can you tell me what an even number is? What are the first few? &lt;em&gt;(Write them down)&lt;/em&gt; How many even numbers are there? &lt;em&gt;Infinity!&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Are there more numbers than there are even numbers? &lt;em&gt;Yes, hmm, they're both infinity&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;But look, if I write the even ones under the ordinary numbers like this...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So we've proved that there are the same amount of even numbers as there are of all numbers. And we've also introduced the idea of infinite sets of numbers. So now we can say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What other sequences of numbers can you think of that are infinitely long like this? &lt;em&gt;All the numbers divisible by 3. All the numbers divisible by 4.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Good. What other kinds? &lt;em&gt;All the square numbers. All the prime numbers. Just 0,1,0,1 repeating forever.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;How many different infinite sequences of numbers do you think there are? &lt;em&gt;Infinity.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The same infinity as the number of even numbers was? &lt;em&gt;Sure, probably.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;But look, if I write all the sequences you can think of alongside the ordinary numbers like this... &lt;em&gt;(Writes Cantor's diagonal argument)&lt;/em&gt; Now I'm going to look down the diagonal and write a new sequence of integers by adding 1 to each... This sequence doesn't appear anywhere in the matching, does it? &lt;em&gt;No.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;And I can do this no matter what matching-up you try to do, right? &lt;em&gt;Yeah, I guess.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;So you'll never be able to match up all the sequences of integers with all the integers. &lt;em&gt;No, there will always be at least one I didn't match up.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;So &lt;em&gt;(this is the leap)&lt;/em&gt; we say that the number of integer sequences is &lt;em&gt;bigger&lt;/em&gt; than the number of integers. It's a &lt;em&gt;bigger&lt;/em&gt; kind of infinity than the other kind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="23021" LastActivityDate="2023-10-10T00:57:14.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26885" PostTypeId="2" ParentId="26878" CreationDate="2023-10-10T01:53:29.580" Score="11" Body="&lt;p&gt;I deal with a dash of this phenomenon in the context of teaching an &amp;quot;advanced programing techniques&amp;quot; course (i.e., 2nd semester programming) in a community-college CS program. There's frequently pretty poor preparation in the first semester, and I'm considered to have rigorous standards for our environment. Compare the current question to mine from four years ago: &lt;a href=&quot;https://academia.stackexchange.com/questions/131849/how-to-deal-with-radically-different-prerequisite-preparation&quot;&gt;How to deal with radically different prerequisite preparation?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here's a few things I do to try to help students get up to speed. They're not all usable in your situation, and none are a silver bullet, but together they give some incremental improvement.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Give a brief overview of expected prerequisite knowledge in the first week (no more than that), referencing where to go for a deeper dive.&lt;/li&gt;&#xA;&lt;li&gt;Provide any summary material from prior courses (e.g., formula card or programming language quick reference card).&lt;/li&gt;&#xA;&lt;li&gt;Make digital lecture slides from prerequisite courses available for optional review.&lt;/li&gt;&#xA;&lt;li&gt;In early assignment specifications, explicitly call out book sections from prior courses they should review (e.g., &amp;quot;For proper input validation, you may wish to review Gaddis C++ Sec. 5.3&amp;quot;).&lt;/li&gt;&#xA;&lt;li&gt;Find digital tools that can double-check work in some fashion (e.g., I wrote a basic C++ style checker and require students to use it, &lt;a href=&quot;https://github.com/danielrcollins1/StyleScanner&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;li&gt;Uphold standards in grading, and give clear feedback to students about the parts at which they're not succeeding (again, referencing prior-course text sections as appropriate).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Full disclosure: One advantage I have is that the 1st-semester and my 2nd-semester course use the same textbook, and I've taught the prior course myself, so it's completely straightforward to expect students have access to the textbook and I can reference earlier sections at will. Maybe this convenience sample points the way to some things that help.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-10-10T01:53:29.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26886" PostTypeId="5" CreationDate="2023-10-10T04:36:57.083" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2023-10-10T04:36:57.083" LastActivityDate="2023-10-10T04:36:57.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26887" PostTypeId="4" CreationDate="2023-10-10T04:36:57.083" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2023-10-10T04:36:57.083" LastActivityDate="2023-10-10T04:36:57.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26888" PostTypeId="2" ParentId="26878" CreationDate="2023-10-10T05:42:38.970" Score="14" Body="&lt;p&gt;I'd like to offer a bit of a frame challenge here. Whenever I made these sorts of mistakes during my mathematics education it was normally because I didn't understand what the symbols in the more advanced course were referring to. This often occurred when switching between courses which used different notations for the same material.&lt;/p&gt;&#xA;&lt;p&gt;In all fields there is an implicit understanding of what type of objects are referred to by different classes of symbols. There is also a set of conventions about what symbols and operations do not need to be explicitly written down. Both of these seem obvious to experienced practitioners, but are really confusing to newcomers.&lt;/p&gt;&#xA;&lt;p&gt;I'd go over all the symbols you use, reiterate what type of objects each one refers to and let your students know how you determined that.&lt;/p&gt;&#xA;" OwnerUserId="23024" LastEditorUserId="23024" LastEditDate="2023-10-10T19:53:39.823" LastActivityDate="2023-10-10T19:53:39.823" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26889" PostTypeId="2" ParentId="26519" CreationDate="2023-10-10T05:43:42.517" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;I'm doing this as my passion without receiving any kind of&#xA;encouragement from local bodies [...]&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That is evidence that you have passion for mathematics.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is there a system to encourage mathematics teachers to be passionate?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;External rewards offered by bureaucratic organizations to mathematics teachers would create a dilemma: are the teachers motivated primarily by the external rewards, or primarily by mathematics itself?&lt;/p&gt;&#xA;&lt;p&gt;Andrew Wiles is an example of somebody who &lt;em&gt;&lt;strong&gt;is truly passionate about mathematics&lt;/strong&gt;&lt;/em&gt;.  However, among those who are passionate about mathematics, Andrew Wiles is unusual.  Andrew Wiles has been both unusually ambitious and unusually successful.&lt;/p&gt;&#xA;&lt;p&gt;From our retrospective point of view, we need more people who will take the first step along a similar path.  However, Andrew Wiles didn't know what the outcome would be when he took the first step.  If he had openly disclosed what he was trying to do during the first few months or years of his seven-year-journey to reach a proof of FLT, then he may have appeared to be a sad example of somebody caught up in a quixotic and doomed-to-fail undertaking.&lt;/p&gt;&#xA;&lt;p&gt;To avoid the trap known as &amp;quot;survivorship bias&amp;quot;, we might think of paying attention to people who tried and failed to resolve Fermat's Last Theorem.  However, they would &lt;em&gt;&lt;strong&gt;also&lt;/strong&gt;&lt;/em&gt; not be typical or representative examples of people who are passionate about math.  They were too ambitious to be typical examples.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In December 2021, I created my own combinatorics challenge and sent it&#xA;to many of experienced and qualified university lecturers but none of&#xA;them could sense the importance of the issue. Finally an overseas&#xA;professor told the problem is good enough for a university research&#xA;project. That encouraged me a lot, and after working with it for&#xA;couple of months I was able to solve the problem and generate 12 new&#xA;sequences which are published on OEIS. After that I have created&#xA;another 4 new problems related to grid paths and generated yet another&#xA;4 completely new sequences which are already published on OEIS.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If there were a venue where you could share your story in detail, then perhaps it could be fully understood by a mass audience. If mathematical passion is contagious, then your story could increase passion for mathematics.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, from the point of view of a mass audience, the mathematical thinking of Andrew Wiles is a black box.  People are impressed by the facts of his life, but naturally they aren't likely to acquire passion for mathematics by hearing vague descriptions of concepts beyond their understanding.&lt;/p&gt;&#xA;&lt;p&gt;Passion for mathematics isn't a byproduct of hearsay, or superficial visual inspection of math.  It is a byproduct of direct tasting, chewing on, and absorption of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="12816" LastActivityDate="2023-10-10T05:43:42.517" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26890" PostTypeId="2" ParentId="7336" CreationDate="2023-10-10T06:45:59.923" Score="2" Body="&lt;p&gt;Just as you crossposted from MSE, let me crosspost &lt;a href=&quot;https://math.stackexchange.com/users/118193/heropup&quot;&gt;heropup&lt;/a&gt;'s &lt;a href=&quot;https://math.stackexchange.com/a/4724011&quot;&gt;tremendous answer&lt;/a&gt;, which I improved.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Imagine the following Socratic dialogue.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  What does  &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{x\to a} f(x) = L$&lt;/span&gt; mean?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  It means that the limit of the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approaches &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, equals &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  Yes, but what does that actually MEAN?  What are we saying about the behavior of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  [Pauses to think.]  I guess what we are saying is that for values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &amp;quot;close to&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; becomes &amp;quot;close to&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  OK.  So how are you defining the concept of &amp;quot;close to?&amp;quot;  In particular, how does math quantify the notion of &amp;quot;closeness&amp;quot;?  Does &amp;quot;close to&amp;quot; mean &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  No — well — maybe sometimes!  Of course, if &lt;span class=&quot;math-container&quot;&gt;$f(a)$&lt;/span&gt; is well-defined, then we just have &lt;span class=&quot;math-container&quot;&gt;$f(a) = L$&lt;/span&gt; — but this is plain vanilla,  and trivial.  The whole point of limits is to describe the function's behavior around the point &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt;, even when &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; ISN'T defined at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  Right, but you didn't answer my question. So how would you mathematically define  &amp;quot;closeness&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  [Long pause.]  I'm not sure.  Well, hold on.  Let me try a geometric argument.  When a number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &amp;quot;close to&amp;quot; another number &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, we are really talking about the &lt;em&gt;distance&lt;/em&gt; between these numbers being small.  Like &lt;span class=&quot;math-container&quot;&gt;$2.00001$&lt;/span&gt; is &amp;quot;close to&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, because the difference is &lt;span class=&quot;math-container&quot;&gt;$0.00001$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  But that difference — which you call &amp;quot;distance&amp;quot; — isn't necessarily &amp;quot;small&amp;quot; in and of itself, is it?  After all, isn't &lt;span class=&quot;math-container&quot;&gt;$10^{-10^{100}}$&lt;/span&gt; much smaller than &lt;span class=&quot;math-container&quot;&gt;$10^{-5}$&lt;/span&gt;?  &amp;quot;Small&amp;quot; is relative.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  [With irritation] Yeah, but you know what I mean!  If the difference is small enough, then the limit exists!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  [Chuckles] Yes, I see what you're getting at! But so far, all you've been doing is choosing different vocabulary to describe the same concept.  What is &amp;quot;distance&amp;quot;?  What is &amp;quot;small enough?&amp;quot;  We are mathematicians — how can improve this inaccurate and imprecise words?  Take your time to think about this.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  [Sighs] What I was doing before, I was calculating a difference between &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, and calling it &amp;quot;small&amp;quot; if it looked like a small number.  But what matters ISN'T the signed difference, but the &lt;em&gt;absolute&lt;/em&gt; difference &lt;span class=&quot;math-container&quot;&gt;$|x - a|$&lt;/span&gt;. Since (as you put it) &amp;quot;small is relative,&amp;quot; let's instead use a variable, say &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt; (to abbreviate &amp;quot;difference&amp;quot;), to represent some bound... [trails off]&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  Go on...&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  All right.  So if &lt;span class=&quot;math-container&quot;&gt;$|x-a| &amp;lt; δ$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &amp;quot;close to&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.  We choose some number &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt;, in some way that quantifies the extent of closeness.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  OK.  Is &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt; allowed to be zero?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  Oh, of course not, no!  I forgot.  No, we need &lt;span class=&quot;math-container&quot;&gt;$\color{red}{0 &amp;lt; |x - a| &amp;lt; δ}$&lt;/span&gt;.  Then &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &amp;quot;delta-close&amp;quot; to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, or in a &amp;quot;delta-&lt;em&gt;neighborhood&lt;/em&gt;&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  All right.  Now how are you going to tie &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt; to the behavior of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  [Exasperated] Yes, yes, I'm getting to that part. As I  said, the limit is something where if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &amp;quot;close to&amp;quot; to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is &amp;quot;close to&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;.  Obviously, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; can have a different extent of &amp;quot;closeness&amp;quot; to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;, as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; does to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, if &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt;, then when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is within &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt; units of (for example) &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is only bounded within &lt;span class=&quot;math-container&quot;&gt;$2δ$&lt;/span&gt; units of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, since &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; |x-1| &amp;lt; δ$&lt;/span&gt; implies that &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; |2x - 2| = |f(x) - 2| &amp;lt; 2δ$&lt;/span&gt;.  But functions can be arbitrarily (although not infinitely) steep.  How can we quantify the relationship between the closeness of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, as this closeness impacts the closeness of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  You actually touched on it, when you said that functions can be arbitrarily but not infinitely steep.  Stated informally another way, it means that the function's value can change very rapidly — in fact, as rapidly as you please — but only finitely so, for some fixed change in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  So if you wanted to guarantee shrinking the difference between &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; &lt;em&gt;as small as you please&lt;/em&gt;, while not necessarily zero, how would you do it?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  [Long pause.]  I need help.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  So far, you've been thinking about using (as you put it) &amp;quot;delta-closeness&amp;quot; to force &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; to be &amp;quot;close to&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;.  But what if you turned it around and instead said, I'll force &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; to be as close as I please to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;? Then what does this closeness of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; say about how close &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;?  That way, you are &lt;em&gt;guaranteeing&lt;/em&gt; that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; becomes close to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;, but the cost of that guarantee is that we need to guarantee that...&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  [Interrupts] Oh, oh!  I get it now!  Yes.  What we need to say is that for a given amount of &amp;quot;closeness&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;, a &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt;-neighborhood around &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; where (if you pick any &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in that neighborhood) will guarantee &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; to be &amp;quot;close enough&amp;quot; to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; — that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; will be within that given amount of closeness.  In other words, we pick some &amp;quot;tolerance&amp;quot; or &lt;em&gt;error&lt;/em&gt; bound between &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and the limit &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; that is our criterion for &amp;quot;close enough.&amp;quot;  And for that closeness, some set of corresponding &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-values close to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; will guarantee that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; meets the closeness criterion.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  Good, good.  But how do we formalize this?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  Well, we need another variable to describe the extent of closeness between &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;...let's use &lt;span class=&quot;math-container&quot;&gt;$ε$&lt;/span&gt;, to abbreviate &amp;quot;error.&amp;quot;  As we did before, we use the absolute difference &lt;span class=&quot;math-container&quot;&gt;$|f(x) - L|$&lt;/span&gt; to describe the &amp;quot;distance&amp;quot; between &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;.  So our criterion has to be &lt;span class=&quot;math-container&quot;&gt;$\color{lightseagreen}{|f(x) - L| &amp;lt; ε}$&lt;/span&gt;. This time, we get to pick &lt;span class=&quot;math-container&quot;&gt;$ε$&lt;/span&gt; freely, because it represents how much error we will tolerate between the function's value and its limit. We must be able to choose this tolerance to be arbitrarily small, but not zero.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  [Looks on silently, smiling]&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  So let's define a procedure.  Pick some &lt;span class=&quot;math-container&quot;&gt;$ε &amp;gt; 0$&lt;/span&gt;.  Then whenever &lt;span class=&quot;math-container&quot;&gt;$\color{red}{0 &amp;lt; |x - a| &amp;lt; δ}$&lt;/span&gt; (in other words, for every &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in a &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-neighborhood of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;),  then &lt;span class=&quot;math-container&quot;&gt;$\color{lightseagreen}{|f(x) - L| &amp;lt; ε}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But I feel like something is missing, because there might not be such a &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt;.  For example, if &lt;span class=&quot;math-container&quot;&gt;$$f(x) = \begin{cases}-1, &amp;amp; x &amp;lt; 0 \\ 1, &amp;amp; x &amp;gt; 0 \end{cases}$$&lt;/span&gt;, then if I pick &lt;span class=&quot;math-container&quot;&gt;$ε = 1/2$&lt;/span&gt;, the &amp;quot;jump&amp;quot; in &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x = 0$&lt;/span&gt; has size &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.  So no matter how small I make the &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt;-neighborhood around &lt;span class=&quot;math-container&quot;&gt;$a = 0$&lt;/span&gt;, this neighborhood will always contain &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-values that are negative, as well as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-values that are positive, which means any such &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-neighborhood will have points where the function has values &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;.  It would be impossible to pick a limit &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; that is simultaneously within &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; unit of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;, let alone simultaneously arbitrarily close to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  Correct. Good job on finding a function that lacks such a &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt;. But why does this function lack such a &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  I don't get what you mean.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt;  Remember how we were talking about guaranteeing the (absolute) difference between &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; to be shrunk as small as you please?  What consequence does this guarantee have on the &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt;-neighborhood?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt;  Well, there has to be some relationship.  As our error tolerance decreases, fewer &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-values around &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; will satisfy that tolerance, right?  So &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt; must depend in some way on our choice of &lt;span class=&quot;math-container&quot;&gt;$ε$&lt;/span&gt;.  Well, except in trivial cases like if &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is a constant, then any &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt; works.  But the point is the EXISTENCE of a &lt;span class=&quot;math-container&quot;&gt;$δ$&lt;/span&gt;.  It doesn't have to be the largest, or even unique.  We merely have to be able to find a sufficiently &amp;quot;small&amp;quot; neighborhood, for which all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-values in that neighborhood around &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; will have function values &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, within the error tolerance we specified to &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; Right.  So if you were to put all of this together, how would you define the concept of a limit?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; I'd say that &lt;span class=&quot;math-container&quot;&gt;$$\lim_{x \to a} f(x) = L$$&lt;/span&gt; if, for any &lt;span class=&quot;math-container&quot;&gt;$\epsilon &amp;gt; 0$&lt;/span&gt;, there exists some &lt;span class=&quot;math-container&quot;&gt;$δ &amp;gt; 0$&lt;/span&gt; such that for every &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; satisfying &lt;span class=&quot;math-container&quot;&gt;$\color{red}{0 &amp;lt; |x - a| &amp;lt; δ}$&lt;/span&gt;, one also has &lt;span class=&quot;math-container&quot;&gt;$\color{lightseagreen}{|f(x) - L| &amp;lt; ε}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="23025" LastEditorUserId="23025" LastEditDate="2023-10-10T06:49:45.497" LastActivityDate="2023-10-10T06:49:45.497" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26891" PostTypeId="2" ParentId="26878" CreationDate="2023-10-10T14:34:54.293" Score="4" Body="&lt;p&gt;The problem is with the prerequisite courses, so if you want to fix it, start by talking to the people who teach the prerequisite courses. If you're not comfortable with that, raise the issue with a trusted colleague who has more seniority and/or is on an appropriate committee (e.g. undergraduate education committee).&lt;/p&gt;&#xA;" OwnerUserId="4939" LastActivityDate="2023-10-10T14:34:54.293" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26892" PostTypeId="1" CreationDate="2023-10-10T14:34:56.243" Score="13" ViewCount="5039" Body="&lt;p&gt;This is more specific than &lt;a href=&quot;https://math.stackexchange.com/questions/365770/what-is-a-simple-example-of-a-limit-in-the-real-world&quot;&gt;this similar question&lt;/a&gt; on math.SE, since I'm not satisfied with the answers there.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&#xA;Can you provide an interesting, natural and simple example of some physical/geometrical etc. system, which has two continuous quantities &lt;span class=&quot;math-container&quot;&gt;$y,x$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt;, such that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is not defined at a finite point &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt;, but we would still find it interesting to ask for &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to a}y$&lt;/span&gt;? Ideally the example should be such that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is an elementary function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; whose expression can be derived from the description of the system and &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to a}f(x)$&lt;/span&gt; is finite.&lt;/p&gt;&#xA;&lt;p&gt;My target audience are beginning engineering students, so &amp;quot;interesting, natural and simple&amp;quot; for them.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt;&#xA;I teach an introductory calculus course to engineers. The main focus is not limits, but we have a short section on limits of functions (not sequences) before defining derivatives. We don't do &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; but want students to be aquatinted with the notation and idea of &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to a} f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Most books &amp;amp; resources I've found do a good job of illustrating the notion graphically, numerically and of course with formula examples like &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$&lt;/span&gt;, where one can do some clever rearranging to see what the limit is. But I haven't seen any motivating example that shows how we might arrive at an expression like &lt;span class=&quot;math-container&quot;&gt;$\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$&lt;/span&gt; from a real life application.&lt;/p&gt;&#xA;&lt;p&gt;Of course I could (and currently do) tell them: we'll later see examples where such things appear, like when dealing with derivatives, integrals etc., but I'd like to give them some applied example right when the notion is introduced, without having to look ahead at derivatives. So please no examples that say &amp;quot;derivatives&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2023-10-18T13:47:20.993" Title="&quot;Real life&quot; examples of limits of functions at finite points" Tags="|calculus|examples|limits|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26893" PostTypeId="2" ParentId="26892" CreationDate="2023-10-10T15:21:15.900" Score="32" Body="&lt;p&gt;First thing that comes to my mind is the limit &lt;span class=&quot;math-container&quot;&gt;$$\lim\limits_{x \to 0} \dfrac{\sin x}{x} = 1.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This limit justifies the &lt;em&gt;small-angle approximation&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\sin \theta \approx \theta$&lt;/span&gt; (for &lt;span class=&quot;math-container&quot;&gt;$\theta \approx 0$&lt;/span&gt;) that is widely leveraged in engineering/physics classes to tame unwieldy equations.&lt;/p&gt;&#xA;&lt;p&gt;(Often, the existence of a sine term prevents the existence of a closed-form solution, but if you replace that sine term with a linear term, then you can get a closed-form first-order approximation of the solution that is sufficiently accurate for practical purposes.)&lt;/p&gt;&#xA;&lt;p&gt;A concrete and accessible example would be calculating the period of a pendulum: &lt;a href=&quot;http://www.physicslab.org/Document.aspx?doctype=3&amp;amp;filename=OscillatoryMotion_PendulumSHM.xml&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.physicslab.org/Document.aspx?doctype=3&amp;amp;filename=OscillatoryMotion_PendulumSHM.xml&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Note that if you want to avoid the need for derivatives and differential equations, then you can ask the question &lt;em&gt;&amp;quot;can you model a pendulum like a spring&amp;quot;&lt;/em&gt; instead of &lt;em&gt;&amp;quot;what is the period of a pendulum&amp;quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Can you model a pendulum like a spring?&lt;/p&gt;&#xA;&lt;p&gt;Springs follow Hooke's law: the restoring force is proportional to the displacement. But in a pendulum, the restoring force is &lt;span class=&quot;math-container&quot;&gt;$mg \sin \theta.$&lt;/span&gt; So, you can't model it like a spring. Right?&lt;/p&gt;&#xA;&lt;p&gt;... actually, for small angles, &lt;span class=&quot;math-container&quot;&gt;$\sin \theta \approx \theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\textrm{displacement} \approx L\theta,$&lt;/span&gt; so the restoring force is &lt;span class=&quot;math-container&quot;&gt;$\approx mg/L \cdot \textrm{displacement}.$&lt;/span&gt; So, you CAN model it like a spring when the angle is small.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-13T14:16:11.610" LastActivityDate="2023-10-13T14:16:11.610" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26894" PostTypeId="2" ParentId="26892" CreationDate="2023-10-10T15:32:46.113" Score="4" Body="&lt;p&gt;Here are some examples from a &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/0020739X.2014.941427&quot; rel=&quot;nofollow noreferrer&quot;&gt;2015 paper by Steven Jones&lt;/a&gt; on students' reasoning about limits.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you pluck a stringed instrument (like a guitar or violin), the pitch of the note (&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;) depends on the length of the string (&lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;), the tension of the string (&lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;), and the thickness of the string (&lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt;). (A picture of a guitar is provided to the students.) Here is the formula that relates pitch, length, tension, and thickness: &lt;span class=&quot;math-container&quot;&gt;$f = \frac{1}{2L} \sqrt{\frac{T}{\rho}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Determine the following limits. What does each one tell us? &lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\lim_{L \to \infty} f$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\lim_{L \to 0^+} f$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\lim_{T \to \infty} f$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\lim_{T \to 0^+} f$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\lim_{\rho \to \infty} f$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\lim_{\rho \to 0^+} f$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In physics, the theory of relativity says an object's mass, &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, is related to the object's velocity, &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, especially as speeds close to the speed of light, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;. The following equation relates these quantities. (&lt;span class=&quot;math-container&quot;&gt;$m_0$&lt;/span&gt; is the 'resting mass' of the object.) &lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$m = \frac{m_0}{1-v^2/c^2}$&lt;/span&gt; &lt;br&gt;&#xA;Work out &lt;span class=&quot;math-container&quot;&gt;$\lim_{v \to c^-} m$&lt;/span&gt;. How did you work out your answer? What does this limit tell you about mass?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately, with the exception of &lt;span class=&quot;math-container&quot;&gt;$\lim_{T \to 0^+} f$&lt;/span&gt;, all of these are either infinite limits or limits at infinity. I think the main reason that it's difficult to find interesting, physical, non-rate-related examples of finite limits at finite points is that physical quantities are, in general, continuous. This means that, unless we are approaching a &lt;em&gt;limiting case&lt;/em&gt;, there's nothing stopping us from substituting values into a formula.&lt;/p&gt;&#xA;&lt;p&gt;Here's another example that I'm adapting from Serway/Jewett's &lt;em&gt;Physics for Scientists and Engineers&lt;/em&gt;. It's rate-related, but the derivative is hidden in the formula already.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An object at a distance &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; from a spherical mirror is moving towards the mirror at speed &lt;span class=&quot;math-container&quot;&gt;$v_p$&lt;/span&gt;. The focal length of the mirror is &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. The velocity of the mirror image of the object is given by &lt;span class=&quot;math-container&quot;&gt;$v_q = -\frac{f^2 v_p}{(p-f)^2}$&lt;/span&gt;. What is &lt;span class=&quot;math-container&quot;&gt;$\lim_{p \to 0^+} v_q$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-10-18T09:51:48.163" LastActivityDate="2023-10-18T09:51:48.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26895" PostTypeId="2" ParentId="26892" CreationDate="2023-10-10T18:12:55.980" Score="20" Body="&lt;p&gt;&lt;em&gt;&amp;quot;interesting, natural and simple&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Illustrating something dynamically might make things interesting. For example, a geometry problem involving a limit (from an old calculus book):&lt;/p&gt;&#xA;&lt;p&gt;Consider a circle of variable radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, centered at the origin &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt;, and also the unit circle centered at &lt;span class=&quot;math-container&quot;&gt;$(1,0)$&lt;/span&gt;. If the circles intersect in the first quadrant at point &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;, then draw the line through &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; and the positive vertical intercept, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, of the first circle. This line will intersect the positive &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis at a point &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Question: If &lt;span class=&quot;math-container&quot;&gt;$r \to 0$&lt;/span&gt;, does the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-coordinate of point &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; have a limiting value?&lt;/p&gt;&#xA;&lt;p&gt;Students can guess the answer, or calculate it using a limit. Then show them an animation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/v3IZy.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/v3IZy.gif&quot; alt=&quot;Animation of intersecting circles and line as described above. As&#xA;the variable radius approaches zero, the intersection point E&#xA;approaches the x value of 4.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2023-10-10T18:34:59.293" LastActivityDate="2023-10-10T18:34:59.293" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26896" PostTypeId="2" ParentId="26892" CreationDate="2023-10-10T18:30:31.760" Score="1" Body="&lt;p&gt;I'll share an idea, but it requires geometric series, a topic typically covered at the end of a Calculus II course. Still, you might consider using this as a way to preview some future work in calculus for the students, while also providing another motivating context for limits, i.e. the sum of a series. (What I describe below is given to students as a problem set in my Calculus II course.)&lt;/p&gt;&#xA;&lt;p&gt;Consider a geometric series with first term 2 and multiplicative ratio &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, so we have &lt;span class=&quot;math-container&quot;&gt;$2+2r+2r^2+2r^3+\cdots$&lt;/span&gt; What possible sum value &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; could this series achieve? After deriving the &lt;a href=&quot;https://en.wikipedia.org/wiki/Geometric_series#Sum&quot; rel=&quot;nofollow noreferrer&quot;&gt;formula for the sum&lt;/a&gt;, namely &lt;span class=&quot;math-container&quot;&gt;$S=\frac{2}{1-r}$&lt;/span&gt;, you can ask some leading questions about what &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; could possibly be. I suggest working with the students to consider at least the cases where &lt;span class=&quot;math-container&quot;&gt;$S=10$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$r=4/5$&lt;/span&gt;), &lt;span class=&quot;math-container&quot;&gt;$S=4$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$r=1/2$&lt;/span&gt;), &lt;span class=&quot;math-container&quot;&gt;$S=2$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$r=0$&lt;/span&gt;), &lt;span class=&quot;math-container&quot;&gt;$S=1$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$r=-1$&lt;/span&gt;), and &lt;span class=&quot;math-container&quot;&gt;$S=0$&lt;/span&gt; (impossible).&lt;/p&gt;&#xA;&lt;p&gt;Then, look back at your derivation of the sum formula and remind students of the domain: &lt;span class=&quot;math-container&quot;&gt;$-1&amp;lt;r&amp;lt;1$&lt;/span&gt;. &lt;a href=&quot;https://www.desmos.com/calculator/jqinbbydgx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graph the function&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$S(r)=\frac{2}{1-r}$&lt;/span&gt; on the domain &lt;span class=&quot;math-container&quot;&gt;$-1&amp;lt;r&amp;lt;1$&lt;/span&gt;. Note that the point &lt;span class=&quot;math-container&quot;&gt;$(-1,1)$&lt;/span&gt; is &lt;strong&gt;not&lt;/strong&gt; on the graph, because of the domain issue (when &lt;span class=&quot;math-container&quot;&gt;$r=-1$&lt;/span&gt;, the geometric series does not converge). However, it's visually clear that the limit exists: &lt;span class=&quot;math-container&quot;&gt;$\lim_{r\to (-1)^+} S(r) = 1$&lt;/span&gt;. All of this can help students grapple with the idea that this infinite series can converge to any sum &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; that satisfies &lt;span class=&quot;math-container&quot;&gt;$S&amp;gt;1$&lt;/span&gt;, but not &lt;span class=&quot;math-container&quot;&gt;$S=1$&lt;/span&gt; itself. I hope this will also provide them with a fairly natural example of where a function clearly has a limiting value at a finite input, even though that point isn't really &amp;quot;achievable,&amp;quot; in some sense.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2023-10-10T18:30:31.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26897" PostTypeId="2" ParentId="26878" CreationDate="2023-10-11T09:47:10.627" Score="6" Body="&lt;p&gt;In my experience mathematical tools are pretty much like learning a language. You won't get fluent if you don't practice a LOT. In high school this is enforced with constant homework, tests, solving exercises in class etc. At university level there is no such baby-sitting, but the students still have more or less the same mindset and maturity level so they do the bare minimum to pass the classes. If the only solution the university has is to inflate grades and make the exams easy, you have your problem.&lt;/p&gt;&#xA;&lt;p&gt;Give them a homework sheet at every class + a short test similar to the previous sheet (making sure they don't rely on sharing answers for the home stuff), both rigorously graded. I see no way around. A few months of this treatment should help them tremendously.&lt;/p&gt;&#xA;" OwnerUserId="23035" LastEditorUserId="23035" LastEditDate="2023-10-11T09:57:42.163" LastActivityDate="2023-10-11T09:57:42.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26899" PostTypeId="1" AcceptedAnswerId="26903" CreationDate="2023-10-11T13:02:37.183" Score="27" ViewCount="16459" Body="&lt;p&gt;Back in high school—back in the 1900s, as my sons say—when our calculus teacher was introducing the chain rule...&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;...he made a special point of saying that we shouldn't think of the &lt;span class=&quot;math-container&quot;&gt;$dt$&lt;/span&gt; as cancelling out in this equation. He said it was just happened to look that way. We were supposed to consider &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial}{dx}$&lt;/span&gt; as an indivisible operator. (That was probably not his term.)&lt;/p&gt;&#xA;&lt;p&gt;So I was confused when my son's calculus book introduced differentials, and explicitly encouraged algebraic manipulation of expressions like &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; — multiplying both sides of an equation by &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;, and things like that. I've looked into it somewhat, and as far as I can tell this is regarded as mathematically legitimate. (Feel free to correct me on that.)&lt;/p&gt;&#xA;&lt;p&gt;I'm mostly just curious now why our calculus teacher warned us off from thinking of Leibniz notation in this way. I could imagine that when students think about &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt;, then there could be a lot of algebraic rabbit trails. Are there other reasons for discouraging high school students from thinking about differentials?&lt;/p&gt;&#xA;" OwnerUserId="23039" LastEditorUserId="590" LastEditDate="2023-10-20T11:18:07.263" LastActivityDate="2023-10-25T11:43:30.557" Title="Why not think of derivatives as fractions?" Tags="|secondary-education|calculus|" AnswerCount="7" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="26900" PostTypeId="2" ParentId="689" CreationDate="2023-10-11T13:19:30.820" Score="1" Body="&lt;p&gt;My advice is to only do this for high track kids.  And within this just teach topical awareness. I.e. just mention that there can be different sizes of infinities.  And leave it at it. No not rigor. And no, not even intuition about it.&lt;/p&gt;&#xA;&lt;p&gt;I think there is a sort of pedagogical blind spot that many math types here have, where the only way to learn something is in detail. Or worse, rigorous proof. But you can actually know about something, without knowing it in detail. Think of things like how a refrigerator works, that you have an idea on. But would get tripped up tracing temp entropy or describing the key components.  It can be a nice little enrichment to know ABOUT something without knowing IT.&lt;/p&gt;&#xA;&lt;p&gt;For example, I learned in a magnet school in 5th or 6th grade that there were different levels of infinity.  And I never learned more than that. It was just an interesting assertion slash factoid of general knowledge. Like Bushmen of the Kalahari. To this date, that is my level of knowledge of the topic. Or consider the inability to solve general quintics.  It was an aside in algebra class. Just magpie knowledge like the battle of Khe Sahn was in VN War. But no clue how or why USMC got tied down there. But at least I have the heard of it, level of awareness.&lt;/p&gt;&#xA;" OwnerUserId="23040" LastActivityDate="2023-10-11T13:19:30.820" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26901" PostTypeId="2" ParentId="26899" CreationDate="2023-10-11T13:51:30.633" Score="12" Body="&lt;p&gt;There is a nice write up at the included link with an application to thermodynamics. I've included two of the images (subsequently transcribed) that give the core of the argument. Single variable functions and differentials (derivatives with &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;) are safe to use as fractions, but multivariate functions and partial derivatives (with a &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt;) aren't.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://johncarlosbaez.wordpress.com/2021/09/13/the-cyclic-identity-for-partial-derivatives/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://johncarlosbaez.wordpress.com/2021/09/13/the-cyclic-identity-for-partial-derivatives/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;WHEN YOU REALIZE, SADLY, THAT DERIVATIVES DON'T ALWAYS ACT LIKE FRACTIONS&lt;/h3&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; are functions on the plane and you can take any&#xA;two as coordinates and write the third as a smooth function of those&#xA;two. Then&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{\partial u}{\partial v}\Bigr|_w \frac{\partial v}{\partial  w}\Bigr|_u \frac{\partial w}{\partial u}\Bigr|_v = \color{red}{-1} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;AN EXAMPLE&lt;/h3&gt;&#xA;&lt;p&gt;Say &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; are these functions on the plane:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ u = x \quad v = y \quad w = x + y $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then each is a function of the other two:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ u = w - v \quad v = w - u \quad w = u + v $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and we get&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \underbrace{\frac{\partial u}{\partial v}\Bigr|_w}_{= -1} \underbrace{\frac{\partial v}{\partial  w}\Bigr|_u}_{= 1} \underbrace{\frac{\partial w}{\partial u}\Bigr|_v}_{= 1} = \color{red}{-1} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2023-10-17T14:48:43.803" LastActivityDate="2023-10-17T14:48:43.803" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26902" PostTypeId="2" ParentId="26878" CreationDate="2023-10-11T15:17:50.130" Score="4" Body="&lt;p&gt;A lot of students can graduate STEM having taken only a single Linear Algebra course. Yes there’s some LA in differential equations or abstract algebra (less sure about the engineering side), but suffice to say this is not a lot of practice. And it’s practice from a single main source. It could be that almost every matrix in their differential equation course was invertible, their LA text focused on symmetric matrices for a whole chapter, they worked heavily in Matlab or similar, etc.&lt;/p&gt;&#xA;&lt;p&gt;I was never taught what a matrix was until college, not a single lesson. I’m about to finish STEM undergrad and that single LA class is doing a lot of heavy lifting. Since all of your points revolve around LA, I’m willing to believe they just need a refresher/reinforcement, especially since the points you make aren’t too advanced. It’s a lack of exposure. Another answer gave a quick fix for most of your points. That’s probably all they need, a small refresher and transition to new your class’ notation.&lt;/p&gt;&#xA;" OwnerUserId="23042" LastEditorUserId="23042" LastEditDate="2023-10-11T15:25:24.563" LastActivityDate="2023-10-11T15:25:24.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26903" PostTypeId="2" ParentId="26899" CreationDate="2023-10-11T15:28:46.883" Score="36" Body="&lt;p&gt;If you have a function &lt;span class=&quot;math-container&quot;&gt;$f(x,y)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$x=x(t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=y(t)$&lt;/span&gt; are themselves functions of a parameter &lt;span class=&quot;math-container&quot;&gt;$t,$&lt;/span&gt; and you blindly cancel out differentials, then you can get to incorrect statements like&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\require{cancel}\dfrac{\partial f}{\partial t} = \dfrac{\partial f}{\cancel{\partial x}} \cdot \dfrac{\cancel{\partial x}}{\partial t} = \dfrac{\partial f}{\cancel{\partial y}} \cdot \dfrac{\cancel{\partial y}}{\partial t}, \quad {\color{red}\times}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;whereas what's actually true is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{\partial f}{\partial t} = \dfrac{\partial f}{\partial x} \cdot \dfrac{\partial x}{\partial t} + \dfrac{\partial f}{\partial y} \cdot \dfrac{\partial y}{\partial t}. \quad {\color{green}\checkmark}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can't cancel because the &lt;span class=&quot;math-container&quot;&gt;$\partial f$&lt;/span&gt;'s in the numerators of &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial f}{\partial t},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial f}{\partial x},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$ \dfrac{\partial f}{\partial y}$&lt;/span&gt; all mean different things.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$\partial f$&lt;/span&gt; in the numerator of &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial f}{\partial t},$&lt;/span&gt; represents the change in &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; &lt;em&gt;attributed to the change in &lt;span class=&quot;math-container&quot;&gt;$t.$&lt;/span&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$\partial f$&lt;/span&gt; in the numerator of &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial f}{\partial x}$&lt;/span&gt; represents the change in &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; &lt;em&gt;attributed to the change in &lt;span class=&quot;math-container&quot;&gt;$x.$&lt;/span&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$\partial f$&lt;/span&gt; in the numerator of &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial f}{\partial y}$&lt;/span&gt; represents the change in &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; &lt;em&gt;attributed to the change in &lt;span class=&quot;math-container&quot;&gt;$y.$&lt;/span&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But in single-variable calculus (which is likely the focus of your son's textbook), you're working exclusively with functions that have only one input variable. And if you have a function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$x=x(t)$&lt;/span&gt; is itself a function of a parameter &lt;span class=&quot;math-container&quot;&gt;$t,$&lt;/span&gt; then it's true that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{\partial f}{\partial t} = \dfrac{\partial f}{\partial x} \cdot \dfrac{\partial x}{\partial t}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The above is conventionally written with &amp;quot;total&amp;quot; derivative symbols (&lt;span class=&quot;math-container&quot;&gt;$\mathrm d$&lt;/span&gt; means &amp;quot;total&amp;quot;, &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt; means &amp;quot;partial&amp;quot;) since the change attributed to the single variable is the same as the total change of the function.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{\mathrm df}{\mathrm dt} = \dfrac{\mathrm df}{\mathrm dx} \cdot \dfrac{\mathrm dx}{\mathrm dt}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So in general, you can manipulate total derivatives (&lt;span class=&quot;math-container&quot;&gt;$\mathrm d$&lt;/span&gt;) like fractions, but you can't do the same with partial derivatives (&lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;\require{cancel}&#xA;\textrm{valid:} \quad &amp;amp;\dfrac{\mathrm df}{\mathrm dt} = \dfrac{\mathrm df}{\cancel{\mathrm dx}} \cdot \dfrac{\cancel{\mathrm dx}}{\mathrm dt} \quad \color{green}\checkmark \\[5pt]&#xA;\textrm{NOT valid:} \quad &amp;amp;\dfrac{\partial f}{\partial t} = \dfrac{\partial f}{\cancel{\partial x}} \cdot \dfrac{\cancel{\partial x}}{\partial t} \quad \color{red}\times&#xA;\end{align*}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Response to comments about notation:&lt;/strong&gt; The streamlined notation makes this answer is friendly for non-experts. Yes, it relies on some intuitive inference as to what is meant -- for instance, given &lt;span class=&quot;math-container&quot;&gt;$f(x,y),$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$x(t),$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$y(t),$&lt;/span&gt; the streamlined notation &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial f}{\partial t}$&lt;/span&gt; refers to &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\partial f(x(t),y(t))}{\partial t}.$&lt;/span&gt; But non-experts generally find it easier to make reasonably intuitive inferences like this than to work with technically complete notation that overwhelms them. (Those who disagree: maybe it was different for you when you were a student, but that's how it is for 99% of students.)&lt;/p&gt;&#xA;&lt;p&gt;If anyone has a suggested improvement that will make the notation more technically unambiguous without making the answer less friendly for non-experts, then feel free to make an edit suggestion.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-14T21:07:10.377" LastActivityDate="2023-10-14T21:07:10.377" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="26904" PostTypeId="2" ParentId="26878" CreationDate="2023-10-11T19:19:45.633" Score="-1" Body="&lt;p&gt;Have you actually looked at linear algebra textbooks? As far as I can tell, they are uniformly bad. It is a really easy subject made incomprehensible by authors. I never took the course, but I got a perfect score on my prequalification test by teaching myself—mostly by solving lots of problems.&lt;/p&gt;&#xA;" OwnerUserId="23044" LastEditorUserId="851" LastEditDate="2023-10-13T14:22:22.487" LastActivityDate="2023-10-13T14:22:22.487" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26905" PostTypeId="2" ParentId="26873" CreationDate="2023-10-11T21:43:06.337" Score="4" Body="&lt;p&gt;I am not a mathematics educator. I might have become one had I not discovered the world of computers in 1981. This put me on a computer science trajectory and  ultimately resulted in a successful Silicon Valley career. I can therefore not talk to theories of learning and if or how they should inform classroom instruction. I have not had to face the challenges of fitting various teaching techniques into a prescribed time frame outside of acting as a private math tutor during my high school years and as a TA for computer science at university.&lt;/p&gt;&#xA;&lt;p&gt;My mathematical educational consisted of the classical chalk and talk plus textbook approach, which worked well enough through high school (I had a STEM focus in high school and graduated in the top ten percent of my class), but I struggled terribly with the math-heavy undergraduate computer science studies at university. I eventually realized which learning style best suited me: Study of literature accompanied by  experimentation &lt;sup&gt;1&lt;/sup&gt;, always with a view to &lt;em&gt;practical applicability&lt;/em&gt; of whatever mathematical technique is being acquired. Being able to put mathematics into a &lt;em&gt;real-life context&lt;/em&gt; has always been a &lt;em&gt;motivating force&lt;/em&gt; for me. Conversely, I have had a certain visceral aversion to acquiring knowledge when I could not perceive its practical utility.&lt;/p&gt;&#xA;&lt;p&gt;One aspect of the history of mathematics that has been a motivating factor for me personally are &lt;em&gt;biographies&lt;/em&gt; of famous mathematicians. This started around age 13 when our math teacher told us the famous story of how Gauss' mathematical talents were first discovered. Shortly thereafter I found an old book in the school library relating lives of some important mathematicians. I specifically remember Cardano and Galois. As I recall it was a very Euro-centric and exclusively male selection. Not until a considerable time later did I learn about Ramanujan or Agnesi, for example. The fact that mathematics was advanced by people from all walks of life made it &lt;em&gt;more relatable&lt;/em&gt; to me: Mersenne was a monk, Fermat was a lawyer, Cholesky a military officer. The humble origins of Gauss, Lambert, Bessel, and others particularly resonated with me.&lt;/p&gt;&#xA;&lt;p&gt;I doubt that adding elements of the history of mathematics to a classical math curriculum will improve test scores. But based on my personal experience I think it can make math a bit more &lt;em&gt;enjoyable&lt;/em&gt;, with the effect lasting beyond school and university. As a practical aspect, it &lt;em&gt;may&lt;/em&gt; contribute to students not dropping math classes at the first opportunity. Because math was such a painful struggle during my undergraduate years (I almost flunked out of the CS program because of it), I elected not to take optional classes in differential equations, a decision I regretted twenty years later when that knowledge would have come in handy in my professional career.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt;&lt;br /&gt;&#xA;S. Breuer and G. Zwas, &lt;em&gt;Numerical Mathematics: A Laboratory Approach.&lt;/em&gt; Cambridge University Press 1993&lt;br /&gt;&#xA;J. Borwein and D. Bailey, &lt;em&gt;Mathematics by Experiment.&lt;/em&gt; A. K. Peters 2004&lt;br /&gt;&#xA;J. Borwein, D. Bailey, and R. Girgensohn, &lt;em&gt;Experimentation in Mathematics.&lt;/em&gt; A. K. Peters 2004&lt;/p&gt;&#xA;" OwnerUserId="17715" LastActivityDate="2023-10-11T21:43:06.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26906" PostTypeId="2" ParentId="26899" CreationDate="2023-10-11T22:53:16.657" Score="3" Body="&lt;p&gt;Mathematics is a formal language, which people often use in an informal way. Think of the symbol ×. It is a binary operator - it needs two numbers to 'work':&lt;/p&gt;&#xA;&lt;p&gt;3 × 2 = 6&lt;/p&gt;&#xA;&lt;p&gt;If we wrote × with only one number (&amp;quot;3 ×&amp;quot; or &amp;quot;× 2&amp;quot;) it doesn't make sense.&lt;/p&gt;&#xA;&lt;p&gt;Differentials are the same. The symbol represents the relative change of one quantity when compared to the change in another quantity. &lt;span class=&quot;math-container&quot;&gt;$\frac {dy}{dx} $&lt;/span&gt; means &amp;quot;how much does y change when compared to a change in x&amp;quot;. Writing just &amp;quot;dy&amp;quot; is writing half a statement, and in formal understanding is meaningless.&lt;/p&gt;&#xA;&lt;p&gt;I've had students say &amp;quot;but we use 'dx' alone in integral calculus&amp;quot;. No we don't. Every integral that ends with a '.dx' starts with a &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt;. They are effectively the two ends of a set of brackets, and writing either part alone doesn't make sense.&lt;/p&gt;&#xA;&lt;p&gt;That doesn't stop people from doing it, however. Derivatives look like fractions, but are not fractions. People can manipulate them in a way similar to fractions, and in many cases will get the right answer, but what they are doing and what they think they are doing are actually two different things. When you use the Chain Rule to make an expression simpler, you are not &amp;quot;cancelling out&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="23049" LastActivityDate="2023-10-11T22:53:16.657" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26907" PostTypeId="2" ParentId="26892" CreationDate="2023-10-11T22:58:06.323" Score="2" Body="&lt;p&gt;Polynomials are everywhere, and in engineering or physics, it is extremely common to use linear approximations to them&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;\lim_{x \rightarrow 0} \frac{(x + 1)^d - 1}{x} = d&#xA;\end{align}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Granted, this is just the derivative, but it need not be presented as such and may in fact motivate derivatives later.&#xA;Armed with this identity, high school knowledge is enough to&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Approximate square roots, e.g. &lt;span class=&quot;math-container&quot;&gt;$\sqrt{1.002} \approx 1.001$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Compute how far the horizon is. For example, suppose you're on board a ship &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;m above the water, how far can you see? What if you're in a crow's nest that's also &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;m high?&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;\text{horizon distance}&#xA;&amp;amp;= \sqrt{(\text{radius of earth + height})^2 - \text{radius of earth}^2} \\&#xA;&amp;amp;\approx \sqrt{2 * \text{radius of earth} * \text{height}}&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;Even if you could just type everything into a calculator, this approximation shows that the height of the crow's nest yields diminishing returns.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Show that tidal forces scales inversely with distance &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; cubed and proportional to displacement &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; from the center&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;\frac{Gm}{r^2} - \frac{Gm}{(r \pm d)^2}&#xA;&amp;amp;= \frac{Gm}{r^2}(1 - (1 \pm \frac{d}{r})^{-2}) \\&#xA;&amp;amp;\approx \mp\frac{2Gmd}{r^3}&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;Thus you can show that a) the sun's tidal force is smaller than the moon's on earth and b) tides are only obvious in large bodies of water.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Derive all sorts of simple harmonic oscillators.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The list goes on.&lt;/p&gt;&#xA;" OwnerUserId="15604" LastEditorUserId="15604" LastEditDate="2023-10-11T23:06:34.993" LastActivityDate="2023-10-11T23:06:34.993" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26908" PostTypeId="2" ParentId="26899" CreationDate="2023-10-12T00:41:44.213" Score="1" Body="&lt;p&gt;Remember that &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial y}{\partial t}$&lt;/span&gt; is actually &lt;span class=&quot;math-container&quot;&gt;$ \lim\limits_{h \to 0} \frac{y(t+h) - y(t)}{h} = \lim\limits_{h \to 0} \frac{y' - y}{h}$&lt;/span&gt; (I'm using &lt;span class=&quot;math-container&quot;&gt;$y'$&lt;/span&gt; to make it less busy). It's not a fraction but the limit of a fraction.&lt;/p&gt;&#xA;&lt;p&gt;If you rewrite the original:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{h \to 0} \frac{y' - y}{x' - x} = \lim\limits_{h \to 0} \frac{y' - y}{t' - t} \cdot \lim\limits_{h \to 0} \frac{t' - t}{x' - x} $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$t'-t$&lt;/span&gt; is inside each limit, so you can't cancel them out.&lt;/p&gt;&#xA;" OwnerUserId="23050" LastActivityDate="2023-10-12T00:41:44.213" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26909" PostTypeId="2" ParentId="26878" CreationDate="2023-10-12T03:38:41.113" Score="-3" Body="&lt;p&gt;I am an engineer who drove math professors and tutors insane. I understood some difficult concepts but struggled with easy details. I did not know it was possible to simultaneously be gifted and disabled.&lt;/p&gt;&#xA;&lt;p&gt;You will have an easier time teaching engineers if you can identify their learning style(s).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Learning Styles—Visual&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here's one technique based in neuroscience:&lt;/p&gt;&#xA;&lt;p&gt;Ask the student to visualize a tree and describe their thought process out loud. What did they see or hear?&lt;/p&gt;&#xA;&lt;p&gt;If they are like me, they saw nothing. One of my disabilities is &lt;a href=&quot;https://en.wikipedia.org/wiki/Aphantasia&quot; rel=&quot;nofollow noreferrer&quot;&gt;aphantasia&lt;/a&gt;. I have never seen an image in my mind. It is impossible for me to visualize any math concept like you or 97% of other people.&lt;/p&gt;&#xA;&lt;p&gt;Before I knew I had aphantasia, I would've described a symbolic tree. I &amp;quot;visualize&amp;quot; by feeling coordinates in the space around me. The result is similar to an outline minus visual detail.&lt;/p&gt;&#xA;&lt;p&gt;To visualize a math concept, I need an external tool. I could draw it on graph paper, use software to model the result, or watch a video which shows visual models of the concepts. Interactive three-dimensional models online seem especially helpful.&lt;/p&gt;&#xA;&lt;p&gt;If the student describes a &lt;em&gt;super&lt;/em&gt; detailed individual tree, they are probably a visual thinker. An engineer who thinks this way can potentially imagine an entire machine in their mind and disassemble it to test each part. If this is your student, try to describe every concept in significant visual detail.&lt;/p&gt;&#xA;&lt;p&gt;If the student begins to visualize a tree by hearing their own voice describing the tree in words—or they begin talking to themselves and are clearly thinking in words first—they may be a verbal thinker. People who think in words tend to be linear, sequential, and detail-oriented—Step A goes to Step B. They may need you to specify every detail in a step-by-step order, including details which might be obvious to most other people.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Learning Styles—Kinesthetic&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;My primary learning style is &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Proprioception&quot; rel=&quot;nofollow noreferrer&quot;&gt;kinesthetic&lt;/a&gt;&amp;quot;—I learn with my hands. Here is an example of a kinesthetic method for learning calculus—&lt;a href=&quot;https://www.makershed.com/products/make-calculus-print&quot; rel=&quot;nofollow noreferrer&quot;&gt;three-dimensional printable calculus models&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Kinesthetic learning could involve using the human body to understand math concepts. Here are some examples: I might draw math with my hands in the air, form an odd shape with my hands or legs, or dance the patterns. I could play an instrument like a piano or keyboard in a specific way related to math concepts. I could make visual art which is based in mathematics like &lt;a href=&quot;https://en.wikipedia.org/wiki/Origami&quot; rel=&quot;nofollow noreferrer&quot;&gt;origami&lt;/a&gt; or data visualizations.&lt;/p&gt;&#xA;&lt;p&gt;Video games where my character walks around in multiple dimensions (three-dimensional, etc.) to explore concepts are very helpful. I enjoyed studying math concepts which related to the physics in video game environments.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Learning Styles—Visual Spatial&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;My thought process is apparently called &amp;quot;visual spatial&amp;quot; or &amp;quot;pattern thinking&amp;quot;—common in engineers.&lt;/p&gt;&#xA;&lt;p&gt;If the engineer happens to be female like I am, she may also have an inverse learning style. Women are more likely to be &amp;quot;holistic thinkers&amp;quot;. I need to understand the big picture of any topic &lt;em&gt;before&lt;/em&gt; the details.&lt;/p&gt;&#xA;&lt;p&gt;I expected my first programming class to start with an overview.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What are all the languages we could use?&lt;/li&gt;&#xA;&lt;li&gt;When would we choose X language over Y?&lt;/li&gt;&#xA;&lt;li&gt;Which approach is the most efficient to solve the problem?&lt;/li&gt;&#xA;&lt;li&gt;Which is easiest for another person to understand?&lt;/li&gt;&#xA;&lt;li&gt;How difficult are these languages compared to each other?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I did not begin to understand math until I read highly theoretical math proofs and lectures by PhDs about topics which are far beyond my level of formal education or knowledge. Math classes did teach me some details, but I hadn't understood how those individual symbols and concepts connected to the big picture.&lt;/p&gt;&#xA;&lt;p&gt;If engineers do not understand the patterns which unite &lt;em&gt;all&lt;/em&gt; concepts you are trying to teach them, some may have an unusually difficult time remembering easy details.&lt;/p&gt;&#xA;&lt;p&gt;How did I end up learning any advanced concepts without a foundation in the basics? Funny story. I'd never taken math beyond algebra. On my first college placement test, the result said calculus.&lt;/p&gt;&#xA;&lt;p&gt;I can almost always pick the correct multiple choice answer on even an advanced math test. If you ask me the answer to a difficult math problem, I can sometimes tell you the correct result.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, I frequently cannot explain &lt;em&gt;how&lt;/em&gt; I know the answer. I don't understand the details of math languages very well yet.&lt;/p&gt;&#xA;&lt;p&gt;I also have &lt;a href=&quot;https://en.wikipedia.org/wiki/Dyscalculia&quot; rel=&quot;nofollow noreferrer&quot;&gt;dyscalculia&lt;/a&gt;. If a student often accidentally transposes numbers, they may have this challenge too. It's frequently impossible for me to do basic easy math in my head. Dyscalculia can result in some challenges with understanding left / right or east / west concepts and aspects of directions in the real world. Since I used software for basic math, like calculating tips at a restaurant, I didn't understand that this was one of my challenges.&lt;/p&gt;&#xA;&lt;p&gt;You can thank &lt;a href=&quot;https://en.wikipedia.org/wiki/Hyperlexia&quot; rel=&quot;nofollow noreferrer&quot;&gt;hyperlexia&lt;/a&gt; for the length of this post—&amp;quot;too many words&amp;quot;. If your student has hyperlexia, they usually talk a lot. They may struggle to summarize math concepts in a linear way like Step A leads to Step B. My thought process is associative, meta, and theoretical. It is a bit similar to the branches of a tree or a fractal pattern. To summarize in any language, I often need to see every detail visually first. Afterwards, I try to arrange those details in order like solving a puzzle.&lt;/p&gt;&#xA;&lt;p&gt;Neuroscience research indicates that disabilities are highly related to stress. The more your student feels like they are failing, the more they might struggle with any unknown learning disability. Unfortunately, I was so frustrating to educators that even incredibly calm and patient TAs and professors sometimes began shouting profanities at the top of their lungs. I used to joke to friends that I must have a superpower which made otherwise rational mathematicians lose their minds. Please try not to give up on the idiot engineers because apparently a few of us might be secretly good at math—we just think differently than most. ;)&lt;/p&gt;&#xA;&lt;p&gt;I hope any of this was helpful. If anyone reading this might like feedback on specific math learning materials, feel free to contact me. Nobody knew I had any learning disability, but I have at least six, if not more.&lt;/p&gt;&#xA;&lt;p&gt;I think more students might love math if they were asked to write their own equation. I fell in love with math last year when I tried to write a symbolic equation which physicists apparently call the &amp;quot;theory of everything&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Since I'm challenging for math educators, I'm hoping to eventually design a custom degree and teach myself with expert guidance. However, if anyone reading this might be interested in trading your math teaching skills for a website / social media / marketing or other technology-related tasks, I love the idea of a trade with a real mathematician. I am a usability engineer, but I studied many topics, so I could likely find something useful to offer you. =]&lt;/p&gt;&#xA;" OwnerUserId="23052" LastEditorUserId="851" LastEditDate="2023-10-13T07:59:20.287" LastActivityDate="2023-10-13T07:59:20.287" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26910" PostTypeId="1" CreationDate="2023-10-12T14:32:03.630" Score="1" ViewCount="168" Body="&lt;p&gt;I am trying to verfy the formula for &amp;quot;cone volume&amp;quot; calculation. It is not clear why cone volume is 1/3 of a cylinder volume with the same bottom size and height. Is there any proof of the fact that a cone's volume is one third of a cylinder's volume of the same height and bottom area?&lt;/p&gt;&#xA;" OwnerUserId="23064" LastEditorUserId="20854" LastEditDate="2023-10-14T20:50:35.597" LastActivityDate="2023-10-14T20:50:35.597" Title="Proof that volume of cone is 1/3 that of a cylinder" Tags="|geometry|" AnswerCount="1" CommentCount="1" ClosedDate="2023-10-12T18:07:37.183" ContentLicense="CC BY-SA 4.0" />
  <row Id="26911" PostTypeId="2" ParentId="26910" CreationDate="2023-10-12T16:02:14.653" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/644/1203732&quot;&gt;Here is an answer on Math.SE&lt;/a&gt; that proves that the volume of a cone is &lt;span class=&quot;math-container&quot;&gt;$\dfrac{1}{3}$&lt;/span&gt; the volume of a cylinder using only algebra and geometry. No calculus required.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; The proof relies on Cavalieri's Principle, which is probably intuitive enough for your purposes. But technically speaking, Cavalieri's Principle is usually proved using calculus (specifically, integrals). That said, there is an elementary proof of Cavalieri's Principle using only limits (&lt;a href=&quot;https://math.stackexchange.com/a/2556933/1203732&quot;&gt;also on Math.SE&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-10-12T16:02:14.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26913" PostTypeId="2" ParentId="26899" CreationDate="2023-10-12T16:59:18.343" Score="5" Body="&lt;p&gt;First: &lt;span class=&quot;math-container&quot;&gt;${\partial}\over{\partial x}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\neq$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;${d}\over{d x}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The difference being the concepts in &lt;strong&gt;multivariable calculus&lt;/strong&gt; vs &lt;strong&gt;single variable calculus&lt;/strong&gt; are different and for different purposes. But by the context of your question I'm pretty sure you meant &lt;span class=&quot;math-container&quot;&gt;${d}\over{d x}$&lt;/span&gt;, so let's go with that.&lt;/p&gt;&#xA;&lt;p&gt;Second: The cancel is actual a &lt;strong&gt;pretty helpful tool&lt;/strong&gt; when solving &lt;strong&gt;differential equations&lt;/strong&gt;, so I do love it. Buuuut, I 100% understand why &lt;strong&gt;you should be taught it last&lt;/strong&gt; and be dissuaded from using it.&lt;/p&gt;&#xA;&lt;p&gt;The reason why he said not to think of it like that is because its &lt;strong&gt;lazy, without conceptual meaning, and an abuse of notation&lt;/strong&gt;. However, it's ironically right unless you go further with the thought. No seriously here's a simple proof.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x+dx)=f(x)+df(x)$$&lt;/span&gt;&#xA;So the chain rule comes naturally from:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$df(g(x))+f(g(x))=$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(g(x+dx))=f(g(x)+dg(x))$$&lt;/span&gt;&#xA;And the cancel:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{dg(x)}\frac{dg(x)}{dx}=\frac{1}{dx}$$&lt;/span&gt;&#xA;And all together is the chain rule as you know and love:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{df}{dx}=\frac{df}{dg}\frac{dg}{dx}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So from this you may tell that, yes, &lt;strong&gt;the cancel is really there, but also an erroneously incomplete perception on the subject&lt;/strong&gt;. In fact, you can do &lt;strong&gt;calculus without the fractions&lt;/strong&gt; at all using &lt;strong&gt;lie algebra&lt;/strong&gt; or using the techniques I used above.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$[D,x]=Dx-xD=1 \implies [D,f(x)]=f'(x)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But probably the most horrendous abuse of notation is with the second derivative.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d^2f}{dx^2} \neq \frac{d^2f}{dg^2}\frac{dg^2}{dx^2}=\frac{d^2f}{dg^2}\left(\frac{dg}{dx}\right)^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Nor is it anything remotely similar because if you start treating it as a &lt;strong&gt;linear algebra&lt;/strong&gt; you should understand that the algebra is &lt;strong&gt;non-commutative&lt;/strong&gt; which means &lt;span class=&quot;math-container&quot;&gt;$ab \neq ba$&lt;/span&gt; and implies &lt;span class=&quot;math-container&quot;&gt;$(ab)^c \neq a^c b^c$&lt;/span&gt;. &lt;strong&gt;The product rule&lt;/strong&gt; makes it more complicated:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\left(g'\frac{d}{dg}\right)^2=$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$g'\frac{d}{dg}g'\frac{d}{dg}=$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$g'\left(\frac{d}{dg} \cdot g'\right)\frac{d}{dg}=$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$g'\left(g''+g'\frac{d}{dg}\right)\frac{d}{dg}=$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$g'g''\frac{d}{dg}+(g')^2\frac{d^2}{dg^2}=$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that this, as expected from the product rule, doesn't equal the equation that is wrongly created using just the cancel.&lt;/p&gt;&#xA;" OwnerUserId="23069" LastEditorUserId="23069" LastEditDate="2023-10-17T05:52:46.127" LastActivityDate="2023-10-17T05:52:46.127" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26915" PostTypeId="2" ParentId="2279" CreationDate="2023-10-12T21:42:17.093" Score="0" Body="&lt;p&gt;It is also well worth checking out the very powerful and entirely free &lt;a href=&quot;https://obsproject.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Open Broadcaster Software&lt;/a&gt;.  Available on all three operating systems.&lt;/p&gt;&#xA;" OwnerUserId="213" LastActivityDate="2023-10-12T21:42:17.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26917" PostTypeId="2" ParentId="26850" CreationDate="2023-10-14T04:58:24.877" Score="0" Body="&lt;p&gt;Why not talk in terms of time series, say the annual population of some city over time. Show them a graph with some points lying on or near some curve in a neighborhood left of a point where the derivative is, say 1,000 (and the second derivative is fairly small, so the trend is fairly obvious), and ask them to predict what the population will be next year. When they find themselves confidently offering guesses, ask them how they arrived at them. Then do the same where the derivative is 2,000.&lt;/p&gt;&#xA;&lt;p&gt;This kind of experience extrapolating should provide an entry point to discuss “where curves seem to be heading“ at various points.&lt;/p&gt;&#xA;" OwnerUserId="23086" LastActivityDate="2023-10-14T04:58:24.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26918" PostTypeId="2" ParentId="26792" CreationDate="2023-10-14T19:03:51.670" Score="0" Body="&lt;p&gt;The goal IS NOT &amp;quot;to teach cofunctions&amp;quot;.&#xA;Trig functions are taught as a package and the relationships between them are usually taught as Pythagorean identities AND NOT as defining a cofun in terms of the original fun.  Standardized tests are not always competently designed -- the &amp;quot;in a right triangle&amp;quot; context has no bearing on the problem or its  solution -- sin and cos retain an inherently Pythagorean relationship to each other even when they are defined by other methods.&#xA;Moreover, the algebra in 2x+4 isn't there to obscure anything -- &amp;quot;Algebra 1&amp;quot; is normally taught in the semester &lt;em&gt;immediately&lt;/em&gt; preceding geometry, and  a standardized test can be trying to confirm the progress through one to the other -- it can be testing whether you know how to solve 2x-4 = whatever, AND whether you know how to compute whatever from 46°. Also, the fact the question is quoted without SAYING ° opens a completely separate can of worms, which becomes a barrel if the test itself didn't clarify that in the original.&lt;/p&gt;&#xA;" OwnerUserId="23092" LastEditorUserId="23092" LastEditDate="2023-10-14T19:08:22.767" LastActivityDate="2023-10-14T19:08:22.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26919" PostTypeId="1" AcceptedAnswerId="26920" CreationDate="2023-10-14T21:45:03.907" Score="15" ViewCount="13518" Body="&lt;p&gt;I transcribed &lt;a href=&quot;https://youtu.be/6jNkYpUL0vU?si=ZAOi9vvaGF5Sv0Nd&amp;amp;t=1320&quot; rel=&quot;noreferrer&quot;&gt;this excerpt starting at the 22-minute mark&lt;/a&gt;, of Okinawa Institute of Science and Technology’s May 19 2020 podcast with Professor Tadashi Tokieda:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For example, this is a bit too technical, but there's a definition.... a very very precise way of thinking about the limits and continuity and so on, which is goes on the name of Epsilon and Delta. So for every epsilon there exists a delta such that and blah blah blah. And this is a stumbling block for almost everyone. But when I came into         mathematics as an adult already (you know I taught myself mathematics), and when I   came to Epsilon-Delta, I felt no difficulty whatsoever. In fact I didn't even notice that it was supposed to be difficult.   That's because I had been very rigorously trained in the use of languages as a linguist.  And so, the idea that, you know, if you change the order quantifiers, of course,    the meaning changes completely. It's comp. It was trivial, of course, I mean, compared with the task a difficult task of taking    apart the syntax of, say something somebody, like Thucydides you know whose     sentence can continue for a page, with subordinate clause upon subordinate clause. By the way, he writes really     clearly, but in a complicated syntax. Well compared with that kind of thing, language in mathematics was very very    easy. I mean there's nothing to it. I think the fact of the matter is, most people don't have a sufficient mastery of their native language. They never had the experience, they don't   have had enough, shall I say a bit more gently, enough practice of careful use of   their own native language. You know, do you speak really carefully and making sure that you understand  absolutely everything that you are saying and every word and every phrase counts? The answer is no. The people just    blah blah blah, just talk away. So, if you have a really careful   habit of careful use of language, it's my personal belief that most of the difficulties in mathematics will go away.     And it's just that mathematics is an unforgiving subject, where any misunderstanding any lack of understanding shows immediately, whereas in the rest of human endeavors, you can keep going by faking for quite a long time. So in that  way, yes the language frames how you understand mathematics, but in that very very practical way. I think the best way to improve your chance of your future advancing mathematics is to practice and   improve your native language.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Does &lt;a href=&quot;https://matheducators.stackexchange.com/questions/2162/evaluating-the-reception-of-epsilon-delta-definitions?noredirect=1&amp;amp;lq=1#comment71381_2162&quot;&gt;research&lt;/a&gt; on &lt;a href=&quot;https://math.stackexchange.com/q/382757&quot;&gt;students’ difficulties with the δ-ε definition of limits and continuity&lt;/a&gt; substantiate Tokieda’s  asseveration that a primary cause is &lt;strong&gt;insufficient mastery—or at least lack of a habit of careful use—of their native languages&lt;/strong&gt;? Does empty-headedness of linguistic syntax bedevil undergraduates on, and thwart them from, δ-ε definitions?&lt;/p&gt;&#xA;" OwnerUserId="23094" LastEditorUserId="372" LastEditDate="2023-10-16T15:43:15.943" LastActivityDate="2023-10-20T04:12:06.670" Title="Do undergraduates struggle with δ-ε definitions because they lack a habit of careful use of their native language?" Tags="|mathematical-analysis|language-use|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26920" PostTypeId="2" ParentId="26919" CreationDate="2023-10-15T00:06:58.163" Score="20" Body="&lt;p&gt;I wouldn't say so. By studying linguistics on a deep level, this person learned to parse complicated multi-part statements and extract precise meaning from them. This skill--which people in general are not born with--is transferable to math, and it's no surprise that they didn't struggle with epsilon-delta the way most people do.&lt;/p&gt;&#xA;&lt;p&gt;But that doesn't mean linguistics should be considered a prerequisite to math. It could just as easily go in the other direction; I'm confident that my years of mathematical training would allow me to parse Thucydides much faster than the average college first-year. It's often said that programming and studying pure math help each other, etc. The point is that many of these skills are transferable.&lt;/p&gt;&#xA;&lt;p&gt;To use a sports analogy, if someone is a professional tennis player, they could pick up squash much more quickly than the average person. But we wouldn't say that the only reason people struggle to learn to play squash is because they haven't sufficiently mastered tennis.&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, students struggle with epsilon-delta for two reasons:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It's just hard.&lt;/li&gt;&#xA;&lt;li&gt;It is (unfortunately) often inflicted on students who are still getting used to the idea of rigorous mathematical proof, and they are still working to make sense of a number of aspects of that, while also grappling with the inherent difficulty of epsilon-delta arguments.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="22788" LastEditorUserId="13919" LastEditDate="2023-10-15T23:45:38.663" LastActivityDate="2023-10-15T23:45:38.663" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26921" PostTypeId="2" ParentId="26919" CreationDate="2023-10-15T00:54:29.660" Score="7" Body="&lt;p&gt;The issue you raise is similar to those raised here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/15477/how-to-make-students-comfortable-with-the-use-of-axiom-of-choice-in-analysis/15487#15487&quot;&gt;How to make students comfortable with the use of axiom of choice in analysis&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;[Note: the question at the link has &amp;quot;axiom of choice&amp;quot; in the title but it really has nothing to do with the axiom of choice!]&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can appreciate the linguistic complexity of the definition by listening to people struggle as they try to state the negation. Here is &amp;quot;fun&amp;quot; assignment that illustrates this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Your friend asserts that for every &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;lephant, there is a &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;ay such that the day is rainy and the elephant forgets to bathe. How would you prove your friend incorrect?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note the logical structure of the elephant sentence is very similar to the definition of a limit. Students give a variety of incorrect solutions. It presents a cognitive challenge.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt; We can formulate the scenario like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\forall\epsilon\exists\delta\left[\delta(R)\wedge\neg\epsilon(B)\right]$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The negation is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\exists\epsilon\forall\delta\left[\delta(R)\rightarrow\epsilon(B) \right]$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;To prove our friend wrong, we need to produce a counterexample. We need to find an elephant with a certain property. Our special elephant will be one that always bathes whenever the day is rainy, i.e., one that never forgets to bathe on rainy days.&lt;/p&gt;&#xA;&lt;p&gt;I always find it tricky to formulate this negation, and whenever I need to prove a function is not continuous using the limit definition, I know I am going to have sort through this all over again!&lt;/p&gt;&#xA;&lt;p&gt;But it is interesting to note that large language models (GPT, Bard, LLaMa,etc.) are extremely proficient with the elephant problem! &lt;em&gt;Use the boldface font above as a prompt, and see for yourself&lt;/em&gt;. On the other hand, the language model will not be so helpful when trying to prove that a specific function is not continuous by using the definition, because of the algebra and other mathematical details. This suggests to me that Tadashi Tokieda is onto something. AI in the form of large language models is very proficient with natural language processing, and tends to be much better at the underlaying elephant problem than are humans like me. The difficult logic of the elephant problem is the same as that of showing a function is not continuous using the definition. This is why we like to have lots of necessary conditions to check continuity--using them can often help us efficiently detect discontinuity without needing to grapple with the elephant.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="408" LastEditDate="2023-10-20T04:12:06.670" LastActivityDate="2023-10-20T04:12:06.670" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="26922" PostTypeId="1" AcceptedAnswerId="26927" CreationDate="2023-10-15T01:10:38.470" Score="0" ViewCount="295" Body="&lt;p&gt;Michael Spivak, &lt;em&gt;Calculus&lt;/em&gt; (4th edn 2008), p 13.  I know this monograph is aimed at undergraduates (not middle schoolers), but this kind of multiple-part question resurfaces on standardized tests for 13 year olds (henceforth y.o.).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Prove the following:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(v) &lt;span class=&quot;math-container&quot;&gt;$x^n - y^n = (x - y)(x^{n - 1} + x^{n - 2}y + \cdots + xy^{n - 2} + y{^n - 1}).$&lt;/span&gt;&lt;br /&gt;&#xA;(vi) &lt;span class=&quot;math-container&quot;&gt;$x^3 + y^3 = (x + y)(x^ 2- xy + y^2)$&lt;/span&gt;. (There is a particularly easy way to do this, using (iv), and it will show you how to find a factorization for &lt;span class=&quot;math-container&quot;&gt;$x^n + y^n$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\color{peru}{\; whenever  \; n \; is \; odd}.$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ibid, p 619.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(vi) Replace &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$-y$&lt;/span&gt; in (iv).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Answers that use &lt;a href=&quot;https://math.stackexchange.com/a/3684657&quot;&gt;Cyclotomic Polynomials&lt;/a&gt;, &lt;a href=&quot;https://math.stackexchange.com/a/3860053&quot;&gt;Modular Arithmetic&lt;/a&gt;, &lt;a href=&quot;https://math.stackexchange.com/a/3860075&quot;&gt;Complex Numbers&lt;/a&gt; are too Delphic for 13 y.o.!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/users/12042/brian-m-scott&quot;&gt;Brian M. Scott&lt;/a&gt;’s &lt;a href=&quot;https://math.stackexchange.com/q/3860047&quot;&gt;answer&lt;/a&gt; is TOO gruff and sketchy for 13 y.o. to fathom. To wit, 13 y.o. know that &lt;span class=&quot;math-container&quot;&gt;$(a + b)[a^{n – 1} – a^{n – 2}b + . . . + \color{mediumspringgreen}{+} ab^{n – 2} \color{red}{-} b^{n – 1}] \equiv a^n – b^n$&lt;/span&gt;. But this doesn’t answer their question!   How can 13 y.o. intuit why they can't factor out &lt;span class=&quot;math-container&quot;&gt;$(x + y)$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$x^n + y^n$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$n \in \mathbb{N}$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="23094" LastEditorUserId="23094" LastEditDate="2023-10-15T07:30:10.810" LastActivityDate="2023-10-16T07:58:26.213" Title="How to explain why we can’t factor $x^n + y^n$ for all natural numbers n, to 13 year olds?" Tags="|middle-school|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26923" PostTypeId="2" ParentId="26919" CreationDate="2023-10-15T02:55:36.293" Score="5" Body="&lt;p&gt;I think there are two (main) reasons why &amp;quot;people just blah, blah, blah, just talk away.&amp;quot; One is linguistic: They have clear thoughts but cannot accurately express them in words. The other is cognitive: They don't think clearly. Tokieda emphasizes the former because he had been &amp;quot;rigorously trained in the use of languages as a linguist&amp;quot;. He overlooks the cognitive aspect because he probably didn't need rigorous training in clear thought; he's probably one of those (fortunate) people to whom clear thinking just comes naturally.&lt;/p&gt;&#xA;&lt;p&gt;Either linguistic or cognitive difficulties can cause &amp;quot;blah, blah, blah&amp;quot; and can obstruct understanding of epsilon-delta definitions. But I'd seriously doubt any claim that a particular cause (or finite list of causes) accounts for all the problems students have with epsilon-delta definitions. Some students fail to understand such definitions (and mathematically precise reasoning in general) for quite unpredictable reasons.&lt;/p&gt;&#xA;" OwnerUserId="326" LastActivityDate="2023-10-15T02:55:36.293" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26924" PostTypeId="2" ParentId="26872" CreationDate="2023-10-15T05:26:53.067" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;would it be fair to ensure that the course description and course syllabus include the name of the particular system of set theory that all theorems studied in the course are deduced from?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No, because most people simply don't care.  Unless we are talking about a course on set theory, the course description or syllabus is the wrong place entirely to be bringing up such an issue.&lt;/p&gt;&#xA;&lt;p&gt;I think you are overreacting. When a student asks a question that is beyond the scope of the course and virtually nobody else taking the course would understand or even care about the question (it wouldn't even occur to them to &lt;em&gt;ask&lt;/em&gt; such a question), you should not try to put a response to the question in a visible part of your course materials for the future. Maybe have a comment about it in your personal notes, but leave it at that and then forget about it.&lt;/p&gt;&#xA;&lt;p&gt;I am reminded now of an earlier question on this site &lt;a href=&quot;https://matheducators.stackexchange.com/questions/20931/what-can-and-should-an-educator-do-about-ambiguous-terms-like-triangle-squ&quot;&gt;here&lt;/a&gt; in which the OP was very concerned about using &amp;quot;ambiguous&amp;quot; terms when teaching geometry (e.g., it is technically ambiguous that the word triangle means both the boundary and plane region inside the boundary) but in practice the terms are ambiguous to nobody who would ever take the course, so all the concern was largely misplaced.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2023-10-15T05:26:53.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26925" PostTypeId="2" ParentId="26922" CreationDate="2023-10-15T06:14:57.187" Score="0" Body="&lt;p&gt;Maybe show them &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2$&lt;/span&gt; as an example?  It is unfactorable. Let them try.&lt;/p&gt;&#xA;&lt;p&gt;If anything this is a lor easier than proving general quintic has no formula, which you can only assert to students not prove.&lt;/p&gt;&#xA;" OwnerUserId="23096" LastEditorUserId="8190" LastEditDate="2023-10-16T07:58:26.213" LastActivityDate="2023-10-16T07:58:26.213" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26927" PostTypeId="2" ParentId="26922" CreationDate="2023-10-15T10:30:50.290" Score="3" Body="&lt;p&gt;We can use the factor theorem, unless you think it’s also too “Delphic” for your students. The factor theorem tells us that a polynomial &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; has a factor &lt;span class=&quot;math-container&quot;&gt;$(x-r)$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$p(r)=0$&lt;/span&gt;. Equivalently, &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; has a factor &lt;span class=&quot;math-container&quot;&gt;$(x-r)$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is a root of &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let’s consider the polynomial defined as &lt;span class=&quot;math-container&quot;&gt;$p(x) = x^n + a^n$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is non-zero. It has a factor &lt;span class=&quot;math-container&quot;&gt;$(x+a)$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$p(-a)=0$&lt;/span&gt;. But &lt;span class=&quot;math-container&quot;&gt;$$p(-a)=(-a)^n+a^n=&#xA;\begin{cases}&#xA;0 &amp;amp;\text{ if $n$ is odd,} \\&#xA;2a^n &amp;amp;\text{if $n$ is even,}&#xA;\end{cases}$$&lt;/span&gt;&#xA;therefore &lt;span class=&quot;math-container&quot;&gt;$(x+a)$&lt;/span&gt; is a factor of &lt;span class=&quot;math-container&quot;&gt;$x^n+a^n$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is odd.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is even, &lt;span class=&quot;math-container&quot;&gt;$x^n+a^n$&lt;/span&gt; has no linear factors as a polynomial over the real numbers. This is because the roots of &lt;span class=&quot;math-container&quot;&gt;$x^n+a^n$&lt;/span&gt; are the solutions to the equation &lt;span class=&quot;math-container&quot;&gt;$x^n=-a^n$&lt;/span&gt;. This equation has no real solutions.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-10-15T10:30:50.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26928" PostTypeId="2" ParentId="26919" CreationDate="2023-10-15T13:38:32.000" Score="3" Body="&lt;p&gt;The is a built-in irreducible logical complexity in epsilon-delta definitions that has to do with the language of logic, rather than natural language.  This has to do with alternations of quantifiers.   To illustrate the complexity, compare the following two definitions of continuity of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous if for every &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and for every positive &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;, there exists a positive &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; such that, for all &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$|h|&amp;lt;\delta$&lt;/span&gt;, one has &lt;span class=&quot;math-container&quot;&gt;$|f(x+h)-f(x)|&amp;lt;\epsilon$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous if every infinitesimal increment &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; leads to an infinitesimal change &lt;span class=&quot;math-container&quot;&gt;$f(x+\alpha)-f(x)$&lt;/span&gt; (this happens to be Cauchy's definition of continuity).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It does not require great analytic skills to notice the difficulty of definition 1, due to a pair of alternations of quantifiers.&lt;/p&gt;&#xA;&lt;p&gt;It would therefore be silly to explain the difficulty of epsilon-delta in terms of insufficient command of natural language.  If this were the case, calculus would be a breeze for English majors! (this is not known to be the case as far as I know; if anything, the opposite).&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2023-10-16T11:22:38.533" LastActivityDate="2023-10-16T11:22:38.533" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="26930" PostTypeId="2" ParentId="26919" CreationDate="2023-10-15T15:10:24.670" Score="6" Body="&lt;p&gt;I think the linguist is looking at it backwards: to study the formal grammar of a language, you have to apply some complex mathematical (or logical) ideas.&lt;/p&gt;&#xA;&lt;p&gt;Natural languages are imprecise, contextual, and intuitive. The millions of people who speak English fluently, and communicate effectively in it, are not &amp;quot;faking it&amp;quot;, they are using an incredibly sophisticated set of &lt;em&gt;unconscious&lt;/em&gt; skills.&lt;/p&gt;&#xA;&lt;p&gt;Mathematicians have devised &lt;em&gt;artificial&lt;/em&gt; languages, firstly to remove the ambiguities of natural language, and secondly as short-hand to refer to complex ideas. The rules of these artificial languages are rigid and plainly logical.&lt;/p&gt;&#xA;&lt;p&gt;Linguists attempt, among other things, to detect underlying rules in how native speakers construct and understand language. They propose formal models of how the language works, and test them against examples from the real world. Those models treat natural languages as, at some deep level, as rigid as the mathematicians' artificial languages.&lt;/p&gt;&#xA;&lt;p&gt;That level of &amp;quot;mastery of your own language&amp;quot; is like having a &amp;quot;mastery of your own car&amp;quot; that would allow you to strip it down to individual nuts and bolts and build it back up again - admirable, maybe, but for most people a &lt;em&gt;sufficient&lt;/em&gt; mastery would be fixing a flat tyre.&lt;/p&gt;&#xA;" OwnerUserId="17848" LastActivityDate="2023-10-15T15:10:24.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26931" PostTypeId="2" ParentId="26922" CreationDate="2023-10-15T18:23:27.273" Score="3" Body="&lt;p&gt;The question is framed very oddly, so I'm not really sure what you're looking for.&lt;/p&gt;&#xA;&lt;p&gt;If the student understands that p(x) being factorable means that p(x)=0 has roots when any linear factor equals 0, then you can use desmos to graph y=^n + a^n. For all real nonzero values of a and even values of n, this does not cross the x-axis, hence there are no linear factors.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-10-16T04:32:13.097" LastActivityDate="2023-10-16T04:32:13.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26932" PostTypeId="1" AcceptedAnswerId="26937" CreationDate="2023-10-15T19:58:45.883" Score="0" ViewCount="286" Body="&lt;p&gt;You can safely presuppose that 13 year old (y.o.) students learned the &lt;a href=&quot;https://www.youtube.com/watch?v=oPCAI-tRF8o&quot; rel=&quot;nofollow noreferrer&quot;&gt;Difference of Squares&lt;/a&gt; and &lt;a href=&quot;https://redd.it/11v3tlh&quot; rel=&quot;nofollow noreferrer&quot;&gt;of Cubes&lt;/a&gt; identities, before tackling this &lt;a href=&quot;https://math.stackexchange.com/q/4740145&quot;&gt;Difference of Powers Identity (&lt;span class=&quot;math-container&quot;&gt;$x^n – y^n$&lt;/span&gt;)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://math.stackexchange.com/q/117660&quot;&gt;glut&lt;/a&gt; of &lt;a href=&quot;https://math.stackexchange.com/q/188657&quot;&gt;duplicates&lt;/a&gt; proves that students need better motivation. How can teachers demystify the 2 key steps below?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Too many students can’t even forebode the 1st step, which is the &lt;a href=&quot;https://math.stackexchange.com/a/2092564&quot;&gt;Sum of a Geometric Sequence&lt;/a&gt;: &lt;span class=&quot;math-container&quot;&gt;$\color{peru}{r^n -1 ≡ (r – 1)(r^{n-1}+r^{n-2}+...+r^2+r+1)}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Step 2 is even more Delphic! &lt;a href=&quot;https://math.stackexchange.com/a/2601906&quot;&gt;The Difference of Powers Identity has no division! Then how can students foreknow to substitute &lt;span class=&quot;math-container&quot;&gt;$r = x/y$&lt;/span&gt;&lt;/a&gt;, into the above &lt;span class=&quot;math-container&quot;&gt;$\color{darkgoldenrod}{\text{Sum of a Geometric Sequence}}$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I’m NOT asking how to &lt;a href=&quot;https://math.stackexchange.com/a/1812555&quot;&gt;prove&lt;/a&gt; with algebraic expansion. This identity is exercise 1v in Michael Spivak, &lt;em&gt;Calculus&lt;/em&gt; (4th edn 2008), page 13.&lt;/p&gt;&#xA;" OwnerUserId="23094" LastEditorUserId="23094" LastEditDate="2023-10-15T23:36:17.730" LastActivityDate="2023-10-17T21:33:31.197" Title="How to motivate $x^n−y^n ≡ (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$, to 13 year olds?" Tags="|middle-school|" AnswerCount="4" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="26933" PostTypeId="2" ParentId="26932" CreationDate="2023-10-15T22:12:17.150" Score="5" Body="&lt;p&gt;Since you're asking about teaching the general difference-of-&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th-powers formula, I'll assume the students have already learned the difference-of-squares and difference-of-cubes formulas:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;n=2: \quad x^2 - y^2 &amp;amp;= (x - y)(x + y)\\&#xA;n=3: \quad x^3 - y^3 &amp;amp;= (x - y)(x^2 + xy + y^2)&#xA;\end{align*}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;If they haven't learned these yet, then something is very wrong with either 1) their curriculum progression leading up to this point, or 2) the expectation that they should be able to learn the general difference-of-&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th-powers formula.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;To nudge them towards the general formula, I'd rewrite the above with color-coding and all exponents explicit:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;n=2: \quad \color{blue}{x^2} - \color{red}{y^2} &amp;amp;= (\color{blue}{x} - \color{red}{y})(\color{blue}{x^1}\color{red}{y^0} + \color{blue}{x^0}\color{red}{y^1}) \\&#xA;n=3: \quad \color{blue}{x^3} - \color{red}{y^3} &amp;amp;= (\color{blue}{x} - \color{red}{y})(\color{blue}{x^2}\color{red}{y^0} + \color{blue}{x^1}\color{red}{y^1} + \color{blue}{x^0}\color{red}{y^2})&#xA;\end{align*}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then I'd ask them to describe the pattern to me. (If they can't do this, then I'd use Socratic questioning to get them over the hump.)&lt;/p&gt;&#xA;&lt;p&gt;Once they've successfully described the pattern, I'd ask them to take their best guess at the case of &lt;span class=&quot;math-container&quot;&gt;$n=4.$&lt;/span&gt; We'd multiply the RHS out the long way to verify it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$n=4: \quad \color{blue}{x^4} - \color{red}{y^4} = (\color{blue}{x} - \color{red}{y})(\color{blue}{x^3}\color{red}{y^0} + \color{blue}{x^2}\color{red}{y^1} + \color{blue}{x^1}\color{red}{y^2} + \color{blue}{x^0}\color{red}{y^3})$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And then I'd ask them to write down the formula for arbitrary &lt;span class=&quot;math-container&quot;&gt;$n.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\color{blue}{x^n} - \color{red}{y^n} = (\color{blue}{x} - \color{red}{y})(\color{blue}{x^{n-1}}\color{red}{y^0} + \color{blue}{x^{n-2}}\color{red}{y^1} + \cdots + \color{blue}{x^1}\color{red}{y^{n-2}} + \color{blue}{x^0}\color{red}{y^{n-1}})$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-10-15T22:12:17.150" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26934" PostTypeId="1" AcceptedAnswerId="26935" CreationDate="2023-10-15T23:31:35.503" Score="5" ViewCount="16867" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$|a – b| &amp;lt; ε, \forall ε &amp;gt; 0 \iff a = b$&lt;/span&gt; resurfaces on standardized tests to 17 year old (y.o.) students, who can memorize and regurgitate &lt;a href=&quot;https://math.stackexchange.com/a/1249401&quot;&gt;the proof&lt;/a&gt; to earn full marks.&lt;/p&gt;&#xA;&lt;p&gt;But the &lt;a href=&quot;https://math.stackexchange.com/q/1426290&quot;&gt;glut&lt;/a&gt; of &lt;a href=&quot;https://math.stackexchange.com/q/2312613&quot;&gt;duplicates&lt;/a&gt; substantiates that most students can’t intuit it! How can I demystify this equivalence, to 17 y.o.? It's counterintuitive and paradoxical for &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; to become equivalent!&lt;/p&gt;&#xA;&lt;p&gt;How does the &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; uncannily transmute to an &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;?&lt;br /&gt;&#xA;Vice versa, how can &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; eerily transmogrify into &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="23094" LastEditorUserId="851" LastEditDate="2023-10-19T04:41:22.553" LastActivityDate="2023-10-26T14:56:12.440" Title="To 17 year olds, how can I explain that two numbers with arbitrarily small difference are equal?" Tags="|secondary-education|" AnswerCount="13" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="26935" PostTypeId="2" ParentId="26934" CreationDate="2023-10-15T23:42:29.800" Score="36" Body="&lt;p&gt;I'd recommend to play a little game with them. You choose a number &lt;span class=&quot;math-container&quot;&gt;$a.$&lt;/span&gt; They choose a number &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; that they think will satisfy &lt;span class=&quot;math-container&quot;&gt;$|a-b| &amp;lt; \epsilon \,\,\, \forall \epsilon &amp;gt; 0.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You try to state a counterexample, that is, an &lt;span class=&quot;math-container&quot;&gt;$\epsilon &amp;gt; 0$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$|a-b| \geq \epsilon.$&lt;/span&gt; If you can do this, you win. Otherwise, if you can't, then they win.&lt;/p&gt;&#xA;&lt;p&gt;Enough rounds of this, and they should see that the only way they can win is if they choose &lt;span class=&quot;math-container&quot;&gt;$b = a.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Some additional notes:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;To help students develop the right intuition, I'd recommend to refer to &lt;span class=&quot;math-container&quot;&gt;$|a-b|$&lt;/span&gt; as the &lt;em&gt;distance&lt;/em&gt; between &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; throughout the game. (And if they don't understand why it represents distance, then it would be worth pausing the game to review that, since it's such an important part of the intuition.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[Noted by Michal Miśkiewicz] This game not only leads the students to intuitively grasp the equivalence, but also leads them into the  proof: after enough rounds, they should also see that there's a simple rule for producing some &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; for each &lt;span class=&quot;math-container&quot;&gt;$b.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-16T15:14:49.253" LastActivityDate="2023-10-16T15:14:49.253" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="26936" PostTypeId="2" ParentId="26934" CreationDate="2023-10-16T13:13:26.413" Score="3" Body="&lt;p&gt;Here are a few possible leads.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ask your students to prove the following equality, then discuss with them whether they find this intuitive or counterintuitive, and why: &lt;span class=&quot;math-container&quot;&gt;$$0.999999... = 1$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Given two real numbers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, consider the three quantities &lt;span class=&quot;math-container&quot;&gt;$d_☾(a,b)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d_☆(a, b)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d_(a, b)$&lt;/span&gt; defined by:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}d_☾(a, b) &amp;amp; = \min\left\{\varepsilon \geq 0 \,\mid\,\left|{a-b}\right|\leq\varepsilon \right\} \\&#xA;d_☆(a, b) &amp;amp; = \inf\left\{\varepsilon &amp;gt; 0 \,\mid\,\left|{a-b}\right|&amp;lt;\varepsilon \right\} \\&#xA;d_(a, b) &amp;amp; = \left|{a-b}\right|&#xA;\end{align*}&lt;/span&gt; then have your students prove that:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$d_☾ = d_☆ = d_$$&lt;/span&gt; then reword your previous theorem as:&lt;span class=&quot;math-container&quot;&gt;$$\forall a, b \in \mathbb{R},\quad d_☆(a,b) = 0 \iff a = b$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Consider the definition of convergence for a sequence &lt;span class=&quot;math-container&quot;&gt;$(b_n)_{n \in \mathbb{N}}$&lt;/span&gt; converging to a limit &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\forall \varepsilon &amp;gt; 0,\, \exists N \in \mathbb{N},\, \forall n \geq N,\, |a-b_n| &amp;lt; \varepsilon$$&lt;/span&gt;&#xA;then discuss the specific case of a stationary sequence, &lt;em&gt;i.e&lt;/em&gt; the case where you have a number &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\forall n \in \mathbb{N},\, b_n = b$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Consider the following propositions for two converging sequences &lt;span class=&quot;math-container&quot;&gt;$(a_n)_{n \in \mathbb{N}}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(b_n)_{n \in \mathbb{N}}$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\forall n,\, a_n \leq b_n &amp;amp; \implies \lim_n a_n \leq \lim_n b_n \\&#xA;\forall n,\, a_n &amp;lt; b_n &amp;amp; \implies \lim_n a_n \leq \lim_n b_n&#xA;\end{align*}&lt;/span&gt;&#xA;And note how we are forced to accept the possibility of equality appearing in the limit.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Note that I used a lot of formal notation in the second and third points, but the notion of &amp;quot;distance between two points&amp;quot; is meant to be rather intuitive and the fact that we have more than one manner to define it with formal symbols should spark a discussion. As for the definition of convergence of a sequence, you can illustrate it graphically, then ask your students what this graph becomes in the case of a constant sequence.&lt;/p&gt;&#xA;&lt;p&gt;Also, if your students are not familiar with the definition of &lt;span class=&quot;math-container&quot;&gt;$\inf$&lt;/span&gt;, then great! Rewrite the definition of &lt;span class=&quot;math-container&quot;&gt;$d_☆$&lt;/span&gt; using only &lt;span class=&quot;math-container&quot;&gt;$\min$&lt;/span&gt; and discuss the special cases. This will force you to make things more explicit and deal with what the abstract &lt;span class=&quot;math-container&quot;&gt;$\forall \varepsilon &amp;gt; 0$&lt;/span&gt; really hides.&lt;/p&gt;&#xA;&lt;p&gt;There are several possible proofs for the equality &lt;span class=&quot;math-container&quot;&gt;$0.999... = 1$&lt;/span&gt;; one possible takeaway is &amp;quot;they are equal because you cannot squeeze a third number inbetween&amp;quot;, ie there is no &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$0.999... &amp;lt; x &amp;lt; 1$&lt;/span&gt;. Or alternatively they are equal because the distance between them is 0. In fact it's a direct application of your &lt;span class=&quot;math-container&quot;&gt;$\forall \varepsilon$&lt;/span&gt;-proposition: since &lt;span class=&quot;math-container&quot;&gt;$0.9 &amp;lt; 0.999... &amp;lt; 1$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$\left| 1 - 0.999... \right| &amp;lt; 0.1$&lt;/span&gt;, and likewise &lt;span class=&quot;math-container&quot;&gt;$0.99 &amp;lt; 0.999... &amp;lt; 1$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$0.999 &amp;lt; 0.999... &amp;lt; 1$&lt;/span&gt;, etc. with any finite number of 9 after the decimal point, and thus &lt;span class=&quot;math-container&quot;&gt;$\left| 1 - 0.999... \right| &amp;lt; 0.0001$&lt;/span&gt; for any number of 0 before the 1, which can be rewritten as &lt;span class=&quot;math-container&quot;&gt;$\forall n \in \mathbb{N},\, \left| 1 - 0.999... \right| &amp;lt; 10^{-n}$&lt;/span&gt;, which is equivalent to &lt;span class=&quot;math-container&quot;&gt;$\forall \varepsilon &amp;gt; 0,\, \left| 1 - 0.999... \right| &amp;lt; \varepsilon$&lt;/span&gt;, which by your proposition is equivalent to &lt;span class=&quot;math-container&quot;&gt;$0.999... = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16159" LastEditorUserId="16159" LastEditDate="2023-10-18T10:21:34.887" LastActivityDate="2023-10-18T10:21:34.887" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="26937" PostTypeId="2" ParentId="26932" CreationDate="2023-10-16T15:29:42.067" Score="4" Body="&lt;p&gt;Here a slightly different proposal.&lt;/p&gt;&#xA;&lt;p&gt;Ask the students to find &lt;span class=&quot;math-container&quot;&gt;$A_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A_2$&lt;/span&gt;, ... , &lt;span class=&quot;math-container&quot;&gt;$A_n$&lt;/span&gt; such that the formula&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x^n - y^n = (x-y)(A_1+A_2+ \dots + A_n)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;holds.&lt;/p&gt;&#xA;&lt;p&gt;You can get that &lt;span class=&quot;math-container&quot;&gt;$A_1 = x^{n-1}$&lt;/span&gt; such that the product by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt;. But doing this will generate an extra factor &lt;span class=&quot;math-container&quot;&gt;$B_1= -y \cdot A_1 = -x^{n-1}y$&lt;/span&gt; when multiplied by &lt;span class=&quot;math-container&quot;&gt;$-y$&lt;/span&gt;. We can however choose &lt;span class=&quot;math-container&quot;&gt;$A_2$&lt;/span&gt; such that it cancels out &lt;span class=&quot;math-container&quot;&gt;$B_1$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$A_2 = x^{n-2}y$&lt;/span&gt;. When multiplied by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A_2$&lt;/span&gt; cancels out exactly &lt;span class=&quot;math-container&quot;&gt;$B_1$&lt;/span&gt;, but it generates an extra factor &lt;span class=&quot;math-container&quot;&gt;$B_2 = -y \cdot A_2=-x^{n-2}y^2$&lt;/span&gt; when multiplied by &lt;span class=&quot;math-container&quot;&gt;$-y$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This in turns generates &lt;span class=&quot;math-container&quot;&gt;$A_3 = x^{n-3}y^2$&lt;/span&gt; and so on.&lt;/p&gt;&#xA;&lt;p&gt;Eventually we arrive at &lt;span class=&quot;math-container&quot;&gt;$A_n = y^{n-1}$&lt;/span&gt; that cancels out &lt;span class=&quot;math-container&quot;&gt;$B_{n-1}$&lt;/span&gt; when multiplied by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and generates the required factor &lt;span class=&quot;math-container&quot;&gt;$-y^n$&lt;/span&gt; when multiplied by &lt;span class=&quot;math-container&quot;&gt;$-y$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="22544" LastActivityDate="2023-10-16T15:29:42.067" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26938" PostTypeId="2" ParentId="26892" CreationDate="2023-10-16T16:39:09.457" Score="2" Body="&lt;p&gt;Suppose you have an event with tiny but positive probability &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;. As an example, you have &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; ping-pong balls, all white except for &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; golden ball, and you choose among them with equal probability, so the probability of choosing the golden ball is &lt;span class=&quot;math-container&quot;&gt;$p = \frac{1}{100}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now suppose that you repeat the event &lt;span class=&quot;math-container&quot;&gt;$1/p$&lt;/span&gt; times rounded down to the nearest integer, i.e. a number of times equal to &lt;span class=&quot;math-container&quot;&gt;$\lfloor \frac{1}{p}\rfloor$&lt;/span&gt;. In the ping-pong ball example, you replace the chosen ball, mix up the 100 balls, and choose again randomly, repeating this a total of &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; times.&lt;/p&gt;&#xA;&lt;p&gt;What's the probability of observing the event at least &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; time?&lt;/p&gt;&#xA;&lt;p&gt;The answer is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$1 - \bigl(1 - p\bigr)^{\lfloor 1/p \rfloor}&#xA;$$&lt;/span&gt;&#xA;Of course, this makes no sense when &lt;span class=&quot;math-container&quot;&gt;$p=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But as &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; gets closer and closer to zero, this probability converges to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{p \to 0} \left( 1 - \bigl(1 - p\bigr)^{\lfloor 1/p \rfloor}\right) = 1 - e^{-1} \approx 0.63212055882&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Comment:&lt;/strong&gt; My first version of this had an integer variable and the expression &lt;span class=&quot;math-container&quot;&gt;$\lim_{n \to \infty} (1 - (1-1/n)^n) = 1-e^{-1}$&lt;/span&gt;, and then I read more closely to see that you asked for a continuous variable. So I had to fudge the example, which leaves a trace of its discrete origin in the expression &lt;span class=&quot;math-container&quot;&gt;$\lfloor 1/p \rfloor$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Still, though, engineers might like this, they have to deal with probabilities. What's particularly nice is that it challenges some peoples' faulty intuition of probability, which goes something like this: &lt;em&gt;&amp;quot;If you do &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; repetitions of an event of probability &lt;span class=&quot;math-container&quot;&gt;$1/n$&lt;/span&gt;, you should definitely see a success&amp;quot;&lt;/em&gt;... Well, no, but you do have about a &lt;span class=&quot;math-container&quot;&gt;$63\%$&lt;/span&gt; chance of success.&lt;/p&gt;&#xA;" OwnerUserId="7258" LastEditorUserId="7258" LastEditDate="2023-10-18T13:47:20.993" LastActivityDate="2023-10-18T13:47:20.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26939" PostTypeId="2" ParentId="26934" CreationDate="2023-10-16T18:17:19.373" Score="13" Body="&lt;p&gt;If a student sees &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; somehow becoming equivalent here, they exhibit a common beginner mistake:&lt;/p&gt;&#xA;&lt;p&gt;They don't actually read the statement they are considering.&lt;/p&gt;&#xA;&lt;p&gt;Quantifiers are a particularly common victim of this: They just get treated as decorations, rather than them (and their order) being as utterly crucial to the meaning of a statement as they are. Other common examples are not distinguishing between &amp;quot;element of&amp;quot; and &amp;quot;subset of&amp;quot;; or whether a property refers to a set or the elements thereof.&lt;/p&gt;&#xA;" OwnerUserId="13471" LastEditorUserId="13471" LastEditDate="2023-10-16T19:52:28.353" LastActivityDate="2023-10-16T19:52:28.353" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26940" PostTypeId="2" ParentId="26934" CreationDate="2023-10-16T21:21:56.743" Score="0" Body="&lt;p&gt;We need to know a couple of properties of the absolute value: that &lt;span class=&quot;math-container&quot;&gt;$|x| \geq 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|x|=0$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$|x|=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then the easy part goes like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a=b \implies |a-b|=|0|=0 \implies \forall ε&amp;gt;0, ε&amp;gt;|a–b|$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We see in this case the &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$ε&amp;gt;|a–b|$&lt;/span&gt; actually comes from the &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$ε&amp;gt;0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now going the other way, the chain of implications is as follows:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall ε&amp;gt;0, ε&amp;gt;|a–b| \implies |a-b| \leq 0$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\implies |a-b|&amp;lt;0 \wedge |a-b|=0 \implies |a-b|=0 \implies a-b=0 \implies a=b$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The first implication is the hard one, and the only part that involves any Real analysis, arising from the Dedekind cut definition of the Reals. For beginners, I'd divide the real number line into two disjoint sets &lt;span class=&quot;math-container&quot;&gt;$x \leq 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x &amp;gt; 0$&lt;/span&gt;, and observe that all the the numbers in the first are safe, but all the numbers in the latter eventually get chopped off by some even tinier &lt;span class=&quot;math-container&quot;&gt;$ε$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The second implication follows from the definition of &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt;, the third from &lt;span class=&quot;math-container&quot;&gt;$|x| \geq 0$&lt;/span&gt;, the fourth from &lt;span class=&quot;math-container&quot;&gt;$|x|=0 \iff |x|=0$&lt;/span&gt;, and the fifth by adding &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; to both sides. This is enough to see that the &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt; is turned into &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; by negation, and then the &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; part dropped, leaving an &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="23115" LastActivityDate="2023-10-16T21:21:56.743" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26941" PostTypeId="2" ParentId="26932" CreationDate="2023-10-16T21:25:34.233" Score="6" Body="&lt;p&gt;Start off by observing that:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$99=9 \times 11$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$999=9 \times 111$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$9999=9 \times 1111$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$99999=9 \times 11111$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The left hand side is a power of ten minus one. The second number on the right hand side is a sum of all the powers of ten up to but not including the one on the left. We obviously used base 10 here, but we can pull this trick in any base. Numbers of the form &lt;span class=&quot;math-container&quot;&gt;$111...1$&lt;/span&gt; are a sum of powers of the base. We multiply by the base minus one to get as high as we can without carrying, and then add one to trigger the carry and get the next base power. We can write this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(b-1)(b^{n-1}+b^{n-2}+\ldots b^2+b^1+b^0)=b^n-1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can prove this by just multiplying out the brackets, and observing that all but the first and last terms cancel.&lt;/p&gt;&#xA;&lt;p&gt;Then point out that the multiplying out brackets proof means this works for &lt;em&gt;any&lt;/em&gt; number &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, not just whole numbers suitable for use as a base. Fractions &lt;span class=&quot;math-container&quot;&gt;$b=x/y$&lt;/span&gt; are then an obvious thing to try out, to demonstrate that. Finally, multiply that by &lt;span class=&quot;math-container&quot;&gt;$y^n$&lt;/span&gt; to cancel the fractions, and get the desired result.&lt;/p&gt;&#xA;" OwnerUserId="23115" LastActivityDate="2023-10-16T21:25:34.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26942" PostTypeId="1" CreationDate="2023-10-16T21:27:11.583" Score="29" ViewCount="6822" Body="&lt;p&gt;I've been a TA across every class in the calculus sequence, under the assignment of professors with different teaching styles and curricula. It's often clear to me ahead of time when a certain subject is going to be a lot easier or harder for students to digest in the allotted amount of time, and in those situations I feel a desire to assuage any frustrations students might have ahead of time. I think it's more common than not, for example, to preempt a discussion of epsilon-delta with disclaimers like &amp;quot;this is a definition with a lot of moving parts,&amp;quot; &amp;quot;if this doesn't make sense right away, that's okay,&amp;quot; &amp;quot;it's a concept most students struggle with,&amp;quot; and &amp;quot;there's no way to get used to it but by grinding through a lot of practice problems.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If I don't offer a warning about a difficult subject, the students may become stressed out, ascribing their slower progress to something about themselves rather than about the material or the circumstance. And pragmatically speaking, it is in a student's best interest to have a keen sense of where they stand relative to their peers, so I sympathize with their concerns.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, students stress out even when I give them warnings. I also worry if this doesn't prime them to give up or become frustrated with the material sooner than if I had said nothing. &amp;quot;Okay, no use trying to understand this lecture, I'll just wait for the practice problems.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Are there any good guidelines for when the warning is worth the effort?&lt;/p&gt;&#xA;" OwnerUserId="14266" LastActivityDate="2023-10-18T23:38:19.553" Title="When is it appropriate to warn about the difficulty of a subject?" Tags="|undergraduate-education|student-motivation|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="26943" PostTypeId="2" ParentId="26942" CreationDate="2023-10-16T23:07:10.290" Score="29" Body="&lt;p&gt;I know this is generally accepted as normal behavior by teachers, but personally, if I ever find myself tempted to attach disclaimers like &amp;quot;don't worry if you feel overwhelmed by this&amp;quot;, then I view it as an indication that I'm about to try to take too big a conceptual leap, and instead, I need to figure out a way to better scaffold the curriculum.&lt;/p&gt;&#xA;&lt;p&gt;As a TA I guess you don't have control over how the teacher introduces the material, but in the off chance that you have enough bandwidth during TA sessions to re-teach from the ground up when needed, here's a rough outline of how I might scaffold &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; limit proofs:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Before I even mention the general definition of &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{x \to a} f(x) = L,$&lt;/span&gt; I instead tell students that &amp;quot;the general definition is easiest to understand if you first understand some concrete examples.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I state the definition in the case of &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{x \to 0} 2x = 0,$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\forall \epsilon &amp;gt; 0 \,\, \exists \delta &amp;gt; 0: |x| &amp;lt; \delta \Rightarrow |2x| &amp;lt; \epsilon,$$&lt;/span&gt;&#xA;and we play a game where I choose an &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; and they win by stating a &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; that makes &lt;span class=&quot;math-container&quot;&gt;$|x| &amp;lt; \delta \Rightarrow |2x| &amp;lt; \epsilon$&lt;/span&gt; true.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Once they find this game easy to win, I ask them to describe their &amp;quot;winning strategy&amp;quot; for general &lt;span class=&quot;math-container&quot;&gt;$\epsilon,$&lt;/span&gt; i.e. create a formula for &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; that will allow them to win regardless of what &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; I choose.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then I say &amp;quot;you guys are too good at this, I'm tired of losing, let's make the game harder,&amp;quot; and state the definition in a &lt;em&gt;slightly&lt;/em&gt; more complicated case like &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{x \to 3} 2x = 6.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Once they find &lt;em&gt;that&lt;/em&gt; game easy to win, we discuss the general definition of &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{x \to a} f(x) = L,$&lt;/span&gt; and then we continue playing the game with progressively more complicated &lt;span class=&quot;math-container&quot;&gt;$f(x),$&lt;/span&gt; discussing the different kinds of strategies that are necessary to win at each &amp;quot;higher level.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And finally I say something like &amp;quot;like any game, you're going to need plenty of practice to become good at it,&amp;quot; which conveys to students that they're going to need to grind practice problems, but doesn't prime them to give up or become frustrated.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-18T23:38:19.553" LastActivityDate="2023-10-18T23:38:19.553" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26944" PostTypeId="2" ParentId="26942" CreationDate="2023-10-17T00:26:48.950" Score="6" Body="&lt;p&gt;I delay that chapter for later in the course. I want them to understand derivatives (which have such great physical meanings) before they try to understand this very technical topic.&lt;/p&gt;&#xA;&lt;p&gt;As a TA, you can't do that. But you could tell them that the reason for this weirdness will be clearer after they've done derivatives.&lt;/p&gt;&#xA;&lt;p&gt;Also, I have a fun exercise that I use when teaching this. I use a (made-up) Cookie Crispness Index, where the x-axis is time in the oven, and the y-axis is crispness of your cookies.  Before we start, I ask everyone their favorite kind of cookie. (Chocolate chip almost always wins.) Then I get everyone to say how crisp they like them, on a scale of 0 to 10. (0 is cookie dough, 10 is charred.) Then I'm hoping someone will be cute and give a decimal. (If no one does that, I ask someone to play the game with me, of they're the pickiest person in the world.) Then we try to find a just right amount of cooking time, but we're never perfect.&lt;/p&gt;&#xA;&lt;p&gt;I draw a curve from (0 min,0 crispness) to (15 min, 10 crispness).&lt;/p&gt;&#xA;&lt;p&gt;They want to be within .01 of the perfect crispness, and I show the epsilon band for that, and we figure out how close the time needs to be. But of course, then they want to be even pickier. Etc.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-10-17T00:26:48.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26945" PostTypeId="2" ParentId="26934" CreationDate="2023-10-17T11:35:40.803" Score="1" Body="&lt;p&gt;Can your audience understand that  &lt;code&gt;p -&amp;gt; q&lt;/code&gt; is equivalent to  &lt;code&gt;non q -&amp;gt; non p&lt;/code&gt; ?&lt;/p&gt;&#xA;&lt;p&gt;And that is easy to prove: suppose non q does not imply non p, therefore non q can coexist with p;&#xA;but p -&amp;gt; q, therefore non q can coexist with q, which is a contradiction.&lt;/p&gt;&#xA;&lt;p&gt;(this can be easily visualized with Venn diagrams: p-&amp;gt;q means that everything that is p is also q, so the circle&#xA;representing p is included in the circle representing q; non p and non q are everything outside of these circles;&#xA;so if circle p is included in circle q, the exterior of circle q is included in the exterior of circle p)&lt;/p&gt;&#xA;&lt;p&gt;Going back to your example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;statement p&lt;/strong&gt; : &amp;quot;&lt;code&gt;two numbers a and b are different&lt;/code&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;implies &lt;strong&gt;statement q&lt;/strong&gt;: &amp;quot;&lt;code&gt;there exists ε&amp;gt;0 such that |a–b| &amp;gt;= ε&lt;/code&gt;&amp;quot;&#xA;(Since a and b are different, therefore their difference differs from 0, the absolute value of&#xA;that difference is a strictly positive number, so pick epsilon anywhere in the interval (0, |a-b|) )&lt;/p&gt;&#xA;&lt;p&gt;the &lt;strong&gt;negation of statement q&lt;/strong&gt; : &amp;quot;&lt;code&gt;there is no ε &amp;gt; 0 such that |a-b| &amp;gt;= ε&lt;/code&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;which is equivalent to your statement &amp;quot;&lt;code&gt;for every ε &amp;gt; 0, |a-b| &amp;lt; ε&lt;/code&gt;&amp;quot;&#xA;(if a certain property does not hold for ANY epsilon, then the opposite(complementary) property&#xA;must hold for EVERY epsilon)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;non q implies non p&lt;/strong&gt; : &amp;quot;&lt;code&gt;numbers a and b are not different &lt;/code&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;An intuitive explanation of the above: I would use the visualization of the numbers on the &lt;strong&gt;real axis&lt;/strong&gt;,&#xA;while making an analogy with &lt;strong&gt;maps&lt;/strong&gt; and &lt;strong&gt;zoom&lt;/strong&gt; levels:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;statement p&lt;/strong&gt;: &amp;quot;&lt;code&gt;points A and B are distinct on the map&lt;/code&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;implies &lt;strong&gt;statement q&lt;/strong&gt; : &amp;quot;&lt;code&gt;there exists a certain zoom level that would clearly show points A and B apart&lt;/code&gt; (separated by a specific distance)&amp;quot; .&#xA;Of course, not all zoom levels would show this, and there is no need to; also, there is not one single zoom level, but once&#xA;we found one, any other zoom level more detailed than that would also be ok.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;non q&lt;/strong&gt;: &amp;quot;&lt;code&gt;for any given zoom level, (no matter how detailed), points A and B still appear to overlap&lt;/code&gt;&amp;quot;&#xA;(with the reals numbers axis, we can zoom indefinitely, unlike real world maps)&lt;/p&gt;&#xA;&lt;p&gt;than students must understand that &lt;strong&gt;non p&lt;/strong&gt; is true: &amp;quot;&lt;code&gt;points A and B are not distinct&lt;/code&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Note : with regard to the fact that the sign &amp;quot;&amp;lt;&amp;quot; magically transformed into an &amp;quot;=&amp;quot;:&#xA;show your students a sequence of strictly negative numbers converging to zero.&#xA;Here, &amp;quot;strictly less than zero&amp;quot; also turns, in the end, into an &amp;quot;equal to zero&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;(EDIT)&lt;/p&gt;&#xA;&lt;p&gt;One more point :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It's counterintuitive and paradoxical for &amp;lt; and = to become equivalent&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No. It would have been paradoxical for &amp;quot;&lt;code&gt;a &amp;lt; b&lt;/code&gt;&amp;quot; to turn, at some point, into &amp;quot;&lt;code&gt;a = b&lt;/code&gt;&amp;quot;.&#xA;But this is not the case. The sign &amp;quot;&amp;lt;&amp;quot; is not applied between a and b, but between&#xA;|a-b|, on one side, and epsilon, on the other side. Your students should pay attention&#xA;to this aspect. Comparing the difference of a and b with some third term really tells&#xA;nothing about the relation between a and b.&lt;/p&gt;&#xA;&lt;p&gt;Please check this: is there any chance that your students read&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;|a–b|&amp;lt;ε,∀ε&amp;gt;0&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;as&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;a &amp;lt; b, ∀ε&amp;gt;0&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;?&lt;/p&gt;&#xA;" OwnerUserId="23126" LastEditorUserId="23126" LastEditDate="2023-10-17T12:28:01.490" LastActivityDate="2023-10-17T12:28:01.490" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="26946" PostTypeId="2" ParentId="26942" CreationDate="2023-10-17T16:17:49.293" Score="15" Body="&lt;p&gt;I can remember two teachers warning us about the difficulty of a subject.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The first warning was by a teacher of measure theory and probabilities.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the first half of the semester we'll study measure theory. There will be a midterm exam on measure theory. Then in the second half we'll study probabilities. The final exam will be entirely focused on probabilities. Every year I warn the students about the second half of the semester. Students seem to interiorise the concepts of measure theory perfectly, and then they understand nothing of its application to probabilities. In the past twenty years I have never managed to understand why. So I'm warning you. The second half of the semester is apparently more difficult than the first half, although I don't know why.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And his prediction turned true; most students struggled with the second half of the semester, and failed the final exam.&lt;/p&gt;&#xA;&lt;p&gt;It wasn't a warning so much as an admission of failure on his part.&lt;/p&gt;&#xA;&lt;p&gt;He was an awesome teacher in my opinion, but this was one big failure in his teaching. So, if you find yourself warning your students about the difficulty of a subject: try to fix it. Warning the students that they'll fail is not a solution, it is not helpful, it is only an admission of failure on your part.&lt;/p&gt;&#xA;&lt;p&gt;In the case of measure theory and probabilities, my understanding is that students were very good at learning all the very formal definitions and theorems. Students were very good at handling measurable functions &lt;span class=&quot;math-container&quot;&gt;$f : A \mapsto B$&lt;/span&gt;. But when we applied measure theory to probabilities, suddenly we were dealing with random variables, which are measurable functions &lt;span class=&quot;math-container&quot;&gt;$X : \Omega \mapsto B$&lt;/span&gt; except space &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; &amp;quot;doesn't matter&amp;quot; or is left unspoken and undefined, so really &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; feels like a function with a codomain but no domain, and students were confused because the very dry formalism of measure theory was now cohabitating with the very blurry vagueness of functions with undefined domains.&lt;/p&gt;&#xA;&lt;p&gt;But the teacher never identified this precise difficulty; so year after year he warned the students about a difficulty without being able to make his class less difficult. The materials covered in the second half of the semester is in my opinion not harder than the material covered in the first half; but it does require a shift in philosophy because of these unspoken domains.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The second warning was by a teacher of the C programming language&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So far we have only introduced variables. Now we will introduce pointers. This is a very hard topic and it makes language C much harder to learn than some other languages. It's okay if you don't get it at first.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Again, the prediction turns out to be true: many students start learning C, but never understand pointers and never become good C programmers, or at least it takes them an awful long time before they understand pointers.&lt;/p&gt;&#xA;&lt;p&gt;Pointers are variables that hold the memory addresses of variables.&lt;/p&gt;&#xA;&lt;p&gt;That's it. Pointers defined in one sentence, and it's actually super simple. So why are pointers so difficult? It's a self-fulfilling prophecy. Most teachers gloss over the topics of variables, which is reputed to be easy, and then dive into pointers and spend a lot of time on it, because it's reputed to be hard. But pointers are actually extremely simple, &lt;em&gt;provided the student really understands variables&lt;/em&gt;. Trying to teach pointers on the shaky foundations of mistaught variables is what makes it hard.&lt;/p&gt;&#xA;&lt;p&gt;Warning the students that &amp;quot;pointers are difficult, it's okay if you don't get it&amp;quot; is not helpful at all. It's laziness on the part of the teacher. Instead of warning the students of the difficulty, the teacher should identify where the difficulty stems from, and do a better job of explaining what a variable is.&lt;/p&gt;&#xA;" OwnerUserId="16159" LastActivityDate="2023-10-17T16:17:49.293" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26947" PostTypeId="1" CreationDate="2023-10-17T16:57:02.750" Score="2" ViewCount="288" Body="&lt;p&gt;A person is standing next to a hot air balloon. At the same time, the person starts moving away from the balloon at 5 ft/sec and the balloon rises straight into the air at a rate of 12 ft/sec. Is the distance between the person and the balloon increasing or decreasing 4 seconds after they start moving?&lt;/p&gt;&#xA;&lt;p&gt;Can this question be used to assess a student's understanding of calculus?&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2023-10-17T17:44:11.317" LastActivityDate="2023-10-18T15:36:05.483" Title="Is this a viable Calculus 1 question?" Tags="|mathematical-pedagogy|calculus|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="26948" PostTypeId="2" ParentId="26878" CreationDate="2023-10-17T18:32:06.240" Score="3" Body="&lt;p&gt;Having taken courses that require linear algebra ~10 times, I can tell you that even performant students need a refresher on linear algebra when they are coming into a class that requires linear algebra, no matter how many times they have had the refresher, or how recent their linear algebra class was.&lt;/p&gt;&#xA;&lt;p&gt;There is so much jargon, gotchas, caveats, and theorems in linear algebra that it isn't reasonable to expect students to recall these the moment they need them months or years after they last heard the words.&lt;/p&gt;&#xA;&lt;p&gt;Every such course should start with one or two weeks of refresher on basic linear algebra, covering every topic needed for the course. This refresher should be based on a concise, clear, well-formatted minimalistic source that they can reference throughout the course. Such linear algebra refreshers are common place. My personal favorite is the first chapter of Griffith's Quantum textbook, but I'm a physicist.&lt;/p&gt;&#xA;" OwnerUserId="23128" LastActivityDate="2023-10-17T18:32:06.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26949" PostTypeId="2" ParentId="26934" CreationDate="2023-10-17T21:08:18.820" Score="0" Body="&lt;p&gt;There's two things I have found which help with the intuition of this.  The first is to break &lt;span class=&quot;math-container&quot;&gt;$|a-b|&amp;lt;\epsilon$&lt;/span&gt; into parts: &lt;span class=&quot;math-container&quot;&gt;$a-b&amp;lt;\epsilon$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a-b&amp;gt;-\epsilon$&lt;/span&gt;.  I think the fact that you can &amp;quot;transmute &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;&amp;quot; is less astonishing when you have both a &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; and a &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt; written out explicitly.&lt;/p&gt;&#xA;&lt;p&gt;The second is a way of thinking about numbers that I found helpful for the famous &lt;span class=&quot;math-container&quot;&gt;$0.9999\ldots=1$&lt;/span&gt; argument.  In this approach, we ask what &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$0.9$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$0.99$&lt;/span&gt; &lt;em&gt;means&lt;/em&gt;, and start to ask what &lt;span class=&quot;math-container&quot;&gt;$0.9999\ldots$&lt;/span&gt; means.  The approach I have found successful is one I borrow from how we handle time: say &amp;quot;1&amp;quot;, &amp;quot;0.9&amp;quot; or &amp;quot;0.999...&amp;quot; is a &amp;quot;name&amp;quot; for a number, not the number itself.  (This is similar to the Peano arithmetic argument that &amp;quot;2&amp;quot; is the name for the number &lt;span class=&quot;math-container&quot;&gt;$S(S(0))$&lt;/span&gt;, where S is the successor function)&lt;/p&gt;&#xA;&lt;p&gt;From there, we can discuss two names of real numbers, &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.  What must be true about them?  What can we say about two named numbers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; that are equal, knowing the above oddness of &lt;span class=&quot;math-container&quot;&gt;$0.999\ldots=1$&lt;/span&gt; above.  Its easy to see how this is just a special case of the general version.  And after going through those steps, it starts to be more intuitive to see that the original &lt;span class=&quot;math-container&quot;&gt;$|a-b|&amp;lt;\epsilon$&lt;/span&gt; handles these odd cases well.&lt;/p&gt;&#xA;&lt;p&gt;Another approach is to turn it around. Because &lt;span class=&quot;math-container&quot;&gt;$|a – b| &amp;lt; ε, \forall ε &amp;gt; 0 \iff a = b$&lt;/span&gt;, we can also write the opposite: &lt;span class=&quot;math-container&quot;&gt;$\exists ε &amp;gt; 0|a – b| \ge ε \iff a \ne b$&lt;/span&gt;.  It becomes the definition of the minimum requirements for two named numbers to be different.&lt;/p&gt;&#xA;" OwnerUserId="5787" LastActivityDate="2023-10-17T21:08:18.820" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26950" PostTypeId="2" ParentId="26932" CreationDate="2023-10-17T21:33:31.197" Score="2" Body="&lt;p&gt;Would long division help? I had something like this in high school&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\require{enclose}&#xA;x - y \enclose{longdiv} {x^2 \qquad- y^2}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We then figured out the answer x + y. You can increase the powers of the dividend to 3, then 4, and eventually n.&lt;/p&gt;&#xA;" OwnerUserId="14854" LastActivityDate="2023-10-17T21:33:31.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26951" PostTypeId="2" ParentId="26934" CreationDate="2023-10-18T00:19:17.487" Score="2" Body="&lt;p&gt;Draw a number line and label a point &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. Then ask them to draw which points &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; satisfy the condition &lt;span class=&quot;math-container&quot;&gt;$|a-b| &amp;lt; \epsilon$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="23135" LastActivityDate="2023-10-18T00:19:17.487" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26952" PostTypeId="2" ParentId="26934" CreationDate="2023-10-18T08:29:36.533" Score="0" Body="&lt;p&gt;I think there are lots of aspects of the original equivalence that distract from the key point.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It's presented as an equivalence; let's just focus on the more conceptually different direction &lt;span class=&quot;math-container&quot;&gt;$(\forall \varepsilon&amp;gt;0:|a-b|&amp;lt;\varepsilon) \Longrightarrow a=b$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;It's all written in symbols: let's use more natural language: &amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$|a-b|&amp;lt;\varepsilon$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$\varepsilon&amp;gt;0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt;&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;There are really three steps here: concluding that &lt;span class=&quot;math-container&quot;&gt;$|a-b|\leq 0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$|a-b|=0$&lt;/span&gt;, and finally &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt;. Again, the first bit is (hopefully) the main difficulty. So let's get rid of this norm business: &amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;\varepsilon$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$\varepsilon&amp;gt;0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x\leq 0$&lt;/span&gt;.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Even the &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; is unnecessary: all you really need is &amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$x\neq\varepsilon$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$\varepsilon&amp;gt;0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x\leq 0$&lt;/span&gt;.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Now it's fairly straightforward: &amp;quot;I'm thinking of a (real) number. I know it's not equal to any positive number. So it must be either negative or zero.&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Once the last line is clear, you can start working upwards. And then you can think about why &lt;span class=&quot;math-container&quot;&gt;$\forall \varepsilon&amp;gt;0:x\leq \varepsilon$&lt;/span&gt; would imply &lt;span class=&quot;math-container&quot;&gt;$\forall \delta&amp;gt;0:x&amp;lt;\delta$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="20327" LastActivityDate="2023-10-18T08:29:36.533" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26953" PostTypeId="2" ParentId="26934" CreationDate="2023-10-18T09:06:03.820" Score="3" Body="&lt;p&gt;I admit that I cannot remember how I thought as a 17 year old, but I am pretty sure my intuitive approach would have been this:&lt;/p&gt;&#xA;&lt;p&gt;From right to left:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Assume &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Hence, &lt;span class=&quot;math-container&quot;&gt;$|a-b|=0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;As given, we know that &lt;span class=&quot;math-container&quot;&gt;$\epsilon&amp;gt;0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$|a-b| &amp;lt; \epsilon$&lt;/span&gt;, qed.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;em&gt;I hope this part is so trivial that it is easy for them. If not, then you probably are fighting with cases of advanced boredom, severe uninterestedness and possibly hormonally induced unstableness in your pupils. &amp;lt;/rant&amp;gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;From left to right:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Preparing a proof by contradiction, assume &lt;span class=&quot;math-container&quot;&gt;$|a-b| &amp;gt; 0$&lt;/span&gt;, i.e. the opposite of the right-hand side.&lt;/li&gt;&#xA;&lt;li&gt;Calculate &lt;span class=&quot;math-container&quot;&gt;$|a-b| / 2$&lt;/span&gt;, and define &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; to be that. Make sure they recognize that this particular choice satisfies &lt;span class=&quot;math-container&quot;&gt;$\epsilon&amp;gt;0$&lt;/span&gt;, and thus conforms to the problem description.&lt;/li&gt;&#xA;&lt;li&gt;Check whether &lt;span class=&quot;math-container&quot;&gt;$|–|&amp;lt;\epsilon$&lt;/span&gt; holds (remember, per the task description, this should be true for &lt;em&gt;any and all&lt;/em&gt; non-0 &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;) by plugging in the value of &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; from the previous step: &lt;span class=&quot;math-container&quot;&gt;$|a-b| &amp;lt; |a-b|/2$&lt;/span&gt;. Clearly, this can not be true for &lt;em&gt;any&lt;/em&gt; combination of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;This completes our proof by contradiction; hence &lt;span class=&quot;math-container&quot;&gt;$|a-b|=0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To sum it up; I used the following concepts, and your 17 year olds would be able to figure this out if they are familiar with them:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;To prove a &lt;span class=&quot;math-container&quot;&gt;$&quot;&amp;lt;=&amp;gt;&quot;$&lt;/span&gt;, you can split it in the two &amp;quot;directions&amp;quot; (i.e. &lt;span class=&quot;math-container&quot;&gt;$&quot;=&amp;gt;&quot;$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$&quot;&amp;lt;=&quot;$&lt;/span&gt;). Divide &amp;amp; conquer.&lt;/li&gt;&#xA;&lt;li&gt;Many proofs use the principle of &amp;quot;proof by contradiction&amp;quot;, i.e. assuming the opposite of what you wish to prove, and proving that from that follows something untrue. Obviously, this is already a slightly higher-level concept, if they have difficulty with that, then it's time to draw the good old boolean diagram for &amp;quot;=&amp;gt;&amp;quot; on the board...&#xA;(And N.B.: an additional pointer would be that proofs of the form &amp;quot;for all ...&amp;quot; &lt;em&gt;often&lt;/em&gt; are amenable to proof by contradiction since the opposite is &amp;quot;there is no ...&amp;quot;, and then you just &lt;em&gt;assume&lt;/em&gt; there &lt;em&gt;is&lt;/em&gt; one &amp;quot;...&amp;quot; and use that &amp;quot;...&amp;quot; in the proof. In other words, it's often mechanically straight-forward to do.)&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;for all &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;&amp;quot; may seem daunting, but the actual proof in this case doesn't require any kind of set-theoretic construct at all; it just requires to pick instances for the numbers which satisfy the conditions from the problem description. It should not be difficult to see that if you can pick any &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; at random, then it holds for all &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;. If this &lt;em&gt;does&lt;/em&gt; provide difficulties, again, you can have them try to prove the opposite, which is &amp;quot;there exists at least &lt;em&gt;one&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; for which the opposite holds&amp;quot;, and this will quickly lead them to the impossibility of that in a more intuitive manner.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am glossing over the Axiom of Choice here when asking them to &amp;quot;pick any &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;&amp;quot; and would reserve that topic for a latter date.&lt;/p&gt;&#xA;" OwnerUserId="8146" LastEditorUserId="8146" LastEditDate="2023-10-19T07:22:25.767" LastActivityDate="2023-10-19T07:22:25.767" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26954" PostTypeId="2" ParentId="26934" CreationDate="2023-10-18T09:18:50.120" Score="0" Body="&lt;p&gt;If for every non-zero &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a - b$&lt;/span&gt; is not &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;, then by deduction &lt;span class=&quot;math-container&quot;&gt;$a - b = 0$&lt;/span&gt;.&#xA;On the other hand, when &lt;span class=&quot;math-container&quot;&gt;$a - b = 0$&lt;/span&gt; it's clearly true that for every non-zero &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a - b$&lt;/span&gt; is not &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;, since &lt;span class=&quot;math-container&quot;&gt;$a - b$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; which is not &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thus, &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon \neq 0,\, a - b \neq \epsilon \iff a = b$&lt;/span&gt;.&#xA;This is the basic principle.&lt;/p&gt;&#xA;&lt;p&gt;Take the absolute value and this becomes &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt; 0,\, |a - b| \neq \epsilon \iff a = b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt; 0,\, |a - b| \neq \epsilon$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\forall \epsilon &amp;gt; 0,\, |a - b| &amp;lt; \epsilon$&lt;/span&gt; are equivalent, so the result follows.&lt;/p&gt;&#xA;" OwnerUserId="23141" LastEditorUserId="23141" LastEditDate="2023-10-18T09:19:43.967" LastActivityDate="2023-10-18T09:19:43.967" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26955" PostTypeId="2" ParentId="26934" CreationDate="2023-10-18T10:12:40.373" Score="3" Body="&lt;p&gt;You could start by using precise language.  Numbers don't have arbitrary small differences.  To get into &amp;quot;arbitrary small&amp;quot; domain, you need at least one constrained variable, like an &lt;span class=&quot;math-container&quot;&gt;$\epsilon&amp;gt;0$&lt;/span&gt;.  If this &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; will be greater than the difference between two numbers even though it can be chosen arbitrarily small within its domain, the two numbers under consideration have a difference of exactly zero.  Not an &amp;quot;arbitrarily small&amp;quot; difference.  What is arbitrarily small is not the difference of the numbers but a variable greater than the difference of the numbers.&lt;/p&gt;&#xA;&lt;p&gt;If you start calling the difference between two numbers &amp;quot;arbitrarily small&amp;quot;, you are teaching pupils to regurgitate meaningless words making the teacher happy.&lt;/p&gt;&#xA;" OwnerDisplayName="user23142" LastActivityDate="2023-10-18T10:12:40.373" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26956" PostTypeId="2" ParentId="26947" CreationDate="2023-10-18T14:33:15.647" Score="3" Body="&lt;p&gt;For me this is a question not for calculus but for right triangles, Pythagoras and not really a calculus question.&lt;/p&gt;&#xA;" OwnerUserId="21051" LastActivityDate="2023-10-18T14:33:15.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26957" PostTypeId="2" ParentId="26947" CreationDate="2023-10-18T14:57:25.377" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Is the distance between the person and the balloon increasing or decreasing 4 seconds after they start moving?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The balloon and the person are always moving further away from each other (balloon moving further away vertically, person moving further away horizontally), so it's just common sense that the distance between them is always increasing.&lt;/p&gt;&#xA;&lt;p&gt;That said, you might see this sort of question being asked as a &amp;quot;first step&amp;quot; in a calculus question consisting of multiple parts (for the purpose of scaffolding the student through the problem-solving procedure):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(i)&lt;/strong&gt; Is the distance between the person and the balloon increasing or decreasing after they start moving?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(ii)&lt;/strong&gt; Draw triangle with the following sides labeled: the height of the balloon, the distance of the person from the balloon launch site, and the distance between the person and the balloon.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(iii)&lt;/strong&gt; Set up an equation relating the sides of that triangle.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(iv)&lt;/strong&gt; Use implicit differentiation to differentiate the equation with respect to time.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(v)&lt;/strong&gt; How fast is the distance between the person and the balloon changing 4 seconds after they start moving? &lt;em&gt;Note: As a sanity check, make sure your answer does not contradict your answer to part (i).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, in conclusion, I would say that the question you've provided would not be a viable calculus question on its own, but could reasonably appear as a &amp;quot;first step&amp;quot; in a viable calculus question that is being scaffolded.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-10-18T14:57:25.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26958" PostTypeId="2" ParentId="26947" CreationDate="2023-10-18T15:36:05.483" Score="0" Body="&lt;p&gt;This question overall looks good, but I'll analyze it more in depth. Let's start at the second line.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the same time, the person starts moving away from the balloon at 5 ft/sec and the balloon starts rising into the air at 12 ft/sec. Is the distance between the person and the balloon increasing or decreasing 4 seconds after they start moving?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;On the surface, this looks like a good question. However, there is only one problem (Mentioned by @Sue VanHattum):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Calculus allows you to find the rate at which that distance is increasing. Logic (no calculus needed) tells you that it's increasing. The scenario is fine. The question (increasing or decreasing) makes no sense to me.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, to understand if this would be viable for a Calculus 1 course (I assume this is a question for a test early in the year), try looking at it yourself &lt;i&gt;from the perspective of one of your Calc 1 students seeing the question on the test&lt;/i&gt;. If there's anything that you find that they might find confusing (i.e., they should already know that the question is stupid, since obviously, as mentioned previously, it's going to be increasing), then try to edit &lt;i&gt;that part specifically&lt;/i&gt; so that it makes sense. In fact, I would probably rewrite the question as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A person is standing next to a hot air balloon. At the same time, the person starts moving away from the balloon at 5 ft/sec and the balloon starts rising into the air at 12 ft/sec. &lt;strong&gt;What is the rate at which the distance between the person and the balloon is going up?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I could probably write my rewrite of the question a bit better, although I'm not exactly sure how if that's the case.&lt;/p&gt;&#xA;&lt;p&gt;So overall, if this is supposed to be a question on a test near the beginning  of the year, then this is a good math question, however I would suggest wording the &lt;i&gt;overall question&lt;/i&gt;  a little better.&lt;/p&gt;&#xA;" OwnerUserId="22983" LastActivityDate="2023-10-18T15:36:05.483" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26959" PostTypeId="1" AcceptedAnswerId="26971" CreationDate="2023-10-18T23:38:33.333" Score="33" ViewCount="7355" Body="&lt;p&gt;When teaching elementary mathematics, it takes a lot of time and effort to teach students that our goal is not to learn the examples, but to learn the concepts first, and then apply them to specific examples or problems. But students have the tendency to just try to learn the examples and skip the concepts. You see this issue when students attempt to do their homework without having learned the concepts first, and then if there is a homework problem that they have not seen before, they get stuck, even if it can be solved by a simple application of definitions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example.&lt;/strong&gt; Consider the concept of continuity, for instance (although my question is broad and is not related to this particular concept). You give your students a problem to determine whether a certain function is continuous at a given point or not. They get stuck and come to you for help. You ask them if they can tell you what it means for a function to be continuous at a given point and you realize that they don't know. It is very strange to see students trying to solve a problem about continuity without knowing what it means for a function to be continuous. But apparently it is not strange to them.&lt;/p&gt;&#xA;&lt;p&gt;The way most students in elementary courses try to learn mathematics is by looking at solved examples and then mimic the solution to solve other similar examples, without necessarily knowing or understanding the concept that is involved.&lt;/p&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is there a name in the literature for the phenomenon that I described? Has this been studied?&lt;/li&gt;&#xA;&lt;li&gt;Does this phenomenon have natural origins, or is it a habit that students develop because of how they have been taught in prior mathematics classes? By &amp;quot;natural origins&amp;quot; I mean does it stem from internal characteristics of human learning?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I received fascinating responses to my question. I would like to make my question more precise by adding one more example.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example.&lt;/strong&gt; Suppose I want to teach my calculus students that if they want to verify whether the limit of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at a number &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; exists, they should check to see if limits from the left and the right of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; are equal or not. I don't have any issues with learning through examples or inductive learning. We can demonstrate this concept through as many examples as necessary. At the core of my question lies the observation that many students seem to focus on the &lt;em&gt;appearance&lt;/em&gt; of the functions in the examples that we solve, rather than what needs to be checked. In the end, when I test my students, I want to see &lt;em&gt;what&lt;/em&gt; they have learned: have they learned that to verify whether the limit of a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at a number &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; exists, they must check to see if limits from the left and the right of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; are equal, or do they only remember the steps that were taken in each &lt;em&gt;appearance&lt;/em&gt; of functions in the examples that we solved? My experience is that, it is often the latter, and the number of examples solved doesn't seem to help. For instance, we solve many types of examples (I emphasize, &lt;em&gt;many&lt;/em&gt;, not just this) including&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x)=\left\{\begin{array}{lcl}\frac{x^2-4}{x+2}&amp;amp; \mathrm{if} &amp;amp;x&amp;gt;-2\\ 2&amp;amp;\mathrm{if}&amp;amp;x=-2\\x^2+x-6&amp;amp;\mathrm{if}&amp;amp; x&amp;lt;-2\end{array}\right.$$&lt;/span&gt;&#xA;When we solve this example, it seems that students don't see the solution as checking the equality of one sided limits of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; (of course, I emphasize that multiple times). Instead, they look at the &lt;em&gt;appearance&lt;/em&gt; of the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and see the solution as follows: first take the first expression &lt;span class=&quot;math-container&quot;&gt;$\frac{x^2-4}{x+2}$&lt;/span&gt; and find the limit, then take the last expression &lt;span class=&quot;math-container&quot;&gt;$x^2+x-6$&lt;/span&gt; and find the limit, then see if the limits are equal to each other. In other words, they create &amp;quot;rules&amp;quot; based on the appearance of functions in each solved example, not realizing that regardless of the appearance of the function, in all these examples we are just checking the equality of the one-sided limits. Consequently, if I give them a similar piecewise-defined function in the test, they can answer the question, because they remember the &lt;em&gt;appearance&lt;/em&gt; of this function and their own &amp;quot;rules&amp;quot; for it, but if I give them a function that has a new &lt;em&gt;appearance&lt;/em&gt;, which they have not seen before, they cannot solve the problem. For instance, if I give them this question: &amp;quot;Given that &lt;span class=&quot;math-container&quot;&gt;$$f(0)=2,\: g(0)=-2,\:\lim_{x\rightarrow0^-}f(x)=-1\:\lim_{x\rightarrow0^+}f(x)=0\:\lim_{x\rightarrow0^-}g(x)=1\:\lim_{x\rightarrow0^+}g(x)=0,$$&lt;/span&gt;&#xA;does limit of &lt;span class=&quot;math-container&quot;&gt;$f(x)+g(x)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x\rightarrow0$&lt;/span&gt; exist?&amp;quot; they cannot solve it, because they do not attempt to check the equality of one-sided limits. That's not what they took away from the examples we solved. They just remember the &amp;quot;algorithmic rules&amp;quot; that they created based on the appearance of each function in each example, and those don't help for this new function.&lt;/p&gt;&#xA;&lt;p&gt;So my question was, is this the nature of human learning, or is it a learning technique that is encouraged by the way we test them (e.g., solve an example involving a piecewise-defined function, then test them by giving another piecewise-defined function)?&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-10-28T15:57:48.543" LastActivityDate="2023-10-28T15:57:48.543" Title="Natural origins or learned habit: Why do students skip concepts before applications?" Tags="|undergraduate-education|" AnswerCount="17" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26961" PostTypeId="2" ParentId="26959" CreationDate="2023-10-19T01:09:44.577" Score="54" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Students have the tendency to just try to learn the examples and skip the concepts. Is there a name for this phenomenon? Has it been studied? Does it have natural origins, or is it taught in prior math classes?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm not familiar with the literature, but I do know that &amp;quot;math people&amp;quot; tend to severely overestimate the degree to which the rest of the population reasons from first principles. Most people exclusively reason by analogy:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; continuous? Well, based on the examples I've seen, continuous functions look &amp;quot;normal&amp;quot; and discontinuous functions look &amp;quot;weird.&amp;quot; In this case &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; looks more normal than weird, so &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is probably continuous.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Side Note: Students who exclusively reason by analogy generally don't see it as problematic that they don't know how to proceed in the absence of similar examples. When that happens, they view it as a &amp;quot;data&amp;quot; problem, not a &amp;quot;model&amp;quot; problem, so they &amp;quot;fix&amp;quot; the problem by obtaining a similar example from someone/something that seems trustworthy and adding it to their dataset of examples for future decisions.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This isn't just in math. It's across life in general. Consider stereotypes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is X true about thing Z? Well, based on the examples I've seen, thing Z seems similar to the type of things for which X is true. So X is probably true about thing Z.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To me, this would suggest that overreliance on analogical reasoning has natural origins (even if the specific decision rule for a particular case of analogical reasoning might be learned from other people).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Response to OP's Addendum:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;. . . they [students] create &amp;quot;rules&amp;quot; for each appearance of functions in each solved example, not realizing that regardless of the appearance of the function in all these examples, we are just checking the equality of the one-sided limits . . . is this the nature of human learning, or is it a learning technique that is encouraged by the way we test them (e.g., solve an example involving a piecewise-defined function, then test them by giving another piecewise-defined function)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I've had countless similar experiences.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;You spend mountains of time and effort providing students with a whole zoo of different-looking examples.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You constantly emphasize that while there are different &amp;quot;rules&amp;quot; for each example, the rules all stem from the same &amp;quot;first principle.&amp;quot; And you demonstrate this by going through exercises where you provide a new example (not already in the zoo) and work with students to solve the exercise from first principles.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You explain to students that understanding that first principle will save them time and effort on homeworks, lead them to better scores on assessments (and hence a better grade in the class), allow them to more easily grasp math that they might have to learn in the future, and empower them to face any related problems they might face that didn't appear in your zoo of examples.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And even after you provide all this support, nothing changes.&lt;/p&gt;&#xA;&lt;p&gt;In my view, the only explanation is that &lt;em&gt;everyone has some level of abstraction beyond which they are incapable of engaging in first-principles reasoning&lt;/em&gt;. That level is different for everyone, and it's not a &amp;quot;hard&amp;quot; threshold, but beyond it the time and mental effort required to perform first-principles reasoning skyrockets until first-principles reasoning becomes completely infeasible. I've seen this happen with my own eyes while working with many students longitudinally as they learned 4+ years of math.&lt;/p&gt;&#xA;&lt;p&gt;We're getting into a really touchy subject, and this may be an uncomfortable (maybe even unpopular) take. But surely there's no issue accepting that not everyone can really understand Topological Quantum Field Theory at its core, right? Why should Calculus be any different?&lt;/p&gt;&#xA;&lt;p&gt;In fact, it would be weird if there were a single level of math that marked the dividing line between &amp;quot;everyone who's learned the prerequisites can do it&amp;quot; versus &amp;quot;reserved for geniuses only.&amp;quot; It's likely a continuous thing that starts in fairly elementary math -- the higher the level of math, the fewer the number of people are capable of really understanding it at its core. Each student eventually hits their &amp;quot;abstraction ceiling,&amp;quot; the amount of time and effort it takes for them to succeed in math class begins to skyrocket, they find other subjects that they enjoy more (which may or may not use the math that they've learned so far), and they take an &amp;quot;off ramp&amp;quot; into those subjects.&lt;/p&gt;&#xA;&lt;p&gt;Of course, this is in no way an argument for giving up on students who are struggling. You support them as much as is feasible given the context of the class (see &lt;a href=&quot;https://matheducators.stackexchange.com/a/26963/22672&quot;&gt;Daniel R. Collins's answer&lt;/a&gt; for a great example of this). But at the end of the day, when you're reflecting on how well you've done as a teacher, you need to have realistic expectations.&lt;/p&gt;&#xA;&lt;p&gt;If you run a summer basketball camp, you might have a couple really impressive kids who make you think &amp;quot;wow, maybe I'll see them play professionally someday,&amp;quot; but you're not going to turn all the attendees into future pro ball players. Most of the kids probably aren't even going to get a basketball scholarship to college. You're definitely not going to get the 5'2&amp;quot; kid to dunk. And that's okay. Hopefully, you'll get each kid to have fun and become the best basketball player that they have the potential to be.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendums to This Answer&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum 1:&lt;/strong&gt; Commenters (ryang &amp;amp; Passer By) have pointed out that analogical reasoning is also known as &lt;em&gt;inductive reasoning&lt;/em&gt; and first-principles reasoning is also known as &lt;em&gt;deductive reasoning&lt;/em&gt;. You may have better luck using those terms when searching the literature.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum 2:&lt;/strong&gt; This is not to claim that analogical reasoning should be avoided entirely. Analogical reasoning generally does a good job of supplying you with a quick decision that is probably correct (or at least more likely correct than incorrect). In most situations in life that's all you need.&lt;/p&gt;&#xA;&lt;p&gt;As so elegantly stated in &lt;a href=&quot;https://matheducators.stackexchange.com/a/26975/22672&quot;&gt;Yakk's insightful answer&lt;/a&gt;, even professional mathematicians &amp;quot;have heuristics and patterns . . . which they are confident they can reduce recursively to first principles.&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A typical professional mathematician doesn't determine if a function is continuous by doing an epsilon-delta test. Given &lt;span class=&quot;math-container&quot;&gt;$\sin(x)+\cos(x),$&lt;/span&gt; they will know it is continuous without thinking; and if asked to verify, it is the sum of two continuous functions (again, not epsilon-delta).&lt;/p&gt;&#xA;&lt;p&gt;They can probably generate an epsilon-delta proof . . . But very few mathematicians will look at &lt;span class=&quot;math-container&quot;&gt;$\sin(x)+\cos(x)$&lt;/span&gt; and generate a first principles proof that it is continuous and use that to answer the question.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There's a difference between &lt;em&gt;using&lt;/em&gt; analogical reasoning and &lt;em&gt;over-relying&lt;/em&gt; on analogical reasoning. When you &lt;em&gt;use&lt;/em&gt; analogical reasoning, you need to have a sense of its confidence (i.e. probability of correctness) and how that measures up against the stakes of the situation you're in. If the confidence does not measure up against the stakes, then you need to fall back to first-principles reasoning (provided that you have enough time to do so). People who &lt;em&gt;over-rely&lt;/em&gt; on analogical reasoning do not do this.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum 3.&lt;/strong&gt; Some hardline pure mathematicians might argue that the entire point of math classes is training in deductive reasoning, so inductive reasoning has no place. While that may be at least partially true from the perspective of some pure math classes, especially those really niche classes focused on the axiomatic foundations of math, this answer interprets math in its most general sense, and in most areas under the gigantic umbrella of math, inductive reasoning has some place. For instance, throughout applied math there's a saying &amp;quot;all models are wrong but some are useful.&amp;quot; And in machine learning specifically (an extreme case for illustrative purposes), the whole point is to use math to build automated systems that use inductive reasoning to make decisions that are correct enough to be useful.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-23T20:02:38.037" LastActivityDate="2023-10-23T20:02:38.037" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="26962" PostTypeId="2" ParentId="26934" CreationDate="2023-10-19T02:30:14.547" Score="1" Body="&lt;p&gt;Ultimately I think these things often come down to a matter of phrasing it in a clear manner, more than needing a new pedagogical approach. Especially since they can regurgitate the correct proof (as you say). I might explain it like this.&lt;/p&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are numbers. Then &lt;span class=&quot;math-container&quot;&gt;$|a-b|$&lt;/span&gt; is a number. If &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are the same, then &lt;span class=&quot;math-container&quot;&gt;$|a-b|$&lt;/span&gt; is obviously zero. If &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are different, then &lt;span class=&quot;math-container&quot;&gt;$|a-b|$&lt;/span&gt; is, in fact, a &lt;em&gt;positive&lt;/em&gt; number. It's not, like, a magically small number, smaller than all other positive numbers, because that doesn't make sense and that doesn't exist. It's just a regular number, and it's positive.&lt;/p&gt;&#xA;&lt;p&gt;Then obviously it's greater than &lt;em&gt;some&lt;/em&gt; positive number, because it's a positive number, and it's bigger than some other positive numbers (actually as the problem is stated, &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; can be &lt;span class=&quot;math-container&quot;&gt;$|a-b|$&lt;/span&gt;, but that's sort of a distraction pedagogically).&lt;/p&gt;&#xA;&lt;p&gt;Note that this uses the contrapositive of the troublesome direction; that is, instead of &lt;span class=&quot;math-container&quot;&gt;$\forall \varepsilon &amp;gt; 0 |a-b|&amp;lt;\varepsilon \Rightarrow a=b$&lt;/span&gt;, I used &lt;span class=&quot;math-container&quot;&gt;$a\ne b \Rightarrow \exists \varepsilon&amp;gt;0 |a-b| \geq \varepsilon$&lt;/span&gt;, which is certainly easier to think about. If your students also don't understand the contrapositive then maybe this is a good time to spend some time there, because it's really helpful.&lt;/p&gt;&#xA;&lt;p&gt;I do think this plays into some confusion about the &lt;span class=&quot;math-container&quot;&gt;$0.999...=1$&lt;/span&gt; theorem, which is often confusing to students who (a) don't understand and won't accept that decimal expansions are infinite series, (b) don't understand or like series, and (c) end up imagining the difference is &lt;em&gt;something&lt;/em&gt; somehow between 1 and every number less than 1.&lt;/p&gt;&#xA;&lt;p&gt;Concern (c) is really the point of both examples, but in that example you have to deal with confusions about (a) and (b); luckily in your example, you don't.&lt;/p&gt;&#xA;" OwnerUserId="23157" LastActivityDate="2023-10-19T02:30:14.547" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26963" PostTypeId="2" ParentId="26959" CreationDate="2023-10-19T03:08:32.733" Score="36" Body="&lt;p&gt;An &lt;a href=&quot;http://www.madmath.com/2014/10/blooms-taxonomy-and-math-education.html&quot; rel=&quot;noreferrer&quot;&gt;observation that I've made in the past&lt;/a&gt; is this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Math is hard because by its nature it's taught at TOO HIGH a level&#xA;compared to other classes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Consider: Does the curriculum make time and space to discuss and &lt;em&gt;directly test&lt;/em&gt; whether students have first learned the definitions? In most cases, no; rather, we're usually running application exercises mere seconds after providing a definition. So this &lt;em&gt;directly signals&lt;/em&gt; to students that the definitions are ignorable boilerplate, and we expect them to jump right into applications.&lt;/p&gt;&#xA;&lt;p&gt;There's a bit of a toxic mix of (a) mathematicians assuming everyone knows how critical knowing the definitions are, and that they'll study/reflect/test themselves on it, and (b) knowing how few people ever learn the definitions, thinking it's a hopeless cause and just not bothering to spend time on them.&lt;/p&gt;&#xA;&lt;p&gt;If we take Bloom's taxonomy as a model, in a math class we generally hand-wave the level 1-2 issues (simple factual recall) as trivial, and jump directly to levels 3-4+ (applications, analysis, etc.). Whereas instructors in almost any other discipline I know constantly bemoan that they can't get students beyond the level 1-2 memorization issues, in math we do the opposite trick and basically spend no time on them at all.&lt;/p&gt;&#xA;&lt;p&gt;I'm teaching college algebra for the first time in several years, and I'm having to dig deep to remind myself to stop on the definition slides, direct students to carefully copy them down, and reflect about what they're telling us. Quick example this week: the standard form of a linear equation in two variables is &lt;span class=&quot;math-container&quot;&gt;$Ax + By = C$&lt;/span&gt;. Here's a whole slew of questions I can pose to my students as we reflect on that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What operations are used here?&lt;/li&gt;&#xA;&lt;li&gt;Are there any exponents or roots?&lt;/li&gt;&#xA;&lt;li&gt;What side of the equation is the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; on?&lt;/li&gt;&#xA;&lt;li&gt;What side of the equation is the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; on?&lt;/li&gt;&#xA;&lt;li&gt;What side of the equation is the constant on?&lt;/li&gt;&#xA;&lt;li&gt;Are there any fractions here?&lt;/li&gt;&#xA;&lt;li&gt;(In any examples) Is this a linear equation?&lt;/li&gt;&#xA;&lt;li&gt;(In any examples) Is this in standard form?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Here's the thing: &lt;em&gt;My students take in literally none of that information on first sight&lt;/em&gt;. They even, as a group, get some of these trivial questions wrong on first inquiry (e.g., after saying the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s are on the left, by symmetry they'll mistakenly say the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;'s are the right, even while looking directly at the definition). I'm re-familiarizing myself that these structural &amp;quot;forms&amp;quot; communicate absolutely nothing to my population of students; every bit of information has to be explicitly called out and highlighted.&lt;/p&gt;&#xA;&lt;p&gt;(To say nothing of a whole slew of key facts that the textbook never bothers to explicitly say: e.g., linear equations have lines for graphs; solving an equation isolates a variable on one side; two points identify a line; solving a linear system requires as many equations as variables; etc.)&lt;/p&gt;&#xA;&lt;p&gt;I think in an ideal world we would model learning definitions and applying them logically by &lt;strong&gt;actually spending time in class reflecting on them&lt;/strong&gt;, and maybe half of our assessments would test for direct recall and identifying example cases -- that is, we'd have a more uniform distribution of time spent on the various levels of Bloom's taxonomy. But most textbooks, curricula, and standardized tests don't provide time or space for that, so most of us are handcuffed to this rather bad model that actively sets an example of skipping over the foundational definitions and principles.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-10-19T18:27:56.213" LastActivityDate="2023-10-19T18:27:56.213" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26964" PostTypeId="2" ParentId="26959" CreationDate="2023-10-19T04:05:41.660" Score="9" Body="&lt;p&gt;I second Justin's comment and want to add to it that if one hasn't been raised with Smullyan's or Carroll's books, one often has trouble comprehending a statement with 2 quantifiers while a typical definition in undergraduate mathematics has 3 to 5 (the definition of continuity at a point &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; has 3: &lt;em&gt;for every&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; &lt;em&gt;there is&lt;/em&gt; a &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; such that &lt;em&gt;for every&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; that is &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-close to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$|f(y)-f(x)|&amp;lt;\epsilon$&lt;/span&gt;). It is just something rather incomprehensible for many beginners, so even if they try to reason from the first principles, they need to go over &lt;em&gt;many&lt;/em&gt; examples to understand what that first principle really is and even then there is a danger that instead of understanding the principle itself, they'll just memorize the algorithm of the continuity/discontinuity verification (which is something very different). The Greek letters traditionally used in the definition don't help much either unless you are teaching in Greece though they can add to the fun if you understand the concept itself rather well. So, what might possibly compensate for that?&lt;/p&gt;&#xA;&lt;p&gt;When I was a student, my undergraduate analysis teacher tried to restate that continuity definition in many ways using just the layperson language. I still remember some analogies like a good and a bad shower temperature control handle, feeders and traps: if a muzzle of an animal &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; enters the feeder (&lt;span class=&quot;math-container&quot;&gt;$x-\delta,x+\delta)$&lt;/span&gt;, its tail &lt;span class=&quot;math-container&quot;&gt;$f(y)$&lt;/span&gt; gets caught in the trap &lt;span class=&quot;math-container&quot;&gt;$(f(x)-\varepsilon,f(x)+\varepsilon)$&lt;/span&gt; (this is pretty much how a humane mousetrap like &amp;quot;mice cube&amp;quot; works, BTW, so the &lt;span class=&quot;math-container&quot;&gt;$\varepsilon-\delta$&lt;/span&gt; question becomes &amp;quot;how close to the back wall should you place the cheese to make sure that the mouse tail won't prevent the trap door to shut fully?&amp;quot;), etc.. In addition he drew the rectangles in which graph should be contained near the point, and whatnot, just to convey one simple idea that formally is fully described by the standard definition. Note that he was very careful to keep all those layperson language descriptions &lt;em&gt;exactly equivalent&lt;/em&gt; to the actual definition (if you interpret them correctly, of course). That was all before we have formally checked the continuity of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (which was done in &lt;em&gt;all&lt;/em&gt; of those languages too).&lt;/p&gt;&#xA;&lt;p&gt;I didn't need any of those restatements, so I'm not qualified to judge how efficient they were for the purpose of conveying the message, but what I have derived from it was that &lt;em&gt;you don't understand the concept fully until you can restate what it means in pretty much every conceivable situation using almost arbitrary underlying objects&lt;/em&gt;. So I'm still preaching to my students that mathematical thinking is mainly just common sense and you absolutely don't need to talk about numbers, functions, equations or geometric shapes to speak &amp;quot;Mathematiese&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;You can try this technique too and see if it works for you. Warning: &lt;em&gt;It takes extra time!&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastEditorUserId="13919" LastEditDate="2023-10-26T20:09:46.647" LastActivityDate="2023-10-26T20:09:46.647" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="26965" PostTypeId="2" ParentId="26873" CreationDate="2023-10-19T07:11:24.577" Score="2" Body="&lt;p&gt;IMHO, there are at least two &amp;quot;Histories of Mathematics&amp;quot;. The first one is temporal. Usually it goes like that.&lt;/p&gt;&#xA;&lt;p&gt;In year A, a not then known mathematician B wrote a letter to a famous specialist C in the field D containing the theorem E, to which the latter replied that it was &amp;quot;of little interest for the contemporary mathematics&amp;quot;, so B switched to something else and was well on his way to develop the grounds of the theory F but at that moment the country G invaded the island H where both B and C lived and both were killed by the common soldiery during the siege of I. The theorem E was then rediscovered in year J by K, who, however left nothing but a cryptic note about it on the margins of the third volume of a twenty volume treatise by L (contemporary of A and B) of which only volume 3 and volume 13 survived the eagerness of the monks of the religious sect M to erase all traces of the heresy L professed in addition to his mathematical studies. Finally, E was rediscovered in year N by O, who accidentally happened to be the first representative of the race P to get the award Q for her work. The content of theory F was completely lost except for a few mentions in the works of R who lived S years after A and B and whose books were found by archeologists in some remote monastery T of the religious sect M where they were piled together with other literature of that time in preparation to be recycled for printing the booklets of prayers. Fortunately, the volcano U erupted just in time and buried the entire monastery T under the ashes before it happened. Some historians, however, maintain that R has never existed and his books were a fake created much later by a well-known alchemist, musician, and mathematician V to sell them as ancient rarities to the manuscript collector W to get out of debts. The most extravagant idea about what F might possibly be belongs to the well-known contemporary physicist X who maintains that it was a primordial version of Y, which was the mathematical foundation of the model Z of the Universe for which X got his Nobel prize.&lt;/p&gt;&#xA;&lt;p&gt;If somebody wants to claim that the actual temporal history of mathematics  is not as twisted, contorted, and random as what I wrote above, I will wholeheartedly agree. It is far crazier. Moreover, the more we learn about it, the crazier it becomes as far as I can tell. If one wants to memorize it, one immediately faces the problem of its Kolmogorov's complexity being proportional to its length. The only lesson one can derive from it in addition to what you can find in your childhood fairy tales is just the usual history mantra: &amp;quot;in addition to working hard, being persistent, having only moderate risk aversion, high power of concentration, and all other virtues from the fairy tales, you need to be in the right place at the right time doing the right thing to get an eternal fame&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The other history is logical. It is based on the idea that certain concepts are naturally related to certain  other ones and the underlying ideas are the same across the results and even across the different fields. It ignores the names, the dates, and the biographies almost entirely (the mathematician names in it are just like the street names in towns with the idea that some person is behind every name, but most of them are just convenient identifiers of the locations used merely for the brevity, and the knowledge of the corresponding biographies is neither necessary, nor sufficient for navigating the map). It just takes all mathematics as a sum of knowledge and tries to build the simplest &amp;quot;Turing machine&amp;quot; (both the code and the tape content, but as we know they are indistinguishable once the universal machine is designed) that outputs all of it (or, at least, noticeable chunks of it). That is what we normally do when preparing to teach some ascending series of courses and that is what we store in our brains when we deal with the areas that we know really well. The Kolmogorov complexity here seems to be much smaller than the length of the output, so we can, say, go to the board and present the full course in measure theory in a non-stop way from the basic definitions to the most advanced topics covered in the graduate courses without a single look at any book or even a sheet of paper. The problem is that we have to communicate the whole protocol of the algorithm execution and the final output rather than the initial tape content to the students, so they have to build the initial tape from it themselves. The &amp;quot;universal Turing machine&amp;quot; here is usually called &amp;quot;mathematical maturity&amp;quot;, which includes the ability to understand arbitrarily complicated mathematical constructs if they are presented from scratch and with full proofs (that is more or less equivalent to the ability of the universal Turing machine to read the tape content) but is not limited to it, and there is currently no way to build it directly in the brain either. They like to say that it &amp;quot;comes with experience&amp;quot;, which doesn't explain much either. I haven't seen any good books on that kind of history and if my description of it seems even more twisted, incomprehensible, and sarcastic than my description of the temporal one, this is by no means surprising though I believe that I can explain what exactly I mean using an example of any topic of your choice I'm reasonably well familiar with.&lt;/p&gt;&#xA;&lt;p&gt;So, what's the moral? For me it is that one can occasionally try to throw in some parts of any of these two (and, possibly, several more) possible &amp;quot;histories of mathematics&amp;quot; into the curriculum and see what effect it has on the students (I usually prefer to play with the second one myself) but it should be done with care and in small quantities. If you overdo the first, you'll not just overload their memory with unnecessary information, but may also interfere with their knowledge compression process by presenting things in the suboptimal for comprehension order and emphasizing random relations rather than natural ones. If you overdo the second, you'll soon discover that what you say is just incomprehensible to your students and the &amp;quot;historical&amp;quot; part of the message you want to deliver doesn't go through at all. Usually neither of them is included. However, a colleague of mine tried once to include elements of the temporal history into the analysis course telling the short stories about the lives of mathematicians whose names appeared in the course (mainly 17, 18, and 19 century ones). His account of the outcome was as follows (I cite his words nearly literally just translating from Russian)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;I told my students the life story of X. When I received my teaching evaluations, one of them read &amp;quot;The teacher's accent is terrible and his lectures are hard, but he is a funny guy who told us many interesting stories. For example, when we learned about Y , he told us how during the Second World War he lived in a village with his grandfather and was sitting on his grandfather knee when the Germans went it and shot the grandfather&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;If somebody can guess correctly what was the subject Y, who was X, and what was actually said of the events in the life of X that inspired this passage in the student evaluations, I'll send the first person with the right answer a small prize (the people who have heard that story before from me or from my colleague himself are, obviously, excepted from this offer).&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-10-19T07:11:24.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26966" PostTypeId="2" ParentId="26959" CreationDate="2023-10-19T07:29:54.360" Score="6" Body="&lt;p&gt;I think there's an issue of time efficiency. Looking at the examples and working the problems is the most bang for the buck. Of course to have 4.0 ability, you should do both. But people are lazy, have other time demands, etc.  Also, prioritization applies to curriculums and advancement paths also.&lt;/p&gt;&#xA;&lt;p&gt;This is not meant to cut but to make a point. I think the above argument should have already occurred to a teacher. I'm continually amazed how people with such strong intellect and math subject knowledge are so babylike new to the world in terms of pedagogy or even basic human psychology.&lt;/p&gt;&#xA;&lt;p&gt;Note that many math people tend to think the definitions are sufficient. As if students were calculating machines. But human brains are imperfect computers. They require practice to remember. In some cases need practice to comprehend.&lt;/p&gt;&#xA;&lt;p&gt;Really one should have some recursion from definitions to problems and back. Because we are imperfect learners. However even here, realize that there is not unlimited time. But also realize that belaboring defintions and side cases too much before doing some drill can lose the class and waste time also.&lt;/p&gt;&#xA;&lt;p&gt;For those who want immediate feedback in terms of reciting definitions, I would recommend to look at programmed learning textbooks.  They tend to introduce a definition snd then immediately test restatement of it by a fill in the blank problem. Very simple snd easy snd immediate. Almost sounds babyish. But works.  They seem to do this well and are very time efficient and neophyte accessible. However, they seem to be disdained by the education establishment as illiberal.&lt;/p&gt;&#xA;" OwnerUserId="23096" LastEditorUserId="23096" LastEditDate="2023-10-19T07:37:40.180" LastActivityDate="2023-10-19T07:37:40.180" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26967" PostTypeId="1" CreationDate="2023-10-19T08:14:30.230" Score="3" ViewCount="230" Body="&lt;p&gt;With &lt;a href=&quot;https://quizlet.com/features/live&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quizlet Live&lt;/a&gt; you can play a flashcard game in classroom, where teams compete against each other. Since Quizlet doesn't seem to support rendering latex formulas, I am looking for a alternative with latex support.&lt;/p&gt;&#xA;&lt;p&gt;Do you have any recommendation?&lt;/p&gt;&#xA;" OwnerUserId="1203" LastActivityDate="2023-10-23T19:11:36.250" Title="Alternative to Quizlet live that supports latex formulas" Tags="|software|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26968" PostTypeId="2" ParentId="17686" CreationDate="2023-10-19T08:18:35.883" Score="2" Body="&lt;p&gt;I know I am late to the party, but I absolutely hate short division. It works for simple division problems, for those that you can almost do in your head. But try dividing 35/43 or 692/37. This is where most students either slow WAY down or get confused and frustrated. A really good example is when you are trying to convert 1/8 into a fraction or percentage. This is what long division is really great for. It takes all of the math that you have to do in your head and puts it onto the paper, making it easier to understand what is going on and what needs to happen next. I am all for teaching short division as a stepping stone for long division, but long division needs to be a priority. I have lost count of how many students I have taught long division to because they got stuck with short division and a more complicated division problem.&lt;/p&gt;&#xA;" OwnerUserId="23161" LastActivityDate="2023-10-19T08:18:35.883" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26970" PostTypeId="2" ParentId="26959" CreationDate="2023-10-19T09:31:58.007" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;One name that is used in the literature is &amp;quot;generalizations&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students (people) naturally use a number of different heuristics to solve problems and make decisions. Reasoning from examples is one of them.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The problem of lack of generalization is both natural and taught. You can teach your students to just match examples, and you can teach your students is such a way that they just match examples, and they can naturally just match examples.&lt;/p&gt;&#xA;&lt;p&gt;But teaching and learning efficiency is demonstrated by teaching &amp;quot;from the concrete to the abstract&amp;quot; -- normal people are much more efficient at learning abstract concepts when they are grounded in previous concrete examples. An attempt to 'teach the concepts first' is sometimes successful, but contrary to research and experience.&lt;/p&gt;&#xA;&lt;p&gt;On the one hand, a common problem for non-teachers is that, once they learn the concepts, the concrete examples just get in the way, and they want to teach for optimum post-learning understanding, rather than for optimum acquisition.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, just teaching examples may never lead to useful generalisation.&lt;/p&gt;&#xA;&lt;p&gt;Students have to learn concepts. You have to teach concepts. Concrete examples are part of the process, but not the end point.&lt;/p&gt;&#xA;" OwnerUserId="20889" LastActivityDate="2023-10-19T09:31:58.007" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26971" PostTypeId="2" ParentId="26959" CreationDate="2023-10-19T11:16:52.413" Score="13" Body="&lt;p&gt;A relevant idea from the literature is the theoretical framework of &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Concept_image_and_concept_definition&quot; rel=&quot;noreferrer&quot;&gt;concept image and concept definition&lt;/a&gt;.&amp;quot; It has been very popular since it was introduced, and you can find studies that use it to analyze students' reasoning about all sorts of mathematical concepts. &lt;a href=&quot;http://wrap.warwick.ac.uk/507/1/WRAP_Tall_dot1981a-concept-image.pdf&quot; rel=&quot;noreferrer&quot;&gt;Tall and Vinner (1981&lt;/a&gt;) introduced the framework by discussing examples of students' reasoning about limits and continuity. Here's an excerpt from that paper.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The human brain is not a purely logical entity. The complex manner in which it functions is often at variance with the logic of mathematics. It is not always pure logic which gives us insight, nor is it chance that causes us to make mistakes. To understand how these processes occur, both successfully and erroneously, we must formulate a distinction between the mathematical concepts as formally defined and the cognitive processes by which they are conceived.&lt;/p&gt;&#xA;&lt;p&gt;Many concepts which we use happily are not formally defined at all, we learn to recognise them by experience and usage in appropriate contexts. Later these concepts may be refined in their meaning and interpreted with increasing subtlety with or without the luxury of a precise definition...&lt;/p&gt;&#xA;&lt;p&gt;We shall use the term &lt;em&gt;concept image&lt;/em&gt; to describe the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes. It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures...&lt;/p&gt;&#xA;&lt;p&gt;The definition of a concept (if it has one) is quite a different matter. We shall regard the &lt;em&gt;concept definition&lt;/em&gt; to be a form of words used to specify that concept. It may be learnt by an individual in a rote fashion or more meaningfully learnt and related to a greater or lesser degree to the concept as a whole. It may also be a personal reconstruction by the student of a definition... In this way a &lt;em&gt;personal&lt;/em&gt; concept definition can differ from a &lt;em&gt;formal&lt;/em&gt; concept definition, the latter being a concept definition which is accepted by the mathematical community at large.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/26963/20719&quot;&gt;As Daniel Collins points out in his answer&lt;/a&gt;, mathematicians often assume that students understand the role of definitions in mathematics. This is an expert blind spot. If we want students to reason using definitions, they must be taught how to do so. When students practice applying the formal definition, it becomes incorporated into their concept images.&lt;/p&gt;&#xA;&lt;p&gt;It is &lt;strong&gt;not&lt;/strong&gt; the case that mathematical thinking requires the replacement of concept images with formal definitions. &lt;a href=&quot;https://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot1991k-psychology-of-amt.pdf&quot; rel=&quot;noreferrer&quot;&gt;Tall (1991)&lt;/a&gt;, in an article on &amp;quot;advanced mathematical thinking,&amp;quot; says the following about how a psychologist might describe the mathematicians' notion of &amp;quot;intuition vs. rigor.&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One may view the usual mathematician's dichotomy between intuition and rigour as one between the holistic, visual thinking characteristic of the right brain, and the rigour of the sequential, logical thinking characteristic of the left. But the psychologist sees the possibility of more sophisticated (secondary) intuitions arising from refined concept images which can include the mental imagery of logic and deduction. Thus aspects of logic too can be honed to become more &amp;quot;intuitive&amp;quot; to the mathematical mind. The development of this refined logical intuition should be one of the major aims of more advanced mathematical education.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So when we ask students &amp;quot;What does it mean for a function to be continuous at a point?&amp;quot; and they struggle to answer in words, it does not mean that they are &amp;quot;skipping concepts.&amp;quot; It means that they do not have a concept definition retrievable from long-term memory. They very likely have a strong concept image of &amp;quot;continuity,&amp;quot; but their concept image does not help or may even actively conflict with the task at hand.&lt;/p&gt;&#xA;&lt;p&gt;One effective response to this, as some other answers have advised, is to test students on recall of formal definitions, model your own reasoning with definitions, and provide worked examples that show how to use definitions. You mention in a comment,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Assume that definitions have been stated and explained and a reasonable number of concrete examples have been solved and demonstrated and in each example, the teacher has emphasized how the example is solved by referring back to the definitions. Now the teacher tests students' learning by giving them a new problem that they have not seen before, but nevertheless can be solved by referring back to definitions... such &amp;quot;new&amp;quot; problems are difficult for most students and they cannot solve them, because they still try to approach the problem by searching their inventory of solved examples.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It seems here that students still do not have a notion of &amp;quot;referring to the definition&amp;quot; as a strategy, even though you may feel that you are emphasizing it. If students have indeed seen a variety of worked examples that involve first referring to the definition, then it may be fruitful to have them compare these examples and synthesize this strategy from the comparison. With worked examples, students often attend to surface-level details and not to the deeper structure that experts can see. There is research on this as well, though I don't have time to add it to the answer right now.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Here are some references on how novices and experts perceive &amp;quot;deep structure&amp;quot; and how this relates to instruction and examples. There have been studies on this in various domains, but I'll focus on references from the mathematics education literature.&lt;/p&gt;&#xA;&lt;p&gt;One early study is Silver (1979) on &lt;a href=&quot;https://www.jstor.org/stable/748807&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Student Perceptions of Relatedness among Mathematical Verbal Problems.&amp;quot;&lt;/a&gt; He had eighth-graders sort various word problems into groups. The problems varied in both problem context and mathematical structure. He found that &amp;quot;the tendency to sort problems on the basis of mathematical structure was significantly positively correlated with all of the general verbal and mathematical ability measures.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Schoenfeld and Hermann (1982) in &lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED200609.pdf&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Problem Perception and Knowledge Structure in Expert and Novice Mathematical Problem Solvers&amp;quot;&lt;/a&gt; asked mathematics professors and undergraduates to sort problems that were designed to have a mathematical &amp;quot;surface structure&amp;quot; and a mathematical &amp;quot;deep structure.&amp;quot; For example, a problem might have a surface structure of &amp;quot;prime numbers,&amp;quot; but the deep structure is &amp;quot;contradiction.&amp;quot; They found that &amp;quot;the sorting by novices appears to depend more on surface characteristics of problems, while the sorting by experts depends on the deep mathematical properties of the problems.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Atkinson et al. (2000) synthesize research in &lt;a href=&quot;https://journals.sagepub.com/doi/pdf/10.3102/00346543070002181&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Learning from Examples: Instructional Principles from the Worked Examples Research.&amp;quot;&lt;/a&gt; They suggest that good design for teaching with examples involves varying surface features to encourage students to search for deep structure, and to encourage students to self-explain deep structure.&lt;/p&gt;&#xA;&lt;p&gt;Rittle-Johnson and Star (2009) discuss the effect of comparing examples in &lt;a href=&quot;https://dash.harvard.edu/handle/1/4889483&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Compared to what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving.&amp;quot;&lt;/a&gt; They state that &amp;quot;Moderately similar, rather than highly similar, examples should help people ignore irrelevant surface features and abstract a more general underlying solution structure... The current findings support this prediction for a problem solving task and extend it to outcomes not previously considered – procedural flexibility and conceptual knowledge.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Lee et al. (2017) in &lt;a href=&quot;https://doi.org/10.1080/00220973.2016.1180277&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Embellishing Problem-Solving Examples with Deep Structure Information Facilitates Transfer&amp;quot;&lt;/a&gt; had participants study examples of &amp;quot;algebra-like&amp;quot; problems, then had them attempt &amp;quot;transfer problems&amp;quot; that required adjustment of their procedures. They found that &amp;quot;students transferred better when they studied with examples that emphasized problem structure rather than solution procedure.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2023-10-19T13:43:05.603" LastActivityDate="2023-10-19T13:43:05.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26972" PostTypeId="2" ParentId="26959" CreationDate="2023-10-19T14:14:31.177" Score="1" Body="&lt;p&gt;The quick answer is that children's brains are still growing and until their early 20s their brains don't function like adults.&lt;/p&gt;&#xA;&lt;p&gt;The longer answer is that yes this has been studied, and you will mostly find the answers in psychological literature, education literature, and more recently neuroscience literature. In psychological literature, district period of brain growth were hypothesized almost a century ago, and have, during the past few decades, been verified through research. Conceptual understanding and task completion go hand in hand with respect forming &amp;quot;knowledge.&amp;quot; For elementary school students and even into middle school consider that they don't even have a real concept of time. When an event occurred in the past they will say things happened variously like &amp;quot;yesterday&amp;quot; or &amp;quot;a week ago&amp;quot; without any hesitation. They aren't making things up or out and out lying- their brains haven't finished the process to fully develop logic. If you search the literature for structural brain development and mathematics education you will find a multitude of studies on this topic.&lt;/p&gt;&#xA;&lt;p&gt;Even when they learn definitions and can connect the skill, they don't really understand in the sense that &amp;quot;continuous&amp;quot; is both a time concept as well as a symbolic concept.&lt;/p&gt;&#xA;&lt;p&gt;At this point many people have asked me:  then why try to teach concepts? The answer is that you are building the logic structure with mathematics. Fundamentally, there are 4 very different periods of brain development marked by an explosion of neuron growth and then a pruning of the least efficient neuronal pathways. Mathematics in early and middle childhood is really for teaching kids &lt;em&gt;how&lt;/em&gt; to think, with the &lt;em&gt;what&lt;/em&gt; being peripheral. This is why the is a big push to stop teaching algebra before the 9th grade- because for the average learner, the ability to think symbolically is just started to emerge.&lt;/p&gt;&#xA;&lt;p&gt;I think that's likely enough info for now, and I think it's great you both noticed this and are looking into it. I think child brain development should be an essential emphasis for teachers- less frustrating for us and the students. One more thing: if you have a younger student that can grasp concepts that other kids can't, then you are likely looking at a student with a higher IQ. It's not a certainly, but it's certainly a big indicator asking with the quickness at which they &amp;quot;get&amp;quot; the skill you are teaching. These are also the kids that will tentatively grasp conceptual information in general- and those kids may sometimes need to know the &amp;quot;why&amp;quot; along with the &amp;quot;how&amp;quot;. For the bulk of the students, however, the &amp;quot;why&amp;quot; will only be parroted until about 9th grade.  The great part is that at that point the concepts will be familiar and that makes it easier for the brain to assimilate and accommodate.  If you want more in depth, check out the bodies of literature I discussed above and look up Jean Piaget as well.  His work is seminal and laid the foundation for our understanding of childhood brain development. Good luck!&lt;/p&gt;&#xA;" OwnerUserId="23170" LastActivityDate="2023-10-19T14:14:31.177" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26973" PostTypeId="2" ParentId="376" CreationDate="2023-10-19T16:42:56.720" Score="0" Body="&lt;p&gt;&lt;strong&gt;I'm just only learning about this stuff but so feel free to fix/improve my answer:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you build a neural network to do some machine learning on predicting financial/economic entities (stock prices, true/false acceptance of loans, etc), you are setting up a function &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^L \to \mathbb{R}^O$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; is the total number of components of your weight matrices &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; is the total number of outputs you're trying to predict. When you test your predicted outputs against the real data, you obtain an error function &lt;span class=&quot;math-container&quot;&gt;$J(\theta)$&lt;/span&gt; which is a multi-variable vector-valued function!&lt;/p&gt;&#xA;&lt;p&gt;The gradient &lt;span class=&quot;math-container&quot;&gt;$\nabla_{J(\theta)}$&lt;/span&gt; which can be approximated numerically, tells us which direction we should alter our components of our weight matrix to maximize (locally) &lt;span class=&quot;math-container&quot;&gt;$J(\theta)$&lt;/span&gt;. We want to minimize error, &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt;, so we walk our &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; in the direction of &lt;span class=&quot;math-container&quot;&gt;$-\nabla_{J(\theta)}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This results in the linear approximation to a path flowing along the vector-field &lt;span class=&quot;math-container&quot;&gt;$-\nabla_{J(\theta)}$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^L$&lt;/span&gt; given by &lt;span class=&quot;math-container&quot;&gt;$\theta_{t+1} = \theta_t - \epsilon\cdot \nabla_{J(\theta)}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; is our length of straight-line walking/flowing. The assumptions that theoretically everything is actually a smooth higher dimensional surface make this method sensible, and so that the smaller &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; is the close our path is to a flow along the field.&lt;/p&gt;&#xA;&lt;p&gt;Now the line integral along this path &lt;span class=&quot;math-container&quot;&gt;$\theta_t$&lt;/span&gt; for the vector field &lt;span class=&quot;math-container&quot;&gt;$-\nabla_{J(\theta)}$&lt;/span&gt; should represent the total change in error, &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt;, as we move from &lt;span class=&quot;math-container&quot;&gt;$\theta_a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\theta_b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Stokes' theorem would say that two different paths we take, flowing along the gradient vector field, if they end at the same place, should end up in equal reductions of the error if our situation is &amp;quot;nice&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="6589" LastActivityDate="2023-10-19T16:42:56.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26974" PostTypeId="2" ParentId="26873" CreationDate="2023-10-19T17:28:38.173" Score="3" Body="&lt;p&gt;This is a narrow reply, limited to one aspect of math history that&#xA;I have found effective in my teaching (college, and some high school, in the US). That aspect is showing that even professional mathematicians make&#xA;mistakes, that various theorem proofs needed later revision.&#xA;(And by implication, it is OK if a student makes a mistake.)&#xA;A few examples:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Kempe's incorrect proof of the &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;-color theorem. But his central&#xA;contribution, Kempe chains, is essential for subsequent proofs.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The discovery of the &lt;em&gt;pseudorhombicuboctahedron&lt;/em&gt;. Lyusternik says: &amp;quot;It is remarkable that in the theory of semiregular polyhedra, which is over 2000 years old, there was a defect, which was recently discovered by the Soviet mathematician V. G. Ashkinuz. He discovered a 14th semiregular polyhedron...&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The flaw in the proof of Cauchy's rigidity theorem, repaired a&#xA;century later by Steinitz.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bricard's incorrect proof that eventually led to the Dehn invariant.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="511" LastActivityDate="2023-10-19T17:28:38.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26975" PostTypeId="2" ParentId="26959" CreationDate="2023-10-19T19:18:58.200" Score="15" Body="&lt;p&gt;Have you heard the story of Von Neumann and the train problem?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Train problem is a classic one.  You have two trains heading towards each other, one going at 30 km/h the other going at 45 km/h, starting 371 km apart.  When they start moving a fly takes off from one train.  It flies to the front of the other train, flying at 150 km/h, lands, takes off instantly and flies back to the first.  It continues to do so until it is squashed between the two trains.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How far does the fly travel before it is squashed?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Actual values don't match the traditional version; it was in miles per hour)&lt;/p&gt;&#xA;&lt;p&gt;The joke is that Von Neumann answered this question instantly.  The person asking the puzzle was disappointed, and said &amp;quot;oh no, you knew the trick&amp;quot;.  Von Neumann answered &amp;quot;what trick?&amp;quot;, as he had just generated the infinite series and summed it in a fraction of a second.&lt;/p&gt;&#xA;&lt;p&gt;(The trick is you can note the trains crash in 371 km/(30+45 km/h) time, and the fly flies at 150 km/h constantly, so the fly travels (150/75) * 371 = 742 km.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Using first principles all the time is a bit like always summing the infinite series.  It works, but it is often much easier to see patterns and skip on the extra work.&lt;/p&gt;&#xA;&lt;p&gt;And to the best of my ability to determine, most professional mathematicians don't reason from first principles either.&lt;/p&gt;&#xA;&lt;p&gt;They have heuristics and patterns that they apply, and they are capable of &lt;em&gt;verifying&lt;/em&gt; their logic using first principles logic (and often not even that; they verify using other patterns which they are confident they can reduce recursively to first principles).&lt;/p&gt;&#xA;&lt;p&gt;A typical professional mathematician doesn't determine if a function is continuous by doing an epsilon-delta test.  Given &lt;span class=&quot;math-container&quot;&gt;$\sin(x)+\cos(x)$&lt;/span&gt;, they will know it is continuous without thinking; and if asked to verify, it is the sum of two continuous functions (again, not epsilon-delta).  They can probably generate an epsilon-delta proof that the sum of continuous functions is continuous; they may even be able to generate a proof that sin and cos are continuous.&lt;/p&gt;&#xA;&lt;p&gt;But very few mathematicians will look at &lt;span class=&quot;math-container&quot;&gt;$\sin(x)+\cos(x)$&lt;/span&gt; and generate a first principles proof that it is continuous and use that to answer the question.&lt;/p&gt;&#xA;&lt;p&gt;We shouldn't expect students learning mathematics to do this either.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Look at the concepts.  Learn some examples.  Go back to the concepts, and see how the concepts relate to the examples.  Look at more examples.&lt;/p&gt;&#xA;&lt;p&gt;A feedback loop, instead of just memorizing concepts.&lt;/p&gt;&#xA;&lt;p&gt;The thing is, the space of proofs is insanely large.  The intuition you gain on what the next step in a proof should be is heavily based on pattern-matching.&lt;/p&gt;&#xA;&lt;p&gt;How do you know how to prove &lt;span class=&quot;math-container&quot;&gt;$f(x) = x$&lt;/span&gt; is continuous?  Because you have seen that proof before.  And you've used the techniques to prove it in a myriad of situations that are similar.  You don't even think about those situations -  you just see the options of where you can go in your proof, and have a good idea which of those lead closer to your goal.&lt;/p&gt;&#xA;&lt;p&gt;On top of that, you nearly instantly see the next step after doing a certain step in a proof.  When you have your epsilon and delta picked, you instantly see the conclusion.  A person more new to this proof has to go and check that the epsilon and delta portions match, and doesn't get it instantly.&lt;/p&gt;&#xA;&lt;p&gt;What more, if they are even less practiced, they can't keep the entire epsilon and delta part in their head all at once because their brain hasn't learned to chunk it into appropriate pieces.  So now they have to literally go back and compare parts of each section to see if they match.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Coming up with a new proof technique is insanely hard!  Piles of mathematics is done using known proof techniques combined in new and interesting ways.  Mathematicians learn those techniques from others, and apply them in new areas - and even that is a hard task.&lt;/p&gt;&#xA;&lt;p&gt;By expecting your students to start with the definition and generate proofs de novo, you are effectively requiring that they come up with new proof techniques.  Sure, to you they are old hat - but they haven't seen it enough to be able to see the patterns.&lt;/p&gt;&#xA;&lt;p&gt;If you have a 5 line proof, and at each step there are 10 different ways to manipulate the previous line, the search space of possible proofs is 100000 large.  Knowing which route to take towards the conclusion is something the student isn't going to know.&lt;/p&gt;&#xA;&lt;p&gt;And in reality, there are way more than 10 ways to go in a proof, and proofs are often longer than 5 lines.&lt;/p&gt;&#xA;&lt;p&gt;Prove &lt;span class=&quot;math-container&quot;&gt;$f(x)=2x$&lt;/span&gt; is continuous.&lt;/p&gt;&#xA;&lt;p&gt;Choice 1: We pick an epsilon.&lt;/p&gt;&#xA;&lt;p&gt;Choice 2: We pick a delta.  Which delta?  Well, we guess.  As someone experienced, we probably guess right!&lt;/p&gt;&#xA;&lt;p&gt;Choice 3: We pick a point &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;.  Note that I was lucky; &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is uniformly continuous. So picking a point afterwards is safe; if I wasn't lucky, my proof fell apart.&lt;/p&gt;&#xA;&lt;p&gt;Choice 4: I describe the points within delta of &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Choice 5: I mention that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is monotonic, so &lt;span class=&quot;math-container&quot;&gt;$\min(f(X)) = f(\min(X))$&lt;/span&gt; for a set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, and similar for &lt;span class=&quot;math-container&quot;&gt;$\max$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Choice 6: I describe the points reached by the min and max of the points within delta of &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Choice 7: I go back and modify my definition of delta (set it equal to half epsilon actually) and show that &lt;span class=&quot;math-container&quot;&gt;$f( \text{points within delta of } x_0 )$&lt;/span&gt; are within epsilon of &lt;span class=&quot;math-container&quot;&gt;$f(x_0)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Without knowing where I am going, finding those 7 steps is simply insanity.  We find it easy, because we have done similar proofs thousands of times.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Don't expect the student to come up with new proof techniques.  Instead, teach them proof techniques through examples, and expect them to mimic.  Then teach them how to recombine multiple proof techniques.  Expect them to mimic the recombining.  Do this in a pile of different domains, so the student knows how to use many proof techniques and how to recombine them.  Teach them how to steal proof techniques from one domain and use them in another.&lt;/p&gt;&#xA;&lt;p&gt;As this builds up and if this works, the student will no longer seem to be merely mimicing you.&lt;/p&gt;&#xA;&lt;p&gt;The worst part of this is the trap of the good student.  Often you'll have students who have no problems when you skip steps.  This is usually because they have been good mathematician students in the past, and they can chunk things better (so their retention is better), and they even sometimes read ahead (so they have seen the new techniques already).&lt;/p&gt;&#xA;&lt;p&gt;Between better chunking (lower load) and reading ahead (or having seen advanced techniques in the past and half-remembering them), they will appear to be able to answer your challenges - when you ask them to solve a problem you haven't taught them and they haven't seen before, they come up with a proof!&lt;/p&gt;&#xA;&lt;p&gt;It is unlikely they came up with the proof technique on the spot.  What you are detecting can be is that some of your students know more than you have taught them, not that good students don't have to be taught proof patterns.&lt;/p&gt;&#xA;" OwnerUserId="9985" LastEditorUserId="9985" LastEditDate="2023-10-24T14:26:21.333" LastActivityDate="2023-10-24T14:26:21.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26976" PostTypeId="2" ParentId="26959" CreationDate="2023-10-19T19:38:21.057" Score="1" Body="&lt;p&gt;Learning is neither learning by analogy or learning the definitions.&lt;/p&gt;&#xA;&lt;p&gt;It is by being exposed to both examples and definitions and then re exposed to both that we come to learn what the definition means.&lt;/p&gt;&#xA;" OwnerUserId="15264" LastActivityDate="2023-10-19T19:38:21.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26977" PostTypeId="2" ParentId="26959" CreationDate="2023-10-19T21:42:03.970" Score="8" Body="&lt;p&gt;Because examples are not only easier to think about than abstract concepts, but learning examples is a &lt;strong&gt;necessary precursor&lt;/strong&gt; to learning the concept which generalises them.&lt;/p&gt;&#xA;&lt;p&gt;A lot of maths educators have observed this from many perspectives, which many other answerers have discussed already. I'd like to add two more perspectives that don't seem to have been covered yet. The first is from Prof. Tim Gowers, who calls it the &lt;a href=&quot;https://gowers.wordpress.com/2007/10/19/my-favourite-pedagogical-principle-examples-first/&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;examples first&amp;quot; principle&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;if a general definition is at all complex, then you will have quite a lot to hold in your head. This can be difficult, but it is much easier if the various aspects of the definition can be related to an example with which you are familiar. Then the words of the definition cease to be free-floating, so to speak, and instead become labels that you can attach to bits of your mental picture of the example.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The second comes from the APOS theory of learning, set out in a 2001 paper by &lt;a href=&quot;https://www.math.kent.edu/%7Eedd/ICMIPaper.pdf&quot; rel=&quot;noreferrer&quot;&gt;Dubinsky &amp;amp; McDonald&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The theory we present begins with the hypothesis that mathematical knowledge consists in an individual’s tendency to deal with perceived mathematical problem situations by constructing mental &lt;em&gt;actions&lt;/em&gt;, &lt;em&gt;processes&lt;/em&gt;, and &lt;em&gt;objects&lt;/em&gt; and organizing them in &lt;em&gt;schemas&lt;/em&gt; to make sense of the situations and solve the problems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The behaviour described in the question, where a student attempts to solve problems &lt;em&gt;&amp;quot;by looking at solved examples and then mimic[king] the solution to solve other similar examples&amp;quot;&lt;/em&gt; is roughly the 'action' level in this theory ─ the student's conception of e.g. continuity is limited to a set of steps to be followed in a set order. Those steps aren't given in the definition of continuity, but the student can learn the steps by seeing examples of the definition being applied.&lt;/p&gt;&#xA;&lt;p&gt;In the question, it's said that students do this &lt;em&gt;&amp;quot;without necessarily knowing or understanding the concept that is involved&amp;quot;&lt;/em&gt;. In terms of APOS theory, we would say that there are different levels or stages of understanding the concept of continuity ─ corresponding to different conceptions or mental constructions which the student possesses ─ and a student who is able to reliably follow the steps according to the given examples &lt;em&gt;does&lt;/em&gt; understand continuity at the 'action' level. The next goal is to get them to understand continuity at the 'process' level, where they might&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Understand how the steps for checking whether a function is continuous relate to the definition,&lt;/li&gt;&#xA;&lt;li&gt;Know why each step is needed (or when certain steps aren't),&lt;/li&gt;&#xA;&lt;li&gt;Have a variety of strategies for checking whether different kinds of functions are continuous, e.g. from first principles or by applying a theorem.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Crucially, the theory predicts that students must reach an 'action'-level understanding &lt;em&gt;before&lt;/em&gt; a 'process'-level understanding. So it's no surprise that many students who don't demonstrate the latter, may still be able to demonstrate the former.&lt;/p&gt;&#xA;" OwnerUserId="19245" LastActivityDate="2023-10-19T21:42:03.970" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26980" PostTypeId="2" ParentId="26959" CreationDate="2023-10-20T08:32:59.927" Score="2" Body="&lt;p&gt;I suggest that this is just human nature, and not immaturity as some of the other answers suggest. Most people learn things because they want to do things with the knowledge. So the most natural way to approach a problem is to try to do something with the knowledge immediately.&lt;/p&gt;&#xA;&lt;p&gt;I would offer as an example the way that computer programmers learn new technology. Every new product will have a “Quick start” guide that allows you do &lt;em&gt;something&lt;/em&gt; with the technology right away—e.g., create a simple application. That engages with people's brains, much more than a technical explanation of how the code works would do.&lt;/p&gt;&#xA;&lt;p&gt;Speaking for myself, I used the Pythagorean Theorem for years successfully, without any concept of why it worked. Eventually I did go back and learn that, and that was intellectually satisfying. But I suspect most people are just content to know that it &lt;em&gt;does&lt;/em&gt; work, not how.&lt;/p&gt;&#xA;&lt;p&gt;(I wonder how many people could give really rigorous definitions of derivatives and integrals—even engineers who use the concepts daily.)&lt;/p&gt;&#xA;" OwnerUserId="23039" LastActivityDate="2023-10-20T08:32:59.927" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="26981" PostTypeId="2" ParentId="26959" CreationDate="2023-10-20T11:33:55.330" Score="3" Body="&lt;p&gt;Undergraduate engineering student here. I like math, but at the end of the day my job is to apply it to a situation. I currently have a professor who just talks theory, and I end up trying to jump through a lot of mental hoops trying to figure out why some theorem or definition is important, how I'll need to use it later on, or just trying to figure out what the point of an hour long lecture was. But I agree that just examples aren't a great way of going about things either, the theory is definitely important as well.&lt;/p&gt;&#xA;&lt;p&gt;Personally I've found what works best for me is to start with a problem, one that has some feature that can't be solved with the current tools I've learned in that class. Then explore why we can't solve it, look at ways we might solve, and then generalize that definition and eventually find a theorem or proof that can be applied to any situation (and then apply it!). In this way, I've been given a reason to care about the theorem, and know why it's important, while still learning the theorems and then being able to apply them to problems I haven't seen before.&lt;/p&gt;&#xA;&lt;p&gt;To go back to the continuity example, where the students ever given a reason to care about continuity? Were they given a reason as to why continuity is important? Were they given any examples where ignoring discontinuity would lead to incorrect results? Do they have a reason to believe it has actual applications and isn't just a cheap party trick?&lt;/p&gt;&#xA;&lt;p&gt;If I just wanted a random stream of theorems and proofs, the internet has been able to do a pretty good job of that for at least the past decade, and textbooks for at least the past couple centuries. The way I view the job of the teacher is to guide me through that stream of human knowledge, to give me a reason why I should care about certain things and to give meaning and reason to a curriculum greater than the individual proofs and theorems it contains.&lt;/p&gt;&#xA;&lt;p&gt;I don't think it's that the students don't care about theory, I think they might have just never been given a reason to care. Hope that helps :)&lt;/p&gt;&#xA;" OwnerUserId="23183" LastActivityDate="2023-10-20T11:33:55.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26982" PostTypeId="2" ParentId="26959" CreationDate="2023-10-20T12:48:45.380" Score="2" Body="&lt;p&gt;In fact this is well studied in the literature and it has to do with concept formation. There are two ways we do this: one is by collecting exemplars and doing a sort of nearest neighbor search to see if we know an example we've seen to help with the new input. This is why you see students making bonkers assumptions they trying to tell you &amp;quot;I thought it was X&amp;quot; when it clearly isn't. A good example of this was from my graduate studies days. It instructor jokingly said she would give us &amp;quot;proof by special example&amp;quot;, by which she meant &amp;quot;here is an example, if you want the proof is longer than the time we have&amp;quot;. However a student in the class later on, when asked to do a proof, asked &amp;quot;can we use proof by special example here?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The other way we gain concepts is by abstracting those examples into something more like what you are looking for. However that only really happens if the concept is very simple or you've seen lots of examples.&lt;/p&gt;&#xA;&lt;p&gt;You can see this when experts solve problems. They essentially &amp;quot;prune the tree&amp;quot;, ignoring irrelevant knowledge and examples to figure out a smaller subset of things that need to be considered.&lt;/p&gt;&#xA;&lt;p&gt;So your students are doing exactly what they need to do: all they have to go on is what they've seen so far, so the best they can do is relate a new problem to an old one and hope for the best. The concepts come later through lots of exposure to examples and discussion, but it almost never happens that we get a concept first then magically can use it.&lt;/p&gt;&#xA;" OwnerUserId="23185" LastActivityDate="2023-10-20T12:48:45.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26983" PostTypeId="2" ParentId="18989" CreationDate="2023-10-20T12:51:14.397" Score="0" Body="&lt;p&gt;As I wrote &lt;a href=&quot;https://matheducators.stackexchange.com/a/26973/6589&quot;&gt;here&lt;/a&gt;, in machine learning applications we are often trying to minimize some cost function &lt;span class=&quot;math-container&quot;&gt;$J(\theta)$&lt;/span&gt; and the vector field induced by the gradient &lt;span class=&quot;math-container&quot;&gt;$-\nabla_{J(\theta)}$&lt;/span&gt; is important in numerical approximations to minimizing error.&lt;/p&gt;&#xA;" OwnerUserId="6589" LastActivityDate="2023-10-20T12:51:14.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26984" PostTypeId="2" ParentId="26959" CreationDate="2023-10-20T13:13:50.457" Score="3" Body="&lt;p&gt;One thing helping me in teaching is homework like: 1. Write down the definition.... apply it to prove the following examples. 2. the announcement: In the next test I will ask not only for proofs or examples but also for the following definitions.&lt;/p&gt;&#xA;" OwnerUserId="21051" LastActivityDate="2023-10-20T13:13:50.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26985" PostTypeId="2" ParentId="26959" CreationDate="2023-10-20T16:28:31.500" Score="0" Body="&lt;p&gt;Apologies up front, I don't have references to the studies I'm about to mention.&lt;/p&gt;&#xA;&lt;p&gt;Psychologists have studied how the brain processes language and math, and found that language and math are processed in two different places in the human brain.&lt;/p&gt;&#xA;&lt;p&gt;Language is processed like a picture.  As children learn language, they learn the sounds of words first, then the letters, the the individual words, followed by sentences.  The reason why children progress through this system and increase their speed is that their brains are memorizing patterns of how the words look and matching them to the sounds for the words they heard.  Inititally, each letter was a picture, but after 1000's of letters, the child can pick out the letter A no matter how stressed it is.  The same pattern follows with words, like CAT, and eventually books containing stories about CATS.&lt;/p&gt;&#xA;&lt;p&gt;When it comes to Math, which is processed in a different region of the brain, the language system passes the picture to the math system and say process this.  Here's where things get interesting.  The math portion of the brain has to break the picture back into it's constituent parts  &amp;quot;1 + 1 = 2&amp;quot;, and then it has to ask the question .. Is this correct? ..  The Math portion of the brain is a difference engine constantly testing the accuracy of the mathematical pictures it's given.&lt;/p&gt;&#xA;&lt;p&gt;The effect that this has on children / humans in general, is that we never know if the mathematical statement we are looking at is true or false until we have enough examples under our belt to say it's true.  This is why mathematical statements like &amp;quot;Continuous Functions&amp;quot; don't immediately register. The students see the definition, read it, understand what it's saying, but can't apply it, because they have no mental picture that says .. true or false.  They have to build that picture by seeing 100's of examples and working through 100s of problems specifically tailored to that concept.&lt;/p&gt;&#xA;&lt;p&gt;Now, consider the problem for someone with Dyscalculia.  They see &amp;quot;1 + 1 = 2&amp;quot;, &amp;quot;2 - 1 = 1&amp;quot;, &amp;quot;1 - 2 = -1&amp;quot;, &amp;quot;1 + 1 -2 = 0&amp;quot; and are asked, are all these statements identical?  The answer is .. NO! .., but not for the reason you are thinking. The answer is .. NO! .. because they all look different.  As a picture, they are all different.  To be processes mathematically, the brain has to break the picture down, pass it to the math processor and ask the question, are these all mathematically identical?  Then, and only after having seen these problems for 100's of times, does the brain maybe say .. Yes! ..&lt;/p&gt;&#xA;&lt;p&gt;Remember, as a difference engine, if garbage data gets in, it will muddle the responses.  The student may be recalling instances where the answer was not correct. Or, were told that the answer was not correct.  These bits of garbage have to be filtered out through careful retraining.  This can be esp frustrating to students who don't trust their notes, and don't have access to resources to check their notes, home work, understanding, .. Which is why I turn to sites like Slader.com to work through problems, at least I can see how someone else solved the problem, reason as to if the problem was solved correctly, and then see if I get the correct answer.&lt;/p&gt;&#xA;&lt;p&gt;So, when considering teaching a topic, look for as many ways to cement the concept into the student as possible ..&lt;/p&gt;&#xA;&lt;p&gt;Discuss a specific topic&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;use 100's of Examples&lt;/li&gt;&#xA;&lt;li&gt;use Text&lt;/li&gt;&#xA;&lt;li&gt;use Video&lt;/li&gt;&#xA;&lt;li&gt;use Interactive Demos&lt;/li&gt;&#xA;&lt;li&gt;use Pictures / Diagrams&lt;/li&gt;&#xA;&lt;li&gt;use Graphs&lt;/li&gt;&#xA;&lt;li&gt;use Practice Problems&lt;/li&gt;&#xA;&lt;li&gt;use Spaced Recall&lt;/li&gt;&#xA;&lt;li&gt;use Physical Experiments&lt;/li&gt;&#xA;&lt;li&gt;check Terminology&lt;/li&gt;&#xA;&lt;li&gt;review Strategy&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The focus is to build the students confidence in the specific topic&lt;/p&gt;&#xA;&lt;p&gt;For higher educational classes .. University Classes .. Have the students&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;identify which section of the book discusses this problem&lt;/li&gt;&#xA;&lt;li&gt;write their strategy before solving the problem.&lt;/li&gt;&#xA;&lt;li&gt;write all the constants needed for the problem&lt;/li&gt;&#xA;&lt;li&gt;identify any equations that seem relevant to the problem&lt;/li&gt;&#xA;&lt;li&gt;draw graphs and diagrams .. I do this for sine and cosine ..&lt;/li&gt;&#xA;&lt;li&gt;transform equations before using them (solve for x)&lt;/li&gt;&#xA;&lt;li&gt;do gap analysis&#xA;-- what variables are unidentified?&#xA;-- do the units match up?&#xA;-- how is the correct answer different from their answer?&#xA;-- does a mathematical transform make sense?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;When I look at problems for Physics, Math, and Engineering, I always ask, how close is the problem in pages to the content that describes the strategy to solve the problem.  This distance is one of many factors I use to determine the difficulty in solving the problem, as it indicates how much effort will be involved to remember / recall where the problem was discussed but what else is built on top of this discussion.&lt;/p&gt;&#xA;&lt;p&gt;Another problem that I'm seeing is how the question is worded.  Some questions introduce new terminology that is not discussed in the text.  Some questions attempt to teach the student something not discussed in the text.  This confuses the student / me as I'm simply looking for the question .. Solve for X .. Instead I get a page and a half description about various components of the problem to be solved, and no real clear definition of what I'm solving for.  Or, worse, the question doesn't seem to match what their expectations of the answer are at all.&lt;/p&gt;&#xA;&lt;p&gt;Example:  Prove that (E) must be real by replacing (E) with (Eo + iT), and show that for Eq 1.20 to be true for all (t), (T) must be zero.&lt;/p&gt;&#xA;&lt;p&gt;Another thing to consider that confuses Dyslexics, is that common letters in math look the same or are distinctly different .. w and W .. are the same when drawn on a board with no other reference points to discern them.  Where as Δx, dx, and ∂x will all mean the same thing until 100's of examples differentiate them.&lt;/p&gt;&#xA;&lt;p&gt;Hopefully this helps inform your instructional practices and gives you ideas on how to approach learning gaps / deficits in math.&lt;/p&gt;&#xA;&lt;p&gt;One Educational Psychologist used the following terms .. Encoding, Generation, Gap Analysis, Recall ..&lt;/p&gt;&#xA;" OwnerUserId="23188" LastActivityDate="2023-10-20T16:28:31.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26986" PostTypeId="2" ParentId="26959" CreationDate="2023-10-20T19:47:43.470" Score="1" Body="&lt;p&gt;Yes, it's natural. Action is prior to conception phylogenetically, ontogenetically and logically, even if entwined temporally.&lt;/p&gt;&#xA;&lt;p&gt;You might class things learned into concepts and skills. A skill is a way of acting in the world, at some level of competence (systematic chance at success), vis-a-vis certain objects and goals. Those goals are most often 'tangible' in the sense of not open-ended, even if the object realm is abstract.&lt;/p&gt;&#xA;&lt;p&gt;Skill is learned through guided repetition. Learning anything entails learning skills, both regarding learning itself and items figuring in it, and the subject matter. At the same time skill development is likely to require some mental structuring by outwardly presented formal or folk concepts or ones the learner makes up.&lt;/p&gt;&#xA;&lt;p&gt;When we humans talk about 'concepts' we mean something capable of expression and transmission by language, something only we can have. Yet many other animals' action is situated and adaptable, and some can be observed cogitating on a challenge (recognized by their going straight through a successful procedure afterwards). Now, the nature of animal conceptualizing is still unclear and there are those who dislike the idea. But the spectrum of specificity and flexibility in various animals' action together with human developmental psychology make palpable that structuring and acting on perceptions is prior to structuring and acting on notions.&lt;/p&gt;&#xA;&lt;p&gt;Sorry I'm not a scholar of these things, but I believe none of the above is controversial.&lt;/p&gt;&#xA;" OwnerUserId="23189" LastActivityDate="2023-10-20T19:47:43.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26987" PostTypeId="2" ParentId="26959" CreationDate="2023-10-20T21:50:57.033" Score="1" Body="&lt;p&gt;The answer to your question is teachers want to rush the learning and associate student struggle with laziness.  Students have to spend a sufficient amount of time in the knowledge phase.  They need time with vocabulary and opportunities to associate the vocabulary with their acquired skills.&lt;/p&gt;&#xA;&lt;p&gt;If teachers move too fast then student learning is compromised.  Some students will be ready to take a deeper after a few days…some students may need longer.  Respect the learning process and your kids will begin performing at higher levels.&lt;/p&gt;&#xA;&lt;p&gt;If you teach where you wants kids to be, then many of your students will never get there, or will take them much longer than needed.  This is my 2 cents‍♂️&lt;/p&gt;&#xA;" OwnerUserId="23191" LastActivityDate="2023-10-20T21:50:57.033" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26990" PostTypeId="1" CreationDate="2023-10-21T14:38:06.977" Score="1" ViewCount="206" Body="&lt;p&gt;How can we identify trigonometrical proofs from geometrical proofs, do we have purely trigonometrical proof of Pythagoras theorem as claimed by two high school students ?&lt;br/&gt; &lt;a href=&quot;https://www.scientificamerican.com/article/2-high-school-students-prove-pythagorean-theorem-heres-what-that-means/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.scientificamerican.com/article/2-high-school-students-prove-pythagorean-theorem-heres-what-that-means/&lt;/a&gt;&lt;br/&gt;&#xA;As mathematics teachers we need to encourage students to find alternative proofs and for that purpose I think it's important to let them discuss positive and negative points of new approaches. I have following issues related to the high school students achievement. &lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How can you apply above mentioned proof for right angled isosceles triangles?&lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Can you say that approach is trigonometrical because of using trigonometrical formula &lt;span class=&quot;math-container&quot;&gt;$\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$&lt;/span&gt; since it also can be derived geometrically by right angled triangles?&lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Is it not possible to complete the proof by just adding the squares of two perpendicular sides without using the formula &lt;span class=&quot;math-container&quot;&gt;$\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$&lt;/span&gt; and say that proof is geometrical? &lt;br/&gt;&#xA;Here my question is related to how we have to update students using new developments in mathematics, therefore I hope you will not misunderstand this as a mathematics question.&lt;br/&gt;&#xA;&lt;br/&gt;&#xA;Edited to add following links to give further references,&#xA;&lt;a href=&quot;https://youtu.be/Ka1k4i1ueNU&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/Ka1k4i1ueNU&lt;/a&gt; &lt;br/&gt;&#xA;&lt;a href=&quot;https://youtu.be/p6j2nZKwf20?si=R-G00I5vyKsFalba&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/p6j2nZKwf20?si=R-G00I5vyKsFalba&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="8190" LastEditDate="2023-10-23T13:44:40.623" LastActivityDate="2023-10-24T02:08:35.630" Title="Identifying Trigonometrical proofs" Tags="|secondary-education|mathematical-pedagogy|proofs|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="26991" PostTypeId="1" AcceptedAnswerId="27030" CreationDate="2023-10-21T16:06:12.447" Score="2" ViewCount="400" Body="&lt;p&gt;I always showcase &lt;em&gt;separate&lt;/em&gt; pictures of &lt;a href=&quot;https://math.stackexchange.com/a/3936145&quot;&gt;Triangle Inequality&lt;/a&gt;, and &lt;a href=&quot;https://math.stackexchange.com/a/774233&quot;&gt;Reverse&lt;/a&gt;, to 16-years-old students in 1st  class. I reshow pictures in 2nd class.  I preachify&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Please remember these 4 inequalities. Save these pictures as screen savers or wallpapers, hang them by your toilet and bedroom, do what you got to do.  In my 10 years of teaching, someone  YEARLY always forget 1 of below 4 inequalities ON MULTIPLE TESTS! Do not be this addlepate!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But no soap! Yearly, someone still forgets! After they lost  marks, they and I always scratch our heads! They wonder why they forgot. And  I wonder why, after ten years of teaching, I again fail to drum all these 4 inequalities into ALL students' heads.  I need new approaches! Rather than separate picture for each inequality, how can I pictorialize below 4 inequalities concurrently?&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.imgur.com/DBnIGdb.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;h4&gt;How can 1 sole picture prove all 4 inequalities at once? Please contrast &lt;span class=&quot;math-container&quot;&gt;$\vec{x} ≠ \vec{y}$&lt;/span&gt;, and draw &lt;span class=&quot;math-container&quot;&gt;$|\vec{x} -  \vec{y}| \neq |\vec{x} + \vec{y}|$&lt;/span&gt; with different lengths.&lt;/h4&gt;&#xA;&lt;p&gt;The four inequalities to be remembered are the ones in the following excerpts from Spivak's &lt;em&gt;Calculus Fourth Edition&lt;/em&gt; (2008):&lt;/p&gt;&#xA;&lt;p&gt;Page 12 proved &lt;span class=&quot;math-container&quot;&gt;$|x + y| ≤ \color{YellowGreen}{|x| + |y|}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Page 16, exercise 12.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(iv) &lt;span class=&quot;math-container&quot;&gt;$\color{red}{|x−y|}≤\color{YellowGreen}{|x|+|y|}$&lt;/span&gt;. (Give a very short proof.)&lt;br /&gt;&#xA;(v) &lt;span class=&quot;math-container&quot;&gt;$\color{deepskyblue}{|x|−|y|}≤\color{red}{|x−y|}$&lt;/span&gt;. (A very short proof is possible, if you write things in the right way).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(vi) &lt;span class=&quot;math-container&quot;&gt;$| \; \color{deepskyblue}{|x|−|y|} \; |≤|x−y|$&lt;/span&gt; (Why does this follow immediately from (v)?)&lt;/p&gt;&#xA;&lt;p&gt;I shalln't blockquote this (vi), the Reverse △ ≠, as I deleted Spivak's superfluous round brackets.&lt;/p&gt;&#xA;" OwnerUserId="23199" LastEditorUserId="1737" LastEditDate="2023-10-26T06:38:30.740" LastActivityDate="2023-10-29T15:25:53.250" Title="How can one lone picture prove the Triangle Inequality, $|x−y|≤|x|+|y|$, $|x|−|y|≤|x−y|$, and the Reverse Triangle Inequality?" Tags="|secondary-education|vectors|absolute-value|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="26992" PostTypeId="2" ParentId="26991" CreationDate="2023-10-21T18:06:53.117" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Please remember these 4 inequalities. Save these pictures as screen savers or wallpapers, hang them by your toilet and bedroom, do what you got to do.&lt;/p&gt;&#xA;&lt;p&gt;I always showcase &lt;em&gt;separate&lt;/em&gt; pictures . . . I wonder why, after ten years of teaching, I again fail to drum all these 4 inequalities into ALL students' heads. I need new approaches! Rather than separate picture for each inequality, how can I pictorialize below 4 inequalities concurrently?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Your goal is for students to remember these inequalities, and you're showing pictures because pictures generally make things easier to remember (not because you want the students to prove the inequalities).&lt;/p&gt;&#xA;&lt;p&gt;However, in this case, pictures are not the best memory aid. You have to remember the pictures and think critically about them in order to translate them into symbols. Critical thinking is always going to be a struggle for some students.&lt;/p&gt;&#xA;&lt;p&gt;The easiest way to remember these inequalities is to combine them into a single compound inequality that looks so remarkably intuitive that it just &amp;quot;feels&amp;quot; right and is easy to reconstruct based on simple patterns, without having to engage in critical thinking.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;\begin{matrix} \\ |x| - |y|, \\ | \,\, |x| - |y| \,\, | \phantom{,} \end{matrix} \, \leq \, \begin{matrix} \\ |x+y|, \\ |x-y| \phantom{,} \\ \end{matrix} \, \leq |x| + |y|&#xA;\\ \\&#xA;\end{align*}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here's how to memorize the inequality:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Put &lt;span class=&quot;math-container&quot;&gt;$\,\, \begin{matrix} |x+y|, \\ |x-y| \phantom{,} \\ \end{matrix} \,\,$&lt;/span&gt; in the middle.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;Distribute&amp;quot; the absolute value to get the top LHS and RHS. (LHS becomes &amp;quot;-&amp;quot; because left is negative direction; RHS changes to &amp;quot;+&amp;quot; because right is positive direction.)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;|x| - |y| \, \leq \, \begin{matrix} \\ |x+y|, \\ |x-y| \phantom{,} \\ \end{matrix} \, \leq |x| + |y|&#xA;\\ \\&#xA;\end{align*}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you put another set of absolute value bars around the LHS, the inequality also holds true. (You can do this to the RHS too; it's just not noteworthy because it's mathematically equivalent to the existing RHS.)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;\begin{matrix} \\ |x| - |y|, \\ | \,\, |x| - |y| \,\, | \phantom{,} \end{matrix} \, \leq \, \begin{matrix} \\ |x+y|, \\ |x-y| \phantom{,} \\ \end{matrix} \, \leq |x| + |y|&#xA;\\ \\ \end{align*}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-28T18:41:23.257" LastActivityDate="2023-10-28T18:41:23.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26993" PostTypeId="2" ParentId="26991" CreationDate="2023-10-22T07:19:11.950" Score="5" Body="&lt;p&gt;First of all, you want them them to &amp;quot;get it right&amp;quot; on a test, rather than be able to use this to more deeply understand vectors. In fact, it seems to be about memorizing this, which is totally inappropriate for something like this. Please approach math with the goal of sharing its beauty and depth. (Not &amp;quot;no addlepates&amp;quot;.)&lt;/p&gt;&#xA;&lt;p&gt;I recommend that you stop trying to make one perfect handout, and start thinking about how your students learn. We learn through play. Make a handout that encourages them to play. Saving your pictures is not going to help.  Walk them through the details on a few problems, and then have them do other problems. Maybe have them make up their own vectors.&lt;/p&gt;&#xA;&lt;p&gt;Next, use a notation that helps students discuss this. There are x and y coordinates. Please don't make the vectors x and y also. Let's use a and b for vectors. Then &lt;span class=&quot;math-container&quot;&gt;$ \left | a \right | = \sqrt{{a_{x}}^{2}+{a_{y}}^{2}} $&lt;/span&gt;. etc.&lt;/p&gt;&#xA;&lt;p&gt;(If you are teaching 16-year-olds, why are you quoting Spivak?)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2023-10-22T15:59:18.963" LastActivityDate="2023-10-22T15:59:18.963" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="26994" PostTypeId="2" ParentId="20687" CreationDate="2023-10-23T08:02:28.493" Score="0" Body="&lt;p&gt;A very unique kind of book is “Machine Proofs in Geometry: Automated production of readable proofs for geometry theorems” by Shang-Ching Chou, Xiao-Shan Gao and Jing-Zhong Zhang. (It is of interest to people studying automated reasoning as well).&lt;/p&gt;&#xA;" OwnerUserId="16420" LastActivityDate="2023-10-23T08:02:28.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26995" PostTypeId="2" ParentId="26991" CreationDate="2023-10-23T08:03:25.073" Score="1" Body="&lt;p&gt;You are telling sixteen-year olds that:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$|x - y| \le |x| + |y|$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;... and you call that &amp;quot;triangle inequality&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;... and your teenage students don't get that?&lt;/p&gt;&#xA;&lt;p&gt;Well, I'm a mathematician and I have troubles understanding that.&lt;/p&gt;&#xA;&lt;p&gt;However, let's look at this in the original way (based on vectors):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$|\vec{x} - \vec{y}| \le |\vec{x}| + |\vec{y}|$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Well: this makes perfectly sense, as you can see from this easy drawing (made in Geogebra, I don't know how to put arrows on the labels of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/i7mht.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/i7mht.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The distance from &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(8,0)$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$|\vec{u} + \vec{v}|$&lt;/span&gt;) is larger than the distance from &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(2,1)$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$\vec{u}$&lt;/span&gt;) plus the distance from &lt;span class=&quot;math-container&quot;&gt;$(2,1)$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$(8,0)$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$\vec{v}$&lt;/span&gt;), which is quite obvious because the smallest distance between two points is a straight line.&lt;/p&gt;&#xA;&lt;p&gt;Now: set &lt;span class=&quot;math-container&quot;&gt;$\vec{u}=\vec{x}-\vec{y}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec{v}=\vec{y}$&lt;/span&gt;, and you have your formula in vector-form.&lt;br /&gt;&#xA;Next you just use 1D-vectors and you have your formula in number-form.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-10-23T08:03:25.073" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="26996" PostTypeId="2" ParentId="26899" CreationDate="2023-10-23T12:51:05.417" Score="4" Body="&lt;p&gt;The reason given in the accepted answer is valid, but I'd like to point out something that Ben Crowell (user507) &lt;a href=&quot;https://matheducators.stackexchange.com/a/4285/590&quot;&gt;said&lt;/a&gt; in a comment several years back: that problem has got nothing to do with calculus, infinitesimals, or treating derivatives as fractions, but more with an incomplete notation used to denote the fractions involved. Because the same problem arises if we look at a linear function &lt;span class=&quot;math-container&quot;&gt;$z=f(x,y)$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; themselves being linear functions of &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, and ask for &lt;span class=&quot;math-container&quot;&gt;$\frac{\Delta z}{\Delta t}$&lt;/span&gt; (no calculus, no infinitesimals, true fractions).&#xA;In that case we also have (with incomplete notation)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \Delta z = \frac{\Delta z}{\Delta x}\Delta x + \frac{\Delta z}{\Delta y}\Delta y$$&lt;/span&gt;&#xA;and dividing by &lt;span class=&quot;math-container&quot;&gt;$\Delta t$&lt;/span&gt; gives the 'chain rule'. Here we could literally cancel &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; in the first summand and &lt;span class=&quot;math-container&quot;&gt;$\Delta y$&lt;/span&gt; in the second, but then we arrive at the nonsense &lt;span class=&quot;math-container&quot;&gt;$\Delta z= 2\Delta z$&lt;/span&gt;. How come? The reason is that in the first summand &lt;span class=&quot;math-container&quot;&gt;$\Delta z$&lt;/span&gt; denotes the change of &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; &lt;em&gt;when &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is held constant&lt;/em&gt;, while in the second it denotes the change of &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; &lt;em&gt;while &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is held constant&lt;/em&gt;. If the notation made explicit that those two &lt;span class=&quot;math-container&quot;&gt;$\Delta z$&lt;/span&gt; are different, we could in principle avoid the problem. For instance writing&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \Delta z = \frac{\Delta z|_{y=\text{const.}}}{\Delta x}\Delta x + \frac{\Delta z|_{x=\text{const.}}}{\Delta y}\Delta y$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If we now cancel we arrive at&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \Delta z = \Delta z|_{y=\text{const.}}+ \Delta z|_{x=\text{const.}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which might seem unusual but at least we wouldn't reduce it to &lt;span class=&quot;math-container&quot;&gt;$\Delta z = 2 \Delta z$&lt;/span&gt;. Similarly we could explain to a student that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{\Delta z}{\Delta t} \neq \frac{\Delta z|_{y=\text{const.}}}{\Delta x}\frac{\Delta x}{\Delta t}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;not because we are not allowed to cancel, but simply because in general &lt;span class=&quot;math-container&quot;&gt;$\Delta z\neq \Delta z|_{y=\text{const.}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Returning to partial derivatives &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial z}{\partial x}$&lt;/span&gt; etc. the same problem arises there: the notation does not show explicitly which variables are held fixed when we take the derivative with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Jacobi was the first to &lt;a href=&quot;https://mathoverflow.net/q/336309/745&quot;&gt;draw attention&lt;/a&gt; to that problem, but his proposed solution was silly and its adoption probably caused more confusion. (It's  &lt;a href=&quot;https://mathoverflow.net/q/252928/745&quot;&gt;likely the reason&lt;/a&gt; for the discussion under the accepted answer). If instead we use a more cumbersome but explicit notation like &lt;span class=&quot;math-container&quot;&gt;$\left.\frac{\partial z}{\partial x}\right|_{y=\text{const.}}$&lt;/span&gt;, which physicist sometimes write as &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{\partial z}{\partial x}\right)_y$&lt;/span&gt;, then we could again explain why that false chain rule in the accepted answer is wrong: it becomes&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \left.\frac{\partial z}{\partial t}\right|_{u=\text{const.}}&#xA;\neq &#xA;\left.\frac{\partial z}{\partial x}\right|_{y=\text{const.}}&#xA;\left.\frac{\partial x}{\partial t}\right|_{u=\text{const.}}&#xA;$$&lt;/span&gt;&#xA;and we are not allowed to cancel the &lt;em&gt;changes in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/em&gt; (i.e. &lt;span class=&quot;math-container&quot;&gt;$\partial x$&lt;/span&gt;), since in the first factor &lt;span class=&quot;math-container&quot;&gt;$\partial x$&lt;/span&gt; is considered under the assumption that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is constant while in the second factor it is considered under the assumption that &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; is constant. (I added the additional variable &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;, like Justin did in a comment, to avoid a discussion about the difference between &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;. So I'm assuming &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; are both functions of &lt;span class=&quot;math-container&quot;&gt;$t,u$&lt;/span&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;The other problem, mentioned in davdan angelo's answer, relating to second derivative notation also disappears, if we understand where that notation for second derivatives comes from: the equality&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{d\frac{dy}{dx}}{dx}=\frac{d^2y}{dx^2}&#xA;$$&lt;/span&gt;&#xA;was historically only considered correct, under the assumption that &lt;span class=&quot;math-container&quot;&gt;$dx=\text{const.}$&lt;/span&gt; See &lt;a href=&quot;https://hsm.stackexchange.com/a/7857/3462&quot;&gt;here&lt;/a&gt;. Under that assumption (and further assumptions necessary to make sense of the involved operations), the chain rule-like equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d^2f}{dx^2}= \frac{d^2f}{dg^2}\left(\frac{dg}{dx}\right)^2$$&lt;/span&gt;&#xA;in davdan angelo's answer is true: we assume &lt;span class=&quot;math-container&quot;&gt;$dx=\text{const.}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$dg=\text{const.}$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; which is a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. These assumptions imply that &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is a linear function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and hence the equations holds.&lt;/p&gt;&#xA;&lt;p&gt;(For people still worrying about what the differentials actually denote: one could make them precise by interpreting variable quantities &lt;span class=&quot;math-container&quot;&gt;$z,x,y,f,g$&lt;/span&gt; etc. as real valued smooth functions on a manifold &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; and differentials &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; etc. as differentials in the sense of differential geometry. The operation &lt;span class=&quot;math-container&quot;&gt;$d^2 x$&lt;/span&gt; would additionally require a connection on the manifold in order to talk about the differential of a differential.)&lt;/p&gt;&#xA;" OwnerUserId="590" LastActivityDate="2023-10-23T12:51:05.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26997" PostTypeId="2" ParentId="3858" CreationDate="2023-10-23T14:06:18.000" Score="0" Body="&lt;p&gt;This is more physics than mathematics, but it can do:&lt;/p&gt;&#xA;&lt;p&gt;In Belgium it is not allowed to pass a vehicle from the right side. Now imagine that you are driving your car on the most right lane, and one the second lane, a car is slowing down and suddenly turns to the right, hitting your car.&lt;/p&gt;&#xA;&lt;p&gt;He might argue that you are passing from the right side, so you are wrong.&lt;br /&gt;&#xA;You might argue that you were doing an ERB (eenparig rechtlijnige beweging, in English you can call this a perfectly straight movement without speed modification). As you are not altering, nor your direction, nor your speed, you are not even performing a manoeuvre, so the word &amp;quot;passing&amp;quot; can't even be used and you are the one being right!&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure if this argument might hold in court, but if ever I get into such an accident, I'll already know my argument :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-10-23T14:06:18.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26998" PostTypeId="2" ParentId="19546" CreationDate="2023-10-23T14:43:50.890" Score="1" Body="&lt;p&gt;Let me tell you the story of my father and the number 5.8:&lt;/p&gt;&#xA;&lt;p&gt;My mother had a shop in Belgium, very near to the French border. This happened in the time before the Euro currency was introduced.&lt;/p&gt;&#xA;&lt;p&gt;Normally the currency change between French and Belgian Frank (&amp;quot;Frank&amp;quot; was the name of the currency) was 6: an article, costing 1800BEF (Belgian Franks) costed 300 FF (French Francs), but after some time the change had dropped to 5.8, and the French customers were very strict on this.&lt;/p&gt;&#xA;&lt;p&gt;One day, a woman came to me and asked me the French price of an article, which costed 300 BEF, and I answered &amp;quot;51 Francs and 70 cents&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;My father overheard this and came running with the calculator in his hand:&lt;br /&gt;&#xA;&amp;quot;&lt;em&gt;My apologies for my son: he's giving us a hand in the shop, but he's not aware of the correct currency change. Let me calculate this for you.&lt;/em&gt;&amp;quot;&lt;br /&gt;&#xA;I said &amp;quot;&lt;em&gt;But dad ...&lt;/em&gt;&amp;quot;&lt;br /&gt;&#xA;My father's immediate answer: &amp;quot;&lt;strong&gt;SH!&lt;/strong&gt; &lt;em&gt;... Ma'm, the correct price is ...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;At that moment, his eyes widened, he didn't know what to say for some seconds, until:&lt;br /&gt;&#xA;&amp;quot;&lt;strong&gt;... he studies mathematics!!!&lt;/strong&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Believe me: my pride was &lt;strong&gt;priceless&lt;/strong&gt;!!!&lt;/p&gt;&#xA;&lt;p&gt;Oh, how did I do this?&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ 300 / 5.8 = 300 / (6 - 0.2) = \frac{300}{6 \cdot (1 - \frac{1}{30})} $&lt;/span&gt;, which can be written as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$300 / 6 \cdot (1 + \frac{1}{30} + (\frac{1}{30})^2 + (\frac{1}{30})^3 ...)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;, which comes down to: &lt;span class=&quot;math-container&quot;&gt;$ 50 + \frac{50}{30} + ...$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$...$&lt;/span&gt; are just some very small numbers, meaning that you need to round up).&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-10-23T14:43:50.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="26999" PostTypeId="2" ParentId="3858" CreationDate="2023-10-23T14:54:36.817" Score="0" Body="&lt;p&gt;Here is a nice article (written for undergraduates)&lt;/p&gt;&#xA;&lt;p&gt;Gray, Mary W.&#xA;&amp;quot;Statistics and the Law&amp;quot;.&#xA;&lt;em&gt;Mathematics Magazine&lt;/em&gt; 56(1983), no.2, 67–81.&lt;/p&gt;&#xA;&lt;p&gt;Mary Gray was both a mathematician and a lawyer.  In this paper, she discusses court cases where mathematics (in particular, statistics) was incorrectly applied.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2023-10-23T14:54:36.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27000" PostTypeId="1" CreationDate="2023-10-23T15:18:48.800" Score="4" ViewCount="179" Body="&lt;p&gt;What options are out there for software that supports interactively constructing, editing, and manipulating &lt;a href=&quot;https://en.wikipedia.org/wiki/Hasse_diagram&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hasse diagrams&lt;/a&gt;? Their semantics is significantly constrained, so garden-variety &lt;em&gt;Let’s-draw-DAGs&lt;/em&gt; apps aren’t all that useful.&lt;/p&gt;&#xA;&lt;p&gt;Desirable features would include&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;automatic monitoring and guided maintenance of transitive reduction&lt;/li&gt;&#xA;&lt;li&gt;dynamic enforcement of the constraint that whenever &lt;span class=&quot;math-container&quot;&gt;$a \leq b$&lt;/span&gt;, the drawing of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is never above the drawing of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. Perhaps if the user tries to drag &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; up past &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, the app rigidly shoves the entire drawing of the induced subposet with minimum &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; upward to make room.&lt;/li&gt;&#xA;&lt;li&gt;support for such fairly standardized queries as “Highlight the (subposet induced by) elements above (resp. below) &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;,” “Highlight the elements incomparable to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;,” “Highlight all the covers of (resp. elements covered by) &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;,” “Highlight the meets (resp. joins) of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;,” and “Highlight successively the maximal (or, in particular, maximum) chains (resp. antichains),” as well as “Show me all the critical pairs.”&lt;/li&gt;&#xA;&lt;li&gt;support for such more general queries as “Here’s a substructure I’m interested in; show me all of the induced subposets isomorphic to it.”&lt;/li&gt;&#xA;&lt;li&gt;Support for labels and annotations in &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; syntax.&lt;/li&gt;&#xA;&lt;li&gt;And of course, an &lt;code&gt;export as PDF&lt;/code&gt; function.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Some of these desired features are computationally intensive. So clearly, as the size of the poset grew, such an app’s performance would inevitably become sluggish. But it would be very useful to have a tool with these capabilities even for only smallish posets.&lt;/p&gt;&#xA;" OwnerUserId="23086" LastActivityDate="2023-10-24T15:24:21.070" Title="What are some decent apps for Hasse diagrams?" Tags="|technology-in-education|software|topology|graph-theory|combinatorics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27001" PostTypeId="2" ParentId="26990" CreationDate="2023-10-23T15:39:30.833" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;The proof: &lt;a href=&quot;https://youtu.be/p6j2nZKwf20&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/p6j2nZKwf20&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For context, here's the main idea of the proof. Using the definition of sine, we have &lt;span class=&quot;math-container&quot;&gt;$c^2 = \dfrac{2ab}{\sin 2\alpha}.$&lt;/span&gt; Our goal is to show that the RHS is equivalent to &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2.$&lt;/span&gt; If &lt;span class=&quot;math-container&quot;&gt;$a=b,$&lt;/span&gt; then this is obvious since &lt;span class=&quot;math-container&quot;&gt;$\alpha = 45^\circ.$&lt;/span&gt; Otherwise, if &lt;span class=&quot;math-container&quot;&gt;$a \neq b,$&lt;/span&gt; then we perform a geometric construction called the &amp;quot;waffle cone&amp;quot; and do some hairy algebra to get to the desired result (this forms the bulk of the proof).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How can you apply above mentioned proof for right angled isosceles triangles?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The proof is for general right triangles. An isosceles right triangle is a particular case of a general right triangle, so the proof already applies to isosceles right triangles (let &lt;span class=&quot;math-container&quot;&gt;$b=a$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;While it's true that the &amp;quot;waffle cone&amp;quot; construction requires &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;b,$&lt;/span&gt; you don't need to do the construction if &lt;span class=&quot;math-container&quot;&gt;$a=b.$&lt;/span&gt; The whole point of the construction is to obtain &lt;span class=&quot;math-container&quot;&gt;$$a^2 + b^2 = \dfrac{2ab}{\sin 2\alpha},$$&lt;/span&gt; but if &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\alpha = 45^\circ$&lt;/span&gt; and the above reduces to &lt;span class=&quot;math-container&quot;&gt;$$a^2 + a^2 = \dfrac{2 \cdot a \cdot a}{\sin 90^\circ}$$&lt;/span&gt; which is true.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Can you say that approach is trigonometrical because of using trigonometrical formula &lt;span class=&quot;math-container&quot;&gt;$\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$&lt;/span&gt; since it also can be derived geometrically by right angled triangles?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When they call it a trigonometric proof, all they're saying is that it uses trigonometric functions. That's it.&lt;/p&gt;&#xA;&lt;p&gt;Sure, you could write the proof without using trigonometric functions. You could reduce everything down to simple algebra on similar triangles, explicitly writing &lt;span class=&quot;math-container&quot;&gt;$b/c$&lt;/span&gt; in place of &lt;span class=&quot;math-container&quot;&gt;$\sin \beta.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But does that mean it's not trigonometric? No. You could apply the same reasoning to argue ridiculous things, e.g. that the Fundamental Theorem of Calculus&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_a^b F'(x) \, \textrm dx = F(b) - F(a)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;is not actually differential/integral calculus because you can reduce it down to just limits:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{n \to 0} \sum_{i=0}^{n-1} \left( \lim_{h \to 0} \frac{F(x_i +h)-F(x_i)}{h} \right) \Delta x = F(b) - F(a)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$\Delta x = \dfrac{b-a}{n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_i = a + i \Delta x.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, when we say the proof is trigonometric, we mean that it uses trigonometric notation to describe symbolic patterns leveraged in trigonometry, not that it is impossible to state without trigonometric notation.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Is it not possible to complete the proof by just adding the squares of two perpendicular sides without using the formula &lt;span class=&quot;math-container&quot;&gt;$\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$&lt;/span&gt; and say that proof is geometrical? &lt;br/&gt;&#xA;Here my question is related to how we have to update students using new developments in mathematics, therefore I hope you will not misunderstand this as a mathematics question.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This question is impossible to answer because the double-angle formula &lt;span class=&quot;math-container&quot;&gt;$\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$&lt;/span&gt; is not even used in the proof.&lt;/p&gt;&#xA;&lt;p&gt;The only trigonometry that's used is the definition of sine and the law of sines. And the law of sines does not rely on the double-angle formula -- it follows directly from the definition of sine.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-24T02:08:35.630" LastActivityDate="2023-10-24T02:08:35.630" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27002" PostTypeId="2" ParentId="27000" CreationDate="2023-10-23T18:04:33.410" Score="2" Body="&lt;p&gt;This is such a niche request that I doubt you're going to be able to find an off-the-shelf app that does everything you want (though I'd love to be proven wrong).&lt;/p&gt;&#xA;&lt;p&gt;That said, Mathematica has a &lt;code&gt;HasseDiagram&lt;/code&gt; function that can be used to draw and manipulate Hasse Diagrams: &lt;a href=&quot;https://resources.wolframcloud.com/FunctionRepository/resources/HasseDiagram/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://resources.wolframcloud.com/FunctionRepository/resources/HasseDiagram/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;While it doesn't have all the bells and whistles that you mention, looking at the variety of examples in that link, I get the sense it might be feasible to write a wrapper around that function to do all the things that you want.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-10-23T18:04:33.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27004" PostTypeId="2" ParentId="26967" CreationDate="2023-10-23T19:05:06.417" Score="1" Body="&lt;p&gt;In case you're not able to find anything that natively supports the functionality you want, here's a hacky-yet-functional workaround:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Write the questions in LaTeX in an Overleaf document as multiple-choice questions with answer choices A, B, C, D.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;On each Quizlet card, just enter &amp;quot;Question #_&amp;quot; in the question textbox and &amp;quot;A&amp;quot;, &amp;quot;B&amp;quot;, &amp;quot;C&amp;quot;, &amp;quot;D&amp;quot; in the answer choice textboxes.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When you do the Quizlet during class, do a split-screen and put the Overleaf document side-by-side with the Quizlet, scrolling down to the next question as needed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Essentially, the Overleaf document manages the questions and the Quizlet Live manages the competition.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-23T19:11:36.250" LastActivityDate="2023-10-23T19:11:36.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27005" PostTypeId="2" ParentId="620" CreationDate="2023-10-23T19:56:44.923" Score="3" Body="&lt;p&gt;To answer your question, this practice can be prevented by promoting Carole Dweck's &lt;em&gt;Growth Mindsets&lt;/em&gt; or &lt;em&gt;Self-theories&lt;/em&gt;. Mindsets or Self-theories are people’s beliefs about human attributes, including abilities. In a fixed mindset, people believe that basic talents and abilities are fixed traits. Some people are well-endowed and some aren’t, and you can’t do much to change that. However, in a growth mindset, people believe that basic abilities can be developed through hard work, good strategies and good mentoring. People can have different mindsets in different areas, believing that some abilities are fixed, but others can be developed. I recommend referring to Carole Dweck's research for more details, and her findings.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastActivityDate="2023-10-23T19:56:44.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27008" PostTypeId="2" ParentId="27000" CreationDate="2023-10-24T15:24:21.070" Score="1" Body="&lt;p&gt;You may want to use Sage. Sage is a mathematics toolkit built upon Python and is extremely powerful, if occasionally slow due to its generality. The &lt;a href=&quot;https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/posets/hasse_diagram.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;documentation&lt;/a&gt; of Hasse diagrams doesn't include how to view them, but there are many methods to do so. If an object is a &lt;a href=&quot;https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/posets/posets.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Poset object&lt;/a&gt; in Sage, then the .plot() method creates a visual of it.&lt;/p&gt;&#xA;&lt;p&gt;There is a &lt;a href=&quot;https://ask.sagemath.org/question/63200/latex-for-hasse-diagram-of-poset-not-properly-laid-out/&quot; rel=&quot;nofollow noreferrer&quot;&gt;discussion question&lt;/a&gt; on the Sage forums which asks about how to generate tikz &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; for the presentation of a Hasse diagram. The solution, pulled from this discussion post, appears to be as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Install sage, compiled from source.&lt;/li&gt;&#xA;&lt;li&gt;Install graphviz to the OS (using the appropriate package manager)&lt;/li&gt;&#xA;&lt;li&gt;Install dot2tex to the Sage directory using ./sage --pip install dot2tex&lt;/li&gt;&#xA;&lt;li&gt;Define a poset object, named B, in any appropriate manner.&lt;/li&gt;&#xA;&lt;li&gt;Call view(B, debug=True)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There are many methods for editing and highlighting, though I don't think they are interactive in the way you want.&lt;/p&gt;&#xA;" OwnerUserId="724" LastActivityDate="2023-10-24T15:24:21.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27009" PostTypeId="1" AcceptedAnswerId="27075" CreationDate="2023-10-24T16:09:35.037" Score="1" ViewCount="514" Body="&lt;p&gt;Oregon appears to have suspended the &amp;quot;basic skills&amp;quot; requirement for graduation; see &lt;a href=&quot;https://oregoncapitalchronicle.com/2023/10/19/oregon-students-wont-have-to-show-added-proficiency-in-basic-skills-through-2028-to-graduate/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;.  What will be the effect of this on the mathematical proficiency of the graduating class?&lt;/p&gt;&#xA;&lt;p&gt;Follow-up question: I noticed a piece of information at the news item &lt;a href=&quot;https://www.foxnews.com/media/oregon-moms-union-slams-removal-standardized-test-high-school-graduation-requirement-not-surprising&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oregon Moms&lt;/a&gt;. Apparently part of the reason for the suppression of the requirements is the fact that the results of the tests can allegedly be predicted by the socio-economic status of the family, and therefore &amp;quot;basically racist&amp;quot; (term used by Farley). I am wondering what can be predicted here exactly: the individual scores of each student (seems unlikely), or rather the average computed per minority?&lt;/p&gt;&#xA;&lt;p&gt;Question concerning constitutionality: The reasoning behind the suppression of the requirements can be summarized as follows: since minorities will fail such tests, the tests are unfair and therefore it is preferable that everybody should fail (in the sense of the diplomas becoming a participation prize). This is consistent with the spirit of Affirmative Action, but wouldn't this be at odds with recent rulings of the Supreme Court as to the unconstitutionality of Affirmative Action?&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="1385" LastEditDate="2023-10-29T12:07:14.037" LastActivityDate="2023-11-02T16:03:01.557" Title="Basic skill requirement suspension" Tags="|secondary-education|usa|" AnswerCount="3" CommentCount="23" ContentLicense="CC BY-SA 4.0" />
  <row Id="27010" PostTypeId="2" ParentId="27009" CreationDate="2023-10-24T18:12:47.993" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;What will be the effect of this [suspending the &amp;quot;basic skills&amp;quot; requirement, which includes some math] on the mathematical proficiency of the graduating class?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't think anyone is denying that lowering math requirements will cause mathematical proficiency to decrease.&lt;/p&gt;&#xA;&lt;p&gt;The point of contention here, and in general, is whether it will lead to a net improvement or decline in the future life outcomes of students when you weight the advantages against the disadvantages.&lt;/p&gt;&#xA;&lt;p&gt;Here are some examples of advantages and disadvantages.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Advantages&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Higher graduation rates.&lt;/em&gt; Lots of career/education opportunities (even those that don't use any math) are off-limits to people who don't have a high school degree.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;More focus on life/career skills.&lt;/em&gt; Most students will never need to solve a quadratic equation or system of linear equations again in their lives. Even if they don't know how to work with variables at all, it might not hinder them too much (again we're talking about &amp;quot;most&amp;quot; students, not the honors students). But they will definitely need to budget their finances, take out and pay back loans, prepare resumes, etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Disadvantages&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Lack of preparation for college.&lt;/em&gt; Many colleges require math courses. When college-bound students graduate high school with lower mathematical proficiency, they are far more likely to struggle in college.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Many lucrative careers become off-limits.&lt;/em&gt; For lots of STEM careers, you actually do need to use algebra, sometimes calculus, and sometimes even math that's beyond calculus (e.g. in engineering). And even for those careers where you don't use much math on a daily basis, you often still need to know math to pass pre-career requirements (e.g. most medical schools have course requirements and standardized testing requirements that cover algebra-based physics). If you want to go into medicine, finance, technology, engineering, law, banking, or some other field generally seen as lucrative, and you didn't develop a firm grasp of at least algebra / trigonometry / precalculus in high school, then you're up against a steep uphill battle.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Decline in course quality.&lt;/em&gt; There's always pressure from students and parents to make courses easier and inflate grades. If there's an external validation metric like a standardized test, then teachers can point to that as justification for &amp;quot;holding the line&amp;quot; in their classroom -- and they're also more incentivized to hold the line because if they don't, it will show in their students' standardized test scores, and they'll get chewed out by school administrators. When you remove the external validation metric, you remove accountability for learning. If teachers aren't held accountable for student learning, many (most?) teachers won't hold students accountable for learning.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Group Differences&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Additionally, standardized test scores generally vary across groups, but people often disagree on how to interpret that, so I won't put it under either list above.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Some believe this indicates bias in the test itself, since they believe all groups would have the same score distribution if the test were unbiased. These people generally want test-based requirements to be removed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Others believe that the test is accurately measuring subject ability, which, due to other factors (quality of schools, access to tutoring, cultural value of academic achievement, kids' available bandwidth for studying, parents' available bandwidth to keep their kids in line, presence of role models, etc) is not distributed equally across groups.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;↑ This side shares a common belief that the tests expose real group differences. But even still, there is controversy: some think such metrics ought to be removed (&amp;quot;group differences are uncomfortable and maybe if we eliminate ways of measuring the differences then the differences will go away&amp;quot;), while others disagree (&amp;quot;it's better to face reality with full knowledge of the situation&amp;quot;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(I'm happy to add to either side of this list if anyone comments with a good suggestion.)&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-26T16:49:25.617" LastActivityDate="2023-10-26T16:49:25.617" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="27012" PostTypeId="2" ParentId="26899" CreationDate="2023-10-25T11:43:30.557" Score="2" Body="&lt;p&gt;There are two reasons, and they both stem from problematic notations being used in calculus.&lt;/p&gt;&#xA;&lt;p&gt;The first reason is higher-order derivatives.  The common notation for the second derivative, &lt;span class=&quot;math-container&quot;&gt;$\frac{d^2y}{dx^2}$&lt;/span&gt;, cannot be used as a fraction.  However, this is because it is problematic notationally.  If you consider &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; a fraction, and then use the quotient rule to take its derivative, the second derivative of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2}$&lt;/span&gt;, which does allow for straightforward algebraic manipulations of differentials.&lt;/p&gt;&#xA;&lt;p&gt;The second reason is partial differentials.  Unfortunately, partial differentials are individually completely ambiguous as to what they refer to.  If you clean up the notation, however, and make each different unambiguous, the problem goes away.&lt;/p&gt;&#xA;&lt;p&gt;You can check out more details about both of these in the paper &amp;quot;&lt;a href=&quot;https://arxiv.org/abs/2210.07958&quot; rel=&quot;nofollow noreferrer&quot;&gt;Total and Partial Differentials as Algebraically Manipulable Entities&lt;/a&gt;&amp;quot; by Maria Isabelle Fite and myself.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2023-10-25T11:43:30.557" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27017" PostTypeId="2" ParentId="26878" CreationDate="2023-10-26T01:35:40.830" Score="1" Body="&lt;p&gt;There are many good suggestions made concerning this topic .. What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations? ..&lt;/p&gt;&#xA;&lt;p&gt;Some common problems:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;No respect for dimensionality. Suppose Av is a column vector, students will equate it to a row vector, a matrix, or even a scalar. This is by far the most common one.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Wild manipulations, such as freely dividing a matrix 1/A and basically treating a matrix as a scalar. Similarly, freely dividing vectors, raising vectors to powers, etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Incorrect beliefs, such as, &amp;quot;if a matrix A is symmetric, then it has an inverse.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some students are coming into advanced mathematics heavy courses without basically any background.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am really at a loss as to why this is happening and what should be the correct remedy, as it seems that it is not easy to address this late into their study.&lt;/p&gt;&#xA;&lt;p&gt;=====&lt;/p&gt;&#xA;&lt;p&gt;As I have ADHD, I have found that I have had to build my own math books.  There are symbols that I don't understand in math, and converting them takes time, thus by building my own math book using a clearer / cleaner language of symbols makes my life easier when it comes to math.&lt;/p&gt;&#xA;&lt;p&gt;It takes longer to learn math concepts.  This is due to how my brain processes math information.  The brain processes language in one area of the brain, and math in another.  Language is processed as a series of pictures, like asking a child to spell a word and sound out the word. They are deconstructing the word to understand how to say it and what it means.&lt;/p&gt;&#xA;&lt;p&gt;Math requires the pictures of equations to be deconstructed. The more letters in the picture / equation, the more frightening the equation becomes.  Most programmers misunderstand math equations as assignment statements, and should be reminded that they are relationships.  This will confuse them until you show how the equation can be solved for different questions based on which variables are defined.&lt;/p&gt;&#xA;&lt;p&gt;Another difference between the math center and language center is that the math center is a true false engine.  If you give a student a rule and ask them to apply it, they won't know if they are applying it correctly until after several hundred problems have been completed. You can see this with 1st and 2nd graders working out simple math problems, ask them .. is &amp;quot;1 = 2 - 1&amp;quot; the same as &amp;quot;2 = 1 + 1&amp;quot; or &amp;quot;1 + 1 = 2&amp;quot;? ..&lt;/p&gt;&#xA;&lt;p&gt;The next part is that physics problems require multiple equations manipulated in the right order to get the right answer.  This requires a strategic thinking, which many students are not exposed to at all.  To help with this, ask the student to write down the problem, the equations they think are needed to solve the problem, all the constants that they know, and their strategy to use this information to solve the problem.&lt;/p&gt;&#xA;&lt;p&gt;As for linear algebra, I found this to be fun.  I liked the repetition of doing the problems using the same basic strategies, over and over again.  One of the things I really wanted to explore with linear algebra was kirkoffs rules (sp).  I had a horrible time trying to complete the equations by substitution, but you needed calculus before you could take linear algebra.&lt;/p&gt;&#xA;&lt;p&gt;To summarize ..&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;ask the student to write down a strategy to solve the problem&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;realize that math center in our brain is a true false engine that requires reinforcement learning&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;provide a lot of similar problems for the student to work on&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;help them to learn and differentiate the aspects you are seeing lacking.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;build a separate book for them to work through, if you're going to be teaching the same class over and over for a few years.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Hopefully, this will help you understand why students are behind and how to bring them up to speed.&lt;/p&gt;&#xA;" OwnerUserId="23188" LastActivityDate="2023-10-26T01:35:40.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27018" PostTypeId="2" ParentId="26188" CreationDate="2023-10-26T06:30:36.107" Score="2" Body="&lt;p&gt;There exist research on online communities for teacher professional development. I am not a scholar of this, having read one example research proposal in an appendix to [1], but the search term &amp;quot;teacher online learning community&amp;quot; seems to be a good start: &lt;a href=&quot;https://scholar.google.com/scholar?q=teacher+online+learning+community&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://scholar.google.com/scholar?q=teacher+online+learning+community&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I recommend doing an informal literature search on the subject to get a sense of what is known. Hopefully something useful comes up.&lt;/p&gt;&#xA;&lt;p&gt;My general experience is that getting any online community up and going requires a committed group of people who are already into it, as a minimum requirement.&lt;/p&gt;&#xA;&lt;p&gt;I know that teachers in Norway have a number of Facebook groups and such, for example one on using Minecraft for mathematics teaching (since master students of mine wrote on that), but I have not delved deeper into these.&lt;/p&gt;&#xA;&lt;p&gt;[1] MAXWELL, Joseph A. Qualitative research design: An interactive approach. Sage publications, 2013.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-10-26T06:30:36.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27019" PostTypeId="1" CreationDate="2023-10-26T08:19:33.123" Score="2" ViewCount="373" Body="&lt;p&gt;There are several AI based adaptive learning environments out there like &lt;a href=&quot;http://squirrelai.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Squirrel&lt;/a&gt;, &lt;a href=&quot;https://www.wiley.com/en-us/education/alta&quot; rel=&quot;nofollow noreferrer&quot;&gt;KNewton&lt;/a&gt; or &lt;a href=&quot;https://mathacademy.com/how-it-works&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathAcademy&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Are there any studies which try to extract from the (big) data of those learning systems implications of how human to human math teaching may be (significantly) improved? If so, what are the results so far?&lt;/p&gt;&#xA;" OwnerUserId="1203" LastActivityDate="2023-10-26T19:42:47.073" Title="Feedback from AI based learning to human based learning in math?" Tags="|technology-in-education|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27020" PostTypeId="2" ParentId="26934" CreationDate="2023-10-26T14:56:12.440" Score="0" Body="&lt;p&gt;One way to understand something is to think about what it means for it to be false.  I'd also argue there is three concepts being connected.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$|a – b| &amp;lt; ε, \forall ε &amp;gt; 0 \iff a = b$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The first bit is&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$y &amp;lt; ε, \forall ε &amp;gt; 0 \iff y \leq 0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;and the second is&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$0 \leq y \leq 0 \iff y = 0$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;finally&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$|a - b| = 0 \iff a = b$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;and we can break those three down in turn.&lt;/p&gt;&#xA;&lt;p&gt;For 1, this is a property of the reals.  It isn't a given.  If ε was a real number, but &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; was from a space that had a value greater than 0 but less than every real number, then it wouldn't hold.  There are extensions of the real numbers that act exactly like that, with a variety of infinitesimals.&lt;/p&gt;&#xA;&lt;p&gt;In short, it isn't a given that 1 is true.  So you &lt;em&gt;have&lt;/em&gt; to talk about it being true in the context of the real numbers before you can expect someone to believe it is true.   You don't even have to talk about infinitesimals, but you do have to make this fact intuitive to the student.&lt;/p&gt;&#xA;&lt;p&gt;Once you have that, the second connects &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\geq$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Finally, the 3rd is a property of &lt;span class=&quot;math-container&quot;&gt;$||$&lt;/span&gt; - that it is 0 if and only if the elements are the same.  This isn't a given, there are spaces where distance doesn't work like that.  But it holds in the real numbers.&lt;/p&gt;&#xA;&lt;p&gt;In effect, we convert &amp;quot;how do I go from &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;&amp;quot; into &amp;quot;how do I go from &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt;, and how do I go from &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\geq$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt;, and how do I go from distance to elements&amp;quot;, which is much more natural.&lt;/p&gt;&#xA;&lt;p&gt;The hard one here, I think, is step 1.  But by removing the need to fold in the next two steps into the intuition for step 1, you can focus on the real problem.  And the conversion from &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; is far easier to grasp; in effect, the interesting part is that we &lt;em&gt;lose&lt;/em&gt; the power of &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; and get a weaker &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I'd start with &lt;span class=&quot;math-container&quot;&gt;$y \leq 0$&lt;/span&gt; and show &lt;span class=&quot;math-container&quot;&gt;$y &amp;lt; ε, \forall ε &amp;gt; 0$&lt;/span&gt; because that is easy.&lt;/p&gt;&#xA;&lt;p&gt;For the next part, I'd look at this negation trick:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$!( \exists ε &amp;gt; 0, y \geq ε ) \iff ( y &amp;lt; ε, \forall ε &amp;gt; 0 )$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This exists makes your situation concrete, and implies an adversarial game.  Your y has the property that someone &lt;em&gt;cannot&lt;/em&gt; find an ε greater than zero that is equal to, or smaller than y.&lt;/p&gt;&#xA;&lt;p&gt;So we need to show&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$!( \exists ε &amp;gt; 0, y \geq ε ) \implies y \leq 0$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And, once you actually describe what this means, it should be really obvious.  This is the adversarial game -- y has the property that you cannot find an epsilon greater than zero that is equal or smaller than y.&lt;/p&gt;&#xA;&lt;p&gt;If they have some maturity, you can also use &lt;span class=&quot;math-container&quot;&gt;$(!x \implies !y) \iff (y \implies x)$&lt;/span&gt;, but that might be stretching it:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$y &amp;gt; 0 \implies \exists ε &amp;gt; 0, y \geq ε$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which is getting to be a rather trivial claim (set ε equal to y).&lt;/p&gt;&#xA;" OwnerUserId="9985" LastActivityDate="2023-10-26T14:56:12.440" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27021" PostTypeId="2" ParentId="27019" CreationDate="2023-10-26T15:19:24.117" Score="4" Body="&lt;p&gt;I'm the Director of Analytics &amp;amp; Algorithms for &lt;a href=&quot;https://mathacademy.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Academy&lt;/a&gt;; of particular relevance to this question, I write all our adaptive learning algorithms, manage our student learning analytics, and calibrate our system to student learning data.&lt;/p&gt;&#xA;&lt;p&gt;Yes, there are plenty of ways that human-to-human teaching can be significantly improved.&lt;/p&gt;&#xA;&lt;p&gt;But these have been known in the literature for a long time -- for instance, &lt;a href=&quot;https://en.wikipedia.org/wiki/Mastery_learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;mastery learning&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Spaced_repetition&quot; rel=&quot;nofollow noreferrer&quot;&gt;spaced repetition&lt;/a&gt;, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Testing_effect&quot; rel=&quot;nofollow noreferrer&quot;&gt;testing effect&lt;/a&gt;, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Varied_practice&quot; rel=&quot;nofollow noreferrer&quot;&gt;varied practice&lt;/a&gt; have been researched extensively since the early to mid 1900s, with key findings being successfully reproduced over and over again since then.&lt;/p&gt;&#xA;&lt;p&gt;The problem is that using these strategies systematically to their fullest extent requires an infeasible amount of effort for any human teacher.&lt;/p&gt;&#xA;&lt;p&gt;For instance, consider spaced repetition. You have to keep track of a repetition schedule for every topic for every student, and each time a student learns (or reviews) an advanced topic, they're implicitly reviewing many simpler topics, all of whose repetition schedules need to be adjusted as a result.&lt;/p&gt;&#xA;&lt;p&gt;In fact, before building our online system, we actually did a very loose approximation of spaced repetition while teaching in a human-to-human classroom. It turned out that, teaching just two classes with only a handful of students in each class, it took more time and effort than a full-time job to implement a very loose approximation of spaced repetition (for the class as a whole -- not even personalized to individual students). And that's just one of many strategies that are necessary for effective teaching!&lt;/p&gt;&#xA;&lt;p&gt;Compared to the scenario described above, a standard teaching load consists of about 3x as many classes and 4x as many students per class. Let's say optimal teaching requires (conservatively) 5 different cognitive learning strategies to be implemented on a fully personalized level. Then, for a typical teacher, we can ballpark-estimate that optimal teaching would require the time and effort of about 3 x 4 x 5 = 60 full-time jobs. Which is totally infeasible.&lt;/p&gt;&#xA;&lt;p&gt;Totally infeasible for a human, at least. But not for a machine. Our solution was to take all of these strategies (including plenty of others not mentioned above such as interleaving, layering, cognitive noninterference, cognitive load minimization) and build an adaptive automated online learning system that leverages them to their full extent. Our goal is for our system to emulate the decisions of a perfect human tutor who knows everything about their student and everything about math.&lt;/p&gt;&#xA;&lt;p&gt;So, to summarize -- for Math Academy, at least, it's not about extracting insights from data that can inform human-to-human teaching, but rather about developing software that can actually leverage the learning strategies that have been known about for decades but require an infeasible amount of effort to implement in a human-to-human classroom.&lt;/p&gt;&#xA;&lt;p&gt;(I'm happy to answer any follow-up questions anyone might have.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Follow-Up Questions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The question was the other way around. For example it might be that you see some invariants from the data, i.e. that a certain way to present a topic works best for most students or you can categorize some aspects of individualization and learn from the data that in some aspects the individualization is significant and in other cases not relevant etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I realize your question was the other way around, but what I'm trying to get at is: in our experience, the most significant way to improve human-to-human teaching is to offload as much as possible to an automated system so that the single human teacher is no longer a bottleneck.&lt;/p&gt;&#xA;&lt;p&gt;That said, we have learned a lot by monitoring where students get stuck and revising our content in those places to provide better scaffolding without lowering standards. The same goes for algorithms / individualization.&lt;/p&gt;&#xA;&lt;p&gt;Our main takeaway is this: learning is like climbing a staircase, and students get stuck at any individual stair that is too tall for them to climb, so the smaller you make the individual stairs, the more students can climb all the way to the top.&lt;/p&gt;&#xA;&lt;p&gt;Our specific improvements are part of a continual feedback loop, and there are too many to enumerate here, but if there is a particular topic or individualization in mind that you'd like me to comment on, I can probably say more about it.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-10-26T19:42:47.073" LastActivityDate="2023-10-26T19:42:47.073" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27022" PostTypeId="1" CreationDate="2023-10-26T17:59:44.880" Score="13" ViewCount="3722" Body="&lt;p&gt;In view of recent developments of AI-based adaptive math learning systems like &lt;a href=&quot;http://squirrelai.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Squirrel&lt;/a&gt;, &lt;a href=&quot;https://www.aleks.com/?_s=6802339836316549&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aleks&lt;/a&gt;, &lt;a href=&quot;https://www.wiley.com/en-us/education/alta&quot; rel=&quot;nofollow noreferrer&quot;&gt;Knewton Alta&lt;/a&gt;, or &lt;a href=&quot;https://mathacademy.com/how-it-works&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Academy&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Does a human math teacher play any reasonable role, or will those systems make human math teachers completely unnecessary?&lt;/p&gt;&#xA;&lt;p&gt;It seems to be clear that human pedagogic, psychological, and technical support may needed. But what is or will be the role of someone who knows math and knows how to teach it?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt; in response of the comment of Adam Rubinson:&#xA;I might differentiate two versions of my question:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Based on how good those systems actually work today.&lt;/li&gt;&#xA;&lt;li&gt;Based on what is advertised, of how good they work.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For example Math Academy advertises:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Our adaptive, fully-automated platform is 4x more efficient than a traditional math class&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I don't know any good, independent peer reviewed papers, that evaluate in depth of how well those systems work.&lt;/p&gt;&#xA;&lt;p&gt;Some work that indicates that it might be quite effective:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/10494820.2020.1808794&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tandfonline.com/doi/full/10.1080/10494820.2020.1808794&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.researchgate.net/profile/Kp-Thai-2/publication/331426195_Performance_Comparison_of_AI-Based_Adaptive_Learning_Systems_China/links/635325c18d4484154a1f804e/Performance-Comparison-of-AI-Based-Adaptive-Learning-Systems-in-China.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.researchgate.net/profile/Kp-Thai-2/publication/331426195_Performance_Comparison_of_AI-Based_Adaptive_Learning_Systems_China/links/635325c18d4484154a1f804e/Performance-Comparison-of-AI-Based-Adaptive-Learning-Systems-in-China.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1203" LastEditorUserId="1203" LastEditDate="2023-11-04T11:01:47.173" LastActivityDate="2023-11-04T11:01:47.173" Title="Role of human math teachers in the century of AI learning tools" Tags="|technology-in-education|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27023" PostTypeId="2" ParentId="17080" CreationDate="2023-10-26T18:42:06.853" Score="2" Body="&lt;p&gt;I was taught &amp;quot;numeral&amp;quot; in the New Math era, and if you had asked me in my youth what it meant, I would have probably replied, &amp;quot;What? You mean like zero through nine?&amp;quot; In my mind, I would have been thinking of the graphemes for zero through nine rather than the words or numbers, although I would not have had the pedantic precision to express that thought.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Number&amp;quot; meant to me any quantity, whether numbers with units (signifying length, area, weight, money, etc.) or not. The fact that you could equate different numbers — eight quarters are two dollars — was just a cool game you could play with numbers.&lt;/p&gt;&#xA;&lt;p&gt;Thus, &amp;quot;numeral&amp;quot; meant for me (and still does) the tokens used to express numbers (in the decimal system at first, and later in other bases). &amp;quot;Numeral&amp;quot; did not mean, nor does it now, &amp;quot;a word that describes a numerical quantity.&amp;quot; If pressed, I would say that such a word is called the name of the number, but that's me searching for the nearest word I could quickly come up with. But I would not say that &amp;quot;zero point nine repeating,&amp;quot; which describes the number 1 in the real-number decimal system, is the name of 1.  The name of 1 is &amp;quot;one.&amp;quot; The &amp;quot;digits of a number,&amp;quot; which phrase I also learned in school, meant the (sequence of) numerals used to express the number. &amp;quot;A decimal point followed by infinitely many nines,&amp;quot; for example.&lt;/p&gt;&#xA;&lt;p&gt;Later, study in mathematics, computer science, and linguistics challenged some aspects of these conceptualizations for me. They allowed me to talk about numbers with enough pedantic precision (1) to wonder whether I'm speaking the same language as the rest of English-speaking Americans and (2) to be wrong instead of sort-of-right, esp. with regard to linguistics. Back to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/17080/the-word-numeral-is-it-being-taught-and-does-the-word-exist-for-it-in-your-la#comment44649_17080&quot;&gt;Feynman's execration&lt;/a&gt; of pedantic precision, excessively precise language does not sound like ordinary English to most teenagers in the U.S. It makes them doubt what you are saying, which is the opposite of what you're hoping to achieve.&#xA;In mathematics, there are (it seems to me, if I'm not making a linguistics gaffe) different language-games for talking about &amp;quot;numbers.&amp;quot; A discussion sometimes begins switching between them until the speakers find a common language-game. For instance, speaking with a student beginning calculus versus talking with someone who has had real analysis.&lt;/p&gt;&#xA;&lt;p&gt;Finally, with respect to an implicit question in the OP's statement, &amp;quot;I believe that for this lack of terminology children are very likely to believe that a number is nothing more than a scribble on a piece of paper.&amp;quot;   Coincidentally, yesterday I was talking with a neuroscientist about mathematical concept development. It's not their area, and I was just looking for pointers on how to find out what is known. They made the point that there is evidence that the concepts that are learned early, even if not mastered, form a framework on which future knowledge will be built. Early learning informs their later learning. And those who are taught mathematical concepts underlying arithmetic do better later than those who are taught only how to do arithmetic, even if the ones taught concepts are not as proficient at calculating. So the question of whether a certain term could help lay a good foundation for understanding the concept of number seems important. I don't know that such a question has been investigated, so I can't give a scientific answer to it.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2023-10-28T19:45:42.010" LastActivityDate="2023-10-28T19:45:42.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27024" PostTypeId="2" ParentId="27022" CreationDate="2023-10-26T19:41:40.260" Score="28" Body="&lt;p&gt;I'm the Director of Analytics &amp;amp; Algorithms for &lt;a href=&quot;https://mathacademy.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Academy&lt;/a&gt;; of particular relevance to this question, I also spent about a decade teaching/tutoring math human-to-human, as well as several years working hands-on with students who were using the Math Academy system.&lt;/p&gt;&#xA;&lt;p&gt;I don't think math teachers will be made completely unnecessary. At least, not for all students.&lt;/p&gt;&#xA;&lt;p&gt;It's true that as we've refined our content and algorithms over the years to better scaffold the instruction, personalize the learning experience, and adapt to the needs of individual students, we've enabled more and more students to achieve comprehensive mastery without the need for a human math teacher in any capacity. (For instance, many students get a 5 on the AP Calculus BC exam working solely on our system, no human teacher needed.)&lt;/p&gt;&#xA;&lt;p&gt;This works for surprisingly many students. But the thing is -- despite our best efforts, there are also many students who will go off the rails without a human teacher or tutor to keep them on track. For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Some students have major attention issues and need a human to reel them back in when they get distracted every minute.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some students have severe math anxiety and will totally freeze up if they don't have a human next to them providing encouragement.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some students find basic math extremely hard and need a human to help them work with physical manipulatives (e.g. counting blocks) and/or explain a concept a hundred different ways until something sticks.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;There are also students who are &amp;quot;on the edge&amp;quot; in the sense that most of the time, they are fine learning without a human teacher, but every once in a while, they need some human support. Some teachers whose students are like this use Math Academy as the primary resource in their classroom and walk around checking in and helping students as needed.&lt;/p&gt;&#xA;&lt;p&gt;Additionally, there are other students who are capable of learning without a human teacher but who enjoy the environment of a classroom with a human teacher where they can have ad-hoc enrichment discussions about various aspects of math that interest them.&lt;/p&gt;&#xA;&lt;p&gt;All this to say, when it comes to teaching math, I think humans and machines can enjoy a symbiotic existence. There is so much demand for math learning that goes unserved -- so much that even if machines are capable of serving a lot (or even most) of it, there will still be more than enough demand to require a supply of human teachers.&lt;/p&gt;&#xA;&lt;p&gt;Human-vs-machine reliance in education is on a spectrum, and while the distribution will likely drift towards less human involvement over time, I think the following things will always be true:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Some students will learn entirely from machines.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some students will learn their core curriculum entirely from machines but will enjoy ad-hoc enrichment discussions with humans.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some students will learn primarily from machines but will need human support every once in a while.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some students will need humans throughout most or all of their learning.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And if you replace the word &amp;quot;machines&amp;quot; with &amp;quot;books&amp;quot; in the above, they &lt;em&gt;have&lt;/em&gt; always been true. The distribution has always existed; new technology (books ⟶ computers ⟶ adaptive learning software) just shifts it upwards.&lt;/p&gt;&#xA;&lt;p&gt;(I'm happy to answer any follow-up questions anyone might have.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Response to Follow-Up Question 1&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Doesn't help from humans during ai learning confuse the ai feedback loop?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Sure, if a human is sitting down with a student and carrying them / correcting them on every problem so that they get everything right, then without any explicit mechanism for detecting this, the AI is going to pass them through and treat them as stronger than they really are. The same will happen if the student is relying on online calculators as a crutch (e.g. WolframAlpha, SymboLab).&lt;/p&gt;&#xA;&lt;p&gt;However, in practice, students' overreliance on outside resources isn't as huge an issue as it may seem in theory. There are a handful of forces working in our favor:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;It's rare that a student would be getting that much help from another human.&lt;/em&gt; Teachers usually have too many kids in their class to spend an inordinate amount of time with any particular student, and it would be unusual for a parent to work with their kid on the system all the time (or hire an expensive tutor to do so) because one of the big value propositions of the system is that it frees up the parent's bandwidth while remaining way less expensive than tutoring.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;We measure knowledge repeatedly into the future.&lt;/em&gt; Our system does not take a &amp;quot;one-and-done&amp;quot; approach to learning. Even if a student gets carried (by another human) through a lesson, they'll still receive spaced reviews and quizzes on the topic in the future, and if they do poorly on those, they'll move backwards in the spaced repetition process, potentially back to the very beginning when they initially got carried through the lesson.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;We are often able to structure questions in a way that prevents or makes it much harder to cheat using online calculators.&lt;/em&gt; Here's the first example that comes to mind: instead of asking a student to select the graph of the transformed function &lt;span class=&quot;math-container&quot;&gt;$2\sin(-3x)+4,$&lt;/span&gt; we might give them the graph, tell them the graphed function takes the form &lt;span class=&quot;math-container&quot;&gt;$a \sin (bx) + c,$&lt;/span&gt; and ask them to compute &lt;span class=&quot;math-container&quot;&gt;$a + b + c.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Because math is so hierarchical, knowledge debts quickly come due.&lt;/em&gt; Most mathematical topics feed into more advanced topics. If a student manages to get through topic X by relying on an online calculator, then they're not going to be able to get through the more advanced topics that depend on X unless they find an online calculator for those too. They quickly run into topics where there is no online calculator and they're unable to do the topic by hand because they never learned how to do the prerequisites by hand. So the only way forward is to go back to the prerequisites and actually learn those properly.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Interestingly, item 4 above is such a barrier to continued overreliance on outside resources that the overreliance tends to manifest primarily as a customer support issue:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;student uses an online calculator for topic X,&lt;/li&gt;&#xA;&lt;li&gt;student runs into &amp;quot;post&amp;quot;-requisite topic Y that they're unable to do by hand but there is no online calculator,&lt;/li&gt;&#xA;&lt;li&gt;student tells their parent/teacher that topic Y is too hard for them,&lt;/li&gt;&#xA;&lt;li&gt;parent/teacher sends a support message,&lt;/li&gt;&#xA;&lt;li&gt;we look into it and pick up on clues indicating that the student used an online calculator for prerequisite topic X,&lt;/li&gt;&#xA;&lt;li&gt;we explain this to the parent/teacher and help them get their student back on track.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;What a headache!&lt;/p&gt;&#xA;&lt;p&gt;To avoid these kinds of support issues, we have recently started working on explicit mechanisms for detecting overreliance on outside resources. (This will also increase learning integrity, which is always a plus, though it wasn't too much of a problem for us in the first place.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Response to Follow-Up Question 2&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Teaching is much more than delivering knowledge to a student's brain. A teacher is a secondary parent. Children imitate their role models and become like them. 90% of being a math teacher is just being (at whatever level) an adult who likes math. I don't think AI will ever be able to act in the capacity of a passer-on of passion. We can't enculturate children without human adults. Optimizing instruction strategies and holding students accountable is the easy, mindless part of teaching. And these things are rarely neglected. Instruction strategies and accountability don't need to be optimized. They just need to be good. Trying to fine tune them is being penny wise and pound foolish.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I do agree that there is a dimension of &amp;quot;passion&amp;quot; along which teachers can be very positive or negative, while AI (at least in the current conception) is more neutral.&lt;/p&gt;&#xA;&lt;p&gt;However, if a math teacher spends &amp;quot;90% of their time/effort just being an adult who likes math&amp;quot; at the exclusion of optimizing instruction strategies and holding students accountable for learning, then the outcome is still a disaster.&lt;/p&gt;&#xA;&lt;p&gt;I've seen this on multiple occasions -- a well-intentioned teacher focuses all their energy on class discussions about mathematical beauty and cool applications, thinks that because they're so good at that they don't have to optimize instruction strategies and hold students accountable, and graduates students who can't solve even the most basic kinds of problems that they were supposed to have learned in the class.&lt;/p&gt;&#xA;&lt;p&gt;And that leads to situations like the one that came up a few weeks ago: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/26878/what-can-i-do-when-advanced-undergraduate-and-or-early-graduate-stem-students-ca&quot;&gt;What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In my experience, when you weight instruction/accountability versus passion, you get the following outcomes:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;best outcome:&lt;/em&gt; HIGH instruction/accountability, HIGH passion&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;good outcome:&lt;/em&gt; HIGH instruction/accountability, NEUTRAL passion&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;bad outcome:&lt;/em&gt; HIGH instruction/accountability, NEGATIVE passion&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;bad outcome:&lt;/em&gt; LOW instruction/accountability, HIGH passion&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;very bad outcome:&lt;/em&gt; LOW instruction/accountability, NEUTRAL passion&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;worst outcome:&lt;/em&gt; LOW instruction/accountability, NEGATIVE passion&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So I would argue that, even though there is value to be had from a passionate human teacher, even students who learn solely from an AI machine can still enjoy good educational outcomes (which is better than most students receive from most human teachers).&lt;/p&gt;&#xA;&lt;p&gt;And that's even an understatement. In our experience, optimizing instruction/accountability has allowed us to accelerate student learning by 4x -- meaning that on our system, serious students learn 4x the amount of material in the same time (or the same amount of material in a quarter of the time) as compared to traditional classrooms. And that's being conservative, since our courses tend to be even more comprehensive than what you'd find in a traditional classroom (our courses aim to cover the superset of all content that one could reasonably expect to find in any major textbook or standard class syllabus).&lt;/p&gt;&#xA;&lt;p&gt;Of course, it's hard for a human teacher to achieve this level of effectiveness by optimizing instruction/accountability. Using long-known learning strategies (like mastery learning, spaced repetition, the testing effect, varied practice, interleaving, layering, cognitive noninterference, cognitive load minimization) to their fullest extent requires an &lt;a href=&quot;https://matheducators.stackexchange.com/a/27021/22672&quot;&gt;infeasible amount of effort for any human teacher.&lt;/a&gt; But just because a human can't do that, doesn't mean that there's little to gain from it.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-04T07:47:06.770" LastActivityDate="2023-11-04T07:47:06.770" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="27025" PostTypeId="2" ParentId="27022" CreationDate="2023-10-27T05:11:05.553" Score="16" Body="&lt;p&gt;Many (most?) mathematics learners benefit immensely from social, mathematical discussions, and often multi-modal ones where people speak, write, point at things, explain what is happening by using their hands and bodies, and so on.&lt;/p&gt;&#xA;&lt;p&gt;There is still some way to go before an AI manages to be a partner in such a discussion, and even longer way before they manage to moderate such a mathematical discussion, which requires skills also in mathematics didactics. The psychology and the mathematics are not orthogonal.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-10-27T05:11:05.553" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27026" PostTypeId="1" CreationDate="2023-10-27T06:27:32.923" Score="8" ViewCount="157" Body="&lt;p&gt;During my teaching training in Germany and in many professional development sessions, there has been repeated emphasis on the importance of cognitive activation and challenging tasks for effective mathematics (and physics) instruction. For instance, the Freiburg COACTIV research or the works of Hungerbühler from Switzerland have highlighted this aspect. You can find more information about COACTIV research &lt;a href=&quot;https://www.kobo.com/ww/en/ebook/cognitive-activation-in-the-mathematics-classroom-and-professional-competence-of-teachers?utm_campaign=shopping_feed_dk_en&amp;amp;utm_source=google&amp;amp;utm_medium=cpc&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, and further details about Hungerbühler's work &lt;a href=&quot;https://data.snf.ch/grants/grant/179186&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In studies by &lt;a href=&quot;https://www.researchgate.net/publication/222830815_Cognitive_load_theory_Implications_of_cognitive_load_theory_on_the_design_of_learning&quot; rel=&quot;noreferrer&quot;&gt;Paul Kirschner&lt;/a&gt; and other sources, particularly from the English-speaking community, I have come across the idea that it is crucial to minimize &lt;em&gt;cognitive load&lt;/em&gt;. Can these two approaches be reconciled without contradiction? If so, I would be interested in specific examples.&lt;/p&gt;&#xA;" OwnerUserId="1203" LastActivityDate="2023-10-31T16:20:24.823" Title="Cognitive activation vs cognitive load reduction" Tags="|education-research|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27027" PostTypeId="1" AcceptedAnswerId="27028" CreationDate="2023-10-27T10:40:01.067" Score="9" ViewCount="3924" Body="&lt;p&gt;When writing &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; proofs, it's common that the ''natural'' choice of &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; leads to the final inequality in the form, say, &lt;span class=&quot;math-container&quot;&gt;$|\ldots| &amp;lt; \varepsilon+\varepsilon+\varepsilon$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$|\ldots| &amp;lt; \varepsilon$&lt;/span&gt;. It's always possible to correct the choice of &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;, but this has its disadvantages (see MSE links below). Instead, one can refer to the general principle (the ''&lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; principle'') which says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;       Instead of doing this, let’s once and for all agree that if you come out in the end with &lt;span class=&quot;math-container&quot;&gt;$2\varepsilon$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$22\varepsilon$&lt;/span&gt;, that’s just as good as coming out with &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; is an arbitrary small number, then so is &lt;span class=&quot;math-container&quot;&gt;$2\varepsilon$&lt;/span&gt;. Therefore, if you can prove something is less than &lt;span class=&quot;math-container&quot;&gt;$2\varepsilon$&lt;/span&gt;, you have shown it can be made as small as desired.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h4&gt;However, why do standard textbooks never teach, or prove &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; using, this &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; principle?&lt;/h4&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/q/2309392&quot;&gt;Every analysis monograph in my library&lt;/a&gt; shirked this K-ε principle, including &lt;a href=&quot;https://math.stackexchange.com/a/3589941&quot;&gt;Abbott’s &lt;em&gt;Understanding Analysis&lt;/em&gt;, Apostol’s &lt;em&gt;Calculus&lt;/em&gt;, Bartle and Sherbert’s &lt;em&gt;Introduction to Real Analysis&lt;/em&gt;, Michael Spivak’s &lt;em&gt;Calculus&lt;/em&gt;&lt;/a&gt;, and &lt;a href=&quot;https://math.stackexchange.com/q/2561232&quot;&gt;Rudin’s &lt;em&gt;Principles of Mathematical Analysis&lt;/em&gt;&lt;/a&gt;. Google found merely 2 books that tout this K-ε principle ―&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/3797955&quot;&gt;Arthur Mattuck’s &lt;em&gt;Introduction to Analysis&lt;/em&gt; (1998), pages 39-40&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.imgur.com/4sKZ3hO.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/q/4700888&quot;&gt;Frank Morgan’s &lt;em&gt;Real Analysis&lt;/em&gt; (2005), pages 17-8&lt;/a&gt;. But Morgan doesn’t headline Mattuck's &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; principle as leading news, or teach it as a godsend to streamline &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; proofs.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.imgur.com/dtqjlAD.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Most students prefer this &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; principle for being MORE forthright and straightforward, because in actuality,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/1961732&quot;&gt;“I prove the limit first, find δ then write it at the beginning in the final proof to make it seem like I knew all along.&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/2700620&quot;&gt;“no one actually magically knows the right value of δ at the start (although with experience you might be able to make a reasonable guess). They work through the proof without saying what δ is, figure out what it needs to be from the end of the proof, and then insert that definition back at the beginning.”&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Math S.E. users kvetch that traditional &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; proofs are&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/780096&quot;&gt;“intimidating and confusing to students; not so much the actual idea, which is simply that you can &lt;em&gt;arbitrarily constrain the output by suitably constraining the input.&lt;/em&gt;”&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/q/1677604&quot;&gt;“a magic ball”&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/q/4126198&quot;&gt;“seems like magic to me”&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/q/101441&quot;&gt;“algebra magic”&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/q/3333009&quot;&gt;“weird”, “circular reasoning”&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/4726169&quot;&gt;“odd challenge-response nature of the limit definition”, “min-max game semantics of logic”&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/3659232&quot;&gt;“stumbling blocks”&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="23199" LastEditorUserId="23199" LastEditDate="2023-10-28T01:13:52.263" LastActivityDate="2023-11-07T10:57:30.800" Title="Why do most Analysis textbooks overlook, and fail to teach delta-epsilon proofs, using the K-ε principle?" Tags="|mathematical-analysis|" AnswerCount="8" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="27028" PostTypeId="2" ParentId="27027" CreationDate="2023-10-27T14:14:27.940" Score="7" Body="&lt;p&gt;I think there are very few textbooks that concentrate on how to write proofs. The &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; principle is a bit helpful in reading proofs, but it usually isn't too hard for an experienced author to choose a &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; at the start in order to get &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Where the &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; principle is invaluable is for a student who is learning to write &lt;span class=&quot;math-container&quot;&gt;$(\delta, \epsilon)$&lt;/span&gt; proofs and can't see which &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; to choose at the start. (Especially if they are writing these proofs on timed tests!)&lt;/p&gt;&#xA;&lt;p&gt;I agree with the OP's implication that it would be a good idea for more books to emphasize this, but this is my literal answer  as to why I think they don't.&lt;/p&gt;&#xA;&lt;p&gt;Also, &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; principle&amp;quot; isn't a very mnemonic name, and I've never heard it before. Perhaps the lack of a catchy name is part of the problem?&lt;/p&gt;&#xA;" OwnerUserId="51" LastActivityDate="2023-10-27T14:14:27.940" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27029" PostTypeId="2" ParentId="27026" CreationDate="2023-10-27T16:14:31.320" Score="5" Body="&lt;p&gt;Here's the way I think about it:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The purpose of &lt;em&gt;cognitive activation&lt;/em&gt; is to get students to think deeply about what they're learning so that they incorporate it into their mental &amp;quot;tree of knowledge&amp;quot;, as opposed to just going through the motions and memorizing facts/procedures.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The purpose of &lt;em&gt;cognitive load minimzation&lt;/em&gt; is to make lessons accessible to all students by avoiding undue stress on working memory (since students have different working memory capacities).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I think there's a strong analogy to sports practice where athletes do drills, scrimmage against each other, and critique tape recordings of prior games.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;During drills, athletes are generally focusing on improving a particular skill. If you want to improve your 3-pointer accuracy, the first thing you should do is drill shooting from the 3-point line. Once you're good at that, you can run progressively more advanced drills: shooting immediately after receiving a pass, shooting fadeaway shots, shooting against an actual defender who's trying to get in your face and make you screw up, shooting against a defender who's actually trying to block you, etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This drill progression is like cognitive load minimization. If you take someone who can't even shoot a 3-pointer and immediately have them practice shooting against a hardcore defender, then it's going to be too overwhelming and they're not going to improve. That's what happens when a learning task exceeds a student's working memory.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But what happens if you do drills all day and never scrimmage? You end up getting pretty good at the skills practiced in the drills but you have no idea how or when to apply them during an actual game. Should I try to go in for the layup, or should I hang back and try to do a fadeaway shot, or should I pass to one of my open teammates (if so, who?), or should I try to move around the court and look for a better opportunity to open up?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In order to answer these questions and perform well during an actual game, you need a good meta-level understanding of the game. So, to force yourself to grapple with these questions, make decisions on the fly, and improve your general meta-level understanding, you also scrimmage during practice and critique tape recordings of prior games.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;It might also be worth mentioning a looser &amp;quot;staircase analogy&amp;quot;: learning is like climbing a staircase, and students get stuck at any individual stair that is too tall for them to climb, so the smaller you make the individual stairs, the more students can climb all the way to the top.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The size of a stair that a student can climb is like their working memory. By minimizing cognitive load, you're making the individual stairs smaller so that more students can climb them.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cognitive activation is meta-reasoning about the &amp;quot;why&amp;quot; and &amp;quot;where&amp;quot;. Why are you climbing the staircase? Where are you trying to go?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-10-27T16:14:31.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27030" PostTypeId="2" ParentId="26991" CreationDate="2023-10-27T17:47:25.313" Score="2" Body="&lt;p&gt;Another possibility is to show a sequence of diagrams explaining how the inequalities relate to one another.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GB3Mx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GB3Mx.png&quot; alt=&quot;four images of triangles&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The top left diagram shows the triangle inequality, &lt;span class=&quot;math-container&quot;&gt;$\lvert \mathbf{a}+\mathbf{b}\rvert \le \vert\mathbf{a}\rvert + \lvert\mathbf{b}\rvert$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The top right diagram shows that &lt;span class=&quot;math-container&quot;&gt;$-\mathbf{b}$&lt;/span&gt; is just as good a vector as &lt;span class=&quot;math-container&quot;&gt;$\mathbf{b}$&lt;/span&gt;, implying that the triangle &lt;span class=&quot;math-container&quot;&gt;$(\mathbf{a},-\mathbf{b},\mathbf{a}-\mathbf{b})$&lt;/span&gt; satisfies the inequality just as much as the triangle &lt;span class=&quot;math-container&quot;&gt;$(\mathbf{a},\mathbf{b},\mathbf{a}+\mathbf{b})$&lt;/span&gt; does. This gives &lt;span class=&quot;math-container&quot;&gt;$\lvert \mathbf{a}-\mathbf{b}\rvert \le \vert\mathbf{a}\rvert + \lvert\mathbf{b}\rvert$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Returning, in the bottom left, to the original triangle, with the sum &lt;span class=&quot;math-container&quot;&gt;$\mathbf{a}+\mathbf{b}$&lt;/span&gt; denoted by &lt;span class=&quot;math-container&quot;&gt;$\mathbf{s}$&lt;/span&gt;, we see that &lt;span class=&quot;math-container&quot;&gt;$\lvert \mathbf{s}\rvert \le \vert\mathbf{a}\rvert + \lvert\mathbf{s}-\mathbf{a}\rvert$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\lvert \mathbf{s}\rvert - \vert\mathbf{a}\rvert \le \lvert\mathbf{s}-\mathbf{a}\rvert$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the bottom right diagram, the roles of &lt;span class=&quot;math-container&quot;&gt;$\mathbf{a}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{s}$&lt;/span&gt; are swapped (which reverses the direction of the red vector), showing that &lt;span class=&quot;math-container&quot;&gt;$\lvert \mathbf{a}\rvert - \vert\mathbf{s}\rvert \le \lvert\mathbf{a}-\mathbf{s}\rvert = \lvert\mathbf{s}-\mathbf{a}\rvert$&lt;/span&gt;. Since at least one of &lt;span class=&quot;math-container&quot;&gt;$\lvert\mathbf{s}\rvert-\lvert\mathbf{a}\rvert$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\lvert\mathbf{a}\rvert-\lvert\mathbf{s}\rvert$&lt;/span&gt; is nonnegative, &lt;span class=&quot;math-container&quot;&gt;$\lvert\mathbf{s}-\mathbf{a}\rvert \ge \lvert \lvert \mathbf{s}\rvert - \vert\mathbf{a}\rvert \rvert$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the comments there there was a request for a way of seeing the inequality in the fourth diagram, &lt;span class=&quot;math-container&quot;&gt;$\lvert \mathbf{a}\rvert - \vert\mathbf{s}\rvert \le \lvert\mathbf{a}-\mathbf{s}\rvert$&lt;/span&gt;, &amp;quot;without algebra&amp;quot;. The following visualization helps me. Here two sides of the triangle gradually &amp;quot;collapse&amp;quot; onto the third side, with &lt;span class=&quot;math-container&quot;&gt;$\lvert\mathbf{a}-\mathbf{s}\rvert$&lt;/span&gt; clearly covering &lt;span class=&quot;math-container&quot;&gt;$\lvert \mathbf{a}\rvert - \vert\mathbf{s}\rvert$&lt;/span&gt; in the final position.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nnPbS.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nnPbS.gif&quot; alt=&quot;collapsing triangle visualization of triangle inequality&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The inequality in the third diagram is evident because &lt;span class=&quot;math-container&quot;&gt;$\lvert\mathbf{s}\rvert - \vert\mathbf{a}\rvert$&lt;/span&gt; is negative.&lt;/p&gt;&#xA;&lt;p&gt;A different visualization of the four inequalities is obtained by drawing the usual parallelogram showing vector addition, that is, the parallelogram two of whose opposite sides are copies of &lt;span class=&quot;math-container&quot;&gt;$\mathbf{a}$&lt;/span&gt;, the other two, copies of &lt;span class=&quot;math-container&quot;&gt;$\mathbf{b}$&lt;/span&gt;. One diagonal of this parallelogram is &lt;span class=&quot;math-container&quot;&gt;$\mathbf{a}+\mathbf{b}$&lt;/span&gt;; the other is &lt;span class=&quot;math-container&quot;&gt;$\mathbf{a}-\mathbf{b}$&lt;/span&gt; (or &lt;span class=&quot;math-container&quot;&gt;$\mathbf{b}-\mathbf{a}$&lt;/span&gt; if the arrow is reversed). Each diagonal divides the parallelogram into two congruent triangles. For any of these four triangles, the sum of lengths of two sides equals or exceeds the length of the third. This produces six distinct inequalities. The two omitted from your post are &lt;span class=&quot;math-container&quot;&gt;$\lvert\mathbf{a}+\mathbf{b}\rvert\ge\lvert\mathbf{a}\rvert-\lvert\mathbf{b}\rvert$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\lvert\mathbf{a}+\mathbf{b}\rvert\ge\lvert\mathbf{b}\rvert-\lvert\mathbf{a}\rvert$&lt;/span&gt;, which, together, give &lt;span class=&quot;math-container&quot;&gt;$\lvert\mathbf{a}+\mathbf{b}\rvert\ge\lvert \lvert\mathbf{a}\rvert-\lvert\mathbf{b}\rvert  \rvert$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xHTyz.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xHTyz.png&quot; alt=&quot;parallelogram from which six inequalities can be derived&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A sample of the &lt;a href=&quot;https://www.sagemath.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;sagemath&lt;/a&gt; code used to generate these diagrams is below.&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt;&lt;pre&gt;&#xA;topleft = dict(horizontal_alignment='left', vertical_alignment='top')&#xA;topright = dict(horizontal_alignment='right', vertical_alignment='top')&#xA;botleft = dict(horizontal_alignment='left', vertical_alignment='bottom')&#xA;botright = dict(horizontal_alignment='right', vertical_alignment='bottom')&#xA;&#xA;a = vector([1, 4])&#xA;b = vector([-3, -1])&#xA;plot2 = (plot(a, color='blue') +&#xA;    plot(b, rgbcolor=(1,0.8,0.8), start=a) +&#xA;    plot(a + b, rgbcolor=(.8,.8,.8)) +&#xA;    plot(-b, color='red', start=a) + plot(a - b, color='black') +&#xA;    plot(point((1, 4), size=52, color='red')) +&#xA;    text(r'&lt;span class=&quot;math-container&quot;&gt;$\mathbf{a}$&lt;/span&gt;', (0.5, 2), **topleft, fontsize=12, color='blue') +&#xA;    text(r'&lt;span class=&quot;math-container&quot;&gt;$\mathbf{-b}$&lt;/span&gt;', (2.5, 4.5), **topleft, fontsize=12, color='red') +&#xA;    text(r'&lt;span class=&quot;math-container&quot;&gt;$\mathbf{a-b}$&lt;/span&gt;', (2, 2.5), **topleft, fontsize=12, color='black') +&#xA;    text(r'&lt;span class=&quot;math-container&quot;&gt;$\mathbf{b}$&lt;/span&gt;', (-0.5, 3.4), **topleft, fontsize=12, rgbcolor=(1,0.8,0.8)) +&#xA;    text(r'&lt;span class=&quot;math-container&quot;&gt;$\mathbf{a+b}$&lt;/span&gt;', (-1, 1.5), **topright, fontsize=12, rgbcolor=(0.8,0.8,0.8)))&#xA;plot2.show(aspect_ratio=1, xmin=-2, xmax=4, ymin=0, ymax=5)&#xA;&lt;/pre&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1737" LastEditorUserId="1737" LastEditDate="2023-10-29T15:25:53.250" LastActivityDate="2023-10-29T15:25:53.250" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27031" PostTypeId="2" ParentId="27027" CreationDate="2023-10-27T17:59:41.977" Score="22" Body="&lt;p&gt;My two cents on this is that it's related to two key issues that I stress in my undergraduate analysis course:&lt;/p&gt;&#xA;&lt;p&gt;Key #1: Existence proofs have two parts: First you exhibit/construct the object, then you show that it works. When proving a quantified statement like &lt;span class=&quot;math-container&quot;&gt;$(\forall \epsilon &amp;gt;0 )(\exists \delta&amp;gt;0)(P(\epsilon, \delta))$&lt;/span&gt;, your proof should start by establishing an arbitrary positive real (&amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$\epsilon &amp;gt;0$&lt;/span&gt;&amp;quot;) and then give an existence proof for &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;. So when writing a &amp;quot;proper&amp;quot; proof, you must give/define the &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; (&amp;quot;Consider &lt;span class=&quot;math-container&quot;&gt;$\delta = 25\epsilon$&lt;/span&gt;&amp;quot;) and then show that it works. You can't have &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; being some sort of place holder.&lt;/p&gt;&#xA;&lt;p&gt;Key #2: Satisfy the definition. If the definition of a limit is (yada yada yada)(stuff &lt;span class=&quot;math-container&quot;&gt;$&amp;lt; \epsilon$&lt;/span&gt;), then in a limit proof you'd better end with showing the stuff is less than &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;, not some other quantity. Again, two things to note: (1) a good instructor will point out that, yes, (stuff &lt;span class=&quot;math-container&quot;&gt;$&amp;lt; 4\epsilon$&lt;/span&gt;) is equivalent and (2) there are plenty of times when immediately after giving a formal definition you give a proposition that says something easier is equivalent. For this particular concept, it's my experience that formalizing the idea in the form of a &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; Principle is overkill leading to student confusion. As the instructor, either accept a proof that ends with (stuff &lt;span class=&quot;math-container&quot;&gt;$&amp;lt; 4\epsilon$&lt;/span&gt;) because you have confidence in your students' understanding, or don't introduce a &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; Principle that is only going to add an extra justification step in the proof that they are struggling with for other reasons.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2023-10-27T17:59:41.977" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27032" PostTypeId="1" CreationDate="2023-10-28T10:12:37.810" Score="1" ViewCount="125" Body="&lt;p&gt;What do you guys think about the ideas presented in this short text: &lt;a href=&quot;https://github.com/yougetyourmanwww/AI-for-math/blob/main/AI.md&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://github.com/yougetyourmanwww/AI-for-math/blob/main/AI.md&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The text is about how LLMs like chatGPT can be used for when doing math. I've never seen anyone with the same opinions as me on that topic, so I'd like some opinions on the ideas.&lt;/p&gt;&#xA;&lt;p&gt;TLDR: LLMs can be used to help communicate math ideas, for example improve the wording in your informal proof or convert the text in a research paper to simpler text. Really the point is that the text generation ability of LLMs can be useful when doing math.&lt;/p&gt;&#xA;" OwnerUserId="23248" LastActivityDate="2023-10-28T13:08:49.743" Title="Idea of using LLMs to help communicate ideas in math" Tags="|proofs|terminology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27033" PostTypeId="2" ParentId="27032" CreationDate="2023-10-28T13:08:49.743" Score="1" Body="&lt;p&gt;If people find it helpful, that's fine. The author of a text is still responsible for the final version, whether it was aided by an LLM or not.&lt;/p&gt;&#xA;&lt;p&gt;I have some quibbles with the specific examples in the linked article (e.g. changing &amp;quot;rigorous&amp;quot; to &amp;quot;strict&amp;quot; veers a bit from what the authors want to say) but that may be beside the point. One can (and should) proofread and edit the output of the LLM before finalizing the document, and errors like that are easy to catch.&lt;/p&gt;&#xA;&lt;p&gt;One downside to keep in mind: there is value in learning to write clearly on one's own. Learning to communicate clearly helps one learn to think clearly. So I would not support students using something like this as they learn to write proofs.&lt;/p&gt;&#xA;" OwnerUserId="22788" LastActivityDate="2023-10-28T13:08:49.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27034" PostTypeId="2" ParentId="11971" CreationDate="2023-10-28T15:17:23.920" Score="0" Body="&lt;p&gt;I the following link has detailed explanation of requirements for Calculus:&#xA;&lt;a href=&quot;https://math.clarku.edu/%7Ema120/background.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://math.clarku.edu/~ma120/background.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Same content saved here if original link will stop working: &lt;a href=&quot;https://web.archive.org/web/20231029160656/https://math.clarku.edu/%7Ema120/background.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://web.archive.org/web/20231029160656/https://math.clarku.edu/~ma120/background.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="23250" LastEditorUserId="23250" LastEditDate="2023-10-29T16:07:41.467" LastActivityDate="2023-10-29T16:07:41.467" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27035" PostTypeId="2" ParentId="27027" CreationDate="2023-10-28T23:47:59.537" Score="16" Body="&lt;p&gt;I can't comment since I lack reputation, so I'll post an answer here.  I largely agree with the other answers, and can say that every time I've taught Introductory analysis after a day or so a student will ask whether &lt;span class=&quot;math-container&quot;&gt;$2\epsilon$&lt;/span&gt; is allowed and usually a very healthy discussion ensues and we talk about what the definition says and doesn't say.  Because this is the first time students have likely proven anything like this, I ask them as first to be very specific and match the definition in their proofs. I find it's usually not that hard to get buy-in about this (after all, this is a class where students are asked to prove things that before they would have said were obvious: (&amp;quot;if &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;0$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$1/a&amp;gt;0$&lt;/span&gt;&amp;quot;, or &amp;quot;if a sequence of integers converges it converges to an integer&amp;quot;). I think students understand that the point of the class is to think about what needs to be shown, and how we can fit pieces together like a puzzle to do so.&lt;/p&gt;&#xA;&lt;p&gt;One answer to your question might be &amp;quot;the way the words appear on the page is NOT the way the proof was written.&amp;quot; The way I think many of us recommend students write these proofs is to leave gaps and to write in pencil expecting to change &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; in their proofs (if you want to see the style I mean, look &lt;a href=&quot;https://www.tiktok.com/t/ZT8BBR9pe/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; or &lt;a href=&quot;https://www.tiktok.com/@rx_math/video/7283665419024583967?_r=1&amp;amp;u_code=di1d525bfm57e6&amp;amp;preview_pb=0&amp;amp;sharer_language=en&amp;amp;_d=e763j9120g65eg&amp;amp;share_item_id=7283665419024583967&amp;amp;source=h5_m&amp;amp;timestamp=1698533399&amp;amp;user_id=6948436480494699525&amp;amp;sec_user_id=MS4wLjABAAAAZMLplqJxAlGLPBKD2sXw6oQeEGWI6j4OyogQB_aNPAEJ-XMYg4C7dtYtXhFw9xXg&amp;amp;social_share_type=0&amp;amp;utm_source=copy&amp;amp;utm_campaign=client_share&amp;amp;utm_medium=android&amp;amp;share_iid=7294541406251697963&amp;amp;share_link_id=5dabba4c-9b41-4b68-9fb7-bd7e53fdc2ca&amp;amp;share_app_id=1233&amp;amp;ugbiz_name=MAIN&amp;amp;ug_btm=b8727%2Cb2878&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; or &lt;a href=&quot;https://www.tiktok.com/t/ZT8BBFXyk/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;) and so in truth these are the last things the student will write and so writing &lt;span class=&quot;math-container&quot;&gt;$\delta=\epsilon/9$&lt;/span&gt; isn't much more writing and has the benefit of matching the definition. I imagine many teachers use this method as well. A student who is reading a proof that &amp;quot;if &lt;span class=&quot;math-container&quot;&gt;$b_n$&lt;/span&gt; converges to &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$1/b_n$&lt;/span&gt; converges to &lt;span class=&quot;math-container&quot;&gt;$1/3$&lt;/span&gt;&amp;quot; probably isn't surprised that this is true (it will feel obvious to them) but presumably they want to know why all the stuff is in there. And so long as the unkempt stuff is explained I think this can just be help for students trying to understand what is going on.  Of course if no care is taken and everything is just splattered in the page, then alas. Part of my reason for making these videos is to show the process, and to show that one need not be afraid of these proofs.&lt;/p&gt;&#xA;&lt;p&gt;As far as which is better - I think it might actually end up being more confusing to students to have different sorts of &amp;quot;rules&amp;quot; that are further removed from the definition.  Sure it's ok for a proof to end &lt;span class=&quot;math-container&quot;&gt;$&amp;lt; 9\epsilon$&lt;/span&gt; or even &lt;span class=&quot;math-container&quot;&gt;$N\epsilon$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is &amp;quot;fixed&amp;quot;. But what about &lt;span class=&quot;math-container&quot;&gt;$n\epsilon$&lt;/span&gt; in a proof of convergence for a sequence &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt;? or what if a student invokes the Archimedean principle to say &amp;quot;There exists an &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; such that blah is &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;N\epsilon$&lt;/span&gt;&amp;quot; at the end of their proof?  These all look distinct to us, but surely seem similar to the student. Aren't they all just &amp;quot;A number times &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;?&amp;quot; And if your reply is that well the quantifiers are different, or that &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is bound to &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; - then you are admitting that adding this extra rule has lots of thorny corners and isn't as simple as &amp;quot;a number times epsilon&amp;quot;.  I'm not saying that I don't talk about whether &lt;span class=&quot;math-container&quot;&gt;$2\epsilon$&lt;/span&gt; is ok, but I let that be student-driven and almost always their reason is &amp;quot;oh it's easy let's just go change that thing and divide it by two.&amp;quot; And then we get a chance to say &amp;quot;Can we do that? Why? Are you sure?&amp;quot; Reminding students of these sorts of things daily has benefits that I think outweigh the benefits from allowing constants. I guess I am saying that spending 20 seconds each day reminding students what freedoms we have (or don't have) is better than spending 5 minutes &amp;quot;once and for all&amp;quot; to prove some lemma taking care of these things.&lt;/p&gt;&#xA;&lt;p&gt;I once needed a bank letter for a visa application to affirm that for each day in the past twelve months my balance was at least \$2000. The banker wrote me a letter saying &amp;quot;Adam's average balance over the last twelve months was \$5,000&amp;quot;.  I had to explain that this was not sufficient and that a bureaucrat could certainly deny my visa for this reason. I don't think he understood the distinction. Indeed he said &amp;quot;but \$5000 is bigger than \$2000.&amp;quot; My point is that when we're first learning things, things that may seem &amp;quot;adjacent&amp;quot; to some are actually not adjacent in the ways that matter.&lt;/p&gt;&#xA;&lt;p&gt;As a final note - in my school, a lot of CS majors take real analysis and I emphasize that writing proofs in real analysis is very similar to writing a computer program.  If your code is supposed to return an integer, but returns the string &amp;quot;1&amp;quot; this is actually a problem. Perhaps now I realize that maybe bureaucrats are also like computer programs.&lt;/p&gt;&#xA;" OwnerUserId="10028" LastActivityDate="2023-10-28T23:47:59.537" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27036" PostTypeId="2" ParentId="26805" CreationDate="2023-10-29T05:32:16.500" Score="1" Body="&lt;p&gt;Unless you're specifically asking about college/superior degrees (which I have none), as posed right now, this question has no possible answer.&lt;/p&gt;&#xA;&lt;p&gt;It depends way too much on your public, their background, their goals and their interests.&lt;/p&gt;&#xA;&lt;p&gt;E.g., in Brazil, there are something like the SATs. I don't know much about US's higher education logistics, but that's not the point. The key differences here are 1) anyone can take the tests/exams however many times they'd like (although they're applied only once or twice a year, in order to properly size classes); 2) only federal universities are required to use the same default exam, elaborated by the ministry of education, although they are somewhat free to define how the results will be calculated (weighting averages). State universities and private universities are almost free to elaborate their own exams. And 3) Some courses and universities will have more competition: Medicine or Law will require higher grades, as well as any course at USP, a world top-100 university. Public vs. private also play a big role.&lt;/p&gt;&#xA;&lt;p&gt;I've seen people who, after graduating from high school, spent 6 to 8 &lt;strong&gt;years&lt;/strong&gt; (which means they took 6 to 8 exams (per non-federal institutions)) in order to get into a Medical school.&lt;/p&gt;&#xA;&lt;p&gt;This means they're likely to have parents/relatives capable of paying all of their expenses and paying for top-grade &amp;quot;prep-schools&amp;quot; (most of those are located in the state's capital, have some of the country's absolutely best teachers, and are progressively expensive).&lt;/p&gt;&#xA;&lt;p&gt;Getting &lt;em&gt;into&lt;/em&gt; higher education is a big deal (and an enormous business). At least for those born in wealthy families who could afford the best education from kindergarten to prep-school, which means they'll be fighting for the absolute best courses.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(I'm circling a lot, but bear with me). These prep-school will have the most meticulously designed classes, textbooks written by their very teachers and highly per course directed schedules/curriculums/timetables (couldn't translate this one, haha): those applying to engineering will have way more Math and Physics classes and so on.&lt;/p&gt;&#xA;&lt;p&gt;Competing against this top-tier education is where most high school graduates are, minus the money. They tend to have had a worse elementary education, they rarely apply for the highly competitive &amp;quot;slots&amp;quot;(?), etc. Still, they focus on the federal universities. Their exams tend to take the whole country's general education and also implement way more inclusion policies.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;That's where I actually start my answer: I happened to be lucky enough to have had an excellent education. During 2020, we had the idea to create an online social prep-school. We had over a hundred volunteers and about 40 students, for whom we had personal lessons (no big classroom problems, such as being afraid to ask, etc.). I managed to make available for all teachers, on every school subject the whole exercise/textbook set, 1 through 8, of a top-tier prep-school.&lt;/p&gt;&#xA;&lt;p&gt;For some subjects, there were also &amp;quot;book 0s&amp;quot;, for actually creating and checking the &lt;strong&gt;common grounds&lt;/strong&gt; before starting.&lt;/p&gt;&#xA;&lt;p&gt;I still recall my first class (in this project) for my student, Lyon. I intended to assess his knowledge and how we would progress the lessons through the year. Lyon had a lot of trouble with book 0 already. So that's what I focused on for months, the proper grounding. I made a big effort to have him master the basics. I actually started with multiplication and how potentiating related to it, I delved deep into how fractions are always divisions or ratios, I also focused on averages and medians, which were common topics for that exam.&lt;/p&gt;&#xA;&lt;p&gt;As such, I knew I'd never have the time to get into analytical geometry, matrixes or polynomial equations, so I tried to create a good foundation, so he could actually &lt;em&gt;know&lt;/em&gt; and &lt;em&gt;understand&lt;/em&gt; the questions, instead of throwing some poorly memorized formula and hoping for the best.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This means I understood where my student was coming from, whether he had actual interest in learning or not, his goal, and, e.g., he would engage his homework or not.&lt;/p&gt;&#xA;&lt;p&gt;I'm very happy to say he got in the federal university he desired (in his own city, so he could keep living with his parents), and the course he chose.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I believe the students shape the teaching as much as the educator, you'll surely not put the same effort if you were to teach a bunch of stupid kids pranking each other and not paying attention at all.&lt;/p&gt;&#xA;&lt;p&gt;As Paulo Freire's (whose workings are a must in Pedagogy, used even in Harvard and the likes) words, teaching is not a full jar filling an empty glass. Teaching is a trade of knowledge and experience, and I'm sure I learned a lot by teaching Lyon during those days!&lt;/p&gt;&#xA;&lt;p&gt;Oh, and BTW, there is no such thing as enforcing ethical practices for online teaching. Netflix, HBO, Disney+ etc. are not able to attain immunity against piracy, you'll certainly not be able to enforce it yourself. Once again, the students goals will shape the teaching.&lt;/p&gt;&#xA;" OwnerUserId="23252" LastEditorUserId="2083" LastEditDate="2023-10-29T08:49:42.617" LastActivityDate="2023-10-29T08:49:42.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27037" PostTypeId="1" CreationDate="2023-10-29T06:56:36.303" Score="14" ViewCount="5719" Body="&lt;p&gt;My first grader is very advanced in math. Rather than doing more and more math and making school math even more boring for him, I recently decided to start going &amp;quot;deeper&amp;quot; rather than &amp;quot;faster.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Some of the questions we've explored recently are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why do we use a numbering system? (Because otherwise we'd need an&#xA;infinite number of names)&lt;/li&gt;&#xA;&lt;li&gt;Why is multiplication commutative? (Because you could switch the&#xA;number of items and the number of people and you'd still need the&#xA;same number of items in the 'rectangle')&lt;/li&gt;&#xA;&lt;li&gt;Why are 2, 5, and 10 the only numbers whose divisibility tests only&#xA;need the last digit? (because they're the factors of 10)&lt;/li&gt;&#xA;&lt;li&gt;Why is a number divisible by 9 if its digit sum is divisible by 9?&#xA;(becuase every power of 10 has remainder 1 when divided by 9)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Are there any resources I can use to find more questions/explorations like this?&lt;/p&gt;&#xA;" OwnerUserId="16497" LastEditorUserId="2083" LastEditDate="2023-10-29T09:02:02.493" LastActivityDate="2024-02-17T18:48:08.303" Title="What are some &quot;deep&quot; questions to explore in elementary school math?" Tags="|primary-education|resource-request|gifted-students|" AnswerCount="18" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="27038" PostTypeId="2" ParentId="27037" CreationDate="2023-10-29T09:01:34.280" Score="5" Body="&lt;ul&gt;&#xA;&lt;li&gt;Matte-LIST is one resource we suggest to our students. In Norwegian. &lt;a href=&quot;https://www.mattelist.no/&quot; rel=&quot;noreferrer&quot;&gt;https://www.mattelist.no/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;I think some of the tasks there are from Youcubed, but I have not looked deeper into it. In English. &lt;a href=&quot;https://www.youcubed.org/tasks/&quot; rel=&quot;noreferrer&quot;&gt;https://www.youcubed.org/tasks/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Tall og tanke aktivitetsbok has exercises for teacher students, but many of them are also suitable for children, or imply such. In Norwegian.&lt;/li&gt;&#xA;&lt;li&gt;Relevant books by Jo Boaler have nice, open exercises, that might be suitable. In English.&lt;/li&gt;&#xA;&lt;li&gt;In general, any good books on mathematics didactics or mathematics education in general are worth a look. They should have useful principles and also often activites.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-10-29T09:01:34.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27039" PostTypeId="2" ParentId="27037" CreationDate="2023-10-29T11:57:03.987" Score="7" Body="&lt;p&gt;Following up on David Raveh's comment, geometric manipulatives would offer&#xA;a wealth of interesting questions.&lt;/p&gt;&#xA;&lt;p&gt;(1) Polydrons.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/rIDxo.jpg&quot; alt=&quot;PolydronsPlatonic&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;(2) Snap / Linking cubes.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/U8D4I.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/U8D4I.jpg&quot; alt=&quot;SnapCubes&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(3) Tangram puzzles&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DOUjx.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DOUjx.jpg&quot; alt=&quot;Tanagrams&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2023-10-29T11:57:03.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27040" PostTypeId="2" ParentId="27022" CreationDate="2023-10-29T12:51:01.800" Score="13" Body="&lt;p&gt;I joined this community just so that I could answer this question.&lt;/p&gt;&#xA;&lt;p&gt;I work in AI and I am big proponent for it. However I do not believe that it is ready to replace math teachers, nor do I think it is a good idea in the first place.&lt;/p&gt;&#xA;&lt;p&gt;Learning mathematics is not about finding the right answers. It is about a journey of discovery! Some of my best math teachers, inspired me with their knowledge, ideas and creativity. They motivated me to explore and find new and exciting ideas. They challenged my beliefs and showed me new viewpoints. They supported me and encouraged me. There was a lot of human emotion and human touch involved. Frankly I don't think that machines are yet capable of any of this.&lt;/p&gt;&#xA;" OwnerUserId="23254" LastActivityDate="2023-10-29T12:51:01.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27041" PostTypeId="2" ParentId="27037" CreationDate="2023-10-29T14:46:13.183" Score="16" Body="&lt;p&gt;Here is a game I stole from Marty Weissman:&lt;/p&gt;&#xA;&lt;p&gt;You start at zero on the number line. You are allowed &amp;quot;hop&amp;quot; forward or backwards by &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; and &amp;quot;jump&amp;quot; forwards or backwards by &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;.  Which numbers can you reach?  Are there any numbers you cannot reach?&lt;/p&gt;&#xA;&lt;p&gt;Variations:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Change &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; to other numbers.&lt;/li&gt;&#xA;&lt;li&gt;Use &amp;quot;hop&amp;quot;, &amp;quot;skip&amp;quot;, and &amp;quot;jump&amp;quot; (3 numbers) instead of just two.&lt;/li&gt;&#xA;&lt;li&gt;Now you are only allowed to hop or jump forwards, never backwards.  How does that change the numbers are you able to reach?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2023-10-30T19:58:49.103" LastActivityDate="2023-10-30T19:58:49.103" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27042" PostTypeId="2" ParentId="27022" CreationDate="2023-10-29T15:47:35.073" Score="8" Body="&lt;p&gt;I'll address a narrow aspect: Access to quality, minimum-standard instruction in math. Two hundred years ago, access to quality math instruction was limited, at least in the US.  General public education was still decades off. By 1918, all US states had passed a compulsory education law, and most had done so by the 1870s or 1880s. The ages for which attendance was required ranged from 7 or 8 to 14–16 years old, and attendance rates ranged from 80-99%, tailing off considerably after age 13. One significant factor was the need for children to work, a need that has greatly diminished from then until now, it seems. Another is the affordability, not so much to individual persons, which was a factor 200 years ago, but to communities, some of which struggled to have good school facilities and teachers. This factor has evolved but persists today. If we look outside the US, it is not hard to find many places in which facilities and teachers are poor or absent.&lt;/p&gt;&#xA;&lt;p&gt;The potential accessibility of the internet would address, or ameliorate, poor facilities by giving cheap access to texts, if nothing else.  The potential of AI teachers would ameliorate the lack of good teachers, which is the relevant factor in this Q&amp;amp;A.  (That is, I wish to set aside other important factor underlying the internet hypothesis, even though lack of access to electricity and computing prevents access to AIs.)  Even if AI teachers are merely mediocre, they probably represent an improvement that could not be accomplished in any other way, simply because we will never train so many teachers as needed and place them in the locations where they are required.&lt;/p&gt;&#xA;&lt;p&gt;In other words, instead comparing AI to the best (or good) teachers, compare them to the other end, where teachers are lacking or poorly trained in mathematics.  My child's Algebra I teacher started a lesson on inequalities by asking the bunch of 14-year-olds in the class, &amp;quot;Does anyone in here know how to teach this?&amp;quot; I give them credit for &amp;quot;doing no harm&amp;quot; instead barging ahead mis-teaching them. One friend's child was told by their teacher that the placement of parentheses don't matter in algebra.  The teacher of another friend's child was so concerned with maintaining control by suppressing the curiosity of the child, that the child got angry and refused to learn (in Algebra I in eighth grade). No doubt an AI can be trained better than this, but then so can a teacher. My doubt is whether everyone will pay the expense for well-trained AIs, or will they opt for a cheaper one in the face of imperfect knowledge about whatever AIs happen to be available.&lt;/p&gt;&#xA;&lt;p&gt;I also think an important consideration, not central to the above remarks but relevant to the OP's question, is that most (or all) learning is accomplished by the individual person.  I was more-or-less told this by one of my high school English teachers.  It's more or less obvious, on a literal level, when considering the biology of the brain.  A teacher has a role in the individual's learning, and it's good to think about what that role is.  I spent a few years, ages 12–14, learning math from books without the aid of a teacher.  I have a student currently who did something similar and is very good and interested in mathematics.  Maybe we're rare exceptions.  Still it is possible to learn by reading, calculating, and thinking.  I wonder if books that always say the same thing, no matter what question you ask them, are so much better than an AI that can interact with you.  It doesn't seem likely that a better AI is impossible.&lt;/p&gt;&#xA;&lt;p&gt;People once asked me, &amp;quot;Won't the internet make teachers unnecessary?&amp;quot; I'd say, &amp;quot;They've been unnecessary for a long time. You can always go the library and find out what you want to learn. But people prefer teachers for some reason.&amp;quot; I guess they always will. But if you don't have access to them, then libraries, the internet, and AIs may be the best alternatives.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2023-10-30T10:53:09.647" LastActivityDate="2023-10-30T10:53:09.647" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27043" PostTypeId="1" CreationDate="2023-10-29T16:39:41.267" Score="1" ViewCount="135" Body="&lt;p&gt;I would like to know the best high school math books from Japan and China. Can you suggest some books or free resources? I would like to compare the different approach betweeen China and Japan and Europe. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="23256" LastActivityDate="2023-10-30T00:41:34.320" Title="Chinese and Japanese most important high school textbooks" Tags="|precalculus|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27044" PostTypeId="2" ParentId="27037" CreationDate="2023-10-29T17:01:26.533" Score="3" Body="&lt;p&gt;Some more:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Why does division by zero not work?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Why is the order of operations important in math?&lt;/strong&gt;&lt;br /&gt;&#xA;(Necessity for a standardized order to get consistent results)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Why are there infinitely many prime numbers?&lt;/strong&gt;&lt;br /&gt;&#xA;(The proof is actually simple enough for a first grader to understand)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;What does it mean for a number to be &amp;quot;irrational&amp;quot;?&lt;/strong&gt;&lt;br /&gt;&#xA;(Discuss how some numbers can't be neatly expressed as a fraction. You don't have to use the vocabulary of &amp;quot;irrational.&amp;quot;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;What is a negative number?&lt;/strong&gt;&lt;br /&gt;&#xA;(Discuss the concept of &amp;quot;less than zero&amp;quot; and where it’s useful. For instance, instead of asking &amp;quot;What is the sum of 10 plus 10?&amp;quot;, you could ask &amp;quot;The sum is 20. What could the added numbers be? Could they be negative?&amp;quot;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="23257" LastActivityDate="2023-10-29T17:01:26.533" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27045" PostTypeId="2" ParentId="27037" CreationDate="2023-10-29T18:00:16.577" Score="1" Body="&lt;p&gt;He can play with blocks. Kapla/Keva/Citi blocks are great, as are pattern blocks.&lt;/p&gt;&#xA;&lt;p&gt;You can read him mathy stories. I love &lt;em&gt;&lt;strong&gt;The Cat in Numberland&lt;/strong&gt;&lt;/em&gt; (may be hard to find); &lt;em&gt;&lt;strong&gt;Quack and Count&lt;/strong&gt;&lt;/em&gt; is most likely too young for him, but he might like to write his own book like it (especially if he has any younger siblings); &lt;em&gt;&lt;strong&gt;Two of Everything&lt;/strong&gt;&lt;/em&gt; by Lily Toy Hong; &lt;em&gt;&lt;strong&gt;How Hungry Are You?&lt;/strong&gt;&lt;/em&gt; by Donna Jo Napoli;  &lt;em&gt;&lt;strong&gt;One Grain of Rice&lt;/strong&gt;&lt;/em&gt; by Demi; &lt;em&gt;&lt;strong&gt;You Can Count on Monsters&lt;/strong&gt;&lt;/em&gt; by Richard Evan Schwartz.&lt;/p&gt;&#xA;&lt;p&gt;You can play mathy games together. Nim is a good one, with tons of variations. I play it with stones, using 3 uneven piles. You can take as many stones from one pile as you want. The one who picks up the last stone wins (or loses, for one variation). Check out Blockus, Katamino, Rush Hour, Chocolate Fix, and Set.&lt;/p&gt;&#xA;&lt;p&gt;Have fun!&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-10-29T18:00:16.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27046" PostTypeId="2" ParentId="10226" CreationDate="2023-10-29T19:28:05.450" Score="0" Body="&lt;p&gt;If we examine the distributive property; we will see it does not apply to your algebraic expression:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://photos.app.goo.gl/UCUCsyTno6j14TVv8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://photos.app.goo.gl/UCUCsyTno6j14TVv8&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If the terms 'ab' and 'ac' have a common factor of 'a' then the sum of the two terms is &amp;quot;factorised&amp;quot; by the following steps:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Remove the common factor 'a' from each expression ie 'ab and ac' become 'b and c'.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sum 'b and c' together: b + c and put parentheses to indicate consolidation: (b + c)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;multiply the term (b+c) with the &amp;quot;common factor&amp;quot; 'a': a(b+c)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Thus &lt;span class=&quot;math-container&quot;&gt;$ab + ac = a(b+c)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In summary; the consolidation of the sum of two terms multiplied with their common factor outside the parentheses as opposed to attaching each term with said common factor can be justified using the definition of multiplication:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$a = 2; b,c \in \mathbb{R} \mid$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2b + 2c = \sum_{i=1}^{2} b + \sum_{i=1}^{2} c    = b + b + c + c$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Apply the commutative property of addition:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$b + b + c + c = b + c + b + c$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Apply the associative property of addition:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$b + c + b + c = (b + c) + (b + c)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But we know:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2(b + c) = \sum_{i=1}^{2} (b + c) = (b + c) + (b + c)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2(b + c) = b + c + b + c = 2b + 2c$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Therefore: &lt;span class=&quot;math-container&quot;&gt;$2(b + c) = 2b + 2c$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you attempt to apply these steps to any two of the terms from your expression you will find that the sums will not consolidate like they do here.&lt;/p&gt;&#xA;" OwnerUserId="22858" LastActivityDate="2023-10-29T19:28:05.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27047" PostTypeId="2" ParentId="27037" CreationDate="2023-10-29T20:25:27.100" Score="3" Body="&lt;p&gt;Take whatever questions the child has found interesting to study, and expand upon them. Some directions will be more fun than others: pay attention to that.&lt;/p&gt;&#xA;&lt;p&gt;For the questions you've mentioned:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What happens to divisibility tests in other numerical bases? (That's where we use a different number of digits: we count up to a different number before deciding we've reached one-zero.)&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Can you find one? (That digit sum insight might help!)&lt;/li&gt;&#xA;&lt;li&gt;Why does it work? Can you prove to yourself that you won't ever find any counterexamples?&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Can you convince somebody else? (Even somebody who's being totally unreasonable about what they'll count as proof?)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;(If it doesn't work:) Why did it look like it worked? What's actually going on here?&lt;/li&gt;&#xA;&lt;li&gt;Can we find base-10 divisibility tests for the &lt;em&gt;hard&lt;/em&gt; numbers, like 7?&lt;/li&gt;&#xA;&lt;li&gt;Why the similarity between base-9's 4-test and base-10's 3-test?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;What happens if we only use the last digit of calculations (like addition, multiplication, subtraction)?&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This is the same as using remainder-10: what if we use remainder other things? (5, 7 and 8 are good numbers to look at.)&lt;/li&gt;&#xA;&lt;li&gt;If division is the opposite of multiplication, can we do remainder-5 division?&lt;/li&gt;&#xA;&lt;li&gt;What about remainder-10 division? Which numbers don't work?&lt;/li&gt;&#xA;&lt;li&gt;For which remainder is 0 the only number you can't divide by?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Using techniques you'll already have studied, like “fill in the blank equations” (algebra), can really help with these problems! (No need to use letters, if squares and circles and different colours do the trick – but equally, no need to use those if letters work fine.)&lt;/p&gt;&#xA;&lt;p&gt;If you want to make sure you're not making future school-maths boring, try university-level mathematics. (No, I'm not joking: the only reason half that stuff is “university-level” is because it's currently not in the curriculum for younger people, often because it's hard to write exam questions for.) Those questions you asked about operations lead nicely on to the study of &lt;a href=&quot;https://en.wikipedia.org/wiki/Magma_(algebra)#Types_of_magma&quot; rel=&quot;nofollow noreferrer&quot;&gt;binars of various kinds&lt;/a&gt;. Yes, this is &lt;em&gt;incredibly&lt;/em&gt; dull – but the maths isn't! &lt;a href=&quot;https://en.wikipedia.org/wiki/Cyclic_group#Additional_properties&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clock addition&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Dihedral_group&quot; rel=&quot;nofollow noreferrer&quot;&gt;flipping shapes&lt;/a&gt;, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Abstract_rewriting_system&quot; rel=&quot;nofollow noreferrer&quot;&gt;rules for transforming strings&lt;/a&gt; can all be used as examples to make &lt;a href=&quot;https://en.wikipedia.org/wiki/Cayley_table&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cayley tables&lt;/a&gt; from, for asking questions like:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What if we know that &lt;span class=&quot;math-container&quot;&gt;$a+b=b+a$&lt;/span&gt;, but &lt;a href=&quot;https://en.wikipedia.org/wiki/Commutative_magma&quot; rel=&quot;nofollow noreferrer&quot;&gt;nothing else&lt;/a&gt;?&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Apparently rock-paper-scissors looks like that – can you make rock-paper-scissors &lt;a href=&quot;https://en.wikipedia.org/wiki/Rock_paper_scissors#Additional_weapons&quot; rel=&quot;nofollow noreferrer&quot;&gt;bigger&lt;/a&gt;?&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What makes rock-paper-scissors, or its variants, fair? (Yes, this is no longer abstract algebra: sue me, it's interesting.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;What if we also have a value that behaves like &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$0+a=a+0=a$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;What's the fewest different values where we can have a &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, but we can define &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt; doesn't always equal &lt;span class=&quot;math-container&quot;&gt;$b+a$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Alternatively, you could go for number theory along those lines:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;We have &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; – can we go bigger? Do those operations have any surprising properties?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Don't invest too much in any one idea, though: it's no use putting together a six month curriculum if the child isn't actually interested in this area of mathematics right now. (You wouldn't want to just re-create school at home.)&lt;/p&gt;&#xA;" OwnerUserId="13158" LastEditorUserId="13158" LastEditDate="2023-10-30T14:01:34.580" LastActivityDate="2023-10-30T14:01:34.580" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27048" PostTypeId="2" ParentId="27037" CreationDate="2023-10-29T20:39:38.180" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://nrich.maths.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;NRICH&lt;/a&gt; has a lot of resources. I usually go through adding 1 to the number in the URL until I find an interesting-looking problem, like &lt;a href=&quot;https://nrich.maths.org/2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geoboards&lt;/a&gt; or &lt;a href=&quot;https://nrich.maths.org/9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cereal Packets&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here's an extract from the end of &lt;a href=&quot;https://nrich.maths.org/50&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clock Hands&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you are the sort of person who prefers to work with a clock that the hour hand moves properly with the minutes hand then you might like to look at these:-&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NPEFj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NPEFj.png&quot; alt=&quot;The hands on the clocks are replaced with squares (the squares' diagonals as the clock's radius), equilateral triangles (the altitude as the radius) and circles (the diameters are radiuses). The clocks show 10 past 12, 20 past 12, half past 12, 20 to 1 and 10 to 1 for each shape, for a total of 15 clocks.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Good luck.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13158" LastEditorUserId="13158" LastEditDate="2023-10-31T00:41:45.923" LastActivityDate="2023-10-31T00:41:45.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27049" PostTypeId="2" ParentId="27037" CreationDate="2023-10-29T23:30:53.740" Score="6" Body="&lt;p&gt;As a child I was fascinated with numbers. As part of our fourth-grade curriculum we were familiarized with conversion to and from, as well as &lt;em&gt;performing basic arithmetic&lt;/em&gt; in, different bases. I remember having fun with that and working on some additional problems on my own beyond assigned homework.&lt;/p&gt;&#xA;&lt;p&gt;I clearly recall that the bases covered were both smaller and larger than ten. I would claim that this does improve mental flexibility and provides a bit deeper insight into the nature of positional number systems. It even has some practical relevance in the world of computers, but that part I was not aware of at the time. Obviously computations base-60 are common when dealing with time, and here one might add a bit of mathematical history by mentioning the use of the sexagesimal system by Sumerians and Babylonians.&lt;/p&gt;&#xA;" OwnerUserId="17715" LastActivityDate="2023-10-29T23:30:53.740" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27050" PostTypeId="2" ParentId="27043" CreationDate="2023-10-30T00:41:34.320" Score="4" Body="&lt;p&gt;The only full-length English translations of Chinese or Japanese secondary math textbooks that I'm aware of are &lt;a href=&quot;https://ucsmp.uchicago.edu/resources/translations/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the ones listed here&lt;/a&gt;, which were produced by the University of Chicago School Mathematics Project. There are seven books in total, covering grades 7 to 11. The original textbooks were from the 1980s.&lt;/p&gt;&#xA;&lt;p&gt;There isn't a very large market for translated school textbooks to begin with, and especially not at the secondary level. If you're interested in comparing different approaches, I would instead look for math education research articles on international textbook comparison. For example, &lt;a href=&quot;https://link.springer.com/article/10.1007/s11858-018-0977-6&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article by Wang and Xu&lt;/a&gt; presents results from a study commissioned by the Chinese Ministry of Education during their revision of the national high school mathematics standards. The authors compare the content of high school textbooks in China, France, Germany, Japan, Russia, the United Kingdom, and the United States.&lt;/p&gt;&#xA;&lt;p&gt;There are many more articles on international textbook comparison—it's a popular area of research. &lt;a href=&quot;https://scholar.google.com/scholar?&amp;amp;q=secondary%20mathematics%20textbook%20comparison&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's a Google Scholar search to help you find more articles.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-10-30T00:41:34.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27052" PostTypeId="2" ParentId="27027" CreationDate="2023-10-30T04:56:02.007" Score="13" Body="&lt;p&gt;The goal of teaching &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; proofs is not necessarily to make students as efficient as possible in writing &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; proofs. The goal is to instill understanding, of which the ability to write &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; proofs is a symptom.&lt;/p&gt;&#xA;&lt;p&gt;If we add tricks that make it easier to write a proof &amp;quot;in order&amp;quot;, starting from the first line and ending at the last line, then we are not doing our students a service, because proofs are not generally written in order. The typical thing to do, when writing a proof, is to explore the problem, see how the parameters interact, figure out what's going on (maybe by solving some equations on scratch paper), and &lt;em&gt;then&lt;/em&gt; write a proof.&lt;/p&gt;&#xA;&lt;p&gt;It so happens that &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; proofs are a good place to teach this skill, because the exploration that needs to be done is relatively routine. It is routine enough that something like a &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; principle&amp;quot; can be invented to eliminate the exploration. But we shouldn't give in to this temptation, because then the students will be stuck learning how to explore &lt;em&gt;later&lt;/em&gt;, on a &lt;em&gt;harder&lt;/em&gt; problem.&lt;/p&gt;&#xA;&lt;p&gt;In addition, a real analysis class is often the students' first class in formal proof-writing. They are learning skills like &amp;quot;if the definition says 'there exists a &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;', then I will have to figure out what &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; to choose to satisfy the definition, and pick that at the beginning of the proof&amp;quot;. We should not muddy the waters while the students are still learning this basic logical structure. Once they are comfortable with it, &lt;em&gt;then&lt;/em&gt; (if they are still writing &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; proofs) we can teach them how to get away with being sloppy.&lt;/p&gt;&#xA;" OwnerUserId="9609" LastActivityDate="2023-10-30T04:56:02.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27053" PostTypeId="2" ParentId="27037" CreationDate="2023-10-30T05:49:05.130" Score="1" Body="&lt;p&gt;one possible thing to look into is knot theory. most of the modern bits are horribly complex, but you can do a lot of the basics by playing around with some string and permanent markers&lt;/p&gt;&#xA;" OwnerUserId="13845" LastActivityDate="2023-10-30T05:49:05.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27054" PostTypeId="2" ParentId="27037" CreationDate="2023-10-30T06:48:07.560" Score="1" Body="&lt;p&gt;There are a lot of accessible ideas from number theory in 'the housekeeper and the professor'. It's also a lovely story in its own right and beautifully written.&lt;/p&gt;&#xA;&lt;p&gt;I read it to my daughter when she was about 10 and we had some good conversations about maths. It complemented well what she was doing at school at the time.&lt;/p&gt;&#xA;&lt;p&gt;She is not hugely interested in maths but it still held her interest both as a story and for the maths ideas.&lt;/p&gt;&#xA;&lt;p&gt;I'll update this answer with some specifics if I can dig out my copy.&lt;/p&gt;&#xA;" OwnerUserId="20916" LastActivityDate="2023-10-30T06:48:07.560" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27055" PostTypeId="2" ParentId="27037" CreationDate="2023-10-30T07:48:03.303" Score="2" Body="&lt;p&gt;There are some really hands on &amp;quot;basic&amp;quot; topology things you could do. The most basic is the Mobius strip: watch this Numberphile video &lt;a href=&quot;https://www.youtube.com/watch?v=wKV0GYvR2X8&amp;amp;ab_channel=Numberphile&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=wKV0GYvR2X8&amp;amp;ab_channel=Numberphile&lt;/a&gt; for some fun hands-on demonstrations you can do.&lt;/p&gt;&#xA;&lt;p&gt;In general, one can draw some arrows on a polygon, and try to connect it up into a surface. For the square, we have the following 3: Mobius strips are the easiest to make, and then Klein bottles, and then the real projective plane.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/fDto2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fDto2.png&quot; alt=&quot;fundamental polygons&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The Klein bottle can be understood via this diagram &lt;a href=&quot;https://en.wikipedia.org/wiki/Klein_bottle&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Klein_bottle&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/PrGuu.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PrGuu.png&quot; alt=&quot;Klein bottle step by step&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The real projective plane is the hardest of the three, but one can make one hands-on via a paper foldable: &lt;a href=&quot;https://divisbyzero.com/2020/04/08/make-a-real-projective-plane-boys-surface-out-of-paper/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://divisbyzero.com/2020/04/08/make-a-real-projective-plane-boys-surface-out-of-paper/&lt;/a&gt;. I made one today! Smoothing this construction out, one gets a Boy's surface. They look very nice &lt;a href=&quot;http://wordpress.discretization.de/ddg2019/2019/05/06/tutorial-4-boys-surface/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://wordpress.discretization.de/ddg2019/2019/05/06/tutorial-4-boys-surface/&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For polygons more complicated than a square, Jos Leys &lt;a href=&quot;https://www.youtube.com/@josleys/videos&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/@josleys/videos&lt;/a&gt; has a lot of videos connecting up various polygons. Mesmerizing to watch! Here are a couple:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The Cross Cap: &lt;a href=&quot;https://www.youtube.com/watch?v=W-sKLN0VBkk&amp;amp;ab_channel=JosLeys&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=W-sKLN0VBkk&amp;amp;ab_channel=JosLeys&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;The Klein Quartic: &lt;a href=&quot;https://www.youtube.com/watch?v=YhVsqnXhVWc&amp;amp;ab_channel=JosLeys&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=YhVsqnXhVWc&amp;amp;ab_channel=JosLeys&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;An octagon: &lt;a href=&quot;https://www.youtube.com/watch?v=G1yyfPShgqw&amp;amp;ab_channel=JosLeys&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=G1yyfPShgqw&amp;amp;ab_channel=JosLeys&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;A 16-gon: &lt;a href=&quot;https://www.youtube.com/watch?v=U5N5mg3MePM&amp;amp;ab_channel=JosLeys&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=U5N5mg3MePM&amp;amp;ab_channel=JosLeys&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;A 40-gon: &lt;a href=&quot;https://www.youtube.com/watch?v=RBWwlfcftYM&amp;amp;ab_channel=JosLeys&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=RBWwlfcftYM&amp;amp;ab_channel=JosLeys&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Another medium of hands-on activity: you can crochet hyperbolic surfaces &lt;a href=&quot;https://www.theiff.org/oexhibits/oe1e.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.theiff.org/oexhibits/oe1e.html&lt;/a&gt;! See this TedTalk &lt;a href=&quot;https://www.youtube.com/watch?v=D-AHvZqbMT4&amp;amp;t=0s&amp;amp;ab_channel=TEDxTalks&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=D-AHvZqbMT4&amp;amp;t=0s&amp;amp;ab_channel=TEDxTalks&lt;/a&gt; (&#xA;Crocheting Hyperbolic Planes: Daina Taimiņa) and this tutorial video &lt;a href=&quot;https://www.youtube.com/watch?v=xtlDND7NVp8&amp;amp;ab_channel=CodeParade&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=xtlDND7NVp8&amp;amp;ab_channel=CodeParade&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If 2 dimensions gets too boring, one can go to the third &lt;a href=&quot;https://www.youtube.com/watch?v=yqUv2JO2BCs&amp;amp;ab_channel=ZenoRogue&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=yqUv2JO2BCs&amp;amp;ab_channel=ZenoRogue&lt;/a&gt;. There are now lots of games that have the player play in non-Euclidean (hyperbolic) 3-space.&lt;/p&gt;&#xA;&lt;p&gt;Hyperbolic geometry in 3-space has no shortage of difficulty: &lt;a href=&quot;https://www.youtube.com/watch?v=IrlaVaATiOY&amp;amp;ab_channel=LastGinger&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=IrlaVaATiOY&amp;amp;ab_channel=LastGinger&lt;/a&gt; ([HD Upscale] Not Knot - A guide to mathematical knots in hyperbolic space).&lt;/p&gt;&#xA;&lt;p&gt;There's also this super classic and super awesome video: &lt;a href=&quot;https://www.youtube.com/watch?v=OI-To1eUtuU&amp;amp;ab_channel=LastGinger&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=OI-To1eUtuU&amp;amp;ab_channel=LastGinger&lt;/a&gt; ([HD Upscale] Outside In - How to turn a sphere inside out). Mathologer also has a video on sphere eversion that I find a bit easier to visualize &lt;a href=&quot;https://www.youtube.com/watch?v=ixduANVe0gg&amp;amp;ab_channel=Mathologer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=ixduANVe0gg&amp;amp;ab_channel=Mathologer&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Also one can play around with visualizing the 4th dimension &lt;a href=&quot;https://www.youtube.com/watch?v=0t4aKJuKP0Q&amp;amp;ab_channel=%5Bmtbdesignworks%7BMiegakure%2C4DToys%7D%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=0t4aKJuKP0Q&amp;amp;ab_channel=%5Bmtbdesignworks%7BMiegakure%2C4DToys%7D%5D&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;EDIT: there's also Henry Segerman's Youtube channel &lt;a href=&quot;https://www.youtube.com/@henryseg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/@henryseg&lt;/a&gt;. There are several videos&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;on gears (if one has access to a 3D printer, e.g. via a public library/makerspace, one can print them out), e.g. &lt;a href=&quot;https://www.youtube.com/watch?v=0ZPo3HxR0KI&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=0ZPo3HxR0KI&lt;/a&gt; (screw gears), &lt;a href=&quot;https://www.youtube.com/watch?v=T3CkqXycT9w&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=T3CkqXycT9w&lt;/a&gt;, &lt;a href=&quot;https://www.youtube.com/watch?v=l4eL1MvEbN0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=l4eL1MvEbN0&lt;/a&gt;, &lt;a href=&quot;https://www.youtube.com/watch?v=dkguSyeQXjc&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=dkguSyeQXjc&lt;/a&gt; (5 axis racks)&lt;/li&gt;&#xA;&lt;li&gt;on &amp;quot;lever mechanisms&amp;quot;: &lt;a href=&quot;https://www.youtube.com/watch?v=uVS7YGSKmJM&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=uVS7YGSKmJM&lt;/a&gt;, &lt;a href=&quot;https://www.youtube.com/watch?v=76XIrm91Ra0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=76XIrm91Ra0&lt;/a&gt; (a &amp;quot;spiraling&amp;quot; mechanism)&lt;/li&gt;&#xA;&lt;li&gt;very cute holonomy mazes: &lt;a href=&quot;https://www.youtube.com/watch?v=wzjUTPCAF4Y&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=wzjUTPCAF4Y&lt;/a&gt; (dodecahedron holonomy), &lt;a href=&quot;https://www.youtube.com/watch?v=LG5HUd0hzzo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=LG5HUd0hzzo&lt;/a&gt; (octahedral)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Another video I found very fun was this one &lt;a href=&quot;https://www.youtube.com/watch?v=Cyhqc8l03GE&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=Cyhqc8l03GE&lt;/a&gt; (Using topology to close a rubber band bracelet), which spawned from a discussion on Math Overflow (a Q&amp;amp;A site for professional mathematicians; this video's construction of a rubber band bracelet is cutting edge research!)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;EDIT 2/17/24: I came across a beautiful beautiful set of videos &lt;a href=&quot;https://www.youtube.com/watch?v=6-Z0qgYjVjU&amp;amp;list=PLJHszsWbB6hq40r_aSVlCXDvTT0VcrgcT&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=6-Z0qgYjVjU&amp;amp;list=PLJHszsWbB6hq40r_aSVlCXDvTT0VcrgcT&lt;/a&gt; (also found here &lt;a href=&quot;https://archive.org/details/M435Ep8Of8FlowsTopology&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/M435Ep8Of8FlowsTopology&lt;/a&gt;), where people show lots and lots of models of surfaces, and prove some wonderful things. I feel like working with children to make these models, and playing around with deforming them (making many models to illustrate a deformation), would be really cool, and also introduce them to really deep ideas!&lt;/p&gt;&#xA;&lt;p&gt;One idea that appears over and over again is the Euler characteristic, which is also a great theorem to study in a &amp;quot;hands on&amp;quot; manner with children, making different polyhedron and so forth.&lt;/p&gt;&#xA;" OwnerUserId="13084" LastEditorUserId="13084" LastEditDate="2024-02-17T18:48:08.303" LastActivityDate="2024-02-17T18:48:08.303" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27056" PostTypeId="2" ParentId="27037" CreationDate="2023-10-30T08:07:28.910" Score="2" Body="&lt;p&gt;These are great questions that find patterns within concept. They are also all closed questions with a set answer. To challenge your young mathematician further, make concepts applicable in daily life, with open-ended questions to exercise fluid thinking and reasoning. The grocery store is an easy one. There are always incongruous unit rate prices for the same items. &amp;quot;Hmmm...I wonder which sandwich bread I should buy. What do you think?&amp;quot; By not identifying any values to the various options, your child gets to identify relationships and reason their decision based on cheapest, or best value, or best unit price, etc. They are playing with the math. My credentials: I taught math for ten years, wrote math curriculum for seven years, wrote a math book, and currently train educators on how to teach reasoning from a pedagogical perspective.&lt;/p&gt;&#xA;" OwnerUserId="23268" LastActivityDate="2023-10-30T08:07:28.910" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27057" PostTypeId="2" ParentId="27037" CreationDate="2023-10-30T09:59:07.497" Score="2" Body="&lt;p&gt;One of the classic responses to this question that I like is to ask the student to invent a new math concept (perhaps an operation, a new way of doing their curriculum problems, or a way of writing a number, etc.). See what they can explain about their new idea.&lt;/p&gt;&#xA;&lt;p&gt;Also, counting problems make interesting activities. For example, &amp;quot;Can you count how many ways you can rearrange these four blocks?&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="23270" LastActivityDate="2023-10-30T09:59:07.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27058" PostTypeId="2" ParentId="27037" CreationDate="2023-10-30T10:18:39.827" Score="6" Body="&lt;p&gt;When I was around this age (I dont remember exactly when, but I could read but only slowly) my parents introduced me to a book &amp;quot;What Is the Name of This Book?&amp;quot; by Raymond M. Smullyan. It is a colection of progressivly more complex logic riddles in the likes of:&#xA;A says &amp;quot;all of us are lying&amp;quot;.&#xA;B says &amp;quot;exactly one of us is truthful&amp;quot;.&#xA;C doesn't say anything.&lt;/p&gt;&#xA;&lt;p&gt;I remember finding it really fun at that time. Formal logic is usually taught much later in math curriculum, but I find it not necessary to postpone, since it felt to be very natural and intuitive. Looking back the very early exposure to evaluating statements as true and false and learning to explicitly think about contradictions and implications has helped me in seeing underlying logical structure in many math problems later on.&lt;/p&gt;&#xA;" OwnerUserId="15467" LastActivityDate="2023-10-30T10:18:39.827" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27059" PostTypeId="2" ParentId="27037" CreationDate="2023-10-30T11:34:58.543" Score="1" Body="&lt;p&gt;There are a number of games based around Venn diagrams.  I once saw a guessing game made up of three loops of yarn and several colored blocks.  The blocks could be any of four colors, any of three shapes, and either of two sizes.  The loops are laid out so as to form a venn diagram of three sets.&lt;br /&gt;&#xA;The teacher has a secret rule that governs each of the loops.  One loop might contain all the green blocks, while a second loop contains all the square blocks,  and the third loop contains all the small blocks.  Each kid, in turn, gets to pick out a block and guess where it goes.  If the kid guesses right,  the block stays, and it scores a point.  Sooner or later, the kids figure out the rule.&lt;br /&gt;&#xA;It might be good to play a practice round with just one loop,  and then another practice round with just two loops.&lt;br /&gt;&#xA;Every time I've seen this played, there was at least one kid who was faster than the teacher at figuring out the three loop case.&lt;br /&gt;&#xA;I've included a diagram from an unrelated web page just to show how three overlapping loops form a venn diagram with eight zones in it, if you include the &amp;quot;outside zode&amp;quot;.  With last zone is where blocks go if they don't belong in any of the loops.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://3.bp.blogspot.com/_r94Ze5inyTo/TKSGchMom9I/AAAAAAAAAfo/hXQHKb9AAjo/s1600/kids+venn.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://3.bp.blogspot.com/_r94Ze5inyTo/TKSGchMom9I/AAAAAAAAAfo/hXQHKb9AAjo/s1600/kids+venn.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here's another image from a product that facilitates Venn games.  I've never used the product, so I have no opinion of it's value.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.pinimg.com/originals/36/18/42/36184276d7f286cb44858ac994ac4609.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://i.pinimg.com/originals/36/18/42/36184276d7f286cb44858ac994ac4609.jpg&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1707" LastEditorUserId="1707" LastEditDate="2023-10-30T11:43:37.433" LastActivityDate="2023-10-30T11:43:37.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27060" PostTypeId="2" ParentId="27037" CreationDate="2023-10-30T11:35:02.527" Score="2" Body="&lt;p&gt;Not a professional teacher but around math competitions for a long time. I love exploring combinatorics with children since the concepts are easy to imagine/visualize but problems are very deep (e.g. pigeonhole principle). Here's a sample problem from a 4-5 grade competition: &amp;quot;15 guests at a party sat down around a round table where each chair had a guest name tag. They noticed that no one sits on their assigned chair, so each tried shifting to the chair on their right. Show that if they continue shifting, at some point at least two guests will be sitting on their assigned chairs.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="23272" LastActivityDate="2023-10-30T11:35:02.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27061" PostTypeId="2" ParentId="27037" CreationDate="2023-10-30T15:21:50.450" Score="1" Body="&lt;p&gt;A classic one I heard from Janos Komlos to keep his kids busy was &amp;quot;&lt;em&gt;can you find a square number whose remainder when divided by 3 is 2?&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Once you have exposure to modular arithmetic this is trivial but without it the result can seem very mysterious. Also right around the corner from this result is the starting point for things like Algebraic Number Theory and yet this is a very very elementary question to ask.&lt;/p&gt;&#xA;&lt;p&gt;Another related one is &lt;em&gt;can you find a fraction that squares to two&lt;/em&gt;? The proof of this is understandable by a grade schooler but requires a lot of creativity to come up with, and hints towards a lot of interesting math. Also the history around it (people died for this proof!) would be very engaging to also discuss.&lt;/p&gt;&#xA;&lt;p&gt;For high schoolers. After having shown the Geometric Series, discussing &lt;span class=&quot;math-container&quot;&gt;$1-1+1 ... = \frac{1}{2}$&lt;/span&gt; and the notion of Cessaro Summation can be very interesting. It challenges our intuitions of &amp;quot;what it means to sum&amp;quot; if there are procedures that sum all convergent sums correctly and yet also can assign values to divergent sums. To me that felt insanely profound the first time I was exposed to it.&lt;/p&gt;&#xA;" OwnerUserId="1909" LastEditorUserId="1909" LastEditDate="2023-10-30T15:34:13.787" LastActivityDate="2023-10-30T15:34:13.787" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27063" PostTypeId="2" ParentId="27027" CreationDate="2023-10-31T00:37:23.930" Score="4" Body="&lt;p&gt;By now, as somewhat experienced in what little of mathematics we use in economics and statistics (and limits are certainly a part), I recognize what the OP is raging about, and in practice, I always skip over it by the verbal &amp;quot;since we can make it arbitrarily small...&amp;quot;. And I am fine with that, and everybody is fine with that.&lt;/p&gt;&#xA;&lt;p&gt;But I am a &lt;em&gt;user&lt;/em&gt; of mathematics at this point, not a &lt;em&gt;student&lt;/em&gt;. And I suspect that the concept&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is fixed and &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; can be made arbitrarily small, then it&#xA;doesn't matter if we arrive at &lt;span class=&quot;math-container&quot;&gt;$C\epsilon$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is not an easy one to grasp for students, because, mentally it &amp;quot;&lt;em&gt;equates&lt;/em&gt;&amp;quot; (qualitatively) &lt;span class=&quot;math-container&quot;&gt;$C\epsilon$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;, and this hits at the heart of what is truly difficult to understand, the consequences of the very small (and of the very large, analogously).&lt;/p&gt;&#xA;&lt;p&gt;So I am not sure that this proposed principle would be helpful upfront as a tool to have. &lt;em&gt;Realizing that this is so&lt;/em&gt; eventually, I believe would be a great mental satisfaction and intuitive milestone for the students.&lt;/p&gt;&#xA;" OwnerUserId="689" LastActivityDate="2023-10-31T00:37:23.930" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27064" PostTypeId="1" CreationDate="2023-10-31T13:05:09.427" Score="3" ViewCount="74" Body="&lt;p&gt;Are there examples of financial institutions that compound their interest more frequently than once-a-month? Are there examples of financial institutions that consider continually compounded interest in practice?&lt;/p&gt;&#xA;" OwnerUserId="5220" LastActivityDate="2023-10-31T13:05:09.427" Title="Examples of Financial Institutions that Compute Interest Atypically?" Tags="|algebra|examples|precalculus|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27065" PostTypeId="1" AcceptedAnswerId="27074" CreationDate="2023-10-31T14:18:38.393" Score="5" ViewCount="354" Body="&lt;p&gt;Back in 2018, I wrote a &lt;a href=&quot;https://www.justinmath.com/graphing-rational-functions-with-slant-and-polynomial-asymptotes/&quot; rel=&quot;nofollow noreferrer&quot;&gt;post&lt;/a&gt; about asymptotes of rational functions in which I addressed not only horizontal and slant/oblique asymptotes, but also the general case of &amp;quot;polynomial asymptotes.&amp;quot; Fast-forward over 5 years later, the post gets plenty of traffic from the query &amp;quot;polynomial asymptote&amp;quot; because apparently polynomial asymptotes are not covered anywhere else! (Below is a snippet of the post that describes what I mean by &amp;quot;polynomial asymptotes.&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Yc80F.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Yc80F.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It strikes me as really weird that polynomial asymptotes are not covered elsewhere. They are a natural generalization of slant asymptotes, they help develop intuition about why graphs of rational functions look like they do, and they don't require any more advanced computation techniques –- it's still just polynomial long division (or even synthetic division if your divisor is linear).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;1. Why are polynomial asymptotes not normally covered along by resources that teach slant asymptotes?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;2. I can't find the phrase &amp;quot;polynomial asymptote&amp;quot; elsewhere on the web. I tried searching images too and don't see any other images of graphs illustrating this concept. Is the phrase &amp;quot;polynomial asymptote&amp;quot; truly nonexistent outside of this post? If so, why? If not, what other resource(s) mention it?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;My intention with these questions is less about critiquing curriculum and more about just trying to make sense of a phenomenon that strikes me as really weird.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Falsified Hypotheses&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Below are some explanations that sounded initially sounded promising, but upon deeper analysis, turned out not to be true.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Maybe it's because people just refer to &amp;quot;slant asymptotes&amp;quot; in the most general sense, not just lines but also polynomials of degree greater than 1.&lt;/em&gt; False. Resources that talk about slant asymptotes generally say that &amp;quot;a slant asymptote occurs when the degree of the numerator is 1 more than the degree of the denominator.&amp;quot; So I don't think that's the case.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Maybe it's because polynomial asymptotes are just one special case in &lt;a href=&quot;https://en.wikipedia.org/wiki/Asymptotic_analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;general asymptotic analysis&lt;/a&gt;.&lt;/em&gt; False. In the context of rational functions, polynomial asymptotes fully describe the general case -- they're not just a special case. And by this reasoning, it should also be uncommon to address slant/oblique asymptotes, which are actually special cases.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Non-Falsified but Non-Definitive Hypotheses&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Below are explanations that sound promising and have not been falsified, but have not accumulated enough evidence to be considered definitively &amp;quot;the answer.&amp;quot;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;It's less useful in practice than it is in theory. In practice, sketching a general polynomial is not that much easier than sketching a general rational function. And often we only care about behavior for large x, which we can easily determine by just dividing the highest-degree term of the numerator by the highest-degree term of the denominator.&lt;/em&gt; True, but not definitive because educational resources often cover other things that are less useful in practice than they are in theory.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt; The accepted answer makes the above hypothesis definitive by pointing out that there's actually not much theory that builds on polynomial asymptotes, whereas there is more theory that builds on related topics of little-o and big-O notation. So if you start with polynomial asymptotes and follow that line of thought deeper, you end up at little-o and big-O notation, for which polynomial asymptotes are not actually a prerequisite.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-01T22:52:15.413" LastActivityDate="2023-11-09T00:32:08.127" Title="Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?" Tags="|secondary-education|mathematical-pedagogy|terminology|precalculus|functions|" AnswerCount="4" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="27066" PostTypeId="2" ParentId="27026" CreationDate="2023-10-31T16:20:24.823" Score="2" Body="&lt;p&gt;The International Mathematical Olympiad's curriculum is designed in such a way that a 12th-grade student would not have an advantage over a 9th-grade student. Therefore, topics such as higher algebra and calculus are avoided. This is opposed to international olympiads at other subjects such as Chemistry or Physics, where a participant is supposed to learn topics that are at university level.&lt;/p&gt;&#xA;&lt;p&gt;The cognitive activation is certainly very high in all the above mentioned competitions. While the cognitive load is also high, it is certainly lower in the case of Mathematics, compared to Physics or Chemistry, due to the choice of the curriculum.&lt;/p&gt;&#xA;" OwnerUserId="23126" LastActivityDate="2023-10-31T16:20:24.823" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27067" PostTypeId="2" ParentId="26878" CreationDate="2023-10-31T17:49:42.753" Score="2" Body="&lt;p&gt;It seems to me that your students have not performed too much matrix multiplication by hand (using pen and paper). Perhaps it is time for some computer assisted algebraic operations.&lt;/p&gt;&#xA;&lt;p&gt;Also, the fact that students mix so easily different types such as scalar, vector, matrix, made me think of an analogy to &lt;strong&gt;data types&lt;/strong&gt; in programming languages. In a programming language, one cannot mix different data types, unless these types are somehow compatible.&lt;/p&gt;&#xA;&lt;p&gt;This leads to the suggestion of having some practical sessions on matrix multiplication in a programming language. (Certainly, your students must have come across at least one programming language, given their technical background)&lt;/p&gt;&#xA;&lt;p&gt;The advantage of a programming environment is that &lt;strong&gt;you get an error&lt;/strong&gt; when performing some illegal operation, such as multiplying matrices of incompatible dimensions, or the division 1/A, where A is a matrix.  Basically, it is the software that tells you when you are doing something wrong.&lt;/p&gt;&#xA;&lt;p&gt;A second advantage is that, after declaring a variable of some type, you can &lt;strong&gt;display&lt;/strong&gt; it (print it to the console),  thus allowing students to notice that a matrix is a totally different thing from a scalar.&lt;/p&gt;&#xA;&lt;p&gt;Most importantly, you don't even need to go deep into the language syntax. In fact, you barely need the basics of a language in order to perform these algebraic operations.&lt;/p&gt;&#xA;&lt;p&gt;A demo with Julia:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/YxR6V.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YxR6V.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6bx23.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6bx23.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DK916.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DK916.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="23126" LastActivityDate="2023-10-31T17:49:42.753" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27068" PostTypeId="2" ParentId="5614" CreationDate="2023-10-31T18:54:04.400" Score="0" Body="&lt;p&gt;If you think about y and f(x) as two operate entities then it makes it much easier to understand. When y = f(x+2), the value of y is the value of f(x) in two places time- in the equation y = f(x+2), the y value steals the y = f(x) value from two x units in front, hence where the (x+2) comes from. As all of the y values are two units behind their f(x) position because they have stolen the original f(x) value from two places ahead, it causes the graph to shift two units backwards.&#xA;For y = f(2x), the y value steals the y = f(x) value from double the space away, e.g. if x = 3, then y = f(2x) will steal the y = f(x) value from x = 6. This causes the graph to shrink&lt;/p&gt;&#xA;" OwnerUserId="23292" LastActivityDate="2023-10-31T18:54:04.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27069" PostTypeId="1" CreationDate="2023-11-01T04:06:44.157" Score="1" ViewCount="139" Body="&lt;p&gt;I'm teaching Grade 9 Probability. We need to compare Experiment Probabilities (Relative Frequencies) with Theoretical Probabilities for Simple and Compound Events.&#xA;I want to come up with and Online tool to generate such data for compound events made from a variety of simple events. Eg the result of flipping a coin and rolling a dice 10 times. Other simple events could be spinners  or drawing balls.&#xA;It is assumed that the two events are independent and done with replacement where necessary (eg drawing 1 cards and returning it and then drawing another card).&lt;/p&gt;&#xA;&lt;p&gt;I would expect sequences like (H,1),(H,4),(T,1),(H,6),...&lt;/p&gt;&#xA;&lt;p&gt;After searching online I've found the following online tools for generating random selections:&#xA;&lt;a href=&quot;http://www.shodor.org/interactivate/activities/ExpProbability/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.shodor.org/interactivate/activities/ExpProbability/&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.mathsisfun.com/data/spinner.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathsisfun.com/data/spinner.php&lt;/a&gt;,&#xA;&lt;a href=&quot;https://wheelofnames.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://wheelofnames.com/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The problem is these tools only seem to allow simple events (eg a spinner Red) or sometimes particular compound events (eg two dice (1,3) or multiple flips of a coin (H,T) ).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Does anyone know of any online tools to generate multiple random selections of compound events?&lt;/strong&gt; I'm after something which runs in the browser with a very accessible interface. It would be great to allow them to drag and drop icons representing the elements.&lt;/p&gt;&#xA;" OwnerUserId="998" LastEditorUserId="998" LastEditDate="2023-11-01T05:55:37.353" LastActivityDate="2023-11-01T14:32:50.223" Title="Online Probability Simulation for Compound Events" Tags="|probability|online|simulation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27070" PostTypeId="2" ParentId="27069" CreationDate="2023-11-01T05:33:33.833" Score="1" Body="&lt;p&gt;It's really easy to write a script to do this, so I wrote one for you that runs entirely in the browser and allows you to edit the sample space elements: &lt;a href=&quot;https://www.pythonmorsels.com/p/24dps/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.pythonmorsels.com/p/24dps/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Just go to that link, click &amp;quot;Run&amp;quot;, and it will spit out a sequence just as you described. If you want to change the sample spaces of the events or the number of samples, you can just edit those fields in the code and click &amp;quot;Run&amp;quot; again.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1J20Z.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1J20Z.png&quot; alt=&quot;compound event tool&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I know this is not &amp;quot;drag-and-drop&amp;quot; as would be ideal but I would be surprised if 9th graders had trouble doing something as simple as editing a list of elements and clicking a big green &amp;quot;Run&amp;quot; button.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-01T14:32:50.223" LastActivityDate="2023-11-01T14:32:50.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27071" PostTypeId="2" ParentId="27069" CreationDate="2023-11-01T07:08:04.240" Score="0" Body="&lt;p&gt;Here is an example in Mathematica:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;In[7]:= RandomChoice[Tuples[{Range[6], {H, T}}], 10]&#xA;&#xA;Out[7]= {{5, T}, {3, T}, {4, H}, {6, T}, {3, T}, {4, H}, {1, H}, {6, H}, {5, H}, {3, T}}&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;The sample spaces of the simple events can easily be swapped out.&lt;/p&gt;&#xA;&lt;p&gt;Though this is not online nor accessible to the average grade 9 student!&lt;/p&gt;&#xA;" OwnerUserId="998" LastActivityDate="2023-11-01T07:08:04.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27072" PostTypeId="2" ParentId="27065" CreationDate="2023-11-01T16:24:41.233" Score="5" Body="&lt;p&gt;This is kind of a joke answer, but in my &lt;a href=&quot;https://groups.google.com/g/sci.physics.research/c/aiMUJrOjE8A/m/jGy2N3IaajwJ&quot; rel=&quot;noreferrer&quot;&gt;favorite math story ever&lt;/a&gt; we have the following exchange:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Eric pondered a moment. &amp;quot;But... but if it's &lt;em&gt;that&lt;/em&gt; simple, why&#xA;don't my textbooks talk about it?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Grinning evilly, the Wiz replied: &amp;quot;They certainly drop the&#xA;necessary clues. As for why they don't &lt;em&gt;emphasize&lt;/em&gt; this stuff,&#xA;well, this is just one of those tricks we Wizards use to distinguish&#xA;the people who think for themselves from those who fall for any&#xA;plausible line of claptrap. In fact, every time Halloween falls&#xA;on a full moon - like this year - we get together and agree on&#xA;what facts like this we will keep secret, precisely to see who&#xA;rediscovers it for themselves. This year we....&amp;quot; At this point&#xA;the absent-minded Wiz caught himself. &amp;quot;Whoops! This is too&#xA;secret for you! Anyway, you just failed one of these tests.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I certainly enjoyed discovering polynomial asymptotes as a high school student, and it looks like several people in the comments to the OP did as well.&lt;/p&gt;&#xA;&lt;p&gt;Maybe it has not made its way into the standard curriculum because it is not essential to any further coursework, but is &amp;quot;low hanging&amp;quot; enough that talented students get to discover it for themselves.  The marginal utility of introducing it to the curriculum is negative since:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Students who don't care about math will not be excited by one more thing to memorize, so less utils for them.&lt;/li&gt;&#xA;&lt;li&gt;Students who do care about math will get less utils out of it when presented as &amp;quot;known math&amp;quot; vs. if they discover it for themselves.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-11-01T16:24:41.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27073" PostTypeId="2" ParentId="27065" CreationDate="2023-11-01T17:08:48.487" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;For issue 1, I suspect that it is just an issue of time benefit, like we could think of many enrichments.   I don't think it is an issue of nobody thought of it. Probably also contributing are the issues that the solutions are not a simple generalization of the line assymptote finding.&lt;/p&gt;&#xA;&lt;p&gt;It may not be exactly be in analytic geometry class, but I do think students look at this behavior in calculus and natural sciences. So, the limit of x plus e to the x is e to the x for instance, as x grows large. Also in ODEs there is some looking at solutions that we can't express analytically but that we can bound or show approaching curves.&lt;/p&gt;&#xA;&lt;p&gt;Possibly another issue is drawing the parabola or what have you that is the limit curve. The whole point of curve sketching is to abstract a few features, intercepts snd assymptotes snd symmetry snd the like, that allow sketching a curve without just plotting z gazillion points. But if your bounding limit is a parabola, to use it well, you'd need to have a good sketch of the parabola itself.  Very different than physically drawing a dashed line on graph paper with a straight edge.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm not so sure about the term  polynomial assymptote being appropriate. If you look at the wiki article on assymptores they talk about curve assymptotes. And you think of hyperbolic or exponentials or trig functions or the like that we might approach with given expressions.  Anyhow, I would loosen up the search terms and see what you get.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P.s.  Just sharing a few thoughts, not a definitive answer, so mods feel free to commentize, although really the question is multifaceted.  And honestly I chafe at the whole idea of lock and key answers and a semantic web. Think forum style discussion is a feature not a bug.&lt;/p&gt;&#xA;" OwnerUserId="23303" LastEditorUserId="8931" LastEditDate="2023-11-05T14:21:30.237" LastActivityDate="2023-11-05T14:21:30.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27074" PostTypeId="2" ParentId="27065" CreationDate="2023-11-01T22:30:12.060" Score="3" Body="&lt;p&gt;I think it is just that the related concepts encompassed by little-&lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt; and big-&lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; notation are more important than &amp;quot;polynomial asymptotes&amp;quot; and do find many applications. We do teach Landau notation in analysis, computer science, etc.&lt;/p&gt;&#xA;&lt;p&gt;When we talk about asymptotics, we typically reference just the dominant term. For examples, if a function is &lt;span class=&quot;math-container&quot;&gt;$O(n^2+n)$&lt;/span&gt; then it is &lt;span class=&quot;math-container&quot;&gt;$O(n^2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I remember when I first taught college algebra after my PhD. I don't think I had ever encountered slant asymptotes in graduate studies, and I embarrassed myself in front of that class many years ago by saying that if &lt;span class=&quot;math-container&quot;&gt;$y=x+1$&lt;/span&gt; is a slant asymptote then saying that &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt; is also a slant asymptote is equally correct. But of course I was confusing the statement that &lt;span class=&quot;math-container&quot;&gt;$O(x+1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$O(x)$&lt;/span&gt; are the same with the more geometrical notion of a unique slant asymptotes.&lt;/p&gt;&#xA;&lt;p&gt;It also occurs to me that in algebra, meaning the world of polynomials and rational functions, the notion of a polynomial asymptote is not very helpful since it says nothing about polynomials. It does have some teeth with rational functions, but then with rational functions we typically focus only on &amp;quot;end behavior.&amp;quot; For example, we look at &lt;span class=&quot;math-container&quot;&gt;$y=\frac{x^2+1}{x+1}$&lt;/span&gt; and see that it behaves like &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt; at infinity, and don't bother with the refinement that it behaves more like &lt;span class=&quot;math-container&quot;&gt;$x-1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-11-01T22:30:12.060" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27075" PostTypeId="2" ParentId="27009" CreationDate="2023-11-02T15:30:48.283" Score="3" Body="&lt;p&gt;My feeling is that it is very difficult to predict what the effect on proficiency will be. Unfortunately, I am not very knowledgeable about this particular issue, but I pointed out some relevant background in the comments, which have by now grow overly long and may disappear at some point, so I think it worth summarizing what was said there, as well as some other things I ran across in my reading but refrained from mentioning earlier. If anyone with actual expertise has corrections, or knows of additional references, please let me know. For now, I am just recording what I have learned; I will come back to add references, links, and quotations as I find the time.&lt;/p&gt;&#xA;&lt;p&gt;In most US states it was for a long time traditional not to require exit exams for high school graduation. Diplomas were generally awarded for having attended grades 9-12 and having passed a certain number of courses in each of several specified subject areas. A few states, such as New York, did give exit exams, but prior to the last couple of decades, these had been used to confer extra distinction and were not a general requirement.&lt;/p&gt;&#xA;&lt;p&gt;Starting in the 1970s there was a movement to introduce such exams, which reached a high point in the mid-1990s, at which point 27 states had done so (or planned to). Subsequently, exit exams have fallen out of favor and only eight states still require them (according to the anti-test advocacy site FairTest). This is down from 11 states last year. (Maryland, New Mexico, and Mississippi dropped their test requirement in 2022.) Some states have retroactively awarded diplomas to students who were denied them due to failure on exit exams. Interestingly, this period of declining exit test requirements has coincided with the No Child Left Behind/Race to the Top/Every Student Succeeds era, which is known for its standardized exam mandates, set by the federal government (and for the ensuing, so far ineffectual, anti-test backlash). The primary rationale for the NCLB tests, however, was for evaluating schools and teachers, not for determining student eligibility to graduate. At certain times and places, some tests have been used for both purposes.&lt;/p&gt;&#xA;&lt;p&gt;For some reason FairTest does not list Oregon either as a state that requires exit exams or as a state that has recently dropped exit exam requirements. I'm not sure why this is. It may be because the Oregon policy allowed students to meet the requirement in other ways, such as through work samples. One state legislator quoted in a couple of news articles expresses the belief the work sample requirement was the original intent of the policy, and that the way it has worked out in practice, with the majority of students fulfilling the requirement through standardized test scores, does not adhere to the real-world experience aspect originally envisioned. Other sources, however, indicate that work samples have been used as a way of making sure those students who failed their 11th grade exams are still able to graduate, by having them enroll in a special course during grade 12 in which the work samples were prepared. One principal states that no student in his school was ever denied a diploma for failing the 11th grade exams. (It is, of course, possible that some students dropped out of school after failing the exam in 11th grade and did not return to complete the work sample requirement in 12th grade.)&lt;/p&gt;&#xA;&lt;p&gt;Oregon's essential skills requirement was passed by the legislature in 2009 and phased in during the years 2012-14, with mathematics being the last subject to be implemented. The policy was suspended during the pandemic. During this halt, a review of the policy and other educational policies was commissioned, and the recommendations of the committee were published in 2022. One of the recommendations was to discontinue the essential skills requirement. In October 2023 the State Board of Education voted to suspend the requirement for a further five years, until 2028.&lt;/p&gt;&#xA;&lt;p&gt;One finding of the committee that the board used to support its decision was that the introduction of the essential skills requirement did not result in a significant change in the level of preparedness of high school graduates enrolling in community college or a four-year university. (About 62% of American high school graduates now do so.) A critical article in the magazine &lt;em&gt;Reason&lt;/em&gt; points out that this analysis may have missed a positive effect for those students not enrolling in post-secondary education. Another factor, mentioned by the committee, is that the required level needed to pass was not set every high, and may have been too low to expect it to have much of an effect on the skills of students enrolling in higher education.&lt;/p&gt;&#xA;&lt;p&gt;In summary, the main effect of the essential skills requirement would seem to be expected among those students not planning to pursue higher education, and, in particular, on those with the very lowest exam scores. The arguments in favor of the requirement are that it would allow employers to more easily distinguish between high school attendees who had attained a certain level of achievement from those who had not, and that it would force schools to give extra instruction to those 12th grade students who had failed their 11th year exams. The argument against is that the requirement seems to have had very little effect on preparedness for higher education, but that it could have a lifelong detrimental effect on students denied a diploma because of a low exam score by making it more difficult for them to find employment.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2023-11-02T15:30:48.283" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27077" PostTypeId="2" ParentId="27009" CreationDate="2023-11-02T16:03:01.557" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Question concerning constitutionality: The reasoning behind the suppression of the requirements can be summarized as follows: since minorities will fail such tests, the tests are unfair and therefore it is preferable that everybody should fail (in the sense of the diplomas becoming a participation prize). This is consistent with the spirit of Affirmative Action, but wouldn't this be at odds with recent rulings of the Supreme Court as to the unconstitutionality of Affirmative Action?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think that what you are looking for is not things related to Affirmative Action, but rather &amp;quot;disparate impact&amp;quot;. I am not a lawyer and what follows is not legal advice.&lt;/p&gt;&#xA;&lt;p&gt;Here is a link to the US Justice department's guidance on the topic:&#xA;&lt;a href=&quot;https://www.justice.gov/crt/fcs/T6Manual7&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.justice.gov/crt/fcs/T6Manual7&lt;/a&gt; It is far too long to summarize beyond the very basics here.&lt;/p&gt;&#xA;&lt;p&gt;In order to go forward with a disparate impact claim, you have to do things like show damages, show that they are centered at or around a particular group, show causation, and show that there isn't a substantial legitimate justification for the policy and that there is a less discriminatory alternative.&lt;/p&gt;&#xA;&lt;p&gt;One could probably argue that unnecessarily tough or poorly targeted basic skills tests might fail these tests and be deemed unconstitutional. It all depends on the court. That said, this doesn't appear to be a court case, but rather a preemptive attempt to deal with disparate outcomes. I'll leave it open as to whether this is a good way to do that.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2023-11-02T16:03:01.557" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27078" PostTypeId="1" CreationDate="2023-11-02T21:09:05.210" Score="10" ViewCount="1182" Body="&lt;p&gt;This is a close cousin of the &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5614/how-does-one-explain-that-transformations-inside-a-function-operate-in-the-opp&quot;&gt;previous question asked here&lt;/a&gt; about transformations inside and outside a function and how they switch things around. I think some of the perspectives there will help here, but I'd also be interested in something specific to this issue.&lt;/p&gt;&#xA;&lt;p&gt;Consider a question like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Graph &lt;span class=&quot;math-container&quot;&gt;$y=\sqrt{2x-6}$&lt;/span&gt; by starting with the graph of &lt;span class=&quot;math-container&quot;&gt;$y=\sqrt{x}$&lt;/span&gt; and applying a series of transformations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For many of my students, their thought process goes like &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$2x-6$&lt;/span&gt; means 'multiply by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; then add &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;', so my transformations should go&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{x} \rightarrow \sqrt{2x} \rightarrow \sqrt{2x-6}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;meaning I should start with my graph of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt;, shrink it horizontally by a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, and then to the right by &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;While I can explain to them where their second transformation goes wrong, I have a harder time helping them understand how to figure out proper ordering of steps.  Whether you write things as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{x} \rightarrow \sqrt{2x} \rightarrow \sqrt{2(x-3)}$$&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{x} \rightarrow \sqrt{x-6} \rightarrow \sqrt{2x-6},$$&lt;/span&gt;&#xA;either way goes in the reverse of the usual order of operations.&lt;/p&gt;&#xA;&lt;p&gt;Is there a good intuitive explanation for how to think about the order of steps when constructing these compound transformations?&lt;/p&gt;&#xA;" OwnerUserId="4832" LastActivityDate="2023-11-09T18:17:55.017" Title="Intuition for order of operations in compound transformations" Tags="|algebra|precalculus|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27079" PostTypeId="2" ParentId="27078" CreationDate="2023-11-02T22:40:36.190" Score="4" Body="&lt;p&gt;One way to build intuition around this is to think about the effects on the intercepts.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Vertical Shifts/Stretches: Effect on &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-Intercept&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Suppose you want to graph &lt;span class=&quot;math-container&quot;&gt;$y=2\sqrt{x}-6$&lt;/span&gt; using transformations.&lt;/p&gt;&#xA;&lt;p&gt;You know the result needs to have a &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept of &lt;span class=&quot;math-container&quot;&gt;$-6$&lt;/span&gt; (which comes from evaluating &lt;span class=&quot;math-container&quot;&gt;$2\sqrt{0}-6$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If you stretch vertically by a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and then shift &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; down, you get a &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept of &lt;span class=&quot;math-container&quot;&gt;$-6$&lt;/span&gt; as desired. &lt;span class=&quot;math-container&quot;&gt;$\color{green}\checkmark$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you shift &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; down and then stretch vertically by a factor of &lt;span class=&quot;math-container&quot;&gt;$2,$&lt;/span&gt; you get a &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-intercept of &lt;span class=&quot;math-container&quot;&gt;$-12$&lt;/span&gt; which is incorrect. &lt;span class=&quot;math-container&quot;&gt;$\color{red}\times$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So vertical stretches come before vertical shifts.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Horizontal Shifts/Shrinks: Effect on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-Intercept&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Suppose you want to graph &lt;span class=&quot;math-container&quot;&gt;$y=\sqrt{2x-6}$&lt;/span&gt; using transformations.&lt;/p&gt;&#xA;&lt;p&gt;You know the result needs to have an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-intercept of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; (which comes from solving &lt;span class=&quot;math-container&quot;&gt;$0=\sqrt{2x-6}$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If you shrink horizontally by a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and then shift &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; right, you get an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-intercept of &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; which is incorrect. &lt;span class=&quot;math-container&quot;&gt;$\color{red}\times$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you shift &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; right and then shrink horizontally by a factor of &lt;span class=&quot;math-container&quot;&gt;$2,$&lt;/span&gt; you get an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-intercept of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; as desired. &lt;span class=&quot;math-container&quot;&gt;$\color{green}\checkmark$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So horizontal shifts come before horizontal shrinks.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;More Hand-Wavy (But Potentially More Memorable) Intuition&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;When it comes to horizontal transformations, everything works the opposite way as vertical transformations.&lt;/p&gt;&#xA;&lt;p&gt;The transformations themselves are opposite:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;shifts right (the negative horizontal direction) instead of shifts up (the positive vertical direction)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;horizontal shrinks instead of vertical stretches&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So it's kind of intuitive that the order of transformations is also opposite.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-03T04:56:54.043" LastActivityDate="2023-11-03T04:56:54.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27080" PostTypeId="2" ParentId="27078" CreationDate="2023-11-03T02:45:33.550" Score="9" Body="&lt;p&gt;For teaching this concept, I always come back to tables of values. In Calculus Reform, there was the &amp;quot;Rule of Three&amp;quot; which emphasized that learning about functions should involve tables, graphs, and algebraic formulas. We should always be able to see a function concept from all three angles. In teaching, we tend to skimp on the &amp;quot;table&amp;quot; perspective.&lt;/p&gt;&#xA;&lt;p&gt;Scaffold this for your students. Let &lt;span class=&quot;math-container&quot;&gt;$f(x)=\sqrt{x}$&lt;/span&gt; with domain &lt;span class=&quot;math-container&quot;&gt;$0\le x\le 4$&lt;/span&gt;. Draw the square root curve with points &lt;span class=&quot;math-container&quot;&gt;$(0,0),\ (1,1),\ (4,2)$&lt;/span&gt; emphasized. Notice the graph ends at &lt;span class=&quot;math-container&quot;&gt;$(4,2)$&lt;/span&gt;!&lt;/p&gt;&#xA;&lt;p&gt;Next make a table of values for &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; like this&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{c|c}&#xA;x &amp;amp; f(x)=\sqrt{x} \\&#xA;\hline&#xA;0 &amp;amp; 0 \\&#xA;1 &amp;amp; 1 \\&#xA;4 &amp;amp; 2 \\&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The next step is a table of values for &lt;span class=&quot;math-container&quot;&gt;$g(x)=f(2x)=\sqrt{2x}$&lt;/span&gt;. A learning moment ensues while we thrash around, trying to figure out the domain for &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;. It is &lt;span class=&quot;math-container&quot;&gt;$0\le x\le 2$&lt;/span&gt;.  We reward ourselves with the template for our next table:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{c|c}&#xA;x &amp;amp; y=\sqrt{2x} \\&#xA;\hline&#xA;0 &amp;amp;  \\&#xA;0.5 &amp;amp;  \\&#xA;2 &amp;amp;  \\&#xA;\end{array}&lt;/span&gt;&#xA;Filling in the second column feels &amp;quot;just right&amp;quot; because it links so nicely with the first table.&lt;/p&gt;&#xA;&lt;p&gt;Next, we turn to &lt;span class=&quot;math-container&quot;&gt;$h(x)=f(2x-6)$&lt;/span&gt;. Another learning moment ensues as we thrash around to figure out the domain for &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;. It is &lt;span class=&quot;math-container&quot;&gt;$3\le x\le5$&lt;/span&gt;. We get rewarded again with the template for our next table:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{c|c}&#xA;x &amp;amp; y=\sqrt{2x-6} \\&#xA;\hline&#xA;3 &amp;amp;  \\&#xA;3.5 &amp;amp;  \\&#xA;5 &amp;amp;  \\&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Filling in the second column is again satisfying, because it just clicks. &lt;strong&gt;But&lt;/strong&gt; it does not link very nicely with the previous table.&lt;/p&gt;&#xA;&lt;p&gt;So now we regroup and take a tricky turn and try to relate the column for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in the second table to the column of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for the third table. Upon reflection, we see the differences of 3. Point out how it might have been slicker to have used something else for our second table, namely a table for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x-3}$&lt;/span&gt; instead:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{c|c}&#xA;x &amp;amp; y=\sqrt{x-3} \\&#xA;\hline&#xA;3 &amp;amp;  \\&#xA;4 &amp;amp;  \\&#xA;7 &amp;amp;  \\&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;then followed with our third table (already existing!) for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2(x-3)}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The moral of the discovery is that the insightful sequence is &lt;span class=&quot;math-container&quot;&gt;$x\to (x-3)\to 2(x-3)$&lt;/span&gt;, as opposed to the first sequence that started off as &lt;span class=&quot;math-container&quot;&gt;$x\to 2x\to\  .$&lt;/span&gt; So the function concept from the &lt;em&gt;graphical&lt;/em&gt; perspective (Rule of Three) is to first shift the graph by 3, then compress.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-11-03T16:48:00.143" LastActivityDate="2023-11-03T16:48:00.143" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27081" PostTypeId="2" ParentId="27078" CreationDate="2023-11-03T11:15:24.137" Score="1" Body="&lt;p&gt;You can perform the transformations in any order you want as long as you respect this rule:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Always transform completely inside, affecting &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; directly, or completely outside, affecting &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; directly.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If you're only teaching about horizontal stretches and shifts and don't care about vertical stretches and shifts yet, then you can discard the second part, and keep only this one rule:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Always transform completely inside, affecting &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; directly.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As long as the students respect this rule, you don't have to teach them a procedure to deduce a possibly-counterintuitive correct order of operations. Just let them work on examples and make sure they respect this one rule.&lt;/p&gt;&#xA;&lt;p&gt;For instance, to go from &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt{2x - 6}$&lt;/span&gt;, if a student tries to replace &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; first, then they cannot replace &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2x-6$&lt;/span&gt; next, because that's not a known transformation. The only thing they're allowed to do is replace &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x-...$&lt;/span&gt; or with &lt;span class=&quot;math-container&quot;&gt;$...\times x$&lt;/span&gt;. Replacing &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$2x-6$&lt;/span&gt; is not part of the transformations we have learned.&lt;/p&gt;&#xA;&lt;p&gt;So the student can either do:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y=f(x) \quad\to\quad y=f(2x) \quad\to\quad y=f(2(x-3))$$&lt;/span&gt;&#xA;or:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y=f(x) \quad\to\quad y=f(x-6) \quad\to\quad y=f(2x-6)$$&lt;/span&gt;&#xA;but this doesn't work:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y=f(x) \quad\to\quad y=f(2x) \quad\not\to\quad y=f(2x-6)$$&lt;/span&gt;&#xA;This is not because of a rule about order of operations, it's because of the one rule about only ever transforming &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; directly and never transforming &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;. If you try to do the &lt;span class=&quot;math-container&quot;&gt;$2\times...$&lt;/span&gt; first, then you cannot do the &lt;span class=&quot;math-container&quot;&gt;$...-6$&lt;/span&gt; because the &lt;span class=&quot;math-container&quot;&gt;$...-6$&lt;/span&gt; is not applied to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, it's applied to &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Note that if the expression of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; contains more than one occurrence of symbol &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, you must apply the transformation to every occurrence of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. For instance, to shift the graph of &lt;span class=&quot;math-container&quot;&gt;$y=x(2x+1)$&lt;/span&gt; horizontally by 5 right you can do:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y=x(2x+1) \quad\to\quad y=(x-5)(2(x-5)+1)$$&lt;/span&gt;&#xA;or:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$y=2x^2 + x \quad\to\quad y = 2(x-5)^2 + (x-5)$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="16159" LastEditorUserId="16159" LastEditDate="2023-11-03T11:27:55.507" LastActivityDate="2023-11-03T11:27:55.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27082" PostTypeId="2" ParentId="2338" CreationDate="2023-11-03T12:01:45.263" Score="1" Body="&lt;p&gt;I may provide yet a third detailed answer to this question, but I am not ready yet. However, this remark is too long for a comment.&lt;/p&gt;&#xA;&lt;p&gt;Historically, there was indeed a notion of &amp;quot;indefinite integral&amp;quot; used, for example, by Cauchy. At the time, the notions of &amp;quot;functions&amp;quot; and &amp;quot;variables&amp;quot; were unlike what they are now.&lt;/p&gt;&#xA;&lt;p&gt;I believe people are trying to include the notion of &amp;quot;indefinite integral&amp;quot; in the sense of Cauchy in modern textbooks. They invent interpretations like &amp;quot;an unknown primitive&amp;quot; or &amp;quot;the set of all primitives.&amp;quot; But I believe to understand that for Cauchy the indefinite integral &lt;span class=&quot;math-container&quot;&gt;$\int f(x)\,dx$&lt;/span&gt; was something like a &lt;em&gt;variable that depends on the variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, but the dependence is only determined up to an additive constant.&lt;/em&gt; So the identity of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int f(x)\,dx = F(x) + C&#xA;$$&lt;/span&gt;&#xA;was making sense.&lt;/p&gt;&#xA;&lt;p&gt;Of course, if &lt;span class=&quot;math-container&quot;&gt;$\int f(x)\,dx$&lt;/span&gt; is an unknown primitive of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;?) or the set of all primitives, this identity is not even false, it is absurd (there is a &amp;quot;type error&amp;quot; in it).&lt;/p&gt;&#xA;&lt;p&gt;Good luck formalizing this in the modern mathematics!&lt;/p&gt;&#xA;" OwnerUserId="6987" LastEditorUserId="6987" LastEditDate="2023-11-03T12:16:58.880" LastActivityDate="2023-11-03T12:16:58.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27083" PostTypeId="2" ParentId="2338" CreationDate="2023-11-03T12:28:07.453" Score="1" Body="&lt;p&gt;Question: &lt;em&gt;What is the function, having &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt; as its derivate?&lt;/em&gt;&lt;br /&gt;&#xA;Answer: &lt;em&gt;There are many of them. Here they are (some of them):&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VBFfY.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VBFfY.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Question: &lt;em&gt;So many? But there are resemblances.&lt;/em&gt;&lt;br /&gt;&#xA;Answer: &lt;em&gt;Indeed: they must all be (possible) vertical shifts of &lt;span class=&quot;math-container&quot;&gt;$-\cos(x)$&lt;/span&gt;, because taking the derivative will remove that vertical shift. Obviously, this vertical shift gives you the opportunity to choose a particular one, e.g. if you want to choose one for which &lt;span class=&quot;math-container&quot;&gt;$D^{-1}(\sin)(x)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; being &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; (I have used &lt;span class=&quot;math-container&quot;&gt;$D^{-1}$&lt;/span&gt; as the anti-derivative operator). As you can see on the drawing, this happens when &lt;span class=&quot;math-container&quot;&gt;$k=1$&lt;/span&gt; (the blue one).&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-11-03T12:28:07.453" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27084" PostTypeId="2" ParentId="27078" CreationDate="2023-11-03T14:44:40.693" Score="3" Body="&lt;p&gt;I'd say the root cause of your problem is that you're forgetting to include the square root in the sequence of operations transforming &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2x-6}$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x&#xA;\overset{\times 2}{\ \to\ } 2x&#xA;\overset{-6}{\ \to\ } 2x-6&#xA;\overset{\sqrt\cdot}{\ \to\ } \sqrt{2x-6}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Having written out the &lt;em&gt;entire&lt;/em&gt; chain of transformations like this, you can then obtain the graph of the compound transformation by starting with the trivial graph of &lt;span class=&quot;math-container&quot;&gt;$y = x$&lt;/span&gt; and going &lt;em&gt;backwards&lt;/em&gt; along this chain, &lt;em&gt;undoing&lt;/em&gt; each operation one at a time:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Conveniently, we've already been given the graph of &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt x$&lt;/span&gt;, so that's the last step in the chain already taken care of.  (Which is nice, because the inverse of taking the square root is squaring, and that's a non-linear transformation that changes the shape of the graph in more complicated ways than just simple shifting and scaling.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Now, the &lt;em&gt;previous&lt;/em&gt; operation in the chain, just before the square root, is the subtraction of &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;.  Thus, to turn the graph of &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt x$&lt;/span&gt; into the graph of &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt{x-6}$&lt;/span&gt;, we need to undo the subtraction by &lt;em&gt;adding&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; to the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-coordinate of each point on the graph, i.e. by moving the entire graph &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; units to the right.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The step &lt;em&gt;before&lt;/em&gt; that (and the first step in the whole chain) is then the multiplication by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.  Thus, to turn the graph of &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt{x-6}$&lt;/span&gt; into the graph of &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt{2x-6}$&lt;/span&gt;, we need to undo the multiplication by halving the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-position of each point on the graph, i.e. shrinking the graph horizontally by a factor of 2 towards the origin.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A pedagogical note that should be emphasized here, in order to turn this method from rote memorization into actual understanding, is that each step in the &lt;em&gt;forwards&lt;/em&gt; chain above corresponds to applying a transformation to the &lt;em&gt;result&lt;/em&gt; of the previous transformations (i.e. to the entire expression).  On the other hand, in the &lt;em&gt;backwards&lt;/em&gt; chain:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sqrt{2x-6} \overset{\times 2}{\ \leftarrow\ }&#xA;\sqrt{x-6} \overset{-6}{\ \leftarrow\ }&#xA;\sqrt{x} \overset{\sqrt\cdot}{\ \leftarrow\ }&#xA;x&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;we're applying each operation to the &lt;em&gt;inputs&lt;/em&gt; of the later transformations, i.e. to each &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in the expression.  And this replacement of each &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in the expression with something else (e.g. &lt;span class=&quot;math-container&quot;&gt;$\sqrt x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x-6$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;) is graphically equivalent scaling the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis of the graph.&lt;/p&gt;&#xA;&lt;p&gt;Another useful way of looking at this is that if we change the expression by replacing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with something else, and &lt;em&gt;still want to get the same values out of it&lt;/em&gt;, we need to adjust the values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; we're feeding into the expression in the opposite way.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;So, for example, if we change &lt;span class=&quot;math-container&quot;&gt;$\sqrt x$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x-6}$&lt;/span&gt; (or any &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$f(x-6)$&lt;/span&gt;), we now need to make each &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; we input into the expression bigger by 6 to compensate, which is equivalent to moving the graph 6 units to the right.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And similarly, when we change &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x-6}$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2x-6}$&lt;/span&gt; (or any &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$f(2x)$&lt;/span&gt;), we now need to make each &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; we input into the expression only half as big to compensate, which is equivalent to horizontally shrinking the graph by a factor of 2 towards the origin.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;There's also a parallelism here between changes to the input and the output of the expression: transforming the input (i.e. replacing each &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in the expression with something else) is equivalent to scaling the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; axis of the graphs, whereas transforming the output (i.e. applying an operation to the value of the entire expression as a while) is equivalent to scaling the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; axis.&lt;/p&gt;&#xA;&lt;p&gt;Indeed, as an advanced exercise / demonstration, you could try obtaining the graph of something like &lt;span class=&quot;math-container&quot;&gt;$y = 3\sqrt{2x - 6} + 1$&lt;/span&gt; starting from the graph of &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt x$&lt;/span&gt;, where you now need to apply transformations both to the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; axis (in &amp;quot;reverse&amp;quot; order) &lt;em&gt;and&lt;/em&gt; to the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; axis (in &amp;quot;forward&amp;quot; order).&lt;/p&gt;&#xA;&lt;p&gt;(Also useful may be giving the students the graph of e.g. &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt{x} - x$&lt;/span&gt; and having them transform it into the graph of e.g. &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt{2x} - 2x$&lt;/span&gt;, just to illustrate the point that the reverse transformations need to be applied to &lt;em&gt;each&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; — and that this trick straight up doesn't work for something like, say, &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt{2x} - 3x$&lt;/span&gt;, at least not without using transformations more complex than just simple orthogonal scaling and shifting).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Also, if your students are already familiar with function notation, you can and should tie this in with function composition.&#xA;For example, in your original example, &lt;span class=&quot;math-container&quot;&gt;$y = \sqrt{2x - 6}$&lt;/span&gt; can be written as &lt;span class=&quot;math-container&quot;&gt;$y = h(g(f(x)))$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g(x) = x - 6$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h(x) = \sqrt x$&lt;/span&gt;, or equivalently drawn as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x&#xA;\overset{f}{\ \to\ } 2x&#xA;\overset{g}{\ \to\ } 2x-6&#xA;\overset{h}{\ \to\ } \sqrt{2x-6}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And then you can point out that going from, say, &lt;span class=&quot;math-container&quot;&gt;$f(x) = 2x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$g(f(x)) = 2x-6$&lt;/span&gt; means applying &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; to the &lt;em&gt;result&lt;/em&gt; of evaluating &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, whereas going from &lt;span class=&quot;math-container&quot;&gt;$h(x) = \sqrt x$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$h(g(x)) = \sqrt{x - 6}$&lt;/span&gt; means applying &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; to the &lt;em&gt;input&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; &lt;em&gt;before&lt;/em&gt; evaluating it.&lt;/p&gt;&#xA;" OwnerUserId="526" LastActivityDate="2023-11-03T14:44:40.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27085" PostTypeId="2" ParentId="27078" CreationDate="2023-11-03T14:50:58.047" Score="2" Body="&lt;p&gt;I think that my &lt;a href=&quot;https://matheducators.stackexchange.com/a/5621/117&quot;&gt;previous answer&lt;/a&gt; addresses this.  If we name the intermediate functions, and make the relationships between these functions explicit, things become a bit easier to understand.&lt;/p&gt;&#xA;&lt;p&gt;In your example you have &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sqrt{2x - 6}$&lt;/span&gt;.  We want to start with &lt;span class=&quot;math-container&quot;&gt;$g_0(x) = \sqrt{x}$&lt;/span&gt; and find an intermediate function &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; so that the graphs of &lt;span class=&quot;math-container&quot;&gt;$g_0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; are related by a basic transformation and the graphs of &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; are related by a basic transformation.&lt;/p&gt;&#xA;&lt;p&gt;If you try what your students suggest then our intermediate function is &lt;span class=&quot;math-container&quot;&gt;$g_1(x) = \sqrt{2x}$&lt;/span&gt;.  We have &lt;span class=&quot;math-container&quot;&gt;$g_1(x) = g_0(2x)$&lt;/span&gt;, so the graph of &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; is &amp;quot;shrunk horizontally&amp;quot; by a factor of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; relative to &lt;span class=&quot;math-container&quot;&gt;$g_0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now we want to try to relate &lt;span class=&quot;math-container&quot;&gt;$g_1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$g_1(x) = \sqrt{2x}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(x) = \sqrt{2x - 6}$&lt;/span&gt;. Your student now wants to use a horizontal translation.  It might be helpful to get this up as a fill in the blank.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;f(x) &amp;amp;= g_1(x - \square )\\&#xA;\sqrt{2x-6} &amp;amp;= \sqrt{2(x - \square)}\\&#xA;\sqrt{2x-6} &amp;amp;= \sqrt{2x - 2\square)}\\&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is now clear that we should be taking &lt;span class=&quot;math-container&quot;&gt;$\square = 3$&lt;/span&gt; not &lt;span class=&quot;math-container&quot;&gt;$\square = 6$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-11-03T14:50:58.047" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27086" PostTypeId="1" CreationDate="2023-11-03T21:32:46.833" Score="6" ViewCount="188" Body="&lt;p&gt;Throughout this year, I have been in charge of a linear algebra course aimed at engineering school. In this course, I asked my students to work on a final project involving finding a winning strategy for the &amp;quot;Lights Out&amp;quot; game - &lt;a href=&quot;https://en.wikipedia.org/wiki/Lights_Out_(game)&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Lights_Out_(game)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This project was very beneficial for them because the following was required:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Solve the problem using linear algebra techniques for the 3x3 case.&lt;/li&gt;&#xA;&lt;li&gt;Program the solution for an nxn board in Python.&lt;/li&gt;&#xA;&lt;li&gt;Implement it physically using Arduino for the 4x4 case.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I would like to repeat the experience, but not exactly the same one.&lt;/p&gt;&#xA;&lt;p&gt;My question is, does anyone know of any games that can be solved using linear algebra tools? Or do you know of any interesting projects to work on?&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance.&lt;/p&gt;&#xA;" OwnerUserId="23313" LastActivityDate="2023-11-08T02:47:10.373" Title="Game project using linear algebra" Tags="|linear-algebra|math-puzzle|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27087" PostTypeId="2" ParentId="27022" CreationDate="2023-11-04T09:54:16.200" Score="3" Body="&lt;p&gt;Interesting question. But, is it at all a new question ? If you look at the way that online education is delivered then isn't it already the case that &amp;quot;instructors&amp;quot; are just there for &amp;quot;pedagogic, psychological, and technical support&amp;quot;. If your school has online classes where you're just using textbook resources to funnel the kids through homework, then they take online tests and you never even have so much as a required online synchronous meeting with the class... are you teaching ?&lt;/p&gt;&#xA;&lt;p&gt;At least I feel like I'm not teaching when I &amp;quot;teach&amp;quot; online. At least not yet.&lt;/p&gt;&#xA;&lt;p&gt;So, have online classes supplanted the need for residential classes ? Sure, for what, maybe 20% of all US higher education credits ? ( I made that up, I'd be curious the real number if it's available). Do these credits earn degrees ? Sure. Do these credits represent actual learning on the part of the students who are enrolled in such classes ? Sometimes. Do these credits represent educational success in view of metrics created and referenced by the progenitors of such products ? Oh, most certainly. We can always prove a new educational product by checking appropriate boxes. Accreditation is the ultimate example of this scam.&lt;/p&gt;&#xA;&lt;p&gt;So, to turn the question over, can AI do a better job at scamming education then humans ? Probably. Given time we'll be able to pretend we're getting an education more efficiently and convincingly with AI tools than with human tools. But it seems to me now that I think about it for a bit, the real endangered part is the more counterfeit realms of education. Those programs which are not authentically thinking for themselves and requiring their students to do the same.&lt;/p&gt;&#xA;&lt;p&gt;When will an AI earn the first online degree ? Someone should offer a prize and see how many months it takes.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2023-11-04T09:54:16.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27088" PostTypeId="2" ParentId="27027" CreationDate="2023-11-04T11:06:31.567" Score="4" Body="&lt;p&gt;Because inventing &lt;em&gt;rules&lt;/em&gt; when teaching this topic does more harm than good.&lt;/p&gt;&#xA;&lt;p&gt;The reason to make some principle a &lt;em&gt;formal rule&lt;/em&gt; is to facilitate its operational use &lt;em&gt;without remembering/understanding why is it true&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, to teach differentiation, one often just need to give the student a set of rules and a lot of practice with them. This is independent from them understanding the proofs of those rules; having to keep in mind the reasons behind Leibnitz rule when differentiating &lt;span class=&quot;math-container&quot;&gt;$x\sin x$&lt;/span&gt; would be harmful.&lt;/p&gt;&#xA;&lt;p&gt;This is, however, exactly the opposite of what we want to do when teaching &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; proofs: it's not like the students have an engineering course ahead where they will need operational fluency with writing &lt;span class=&quot;math-container&quot;&gt;$\epsilon-\delta$&lt;/span&gt; proofs.  There's no other reason to teach those proofs than to make students &lt;em&gt;understand&lt;/em&gt; them.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2023-11-04T11:06:31.567" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27089" PostTypeId="2" ParentId="27086" CreationDate="2023-11-04T13:13:06.390" Score="1" Body="&lt;p&gt;Since this is for engineering students, perhaps the realm of Markov Decision  Processes would be of interest. The canonical game is gridworld.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5lyHd.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5lyHd.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A standard source is the textbook &lt;strong&gt;Artificial Intelligence: A Modern Approach&lt;/strong&gt; by Russell and Norvig. In the fourth edition, gridword is analyzed in Chapter 17. There are many, many resources on the web. Many resources!&lt;/p&gt;&#xA;&lt;p&gt;The game involves an agent that attempts to navigate to the reward state in a probabilistic environment, where sometimes it stochastically veers off at right angles.&lt;/p&gt;&#xA;&lt;p&gt;Finding the optimal policy to maximize rewards uses the Bellman Equation. If we have &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; states &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; and an arbitrary policy &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, then we have a linear system of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; equations in &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; unknowns for the state-values &lt;span class=&quot;math-container&quot;&gt;$V^{\pi}(s)$&lt;/span&gt; of the policy.&lt;/p&gt;&#xA;&lt;p&gt;The linear system looks like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$V^{\pi}(s) = R(s, \pi(s)) + \gamma \sum_{s' \in S} P(s' | s, \pi(s)) V^{\pi}(s')&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here the coefficients &lt;span class=&quot;math-container&quot;&gt;$ P(s' | s, \pi(s))$&lt;/span&gt; are related to the transition probabilities of the environment, and &lt;span class=&quot;math-container&quot;&gt;$R(s, \pi(s))$&lt;/span&gt; are inhomogeneous terms related to the rewards for state-action pairs. The so-called &amp;quot;discount factor&amp;quot; is denoted by &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Policy iteration is an algorithm to find the optimal policy. We start with a random policy &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; and solve the linear system for &lt;span class=&quot;math-container&quot;&gt;$V^{\pi}$&lt;/span&gt;. Then follows &amp;quot;policy improvement&amp;quot; that involves adjusting &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; so the actions point to the highest value &lt;span class=&quot;math-container&quot;&gt;$V^{\pi}$&lt;/span&gt;. Now that we have a new and improved policy &lt;span class=&quot;math-container&quot;&gt;$\pi_1$&lt;/span&gt;, we solve the resulting linear system for the new policy. We repeat until the policy converges.&lt;/p&gt;&#xA;&lt;p&gt;When the number of states is large, the system of linear inhomogeneous equations is solved using iterative numerical methods. But when the number of states is not overwhelming, we can solve the linear system using methods taught in a standard linear algebra course.&lt;/p&gt;&#xA;&lt;p&gt;Admittedly, it takes some effort to understand the context of the problem because of the notation and concepts. But it is a very nice application of solving linear systems, and gridworld is the &amp;quot;hello world&amp;quot; for reinforcement learning.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-11-04T13:13:06.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27090" PostTypeId="2" ParentId="27086" CreationDate="2023-11-04T16:38:55.493" Score="1" Body="&lt;p&gt;This one is similar, but maybe slightly more general: &lt;a href=&quot;https://www.codewars.com/kata/63306fdffa185d004a987b8e&quot; rel=&quot;nofollow noreferrer&quot;&gt;light switch subsets&lt;/a&gt;.   One feature of these problems is that a solution can be found just using row reduction, however, one has to work over &lt;span class=&quot;math-container&quot;&gt;$Z/2Z$&lt;/span&gt; to reflect the on/off properties.  Some possible benefits of this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;By translating each row operation to &amp;quot;switches and bulbs&amp;quot; this can help visualize what these row operations are doing.  I could imagine making a cool animation as well.&lt;/li&gt;&#xA;&lt;li&gt;It can help students to think about whether or not their favorite RREF algorithm will work with a number system other than the real numbers.  For &lt;span class=&quot;math-container&quot;&gt;$Z/2Z$&lt;/span&gt; there is never any need for division, but what would happen for &lt;span class=&quot;math-container&quot;&gt;$Z/4Z$&lt;/span&gt;? What would a &amp;quot;reduced&amp;quot; form look like over this system.&lt;/li&gt;&#xA;&lt;li&gt;Implementing a RREF algorithm over &lt;span class=&quot;math-container&quot;&gt;$Z/2Z$&lt;/span&gt; in Python can be a fun project.&lt;/li&gt;&#xA;&lt;li&gt;If the students have taken an algorithms class this helps them experience how &amp;quot;fast&amp;quot; RREF is as compared with taking all subsets of lightswitches.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10028" LastActivityDate="2023-11-04T16:38:55.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27091" PostTypeId="2" ParentId="27027" CreationDate="2023-11-04T18:05:27.813" Score="2" Body="&lt;p&gt;One might consider the purpose or goal of a particular book. Each author might have their own. [I am slow to answer because my real job takes time. Answer have been trickling in since I started to formulate mine. Forgive me if answers of some other's have escaped my notice or memory.]&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;To understand &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; (real functions) or &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; (sequences) definitions —&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then some attention to the technical difficulties in satisfying a definition probably seems appropriate to the author. Compare with &lt;a href=&quot;https://matheducators.stackexchange.com/a/27052&quot;&gt;Misha Lavrov's&lt;/a&gt; and &lt;a href=&quot;https://matheducators.stackexchange.com/a/27088&quot;&gt;Kostya_I's&lt;/a&gt; answers. A criticism I have of this approach is that sometimes students come out with a heavy, pedantic approach to analysis.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;To present efficiently the theory of real analysis —&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This seems Rudin's purpose, for instance, AFAIR. It also seemed the goal in Marsden and Tromba's &lt;em&gt;Vector Calculus&lt;/em&gt;, at least as far their presentation of proofs. They tend to avoid &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; when a hypothesis of continuity makes it unnecessary or by a form of reasoning I likened to an unstated squeeze theorem — not the dinosaur in  typical calculus texts, but this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;If &lt;span class=&quot;math-container&quot;&gt;$|f(x)-L| \le g(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x) \rightarrow 0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f(x) \rightarrow L$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proof of this simple and at the same time intuitive (imho). This is essentially &lt;a href=&quot;https://matheducators.stackexchange.com/questions/27027/why-do-most-analysis-textbooks-overlook-and-fail-to-teach-delta-epsilon-proofs#comment71884_27027&quot;&gt;Steve Gubkin's&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$g(\epsilon)=\sqrt{\epsilon}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(\epsilon)=\sin \epsilon$&lt;/span&gt; principles all rolled into one.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;To teach students to prove theorems in real analysis —&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This can be distinct from understanding &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proofs,&#xA;as well as in addition to it. It also seems a goal of Rudin. This goal can be accompanied by an assumption that students have some general understanding how to construct a proof based on axioms, hypotheses, and other theorems. The author need not include a narrower focus on proficiency in applying the definition of a limit.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;To give a gentle introduction to analysis —&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This might mean the author wishes to teach students how to prove theorems in analysis but wants to avoid some of the technical difficulties in &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proofs, which the author assumes the student will master at some later date, should they ever need such mastery. Possibly that describes Marsden &amp;amp; Tromba, but I don't think they are trying to teach students to prove theorems.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; principle in the OP brings to mind the following version of the squeeze theorem (a &amp;quot;pointwise Lipschitz&amp;quot; condition):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;If &lt;span class=&quot;math-container&quot;&gt;$|f(x)-L| \le M\,|x-c|$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f(x) \rightarrow L$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x\rightarrow c$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course not all functions satisfy such a condition, but most &amp;quot;calculus-level&amp;quot; formulas do. Notable exceptions are roots like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$c=0$&lt;/span&gt;, which are easy to handle with &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;. It probably doesn't belong in a general analysis course, but it's helpful in explanations at a lower level. And it prepares students for the inequalities found in the &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; definition. It's not a point closely related to the OP's question, but sharing it felt more generous than not.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2023-11-04T18:05:27.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27092" PostTypeId="1" CreationDate="2023-11-05T13:57:59.617" Score="2" ViewCount="193" Body="&lt;p&gt;Hello fellow educators and math enthusiasts,&lt;/p&gt;&#xA;&lt;p&gt;I am on the hunt for entertaining and thought-provoking mathematical riddles suitable for a classroom setting. My objective is to find riddles that not only captivate the students' interest but also sharpen their problem-solving skills.&lt;/p&gt;&#xA;&lt;p&gt;Could you please share the most humorous or intriguing math riddle you've encountered? Additionally, I'm interested in any books or resources that compile such mathematical puzzles. If you know of any &amp;quot;What is...?&amp;quot; type riddles that are particularly engaging, I'd love to hear about those as well.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, I'm curious about the pedagogical impact of using mathematical riddles as a teaching tool. Are there any studies or research findings on how such riddles can enhance learning or engagement in the classroom?&lt;/p&gt;&#xA;&lt;p&gt;Thank you for your insights and recommendations!&lt;/p&gt;&#xA;&lt;p&gt;HERE IS A REPRESENTATIVE EXAMPLE: hat is what is: hat i has a head and it has a tail but no body. Answer: a coin.&lt;/p&gt;&#xA;&lt;p&gt;being a little bit mor precise: I am searching for clever riddles that would be suitable for use in a stand-up comedy routine.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="15254" LastEditDate="2023-11-13T07:09:14.300" LastActivityDate="2023-11-13T21:16:45.230" Title="Engaging Mathematical Riddles for Classroom Enrichment: examples and impact studies references" Tags="|activities|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="27093" PostTypeId="1" AcceptedAnswerId="27094" CreationDate="2023-11-05T23:32:45.583" Score="6" ViewCount="5144" Body="&lt;p&gt;In every of my ten years as course instructor &lt;a href=&quot;https://math.stackexchange.com/a/606590&quot;&gt;introducing first year undergraduates to proofs&lt;/a&gt;, I always project this quotation on the lecture screen, then dictate the whole caboodle word for word! But for the tenth straight year, this quotation failed to teach intuition why &lt;span class=&quot;math-container&quot;&gt;$\color{mediumseagreen}{\text{If P then Q}}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\color{darkGoldenrod}{\text{P only if Q}}$&lt;/span&gt; mean the same! On the midterm that I am marking now, some students always answer false to this question&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;True or false ?   &lt;span class=&quot;math-container&quot;&gt;$\color{mediumseagreen}{\text{'If P then Q'}}$&lt;/span&gt; is logically equivalent to &lt;span class=&quot;math-container&quot;&gt;$\color{darkGoldenrod}{\text{'P only if Q'}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h4&gt;How else can I teach intuition for &lt;span class=&quot;math-container&quot;&gt;$\color{MediumAquamarine}{\text{If P then Q}}$&lt;/span&gt; = &lt;span class=&quot;math-container&quot;&gt;$\color{ Goldenrod}{\text{P only if Q}}$&lt;/span&gt;? How can I better assist students to intuit this equivalence? I don't want them to memorize anything. Truth Tables and proofs don't teach intuition.&lt;/h4&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;        The obvious question: Why do&#xA;our intuitions have to be dragged kicking and screaming&#xA;before they will acknowledge that ‘If P then Q’ and&#xA;‘P only if Q’ have the exact same meaning? &lt;br /&gt;&#xA;        The non-obvious answer: In&#xA;the course of using conditionals in everyday conversation,&#xA;we presume that the speaker (or writer) believes that&#xA;there is a &lt;em&gt;connection&lt;/em&gt; of some sort between the state&#xA;of affairs to which the antecedent of the conditional&#xA;refers and the state of the affairs to which the&#xA;consequent refers. For example, the connection might&#xA;be of the causal sort, of the definitional sort, or&#xA;of the logical sort—and the context will normally&#xA;make clear exactly which sort of conditional the&#xA;speaker (or writer) is using. &lt;br /&gt;&#xA;        Suppose that a dog-walker&#xA;reprimands you: “If you pull Old Fido’s tail one more&#xA;time then he’ll bite you.” Your presumption is that the&#xA;dog-walker believes that there is a &lt;em&gt;causal connection&lt;/em&gt;&#xA;between the former state of affairs—your pulling&#xA;Old Fido’s tail—and the latter state of affairs—Old Fido’s&#xA;biting you—such that the former will be the &lt;em&gt;cause&lt;/em&gt; of the&#xA;latter. But here’s the crucial point: Your &lt;em&gt;presumption&lt;/em&gt;,&#xA;as well as the dog-walker’s &lt;em&gt;belief&lt;/em&gt;, are distinct from&#xA;the &lt;em&gt;meaning&lt;/em&gt; of the conditional sentence itself. The&#xA;&lt;em&gt;meaning&lt;/em&gt; of the conditional sentence itself is what&#xA;it shares with &lt;em&gt;all&lt;/em&gt; typical conditionals. Because a&#xA;reference to causality doesn’t characterize the&#xA;other sorts of conditionals, a reference to&#xA;causality can be no part of its own meaning. &lt;br /&gt;&#xA;         Its &lt;em&gt;meaning&lt;/em&gt;—what it shares&#xA;(once again) with all typical conditionals—is precisely this:&#xA;&lt;em&gt;It isn’t the case that its antecedent is true and its&#xA;consequent isn’t true.&lt;/em&gt; Once you abandon the belief&#xA;that there is a &lt;em&gt;connection&lt;/em&gt; of some sort between&#xA;the state of affairs to which the antecedent of&#xA;the conditional refers and the state of affairs&#xA;to which the consequent refers, then you should&#xA;have no difficulty seeing that the &lt;em&gt;meaning&lt;/em&gt; of&#xA;a conditional consists exclusively in its not&#xA;being the case that its antecedent is true&#xA;and its consequent is not true. &lt;br /&gt;&#xA;        Think of a conjunction.&#xA;Suppose that on the first day of the semester, your&#xA;instructor had walked into your class and said,&#xA;“This is a course in formal logic and I shall now&#xA;be taking roll.” You would have found that entirely&#xA;unsurprising. Suppose instead that on the first day&#xA;of the semester your instructor had walked into&#xA;your class and said, “This is a course in formal&#xA;logic, and Lenin suffered his first stroke in&#xA;May 1922.” You would have found this more than a&#xA;bit odd. Suppose that a short while later in the&#xA;same class your instructor had then gone on to&#xA;say, “There will be a quiz every other week, and&#xA;Alexandria, Egypt, is named after Alexander the Great.”&#xA;At this point you would have begun to feel a bit&#xA;uneasy and you would have looked around at the&#xA;other students. Suppose finally that somewhat&#xA;later, your instructor had then gone on to say,&#xA;“The final exam will count for one-third of&#xA;your course grade, and Euclid is credited with&#xA;the proof that the square root of 2 is an&#xA;irrational number.” My guess is that at that point&#xA;you and your fellow classmates would have started&#xA;tiptoeing toward the exit. The collective bubble&#xA;over all of your heads would have read: “What&#xA;does the one thing have to do with the other?&#xA;What’s the &lt;em&gt;connection&lt;/em&gt; between the &lt;em&gt;first&lt;/em&gt; half&#xA;of each of this instructor’s sentences and the&#xA;&lt;em&gt;second&lt;/em&gt; half?” Or (if on that day you had known&#xA;the terminology) your collective bubble would have&#xA;read: “In each of the preceding conjunctions the&#xA;two conjuncts have no &lt;em&gt;relation&lt;/em&gt; to one another.&#xA;Why, then, is this instructor conjoining such&#xA;conjuncts?” &lt;br /&gt;&#xA;        Your unease, however, would&#xA;have concerned &lt;em&gt;psychology&lt;/em&gt; (your instructor’s) and&#xA;not &lt;em&gt;logic&lt;/em&gt; as such. Your confusion concerned not the&#xA;&lt;em&gt;meaning&lt;/em&gt; of your instructor’s statements but rather&#xA;your instructor’s &lt;em&gt;reasons&lt;/em&gt; for uttering them. The&#xA;point is that you understood each of the sentences&#xA;and you could have determined their truth-values&#xA;without too much difficulty. Consider the sentence&#xA;‘This is a course in formal logic, and Lenin suffered&#xA;his first stroke in May 1922’. Had you known that each&#xA;conjunct is true, you would have known in a jiffy that&#xA;the entire conjunction is true. Whether the sentence&#xA;is true or false is one thing; whether it’s appropriate&#xA;or inappropriate (i.e., bizarre) to utter it is another&#xA;thing altogether. In logic our concern is exclusively&#xA;with truth and falsehood, rather than with&#xA;appropriateness and inappropriateness. &lt;br /&gt;&#xA;          It simply doesn’t matter—at&#xA;least where the truth-value of the sentence is&#xA;concerned—whether there’s any connection between the&#xA;left conjunct and the right conjunct in the conjunction&#xA;‘This is a course in formal logic, and Lenin suffered&#xA;his first stroke in May 1922’. By the same token, it&#xA;simply doesn’t matter—again, at least where the truthvalue&#xA;of the sentence is concerned—whether there’s any connection&#xA;between the antecedent and the consequent in the conditional&#xA;‘If Fido wrote the Iliad then the Moon is made of pink fluff ’. &lt;br /&gt;&#xA;        Once you divest yourself&#xA;of the view that there &lt;em&gt;has&lt;/em&gt; to be a connection,&#xA;it should become somewhat easier to see that&#xA;there’s no difference either meaning-wise or&#xA;truth-value-wise between the sentence ‘&lt;em&gt;If&lt;/em&gt;&#xA;Fido wrote the Iliad &lt;em&gt;then&lt;/em&gt; the Moon is made&#xA;of pink fluff ’ and the sentence ‘Fido wrote&#xA;the Iliad &lt;em&gt;only if&lt;/em&gt; the Moon is made of pink fluff ’.&#xA;Each of these sentences has the exact same meaning&#xA;as the sentence ‘It is not the case both that Fido&#xA;wrote the Iliad and that the Moon is &lt;em&gt;not&lt;/em&gt; made&#xA;of pink fluff ’. Now, since &lt;em&gt;this&lt;/em&gt; sentence is&#xA;true—Fido did &lt;em&gt;not&lt;/em&gt; write the Iliad—each of the two&#xA;former sentences is true as well. And once you see&#xA;&lt;em&gt;that&lt;/em&gt;, it should become easy-ish to see that&#xA;there’s no difference &lt;em&gt;truth-value-wise&lt;/em&gt; between&#xA;the sentence ‘If you work hard next semester then&#xA;you’ll pass’ and the sentence ‘You’ll work hard&#xA;next semester only if you’ll pass’. They both&#xA;mean that it’s not the case that you’ll work&#xA;hard and yet that you &lt;em&gt;won’t&lt;/em&gt; pass.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Lande NP, &lt;a href=&quot;https://hackettpublishing.com/formal-logic-and-its-rabbit-holes&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Classical logic and its rabbit-holes: A first course&lt;/em&gt; (Hackett Publishing 2013)&lt;/a&gt;, pages 53-5.&lt;/p&gt;&#xA;" OwnerUserId="23025" LastEditorUserId="851" LastEditDate="2023-11-09T18:28:48.027" LastActivityDate="2023-11-09T18:32:45.920" Title="How can I teach intuition why ‘If P then Q’ and ‘P only if Q’ mean the same, to first year undergraduates?" Tags="|logic|" AnswerCount="11" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="27094" PostTypeId="2" ParentId="27093" CreationDate="2023-11-06T00:28:37.747" Score="17" Body="&lt;p&gt;I would totally tune out if you dictated that massive quote to me. I can't even bring myself to properly skim it. You say teaches intuition, but for me, at least, the whole point of intuition is to make something memorable with a concise obvious-sounding explanation (or possibly just an example that extrapolates well). And an 8-paragraph, 1000+ word wall of text is definitely not that.&lt;/p&gt;&#xA;&lt;p&gt;Here's what feels intuitive to me:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppose we're sitting next to each other in class. If you say &amp;quot;hello&amp;quot; then I hear &amp;quot;hello.&amp;quot; You say &amp;quot;hello&amp;quot; only if I hear &amp;quot;hello&amp;quot; (because if I didn't hear it then you didn't say it). But other people can also say hello to me so it's not true that I hear &amp;quot;hello&amp;quot; only if you say &amp;quot;hello.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But personally, that intuition isn't what I use in practice. In practice, I just think about this visually with inference arrows:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the phrase &amp;quot;if X&amp;quot; tells you the arrow starts at X,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;but when the word &amp;quot;only&amp;quot; precedes it, that indicates to flip the arrow.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For the phrases you mentioned:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;if P then Q&amp;quot; starts at P: &lt;span class=&quot;math-container&quot;&gt;$P \rightarrow Q$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;P if Q&amp;quot; starts at Q: &lt;span class=&quot;math-container&quot;&gt;$P \leftarrow Q$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;P only if Q&amp;quot; is like the above but flipped: &lt;span class=&quot;math-container&quot;&gt;$P \rightarrow Q$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Conveniently, these inference arrows can be replaced with implications:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;if P then Q&amp;quot; means &lt;span class=&quot;math-container&quot;&gt;$P \Rightarrow Q$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;P if Q&amp;quot; means &lt;span class=&quot;math-container&quot;&gt;$P \Leftarrow Q$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;P only if Q&amp;quot; means &lt;span class=&quot;math-container&quot;&gt;$P \Rightarrow Q$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-08T15:58:44.520" LastActivityDate="2023-11-08T15:58:44.520" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="27095" PostTypeId="2" ParentId="27093" CreationDate="2023-11-06T08:26:56.153" Score="6" Body="&lt;p&gt;Two expressions are equivalent if they produce the same true/false result under the same circumstances.&lt;/p&gt;&#xA;&lt;p&gt;For &amp;quot;if&amp;quot; sentences, to match intuition it's easier to concentrate on counter-examples, because the mathematical definition of &amp;quot;if P then Q&amp;quot; being true for P=false does not match with intuition so well.&lt;/p&gt;&#xA;&lt;p&gt;Let the students create counter-examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Under which circumstances can &amp;quot;if P then Q&amp;quot; be falsified?&lt;/li&gt;&#xA;&lt;li&gt;Under which circumstances can &amp;quot;P only if Q&amp;quot; be falsified?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;They'll (hopefully) come to the very same situation for both cases: both sentences can be falsified only with P=true and Q=false.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(1) Maybe, you want to start with real-world sentences instead of the symbols P and Q, depending on the expected abstraction capabilities of your students.&lt;/p&gt;&#xA;&lt;p&gt;An example that comes to my mind:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If the lamp shines (P), the switch is on (Q). &lt;em&gt;[Not the other way round, there might be a power outage, the fuse might be blown etc.]&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;The lamp shines (P) only if the switch is on (Q).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15862" LastActivityDate="2023-11-06T08:26:56.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27096" PostTypeId="2" ParentId="27093" CreationDate="2023-11-06T08:56:01.783" Score="5" Body="&lt;p&gt;True or false ? 'If P then Q'&#xA;is logically equivalent to 'P only if Q'&lt;/p&gt;&#xA;&lt;p&gt;You then provide the example for if P then Q - if you pull Fido's tail (P) then he'll bite you (Q). The 'P only if Q' version would be 'Only if Fido bites you (Q) you'll pull his tail (P)' which to a &amp;quot;normal&amp;quot; person (ie me) isn't the same thing at all - perhaps a better example would help&lt;/p&gt;&#xA;" OwnerUserId="23322" LastActivityDate="2023-11-06T08:56:01.783" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27097" PostTypeId="2" ParentId="27093" CreationDate="2023-11-06T10:41:28.053" Score="4" Body="&lt;p&gt;I want to challenge the frame:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;if you want people to answer this question correctly on a test, repeat the correct answer 5 times and insist that it will be on a test.&lt;/li&gt;&#xA;&lt;li&gt;if you rather want to check whether people really understand this, do it in a different way. For example, give them two actual mathematical statements and ask whether they have the same meaning. (As an aside, I would struggle to find and example for the &amp;quot;Q only if P&amp;quot; since people rarely phrase math like this.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;There are multiple problems with the question as it is asked. First one is that it is posed in an abstract form, with &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, which adds an additional layer of difficulty for the students. More importantly, it is likely that &lt;em&gt;you never explained to your students properly what questions like this even mean&lt;/em&gt;. For example, what are &lt;span class=&quot;math-container&quot;&gt;$P,Q$&lt;/span&gt; and &amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;&amp;quot; - are they sentences or predicates? In the long text you've quoted, the two are badly conflated, without ever acknowledging the difference: there are examples that are sentences, but old Fido one is secretly a predicate - truth/or falsity of statements depend on an additional input (your behavior).&lt;/p&gt;&#xA;&lt;p&gt;Now, suppose it is presumed they are sentences. I think it is safe to assume you never explained what &amp;quot;sentence&amp;quot; means (Is &amp;quot;this sentence is false&amp;quot; a sentence? Is a sentence like &amp;quot;Fido will bite you&amp;quot; a sentence, and if it is, does it (yet?) have a true/false value?). Maybe you have at least explained what it means for two sentences to be &amp;quot;logically equivalent&amp;quot;. Did you tell them that it means that either both are true, or both are false? Did you go through the examples like &amp;quot;2+2=4&amp;quot; and &amp;quot;Joe Biden is the POTUS&amp;quot;, insisting they are &amp;quot;logically equivalent&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;But even assuming you did, the question as you asked still does not fully make sense, since it contains placeholders for unknown statements &lt;span class=&quot;math-container&quot;&gt;$P,Q$&lt;/span&gt;. Is &amp;quot;If &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;&amp;quot; still a statement, or is it now a predicate taking two statements as arguments?&lt;/p&gt;&#xA;&lt;p&gt;How are students suppose to &amp;quot;intuit&amp;quot; a jargon that you never give a precise meaning to?&lt;/p&gt;&#xA;" OwnerUserId="10349" LastEditorUserId="10349" LastEditDate="2023-11-06T11:04:22.580" LastActivityDate="2023-11-06T11:04:22.580" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27098" PostTypeId="2" ParentId="27093" CreationDate="2023-11-06T15:00:59.833" Score="13" Body="&lt;p&gt;I'll do a frame challenge answer that is closely related to the others: the phrase &amp;quot;only if&amp;quot; is generally only useful inside the larger phrase &amp;quot;if and only if&amp;quot;. It's not hard to intuit the idea that &amp;quot;if and only if&amp;quot; means &amp;quot;is logically equivalent to&amp;quot;. Once students are okay with that, since &amp;quot;if&amp;quot; is clearly one of the implications, &amp;quot;only if&amp;quot; must be the other one. This doesn't need to cause a ton of confusion.&lt;/p&gt;&#xA;&lt;p&gt;Aside from that, I agree with the people who say there's not much benefit to using the phrase &amp;quot;only if&amp;quot; at all.&lt;/p&gt;&#xA;" OwnerUserId="22788" LastActivityDate="2023-11-06T15:00:59.833" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27099" PostTypeId="2" ParentId="27093" CreationDate="2023-11-06T17:25:14.100" Score="10" Body="&lt;p&gt;To avoid confusion, in examples of implication, it may help to avoid any suggestion of causality, and to have both antecedent and consequent expressed in the present tense.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EXAMPLE&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Consider the implication: If it is raining (R), then it is cloudy (C).&lt;/p&gt;&#xA;&lt;p&gt;R =&amp;gt; C&lt;/p&gt;&#xA;&lt;p&gt;Clearly, rain does not cause cloudiness. And it is raining &lt;strong&gt;ONLY IF&lt;/strong&gt; it is cloudy.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-11-06T19:24:22.363" LastActivityDate="2023-11-06T19:24:22.363" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27100" PostTypeId="1" CreationDate="2023-11-07T05:03:08.760" Score="1" ViewCount="98" Body="&lt;p&gt;A few years back I had a student attend the MASS semester at Penn State. It was a fantastic experience for my student and it certainly helped him find a place in graduate school and I would wager it figured significantly in terms of his mathematical formation.&lt;/p&gt;&#xA;&lt;p&gt;I currently have a student who is in an usual spot where he will graduate in the middle of the year. I know it is possible to apply for graduate school to begin in the Spring semester, but it seems it might be better for him to take a semester to attend some bridge program like MASS. From what I can tell, it appears MASS stopped in 2018 from their website.&lt;/p&gt;&#xA;&lt;p&gt;Now, there is thankfully some flexibility in his future plans. It is possible to attend some bridge program either in the Fall or Spring terms of 2024-2025 and I am trying to give him guidance on seeking some REU experience in either summer 2024 or summer 2025.&lt;/p&gt;&#xA;&lt;p&gt;Much like myself, this student found math in the midst of his undergraduate work, he has only recently understood that he would like to make math his career since, as most of us know, math is totally awesome.&lt;/p&gt;&#xA;&lt;p&gt;My question is thus as follows, and I think this has relevance beyond my current student,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What are the best bridge programs for graduate math for a junior in math ? Let us suppose the hypothetical student already has basic proof techniques well under control and is seeking something like a token semester in graduate math. Partly as an opportunity to stretch themselves beyond the standards of their undergraduate institution, but also as an opportunity to network and forge relationships for future mathematical endeavors.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Again, MASS was such a program, but I am curious what else is out there ? Thanks in advance for your help in advising this student, and I hope this question can help others who are seeking such experience.&lt;/p&gt;&#xA;" OwnerUserId="128" LastEditorUserId="2083" LastEditDate="2023-11-07T13:30:42.837" LastActivityDate="2023-11-07T13:30:42.837" Title="Summer or Semester Programs which bridge to Graduate Mathematics" Tags="|undergraduate-education|graduate-education|usa|advising|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27101" PostTypeId="2" ParentId="27100" CreationDate="2023-11-07T05:51:13.213" Score="3" Body="&lt;p&gt;I see how it looks like they stopped in 2018, but you might at least contact PSU Math. Call the department secretary and ask for a point of contact. Perhaps they are still running it, after all there's still an application form on their website. Also, note that MASS runs in fall, but PMASS runs in spring.  So if they are running PMASS, still, that would be ideal for your student.  In addition to confirming if they've stopped MASS/PMASS, they might have a suggestion for an alternate for your kid (if they have stopped it).&lt;/p&gt;&#xA;&lt;p&gt;I did a search for math semesters or years or summer programs and there wasn't much (some stuff in Budapest and Moscow):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://maa.org/member-communities/students/semester-and-summer-programs&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://maa.org/member-communities/students/semester-and-summer-programs&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The no such agency also came up as a sponsored link, so you could get the kid to look into that.  &lt;a href=&quot;https://www.intelligencecareers.gov/NSA/students-and-internships&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.intelligencecareers.gov/NSA/students-and-internships&lt;/a&gt; (bunch of options, so weed through it and probably have the kid contact them for more details).&lt;/p&gt;&#xA;&lt;p&gt;Slightly joking, but kid could maybe double dip and sign up with NSA and have them pay to send him to Moscow math program.  Hmmm...  ;-)&lt;/p&gt;&#xA;" OwnerUserId="23327" LastEditorUserId="23327" LastEditDate="2023-11-07T06:05:44.913" LastActivityDate="2023-11-07T06:05:44.913" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27102" PostTypeId="2" ParentId="27027" CreationDate="2023-11-07T10:57:30.800" Score="0" Body="&lt;p&gt;This answer might involve more math than talk about math education, but nevertheless:&lt;/p&gt;&#xA;&lt;p&gt;I am currently a student, and this precise issue has oftentimes plagued me, it started with me being frustrated in an exam when I would guess some expression for delta, and end up with something that is not quite epsilon, and then have to correct the whole ordeal as you describe, but recently I pondered it more deeply and realised that it's not just for expressions of the form &lt;span class=&quot;math-container&quot;&gt;$K\varepsilon$&lt;/span&gt; that I can go back and apply a suitable correction, but also for an expression such as &lt;span class=&quot;math-container&quot;&gt;$\varepsilon^2$&lt;/span&gt; although the correction is oftentimes more complicated, but then I asked myself, &amp;quot;How did I know that I can correct this, before seeing precisely how to do it?&amp;quot; Which lead me to realise that actually, if you end up with &lt;em&gt;any&lt;/em&gt; surjective function (from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R^+}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^+$&lt;/span&gt;) of &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;, then in principle, you should be able to correct it, the &amp;quot;correction&amp;quot; that works, in principle, amounts to taking the specific &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; which maps to the desired &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;, guaranteed to exist by surjectivity. So actually, we can rewrite the condition&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\forall_{\varepsilon&amp;gt;0}\exists_{\delta&amp;gt;0}\forall_{x\in \mathbb{R}}[|x-c|&amp;lt;\delta \implies |f(x)-f(c)|&amp;lt;\varepsilon ]&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As: There exists a surjective function &lt;span class=&quot;math-container&quot;&gt;$g:\mathbb{R}^+ \to \mathbb{R}^+$&lt;/span&gt; such that for any &lt;span class=&quot;math-container&quot;&gt;$\delta&amp;gt;0$&lt;/span&gt; and any &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; we have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$|x-c|&amp;lt;\delta \implies |f(x)-f(c)|&amp;lt;g(\delta) $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Which generalises the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$K-\varepsilon$&lt;/span&gt;&amp;quot; principle.&lt;/p&gt;&#xA;&lt;p&gt;To me this condition is much simpler, as you exhibit something first (the function &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;) before making a universal statement involving it, as opposed to making a universal statement, followed by an existential statement, but when I showed this to my lecturer, it took significant convincing, and eventually a very formal and overly detailed proof of equivalence of the two definitions to convince them that this is correct, although it is more abstract than the &lt;span class=&quot;math-container&quot;&gt;$K-\varepsilon$&lt;/span&gt; principle, I think it illustrates an important point, which is that many educators probably are not too comfortable with things that don't look like what they learnt themselves. I also showed it to some of my friends, and they were mostly confused, which I think also illustrates that sometimes a more &amp;quot;elegant&amp;quot; definition is not the easiest to understand, in fact sometimes something which is &amp;quot;elegant&amp;quot; is just very &amp;quot;clever&amp;quot;, and oftentimes when students start out with course material, cleverness to them seems like daunting magic, especially if they're not able to fully justify it without assistance. Anyhow, I know this doesn't quite answer the question, but I thought it was interesting&lt;/p&gt;&#xA;" OwnerUserId="23186" LastActivityDate="2023-11-07T10:57:30.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27103" PostTypeId="2" ParentId="27093" CreationDate="2023-11-07T18:05:01.410" Score="3" Body="&lt;p&gt;Point out that the question is designed to be deceptive by employing a sense of causality that doesn't exist in formal logic, which describes relationships but not necessarily causality.&lt;/p&gt;&#xA;&lt;p&gt;It's fine to say &amp;quot;&lt;strong&gt;If&lt;/strong&gt; light &lt;strong&gt;P&lt;/strong&gt; is on &lt;strong&gt;then&lt;/strong&gt; switch &lt;strong&gt;Q&lt;/strong&gt; must be closed,&amp;quot; even though it's the state of the switch which controls the state of the light.  One could also say &amp;quot;light &lt;strong&gt;P&lt;/strong&gt; is on &lt;strong&gt;only if&lt;/strong&gt; switch &lt;strong&gt;Q&lt;/strong&gt; is closed.&amp;quot;  You haven't affected the causality, just looked at the relationship another way.&lt;/p&gt;&#xA;" OwnerUserId="13286" LastActivityDate="2023-11-07T18:05:01.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27104" PostTypeId="2" ParentId="18504" CreationDate="2023-11-07T22:48:59.303" Score="3" Body="&lt;p&gt;With exponential functions, the important thing to note is that the RATE of change is proportional.  In a calculus class we might approach this in terms of taking the derivative and noticing that &lt;span class=&quot;math-container&quot;&gt;$f'(x)=a\cdot f(x)$&lt;/span&gt;.  While precalculus students haven't yet learned derivatives we can still take the same approach even if the tools are a bit more unwieldy, namely by looking at a difference quotient over an arbitrary interval.&lt;/p&gt;&#xA;&lt;p&gt;Given &lt;span class=&quot;math-container&quot;&gt;$f(x)=ab^x+k$&lt;/span&gt; we can show that across two consecutive intervals of arbitrary size &lt;span class=&quot;math-container&quot;&gt;$h&amp;gt;0$&lt;/span&gt; the proportion of growth of the 2nd interval over the first remains the same independent of your choice in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{f(x+2h)-f(x+h)}{f(x+h)-f(x)}=\frac{(ab^{x+2h}+k)-(ab^{x+h}+k)}{(ab^{x+h}+k)-(ab^x+k)}=\frac{ab^h(b^h-1)b^x}{a(b^h-1)b^x}=b^h&#xA;$$&lt;/span&gt;&#xA;Since the choice of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is arbitrary, you may as well choose &lt;span class=&quot;math-container&quot;&gt;$h=1$&lt;/span&gt; for simplicity's sake, and show the precalculus student that &lt;span class=&quot;math-container&quot;&gt;$\frac{f(2)-f(1)}{f(1)-f(0)}=b$&lt;/span&gt; and for rigor's sake also show that &lt;span class=&quot;math-container&quot;&gt;$\frac{f(3)-f(2)}{f(2)-f(1)}=b$&lt;/span&gt;.  If the student can convince themselves that any three &amp;quot;consecutive&amp;quot; terms they choose all demonstrate a proportional growth rate, then the function must be exponential.&lt;/p&gt;&#xA;&lt;p&gt;I consider this approach to be a more generalized and inclusive approach to categorizing the function as exponential than the terse (and in my opinion naive) definition as that fitting the pattern of &lt;span class=&quot;math-container&quot;&gt;$f(x)=ab^x$&lt;/span&gt;.  But therein lies the rub.  As one moves through the landscape of mathematics one must discover and grapple with the fact that the most important definitions that one should adhere to are those either explicitly stated by an author, or inherently assumed by the author.  Yes, there are conventions and definitions that many/most mathematicians adhere to that make communicating math hella convenient (if laden with burdensome notation from time to time), but it would be erroneous to assume that there is an absolute authority we all do/must ascribe to.&lt;/p&gt;&#xA;" OwnerUserId="23332" LastEditorUserId="23332" LastEditDate="2023-11-08T12:44:39.683" LastActivityDate="2023-11-08T12:44:39.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27105" PostTypeId="2" ParentId="27093" CreationDate="2023-11-08T00:41:19.183" Score="4" Body="&lt;p&gt;If all you are trying to convey is an intuition, could a Venn diagram work, with a region P completely inside a region Q?&lt;/p&gt;&#xA;&lt;p&gt;This would intuitively show that if P is true, then Q is true (if P, then Q). It would also show that if Q is not true, then P cannot be true (P only if Q).&lt;/p&gt;&#xA;&lt;p&gt;If you wanted to make this less abstract (more intuitive) than a Venn diagram, you could express regions P and Q as locations on a map, for example Kansas and the USA (or wherever you may be located): If you are in Kansas, then you are in the USA. Also, if you are not in the USA, then you are not in Kansas.&lt;/p&gt;&#xA;" OwnerUserId="23335" LastEditorUserId="23335" LastEditDate="2023-11-09T18:32:45.920" LastActivityDate="2023-11-09T18:32:45.920" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27106" PostTypeId="2" ParentId="27086" CreationDate="2023-11-08T02:47:10.373" Score="1" Body="&lt;p&gt;When I was teaching linear algebra, I made 3 HTML/javascript games for students:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.math.kent.edu/%7Enazarov/switches/switchgame.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Switches and lightbulbs&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This is pretty much what Adam described and the interface should be self-explanatory&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.math.kent.edu/%7Enazarov/T-intersection/tint.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Intersection traffic counters&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The goal is to determine how many vehicles went through the intersection in each particular way from the entrance and exit counter readings. The point was that it is generally impossible if the left turn from the side road into the main horizontal road is allowed but if you prohibit it (by clicking on the sign), the linear system becomes well-posed.&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.math.kent.edu/%7Enazarov/Hillcipher/hill.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hill cipher&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The allowed text symbols are the 26 lowercase letters of the Latin alphabet, period, question mark, and star (The last one will replace everything else when parsing and will be added if the text length is not divisible by 3). The ciphering/deciphering is done in groups of 3 using the matrix specified by the user (all computations are modulo 29).&lt;/p&gt;&#xA;&lt;p&gt;This is just to give you some ideas. You can play with these in various ways or you can ask students to write something similar from scratch or mimicking the existing code. The choice is yours. :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-11-08T02:47:10.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27107" PostTypeId="2" ParentId="27093" CreationDate="2023-11-08T04:12:01.660" Score="3" Body="&lt;p&gt;The expressions “if &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;” and “&lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; only if &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;” have the same literal meaning, but they carry different connotations—different enough that in many situations, only one of them sounds natural. To internalize their shared literal meaning, it might help to practice using them in situations where they both sound natural.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The king frowned. “The witch? She’s dangerous—too dangerous to ever set foot on this island again. I’m calling on her only if I have no one else.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“And you’re going? After what he did to you? Let someone else clean up his mess.”&lt;/p&gt;&#xA;&lt;p&gt;The witch shook her head. “If he’s calling on me, he has no one else.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“Don’t worry about the gumshoe,” the DA chuckled. “If she’s as smart as they say, then she drops that ring down a storm drain.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He tossed the ring on my desk, followed by a stack of twenties. “I hope you’re as smart as they say.”&lt;/p&gt;&#xA;&lt;p&gt;I shrugged. “I’m as smart as they say only if I drop this ring down a storm drain.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The geologist broke the silence. “He’s alive only if there’s oxygen down there somewhere. Oxygen we’ve seen no trace of.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The captain’s voice crackled in her ear. “Tjio, continue to base. Your air reserves are gone. If you stay out, you’ll never make it back.”&lt;/p&gt;&#xA;&lt;p&gt;Tjio was already wheeling around toward the signal. “If he’s alive, there’s oxygen down there somewhere.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“Can’t you”—she sobbed—“can’t you see? You’ll &lt;em&gt;never&lt;/em&gt; be safe from your past, until—” She whirled, eyes blazing. “The wedding goes ahead only if—only if you never see him again!”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He gazed across the table. The firelight vanished into the depths of his brown eyes. “The day you met her,” he said quietly, “was the first day you looked happy.”&lt;/p&gt;&#xA;&lt;p&gt;“If the wedding goes ahead... I never see you again.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5119" LastEditorUserId="5119" LastEditDate="2023-11-08T06:35:14.660" LastActivityDate="2023-11-08T06:35:14.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27108" PostTypeId="1" CreationDate="2023-11-08T06:28:55.540" Score="3" ViewCount="2693" Body="&lt;p&gt;Many students question the importance of complex numbers in real life. We can find many important applications of imaginary numbers in Engineering field and physics. This question is not related to that kind of general issue. Here what I want to know is how we can emphasize the importance of imaginary numbers when we are dealing with real numbers by using examples where we need knowledge of complex numbers to work with real numbers. &lt;br/&gt;&#xA;Here you have one example, there is a formula derived in finding roots of depressed cubic equation &lt;span class=&quot;math-container&quot;&gt;$x^3 - 3px - 2q = 0$&lt;/span&gt; and it can be stated as , &lt;br/&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$x = \root3\of{q+ \sqrt{q^2 - p^3}}+ \root3\of{q-\sqrt{q^2 - p^3}}$&lt;/span&gt;&lt;br/&gt;&#xA;Now if you use the formula to obtain roots of &lt;span class=&quot;math-container&quot;&gt;$x^3 - 15x - 4 = 0$&lt;/span&gt; you have to substitute &lt;span class=&quot;math-container&quot;&gt;$p = 5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q = 2$&lt;/span&gt; , which gives roots in complex form such as ,&lt;br/&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$x = \root3\of{2+ 11\sqrt{-1}}+ \root3\of{2-11\sqrt{-1}}$&lt;/span&gt; which can be simplified using the knowledge of complex numbers to obtain the real root 4 of the above mentioned cubic equation. Here we need to know &lt;span class=&quot;math-container&quot;&gt;$(2+i)^3 = 2 + 11i$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(2-i)^3=2-11i$&lt;/span&gt;&lt;br/&gt;&#xA;Actually this real root 4 can be directly found by applying factor theorem and without having idea of complex numbers we cannot verify that formula. I think this kind of awareness may be helpful for students to understand the value of studying complex numbers. What would you suggest in parallel to this?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2023-11-11T19:18:31.320" Title="Importance of complex numbers knowledge in real roots" Tags="|secondary-education|mathematical-pedagogy|complex-numbers|" AnswerCount="11" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="27109" PostTypeId="2" ParentId="27093" CreationDate="2023-11-08T06:41:34.250" Score="2" Body="&lt;p&gt;Following on from @Vectornaut's answer and the comment by @John Bollinger: I think the issue is that 'only if' can be interpreted differently in everyday language. For instance, suppose I said the following to my housemate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I will do the washing-up only if you clean the living room.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Upon hearing this utterance, my housemate thinks:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If I clean the living room, then he will do the washing-up.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In other words, the everyday interpretation of 'only if' in the above utterance is the &lt;em&gt;opposite&lt;/em&gt; to its interpretation in logic. Now, there are plenty of linguistic and logical nuances around the meaning and interpretation of the above utterance that one could discuss, but I think the immediate thought of my hypothetical housemate alone explains why students find the logical equivalence of 'if P, then Q' and 'P only if' confusing. As @Greg Martin, @Jochen Glueck and others have commented, using 'only if' outside of the phrase 'if and only if' is generally avoided, even by professional mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="20481" LastEditorUserId="20481" LastEditDate="2023-11-08T06:48:41.083" LastActivityDate="2023-11-08T06:48:41.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27110" PostTypeId="2" ParentId="27108" CreationDate="2023-11-08T07:26:58.397" Score="8" Body="&lt;p&gt;Another topic that uses complex numbers in service of real numbers, and is immediately accessible to a high schooler who has learned or is learning calculus, is &lt;a href=&quot;https://en.wikipedia.org/wiki/Integration_using_Euler%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;integration using Euler's formula&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here's a decent example from that article. (There are also some simpler examples and some more advanced examples in that article.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RqAlg.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RqAlg.png&quot; alt=&quot;integration using Euler's formula&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some extra references on this topic, courtesy of Dave L Renfro:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Paul Glaister, &lt;a href=&quot;https://www.tandfonline.com/toc/tmes20/22/3&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Complex integrals&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;International Journal of Mathematical Education in Science and Technology&lt;/strong&gt; 22 #3 (1991), pp. 476−477&lt;/p&gt;&#xA;&lt;p&gt;Charles Otto Gunther (1879−1958), &lt;strong&gt;Integration by Trigonometric and Imaginary Substitution&lt;/strong&gt;, D. Van Nostrand Company, 1907 (2nd edition -- only a corrected reprint -- is 1920), vi + 79 pages (&lt;a href=&quot;https://archive.org/details/integrationbytri00gunt&quot; rel=&quot;nofollow noreferrer&quot;&gt;internet archive copy&lt;/a&gt; and &lt;a href=&quot;https://catalog.hathitrust.org/Record/008857537&quot; rel=&quot;nofollow noreferrer&quot;&gt;HathiTrust copy&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books?id=kf4SAQAAMAAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google-Books copy&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-09T16:03:58.163" LastActivityDate="2023-11-09T16:03:58.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27111" PostTypeId="2" ParentId="27092" CreationDate="2023-11-08T10:53:49.060" Score="3" Body="&lt;p&gt;I like the following puzzle because the answer is unexpected and also because it doesn’t require any advanced mathematics.&lt;/p&gt;&#xA;&lt;p&gt;What is the next number in the following sequence?&lt;/p&gt;&#xA;&lt;p&gt;0, 4, 8, 21, 52, 65, 96, ?&lt;/p&gt;&#xA;&lt;p&gt;Answer:&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; The next number is 1.  The rule is to add 4 and then reverse the digits to get the next number.  For example 21+4=25 and reverse it to get 52.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I gave this puzzle to my daughter’s grade six class and as a group they solved it in about five minutes.&lt;/p&gt;&#xA;" OwnerUserId="23338" LastActivityDate="2023-11-08T10:53:49.060" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27112" PostTypeId="2" ParentId="27108" CreationDate="2023-11-08T12:00:41.960" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;we need knowledge of complex numbers to work with real numbers&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We need to be honest that at the level of your students, there are &lt;strong&gt;no examples of this&lt;/strong&gt;: in all sufficiently elementary uses of complex numbers, one can get by without them.&lt;/p&gt;&#xA;&lt;p&gt;Consider your example: if &lt;span class=&quot;math-container&quot;&gt;$p,q$&lt;/span&gt; are rational and &lt;span class=&quot;math-container&quot;&gt;$q^2&amp;lt;p^3$&lt;/span&gt;, then either your polynomial has a rational root (in which case it can be found directly using Eisenstein rule, for example), or evaluating the third root of a complex number will proceed via calculating its argument, and dividing by 3. Hence, your final answer will involve &lt;span class=&quot;math-container&quot;&gt;$\arccos$&lt;/span&gt; or some other similar function. It is a theorem that it does not simplify further, e.g., it's cannot be expressed in &lt;em&gt;real&lt;/em&gt; radicals. But, you can also reduce your equation to the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;4\cos^3t-3\cos t =q/p^\frac{3}{2},&#xA;$$&lt;/span&gt;&#xA;by substituting &lt;span class=&quot;math-container&quot;&gt;$x=2 p^\frac12 \cos t$&lt;/span&gt;. Since the left-hand side is now &lt;span class=&quot;math-container&quot;&gt;$\cos 3t$&lt;/span&gt; and the right-hand side is &lt;span class=&quot;math-container&quot;&gt;$\leq 1$&lt;/span&gt; in absolute value, we find three distinct solutions, corresponding to &lt;span class=&quot;math-container&quot;&gt;$$t_1=\frac13 \arccos(q/p^\frac{3}{2}), \quad t_{2,3}=\frac13 \arccos(q/p^\frac{3}{2})\pm\frac{2\pi}{3}.$$&lt;/span&gt; This is of course the same solution in disguise, and the same &lt;span class=&quot;math-container&quot;&gt;$\arccos$&lt;/span&gt; that you would have calculated in the approach with the complex numbers. So, in this case, not only aren't complex numbers &lt;em&gt;needed&lt;/em&gt;, they actually only complicate matters.&lt;/p&gt;&#xA;&lt;p&gt;That said, there are many elementary examples where complex numbers are sufficiently useful as a &lt;em&gt;notational device&lt;/em&gt; to motivate their introduction:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Euler's formulae for &lt;span class=&quot;math-container&quot;&gt;$\cos t$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin t$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$e^{it}$&lt;/span&gt; reduce trigonometric identities to algebraic ones. One example is given in Justin's answer; another one: calculate&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sin \frac{\pi}{100}+\sin\frac{2\pi}{100}+\dots+\sin\frac{50\pi}{100}.&#xA;$$&lt;/span&gt;&#xA;Of course, you can do it by any number of tricks, but if you use Euler's formula, no tricks are needed: this is just geometric series. The entire scope of &amp;quot;application&amp;quot; of complex numbers in electrical engineering, signal processing, etc., boils down to this. Another example: find &lt;span class=&quot;math-container&quot;&gt;$\int\cos x e^{x}$&lt;/span&gt;. You can do it by integrating by parts two times, but Euler's formula immediately gives &lt;span class=&quot;math-container&quot;&gt;$\Re(\frac{1}{1+i}e^{(1+i)x}).$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Polynomial factorization, in particular, e.g., as applied to partial fraction decomposition.  Suppose you want to integrate &lt;span class=&quot;math-container&quot;&gt;$$\int \frac{2x^2+x+1}{x^3+x}.\,dx$$&lt;/span&gt; The usual way to do it is to decompose this into simple fractions. Over the reals, the decomposition reads &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x}+\frac{x}{x^2+1}+\frac{1}{x^2+1},$&lt;/span&gt; but if you don't know that, you will need to solve a system of equiations. Then, you will have to remember how to integrate each of the terms. Over the complex numbers, on the other hand, we know a priori that the decomposition will be &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{x}+\frac{b}{x-i}+\frac{c}{x+i}$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; real and &lt;span class=&quot;math-container&quot;&gt;$b,c$&lt;/span&gt; complex conjugate. You can now observe that, e.g., &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is the limit of the above expression, multiplied by &lt;span class=&quot;math-container&quot;&gt;$(x-i)$&lt;/span&gt;, as &lt;span class=&quot;math-container&quot;&gt;$x\to i$&lt;/span&gt;. So, it can be found simply by removing &lt;span class=&quot;math-container&quot;&gt;$(x-i)$&lt;/span&gt;  from the denominator of the original fraction and plugging in &lt;span class=&quot;math-container&quot;&gt;$x=i$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$$b=\frac{3i^2+i+1}{i(i+1)}.$$&lt;/span&gt; No solving linear system is needed. To calculate integrals, you only need to remember that &lt;span class=&quot;math-container&quot;&gt;$\int \frac{dx}{x-a}=\log (x-a)+C$&lt;/span&gt; for any complex &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;; of course if you want an answer not involving complex logarithms, you will need to work a bit in the end.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2023-11-08T12:00:41.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27113" PostTypeId="2" ParentId="27108" CreationDate="2023-11-08T14:27:47.480" Score="1" Body="&lt;p&gt;This might be the oldest question in mathematics education! But  you are probably not going to like Euclid’s answer.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, &amp;quot;What do I get by learning these things?&amp;quot; So Euclid said &amp;quot;Give him threepence, since he must make a gain out of what he learns.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;When I encounter the &amp;quot;why do I need to learn this&amp;quot; sentiment of this question in the classroom, I turn the focus to a discussion of growth mindset and delayed gratification.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2023-11-08T14:27:47.480" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27114" PostTypeId="2" ParentId="27108" CreationDate="2023-11-08T16:52:26.597" Score="6" Body="&lt;p&gt;If series and radius of convergence are within the reach of your syllabus, then a standard example of &amp;quot;needing to go through the complex numbers&amp;quot; is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Radius_of_convergence#Radius_of_convergence_in_complex_analysis&quot; rel=&quot;noreferrer&quot;&gt;radius of convergence&lt;/a&gt; of the series expansion of&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x)=\frac{1}{x^2+1}&#xA;$$&lt;/span&gt;&#xA;around 0. Said radius is 1, even though &lt;span class=&quot;math-container&quot;&gt;$x^2+1$&lt;/span&gt; has no (real!) roots.&lt;/p&gt;&#xA;" OwnerUserId="12521" LastActivityDate="2023-11-08T16:52:26.597" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27115" PostTypeId="2" ParentId="5983" CreationDate="2023-11-08T17:34:42.973" Score="0" Body="&lt;p&gt;One point I think is worth emphasising is that working forward is not just a good method for derivation questions. When formulas for the final answer are not known, it is an &lt;em&gt;essential&lt;/em&gt; method. Consider the following toy example as an illustration of this. Say a confused student has a crazy new theory that 1=2. Working backwards, they multiply both sides by 0 to get 0=0. Seeing this is a true statement, they may think they have shown that 1=2.&lt;/p&gt;&#xA;" OwnerUserId="22233" LastActivityDate="2023-11-08T17:34:42.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27118" PostTypeId="2" ParentId="27108" CreationDate="2023-11-09T00:27:15.663" Score="2" Body="&lt;p&gt;AThe book 'Visual Complex Analysis' by Tristan Needham highlights the significance of complex numbers, emphasizing their historical importance in mathematics. It details how the real solutions of cubic equations were originally discovered using complex numbers.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;book excerpt:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/k8I9j.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/k8I9j.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2023-11-09T00:27:15.663" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27119" PostTypeId="2" ParentId="27065" CreationDate="2023-11-09T00:32:08.127" Score="0" Body="&lt;p&gt;A guess: Could it be a matter of time constraints? Thoroughly covering all potential outcomes and their implications is a time-intensive endeavor.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2023-11-09T00:32:08.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27120" PostTypeId="2" ParentId="27108" CreationDate="2023-11-09T05:17:32.783" Score="4" Body="&lt;p&gt;In the setting of single-variable calculus, here is one of the most striking uses of complex numbers: the radius of convergence of the power series at &lt;span class=&quot;math-container&quot;&gt;$x = 0$&lt;/span&gt; of a rational function &lt;span class=&quot;math-container&quot;&gt;$p(x)/q(x)$&lt;/span&gt;, written in reduced form, is the distance from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to the nearest root of &lt;span class=&quot;math-container&quot;&gt;$q(x)$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbf C$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example&lt;/strong&gt;.  Consider &lt;span class=&quot;math-container&quot;&gt;$1/(x^2+x+2)$&lt;/span&gt;.  The denominator has no real roots, but it has two complex roots, both with absolute value &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;, so the power series at &lt;span class=&quot;math-container&quot;&gt;$x = 0$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$1/(x^2+x+2)$&lt;/span&gt; has radius of convergence is &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;.  Good luck trying to explain that in a simple way using only real numbers.&lt;/p&gt;&#xA;&lt;p&gt;Complex numbers also reveal the tight connection between trigonometric functions and exponential functions. Through the power series of  &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;, we are led to define &lt;span class=&quot;math-container&quot;&gt;$e^z$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos z$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\sin z$&lt;/span&gt; at complex &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; and then are led to Euler's remarkable formula &lt;span class=&quot;math-container&quot;&gt;$e^{i\theta} = \cos\theta + i\sin\theta$&lt;/span&gt;, which reveals the &lt;em&gt;imaginary periodicity&lt;/em&gt; of the exponential function.  This is totally opaque when you only work with real numbers.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="590" LastEditDate="2023-11-09T13:45:37.250" LastActivityDate="2023-11-09T13:45:37.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27121" PostTypeId="1" AcceptedAnswerId="27123" CreationDate="2023-11-09T15:31:49.083" Score="-6" ViewCount="119" Body="&lt;p&gt;How to solve this probability question?f a, b, and c are in the range [0, 1], what is the probability that the quadratic equation ax^2 + bx + c has real solutions? justify your answer!&lt;/p&gt;&#xA;&lt;p&gt;This question was posed by the esteemed Portuguese mathematics educator, José Paulo Viana, during a session of the APM - Mathematics Teachers Association of Portugal(&lt;a href=&quot;https://www.facebook.com/photo/?fbid=718336196985035&amp;amp;set=a.415008250651166&amp;amp;__cft__%5B0%5D=AZWSIZa5VPhQfEw1WoyJo9yDPREx3Y5eBshgw9sHI9RYRnobpRh_nvBUEKPwW_U4pt2Z66dX39xJa7ZJIzRoYEIYugSvksorwI-1qIymHA5CeMGqmJR-zlYWC1cxpIXQk4VExGzEeSb8ZFFUAvAn11Do91iyHUYNi9kGLiQiFBEIw0zootIC61uUahWmo8TX6OQ&amp;amp;__tn__=%2CO*F&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.facebook.com/photo/?fbid=718336196985035&amp;amp;set=a.415008250651166&amp;amp;__cft__[0]=AZWSIZa5VPhQfEw1WoyJo9yDPREx3Y5eBshgw9sHI9RYRnobpRh_nvBUEKPwW_U4pt2Z66dX39xJa7ZJIzRoYEIYugSvksorwI-1qIymHA5CeMGqmJR-zlYWC1cxpIXQk4VExGzEeSb8ZFFUAvAn11Do91iyHUYNi9kGLiQiFBEIw0zootIC61uUahWmo8TX6OQ&amp;amp;__tn__=%2CO*F&lt;/a&gt;). It serves as an exemplary case of an engaging problem in the field of probability and has indeed captured my attention.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1m07h.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1m07h.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="8828" LastEditDate="2023-11-11T12:50:10.110" LastActivityDate="2023-11-11T12:50:10.110" Title="PROBABILITY QUESTION" Tags="|probability|" AnswerCount="1" CommentCount="5" ClosedDate="2023-11-10T05:52:30.860" ContentLicense="CC BY-SA 4.0" />
  <row Id="27122" PostTypeId="2" ParentId="27093" CreationDate="2023-11-09T15:44:38.540" Score="0" Body="&lt;p&gt;An example would be helpful.   As someone that might have answered 'false' on that question, I will tell you what I am thinking...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If P then Q&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This says to me that if proposition P is true, then Q must also be true, but there is no reason to say that P is true.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;P only if Q&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This seems to start with the assumption that &lt;em&gt;given no other information&lt;/em&gt;, P would be true, &lt;em&gt;but&lt;/em&gt; only if Q is true, and Q is the only thing required to make P true.&lt;/p&gt;&#xA;&lt;p&gt;The second is much less common in normal English usage.  Adding a comma when written or only a short pause when spoken changes the meaning.  &amp;quot;P, only if Q&amp;quot; makes the statement that P is definitely true as long as Q is true.  If spoken out loud, the comma would only be a short pause.&lt;/p&gt;&#xA;&lt;p&gt;Say you are looking at a chicken and make the statement &amp;quot;That animal is a mammal only if it is warm blooded&amp;quot;.   Now add a short pause &amp;quot;That animal is a mammal, only if it is warm blooded&amp;quot;.   Maybe that's part of it as well, the statement doesn't seem so strange to me when talking in abstracts: &amp;quot;An animal is a mammal only if it is warm blooded&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="14840" LastActivityDate="2023-11-09T15:44:38.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27123" PostTypeId="2" ParentId="27121" CreationDate="2023-11-09T16:17:53.510" Score="2" Body="&lt;p&gt;This sounds like a homework question, but it seems you're a &lt;a href=&quot;http://www.professores.im-uff.mat.br/hjbortol/&quot; rel=&quot;nofollow noreferrer&quot;&gt;professor&lt;/a&gt; with &lt;a href=&quot;https://scholar.google.com/citations?user=0v4HEQoAAAAJ&amp;amp;hl=pt-BR&quot; rel=&quot;nofollow noreferrer&quot;&gt;publications&lt;/a&gt; and years of &lt;a href=&quot;https://matheducators.stackexchange.com/users/8828/humberto-jos%C3%A9-bortolossi&quot;&gt;history&lt;/a&gt; on the site, so I'll give you the benefit of the doubt and assume you're asking for a good reason (though it would probably be worth editing your post to explain what that reason is).&lt;/p&gt;&#xA;&lt;p&gt;This question has been asked before on Math.SE (&lt;a href=&quot;https://math.stackexchange.com/questions/29242/probability-that-a-quadratic-polynomial-with-random-coefficients-has-real-roots&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/questions/1607395/probability-that-a-quadratic-equation-with-random-coefficients-has-real-roots&quot;&gt;here&lt;/a&gt;) where it has received plenty of thorough answers, all arriving at the same result. (If there are specific questions you have about those solutions, it would also be worth editing your post to explain what your specific questions are.)&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-11-09T16:17:53.510" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27124" PostTypeId="2" ParentId="27092" CreationDate="2023-11-09T16:48:13.247" Score="3" Body="&lt;p&gt;Here's a puzzle that I used to give to all grades Algebra through Calculus on the first day of class as something fun to do while many students were waiting to sort out schedule and laptop issues.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Draw an &lt;span class=&quot;math-container&quot;&gt;$10 \times 10$&lt;/span&gt; square grid. How many squares are there in total? Not just &lt;span class=&quot;math-container&quot;&gt;$1 \times 1$&lt;/span&gt; squares, but also &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; squares, &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; squares, and so on.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I liked this puzzle in particular because the solution was not immediately obvious to even the brightest / oldest students, yet all students (even the younger grades / weaker students) were able to understand the goal of the problem and work through concrete examples to notice a pattern that they could extrapolate to find the solution. (The higher grades / brighter students just picked up on the pattern faster.)&lt;/p&gt;&#xA;&lt;p&gt;Whenever students needed some guidance getting started, I would explicitly walk them through the process of drawing up a &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; grid and counting the number of &lt;span class=&quot;math-container&quot;&gt;$1 \times 1$&lt;/span&gt; squares (there are &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;), &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; squares (there are &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;), and &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; squares (there is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;), for a total of &lt;span class=&quot;math-container&quot;&gt;$9 + 4 + 1 = 14$&lt;/span&gt; squares. I would then leave them alone to do this for a &lt;span class=&quot;math-container&quot;&gt;$4 \times 4$&lt;/span&gt; grid and then a &lt;span class=&quot;math-container&quot;&gt;$5 \times 5$&lt;/span&gt; grid, after which I would help them notice the following pattern:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;3 \times 3 \textrm{ grid} \quad \to &amp;amp; \quad 9 + 4 + 1 \\&#xA;&amp;amp; \quad 3^2 + 2^2 + 1^2 \\[5pt]&#xA;4 \times 4 \textrm{ grid} \quad \to &amp;amp; \quad 16 + 9 + 4 + 1 \\&#xA;&amp;amp; \quad 4^2 + 3^2 + 2^2 + 1^2 \\[5pt]&#xA;5 \times 5 \textrm{ grid} \quad \to &amp;amp; \quad 25 + 16 + 9 + 4 + 1 \\&#xA;&amp;amp; \quad 5^2 + 4^2 + 3^2 + 2^2 + 1^2 \\[5pt]&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This pattern is almost a &amp;quot;punch line&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What's the total number of squares (square shapes)? Well, it's the total sum of squares (square numbers).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Extrapolating the pattern to solve the puzzle:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;10 \times 10 \textrm{ grid} \quad \to &amp;amp; \quad 10^2 + 9^2 + 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2 \\&#xA;&amp;amp; \quad = 385 \textrm{ squares in total} \\[5pt]&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-13T21:16:45.230" LastActivityDate="2023-11-13T21:16:45.230" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27125" PostTypeId="2" ParentId="27078" CreationDate="2023-11-09T17:42:58.717" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Is there a good intuitive explanation for how to think about the order of steps when constructing these compound transformations?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here's a stab at what it would mean to obey the order of operations. Note: In what follows, I am using the interpretation of curve transformations to mean we are transforming the underlying axis, not moving a curve around on the plane. I don't think I would ever teach this interpretation like this, but here is how one might just follow the order of operations to correctly draw a transformed function.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example&lt;/strong&gt;:&lt;sup&gt;*&lt;/sup&gt; Suppose we want to graph the curve &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}y - 1 = \sqrt{2x-6}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let's first graph the standard function &lt;span class=&quot;math-container&quot;&gt;$y=\sqrt{x}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Pgp3Gm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Pgp3Gm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now start changing the horizontal axis. We multiply the values on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis by a factor of 2 (making it the &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;-axis). Redraw the curve:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8fjJDm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8fjJDm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Next, subtract 6 from the values on the &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;-axis (making it the (&lt;span class=&quot;math-container&quot;&gt;$2x-6$&lt;/span&gt;)-axis). Redraw the curve:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kQr0km.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kQr0km.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now start changing the vertical axis. We multiply the values on the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis by &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; (making it the &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}y$&lt;/span&gt;-axis). Redraw the curve:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4mFGAm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4mFGAm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Next, subtract 1 from the values on the &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}y$&lt;/span&gt;-axis, (making it the (&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}y-1$&lt;/span&gt;)-axis):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Kgdmim.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Kgdmim.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Finally, put this curve back on the original &lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt;-plane:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xKFZam.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xKFZam.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is the correct curve which we found by altering the two coordinate axes by just using the order of operations. Note: This is entirely different from how I have ever taught function transformations -- I just wanted to think through how one might obey the order of operations and get the right curve.&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, it made me think of a simpler task: Solving a linear equation. Try this same thing out with one axis in order to solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Example&lt;/strong&gt;: Solve the equation &lt;span class=&quot;math-container&quot;&gt;$4x-5=-3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;First, just plot the number &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt; on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TlKih.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TlKih.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Next, multiply the values on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis by 4 (making it...you guessed it, the (&lt;span class=&quot;math-container&quot;&gt;$4x$&lt;/span&gt;)-axis). Redraw the point:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NYvY7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NYvY7.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This updates the equation, which now says that &lt;span class=&quot;math-container&quot;&gt;$4x=-3$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$x=-\frac{3}{4}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then subtract 5 to the values on the (&lt;span class=&quot;math-container&quot;&gt;$4x$&lt;/span&gt;)-axis (making it the (&lt;span class=&quot;math-container&quot;&gt;$4x-5$&lt;/span&gt;)-axis). Redraw the point:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oLPDV.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oLPDV.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This again updates the equation, which says that &lt;span class=&quot;math-container&quot;&gt;$4x-5=-3$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$x = \frac{1}{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Summary&lt;/strong&gt;: There's probably nothing novel about this, but I just wanted to see what it would mean to strictly follow the order of operations in a transformation, since that's what your students &lt;em&gt;feel&lt;/em&gt; we should be doing. My takeaway here is that while it does work, it is perhaps more work than we want to take on in class.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;[&lt;sup&gt;*&lt;/sup&gt;I am purposefully &lt;em&gt;not&lt;/em&gt; isolating &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; here in order to avoid the inconsistencies we have to explain to students -- the first being that when we solve for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, then any &lt;em&gt;outside&lt;/em&gt; change to a function behaves opposite to an &lt;em&gt;inside&lt;/em&gt; change. I.e. &lt;span class=&quot;math-container&quot;&gt;$y=(x-5)^2 + 3$&lt;/span&gt; involves positive movements in both &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, though one is brought about by subtraction, and the other by addition.]&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="470" LastEditorUserId="470" LastEditDate="2023-11-09T18:17:55.017" LastActivityDate="2023-11-09T18:17:55.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27126" PostTypeId="2" ParentId="27108" CreationDate="2023-11-10T00:55:20.650" Score="6" Body="&lt;p&gt;Consider the &lt;strong&gt;real&lt;/strong&gt; initial value problem&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{cases}&#xA;y'' + 2y' + 10y = 0\\&#xA;y(0) = 0\\&#xA;y'(0) = 1&#xA;\end{cases}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;One nice way to do it is to first finding generic solutions with real inputs but complex outputs!  Guessing that we may have some solutions of the form &lt;span class=&quot;math-container&quot;&gt;$y = e^{\alpha t}$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; complex, we obtain&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;&amp;amp;y'' + 2y' + 10y = 0\\&#xA;&amp;amp;(\alpha^2 + 2\alpha + 10) e^{\alpha t} = 0\\&#xA;&amp;amp;\alpha^2 + 2\alpha + 10 = 0\\&#xA;&amp;amp;\alpha^2 + 2\alpha + 1 + 9 = 0\\&#xA;&amp;amp;(\alpha + 1)^2 = -9\\&#xA;&amp;amp;\alpha + 1 = \pm -3i\\&#xA;&amp;amp;\alpha = -1 \pm -3i&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So we have solutions&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$e^{(-1 - 3i)t} = e^{-t}(\cos(3t) - i\sin(3t))$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$e^{(-1 + 3i)t} = e^{-t}(\cos(3t) + i\sin(3t))$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;These are complex valued solutions, but we can see that adding/subtracting and rescaling them leads to two real valued solutions:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;&amp;amp;e^{-t}\cos(3t)\\&#xA;&amp;amp;e^{-t}\sin(3t)&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So our general solution is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;ae^{-t}\cos(3t) + be^{-t}\sin(3t)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Using our initial values we find that the solution to our IVT is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;y = \frac{1}{3}e^{-t}\sin(3t)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;&lt;a href=&quot;https://i.stack.imgur.com/YvPAw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YvPAw.png&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;This is an important example of the solution to a problem with &lt;strong&gt;real&lt;/strong&gt; numbers being more easily solved when we expand our notion of solution to permit &lt;strong&gt;complex&lt;/strong&gt; values.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2023-11-11T15:01:01.003" LastActivityDate="2023-11-11T15:01:01.003" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27127" PostTypeId="2" ParentId="27108" CreationDate="2023-11-10T01:10:37.760" Score="1" Body="&lt;p&gt;Students who are interested in geometry might find &lt;a href=&quot;https://en.wikipedia.org/wiki/Ptolemy%27s_theorem#Proof_using_complex_numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;this proof&lt;/a&gt; of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ptolemy%27s_theorem#&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ptolemy's theorem&lt;/a&gt; to be a compelling application of complex numbers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;AC\cdot BD = AB\cdot CD+BC\cdot AD&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th style=&quot;text-align: center;&quot;&gt;&lt;a href=&quot;https://i.stack.imgur.com/hQJlzm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hQJlzm.png&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;" OwnerUserId="117" LastActivityDate="2023-11-10T01:10:37.760" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27128" PostTypeId="2" ParentId="27108" CreationDate="2023-11-10T15:54:08.540" Score="2" Body="&lt;p&gt;The complex numbers are really just the scale-rotations in 2D geometry. So you can get lots of examples from geometry.&lt;/p&gt;&#xA;&lt;p&gt;We start with two particular scale-rotations that form a basis for building up the rest. These are:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1=\pmatrix{1 &amp;amp; 0 \\ 0 &amp;amp; 1}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$i=\pmatrix{0 &amp;amp; 1 \\ -1 &amp;amp; 0}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then a general scale-rotation can be written:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a+bi=\pmatrix{a &amp;amp; b \\ -b &amp;amp; a}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Scale-rotations obey all the usual rules of algebra. We can add them. We can multiply them. Multiplication distributes over addition. There are additive inverses for all of them, and multiplicative inverses for all but zero. And they're &lt;em&gt;better&lt;/em&gt; than matrices, because multiplication is commutative!&lt;/p&gt;&#xA;&lt;p&gt;Rotation matrices are obviously of this form.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos \theta + i \sin \theta =\pmatrix{\cos \theta &amp;amp; \sin \theta \\ -\sin \theta &amp;amp; \cos \theta}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Take a simple problem from geometry, add up the three angles shown here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YAjd9.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YAjd9.png&quot; alt=&quot;Add angles of lines from origin to three points (1,1),(2,1),(3,1)&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can transform the unit line segment shown in purple to any of the three other lines with a scale-rotation, as shown. (We can thus label every point of the plane with the particular scale-rotation that maps a selected unit line segment to it.) Each scale-rotation rotates its input through one of the three angles illustrated. So if we do the three scale-rotations in succession, the result will be a scale-rotation that rotates through the sum of the three angles.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\pmatrix{1 &amp;amp; 1 \\ -1 &amp;amp; 1} \pmatrix{2 &amp;amp; 1 \\ -1 &amp;amp; 2} \pmatrix{3 &amp;amp; 1 \\ -1 &amp;amp; 3}=\pmatrix{0 &amp;amp; 10 \\ -10 &amp;amp; 0}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can write this more briefly and clearly as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(1+i)(2+i)(3+i)=10 i$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This implies that the sum of the three angles is &lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We have turned geometry into algebra! We can use all the normal algebraic rules we just spent years learning, and adapt them to simplifying geometry problems. Geometry only involves real numbers - real number lengths, real number angles, real number areas, and so on. Scale-rotations are easy enough to grasp geometrically, but they have this unfamiliar property that &lt;span class=&quot;math-container&quot;&gt;$i^2=-1$&lt;/span&gt;, which we have previously always said is impossible. We now discover that it &lt;em&gt;is&lt;/em&gt; impossible for the scaling transformations, but &lt;em&gt;not&lt;/em&gt; impossible for the scale-rotations. You can build a &lt;span class=&quot;math-container&quot;&gt;$180^\circ$&lt;/span&gt; rotation from two identical &lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt; rotations, but you can't build it from two identical scalings (i.e. by squaring real numbers).&lt;/p&gt;&#xA;&lt;p&gt;Historically, the mathematicians found the algebra first, and only spotted the geometrical connection to scale-rotations much later. This misled everyone into thinking the rotations part of the algebra were made up of nonsensical, or 'imaginary' numbers. They worked, but people didn't know why.&lt;/p&gt;&#xA;&lt;p&gt;Mathematics has lots of situations like this: where two completely separate areas of study turn out to be about exactly the same thing, in disguise. Then all the tools we developed for solving problems in one subject can suddenly be used to solve problems in the other subject, and vice versa. This often gives us new avenues for solving problems that have previously resisted all efforts.&lt;/p&gt;&#xA;" OwnerUserId="23355" LastActivityDate="2023-11-10T15:54:08.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27129" PostTypeId="2" ParentId="27037" CreationDate="2023-11-10T15:57:20.953" Score="0" Body="&lt;p&gt;This kind of approach may be helpful to improve exploring skills in mathematics and if you can start by giving hints for elementary students later on they can buildup lateral thinking which is the key point in creating new conjectures and solving global challenges. Here I share my suggestions, &lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How to illustrate geometrically that, sum of two consecutive triangle numbers is a square number and what are the other cases relevant to triangle numbers which you can be proved geometrically?&lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Many of the students know how to obtain the product when single digit number is multiplied by 9 using the pattern. How can you describe the pattern when two digit or three digit numbers are multiplied by 9 ?&lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;In 9 th century Al-Khuwarizmi discovered completing square method by geometrical approach. Since negative numbers were not used at that time he described his approach for six different types of quadratic equations. How can you apply his geometrical approach for negative coefficients without using negative values as side lengths?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2023-11-10T15:57:20.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27130" PostTypeId="1" CreationDate="2023-11-10T17:02:37.883" Score="9" ViewCount="1175" Body="&lt;p&gt;Consider the expression&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$|x + 2|x + 3|x + 4|.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;One way to interpret this is that there are two products being added together:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$|x+2|x \hspace{1cm} + \hspace{1cm} 3|x+4|$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But you could also interpret it as the absolute value of an expression that itself contains an absolute value:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$|x \hspace{1cm} +2|x+3|x \hspace{1cm} +4|$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;These two interpretations are not equivalent. For instance, substituting &lt;span class=&quot;math-container&quot;&gt;$x=0{:}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;|0+2|0 \hspace{1cm} &amp;amp;+ \hspace{1cm} 3|0+4| \hspace{1cm} = 12 \\[7pt]&#xA;|0 \hspace{1cm} +2|0&amp;amp;+3|0 \hspace{1cm} +4| \hspace{1cm} = 4 \\[7pt]&#xA;|0 + 2|0 &amp;amp;+ 3|0 + 4| \hspace{2cm} = \,\, ???&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Granted, this is a contrived example and I've never actually seen an ambiguous case come up in real life (or in any math textbook). I only stumbled upon this while developing algorithms to handle edge cases in a free response grader a couple years ago. (And even then, behavior on this edge case doesn't make a difference in practice since none of the correct answers that would be graded against involve ambiguous notation.)&lt;/p&gt;&#xA;&lt;p&gt;I also realize that the expression could be made un-ambiguous by explicitly writing multiplication symbols, or by tweaking the absolute value notation to distinguish between left bars and right bars (e.g. via sizing/spacing/padding, or by writing &lt;span class=&quot;math-container&quot;&gt;$\textrm{abs}(\cdot)$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$|\cdot|$&lt;/span&gt;). And I realize that there are obvious ways to solve this issue in the context of software (e.g. designing a user interface that avoids ambiguous notation, or using heuristics like choosing the interpretation with the lowest nesting depth).&lt;/p&gt;&#xA;&lt;p&gt;But I'm still curious to know: &lt;strong&gt;given an ambiguous absolute value expression, is there a standard convention for interpreting it?&lt;/strong&gt; In other words, loosely speaking, is there a standard &amp;quot;order of operations&amp;quot; for parallel vs nested absolute value expressions, in the absence of clarifying notation?&lt;/p&gt;&#xA;&lt;p&gt;(The answer may be that there is no agreed-upon rule.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt; Dave L Renfro posted a comment &lt;em&gt;&amp;quot;If there is [a standard convention], then it almost certainly would only be applied in a computer coding (or calculator) setting, and it would not generally be known in the mathematical community&amp;quot;&lt;/em&gt; which seems convincing enough that there is no agreed-upon rule: mathematicians use symbol sizing (or other notational means) to avoid ambiguity and therefore have no need for a rule to interpret ambiguous cases (since ambiguous cases shouldn't exist in a mathematical text).&lt;/p&gt;&#xA;&lt;p&gt;I will accept if this comment is posted as an answer or integrated to the parent answer.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-15T02:38:20.953" LastActivityDate="2023-11-15T02:38:20.953" Title="Is there a standard convention for interpreting ambiguous absolute value expressions?" Tags="|secondary-education|algebra|notation|absolute-value|" AnswerCount="3" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="27131" PostTypeId="2" ParentId="27130" CreationDate="2023-11-10T17:24:45.037" Score="8" Body="&lt;p&gt;If I want to have nested absolute-value expressions, I would use different sizes&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\big|x + 2|x + 3|x + 4\big|,&#xA;$$&lt;/span&gt;&#xA;with variations possible&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\bigg|x + 2|x + 3|x + 4\bigg|.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="127" LastEditorUserId="127" LastEditDate="2023-11-14T02:21:44.280" LastActivityDate="2023-11-14T02:21:44.280" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27132" PostTypeId="2" ParentId="27130" CreationDate="2023-11-10T21:21:37.133" Score="8" Body="&lt;p&gt;I don't know about standards, but I read these things using a left to right, greedy algorithm. More specifically, the bars are like parentheses but you don't automatically know if they are opening or closing parens. The first one, though, hast to be an opener. The next one going to the right could be an opener or a closer. Try thinking of the expression as if it were a closer first. Does this make sense? Inside your pair of bars and to the right of them? If so, those were a pair. Otherwise, the bar was another opener. Continue this in a recursive fashion.&lt;/p&gt;&#xA;&lt;p&gt;This procedure would give you the first of the two options and tries to make the nesting depth shallow.&lt;/p&gt;&#xA;&lt;p&gt;The second of the two options is cursed, blech!&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2023-11-10T21:21:37.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27133" PostTypeId="1" CreationDate="2023-11-10T22:46:43.927" Score="6" ViewCount="1903" Body="&lt;p&gt;It is known that &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\sum_1^{\infty} \frac{1}{n^{1.000001}}$&lt;/span&gt; converges while &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\sum_{n\text{ is a prime number}}\frac{1}{n}$&lt;/span&gt; diverges. Though we can logically prove these results, I found it difficult to explain their intuition to my students. How could it be done?&lt;/p&gt;&#xA;" OwnerUserId="6166" LastEditorUserId="851" LastEditDate="2023-11-13T15:32:19.470" LastActivityDate="2023-11-14T21:32:56.953" Title="How can we explain intuitively the convergence and divergence of these two series?" Tags="|undergraduate-education|calculus|intuition|series|prime-numbers|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27134" PostTypeId="2" ParentId="27133" CreationDate="2023-11-10T23:36:31.660" Score="6" Body="&lt;p&gt;For me, the intuition just comes from the integral test (which is itself intuitive since a series is just a Riemann sum of rectangles with unit width).&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th prime is asymptotically &lt;span class=&quot;math-container&quot;&gt;$n \ln n$&lt;/span&gt; (see &lt;a href=&quot;https://math.stackexchange.com/a/2853839/1203732&quot;&gt;here&lt;/a&gt; for intuition on that), and the integral of &lt;span class=&quot;math-container&quot;&gt;$\dfrac{1}{x \ln x}$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\ln \ln x,$&lt;/span&gt; whose end behavior is divergence as &lt;span class=&quot;math-container&quot;&gt;$x \to \infty.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But as soon as that exponent in the denominator exceeds &lt;span class=&quot;math-container&quot;&gt;$1,$&lt;/span&gt; the integral is a reciprocal power function, whose end behavior is convergence as &lt;span class=&quot;math-container&quot;&gt;$x \to \infty.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-12T00:49:31.083" LastActivityDate="2023-11-12T00:49:31.083" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27136" PostTypeId="2" ParentId="27108" CreationDate="2023-11-11T04:37:04.303" Score="1" Body="&lt;p&gt;Complex numbers don't have an ordering like real numbers, i.e. &lt;span class=&quot;math-container&quot;&gt;$a + i b\ngtr c + id$&lt;/span&gt;. This is both subtle and profound. To explain, assign north as imaginary and east as real and then talk about locations in a classroom as complex numbers. Perform distance calculations between those locations using complex numbers to get real-valued distances. If you're there, figure out the angle between two lines from their point of intersection. Will the answers to either of these questions change if we move or rotate the origin? Does the origin matter for complex numbers?&lt;/p&gt;&#xA;" OwnerUserId="23087" LastActivityDate="2023-11-11T04:37:04.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27137" PostTypeId="2" ParentId="27133" CreationDate="2023-11-11T05:27:45.377" Score="11" Body="&lt;p&gt;Look at a simpler example first:  &lt;span class=&quot;math-container&quot;&gt;$(1.000000000001)^n$&lt;/span&gt; compared to &lt;span class=&quot;math-container&quot;&gt;$0.9999999999^n$&lt;/span&gt;.  Do they accept that the first sequence tends to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; and the second to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; even though it would take quite a while to see either such behavior happening?  Not all interesting mathematical properties in a sequence have to occur among the first 100, 1000, or even million terms.&lt;/p&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2023-11-12T21:16:50.147" LastActivityDate="2023-11-12T21:16:50.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27138" PostTypeId="1" CreationDate="2023-11-11T12:57:00.467" Score="-4" ViewCount="262" Body="&lt;p&gt;Why was the order of operations established in mathematics with multiplication taking precedence over addition, as dictated by the PEMDAS rule? What historical or practical factors influenced this choice? Additionally, what would be the implications for mathematical consistency and computation if this hierarchical order were changed?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iLYTC.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iLYTC.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2023-11-21T18:38:18.020" Title="Order of operations pemdas" Tags="|notation|arithmetic-operations|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27139" PostTypeId="2" ParentId="27138" CreationDate="2023-11-11T15:27:18.883" Score="3" Body="&lt;p&gt;I don't have historical proof but the following explanation feels so intuitive to me that I'd be shocked if there were a different reason.&lt;/p&gt;&#xA;&lt;p&gt;When you use multiplication and addition for simple real-life applications, you're usually trying to find the total of &amp;quot;some number of this and some number of that.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;For instance, if you buy two &lt;span class=&quot;math-container&quot;&gt;$3 \, \textrm{kg}$&lt;/span&gt; bags of rice and four &lt;span class=&quot;math-container&quot;&gt;$5 \, \textrm{kg}$&lt;/span&gt; bags of wheat, then how many kilograms of grain do you have in all?&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(2 \times 3) + (4 \times 5),$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;not&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2 \times (3 + 4) \times 5.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Because we do multiplication first by convention, we can just write&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2 \times 3 + 4 \times 5$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;without parentheses and by default it will still represent the real-life situation that we are modeling.&lt;/p&gt;&#xA;&lt;p&gt;But if we did addition first, then we would have to introduce explicit parentheses around the multiplications &lt;span class=&quot;math-container&quot;&gt;$(2 \times 3) + (4 \times 5)$&lt;/span&gt; pretty much every time that we wanted to write down an expression for a real-life modeling situation, because &lt;span class=&quot;math-container&quot;&gt;$2 \times 3 + 4 \times 5$&lt;/span&gt; by default would not represent what we are intending.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-11-11T15:27:18.883" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27140" PostTypeId="2" ParentId="27108" CreationDate="2023-11-11T19:18:31.320" Score="0" Body="&lt;p&gt;My favorite goes approximately like this:&lt;/p&gt;&#xA;&lt;p&gt;The geometric series &lt;span class=&quot;math-container&quot;&gt;$\sum\limits_{i=0}^{\infty}\frac{1}{x^n}$&lt;/span&gt; converges to a function &lt;span class=&quot;math-container&quot;&gt;$f =\frac{1}{1-x}$&lt;/span&gt; only on &lt;span class=&quot;math-container&quot;&gt;$-1\lt x \lt 1$&lt;/span&gt;. On the other hand, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is well defined way beyond the interval of convergence. Is it a coincidence, a freak of nature, or what?&lt;/p&gt;&#xA;&lt;p&gt;It is more or less easy to demonstrate how it can be continued towards &lt;span class=&quot;math-container&quot;&gt;$-\infty$&lt;/span&gt;. However, there is no way - as long as we stay in a domain of reals - to continue it into &lt;span class=&quot;math-container&quot;&gt;$x \gt 1$&lt;/span&gt;. We cannot jump over the singularity.&lt;/p&gt;&#xA;&lt;p&gt;In the complex domain we don't need to jump over the singularity. We can walk around it, and lo and behold, we have a continuation.&lt;/p&gt;&#xA;" OwnerUserId="17936" LastActivityDate="2023-11-11T19:18:31.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27141" PostTypeId="2" ParentId="20649" CreationDate="2023-11-11T20:06:19.503" Score="0" Body="&lt;p&gt;Being a graduate student and having studied mathematics for almost all my life, I think I am in position to answer this question. Well I love mathematics but that necessarily don’t mean everyone does the same. There could be no reason for anyone to hate maths unless the followings:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;One had practiced memorisations while doing mathematics&lt;/li&gt;&#xA;&lt;li&gt;One didn’t had a good teacher in high school&lt;/li&gt;&#xA;&lt;li&gt;One couldn’t grasp the intuitive ideas behind any mathematical proof.&lt;/li&gt;&#xA;&lt;li&gt;One doesn’t questions the theory.&lt;/li&gt;&#xA;&lt;li&gt;One is scared of the notations used in mathematics.&lt;/li&gt;&#xA;&lt;li&gt;One doesn’t have fascinations for reasonings.&lt;/li&gt;&#xA;&lt;li&gt;One has a gap in knowledge.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The list could go on.&lt;/p&gt;&#xA;&lt;p&gt;These are the fundamental reasons why many people are scared of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="23360" LastActivityDate="2023-11-11T20:06:19.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27143" PostTypeId="2" ParentId="27133" CreationDate="2023-11-12T01:37:15.377" Score="1" Body="&lt;p&gt;Consider the fact that &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty n^x$&lt;/span&gt; converges if &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt;0$&lt;/span&gt;, diverges if &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0$&lt;/span&gt;. Clearly the transition from just a little bit negative to just a little bit positive makes a big change to the sum. This may not help your students see &lt;em&gt;why&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; diverges, but it should at least help them not to be surprised that it behaves differently.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Afterword&lt;/strong&gt;. I forgot to mention that &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty n^x = \zeta(-x)$&lt;/span&gt;, and the Riemann zeta function has a pole at &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt;. That may put it in conext for students who want to explore.&lt;/p&gt;&#xA;" OwnerUserId="19663" LastEditorUserId="19663" LastEditDate="2023-11-13T00:22:43.047" LastActivityDate="2023-11-13T00:22:43.047" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27144" PostTypeId="2" ParentId="5614" CreationDate="2023-11-12T13:33:24.497" Score="2" Body="&lt;p&gt;I think the best way to think about this is that is to use the concept &amp;quot;as if&amp;quot;.  For an example, for &lt;span class=&quot;math-container&quot;&gt;$f(x+2)$&lt;/span&gt;, it is like &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, but &lt;em&gt;as if&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; were two values ahead.  That's why the graph shifts to the left, because the other values are operating &lt;em&gt;as if&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is two greater than it actually is.&lt;/p&gt;&#xA;&lt;p&gt;This works for the other transformations as well.  &lt;span class=&quot;math-container&quot;&gt;$f(2x)$&lt;/span&gt; shrinks the graph on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; axis because we are operating &lt;em&gt;as if&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is two times larger than it actually is.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2023-11-12T13:33:24.497" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27145" PostTypeId="1" CreationDate="2023-11-13T01:56:55.950" Score="1" ViewCount="114" Body="&lt;p&gt;There are many &amp;quot;rules&amp;quot; for speed arithmetic.&lt;/p&gt;&#xA;&lt;p&gt;List of some reference links showing speed methods or rules:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ofpad.com/multiplication-tricks-for-mental-math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ofpad.com/multiplication-tricks-for-mental-math/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ofpad.com/mental-math-tricks-speed-math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ofpad.com/mental-math-tricks-speed-math/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://byjus.com/maths/multiplication-tricks/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://byjus.com/maths/multiplication-tricks/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.udemy.com/course/master-speed-math-multiplication-division-techniques/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.udemy.com/course/master-speed-math-multiplication-division-techniques/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Vedic_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Vedic_Mathematics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mental_abacus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Mental_abacus&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Chisanbop&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Chisanbop&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Many of these are clever, some are insightful, some are limited.&lt;/p&gt;&#xA;&lt;p&gt;Are there any authoritative guides that show which types of speed arithmetic methods are actually valuable for students in higher mathematics?&lt;/p&gt;&#xA;" OwnerUserId="23366" LastActivityDate="2023-11-13T13:13:03.907" Title="Speed math appropriate for middle-school students" Tags="|secondary-education|teaching|arithmetic-operations|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27146" PostTypeId="2" ParentId="27092" CreationDate="2023-11-13T08:29:49.507" Score="3" Body="&lt;p&gt;This one is a mathematics killer:&lt;br /&gt;&#xA;What's the next line, knowing that the numbers can only be &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; (you can never have &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; or higher):&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$1 1$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$2 1$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$1 2 1 1$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$1 1 1 2 2 1$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$3 1 2 2 1 1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; You just need to read what it says:&#xA; &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; reads as &amp;quot;one one&amp;quot; (one time the number one)&lt;br /&gt;&#xA; &lt;span class=&quot;math-container&quot;&gt;$1 1$&lt;/span&gt; reads as &amp;quot;two ones&amp;quot; (two times the number one)&#xA; &lt;span class=&quot;math-container&quot;&gt;$2 1$&lt;/span&gt; reads as &amp;quot;one two one one (one time the number two and one time the number one)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Why do I call this a mathematics killer? Well, the better you know mathematics, the more you will look for additions, subtractions and other multiplications, powers, primes, ..., and you'll never find it, although this has nothing to do with the actual solution.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-11-13T08:29:49.507" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27147" PostTypeId="1" AcceptedAnswerId="27148" CreationDate="2023-11-13T09:04:25.583" Score="0" ViewCount="1793" Body="&lt;p&gt;I am a maths enthusiast and have been exposed to maths olympiad problems for some time. I wanted to know how does a math Olympian do in undergraduate courses of mathematics, statistics or computer science. Are the problems given on the courses even harder than maths Olympiad? And do the maths olympians score full marks in their semester exam? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="23368" LastEditorUserId="2083" LastEditDate="2023-11-17T10:02:17.767" LastActivityDate="2023-11-17T10:02:17.767" Title="How does a math Olympian fare in undergraduate maths courses?" Tags="|undergraduate-education|math-contests|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27148" PostTypeId="2" ParentId="27147" CreationDate="2023-11-13T10:09:02.007" Score="15" Body="&lt;p&gt;Mathematics is many things. Solving tricky problems is part of it, but there is also learning theory and new concepts and ways of thinking, and even developing routine.&lt;/p&gt;&#xA;&lt;p&gt;I would expect a person with good results of mathematics olympiads and similar contests to have a lot of skills that are useful in studying mathematics, like problem-solving skills (including persistence and not giving up in the face of hard problems), good routine and good basic knowledge of certain fields of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;However, this does not guarantee success in understanding new theories and ways of thinking. These are specific to a given field or subfield.&lt;/p&gt;&#xA;&lt;p&gt;All other things being equal, I would generally expect nice results, but only if the student does the required work. There is still a lot to learn. Thinking that one is already finished and can coast along leads to failure within a couple of years, and is a definite risk if everything has always been easy and one is &amp;quot;clever&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="376" LastEditDate="2023-11-15T06:45:03.467" LastActivityDate="2023-11-15T06:45:03.467" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27149" PostTypeId="2" ParentId="27145" CreationDate="2023-11-13T13:13:03.907" Score="4" Body="&lt;p&gt;I agree with @DanFox's comment, that speed in arithmetic has no bearing on how well someone can learn higher mathematics. And yet, there are some methods for mental math that help one see relationships.&lt;/p&gt;&#xA;&lt;p&gt;Beast Academy, an excellent curriculum for elementary school level math, teaches students to use methods that work best when you visualize. That visualizing is helpful with higher mathematics. [Until a few weeks ago, I had no connection with them. Now I am in the middle of applying to teach a course for Art of Problem Solving, the company that produced this curriculum.]&lt;/p&gt;&#xA;&lt;p&gt;For example, how would you find 65&lt;sup&gt;2&lt;/sup&gt; in your head? I imagine a square, 65 by 65. Now I take a strip, 5 by 65, and move it from the top to the right side (rotated to lie alongside the big rectangle). If I take off a 5 by 5 square at the top of that strip, I now have a rectangle that is 60 by 70. So 65&lt;sup&gt;2&lt;/sup&gt; equals 60*70 +25, which is 4225.&lt;/p&gt;&#xA;&lt;p&gt;Doing mental math where you understand why each step works will help with algebraic reasoning.&lt;/p&gt;&#xA;&lt;p&gt;So you would want to evaluate each method. If it feels like a trick, it might be a fun way to do the arithmetic, but probably is of no use further along. It it feels like a method that makes sense, perhaps it will be helpful long-term.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2023-11-13T13:13:03.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27150" PostTypeId="2" ParentId="27133" CreationDate="2023-11-13T13:58:45.243" Score="3" Body="&lt;p&gt;Intuitively, to me, it means that if you take the positive number line, put a blue dot at every prime, and a red dot on all the the numbers of the form &lt;span class=&quot;math-container&quot;&gt;$n^{1.000000000001}$&lt;/span&gt;, then eventually, very far out on the number line, there will be more blue dots than red dots.&lt;/p&gt;&#xA;" OwnerUserId="7624" LastActivityDate="2023-11-13T13:58:45.243" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27151" PostTypeId="2" ParentId="27130" CreationDate="2023-11-13T16:59:04.917" Score="4" Body="&lt;p&gt;As I was looking at your expression, something just seemed typographically off, and then I realized that it was the missing padding around the bars that you see when mathematics is well-typeset. This helps indicate where juxtaposition-as-multiplication is being used. It's subtle, but compare:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$| x + 2| x + 3| x + 4|$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\left|x + 2\right|x + 3\left|x + 4\right|$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\left|x + 2\left|x + 3\right|x + 4\right|$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't call it any kind of standard, but as a case study I'll mention it. The mathematics library for Lean (mathlib) also noted this parsing ambiguity problem, and the solution they have at the moment is that this notation is parsed outside-in with the longest possible parse, but you &lt;em&gt;must not&lt;/em&gt; include any whitespace after the first vertical bar or before the second. This second rule gives you the freedom to control intent. So, to get one interpretation you can write &lt;code&gt;|x + 2| x + 3 |x + 4|&lt;/code&gt; and to get the other you can write &lt;code&gt;|x + 2 |x + 3| x + 4|&lt;/code&gt;. If you write &lt;code&gt;|x+2|x+3|x+4|&lt;/code&gt;, I think it would come out as the second of these, but mathlib does not have juxtaposition for multiplication, so I can't check.&lt;/p&gt;&#xA;&lt;p&gt;It's worth noting that mathematical notation in general is not really designed to be mechanically interpretable. The conventions we have make it convenient to work with polynomials, rational functions, and expressions to the power of a polynomial. But it's a 2D layout, which is not generally what you'd enter into a computer or calculator, and furthermore it's never in isolation. Notation is a language for communication, and, just like for other human languages, if an author detects what they are saying might be ambiguous given the context, they should disambiguate it in some way.&lt;/p&gt;&#xA;&lt;p&gt;In an educational setting, I would aim for flagging this ambiguity in student work.&lt;/p&gt;&#xA;" OwnerUserId="8981" LastEditorUserId="8981" LastEditDate="2023-11-13T18:26:08.317" LastActivityDate="2023-11-13T18:26:08.317" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27152" PostTypeId="2" ParentId="20649" CreationDate="2023-11-14T13:20:43.380" Score="2" Body="&lt;p&gt;In addition to the standard reasons:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Bad teachers,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's hard,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;there is an interesting reason why one hears so much about people hating math that's due to&lt;span class=&quot;math-container&quot;&gt;${\ldots}$&lt;/span&gt;  Conan Doyle.  Arguably he created the fad; see &lt;a href=&quot;https://doi.org/10.1007/s00283-022-10170-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Blowing Off Steam: Victorian Narratives of Frustration About Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1385" LastActivityDate="2023-11-14T13:20:43.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27155" PostTypeId="1" AcceptedAnswerId="27157" CreationDate="2023-11-15T16:14:07.050" Score="0" ViewCount="285" Body="&lt;p&gt;I am highly interested in mathematics, and solving Olympiad maths problems has been a type of hobby for me. But due to my age, I will never be able to give the olympiads a go again. I want to know whether I will still be able to become a sharp problem solver like the IMO toppers if I regularly practice olympiad problems with full dedication?&lt;/p&gt;&#xA;" OwnerUserId="23368" LastEditorUserId="2083" LastEditDate="2023-11-18T14:20:11.647" LastActivityDate="2023-11-18T14:20:11.647" Title="What will happen if someone doesn’t make it to the IMO team?" Tags="|problem-solving|math-contests|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27156" PostTypeId="2" ParentId="27155" CreationDate="2023-11-15T16:55:43.453" Score="15" Body="&lt;p&gt;You have posted two questions on this website so far.  Both of them show what is (to me) an unhealthy focus on Olympiad problems.&lt;/p&gt;&#xA;&lt;p&gt;Firstly I want to remind you that you have value as a living being.  Please don't lose sight of that!&lt;/p&gt;&#xA;&lt;p&gt;Secondly, there are many ways to contribute to mathematics.  Some people are really good at solving tricky problems.  Some people are really good at noticing when tools from one field might help in another field.  Some people are really good at learning a subject deeply, figuring out exactly what makes it tick, and then abstracting away the irrelevant parts to make a more abstract theory which can be applied in greater generality.  Some people contribute to math by being really good teachers.  Some people contribute to math by becoming rich working for a hedge fund and then donating to mathematical non-profits!  These are just a small fraction of the ways people contribute to math.&lt;/p&gt;&#xA;&lt;p&gt;If you find Olympiad problems to be fun and satisfying, by all means keep solving them.  I am sure with practice you can improve.  However, if you find yourself feeling sad, overwhelmed, or anxious when you are practicing you should probably just stop.  Try to find ways to engage with mathematics which feel good for you.&lt;/p&gt;&#xA;&lt;p&gt;Please do not make the mistake of thinking that you &lt;em&gt;need&lt;/em&gt; to be good at these kinds of problems to make important contributions to mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Finally, please do not make the mistake of tying your self-worth to your mathematical performance.  Think of the people who you love most in your life:  probably their mathematical performance is not at the root of why you love them.  It is the same for the people who love you.  Your worth is unquestionable and is completely independent of your mathematical accomplishment.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2023-11-15T16:55:43.453" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27157" PostTypeId="2" ParentId="27155" CreationDate="2023-11-15T17:38:11.570" Score="5" Body="&lt;p&gt;Based on this question and your &lt;a href=&quot;https://matheducators.stackexchange.com/questions/27147/how-does-a-math-olympian-fare-in-undergraduate-maths-courses&quot;&gt;previous question&lt;/a&gt;, my impression (correct me if I'm wrong) is that you are driven to become a hardcore problem-solver, and you've had fun participating in Math Olympiad, which you view as the pinnacle of problem-solving -- but you didn't end up topping Olympiad, you're now too old to compete, you're disappointed that you can't try again, and you're maybe a bit regretful in feeling that you could have practiced with fuller dedication.&lt;/p&gt;&#xA;&lt;p&gt;Well, I have good news for you! This may feel a bit hopeless at the moment, but it's actually the opposite. And when I say that, I'm &lt;em&gt;not&lt;/em&gt; just talking about opportunities to mentally reframe the situation to help cope with the pain (such as loosening psychological ties between problem-solving ability and self-worth, which would also be healthy but is not the subject of this answer). I mean that, in an absolute sense, you can still become an amazing problem-solver and get recognized for it. This is just the beginning.&lt;/p&gt;&#xA;&lt;p&gt;First of all, if you're at university, then there might be even higher competition math that you are eligible for -- for instance, in the USA/Canada, any undergraduate can take the &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition&quot; rel=&quot;nofollow noreferrer&quot;&gt;Putnam Exam&lt;/a&gt;, the topping of which is (in my experience) generally considered even more impressive than topping any high school Math Olympiad (even International Math Olympiad).&lt;/p&gt;&#xA;&lt;p&gt;But second, &lt;strong&gt;and more importantly&lt;/strong&gt;, math competitions like Olympiad and Putnam are &lt;em&gt;not&lt;/em&gt; the pinnacle of problem-solving. It just seems that way because in school, that's what lots of &amp;quot;math&amp;quot;-y people focus on and get recognized for, so it's always in your face. But think about it -- of all the world's famous problem-solvers, how many of them gained their reputation from topping math competitions? None of them. Even amongst the minority that did top math competitions, that's not what they're known for. They're generally known for their problem-solving success on more widely branching paths that they pursued after their initial schooling. Below are some of the most well-known of these paths, along with links to further reading about some of their &amp;quot;toppers&amp;quot;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Research in math and associated disciplines.&lt;/em&gt; &amp;quot;Toppers&amp;quot; of this path include &lt;a href=&quot;https://en.wikipedia.org/wiki/Fields_Medal&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fields Medalists&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Abel_Prize&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abel Prize Winners&lt;/a&gt;, and awardees of &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_prizes_known_as_the_Nobel_or_the_highest_honors_of_a_field&quot; rel=&quot;nofollow noreferrer&quot;&gt;other highest honors&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Application of math to solve hard practical problems in industry.&lt;/em&gt; I don't know of any coordinated awards for this kind of thing (besides becoming very wealthy!) but some individual &amp;quot;toppers&amp;quot; of this path include &lt;a href=&quot;https://en.wikipedia.org/wiki/Jeff_Dean&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Dean&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Jim_Simons_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jim Simons&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Demis_Hassabis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Demis Hassabis&lt;/a&gt;, and pretty much any &amp;quot;math&amp;quot;-y person who was a founder or early employee of a company that has become widely known (especially in tech / engineering / finance).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-15T18:04:14.013" LastActivityDate="2023-11-15T18:04:14.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27158" PostTypeId="1" AcceptedAnswerId="27163" CreationDate="2023-11-16T04:02:13.187" Score="11" ViewCount="2773" Body="&lt;p&gt;I tutor a student (9th grade, United States) who is in an algebra class. She consistently makes mistakes when dealing with coefficients.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The most common one is attempting to subtract away a coefficient, e.g. if I ask how to isolate &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$3x = 9$&lt;/span&gt; she'll try to subtract &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. This will occur if I use a vague question such as &amp;quot;How do we isolate &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;?&amp;quot;. If I point out that the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; is multiplying the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and ask how to reverse that, she will correctly identify division as the answer. Then, the next problem, she will try to subtract the coefficient again...&lt;/li&gt;&#xA;&lt;li&gt;She will also try to keep variables around when their coefficients become &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, e.g. she did &lt;span class=&quot;math-container&quot;&gt;$2x - 2x = x$&lt;/span&gt;. When pressed she said that the 2's canceled out and left the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; behind.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This clearly represents some sort of conceptual issue with how to handle coefficients, and I'm lost as to what more I can do. Things that I have tried include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Solve a similar problem in front of her, explaining each step. She'll make the same mistake on the next problem, or if not the next one, the one after that.&lt;/li&gt;&#xA;&lt;li&gt;Repeat that we need to do the opposite of the operation to undo it. This works only if I also point out each time that the operation a coefficient represents is multiplication. Otherwise, just saying to reverse the operation has her trying to subtract coefficients.&lt;/li&gt;&#xA;&lt;li&gt;Get out a virtual (we meet online) set of algebra tiles. She said they were too confusing; her class doesn't use them.&lt;/li&gt;&#xA;&lt;li&gt;Use word problems. Even when she's just written down an equation where she &lt;em&gt;knows&lt;/em&gt; that, e.g., &lt;span class=&quot;math-container&quot;&gt;$3x$&lt;/span&gt; represents 3 chocolates of unknown cost, once the problem is turned into numbers and letters she will lose sight of the original meaning.&lt;/li&gt;&#xA;&lt;li&gt;Use past problems, e.g. &amp;quot;What did we do with the coefficient last time?&amp;quot;. This has the best track record but it's not addressing the actual issue, it's just her blindly applying a prior method.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If I can figure out what's at the root of this issue and address it, I think it will go a long way towards her being able to do algebra independently. (Again, currently I need to prompt each step with leading questions - &amp;quot;The 3 is multiplying the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. What is the opposite of that? How do we reverse it?&amp;quot;.) I get the sense sometimes that when she's stuck, she just guesses operations she knows are possible and hopes they're the correct thing to do; she doesn't appear to have a sense of what makes sense to try.&lt;/p&gt;&#xA;&lt;p&gt;What else can I do to attack her issue with coefficients?&lt;/p&gt;&#xA;" OwnerUserId="19777" LastActivityDate="2023-11-19T11:48:51.983" Title="Fixing wrong ideas about coefficients (e.g. subtract 3 from 3x to isolate x)" Tags="|algebra|tutoring|" AnswerCount="12" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27159" PostTypeId="2" ParentId="27158" CreationDate="2023-11-16T04:53:53.360" Score="10" Body="&lt;p&gt;If it were me, I'd try building on what you mentioned here:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Use word problems. Even when she's just written down an equation where she knows that, e.g., &lt;span class=&quot;math-container&quot;&gt;$3x$&lt;/span&gt; represents &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; chocolates of unknown cost, once the problem is turned into numbers and letters she will lose sight of the original meaning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You mention that she loses the connection between the symbols and their meaning. This sounds like the root cause behind all her issues. Without that connection, she's just pattern matching, which means everything comes down to purely stimulus-response muscle memory.&lt;/p&gt;&#xA;&lt;p&gt;What I would try first is having her replace the variable with an actual word. If &lt;span class=&quot;math-container&quot;&gt;$3x$&lt;/span&gt; represents &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; chocolates of unknown cost, try having her write &lt;span class=&quot;math-container&quot;&gt;$3 \textrm{ cost}$&lt;/span&gt; (maybe even more explicitly &lt;span class=&quot;math-container&quot;&gt;$3 \cdot \textrm{cost}$&lt;/span&gt;) or something similar that bridges the gap between the verbal and symbolic representations of the problem. Maybe some practice doing algebraic manipulations with that sort of &amp;quot;mid-way&amp;quot; notation will help her make the connection between the symbols and their meaning.&lt;/p&gt;&#xA;&lt;p&gt;If that doesn't work, then unfortunately you might just need to help her build muscle memory on valid manipulations. Of course that kind of approach can't carry a student through advanced math, but if she's having this much difficulty with early algebra, then future-proofing strategies for advanced math might not be too big a concern...&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-18T19:48:17.627" LastActivityDate="2023-11-18T19:48:17.627" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27161" PostTypeId="2" ParentId="27158" CreationDate="2023-11-16T11:23:20.040" Score="8" Body="&lt;p&gt;This is a classical case of having learnt algebraic manipulation without meaning. The first thing to try is to build up algebra again from the foundations. You already tried using (virtual) concrete representations; I would recommend finding some such that the pupil accepts. A second is teaching flexible thinking and broad strategy use.&lt;/p&gt;&#xA;&lt;p&gt;In any case, doing more drills is unlikely to help - the pupil, most likely, has eight grades of drills in their history, and they have not worked thus far, so they are unlikely to work in the future.&lt;/p&gt;&#xA;&lt;h1&gt;Different levels of abstraction&lt;/h1&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Think of the problems that have a scale and three apples on one side, one watermelon on the other, and price or weight or whatever is given to one side, and you are asked for the price or weight or a single apple. I think the mistake you note is very unlikely to be made here. Learning to draw an equation or a word problem like this can be helpful. Other concrete representations are fine.&lt;/li&gt;&#xA;&lt;li&gt;As a next step, you can write an equation with three apples, equality sign, 13 money (3 apples = 13 money). How much does a single apple cost? Maybe the pupil solves this in the head without writing anything down, but then you can ask questions and have them write down the reasoning or step, same as would be done with an equation.&lt;/li&gt;&#xA;&lt;li&gt;Next step might be to write &lt;span class=&quot;math-container&quot;&gt;$3 \text{apple} = 13$&lt;/span&gt;. Here you treat apple as a variable.&lt;/li&gt;&#xA;&lt;li&gt;The next step would be to use &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; for apples.&lt;/li&gt;&#xA;&lt;li&gt;And then go to using &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; or other arbitrary letters.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The difficult skill the pupil needs to learn is recognizing their own uncertainty. Then they can and should move to the less abstract representation and operate there until they know what can and can not be done. This is an important metacognitive skill.&lt;/p&gt;&#xA;&lt;h1&gt;Inflexibility in tactic use&lt;/h1&gt;&#xA;&lt;p&gt;Pupils who have difficulty with mathematics are often very inflexible in their strategy (use of different tactics). This can already manifest in very simple problems, like calculating &lt;span class=&quot;math-container&quot;&gt;$5+8$&lt;/span&gt;. A young child would first count from one to five, then from one to eight, and then from one to thirteen. A more advanced tactic would be to count from six to thirteen, and even more from nine to thirteen. And then one might do &lt;span class=&quot;math-container&quot;&gt;$5+8 = 3 + 2 + 8 = 3 + 10 = 13$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You can find similar inflexibility in pupils who tend to have problems with mathematics later. They see mathematics as a collection of arbitrary rules, rather than things that one can use in several ways and play around with.&lt;/p&gt;&#xA;&lt;p&gt;The recommended best practice in classroom use is to get several different pupils show how they have  solved a given problem In one-on-one tutoring you would have to provide alternate tactics yourself. They key point is to not suggest that the pupil's own solution was bad, but rather show that there are many ways ahead.&lt;/p&gt;&#xA;&lt;h1&gt;Varying types of exercises&lt;/h1&gt;&#xA;&lt;p&gt;The pupil is likely to meet pure calculations sans context, and also word problems where they are supposed to extract a calculation from and then proceed with the calculation.&lt;/p&gt;&#xA;&lt;p&gt;How about giving them a solution (it could be the value of the unknown, or even an equation they should give a corresponding word problem to) and asking them to produce a problem where the solution is, well, the solution? How many different ones can they create?&lt;/p&gt;&#xA;&lt;p&gt;Or give them an equation and ask to provide a visualization or concretization of it?&lt;/p&gt;&#xA;&lt;p&gt;Or give some information, like the menu with prices at a restaurant, and ask them to formulate equations based on a family visiting that restaurant. A more relevant scenario is of course better.&lt;/p&gt;&#xA;&lt;p&gt;These types of exercises require understanding of algebra and equations, but are more free and playful than just solving an equation, and do not have an algorithm one can follow to solve them. They are likely to lead to a more robust understanding of the material. They should not completely take over; variety is the goal.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-11-16T11:23:20.040" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27162" PostTypeId="2" ParentId="27158" CreationDate="2023-11-16T15:04:21.000" Score="5" Body="&lt;p&gt;&lt;strong&gt;&lt;em&gt;tl;dr&lt;/em&gt;–&lt;/strong&gt;  You might just work out whatever algebraic manipulations the student suggests, showing them the results of their ideas until they get a hang of it.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;Show them how the math would work out.&lt;/h3&gt;&#xA;&lt;p&gt;It might help to show the results of the student's ideas.&lt;/p&gt;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The most common one is attempting to subtract away a coefficient, e.g. if I ask how to isolate &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$3x = 9$&lt;/span&gt; she'll try to subtract &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Okay, so starting with the equation and then subtracting &lt;span class=&quot;math-container&quot;&gt;$3 ,$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;3x&amp;amp;=9 \,,&#xA;\\&#xA;3x \color{red}{-3} &amp;amp; = 9 \color{red}{-3} \,,&#xA;\\&#xA;3x - 3 &amp;amp; = 6 \,,&#xA;\end{align*}&#xA;$$&lt;/span&gt;&#xA;so what next?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;She will also try to keep variables around when their coefficients become &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, e.g. she did &lt;span class=&quot;math-container&quot;&gt;$2x - 2x = x$&lt;/span&gt;. When pressed she said that the 2's canceled out and left the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; behind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Okay, so let's cancel those &lt;span class=&quot;math-container&quot;&gt;$2's ,$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2x - 2x \,,&#xA;\\&#xA;\left(2 - 2\right) x \,,&#xA;\\&#xA;\left( 0 \right) x \,,&#xA;\\&#xA;0x \,,&#xA;$$&lt;/span&gt;&#xA;so what next?&lt;/p&gt;&#xA;&lt;p&gt;Or, did you want to cancel the &lt;span class=&quot;math-container&quot;&gt;$2's$&lt;/span&gt; by division?  If so:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;2x - 2x \,,&#xA;\\&#xA;\frac{2x - 2x}{2} \,,&#xA;\\&#xA;\frac{2x}{2} - \frac{2x}{2} \,,&#xA;\\&#xA;x - x \,,&#xA;$$&lt;/span&gt;&#xA;and then what next?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Basically:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Start with the problem-to-solve.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ask the student what to do next.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Interpret the student's idea as something mathematically valid.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If there're multiple reasonable interpretations, maybe choose a few good ones to demonstrate.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Show how that mathematically-valid idea works out.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Loop back to Step-(2).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;They might end up bumbling around a bit, suggesting weird ideas that might make some busy-work for you if the results of their ideas don't reduce so well.  Still, if you walk them through the steps and help them to understand their options, even seeing how ugly the math can in a wrong direction might help them understand why such a direction wouldn't be one to go in.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;Don't do fake math.&lt;/h3&gt;&#xA;&lt;p&gt;They might suggest something that wouldn't make sense.  For example, from&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The most common one is attempting to subtract away a coefficient, e.g. if I ask how to isolate &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$3x = 9$&lt;/span&gt; she'll try to subtract &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;, they might've been thinking something like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;3x&amp;amp;=9 \,,&#xA;\\&#xA;3x \color{red}{-3} &amp;amp; = 9 \color{red}{-3} \,,&#xA;\\&#xA;x &amp;amp; \neq 6 \,.&#xA;\end{align*}&#xA;$$&lt;/span&gt;&#xA;It might be best to ignore such interpretations entirely.&lt;/p&gt;&#xA;&lt;p&gt;If the student presses for such an interpretation, you might just tell them that you don't understand what they mean.  You only understand math, and you just need them to tell you what they want to do in terms of math.&lt;/p&gt;&#xA;" OwnerUserId="8943" LastActivityDate="2023-11-16T15:04:21.000" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27163" PostTypeId="2" ParentId="27158" CreationDate="2023-11-16T17:44:58.213" Score="14" Body="&lt;p&gt;From very limited experience: have you tried writing &lt;span class=&quot;math-container&quot;&gt;$$3 \times x$$&lt;/span&gt; (or using one of the various stars, dots, etc.)?&lt;/p&gt;&#xA;&lt;p&gt;It might be that it’s not obvious that there is a multiplication here: you say that pointing that out leads immediately to division. This coefficient notation hides some things that no other core operation (addition, subtraction, division) has.&lt;/p&gt;&#xA;&lt;p&gt;If that works, then maybe you can focus on connecting coefficient notation to operator notation.&lt;/p&gt;&#xA;" OwnerUserId="23392" LastActivityDate="2023-11-16T17:44:58.213" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27164" PostTypeId="2" ParentId="27158" CreationDate="2023-11-16T19:37:55.757" Score="0" Body="&lt;p&gt;I think you need to work on muscle memory, more than concepts.  You have given her the concepts before, but clearly it wasn't sufficient.  No problem with repeating the &amp;quot;why&amp;quot; also, while doing the drill.  But as a &amp;quot;side&amp;quot;, not the main course.  There's a decent chance that the concepts start to sink in AFTER she masters the monkey skills, not before.&lt;/p&gt;&#xA;&lt;p&gt;Do imitation and practice.  Including very small step (within a problem) and immediate imitation by her.  And very gradual removing of scaffolding (say unimitated problems, but still step by step.  Consider also, a very easy problem type and only gradual increase in difficulty (more gradual than you think reasonable).&lt;/p&gt;&#xA;&lt;p&gt;When she makes a mistake and gets it wrong, make her repeat the same problem.  Not just the step she missed, but the whole thing (line by line writing it out).  I did this to myself in AP Calculus and it helped immensely.  It's much to easy to assume that a failure (&amp;quot;dumb mistake&amp;quot;) was corrected, otherwise, but often it's not.&lt;/p&gt;&#xA;&lt;p&gt;The tiles were a decent try, but since you have low contact hours, I would not push something that is a distraction from what she's doing in school.  Word problems are even more contraindicated (1. Word problems are often harder in terms of cognitive load.  2. you tried it and it didn't help.)&lt;/p&gt;&#xA;&lt;p&gt;I don't know everything about how you are teaching her, but there were a few things that made my spidey sense tingle.  Bullet one you're actually getting partial success.  Lean into that and do things that build towards longer cycles efore she forgets.  (Probably keeping the problems more similar).  Bullet two of the things tried reads too abstract to me (and I'm good at math).  And then on bullet five, you have something good happening and want to dismiss it since it's &amp;quot;blind procedure following&amp;quot;.  Would you rather succeed the wrong way or fail the right way!?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Good luck and stick with it.  Half a loaf is better than none.  This is a big transition for the kid and harder than many people realize.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  Take a look at this essay. (Ignore the discovery based learning...that's not the key issue here...but other things like cognitive load and prealgebra difficulty should resonate. If they do, I would read more of what he has to say in favor of smaller increment instruction.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://fillingthepail.substack.com/p/peter-liljedahl-wants-to-make-kids&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://fillingthepail.substack.com/p/peter-liljedahl-wants-to-make-kids&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="23393" LastEditorUserId="23393" LastEditDate="2023-11-16T19:40:44.110" LastActivityDate="2023-11-16T19:40:44.110" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27165" PostTypeId="2" ParentId="27158" CreationDate="2023-11-17T00:54:43.857" Score="7" Body="&lt;p&gt;Unfortunately I do not have experience in your (the OP’s) role, but at one time I was in the shoes of your student.&#xA;When I encountered rudimentary algebra for the first time I was completely lost and none of those weird rules made sense to me. By a lucky coincidence, I took up (computer) programming a few months later and that made exactly this conceptual layer of what it means to do arithmetic operations with letters instead of numbers clear to me in very short order. I proceeded to do very well in middle and high school math with very little effort.&lt;/p&gt;&#xA;&lt;p&gt;I think the core principle behind it is that one can very easily explore the relationship between abstractly described operations (the code) and the concrete inputs and outputs (and even the intermediate values in variables if your programming environment allows that) by running a program many times, seeing how the output changes when you vary the inputs or tinker with the intermediate steps. It is like bringing equations to life.&lt;/p&gt;&#xA;&lt;p&gt;On paper this is much more cumbersome; since each “run” requires a lot of arithmetics I find it hard to imagine a student who would actually calculate everything out more than maybe two or three times while a computer does all the arithmetics instantly and flawlessly (and seeing stuff appear or move on screen can of course be fun and exciting).&lt;/p&gt;&#xA;&lt;p&gt;Also, if you expect an operation (such as subtracting coefficients) to work differently than they actually do, the computer immediately shows you the difference between your expectations and reality with cold hard calculation.&lt;/p&gt;&#xA;&lt;p&gt;Of course, it doesn’t seem very likely for your student to be particularly interested in programming, so this answer may very well be for the exclusive benefit of hypothetical future readers. But perhaps there is a chance you could still make use of this idea: there are quite a few graphical programming environments out there – &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Visual_programming_language&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia lists&lt;/a&gt; a whole bunch (maybe Scratch seems plausible? I have no experience with any of them) – that are supposed to be extremely accessible and easy to use. You could make a program relating to one of the practice problems (e.g. a program that calculates the price of various fruits or chocolates) and study together what happens with the results when the inputs change or when you mess with the intermediate steps.&lt;/p&gt;&#xA;" OwnerUserId="12697" LastEditorUserId="12697" LastEditDate="2023-11-17T01:00:34.860" LastActivityDate="2023-11-17T01:00:34.860" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27166" PostTypeId="2" ParentId="27158" CreationDate="2023-11-17T03:03:41.513" Score="4" Body="&lt;p&gt;A thing that I do in my low-level college courses is to spend a beat with each new example/exercise to inspect the statement and make sure we all agree with what we're looking at. &amp;quot;What operations are here?&amp;quot; is among the most important questions; even pointing at each connection as needed.&lt;/p&gt;&#xA;&lt;p&gt;I notice in the OP's question they lean on the word &amp;quot;coefficient&amp;quot; many times (10), and &amp;quot;multiply&amp;quot; not so much (3). I'd suggest, with this student, that they lean more on the latter word to make sure they're reading the statements correctly.&lt;/p&gt;&#xA;&lt;p&gt;For the next several equations they look at, ask the student, &amp;quot;What operation connects the &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; to the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;?&amp;quot;, and so on, and see if the expected decoding gets more natural.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2023-11-17T03:03:41.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27167" PostTypeId="2" ParentId="26660" CreationDate="2023-11-17T07:08:44.503" Score="4" Body="&lt;p&gt;Peter answered how to explain the ideas to students. So I'll answer your second question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is &amp;quot;square meter&amp;quot; a badly-phrased term? What would be a more appropriate term?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It's not badly-phrased, but I see what you're getting at: &amp;quot;square meter&amp;quot; is in reference to a particular shape, whereas something like &amp;quot;area meter&amp;quot; would sound more general to other shapes.&lt;/p&gt;&#xA;&lt;p&gt;However, the problem with a phrase like &amp;quot;area meter&amp;quot; that is intended to be general to other shapes, is that not knowing what specific shape you're talking about actually creates &lt;em&gt;more&lt;/em&gt; confusion. Say I ask you to shade in 1 &amp;quot;area meter&amp;quot;. What do you do? Do you draw a triangle with sides of length 1 and then shade it in? Or do you do that with a square? A pentagon? A circle with radius 1? The nice thing about &amp;quot;square meter&amp;quot; is that its name tells you exactly how to measure it.&lt;/p&gt;&#xA;&lt;p&gt;More abstractly, you can view units of area as being a combination of two independent units: &amp;quot;square&amp;quot; tells you the shape, and &amp;quot;meter&amp;quot; tells you the side length. In that view, &amp;quot;square&amp;quot; is the conventional unit for shape, but you could talk about a thing like &amp;quot;1 triangle inch&amp;quot;. But using &amp;quot;square&amp;quot; usually makes things way easier for our brains, just like using a decimal number system usually makes things way easier than working in binary.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-27T16:29:53.693" LastActivityDate="2023-11-27T16:29:53.693" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27170" PostTypeId="2" ParentId="1589" CreationDate="2023-11-17T15:40:44.777" Score="1" Body="&lt;p&gt;Just reposting: &lt;a href=&quot;https://math.stackexchange.com/a/4183739/303300&quot;&gt;Proof&lt;/a&gt; (can be found in Lang’s analysis book and Vinberg’s algebra book)&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;${ f(z) }$&lt;/span&gt; be a complex polynomial of positive degree. The approach is to show that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;${ \min _{z \in \mathbb{C}} \vert f(z) \vert }$&lt;/span&gt; exists, i.e. there is a &lt;span class=&quot;math-container&quot;&gt;${ z _0 \in \mathbb{C} }$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;${ \vert f(z _0) \vert \leq \vert f(z) \vert }$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;${ z \in \mathbb{C} }.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Supposing &lt;span class=&quot;math-container&quot;&gt;${ f(z _0) \neq 0 ,}$&lt;/span&gt; by a choice of &lt;span class=&quot;math-container&quot;&gt;${ t \in (0,1) }$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;${ \delta \in \mathbb{C} }$&lt;/span&gt; one has &lt;span class=&quot;math-container&quot;&gt;${ \vert f(z _0 + t \delta ) \vert &amp;lt; \vert f(z _0) \vert },$&lt;/span&gt; a contradiction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16420" LastActivityDate="2023-11-17T15:40:44.777" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27171" PostTypeId="2" ParentId="27158" CreationDate="2023-11-17T16:15:32.627" Score="1" Body="&lt;p&gt;Following up on &lt;a href=&quot;https://matheducators.stackexchange.com/users/5563&quot;&gt;Daniel R. Collins&lt;/a&gt;' &lt;a href=&quot;https://matheducators.stackexchange.com/questions/27158#27166&quot;&gt;suggestion&lt;/a&gt;,&#xA;teach the student a step-by-step way to solve and check problems.&#xA;For problems that are given to her in the terse algebraic format (like &amp;quot;3x = 9&amp;quot;, which omits the multiplication signs), the first step needs to be:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Always translate the problem into a format she understands.  And write out the translated problem.  Specifically, show the multiplication signs.  For example, &amp;quot;3 ˟ x = 9&amp;quot; or &amp;quot;3 · x = 9&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;In all subsequent steps, always write out the multiplication signs.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This will cause her to always be slower than her classmates.  But it is better to be slow and right, than fast and very wrong.&lt;/p&gt;&#xA;" OwnerUserId="1147" LastActivityDate="2023-11-17T16:15:32.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27174" PostTypeId="1" AcceptedAnswerId="27175" CreationDate="2023-11-18T03:05:05.273" Score="12" ViewCount="4421" Body="&lt;p&gt;I teach high school math.&lt;/p&gt;&#xA;&lt;p&gt;Some of my colleagues insist that a proof by induction should explicitly refer to the principle of mathematical induction, i.e. it must include the words &amp;quot;by the principle of mathematical induction&amp;quot; or words of equivalent meaning.&lt;/p&gt;&#xA;&lt;p&gt;Others say that (given a proposition &lt;span class=&quot;math-container&quot;&gt;$H_n$&lt;/span&gt;) it is enough to show that the base case is true and that &lt;span class=&quot;math-container&quot;&gt;$H_k\implies H_{k+1}$&lt;/span&gt;; it is not necessary to explicitly refer to the principle of mathematical induction.&lt;/p&gt;&#xA;&lt;p&gt;Solutions in the mark schemes for the exams (IB and A-level) are inconsistent: sometimes they explicitly refer to the principle of mathematical induction, sometimes they do not, and sometimes they refer to it in brackets (indicating that it is optional).&lt;/p&gt;&#xA;&lt;p&gt;If a proof by induction has to explicitly refer to the principle of mathematical induction, then do proofs using the pigeonhole principle have to explicitly refer to the pigeonhole principle? Do proofs by contradiction have to explicitly refer to the principle of proof by contradiction (if there is such a thing)?&lt;/p&gt;&#xA;&lt;p&gt;Possibly relevant: &lt;a href=&quot;https://math.stackexchange.com/q/2817147/398708&quot;&gt;Why does induction have to be an axiom?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20101" LastEditorUserId="20101" LastEditDate="2023-11-18T06:38:45.910" LastActivityDate="2023-11-28T13:04:41.040" Title="Does a proof by induction have to explicitly refer to the principle of mathematical induction?" Tags="|proofs|induction|" AnswerCount="11" CommentCount="29" ContentLicense="CC BY-SA 4.0" />
  <row Id="27175" PostTypeId="2" ParentId="27174" CreationDate="2023-11-18T03:25:19.283" Score="26" Body="&lt;p&gt;The appropriate level of granularity for a proof depends on the audience.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If you're taking an &amp;quot;Intro to Proofs&amp;quot; class and your homework is to do some proofs by induction, then yeah, you will be expected to state when, where, and how you use the principle of mathematical induction. You should explicitly label the base case, the inductive hypothesis, etc. Not because your grader needs that information to follow your proof, but because the whole point of the homework is to demonstrate that you know this stuff. (Likewise, if you're teaching an &amp;quot;Intro to Proofs&amp;quot; class then obviously you need to label all those steps in your example proofs to provide proper scaffolding for students.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you're a full-blown mathematician who is writing a research paper, then your audience already knows this stuff like the back of their hand, so most of it doesn't need to be said explicitly. Sure, it will make the reading a little gentler if you precede the inductive proof with &amp;quot;By induction, ...&amp;quot;, and that would probably be a nice thing to do for your audience, but you're not going to lose any other research mathematician in your audience if you omit the explicit reference to induction, and nobody is going to question your understanding of induction because you didn't explicitly label something (you'd have to make a legitimate and egregious error in the basic logic for that to happen).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Your case sounds closer to the first one, so I would say sure, as long as your colleagues tell their students what features of the induction proof they are required to explicitly state/label, it's appropriate for them to expect a student's induction proof to include those features.&lt;/p&gt;&#xA;&lt;p&gt;... though if they're reading a math research publication and claiming that it should be edited to include the include the words &amp;quot;by the principle of mathematical induction&amp;quot;, then that's a different story ;)&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-18T15:53:49.357" LastActivityDate="2023-11-18T15:53:49.357" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="27176" PostTypeId="2" ParentId="27174" CreationDate="2023-11-18T04:31:16.227" Score="-3" Body="&lt;p&gt;The Principle of Mathematical Induction states:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\forall P\subset N:[0 \in P \land \forall x\in P: [S(x) \in P] \implies P=N]$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is the successor function on &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;IMHO high-school students in North America will not be able to grasp the meaning of this statement, never mind apply it in proofs. I'm guessing most high-school teachers there won't understand it either.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is not to say that proof by induction should not be taught in high schools. Just don't dwell on the above axiom as seems to be suggested here. A formalized approach, even if translated into words, will only create confusion at that level. If students cannot recall the exact wording of the axiom, they may think the cannot do proof by induction.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-11-18T13:32:24.573" LastActivityDate="2023-11-18T13:32:24.573" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27179" PostTypeId="2" ParentId="27174" CreationDate="2023-11-18T08:47:28.317" Score="3" Body="&lt;p&gt;Typically you want to name things. This makes them visible and something you can discuss. So, while teaching, you do want to say that this thing her is induction, so we have to remember to check the base case, and that from every case we can deduce the next one. If you do not name this thing, then it is likely that many students do not know what they should be paying attention to.&lt;/p&gt;&#xA;&lt;p&gt;An exam requiring particular magical incantations seems silly, but if succeeding at them is important, then you should teach the magical incantations to the students. You may want to say that in general it is a good idea to say that you using induction, so that the reader is oriented, but whether it is called the principle of induction, or one just writes &amp;quot;By induction...&amp;quot;, or explains the idea in the proof itself; not really essential.&lt;/p&gt;&#xA;&lt;p&gt;So:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Make the proof as easy to understand as possible.&lt;/li&gt;&#xA;&lt;li&gt;Tell students to use any magic words an exam requires &lt;em&gt;in that exam&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Writing like mathematics articles are written is not always a good idea, as many mathematics articles are poorly written, just like many academic texts in general. There are of course many good ideas there, too, but one has to think communication first, not just copy and paste expressions from other examples of the genre (mathematics research article or textbook).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2023-11-18T08:47:28.317" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27180" PostTypeId="1" CreationDate="2023-11-18T13:27:05.073" Score="6" ViewCount="303" Body="&lt;p&gt;So in an advanced mathematics course for engineers, there are often problems of the type:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove claim A.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given equation A, show that you can obtain equation Z.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am frequently faced with a problem where students would write possibly large amount of derivations/calculations, essentially taking you on a wild goose chase and then at the end (&lt;strong&gt;no matter if what they wrote is correct, logical or even coherent&lt;/strong&gt;), they will always write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;We have proven claim A&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;or &amp;quot;We obtained equation Z&amp;quot; (directly copy&#xA;equation Z from the question itself)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The worst is that all steps are correct, but then a quantum leap happens (key steps skipped), and then the &amp;quot;We have solved the question.&amp;quot; It always make you think if going from their final correct step to the conclusion is trivial, therefore it is myself who is unable to see such an obvious conclusion.&lt;/p&gt;&#xA;&lt;p&gt;What do you do with this type of behavior?&lt;/p&gt;&#xA;" OwnerUserId="3560" LastActivityDate="2023-11-25T21:36:23.740" Title="What to do with &quot;wild goose chase&quot; or &quot;quantum leap&quot;-types of incorrect solutions when you ask students to prove/show something?" Tags="|proofs|students-mistakes|problem-design|test-design|engineering-mathematics|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27181" PostTypeId="2" ParentId="27180" CreationDate="2023-11-18T15:43:24.253" Score="2" Body="&lt;p&gt;I think the easiest way to deal with behavior like this is to tweak the grading method to disincentivize it (and of course have a conversation with the class where you explain the behavior, why it's bad, and why/how the grading is going to be tweaked).&lt;/p&gt;&#xA;&lt;p&gt;For the specific situation that you describe, I would probably make a list of the key components that I want to see in the proof, and then grade using some kind of plus-minus balance for relevant vs irrelevant information in their proof.&lt;/p&gt;&#xA;&lt;p&gt;For instance, if there are 3 key components, and they addressed 2 of those components, but also threw in 1 unnecessary component, then the amount of credit would be &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{2 - 1}{3} = \dfrac{1}{3} = 33\%.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This generalizes naturally to weighting: if there are 2 minor components that they addressed and 1 major component that they did not, and they threw in 1 large unnecessary component, then the amount of credit would be &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{0.5 + 0.5 - 1}{0.5 + 0.5 + 1} = \dfrac{0}{2} = 0\%.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And you can of course tweak bits of the calculation as desired to get more nuanced behavior:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Maybe you have a ceiling of &lt;span class=&quot;math-container&quot;&gt;$50\%$&lt;/span&gt; on the partial credit that can be awarded, so that if there are actually 5 equally-important components to the proof and the the student got 4 right, then they get &lt;span class=&quot;math-container&quot;&gt;$50\%$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$80\%.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If the student also inserted 1 component of junk, then the &amp;quot;ceiling&amp;quot; approach could generalize to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\min\left( \dfrac{4}{5}, 50\% \right) - \dfrac{1}{5} = 50\% - 20\% = 30\%.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-18T16:18:27.397" LastActivityDate="2023-11-18T16:18:27.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27182" PostTypeId="1" CreationDate="2023-11-18T16:13:14.737" Score="-1" ViewCount="46" Body="&lt;p&gt;I have 3 puppies and 2 piglets.&#xA;Can you that ratio be written in this fractional form 5/2?&#xA;If yes, how would you describe this ratio as 5/2?&lt;/p&gt;&#xA;" OwnerUserId="23404" LastActivityDate="2023-11-18T16:24:43.200" Title="Ratio in Fractional Form" Tags="|fractions|rational-numbers|" AnswerCount="1" CommentCount="1" ClosedDate="2023-11-18T19:05:36.733" ContentLicense="CC BY-SA 4.0" />
  <row Id="27183" PostTypeId="2" ParentId="27182" CreationDate="2023-11-18T16:24:43.200" Score="1" Body="&lt;p&gt;In order to judge whether &lt;span class=&quot;math-container&quot;&gt;$5/2$&lt;/span&gt; is correct, you need to specify which ratio you're talking about. There are a lot of ratios that can be made here, one of which is &lt;span class=&quot;math-container&quot;&gt;$5/2.$&lt;/span&gt; (There are &lt;span class=&quot;math-container&quot;&gt;$3+2=5$&lt;/span&gt; animals in total, so the ratio of total animals to piglets is &lt;span class=&quot;math-container&quot;&gt;$5/2.$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;p&gt;I'm guessing the correct answer you have in mind is &lt;span class=&quot;math-container&quot;&gt;$3/2,$&lt;/span&gt; which is indeed one of the possible ratios here (and probably the most natural ratio), but the question would need to specify that it's asking for the ratio of puppies to piglets.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-11-18T16:24:43.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27184" PostTypeId="2" ParentId="27158" CreationDate="2023-11-18T19:45:21.773" Score="1" Body="&lt;p&gt;I'm not a teacher, so maybe I don't get your advanced teaching techniques, but why isn't anyone explaining what 3x means in reality? When I have 3 apples, I can show 3 apples. Why aren't you breaking down the x? Instead of 3 dot x or 3 star x, why not show them X + X + X = 6 then point out they need to divide the 6 into 3 pieces to match the Xs, so you get X+X+X = 2+2+2.&lt;/p&gt;&#xA;" OwnerUserId="23406" LastActivityDate="2023-11-18T19:45:21.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27187" PostTypeId="2" ParentId="27158" CreationDate="2023-11-18T22:09:56.070" Score="0" Body="&lt;p&gt;Have you tried to use the balance metaphor to solve the equation?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2aSjg.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2aSjg.jpg&quot; alt=&quot;unbalanced scales with moneybags on one side and coins on the other&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;To solve the equation &lt;span class=&quot;math-container&quot;&gt;$3x=9$&lt;/span&gt; using the balance metaphor, you can imagine a scale with two sides in balance. On one side of the scale, there are three bags (each containing a quantity 'x'), and on the other side, there are nine coins.&lt;/p&gt;&#xA;&lt;p&gt;To maintain the balance while finding out how much is in one bag (one 'x'), you would take away the same number of coins from each side. So, you would remove one bag (containing 'x') from one side and three coins from the other side, doing this three times.&lt;/p&gt;&#xA;&lt;p&gt;Each time you remove a bag and three coins, the balance is maintained. After three repetitions, you end up with one bag on one side and three coins on the other side, showing that the contents of one bag (one 'x') is equal to three coins, hence &lt;span class=&quot;math-container&quot;&gt;$x=3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There are some interactive applets that implements this metaphor on the web. See, for instance: &lt;a href=&quot;https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Pan-Balance----Numbers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Pan-Balance----Numbers/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="19777" LastEditDate="2023-11-19T02:03:26.440" LastActivityDate="2023-11-19T02:03:26.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27188" PostTypeId="2" ParentId="27174" CreationDate="2023-11-18T22:12:16.840" Score="2" Body="&lt;p&gt;From a traditional standpoint in teaching high school math, clarity and pedagogy are paramount. A proof by induction traditionally includes an explicit reference to the principle of mathematical induction to ensure that students understand not just the mechanics of the proof but also the underlying logical framework that validates the approach. While technically it may be sufficient to show that the base case holds and the inductive step follows (Hk ⟹ Hk+1), explicitly stating that one is using the principle of mathematical induction reinforces the method and its formal structure, which is crucial for educational purposes.&lt;/p&gt;&#xA;&lt;p&gt;Regarding other principles like the pigeonhole principle or proof by contradiction, traditionally, it is also considered good practice to name the principle being used. This practice helps students to distinguish between various proof techniques and understand their applications. However, as with the principle of mathematical induction, it may not always be necessary to name the principle as long as the logical structure of the proof is clear and correct.&lt;/p&gt;&#xA;&lt;p&gt;In educational settings, it is often beneficial to be explicit about these principles to aid learning, even if, in professional mathematics, such explicit references may sometimes be omitted.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2023-11-18T22:12:16.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27190" PostTypeId="2" ParentId="27158" CreationDate="2023-11-18T23:28:47.600" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Without math, you can ask what is left 8f you take one apple from 2 apples. Write down algebraic expressions for one apple and two apples, and abbreviate apples by a. So she may know how to compute 2a-a.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;With math, it simply distributive rule. It tells you why 30-10=20, why 2 mill-1 mill=1 mill. And why 2a-a=a.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;More rigorously, American kids need to write down, at least once in their whole life, that 2x-x=2•x-1•x(also read out 2 copies of x minus one copy of x)=(2-1)•x=x.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;After all these, never be mad at kids if they keep making similar mistakes. They may just want to tease you.You just repeat the above for them, say, 10 times, like how we train kids to learn Chinese characters. Good luck with your journey and good luck with your students. In particular, if she starts to learn Algebra I in 9th grade.&lt;/p&gt;&#xA;" OwnerUserId="23408" LastActivityDate="2023-11-18T23:28:47.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27191" PostTypeId="2" ParentId="27174" CreationDate="2023-11-19T05:26:14.197" Score="5" Body="&lt;p&gt;I am (one of the) colleagues David refers to in his post. The reason I am doing this lies in some of the answers/comments posted here already. For example, Humberto sais: &amp;quot;While technically it may be sufficient to show that the base case holds and the inductive step follows (Hk ⟹ Hk+1), ...&amp;quot;. My question is why? In my opinion, the answer is &amp;quot;because of the principle of mathematical induction&amp;quot;. In my opinion, this is not at all obvious. Nothing else (except the well-ordering principle) implies this. This is why (in my opinion) it needs to be stated. Otherwise students may think that this is obviously some consequence of some theorem(s) that mathematicians who deal with the formal foundation of mathematics proved already. I may be wrong, but I think that it is not. It is an intuitive belief that cannot be proven from the other properties we take for granted (the axioms, if we want to use more formal language). So, back to Humberto's comment, I do not think that technically it is enough to check the base case and the induction step. Intuitively I agree it is enough, but technically we need a concluding statement (with or without the explicit reference to the principle).&lt;/p&gt;&#xA;&lt;p&gt;Whether we want to insist on students mentioning the point where they use the principle is of course a matter of opinion. An example is posted in &lt;a href=&quot;https://matheducators.stackexchange.com/questions/20666/grade-on-proving-a-1-a-2-a-n-le-a-1a-2-a-n/20675#comment56005_20675&quot;&gt;Grade on proving |$a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$&lt;/a&gt; . In this post the student used practically an inductive proof, but instead of formally following the steps (required by most IB and A-level high school markschemes) the student wrote &amp;quot;repeating this ...&amp;quot;. He clearly understood the intuition, but not what induction is about. It is not just about formally using a certain method. It is also about why to use the method instead of intuitive arguments.&lt;/p&gt;&#xA;" OwnerUserId="14957" LastActivityDate="2023-11-19T05:26:14.197" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27192" PostTypeId="2" ParentId="2157" CreationDate="2023-11-19T06:00:06.780" Score="1" Body="&lt;p&gt;Consider two concentric coplanar unit squares (their sides do not have to be coincident).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OwuMt.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OwuMt.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Show that the area of the region inside both squares is greater than &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{4}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It's easier if we show something stronger.&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt;  &lt;a href=&quot;https://i.stack.imgur.com/8WU0f.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8WU0f.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt; The area is greater than &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}{4}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="20101" LastActivityDate="2023-11-19T06:00:06.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27193" PostTypeId="2" ParentId="27158" CreationDate="2023-11-19T11:48:51.983" Score="0" Body="&lt;p&gt;Maybe a bit of a lateral solution compared to the other answer here, but given you're meeting online, have you considered using emojis to represent your unknowns? This can help keep things tangible.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3x$&lt;/span&gt; doesn't look like 3 apples, but &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; seems much easier to intuitively read as &amp;quot;three apples&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Some people are good at abstraction. You could use virtually &lt;em&gt;anything&lt;/em&gt; to represent &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and they'll follow along. Others get caught up in the symbology even though the specific symbol is irrelevant (i.e. only the consistent usages between instances of the same unknown value is relevant).&lt;/p&gt;&#xA;&lt;p&gt;Given her lack of intuitive understanding of the concepts you've listed, it seems highly likely that she's not seeing the wood for the trees and instead fixating on the symbols you're using. A more intuitive syntax might help lower the entry threshold for her.&lt;/p&gt;&#xA;" OwnerUserId="13169" LastActivityDate="2023-11-19T11:48:51.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27194" PostTypeId="2" ParentId="27174" CreationDate="2023-11-19T17:59:52.593" Score="3" Body="&lt;p&gt;In a pedagogical context I can see four types of situation where it may reasonably be required for students to explicitly state their use of the principle of mathematical induction:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If writing proofs by induction is part of the material being tested, students may be required to state its use explicitly.&lt;/li&gt;&#xA;&lt;li&gt;If the material being tested itself distinguishes in a meaningful way between proofs that use induction and proofs that don't (e.g. a class covering different logical systems and what can/can't be proven in each), students may be required to state where any such proof uses the principle of mathematical induction explicitly.&lt;/li&gt;&#xA;&lt;li&gt;If the students need to write proofs in an extremely standardized format (e.g. so that it can be automatically graded) they may be required to explicitly state whether or not the principle of mathematical induction is used.&lt;/li&gt;&#xA;&lt;li&gt;If the students are generally not expected to already know the principle of mathematical induction and it is not part of the material covered in the class, then any student who does happen to know it and wishes to use it in a proof nonetheless should probably state explicitly that they are using it (although this situation is unlikely to appear in any marking schemes as it is very likely not the kind of answer that the person writing the marking scheme is anticipating).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;To summarize, it may be required if it is used to demonstrate the students' understanding of induction as a principle/proof method, if it is relevant to the metamathematical material being tested, if strict formatting guidelines must be followed for some reason, or if students are not expected to know it or learn it in the course at hand but are allowed to use it if they happen to already understand it. Outside of these situations I don't see why students should be required to state its use explicitly.&lt;/p&gt;&#xA;" OwnerUserId="23413" LastActivityDate="2023-11-19T17:59:52.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27195" PostTypeId="2" ParentId="27174" CreationDate="2023-11-19T22:24:59.297" Score="3" Body="&lt;p&gt;Peano's axioms without the axiom  of induction has some models that do not correspond to the natural numbers that we have in our minds. In order to prove that every natural number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; other than &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is the successor of some other natural number &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, i.e., for all &lt;span class=&quot;math-container&quot;&gt;$n\ne0$&lt;/span&gt; there exists &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$S(k)=n$&lt;/span&gt;, we must harness the axiom of induction. Think of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Consider the model &lt;span class=&quot;math-container&quot;&gt;${\bf N}_\text{strange}$&lt;/span&gt; of Peano's axioms without induction that looks like &lt;span class=&quot;math-container&quot;&gt;$$\{0,1,2,3,\ldots, \omega, \omega+1,\omega+2,\omega+3&#xA;\ldots \}$$&lt;/span&gt; This system satisfies the Peano axioms without induction, although I am not going to digress and give details (use your intuition). In this system, &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; is not the successor of anything. That this is possible is hardly surprising, because we need the axiom of induction to prove that every number other than &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; has a predecessor.&lt;/p&gt;&#xA;&lt;p&gt;So why do we want high school students learning about induction to chat &amp;quot;&lt;em&gt;by mathematical induction blah blah blah&lt;/em&gt;? This is a tricky question. But under the hood, given a proposition &lt;span class=&quot;math-container&quot;&gt;$H(n)$&lt;/span&gt;  and showing (i) that the base case is true and (ii)  &lt;span class=&quot;math-container&quot;&gt;$H_k\implies H_{k+1}$&lt;/span&gt; is &lt;strong&gt;not&lt;/strong&gt; sufficient to deduce &lt;span class=&quot;math-container&quot;&gt;$H(k)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;${\bf N}_\text{strange}$&lt;/span&gt;. The truthiness of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; would not extend to &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;. One justification for the required chat invoking induction is &amp;quot;mathematical honesty,&amp;quot; although I don't think this will be appreciated at scale!&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-11-22T16:23:07.283" LastActivityDate="2023-11-22T16:23:07.283" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27196" PostTypeId="1" AcceptedAnswerId="27204" CreationDate="2023-11-20T02:54:29.657" Score="0" ViewCount="248" Body="&lt;p&gt;I remember watching a youtube playlist on a video about equation of a plane and about vectors in 3D. The video was fully animated in 3D and had good sound effects too. I feel like it had a clock mascot somewhere but I could be wrong. The video was narrated by a professor.&lt;/p&gt;&#xA;" OwnerUserId="23416" LastActivityDate="2023-12-28T22:33:46.997" Title="Help finding a YouTube channel for Linear Algebra" Tags="|linear-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27197" PostTypeId="1" AcceptedAnswerId="27198" CreationDate="2023-11-20T13:16:29.250" Score="10" ViewCount="1673" Body="&lt;p&gt;I am teaching a course in commutative algebra, and it includes a project where the students research on a particular topic, solve a small problem and present it to the class.&lt;/p&gt;&#xA;&lt;p&gt;I usually give my students some project proposals which they can choose from, in case they cannot come up with an interesting problem statement themselves. I try to be as varied as possible in the problems I propose, covering most areas of commutative algebra.&lt;/p&gt;&#xA;&lt;p&gt;One such area I wanted to include is Gröbner bases and Elimination theory. I wanted to ask the Mathematics StackExchange community if they knew of any interesting polynomial systems of equations, perhaps arising from physics or chemistry, that are not completely straightforward but are also approachable for a graduate student of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="23421" LastActivityDate="2023-11-25T23:39:05.507" Title="Requesting a Polynomial System of Equations" Tags="|abstract-algebra|projects|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27198" PostTypeId="2" ParentId="27197" CreationDate="2023-11-20T17:06:22.160" Score="10" Body="&lt;p&gt;Here's a really nice textbook chapter that covers using algebraic geometry to compute equilibria in economics. It looks pretty approachable for students while simultaneously getting into plenty of nontrivial stuff. There are also exercises at the end of the chapter, which may serve as inspiration for project ideas.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sturmfels, B. (2002). Polynomial Systems in Economics. In &lt;a href=&quot;https://math.berkeley.edu/%7Ebernd/cbms.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Solving systems of polynomial equations&lt;/em&gt;&lt;/a&gt; (pp. 235-246). American Mathematical Soc.&lt;/p&gt;&#xA;&lt;p&gt;Teaser snippet:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MBrM7.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MBrM7.png&quot; alt=&quot;econ teaser&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here's another pedagogical article, this time from robotics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wampler, C. W., &amp;amp; Sommese, A. J. (2011). &lt;a href=&quot;https://www.researchgate.net/publication/281313535_Numerical_algebraic_geometry_and_algebraic_kinematics&quot; rel=&quot;noreferrer&quot;&gt;Numerical algebraic geometry and algebraic kinematics.&lt;/a&gt; &lt;em&gt;Acta Numerica, 20&lt;/em&gt;, 469-567.&lt;/p&gt;&#xA;&lt;p&gt;Teaser snippet:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Tnq4u.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Tnq4u.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And here's a research paper that uses algebraic geometry to compute protein structures. Not as friendly as the above, but potentially more interesting in terms of project ideas.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wang, L., Mettu, R. R., &amp;amp; Donald, B. R. (2005, August). &lt;a href=&quot;https://users.cs.duke.edu/%7Ebrd/papers/src-papers/2005/csb05-polytime.pdf&quot; rel=&quot;noreferrer&quot;&gt;An algebraic geometry approach to protein structure determination from NMR data.&lt;/a&gt; In &lt;em&gt;2005 IEEE Computational Systems Bioinformatics Conference (CSB'05)&lt;/em&gt; (pp. 235-246). IEEE.&lt;/p&gt;&#xA;&lt;p&gt;Teaser snippet:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Kh4pz.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Kh4pz.png&quot; alt=&quot;bio teaser&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-20T17:23:26.873" LastActivityDate="2023-11-20T17:23:26.873" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27200" PostTypeId="2" ParentId="27180" CreationDate="2023-11-20T22:48:31.073" Score="6" Body="&lt;p&gt;It's quite likely to be a consequence of the belief that they &lt;em&gt;have&lt;/em&gt; to answer the question. When they can't work it out, when they've gone around in circles and got lost, but still they &lt;em&gt;have&lt;/em&gt; to give an answer, what are they supposed to do? They believe that at least giving something in the &lt;em&gt;form&lt;/em&gt; of an answer is preferable to saying: &amp;quot;I don't know. I couldn't fill in this step. I went around in circles and got lost.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;It means they haven't understood the &lt;em&gt;purpose&lt;/em&gt; of doing exercises, which is &lt;em&gt;not&lt;/em&gt; to get a high score, or a passing grade, or to 'get the answer', or to 'follow the rules'. It is to enable both you and them to &lt;em&gt;identify gaps in their understanding&lt;/em&gt;, so you can go back and &lt;em&gt;fix&lt;/em&gt; them. The goal of education is not to get good grades, but to learn a skill. If they understand this, then it should be obvious that the right response is to say you can't do it, identify the gap, and perhaps try to explain what it is about it that you find so confusing.&lt;/p&gt;&#xA;&lt;p&gt;They should suffer no detriment from doing so. Only apply tests that they're going to be graded on once they have had a chance to fix their understanding as far as it can be fixed. With graded tests, the purpose then is to &lt;em&gt;measure/demonstrate&lt;/em&gt; their ability having been educated (or not), and here they should simply skip any question they can't do, and not waste time. Here again, they need to understand that there is no point or benefit in pretending to be able to do something you can't actually do. You might get a pass, and even a job, but you'll get caught out as soon as somebody asks you to do it for real. And the consequences of getting caught lying/cheating are infinitely worse than they consequences of saying &amp;quot;I don't know. I can't do this.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, the belief that grades are the goal is an &lt;em&gt;incredibly&lt;/em&gt; common misunderstanding, which the structure of school/university education often makes worse.&lt;/p&gt;&#xA;&lt;p&gt;--&lt;/p&gt;&#xA;&lt;p&gt;Very occasionally, you might get someone who picked up the lesson that because &lt;em&gt;you&lt;/em&gt; sometimes perform leaps that they don't understand, that seem to have no rhyme or reason, but just jump to the conclusion, &lt;em&gt;that this is how mathematics is done&lt;/em&gt;. Again, they need to be given the confidence and moral licence to say &amp;quot;I don't understand&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;It used to drive me mad when students obviously didn't understand something, but nobody dared ask for fear of looking stupid in front of their peers, or making themselves unpopular by delaying the class escaping at the end of the lecture. There is no point in having a live person there doing the lecture, rather than watching a video, if you're not going to ask questions. Nevertheless, the attitude was nearly ubiquitous.&lt;/p&gt;&#xA;" OwnerUserId="23425" LastActivityDate="2023-11-20T22:48:31.073" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27201" PostTypeId="2" ParentId="27197" CreationDate="2023-11-20T22:52:48.237" Score="6" Body="&lt;p&gt;Boolean networks (i.e. &lt;span class=&quot;math-container&quot;&gt;$F_2$&lt;/span&gt; valued systems) are a popular tool in mathematical biology for analysing gene regulatory networks. Fixed points correspond (roughly) to phenotypes and can be computed using Gröbner bases.&lt;/p&gt;&#xA;&lt;p&gt;Here is a database of examples: &lt;a href=&quot;https://booleangenenetworks.math.iastate.edu/&quot; rel=&quot;noreferrer&quot;&gt;https://booleangenenetworks.math.iastate.edu/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2023-11-20T22:52:48.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27202" PostTypeId="2" ParentId="27174" CreationDate="2023-11-21T02:22:29.947" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Does a proof by induction have to explicitly refer to the principle of mathematical induction?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There's no obligation of course, but it seems to be good pedagogy, especially at &lt;a href=&quot;https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;ealy(er) stages&lt;/a&gt;, to ask for and foster explicitness on argumentation. On the matter of induction specifically, one could do wonders by discussing &lt;em&gt;the different forms&lt;/em&gt; of induction:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;'Sucessor induction', the best known, that holds in the natural numbers;&lt;/li&gt;&#xA;&lt;li&gt;'Well-ordered induction', that also holds in, say, &lt;span class=&quot;math-container&quot;&gt;$\omega+\omega$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\omega + \omega + \omega$&lt;/span&gt;,..., with these lacking 'sucessor induction' &lt;a href=&quot;https://matheducators.stackexchange.com/a/27195/15843&quot;&gt;as a previous answer noted&lt;/a&gt;, but otherwise being models for sucessor-addition-multiplication arithmetic;&lt;/li&gt;&#xA;&lt;li&gt;'&lt;a href=&quot;https://en.wikipedia.org/wiki/Well-founded_relation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Well-founded induction&lt;/a&gt;', (sometimes called either 'weak' or 'strong' [!] induction in the context of the naturals) that holds for binary strings &lt;span class=&quot;math-container&quot;&gt;$2^{&amp;lt;\omega}$&lt;/span&gt; ordered by the substring relation, or for &lt;a href=&quot;https://en.wikipedia.org/wiki/Well-formed_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;well-formed formulas&lt;/a&gt; ordered by the subformula relation, each of these &lt;em&gt;not&lt;/em&gt; being well-orders&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Of course each case implies the next, and the examples show the implications are not reversible. The last one's likely to be more interesting/useful to students with interests in computing sciences, but really, just talk to them about these things&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Do proofs by contradiction have to explicitly refer to the principle of proof by contradiction (if there is such a thing)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes, please, let's try to teach people that &lt;a href=&quot;https://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/&quot; rel=&quot;nofollow noreferrer&quot;&gt;proofs of negations are not the same thing as proofs by contradiction&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15843" LastActivityDate="2023-11-21T02:22:29.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27203" PostTypeId="2" ParentId="27197" CreationDate="2023-11-21T02:53:34.510" Score="6" Body="&lt;p&gt;Two resources:&lt;/p&gt;&#xA;&lt;p&gt;(1) A really nice website &lt;a href=&quot;https://mattpap.github.io/masters-thesis/html/src/groebner.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Gröbner bases and their applications&lt;/em&gt;&lt;/a&gt;.&#xA;Includes many elementary examples coded in a Python package.&lt;/p&gt;&#xA;&lt;p&gt;(2) Automatic theorem discovery in elementary geometry. The paper below&#xA;applies this to several examples, one of which characterizes when a triangle's orthic triangle is isosceles.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Montes, Antonio, and Tomás Recio. &amp;quot;Automatic discovery of geometry theorems using minimal canonical comprehensive Gröbner systems.&amp;quot; In &lt;em&gt;Automated Deduction in Geometry: 6th International Workshop&lt;/em&gt;, ADG 2006, Pontevedra, Spain, August 31-September 2, 2006. Revised Papers 6, pp. 113-138. Springer Berlin Heidelberg, 2007.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="511" LastActivityDate="2023-11-21T02:53:34.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27204" PostTypeId="2" ParentId="27196" CreationDate="2023-11-21T04:34:55.263" Score="7" Body="&lt;p&gt;Just to provide this question with an answer, as my suggestion was confirmed in the comments, the Youtube playlist you're looking for is Robert Ghrist's &lt;a href=&quot;https://youtube.com/playlist?list=PL8erL0pXF3JYm7VaTdKDaWc8Q3FuP8Sa7&amp;amp;si=LUJdBDbwshsPzJNQ&quot; rel=&quot;noreferrer&quot;&gt;Calculus Blue Volume 1&lt;/a&gt; on vectors and matrices.&lt;/p&gt;&#xA;&lt;p&gt;(Post made Community wiki.)&lt;/p&gt;&#xA;" OwnerUserId="376" LastActivityDate="2023-11-21T04:34:55.263" CommentCount="0" CommunityOwnedDate="2023-11-21T04:34:55.263" ContentLicense="CC BY-SA 4.0" />
  <row Id="27205" PostTypeId="2" ParentId="26872" CreationDate="2023-11-21T16:48:19.200" Score="2" Body="&lt;p&gt;[&lt;strong&gt;Personal prelude&lt;/strong&gt;: once in a model theory course we were being presented a proof of Ramsey's theorem, which I found very similar in spirit to some proofs of Bolzano-Weierstrass, so I mentioned it and asked &amp;quot;Are they equivalent?&amp;quot; (in the non-tautological sense). The professor replied they could quite see it too, but didn't know for sure, so I rapily downloaded Simpson's &lt;em&gt;Subsystems of second order arithmetic&lt;/em&gt;, and there it was: both statements are indeed equivalent to arithmetic comprehension, over the weaker base theory. It's nice to know there is in fact a very definite way to examine such intuitive guesses, even if one doesn't follow all the details. So:]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How should an educator answer a student who asks &amp;quot;Can this theorem be&#xA;deduced in other systems of set theory?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Ideally&lt;/em&gt; with something along the lines of &amp;quot;There's &lt;strong&gt;a lot&lt;/strong&gt; of research about provability in set theories weaker than ZFC, the most authoritative reference for theorems of 'ordinary mathematics' certainly being Simpson's SSSO, take it from there!&amp;quot;, but this is rather unrealistic, as we can guess most people simply haven't ever thought about such things, in which case an honest answer is the best, followed by the suggestion to look it up on the internet. Just anything that's supportive of the student's curiosity, and not dismissive/hostile, really&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] would it be fair to ensure that the course description and&#xA;course syllabus include the name of the particular system of set&#xA;theory that all theorems studied in the course are deduced from?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some texts do so - most famously Kelley, Bourbaki, and SGA4, but also, say, MacLane's CFTWM and Gabriel &amp;amp; Demazure -, so that one may list the same system as their chosen reference. If such questions keep popping up, that's a sign the students consider it relevant, and that &lt;em&gt;maybe&lt;/em&gt; it's not totally beyond the scope of the course after all :)&lt;/p&gt;&#xA;" OwnerUserId="15843" LastActivityDate="2023-11-21T16:48:19.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27206" PostTypeId="2" ParentId="20649" CreationDate="2023-11-21T18:33:17.873" Score="5" Body="&lt;p&gt;I'm very fond of Eugenia Cheng's perspective: many people dislike certain parts and/or aspects of mathematics, and it just happens that these are precisely the ones emphasized by most (all?) school curricula. To quote from her wonderful &amp;quot;Joy of abstraction&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Some people do need to build up gradually through concrete examples&#xA;towards abstract ideas. But not everyone is like that. For some&#xA;people, the concrete examples don’t make sense until they’ve grasped&#xA;the abstract ideas or, worse, the concrete examples are so offputting&#xA;that they will give up if presented with those first. [...]&lt;/p&gt;&#xA;&lt;p&gt;I have confirmed from several years of teaching abstract mathematics&#xA;to art students that I am not the only one who prefers to use abstract&#xA;ideas to illuminate concrete examples rather than the other way round.&#xA;Many of these art students consider that they’re bad at math because&#xA;they were bad at memorizing times tables, because they’re bad at&#xA;mental arithmetic, and they can’t solve equations. But this doesn’t&#xA;mean they’re bad at math — it just means they’re not very good at&#xA;times tables, mental arithmetic and equations, an absolutely tiny part&#xA;of mathematics that hardly counts as abstract at all. It turns out&#xA;that they do not struggle nearly as much when we get to abstract&#xA;things such as higher-dimensional spaces, subtle notions of&#xA;equivalence, and category theory structures. Their blockage on mental&#xA;arithmetic becomes irrelevant.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me that we are denying students entry into abstract&#xA;mathematics when they struggle with non-abstract mathematics, and that&#xA;this approach is counter-productive. Or perhaps some students&#xA;self-select out of abstract mathematics if they did not enjoy&#xA;non-abstract mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Sadly, there doesn't seem to be any easy way out of this problem. Notice that 'New Math' doesn't necessarily aling with what's called &amp;quot;abstract&amp;quot; here.&lt;/p&gt;&#xA;" OwnerUserId="15843" LastActivityDate="2023-11-21T18:33:17.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27207" PostTypeId="2" ParentId="27138" CreationDate="2023-11-21T18:38:18.020" Score="0" Body="&lt;p&gt;It &lt;em&gt;might&lt;/em&gt; be related to the order (some of) these operations appear on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hyperoperation&quot; rel=&quot;nofollow noreferrer&quot;&gt;hyperoperation sequence&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15843" LastActivityDate="2023-11-21T18:38:18.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27208" PostTypeId="2" ParentId="27174" CreationDate="2023-11-21T22:48:21.517" Score="6" Body="&lt;p&gt;A blast from the past comment, for the consolation of your students, of a mathematician being marked down by one of the most influential mathematicians of his day:&lt;/p&gt;&#xA;&lt;p&gt;John Wallis in his &lt;em&gt;Arithmetica Infinitorum&lt;/em&gt; (1656) in Prop. XIX&#xA;examines the sum of squares up to &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; divided by &lt;span class=&quot;math-container&quot;&gt;$n^2(n+1)$&lt;/span&gt;.&#xA;He computes them for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; up to &lt;span class=&quot;math-container&quot;&gt;$n=6$&lt;/span&gt; and proclaims&#xA;that by way of induction (&amp;quot;&lt;em&gt;per modum inductionis&lt;/em&gt;&amp;quot;),&#xA;that the sum is given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$${\,0\;+\,1\,+\,4\;+\cdots+n^2 \over n^2+n^2+n^2+\cdots+n^2}={1\over3}+{1\over6n} \,.$$&lt;/span&gt;&#xA;Wallis is the first written source to use the word &amp;quot;induction&amp;quot;&#xA;to describe and justify the proof of a proposition.&#xA;Wallis used his form of induction frequently.&#xA;Wallis was criticized, but he defended his method.&#xA;Jacob Bernoulli suggested that one of Wallis's proofs could be&#xA;improved by showing how the &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; case followed from the case for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; —&#xA;an explicit statement of the induction step.&#xA;Here is Wallis defending his method in 1685, &lt;em&gt;A Treatise of Algebra&lt;/em&gt;,&#xA;Ch.78, p.298:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Those propositions in my Arithmetick of Infinites, are (some of them) demonstrated by way of Induction: Which is plain, obvious, and easy; and where things proceed in a clear regular Order, (as here they do,) very satisfactory, (to any who hath not a mind to cavil;) and shews the true natural investigation. Which to me, is much more grateful and agreeable, than the Operose Apagogical Demonstrations, (by reducing to Absurdities or Impossibilities,) which some seem to affect;...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Long before the Principle of Mathematical Induction had been formalized, there was controversy over how pedantically formal proofs should be.&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2023-11-22T14:04:33.417" LastActivityDate="2023-11-22T14:04:33.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27209" PostTypeId="2" ParentId="27197" CreationDate="2023-11-23T17:53:13.397" Score="3" Body="&lt;p&gt;This is great to include such problems. Here are a few examples of nice doable problems:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Any problem that involves Lagrange multipliers, a constraint curve given as a multivariate polynomial, and a function to be maximised/minimised also given as such polynomial.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Comment: If one has trig functions, e.g. sine and cosine, one can always replace them with new indeterminates  say &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; provided this extra polynomial &lt;span class=&quot;math-container&quot;&gt;$X^2+Y^2-1$&lt;/span&gt; is added to the list of polynomials that give a basis in some ideal &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; for which later Gröbner basis will be computed.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Let's ask this question: let's take an ellipse (or any other quadratic like a parabola or a hyperbola) say:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2/a^2 + y^2/b^2 = 1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;with &lt;span class=&quot;math-container&quot;&gt;$a \ne b$&lt;/span&gt; and let &lt;span class=&quot;math-container&quot;&gt;$r&amp;gt;0$&lt;/span&gt; be a parameter. Imagine a circle of radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; whose center &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; is located on the ellipse at a point &lt;span class=&quot;math-container&quot;&gt;$(x_0,y_0)$&lt;/span&gt;. Now move the circle along the ellipse so that its center &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; remains on the ellipse. At each point &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; draw a line through &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; perpendicular to the tangent at &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; to the ellipse. Each such line intersects the circle at &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; of radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; at two points &lt;span class=&quot;math-container&quot;&gt;$P_1(x,y)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P_2(x,y)$&lt;/span&gt;. Find two polynomials &lt;span class=&quot;math-container&quot;&gt;$p_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p_2$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$x, y, a, b, x_0, y_0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; such that the equations &lt;span class=&quot;math-container&quot;&gt;$p_1=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p_2=0$&lt;/span&gt; give two curves traced by the points &lt;span class=&quot;math-container&quot;&gt;$P_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P_2$&lt;/span&gt; when the center &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; of the circle traces the ellipse. These curves will be equidistant to the ellipse at the distance &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;. Analyse the shape of these two curves for the given &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; as parameterised by &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;. For small enough &lt;span class=&quot;math-container&quot;&gt;$r &amp;lt; r_0$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$r_0$&lt;/span&gt; versus the values of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, the two curves will be smooth. When &lt;span class=&quot;math-container&quot;&gt;$r=r_0$&lt;/span&gt;, two singular point will appear. When &lt;span class=&quot;math-container&quot;&gt;$r&amp;gt;r_0$&lt;/span&gt;, each singular point will become a dovetail with three singular points. Find a formula for &lt;span class=&quot;math-container&quot;&gt;$r_0$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. Relate it to the curvature of the ellipse.&lt;/p&gt;&#xA;&lt;p&gt;In another words, find an envelope of the ellipse for each &lt;span class=&quot;math-container&quot;&gt;$r&amp;gt;0$&lt;/span&gt;. What happens in the special case when &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt;? Geometrically and algebraically?&lt;/p&gt;&#xA;&lt;p&gt;Problems like these two above require a CAS like Maple, for example, to compute Gröbner bases. The way I have formulated problem 2 is a hint how to find the two polynomial curves for the ellipse or for any other quadratic. Nothing else is needed except the understanding of a tangent line and a normal line to a curve in a plane.&lt;/p&gt;&#xA;&lt;p&gt;Warning: The polynomial curves equidistant do a true ellipse (&lt;span class=&quot;math-container&quot;&gt;$a \ne b$&lt;/span&gt;) are not ellipses (of course). It is easy to predict what they are when &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;For the given ellipsoid&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2/a^2+y^2/b^2+z^2/c^2=1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and a point &lt;span class=&quot;math-container&quot;&gt;$P(x_0,y_0,z_0)$&lt;/span&gt; not on the ellipsoid, find:&lt;/p&gt;&#xA;&lt;p&gt;(a) a point &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; on the ellipsoid closest to &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(b) points &lt;span class=&quot;math-container&quot;&gt;$S_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$S_2$&lt;/span&gt; on the ellipsoid such that the lines &lt;span class=&quot;math-container&quot;&gt;$PS_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$PS_2$&lt;/span&gt; are normal to the ellipsoid.&lt;/p&gt;&#xA;&lt;p&gt;As far as a reference, I recommend the book &lt;em&gt;Ideals, Varieties, and Algorithms&lt;/em&gt; by Cox, Little &amp;amp; O'Shea on elementary Algebraic Geometry. It is a fantastic book whose initial chapters through Gröbner bases and the theory of elimination should be accessible to algebraically advanced high school students.&lt;/p&gt;&#xA;&lt;p&gt;Good luck!&lt;/p&gt;&#xA;&lt;p&gt;Rafal Ablamowicz&#xA;rablamowicz@gmail.com&lt;/p&gt;&#xA;" OwnerUserId="23434" LastEditorUserId="376" LastEditDate="2023-11-24T15:34:52.523" LastActivityDate="2023-11-24T15:34:52.523" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27210" PostTypeId="2" ParentId="27197" CreationDate="2023-11-24T00:29:46.593" Score="2" Body="&lt;p&gt;Code theory, from the book &amp;quot;A course in error correcting codes&amp;quot;, from Justesen and Hohold. Chapter 5 and 6 tell you how to relate a linear codes, Reed Solomon and cyclic codes, with special polynomials. It is enough the first 2 sections of those chapters and chapter 1, fist 2 sections, too.&lt;/p&gt;&#xA;" OwnerUserId="23435" LastActivityDate="2023-11-24T00:29:46.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27212" PostTypeId="1" AcceptedAnswerId="27213" CreationDate="2023-11-25T05:55:03.050" Score="3" ViewCount="1692" Body="&lt;p&gt;Though not an undergraduate student , I just wanted to know where can I find hard new types of problems regarding the problems in graduate level mathematics. As per my information , standard books from renowned authors and publications are the only sources of problems but still I want to know some sources where I can get difficult problems which can be useful for  acquiring some skill set. Also I wanted to know where can I find all kinds of research and open problem? I know that Wikipedia gives a list of problems but I still doubt that there are many problems on some authentic math-research websites. Can anyone provide any information regarding this? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="23368" LastActivityDate="2023-11-25T17:32:56.077" Title="Where can I find new types of problems regarding graduate level mathematics?" Tags="|undergraduate-education|education-research|graduate-education|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27213" PostTypeId="2" ParentId="27212" CreationDate="2023-11-25T06:47:44.517" Score="9" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;As per my information , standard books from renowned authors and publications are the only sources of problems but still I want to know some sources where I can get difficult problems which can be useful for acquiring some skill set.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't really understand this first question. If you want difficult problems that are useful for acquiring some skill set, wouldn't standard textbooks be the optimal resource for that? What is a textbook, if not a collection of information and exercises that the author believes is optimal to acquire some skill set? (If I'm misinterpreting this question, then perhaps you can edit your post to clarify it and leave a comment to let me know.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;where can I find all kinds of research and open problem? I know that Wikipedia gives a list of problems but I still doubt that there are many problems on some authentic math-research websites.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This question I understand and am happy to answer, though I need to preface my answer with a couple disclaimers:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It's really unlikely that you'll be able to rigorously solve any of these problems without a thorough background in both undergraduate and graduate-level math. Even graduate students generally need to spend some time specializing before they're in a position where they can make a contribution towards the &amp;quot;edge&amp;quot; of humanity's collective mathematical knowledge.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm not a professional research mathematician. I just had a phase during undergrad when I was really interested in learning more about what kinds of problems are &amp;quot;open&amp;quot; in math today -- in particular, the &amp;quot;typical&amp;quot; kinds of open problems, not just genius-level Millennium Prize problems and the like. (I was curious to see what the &amp;quot;edge&amp;quot; of math looks like nowadays, since today's math goes so deep that it can be hard to know just how deep.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Anyway, here's my answer.&lt;/p&gt;&#xA;&lt;p&gt;The site &lt;a href=&quot;http://www.openproblemgarden.org/&quot; rel=&quot;noreferrer&quot;&gt;http://www.openproblemgarden.org/&lt;/a&gt; gets mentioned a lot for these kinds of questions, but it's heavily slanted towards graph theory and isn't exactly comprehensive and up-to-date.&lt;/p&gt;&#xA;&lt;p&gt;I think you'll have better luck searching &lt;a href=&quot;https://scholar.google.com/&quot; rel=&quot;noreferrer&quot;&gt;Google Scholar&lt;/a&gt; with the query &lt;code&gt;&amp;quot;unsolved open problems in &amp;lt;&amp;lt;insert specific field of math here&amp;gt;&amp;gt;&amp;quot;&lt;/code&gt; and putting some date constraint to limit the search to fairly recent results.&lt;/p&gt;&#xA;&lt;p&gt;For example, you might &lt;a href=&quot;https://scholar.google.com/scholar?hl=en&amp;amp;as_sdt=0%2C22&amp;amp;as_ylo=2020&amp;amp;q=unsolved%20open%20problems%20in%20combinatorics&amp;amp;btnG=&quot; rel=&quot;noreferrer&quot;&gt;search&lt;/a&gt; &amp;quot;unsolved open problems in combinatorics&amp;quot; and limit the results since 2020. (Of course, this is just a starting point, and you'll want to play around with the date range and the specificity of the query to get better results.)&lt;/p&gt;&#xA;&lt;p&gt;Note that if you identify an open problem that you're interested in learning more about, you'll also need to do follow-up searches to learn more about the work that has been done on that problem, especially any work that has been done since the &amp;quot;open problem&amp;quot; article that you're reading was published (who knows, the problem may have been solved since then, or some other kind of progress may have been made).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zTgdw.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zTgdw.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-11-25T07:12:15.567" LastActivityDate="2023-11-25T07:12:15.567" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27216" PostTypeId="2" ParentId="27212" CreationDate="2023-11-25T17:32:56.077" Score="9" Body="&lt;p&gt;Nowadays, you will be able to find many research mathematicians' websites with lecture notes for advanced courses, many with exercises/examples, many with solutions/discussions. I myself certainly put lots of things on-line.&lt;/p&gt;&#xA;&lt;p&gt;If you haven't studied much graduate-level mathematics, I'd anticipate that the main obstacle for you would be understanding the words used in the very formulation of the questions. And it's not just about &amp;quot;parsing definitions&amp;quot;... very often the definition itself is unintelligible without knowing lots of stuff... but, awkwardly, most definitions do not acknowledge or explain the necessary background to understand them! :)&lt;/p&gt;&#xA;&lt;p&gt;In my own experience, the most productive mental viewpoint is to think about simply working-to-understand various things, rather than problem-solving. Sometimes, you'll happen to come to an understanding that no one had previously. Maybe not describable as &amp;quot;solving a problem&amp;quot;. :)&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2023-11-25T17:32:56.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27217" PostTypeId="2" ParentId="27174" CreationDate="2023-11-25T18:52:27.363" Score="3" Body="&lt;p&gt;Let's not overlook an obvious application of induction is to turn it around and go from general to specific.&lt;/p&gt;&#xA;&lt;p&gt;Let's say we have established &lt;span class=&quot;math-container&quot;&gt;$H(1)$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$H(k+1)$&lt;/span&gt; follows from &lt;span class=&quot;math-container&quot;&gt;$H(k)$&lt;/span&gt; and we are perfectly content with whatever algebra and logic was involved to establish this. But we simply refuse to recognize or accept induction as method of proof. We just refuse to chant &amp;quot;&lt;em&gt;by induction, &lt;span class=&quot;math-container&quot;&gt;$H(n)$&lt;/span&gt; is true for all &lt;span class=&quot;math-container&quot;&gt;$n\in{\bf N}$&lt;/span&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Now then, how would we actually write down a proof of &lt;span class=&quot;math-container&quot;&gt;$H(10,000)$&lt;/span&gt;? Well, we would start our proof by establishing &lt;span class=&quot;math-container&quot;&gt;$H(2)$&lt;/span&gt;, then we would continue writing by establishing &lt;span class=&quot;math-container&quot;&gt;$H(3)$&lt;/span&gt;, etc. Eventually we would have written down a very long, boring and stupid proof that establishes &lt;span class=&quot;math-container&quot;&gt;$H(10,000)$&lt;/span&gt;. If only we not so stubborn, and were willing to invoke induction to &lt;strong&gt;first&lt;/strong&gt; conclude &lt;span class=&quot;math-container&quot;&gt;$H(n)$&lt;/span&gt; is true for all &lt;span class=&quot;math-container&quot;&gt;$n\in{\bf N}$&lt;/span&gt;, and &lt;em&gt;then&lt;/em&gt; go from general to specific: apply the validity of &lt;span class=&quot;math-container&quot;&gt;$H(n)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$n\in{\bf N}$&lt;/span&gt; to the specific case &lt;span class=&quot;math-container&quot;&gt;$n=10,000$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In a similar vein, how can we establish &lt;span class=&quot;math-container&quot;&gt;$$\sum_{n=1}^{10,000} n^2 = 333383335000$$&lt;/span&gt; if we refuse to accept induction as a method of proof?&lt;/p&gt;&#xA;&lt;p&gt;No problem. We just have a symbolic computational system like Wolfram perform the sum. Perhaps for the first &lt;span class=&quot;math-container&quot;&gt;$N=10,000$&lt;/span&gt; terms the system is actually doing the addition by brute force. But at higher values of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;, the system is smart enough to &amp;quot;cheat&amp;quot; and skip the onerous addition, and instead substitute &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; into the formula &lt;span class=&quot;math-container&quot;&gt;$\frac{N(N + 1)(2N + 1)}{6}$&lt;/span&gt;. So again, induction is being used to go from general to specific. The general result is the statement&#xA;&lt;span class=&quot;math-container&quot;&gt;$$H(N):\ \sum_{n=1}^N n^2=\frac{N(N+1)(2N+1)}{6}$$&lt;/span&gt; which is established by induction, and we go to the specific case by substituting a numerical value into the formula.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-11-25T20:53:31.207" LastActivityDate="2023-11-25T20:53:31.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27218" PostTypeId="2" ParentId="27180" CreationDate="2023-11-25T21:36:23.740" Score="3" Body="&lt;p&gt;The answer to &amp;quot;what to do?&amp;quot; is to discourage it as much as you can, arguing from the position of authority, if normal explanations of why it doesn't work fail (but certainly you should try the latter too).&lt;/p&gt;&#xA;&lt;p&gt;The first step is to get rid of the idea of &amp;quot;partial credit&amp;quot; entirely or to reduce it to the minimum. In undergraduate courses I usually announce in the very beginning that every problem will be scored from 0 to 3, 3 being a full solution, 2 being a full solution with a minor glitch (like a stupid numerical error in an otherwise correct computation scheme or an erroneous sign flip near the end of the derivation), 1 being an essential progress in the right direction, 0 being everything else and that I'm the sole authority to identify all the adjectives here (full, minor, essential, right, etc.).&lt;/p&gt;&#xA;&lt;p&gt;That may sound a bit harsh, but that allows me to shut down all the potential whining of the type &amp;quot;I wrote 5 pages of correct computations/derivations; don't I deserve some points for that?&amp;quot; from the beginning. Note that I never give a problem on a written test that would require more than half a page to explain the solution in full if one knows what he or she is doing and I tell students that, though occasionally long solutions are correct and will be given a full credit in such cases, going beyond 2 pages without any clear ending in sight should raise some red flags in their heads on an exam.&lt;/p&gt;&#xA;&lt;p&gt;I fully realize that spending a huge amount of time going in a completely wrong direction is an essential part of every problem solving attempt both on an exam and in the real research. So, I believe that one needs to explain to the students that it is entirely normal and I would not introduce any punishments for that (especially if they are just trying to get that 1 point in my grading scheme). However, one needs also to make it clear to them that if somebody wants to get from New York to San-Francisco and goes 4000 miles &lt;em&gt;south&lt;/em&gt;, that wouldn't count as progressing towards the goal &lt;em&gt;regardless&lt;/em&gt; of the time, effort, or money spent on that part of the trip. As working mathematicians, ninety percent of the time (if not more) we do find ourselves exactly in the situation of that hapless traveler, and we often have to scrap it all and return to square one in such cases. The students should understand that they can hardly expect anything else from mathematical (and not only) problem solving beyond the most routine exercises where the answer or a solution algorithm is immediately clear from the problem formulation itself and get used to it if they want to succeed in the course or beyond it.&lt;/p&gt;&#xA;&lt;p&gt;As to what to do in the cases of completely routine problems whose way of solving they should just &lt;em&gt;know&lt;/em&gt;, I usually merely tell them that such things should become like a knee reflex and if they fail to recognize and follow the standard routine, they just demonstrate that they failed to learn the routine stuff, so they are getting nowhere in the subject and, unless they are able to compensate for that with ingenuity (i.e., to solve the problem &lt;em&gt;completely&lt;/em&gt; by some alternative method), they don't really deserve a decent grade either.&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents :-).&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2023-11-25T21:36:23.740" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27219" PostTypeId="1" AcceptedAnswerId="27226" CreationDate="2023-11-25T22:33:57.663" Score="4" ViewCount="253" Body="&lt;p&gt;I have created the following simple tiling puzzle for children.&lt;/p&gt;&#xA;&lt;p&gt;The puzzle consists of 9 tiles shown below (4 pieces of type A, 4 pieces of type B and 1 piece of type C):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lQOja.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lQOja.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The goal of the puzzle is to arrange the 9 tiles to form a 3x3 square such that the border of the square is all blue.  You are allowed to rotate the tiles.&lt;/p&gt;&#xA;&lt;p&gt;The solution is:&lt;/p&gt;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;&lt;p&gt; &lt;a href=&quot;https://i.stack.imgur.com/ecUO1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ecUO1.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question is what age group is this puzzle appropriate for (not too easy, not too tough)?&lt;/p&gt;&#xA;" OwnerUserId="23338" LastActivityDate="2023-11-27T20:36:54.847" Title="What age group is this puzzle appropriate for?" Tags="|math-puzzle|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27220" PostTypeId="2" ParentId="1860" CreationDate="2023-11-26T19:36:54.767" Score="2" Body="&lt;p&gt;When we work with numbers of the form &lt;span class=&quot;math-container&quot;&gt;$a+b\sqrt{2}$&lt;/span&gt; (with &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; rational), it is useful to rationalize.  For example, to determine that the set of numbers of that form is closed under division.  Or to determine whether &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{7}{1+5\sqrt{2}}\quad\text{and}\quad -\frac{1}{7}+\frac{5}{7}\sqrt{2}&#xA;$$&lt;/span&gt;&#xA;are equal.&lt;/p&gt;&#xA;&lt;p&gt;This is a similar consideration to re-writing complex numbers in the Argand form &lt;span class=&quot;math-container&quot;&gt;$a+bi$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="127" LastActivityDate="2023-11-26T19:36:54.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27221" PostTypeId="1" CreationDate="2023-11-27T11:40:36.147" Score="7" ViewCount="675" Body="&lt;p&gt;Seeking guidance on how to assist students who possess a solid grasp of problem-solving concepts, allowing them to intuitively arrive at solutions, yet encounter difficulties when it comes to communicating their understanding through formal proofs and written formats.&lt;/p&gt;&#xA;&lt;p&gt;These students exhibit a deep comprehension of the underlying principles and can often visualize the solution or approach to a problem. However, they face challenges in articulating their ideas in a structured and rigorous manner. It is crucial to help these students bridge the gap between their intuitive problem-solving abilities and their ability to express their reasoning formally.&lt;/p&gt;&#xA;&lt;p&gt;I am interested in hearing your suggestions regarding effective strategies or techniques to support these students. How can we help them refine their problem-solving skills by guiding them to articulate their intuitive ideas in a more formal and coherent manner? Are there specific exercises, resources, or approaches that have proven successful in facilitating this transition?&lt;/p&gt;&#xA;&lt;p&gt;While my primary preference is to focus on undergraduate students, I am also interested in insights and strategies that are applicable to high school-level students. Hearing perspectives from both levels would provide valuable insights for supporting students in their transition towards more formal proof-writing.&lt;/p&gt;&#xA;&lt;p&gt;Please note that the focus here is not solely on teaching proof-writing techniques but also on empowering students to leverage their existing intuitive understanding to construct sound and logical arguments in written form.&lt;/p&gt;&#xA;" OwnerUserId="23442" LastEditorUserId="23442" LastEditDate="2023-11-27T15:04:52.703" LastActivityDate="2023-11-27T17:16:28.963" Title="Bridging the gap between students' intuitive problem-solving abilities and expressing ideas through formal writing" Tags="|undergraduate-education|mathematical-pedagogy|proofs|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27223" PostTypeId="1" CreationDate="2023-11-27T14:44:44.413" Score="9" ViewCount="442" Body="&lt;p&gt;Context: I am an associate professor at a small liberal arts institution in the US.&lt;/p&gt;&#xA;&lt;p&gt;I find in my introductory business math course that students sometimes fail to buy a calculator for the course, despite my frequent reminders. Rather, they get by with their phone calculator or with help from their friends during lessons and on homework. This means that when an exam comes, they are not prepared to use a proper calculator, nor do they have one for the exam.&lt;/p&gt;&#xA;&lt;p&gt;Typically it is my weakest students who fall into this category, so this problem exacerbates the gap between the strongest and weakest students.&lt;/p&gt;&#xA;&lt;p&gt;Ideally, students would be confident using a real calculator to accurately compute functions involving parentheses, logs, and exponents. I recommend that students purchase a TI-30XIIS or similar, although they are allowed to use any calculator that doesn't have CAS capabilities.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have any suggestions for how to deal with this? Here's what I see as my options:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Keep a few spare calculators for students to borrow during exams.  This has the added benefit of helping those who legitimately forget their calculator on exam day, have their batteries die, etc. (This is what I currently do. I tell students that I am happy to show them any computations before the exam, but I can't show them anything about a calculator once the exam is started.)&lt;/li&gt;&#xA;&lt;li&gt;Do the same thing as #1, but &amp;quot;tax&amp;quot; students some amount of points or provide some other disincentive for borrowing one of my calculators. (I had a high school teacher who would make you leave one of your shoes at her desk if you wanted to borrow something from her--the embarrassment/inconvenience incentivized students to remember their stuff.)&lt;/li&gt;&#xA;&lt;li&gt;Require that students bring a calculator to class every day, and penalize them some small amount of participation credit for not doing so.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5988" LastEditorUserId="22983" LastEditDate="2024-01-04T17:52:44.613" LastActivityDate="2024-01-06T01:30:19.413" Title="Loaning students calculators during exams" Tags="|undergraduate-education|student-motivation|exams|calculators|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27224" PostTypeId="2" ParentId="27221" CreationDate="2023-11-27T17:16:28.963" Score="7" Body="&lt;p&gt;I've worked with some of these types of students in the past. One trend I noticed was that these students often have experience with coding, which they tend to enjoy and excel at (since the computer tells them exactly how precisely they have to specify their ideas and will error in obvious ways if they're not being precise enough or they're skipping over important nuances).&lt;/p&gt;&#xA;&lt;p&gt;When teaching proofs to students who fit this trend, I've had success leaning on analogies to coding as much as possible as a way to get through to them.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I tell them to think of the proof as a program that runs in mathematicians' brains. The theorem being proved is like a new &amp;quot;software feature&amp;quot; that can be used in the future, and the proof is like the code supporting that feature. The goal is to &amp;quot;install&amp;quot; this theorem in somebody else's mind by having them &amp;quot;run&amp;quot; the proof.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When you're coding at your computer, as long as you've got the right intuition/principles in mind, you can start with a vague, imprecise program and then get it working by running it, getting an error, and tightening up the code in the place where the error occurred. But you don't have an unlimited instant feedback loop like that when you're writing a proof. With human-to-human writing, you generally have one shot, and you have to make that shot count. So you need to be completely explicit with every step of the proof. It's like the code that you're writing is going straight into production, without testing beforehand.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As a general rule, if someone could ask &amp;quot;why&amp;quot; or &amp;quot;how&amp;quot; about some assertion in your proof, and the answer to that question is not already contained in your proof, then an error is going to occur when that person tries to &amp;quot;run&amp;quot; the proof (and you won't be around to fix the error). For instance, if you have a mental image that makes an assertion obvious in your mind, you need to communicate that mental image in your proof so that it can be used by the reader as well. Otherwise it's like your program is attempting to import code from a package that is not installed in the environment where the code is running.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-11-27T17:16:28.963" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27225" PostTypeId="1" CreationDate="2023-11-27T20:05:04.340" Score="2" ViewCount="125" Body="&lt;p&gt;Since a few weeks, I teach as a tutor (not from that school) a support course in a German 9/10 class. I quickly noticed a horrible lack of basics. (Partly based on just different names - I had to learn the hard way that they, say, use |-1 to subtract 1 from both sides of an equation. Of course I know the notation, but the operation was too obvious for me.)&lt;/p&gt;&#xA;&lt;p&gt;My idea was to increase their mental capacity of using the toolbox I will teach by literarily naming the tools with a, duh, similar household tool. (Or other things.) For example, |-1 would be the saw, since you saw off the same on both sides.&lt;/p&gt;&#xA;&lt;p&gt;My course at the moment deals with solving quadratic equations but I first have to re-teach them all the forgotten stuff...&lt;/p&gt;&#xA;&lt;p&gt;My list so far:&lt;/p&gt;&#xA;&lt;p&gt;stop sign - to remember traffic rules: +- yields to */ yields to ^ yields to ([])&lt;/p&gt;&#xA;&lt;p&gt;worker helmet - Danger! E.g. a root is defined as positive, but still a quadratic equation has (up to) two solutions&lt;/p&gt;&#xA;&lt;p&gt;= reflexivity - ???&lt;/p&gt;&#xA;&lt;p&gt;= symmetry - Mirror&lt;/p&gt;&#xA;&lt;p&gt;= transitivity - Chain&lt;/p&gt;&#xA;&lt;p&gt;Commutativity of *,+ - ???&lt;/p&gt;&#xA;&lt;p&gt;Associativity of *,+ - ???&lt;/p&gt;&#xA;&lt;p&gt;Distributivity - wrench (screws ([]) off or on)&lt;/p&gt;&#xA;&lt;p&gt;Inserting known values - funnel&lt;/p&gt;&#xA;&lt;p&gt;Simplifying equation - rake&lt;/p&gt;&#xA;&lt;p&gt;same action on both sides of = sign - saw&lt;/p&gt;&#xA;&lt;p&gt;collect (x+a/2)^2 from x^2+a*x+b - hammer&lt;/p&gt;&#xA;&lt;p&gt;extract square root - tongs&lt;/p&gt;&#xA;&lt;p&gt;general catch-em-all formula - Robot&lt;/p&gt;&#xA;&lt;p&gt;And when they are done with this, Vieta's theorem dynamite, blows&#xA;&lt;em&gt;every&lt;/em&gt; equation with integer solutions skyhigh! (This theorem&#xA;is so useful, yet so little known or taught...)&lt;/p&gt;&#xA;&lt;p&gt;If you like to improve or fill in on some of my random associations (the hammer is...silly), feel free to use comments, but my question is rather:&lt;/p&gt;&#xA;&lt;p&gt;Can you suggest a reference where &amp;quot;my&amp;quot; concept (make it stick by association),&#xA;best specific for quadratic equations (otherwise I can google myself, since the general concept is well known), already has been fleshed out, even tested?&lt;/p&gt;&#xA;&lt;p&gt;Vaguely related: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13661/how-to-write-steps-of-solving-equations&quot;&gt;since I have to do &lt;em&gt;everything&lt;/em&gt; step by step&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="23446" LastEditorUserId="2083" LastEditDate="2023-11-27T20:24:02.367" LastActivityDate="2023-11-27T20:24:02.367" Title="&quot;Tools&quot; (literarily) for solving linear or quadratic equations" Tags="|undergraduate-education|mathematical-pedagogy|reference-request|germany|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27226" PostTypeId="2" ParentId="27219" CreationDate="2023-11-27T20:36:54.847" Score="6" Body="&lt;p&gt;I'd say it depends highly on how it's presented.&#xA;&amp;quot;What's the prettiest picture you can make with these nine tiles?&amp;quot; will probably go with any age.&#xA;&amp;quot;Bad&amp;quot; solutions can then be elaborated on by &amp;quot;But what if ...&amp;quot;s.&#xA;Why not initially test it on a sample of different ages in this form, and evaluate how it's handled by different ages?&lt;/p&gt;&#xA;" OwnerUserId="13868" LastActivityDate="2023-11-27T20:36:54.847" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27227" PostTypeId="2" ParentId="27174" CreationDate="2023-11-28T13:04:41.040" Score="3" Body="&lt;p&gt;To be a proof, an argument &lt;em&gt;needs to be explicit about the logical structure&lt;/em&gt;.  An induction proof won't be any different.  Take a very standard task like: such as showing that, for all &lt;span class=&quot;math-container&quot;&gt;$n\in \mathbb{N}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{1 - x^{n+1}}{1-x} = 1 + x + x^2 + \cdots + x^n&#xA;$$&lt;/span&gt;&#xA;(i.e., that the rational function on the left is the polynomial on the right).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Answer 1:&lt;/strong&gt; Let &lt;span class=&quot;math-container&quot;&gt;$n\in \mathbb{N}$&lt;/span&gt; be given. We  compute&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; (1 - x)(1 + x + x^2 + \cdots + x^n) = &#xA;  (1 - x) + (x - x^2) + \cdots + (x^n - x^{n+1})&#xA;$$&lt;/span&gt;&#xA;Rebracketing the rhs we get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  1 + (-x + x) + (-x^2 + x^2) + \cdots + (-x^n + x^n) - x^{n+1} = 1 - x^{n+1}&#xA;$$&lt;/span&gt;&#xA;and so&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  (1 - x)(1 + x + x^2 + \cdots + x^n) = 1 - x^{n+1}&#xA;$$&lt;/span&gt;&#xA;Dividing both sides by &lt;span class=&quot;math-container&quot;&gt;$1-x$&lt;/span&gt; gives the identity we wanted for this &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&#xA;As &lt;span class=&quot;math-container&quot;&gt;$n\in \mathbb{N}$&lt;/span&gt; was arbitrary, the identity holds for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Comments:&lt;/em&gt; I picked this example because it is a standard&#xA;school induction exercise, but this isn't a proof by induction.&#xA;The logical structure is different.  (Though it is the same structure you see in &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; proofs from calculus class. A  weird thing to university level instructors is that some students think that &amp;quot;proofs&amp;quot; are &amp;quot;proof by induction&amp;quot; when they arrive, and then struggle with simpler direct proofs like this one.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Answer 2:&lt;/strong&gt; The proof is by induction on &lt;span class=&quot;math-container&quot;&gt;$n\in \mathbb{N}$&lt;/span&gt;. The base case is &lt;span class=&quot;math-container&quot;&gt;$n=0$&lt;/span&gt;. We  verify it directly:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{1 - x^{0 + 1}}{1 - x} = \frac{1 - x}{1 - x} = 1&#xA;$$&lt;/span&gt;&#xA;Now we suppose that, for some &lt;span class=&quot;math-container&quot;&gt;$n\ge 0$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{1 - x^{n+1}}{1-x} = 1 + x + x^2 + \cdots + x^n&#xA;$$&lt;/span&gt;&#xA;Consider now the sum&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;1 + x + \cdots + x^n + x^{n+1}&#xA;$$&lt;/span&gt;&#xA;By the induction hypothesis, this is equal to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{1 - x^{n+1}}{1-x} + x^{n+1} = &#xA;\frac{1 - x^{n+1} + (1 - x)x^{n+1}}{1-x} = &#xA;\frac{1 - x^{n+1} +  x^{n+1} - x^{n+2}}{1-x} =&#xA;\frac{1- x^{(n+1) + 1}}{1-x}&#xA;$$&lt;/span&gt;&#xA;Hence the induction hypothesis implies that the identity holds for &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt;.  This closes the induction, and so we conclude that the identity holds for all &lt;span class=&quot;math-container&quot;&gt;$n\in \mathbb{N}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Comments:&lt;/em&gt; Whether a version of this  argument uses the specific template I have (or some other one), it would need to say that induction is being used, what the parameter is, what the IH is and so on. By  itself, the final computation is pretty meaningless. Even a very terse version like:&#xA;&lt;em&gt;we have &lt;span class=&quot;math-container&quot;&gt;$\frac{1 - x^{n+1}}{1-x} + x^{n+1} = \frac{1 - x^{n+2}}{1-x}$&lt;/span&gt; so the identity holds for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; by induction&lt;/em&gt; is clear about the logical structure.&lt;/p&gt;&#xA;&lt;p&gt;You also ask:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If a proof by induction has to explicitly refer to the principle of&#xA;mathematical induction, then do proofs using the pigeonhole principle&#xA;have to explicitly refer to the pigeonhole principle? Do proofs by&#xA;contradiction have to explicitly refer to the principle of proof by&#xA;contradiction (if there is such a thing)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The answer is (obviously?) &amp;quot;yes&amp;quot; to both of these questions.  Much like with induction, there is even a standard template for proofs by contradiction, which begins: &amp;quot;&lt;em&gt;Suppose, for a contradiction, that ...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="22828" LastActivityDate="2023-11-28T13:04:41.040" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="27228" PostTypeId="1" CreationDate="2023-11-28T14:19:28.967" Score="2" ViewCount="250" Body="&lt;p&gt;Around the 1960s a collection of math books by authors such as Mary P. Dolciani, William Wooton, Edwin F. Beckenbach, Ray C. Jurgensen etc. started to appear which were part of a series of books called &amp;quot;Modern Mathematics Series&amp;quot; by the company Houghton Mifflin.&lt;/p&gt;&#xA;&lt;p&gt;What I want to know is:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What are the books that make up this series? There are several books, each with multiple editions and revisions. They have similar names, but some have the prefix &amp;quot;Modern&amp;quot; and some don't. Some have the subtitle &amp;quot;Structure and method&amp;quot;, and some don't.&lt;/li&gt;&#xA;&lt;li&gt;What are the levels that each book is intended to (middle school, high school, first year of college, etc.)?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20015" LastActivityDate="2023-12-12T15:20:05.660" Title="Which Books Comprised The Houghton Mifflin Modern Mathematics Series" Tags="|textbooks|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27229" PostTypeId="1" CreationDate="2023-11-28T22:08:26.437" Score="8" ViewCount="2801" Body="&lt;p&gt;I'm considering creating a series of classes that explore deeper ideas in calculus without overemphasizing the various computational tricks used in integration and differentiation.&lt;/p&gt;&#xA;&lt;p&gt;My vision is essentially akin to a &amp;quot;Survey of Calculus&amp;quot; kind of instruction.&#xA;Nonetheless, i intend to cover a significant part of the basic differential equations and the conventional calculus sequence.&lt;/p&gt;&#xA;&lt;p&gt;Are there any examples of these classes that I could look up?&lt;/p&gt;&#xA;" OwnerUserId="19807" LastEditorUserId="2083" LastEditDate="2023-11-29T08:59:20.457" LastActivityDate="2023-12-21T00:49:27.573" Title="Any examples of calculus sequence that deemphasizes calculation tricks?" Tags="|undergraduate-education|calculus|vector-calculus|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27230" PostTypeId="2" ParentId="27229" CreationDate="2023-11-29T02:14:02.147" Score="16" Body="&lt;p&gt;As you undertake the journey, you should read about the Calculus Reform movement of the 1990s. A unifying idea of the efforts was to increase conceptual understanding by using symbolic computational systems (Maple, Mathematica, TI-89, etc.) to do calculations, thereby freeing up time and mental bandwidth for conceptual understanding.&lt;/p&gt;&#xA;&lt;p&gt;An excellent summary of these efforts is found in the article by Deborah Hughes Hallett titled &lt;strong&gt;What Have We Learned from Calculus Reform? The Road to Conceptual Understanding&lt;/strong&gt;. This is chapter 5 in the book &lt;a href=&quot;https://maa.org/sites/default/files/pdf/pubs/books/members/NTE69.pdf&quot; rel=&quot;noreferrer&quot;&gt;A Fresh Start for Collegiate Mathematics&lt;/a&gt; book published by the MAA. The article is also available at this &lt;a href=&quot;https://www.math.arizona.edu/%7Edhh/NOVA/calculus-conceptual-understanding.pdf&quot; rel=&quot;noreferrer&quot;&gt;University of Arizona website&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In many ways, Calculus Reform took us three steps forward, and then we took one or two steps back. Hughes Hallett provides an excellent summary of the positive changes that persist to this day, including increased conceptual understanding.  But as always with reform, there was overreach. Deprecated calculation skills come with a cost. &amp;quot;Calculus without calculation&amp;quot; borders on being an oxymoron.&lt;/p&gt;&#xA;&lt;p&gt;Another outcome of Calculus Reform was research mathematicians teaching undergraduate mathematics coming to realize how important it is to focus not just on the usual themes of content, textbooks, and curriculum. But we learned how important it is to focus on &lt;em&gt;how people learn&lt;/em&gt; mathematics. I think the work of Ed Dubinsky is emblematic of this lesson. He headed a Calculus Reform project at Purdue titled &lt;strong&gt;&lt;span class=&quot;math-container&quot;&gt;$C^4L$&lt;/span&gt;: Calculus, &lt;em&gt;Concepts&lt;/em&gt;, Computers and Cooperative Learning&lt;/strong&gt;. Dubinsky was a constructivist in the sense of Piaget, and gave focus to how  learning mathematics involves construction of concepts and knowledge, not only socially (hence the cooperative learning part of &lt;span class=&quot;math-container&quot;&gt;$C^4L$&lt;/span&gt;) but also in their individual minds. Dubinsky referred to a process he called APOS, which stands for &amp;quot;Action, Process, Object, Schema.&amp;quot; See &lt;a href=&quot;https://doi.org/10.1007/978-94-007-4978-8_3&quot; rel=&quot;noreferrer&quot;&gt;Actions, Processes, Objects, Schemas (APOS) in Mathematics Education&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Curriculum innovation alone will not guarantee deeper conceptual understanding. We must also focus on pedagogy and the messiness of how knowledge is constructed.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-11-29T02:56:38.493" LastActivityDate="2023-11-29T02:56:38.493" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27231" PostTypeId="2" ParentId="27223" CreationDate="2023-11-29T20:44:57.400" Score="-1" Body="&lt;p&gt;Frame change.&lt;/p&gt;&#xA;&lt;p&gt;Can you imagine a way to let them use their phones without allowing/encouraging cheating? Then they have access to a calculator, even to google for calculations. That would match what would happen with a real problem at their work. I understand that it would make constructing and grading difficult. But perhaps worth the effort.&lt;/p&gt;&#xA;" OwnerUserId="4984" LastActivityDate="2023-11-29T20:44:57.400" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27232" PostTypeId="2" ParentId="27223" CreationDate="2023-11-29T22:05:53.733" Score="4" Body="&lt;p&gt;My intuition is that further penalizing already disengaged students will likely lead them to become even less engaged.&lt;/p&gt;&#xA;&lt;p&gt;It also sounds like some of these students might not know how to use scientific calculators.&lt;/p&gt;&#xA;&lt;p&gt;Freely lending out calculators for student use during exams, providing them instructions on how to use the calculators, and giving them in-class time to practice using them should help clear these obstacles to engagement.&lt;/p&gt;&#xA;&lt;p&gt;If you get a calculator per student, you also get the added benefit of having calculator uniformity. Small differences like whether a calculator can do arbitrary logs or just logs base 10 and &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; can affect students' grades, reducing the quality of their measurement of student learning outcomes.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2023-11-29T22:05:53.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27233" PostTypeId="2" ParentId="27223" CreationDate="2023-11-29T23:13:19.197" Score="3" Body="&lt;p&gt;Allow the students to use &lt;b&gt;Desmos Test Mode&lt;/b&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The app allows students to lock a phone or other android/iOS device in single-app mode and use the Desmos graphing calculator that we know and love in an exam environment.&lt;/p&gt;&#xA;&lt;p&gt;When the student leaves single-app mode, the app displays the amount of time that the app was locked. If the time is too short, it means the student closed the app in the middle of the assessment, so you need to give them a new one or accuse them of cheating with the device.&lt;/p&gt;&#xA;&lt;p&gt;Some of my students bring spare old iPads; most students just use the device they have with them. A few stick with their TI-84.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2023-11-29T23:13:19.197" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27234" PostTypeId="2" ParentId="27223" CreationDate="2023-11-29T23:26:21.387" Score="10" Body="&lt;p&gt;To summarize what I arrived at, based on the top answer on my prior parallel &lt;a href=&quot;https://academia.stackexchange.com/questions/181561/students-lacking-test-material&quot;&gt;question on SE Academia&lt;/a&gt;, OP's #2 is the way to go -- keep a small supply of calculators on hand, but assess a usage penalty to incentivize students to bring their own.&lt;/p&gt;&#xA;&lt;p&gt;I'm guessing that I'm in a roughly analogous situation -- urban community college, many liberal-arts math courses, and a significant proportion of disengaged and/or challenged students. The constraints are something like this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Many of our liberal-arts students (the largest major at our college, in fact the default for incoming freshmen) don't even know their basic times tables, negative numbers, estimations, etc., so calculators are required for them to do anything.&lt;/li&gt;&#xA;&lt;li&gt;Even if that weren't the case, the received, accredited, and articulation-approved curriculum from the larger institution expects and requires work with calculators, and we can't change that (e.g., real-world statistical problems with square roots, compound interest, etc.).&lt;/li&gt;&#xA;&lt;li&gt;Phones can't be permitted for any reason by department policy -- and it's sensible, because students would absolutely cheat with internet access. We also need to scrupulously demand removal of earbuds, watches, etc.&lt;/li&gt;&#xA;&lt;li&gt;If there were an open-access provided calculator policy, students would absolutely come to depend on that, and the majority of a class would expect to receive one on tests.&lt;/li&gt;&#xA;&lt;li&gt;The college and department have zero money to provide any equipment whatsoever, to say nothing of a large supply of calculators.&lt;/li&gt;&#xA;&lt;li&gt;There's no instructor time, and no TA's, with which to manage a large supply of equipment, or monitor device usage for one function but not another.&lt;/li&gt;&#xA;&lt;li&gt;A notable number of students will simply not read or listen to anything you say, until informed that they've lost points for something.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So a small number of calculators to fill in some gaps, with a points-penalty to discourage use, is the best option I've seen. I picked up 3 calculators to cover classes of 25 registered students, with about half that usually attending, and that's covered my needs in the last few years. You can find packs of around that number via online retailers.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-11-30T21:52:02.060" LastActivityDate="2023-11-30T21:52:02.060" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27235" PostTypeId="2" ParentId="27223" CreationDate="2023-11-30T09:07:09.170" Score="-2" Body="&lt;p&gt;&amp;quot;Giving&amp;quot; calculators to &amp;quot;weak&amp;quot; students is one of the worst things you can do: it's not just about bringing a calculator to the exam, but they also need to learn how to operate it. If a student does not master this, (s)he might have a serious disadvantage compared with other students during the exam. (So in fact, while trying to help &amp;quot;weak&amp;quot; students, you're discriminating them even more.)&lt;br /&gt;&#xA;Therefore I advise you to stand your ground: all students must have a calculator, and it must be the type you have mentioned (you might give a choice between 2-3 types, and keep that for several years, so students can pass it on to younger ones after they've finished school).&lt;br /&gt;&#xA;You check if students  actually have a calculator during the year and in case they don't, you punish them for that. At the exam, if a student has &amp;quot;forgotten&amp;quot; his or her calculator, you might loan a spare, but you only do this for students who clearly own one.&lt;/p&gt;&#xA;&lt;p&gt;Remember: fighting poverty might sound like a noble goal, but IT'S NOT YOUR JOB!&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-11-30T09:07:09.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27236" PostTypeId="2" ParentId="27229" CreationDate="2023-11-30T20:17:33.523" Score="2" Body="&lt;p&gt;Take a look at this:&#xA;&lt;a href=&quot;https://calculus.ucmerced.edu/wwh-calculus-project/project-mission-and-goals&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://calculus.ucmerced.edu/wwh-calculus-project/project-mission-and-goals&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You may find other examples that the California Learning Lab is funding at:&#xA;calearninglab.org&lt;/p&gt;&#xA;" OwnerUserId="23462" LastActivityDate="2023-11-30T20:17:33.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27237" PostTypeId="1" AcceptedAnswerId="27240" CreationDate="2023-12-01T20:54:59.940" Score="7" ViewCount="1296" Body="&lt;p&gt;I am helping to design a low-level college course whose purpose is to teach critical thinking, logic, finance and probability. I have been tasked with developing the probability section. I am following chapter 4 from &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/book/10.1002/9780470181188&quot; rel=&quot;noreferrer&quot;&gt;Bolstad's Intro to Bayesian Statistics&lt;/a&gt;. This goes from the definition of probability through Bayes' rule.&lt;/p&gt;&#xA;&lt;p&gt;I was hoping to add one or two practical applications of these ideas. Types of errors is one idea. Another would be basic AI. I want things that are relevant to today yet easy enough for a student with only an algebra background to understand. Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="23465" LastEditorUserId="23465" LastEditDate="2023-12-01T21:20:20.663" LastActivityDate="2023-12-04T15:03:35.447" Title="Special topics for introductory probability" Tags="|undergraduate-education|secondary-education|algebra|course-design|probability|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27238" PostTypeId="2" ParentId="27237" CreationDate="2023-12-01T21:39:06.713" Score="6" Body="&lt;p&gt;You might already be aware of this one, given how famous it is, but the first thing that comes to my mind is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Monty_Hall_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Monty Hall Problem&lt;/a&gt;. It doesn't require any fancy mathematical machinery, just a basic understanding of conditional probability, and there are a handful of factors that make it a particular fun topic to cover during class:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It's easy to design a class activity around it -- for instance, you could put three textbooks on a desk, one of which you've inserted a sticky note behind the cover, and tell the class that if they pick the book with that piece of paper on it, then they get an extra couple percent added to their next assignment grade. (Also, you might want to tell them beforehand that once they choose the book to open, you will open a different book and ask if they want to change their decision, to make it clear that your action of opening another book will be independent of whether their initial choice is correct.) The class collectively decides on a book to open, and then you open a different book that does not contain the sticky note and ask them to collectively decide whether they want to change their decision. After giving them some time to debate, you can launch into the lesson.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You can talk about intuition behind the result: &amp;quot;What if there were 100 textbooks, and I opened 98 of them that did not have the sticky note? Would you change your decision then?&amp;quot; Even students who don't fully grok the conditional probability aspect should be able to grasp the intuition, and the intuition might even help them develop a more tangible sense of what conditional probability is and why it's important.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There's a lot of interesting history behind this problem that might be entertaining for the class -- &lt;a href=&quot;https://web.archive.org/web/20140413131827/http://www.decisionsciences.org/DecisionLine/Vol30/30_1/vazs30_1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;apparently&lt;/a&gt;, even the famed Paul Erdős got it wrong and remained unconvinced of the correct choice until he saw otherwise in a computer simulation. (&lt;a href=&quot;https://www.nytimes.com/1991/07/21/us/behind-monty-hall-s-doors-puzzle-debate-and-answer.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thousands&lt;/a&gt; of other math PhDs got it wrong too and were so confident in their wrong answers that they wrote letters to a magazine that published a response advocating the correct result.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-12-04T15:03:35.447" LastActivityDate="2023-12-04T15:03:35.447" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27240" PostTypeId="2" ParentId="27237" CreationDate="2023-12-02T00:07:43.263" Score="10" Body="&lt;p&gt;A classic application of Bayes' Theorem is in medical testing, and the difference/conversion between &amp;quot;what is the probability I test positive, given I have the condition&amp;quot; vs. &amp;quot;what is the probability I have the condition, given I tested positive&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;If you want a specific example to use, one would be the use of lateral flow (rapid) CoViD tests in the UK in 2021.  The government proposed a &lt;a href=&quot;https://www.liverpool.ac.uk/research/research-themes/infectious-diseases/coronavirus-research/covid-smart-pilot/&quot; rel=&quot;noreferrer&quot;&gt;community testing&lt;/a&gt; program where even asymptomatic people would be tested regularly in an attempt to limit transmission.  In April 2021 (when the underlying case rate was very low) there were calls to &lt;a href=&quot;https://www.theguardian.com/world/2021/apr/15/rapid-covid-testing-in-england-may-be-scaled-back-over-false-positives&quot; rel=&quot;noreferrer&quot;&gt;scale back&lt;/a&gt; the testing because almost all of the positive test results were false positive.  In January 2022 (when Omicron was in full swing), widespread testing &lt;a href=&quot;https://www.theguardian.com/world/2022/jan/07/the-rise-of-lateral-flow-tests-are-these-heroes-of-the-pandemic-here-to-stay&quot; rel=&quot;noreferrer&quot;&gt;became useful again&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Although Bayes' Theorem is never mentioned outright in the April 2021 article, the article contains enough information on underlying case rates, rates of false positives, etc. to be turned into a Bayes' Theorem example (when I taught probability this fall, I used that example as my introduction to Bayes).&lt;/p&gt;&#xA;" OwnerUserId="4832" LastActivityDate="2023-12-02T00:07:43.263" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27241" PostTypeId="2" ParentId="27237" CreationDate="2023-12-02T00:23:17.000" Score="0" Body="&lt;p&gt;I would suggest geometric probability and applications in stereometry!&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2023-12-02T00:23:17.000" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27242" PostTypeId="2" ParentId="27237" CreationDate="2023-12-02T10:15:18.547" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)&quot; rel=&quot;noreferrer&quot;&gt;Bertrand's Paradox&lt;/a&gt; is an old saw. The point is that trying to randomize an experiment is tricky since there can be different points of view.&lt;/p&gt;&#xA;" OwnerUserId="13111" LastActivityDate="2023-12-02T10:15:18.547" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27243" PostTypeId="2" ParentId="27237" CreationDate="2023-12-02T10:26:35.263" Score="2" Body="&lt;p&gt;I would look ar some of the basic six sigma literature and at doe.  It is connected to all kinds of factory snd other process improvement. Very clear business connection.&lt;/p&gt;&#xA;&lt;p&gt;I would eschew the Bayesian emphasis as that is sort of a flourish and doing for you, not optimal for a low level and business emphasis course.  I have consulted in pharma manufacturing and frequentist approach was fine and still an upgrade for the middle managers doing annual product reviews versus no stats at all.  And simpler for a new learner.  Just picking colored balls from a bag.   Along of course with basic data analysis like charting, anova, etc. This is not to say you can't cover Bayes thereom at one time and briefly. And I realize it's an interesting way to think about all probability.  But more for you not for students.&lt;/p&gt;&#xA;&lt;p&gt;And please no Monty Haul. Fun for brainy already know something types. A distraction from basic learning of neophytes.&lt;/p&gt;&#xA;" OwnerUserId="23303" LastEditorUserId="23303" LastEditDate="2023-12-02T10:27:49.910" LastActivityDate="2023-12-02T10:27:49.910" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27244" PostTypeId="2" ParentId="17454" CreationDate="2023-12-02T14:22:10.950" Score="4" Body="&lt;p&gt;Take any triangle, and draw any number of cevians from the top vertex to the base, with any spacing between the cevians.&lt;/p&gt;&#xA;&lt;p&gt;In each sub-triangle thus formed, inscribe a circle.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BWM3w.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BWM3w.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now rearrange the order of the circles from left to right (but don't change their radii).&lt;/p&gt;&#xA;&lt;p&gt;Adjust the cevians so that each cevian touches two neighboring circles.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/d3wBJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/d3wBJ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The circles always fit perfectly in the sub-triangles, with no gaps, no matter what their order&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/4711623/398708&quot;&gt;Here&lt;/a&gt; is a proof.&lt;/p&gt;&#xA;" OwnerUserId="20101" LastActivityDate="2023-12-02T14:22:10.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27245" PostTypeId="2" ParentId="27237" CreationDate="2023-12-02T17:40:38.080" Score="1" Body="&lt;p&gt;[Additional to previous answer--can't edit, sorry.]&lt;/p&gt;&#xA;&lt;p&gt;dt688:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I would be very wary about being too difficult or particular, when teaching in a corporate environment.  I.e. if GMers are your target audience.  &amp;quot;What is interesting, to me&amp;quot; can lead to wrong choices, for the audience.  I have learned this the hard way...even with beautiful, hard prepped lectures.  And that I thought dumbed down enough.  And I know a lot more about GM-like engineers, marketers, managers, finance types than I do about college teaching!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You don't need to jump off the whole Six Sigma cliff.  I dislike the buzzword and the large programs.  But they do have a very good approach for basic teaching to adult professionals in industry setting.  Look into the &amp;quot;catapult training&amp;quot; workshops.  Google it, see YT videos...but here is one link:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=VhbbNkDathE&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=VhbbNkDathE&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;When I got this training (in an industrial setting), I remember how even some simple things like accuracy of measurement were revealed.  Like first, they made us estimate the landing position and churn with that data.  Then they allowed us to use a carbon paper that marked the location and so got more accurate data.  This is something that clearly makes sense in the manufacturing environment.  But is not drummed in, in the average academic stats course, with lots of equations and the like.  Remember Pearson (or Dewey or whoever) who said to his stats grad students, &amp;quot;Never forget that your municipal data came from the night watchmen.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Of course, you are a quality engineer, and thus know the drill.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Tommi:&lt;/p&gt;&#xA;&lt;p&gt;DOE, Six Sigma and ANOVA should be capitalized.  And &amp;quot;snd&amp;quot; is &amp;quot;and&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="23470" LastEditorUserId="23470" LastEditDate="2023-12-02T18:10:08.200" LastActivityDate="2023-12-02T18:10:08.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27246" PostTypeId="2" ParentId="27223" CreationDate="2023-12-03T14:56:42.520" Score="2" Body="&lt;p&gt;In my opinion, all of your decisions should be based on the following question: &lt;strong&gt;What is the pedagogical goal?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In my classes, I generally try to focus on the mathematics and on basic (&lt;em&gt;basic&lt;/em&gt;) &amp;quot;how to college&amp;quot; skills.  I assign my final grades based on a combination of (1) hard skills (i.e. being able to solve problems), and (2) soft skills (such as attending office hours, turning in homework on time, etc).&lt;/p&gt;&#xA;&lt;p&gt;However, many (if not most) of my students are quite poor.  A significant number of my students do not have running water or electricity at home, let alone reliable internet access or cell phone coverage.  I have students who &lt;em&gt;hitchhike&lt;/em&gt; for several hours a day to get to one of our campuses.&lt;/p&gt;&#xA;&lt;p&gt;As such, I don't think that it is reasonable for me to base any portion of my grading scheme on something which many students are going to have a hard time with for reasons beyond their control (e.g. getting a calculator for my class).  So &lt;em&gt;if&lt;/em&gt; I were in a position similar to the original asker's (I am not, as I write exams to be done without a calculutor), I would simply have several calculators ready to loan to students with no penalty.  In the &amp;quot;real world&amp;quot;, someone with a job should have the tools they need provided to them by their employer.  In as much as it is possible, I would prefer to model that.  In &lt;em&gt;my&lt;/em&gt; classroom, I would use option (1).&lt;/p&gt;&#xA;&lt;p&gt;That being said, I know instructors who believe that more advanced &amp;quot;how to college&amp;quot; and &amp;quot;life skills&amp;quot; should be part of their pedagogical goals.  These instructors &lt;em&gt;will&lt;/em&gt; penalize students who don't have the right gear.  This is reasonable, &lt;em&gt;if&lt;/em&gt; your learning objectives include these skills.  In this case, I would prefer (3) over (2) (so that students are not surprised at the last minute in a high-stakes situation),&lt;/p&gt;&#xA;&lt;p&gt;So, again:  &lt;strong&gt;What is the pedagogical goal?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2023-12-03T14:56:42.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27247" PostTypeId="2" ParentId="27237" CreationDate="2023-12-03T15:01:06.993" Score="6" Body="&lt;p&gt;One example of elementary probability is the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Birthday_problem&quot; rel=&quot;noreferrer&quot;&gt;Birthday problem&lt;/a&gt; which asks for the probability that in a room of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; people two will share the same birthday. Sometimes formulated as a paradox that if there are 23 or more the probability exceeds one-half. Assuming your class has about that number you can try it experimentally. Whatever the result of the experiment you have a discussion topic.&lt;/p&gt;&#xA;" OwnerUserId="23475" LastActivityDate="2023-12-03T15:01:06.993" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27249" PostTypeId="2" ParentId="19069" CreationDate="2023-12-04T09:40:25.460" Score="4" Body="&lt;p&gt;Are you aware you can choose not to show objects or labels with Geogebra? You can draw arcs, but at first the bounding points will be shown too, but you can hide them afterwards, as in this example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/P5lfu.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/P5lfu.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2023-12-04T09:40:25.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27250" PostTypeId="1" CreationDate="2023-12-05T16:21:38.953" Score="4" ViewCount="377" Body="&lt;p&gt;&lt;a href=&quot;http://ramanujan.math.trinity.edu/bmiceli/problemoftheweek/index.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trinity's Problem of the Week&lt;/a&gt; and &lt;a href=&quot;https://www.math.purdue.edu/pow2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Purdue's Problem of the Week&lt;/a&gt; were both weekly problems that were excellent for college math clubs to tangle with at their weekly meetings. However, both of them are no longer posting new problems.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What is a good replacement?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A good answer would involve contest-type questions but they would &lt;strong&gt;not&lt;/strong&gt; be part of a contest. Or, if they are part of a contest, the competition part of it should be clearly de-emphasized. The point is to engage with interesting problems for the sake of themselves.&lt;/p&gt;&#xA;" OwnerUserId="23487" LastActivityDate="2023-12-05T17:39:07.077" Title="Problem of the Week for College Students" Tags="|undergraduate-education|problem-solving|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27251" PostTypeId="2" ParentId="27250" CreationDate="2023-12-05T17:01:03.920" Score="4" Body="&lt;p&gt;I don't understand the aversion to contest problems. If you want contest-type questions, then why not just pull from well-known contests (&lt;a href=&quot;https://artofproblemsolving.com/wiki/index.php/AMC_Problems_and_Solutions&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMC&lt;/a&gt;, &lt;a href=&quot;https://artofproblemsolving.com/wiki/index.php/IMO_Problems_and_Solutions&quot; rel=&quot;nofollow noreferrer&quot;&gt;IMO&lt;/a&gt;, &lt;a href=&quot;https://kskedlaya.org/putnam-archive/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Putnam&lt;/a&gt;, etc) where a variety of interesting, engaging problems can be found along with solutions?&lt;/p&gt;&#xA;&lt;p&gt;Competition is not inherent to these problems. Rather, it depends entirely on the environment in which the problems are solved. If you take a contest problem and present it to a math club as as something to engage with for the sake of itself, then students will not be engaged in competition.&lt;/p&gt;&#xA;&lt;p&gt;Avoiding problems because they appear in contests and you don't want competition, is like avoiding jelly because it appears in peanut-butter-and-jelly sandwiches and you're allergic to peanuts.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-12-05T17:39:07.077" LastActivityDate="2023-12-05T17:39:07.077" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27253" PostTypeId="1" CreationDate="2023-12-06T19:18:46.440" Score="12" ViewCount="377" Body="&lt;p&gt;I am almost surely ADHD, although undiagnosed. I have colleagues in mathematics who also are, but diagnosed. I have an extremely talented and successful graduate student who is (diagnosed) ADHD.&lt;/p&gt;&#xA;&lt;p&gt;It is almost surely the case that I would have failed to do even remotely well in mathematics if I had not been intrinsically interested in it, and willing to &amp;quot;superfocus&amp;quot; on it for long periods. I was lucky enough to have been homeschooled in order to do this. Others are not so lucky. The idea of trying to learn this subject using the suggested reward structures found online that are supposed to be effective for ADHD people sounds like the worst kind torture to me. The purpose of this question is to collect some data to clarify my thoughts on the topic, and my own experiences with it.&lt;/p&gt;&#xA;&lt;p&gt;ADHD people have, I have read, a deficit in working memory. (This suggests that math would be a very bad thing for students with ADHD to get involved with...since problem solving relies so much on working memory.) Growing up, I found that I did have a difficulty in short-term memory of some kind, but that obsessive drilling and overlearning could move ideas effectively into permanent memory where they could be accessed and used readily. (This aspect makes it seem like an ADHD person &lt;em&gt;could&lt;/em&gt; be successful in math, if, for example, Keith Devlin's claims that math requires overlearning of concepts and procedures.) My experience has given a handful of examples showing me that the latter situation can, in fact, occur. The need to overlearn material, and the resulting expertise gained by ADHD people (often homeschooled, since this sometimes gives these people the freedom to take the time to obsess and superfocus) can make difficult mathematical topics and the grit needed to internalize them &lt;em&gt;more&lt;/em&gt; natural than for neurotypical people.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there any writing about ADHD mathematicians and their experiences? Is there anything out there in the literature discussing the phenomena I observed in these few cases?&lt;/p&gt;&#xA;&lt;p&gt;Note: I'm not interested in the dystopian approaches to bring ADHD people into tow with traditional math education. I'm interested in the experiences of mathematicians and successful math students who are on the ADHD spectrum, their coping mechanisms and experiences, and writing closely related to these.&lt;/p&gt;&#xA;&lt;p&gt;Some mathematicians often complain of having a poor memory...maybe some of this may relate to the above phenomena? Michael Atiyah veered away from studying chemistry due to a self-reported memory deficit, for example.&lt;/p&gt;&#xA;&lt;p&gt;Problems can serve as instruments to manage attention. I'd be interested in reading about how this affects ADHD people.&lt;/p&gt;&#xA;&lt;p&gt;(Sorry for a completely rambling question, totally confirming my suspected condition!)&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2023-12-13T23:21:05.147" Title="ADHD superfocus and motivation for mathematics" Tags="|reference-request|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="27254" PostTypeId="1" AcceptedAnswerId="27256" CreationDate="2023-12-07T14:28:23.617" Score="5" ViewCount="223" Body="&lt;p&gt;In the past semester, I taught two 7-week courses: discrete math and algorithms designs (which is essentially still math) for undergraduate CS students. I implemented weekly 25-minute quizzes containing three problems based on the week's content. (This is due to the observation that the majority of students do assignment via Google or ChatGPT, or copy from each other.)&lt;/p&gt;&#xA;&lt;p&gt;My objective was to challenge students while ensuring the problems were solvable within the given time limit. However, I encountered several issues:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Cheat Sheet Utilization:&lt;/strong&gt; I allowed cheat sheets, but some students crammed many solutions of exercises on them. This necessitated the creation of original problems, which varied significantly in difficulty. Crafting these problems often required a substantial amount of preparation time.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Complexity of Upper-Level Content:&lt;/strong&gt; The challenge is more pronounced in upper-level courses, which involve intricate proofs, as opposed to lower-level courses with more computational tasks. At times, a difficult problem would stump the entire class, while other questions felt too trivial.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Given these challenges, I would appreciate your insights on the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How can I develop quiz problems that are appropriately challenging for upper-level math students but still achievable within a 25-minute quiz?&lt;/li&gt;&#xA;&lt;li&gt;Or maybe I should simply go back to assignment rely solely on one midterm and one final exam as the only true assessment? (One can argue that force students who are not interested in the topic to study by using grades as sticks is futile in the end.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user19945" LastEditorUserId="2083" LastEditDate="2023-12-08T05:46:18.520" LastActivityDate="2023-12-08T05:46:18.520" Title="Strategies for Designing Challenging Yet Feasible Quiz Problems for Upper-Level Math Courses" Tags="|undergraduate-education|exams|graduate-education|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27255" PostTypeId="2" ParentId="27253" CreationDate="2023-12-07T18:43:02.270" Score="3" Body="&lt;p&gt;Based on a cursory investigation of the literature, it seems like ADHD can loosely be described as providing an extreme combination of &amp;quot;superfocus&amp;quot; on things that one finds interesting, and &amp;quot;super-distractability&amp;quot; on things that one doesn't find interesting. Different people with ADHD have different focus/distractability profiles across different things, so depending on who the person is and what you're asking them to do, ADHD might help or hinder their performance.&lt;/p&gt;&#xA;&lt;p&gt;(As Sue VanHattum points out in the comments, plenty of non-ADHD people have attention &amp;quot;differences&amp;quot; where they can focus better on things that they are interested in than things they find boring. With that in mind, an attention &amp;quot;disorder&amp;quot; like ADHD might be characterized as an attention difference that is simultaneously extreme enough and unmanageable enough to cause severe distress and impediments in one's life.)&lt;/p&gt;&#xA;&lt;p&gt;In particular, when ADHD people have trouble with math, it the issue is mainly due to impaired central executive functioning, which would suggest the issue primarily boils down to things like getting distracted and losing focus of the goal of a problem when one does not find it super interesting.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Friedman, L. M., Rapport, M. D., Orban, S. A., Eckrich, S. J., &amp;amp; Calub, C. A. (2018). &lt;a href=&quot;https://stars.library.ucf.edu/cgi/viewcontent.cgi?article=6457&amp;amp;context=etd&quot; rel=&quot;nofollow noreferrer&quot;&gt;Applied problem solving in children with ADHD: The mediating roles of working memory and mathematical calculation.&lt;/a&gt; &lt;em&gt;Journal of Abnormal Child Psychology, 46&lt;/em&gt;, 491-504.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;CE [central executive working memory processes] ability fully mediated between-group differences in applied problem solving whereas math calculation ability partially mediated the relation. Neither PH STM [phonological short-term memory] nor VS STM [visuospatial short-term memory] was a significant mediator.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;Coordination within and between the two STM subsystems is superintended by the domain general CE to (a) determine the task-relevance of the information contained in the mathematical word problem; (b) update information in PH/VS STM with newer, more relevant information; (c) connect information contained in the mathematical word problem with knowledge stored in long-term memory regarding math rules and potential mathematical algorithms to be applied in the current problem; (d) maintain the overall ‘goal’ of the applied problem; and (e) sustain attentional focus while concomitantly inhibiting irrelevant information from entering/competing with temporarily stored information&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Raciquel posted a comment to a video where an ADHD mathematician describes his experience, and it sounds just like you'd expect based on the above -- basically, he's really &amp;quot;spikey&amp;quot; as opposed to well-rounded, and whether he feels his ADHD symptoms depends on whether he's doing things that are within his &amp;quot;spikes&amp;quot; of interest/ability/motivation.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=lAdVN4yzJCs&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How Getting a Math PhD Cured My ADHD (NOT)&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;eventually I found something I felt so passionate about, I couldn't think of doing anything else. And that was applied math for me. And that ended up masking a lot of symptoms through grad school, and it wasn't until I left grad school recently in 2018 that my ADHD symptoms came back in full force.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He (Youngmin Park) describes how he actually struggled a bit with K-12 math due to the emphasis on memorizing formulas -- he didn't find that interesting, so he just rushed through homework, putting in the bare minimum amount of work for a passing grade. But he could be incredibly hyperfocused on other things (e.g. as a toddler, he would play with Legos for hours and hours and hours, forgetting to eat and sleep).&lt;/p&gt;&#xA;&lt;p&gt;The thing that really drew him into math later on was a professor who sparked his interest in some particular research problems. He found grad school liberating because he had tons of freedom to follow his natural motivations/interests:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;I could work or not work whenever I wanted. I could work 12-hour days one day. I could work a 6-hour day the next day. I could take a day off the third day and go back to like a 14-hour day the day after that. And that was very liberating because my motivation doesn't start in the morning and fade off at night, it actually starts at very random times throughout the day and I can't predict when it comes about.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Basically, he's just really good at doing what he wants to do, when he wants to do it, on his own schedule. When he's in an environment that allows him to funnel that kind of effort into something productive, then he does really well. His impatience, impulsiveness, and obsessiveness become a superpower because he can just floor the gas and charge full-speed ahead on the stuff that he's interested in. But whenever he's in a context where this kind of behavior isn't a superpower, it creates tons of friction and becomes a super-weakness (he mentioned some examples in the video, e.g. romantic relationships).&lt;/p&gt;&#xA;&lt;p&gt;As far as other ADHD mathematicians and their experiences, there are a bunch of famous academics in math and adjacent fields who are known not only for their contributions, but also for amusing backstories and habits that seem strikingly aligned with what Youngmin Park describes in his video above. Erdős, Einstein, Tesla, you get the idea. They all clearly have a high degree of attention &amp;quot;difference&amp;quot;, and I would venture that at least some of them would have been properly diagnosed with ADHD if tested today.&lt;/p&gt;&#xA;&lt;p&gt;Courtesy of Dave L Renfro, there are also plenty of relevant and humorous &lt;a href=&quot;https://www.technologyreview.com/2011/06/21/193920/the-original-absent-minded-professor/&quot; rel=&quot;nofollow noreferrer&quot;&gt;anecdotes&lt;/a&gt; about mathematician Norbert Wiener:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;... the time he reported the theft of his car to the police, only to discover that he had driven it to Providence for a talk and taken the train back; the conversation in an MIT hallway that he concluded by asking his interlocutor which way he had been heading when he stopped to chat, greeting the answer with 'Good! That means I’ve already had lunch.'&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-12-07T22:32:12.963" LastActivityDate="2023-12-07T22:32:12.963" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27256" PostTypeId="2" ParentId="27254" CreationDate="2023-12-07T23:31:18.023" Score="4" Body="&lt;p&gt;I get a lot of clarity for designing assessments by writing out a list of learning objectives for the course. To get started, I write out three lists:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Things I want my students to have seen (only need to show up in lecture, I don't care if students can replicate it)&lt;/li&gt;&#xA;&lt;li&gt;Things I want my students to have done (with reference and help, so on homeworks, in-class assignments)&lt;/li&gt;&#xA;&lt;li&gt;Key take-away skills (things students should be able to do on their own, so on in-class assessment)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In general, but particularly for in-class assessment, I go through the list of relevant skills and identify the simplest problem that will assess whether students have proficiency in that skill (as per Gerald Edgar's comment, although I would simplify further if your text has complicated questions).&lt;/p&gt;&#xA;&lt;p&gt;It helps to get really specific about your goals for the course:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Do you want your students to have a specific proof memorized? A good approach here is to give students a list of ~10 proofs you want them to know, pick a random 4 of them for the assessment and require that students prove 3 out of the 4 of them. Obviously reference sheets are no good in this case.&lt;/li&gt;&#xA;&lt;li&gt;Do you want your students to be able to be able to write out a proof by induction given another proof by induction as reference? Reference sheets are okay, but as Dave L Renfro suggests in the comments, giving your students your own reference proof gives you more control over the situation.&lt;/li&gt;&#xA;&lt;li&gt;Do you want your students to be able to be able to come up with a proof by induction not given any reference? Pick the easiest proof by induction you can think of.&lt;/li&gt;&#xA;&lt;li&gt;Do you want to use quizzes to enhance the effectiveness of homework? As Aeryk suggests in the comments, tell students in advance that you will pick a random homework problem to be the quiz. You won't be able to tell if they came up with the solution on their own, but this improves the chances students will actually get something from the homework.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2023-12-07T23:31:18.023" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27257" PostTypeId="1" CreationDate="2023-12-08T16:12:49.623" Score="-4" ViewCount="243" Body="&lt;p&gt;I did not win the International Mathematics Olympiad, so I was very curious to know the standard of questions asked in the graduate and undergraduate courses of maths and computer science in institutions like MIT, Stanford, Princeton, Oxford, etc. For example, suppose someone is taking a class in topology. Will the institution give problems related to the theory but the difficulty standard of the problems equivalent to that of an IMO or Putnam problem?&lt;/p&gt;&#xA;" OwnerUserId="23368" LastEditorUserId="376" LastEditDate="2023-12-10T07:46:53.493" LastActivityDate="2023-12-10T07:46:53.493" Title="Are the standard of questions provided in reputed institutions like MIT, Stanford, Oxford, etc., as good as the problems of IMO?" Tags="|undergraduate-education|graduate-education|math-contests|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="27259" PostTypeId="2" ParentId="27257" CreationDate="2023-12-08T16:47:35.173" Score="8" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Will the institution give problems related to the theory but the difficulty standard of the problems equivalent to that of an imo or Putnam problem?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Probably not.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The whole point of a class (or textbook) is to get students to be able to solve most if not all of the problems that they are presented. Class problems are meant to be organized and scaffolded in a way that makes them fairly solvable if you paid attention to the underlying content and techniques covered in class or in the textbook.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The whole point of a competition problem (e.g. on IMO or Putnam) is for it to be very difficult to solve even if you know the underlying content. The goal is to &amp;quot;spread out&amp;quot; students' performance on the basis of their ability to think insightfully about these kinds of tricky problems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As supporting evidence, just think about the typical grade distribution in a class vs the typical score distribution on one of these competitive exams.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In a typical class (in the USA, at least), the vast majority of students are getting 70%+ of the problems right on homework, exams, etc. Granted, if a class is covering really hardcore content, then the grades might be extremely curved (the most extreme case I've heard of was a something like 30% being passing and 50% being an A). But even still, this is no comparison for the typical score distribution on a competitive exam.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;On the Putnam, for instance, the top scores are generally somewhere around 100 points out of 120 possible, and the median score is usually... wait for it... usually no higher than 2 points out of 120 possible. Typically the median is 1 point, and sometimes it's literally 0 points, meaning that if you scored &lt;em&gt;any&lt;/em&gt; points, you did better than half of the people taking the test.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, where can you find very very tough problems in graduate-level math? That's called research ;) After Putnam, there aren't really any more competitive exams. There are PhD qualifications, but those just serve to ensure that PhD candidates have a baseline level of expertise in their field. After undergrad, mathematicians don't compete on exams -- they compete on research impact. And as Raciquel rightly points out in the comments, the research phase is when the game begins to lend itself to more cooperative play.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-12-10T00:15:02.233" LastActivityDate="2023-12-10T00:15:02.233" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27260" PostTypeId="2" ParentId="26304" CreationDate="2023-12-10T05:35:46.830" Score="4" Body="&lt;p&gt;With some supplementation of your own, I think the second chapter from Hilton and Stammbach's &amp;quot;Homological Algebra&amp;quot; is a great place to begin when it comes to category theory. The presentation is clear and precise. But of course, it is a graduate book, and will require some supplementation of your own. Lang's &amp;quot;Algebra&amp;quot; might work as well, but the order in which the topics are covered in Lang's book seems somewhat unnatural to me. Hungerford's &amp;quot;Algebra&amp;quot; also has some good content pertaining to category theory.&lt;/p&gt;&#xA;" OwnerUserId="23421" LastActivityDate="2023-12-10T05:35:46.830" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27262" PostTypeId="1" CreationDate="2023-12-10T18:17:28.047" Score="10" ViewCount="195" Body="&lt;p&gt;&lt;strong&gt;Background.&lt;/strong&gt; I teach math at a German university (both undergraduate courses and courses for Master programmes), mostly for students who major in mathematics or very similar programmes. Students typically choose their major before starting their studies. Courses aimed at students who major in mathematics are proof-based from the very beginning and proofs are an essential part of both homework and exam problems.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Context.&lt;/strong&gt; I regularly find myself in discussions about whether every course should be designed around measurable learning objectives.&#xA;When I point out that I don't know how I should do this for any but the most elemantary math courses that I teach, this usually leads to one of the following two points:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The very generic comment that it's of course possible, if one just commits enough to doing it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some examples to show how it should be done, but which are extremely elementary (along the lines of &amp;quot;How to prove a formula by induction&amp;quot;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Both answers don't help me to see how this could be done in my courses in practice.&lt;/p&gt;&#xA;&lt;p&gt;Empirically, when I look at descriptions of math courses (in Germany, those are collected in so-called &amp;quot;module descriptions&amp;quot; which are supposed to have a section that lists the learning objectives) I always see one of two patterns: Either the objectives are formulated extremely vaguely (which contradicts the goal to make them measurable) or they are essentially a rewording of the table of contents (sometimes enriched by a few verbs to make it sound more like objectives rather than contents).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; I'm looking for concrete examples of somewhat advanced math courses that are based on explicitly stated measurable learning objectives.&lt;/p&gt;&#xA;&lt;p&gt;Criteria for a good answer:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It should be a proof-based course which requires a certain level of abstraction. I'm thinking, for instance, of an introduction to point set topology or measure theory or group theory or functional analysis, or anything of a similar level.&lt;/p&gt;&#xA;&lt;p&gt;I am &lt;strong&gt;not&lt;/strong&gt; interested in &amp;quot;Introduction to proofs&amp;quot; like courses, because for such courses I can see myself how this approach can work.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The description of the learning objectives should be sufficiently concrete to make it clear that and how they can be measured. Moreover, they should not merely be a slight rephrasing of the course contents (because if they were, I wouldn't see the point of it).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's not so important for me to have access to all the courses materials. I'd mainly like to see the description of the learning objectives and a (maybe brief) description of the course contents.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bonus points if the course is even part of an entire math programme that is designed around measurable learning objectives (but that's not a requirement).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I'd be happy with sources in English or German (or, if there's no help otherwise, in French).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note.&lt;/strong&gt; This question is &lt;strong&gt;not&lt;/strong&gt; about the following things:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm not asking about opinions on whether basing courses on measurable learning oucomes is a good or bad idea. I'm only asking for concrete examples where it has been done.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I'm not looking for advice on how I could create such a list of measurable learning objectives myself for a course. This would also be an interesting question, but it is not part of &lt;strong&gt;this&lt;/strong&gt; question.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2023-12-10T18:17:28.047" Title="Examples for advanced math courses based on measurable learning objectives" Tags="|course-design|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="27263" PostTypeId="1" AcceptedAnswerId="27267" CreationDate="2023-12-11T06:50:13.267" Score="11" ViewCount="1506" Body="&lt;p&gt;I have been assigned to teach a first year course in geometry the next academic year. This course has been running for quite a while in the university, but of late, has been thought of as redundant because the material being covered in the course is something they have already seen in lower levels during their course of study. My task is to redesign the course. In doing so, I have essentially full freedom except for the fact that it is a first year course, and I cannot assume much in terms of prerequisites.&lt;/p&gt;&#xA;&lt;p&gt;I want to include the proofs of anything that I cover (consistent with good pedagogy of course). I also have absolutely no problem with having to build a concept from the ground up, if it leads to an important result or is an interesting technique in its own right. My students are going to be sharp; they're some of the most gifted freshmen in the country when it comes to mathematics. I want to make this a rigorous course for them.&lt;/p&gt;&#xA;&lt;p&gt;Can anyone suggest a good textbook that satisfies my requirements that I can follow for the course?  I hope I have been (at least somewhat) clear on what I need. I will clarify anything else in the comments.&lt;/p&gt;&#xA;" OwnerUserId="23421" LastEditorUserId="2083" LastEditDate="2023-12-11T12:48:35.520" LastActivityDate="2023-12-26T20:53:50.717" Title="Textbook For a Course on Classical Geometry" Tags="|undergraduate-education|geometry|course-design|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27264" PostTypeId="2" ParentId="27263" CreationDate="2023-12-11T09:11:38.447" Score="5" Body="&lt;p&gt;It depends on how much of prerequisites are to be assumed and which materials should be included.&#xA;I have taught a geometry course (one semester course) for years that is aimed for sophomores.&#xA;It includes Euclidean, hyperbolic and spherical geometry in a context of transformation groups (the concept of group is not introduced explicitly though).&#xA;For some years, I used  &lt;em&gt;Geometry of Surfaces by J. Stillwell&lt;/em&gt; and the course covers its first four chapters.&#xA;I liked the book but I felt the explanation gets sloppy in later chapters(especially chapter four). And it uses some elementary properties of complex numbers, which causes some difficulties to the students.&lt;/p&gt;&#xA;&lt;p&gt;Later, I changed the textbook to &lt;em&gt;Geometry: from Isometries to Special Relativity by N. Lee&lt;/em&gt;.&#xA;This book is somewhat similar to the one by Stillwell but It assumes only elementary calculus as prerequisites.&#xA;The first five chapters were covered in the course. The book focuses on the &amp;quot;three-reflection-theorem&amp;quot; which appears in very similar forms in all of Euclidean, hyperbolic and spherical geometry.&#xA;The chapter 6 is about the geometry of special relativity, for which some background in physics can be helpful. But I have never been able to cover chapter 6 due to the time limit.&lt;/p&gt;&#xA;" OwnerUserId="23512" LastEditorUserId="23512" LastEditDate="2023-12-12T03:57:50.987" LastActivityDate="2023-12-12T03:57:50.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27265" PostTypeId="2" ParentId="27263" CreationDate="2023-12-11T17:02:19.137" Score="3" Body="&lt;p&gt;I recommend &lt;strong&gt;Geometry: A Metric Approach with Models&lt;/strong&gt; by Richard Millman and George Parker. I have used it as a textbook in the past and liked it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum.&lt;/strong&gt; A little bit about this book, in response to one of the comments: It offers an axiomatic and rigorous development of classical and non-Euclidean geometries. The axiomatic approach is very effective and well-explained, in my opinion. Here is a quote from the beginning of the text:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the early development of geometry the point of view was that an axiom was a statement that described the true state of the universe. Axioms were thought of as &amp;quot;basic truths&amp;quot;. Such axioms should be &amp;quot;self-evident&amp;quot;. Of the basic axioms stated by Euclid in his &lt;em&gt;Elements&lt;/em&gt;, all but one was accepted by the mathematical community as &amp;quot;true&amp;quot; and self-evident. However, his fifth axiom, which dealt with parallel lines, was not as well received. While everyone agreed it was true (whatever that meant) it was by no means obvious. For over two thousand years mathematicians tried to show that the fifth axiom was a theorem which could be proved on the basis of the remaining axioms. As we shall see, such efforts were doomed to fail. The modern view is that an axiom is a statement of a useful property. When we assume an axiom holds, usually in a definition, we are saying that we want to discuss those objects which possess this special property. We are making no statement as to whether the axiom is a statement about the real world. Rather, we are saying &amp;quot;accept the following as a hypothesis.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As the title suggests, the book takes the &lt;em&gt;metric approach&lt;/em&gt; to developing and studying geometry. This approach is due to George Birkhoff. In this approach, the concept of distance (or a &lt;em&gt;metric&lt;/em&gt;) and angle measurement is added to that of an incidence geometry to obtain basic ideas of betweenness, line segments and congruence. Another possible approach to geometry (not treated in this book) is Davis Hilbert's &lt;em&gt;synthetic approach&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The book works with a few models such as the Cartesian Plane, The Poincare Upper Half Plane and The Taxicab Plane. When a new axiom is introduced, it is shown which models do (or do not) satisfy that axiom. The authors say in the Preface:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We hope that through an intimate acquaintance with examples (and a model is just an example), the reader will obtain a real feeling and intuition for non-Euclidean (and in particular, hyperbolic) geometry. From a pedagogical viewpoint this approach has the advantage of reducing the reader's tendency to reason from a picture. In addition, our students have found the strange new world of the non-Euclidean geometries both interesting and exciting.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I can confirm the last statement about students finding the strange new world of the non-Euclidean geometries both interesting and exciting. The text also has an adequate number of exercises at the end of each section.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastEditorUserId="20488" LastEditDate="2023-12-16T00:16:53.067" LastActivityDate="2023-12-16T00:16:53.067" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27266" PostTypeId="2" ParentId="19069" CreationDate="2023-12-11T17:52:44.847" Score="1" Body="&lt;p&gt;you can consider using &lt;a href=&quot;https://cabricloud.com/cabriexpress/secondary/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cabricloud.com/cabriexpress/secondary/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is an on screen compass, protractor, ruler and set square.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4rlmb.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4rlmb.png&quot; alt=&quot;construction&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="23514" LastActivityDate="2023-12-11T17:52:44.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27267" PostTypeId="2" ParentId="27263" CreationDate="2023-12-11T21:17:45.933" Score="6" Body="&lt;p&gt;A fine undergraduate text is:&lt;/p&gt;&#xA;&lt;p&gt;H. S. M. Coxeter, &lt;em&gt;Introduction to Geometry&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This classic work is now available in an unabridged paperback edition. The Second Edition retains all the characterisitcs that made the first edition so popular: brilliant exposition, the flexibility permitted by relatively self-contained chapters, and broad coverage ranging from topics in the Euclidean plane, to affine geometry, projective geometry, differential geometry, and topology. The Second Edition incorporates improvements in the text and in some proofs, takes note of the solution of the 4-color map problem, and provides answers to most of the exercises.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="127" LastActivityDate="2023-12-11T21:17:45.933" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27268" PostTypeId="2" ParentId="27228" CreationDate="2023-12-12T15:20:05.660" Score="1" Body="&lt;p&gt;They were high school textbooks (maybe used in middle school if people took algebra in middle school).  My father, who taught high school math for many years, has some. Just searching on the series name and author names you can easily locate some online about &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B000J4N96Y&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;trigonometry&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0395143853&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;pre-algebra&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B001JYSMKG&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;algebra 1&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0007GU5S4&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;algebra&lt;/a&gt;,&#xA;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0007E5VHQ&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;geometry&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="893" LastActivityDate="2023-12-12T15:20:05.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27270" PostTypeId="2" ParentId="21048" CreationDate="2023-12-13T20:39:42.530" Score="2" Body="&lt;p&gt;Here's one of my favorite examples of an abstract vector space:  &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt; with the prime numbers as the basis and arithmetic multiplication as vector addition.  For example, &lt;span class=&quot;math-container&quot;&gt;$5 = 2^0\cdot 3^0\cdot 5^1 \cdot 7^0 \cdot \cdots \longrightarrow [0,0,1,0,0,\ldots]$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$24 = 2^3\cdot 3^1\cdot 5^0 \cdot 7^0\cdot \cdots \longrightarrow [3,1,0,0,0,\ldots]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The number &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; becomes the zero vector, and two vectors being orthogonal means that the numbers are relatively prime.  Scalar multiples of vectors are powers of the numbers.&lt;/p&gt;&#xA;&lt;p&gt;We can expand the numbers to include all positive rational numbers and let the vector coefficients be all integers (positive and negative).  For example, &lt;span class=&quot;math-container&quot;&gt;$\dfrac{14}{45} \longrightarrow [1, -2, -1, 1, 0, \ldots]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Note that &lt;em&gt;technically&lt;/em&gt; this is a &amp;quot;module&amp;quot; rather than a true abstract vector space because we are taking our scalars to be elements of a ring (&lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt;) rather than elements of a field (like &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="23527" LastEditorUserId="23527" LastEditDate="2023-12-15T20:05:43.947" LastActivityDate="2023-12-15T20:05:43.947" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27271" PostTypeId="2" ParentId="27253" CreationDate="2023-12-13T23:01:45.977" Score="5" Body="&lt;p&gt;As often happens: not an answer to the literal question (about references), but just some (substantial?) anecdotal stuff over 50 years of observing mathematicians:&lt;/p&gt;&#xA;&lt;p&gt;I'd agree that being able to focus for long periods is unusual, whatever label.&lt;/p&gt;&#xA;&lt;p&gt;I'd in fact &lt;em&gt;disagree&lt;/em&gt; with the idea that people should (obediently?) focus on whatever they're told to. As a corollary, if/when people don't necessarily focus on some random thing... well, that's probably a wise choice.&lt;/p&gt;&#xA;&lt;p&gt;Which leads me to think about &lt;em&gt;prioritization&lt;/em&gt;: not everything is of equal importance. Sure, sometimes prioritization is just a more marketable name for &lt;em&gt;procrastination&lt;/em&gt;, but, really, there is a meaningful notion of prioritization (all the more in our &amp;quot;modern times&amp;quot; of over-communication).&lt;/p&gt;&#xA;&lt;p&gt;Another point, which may muddle things in terms of labels for neurodiverse people: some people do have abnormally good memories for &lt;em&gt;everything&lt;/em&gt;, whether they're interested or not. So, one way or another, this includes remembering garbage and dumb stuff... which may require some effort to separate from the useful things. In effect, sometimes the prioritization occurs long enough &lt;em&gt;after&lt;/em&gt; the intake that the crap is still &amp;quot;in the system&amp;quot;, taking up space, and perhaps time and energy.&lt;/p&gt;&#xA;&lt;p&gt;Still, as I've said to my students many times, it is very convenient to be able to remember things that one does not yet understand, so as to be able to think about them... as opposed to the most common situation that people can only remember things they understand (=fit into their worldview...)&lt;/p&gt;&#xA;&lt;p&gt;In that vein, I myself do not understand the extent to which some sort of &amp;quot;rigidity&amp;quot; is part of some definition of some neurodiversity... but, yes, one of the most inconvenient impulses that I've seen is an unwillingness/incapacity to contemplate new/unanticipated things. I've certainly observed that quite a few people &amp;quot;like mathematics&amp;quot; because (supposedly) it's all settled and perfectly whatever... So then they go to grad school, and after some time encounter what T. Tao has labelled &amp;quot;post-rigorous&amp;quot; math (wherein not all details are of interest, to say the least!), and are unhappy... because they thought that the pedantic ultra-rigorous (painfully pedantic?) undergrad upper-division math classes were what math is all about.&lt;/p&gt;&#xA;&lt;p&gt;So, yes, &amp;quot;undergrad math&amp;quot; (as often taught) is a trap for certain people.&lt;/p&gt;&#xA;&lt;p&gt;Oppositely, as for me, &amp;quot;higher-level&amp;quot; (a touch post-rigorous?) math can be a great relief, aesthetically, etc., for some people. Not a moral virtue, for sure! ... but convenient (in my opinion) if one wants to find out new things.&lt;/p&gt;&#xA;&lt;p&gt;:)&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2023-12-13T23:21:05.147" LastActivityDate="2023-12-13T23:21:05.147" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27272" PostTypeId="1" AcceptedAnswerId="27273" CreationDate="2023-12-14T12:55:45.480" Score="6" ViewCount="627" Body="&lt;p&gt;In my undergraduate courses (discrete math, linear algebra), students' backgrounds vary significantly. One group has a strong math foundation from high school, while the other, less prepared, struggles with rigorous assessments. Making exams too challenging leads to higher dropout rates among the latter, but easier exams result in too many high grades, blurring distinctions between these groups.&lt;/p&gt;&#xA;&lt;p&gt;I'm seeking advice on balancing assessment rigor and fairness without using grading curves. How can I fairly assess a diverse student body while maintaining academic integrity?&lt;/p&gt;&#xA;" OwnerDisplayName="user19945" LastActivityDate="2023-12-16T13:23:28.963" Title="Strategies for Fair Assessment in Mixed-Ability Undergraduate Math Classes" Tags="|exams|assessment|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27273" PostTypeId="2" ParentId="27272" CreationDate="2023-12-14T16:13:08.303" Score="9" Body="&lt;p&gt;This is essentially not a problem that a single instructor can solve on their own.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The current top pedagogical theory in these cases is to provide &amp;quot;corequisite&amp;quot; or some kind of outside supplemental support to help the students who need to catch up. (IMO, the in-practice results of this have yet to be proven.)&lt;/li&gt;&#xA;&lt;li&gt;I would look closely at requirements for the next courses to follow (if any), and where students need to be at the end of your course to succeed at the next step.&lt;/li&gt;&#xA;&lt;li&gt;You should consult with colleagues, chair, program coordinators, etc., on the expected proficiency level at the end of your course, historical and expected pass rates, how the institution responds to high pass or fail rates, etc.&lt;/li&gt;&#xA;&lt;li&gt;Once this requirements-gathering phase has been done, my core advice would be to set reasonable standards for success (in your class and to enter the next level), and hold fairly to those expected standards.&lt;/li&gt;&#xA;&lt;li&gt;Give test questions which demonstrate the basic, essential, expected skills. Avoid questions which are overly clever or &amp;quot;interesting&amp;quot; from the instructor's POV.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Ultimately if the context and institution are letting low-skills students into your courses, who cannot possibly succeed at the expected level, and there aren't the needed outside supports to help, then the individual instructor cannot fix this. Sometimes the job is to give students the taste and opportunity to try a certain step, and give the difficult lesson that they're not ready to digest it at the current time. There's a passage from Ken Bain's &lt;em&gt;What the Best College Teachers Do&lt;/em&gt; that sticks with me:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Susan Wiltshire, a classics professor at Vanderbilt, captured a&#xA;sentiment we heard often. Her classes, she explained, were in her view&#xA;like a great meal she had prepared, and she simply wished to invite&#xA;her students to the dinner table.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ultimately, and especially in a scaffolded STEM discipline like mathematics, the instructor needs to maintain certain standards for students to succeed at the next level. If the institution is funneling students into a situation where they can't succeed -- then, unfortunately, they won't succeed.&lt;/p&gt;&#xA;&lt;p&gt;As a brief example, I teach at an open-admissions community college which is simultaneously part of an R1 university (every credit we award is a full credit for both). We've been told in official guidance, &amp;quot;The college must allow enrollment of students who are &lt;em&gt;not&lt;/em&gt; skills proficient&amp;quot; in our lower-level, corequisite credit courses (emphasis in original). Failure rates in courses throughout our program are about half. In some cases of remedial courses I've had semesters where the median grade for all students in the semester was circa 5%. So there are certain contexts where high failure rates are simply an expected part of the institution's operation; you should find out your context and calibrate accordingly.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2023-12-16T13:23:28.963" LastActivityDate="2023-12-16T13:23:28.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27275" PostTypeId="1" CreationDate="2023-12-15T21:57:16.687" Score="8" ViewCount="1489" Body="&lt;p&gt;How do you explain that the implication &lt;span class=&quot;math-container&quot;&gt;$p \implies q$&lt;/span&gt; has the value TRUE&#xA;when &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is FALSE and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is TRUE? It is difficult for students to accept&#xA;that FALSE &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; TRUE should have the value TRUE.&lt;/p&gt;&#xA;&lt;p&gt;I can't say I've developed a compelling argument.&#xA;When &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is FALSE, there is no implication, so &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is unaffected and can be either&#xA;TRUE or FALSE---both compatible with the expression value of TRUE.&lt;/p&gt;&#xA;&lt;p&gt;I'd be interested to hear more convincing approaches.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2023-12-18T09:41:37.347" Title="How to explain (FALSE =&gt; TRUE) is TRUE" Tags="|logic|" AnswerCount="10" CommentCount="5" ClosedDate="2023-12-17T00:33:25.693" ContentLicense="CC BY-SA 4.0" />
  <row Id="27276" PostTypeId="2" ParentId="27275" CreationDate="2023-12-15T22:57:52.123" Score="15" Body="&lt;p&gt;It makes more intuitive sense if you view an implication as a &amp;quot;promise&amp;quot;. The truth value of the expression represents whether the promise can be broken (the promise is considered true unless broken).&lt;/p&gt;&#xA;&lt;p&gt;From that point of view, it makes intuitive sense that in a situation when the antecedent is false, the promise is irrelevant to the situation, so the promise remains unbroken (i.e. true) regardless of the consequent.&lt;/p&gt;&#xA;&lt;p&gt;For instance, suppose the promise is that &amp;quot;if you eat your vegetables then you can have dessert.&amp;quot; Today you didn't eat your vegetables, but I still let you have dessert. Have I broken my promise? No. I have been true to my word. My promise has nothing to do with the situation at hand. I never said anything about what would happen if you didn't eat your vegetables.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2023-12-15T23:03:15.327" LastActivityDate="2023-12-15T23:03:15.327" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27277" PostTypeId="2" ParentId="27275" CreationDate="2023-12-15T23:16:26.993" Score="2" Body="&lt;p&gt;A nice thing about the convention we use for the Booleans of &lt;span class=&quot;math-container&quot;&gt;$p\rightarrow q$&lt;/span&gt; is that when &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is true, then  &lt;span class=&quot;math-container&quot;&gt;$p\rightarrow q$&lt;/span&gt; is &amp;quot;true&amp;quot; independent of the truthiness of the premises &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If we know that &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is true, for reasons independent of the premises &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, it would be awkward for the Boolean of &lt;span class=&quot;math-container&quot;&gt;$p\rightarrow q$&lt;/span&gt; to shift from true to false if we reconsidered the premises &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and decided they were flawed.&lt;/p&gt;&#xA;&lt;p&gt;If we wanted &amp;quot;false implies true&amp;quot; to be false, then we might as well just use the logical operation of &amp;quot;and&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2023-12-15T23:21:53.667" LastActivityDate="2023-12-15T23:21:53.667" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27278" PostTypeId="2" ParentId="27275" CreationDate="2023-12-15T23:40:35.720" Score="5" Body="&lt;p&gt;I convert &lt;span class=&quot;math-container&quot;&gt;$p \Rightarrow q$&lt;/span&gt; to conjunctions and disjunctions as follows.&lt;/p&gt;&#xA;&lt;p&gt;Consider a simple, familiar theorem of the form &lt;span class=&quot;math-container&quot;&gt;$p(x) \Rightarrow q(x)$&lt;/span&gt;. For my students, who have had at least a year of calculus, the following has seemed perfect:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is differentiable at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuuous at &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I then ask them to come up with example for &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; such &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; exhibit all four combinations of TRUE and FALSE, or explain why a particular combination is impossible. This in effect fills in the truth table with possible/impossible instead of TRUE/FALSE; but let's not dwell on that.&lt;/p&gt;&#xA;&lt;p&gt;Instead, we agree (the class and I) that &lt;span class=&quot;math-container&quot;&gt;$p \Rightarrow q$&lt;/span&gt; is equivalent to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(p \wedge q) \mathrel{\vee} (\lnot p \wedge q) \mathrel{\vee} (\lnot p \wedge \lnot q) \,.$$&lt;/span&gt;&#xA;And we agree that if &lt;span class=&quot;math-container&quot;&gt;$p \Rightarrow q$&lt;/span&gt; is a theorem then &lt;span class=&quot;math-container&quot;&gt;$p \wedge \lnot q$&lt;/span&gt; cannot be TRUE.&#xA;And that fills in the truth table.&lt;/p&gt;&#xA;&lt;p&gt;I used to use more natural language example, but biological examples, including humans, seem full of exceptions. It seems safer to build on mathematics they know. And it seems good for the student to reinforce mathematics they should know, in case they don't.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2023-12-15T23:40:35.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27279" PostTypeId="2" ParentId="27275" CreationDate="2023-12-15T23:58:30.383" Score="9" Body="&lt;p&gt;I use the example&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If your flight is on time, then I will pick you up from the airport.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If your friend tells you this, the &lt;em&gt;only&lt;/em&gt; time they have lied is if the premise is true and the conclusion false. If your flight is late, no matter what happens afterwards, they won't have lied.&lt;/p&gt;&#xA;&lt;p&gt;It is the same with formal statements: the only time &lt;span class=&quot;math-container&quot;&gt;$P \to Q$&lt;/span&gt; is false is when the premise is true and the conclusion false. If you don't want them to be false, logic demands that the other situations must be true.&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2023-12-15T23:58:30.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27280" PostTypeId="2" ParentId="27275" CreationDate="2023-12-16T04:03:42.963" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;How to explain (FALSE =&amp;gt; TRUE) is TRUE&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You might try to avoid using the literals &amp;quot;TRUE&amp;quot; AND &amp;quot;FALSE&amp;quot; as logical propositions as here.&lt;/p&gt;&#xA;&lt;p&gt;Try to clarify the meaning in propositional logic of &amp;quot;implies&amp;quot; with an example:&lt;/p&gt;&#xA;&lt;p&gt;If it is raining (&lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;), then it is cloudy (&lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$R \implies C$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This does not mean that rain causes cloudiness. It does not even mean that it is always cloudy when it is raining. (That would require some form of predicate logic, e.g. quantifying over time.) It means only that, &lt;strong&gt;at the moment&lt;/strong&gt;, it is not both raining and not cloudy.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$R\implies C ~~\equiv ~~ \neg (R \land \neg C)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It also means that if it is not raining, then it may or may not be cloudy. Both are still possibilities.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2023-12-16T19:05:12.110" LastActivityDate="2023-12-16T19:05:12.110" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27281" PostTypeId="2" ParentId="27275" CreationDate="2023-12-16T07:33:40.520" Score="5" Body="&lt;p&gt;One informal interpretation is that &lt;span class=&quot;math-container&quot;&gt;$p \implies q$&lt;/span&gt; means &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is at least as true as &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Since TRUE is &amp;quot;at least as true as&amp;quot; FALSE, the implication FALSE &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; TRUE is true.&lt;/p&gt;&#xA;" OwnerUserId="23536" LastEditorUserId="23536" LastEditDate="2023-12-16T07:39:35.963" LastActivityDate="2023-12-16T07:39:35.963" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27282" PostTypeId="2" ParentId="27275" CreationDate="2023-12-16T10:12:21.660" Score="2" Body="&lt;p&gt;Logical implication is a binary operator &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; defined for booleans &lt;span class=&quot;math-container&quot;&gt;$p,\ q$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$p \implies q$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$\neg\,(\,p \land \neg\,q)$&lt;/span&gt; is true. But by the time I was introduced to this definition, I already had a strong intuitive sense of the meaning of &amp;quot;implication&amp;quot; which did not align with this mathematical definition and confused me.&lt;/p&gt;&#xA;&lt;p&gt;Suppose that &lt;span class=&quot;math-container&quot;&gt;$\mathrm{rain} \implies \mathrm{clouds}$&lt;/span&gt; is true. If it is not raining, then the truth of the implication tells us nothing about whether there are clouds. &lt;span class=&quot;math-container&quot;&gt;$\neg\,\mathrm{rain}$&lt;/span&gt; does not &amp;quot;imply&amp;quot; anything. Yet the mathematical definition assigns truth value to both expressions, &lt;span class=&quot;math-container&quot;&gt;$\neg\,\mathrm{rain} \implies \mathrm{clouds}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\neg\,\mathrm{rain} \implies \neg\,\mathrm{clouds}$&lt;/span&gt;. This way of expressing the lack of information in the proposition &lt;span class=&quot;math-container&quot;&gt;$\neg\,\mathrm{rain}$&lt;/span&gt; can feel like a contradiction.&lt;/p&gt;&#xA;&lt;p&gt;The source of my confusion was the result in classical logic that &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; entails &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$p \implies q$&lt;/span&gt;. In fact, there is a &lt;a href=&quot;https://philosophy.stackexchange.com/questions/12816/difference-between-implication-conditional-and-logical-entailment&quot;&gt;distinction between implication and entailment&lt;/a&gt; in philosophy. I was able to accept &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; as a simple mathematical function on booleans which could be meaningfully applied to false &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, once I realised that the source of my confusion was an significant problem in philosophy which I did not need to resolve in a mathematics class.&lt;/p&gt;&#xA;" OwnerUserId="1500" LastActivityDate="2023-12-16T10:12:21.660" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27283" PostTypeId="2" ParentId="27275" CreationDate="2023-12-16T13:14:38.680" Score="1" Body="&lt;p&gt;Are they familiar with the principle of explosion, or could it be introduced? The idea that deriving a contradiction (or a proof of &lt;span class=&quot;math-container&quot;&gt;$\bot$&lt;/span&gt;, if you prefer) means you can prove anything is slightly more general, but then explains that &lt;span class=&quot;math-container&quot;&gt;$p \implies q$&lt;/span&gt; makes sense whenever &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is false (since any value of &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; could be derived).&lt;/p&gt;&#xA;" OwnerUserId="23392" LastActivityDate="2023-12-16T13:14:38.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27285" PostTypeId="2" ParentId="27275" CreationDate="2023-12-16T22:35:26.030" Score="2" Body="&lt;p&gt;Given a relationship : PREMISE ⟶ CONSEQUENCE&lt;/p&gt;&#xA;&lt;p&gt;The truthy value for that relationship is: &lt;strong&gt;is it possible to get the consequence given the premise?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This relationship is easy to understand for true premises.&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;PREMISE ⟶ CONSEQUENCE&lt;/th&gt;&#xA;&lt;th&gt;CAN WE GET THAT CONSEQUENCE?&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;TRUE ⟶ TRUE&lt;/td&gt;&#xA;&lt;td&gt;YES - we get truth from true premises&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;TRUE ⟶ FALSE&lt;/td&gt;&#xA;&lt;td&gt;NO - we do not get falsehood from true premises&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;The problem is that we want false premises to work the same way as true ones. But they don’t.&lt;/p&gt;&#xA;&lt;div class=&quot;s-table-container&quot;&gt;&#xA;&lt;table class=&quot;s-table&quot;&gt;&#xA;&lt;thead&gt;&#xA;&lt;tr&gt;&#xA;&lt;th&gt;PREMISE ⟶ CONSEQUENCE&lt;/th&gt;&#xA;&lt;th&gt;CAN WE GET THAT CONSEQUENCE?&lt;/th&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/thead&gt;&#xA;&lt;tbody&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;FALSE ⟶ TRUE&lt;/td&gt;&#xA;&lt;td&gt;&lt;strong&gt;YES&lt;/strong&gt;&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;tr&gt;&#xA;&lt;td&gt;FALSE ⟶ FALSE&lt;/td&gt;&#xA;&lt;td&gt;&lt;strong&gt;YES&lt;/strong&gt;&lt;/td&gt;&#xA;&lt;/tr&gt;&#xA;&lt;/tbody&gt;&#xA;&lt;/table&gt;&#xA;&lt;/div&gt;&#xA;&lt;p&gt;The reason FALSE ⟶ ANYTHING is valid is because FALSE does not really imply anything — the consequence could be either TRUE or FALSE. That is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A FALSE premise cannot imply any consequence.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Written another way:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A FALSE premise does not have the power to invalidate either TRUE or FALSE consequences&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14983" LastActivityDate="2023-12-16T22:35:26.030" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27286" PostTypeId="2" ParentId="27275" CreationDate="2023-12-16T23:19:30.323" Score="2" Body="&lt;p&gt;I think one possible difficulty is the idea itself that &lt;span class=&quot;math-container&quot;&gt;$p \Rightarrow q$&lt;/span&gt; might have a value TRUE or FALSE. Reading &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;&amp;quot; some might interpret it as what programmers would call a &amp;quot;statement&amp;quot; rather than an &amp;quot;expression&amp;quot; that evaluates to something. I think a way to help students make sense of this is to interpret the truth value of &lt;span class=&quot;math-container&quot;&gt;$p \Rightarrow q$&lt;/span&gt; as an assessment of whether &amp;quot;the data is consistent with the implication statement&amp;quot;. So if &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;=FALSE and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;=TRUE, the data is consistent with &lt;span class=&quot;math-container&quot;&gt;$p \Rightarrow q$&lt;/span&gt; and so its value is TRUE. If instead &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;=TRUE and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;=FALSE, the data violates &lt;span class=&quot;math-container&quot;&gt;$p \Rightarrow q$&lt;/span&gt; which must be FALSE.&#xA;It is also important that we stop at assessing whether the data is &lt;em&gt;consistent&lt;/em&gt; with the implication without trying to assess the truthiness of the implication (in everyday language); as a result the expression &amp;quot;3 is odd &lt;span class=&quot;math-container&quot;&gt;$\Rightarrow$&lt;/span&gt; 3 &amp;lt; 12&amp;quot; is true because the data is consistent with the implication albeit the implication is false in the everyday meaning of the term (where implication suggests some form of causality).&lt;/p&gt;&#xA;" OwnerUserId="23546" LastEditorUserId="23546" LastEditDate="2023-12-18T09:41:37.347" LastActivityDate="2023-12-18T09:41:37.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27287" PostTypeId="1" CreationDate="2023-12-17T04:10:36.930" Score="4" ViewCount="208" Body="&lt;p&gt;I asked this question at &lt;a href=&quot;https://mathoverflow.net/questions/460493/the-game-related-to-ring-theory&quot;&gt;'mathoverflow'&lt;/a&gt;. However, the character of this question does not match mathoverflow, so I am asking this question again on this site.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Motivation:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Among abstract algebra, I am interested in ring theory, which is closely related to algebraic geometry. And among them, I am very interested in ring of integers, which is familiar because it is related to number theory and interested in Ideal theory, which Dedekind first formalized in detail. Among various ideals, the ones of particular interest are ‘maximal ideal’ and ‘prime ideal’.&lt;/p&gt;&#xA;&lt;p&gt;I saw &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486284247&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;the book &amp;quot;Concepts of Modern Mathematics&amp;quot; by Ian Stewart&lt;/a&gt;. In this book, I saw the game, &lt;strong&gt;'solitaire game'&lt;/strong&gt;. the game's rule is like this below contents.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The game of solitaire as you probably know, is played on a board with holes in the pattern&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GArF9.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GArF9.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the start, each hole except the centre contains a peg. The player may jump any peg over another horizontally or vertically adjacent peg into an empty hole, removing the peg jumped over. Diagonal moves are not allowed. The aim is to remove all pegs but one. Usually, it is required that the last peg should end up in the centre. Anyone playing the game will observe, if he plays it enough, that although the final peg need not end up in the centre, it does not seem possible to make it finish anywhere. There is a limited number of final positions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;We shall ask:&lt;/strong&gt; In what positions is it possible for the last peg to be? And we shall answer it by a method due to de Bruijn, using a certain field with 4 elements.&lt;/p&gt;&#xA;&lt;p&gt;This book says that this game is a game that utilizes field theory. At this point, I became curious about whether there were any games that utilized ring theory.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Why this game is related to field theory?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;We shall answer it by a method due to de Bruijn, using a certain field with 4 elements. The elements will be &lt;span class=&quot;math-container&quot;&gt;$0,1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;: and addition and multiplications are defined by the tables&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/dRkFI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dRkFI.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;We observe that the equation &lt;span class=&quot;math-container&quot;&gt;$p^2+p+1=0$&lt;/span&gt; holds.(This is crucial, and it is why we use the field chosen.) For &lt;span class=&quot;math-container&quot;&gt;$$p^2+p+1=q+p+1$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$=1+1$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$=0.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We assign integer coordinates to the holes on the board, like this:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/geMr2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/geMr2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A situation is defined to be a set of pegs on the board. For any situation &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; we define a value &lt;span class=&quot;math-container&quot;&gt;$A(S)=\sum p^{k+l}$&lt;/span&gt; where the &lt;span class=&quot;math-container&quot;&gt;$\sum$&lt;/span&gt; sign indicates that we add up all the &lt;span class=&quot;math-container&quot;&gt;$p^{k+l}$&lt;/span&gt; for all the coordinates &lt;span class=&quot;math-container&quot;&gt;$(k,l)$&lt;/span&gt; of pegs in the set S. (This, you may note, makes A a function with domain the set of possible situations and target the field with 4 elements). Thus for the situation (pegs marked in black)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/L6j9a.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/L6j9a.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;S is the set &lt;span class=&quot;math-container&quot;&gt;${(-2,-1),(-1,0),(0,0),(0,-1),(2,0),(1,2)}$&lt;/span&gt; and we have &lt;span class=&quot;math-container&quot;&gt;$$A(S)=p^{-2-1}+p^{-1+0}+p^{0+0}+p^{0-1}+p^{2+0}+p^{1+2}$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$=p^{-3}+p^{-1}+p^0+p^{-1}+p^2+p^3$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$=1+q+1+q+q+1$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$=1+q$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$=p.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The function A is defined to have the very nice property: if a legal move changes situation &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; into situation &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$A(S)=A(T)$&lt;/span&gt;. The value of a situation is unchanged by legal moves, and therefore remains constant throughout the game.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&#xA;Are there any games or examples that can familiarize us with 'ring of integers' and 'maximal ideal' or 'prime ideal'?&lt;/p&gt;&#xA;" OwnerUserId="23548" LastEditorUserId="23548" LastEditDate="2023-12-17T06:36:55.857" LastActivityDate="2023-12-17T06:36:55.857" Title="Learn ring theory more fun" Tags="|abstract-algebra|applied-mathematics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27288" PostTypeId="2" ParentId="13032" CreationDate="2023-12-17T15:58:08.993" Score="7" Body="&lt;p&gt;Giving examples is not the only thing I would do in this case.&lt;/p&gt;&#xA;&lt;p&gt;I would also introduce new vocabulary. Having precise words helps when talking about precise concepts.&lt;/p&gt;&#xA;&lt;p&gt;If each step of an algorithm can be executed in a unique way, then the algorithm is &lt;em&gt;deterministic&lt;/em&gt;. This means that if two different people execute this algorithm, they will get the same result, even if they execute it on different days with different weathers and while feeling in different moods.&lt;/p&gt;&#xA;&lt;p&gt;Yes, the output of the algorithm is unique. This does not imply that the solution to the problem is unique.&lt;/p&gt;&#xA;" OwnerUserId="16159" LastActivityDate="2023-12-17T15:58:08.993" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27289" PostTypeId="1" CreationDate="2023-12-18T09:42:07.380" Score="18" ViewCount="3572" Body="&lt;p&gt;I'm going to give a class to highschoolers about topology. I've prepared the beginning of the class where I introduce what is topology and give them different ways we use to describe spaces, but everything is described very intuitively since they are in highschool. For example, I describe compactness as can we put this space into a ball or dimension as the number of different direction we can go,...&lt;/p&gt;&#xA;&lt;p&gt;Now I want to tell them some weird and fun fact about topology, and maybe make them do some intuitive exercises, like that the Klein bottle has no inside or outside or the seven bridges of Königsberg problem.&lt;/p&gt;&#xA;&lt;p&gt;So my question is do you know any topological problem or fun fact that highschooler can understand and find interesting enough that they would want to do topology ?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt; I found a great book and wanted to share it here: intuitive topology by V.V. Prasolov&lt;/p&gt;&#xA;" OwnerUserId="23555" OwnerDisplayName="marilou64" LastEditorUserId="23555" LastEditDate="2024-01-10T15:48:00.203" LastActivityDate="2024-01-10T15:48:00.203" Title="Topological fun facts for high school students" Tags="|secondary-education|topology|" AnswerCount="11" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="27290" PostTypeId="2" ParentId="27289" CreationDate="2023-12-19T01:06:06.520" Score="5" Body="&lt;p&gt;The organization &lt;a href=&quot;https://www.mappmath.org/poster&quot; rel=&quot;noreferrer&quot;&gt;Mathematical Puzzle Programs&lt;/a&gt; organizes puzzle events aimed at connecting high school students with contemporary/research mathematics.&lt;/p&gt;&#xA;&lt;p&gt;This poster includes three puzzles, two of which are connected to topology (one boils down to the concept of homeomorphism, the other is one dimensional persistent homology). &lt;a href=&quot;https://www.mappmath.org/poster&quot; rel=&quot;noreferrer&quot;&gt;https://www.mappmath.org/poster&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15209" OwnerDisplayName="Steven Clontz" LastActivityDate="2023-12-19T01:06:06.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27291" PostTypeId="2" ParentId="27289" CreationDate="2023-12-19T16:54:17.353" Score="7" Body="&lt;p&gt;Not so much a &amp;quot;topological fact&amp;quot; but more of a hands-on exploration with two topological objects:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://eugeniacheng.com/&quot; rel=&quot;noreferrer&quot;&gt;Eugenia Cheng&lt;/a&gt; suggests cutting a bagel (torus) along a Mobius strip to increase the available surface area for cream cheese!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.southernliving.com/food/kitchen-assistant/how-to-cut-bagel-according-mathematician&quot; rel=&quot;noreferrer&quot;&gt;This link&lt;/a&gt; describes the method and has a video demonstration.&lt;/p&gt;&#xA;" OwnerUserId="80" LastActivityDate="2023-12-19T16:54:17.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27293" PostTypeId="2" ParentId="27289" CreationDate="2023-12-19T20:18:50.803" Score="5" Body="&lt;p&gt;This can be presented in ordinary terminology to high school students and even lower. I made it to first and second year college students. While I think a student can appreciate the connection to maxima, minima, and saddle points better if they have had multivariable calculus, I don't think it is necessary. Only some in my audience had had it.  Pretty much everyone had seen a donut before.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;When you dunk a donut in coffee (or cider), the wet surface morphs.  The surface transitions between homeomorphically distinct objects when the donut passes a critical value of a certain distance function: the depth the donut has been dunked.&lt;/p&gt;&#xA;&lt;p&gt;There are three types of transitions that can be characterized by adding one of three elements, whose boundary is connected to the current boundary:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a point (0D)&lt;/li&gt;&#xA;&lt;li&gt;a line segment (1D)&lt;/li&gt;&#xA;&lt;li&gt;a disk (2D)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Once added, the element can spread out and form a (bigger) surface.&lt;/p&gt;&#xA;&lt;p&gt;For instance, as the donut approaches the coffee, no part is wet and the wet &amp;quot;surface&amp;quot; is the &amp;quot;empty set.&amp;quot;  As the donut makes contact with the coffee, a point gets wet which spreads out in a bigger wet surface as the donut is dunked deeper. The boundary remains a loop until the next transition.&lt;/p&gt;&#xA;&lt;p&gt;The next two transitions are a bit tricky to describe. They occur when the donut hole first reaches the surface of the coffee and when it first becomes completely submerged.  In between the transitions the boundary consists of two loops.&lt;/p&gt;&#xA;&lt;p&gt;What one sees at the second critical value (depth = lower inside of the donut hole) is the (single) boundary loop pinching together at a point and immediately separating into two loops.&#xA;This is not what happens if you connect a point to the single loop boundary. Connecting a point changes nothing. However, it is what happens when you add a line segment, which immediately spreads out.  Topologically, this is the correct construction. Unfortunately, most people, even high-schoolers, do not have the appetite to experiment enough times to convince themselves that the construction is good description of what they see happening.&lt;/p&gt;&#xA;&lt;p&gt;At the third critical value (depth = upper inside of the donut hole),&#xA;a line connects the two boundary loops and spreads out so that the boundary is again a single loop.&lt;/p&gt;&#xA;&lt;p&gt;At the fourth critical value (depth = complete submersion), a disk is added. Again this may be hard to see. The boundary appears to &amp;quot;shrink to a point&amp;quot; as the donut goes under. However, at no depth is the boundary at the top a point. Until the top is level with the surface of the coffee, the boundary is a loop. When level, the whole surface of the donut is included.  The only way to close the hole is to connect the boundary of a disk to the boundary of the wet surface.&lt;/p&gt;&#xA;&lt;p&gt;A theoretical connection is that the dimension of the element that is added is equal to the index of the critical value.  At each critical point, one of three things happen:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a &lt;em&gt;&lt;strong&gt;cup&lt;/strong&gt;&lt;/em&gt;: the principal curvatures are up (all sections by a vertical plane are concave up)&lt;/li&gt;&#xA;&lt;li&gt;a &lt;em&gt;&lt;strong&gt;saddle&lt;/strong&gt;&lt;/em&gt;: one of the principal curvatures is up, the other down (some sections by a vertical plane are concave up, some concave down)&lt;/li&gt;&#xA;&lt;li&gt;a &lt;em&gt;&lt;strong&gt;cap&lt;/strong&gt;&lt;/em&gt;: the principal curvatures are down (all sections by a vertical plane are concave down)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The index is equal to the number of principal curvatures that are down.&#xA;The transitions have indices &lt;span class=&quot;math-container&quot;&gt;$0,1,1,2$&lt;/span&gt; in order (hence point, line, line, disk).&#xA;That in turn implies an Euler characteristic of &lt;span class=&quot;math-container&quot;&gt;$\chi=(-1)^0+(-1)^1+(-1)^1+(-1)^2 = 0$&lt;/span&gt;, should you want to make a connection there.&lt;/p&gt;&#xA;&lt;p&gt;Finally, it's an argument to get your department to spring for donuts for your class.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I believe I got the idea from Milnor's &lt;em&gt;Morse Theory&lt;/em&gt;, but I'm on vacation and cannot check.&lt;/p&gt;&#xA;&lt;p&gt;Marston Morse himself gives a demo (of a different surface, not a donut) which I was shown in college: &lt;a href=&quot;https://www.youtube.com/watch?v=QVMnv-KpQlM&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=QVMnv-KpQlM&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="498" LastEditorUserId="498" LastEditDate="2023-12-20T06:29:47.057" LastActivityDate="2023-12-20T06:29:47.057" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27294" PostTypeId="2" ParentId="27289" CreationDate="2023-12-19T22:59:04.577" Score="3" Body="&lt;p&gt;Here are a few ideas for exercises that hopefully lead to &amp;quot;fun&amp;quot; facts in (algebraic) topology. If I remember correctly, most (but not all) of them are contained in Prasolov's book you mentioned.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;a) Let them try to solve the &lt;a href=&quot;https://en.wikipedia.org/wiki/Three_utilities_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;three utilities problem&lt;/a&gt;. This boils down to embedding the graph &lt;span class=&quot;math-container&quot;&gt;$K_{3,3}$&lt;/span&gt; in the plane, which is impossible. b) Explain that this is impossible. If you take Euler's theorem&#xA;(about the Euler characteristic) for granted, the proof is quite&#xA;accessible. c) Let them try to solve the same problem on the surface&#xA;of a mug, which is a torus. I'd expect them to succeed with some&#xA;trial and error, and possibly some hints (the handle lets you avoid one intersection). d) Discuss the difference between the two cases, and how Euler characteristic works for different surfaces.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let them discover the &lt;a href=&quot;https://en.wikipedia.org/wiki/Borromean_rings&quot; rel=&quot;nofollow noreferrer&quot;&gt;Borromean rings&lt;/a&gt; by themselves. That is, give them three loops (that can be open and closed at will) and tell them to arrange it so that the three loops together are tangled, but removing one loop makes the other two untangled. The solution is not unique, but there's a good chance they will arrive at this one, because it's the simplest.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a) Consider the following puzzle. There are two nails in the wall, and the goal is to hang a picture on them (using a loop attached to the picture), according to two rules: the picture hangs if both nails are in place, but if either nail is removed, it falls down. b) After they arrive at a solution, discuss why it's the same puzzle as the Borromean rings. In both cases we're studying the same fundamental group: of the space minus two untangled circles (in 2) or of the plane minus two points (in 3).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a) Let &lt;span class=&quot;math-container&quot;&gt;$A,B,C$&lt;/span&gt; be Borromean rings. Find a surface spanned by &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; (i.e. a surface with &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; as its boundary) which doesn't intersect &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;. b) Discuss why this surface cannot be a disk (because you could shrink &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; and remove it without touching &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;). c) If possible, introduce Hurewicz's theorem. It explains the difference between spanning a disk (i.e. being trivial in the fundamental group) and spanning a surface (i.e. being trivial in the first homology group).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15367" LastEditorUserId="15367" LastEditDate="2023-12-20T09:54:09.030" LastActivityDate="2023-12-20T09:54:09.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27295" PostTypeId="2" ParentId="19183" CreationDate="2023-12-20T10:52:09.537" Score="2" Body="&lt;p&gt;I would like to suggest following challenges,&lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Geometric proof of sum and product of three tangents of a triangle are equal for obtuse-angled triangles.&lt;br/&gt;&#xA;(&lt;a href=&quot;https://www.janakasrodrigo.com/wp-content/uploads/2023/12/Proof-7-1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.janakasrodrigo.com/wp-content/uploads/2023/12/Proof-7-1.pdf&lt;/a&gt;)&lt;br/&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Trigonometric proof of Pythagoras theorem without applying properties of similar triangles and any form of Pythagorean Identity. (&lt;a href=&quot;https://www.janakasrodrigo.com/wp-content/uploads/2023/12/Proof-4.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.janakasrodrigo.com/wp-content/uploads/2023/12/Proof-4.pdf&lt;/a&gt; )&lt;br/&gt; Editor board of the journal American Mathematical Monthly acknowledged this as a new proof of Pythagoras theorem. &lt;br/&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simultaneous single proof of the Sine Law and Cosine Law covering all three types of triangles.(&lt;a href=&quot;https://www.janakasrodrigo.com/wp-content/uploads/2023/12/Proof-5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.janakasrodrigo.com/wp-content/uploads/2023/12/Proof-5.pdf&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2023-12-23T15:20:36.163" LastActivityDate="2023-12-23T15:20:36.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27296" PostTypeId="2" ParentId="27289" CreationDate="2023-12-20T13:21:39.183" Score="12" Body="&lt;p&gt;Cutting a Möbius strip &lt;a href=&quot;https://math.stackexchange.com/a/67564/237&quot;&gt;in half&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lgomh.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lgomh.png&quot; alt=&quot;J.M.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;or at &lt;a href=&quot;https://matheducators.stackexchange.com/q/7392/511&quot;&gt;one-third&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/DbCxU.jpg&quot; alt=&quot;MobiusThird&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;I would bring a collection of ~20 scissors and tape dispensers,&#xA;and have the students form Möbius strips (sharing equipment), predict what will happen, and then cut the strips.&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2023-12-21T18:47:15.617" LastActivityDate="2023-12-21T18:47:15.617" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27297" PostTypeId="2" ParentId="27229" CreationDate="2023-12-20T13:41:11.303" Score="0" Body="&lt;p&gt;I don't have a class recommendation, but I think the historical development of calculus is a useful tool for an educator with this goal, and I can recommend Bressoud's recent book &lt;em&gt;Calculus Reordered: A History of the Big Ideas&lt;/em&gt; for inspiration and guidance.&lt;/p&gt;&#xA;" OwnerUserId="22304" LastActivityDate="2023-12-20T13:41:11.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27298" PostTypeId="2" ParentId="27289" CreationDate="2023-12-20T22:53:18.253" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sphere_eversion&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sphere eversion&lt;/a&gt; is a good one for several reasons.&#xA;First, it's very counter-intuitive, and yet you can show them this &lt;a href=&quot;https://www.youtube.com/watch?v=OI-To1eUtuU&quot; rel=&quot;nofollow noreferrer&quot;&gt;beautiful animation&lt;/a&gt; of it.&#xA;Second, it's a good way to introduce a lot of important higher-level concepts -- what does it mean to deform something without cutting or tearing it and how does that relate to things like continuity or differentiability?&#xA;Finally, at a personal level, the characters involved in the story could have walked out of a movie.&#xA;Stephen Smale gave a non-constructive proof of eversion that plenty of people thought was just wrong at first.&#xA;There's the famous story about him showing up to a lecture late and copying down a list of problems the instructor wrote down on the board thinking they were homework and solving a couple of them, not realizing that those were unsolved problems in the field.&#xA;Then there's Bernard Morin, who gave a proof of sphere eversion by constructing the sequence of transformations; Morin went blind at the age of 6.&#xA;At that rate, you can even talk about the differences between constructive and non-constructive proof.&lt;/p&gt;&#xA;" OwnerUserId="19634" LastActivityDate="2023-12-20T22:53:18.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27299" PostTypeId="2" ParentId="27229" CreationDate="2023-12-21T00:49:27.573" Score="1" Body="&lt;p&gt;You may be interested in the approach taken in the Mathematics for Life Sciences course series (LS 30a and LS 30b) at UCLA, meant to be taken in place of Calculus I/II. It's designed to efficiently teach the calculus and linear algebra needed to understand dynamical systems and to emphasize numerical and qualitative methods over analytic methods. You can find more information on &lt;a href=&quot;https://modelinginbiology.github.io&quot; rel=&quot;nofollow noreferrer&quot;&gt;this website&lt;/a&gt;, made to support the textbook written for the course.&lt;/p&gt;&#xA;&lt;p&gt;I don't think this is the &amp;quot;survey of calculus&amp;quot; that you're looking for. Nevertheless, one of the tenets of the course design is to eschew symbolic tricks, if that's what you mean by &amp;quot;the various computational tricks used in integration and differentiation.&amp;quot; For example, rather than spending time on integration techniques, separation of variables, etc., the course has students implement Euler's method in Python to integrate systems of nonlinear differential equations in the first semester.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2023-12-21T00:49:27.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27300" PostTypeId="2" ParentId="27289" CreationDate="2023-12-21T13:59:55.913" Score="3" Body="&lt;p&gt;I have given talks to high school students about topology previously. I often talk about the surfaces that can be obtained by identifying sides of a square in different ways - this is quite accessible and there are nice animations that show how to obtain the immersion of the Klein bottle in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;, beginning with a square with edges identified appropriately.&lt;/p&gt;&#xA;&lt;p&gt;From there, you could go on to talk about 3-manifolds such as the 3-torus and other manifolds obtained by identifying faces in a cube. You could also show them how to play tic-tac-toe on a torus, Klein bottle or projective plane, and how there are new ways to win as a result of the gluing.  Jeff Weeks has some relevant games on his &lt;a href=&quot;https://www.geometrygames.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt;. You could also look at gluings of other polygons, such as gluings of an octagon that give a double torus.&lt;/p&gt;&#xA;" OwnerUserId="8900" LastActivityDate="2023-12-21T13:59:55.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27301" PostTypeId="2" ParentId="27289" CreationDate="2023-12-21T22:05:05.803" Score="2" Body="&lt;p&gt;Proof&lt;/p&gt;&#xA;&lt;p&gt;Let S be a square with bottom left point at (0,0) and top right point at (1,1)&lt;/p&gt;&#xA;&lt;p&gt;Let (0,y) be equivalent to (1,y) for any y&lt;/p&gt;&#xA;&lt;p&gt;Let (x,0) be equivalent to (x,1) for any x&lt;/p&gt;&#xA;&lt;p&gt;Thus, Pacman is played on the surface of a donut, where Pacman is a member of the set of games which high schoolers don't play these days.&lt;/p&gt;&#xA;&lt;p&gt;QED&lt;/p&gt;&#xA;" OwnerUserId="23561" LastActivityDate="2023-12-21T22:05:05.803" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27302" PostTypeId="2" ParentId="27289" CreationDate="2023-12-22T11:16:31.630" Score="4" Body="&lt;p&gt;Speaking from a high school student's perspective who has attended a few talks about topology before, I think it was really interactive and fun when we created a Mobius strip and attempted to cut it in half like how Joseph O'Rourke mentioned in his answer. One of my teachers also brought a Klein Bottle in so that we could touch it and attempt to trace the path around it, this was helpful since visualising it from 3D images was trickier, however this depends on whether you have a Klein Bottle at home or not... To end the talk, you could recommend a few books for anyone interested in topology, for example, one of my teachers recommended the book 'The Shape of Space' by Jeff R Weeks.&lt;/p&gt;&#xA;&lt;p&gt;Some of the answers on here are addressing more advanced topics and whilst I think this is great, it seems like OP would want ideas which cater to introductory topics.&lt;/p&gt;&#xA;&lt;p&gt;Hope the above helps and good luck with your presentation! :)&lt;/p&gt;&#xA;" OwnerUserId="22471" LastEditorUserId="22471" LastEditDate="2023-12-22T16:29:54.383" LastActivityDate="2023-12-22T16:29:54.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27304" PostTypeId="2" ParentId="27289" CreationDate="2023-12-22T17:31:04.910" Score="3" Body="&lt;p&gt;I like Dirac's string trick.&lt;/p&gt;&#xA;&lt;p&gt;It shows that a full rotation doesn't bring an object back to its initial state, a second rotation is required.  Any object has some kind of invisible parity such that a full rotation in any direction flips the parity.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.av8n.com/physics/dirac-string-trick.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.av8n.com/physics/dirac-string-trick.htm&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12787" LastActivityDate="2023-12-22T17:31:04.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27306" PostTypeId="2" ParentId="27289" CreationDate="2023-12-23T19:44:12.803" Score="5" Body="&lt;p&gt;I gave a well received talk about the topology of spaces of planar polygons with specified side lengths, considered up to rotation. E.g. the space of triangles is empty or a point, of quadrilaterals is usually a circle, but of nearly regular pentagons starts to get you surfaces of various genera.&lt;/p&gt;&#xA;&lt;p&gt;The main point I was hoping to stress was that mathematicians routinely mix levels - what is a polygon here is a point there.&lt;/p&gt;&#xA;" OwnerUserId="23569" LastActivityDate="2023-12-23T19:44:12.803" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27307" PostTypeId="1" AcceptedAnswerId="27311" CreationDate="2023-12-25T14:14:46.723" Score="2" ViewCount="189" Body="&lt;p&gt;My son was working the other day with exercises such as: Find all the mappings &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{N}\rightarrow\mathbb{Z}$&lt;/span&gt; verifying&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\forall m,n \in \mathbb{N}, f(m+n)=f(n)+f(m).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As another example: Find all the mappings &lt;span class=&quot;math-container&quot;&gt;$f:\mathbb{R}\rightarrow\mathbb{R}$&lt;/span&gt; such that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\forall x,y \in \mathbb{R}, f(x)f(y)=f(xy)+x+y.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I know that there several equivalent ways to introduce the natural logarithmic function but I was wondering how one can define it properly and consicely by means of the functional relation&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(xy)=f(x)+f(y).\qquad (1)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;That is starting from this relation and not any prior knowledge of &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\exp$&lt;/span&gt;.&#xA;Find all applications &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}\rightarrow\mathbb{R}$&lt;/span&gt; such that (1) is satisfied, posing all the necessary restrictions.&lt;/p&gt;&#xA;&lt;p&gt;Thank you very much for any help, or pointing me out if the question is a duplicate.&lt;/p&gt;&#xA;" OwnerUserId="18147" LastEditorUserId="18147" LastEditDate="2023-12-25T14:20:34.400" LastActivityDate="2023-12-27T05:46:42.620" Title="Define logarithmic function by functional relation" Tags="|calculus|functions|definitions|logarithm|" AnswerCount="1" CommentCount="6" ClosedDate="2024-01-03T12:15:42.100" ContentLicense="CC BY-SA 4.0" />
  <row Id="27308" PostTypeId="2" ParentId="27263" CreationDate="2023-12-25T17:31:18.377" Score="2" Body="&lt;p&gt;Given that this is a first course, I'd use a combination of &lt;em&gt;Harold Jacobs Geometry: Seeing, Doing, Understanding&lt;/em&gt; (first edition is my favorite) and I'd also mix in some sections of &lt;em&gt;Euclid's Elements by Oliver Byrne&lt;/em&gt;, which is illustrated with color — see &lt;a href=&quot;https://www.c82.net/euclid/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.c82.net/euclid/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Lastly, I'd incorporate some philosophy that was happening at the time. Not a lot, but some interesting aspects that will help raise curiosity and wonder. It's important to put yourself in the context of what was happening in classical mathematics. It was not simply &amp;quot;Hey guys, let's do math!&amp;quot; These people were not doing math just to math it up. Classical geometry was heavily rooted to philosophy and all of what they were doing was to explore philosophical problems, not mathematical problems. I do not see a point in classical math, unless it is to include the philosophy side.&lt;/p&gt;&#xA;&lt;p&gt;Elements is a great introduction to proofs, so I'd cherry-pick some examples that will help the student ease into proving theorems. Contemporary students cannot handle this kind of book solely. It should be used more like a snack these days. I believe it is important to remind them when they're actually looking at the Elements, to remind them of the difficulties of such a book, so that they can feel a sense of accomplishment when they do get through portions of it. Elements deserves to be mythologized and should be propped up as if it were a book written by an ancient wizard, IMO.&lt;/p&gt;&#xA;" OwnerUserId="23572" LastActivityDate="2023-12-25T17:31:18.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27310" PostTypeId="2" ParentId="27263" CreationDate="2023-12-26T20:53:50.717" Score="0" Body="&lt;p&gt;[Too long to comment]&lt;/p&gt;&#xA;&lt;p&gt;Agreed with Krisman's comment (need to understand context better).  But with that in mind, some thoughts:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If the target is prospective math teachers (perhaps not all stars, and some weak, but still not completely remedial as in next bullet), then I would design the course more around teaching geometry (there are texts that have this slant).  Perhaps a few intuitions and insights that go beyond a classical US high school text, that might enrich their teaching...but really...not much in that direction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If the target is remedial instruction, and this is then extremely remedial, then I would used something that mimics a normal US high school class.  Perhaps with some packaging, like the title, so that it doesn't appear to be high school.  But high school content in sheep's clothing.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In both cases, I would be very wary of being too enriched or proof-y.  It's a natural instinct of new teachers, emerging from their own research experience. We see it all the time no this forum.&lt;/p&gt;&#xA;&lt;p&gt;specially in #2, be wary.  Simple SAS two column stuff is enough.&lt;/p&gt;&#xA;&lt;p&gt;Either case is not dealing with students that are prospective grad students.  You probably already think you're aiming a notch below that, but take it down another 1-2 notches.&lt;/p&gt;&#xA;" OwnerUserId="23576" LastActivityDate="2023-12-26T20:53:50.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27311" PostTypeId="2" ParentId="27307" CreationDate="2023-12-26T21:06:49.210" Score="6" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Find all applications &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} \to \mathbb{R}$&lt;/span&gt;&#xA;such that (1) is satisfied, posing all the necessary restrictions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From the textbook Calculus 9th edition by Salas, Hille, Etgen (section 7.2):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;From our point of view the fundamental property of logarithms is that they transform multiplication into addition: the log of a product = the sum of the logs. Taking this as the central idea, we are led to a general notion of logarithm [...]&lt;/p&gt;&#xA;&lt;p&gt;DEFINITION 7.2.1 A logarithm function is a nonconstant differentiable function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; defined on a set of positive numbers such that for all &lt;span class=&quot;math-container&quot;&gt;$a &amp;gt; 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b &amp;gt; 0$&lt;/span&gt; , &lt;span class=&quot;math-container&quot;&gt;$f(ab) = f(a) + f(b)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let's assume for the time being that such logarithm functions exist, and let's see what we can find out about them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After some work they get &lt;span class=&quot;math-container&quot;&gt;$f'(x) = \dfrac{1}{x} f'(1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The most natural choice, the one that will keep calculations as simple as possible, is to set &lt;span class=&quot;math-container&quot;&gt;$f'(1)=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;more discussion&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This function, which takes on the value 0 at 1 and has derivative &lt;span class=&quot;math-container&quot;&gt;$1/x$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0$&lt;/span&gt;, must, by Theorem 5.3.5, take the form &lt;span class=&quot;math-container&quot;&gt;$\int_1^x \dfrac{dt}{t}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;DEFINITION 7.2.3 The function &lt;span class=&quot;math-container&quot;&gt;$L(x) = \int_1^x \dfrac{dt}{t}, x &amp;gt; 0$&lt;/span&gt;, is called the &lt;em&gt;(natural) logarithm function&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then pages of developing the properties of &lt;span class=&quot;math-container&quot;&gt;$L(x)$&lt;/span&gt; before defining the exponential function in section 7.4.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit (addendum):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;THEOREM 5.3.5 Let &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; be continuous on &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt;. The function &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; defined on &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$F(x) = \int_a^x f(t) dt$&lt;/span&gt; is continuous on &lt;span class=&quot;math-container&quot;&gt;$[a,b]$&lt;/span&gt;, differentiable on &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; and has derivative &lt;span class=&quot;math-container&quot;&gt;$F'(x) = f(x)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="23575" LastEditorUserId="23575" LastEditDate="2023-12-27T05:46:42.620" LastActivityDate="2023-12-27T05:46:42.620" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27315" PostTypeId="1" AcceptedAnswerId="27317" CreationDate="2023-12-27T11:43:41.687" Score="-4" ViewCount="102" Body="&lt;p&gt;I have a function&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x) = {1 \over 1 + e^{-k(x-0.5)}}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;that plots a logistic curve that is symmetric around the point &lt;span class=&quot;math-container&quot;&gt;$0.5 / 0.5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; defines the steepness of the curve.&lt;/p&gt;&#xA;&lt;p&gt;I would now like to add another parameter (let's call it &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;) that stretches/condenses the curve on the y-axis while it stays symmetric around its midpoint &lt;span class=&quot;math-container&quot;&gt;$0.5 / 0.5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;How/where would I add &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; to this function?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="23581" LastEditorUserId="23581" LastEditDate="2024-02-03T12:48:34.323" LastActivityDate="2024-02-03T12:48:34.323" Title="Condense a logistic function around its midpoint" Tags="|self-learning|linear-algebra|" AnswerCount="1" CommentCount="4" ClosedDate="2023-12-27T16:22:03.540" ContentLicense="CC BY-SA 4.0" />
  <row Id="27317" PostTypeId="2" ParentId="27315" CreationDate="2023-12-27T16:17:13.570" Score="0" Body="&lt;p&gt;Shift it down so it's symmetric to &lt;span class=&quot;math-container&quot;&gt;$y=0,$&lt;/span&gt; then stretch it, then shift it back up so that it's symmetric to &lt;span class=&quot;math-container&quot;&gt;$y=0.5$&lt;/span&gt; again.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x) = c \left( \dfrac{1}{1 + e^{-k(x-0.5)}} - 0.5 \right) + 0.5$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2023-12-27T16:17:13.570" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27320" PostTypeId="2" ParentId="26510" CreationDate="2023-12-28T00:27:22.843" Score="2" Body="&lt;p&gt;This responds to comments in &lt;a href=&quot;https://matheducators.stackexchange.com/a/26513/511&quot;&gt;@AGoodier's answer&lt;/a&gt;.&#xA;@Shay_18 says,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Out of interest, what website do you use to program knots? Most programs or softwares I find online either require money or do not work.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.sagemath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;SageMath&lt;/a&gt; is&#xA;similar to (expensive) Mathematica, but it is free.&#xA;It has substantial knot theory capabilities:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Uoa07.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Uoa07.png&quot; alt=&quot;Sage&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;They just updated the &lt;a href=&quot;https://doc.sagemath.org/html/en/reference/knots/sage/knots/knot.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;knot functions&lt;/a&gt; this month (Dec 2023).&lt;/p&gt;&#xA;" OwnerUserId="511" LastEditorUserId="511" LastEditDate="2023-12-28T00:36:23.073" LastActivityDate="2023-12-28T00:36:23.073" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27323" PostTypeId="1" CreationDate="2023-12-29T13:41:59.993" Score="3" ViewCount="323" Body="&lt;p&gt;I have to do a short presentation using beamer for 25 minutes about a topic.&#xA;In the presentation i want to discuss the following proposition with a lot of conditions&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proposition 2.8.5.8.&lt;/strong&gt;  Let &lt;span class=&quot;math-container&quot;&gt;$\mathscr{C}$&lt;/span&gt; be a monoidal &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;-category.  Assume that &lt;span class=&quot;math-container&quot;&gt;$\mathscr{C}$&lt;/span&gt; admits geometric realizations of simplicial objects and that the tensor product &lt;span class=&quot;math-container&quot;&gt;$\mathscr{C}\times\mathscr{C}\to\mathscr{C}$&lt;/span&gt; preserves geometric realizations.  Let &lt;span class=&quot;math-container&quot;&gt;$\mathscr{M}$&lt;/span&gt; be an &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;-category left-tensored over &lt;span class=&quot;math-container&quot;&gt;$\mathscr{C}$&lt;/span&gt; and let &lt;span class=&quot;math-container&quot;&gt;$M \in \mathscr{M}$&lt;/span&gt; be an object.  There exists an algebra object &lt;span class=&quot;math-container&quot;&gt;$A \in \operatorname{Alg}(\mathscr{C})$&lt;/span&gt; and an equivalence &lt;span class=&quot;math-container&quot;&gt;$\operatorname{RMod}_{A}(\mathscr{C}) \simeq \mathscr{M}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;-categories left-tensored over &lt;span class=&quot;math-container&quot;&gt;$\mathscr{C}$&lt;/span&gt; which carries &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; if and only if the following conditions are satisfied:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;-category &lt;span class=&quot;math-container&quot;&gt;$\mathscr{M}$&lt;/span&gt; admits geometric realizations of simplicial objects.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The action map &lt;span class=&quot;math-container&quot;&gt;$\mathscr{C}\times\mathscr{M} \to \mathscr{M}$&lt;/span&gt; preserves geometric realizations of simplicial objects.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The function &lt;span class=&quot;math-container&quot;&gt;$F:\mathscr{C}\to\mathscr{M}$&lt;/span&gt; given by &lt;span class=&quot;math-container&quot;&gt;$F(C)=C\otimes M$&lt;/span&gt; admits a right adjoint &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The functor &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; preserves geometric realizations of simplicial objects.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The functor &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is conservative.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For every object &lt;span class=&quot;math-container&quot;&gt;$N\in\mathscr{M}$&lt;/span&gt; and every object &lt;span class=&quot;math-container&quot;&gt;$C\in\mathscr{C}$&lt;/span&gt;, the evident map&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ F(C\otimes G(N)) \simeq C\otimes G(N)\otimes M \simeq C\otimes FG(N) \to C\otimes N $$&lt;/span&gt;&#xA;is adjoint to an equivalence &lt;span class=&quot;math-container&quot;&gt;$C \otimes G(N) \to G(C\otimes N)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am not sure if it is better to write out the proposition in full, or instead just write &amp;quot;satisfying some conditions&amp;quot;. Either way i would appreciate some advice&lt;/p&gt;&#xA;&lt;p&gt;Edit:&#xA;To give some further context: this is for a purpose of an interview, so the audience are expected to know the background of the topic well. I do not expect the audience to know the proposition beforehand, but that they have seen similar results, and will find it reasonable if they read the statement.&lt;/p&gt;&#xA;&lt;p&gt;I use the proposition on a specific case which satisfies the criteria to show two objects are the same, which is a vital part of the presentation. The focus on the talk is not on the criteria however.&lt;/p&gt;&#xA;&lt;p&gt;I would say conditions (1),(2),(4) are not very meaningful while (3),(5) are meaningful enough for me to mention, and (6) somewhere in between.&lt;/p&gt;&#xA;" OwnerUserId="23589" LastEditorUserId="8571" LastEditDate="2023-12-30T13:40:35.170" LastActivityDate="2023-12-31T10:29:21.613" Title="How to mention a long theorem in a presentation/beamer" Tags="|presentation|" AnswerCount="2" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="27324" PostTypeId="2" ParentId="26575" CreationDate="2023-12-30T04:02:47.647" Score="2" Body="&lt;p&gt;It is &lt;em&gt;possible&lt;/em&gt; that the argument&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] since if there was a countable basis the real numbers would be countable too, since every real number could be written as a finite linear combination of countably many basis elements over a countable field.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is not correct, for there are models of &lt;span class=&quot;math-container&quot;&gt;$ZF(U)$&lt;/span&gt; that satisfy the existence of a Hamel basis for &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt; (called &lt;em&gt;&lt;strong&gt;FORM 367.&lt;/strong&gt;&lt;/em&gt; in &lt;a href=&quot;https://bookstore.ams.org/surv-59&quot; rel=&quot;nofollow noreferrer&quot;&gt;Howard &amp;amp; Rubin&lt;/a&gt;) while falsifying the general statement&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;FORM 31.&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$UT(\aleph_0, \aleph_0, \aleph_0)$&lt;/span&gt;: The countable union theorem: The union of a denumerable set of denumerable sets is denumerable.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;for example, the models &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}2(n)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}2^*(3)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}2(LO)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}6$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}18$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}22(K, p)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}35$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}41$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}43$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{N}50(E)$&lt;/span&gt;, loc. cit., &lt;em&gt;&lt;strong&gt;but&lt;/strong&gt;&lt;/em&gt; all these, being permutation models, satisfy the more specific&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;FORM 6.&lt;/strong&gt;&lt;/em&gt; &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$UT(\aleph_0, \aleph_0, \aleph_0, \mathbb{R})$&lt;/span&gt;: The union of a denumerable family of denumerable subsets of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; is denumerable.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;so I'm not sure&lt;/p&gt;&#xA;" OwnerUserId="15843" LastEditorUserId="15843" LastEditDate="2023-12-30T05:13:38.183" LastActivityDate="2023-12-30T05:13:38.183" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27325" PostTypeId="2" ParentId="27323" CreationDate="2023-12-30T13:07:01.040" Score="12" Body="&lt;p&gt;Alright, I hope you don't mind if I'm a bit blunt. Given the information in the post we can't know for sure, of course, but it seems unlikely that what you are planning to do will work well.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;You say you want &lt;em&gt;&amp;quot;use the proposition on a specific case which satisfies the criteria to show two objects are the same&amp;quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;How are you going to do this in practice? As your question indicates, you have noted that even the proposition itself hardly fits on a slide. As soon as the slide with the proposition is gone, nobody in the audience will remember the details of the proposition. The precise setting of the proposition, the mathematical objects and structures that occur in the proposition, the assumptions, the six conditions - almost all of it will be gone at the very moment you switch the slide.&lt;/p&gt;&#xA;&lt;p&gt;If you're skeptical, you can test this yourself: ask a friend or colleague who's similarly familiar with the topic as the expected audience to assist you with the following little experiment: Show and explain them the slide with the proposition in the same way as you would to it with your audience, then do the same with the next slide - then stop and ask them to recall all of the proposition that they remember. I promise it will be heart-breaking.&lt;/p&gt;&#xA;&lt;p&gt;I can see one potential way, though, how to make this work: After discussing the proposition (slowly and in detail), you could proceed as follows: on the next slide, copy only the first part of the proposition without the conditions (1)-(6), then add the situation where you want to use the proposition and make sure that there is still sufficient space left at the bottom of the slide. Then go through the conditions (1)-(6) step by step in the following way: always leave the upper two parts of the slide the same, show one of the conditions (1)-(6) (but not the others!), and explain (slowly!) why this condition is satisfied here. Give the audience time to digest, then replace the condition with the next one and iterate.&lt;/p&gt;&#xA;&lt;p&gt;This is something that &lt;em&gt;can&lt;/em&gt; work if you do it really well and if you take enough time to do it. It will, however, use up most of your 25 minutes. For a job interview I would certainly not recommend it, though (even less if you don't have practice in doing this kind of thing in other talks where the stakes are lower). In any case, this strategy only makes sense if showing the application of this proposition is the main point of your talk.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Regarding &lt;em&gt;&amp;quot;I use the proposition on a specific case which satisfies the criteria to show two objects are the same, which is a vital part of the presentation. The focus on the talk is not on the criteria however&amp;quot;&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;While I don't know any details of your planned talk, those two sentences make me worried that you are heavily overestimating how much you can do in 25 minutes. This is a very common problem with mathematical talks: Many mathematicians seem, for reasons that are beyond me, unable or unwilling to apply basic arithmetic when planning a talk. So here's the arithmetic for your proposition:&lt;/p&gt;&#xA;&lt;p&gt;I read out the proposition loud and clearly and measured the time: it took me about 1 minute and 45 seconds - just for reading it aloud, without any explanations or context, without rhetoric pauses, without any rephrasing, without using any additional measures that would give the audience time to think and to process what is being discussed. From experience I can say that adding all those things to reach at least an acceptable level of comprehensibility will more than double the required time - so you'll need at least about four minutes for this single slide (and this is really the minimum - I'd personally plan more time than that for such a slide). So for a 25 minute talk you can, at the very most, have six times the content that is in the proposition (assuming that you want the audience to have a serious chance of really understanding what you are talking about). If this fits with your plan, you might be fine. If not, it shows a problem that you need to solve before giving your talk.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You also write &lt;em&gt;&amp;quot;this is for a purpose of an interview, so the audience are expected to know the background of the topic well.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I obviously don't know your audience, so please take the following with a grain of salt. But another very common mistake in mathematical talks is that the speaker heavily overestimates how familiar the audience is with (the foundations of) the material in the talk. To a certain degree, this even seems to remain true if one is aware of this fact and tries to compensate for it. (I'd say that I'm personally very well aware of this problem, and still I regularly happen to fall prey to this bias.)&lt;/p&gt;&#xA;&lt;p&gt;In the end, it's you who knows the precise context of the talk, but just to be on the safe side, I'd suggest that you reconsider why you assume that the fact that this is for an interview implies that the audience knows the background well. Do you know who precisely is in the audience? And do you know that their field is really so close to yours that they know all the things well which you assume they know well?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I hope this is of some help to you. Best of luck for your interview!&lt;/p&gt;&#xA;" OwnerUserId="16250" LastEditorUserId="16250" LastEditDate="2023-12-31T10:29:21.613" LastActivityDate="2023-12-31T10:29:21.613" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27326" PostTypeId="2" ParentId="27323" CreationDate="2023-12-31T04:30:43.790" Score="7" Body="&lt;p&gt;As others have speculated/commented, attempting explicit enunciation of such a  long &amp;quot;proposition&amp;quot; is almost never the right thing to do, apart from formatting issues: the people who already know it more-or-less are not enlightened, and people who've never seen it before will definitely not assimilate it in a minute or two and be ready to use it.&lt;/p&gt;&#xA;&lt;p&gt;(I do understand the impulse to &amp;quot;explain&amp;quot; how technicalities work out just right... but sometimes it just doesn't fit... I've had my share of ghastly failures in such regard, and never had a good feeling about it.)&lt;/p&gt;&#xA;&lt;p&gt;I'd try thinking of a &lt;em&gt;qualitative&lt;/em&gt; (and shorter) description, even if imprecise, of what that proposition asserts, ... and how happy/fortunate that is for your larger story.&lt;/p&gt;&#xA;" OwnerUserId="63" LastActivityDate="2023-12-31T04:30:43.790" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27327" PostTypeId="2" ParentId="4351" CreationDate="2024-01-01T03:02:26.217" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Does anyone have any simple, natural examples where there is a &amp;quot;unique candidate for a thing satisfying properties&amp;quot;, but that candidate does not actually work?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A bit of a loose(r) answer, but one coud say that &lt;span class=&quot;math-container&quot;&gt;$\aleph_1$&lt;/span&gt; was (is?) the 'unique &lt;em&gt;natural&lt;/em&gt; candidate' for the property of &amp;quot;being the cardinality of &lt;span class=&quot;math-container&quot;&gt;$2^{\aleph_0}$&lt;/span&gt;&amp;quot;, this both from the very naïve standpoint of &amp;quot;the only operation increasing cardinalities is 'taking powers', so it must be that &lt;span class=&quot;math-container&quot;&gt;$\aleph_1 = |2^{\aleph_0}|$&lt;/span&gt;!&amp;quot; and from the more elaborate one of &amp;quot;uncountable projective subsets of reals have cardinality continuum&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="15843" LastActivityDate="2024-01-01T03:02:26.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27328" PostTypeId="1" AcceptedAnswerId="27330" CreationDate="2024-01-01T11:26:17.157" Score="9" ViewCount="567" Body="&lt;p&gt;I'm teaching a class small enough that I'm considering encouraging proof revisions (i.e. students taking a second try on proof based homework problems after getting feedback) for the first time.  I'd love advice from instructors who have tried it about how it went and how your course and the revisions were structured.  How did you award points?  What did you do to balance the increased grading workload?  How did you spread out the grading work? Did you think it was useful?&lt;/p&gt;&#xA;&lt;p&gt;I remember benefiting from a similar class as a student, but I'm not the typical student and as an instructor I can now see some challenges with this approach.  Any experiences (positive or negative) are appreciated.&lt;/p&gt;&#xA;" OwnerUserId="9542" LastEditorUserId="9542" LastEditDate="2024-01-01T17:45:30.877" LastActivityDate="2024-01-04T18:14:46.353" Title="Best practices for Proof Revision/ Proof Portfolio?" Tags="|mathematical-pedagogy|proofs|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27329" PostTypeId="2" ParentId="27328" CreationDate="2024-01-01T19:54:13.707" Score="9" Body="&lt;p&gt;I taught an abstract algebra course this way.&lt;/p&gt;&#xA;&lt;p&gt;Each student made an &lt;a href=&quot;https://www.overleaf.com/&quot; rel=&quot;noreferrer&quot;&gt;overleaf&lt;/a&gt; account which I required to be something like &amp;quot;Firstname-Lastname-MTH358-Spring-2024&amp;quot;.  They shared this folder with me.&lt;/p&gt;&#xA;&lt;p&gt;Each homework assignment had a few conceptual questions, a few computations, and a few proofs.  I would have them complete most of the homework using pencil/paper.  One proof problem per homework was marked as a &amp;quot;portfolio problem&amp;quot;:  they had to type this up on overleaf.&lt;/p&gt;&#xA;&lt;p&gt;I would make comments using the overleaf comment system and send them an email when I was finished commenting on their proof.  They then had one week to revise and email me to check their revision to update their grade.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What did you do to balance the increased grading workload?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I didn't:  I massively overworked myself. Burnout from this kind of experiment was one of the contributing factors which led me to leave the profession.  I wouldn't say that burnout was the primary contributing for me, but it is still something to think about carefully.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Did you think it was useful?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes, I saw a consistent improvement in the mathematical writing of my students throughout the semester.  Some of them thanked me for being &amp;quot;the only professor who actually showed them how to write proofs&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastActivityDate="2024-01-01T19:54:13.707" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27330" PostTypeId="2" ParentId="27328" CreationDate="2024-01-01T23:12:00.527" Score="6" Body="&lt;p&gt;Here's a thing I did stepping in that direction. In a discrete mathematics course, I made use of the discussion forum in the LMS in this way -- require each student weekly to pick a distinct homework problem from that week's section, post an answer to the discussion board, give a specific critique/advice for improvement on another person's post, and edit their initial post incorporating any feedback.&lt;/p&gt;&#xA;&lt;p&gt;So in some sense I was trying to model the activity of reading and writing other people's (possibly substandard) math writing, and also leverage the students to engage in doing the feedback for me.&lt;/p&gt;&#xA;&lt;p&gt;This being on Blackboard, there's a math editor for symbolic expressions, and optionally you can directly write LaTeX for those bits. I required students to learn the minimal LaTeX and write symbolic pieces that way.&lt;/p&gt;&#xA;&lt;p&gt;This all occurred within a week cycle -- e.g., class Mon-Thu, suggested initial post Friday, feedback Saturday, improved solution Sunday. Grading was kept to either 0-1-2 points for both the finalized post-feedback solution, and also the critique to another person's work.&lt;/p&gt;&#xA;&lt;p&gt;With practice I could get a weekly grading cycle done (along with some feedback of my own), for both the solutions &amp;amp; critiques, in about an hour for a class of up to 25 students. (Data shows for that number of registered students, I'd get about 20 submits weekly; taking 2-5 minutes, average 3, per submission.) That's in line with time grading for my other courses, but it felt much more mentally taxing -- instead of one problem I was grading longitudinally, for every submission I had to parse a unique problem and think about whether the solution &amp;amp; feedback made any sense (with assistance from the instructor solution guide), and any feedback from me had to be customized to that problem (e.g., no copy-paste from a FAQ document).&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2024-01-01T23:12:00.527" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27331" PostTypeId="1" CreationDate="2024-01-02T14:50:05.860" Score="6" ViewCount="189" Body="&lt;p&gt;I am currently working on a project to secure funding from our city council for our university. Participants from various academic disciplines are invited to contribute. The primary challenge I'm facing is convincing policymakers and public administrators of the value of investing in mathematics education. While it is generally acknowledged that mathematics is an indispensable part of education, I aim to go beyond this common understanding by presenting evidence-based justifications.&lt;/p&gt;&#xA;&lt;p&gt;In this context, I've observed some movements, notably in certain U.S. states, advocating for the removal of mathematics from school curricula, citing the subject's potential to cause (math anxiety) trauma among students.&lt;/p&gt;&#xA;&lt;p&gt;My question to the community is: Are there any references or studies that discuss the importance of mandating mathematics education for children and adolescents? Specifically, I am looking for research that highlights the cognitive, developmental, and practical benefits of learning mathematics, which could effectively counter the arguments for its removal from school curricula.&lt;/p&gt;&#xA;&lt;p&gt;Any insights or guidance on where to find such studies would be greatly appreciated.&lt;/p&gt;&#xA;&lt;p&gt;there is already a  related question posted in this forum: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/8052/justifications-for-why-learn-mathematics&quot;&gt;Justifications for: Why learn mathematics?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance for your assistance.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2024-01-06T04:23:16.247" Title="are there Evidence-Based Justifications for Mandatory Mathematics Education in School Curricula?" Tags="|value|value-of-mathematics-education|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27332" PostTypeId="2" ParentId="25979" CreationDate="2024-01-02T14:54:53.003" Score="-1" Body="&lt;p&gt;I believe the reason people often omit units in their variable names is rooted in the principle of economy in notation. This principle suggests that when conveying information, especially in scientific and mathematical contexts, one should strive for the most concise and efficient representation. In cases where the units of a variable are clear from the context or are universally understood, adding units to the variable name may be considered redundant. Therefore, to save time and simplify the notation, it's common to choose the shortest, yet still clear, representation. This approach prioritizes ease of writing and reading while maintaining sufficient clarity, provided the context supports it.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2024-01-02T14:54:53.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27333" PostTypeId="1" CreationDate="2024-01-02T20:53:28.527" Score="6" ViewCount="2280" Body="&lt;p&gt;Take this question for an example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A smartphone is now &lt;span class=&quot;math-container&quot;&gt;$\$500$&lt;/span&gt; after a &lt;span class=&quot;math-container&quot;&gt;$20\%$&lt;/span&gt; discount. What was its original price?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, this is an example of a type of math problem that students face usually in 7th or 8th grade&lt;sup&gt;1&lt;/sup&gt;, and usually get the question incorrect due to not being careful.&lt;/p&gt;&#xA;&lt;p&gt;The solution to this specific one would be as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since it is &lt;span class=&quot;math-container&quot;&gt;$500$&lt;/span&gt; dollars after a &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; percent discount, that means that &lt;span class=&quot;math-container&quot;&gt;$500$&lt;/span&gt; dollars is going to be &lt;span class=&quot;math-container&quot;&gt;$80$&lt;/span&gt; percent of the original price. Therefore, we can set up this equation&lt;span class=&quot;math-container&quot;&gt;$$0.8x=500$$&lt;/span&gt;which then we can multiply both sides by &lt;span class=&quot;math-container&quot;&gt;$10/8$&lt;/span&gt; (or &lt;span class=&quot;math-container&quot;&gt;$5/4$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$1.25$&lt;/span&gt;) to get that the original price of the smartphone would be&lt;span class=&quot;math-container&quot;&gt;$$x=\$625$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;While I am not a teacher, I do remember a few specific mistakes that me and my friends would make on these types of problems back in the day:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Using the example question as an example, &lt;span class=&quot;math-container&quot;&gt;$\$500$&lt;/span&gt; after a &lt;span class=&quot;math-container&quot;&gt;$20\%$&lt;/span&gt; discount implies &lt;span class=&quot;math-container&quot;&gt;$0.2x=500$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;When tasked to find the discount, for example a price that was knocked down to &lt;span class=&quot;math-container&quot;&gt;$\$300$&lt;/span&gt; when it was originally &lt;span class=&quot;math-container&quot;&gt;$\$700$&lt;/span&gt;, doing &lt;span class=&quot;math-container&quot;&gt;$300x=700\implies x=\dfrac73\approx233\%$&lt;/span&gt; discount.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;However that is all that I can think of, so my question is, what are other common mistakes that students will make when solving &amp;quot;What's the original price&amp;quot; percentage problems?&lt;/p&gt;&#xA;&lt;p&gt;While I have found a &lt;a href=&quot;https://youtu.be/IYmiFQUsb2Q?si=0LSiwXootaXbLTL9&quot; rel=&quot;noreferrer&quot;&gt;Blackpenredpen video on this&lt;/a&gt;, any other references would be appreciated.&lt;/p&gt;&#xA;&lt;p&gt;I did attempt a few Google searches, but nothing turned up.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt;In America, this is students age 12-13, although it might be different in other places.&lt;/p&gt;&#xA;" OwnerUserId="22983" LastActivityDate="2024-01-03T19:04:35.017" Title="What are common mistakes that students will make when solving &quot;What's the original price&quot; percentage problems?" Tags="|secondary-education|algebra|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27334" PostTypeId="2" ParentId="27333" CreationDate="2024-01-02T21:09:26.557" Score="9" Body="&lt;p&gt;The most common one I think is going from&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The product costs &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; after a &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; percent discount&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;to trying to calculate the original price by&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;original.price &lt;span class=&quot;math-container&quot;&gt;$ = x \times (100+y)/100$&lt;/span&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;It &lt;em&gt;possibly&lt;/em&gt; originates from a certain failure in realising that talking about discounts commonly involves a certain mixture of addition/subtraction &lt;em&gt;and&lt;/em&gt; multiplication/division, and this subsequent mistaken application of 'inverse rules/operations'&lt;/p&gt;&#xA;" OwnerUserId="15843" LastActivityDate="2024-01-02T21:09:26.557" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27335" PostTypeId="2" ParentId="20962" CreationDate="2024-01-03T02:44:13.707" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Response to Students Who Say &amp;quot;This Is Not Important&amp;quot;. Lately, my students keep telling me why what we are learning is not important.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Why not go full on Lakatos-ian, ask them &amp;quot;What is it that you mean by 'important'? Can you try to give a clear cut definition?&amp;quot;. Then you, collectively, try to see what things that are 'intuitively' thought of as 'important' do not fit into the definition, and, dualy, what 'non-important' things do, and refine it&lt;/p&gt;&#xA;&lt;p&gt;After some rounds of refinement, try to explain to them that this activity you've been doing mirrors, to some extent, an important (!) part of mathematical activity, that of precisifying intuitive notions, capturing as many 'essential' examples and barring as many monsters as possible. Tell them we have to start somewhere, and we start with numbers and shapes because, believe it or not, such abstracted things are much less complicated than basically everything 'out there in &amp;quot;real life&amp;quot;/&amp;quot;the real world&amp;quot;'&lt;/p&gt;&#xA;&lt;p&gt;This may be hopelessly naïve, impractical suggestion, that can't be brought to bear in most places, times and situations, but it's something I would really like to see&lt;/p&gt;&#xA;" OwnerUserId="15843" LastActivityDate="2024-01-03T02:44:13.707" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27336" PostTypeId="2" ParentId="27333" CreationDate="2024-01-03T06:59:22.450" Score="7" Body="&lt;p&gt;I agree with ac15's answer (and fedja's comment) describing two specific instances of mistake categories, and I would like to propose a more general framing that explains why students make those mistakes and also suggests other mistakes.&lt;/p&gt;&#xA;&lt;p&gt;The idea is that in general, many students who pattern-match (as opposed to reasoning from first principles) will tend to prefer solutions requiring fewer and simpler operations, especially if those solutions are&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;directionally correct&amp;quot; in the sense that they make a quantity increase or decrease as expected, and&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;order-of-magnitude correct&amp;quot; in the sense that the final result is on a sensible order of magnitude compared to expectations.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;It's essentially the result of applying &lt;a href=&quot;https://en.wikipedia.org/wiki/Occam%27s_razor&quot; rel=&quot;nofollow noreferrer&quot;&gt;Occam's razor&lt;/a&gt; in the case when common sense has granted you some significant but imperfect level of predictive ability. (As ac15 points out in the comments, this might be better characterized as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Principle_of_least_effort&quot; rel=&quot;nofollow noreferrer&quot;&gt;principle of least effort&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Explaining Previously Mentioned Mistake Categories&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;If the price is &lt;span class=&quot;math-container&quot;&gt;$\$500$&lt;/span&gt; after a &lt;span class=&quot;math-container&quot;&gt;$20\%$&lt;/span&gt; discount, then what was the original price?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Well, the original price was obviously higher, and the simplest way to get a sensibly higher number using the quantities given in the problem is to just add &lt;span class=&quot;math-container&quot;&gt;$20\%$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$\$500$&lt;/span&gt; to the original price.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;If a &lt;span class=&quot;math-container&quot;&gt;$\$500$&lt;/span&gt; price is discounted by &lt;span class=&quot;math-container&quot;&gt;$10\%$&lt;/span&gt; today and again by &lt;span class=&quot;math-container&quot;&gt;$20\%$&lt;/span&gt; tomorrow, what is the price tomorrow?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Well, the discounted price is obviously lower, and a simple way to get a sensibly lower number using the quantities given in the problem is just to total up the discounts (&lt;span class=&quot;math-container&quot;&gt;$10\% + 20\% = 30\%$&lt;/span&gt;) and then apply that discount.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Suggesting Another Mistake Category&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A third category of mistake suggested by this framing (that nobody has mentioned yet) is that when applying a discount, you might also see a student divide by the markup multiplier. For instance, if a &lt;span class=&quot;math-container&quot;&gt;$\$500$&lt;/span&gt; price is discounted by &lt;span class=&quot;math-container&quot;&gt;$20\%,$&lt;/span&gt; you might see a student divide &lt;span class=&quot;math-container&quot;&gt;$\$500 \div 1.2.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I used to see this occasionally with students who were first learning about percents. If a student gets used to solving two-step problems about percent increase (write &lt;span class=&quot;math-container&quot;&gt;$1.\textrm{something}$&lt;/span&gt; and then multiply or divide), then it's easy for them to mistakenly apply the same technique to get a directionally-correct, order-of-magnitude correct answer to a percent decrease problem. That said, this mistake tends to get hammered out quicker than the previous ones because the corresponding problems are more basic / foundational and therefore students get way more practice with them.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-01-03T19:04:35.017" LastActivityDate="2024-01-03T19:04:35.017" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27337" PostTypeId="2" ParentId="27333" CreationDate="2024-01-03T14:47:08.280" Score="3" Body="&lt;p&gt;Whatever their mistake, they should learn that working with percentages is &lt;strong&gt;always&lt;/strong&gt; about performing multiplications:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Take &lt;span class=&quot;math-container&quot;&gt;$20\%$&lt;/span&gt; of 500 =&amp;gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{20}{100} \cdot 500 = 0.2 \cdot 500 = 100$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Add &lt;span class=&quot;math-container&quot;&gt;$20\%$&lt;/span&gt; to 500 =&amp;gt; &lt;span class=&quot;math-container&quot;&gt;$(1+\frac{20}{200}) \cdot 500 = 1.2 \cdot 500 = 600$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Subtract &lt;span class=&quot;math-container&quot;&gt;$20\%$&lt;/span&gt; from 500 =&amp;gt; &lt;span class=&quot;math-container&quot;&gt;$(1 - \frac{20}{100}) \cdot 500 = 0.8 \cdot 500 = 400$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This will make it more or less &amp;quot;obvious&amp;quot; to not make the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$20\% + 30\%$&lt;/span&gt;&amp;quot; mistake, as: &lt;span class=&quot;math-container&quot;&gt;$500 + 20\% + 30\% = 500 \cdot 1.2 \cdot 1.3 = 500 \cdot 1.56 = 780$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2024-01-03T14:47:08.280" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27338" PostTypeId="1" CreationDate="2024-01-03T20:58:01.447" Score="4" ViewCount="66" Body="&lt;p&gt;I'd like to design a course with a unit on fair division, and I'm looking for an OER text to use. The course will be a general education seminar, so students won't be expected to have any mathematical background beyond high school. Thus, I'm looking for a very accessible text suited for a beginning learner. I've done some basic searches and found a few resources, but they are hard for me to assess for my purposes, and am hoping someone here can recommend something to me based on their experience. Thank you!&lt;/p&gt;&#xA;" OwnerUserId="21" LastActivityDate="2024-01-03T20:58:01.447" Title="OER text for the mathematics of fair division" Tags="|textbooks|course-design|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27340" PostTypeId="1" AcceptedAnswerId="27341" CreationDate="2024-01-04T01:47:59.413" Score="19" ViewCount="3405" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;: Most textbooks end a section with a set of questions ranked either by topic or by difficulty. A &lt;a href=&quot;https://math.libretexts.org/Courses/College_of_the_Desert/College_of_the_Desert_MATH_011%3A_Math_Concepts_for_Elementary_School_Teachers__Number_Systems/01%3A_Introduction/1.03%3A_Problem_or_Exercise&quot; rel=&quot;noreferrer&quot;&gt;distinction&lt;/a&gt; is often made between &amp;quot;exercises&amp;quot;, which are for directly practicing a known skill, and &amp;quot;problems&amp;quot;, which are generally some kind of application questions where the math expected to be used to answer is not spelled out in the question text. Of course, some books also have sets of questions relating to &lt;em&gt;definitions&lt;/em&gt; or &lt;em&gt;big ideas&lt;/em&gt; or &lt;em&gt;concepts&lt;/em&gt; or &lt;em&gt;challenges&lt;/em&gt;, and other texts use the terms &amp;quot;exercises&amp;quot; and &amp;quot;problems&amp;quot; interchangeably -- though the harder questions generally come after the easier ones. So, the student who carefully reads the headers for the end-of-section questions has an idea whether the questions should be easy or hard(er) to answer.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;: In a setting where students aren't working from a book with these labels on questions, is it worthwhile for the instructor to indicate to students where the work they are asking them to do falls on a scale of difficulty? Do you have experience or know of any research done on the effects (positive or negative) of including such commentary on questions?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Motivation&lt;/strong&gt;: I write and assemble questions for an online college math course and tend to put the easier/basic-definition/single-skill questions first, followed by word &amp;quot;problems&amp;quot; and applications. I have been considering breaking this into two separate assignments per topic (or just in sections) with labels indicating whether (I consider) the problems are basic/foundational and should be learned to the point of automaticity, or if they are harder and will require some problem solving. I am wondering if there are any down sides to a plan like this.&lt;/p&gt;&#xA;" OwnerUserId="470" LastActivityDate="2024-01-05T05:27:33.613" Title="Whether to tell students how difficult (you think) a problem is" Tags="|mathematical-pedagogy|textbooks|problem-design|writing|community-colleges|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27341" PostTypeId="2" ParentId="27340" CreationDate="2024-01-04T06:04:11.803" Score="32" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;In a setting where students aren't working from a book with these labels on questions, is it worthwhile for the instructor to indicate to students where the work they are asking them to do falls on a scale of difficulty?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would say so. One of the things I learned from experience while teaching was that the better I can communicate to students what an intended learning experience should feel like for them, the smoother it will run, and the quicker we can identify any issues that would otherwise slide under the radar. This is especially true in an online course.&lt;/p&gt;&#xA;&lt;p&gt;For instance, one time I assigned about 15 exercises to be completed for homework, with each exercise intended to take a minute or two.  But I didn't explicitly tell this to the students. While most students completed the work just fine, there was one student who became concerned about the workload of the course. Upon further investigation, it turned out that they had a misconception about what was covered during class and spent hours trying to solve the problems in a weird convoluted way. I asked why they didn't just post on the class forum to get help, and they said they didn't realize they were going about it wrong. From that point onward, if I ever assigned exercises that were meant to be quick, I explicitly told the students &amp;quot;if you find yourself working on any of these exercises for more than 5 minutes, and definitely if it's more than 10 minutes, then you've probably got a misconception that is causing you to go about it wrong and you should ask for help so that you don't waste time grinding away unproductively.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Of course, if you tell students not to spend too long on some exercises, you'll also need to give them a heads up when they are expected to spend a longer time on a problem, otherwise they might misunderstand and ask for help too early.&lt;/p&gt;&#xA;&lt;p&gt;Basically, it's good to give students an idea of how long they should wrestle with a problem before asking for help. This will prevent them from wasting their time (grinding away unproductively), and also from wasting your time (asking for help too early).&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-01-05T05:27:33.613" LastActivityDate="2024-01-05T05:27:33.613" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27342" PostTypeId="2" ParentId="20962" CreationDate="2024-01-04T08:03:53.127" Score="1" Body="&lt;h1&gt;Have actual applications in class&lt;/h1&gt;&#xA;&lt;p&gt;Some examples:&lt;/p&gt;&#xA;&lt;p&gt;At ISCED level 1 (elementary school, barneskole, alakoulu), maybe you can have pupils plan and budget for some kind of party or trip. There will be plenty of training on addition, multiplication, what prices mean, being systematic, digital tools (maybe a spreadsheet), gathering data (who are coming, are there allergies or other concerns) and handling uncertainty (someone might not come after all and we might not know how many cookies everyone eats).&lt;/p&gt;&#xA;&lt;p&gt;At ISCED level 3 (high school, lukio, gymnasium, videregående) you might ask whether it is useful to have solar panels on electric cars.&lt;/p&gt;&#xA;&lt;h1&gt;Some related theories&lt;/h1&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Olav Skovsmose has written about landscapes of investigation (undersøgelseslandskaber), where mathematics tasks are classified along two axes: first is whether the task is open or closed, and the second is whether it lies within pure mathematics (what kinds of patterns are there in Pascal's triangle), semi-reality (Kauko is 32 years elder than Teija and next year Kauko will be five times as old as Teija will; how old are they?) or reality (the examples above).&lt;/li&gt;&#xA;&lt;li&gt;Etnomathematics is all about the mathematics that a given group (maybe an ethnicity, or maybe a subculture, a profession or a hobby group) actually uses; the mathematical content, the expressions used, and so on. Often the pupils might do very poorly at school but fluently handle the same calculations in some other context. Bridging the gap and showing how the pupils actually already have lots of competency and do use what can be abstracted as school mathematics in their everyday life might be helpful, and then the abstract mathematics might bleed back in and help in their everyday context, too.&lt;/li&gt;&#xA;&lt;li&gt;Some theories of interdisciplinarity (my translation from Norwegian tverrfaglighet, might not be the jargon actually in use in English) classify things according to whether the non-mathematical discipline is just mentioned (a farmer has 100 meters of fence and wants to fence off as large an area as possible; a straight cliff edge already works as a fence...), up to actual integration of the subject matters, like in the examples above.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;All of these have the shared idea of not just touching upon some real-life context, but actually using it in a way natural to it, and seeing the mathematics there.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2024-01-04T08:03:53.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27343" PostTypeId="2" ParentId="27340" CreationDate="2024-01-04T13:24:17.800" Score="4" Body="&lt;p&gt;That depends on the student level and on the objective of problem solving. For rookies just learning elementary skills, my answer would be &amp;quot;definitely yes&amp;quot;. For graduate students taking a topics course with a view at some potential research in the area later, my answer would be &amp;quot;under no circumstances&amp;quot;. In the intermediate cases I would say &amp;quot;to some extent, but not entirely&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The reason is that indicating the difficulty level creates a prejudice in the student mind about his/her ability to solve the problem or, at the very least, about the amount of effort needed for it. Such prejudice by itself is useful and can save one's day when getting exposed to some unfamiliar environment (just like any other prejudice in real life) but becomes a hindrance when you navigate the ground with (justified!) confidence.&lt;/p&gt;&#xA;&lt;p&gt;Just my two cents :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2024-01-04T13:24:17.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27344" PostTypeId="2" ParentId="27328" CreationDate="2024-01-04T18:14:46.353" Score="2" Body="&lt;p&gt;I've done this multiple times for our intro to proofs class. I used to do it all with LaTeX files, etc. but, like the others, have moved to using Overleaf. (I just have them use link sharing to avoid having to pay for a premium account.) I could say a lot about this (in fact I gave a talk at JMM in 2016 about it, &lt;a href=&quot;https://course1.winona.edu/eerrthum/JointTalk16.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;slides here&lt;/a&gt;.) For brevity I will try to answer your specific questions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;advice from instructors who have tried it about how it went&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It went great! Do it! I'm doing it again this semester.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;how your course and the revisions were structured... How did you award points?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See the aforementioned slides. Basically problems were reserved by students, submitted solutions were hashed over in class, and then either deemed accepted or needed to be revised. I award regular points for work base on a rubric but I also give &amp;quot;reputation&amp;quot; points for good answers and participation in the critiquing process. I got the idea from... Stack Exchange! More details in &lt;a href=&quot;http://course1.winona.edu/eerrthum/24Spring/24SpringMath327.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;my syllabus&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What did you do to balance the increased grading workload? How did you spread out the grading work?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This was the main key to this: Students do far fewer exercises, perhaps as few as one or none in a given section. But it's quality over quantity. Since this course is primarily focused on proof writing, I feel like, if you can write one really good proof in a given concept, that's enough for me to believe you know how to write a proof with those ideas. Anything beyond that is just learning the mathematics concept better as opposed to learning about the process of writing a proof. (Admittedly, this part would need to be altered in a course that actually is about content.) Most of the grading is done live in front of the class so everyone can learn from it. So the extra work is just the bookkeeping, compiling, prepping, etc.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Did you think it was useful?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes. The students who knew what they were doing did great. The students who struggled were given the chance to do it until it was right and significantly improved over the semester. In fact, it was the success with revisions in this class that led me to embrace standards-based grading in other courses.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2024-01-04T18:14:46.353" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27345" PostTypeId="2" ParentId="27340" CreationDate="2024-01-04T19:14:09.143" Score="7" Body="&lt;p&gt;Donald Knuth certainly thought so: he graded the exercises in his monumental work &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Art_of_Computer_Programming&quot; rel=&quot;noreferrer&quot;&gt;The Art Of Computer Programming&lt;/a&gt; from &lt;em&gt;00&lt;/em&gt; ‘An extremely easy exercise that can be answered immediately if the material of the text has been understood’ up to &lt;em&gt;50&lt;/em&gt; ‘A research problem that has not yet been solved satisfactorily’.&lt;/p&gt;&#xA;&lt;p&gt;To illustrate, the section explaining this has its own set of exercises:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;[&lt;em&gt;00&lt;/em&gt;] What does the rating “&lt;em&gt;M20&lt;/em&gt;” mean?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[&lt;em&gt;10&lt;/em&gt;] Of what value can the exercises in a textbook be to the reader?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[&lt;em&gt;14&lt;/em&gt;] Prove that 13³ = 2197. Generalize your answer. [This is an example of a horrible kind of problem that the author has tried to&#xA;avoid.]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[&lt;em&gt;HM45&lt;/em&gt;] Prove that when &lt;em&gt;n&lt;/em&gt; is an integer, &lt;em&gt;n&lt;/em&gt; &amp;gt; 2, the equation &lt;em&gt;xⁿ&lt;/em&gt; + &lt;em&gt;yⁿ&lt;/em&gt; = &lt;em&gt;zⁿ&lt;/em&gt; has no solution in positive integers &lt;em&gt;x, y, z&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Note that that &lt;a href=&quot;https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem&quot; rel=&quot;noreferrer&quot;&gt;last exercise&lt;/a&gt; hadn't yet been proved when that first volume was originally published…  Interestingly, it has two problems rated even harder than that!)&lt;/p&gt;&#xA;&lt;p&gt;If you're going to include problems with a range of difficulty even a fraction of that, then I think it's only fair to your students to give them some indication, so they won't give up too easily on foundational exercises, nor waste too much time and get discouraged on really advanced problems.&lt;/p&gt;&#xA;&lt;p&gt;After all, they won't all be as capable as &lt;a href=&quot;https://en.wikipedia.org/wiki/George_Dantzig&quot; rel=&quot;noreferrer&quot;&gt;George Dantzig&lt;/a&gt;…  (Who arrived late to a statistics lecture and, not realising that blackboard was showing two famous unsolved problems, took them for a homework assignment and handed in solutions a few days later!)&lt;/p&gt;&#xA;" OwnerUserId="12762" LastActivityDate="2024-01-04T19:14:09.143" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27346" PostTypeId="2" ParentId="27340" CreationDate="2024-01-04T21:24:39.060" Score="8" Body="&lt;h3&gt;A Frame Challenge&lt;/h3&gt;&#xA;&lt;p&gt;I think that the premise of the question is slightly flawed:  I do not think that the distinction between &amp;quot;exercises&amp;quot; and &amp;quot;problems&amp;quot; is one of difficulty—an exercise is not &amp;quot;easy&amp;quot; and an problem &amp;quot;hard&amp;quot;.  Instead, the distinction between the two is &lt;em&gt;qualitative&lt;/em&gt;:  exercises are meant to drill, while problems are meant to provoke thinking.  This is even pointed out in the &lt;a href=&quot;https://math.libretexts.org/Courses/College_of_the_Desert/College_of_the_Desert_MATH_011%3A_Math_Concepts_for_Elementary_School_Teachers__Number_Systems/01%3A_Introduction/1.03%3A_Problem_or_Exercise&quot; rel=&quot;noreferrer&quot;&gt;websited linked in the question&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is entirely appropriate to describe this distinction to students.  When a student is presented with an &amp;quot;exercise&amp;quot;, they should expect that they should have a good sense of &lt;em&gt;how&lt;/em&gt; to solve the problem from the start—the expectation is that an exercise will be an application of skills already learned.  This does &lt;em&gt;not&lt;/em&gt; mean that the exercise will be &amp;quot;easy&amp;quot;.  It might involve some tedious computation, or a subtle use of some technique already learned (e.g. I could come up with a very difficult integration by parts exercise by building a function which requires two applications of IBP, with the correct choice of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;&amp;quot; and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;&amp;quot; not being immediately obvious from the start—but such a problem would still be intended as practice with IBP).&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, if a student is presented with a &amp;quot;problem&amp;quot;, the expectation should be that the difficulty of the problem is not the mathematical concept, but the process of getting to the point where whatever techniques have been learned in the class can actually be applied.  Problems often require lateral thinking, translation from English to mathematics (e.g. &amp;quot;application&amp;quot; problems), or the integration of multiple ideas which have been taught in the class.  A problem that I often give to my precalculus students, for example, is something like the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\ell_1$&lt;/span&gt; be the line given by &lt;span class=&quot;math-container&quot;&gt;$y = ax + b$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\ell_2$&lt;/span&gt; be the line given by &lt;span class=&quot;math-container&quot;&gt;$y = -\frac{1}{a} x + c$&lt;/span&gt;; note that these lines are perpendicular.  Let &lt;span class=&quot;math-container&quot;&gt;$P = (d,e)$&lt;/span&gt; (with &lt;span class=&quot;math-container&quot;&gt;$(d,e)$&lt;/span&gt; chosen so that &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is on one of the two given lines), and let &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; denote the point where &lt;span class=&quot;math-container&quot;&gt;$\ell_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\ell_2$&lt;/span&gt; intersect.  Find two points &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$OPQR$&lt;/span&gt; is a square.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this problem, the constants (&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;) are chosen so that the arithmetic is pretty simple.  This is not a &amp;quot;hard&amp;quot; problem once you see how to set it up (by this point in the semester, they have seen vectors in plane, so the computation ends up being a little bit of subtraction to obtain the vector &lt;span class=&quot;math-container&quot;&gt;$\vec{v} = P-O$&lt;/span&gt;, swap the coordinates of &lt;span class=&quot;math-container&quot;&gt;$\vec{v}$&lt;/span&gt; and make one of them negative to get &lt;span class=&quot;math-container&quot;&gt;$\vec{v}_{\perp}$&lt;/span&gt;, then compute &lt;span class=&quot;math-container&quot;&gt;$P+\vec{v}_{\perp}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$O+\vec{v}_{\perp}$&lt;/span&gt;).  This is not a hard problem, but it does require some work.&lt;/p&gt;&#xA;&lt;h3&gt;NEVER Call a Problem Easy&lt;/h3&gt;&#xA;&lt;p&gt;Contrary to the other answers, I also think that it is a Bad Idea™ to make an attempt to describe the difficulty level of a problem, particularly if &amp;quot;easy&amp;quot; or &amp;quot;simple&amp;quot; or &amp;quot;routine&amp;quot; or &amp;quot;trivial&amp;quot; or anything like that is one of the possible difficulty levels.&lt;/p&gt;&#xA;&lt;p&gt;The problem is that you will often have students who get tripped up by problems which you think are simple.  Or you think that you have presented the students with a simple problem, and it turns out not to be that simple (for example, after introducing students to the change of variables formula (&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;-substitution), I asked them to integrate something which involved a trigonometric substitution (in what I thought was a relatively simple and direct way)—I thought it was a very easy problem—but students lacked some of the necessary knowledge about trigonometric functions, and a large portion of the class thought the problem was very, very hard).  If you call a problem &amp;quot;easy&amp;quot; and a student struggles with it, they are going to disengage.&lt;/p&gt;&#xA;&lt;p&gt;I &lt;em&gt;do&lt;/em&gt; think it is fair to give some estimate about how &lt;em&gt;long&lt;/em&gt; you think it will take to get through a particular set of problems, and make it clear that if a student significantly exceeds that time, then they should talk to you (since it is possible that they are simply approaching the problems incorrectly, or that they are missing some prerequisite knowledge).&lt;/p&gt;&#xA;&lt;h3&gt;My Own Practice&lt;/h3&gt;&#xA;&lt;p&gt;In my own classes, I classify problems into three rough categories, with the categorizations related to &lt;a href=&quot;https://cft.vanderbilt.edu/guides-sub-pages/blooms-taxonomy/&quot; rel=&quot;noreferrer&quot;&gt;Bloom's Taxonomy&lt;/a&gt;.  From my syllabi:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;[R] (recall; remember, understand, explain): Demonstrating R-level proficiency typically requires you to recall a definition, complete a short computation, or provide an example.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[A] (analyze; apply, evaluate, connect): Demonstrating A-level proficiency typically requires you to complete a more complicated computation, apply a definition or result to solve a problem, or use mathematical tools to solve a “real world” problem.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[S] (synthesize; create): Demonstrating S-level proficiency typically requires you to combine several ideas in order to solve a single problem—this could include writing a proof, solving more complicated “real world” problems, or using multiple tools (computers, theorems, etc) in order to generate results.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the language of &amp;quot;problems&amp;quot; and &amp;quot;exercises&amp;quot;, R questions are &amp;quot;exercises&amp;quot;, S questions are &amp;quot;problems&amp;quot;, and A questions are a mixture of both.  But note that these levels are not a about *difficulty—the classifications are based on the kinds of thinking that students are expected to have to utilize in order to finish the problems.  In general, R questions should probably be easy, and S questions might be nearly impossible (and I &lt;em&gt;do&lt;/em&gt; set the expectation with students that they probably won't be able to solve S problems without help), but these are not hard-and-fast rules.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastEditorUserId="8571" LastEditDate="2024-01-04T22:43:08.787" LastActivityDate="2024-01-04T22:43:08.787" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27347" PostTypeId="2" ParentId="27223" CreationDate="2024-01-06T01:30:19.413" Score="-1" Body="&lt;p&gt;Departmental Fund Raiser!&lt;/p&gt;&#xA;&lt;p&gt;Charge a small fee, maybe $2, to borrow a calculator during the exam.&lt;/p&gt;&#xA;&lt;p&gt;This is my suggestion. We've been loaning them out for free for a couple years, but it occurred to me we're wasting an opportunity to capitalize on the situation. You could even offer a subscription option for frequent forgetters. Much to think about here.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2024-01-06T01:30:19.413" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27348" PostTypeId="2" ParentId="27331" CreationDate="2024-01-06T04:23:16.247" Score="1" Body="&lt;p&gt;I am not sure about using the adjective &amp;quot;mandatory&amp;quot; just for mathematics, as most other subjects that are in school curricula are also mandatory. But you can refer to findings in the book titled:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1786340682&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Does Mathematical Study Develop Logical Thinking?&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://ro.uow.edu.au/cgi/viewcontent.cgi?referer=&amp;amp;httpsredir=1&amp;amp;article=1480&amp;amp;context=eispapers1#:%7E:text=This%20book%20is%20recommended.,is%20clear%20and%20well%20written.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; and &lt;a href=&quot;https://maa.org/press/maa-reviews/does-mathematical-study-develop-logical-thinking-testing-the-theory-of-formal-discipline&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; are two different reviews of this book, if you want to know what is in the book.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastActivityDate="2024-01-06T04:23:16.247" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27349" PostTypeId="1" CreationDate="2024-01-06T15:40:08.403" Score="2" ViewCount="977" Body="&lt;p&gt;When I was a grad student, I taught a Calculus I course during the summer. Somehow I came across Robert Moore and I ended up reading his book which I connected with. Rather than being a great student, I learned slowly but deeply—a lot of experimentation because I do not have a good memory.&lt;/p&gt;&#xA;&lt;p&gt;I tried that summer course  using the Moore Method and it was semi-successful. Most of the students I could see were willing to try to cooperate. Now in high school it seems the Moore Method can morph into a project-orientated program using Desmos to experiment and create effectively &amp;quot;products&amp;quot; for the real-world. One difference (and also the problem) between college and high school seems to be this mandated passing of the state test.&lt;/p&gt;&#xA;&lt;p&gt;Do you think with the advent of Desmos/GeoGebra, the Moore Method is more accessible to high school?&lt;/p&gt;&#xA;&lt;p&gt;But of course this requires teachers to be well-versed in programming with Desmos.&lt;/p&gt;&#xA;" OwnerUserId="23584" LastEditorUserId="17865" LastEditDate="2024-01-14T15:51:11.190" LastActivityDate="2024-01-14T15:51:11.190" Title="Do you think with the advent of Desmos/GeoGebra, the Moore Method is more accessible to high school?" Tags="|secondary-education|teaching|projects|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27350" PostTypeId="2" ParentId="27349" CreationDate="2024-01-06T16:18:06.360" Score="10" Body="&lt;p&gt;It seems like there are several questions here. I'll try to untangle them and address each one individually.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Do online graphing/visualization tools like Desmos and GeoGebra make projects more accessible?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yeah, I would say so. There's a quick feedback loop, and it can kind of feel like a video game where your joystick is a mathematical expression. These tools can make projects fast &amp;amp; fun.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Can you teach a math class on the basis of projects only, as opposed to more traditional exercises?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Not really. Sure, projects can be useful for pulling a lot of skills together to do something cool and exciting. That's valuable. But the thing is, students need to acquire the component skills first. Most student's can't learn component skills on the fly in the context of a bigger project -- they need plenty of practice on each particular skill, where lots of scaffolding is provided at the beginning and then gradually removed.&lt;/p&gt;&#xA;&lt;p&gt;For example, let's say you want to do a project where students sketch an image by graphing a bunch of straight lines in Desmos. Obviously, going into the project, students are going to need to know how to represent points in the coordinate plane and find the equation of the line between two points. They should also know how to manually create the graph of a given linear equation (otherwise they're not going to understand what Desmos is actually doing for them, and they won't be able to debug their equations if they run into issues). They'll need to be comfortable different formats of equations of lines, including &lt;span class=&quot;math-container&quot;&gt;$x=\textrm{constant},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$y=\textrm{constant},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$y=mx + b,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$y-h = m(x-k).$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are tons of component skills here and you can't teach this all in the context of the project. Each of those component skills has to be built up beforehand. And it takes plenty of time and practice to build them up. Most students are going to need a worked example and a handful of practice problems (with feedback) on each separate case of each component skill (separate cases involving positive numbers, negative numbers, fractions, decimals, etc.). If they don't get this practice, then they're going to be totally overwhelmed by the project, spend a ton of time working unproductively, and ultimately learn little to nothing from it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Can you run a successful high school class using the Moore Method?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Regardless of the tools at your disposal (Desmos, GeoGebra, etc.), a successful implementation of the Moore Method will require extremely motivated students who really enjoy intense, effortful thinking about math and are bright enough to construct a subject from the ground up with minimal guidance. Needless to say, this is typically a tiny proportion of students, which becomes vanishingly small as you climb down the ladder from graduate --&amp;gt; undergraduate --&amp;gt; high school --&amp;gt; middle school --&amp;gt; elementary school math courses.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I spent several years teaching at a highly advanced, opt-in math program where students entered in 6th grade (generally starting at prealgebra), took AP Calculus BC in 8th grade (with many students earning 5's), and studied undergraduate math during high school. I worked with something like 100 students across that program but can count on a single hand those students who I think would have been able to succeed with something like the Moore Method. Even in the highest courses within that program, which filtered down to the most motivated and capable students, I would estimate that the proportion capable of succeeding with the Moore Method was something like 10% -- and this an incredibly optimistically biased subsample, within an already incredibly optimistically biased sample.&lt;/p&gt;&#xA;&lt;p&gt;So while I suppose I can't claim that it's impossible in theory for a Moore Method course to be successful at the high school level, I think the likelihood of having a group of students for which the Moore Method could work (leaving students not only with a decent grade in the class, but also the ability to solve actual problems that are standard for the subject, and not just the simplest cases) is so vanishingly small as to be effectively impossible in practice.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-01-06T22:31:28.770" LastActivityDate="2024-01-06T22:31:28.770" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="27351" PostTypeId="1" CreationDate="2024-01-06T16:39:20.517" Score="0" ViewCount="128" Body="&lt;p&gt;Do you think having students turn in homework in &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; say for extra credit is helpful for them at the college level? At the high school level? There could also be a few assignments that have to be in &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="23584" LastActivityDate="2024-01-07T12:10:43.183" Title="In what ways could or couldn't having students do homework in $\LaTeX$ for extra credit be helpful for them at the college level? High school level?" Tags="|secondary-education|homework|latex|" AnswerCount="2" CommentCount="2" ClosedDate="2024-01-07T12:10:08.937" ContentLicense="CC BY-SA 4.0" />
  <row Id="27352" PostTypeId="2" ParentId="27351" CreationDate="2024-01-06T16:39:20.517" Score="0" Body="&lt;p&gt;In the Age of Information, but before the Age of AI, it seems like it would be very helpful for the students--especially since we have Overleaf and don't have to download a whole editor. Learning a neat type-setting language, maybe even some programming that could be helpful for future employment. Having a neat organized solution to a problem to read. The only problem is they can use ChatGPT, and while it is not great at math, it is great at programming, so it seems code can be copied. Though, prompting the &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; code would still be non-trivial as you must type in your solution for ChatGPT and there will still be several follow-ups.&lt;/p&gt;&#xA;" OwnerUserId="23584" LastActivityDate="2024-01-06T16:39:20.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27353" PostTypeId="1" CreationDate="2024-01-06T17:26:09.330" Score="0" ViewCount="362" Body="&lt;p&gt;The Poincaré conjecture is one of the millennium problems which have been solved by the famous mathematician Grigori Perelman . As per I know he worked on the problem in isolation for 8 years. I wanted to know from where did Perelman got the study materials and other necessary theory needed to prove the conjectures as he was not connected to any university or institution. So from where did he get the work of Richard Hamilton and all the other advanced knowledge which a can only be gathered from college and money paid journals . Thank you.&lt;/p&gt;&#xA;" OwnerUserId="23368" LastActivityDate="2024-01-06T19:55:13.227" Title="How did Grigori Perelman , got all the necessary theory and materials to prove the Poincaré conjecture?" Tags="|reference-request|mathematician|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27354" PostTypeId="2" ParentId="27353" CreationDate="2024-01-06T19:55:13.227" Score="7" Body="&lt;p&gt;Perelman had a conventional academic education through and well past his Ph.D., albeit in Russia.  He was exposed to and was a part of the mainstream of research mathematics.  He was not a Ramanujan, with very limited exposure to hard core math research.&lt;/p&gt;&#xA;&lt;p&gt;Note, that all of this, you could have easily learned by reading the &amp;quot;Early Life and Education&amp;quot; section of the Wikipedia article on Perelman.  If you want to go out and make some amazing discoveries, you will need to learn the resourcefulness to at least (!) Google.  Not just dream about things and ask questions on forums.&lt;/p&gt;&#xA;" OwnerUserId="23609" LastActivityDate="2024-01-06T19:55:13.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27355" PostTypeId="2" ParentId="27351" CreationDate="2024-01-06T20:10:45.047" Score="5" Body="&lt;p&gt;I would say that one should &lt;em&gt;allow&lt;/em&gt; submitting homework, etc. in &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt;, perhaps, one can even go as far as encouraging it, but &lt;em&gt;demanding&lt;/em&gt; it will be way overboard.&lt;/p&gt;&#xA;&lt;p&gt;In my humble opinion, if your subject and class format do not strictly require working on a computer or knowing how to program, then there is no reason why you should assume that the students have either the equipment, or the necessary skills. If they can handle pen and paper, they should be already qualified to get into such a class assuming they have completed all necessary prerequisites. The new technology should &lt;em&gt;give new opportunities&lt;/em&gt;, not &lt;em&gt;enforce new restrictions&lt;/em&gt;. Whether to use those opportunities or not should be left to the user to choose.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2024-01-06T20:10:45.047" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27356" PostTypeId="1" CreationDate="2024-01-07T01:49:22.923" Score="0" ViewCount="85" Body="&lt;p&gt;I have learned some useful tools/books to visualize some MATH concepts:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/088385757X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Visual Group Theory (MAA Problem Book Series) 1st Edition&lt;/a&gt;. The paired website with this book is &lt;a href=&quot;https://nathancarter.github.io/group-explorer/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;GroupExplorer&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0198534469&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Visual Complex Analysis New Ed Edition&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cdots$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I'm Looking for more useful resources about visualization in math. Any supplementation would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="21068" LastActivityDate="2024-01-07T01:49:22.923" Title="Looking for resources about visualizing in mathematics" Tags="|textbooks|visual-proof|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27358" PostTypeId="2" ParentId="26432" CreationDate="2024-01-08T04:12:15.253" Score="-1" Body="&lt;p&gt;A dictionary like Cambridge or Merriam-Webster defines the word &amp;quot;cancel&amp;quot;.  Why not use the word &amp;quot;vaporize&amp;quot; instead of &amp;quot;cancel&amp;quot;?  At least vaporize would be interesting and closer to what truly happens in the operation of removing common factors in a fraction.&lt;/p&gt;&#xA;" OwnerUserId="23616" LastActivityDate="2024-01-08T04:12:15.253" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27359" PostTypeId="2" ParentId="19164" CreationDate="2024-01-08T07:52:36.673" Score="0" Body="&lt;p&gt;'Bout the most important thing you take away from your K-12 years is the ability to learn new things and to learn them quickly.&lt;/p&gt;&#xA;&lt;p&gt;As far as I can tell, math is the &amp;quot;gold standard&amp;quot; for teaching us to learn and to learn quickly. Music and foreign language study will do this for some of us, but, by and large, math is king when it comes to developing a knack for learning.&lt;/p&gt;&#xA;&lt;p&gt;Some of us will actually use calculus and above in our lives, but most won't. We all, however, benefit from developing and exercising our &amp;quot;learning muscles!&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="23617" LastActivityDate="2024-01-08T07:52:36.673" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27360" PostTypeId="2" ParentId="15366" CreationDate="2024-01-08T09:33:23.973" Score="0" Body="&lt;p&gt;In writing &lt;span class=&quot;math-container&quot;&gt;$f(z)=z^{5x+1}$&lt;/span&gt;, the student is (incorrectly) taking &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to be a constant.&lt;/p&gt;&#xA;&lt;p&gt;Now, given also &lt;span class=&quot;math-container&quot;&gt;$g(z)=3$&lt;/span&gt; (for all &lt;span class=&quot;math-container&quot;&gt;$z \in \mathbb R$&lt;/span&gt;), we have &lt;span class=&quot;math-container&quot;&gt;$f(g(z))=f(3)=3^{5x+1}$&lt;/span&gt; (where again &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a constant).&lt;/p&gt;&#xA;&lt;p&gt;Observe that given a small change in &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(g(a))$&lt;/span&gt; doesn't change. That is, &lt;span class=&quot;math-container&quot;&gt;$(fg)^\prime(a)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;And this is indeed what we get when we apply the Chain Rule:&lt;/p&gt;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$f^\prime(z)=(5x+1)z^{5x}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g^\prime(z)=0$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$$(fg)^\prime(z)=f^\prime(g(z))g^\prime(z)=(5x+1)[g(z)]^{5x}\times0=0.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="23536" LastEditorUserId="23536" LastEditDate="2024-01-09T00:49:29.003" LastActivityDate="2024-01-09T00:49:29.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27361" PostTypeId="2" ParentId="15366" CreationDate="2024-01-08T16:33:52.360" Score="1" Body="&lt;p&gt;This answer may not be a lot different from the other answers, but here is how I would phrase it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When we use the chain rule to compute &lt;span class=&quot;math-container&quot;&gt;$\tfrac{d}{dx} f(g(x))$&lt;/span&gt;, the dependence on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; should only be through the input &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;. In your function, &lt;span class=&quot;math-container&quot;&gt;$f(z) = z^{5x+1}$&lt;/span&gt;, the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; depends on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; not only through &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;, but also through the &lt;span class=&quot;math-container&quot;&gt;$5x+1$&lt;/span&gt; in the exponent. It would be clearer to write it as &lt;span class=&quot;math-container&quot;&gt;$f(x,z)$&lt;/span&gt;, so this issue would be clear. When you use the chain rule, your &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; should be big enough to include everything that depends on &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; inside it.&lt;/p&gt;&#xA;&lt;p&gt;(Optional second paragraph) If you need to compute &lt;span class=&quot;math-container&quot;&gt;$\tfrac{d}{dx} f(x,g(x))$&lt;/span&gt;, there is a formula for that: &lt;span class=&quot;math-container&quot;&gt;$$\frac{d}{dx} f(x,g(x)) = \frac{\partial f}{\partial x} (x,g(x)) +\frac{dg}{dx}(x) \cdot \frac{\partial f}{\partial y} (x,g(x)).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="51" LastActivityDate="2024-01-08T16:33:52.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27362" PostTypeId="2" ParentId="26432" CreationDate="2024-01-08T16:50:39.050" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;In addition to the other pro using &amp;quot;cancel&amp;quot; answers, it's important to realize that there's a vast amount of dimensional analysis (&amp;quot;railroad track problems&amp;quot;) in science and engineering classes.  Especially and extremely prominent in college or even high school &lt;strong&gt;general chemistry&lt;/strong&gt; classes (stoichiometry, gas laws, etc.)  &amp;quot;Cancel&amp;quot; is a completely normal term in those contexts.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;E.g. &lt;a href=&quot;https://chem.libretexts.org/Courses/Grand_Rapids_Community_College/CHM_110%3A_Chemistry_of_the_Modern_World_(Neils)/2%3A_Measurements/2.3_Dimensional_Analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://chem.libretexts.org/Courses/Grand_Rapids_Community_College/CHM_110%3A_Chemistry_of_the_Modern_World_(Neils)/2%3A_Measurements/2.3_Dimensional_Analysis&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;I think there's a tendency of the &amp;quot;humans as computers&amp;quot; thinking to propose that errors (sign errors, linearity, transcribing, etc.) can be prevented by alternate terminology.  This is generally NOT the case.  It's not how the human brain works.  Drill and repetition and rapid feedback are required to learn things, whether it is math procedures or rebuilding an engine.  Those errors will come out, especially with new or weak (or new AND weak) students, regardless of instruction.  In fact, making the language too formal and precise can overcomplicate things for new learners.  It's a balance, sure, and I'm not &amp;quot;Every Which Way but Loose&amp;quot; on instructions.  But in general, the errors students come up with are not so much instruction errors, but natural (human, flesh brain) errors, which need to be drilled out.  So thinking you can preempt them is naive.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One term that does bug me is &amp;quot;flip to other side of equation and change sign&amp;quot;, when used for new learners of solving linear equations of x.  This is a profound beginning stage of algebra.  And at that time, it is better to be very clear to the students, that the thing about an equality*, is that you can do same thing to both sides.  So, don't &amp;quot;flip to other side of equation and change sign&amp;quot;, but subtract the quantity from both sides and then cancel (addition cancel).  This is not a problem say for 11th grade physics students, who have extensive experience in algebraic manipulation.  Fine to &amp;quot;flip to other side of equation and change sign&amp;quot; for them.  But for 7th or 8th graders, first &amp;quot;doing math letters&amp;quot;, don't be so cavalier.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;*Later on, in a precalculus class, they will learn about how to manipulate inequalities.&lt;/p&gt;&#xA;" OwnerUserId="23609" LastActivityDate="2024-01-08T16:50:39.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27363" PostTypeId="1" CreationDate="2024-01-09T01:13:58.893" Score="5" ViewCount="267" Body="&lt;p&gt;I don't know if this is the best stack exchange for this question, but I couldn't think of a better one. Please migrate it to an appropriate stack exchange, if this one is not appropriate. Anyway, my question is, are some human languages better for expressing mathematical concepts than other languages? I am asking because it is my belief that German is better than English for expressing mathematical concepts and doing mathematics. Even if that is not true, I am certain that not all languages are equally suited for every single purpose. Also, has anyone in the academic literature discussed this question? I would like some references.&lt;/p&gt;&#xA;" OwnerUserId="23622" LastActivityDate="2024-01-10T17:22:45.277" Title="Are some natural languages better for expressing mathematical concepts than others?" Tags="|reference-request|language-use|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27364" PostTypeId="2" ParentId="27363" CreationDate="2024-01-09T07:47:37.217" Score="1" Body="&lt;p&gt;Let me give you an example in two languages: English (Dutch), and let's talk about tables (tafels [ˈtafəls]) and tables (tabellen [taˈbɛllən]):&lt;br /&gt;&#xA;A table (tafel) is the piece of furniture at which you can sit and eat and pile stuff on.&lt;br /&gt;&#xA;In primary school, you learn about multiplication tables (tafels van vermenigvuldiging), and apparently, children have no problems accepting that the word &amp;quot;table (tafel)&amp;quot; is used for this purpose.&lt;br /&gt;&#xA;Later on, you learn about tables (tabellen) as grids on a piece of paper, containing information, grouped by columns and rows.&lt;br /&gt;&#xA;In fact, the tables of multiplication (tafels van vermenigvuldiging) are not actual tables (tafels) but more like tables (tabellen), so that's a very fine nuance, getting &amp;quot;lost in translation&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2024-01-09T07:47:37.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27365" PostTypeId="2" ParentId="27363" CreationDate="2024-01-09T08:39:54.977" Score="1" Body="&lt;p&gt;There is mathematics and then there is mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Languages do presumably affect the speed at which children learn basic number words.&lt;/p&gt;&#xA;&lt;p&gt;It is known that children learn Danish (in general, not specific to mathematics) slower than Norwegian and Swedish while very young, but then catch up and soon there is no difference. I also just now read a book about mathematical difficulties in the Norwegian context, which specifically names the &amp;quot;-ten -numbers&amp;quot;; tretten (13), fjorten (14), and so on, up to nitten (19). Children might instead write these as for example 31, since the three (tre) is named first.&lt;/p&gt;&#xA;&lt;p&gt;What follows is guessing on my part based on the two known things above. I guess that most children understand and get past these issues fairly quickly, and afterwards there are no significant differences. In particular, once one gets all the way to research mathematics, there is almost certainly no difference. However, there might always be individual issues that might be caused by the language to a single person, or maybe at a given age a given language has a minor edge.&lt;/p&gt;&#xA;&lt;p&gt;In addition to number words, we might consider a word like &amp;quot;polygon&amp;quot; in English, which is Greek (also to most English-speakers), and compare it to monikulmio, mangekant, månghörning, and so on, which specify that there are many angles or sides, and do it in a language the speakers understand.&lt;/p&gt;&#xA;&lt;p&gt;Some languages might also have concepts and ways of stating things others lack: like &amp;quot;puolet enemmän&amp;quot; and &amp;quot;puolet vähemmän&amp;quot; in Finnish and the corresponding expressions in at least one Sami language. But, again, I doubt these make a difference in the long run.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2024-01-09T08:39:54.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27366" PostTypeId="2" ParentId="27363" CreationDate="2024-01-09T14:29:28.227" Score="3" Body="&lt;p&gt;Perhaps taking things to their extreme there are languages which lack simple numbering like the language of the Brazilian Amazonian people the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pirah%C3%A3_language&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pirahã&lt;/a&gt; which appears to have only words for one and two (or small quantity and large quantity) according to some authorities. This obviously restricts their ability to count and to compare sizes of groups.&lt;/p&gt;&#xA;" OwnerUserId="23475" LastActivityDate="2024-01-09T14:29:28.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27367" PostTypeId="2" ParentId="27363" CreationDate="2024-01-10T17:22:45.277" Score="3" Body="&lt;p&gt;This question reminds me of the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Hopi_time_controversy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hopi time controversy.&lt;/a&gt; In 1958, Stuart Chase &lt;a href=&quot;https://cs.nyu.edu/%7Edavise/papers/Whorf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;asserted&lt;/a&gt; (see page 3) that a person whose native language is Hopi has less trouble grappling with the &amp;quot;fourth dimension&amp;quot; than does somebody whose native language is, for example, English. Although intriguing, the theory is unfounded and not widely accepted by linguists.&lt;/p&gt;&#xA;&lt;p&gt;I think the same negative conclusions will apply to any speculation that there is a particular language which is best suited for learning mathematics.&lt;/p&gt;&#xA;&lt;p&gt;A narrower but well supported phenomenon is Spatial-Numerical Association of Response Codes (SNARC effect) of Dehaene &lt;em&gt;et al.&lt;/em&gt; Admittedly, this has more to do with cognitive parsing of numerical data than high-level mathematics. The idea is that when presented with small numbers, a person will parse them faster if they are presented on the left, while larger numbers should be presented on the right, as we think about a number line. However, there is &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7227642/&quot; rel=&quot;nofollow noreferrer&quot;&gt;research&lt;/a&gt; explaining that this effect is reversed for people whose native language is one where writing proceeds from right to left, i.e., &amp;quot;right-starters.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2024-01-10T17:22:45.277" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27368" PostTypeId="1" CreationDate="2024-01-12T19:56:03.203" Score="7" ViewCount="2111" Body="&lt;p&gt;If you teach &lt;a href=&quot;https://en.wikipedia.org/wiki/Newton%27s_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's method&lt;/a&gt; for finding roots of real functions on the high school (or freshmen) level, I think some students may reason like a variant of the following:&lt;/p&gt;&#xA;&lt;p&gt;Why do I need learn such a &amp;quot;complicated&amp;quot; method if I simply can use the following?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;plot it, for example, via Geogebra, and zoom in and just read off the root to the desired precision&lt;/li&gt;&#xA;&lt;li&gt;or just use the table value function of my calculator and do some kind of interval nesting&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;What would be good examples or activities to make it clear that the Newton method is useful and in some ways better than the approaches above? How can I explain where it makes sense to use Newton's method over the more simple methods above?&lt;/p&gt;&#xA;&lt;p&gt;I should make it clear that I don't compare doing one method by hand and the other using a computer. At the end, I want to compare both using a computer, but for pedagogic purposes, both could be compared manually.&lt;/p&gt;&#xA;&lt;p&gt;For comparing speed of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bisection_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;bisection method&lt;/a&gt; with Newton's method, I just tried the corresponding &lt;a href=&quot;https://en.wikipedia.org/wiki/Python_%28programming_language%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Python&lt;/a&gt; functions like this:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;import numpy as np&#xA;from scipy.optimize import bisect, newton&#xA;import time&#xA;&#xA;# Define an example function&#xA;def f(x):&#xA;    return np.exp(x) - x**3 + 5&#xA;&#xA;# Bisection method&#xA;start_time_bisection = time.time()&#xA;root_bisection = bisect(f, 2, 3)&#xA;end_time_bisection = time.time()&#xA;time_bisection = end_time_bisection - start_time_bisection&#xA;&#xA;# Newton's method&#xA;start_time_newton = time.time()&#xA;root_newton = newton(f, 2.0)&#xA;end_time_newton = time.time()&#xA;time_newton = end_time_newton - start_time_newton&#xA;&#xA;# Print results&#xA;print(&amp;quot;Bisection Method Result:&amp;quot;, root_bisection)&#xA;print(&amp;quot;Newton's Method Result:&amp;quot;, root_newton)&#xA;print(&amp;quot;Bisection Method Time:&amp;quot;, time_bisection)&#xA;print(&amp;quot;Newton's Method Time:&amp;quot;, time_newton)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;The results on my laptop were:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;Bisection Method Result: 2.728073969030447&#xA;Newton's Method Result: 2.72807396903125&#xA;Bisection Method Time: 0.00011157989501953125&#xA;Newton's Method Time: 0.00045108795166015625&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;In this case, the bisection method was faster.&lt;/p&gt;&#xA;&lt;p&gt;If I run it with the function &lt;span class=&quot;math-container&quot;&gt;$x^3 - 2$&lt;/span&gt; from &lt;a href=&quot;https://matheducators.stackexchange.com/a/27369/1203&quot;&gt;Justin Skycak's&lt;/a&gt; answer, the results look like:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;Bisection Method Result: 1.2599210498951834&#xA;Newton's Method Result: 1.2599210498947795&#xA;Bisection Method Time: 2.0265579223632812e-05&#xA;Newton's Method Time: 0.00040721893310546875&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="1203" LastEditorUserId="851" LastEditDate="2024-01-14T15:51:40.733" LastActivityDate="2024-02-12T21:35:04.540" Title="How can we motivate that Newton's method is useful?" Tags="|undergraduate-education|secondary-education|concept-motivation|mathematical-analysis|" AnswerCount="6" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="27369" PostTypeId="2" ParentId="27368" CreationDate="2024-01-12T20:33:27.457" Score="7" Body="&lt;p&gt;I only taught this to students who had some baseline level of interest in math/CS and were amenable to &amp;quot;here's why this is cool/powerful&amp;quot; examples (not just &amp;quot;here's why you might need to know this in the future&amp;quot; examples), so YMMV, but what worked for me was to start off with the following question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How can you estimate &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{2}$&lt;/span&gt; without directly using fractional exponents?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The most straightforward option is to use bisection search, but Newton's method (finding the root of &lt;span class=&quot;math-container&quot;&gt;$x^3-2$&lt;/span&gt; is faster). &lt;a href=&quot;https://www.justinmath.com/estimating-roots-via-bisection-search-and-newton-raphson-method/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; is the problem set I had them do.&lt;/p&gt;&#xA;&lt;p&gt;(In the broader scope of that course, Newton's method helped instill some intuition that the slope of a function is a key property that can be leveraged to computationally search the graph of the function for places of interest. That way, by the time we got to gradient descent, the students already had some intuition about how the gradient can be useful.)&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum:&lt;/strong&gt; OP notes that they tested out Newton's method vs bisection method and found that the bisection method was faster. However, this is not a fair comparison for two reasons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Newton's method produced a more accurate result by at least several orders of magnitude. (Since the level of precision is not being held constant, it's not an apples-to-apples comparison).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Additionally, Newton's method was not supplied any information about the derivative, so it's being artificially hamstrung it to use a difference approximation (which is slower).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;A proper comparison would be to compare iterating the function &lt;code&gt;update_newton(x) = 2/3 * (x + 1/x^2)&lt;/code&gt; versus &lt;code&gt;update_bisection(bounds) = [bounds[0], mid] if mid**3 &amp;gt; 2 else [mid, bounds[1]]&lt;/code&gt; (where &lt;code&gt;mid = (bounds[0]+bounds[1])/2&lt;/code&gt;) until a desired level of convergence.&lt;/p&gt;&#xA;&lt;p&gt;Doing this, we see that Newton's method is in fact faster.&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;Num Trials:              1000&#xA;True Root:               1.259921049894873191&#xA;Epsilon:                 0.000000000001000000&#xA;&#xA;Bisection Method Result: 1.259921049895638134&#xA;Newton's Method Result:  1.259921049895406320&#xA;&#xA;Bisection Method Time:   0.000012086868286133&#xA;Newton's Method Time:    0.000001034021377563&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Here's the script:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;import time&#xA;&#xA;def update_bisection(bounds):&#xA;    mid = (bounds[0] + bounds[1]) / 2&#xA;    if mid**3 &amp;gt; 2:&#xA;        return [bounds[0], mid]&#xA;    else:&#xA;        return [mid, bounds[1]]&#xA;&#xA;def update_newton(x):&#xA;    # f(x) = x^3 - 2&#xA;    # x - f(x) / f'(x)&#xA;    # x - (x^3 - 2) / 3x^2&#xA;    # x - 1/3 x + 2/3 x^2&#xA;    # 2/3 x + 2/3 x^2&#xA;    return 2/3 * (x + 1/x**2)&#xA;&#xA;root = 2**(1/3) # = 1.2599210498948732&#xA;epsilon           = 0.000000000001&#xA;num_trials = 1000&#xA;&#xA;sum_results_bisection = 0&#xA;sum_results_newton = 0&#xA;&#xA;def run_bisection():&#xA;    global sum_results_bisection&#xA;    guess_bisection = [1, 2]&#xA;    while abs((guess_bisection[0] + guess_bisection[1]) / 2 - root) &amp;gt; epsilon:&#xA;        guess_bisection = update_bisection(guess_bisection)&#xA;    sum_results_bisection += (guess_bisection[0] + guess_bisection[1]) / 2&#xA;&#xA;def run_newton():&#xA;    global sum_results_newton&#xA;    guess_newton = 1.5&#xA;    while abs(guess_newton - root) &amp;gt; epsilon:&#xA;        guess_newton = update_newton(guess_newton)&#xA;    sum_results_newton += guess_newton&#xA;&#xA;start_time_bisection = time.time()&#xA;for _ in range(num_trials):&#xA;    run_bisection()&#xA;end_time_bisection = time.time()&#xA;&#xA;start_time_newton = time.time()&#xA;for _ in range(num_trials):&#xA;    run_newton()&#xA;end_time_newton = time.time()&#xA;&#xA;time_bisection = (end_time_bisection - start_time_bisection) / num_trials&#xA;time_newton = (end_time_newton - start_time_newton) / num_trials&#xA;result_bisection = sum_results_bisection / num_trials&#xA;result_newton = sum_results_newton / num_trials&#xA;&#xA;num_decimals = len(str(root))&#xA;format_str = &amp;quot;{:.&amp;quot; + str(num_decimals) + &amp;quot;f}&amp;quot;&#xA;def format_decimal(x):&#xA;    return format_str.format(x)&#xA;&#xA;print(&amp;quot;Num Trials:             &amp;quot;, num_trials)&#xA;print(&amp;quot;True Root:              &amp;quot;, format_decimal(root))&#xA;print(&amp;quot;Epsilon:                &amp;quot;, format_decimal(epsilon))&#xA;print(&amp;quot;&amp;quot;)&#xA;print(&amp;quot;Bisection Method Result:&amp;quot;, format_decimal(result_bisection))&#xA;print(&amp;quot;Newton's Method Result: &amp;quot;, format_decimal(result_newton))&#xA;print(&amp;quot;&amp;quot;)&#xA;print(&amp;quot;Bisection Method Time:  &amp;quot;, format_decimal(time_bisection))&#xA;print(&amp;quot;Newton's Method Time:   &amp;quot;, format_decimal(time_newton))&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-01-13T21:47:37.450" LastActivityDate="2024-01-13T21:47:37.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27371" PostTypeId="2" ParentId="27368" CreationDate="2024-01-13T07:13:24.900" Score="10" Body="&lt;p&gt;&lt;strong&gt;Mild challenge to the framing of the problem&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It seems that a by-hand Newton's method is being compared to a graphing program. That does not seem a fair comparison. Why not compare an equation solver to a graphing program (whether on a calculator, computer or website)? The equation solver seems simpler than zooming in on a graph. And some equation solvers use Newton's method (see &lt;a href=&quot;https://support.casio.com/global/en/calc/manual/fx-115ESPLUS_991ESPLUSC_en/function_calculations/using_SOLVE.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://education.ti.com/en/customer-support/knowledge-base/other-graphing/product-usage/11819&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, or &lt;a href=&quot;https://education.ti.com/en/customer-support/knowledge-base/ti-83-84-plus-family/product-usage/11576&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Newton's method is to improve an estimate &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; of a root by finding the x intercept of the tangent line at a point &lt;span class=&quot;math-container&quot;&gt;$(x_1, f(x_1))$&lt;/span&gt;. Two function evaluations and solve a linear equation. It's not that complicated, imo. One can repeat with the new estimate until the root stops changing much. Much easier than recalculating a 100 points or so for a graph again and again.&lt;/p&gt;&#xA;&lt;p&gt;The secant method (1+N function evaluations for N steps) is often as good or better than Newton's method (2N function evaluations for N steps); used &lt;a href=&quot;https://education.ti.com/en/customer-support/knowledge-base/ti-nspire-family/product-usage/12065&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Usefulness&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Newton's method is actually used a lot and is therefore, by a pragmatic definition, useful. Of course, most of its uses lie in off-the-shelf software libraries. Many people use it without knowing how useful it is. They just need to know how to set up the equation solver...or use the app that sets up the equation solver.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Treasure Hunt&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Have them search the internet for applications, and each student is to share one use that interested them. Or ask ChatGPT &amp;quot;What is Newton's method actually used for? Please give three examples from real life.&amp;quot; (BTW, ChatGPT's response to my insisting on specific examples was basically, IDK.)&lt;/p&gt;&#xA;&lt;p&gt;Searching&#xA;scholar.google.com was more productive&#xA;with the search terms&#xA;&amp;quot;newton's method&amp;quot; plus some field such as the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;economics&lt;/li&gt;&#xA;&lt;li&gt;chemistry&lt;/li&gt;&#xA;&lt;li&gt;glass&lt;/li&gt;&#xA;&lt;li&gt;engineering&lt;/li&gt;&#xA;&lt;li&gt;robotics&lt;/li&gt;&#xA;&lt;li&gt;physics&lt;/li&gt;&#xA;&lt;li&gt;machine learning&lt;/li&gt;&#xA;&lt;li&gt;control systems&lt;/li&gt;&#xA;&lt;li&gt;investment portfolios&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;spectroscopy&amp;quot; [quotation marks needed]&lt;/li&gt;&#xA;&lt;li&gt;cancer&lt;/li&gt;&#xA;&lt;li&gt;COVID&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The students probably cannot completely understand the hits, and maybe they can't sift out the gobbledygook to reach the summary, &amp;quot;Newton's method is used in breast-cancer detection.&amp;quot; I think with the right framing of the assignment, they could identify the topic of a paper and come up with a short summary. One warning, many applications of Newton's method are to multivariate systems of equations, not the univariate one in first-year calculus. This would be acceptable to me, as it is the same idea applied to more variables. Others may feel differently.&lt;/p&gt;&#xA;" OwnerUserId="15254" LastActivityDate="2024-01-13T07:13:24.900" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27372" PostTypeId="2" ParentId="27368" CreationDate="2024-01-13T14:28:53.440" Score="17" Body="&lt;p&gt;I'm going to respond from an applied math (or maybe CS) perspective: Part of the problem is that the functions that you look at in high school and the standard calculus sequence are unrealistically simple. Not only do you know what their formulas are, but you can usually write them down on a single line and maybe even evaluate them by hand in a reasonable amount of time. (e.g. rational functions)&lt;/p&gt;&#xA;&lt;p&gt;Because of that, it's pretty easy and quick to ask the computer to evaluate the function at thousands and thousands of points without any particular system beyond regular spacing. That is, after all, what you are asking the computer to do when you make it graph something.&lt;/p&gt;&#xA;&lt;p&gt;Now, outside of the classroom, there are many functions that aren't so simple. You might literally have to go to a lab and do an experiment to find a single value of the function. Now you have both time and money costs to evaluating the function and want to minimize these costs. (I'm actually thinking of the secant method variant of Newton's method here; where you approximate the derivative.)&lt;/p&gt;&#xA;&lt;p&gt;Even if getting function values isn't that expensive, sometimes you want to find zeros very quickly and automatically. For example, in computer graphics.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, I guess that my point is that graphing is an excellent intuition builder but a horrible algorithm. If you made one group of students do the graph method entirely by hand (the same way the computer does) and another group do Newton's method, the difference would be obvious.&lt;/p&gt;&#xA;&lt;p&gt;(Also, Newton's method has a variant that works in higher dimensions where you can't graph things to begin with.)&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2024-01-13T14:28:53.440" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27374" PostTypeId="1" CreationDate="2024-01-14T08:02:38.637" Score="4" ViewCount="176" Body="&lt;p&gt;What are the examples we can use to explain infinite descent as an efficient method of proofs in geometry? &lt;br/&gt;&#xA;I think one of the best may be proving medians of a triangle are concurrent by the infinite descent method when compared with other approaches. If we have a few more, it can be useful to make the topic much more interesting for students.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2024-01-20T23:37:52.153" Title="Infinite descent method in geometry" Tags="|secondary-education|geometry|contradiction|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27375" PostTypeId="2" ParentId="27368" CreationDate="2024-01-14T12:23:54.820" Score="14" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In video game animation, each second thousands of equations need to be solved with &lt;em&gt;code&lt;/em&gt;. You can't tell a computer &amp;quot;look at a graph&amp;quot;!  Your students need to think beyond the setting of solving one problem one time to realize the inadequacy of &amp;quot;plot and look with your eyes&amp;quot; as a serious solution method.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Newton’s method illustrates a fantastic &lt;em&gt;new&lt;/em&gt; idea: give up on exact solution formulas and instead find a solution as a limit of an iterative process (and iterations are amenable to being coded up). More iterative methods are used in machine learning: gradient descent and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-means clustering. The Numberphile video &amp;quot;Math and Movies (Animation at Pixar)&amp;quot; shows the role of iterative methods in computer animation. Newton's method is simply a basic example showing what iterative methods can do.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When Newton's method works, it does so &lt;em&gt;very&lt;/em&gt; quickly, as the number of correct digits roughly doubles at each step (exponential convergence).  Many numerical algorithms, when they work, only give you one new correct digit at each step (geometric convergence).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Division is an &amp;quot;expensive&amp;quot; numerical operation. Newton's method reveals a way to do division without actually doing any division. Say we have a number &lt;span class=&quot;math-container&quot;&gt;$a &amp;gt; 0$&lt;/span&gt; and want to compute &lt;span class=&quot;math-container&quot;&gt;$1/a$&lt;/span&gt;. Setting this up as the equation &lt;span class=&quot;math-container&quot;&gt;$a - 1/x = 0$&lt;/span&gt; (&lt;em&gt;not&lt;/em&gt; as &lt;span class=&quot;math-container&quot;&gt;$x - 1/a = 0$&lt;/span&gt;), the recursion with Newton's method is &lt;span class=&quot;math-container&quot;&gt;$x_{n+1} = 2x_n - ax_n^2$&lt;/span&gt;.  When the initial value &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; is close enough to &lt;span class=&quot;math-container&quot;&gt;$1/a$&lt;/span&gt; (more precisely, when &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; x_1 &amp;lt; 2/a$&lt;/span&gt;), this recursion converges rapidly (try it with &lt;span class=&quot;math-container&quot;&gt;$a = 7$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_1 = .25$&lt;/span&gt;), but the main point I want to make is that calculating &lt;span class=&quot;math-container&quot;&gt;$x_{n+1}$&lt;/span&gt; only involves adding and multiplying: no actual division to estimate &lt;span class=&quot;math-container&quot;&gt;$1/a$&lt;/span&gt;!&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Show your students the 3Blue1Brown video &amp;quot;Newton’s Fractal (which Newton knew nothing about)&amp;quot;, esp. starting at 8:45. This shows how Newton's method with complex numbers and varying initial seed values leads to remarkable images.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Newton's method can be extending from solving a single equation in one variable to solving &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; equations in &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; variables: a &lt;em&gt;multivariable&lt;/em&gt; Newton's method. It generalizes the recursion &lt;span class=&quot;math-container&quot;&gt;$x_{n+1} = x_n - f(x_n)/f'(x_n)$&lt;/span&gt; by using multivariable calculus (partial derivatives) and linear algebra (inverting a matrix). Once you get beyond a single unknown, or especially 2 unknowns, the student's reliance on Geogebra totally breaks down since they can't stare at a graph when the equations have 5 variables. But mathematical algorithms still works in 5 variables even when we have no direct picture to look at.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="893" LastEditorUserId="893" LastEditDate="2024-01-14T12:34:50.770" LastActivityDate="2024-01-14T12:34:50.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27376" PostTypeId="2" ParentId="27368" CreationDate="2024-01-14T13:20:43.993" Score="1" Body="&lt;p&gt;In a calculus course, one of the key concepts students encounter is the necessity of solving equations to find maxima and minima, which are critical points in various functions. in this context of solving equations, I provide a comprehensive overview of solving polynomial equations. This includes:&lt;/p&gt;&#xA;&lt;p&gt;Linear Equations: The simplest form, showcasing how a single variable can determine the slope and position of a line.&#xA;Quadratic Equations: Introducing the quadratic formula and the concept of roots, and how they relate to the graph's vertex and axis of symmetry.&#xA;Cubic Equations: I delve into Cardano's method, explaining its historical significance and application in finding the roots of cubic equations.&#xA;Quartic Equations: Here, Ferrari's technique is introduced, demonstrating a more complex method for finding roots of fourth-degree equations.&#xA;After these, I segue into Galois Theory, highlighting its profound conclusion: there is no general closed-form solution for polynomial equations of degree five or higher. This revelation leads us into the realm of approximation and numerical methods.&lt;/p&gt;&#xA;&lt;p&gt;A key method we explore is Newton's method (also known as the Newton-Raphson method), a powerful technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. I ensure to explain the significance of these numerical methods not just in theoretical mathematics, but also in their practical applications across various scientific and engineering disciplines. This approach helps students appreciate the breadth and depth of calculus and its problem-solving potential in real-world scenarios.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This revised answer provides a more structured and detailed explanation, enhancing the educational value of your response and making it more engaging and informative for readers seeking to understand the application of these concepts in calculus&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2024-01-14T13:20:43.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27377" PostTypeId="2" ParentId="24445" CreationDate="2024-01-14T14:38:07.677" Score="0" Body="&lt;p&gt;&lt;em&gt;Can anyone provide a relatively simple (for educational purposes) &amp;quot;true&amp;quot; proof by contradiction of a proposition &amp;quot;P =&amp;gt; Q&amp;quot; in which one would really need BOTH the assumption (not Q) and the assumption (P) to actually reach some kind of contradiction?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem:&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$x\in(0,\infty)$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$x+\frac{4}{x}\geq 4$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Proof by contradiction:&lt;/em&gt; Suppose that &lt;span class=&quot;math-container&quot;&gt;$x+\frac{4}{x}&amp;lt;4$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0$&lt;/span&gt; (by hypothesis), it follows that &lt;span class=&quot;math-container&quot;&gt;$(x+\frac{4}{x})\cdot x&amp;lt;4\cdot x$&lt;/span&gt;. This implies that &lt;span class=&quot;math-container&quot;&gt;$(x-2)^2&amp;lt;0$&lt;/span&gt;, which is a contradiction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is the assertion &lt;span class=&quot;math-container&quot;&gt;$x\in(0,\infty)$&lt;/span&gt;, that is, &lt;span class=&quot;math-container&quot;&gt;$x&amp;gt;0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; is the assertion &lt;span class=&quot;math-container&quot;&gt;$x+\frac{4}{x}\geq 4$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\neg Q$&lt;/span&gt; is the assertion &lt;span class=&quot;math-container&quot;&gt;$x+\frac{4}{x}&amp;lt; 4$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The contradiction is &lt;span class=&quot;math-container&quot;&gt;$(x-2)^2&amp;lt;0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Note that the assumptions &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\neg Q$&lt;/span&gt; were used together in order to reach the contradiction. And the contradiction is different from the negation of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. Therefore, it is a &amp;quot;true&amp;quot; proof by contradiction.&lt;/p&gt;&#xA;&lt;p&gt;However (and this is a fact that should be clear to the people you are educating), we use contradiction because, in general, it is easier (requires less knowledge) and simpler (requires less creativity) than the contrapositive and not, necessarily, because the use of the contrapositive is impossible. For example, the same theorem above can be proved by contrapositive. Write this proof. Possibly, we can argue that the proof by contrapositive is more elegant. But which one is more &amp;quot;naturtal&amp;quot; (especially for students)?&lt;/p&gt;&#xA;" OwnerUserId="13585" LastActivityDate="2024-01-14T14:38:07.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27379" PostTypeId="1" AcceptedAnswerId="27382" CreationDate="2024-01-15T14:10:59.237" Score="5" ViewCount="2491" Body="&lt;p&gt;Sometimes awareness of alternative approaches of different origins may be helpful for students to improve their creative skills in mathematics. What would you suggest as alternative methods in basic mathematical operations such as addition, subtraction, multiplication, division, squaring, and finding square roots? &lt;br/&gt;&lt;/p&gt;&#xA;&lt;p&gt;As an example when it comes to the multiplication of numbers, there are alternative ways such as the Japanese method using straight lines and the Chinese method using lattice. There should be much more to explain to our students and it would be great if you could share your experience and knowledge related to this issue. &lt;br/&gt;&#xA;P.S.&lt;br/&gt;&#xA;I thought it would be better to add further information to give a much clearer idea about this issue. &lt;br/&gt;&#xA;1)There can be some special benefits when applying a particular approach, as example when you consider the products 11×11, 111×111,..... it is much easier to understand the pattern using a straight-line approach rather than the so-called standard method. &lt;br/&gt;&#xA;2) As I stated in my original question, what I'm interested in here is not just alternative approaches but approaches from different origins. &lt;br/&gt;&#xA;3)If talking about alternative approaches is not good in classroom for all students as a common topic, what about discussing these as activities in other platforms like maths clubs ,&#xA;considering as an option for students who are interested.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2024-01-20T03:23:42.060" LastActivityDate="2024-01-20T21:42:18.693" Title="Alternative approaches in basic mathematical operations" Tags="|mathematical-pedagogy|" AnswerCount="7" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="27380" PostTypeId="2" ParentId="8173" CreationDate="2024-01-15T14:44:29.503" Score="1" Body="&lt;p&gt;The best single book for preparing for mathematical olympiads in my opinion is&lt;/p&gt;&#xA;&lt;h2&gt;Arthur Engel, &lt;em&gt;Problem-Solving Strategies&lt;/em&gt; (1998).&lt;/h2&gt;&#xA;&lt;p&gt;It has chapters on the invariance principle, colouring proofs, the extremal principle, the box principle, enumerative combinatorics, number theory, inequalities, induction, sequences, polynomials, functional equations, geometry, games, and further strategies.&lt;/p&gt;&#xA;&lt;p&gt;Engel was leader of the German team at the International Mathematical Olympiad. Sadly he passed away a little more than a year ago. You can read more about him on his Wikipedia page &lt;a href=&quot;https://en.wikipedia.org/wiki/Arthur_Engel_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="23642" LastActivityDate="2024-01-15T14:44:29.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27381" PostTypeId="1" CreationDate="2024-01-15T15:04:17.363" Score="2" ViewCount="152" Body="&lt;p&gt;Educators in many countries have found that preparing highly motivated students for national and international mathematical olympiads can be useful in training them in mathematical ways of thinking.&lt;/p&gt;&#xA;&lt;p&gt;Successfully answering the questions in such competitions is said to require only a pre-university level of mathematical knowledge. But it often requires an advanced use of it. Moreoever, it's made easier by familiarity and practical experience with topics that aren't on many pre-university syllabuses in any country - or even, for that matter, at university level.&lt;/p&gt;&#xA;&lt;p&gt;My question is whether there are any international mathematical olympiad-style competitions at a lower level. If so, what are they, what are the age restrictions, at how low a level do they start (and what does that mean), and how useful have mathematics educators found them?&lt;/p&gt;&#xA;&lt;p&gt;I'm especially interested in any international competitions in which candidates' and countries' scores are made public and can be ranked, as is true for the IMO.&lt;/p&gt;&#xA;" OwnerUserId="23642" LastEditorUserId="2083" LastEditDate="2024-01-16T06:35:55.467" LastActivityDate="2024-01-17T12:12:33.433" Title="International mathematical olympiad-type competitions at lower levels, and if they exist, their educational usefulness?" Tags="|secondary-education|primary-education|math-contests|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27382" PostTypeId="2" ParentId="27379" CreationDate="2024-01-15T15:58:28.223" Score="18" Body="&lt;p&gt;I would be very cautious when introducing any algorithms other than the standard methods. The standard algorithms are standard for a reason: they're easy to set up and hard to mess up. (The standard algorithm for multiplication by hand is &lt;a href=&quot;https://mathworld.wolfram.com/LongMultiplication.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;long multiplication&lt;/a&gt;, and, as Xander mentions in the comments, it has been standard for the past century or so in the USA.)&lt;/p&gt;&#xA;&lt;p&gt;The lattice method, for instance, is hard to set up (it takes a lot of time and effort to draw the entire grid with diagonals) and easy to mess up (if I had a dollar for every time a student drew a sloppy grid with misaligned diagonals and screwed up the calculation as a result, I'd be filthy rich).&lt;/p&gt;&#xA;&lt;p&gt;Kids will often latch onto whatever method they &amp;quot;like&amp;quot; best, as though it were a flavor of ice cream, regardless of its practicality -- and their incentives are often misaligned. For instance, I've tutored students who straight-up told me they preferred the lattice method because they liked being able to take a break from math to draw the grid (and believe me, they took their sweet time drawing the grid and making it perfect). Of course, it took these students forever to complete their problems because they were working with incredibly low efficiency, and that frustrated them, but another factor leading them to resist switching to the more efficient standard method was they had completely forgotten it (as a result of using the lattice method for so long) and relearning it would require some additional up-front time and effort on top of what already felt like an overwhelming workload.&lt;/p&gt;&#xA;&lt;p&gt;I'm not against alternatives, but as a teacher you have to run through a simulation in your mind: &amp;quot;what will happen to students who latch onto this method indefinitely and resist using other methods?&amp;quot; If the alternative method is just as efficient and just as general, then sure, introduce it. But if not, then I wouldn't introduce it, because students who latch onto it and resist letting go are going to be in for a world of hurt. (Even if you introduce an alternative method as a fun, temporary vacation away from standard techniques, some students will try to stay on that vacation forever.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Note that while I be focused on the lattice method in this answer, the same argument applies to the &amp;quot;straight lines&amp;quot; method, which seems even harder to set up and even easier to mess up.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-01-17T22:16:26.797" LastActivityDate="2024-01-17T22:16:26.797" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27383" PostTypeId="2" ParentId="27379" CreationDate="2024-01-15T18:25:39.450" Score="10" Body="&lt;p&gt;It is absolutely a good idea to give students different ways of thinking about things, but I do not think that the examples in the question are good examples of what this should look like.&lt;/p&gt;&#xA;&lt;h3&gt;What is wrong with the given examples?&lt;/h3&gt;&#xA;&lt;p&gt;The examples in the question are all examples of algorithms which students might use to &lt;em&gt;compute&lt;/em&gt; a product, not examples of how students might think about what a product &lt;em&gt;is&lt;/em&gt;.  Generally speaking, having more algorithms available doesn't really help students to understand what they are doing any better—it just gives them more ways of doing computations (which, honestly, might cause confusion).&lt;/p&gt;&#xA;&lt;p&gt;Teaching more algorithms also leads students into the trap of thinking that mathematics is nothing more than a bunch of computational &amp;quot;tricks&amp;quot;, which &lt;em&gt;must&lt;/em&gt; be applied in certain circumstances.  Students get into the mindset of looking at a problem, trying to decide what &amp;quot;type&amp;quot; of problem it is, and then implementing whichever tool or algorithm they determine is the &amp;quot;correct&amp;quot; tool for the job.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I would avoid teaching too many algorithms—keep it to the minimal set of techniques which will be needed to get the job done.&lt;/p&gt;&#xA;&lt;h3&gt;What could one try instead?&lt;/h3&gt;&#xA;&lt;p&gt;Thinking about multiplication, there are a number of ways of &lt;em&gt;describing&lt;/em&gt; what multiplication &amp;quot;is&amp;quot; which might give students different approaches for solving problems.  For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Multiplication is Area:&lt;/strong&gt;  To the ancient Greek way of thinking, number was a length and a product was an area.  To multiply two numbers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, lay one length horizontally and the other perpendicular in order to build a rectangle with width &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and height &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.  The area of this rectangle is the product.&lt;/p&gt;&#xA;&lt;p&gt;This isn't a &lt;em&gt;great&lt;/em&gt; model for computation, since it can be hard to figure out what the area of a rectangle is, but multiplication over the natural numbers can be done with physical manipulatives.  To multiply &lt;span class=&quot;math-container&quot;&gt;$a \times b$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;lay down a horizontal line of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; stones (or coins, or poker chips, or whatever),&lt;/li&gt;&#xA;&lt;li&gt;starting at the far left, lay down a vertical line of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; stones, arranged so that the left-most stone of the horizontal line is the bottom-most stone of the vertical line, and then&lt;/li&gt;&#xA;&lt;li&gt;continue laying down stones until a rectangle is formed.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The product &lt;span class=&quot;math-container&quot;&gt;$a\times b$&lt;/span&gt; is equal to the area of the rectangle, which can be computed by counting the number of stones.  I have put together a small Desmos model for this:  &lt;a href=&quot;https://www.desmos.com/calculator/n2e06dhgdq&quot; rel=&quot;noreferrer&quot;&gt;https://www.desmos.com/calculator/n2e06dhgdq&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;p&gt;I like this model because it reverses the usual approach:  what &lt;em&gt;often&lt;/em&gt; happens in elementary school is that students are first taught algorithms for multiplying numbers, and then taught to use these algorithms to find areas.  Here, area &lt;em&gt;is&lt;/em&gt; a product, from the start.  A student who understands this model doesn't have to ask if they should multiply or add (or whatever) to find an area—it should fairly quickly be internalized as part of the definition.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Multiplication is Scaling:&lt;/strong&gt;  Imagine a real number line sitting in a plane, made out of some rigid material (e.g. imagine a wooden ruler).  Attach the ray &lt;span class=&quot;math-container&quot;&gt;$[0,\infty)$&lt;/span&gt; to this rigid number line at zero, and imagine that this half-ray is made out of some elastic material which can be stretched or compressed.&lt;/p&gt;&#xA;&lt;p&gt;The product &lt;span class=&quot;math-container&quot;&gt;$a \times b$&lt;/span&gt; can be thought of as scaling &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; by a factor of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.  To visualize this, imagine the &lt;em&gt;entire&lt;/em&gt; ray &lt;span class=&quot;math-container&quot;&gt;$[0,\infty)$&lt;/span&gt; being scaled by this factor—stretch the elastic ray so that the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; on the ray aligns with &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; on the rigid number line; the product &lt;span class=&quot;math-container&quot;&gt;$a\times b$&lt;/span&gt; can be found by locating the &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; on the elastic number line and determining where it lines up.&lt;/p&gt;&#xA;&lt;p&gt;To give an idea of what this looks like, I've whipped up a very basic GeoGeobra model:  &lt;a href=&quot;https://www.geogebra.org/calculator/pk3d65ax&quot; rel=&quot;noreferrer&quot;&gt;https://www.geogebra.org/calculator/pk3d65ax&lt;/a&gt; .  To multiply &lt;span class=&quot;math-container&quot;&gt;$a \times b$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Set &lt;span class=&quot;math-container&quot;&gt;$a=1$&lt;/span&gt; (this aligns the blue ray with the real number line on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis).&lt;/li&gt;&#xA;&lt;li&gt;Set &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.  Note that the purple point labeled &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; will align with the value of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; on the real number line.&lt;/li&gt;&#xA;&lt;li&gt;Set &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to the actual value of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The purple point on the ray will now align with the product &lt;span class=&quot;math-container&quot;&gt;$a \times b$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;One of the advantages of this model is that it very quickly justifies the rules for multiplication by negative numbers (e.g. &amp;quot;negative times positive is negative&amp;quot;):  if &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is negative, then the ray initially points along the negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis; scaling by a negative &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; reverses the orientation of the ray when &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; on the ray aligns with the negative value of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Multiplication is Repeated Addition:&lt;/strong&gt;  First off, I side very much with Keith Devlin's assertion that &lt;a href=&quot;https://profkeithdevlin.org/devlins-angle/2008-posts/#jun08&quot; rel=&quot;noreferrer&quot;&gt;Multiplication ain't no repeated addition&lt;/a&gt;, and I think that it is important to teach multiplication as being a distinct operation.  However, there are versions of multiplication-as-repeated-addition which pop up in other places&lt;sup&gt;[1]&lt;/sup&gt;, and treating it as repeated multiplication does give rise to an algorithm for computation.  And this model can be justified for natural numbers by looking at the area model (add together &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; rows, each of which contains &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; stones—this is really just &lt;span class=&quot;math-container&quot;&gt;$a+a+\dotsb+a$&lt;/span&gt; (with &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; copies of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I am not a huge fan of this model, but I am not sure that it hurts to introduce it to students alongside other models.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h3&gt;Summary&lt;/h3&gt;&#xA;&lt;p&gt;I think that it is very important to give students different ways of thinking about the tools, definitions, and results which they being introduced to.  But this different ways of thinking need to be about the underlying logical model, and not just distinct algorithmic or computational tricks.&lt;/p&gt;&#xA;&lt;p&gt;The idea is that students should have multiple internal models, which gives them multiple lines of attack when trying to solve a problem.  This kind of mental flexibility should be supported and encouraged.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;[1] For example, in the study of groups, one might write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ ng = \underbrace{g+g+g+\dotsb+g}_{\text{$n$ times}} $$&lt;/span&gt;&#xA;in order to denote adding the group element &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; to itself &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; times, where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a natural number.  However, that is &lt;em&gt;way&lt;/em&gt; beyond the scope of an elementary school math curriculum.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2024-01-15T18:25:39.450" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27384" PostTypeId="2" ParentId="27379" CreationDate="2024-01-16T08:47:53.030" Score="3" Body="&lt;p&gt;I am not a professional educator. I have, however, tutored students in math that needed to improve their grades. My personal observation in that context is that presenting alternative explanations or algorithms is confusing to students if they have not yet mastered &lt;em&gt;one&lt;/em&gt; particular algorithm based on &lt;em&gt;one&lt;/em&gt; particular set of explanations. For this reason I always consulted a students' textbook first to make sure I would be in line with it.&lt;/p&gt;&#xA;&lt;p&gt;Assuming you are working with gifted and / or highly motivated students that are not overwhelmed or confused by the additional information, I think covering a wider array of algorithm options is possible within the realm of Euro-centric mathematics, which is the only mathematical tradition I am familiar with. I do not have a good overview of secondary literature that covers this well. It may or may not exist. Students might want to take note that having knowledge of French, German, Italian, and Latin is useful if one wants to consult historical mathematical literature.&lt;/p&gt;&#xA;&lt;p&gt;When European countries switched from abacus-like computation (by manipulating tokens on lines drawn on a desk) to pen-and-paper calculation during the 16&lt;sup&gt;th&lt;/sup&gt; century, there was initially great diversity in the arithmetic algorithms used. This is covered in much detail in&lt;/p&gt;&#xA;&lt;p&gt;P. Treutlein, &lt;a href=&quot;https://www.google.com/books/edition/Abhandlungen_zur_Geschichte_der_Mathemat/RIRtAAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA1&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Das Rechnen im 16. Jahrhundert.&lt;/em&gt;&lt;/a&gt; Leizpig: B.G. Teubner 1877&lt;/p&gt;&#xA;&lt;p&gt;The following book for example recommended a subtraction methods that proceeds from most significant to least significant digit on p. 9, employing a one digit look-ahead:&lt;/p&gt;&#xA;&lt;p&gt;Peter Ramus, &lt;a href=&quot;https://www.google.com/books/edition/Petri_Rami_Arithmetices_Libri_Dvo_Et_Alg/JL1pAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=Petri%20Rami%20Arithmetices&amp;amp;pg=PA9&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Arithmetices Libri Duo, et Algebræ totidem&lt;/em&gt;&lt;/a&gt;, Frankfurt 1586&lt;/p&gt;&#xA;&lt;p&gt;As another example, in the first printed work on algebra, the author presented a total of eight different methods of performing multiplication, many of which originated in India from what I understand.&lt;/p&gt;&#xA;&lt;p&gt;Luca Pacioli, &lt;a href=&quot;https://www.google.com/books/edition/Summa_de_Arithmetica_geometria_proportio/iqgPe49fhrsC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA2&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Summa de Arithmetica geometria, proportioni : et proportionalita&lt;/em&gt;&lt;/a&gt;, Venice 1494&lt;/p&gt;&#xA;&lt;p&gt;The description of the multiplication algorithms is at pp. 26ff and includes the &lt;em&gt;gelosia&lt;/em&gt; method as the sixth method, which is a lattice method named after the grids installed in front of the windows of high-ranking ladies so they could look out but not be seen from the street. The first method covered is the distributive method which is the standard method used in long-hand multiplication today. The &lt;em&gt;crocetta&lt;/em&gt; method, which is based on the column-wise addition of partial products is one that is relevant to performing multiplications with computers.&lt;/p&gt;&#xA;&lt;p&gt;The following influential French work starts with the current standard long-hand multiplication method but also provides multiple alternative algorithms, however there are fewer than in Pacioli's earlier work, and the &lt;em&gt;gelosia&lt;/em&gt; method is not among them:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/books/edition/Larismetique_geometrie_de_maistre_Estien/4EJ9a5bap-8C?hl=en&amp;amp;gbpv=1&amp;amp;pg=PP7&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Larismetique &amp;amp; Geometrie de maistre Estienne de la Roche dict Ville Franche, Nouuellemente Imprimee &amp;amp; des faultes corrigee&lt;/em&gt;&lt;/a&gt;, Lyon 1538&lt;/p&gt;&#xA;&lt;p&gt;If one looks at further developments during the 16&lt;sup&gt;th&lt;/sup&gt; century, it is obvious that there was a rapid convergence of arithmetic algorithms towards the standard long-hand methods taught today. This seems to be a clear indication that the alternative methods were considered impractical, cumbersome, or overly difficult. If one looks at a standard introduction to basic arithmetic from the late 1600s, the algorithms are as I learned them in elementary school in the early 1970s in a non-English speaking country, representing multiple centuries of continuity:&lt;/p&gt;&#xA;&lt;p&gt;William Leybourn, &lt;a href=&quot;https://www.google.com/books/edition/Arithmetick_Vulgar_Decimal_Instrumental/4qk2AAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PP9&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Arithmetick, Vulgar, Decimal, Instrumental, Algebraical, 7th ed.&lt;/em&gt;&lt;/a&gt;, London 1700&lt;/p&gt;&#xA;&lt;p&gt;As for square roots, the standard long-hand method in use today (if it is taught at all, that is) was already the dominant method in the 16&lt;sup&gt;th&lt;/sup&gt; century, as for example in this book:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=7sVZAAAAcAAJ&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Die Coss Christoffs Rudolffs, Mit schönen Exempeln der Cosz. Durch Michael Stifel Gebessert vnd sehr gemehrt&lt;/em&gt;&lt;/a&gt;, Königsberg 1571&lt;/p&gt;&#xA;&lt;p&gt;Treutlein notes, however, that in addition to this algorithm some authors also provided methods for the approximate computation of square roots.&lt;/p&gt;&#xA;" OwnerUserId="17715" LastEditorUserId="17715" LastEditDate="2024-01-20T21:42:18.693" LastActivityDate="2024-01-20T21:42:18.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27385" PostTypeId="2" ParentId="27379" CreationDate="2024-01-16T13:56:07.990" Score="9" Body="&lt;p&gt;I think that introducing new algorithms just for the sake of diversity is not such a great idea &lt;strong&gt;especially&lt;/strong&gt; if you have the intention of getting students to be proficient in multiple algorithms.  Achieving proficiency in one algorithm is hard enough.&lt;/p&gt;&#xA;&lt;p&gt;I &lt;strong&gt;do&lt;/strong&gt; think that the &amp;quot;standard algorithms&amp;quot; are suboptimal for understanding and for checking your work.  Here are &amp;quot;alternative&amp;quot; algorithms which I would actually make the &lt;strong&gt;new standard&lt;/strong&gt; if I had the power.  These are not my own inventions:  they are known in the education world as &amp;quot;partial sum&amp;quot;, &amp;quot;partial product&amp;quot;, or &amp;quot;scaffold&amp;quot; algorithms.&lt;/p&gt;&#xA;&lt;h3&gt;Addition&lt;/h3&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AacxZb.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AacxZb.png&quot; alt=&quot;partial sums addition algorithm&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This algorithm reinforces place-value concepts.  Instead of placing a random &amp;quot;one&amp;quot; above the 8 (not understanding that this one actually represents ten!), we explicitly show the partial sums.&lt;/p&gt;&#xA;&lt;p&gt;We can &lt;strong&gt;verbalize&lt;/strong&gt; our reasoning:  &amp;quot;I need to combine 789 units with 62 units.  First I combine the 9 singles with the 2 singles to get 11 singles.&#xA;Then I combine the 8 tens with the 6 tens to get 14 tens (which is 140)...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;We can also check our work.  The standard algorithm is hard to &amp;quot;debug&amp;quot;: you can get single digit sum wrong, forget to carry, carry to the wrong place value, etc.  Looking back at your work it is very hard to tell where an error happened.&lt;/p&gt;&#xA;&lt;p&gt;With this algorithm, if the student made an error like 80 + 60 = 120 they can see exactly where the mistake was made and how to fix it.&lt;/p&gt;&#xA;&lt;p&gt;The only disadvantage is a few extra lines of writing.&lt;/p&gt;&#xA;&lt;h3&gt;Multiplication&lt;/h3&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/hIlUQt.png&quot; width=&quot;230&quot; /&gt;&#xA;&lt;p&gt;Again we reinforce place value.  We also make the use of the distributive property explicit here.&lt;/p&gt;&#xA;&lt;p&gt;We can &lt;strong&gt;verbalize&lt;/strong&gt; our reasoning coherently:  &amp;quot;I want 24 groups of 357 units.  I start with 4 groups of 357 units.  So I need 4 groups of 7, 4 groups of 50, and 4 groups of 300.  Then ...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;It is also nice to pair this with an area model to see where the partial products come from geometrically.&lt;/p&gt;&#xA;&lt;p&gt;Again we can check our work:  instead of crossing things out and carrying we explicitly show each partial product.  If we make a mistake on a single digit multiplication fact (like  &lt;span class=&quot;math-container&quot;&gt;$4 \times 7 = 24$&lt;/span&gt;), or make a place value error (like &lt;span class=&quot;math-container&quot;&gt;$20 \times 50 = 100$&lt;/span&gt;), we can pinpoint exactly where the error was made and correct it.&lt;/p&gt;&#xA;&lt;h3&gt;Division&lt;/h3&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Pb12sm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Pb12sm.png&quot; alt=&quot;scaffold division algorithm&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can &lt;strong&gt;verbalize&lt;/strong&gt; our reasoning coherently: &amp;quot;I have 937 units which I want to share equally with 4 groups.  First I put 200 units in each group which uses 800 units, leaving me with 137 units remaining.  Then I put 30 units in each group...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Note that this reasoning continues to make sense even with decimal quotients!  &amp;quot;I have 1 unit left.  I break that unit into tenths, so I have ten tenths.  I put 2 tenths of a unit in each group, leaving me with 2 tenths of a unit remaining...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Again we can check our work and see if the mistake was with a division fact, a multiplication fact, or subtracting incorrectly.&lt;/p&gt;&#xA;&lt;h3&gt;Any use for multiple algorithms?&lt;/h3&gt;&#xA;&lt;p&gt;I do think that, if you have the time, it could be nice to &lt;strong&gt;very occasionaly&lt;/strong&gt; give students the challenge of &lt;strong&gt;figuring out why an alternative algorithm works&lt;/strong&gt;.  The goal is not to teach them to use a new algorithm:  it is to show that many methods are possible and that they are all based on reasoning.&lt;/p&gt;&#xA;&lt;p&gt;One I like is the following alternative algorithm for subtraction:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LwiNQt.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LwiNQt.png&quot; alt=&quot;two subtraction algorithms&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The standard algorithm is on the left, while the &amp;quot;novel&amp;quot; algorithm is on the right.&lt;/p&gt;&#xA;&lt;p&gt;In the standard algorithm we think about breaking down one ten into ten ones so that we have &amp;quot;enough ones&amp;quot;.  This is an important idea!&lt;/p&gt;&#xA;&lt;p&gt;You could challenge your students to explain the reasoning behind the second algorithm.   Can you do some more subtraction problems this funny way?  Do you think it will always work?  Why?  Is this method easier or harder for you to use?  Are there any situations where choosing this algorithm might make it easier?&lt;/p&gt;&#xA;&lt;p&gt;This algorithm relies on the principle that &lt;span class=&quot;math-container&quot;&gt;$A - B = (A + C) - (B + C)$&lt;/span&gt; which is also an important idea:  the difference is invariant under shifts.&lt;/p&gt;&#xA;&lt;p&gt;For an interesting situation, consider &lt;span class=&quot;math-container&quot;&gt;$1000 - 1$&lt;/span&gt; using the standard algorithm vs. this new algorithm.  Of course, if you really understand our number system you should be able to compute the difference without an algorithm.  However, it is instructive to compare the &amp;quot;multiple borrowing&amp;quot; in the standard algorithm to the relatively clean treatment with this alternative algorithm.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2024-01-18T13:53:19.363" LastActivityDate="2024-01-18T13:53:19.363" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27386" PostTypeId="1" AcceptedAnswerId="27496" CreationDate="2024-01-16T16:24:29.703" Score="3" ViewCount="236" Body="&lt;p&gt;Answers to the &lt;a href=&quot;https://matheducators.stackexchange.com/questions/18030/online-whiteboard-application-with-simple-latex-support?newreg=b72e64a72b7f4f1fb764e498d8381b83&quot;&gt;Previou question&lt;/a&gt; seem to suggest some whiteboard programs with latex support.&lt;/p&gt;&#xA;&lt;p&gt;However, my impression is that they are mostly designed for &amp;quot;handwriting&amp;quot; with some electrical pen.&lt;/p&gt;&#xA;&lt;p&gt;I wonder if there exists some program that resembles the webpage where I am writing my question now. That is, I &amp;quot;type&amp;quot; my question here, including formulas, and the output is instantly shown below in the latex format.&lt;/p&gt;&#xA;&lt;p&gt;Could anyone pleaes help me?&lt;/p&gt;&#xA;" OwnerUserId="23645" LastActivityDate="2024-02-19T04:48:51.727" Title="Some whiteboard program specialized for &quot;typing&quot; with latex support" Tags="|reference-request|computer-programs|latex|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27387" PostTypeId="2" ParentId="27381" CreationDate="2024-01-16T22:54:26.787" Score="1" Body="&lt;p&gt;I'm not sure if this qualifies as &amp;quot;IMO-style&amp;quot;, but there is the International Math Youth Challenge &lt;a href=&quot;https://iymc.info/en/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://iymc.info/en/&lt;/a&gt;, founded in 2018. Regarding age limit, the FAQ state&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;All students (high school or university) are allowed to participate. Students from all grades and all countries are invited to participate! However, there are two age categories (under 18 years and over 18 years).&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I think you can estimate &amp;quot;level&amp;quot; best when having a look at previous problems: &lt;a href=&quot;https://iymc.info/en/training&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://iymc.info/en/training&lt;/a&gt;. Note that all participants are given the same problem sets, but the over 18 year olds are required to achieve more points in order to qualify for the next round.&lt;/p&gt;&#xA;&lt;p&gt;There is no ranking of all participants, but for the final round the winners (without score) are published: &lt;a href=&quot;https://iymc.info/en/announcement2023&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://iymc.info/en/announcement2023&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user23606" LastActivityDate="2024-01-16T22:54:26.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27388" PostTypeId="2" ParentId="27381" CreationDate="2024-01-17T12:12:33.433" Score="2" Body="&lt;p&gt;I know that at least in Germany the qualification for the International Math Olympiad consist of an entire pyramid of math competition exams at different levels and age groups. The lowest level is open to all and you qualify for the next higher level by doing well enough. All of them are in a format very similar to the IMO, very small number of problems, several hours long written exam, no knowledge beyond the age apropriate school knowledge is required.&lt;/p&gt;&#xA;&lt;p&gt;The problems and solutions are all available &lt;a href=&quot;https://www.mathematik-olympiaden.de/moev/aufgaben?view=aktaufg&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt; (in German) and range from 3rd grade (8 to 9 year olds) to 11/12 year of school (16 to 18 years old) and have 4 levels: school, regional, statewide, federal with a significantly increasing level of difficulty.&lt;/p&gt;&#xA;" OwnerUserId="13433" LastActivityDate="2024-01-17T12:12:33.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27389" PostTypeId="1" AcceptedAnswerId="27402" CreationDate="2024-01-17T13:47:45.143" Score="6" ViewCount="314" Body="&lt;p&gt;I am curious about the effects of certain phrasings in educational contexts, particularly in mathematics. In fields like physiotherapy, using phrases like &amp;quot;it's just lifting your leg&amp;quot; can seem dismissive and fail to empathize with the patient's experience. Similarly, in education, saying &amp;quot;it's just using the chain rule&amp;quot; might undermine the student's challenges.&lt;/p&gt;&#xA;&lt;p&gt;Are there any studies or research that explore the impact of such seemingly trivializing language in the context of teaching mathematics? How does this language affect student perception, motivation, and understanding of mathematical concepts? I am looking for references about this topic.&#xA;i've done an animation trying to explain the context: &lt;a href=&quot;https://youtu.be/vnhK3bTVYLk&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/vnhK3bTVYLk&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="20488" LastEditDate="2024-01-20T00:31:30.897" LastActivityDate="2024-01-20T00:31:30.897" Title="Is there any research on the impact of dismissive language (e.g., 'it's just...') in mathematics education?" Tags="|language-use|" AnswerCount="3" CommentCount="5" ClosedDate="2024-01-22T18:40:05.830" ContentLicense="CC BY-SA 4.0" />
  <row Id="27391" PostTypeId="2" ParentId="26510" CreationDate="2024-01-17T21:14:11.917" Score="0" Body="&lt;p&gt;For olympiad preparation, the best book in my opinion is&lt;/p&gt;&#xA;&lt;h2&gt;Arthur Engel, Problem-Solving Strategies.&lt;/h2&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://matheducators.stackexchange.com/questions/8173/self-teaching-theory-for-olympiad-need-advice/27380#27380&quot;&gt;this other answer&lt;/a&gt; I posted.&lt;/p&gt;&#xA;" OwnerUserId="23642" LastEditorUserId="23642" LastEditDate="2024-01-17T21:32:03.550" LastActivityDate="2024-01-17T21:32:03.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27392" PostTypeId="2" ParentId="27374" CreationDate="2024-01-18T01:09:53.790" Score="1" Body="&lt;p&gt;This is an example for what you are looking for. I didn't include the solution!&lt;/p&gt;&#xA;&lt;p&gt;Four grasshoppers sit at four vertices of a square. Every second one&#xA;of the grasshoppers jumps over another one and lands at the symmetric&#xA;point (that is, if a grasshopper jumps from point &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and lands&#xA;at &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, then vectors &lt;span class=&quot;math-container&quot;&gt;$\vec{AB}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\vec{BC}$&lt;/span&gt; are equal). Prove that the four of them will never form a square bigger than the original one.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastActivityDate="2024-01-18T01:09:53.790" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27393" PostTypeId="1" CreationDate="2024-01-18T02:23:23.877" Score="1" ViewCount="128" Body="&lt;p&gt;I'm not a math teacher but I've stepped into the role to assist a small private school who lost their math teacher during the holidays (I'm a mechanical engineer by trade). The course is Algebra 1 (9th,10th grade) using the Pearson Common Core Algebra 1 textbook (Chapter 5 test). I wasn't given prior tests to go off of, so I've made one with the intent that it cover the essentials, but not be unnecessarily difficult. The students had some trouble keeping up in the first semester, and some have developed some math anxiety because of it, so I want this test to help them regain some confidence with the subject. Any feedback or suggestions from experienced math educators on the content, length, formatting, or difficulty are appreciated.&lt;/p&gt;&#xA;&lt;p&gt;Link to google photos in case the images don't appear correctly in the post&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://photos.google.com/share/AF1QipPmwbeooSRo3-RVOHKX4L6old-hEdpye9yJosSbKyJtMmKlXoN9nU7_kv1A3AC3yQ?key=Zmx2dW9UTUpyWmFwY0NjY1VGR0Vad0xXdkNwRmd3&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://photos.google.com/share/AF1QipPmwbeooSRo3-RVOHKX4L6old-hEdpye9yJosSbKyJtMmKlXoN9nU7_kv1A3AC3yQ?key=Zmx2dW9UTUpyWmFwY0NjY1VGR0Vad0xXdkNwRmd3&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://lh3.googleusercontent.com/pw/ABLVV86TVp0JKPpA4u3fR9dCeTGFhcHDWQsyBzPCpE9en6ShR9CPFriRLCW2IYTWGu6BxMJRFRzPMWKKLmCNpNKM8Nhg2h3vw33uav-ACB4XOslcD_JsjHGjk7-CCtEsKmIUlSbWxBO50Yg_XPPLgS7_JF4TBA=w962-h1271-s-no-gm?authuser=0&quot; alt=&quot;Text&quot; /&gt;&#xA;&lt;img src=&quot;https://lh3.googleusercontent.com/pw/ABLVV87zEitJj60lAxhI54-31-wKJxVRG4VUO6L0IDS-bVK1JVcaZFHlE1CpJhCbdWjtcdUG69mRY4c01ZsdbjNZxvISASu5Fh8ik_Az3spxcR4ZFh4jdrTcIupi3f3hBGNiMueVvqtWbvgk06SSHu_Q9CisJQ=w954-h1271-s-no-gm?authuser=0&quot; alt=&quot;Text&quot; /&gt;&#xA;&lt;img src=&quot;https://photos.google.com/share/AF1QipPmwbeooSRo3-RVOHKX4L6old-hEdpye9yJosSbKyJtMmKlXoN9nU7_kv1A3AC3yQ/photo/AF1QipNhKAvv5RIguil1HmE9UggJbNnznYn4Vl35FgA9?key=Zmx2dW9UTUpyWmFwY0NjY1VGR0Vad0xXdkNwRmd3&quot; alt=&quot;Text&quot; /&gt;&lt;/p&gt;&#xA;" OwnerUserId="23650" LastEditorUserId="2083" LastEditDate="2024-01-19T18:24:57.597" LastActivityDate="2024-01-19T18:24:57.597" Title="1st time math teacher 1st test feedback appreciated" Tags="|secondary-education|algebra|usa|testing|test-design|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27394" PostTypeId="2" ParentId="27379" CreationDate="2024-01-18T03:03:09.443" Score="1" Body="&lt;p&gt;My favorite alternate method is what one of my students called &amp;quot;adding by subtracting&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Taking a problem from above: &lt;strong&gt;789 + 62&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;First, subtract 1 from 62 and add it to 789 to turn the problem into &lt;strong&gt;790 + 61&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then, subtract 10 from 61 and add it to 790 to turn the problem into &lt;strong&gt;800 + 51&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then it is hopefully easy to get &lt;strong&gt;851&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You can also &amp;quot;subtract by adding&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;789 - 62 = 789+8 - 62+8 = 797 - 70 = 727&lt;/p&gt;&#xA;&lt;p&gt;Or:&lt;/p&gt;&#xA;&lt;p&gt;789 - 62 = 789+1 - 62+1 = 790 - 63 = 730 - 3 = 727&lt;/p&gt;&#xA;&lt;p&gt;Many of you might be thinking &amp;quot;neither of those is easier&amp;quot;. Of course not, &lt;strong&gt;for you&lt;/strong&gt;. A very small portion of my tutoring students have serious lightbulb moments when I mention other ways to think about adding and subtracting. Some brains are just &lt;em&gt;different&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Another thing about these methods is they are more catered to doing the math in your head without paper. Again, some students find the writing of math to be the most onerous part and giving them some mental math techniques can really help make them hate math class a lot less.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This is a little more esoteric, but I like to divide through prime factorization. So... how about 489/48:&lt;/p&gt;&#xA;&lt;p&gt;489 = 163 * 3 and 48 = 12 * 4 = 3 * 2^4 (not sure if 163 is prime but it's coprime with 2^4 of course)&lt;/p&gt;&#xA;&lt;p&gt;So 489/48 = 163/16 = 10 3/16&lt;/p&gt;&#xA;" OwnerUserId="5176" LastEditorUserId="5176" LastEditDate="2024-01-18T03:13:39.767" LastActivityDate="2024-01-18T03:13:39.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27395" PostTypeId="2" ParentId="27379" CreationDate="2024-01-18T07:10:14.777" Score="1" Body="&lt;p&gt;In basic mathematical operations, alternative approaches can include mental math shortcuts, the use of different number bases, or applying algebraic manipulations. Strategies like estimation, breaking down problems into simpler steps, or leveraging patterns and properties offer alternative ways to approach addition, subtraction, multiplication, and division.&lt;/p&gt;&#xA;" OwnerUserId="23651" LastActivityDate="2024-01-18T07:10:14.777" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27396" PostTypeId="2" ParentId="27379" CreationDate="2024-01-18T16:13:35.237" Score="2" Body="&lt;p&gt;I would like to challenge the view that knowing and using more than one method is necessarily harmful. I shall take subtraction as an example.&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Subtraction&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Wikipedia page on subtraction&lt;/a&gt; there are two main methods here, the American and the Austrian. I remember confusion at secondary school when boys had been taught different methods at their primary school. While trying to clear this up the maths master remarked to me &amp;quot;There is a third way, Dewey, isn't there? The right way.&amp;quot;. I answered yes as tactically the best answer. I was not challenged on what it was but there is indeed another way which I suspect I learned from my father and which I still use now for mental arithmetic alongside the other method for bigger numbers. For small numbers I use counting up. So to take 37 from 95 I would go 3 (to make 37 up to 40) 53 (to make 37 up to 90) 58. I suspect many people in fact use multiple methods depending on circumstances and I can see no real disadvantage as long as they have different areas of application.&lt;/p&gt;&#xA;&lt;p&gt;If our schoolmasters had been unkind enough to make us do subtraction in the mixed radix systems we used for distance and weight at the time I suspect my method would have been better there. Of course it was used in a process for currency called giving change but that is obsolete now we have almost abandoned cash&lt;/p&gt;&#xA;" OwnerUserId="23475" LastActivityDate="2024-01-18T16:13:35.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27397" PostTypeId="2" ParentId="27389" CreationDate="2024-01-18T16:54:51.740" Score="5" Body="&lt;p&gt;In my experience (obviously anecdotal, but take from it what you will), it all comes down to whether you're dismissing the struggles of the student (i.e., &amp;quot;why is this not easy for you&amp;quot;) vs the complexity of the problem (i.e., &amp;quot;the problem looks intimidating until you realize this key insight that makes it more routine based on what you already know&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;For instance, suppose a student is struggling to use the chain rule to compute &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\textrm d}{\textrm dx} \left[ \sin (x^2) \right].$&lt;/span&gt; In that case, it would be unhelpful to say &amp;quot;it's just the chain rule&amp;quot; because that's what they're trying to do.&lt;/p&gt;&#xA;&lt;p&gt;But now imagine that a student has learned the chain rule and is comfortable applying it, and you're teaching them implicit differentiation. To show them how to compute &lt;span class=&quot;math-container&quot;&gt;$\dfrac{\textrm d}{\textrm dx} \left[ y^2 \right],$&lt;/span&gt; you might say something like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I know it looks weird because there's no &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in that expression being differentiated. However, if you just write &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as a function &lt;span class=&quot;math-container&quot;&gt;$y(x),$&lt;/span&gt; then it's easier to see that this is ultimately just boils down to a straightforward application of the chain rule:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{\textrm d}{\textrm dx} \left[ y(x)^2 \right] = 2y(x) \dfrac{\textrm dy(x)}{\textrm dx} = 2y \dfrac{\textrm dy}{\textrm dx}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this case, the point the phrases &amp;quot;if you just,&amp;quot; &amp;quot;it's easier to see,&amp;quot; and &amp;quot;a straightforward application&amp;quot; is to highlight key insights that can be used to reduce the complexity of the problem. You're trying to convey that a change of perspective can make this problem feel less intimidating and more routine to the student based on what they already know.&lt;/p&gt;&#xA;&lt;p&gt;That said, it's easy to overestimate the prior knowledge of the student, which can lead an expert to &lt;em&gt;think&lt;/em&gt; they're dismissing the complexity of the problem when they're actually dismissing the struggles of the student.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2024-01-18T16:54:51.740" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27398" PostTypeId="2" ParentId="27389" CreationDate="2024-01-19T05:28:20.007" Score="9" Body="&lt;p&gt;Hypothetically, a study researching the effect of a single word (&amp;quot;just&amp;quot;) within a complex educational context would fall under the heading of a psychological &lt;a href=&quot;https://en.wikipedia.org/wiki/Priming_(psychology)&quot; rel=&quot;noreferrer&quot;&gt;priming effect&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In some sense, these studies are easy to do. But the problem is, these kinds of studies are infamous for being poorly designed, very subjective, easily biased by researchers, not generalizable to real-world classrooms, and routinely fail under larger replication attempts.&lt;/p&gt;&#xA;&lt;p&gt;Psychologist and Nobel laureate Daniel Kahneman wrote a letter to the community in &lt;em&gt;Nature&lt;/em&gt; in 2012, &lt;a href=&quot;https://www.nature.com/news/polopoly_fs/7.6716.1349271308!/suppinfoFile/Kahneman%20Letter.pdf&quot; rel=&quot;noreferrer&quot;&gt;saying this&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As all of you know, of course, questions have been raised about the&#xA;robustness of priming results. The storm of doubts is fed by several&#xA;sources, including the recent exposure of fraudulent researchers,&#xA;general concerns with replicability that affect many disciplines,&#xA;multiple reported failures to replicate salient results in the priming&#xA;literature, and the growing belief in the existence of a pervasive&#xA;file drawer problem that undermines two methodological pillars of your&#xA;field: the preference for conceptual over literal replication and the&#xA;use of meta-analysis. Objective observers will point out that the&#xA;problem could well be more severe in your field than in other branches&#xA;of experimental psychology, because every priming study involves the&#xA;invention of a new experimental situation.&lt;/p&gt;&#xA;&lt;p&gt;For all these reasons, right or wrong, your field is now the poster&#xA;child for doubts about the integrity of psychological research.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;While it's unlikely that any research exists on such a fine-grained linguistic issue in math education, if any such research exists, we should definitely not trust its findings, nor expect that its findings would be useful in real classrooms.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2024-01-19T05:28:20.007" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27400" PostTypeId="1" CreationDate="2024-01-19T17:06:00.790" Score="11" ViewCount="914" Body="&lt;p&gt;In the topic of integration and anti-derivatives in Calculus we come across cases where the attempt at integration by parts brings us back to the original integral, the most basic example being &lt;span class=&quot;math-container&quot;&gt;$\int e^{a x} \cos(b x) \,dx$&lt;/span&gt;. Is there a standard name for this type of integrals? I have been calling them &amp;quot;self-referral&amp;quot;, but the only usage of this expression that I can find is in health insurance! Have  you seen a name in standard textbooks? I have seen &amp;quot;circular integrals&amp;quot; online. What do you call them? Does anyone call it &amp;quot;self-referential&amp;quot;? (Stewart does not have a name for it. Self-referential is an expression in logic. Circular integral might be used for contour integrals.)&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="205" LastEditDate="2024-02-10T23:17:59.177" LastActivityDate="2024-02-12T01:08:56.490" Title="What is the terminology for &quot;self-referral&quot; integrals in calculus?" Tags="|calculus|terminology|integration|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="27401" PostTypeId="2" ParentId="27400" CreationDate="2024-01-19T19:22:17.317" Score="10" Body="&lt;p&gt;To my knowledge, this is most commonly known as &lt;a href=&quot;https://www.google.com/search?q=cyclic%20integration%20by%20parts&amp;amp;tbm=vid&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;cyclic&amp;quot; integration by parts&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2024-01-19T19:22:17.317" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27402" PostTypeId="2" ParentId="27389" CreationDate="2024-01-20T00:22:59.697" Score="2" Body="&lt;p&gt;You can find one such study &lt;a href=&quot;https://drive.google.com/file/d/1yIvy9951kk4bxjjJ40deT4QsFfYpcM7K/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Kroeper, K. M., Muenks, K., Canning, E. A., &amp;amp; Murphy, M. C. (2022). An exploratory study of the behaviors that communicate perceived instructor mindset beliefs in college stem classrooms. &lt;em&gt;Teaching and Teacher Education&lt;/em&gt;, 114.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Take a look at Table 1 on page 5, for example.&lt;/p&gt;&#xA;" OwnerUserId="20488" LastActivityDate="2024-01-20T00:22:59.697" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27403" PostTypeId="2" ParentId="27386" CreationDate="2024-01-20T02:35:58.843" Score="0" Body="&lt;p&gt;Xournal++ supports inline Latex. Would that do what you want?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://xournalpp.github.io/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://xournalpp.github.io/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15606" LastActivityDate="2024-01-20T02:35:58.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27404" PostTypeId="1" CreationDate="2024-01-20T15:24:16.370" Score="-2" ViewCount="193" Body="&lt;p&gt;I found the &lt;a href=&quot;https://play.google.com/store/apps/details?id=com.rvappstudios.math.games.kids.addition.subtraction.multiplication.division&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Games: Math for Kids&lt;/a&gt; to be excellent (if not extremely, excellent) at teaching elementary school math.&lt;/p&gt;&#xA;&lt;p&gt;But it made me think of some tricks, that could be displayed when teaching at a blackboard or on some apps.&lt;/p&gt;&#xA;&lt;p&gt;Here is one of them:&lt;/p&gt;&#xA;&lt;p&gt;I was wondering whether anyone can improve on the following method of doing addition for someone who has learned multiplication tables.&lt;/p&gt;&#xA;&lt;p&gt;Suppose one needs to do&lt;/p&gt;&#xA;&lt;p&gt;7+3&lt;/p&gt;&#xA;&lt;p&gt;The user draws a statisíce on top of a staircase. The staircase at the top is there steps long, the one at the bottom seven steps long.&lt;/p&gt;&#xA;&lt;p&gt;The user sees that the bottom three steps of the three step staircase are stacked upon the first three steps of the bottom step staircase.&lt;/p&gt;&#xA;&lt;p&gt;So the user can see the staircases coming together, as though, being, superimposed.&lt;/p&gt;&#xA;&lt;p&gt;They can see that&lt;/p&gt;&#xA;&lt;p&gt;3+3 = 3's timetable carried forward two steps (3*2) = 6&lt;/p&gt;&#xA;&lt;p&gt;Just like the first three stairs can be visually seen to be 3, the remaining stairs can be visually seen to be 4&lt;/p&gt;&#xA;&lt;p&gt;So the user must do&lt;/p&gt;&#xA;&lt;p&gt;6+4 = counting up yields 10&lt;/p&gt;&#xA;&lt;p&gt;The full equation is&lt;/p&gt;&#xA;&lt;p&gt;7+3=(3+4)+3=6+4=10&lt;/p&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;p&gt;Second point, another trick is to find the complement of 6 in ten, in this case it is just 4, which together with 6, fits in exactly in 10. Noticing this, you can just fill in the sum.&lt;/p&gt;&#xA;&lt;p&gt;But if you had to do 6+5 you would use the staircases method to do 1+(5+5)=1+10=11.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/COf0Y.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/COf0Y.jpg&quot; alt=&quot;staircase calculation example&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I wonder whether anyone works like to implement any of these tricks and others in a fast and mentally comfortable math trick teaching app for elementary school kids I was thinking. I would like to invite anyone with tricks and ideas and feels of what feels comfortable (not just efficient) and, hopefully, fast, to do so.&lt;/p&gt;&#xA;&lt;p&gt;Meanwhile, this question would like to ask for any observations or improvements to the staircase method.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="12214" LastEditorUserId="12214" LastEditDate="2024-01-20T15:32:17.813" LastActivityDate="2024-01-21T04:23:20.687" Title="Staircase method of small integer addition" Tags="|primary-education|applications|apps|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27405" PostTypeId="2" ParentId="27404" CreationDate="2024-01-20T16:22:06.183" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I would like to invite anyone with tricks and ideas and feels of what feels comfortable (not just efficient) and, hopefully, fast, to do so.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Maybe this is an unpopular answer nowadays, but honestly I think the best &amp;quot;trick&amp;quot; is to just memorize your single-digit addition facts (while conceptually understanding that addition is ultimately just &amp;quot;counting up&amp;quot; and the order doesn't matter).&lt;/p&gt;&#xA;&lt;p&gt;Yeah, it takes some up-front work to memorize them, but it's really not &lt;em&gt;that&lt;/em&gt; much work, and once you can recall these automatically, addition in general becomes super easy, comfortable, efficient, and fast.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, if you don't memorize them, then your limited working memory will continually get hijacked and wasted on low-level processing, which will make it difficult to build more advanced skills of which single-digit addition is a component. At best, you'll take longer to learn/execute more advanced skills and have an elevated error rate while executing them; at worst, you'll experience total cognitive overload and not be able to learn/execute more advanced skills at all. Everything becomes super hard, uncomfortable, inefficient, and slow.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2024-01-20T16:22:06.183" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="27406" PostTypeId="1" CreationDate="2024-01-20T18:09:24.300" Score="4" ViewCount="370" Body="&lt;p&gt;I am a med student and left maths in 10th class. Log and antilog and calculus etc are taught in 11th and 12th. Every time I have any exam that includes usage of these topics I go to Youtube and search for a video over log antilog and there are tons of videos with this common title: solve xyz in 2 sec.&lt;/p&gt;&#xA;&lt;p&gt;I watch them but when my next exams come I have to watch them again and again and repeat.&lt;/p&gt;&#xA;&lt;p&gt;Is repetition every week the only way out of this problem or is there any faster (not necessarily easier) thing that could be done?&lt;/p&gt;&#xA;" OwnerUserId="23655" LastEditorUserId="7365" LastEditDate="2024-01-26T10:29:31.140" LastActivityDate="2024-01-26T10:29:31.140" Title="Why don't these 'solve in 2 sec' tricks work?" Tags="|self-learning|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27407" PostTypeId="1" CreationDate="2024-01-20T23:01:26.880" Score="5" ViewCount="166" Body="&lt;p&gt;I'm reading a book chapter (written in 1992) about the teaching of mental arithmetic and estimation techniques in elementary school. The author refers to a particular series of books as &amp;quot;Out-of-print American Program,&amp;quot; with a grade 3 book dated 1962 and a grade 5 book dated 1956.&lt;/p&gt;&#xA;&lt;p&gt;The third-grade text has a story of a boy named Joe Bates who estimates that a 49-cent hat and a 29-cent ball will together cost about 80 cents. Then it has a little rhyme: &amp;quot;A wise boy is Joe Bates. When he adds, he estimates.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The fifth-grade text suggests three methods of estimating the quotient 1215 ÷ 52.&lt;/p&gt;&#xA;&lt;p&gt;I'd like to know the name of the program so that I can investigate it further. I've tried searching Google and the Internet Archive for this text, but to no avail.&#xA;Thanks for any help!&lt;/p&gt;&#xA;" OwnerUserId="23656" LastEditorUserId="23656" LastEditDate="2024-01-20T23:05:16.957" LastActivityDate="2024-01-20T23:05:16.957" Title="Please identify this American elementary textbook series from the 1950s" Tags="|textbooks|curriculum|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27408" PostTypeId="2" ParentId="27374" CreationDate="2024-01-20T23:37:52.153" Score="0" Body="&lt;p&gt;If the vertices of a regular &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-gon have integer coordinates, then &lt;span class=&quot;math-container&quot;&gt;$n \leq 6$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Proof: Draw the vectors representing the sides of the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-gon so that their initial point is the origin. Then you have drawn a new regular &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-gon whose vertices have integer coordinates and that is &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; times the size of the previous one, where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is a constant depending only on &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. And if &lt;span class=&quot;math-container&quot;&gt;$n &amp;gt; 6$&lt;/span&gt;, then it is easily seen that &lt;span class=&quot;math-container&quot;&gt;$a &amp;lt; 1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="23656" LastActivityDate="2024-01-20T23:37:52.153" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27409" PostTypeId="2" ParentId="27404" CreationDate="2024-01-21T04:23:20.687" Score="-3" Body="&lt;p&gt;I thought of the concentric rotating polygon method.&lt;/p&gt;&#xA;&lt;p&gt;In this method, which could prove extremely strong, for some users, in some cases, or for an extremely vast variety of enthusiasts and enthusiastic aficionados, also intriguing, and you might think gotta love and try it:&lt;/p&gt;&#xA;&lt;p&gt;In this method, you envision one larger polygon with a smaller polygon inside it. The inner polygon rotates and covers the edges of the outer polygon.&lt;/p&gt;&#xA;&lt;p&gt;As it does so, you can see where it touches, to paint the outer polygon with all of it's sides paint.&lt;/p&gt;&#xA;&lt;p&gt;The rest of the sides is the rest you need to add.&lt;/p&gt;&#xA;&lt;p&gt;And, then, you get&lt;/p&gt;&#xA;&lt;p&gt;3+7 = (3+3)+4 = 6+4 = 10&lt;/p&gt;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&lt;p&gt;You must get comfortable with the case n=2, as well. Because, here, you can't have a polygon (you must consider a two sided line segment).&lt;/p&gt;&#xA;&lt;p&gt;It, turns over itself, twice (painting, twice).&lt;/p&gt;&#xA;&lt;p&gt;So, you may do&lt;/p&gt;&#xA;&lt;p&gt;2+7 = (2+2)+5 = 4 + 5 = 9&lt;/p&gt;&#xA;&lt;p&gt;You, might, get more entusiastic, doing it this way (no, you wouldn't want, to just add two at the end).&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ote3F.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ote3F.jpg&quot; alt=&quot;concentric rotating polygon method&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12214" LastActivityDate="2024-01-21T04:23:20.687" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27410" PostTypeId="1" CreationDate="2024-01-21T05:31:31.297" Score="-5" ViewCount="103" Body="&lt;p&gt;Here are some suggestions for memorizing the difference between an open interval and a closed interval.&lt;/p&gt;&#xA;&lt;p&gt;[a,b]={x|a&amp;lt;=x&amp;lt;=b}&lt;/p&gt;&#xA;&lt;p&gt;(a,b)={x|a&amp;lt;x&amp;lt;b}&lt;/p&gt;&#xA;&lt;p&gt;The first thing I would notice is that the bracket (square bracket), looks a little bit like the letter&lt;/p&gt;&#xA;&lt;p&gt;1.&lt;/p&gt;&#xA;&lt;p&gt;|= (the letter C, but, draw a square C, otherwise you get confused. I cannot do it in ascii.&lt;/p&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;p&gt;2.&lt;/p&gt;&#xA;&lt;p&gt;Remember, that it's a square C&lt;/p&gt;&#xA;&lt;p&gt;The parenthesis, is a little bit like the letter O (notice the roundness). O, is like, &amp;quot;omit&amp;quot;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;To envision that the dot near the equal, part, is included, envision a door closing (this is the square bracket, with the door being the vertical edge and the doors by the door being the other two horizontal edges. The edges, close, onto the door, and include (by squishing it), the point that is in it.&lt;/p&gt;&#xA;&lt;p&gt;|=  ...   |•|&lt;/p&gt;&#xA;&lt;p&gt;Again, this is hard to draw. A video would be needed.&lt;/p&gt;&#xA;&lt;p&gt;2.&lt;/p&gt;&#xA;&lt;p&gt;For the open interval, envision a rubber band, like a spring. The rubber band pushes the dot on the edge, out, (onto the points not included in the endpoint).&lt;/p&gt;&#xA;&lt;p&gt;Do things right.&lt;/p&gt;&#xA;&lt;p&gt;Use this mnemonic from the beginning.&lt;/p&gt;&#xA;&lt;p&gt;You will never get confused.&lt;/p&gt;&#xA;&lt;p&gt;....&lt;/p&gt;&#xA;&lt;p&gt;For those without textbook, let me read the notation for you.&lt;/p&gt;&#xA;&lt;p&gt;[a,b]={x|a&amp;lt;=x&amp;lt;=b}&lt;/p&gt;&#xA;&lt;p&gt;The closed interval a b is the set of all x, such that (| is read such that), a is less than or equal to x is less than or equal to b.&lt;/p&gt;&#xA;&lt;p&gt;(a,b)={x|a&amp;lt;x&amp;lt;b}&lt;/p&gt;&#xA;&lt;p&gt;The open interval a b is the set of all x, such that, a is &lt;em&gt;strictly&lt;/em&gt; less than x is strictly less than b.&lt;/p&gt;&#xA;&lt;p&gt;....&lt;/p&gt;&#xA;&lt;p&gt;x is a variable&lt;/p&gt;&#xA;&lt;p&gt;The stuff inside the { } (curly brackets), denotes a set.&lt;/p&gt;&#xA;&lt;p&gt;a and b are points (constants on the real line).&lt;/p&gt;&#xA;&lt;p&gt;.....&lt;/p&gt;&#xA;&lt;p&gt;How would you make improvements to this explanation, and, more importantly, what mnemonics would you utilize to make sure the students get it right, and, then, get it right for the rest of their life.&lt;/p&gt;&#xA;&lt;p&gt;This post is about notation, and, about, making it such, that the student does not get confused by notation.&lt;/p&gt;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="12214" LastActivityDate="2024-01-21T06:15:34.940" Title="Mnemonics for memorizing the difference between a closed interval and an open interval" Tags="|notation|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27411" PostTypeId="2" ParentId="27410" CreationDate="2024-01-21T05:58:04.537" Score="6" Body="&lt;p&gt;Honestly this all feels a bit over the top to me. I think all that needs to be said is that the open interval &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; fits inside the closed interval &lt;span class=&quot;math-container&quot;&gt;$[a,b].$&lt;/span&gt; That's what I've always thought was the intuition behind the notation anyway. An oval &lt;span class=&quot;math-container&quot;&gt;$()$&lt;/span&gt; fits inside a rectangle &lt;span class=&quot;math-container&quot;&gt;$[\,\,]$&lt;/span&gt; with the same width and height.&lt;/p&gt;&#xA;&lt;p&gt;It even looks that way visually with inequalities. Literally, the physical text &lt;span class=&quot;math-container&quot;&gt;$a &amp;lt; x &amp;lt; b$&lt;/span&gt; fits inside the text &lt;span class=&quot;math-container&quot;&gt;$a \leq x \leq b.$&lt;/span&gt; The latter is just the former with some extra lines underneath.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-01-21T06:15:34.940" LastActivityDate="2024-01-21T06:15:34.940" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27412" PostTypeId="1" AcceptedAnswerId="27433" CreationDate="2024-01-21T15:49:59.827" Score="5" ViewCount="2181" Body="&lt;p&gt;I am planning to become a math high-school teacher and have the following question:&lt;/p&gt;&#xA;&lt;p&gt;What Probability Theory should I base myself on to teach probabilities to students ?&lt;/p&gt;&#xA;&lt;p&gt;The classical approach is via the Kolmogorov axioms. The second one would be the Probability Theory via the Cox theorem, Probability Theory whose foundations are described in Jaynes' book &lt;a href=&quot;http://www.med.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/JaynesProbabilityTheory.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Probability Theory: the Logic of Science&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;These two SE questions (&lt;a href=&quot;https://stats.stackexchange.com/questions/126056/do-bayesians-accept-kolmogorovs-axioms&quot;&gt;this&lt;/a&gt; and  &lt;a href=&quot;https://math.stackexchange.com/questions/2310252/bayesian-probability-and-measure-theory&quot;&gt;this&lt;/a&gt;) summarize the antagonism between the two approaches.&lt;/p&gt;&#xA;&lt;p&gt;The thing is that I am much more confortable with the second approach, and (since I know almost nothing about Measure Theory) I am unconfortable with the classical Kolmogorov approach.&lt;/p&gt;&#xA;&lt;p&gt;Yet, I am afraid to teach the second approach (the one taught by Jaynes) because:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I am afraid to teach the &amp;quot;wrong math&amp;quot; to students (see why below).&lt;/li&gt;&#xA;&lt;li&gt;even if it is no &amp;quot;wrong math&amp;quot;, I am afraid that an inspector comes along and punishes me for &lt;em&gt;teaching what he thinks are wrong math&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am totally unable to say whether Cox theorem derivations are weaker than the Kolmogorov approach, whether they &lt;em&gt;are&lt;/em&gt; but it doesn't quite matter to teach Cox-based Probability Theory.&lt;/p&gt;&#xA;&lt;p&gt;Yet from what I have understood, countable additivity is not ensured by Cox-based Probability Theory even though &lt;a href=&quot;https://arxiv.org/abs/1411.7920&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt; seems to claim that both approaches are equivalent.&lt;/p&gt;&#xA;&lt;p&gt;I am aware that probably the best thing I could do is to teach the Probability Theory I am most confortable with and prepare to elaborate on why I chose it to the hypothetical inspector ?&#xA;However, could there be something wrong with that ?&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;&lt;p&gt;PS: I am of course not going to teach the whole theory (be it Kolmogorov or the other one) to students, but I want to know what should I derive from what (and what I can use to prove these derivations).&lt;/p&gt;&#xA;" OwnerUserId="22167" LastEditorUserId="2083" LastEditDate="2024-01-21T19:41:31.197" LastActivityDate="2024-02-16T08:07:52.177" Title="Should I choose Cox-based Jaynes' approach or Kolmogorov approach to base myself on to teach probabilities to high-school students?" Tags="|secondary-education|teaching|probability|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27413" PostTypeId="2" ParentId="26811" CreationDate="2024-01-21T17:19:02.650" Score="0" Body="&lt;p&gt;Is there a resource for learning to read mathematical notation/equations/formulae?&lt;/p&gt;&#xA;&lt;p&gt;Yes, there are several resources available for learning to read mathematical notation, equations, and formulae. Here are some suggestions:&lt;/p&gt;&#xA;&lt;p&gt;Khan Academy:&lt;/p&gt;&#xA;&lt;p&gt;Website: Khan Academy&#xA;Khan Academy offers a comprehensive set of lessons on various math topics, including how to read and understand mathematical notation. The lessons are organized in a progressive manner, making it easy for beginners to follow.&#xA;Purplemath:&lt;/p&gt;&#xA;&lt;p&gt;Website: Purplemath&#xA;Purplemath provides lessons and resources to help students understand and navigate mathematical concepts, including notations and equations. The website is particularly useful for algebra-related topics.&#xA;MIT OpenCourseWare:&lt;/p&gt;&#xA;&lt;p&gt;Website: MIT OCW&#xA;MIT OpenCourseWare offers free access to a wide range of math courses. You can find lecture notes, assignments, and other materials that cover mathematical notation and equations.&#xA;Paul's Online Math Notes:&lt;/p&gt;&#xA;&lt;p&gt;Website: Paul's Online Math Notes&#xA;This resource provides comprehensive notes and tutorials on various math topics, including algebra, calculus, and more. It can be a helpful reference for understanding mathematical notation.&#xA;Mathway:&lt;/p&gt;&#xA;&lt;p&gt;Website: Mathway&#xA;Mathway is an online tool that allows you to input mathematical problems, and it provides step-by-step solutions. It can be a helpful resource for understanding how different notations and equations are solved.&#xA;Wolfram Alpha:&lt;/p&gt;&#xA;&lt;p&gt;Website: Wolfram Alpha&#xA;Wolfram Alpha is a computational search engine that can solve mathematical problems and provide detailed explanations. It's a valuable tool for understanding mathematical notations and exploring different mathematical concepts.&#xA;Books:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;How to Read and Do Proofs: An Introduction to Mathematical Thought Processes&amp;quot; by Daniel Solow&#xA;&amp;quot;How to Solve It: A New Aspect of Mathematical Method&amp;quot; by George Pólya&#xA;Remember that practice is crucial when it comes to understanding mathematical notation. Work through exercises and problems regularly to reinforce your understanding. Additionally, don't hesitate to ask for help from teachers, tutors, or online communities if you encounter difficulties with specific concepts.&lt;/p&gt;&#xA;" OwnerUserId="23659" LastActivityDate="2024-01-21T17:19:02.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27414" PostTypeId="2" ParentId="27412" CreationDate="2024-01-21T19:22:02.577" Score="9" Body="&lt;p&gt;I'm not familiar with Cox approach at all, so I cannot provide a qualified comparison, but I find the Kolmogorov's axioms pretty easy to comprehend and use and I'll try to explain why.&lt;/p&gt;&#xA;&lt;p&gt;No matter what approach you take, you should probably start with the finite discrete setting when you have just finitely many possible outcomes &lt;span class=&quot;math-container&quot;&gt;$\omega_1,\dots,\omega_n$&lt;/span&gt; forming the set of elementary events &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; with some probabilities &lt;span class=&quot;math-container&quot;&gt;$p_j$&lt;/span&gt; assigned to them, so that the probability of a compound event &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is just &lt;span class=&quot;math-container&quot;&gt;$\sum_{j:\omega_j\in E}p_j$&lt;/span&gt;. You can then teach almost all relevant concepts up to independence and conditioning in this simple setting and I wouldn't move any further until this part is mastered entirely.&lt;/p&gt;&#xA;&lt;p&gt;The second useful initial setting to cover is the &amp;quot;geometric&amp;quot; probability, when you have a random point distributed in some domain in &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt; (you start teaching it for &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, of course) according to some nice continuous density and the students learn to find the probability of an event by figuring out what part of the domain corresponds to it and integrating the density over that part. Here you may have a gentle nab at the countable additivity already (like asking &amp;quot;what is a probability that a random variable uniformly distributed on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; is rational&amp;quot;). Again, before that part is mastered, I would not proceed to any general axiomatic setup.&lt;/p&gt;&#xA;&lt;p&gt;Kolmogorov's approach then just generalizes these settings in (IMHO) the most direct way. You still have the set &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; of elementary events &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; and the compound events are just the &amp;quot;reasonable&amp;quot; (measurable) subsets of &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;. You don't need to know what exactly &amp;quot;measurable&amp;quot; means, only to understand that all reasonable events (like &lt;span class=&quot;math-container&quot;&gt;$X\in[a,b]$&lt;/span&gt; for random variables &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;) are measurable and that you can perform all finite set-theoretic operations and countable unions and intersections without destroying measurability. Also, you can postulate the countable additivity of the probability. It, of course, raises the question why at least one non-trivial (not discrete and finite) probability space exists, but I doubt your students will care of it any more than about why the set of real numbers exists and I don't recommend that you care about that too much yourself in the beginning. At this point it is important to emphasize that not everything is an event and that one can express a lot of things using only countable operations, but not everything.&lt;/p&gt;&#xA;&lt;p&gt;Once that is understood, you can learn/teach pretty much anything you want, just sweeping under the rug the related existence questions (like why can one construct a probability space with an infinite sequence of independent random variables uniformly distributed on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt;, etc.) but not the questions why something is an event.&lt;/p&gt;&#xA;&lt;p&gt;That is a brief summary of the Kolmogorov's approach as I see it. The only essential thing that I didn't mention in the above outline is the conditional expectation with respect to a &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-subfield of the original &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-field. However, that concept is not hard too, plus I doubt it very-very much that you would ever have an opportunity to reach that far in the high school probability course.&lt;/p&gt;&#xA;&lt;p&gt;As to the &amp;quot;inspectors punishing teachers for teaching wrong math&amp;quot;, yeah, that is always a possibility.&lt;/p&gt;&#xA;" OwnerUserId="19946" LastEditorUserId="60" LastEditDate="2024-01-23T03:31:18.750" LastActivityDate="2024-01-23T03:31:18.750" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27415" PostTypeId="2" ParentId="27412" CreationDate="2024-01-21T20:08:44.817" Score="15" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I think you should (and likely will have to) use the assigned text and approach.  It's incredibly unlikely you will just derive some new approach.  That's not how high school teaching works.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And on top of everything else, you've got other things to concentrate on, not just curriculum design, but pedagogy, behavior management, etc. that will be higher priorities.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And it's not even clear that you would be assigned a probability course, as a new teacher.  You might be teaching more traditional courses.  For that matter, often the modern approach is to slide a little probability (and stats) into the basic courses themselves.  In which case, the content will be very clearly constrained.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;FWIW, even at the &amp;quot;not the whole theory&amp;quot;, your approach seems way TOO HARD for an initial approach.  You should instead do VERY basic frequentist stuff.  And the approach is not &amp;quot;what it's derived from&amp;quot;, but descriptive:  &amp;quot;Here's a permutation and the formula for it.&amp;quot;  Rigorous justifications can wait for college and will only be needed for math majors...not the vast amount of STEM students in hard and soft sciences and in engineering.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="23660" LastActivityDate="2024-01-21T20:08:44.817" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27417" PostTypeId="2" ParentId="27406" CreationDate="2024-01-22T16:50:51.270" Score="9" Body="&lt;p&gt;I think the mistake here is using videos to learn skills. I suggest you obtain a text which suits you, sit down on a hard chair (not a comfortable arm chair), at a table, with a pen and a piece of paper to make notes. Be sure to do all the examples. Then either find a new text with different examples or go back to the beginning of your current one and go through yet more examples.&lt;/p&gt;&#xA;" OwnerUserId="23475" LastActivityDate="2024-01-22T16:50:51.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27418" PostTypeId="2" ParentId="27412" CreationDate="2024-01-23T03:54:18.237" Score="5" Body="&lt;p&gt;&amp;quot;I am planning to become a math high-school teacher&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So it seems to me that you expect too much from high school students. I made the same mistake, the true is the vast majority of them will not understand (or even care for) a derivation of the basic probability results. Be grateful if they show interest to use these results to solve easy problems.&lt;/p&gt;&#xA;&lt;p&gt;So what is truly important at this level is the pedagogy, You have to engage them to the themes with interesting examples, remember that at this age the life is not around maths. In my case, I try to center my class in showing that solving math problems can be very useful in exercising the brain and learn to deal with stressing situations.&lt;/p&gt;&#xA;" OwnerUserId="23666" LastEditorUserId="23666" LastEditDate="2024-01-23T23:04:34.283" LastActivityDate="2024-01-23T23:04:34.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27420" PostTypeId="2" ParentId="27406" CreationDate="2024-01-23T10:49:06.910" Score="7" Body="&lt;p&gt;Watching videos is a deceptive learning activity because it's very passive. It is incredibly easy to watch a video, about any topic, and think you've deeply understood everything that was mentioned in the video. But then it turns out your understanding is actually not deep enough that you could do it yourself.&lt;/p&gt;&#xA;&lt;p&gt;This is true for videos about solving math equations, just like it's true for videos about learning to dance Argentine tango.&lt;/p&gt;&#xA;&lt;p&gt;The only way to face head-on your lack of mastering of a topic is to practice. That doesn't mean sitting passively and watching someone else do it. In the case of solving math equations, it means taking a pen and paper and solving equations yourself.&lt;/p&gt;&#xA;&lt;p&gt;Do you have access to past exam papers? If you can get your hand on those, then solve the equations from past exams.&lt;/p&gt;&#xA;&lt;p&gt;And when you watch youtube videos about solving an equation: pause the video, solve the equation yourself, and do not watch the remainder of the video until you've successfully solved the equation yourself.&lt;/p&gt;&#xA;" OwnerUserId="16159" LastActivityDate="2024-01-23T10:49:06.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27422" PostTypeId="1" CreationDate="2024-01-25T06:32:42.167" Score="2" ViewCount="165" Body="&lt;p&gt;I need advice on methods and techniques for studying mathematics that are commonly used at undergraduate and masters level in mathematics.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What are some strategies that you find useful in &lt;strong&gt;coping with difficult mathematical topics and exams&lt;/strong&gt;?&lt;/li&gt;&#xA;&lt;li&gt;Are &lt;strong&gt;flashcards or cue cards&lt;/strong&gt; a common or effective tool for memorising and understanding mathematical concepts and equations?&lt;/li&gt;&#xA;&lt;li&gt;I would also appreciate hearing about &lt;strong&gt;personal experiences&lt;/strong&gt; of mathematics study techniques that have been particularly effective in your mathematical journey.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Thanks a lot!&lt;/p&gt;&#xA;" OwnerUserId="23678" LastActivityDate="2024-02-06T22:11:26.060" Title="Flashcards and Study Methods for Undergraduate and Masters Degrees" Tags="|mathematical-analysis|struggling-students|student-question|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27423" PostTypeId="2" ParentId="27422" CreationDate="2024-01-25T08:42:12.460" Score="9" Body="&lt;p&gt;I think it all comes down to four things.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;1. Solving Problems&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The most important thing is to solve challenging problems successfully while relying on as little assistance as possible (from people, internet, reference books, worked examples, hints, etc.). Most of the things you learn will of course come from those kinds of external resources, but you need to challenge yourself to apply things independently once you learn them. This may take some work, and you may initially need some help (e.g. studying some example solutions) to actually get through problems, but the goal is to strip away that scaffolding as soon as you're able to do so while continuing to get through problems.&lt;/p&gt;&#xA;&lt;p&gt;Some additional pointers:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Make sure that you're actually solving the problems correctly (you need to check your work against the correct solution, even if you think you got it right).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Try to actually understand what you're doing -- talk yourself through the &lt;em&gt;why&lt;/em&gt; behind each step you carry out; if you find yourself at a loss for words at any step, then you need to spend some time drilling down to the purpose of that step. (Some mental fuzziness is natural when you first learn a new topic and begin working problems, but that fuzziness should fade as you work more problems and you should actively take steps to attack the fuzziness if continues hanging around.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Worked examples (in lecture, in a video or textbook section, etc) are not just for viewing. You should actively work them yourself on a piece of paper. If there's a step that you don't understand, pause and think about it. If you still can't figure it out, ask the instructor. Or if you're on your own, do some Google searches to look for explanations and/or worked examples from different resources (sometimes a different example will help clarify your understanding) -- and if you still can't resolve your confusion even after trying that, then ask a question about it on some forum like StackExchange or Reddit.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;2. Building On Top of What You've Learned&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Once you're able to get through some types of problems successfully, just keep doing this with more advanced problems, building on top of what you've learned. If there are optional challenge problems, do them. You can also read ahead to learn more advanced content and work on the corresponding problems. The goal is to continually layer on top of existing knowledge -- that is, continually acquire new knowledge that exercises prerequisite or component knowledge.&lt;/p&gt;&#xA;&lt;p&gt;Layering produces a kind of &amp;quot;structural integrity&amp;quot; in your knowledge, just like what happens in engineering. When advanced features are built on top of a system, they sometimes fail in ways that reveal previously-unknown foundational weaknesses in the underlying structure. This forces engineers to fortify the underlying structure so that the system can accommodate new elements without compromising its integrity. Fortifying the underlying structure often requires improving its organization and elegance, which, when applied to your knowledge, produces deep understanding and insight. (When the structural integrity of a system is increased, it also becomes easier to add more advanced features in general -- and in the same way, when the structural integrity of your knowledge is increased, it becomes easier to assimilate new knowledge in general.)&lt;/p&gt;&#xA;&lt;p&gt;One key decision that you'll have to make here is when to move on to more advanced problems. You don't want to move on to more advanced problems until you're able to do the simpler problems, but you also don't want to continue working simple problems after you're (consistently) able to do them correctly. As a loose rule of thumb, if you're working out of a textbook where similar problems are grouped into categories, I'd recommend to work out problems in each category until you've gotten at least 2 or 3 correct in a row and you're comfortable and confident that you could successfully complete another problem if you kept going. At that point, it's time to move on to the next category (unless your homework requires you to solve additional problems in the same category).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;3. Reviewing What You've Learned&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;You should also make sure to do review problems from prior topics once in a while. As a rough rule of thumb, whenever you learn a new topic, you should do a few more problems the next day or so (again, continuing until you've gotten at least 2 or 3 correct in a row and you're comfortable and confident that you could successfully complete another problem if you kept going -- though you should reach this point pretty quickly during review since you've already managed to reach that point on a previous occasion).&lt;/p&gt;&#xA;&lt;p&gt;Roughly, every time you do a review, you can double the length of time until the next review. So if you learn a new thing on day 1, then you should review it on days 2, 4, 8, 16, 32. (Again, to be clear: when you review, you need to work out actual problems, not just re-read.) This is of course a loose approximation – you don't have to get the review schedule perfect, but you do need to get the &amp;quot;spirit&amp;quot; of it right in the sense that you should review stuff again soon after learning it, and continue reviewing it into the future every once in a while, but you can wait longer between reviews as you grow more comfortable with it. (The general name for this kind of systematic review process is called &lt;a href=&quot;https://en.wikipedia.org/wiki/Spaced_repetition&quot; rel=&quot;noreferrer&quot;&gt;spaced repetition&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;However, if you're ever looking through topics to review and your reaction to some topic is &amp;quot;I barely remember that, but I don't really want to review it now, I'll come back to it later&amp;quot; then that actually means you need to review it right now. When you run into situations like that, don't wait. If you ever feel yourself forgetting something, especially when it's something that blindsided you when it came up as a component skill in a new thing you were trying to learn, then that's an indication you're due for a review.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;4. Quality, Quantity, and Spacing of Practice&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Get quality practice, get enough of it (quantity), and space it out. The quality/quantity part should be pretty obvious: 1 hour of fully-focused practice, successfully getting through 10 problems (or whatever), is totally different from 1 hour of distracted practice while simultaneously watching TV or goofing off with friends, only getting through a couple problems successfully (and by the way, solving problems successfully as a group is not at all the same thing as solving problems successfully on your own). And, of course, no matter how focused you are, you won't get enough practice if all you do is study before the day of the test.&lt;/p&gt;&#xA;&lt;p&gt;But a lot of people don't realize that you really need to space out the practice. Don't do 7 hours of practice for a class one day and then take a week off. It would be way better to space those 7 hours out over the week, doing 1 hour every day. Why?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;First, if your practice is as intensely focused as it should be, you're going to be getting tired after an hour. Even if you try your hardest, you're just not as productive when you're tired. It's diminishing returns: each successive hour of practice is worth less than the previous. Adjusting for productivity, 7 hours of crammed practice is effectively less than 7 hours of spaced practice.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Second, it's well-known phenomenon in cognitive psychology that the spacing itself drastically increases retention. This is the whole idea behind &lt;a href=&quot;https://en.wikipedia.org/wiki/Spaced_repetition&quot; rel=&quot;noreferrer&quot;&gt;spaced repetition&lt;/a&gt;, which I mentioned earlier. &lt;a href=&quot;https://psychology.ucsd.edu/undergraduate-program/undergraduate-resources/academic-writing-resources/effective-studying/spaced-practice.html&quot; rel=&quot;noreferrer&quot;&gt;Here's some additional follow-up reading.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-01-27T15:22:40.650" LastActivityDate="2024-01-27T15:22:40.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27424" PostTypeId="2" ParentId="27406" CreationDate="2024-01-25T09:17:49.333" Score="3" Body="&lt;p&gt;&amp;quot;Solve in 2 seconds&amp;quot; tricks often oversimplify complex problems. Real problem-solving requires understanding nuances, critical thinking, and adapting methods. Quick fixes may ignore important details, leading to inaccurate or incomplete solutions.&lt;/p&gt;&#xA;" OwnerUserId="23651" LastActivityDate="2024-01-25T09:17:49.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27425" PostTypeId="1" CreationDate="2024-01-25T14:04:27.007" Score="5" ViewCount="186" Body="&lt;p&gt;I'm looking for sample data to give algebra 2 students to teach about using Desmos to do regressions.&lt;/p&gt;&#xA;&lt;p&gt;Some example data sets folks at my school already have compiled are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Number of Lego pieces vs Price of set (linear)&lt;/li&gt;&#xA;&lt;li&gt;Arm span vs Height of person (linear)&lt;/li&gt;&#xA;&lt;li&gt;Number of years since 1948 vs Average movie ticket price (linear or exponential)&lt;/li&gt;&#xA;&lt;li&gt;Dollars spent on advertisements vs Dollars earned as revenue (square root)&lt;/li&gt;&#xA;&lt;li&gt;NBA draft pick number vs Salary in dollars (exponential decay)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Just to name a few. We would like to add to our collection for each of the following function families:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Linear&lt;/li&gt;&#xA;&lt;li&gt;Absolute Value&lt;/li&gt;&#xA;&lt;li&gt;Quadratic&lt;/li&gt;&#xA;&lt;li&gt;Cubic (and higher degree polynomials)&lt;/li&gt;&#xA;&lt;li&gt;Square root&lt;/li&gt;&#xA;&lt;li&gt;Reciprocal (or other rational functions)&lt;/li&gt;&#xA;&lt;li&gt;Exponential (growth and decay)&lt;/li&gt;&#xA;&lt;li&gt;Logarithmic&lt;/li&gt;&#xA;&lt;li&gt;Sine or Cosine&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;An ideal answer would provide sources of bivariate data from the various function families. Does such a compilation exist online already? Is there a good textbook that has this type of thing?&lt;/p&gt;&#xA;&lt;p&gt;In particular, we're having trouble finding data modeled by absolute value functions. I thought of vehicle MPG vs Speed, but couldn't find much real data.&lt;/p&gt;&#xA;&lt;p&gt;It is not critically important that the data be real. Thank you!&lt;/p&gt;&#xA;" OwnerUserId="16289" LastActivityDate="2024-01-25T17:40:20.623" Title="Sources of sample data for regressions" Tags="|technology-in-education|functions|resource-request|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27426" PostTypeId="1" AcceptedAnswerId="27430" CreationDate="2024-01-25T14:17:28.617" Score="14" ViewCount="1887" Body="&lt;p&gt;I am a qualified mathematics teacher but I have left teaching because I could not tolerate the behaviour of students. Now I am a mathematics tutor and I love that I get to teach students who are eager to learn. I had fantastic results with tutoring. Some of my students had failed repeatedly and after working with me they got top grades. These were master's students. I also have a couple of school students who are like little scientists and love to work with them. I know that success comes down to enabling them to understand the intuition behind the concepts.&lt;/p&gt;&#xA;&lt;p&gt;However, I do not want to help my students to only improve their grades but I want them to learn to think mathematically. I would like to create tutoring groups that focus exactly on teaching university-level students how to think mathematically. However, I am finding it tricky to find an appropriate book to use as a foundation for tutoring. I do not want to assume too much knowledge. A first-semester university student should be able to follow it. So far I have thought of using a book about abstract algebra for this purpose. This is because that was the subject that changed the way I saw mathematics and it got me comfortable with understanding and writing proofs. I am not thinking of just going over the abstract algebra but concentrating on understanding the reasoning and motivation behind it all. Enabling them to see how theorems are born etc. Now I am thinking that maybe abstract algebra might be too much. I am looking for a book that helps university students learn how to think mathematically and most importantly how to write proofs. I want to emphasize that it mustn't assume too much knowledge. Thank you in advance.&lt;/p&gt;&#xA;" OwnerUserId="23680" LastActivityDate="2024-02-07T17:02:12.113" Title="A good book about mathematical thinking" Tags="|mathematical-pedagogy|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27427" PostTypeId="2" ParentId="27425" CreationDate="2024-01-25T14:38:38.500" Score="1" Body="&lt;p&gt;This might be an interesting task. It is based on what is called Zipf's Law. Roughly it states that the frequency distribution of words in a typical passage of text of natural language follows a curve &lt;span class=&quot;math-container&quot;&gt;$\displaystyle{f(r)=\frac{1}{r^\alpha}}$&lt;/span&gt; Here &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the &amp;quot;rank&amp;quot; of the word.  For example, the most frequently occurring word tends to be &amp;quot;the&amp;quot; so &amp;quot;the&amp;quot; is first in rank. Typically the exponent &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is slightly larger than 1. You can find &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; by regression.&lt;/p&gt;&#xA;&lt;p&gt;Now to collect data, go to a source like &lt;a href=&quot;https://www.gutenberg.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Project Gutenberg&lt;/a&gt; and find a passage of something like five paragraphs. You can of course use a passage from anywhere, but Project Gutenberg seems appropriate for education.&lt;/p&gt;&#xA;&lt;p&gt;Next go to ChatGPT and say&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;strong&gt;Give me word frequencies for the follow passage, ignoring capitalization. Sort into decreasing order.&lt;/strong&gt; and paste the copied text.&lt;/p&gt;&#xA;&lt;p&gt;You will get something like this. You can ask for the data in a format that you can manipulate with a spreadsheet, such as csv.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gV7s6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gV7s6.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;At the bottom of the list will be the words that occurred just once, the so-called &lt;em&gt;hapaxes&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[Single occurrence words]: 'used', 'so', 'frightened', 'when', 'my', 'turn', 'big', 'crown', 'on', 'see', 'give', 'presents', 'kiss', 'liked', 'kisses', 'dreadful', 'have', 'looking', 'while', 'opened', 'bundles',&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2024-01-25T14:52:51.260" LastActivityDate="2024-01-25T14:52:51.260" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27428" PostTypeId="2" ParentId="27426" CreationDate="2024-01-25T14:46:16.850" Score="6" Body="&lt;p&gt;You might take a look at Mathematics and Plausible Reasoning by Polya.  First volume is probably sufficient for a starter.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0691025096&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's not 100% what you are looking for, but I think a vector in similar direction.  IOW, the emphasis not on proofiness but on how to make good guesses.  The subject he mostly uses is number theory and of a easily grasped slant (not abstractions, but observations on primes and the like).&lt;/p&gt;&#xA;&lt;p&gt;I had it as a college freshman in a special course.  I didn't put anything like full effort into the course...but I still remembered it being fun and motivational and special and different (from calc 3 I was in for regular math).&lt;/p&gt;&#xA;" OwnerUserId="23681" LastEditorUserId="23681" LastEditDate="2024-01-25T14:58:40.470" LastActivityDate="2024-01-25T14:58:40.470" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27429" PostTypeId="2" ParentId="27425" CreationDate="2024-01-25T16:50:37.697" Score="1" Body="&lt;p&gt;Here are two resources off the top of my head that exercise a variety of function families:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.worldometers.info/coronavirus/coronavirus-cases/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Infections at the beginning of the COVID-19 pandemic&lt;/strong&gt;&lt;/a&gt;&#xA;&lt;br&gt;&lt;em&gt;linear, exponential growth/decay, sine/cosine&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Spring 2020 - Summer 2020&lt;/em&gt; | the number of daily new cases increased linearly, and the total cumulative number of cases grew exponentially.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Fall 2020 - Fall 2021&lt;/em&gt; | the number of daily new cases oscillated sinusoidally, and the total cumulative number of cases increased roughly linearly.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Winter 2022 - Summer 2023&lt;/em&gt; | the number of daily new cases decayed roughly exponentially.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.justinmath.com/overfitting-underfitting-cross-validation-and-the-bias-variance-tradeoff/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A made-up rocket launch data set (height of rocket vs time elapsed since launch)&lt;/strong&gt;&lt;/a&gt;&#xA;&lt;br&gt;&lt;em&gt;linear, quadratic, higher-degree polynomial&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;This is a problem that I made for a modeling / machine learning class to illustrate the idea that &amp;quot;closer to the data points&amp;quot; is not always better when it comes to fitting models. You don't want to underfit, but you don't want to overfit either.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In this problem, we fit linear, quadratic, and high-degree polynomial models to the data. The linear model slightly underfits, the high-degree polynomial way overfits, and the quadratic provides a good fit.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2024-01-25T16:50:37.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27430" PostTypeId="2" ParentId="27426" CreationDate="2024-01-25T17:02:35.280" Score="6" Body="&lt;p&gt;Here are some really friendly proof books (with PDFs available online):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://github.com/carlosal1015/Books/blob/master/How%20to%20Read%20and%20Do%20Proofs%20by%20Daniel%20Solow.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to Read and Do Proofs: An Introduction to Mathematical Thought Processes&lt;/a&gt; (Solow)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.people.vcu.edu/%7Erhammack/BookOfProof/BookOfProof.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Book of Proof&lt;/a&gt; (Hammack)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://users.metu.edu.tr/serge/courses/111-2011/textbook-math111.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;How to Prove It&lt;/a&gt; (Velleman)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;These books do a good job of not just asking students to prove things, but also gently walking students through the whole idea of what it means to prove something and how one actually goes about proving things in a variety of different contexts.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-01-26T20:07:16.140" LastActivityDate="2024-01-26T20:07:16.140" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27431" PostTypeId="2" ParentId="27425" CreationDate="2024-01-25T17:40:20.623" Score="1" Body="&lt;p&gt;Think about what we use absolute value for: measuring distance when we don't know which of the two values is bigger.&lt;/p&gt;&#xA;&lt;p&gt;One activity I created with my students involved guessing the number of origami stars in a jar. (Students put their initials and their guess onto a spreadsheet.) We counted them after all the guesses were in.&lt;/p&gt;&#xA;&lt;p&gt;Then graph them all as x=guess, y=|guess - true value|. This let us think about properties of absolute value graphs.&lt;/p&gt;&#xA;&lt;p&gt;But the regression would be perfect. So maybe this doesn't work for your purposes?&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2024-01-25T17:40:20.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27432" PostTypeId="2" ParentId="27422" CreationDate="2024-01-26T08:26:23.247" Score="3" Body="&lt;p&gt;Let me tell you my story on &amp;quot;memorising formulae&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;When I was in secondary school, I was awfully bad at memorising (I still am), and there was no way for me to remember the Simpson formulae. In fact, the only formula I could remember was the sum-formula for the sinus and the cosinus-related double switch.&lt;br /&gt;&#xA;So, at every exam, this is what I did: I wrote down the sum-formula for the sinus:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(a+b) = \sin a \cos b + \cos a \sin b$&lt;/span&gt; (1)&lt;/p&gt;&#xA;&lt;p&gt;Then I replaced &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$-b$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(a+(-b)) = \sin a \cos (-b) + \cos a \sin (-b)$&lt;/span&gt;&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\sin(a-b) = \sin a \cos b - \cos a \sin b$&lt;/span&gt; (2)&lt;/p&gt;&#xA;&lt;p&gt;There also is the double switch for the cosinus:&lt;br /&gt;&#xA;(not &lt;span class=&quot;math-container&quot;&gt;$sc \pm cs$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$cc \mp ss$&lt;/span&gt; :-)):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(a+b) = \cos a \cos b - \sin a \sin b$&lt;/span&gt; (3)&lt;/p&gt;&#xA;&lt;p&gt;Hence (replace &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$-b$&lt;/span&gt;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(a-b) = \cos a \cos b + \sin a \sin b$&lt;/span&gt; (4)&lt;/p&gt;&#xA;&lt;p&gt;The Simpson formulae are derived by taking the &amp;quot;plusminus&amp;quot; averages of &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; and the &amp;quot;plusminus&amp;quot; averages of &lt;span class=&quot;math-container&quot;&gt;$(3)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(4)$&lt;/span&gt; :-)&lt;/p&gt;&#xA;&lt;p&gt;Once I told this to a friend and he said &amp;quot;&lt;em&gt;Are you crazy??? You depend your entire exam on just one formula. If you forget about that, you will fail the entire exam!&lt;/em&gt;&amp;quot;.&lt;br /&gt;&#xA;I replied &amp;quot;&lt;em&gt;Why? In case I forget my first formula, I know it can be derived from the internal products of two vectors on the unity circle. That gives me &lt;span class=&quot;math-container&quot;&gt;$\cos(a-b)$&lt;/span&gt; and I might start again.&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I've never seen a person having his jaw dropping so widely :-)&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2024-01-26T08:26:23.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27433" PostTypeId="2" ParentId="27412" CreationDate="2024-01-26T09:45:21.763" Score="9" Body="&lt;p&gt;I would recommend avoiding foundational issues when teaching probability at a low level. At the high school level one mostly deals with finite probability spaces and the normal distribution. The Kolmogorov axioms, which in any case you will probably not teach fully to (normal) high school students, are the safest approach. Kolmogorov explicitly avoided justifying his axioms very much beyond pragmatic considerations - they work and they encompass the important examples known then (why won't you teach the Kolmogorov axioms? The key one, the continuity axiom, is about passing to infinite limits ...)&lt;/p&gt;&#xA;&lt;p&gt;While relative frequency and area do not provide an adequate conceptual basis for the foundations of a probabilistic theory that encompasses processes such as Brownian motion, they work just fine at the introductory level.&lt;/p&gt;&#xA;&lt;p&gt;I have sympathy with the de Finetti/Cox/Jaynes/etc style approaches to foundations (by the way, the three mentioned are different), but these are hard to find written rigorously and their practical implications are mainly relevant when trying to formalize how one does inference. If you look at Cox's nice book you won't find much directly useful for teaching at the high school level (although it might change how you think about what you teach). For a teacher better to read Polya's book on plausible inference (I can't remember the title) and follow the line he advanced there. An approach like that of de Finetti starts with the theorem on exchangeable sequences - but this is quite technical even at the introductory university level - the idea of exchangeability as more fundamental than independence and perhaps as justifying (via de Finetti's theorem) in practice the assumption of independence is quite appealing, and it might make sense to mention it in an introductory university level course aimed at mathematics students, but it will be totally lost on high school students who in general will have considerable difficulty to even understand the definition of independence, much less that it is a defined concept, rather than a fact of nature.&lt;/p&gt;&#xA;&lt;p&gt;I like Jaynes's book, but it is intentionally provocative and argumentative and also is itself not unproblematic. Many professional probabilists and statisticians disagree strongly with his view point, and almost all will quibble with some of his more extreme claims. In the high school setting one wants students to understand something far more basic than what Jaynes is arguing about - namely that while deductive inference tells us that just because A implies B there is no need that B implies A, Bayes tells us that when A implies B and we observe B we gain more confidence in A. This point, which is often lost on professional mathematicians, is very basic, but it does not depend on whether one follows Jaynes or Kolmogorov or someone else.&lt;/p&gt;&#xA;" OwnerUserId="672" LastEditorUserId="672" LastEditDate="2024-02-16T08:07:52.177" LastActivityDate="2024-02-16T08:07:52.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27434" PostTypeId="2" ParentId="27412" CreationDate="2024-01-26T14:30:02.907" Score="3" Body="&lt;p&gt;In Economics of past decades, an analogous issue arose as regards teaching &amp;quot;the neoclassical paradigm&amp;quot; versus teaching the &amp;quot;Keynsian paradigm&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In a pretty realistic approach prominent neo-classical economist Barro in his textbook wrote that Keynsian economics is an &lt;em&gt;advanced&lt;/em&gt; topic, and students should first be exposed to the neo-classical paradigm, as a base to pivot afterwards.&lt;/p&gt;&#xA;&lt;p&gt;The same holds here: &amp;quot;Kolmogorov axioms&amp;quot; align with the most basic intuition we can have about chance and probabilities, while the Cox/Jayne approach, even though Jayne did build his theory from scratch, is an advanced stage.&lt;/p&gt;&#xA;&lt;p&gt;I also mention that in his main book Jayne finds himself in agreement with the Kolmogorov approach on many aspects, and he does not hesitate to explicitly say so. For example he writes on p. 49:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In Appendix A we describe the Kolmogorov system and show that, for all&#xA;practical purposes, the four axioms concerning his probability&#xA;measure, first stated arbitrarily (for which Kolmogorov has been&#xA;criticized), have all been derived in this chapter as necessary to&#xA;meet our consistency requirements. As a result, we shall find&#xA;ourselves defending Kolmogorov against his critics on many technical points.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And then on p. 50 he concludes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Our system of probability, however, differs conceptually from that of&#xA;Kolmogorov in that we do not interpret propositions in terms of sets,&#xA;but we do interpret probability distributions as carriers of&#xA;incomplete information. Partly as a result, our system has analytical&#xA;resources not present at all in the Kolmogorov system. This enables us&#xA;to formulate and solve many problems – particularly the so-called ‘ill&#xA;posed’ problems and ‘generalized inverse’ problems – that would be&#xA;considered outside the scope of probability theory according to the&#xA;Kolmogorov system. These problems are just the ones of greatest&#xA;interest in current applications.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Namely: advanced stage.&lt;/p&gt;&#xA;" OwnerUserId="689" LastEditorUserId="689" LastEditDate="2024-01-26T17:39:17.773" LastActivityDate="2024-01-26T17:39:17.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27435" PostTypeId="2" ParentId="27426" CreationDate="2024-01-26T18:55:44.753" Score="4" Body="&lt;p&gt;Many of us have had good results doing this with a class run in Inquiry-Based style.  There are many, many variations but basically the students are responsible for working their way through the material.  The teacher is more of a guide than a lecturer.&lt;/p&gt;&#xA;&lt;p&gt;I'll describe the classes I have run, to give a feel.  On the first day I give them notes that are pretty much just definitions and statements of results.  I ask them as homework to prove the first four results working entirely on their own, without books or the internet, and when they come back I assign four people at random to present at the board.  Other class members gently discuss, criticize, give examples and counter examples, and gradually come to a consensus about how to do it correctly, usually by modifying the initial argument.  I say as little as possible, including not nodding or head-shaking, so that &lt;em&gt;they&lt;/em&gt; do it instead of looking to me.  But if the group agrees on something that is wrong then I do not let it pass.  I might say &amp;quot;Giving three examples that the square of an odd is odd is not a proof, we need to show it holds in all cases,&amp;quot; and it is up to the group to go at it again.&lt;/p&gt;&#xA;&lt;p&gt;You need a right-sized group (maybe ten to twenty) of students who have seen at least some reasonably hard mathematics (maybe Calc II?). However, under the right circumstances it is beautiful-- you can see them becoming mathematically-minded people in front of your eyes.&lt;/p&gt;&#xA;&lt;p&gt;For a taste, I have an &lt;a href=&quot;https://hefferon.net/proofs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Intro to Proofs&lt;/em&gt;&lt;/a&gt; book and there are many other suitable ones at the &lt;a href=&quot;http://jiblm.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Journal of Inquiry-Based Learning in Mathematics&lt;/em&gt;&lt;/a&gt;.  For more see the web site of &lt;a href=&quot;https://www.comathinquiry.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the national organization, COMMIT&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="213" LastEditorUserId="213" LastEditDate="2024-01-27T17:15:32.383" LastActivityDate="2024-01-27T17:15:32.383" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27436" PostTypeId="2" ParentId="27412" CreationDate="2024-01-27T02:44:50.023" Score="6" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Should I choose Cox-based Jaynes' approach or Kolmogorov approach to&#xA;base myself on to teach probabilities to high-school students?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My answer is: No, you should not do that.&lt;/p&gt;&#xA;&lt;p&gt;These theoretical foundational issues are &lt;em&gt;way&lt;/em&gt; outside any high school curriculum, and would frankly be a waste of time for you to spend working on.&lt;/p&gt;&#xA;&lt;p&gt;There should be a received curriculum and an established textbook for the course. The authors have likely spent years or decades refining the best scaffolding for students of the given class level. The probability concepts in that book are almost surely predicated on an equal-likelihood model and the frequentist interpretation.&lt;/p&gt;&#xA;&lt;p&gt;Read that book and follow it closely (sequencing, terminology, notation, exercises, and problem sets). You will have your hands full interpreting the book for students, establishing a schedule, writing and grading assignments and tests, answering questions, behavioral issues and grading disputes, etc., etc. Don't add outside work that doesn't benefit that, especially when you're a new teacher.&lt;/p&gt;&#xA;&lt;p&gt;Side piece of advice for any teacher: You're almost certainly assigned to a course in &lt;em&gt;statistics&lt;/em&gt;. Don't turn it into a course essentially in &lt;em&gt;probability&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I am aware that probably the best thing I could do is to teach the&#xA;Probability Theory I am most confortable with and prepare to elaborate&#xA;on why I chose it to the hypothetical inspector?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If this a thing where you are, then I'd guess that the most likely risk from any inspector would be if they see that you are wildly off-book and off-curriculum in the classroom; for example, if you're spending time on foundational probability theory that is outside the book and inappropriate for that class level.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2024-01-27T06:19:52.813" LastActivityDate="2024-01-27T06:19:52.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27437" PostTypeId="1" CreationDate="2024-01-27T10:10:58.457" Score="1" ViewCount="555" Body="&lt;p&gt;The first high school I went to I did Integrated Math 1 my Freshman year, Integrated 2 my Sophomore year, and Integrated 3 my junior year. My junior year I transferred to a continuation school where I was told I no longer needed to continue math classes since I did 3 years at my previous High School. I continued with my junior year and Senior year at this continuation school but did not end up graduating due to insufficient credits on my end. The next year I started Adult education where they helped build my classes and schedule and specifically told me I do not need math since I finished at my first high school. I am now starting my second semester at this adult education, where I have to set up new classes. The lady helping me out now tells me I need 9 credits in math, which I was confused and told her I finished 3 years of math so I should be good. She told me Integrated math is ONLY Algebra, but I need general mathematics. Is she right? I tried to do some research but I don’t really know where to go, I thought integrated math was just general math not only algebra is taught. It leaves me to wonder why my first high school only had me in Algebra classes for 3 years if I needed mathematics to graduate?&lt;/p&gt;&#xA;" OwnerUserId="23686" LastActivityDate="2024-01-27T20:16:58.627" Title="Is integrated math class the same as mathematics class?" Tags="|secondary-education|mathematics-in-daily-life|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27438" PostTypeId="2" ParentId="27437" CreationDate="2024-01-27T14:46:42.980" Score="6" Body="&lt;p&gt;Integrated mathematics means a connected mixture of topics, not just algebra. &lt;a href=&quot;https://en.wikipedia.org/wiki/Integrated_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia says&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Integrated mathematics&lt;/strong&gt;&lt;/em&gt; is the term used in the United States to describe the style of mathematics education which integrates many&#xA;topics or strands of mathematics throughout each year of secondary&#xA;school. Each math course in secondary school covers topics in algebra,&#xA;geometry, trigonometry and functions. Nearly all countries throughout&#xA;the world, except the United States, follow this type of curriculum.&lt;/p&gt;&#xA;&lt;p&gt;In the United States, topics are usually integrated throughout&#xA;elementary school up to the seventh or sometimes eighth grade.&#xA;Beginning with high school level courses, topics are usually separated&#xA;so that one year a student focuses entirely on algebra (if it was not&#xA;already taken in the eighth grade), the next year entirely on&#xA;geometry, then another year of algebra (sometimes with trigonometry),&#xA;and later an optional fourth year of precalculus or calculus.&#xA;Precalculus is the exception to the rule, as it usually integrates&#xA;algebra, trigonometry, and geometry topics. Statistics may be&#xA;integrated into all the courses or presented as a separate course.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That said, the OP's primary problem sounds more administrative than math education (contradictory claims from intake to current time). Given the above, it sounds like the &amp;quot;lady helping me out now&amp;quot; is the one more likely to be mistaken, and should be joined in a meeting or conference call with the person or office who set the initial schedule to straighten things out.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2024-01-27T20:16:58.627" LastActivityDate="2024-01-27T20:16:58.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27439" PostTypeId="2" ParentId="13367" CreationDate="2024-01-28T02:53:01.903" Score="8" Body="&lt;p&gt;The student in the question here is very similar to many (most?) of the students that I get in remedial college elementary algebra courses. In particular, if they don't have basic &lt;em&gt;arithmetic&lt;/em&gt; skills, including primarily signed numbers, and secondarily fractions, then it's exceedingly difficult for them to get traction -- and we definitely don't have time at that level to remediate those even-more-basic skills.&lt;/p&gt;&#xA;&lt;p&gt;In response, I developed a quiz website (free, no ads, no login), &lt;a href=&quot;http://automatic-algebra.org/&quot; rel=&quot;noreferrer&quot;&gt;Automatic-Algebra.org&lt;/a&gt;, with quick timed quizzes on what I'd identify as the prerequisite &amp;quot;automatic&amp;quot; skills needed for algebra and other courses. While mechanical drilling is not the majority of what you should be doing in a math class, I do think it's necessary for a small core of math-facts that (like the basic vocabulary in a new language) the user needs to have automated, so as to not spend any mental energy in the middle of more complex work.&lt;/p&gt;&#xA;&lt;p&gt;In particular, for elementary algebra, I have quizzes on the arithmetic skills of (1) times tables, (2) negative numbers, and (3) order of operations (quizzes 5 questions each, time 15, 30, 60 seconds respectively; based on practical testing and some outside research). I do feel that to succeed a student needs to be able to complete these quizzes, without any errors, within the indicated times. I tell my students that they should practice every day until they never get any wrong.&lt;/p&gt;&#xA;&lt;p&gt;Full disclosure: I think this only gets a small amount of traction in my classes. I'd guess that the students who arrive with the given level of prior gaps usually don't have the study skills to do daily drills like that, and it's overshadowed by the actual algebra content coming at them that we're working on in class, homework, and graded tests. I certainly don't have time to spend on it in class, so the student needs to be sufficiently self-motivated to do it on their own.&lt;/p&gt;&#xA;&lt;p&gt;That said, for someone in the OP's position as a weekly one-on-one tutor, I think they'd have a much greater opportunity to check in and monitor this. Start every session with a run through of the basic quizzes (could take about 2 minutes). If the student is still deficient, encourage them again to practice more next week. Keep checking in every week like that. (And I'd also add in a rapid-fire verbal quiz of &amp;quot;What's the inverse of the ____ operation?&amp;quot; every week.)&lt;/p&gt;&#xA;&lt;p&gt;Admittedly, some students will never be able to perform these basic arithmetic skills proficiently. (In fact, I've had students who &lt;em&gt;couldn't believe that anyone could ever&lt;/em&gt; complete these quizzes in the indicated times.) Unfortunately, I don't think students in that category will ever really succeed at algebra or higher courses.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2024-01-28T06:10:27.627" LastActivityDate="2024-01-28T06:10:27.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27440" PostTypeId="2" ParentId="18852" CreationDate="2024-01-29T13:20:01.697" Score="3" Body="&lt;p&gt;What's about mixing problems?&lt;/p&gt;&#xA;&lt;p&gt;For example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;From 80% and 30% alcohol, a mixture of 40 liters is to be produced,&#xA;containing 50% alcohol. How many liters of each type are needed?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Of 63 kg of brass containing 42% copper, brass with 70% copper is to&#xA;be melted in order to obtain brass with 52% copper. How much brass&#xA;with 70% copper is necessary? (Brass is an alloy of copper (Cu) and&#xA;zinc (Zn)).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The examples above are taken from &lt;a href=&quot;http://bk-unterricht.de/wp-content/uploads/2013/12/x04_n6_LGS-Mischaufgaben-Loesung.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; document (in german).&lt;/p&gt;&#xA;&lt;p&gt;However I am not quite sure which of those mixture problems are indeed &amp;quot;practical&amp;quot; and authentic (i.e. the problem and solutions arises in real world as described in the exercise) or only &amp;quot;pseudo-practical&amp;quot; (i.e. it &lt;em&gt;seems&lt;/em&gt; like a real world problem, but it is constructed and nobody would do it in real world like this). Please comment about this, if you know more.&lt;/p&gt;&#xA;" OwnerUserId="1203" LastActivityDate="2024-01-29T13:20:01.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27441" PostTypeId="2" ParentId="18852" CreationDate="2024-01-29T14:19:18.643" Score="1" Body="&lt;p&gt;Let's play a game: you say a number of two digits. Then I say a number, also with two digits, and I will multiply both of them. Ready? Go!&lt;/p&gt;&#xA;&lt;p&gt;He: &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$71$&lt;/span&gt;&amp;quot;&lt;br /&gt;&#xA;You: &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$69$&lt;/span&gt;, that makes &lt;span class=&quot;math-container&quot;&gt;$4899$&lt;/span&gt;.&amp;quot;&lt;br /&gt;&#xA;He: &amp;quot;Ooh, that's fast! &lt;span class=&quot;math-container&quot;&gt;$63$&lt;/span&gt;&amp;quot;&lt;br /&gt;&#xA;You: &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$57$&lt;/span&gt;, that makes &lt;span class=&quot;math-container&quot;&gt;$3591$&lt;/span&gt;.&amp;quot;&lt;br /&gt;&#xA;He: &amp;quot;Waw, quite fast indeed. &lt;span class=&quot;math-container&quot;&gt;$94$&lt;/span&gt;?&amp;quot;&lt;br /&gt;&#xA;You: &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$86$&lt;/span&gt;, that makes &lt;span class=&quot;math-container&quot;&gt;$8084$&lt;/span&gt;. Wanna learn how?&amp;quot;&lt;br /&gt;&#xA;He: &amp;quot;YES! YES!!!&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(a-b) \cdot (a+b) = a^2-b^2$&lt;/span&gt;, which means that you can write a product as the subtraction of two squares. In order to do that, you need to write the numbers as a sum and as a subtraction of the same numbers. I know, sounds complicated. Let's take the first example: &lt;span class=&quot;math-container&quot;&gt;$71$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$69$&lt;/span&gt;. Write them as &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a-b$&lt;/span&gt;. That makes a system of two equations, leading to &lt;span class=&quot;math-container&quot;&gt;$a=70$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b=1$&lt;/span&gt;. Calculating the squares and subtracting them is easy.&lt;br /&gt;&#xA;Doing the same for &lt;span class=&quot;math-container&quot;&gt;$a+b=63$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a-b=57$&lt;/span&gt;. Similar system of equation leads to &lt;span class=&quot;math-container&quot;&gt;$a=60$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b=3$&lt;/span&gt;, leading to &lt;span class=&quot;math-container&quot;&gt;$63 \cdot 57 = 3591$&lt;/span&gt;.&lt;br /&gt;&#xA;... and again the same for &lt;span class=&quot;math-container&quot;&gt;$86$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$94$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;He:&amp;quot;But how do you know which number you should choose?&amp;quot;&lt;br /&gt;&#xA;You: &amp;quot;That's the trick of the game. As you can see, the product of two numbers is equal to the subtraction of two other numbers, &lt;strong&gt;more exactly the average and the distance towards that average&lt;/strong&gt;, you see? &lt;span class=&quot;math-container&quot;&gt;$71$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$69$&lt;/span&gt;, leading to &lt;span class=&quot;math-container&quot;&gt;$70$&lt;/span&gt; (the average) and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; (the distance)? &lt;span class=&quot;math-container&quot;&gt;$63$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$57$&lt;/span&gt;, leading to &lt;span class=&quot;math-container&quot;&gt;$60$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;?&lt;br /&gt;&#xA;Well, when you say a number (e.g. &lt;span class=&quot;math-container&quot;&gt;$48$&lt;/span&gt;), then I calculate the nearest easy number (which is &lt;span class=&quot;math-container&quot;&gt;$50$&lt;/span&gt; here). I calculate the distance (which is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;) and as &lt;span class=&quot;math-container&quot;&gt;$50$&lt;/span&gt; must be the average, the other number needs to be &lt;span class=&quot;math-container&quot;&gt;$50+2$&lt;/span&gt;, which is &lt;span class=&quot;math-container&quot;&gt;$52$&lt;/span&gt;. From there, you say &lt;span class=&quot;math-container&quot;&gt;$52$&lt;/span&gt;, you calculate &lt;span class=&quot;math-container&quot;&gt;$(50 \cdot 50) - (2 \cdot 2)$&lt;/span&gt; and everybody will be amazed by your mathematical genius!&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2024-01-29T14:19:18.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27443" PostTypeId="2" ParentId="25386" CreationDate="2024-01-30T15:57:19.653" Score="-1" Body="&lt;p&gt;Percentages are indeed a specific type of fraction, where the denominator is implicitly 100. Understanding fractions greater than one is key in grasping this concept. Let's delve into the example of 4/3 of a pizza to clarify this.&lt;/p&gt;&#xA;&lt;p&gt;When dealing with fractions like 4/3, we're essentially talking about a quantity that is more than one whole. In the context of a pizza, imagine having a whole pizza (which represents the 'unity' or 'one whole') and then some part of another pizza. To visualize 4/3 of a pizza, you would follow these steps:&lt;/p&gt;&#xA;&lt;p&gt;Divide Two Pizzas into Thirds: Start with two identical pizzas. Each of these pizzas should be divided into three equal parts. This division ensures that each part is one third (1/3) of a pizza.&lt;/p&gt;&#xA;&lt;p&gt;Select Four Parts: From these divided pizzas, select four of the one-third pieces.&lt;/p&gt;&#xA;&lt;p&gt;Combine to Form 4/3: By combining these four pieces, you now have 4/3 of a pizza. This represents one full pizza (which is 3/3 or 100%) plus an additional third (1/3) from the second pizza.&lt;/p&gt;&#xA;&lt;p&gt;In percentage terms, 4/3 translates to 133.33%. This is because 100% represents a whole (3/3 in this case), and the additional third (1/3) is equivalent to 33.33% more than the whole. Therefore, adding this to 100% gives you 133.33%.&lt;/p&gt;&#xA;&lt;p&gt;This approach helps in understanding fractions greater than one and their equivalent percentage values, providing a clear and practical visualization of the concept.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2024-01-30T15:57:19.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27444" PostTypeId="2" ParentId="18852" CreationDate="2024-01-30T16:03:59.363" Score="1" Body="&lt;p&gt;dWhen teaching students about solving systems of equations, it's beneficial to approach the topic from both an algebraic and geometric perspective. This dual approach aligns well with Duval's Multiple Representation Theory, which emphasizes the importance of understanding mathematical concepts through various forms of representation.&lt;/p&gt;&#xA;&lt;p&gt;First, ensure your students are familiar with the concept of line equations. These equations are not just sets of numbers and variables; they geometrically represent lines on a graph.&lt;/p&gt;&#xA;&lt;p&gt;Now, consider the concept of solving systems of equations. Algebraically, this involves finding a set of values that satisfy all equations in the system simultaneously. Geometrically, however, we are looking for the point where the lines, represented by these equations, intersect. This is where dynamic geometry software like GeoGebra becomes incredibly useful.&lt;/p&gt;&#xA;&lt;p&gt;GeoGebra (see examples at GeoGebra Systems of Equations) allows students to visually explore these intersections. By inputting the equations into the software, students can see the lines drawn on a coordinate plane and observe where they intersect. This visual representation helps in understanding that solving a system of equations is essentially finding the coordinates where the lines meet.&lt;/p&gt;&#xA;&lt;p&gt;This approach, combining algebraic problem-solving with geometric visualization, resonates with Duval's theory. It suggests that students' understanding of mathematical concepts is deepened when they engage with multiple representations of the same concept. In this case, the algebraic and geometric interpretations of systems of equations provide a more rounded and comprehensive understanding.&lt;/p&gt;&#xA;&lt;p&gt;By integrating dynamic geometry software into your teaching, you offer students a tangible and interactive way to grasp the concept of systems of equations. This method not only makes the learning process more engaging but also helps in building a stronger conceptual foundation in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2024-01-30T16:03:59.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27445" PostTypeId="1" CreationDate="2024-01-31T05:41:53.793" Score="0" ViewCount="92" Body="&lt;p&gt;Can anyone share the books that a Chinese maths undergraduate follows during their course? Also what are some of the problem books and exam papers that the Chinese students use to become better in mathematics? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="23368" LastEditorUserId="2083" LastEditDate="2024-01-31T06:12:29.203" LastActivityDate="2024-01-31T06:12:29.203" Title="What books does a Chinese institution use in the undergraduate maths courses?" Tags="|undergraduate-education|textbooks|china|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27446" PostTypeId="2" ParentId="27426" CreationDate="2024-01-31T11:06:33.960" Score="2" Body="&lt;p&gt;I think the following books can help.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.google.com.br/books/edition/How_to_Think_Like_a_Mathematician/sdgu2IICzu4C?hl=pt-BR&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How to Think Like a Mathematician: A Companion to Undergraduate Mathematics&lt;/em&gt;&lt;/a&gt; by Kevin Houston.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.google.com.br/books/edition/Theorems_Corollaries_Lemmas_and_Methods/kSwVGbBtel8C?hl=pt&amp;amp;gl=BR&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Theorems, Corollaries, Lemmas, and Methods of Proof&lt;/em&gt;&lt;/a&gt; by Richard J. Rossi.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13585" LastActivityDate="2024-01-31T11:06:33.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27448" PostTypeId="2" ParentId="4351" CreationDate="2024-01-31T16:06:41.547" Score="4" Body="&lt;p&gt;I stumbled on one of these today:&lt;/p&gt;&#xA;&lt;p&gt;The results of partial fraction decomposition, when used incorrectly, provide a unique candidate that fails.&lt;/p&gt;&#xA;&lt;p&gt;For example, if you try to decompose &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x^2+x^3}$&lt;/span&gt; as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{x^2+x^3} = \frac{A}{x^2}+\frac{B}{x+1}$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;you can clear fractions to get&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1 = A(x+1) + Bx^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course there are no &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; that satisfy this, but if you don't notice this and instead try any two values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x=0 \implies 1 = A$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x = -1 \implies 1 = B$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;you get a unique candidate for &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, and it fails.&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2024-01-31T16:06:41.547" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27449" PostTypeId="1" CreationDate="2024-01-31T17:53:03.893" Score="-1" ViewCount="544" Body="&lt;p&gt;Imagine a course barely getting into some topics more theoretical than what is done in the typical very staid first-year calculus course, and the kind of students for whom such a course is appropriate. Here the student might learn a bit about the role of the least-upper-bound axiom for the first time.&lt;/p&gt;&#xA;&lt;p&gt;In the typical first-year calculus course students are asked to do exercises that say &amp;quot;Why is the following not a counterexample to the mean value theorem?&amp;quot;, and are given a function whose domain clearly fails to include one point within the given interval.&lt;/p&gt;&#xA;&lt;p&gt;A number of theorems of a slightly more &amp;quot;theoretical&amp;quot; account of calculus depend on intervals having no &amp;quot;gaps&amp;quot;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The intermediate value theorem;&lt;/li&gt;&#xA;&lt;li&gt;The extreme value theorem (e.g. the function &lt;span class=&quot;math-container&quot;&gt;$x\mapsto|x|$&lt;/span&gt; on the set &lt;span class=&quot;math-container&quot;&gt;$\mathbb R\smallsetminus\{0\}$&lt;/span&gt;);&lt;/li&gt;&#xA;&lt;li&gt;The mean value theorem;&lt;/li&gt;&#xA;&lt;li&gt;The fact that if &lt;span class=&quot;math-container&quot;&gt;$f'&amp;gt;0$&lt;/span&gt; on an interval then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is increasing on that interval (&lt;span class=&quot;math-container&quot;&gt;$x\mapsto-1/x$&lt;/span&gt; has a positive derivative on its whole domain but its value at &lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt; is less than its value at &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;);&lt;/li&gt;&#xA;&lt;li&gt;The fact that derivatives have the intermediate value property even if they are not continuous;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;etc.&lt;/p&gt;&#xA;&lt;p&gt;In first-year calculus a &amp;quot;gap&amp;quot; is merely a real number not in the domain of the function, so that, in the exercises mentioned above, the domain is &lt;span class=&quot;math-container&quot;&gt;$[a,b)\cup(b,c].$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, in a theoretical course of the kind referred to in the first paragraph above, as opposed to a first-year calculus course, one wants to define a &lt;b&gt;gap&lt;/b&gt; without referring to a point such as the one called &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; above, which is not in the domain.&lt;/p&gt;&#xA;&lt;p&gt;Thus: Suppose the domain of a function is the union of two nonempty subsets &lt;span class=&quot;math-container&quot;&gt;$A,B$&lt;/span&gt; with the property that every member of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is less than every member of &lt;span class=&quot;math-container&quot;&gt;$B.$&lt;/span&gt; For every member of &lt;span class=&quot;math-container&quot;&gt;$A.$&lt;/span&gt; Suppose further that for every member of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; there is a larger member of &lt;span class=&quot;math-container&quot;&gt;$A,$&lt;/span&gt; and for every member of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; there is a smaller member of &lt;span class=&quot;math-container&quot;&gt;$B.$&lt;/span&gt; &lt;b&gt;That&lt;/b&gt; would be a &lt;b&gt;“gap.”&lt;/b&gt;&lt;/p&gt;&#xA;&lt;p&gt;To assert the existence of a gap thus defined is of course logically equivalent, but &lt;b&gt;not pedagogically equivalent&lt;/b&gt; (for the kind of students I have in mind) to saying that some non-empty bounded-above set has no smallest upper bound. If you don't see why it is not pedagogically equivalent then I suspect you will entirely miss the point of this question, and maybe you could post a second question asking about that. But to those who do understand the pedagogical non-equivalence I have &lt;b&gt;this question:&lt;/b&gt; How does one explain to students in an easy-to-understand way this definition is preferred to one mentioned above involving the point called &lt;span class=&quot;math-container&quot;&gt;$b,$&lt;/span&gt; where a &amp;quot;gap&amp;quot; is defined via the existence of an omitted point?&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Postscript in response to all (or most of?) the comments:&lt;/b&gt; The least-upper-bound axiom and the absence of &amp;quot;gaps&amp;quot; in the sense here defined are plainly equivalent to each other, and both avoid any mention of points that do not belong to the linearly ordered set alleged not to have gaps. The reason I prefer the version involving &amp;quot;gaps&amp;quot; as defined here is that I know that for many students that is easier to understand. That is why I said the two are not pedagogically equivalent. But the essence of the question is: How to explain to students why statements avoiding all mention of points not within the linearly ordered set being considered are better than statements that mention such points?&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2024-02-05T05:39:51.450" LastActivityDate="2024-02-06T16:14:39.840" Title="Motivating a definition of &quot;gap&quot; in a line just barely more advanced than the one used in the typical first-year calculus course" Tags="|calculus|mathematical-analysis|" AnswerCount="4" CommentCount="34" ContentLicense="CC BY-SA 4.0" />
  <row Id="27450" PostTypeId="2" ParentId="27412" CreationDate="2024-01-31T18:06:10.963" Score="3" Body="&lt;p&gt;How probable is it that a &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; will appear when you throw a die?&lt;/p&gt;&#xA;&lt;p&gt;How probable is it that there was life on Mars a billion years ago?&lt;/p&gt;&#xA;&lt;p&gt;For the first question, one can say &lt;span class=&quot;math-container&quot;&gt;$1/6$&lt;/span&gt; because it happens one-sixth of the time.&lt;/p&gt;&#xA;&lt;p&gt;For the second question, if the answer is &lt;span class=&quot;math-container&quot;&gt;$1/6,$&lt;/span&gt; that does &lt;em&gt;not&lt;/em&gt; mean it happens one-sixth of the time.&lt;/p&gt;&#xA;&lt;p&gt;You can use the axioms of mathematical probability to model relative frequency, as in the first example, or degree of belief, as in the second (that's the Cox–Jaynes approach, usually called &amp;quot;Bayesian&amp;quot; although Bayes himself probably had nothing to do with it).&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Either way, you'll use the Kolmogorov axioms.&lt;/b&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A probability space, or &amp;quot;sample space&amp;quot;, is a set of all possible &amp;quot;outcomes.&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;An &amp;quot;event&amp;quot; is a (measurable) subset of the sample space.&lt;/li&gt;&#xA;&lt;li&gt;A probability measure is a function assigning probabilities to events in a way that is (countably) additive and assigns measure &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to the whole space.&lt;/li&gt;&#xA;&lt;li&gt;A random variable is a (measurable) function whose domain is the sample space.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;You'll use that regardless of whether you take a frequentist or a Bayesian approach.&lt;/p&gt;&#xA;&lt;p&gt;And I probably wouldn't talk about frequentism versus Bayesianism in a high-school course unless the topic somehow comes up when I didn't initially intend it to (e.g. a student mentions it or it becomes relevant to an particular exercise).&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2024-02-07T18:54:54.683" LastActivityDate="2024-02-07T18:54:54.683" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27451" PostTypeId="2" ParentId="27449" CreationDate="2024-02-01T03:38:38.190" Score="12" Body="&lt;p&gt;In my experience, the only way to successfully teach a more sophisticated technique is to present a problem where known simpler techniques fail.&lt;/p&gt;&#xA;&lt;p&gt;For instance, anyone who's taught algebra to kids will know that if you try to teach &lt;span class=&quot;math-container&quot;&gt;$ax+b=c$&lt;/span&gt; equations by demonstrating on a problem that has a single-digit solution, like &lt;span class=&quot;math-container&quot;&gt;$2x-1=5,$&lt;/span&gt; then a portion of the students with decent number sense will tune out because they can get by just fine using guess-and-check:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$-1,$&lt;/span&gt; too low, go up a bit &lt;span class=&quot;math-container&quot;&gt;$x=2$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$3,$&lt;/span&gt; tiny bit higher &lt;span class=&quot;math-container&quot;&gt;$x=3$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$5,$&lt;/span&gt; BAM im so smart i dont even need algebra&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The way to avoid this is to present a problem like &lt;span class=&quot;math-container&quot;&gt;$9x-13=18$&lt;/span&gt; and say something like &amp;quot;take a moment to see if you can guess the solution to that... yeah, not so easy, huh? Okay, now you see that guess-and-check is not going to work for some of these trickier equations, so let me show you a trick that does work.&amp;quot; This way, the kids understand the value in the more sophisticated solution technique that you're about to show them.&lt;/p&gt;&#xA;&lt;p&gt;It's the same thing at all levels of math. If you want your students to understand why a more sophisticated definition is preferable, then you have to introduce them to a situation where the simpler definition breaks down and the sophisticated definition saves the day.&lt;/p&gt;&#xA;&lt;p&gt;(And if you can't find such a situation, then I think that's an indication that the definition is not appropriate to use in the class -- though perhaps it could be made appropriate by expanding the syllabus of the class to cover an area where such a situation arises.)&lt;/p&gt;&#xA;&lt;p&gt;To find such a situation, here's the line of thought I'd start following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One situation might be when gaps are missing a whole interval of numbers, such as &lt;span class=&quot;math-container&quot;&gt;$[1,2) \cup (3,4]$&lt;/span&gt; missing the entire interval &lt;span class=&quot;math-container&quot;&gt;$[2,3].$&lt;/span&gt; Your abstract definition accommodates this case, whereas &lt;span class=&quot;math-container&quot;&gt;$[a,b) \cup (b,c]$&lt;/span&gt; doesn't.&lt;/p&gt;&#xA;&lt;p&gt;However, we can easily accommodate this case by tweaking the interval notation definition: &lt;span class=&quot;math-container&quot;&gt;$[a,b) \cup (c,d]$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$b \leq c.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, carrying this line of thought further, is there a more nuanced case that your abstract definition accommodates, that we can't accommodate with by tweaking the interval notation definition?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt; OP has clarified that the key feature of their abstract definition is that it defines a gap without referring to any specific missing points (like &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; in the interval notation definition).&lt;/p&gt;&#xA;&lt;p&gt;I think a concrete example is still needed to illustrate why it's problematic to refer to such points. Sure, that's a feature of the abstract definition, but it's unclear how that feature &amp;quot;saves the day.&amp;quot; (Not saying the feature is useless, just that its utility is not obvious, so an example is needed to illustrate the utility of the feature.)&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-02-02T15:55:48.023" LastActivityDate="2024-02-02T15:55:48.023" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="27453" PostTypeId="2" ParentId="27449" CreationDate="2024-02-02T03:23:36.207" Score="6" Body="&lt;p&gt;Your definition of a &amp;quot;gap&amp;quot; is reminiscent of a Dedekind cut. If we further assume &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are sets of rational numbers and that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; form a partition of &lt;span class=&quot;math-container&quot;&gt;${\bf Q}$&lt;/span&gt;, then we have a Dedekind cut. The cut corresponds to &amp;quot;gap&amp;quot; which in turn corresponds to a new number not in &lt;span class=&quot;math-container&quot;&gt;${\bf Q}$&lt;/span&gt;. In this way, we fill all the gaps in &lt;span class=&quot;math-container&quot;&gt;${\bf Q}$&lt;/span&gt; with these newly created irrational numbers.&lt;/p&gt;&#xA;&lt;p&gt;Your approach seems appropriate to students in a first course in mathematical analysis. To pursue your approach, you could tell students about Dedekind cuts and why the construction was such an important mathematical development.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2024-02-02T04:05:23.087" LastActivityDate="2024-02-02T04:05:23.087" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27454" PostTypeId="2" ParentId="27426" CreationDate="2024-02-03T00:36:00.577" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Byers, William. &lt;em&gt;How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics&lt;/em&gt;. Princeton University Press, 2010.&#xA;&lt;a href=&quot;https://press.princeton.edu/books/paperback/9780691145990/how-mathematicians-think&quot; rel=&quot;nofollow noreferrer&quot;&gt;PUP link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/EIFpp.jpg&quot; width=&quot;300&quot; /&gt;&#xA;&lt;p&gt;Emphasizes the creativity in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2024-02-03T00:36:00.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27455" PostTypeId="2" ParentId="12024" CreationDate="2024-02-05T04:24:37.590" Score="1" Body="&lt;p&gt;I administer individual math assessments to students. I think students need to learn to understand basic math concepts and apply them to every day situations. The best way to do that is to learn basic calculation, estimating, and predicting a range for possible answers. I think students grasp math concepts better when they have to practice doing the calculations.&lt;/p&gt;&#xA;" OwnerUserId="23707" LastActivityDate="2024-02-05T04:24:37.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27456" PostTypeId="2" ParentId="27449" CreationDate="2024-02-05T07:51:04.700" Score="6" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;How to explain to students why statements avoiding all mention of&#xA;points not within the linearly ordered set being considered are better&#xA;than statements that mention such points?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You can't. Because they're not.&lt;/p&gt;&#xA;&lt;p&gt;To elaborate: You can't make up a new, unconventional definition and expect others to explain why it's better. That burden of motivating it is on you. Aside from classroom math education, perhaps you can have a conversation with a fellow mathematician and see if you can illuminate to them why your definition might be preferred. If that succeeds, then you could cycle back and think about how to repackage it for students.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2024-02-05T16:58:07.537" LastActivityDate="2024-02-05T16:58:07.537" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27457" PostTypeId="2" ParentId="27449" CreationDate="2024-02-06T16:14:39.840" Score="3" Body="&lt;p&gt;A possibly motivating example might be the following: let &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; be the set of all positive numbers whose square is not &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. This description of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; seems operationally more straightforward than &amp;quot;the set of all positive numbers not equal to &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3}$&lt;/span&gt;&amp;quot;. To use the latter to decide whether a given number is in &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;, you have to imagine comparing the given number with some number whose decimal expansion most students probably don't remember beyond the first couple of digits and which, in any case, might involve the unpleasant clerical task of comparing long strings of rather unpredictable digits; to use the former, you ask whether the given number squares to &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; or to a number not equal to &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, that is, to a number either less than or greater than &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. While in practice this might also involve a lot of unpleasant work, it's conceptually very clean, and it's easy to visualize what you need to do. It also naturally gives the partition mentioned in your definition.&lt;/p&gt;&#xA;&lt;p&gt;At the very least this example shows that having to talk about elements not in the set is not always advantageous, and can be disadvantageous.&lt;/p&gt;&#xA;" OwnerUserId="1737" LastActivityDate="2024-02-06T16:14:39.840" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27458" PostTypeId="1" CreationDate="2024-02-06T17:33:34.823" Score="4" ViewCount="434" Body="&lt;p&gt;I am using the textbook Linear Algebra done right by Sheldon Axler (fourth edition)  to teach an undergraduate linear algebra course.&#xA;Please find here a link to &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/978-3-031-41026-0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;the book here.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now Axler does not introduce the idea of determinants and characteristic polynomials while discussing eigenvalues and minimal polynomials. So after section 5B if one needs to solve the following question, how should one solve the question without using determinants?&lt;/p&gt;&#xA;&lt;p&gt;The question is as follows: Find the eigenvalues and minimal polynomial of the given operator &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;, defined by&#xA;&lt;span class=&quot;math-container&quot;&gt;$T(x,y,z)=(x+y, x+2y, 3z)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note: I am aware of &lt;a href=&quot;https://matheducators.stackexchange.com/questions/21197/computing-eigenvalues-by-hand-without-determinants&quot;&gt;this question&lt;/a&gt;, but here they are discussing only &lt;span class=&quot;math-container&quot;&gt;$2\times 2$&lt;/span&gt; matrices. My operator above involves a &lt;span class=&quot;math-container&quot;&gt;$3\times 3$&lt;/span&gt; matrix.&lt;/p&gt;&#xA;&lt;p&gt;So basically I need to find values of &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; such that there exists &lt;span class=&quot;math-container&quot;&gt;$(x,y,z)\neq 0\in \mathbb{R}^3$&lt;/span&gt;, such that &lt;span class=&quot;math-container&quot;&gt;$T(x,y,z)=\lambda(x,y,z)$&lt;/span&gt;.&#xA;i.e if the equation &lt;span class=&quot;math-container&quot;&gt;$(T-\lambda I)(x,y,z)=0$&lt;/span&gt; has a non trivial solution in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;&#xA;i.e does the system &lt;span class=&quot;math-container&quot;&gt;$$\begin{bmatrix}&#xA;1-\lambda &amp;amp; 1 &amp;amp; 0 \\&#xA;1 &amp;amp; 2-\lambda &amp;amp; 0\\&#xA;0 &amp;amp; 0 &amp;amp; 3-\lambda&#xA;\end{bmatrix}\begin{bmatrix}&#xA;x \\&#xA;y \\&#xA;z&#xA;\end{bmatrix} = \begin{bmatrix}&#xA;0 \\&#xA;0 \\&#xA;0&#xA;\end{bmatrix} $$&lt;/span&gt;&#xA;have a non trivial solution.&lt;/p&gt;&#xA;&lt;p&gt;Now for &lt;span class=&quot;math-container&quot;&gt;$(x,y,z)$&lt;/span&gt; to be a non trivial solution , at least one of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; has to be non-zero.&lt;/p&gt;&#xA;&lt;p&gt;Suppose I take &lt;span class=&quot;math-container&quot;&gt;$x\neq 0$&lt;/span&gt; I get &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; to be the values &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}-\frac{\sqrt{5}}{2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}+\frac{\sqrt{5}}{2}$&lt;/span&gt;.&#xA;So I have got two eigenvalues out of maximum three. I see it is very evident that &lt;span class=&quot;math-container&quot;&gt;$\lambda=3$&lt;/span&gt; is also an eigenvalue. I will get that if I assume &lt;span class=&quot;math-container&quot;&gt;$z\neq 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My question is can I consider the two cases? i.e Case 1: Since &lt;span class=&quot;math-container&quot;&gt;$(x,y,z)\neq 0$&lt;/span&gt; so I can assume &lt;span class=&quot;math-container&quot;&gt;$x\neq 0$&lt;/span&gt; and Case 2: Since &lt;span class=&quot;math-container&quot;&gt;$(x,y,z)\neq 0$&lt;/span&gt; so suppose &lt;span class=&quot;math-container&quot;&gt;$z\neq 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Or is it, as I suppose I can consider only one case?&lt;/p&gt;&#xA;" OwnerUserId="23718" LastEditorUserId="23718" LastEditDate="2024-02-06T17:44:51.687" LastActivityDate="2024-02-10T15:55:23.007" Title="Regarding finding eigenvalues and minimal polynomial of an operator with limited tools while following Sheldon Axler" Tags="|undergraduate-education|linear-algebra|eigenvalues|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27459" PostTypeId="2" ParentId="27458" CreationDate="2024-02-06T20:10:50.783" Score="6" Body="&lt;p&gt;Let's assume I don't know anything about determinants and characteristic polynomials, but I understand that I'm interested in non-trivial solutions to &lt;span class=&quot;math-container&quot;&gt;$T(x,y,z)=\lambda(x,y,z)$&lt;/span&gt; with the &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; you provide. Then I proceed as follows.&lt;/p&gt;&#xA;&lt;p&gt;I start with the system of equations&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{rl} x+y &amp;amp; =\lambda x\\&#xA; x+2y &amp;amp;=\lambda y \\ &#xA;3z &amp;amp;= \lambda z\end{array}&lt;/span&gt;&#xA;Immediately I notice that if &lt;span class=&quot;math-container&quot;&gt;$z \neq 0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\lambda =3$&lt;/span&gt;. This reduces the system to&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{rl} -2x+y &amp;amp;=0\\&#xA;x-y &amp;amp;=0 \end{array}&lt;/span&gt;&#xA;which pretty quickly I see has &lt;span class=&quot;math-container&quot;&gt;$x=y=0$&lt;/span&gt; (but &lt;span class=&quot;math-container&quot;&gt;$z \neq 0$&lt;/span&gt;, so not a trivial solution).&lt;/p&gt;&#xA;&lt;p&gt;So now I return to the possibility that &lt;span class=&quot;math-container&quot;&gt;$z=0$&lt;/span&gt;. Then from the first equation I have &lt;span class=&quot;math-container&quot;&gt;$x+y=\lambda x$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$$y=(\lambda-1)x.$$&lt;/span&gt; Substituting into the second gives &lt;span class=&quot;math-container&quot;&gt;$$x+2(\lambda-1)x=\lambda(\lambda-1)x.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now I notice that if &lt;span class=&quot;math-container&quot;&gt;$x \neq 0$&lt;/span&gt;, then I have &lt;span class=&quot;math-container&quot;&gt;$$1+2(\lambda-1)=\lambda(\lambda-1).$$&lt;/span&gt; From which I solve &lt;span class=&quot;math-container&quot;&gt;$$\lambda=\frac{1}{2} \left(3\pm\sqrt{5}\right).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Lastly I check if &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt; and this is the trivial solution.&lt;/p&gt;&#xA;&lt;p&gt;The take-away, from a pedagogical point of view, is this: This method is annoying (and look how lucky we got with the first step with &lt;span class=&quot;math-container&quot;&gt;$z \neq 0$&lt;/span&gt; being such an easy case). This is why we go off and develop more theory (determinants, etc.) to find a better way.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2024-02-06T20:10:50.783" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27460" PostTypeId="2" ParentId="27422" CreationDate="2024-02-06T22:11:26.060" Score="1" Body="&lt;p&gt;Ultimately, I believe you need to find what works for you and enables you to make sure that, when you step into the exam room, you have the skills and knowledge you need to achieve the mark you want. To do this you need to:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Know what you know&lt;/li&gt;&#xA;&lt;li&gt;Know what you don't know&lt;/li&gt;&#xA;&lt;li&gt;Have a plan for how you are going to learn what you don't know&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Justin's answer above goes into great detail about how to achieve these points. I would add that I believe there is a lot of room for memorisation in mathematics that can help you achieve success.&lt;/p&gt;&#xA;&lt;p&gt;In my own undergraduate study, I condensed each of my modules into a set of summary notes, detailing just Theorems, Corollaries, Propositions, Lemmas, etc. and their associated proofs. Each module came to about 30 pages. Prior to each exam, I ensured I had fully memorised the contents of all of these 30 pages. I did this one page at a time, reading the page, turning it over and attempting to rewrite it's contents without looking. I repeated this until I had it. This allowed me not only to memorise the results and proof, but also to build a much heightened understanding of the content.&lt;/p&gt;&#xA;&lt;p&gt;Often, exam questions would be variants of results from lectures, requiring me to adapt the memorised proof to the unique circumstance presented in the exam. This is where the knowledge and understanding came in. However, as I had the line of argument and reasoning of the proof memorised, I was able to focus my attention on this adaptation, without having to concentrate too much on the line of argument.&lt;/p&gt;&#xA;&lt;p&gt;If you want to be sure of success, you can't walk into an exam relying on a flash of inspiration - you will be disappointed!&lt;/p&gt;&#xA;" OwnerUserId="23719" LastActivityDate="2024-02-06T22:11:26.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27461" PostTypeId="2" ParentId="27458" CreationDate="2024-02-07T14:01:01.513" Score="1" Body="&lt;p&gt;One can get a fair amount of information by using row reduction. Subtracting &lt;span class=&quot;math-container&quot;&gt;$1-\lambda$&lt;/span&gt; times the second row from the first one and then exchanging the first two rows,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{vmatrix}&#xA;1-\lambda &amp;amp; 1 &amp;amp; 0 \\&#xA;1 &amp;amp; 2-\lambda &amp;amp; 0\\&#xA;0 &amp;amp; 0 &amp;amp; 3-\lambda&#xA;\end{vmatrix}&#xA;\xrightarrow{\phantom{xxxxxx}}&#xA;\begin{bmatrix}&#xA;1 &amp;amp; 2-\lambda &amp;amp; 0\\&#xA;0&amp;amp;1-(1-\lambda)(2-\lambda)&amp;amp;0\\&#xA;0 &amp;amp; 0 &amp;amp; 3-\lambda&#xA;\end{bmatrix}&#xA;$$&lt;/span&gt;&#xA;Now the system is in echelon form. For the system to have a non-trivial solution, one of the diagonal entries needs to be zero. We are left with &lt;span class=&quot;math-container&quot;&gt;$\lambda=3$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$0=1-(1-\lambda)(2-\lambda)=-\lambda^2+3\lambda-1$&lt;/span&gt;, which gives the roots&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lambda=\frac{3\pm\sqrt5}2.&#xA;$$&lt;/span&gt;&#xA;As all eigenvalues are simple, the minimal polynomial agrees with the characteristic polynomial,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;p(x)=(\lambda-3)(\lambda^2-3\lambda+1).&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="475" LastActivityDate="2024-02-07T14:01:01.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27462" PostTypeId="2" ParentId="27426" CreationDate="2024-02-07T17:02:12.113" Score="1" Body="&lt;p&gt;My favorite book so far is Jay Cummings &amp;quot;Proofs: A Long-Form Mathematics Textbook.&amp;quot;  It not only has great information, it is well-written and an enjoyable read.&lt;/p&gt;&#xA;" OwnerUserId="9594" LastActivityDate="2024-02-07T17:02:12.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27463" PostTypeId="1" AcceptedAnswerId="27464" CreationDate="2024-02-08T20:51:33.003" Score="6" ViewCount="457" Body="&lt;p&gt;I'd like to know where I can find resources which are helpful when one has to design or improve grad and undergrad degree programs in pure and applied mathematics.&#xA;In particular, I'm searching for up-to-date field studies containing data and/or analyses on career opportunities for math graduates who focused on dynamical systems and mathematical physics.&lt;/p&gt;&#xA;&lt;p&gt;I know SIAM has released something of this kind, I'm searching some more material.&lt;/p&gt;&#xA;&lt;p&gt;Just for clarity: I'm not necessarily asking for material released from Universities, rather for resources produced by international societies for the development of mathematics, governments, non-profit companies etc...&lt;/p&gt;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;&lt;p&gt;(I also asked this on Academia Stack Exchange, but someone suggested this is a better place.)&lt;/p&gt;&#xA;" OwnerUserId="23728" LastEditorUserId="6053" LastEditDate="2024-02-12T14:47:21.820" LastActivityDate="2024-02-12T14:47:21.820" Title="Resources for designing math degree programs" Tags="|undergraduate-education|graduate-education|resource-request|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27464" PostTypeId="2" ParentId="27463" CreationDate="2024-02-08T21:05:10.107" Score="10" Body="&lt;p&gt;For undergrad, the MAA put out &amp;quot;Guidelines for Programs and Departments in Undergraduate Mathematical Sciences&amp;quot; about 20 years ago. We used this when preparing for program review about a decade ago and it was still relevant then.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://maa.org/programs-and-communities/member-communities/committee-on-departmental-review/guidelines-programs-departments&quot; rel=&quot;noreferrer&quot;&gt;https://maa.org/programs-and-communities/member-communities/committee-on-departmental-review/guidelines-programs-departments&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt; Lest it gets buried in the comments, Brian Borchers points out:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There's a 2015 update that is still quite relevant. Presumably a new&#xA;report will come out in 2025.&#xA;&lt;a href=&quot;https://maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/cupm/2015-cupm-curriculum-guide&quot; rel=&quot;noreferrer&quot;&gt;https://maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/cupm/2015-cupm-curriculum-guide&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thanks for pointing this out!!&lt;/p&gt;&#xA;" OwnerUserId="401" LastEditorUserId="401" LastEditDate="2024-02-09T14:55:36.397" LastActivityDate="2024-02-09T14:55:36.397" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27465" PostTypeId="2" ParentId="27368" CreationDate="2024-02-09T04:01:44.170" Score="5" Body="&lt;p&gt;Within a larger computer program it may be necessary to solve many problems within a fraction of a second, each of which is done by Newton's method.&lt;/p&gt;&#xA;&lt;p&gt;You cannot simply plot a function and zoom in if finding the root needs to be done within a software program that will immediately report the value and will be used 10000 times within a fraction of a second before reporting output that does not tell you the value that the root-finding algorithm seeks, but nonetheless uses that value in the computation of something else.&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2024-02-12T21:35:04.540" LastActivityDate="2024-02-12T21:35:04.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27466" PostTypeId="1" CreationDate="2024-02-09T18:06:48.537" Score="5" ViewCount="1546" Body="&lt;p&gt;I have noticed that beginning algebra (or pre-algebra) students can easily make the following type of error, confusing variables with units when being introduced to equations for proportional relationships.&lt;/p&gt;&#xA;&lt;p&gt;Say they are presented with an equation &lt;span class=&quot;math-container&quot;&gt;$d=6t$&lt;/span&gt; and told that it represents how fast a person can run, with &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; representing distance in miles and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; representing time in hours. The student then reasons as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is miles and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is hours, so the equation says that 1 mile is the&#xA;same as 6 hours, or in other words the person runs 1 mile for every&#xA;six hours.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They might make the same type of error in the other direction: if told that someone runs 10 miles for every two hours they might reason&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is miles and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is hours, so the person runs &lt;span class=&quot;math-container&quot;&gt;$10d$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$2t$&lt;/span&gt;&#xA;and the equation is &lt;span class=&quot;math-container&quot;&gt;$10d=2t$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What are some ways of helping them correct this mistake?&lt;/p&gt;&#xA;" OwnerUserId="20667" LastEditorUserId="20667" LastEditDate="2024-02-09T21:11:45.310" LastActivityDate="2024-02-13T09:37:47.047" Title="Confusing variables with units in simple equations" Tags="|algebra|units|" AnswerCount="8" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27467" PostTypeId="2" ParentId="27466" CreationDate="2024-02-09T19:19:20.853" Score="4" Body="&lt;p&gt;I think the way to help this kind of student is to get them in the habit of reading the equation correctly. The equation &lt;span class=&quot;math-container&quot;&gt;$d=6t$&lt;/span&gt; does not say that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1 mile is the same as 6 hours,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;it says that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;the number of&lt;/em&gt; miles is the same as 6 times &lt;em&gt;the number of&lt;/em&gt; hours.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In other words, if you want to calculate the number of miles, take the number of hours and multiply by 6.&lt;/p&gt;&#xA;&lt;p&gt;So in one hour, the person runs 6 miles.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In the other direction: if someone runs 10 miles for every 2 hours, then they run 5 miles each hour, so&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;the number of&lt;/em&gt; miles is the same as 5 times &lt;em&gt;the number of&lt;/em&gt; hours,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and the equation is &lt;span class=&quot;math-container&quot;&gt;$d=5t.$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-02-09T19:27:21.817" LastActivityDate="2024-02-09T19:27:21.817" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27468" PostTypeId="2" ParentId="27466" CreationDate="2024-02-09T21:19:54.107" Score="6" Body="&lt;p&gt;There are two things that can be done to really drive this point to someone first learning algebraic equations.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Focus on how the units of both sides of the equation must be equal. This might seem elementary, but you've noticed your students reasoning that somehow time can be equal to distance which doesn't really make sense. It's just not obvious to someone first learning algebra. Try to make this a fundamental concept that they consider when they approach any problem. This is incredibly useful for learning how to approach more complex mathematical expressions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Make the units of the problem very explicit in the statement. This supports #1 so the student can use that information to check their solution.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Here's an example problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Alice is late for school and is running to make it to class before her math exam begins. An expression for how far Alice has traveled is given by the equation d = 6t where d is the distance in miles, t is the time in hours. What units must the number 6 in the equation have to satisfy the equality? How fast is Alice running? (answer: miles/hour, 6 miles/hour).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here's how I would approach the other example from your second problem.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;if told that someone runs 10 miles for every two hours they might reason &amp;quot;d is miles and t is hours, so the person runs 10d for every 2t and the equation is &lt;span class=&quot;math-container&quot;&gt;$$10d=2t$$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Let's say they try to make this assertion. Now the student should check their units. If the number 10 has units of miles and the number 2 has units of hours as posed by the problem, the units of the expression they created are &lt;span class=&quot;math-container&quot;&gt;$$miles^2 = hours^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This doesn't match up, so we must have gone wrong somewhere. Then I would approach the problem by trying to determine where the constants and the variables need to be in the expression to create equal units on both sides.&lt;/p&gt;&#xA;" OwnerUserId="23731" LastEditorUserId="23731" LastEditDate="2024-02-09T23:02:24.477" LastActivityDate="2024-02-09T23:02:24.477" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27469" PostTypeId="2" ParentId="27466" CreationDate="2024-02-09T22:08:40.723" Score="4" Body="&lt;p&gt;One possible method is to have students simply do t vs d value tables.&#xA;&amp;quot;If t is 1, what value does d have to have to make the equation true?&amp;quot;  So starting with given values for t, figuring out d is just a multiplication by 6.&lt;/p&gt;&#xA;&lt;p&gt;Assuming they are comfortable with that, then start with a value for d, and ask what value of t will make the equation true. &amp;quot;If d is 12, what value of t will make the equation true?&amp;quot; &amp;quot;What number times 6 equals 12?&amp;quot; Obviously these would then all be division problems, and hopefully the students remember their 2nd and 3rd grade &amp;quot;fact families&amp;quot;!&lt;/p&gt;&#xA;&lt;p&gt;This would, I hope, avoid worries I have about whether the students understand concepts like conversion factors and unit rates.&lt;/p&gt;&#xA;" OwnerUserId="15606" LastActivityDate="2024-02-09T22:08:40.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27470" PostTypeId="2" ParentId="27466" CreationDate="2024-02-10T01:51:58.597" Score="2" Body="&lt;p&gt;I posted an answer earlier, but after more thought, here's a different approach that I think might be better.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$d=6t$&lt;/span&gt; means 1 mile is the same as 6 hours.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; Really? So if you plug in 1 mile for &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; and 6 hours for &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; the equation comes out true?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$1=6\cdot 6 =36$&lt;/span&gt;... Wait that's not right.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; Plug in 1 hour for &lt;span class=&quot;math-container&quot;&gt;$t.$&lt;/span&gt; What comes out for &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Student:&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$d=6 \cdot 1 = 6.$&lt;/span&gt; So 1 hour is 6 miles.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Teacher:&lt;/strong&gt; Exactly. It's the opposite of what you said initially.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-02-10T03:27:04.833" LastActivityDate="2024-02-10T03:27:04.833" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27471" PostTypeId="2" ParentId="27466" CreationDate="2024-02-10T01:54:47.840" Score="3" Body="&lt;p&gt;Personally, I would emphasize translating natural statements very carefully, one word at a time. Starting with the fact that the verb &amp;quot;to be&amp;quot; (is) is the thing that translates to the equals sign in an assertive statement. So:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;someone runs 10 miles for every two hours&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The first thing I'd note is that there's no &amp;quot;is&amp;quot; verb in this statement. It's somewhat vaguely/sloppily expressed. We can improve on it like so:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;someone's running speed is 10 miles for every two hours&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Okay, now we have an assertive statement with &amp;quot;is&amp;quot;. Ideally we'd carefully define a variable for the quantity in question: &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; = speed, in miles per hour. Importantly we note that the &amp;quot;is&amp;quot; translates to equals, and &amp;quot;for every&amp;quot; (or &amp;quot;per&amp;quot;) translates to division. Hence:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$s = 10 \text{ miles} / 2 \text{ hours} = 5 \text{ mph}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Most elementary textbooks have blocks of exercises on this translation skill, so you can lean on that resource for practice. Generally the statements there are more clear then in the OP example, and I'm likewise careful to phrase my examples precisely for beginning students.&lt;/p&gt;&#xA;&lt;p&gt;If the instructor's &lt;em&gt;intent&lt;/em&gt; is to present more vaguely-formed statements and let the students grapple with that, then it's by nature more challenging. I will also say that I've found proportional problems in a basic arithmetic class to be among the most troublesome, precisely because no book or exercise ever explains that things are proportional -- students are expected to just intuit the proportionality in each separate case.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2024-02-10T05:02:09.797" LastActivityDate="2024-02-10T05:02:09.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27472" PostTypeId="1" CreationDate="2024-02-10T05:00:34.450" Score="2" ViewCount="139" Body="&lt;p&gt;It is known that the meaning of a conditional statement in fuzzy logic can vary depending on the interpretation and context. As we know, some ones interpret &amp;quot;if A, then B&amp;quot; as &amp;quot;A &lt;em&gt;coupled&lt;/em&gt; with B&amp;quot; instead of &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\neg A \lor B$&lt;/span&gt;&amp;quot; when dealing with fuzzy rules and fuzzy inference systems; for example see the below update part. However, for students who are solely familiar with classical logic, this unconventional perspective may seem puzzling and give rise to various questions such as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;in which context this interpretation is applicable?&lt;/li&gt;&#xA;&lt;li&gt;why we have to use &lt;span class=&quot;math-container&quot;&gt;$A \Rightarrow B$&lt;/span&gt; notation and symbols when it means &lt;span class=&quot;math-container&quot;&gt;$A \land B$&lt;/span&gt;?&lt;/li&gt;&#xA;&lt;li&gt;whether it has any relevance in classical logic?&lt;/li&gt;&#xA;&lt;li&gt;whether this interpretation can be rational and relevant in everyday human speech?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Can you help me to find answers of above questions? I think that first and last questions are more important and can cause better understanding.&lt;/p&gt;&#xA;&lt;h2&gt;Update&lt;/h2&gt;&#xA;&lt;p&gt;Mamdani in &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/S0020737375800022&quot; rel=&quot;nofollow noreferrer&quot;&gt;An experiment in linguistic synthesis with a fuzzy logic controller&lt;/a&gt; has said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The control rules were implemented by using fuzzy conditional statements, for example &amp;quot;&lt;em&gt;If PE is &lt;span class=&quot;math-container&quot;&gt;$NB$&lt;/span&gt; then HC is &lt;span class=&quot;math-container&quot;&gt;$PB$&lt;/span&gt;&lt;/em&gt;&amp;quot;. The implied relation between the two fuzzy variables &lt;code&gt;PE&lt;/code&gt; and &lt;code&gt;HC&lt;/code&gt; is expressed in terms of the cartesian product of the two subsets &lt;span class=&quot;math-container&quot;&gt;$NB$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$PB$&lt;/span&gt;. The cartesian product of two sets &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is denoted &lt;span class=&quot;math-container&quot;&gt;$A \times B$&lt;/span&gt; and is defined by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;A \times B = \sum\limits_{i} \sum\limits_{j} \min \{ \mu_{A}(u_i), \mu_{B}(v_j) \} \\&#xA;= \sum\limits_{ij} \min u_i v_j&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; are generic elements of the universes of discourse of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; respectively.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As you can see, he used &lt;span class=&quot;math-container&quot;&gt;$\min (\mu_{A}(x), \mu_{B}(y))$&lt;/span&gt; as interpretation for &lt;span class=&quot;math-container&quot;&gt;$A \rightarrow B$&lt;/span&gt;, which is&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;equal to &lt;span class=&quot;math-container&quot;&gt;$A \land B$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;a special case for &lt;span class=&quot;math-container&quot;&gt;$T_{norm} (\mu_{A}(x), \mu_{B}(y))$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;not equal to &lt;span class=&quot;math-container&quot;&gt;$\neg A \lor B$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In some books, for example &amp;quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0132610663&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Neuro-Fuzzy and Soft Computing: A omputational Approach to Learning and Machine Intelligence&lt;/a&gt;&amp;quot; the &lt;span class=&quot;math-container&quot;&gt;$T_{norm} (\mu_{A}(x), \mu_{B}(y))$&lt;/span&gt; interpretation is called as &amp;quot;A coupled with B&amp;quot;. Although, here the word &amp;quot;coupled&amp;quot; is not so important for me. I wanted to know when that interpretation is rational?&lt;/p&gt;&#xA;" OwnerUserId="5649" LastEditorUserId="5649" LastEditDate="2024-02-14T05:32:16.463" LastActivityDate="2024-02-14T05:32:16.463" Title="When Interpreting &quot;If A, then B&quot; as &quot;A coupled with B&quot; is rational?" Tags="|mathematical-pedagogy|logic|student-question|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27473" PostTypeId="2" ParentId="27458" CreationDate="2024-02-10T15:55:23.007" Score="0" Body="&lt;p&gt;The matrix form of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{bmatrix}&#xA;1 &amp;amp; 1 &amp;amp; 0 \\&#xA;1 &amp;amp; 2 &amp;amp; 0 \\&#xA;0 &amp;amp; 0 &amp;amp; 3 \\&#xA;\end{bmatrix}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;At this point in Axler's book (linked in the OP), it should be obvious that &lt;span class=&quot;math-container&quot;&gt;$\lambda=3$&lt;/span&gt; is an eigenvalue and that &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; has an invariant subspace &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$T\vert_W$&lt;/span&gt; has matrix&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{bmatrix}&#xA;1&amp;amp;1\\&#xA;1&amp;amp;2\\&#xA;\end{bmatrix}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Exercise 5B#11 page 151 says that for a 2x2 matrix&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{bmatrix}&#xA;a&amp;amp;c\\&#xA;b&amp;amp;d\\&#xA;\end{bmatrix}&lt;/span&gt;&#xA;the minimal polynomial of a linear operator on a two-dimensional vector space with this matrix representation is (except for a degenerate case) &lt;span class=&quot;math-container&quot;&gt;$$z^2-(a+d)z+(ad-bc)$$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Notice that the constant term of the quadratic is the determinant, and the linear term is the negative of the trace. In effect, this result (Exercise #11) gives us the green light to use &lt;span class=&quot;math-container&quot;&gt;$\det(A-\lambda I)$&lt;/span&gt; to calculate eigenvalues of a 2x2 matrix, even if we are not formally allowed to utter the words &amp;quot;trace&amp;quot; and &amp;quot;determinant.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2024-02-10T15:55:23.007" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27474" PostTypeId="1" CreationDate="2024-02-10T16:43:12.953" Score="2" ViewCount="215" Body="&lt;p&gt;You have to go from a point &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; (start) to a point &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; (arrival) by crossing a river &lt;span class=&quot;math-container&quot;&gt;$(d)$&lt;/span&gt; and traveling as little distance as possible.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7CGBs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7CGBs.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Pupils first do a search by trying several paths &lt;span class=&quot;math-container&quot;&gt;$1,2,3,4$&lt;/span&gt; and completing a table with the different perimeters &lt;span class=&quot;math-container&quot;&gt;$p_1, p_2, p_3, p_4$&lt;/span&gt;. I use perimeter as on geobra for the length of a broken line. Out of two classes to which I proposed the exercise, none of the pupils thought of using symmetry on their own, although the exercise is part of the chapter Symmetry; On the other hand, they all very naturally thought of the straight line to reach the river from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. Once the indication was given: &amp;quot;perhaps  symmetry could be used&amp;quot;, the solution was quickly found by some of the students. On the other hand, doubts about this solution have arisen, with some students thinking that they have measured a shorter path than the red path. In addition, with a geogebra animation, because of the rounding, students pointed out that there were perhaps several possibilities...&#xA;From then on, I was forced to give a demonstration.&lt;/p&gt;&#xA;&lt;p&gt;The knowledge that &amp;quot;the shortest path from one point to another is the straight line&amp;quot; having been implicitly accepted, the triangular inequality &lt;span class=&quot;math-container&quot;&gt;$\color{red}{(1)}$&lt;/span&gt; flowed from it.&lt;/p&gt;&#xA;&lt;p&gt;In addition, the students knew that &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$B'$&lt;/span&gt; symmetrical of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;&amp;quot; was equivalent to writing &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; mediatrix of &lt;span class=&quot;math-container&quot;&gt;$BB'$&lt;/span&gt;&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\color{green}{(2)}$&lt;/span&gt;;&lt;/p&gt;&#xA;&lt;p&gt;that if a point is on the mediatrix of a segment, then it is equidistant from the endpoints&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\color{orange}{(3)}$&lt;/span&gt;;&lt;/p&gt;&#xA;&lt;p&gt;that &amp;quot;if &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$a\color{red}{+c}=b\color{red}{+c}$&lt;/span&gt;&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$ (4)$&lt;/span&gt;, e.g. through the example &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$5=2+3$&lt;/span&gt; therefore &lt;span class=&quot;math-container&quot;&gt;$5\color{red}{+7}=2+3\color{red}{+7}$&lt;/span&gt;&amp;quot;;&lt;/p&gt;&#xA;&lt;p&gt;that if &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; is on &lt;span class=&quot;math-container&quot;&gt;$[AB]$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$AB=AM+MB$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\color{Magenta}{(5)}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We then had to formalize the problem: we are looking for &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; as for any point &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$d,AM+MB&amp;gt;AH+HB$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then prove that &lt;span class=&quot;math-container&quot;&gt;$H=I$&lt;/span&gt; is the only point that fits.&lt;/p&gt;&#xA;&lt;p&gt;Here is the demonstration I provided, with obvious notations that I obviously spared the pupils: let &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; any point other than &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$[AB]$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B'$&lt;/span&gt; the symmetric of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; with respect to &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$I:=[AB']\cap d $&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$\color{red}{AM+}MB=\color{red}{AM+}MB'&amp;gt;AB'=\color{red}{AI+}IB'=\color{red}{AI+}IB$$&lt;/span&gt; according to,  in order of use, &lt;span class=&quot;math-container&quot;&gt;$\color{green }{(2)}$&lt;/span&gt;,&lt;span class=&quot;math-container&quot;&gt;$\color{orange}{(3)}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$ (4)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\color{red}{(1)}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\color{Magenta} {(5)}$&lt;/span&gt; and finally again  &lt;span class=&quot;math-container&quot;&gt;$\color{green }{(2)}$&lt;/span&gt;,&lt;span class=&quot;math-container&quot;&gt;$\color{orange}{(3)}$&lt;/span&gt; et  &lt;span class=&quot;math-container&quot;&gt;$(4)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Aware that the demonstration was not envisaged by the authors of the exercise and that only the first research was, giving the demonstration to twelve-year-old students, some of whom have very sharp minds, did not seem to me useless or too problematic for the more difficult students.&lt;/p&gt;&#xA;&lt;p&gt;Is the proof mathematically correct?&lt;/p&gt;&#xA;&lt;p&gt;What about its relevance to twelve-year-olds?&lt;/p&gt;&#xA;" OwnerUserId="23732" LastEditorUserId="23732" LastEditDate="2024-02-10T17:09:49.120" LastActivityDate="2024-02-12T04:45:53.563" Title="About a difficult exercise for 12 years pupils" Tags="|secondary-education|mathematical-pedagogy|proofs|geometry|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27475" PostTypeId="2" ParentId="27466" CreationDate="2024-02-10T22:40:06.210" Score="16" Body="&lt;p&gt;A practical suggestion: make sure your own equations are dimensionally consistent, and your students will be more likely to make theirs consistent as well, thus avoiding mistakes like these.&lt;/p&gt;&#xA;&lt;p&gt;In particular, your equation &lt;span class=&quot;math-container&quot;&gt;$d = 6t$&lt;/span&gt; is not dimensionally consistent.  If I plug in, say, &lt;span class=&quot;math-container&quot;&gt;$d = 1\text{ mile}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t = 1\text{ hour}$&lt;/span&gt;, I get the obviously nonsensical equation &lt;span class=&quot;math-container&quot;&gt;$1\text{ mile} = 6\text{ hours}$&lt;/span&gt;.  Which is what the student in your example got too, and then tried to somehow make sense of.  Arguably their mistake was in trying to make sense of nonsense, but it's an understandable mistake if they haven't been taught any better.&lt;/p&gt;&#xA;&lt;p&gt;OK, but enough polemic.  If &lt;span class=&quot;math-container&quot;&gt;$d = 6t$&lt;/span&gt; is nonsense, then how &lt;em&gt;should&lt;/em&gt; you correctly express the idea that a person runs 6 miles in an hour?  Simply like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$d = 6\frac{\text{miles}}{\text{hour}}\, t$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now you, or your students, can in fact plug in, say, &lt;span class=&quot;math-container&quot;&gt;$t = 1\text{ hour}$&lt;/span&gt; and solve for &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\require{cancel} d = 6\frac{\text{miles}}{\cancel{\text{hour}}} \cdot 1\,\cancel{\text{hour}} = 6\text{ miles}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Better yet, they can check that the result they got in fact fact has units that make sense.  If they had instead obtained a result of, say, &lt;span class=&quot;math-container&quot;&gt;$d = 6\text{ hours}$&lt;/span&gt;, they would've immediately known that they'd made a mistake somewhere, since &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is supposed to be a distance, and an hour is &lt;em&gt;not&lt;/em&gt; a unit of distance!&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I do understand that my suggestion above might not seem practical at a glance, since it basically amounts to completely changing the way you teach basic algebra whenever physical quantities with units are involved.  You might even have a textbook that teaches things the same way you currently do, and you'd have to either find a new textbook or tell your students &amp;quot;don't do it the way the book says, do it the way I say&amp;quot; (which, while sometimes a possible solution if you must work with a subpar textbook, is far from pedagogically ideal).  Your students may also have previously learned to do things differently, and you might need to get them to unlearn bad habits like leaving units out of equations.  Heck, you might have to force &lt;em&gt;yourself&lt;/em&gt; to unlearn these habits first, too.&lt;/p&gt;&#xA;&lt;p&gt;But I would still suggest at least considering this approach.  Maybe try it out on one class with some particularly struggling students as an experiment.  (Do &lt;em&gt;tell&lt;/em&gt; your students that you're doing an experiment, especially if you've previously taught the same class to do things differently, or they'll just be confused!)  I am quite confident that, despite whatever initial difficulties you might have, in the long run this &lt;em&gt;will&lt;/em&gt; get you better results and help your students better understand what's happening with equations like this.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Ps. Also make sure that &lt;a href=&quot;https://matheducators.stackexchange.com/questions/5877/should-word-problems-be-reasonable/5883#5883&quot;&gt;your word problems actually have answers that make sense&lt;/a&gt;, so that your students can use their common sense to check their answers.  And teach them to refine that common sense using techniques such as dimensional analysis (as described above), order-of-magnitude estimation and other approximation and consistency checking methods.  This is a valuable skill that will serve your students well in their life, and one that mathematics education all too often neglects (or sometimes even actively harms!).&lt;/p&gt;&#xA;" OwnerUserId="526" LastActivityDate="2024-02-10T22:40:06.210" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27476" PostTypeId="2" ParentId="27400" CreationDate="2024-02-12T01:08:56.490" Score="1" Body="&lt;p&gt;From &lt;a href=&quot;https://math.stackexchange.com/questions/26325&quot;&gt;Does integration by parts with &amp;quot;deja vu&amp;quot; have a name?&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sheard (&amp;quot;Trick or Technique?&amp;quot;, &lt;a href=&quot;https://maa.org/sites/default/files/Sheard-CMJ0917264.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2009&lt;/a&gt;) calls it the &lt;em&gt;one-step algebra trick&lt;/em&gt;;&lt;/li&gt;&#xA;&lt;li&gt;OP says he saw it called &lt;em&gt;integration by parts with &amp;quot;déjà vu&amp;quot;&lt;/em&gt;;&lt;/li&gt;&#xA;&lt;li&gt;Glen Wheeler suggests calling it &lt;em&gt;absorption&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="23536" LastActivityDate="2024-02-12T01:08:56.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27477" PostTypeId="2" ParentId="27474" CreationDate="2024-02-12T04:45:53.563" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;The exercise is part of the chapter Symmetry... Is the proof mathematically correct? What about its relevance to twelve-year-olds?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The proof seems fine, but I think there is a more intuitive way to explain the relevance of symmetry to twelve-year-olds using this problem as an example.&lt;/p&gt;&#xA;&lt;p&gt;If I were to explain the solution to this problem to a twelve-year-old, I'd do the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;First establish baseline intuition about the relevance of symmetry to optimization problems. In my experience, kids will typically pick up on this pretty quickly if you give them a few simple yet illustrative problems. For instance, if you have &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; meters of fencing and you want to fence off the largest possible area for a rectangular garden, then what should the dimensions of the garden be? The kids might need to try a few configurations but they'll see the trend and arrive at &lt;span class=&quot;math-container&quot;&gt;$5 \times 5$&lt;/span&gt; pretty quickly.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Then, connect that intuition to the problem at hand. In the problem you've given, the core idea is that you want to minimize the sum of two diagonal distances, and each distance is the hypotenuse of a right triangle (the legs of the triangle are parallel and perpendicular to the river). Just as no side of the rectangular garden is any more special than the other, neither triangle in this problem is any more special than the other. So the triangles should have the same relative dimensions (one triangle might be bigger than the other, but their angles should all be the same).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Of course, it remains to actually construct the two triangles that have the same relative dimensions. As you show in your diagram, we can use symmetry here too: reflect &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; over the river to &lt;span class=&quot;math-container&quot;&gt;$B',$&lt;/span&gt; draw &lt;span class=&quot;math-container&quot;&gt;$AIB',$&lt;/span&gt; and then draw &lt;span class=&quot;math-container&quot;&gt;$IB.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But the relevance of symmetry here, and especially its relevance to twelve-year-olds, is not so much this specific construction but rather the general idea that an &amp;quot;optimal configuration of things,&amp;quot; when no particular thing is substantively different from the others, will exhibit symmetry.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2024-02-12T04:45:53.563" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27478" PostTypeId="2" ParentId="377" CreationDate="2024-02-13T03:51:33.883" Score="3" Body="&lt;p&gt;I understand that this is an old question, but still a good topic. I was an elementary student in the ‘60sf when, in 5th grade, they changed to teaching “New Math”. Problem was, the teacher didn’t understand it, had taken some training over the summer but was confused at the chalkboard. So were we, the entire class. Because she felt so insecure, she became impatient with our lack of understanding. I’d loved math until then, but became hopelessly confused, lost, and math-phobic. I believe a big reason it was dropped was the abysmal job of teaching it exhibited by most elementary math teachers. They didn’t really understand it, so couldn’t teach it to us. Math scores continued to decline in the U.S. until this was dropped. Unsure what replaced it.&lt;/p&gt;&#xA;" OwnerUserId="23739" LastActivityDate="2024-02-13T03:51:33.883" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27479" PostTypeId="2" ParentId="27466" CreationDate="2024-02-13T06:44:54.083" Score="1" Body="&lt;p&gt;I'm coming from a physics background. I avoid writing numbers for any measured quantities into equations until the last step where I plug in values to get a final answer. This is because it can be difficult to remember which number have quantities, which can make finding errors harder. In the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$d=6t,$$&lt;/span&gt;&#xA;it is not immediately obvious that the 6 has units of miles per hour. Bare numbers usually don't have units unless explicitly written out. The left side looks like it has units of distance, while the right side looks like it has units of time, which doesn't work [1]. Even if I knew that the speed of the running child was 6 miles per hour, I would still write&#xA;&lt;span class=&quot;math-container&quot;&gt;$$d=v\times t.$$&lt;/span&gt;&#xA;The natural way a student would read this is &amp;quot;distance is velocity times time,&amp;quot; which would be correct. All measured quantities (especially those with units) are given symbols to distinguish them from bare numbers that don't. Another advantage to using only symbols for quantities is that it allows for error checking during algebraic manipulation. If I'm doing a long derviation and I find that I've written &lt;span class=&quot;math-container&quot;&gt;$x + t$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is position and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is time, I know I've made a mistake earlier and can go looking for it. If I write &lt;span class=&quot;math-container&quot;&gt;$5 + t$&lt;/span&gt;, it's much less obvious.&lt;/p&gt;&#xA;&lt;p&gt;Another example is the acceleration equation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$d = \frac{1}{2}at^2.$$&lt;/span&gt;&#xA;The &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; is a mathematical constant with no units, so it needs no symbol. A quick glance shows that both sides have units of distance, which makes subsequent algebraic manipulation easier to reason about.&lt;/p&gt;&#xA;&lt;p&gt;[1] I'm ignoring the high-energy physicists who set &lt;span class=&quot;math-container&quot;&gt;$c=1$&lt;/span&gt; in their equations.&lt;/p&gt;&#xA;" OwnerUserId="3344" LastActivityDate="2024-02-13T06:44:54.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27480" PostTypeId="2" ParentId="27466" CreationDate="2024-02-13T09:37:47.047" Score="1" Body="&lt;p&gt;I would say that your students are completely right:&lt;/p&gt;&#xA;&lt;p&gt;In general life it's completely normal to write six seconds as &lt;span class=&quot;math-container&quot;&gt;$6s$&lt;/span&gt; or thirteen meter as &lt;span class=&quot;math-container&quot;&gt;$13m$&lt;/span&gt;, so thinking that &lt;span class=&quot;math-container&quot;&gt;$6t$&lt;/span&gt; is some way of writing &amp;quot;six (t)s&amp;quot; where &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is some time or space unit, that's just common sense.&lt;/p&gt;&#xA;&lt;p&gt;In order for you to avoid this, you might write the dot operator as the multiplication, so not &lt;span class=&quot;math-container&quot;&gt;$6s$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$6 \cdot s$&lt;/span&gt;. It's a very small burden for you, but a lot clearer for your students.&lt;br /&gt;&#xA;Once you start dealing with more complex equations, like &lt;span class=&quot;math-container&quot;&gt;$y=mx+q$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$ax^2+bx+c$&lt;/span&gt;, you might start dropping the dot (but in case of doubt, you might always choose to put it back).&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2024-02-13T09:37:47.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27481" PostTypeId="2" ParentId="7608" CreationDate="2024-02-14T01:06:32.673" Score="-2" Body="&lt;p&gt;Instruct students to create drawings that incorporate memes using words beginning with the syllable &amp;quot;pi.&amp;quot; This activity offers an engaging combination of language —focusing on word selection—)and visual art(whether through traditional hand-drawing or the use of generative AI technologies) the collective work can compose a book. one example ( have much ore in Portuguese:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VWFcn.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VWFcn.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ssqw5.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ssqw5.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6BFoS.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6BFoS.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TxlsJ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TxlsJ.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;this activity is specially convinient for multilianguage classes.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="8828" LastEditDate="2024-02-16T16:19:09.377" LastActivityDate="2024-02-16T16:19:09.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27482" PostTypeId="1" AcceptedAnswerId="27483" CreationDate="2024-02-14T15:26:10.680" Score="5" ViewCount="373" Body="&lt;h3&gt;Background&lt;/h3&gt;&#xA;&lt;p&gt;I am a class 10 student who is fond of maths. I like making math questions.&lt;/p&gt;&#xA;&lt;h3&gt;Question&lt;/h3&gt;&#xA;&lt;p&gt;I do not know of any platform where I can post these questions for others to practice and learn. I really want not to let them go unnoticed. Is there any well-known website or any other platform for this?&lt;/p&gt;&#xA;&lt;p&gt;My questions require some critical thinking and are different from the ones which are found the textbooks that follow the syllabus.&lt;/p&gt;&#xA;&lt;h3&gt;Example Questions&lt;/h3&gt;&#xA;&lt;p&gt;Q - 1 Let the function &lt;span class=&quot;math-container&quot;&gt;$S(n)$&lt;/span&gt; denote the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; terms of Arithematic Progression. Given that the degree of &lt;span class=&quot;math-container&quot;&gt;$S(n)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and the first term of the AP is 10, find the &lt;span class=&quot;math-container&quot;&gt;$10^2$&lt;/span&gt;th term of the AP.&lt;/p&gt;&#xA;&lt;p&gt;Q - 2 Let there be 2 squares, whose area is given by &lt;span class=&quot;math-container&quot;&gt;$[p(x)]^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$[g(x)]^2$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$x\geq0$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; are quadratic polynomials.&lt;/p&gt;&#xA;&lt;p&gt;It is given that &lt;span class=&quot;math-container&quot;&gt;$p(x) &amp;gt; g(x) \quad \forall x\geq0$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;When the smaller square is placed on the larger square, such that it overlaps completely with the larger square then the area not overlapped by the smaller square is given by the polynomial &lt;span class=&quot;math-container&quot;&gt;$r(x) = 3x^4 + 20x^3 + 46x^2 + 44x + 15$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If the difference in side lengths of both squares is &lt;span class=&quot;math-container&quot;&gt;$3,8,15$&lt;/span&gt; for values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$0,1,2$&lt;/span&gt;, then find how many times &lt;span class=&quot;math-container&quot;&gt;$p(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; intersect on the graph. Also evaluate &lt;span class=&quot;math-container&quot;&gt;$\frac{p(12) - 1}{g(10)}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Q - 3 Prove that for any natural number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$1101$&lt;/span&gt; cannot be the sum of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and the number obtained by reversing the digits of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Q - 4 Prove that for natural numbers &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$HCF(a^n, b^n) = [HCF(a,b)]^n$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h3&gt;Edit&lt;/h3&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit 1 -&lt;/strong&gt; My problems are relatively straightforward for students who have taken specific courses that cover the required problem-solving techniques. My original intent was to look for an audience that had just learnt the basics and for whom these questions would require critical thinking, but now I realize I should leave that idea and post my problems for a general maths audience.&lt;/p&gt;&#xA;" OwnerUserId="23742" LastEditorUserId="23742" LastEditDate="2024-02-15T09:07:20.853" LastActivityDate="2024-02-15T09:07:20.853" Title="Any known platform to post self made math questions" Tags="|secondary-education|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27483" PostTypeId="2" ParentId="27482" CreationDate="2024-02-14T16:07:33.333" Score="4" Body="&lt;p&gt;Perhaps &lt;a href=&quot;https://www.reddit.com/r/mathriddles/&quot; rel=&quot;nofollow noreferrer&quot;&gt;reddit.com/r/mathriddles&lt;/a&gt;. The problems in the tag &lt;a href=&quot;https://www.reddit.com/r/mathriddles/?f=flair_name%3A%22Easy%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;easy&amp;quot;&lt;/a&gt; seem somewhat similar to the type of problems you posted, and they do get upvoted, which indicates that the community is receptive to them.&lt;/p&gt;&#xA;&lt;p&gt;And of course nothing is stopping you from putting up a blog and posting the problems there, and/or making some YouTube videos about them.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;Something to aspire to:&lt;/em&gt; Math.SE has some tags where people often post problems that they believe are interesting &amp;amp; non-trivial in some way.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/tagged/recreational-mathematics&quot;&gt;&amp;quot;recreational-mathematics&amp;quot;&lt;/a&gt;&lt;/strong&gt;&#xA;&lt;br&gt;For instance: &lt;a href=&quot;https://math.stackexchange.com/questions/2855975/what-is-the-maximum-volume-that-can-be-contained-by-a-sheet-of-paper&quot;&gt;What is the maximum volume that can be contained by a sheet of paper?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/tagged/puzzle&quot;&gt;&amp;quot;puzzle&amp;quot;&lt;/a&gt;&lt;/strong&gt;&#xA;&lt;br&gt;For instance: &lt;a href=&quot;https://math.stackexchange.com/questions/3972098/how-few-42-circ-60-circ-78-circ-triangles-can-an-equilateral-triangle-be&quot;&gt;How few &lt;span class=&quot;math-container&quot;&gt;$(42^\circ,60^\circ,78^\circ)$&lt;/span&gt; triangles can an equilateral triangle be divided into?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/tagged/contest-math&quot;&gt;&amp;quot;contest-math&amp;quot;&lt;/a&gt;&lt;/strong&gt;&#xA;&lt;br&gt;For instance: &lt;a href=&quot;https://math.stackexchange.com/questions/772665/find-all-functions-f-such-that-if-ab-is-a-square-then-fafb-is-a-squ&quot;&gt;Find all functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; such that if &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt; is a square, then &lt;span class=&quot;math-container&quot;&gt;$f(a)+f(b)$&lt;/span&gt; is a square&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I think the problems you provided here are too straightforward for Math.SE -- Q1 and Q4 look like standard textbook exercises, and I'll elaborate on Q2 and Q3 below. But perhaps you could try to create some more challenging problems to post there as well. It could be a good source of inspiration to continue developing your math chops! :)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Q2 seems straightforward if you know how solve systems of arbitrarily many linear equations (which is covered at the beginning of Linear Algebra, and sometimes the end of Precalculus). You have &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; unknowns (&lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; coefficients for each of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;) and &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; linear equations (&lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; equations from equating the coefficients in &lt;span class=&quot;math-container&quot;&gt;$p(x)^2 - g(x)^2 = r(x),$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; more equations from the differences in side lengths for &lt;span class=&quot;math-container&quot;&gt;$x=0,1,2$&lt;/span&gt;), so you can just solve the system to find &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; exactly, and then the problem is essentially solved.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Q3 looks a little bit more interesting but it's still straightforward for people who have taken an Introduction to Proofs course -- they'll naturally think of this number as having digits &lt;span class=&quot;math-container&quot;&gt;$abc,$&lt;/span&gt; and arrive at&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;1101 &amp;amp;= (a + 10b + 100c) + (100a + 10b + c) \\&#xA;&amp;amp;= 101(a+c) + 20b,&#xA;\end{align*}&lt;/span&gt;&#xA;and notice a contradiction: &lt;span class=&quot;math-container&quot;&gt;$a+c$&lt;/span&gt; must end in &lt;span class=&quot;math-container&quot;&gt;$1,$&lt;/span&gt; yet also &lt;span class=&quot;math-container&quot;&gt;$a+c=10$&lt;/span&gt; (because &lt;span class=&quot;math-container&quot;&gt;$\frac{1101-(9)(20)}{101} \leq a+c \leq \frac{1101}{101}$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-02-14T22:44:35.403" LastActivityDate="2024-02-14T22:44:35.403" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27484" PostTypeId="1" CreationDate="2024-02-14T17:32:50.797" Score="4" ViewCount="83" Body="&lt;p&gt;I am a teacher at a gifted center up to middle school grades, and I am looking for interesting tasks and activities for gifted 6th grade students for a course on problem solving and using varied strategies. The emphasis is on developing different skills such as creative thinking, mental flexibility, and dealing with challenges. Thanks to all the helpers.&lt;/p&gt;&#xA;" OwnerUserId="23743" LastEditorUserId="15367" LastEditDate="2024-02-16T00:01:54.783" LastActivityDate="2024-02-16T00:38:14.237" Title="Looking for interesting tasks and activities for 6th grade students for a course on problem solving" Tags="|logic|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27485" PostTypeId="2" ParentId="27484" CreationDate="2024-02-14T17:41:10.620" Score="2" Body="&lt;p&gt;Here's a puzzle that I used to give to all grades 6th grade and up on the first day of class as something fun to do while many students were waiting to sort out schedule and laptop issues.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Draw an &lt;span class=&quot;math-container&quot;&gt;$10 \times 10$&lt;/span&gt; square grid. How many squares are there in total? Not just &lt;span class=&quot;math-container&quot;&gt;$1 \times 1$&lt;/span&gt; squares, but also &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; squares, &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; squares, and so on.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I liked this puzzle in particular because the solution was not immediately obvious to even the brightest / oldest students, yet all students (even the younger grades / weaker students) were able to understand the goal of the problem and work through concrete examples to notice a pattern that they could extrapolate to find the solution. (The higher grades / brighter students just picked up on the pattern faster.)&lt;/p&gt;&#xA;&lt;p&gt;Whenever students needed some guidance getting started, I would explicitly walk them through the process of drawing up a &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; grid and counting the number of &lt;span class=&quot;math-container&quot;&gt;$1 \times 1$&lt;/span&gt; squares (there are &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;), &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; squares (there are &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;), and &lt;span class=&quot;math-container&quot;&gt;$3 \times 3$&lt;/span&gt; squares (there is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;), for a total of &lt;span class=&quot;math-container&quot;&gt;$9 + 4 + 1 = 14$&lt;/span&gt; squares. I would then leave them alone to do this for a &lt;span class=&quot;math-container&quot;&gt;$4 \times 4$&lt;/span&gt; grid and then a &lt;span class=&quot;math-container&quot;&gt;$5 \times 5$&lt;/span&gt; grid, after which I would help them notice the following pattern:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;3 \times 3 \textrm{ grid} \quad \to &amp;amp; \quad 9 + 4 + 1 \\&#xA;&amp;amp; \quad 3^2 + 2^2 + 1^2 \\[5pt]&#xA;4 \times 4 \textrm{ grid} \quad \to &amp;amp; \quad 16 + 9 + 4 + 1 \\&#xA;&amp;amp; \quad 4^2 + 3^2 + 2^2 + 1^2 \\[5pt]&#xA;5 \times 5 \textrm{ grid} \quad \to &amp;amp; \quad 25 + 16 + 9 + 4 + 1 \\&#xA;&amp;amp; \quad 5^2 + 4^2 + 3^2 + 2^2 + 1^2 \\[5pt]&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This pattern is almost a &amp;quot;punch line&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What's the total number of squares (square shapes)? Well, it's the total sum of squares (square numbers).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Extrapolating the pattern to solve the puzzle:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;10 \times 10 \textrm{ grid} \quad \to &amp;amp; \quad 10^2 + 9^2 + 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2 \\&#xA;&amp;amp; \quad = 385 \textrm{ squares in total} \\[5pt]&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2024-02-14T17:41:10.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27486" PostTypeId="1" CreationDate="2024-02-14T20:43:10.770" Score="-1" ViewCount="223" Body="&lt;p&gt;Imagine an introductory probability course that assumes the students have had first-year calculus and understand mathematical reasoning. At some point in such a course has explicated several families of discrete probability distributions, including the binomial distribution as the number of &amp;quot;successes&amp;quot; needed to get a specified non-random number of trials, and the negative binomial distribution as the number of failures before a specified non-random number of successes. Then one develops the Poisson distribution as a limit of binomial distributions with the expected number of successes remaining constant as the number of trials grows and the probability of success on each trial decreases, being inversely proportional to the number of trials.&lt;/p&gt;&#xA;&lt;p&gt;One reaches a point in the exposition where to different Poisson-distributed random variables correspond to different subsets of the line, the expected value being &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; times the measure of the subset, where &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; is the same for all such subsets, and these random variables are independent when their corresponding subsets are essentially disjoint, and otherwise dependent and positively correlated.&lt;/p&gt;&#xA;&lt;p&gt;At this point all probability distributions one has dealt with are &lt;b&gt;discrete&lt;/b&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It seems that in &lt;b&gt;conventional&lt;/b&gt; textbooks, one does not go from there do dealing with &lt;b&gt;particular continuous&lt;/b&gt; probability distributions until one has dealt with continuous distributions in &lt;b&gt;general&lt;/b&gt;, stating that they have continuous cumulative distribution functions, and in absolutely continuous cases, are characterized by a density function, the value of whose integral over a set is the probability assigned to that set.&lt;/p&gt;&#xA;&lt;p&gt;But alternatively, suppose one has not yet done what is described in the paragraph above, but one has done what is described in the paragraph before that.&lt;/p&gt;&#xA;&lt;p&gt;One can then let &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; be the time until the first Poisson arrival after time &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and let &lt;span class=&quot;math-container&quot;&gt;$N_t$&lt;/span&gt; be the number of arrivals before time &lt;span class=&quot;math-container&quot;&gt;$t,$&lt;/span&gt; and observe this:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\Pr(T&amp;gt;t) = {} \, \underbrace{\Pr( N_t=0) = \frac{(\mu t)^0 e^{-\mu t}}{0!}}_\text{This part has already been established.} \,= e^{-\mu t}.&#xA;$$&lt;/span&gt;&#xA;From there, it is easy to show that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\Pr(T\in S) = \int_S e^{-\mu t} \big( \mu\, dt\big).&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then one is dealing with a continuous distribution without having done what I reported to be done in conventional textbooks.&lt;/p&gt;&#xA;&lt;p&gt;This has pedagogical advantages, upon which I may remark in comments if anybody cares.&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;My question is&lt;/b&gt; whether there is some published textbook that makes this kind of transition from discrete to continuous &lt;b&gt;without&lt;/b&gt; first treating continuous distributions in general?&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2024-02-15T18:13:27.333" LastActivityDate="2024-02-24T11:44:56.500" Title="Does some textbook on probability use a particular way (described here) of accomplishing the segue into continuous distributions?" Tags="|textbooks|probability|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27488" PostTypeId="2" ParentId="27484" CreationDate="2024-02-16T00:38:14.237" Score="3" Body="&lt;p&gt;Justin's problem is a counting problem, and there are lots of those in any discrete math textbook. Many of those are approachable at 6th grade. Here's a free textbook that I like: &lt;a href=&quot;https://discrete.openmathbooks.org/dmoi3.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://discrete.openmathbooks.org/dmoi3.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I also recommend geometric puzzles. &lt;a href=&quot;https://mathwithbaddrawings.com/2019/12/09/geometry-puzzles-for-a-winters-day/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math with Bad Drawings posted a collection&lt;/a&gt;. Catriona Shearer makes up great ones, but they might all be too hard. I got ebooks of &lt;em&gt;&lt;strong&gt;Geometry Snacks&lt;/strong&gt;&lt;/em&gt; and &lt;a href=&quot;https://bookshop.org/p/books/more-geometry-snacks-bite-size-problems-and-how-to-solve-them-ed-southall/15079858?ean=9781911093961&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;More Geometry Snacks&lt;/strong&gt;&lt;/em&gt;, by Vincent Pantalloni and Ed Southall&lt;/a&gt;. There are loads of good puzzles in those two books.&lt;/p&gt;&#xA;&lt;p&gt;Beast Academy is a wonderful curriculum put out by Art of Problem Solving. [Disclaimer: I currently work part-time for AOPS.] BA has some pages available to print out for free. Here's a link to &lt;a href=&quot;https://beastacademy.com/resources/printables&quot; rel=&quot;nofollow noreferrer&quot;&gt;the level 5 printables&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I love &lt;em&gt;&lt;strong&gt;The Art and Craft of Problem Solving&lt;/strong&gt;&lt;/em&gt;, by Paul Zeitz. Most of the problems might be too hard for 6th graders, but I think you'll find a few that would work. (You might have to get that from a library. Used copies are currently priced way too high.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2024-02-16T00:38:14.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27490" PostTypeId="1" CreationDate="2024-02-18T01:45:11.867" Score="9" ViewCount="3231" Body="&lt;p&gt;Sometimes I struggle to give my students a sufficient number of reasons why they should study Geometry in high school, other than that it helps them think and increases their understanding of the world.  Aside from sports and teaching itself, what are examples of jobs/careers that apply this material?&lt;/p&gt;&#xA;" OwnerUserId="23752" LastActivityDate="2024-02-24T09:55:37.720" Title="Applications of High School Geometry" Tags="|secondary-education|applications|" AnswerCount="11" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="27491" PostTypeId="2" ParentId="27490" CreationDate="2024-02-18T03:50:18.043" Score="31" Body="&lt;p&gt;Every time I'm doing carpentry/DIY remodeling on my house, I use so much high school geometry that it makes me chuckle.&lt;/p&gt;&#xA;&lt;p&gt;I've had instances where I need to cut angles to make pieces parallel and used Alternate Interior reasoning to pull it off. I've had times where the angle I can measure is complimentary to the angle I need to cut, so I have to figure out how to reflect the piece or the cut line to get it right. I've needed pieces of plywood cut that were general quadrilaterals (because nothing is square in old homes) and was able to pull it off by measuring diagonals and marking equidistant points using a string compass. And of course there's all the tricks for actually getting a square corner.&lt;/p&gt;&#xA;&lt;p&gt;Of all the subjects taught in school from algebra to calculus to abstract algebra, I've used high school geometry far above and beyond the others in these types of &amp;quot;real&amp;quot; world situations. Set up an equation and solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;? Not sure I've ever really had the need to do that outside of my professional life (okay, maybe a few times when I was baking). Need to literally construct a congruent triangle? I can't even begin to count how many times.&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2024-02-18T03:50:18.043" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27492" PostTypeId="2" ParentId="27490" CreationDate="2024-02-18T16:38:35.690" Score="3" Body="&lt;p&gt;Before you cut a tree down,&#xA;it is quite useful to know where it will land,&#xA;which requires calculating its height.&#xA;There are several methods depending on similar triangles, e.g.,&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Tree_height_measurement&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A variation was depicted in this classic &lt;a href=&quot;https://www.youtube.com/watch?v=LMgE95saBx0&amp;amp;ab_channel=Abakcus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tom &amp;amp; Jerry cartoon&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/9ifzn.png&quot; width=&quot;300&quot; /&gt; &#xA;" OwnerUserId="511" LastActivityDate="2024-02-18T16:38:35.690" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27493" PostTypeId="2" ParentId="27490" CreationDate="2024-02-18T17:08:45.950" Score="6" Body="&lt;p&gt;The problem here is usually less a matter of ‘I know of no instances where I will use this’ and more ‘I cannot relate things I actually do to the abstract concepts here’. A very large number of things people do regularly actually use a huge amount of math, but they never think about that because they have trouble translating from the abstract aspects to concrete applications.&lt;/p&gt;&#xA;&lt;p&gt;The best example of this that I can think of for geometry is actually shooter games. You can’t do accurate target-lead calculations without using trigonometry, and thus you need at least a basic understanding (be it formally taught or simply learned intuition) of trig to be able to reliably hit moving targets. Most people who play shooters regularly have such an understanding, but in many cases it’s an intuitive skill and they can’t explain it easily in formal mathematical terms, if at all. But, on the other hand, having a knowledge of trig can greatly accelerate learning that as an intuitive skill.&lt;/p&gt;&#xA;" OwnerUserId="9473" LastActivityDate="2024-02-18T17:08:45.950" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27494" PostTypeId="2" ParentId="27490" CreationDate="2024-02-18T17:50:40.260" Score="5" Body="&lt;p&gt;[This may seem indirect, but I want to give you something different, therefore additive.]&lt;/p&gt;&#xA;&lt;p&gt;Nate:  For geometry, specifically, I think you have to tell the kids something along the lines of&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;This course is going to feel different from algebra class...and it will feel different from the classes after, where you get back to algebra.  In&#xA;this class, we are going to do a lot of proofs.  It's important that&#xA;you see what it means to prove something logically.&#xA;Even if you don't remember every picky part of geometry class, you will know what it means to prove something and know what that feels like.&#xA;Geometry was a wonderful invention by the Greek philosophers and&#xA;something that many important thinkers like Abraham Lincoln loved.&#xA;Because they loved to think logically.&lt;/p&gt;&#xA;&lt;p&gt;Some of you are going to love how this class is different&#xA;from last year's algebra.  And some of you are going to hate it...and&#xA;want to get back to equations with letters and numbers.  But...I'm&#xA;just warning you so it won't be such a shock, this year is going to be very different from algebra.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note:  I'm actually way less proof-y than the average &amp;quot;want to teach real analysis in calculus class&amp;quot; MSE/MESE type.  So this is a sinner counseling morality.  But geometry is an exposure to proof.  It just is.  And even if it's wrong...that's what it is.  And I don't really even think it's wrong or if it is, very wrong, to have an exposure to proofs.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Realistically, I think straining to find direct applications in daily work (or personal) life is not the right rationale for the course.  It's a part of the overall math program, which is mostly (i.e. well over 50%) a tool for science and engineering, which there are a lot of jobs in.&lt;/p&gt;&#xA;&lt;p&gt;For students that are college-bound in STEM, they need to learn a lot of math...and later math has some dependencies on geometry.  Especially trigonometry, which is all over the place in ODEs, calculus, kinematics, electrical engineering, etc.&lt;/p&gt;&#xA;&lt;p&gt;For kids that are college-bound in non-STEM, they will need less math overall, but still may need some (e.g. medical fields).  And geometry will be a small part of that.&lt;/p&gt;&#xA;&lt;p&gt;And all of them, college-bound or not, will need some basic grounding in math/science to be citizens and read the newspaper, with some amount of perspective.  Yes, they won't all have perfect or deep mastery...and will forget a lot of it.  Same could be said of history and geography...how many people have read Mahan and have some insights from him on modern geopolitics?  All that said, if you don't know France is west of Germany is west of Poland...or that the Germans invaded Russia in WW2, you can't thoughtfully read NYT articles about NATO security.&lt;/p&gt;&#xA;&lt;p&gt;P.s. I wouldn't transfer this explanation as is.  But you need to be informed by it.  Often I find these questions are more coming from instructors who are concerned themselves about the relevance of the courses, rather than because they really are too often challenged by students (95% of whom will take direction and intuitively know that the course content is fixed) and can't reply well.  I would probably avoid getting too much into these relevancy debates.  Especially don't open them if you don't know how to handle them!  The kids have to do it because it is a part of school.  Someone made a decision what is needed and that's that.  You don't need to say it so brutally, but at least in your mind, know it is true.  And don't let the one doubting Thomas derail your class.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit:  (Hope I don't lose my, surprising, +1s for this)  I'm not sure that you want to say this...or if you do, be careful how you package it.  This is a little Machiavellian...but geometry is a pretty core part of the SAT-Math exam.&lt;/p&gt;&#xA;&lt;p&gt;If we want to get meta, we can argue what's the point of the SAT (and there are good arguments pro/con).  But if we just live in the rat race we live in...doing better is helpful to kids.&lt;/p&gt;&#xA;&lt;p&gt;Now it's not really a pure subject matter test...has aspects of aptitude (in the title even).  But it's not a pure aptitude test either.  It certainly is a lot easier to take that test if you've had basic geometry (and basic algebra) than if you have to nuke everything out from pure smarts.  I mean look at vertical angle theorem.  ETS loves that...wants to make babies with that theorem.&lt;/p&gt;&#xA;" OwnerUserId="23755" LastEditorUserId="23755" LastEditDate="2024-02-20T15:09:59.673" LastActivityDate="2024-02-20T15:09:59.673" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27495" PostTypeId="2" ParentId="27490" CreationDate="2024-02-18T22:20:24.217" Score="3" Body="&lt;p&gt;In general, it makes it more difficult for someone to deceive you. It also lets you better recognize when you're getting a good deal.&lt;/p&gt;&#xA;&lt;p&gt;It helps when you're buying something based on dimensions to determine whether the price is reasonable or not. The size might not be super critical so you might be willing to spend a bit more to go up a size to have some extra material if the cost per volume gets better for the larger size, or save money by going down a size because the cost per volume sky rockets for the larger size:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Why are these 1/2&amp;quot; rods so expensive? They're nearly double the price of the 3/8&amp;quot;! What a rip off! Oh wait, 1/2&amp;quot; is only 33% more diameter than 1/2&amp;quot; but that means the cross sectional area is 76% more. I guess that's why it costs so much.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I've also seen people complain about that a 5&amp;quot; vise is a rip off because it costs double the 4&amp;quot; version of the vise. You can look at the photos of the two vises and notice that the 4&amp;quot; and 5&amp;quot; vises look identical. The 5&amp;quot; vise doesn't have any ungainly proportions because there is no difference in proportions. It might be obvious if I point that out, but how is a person uneducated in geometry going to notice this? That means the 5&amp;quot; vise isn't just like the 4&amp;quot; vise except only the jaws are 25% wider. It's that the 25% larger in all 3 dimensions which means it's 1.95x more material. THAT's why it costs double. Incidentally, that also means it's weighs double and you might not want to deal with that for only a 25% increase in jaw width.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;It comes up when buying a house. It seems a lot of people only know of square footage by feel, or comparative numbers, or looking at other houses of known square footage. Especially for specific rooms, can just go to a kitchen, look down, notice that the tile size is 12&amp;quot; x 12&amp;quot; which makes it one square foot and just counting up a set number of those in particular sized rectangle just looking at that.&lt;/p&gt;&#xA;&lt;p&gt;It also helps you calculate the best pizza size to buy to get the most pizza for your money.&lt;/p&gt;&#xA;&lt;p&gt;It helps you figure out the shortest wood plank you can get away with while still being at an angle where the mouse can easily climb up it to get into your bucket mouse trap.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Oh, and perhaps the most important thing it does for you as far as your students are concerned:&lt;/p&gt;&#xA;&lt;p&gt;For the girls, it helps prevent them from being easily impressed by rudimentary knowledge when, for their elective, guys from the second and third year of the engineering program start taking introductory STEM courses meant for first year art students as a way to meet girls. They would get closer to them by helping them with their homework and impressing them with basic knowledge. Your students may laugh, but this actually worked for multiple guys when it happened to the point there were complaints about other students about showboating. It's one thing to be impressed when someone solves the heat equation. It's another to be impressed when someone finds the area of a circle.&lt;/p&gt;&#xA;&lt;p&gt;For the guys, it helps them meet and impress girls in the arts program who did not pick up this knowledge.&lt;/p&gt;&#xA;" OwnerUserId="13250" LastEditorUserId="13250" LastEditDate="2024-02-18T23:43:48.823" LastActivityDate="2024-02-18T23:43:48.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27496" PostTypeId="2" ParentId="27386" CreationDate="2024-02-19T03:30:06.607" Score="2" Body="&lt;p&gt;Gosh, the very platform where you are asking your question supports MathJax.  As you start writing an answer like I am doing now, &lt;span class=&quot;math-container&quot;&gt;$\LaTeX$&lt;/span&gt; and math equations &lt;span class=&quot;math-container&quot;&gt;$$x^2+x-1=0$$&lt;/span&gt; get rendered in a preview box below. Try it!&lt;/p&gt;&#xA;&lt;p&gt;Now look at the MathJax demo page.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.mathjax.org/#demo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathjax.org/#demo&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can make your own interactive LaTeX MathJax interface like what you find on StackExchange by making an html file on your local computer with the syntax shown in the image below, then open the file with your browser. Make sure that when you make the file with your text editor, you use plain text, not RTF. Make sure the extension for your file is .html&lt;/p&gt;&#xA;&lt;p&gt;After you open the file with your browser and start typing math code, use   \( and \) for delimiters, not dollar signs!&lt;/p&gt;&#xA;&lt;p&gt;Here is the image. Note: I tried to paste the text into here, but the indentations got lost, so I will settle for a screenshot of the html file. If you are interested, ask me in a comment and I can post a link to the html file so you can save it to your computer.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vxCww.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vxCww.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is the simple interface I get when I open the file with a web browser and start typing math.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FbH2D.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FbH2D.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But when I need to do what you are describing, I usually use Jupyter notebooks in Google Colab.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2024-02-19T04:48:51.727" LastActivityDate="2024-02-19T04:48:51.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27497" PostTypeId="1" AcceptedAnswerId="27498" CreationDate="2024-02-19T10:07:56.783" Score="4" ViewCount="442" Body="&lt;p&gt;I'm reading an introductory course on groups. In this course, the author illustrates concepts using the vectors of the plane. For example, &amp;quot;the set of vectors in the plane(or in space) is a group for &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;&amp;quot;. Thus, it refers to the notion of vector as taught in secondary education. However, everyone knows that this concept is quite complex. Why not settle for the additive group of real vector space &lt;span class=&quot;math-container&quot;&gt;$(\mathbb R^2,+,.)$&lt;/span&gt; and a fortiori &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^3$&lt;/span&gt;? Even if it means illustrating the elements of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^2$&lt;/span&gt; with vectors of the plane to refer to what the student already knows. It's mathematically satisfying, as well as enriching for the beginner focused on group theory.&lt;/p&gt;&#xA;&lt;p&gt;To put it simply, we could just as easily illustrate the notion of an abstract set by talking about the set &lt;em&gt;underlying&lt;/em&gt; a group &lt;span class=&quot;math-container&quot;&gt;$(G,.)$&lt;/span&gt; such as &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{S}_3$&lt;/span&gt;. Why do we do with the complex geometrical notion of &amp;quot;vector&amp;quot; what we would not reasonably allow ourselves to do elsewhere?&lt;/p&gt;&#xA;" OwnerUserId="23732" LastEditorUserId="205" LastEditDate="2024-02-25T17:06:04.657" LastActivityDate="2024-02-25T17:06:04.657" Title="Why use the vague notion of &quot;vector&quot; when you have $\mathbb R^2,\mathbb R^3,\mathbb R^4,\ldots$?" Tags="|secondary-education|mathematical-pedagogy|vectors|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27498" PostTypeId="2" ParentId="27497" CreationDate="2024-02-19T12:30:08.050" Score="6" Body="&lt;p&gt;In the euclidian spaces vectors and points are the same thing (there is a bijection between them and everything works as it should).&lt;/p&gt;&#xA;&lt;p&gt;In physics and in differential geometry vectors and points have different roles.&lt;/p&gt;&#xA;&lt;p&gt;You will lose nothing in an introductory group theory course if you understand vectors as corresponding to points. However, a physicist taking the course might indeed want to think about vectors, as adding them and scalar multiplication makes sense.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="2083" LastEditDate="2024-02-20T15:09:30.493" LastActivityDate="2024-02-20T15:09:30.493" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27499" PostTypeId="2" ParentId="27490" CreationDate="2024-02-19T22:10:54.613" Score="15" Body="&lt;p&gt;Here's my rant that I gave to my students:&lt;/p&gt;&#xA;&lt;p&gt;You are learning this right now because some of you will need this later in life. You may need this information for your daily job. You may need this information to check off a requirements to GET your job.&lt;/p&gt;&#xA;&lt;p&gt;We don't know who will actually need to use this yet.&lt;br /&gt;&#xA;We do know that it will be MUCH harder to start learning this when you are all older.&lt;br /&gt;&#xA;We teach it to all of you now, while you are young and have an easier time learning.&lt;/p&gt;&#xA;&lt;p&gt;Math education is a tower. You have to build one floor before you can start on the next. The floor of Geometry opens the door to higher math.&lt;/p&gt;&#xA;&lt;p&gt;There are many jobs where you WILL need to reach for the top of this tower of mathematics. Many of them are good jobs.&lt;/p&gt;&#xA;&lt;p&gt;We all benefit from the things those people create with their knowledge and experience.&lt;/p&gt;&#xA;&lt;p&gt;Your phone could not exist without very intelligent, hard working people who most likely learned this math when they were young.&lt;/p&gt;&#xA;&lt;p&gt;The internet, streaming video, video games... all require a ton of math to work.&lt;/p&gt;&#xA;&lt;p&gt;Some of you may never need to worry about internal angles or parallel lines again after this class, but the world is better when everyone learns this now.&lt;/p&gt;&#xA;&lt;p&gt;Even if you don't think you'll ever go into a career that requires it, you do not know what your future holds for you either.&lt;/p&gt;&#xA;&lt;p&gt;If you do end up needing this, it really, REALLY sucks having to go back and start learning it in college.&lt;/p&gt;&#xA;" OwnerUserId="13513" LastActivityDate="2024-02-19T22:10:54.613" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27500" PostTypeId="2" ParentId="27490" CreationDate="2024-02-20T01:33:30.423" Score="3" Body="&lt;p&gt;The answer to most &amp;quot;when am I gonna use this&amp;quot; questions is the same.  The best analogy that I've heard is a gym.  People go to the gym and do bicep curls with weights.  Why?  Are they ever going to need to lift something while they're somehow restricted to moving only the lower half of one arm?  Of course not.  They do it because that exercise isolates and strengthens a specific muscle group.  By doing many such exercises, they develop a body that can use all of those strong muscles together to do all sorts of things that &lt;em&gt;weren't&lt;/em&gt; part of their exercise routine.  You don't go the gym and practice moving a couch up the stairs, you build up all the individual muscles so that when you need to move a couch up the stairs you can do it.&lt;/p&gt;&#xA;&lt;p&gt;School is the same way.  The classes you take and the activities you do isolate and strengthen specific parts of the &lt;em&gt;mind&lt;/em&gt;.  A strong mind can tackle tasks that are far beyond anything you could ever learn in school.  To get there, though, you have to have a well-balanced exercise routine that increases in intensity as you grow.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Speaking specifically about high-school geometry, I've always ranked that class as &lt;strong&gt;the&lt;/strong&gt; most important class out of my entire primary education.  At one point, my teacher explained that the course really didn't have anything to do with shapes or angles or any of that stuff.  The purpose of the course was to learn &lt;em&gt;proofs&lt;/em&gt; and by extension, developing deductive reasoning skills.  All the stuff about shapes and the other things we typically associate with geometry are all just a delivery mechanism for learning this core concept.  They're easy to draw and visualize, which makes it easier to learn than if you were doing the same with a more abstract subject matter like algebra or philosophy.  My teacher said that she didn't care if we left the class and instantly forgot how to calculate the volume of a cylinder.  She was more concerned with whether we could approach a scenario by taking observations, applying logic and facts that we know to be true, and then arriving at a solution for a problem that we've never encountered before.&lt;/p&gt;&#xA;&lt;p&gt;Looking at geometry from that lens, when does any of this apply to real life?  Every time you read or hear something and want to figure out whether it's true.  Every time something in your house breaks and you need to get it working again.  Every time you need to navigate &lt;em&gt;any&lt;/em&gt; sort of situation that's new or unusual.  In other words, &lt;em&gt;all day, every day&lt;/em&gt;.  Many professions (medicine, law, engineering, software development, auto mechanics, etc.) are directly based on this type of deductive reasoning loop.  Even outside specific job scenarios, it's a process that (hopefully) runs inside your head every time that you have to think your way through a problem.  If you don't learn how to reason effectively, you'll be prone to making poor decisions and find it harder to continue to learn as you go through life.  Or even worse, you might not have the courage to try and tackle new problems at all.&lt;/p&gt;&#xA;" OwnerUserId="20163" LastActivityDate="2024-02-20T01:33:30.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27501" PostTypeId="2" ParentId="27490" CreationDate="2024-02-20T01:54:11.080" Score="1" Body="&lt;p&gt;I think a big part of why I was successful with math in junior high and high school is because I had a strong interest in video game programming. It turns out that 3D rendering is a lot of just algebra, linear algebra, geometry, and trigonometry. There are many different career fields where this is relevant (video games, movies, architecture, etc), and the nice thing is that it can be a fun topic for kids to learn.&lt;/p&gt;&#xA;&lt;p&gt;I do a lot of spatial computing now, and interestingly, many of the math concepts underpinning 3D rendering are applicable to GIS (Geographic Information Systems), which made learning spatial computing very easy for me. Specifically, one of the major types of data used in GIS is &amp;quot;vector&amp;quot; data, which consists of points, lines, and polygons. GIS is used in many professional fields (environmental sciences, infrastructure planning, economics, etc.). GIS expertise can lead to very well-paying jobs.&lt;/p&gt;&#xA;" OwnerUserId="13207" LastActivityDate="2024-02-20T01:54:11.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27502" PostTypeId="2" ParentId="27497" CreationDate="2024-02-20T03:03:03.987" Score="6" Body="&lt;p&gt;I don't think the concept of vector addition of geometric vectors is especially more complicated than the concept of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt;. In particular, the tip-to-tail addition and scalar multiplication of vectors is geometrically natural and it is for that reason it is likely given as an example. In terms of ease of use, formulation and concreteness, sure &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; is easier.&lt;/p&gt;&#xA;&lt;p&gt;But, in the same way many authors find it interesting to illustrate the dihedral group in terms of physical motions to defining objects. Such rotations and reflections are stupidly complicated, especially as the number of sides increases. Why not just teach the generators and relations formulation of &lt;span class=&quot;math-container&quot;&gt;$D_n$&lt;/span&gt;? I'm guessing the motivation is to ground the students intuition in the idea that groups can be geometric in their origin. In other words, groups are not just a formal system with neat and tidy rules. Rather, a group may arise from a nasty and somewhat complicated geometric question which is inconvient in it's formulation and certainly far from minimal in terms of the economy of its logic. Yet we teach the geometry. Why ? Because there exist geometers.&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2024-02-20T03:03:03.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27503" PostTypeId="1" AcceptedAnswerId="27506" CreationDate="2024-02-20T05:38:04.483" Score="11" ViewCount="3569" Body="&lt;p&gt;In teaching mathematics, I've avoided asking students to replicate proofs I've demonstrated in class, believing this approach primarily tests memorization rather than understanding or critical thinking. However, I've observed a potential downside: students might pay less attention to the proofs when they know they won't have to reproduce them. While my goal is to cultivate an environment that encourages students to independently engage with and think through proofs, I'm reconsidering the strategy.&lt;/p&gt;&#xA;&lt;p&gt;Is there pedagogical value in occasionally requiring students to repeat simpler proofs that were covered in class? How can this practice be balanced with the objective of promoting deep understanding and independent problem-solving skills?&lt;/p&gt;&#xA;" OwnerDisplayName="user19945" LastActivityDate="2024-02-22T14:24:35.810" Title="Effectiveness of Requiring Students to Repeat Proofs Presented in Class" Tags="|exams|assessment|" AnswerCount="9" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27504" PostTypeId="2" ParentId="27503" CreationDate="2024-02-20T12:21:18.350" Score="6" Body="&lt;p&gt;You will never be sure that the students really grasp a proof, if they can not express it or at least its essential part in there own words. Knowing proof really helps them to understand, what they do. So it's not just &amp;quot;memorizing&amp;quot; encourage them to talk about the essential idea of the proof, not just a formal memorized text.&lt;/p&gt;&#xA;" OwnerUserId="21051" LastActivityDate="2024-02-20T12:21:18.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27505" PostTypeId="2" ParentId="27503" CreationDate="2024-02-20T13:55:17.297" Score="3" Body="&lt;p&gt;I vote yes, do it.  If you want them to know the proofs (and you do), demonstrated recall will help assure that it happens.&lt;/p&gt;&#xA;&lt;p&gt;I think your hesitancy to have students duplicate work, and do something new instead, is misplaced.  Memorization of math proofs (or algorithms) is not either as evil or as simple some people think.  Students still incorporate a lot of the knowledge of step by step workings, if they spit out a memorized proof.&lt;/p&gt;&#xA;&lt;p&gt;It's not like you evaluate them based on word for word text recall (but for the key ideas logically presented).  I mean do you think it is wrong (evil &amp;quot;memorization&amp;quot;) if an AP Bio test has kids write essays summarizing a biological system?&lt;/p&gt;&#xA;&lt;p&gt;It won't be exactly memorization, say in the sense of a meaningless random number sequence, to &amp;quot;memorize&amp;quot; a proof.  Part of the memorization may be short term like that.  But they will also get at least some long term schema in their memorization.  And, really having a deeper stock of memorized proofs, makes working on fresh problems easier (have more tools in your kit).&lt;/p&gt;&#xA;&lt;p&gt;This doesn't mean you can't have any fresh problems.  But they should be slight variations.  You can't expect students to make huge leaps.  Not in general and especially not in exam situations.&lt;/p&gt;&#xA;" OwnerUserId="23755" LastEditorUserId="23755" LastEditDate="2024-02-20T14:38:29.750" LastActivityDate="2024-02-20T14:38:29.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27506" PostTypeId="2" ParentId="27503" CreationDate="2024-02-20T14:06:35.090" Score="21" Body="&lt;p&gt;I view a course as a story.  The definitions introduce the characters, the theorems are the plot, and hopefully there is some sort of overarching message that plot is trying to convey.&lt;/p&gt;&#xA;&lt;p&gt;If you were teaching Beowulf it would not be unreasonable to ask who Grendel is and how he dies.  It is just a basic plot point:  if you don't remember this you are not really following the story and have no hope of learning anything deeper.  You might want them to develop their own interesting take on what the story is trying to convey but they cannot do this if they do not follow the plot.&lt;/p&gt;&#xA;&lt;p&gt;It is the same if you cannot prove that every subgroup of a cyclic group is cyclic:  you are just not really following the story.  It is necessary &lt;strong&gt;but not sufficient&lt;/strong&gt; for a student to know how to tell that part of the story if they are to have any hope of learning group theory.&lt;/p&gt;&#xA;&lt;p&gt;Beginning students of mathematics have literally no idea what they are doing.  The case is similar to people who have done very well in spelling competitions taking an English literature class and expecting to do well because of their spelling prowess. The experience is so completely unlike what they are expecting that they do not even know what to focus on.  So a large part of your job will be to convey what it is that they are even trying to do.  While it is a baby step, parroting back the basic proofs is a step towards comprehension.&lt;/p&gt;&#xA;&lt;p&gt;When I was teaching these kinds of courses I would make sure to have a small collection of &amp;quot;core theorems&amp;quot;.  I expected the students to be able to reproduce the proofs.  I made it clear that my intention was not for them to memorize the proofs.  I asked them to find a personal narrative for the kind of discovery process someone would have gone through to discover the proof and to retrace this discovery path until it felt familiar.  Some students, undoubtedly, did just memorize.  However some of them really did learn something!&lt;/p&gt;&#xA;&lt;p&gt;EDIT:  I should also mention the research of Lara Alcock.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://laraalcock.com/papers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://laraalcock.com/papers/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In particular I would like to point to &lt;a href=&quot;https://personal.utdallas.edu/%7Ezweck/SEbooklet.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;self explanation training&lt;/a&gt; as a strategy for increasing proof comprehension.  I wouldn't implement this kind of work as part of a summative assessment (a bit too subjective), but it is a valuable component of &amp;quot;helping students realize what they are even supposed to be doing&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2024-02-21T15:10:38.843" LastActivityDate="2024-02-21T15:10:38.843" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27507" PostTypeId="2" ParentId="27503" CreationDate="2024-02-20T14:58:26.440" Score="6" Body="&lt;p&gt;If you're training musicians, you might have them do exercises in 2 different categories:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;improvising from scratch in real time, on the spot, without prior rehearsal&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;rehearsing and analyzing difficult pieces by master composers&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Category 1 is about developing creativity within your repertoire, completely mastering the techniques that you're already familiar/comfortable with.&lt;/p&gt;&#xA;&lt;p&gt;Category 2 is about extending your repertoire to include new techniques that are initially very unfamiliar/uncomfortable. Initially, these techniques are advanced relative to your ability, so the only way to practice them properly is by imitating and analyzing them. But over time, as you build a baseline level of competence imitating/analyzing these techniques, they can gradually be moved into category 1 (and replaced with even more advanced techniques in category 2).&lt;/p&gt;&#xA;&lt;p&gt;Basically every skill domain is the same way, including math.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Category 1 contains general techniques like proof by induction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Category 2 contains key theorems like the fundamental theorem of calculus.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;On an exam, it would be reasonable to ask students to&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;solve previously-unseen problems in category 1, and&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;imitate and analyze known solutions to previously-seen problems in category 2. (&amp;quot;Analyze&amp;quot; could mean, for example, identifying an error in a given proof.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Lastly, I should emphasize that moving techniques from category 2 to category 1 requires both imitation and analysis, not just one or the other.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If you don't imitate, then you won't be able to execute the mechanics of the technique. One who analyzes but doesn't imitate is called a critic.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If you don't analyze, then you won't know how to tweak the mechanics of the technique to new situations, and you also won't know whether it's even appropriate to apply the technique in a new situation unless you're explicitly told to use the technique.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Imitating without analyzing produces a robot / ape who can't think critically; analyzing without imitating produces a critic who can't act on their own advice.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-02-21T20:25:23.763" LastActivityDate="2024-02-21T20:25:23.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27508" PostTypeId="2" ParentId="27503" CreationDate="2024-02-20T15:13:43.240" Score="8" Body="&lt;p&gt;I used to often ask questions of the form &amp;quot;State and prove Theorem X&amp;quot; in proof-based courses. My hope was that, instead of memorizing verbatim, students would boil the proof down to a few key ideas, memorize those, and trust themselves to fill in the details on the fly during the test. And the very best students did this. But the majority memorized verbatim without seeming to understand much. In fact, I once asked for a proof where the textbook's version had a typo, and many students reproduced the typo on the exam!&lt;/p&gt;&#xA;&lt;p&gt;Now, I prefer to look at specific proof techniques that I want them to know, and write questions that test their ability to do that technique in a (slightly) unfamiliar setting.&lt;/p&gt;&#xA;" OwnerUserId="23762" LastEditorUserId="23762" LastEditDate="2024-02-20T15:14:25.193" LastActivityDate="2024-02-20T15:14:25.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27509" PostTypeId="2" ParentId="27503" CreationDate="2024-02-20T15:44:21.327" Score="9" Body="&lt;ol&gt;&#xA;&lt;li&gt;As a first intuition, I would rather ask for a proof of a special case or a different case; something which asks for some level of understanding (of what to alter), not only copying and pasting. It can be essentially the same proof for you and still a significant challenge for the students, and one that requires engagement with the original proof.&lt;/li&gt;&#xA;&lt;li&gt;However, just basic mathematical literacy (reading and writing) is challenging for many students, so copying a proof might turn out to be somewhat challenging to some of them. With this, you would be training them in copying, rather than thinking, so I would be wary of this as a repeated practice.&lt;/li&gt;&#xA;&lt;li&gt;In a situation without notes available or in an oral situation, asking to repeat a proof is okay. Memorizing a proof usually happens by the way of understanding, though some people do brute-force memorize complicated things. However, I would still consider alternatives like presenting a proof attempt or a sketch of one and asking the student to find mistakes there and fix them, if possible, or at least explain what the problem is.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2083" LastEditorUserId="376" LastEditDate="2024-02-20T19:07:07.767" LastActivityDate="2024-02-20T19:07:07.767" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27510" PostTypeId="2" ParentId="27490" CreationDate="2024-02-20T16:55:55.727" Score="2" Body="&lt;p&gt;My wife is a quilter.  We are constantly scaling shapes, figuring out how much fabric is needed to make something, finding the best way to cut fabric to make a given shape, computing how many squares to make a quilt of a desired size or the border to stretch a center to a desired size, and many more.&lt;/p&gt;&#xA;" OwnerUserId="12340" LastActivityDate="2024-02-20T16:55:55.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27511" PostTypeId="2" ParentId="27497" CreationDate="2024-02-20T18:05:18.287" Score="5" Body="&lt;p&gt;Because vectors are &lt;em&gt;familiar&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Sure, &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; is simpler to construct &amp;quot;from scratch&amp;quot;. But when teaching you're not constructing things from scratch: you're doing it from &lt;em&gt;the ideas students already understand&lt;/em&gt;. And with those as your primitives, &amp;quot;vectors&amp;quot; &lt;em&gt;are&lt;/em&gt; simpler than &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;, equipped with the operation of addition of corresponding components&amp;quot; -- because  the concept of &amp;quot;vectors, which get combined with vector addition&amp;quot; is &lt;em&gt;already available&lt;/em&gt; for you to use!&lt;/p&gt;&#xA;&lt;p&gt;This is especially true when &amp;quot;equipping a set with an operation&amp;quot; is still a hazy concept for most students: having a familiar set, with an operator that comes pre-equipped, is very effective in helping students &amp;quot;get their footing&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="8248" LastActivityDate="2024-02-20T18:05:18.287" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27512" PostTypeId="2" ParentId="27503" CreationDate="2024-02-21T12:54:41.057" Score="1" Body="&lt;p&gt;I want to emphasize an alternative approach mentioned by &lt;em&gt;Tommi&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I would rather ask for a proof of a special case or a different case; something which asks for some level of understanding (of what to alter), not only copying and pasting. It can be essentially the same proof for you and still a significant challenge for the students, and one that requires engagement with the original proof.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In my opinion this approach &lt;em&gt;together with&lt;/em&gt; an &lt;strong&gt;open-book exam&lt;/strong&gt; is the best strategy to encourage true engagement and retention of the proofs that you have covered. When the students can refer to the proofs covered in class, but have to adapt them to solve new problems, they will definitely have to pay attention to them to get to know them well!&lt;/p&gt;&#xA;" OwnerUserId="1550" LastActivityDate="2024-02-21T12:54:41.057" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27513" PostTypeId="2" ParentId="27497" CreationDate="2024-02-21T14:03:09.437" Score="2" Body="&lt;p&gt;Perhaps the more general context of an affine space can be helpful. When we discuss &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt;, we typically think of the origin as a special point. We think of vectors as ordered tuples of real numbers. The we can add on the image of an arrow that has its tail at the special point (origin) and its tip at the point corresponding to the ordered tuple.&lt;/p&gt;&#xA;&lt;p&gt;But in physics, we tend to think of vectors as arrows that go between any two points, and consider two such entities as equal if they are just translations of one another. So the vector tailed at &lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt; with tip at &lt;span class=&quot;math-container&quot;&gt;$(4,2)$&lt;/span&gt; is equal to the one tailed at &lt;span class=&quot;math-container&quot;&gt;$(3,-1)$&lt;/span&gt; and tipped at &lt;span class=&quot;math-container&quot;&gt;$(6,0)$&lt;/span&gt;. We go from the first to the second by translating the plane by isometry &lt;span class=&quot;math-container&quot;&gt;$(x,y)\rightarrow (x+3,y+1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Notice that the translation &lt;span class=&quot;math-container&quot;&gt;$(x,y)\rightarrow (x+3,y+1)$&lt;/span&gt; is technically not a transformation of &lt;span class=&quot;math-container&quot;&gt;${\mathbb R^2}$&lt;/span&gt; as a vector space because it does not preserve the identity for addition, i.e., it moves the origin. The transformation group is the general linear group &lt;span class=&quot;math-container&quot;&gt;$GL(2,{\mathbb R})$&lt;/span&gt;, which can be represented as the 2x2 matrices with non-zero determinant.&lt;/p&gt;&#xA;&lt;p&gt;So what is going on? The collection of arrows that are vectors for physicists form what is called an affine space, as opposed to a vector space. The geometer Marcel Berger said &amp;quot;An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps.&amp;quot; The definition of an affine space is kind of tricky. &lt;a href=&quot;https://ncatlab.org/nlab/show/affine+space&quot; rel=&quot;nofollow noreferrer&quot;&gt;nLab&lt;/a&gt; might give you some ideas. One characterization found there is that &lt;em&gt;an affine space is simply a vector space, but with different morphisms; an affine linear map is a function that is the difference between a linear map and a constant function&lt;/em&gt;. There are many ideas on the nLab site, with varying degrees of abstraction.&lt;/p&gt;&#xA;&lt;p&gt;In Newtonian physics, we inhabit a &amp;quot;big cube&amp;quot; that is three-dimensional. Sometimes we over-step and think of it as the vector space &lt;span class=&quot;math-container&quot;&gt;${\mathbb R^3}$&lt;/span&gt;. But in Newtonian physics, there is no distinguished point like the origin, e.g., the &amp;quot;me reference frame&amp;quot; which puts me at the center of the universe. So the big cube of Newtonian physics is really an affine space, not a vector space. The symmetry group of Newtonian physics is larger than the symmetry group for an associated vector space.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2024-02-21T14:08:56.350" LastActivityDate="2024-02-21T14:08:56.350" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27514" PostTypeId="2" ParentId="27503" CreationDate="2024-02-21T14:11:27.603" Score="1" Body="&lt;p&gt;Asking students to reproduce proofs  &lt;em&gt;can&lt;/em&gt; be very effective. The degree of effectiveness is largely determined by &lt;em&gt;how&lt;/em&gt; you ask students to do it. &lt;a href=&quot;https://matheducators.stackexchange.com/a/27508/20719&quot;&gt;As this answer describes&lt;/a&gt;, some reproduction tasks can lead students to focus only on the words and symbols of a proof, missing the underlying structure and ideas. Here's a practice that was effective at getting me to think about the  latter when I was a beginning student.&lt;/p&gt;&#xA;&lt;p&gt;In my first course with proofs, I had to memorize and present important proofs that had been shown in lecture. It wasn’t sufficient to simply memorize the words, because the instructor would always ask follow-up questions to probe for understanding. If they weren’t satisfied with my understanding of a certain proof, I would have to go back and study more, then present the proof again. Each presentation was essentially a small, low-stakes oral exam. (This was a large course with many students, so the logistics were more complex than what I've described here. There was a peer-to-peer aspect as well, which I think had its own benefits. )&lt;/p&gt;&#xA;&lt;p&gt;This supported my learning in several ways. Studying proofs in order to present them helped me learn the structure and style of mathematical arguments. Preparing for potential questions helped me understand the proofs and concepts from lecture more deeply. And through the presentations, I practiced talking about mathematics and became more comfortable participating in mathematical conversations.&lt;/p&gt;&#xA;&lt;p&gt;This can't replace students coming up with proofs on their own. But writing a proof is challenging, even when one is comfortable with mathematical language and understands related proofs. Beginning students haven't reached this threshold yet. Familiar proofs are a context in which students can practice the mechanics of proving without the added complexity of having to construct a novel argument. And even once students become more proficient with proof-writing, they still benefit from studying important proofs outside of class to solidify their understanding.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2024-02-21T14:11:27.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27515" PostTypeId="1" CreationDate="2024-02-21T21:18:23.117" Score="22" ViewCount="3525" Body="&lt;p&gt;In teaching multi-variable calculus, it's insightful to discuss with students not only how certain concepts from single-variable calculus extend to multiple variables but also where these extensions fail. For example, while in single-variable calculus, a continuously differentiable function defined on an open interval that has a unique critical point, which is a local minimum, also has a global minimum at that point. However, this is not necessarily the case in multi-variable calculus, as there are known counterexamples. here is one for a two-variable function:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$z= f(x, y) = x^2 + (1 − x)^3y^2$&lt;/span&gt;&#xA;for this function, &lt;span class=&quot;math-container&quot;&gt;$(0, 0)$&lt;/span&gt; is the single critical point that is a local minimum but it is not global&#xA;&lt;a href=&quot;https://i.stack.imgur.com/3CoDA.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3CoDA.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/i3SFf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/i3SFf.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am compiling a list of such instances to enhance my curriculum. Could you share other specific results or theorems that hold in single-variable calculus but do not necessarily apply in the multi-variable setting? Examples where the multi-variable context introduces significant complexity or entirely new phenomena would be particularly valuable.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="205" LastEditDate="2024-02-28T18:44:16.337" LastActivityDate="2024-02-28T22:02:13.970" Title="What theorems from single-variable calculus break down in the multi-variable context?&quot;" Tags="|calculus|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27516" PostTypeId="2" ParentId="27490" CreationDate="2024-02-21T21:32:12.097" Score="1" Body="&lt;p&gt;&lt;strong&gt;you can use history of mthematics for that&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Geometry has played a pivotal role throughout history in various fields, and understanding its applications can inspire students to appreciate its value beyond the classroom. Here are refined examples to illustrate this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Ancient Land Measurement&lt;/strong&gt;: In ancient Egypt, geometry was vital in calculating the area of land for taxation and re-establishing boundaries after the Nile's annual floods. This demonstrates geometry's role in governance and civil administration.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Astronomical Discoveries&lt;/strong&gt;: The Greeks used geometric principles to estimate large distances, such as the gap between the Earth and the Sun. These methods laid the groundwork for modern astronomy and space exploration.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Cartography and Navigation&lt;/strong&gt;: During the Age of Discovery, geometry was crucial for mapmaking and navigation. It helped explorers chart courses and map the Earth's surface, leading to the world as we know it today.&lt;/p&gt;&#xA;&lt;p&gt;By presenting geometry as a historical tool that solved real-world problems and paved the way for current technologies, we can inspire students with tangible examples of how geometry shapes our understanding of the world. These historical applications not only show geometry’s practicality but also its enduring legacy in advancing human knowledge and society.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2024-02-21T21:32:12.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27517" PostTypeId="2" ParentId="27515" CreationDate="2024-02-21T21:37:12.617" Score="22" Body="&lt;p&gt;Critical points of smooth functions in one-dimension tend to be local extrema. This is because &lt;span class=&quot;math-container&quot;&gt;$f''(c)=0$&lt;/span&gt; is an unlikely event. When &lt;span class=&quot;math-container&quot;&gt;$f'(c)=0$&lt;/span&gt;, we  expect a local min or max, depending on the sign of the second derivative.&lt;/p&gt;&#xA;&lt;p&gt;In higher dimensions, the critical points correspond to the simultaneous vanishing of all the first-partial derivatives. The second derivative becomes a matrix of second partial derivatives called the Hessian.  For a critical point to be a local maximum or minimum, the eigenvalues of this matrix all have to be the same sign.&lt;/p&gt;&#xA;&lt;p&gt;Since there are &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; eigenvalues, and each can be either positive or negative, we have &lt;span class=&quot;math-container&quot;&gt;$2^n$&lt;/span&gt; types of critical point behavior.  Only 2 of these are a max or a min.  So we should expect roughly &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2^{n-1}}$&lt;/span&gt; of all critical points to be local extrema, with the rest saddles.  This is imprecise since we have not clarified what &amp;quot;picking a function at random&amp;quot; means, but it is a solid intuition.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="117" LastEditDate="2024-02-22T19:48:53.663" LastActivityDate="2024-02-22T19:48:53.663" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27518" PostTypeId="2" ParentId="27515" CreationDate="2024-02-21T22:24:41.317" Score="36" Body="&lt;p&gt;Path-dependency of limits can lead to very counterintuitive results. For instance, consider the following limit:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}\lim\limits_{(x,y) \to (0,0)} \frac{x^2y}{x^4+y^2}\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;An intuitive-but-incorrect way to think about this limit is to imagine it only has one variable:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}\lim\limits_{x \to 0} \frac{x^2x}{x^4+x^2} = \lim\limits_{x \to 0} \frac{x}{x^2+1} = \frac{0}{1} = 0\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The problem is that in the computation above, we evaluated the limit along the path &lt;span class=&quot;math-container&quot;&gt;$y=x,$&lt;/span&gt; but this is just one of many possible paths.&lt;/p&gt;&#xA;&lt;p&gt;Okay, then why don't we evaluate the limit along all lines through the origin? We can handle that in three cases: the general case &lt;span class=&quot;math-container&quot;&gt;$y=mx$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is a nonzero constant, and then the cases &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; separately. The limit comes out to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; in all these cases:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}\lim\limits_{x \to 0} \frac{x^2(mx)}{x^4+(mx)^2} = \lim\limits_{x \to 0} \frac{mx}{x^2+m^2} = \frac{0}{m^2} = 0,\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}\lim\limits_{x \to 0} \frac{x^2(0)}{x^4+0^2} = 0, \quad \quad \lim\limits_{y \to 0} \frac{(0^2)y}{0^4+y^2} = 0. \end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, this is still not correct! Look what happens if you approach along the path &lt;span class=&quot;math-container&quot;&gt;$y=x^2.$&lt;/span&gt; We get a different result:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}\lim\limits_{x \to 0} \frac{x^2(x^2)}{x^4+(x^2)^2} = \lim\limits_{x \to 0} \frac{x^4}{x^4+x^4} = \frac{x^2}{2x^4} = \dfrac{1}{2}\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It turns out our limit &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{(x,y) \to (0,0)} \frac{x^2y}{x^4+y^2}$&lt;/span&gt; evaluates to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; along all linear paths, but not along all paths in general (for instance, it came out to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; along a quadratic path). So generally speaking, our limit does not exist.&lt;/p&gt;&#xA;&lt;p&gt;This is super weird and counterintuitive, but it's true, and you can get a better sense of why it happens if you look at the graph of &lt;span class=&quot;math-container&quot;&gt;$f(x,y) = \frac{x^2y}{x^4+y^2}{:}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/q5Y3O.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/q5Y3O.png&quot; alt=&quot;multivariable-surface-graph&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-02-21T22:35:22.327" LastActivityDate="2024-02-21T22:35:22.327" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="27519" PostTypeId="1" CreationDate="2024-02-21T23:36:57.957" Score="7" ViewCount="101" Body="&lt;p&gt;I've got a student in my introductory calculus course. They're failing because they lack algebra skills. They understand the concepts just fine, and can articulate their understanding fine, but get hung up on algebraic mechanics — factoring quadratic polynomials, manipulating rational expressions, conventions of exponent notation, etc — which then manifests poorly on exams. The natural advice to give them is to &amp;quot;&lt;em&gt;take a semester of pre-cal to buff your algebra skills&lt;/em&gt;&amp;quot;. However they are unwilling to do this; they've already taken and failed this class once (at least?) and feel committed to this calculus class now. Okay. Then how do I advise them?&lt;/p&gt;&#xA;&lt;p&gt;I seem to be having students year-after-year in the same situation: they lack algebra proficiency but otherwise appear to be mathematically capable, and have already failed calculus at least once and are now unwilling to enroll in a &amp;quot;lower&amp;quot; course instead. I suppose the root cause of this is out of my control, but the advise I give to such students in my class now is in my control. These students usually understand that their algebra skills need attention, so my advice has been to work though a free online algebra or pre-cal course in tandem with my calculus course. But I don't know &lt;em&gt;which&lt;/em&gt; online course to point them towards, or if there is a better idea I'm not considering.&lt;/p&gt;&#xA;&lt;p&gt;What resources would you point these students towards? How would you advise these students?&lt;/p&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2024-02-21T23:42:55.947" LastActivityDate="2024-02-22T01:56:40.080" Title="Advice and Remedial Algebra Resources for Students Committed to Calculus" Tags="|calculus|algebra|precalculus|advising|remedial-courses|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27520" PostTypeId="2" ParentId="27519" CreationDate="2024-02-22T01:21:09.283" Score="-2" Body="&lt;p&gt;My (paradoxical and politically incorrect) advice is to stick with the class they are accelerated into.  but to do every single problem.  Yes, even when the soft Millenials complain.  Do every single problem AND lots of the &amp;quot;easy&amp;quot; ones.  They will teach you algebra en passant.  Tell him (or her...or it) to Drill, baby, drill!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://theoildrum.com/files/Slide20.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://theoildrum.com/files/Slide20.jpg&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="23769" LastEditorUserId="23769" LastEditDate="2024-02-22T01:25:45.963" LastActivityDate="2024-02-22T01:25:45.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27521" PostTypeId="2" ParentId="27519" CreationDate="2024-02-22T01:56:40.080" Score="2" Body="&lt;p&gt;I know this doesn't directly answer your question which is about helping &lt;em&gt;one&lt;/em&gt; student. But I would be wanting to see my institution implement sections of calculus with support. &lt;a href=&quot;https://accelerationproject.org/adventures-in-corequisite-stem-calculus-warm-up-routines-that-build-curiosity-confidence-conceptual-connections-and-community/&quot; rel=&quot;nofollow noreferrer&quot;&gt;This blog addresses that situation&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My students liked &lt;a href=&quot;https://www.khanacademy.org/math/algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;Khan Academy&lt;/a&gt;, but I'm not sure how well it would do with helping them shift their mindset. (I just went and looked at it. First time ever, because I have a bias against it.) I guess I'd recommend the Algebra 1 and 2, and the Precalculus, if this was their preference. Free, friendly, and they get to watch their progress.&lt;/p&gt;&#xA;&lt;p&gt;I like the workbook style of AOPS textbooks. But they aren't cheap. I wonder if the student would find their free site useful. It's called &lt;a href=&quot;https://artofproblemsolving.com/alcumus/problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alcumus&lt;/a&gt;. They have algebra and intermediate algebra levels. They are meant to be challenging, but they do have good explanations.&lt;/p&gt;&#xA;&lt;p&gt;I'm not satisfied with my own answer. I hope other people can offer even better resources. This is such a common problem.&lt;/p&gt;&#xA;&lt;p&gt;I will add that most students like this that I've encountered also have problems with some of their arithmetic skills, especially fractions and negative numbers. It's not just algebra skills, usually.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2024-02-22T01:56:40.080" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27522" PostTypeId="2" ParentId="27503" CreationDate="2024-02-22T14:24:35.810" Score="1" Body="&lt;p&gt;Allow me to add an answer from a student's perspective (many many years ago).&lt;/p&gt;&#xA;&lt;p&gt;In high school my maths marks were pretty mediocre, not bad enough to bar me from college, but still. Proofs were always something I considered I just need to glue myself to a chair and memorize - and it wasn't a pleasant experience... nor very successful when under pressure. Getting to college to study computer science, I added some maths subjects - which I approached with the same mindset. In my second year of trudging along like this, a (well-loved) professor took pity on us and took one whole hour's class, not in the curriculum, to talk about the whole logic behind proofs; how things start with the basic, given building blocks (axioms), and given a set of operations (previous theorems), you'd follow a number of logic steps to get to the proof.&lt;/p&gt;&#xA;&lt;p&gt;It was like an epiphany. I never again memorized a single proof. Exam prep might involve working them out, just for practice. Also, my maths marks (and enjoyment) considerably improved from there on. As a more useful side effect, the same &amp;quot;way of doing logic&amp;quot; also helped me in other subjects as well as general life. Even though I haven't done any pure math since then.&lt;/p&gt;&#xA;&lt;p&gt;To answer your question: I believe asking proofs (of some or other variation) is useful, BUT as an educator you need to ensure that your students understand the reason for proofs and are able to work these basic tools of their (temporary?) trade without memorization. Not asking for proofs leaves you with an assumption of their knowledge (which is probably wrong in many cases - students are masters at minimizing effort for getting acceptable results). Asking proofs tells you they either (A) understand or (B) are good at memorizing (without deep understanding - which is not desirable). Equip them to understand, then challenge them to apply the understanding.&lt;/p&gt;&#xA;" OwnerUserId="23771" LastActivityDate="2024-02-22T14:24:35.810" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27523" PostTypeId="2" ParentId="27490" CreationDate="2024-02-22T15:15:33.660" Score="1" Body="&lt;p&gt;As is fashionable, I talked to my pupils about careers in artificial intelligence. I think they were interested. I will be told that logic that comes into play in AI is not geometry. I'll get to that. Today's mathematics doesn't pay much attention to classical geometry because we have so much to teach that we often don't have the time to do it. But getting students to study classical geometry is never a waste of time.&lt;/p&gt;&#xA;&lt;p&gt;Everyone knows or should know that if classical geometry does not play a major role in modern mathematics, on the other hand the spirit of geometry reigns everywhere (the sentence is not mine, I don't remember who said it). It is perfectly illusory to learn anything in mathematics without some basic geometry. For example, if we consider one of the simplest structures, that of vector space, do not connect the elementary results from linear algebra to elementary geometry, it is to build on sand and condemn oneself to psitassism.&lt;/p&gt;&#xA;&lt;p&gt;And how can we learn logic, in order to return to it, without having some geometrical results to illustrate its principles? (see &lt;a href=&quot;https://math.stackexchange.com/questions/1193548/prove-by-contradiction-that-a-circle-chord-is-no-longer-than-its-diameter/4868716#4868716&quot;&gt;here&lt;/a&gt; for example).&lt;/p&gt;&#xA;&lt;p&gt;In short, there is no math without geometry. And since I've heard that &lt;span class=&quot;math-container&quot;&gt;$75\%$&lt;/span&gt; of the jobs offered today more or less use math, I don't hesitate to repeat it to the students.&lt;/p&gt;&#xA;" OwnerUserId="23732" LastEditorUserId="23732" LastEditDate="2024-02-24T09:55:37.720" LastActivityDate="2024-02-24T09:55:37.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27524" PostTypeId="2" ParentId="27515" CreationDate="2024-02-22T17:54:24.637" Score="17" Body="&lt;p&gt;&lt;strong&gt;Existence of antiderivatives.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Every continuous function &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{R} \to \mathbb{R}$&lt;/span&gt; has an antiderivative.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, a continuous vector field &lt;span class=&quot;math-container&quot;&gt;$f: \mathbb{R}^d \to \mathbb{R}^d$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$d \ge 2$&lt;/span&gt; need not be the gradient of a function &lt;span class=&quot;math-container&quot;&gt;$u: \mathbb{R}^d \to \mathbb{R}$&lt;/span&gt;, in general.&#xA;If the vector field &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuously differentiable, then the existence of a differentiable function &lt;span class=&quot;math-container&quot;&gt;$u: \mathbb{R}^d \to \mathbb{R}$&lt;/span&gt; that satisfies &lt;span class=&quot;math-container&quot;&gt;$\nabla u = f$&lt;/span&gt; is equivalent to the Jacobi matrix of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; being symmetric.&lt;/p&gt;&#xA;&lt;p&gt;Another obstacle to the existence of such a &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; that occurs in dimension &lt;span class=&quot;math-container&quot;&gt;$d \ge 2$&lt;/span&gt; is the existence of holes in the domain of the function: let &lt;span class=&quot;math-container&quot;&gt;$\Omega \subseteq \mathbb{R}^d$&lt;/span&gt; be non-empty and open and let &lt;span class=&quot;math-container&quot;&gt;$f: \Omega \to \mathbb{R}^d \to \mathbb{R}^d$&lt;/span&gt; be a continuously differentiable vector field whose Jacobi matrix is symmetric. If &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is, for instance, simply connected, then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a gradient, but there are more general domains &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; where this is not always true.&lt;/p&gt;&#xA;" OwnerUserId="16250" LastActivityDate="2024-02-22T17:54:24.637" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27525" PostTypeId="2" ParentId="27515" CreationDate="2024-02-23T08:54:49.743" Score="4" Body="&lt;p&gt;One elementary property that fails already for functions of one variable with values in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; is the chord-tangent property. For example, for the helix &lt;span class=&quot;math-container&quot;&gt;$$&#xA;h:t\mapsto (\sin t,\cos t, t),\quad t\in[0, 2\pi], &#xA;$$&lt;/span&gt;&#xA;we have &lt;span class=&quot;math-container&quot;&gt;$h(2\pi)-h(0)=(0,0,2\pi)$&lt;/span&gt;, however at no point on &lt;span class=&quot;math-container&quot;&gt;$[0,2\pi]$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$h'$&lt;/span&gt; equal, or even parallel, to &lt;span class=&quot;math-container&quot;&gt;$(0,0,2\pi)$&lt;/span&gt;. Of course, we mostly care about the chord-tangent property because it gives the bound &lt;span class=&quot;math-container&quot;&gt;$|h(b)-h(a)|\leq |b-a|\max_{(a,b)}|h'|$&lt;/span&gt;, which still holds in higher dimensions.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastEditorUserId="1680" LastEditDate="2024-02-23T14:15:08.140" LastActivityDate="2024-02-23T14:15:08.140" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27526" PostTypeId="2" ParentId="14034" CreationDate="2024-02-23T09:14:45.023" Score="1" Body="&lt;p&gt;On a practical level, give extra credit assignments about things she is enthusiastic about. Give generous partial credit on these assignments for original ideas that don't necessarily work out. Let her pass the course based on the marks you give her for those assignments.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2024-02-23T09:14:45.023" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27527" PostTypeId="2" ParentId="27515" CreationDate="2024-02-23T23:08:52.143" Score="3" Body="&lt;p&gt;What about the generalized Stokes' Theorem ? In some sense this is a story about how Calculus III with it's three dimensional-centric formulae fails to generalize that story for arbitrary vector fields in higher dimensions. Why? Because something beyond vector fields are needed to understand the generalization of the FTC for higher dimensions.&lt;/p&gt;&#xA;&lt;p&gt;In particular, we must introduce differential forms in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;. The degrees of freedom in a differential &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-form in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\frac{n!}{p!(n-p)!}$&lt;/span&gt;. We need that many scalar functions to define a &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-form. It just happens that &lt;span class=&quot;math-container&quot;&gt;$p=2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p=3$&lt;/span&gt; forms in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; are both in one-to-one correspondance with vector fields. No such luck in higher dimensions in general, only the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-forms and &lt;span class=&quot;math-container&quot;&gt;$(n-1)$&lt;/span&gt;-forms are in direct correspondance with vector fields. Otherwise, the other objects described by &lt;span class=&quot;math-container&quot;&gt;$p=2,\dots , (n-2)$&lt;/span&gt;-forms are new objects which are not just another take on vector fields.&lt;/p&gt;&#xA;&lt;p&gt;These new objects are fluxes. Yet, these play the role vector fields did in lower dimensions. For example, in electromagnetism in four spatial dimensions you'd find a Faraday tensor formed by more magnetic components than electric components.In short, the FTC does not generalize for vector fields, it generalized for differential forms. It is a happy accident of low dimensions that we didn't need differential forms to understand the FTC.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int_M d\alpha = \int_{\partial M} \alpha $$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="128" LastActivityDate="2024-02-23T23:08:52.143" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27528" PostTypeId="1" CreationDate="2024-02-24T07:58:51.167" Score="3" ViewCount="247" Body="&lt;p&gt;I will teach a special class of 10 high school students who are aiming to study math at top universities. The purpose of the class is to increase students' interest and ability in math. There is competition to be admitted into this class. [EDIT: I will teach this class six lessons, each lasting 90 minutes.]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What are pros and cons of requiring students to create their own math questions, and what are some tips to make it work well?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h2&gt;My plan&lt;/h2&gt;&#xA;&lt;p&gt;Each week I will ask each student to create their own math question (and solution). The question  should be challenging but the solution should be explanable in a few minutes using high school math. Examples will be provided.&lt;/p&gt;&#xA;&lt;p&gt;I will collect and distribute the questions to the students, who will then have a few days to work on them. During the next lesson, each student will explain their solution to a randomly chosen question.&lt;/p&gt;&#xA;&lt;p&gt;To deter students from just copying a question from somewhere, sometimes I will provide more requirements for their question. For example, their question must be:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;related to one of several geometrical diagrams that I provide;&lt;/li&gt;&#xA;&lt;li&gt;posed as an extension question to one of several questions that I provide; or&lt;/li&gt;&#xA;&lt;li&gt;inspired by a &lt;a href=&quot;https://www.maths.cam.ac.uk/undergrad/admissions/step&quot; rel=&quot;nofollow noreferrer&quot;&gt;STEP&lt;/a&gt; question of their choice.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am looking for more such ways to deter students from copying a question from somewhere.&lt;/p&gt;&#xA;&lt;h2&gt;Possible pros&lt;/h2&gt;&#xA;&lt;p&gt;This practice might generate interest in math.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Students will be encouraged to &amp;quot;play&amp;quot; with math, instead of just trying to answer a question.&lt;/li&gt;&#xA;&lt;li&gt;There may be tacit competition to see who can come up with the most interesting question.&lt;/li&gt;&#xA;&lt;li&gt;Students personally know the creators of the questions, so they may be more interested in finding the answer (compared to, say, facing Exercise 14.3 in a textbook).&lt;/li&gt;&#xA;&lt;li&gt;The questions (which the students will attempt to answer) will probably be neither too easy nor too difficult.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Speaking from personal experience, I have created my own &lt;a href=&quot;https://math.stackexchange.com/users/398708/dan?tab=questions&amp;amp;sort=votes&quot;&gt;math questions&lt;/a&gt;, and this has made me more interested in math. However, I'm not sure if requiring students to create their own questions will make them more interested in math.&lt;/p&gt;&#xA;" OwnerUserId="20101" LastEditorUserId="20101" LastEditDate="2024-02-24T23:29:42.077" LastActivityDate="2024-02-25T14:18:06.827" Title="What are pros and cons of requiring students to create their own math questions, and what are some tips to make it work well?" Tags="|teaching|gifted-students|creativity|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27529" PostTypeId="2" ParentId="27528" CreationDate="2024-02-24T09:21:34.060" Score="0" Body="&lt;p&gt;Pardon my French, but this looks like pure horror to me.&lt;/p&gt;&#xA;&lt;p&gt;I admit, I express myself in an extreme way, but I think about the following:&lt;br /&gt;&#xA;You start by saying:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There is competition to be admitted into this class.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, there already &lt;strong&gt;is&lt;/strong&gt; a form of hostility between the students.&lt;/p&gt;&#xA;&lt;p&gt;... and now you let students create questions, which might not be solved by their classmates, so this might increase the interpersonal hostility even more?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;NO!&lt;/strong&gt;&lt;br /&gt;&#xA;Creating math questions is the job of the teacher. If they grow hostile feelings against you, no one cares!&lt;br /&gt;&#xA;But students amongst each other should not be stimulated into disliking each other.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2024-02-24T09:21:34.060" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27530" PostTypeId="1" AcceptedAnswerId="27533" CreationDate="2024-02-24T11:15:32.893" Score="2" ViewCount="166" Body="&lt;p&gt;I consider pupils 12-14 years old, who are new to translation. On the other hand, they have been accustomed to placing points in a coordinate system, especially when studying relative numbers. In addition, they know the addition and subtraction of relatives like &lt;span class=&quot;math-container&quot;&gt;$$ (-2.5)+7$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;They can be offered the following activity and many others of the same type:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/bUBY8.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bUBY8.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\color{Blue} {\begin{pmatrix} 3\\5 \end{pmatrix}}+\begin{pmatrix}\color{Green} 6 \\\color{Red} 7\end{pmatrix}:=\begin{pmatrix}3+6 \\5+7\end{pmatrix}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are real difficulties with the many mathematical notations that should not be overlooked but with the awareness of these difficulties and patience, nothing insurmountable. All the more so if this type of activity in reference points was done in primary school.&lt;/p&gt;&#xA;&lt;p&gt;The other classic way to introduce translation is to use pupils' knowledge of parallelograms:&lt;span class=&quot;math-container&quot;&gt;$$M'=t_{A\to B}(M)\iff AMM'B \text{ is a parallelogram}\iff A'=s_D(A)$$&lt;/span&gt;where &lt;span class=&quot;math-container&quot;&gt;$D:=m(M,B)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s_D $&lt;/span&gt; is symmetry.&lt;/p&gt;&#xA;&lt;p&gt;What do you think? Should one be preferred over the other? Should you practice both? An argument for the former is that it effectively prepares for higher education and here for the &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^2$&lt;/span&gt; affine space(here &lt;span class=&quot;math-container&quot;&gt;$A+u$&lt;/span&gt; is illustrated). But I am often told that secondary education is not intended to train future mathematicians, neither engineers nor ... One argument for the second is that it leaves students in a familiar environment but, in my experience, I wonder: really that familiar?&lt;/p&gt;&#xA;" OwnerUserId="23732" LastEditorUserId="23732" LastEditDate="2024-02-24T13:16:17.047" LastActivityDate="2024-03-10T22:15:33.687" Title="Comparison of two ways to introduce translation to 12-14 year olds" Tags="|mathematical-pedagogy|geometry|analytic-geometry|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27531" PostTypeId="2" ParentId="27486" CreationDate="2024-02-24T11:44:56.500" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;One reaches a point in the exposition where to different Poisson-distributed random variables correspond to different subsets of the line.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't understand how you are supposed to &amp;quot;reach&amp;quot; this point, but assuming you did, the object you have at hand is an uncountably infinite collection of coupled random variables. This is much more complicated than a single continuous random variable, both to grasp intuitively and to construct rigorously (which is why the Poisson process on the real line is usually constructed starting with an infinite sequence of i.i.d. exponential waiting times).&lt;/p&gt;&#xA;&lt;p&gt;Note that if you just have a countable collection of independent coin flips (and what you are doing is clearly more difficult), this is already equivalent to the construction of Lebesgue measure and trivially allows to construct continuous distributions.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2024-02-24T11:44:56.500" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27532" PostTypeId="2" ParentId="27528" CreationDate="2024-02-24T13:05:37.317" Score="3" Body="&lt;p&gt;It is generally a good idea to have students express their mathematical ideas and thinking for others to see and engage with. Having them create exercises is a way of doing this, and is likely to reveal to you a great deal about the thinking and skills of the students in question.&lt;/p&gt;&#xA;&lt;p&gt;However, for the purpose of getting good exercises, consider the mathematical maturity and knowledge base of the students and compare to yours. You hopefully have a bit of an edge there. It is much more likely that you can create interesting exercises that are related to real life, concepts that appear later but you can set the stage for now, that are targeted towards a specific idea or technique, that do not have accidental features like misleading wrong ways of getting to the correct solution, and so forth.&lt;/p&gt;&#xA;&lt;p&gt;Having the students create some exercises, maybe in groups or pairs, and solve each others', can be a good move, but it should be used in moderation.&lt;/p&gt;&#xA;&lt;p&gt;Also, this activity might be threatening to some students, as they need to show their work to their classmates who might judge them for it; even if this is not the case, it might be perceived as a possibility. Also, some students might used to traditional schooling where the teacher tells what to do and the student just does, no thinking or creativity required. A change to that can be resisted.&lt;/p&gt;&#xA;&lt;p&gt;For the two aforementioned reasons, and depending on how you frame the activity, there might be resistance.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2024-02-24T13:05:37.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27533" PostTypeId="2" ParentId="27530" CreationDate="2024-02-24T15:27:56.157" Score="4" Body="&lt;p&gt;My recommendation is to show both concepts, but with the pictorial representation of vector addition more as &amp;quot;motivation&amp;quot;.  Of course, even before doing that, you ought to start even simpler with discussion of vectors, themselves.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;There's these things called vectors.  Vector means magnitude and direction.  Think of a car moving (how fast it goes, but also what direction it goes).  Other vectors can be forces (how hard you push and what direction you push).&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;IOW, don't jump to addition as the first concept...it needs to come after grokking vectors themselves.&lt;/p&gt;&#xA;&lt;p&gt;In my experience, in the US, (and looking at a couple texts), both ways of thinking about it are brought up, pretty close to the same time.  The parallelogram is more of a rationale to help understand a bit of an abstract topic, rather than the retained algorithm.&lt;/p&gt;&#xA;&lt;p&gt;In math class, the student will pretty quickly turn to using the algebraic reasoning for most problems.  But in physics classes, the geometric view of vectors can still be helpful.  At least draw a picture, label it, before starting to crank algebra.  And in many cases, you really do care about the difference between axes (height versus distance over ground, having very different physical implications).&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't worry about &amp;quot;inhibit generalization to 3D or higher&amp;quot;. [comment to head post]  There is an expert bias tendency of many math grad students to think that lower classes should be taught in generalized form.  That it will somehow either save time or eliminate confusion.  But the opposite is true (for mortal non geniuses).  I mean, would you teach multivariable calculus first?  How would that go over?  And while div/grad/curl/and all that are sorta intrinsically hard...do we really think they are easier for kids unexposed to any (single-variable) calculus, versus having had a basic single variable class first?&lt;/p&gt;&#xA;&lt;p&gt;FWIW, in the US, your timing for introducing this topic seems early. Traditionally, it was not even an &amp;quot;algebra 2&amp;quot; (11th grade, 16-17 year olds) topic, but was part of analytic geometry (or &amp;quot;pre-calculus&amp;quot;).  Even if they have moved vectors earlier into the curriculum, than my dated experience/texts, it's not a 13 year old topic.  Not generally.&lt;/p&gt;&#xA;&lt;p&gt;Also, I would be a little wary of how you are approaching this question.  It sounds very theoretical, versus observational.  &amp;quot;I consider pupils 12-14 years old&amp;quot; and &amp;quot;Should one be preferred&amp;quot; and &amp;quot;An argument for&amp;quot;.  Instead of trying to come up with de novo insights from thinking about pedagogy, go look at examples and ask for experiences.  I'm not saying to NEVER try to be deductive in pedagogy, but it probably should NOT be your first or primary frame of reference.  You will miss a lot of insights that you might have gotten by &amp;quot;looking at the animals&amp;quot; instead of &amp;quot;I consider&amp;quot; or just brainstormed possible rationales (&amp;quot;An argument for&amp;quot;).  And you will develop a lot of crazy Ivory Tower fiascos if you try to design things de novo, versus looking at practice/experience.&lt;/p&gt;&#xA;" OwnerUserId="23784" LastActivityDate="2024-02-24T15:27:56.157" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27535" PostTypeId="2" ParentId="27528" CreationDate="2024-02-25T14:18:06.827" Score="0" Body="&lt;p&gt;As a practical matter, I doubt the kids are going to do much out of class work...this sounds like an enrichment activity, ungraded.  So, I think any content they prepare for each other, will be pretty weak.&lt;/p&gt;&#xA;&lt;p&gt;I think you should try to think about what content/activities to do with the kids that is (a) enjoyable, motivational and (b) not too much work for you.  This is not a trivial Venn diagram.  I do realize your suggestion satisfies (b).  I doubt it satisfies (a).&lt;/p&gt;&#xA;&lt;p&gt;I think some sort of &amp;quot;math team&amp;quot; type activities might work.  Hopefully you can find some cheap and easily modified content.&lt;/p&gt;&#xA;&lt;p&gt;Something like 30 minutes (start of class) math problems, done by the kids.  These should by high school math team level problems. Not routine drill from current classes...but not Putnam level craziness. Then exchange papers and grade (maybe have volunteers at the board).  Then discuss (maybe you can generalize or give some aenecdotal context on how they relate to college math course.)&lt;/p&gt;&#xA;&lt;p&gt;Try to keep it interactive (the upfront problems help here, could also bake in some voting or the like...&amp;quot;what question was best, worst&amp;quot;).  I would plan in a 15 minute break also.  90 minutes is a long class.&lt;/p&gt;&#xA;&lt;p&gt;If you can do so, with a modicum of work, you might try to orient the content to some broad themes.  I.e. one day of geometry, one of number theory, one of probability, etc.  Just...when I did math team, it was mixed...and that was fine.  But for your class to look good and to give some focus to the discussion, maybe if you could segregate to themes, it would be good.&lt;/p&gt;&#xA;&lt;p&gt;I'd also consider bringing in a few guests.  E.g. someone in industry (like a stats person from pharma industry) or a math major senior undergrad or senior grad student.  Don't make guest carry the whole class...but if you can have some discussion with the kids, it's nice.  After all these kids are thinking of going into math (you say).  This gives them a chance to hear other voices than your own.  And 11th and 12th grade HS kids are starting to transition to adulthood...so nice to hear perspectives form people further on that road.  Guests can say if math major was good/bad choice (jobs, socializing, workload, interest, etc.)&lt;/p&gt;&#xA;&lt;p&gt;I would also think a little about the level of the kids.  You said they competed to get in (fine).  But what elite are they?  I just think it will help ground you in designing, choosing your content.  I love them all...it's not like being smart makes you closer to God.  And some of the psychology (keep em active, needing a bio break in 90 minutes) applies regardless of brains.  But just, think about your audience.  Are they little Lisa Randalls or little guest trolls?  10 out of a high school or 10 out of a district?&lt;/p&gt;&#xA;" OwnerUserId="23784" LastActivityDate="2024-02-25T14:18:06.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27537" PostTypeId="1" CreationDate="2024-02-27T14:34:09.363" Score="3" ViewCount="220" Body="&lt;p&gt;Recreational mathematics serves as a valuable educational tool by enhancing interest, fostering engagement, and refining mathematical thinking. While it often falls outside traditional curricula, innovative approaches can significantly benefit students. For instance, to grasp binary numbers, educators can employ interactive methods like Magic Cards, prompting students to identify missing numbers. Exploring puzzles, such as determining the minimum door openings to label switches for 100 bulbs without touching them, offers a playful yet instructive way to delve into mathematical concepts.&lt;br/&gt;&#xA;What other examples you would suggest relevant to other topics in mathematics?&lt;/p&gt;&#xA;" OwnerUserId="19814" LastActivityDate="2024-02-27T17:08:35.183" Title="Recreational mathematics to create sense of mathematics" Tags="|mathematical-pedagogy|student-motivation|recreational-learning|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27538" PostTypeId="2" ParentId="27537" CreationDate="2024-02-27T17:08:35.183" Score="2" Body="&lt;p&gt;Personally, my most enjoyable and productive mathematical experiences while growing came from toy research projects. Some examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;When I learned about partial fractions decomposition in high school, I wondered if I could come up with a formula for the coefficients &lt;span class=&quot;math-container&quot;&gt;$c_{jk}$&lt;/span&gt; in the partial fractions expression of &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{b_0 + b_1 x_1 + b_2 x_2 + \cdots + b_n x^n}{(x-a_1)^{k_1} (x-a_2)^{k_2} \cdots (x-a_m)^{k_m}} = \sum_{j=1}^m \sum_{k=1}^{k_j} \dfrac{c_{jk}}{(x-a_j)^k}$$&lt;/span&gt; by writing down and solving the system of equations in the general case, and also conducting numerical experiments to look for relevant patterns that I could prove and leverage in helpful ways. This turned out to be a very difficult and messy project but I did obtain one reasonably neatly-packaged result, &lt;span class=&quot;math-container&quot;&gt;$$\dfrac{x^n}{(x-a)^k} = \sum_{i=0}^{n-k} \binom{n-1-i}{k-1} a^{n-k-i} x^i + \sum_{i=\max(k-n,1)}^k \dfrac{\binom{n}{k-i} a^{n-k+i}}{(x-a)^i},$$&lt;/span&gt; and proving it gave me my first experience with double-induction.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In my first year of college I got interested in neuroscience and I wondered if it was possible to, given a network of neurons, work out how the network's connectivity will change if you pick one neuron and repeatedly &amp;quot;activate&amp;quot; it with a pulse that ripples through the network. (Whenever two neurons activate, the connection strength between them changes according to known biological learning rules -- the simplest and earliest-discovered rule is &amp;quot;what fires together wires together,&amp;quot; known as Hebbian learning, but since then more nuanced rules have been discovered like spike-timing-dependent plasticity, which takes temporal directionality into account.)&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;Similar to the previous problem, my approach was again a combination of working out math by hand and also running computer simulations to look for helpful patterns. The problem was again very difficult and messy, but again I really leveled up my skills by grappling with it and I discovered plenty of interesting things along the way.&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/hJkak.png&quot; width=&quot;300&quot; /&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/upjaR.png&quot; width=&quot;300&quot; /&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;More info about these and a ton of other projects can be found here: &lt;a href=&quot;https://www.justinmath.com/old-projects/&quot; rel=&quot;nofollow noreferrer&quot;&gt;justinmath.com/old-projects&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;They were a lot of hard work, and I can't say that any of them were academically successful in the sense of yielding publishable results, but boy did they level up my mathematical ability more than any puzzle or textbook exercise -- not just my pure mathematical reasoning ability, but also my general ability to wield math as a weapon to fight against really hard problems.&lt;/p&gt;&#xA;&lt;p&gt;That's the thing about research projects versus puzzles. There's only so much you can hone your math skills by working on a problem that someone else has intentionally set up to be well-posed and elegantly solvable if you think about it the right way. You're in a sandbox, and the problems that you fight against have been artificially handicapped to be weak opponents. It's perfectly reasonable for beginners to start here, but people often continue training in that sandbox for way too long and then come out expecting to conquer the world. And then they're faced with the rude awakening that they're actually really weak against problems that have not been artificially subdued.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2024-02-27T17:08:35.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27539" PostTypeId="1" CreationDate="2024-02-27T23:44:38.013" Score="4" ViewCount="127" Body="&lt;p&gt;I'm interested in learning two transitions:&lt;/p&gt;&#xA;&lt;p&gt;(1) When can students reason (intuitively, but accurately) to conclude that two circles in the plane&#xA;could intersect in &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; points, or are coincident so sharing &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;-many points?&lt;/p&gt;&#xA;&lt;p&gt;(2) When can students prove this? E.g., distinguishing between circles and ellipses, which could intersect &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; times?&lt;/p&gt;&#xA;&lt;p&gt;I am hoping that in the US, (1): after a first course in geometry, and (2): after a first course in algebra. So sometime in &amp;quot;middle school&amp;quot; in the US.&#xA;But I only teach college students, so I don't really understand when certain&#xA;pre-college transitions are crossed. Maybe that algebra would need to await&#xA;high school?&lt;/p&gt;&#xA;" OwnerUserId="511" LastActivityDate="2024-02-27T23:44:38.013" Title="When can students understand the intersection of two circles?" Tags="|secondary-education|algebra|geometry|middle-school|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="27540" PostTypeId="1" CreationDate="2024-02-28T19:11:43.170" Score="0" ViewCount="292" Body="&lt;p&gt;Imagine a class of students who don't know what smallest common multiples (LCMs) are, or indeed what multiples are. (Notice I said &amp;quot;smallest&amp;quot; rather than &amp;quot;least&amp;quot; so as to avoid people thinking it means multiples that are less common than others. Actual humans exist who think that.)&lt;/p&gt;&#xA;&lt;p&gt;You tell them the &lt;b&gt;multiples&lt;/b&gt; of &lt;span class=&quot;math-container&quot;&gt;$63$&lt;/span&gt; can be found as follows:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;1\times63 &amp;amp; = 63 \\&#xA;2\times63 &amp;amp; = 126 \\&#xA;3\times63 &amp;amp; = 189 \\&#xA;4\times63 &amp;amp; = 252 \\&#xA;5\times63 &amp;amp; = 315 \\&#xA;6\times63 &amp;amp; = 378 \\&#xA;&amp;amp; \,\,\,\vdots&#xA;\end{align}&lt;/span&gt;&#xA;and also as follows:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;63 \, \overset{\text{add 63}} \longmapsto \, 126 \, \overset{\text{add 63}} \longmapsto \, 189  \, \overset{\text{add 63}} \longmapsto \, 252 \, \overset{\text{add 63}} \longmapsto \, 315 \, \overset{\text{add 63}} \longmapsto \, 378 \, \overset{\text{add 63}} \longmapsto  \,\cdots&#xA;$$&lt;/span&gt;&#xA;Now do this with two numbers&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{array}{cccccccccccccccccccccccc}&#xA;\text{multiples of 63:} &amp;amp; 63, &amp;amp; 126, &amp;amp; 189, &amp;amp; 252, &amp;amp; 315, &amp;amp; 378, &amp;amp; \ldots\ldots \\[4pt]&#xA;\text{multiples of 77:} &amp;amp; 77, &amp;amp; 154, &amp;amp; 231, &amp;amp; 308, &amp;amp; 385, &amp;amp; 462, &amp;amp; \ldots\ldots&#xA;\end{array}&#xA;$$&lt;/span&gt;&#xA;Not ask the class to (silently) &lt;b&gt;vote&lt;/b&gt; &amp;quot;yes&amp;quot; or &amp;quot;no&amp;quot; on the answer to the following question:&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;Might these two sequences of multiples go on forever without ever having any numbers is common, or if not these two, might that happen with some other pair of numbers? Yes or no.&lt;/b&gt;&lt;/p&gt;&#xA;&lt;p&gt;(By &amp;quot;silently&amp;quot; I mean they share no information before casting votes.)&lt;/p&gt;&#xA;&lt;p&gt;Has anyone done this, perhaps with a fairly large number of students? What were the results?&lt;/p&gt;&#xA;" OwnerUserId="205" LastEditorUserId="205" LastEditDate="2024-03-06T22:46:17.137" LastActivityDate="2024-03-08T14:32:24.693" Title="Understanding common multiples" Tags="|assessment|arithmetic|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="27541" PostTypeId="2" ParentId="27515" CreationDate="2024-02-28T22:02:13.970" Score="5" Body="&lt;p&gt;The inverse function theorem on &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; is &lt;strong&gt;global&lt;/strong&gt;: any differentiable function &lt;span class=&quot;math-container&quot;&gt;$f \colon \mathbb{R} \to \mathbb{R}$&lt;/span&gt; satisfying &lt;span class=&quot;math-container&quot;&gt;$f'(x) \neq 0$&lt;/span&gt; everywhere has to be one-to-one, and so it has a global inverse &lt;span class=&quot;math-container&quot;&gt;$f^{-1} \colon f(\mathbb{R}) \to \mathbb{R}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The inverse function theorem on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; is only &lt;strong&gt;local&lt;/strong&gt;: if &lt;span class=&quot;math-container&quot;&gt;$f \colon \mathbb{R}^n \to \mathbb{R}^n$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$C^1$&lt;/span&gt;-regular and &lt;span class=&quot;math-container&quot;&gt;$Df(x)$&lt;/span&gt; is invertible, then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is invertible on some neighorhood of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;One can appreciate the difference by looking at the example of &lt;span class=&quot;math-container&quot;&gt;$\exp \colon \mathbb{C} \to \mathbb{C}$&lt;/span&gt;, which is clearly not one-to-one (we have &lt;span class=&quot;math-container&quot;&gt;$\exp(0) = \exp(2\pi i)$&lt;/span&gt;) but its derivative is everywhere invertible.&lt;/p&gt;&#xA;" OwnerUserId="15367" LastActivityDate="2024-02-28T22:02:13.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27542" PostTypeId="1" CreationDate="2024-02-28T23:19:53.667" Score="5" ViewCount="222" Body="&lt;p&gt;Every year the &lt;a href=&quot;https://maa.org/math-competitions/william-lowell-putnam-mathematical-competition&quot; rel=&quot;noreferrer&quot;&gt;Putnam Competition&lt;/a&gt; features at least one question that describes Alice and Bob playing game, and asks for one player's winning strategy or something similar. Since these questions are usually more comprehensible than the others on the Putnam, many of my students who compete submit responses to these questions, but their submissions usually don't score.&lt;/p&gt;&#xA;&lt;p&gt;How should I teach students to better answer these types of questions? What proof-writing techniques should I demonstrate to them? What problem-solving (thought) pattern(s) should I encourage them to adopt to help them translate their understanding of the game through &amp;quot;playing some examples&amp;quot; to writing a proof?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/q/4872433/167197&quot;&gt;Here's the MathSE analog of this question&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For example, &lt;a href=&quot;https://maa.org/sites/default/files/Programs/2023Putnam_A.pdf&quot; rel=&quot;noreferrer&quot;&gt;problem A6 from the 2023 Putnam Competition&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Alice and Bob play a game in which they take turns choosing integers from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. Before any integers are&#xA;chosen, Bob selects a goal of &amp;quot;odd&amp;quot; or &amp;quot;even&amp;quot;. On the first turn, Alice chooses one of the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; integers. On&#xA;the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the&#xA;integers that has not yet been chosen, until the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th turn, which is forced and ends the game. Bob wins if the&#xA;parity of &lt;span class=&quot;math-container&quot;&gt;$\{k : \text{the number $k$ was chosen on the $k$th turn}\}$&lt;/span&gt; matches his goal. For which values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; does Bob&#xA;have a winning strategy?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5220" LastEditorUserId="5220" LastEditDate="2024-02-28T23:36:41.763" LastActivityDate="2024-03-04T08:54:51.720" Title="How would you prepare students for &quot;Alice and Bob&quot; Putnam problems?" Tags="|undergraduate-education|math-contests|games|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27543" PostTypeId="1" CreationDate="2024-02-29T02:12:51.633" Score="4" ViewCount="70" Body="&lt;p&gt;I've been utilizing edstem.org for facilitating math and logistics discussions in my course. Its support for LaTeX and hand-drawn illustrations has been convenient. Typically, students post questions, and I respond, though occasionally, some students also participate in discussions.&lt;/p&gt;&#xA;&lt;p&gt;I'm contemplating integrating these online discussions into the grading system, specifically allocating 1% of the total grade to students who contribute two accepted answers. My goal is to encourage more engagement and collaborative learning.&lt;/p&gt;&#xA;&lt;p&gt;However, I'm concerned this might lead to an influx of low-quality, or &amp;quot;fake,&amp;quot; questions and answers from students aiming merely to secure the points.&lt;/p&gt;&#xA;&lt;p&gt;Has anyone implemented a similar grading policy? What were your experiences?&lt;/p&gt;&#xA;" OwnerDisplayName="user19945" LastActivityDate="2024-02-29T02:12:51.633" Title="Seeking Advice on Incorporating Online Discussions into Course Grading" Tags="|grading|assessment|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27544" PostTypeId="2" ParentId="27542" CreationDate="2024-02-29T16:13:25.050" Score="2" Body="&lt;p&gt;All games with numbers: teach them to start with very small numbers, like here with n=3,4,5 may be some more they find a rule and then only they start to generalize and prove.&lt;/p&gt;&#xA;" OwnerUserId="21051" LastActivityDate="2024-02-29T16:13:25.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27545" PostTypeId="2" ParentId="27540" CreationDate="2024-02-29T16:20:02.737" Score="1" Body="&lt;p&gt;I think you make it more difficult for students by introducing the &amp;quot;multiples&amp;quot; not as multiples, but by continual adding, also since 77*63 is an easy solution you are not going in the direction of LCM&lt;/p&gt;&#xA;" OwnerUserId="21051" LastActivityDate="2024-02-29T16:20:02.737" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27546" PostTypeId="2" ParentId="17302" CreationDate="2024-02-29T20:15:38.150" Score="5" Body="&lt;p&gt;A given point with polar coordinates &lt;span class=&quot;math-container&quot;&gt;$(r, \theta)$&lt;/span&gt; has a doubly infinite set of representations &lt;span class=&quot;math-container&quot;&gt;$(r,\theta+2n\pi)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(-r,\theta+(2n+1)\pi)$&lt;/span&gt;, for integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. When checking for symmetry they should all be considered.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; be the polar equation &lt;span class=&quot;math-container&quot;&gt;$f(r,\theta)=0$&lt;/span&gt; with a graph &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;. If for some integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; the following equation is equivalent to &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; then the indicated symmetry applies.&lt;/p&gt;&#xA;&lt;p&gt;X1:  &lt;span class=&quot;math-container&quot;&gt;$f(r,-\theta+2n\pi)=0$&lt;/span&gt; or&#xA;X2:  &lt;span class=&quot;math-container&quot;&gt;$f(-r,-\theta+(2n+1)\pi)=0$&lt;/span&gt;&#xA;then &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is symmetric with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis or polar axis.&lt;/p&gt;&#xA;&lt;p&gt;Y1:  &lt;span class=&quot;math-container&quot;&gt;$f(r,-\theta+(2n+1)\pi)=0$&lt;/span&gt; or&#xA;Y2:  &lt;span class=&quot;math-container&quot;&gt;$f(-r,-\theta+2n\pi)=0$&lt;/span&gt;&#xA;then &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is symmetric with respect to &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis or &lt;span class=&quot;math-container&quot;&gt;$\theta=\pi/2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;O1: &lt;span class=&quot;math-container&quot;&gt;$f(r,\theta+(2n+1)\pi)=0$&lt;/span&gt; or&#xA;O2: &lt;span class=&quot;math-container&quot;&gt;$f(-r,\theta+2n\pi)=0$&lt;/span&gt;&#xA;then &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is symmetric with respect to origin or pole.&lt;/p&gt;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$f(r,\theta)=r^2-\sin(\theta)=0$&lt;/span&gt; all three symmetries are present.&lt;/p&gt;&#xA;&lt;p&gt;Apply X2: then &lt;span class=&quot;math-container&quot;&gt;$f(-r,-\theta+\pi)= (-r)^2-\sin(-\theta+\pi)=r^2-\sin(\theta)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Apply Y1: then &lt;span class=&quot;math-container&quot;&gt;$f(r,-\theta+\pi)=(r)^2-\sin(-\theta+\pi)=r^2-\sin(\theta)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Apply O2: then &lt;span class=&quot;math-container&quot;&gt;$f(-r,\theta)=(-r)^2-\sin(\theta)=r^2-\sin(\theta)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For additional examples you may want to consider &lt;span class=&quot;math-container&quot;&gt;$r=\sin(\theta/k)$&lt;/span&gt; for some integer &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2024-02-29T20:35:17.040" LastActivityDate="2024-02-29T20:35:17.040" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27549" PostTypeId="2" ParentId="604" CreationDate="2024-03-02T16:35:13.410" Score="0" Body="&lt;p&gt;In this context, I see &amp;quot;function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;&amp;quot; without any further information to be a reference to an arbitrary function named &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; with unspecified domain, codomain and rule relating its input and output.&lt;/p&gt;&#xA;&lt;p&gt;If we have &amp;quot;functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;,&amp;quot; both with the same domain and codomain then they we can say they are equal if and only if their rules relating input and output are logically equivalent.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; on it own, without further information, suggests only that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is an arbitrary function of one variable with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; being an element of its unspecified domain.&lt;/p&gt;&#xA;" OwnerUserId="6103" LastEditorUserId="6103" LastEditDate="2024-03-02T16:40:55.717" LastActivityDate="2024-03-02T16:40:55.717" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27550" PostTypeId="2" ParentId="24860" CreationDate="2024-03-02T17:03:47.927" Score="1" Body="&lt;p&gt;Maybe start discussion with photos something like:&lt;/p&gt;&#xA;&lt;p&gt;Sign indicating a 10% grade at the base of hill. What does it mean?&#xA;&lt;a href=&quot;https://www.shutterstock.com/image-photo/road-sign-steep-slope-105861944&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.shutterstock.com/image-photo/road-sign-steep-slope-105861944&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Picture of street with very steep grade/slope. Estimate its grade/slope.&#xA;&lt;a href=&quot;https://www.shutterstock.com/image-photo/baldwin-street-steepest-world-dunedin-new-484970863&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.shutterstock.com/image-photo/baldwin-street-steepest-world-dunedin-new-484970863&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Dan&lt;/p&gt;&#xA;" OwnerUserId="6103" LastActivityDate="2024-03-02T17:03:47.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27551" PostTypeId="2" ParentId="17302" CreationDate="2024-03-03T00:29:29.163" Score="1" Body="&lt;p&gt;If a textbook gives symmetry tests for graphs of polar equations without stating that they are sufficient but not necessary, that's a glaring omission, in my opinion. I looked through several books, and I only saw this in Stewart and one other book. The rest either didn't cover symmetry tests or stated that they weren't necessary conditions. Larson seemed the most comprehensive to me. The book gives the following tests.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The graph of a polar equation is symmetric with respect to the following when the given substitution yields an equivalent equation.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The line &lt;span class=&quot;math-container&quot;&gt;$\theta=\pi/2$&lt;/span&gt;: Replace &lt;span class=&quot;math-container&quot;&gt;$(r, \theta)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$(r, \pi-\theta)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(-r, -\theta)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The polar axis:      Replace &lt;span class=&quot;math-container&quot;&gt;$(r, \theta)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$(r, -\theta)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(-r, \pi-\theta)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The pole:               Replace &lt;span class=&quot;math-container&quot;&gt;$(r, \theta)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$(r, \pi+\theta)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(-r, \theta)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Larson also gives two more &amp;quot;quick tests for symmetry.&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The graph of &lt;span class=&quot;math-container&quot;&gt;$r=f(\sin\theta)$&lt;/span&gt; is symmetric with respect to the line &lt;span class=&quot;math-container&quot;&gt;$\theta = \pi/2$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The graph of &lt;span class=&quot;math-container&quot;&gt;$r=g(\cos\theta)$&lt;/span&gt; is symmetric with respect to the polar axis.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The graph of &lt;span class=&quot;math-container&quot;&gt;$r=\theta+2\pi$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$ -4\pi \leq \theta \leq 0$&lt;/span&gt; is given as an example to show that these tests are not necessary for symmetry.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nZmudm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nZmudm.png&quot; alt=&quot;A graph of the polar equation r = θ + 2π between θ = -4π and θ = 0. The graph is symmetric about the line θ = π/2.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Maesumi gave necessary conditions in &lt;a href=&quot;https://matheducators.stackexchange.com/a/27546/20719&quot;&gt;their answer&lt;/a&gt;. The equation &lt;span class=&quot;math-container&quot;&gt;$r=\theta+2\pi$&lt;/span&gt; yields an equivalent equation under the substitution &lt;span class=&quot;math-container&quot;&gt;$(r, \theta) \mapsto (-r, -\theta-4\pi)$&lt;/span&gt;, satisfying Y2.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastActivityDate="2024-03-03T00:29:29.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27552" PostTypeId="2" ParentId="2428" CreationDate="2024-03-03T20:39:44.920" Score="3" Body="&lt;p&gt;I am teaching Calc II for the first time in many years and I am again finding it a pleasure.  I'm having fun, and I believe my students are also, with regions in the plane that are unbounded but have finite area, books that stack infinitely far off the table, and how conics unify circles, parabolas, and hyperbolas.&lt;/p&gt;&#xA;&lt;p&gt;Just to give one specific suggestion though, I find that using Taylor polynomials to motivate the presentation of series gives the chapter a theme, and adds some sense of what we are up to, which can help sustain a student through what in Stewart is an eleven section chapter.&lt;/p&gt;&#xA;" OwnerUserId="213" LastActivityDate="2024-03-03T20:39:44.920" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27553" PostTypeId="2" ParentId="27542" CreationDate="2024-03-04T08:54:51.720" Score="2" Body="&lt;p&gt;To give your students well rounded training, I suggest you use puzzles from as many different sources as possible.&lt;/p&gt;&#xA;&lt;p&gt;As an example, here are some problems you might use (they are from Leningrad Mathematical Olympiads).&lt;/p&gt;&#xA;&lt;p&gt;1982:&lt;/p&gt;&#xA;&lt;p&gt;Anna and Boris play a game with a stack of 100 counters. Anna goes first, and moves alternate thereafter. In each move, a player divides a stack of at least two counters into two smaller piles. The loser is the player without a move, when each stack consists of exactly one counter. Prove that Anna must win this game.&lt;/p&gt;&#xA;&lt;p&gt;1987:&lt;/p&gt;&#xA;&lt;p&gt;Anna and Boris play a game on a 9 × 9 chessboard. Anna goes first and turns alternate thereafter. In each move, Anna puts a red counter on a vacant square while Boris puts a blue counter on a vacant square. When the board is completely filled, a row with more red counters than blue counters is called a red row, and a blue row otherwise. Red and blue columns are similarly defined. The score for Anna is the sum of the numbers of red rows and red columns while that for Boris is the sum of the numbers of blue rows and blue columns. What is the highest possible score for Anna?&lt;/p&gt;&#xA;&lt;p&gt;1990:&lt;/p&gt;&#xA;&lt;p&gt;Anna and Boris play a game starting with the number 1234. Anna goes first, and turns alternate thereafter. In each turn, the player subtracts from the number one of its non-zero digits. A player wins if the number is reduced to 0. Who has a winning strategy, Anna or Boris?&lt;/p&gt;&#xA;" OwnerUserId="23338" LastActivityDate="2024-03-04T08:54:51.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27554" PostTypeId="1" CreationDate="2024-03-04T15:59:51.047" Score="2" ViewCount="102" Body="&lt;p&gt;I am looking for US average national or state grade distribution data for courses that are typically considered as math service courses: precalculus, calculus 1,2,3, linear algebra, differential equations, and statistics, by various classes of universities (for example  public vs private), by year. Does your university make their data public?&lt;/p&gt;&#xA;&lt;p&gt;Do you know of any source?&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="2083" LastEditDate="2024-03-06T09:04:57.720" LastActivityDate="2024-03-06T09:04:57.720" Title="Is there national grade distribution data for introductory service courses?" Tags="|undergraduate-education|course-design|usa|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27555" PostTypeId="2" ParentId="2152" CreationDate="2024-03-06T01:38:05.840" Score="0" Body="&lt;p&gt;I'm not sure if this is a Taylor series, but I know in physical chemistry, the higher &lt;strong&gt;virial coefficients of pressure&lt;/strong&gt; have an intuitive explanation, for the deviations from ideal gas law.  Like the first term is just the law itself (PV=nRT) and then the second and third are deviations based on (a) volume of the molecules themselves (not point masses bouncing around) and (b) attraction between the moleculues when close.&lt;/p&gt;&#xA;&lt;p&gt;I forget which is which, but see:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/16%3A_The_Properties_of_Gases/16.05%3A_The_Second_Virial_Coefficient&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/16%3A_The_Properties_of_Gases/16.05%3A_The_Second_Virial_Coefficient&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Or any physical chemistry, or engineering or physics thermodynamics text.&lt;/p&gt;&#xA;&lt;p&gt;Anyhow, hope it helps since I always felt it was cool that these extra terms had an intuitive physical meaning!&lt;/p&gt;&#xA;" OwnerUserId="23808" LastActivityDate="2024-03-06T01:38:05.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27556" PostTypeId="1" AcceptedAnswerId="27558" CreationDate="2024-03-06T02:56:23.377" Score="10" ViewCount="4005" Body="&lt;p&gt;When grading problems on quizzes and exams, I often break them down into sub-problems, each worth a portion of the total points. I use rubrics to award partial credit for each sub-problem. However, this practice leads to students arguing that their incorrect work deserves more partial credit than what I gave them. For example, a student might claim that their mistake only warrants a deduction of 0.1 points rather than the 0.3 points I deducted.&lt;/p&gt;&#xA;&lt;p&gt;To avoid these disputes, I'm considering switching to a binary grading system where each sub-problem is graded as either completely correct (full points) or completely incorrect (zero points), with no partial credit awarded.&lt;/p&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Do you award partial credit when grading, or do you use a binary grading system?&lt;/li&gt;&#xA;&lt;li&gt;If you do award partial credit, how do you justify and explain the specific point deductions to students who argue that they deserve more partial credit?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Please let me know if this revised phrasing accurately captures your questions and concerns.&lt;/p&gt;&#xA;" OwnerDisplayName="user19945" LastEditorUserId="2083" LastEditDate="2024-03-06T12:02:08.683" LastActivityDate="2024-03-09T18:20:54.883" Title="Is it a good idea to give partial points in grading" Tags="|undergraduate-education|grading|usa|" AnswerCount="7" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27557" PostTypeId="2" ParentId="27556" CreationDate="2024-03-06T05:30:12.063" Score="12" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I always allow the possibility for partial credit. Unless the problems are quite simple, there will be a good amount of room between a perfect solution and a solution that deserves no credit. If you don't take this into account, you lose a lot of the information that grading is supposed to capture, e.g. the difference between a student who knows nothing vs. a student who implements the right overall idea, but with errors, vs. a student with a perfect response.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Students often ask for more points, this is part of teaching. My policy on such requests is: if I have made a clear error in grading your work, such as a misreading or misunderstanding of what you wrote (which does happen sometimes), I'll give you the points back. But I won't revisit judgment calls about how much partial credit a response deserves. This policy is easy to apply, and easy to justify on fairness grounds: reconsidering partial credit for one student who happens to complain is not fair to the other students who may have made similar errors (often, the exact same error) but didn't complain. I have used this policy for a long time, and have never had a problem with it. Students might not be happy, but once they see I won't budge on this, they stop bothering me with meritless requests for more credit.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="23762" LastActivityDate="2024-03-06T05:30:12.063" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27558" PostTypeId="2" ParentId="27556" CreationDate="2024-03-06T05:33:52.640" Score="18" Body="&lt;p&gt;&lt;em&gt;Is it a good idea to give partial points in grading?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, yes because it makes the grading more fair.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Do you give partial points when you grade?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Yes. I set a criterion and adapt it according to unexpected errors that appear (for example: -0.1 if it is a small error; -0.5 if half is right; 0.1 if almost everything is wrong; -0.2 if there is an error X which I didn't even imagine). This criterion is explicitly presented to students.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;How do you justify how much points to deduct when students come to argue for more points?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I use the criteria I presented to them (I usually make printed copies available): these are the points because this is what the criteria say.&lt;/p&gt;&#xA;&lt;p&gt;Also (and this is very important), I tell them upfront that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the same criteria were applied to all students equally.&lt;/li&gt;&#xA;&lt;li&gt;I will only change the grade if they detect an error in the way the criteria were applied (which eventually happens).&lt;/li&gt;&#xA;&lt;li&gt;If they don't agree with me, they can appeal under the terms of the regulation (in this case, the department would set up a commission to decide, but that never happened to me).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In summary, in my experience, the most important thing to avoid complaints is to be completely clear about the criteria (what the points are for each part of each question) and to apply them uniformly (equal errors should receive equal scores).&lt;/p&gt;&#xA;" OwnerUserId="13585" LastEditorUserId="13585" LastEditDate="2024-03-06T14:27:40.850" LastActivityDate="2024-03-06T14:27:40.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27559" PostTypeId="2" ParentId="27556" CreationDate="2024-03-06T06:34:55.433" Score="8" Body="&lt;p&gt;In my experience, binary credit on sub-problems is the way to go. Once you open the door to interpretation on partial credit, you get your time burned by pointless debates (pun intended) with adversarial students.&lt;/p&gt;&#xA;&lt;p&gt;But the thing is, you have to make sure that the sub-problems are&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;sufficient in number, so that there is room to make a few mistakes on the quiz and still get a good grade, and&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;sufficiently atomic, so that it feels reasonable to grade them on a binary scale.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The goal is for the grade calculation to be totally obvious and feel fair enough that the student doesn't argue against its mechanics.&lt;/p&gt;&#xA;&lt;p&gt;In this way, grade calculations are like laws: you want them to be both clear and fair. If you have unclear laws then people will waste your time in court arguing about interpretation. If you have unfair laws then you'll hurt good people and people will rebel against you.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-03-06T06:49:16.763" LastActivityDate="2024-03-06T06:49:16.763" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27560" PostTypeId="2" ParentId="27556" CreationDate="2024-03-06T22:49:15.920" Score="4" Body="&lt;p&gt;I use a binary grading system.&lt;/p&gt;&#xA;&lt;p&gt;A few tips to help students accept a partial-credit-less grading system:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Offer full credit if the student makes a mistake that does not affect their demonstration of the assessed learning outcome (e.g. arithmetic error)&lt;/li&gt;&#xA;&lt;li&gt;Tune your assessments to make up for the dip in grades due to the missing partial credit. Usually this means lower complexity questions. The real advantage of binary grading is that it makes it much easier to assess an assessment: &amp;quot;I expect a B student to definitely be able to get these questions correct, and maybe half of these questions correct, so they'll probably get somewhere around this grade, which should be in the middle of the B range&amp;quot;. Don't be afraid to err on the side of generosity.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If you want to be really generous with your time, you can also make up the dip in grades by offering retakes. Retakes synergize well with binary grading, since:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Binary grading is much faster to balance the extra time grading retakes&lt;/li&gt;&#xA;&lt;li&gt;Students are less likely to retake a question the more partial credit they have on it&lt;/li&gt;&#xA;&lt;li&gt;Students are less devastated by missing out on getting any points for a question if they can retake&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2024-03-06T22:49:15.920" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27561" PostTypeId="2" ParentId="27556" CreationDate="2024-03-07T05:31:26.623" Score="2" Body="&lt;p&gt;Another reason I like partial credit on tests is it reduces student nervousness. They may have the right approach but be completely stuck bringing it home. With partial credit they know they got those points and can move on, coming back if there's time. Without, they can spend far too much time banging their heads, or checking each question is perfect.&lt;/p&gt;&#xA;" OwnerUserId="13766" LastActivityDate="2024-03-07T05:31:26.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27562" PostTypeId="1" AcceptedAnswerId="27578" CreationDate="2024-03-07T07:51:30.570" Score="8" ViewCount="1109" Body="&lt;p&gt;These were two actual exercises given to students I have been tutoring for a college algebra class:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/9Qf77.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9Qf77.png&quot; alt=&quot;Exercise from Pearson: &amp;quot;Use grouping symbols to clarify the meaning of each statement. Then construct a truth table for the statement. p \rightarrow q \lor \sim r \leftrightarrow \sim p \land r&amp;quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VfyMW.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VfyMW.png&quot; alt=&quot;Exercise from Pearson: &amp;quot;Use grouping symbols to clarify the meaning of each statement. Then construct a truth table for the statement. \sim p \leftrightarrow q \leftrightarrow q \rightarrow \sim p \land \sim p \rightarrow q&amp;quot;. A solution is filled in: &amp;quot;The meaning of the statement is (\sim p \leftrightarrow q) \leftrightarrow ((q \implies \sim p) \land (\sim p \implies q))&amp;quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have been working very hard to convince my students of the importance and utility of learning formal logic, something which I do think more non-STEM majors would benefit from understanding, but I really cannot imagine ever needing something like this. These are just horribly written symbolic expressions that no one should ever use and seem to exist just for the purpose of tormenting lower-division math students. Can anyone provide a justification for including this in math curricula?&lt;/p&gt;&#xA;" OwnerUserId="22348" LastActivityDate="2024-03-10T23:30:08.417" Title="Dominance of connectives: Why do we teach this?" Tags="|undergraduate-education|student-motivation|curriculum|logic|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="27563" PostTypeId="2" ParentId="27562" CreationDate="2024-03-07T08:02:48.897" Score="10" Body="&lt;p&gt;When I see exercises like this, I often find that it teaches students to make assumptions about symbolic statements that may not be there - in a real world situation, if a statement is ambiguous, I would expect the response to be asking for it to be provided in a way that's explicit... considering that's the point of reducing something to a symbolic expression.&lt;/p&gt;&#xA;&lt;p&gt;Looking into it, the only reason it would be something that could be useful is as put in this explanation of logical connectives by &lt;a href=&quot;https://sites.millersville.edu/bikenaga/math-proof/logical-connectives/logical-connectives.html&quot; rel=&quot;noreferrer&quot;&gt;Bruce Ikenaga&lt;/a&gt; - when translating a sentence into a symbolic expression, it's uncommon to use parentheses, even though in mathematics it's rare to not make the statement explicit by grouping. So in some applications, it may be a useful thing to know but I honestly can't see it as a useful thing to memorize and test on. I'd guess that the push to memorize the precedence is similar to expecting students to memorize things most people tend to use a reference for (and now look it up online) - curricula is still written as if students won't have internet access in their pockets.&lt;/p&gt;&#xA;" OwnerUserId="23811" LastActivityDate="2024-03-07T08:02:48.897" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27565" PostTypeId="2" ParentId="21048" CreationDate="2024-03-07T13:31:12.520" Score="1" Body="&lt;p&gt;If you have a rigid body, you may apply a force to it, and if you apply two forces &lt;em&gt;at the same point&lt;/em&gt;, they are equivalent to their sum applied at that point, but this is not the case for forces applied at different points. So, mechanics textbooks implicitly consider the (infinitely-dimensional) vector space &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; which is a direct sum of uncountably many copies &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;, indexed by points in the body.&lt;/p&gt;&#xA;&lt;p&gt;Now, this vector space has a (still infinite dimensional) subspace &lt;span class=&quot;math-container&quot;&gt;$V_0$&lt;/span&gt; of forces that do not cause the body to move - for example, when  two persons  both pull a table in the opposite directions. The &lt;em&gt;laws of statics&lt;/em&gt; describe the basis of &lt;span class=&quot;math-container&quot;&gt;$V_0$&lt;/span&gt;, while the &lt;em&gt;theory of torque&lt;/em&gt; deals with constructing an explicit isomorphism &lt;span class=&quot;math-container&quot;&gt;$V/V_0\mapsto \mathbb{R}^6$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastEditorUserId="117" LastEditDate="2024-03-07T14:07:54.737" LastActivityDate="2024-03-07T14:07:54.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27566" PostTypeId="1" AcceptedAnswerId="27569" CreationDate="2024-03-07T13:41:39.513" Score="6" ViewCount="348" Body="&lt;p&gt;In primary school, it's extremely easy to show that &lt;span class=&quot;math-container&quot;&gt;$a \times b = b \times a$&lt;/span&gt;, as follows:&lt;/p&gt;&#xA;&lt;p&gt;The surface of a rectangle can be calculated using the formula &lt;span class=&quot;math-container&quot;&gt;$\text{Basis} \times \text{Height}$&lt;/span&gt;, as in following examples:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GOoBx.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GOoBx.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Flipping the rectangle does not change its surface, you can even elaborate that by drawing unity squares inside the rectangles and count them.)&lt;/p&gt;&#xA;&lt;p&gt;But how can you show that &lt;span class=&quot;math-container&quot;&gt;$a + b = b + a$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2024-03-16T01:42:45.493" Title="Why is $a+b = b+a$?" Tags="|primary-education|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27567" PostTypeId="2" ParentId="27566" CreationDate="2024-03-07T14:14:54.497" Score="14" Body="&lt;p&gt;If you're happy with the given demonstration for multiplication, then take two connected line segments (rulers) and flip them 180 degrees.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2024-03-07T14:14:54.497" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27568" PostTypeId="2" ParentId="27562" CreationDate="2024-03-07T14:41:03.517" Score="4" Body="&lt;p&gt;I think we all agree that there is a commonly accepted precedence to, say, arithmetic operators, and that it's common to write things like &lt;span class=&quot;math-container&quot;&gt;$3x^2$&lt;/span&gt; without parentheses and expect it to be understood. (In fact, I routinely have college students who cannot correctly evaluate &lt;span class=&quot;math-container&quot;&gt;$3(5)^2$&lt;/span&gt; and need correction.)&lt;/p&gt;&#xA;&lt;p&gt;Similarly, for any other type of operator (relational, logical, etc.) it makes sense to have a defined order of operations and communicate that. This is defined and exercised in standard texts such as Rosen's &lt;a href=&quot;https://drive.google.com/file/d/0B073BLiAJ6_gSzYyR1lBMDlfaGc/view?resourcekey=0-bjzV-nJxf-bRCap9m8ZwEg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discrete Mathematics and its Applications&lt;/a&gt; (Sec. 1.1):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We will generally use parentheses to specify the order in which&#xA;logical operators in a compound proposition are to be applied...&#xA;However, to reduce the number of parentheses, we specify that the&#xA;negation operator is applied before all other logical operators....&#xA;Another general rule of precedence is that the conjunction operator&#xA;takes precedence over the disjunction operator... Finally, it is an&#xA;accepted rule that the conditional and biconditional operators... have&#xA;lower precedence than the conjunction and disjunction operators...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As an analog, teaching computer programming in a &lt;a href=&quot;https://en.cppreference.com/w/c/language/operator_precedence&quot; rel=&quot;nofollow noreferrer&quot;&gt;C-like language&lt;/a&gt;, there is necessarily a particular precedence applied by the compiler to translate any expression, and this precedence list is at least 15 levels deep. Again, the C-type language's precedence of the logical operators not-and-or are the same as outlined by Rosen above. It's likewise common to test students on their awareness of this fact.&lt;/p&gt;&#xA;&lt;p&gt;For a place where this knowledge is needed, I just grabbed the first book from my shelf in reach, Casella/Berger's &lt;a href=&quot;https://ia800905.us.archive.org/17/items/casella_berger_statistical_inference1/casella_berger_statistical_inference1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Statistical Inference&lt;/a&gt; (1990). On page 2 we see the first symbolic definition for &amp;quot;containment&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$A \subset B \iff x \in A \implies x \in B$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, which symbol creates the higher-precedence grouping, the logical equivalence, or the implication? If a student doesn't know, then they can't make sense of this definition, or any others.&lt;/p&gt;&#xA;&lt;p&gt;We'll take another example from, say, Manin's &lt;em&gt;A Course in Mathematical Logic&lt;/em&gt;. Manin expresses the Axiom of Choice this way (Sec. 2.4, p. 47):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\forall x (\lnot x = \emptyset \implies \exists y (\text {&quot;y is a function with domain of definition x&quot;} \land \forall u (u \in x \land \lnot u = \emptyset \implies \exists w (w \in u \land &quot;&amp;lt;u, w&amp;gt; \in y&quot;))))$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note in the second-innermost group in the latter half, there are symbols for logical and, not, and implication -- not separated by any parentheses, so the reader is expected to understand a certain precedence to parse this (and other definitions). In Sec. 1.2 (p. 8) he notes, &amp;quot;We shall not specify precisely when parentheses may be omitted; in any case, it must be possible to reinsert them in a way that is unique or is clear from the context without any special effort.&amp;quot; But in Sec. 1.3 he presents a formula that is fully, formally parenthesized, and gives the challenging exercise to find the open parenthesis corresponding with a particular closing parenthesis near the end (so demonstrating that expecting to fully parenthesize formulas is not viable).&lt;/p&gt;&#xA;&lt;p&gt;As a further analog from computer science, we can look to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Backus%E2%80%93Naur_form&quot; rel=&quot;nofollow noreferrer&quot;&gt;Backus-Naur form&lt;/a&gt; for specifying formal languages. In this notation, conjunction (indicated by juxtaposition) binds first, then disjunction (indicated by a vertical bar &amp;quot;|&amp;quot;), and then finally the replacement symbol (indicated by &amp;quot;::=&amp;quot;, or in other notations an implication-style arrow). Of course, that's the same precedence common to all the prior examples. Parentheses are not used to separate these groupings (we can look to the &lt;a href=&quot;https://cs.wmich.edu/%7Egupta/teaching/cs4850/sumII06/The%20syntax%20of%20C%20in%20Backus-Naur%20form.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;C syntax&lt;/a&gt; as an example). This notation is widely used to specify languages such as Java, LISP, SQL, XML, and so forth.&lt;/p&gt;&#xA;&lt;p&gt;The only way to confirm that students have digested these facts is to test them on it. Yes, one might say that the test questions are unrealistically convoluted. But the only other option is to just regurgitate one of the small number of previously-seen, very-simple, real-world examples (and everyone moans about rote memorization, right?).&lt;/p&gt;&#xA;&lt;p&gt;One may as well complain about why ball players do weightlifting (when weightlifting isn't in a ball game), or why musicians practice scales (when scale-playing isn't in any performance). Practice is practice, and we practice on artificially difficult tasks so that things are easy when they count in a more elaborate context.&lt;/p&gt;&#xA;&lt;p&gt;See also: &lt;a href=&quot;https://en.wikipedia.org/wiki/Overlearning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Overlearning&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastEditorUserId="5563" LastEditDate="2024-03-10T23:30:08.417" LastActivityDate="2024-03-10T23:30:08.417" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27569" PostTypeId="2" ParentId="27566" CreationDate="2024-03-07T15:12:06.027" Score="13" Body="&lt;p&gt;Suppose you have &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; crayons in your left hand and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; crayons in your right hand. All together, you have &lt;span class=&quot;math-container&quot;&gt;$a+b$&lt;/span&gt; crayons.&lt;/p&gt;&#xA;&lt;p&gt;Now you switch hands. You now have &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; crayons in your left hand and &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; crayons in your right hand. All together, you have &lt;span class=&quot;math-container&quot;&gt;$b+a$&lt;/span&gt; crayons.&lt;/p&gt;&#xA;&lt;p&gt;You haven't put any of the crayons down, so the number of crayons you have hasn't changed.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2024-03-07T15:12:06.027" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27570" PostTypeId="2" ParentId="27566" CreationDate="2024-03-07T16:25:53.900" Score="8" Body="&lt;p&gt;I would encourage you not to &amp;quot;teach&amp;quot; this (neither multiplication nor addition). Kids discover these things on their own, and it can be an exciting discovery for them.&lt;/p&gt;&#xA;&lt;p&gt;Designing activities that help them to see it is great.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2024-03-07T16:25:53.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27571" PostTypeId="2" ParentId="27566" CreationDate="2024-03-07T16:51:15.373" Score="10" Body="&lt;p&gt;It's pretty straightforward to demonstrate:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xz4tK.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xz4tK.png&quot; alt=&quot;a+b=b+a&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can also get the distributive law &lt;span class=&quot;math-container&quot;&gt;$(a+b)c=ac+bc$&lt;/span&gt; and others this way.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/s2q65.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/s2q65.png&quot; alt=&quot;(a+b)c=ac+bc&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm not directly involved with elementary ed, but my kids (USA, recent) were taught such explanations.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="4791" LastEditDate="2024-03-07T16:56:36.093" LastActivityDate="2024-03-07T16:56:36.093" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27572" PostTypeId="1" CreationDate="2024-03-07T17:11:31.073" Score="2" ViewCount="109" Body="&lt;p&gt;For educational purposes, I am seeking example problems in the realm of natural sciences, engineering, and business that satisfy the following criteria:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Consider a one-dimensional real function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; (potentially with some parameters).&lt;/li&gt;&#xA;&lt;li&gt;The problem should involve finding extrema of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;It should be meaningful in this context to employ the first or second derivative test to confirm that the critical points identified are indeed extrema and to determine their nature (maximum/minimum). Importantly, this determination should not be evident through other means beforehand.&lt;/li&gt;&#xA;&lt;li&gt;The calculations should be manageable manually, suitable for an advanced high school or freshmen level.&lt;/li&gt;&#xA;&lt;li&gt;The problem should represent a realistic scenario in the specified field (natural sciences, engineering, or business), avoiding unrealistic constructions.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1203" LastEditorUserId="2083" LastEditDate="2024-03-09T16:43:12.063" LastActivityDate="2024-03-10T22:01:47.500" Title="Real-World Problems for Teaching Extrema and Derivative Tests in STEM Education" Tags="|undergraduate-education|secondary-education|mathematical-analysis|applications|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27573" PostTypeId="2" ParentId="27556" CreationDate="2024-03-07T17:26:02.340" Score="2" Body="&lt;p&gt;I not only award partial credit when grading, but I actively encourage the students to &amp;quot;dispute&amp;quot; it and I guarantee that the grade won't be decreased after review. I've been applying this system for 8 years.&lt;/p&gt;&#xA;&lt;p&gt;Teaching doesn't end when the exam starts, and the exam is not only a tool for sorting students as either failing or passing. If the student fails, they will retake the exam, and they have to know how bad they did. A mark of 15% is different from a mark of 50% even though both result in the student failing the exam. Some students think they know the subject better than they really do. A very low mark tells them otherwise, and partial grading allows me to convey meaning within the not-pass range.&lt;/p&gt;&#xA;&lt;p&gt;Partial credit also allows me to distinguish between minor errors and fundamental misunderstandings of core concepts. For instance, a student may receive a better grade for a full page of work with minor mistakes than for page where a single line overlooks a crucial topic covered extensively in class. As a professor, it's important that the student knows which topics are crucial (well, I talk about that repeatedly during the course, but they put more attention after they failed the exam, and they put even more attention when they are trying to plea)&lt;/p&gt;&#xA;&lt;p&gt;In practice, many students will neither dispute the grade nor request clarifications. Those who do, usually request clarifications without disputing the grade. There are few cases where a student actively attempts to improve their grade, or even convert a fail to a pass.&lt;/p&gt;&#xA;&lt;p&gt;Very few students require an itemized detail of their grade (as in how many points were deducted by each mistake). If they ask for it, I'll deliver. Usually they stop me midway &amp;quot;Ok, I get it&amp;quot;. If they insist that the grade is not fair, it's always possible to ask another professor for an opinion (which, in my experience, is hardly even needed)&lt;/p&gt;&#xA;&lt;p&gt;If they honestly have no clue about why the mark was so low, or they know they did poorly and they're just trying to game the system, the answer is straighforward: &amp;quot;it was a crucial mistake, we talked a lot about that in the lecture of &lt;code&gt;&amp;lt;date&amp;gt;&lt;/code&gt;, see exercise  of the book.&amp;quot; and the &amp;quot;dispute&amp;quot; doesn't take long.&lt;/p&gt;&#xA;&lt;p&gt;If I cannot provide an explanation, or they present a convincing argument, then it's clear that it was &lt;em&gt;my&lt;/em&gt; mistake in one way or another.&lt;/p&gt;&#xA;" OwnerUserId="23815" LastActivityDate="2024-03-07T17:26:02.340" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27577" PostTypeId="1" AcceptedAnswerId="27581" CreationDate="2024-03-08T10:12:11.557" Score="6" ViewCount="229" Body="&lt;p&gt;Form &lt;a href=&quot;https://matheducators.stackexchange.com/q/17302/10349&quot;&gt;this question&lt;/a&gt;, I was surprised to learn that it is common for calculus textbooks in the US to allow &lt;span class=&quot;math-container&quot;&gt;$r&amp;lt;0$&lt;/span&gt; when discussing polar coordinates. &lt;a href=&quot;https://matheducators.stackexchange.com/a/17305/10349&quot;&gt;This answer&lt;/a&gt; by Dan Fox summarizes some mathematical and pedagogical drawbacks of this. Let me outline them and add some more:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If we restrict to &lt;span class=&quot;math-container&quot;&gt;$r\geq 0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$(r,\theta)$&lt;/span&gt; have a natural geometric meaning - the distance to the origin and the polar angle, the latter defined up to &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt; by necessity. Allowing &lt;span class=&quot;math-container&quot;&gt;$r&amp;lt;0$&lt;/span&gt; makes it much messier and necessarily de-emphasizes the important geometric picture.&lt;/li&gt;&#xA;&lt;li&gt;This aligns well with the view that coordinates are functions on the manifold, or, if you wish, that they are numbers assigned to physical points. Then, equations in polar coordinates, such as &lt;span class=&quot;math-container&quot;&gt;$r^2=\sin\theta$&lt;/span&gt;, simply describe the set of points which satisfy the equation when fed into it (with some choice of a branch for &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;), just as it is the case with equations in Carthesian coordinates.&lt;/li&gt;&#xA;&lt;li&gt;An alternative view, as &lt;a href=&quot;https://matheducators.stackexchange.com/questions/17302/symmetry-in-polar-functions-how-to-explain/17305?noredirect=1#comment74462_17305&quot;&gt;expressed in comments&lt;/a&gt;, is that polar coordinates is a map &lt;span class=&quot;math-container&quot;&gt;$P:(r,\theta)\mapsto (r\cos\theta,r\sin\theta)$&lt;/span&gt;, and the equation &lt;em&gt;denotes&lt;/em&gt; an image of the corresponding set in &lt;span class=&quot;math-container&quot;&gt;$(r,\theta)$&lt;/span&gt; plane. Apart from a need to re-define the meaning of an equation, in practice one is never interested in an image of a given set under &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; - rather, one is given a set &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; in the physical plane and looks for &lt;em&gt;a description&lt;/em&gt; of that set in polar coordinates, i.e., a set &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$P(U)=V$&lt;/span&gt;, e.g., to integrate in polar coordinates. In practice, this is nearly always done with the restriction &lt;span class=&quot;math-container&quot;&gt;$r\geq 0$&lt;/span&gt;, especially since bijectivity on &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; is desirable.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, the question is: what are the benefits of allowing &lt;span class=&quot;math-container&quot;&gt;$r&amp;lt;0$&lt;/span&gt;? If it is in the textbooks by historical reasons, what was the historical point of view that lead to it? What are some applications of polar coordinates where this is useful?&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2024-03-09T03:22:24.557" Title="Why do some (pre-) calculus text allow $r&lt;0$ in polar coordinates?" Tags="|precalculus|vector-calculus|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27578" PostTypeId="2" ParentId="27562" CreationDate="2024-03-08T13:04:03.810" Score="8" Body="&lt;p&gt;&lt;strong&gt;This shouldn't be taught, and those exercises are pointless.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is clearly no intrinsic value in introducing and memorizing precedence of operations. If there is a point, it either is that we would frequently encounter expressions involving so many parenthesis that parsing them becomes difficult otherwise; or it is that a particular convention regarding precedence is so common that without knowing it we struggle reading third parties' mathematics.&lt;/p&gt;&#xA;&lt;p&gt;When it comes to stuff like multiplication binding stronger than addition, both reasons apply. [However, there are still plenty of situations where just putting in a few more parenthesis just makes it easier to read.]&lt;/p&gt;&#xA;&lt;p&gt;However, no such reason applies to putting some precedence between &lt;span class=&quot;math-container&quot;&gt;$\leftrightarrow$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\rightarrow$&lt;/span&gt;. I'm a logician by trade, and I'm unaware of any claimed convention when it comes to these two. Authors I'm reading seem to manage just fine to use parenthesis or other signifiers to clarify their meaning without relying on such conventions.&lt;/p&gt;&#xA;" OwnerUserId="13471" LastActivityDate="2024-03-08T13:04:03.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27579" PostTypeId="2" ParentId="27577" CreationDate="2024-03-08T13:36:58.747" Score="8" Body="&lt;p&gt;It would be nice if we distinguished between &amp;quot;polar coordinates&amp;quot; and &amp;quot;polar parameterizations&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\Omega \subset \mathbb{R}^2$&lt;/span&gt; be an open set which does not contain any loop around the origin.  Then a choice of &lt;strong&gt;polar coordinates&lt;/strong&gt; for &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is a choice of smooth section of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;.  Such a section always exists and, if &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is connected, the section is completely determined up to a sign of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and a multiple of &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;.  You are correct that we never have polar coordinates where &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; changes sign.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, we can describe some subsets &lt;span class=&quot;math-container&quot;&gt;$S_{x,y} \subset \mathbb{R}^2$&lt;/span&gt; conveniently using &lt;strong&gt;polar parameterizations&lt;/strong&gt;.  A polar parameterization is just describing &lt;span class=&quot;math-container&quot;&gt;$S_{x,y}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$S_{x,y} = P(S_{r,\theta})$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$S_{r,\theta} \subset \mathbb{R}^2$&lt;/span&gt;.  Note that polar coordinates are polar parameterizations, but not all polar parameterizations are polar coordinates!&lt;/p&gt;&#xA;&lt;p&gt;When we write &amp;quot;polar equations&amp;quot; are are actually talking about &lt;strong&gt;polar parameterizations&lt;/strong&gt;, not polar coordinates. Let's look at the equation &lt;span class=&quot;math-container&quot;&gt;$r = \cos(\theta)$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$\theta \in [0,\pi]$&lt;/span&gt; to describe a circle.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/uuDvd.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uuDvd.png&quot; alt=&quot;parameterization of circle&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that the restriction of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; from the graph of &lt;span class=&quot;math-container&quot;&gt;$r = \cos(\theta)$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$\theta \in [0,\pi]$&lt;/span&gt;, is smooth through &lt;span class=&quot;math-container&quot;&gt;$r=0$&lt;/span&gt;!  This is as nice as it gets.&lt;/p&gt;&#xA;&lt;p&gt;In general if &lt;span class=&quot;math-container&quot;&gt;$f: (\theta_1, \theta_2) \to \mathbb{R}$&lt;/span&gt; is any smooth function, the map &lt;span class=&quot;math-container&quot;&gt;$\theta \mapsto (f(\theta),\theta) \mapsto P(f(\theta),\theta)$&lt;/span&gt; will also be smooth.  When we write &lt;span class=&quot;math-container&quot;&gt;$r = f(\theta)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\theta \in (\theta_1, \theta_2)$&lt;/span&gt; we are actually thinking about a &lt;em&gt;parameterized&lt;/em&gt; curve in the &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; plane.  We are not (as you claim in the OP) graphing the equation &lt;span class=&quot;math-container&quot;&gt;$r = f(\theta)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$(r,\theta)$&lt;/span&gt; are polar coordinates for some region containing the circle (in fact, no such polar coordinates exist since no region containing the origin has a smooth section).&lt;/p&gt;&#xA;&lt;p&gt;Usually you will not want to include a point where &lt;span class=&quot;math-container&quot;&gt;$r = 0$&lt;/span&gt; even in a parameterization of a 2D region, since &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is locally &amp;quot;two to one&amp;quot; on any punctured neighborhood of a point &lt;span class=&quot;math-container&quot;&gt;$(0,\theta_0)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, there are even circumstances when this is natural.  When I want to find the area of the disk bounded by &lt;span class=&quot;math-container&quot;&gt;$r = \cos(\theta)$&lt;/span&gt;, it is natural to use an integral from &lt;span class=&quot;math-container&quot;&gt;$\theta = 0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\theta = \pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tQR3Z.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tQR3Z.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is still one to one and nice enough that you can write the whole area of the disk as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_0^\pi \cos^2(\theta)\textrm{ d}\theta&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;with no issue (breaking it up from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}{2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}{2}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; gives two nice non-overlapping pieces).&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't usually think about the pre-image when I write down this integral:  I just think about cutting the disk into small pieces.&lt;/p&gt;&#xA;&lt;p&gt;While I agree that &lt;strong&gt;for many purposes&lt;/strong&gt; it is nice to restrict to &lt;span class=&quot;math-container&quot;&gt;$r&amp;gt;0$&lt;/span&gt; it is not always needed.&lt;/p&gt;&#xA;&lt;p&gt;I want to illustrate how a region can have &lt;strong&gt;polar coordinates&lt;/strong&gt; without restricting &lt;span class=&quot;math-container&quot;&gt;$\theta \in [0,2\pi)$&lt;/span&gt;.  For example, &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; restricts to a diffeomorphism between the rectangle &lt;span class=&quot;math-container&quot;&gt;$(0.5,3) \times (0,6\pi)$&lt;/span&gt; in the &lt;span class=&quot;math-container&quot;&gt;$(r,\theta)$&lt;/span&gt; plane and the region &lt;span class=&quot;math-container&quot;&gt;$ 0.5 + \theta &amp;lt; r &amp;lt; 3 + \theta$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$\theta \in (0,6\pi)$&lt;/span&gt;.  I would like to be able to work with this region without breaking it up into chunks which are artificially restricted to only go from &lt;span class=&quot;math-container&quot;&gt;$\theta = 0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\theta = 2\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jN4VFm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jN4VFm.png&quot; alt=&quot;spiral region&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I will also note that contending with these kinds of issues is bread and butter stuff in complex analysis.&lt;/p&gt;&#xA;&lt;p&gt;EDIT:  One more note.  The OP says that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This aligns well with the view that coordinates are functions on the manifold, or, if you wish, that they are numbers assigned to physical points. Then, equations in polar coordinates, such as &lt;span class=&quot;math-container&quot;&gt;$r^2 = \sin(\theta)$&lt;/span&gt;, simply describe the set of points which satisfy the equation when fed into it (with some choice of a branch for &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;&#xA;), just as it is the case with equations in Cartesian coordinates.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This point of view has a few serious drawbacks.&lt;/p&gt;&#xA;&lt;p&gt;First of all &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is &lt;strong&gt;not&lt;/strong&gt; a coordinate map.  It is locally but not globally.  Hence the need for a choice of the branch.&lt;/p&gt;&#xA;&lt;p&gt;More seriously, this interpretation of a &amp;quot;polar equation&amp;quot; never admits &lt;span class=&quot;math-container&quot;&gt;$r=0$&lt;/span&gt; as a solution.  Let's be explicit and attempt to define&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$r(x,y) = \sqrt{x^2 + y^2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\theta(x,y) = \operatorname{atan2}(x,y)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Atan2&quot; rel=&quot;nofollow noreferrer&quot;&gt;this definition&lt;/a&gt; of &lt;span class=&quot;math-container&quot;&gt;$\operatorname{atan2}(x,y)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Then according to OP definition a solution of &lt;span class=&quot;math-container&quot;&gt;$r^2 = \sin(\theta)$&lt;/span&gt; is a point &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; satisfying&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x^2 + y^2 = \sin(\operatorname{atan2}(x,y))&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$\operatorname{atan2}(0,0)$&lt;/span&gt; we must admit that &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt; is not a solution of &lt;span class=&quot;math-container&quot;&gt;$r^2 = \sin(\theta)$&lt;/span&gt; under the definition of the OP.&lt;/p&gt;&#xA;&lt;p&gt;This is not to say that this is a poor definition:  it is good for some purposes and not for others.&lt;/p&gt;&#xA;" OwnerUserId="117" LastEditorUserId="117" LastEditDate="2024-03-08T17:51:56.067" LastActivityDate="2024-03-08T17:51:56.067" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="27580" PostTypeId="2" ParentId="27577" CreationDate="2024-03-08T13:53:29.770" Score="3" Body="&lt;p&gt;Polar curves are a very old subject. This article, published in 1949 in the American Mathematical Monthly, suggests that the subject goes back to Newton. &lt;a href=&quot;https://www.jstor.org/stable/2306162&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton as an Originator of Polar Coordinates&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In this framework, we represent curves in the plane by the scheme &lt;span class=&quot;math-container&quot;&gt;$R=f(\theta)$&lt;/span&gt;. There is simplicity and elegance that would not be present if we decreed that the polar variable &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; must be positive.&lt;/p&gt;&#xA;&lt;p&gt;A simple example is the curve &lt;span class=&quot;math-container&quot;&gt;$R=\cos(\theta)$&lt;/span&gt;. If there is a rule forbidding &lt;span class=&quot;math-container&quot;&gt;$R&amp;lt;0$&lt;/span&gt; then we are stuck with a semicircle. But without the restriction, the graph is the entire circle, traversed for &lt;span class=&quot;math-container&quot;&gt;$\theta\in[0,\pi]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Another example is the equation of a conic in polar coordinates. A very general form is &lt;span class=&quot;math-container&quot;&gt;$$R=\frac{l}{1-e\cos(\theta)}$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is the eccentricity. The equation unifies conic sections. But alas, if we had to live under a regime where &lt;span class=&quot;math-container&quot;&gt;$R&amp;lt;0$&lt;/span&gt; is forbidden, the hyperbolas would be missing one of their branches.&lt;/p&gt;&#xA;&lt;p&gt;It is also amusing to think about what would happen to curves like the four-leaf rose &lt;span class=&quot;math-container&quot;&gt;$R=\cos(2\theta)$&lt;/span&gt;. I suppose we would have to rename it, since it would have just two lobes. Even then it is awkward because you have to wait and look the other way while &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; progresses through the intervals where &lt;span class=&quot;math-container&quot;&gt;$\cos(2\theta)&amp;lt;0$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastActivityDate="2024-03-08T13:53:29.770" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27581" PostTypeId="2" ParentId="27577" CreationDate="2024-03-08T14:44:24.167" Score="10" Body="&lt;p&gt;There is still a simple geometric meaning for polar coordinates when we allow for negative values of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;. (It's how I was taught polar coordinates.) Given an ordered pair &lt;span class=&quot;math-container&quot;&gt;$(r, \theta)$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Start at the origin facing the polar axis.&lt;/li&gt;&#xA;&lt;li&gt;Rotate by the &lt;em&gt;directed angle&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;. (Counterclockwise if &lt;span class=&quot;math-container&quot;&gt;$\theta&amp;gt;0$&lt;/span&gt; and clockwise if &lt;span class=&quot;math-container&quot;&gt;$\theta&amp;lt;0$&lt;/span&gt;.)&lt;/li&gt;&#xA;&lt;li&gt;Travel the &lt;em&gt;directed distance&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;. (Forwards if &lt;span class=&quot;math-container&quot;&gt;$r&amp;gt;0$&lt;/span&gt; and backwards if &lt;span class=&quot;math-container&quot;&gt;$r&amp;lt;0$&lt;/span&gt;.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This does not fit the notion of a &lt;em&gt;coordinate chart&lt;/em&gt;, which should assign to each point a unique list of coordinates. But it makes perfect sense as a &lt;em&gt;coordinate transformation&lt;/em&gt;, which assigns to each list of coordinates a point. If we restrict &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; to the domain &lt;span class=&quot;math-container&quot;&gt;$[0, \infty) \times [0, 2\pi)$&lt;/span&gt;, then we have an (almost) bijection that can be used to define a coordinate chart. But it is sometimes more convenient to restrict to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}\times(-\frac{\pi}{2},\frac{\pi}{2}]$&lt;/span&gt;. And when we don't need a coordinate chart and bijectivity isn't important, why not allow for the domain to be &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure what you mean by needing to &amp;quot;re-define the meaning of an equation.&amp;quot; I was taught that an equation is an open statement of equality involving some number of free variables. The solutions of an equation, given an ordering of the variables, are lists of values that satisfy the equation. To produce a graph of an equation, we interpret the solutions as coordinates, then map coordinates to points.&lt;/p&gt;&#xA;&lt;p&gt;In addition to the examples of circles, spirals, hyperbolas, and roses others have already given, there are many more interesting and beautiful curves generated by simple polar equations that rely on negative values of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and/or values of &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; outside of &lt;span class=&quot;math-container&quot;&gt;$[0, 2\pi)$&lt;/span&gt;. For example, the graph of &lt;span class=&quot;math-container&quot;&gt;$r=\frac{1}{2}+\cos(\theta)$&lt;/span&gt; is a limaçon with an inner loop.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Tvswjm.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Tvswjm.png&quot; alt=&quot;Graph of r=1/2 + cos(theta), a limaçon with an inner loop.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here's another situation where it makes perfect sense to use negative values of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;. Consider a particle traveling down the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis whose position in Cartesian coordinates is described by &lt;span class=&quot;math-container&quot;&gt;$\mathbf{r}_\text{C}(t)=(0, 1-t)$&lt;/span&gt;. Its position in polar coordinates can be described simply as &lt;span class=&quot;math-container&quot;&gt;$\mathbf{r}_\text{P}(t)=(1-t,\frac{\pi}{2})$&lt;/span&gt;. This matches our sense that the particle never &amp;quot;changes direction,&amp;quot; even while passing through the origin. But if negative values of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; are not allowed, the parametrization must be discontinuous at &lt;span class=&quot;math-container&quot;&gt;$t=1$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$\mathbf{r}_\text{P}(t)=(1-t,\frac{\pi}{2})$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$t\leq1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{r}_\text{P}(t)=(t-1,\frac{3\pi}{2})$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$t&amp;gt;1$&lt;/span&gt;. The same applies to almost any other smooth path through the origin.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2024-03-08T14:57:13.727" LastActivityDate="2024-03-08T14:57:13.727" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27582" PostTypeId="2" ParentId="27577" CreationDate="2024-03-08T15:24:08.353" Score="1" Body="&lt;p&gt;There is a clear interpretation of negative angles, and we allow angles to be negative. There is a clear interpretation of negative radii, so it's only natural (or at least &amp;quot;consistent&amp;quot;) to do the same and allow radii to be negative.&lt;/p&gt;&#xA;&lt;p&gt;Sure, there are some more advanced contexts where that causes an issue. But we can resolve those issues when we get to them (by introducing the restriction that radii are to be non-negative in those contexts).&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-03-09T03:22:24.557" LastActivityDate="2024-03-09T03:22:24.557" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27583" PostTypeId="2" ParentId="27577" CreationDate="2024-03-08T15:30:01.720" Score="4" Body="&lt;p&gt;Allowing &lt;span class=&quot;math-container&quot;&gt;$r&amp;lt;0$&lt;/span&gt; certainly does not de-emphasize the geometric picture. The image below is the graph of &lt;span class=&quot;math-container&quot;&gt;$r=\sin(3\theta).$&lt;/span&gt; As &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; goes from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\pi/3,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$3\theta$&lt;/span&gt; goes from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and back down to &lt;span class=&quot;math-container&quot;&gt;$0,$&lt;/span&gt;  &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; goes from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and then back down to &lt;span class=&quot;math-container&quot;&gt;$0,$&lt;/span&gt; and you see the part of this figure that is in the first quadrant. It lies below the line &lt;span class=&quot;math-container&quot;&gt;$\theta=\pi/3,$&lt;/span&gt; the &lt;span class=&quot;math-container&quot;&gt;$60^\circ$&lt;/span&gt; ray. As &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; goes on from &lt;span class=&quot;math-container&quot;&gt;$\pi/3$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$2\pi/3,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; goes from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; and back up to &lt;span class=&quot;math-container&quot;&gt;$0,$&lt;/span&gt; giving us the part of this picture that is below the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis. The third leaf, in the second quadrant, has &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; going from &lt;span class=&quot;math-container&quot;&gt;$2\pi/3$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\pi.$&lt;/span&gt; Then as &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; goes from &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$4\pi/3,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is negative again and we traverse the first leaf, in the first quadrant, for the second time. And so on. Negative values of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; have a natural geometric meaning, just as positive values do.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/n9Glc.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/n9Glc.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2024-03-08T15:30:01.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27584" PostTypeId="1" AcceptedAnswerId="27585" CreationDate="2024-03-08T19:00:22.457" Score="3" ViewCount="72" Body="&lt;p&gt;Does anyone know what is the exact program of the International Baccalaureate in math?&lt;/p&gt;&#xA;&lt;p&gt;I've been looking for the MYP and DP programs in the website of the International Baccalaureate Institute but could not find any detailed program.&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance!&lt;/p&gt;&#xA;&lt;p&gt;Edit: I realize one cannot access these documents unless you are a teacher in an IB school. However, can anyone tell me if there are Markov chains, as well as number theory (Fermat's theorem) in these programs ?&lt;/p&gt;&#xA;" OwnerUserId="22167" LastEditorUserId="22167" LastEditDate="2024-03-08T20:22:26.560" LastActivityDate="2024-03-08T23:32:18.353" Title="International Baccalaureate - where to find the detail of the math programs?" Tags="|secondary-education|middle-school|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27585" PostTypeId="2" ParentId="27584" CreationDate="2024-03-08T23:26:16.270" Score="2" Body="&lt;p&gt;As of 2024, Mathematics: Applications and Interpretation does cover Markov chains, but at the higher level only, not at the standard level. It is listed in the syllabus under subtopic AHL 4.19. Mathematics: Analysis and Approaches does not cover Markov chains at the standard level or at the higher level.&lt;/p&gt;&#xA;&lt;p&gt;Neither of the two mathematics courses currently offered in the IB Diploma Programme cover number theory. Prior to 2021, there was some number theory, up to Fermat's little theorem. It was a part of the discrete math topic, which was an option topic for Mathematics HL and a required topic for Further Mathematics HL.&lt;/p&gt;&#xA;" OwnerUserId="20719" LastEditorUserId="20719" LastEditDate="2024-03-08T23:32:18.353" LastActivityDate="2024-03-08T23:32:18.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27586" PostTypeId="2" ParentId="27572" CreationDate="2024-03-09T18:04:27.363" Score="1" Body="&lt;p&gt;A Google search showed several that I bet won't satisfy you (fence enclosing, box making).  I think you'd have to trawl through a lot of textbooks and look at end of chapter exercises and extract the few decent ones you find.&lt;/p&gt;&#xA;&lt;p&gt;A classic example that is (I think) reasonably realistic is revenue maximizing for a monopolist (assume linear elasticity of demand, although you could have a higher function if you want).  In addition to &amp;quot;working&amp;quot; and being reasonably real world (at least to being used in econ classes, not just math), I like that two competing factors are sort of intuitive and opposed.  Higher price gets you more per purchase, but less purchases.&lt;/p&gt;&#xA;&lt;p&gt;P.s.  See similar question asked here, before (no killer compendium found):  &lt;a href=&quot;https://matheducators.stackexchange.com/questions/1550/optimization-problems-that-todays-students-might-actually-encounter&quot;&gt;Optimization problems that today&amp;#39;s students might actually encounter?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="23808" LastEditorUserId="23808" LastEditDate="2024-03-09T18:09:14.960" LastActivityDate="2024-03-09T18:09:14.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27587" PostTypeId="2" ParentId="27556" CreationDate="2024-03-09T18:20:54.883" Score="2" Body="&lt;p&gt;There are lots of good answers already, but I'll add one thing that works from my experience: add points, don't subtract them.&lt;/p&gt;&#xA;&lt;p&gt;Polya's principles include &lt;em&gt;understand the problem&lt;/em&gt;, &lt;em&gt;devise a plan&lt;/em&gt;, and then &lt;em&gt;carry out the plan&lt;/em&gt; (the fourth, &lt;em&gt;look back&lt;/em&gt;, doesn't necessarily apply to summative exam questions). So I'll award ~30% of the credit for evidence that they understood the problem. For instance, a decent picture, or writing facts or theorems that are relevant. Then ~40% for carrying that understanding through to a strategy for solution, for instance, an integral that needs to be evaluated. Then final ~30% is for carrying out that strategy.&lt;/p&gt;&#xA;&lt;p&gt;These criteria aren't independent, of course. A flawed strategy for setting up an integral may lead to on that can't be evaluated in elementary terms. But scoring with these benchmarks in mind shifts the discussion from “How many points did you lose for doing things wrong?” to “How many points did you earn for doing things right?”&lt;/p&gt;&#xA;" OwnerUserId="446" LastActivityDate="2024-03-09T18:20:54.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27588" PostTypeId="1" CreationDate="2024-03-10T11:40:27.823" Score="6" ViewCount="323" Body="&lt;p&gt;Terence Tao says the following in the preface to his book Analysis I:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;With regard to examinations for a course based on this text, I would recommend either an open-book, open-notes examination with problems similar to the exercises given in the text (but perhaps shorter, with no unusual trickery involved), or else a take home examination that involves problems comparable to the more intricate exercises in the text. The subject matter is too vast to force the students to memorize the definitions and theorems, so I would not recommend a closed-book examination, or an examination based on regurgitating extracts from the book. (Indeed, in my own examinations I gave a supplemental sheet listing the key definitions and theorems which were relevant to the examination problems.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Based on Tao's comments, we can separate the most common types of examination into the following categories:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Closed-book/closed-notes examination without supplemental sheet.&lt;/li&gt;&#xA;&lt;li&gt;Closed-book/closed-notes examination with supplemental sheet.&lt;/li&gt;&#xA;&lt;li&gt;Open-book/open-notes examination.&lt;/li&gt;&#xA;&lt;li&gt;Home examination.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;strong&gt;Based on your experience (or on sources on the subject that you know), in the context of real analysis examination:&lt;br /&gt;&#xA;(a) What is the best category?&lt;br /&gt;&#xA;(b) What are the pros and cons?&lt;br /&gt;&#xA;(c) How similar should the exam questions be to the homework questions?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you use a different category from those listed above, feel free to comment.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Addendum:&lt;/em&gt; I'm talking about an introductory course that covers the basics of natural and real numbers, convergence of sequences and series, limits of functions, continuity, derivatives, integrals, sequences and series of functions.&lt;/p&gt;&#xA;" OwnerUserId="13585" LastEditorUserId="13585" LastEditDate="2024-03-11T23:24:09.630" LastActivityDate="2024-03-29T03:40:56.153" Title="How to assess students in real analysis?" Tags="|undergraduate-education|exams|mathematical-analysis|assessment|" AnswerCount="1" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="27589" PostTypeId="2" ParentId="27572" CreationDate="2024-03-10T22:01:47.500" Score="-2" Body="&lt;p&gt;FThe Problem:&#xA;Given a specific volume, find the dimensions of a cylinder that minimize its surface area. This optimization problem is crucial in designing objects like Le Grand K, where the goal is to reduce air exposure and thereby slow the material's deterioration over time. For Le Grand K, a crucial design feature is that its height equals its diameter.&lt;/p&gt;&#xA;&lt;p&gt;Why This Shape?&#xA;Choosing a cylinder with height equal to its diameter optimizes the ratio of volume to surface area, a principle critical in the design of Le Grand K. This geometric optimization reduces the cylinder's exposure to air, minimizing the oxidation and wear over time, thus preserving its mass and, by extension, the standard of the kilogram.&lt;/p&gt;&#xA;&lt;p&gt;Conclusion:&#xA;This approach highlights the intersection of mathematical optimization and practical application in material science and metrology. By mathematically determining the dimensions that minimize surface area for a given volume, we can design objects like Le Grand K to ensure longevity and stability in physical standards, which are foundational to scientific measurement and research.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eaX8Q.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eaX8Q.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2024-03-10T22:01:47.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27591" PostTypeId="2" ParentId="27530" CreationDate="2024-03-10T22:15:33.687" Score="-1" Body="&lt;p&gt;TTo enhance the understanding and teaching of symmetry and invariants, employing transparency sheets can be a highly effective and interactive method. This technique not only visually demonstrates these concepts but also engages learners in hands-on exploration.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EOUUm.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EOUUm.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;TThis method not only visualizes the concept but also allows learners to physically engage with the material, facilitating a deeper understanding through embodied learning.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2024-03-10T22:15:33.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27592" PostTypeId="2" ParentId="13438" CreationDate="2024-03-11T08:33:53.970" Score="1" Body="&lt;p&gt;I think the substance of the question is about teaching strategies (the choice of which tactics to use).&lt;/p&gt;&#xA;&lt;p&gt;Most problems in mathematics can be solved in several ways and development of strategic flexibility, the ability to choose a reasonable tactic for a given task, is pretty important. Difficulties in mathematics are connected to a narrow scope of tactics and poor strategy in choosing them. [1]&lt;/p&gt;&#xA;&lt;p&gt;A recommended tool is sharing strategies. Groupwork, as Sue VanHattum suggests, is good, especially is the teacher observes and then asks particular solutions to be shown publicly, preferably in a well-considered order. This accomplishes several things:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Students often explain their solutions in a different manner than the teacher does or would. This can be easier for other students to understand, and can especially be a good complement to the teacher explaining the optimal or correct way of doing things.&lt;/li&gt;&#xA;&lt;li&gt;Different solutions bring out different mathematical ideas. The student's solution shows how factorization can be used, whereas I noticed that &lt;span class=&quot;math-container&quot;&gt;$x-2 = \pm 3$&lt;/span&gt; and that could have lead to meeting absolute values or the non-ijectivity of second power in some guise.&lt;/li&gt;&#xA;&lt;li&gt;Different ways of solving a given problem is one place where mathematical thinking and creativity show up, and reinforcing the existence of multiple different and valid strategies also reinforces this idea. In particular, this fights against the idea that mathematics is an arbitrary collection of algorithms to memorize, where each task is always coupled with an specific algorithm that one is supposed to use there.&lt;/li&gt;&#xA;&lt;li&gt;Knowing that there are many valid strategies for finding a solution can help with or mitigate fear of mathematics, as it makes it easier to be correct in different ways.&lt;/li&gt;&#xA;&lt;li&gt;Having students compare the different strategies, or thinking up examples where one of them is better than the others, are fine mathematical activities in and of themselves.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This all suggests that the question should be reframed from &amp;quot;when and if I should correct&amp;quot; to &amp;quot;how to structure the teaching such that students get to share their strategies and get to see, discuss and compare each others' strategies&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;[1] OSTAD, Snorre A. Strategier, strategiobservasjon og strategiopplæring. Fokus på elever med matematikkvansker. Trondheim: Læreboka forlag, 2013.&lt;/p&gt;&#xA;" OwnerUserId="2083" LastActivityDate="2024-03-11T08:33:53.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27594" PostTypeId="1" AcceptedAnswerId="27599" CreationDate="2024-03-12T04:34:05.587" Score="6" ViewCount="1171" Body="&lt;p&gt;why write a conic in this form?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;for any real A, B, C, D, E, and F, a conic may be expressed as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$Ax^2 + Bxy +Cy^2+Dx+Ey+F=0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;why wouldn't it be more natural to write the conic in standard form with some kind of center &lt;span class=&quot;math-container&quot;&gt;$(h, k)$&lt;/span&gt; and all of its other features encoded into &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Why go through the trouble of teaching students to go from the general form to the standard form? Expanding it out from standard form to general form is also a very lengthy process but why might someone want that?&lt;/p&gt;&#xA;&lt;p&gt;(I don't have an issue with completing the square)&lt;/p&gt;&#xA;&lt;p&gt;Is it because of the study of second-order differential equations?&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i0.wp.com/cfdisraelblog.com/wp-content/uploads/2023/01/d8efb-image-1.png?resize=650%2C207&amp;amp;ssl=1&quot; alt=&quot;Text&quot; /&gt;&lt;/p&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="10335" LastEditDate="2024-03-12T22:49:44.183" LastActivityDate="2024-03-16T13:26:49.060" Title="why or when do we use the general form of conic sections?" Tags="|algebra|analytic-geometry|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27596" PostTypeId="2" ParentId="27594" CreationDate="2024-03-12T12:55:07.213" Score="11" Body="&lt;p&gt;The point is to start with the general quadratic equation in two variables, and show that the so-called &amp;quot;non-degenerate&amp;quot; cases lead to the three types of curves (ellipse, parabola, hyperbola).  Note that not all solutions of the general quadratic equation are literally conic sections, i.e., not all are identifiable with the intersection of some plane with a standard cone in 3-space.  For example, a pair of parallel lines can arise as the solution of a quadratic equation, but cannot arise literally as a planar section of a cone.  The same goes for the empty set (a literal conic section is always nonempty).&lt;/p&gt;&#xA;" OwnerUserId="1385" LastEditorUserId="60" LastEditDate="2024-03-13T03:37:22.860" LastActivityDate="2024-03-13T03:37:22.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27597" PostTypeId="2" ParentId="27594" CreationDate="2024-03-12T21:57:58.940" Score="2" Body="&lt;p&gt;Before you start manipulating the general form of a conic equation you should be able to recognize whether it is a circle, ellipse, parabola or hyperbola.&lt;/p&gt;&#xA;&lt;p&gt;In standard form, the two coefficients to examine are A and C.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For circles, &lt;span class=&quot;math-container&quot;&gt;$ B^2 - 4AC &amp;lt; 0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A=C$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For ellipses, &lt;span class=&quot;math-container&quot;&gt;$ B^2 - 4AC &amp;lt; 0 $&lt;/span&gt; and either &lt;span class=&quot;math-container&quot;&gt;$B \neq 0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$A \neq C$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For parabolas, &lt;span class=&quot;math-container&quot;&gt;$B^2 - 4AC = 0$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;For hyperbolas, &lt;span class=&quot;math-container&quot;&gt;$ B^2 - 4AC &amp;gt; 0$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10335" LastEditorUserId="10335" LastEditDate="2024-03-14T01:53:44.047" LastActivityDate="2024-03-14T01:53:44.047" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27598" PostTypeId="2" ParentId="27594" CreationDate="2024-03-13T04:29:35.693" Score="4" Body="&lt;p&gt;&lt;strong&gt;Because the general equation and the standard equation may arise in different contexts&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Context 1: no a priori coordinate system&lt;/em&gt;: for example, you are given a conic section either as a locus of points (e.g. foci and length of axes for an ellipse) or a real-life problem (parabolic bridge, satellite dish, etc) that you want to study.&lt;/p&gt;&#xA;&lt;p&gt;In that case, you are free to set a coordinate system (with perpendicular axes and the same unit on each axis) the way you prefer. In that case, it is a good choice to take one axis parallel (or perpendicular) to the major/transverse/symmetry axis of the conic. In that case, the formulas for distances and a bit of algebra lead to the standard equations.&lt;/p&gt;&#xA;&lt;p&gt;(Good idea to also place the origin at the center of the ellipse/hyperbola or the vertex of the parabola).&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Context 2: a coordinate system is given:&lt;/em&gt; For example, a curve is given by an equation in the plane, and you need to identify it, or study it, or sketch it. In that case, if the equation is equivalent to a quadratic equation, there is no reason for this equation to be in standard form. We need knowledge on the general equation of conic sections to deal with these situations.&lt;/p&gt;&#xA;&lt;p&gt;Examples:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The graph of functions &lt;span class=&quot;math-container&quot;&gt;$f(x)=\frac{ax+b}{cx+d}$&lt;/span&gt; is a hyperbola when &lt;span class=&quot;math-container&quot;&gt;$c\neq 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ad-bc\neq 0$&lt;/span&gt;. I introduce the rotation of axes by considering the graph of &lt;span class=&quot;math-container&quot;&gt;$y=\frac{1}{x}$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$\iff xy=1$&lt;/span&gt;) and discussing with students how to show it is really a hyperbola.&lt;/li&gt;&#xA;&lt;li&gt;The graph of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{x}+\sqrt{y}=1$&lt;/span&gt; is part of a parabola.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1538" LastActivityDate="2024-03-13T04:29:35.693" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27599" PostTypeId="2" ParentId="27594" CreationDate="2024-03-13T21:29:51.977" Score="7" Body="&lt;p&gt;What general form do you propose instead? You seem to think there is a better one, but I think probably the one you are thinking of is not able to encode all the conics.&lt;/p&gt;&#xA;&lt;p&gt;I'm guessing based on your question, but you are probably looking for something like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\pm \frac{(x-h)^2}{a^2} \pm \frac{(y-k)^2}{b^2} = 1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;as a candidate general form instead.&lt;/p&gt;&#xA;&lt;p&gt;However, many simple conic sections cannot be written that way, like for example the hyperbola&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$xy = 1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;has no &lt;span class=&quot;math-container&quot;&gt;$h, k, a, b$&lt;/span&gt; that would express it in the proposed form.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KcSfV.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KcSfV.png&quot; alt=&quot;A graph of &lt;span class=&quot;math-container&quot;&gt;$xy=1$&lt;/span&gt;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2024-03-13T21:29:51.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27600" PostTypeId="1" AcceptedAnswerId="27603" CreationDate="2024-03-14T18:11:53.053" Score="4" ViewCount="123" Body="&lt;p&gt;When teaching an upper-level proof based course, what criteria do you use to determine which and how many problems to assign?  And, as a corollary question: How do you determine the level of success on problems indicating success in the course?&lt;/p&gt;&#xA;&lt;p&gt;In an advanced course, problems are intended to be hard.  A student may struggle, even fail, to do them, yet still master the course:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A word about the problems.  There are a great number of them. It would be an extraordinary student indeed who could solve them all.... Many are introduced not so much to be solved as to be tackled (Herstein).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The homework problems are the most important part of the course. Many of the problems are extensive and difficult, and require hard work. They are not meant to be completed in one sitting. (&lt;a href=&quot;https://ocw.mit.edu/courses/18-701-algebra-i-fall-2010/pages/syllabus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Artin&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It's interesting to note that when teaching a course from his own textbook, &lt;a href=&quot;https://ocw.mit.edu/courses/18-701-algebra-i-fall-2010/pages/readings/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Professor Artin himself&lt;/a&gt; only assigns about 15% of the problems in the book.&lt;/p&gt;&#xA;&lt;p&gt;How do you determine &lt;em&gt;which&lt;/em&gt; problems to assign? And how do you determine how &lt;em&gt;many&lt;/em&gt;? Is it simply a matter of taste (&amp;quot;this one was interesting&amp;quot;)? Do you try to ensure that each subtopic has one problem? How many is too much?&lt;/p&gt;&#xA;&lt;p&gt;And, given the vast range in difficulty (from quick and easy to bordering on research), are there multiple dimension to the criteria? For example:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Ensure that every concept, definition, theorem, or technique appears in at least one problem&lt;/li&gt;&#xA;&lt;li&gt;Have at least one problem that merely tests comprehension, one that requires insight or recognition of the true meaning of problems (in different contexts), and at least one that is too hard to be solved.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Or is it budget based: I need to cover X sections in Y weeks, and can assign Z hours of problem sets per week.&lt;/p&gt;&#xA;&lt;p&gt;What criteria do you use to say &amp;quot;This course has enough problems.  If you do them, you can master the material.&amp;quot;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h1&gt;Update &amp;amp; Clarification&lt;/h1&gt;&#xA;&lt;p&gt;To refine the scope of the question and make it easier to answer:&lt;/p&gt;&#xA;&lt;p&gt;This question does &lt;em&gt;not&lt;/em&gt; seek a &lt;em&gt;universal formula&lt;/em&gt;, but rather &lt;em&gt;individual data points&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Obviously, what's appropriate for one setting is not appropriate for another.  That doesn't imply, however, that it's impossible to learn from experience (your own or others).  We can collect individual data and learn from it; learning from data is &lt;em&gt;not&lt;/em&gt; mimicry.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, an answer should not be &amp;quot;Here is a universal formula for these criteria&amp;quot; but rather &amp;quot;In setting X, I used criteria Y, and got results Z&amp;quot;.  Extrapolation and generalization are the job of the &lt;em&gt;reader&lt;/em&gt; of the answer, not the poster of the answer.&lt;/p&gt;&#xA;" OwnerUserId="23837" LastEditorUserId="23837" LastEditDate="2024-03-15T11:23:10.163" LastActivityDate="2024-03-15T14:36:23.337" Title="When teaching an upper-level proof based course, what criteria do you use to determine which and how many problems to assign?" Tags="|proofs|course-design|homework|problem-design|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="27601" PostTypeId="2" ParentId="27594" CreationDate="2024-03-14T21:20:27.823" Score="4" Body="&lt;p&gt;A major advantage to writing conic sections in standard form is the ease with which you can apply techniques like &lt;strong&gt;Joachimsthal's notation&lt;/strong&gt; to solve problems such as finding tangents. In fact this is selling Joachimsthal short: you can do a &lt;em&gt;lot&lt;/em&gt; more than finding tangents, and if you've not heard of it then it's worth reading &lt;a href=&quot;https://www.cut-the-knot.org/Generalization/JoachimsthalsNotations.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article in &lt;em&gt;Cut-The-Knot&lt;/em&gt;&lt;/a&gt; and &lt;a href=&quot;https://www.maths.gla.ac.uk/wws/cabripages/formula.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this more technical exposition by Wilson Stothers&lt;/a&gt;. This area used to be a major topic in the British A-level syllabus, and &lt;a href=&quot;https://math.stackexchange.com/questions/1176733/fast-method-to-find-the-tangent-line-to-a-conic-section-why-does-it-work/&quot;&gt;this Math SE question&lt;/a&gt; suggests it is still taught at high school level in the Netherlands. Even if this is a bag of tricks you're unfamiliar with, the standard form of conics also makes it straightforward to apply implicit differentiation. In fact in current British A-level textbooks, the chapter on implicit differentiation is where you'd come across finding tangents and normals to conic sections. I'll quote you some of the standard results about circles given in a 1980s A-level textbook, which still gives a sense of the &amp;quot;old skool&amp;quot; way students were taught to do things without resorting to calculus: but even by this time, many conics topics had been removed from the syllabus.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The &lt;strong&gt;general equation of a circle&lt;/strong&gt; is &lt;span class=&quot;math-container&quot;&gt;$$x^2 + y^2 + 2gx + 2fy + c = 0$$&lt;/span&gt; The circle has centre &lt;span class=&quot;math-container&quot;&gt;$(-g, -f)$&lt;/span&gt; and radius &lt;span class=&quot;math-container&quot;&gt;$\sqrt{g^2 + f^2 - c}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;strong&gt;equation of the circle with &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; as diameter&lt;/strong&gt; where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$(x_1, y_1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$(x_2, y_2)$&lt;/span&gt; is given by &lt;span class=&quot;math-container&quot;&gt;$$(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;To find the &lt;strong&gt;equation of a circle passing through three points&lt;/strong&gt;, substitute the coordinates of each point in turn into the general equation, solve the three simultaneous equations for the values of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, then substitute these into the general equation.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;strong&gt;equation of the tangent&lt;/strong&gt; at &lt;span class=&quot;math-container&quot;&gt;$(x_1, y_1)$&lt;/span&gt; to the circle &lt;span class=&quot;math-container&quot;&gt;$x^2 + y^2 + 2gx + 2fy + c = 0$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$$xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$$&lt;/span&gt; Note the relation between the equations: &lt;span class=&quot;math-container&quot;&gt;$x^2 \to xx_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y^2 \to yy_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2x \to (x+x_1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2y \to (y+y_1)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;strong&gt;length of the tangent&lt;/strong&gt; from &lt;span class=&quot;math-container&quot;&gt;$(x_1, y_1)$&lt;/span&gt; to the circle &lt;span class=&quot;math-container&quot;&gt;$x^2 + y^2 + 2gx + 2fy + c = 0$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$$\sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;The &lt;strong&gt;equation of any circle through the intersection of two circles&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$&lt;/span&gt; is given by an equation of the form &lt;span class=&quot;math-container&quot;&gt;$$(x^2 + y^2 + 2g_1x + 2f_1y + c_1) + \lambda(x^2 + y^2 + 2g_2x + 2f_2y + c_2) = 0$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; is a constant that is usually found from the given conditions. If &lt;span class=&quot;math-container&quot;&gt;$\lambda = -1$&lt;/span&gt;, the equation reduces to a straight line. This is the &lt;strong&gt;common chord&lt;/strong&gt; of the two circles and is known as their &lt;strong&gt;radical axis&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The above makes heavy use of the standard form of the equation for a &lt;em&gt;circle&lt;/em&gt;, but a lot of that material will generalise to other conics. For example, using the notation at &lt;em&gt;Cut-The-Knot&lt;/em&gt;, say we have the general conic&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ Ax^2 + 2Bxy + Cy^2 + 2Fx + 2Gy + H=0 \tag{1}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;One advantage of this form is we can (as Stothers does) write our equation succinctly in matrix form as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \begin{bmatrix}x &amp;amp; y &amp;amp; 1 \end{bmatrix} \begin{bmatrix}A &amp;amp; B &amp;amp; F \\ B &amp;amp; C &amp;amp; G \\ F &amp;amp; G &amp;amp; H \end{bmatrix} \begin{bmatrix}x \\ y \\ 1 \end{bmatrix} = 0 \tag{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;or simply &lt;span class=&quot;math-container&quot;&gt;$\mathbf{x'Mx} = 0$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\mathbf{M}$&lt;/span&gt; is a symmetric 3-by-3 matrix. If the coordinates vector &lt;span class=&quot;math-container&quot;&gt;$\mathbf{x}$&lt;/span&gt; looks suspiciously like it should have a &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; but we have for some reason set &lt;span class=&quot;math-container&quot;&gt;$z=1$&lt;/span&gt;, then welcome to the realm of &lt;a href=&quot;https://www.maths.gla.ac.uk/wws/cabripages/homogeneous.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;homogeneous coordinates&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Projective_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;projective geometry&lt;/a&gt; — this setting unifies the treatment of conics and makes sense of many of their features, such as &lt;a href=&quot;https://www.maths.gla.ac.uk/wws/cabripages/tangents.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;tangents and asymptotes&lt;/a&gt;, and &amp;quot;&lt;a href=&quot;https://www.maths.gla.ac.uk/wws/cabripages/ideal.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;ideal points&lt;/a&gt;&amp;quot; at infinity. This matrix form gives us a &lt;strong&gt;test for whether a &lt;a href=&quot;https://en.wikipedia.org/wiki/Degenerate_conic&quot; rel=&quot;nofollow noreferrer&quot;&gt;conic is degenerate&lt;/a&gt;&lt;/strong&gt;: this occurs if and only if &lt;span class=&quot;math-container&quot;&gt;$\det \mathbf{M} = 0$&lt;/span&gt;. The determinant of the top left 2-by-2 sub-matrix is &lt;span class=&quot;math-container&quot;&gt;$\Delta = AC - B^2$&lt;/span&gt; and gives a &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Degenerate_conic#Discriminant&quot; rel=&quot;nofollow noreferrer&quot;&gt;test to classify conics as ellipses, parabolas or hyperbolas&lt;/a&gt;&lt;/strong&gt; depending on whether &lt;span class=&quot;math-container&quot;&gt;$\Delta &amp;gt; 0$&lt;/span&gt; (ellipse, or in the degenerate case, a single point), &lt;span class=&quot;math-container&quot;&gt;$\Delta = 0$&lt;/span&gt; (parabola, or if real solutions exist in the degenerate case, two possibly coincident parallel lines) or &lt;span class=&quot;math-container&quot;&gt;$\Delta &amp;lt; 0$&lt;/span&gt; (hyperbola, or in the degenerate case, two intersecting lines).&lt;/p&gt;&#xA;&lt;p&gt;Writing the conic as &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; lets us generalise the technique for finding the equation of a circle from three points, to &lt;strong&gt;find the equation of a general conic &lt;a href=&quot;https://en.wikipedia.org/wiki/Five_points_determine_a_conic&quot; rel=&quot;nofollow noreferrer&quot;&gt;if given &lt;em&gt;five&lt;/em&gt; points&lt;/a&gt;&lt;/strong&gt;. (There are six coefficients and only five simultaneous equations, but solutions for the coefficients for the general conic can differ by a multiplicative constant. For the standard equation of a circle we fixed the &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$y^2$&lt;/span&gt; coefficients to one, so this issue didn't arise, but for a general conic we aren't guaranteed any particular coefficient is non-zero, so can't necessarily pick one to fix in this way.)&lt;/p&gt;&#xA;&lt;p&gt;The textbook extract showed how to find the &lt;a href=&quot;http://en.wikipedia.org/wiki/Pencil_of_circles&quot; rel=&quot;nofollow noreferrer&quot;&gt;pencil of circles&lt;/a&gt; passing through the intersection points of two given circles, by taking the weighted sum of the circle's equations (in standard form). The textbook weighted the first equation by one and second by &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$\lambda = -1$&lt;/span&gt; gave the degenerate case of the radical axis, &lt;span class=&quot;math-container&quot;&gt;$\lambda = 0$&lt;/span&gt; recovers the first circle, but the second circle is only recovered in the limit &lt;span class=&quot;math-container&quot;&gt;$\lambda \to \infty$&lt;/span&gt;. To keep things real, you may prefer to weight the first equation by &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; and the second by &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt;, not both zero. Then &lt;span class=&quot;math-container&quot;&gt;$(\lambda, \mu)=(1,0)$&lt;/span&gt; recovers the first circle, &lt;span class=&quot;math-container&quot;&gt;$(0,1)$&lt;/span&gt; recovers the second circle, and &lt;span class=&quot;math-container&quot;&gt;$(1,-1)$&lt;/span&gt; gives the radical axis. This doesn't really give us an extra degree of freedom since multiplying &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; by a common factor produces the same circle: all that matters is the ratio &lt;span class=&quot;math-container&quot;&gt;$\lambda:\mu$&lt;/span&gt; and we are once again in the realm of &lt;a href=&quot;https://en.wikipedia.org/wiki/Homogeneous_coordinates&quot; rel=&quot;nofollow noreferrer&quot;&gt;homogeneous coordinates&lt;/a&gt;. (What the textbook denoted by &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; is in our notation the ratio &lt;span class=&quot;math-container&quot;&gt;$\mu/\lambda$&lt;/span&gt;, which is often what we want to work with anyway.) We can similarly &lt;strong&gt;find a &lt;a href=&quot;http://en.wikipedia.org/wiki/Pencil_of_conics&quot; rel=&quot;nofollow noreferrer&quot;&gt;pencil of conics&lt;/a&gt; through the intersection points of two given conics&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$\mathbf{x'Mx} = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{x'Nx} = 0$&lt;/span&gt; by taking the linear combination &lt;span class=&quot;math-container&quot;&gt;$\mathbf{x}'(\lambda \mathbf{M} + \mu \mathbf{N})\mathbf{x} = 0$&lt;/span&gt;. We illustrate this for the circle &lt;span class=&quot;math-container&quot;&gt;$x^2 + y^2 - 5 = 0$&lt;/span&gt; (yellow) and hyperbola &lt;span class=&quot;math-container&quot;&gt;$x^2 - y^2 - 3 = 0$&lt;/span&gt; (blue). The four common points are the &lt;em&gt;base points&lt;/em&gt; of the pencil, and for any other point there is only one conic of the pencil that passes through it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mkbJL.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mkbJL.png&quot; alt=&quot;Pencil of conics through intersection points of circle x^2 + y^2 - 5 = 0 and hyperbola x^2 - y^2 - 3 = 0&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^2$&lt;/span&gt; conics can intersect in 0, 2 or 4 (possibly coincident) points. If two given circles intersect each other at 2 points, the procedure above for the pencil of circles generates &lt;em&gt;all&lt;/em&gt; circles passing through their intersection points, including the degenerate case of the radical axis. If the two points are coincident, the generated circles and radical axis will all be mutually tangential there. The procedure for the pencil of conics will similarly find &lt;em&gt;all&lt;/em&gt; conics passing through the intersection points of two given conics, but only if they intersect in 4 (possibly coincident) points. &lt;a href=&quot;https://math.stackexchange.com/a/324605/72968&quot;&gt;Even if the given conics intersect only in 2 points&lt;/a&gt;, the pencil gives a &lt;strong&gt;&lt;a href=&quot;https://math.stackexchange.com/a/425412/72968&quot;&gt;method to find the intersection points of two conics&lt;/a&gt;&lt;/strong&gt;: find (one of) the degenerate conics in the pencil by obtaining a solution to &lt;span class=&quot;math-container&quot;&gt;$\det (\lambda \mathbf{M} + \mu \mathbf{N}) = 0$&lt;/span&gt;. This involves solving a cubic equation in &lt;span class=&quot;math-container&quot;&gt;$\mu/\lambda$&lt;/span&gt; (or just &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; if you use the A-level textbook's approach and solve &lt;span class=&quot;math-container&quot;&gt;$\det (\mathbf{M} + \lambda \mathbf{N}) = 0$&lt;/span&gt; instead). Find the equations of the lines that make up your degenerate conic: if you have a degenerate parabola or hyperbola, you'll need to factorise. Now find the intersection of your lines with one of your given conics. In the example illustrated above, the degenerate cases occur when &lt;span class=&quot;math-container&quot;&gt;$\lambda:\mu = -1:1$&lt;/span&gt; (giving the degenerate parabola &lt;span class=&quot;math-container&quot;&gt;$2-2y^2-0$&lt;/span&gt; which factorises as &lt;span class=&quot;math-container&quot;&gt;$(y+1)(y-1)=0$&lt;/span&gt; i.e. the parallel lines &lt;span class=&quot;math-container&quot;&gt;$y=\pm 1$&lt;/span&gt;), when &lt;span class=&quot;math-container&quot;&gt;$\lambda:\mu = -3:5$&lt;/span&gt; (giving the degenerate hyperbola &lt;span class=&quot;math-container&quot;&gt;$2x^2 - 8y^2 = 0$&lt;/span&gt; which factorises as &lt;span class=&quot;math-container&quot;&gt;$(x+2y)(x-2y)=0$&lt;/span&gt; i.e. the intersecting lines &lt;span class=&quot;math-container&quot;&gt;$y=\pm \frac{1}{2}x$&lt;/span&gt;) and when &lt;span class=&quot;math-container&quot;&gt;$\lambda:\mu=1:1$&lt;/span&gt; (giving the degenerate parabola &lt;span class=&quot;math-container&quot;&gt;$2x^2-8=0$&lt;/span&gt; which factorises as &lt;span class=&quot;math-container&quot;&gt;$(x+2)(x-2)=0$&lt;/span&gt; i.e. the parallel lines &lt;span class=&quot;math-container&quot;&gt;$x=\pm 2$&lt;/span&gt;). Clearly the intersection points are &lt;span class=&quot;math-container&quot;&gt;$(2,1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(2,-1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(-2,1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(-2,-1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now we introduce the notation of &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Joachimsthal/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ferdinand Joachimsthal&lt;/a&gt;. Define for points &lt;span class=&quot;math-container&quot;&gt;$P(x_i, y_i)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q(x_j, y_j)$&lt;/span&gt; a quantity &lt;span class=&quot;math-container&quot;&gt;$s_{ij} =  \mathbf{x}_i' \mathbf{Mx}_j$&lt;/span&gt;. This is equivalent to taking the standard equation of the conic and transforming the quadratic terms as &lt;span class=&quot;math-container&quot;&gt;$x^2 \to x_i x_j$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y^2 \to y_i y_j$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2xy \to x_i y_j + x_j y_i$&lt;/span&gt;, and the linear terms as &lt;span class=&quot;math-container&quot;&gt;$2x \to x_i + x_j$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2y \to y_i + y_j$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$s_{ij} = Ax_i x_j + B(x_i y_j + x_j y_i) + C y_i y_j + F(x_i + x_j) + G(y_i + y_j) + H$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note the symmetry &lt;span class=&quot;math-container&quot;&gt;$s_{ij} = s_{ji}$&lt;/span&gt;, and that point &lt;span class=&quot;math-container&quot;&gt;$P(x_i,y_i)$&lt;/span&gt; lies on the conic if and only if &lt;span class=&quot;math-container&quot;&gt;$s_{ii}=0$&lt;/span&gt;. The textbook extract above shows that in the special case of a point outside a circle, and with the &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y^2$&lt;/span&gt; coefficients standardised to one, then &lt;span class=&quot;math-container&quot;&gt;$s_{ii}$&lt;/span&gt; is the &lt;em&gt;squared length of the tangent from that point to the circle&lt;/em&gt; so naturally becomes zero if we move that point onto the circle. In fact &lt;span class=&quot;math-container&quot;&gt;$s_{ii}$&lt;/span&gt; is just the &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Power_of_a_point&quot; rel=&quot;nofollow noreferrer&quot;&gt;power of a point&lt;/a&gt;&lt;/strong&gt; with respect to the circle, a definition that also extends to the interior of the circle and ties together the &lt;a href=&quot;https://en.wikipedia.org/wiki/Intersecting_chords_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;intersecting chords theorem&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Intersecting_secants_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;intersecting secants theorem&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Tangent%E2%80%93secant_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;tangent-secant theorem&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The equation of a general conic can be written as &lt;span class=&quot;math-container&quot;&gt;$s=0$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$s =  \mathbf{x'Mx}$&lt;/span&gt; is like &lt;span class=&quot;math-container&quot;&gt;$s_{ii}$&lt;/span&gt; but with the specific point &lt;span class=&quot;math-container&quot;&gt;$P(x_i,y_i)$&lt;/span&gt; replaced by the general point &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; is the left-hand sides of equations &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Keeping &lt;span class=&quot;math-container&quot;&gt;$P(x_i, y_i)$&lt;/span&gt; as a specific point, but replacing &lt;span class=&quot;math-container&quot;&gt;$x_j$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y_j$&lt;/span&gt; associated with point &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; by a completely general &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, we can also define &lt;span class=&quot;math-container&quot;&gt;$s_i = \mathbf{x}_i' \mathbf{Mx}$&lt;/span&gt;, i.e.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ s_{i} = Ax_i x + B(x_i y + x y_i) + C y_i y + F(x_i + x) + G(y_i + y) + H$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then if point &lt;span class=&quot;math-container&quot;&gt;$P(x_i, y_i)$&lt;/span&gt; lies on the conic, the &lt;strong&gt;equation of the tangent to the conic&lt;/strong&gt; at &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$s_i = 0$&lt;/span&gt;, i.e.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ Ax_i x + B(x_i y + x y_i) + C y_i y + F(x_i + x) + G(y_i + y) + H=0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The method for finding the tangent to a circle quoted above was just a special case of this.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;strong&gt;condition for the line &lt;span class=&quot;math-container&quot;&gt;$PQ$&lt;/span&gt; to be a tangent to the conic&lt;/strong&gt; is &lt;span class=&quot;math-container&quot;&gt;$s_{ij}^2 = s_{ii}s_{jj}$&lt;/span&gt;. We can use this condition to &lt;strong&gt;find the equations of the tangents to a conic passing through a given point&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$P(x_i, y_i)$&lt;/span&gt;: we replace &lt;span class=&quot;math-container&quot;&gt;$Q(x_j,y_j)$&lt;/span&gt; by the general point &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; and rewrite the tangency condition as &lt;span class=&quot;math-container&quot;&gt;$s_i^2 - s_{ii}s = 0$&lt;/span&gt;. The left-hand side is a quadratic in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. Factorise it into two linear factors: setting each factor to zero gives the equation of a tangent passing through &lt;span class=&quot;math-container&quot;&gt;$P(x_i,y_i)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$P(x_i,y_i)$&lt;/span&gt; lies on the tangent to the conic at &lt;span class=&quot;math-container&quot;&gt;$Q(x_j,y_j)$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$$s_{ii} = A(x_i - x_j)^2 + 2B(x_i - x_j)(y_i - y_j) + C(y_i - y_j)^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This generalises the tangent-length formula for a circle quoted above: put &lt;span class=&quot;math-container&quot;&gt;$A=C=1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B=0$&lt;/span&gt;, and the right-hand side is the square of the length &lt;span class=&quot;math-container&quot;&gt;$PQ$&lt;/span&gt;. The right-hand side is in general equal to &lt;span class=&quot;math-container&quot;&gt;$s_{ii} - 2s_{ij} + s_{jj}$&lt;/span&gt;: the &lt;a href=&quot;https://math.stackexchange.com/a/4755652&quot;&gt;algebraic proof&lt;/a&gt; boils down to using the symmetry of &lt;span class=&quot;math-container&quot;&gt;$\mathbf{M}$&lt;/span&gt; and factorising &lt;span class=&quot;math-container&quot;&gt;$\mathbf{x}_i' \mathbf{Mx}_i - \mathbf{x}_i' \mathbf{Mx}_j - \mathbf{x}_j' \mathbf{Mx}_i - \mathbf{x}_j' \mathbf{Mx}_j$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$(\mathbf{x}_i - \mathbf{x}_j)' \mathbf{M} (\mathbf{x}_i - \mathbf{x}_j)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$(\mathbf{x}_i - \mathbf{x}_j)'$&lt;/span&gt; is  &lt;span class=&quot;math-container&quot;&gt;$\begin{bmatrix}x_i - x_j &amp;amp; y_i - y_j &amp;amp; 0 \end{bmatrix}$&lt;/span&gt;. Since the third component is zero, only the coefficients &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; from the top left two-by-two submatrix of &lt;span class=&quot;math-container&quot;&gt;$\mathbf{M}$&lt;/span&gt; appear in our final equation. We have &lt;span class=&quot;math-container&quot;&gt;$s_{jj} = 0$&lt;/span&gt; since &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; lies on the conic, and &lt;span class=&quot;math-container&quot;&gt;$s_{ij} = 0$&lt;/span&gt; by the tangency condition, so &lt;span class=&quot;math-container&quot;&gt;$s_{ii} - 2s_{ij} + s_{jj}$&lt;/span&gt; reduces to &lt;span class=&quot;math-container&quot;&gt;$s_{ii}$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$PQ$&lt;/span&gt; is a tangent to the conic at &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; .&lt;/p&gt;&#xA;&lt;p&gt;Joachimsthal's notation gives a powerful way to explore &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pole_and_polar&quot; rel=&quot;nofollow noreferrer&quot;&gt;poles and polars&lt;/a&gt;&lt;/strong&gt; with respect to a conic. For a given point &lt;span class=&quot;math-container&quot;&gt;$P(x_i, y_i)$&lt;/span&gt;, there is an associated &lt;em&gt;polar line&lt;/em&gt; with respect to the conic, whose equation is simply &lt;span class=&quot;math-container&quot;&gt;$s_i = 0$&lt;/span&gt;, and we say &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is the &lt;em&gt;pole&lt;/em&gt; of that line (again, with respect to the given conic). Stothers treats this both &lt;a href=&quot;https://www.maths.gla.ac.uk/wws/cabripages/algebra.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;algebraically&lt;/a&gt; and &lt;a href=&quot;https://www.maths.gla.ac.uk/wws/cabripages/geometry.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;geometrically&lt;/a&gt;. We saw above the special case that when &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; lies on the conic, its polar line with respect to the conic is the tangent to the conic at &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. Vice versa, if a line is tangential to the conic then its pole with respect to that conic is the point of tangency. From the symmetry of &lt;span class=&quot;math-container&quot;&gt;$s_{ij}=s_{ji}$&lt;/span&gt;, it is trivial to prove &lt;a href=&quot;https://www.maths.gla.ac.uk/wws/cabripages/lahire.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;La Hire's theorem&lt;/a&gt; that if &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; lies on the polar of &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; lies on the polar of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; (with respect to the same conic). So as a practical example, if a line intersects the conic at two points &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, and the tangents to the conic at &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; intersect at &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is the pole of the original line &lt;span class=&quot;math-container&quot;&gt;$PQ$&lt;/span&gt; with respect to the conic. This follows since &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; lay on both the polar of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and polar of &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; so, by La Hire's theorem, &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; both lie on the polar of &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, and these two distinct points suffice to define the polar line. Vice versa, given a point through which we can find two tangents to the conic, its polar is the line through the two points of tangency.&lt;/p&gt;&#xA;&lt;p&gt;There are many other uses for this notation, including &lt;a href=&quot;https://math.stackexchange.com/questions/4755465/find-the-common-tangent-to-the-parabolas-y-x2-6x9-and-y-x2-2x-1/4755600#4755600&quot;&gt;finding the common tangents to two conics&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/q/4794224/72968&quot;&gt;determining the minor radius of an ellipse from its major axis and a tangent line&lt;/a&gt;, but I'll leave it there. An excellent book for further reading is &lt;em&gt;Geometry&lt;/em&gt; by David A. Brannan, Matthew F. Esplen and Jeremy J. Gray, Cambridge University Press (1st edition 2002, 2nd edition 2012).&lt;/p&gt;&#xA;" OwnerUserId="5640" LastEditorUserId="5640" LastEditDate="2024-03-16T13:26:49.060" LastActivityDate="2024-03-16T13:26:49.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27602" PostTypeId="2" ParentId="27594" CreationDate="2024-03-15T02:29:50.223" Score="4" Body="&lt;p&gt;Here is one way to think about the implications of the general form of a conic:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We have &lt;strong&gt;six&lt;/strong&gt; parameters (&lt;span class=&quot;math-container&quot;&gt;$A,B, C, D, E, F$&lt;/span&gt;) but since the right hand side of the equation is 0, we effectively have just &lt;strong&gt;five&lt;/strong&gt; parameters. (Think about forcing one of the parameters like &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to be equal to 1.)&lt;/p&gt;&#xA;&lt;p&gt;From this point of view, we can expect that the space of conics in the plane is  five-dimensional.&lt;/p&gt;&#xA;&lt;p&gt;Now let's think about the celebrated theorem of &lt;a href=&quot;https://en.wikipedia.org/wiki/Five_points_determine_a_conic&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Pascal&lt;/strong&gt;&lt;/a&gt;, which tells us that 5 points in the plane determine a conic! So you give me 5 points, and I can find a unique conic that passes through all five points...sort of like two points determine a line. This is really a theorem. It is not obvious!&lt;/p&gt;&#xA;&lt;p&gt;Now let's connect our heuristic counting with Pascal's theorem. If we are given one point &lt;span class=&quot;math-container&quot;&gt;$(x_1,y_1)$&lt;/span&gt; then this imposes one linear constraint on our five variables:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Ax_1^2+Bx_1y_1+Cy_1^2+Dx_1+Ey_1+F=0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then we impose a second constraint by requiring the conic to pass through &lt;span class=&quot;math-container&quot;&gt;$(x_2,y_2)$&lt;/span&gt;, etc. Our counting is this:&lt;/p&gt;&#xA;&lt;p&gt;0 constraints: 5 degrees of freedom (all conics)&lt;/p&gt;&#xA;&lt;p&gt;1 constraint (force 1 point): 4 degrees of freedom&lt;/p&gt;&#xA;&lt;p&gt;2 constraints (force 2 points): 3 degrees of freedom&lt;/p&gt;&#xA;&lt;p&gt;3 constraints (force 3 points): 2 degrees of freedom (called a &amp;quot;net of conics&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;4 constraints (force 4 points): 1 degree of freedom (called a &amp;quot;pencil of conics&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;5 constraints (force 5 points): 0 degrees of freedom, so a unique conic.&lt;/p&gt;&#xA;&lt;p&gt;We see that our intuitive expectation that the space of conics is 5-dimensional based on the general form of a conic is consistent with Pascal's Theorem. Perhaps this is why Pascal suspected the theorem in the first place.&lt;/p&gt;&#xA;&lt;p&gt;Given five points &lt;span class=&quot;math-container&quot;&gt;$(x_1,y_1),\ldots,(x_5,y_5)$&lt;/span&gt;, we could solve the  system of five linear equations in six unknowns to actually determine &lt;span class=&quot;math-container&quot;&gt;$A, B, C, D, E, F$&lt;/span&gt; up to a scalar multiple. We could then look at &lt;span class=&quot;math-container&quot;&gt;$B^2-4AC$&lt;/span&gt; to determine the type of the conic that passes through the five given points.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2024-03-15T12:22:34.830" LastActivityDate="2024-03-15T12:22:34.830" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27603" PostTypeId="2" ParentId="27600" CreationDate="2024-03-15T14:36:23.337" Score="2" Body="&lt;p&gt;The post has many very broad questions. Here are some principles I try to apply.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;(1) Which problems?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If the subject has &amp;quot;classic exercises&amp;quot;, they must be on the list.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The list must contain a sufficient number of &amp;quot;easy&amp;quot; exercises (that is, those that the average hard-working student can solve).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The list must contain exercises that really prepare the student for the tests.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The list can contain interesting results that were not covered in class and/or results (with accessible proofs) used in class without demonstration.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The list can contain some &amp;quot;hardy&amp;quot; problems or challenges (to emphasize that things may not be as simple as they seem). But in this case, I believe the student should know that they are not easy (personally, I like to include hints).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Warning:&lt;/strong&gt;&lt;/em&gt; The meaning of &amp;quot;easy&amp;quot; and &amp;quot;hardy&amp;quot; depends on the audience.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;(2) How many?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;em&gt;all&lt;/em&gt; exercises will be graded, I choose just a few. (I believe this can only be done if the class is very small; I only did this once due to very specific circumstances).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If &lt;em&gt;no&lt;/em&gt; exercises will be graded, I choose many. I seek to ensure, for example, that important techniques are applied more than once.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;(3) How do you determine the level of success on problems indicating success in the course?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Type I problems (requires only mastery of standard demonstration methods and the main concepts/theorems of the theory). I believe that success on this type of problem indicates success in the course.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Type II problems (requires creativity, tricks or very long solutions). I believe that failure in this type of problem does not indicate failure in the course.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Warning:&lt;/strong&gt;&lt;/em&gt; Possibly, not every problem is of type I or II.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;(4) Is it simply a matter of taste (&amp;quot;this one was interesting&amp;quot;)?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;No. See (1).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;(5) Do you try to ensure that each subtopic has one problem?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Not necessarily. But as I think that the main examples worked in class generally emphasize the main points of the theory, for each main example I always try to include at least one exercise that uses the same idea. And since the main subtopics usually have main examples, the main subtopics always have at least one exercise.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;(6) Are there multiple dimension to the criteria?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Certainly yes.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Ensure that every concept, definition, theorem, or technique appears in at least one problem&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Certainly not.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Have at least one problem that merely tests comprehension, one that requires insight or recognition of the true meaning of problems (in different contexts), and at least one that is too hard to be solved.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Desirable, but not mandatory.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;(7) Is it budget based?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is an important issue. I personally do not assign problems for which there is not at least a sketch of solutions available (either in the book or in my class notes). Therefore, the problems I assign are subject to my time in writing the solutions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Warning:&lt;/strong&gt;&lt;/em&gt; The exact moment of making solutions available for the students depends on many factors.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;(8) What criteria do you use to say &amp;quot;This course has enough problems. If you do them, you can master the material.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is enough when (at least) the three &amp;quot;must&amp;quot; criteria in (1) are fulfilled, with the majority of problems being Type I (as defined in (3)).&lt;/p&gt;&#xA;" OwnerUserId="13585" LastActivityDate="2024-03-15T14:36:23.337" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27604" PostTypeId="2" ParentId="21048" CreationDate="2024-03-15T17:27:12.640" Score="2" Body="&lt;p&gt;Here is an example from Nelson's &lt;a href=&quot;https://doi.org/10.1515/9781400882144&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Radically Elementary Probability Theory&lt;/em&gt;&lt;/a&gt; (&lt;a href=&quot;https://zbmath.org/0651.60001&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zbl 0651.60001&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Suppose that &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is a finite set with a strictly positive real function &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\sum_{x \in \Omega} p(x) = 1$&lt;/span&gt;. So, &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is really a probability function on &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;. A random variable &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is then just any function from &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, and we define its expectation &lt;span class=&quot;math-container&quot;&gt;$\mathbf{E}[X]$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\sum_{x \in \Omega} p(x) f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;the collection of all random variables, &lt;span class=&quot;math-container&quot;&gt;$\mathbf{R}^\Omega$&lt;/span&gt;, is a (finite-dimensional, real) vector space;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathbf{E}$&lt;/span&gt; is a linear map into &lt;span class=&quot;math-container&quot;&gt;$\mathbf{R}$&lt;/span&gt; (i.e., a linear functional on &lt;span class=&quot;math-container&quot;&gt;$\mathbf{R}^\Omega$&lt;/span&gt;). In particular, the collection of random variables with mean &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is a hyperplane &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\langle X, Y \rangle := \mathbf{E}[XY]$&lt;/span&gt; is an inner product on &lt;span class=&quot;math-container&quot;&gt;$\mathbf{R}^\Omega$&lt;/span&gt;;&lt;/li&gt;&#xA;&lt;li&gt;the orthogonal complement of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; with respect to this inner product is the space of constant random variables;&lt;/li&gt;&#xA;&lt;li&gt;the typical way of shifting an arbitrary random variable &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; to one with mean &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; by considering &lt;span class=&quot;math-container&quot;&gt;$X - \mathbf{E}[X]$&lt;/span&gt; is the same thing as taking the projection of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; onto the subspace &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;;&lt;/li&gt;&#xA;&lt;li&gt;similarly, computing the mean is the same as finding the length of the projection onto the orthogonal complement of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;, i.e., onto the 1D space of constant random variables;&lt;/li&gt;&#xA;&lt;li&gt;for a mean &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; random variable &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, the standard deviation is just its length;&lt;/li&gt;&#xA;&lt;li&gt;for mean &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; random variables &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt;, the correlation coefficient is the cosine of the angle between them.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Lastly, the &lt;em&gt;radically elementary&lt;/em&gt; idea is that this nice picture can be very useful even when &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is not finite, via &lt;a href=&quot;https://en.wikipedia.org/wiki/Nonstandard_analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;nonstandard&lt;/a&gt; methods.&lt;/p&gt;&#xA;" OwnerUserId="15884" LastEditorUserId="15884" LastEditDate="2024-03-15T23:00:38.727" LastActivityDate="2024-03-15T23:00:38.727" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27605" PostTypeId="2" ParentId="27566" CreationDate="2024-03-15T23:30:38.017" Score="3" Body="&lt;p&gt;Quite similar to @JustinSkycak's idea: take &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; blue marbles and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; red marbles. First, put the marbles in line, the blue ones on the left and the red ones on the right. Then swap them.&lt;/p&gt;&#xA;&lt;p&gt;Add a third color and you can show associativity as well (without needing a third arm!)&lt;/p&gt;&#xA;&lt;p&gt;Note: I read &lt;em&gt;Comment les enfants apprennent a calculer?&lt;/em&gt; by Remi Brissiaud (&lt;em&gt;How children learn to calculate?&lt;/em&gt;, I don't think there is an English translation) a long time ago. It is about the psychological development of the concept of numbers and calculation in young infants, and I was impressed by how complex the process is. If the goal of the questions is to teach your own child, be careful not giving him wrong mental images.&lt;/p&gt;&#xA;" OwnerUserId="1538" LastActivityDate="2024-03-15T23:30:38.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27606" PostTypeId="2" ParentId="27566" CreationDate="2024-03-16T01:42:45.493" Score="8" Body="&lt;p&gt;Perhaps this drawing will make the answer of @DanielR.Collins clearer. The person on the left sees 3+1; the person on the right sees 1+3. They're looking at the same thing.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lSZEA.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lSZEA.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Edit: Oops, it seems that @Taladris has already mentioned something like this. Anyway, I'm keeping my answer so that the drawing can make things clear.&lt;/p&gt;&#xA;" OwnerUserId="77" LastActivityDate="2024-03-16T01:42:45.493" CommentCount="1" CommunityOwnedDate="2024-03-16T01:42:45.493" ContentLicense="CC BY-SA 4.0" />
  <row Id="27607" PostTypeId="1" CreationDate="2024-03-17T07:53:43.013" Score="5" ViewCount="1979" Body="&lt;p&gt;Maths is already quite an abstract subject, so to spend time discussing the quality of a student's or even a research-level mathematician's handwriting seems rather tedious. I proceed nonetheless.&lt;/p&gt;&#xA;&lt;p&gt;When I was in school, among my peers I didn't necessarily notice (although maybe I didn't gather enough statistics on) any (positive or negative) correlation between those who have good handwriting and those who have good understanding of a subject or do well in homework or exams.&lt;/p&gt;&#xA;&lt;p&gt;Obviously there is a trade-off between writing quickly, which allows one to carry out their own thought process quickly in regards to their &lt;em&gt;working memory&lt;/em&gt;, and writing neatly, which makes their handwriting more legible for themselves and for others, i.e. communicating their thoughts when looking back upon their workings out.&lt;/p&gt;&#xA;&lt;p&gt;I have always written as fast as possible because I need my working memory to get through it's thoughts as quickly as possible: this surely maximises my understanding. But my handwriting has suffered as a consequence.&lt;/p&gt;&#xA;&lt;p&gt;One of the girls I currently tutor writes ridiculously neatly, and she doesn't write very slowly, but I think she would benefit from writing faster when getting her thoughts down, then re-writing her scribblings up neatly later.&lt;/p&gt;&#xA;&lt;p&gt;I also feel that, now that I have gotten into a bad habit of my handwriting not being neat, that I need to re-learn how to write neatly again, and then once I have done this, speed will naturally come later.&lt;/p&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;p&gt;Would it benefit me to practise making my handwriting neater and then once I have gotten into this new &amp;quot;good habit&amp;quot; I can start increasing my writing speed? Should my student who writes slowly but neatly sacrifice her neatness for scribbling but with speed, but then she can re-write her answer up later on? Should I encourage other students to write in this manner? Or should I encourage students to focus on neat handwriting and speed will come naturally later? Or does it not matter what I suggest to students because they are going to write how they write regardless of my suggestion anyway?&lt;/p&gt;&#xA;&lt;p&gt;I am not sure if this is important or not, but I have always found that writing as quickly as possible, as close to the pace of your working memory as possible (my working memory is much faster than my writing speed anyway), to be the best way to improve mathematical understanding, and I imagine this is supported by statistical evidence: if you get your thoughts down on paper much slower than the speed at which you are trying to flesh out your thoughts, then this will surely hinder your understanding, right? I have always gone through this process as in the first sentence of this paragraph, and then re-written my answer up neatly if necessary or if I feel it would benefit me. But is this the right or best or a good way of doing things? Or is it personal preference / trial and error?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This is probably more of a convoluted ramble than a legit question, but I just want to see what others think. Maybe it's also a bad question because it is opinion-based, but who knows: maybe there have been statistically significant findings to some of my questions.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, you could argue that this question is not necessarily related to maths as it could be applied to other subjects as well. If others agree with this, what is the more appropriate stackexchange to ask this question on?&lt;/p&gt;&#xA;" OwnerUserId="12691" LastEditorUserId="12691" LastEditDate="2024-03-17T07:59:38.280" LastActivityDate="2024-03-18T15:05:23.217" Title="Trade-off between writing neatly vs scribbling (scribbling aims to maximise working memory)" Tags="|working-memory|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27608" PostTypeId="2" ParentId="27607" CreationDate="2024-03-17T10:57:22.343" Score="10" Body="&lt;p&gt;I think your question already contains the appropriate advice on what to do, but this might not become completely clear since the frameing of the question puts the focus elsewhere. I'd say the main point is not to either write neatly or to scribble - it's rather to adapt the writing style as appropriate to the situation.&lt;/p&gt;&#xA;&lt;p&gt;When I teach at the blackboard in front of a class, I'll take care to have a very well-structured presentation and to write in a legible way (unfortunately I'm not particularly good at the latter, but still). When I discuss something with a colleague at a blackboard or on a sheet of paper, what I write will naturally be much less structured since we don't know the result of our discussion in advance - but it's still important to ensure that the other person can at least read what I write down. When I do math on my own (either on the blackboard or on a sheet of paper) I'll just scribble there whatever comes to my mind without caring much for structure or legibility. If, at the end, it turned out that my thoughts have lead anywhere, I'll rewrite the insights clearly and neatly and leave out all the clutter that turned out not to be useful.&lt;/p&gt;&#xA;&lt;p&gt;Similarly, a student should adapt their writing style to the situation: when they do a problem on their own and the problem is not merely about practicing an algorithm, then they should be spontaneous and have no inhibitions to scribble something on a sheet of paper. When they discuss a problem with other students, they should still feel free to write down their thoughts in a spontaneous way on a sheet of paper (or blackboard or whiteboard), but what they write should of course be legible for the other students with whom they are discussing. And when they have found a solution and, e.g. want to submit it for grading, they should write it down again - this time in a well-structured and neatly written way.&lt;/p&gt;&#xA;&lt;p&gt;To a lot of students this does not come easily. When I teach undergraduate students in their first year, I notice that many of them have a very strong inhibition to just scribble something on a sheet of paper. I guess that this is because many of them were taught math as a mechanical rather than a creative subject in school. They were made to believe that math were a collection of methods or algorithms, so when you &amp;quot;are given a problem&amp;quot; there were a clear path to the solution and solving the problem meant to follow this path in writing.&lt;/p&gt;&#xA;&lt;p&gt;I'm under the impression that, while you're interpreting the situation that you experience as being about writing slowly and neatly vs. scribling quickly, it might in fact not be so much the speed itself that causes issues. Rather your student might feel very inhibited to just &amp;quot;give it a try&amp;quot; and write something down without any expectations that this will become a presentable solution at the first try.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;So my advice&lt;/strong&gt; is not to convince your students to &amp;quot;scribble instead of writing neatly&amp;quot;, but rather to teach them that it is - for all those problems that do not merely practice a method or an algorithm - good and desirable not to write down the solution of a problem at first try, but to be spontaneous and creative, to try several approaches and ideas, and only to turn it into a neat presentable solution at the end of the process.&lt;/p&gt;&#xA;" OwnerUserId="16250" LastEditorUserId="16250" LastEditDate="2024-03-17T11:21:15.363" LastActivityDate="2024-03-17T11:21:15.363" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27609" PostTypeId="2" ParentId="27607" CreationDate="2024-03-17T12:57:52.840" Score="1" Body="&lt;p&gt;I say this as a fellow sinner, probably worse than you, and man to man:  Yes, you should work on your handwriting.&lt;/p&gt;&#xA;&lt;p&gt;I wouldn't sweat it in the immediate world of say lecture note taking, if you are pressed for speed.  (But I find that pre-studying the textbook makes the compulsive &amp;quot;write everything down&amp;quot; much less needed.  And I disdain teachers who want to create classes that don't have strong supporting textbooks...as if they were medieval lecturers at the University of Paris.)&lt;/p&gt;&#xA;&lt;p&gt;But...on drill problems, I think you should have a moderate effort at neatness--even just to &amp;quot;respect&amp;quot; the problem and your demonstration of an answer--but very light...mostly working on the problems themselves.&lt;/p&gt;&#xA;&lt;p&gt;In particular in the modern world, it is important in academia and business to be able to write clearly at the white/black board.  I have personally had issues in the Navy, when my CO looking into a reactor incident I was involved in (which is another story) asked why my handwriting in the log (a legal document!) was so bad.  And he was not even fussing for beauty...we are talking med school undecipherable Sanskrit.  (I told him I would take a strain in the future and do better.)&lt;/p&gt;&#xA;&lt;p&gt;One other note...I have personally have (often) found myself unable to decipher my OWN handwriting from classes or in the corporate world (e.g. interview notes).  I do try to work around this in the corporate world by direct typing (or buying transcripts).  And I'm a pretty intermediate level typer...but I slam away and get most of it down as running notes (not worring about perfection, going and cleaning up later).  [I probably wouldn't do it in a classroom as my clicking can be pretty loud and distracting.]&lt;/p&gt;&#xA;&lt;p&gt;Leave the girl who writes neatly (or others like her) alone.  Find something else to work on.  Teaching is an optimization problem where you (and the pupil) have limited time.  Don't nitpick low priority items when there are much juicier fruit to pick first.&lt;/p&gt;&#xA;&lt;p&gt;There is also an element of different strokes for different folks.  Don't try to use yourself too much as a model, to expect everyone to be like you.  And for all you know, slowing the problem down and being very deliberate may be helpful to the girl.  For example if she typically makes mistakes in the details of multi-step algebra.&lt;/p&gt;&#xA;&lt;p&gt;Same applies for bad handwriting kids.  Leave 'em be, work on the math.  Just deal with it and don't get into it.  Unless the work is truly undecipherable or if it starts to show a lack of respect for the problem/answer (as my messy nuclear logs did).  In which case, tell the kid to take a strain (but don't make some big hairy thing of it, or design a handwriting remediation program.)&lt;/p&gt;&#xA;" OwnerUserId="23843" LastEditorUserId="23843" LastEditDate="2024-03-17T13:05:49.190" LastActivityDate="2024-03-17T13:05:49.190" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27610" PostTypeId="1" AcceptedAnswerId="27611" CreationDate="2024-03-17T14:21:18.590" Score="2" ViewCount="230" Body="&lt;p&gt;The mainstream way to show &lt;span class=&quot;math-container&quot;&gt;$V[aX+b]= a^2 V[X]$&lt;/span&gt; is by using LOTUS. However, LOTUS seems to me too powerful and out-of-reach for a last-year high-school student.&lt;/p&gt;&#xA;&lt;p&gt;Therefore I was wondering if we could derive it from the definition of Variance:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$V[aX+b]=\sum (x-E[aX+b])^2 p(aX+b=x)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We know that &lt;span class=&quot;math-container&quot;&gt;$E[aX+b] = aE[X]+b$&lt;/span&gt; but I don't know how to handle &lt;span class=&quot;math-container&quot;&gt;$p(aX+b=x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I was thinking to do &lt;span class=&quot;math-container&quot;&gt;$p(aX+b=x)=p((aX+b)^{-1} (\{ x\}))=p(aX^{-1} (\{ x\}) +b)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;but I don't know if that makes sense to add to a point in the sample space (&lt;span class=&quot;math-container&quot;&gt;$X^{-1}(\{ x\})$&lt;/span&gt;) a constant &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Do you think it is possible to derive &lt;span class=&quot;math-container&quot;&gt;$V[aX+b]=a^2 V[X]$&lt;/span&gt; without the LOTUS formula ?&lt;/p&gt;&#xA;&lt;p&gt;I am working with finite discrete sample spaces, and a RV is here a function from the sample space to &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is defined as &lt;span class=&quot;math-container&quot;&gt;$E[X]=\sum_i x_i p(X=x_i) $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thanks i.a.&lt;/p&gt;&#xA;" OwnerUserId="22167" LastEditorUserId="22167" LastEditDate="2024-03-17T16:28:16.550" LastActivityDate="2024-03-23T06:00:26.947" Title="How to prove, without the LOTUS formula, to student that $V[aX+b]= a^2 V[X]$?" Tags="|secondary-education|proofs|probability|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27611" PostTypeId="2" ParentId="27610" CreationDate="2024-03-17T16:45:27.993" Score="9" Body="&lt;p&gt;I think you will want to start by convincing the audience that &lt;span class=&quot;math-container&quot;&gt;$p(aX+b = ax_i + b)$&lt;/span&gt; is equal to &lt;span class=&quot;math-container&quot;&gt;$p(X=x_i)$&lt;/span&gt;, probably with examples. I am not an expert statistician so please let me know politely if I am confused by what you mean here.&lt;/p&gt;&#xA;&lt;p&gt;Assuming that introduction is correct, I would then rename that whole mess &lt;span class=&quot;math-container&quot;&gt;$p_i$&lt;/span&gt;, because it is a distraction from where the action is really happening. So the notation I would be using is&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$E[X] = \sum_i x_i p_i$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$V[X] = \sum_i (x_i-E[X])^2p_i$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$V[aX+b] = \sum_i (ax_i + b - E[aX+b])^2p_i$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Then proceed like:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\begin{align*}&#xA;V[aX+b] &amp;amp; = \sum_i (ax_i+b - E[aX+b])^2p_i \\&#xA;        &amp;amp; = \sum_i (ax_i+b - aE[X]-b)^2p_i  \\&#xA;        &amp;amp; = \sum_i (ax_i - aE[X])^2p_i  \\&#xA;        &amp;amp; = a^2 \sum_i (x_i - E[X])^2p_i  \\&#xA;        &amp;amp; = a^2 V[X]  \\&#xA;\end{align*}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="11" LastActivityDate="2024-03-17T16:45:27.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27612" PostTypeId="2" ParentId="27607" CreationDate="2024-03-18T08:28:28.467" Score="1" Body="&lt;p&gt;I see two questions, both about a similar subject.&lt;/p&gt;&#xA;&lt;p&gt;Let me start by your second question: should you advise a girl with a neat handwriting to reduce that quality in order to &amp;quot;be able needing to&amp;quot; write the same information twice (once fast scribble in class, and next neatly written at home)? I believe the answer is in the question: NO WAY! That girl has a competence which you don't have: neat handwriting. And you seem to seek for a reason to criticise this girl, just out of pure jealousy. You should be &lt;strong&gt;ashamed&lt;/strong&gt;!&lt;br /&gt;&#xA;Leave that girl alone and if you see that slowing down a little might be helpful for her, then that's what you do!&lt;/p&gt;&#xA;&lt;p&gt;Second: your own handwriting. I remember in my second year at primary school, I had calligraphy classes, and although my current handwriting is a mess, I do realise the importance of those classes. But let's face it: calligraphy classes are given on paper, not on a blackboard (which is the thing you're writing on now).&lt;br /&gt;&#xA;As you admit that your handwriting is bad, I would advise you to get a blackboard with the typical calligraphic lines and do some calligraphy practices yourself in your free time, in order to improve your handwriting skills on a blackboard.&lt;/p&gt;&#xA;" OwnerUserId="8190" LastActivityDate="2024-03-18T08:28:28.467" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27613" PostTypeId="2" ParentId="27607" CreationDate="2024-03-18T14:58:16.060" Score="6" Body="&lt;h2&gt;&amp;quot;Its me I'm the problem its me&amp;quot;&lt;/h2&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I don't know the correct term to describe someone who tells someone to do a worse job because the other person makes you feel inadequate (in the case writing). But it is not complimentary.&lt;/p&gt;&#xA;&lt;p&gt;You need to relearn how to write.&lt;/p&gt;&#xA;&lt;p&gt;How legible/illegible your handwriting is, is not impacted by how quickly you write. But by how much effort you have put into practicing writing neatly. Source: I know people with perfect... and I mean perfect handwriting who can write faster then thought. and I know people with barely legible chicken scratch writing who are really really slow at writing.&lt;/p&gt;&#xA;&lt;p&gt;Speed at writing will come in time to this girl as she practices writing faster. But writing fast won't help her understand things better.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I am not sure if this is important or not, but I have always found that writing as quickly as possible, as close to the pace of your working memory as possible (my working memory is much faster than my writing speed anyway), to be the best way to improve mathematical understanding&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is wrong. Research suggests the opposite. The longer something stays in your working memory, the better you understand things, and the better your rate of retention is.&lt;/p&gt;&#xA;&lt;p&gt;Hastily scribbled barely legible notes therefore aren't as good as slowly written meticulous notes. Because slowly written legible notes force students to keep that information inside of their working memory for longer.&lt;/p&gt;&#xA;&lt;p&gt;That being said there is &lt;a href=&quot;https://www.pediastaff.com/blog/slp/study-suggests-that-hard-to-read-fonts-may-increase-reading-retention-3749&quot; rel=&quot;noreferrer&quot;&gt;research&lt;/a&gt; that suggests that deciphering hastily scribbled barely legible notes does help with retention. Because bad writing is harder to decode it takes much longer to decipher it. The extra time it takes to decipher the words makes them stick the working memory for a longer period of time. Which helps with retention. SO its not the fast writing that helps you with understanding... its the time wasted trying to interpret your Egyptian hieroglyphics that increase retention. But the same results can be achieved with a lower chance for error by taking things slow and slowly/carefully writing easy clear notes.&lt;/p&gt;&#xA;" OwnerUserId="23110" LastActivityDate="2024-03-18T14:58:16.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27614" PostTypeId="2" ParentId="27607" CreationDate="2024-03-18T15:05:23.217" Score="3" Body="&lt;p&gt;An answer by way of a cautionary tale:&lt;/p&gt;&#xA;&lt;p&gt;In my second year at university (studying mathematics), I too reasoned that it's not worth spending too long writing notes.  However, rather than work on my poor handwriting, I looked at shorthand.  Since neither of the systems I investigated seemed suitable, I had great fun inventing my own — and was pleasantly surprised that I could write it faster than longhand after only a week or so's practice.  So I switched to taking all my lecture notes in shorthand.&lt;/p&gt;&#xA;&lt;p&gt;The folly of this became clear when revising for my end-of-year exams — I'd had far less practice &lt;em&gt;reading&lt;/em&gt; my shorthand than writing it, and even after many weeks I couldn't approach even a fraction of my normal fluency!  Result: revising was hard work, I learned poorly, and did pretty badly in most of my end-of-year exams…&lt;/p&gt;&#xA;&lt;p&gt;(The following year, I abandoned shorthand, and instead redesigned my handwriting from scratch&lt;sup&gt;*&lt;/sup&gt;, optimising for legibility at all speeds.  So my third-year notes are very clear and easy-to-read, and I did rather better in my finals — though not enough to fully make up for the previous year.)&lt;/p&gt;&#xA;&lt;p&gt;Hopefully this demonstrates the importance of being able to easily read what you write, even if it takes slightly longer!&lt;/p&gt;&#xA;&lt;p&gt;(Being able to write clearly and legibly is a life skill with far wider benefits, of course.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;(* In case anyone's interested, after much experimentation I settled on printing instead of joined-up (cursive) writing — with practice, it's not significantly slower, and remains legible even at speeds where cursive degenerates to an unreadable scrawl.  I also reduced the size of ascenders and descenders and instead enlarged the central part of letters — they need hardly any separation — and used simple, easily-distinguished letter forms.  In fact, it turned out very like the &lt;a href=&quot;https://en.wikipedia.org/wiki/Century_Gothic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Century Gothic&lt;/a&gt; font, though that wasn't intentional.  I've written pretty much the same way ever since, and have been complimented on it.)&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="12762" LastActivityDate="2024-03-18T15:05:23.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27615" PostTypeId="2" ParentId="27610" CreationDate="2024-03-18T20:35:26.640" Score="8" Body="&lt;p&gt;This is a consequence of the definition of the variance (1) the linearity of expectation (2) and an algebraic manipulation (3): &lt;span class=&quot;math-container&quot;&gt;$$V(aX+b)\stackrel{(1)}{=}\mathbb{E}(aX+b-\mathbb{E}(aX+b))^2\stackrel{(2)}{=}\mathbb{E}(aX+b-a\mathbb{E}(X) - b))^2\stackrel{(3)}{=}\mathbb{E}[a^2(X-\mathbb{E}(X))^2]\stackrel{(2)}{=}a^2\mathbb{E}(X-\mathbb{E}(X))^2\stackrel{(1)}{=}a^2V(X).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, you don't really need to invoke the definition or formula for the expectation to prove this, just use its properties.&lt;/p&gt;&#xA;" OwnerUserId="10349" LastActivityDate="2024-03-18T20:35:26.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27616" PostTypeId="2" ParentId="17302" CreationDate="2024-03-19T10:12:00.727" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If we applied two tests for each symmetry (one with &#xA;and one with −&#xA;), are there any equations that would still &amp;quot;get away&amp;quot; undetected?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is a question that came up recently in my class as well. The following function seems to get away from both tests for symmetry over the pole undetected:&#xA;&lt;span class=&quot;math-container&quot;&gt;$r=\cos\frac{\theta}{2}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you replace &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$-r$&lt;/span&gt;, it yields &lt;span class=&quot;math-container&quot;&gt;$r=-\cos\frac{\theta}{2}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Instead, if you were to replace &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$\theta+\pi$&lt;/span&gt; (or, alternatively &lt;span class=&quot;math-container&quot;&gt;$\theta-\pi$&lt;/span&gt;), it results in &lt;span class=&quot;math-container&quot;&gt;$r=\sin\frac{\theta}{2}$&lt;/span&gt; (or, alternatively, &lt;span class=&quot;math-container&quot;&gt;$r=-\sin\frac{\theta}{2}$&lt;/span&gt;, for the other substitution).&lt;/p&gt;&#xA;&lt;p&gt;Taking some specific points as examples, we see that &lt;span class=&quot;math-container&quot;&gt;$(1,0)$&lt;/span&gt; is on the graph and satisfies the equation, but the ordered pair &lt;span class=&quot;math-container&quot;&gt;$(-1,0)$&lt;/span&gt; does not satisfy the equation. Neither do &lt;span class=&quot;math-container&quot;&gt;$(1,\pi)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(1,-\pi)$&lt;/span&gt;. The ordered pair that does satisfy the equation and is across the pole from &lt;span class=&quot;math-container&quot;&gt;$(1,0)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$(-1,2\pi)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I suppose we could make a more rigorous version of the tests that also took into account coterminal angles.&lt;/p&gt;&#xA;" OwnerUserId="23849" LastActivityDate="2024-03-19T10:12:00.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27617" PostTypeId="1" CreationDate="2024-03-20T15:25:15.240" Score="10" ViewCount="3287" Body="&lt;p&gt;In many universities there are honors math classes. For example, instead of having five &amp;quot;mixed&amp;quot; Calculus I sections they arrange one &amp;quot;honors&amp;quot; class and four &amp;quot;ordinary&amp;quot; classes.&lt;/p&gt;&#xA;&lt;p&gt;How effective is this? What are the arguments for and against such arrangements? Is there any research showing the effectiveness of such an approach?&lt;/p&gt;&#xA;&lt;p&gt;My personal view (from teaching in a regional university) is generally against the practice. I prefer all &amp;quot;mixed&amp;quot; sections, thinking that the presence of well-prepared and gifted students among the general group of students helps both. I also think honors credit are better to be given to students who work on a project with a professor in a personal setting.&lt;/p&gt;&#xA;&lt;p&gt;Edit:&lt;/p&gt;&#xA;&lt;p&gt;In USA the terminology of &amp;quot;honors&amp;quot; is used to refer to classes of well-prepared and motivated students who take the same course at a higher level. There are no standard terminology for what I referred to as &amp;quot;mixed&amp;quot; or &amp;quot;ordinary&amp;quot;. Typically no adjectives are used, or they may be referred to as &amp;quot;non-honors&amp;quot;.&#xA;Sometimes terminology such as &amp;quot;STEM College Algebra&amp;quot;, or &amp;quot; Calculus for majors&amp;quot; etc. may be used to note a course with higher standards than the average.&lt;/p&gt;&#xA;&lt;p&gt;The expectations from an honors class varies very widely.&lt;/p&gt;&#xA;" OwnerUserId="1472" LastEditorUserId="1472" LastEditDate="2024-03-21T16:44:19.263" LastActivityDate="2024-03-23T03:42:25.820" Title="The effectiveness of &quot;honors&quot; classes" Tags="|gifted-students|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="27618" PostTypeId="2" ParentId="27617" CreationDate="2024-03-20T16:17:34.943" Score="23" Body="&lt;p&gt;My undergrad university had an honors math program and the point was to prepare students for top graduate programs in math. The content was far more rigorous/intense than the regular classes, which I don't think would have provided adequate preparation.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastActivityDate="2024-03-20T16:17:34.943" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27619" PostTypeId="2" ParentId="27617" CreationDate="2024-03-20T18:29:13.900" Score="6" Body="&lt;p&gt;This can play out very differently depending on the setup of the educational system that you're in. When I was an undergrad at Berkeley in the 80's, honors classes were much more rigorous than ordinary classes, they were only intended for students intending to major in the subject, there was an entrance exam to get into them, and they used a different textbook. But at the California community college from which I recently retired, honors classes were expected to use the same book and cover the same material as non-honors classes. Any student could enroll in an honors class, and there were explicit rules saying that an honors class was not supposed to have more work or more difficult work.&lt;/p&gt;&#xA;&lt;p&gt;If one was to measure the effectiveness of such a program, how would it be defined? By whether students gain self-esteem and have transcripts that look more impressive? By how they do on a standardized test? By how many of them end up having research careers?&lt;/p&gt;&#xA;" OwnerUserId="23851" LastActivityDate="2024-03-20T18:29:13.900" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27620" PostTypeId="2" ParentId="19069" CreationDate="2024-03-21T03:10:48.587" Score="0" Body="&lt;p&gt;I have been using &amp;quot;polypad&amp;quot; and it allows you to demonstrate online geometric constructions using a compass and straight edge.  In addition, you can use a graphics tablet with it to ink on your construction with a virtual pen.  It is free and requires no installation.&lt;/p&gt;&#xA;" OwnerUserId="23852" LastActivityDate="2024-03-21T03:10:48.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27623" PostTypeId="1" CreationDate="2024-03-22T17:42:00.627" Score="5" ViewCount="419" Body="&lt;p&gt;I am 25 and have been studying mathematics on my own for several years, but I am still between the middle and high school levels.&lt;/p&gt;&#xA;&lt;p&gt;My main weakness is my dyslexia. I sometimes forget words or confuse them without even realizing it, unless someone points out that I'm talking nonsense.&lt;/p&gt;&#xA;&lt;p&gt;I also have a big lack of intuition.&#xA;For example, I know that an equation like &lt;span class=&quot;math-container&quot;&gt;$x\mapsto ax+b$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$a,b\in R$&lt;/span&gt; represents a line. I can prove it, but when faced with a concrete case, I struggle to see the line behind the symbols. It's the same with the set of points &lt;span class=&quot;math-container&quot;&gt;$(x,ax+b)$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; describes &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If I see written &lt;span class=&quot;math-container&quot;&gt;$(a+b)^2=a^2+b^2$&lt;/span&gt;, I might not necessarily realize the mistake, even though I have known the identity &lt;span class=&quot;math-container&quot;&gt;$(a+b)^2= a^2+2ab+b^2$&lt;/span&gt; for years. It's not intuitive for me compared to &lt;span class=&quot;math-container&quot;&gt;$1=2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c&amp;gt;d$&lt;/span&gt;, I know I can't write &lt;span class=&quot;math-container&quot;&gt;$a/c&amp;gt;b/d$&lt;/span&gt; but once again it's &amp;quot;by heart&amp;quot;. I am perfectly capable of making the mistake and not noticing it despite several reviews.&#xA;Again, I am capable of proving it's false, but it's not obvious like &lt;span class=&quot;math-container&quot;&gt;$1=2$&lt;/span&gt;.&#xA;I've tried to turn it over in my head by visualizing the real line, visualizing a histogram, etc.&lt;/p&gt;&#xA;&lt;p&gt;Not long ago, I got stuck on a problem because I didn't recognize that &lt;span class=&quot;math-container&quot;&gt;$2n + 1$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is an integer, is just the sum of two consecutive integers. It's silly, but that's how it is with everything.&lt;/p&gt;&#xA;&lt;p&gt;Despite my efforts to prove everything, to make analogies, spaced repetitions over time, apply the Feynman technique, and others, I cannot develop a mathematical intuition. I also tried tutoring without noticing any progress.&lt;/p&gt;&#xA;&lt;p&gt;After reading &amp;quot;The Prince of Mathematics&amp;quot; about Gauss, I tried to construct mathematics by myself and derive new results, but my limited intuition and creativity didn't lead to success.&lt;/p&gt;&#xA;&lt;p&gt;My thinking is mainly through an internal monologue. I've tried to slow it down to promote more intuitive thinking, in images or sensations, but it wasn't conclusive.&lt;/p&gt;&#xA;&lt;p&gt;To be precise, I've been back to studying mathematics for four years, dedicating an average of 15 to 20 hours a week to this discipline, but I often revise the same concepts without seeing any evolution. I even feel like I'm regressing at times. I know there are many teachers on this forum. Have you ever encountered a student facing such difficulties? If so, how did you proceed to help them?&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="23859" LastActivityDate="2024-03-22T22:19:32.457" Title="Overcoming Dyslexia and Building Intuition" Tags="|self-learning|intuition|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27624" PostTypeId="2" ParentId="27623" CreationDate="2024-03-22T19:12:02.847" Score="6" Body="&lt;p&gt;This might be an unpopular answer but I'm going to try to be honest and real with you.&lt;/p&gt;&#xA;&lt;p&gt;Math gets hard and unintuitive for everyone at some point, and that point is different for everyone -- for some people that's algebra, for other people it's calculus, for others it's real analysis, for others it's algebraic topology, for others it's research-level math, ... you get the idea. There's this same gradation even within research-level math problems.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes people hit that point early due to ineffective or inconsistent practice techniques. But even if you practice effectively and consistently, a ceiling still exists.&lt;/p&gt;&#xA;&lt;p&gt;It's kind of like sports. Few people practice effectively and consistently enough to reach their athletic potential, but it's just a fact of life that most people could not become professional basketball players even with 100% effective and consistent practice.&lt;/p&gt;&#xA;&lt;p&gt;Are you at your mathematical ceiling? I guess we can't say with 100% certainty. But if you're practicing effectively and consistently (which it sounds like you are), even sometimes with a tutor, and you're still stuck in a plateau, then it sure seems likely.&lt;/p&gt;&#xA;&lt;p&gt;In general, when you feel yourself running up against a ceiling in life, the solution is typically to pivot and into a direction where the ceiling is higher.&lt;/p&gt;&#xA;&lt;p&gt;For instance, the story of many a quantitative software engineer goes like this:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;loved math growing up and wanted to be a mathematician&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;realized during undergrad or grad school that they had lost their &amp;quot;edge&amp;quot; compared to other aspiring mathematicians&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;also realized that they have a knack for coding and interest in some applied domain, and that the problems that need to be solved there boil down to interesting math that most people in software don't have the math chops for&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;pivoted in that direction where their ceiling is higher&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum.&lt;/strong&gt; In the comments, OP mentions that their initial milestone is to solve certain problems from the Math Olympiads, and their overall goal is to actually understand mathematics.&lt;/p&gt;&#xA;&lt;p&gt;One thing that OP might try is learning more advanced math subjects (e.g. calculus, linear algebra, differential equations, etc) as opposed to focusing on problems from Math Olympiads. Competition math is very g-loaded, and the point is separate students who have loads of underlying mathematical talent from students who learned a bunch of math early but don't have as much talent.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps counterintuitively, it's a lot easier to make progress by continuing into more advanced subjects and layering on top of your content knowledge than it is to make progress by focusing on competition math and hoping that some of the insight you see in the solutions will rub off on you.&lt;/p&gt;&#xA;&lt;p&gt;Even for measuring progress in general, I don't think Math Olympiad performance is the right metric to use. It's pretty common for people to learn and apply university-level math yet not be able to solve competition problems. You could very well improve your mathematical knowledge and capability to do things with math, and not have it carry over much to competition problems.&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-03-22T22:19:32.457" LastActivityDate="2024-03-22T22:19:32.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27625" PostTypeId="2" ParentId="27623" CreationDate="2024-03-22T22:04:59.260" Score="3" Body="&lt;p&gt;I am a student in mathematics, and I've always been told that mathematics is not for everyone and that our brain is &amp;quot;weird&amp;quot; or works differently so that we can do maths. But as I read maths books or papers for my classes, I've come to a conclusion. Yes, maths is hard, it's abstract and in order to understand difficult concept, you need to build your knowledge and to practice a lot, but mathematician makes it even harder with phrases such as &amp;quot; it follows directly from...&amp;quot; or &amp;quot; it's easy to see that...&amp;quot; well I'm sorry it's not always that easy to see...&lt;/p&gt;&#xA;&lt;p&gt;I don't consider myself as someone that has a lot of mathematical intuition, when I speak to my peers or with my teachers about maths, I don't understand what they say. I have to write down every concept and dissect them in order to fully understand them. When I see new maths, I don't get it right away. To make an analogy with you, If I see &lt;span class=&quot;math-container&quot;&gt;$(a+b)^2$&lt;/span&gt; I don't see right away that this is &lt;span class=&quot;math-container&quot;&gt;$a^2+2ab+b^2$&lt;/span&gt;, I would have to calculate it. It may take more time, but in the end I can still do the same maths as the people that see it right away.&lt;/p&gt;&#xA;&lt;p&gt;What I'm trying to say is that If you tend to make a lot of mistakes and that you find it hard to have a &amp;quot;mathematical intuition&amp;quot; then you have to find a method to cope with those &amp;quot;weakness&amp;quot;, what I would suggest is that you stop to try to visualize concepts when you can't. Instead, find a method that works for you. For example, when you see an equation like &lt;span class=&quot;math-container&quot;&gt;$y=ax+b$&lt;/span&gt; you start to ask you those questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Have I seen such an equation before ?&#xA;--&amp;gt; yes this is a line&lt;/li&gt;&#xA;&lt;li&gt;What are the properties of a line?&#xA;--&amp;gt; write down everything that you know about lines&lt;/li&gt;&#xA;&lt;li&gt;Now that you have all the properties of the lines in front of you, ask yourself which properties might be helpful in the settings of your particular problem?&lt;/li&gt;&#xA;&lt;li&gt;Resolve your problem!&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I use this technique all the time: when I want to resolve a maths problem, I first analyse the mathematical objects of the problem and I write down everything I know about them (for that I take my textbooks or even Wikipedia since I don't remember everything). Then I have all the tools in front of me to resolve my problem, so it's much easier to see what to use and have this &amp;quot;intuition&amp;quot; you're talking about!&lt;/p&gt;&#xA;&lt;p&gt;Maths is not that hard, it's just a lot of different tools, the hard part is how to use them together to create new tools!&lt;/p&gt;&#xA;" OwnerUserId="23555" LastActivityDate="2024-03-22T22:04:59.260" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27626" PostTypeId="2" ParentId="27617" CreationDate="2024-03-23T03:42:25.820" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;How effective is this?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Mathematicians have difficulty understanding how the practice of requiring nearly everyone to take calculus leads inevitably to disastrous results. In fact, they have difficulty even realizing this is happening, after seeing it every day for over half a century.&lt;/p&gt;&#xA;&lt;p&gt;Since this kind of subject matter is being taught to students who are not there out of any desire to understand this subject, but who are there in order to get an impressive grade, the result is that mathematicians have given up on the impossible and undesirable task of coercing people to understand, and instead they teach them to execute algorithms with no understanding. And it is those who work the hardest at that, who end up in &amp;quot;honors&amp;quot; courses. The mathematicians who organize these see that those students aren't really better than the others and the whole thing isn't working, and at least have enough sense to abolish such &amp;quot;honors&amp;quot; courses.&lt;/p&gt;&#xA;&lt;p&gt;As for gifted students, they learned calculus before their first year at university.&lt;/p&gt;&#xA;&lt;p&gt;It is unethical to design the curriculum for the purpose of extracting the few lumps of gold from a few tons of dust. If mathematics is to be taught to broad masses whose interests are in other subjects, it should be designed to teach them things they will understand and things they will use. Otherwise those broad masses of students are being abused and defrauded.&lt;/p&gt;&#xA;&lt;p&gt;There are &lt;em&gt;some&lt;/em&gt; universities at which the &amp;quot;honors&amp;quot; classes may work they way those who design them hope they will work. And it is very difficult to tell which universities those are without talking to people at each university and knowing which questions to ask. You can't tell from university web sites or any sort of published information.&lt;/p&gt;&#xA;" OwnerUserId="205" LastActivityDate="2024-03-23T03:42:25.820" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="27627" PostTypeId="2" ParentId="27610" CreationDate="2024-03-23T06:00:26.947" Score="3" Body="&lt;p&gt;Do it in two easy steps:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove that &lt;span class=&quot;math-container&quot;&gt;$V[X+b]=V[X]$&lt;/span&gt;. This one is easy to prove since variance is a measure of deviation from the mean, hence change of origin will not affect it.&#xA;Mathematically, &lt;span class=&quot;math-container&quot;&gt;$ E({X+b})=E(X)+b$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$V[X+b]  = E[((X+b)-E(X+b))^2] = E[(X-E(X))^2] = V[X] $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Prove that &lt;span class=&quot;math-container&quot;&gt;$V[aX]=a^2V[X]$&lt;/span&gt;. This one is easy to prove as well. Start with &lt;span class=&quot;math-container&quot;&gt;$E[aX]=aE[X]$&lt;/span&gt;. Then,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ V[aX]  = E[((aX)-E(aX))^2] = E[a^2(X-E(X))^2]  = a^2V[X] $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="20389" LastActivityDate="2024-03-23T06:00:26.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27628" PostTypeId="1" CreationDate="2024-03-23T19:54:45.897" Score="17" ViewCount="2932" Body="&lt;p&gt;There are a lot of sorting algorithms to sort a list on a computer, and a lot of theory about them. However, my problem is not how to sort a list under the quite well defined conditions of a computer, but to sort alphabetically a pile of exams (made on paper) after grading them.&lt;/p&gt;&#xA;&lt;p&gt;I must admit that I'll never amortise the effort of optimising a task that just takes a fraction of an hour a few times a year, but that fraction of an hour is boring and I can't avoid thinking about the problem of optimising the task while I'm at it, and getting interested in that problem. I hope the question is in topic here - more than in ComputerScience.SE or even LifeHacks.SE - and I think I'm more likely to find here users with related experience and expertise.&lt;/p&gt;&#xA;&lt;p&gt;The question, formatted in a general way, is &lt;strong&gt;what's the most efficient and practical way to sort a set of papers&lt;/strong&gt;. If details matter, we can assume we are ordering one hundred to a few hundreds of papers, and that we have a table but not a very large one - we can't put all the papers on the table to see all of them at the same time. However, the order key is in the top of the paper (we can assume the top inch), so we can pile a few papers and still be able to see the key. In any way, the conditions are different than those for a computer algorithm and they are less well defined.&lt;/p&gt;&#xA;&lt;p&gt;This problem has two variants:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The order key is the student's name - that is, given any two exams, there can be more exams that must be placed between them.&lt;/li&gt;&#xA;&lt;li&gt;The order key is a natural number - that is, if we find exams with successive numbers, we can be sure that no other exam will go between them.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So far, I think the most efficient way I use is to order small batches of exams - small enough that I can hold them in my hand or put them on the table while being able to see all order keys or to find them quickly - and then join batches pairwise until I get a single batch.&lt;/p&gt;&#xA;&lt;p&gt;A different approach I've used is to bin the exams in small batches according to initial letter or by another way of making similar sized intervals, and then order every batch.&lt;/p&gt;&#xA;" OwnerUserId="7178" LastEditorUserId="7178" LastEditDate="2024-03-23T21:11:49.387" LastActivityDate="2024-03-27T11:07:22.847" Title="What's the most practical and efficient way to sort exams on paper?" Tags="|computer-science|" AnswerCount="11" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="27629" PostTypeId="2" ParentId="27628" CreationDate="2024-03-23T21:50:59.103" Score="4" Body="&lt;p&gt;I use a set of folders (like this):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B075F1N2DM&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Expanding-Multi-Color-Accordion-Organizer-Expandable/dp/B075F1N2DM/&lt;/a&gt;  (Can kludge your own also...a milk crate works OK.  Or use a filing drawer, that is low enough).&lt;/p&gt;&#xA;&lt;p&gt;I find myself able to file papers alpha within each letter folder as I file by letter.  But you could go back later if you prefer.  This is helped if you have the name field, printed in a prominent location (and last name, first name ordered prompts).&lt;/p&gt;&#xA;" OwnerUserId="23863" LastEditorUserId="23863" LastEditDate="2024-03-23T21:51:48.547" LastActivityDate="2024-03-23T21:51:48.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27630" PostTypeId="2" ParentId="27628" CreationDate="2024-03-23T22:53:30.023" Score="23" Body="&lt;p&gt;An eminently pragmatic question about sorting! :)&lt;/p&gt;&#xA;&lt;p&gt;When I was dealing with 150+ papers, on a not-too-large table-top, after various experiments, I found that subsets of 10 or 12 I could visually sort fairly easily. Then, with 10-ish stacks of already-ordered papers, it was feasable for me to visually choose things in alpha order, to merge them/&lt;/p&gt;&#xA;&lt;p&gt;That is, a two-stage (is it &amp;quot;merge&amp;quot;?) sort is pretty fairly human executable, for 100-ish things. :)&lt;/p&gt;&#xA;&lt;p&gt;EDIT: in particular, after the first sorting of the smaller piles, those smaller piles do not have to have &lt;em&gt;all&lt;/em&gt; the names visible, since the &amp;quot;merge&amp;quot; step only needs to know about the &lt;em&gt;top&lt;/em&gt; item in each sub-collection. So, in the merge step, it does not matter where on the page the students' names are, nor is there any need for delicate physical configuration.&lt;/p&gt;&#xA;" OwnerUserId="63" LastEditorUserId="63" LastEditDate="2024-03-24T16:34:07.693" LastActivityDate="2024-03-24T16:34:07.693" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27631" PostTypeId="2" ParentId="27628" CreationDate="2024-03-24T00:57:53.230" Score="12" Body="&lt;p&gt;For hundreds of exams, I'm a fan of your second proposed approach, specifically, break up the alphabet into &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; intervals where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is the number of piles you can reach simultaneously without moving, such that the number of students who fall into each interval is roughly even, then sort each pile. Depending on the situation, you may be able to get students to hand in their exams into different piles by first letter of their name.&lt;/p&gt;&#xA;&lt;p&gt;For slightly smaller piles of exams (30-70), I prefer to simultaneously enter grades into my spreadsheet and sort. At each stage of the sorting process, every alphabetically consecutive group of exams should form a pile with each exam stacked directly on top of the next. I stack these piles fanning vertically so that for each pile I can see the name on the exam on top of the pile. When I enter the grade for a new exam into my spreadsheet, I branch:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If I haven't yet entered the score for the student before or after this one into my grade book, I make a note of the next student's name whose grade I have already entered, and start a new pile between the pile with that student's name on top and the previous pile (I also check whose name should be on top of that pile to make sure I haven't made any mistakes).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If I have entered the next student's score, but not the previous student's score, the exam goes directly on top of the pile that the next student's exam is currently on top of.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If I have entered the previous students' score, but not the next student's score, the exam goes on the bottom of a consecutive pile: look back through the spreadsheet for the first student in the consecutive group you just added to, then find the pile with their exam on top.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If I have entered both the previous and next students' scores, it joins two consecutive intervals of students within the spreadsheet. I place the exam between the two piles and merge them together.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The amount of physical space this takes up depends on the maximum amount of piles necessary at any one time during the process. If there are &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; students, what is the expected maximum number of piles necessary?&lt;/p&gt;&#xA;" OwnerUserId="15671" LastActivityDate="2024-03-24T00:57:53.230" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27632" PostTypeId="1" CreationDate="2024-03-24T12:43:44.580" Score="-1" ViewCount="432" Body="&lt;p&gt;In a recently viewed educational video focused on the concept of spinors (available at this link:&lt;a href=&quot;https://youtu.be/b7OIbMCIfs4?si=5ZZLxdGotxAj6YwP&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/b7OIbMCIfs4?si=5ZZLxdGotxAj6YwP&lt;/a&gt; ), an intriguing visual representation caught my attention – it features a captivating image of a 'pencil of bands' at the very beginning. This visual is not only aesthetically pleasing but also appears to hold significant mathematical intrigue, potentially offering a tangible exploration path for abstract concepts in multivariate calculus.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/X4YcS.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/X4YcS.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm keen on leveraging this visual by parametrizing the pencil of bands, aiming to integrate this into GeoGebra for an interactive learning experience. My objective is to facilitate a deeper understanding among my students, making abstract mathematical principles more accessible through visual and interactive means.&lt;/p&gt;&#xA;&lt;p&gt;My question is: How can I effectively parametrize this pencil of bands for educational implementation in GeoGebra? I'm looking for a mathematical approach that translates this visual phenomenon into a parametric form, suitable for exploration and manipulation within the GeoGebra environment.&lt;/p&gt;&#xA;&lt;p&gt;This endeavor not only aims at enhancing the learning experience but also at fostering a deeper appreciation and understanding of the interconnectedness of mathematical concepts, as seen through the lens of real-world applications and visual mathematics.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastEditorUserId="63" LastEditDate="2024-03-24T19:19:50.517" LastActivityDate="2024-03-25T02:29:45.880" Title="how to parametrize these bands related with spinors?" Tags="|vector-calculus|" AnswerCount="3" CommentCount="4" ClosedDate="2024-03-25T21:57:32.947" ContentLicense="CC BY-SA 4.0" />
  <row Id="27633" PostTypeId="2" ParentId="27632" CreationDate="2024-03-24T14:35:25.797" Score="3" Body="&lt;p&gt;I strongly suspect that it would be very difficult to do this in geogebra and be unsuitable for most students in multivariable calculus.  Maybe if you have some advanced math students with computer skills you could pull it off?&lt;/p&gt;&#xA;&lt;p&gt;The thing that's being done is to start with a bunch of rays from the origin, parameterized. Then you think about parameterizing 3-space like this: on each sphere &lt;span class=&quot;math-container&quot;&gt;$S_r$&lt;/span&gt; or radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; about the origin, you act by an element of &lt;span class=&quot;math-container&quot;&gt;$SO(3)$&lt;/span&gt;. Call that element &lt;span class=&quot;math-container&quot;&gt;$U_r$&lt;/span&gt;. As long as &lt;span class=&quot;math-container&quot;&gt;$r \to U_r$&lt;/span&gt; is a continuous map &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^+ \to SO(3)$&lt;/span&gt;, you get a continuous transformation of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;. Now, what you want is a family of these things where at &lt;span class=&quot;math-container&quot;&gt;$t=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t=1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(r,t)\to U_{r,t}$&lt;/span&gt; is just the map to the identity. You also want the &lt;span class=&quot;math-container&quot;&gt;$r=0$&lt;/span&gt; map &lt;span class=&quot;math-container&quot;&gt;$(0,t)\to U_{0,t}$&lt;/span&gt; to be two full rotations about an axis and for &lt;span class=&quot;math-container&quot;&gt;$r\gg 0$&lt;/span&gt; the map should be constant at the identity.&lt;/p&gt;&#xA;&lt;p&gt;In other words, you need to provide an explicit parameterization of the homotopy of the square of the generator of &lt;span class=&quot;math-container&quot;&gt;$\pi_1(SO(3),I_3)$&lt;/span&gt; to the identity. I think that this might be the toughest part, but with an appropriate model of &lt;span class=&quot;math-container&quot;&gt;$SO(3)$&lt;/span&gt;, it should be easier. (Maybe view it as a quotient of &lt;span class=&quot;math-container&quot;&gt;$SU(2)=S^3$&lt;/span&gt; to ease things. The homotopy should be one of the hemispheres of an equitorial &lt;span class=&quot;math-container&quot;&gt;$S^2$&lt;/span&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;Once you have that, it's a matter of composing the rays with this thing, thought of as a parameterized family of diffeomorphisms of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Maybe appropriate for a semester long or capstone project for an advanced student. Make sure you do it yourself first.&lt;/p&gt;&#xA;" OwnerUserId="4791" LastActivityDate="2024-03-24T14:35:25.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27634" PostTypeId="2" ParentId="27632" CreationDate="2024-03-24T19:14:21.047" Score="6" Body="&lt;p&gt;I don't know anything about how the paths were parametrized in the video. If the OP could post the formulas, it would help. But the image in the video looks roughly like this:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;(* Wolfram Language *)&#xA;Block[{v, sp},&#xA; v = {1., 2., 3};&#xA; (*sp = SpherePoints[32];*)&#xA; sp = RandomPoint[Sphere[], 80];&#xA; Show[&#xA;  ParametricPlot3D[&#xA;   Evaluate@Table[&#xA;     r*RotationMatrix[4 Pi*Erf[r/2], v] . p,&#xA;     {p, sp}]&#xA;   , {r, 1, 30}&#xA;   , PlotStyle -&amp;gt; Directive[Specularity[2], Tube[0.05]], &#xA;   Background -&amp;gt; Black, Axes -&amp;gt; False, Boxed -&amp;gt; False, &#xA;   PlotRange -&amp;gt; 20&#xA;   ],&#xA;  Graphics3D[{White, Sphere[]}]&#xA;  ]&#xA; ]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/f5cO8.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/f5cO8.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The image changes randomly on re-execution.&lt;/p&gt;&#xA;" OwnerUserId="498" LastActivityDate="2024-03-24T19:14:21.047" CommentCount="2" CommunityOwnedDate="2024-03-24T19:14:21.047" ContentLicense="CC BY-SA 4.0" />
  <row Id="27635" PostTypeId="2" ParentId="27632" CreationDate="2024-03-24T20:17:07.380" Score="4" Body="&lt;p&gt;Here is a way to approach this. First we define&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$R(\alpha,s)=\left[\begin{matrix}\cos{\left(\frac{\alpha}{s} \right)} &amp;amp; - \sin{\left(\frac{\alpha}{s} \right)} &amp;amp; 0\\\sin{\left(\frac{\alpha}{s} \right)} &amp;amp; \cos{\left(\frac{\alpha}{s} \right)} &amp;amp; 0\\0 &amp;amp; 0 &amp;amp; 1\end{matrix}\right]$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Notice that &lt;span class=&quot;math-container&quot;&gt;$R(\alpha,1)$&lt;/span&gt; is rotation around the &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;-axis through an angle of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$R(\alpha,\infty)$&lt;/span&gt; is the identity rotation.&lt;/p&gt;&#xA;&lt;p&gt;Now we concentrate on a ray starting on the unit sphere and going out to infinity. We can parameterize the unit sphere by&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*} X &amp;amp;= \cos(\theta) \sin(\phi) \\ Y &amp;amp;= \sin(\theta) \sin(\phi) \\ Z &amp;amp;= \cos(\phi) \end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where the angle &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; ranges from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A ray starting at a point &lt;span class=&quot;math-container&quot;&gt;$(X,Y,Z)$&lt;/span&gt; on the unit sphere and extending to infinity will be parametrized by &lt;span class=&quot;math-container&quot;&gt;$(s X,&#xA;s Y, s Z)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$1\le s&amp;lt;\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We apply the rotation &lt;span class=&quot;math-container&quot;&gt;$R(\alpha,s)$&lt;/span&gt; to the point &lt;span class=&quot;math-container&quot;&gt;$(sX, sY, sZ)$&lt;/span&gt;. This twists the sphere of radius &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; by the angle &lt;span class=&quot;math-container&quot;&gt;$\alpha/s$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To get a moving curve, we fix &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; to specify a point &lt;span class=&quot;math-container&quot;&gt;$(X,Y,Z)$&lt;/span&gt; on the unit sphere, and therefore also the ray. Then we let the animation parameter be &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, ranging from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$4\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is the equation for the moving curve. We fix &lt;span class=&quot;math-container&quot;&gt;$\theta,\phi$&lt;/span&gt;. This tells us where the curve initially attaches itself to the unit sphere. For fixed &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, we have a parametric curve with parameter &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;. As &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; varies, the point of attachment moves twice around a circle of constant latitude. The point of attachment rotates through an angle &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, whereas points far out on the ray (&lt;span class=&quot;math-container&quot;&gt;$s=\infty$&lt;/span&gt;) are fixed.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\left[\begin{matrix}- s \sin{\left(\phi \right)} \sin{\left(\theta \right)} \sin{\left(\frac{\alpha}{s} \right)} + s \sin{\left(\phi \right)} \cos{\left(\theta \right)} \cos{\left(\frac{\alpha}{s} \right)}\\s \sin{\left(\phi \right)} \sin{\left(\theta \right)} \cos{\left(\frac{\alpha}{s} \right)} + s \sin{\left(\phi \right)} \sin{\left(\frac{\alpha}{s} \right)} \cos{\left(\theta \right)}\\s \cos{\left(\phi \right)}\end{matrix}\right]$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I picked some random values of &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; and set up the animation in Python and all seems OK but I don't want to take time to polish the graphics and post here. The project would be accessible to students in a multi-variable calculus course, assuming they have some fortitude.&lt;/p&gt;&#xA;" OwnerUserId="1680" LastEditorUserId="1680" LastEditDate="2024-03-25T02:29:45.880" LastActivityDate="2024-03-25T02:29:45.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27636" PostTypeId="2" ParentId="27628" CreationDate="2024-03-25T12:11:06.113" Score="5" Body="&lt;p&gt;About four times per year, after correcting them, we had to sort 600+ exams by student numbers, which was a 6 digit number given consecutively to all students at enrolment, so kind of the worst case, since it is not equidistributed in the first digits, and also has unpredictable gaps. Still I think over time we got quite good at it.&lt;/p&gt;&#xA;&lt;p&gt;For me it has always been practical to first do a bin sort (e.g. by first and possibly second digit) until the individual bins are down to &amp;lt;20 exams and then sort the individual stacks by insertion sort. This has the advantage that both steps can be distributed to multiple people (just remember to combine multiple stacks belonging to the same bin before doing the insertion part).&lt;/p&gt;&#xA;&lt;p&gt;One important thing though is that the sort key should be clearly visible at the margin, so the insertion step can be done while only slightly fanning the stack.&lt;/p&gt;&#xA;" OwnerUserId="8988" LastActivityDate="2024-03-25T12:11:06.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27637" PostTypeId="2" ParentId="27628" CreationDate="2024-03-25T14:57:37.743" Score="5" Body="&lt;p&gt;I suggest starting with a small table and a large floor area.  Sort by the first letter of the name into piles on the floor, grouping rare letters (so at least Q in with P and WXYZ together for typical British surnames, which are what I know best).  You can reach a larger area on the floor than at any typical table anyway.&lt;/p&gt;&#xA;&lt;p&gt;This is an intuitive way to divide them into bins.&lt;/p&gt;&#xA;&lt;p&gt;Then each pile should be quite easily sortable by whatever method you choose.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Applying a little Python and curiosity to the problem, and using &lt;a href=&quot;https://one-name.org/modern-british-surnames/statistics/top-surnames/england-wales/top-500-names/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this list of the top 500 surnames in England and Wales&lt;/a&gt; (which accounts for just under 40% of people's surnames) I suggest grouping DE, KL, NO, PQ, TUV, and XYZ for a manageable total of 18 piles, adding up to 0.2--4.2% of surnames based on that list.&lt;/p&gt;&#xA;&lt;p&gt;Making the massive assumption that the long tail of surnames has the same distribution of first letters, you would have something like 0.5 (XYZ)--10.5%(W) of your scripts in each pile.  W, B, H, and S are then your biggest piles, in that order.  These may need further sorting in a big cohort, noting that the distribution of second letters is far more restricted (e.g. anything other than a vowel, H, or R is highly unlikely after W with similar patterns  for the other common first letters, so this 2nd pass would result in less than 10 extra piles at a time).&lt;/p&gt;&#xA;&lt;p&gt;You could then insertion sort each pile before stacking them in order. In a small cohort (or for rare letters in a larger one) you may get away with no further sorting, and rely on searching a small disordered set as required.  This latter approach is sometimes taken in bookshops, where a full author-alphabetical sort is too time-consuming; it's only moderately annoying to those browsing.&lt;/p&gt;&#xA;" OwnerUserId="8180" LastEditorUserId="8180" LastEditDate="2024-03-27T10:40:42.473" LastActivityDate="2024-03-27T10:40:42.473" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27638" PostTypeId="2" ParentId="27628" CreationDate="2024-03-25T15:10:40.520" Score="4" Body="&lt;p&gt;I think that &lt;a href=&quot;https://en.wikipedia.org/wiki/Quicksort&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quicksort&lt;/a&gt; is very suitable for working by hand.  It's reasonably expensive in space (you need a largeish table), but takes only &lt;span class=&quot;math-container&quot;&gt;$O(n\log n)$&lt;/span&gt; comparisons on average, which is pretty good.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Put an arbitrary paper (eg, the top one) in the middle of the table (the ‘pivot’), and form a pile on the left of all those ordered before that one, and on the right a pile of all those ordered after.&lt;/li&gt;&#xA;&lt;li&gt;Recurse.&lt;/li&gt;&#xA;&lt;li&gt;When you end up with a pile containing only a few papers to sort – say half-a-dozen – sort them in-hand.&lt;/li&gt;&#xA;&lt;li&gt;When you have two sorted piles separated by a pivot, combine them into a single one by stacking left-pile above pivot above right-pile.  You now have a larger sorted pile.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It's significantly easier to do than to explain.  It's also highly parallelisable, in the sense that, while one person is working on left-pile, someone else can be working the same process completely independently on right-pile.&lt;/p&gt;&#xA;&lt;p&gt;Just don't anyone sneeze....&lt;/p&gt;&#xA;" OwnerUserId="23866" LastActivityDate="2024-03-25T15:10:40.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27639" PostTypeId="2" ParentId="27628" CreationDate="2024-03-25T17:58:21.930" Score="0" Body="&lt;p&gt;Consider &lt;a href=&quot;https://en.wikipedia.org/wiki/Radix_sort&quot; rel=&quot;nofollow noreferrer&quot;&gt;radix sort&lt;/a&gt; when you have a numerical student ID.&lt;/p&gt;&#xA;" OwnerUserId="4984" LastActivityDate="2024-03-25T17:58:21.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27640" PostTypeId="2" ParentId="27628" CreationDate="2024-03-25T23:09:22.727" Score="4" Body="&lt;p&gt;For the case when the order key is a natural number: use a jig and a hole punch to punch each paper's key into the top of the paper in binary. Encode a 1 by punching a single hole a short distance from the top edge of the paper, and a 0 by making two overlapping punches so that the &amp;quot;hole&amp;quot; extends to the top edge. The punches for each bit position should be in the same place on each paper, and the next step will probably be easier if they're all generally towards the top-middle.&lt;/p&gt;&#xA;&lt;p&gt;Then, stack the papers up and put a needle through the least-significant bit hole. Lift the needle up so that the papers with a 1 in the LSB come with it, and the papers with a 0 are left behind. Slide the &amp;quot;1&amp;quot; papers off of the needle and stack them neatly behind the &amp;quot;0&amp;quot; papers, then repeat with the next-more-significant bit.&lt;/p&gt;&#xA;&lt;p&gt;After you've done this for the most-significant bit, the papers will be in order. This is, of course, radix sort.&lt;/p&gt;&#xA;&lt;p&gt;The hole-punching phase is O(n log m) operations, where n is the number of papers and m is the number of possible keys. The select-and-restack phase is O(log m) operations.&lt;/p&gt;&#xA;&lt;p&gt;In the case where the papers are numbered sequentially 1 to n, this simplifies to O(n log n) for the first phase and O(log n) for the second phase.&lt;/p&gt;&#xA;&lt;p&gt;In the case where the keys are arbitrary and non-sequential, but they're drawn from some finite set that can be considered a priori fixed, then this simplifies to O(n) for the first phase and O(1) for the second phase.&lt;/p&gt;&#xA;" OwnerUserId="10195" LastActivityDate="2024-03-25T23:09:22.727" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="27641" PostTypeId="2" ParentId="27628" CreationDate="2024-03-26T13:33:47.600" Score="7" Body="&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://rpg.meta.stackexchange.com/a/6843/&quot;&gt;Frame Challenge:&lt;/a&gt;&lt;/strong&gt;  Don't bother.&lt;/p&gt;&#xA;&lt;p&gt;For the last five or six years, I have been using &lt;a href=&quot;https://www.gradescope.com/&quot; rel=&quot;noreferrer&quot;&gt;GradeScope&lt;/a&gt; for all of my grading.  Before that, I used &lt;a href=&quot;https://crowdmark.com/&quot; rel=&quot;noreferrer&quot;&gt;CrowdMark&lt;/a&gt;.  Both of these services allow you to upload unsorted exams, then grade them through their system.&lt;/p&gt;&#xA;&lt;p&gt;The trick is that each copy of the written exam (printed on paper) has a small QR code in the corner of each page (the software helps you to do this).  On the font page of each exam, we placed areas for students to write their names and student IDs (rows of boxes with one letter or number per box).  After the exam, we used a heavy duty paper cutter to remove the staples from exams, then a top-loading scanner to scan the exams into a .pdf.  The file(s) produced in this manner can then be uploaded to the grading service, and are automatically grouped (using the QR codes) into per-student exams, and automatically associated with a student using the identification regions on the front page.  For students whose handwriting cannot be recognized by the OCR system (typically less than 3% of exams), the association can be done manually.&lt;/p&gt;&#xA;&lt;p&gt;In addition to &amp;quot;sorting&amp;quot; exams, these systems make it very easy to work with TAs (you can assign TAs to grade certain parts of each exam, for example), and students get much more immediate feedback (as soon as you are done grading, you can publish the scores, and students can see your grading and feedback).  You also have more control over students altering exams after being graded, in that they &lt;em&gt;can't&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The major downsides I see are that these services are proprietary (though they work with universities to maintain FERPA compliance and the like, and they do integrate with most LMSes, you lose some control over the raw files), and the free versions suppress a lot of features and can't be used by very many people at a single institution.&lt;/p&gt;&#xA;&lt;p&gt;A couple of anecdotes about my own experience:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;While in graduate school, I worked closely with one of the faculty teaching precalculus.  Most semesters, he was assigned three sections of 3-400 students each (for a total of around 1000 students).  Creating the exams for printing takes maybe an extra 10-15 minutes over the usual prep time, and we could usually get all of the exams de-stapled, scanned, and uploaded in about an hour.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At my current institution, a lot of the instruction I do is remote (we have classrooms at nine locations spread across our service area; it is not uncommon for me to have students in four or five different classrooms, all at the same time).  When I proctor exams, I have my students upload the work themselves (which also ensures that exams are correctly associated with students).  Granted, my current classes are &lt;em&gt;much&lt;/em&gt; smaller (typically 10–20 students per section make it to the final), but the remote nature of instruction makes these online services invaluable (the alternative is that they are mailed to me via the campus mail system, or students email me exams).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;NB:  I know this answer looks spammy, but I genuinely do not work for either organization, nor have I been paid to advertise either.  I have used both systems, and &lt;em&gt;really&lt;/em&gt; like not having to work with paper exams anymore.&lt;/p&gt;&#xA;" OwnerUserId="8571" LastActivityDate="2024-03-26T13:33:47.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27642" PostTypeId="2" ParentId="27628" CreationDate="2024-03-26T15:31:51.410" Score="1" Body="&lt;p&gt;I find it best to sit on the floor in my living room, where there is plenty of space.  Then I distribute the scripts into about 20 piles in a circle around me according to the initial digits of the registration number.  Then I sort the piles individually and finally combine them. I think it would be significantly slower and less convenient to use a table.&lt;/p&gt;&#xA;" OwnerUserId="76" LastActivityDate="2024-03-26T15:31:51.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27643" PostTypeId="1" CreationDate="2024-03-27T04:51:03.073" Score="1" ViewCount="70" Body="&lt;p&gt;I have done couple of activities and experienced how interestingly students are engaging. At the end I explained the Mathematical concepts related to the activities. Here I'm suggesting two of those, &lt;br/&gt;&lt;/p&gt;&#xA;&lt;p&gt;Engaging Group Activities for Students:&lt;br/&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Tower of Hanoi Challenge:&lt;/strong&gt;&lt;br/&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gather a group of ten students with varying heights.&lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Assign each student a position in front of a Tower of Hanoi puzzle without verbal communication.&lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Each student in front independently decides on their move, representing the discs' movement without communicating &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;The challenge encourages strategic thinking and problem-solving without direct leadership.&lt;br/&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Fifteen Number Puzzle:&lt;/strong&gt;&lt;br/&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Create a 4x4 grid on the floor and position 16 students one in each square.&lt;/li&gt;&#xA;&lt;li&gt;Fifteen students wear caps numbered from  1 to 15 while one remains uncapped.&lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Only students adjacent to the uncapped individual can communicate with the uncapped student and interchange position with the uncapped student. &lt;br/&gt;&lt;/li&gt;&#xA;&lt;li&gt;Through collaborative decision-making, students must rearrange themselves to solve the puzzle without having a leader.&lt;br/&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These activities not only foster teamwork and critical thinking but also provide an opportunity to explore mathematical&#xA;concepts in a practical, hands-on manner. I would like to know what other activities are there to fulfill this requirement.&lt;/p&gt;&#xA;" OwnerUserId="19814" LastEditorUserId="19814" LastEditDate="2024-03-27T06:19:22.107" LastActivityDate="2024-03-27T06:19:22.107" Title="Engaging in a puzzle without a guidance of a leader while simultaneously being a component of that puzzle" Tags="|mathematical-pedagogy|activities|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27644" PostTypeId="1" CreationDate="2024-03-27T05:46:57.537" Score="6" ViewCount="132" Body="&lt;p&gt;Edwards and Penney are the authors of a &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0321796985&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;popular differential equations textbook&lt;/a&gt;.  Bizarrely, there seem to be &lt;span class=&quot;math-container&quot;&gt;$2^4$&lt;/span&gt; permutations of the title:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Some versions have &amp;quot;Elementary&amp;quot; and some don't&lt;/li&gt;&#xA;&lt;li&gt;Some versions have &amp;quot;with [or 'and'] Boundary Value Problems&amp;quot; and some don't&lt;/li&gt;&#xA;&lt;li&gt;Some versions have &amp;quot;and Linear Algebra&amp;quot;, some don't&lt;/li&gt;&#xA;&lt;li&gt;Some versions have &amp;quot;Computing and Modeling&amp;quot;, some don't&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;That's aside from the numerous editions of each.&lt;/p&gt;&#xA;&lt;p&gt;What are the differences between them?  Is there a core book, with the permutations simply adding chapters - or is there a difference in content and writing?&lt;/p&gt;&#xA;" OwnerUserId="23837" LastActivityDate="2024-04-01T15:00:50.207" Title="What are the differences between the $n$ permutations of the Edwards &amp; Penney Differential Equations books?" Tags="|textbooks|differential-equations|" AnswerCount="0" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="27645" PostTypeId="2" ParentId="27628" CreationDate="2024-03-27T11:07:22.847" Score="0" Body="&lt;p&gt;I don't sort my exam papers manually. Instead, I use GradeScope.&lt;/p&gt;&#xA;&lt;p&gt;Before the exam, I design the papers so that students know exactly where to write their answers, by providing designated boxes for them. After the exam, I scan all the papers and upload them to GradeScope.&lt;/p&gt;&#xA;&lt;p&gt;This method has several advantages. One of the most significant benefits is the ability to hire someone to help grade the exams remotely. It makes the process much easier, especially for grading objective questions like multiple choice or numerical answer questions. Further more, I can synchronize the grades to Canvas so I don't have to manually input them.&lt;/p&gt;&#xA;" OwnerUserId="23853" LastActivityDate="2024-03-27T11:07:22.847" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="27646" PostTypeId="2" ParentId="27588" CreationDate="2024-03-29T03:40:56.153" Score="1" Body="&lt;p&gt;I suspect that the rules of assessment are the same for every discipline, calculus not excluded, and can be just derived from first principles and your particular version of common sense without any help from Terence Tao. Mine are the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Decide on a few (3-5) topics you want to test and create one problem for each. Don't give more than 5 problems for a one hour test or include problems that require long solutions with extensive computations.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Tell the students that the exam will be concentrated on these topics and give them sufficient number of practice problems well before the test.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Out of 5 problems make 3 completely routine (for the students who know the subject, of course), 1 with a minor twist and 1 really hard (though requiring nothing beyond the covered material).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Invent your problems from scratch and tune them up exactly to your satisfaction yourself.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Conduct the exam in a fully controlled environment (ideally in a classroom with your own presence).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Grade on some simple scale with minimal partial credit (mine is 6 - full solution, 5-a solution with minor error, 1-just a good idea, 0-everything else; occasionally I do give the scores of 4 and 2, but those are exceptions).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Whether it is open or closed book/notes doesn't matter too much as long as you understand that each of those kinds of exams just requires a different tune-up of the problems. I never give take home exams to undergraduates though: tuning those up properly would require creating open problems based on the elementary calculus material. That &lt;em&gt;is&lt;/em&gt; possible, but the most likely outcome will be that nobody will solve anything. Everything short of that will result in shameless cheating and total mayhem with grading, IMHO.&lt;/p&gt;&#xA;&lt;p&gt;Needless to say, this is just &lt;em&gt;my&lt;/em&gt; approach based on &lt;em&gt;my&lt;/em&gt; common sense. Your common sense may be totally different. Just follow it and don't be afraid to experiment with various formats. Whatever works to your satisfaction is the right way, whatever doesn't is the wrong one. It is always as simple as that.&lt;/p&gt;&#xA;&lt;p&gt;Just my 2 cents :-)&lt;/p&gt;&#xA;" OwnerUserId="19946" LastActivityDate="2024-03-29T03:40:56.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27647" PostTypeId="1" AcceptedAnswerId="27649" CreationDate="2024-03-29T04:59:35.223" Score="1" ViewCount="1986" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/a/27624&quot;&gt;It's pretty common for people to learn and apply university-level math yet not be able to solve competition problems.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h4&gt;Isn't this sentence a contradiction? Please unpack this sentence? Is &lt;a href=&quot;https://redd.it/lpbxos&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://redd.it/lpbxos&lt;/a&gt; relevant ?&lt;/h4&gt;&#xA;&lt;p&gt;I agree, weak  math graduates can fail to solve &lt;strong&gt;competition problems (henceforward CP)&lt;/strong&gt; at their degree level.&lt;/p&gt;&#xA;&lt;p&gt;But I am wildered if B.Sc. grads can't  solve &lt;strong&gt;high school (HS)&lt;/strong&gt; CP, eg International Mathematical Olympiad.&lt;/p&gt;&#xA;&lt;p&gt;I am wildered if M.Sc.    grads can't solve undergrad CP, eg William Lowell Putnam Mathematical Competition.&lt;br /&gt;&#xA;I am more wildered if MSc's can't solve HS CP!&lt;/p&gt;&#xA;&lt;p&gt;I am bewildered if Ph.D.'s can't solve undergrad CP!&lt;br /&gt;&#xA;I am even bewildered if PhD's can't solve HS CP!&lt;/p&gt;&#xA;&lt;h4&gt;If a graduate can't solve CP designed for a &lt;em&gt;LOWER&lt;/em&gt; degree level, then she failed to learn and apply university-level math. Correct? What's wrong with my reasoning?&lt;/h4&gt;&#xA;" OwnerUserId="21013" LastEditorUserId="21013" LastEditDate="2024-03-29T09:31:38.047" LastActivityDate="2024-04-02T04:20:40.797" Title="How can graduates learn and apply university-level math, but fail to solve competition problems?" Tags="|math-contests|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="27648" PostTypeId="2" ParentId="27647" CreationDate="2024-03-29T05:24:49.200" Score="14" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Terrell_Owens&quot; rel=&quot;noreferrer&quot;&gt;Terrell Owens&lt;/a&gt; is one of the all-time greatest wide receivers in American football. In 2009 he competed in the ABC Superstars sporting game show. Here's the clip where he fails miserably to get through the first obstacle course, and so gets eliminated in the very first episode:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/FYJntQVZeKI&quot; rel=&quot;noreferrer&quot;&gt;Terrell Owens on Superstars 2009&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;How could this happen to a world-class athlete like T.O.? They're simply different games, is all.&lt;/p&gt;&#xA;&lt;p&gt;No practicing mathematician really cares about game-show competitions.&lt;/p&gt;&#xA;" OwnerUserId="5563" LastActivityDate="2024-03-29T05:24:49.200" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="27649" PostTypeId="2" ParentId="27647" CreationDate="2024-03-29T06:10:56.823" Score="15" Body="&lt;p&gt;There's a huge difference between math degree programs and math competitions.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Degrees are about content knowledge. The way a student graduates from a degree program is by learning and evidencing a base level of competency in some further fields of math. The problems here are intentionally selected, organized, scaffolded to be fairly solvable if the student paid attention to what was covered in class.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Competitions are about problem-solving insight. It doesn't matter if your base level of competency extends into levels of math further than what's tested in the competition. It's about how good you are with the tools at the level of the competition. The whole point of a competition problem is for it to be very difficult to solve even if you know the underlying content. The goal is to &amp;quot;spread out&amp;quot; students' performance on the basis of their ability to think insightfully about these kinds of tricky problems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And more concretely: have you seen the caliber of some of these competition problems? I mean, just look at &lt;a href=&quot;https://artofproblemsolving.com/wiki/index.php/2023_IMO_Problems/Problem_3&quot; rel=&quot;noreferrer&quot;&gt;Problem 3&lt;/a&gt; on the 2023 IMO. Sure, if you're an average student with an undergrad math degree, then you can probably do basic induction and divisibility proofs on the fly, and you're probably able to (or at least were able to at one point) reproduce proofs of key theorems in real analysis and abstract algebra, but you're probably going to have no chance against these IMO problems.&lt;/p&gt;&#xA;&lt;p&gt;And the Putnam?!?! Don't even get me started on the Putnam. The top scores are generally somewhere around 100 points out of 120 possible, and the median score is usually... wait for it... usually no higher than 2 points out of 120 possible. Typically the median is 1 point, and sometimes it's literally 0 points. And typically the students who even take the Putnam at all have mathematical ability well above the average math major. If you take a graduating math major who scored a 0 on the Putnam and put them through 2 years of master's level mathematics, is that going to bring them to the level of collecting half or more (60+) of the total 120 points? Heck no! It would be a &amp;quot;win&amp;quot; to put &lt;em&gt;any&lt;/em&gt; points on the board at all.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum.&lt;/strong&gt; I realized that I haven't talked explicitly about &amp;quot;application&amp;quot; of math in my answer, and one might argue that being unable to solve competition problems is a symptom of being unable to apply what you've learned. The thing is, there are many many ways to apply math, and the vast vast majority have no bearing on competition problems.&lt;/p&gt;&#xA;&lt;p&gt;Here's a concrete example. Going back to that average student with an undergrad math degree (who can do basic induction, etc.), suppose that student also took a machine learning class and learned how to fit a mathematical model to data using gradient descent. After graduation they become a data scientist and complete a bunch of work projects in which they fit models to data and use those models to make predictions that are useful to their employer. Clearly, they are applying what they've learned -- but have they gotten any better at competition math through this process? Nope!&lt;/p&gt;&#xA;" OwnerUserId="22672" LastEditorUserId="22672" LastEditDate="2024-03-29T15:23:12.303" LastActivityDate="2024-03-29T15:23:12.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27650" PostTypeId="2" ParentId="27647" CreationDate="2024-03-29T10:09:46.113" Score="11" Body="&lt;p&gt;Focusing just on one aspect of the bewilderment....&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But I am wildered if B.Sc. math graduates can't solve high school&#xA;competition problems, eg International Mathematical Olympiad.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Based on &lt;a href=&quot;https://imogeometry.blogspot.com/p/1.html&quot; rel=&quot;noreferrer&quot;&gt;this listing&lt;/a&gt;, recent IMOs have included one or two geometry problems. Anecdotally, most math undergrads I knew had not engaged with synthetic geometry since high school, and an average-case high school geometry experience (freshman year??) is not on par with these Olympiad problems. The specialized terminology (e.g., incenter, circumcircle, concyclic points) in some of the problem statements and lack of diagrams would make even content knowledge / topic recall an issue in addition to the overall difficulty level and required ingenuity.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, there are suggestions online that success at these problems is related not only to &amp;quot;innate problem-solving ability&amp;quot; but also to accumulating key facts/techniques and pattern-recognizing when to apply certain lines of attack -- skills that are highly correlated with time spent working on similar problems. IMO participants spend a significant amount of time practicing past exam problems, likely far more than a math undergrad for the given class of problems. This effort asymmetry is especially acute for the geometry problems.&lt;/p&gt;&#xA;" OwnerUserId="1230" LastEditorUserId="1230" LastEditDate="2024-03-29T10:17:23.567" LastActivityDate="2024-03-29T10:17:23.567" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27651" PostTypeId="1" CreationDate="2024-03-29T13:39:15.697" Score="-4" ViewCount="53" Body="&lt;p&gt;I am on a quest to gather a wide range of cultural examples of myths, legends, fables, tales of heroes, or ethnic stories that incorporate mathematical concepts in a meaningful way. For instance, the legend of Theseus and the Minotaur highlights the mathematical concept of a labyrinth and strategies for navigating or solving it. Another intriguing example is that of Dido of Carthage, which introduces the isoperimetric problem. Additionally, the tale of Hercules and the Hydra touches upon recursive or sequential mathematical strategies. However, these examples predominantly reflect Western perspectives. My objective is to compile a more culturally diverse collection of stories to enrich my students' understanding of ethnomathematics.&lt;/p&gt;&#xA;&lt;p&gt;Could the community share examples from various cultures that intertwine mathematics with folklore, myth, or legend? The goal is to illustrate the universal nature of mathematical thought across different societies and historical periods, thereby broadening our educational approach to encompass a global perspective on mathematics.&lt;/p&gt;&#xA;" OwnerUserId="8828" LastActivityDate="2024-03-29T13:39:15.697" Title="Seeking Multicultural Myths and Legends Rich in Mathematical Concepts" Tags="|culture|" AnswerCount="0" CommentCount="11" ClosedDate="2024-03-29T13:51:03.433" ContentLicense="CC BY-SA 4.0" />
  <row Id="27652" PostTypeId="2" ParentId="2262" CreationDate="2024-03-30T01:55:53.483" Score="0" Body="&lt;p&gt;I wonder if the switch to see logs early is due to the prior use of log and trig tables to handle large numbers, like in Engineering schools of the 60s and before when the other tool was the slide rule with a much poorer resolution. Students had to know what a log was. And the base ten was the standard. The overwhelming use of calculators in the 70s made this requirement obsolete. Even though my father in law, who was a surveyor, did not like the poor resolution of the SR50 and kept using the tables. Yep, this was before the PCs too.&lt;/p&gt;&#xA;" OwnerUserId="23877" LastActivityDate="2024-03-30T01:55:53.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27653" PostTypeId="1" CreationDate="2024-03-30T17:14:07.243" Score="14" ViewCount="125" Body="&lt;p&gt;Does anyone have experience with giving students a collection of problems where they have to assess which ones they have the knowledge to solve?&lt;/p&gt;&#xA;&lt;p&gt;I was exploring the connection between video games and teaching, and noticed that video games provide a valuable meta-reasoning experience I haven't seen in the classroom. Some video games allow the player to select challenges they're not ready for, and so the player has to learn how to assess whether they are ready to face a particular challenge, or should leave and try something else.&lt;/p&gt;&#xA;&lt;p&gt;Specifically, I'm thinking of a game like Elden Ring, which has a wide open world where the player can easily reach a large number of boss fights early on. Some of these are appropriate for a player at such an early level, and others are not. The game doesn't do much to indicate which boss fights are which, the player just has to try to assess for themselves whether they just need a little bit more practice or should come back later.&lt;/p&gt;&#xA;&lt;p&gt;In the classroom, however, we usually present students with problems they know can be solved with what they've learned. One could imagine, instead, having a list of problems in a random order that students have to chip away at as the semester goes on. This would require them to reflect on the totality of what they've learned in the semester and what problems they know how to solve.&lt;/p&gt;&#xA;&lt;p&gt;Has anyone tried anything similar? Specifically, some situation where you present students with problems only some of which they can solve with what they know, and have them sort through them? Does anyone know of any trials of this sort? I imagine it taking a lot of fine-tuning to get right.&lt;/p&gt;&#xA;" OwnerUserId="15671" LastEditorUserId="15671" LastEditDate="2024-03-31T05:34:48.283" LastActivityDate="2024-04-06T19:31:45.107" Title="Having students assess what problems they're ready for" Tags="|problem-solving|critical-thinking|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="27654" PostTypeId="2" ParentId="27647" CreationDate="2024-03-30T23:11:38.457" Score="1" Body="&lt;p&gt;Reminds me of Arnold's diatribe&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A student who takes much more than five minutes to calculate the mean of sin(x)^100 with 10% accuracy has no mastery of mathematics, even if he has studied non-standard analysis, universal algebra, supermanifolds, or embedding theorems.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Arnol'd, V. I. (1991). A mathematical trivium.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;where he prioritizes some arbitrary complicated thing - nevertheless amenable to mechanical solution - in detriment to that which is deeper, more abstract and, dare say, more interesting&lt;/p&gt;&#xA;" OwnerUserId="15843" LastActivityDate="2024-03-30T23:11:38.457" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="27655" PostTypeId="2" ParentId="27647" CreationDate="2024-03-30T23:15:56.917" Score="6" Body="&lt;p&gt;Mathematics competitions involve nonroutine problems and most mathematics courses more heavily emphasize the routine (not to be confused with easy!) components of various subjects. Rather than addressing the original question in its fullness, here is just a fragment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Can students who are doing well in the Calculus sequence (say, they have taken Calc I, Calc II, Calc III, and are currently taking Differential Equations) solve nonroutine Calculus problems?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The answer is mostly no. See, for example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Selden, Annie, et al. &amp;quot;Do Calculus Students Eventually Learn to Solve Non-Routine Problems? Technical Report. No. 1999-5.&amp;quot; Online Submission (1999). &lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED518600.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Link&lt;/strong&gt;&lt;/a&gt; (no paywall!).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even students with full/substantial requisite knowledge of the corresponding mathematical topic, who are presently studying in a relevant mathematics course, are unable to provide completely/substantially correct solutions to nonroutine problems – examples of which are included in the linked study above, and none of which is near the difficulty of a typical Olympiad problem.&lt;/p&gt;&#xA;&lt;p&gt;Readers may find that this answer offers little else by way of insight as pertains to the original question, although I wish to advocate for first answering a weaker version of a question (can students doing well amid the Calculus sequence and who have the requisite knowledge successfully solve nonroutine Calc problems?) before zooming outward to a stronger question (can students who may be far removed from the material – e.g., years away from a geometry course – and/or have never seen the material – e.g., unfamiliar with stars and bars, proof by induction, generating functions – successfully solve Olympiad style problems?).&lt;/p&gt;&#xA;&lt;p&gt;I hope the Selden et al piece is of interest for those who are curious about the manifestation of the routine/nonroutine phenomenon presented by the Putnam, IMO, etc.&lt;/p&gt;&#xA;" OwnerUserId="262" LastActivityDate="2024-03-30T23:15:56.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
</posts>